ABSTRACT
With the rise of quantum mechanical/molecular mechanical (QM/MM) methods, the interest in the calculation of molecular assemblies has increased considerably. The structures and dynamics of such assemblies are usually governed to a large extend by intermolecular interactions. As a result, the corresponding potential energy surfaces are topological rich and possess many shallow minima. Therefore, local structure optimizations of QM/MM molecular assemblies can be challenging, in particular if optimization constraints are imposed. To overcome this problem, structure optimization in normal coordinate space is advocated. To do so, the external degrees of freedom of a molecule are separated from the internal ones by a projector matrix in the space of the Cartesian coordinates. Here we extend this approach to Cartesian constraints. To this end, we devise an algorithm that adds the Cartesian constraints directly to the projector matrix and in this way eliminates them from the reduced coordinate space in which the molecule is optimized. To analyze the performance and stability of the constrained optimization algorithm in normal coordinate space, we present constrained minimizations of small molecular systems and amino acids in gas phase as well as water employing QM/MM constrained optimizations. All calculations are performed in the framework of auxiliary density functional theory as implemented in the program deMon2k.
ABSTRACT
In this paper, we consider a (control) optimization problem, which involves a stochastic dynamic. The model proposes selecting the best control function that keeps bounded a stochastic process over an interval of time with a high probability level. Here, the stochastic process is governed by a stochastic differential equation affected by a stochastic process. This setting becomes a chance-constrained control optimization problem, where the constraint is given by the probability level of infinitely many random inequalities. Since such a model is challenging, we discretize the dynamic and restrict the space of control functions to piecewise mappings. On the one hand, it transforms the infinite-dimensional optimization problem into a finite-dimensional one. On the other hand, it allows us to provide the well-posedness of the problem and approximation. Finally, the results are illustrated with numerical results, where classical model for the growth of a population are considered.
ABSTRACT
BACKGROUND: In this study, a chocolate cake formulation was developed with partial substitution of wheat flour by yacon and maca flour. A simplex-centroid design was applied to determine the proportions of the three flours, and the amount of water was included as a process variable at three distinct levels. According to the overall acceptability of the cakes, the tasters were separated into two groups using k-means. RESULTS: After segmentation, regression models were constructed for overall acceptability of each group; R2adjusted values of 92.5% for group 1 and 98.9% for group 2 were obtained. Using the sequential simplex method an optimized formulation was determined for group 1 (0.49 kgwheat kg-1total flour , 0.37 kgyacon kg-1total flour , 0.14 kgmaca kg-1total flour and 140.0 mL of water) and another for group 2 (0.35 kgwheat kg-1total flour , 0.65 kgyacon kg-1total flour and 120.0 mL of water). In addition to these formulations, a third formulation was proposed with a greater maca proportion (0.32 kgmaca kg-1total flour ), which does not significantly alter the overall acceptability of both groups. The three optimized formulations and two control formulations were evaluated through free-choice profiling. The data were evaluated using the multi-block method common components and specific weights analysis (CCSWA). CONCLUSION: It was observed that a greater proportion of maca intensified brownness and burnt aroma and taste, whereas a larger proportion of yacon produced a better appearance, softness, sweetness and chocolate flavor. © 2017 Society of Chemical Industry.