Improvement of RSM Prediction and Optimization by Using Box-Cox Transformation

Electrocoagulation can be effectively used in the elimination of the colloids from the tailings of the mineral industries. Owing to the numerous operational parameters of this physicochemical process, the best engineering technique for the characterization of the process is RSM. In this chapter, a non-transformed quadratic model is firstly formed considering the supernatant turbidity of the electrocoagulation experiments as a function of temperature, pH, and electrical current. Then, the non-normality and the heteroscedasticity of this initial model was indicated. These drawbacks were improved by using the Box-Cox transformation with λ of -0.32 and a new model with a perfect normality and homoscedasticity was obtained. The R2 value increased from 81.60% to 99.48% and adjusted R2 increased from 48.48% to 99.22% upon the transformation. According to the confirmed optimization results of the Box-Cox transformed model, the maximum desirability was obtained at pH of 5, temperature of 85°C, and electrical current of 0.25A, and the supernatant turbidity decreased down to 2.25 NTU.

A generalization of Tong's theorem and properties of pairwise perfectly normal spaces

In this paper we give some properties of the pairwise perfectly normal spaces defined by Lane. In particular we prove that a space (X, P, Q) is pairwise perfectly normal if and only if every P(Q)–closed set is the zero of a P(Q)–l.s.c. and Q(P)–u.s.c. function. Also we characterize the pairwise perfect normality in terms of sequences of semicontinuous functions by means of a result which contains the known Tong's characterization of perfectly normal topological spaces, whose proof we modify by using the technique of binary relations.