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We intend to study a new class of algebraic approximations, called -approximations, and their properties. We have shown that -approximations can be used for applied problems which cannot be modeled by inclusion based approximations. Also, in this work, we studied a subclass of -approximations, called M -approximations, and showed that this subclass preserves most of the properties of inclusion based approximations but is not necessarily inclusionbased. The paper concludes by studying some basic operations on -approximations and counting the number of -min functions. IntroductionUncertainty is often present in real-life applications. Uncertainty in noncrisp sets is characterized by nonempty boundary regions, in which nothing can be said about their elements with certainty. In classical set theory, a subset of a universe induces a partition { , − } over that universe. This partition can be interpreted as a knowledge on elements of ; that is, elements in are indiscernible to each other and also the same thing holds for items in − . This knowledge may be improved to another partition, for example, P, whose items in each partition are indiscernible to each other. In consequence, for a subset of , the problem of whether belongs to or not, with respect to knowledge P, may become undecidable; that is, some elements indiscernible to with respect to knowledge P may be in , whereas some other indiscernible elements to with knowledge P may not belong to . To cope with such uncertainty, some tools were invented such as the Dempster-Shafer theory of evidence [1], theory of fuzzy sets [2][3][4][5], and theory of rough sets [6][7][8]. Rough set theory and fuzzy set theory are two independent approaches for uncertainty. There is a connection between rough set theory and Dempster-Shafer theory. Strictly speaking, lower and upper approximations of rough set theory correspond to the inner and outer reductions from Dempster-Shafer theory [9].Rough set theory and its generalizations are all based on the inclusion relation [7,8,[10][11][12][13][14][15], which is a limitation in approximations. In this work, we introduce a new concept named -approximation set. This concept is independent of inclusion relation and contains rough sets and their generalizations as special cases. We provide some examples of approximations using this new concept, which cannot be obtained by rough set theory. This paper is organized as follows. The notion ofapproximation sets is proposed in Section 2, followed by considering some operations on them. The definition of M conditioned rough sets is proposed in Section 3 and the number of such sets is counted. Then we conclude the paper. S-ApproximationIn this section, with regard to Dempster's multivalued mappings [16], we propose a new mathematical approach to study approximation spaces and we will show that this concept can be independent of inclusion relations and the rough set and its generalizations are all special cases of this concept. where is the complement of with respect to .Example 2. Let and be none...

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In this paper, we will study neighborhood system S-approximation spaces, i.e., combination of S-approximation spaces with identical elements except that they have different knowledge mappings, e.g., the knowledge mappings differ due to different experimental conditions and/or sampling methodology. In such situations, there is a risk of contradictory knowledge sets which can lead to different decisions by the same query. These situations are studied in this paper in detail. Moreover, neighborhood system S-approximation spaces are investigated from a three-way decisions viewpoint with respect to different deciders. In addition, completeness results are shown for optimistic and pessimistic neighborhood system S-approximation spaces, i.e., these constructions can be represented by an ordinary S-approximation space. Also, the concept of knowledge significance is proposed and studied in detail, and we have shown that computing a minimal set of knowledge mappings for a neighborhood system S-approximation space is NP-hard. Finally, the paper is concluded by two illustrative medical examples.

In this paper, an efficient and accurate computational method based on the hat functions (HFs) is proposed for solving a class of fractional optimal control problems (FOCPs). In the proposed method, the fractional optimal control problem under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved. To this end, the fractional state and control variables are expanded by the HFs with unknown coefficients. Then, the operational matrix of fractional integration of the HFs with some properties of these basis functions are employed to achieve a nonlinear algebraic equation, replacing the performance index and a nonlinear system of algebraic equations, replacing the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.