2,396 research outputs found
An Integro-Differential Structure for Dirac Distributions
We develop a new algebraic setting for treating piecewise functions and
distributions together with suitable differential and Rota-Baxter structures.
Our treatment aims to provide the algebraic underpinning for symbolic
computation systems handling such objects. In particular, we show that the
Green's function of regular boundary problems (for linear ordinary differential
equations) can be expressed naturally in the new setting and that it is
characterized by the corresponding distributional differential equation known
from analysis.Comment: 38 page
Mass concentration in a nonlocal model of clonal selection
Self-renewal is a constitutive property of stem cells. Testing the cancer
stem cell hypothesis requires investigation of the impact of self-renewal on
cancer expansion. To understand better this impact, we propose a mathematical
model describing dynamics of a continuum of cell clones structured by the
self-renewal potential. The model is an extension of the finite
multi-compartment models of interactions between normal and cancer cells in
acute leukemias. It takes a form of a system of integro-differential equations
with a nonlinear and nonlocal coupling, which describes regulatory feedback
loops in cell proliferation and differentiation process. We show that such
coupling leads to mass concentration in points corresponding to maximum of the
self-renewal potential and the model solutions tend asymptotically to a linear
combination of Dirac measures. Furthermore, using a Lyapunov function
constructed for a finite dimensional counterpart of the model, we prove that
the total mass of the solution converges to a globally stable equilibrium.
Additionally, we show stability of the model in space of positive Radon
measures equipped with flat metric. The analytical results are illustrated by
numerical simulations
Adaptation and migration of a population between patches
A Hamilton-Jacobi formulation has been established previously for
phenotypically structured population models where the solution concentrates as
Dirac masses in the limit of small diffusion. Is it possible to extend this
approach to spatial models? Are the limiting solutions still in the form of
sums of Dirac masses? Does the presence of several habitats lead to polymorphic
situations? We study the stationary solutions of a structured population model,
while the population is structured by continuous phenotypical traits and
discrete positions in space. The growth term varies from one habitable zone to
another, for instance because of a change in the temperature. The individuals
can migrate from one zone to another with a constant rate. The mathematical
modeling of this problem, considering mutations between phenotypical traits and
competitive interaction of individuals within each zone via a single resource,
leads to a system of coupled parabolic integro-differential equations. We study
the asymptotic behavior of the stationary solutions to this model in the limit
of small mutations. The limit, which is a sum of Dirac masses, can be described
with the help of an effective Hamiltonian. The presence of migration can modify
the dominant traits and lead to polymorphic situations
Toward a global description of the nucleus-nucleus interaction
Extensive systematization of theoretical and experimental nuclear densities
and of optical potential strengths exctracted from heavy-ion elastic scattering
data analyses at low and intermediate energies are presented.The
energy-dependence of the nuclear potential is accounted for within a model
based on the nonlocal nature of the interaction.The systematics indicate that
the heavy-ion nuclear potential can be described in a simple global way through
a double-folding shape,which basically depends only on the density of nucleons
of the partners in the collision.The poissibility of extracting information
about the nucleon-nucleon interaction from the heavy-ion potential is
investigated.Comment: 12 pages,12 figure
Relativistic Hartree-Bogoliubov theory with finite range pairing forces in coordinate space: Neutron halo in light nuclei
The Relativistic Hartree Bogoliubov (RHB) model is applied in the
self-consistent mean-field approximation to the description of the neutron halo
in the mass region above the s-d shell. Pairing correlations and the coupling
to particle continuum states are described by finite range two-body forces.
Finite element methods are used in the coordinate space discretization of the
coupled system of Dirac-Hartree-Bogoliubov integro-differential eigenvalue
equations, and Klein-Gordon equations for the meson fields. Calculations are
performed for the isotopic chains of Ne and C nuclei. We find evidence for the
occurrence of neutron halo in heavier Ne isotopes. The properties of the 1f-2p
orbitals near the Fermi level and the neutron pairing interaction play a
crucial role in the formation of the halo. Our calculations display no evidence
for the neutron halo phenomenon in C isotopes.Comment: 7 pages, Latex, 5 P.S. Figures, To appear in Phys. Rev. Let
Solving the Dirac equation with nonlocal potential by Imaginary Time Step method
The Imaginary Time Step (ITS) method is applied to solve the Dirac equation
with the nonlocal potential in coordinate space by the ITS evolution for the
corresponding Schr\"odinger-like equation for the upper component. It is
demonstrated that the ITS evolution can be equivalently performed for the
Schr\"odinger-like equation with or without localization. The latter algorithm
is recommended in the application for the reason of simplicity and efficiency.
The feasibility and reliability of this algorithm are also illustrated by
taking the nucleus O as an example, where the same results as the
shooting method for the Dirac equation with localized effective potentials are
obtained
Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology
We discuss a general class of nonlinear mean-field Fokker-Planck equations
[P.H. Chavanis, Phys. Rev. E, 68, 036108 (2003)] and show their applications in
different domains of physics, astrophysics and biology. These equations are
associated with generalized entropic functionals and non-Boltzmannian
distributions (Fermi-Dirac, Bose-Einstein, Tsallis,...). They furthermore
involve an arbitrary binary potential of interaction. We emphasize analogies
between different topics (two-dimensional turbulence, self-gravitating systems,
Debye-H\"uckel theory of electrolytes, porous media, chemotaxis of bacterial
populations, Bose-Einstein condensation, BMF model, Cahn-Hilliard
equations,...) which were previously disconnected. All these examples (and
probably many others) are particular cases of this general class of nonlinear
mean-field Fokker-Planck equations
On the velocity distributions of the one-dimensional inelastic gas
We consider the single-particle velocity distribution of a one-dimensional
fluid of inelastic particles. Both the freely evolving (cooling) system and the
non-equilibrium stationary state obtained in the presence of random forcing are
investigated, and special emphasis is paid to the small inelasticity limit. The
results are obtained from analytical arguments applied to the Boltzmann
equation along with three complementary numerical techniques (Molecular
Dynamics, Direct Monte Carlo Simulation Methods and iterative solutions of
integro-differential kinetic equations). For the freely cooling fluid, we
investigate in detail the scaling properties of the bimodal velocity
distribution emerging close to elasticity and calculate the scaling function
associated with the distribution function. In the heated steady state, we find
that, depending on the inelasticity, the distribution function may display two
different stretched exponential tails at large velocities. The inelasticity
dependence of the crossover velocity is determined and it is found that the
extremely high velocity tail may not be observable at ``experimentally
relevant'' inelasticities.Comment: Latex, 14 pages, 12 eps figure
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