On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models

In this work, we introduce novel postulates to establish fixed figure theorems with a focus on their extension to the domain of $m_v^b-$metric spaces. Consequently, we define conditions ensuring the existence and uniqueness of fixed circles, fixed ellipses, fixed Apollonius circles, fixed Cassini curves, fixed hyperbola, and so on for self mapping. We also partially address an open problem demonstrating that a $JS-$contraction possesses a fixed elliptic disc. This property extends to smaller discs and ellipses within a complete $m_v^b-$metric space. By challenging the conventional assumption of zero self-distance, we pave the way for more accurate mathematical models applicable to real-world scenarios. Consequently, our research contributes not only to a deeper comprehension of mathematical concepts but also to practical utility across various scientific domains. Our findings are supported by illustrative examples. Additionally, we explore the concept of fixed figures in the context of Rectified Linear Unit (ReLU), a widely-used activation function in neural networks and machine learning models. Our exploration of fixed figures in the context of Rectified Linear Units (ReLU) further deepens our understanding of nonlinear systems and their relationship to neural network behavior.