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A stationary boundary-value problem for the Boltzmann equation in a half space is considered for a binary mixture of gases when the indata on the boundary are given for the both species. Under the assumption that one of the species is dominant and close to equilibrium but the density of the other is small, the problem is decomposed into two half-space problems: the so-called Milne problem for the linearized Boltzmann equation with a source term for the dominant species and that for the linear Boltzmann equation for the low-density species. The two problems are coupled in such a way that the source term in the former is specified by the solution of the latter. The existence and uniqueness of the solutions to the two problems are proved, and their accurate asymptotic behavior in the far field is obtained. In particular, the precise rate of approach of the solution to the state at infinity is expressed in terms of the decay rate in the molecular velocity of the boundary data for both species.

Display Omitted We propose a simple and cost-effective method of immersion interference lithography.It allows close contact with the solid coupling medium without index matching liquid.The photoresist is kept free from the contamination of liquid.The solid coupling medium improves the resolution by a factor of its refractive index.We obtained a 58-nm half-pitch grating, which corresponds to a feature size of 0.18λ. This study proposes a simple and cost-effective method of solid immersion interference lithography that uses a conformable phase mask. Perfluoropolyether based polymer was used as the material of the elastomeric phase mask. The proposed method requires no liquid layer to function as the coupling medium or the index matching layer and keeps the photoresist free from the contamination of liquid. The solid coupling medium improves the resolution of interference fringes by a factor of its refractive index. This study presents an exposed interference fringe with a half-pitch of 58nm using a He-Cd laser with λ=325nm, which corresponds to a feature size of λ/5.6.

Design and fabrication of transparent and superhydrophilic self-cleaning thin films on glass substrate by 7 nm TiO 2 /22 nm SiO 2 all-nanoparticle coatings via aqueous electrostatic layer-by-layer assembly processing were demonstrated in this work. Stable and conformal structure of multibilayer (TiO 2 /SiO 2 ) x with x up to 120 was shown to exhibit antireflective, superhydrophilic, and self-cleaning multifunctional properties. This was achieved by using appropriate periodic calcinations rather than only once calcination of the assembled nanoparticulate thin films. Average refractive index of 1.30 ± 0.03 at 633 nm and average bilayer thickness of 20.18 nm were determined for the as-fabricated nanoparticulate thin films. Furthermore, average transmittance values higher than 93.5% were exhibited by all the nanoparticulate thin films and the experimental results were only at the very most of 2% lower than those predicted by the theory. The self-cleaning properties of the nanoparticulate thin films were revealed by the decomposition of two model contaminants under UV illumination. The experimental results showed that the photocatalytic decomposition rates are increased with the increase of bilayer number of the nanoparticulate thin film.

For a class of general quasilinear equations on S 1 , we show that, by a very simple maximum principle technique, as long as the solution stays finite, all of its derivatives also remain finite. Some specific examples are given. Under suitable assumptions, we also derive exponential decay of the derivatives of the solution.