Hierarchical Equations

A hierarchical equation is an equation written using rank–degree derivatives \(D_r^n\). The goal is structural simplification: a nonlinear equation in one representation may become separable (or closer to linear) after choosing the right rank and applying a suitable hierarchical lift.

Notation: D_r^n (rank + degree)

Strict definition uses \(\ln^{(r)}\) on both sides

Main method: lifting via \(u=\ln^{(k)}(f)\)

Ahmed Gossa

Independent Researcher — Hierarchical Calculus

Concept DOI Derivative Relations Rank & Degree

Single-source-of-truth rule: Throughout this website, the rank-\(r\) derivative is always defined by applying \(\ln^{(r)}\) to both \(x\) and \(f(x)\). This prevents contradictions between pages.

Quick Table of Contents

1) Definition of hierarchical equations 2) Notation and strict rank definition 3) Domain assumptions (avoid hidden singularities) 4) Classification by rank (0,1,2,3,...) 5) Canonical examples across ranks 6) Hierarchical lifting (the main method) 7) Practical workflow (mini-solver) 8) Consistency checklist (avoid contradictions) 9) Related pages 10) Citation

1) Definition

A rank–degree hierarchical equation is any relation involving a function \(f\) and a finite set of hierarchical derivatives \(D_{r}^{n}f\), possibly with different ranks and degrees:

\[ \mathcal{E}\Big( x,\ f,\ \{D_{r_i}^{\,n_i}f\}_{i=1}^{m} \Big)=0. \]

A useful “max rank” view: a level-\(k\) equation uses ranks up to \(k\):

\[ \mathcal{E}_k\big(x,\ f,\ D_{0}^{\bullet}f,\ D_{1}^{\bullet}f,\ \dots,\ D_{k}^{\bullet}f\big)=0, \]

where \(D_{r}^{\bullet}f\) means “some degrees of rank \(r\)”.

Why rank matters

Choosing a higher rank means measuring variation on a different scale (log-scale, iterated log-scale, …). Many equations become simpler when their natural scaling is matched by the chosen rank.

4) Quick classification (by rank)

Rank 0 — Differential (Classical)

\[ F\big(x,f,D_0^{\bullet} f\big)=0 \]

Measures additive change in \(x\). Solutions often involve integrals.

Rank 1 — Relative

\[ F\big(x,f,D_1^{\bullet} f\big)=0 \]

Measures scaling/ratio change in \(\ln x\). Solutions often become exponential forms.

Rank 2 — Logarithmic

\[ F\big(x,f,D_2^{\bullet} f\big)=0 \]

Measures variation on \(\ln(\ln x)\). Solutions often involve nested exponentials.

Rank 3+ — Higher hierarchy

\[ F\big(x,f,D_3^{\bullet} f,\dots\big)=0 \]

Nested scales beyond \(\ln(\ln(\ln(\cdot)))\).

2) Notation (D0, D1, and the general form \(D_r^n\))

This website uses the shorthand: D0 for the classical derivative and D1 for the relative derivative. In rank–degree form:

\[ D_0 \equiv D_{0}^{1}, \qquad D_1 \equiv D_{1}^{1}. \]

The strict adopted definition of the rank-\(r\), degree-1 derivative is:

\[ \boxed{ D_{r}^{1} f(x)=\frac{d\,\ln^{(r)}\!\big(f(x)\big)}{d\,\ln^{(r)}\!(x)} } \]

Consistency: We do not use mixed forms such as \(\frac{d(\ln f)}{d\ln^{(r)}x}\). The adopted definition applies \(\ln^{(r)}\) to both sides.

Special cases:

\[ D_{0}^{1} f=\frac{df}{dx}, \qquad D_{1}^{1} f=\frac{d\ln f}{d\ln x}=\frac{x}{f}\frac{df}{dx}, \qquad D_{2}^{1} f=\frac{d\ln(\ln f)}{d\ln(\ln x)}. \]

Key idea

The same equation can be rewritten in different ranks. The “right” rank often exposes a simpler separable or linear structure. This is the heart of hierarchical lifting.

3) Domain assumptions (avoid hidden singularities)

Higher ranks explicitly involve iterated logarithms, so every equation must specify its domain. A safe checklist:

Rank 0: only the classical domain of \(f\) is required.
Rank 1: typically require \(x>0\) and \(f(x)>0\) (because \(\ln x\) and \(\ln f\) appear).
Rank 2: typically require \(x>1\) and \(f(x)>1\) (so \(\ln(\ln x)\) and \(\ln(\ln f)\) exist).
Avoid points where the denominator scale vanishes (e.g., \(\ln x=0\) when differentiating w.r.t. \(\ln(\ln x)\)).

Reviewer-style note: For rank 2 and above, state the stronger conditions \(x>1\) and \(f(x)>1\) explicitly (or their equivalent inequalities).

