Fuzzy Complex Numbers Research Articles

Novel similarity measures under complex pythagorean fuzzy soft matrices and their application in decision making problems

Complex fuzzy soft matrices play a crucial role in various applications, including decision-making, pattern recognition, signals processing, and image processing. The main objective of this study is to introduce the unique notions of complex Pythagorean fuzzy soft matrices (CPFSMs), which provide more flexibility and accuracy in modelling uncertainty. CPFSMs incorporate Pythagorean fuzzy soft matrices, allowing for more sophisticated uncertainty modeling. The key findings of CPFSMs, specific instances, and certain fundamental set-theoretic operations and principles were covered. A set of new distance metrics between two CPFSMs has been defined. In the context of complex Pythagorean fuzzy soft sets and complex Pythagorean fuzzy soft matrices, we created a CPFS decision-making technique. Moreover, the application’s numerical example and comparison analysis have been effectively demonstrated. Thus, by integrating the concepts of Pythagorean fuzzy sets, soft matrices, and complex numbers, CPFSMs provide a robust framework with membership and non-membership degrees for complex decision-making modeling and analyzing uncertain data.

Cognitive and Non-cognitive States in Romeo and Juliet’s Love Model and Its Chaotic Behaviors by Complex Fuzzy Numbers

Cognitive and Non-cognitive States in Romeo and Juliet’s Love Model and Its Chaotic Behaviors by Complex Fuzzy Numbers

Dual rectangular fuzzy complex matrix equations: extended solution, algebraic solution, solution and their calculating1

In this paper, we focus on generalized fuzzy complex numbers and propose a straightforward matrix method to solve the dual rectangular fuzzy complex matrix equations C · Z ˜ + L ˜ = R · Z ˜ + W ˜ , in which C and R are crisp complex matrices and Z ˜ , L ˜ and M ˜ are fuzzy complex number matrices. The existing methods for solving fuzzy complex matrix equations involve separately calculating the extended solution and the corresponding parameters of the real and imaginary parts, whereby we obtain the algebraic solution of the equations. By means of the interval arithmetic and embedding approach, the n × n dual rectangular fuzzy complex linear systems could be converted into 2n × 2n fuzzy linear systems, which are also equivalent to the 4n × 4n real linear systems. By directly solving the 4n × 4n real linear systems, the algebraic solutions can be obtained. The general dual rectangular fuzzy complex matrix equations and dual rectangular fuzzy complex linear systems are investigated by the generalized inverses of matrices. Finally, some examples are given to illustrate the effectiveness of method.

On Some Novel Generalizations of Weak Fuzzy Complex Numbers

The ring of weak fuzzy complex numbers is an extension of real numbers ring by using an algebraic element with fuzzy property. In this paper, we present two novel generalizations of weak fuzzy complex numbers, where the concepts of strong fuzzy complex numbers and split-complex weak fuzzy complex numbers will be defined for the first time with a general study of their elementary properties and special elements. On the other hand, we provide an algorithm to compute the dempotent elements in the ring of split-complex weak fuzzy complex numbers with many related examples that clarify the validity of our work.

On the Geometry of Weak Fuzzy Complex Numbers and Applications to the Classification of Some A-Curves

The concept of A-curves is considered as a novel application of real field extensions in solving some algebraic vectorial equations defined by Euclidean norms. In this paper, we present a novel insight through the classification of A-curves by illustrating many new semi-module isomorphisms between the direct product of weak fuzzy complex numbers with itself and the direct product of classical Euclidean vector spaces multiplied by itself. These isomorphisms will give us a full classification of A-curves that are related to weak fuzzy complex ring. Also, we provide many examples to explain the contribution of our work.

Fuzzy Metric Space of Weak Fuzzy Complex Numbers and Plithogenic Metric Spaces

The main goal of this work is to define for the first time the concept of the metric space of weak fuzzy complex numbers, where we present a suitable metric defined over the ring of weak fuzzy complex numbers, and we study the open balls, closed balls, and torus generated from its structure. On the other hand, we study the symbolic 2-plithogenic and 3-plithogenic4-plithogenic metric space and its ability to be generated from classical metrics, with many interesting properties related to its analytical structure. Also, many examples will be illustrated to clarify and explain the novelty of this work.

