Experimental study on the efficiency and accuracy of a chance-constrained programming algorithm

The CHAPS algorithm (CHAPS = Chance-Constrained Programming System) has proved to be an efficient and accurate method for solving linear optimization problems which have several random variables distributed normally and independently of each other. The CHAPS algorithm is based on the separation, linearization and iterative adjusting of linearization of chance-constrained deterministic equivalents by using the simplex method.According to test results the solution time of the algorithm is directly proportional to the second power of the number of constraints of a linearized model corresponding to the chance-constrained model. The positive result is partly due to the fact that the linearized model is very sparse. The algorithm requires six to eight CHAPS iteration runs in order to achieve sufficient accuracy in practice (10^?5 –10^?6). The algorithm converges linearly and its asymptotic error constant is $\text{1}\text{4}$.     

The enolate anions of chlorophylls a and b as ambident nucleophiles in oxidations with (−)- or (+)-(10-camphorsulfonyl)oxaziridine. Synthesis of 13(S/R)-hydroxychlorophylls a and b

The enolate anions of chlorophylls (Chl) are ambident nucleophiles that are of considerable organic chemical interest in relation to the theory of electron delocalization (aromaticity) and charge-transfer in large conjugated π-systems, as well as for their chemical reactivity. Under deaerated conditions, the (−)- and (+)-enantiomers of (10-camphorsulfonyl)oxaziridine (CSOAI) are effective oxidants for the enolate anions of Chl a and Chl b, when 1,8-diazabicyclo[5.4.0]undec-7-ene (DBU) serves as a base. In this study, the use of these sterically hindered reagents to hydroxylate Chl a and Chl b is described for the first time. The total yield of 13²(S/R)-HO-Chl a was 71 and 90% for the oxidations of Chl a with (−)-CSOAI and (+)-CSOAI, respectively. Chl b, however, behaved clearly differently from Chl a. The total yield of 13²(S/R)-HO-Chl b was 40% in the oxidation with (−)-CSOAI and 60% in the reaction with (+)-CSOAI. A competing side-reaction, which resulted in the 15²-methyl, 17³-phytyl ester of Mg-15¹(S/R)-unstable rhodin, was found to lower the yields of the desired main products. The formation of the side-products was largely avoided and the yield of 13²(S/R)-HO-Chl b was improved by increasing the volume of hexane and using phosphate buffer in the first step of the work-up. With (−)-CSOAI, a 94% diastereomeric excess (de) was achieved for 13²(R)-HO-Chl a, whereas the de for 13²(R)-HO-Chl b was 66%. With (+)-CSOAI, the de was 10% for 13²(R)-HO-Chl a and 8% for 13²(R)-HO-Chl b. The results were interpreted in terms of a nucleophilic reaction mechanism, kinetically controlled by steric hindrance, originating on the one hand in the 17-propionate phytyl ester side-chain, protruding over the isocyclic ring E of the Chl enolate ion, and on the other hand in the bulky camphorsulfonyl unit of CSOAI. Possible reasons for the different results from the Chl b oxidations as compared with those of the Chl a oxidations are discussed. Comparison of the differences in the NMR δ_C-values between 13²(S)- and 13²(R)-HO-Chl a as well as those between 13²(S)- and 13²(R)-HO-Chl b, indicated that the change of stereochemical configuration at C-13² induces only slight differences in the δ_C-values. Of special interest are the δ_C-values of C-13², which are at ca. 91 ppm for the a- and b-series diastereomers. This carbon is deshielded by ca. 25 ppm relative to the C-13² of 13²(R)-Chl a (δ_C=65.5). Owing to this, ¹³C NMR spectroscopy is a good method to distinguish the 13²-hydroxylated chlorophylls from the intact, naturally occurring chlorophylls.     

Modeling of the adsorption and desorption of CO2 on Cu/ZrO2 and ZrO2 catalysts

The interaction between carbon dioxide and two zirconia catalysts-a Cu/ZrO2 catalyst containing 34% copper and a pure ZrO2 catalyst-was studied by pulse adsorption and temperature-programmed desorption methods. Kinetic modeling by nonlinear regression was applied to acquire information on the adsorption and desorption of CO2 relevant in the synthesis of methanol from carbon dioxide. A model that included three types of adsorption sites described well the experimental data for both Cu/ZrO2 and ZrO2. The model assumed first-order kinetics and a Freundlich-type logarithmic dependence of adsorption enthalpy on surface coverage. The parameters of the model were well identified and were in the physically meaningful range. The results indicate that, at 30 degrees C, on both catalysts, carbon dioxide adsorbs reversibly on one type of site and irreversibly on two other types of sites.     

Radial Limits of Mappings of Bounded and Finite Distortion

We give sufficient conditions for mappings defined on the unit ball of ? ⁿ to have radial limits almost everywhere. In particular, we show that if f:B(0,1)→? ⁿ is a mapping with exponentially integrable distortion satisfying the growth condition $$\int_{B(0,r)}J_f(x)\,dx\leq c(1-r)^{-a} $$ for some a∈[0,n?1), then . Here the set E(f) consists of those points in ?B(0,1) where f does not have radial limits. We also give an example which shows the difference between the classes of mappings of bounded distortion and certain integrable distortion in terms of radial limits.     

Large-Scale Analyticity and Unique Continuation for Periodic Elliptic Equations

We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large-scale C^{k, 1} estimate scale exponentially in k , just as for the classical estimate for harmonic functions, and the minimal scale grows at most linearly in k . As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations that exhibit growth like O(exp(δ| x| )) for small δ > 0 . The large-scale analyticity also implies quantitative unique continuation results, namely a three-ball theorem with an optimal error term as well as a proof of the nonexistence of L² eigenfunctions at the bottom of the spectrum. © 2020 Wiley Periodicals LLC.