On rational approximations of functions of complex variable integrable over plane domains

пУстьE — ИжМЕРИМОЕ пО лЕБЕгУ ОгРАНИЧЕННОЕ МНОжЕстВО пОлОжИтЕльНОИ плОЩА ДИ mes₂ E кОМплЕксНОИ плОск ОстИ с. кАк ОБыЧНО, пРИp≧1 ОБОжНАЧИМ ЧЕРЕжL ^p (E) БА НАхОВО пРОстРАНстВО ИжМЕРИ Мых пО лЕБЕгУ НАE кОМплЕксНОжНАЧНых Ф УНкцИИf с сУММИРУЕМО Иp—стЕпЕНьУ Их МОДУль И ОБыЧНОИ НОРМОИ \(\left\| \cdot \right\|_p = \left\| \cdot \right\|_{L_p (E)}\) . ЧЕР ЕжL ^p R _n (f,E) ОБОжНАЧИМ НАИМЕН ьшЕЕ УклОНЕНИЕf?L ^p (E) От РАц ИОНАльНых ФУНкцИИ ст ЕпЕНИ ≦n кОМплЕксНОгО пЕРЕМЕ ННОгОz пО НОРМЕ ∥ · ∥. пОлОжИМf(z)=0 Дльz?¯CE,E _δ —δ-ОкРЕстНОсть МНО жЕстВАE (δ>0), И $$\omega _p (\delta ,f) = \mathop {\sup {\mathbf{ }}}\limits_{\left| h \right|< \delta } \{ \int\limits_{E_\sigma } {\int {{\mathbf{ }}|f(z + h) - f(z)|^p } d\sigma } \} ^{1/p} .$$ тЕОРЕМА.пУсть 1≦p<2,f?L _p (E),n≧4.тОгДА $$\begin{array}{*{20}c} {L^p R_n (f,E) \leqq 12\omega _p \left( {\frac{{\delta + \ln n}}{{\sqrt n }},f} \right){\mathbf{ }}npu{\mathbf{ }}p = 1,} \\ {L^p R_n (f,E) \leqq \frac{{24}}{{(p - 1)(2 - p)}}\omega _p (n^{(p - 2)/2p} ,f){\mathbf{ }}npu{\mathbf{ }}1< p< 2,} \\ {L^1 R_n (\bar z,[0,1] \times [0,1]) \geqq \frac{1}{{32\sqrt n }}.} \\ \end{array} $$ .     

Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain

Letd(;z, t) be the smallest diameter of the arcs of a Jordan curve with endsz andt. Consider the rapidity of decreasing ofd(;)=sup{d(;z, t):z, t , ¦z–t¦} (as 0,0) as a measure of nicety of . Letg(x) (x0) be a continuous and nondecreasing function such thatg(x)x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(x))². LetH(x) be an arbitrary primitive of 1/h ^–1(x). Note that the functionH ^–1 x is positive and increasing on (–, +),H ^–1 0 asx– andH ^–1+ asx +. The following statement is proved in the paper.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 176–184, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

Consecutive hydrogenation of pseudoionon. Application of diffusion kinetic model to predict experimental reactor operation

A method for the calculation of fixed catalyst bed reactors by using a diffusion kinetic model is suggested. Correlation between calculated curves for the time distribution of the reaction mixture composition and experimental data is given.

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Synthesis and biological activity of fluorinated 2-amino-4-aryl-3,4-dihydro[1,3,5]triazino[1,2-a]benzimidazoles

The heterocyclic nucleus s-triazino[1,2-a]benzimidazole has been reported to exhibit antibacterial activity. In this study, seven new 3,4-dihydro[1,3,5]triazino[1,2-a]benzimidazole derivatives were prepared via cyclocondensation between 2-guanidinobenzimidazole and fluorine substituted (including trifluoromethyl) benzaldehydes. The structures of all the compounds were confirmed by ¹H, ¹³C NMR and IR spectral data. Spectral data also suggested the existence of various tautomeric forms of the fluorine-containing s-triazino[1,2-a]benzimidazole compounds. The synthesized compounds were also screened for antibacterial and bovine dihydrofolate reductase (DHFR) inhibitory activities. The compound 3g substituted with a 3′,5′-bis(trifluoromethyl)phenyl moiety demonstrated the best antibacterial activity in the series. None of the tested compounds significantly inhibited bovine DHFR.     

4-Amino-substituted pyrazolo[1,5-a][1,3,5]triazin-2-amines: a new practical synthesis and biological activity

The trichloromethyl moiety was successfully employed as a leaving group in nucleophilic substitutions with various amines for the synthesis of 4-amino-substituted pyrazolo[1,5-a][1,3,5]triazin-2-amines. The key precursor for this reaction, 4-trichloromethylpyrazolo[1,5-a][1,3,5]triazin-2-amine, was prepared via the solvent-dependent condensation of 5-guanidino-3-phenylpyrazole with trichloroacetonitrile. In a broad biological activity screening, some of the prepared compounds were identified as CGRP receptor antagonists.     

Synthesis,structure, and optical properties of iridium(III) complex with 1-benzyl-2-phenylbenzimidazole and 4,4'-dicarboxy-2,2'-bipyridyne

A new cyclometalated iridium(III) complex [Ir(L)₂(Hdcbpy)] (1) has been synthesized, where L is 1-benzyl-2-phenylbenzimidazole and Hdcbpy is monoprotonated 4,4′-dicarboxy-2,2′-bipyridine. The structure of complex 1 has been determined by X-ray diffraction. The optical properties of complex 1 have been studied, and the quantum yield of luminescence has been measured.     

Metric spaces of semicontinuous functions

This work was supported by the Russian Foundation of Fundamental Research under Grant No. 93-011-236.     

Cyclocondensation of Nitromalonaldehyde Sodium Salt with Substituted 1-Amino-3-butanones

Cyclocondensation of nitromalonaldehyde sodium salt with substituted 1-amino-3-butanones afforded 2-dimethyl-, diethylaminomethyl-4-nitrophenols and 2-piperidylo-, morpholylomethyl-4-nitrophenols.