How electrons guard the space: shape optimization with probability distribution criteria

Efficient formulas for computing the probabilities of finding exactly electrons in an arbitrarily chosen volume ³ for Hartree–Fock wavefunctions are presented. These formulas allow the use of shape optimization techniques, such as level set methods, for optimizing with respect to various criteria involving such probabilities. The criterion defined as the difference between the Hartree–Fock and the independent-particle model probabilities of finding electrons in stresses the quantum effects due to the Pauli principle. We have implemented a 2D level set method for optimizing this criterion in order to study spatial separation of electron pairs in linear molecules. The method is described and the illustrative example of the BH molecule is reported.Contribution to the Jacopo Tomasi Honorary Issue     

Static micromixers based on large-scale industrial mixer geometry

Mixing liquids at the micro-scale is difficult because the low Reynolds numbers in microchannels and in microreactors prohibit the use of conventional mixing techniques based on mechanical actuators and induce turbulence. Static mixers can be used to solve this mixing problem. This paper presents micromixers with geometries very close to conventional large-scale static mixers used in the chemical and food-processing industry. Two kinds of geometries have been studied. The first type is composed of a series of stationary rigid elements that form intersecting channels to split, rearrange and combine component streams. The second type is composed of a series of short helix elements arranged in pairs, each pair comprised of a right-handed and left-handed element arranged alternately in a pipe. Micromixers of both types have been designed by CAD and manufactured with the integral microstereolithography process, a new microfabrication technique that allows the manufacturing of complex three-dimensional objects in polymers. The realized mixers have been tested experimentally. Numerical simulations of these micromixers using the computational fluid dynamics (CFD) program FLUENT are used to evaluate the mixing efficiency. With a low pressure drop and good mixing efficiency these truly three-dimensional micromixers can be used for mixing of reactants or liquids containing cells in many microTAS applications.     

Enantioselective Diels-Alder reactions with N-hydroxy-N-phenylacrylamide

[reaction: see text] The use of hydroxamic acids as templates for Lewis acid catalyzed enantioselective Diels-Alder reactions has been examined. A very simple chiral Lewis acid, prepared by mixing optically pure binaphthol with 3 equiv of trimethylaluminum, catalyzes the [4 + 2] cycloaddition of N-hydroxy-N-phenylacrylamide with cyclopentadiene at 0 degrees C in high yield (>96%) and with a fairly good level of enantioselectivity (91% ee). Facile conversion of the products to the corresponding alcohols or aldehydes makes the hydroxamic acid intermediates particularly useful.     

Quantifying uncertainty with ensembles of surrogates for blackbox optimization

Blackbox optimization tackles problems where the functions are expensive to evaluate and where no analytical information is available. In this context, a tried and tested technique is to build surrogates of the objective and the constraints in order to conduct the optimization at a cheaper computational cost. This work introduces an extension to a specific type of surrogates: ensembles of surrogates, enabling them to quantify the uncertainty on the predictions they produce. The resulting extended ensembles of surrogates behave as stochastic models and allow the use of efficient Bayesian optimization tools. The method is incorporated in the search step of the mesh adaptive direct search (MADS) algorithm to improve the exploration of the search space. Computational experiments are conducted on seven analytical problems, two multi-disciplinary optimization problems and two simulation problems. The results show that the proposed approach solves expensive simulation-based problems at a greater precision and with a lower computational effort than stochastic models.

     

Higher Coverings of Racks and Quandles—Part II

Applied Categorical Structures - This article is the second part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in...