Merge pull request #8 from kolosovpetro/MATH-46

MATH-46

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

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}

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}

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)

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}