An Effective Algorithm for Solving Weak Fuzzy Complex Diophantine Equations in Two Variables

Weak fuzzy complex numbers are defined as , with as an extension of real numbers with . This paper is dedicated to studying weak fuzzy complex linear Diophantine equations in two weak fuzzy complex variables, by transforming the weak fuzzy complex Diophantine equation to a classical equivalent Diophantine system and going directly from the solutions of classical system into the desired equation. Algorithms for generating solutions of the previous equation will be presented in terms of theorems with many related examples that clarify the validity of our work.

A Solution of the Complex Fuzzy Heat Equation in Terms of Complex Dirichlet Conditions Using a Modified Crank–Nicolson Method

Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that involves a complex fuzzy heat equation under Hukuhara differentiability. The researchers utilize an implicit finite difference scheme, namely the Crank–Nicolson method, to tackle complex fuzzy heat equations. The problem’s fuzziness arises from the coefficients in both amplitude and phase terms, as well as in the initial and boundary conditions, with the Convex normalized triangular fuzzy numbers extended to the unit disk in the complex plane. The researchers take advantage of the properties and benefits of CFS theory in the proposed numerical methods and subsequently provide a new proof of the stability under CFS theory. A numerical example is presented to demonstrate the proposed approach’s reliability and feasibility, with the results showing good agreement with the exact solution and relevant theoretical aspects.

Fractional derivatives of fuzzy complex-variable functions and application to fuzzy fractional complex differential equations

In this paper, through theory of fuzzy complex analysis, we introduce some representations of r-cuts of trivial fuzzy zero, nontrivial fuzzy zero and fuzzy complex numbers in polar form on complex plane and establish some of their basic properties. Based on this, the addition and scalar multiplication operations of fuzzy complex numbers are established and the calculus of fuzzy complex-variable functions is studied. Moreover, we introduce a notion of fuzzy Ortigueira fractional derivative of fuzzy complex-variable functions and develop a theory of fuzzy fractional calculus on complex plane including some geometric features. As applications, the existence and uniqueness theorem of solutions for nonlinear fuzzy fractional complex-variable differential equations is established and some examples are provided.

Discrete fuzzy complex-valued function and complex fuzzy Caputo fractional difference equations

In this paper, we introduce a notion of complex compact intervals on which Minkowski addition and scalar multiplication operators are well defined to form the space of complex fuzzy numbers whose α-cuts are complex intervals. Some basic fuzzy partial ordering relations among the complex numbers with “≤C”, the complex intervals with “≪”, the complex fuzzy numbers with “⪯C˜” and the discrete fuzzy complex-valued functions with ≲ are studied. Based on these fundamental properties, we introduce a concept of Caputo fractional sum and difference for the discrete fuzzy complex-valued functions and establish some of their basic results. Moreover, by introducing the complex monotone iterative technique combined with the method of upper and lower solutions, the existence of extremal solutions for discrete complex fuzzy Caputo fractional difference equations is investigated. In each section, several examples are presented to show the feasibility of our obtained results.

Notes on Extension of Fuzzy Complex Sets

The aim of this paper is to modify and improve the corresponding weakness results of multi-fuzzy complex numbers as an extension of fuzzy complex numbers, next we introduce and study the generalized multi-fuzzy complex numbers and get some results. Lastly, we discuss the derivative of functions mapping complex numbers into as an extension of fuzzy complex derivatives.

On The Fuzzy Weak Complex Vector Spaces

The objective of this paper is to present for the first time the concept of fuzzy weak complex vector spaces depending on the ring of fuzzy weak complex numbers. Also, we examine some elementary algebraic properties of fuzzy weak complex vector spaces in terms of theorems. On the other hand, we illustrate many related examples to explain the novelty of these spaces.

