422 research outputs found
Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up
We investigate a particle system which is a discrete and deterministic
approximation of the one-dimensional Keller-Segel equation with a logarithmic
potential. The particle system is derived from the gradient flow of the
homogeneous free energy written in Lagrangian coordinates. We focus on the
description of the blow-up of the particle system, namely: the number of
particles involved in the first aggregate, and the limiting profile of the
rescaled system. We exhibit basins of stability for which the number of
particles is critical, and we prove a weak rigidity result concerning the
rescaled dynamics. This work is complemented with a detailed analysis of the
case where only three particles interact
Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion
Recently, there has been a wide interest in the study of aggregation
equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate
diffusion. The focus of this paper is the unification and generalization of the
well-posedness theory of these models. We prove local well-posedness on bounded
domains for dimensions and in all of space for , the
uniqueness being a result previously not known for PKS with degenerate
diffusion. We generalize the notion of criticality for PKS and show that
subcritical problems are globally well-posed. For a fairly general class of
problems, we prove the existence of a critical mass which sharply divides the
possibility of finite time blow up and global existence. Moreover, we compute
the critical mass for fully general problems and show that solutions with
smaller mass exists globally. For a class of supercritical problems we prove
finite time blow up is possible for initial data of arbitrary mass.Comment: 31 page
Mathematical description of bacterial traveling pulses
The Keller-Segel system has been widely proposed as a model for bacterial
waves driven by chemotactic processes. Current experiments on {\em E. coli}
have shown precise structure of traveling pulses. We present here an
alternative mathematical description of traveling pulses at a macroscopic
scale. This modeling task is complemented with numerical simulations in
accordance with the experimental observations. Our model is derived from an
accurate kinetic description of the mesoscopic run-and-tumble process performed
by bacteria. This model can account for recent experimental observations with
{\em E. coli}. Qualitative agreements include the asymmetry of the pulse and
transition in the collective behaviour (clustered motion versus dispersion). In
addition we can capture quantitatively the main characteristics of the pulse
such as the speed and the relative size of tails. This work opens several
experimental and theoretical perspectives. Coefficients at the macroscopic
level are derived from considerations at the cellular scale. For instance the
stiffness of the signal integration process turns out to have a strong effect
on collective motion. Furthermore the bottom-up scaling allows to perform
preliminary mathematical analysis and write efficient numerical schemes. This
model is intended as a predictive tool for the investigation of bacterial
collective motion
Symplectically degenerate maxima via generating functions
We provide a simple proof of a theorem due to Nancy Hingston, asserting that
symplectically degenerate maxima of any Hamiltonian diffeomorphism of the
standard symplectic 2d-torus are non-isolated contractible periodic points or
their action is a non-isolated point of the average-action spectrum. Our
argument is based on generating functions.Comment: 25 pages, thoroughly revised version, new titl
Analysis of symmetries in models of multi-strain infections
In mathematical studies of the dynamics of multi-strain diseases caused by antigenically diverse pathogens, there is a substantial interest in analytical insights. Using the example of a generic model of multi-strain diseases with cross-immunity between strains, we show that a significant understanding of the stability of steady states and possible dynamical behaviours can be achieved when the symmetry of interactions between strains is taken into account. Techniques of equivariant bifurcation theory allow one to identify the type of possible symmetry-breaking Hopf bifurcation, as well as to classify different periodic solutions in terms of their spatial and temporal symmetries. The approach is also illustrated on other models of multi-strain diseases, where the same methodology provides a systematic understanding of bifurcation scenarios and periodic behaviours. The results of the analysis are quite generic, and have wider implications for understanding the dynamics of a large class of models of multi-strain diseases
The Origins of Concentric Demyelination: Self-Organization in the Human Brain
Baló's concentric sclerosis is a rare atypical form of multiple sclerosis characterized by striking concentric demyelination patterns. We propose a robust mathematical model for Baló's sclerosis, sharing common molecular and cellular mechanisms with multiple sclerosis. A reconsideration of the analogies between Baló's sclerosis and the Liesegang periodic precipitation phenomenon led us to propose a chemotactic cellular model for this disease. Rings of demyelination appear as a result of self-organization processes, and closely mimic Baló lesions. According to our results, homogeneous and concentric demyelinations may be two different macroscopic outcomes of a single fundamental immune disorder. Furthermore, in chemotactic models, cellular aggressivity appears to play a central role in pattern formation
The Conley Conjecture and Beyond
This is (mainly) a survey of recent results on the problem of the existence
of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb
flows. We focus on the Conley conjecture, proved for a broad class of closed
symplectic manifolds, asserting that under some natural conditions on the
manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic
orbits. We discuss in detail the established cases of the conjecture and
related results including an analog of the conjecture for Reeb flows, the cases
where the conjecture is known to fail, the question of the generic existence of
infinitely many periodic orbits, and local geometrical conditions that force
the existence of infinitely many periodic orbits. We also show how a recently
established variant of the Conley conjecture for Reeb flows can be applied to
prove the existence of infinitely many periodic orbits of a low-energy charge
in a non-vanishing magnetic field on a surface other than a sphere.Comment: 34 pages, 1 figur
Public health insurance and entry into self-employment
We estimate the impact of a differential treatment of paid employees versus
self-employed workers in a public health insurance system on the entry rate
into entrepreneurship. In Germany, the public health insurance system is
mandatory for most paid employees, but not for the selfemployed, who usually
buy private health insurance. Private health insurance contributions are
relatively low for the young and healthy, and until 2013 also for males, but
less attractive at the other ends of these dimensions and if membership in the
public health insurance allows other family members to be covered by
contribution-free family insurance. Therefore, the health insurance system can
create incentives or disincentives to starting up a business depending on the
family’s situation and health. We estimate a discrete time hazard rate model
of entrepreneurial entry based on representative household panel data for
Germany, which include personal health information, and we account for non-
random sample selection. We estimate that an increase in the health insurance
cost differential between self-employed workers and paid employees by 100 euro
per month decreases the annual probability of entry into selfemployment by
0.38 percentage points, i.e. about a third of the average annual entry rate.
The results show that the phenomenon of entrepreneurship lock, which an
emerging literature describes for the system of employer provided health
insurance in the USA, can also occur in a public health insurance system.
Therefore, entrepreneurial activity should be taken into account when
discussing potential health care reforms, not only in the USA and in Germany
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