The Pythagorean triples of integers satisfying x + y = z have been studied and enumerated since Babylonian times. Since Diophantus, it has been known that this set of triples is related to the standard rational parameterization of the unit circle, ( t 2−1 t2+1 , 2t t2+1 ). The Pythagorean triple solutions, which are relatively prime, have the elementary and beautiful characterization as integer...

there are many situations for statistical process in which we have both random and vagueinformation. when uncertainty is due to fuzziness of information, fuzzy statistical control charts play animportant role in the monitoring process, because they simultaneously deal with both kinds of uncertainty.dealing with fuzzy characteristics using classical methods may cause the loss of information and ...

The aim of each company/industry is to provide a final product customers at the minimum possible cost, as well protect environment from degradation. Ensuring shortest travel distance between involved locations plays an important role in achieving company’s/industry’s objective (i) cost can be minimized by minimizing total travelled (ii) finding will require less fuel than longest locations. Thi...

As an extension of Pythagorean fuzzy sets and linguistic term sets, (PFSs) are powerful to describe decision-making information quantificational qualitatively, which have received much scholars’ attention. The purpose this paper is propose a new multiple attribute group (MAGDM) approach with (PFL) information. To end, we firstly analyze the drawbacks existing operations PFL numbers opera...

in the mathematical analysis, there are some theorems and definitions that established for both real and fuzzy numbers. in this study, we try to prove  bernoulli's inequality in fuzzy real numbers with some of its applications. also, we prove two other theorems in fuzzy real numbers which are proved before, for real numbers.

Ranking fuzzy numbers plays a main role in many applied models inreal world and in particular decision-making procedures. In manyproposed methods by other researchers may exist some shortcoming.The most commonly used approaches for ranking fuzzy numbers isbased on defuzzification method. Many ranking fuzzy numberscannot discriminate between two symmetric fuzzy numbers withidentical core. In 200...

in this paper,  we determine necessary and sufficient tauberian conditions under which convergence in pringsheim's sense of a double sequence of fuzzy numbers follows from its $(c,1,1)$ summability. these conditions are satisfied if the double sequence of fuzzy numbers is slowly oscillating in different senses. we also construct some interesting double sequences of fuzzy numbers.

As general concepts of sup-hesitant fuzzy right (resp., left, interior, two-sided) ideals semigroups, the sup+α-hesitant and sup-β-hesitant are introduced their properties investigated. Then, established by sets, Łukasiewicz anti-fuzzy Pythagorean hesitant hybrid interval-valued sets cubic sets. Finally, we characterize which is intra-regular, completely regular, simple semigroups or another ty...

in this paper we introduce the root of a fuzzy number, and we present aniterative method to nd it, numerically. we present an algorithm to generatea sequence that can be converged to n-th root of a fuzzy number.