A Historical Account of Types of Fuzzy Sets and Their Relationships

In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used

New Challenges in Neutrosophic Theory and Applications

Neutrosophic theory has representatives on all continents and, therefore, it can be said to be a universal theory. On the other hand, according to the three volumes of “The Encyclopedia of Neutrosophic Researchers” (2016, 2018, 2019), plus numerous others not yet included in Encyclopedia book series, about 1200 researchers from 73 countries have applied both the neutrosophic theory and method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics lies in the introduction of the degree of indeterminacy/neutrality (I) as an independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus, etc., and their applications in multiple fields have been extended and applied in various fields, such as communication, management, and information technology. We believe that this book serves as useful guidance for learning about the current progress in neutrosophic theories. In total, 22 studies have been presented and reflect the call of the thematic vision. The contents of each study included in the volume are briefly described as follows. The first contribution, authored by Wadei Al-Omeri and Saeid Jafari, addresses the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets in neutrosophic topological spaces. In the article “Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution”, the authors Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, and Abdur Razzaque Mughal discuss the use of probability distribution function of Birnbaum–Saunders distribution as a proportion of defective items and the acceptance probability in a fuzzy environment. Further, the authors Derya Bakbak, Vakkas Uluc¸ay, and Memet S¸ahin present the “Neutrosophic Soft Expert Multiset and Their Application to Multiple Criteria Decision Making” together with several operations defined for them and their important algebraic properties. In “Neutrosophic Multigroups and Applications”, Vakkas Uluc¸ay and Memet S¸ahin propose an algebraic structure on neutrosophic multisets called neutrosophic multigroups, deriving their basic properties and giving some applications to group theory. Changxing Fan, Jun Ye, Sheng Feng, En Fan, and Keli Hu introduce the “Multi-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environment” and test the effectiveness of their new methods. Another decision-making study upon an everyday life issue which empowered us to organize the key objective of the industry developing is given in “Neutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Method” written by Khaleed Alhazaymeh, Muhammad Gulistan, Majid Khan, and Seifedine Kadry

Planarity in cubic intuitionistic graphs and their application to control air traffic on a runway

Fuzzy modeling plays a pivotal role in various fields, including science, engineering, and medicine. In comparison to conventional models, fuzzy models offer enhanced accuracy, adaptability, and resemblance to real-world systems and help researchers to always make the best choice in complex problems. A type of fuzzy graph that is widely used in medical and psychological sciences is the cubic intuitionistic fuzzy graph, which plays an important role in various fields such as computer science, psychology, medicine, and political sciences. It is also used to find effective people in an organization or social institution. In this research endeavor, we embark upon elucidating the innovative notion of a cubic intuitionistic planar graph, delving into its intricate properties and attributes. Additionally, we unveil the novel concept of a cubic intuitionistic dual graph, thus enriching the realm of graph theory with further profundity. Furthermore, our exploration encompasses the elucidation of other pertinent terminologies, such as cubic intuitionistic multi-graphs, along with the categorization of edges into the distinct classifications of strong and weak edges. Moreover, we discern the concept of the degree of planarity within the context of CIPG and unveil the notion of strong and weak faces. Additionally, we delve into the construction of cubic intuitionistic dual graphs, which can be realized in cases where the initial graph is planar or possesses a degree of planarity ≥0.67. Notably, we furnish the exposition with a comprehensive discussion on noteworthy findings and substantial results pertaining to these captivating topics, contributing valuable insights on the field of graph theory. Last, we shall endeavor to exemplify the practical relevance and importance of our research by presenting an illuminating real-world application, thus demonstrating the tangible impact and significance of our endeavors in this research article

Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity {i.e. element, concept, idea, theory, logical proposition, etc.}, is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor .Based on neutrosophy, the neutrosophic triplets were founded, which have a similar form (x, neut(x), anti(x)), that satisfy several axioms, for each element x in a given set.This collective book presents original research papers by many neutrosophic researchers from around the world, that report on the state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets and their algebraic structures – that have been defined recently in 2016 but have gained interest from world researchers. Connections between classical algebraic structures and neutrosophic triplet / duplet / multiset structures are also studied. And numerous neutrosophic applications in various fields, such as: multi-criteria decision making, image segmentation, medical diagnosis, fault diagnosis, clustering data, neutrosophic probability, human resource management, strategic planning, forecasting model, multi-granulation, supplier selection problems, typhoon disaster evaluation, skin lesson detection, mining algorithm for big data analysis, etc

