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We investigate the Cauchy problem for ordinary differential equation (ODE) with fuzzy initial condition. We define a solution to this problem and propose a new method of how an approximate solution can be constructed. The proposed method is based on the technique of fuzzy transforms and ends up with a solution of a system of fuzzy relation equations. The approximate solution is expressed symbolically...
In this paper, the transitivity properties of reciprocal relations, also called probabilistic relations, are investigated within the framework of cycle-transitivity, which generalizes the concepts of T-transitivity, stemming from fuzzy set theory, and of stochastic transitivity, common to many mathematical models in psychology, social choice and welfare, financial mathematics, etc. It is emphasized...
We establish a general framework for constructing copulas that can be regarded as a patchwork-like assembly of arbitrary copulas, with nonoverlapping rectangles as patches. We derive a family of construction methods that require the choice of a single background copula. When this background copula is the greatest copula , we retrieve the well-known ordinal sum construction, while in the case of the...
In this paper, we study identities and inequalities on fuzzy set cardinalities. An important such identity is the valuation property. Other identities are related to the fuzzification of the (symmetric) difference of fuzzy sets. We show that using a stable, commutative quasi-copula and its dual operator for modelling intersection and union of fuzzy sets, identities which are valid in the crisp case,...
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