Cerium oxide nanoparticles have found numerous applications in the biomedical industry due to their strong antioxidant properties. In the current study, we report the influence of nine different physical and chemical parameters: pH, aeration and, concentrations of MgSO4, CaCl2, KCl, natural organic matter, fructose, nanoparticles and Escherichia coli, on the antibacterial activity of dextran coated cerium oxide nanoparticles. A least-squares quadratic regression model was developed to understand the collective influence of the tested parameters on the anti-bacterial activity and subsequently a computer-based, interactive visualization tool was developed. The visualization allows us to elucidate the effect of each of the parameters in combination with other parameters, on the antibacterial activity of nanoparticles. The results indicate that the toxicity of CeO2 NPs depend on the physical and chemical environment; and in a majority of the possible combinations of the nine parameters, non-lethal to the bacteria. In fact, the cerium oxide nanoparticles can decrease the anti-bacterial activity exerted by magnesium and potassium salts.

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Cutinase enzymes from fungi have found diverse applications in industry. However, most of the available literature on cutinase production is related to the cultivation of genetically engineered bacteria or yeast cells. In the present study, we use mixture design experiments to evaluate the influence of six nutrient elements on production of cutinase from the fungus Colletotrichum lindemuthianum. The nutritional elements were starch, glucose, ammonium sulfate, yeast extract, magnesium sulfate, and potassium phosphate. In the experimental design, we imposed the constraints that exactly one factor must be omitted in each set of experiments and no factor can account for more than one third of the mixture. Thirty different sets of experiments were designed. Results obtained showed that while starch is found to have negative influence on the production of the enzyme, yeast extract and potassium phosphate have a strong positive influence. Magnesium sulfate, ammonium sulfate, and glucose have low positive influence on the enzyme production. Contour plots have also been created to obtain information concerning the interaction effects of the media components on enzyme production.

We investigate the diameter of the polytope arising in the n-city symmetric traveling salesman problem (TSP) and perfect matching polytopes. Grötschel and Padberg [The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley-Intersci. Ser. Discrete Math., E. Lawler et al., eds., John Wiley, Chichester, 1985, pp. 251-305] conjectured that the diameter of the symmetric TSP polytope is 2, independent of n. We constructively show that its diameter is at most 4, for all n ≥ 3. Our result also shows that the diameter of the perfect 2-matching polytope is at most 6, for every n ≥ 3.