Functions of Pairs of Unbounded Noncommuting Self-Adjoint Operators under Perturbation

Doklady Mathematics - For a pair $$(A,B)$$ of not necessarily bounded and not necessarily commuting self-adjoint operators and for a function f on the Euclidean space $${{\mathbb{R}}^{2}}$$ that...     

Nitroazines 24. stable anionic σ-adducts of condensed nitrotriazines

Stable anionic -adducts of azolo-ring condensed 1, 2, 4-traizines with water were synthesized, and their x-ray structural analysis was carried out.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 1, pp. 52–57, January, 1994.     

Numerical Simulation of Rayleigh–Bènard Instability in Solutions Containing Ions of Three Types

The conditions for appearance of a Rayleigh–Benard instability (RBI) induced by a constant current between two horizontal electrodes in solution containing electroactive ions and indifferent electrolyte (system with three kinds of ions) are analyzed. It is taken into account that all the components take part in the formation of buoyant forces. The problem is solved for a stationary case (as an expansion to a series by a small parameter), and then equations for low perturbations are solved by the Galerkin method. The calculation is performed with a program that accounts for any number of members in a Galerkin expansion. A diagram is obtained, which determines conditions for the RBI emergence in terms of ratios of partial Raleigh numbers. The ratio of diffusion coefficients for supporting cations and anions significantly affects position of the boundary dividing ranges of convective stability and RBI in a diagram of nonequilibrium phase transitions.     

Synthesis of enaminoamides of the 1,2,3,4-tetrahydroisoquinoline series

Enaminoamides of the 1,2,3,4-tetrahydroisoquinoline series having the Z-configuration were synthesized by the reaction of dialkylbenzylcarbinols cyanaceamides.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 9, pp. 1239–1242, September, 1989.     

Relative rate constants of reactions of alkyl radicals of methacrylates and acrylates with oxygen and stable nitroxyl radicals

Conclusions In methacrylates and acrylates undergoing oxidation, stable nitroxyl radicals react with the alkyl radicals of the monomers. The relative rate constants of the reactions of and O₂ with alkyl radicals for the different esters range from 0.36 to 5.20.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 4, pp. 753–756, April, 1977.We thank V. A. Golubev for providing the stable nitroxyl radicals.     

Synthesis and crystal structure analysis of binary molybdate CsNamoO4 · 2H2O

Single crystals of CsNaMoO₄ · 2H₂O were synthesized, and the crystal structure of this compound was determined from X- ray diffraction data (SYNTEX P2₁ automatic diffractometer, MoK_α radiation, 2252 reflections, R =0.037). Crystals are orthorhombic, space group P2₁2₁2₁ with unit cell dimensions a =6.379(1), b =8.625(1), c =13.671(2) å, Z =4. The structure of the binary molybdate consists of continuous columns of face- sharing Na octahedra with attached MoO₄ tetrahedra. The neighboring columns are linked by the Cs cations and the hydrogen bonds of the water molecules entering into the coordination polyhedra of the Na and Cs atoms.     

Cubane derivatives

New substituted cubane-1,4-dicarboxamides and 4-bromocubanecarboxamides containing the nitroxyl group and amino acid fragments along with the cubane skeleton were synthesized. The cardiac pharmacological activity (both the effect on aorta relaxation and the calcium channel blocking activity) of most of nitroxyalkyl derivatives is equal to or higher than that of N-(2-nitroxyethyl)nicotinamide (nicorandil).     

Formation and Transformations of Polynuclear Hydroxo- and Oxotrimethylacetate Complexes of Ni(II) and Co(II)

Transformations of polymeric trimethylacetate complexes [M(OH) _n (OOCCMe₃)_{2 – n} ] _m (M = Ni (I) and Co (II)) and clusters Ni₉(₄-OH)₃(₃-OH)₃(-O,O-OOCCMe₃)(-O,O"-OOCCMe₃)₇(₃-O,O,O"-OOCCMe₃)₃(₄-O,O,O",O"-OOCCMe₃)(HOOCCMe₃)₄(III) and Co₆(₃-OH)₂(-OOCCMe₃)₁₀(HOOCCMe₃)₄(VIII), which are formed from Iand IIupon their recrystallization from nonpolar solvents, were studied. It was shown that the action of N-phenyl-o-phenylenediamine (L) on Ior IIIresults, depending on the solvent, in different tetranuclear clusters with the hydroxo bridges. For example, in benzene, the L₂Ni₄(₃-OH)₂(HOOCCMe₃)₄(-OOCCMe₃)₆complex (IX) is formed; its L molecules are coordinated in a monodentate way, whereas in acetonitrile, they chelate to give the {[o-C₆H₄(NH₂)(NHPh)]₂Ni₄(₃-OH)₂(MeCN)₂(OOCCMe₃)₂(-OOCCMe₃)₄} compound (X). Heating of Xin the presence of atmospheric oxygen yields IX, the mononuclear bissemiquinonediimine [o-C₆H₄(NH)(NPh)]₂Ni complex (XI), and water. It was noted that the use of aniline in these reactions affords, independent of the nature of the solvent, only one (NH₂C₆H₅)₂Ni₄(₃-OH)₂(HOOCCMe₃)₄(-OOCCMe₃)₆cluster (VI); in acetonitrile, this cluster is formed as the solvate VI· 2HOOCCMe₃(VIa). When treated with ethanol, Iand IIIgive the Ni₄(EtOH)₆(₃-OH)₂(₂-OOCCMe₃)₄(OOCCMe₃)₂cluster (V), which is structurally close to the known cobalt-containing analog IV. Thermolysis of IVin decalin at 170° causes its dimerization, giving the octanuclear Co₈(₄-O)₂( _n -OOCCMe₃)₁₂complex (VII) with the tetradentate oxo bridges.     

Invariant subspaces of the shift operator. Axiomatic approach

There is axiomatically described the class of spaces (resp. ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) z f (z) (resp. the inverse shift f(z)z^–1(f(z)–f (0))) are constructed just like the Hardy space H². It is proved that as one can take, for example, the space H¹, the disk-algebra C_A, the space U_A of all uniformly convergent power series; and as the space of integrals of Cauchy type L¹/H _– ¹ , the space VMO_A. There is also obtained an analog for the space U_A of W. Rudin's theorem on z-invariant subspaces of the space C_A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 113, pp. 7–26, 1981.