Computation of limit periodic continued fractions. A survey

Over the last 20 years a large number of algorithms has been published to improve the speed and domain of convergence of continued fractions. In this survey we show that these algorithms are strongly related. Actually, they essentially boil down to two main principles.We also prove some results on asymptotic expansions of tail values of limit periodic continued fractions.Dedicated to Luigi Gatteschi on his seventieth birthdayThis research was partially supported by The Norwegian Research Council and by the HMC project ROLLS, under contract CHRX-CT93-0416.     

On the group extension of the transformation associated to non-archimedean continued fractions

Summary Let F_q be a finite field with q elements. We consider formal Laurent series of F_q -coefficients with their continued fraction expansions by F_q -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors.     

因子分析在企业竞争力评价中的应用

根据因子分析原理,建立了企业竞争力的规模因子、效益因子和成长因子得分模型。通过现实竞争力和潜在竞争力分析,实现了企业竞争力和可持续发展的评价。     

Size exclusion chromatographic characterization of sodium hyaluronate fractions prepared by high energetic sonication

Summary A high molecular weight sodium hyaluronate isolated from rooster combs was degraded by ultrasonication. High-performance size exclusion chromatography allowed rapid and accurate determination of molecular weight parameters (M_z, M_w, M_n) and distributions. The time dependence of hyaluronan ultrasonication to the molecular characteristics of the polymer was investigated. A non-random nature of the degradation process was demonstrated and the reciprocal M_n value was found to be linearly proportional to the time of ultrasonication.     

Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Padé approximants occupying the diagonal of the Padé table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.     

Molecular properties of hemicelluloses located in the surface and inner layers of hardwood and softwood pulps

The molecular properties of hemicelluloses located in the surface and inner layers of fibers present in hardwood and softwood pulps, together with the effects of different bleaching processes on these properties, have been investigated in this study. In order to separate the hemicelluloses located in these two layers, fibers were subjected to mechanical peeling and then separated by filtration into surface (filtrate) and inner layer materials. The materials thus obtained were characterized with respect to their polysaccharide compositions and uronic acid contents. The molar mass parameters of the hemicelluloses (extracted by alkali) were determined by employing size-exclusion chromatography in combination with off-line MALDI mass spectrometry. For all of the pulps examined, the relative content of xylan was found to be greater in the surface layer of the fiber than in the corresponding inner layer. The xylan polymers of the surface layer exhibited higher molar masses and lower frequencies of uronic acid side groups than did the xylans in the inner fiber layer. In connection with ozone treatment, hexenuronic acid residues in the surface layer xylan were removed to a greater extent than in the case of the inner layers, indicating a gradient for the reaction with ozone across the fiber wall. The xylan polymer remaining on the surface of the softwood pulps after completion of the chlorine dioxide bleaching process was predominantly uncharged.     

Brownian motion of nonlinear oscillators: The projection operator method and its application to a double-well-potential oscillator

A projection operator method is presented, which provides the most efficient way for calculating the stationary behavior of nonlinear Brownian motion. A continued-fraction expansion of the Fourier-Laplace transform of the displacement correlation function or the spectral density is used. This method utilizes a successive optimization procedure on the nonlinear terms and includes the method of statistical linearization as the lowest order approximation. A systematic way to calculate the continued fraction numerically up to sufficient order for convergence is developed, which enables us to obtain the spectral density of a system previously uncomputable.Numerical computations of the spectral density of a nonlinear oscillator with a double-well potential are presented and compared with the results obtained by statistical linearization.This work was supported in part by the National Science Foundation under Grant CHE 75-20624.     

Symbiotic N2 Fixation and N Uptake of Yellow and White Lupines Influenced by Spontaneous Rhizobia Infection and Substrate

Abstract

By using soil as substrate, white and yellow lupines (Lupinus albus L., Lupinus luteus L.) assimilated higher N amounts than under quartz sand conditions. This was caused by spontaneous infection of lupines with wild Rhizobia strains and also by an additional N uptake from the soil. In yellow lupines without inoculation in non-sterile soil, only the additional N uptake played a role. Differences in P and K supply as the cause of different N acquisition from soil and quartz sand could be excluded. As compared with white lupines, yellow lupines inoculated with Rhizobia had a high N₂ fixation that exceeded the effect of spontaneous infections. This result as well as the positive effect of spontaneous infections with soil-borne Rhizobia on white lupines indicates insufficient effectiveness of the strains used for inoculation on this plant species.     

反应温度对加氢残渣油四组分含量和结构的影响

以沙轻减渣为原料,在高压釜内研究了不同反应温度下加氢反应前后渣油的四组分含量及其结构组成变化。结果表明,加氢残渣油中的饱和分含量明显增加,而芳香分和胶质的含量均降低,四组分含量随反应温度的升高均呈现规律性变化。加氢后四组分的H/C摩尔比和平均相对分子质量均降低,芳碳分率增加。随反应温度升高,四组分的H/C摩尔比和平均相对分子质量降低,烷基碳分率降低;芳香分、胶质和沥青质的芳碳分率增加;胶质和沥青质的总环数和芳环数均降低。渣油加氢过程中四组分都发生了明显的氢解和脱烷基反应。加氢反应中,胶质和沥青质结构单元间的各种桥键可发生明显地断裂,导致其结构单元数减少,且结构单元数随反应温度的升高而减少。     

Ramanujan-Göllnitz-Gordon Continued Fraction

We derive many new identities involving the Ramanujan-Göllnitz-Gordon continued fraction H(q). These include relations between H(q) and H(q ⁿ ) , which are established using modular equations of degree n. We also evaluate explicitly H(q) at for various positive integers n. Using results of M. Deuring, we show that are units for all positive integers n.