Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions

This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave connecting and for the scalar viscous conservation law in two space dimensions. We assume that the initial data tends to constant states as , respectively. Then, the convergence rate to of the solution is investigated without the smallness conditions of and the initial disturbance. The proof is given by elementary -energy method.

     

Integral analogues of almost sure limit theorems

Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.     

The state of dissolved benzene in aqueous solution

Henry's law constants for benzene in water have been measured by bringing water layers into equilibrium with solutions of benzene in carbon tetrachloride or in cyclohexane. The mole fraction of benzene in the aqueous layer was determined by ultraviolet absorption spectrophotometry, and its fugacity was taken as equal to that in the nonaqueous phase, reliable data for the C₆H₆–CCl₄ and C₆H₆–c–C₆H₁₂ systems being available in the literature. Measurements were made at 5^o intervals from 10 to 30°C inclusive, at mole fractions down to from 10% to 20% of saturation. In no case did Henry's law constant depart significantly from constancy, and it was in reasonable agreement with some representative literature values based on saturated solubility. The constancy and the magnitudes of our K_h values indicate that appreciable dimerization does not occur in the temperature range examined here. This conclusion contrasts with the suggestion of Reid, Quickenden, and Franks that their calorimetrically measured heat of solution of benzene in water is different enough from the van't Hoff heat to imply possible dimerization of the solute; it also contrasts with the hydrophobic-bond-forming tendency which Ben-Naim, Wilf, and Yaacobi ascribe to benzene on the basis of their studies of the solubilities of benzene and biphenyl. The results of the latter study, when combined with the known second virial coefficient of benzene vapor, predict that more than 20% of the benzene in saturated aqueous solution at 25°C should be present as dimer, in clear contradiction to the results of the present work.     

Example of diffusion in a disordered Lorentz gas

We prove a diffusion law for a disordered Lorentz gas obtained by modification of a model of Gates, Gerst, Kac in Ref. 1, even though the motion is not a Markovian one in the technical sense of the word.     

二维带非线性源项单个守恒律的有限体积方法的收敛性

本文讨论在无结构网格下用有限体积方法离散二维带非线性源项的单个守恒律,在测度值解与Diperna唯一性结果的框架下,证明了估计解在L_(loc)~l(R~2×(0,T))意义下收敛到单个守恒律的熵解。     

Nonequilibrium expressions for entropy and other thermodynamic quantities

The entropy is written as a density series expansion involving a new kind of cumulant. These are defined as usual from the so-called reduced distribution functions. The first four terms of the series expansion of the entropy are shown to be identical to the known result. When density corrections are retained up to theuth order, the entropy is proved to obey approximately a conservation theorem. Finally, a discussion of nonequilibrium and equilibrium properties of the grand canonical ensemble is presented.     

求无穷级数和以及多重积分极限的概率方法

文 [1 ]举例说明了概率思想在求无穷级数的和以及多重积分极限方面的应用 ,本文将对文 [1 ]中的例 1加以推广 ,拓展这一类题的解题思路 ,并对例 2用不同的概率方法加以证明 ,从而使数学分析与概率统计的有关知识联系起来 ,达到知识的融汇贯通 .     

Pseudopotential study of lattice parameter and heat of formation for substitutional alloys

Ashcroft’s empty core pseudopotential is applied to the substitutional alloy (K-CS) to calculate the heat of formation and lattice parameter over the entire concentration range. At any concentration the defect crystal is considered to be equivalent to a perfect crystal with a modified lattice parameter and the potential parameter for the defect crystal is calculated by using some suitable interpolation formula. The calculated results agree well with the available experimental results.     

Dynamics of liquid silane

Neutron scattering of cold neutrons from liquid silane at 137° K and 98°K is explained on the basis of a simple model. The rotational diffusion constant,D _r, and the delay time,τ ₀, after which rotational diffusion may be said to occur are derived on the basis of this model. At 137° K we getD _r=0.22×10¹³ sec⁻¹ andτ ₀=0.68×10⁻¹³ sec. At 98°KD _r (=0.06×10¹³ sec⁻¹) is down by a factor of more than three butτ ₀=(0.54×10⁻¹³ sec) shows only a small change. By comparison with data on liquid CH₄ it is concluded that the law of corresponding states is not applicable for describing rotational dynamics of CH₄ and SiH₄. Rotational motions in SiH₄ are more hindered than in CH₄ at the same reduced temperature.