Hierarchical Calculus Official website + DOI (Zenodo)
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Hierarchical Calculus generalizes classical derivatives by introducing a rank parameter. Rank 0 corresponds to the usual derivative, rank 1 to the relative derivative, rank 2 to the logarithmic derivative, and so on. This site provides an academic reference with DOI-archived releases on Zenodo.

Core Theory Definitions, axioms, and notation.
Open theory
One Simple Example D₀ vs D₁ in a clear case.
Open example
Equations Hierarchical equations & transformations.
Open equations
FAQ Common questions and clarifications.
Open FAQ

Hierarchical Calculus

Ahmed Gossa
Ahmed Gossa
Independent Researcher — Hierarchical Calculus

Abstract: Hierarchical Calculus is a mathematical framework introducing multi-level derivatives and equations by extending classical calculus using a rank-based operator \(D^{k}_{n}\), where \(n\) denotes rank and \(k\) denotes degree. The framework includes differential (rank 0), relative (rank 1), logarithmic (rank 2), tetrational (rank 3), and the general rank-\(n\) derivative.

Concept DOI (main reference): 10.5281/zenodo.17917302
Version DOI (v1.0.0): 10.5281/zenodo.17917303 — website release v1.0.0
GitHub: GOSSAAHMED/gossa-math · Contact: ahmedgossa50@gmail.com
Zenodo (CERN) Archive MathJax Typesetting Arabic + English Release v1.0.0

Core Definitions

Official notation: degree above, rank below. We write the hierarchical derivative as \(D^{k}_{n}\) where \(n\) is rank and \(k\) is degree. The most common case is degree \(k=1\).

Definition General

The rank-\(n\), degree-1 hierarchical derivative (domain conditions apply):

\[ D^{1}_{n} f(x)=\frac{d\,\ln^{(n)}(f(x))}{d\,\ln^{(n)}(x)} \]

where \(\ln^{(n)}\) denotes the iterated logarithm (e.g., \(\ln^{(3)}(x)=\ln(\ln(\ln x))\)).

Special Ranks D₀–D₃
\[ D^{1}_{0} f(x)=\frac{df(x)}{dx} \]
\[ D^{1}_{1} f(x)=\frac{d\ln f(x)}{d\ln x} \]
\[ D^{1}_{2} f(x)=\frac{d\ln(\ln f(x))}{d\ln(\ln x)} \]
\[ D^{1}_{3} f(x)=\frac{d\ln^{(3)} f(x)}{d\ln^{(3)} x} \]
Key Points Summary

• Rank controls the logarithmic “level” of the derivative.
• Degree controls iteration or higher-order behavior.
• Hierarchical equations generalize differential equations by replacing \(d/dx\) with \(D_n\).
• Official releases are archived on Zenodo with DOI.

Hierarchical Equations

Hierarchical equations generalize differential equations by using derivatives of rank \(n\). General form:

\[ D^{1}_{n}y = F(x,y) \]
Example Sketch

A guiding example of a blow-up type equation at the differential level:

\[ \frac{dy}{dx}=y\ln y \]

With an appropriate hierarchical lifting, one may obtain a simpler expression in a higher rank (details in the equations pages).

Citation

Copy/paste-ready references.

Text Citation Text
Gossa, Ahmed. Hierarchical Calculus — Official Website. Zenodo. https://doi.org/10.5281/zenodo.17917302 (2025).
BibTeX LaTeX
@software{gossa_hierarchical_calculus_2025, author = {Gossa, Ahmed}, title = {Hierarchical Calculus}, year = {2025}, doi = {10.5281/zenodo.17917302}, url = {https://gossaahmed.github.io/gossa-math/} }