 Longitudinal studies show that positive emotions play a role in the development of long-term resources such as psychological resilience and flourishing.Fredrickson's 2005 paper, co-authored with Marcial Losada, argued that there exist precise values of an individual's emotional positivity-to-negativity ratio, outside of which they will fail to flourish.Fredrickson and Losada's use of nonlinear dynamics modelling, taken from fluid dynamics, to derive these values, has been strongly criticised by Nicholas Brown, Alan Sokal, and Harris Friedman, who point out numerous fundamental mathematical errors in the article (see Losada line).