In this research, we first offer unique notions of averaging and geometric aggregation operators with confidence level by employing a probabilistic neutrosophic hesitant fuzzy rough framework. Then, we look into other descriptions of the suggested operators, such as idempotency, boundedness, and monotonicity. Additionally, for the derived operators, we establish the score and accuracy functions. We also provide a novel approach to assessing the selection procedure for smart medical devices (SMDs). The selection criteria for SMDs are quite complex, which is the most noteworthy feature of this investigation. It is suggested that these processes be simulated using a method utilizing a hesitant fuzzy set, a rough set, and a probabilistic single-valued neutrosophics set. The proposed approach is employed in the decision-making process, while taking into consideration the decision-makers’ (DMs’) level of confidence in the data they have obtained in order to deal with ambiguity, incomplete data, and uncertainty in lower and upper approximations. The major goal was to outline the issue’s complexities in order to pique interest among experts in the health care sector and encourage them to evaluate SMDs using various evaluation standards. The analysis of the technique’s outcomes demonstrated that the rankings and the results themselves were adequate and trustworthy. The effectiveness of our suggested improvements is also demonstrated through a symmetrical analysis. The symmetry behavior shows that the current techniques address more complex and advanced data.

Euclidean geometry is utilized to establish the Sombor graph parameter and its invariants. It is sum of all adjacent vertices in graph theory $\sqrt{d_{\mathrm{\Upsilon }}^2+d_{\mathrm{\Gamma }}^2}$ where $d_{\mathrm{\Upsilon }}$ is the degree of the vertex ϒ. Geometrical interpretation is used to describe the new Sombor indices types. We examined, recently developed Sombor indices for various nanotube Y-junctions in this article. In specifically, the first area-based Sombor index was introduced by Euclidean geometry. Angular orientation concept to construct the second, fourth, and sixth Sombor graph parameters, while third and fifth Sombor graph parameters are constructed by perimeter.

A computer vision model known as a generative adversarial network (GAN) creates all the visuals, including images, movies, and sounds. One of the most well-known subfields of deep learning and machine learning is generative adversarial networks. It is employed for text-to-image translations, as well as image-to-image and conceptual image-to-image translations. Different techniques are used in the processing and generation of visual data, which can lead to confusion and uncertainty. With this in mind, we define some solid mathematical concepts to model and solve the aforementioned problem. Complex picture fuzzy soft relations are defined in this study by taking the Cartesian product of two complex picture fuzzy soft sets. Furthermore, the types of complex picture fuzzy soft relations are explained, and their results are also discussed. The complex picture fuzzy soft relation has an extensive structure comprising membership, abstinence, and non-membership degrees with multidimensional variables. Therefore, this paper provides modeling methodologies based on complex picture fuzzy soft relations, which are used for the analysis of generative adversarial networks. In the process, the score functions are also formulated. Finally, a comparative analysis of existing techniques was performed to show the validity of the proposed work.

The use of a single-valued neutrosophic set (svns) makes it much easier to manage situations in which one must deal with incorrect, unexpected, susceptible, faulty, vulnerable, and complicated information. This is a result of the fact that the specific forms of material being discussed here are more likely to include errors. This new theory has directly contributed to the expansion of both the concept of fuzzy sets and intuitionistic fuzzy sets, both of which have experienced additional development as a direct consequence of the creation of this new theory. In svns, indeterminacy is correctly assessed in a way that is both subtle and unambiguous. Furthermore, membership in the truth, indeterminacy, and falsity are all completely independent of one another. In the context of algebraic analysis, certain binary operations may be regarded as interacting with algebraic modules. These modules have pervasive and complicated designs. Modules may be put to use in a wide variety of different applications. Modules have applications in a diverse range of industries and market subsets due to their adaptability and versatility. Under the umbrella of the triplet $(\mu ,\nu ,\omega )$ structure, we investigate the concept of svns and establish a relationship between it and the single-valued neutrosophic module and the single-valued neutrosophic submodule, respectively. The purpose of this study is to gain an understanding of the algebraic structures of single-valued neutrosophic submodules under the triplet structure of a classical module and to improve the validity of this method by analyzing a variety of important facets. In this article, numerous symmetrical features of modules are also investigated, which demonstrates the usefulness and practicality of these qualities. The results of this research will allow for the successful completion of both of these objectives. The tactics that we have devised for use in this article are more applicable to a wide variety of situations than those that have been used in the past. Fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets are some of the tactics that fall under this category.

The notion of fuzzy graph (FG) is widely used in many problems arising from partial or incomplete descriptions of the real world and in particular from fields such as engineering, economics, computer science, social disciplines, or medical diagnostics, and has been used in many fields of pure mathematics as well as in several areas of applied sciences such as decision making, statistics and networking. In this paper we will deal with the graph of the picture fuzzy(symmetric) set using the notion of domination in picture fuzzy graph (PFG) as a generalization of both the concept of fuzzy graph domination and intuitionistic fuzzy graph (IFG) domination. The concepts of domination theory (DT) and double domination theory (DDT) of a PFG are introduced, studied and concretely applied to the real case of an election competition to determine the minimum number of citizens a politician should meet in person in order to win the election. The choice of fuzzification (symmetric) and defuzzification (anti-symmetric) methods depends on the specific application and the type of fuzzy sets being used, whether they are symmetric or anti-symmetric. There are various methods for each process, such as centroid, max-min, and weighted average methods for defuzzification. Finally, in the last section, drawing from the application example, the features and benefits of PFGs with respect to fuzzy graphs and intuitionistic fuzzy graphs are compared and discussed.

Multidimensional datasets pose a considerable problem to machine learning models (especially in their interpretability, computation speed, and predictability). The paper explores how statistical feature selection methods e.g. LASSO, Ridge Regression, Elastic Net and Mutual Information can be used to improve the performance and transparency of the machine learning algorithms used in genomics and financial data. We compare the results of the Random Forest, Support Vector Machine, and Neural Networks, using gene expression profiles of The Cancer Genome Atlas (TCGA), and credit scoring data of Home Credit Default Risk dataset, on several metrics such as accuracy, F1-score, SHAP-based interpretability, and resources. Findings indicate that Elastic Net is always better than other approaches in processing correlated features as well as balancing between sparsity and stability, whereas Mutual Information is effective in revealing non-linear relationships. By up to 40% reducing training time and selecting features to improve model generalization and 30 reducing memory use, machine learning pipelines will be more interpretable and scalable. These results highlight the importance of statistical rigor in high dimensional machine learning processes to achieve robust and explainable AI.