A Unified Framework for Hyperbolic and Ambiguous Spaces
DOI:
https://doi.org/10.26740/vubeta.v3i2.47699Keywords:
Hyperbolic space, gyrovector space, Ambiguous set (AS), ambiguous spaceAbstract
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.
References
[1] J. Gray, “Euclidean And Non-Euclidean Geometry,” Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, vol. 2, p. 877, 2004.
[2] J. W. Milnor, “Hyperbolic Geometry: The First 150 Years,” Bulletin of the American Mathematical Society, vol. 6, no. 1, pp. 9–24, 1982. https://doi.org/10.1090/S0273-0979-1982-14958-8.
[3] J. Stillwell, “Sources Of Hyperbolic Geometry,” American Mathematical Society, 1996, no. 10. https://doi.org/10.1090/hmath/010.
[4] B. Iversen, “Hyperbolic Geometry,” Cambridge University Press, 1992, no. 25. https://doi.org/10.1017/CBO9780511569333.
[5] W. F. Reynolds, “Hyperbolic Geometry on a Hyperboloid,” The American Mathematical Monthly, vol. 100, no. 5, pp. 442–455, 1993. https://doi.org/10.1080/00029890.1993.11990430.
[6] D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, and M. Bogun´a, “Hyperbolic Geometry Of Complex Net- Works,” Physical Review E, vol. 82, no. 3, p. 036106, 2010. https://doi.org/10.1103/PhysRevE.82.036106.
[7] J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, “Hyperbolic Geometry,” Flavors of geometry, vol. 31, pp. 59–110, 1997.
[8] J. M. Lee, “Smooth Maps. Springer,” New York, 2012, pp. 32–49. https://doi.org/10.1007/978-1-4419-9982-5_2.
[9] A. A. Ungar, “Analytic Hyperbolic Geometry: Mathematical Foundations And Applications,” World Scientific, 2005. https://doi.org/10.1142/5914.
[10] F. Papadopoulos, M. Kitsak, M. A´. Serrano, M. Boguna, and D. Krioukov, “Popularity Versus Similarity In Growing Networks,” Nature, vol. 489, no. 7417, pp. 537–540, 2012. https://doi.org/10.1038/nature11459.
[11] I. Athal et al., “Combined Impact of Radiation and Chemical Reaction on MHD Hyperbolic Tangent Nanofluid Boundary Layer Flow Past a Stretching Sheet,” Modern Physics Letters B, vol. 38, no. 16, p. 2341010, 2024. https://doi.org/10.1142/S0217984923410105.
[12] N. Khan, M. Zeeshan, M. Hashmi, and M. Inc, “Variable Thermal Conductivity And Chemical Reaction Aspects In MHD Tangent Hyperbolic Nanofluid Flow Over An Exponentially Stretching Surface,” Modern Physics Letters B, vol. 38, no. 11, p. 2450066, 2024. https://doi.org/10.1142/S0217984924500660.
[13] H. Wu et al., “Tunable Biaxial Hyperbolic Dispersion and Negative Refraction In Graphite,” Modern Physics Letters B, vol. 34, no. 11, p. 2050110, 2020. https://doi.org/10.1142/S0217984920501109.
[14] Y.-M. Chu et al., “Joule Heating, Activation Energy And Modified Diffusion Analysis For 3D Slip Flow Of Tangent Hyperbolic Nanofluid with Gyrotactic Microorganisms,” Modern Physics Letters B, vol. 35, no. 16, p. 2150278, 2021. https://doi.org/10.1142/S021798492150278X.
[15] L. Mari, J. R. de Oliveira, A. Savas-Halilaj, and R. S. de Sena, “Conformal Solitons For The Mean Curvature Flow In Hyperbolic Space,” Annals of Global Analysis and Geometry, vol. 65, no. 2, p. 19, 2024. https://doi.org/10.1007/s10455-024-09947-y.
[16] A. Khaldi, M. Maouni, and A. Ouaoua, “Asymptotic Behavior of Solution for a Fractional P-Kirchhoff Type Hyperbolic Equation with Variable Exponent,” International Journal of Applied and Computational Mathematics, vol. 10, no. 2, p. 51, 2024. https://doi.org/10.1007/s40819-023-01657-6.
[17] G. Mandal, S. Das, S. Dutta, L. N. Guin, and S. Chakravarty, “A Comparative Study Of Allee Effects and Fear- Induced Responses: Exploring Hyperbolic and Ratio-Dependent Models,” International Journal of Applied and Computational Mathematics, vol. 10, no. 5, p. 147, 2024. https://doi.org/10.1007/s40819-024-01773-x.
[18] A. A. Mahmud, T. Tanriverdi, K. A. Muhamad, and H. M. Baskonus, “An Investigation Of The Influence Of Time Evolution on The Solution Structure Using Hyperbolic Trigonometric Function Methods,” International Journal of Applied and Computational Mathematics, vol. 10, no. 4, p. 137, 2024. https://doi.org/10.1007/s40819-024-01769-7.
[19] A. A. Ungar, “Gyrovector Spaces and Their Differential Geometry,” Nonlinear Funct. Anal. Appl., vol. 10, no. 5, pp. 791–834, 2005.
[20] P. Singh, “From Ambiguous Sets to Single-Valued Ambiguous Complex Numbers: Applications In Mandelbrot Set Generation and Vector Directions,” Modern Physics Letters B, vol. 39, no. 14, p. 2450509, 2025. https://doi.org/10.1142/S0217984924505092.
