In this paper we explored the algebraic and topological underpinnings of non-Newtonian analysis through the framework of a generator function $ \alpha $. We introduced the $ \alpha $-number systems and the associated arithmetic operations, and we established the corresponding algebraic structures, including groups, fields, and vector spaces defined over $ \alpha $-real numbers. On the topological side, we developed the notions of $ \alpha $-metric spaces and $ \alpha $-sequences, thereby extending core concepts of analysis to the non-Newtonian setting. The study culminates with the formulation of star ($ \star $) analysis, which provides a systematic mechanism for transitioning between distinct arithmetic systems, together with a rigorous treatment of $ \star $-vector spaces and linear operators.
Citation: Emre Civgin, Numan Yalcin. Algebraic and topological foundations of non-Newtonian analysis via generator functions[J]. AIMS Mathematics, 2025, 10(11): 26633-26661. doi: 10.3934/math.20251171
In this paper we explored the algebraic and topological underpinnings of non-Newtonian analysis through the framework of a generator function $ \alpha $. We introduced the $ \alpha $-number systems and the associated arithmetic operations, and we established the corresponding algebraic structures, including groups, fields, and vector spaces defined over $ \alpha $-real numbers. On the topological side, we developed the notions of $ \alpha $-metric spaces and $ \alpha $-sequences, thereby extending core concepts of analysis to the non-Newtonian setting. The study culminates with the formulation of star ($ \star $) analysis, which provides a systematic mechanism for transitioning between distinct arithmetic systems, together with a rigorous treatment of $ \star $-vector spaces and linear operators.
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Emre Civgin, Numan Yalcin. Algebraic and topological foundations of non-Newtonian analysis via generator functions[J]. AIMS Mathematics, 2025, 10(11): 26633-26661. doi: 10.3934/math.20251171
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