Even the first iterate of a Markov operator is contracting in anL 2 norm

A weightedL ₂ norm is introduced in which Markov operators, e.g., associated with noisy maps, are contracting provided the kernel (i.e., the transitional distribution) is smooth enough. This results in strong relaxational properties of noisy maps. Similar to this norm, integral functionals appear useful when studying spatiotemporal chaos and random fields.     

Synthesis of new types of aminomethylenediphosphorus-containing acids and their derivatives

Convenient methods for synthesis of various aminomethylenediphosphorus-containing acids and their derivatives starting from available trimethylsilyl esters of hypophosphorous and phosphorous acids, ethoxymethyleneimine hydrochlorides, and N-substituted formamides have been proposed. Selected properties of the obtained compounds have been examined.

     

Synthesis and optical properties of new coumarin derivatives based on 2-(2-chlorobenzylidene)malononitrile

A procedure has been developed for the conversion of the CS gas [2-(2-chlorobenzylidene)-malononitrile] to new 3-cyanocoumarin derivatives, 4-(2-chlorophenyl)-2-oxo-2H-chromene-3-carbonitriles. Study of the optical properties of the synthesized compounds has revealed strong fluorescence in the violet and blue regions.     

Synthesis of fused derivatives of 1,8-naphthyridine

Reactions of aromatic aldehyde, 2-aminoprop-1-ene-1,1,3-tricarbonitrile and cyclic enhydrazinoketones led to the formation of N-substituted 5-aryl-2,4-diamino-6-oxo-5,6,7,8,9,10-hexahydrobenzo[b][1,8]- naphthyridine-3-carbonitriles     

Synthesis of aminium 23,24,24-tricyano-3β-hydroxy-20-oxo-21-nor-17β-cholane-5,21-dien-24-ides from tetracyanoethylated pregnenolone

Russian Journal of Organic Chemistry - Reactions of tetracyanoethylated pregnenolone with ammonia and amines in anhydrous medium at reduced temperature afforded new ionic pregnenolone derivatives,...     

Synthesis of 2-(5-aryl-4-methyl-2-oxo-1,2-dihydro- 3<Emphasis Type="Italic">H</Emphasis>-pyrrol-3-ylidene)malononitriles

4-Oxoalkane-1,1,2,2-tetracarbonitriles in the presence of an equimolar amount of morpholine were converted to previously unknown 2-(5-aryl-4-methyl-2-oxo-1,2-dihydro-3H-pyrrol-3-ylidene)malononitriles.