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**10 July 2017**

**08:00AM - 10:00AM**

##### Room: **Catalana**

##### Chair: **Stefano Aguzzoli**

## Mathematical and Theoretical Foundations of Fuzzy sets and Systems

**Abstract - ** A category of fuzzy type automata in monads (in clone form) is introduced. A monad of extensional fuzzy sets in sets with similarity relations and a monad of fuzzy objects in spaces with fuzzy partitions are introduced and relationships between fuzzy automata in sets with similarity relation or in spaces with fuzzy partitions, on one hand, and fuzzy type automata in corresponding monads, on the other, are investigated.

**Abstract - ** In this paper, we approximate an arbitrary fuzzy number by a polynomial fuzzy number through minimizing the distance between them. Throughout this work, we used a distance that is a meter on the set of all fuzzy numbers with continuous left and right spread functions. To support our claims analytically, we have proven some theorems and given supplementary corollaries.

**Abstract - ** In this paper, we propose a numerical method based on the F-transform for an approximate solution of a certain type of partial differential equations with Dirichlet boundary conditions and initial conditions. The F- transform is an efficient method for the approximation of multivariate functions, where a transformation of a space of locally square integrable functions into a simple vector space of F-transform components is used. The F-transform is based on a fuzzy partition of the domain of multivariate functions. We show how the partial differential equations after the application of the Crank-Nicolson scheme for time discretization can be approximated by a system of linear equations with the direct F-transform components as their variables. Then, we derive elements of matrices using them the system of linear equations can be simply formulated. The F-transform components are then obtained as solutions of this system. The numerical solution of partial differential equations is a function which is found by the inverse F-transform. This function approximates the analytical solution, if there exists, with respect to the boundary and initial conditions. The proposed numerical method is concretely adjusted to the 2-dimensional boundary value problem, which is then used for solving two examples of partial differential equations, where one is artificial and the second one is real, namely, the Black-Scholes equation well-known in financial modeling.

**Abstract - ** We present a way to compute the set of fixpoints of a given fuzzy closure operator via algorithms for computing sets of fixpoints of ordinary closure operators. We assume that the fuzzy closure operator is given by a set of fuzzy sets generating this operator. The proposed way is based on certain reduction theorems which we provide and which relate fuzzy and ordinary closure operators and the sets of their fixpoints. We also present explicit description of selected algorithms which result using the presented approach.

**Abstract - ** The problem of studying the existence of a right adjoint to a mapping defined between sets with different fuzzy structure naturally leads to the search of new notions of adjunction which fit better with the underlying structure of domain and codomain. In this work, we introduce a version of relational fuzzy adjunction between fuzzy preposets which generalizes previous approaches in that its components are fuzzy relations. We also prove that the construction behaves properly with respect to the formation of quotient with respect to the symmetric kernel relation and, thus, giving rise to a relational fuzzy adjunction between fuzzy posets.

**Abstract - ** In computing the similarity of intervals, current similarity measures such as the commonly used Jaccard and Dice measures are at times not sensitive to changes in the width of intervals, producing equal similarities for substantially different pairs of intervals. To address this, we propose a new similarity measure that uses a bi-directional approach to determine interval similarity. For each direction, the overlapping ratio of the given interval in a pair with the other interval is used as a measure of uni- directional similarity. We show that the proposed measure satisfies all common properties of a similarity measure, while also being invariant in respect to multiplication of the interval endpoints and exhibiting linear growth in respect to linearly increasing overlap. Further, we compare the behavior of the proposed measure with the highly popular Jaccard and Dice similarity measures, highlighting that the proposed approach is more sensitive to changes in interval widths. Finally, we show that the proposed similarity is bounded by the Jaccard and the Dice similarity, thus providing a reliable alternative.