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z!*^ZshX>oXO(6c{K@yrTDufhJ#H2Y2CMA>)Hqef)2Ei0eh8_82K;%#u1=>$6p|H;u zG)I%@Y6CP1-LHWLPj(axG=?UPhXWk^82EtLeiixl85|;tRtX1nN6U{EO6+El4)F1e zupe#D@ph0(zlviRVGI;1We3pB+1ZaY9`XOOco~lzsSzCFvp6JdvHu(mSmFqg)G#$2 I Date: Wed, 18 Sep 2024 17:18:46 +0200 Subject: [PATCH 3/5] remove redundant sections --- src/AStudyOnDynamicEquations.tex | 8 --- .../conclusion-and-future-research.tex | 14 ---- src/sections/proof-of-main-theorem.tex | 72 ------------------- 3 files changed, 94 deletions(-) delete mode 100644 src/sections/conclusion-and-future-research.tex delete mode 100644 src/sections/proof-of-main-theorem.tex 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/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 From f63f29f1e7c7dae149a8ac5da1b62737825279fa Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Wed, 18 Sep 2024 17:19:40 +0200 Subject: [PATCH 4/5] update changelog --- CHANGELOG.md | 9 +++++++++ 1 file changed, 9 insertions(+) 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 From 5d620021fb5684613bbea4853388af449c3d18b5 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Wed, 18 Sep 2024 17:20:14 +0200 Subject: [PATCH 5/5] recompile pdf --- out/AStudyOnDynamicEquations.pdf | Bin 377175 -> 356265 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index d0183701fac671046be74586b20ae6e8d95f0db6..eedee0d33ee2959cde49b6756ab294a47da86cb0 100644 GIT binary patch delta 106481 zcmZ^rby!qu_xA}2k?v;bju{3RTBK828tIU35ExpzhDI7B1u5wUDM4xJP&x$Zen&m$ 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