On Projective Mappings

Mathematical Notes - Let X,Y be Polish spaces, and let $\mathcal{B}_k $ be the σ-algebra generated by the projective class $L_{2k + 1} $ . A mapping $f:X \mapsto Y$ is called $k$...     

On the contribution of interference of electromagnetic and chromomagnetic dipole operators in the $$ \bar B $$ → Xsγ decay at O(αs2)

The contribution of c-quark loops into the interference of the electromagnetic and chromomagnetic dipole operators in the \(\bar B\) → X _s γ decay is calculated to O(α _s ² ) within the framework of the Standard Model (SM). Several new calculation techniques, such as Laporta’s automated algorithm, the second order decomposition method, and Mellin-Barnes representation, have been used.     

Physical and physicochemical processes accompanying powder synthesis,growth of PbMoO<Subscript>4</Subscript> crystals,and their annealing in various media: II. Shape of the crystal-melt interface caused by melt evaporation and supercooling

The temperature distribution in a melt of PbMoO₄ is investigated using a modified thermocouple and a scanning device. It is shown that the PbMoO₄ melt whose thermal field is significantly inhomogeneous at different depths demonstrates a strong (below the melting temperature) supercooling at the surface due to the evaporation. The position of the isotherm corresponding to the melting temperature of PbMoO₄ in the melt is determined by the degree of the melt supercooling at different growth parameters. The results obtained show that the conditions of mass and heat exchange in the melt are mainly determined by convection. The free convection remains dominant even at intense rotation of a crystal and the most thorough thermal isolation of the crystallization unit.     

Rearrangement–cyclization of dialkyl(4-hydroxybut-2-ynyl)meth-allyl- and dialkylcrotyl(4-hydroxy-but-2-ynyl) ammonium halides

It was found that dialkyl(4-hydroxybut-2-ynyl)methallyl- and dialkylcrotyl(4-hydroxybut-2-ynyl)- ammonium halides undergo a Stevens rearrangement under the influence of aqueous alkali with transfer of the reaction center in the receiving group and inversion of the migrating group followed by cyclization to amino derivatives of furan.     

Behavior of dialkylallyl(4-hydroxybut-2-ynyl)ammonium bromides toward powdered sodium hydroxide and aqueous alkali

Under the influence of a twofold molar quantity of powdered sodium hydroxide in the presence of a few drops of methanol at room temperature dialkylallyl(4-hydroxybut-2-ynyl)ammonium bromides undergo a Stevens rearrangement with transfer of the reaction center, forming substituted amino alcohols with an allene group. Intramolecular cyclization of the products and concurrent hydration lead to the formation of a mixture of dialkyl(4-allyl-2,5-dihydrofuran-2-yl)amines and dialkylamino-3-allyl-4-hydroxybutanones with overall yields of 37-41%. During the aqueous-alkali cleavage of dialkylallyl(4-hydroxybut-2-ynyl)ammonium bromides the products from intramolecular cyclization were obtained with yields of 38-41%. Under the conditions both of Stevens rearrangement and of aqueous-alkali cleavage the secondary amines are also formed with yields of 15-17%. As nonamine products mixtures of compounds, which according to the IR spectra contain unconjugated and conjugated carbonyl groups, were obtained. The presence of an aldehyde group was established by a silver mirror reaction. Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 3, pp. 354-360, March, 2009.     

Tunneling of electromagnetic waves through a barrier with a periodic structure

The propagation of an electromagnetic wave through a barrier with a periodic structure is considered. An interesting manifestation of electromagnetic wave tunneling through a barrier with a periodic structure is revealed. Specifically, it is theoretically and experimentally shown that, when an electromagnetic wave passes through a medium with a periodic structure in the diffraction mode within a photonic band gap, the field in the medium is localized near its boundaries. Theoretical calculations have been performed for a cholesteric liquid crystal layer of finite thickness and for a 1D photonic crystal. The experiment was carried out with perfect Si single crystals in reflection from the $ (2\bar 20)$ (2\bar 20) and $ (4\bar 40)$ (4\bar 40) planes using MoK _α X rays.     

Physical and physicochemical processes accompanying powder synthesis,growth of PbMoO<Subscript>4</Subscript> crystals,and their annealing in various media: III. Methods of controlling the crystallization front shape

Different methods of controlling the crystallization front shape are considered: variation in a crucible position in a heater, rotation of a crystal, introduction of impurities absorbing melt radiation, and formation of forced convection under dominant free convection in the melt.     

Transformations of dialkyl(4-hydroxy-2-butynyl)-(3-phenylallyl)ammonium bromides in an KOH aqueous solution or in the presence of powdered KOH

Under the action of a twofold excess of KOH and heating in aqueous solution, and also under the conditions of the Stevens rearrangement (with KOH powder and a small amount of methanol) dialkyl-(4-hydroxy-2-butynyl)(3-phenylallyl)ammonium bromides form dialkyl[4-(1-phenylallyl)-2,5-dihydro-2-furyl]amines. Rearrangement–cleavage reaction also occurs under the same conditions.