From 95e99bafdb27e520c243ceaba364344bfab24c9e Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 15:31:06 +0200 Subject: [PATCH 01/22] initial updates --- .editorconfig | 12 ++ .gitattributes | 11 ++ .github/workflows/build-and-deploy-pdf.yml | 93 +++++++++--- .github/workflows/build-pdf.yml | 73 +++++++-- BIBTEX-CITATION.bib | 6 + CHANGELOG.md | 10 ++ GitVersion.yml | 1 + README.md | 73 +++------ img/abstract.PNG | Bin 79898 -> 0 bytes out/AStudyOnDynamicEquations.bbl | 138 +++++++++--------- out/AStudyOnDynamicEquations.pdf | Bin 408255 -> 407096 bytes ..._refs.bib => AStudyOnDynamicEquations.bib} | 0 src/AStudyOnDynamicEquations.tex | 21 +-- src/sections/version.tex | 1 + 14 files changed, 272 insertions(+), 167 deletions(-) create mode 100644 .editorconfig create mode 100644 .gitattributes create mode 100644 BIBTEX-CITATION.bib create mode 100644 CHANGELOG.md create mode 100644 GitVersion.yml delete mode 100644 img/abstract.PNG rename src/{time_scale_calculus_refs.bib => AStudyOnDynamicEquations.bib} (100%) create mode 100644 src/sections/version.tex diff --git a/.editorconfig b/.editorconfig new file mode 100644 index 0000000..8389293 --- /dev/null +++ b/.editorconfig @@ -0,0 +1,12 @@ +root = true + +[*] +charset = utf-8 +indent_style = space +indent_size = 4 +insert_final_newline = true +trim_trailing_whitespace = true +end_of_line = lf + +[*.{xml,yml}] +indent_size = 2 diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6656588 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,11 @@ +* text=auto eol=lf +*.cs linguist-detectable=true +*.js linguist-detectable=false +*.cshtml linguist-detectable=false +*.razor linguist-detectable=false +*.html linguist-detectable=false +*.htm linguist-detectable=false +*.xml linguist-detectable=false +*.css linguist-detectable=false +*.m linguist-detectable=false +*.nb linguist-detectable=false \ No newline at end of file diff --git a/.github/workflows/build-and-deploy-pdf.yml b/.github/workflows/build-and-deploy-pdf.yml index 9bfb8d6..f0c28bc 100644 --- a/.github/workflows/build-and-deploy-pdf.yml +++ b/.github/workflows/build-and-deploy-pdf.yml @@ -2,39 +2,90 @@ name: Build and Deploy PDF on: push: - branches: - - master - - main + tags: + - v* + workflow_dispatch: + +env: + FILE_NAME: "AStudyOnDynamicEquations" + ACTIONS_RUNNER_DEBUG: false jobs: - build-resume: + build-pdf: runs-on: ubuntu-latest - name: Create document + name: Build and Deploy PDF + steps: - name: Checkout - uses: actions/checkout@v1 - - name: Convert to pdf - id: convert-to-pdf - uses: xu-cheng/latex-action@v2 + uses: actions/checkout@v4 with: - root_file: AStudyOnDynamicEquations.tex + fetch-depth: 0 + + - name: Install GitVersion + uses: gittools/actions/gitversion/setup@v1.1.1 + with: + versionSpec: '5.x' + + - name: Determine Version + uses: gittools/actions/gitversion/execute@v1.1.1 + + - name: Print SemVer + run: | + echo "SemVer: ${{ env.GitVersion_SemVer }}" + echo "BranchName: ${{ env.GitVersion_BranchName }}" + echo "ShortSha: ${{ env.GitVersion_ShortSha }}" + newVersion="${{ env.GitVersion_SemVer }}+${{ env.GitVersion_BranchName }}.${{ env.GitVersion_ShortSha }}" + echo "Next version: $newVersion" + + - name: Update version.tex + shell: bash + run: | + newVersion="${{ env.GitVersion_SemVer }}+${{ env.GitVersion_BranchName }}.${{ env.GitVersion_ShortSha }}" + sed -i "s|Local-0.1.0|$newVersion|" "src/sections/version.tex" + + - name: Build PDF + uses: xu-cheng/latex-action@v3 + with: + root_file: "${{ env.FILE_NAME }}.tex" working_directory: src - - uses: actions/upload-artifact@v2 + + - name: List src + run: | + ls -lsa src + + - name: Upload artifacts + uses: actions/upload-artifact@v4 with: - name: PDF - path: AStudyOnDynamicEquations.pdf - - uses: actions/checkout@v2 + name: "${{ env.FILE_NAME }}-${{ env.GitVersion_InformationalVersion }}" + path: | + src/${{ env.FILE_NAME }}.tex + src/${{ env.FILE_NAME }}.bbl + src/${{ env.FILE_NAME }}.bib + src/sections + + - name: Upload artifacts PDF + uses: actions/upload-artifact@v4 with: - token: ${{ secrets.GH_ACCESS_TOKEN }} + name: "${{ env.FILE_NAME }}-PDF-${{ env.GitVersion_InformationalVersion }}" + path: | + src/${{ env.FILE_NAME }}.pdf + - name: Clone repository and add document run: | - git clone https://${{ secrets.GH_ACCESS_TOKEN }}@github.com/kolosovpetro/kolosovpetro.github.io.git ~/kolosovpetro.github.io - cp out/AStudyOnDynamicEquations.pdf ~/kolosovpetro.github.io/pdf/AStudyOnDynamicEquations.pdf + git clone https://$TOKEN@github.com/kolosovpetro/kolosovpetro.github.io.git ~/kolosovpetro.github.io + cp src/${{ env.FILE_NAME }}.pdf ~/kolosovpetro.github.io/pdf/${{ env.FILE_NAME }}.pdf + env: + TOKEN: ${{ secrets.GH_ACCESS_TOKEN }} + - name: Commit document changes if any run: | cd ~/kolosovpetro.github.io - git config --global user.name '${{ secrets.GH_NAME }}' - git config --global user.email '${{ secrets.GH_EMAIL }}' + git config --global user.name "$USERNAME" + git config --global user.email "$EMAIL" git add . - git commit -am "CICD deploy from main repo" - git push \ No newline at end of file + git commit -m "$MESSAGE" + git push + env: + USERNAME: "kolosovpetro" + EMAIL: "kolosovp94@gmail.com" + MESSAGE: "CICD deploy of ${{ env.FILE_NAME }} PDF document" diff --git a/.github/workflows/build-pdf.yml b/.github/workflows/build-pdf.yml index 94d75b9..4d63e44 100644 --- a/.github/workflows/build-pdf.yml +++ b/.github/workflows/build-pdf.yml @@ -4,24 +4,75 @@ on: push: branches: - develop + - master + - main + - release/* pull_request: branches: - - develop + - main + - master + workflow_dispatch: + +env: + FILE_NAME: "AStudyOnDynamicEquations" + ACTIONS_RUNNER_DEBUG: false jobs: - build-resume: + build-pdf: runs-on: ubuntu-latest - name: Create document + name: Build PDF + steps: - name: Checkout - uses: actions/checkout@v1 - - name: Convert to pdf - id: convert-to-pdf - uses: xu-cheng/latex-action@v2 + uses: actions/checkout@v4 + with: + fetch-depth: 0 + + - name: Install GitVersion + uses: gittools/actions/gitversion/setup@v1.1.1 + with: + versionSpec: '5.x' + + - name: Determine Version + uses: gittools/actions/gitversion/execute@v1.1.1 + + - name: Print SemVer + run: | + echo "SemVer: ${{ env.GitVersion_SemVer }}" + echo "BranchName: ${{ env.GitVersion_BranchName }}" + echo "ShortSha: ${{ env.GitVersion_ShortSha }}" + newVersion="${{ env.GitVersion_SemVer }}+${{ env.GitVersion_BranchName }}.${{ env.GitVersion_ShortSha }}" + echo "Next version: $newVersion" + + - name: Update version.tex + shell: bash + run: | + newVersion="${{ env.GitVersion_SemVer }}+${{ env.GitVersion_BranchName }}.