[21] P. Singh, “A Journey into Ambiguous Set Theory: Exploring Ambiguity,” Cambridge Scholars Publishing, 2025.
[22] P. Singh and S. S. Bose, “Ambiguous D-Means Fusion Clustering Algorithm Based On The Ambiguous Set Theory: Special Application in Clustering of CT Scan Images Of COVID-19,” Knowledge-Based Systems, vol. 231, p. 107432, 2021. https://doi.org/10.1016/j.knosys.2021.107432.
[23] P. Singh and Y.-P. Huang, “Membership Functions, Set-Theoretic Operations, Distance Measurement Methods Based on Ambiguous Set Theory: A Solution to a Decision-Making Problem In Selecting The Appropriate Colleges,” Int. J. Fuzzy Syst., vol. 25, pp. 1311–1326, 2023. https://doi.org/10.1007/s40815-023-01468-3.
[24] P. Singh, “Data-Driven Ambiguous Cognitive Map for Complex Decision-Making In Supply Chain Management,” Journal of Computational Mathematics and Data Science, vol. 14, p. 100110, 2025. https://doi.org/10.1016/j.jcmds.2025.100110.
[25] P. Singh, “Ambiguous Sets and Various Distance Measurements,” Soft Computing and Its Engineering Appli- cations, ser. Communications in Computer and Information Science, vol. 2430, 2025.
[26] P. Singh, “A Novel Model to Deal with Ambiguous and Complex Time Series: Application to Sunspots Forecasting,” Knowledge-Based Systems, vol. 329, p. 114257, 2025. https://doi.org/10.1016/j.knosys.2025.114257.
[27] P. Singh, B. Saini, and Y.-P. Huang, “AECA: An Ambiguous-Entropy Clustering Algorithm For The Analysis of Resting-State Fmriss of Human Brain and Their Functional Connections,” Modern Physics Letters B, vol. 39, no. 18, p. 2550023, 2025. https://doi.org/10.1142/S021798492550023X.
[28] P. Singh and T. Liao, “Multi-Criteria Group Decision-Making Using Ambiguous Sets, Weibull Distribution, And Aggregation Operators: A Case Study in Optimal Vendor Selection For Office Supplies,” Systems and Soft Computing, vol. 7, p. 200283, 2025. https://doi.org/10.1016/j.sasc.2025.200283.
[29] P. Singh, Y.-P. Huang, and T.-T. Lee, “A Novel Ambiguous Set Theory To Represent Uncertainty and Its Application to Brain MR Image Segmentation,” Proc. of IEEE Int. Conf. on Systems, Man and Cybernetics (SMC), Bari, Italy, 2019, pp. 2460–2465. https://doi.org/10.1109/SMC.2019.8914080.
[30] P. Singh, “An Investigation of Ambiguous Sets and Their Application to Decision-Making from Partial Order to Lattice Ambiguous Sets,” Decision Analytics Journal, vol. 8, p. 100286, 2023. https://doi.org/10.1016/j.dajour.2023.100286.
[31] P. Singh, “A General Model of Ambiguous Sets to A Single-Valued Ambiguous Numbers with Aggregation Operators,” Decision Analytics Journal, vol. 8, p. 100260, 2023. https://doi.org/10.1016/j.dajour.2023.100260.
[32] P. Singh, “Ambiguous set theory: A New Approach to Deal with Unconsciousness and Ambiguousness Of Human Perception,” Journal of Neutrosophic and Fuzzy Systems, vol. 5, no. 1, pp. 52–58, 2023. https://doi.org/10.54216/JNFS.050106.
[33] P. Singh and Y.-P. Huang, “An Ambiguous Edge Detection Method for Computed Tomography Scans Of Coronavirus Disease 2019 Cases,” IEEE Trans. on Systems, Man, and Cybernetics: Systems, vol. 54, no. 1, pp. 352–364, 2024. https://doi.org/10.1109/TSMC.2023.3307393.
[34] P. Singh and Y.-P. Huang, “AKDC: Ambiguous Kernel Distance Clustering Algorithm For COVID-19 CT Scans Analysis,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 54, no. 10, pp. 6218–6229, 2024. https://doi.org/10.1109/TSMC.2024.3418411.
[35] P. Singh and Y.-P. Huang, “A four-valued ambiguous logic: application in designing ambiguous inference system for control systems,” International Journal of Fuzzy Systems, vol. 25, no. 8, pp. 2921–2938, 2023. https://doi.org/10.1007/s40815-023-01582-2.
[36] A. Fitriani, A. Amirullah, and K. Bhumkittipich, “Power quality enhancement using single phase shunt active filter based ANFIS supplied by photovoltaic,” Vokasi Unesa Bull. Eng. Technol. Appl. Sci., vol. 2, no. 3, pp. 558–580, 2025. https://doi.org/10.26740/vubeta.v2i3.39071.
[37] J. A. Obari, S. T. A, I. M. A, and A. B. H, “An emission and weight of vehicles-based road traffic congestion pricing system and control with consideration of investment worthiness,” Vokasi Unesa Bull. Eng. Technol. Appl. Sci., vol. 2, no. 3, pp. 401–411, 2025. https://doi.org/10.26740/vubeta.v2i3.38528
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