The stochastic Boltzmann equation and hydrodynamic fluctuations

Based on the assumption of a kinetic equation in space, a stochastic differential equation of the one-particle distribution is derived without the use of the linear approximation. It is just the Boltzmann equation with a Langevin-fluctuating force term. The result is the general form of the linearized Boltzmann equation with fluctuations found by Bixon and Zwanzig and by Fox and Uhlenbeck. It reduces to the general Landau-Lifshitz equations of fluid dynamics in the presence of fluctuations in a similar hydrodynamic approximation to that used by Chapman and Enskog with respect to the Boltzmann equation.This work received financial support from the Alexander von Humboldt Foundation.     

Field-theoretic approach to second-order phase transitions in two- and three-dimensional systems

We review the physical principles which are at the basis of recent field-theoretic computations of the critical exponents in two- and three-dimensional systems. We concentrate on those points that do not show up explicitly in the more standard-expansion: they must be discussed with care if one uses a perturbative approach at fixed space dimensions (the loop expansion). We present in detail simple computations of the critical exponents, while we summarize the results of longer and more accurate computations.     

Conjugate-gradient method for computing the Moore-Penrose inverse and rank of a matrix

A conjugate-gradient method is developed for computing the Moore-Penrose generalized inverseA of a matrix and the associated projectors, by using the least-square characteristics of both the method and the inverseA . Two dual algorithms are introduced for computing the least-square and the minimum-norm generalized inverses, as well asA . It is shown that (i) these algorithms converge for any starting approximation; (ii) if they are started from the zero matrix, they converge toA ; and (iii) the trace of a sequence of approximations multiplied byA is a monotone increasing function converging to the rank ofA. A practical way of compensating the self-correcting feature in the computation ofA is devised by using the duality of the algorithms. Comparison with Ben-Israel's method is made through numerical examples. The conjugate-gradient method has an advantage over Ben-Israel's method.After having completed the present paper, the author received from Professor M. R. Hestenes his paper entitledPseudo Inverses and Conjugate Gradients. This paper treated the same subject and appeared in Communications of the ACM, Vol. 18, pp. 40–43, 1975.     

Derivation of the transport equation for electrons moving through random impurities

A single (nonrelativistic, spinless) electron subject to a constant external electric field interacts with impurities located on an infinitely extended lattice by a potential of random strength. The random strength is given by a field of Gaussian random variables. We show the existence of the averaged dynamics and prove that in the weak coupling limit, 0, ₂ t= fixed, one obtains the usual transport equation for the velocity distribution.Work supported by a Max Kade Foundation fellowship.On leave of absence of the Fachbereich Physik der Universität München.     

Some model theory for monotone quantifiers

A generalization of the existential and universal quantifier, the monotone quantifiers, are studied. It is shown that the model theory for monotone quantifiers behaves very much like classical model theory. Completeness theorems, definability theorems and preservation theorems are given. Ultraproducts, reduced products and Back and Forth arguments are studied.This work is part of the research program of the G.N.S.A.G. group of the Italian C.N.R.     

4fN-15d组态自旋允许与禁戒跃迁能级的研究

通过稀土光谱理论，计算了三价稀土离子的4f^N-15d组态能级的f与d电子间的库仑相互作用，结果表明对于重稀土元素，其自旋允许跃迁能级位置高于自旋禁戒跃迁能级位置，并且两种能级间的能级差随f电子数的增加而依次减小。对于轻稀土元素,则自旋禁戒跃迁能级位置高于自旋允许跃迁能级位置。结果很好地解释了自旋禁戒跃迁能级的光谱现象。     

Double Pore Silica Gel Monolith Applied to Liquid Chromatography

Silica gels retaining double pore structure in the size ranges of micrometer and nanometer have been applied to the rod-shaped monolithic column for liquid chromatography. The macropore structure was designed by controlling the phase separation process induced by the hydrolysis and polycondensation of alkoxysilane, whereas the mesopore structure was tailored by the solvent exchange treatments on wet gels. The size exclusion chromatograms on polystyrene standards exhibited almost similar features for octadecyl-modified rod and conventional packed beads columns. The dependence of plate height on the velocity of mobile phase determined for amylbenzene was by far weaker in the rod column than in the packed beads column, suggesting that additional geometrical factors should be considered in describing the separation mechanism in the rod column.     

Properties of monolayers of crown ethers bearing an azo and azoxy group in the macrocycle

Two methods are described for preparing monomolecular layers of crown ethers with an azo or azoxy group in the macrocycle. When the molecules used to build the monolayer are soluble in aqueous solutions, adsorptive preconcentration on mercury electrodes was used to prepare the monolayer coating. The monolayer was electroactive due to the presence of the azo or azoxy unit in the molecules. Monolayers of crown ethers bearing an azo group in the macrocycle were shown to recognize alkali metal cations present in the solution. Changes of the parameters of the voltammetric reduction peaks - peak potential and peak width, served as an indication of specific interactions of the monolayer of 13-membered and 16-membered azocrown ethers with Na⁺ and K⁺ cations, respectively.The monolayers capable of recognizing cations have also been prepared on the aqueous solution-air interface, using the Langmuir technique. In this approach, amphiphilic derivatives of the azocrowns were synthesized and the monolayer has been assembled on the subphase containing metal cations. Binding of the cation by the macrocycle has a stabilizing effect on the monolayer and higher collapse pressures are achieved than on the pure water subphase. The monolayer was transferred from the air-water interface on the solid substrate using the Langmuir-Blodgett technique. Thin mercury film electrodes on the Ag substrate, or An films evaporated on glass slides were employed as the electrode substrates. The former gave monolayer modified electrodes of higher stability.Presented at the Sixth International Seminar on Inclusion Compounds, Istanbul, Turkey, 27–31 August, 1995.     

Spatial relaxation of electrons in nonisothermal plasmas

The spatial relaxation of electrons to homogeneous states under the action of space-independent electric fields is investigated in helium, krypton, and N₂ plasmas for various electric field strengths. These investigations are based on a new method recently developed for solving the one-dimensional inhomogeneous electron Boltzmann equation in weakly ionized, collision-dominated plasmas. Elastic as well as conservative inelastic collisions of electrons with gas atoms have been included in the kinetic treatment. The spatial relaxation is caused by an imposed direct disturbance in the velocity distribution of the electrons on a spatial boundary. A pronounced dependence of the relaxation structure and the resultant relaxation length on the atomic data of the electron collision processes in different gases has been found. Furthermore the relaxation process sensitively depends on the electric field strength in the region of medium field values.     

Synthesis of molecular ferromagnetics [R3R′X]MCr(C2O4)3 (X = N,R = Bun,R′ = Prn,Et, Me; X = P,R = Ph,R′ = Bun; M = Mn,Fe, Co,Ni, Cu)

The synthesis, IR spectra, and the temperatures of the transition into a ferromagnetic state (T _c) of layered ferromagnetics [R₃RX[MCr(C₂O₄)₃ (M = Mn, Fe, Co, Cu, and Ni) with the [Ph₃BuP]⁺, [Bu₃RN]⁺ (R = Pr, Et, and Me) cations capable of subsequently changing the distances between metallooxalate layers have been considered. The temperatureT _c has been found to be independent of the size of the organic cation. It is believed that the determining factors in the transition to a ferromagnetic state are exchange interactions inside the metallooxalate layer.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 2327–2330, September, 1996.