On the Askey-Wilson polynomials

Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalization of Hahn's approach.     

Renormalons and analytic properties of the β function

The presence or absence of renormalon singularities in the Borel plane is shown to be determined by the analytic properties of the Gell-Mann-Low function β(g) and some other functions. A constructive criterion for the absence of singularities consists in the proper behavior of the β function and its Borel image at infinity, β(g) ∝ g^α and B(z) ∝ z^α with α ≤ 1. This criterion is probably fulfilled for the ?⁴ theory, quantum electrodynamics, and quantum chromodynamics, but is violated in the O(n)-symmetric sigma model with n → ∞.     

Higher order corrections to the Lipatov asymptotics in the ϕ<Superscript>4</Superscript> theory

Higher orders in perturbation theory can be calculated by the Lipatov method [1]. For most field theories, the Lipatov asymptotics has the functional form ca ^N Γ(N+b (N is the perturbation theory order); relative corrections to this asymptotics have the form of a power series in 1/N. The coefficients of higher order terms of this series can be calculated using a procedure analogous to the Lipatov approach and are determined by the second instanton in the field theory in question. These coefficients are calculated quantitatively for the n-component ?⁴ theory under the assumption that the second instanton is (i) a combination of elementary instantons and (ii) a spherically asymmetric localized function. A technique of two-instanton computations, as well as the method for integrating over rotations of an asymmetric instanton in the coordinate state, is developed.     

Calculation of convective heat flux in the vicinity of the stagnation point for partially ionized gas flow past a body

From numerical solutions of the boundary layer equations for a four-component gas mixture (E, N⁺, N₂, and N) with gas injection, approximate formulas for the heat flux as a function of the variation of λρ/c_p and h^* across the boundary layer and the magnitude of the objection are obtained (λ is the thermal conductivity of the mixture,ρ is density, c_p is the specific heat, and h^* is the enthalpy of the ideal gas state of the mixture). An effective ambipolar diffusion coefficient D^(a)(i) is introduced, making possible finite formulas for the convective heat fluxes in the “frozen” boundary layer. We study the behavior of these coefficients within the boundary layer. A formula is obtained for convective heat flux to the wall from partially ionized air for a nine-component mixture (E, O⁺, N⁺, NO⁺, O, N, NO, O₂ N₂). Even for simpler four-component gas model three effective ambipolar diffusion coefficients are necessary: $$\begin{gathered} D^{(a)} (A) = D (A, M) D^{(a)} (I) = 2D (A, M), \hfill \\ D^{(a)} (M) = [ 1 + c_e (I)] D(A, M). \hfill \\ \end{gathered} $$ Here D(A, M) is the binary diffusion coefficient of the atoms into molecules, and c_e(I) is the ion concentration at the outer edge of the boundary layer. The assumption of an infinitely large charge-exchange cross section and the other simplifying assumptions used in [1] lead to overestimation of the magnitude of the dimensionless heat flux by 7–15% for the “frozen” boundary layer case.     

Localization theory in zero dimension and the structure of the diffusion poles

The 1/[-iω + D(ω, q)q ²] diffusion pole in the localized phase transfers to the 1/ω Berezinskii-Gorkov singularity, which can be analyzed by the instanton method {M. V. Sadovski?, Zh. Éksp. Teor. Fiz. 83, 1418 (1982) [Sov. Phys. JETP 56, 816 (1982)] and J. L. Cardy, J. Phys. C 11, L321 (1978)}. When this approach is used directly, contradictions arise and do not disappear even if the problem is extremely simplified by taking the zero-dimensional limit. On the contrary, they are extremely sharpened in this case and become paradoxes. The main paradox is specified by the following statements: (i) the 1/ω singularity is determined by high orders of perturbation theory, (ii) the high-order behaviors for Φ^RA and U ^RA are the same, and (iii) Φ^RA has the 1/ω singularity, whereas U ^RA does not have it. Solution to the paradox indicates that the instanton method makes it possible to obtain only the 1/(ω + 2iγ) singularity, where the parameter γ remains indefinite and must be determined from additional conditions. This conceptually confirms the necessity of the self-consistent treatment of the diffusion coefficient used in the Vollhardt-Wölfle-type theories.     

