In recent years, numerous scholars have investigated the relationship between symmetry and generalized convexity. Due to this close relationship, generalized convexity and symmetry have become new areas of study in the field of inequalities. With the help of fuzzy up and down relation, the class of up and down $\lambda$-convex fuzzy-number valued mappings is introduced in this study; and weighted Hermite-Hadamard type fuzzy inclusions are demonstrated for these functions. The product of two up and down $\lambda$-convex fuzzy-number valued mappings also has Hermite-Hadamard type fuzzy inclusions, which is another development. Additionally, by imposing some mild restrictions on up and down $\lambda$-convex ( $\lambda$-concave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down $\lambda$-convexity ( $\lambda$-concavity), referred to as lower up and down $\lambda$-convex (lower up and down $\lambda$-concave) and upper up and down $\lambda$-convex ( $\lambda$-concave) fuzzy number valued mappings. Using these definitions, we have amassed many classical and novel exceptional cases that implement the key findings. Our proven results expand and generalize several previous findings in the literature body. Additionally, we offer appropriate examples to corroborate our theoretical findings.

To create various kinds of inequalities, the idea of convexity is essential. Convexity and integral inequality hence have a significant link. This study's goals are to introduce a new class of generalized convex fuzzy-interval-valued functions (convex 𝘍𝘐𝘝𝘍s) which are known as $\left(\mathfrak{p},\mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s and to establish Jensen, Schur and Hermite-Hadamard type inequalities for $\left(\mathfrak{p},\mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s using fuzzy order relation. The Kulisch-Miranker order relation, which is based on interval space, is used to define this fuzzy order relation level-wise. Additionally, we have demonstrated that, as special examples, our conclusions encompass a sizable class of both new and well-known inequalities for $\left(\mathfrak{p},\mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s. We offer helpful examples that demonstrate the theory created in this study's application. These findings and various methods might point the way in new directions for modeling, interval-valued functions and fuzzy optimization issues.

There are many schools of Chinese martial arts routines and complex movements; research on this topic is more geared toward Taijiquan (a kind of traditional Chinese shadow boxing), which is a more well-known type of martial arts. Therefore, the purpose of this paper is to visually analyze the research of Chinese martial arts routines based on the knowledge graph method and to propose a knowledge graph method based on the fuzzy set theory, which is called the transF model throughout this paper. The transF model used the fuzzy relational operation of vectors to not only reduce the computational complexity, but to also better integrate multi-dimensional data, especially when the training set is not particularly sufficient. For the visual analysis of Chinese martial arts routines, this paper selected the 16-year data from 2005 to 2020 as the analysis sample, analyzed high-yield institutions and high-yield authors, and conducted a centrality analysis of the whole dataset. From the structure of the knowledge graph, traditional martial arts are the core part of Chinese martial arts, with a centrality of 0.14. Competitive martial arts are the main branch of Chinese martial arts and the third core after Tai Chi and traditional martial arts, with a centrality of 0.41, which is higher than that of traditional martial arts. This shows its importance in martial arts research.

In this manuscript, we generalized the notions of three-way decisions (3WD) and decision theoretic rough sets (DTRS) in the framework of Complex q-rung orthopair 2-tuple linguistic variables (CQRO2-TLV) and then deliberated some of its important properties. Moreover, we considered some very useful and prominent aggregation operators in the framework of CQRO2-TLV, while further observing the importance of the generalized Maclurin symmetric mean (GMSM) due to its applications in symmetry analysis, interpolation techniques, analyzing inequalities, measuring central tendency, mathematical analysis and many other real life problems. We initiated complex q-rung orthopair 2-tuple linguistic (CQRO2-TL) information and GMSM to introduce the CQRO2-TL GMSM (CQRO2-TLGMSM) operator and the weighted CQRO2-TL GMSM (WCQRO2-TLGMSM) operator, and then demonstrated their properties such as idempotency, commutativity, monotonicity and boundedness. We also investigated a CQRO2-TL DTRS model. In the end, a comparative study is given to prove the authenticity, supremacy, and effectiveness of our proposed notions.