${{ env.GitVersion_ShortSha }}" + sed -i "s|Local-0.1.0|$newVersion|" "src/sections/version.tex" + + - name: Build PDF + uses: xu-cheng/latex-action@v3 with: - root_file: AStudyOnDynamicEquations.tex + root_file: "${{ env.FILE_NAME }}.tex" working_directory: src - - uses: actions/upload-artifact@v2 + + - name: List src + run: | + ls -lsa src + + - name: Upload artifacts + uses: actions/upload-artifact@v4 + with: + name: "${{ env.FILE_NAME }}-${{ env.GitVersion_InformationalVersion }}" + path: | + src/${{ env.FILE_NAME }}.tex + src/${{ env.FILE_NAME }}.bbl + src/${{ env.FILE_NAME }}.bib + src/sections + + - name: Upload artifacts PDF + uses: actions/upload-artifact@v4 with: - name: PDF - path: AStudyOnDynamicEquations.pdf \ No newline at end of file + name: "${{ env.FILE_NAME }}-PDF-${{ env.GitVersion_InformationalVersion }}" + path: | + src/${{ env.FILE_NAME }}.pdf diff --git a/BIBTEX-CITATION.bib b/BIBTEX-CITATION.bib new file mode 100644 index 0000000..350007f --- /dev/null +++ b/BIBTEX-CITATION.bib @@ -0,0 +1,6 @@ +@MISC{GithubSource_2022, + author = {Petro Kolosov}, + title = {{"Github Template" Source files}}, + howpublished = {Available electronically at \url {https://github.com/kolosovpetro/github-latex-template}}, + year = {2022} +} \ No newline at end of file diff --git a/CHANGELOG.md b/CHANGELOG.md new file mode 100644 index 0000000..356cc04 --- /dev/null +++ b/CHANGELOG.md @@ -0,0 +1,10 @@ +# Changelog + +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). + +## [1.0.0] - In progress + +### Changed diff --git a/GitVersion.yml b/GitVersion.yml new file mode 100644 index 0000000..c311488 --- /dev/null +++ b/GitVersion.yml @@ -0,0 +1 @@ +mode: Mainline \ No newline at end of file diff --git a/README.md b/README.md index 7c26987..690a61d 100644 --- a/README.md +++ b/README.md @@ -1,58 +1,19 @@ # A study on partial dynamic equation on time scales involving derivatives of polynomials -[![Build PDF](https://github.com/kolosovpetro/AStudyOnDynamicEquations/actions/workflows/build-pdf.yml/badge.svg)](https://github.com/kolosovpetro/AStudyOnDynamicEquations/actions/workflows/build.yml/badge.svg) -[![Build and Deploy PDF](https://github.com/kolosovpetro/AStudyOnDynamicEquations/actions/workflows/build-and-deploy-pdf.yml/badge.svg)](https://github.com/kolosovpetro/AStudyOnDynamicEquations/actions/workflows/build-and-deploy.yml/badge.svg) -![contributors count](https://img.shields.io/github/contributors/kolosovpetro/AStudyOnDynamicEquations) - -

- abstract -

- -## What is all about - -Source code of the manuscript entitled -"A study on partial dynamic equation on time scales involving derivatives of polynomials" -along with Mathematica programs in order to verify results. - -## How to use Mathematica package - -- Open the package file `AStudyOnDinamicEquationsPackage.m` in Wolfram Mathematica, I use version 13.0 -- Execute the package using `Shift+Enter` -- Open the notebook file `AStudyOnDynamicEquationsNotebook.nb` -- Execute the line: `Needs["AStudyOnDynamicEquations"]` -- Execute the line: `mainTheorem[m_] := Expand[timeScaleDerivativeX[m, t, sigma[t]] + timeScaleDerivativeB[m, t, t]]` -- Continue executing according to the guideline below - -## Reproduce the results using Mathematica - -Few examples of the outcomes of the manuscript and how to reproduce them - -### Example for Z x Z time scale. - -

- timescale-z -

- -To reproduce example 4.2 proceed as follows with Mathematica: - -- Set `sigma[x_] := x + 1` in Mathematica package and execute definition. -- Execute `timeScaleDerivativeX[1, x, b]` which produces `-3 b + 3 b^2`. -- Execute `Expand[timeScaleDerivativeX[1, t, sigma[t]]]` which produces `3 t + 3 t^2`. -- Execute `timeScaleDerivativeB[1, x, b]` which produces `1 - 6 b^2 + 6 b x`. -- Execute `timeScaleDerivativeB[1, t, t]` which produces `1`. -- Execute `mainTheorem[1]` which produces `1 + 3 t + 3 t^2`. - -### Example for R x R time scale. - -

- timescale-z -

- -To reproduce example 4.7 proceed as follows with Mathematica: - -- Set `sigma[x_] := x + Global`dx` in Mathematica package and execute definition. -- Execute `timeScaleDerivativeX[1, x, b]` which produces `-3 b + 3 b^2`. -- Execute `Limit[Expand[timeScaleDerivativeB[1, x, b]], dx -> 0]` which produces `6 b - 6 b^2 - 3 x + 6 b x`. -- Execute `timeScaleDerivativeX[1, t, t]` which produces `-3 t + 3 t^2`. -- Execute `Limit[Expand[timeScaleDerivativeB[1, t, t]], dx -> 0]` which produces `3t`. -- Execute `Limit[mainTheorem[1], dx -> 0]` which produces `3t^2`. \ No newline at end of file +## Build and run in Intellij IDEA + +- Install `MikTeX`: https://miktex.org/download +- Update `MikTeX` +- Install `SumatraPDF` viewer: https://www.sumatrapdfreader.org/download-free-pdf-viewer +- Path to SumatraPDF: `C:\Program Files\SumatraPDF` +- Install `Intellij IDEA Ultimate`: https://www.jetbrains.com/idea/download/#section=windows +- Activate `Intellij IDEA Ultimate` +- Install `TeXiFy IDEA` plugin: https://plugins.jetbrains.com/plugin/9473-texify-idea +- Clone this repository locally: `https://github.com/kolosovpetro/github-latex-template.git` +- Open `github-latex-template` folder in `Intellij IDEA Ultimate` and configure as follows + - LaTeX Configuration + ![LaTeX Configuration](./img/latex_configuration.PNG "LaTeX Configuration") + - BibTeX Configuration + ![BibTeX Configuration](./img/bibtex_configuration.PNG "BibTeX Configuration") +- Configure Inverse Search in `Intellij IDEA` for SumatraPDF: `Tools -> LaTeX -> Configure Inverse Search` +- Compile document using `Shift + F10` diff --git a/img/abstract.PNG b/img/abstract.PNG deleted file mode 100644 index 9190cff1c815ef9320a8c521daf611c34fbd2c34..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 79898 zcmdSA^;c9;+xU%=(%nN10@5W6-6co}(v5VEbTL!KB1QLPEk(Qj~p%goLt#g!HTj0}Z%RBvO?O 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Agnieszka~B Malinowska, and Delfim~FM Torres. -\newblock The power quantum calculus and variational problems. -\newblock {\em arXiv preprint arXiv:1107.0344}, 2011. -\newblock \url {https://arxiv.org/abs/1107.0344}. +\bibitem{Bohner2001DynamicEO} +Martin Bohner and Allan Peterson. +\newblock {\em {Dynamic equations on time scales: An introduction with + applications}}. +\newblock Springer Science \& Business Media, 2001. +\newblock \url {https://web.mst.edu/~bohner/sample.pdf}. -\bibitem[BG04]{bohner2004partial} +\bibitem{bohner2004partial} Martin Bohner and Gusein~Sh Guseinov. \newblock Partial differentiation on time scales. \newblock {\em Dynamic systems and applications}, 13(3-4):351--379, 2004. \newblock \url {http://web.mst.edu/~bohner/papers/pdots.pdf}. -\bibitem[BHT16]{benkhettou2016conformable} -Nadia Benkhettou, Salima Hassani, and Delfim~FM Torres. -\newblock A conformable fractional calculus on arbitrary time scales. -\newblock {\em Journal of King Saud University-Science}, 28(1):93--98, 2016. - -\bibitem[BHT17]{bayour2017truly} -Benaoumeur Bayour, Ahmed Hammoudi, and Delfim~FM Torres. -\newblock A truly conformable calculus on time scales. -\newblock {\em arXiv preprint arXiv:1705.08928}, 2017. -\newblock \url {https://arxiv.org/abs/1705.08928}. +\bibitem{ahlbrandt2002partial} +Calvin~D Ahlbrandt and Christina Morian. +\newblock Partial differential equations on time scales. +\newblock {\em Journal of Computational and Applied Mathematics}, + 141(1-2):35--55, 2002. +\newblock \url {https://doi.org/10.1016/S0377-0427(01)00434-4}. -\bibitem[BP01]{Bohner2001DynamicEO} -Martin Bohner and Allan Peterson. -\newblock {\em {Dynamic equations on time scales: An introduction with - applications}}. -\newblock Springer Science \& Business Media, 2001. -\newblock \url {https://web.mst.edu/~bohner/sample.pdf}. +\bibitem{JACKSON2006391} +B.