On mean values of multiplicative functions on the symmetric group

Mean values of nonnegative multiplicative functions defined on the symmetric group are explored in the paper. The result gives a sharp quantitative upper bound for their Cesàro mean. An approach that originated in number theory is adopted. It can be further applied for mappings defined on general decomposable structures, in particular, for estimating mean values with respect to multiplicative measures defined on additive partitions of a natural number.     

Variance of additive functions defined on random assemblies

We establish an inequality for the variance of an additive function defined on random decomposable structures called assemblies. The result generalizes estimates obtained earlier in the cases of permutations and mappings of a finite set into itself. It is analogous to the Turán–Kubilius inequality for additive number-theoretic functions.     

Total variation approximation for random assemblies and a functional limit theorem

A class of random weakly logarithmic combinatorial assemblies is explored in the paper. We extend total variation approximations for the distribution of component vector of a random structure. That leads to the probabilistic approach suitable to examine the asymptotic value distribution of additive functions defined on such assemblies with the component sizes restricted to a given set. The results generalize several investigations of random Λ-permutations and their extensions to other structures obtained mainly by the Russian mathematicians. Instead of the most popular approach based upon the Tauber type theorems, we develop a comparative asymptotical analysis of coefficients of two Taylor series. Demonstrating possible applications, we obtain necessary and sufficient conditions for the weak convergence of processes defined via partial sums of an additive function to the Brownian motion.     

Strong convergence on weakly logarithmic combinatorial assemblies

We deal with the random combinatorial structures called assemblies. Instead of the traditional logarithmic condition which assures asymptotic regularity of the number of components of a given order, we assume only lower and upper bounds of this number. Using the author’s analytic approach, we generalize the independent process approximation in the total variation distance of the component structure of an assembly. To evaluate the influence of strongly dependent large components, we obtain estimates of the appropriate conditional probabilities by unconditioned ones. The estimates are applied to examine additive functions defined on a new class of structures, called weakly logarithmic. Some analogs of Major’s and Feller’s theorems which concern almost sure behavior of sums of independent random variables are proved.     

Asymptotic value distribution of additive functions defined on the symmetric group

We examine the asymptotic value distribution of additive functions defined via the multiplicities of lengths of the cycles comprising a random permutation taken from the symmetric group with equal probability. We establish necessary and sufficient conditions for the weak law of large numbers and for the relative compactness of the sequence of distributions. Considering particular cases, we demonstrate that long cycles play an exceptional role and that, sometimes, in order to obtain a Poisson limit law, their influence must be negligible. The proofs are based on the ideas going back to the seminal papers of I.Z. Ruzsa on the classical additive arithmetic functions.      

Syntheses of symmetric and unsymmetric 2,6‐bis(phosphino)phenols

Compounds bearing the structural motif of 2,6‐bis(phosphino)phenol have been synthesized via two general methods. Double lithium‐halogen exchange occurred in low‐temperature reactions of O‐protected (by methyl‐ or tetrahydropyranyl groups) 2,6‐dibromo‐4‐methylphenol derivatives with BuLi (2 equivalents); quenching the reaction mixtures with chlorophosphines ClPR₂ (R = Ph, ⁱPr) and corresponding O‐deprotection yielded symmetrically substituted 2,6‐bis(phosphino)phenols. Sequential incorporation of  PR₂ functionalities was accomplished via single lithium‐halogen exchange (1 eq. of BuLi) of tetrahydropyranyl‐protected 2,6‐dibromo‐4‐methylphenol followed by ClPR₂ quenches, thus enabling the syntheses of unsymmetric 2,6‐bis(phosphino)phenols. Such compounds were also obtained via sequential ortho‐lithiations of tetrahydropyranyl‐protected 4‐tert‐but ylphenol, followed by ClPR₂ quenches. All of the new compounds have been characterized by spectrometric methods (¹H and ³¹P NMR, and mass spectrometry). In addition, two of the compounds, 1‐(diphenylphosphino)‐3‐(diphenylphosphoryl)‐2‐methoxy‐5‐methylbenzene ( 3a‐ox ) and 1,3‐bis(diphenylphosphino)‐2‐methoxy‐5‐methylbenzene ( 6a ) have also been characterized via single crystal X‐ray diffraction experiments. © 2006 Wiley Periodicals, Inc. Heteroatom Chem 17:656–663, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20251     

Ring contraction/transannular cyclization of chiral bicyclo[3.3.1]nonanediones mediated by thallium(III) nitrate

The reaction of several chiral bicyclo[3.3.1]nonane ketones mediated by thallium(III) nitrate to afford ring contraction products is investigated. The effect of solvent on the oxidation is discussed, and the use of thallium(III) nitrate for the oxidative rearrangement of regioisomers of bicyclononanones is outlined. In compounds with proximately located carbonyl groups in the bicyclic framework, unexpected transannular ring closure was observed.