In this section we will consider the general properties of the solution p(r, t) of the Smoluchowski equation in the case that the force field is derived from a potential, i.e., F (r) = −∇U(r) and that the potential is finite everywhere in the diffusion doamain Ω. We will demonstrate first that the solutions, in case of reaction-free boundary conditions, obey an extremum principle, namely, that ...

An expression is derived that relates the average population of a particular type of cluster in a metastable vapor phase of volume V tot to the probability, estimated by simulation, of finding this cluster in a system of volume V taken inside V tot , where V!V tot . Correct treatment of the translational free energy of the cluster is crucial for this purpose. We show that the problem reduces to...

We consider the van der Waals’ free energy functional, with scaling parameter ε, in the plane domain R+ × R+, with inhomogeneous Dirichlet boundary conditions. We impose the two stable phases on the horizontal boundaries R+×{0} and R+×{∞}, and free boundary conditions on {∞}×R+. Finally, the datum on {0} ×R+ is chosen in such a way that the interface between the pure phases is pinned at some po...

— We prove: 1) for any linear language L, it is recursively unsolvable to détermine whether L is an 1RS language; 2) for any deterministic context-free language L, it is recursively solvable to détermine whether L is an 1RS language; 3) there exists an 1RS linear language containing the concaténation product oftwo infinité contextfree languages.

ABSTRACT Axisymmetric magnetohydrodynamic (MHD) models are useful in studies of magnetized winds and non-linear Alfvén waves solar stellar atmospheres. We demonstrate that a condition often used these for the determination nearly vertical magnetic field is applicable to radial instead. A general divergence-free curvilinear coordinates self-consistently derived obtain correct variation field. Th...

We consider a family of explicit compact schemes for advection in one dimension. The order is arbitrarily high. These stencils may be called Strang’s stencils after the seminal work of Strang [J. Math. Phys., 41 (1962), pp. 147–154]. We prove that odd order schemes are stable in all Lq under CFL one. The strategy of the proof is similar to the one of Thomée [J. Differential Equations, 1 (1965),...