A Computer Program For The System Of Weak Fuzzy Complex Numbers And Their Arithmetic Operations Using Python

Weak fuzzy complex number system is considered as a generalization of complex numbers by using an algebraic element with fuzzy property. It is believed that this new numerical system will play many important roles in various computer science disciplines, proceeding from this importance this paper aims to introduce for the first time the algebraic operations defined on weak fuzzy complex numbers using Python and Jupyter Notebook. Hence, by building a new class and its functions in Python for this new set, we can now use weak fuzzy complex numbers and the main arithmetic operations on them.

The Geometrical Characterization for The Solutions of a Vectorial Equation By Using Weak Fuzzy Complex Numbers and Other Generalizations Of Real Numbers

The main goal of this paper is to study the geometrical characterization of the solutions for a vectorial equation defined in the twothree dimensional Euclidean spaces. The geometrical characterization of the solutions for the desired vectorial equation is obtained for many different values of t based on the circles and spheres in some generalizations of the real field, especially dual numbers, weak fuzzy complex numbers split-complex numbers, and complex numbers.

An Introduction to Weak Fuzzy Complex Numbers

The objective of this paper is to define for the first time the concept of weak fuzzy complex numbers as a novel generalization of real numbers. Also, it presents some of their elementary properties and algorithms for solving some related algebraic equations.

Some Properties of The Difference Quadruple Sequences Spaces of Fuzzy Complex Numbers Related to The Space (l_p )^4 Defined by Triple Maximal Orlicz Functions

In this paper, we present some properties of the difference for quadruple sequences spaces of fuzzy complex numbers related to the space defined by triple maximal Orlicz functions and discuss some properties such as completeness, solidity, monotonicity, symmetric, convergence-freeness, and so on.

The Fuzzy Complex Linear Systems Based on a New Representation of Fuzzy Complex Numbers

Since the product of complex numbers and rectangular fuzzy complex numbers (RFCN) is not necessarily a RFCN in the former fuzzy complex linear system (FCLS), the scalar multiplication and addition operations of complex numbers and fuzzy complex numbers (FCN) based on a new representation of FCN are proposed. We also introduce a new method for solving FCLS, which can convert FCLS into two distinct linear systems. One is an n×n complex linear system, and the other is an (mn)×(mn) real linear system, where n is the number of unknown variables, and m is the number of substitutional cyclic sets composed of coefficients of FCLS. In particular, using this method to solve one-dimensional fuzzy linear systems, a (2n)×(2n) RLS is obtained, which is consistent with Friedman’s method. Finally, FCLS based on the RFCN as a special case are also investigated.

Interval Type-2 Fuzzy Differential Equations and Stability

This article is devoted to present a framework for studying type-2 fuzzy calculus. Some new concepts pertaining to interval type-2 fuzzy functions (IT2FF) and interval type-2 fuzzy differential equations (IT2FDE) with their stability analysis are introduced. A new definition of type-2 fuzzy derivative called type-2 granular derivative (T2gr-derivative) is defined to overcome the disadvantages of the previous definitions of type-2 fuzzy derivatives. Furthermore, the definitions of elementary arithmetic on the set of interval type-2 fuzzy numbers (IT2FNs) and the set of interval type-2 fuzzy complex numbers are presented. The notion of granular metric on the set of IT2FNs is also defined. Besides, the concepts of continuity, integral, and limit of IT2FFs are introduced. Using the concept of type-2 granular fuzzy Laplace transform and the approximations of T2gr-derivative proposed in this article, the exact and approximate solutions of interval T2FDE are presented. Then, some new concepts such as fuzzy node, fuzzy saddle spiral point, fuzzy star, and so forth lay the groundwork for the analysis of interval T2FDE stability. Several examples are given to illustrate the presented notions.

RETRACTED: Approximation of fuzzy complex numbers by the form x ˜ + i y ˜ 1

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219433.

On The Quadruple Sequence Spaces Of Fuzzy Complex Numbers

In this paper, the quadruple sequence spaces of fuzzy complex numbers are shown, and several features such as solidity, symmetry, monotonicity, and convergence-free are discussed.