Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications

Topic Information

Dear Colleagues,

With the rapid development of new technologies such as artificial intelligence and big date, as the mathematical basis of artificial intelligence, fuzzy set theory has always been concerned by scholars. And fuzzy analysis theory is an important part of it. As the basic concept of fuzzy analysis, fuzzy number, fuzzy difference, fuzzy differential, etc. involve many related important theories, such as: fuzzy difference equation, fuzzy differential equation, fuzzy optimization, fuzzy game, fuzzy decision-making, fuzzy control, etc. Therefore, it is necessary to study the fuzzy number, fuzzy difference, fuzzy differential and related theories and applications.

This Topic, “Fuzzy Number, Fuzzy Difference, Fuzzy Differential：Theory and  Applications”, invites papers on theoretical and applied issues, including, but not limited to, the following:

• Theories and applications of fuzzy Number.
• Theories and applications of fuzzy difference equation.
• Theories and applications of fuzzy differential equation. 
• Techniques and algorithms of fuzzy optimization.
• Strategy of fuzzy game.
• Methods and analysis fuzzy decision-making.  
• Techniques and methods of fuzzy control.
• This Topic will present the results of research describing recent advances in fuzzy number, fuzzy difference, and fuzzy differential.

Prof. Dr. Changyou Wang
Prof. Dr. Dong Qiu
Dr. Yonghong Shen
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Algorithms algorithms	2.3	3.7	2008	15 Days	CHF 1600	Submit
Axioms axioms	2.0	-	2012	21.8 Days	CHF 2400	Submit
Information information	3.1	5.8	2010	18 Days	CHF 1600	Submit
Mathematics mathematics	2.4	3.5	2013	16.9 Days	CHF 2600	Submit
Symmetry symmetry	2.7	4.9	2009	16.2 Days	CHF 2400	Submit

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Published Papers (11 papers)

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24 pages, 522 KiB  

Open AccessArticle

The Enumeration of (⊙,∨)-Multiderivations on a Finite MV-Chain

by Xueting Zhao, Kai Duo, Aiping Gan and Yichuan Yang

Axioms 2024, 13(4), 250; https://doi.org/10.3390/axioms13040250 - 10 Apr 2024

Viewed by 297

Abstract

In this paper, $(\odot ,\vee )$-multiderivations on an MV-algebra A are introduced, the relations between $(\odot ,\vee )$-multiderivations and $(\odot ,\vee )$-derivations are discussed. The set $\mathrm{MD}\left(A\right)$ of  [...] Read more.

In this paper, $(\odot ,\vee )$-multiderivations on an MV-algebra A are introduced, the relations between $(\odot ,\vee )$-multiderivations and $(\odot ,\vee )$-derivations are discussed. The set $\mathrm{MD}\left(A\right)$ of $(\odot ,\vee )$-multiderivations on A can be equipped with a preorder, and $\left(\mathrm{MD}\right(A)/\sim ,\preccurlyeq )$ can be made into a partially ordered set with respect to some equivalence relation ∼. In particular, for any finite MV-chain $L_n$, $(\mathrm{MD}\left(L_n\right)/\sim ,\preccurlyeq )$ becomes a complete lattice. Finally, a counting principle is built to obtain the enumeration of $\mathrm{MD}\left(L_n\right)$. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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18 pages, 1844 KiB  

Open AccessArticle

Two Extensions of the Sugeno Class and a Novel Constructed Method of Strong Fuzzy Negation for the Generation of Non-Symmetric Fuzzy Implications

by Maria N. Rapti, Avrilia Konguetsof and Basil K. Papadopoulos

Symmetry 2024, 16(3), 317; https://doi.org/10.3390/sym16030317 - 06 Mar 2024

Viewed by 480

Abstract

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method [...] Read more.

