This article discusses the comments made by some scientists that multidimensional Interval arithmetic (MIA) is the same as constraint Interval arithmetic (CIA) and multidimensional fuzzy arithmetic (MFA) is the same as constraint fuzzy arithmetic (CFA). Both types of arithmetic are briefly presented and then the difference in their dimensions, calculation methods, differences in the obtained results and the way they are used in complex calculations are shown. The answer to the question posed is presented in the conclusions.

The article presents a new, multidimensional arithmetic of type 2 fuzzy numbers (M-IT2-F arithmetic) in which the result is a multidimensional fuzzy set. This arithmetic increases the accuracy of calculations and the scope of problems solved in relation to the currently used Interval type 2 standard fuzzy arithmetic (IT2-SF arithmetic). The proposed M-IT2-F arithmetic has mathematical properties that IT2-SF arithmetic does not have. Thanks to these properties, it provides accurate calculation results that are not over-or under-estimated in terms of uncertainty. The paper contains comparisons of both types of arithmetic in application to two problems. Fuzzy arithmetic is not a finished work and is in a phase of continuous improvement and development. M-IT2-F arithmetic is a higher form of M-IT2 (non-fuzzy) arithmetic.

The VIKOR method was developed for Multi-Criteria Decision Making (MCDM). It determines the compromise ranking list and the compromise solution obtained with the initial weights. This method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria. It introduces the multi-criteria ranking index based on the particular measure of ‘ ‘ closeness” to the “ Ideal” solution. The aim of this paper is to extend the VIKOR method for decision making problems with Interval number. The extended VIKOR method’ s ranking is obtained through comparison of Interval numbers and for doing the comparisons between Intervals. In the end, a numerical example illustrates and clarifies the main results developed in this paper.

This work studies Interval Initial Value Problems (IIVPs), where the derivative is given by the generalized Hukuhara derivative ($gH$-derivative) and the initial condition is given by an Interval. The focus of the paper is to provide the numerical approximations for the solutions associated with the $gH$-derivative of IIVPs. This article considers the Euler numerical method, where the classical arithmetic operation is adapted for Intervals. The arithmetic considered here is obtained using sup-$J$ extension principle, where $J$ is a particular family of joint possibility distributions. This family gives raise to different types of interactivity and this work shows what kind of interactivity is necessary in the numerical method, in order to approximate the solution via $gH$-derivative. To illustrate the results, the paper focuses in the decay Malthusian model.

Solution of the static attitude determination of the satellite leads to the Wahba problem. The Wahba problem uses a set of at least two independent sensor measurements and reference vectors. These input vectors are not accurate due to sensor noises, misalignment, and low-order approximations. But these uncertainties do not view in the classic Wahba problem directly. Hence, the estimation error of the Wahba problem depends on the accuracy of the input vectors. In this paper, modeling error, measurement noise, and biases are proposed unknown but bounded. These errors are modeled using Interval analysis. The innovations of this research are considering the uncertainties in the input vectors in the Wahba problem using Interval arithmetic and transforming the solution of the attitude determination problem from a single-objective problem to a multi-objective robust optimization problem. Then the multi-objective problem is optimized using an NSGA solver. The results indicate a lower attitude estimation error of the proposed method in attitude determination of the satellite.

The intuitionistic linguistic set (ILS) is an extension of linguisitc variable. To overcome the drawback of using single real number to represent membership degree and non-membership degree for ILS, the concept of Interval-valued intuitionistic linguistic set (IVILS) is introduced through representing the membership degree and non-membership degree with Intervals for ILS in this paper. The operation law, score function, accuracy function, and certainty function for Interval-valued intuitionistic linguistic varibales (IVILVs) are defined. Hereby a lexicographic method is proposed to rank the IVILVs. Then, three kinds of Interval-valued intuitionistic linguistic arithmetic average operators are defined, including the Interval-valued intuitionistic linguistic weighted arithmetic average (IVILWAA) operator, Interval-valued intuitionistic linguistic ordered weighted arithmetic (IVILOWA) operator, and Interval-valued intuitionistic linguistic hybrid arithmetic (IVILHA) operator, and their desirable properties are also discussed. Based on the IVILWAA and IVILHA operators, two methods are proposed for solving multi-attribute group decision making problems with IVILVs. Finally, an investment selection example is illustrated to demonstrate the applicability and validity of the methods proposed in this paper.

Matrix functions play important roles in various branches of science and engineering. In numerical computations and physical measurements there are several sources of error which significantly affect the main results obtained from solving the problems. This effect also influences the matrix computations. In this paper, we propose some approaches to enclose the matrix functions. We then present some analytical arguments to ensure that the obtained enclosures contain the exact result. Numerical experiments are given to illustrate the performance and effectiveness of the proposed approaches.

In this paper, first, we propose Kaucher arithmetic for Interval number and then solve the system A2n×nXn×1=b2n×1 where aij= [ai,j,`ai,j] and xj= [xj,`xj] and bi= [bi,`bi] based on Kaucher arithmetic such that ai,j .`ai,j ³ 0. Then, we extend this method by use of a- level set for fully fuzzy system of A~2n×nX~n×1= b~2n×1.

By Grothendieck's anabelian conjectures, Galois representa-tions landing in outer automorphism group of the algebraic fundamen-tal group which are associated to hyperbolic smooth curves de ned over number- elds encode all the arithmetic information of these curves. The Goal of this paper is to develop an arithmetic Teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arith-metic information coming from all curves of the same topological type de ned over number- elds. We also introduce Hecke-Teichmuller Lie al-gebra which plays the role of Hecke algebra in the anabelian framework.