Krasner hyperrings are a generalization of rings. Indeed, in a Krasner hyperring the addition is a hyperoperation, while the multiplication is an ordinary operation. On the other hand, a generalization of rough set theory is the near set theory. Now, in this paper we are interested in combining these concepts. We study and investigate the notion of near Krasner  hyperrings on a nearness APPROXIMATION SPACE. Also, we define near subhyperring, near hyperideal, near homomorphism  and prove some results and present several examples in this respect.

If there is a fuzzy relation $\widetilde{R}$ between two SPACEs $U$ and $V$, the fuzzy APPROXIMATIONs in both SPACEs based on $\widetilde{R}$ are widely studied, and they basically reflect only the influence from one SPACE to another. In this paper, on each SPACE of $U$ and $V$, two new fuzzy relations are derived from $\widetilde{R}$, a positive low-value relation and a conservative high-value relation, to reflect the interaction and feedback between the two SPACEs. So, the fuzzy APPROXIMATIONs based on them can reflect the combination of the action and the reaction from one SPACE to another. Therefore, two SPACEs $U$ and $V$ are closely accompanied, and $(U, V, \widetilde{R})$ is a whole, so it is called a fuzzy accompanied APPROXIMATION SPACE (FAAS). In an FAAS, the properties of the fuzzy APPROXIMATION models on each SPACE are studied, the relationships between fuzzy APPROXIMATION models of two SPACEs are researched, and examples to show how the APPROXIMATION operator models in the FAAS to solve practical problems from multiple perspectives are also illustrated. More importantly, when the fuzzy relation $\widetilde{R}$ is a binary relation $R$ or the two SPACEs are the same, the special cases of FAAS are investigated and some important new models and new results are obtained, which add new ideas and methods to the current research.

To tackle the problem with inexact, uncertainty and vague knowledge, constructive method is utilized to formulate lower and upper APPROXIMATION sets. Rough set model over dual-universes in fuzzy APPROXIMATION SPACE is constructed. In this paper, we introduce the concept of rough set over dual-universes in fuzzy APPROXIMATION SPACE by means of cut set. Then, we discuss properties of rough set over dual-universes in fuzzy APPROXIMATION SPACE from two viewpoints: APPROXIMATION operators and cut set of fuzzy set. Reduction of attributes and rules extraction of rough set over dual-universes in fuzzy APPROXIMATION SPACE are presented. Finally, an example of disease diagnoses expert system illustrates the possibility and eciency of rough set over dual-universes in fuzzy APPROXIMATION SPACE.

The aim of this paper is to present the new and interesting notion of ascending family of a- n-norms corresponding to an intuitionistic fuzzy nnormed linear SPACE. The notion of best aproximation sets in an a- n-normed SPACE corresponding to an intuitionistic fuzzy n-normed linear SPACE is also defined and several related results are obtained.

Classical structure of rough set theory was first formulated by Z. Pawlak in [6]. The foundation of its object classification is an equivalence binary relation and equivalence classes. The upper and lower APPROXIMATION operations are two core notions in rough set theory. They can also be seen as a closure operator and an interior operator of the topology induced by an equivalence relation on a universe. There are many studies on the relations between generalized rough set APPROXIMATION and rough topological SPACE. In this paper, We are defined some properties of an n-ary relation such as reflexive, symmetry, strongly symmetry, quasi-transitive, n-transitive and n-equivalence on a set. We introduce topological method to generalized rough set and study the relationship between topological theory and rough set theory. By using an n-ary tolerance relation and quasi-transitive relation, we define a topology on nonempty set. In the end, we prove some topological properties such as quasi-discreteness, connectivity, compactness, quasi-metric able, separateness.

In this paper, the structural optimization of a SPACE capsule has been discussed by approximating a thin-walled cylindrical shell with a certain length under the axial compression force and constant lateral pressure. Design variables include the outer diameter and cylinder thickness. The purpose of optimization is to minimize the mass and maximize the frequency of the first vibration shape mode of the cylinder. The design constraints include the buckling load multiplier (buckling safety factor) above 1. 5 and Von Mises stress below 100 MPa. In this article, first, according to the permissible limits of the design variables, a design of experiment (DOE) and then a sensitivity analysis was carried out to check the sensitivity of the objective functions and constraints to the design variables. After numerically solving the output values with the help of Ansys software and preparing the response surface, the optimal design point has been identified with the help of the two objectives optimization Genetic algorithm. Then, with the numerical simulation of the optimal point, the accuracy of the values obtained from the response surface method was checked and their accuracy was confirmed. The results show that at the selected design point, Von Mises stress becomes less than its allowed value, i. e. 100 MPa, and also the buckling load factor is more than twice its minimum allowed value. However, this point has the smallest distance from the origin and the optimum point has been chosen as the knee point.

We study the thoery of best APPROXIMATION in tensor product SPACE and the direct sum of some lattice normed SPACEs Xi: We introduce quasi tensor product SPACE and discuss about the relation between tensor product SPACE and this new SPACE which we denote it by X⊠ Y. We investigate best APPROXIMATION in direct sum of lattice normed SPACEs by elements which are not necessar-ily downward or upward and we call them Im􀀀 quasi downward or Im􀀀 quasi upward. We show that these sets can be interpreted as downward or upward sets. The relation of these sets with down-ward and upward subsets of the direct sum of lattice normed SPACEs Xi is discussed. This will be done by homomorphism functions. Fi-nally, we introduce the best APPROXIMATION of these sets.