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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20 pages, 1847 KiB  

Open AccessArticle

Special Discrete Fuzzy Numbers on Countable Sets and Their Applications

by Na Qin and Zengtai Gong

Symmetry 2024, 16(3), 264; https://doi.org/10.3390/sym16030264 - 21 Feb 2024

Viewed by 569

Abstract

There are some drawbacks to arithmetic and logic operations of general discrete fuzzy numbers, which limit their application. For example, the result of the addition operation of general discrete fuzzy numbers defined by the Zadeh’s extension principle may not satisfy the condition of [...] Read more.

There are some drawbacks to arithmetic and logic operations of general discrete fuzzy numbers, which limit their application. For example, the result of the addition operation of general discrete fuzzy numbers defined by the Zadeh’s extension principle may not satisfy the condition of becoming a discrete fuzzy number. In order to solve these problems, special discrete fuzzy numbers on countable sets are investigated in this paper. Since the representation theorem of fuzzy numbers is the basic tool of fuzzy analysis, two kinds of representation theorems of special discrete fuzzy numbers on countable sets are studied first. Then, the metrics of special discrete fuzzy numbers on countable sets are defined, and the relationship between these metrics and the uniform Hausdorff metric (i.e., supremum metric) of general fuzzy numbers is discussed. In addition, the triangular norm and triangular conorm operations (t-norm and t-conorm for short) of special discrete fuzzy numbers on countable sets are presented, and the properties of these two operators are proven. We also prove that these two operators satisfy the basic conditions for closure of operation and present some examples. Finally, the applications of special discrete fuzzy numbers on countable sets in image fusion and aggregation of subjective evaluation are proposed. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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12 pages, 1442 KiB  

Open AccessArticle

Simulation-Enhanced MQAM Modulation Identification in Communication Systems: A Subtractive Clustering-Based PSO-FCM Algorithm Study

by Zhi Quan, Hailong Zhang, Jiyu Luo and Haijun Sun

Information 2024, 15(1), 42; https://doi.org/10.3390/info15010042 - 12 Jan 2024

Viewed by 925

Abstract

Signal modulation recognition is often reliant on clustering algorithms. The fuzzy c-means (FCM) algorithm, which is commonly used for such tasks, often converges to local optima. This presents a challenge, particularly in low-signal-to-noise-ratio (SNR) environments. We propose an enhanced FCM algorithm that incorporates [...] Read more.

Signal modulation recognition is often reliant on clustering algorithms. The fuzzy c-means (FCM) algorithm, which is commonly used for such tasks, often converges to local optima. This presents a challenge, particularly in low-signal-to-noise-ratio (SNR) environments. We propose an enhanced FCM algorithm that incorporates particle swarm optimization (PSO) to improve the accuracy of recognizing M-ary quadrature amplitude modulation (MQAM) signal orders. The process is a two-step clustering process. First, the constellation diagram of the received signal is used by a subtractive clustering algorithm based on SNR to figure out the initial number of clustering centers. The PSO-FCM algorithm then refines these centers to improve precision. Accurate signal classification and identification are achieved by evaluating the relative sizes of the radii around the cluster centers within the MQAM constellation diagram and determining the modulation order. The results indicate that the SC-based PSO-FCM algorithm outperforms the conventional FCM in clustering effectiveness, notably enhancing modulation recognition rates in low-SNR conditions, when evaluated against a variety of QAM signals ranging from 4QAM to 64QAM. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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18 pages, 371 KiB  

Open AccessArticle

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

by Abdullah Assiry, Sabeur Mansour and Amir Baklouti

Axioms 2024, 13(1), 2; https://doi.org/10.3390/axioms13010002 - 19 Dec 2023

Viewed by 852

Abstract

This paper performed an investigation into the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1\mid 1})$, a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively [...] Read more.