~Jackson. +\newblock Partial dynamic equations on time scales. +\newblock {\em Journal of Computational and Applied Mathematics}, 186(2):391 -- + 415, 2006. +\newblock \url {https://doi.org/10.1016/j.cam.2005.02.011}. -\bibitem[Cap09]{caputo2009time} -M~Cristina Caputo. -\newblock Time scales: from nabla calculus to delta calculus and vice versa via - duality. -\newblock {\em arXiv preprint arXiv:0910.0085}, 2009. +\bibitem{jackson_1909} +Frederick~H Jackson. +\newblock Xi.—on q-functions and a certain difference operator. +\newblock {\em Earth and Environmental Science Transactions of the Royal + Society of Edinburgh}, 46(2):253--281, 1909. -\bibitem[Ern00]{ernst2000history} +\bibitem{ernst2000history} Thomas Ernst. \newblock {\em The history of q-calculus and a new method}. \newblock Citeseer, 2000. -\bibitem[Ern08]{ernst2008different} +\bibitem{ernst2008different} Thomas Ernst. \newblock The different tongues of q-calculus. \newblock {\em Proceedings of the Estonian Academy of Sciences}, 57(2), 2008. -\bibitem[Hil88]{hilger1988ein} -S~Hilger. -\newblock {\em {E {\i} n Ma {\ss} kettenkalk {\"u} l m {\i} t Anwendung auf - Zentrumsmann {\i} gfalt {\i} gke {\i} ten}}. -\newblock PhD thesis, Ph. D. Thesis, Universt {\"a} t W {\"u} rzburg, 1988. - -\bibitem[Jac09]{jackson_1909} -Frederick~H Jackson. -\newblock Xi.—on q-functions and a certain difference operator. -\newblock {\em Earth and Environmental Science Transactions of the Royal - Society of Edinburgh}, 46(2):253--281, 1909. - -\bibitem[Jac06]{JACKSON2006391} -B.~Jackson. -\newblock Partial dynamic equations on time scales. -\newblock {\em Journal of Computational and Applied Mathematics}, 186(2):391 -- - 415, 2006. -\newblock \url {https://doi.org/10.1016/j.cam.2005.02.011}. - -\bibitem[KC01]{kac2001quantum} +\bibitem{kac2001quantum} Victor Kac and Pokman Cheung. \newblock {\em Quantum calculus}. \newblock Springer Science \& Business Media, 2001. -\bibitem[Kol16]{kolosov2016link} +\bibitem{aldwoah2011power} +Khaled~A Aldwoah, Agnieszka~B Malinowska, and Delfim~FM Torres. +\newblock The power quantum calculus and variational problems. +\newblock {\em arXiv preprint arXiv:1107.0344}, 2011. +\newblock \url {https://arxiv.org/abs/1107.0344}. + +\bibitem{kolosov2016link} Petro Kolosov. \newblock {On the link between Binomial Theorem and Discrete Convolution of Polynomials}. \newblock {\em arXiv preprint arXiv:1603.02468}, 2016. \newblock \url {https://arxiv.org/abs/1603.02468}. -\bibitem[Kol20]{kolosov2022mathematica} -Petro Kolosov. -\newblock {Supplementary Mathematica Programs}. -\newblock 2020. -\newblock \url - {https://github.com/kolosovpetro/AStudyOnDynamicEquations/tree/develop/mathematica}. +\bibitem{WeissteinBernoulli} +Eric~W Weisstein. +\newblock {"Bernoulli Number." From MathWorld -- A Wolfram Web Resource.} +\newblock \url {http://mathworld.wolfram.com/BernoulliNumber.html}. -\bibitem[MT09]{martins2009calculus} +\bibitem{hilger1988ein} +S~Hilger. +\newblock {\em {E {\i} n Ma {\ss} kettenkalk {\"u} l m {\i} t Anwendung auf + Zentrumsmann {\i} gfalt {\i} gke {\i} ten}}. +\newblock PhD thesis, Ph. D. Thesis, Universt {\"a} t W {\"u} rzburg, 1988. + +\bibitem{bayour2017truly} +Benaoumeur Bayour, Ahmed Hammoudi, and Delfim~FM Torres. +\newblock A truly conformable calculus on time scales. +\newblock {\em arXiv preprint arXiv:1705.08928}, 2017. +\newblock \url {https://arxiv.org/abs/1705.08928}. + +\bibitem{benkhettou2016conformable} +Nadia Benkhettou, Salima Hassani, and Delfim~FM Torres. +\newblock A conformable fractional calculus on arbitrary time scales. +\newblock {\em Journal of King Saud University-Science}, 28(1):93--98, 2016. + +\bibitem{caputo2009time} +M~Cristina Caputo. +\newblock Time scales: from nabla calculus to delta calculus and vice versa via + duality. +\newblock {\em arXiv preprint arXiv:0910.0085}, 2009. + +\bibitem{martins2009calculus} Nat {\'a}~lia Martins and Delfim~FM Torres. \newblock Calculus of variations on time scales with nabla derivatives. \newblock {\em Nonlinear Analysis: Theory, Methods \& Applications}, 71(12):e763--e773, 2009. -\bibitem[Wei]{WeissteinBernoulli} -Eric~W Weisstein. -\newblock {"Bernoulli Number." From MathWorld -- A Wolfram Web Resource.} -\newblock \url {http://mathworld.wolfram.com/BernoulliNumber.html}. +\bibitem{kolosov2022mathematica} +Petro Kolosov. +\newblock {Supplementary Mathematica Programs}. +\newblock 2020. +\newblock \url + {https://github.com/kolosovpetro/AStudyOnDynamicEquations/tree/develop/mathematica}. \end{thebibliography} diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index 77b90f5baefaa09dc17bfb72290d811834fdb378..ece4b353a945825955c188e595e9b4b7b1ee0b58 100644 GIT binary patch delta 101368 zcmZ_VWmKEnwlHAarMN@Tq5*=t6nA%bcP*4q+%33Uai_Qxr?{8mZpDii``G)ObMHRm z8~0B#-!*2k)|($0^I3Df-);z7PYCO5(4hI!C_`cC!KZq<>Z>x?0SDTLs$@IB#p;TT zi=hVYT$5o0G$9Eep({W;{IvuHpxp85oacC2^_QIu=}G_487fPPT86IgLK=gM8)S2L z5w>&Zjsgf1Cj5)zV44uNi-tQd-^3wmP28@-1MPnWcME1=jZ#=qJa0RzC&B4Wfbk;^(Bp7ktgHZDVSPFJDiyp4>lh*-VRbRkeOpk5&VG-6$E8I- zCh@_2pPo<&vVl$r);=2)^$&xSDV0(E74gyo;|E&2}K9@^nvy} zX)rXZH>87tc!xESs>&Bkgau{F3|dfmvbg`9?4#AGOmCuGtPa0(*v)ZKSek*yW4Yy- 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a/src/AStudyOnDynamicEquations.tex b/src/AStudyOnDynamicEquations.tex index ae5b7c6..b707131 100644 --- a/src/AStudyOnDynamicEquations.tex +++ b/src/AStudyOnDynamicEquations.tex @@ -100,7 +100,7 @@ } \begin{document} \begin{abstract} - \input{sections/abstract.tex} + \input{sections/abstract} \end{abstract} \maketitle @@ -109,33 +109,34 @@ \section{Definitions} \label{sec:definitions} - \input{sections/definitions-notations-and-conventions.tex} + \input{sections/definitions-notations-and-conventions} \section{Introduction} \label{sec:introduction} - \input{sections/introduction.tex} + \input{sections/introduction} \section{Main results} \label{sec:main_results} - \input{sections/main-results.tex} + \input{sections/main-results} \section{Discussion and examples} \label{sec:discussion_and_examples} - \input{sections/discussion-and-examples.tex} + \input{sections/discussion-and-examples} \section{Proof of main theorem} \label{sec:proof_main_theorem} - \input{sections/proof-of-main-theorem.tex} + \input{sections/proof-of-main-theorem} \section{Mathematica scripts} \label{sec:mathematica_scripts} - \input{sections/mathematica-scripts.tex} + \input{sections/mathematica-scripts} \section{Conclusion and future research} \label{sec:conclusion} - \input{sections/conclusion-and-future-research.tex} + \input{sections/conclusion-and-future-research} - \bibliographystyle{alpha} - \bibliography{time_scale_calculus_refs} + \bibliographystyle{unsrt} + \bibliography{AStudyOnDynamicEquations} + \noindent \textbf{Version:} \input{sections/version} \end{document} diff --git a/src/sections/version.tex b/src/sections/version.tex new file mode 100644 index 0000000..5c3d722 --- /dev/null +++ b/src/sections/version.tex @@ -0,0 +1 @@ +\texttt{Local-0.1.0} \ No newline at end of file From c1829e9a7643b78fed1ee2ecaf49754e6acad454 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 15:33:32 +0200 Subject: [PATCH 02/22] update changelog --- CHANGELOG.