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.

It is renowned that the immune reaction in the tumour micro environment is a complex cellular process that requires additional research. Therefore, it is important to interrogate the tracking path behaviour of tumor-immune dynamics to alert policy makers about critical factors of the system. Here, we use fractional derivative to structure tumor-immune interactions. Furthermore, in our research, we concentrated on the qualitative investigation and time series analysis of tumor-immune cell interactions. The solution routes are examined using a new numerical technique to emphasis the impact of the factors on tumor-immune system. We focused on the behaviour of the system with fluctuation of different values. The most crucial components of the proposed system are identified and policymakers are advised. The outcomes of the present study are the strong predictor of clinical success and the in-out of immune cells in a tumour is also critical to treatment efficacy. As a result, studying the behaviour of tumor-immune cell interactions is important to predict crucial factors for the prevention and management to the health officials.

In this work, we investigate a transmission problem for the Timoshenko system with distributed delay terms acting on the rotation-angle equations of a beam. The model describes a structure composed of two different materials connected at an interface, which introduces transmission conditions that complicate the stability analysis. Distributed delays are incorporated into the internal feedback laws associated with the variables \(v_i\), which may lead to destabilizing effects if they are not properly balanced with non-delayed damping mechanisms. Under suitable assumptions on the relative weights of the delayed and non-delayed feedback terms, as well as on the wave propagation speeds, we first establish the well-posedness of the system by using semigroup theory. In the case of equal propagation speeds, we construct a suitable Lyapunov functional and apply the energy method to derive an exponential decay estimate for the total energy of the system. This result shows that the combined action of distributed delay and non-delayed feedback is capable of guaranteeing uniform stability, provided that the delay contribution remains sufficiently small. Finally, we highlight the importance of the equal wave speed condition, which plays a crucial role in ensuring exponential convergence to equilibrium.

This study on the negative-order fractional space-time modified KdV (nfmKdV) equation provides a comprehensive analysis of how fractional differentials affect the dynamics of solitons in non-linear wave models. We are referring to introduces the nfmKdV equation, a significant extension of the traditional KdV equation, which is commonly used to model wave propagation in non-linear dispersive media. By developing both focusing and defocusing solutions and employing the Hirota technique to construct multisoliton solutions, the study opens new avenues for the exploration of fractional wave equations in diverse physical contexts. The use of fractional calculus, and specifically negative-order derivatives, enhances the model?s ability to describe real-world phenomena with long-range interactions and memory effects, offering significant potential for future research in non-linear and fractional dynamics. This newly established result warrants further investigation determine its applicability to other non-linear fractional order models, and other existing methods may be employed to explore this new development. As the fractional order approaches one, the results align with well-established findings in the literature. This study provides a deeper understanding of the dynamics of solitons in fractional media, which could be useful for modelling soliton propagation in systems where traditional integer-order models fail to capture essential behavior.