A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.

Nonlinear waves in deformed optical honeycomb lattices are investigated. Discrete couple mode equations are used to find higher order continuous nonlinear Dirac systems which are employed to describe key underlying phenomena. For weak deformation and nonlinearity the wave propagation is circular–ellliptical. At strong nonlinearity the diffraction pattern becomes triangular in structure which is traced to appropriate nonequal energy propagation in momentum space. At suitably large deformation the dispersion structure can have nearby Dirac points or small gaps. The effective dynamics of the wave packets is described by two maximally balanced nonlocal nonlinear Schrödinger type equations.

In section 2 we introduce and study the independence property for a sequence of two-dimensional random variables and by means of this property we define independent motion in section 3. Section 4 is mainly a survey of known results about the convergence of the spatial distribution of the point system as the time$t$→∞. In theorem 5.1 we show that the only distributions which are time-invariant under given reversible motion of non-degenerated type are the weighted Poisson ones. Lastly in section 6 we study a more general type of random motion where the position of a point after translation is a function$f$of its original position and its motion ability. We consider functions$f$which are monotone in the starting position. Limiting ourselves to the case when the point system initially is weighted Poisson distributed with independent motion abilities, we prove in theorem 6.1 that this is the case also after the translations, if and only if the function$f$is linear in the starting position. In the paper also some implications of our results to the theory of road traffic with free overtaking are given.

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The varying coefficient model is an important class of nonparametric statistical model, which allows us to examine how the effects of covariates vary with exposure variables. When the number of covariates is large, the issue of variable selection arises. In this article, we propose and investigate marginal nonparametric screening methods to screen variables in sparse ultra-high-dimensional varying coefficient models. The proposed nonparametric independence screening (NIS) selects variables by ranking a measure of the nonparametric marginal contributions of each covariate given the exposure variable. The sure independent screening property is established under some mild technical conditions when the dimensionality is of nonpolynomial order, and the dimensionality reduction of NIS is quantified. To enhance the practical utility and finite sample performance, two data-driven iterative NIS (INIS) methods are