diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9c7bf02..18d17b2 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -5,9 +5,18 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec/v2.0.0.html).
 
+## [1.0.1] - 2024-09-18
+
+### Changed
+
+- Remove function `mu` from definitions
+- Remove redundant sections
+- Typo fixes
+
 ## [1.0.0] - 2024-09-18
 
 ### Changed
+
 - Change bibliography style
 - Update CI/CD
 - Update version inside document
diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf
index 7f41a61..eedee0d 100644
Binary files a/out/AStudyOnDynamicEquations.pdf and b/out/AStudyOnDynamicEquations.pdf differ
diff --git a/src/AStudyOnDynamicEquations.tex b/src/AStudyOnDynamicEquations.tex
index 3c3dbb5..1a2dd30 100644
--- a/src/AStudyOnDynamicEquations.tex
+++ b/src/AStudyOnDynamicEquations.tex
@@ -128,14 +128,6 @@
     \section{Discussion and examples} \label{sec:discussion_and_examples}
     \input{sections/discussion-and-examples}
 
-
-    \section{Proof of main theorem} \label{sec:proof_main_theorem}
-    \input{sections/proof-of-main-theorem}
-
-
-    \section{Conclusion and future research} \label{sec:conclusion}
-    \input{sections/conclusion-and-future-research}
-
     \bibliographystyle{unsrt}
     \bibliography{AStudyOnDynamicEquations}
     \noindent \textbf{Version:} \input{sections/version}
diff --git a/src/sections/conclusion-and-future-research.tex b/src/sections/conclusion-and-future-research.tex
deleted file mode 100644
index ad9aeef..0000000
--- a/src/sections/conclusion-and-future-research.tex
+++ /dev/null
@@ -1,14 +0,0 @@
-In this manuscript we have discussed partial time scale differential equation involving derivatives of polynomials
-in context of time scale $\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2$ where $\mathbb{T}_1 = \mathbb{T}_2$.
-Future research can be conducted to study the case $\mathbb{T}_1 \neq \mathbb{T}_2$,
-which makes the theorem ~\ref{main_theorem} to be generalised
-\[
-    \pTsDerivative{\polynomialP{m}{b}{x}}{x} +
-    \pTsDerivative{\polynomialP{m}{b}{x}}{b}
-    = \alpha_m(x,b) (x^{2m+1})^{\Delta}
-\]
-where $\alpha_m(x,b)$ is arbitrary differentiable function.
-Also, it is worth to discuss the theorem~\ref{main_theorem} in context of high order derivatives on time scales.
-We have established a few power identities, and shown the theorem~\ref{main_theorem} for different 2-dimensional
-time scales $\Lambda^2$ like integer time scale $\mathbb{Z} \times \mathbb{Z}$, real time scale $\mathbb{R} \times \mathbb{R}$,
-quantum time scale $q^{\mathbb{R}} \times q^{\mathbb{R}}$ and quantum power time scale $\mathbb{R}^q \times \mathbb{R}^q$.
diff --git a/src/sections/definitions-notations-and-conventions.tex b/src/sections/definitions-notations-and-conventions.tex
index 65999b8..8233782 100644
--- a/src/sections/definitions-notations-and-conventions.tex
+++ b/src/sections/definitions-notations-and-conventions.tex
@@ -45,16 +45,16 @@
     \end{equation}
     where $\coeffA{m}{r}$ is a real coefficient defined recursively, see~\cite{kolosov2016link}.
 
-    \item $\mathbb{Z}$ is an integer timescale such that $\sigma(t) = t+1$ and $\mu(t) = 1$.
+    \item $\mathbb{Z}$ is an integer timescale such that $\sigma(t) = t+1$.
 
-    \item $\mathbb{R}$ is a real timescale such that $\sigma(t) = t+\Delta t$ and $\mu(t) = \Delta t, \; \Delta t \to 0$.
+    \item $\mathbb{R}$ is a real timescale such that $\sigma(t) = t+\Delta t$ where $\Delta t \to 0$.
 
