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- W2273331087 abstract "A one parameter family of a linearly ordered, complete nonarchimedean field extensions ${bf R^*}_{delta}, delta>0$ of the real number system $bf R$ is constructed using a novel asymptotic concept of duality structure. The parameter $delta$ is called an asymptotic scale. The extension is facilitated by an ultrametric absolute value, called the asymptotic visibility norm $v(cdot)$, that awards a vanishing (or divergent) quantity $xrightarrow 0 ({rm or} infty)$ a nontrivial effective renormalized value $v(x)$..... The framework of ordinary analysis enriched with this nonlinear asymptotic structure is shown to accommodate, apart from linear shifts, new nonlinear increments in the form of {em smooth jumps} in the asymptotic neighbourhood ${cal N}_{delta}(0)$.. ......Linear differential equations are shown to proliferate asymptotically on ${bf R^*}_{delta}$ as jump differential equations leading to nontrivial scaling laws typical of emergent, complex systems. As a particular example the power law attenuation of dispersive, lossy wave equation is derived invoking a principle of coherent correlated spatio-temporal scaling. As a second application of the formalism, the renormalized variable, equipped with the calculus of jump derivative $D_J$, is shown to award a smooth differentiable structure on the triadic Cantor set. The associated Cantor's staircase function $f_C(x)$ is shown to satisfy the jump differential equation $D_J f_C(x)=chi_C(x)$ where $chi_C(x)$ is the characteristic function with support $C$, the triadic Cantor set." @default.
- W2273331087 created "2016-06-24" @default.
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- W2273331087 date "2016-01-12" @default.
- W2273331087 modified "2023-09-27" @default.
- W2273331087 title "Duality Structure, Non-archimedean Extension of Real Number System and Emergent Fractal sets" @default.
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