This paper performed an investigation into the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1\mid 1})$, a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( $S\psi D\odot$) residing on the super-circle $S^{1|1}$. We also introduce and rigorously define the central charge within the framework of $\overrightarrow{(}{S}^{1\mid 1})$, leveraging the canonical central extension of $S\psi D\odot$. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1|1})$ is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

17 pages, 341 KiB  

Open AccessArticle

Multi-Attribute Group Decision-Making Methods Based on Entropy Weights with q-Rung Picture Uncertain Linguistic Fuzzy Information

by Mengran Sun, Yushui Geng and Jing Zhao

Symmetry 2023, 15(11), 2027; https://doi.org/10.3390/sym15112027 - 08 Nov 2023

Viewed by 652

Abstract

This paper introduces a new concept called q-rung picture uncertain linguistic fuzzy sets (q-RPULSs). These sets provide a reliable and comprehensive method for describing complex and uncertain decision-making information. In addition, q-RPULSs help to integrate the decision maker’s quantitative assessment ideas with qualitative [...] Read more.

This paper introduces a new concept called q-rung picture uncertain linguistic fuzzy sets (q-RPULSs). These sets provide a reliable and comprehensive method for describing complex and uncertain decision-making information. In addition, q-RPULSs help to integrate the decision maker’s quantitative assessment ideas with qualitative assessment information. For the q-RPUL multi-attribute group decision-making problem with unknown weight information, an entropy-based fuzzy set method for q-rung picture uncertainty language is proposed. The method considers the interrelationships among attributes and builds a q-rung picture uncertain language model. In addition, the q-RPULMSM operator and its related properties are discussed in this paper. This operator enables the fusion of q-RPULSs and helps to reach consensus in decision-making scenarios. To demonstrate the validity of the methodology, we provide a real case study involving commodity selection. Based on this case study, the reasonableness and superiority of the method are evaluated, highlighting the practical advantages and applicability of q-RPULSs in decision-making processes. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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29 pages, 1203 KiB  

Open AccessArticle

Defuzzification of Non-Linear Pentagonal Intuitionistic Fuzzy Numbers and Application in the Minimum Spanning Tree Problem

by Ali Mert

Symmetry 2023, 15(10), 1853; https://doi.org/10.3390/sym15101853 - 02 Oct 2023

Cited by 1 | Viewed by 790

Abstract

In recent years, with the variety of digital objects around us becoming a source of information, the fields of artificial intelligence (AI) and machine learning (ML) have experienced very rapid development. Processing and converting the information around us into data within the framework [...] Read more.

In recent years, with the variety of digital objects around us becoming a source of information, the fields of artificial intelligence (AI) and machine learning (ML) have experienced very rapid development. Processing and converting the information around us into data within the framework of the information processing theory is important, as AI and ML techniques need large amounts of reliable data in the training and validation stages. Even though information naturally contains uncertainty, information must still be modeled and converted into data without neglecting this uncertainty. Mathematical techniques, such as the fuzzy theory and the intuitionistic fuzzy theory, are used for this purpose. In the intuitionistic fuzzy theory, membership and non-membership functions are employed to describe intuitionistic fuzzy sets and intuitionistic fuzzy numbers (IFNs). IFNs are characterized by the mathematical statements of these two functions. A more general and inclusive definition of IFN is always a requirement in AI technologies, as the uncertainty introduced by various information sources needs to be transformed into similar IFNs without neglecting the variety of uncertainty. In this paper, we proposed a general and inclusive mathematical definition for IFN and called this IFN a non-linear pentagonal intuitionistic fuzzy number (NLPIFN), which allows its users to maintain variety in uncertainty. We know that AI technology implementations are performed in computerized environments, so we need to transform the IFN into a crisp number to make such IFNs available in such environments. Techniques used in transformation are called defuzzification methods. In this paper, we proposed a short-cut formula for the defuzzification of a NLPIFN using the intuitionistic fuzzy weighted averaging based on levels (IF-WABL) method. We also implemented our findings in the minimum spanning tree problem by taking weights as NLPIFNs to determine the uncertainty in the process more precisely. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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20 pages, 330 KiB  

Open AccessArticle

Interval-Valued Topology on Soft Sets

by Sadi Bayramov, Çiğdem Gündüz Aras and Ljubiša D. R. Kočinac

Axioms 2023, 12(7), 692; https://doi.org/10.3390/axioms12070692 - 16 Jul 2023

Cited by 1 | Viewed by 758

Abstract

In this paper, we study the concept of interval-valued fuzzy set on the family $\mathsf{SS}\left(X,E\right)$ of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy [...] Read more.

