A Unified Framework for Hyperbolic and Ambiguous Spaces - Anoop Kumar Tiwari (Universitas Negeri Surabaya or The State University of Surabaya, 2026)
The concept of space has been a fundamental aspect of mathematical and computational modeling, with Euclidean and hyperbolic spaces serving as classical frameworks. This article explores the transition from hyperbolic space to ambiguous space, establishing a novel comparative framework that integrates gyrovector spaces and ambiguous set theory. Hyperbolic space, characterized by non-Euclidean geometry, forms the foundation for many applications, including machine learning and network analysis. Gyrovector space, an extension of vector space under Möbius addition, provides a computationally efficient model for hyperbolic geometry. In contrast, ambiguous sets introduce a four-dimensional membership structure, enabling more nuanced representations of uncertainty and vagueness in decision-making contexts. The concept of ambiguous space is then developed as a generalized mathematical structure that incorporates elements from both hyperbolic geometry and ambiguous set theory. Finally, we demonstrate the applicability of ambiguous space in customer segmentation, where traditional clustering methods often fail to capture complex consumer behavior.