Single‐Electron Transfer in σ‐Complexation Reactions of 2,6‐Dimethoxy‐3,5‐dinitropyridine with Para‐X‐phenoxide Anions in Aqueous Solution

Second‐order rate constants have been measured spectrophotometrically for reactions of 2,6‐dimethoxy‐3,5‐dinitropyridine 1 with 4‐X‐substituted phenoxide anions (X = OMe, Me, H, Cl, and CN) 2a–e in aqueous solution at various temperatures. The effect of phenoxide substituents on the reaction rate was examined quantitatively on the basis of kinetic measurements, leading to nonlinear correlations of Δ^≠H and Δ^≠S with Hammett's substituent constants (σ). Each Hammett plots exhibits two intersecting straight lines for the reactions of 1 with the phenoxide anions 2a–e , whereas the Yukawa–Tsuno plots for the same reactions are linear. The large negative ρ values (?4.03 to ?3.80) obtained for the reactions of 1 with the phenoxide anions possessing an electron‐donating group supports the proposal that the reactions proceed through a single‐electron transfer mechanism.     

Electrophilic reactivities of 7-L-4-nitrobenzofurazans in σ-complexation processes: Kinetic studies and structure–reactivity relationships

The second-order rate constants (k) for reaction of 7-chloro-4-nitrobenzofurazan 1 and 7-methoxy-4-nitrobenzofurazan 2 with a series of nitroalkyl anions and several of para-substituted phenoxide anions in aqueous solution at 20 °C have been reported. On the basis of the linear novel approach recently designed by Mayr and coworkers, the electrophilicity parameters E at the C-5 position of the two nitrobenzofurazans 1 and 2 have been quantified and ranked on the comprehensive electrophilicity scale. Mayr's approach was found to correctly predict the rate constants for the addition of phenoxide anions at the C-5 position of 1 and 2 witting a factor of <2. Analysis of the kinetic measurements using Brønsted's model shows that β_nuc values remain remarkably constant for changes in the nature of the substituent and that the σ-complexation process is associated with high Marcus intrinsic barriers. In addition, satisfactory correlations between the log k_exp (k_exp values measured in this work for reactions of benzofurazans 1 and 2 with a series of phenoxide anions in aqueous solution at 20 °C) and log k_calcd (k_calcd values calculated from equation 1 using the electrophilicity parameters E of benzofurazans 1 and 2 and the previously published nucleophilicity parameters N and s_N of the phenoxide anions) with a slope very close to unity have been obtained and discussed.     

On the Spectrum of the Poincaré Variational Problem for Two Close-to-Touching Inclusions in 2D

We study the spectrum of the Poincaré variational problem for two close to touching inclusions in R ². We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.     

Stochastic multi-site supply chain planning in textile and apparel industry under demand and price uncertainties with risk aversion

A multi-product, multi-period, multi-site supply chain production and transportation planning problem, in the textile and apparel industry, under demand and price uncertainties is considered in this paper. The problem is formulated using a two-stage stochastic programming model taking into account the production amount, the inventory and backorder levels as well as the amounts of products to be transported between the different plants and customers in each period. Risk management is addressed by incorporating a risk measure into the stochastic programming model as a second objective function, which leads to a multi-objective optimization model. The objectives aim to simultaneously maximize the expected net profit and minimize the financial risk measured. Two risk measures are compared: the conditional-value-at-risk and the downside risk. As the considered objective functions conflict with each other’s, the problem solution is a front of Pareto optimal robust alternatives, which represents the trade-off among the different objective functions. A case study using real data from textile and apparel industry in Tunisia is presented to illustrate the effectiveness of the proposed model and the robustness of the obtained solutions.     

ERROR ESTIMATES FOR THE RECURSIVE LINEARIZATION OF INVERSE MEDIUM PROBLEMS

This paper is devoted to the mathematical analysis of a general recursive linearization algorithm for solving inverse medium problems with multi-frequency measurements. Under some reasonable assumptions, it is shown that the algorithm is convergent with error estimates. The work is motivated by our effort to analyze recent significant numerical results for solving inverse medium problems. Based on the uncertainty principle, the recursive linearization allows the nonlinear inverse problems to be reduced to a set of linear problems and be solved recursively in a proper order according to the measurements. As an application, the convergence of the recursive linearization algorithm [Chen, Inverse Problems 13（1997）, pp.253-282] is established for solving the acoustic inverse scattering problem.     

Three-Period Quasi-Periodic Solutions in the Self-Excited Quasi-Periodic Mathieu Oscillator

In this work, we investigate three-period quasi-periodic (QP) oscillations in the vicinity of 2:2:1 resonance in a self-excited QP Mathieu equation using perturbation method. Two successive averaging are performed to reduce the original QP equation to an autonomous amplitude and phase system describing the modulation of the slow flow dynamic. Approximation of three-period QP solution is obtained via the study of limit cycle of the reduced autonomous system. The efficiency of the method is illustrated by comparison between analytical approximations and numerical integration. The double reduction procedure, applied in previous works to construct two-period QP solution, can be implemented to approximate excplicit analytical three-period QP solutions.     

Synthesis and characterization of new hexahelicene derivatives

The synthesis and structural characterization of a new hexahelicene derivative 2 is reported. This compound features a new type of structure bearing a bromine atom and a methoxy group. Suitable crystals of the latter indicate that its conformation closely resembles that of the unsubstituted [6]helicene, whose idealized symmetry is C2. This 3-bromo-14-methoxyhexahelicene was subjected to lithiation/phosphinilation and yielded the new 3-methoxy-14-(diphenylphosphino)hexahelicene 3.     

Synthesis and characterisation of a new helically chiral ruthenium complex

A new helically chiral hexacyclic phosphine, containing one thiophene ring was prepared in good yield via a five-step sequence involving a palladium-catalysed Mizoroki-Heck coupling reaction and classical oxidative photocyclisation. A mononuclear air stable ruthenium complex of this phosphine was also prepared and characterised.     

L 2 -discrete hedging in a continuous-time model

In the setting of the Black-Scholes option pricing market model, the seller of a European option must trade continuously in time. This is, of course, unrealistic from the practical viewpoint. He must then follow a discrete trading strategy. However, it does not seem natural to hedge at deterministic times regardless of moves of the spot price. In this paper, it is supposed that the hedger trades at a fixed number N of rebalancing (stopping) times. The problem (P^N) of selecting the optimal hedging times and ratios which allow one to minimize the variance of replication error is considered. For given N rebalancing, the discrete optimal hedging strategy is identified for this criterion. The problem (P^N) is then transformed into a multidimensional optimal stopping problem with boundary constraints. The restrictive problem (P^N _BS) of selecting the optimal rebalancing for the same criterion is also considered when the ratios are given by Black-Scholes. Using the vector-valued optimal stopping theory, the existence is shown of an optimal sequence of rebalancing for each one of the problems (P^N) and (P^N _BS). It also shown BS that they are asymptotically equivalent when the number of rebalances becomes large and an optimality criterion is stated for the problem (P^N). The same study is made when more realistic restrictions are imposed on the hedging times. In the special case of two rebalances, the problem (P² _BS) is solved and the problems (P² _BS) and (P²) are transformed into two optimal stopping problems. This transformation is useful for numerical purposes.