Conformal blocks and generalized Selberg integrals

Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special “conservation” relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko–Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an n-point conformal block on Riemann sphere, one reproduces the earlier conjectured β-ensemble representation of conformal blocks. The statement can also be regarded as a relation between the 3j  -symbols of the Virasoro algebra and the slightly generalized Selberg integrals IY$I_Y$, associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.     

Tellurium-containing rhenium cluster complexes

Synthetic procedures as well as the structures and some properties of tetrahedral and octahedral tellurium-containing rhenium cluster complexes are reported. Crystal chemistry of various types of structure is discussed.     

Analysis of the spatial orientation distribution of building blocks in polycrystals as determined using transmission electron microscopy and a backscattered electron beam in a scanning electron microscope

The advantages and disadvantages of the method of automatic analysis of electron backscattering diffraction (EBSD) patterns for studying spatial orientation distributions are considered as compared to transmission electron microscopy (TEM). A misorientation spectrum in a test alloy (Kh20N80 alloy) having a high content of annealing twins is experimentally studied using both TEM and EBSD, and the results obtained are compared.     

Some algebraic examples of Frobenius manifolds

Using analytic methods of finite-gap integration, we construct quasihomogeneous algebraic solutions of the WDVV associativity equations and the nonsemisimple Frobenius manifolds associated with them. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 195–206, May, 2007.     

Self-Action of a Supershort Laser Pulse in a Medium with Normal Group-Velocity Dispersion

We present the results of numerical and analytical analysis of solutions of the three-dimensional (3D) nonlinear Schröodinger equation with hyperbolic spatial operator. Evolution of the system is considered in separate for two types of the initial field: a Gaussian distribution and a hollow-type (tubular or horseshoe) distribution. The effect of the nonlinear dispersion on wave-packet splitting during self-compression toward the system axis is studied. It is shown that additional focusing of Gaussian wave packets takes place in a wide range of the nonlinear-dispersion parameter. This effect results in a noticeable amplitude growth of one of the two secondary pulses formed as a result of the splitting. For hollow-type distributions, we note the formation of moving inhomogeneities and the excitation of secondary wave fields typical of the hyperbolic system.     

Tris(fluoroalkoxy)methylphosphonium salts in the synthesis of methylphosphoranes

The reactions of phosphites containing electron-withdrawing fluoroalkyl groups with methyl triflate gave tris(fluoroalkoxy)methylphosphonium triflates, which react with alcohols to give either tetrakis(fluoroalkoxy)methylphosphoranes (in the case of fluoroalcohols) or alkylation products — ethers and bis(fluoroalkoxy)methylphosphonates (in the case of unsubstituted alcohols). The structures of the phosphonium salts and phosphoranes were supported by¹H and³¹P NMR spectroscopy.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 4, pp. 946–948, April, 1991.     

Thermally stimulated currents in ice. Correction for orientational defects

The Bernal and Fowler model of ice [1] has been supplemented with modern ideas about relaxing defects arising as a result of migration of a proton along hydrogen bonds [2, 3]. We develop a theory for thermally stimulated currents (TSC) in hexagonal ice. We propose a kinetic equation taking into account relaxation of the structural defects supporting proton conductivity in the crystals. We obtain formulas qualitatively describing thermally stimulated depolarization currents and also a formula for calculating the activation energy for structural defects, allowing us to compare with experiment. Due to the unwieldiness of the calculations, we present only the derivation of the kinetic equations and analysis of the solution.Karaganda Polytechnical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 93–98, February, 1994.     

New Crystalline Lasers on the Base of Monoclinic KR(W04)2:Ln3+ Tungstates (R=Y and Ln)

Monoclinic KY(WO₄)₂:Pr₃₊, Tm₃₊:KY(WO₄)₂:Er₃₊, Yb₃₊ single crystals are grown by low-gradient Czochralski technique. Stimulated emission at 1.0223μm wavelength in the new lasing ¹D₂→³F₃ channel of Pr³⁺ ions in KY(WO₄)₂ at room temperature under Xe-flashlamp pumping is excited. Full sets of Stark-level energies for Pr₃₊ ions in KY(WO₄)2 and KGd(WO₄)₂ crystals at 77 K are determined. 4f²-4f² intensity-transitions for the KY(WO₄)₂:Pr³⁺ are analyzed preliminarily. All observed at present and earlier one-micron Pr³⁺-ion, induced transitions in the ¹D₂→³F₃ and ¹D₂→³F₄ channels are identified. In yttrium and gadolinium sensitized-tungstates a two-micron (³H₄→³H₆) low-threshold generation of Tm³⁺ ions at cryogenic temperatures is achieved.All authors cooperate with the Joint Open Laboratory for Laser Crystals and Precise Laser Systems at the Institute of Crystallography and Institute of Laser Physics, Russian Academy of Sciences.