Mathematical modeling of nonlinearity, uncertainty and decision

The main problem in today mathematical modeling is related to the nonlinearity of the considered problems mostly under uncertainty, and usualy looking for optimal solution. Mathematical mashinery initiated by many different applications, which cover all three mentioned aspects is pseudo-analysis, which requires an introduction of new real operations. Besides several properties and construction methods, also several kinds of aggregation functions will be introduced and examined. It will be investigated and developed further the notion of universal integral and many of important cases of it. There will be proved inequalities for nonlinear integrals (Jensen, Chebyshev, Hölder, Minkowski, Stolarsky). There will be introduced new types of integrals with respect to the absolutely monotone set functions. Further applications of of the pseudo-superposition principle for some new nonlinear PDEs will be managed. Based on countable extensions of the triangular norms fixed point theorems in probabilistic metric and fuzzy metric spaces will be proved.There will be managed new models, based on pseudo-analysis, in social sciences, transportations and imprecise spatial objects modelling. Using semirings of bistochastic matrices there will be investigated the mobility index and related orderings, and the extension of the whole theory to continuous state Markov processes. Comparative analysis of existing causality measures applied to problems of brain connectivity will be performed.