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/CHANGELOG.md b/CHANGELOG.md index 356cc04..14140ba 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -8,3 +8,7 @@ and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec ## [1.0.0] - In progress ### Changed +- Change bibliography style +- Update CI/CD +- Update version inside document +- Update README.md From cfd3c407b2e0ae10feb67c5b3d705642a13722ee Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 15:40:22 +0200 Subject: [PATCH 03/22] work in progress --- CHANGELOG.md | 2 ++ out/AStudyOnDynamicEquations.pdf | Bin 407096 -> 412891 bytes src/AStudyOnDynamicEquations.tex | 15 +++++++++------ 3 files changed, 11 insertions(+), 6 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 14140ba..cccd093 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -12,3 +12,5 @@ and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec - Update CI/CD - Update version inside 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+\apptocmd{\sloppy}{\hbadness 10000\relax}{}{} +\raggedbottom \newcommand \anglePower [2]{\langle #1 \rangle \sp{#2}} \newcommand \bernoulli [2][B] {{#1}\sb{#2}} @@ -128,10 +132,6 @@ \input{sections/proof-of-main-theorem} - \section{Mathematica scripts} \label{sec:mathematica_scripts} - \input{sections/mathematica-scripts} - - \section{Conclusion and future research} \label{sec:conclusion} \input{sections/conclusion-and-future-research} @@ -139,4 +139,7 @@ \bibliography{AStudyOnDynamicEquations} \noindent \textbf{Version:} \input{sections/version} + \section{Addendum 1: Mathematica scripts} \label{sec:mathematica_scripts} + \input{sections/mathematica-scripts} + \end{document} From 7279be4aa8d0a1a20cfc1a5a604118aacb77e3e5 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 15:50:34 +0200 Subject: [PATCH 04/22] work in progress --- out/AStudyOnDynamicEquations.pdf | Bin 412891 -> 394531 bytes src/AStudyOnDynamicEquations.tex | 3 ++- src/sections/abstract.tex | 18 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cdbc9a8..32f96e3 100644 --- a/src/AStudyOnDynamicEquations.tex +++ b/src/AStudyOnDynamicEquations.tex @@ -21,7 +21,7 @@ \newcommand \bernoulli [2][B] {{#1}\sb{#2}} \newcommand \curvePower [2]{\{#1\}\sp{#2}} \newcommand \coeffA [3][A] {{\mathbf{#1}} \sb{#2,#3}} -\newcommand \polynomialP [4][P]{{\mathbf{#1}}\sp{#2} \sb{#3}(#4)} +\newcommand \polynomialP [4][P] {#1 ( #2, #3, #4 )} % ordinary derivatives \newcommand \derivative [2] {\frac{d}{d #2} #1} % 1 - function; 2 - variable; @@ -139,6 +139,7 @@ \bibliography{AStudyOnDynamicEquations} \noindent \textbf{Version:} \input{sections/version} + \section{Addendum 1: Mathematica scripts} \label{sec:mathematica_scripts} \input{sections/mathematica-scripts} diff --git a/src/sections/abstract.tex b/src/sections/abstract.tex index f14d360..c83c056 100644 --- a/src/sections/abstract.tex +++ b/src/sections/abstract.tex @@ -1,14 +1,14 @@ -Let $\mathbf{P}_b^m(x)$ be a $2m+1-$degree integer-valued polynomial in $x,b$. -Let be a two-dimensional time scale -$\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$. -Let be $\mathbb{T}_1 = \mathbb{T}_2$. +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 $\mathbb{T}_1 = \mathbb{T}_2$. In this manuscript we derive and discuss the following partial dynamic equation on time scales. -For every $t\in\mathbb{T}_1, \; x,b\in \Lambda^2, \; m = const, \; m\in\mathbb{N}$ +For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$ \[ (t^{2m+1})^{\Delta} = - \frac{\partial \mathbf{P}_b^m(x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)}+ - \frac{\partial \mathbf{P}_b^m(x)}{\Delta b}\bigg |_{x = t, \; b = t}, + \frac{\partial P(m,b,x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)} + + \frac{\partial P(m,b,x)}{\Delta b}\bigg |_{x = t, \; b = t} \] -where $\sigma(t) > t$ is forward jump operator. +such that $\sigma(t) > t$ is forward jump operator. In addition, we discuss various derivative operators in context of partial cases of above equation, -we show finite difference, classical derivative, $q-$derivative, $q-$power derivative on behalf of it. \ No newline at end of file +we show finite difference, classical derivative, $q-$derivative, $q-$power derivative on behalf of it. From 9b75d5bfea5bfacf645962fe53b99f5037827575 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 16:03:59 +0200 Subject: [PATCH 05/22] update git ignore | update citation --- .gitignore | 3 ++- BIBTEX-CITATION.bib | 13 +++++++------ 2 files changed, 9 insertions(+), 7 deletions(-) diff --git a/.gitignore b/.gitignore index 048ace8..caaaf09 100644 --- a/.gitignore +++ b/.gitignore @@ -29,6 +29,7 @@ *-blx.aux *-blx.bib *.run.xml +*.annotated.pdf ## Build tool auxiliary files: *.fdb_latexmk @@ -635,4 +636,4 @@ ASALocalRun/ healthchecksdb # code coverage temp -coverage.opencover.xml \ No newline at end of file +coverage.opencover.xml diff --git a/BIBTEX-CITATION.bib b/BIBTEX-CITATION.bib index 350007f..000f030 100644 --- a/BIBTEX-CITATION.bib +++ b/BIBTEX-CITATION.bib @@ -1,6 +1,7 @@ -@MISC{GithubSource_2022, - author = {Petro Kolosov}, - title = {{"Github Template" Source files}}, - howpublished = {Available electronically at \url {https://github.com/kolosovpetro/github-latex-template}}, - year = {2022} -} \ No newline at end of file +@article{kolosov2016study, + title = {{A study on partial dynamic equation on time scales involving derivatives of polynomials}}, + author = {Kolosov, Petro}, + journal = {{arXiv preprint arXiv:1608.00801}}, + note = {\url {https://arxiv.org/abs/1608.00801}}, + year = {2016} +} From f6ea0c07435388e0495871fe6d2770b73a7cf00f Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 16:30:53 +0200 Subject: [PATCH 06/22] update definitions --- out/AStudyOnDynamicEquations.pdf | Bin 394531 -> 372710 bytes src/sections/abstract.tex | 10 ++- .../definitions-notations-and-conventions.tex | 80 ++++++++---------- 3 files changed, 40 insertions(+), 50 deletions(-) diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index eeef184f4d1a8a29a5256291933f85fa6afa9c43..3eb68679c32033a6272c447e80e086d9159285d1 100644 GIT binary patch delta 101278 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manuscript we derive and discuss the following partial dynamic equation on time scales. +In this manuscript we derive and discuss an identity that connects the timescale derivative of odd-power polynomial +with partial derivatives of polynomial $P(m,b,x)$ evaluated in particular points. For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$ \[ - (t^{2m+1})^{\Delta} = - \frac{\partial P(m,b,x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)} + - \frac{\partial P(m,b,x)}{\Delta b}\bigg |_{x = t, \; b = t} + \frac{\Delta t^{2m+1}}{\Delta t} = +% \frac{\partial P(m,b,x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)} + + \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] such that $\sigma(t) > t$ is forward jump operator. In addition, we discuss various derivative operators in context of partial cases of above equation, diff --git a/src/sections/definitions-notations-and-conventions.tex b/src/sections/definitions-notations-and-conventions.tex index c47ea37..581be58 100644 --- a/src/sections/definitions-notations-and-conventions.tex +++ b/src/sections/definitions-notations-and-conventions.tex @@ -1,73 +1,61 @@ -We now set the following notation, which remains fixed for the remainder of this paper: +We now set the following notation such that remains fixed for the remainder of this manuscript \begin{itemize} - \setlength\itemsep{1em} + \setlength\itemsep{1.6em} \item Let be a function $f\colon \mathbb{T} \to \mathbb{R}$ and $t\in\mathbb{T}^{\kappa}$ then $f^{\Delta}(t)$ - is delta time-scale derivative~\cite{Bohner2001DynamicEO} of $f$ - \[ - f^{\Delta} (t) = \frac{f(\sigma(t)) - f(t)}{\mu(t)} = \frac{f(\sigma(t)) - f(t)}{\sigma(t) - t}, - \] - where $\mu(t) = \sigma(t) - t, \; \mu(t) \neq 0$ and $\sigma(t) > t$ is forward jump operator. - - \item $\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}, \; f^{\Delta_i}_{t_i}(t)$ is delta partial derivative - of $f\colon \Lambda^n \to \mathbb{R}$ on $n-$dimensional time scale - $\Lambda^n$~\cite{bohner2004partial, ahlbrandt2002partial,JACKSON2006391}, - defined as a limit - \[ - f^{\Delta_i}_{t_i}(t) = \lim \limits_{\substack{s_i \to t_i \\ s_i \neq \sigma_i(t_i)}} + is delta timescale derivative~\cite{Bohner2001DynamicEO} + \begin{align*} + f^{\Delta} (t) = \frac{f(\sigma(t)) - f(t)}{\sigma(t) - t} + \end{align*} + where $\sigma(t) - t \neq 0$ and $\sigma(t) > t$ is forward jump operator. + + \item $\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}$ is the delta partial derivative + of $f\colon \Lambda^n \to \mathbb{R}$ on $n$-dimensional timescale + $\Lambda^n$ defined via the limit~\cite{bohner2004partial, ahlbrandt2002partial,JACKSON2006391} + \begin{align*} + f^{\Delta_i}_{t_i}(t) = \lim \limits_{\substack{s_i \to t_i}} \frac{ f(t_1, \ldots, t_{i-1}, \sigma_i(t_i), t_{t+1}, \ldots, t_n) - f(t_1, \ldots, t_{i-1}, s_i, t_{t+1}, \ldots, t_n) - }{\sigma_i(t_i) - s_i}, - \] + }{\sigma_i(t_i) - s_i} + \end{align*} where $\sigma_i(t_i) > t_i$ and $\sigma_i(t_i) - s_i \neq 0$. \item $\qderivative{f(x)}$ is $q-$derivative~\cite{jackson_1909,ernst2000history,ernst2008different,kac2001quantum} - \[ - \qderivative{f(x)} = \frac{f(qx)-f(x)}{qx-x}, - \] + \begin{align*} + \qderivative{f(x)} = \frac{f(qx)-f(x)}{qx-x} + \end{align*} where $x\neq 0, \; x\in\mathbb{R}, \; q\in\mathbb{R}$. \item $\nqderivative{f(t)}$ is $q-$power derivative~\cite{aldwoah2011power} - \[ - \nqderivative{f(t)} = \frac{f(qt^n) - f(t)}{qt^n - t}, - \] + \begin{align*} + \nqderivative{f(t)} = \frac{f(qt^n) - f(t)}{qt^n - t} + \end{align*} where $qt^n - t \neq 0$ and $n$ is odd positive integer and $0 < q < 1$. \item $\qpowerDerivative{f(x)}$ is $q-$power derivative - \[ - \qpowerDerivative{f(x)} = \frac{f(x^q)-f(x)}{x^q-x}, - \] + \begin{align*} + \qpowerDerivative{f(x)} = \frac{f(x^q)-f(x)}{x^q-x} + \end{align*} where $x^q \neq x, \; x\in\mathbb{R}, \; q\in\mathbb{R}$. - \item $\polynomialP{m}{b}{x}, \; x,b\in\mathbb{R}, \; m\in\mathbb{N}$ is $2m+1-$degree integer-valued polynomial~\cite{kolosov2016link} + \item $\polynomialP{m}{b}{x}, \; x,b\in\mathbb{R}, \; m\in\mathbb{N}$ is $2m+1$-degree + polynomial in $x,b$ \begin{equation} - \polynomialP{m}{b}{x} = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r, + \polynomialP{m}{b}{x} = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \label{eq:polynomial_p} \end{equation} - where $\coeffA{m}{r}, \; m\in\mathbb{N}$ is a real coefficient defined recursively - \[ - \coeffA{m}{r} = - \begin{cases} - (2r+1) - \binom{2r}{r}, & \text{if } r=m,\\ - (2r+1) \binom{2r}{r} \sum_{d=2r+1}^{m} \coeffA{m}{d} \binom{d}{2r+1} \frac{(-1)^{d-1}}{d-r} - \bernoulli{2d-2r}, & \text{if } 0 \leq rm, - \end{cases} - \] - where $\bernoulli{t}$ are Bernoulli numbers~\cite{WeissteinBernoulli}. - It is assumed that $\bernoulli{1}=\frac{1}{2}$. + where $\coeffA{m}{r}$ is a real coefficient defined recursively, see~\cite{kolosov2016link}. - \item $\mathbb{Z}$ is an integer time scale such that $\sigma(t) = t+1$ and $\mu(t) = 1$. + \item $\mathbb{Z}$ is an integer timescale such that $\sigma(t) = t+1$ and $\mu(t) = 1$. - \item $\mathbb{R}$ is a real time scale 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 $\sigma(t) = t+\Delta t$ and $\mu(t) = \Delta t, \; \Delta t \to 0$. - \item $q^\mathbb{R}$ is a quantum time scale such that $\sigma(t) = qt$ and $\mu(t) = qt - t$, + \item $q^\mathbb{R}$ is a quantum timescale such that $\sigma(t) = qt$ and $\mu(t) = qt - t$, [page 18~\cite{Bohner2001DynamicEO}]. - \item $\mathbb{R}^q$ is a quantum power time scale such that $\sigma(t) = t^q$ and $\mu(t) = t^q - t$. + \item $\mathbb{R}^q$ is a quantum power timescale such that $\sigma(t) = t^q$ and $\mu(t) = t^q - t$. - \item $q^{\mathbb{R}^n}$ is a pure quantum power time scale + \item $q^{\mathbb{R}^n}$ is a pure quantum power timescale such that $\sigma(t) = qt^n > t, \; 0 Date: Mon, 16 Sep 2024 16:38:56 +0200 Subject: [PATCH 07/22] work in progress --- out/AStudyOnDynamicEquations.pdf | Bin 372710 -> 372149 bytes src/sections/abstract.tex | 1 - src/sections/introduction.tex | 2 +- src/sections/main-results.tex | 32 +++++++++++++++---------------- 4 files changed, 17 insertions(+), 18 deletions(-) diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index 3eb68679c32033a6272c447e80e086d9159285d1..4a29e0833947adbe5f88b15614f8a49a5ca3a9eb 100644 GIT binary patch delta 44934 zcmZ^KWk6KX);2?Thje$&P(y<>(jp+;C7^W5kRqZI2N|RU2?eCPyQD+9I|L-8;~Vte z``-KB_xrQYI%n27``K$fd&QZVjYaaxS@IGi8Gu?17f0RJ)!x&A0U`_*0^DRcVN|p3 zvo*ix&0Ce7s2DqF9oW+9sO`Jc<|an`?L>P~ct=F4KXU*Hd+M+z3?$V!QWoR)yUx{D z3WSo9&@*aEGzvM=Yu{fny{*TS@#xr&jD`YY1fc zc*E)K0H!rvIupj#&eq!b_&!ZCz}Ec*@A_&|80{Cj 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zZ*am?_=j4FZ{!!2l9CC(fgx$a8u^7J$ELtvpk4i6ftr#2-L-75Krtu~^xKP~d-ezl zWO%+uPnu`A9zoq1sjKGyo#IwIXOsrM`8y*sDm^nhJtnph9Dw0^rjeUluK_`g{tv1j BYS#b& diff --git a/src/sections/abstract.tex b/src/sections/abstract.tex index 3d43b0c..88b7f05 100644 --- a/src/sections/abstract.tex +++ b/src/sections/abstract.tex @@ -7,7 +7,6 @@ For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$ \[ \frac{\Delta t^{2m+1}}{\Delta t} = -% \frac{\partial P(m,b,x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)} + \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] diff --git a/src/sections/introduction.tex b/src/sections/introduction.tex index f0e590e..fd40cb0 100644 --- a/src/sections/introduction.tex +++ b/src/sections/introduction.tex @@ -16,4 +16,4 @@ In context of Computer Science, namely object oriented programming paradigm, the time scale calculus may be thought as unified interface of derivative operator. Furthermore, the idea of time-scale calculus was slightly extended -in~\cite{bayour2017truly,benkhettou2016conformable,caputo2009time,martins2009calculus}. \ No newline at end of file +in~\cite{bayour2017truly,benkhettou2016conformable,caputo2009time,martins2009calculus}. diff --git a/src/sections/main-results.tex b/src/sections/main-results.tex index b36a653..e096b11 100644 --- a/src/sections/main-results.tex +++ b/src/sections/main-results.tex @@ -1,31 +1,31 @@ -Time scale derivative of the polynomial $t^{2m+1}$ may be expressed as follows +Timescale derivative of the polynomial $t^{2m+1}$ may be expressed as follows \begin{thm} \label{main_theorem} - Let $\polynomialP{m}{b}{x}$ be a $2m+1-$degree integer-valued polynomial defined