Adamantane-Monoterpenoid Conjugates Linked via Heterocyclic Linkers Enhance the Cytotoxic Effect of Topotecan

Inhibiting tyrosyl-DNA phosphodiesterase 1 (TDP1) is a promising strategy for increasing the effectiveness of existing antitumor therapy since it can remove the DNA lesions caused by anticancer drugs, which form covalent complexes with topoisomerase 1 (TOP1). Here, new adamantane–monoterpene conjugates with a 1,2,4-triazole or 1,3,4-thiadiazole linker core were synthesized, where (+)-and (−)-campholenic and (+)-camphor derivatives were used as monoterpene fragments. The campholenic derivatives 14a–14b and 15a–b showed activity against TDP1 at a low micromolar range with IC₅₀ ~5–6 μM, whereas camphor-containing compounds 16 and 17 were ineffective. Surprisingly, all the compounds synthesized demonstrated a clear synergy with topotecan, a TOP1 poison, regardless of their ability to inhibit TDP1. These findings imply that different pathways of enhancing topotecan toxicity other than the inhibition of TDP1 can be realized.     

Novel Bispidine-Monoterpene Conjugates—Synthesis and Application as Ligands for the Catalytic Ethylation of Chalcones

A number of new chiral bispidines containing monoterpenoid fragments have been obtained. The bispidines were studied as ligands for Ni-catalyzed addition of diethylzinc to chalcones. The conditions for chromatographic analysis by HPLC-UV were developed, in which the peaks of the enantiomers of all synthesized chiral products were separated, which made it possible to determine the enantiomeric excess of the resulting mixture. It was demonstrated that bispidine-monoterpenoid conjugates can be used as the ligands for diethylzinc addition to chalcone C=C double bond but not as inducers of chirality. Besides products of ethylation, formation of products of formal hydrogenation of the chalcone C=C double bond was observed in all cases. Note, that this formation of hydrogenation products in significant amounts in the presence of such catalytic systems was found for the first time. A tentative scheme explaining the formation of all products was proposed.     

Structure of higher order corrections to the Lipatov asymptotic form

High orders of perturbation theory can be calculated by the Lipatov method, whereby they are determined by saddle-point configurations, or instantons, of the corresponding functional integrals. For most field theories, the Lipatov asymptotic form has the functional form ca ^NΓ(N+b) (N is the order of perturbation theory) and the relative corrections to it are series in powers of 1/N. It is shown that this series diverges factorially and its high-order coefficients can be calculated using a procedure similar to the Lipatov one: the Kth expansion coefficient has the form const[ln(S ₁/S ₀)]^?K Γ(K+(r ₁? r ₀)/2), where S ₀ and S ₁ are the values of the action for the first and second instantons of this particular field theory, and r ₀ and r ₁ are the corresponding number of zeroth-order modes; the instantons satisfy the same equation as in the Lipatov method and are assumed to be renumbered in order of their increasing action. This result is universal and is valid in any field theory for which the Lipatov asymptotic form is as specified above.     

Asymptotic behavior of the β function in the ϕ<Superscript>4</Superscript> theory: A scheme without complex parameters

The previously-obtained analytical asymptotic expressions for the Gell-Mann-Low function β(g) and anomalous dimensions in the ϕ⁴ theory in the limit g → ∞ are based on the parametric representation of the form g = f(t), β(g) = f ₁(t) (where t ∝ g ₀^−1/2 is the running parameter related to the bare charge g ₀), which is simplified in the complex t plane near a zero of one of the functional integrals. In this work, it has been shown that the parametric representation has a singularity at t → 0; for this reason, similar results can be obtained for real g ₀ values. The problem of the correct transition to the strong-coupling regime is simultaneously solved; in particular, the constancy of the bare or renormalized mass is not a correct condition of this transition. A partial proof has been given for the theorem of the renormalizability in the strong-coupling region.