Inequalities in probability theory and turán-type problems for graphs with colored vertices

The “best” inequalities of type P{(ζ, η)? E} ≧f(P{η? D₁}P{η?Dm}) for independent and identically distributed random elements ζ and η can be reduced to Turán-type problems for graphs with colored vertices. In the present work we describe a finite algorithm for obtaining the asymptotical solution for an arbitrary problem of such type. In the case of two colors we obtain the final form of asymptotic solution without using the algorithm.     

A general strategy for the synthesis of oxoisoindolo[2,1-a]quinoline derivatives: the first efficient synthesis of 5,6,6a,11-tetrahydro-11-oxoisoindolo[2,1-a]quinoline-10-carboxylic acids

An efficient two-step synthesis of new isoindolo[2,1-a]quinoline-10-carboxylic acids via [4+2] cycloaddition of the 4-α-furyl-4-N-arylaminobut-1-enes and maleic anhydride is described.     

Correcting an experimental absorption spectrum for the core level width

The problem of correcting EXAFS spectra by the core level width is discussed. A method of correcting experimental EXAFS spectra is developed. An iteration algorithm for solving the integral equation is proposed. This method has a number of advantages over other methods, namely, high stability against statistical error and fast convergence. Model numerical calculations are performed. An example is the EXAFS spectrum of polycrystalline Pd for which the broadening effect is significant. Translated fromZhumal Struktumoi Khimii, Vol. 39, No. 6, pp. 1018–1024, November–December, 1998.     

Muonic study of type II superconductors

A theory of μSR method is developed for uniaxial anisotropic high-T _c superconductors. In two extreme cases ofH _ext‖c andH _ext⊥c analytical formulas are obtained which makes it possible to determine from the position of van Hove singularities in the Fourier-spectrum of muonium polarization, the type and parameters of the vortex lattice, such as λ _ab (λ _c ). In caseH _ext≲H _c2 we obtained the shape of the Fourier-spectrum in analytical form and the simple method of determining the Ginsburg-Landau parameter κ. Convenient expressions for numerical calculations are obtained for the arbitrary orientation of the external field, and an algorithm is provided to compute the mean fieldB, the vortex lattice parameters and the bulk field distribution in anisotropic superconductor. The Fourier spectrum of polarization based on these calculations can be used to independently check the validity of the high-T _c parameters determination for “appropriate” orientations.     

What we know and what we do not know about Turán numbers

The numbers which are traditionally named in honor of Paul Turán were introduced by him as a generalization of a problem he solved in 1941. The general problem of Turán having anextremely simple formulation but beingextremely hard to solve, has become one of the most fascinatingextremal problems in combinatorics. We describe the present situation and list conjectures which are not so hopeless.     

Impact of dangling bonds on the surface of silicon nanocrystals doped with shallow donors upon radiative interband transitions

It is found that the radiative recombination rate rises when silicon nanocrystals are doped with donors. This rise turns out to be stronger when a nanocrystal surface contains abundant defects capable of capturing electrons emitted by donors.     

4-hydroxy-2-quinolones 172*. Synthesis and structure of 4,3'-spiro[(6-allyl-2-amino- 5-oxo-5,6-dihydro-4h-pyrano- [3,2-<Emphasis Type="Italic">c</Emphasis>]quinoline-3-carbo- nitrile)-2'-oxindole]

A three-component condensation of 1-allyl-4-hydroxy-2-oxo-1,2-dihydroquinoline, isatin, and malononitrile gave a satisfactory yield of 4,3'-spiro[(6-allyl-2-amino-5-oxo-5,6-dihydro-4H-pyrano-[3,2-c]quinoline-3-carbonitrile)-2'-oxindole], which structure was confirmed by X-ray analysis.     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.