Fuzzy Number, Fuzzy Difference, Fuzzy Differential

The purpose of this reprint is to offer readers a deeper comprehension of the latest research advancements in fuzzy numbers, fuzzy differences, and fuzzy differentials. It covers fuzzy reasoning, optimization, decision-making, and control. A method based on symmetric quintuple-implication principles is introduced for pattern recognition in fuzzy reasoning. Fuzzy analysis focuses on trapezoidal-type inequalities, while fuzzy optimization and decision-making incorporate fixed-point theorems and intuitionistic fuzzy granular matrices. Combining fuzzy time series analysis with neural networks enhances integer time series model estimations. The reprint also discusses MV-algebra structures, fuzzy negations, special discrete fuzzy numbers, and non-linear pentagonal intuitionistic fuzzy numbers. In communication systems, an enhanced fuzzy c-means algorithm is developed for modulation identification. A structured multi-attribute group decision-making method has been introduced. Furthermore, the reprint explores the best approximation results for fuzzy-number-valued continuous functions, contributing to the theoretical and practical foundations of fuzzy mathematics. These studies collectively push the boundaries of fuzzy mathematics and broaden its applications, making significant contributions to the field.