Numerical solution of fractional order differential equations by extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples. This revised version was published online in August 2006 with corrections to the Cover Date.     

Trimethylsilylation and methylation of 1-hydroxybenzotriazole 3-oxide

Trimethylsilylation and methylation of 1 -hydroxybenzotriazole 3-oxide (1) proceed regioselectively and afford 1-trimethylsilyloxy- and 1-methoxybenzotriazole 3-oxides, respectively. The first compound easily undergoes fast hydrolysis and methanolysis, but does not react with nitroethane at 20 °C. A fast reversible 1,5-O,O-migration of the SiMe₃ group is observed for this compound.The authors thank V. I. Gulevskaya for DTA analysis of compound4.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 2283–2285, November, 1995.The present work was supported by the Russian Foundation for Basic Research (project No. 93-03-18461).     

Composition operators acting on holomorphic Sobolev spaces

We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

     

Properties of an amphoteric cellulose derivative containing anionic carboxymethyl and cationic 2-hydroxy-3-(trimethylammonio) propyl substituents

An amphoteric cellulose derivative, O-carboxymethyl-O-2-hydroxy-3-(trimethylammonio) propylcellulose (CM-HTMAPC), was prepared by the etherification of O-carboxymethylcellulose (CMC) with (3-chloro-2-hydroxy-n-propyl)-trimethylammonium chloride in a NaOH solution. Apparent molecular sizes of the amphoteric cellulose derivative in aqueous solutions of different NaCl concentrations under various pH conditions were investigated by gel permeation chromatography (GPC), and results were evaluated in terms of inter- and intramolecular ionic interactions. Relation between GPC and viscosity results was also discussed. A titration method to determine the apparent acidic dissociation constants of carboxymethyl substituents was developed and the effect of cationic substituents on apparent acidic dissociation constants was examined. ©1995 John Wiley & Sons, Inc.     

Embedding theorems in the Sobolev-Morrey type spaces S p,a,ℵτ W W(G) with dominant mixed derivatives

We construct Sobolev-Morrey type spaces with dominant mixed derivatives and, using the obtained integral representation, prove some embedding theorems in these spaces.     

Multicomponent One-pot Diastereoselective Synthesis ofBiologically Important Scaffold under Microwaves

A simple and fast three-component diastereoselective synthesis of biologically important spiro scaffold 4 wascarried out in reasonable purity starting from readily available 1H-indole-2,3-dione 1,ethyl cyanoacetate 2 and4-hydroxycoumarin 3 under microwave in high yield(88%-92%)and short time.     

自由亚交换群的直积的检验元素

设Sri(i=1,2,…,n)为秩ri的自由亚交换群,G=Sr1×Sr2…×Srn为自由亚交换群的直积,本文证明了G有检验元素的充分必要条件为ri=2(i=1,2,…,n)．同时,还证明了g=(g1,g2,…,gn)为G的检验元素的充分必要条件是:gi∈S′2-1(i= 1,2,…,n),且{g1,g2,…,gn}为独立集．此外,我们给出了一类具体的检验元素．     

矢量场方向导数和时标正则曲线

研究了一类参数方程的曲线，其参变量表示为所谓时标（time scale），该时标是所有实数的集的一个任意闭子集．引入了相应于矢量场的方向导数．     

2—苯基接三唑衍生物的合成研究

本文报道了用D-阿拉伯-己醛糖脎合成糖接三唑的基本方法,并由此合成了一系列接三唑衍生物,元素分析、~1H-NMR,~(13)C-NMR确证了它们的结构。     

Effect of the structure of mesophyll in a leaf on the form of the spectral curve for the first derivatives of the reflection coefficients of leaves over the range 0.68–0.75 μm

It is shown that replacement of the investigated surface of a leaf by the opposite one, as well as nonuniform content of chlorophyll over a pile of leaves, may lead to a change in the form of spectra for the first derivatives of reflection coefficients of leaves. A simplified model of the reflection of a leaf is used to explain the experimental results obtained. B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, 68, F. Skorina Ave., Minsk, 220072, Belarus. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 65, No. 6, pp. 921–925, November–December, 1998.