11 research outputs found
Coupled Systems of Differential-Algebraic and Kinetic Equations with Application to the Mathematical Modelling of Muscle Tissue
We consider a coupled system composed of a linear differential-algebraic
equation (DAE) and a linear large-scale system of ordinary differential
equations where the latter stands for the dynamics of numerous identical
particles. Replacing the discrete particles by a kinetic equation for a
particle density, we obtain in the mean-field limit the new class of partially
kinetic systems. We investigate the influence of constraints on the kinetic
theory of those systems and present necessary adjustments.
We adapt the mean-field limit to the DAE model and show that index reduction
and the mean-field limit commute. As a main result, we prove Dobrushin's
stability estimate for linear systems. The estimate implies convergence of the
mean-field limit and provides a rigorous link between the particle dynamics and
their kinetic description.
Our research is inspired by mathematical models for muscle tissue where the
macroscopic behaviour is governed by the equations of continuum mechanics,
often discretised by the finite element method, and the microscopic muscle
contraction process is described by Huxley's sliding filament theory. The
latter represents a kinetic equation that characterises the state of the
actin-myosin bindings in the muscle filaments. Linear partially kinetic systems
are a simplified version of such models, with focus on the constraints.Comment: 32 pages, 18 figure
On Landau damping
Going beyond the linearized study has been a longstanding problem in the
theory of Landau damping. In this paper we establish exponential Landau damping
in analytic regularity. The damping phenomenon is reinterpreted in terms of
transfer of regularity between kinetic and spatial variables, rather than
exchanges of energy; phase mixing is the driving mechanism. The analysis
involves new families of analytic norms, measuring regularity by comparison
with solutions of the free transport equation; new functional inequalities; a
control of nonlinear echoes; sharp scattering estimates; and a Newton
approximation scheme. Our results hold for any potential no more singular than
Coulomb or Newton interaction; the limit cases are included with specific
technical effort. As a side result, the stability of homogeneous equilibria of
the nonlinear Vlasov equation is established under sharp assumptions. We point
out the strong analogy with the KAM theory, and discuss physical implications.Comment: News: (1) the main result now covers Coulomb and Newton potentials,
and (2) some classes of Gevrey data; (3) as a corollary this implies new
results of stability of homogeneous nonmonotone equilibria for the
gravitational Vlasov-Poisson equatio
On the rate of convergence to equilibrium in the Becker-Doring equations
AbstractWe provide an explicit rate of convergence to equilibrium for solutions of the BeckerâDöring equations using the energy/energy-dissipation relation. The main difficulty is the structure of equilibria of the BeckerâDöring equations, which do not correspond to a Gaussian measure, such that a logarithmic Sobolev-inequality is not available. We prove a weaker inequality which still implies for fast decaying data that the solution converges to equilibrium as eâct1/3
Discontinuities, generalized solutions and (dis)agreement in opinion dynamics
International audienceThis chapter is devoted to the mathematical analysis of some continuous-time dynamical systems defined by ordinary differential equations with discontinuous right-hand side, which arise as models of opinion dynamics in social networks. Discontinuities originate because of specific communication constraints, namely quantization or bounded confidence. Solutions of these systems may or may not converge to a state of agreement, where all components of the state space are equal. After presenting three models of interest, we elaborate on the properties of their solutions in terms of existence, completeness, and convergence
Agricultural experiments
A good experiment must be designed and analyzed so that, so far as is possible, the treatments under investigation are the only factors which influence variation in the characteristic being investigated.
'Design and Analysis of Small Agricultural Experiments', by M Morad of the Department of Geography, University of Wales, UK, is a short manual which outlines the principles and procedures for conducting small agricultural experiments in a scientific manner. The mathematics have been kept as simple as possible. There are two appendices, one offering researchers the chance to calculate the significance of the results of their own experiments, and the other exploring how to assess the reliability of an experiment
STOAS ( Agricultural Teacher Training Institute)
Salverda Plein 10
6701 DB Wageningen The NEDERLANDS'Design and Analysis of Small Agricultural Experiments', by M Morad
STOAS ( Agricultural Teacher Training Institute)
Salverda Plein 10
6701 DB Wageningen The NEDERLAND