by~\eqref{eq:polynomial_p}. - Let be a two-dimensional time scale - $\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$. - Let be $\mathbb{T}_1 = \mathbb{T}_2$. - For every $t\in\mathbb{T}_1, \; x,b\in \Lambda^2, \; m\in\mathbb{N}, \; m = const$ + 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 $x,b\in \Lambda^2$ \[ - (t^{2m+1})^{\Delta} = - \frac{\partial \polynomialP{m}{b}{x}}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)}+ - \frac{\partial \polynomialP{m}{b}{x}}{\Delta b} \bigg |_{x = t, \; b = t}, + \frac{\Delta t^{2m+1}}{\Delta t} = + \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] where \begin{itemize} \setlength\itemsep{1em} - \item $\sigma(t) > t$ is forward jump operator, + \item $\sigma(t) > t$ -- is forward jump operator - \item $\frac{\partial \polynomialP{m}{b}{x}}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)}$ + \item $\frac{\partial \polynomialP{m}{b}{x}}{\Delta x} (m, \sigma(t), t)$ -- is the value of the partial derivative on time scales of - $\polynomialP{m}{b}{x}$ with respect to the variable $x$, evaluated at $x = t, \; b = \sigma(t)$, + $\polynomialP{m}{b}{x}$ with respect to the variable $x$ evaluated in point $x = t, \; b = \sigma(t)$ - \item $\frac{\partial \polynomialP{m}{b}{x}}{\Delta b} \bigg |_{x = t, \; b = t}$ + \item $\frac{\partial \polynomialP{m}{b}{x}}{\Delta b} (m, t, t)$ -- is the value of the partial derivative on time scales of - $\polynomialP{m}{b}{x}$ with respect to the variable $b$, evaluated at $x = t, \; b = t$. + $\polynomialP{m}{b}{x}$ with respect to the variable $b$, evaluated at $x = t, \; b = t$ \end{itemize} \end{thm} -In other words, theorem ~\ref{main_theorem} says +In simpler words, the theorem ~\ref{main_theorem} says \begin{center} \begin{quotation} For every odd-exponent polynomial $t^{2m+1}$, its derivative on time scales equals to the sum @@ -51,4 +51,4 @@ ^{\Delta}_{t} = \frac{\partial}{\Delta x} \left( \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \right) \Bigg |_{x = t, \; b = \sigma(t)} + \frac{\partial}{\Delta b} \left( \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \right) \Bigg |_{x = t, \; b = t} -\end{align*} \ No newline at end of file +\end{align*} From 64eac96f0c0b5cdf75a0be5e0b340ad95296b7bc Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 16:56:47 +0200 Subject: [PATCH 08/22] minor edits --- out/AStudyOnDynamicEquations.pdf | Bin 372149 -> 372492 bytes src/sections/proof-of-main-theorem.tex | 5 +---- 2 files changed, 1 insertion(+), 4 deletions(-) diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index 4a29e0833947adbe5f88b15614f8a49a5ca3a9eb..b310febb2ae89ff067c9e84f296c8ad61a185e17 100644 GIT binary patch delta 23261 zcmaI71yozz*7u9MJH@p)BqYII3dOCsySo($4lNXEfB*&B;?Uw&oB~CP7k4f06t|l` z=R5EBo^!{&V`OA!XRo#9+-qlK{(p0>Rlh)yuuM@!#(-Q{OU`hKjs}V0eat6hONE*C 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z#=E}+F%qu>hz2oI;0ilWy}%JogWP-s?RF@`F{@zl_s0K;lHijlhQ=M7npZyylv3zV1&AEpv& z)&=tZGK1@iI(NiRjwoV3VPf+MTn#n-grWW_#KeiPQ?^Qde*?9qL@9~}X@#cg?|AzC vqVy=L^<`0L>mM2R%1@$<35$&M%}9GYHSM>lNfUffvrtC9jvf2Q4)FaSp9}zs diff --git a/src/sections/proof-of-main-theorem.tex b/src/sections/proof-of-main-theorem.tex index 49d45b6..695e5c7 100644 --- a/src/sections/proof-of-main-theorem.tex +++ b/src/sections/proof-of-main-theorem.tex @@ -46,13 +46,10 @@ \subsection*{Proof of theorem ~\ref{main_theorem}} \] 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, @@ -70,4 +67,4 @@ \subsection*{Proof of theorem ~\ref{main_theorem}} + \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t} \end{equation*} -This completes the proof. \qed \ No newline at end of file +This completes the proof. \qed From 3c865fad8421a6df2c41b063fba8e057da35c0f5 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 17:37:03 +0200 Subject: [PATCH 09/22] work in progress --- out/AStudyOnDynamicEquations.pdf | Bin 372492 -> 371699 bytes src/sections/discussion-and-examples.tex | 20 +++++++++++--------- src/sections/main-results.tex | 8 ++++---- 3 files changed, 15 insertions(+), 13 deletions(-) diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index b310febb2ae89ff067c9e84f296c8ad61a185e17..21a43209ebe6b283bfc5e61025fc56af51cbc5e4 100644 GIT binary patch delta 28960 zcmZsDbwE^G_qIcKcXxLW-Q5U+beA*&QUij3gv0@ck`Sell17jal#=dl0qM?f(EEPB z``-KV$3E-qvuEvRt@W(6_P`9W3q;40M3ros0E-a@kQIsoFT^hh$Y`=dyvT*|f+s7| zg2+?lNY1Oco|FsM>8gS~>AXzI!7p;V68iY*AZ?rx)kyKA|6BhYtar|or#dVetCeJn*A;53Wn`CMZsev2*1hK_YjPTUJ$

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b/src/sections/abstract.tex @@ -6,7 +6,7 @@ with partial derivatives of polynomial $P(m,b,x)$ evaluated in particular points. For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$ \[ - \frac{\Delta t^{2m+1}}{\Delta t} = + \frac{\Delta x^{2m+1}}{\Delta x}(t) = \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] diff --git a/src/sections/definitions-notations-and-conventions.tex b/src/sections/definitions-notations-and-conventions.tex index 581be58..6d7b3dc 100644 --- a/src/sections/definitions-notations-and-conventions.tex +++ b/src/sections/definitions-notations-and-conventions.tex @@ -12,7 +12,7 @@ of $f\colon \Lambda^n \to \mathbb{R}$ on $n$-dimensional timescale $\Lambda^n$ defined via the limit~\cite{bohner2004partial, ahlbrandt2002partial,JACKSON2006391} \begin{align*} - f^{\Delta_i}_{t_i}(t) = \lim \limits_{\substack{s_i \to t_i}} + \dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i} = \lim \limits_{\substack{s_i \to t_i}} \frac{ f(t_1, \ldots, t_{i-1}, \sigma_i(t_i), t_{t+1}, \ldots, t_n) - f(t_1, \ldots, t_{i-1}, s_i, t_{t+1}, \ldots, t_n) diff --git a/src/sections/main-results.tex b/src/sections/main-results.tex index bbe577f..858b6ca 100644 --- a/src/sections/main-results.tex +++ b/src/sections/main-results.tex @@ -1,4 +1,4 @@ -Timescale derivative of the polynomial $t^{2m+1}$ may be expressed as follows +Timescale derivative of odd-powered polynomial $t^{2m+1}$ may be expressed as follows \begin{thm} \label{main_theorem} Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$. @@ -7,7 +7,7 @@ such that $\mathbb{T}_1 = \mathbb{T}_2$. For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$ \[ - \frac{\Delta t^{2m+1}}{\Delta t} = + \frac{\Delta x^{2m+1}}{\Delta x}(t) = \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] @@ -28,27 +28,24 @@ In simpler words, the theorem ~\ref{main_theorem} says \begin{center} \begin{quotation} - For every odd-powered polynomial $t^{2m+1}$, its derivative on time scales equals to the sum - of the value of the partial derivative on time scales of + For every odd-powered polynomial $x^{2m+1}$, the derivative on time scales $\frac{\Delta x^{2m+1}}{\Delta x}$ + evaluated in point $t\in\mathbb{T}_1$ equals to partial derivative on time scales of the polynomial $\polynomialP{m}{b}{x}$ - with respect to the variable - $x$, - evaluated at - $x = t, \; b = \sigma(t)$ - and the value of the partial derivative on time scales of + with respect to $x$ + evaluated in point + $(x,b) = (t, \sigma(t))$ + plus the value of the partial derivative on time scales of the polynomial $\polynomialP{m}{b}{x}$ - with respect to the variable - $b$, - evaluated at - $x = t, \; b = t$. + with respect to $b$, + evaluated in point + $(x,b)=(t,t)$. \end{quotation} \end{center} In its extended form the theorem ~\ref{main_theorem} is as follows \begin{align*} -(t^{2m+1}) - ^{\Delta}_{t} = + \frac{\Delta x^{2m+1}}{\Delta x}(t) = \frac{\partial}{\Delta x} \left( \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \right) (m, \sigma(t), t) + \frac{\partial}{\Delta b} \left( \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \right) (m, t, t) \end{align*} From 7a1f65bc7d8b8fe2145a933ceee2ce5d14f528cb Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 19:46:07 +0200 Subject: [PATCH 11/22] work in progress --- out/AStudyOnDynamicEquations.pdf | Bin 380897 -> 380947 bytes src/sections/main-results.tex | 6 +++--- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/out/AStudyOnDynamicEquations.pdf 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zaVP`_$AXH*!NP>aVTkZBfw2K;$Tn$!c@>MpW6|KeU;h7(`>{B{5nv?