145 research outputs found
Boundary Value Problem for an Oblique Paraxial Model of Light Propagation
We study the Schr\"odinger equation which comes from the paraxial
approximation of the Helmholtz equation in the case where the direction of
propagation is tilted with respect to the boundary of the domain. This model
has been proposed in (Doumic, Golse, Sentis, CRAS, 2003). Our primary interest
here is in the boundary conditions successively in a half-plane, then in a
quadrant of R2. The half-plane problem has been used in (Doumic, Duboc, Golse,
Sentis, JCP, to appear) to build a numerical method, which has been introduced
in the HERA plateform of CEA
Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations
Fragmentation and growth-fragmentation equations is a family of problems with
varied and wide applications. This paper is devoted to description of the long
time time asymptotics of two critical cases of these equations, when the
division rate is constant and the growth rate is linear or zero. The study of
these cases may be reduced to the study of the following fragmentation
equation:Using the Mellin transform of the equation, we
determine the long time behavior of the solutions. Our results show in
particular the strong dependence of this asymptotic behavior with respect to
the initial data
Explicit Solution and Fine Asymptotics for a Critical Growth-Fragmentation Equation
We give here an explicit formula for the following critical case of the
growth-fragmentation equation for some constants , and - the case being the emblematic binary fission case. We discuss
the links between this formula and the asymptotic ones previously obtained in
(Doumic, Escobedo, Kin. Rel. Mod., 2016), and use them to clarify how
periodicity may appear asymptotically
Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
We study the asymptotic behaviour of the following linear
growth-fragmentation equation and prove that under fairly general assumptions on the division
rate its solution converges towards an oscillatory function,explicitely
given by the projection of the initial state on the space generated by the
countable set of the dominant eigenvectors of the operator. Despite the lack of
hypo-coercivity of the operator, the proof relies on a general relative entropy
argument in a convenient weighted space, where well-posedness is obtained
via semigroup analysis. We also propose a non-dissipative numerical scheme,
able to capture the oscillations
Long-time asymptotics for polymerization models
This study is devoted to the long-term behavior of nucleation, growth and
fragmentation equations, modeling the spontaneous formation and kinetics of
large polymers in a spatially homogeneous and closed environment. Such models
are, for instance, commonly used in the biophysical community in order to model
in vitro experiments of fibrillation. We investigate the interplay between four
processes: nucleation, polymeriza-tion, depolymerization and fragmentation. We
first revisit the well-known Lifshitz-Slyozov model, which takes into account
only polymerization and depolymerization, and we show that, when nucleation is
included, the system goes to a trivial equilibrium: all polymers fragmentize,
going back to very small polymers. Taking into account only polymerization and
fragmentation, modeled by the classical growth-fragmentation equation, also
leads the system to the same trivial equilibrium, whether or not nucleation is
considered. However, also taking into account a depolymer-ization reaction term
may surprisingly stabilize the system, since a steady size-distribution of
polymers may then emerge, as soon as polymeriza-tion dominates depolymerization
for large sizes whereas depolymerization dominates polymerization for smaller
ones-a case which fits the classical assumptions for the Lifshitz-Slyozov
equations, but complemented with fragmentation so that " Ostwald ripening "
does not happen.Comment: https://link.springer.com/article/10.1007/s00220-018-3218-
Statistical estimation of a growth-fragmentation model observed on a genealogical tree
We model the growth of a cell population by a piecewise deterministic Markov
branching tree. Each cell splits into two offsprings at a division rate
that depends on its size . The size of each cell grows exponentially in
time, at a rate that varies for each individual. We show that the mean
empirical measure of the model satisfies a growth-fragmentation type equation
if structured in both size and growth rate as state variables. We construct a
nonparametric estimator of the division rate based on the observation of
the population over different sampling schemes of size on the genealogical
tree. Our estimator nearly achieves the rate in squared-loss
error asymptotically. When the growth rate is assumed to be identical for every
cell, we retrieve the classical growth-fragmentation model and our estimator
improves on the rate obtained in \cite{DHRR, DPZ} through
indirect observation schemes. Our method is consistently tested numerically and
implemented on {\it Escherichia coli} data.Comment: 46 pages, 4 figure
Toward an integrated workforce planning framework using structured equations
Strategic Workforce Planning is a company process providing best in class,
economically sound, workforce management policies and goals. Despite the
abundance of literature on the subject, this is a notorious challenge in terms
of implementation. Reasons span from the youth of the field itself to broader
data integration concerns that arise from gathering information from financial,
human resource and business excellence systems. This paper aims at setting the
first stones to a simple yet robust quantitative framework for Strategic
Workforce Planning exercises. First a method based on structured equations is
detailed. It is then used to answer two main workforce related questions: how
to optimally hire to keep labor costs flat? How to build an experience
constrained workforce at a minimal cost
Microscopic approach of a time elapsed neural model
The spike trains are the main components of the information processing in the
brain. To model spike trains several point processes have been investigated in
the literature. And more macroscopic approaches have also been studied, using
partial differential equation models. The main aim of the present article is to
build a bridge between several point processes models (Poisson, Wold, Hawkes)
that have been proved to statistically fit real spike trains data and
age-structured partial differential equations as introduced by Pakdaman,
Perthame and Salort
A purely mechanical model with asymmetric features for early morphogenesis of rod-shaped bacteria micro-colony
To model the morphogenesis of rod-shaped bacterial micro-colony, several
individual-based models have been proposed in the biophysical literature. When
studying the shape of micro-colonies, most models present interaction forces
such as attraction or filial link. In this article, we propose a model where
the bacteria interact only through non-overlapping constraints. We consider the
asymmetry of the bacteria, and its influence on the friction with the
substrate. Besides, we consider asymmetry in the mass distribution of the
bacteria along their length. These two new modelling assumptions allow us to
retrieve mechanical behaviours of micro-colony growth without the need of
interaction such as attraction. We compare our model to various sets of
experiments, discuss our results, and propose several quantifiers to compare
model to data in a systematic way
Relative Entropy Method for Measure Solutions of the Growth-Fragmentation Equation
The aim of this study is to generalise recent results of the two last authors
on en-tropy methods for measure solutions of the renewal equation to other
classes of structured population problems. Specifically, we develop a
generalised relative entropy inequality for the growth-fragmentation equation
and prove asymptotic convergence to a steady-state solution, even when the
initial datum is only a non-negative measure
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