280 research outputs found
Optimal Regularizing Effect for Scalar Conservation Laws
We investigate the regularity of bounded weak solutions of scalar
conservation laws with uniformly convex flux in space dimension one, satisfying
an entropy condition with entropy production term that is a signed Radon
measure. The proof is based on the kinetic formulation of scalar conservation
laws and on an interaction estimate in physical space.Comment: 24 pages, assumption (11) in Theorem 3.1 modified together with the
example on p. 7, one remark added after the proof of Lemma 4.3, some typos
correcte
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-
Dirac mass dynamics in multidimensional nonlocal parabolic equations
Nonlocal Lotka-Volterra models have the property that solutions concentrate
as Dirac masses in the limit of small diffusion. Is it possible to describe the
dynamics of the limiting concentration points and of the weights of the Dirac
masses? What is the long time asymptotics of these Dirac masses? Can several
Dirac masses co-exist? We will explain how these questions relate to the
so-called "constrained Hamilton-Jacobi equation" and how a form of canonical
equation can be established. This equation has been established assuming
smoothness. Here we build a framework where smooth solutions exist and thus the
full theory can be developed rigorously. We also show that our form of
canonical equation comes with a structure of gradient flow. Numerical
simulations show that the trajectories can exhibit unexpected dynamics well
explained by this equation. Our motivation comes from population adaptive
evolution a branch of mathematical ecology which models darwinian evolution
Super-linear propagation for a general, local cane toads model
We investigate a general, local version of the cane toads equation, which
models the spread of a population structured by unbounded motility. We use the
thin-front limit approach of Evans and Souganidis in [Indiana Univ. Math. J.,
1989] to obtain a characterization of the propagation in terms of both the
linearized equation and a geometric front equation. In particular, we reduce
the task of understanding the precise location of the front for a large class
of equations to analyzing a much smaller class of Hamilton-Jacobi equations. We
are then able to give an explicit formula for the front location in physical
space. One advantage of our approach is that we do not use the explicit
trajectories along which the population spreads, which was a basis of previous
work. Our result allows for large oscillations in the motility
Employee turnover prediction and retention policies design: a case study
This paper illustrates the similarities between the problems of customer
churn and employee turnover. An example of employee turnover prediction model
leveraging classical machine learning techniques is developed. Model outputs
are then discussed to design \& test employee retention policies. This type of
retention discussion is, to our knowledge, innovative and constitutes the main
value of this paper
Distributed synaptic weights in a LIF neural network and learning rules
Leaky integrate-and-fire (LIF) models are mean-field limits, with a large
number of neurons, used to describe neural networks. We consider inhomogeneous
networks structured by a connec-tivity parameter (strengths of the synaptic
weights) with the effect of processing the input current with different
intensities. We first study the properties of the network activity depending on
the distribution of synaptic weights and in particular its discrimination
capacity. Then, we consider simple learning rules and determine the synaptic
weight distribution it generates. We outline the role of noise as a selection
principle and the capacity to memorized a learned signal.Comment: Physica D: Nonlinear Phenomena, Elsevier, 201
Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result
We study two equations of Lotka-Volterra type that describe the Darwinian
evolution of a population density. In the first model a Laplace term represents
the mutations. In the second one we model the mutations by an integral kernel.
In both cases, we use a nonlinear birth-death term that corresponds to the
competition between the traits leading to selection. In the limit of rare or
small mutations, we prove that the solution converges to a sum of moving Dirac
masses. This limit is described by a constrained Hamilton-Jacobi equation. This
was already proved by B. Perthame and G. Barles for the case with a Laplace
term. Here we generalize the assumptions on the initial data and prove the same
result for the integro-differential equation
Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway
Kinetic-transport equations are, by now, standard models to describe the
dynamics of populations of bacteria moving by run-and-tumble. Experimental
observations show that bacteria increase their run duration when encountering
an increasing gradient of chemotactic molecules. This led to a first class of
models which heuristically include tumbling frequencies depending on the
path-wise gradient of chemotactic signal.
More recently, the biochemical pathways regulating the flagellar motors were
uncovered. This knowledge gave rise to a second class of kinetic-transport
equations, that takes into account an intra-cellular molecular content and
which relates the tumbling frequency to this information. It turns out that the
tumbling frequency depends on the chemotactic signal, and not on its gradient.
For these two classes of models, macroscopic equations of Keller-Segel type,
have been derived using diffusion or hyperbolic rescaling. We complete this
program by showing how the first class of equations can be derived from the
second class with molecular content after appropriate rescaling. The main
difficulty is to explain why the path-wise gradient of chemotactic signal can
arise in this asymptotic process.
Randomness of receptor methylation events can be included, and our approach
can be used to compute the tumbling frequency in presence of such a noise
Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient
Several mathematical models of tumor growth are now commonly used to explain
medical observations and predict cancer evolution based on images. These models
incorporate mechanical laws for tissue compression combined with rules for
nutrients availability which can differ depending on the situation under
consideration, in vivo or in vitro. Numerical solutions exhibit, as expected
from medical observations, a proliferative rim and a necrotic core. However,
their precise profiles are rather complex, both in one and two dimensions.Comment: 25 page
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