-    \item $q^\mathbb{R}$ is a quantum timescale such that $\sigma(t) = qt$ and $\mu(t) = qt - t$,
+    \item $q^\mathbb{R}$ is a quantum timescale such that $\sigma(t) = qt$,
     see~\cite[p. 18]{Bohner2001DynamicEO}.
 
-    \item $\mathbb{R}^q$ is a quantum power timescale such that $\sigma(t) = t^q$ and $\mu(t) = t^q - t$.
+    \item $\mathbb{R}^q$ is a quantum power timescale such that $\sigma(t) = t^q$.
 
     \item $q^{\mathbb{R}^n}$ is a pure quantum power timescale
-    such that $\sigma(t) = qt^n > t, \; 0<q<1, \; \mu(t) = qt^n - t$ and $n$ is positive
+    such that $\sigma(t) = qt^n > t, \; 0<q<1$ where $n$ is a positive
     odd integer~\cite{aldwoah2011power}.
 \end{itemize}
diff --git a/src/sections/main-results.tex b/src/sections/main-results.tex
index 7ace002..fed2c08 100644
--- a/src/sections/main-results.tex
+++ b/src/sections/main-results.tex
@@ -1,16 +1,16 @@
-Timescale derivative of odd-powered polynomial $t^{2m+1}$ may be expressed as follows
+Timescale derivative of odd-powered polynomial $x^{2m+1}$ may be expressed as follows
 \begin{thm}
     \label{main_theorem}
     Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$.
     Let be a two-dimensional timescale
     $\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$
     such that $\mathbb{T}_1 = \mathbb{T}_2$.
-    For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$
-    \[
+    For every $t\in\mathbb{T}_1$ and $(x,b) \in \Lambda^2$
+    \begin{align*}
         \frac{\Delta x^{2m+1}}{\Delta x}(t) =
         \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) +
         \frac{\partial P(m,b,x)}{\Delta b} (m, t, t)
-    \]
+    \end{align*}
     where
     \begin{itemize}
         \setlength\itemsep{1em}
@@ -42,7 +42,7 @@
     \end{quotation}
 \end{center}
 
-In its extended form the theorem ~\ref{main_theorem} is as follows
+In its extended form the theorem ~\eqref{main_theorem} is as follows
 
