but also that it was difficult to draw conclusions from Losada's previous cited studies because critical details were omitted, "interpretations of results [were] made with little or no justification", and that elementary errors were made in the application of differential equations. Among the severe flaws claimed by Brown et al. in the positivity-ratio theory and its presentation were that: data used by Losada in several analyses do not meet a basic criteria for the use of differential equations (such as the use of continuous variables that evolve smoothly and deterministically over time); differential equations used by Losada to calculate the critical positivity ratio use parameters taken directly from Lorenz's simplified, illustrative, and arbitrary models for fluid dynamics, with Losada giving no rationale for his choice of parameters; use of different arbitrary parameters would give different positivity ratios, thus the precise values for the lower and upper critical ratios based on the arbitrary parameters, Fredrickson and Losada's 2.9013 to 11.6346 ratios, are meaningless; the butterfly-like first figure provided by Fredrickson and Losada is not a model of the data taken from their human participants, but "the results of computer simulations of the Lorenz equations, nothing more"; and based on the maths, even if precise positivity/negativity ratios could be derived, several "windows" of desirable and undesirable positivity/negativity ratios above a certain value should exist, rather than a simple range of ratios in which "flourishing" should occur. With regard to these, and especially the last, the Brown-Sokal-Friedman rebuttal argues that it is likely that Fredrickson and Losada did not fully grasp the implications of applying nonlinear dynamics to their data. Brown, Sokal, and Friedman state that one can:only marvel at the astonishing coincidence that human emotions should turn out to be governed by exactly the same set of equations that were derived in a celebrated article several decades ago as a deliberately simplified model of convection in fluids, and whose solutions happen to have visually appealing properties. An alternative explanation – and, frankly, the one that appears most plausible to us – is that the entire process of "derivation" of the Lorenz equations has been contrived to demonstrate an imagined fit between some rather limited empirical data and the scientifically impressive world of nonlinear dynamics. They "urge future researchers to exercise caution in the use of advanced mathematical tools, such as nonlinear dynamics". Responses to the Brown-Sokal-Friedman rebuttal Fredrickson and Am. Psychol. retraction