The importance of convex and non-convex functions in the study of optimization is widely established. The concept of convexity also plays a key part in the subject of inequalities due to the behavior of its definition. The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this study, first, Hermite-Hadamard type inequalities for LR- $p$-convex interval-valued functions (LR- $p$-convex-I-V-F) are constructed in this study. Second, for the product of p-convex various Hermite-Hadamard (HH) type integral inequalities are established. Similarly, we also obtain Hermite-Hadamard-Fejér (HH-Fejér) type integral inequality for LR- $p$-convex-I-V-F. Finally, for LR- $p$-convex-I-V-F, various discrete Schur's and Jensen's type inequalities are presented. Moreover, the results presented in this study are verified by useful nontrivial examples. Some of the results reported here for be LR- $p$-convex-I-V-F are generalizations of prior results for convex and harmonically convex functions, as well as $p$-convex functions.

In this paper, we determine the sufficient Karush-Kuhn-Tucker (KKT) conditions of optimality of a set-valued fractional programming problem (in short, SVFP) $(\mathrm{F}\mathrm{P})$ under the suppositions of contingent epidifferentiation and $\sigma$-arcwisely connectivity. We additionally explore the results of duality of parametric $(\mathrm{P}\mathrm{D})$, Mond-Weir $(\mathrm{M}\mathrm{W}\mathrm{D})$, Wolfe $(\mathrm{W}\mathrm{D})$, and mixed $(\mathrm{M}\mathrm{D})$ kinds for the problem $(\mathrm{F}\mathrm{P})$.

For figuring out general variational inequalities, we propose a novel and innovative iterative method. First, we demonstrate that the fixed point formulation and general vaiational inequality are equivalent. The fixed point formulation is used to formulate the explicit and implicit schemes. The general variational inequalities are the basis for the new algorithms. The newly developed algorithm is demonstrated numerically. For figuring out general variational inequalities, these new methods are innovative. Additionally, the convergence analysis is provided under certain favorable conditions.

Reference parameter mapping (passing arguments by reference) is a technique where the reference (like to find physical meaning, memory address) of a parameter is passed to a function or procedure, rather than a copy of the parameter's value. This approach enables changes made to the parameter within the function to affect the original data. In decision-making systems, reference parameter mapping (passing arguments by reference) offers several key advantages that enhance flexibility, consistency, and efficiency. This is especially useful in scenarios where decisions are based on shared data, complex interactions, and iterative updates. In this paper, a new class of fuzzy set was introduced that is known as the $\left(q_1^{},q_2^{}\right)$-rung Diophantine fuzzy set, where $q_1$ and $q_2$ are reference parameter mappings. Most of the classical and new generalized fuzzy sets are exceptional classes of $\left(q_1^{},q_2^{}\right)$-rung Diophantine fuzzy set ( $\left(q_1^{},q_2^{}\right)$- $RDFS$) like intuitionistic fuzzy set ( $IFS$), Pythagorean fuzzy Sets ( $P\mathrm{y}FS$s) and $\mathfrak{q}$-rung Orthopair fuzzy sets ( $\mathfrak{q}$- $ROFS$s), linear Diophantine fuzzy sets ( $LDFS$), and so on. It is commonly seen in multi-criteria decision-making ( $MCDM$) scenarios that the presence of imprecise information and ambiguity in the decision maker's judgment affects the resolution technique. Fuzzy models that are now in use are unable to effectively manage these uncertainties to provide an appropriate balance during the decision-making process. Using control (reference) parameter mappings, $\left(q_1^{},q_2^{}\right)$- $RDFS$s are potent fuzzy model that can handle these challenging problems. Two more novel ideas are presented in this work: $\left(q_1^{},q_2^{}\right)$-rung Diophantine fuzzy averaging and geometric aggregation operators with newly defined score and accuracy functions. An agricultural field robot $MCDM$ framework was proposed, incorporating $\left(q_1^{},q_2^{}\right)$-rung Diophantine fuzzy averaging and geometric aggregation operators. This strategy's efficacy and adaptability in addressing real-world issues were demonstrated by its application to get more benefits. This study has a lot of potential to handle difficult socioeconomic issues and offer vital information to academic, government, and analysts searching for fresh approaches in a variety of fields.