 \begin{align*}
     \frac{\Delta x^{2m+1}}{\Delta x}(t)
diff --git a/src/sections/proof-of-main-theorem.tex b/src/sections/proof-of-main-theorem.tex
deleted file mode 100644
index b4c259b..0000000
--- a/src/sections/proof-of-main-theorem.tex
+++ /dev/null
@@ -1,72 +0,0 @@
-%! suppress = SuspiciousSectionFormatting
-By~\cite[Lemma 3.1]{kolosov2016link}, for every $x\in\mathbb{R}, \; m\in\mathbb{N}$ it is true that
-\begin{equation}
-    \label{eq:odd_power_identity}
-    \polynomialP{m}{x}{x} = x^{2m+1}
-\end{equation}
-
-\subsection{Proof of theorem~\ref{main_theorem}}
-\label{subsec:proof-of-theorem}
-Let be
-$x,b\in \Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2
-\colonequals \{t = (x,b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2\}$.
-Let be $\mathbb{T}_1 = \mathbb{T}_2$.
-Assume that timescale derivative $(x^{2m+1})^{\Delta}$ is
-\begin{equation}
-    \label{eq:proof1}
-    (x^{2m+1})^{\Delta}
-    = \lim \limits_{b \to x}
-    \lim \limits_{t \to x}
-    \frac{\polynomialP{m}{\sigma(b)}{\sigma(x)} - \polynomialP{m}{b}{t}}{\sigma(x) - t}
-\end{equation}
-where $\sigma(x) > x$ is forward jump operator.
-However, equation ~\eqref{eq:proof1} is not a timescale derivative of $\polynomialP{m}{b}{x}$ over $x$
-how it might seem because of denominator $\sigma(x) - t$.
-Parameter $b$ of $\polynomialP{m}{b}{x}$ is implicitly incremented as well.
-Let's try to express nominator of ~\eqref{eq:proof1} in terms of
-partial derivative $\pTsDerivative{\polynomialP{m}{b}{x}}{b}$ on timescales.
-Let be the following equation
-\[
-    \polynomialP{m}{\sigma(b)}{x} - \polynomialP{m}{b}{x}
-    = \polynomialP{m}{b}{x}^{\Delta}_{b} \cdot \Delta b
-\]
-Let $t \to x$ in ~\eqref{eq:proof1}.
-Then nominator of ~\eqref{eq:proof1} equals to
-\[
-    \polynomialP{m}{\sigma(b)}{\sigma(x)} - \polynomialP{m}{b}{x}
-    = \polynomialP{m}{\sigma(b)}{x} - \polynomialP{m}{b}{x} + A
-\]
-where $A$ is yet implicit term.
-Let's now collapse the terms $f_m (x, b)$ from both sides of above equation, such that
-\[
-    \polynomialP{m}{\sigma(b)}{\sigma(x)} = \polynomialP{m}{\sigma(b)}{\sigma(x)} + A
-\]
-Therefore,
-\[
-    A = \polynomialP{m}{\sigma(b)}{\sigma(x)} - \polynomialP{m}{\sigma(b)}{\sigma(x)}
-    = \polynomialP{m}{b}{x}^{\Delta}_{x} (x, \sigma(b)) \cdot \Delta x
-\]
-Now, let's express the nominator of ~\eqref{eq:proof1} as follows
-\begin{align*}
-    \polynomialP{m}{\sigma(b)}{\sigma(x)} - \polynomialP{m}{b}{x}
-    &= \polynomialP{m}{b}{x}^{\Delta}_{x} (x, \sigma(b)) \cdot \Delta x + \polynomialP{m}{b}{x}^{\Delta}_{b} (x,b) \cdot \Delta b \\
-    \polynomialP{m}{\sigma(b)}{\sigma(x)} - \polynomialP{m}{b}{x}
-    &= \polynomialP{m}{b}{x}^{\Delta}_{x} (x, \sigma(b)) \cdot (\sigma(x) - x) + \polynomialP{m}{b}{x}^{\Delta}_{b} (x,b) \cdot (\sigma(b) - b)
-\end{align*}
-We can collapse the terms $(\sigma(x) - x), \; (\sigma(b) - b)$ in above expressions, as $b\to x$.
-Therefore,
-\begin{align*}
-    \frac{\polynomialP{m}{\sigma(x)}{\sigma(x)} - \polynomialP{m}{x}{x}}{\sigma(x) - x}
-    = \polynomialP{m}{b}{x}^{\Delta}_{x} (m, \sigma(x), x)
-    + \polynomialP{m}{b}{x}^{\Delta}_{b} (m, x, x)
-\end{align*}
-Finally, by the identity ~\eqref{eq:odd_power_identity} we can express
-timescale derivative of $x^{2m+1}, \; x\in \Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2, \; m\in\mathbb{N}$
-as
-\begin{equation*}
-(x^{2m+1})
-    ^{\Delta}(t)= \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(x), x)
-    + \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, x, x)
-\end{equation*}
-
-This completes the proof. \qed
diff --git a/src/sections/time-scale-r.tex b/src/sections/time-scale-r.tex
index 6e58313..70b1859 100644
--- a/src/sections/time-scale-r.tex
+++ b/src/sections/time-scale-r.tex
@@ -9,7 +9,7 @@
         = \pdv{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t)
         + \pdv{\polynomialP{m}{b}{x}}{b} (m, t, t)
     \end{align*}
-    where $\sigma(t) = t + \Delta t, \; \Delta t \to 0.$
+    where $\sigma(t) = t + \Delta t$ such that $ \Delta t \to 0.$
 \end{cor}
 \begin{examp}
     \label{time_scale_r_example_1}