-PzJp6!nDQ# zM1s2&2e=bl22kOTCJOKY3n=q%$qN1bNQ?*mM{EWt3ik{i2jCi}3`jxbUIGT*!~m3G z;0cVyD*)DjIFJlV6m|&#gcGhHpqiA$GP->boQZ?hOok%?Acu&T10egGT>gfeKt_U- zRWK1F{RrM6|Iz&49tsxX;Ik%>0Y%~F09=O$l>jI)VjXy$M9RS6`}=tyhp+@NOClE% zk(Llh@QwoZ{`0#0Z}$Kk9?<09FWJ5W(4{{v{w2fU#SoSdk@wM~zz2NQS%m6AIOvntv K!m9de4F3mOkUnt$ diff --git a/src/sections/definitions-notations-and-conventions.tex b/src/sections/definitions-notations-and-conventions.tex index 6d7b3dc..96639bd 100644 --- a/src/sections/definitions-notations-and-conventions.tex +++ b/src/sections/definitions-notations-and-conventions.tex @@ -38,8 +38,7 @@ \end{align*} where $x^q \neq x, \; x\in\mathbb{R}, \; q\in\mathbb{R}$. - \item $\polynomialP{m}{b}{x}, \; x,b\in\mathbb{R}, \; m\in\mathbb{N}$ is $2m+1$-degree - polynomial in $x,b$ + \item $\polynomialP{m}{b}{x}$ is $2m+1$-degree polynomial in $x,b$ \begin{equation} \polynomialP{m}{b}{x} = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \label{eq:polynomial_p} diff --git a/src/sections/time-scale-z.tex b/src/sections/time-scale-z.tex index 446368b..3535e02 100644 --- a/src/sections/time-scale-z.tex +++ b/src/sections/time-scale-z.tex @@ -1,52 +1,52 @@ \begin{cor} \label{finite_difference_case} (Finite difference.) - Let be a two-dimensional time scale - $\Lambda^2 = \mathbb{Z} \times \mathbb{Z} \colonequals \{t=(x, b) \colon \; x\in \mathbb{Z}, \; b\in \mathbb{Z} \}$. - For every $t\in\mathbb{Z}, \; x,b\in \Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ + Let be a two-dimensional timescale + $\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$. + For every $t\in\mathbb{Z}$ and $x,b\in \Lambda^2$ \[ \finiteDifference{t^{2m+1}} - = \pTsDerivative{\polynomialP{m}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{m}{b}{x}}{b}(t, t) \bigg |_{x = t, \; b = t}, + = \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) + + \pTsDerivative{\polynomialP{m}{b}{x}}{b}(m, \sigma(t), t) \] - where the forward jump operator $\sigma(t)$ is defined as $\sigma(t) = t+1$. + where $\sigma(t)$ is the forward jump operator defined as $\sigma(t) = t+1$. \end{cor} \begin{examp} \label{time_scale_z_example_1} - Let be $t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ and let $m=1$, then + Let be $t \in \mathbb{Z}, \; x,b \in \Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ and let $m=1$, then \begin{align*} - &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} = 3 t + 3 t^2\\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b}= 1 - 6 b^2 + 6 b x\\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b}\bigg |_{x = t, \; b = t}= 1 + &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ + &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (t, \sigma(t)) = 3 t + 3 t^2 \\ + &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 1 - 6 b^2 + 6 b x \\ + &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(t,t) = 1 + \end{align*} + Summing up previously obtained partial timescale derivatives, we get ordinary finite difference of odd-powered polynomial + $x^{3}$ evaluated in point $ t\in\mathbb{Z}, \; x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$ + \begin{align*} + \finiteDifference{x^{3}}(t) + = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (t, \sigma(t)) + + \pTsDerivative{\polynomialP{1}{b}{x}}{b}(t,t) + = 3 t + 3 t^2 + 1 \end{align*} - Summing up previously obtained partial time-scale derivatives, we get the ordinary finite difference of odd polynomial - $t^{2m+1}, \; t\in\mathbb{Z}, \; x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ - \[ - \finiteDifference{t^{3}} - = \pTsDerivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} - = 3 t^2 + 3 t + 1. - \] \end{examp} \begin{examp} \label{time_scale_z_example_2} Let be $t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ and let $m=2$, then \begin{align*} - &\pTsDerivative{\polynomialP{2}{b}{x}}{x} = 5 b - 30 b^2 + 40 b^3 - 15 b^4 + 10 b x - 30 b^2 x + 20 b^3 x \\ - &\pTsDerivative{\polynomialP{2}{b}{x}}{b} = 1 + 30 b^4 - 60 b^3 x + 30 b^2 x^2 \\ - &\pTsDerivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ - &\pTsDerivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} = 1 + &\pTsDerivative{\polynomialP{2}{b}{x}}{x} = 5 b - 30 b^2 + 40 b^3 - 15 b^4 + 10 b x - 30 b^2 x + 20 b^3 x \\ + &\pTsDerivative{\polynomialP{2}{b}{x}}{b} = 1 + 30 b^4 - 60 b^3 x + 30 b^2 x^2 \\ + &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (t, \sigma(t)) = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ + &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (t, t) = 1 \end{align*} Summing up previously obtained partial time-scale derivatives, we get time ordinary finite difference of odd polynomial $t^{2m+1}, \; t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ - \[ - \finiteDifference{t^{5}} - = \pTsDerivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} - = 1 + 5 t + 10 t^2 + 10 t^3 + 5 t^4. - \] + \begin{align*} + \finiteDifference{x^{5}} (t) + = \pTsDerivative{\polynomialP{2}{b}{x}}{x} (t, \sigma(t)) + + \pTsDerivative{\polynomialP{2}{b}{x}}{b} (t, t) + = 1 + 5 t + 10 t^2 + 10 t^3 + 5 t^4 + \end{align*} \end{examp} \begin{cor} \label{time_scale_z_corollary_1} @@ -62,4 +62,4 @@ \[ \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t} = 1 \] -\end{cor} \ No newline at end of file +\end{cor} From 82c71358da4ae4e3890f5f8e562787c456c22edd Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 21:08:07 +0200 Subject: [PATCH 13/22] work in progress --- out/AStudyOnDynamicEquations.pdf | Bin 380767 -> 380766 bytes src/AStudyOnDynamicEquations.tex | 2 ++ 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zS?C%|Fl7YQd)0#*dE-#SxbPz^B;@iF(MBYeWbIQxlR7?