Sub-Riemannian distance on the Lie group SL(2)

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ? SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.     

Compression of a Runaway Electron Flow in an Air Gap with a Nonuniform Magnetic Field

JETP Letters - The first results of study of the compression of a picosecond runaway electron flow in an air electrode gap by a pulsed guiding magnetic field increasing along the electron...     

Exact Solutions to the Problem of Dynamics of a Liquid with a Free Surface between Two Approaching Vertical Walls

Doklady Physics - Exact solutions of the classical problem of a plane unsteady potential flow of an ideal incompressible fluid with a free boundary are presented. The fluid occupies a semi-infinite...     

Equilibrium configurations of the conducting liquid surface in a nonuniform electric field

Possible equilibrium configurations of the free surface of a conducting liquid deformed by a nonuniform external electric field are investigated. The liquid rests on an electrode that has the shape of a dihedral angle formed by two intersecting equipotential half-planes (conducting wedge). It is assumed that the problem has plane symmetry: the surface is invariant under shift along the edge of the dihedral angle. A one-parametric family of exact solutions for the shape of the surface is found in which the opening angle of the region above the wedge serves as a parameter. The solutions are valid when the pressure difference between the inside and outside of the liquid is zero. For an arbitrary pressure difference, approximate solutions to the problem are constructed and it is demonstrated the approximation error is small. It is found that, when the potential difference exceeds a certain threshold value, equilibrium solutions are absent. In this case, the region occupied by the liquid disintegrates, the disintegration scenario depending on the opening angle.     

Catalysis of the alkaline hydrolysis of 4-nitrophenyl diethyl phosphonate by cationic dimeric surfactant micelles

Dimeric (gemini) surfactants containing hydroxyl functional groups have been synthesized and studied. These new surfactants have low critical micelle formation concentrations (<5·10⁻⁵ mol/L) and Krafft temperatures (≤0 °C). Depending on the pH of the medium, the alkaline hydrolysis of 4-nitrophenyl diethyl phosphonate, which is a model of organophosphorus ecotoxicants, proceeds 20–80 times more rapidly in the presence of gemini surfactant micelles than in water.     

Stereoselective hydrogenation of a menthone-isomenthone mixture on heterogeneous nickel,nickel-cobalt,and cobalt catalysts

A study was carried out on the effect of the nature of the catalysts, additives, and solvents on the stereoselectivity of the liquid-phase hydrogenation of an equilibrium menthone-isomenthone mixture to give menthols. Neoisomenthol, which is the least stable of all the menthol isomers, was predominantly formed on a cobalt catalyst modified by (+)-tartaric acid in ethyl acetate at 130°C and 10 MPa.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 8, pp. 1920–1923, August, 1989.     

Abnormal Extremals of Left-Invariant Sub-Finsler Quasimetrics on Four-Dimensional Lie Groups with Three-Dimensional Generating Distributions

Siberian Mathematical Journal - We find the three-dimensional subspaces of four-dimensional Lie algebras which generate the algebras, as well as abnormal extremals on the connected Lie groups...     

A study of sodium bis(2-ethylhexyl) sulfosuccinate adsorption on silica from silver and gold organosols

Adsorption of sodium bis(2-ethylhexyl) sulfosuccinate (Aerosol OT) and silver and gold nanoparticles on SiO₂ from heptane solutions has been investigated by spectophotometery and CHN analysis. The adsorption isotherm of Aerosol OT is described by the Langmuir equation. Sorption capacity (2.4 × 10^–4 mol/g), sorption constant (9.0 m³/mol), and area per Aerosol OT molecule in an adsorption layer (0.83 nm²) have been determined. It has been shown that, at Aerosol OT concentrations lower than 6 × 10^–4 M, gold and silver ((0.3–6) × 10^–4 M) are simultaneously extracted by 98%; however, they have no effect on the adsorption and determination of the surfactant.     

Synthesis,some properties,and crystalline modifications of <Emphasis Type="Italic">fac</Emphasis>-[Ru(NO)(Py)<Subscript>2</Subscript>Cl<Subscript>3</Subscript>]

According to the data of ¹H NMR spectroscopy, trans-hydroxochloro complexes containing from two to four pyridine molecules in the internal sphere are formed on the heating of a dilute aqueous solution of K₂[Ru(NO)Cl₅] with pyridine. The evaporation of the reaction solution with concentrated hydrochloric acid gives fac-[Ru(NO)(Py)₂Cl₃] (I) in a yield of ~90%. The structures of two crystalline modifications of this complex are determined by X-ray diffraction analysis (CIF files ССDС nos. 1452208 (Ia) and 1452207 (Ib)). IR spectroscopy shows that the irradiation of complex I (λ ~ 450 nm, T = 80 K) results in photoisomerization with the formation of the metastable state MS1 in which the nitroso group is coordinated by the oxygen atom. The activation parameters of the photoisomerization are determined from the data of differential scanning calorimetry (DSC). Compound trans-[Ru(NO)Py₄(OH)]Cl₂ ? H₂O is isolated in a yield of ~70% on reflux of complex I with a pyridine excess in an aqueous solution, and the presence of molecules of water of crystallization in this compound is confirmed by thermal gravimetry (TG) and IR spectroscopy.