Wearable technology allows users to monitor and track various fitness-related activities and data, including calories burned, distance traveled, heart rate, and sleep patterns. To choose best app for fitness, here introduced a new innovation in fuzzy algebra named as complex interval valued picture fuzzy soft relations (CIVPFSR) by defining the cartesian product (CP) of two complex interval valued picture fuzzy soft sets (CIVPFSS). Additionally, the many kinds of CIVPFS relations are described, and their outcomes are also mentioned in this article. The CIVPFSS has a complex structure that includes multidimensional variables for membership, abstinence, and non-membership degrees. As a result, this research offers modeling techniques for wearable technology that are based on CIVPFSR. The score function for best human decision making is also formulated in this process. Lastly, a comparison of current structures is done to demonstrate the viability of the suggested work with the conclusion highlighting its potential to significantly advance wearable technology. Additionally, a topological space is developed for CIVPFSR, providing a geometric interpretation of these relations and offering insights into their behavior in multi-dimensional decision spaces. The basic operations, operators, related theorems, and results are discussed. Results concerning the interiors and closures are also addressed.

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.

Neutrosophic set theory, an advanced framework for error reduction, extends fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). It enhances effectiveness by refining the definition of indeterminacy, a concept for situations where values cannot be precisely determined. In this paper, we propose dividing indeterminacy into two components based on membership: relative truth (RT), which leans toward truth, and relative falsehood (RF), which leans toward falsehood. This approach improves accuracy by considering, relative truth and relative falsehood, rather than a single indeterminate value. The modified neutrosophic set, called the quadri-portioned neutrosophic soft set (QPNSS), includes four membership attributes: absolute truth, relative truth, relative falsehood, and absolute falsehood, offering greater clarity in uncertain situations. New operations are introduced on QPNSS, such as the quadri-partitioned neutrosophic soft set, subsets, complement, absolute set, set difference, and null set. AND and OR operations are also defined. Additionally, a quadri-partitioned neutrosophic soft topological space (QPNSTS) is defined, with key results presented. We define and explore new concepts such as pre-open (p-open) sets, interior, and closure. The paper also examines QPNS compactness, reducibility to finite sub-covers, and other important properties like the intersection of QPNS p-closed sets and QPNS p-compact spaces. This work contributes to the theoretical understanding of QPNS spaces, particularly in soft topological spaces.

We rigorously examine the concept of bipolar vague soft expert sets (BPVSESs) and their defining characteristics. Fundamental operations such as complement, union, and intersection are firmly established as foundational elements of the framework. Additionally, the notion of bipolar vague soft expert topology (BPVSET) is introduced, along with eight innovative definitions. Among these, the definition of the bipolar vague soft expert pre-open set, often abbreviated as the p-open set, is particularly significant for constructing diverse structures. This study also provides a strong and healthy articulation of the concepts of interior and closure, offering a detailed exploration of their interactions. Furthermore, it develops foundational topological concepts in bipolar vague soft expert topology by introducing and analyzing bases, sub-bases, and local bases. The notions of first and second countability in the bipolar vague soft expert topology context are formally defined, while separability is explored via countable dense sets. These results enhance the theoretical framework of bipolar vague soft expert topological space, supporting soft topological modeling under uncertainty and parameterization. A comprehensive investigation into these foundational concepts culminates in a series of compelling results concerning the basis of bipolar vague soft expert topological spaces. Finally, it introduces a decision-making framework based on bi-polar vague soft expert sets to support cancer diagnosis.

American scientific publishing group

This study uses important findings from Hilbert Space (HS) theory to suggest new fixed-point theorems for Three Self-Mappings (3-SMs) that apply to Generalized Inner Product Spaces (GIPS). We made it apparent when Common Fixed Points (CFPs) can exist and be unique, even when the contraction criteria aren’t as strict. This improvement makes the theory better for systems with more than one operator. Our results are important in two main areas: (1) Artificial Intelligence (AI), where we use fixed-point analysis to make sure that Deep Equilibrium Models (DEMs) will work correctly, and (2) Cryptography (Crypto), where we use the mathematical properties of inner product spaces to make new lattice-based designs for Post-Quantum Protocols (PQPs). Real-world examples from nonlinear analysis and computational proof of the presented theorems support the theoretical contributions. This work combines advanced functional analysis with modern problems in Machine Learning (ML) and cybersecurity, giving it both mathematical depth and real-world usefulness.