their model predicted cut-off points for the minimum and maximum positivity ratios within which one should observe qualitative changes in an individual's level of flourishing, specifically, that those within this range of ratios would "flourish", and those outside would "languish". As of January 2014, the 2005 Fredrickson and Losada's paper had been cited more than 320 times in the psychology literature. Criticism Initially ignored questioning The first critical evaluation of the mathematical modeling behind the critical positivity ratio was published by a group of Finnish researchers—Luoma, Hämäläinen, and Saarinen of the Systems Analysis Laboratory at Aalto University—in 2008. The authors noted that "[o]nly very limited explanations are given about the modeling process and the meaning and interpretation of its parameters... [so that] the reasoning behind the model equations remains unclear to the reader"; moreover, they noted that "the model also produces strange and previously unreported behavior under certain conditions... [so that] the predictive validity of the model also becomes problematic." Not widely impactful at the time, Losada's earlier modeling article was also critiqued by Andrés Navas in a French language publication, a note in the CNRS publication, "Images des Mathématiques", which also failed to attract a wide readership. In their followup to Fredrickson's immediate response to the rebuttal, Brown, Sokal, and Friedman note as a footnote to their submission:After the publication of Brown et al. (2013), Andrés Navas kindly drew our attention to his article (Navas, 2011) in which a very similar (though briefer) critique of Losada (1999) was made. [This footnote was unfortunately omitted from the published version of this article, due to space limitations.] The Brown-Sokal-Friedman rebuttal The Fredrickson and Losada work on modeling the positivity ratio aroused the skepticism of Nick Brown, a graduate student in applied positive psychology, who questioned whether such work could reliably make such broad claims, and perceived that the paper's mathematical claims were suspect. Brown contacted and ultimately collaborated with physics and maths professor Alan Sokal and psychology professor Friedman on a re-analysis of the paper's data. The result was a strong critique of the critical positivity ratio in its entirety by Brown, Sokal, and Friedman, that appeared in a 2013 article in American Psychologist, here referred to as the Brown-Sokal-Friedman rebuttal. These authors argued that Losada's conclusions in previous papers using modelling from fluid dynamics, and those in his paper co-authored with Fredrickson, were not only based on poorly reported experiments,