Mapping Analyte-Signal Relations in LC-MS Based Untargeted Metabolomics

The goal of untargeted metabolomics is to profile metabolism by measuring as many metabolites as possible. A major advantage of the untargeted approach is the detection of unexpected or unknown metabolites. These metabolites have chemical structures, metabolic pathways, or cellular functions that have not been previously described. Hence, they represent exciting opportunities to advance our understanding of biology. This beneficial approach, however, also adds considerable complexity to the analysis of metabolomics data - an individual signal cannot be readily identified as a unique metabolite. As such, a major challenge faced by the untargeted metabolomic workflow is extracting the analyte content from a dataset. Successful applications of metabolomics bypass this limitation by throwing away the 99% of the dataset that is not statistically altered between sample groups.1 This widely accepted approach to untargeted metabolomics is functional for a very narrow set of applications, but critically, it fails to provide a comprehensive view of metabolism

New Development of Neutrosophic Probability, Neutrosophic Statistics, Neutrosophic Algebraic Structures, and Neutrosophic & Plithogenic Optimizations

This Special Issue puts forward for discussion state-of-the-art papers on new topics related to neutrosophic theories, such as neutrosophic algebraic structures, neutrosophic triplet algebraic structures, neutrosophic extended triplet algebraic structures, neutrosophic algebraic hyperstructures, neutrosophic triplet algebraic hyperstructures, neutrosophic n-ary algebraic structures, neutrosophic n-ary algebraic hyperstructures, refined neutrosophic algebraic structures, refined neutrosophic algebraic hyperstructures, quadruple neutrosophic algebraic structures, refined quadruple neutrosophic algebraic structures, neutrosophic image processing, neutrosophic image classification, neutrosophic computer vision, neutrosophic machine learning, neutrosophic artificial intelligence, neutrosophic data analytics, neutrosophic deep learning, neutrosophic symmetry, and their applications in the real world. This book leads to the further advancement of the neutrosophic and plithogenic theories of NeutroAlgebra and AntiAlgebra, NeutroGeometry and AntiGeometry, Neutrosophic n-SuperHyperGraph (the most general form of graph of today), Neutrosophic Statistics, Plithogenic Logic as a generalization of MultiVariate Logic, Plithogenic Probability and Plithogenic Statistics as a generalization of MultiVariate Probability and Statistics, respectively, and presents their countless applications in our every-day world

Searching the space of representations: reasoning through transformations for mathematical problem solving

The role of representation in reasoning has been long and widely regarded as crucial. It has remained one of the fundamental considerations in the design of information-processing systems and, in particular, for computer systems that reason. However, the process of change and choice of representation has struggled to achieve a status as a task for the systems themselves. Instead, it has mostly remained a responsibility for the human designers and programmers. Many mathematical problems have the characteristic of being easy to solve only after a unique choice of representation has been made. In this thesis we examine two classes of problems in discrete mathematics which follow this pattern, in the light of automated and interactive mechanical theorem provers. We present a general notion of structural transformation, which accounts for the changes of representation seen in such problems, and link this notion to the existing Transfer mechanism in the interactive theorem prover Isabelle/HOL. We present our mechanisation in Isabelle/HOL of some specific transformations identified as key in the solutions of the aforementioned mathematical problems. Furthermore, we present some tools that we developed to extend the functionalities of the Transfer mechanism, designed with the specific purpose of searching efficiently the space of representations using our set of transformations. We describe some experiments that we carried out using these tools, and analyse these results in terms of how close the tools lead us to a solution, and how desirable these solutions are. The thorough qualitative analysis we present in this thesis reveals some promise as well as some challenges for the far-reaching problem of representation in reasoning, and the automation of the processes of change and choice of representation