Local probabilities and total variation distance for random permutations

We are concerned with permutations taken uniformly at random from the symmetric group. Firstly, we study the probability of a permutation missing short cycles. Secondly, the result is employed to establish a formula for total variation distance between the process of multiplicities of cycle lengths in a random permutation and a process of independent Poisson random variables. We apply an analytic approach originated in number theory (K. Gyory et al. (Eds.) in Number Theory in Progress, 1999).     

On a certain class of infinite products with an application to arithmetical semigroups

This work was done while the second author held a visiting professorship at the University of Paderborn. Financial support has been provided by DFG.     

A sharp inequality for the variance with respect to the Ewens sampling formula

Lithuanian Mathematical Journal - We examine the variance of a linear statistic defined on the symmetric group endowed with the Ewens probability. Despite the dependence of the summands, it can be...     

2-Methyl-1,4-naphthoquinones containing 3-[N-(ω-mercaptoalkyl)alkanamide] chains: synthesis, self-assembling, and electrochemical properties

2-Methyl-1,4-naphthoquinone derivatives containing 3-[N-(ω-mercaptoalkyl)alkanamide] chains were synthesized from ω-bromoalkylamine salts of 2-methyl-3-carboxyalkyl-1,4-naphthoquinones in the presence of N,N′-dicyclohexylcarbodiimide at ambient temperature, and then transformed into the corresponding mercapto derivatives. Their self-assembling and electrochemical properties on gold were studied. The influence of an intrachain amide group on the structure and electron transfer properties of self-assembled monolayers were evaluated by comparison with analogous ester and alkyl chain-containing 2-methyl-1,4-naphthoquinones.     

1,4-Bis(phosphine)-2,5-difluoro-3,6-dihydroxybenzenes and their P-oxides: Syntheses, structures, ligating and electronic properties

Reactions of 1,4-difluoro-2,5-dimethoxybenzene with LDA (1:2) at low temperatures generated organodilithio intermediates; quenching the reaction mixtures with chlorophosphines ClPR₂ produced 1,4-bis(phosphino)-2,5-difluoro-3,6-dimethoxybenzenes 1a (R = Ph) and 1b (R = iPr). Demethylation of 1a–b was accomplished by BBr₃, yielding bis(phosphino)hydroquinones 2a–b. Treating 2a–b with excess hydrogen peroxide produced bis(phosphinyl)hydroquinones 3a–b. The binucleating properties of 2a were established by the formation of a bimetallic nickel complex upon reaction with Ph₂Ni(PMe₃)₂. Electrochemical activity of hydroquinones 2a–b and 3a–b was examined by cyclic voltammetry. In addition, compounds 2a, 3a and 3b were obtained in crystalline form and characterized by single-crystal X-ray diffraction. The influence of the fluorine substituents on the composition of the frontier orbitals of 2a and 3a was examined by computational methods.     

Triphosphane formation from the terminal zirconium phosphinidene complex [Cp2Zr=PDmp(PMe3)] (Dmp=2,6-Mes2C6H3) and crystal structure of DmpP(PPh2)2

The new terminal phosphinidene complex [Cp₂Zr=PDmp(PMe₃)] (Dmp=2,6-Mes₂C₆H₃; 1) was prepared in 81% yield by the reaction of [Li(Et₂O)][P(H)Dmp] with [Cp₂Zr(Me)Cl] in the presence of excess PMe₃. Compound 1 reacts with Ph₂PCl to produce selectively the sterically congested triphosphane DmpP(PPh₂)₂ (2) and [Cp₂ZrCl₂] in high yields. The structure of 2 obtained by X-ray diffraction analysis of a single crystal reveals phosphorus–phosphorus bond lengths of 2.251(2) and 2.234(2) Å and a PPP bond angle of 105.46(6)°.     

Syntheses and structural characterizations of the unsymmetrical diphosphene DmpP=PMes* (Dmp =2,6-Mes(2)C(6)H(3), Mes* = 2,4,6-(t)Bu(3)C(6)H(2)) and the cyclotetraphosphane [DmpPPPh](2)

The new diphosphene DmpP=PMes* (Dmp = 2,6-Mes(2)C(6)H(3); Mes* = 2,4,6-(t)Bu(3)C(6)H(2), 1) having two different classes of sterically demanding aryls has been prepared and structurally characterized. This structure appears to be the first featuring both types of sterically demanding groups (a meta-terphenyl and Mes*) in a single molecule about a multiply bonded unit. Compound 1 features a P=P bond length of 2.024(13) A. The structure of 1 also allows comparisons to the two previously structurally characterized symmetric diphosphenes DmpP=PDmp and Mes*P=PMes*. The crystal structure of the cyclotetraphosphane [DmpPPPh](2) (3), the product of self-dimerization of the unstable diphosphene DmpP=PPh (2), has been determined. The structure of 3 demonstrates that a single bulky Dmp group is insufficient to prevent dimerization of 2. (31)P NMR data for all three compounds are also reported.     

Oxidation of cobalt(II) with air oxygen in aqueous ethylenediamine solutions

Oxidation of aqueous Co(NO₃)₂–ethylenediamine (En) solutions with air oxygen was investigated at 20 °C and pH 5.2–7.0, with and without mechanical stirring, by measuring the Co^II concentration, pH and redox potential on an Au electrode. In most cases, the oxidation rate was proportional to the concentration of CoEn ²⁺ _n (n = 2, 3) complexes, and the influence of the solution pH on the rate of reaction was accounted for by the pH dependence of the Co^II complex distribution. It was found that sulphate inhibits and bromide accelerates the oxidation process. Possible oxidation routes are discussed. The oxidation process is limited to some extent by O₂ transport from the air to the bulk solution.     

Carbon paste electrodes modified with Bi(2)O(3) as sensors for the determination of Cd and Pb

Carbon paste electrodes bulk-modified with Bi(2)O(3)were used for the determination of Cd(II) and Pb(II). The best composition was 1% (wt%) Bi(2)O(3) in the paste. The measurements were made by differential pulse voltammetry in the potential range from -1.2 V to -0.3 V. The peak potential of the reoxidation of Cd is -0.85 V, and of Pb -0.60 V vs. SCE. The lowest concentration that could be determined was 5 microg L(-1) of both metals (preconcentration time 240 s), the relative standard deviation was 3.5%-5.0% (four determinations). The correlation coefficient ( r(2)) of the calibration curves was 0.9966 (for Cd) and 0.9971 (for Pb). The Bi(2)O(3)-modified electrode could be used for the analysis of drinking water, mineral water and urine.