13,609 research outputs found
A -uniform quantitative Tanaka's theorem for the conservative Kac's -particle system with Maxwell molecules
This paper considers the space homogenous Boltzmann equation with Maxwell
molecules and arbitrary angular distribution. Following Kac's program, emphasis
is laid on the the associated conservative Kac's stochastic -particle
system, a Markov process with binary collisions conserving energy and total
momentum. An explicit Markov coupling (a probabilistic, Markovian coupling of
two copies of the process) is constructed, using simultaneous collisions, and
parallel coupling of each binary random collision on the sphere of collisional
directions. The euclidean distance between the two coupled systems is almost
surely decreasing with respect to time, and the associated quadratic coupling
creation (the time variation of the averaged squared coupling distance) is
computed explicitly. Then, a family (indexed by ) of -uniform
''weak'' coupling / coupling creation inequalities are proven, that leads to a
-uniform power law trend to equilibrium of order , with constants depending on moments of the velocity
distributions strictly greater than . The case of order
moment is treated explicitly, achieving Kac's program without any chaos
propagation analysis. Finally, two counter-examples are suggested indicating
that the method: (i) requires the dependance on -moments, and (ii) cannot
provide contractivity in quadratic Wasserstein distance in any case.Comment: arXiv admin note: text overlap with arXiv:1312.225
Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision
This paper considers space homogenous Boltzmann kinetic equations in
dimension with Maxwell collisions (and without Grad's cut-off). An explicit
Markov coupling of the associated conservative (Nanbu) stochastic -particle
system is constructed, using plain parallel coupling of isotropic random walks
on the sphere of two-body collisional directions. The resulting coupling is
almost surely decreasing, and the -coupling creation is computed
explicitly. Some quasi-contractive and uniform in coupling / coupling
creation inequalities are then proved, relying on -moments () of velocity distributions; upon -uniform propagation of moments of the
particle system, it yields a -scalable -power law trend to
equilibrium. The latter are based on an original sharp inequality, which bounds
from above the coupling distance of two centered and normalized random
variables in , with the average square parallelogram area spanned
by , denoting an independent copy. Two
counter-examples proving the necessity of the dependance on -moments and
the impossibility of strict contractivity are provided. The paper, (mostly)
self-contained, does not require any propagation of chaos property and uses
only elementary tools.Comment: 29 page
A simple criterion of transverse linear instability for solitary waves
We prove an abstract instability result for an eigenvalue problem with
parameter. We apply this criterion to show the transverse linear instability of
solitary waves on various examples from mathematical physics.Comment: The main result has been improved and its proof simplifie
Geometric optics and boundary layers for Nonlinear Schrodinger equations
We justify supercritical geometric optics in small time for the defocusing
semiclassical Nonlinear Schrodinger Equation for a large class of
non-necessarily homogeneous nonlinearities. The case of a half-space with
Neumann boundary condition is also studied.Comment: 44 page
Dismantling the Mantel tests
The simple and partial Mantel tests are routinely used in many areas of
evolutionary biology to assess the significance of the association between two
or more matrices of distances relative to the same pairs of individuals or
demes. Partial Mantel tests rather than simple Mantel tests are widely used to
assess the relationship between two variables displaying some form of
structure.
We show that contrarily to a widely shared belief, partial Mantel tests are
not valid in this case, and their bias remains close to that of the simple
Mantel test.
We confirm that strong biases are expected under a sampling design and
spatial correlation parameter drawn from an actual study.
The Mantel tests should not be used in case auto-correlation is suspected in
both variables compared under the null hypothesis. We outline alternative
strategies. The R code used for our computer simulations is distributed as
supporting material
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