Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.Comment: 23 pages; this is an essentially extended version of the previous preprint: the construction of an equivariant cycle (co)homology and the spectral sequence (generalizing the long exact localization sequence) are adde

Multicomponent Electrocatalytic Selective Approach to Unsymmetrical Spiro[furo[3,2-c]pyran-2,5′-pyrimidine] Scaffold under a Column Chromatography-Free Protocol at Room Temperature

Electrochemical synthesis suggested a mild, green and atom-efficient route to interesting and useful molecules, thus avoiding harsh chemical oxidizing and reducing agents used in traditional synthetic methods. Organic electrochemistry offers an excellent alternative to conventional methods of organic synthesis and creates a modern tool for carrying out organic synthesis, including cascade and multicomponent ones. In this research, a novel electrocatalytic multicomponent transformation was found: the electrochemical multicomponent assembly of arylaldehydes, N,N′-dimethylbarbituric acid and 4-hydroxy-6-methyl-2H-pyran-2-one in one pot reaction was carried out in alcohols in an undivided cell in the presence of alkali metal halides with the selective formation of substituted unsymmetrical 1′,3′,6-trimethyl-3-aryl-2′H,3H,4H-spiro[furo[3,2-c]pyran-2,5′-pyrimidine]-2′,4,4′,6′(1′H,3′H)-tetraones in 73–82% yields. This new electrocatalytic process is a selective, facile and efficient way to obtain spiro[furo[3,2-c]pyran-2,5′-pyrimidines]. According to screening molecular docking data using a self-made Python script in Flare, all synthesized compounds may be prominent for different medical applications, such as breast cancer, neurodegenerative diseases and treatments connected with urinary tract, bones and the cardiovascular system

Multicomponent Electrocatalytic Selective Approach to Unsymmetrical Spiro[furo[3,2-<i>c</i>]pyran-2,5′-pyrimidine] Scaffold under a Column Chromatography-Free Protocol at Room Temperature

Electrochemical synthesis suggested a mild, green and atom-efficient route to interesting and useful molecules, thus avoiding harsh chemical oxidizing and reducing agents used in traditional synthetic methods. Organic electrochemistry offers an excellent alternative to conventional methods of organic synthesis and creates a modern tool for carrying out organic synthesis, including cascade and multicomponent ones. In this research, a novel electrocatalytic multicomponent transformation was found: the electrochemical multicomponent assembly of arylaldehydes, N,N′-dimethylbarbituric acid and 4-hydroxy-6-methyl-2H-pyran-2-one in one pot reaction was carried out in alcohols in an undivided cell in the presence of alkali metal halides with the selective formation of substituted unsymmetrical 1′,3′,6-trimethyl-3-aryl-2′H,3H,4H-spiro[furo[3,2-c]pyran-2,5′-pyrimidine]-2′,4,4′,6′(1′H,3′H)-tetraones in 73–82% yields. This new electrocatalytic process is a selective, facile and efficient way to obtain spiro[furo[3,2-c]pyran-2,5′-pyrimidines]. According to screening molecular docking data using a self-made Python script in Flare, all synthesized compounds may be prominent for different medical applications, such as breast cancer, neurodegenerative diseases and treatments connected with urinary tract, bones and the cardiovascular system

Electrocatalytic Cascade Selective Approach to 3-Aryl-2'<i>H</i>,3<i>H</i>,4<i>H</i>-Spiro{Furo[2,3-<i>с</i>]Chromene-2,5'-Pyrimidine}-2',4,4',6'(1'<i>H</i>,3'<i>H</i>)Tetraones and Its Automatic Screening Docking Studies

The general electrocatalytic cascade approach was found to medicinally relevant spirobarbituric 2,3-dihydro-4H-furo[2,3-c]chromen-4-one scaffold. This new electrocatalytic fast and efficient reaction of benzylidenebarbiturates and 4-hydroxycoumarin in alcohols in the presence of alkali metal halides as mediators in an undivided cell results in the formation of the substituted unsymmetric spirobarbituric 2,3-dihydro-4H-furo[2,3-с]chromen-4-ones in 72–83% yields. This general and efficient approach avoids traditional recrystallization and chromatographic purification stages. According to screening molecular docking data, all synthesized compounds may be prominent for different medical applications, such as control pain, breast cancer therapies, and treatment of obesity, epilepsy, hypertension, cardiovascular disease and diabetes.</p