In this paper, we study the concept of interval-valued fuzzy set on the family $\mathsf{SS}\left(X,E\right)$ of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy topology (cotopology) $\tau$ on $\mathsf{SS}\left(X,E\right)$. We obtain that each interval-valued fuzzy topology is a descending family of soft topologies. In addition, we study some topological structures such as interval-valued fuzzy neighborhood system of a soft point, base and subbase of $\tau$ and investigate some relationships among them. Finally, we give some concepts such as direct sum, open mapping and continuous mapping and consider connections between them. A few examples support the presented results. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

16 pages, 313 KiB  

Open AccessArticle

Multiple-Attribute Decision Making Based on the Probabilistic Dominance Relationship with Fuzzy Algebras

by Amir Baklouti

Symmetry 2023, 15(6), 1188; https://doi.org/10.3390/sym15061188 - 02 Jun 2023

Cited by 4 | Viewed by 773

Abstract

In multiple-attribute decision-making (MADM) problems, ranking the alternatives is an important step for making the best decision. Intuitionistic fuzzy numbers (IFNs) are a powerful tool for expressing uncertainty and vagueness in MADM problems. However, existing ranking methods for IFNs do not consider the [...] Read more.

In multiple-attribute decision-making (MADM) problems, ranking the alternatives is an important step for making the best decision. Intuitionistic fuzzy numbers (IFNs) are a powerful tool for expressing uncertainty and vagueness in MADM problems. However, existing ranking methods for IFNs do not consider the probabilistic dominance relationship between alternatives, which can lead to inconsistent and inaccurate rankings. In this paper, we propose a new ranking method for IFNs based on the probabilistic dominance relationship and fuzzy algebras. The proposed method is able to handle incomplete and uncertain information and can generate consistent and accurate rankings. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

17 pages, 326 KiB  

Open AccessArticle

New Arithmetic Operations of Non-Normal Fuzzy Sets Using Compatibility

by Hsien-Chung Wu

Axioms 2023, 12(3), 277; https://doi.org/10.3390/axioms12030277 - 07 Mar 2023

Cited by 2 | Viewed by 1084

Abstract

The new arithmetic operations of non-normal fuzzy sets are studied in this paper by using the extension principle and considering the general aggregation function. Usually, the aggregation functions are taken to be the minimum function or t-norms. In this paper, we considered a [...] Read more.

The new arithmetic operations of non-normal fuzzy sets are studied in this paper by using the extension principle and considering the general aggregation function. Usually, the aggregation functions are taken to be the minimum function or t-norms. In this paper, we considered a general aggregation function to set up the arithmetic operations of non-normal fuzzy sets. In applications, the arithmetic operations of fuzzy sets are always transferred to the arithmetic operations of their corresponding $\alpha$-level sets. When the aggregation function is taken to be the minimum function, this transformation is clearly realized. Since the general aggregation function was adopted in this paper, the concept of compatibility with $\alpha$-level sets is needed and is proposed, which can cover the conventional case using minimum functions as the special case. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

9 pages, 273 KiB  

Open AccessArticle

Best Approximation Results for Fuzzy-Number-Valued Continuous Functions

by Juan J. Font and Sergio Macario

Axioms 2023, 12(2), 192; https://doi.org/10.3390/axioms12020192 - 11 Feb 2023

Viewed by 891

Abstract

In this paper, we study the best approximation of a fixed fuzzy-number-valued continuous function to a subset of fuzzy-number-valued continuous functions. We also introduce a method to measure the distance between a fuzzy-number-valued continuous function and a real-valued one. Then, we prove the [...] Read more.

In this paper, we study the best approximation of a fixed fuzzy-number-valued continuous function to a subset of fuzzy-number-valued continuous functions. We also introduce a method to measure the distance between a fuzzy-number-valued continuous function and a real-valued one. Then, we prove the existence of the best approximation of a fuzzy-number-valued continuous function to the space of real-valued continuous functions by using the well-known Michael selection theorem. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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