5) Canonical examples across ranks

These examples show how the same “shape” of equation leads to different natural solution forms depending on rank. They are written using the strict definition from the Notation section.

Example A — Rank 0 (classical): \(D_{0}^{1} f=g(x)\)

\[ D_{0}^{1} f(x)=g(x) \quad\Longleftrightarrow\quad \frac{df}{dx}=g(x) \quad\Longrightarrow\quad f(x)=\int g(x)\,dx + C. \]

Example B — Rank 1 (relative): \(D_{1}^{1} f=g(x)\)

\[ D_{1}^{1} f(x)=g(x) \quad\Longleftrightarrow\quad \frac{d\ln f}{d\ln x}=g(x). \]

\[ \frac{d\ln f}{dx}=\frac{g(x)}{x} \quad\Longrightarrow\quad \ln f(x)=\int \frac{g(x)}{x}\,dx + C \quad\Longrightarrow\quad f(x)=C_0\exp\!\left(\int \frac{g(x)}{x}\,dx\right). \]

Interpretation

Rank 1 naturally produces multiplicative/exponential solutions because the unknown becomes \(\ln f\). This is why rank 1 aligns with scaling laws and power-like behaviors.

Example C — Rank 1 logistic-type: \(D_{1}^{1} f = a - b f\)

This is structurally simple in rank 1 because it becomes separable in \(f\) versus \(\ln x\):

\[ \frac{d\ln f}{d\ln x} = a-bf \quad\Longleftrightarrow\quad \frac{1}{f}\frac{df}{d\ln x}=a-bf \quad\Longrightarrow\quad \frac{df}{f(a-bf)}=d\ln x. \]

Practical tip: You may keep this as an implicit solution here and link to a full explicit solution on a “Solved Equations” page.

Example D — Rank 2: \(D_{2}^{1} f=g(x)\)

\[ D_{2}^{1} f(x)=g(x) \quad\Longleftrightarrow\quad \frac{d\ln(\ln f)}{d\ln(\ln x)}=g(x). \]

\[ \ln(\ln f)=\int g(x)\,d\ln(\ln x) + C \quad\Longrightarrow\quad \ln f = C_1 \exp\!\left(\int g(x)\,d\ln(\ln x)\right) \quad\Longrightarrow\quad f=\exp\!\Big( C_1 \exp(\cdots)\Big). \]

Interpretation

Rank 2 naturally generates nested exponential behavior because the transformed unknown is \(\ln(\ln f)\). This is ideal when the phenomenon lives on nested scales.

6) Hierarchical lifting (the main method)

Lifting means rewriting the same problem at a rank where its structure becomes simpler. A standard lift is:

\[ u=\ln^{(k)}(f) \quad\Rightarrow\quad D_{k}^{1} f = \frac{du}{d\ln^{(k)}(x)}. \]

\[ \mathcal{E}_k\big(x,f,D_{0}^{\bullet} f,\dots,D_{k}^{\bullet} f\big)=0 \quad \xrightarrow{\;\text{Lift}\;} \widetilde{\mathcal{E}}_{k}\big(\ln^{(k)}x,\ u,\ \tfrac{du}{d\ln^{(k)}x}\big)=0. \]

Main message

The goal is not to replace classical calculus, but to choose a rank that matches the natural scaling so the solver sees a simpler pattern (separable, linear, constant-coefficient, etc.).

7) Practical workflow (mini-solver)

Step 1: Identify the growth type (power? exponential? nested? multi-scale?).
Step 2: Choose a rank \(k\) matching the scale (0, 1, 2, ...).
Step 3: Transform the unknown: \(u=\ln^{(k)}(f)\).
Step 4: Rewrite the equation in \(u\) and \(\ln^{(k)}(x)\).
Step 5: Solve in \(u\) using classical methods (separable/linear/etc.).
Step 6: Invert the transform (iterated exponentials) to recover \(f\).
Step 7: State the domain where all logs are valid.

Tip: If you create a future page like equations-solved-en.html, link each example above to its full solution.

8) Consistency checklist (avoid contradictions)

Always write derivatives in explicit form \(D_r^n\) when explaining rank/degree concepts.
Use the strict definition: \(\ln^{(r)}\) applied to both \(x\) and \(f(x)\).
State domain conditions whenever \(\ln\) or iterated \(\ln\) appears.
Keep rank fixed when raising degree \(n\) (do not mix “rank change” with “degree change”).
Link to the Rank & Degree page for the global definition used on the site.

10) Citation

Official reference DOI: 10.5281/zenodo.17917302

@software{gossa_hierarchical_calculus_2025,
  author = {Gossa Ahmed},
  title  = {Hierarchical Calculus},
  year   = {2025},
  doi    = {10.5281/zenodo.17917302},
  url    = {https://github.com/GOSSAAHMED/gossa-math}
}