_S?Ib;rrc8&=Db-NZ)sICuK&*CU09qq3RX5thgT3}eDTwC;UanC wVTp$a>B2=%n5~x%+oe|gVJinz6#e3hr=S17A>+c&)2T=EiL+-fy>fWsU(mXfaR2}S diff --git a/src/AStudyOnDynamicEquations.tex b/src/AStudyOnDynamicEquations.tex index 32f96e3..8d2ea50 100644 --- a/src/AStudyOnDynamicEquations.tex +++ b/src/AStudyOnDynamicEquations.tex @@ -139,6 +139,8 @@ \bibliography{AStudyOnDynamicEquations} \noindent \textbf{Version:} \input{sections/version} + \clearpage + \section{Addendum 1: Mathematica scripts} \label{sec:mathematica_scripts} \input{sections/mathematica-scripts} diff --git a/src/sections/time-scale-z.tex b/src/sections/time-scale-z.tex index 3535e02..a9d4aa2 100644 --- a/src/sections/time-scale-z.tex +++ b/src/sections/time-scale-z.tex @@ -4,11 +4,11 @@ Let be a two-dimensional timescale $\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$. For every $t\in\mathbb{Z}$ and $x,b\in \Lambda^2$ - \[ - \finiteDifference{t^{2m+1}} + \begin{align*} + \finiteDifference{x^{2m+1}} (t) = \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) + \pTsDerivative{\polynomialP{m}{b}{x}}{b}(m, \sigma(t), t) - \] + \end{align*} where $\sigma(t)$ is the forward jump operator defined as $\sigma(t) = t+1$. \end{cor} \begin{examp} @@ -16,8 +16,15 @@ Let be $t \in \mathbb{Z}, \; x,b \in \Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ and let $m=1$, then \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ + &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 1 - 6 b^2 + 6 b x +% &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (t, \sigma(t)) = 3 t + 3 t^2 \\ +% &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(t,t) = 1 + \end{align*} + Evaluating in points yields + \begin{align*} +% &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ +% &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 1 - 6 b^2 + 6 b x \\ &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (t, \sigma(t)) = 3 t + 3 t^2 \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 1 - 6 b^2 + 6 b x \\ &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(t,t) = 1 \end{align*} Summing up previously obtained partial timescale derivatives, we get ordinary finite difference of odd-powered polynomial @@ -35,12 +42,17 @@ and let $m=2$, then \begin{align*} &\pTsDerivative{\polynomialP{2}{b}{x}}{x} = 5 b - 30 b^2 + 40 b^3 - 15 b^4 + 10 b x - 30 b^2 x + 20 b^3 x \\ - &\pTsDerivative{\polynomialP{2}{b}{x}}{b} = 1 + 30 b^4 - 60 b^3 x + 30 b^2 x^2 \\ + &\pTsDerivative{\polynomialP{2}{b}{x}}{b} = 1 + 30 b^4 - 60 b^3 x + 30 b^2 x^2 +% &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (t, \sigma(t)) = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ +% &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (t, t) = 1 + \end{align*} + Evaluating in points yields + \begin{align*} &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (t, \sigma(t)) = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (t, t) = 1 \end{align*} - Summing up previously obtained partial time-scale derivatives, we get time ordinary finite difference of odd polynomial - $t^{2m+1}, \; t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ + Summing up previously obtained partial time-scale derivatives, we get time ordinary finite difference of odd-powered + polynomial $x^{2m+1}$ and $t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ \begin{align*} \finiteDifference{x^{5}} (t) = \pTsDerivative{\polynomialP{2}{b}{x}}{x} (t, \sigma(t)) From 05fec59c4db73182579bc5ec6936cbcc493c0194 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 21:25:39 +0200 Subject: [PATCH 14/22] work in progress --- out/AStudyOnDynamicEquations.pdf | Bin 380766 -> 380718 bytes src/sections/time-scale-z.tex | 20 ++++++++++---------- 2 files changed, 10 insertions(+), 10 deletions(-) diff --git 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x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$ \begin{align*} \finiteDifference{x^{3}}(t) - = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (t, \sigma(t)) + = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (\sigma(t), t) + \pTsDerivative{\polynomialP{1}{b}{x}}{b}(t,t) = 3 t + 3 t^2 + 1 \end{align*} @@ -48,10 +48,10 @@ \end{align*} Evaluating in points yields \begin{align*} - &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (t, \sigma(t)) = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ + &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (\sigma(t), t) = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (t, t) = 1 \end{align*} - Summing up previously obtained partial time-scale derivatives, we get time ordinary finite difference of odd-powered + Summing up previously obtained partial timescale derivatives, we get time ordinary finite difference of odd-powered polynomial $x^{2m+1}$ and $t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ \begin{align*} \finiteDifference{x^{5}} (t) @@ -63,15 +63,15 @@ \begin{cor} \label{time_scale_z_corollary_1} For every $t\in\mathbb{Z}, \; x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ - \[ - \pTsDerivative{\polynomialP{m}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} + \begin{align*} + \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) = \sum_{r=1}^{2m} \binom{2m+1}{r} t^{r} - \] + \end{align*} \end{cor} \begin{cor} \label{time_scale_z_corollary_2} For every $t\in\mathbb{Z}, \; x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ - \[ - \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t} = 1 - \] + \begin{align*} + \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = 1 + \end{align*} \end{cor} From 3c3e1e5c0c8089bdf02fb74c970df2f8775f0ce6 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 21:35:04 +0200 Subject: [PATCH 15/22] bibliography update --- out/AStudyOnDynamicEquations.bbl | 71 +++++++++++------------ out/AStudyOnDynamicEquations.pdf | Bin 380718 -> 379631 bytes src/AStudyOnDynamicEquations.bib | 94 ++++++++++++------------------- 3 files changed, 68 insertions(+), 97 deletions(-) diff --git a/out/AStudyOnDynamicEquations.bbl b/out/AStudyOnDynamicEquations.bbl index 0f6d3fe..201906b 100644 --- a/out/AStudyOnDynamicEquations.bbl +++ b/out/AStudyOnDynamicEquations.bbl @@ -1,22 +1,22 @@ \begin{thebibliography}{10} \bibitem{Bohner2001DynamicEO} -Martin Bohner and Allan Peterson. +{Bohner, Martin and Peterson, Allan}. \newblock {\em {Dynamic equations on time scales: An introduction with applications}}. -\newblock Springer Science \& Business Media, 2001. +\newblock {Springer Science \& Business Media}, 2001. \newblock \url {https://web.mst.edu/~bohner/sample.pdf}. \bibitem{bohner2004partial} -Martin Bohner and Gusein~Sh Guseinov. -\newblock Partial differentiation on time scales. -\newblock {\em Dynamic systems and applications}, 13(3-4):351--379, 2004. +{Bohner, Martin and Guseinov, Gusein Sh.} +\newblock {Partial differentiation on time scales}. +\newblock {\em {Dynamic systems and applications}}, 13(3-4):351--379, 2004. \newblock \url {http://web.mst.edu/~bohner/papers/pdots.pdf}. \bibitem{ahlbrandt2002partial} -Calvin~D Ahlbrandt and Christina Morian. +Calvin~D. Ahlbrandt and Christina Morian. \newblock Partial differential equations on time scales. -\newblock {\em Journal of Computational and Applied Mathematics}, +\newblock {\em {Journal of Computational and Applied Mathematics}}, 141(1-2):35--55, 2002. \newblock \url {https://doi.org/10.1016/S0377-0427(01)00434-4}. @@ -28,71 +28,66 @@ B.~Jackson. \newblock \url {https://doi.org/10.1016/j.cam.2005.02.011}. \bibitem{jackson_1909} -Frederick~H Jackson. -\newblock Xi.—on q-functions and a certain difference operator. -\newblock {\em Earth and Environmental Science Transactions of the Royal - Society of Edinburgh}, 46(2):253--281, 1909. +{Jackson, Frederick H.} +\newblock {XI.—on q-functions and a certain difference operator}. +\newblock {\em {Earth and Environmental Science Transactions of the Royal + Society of Edinburgh}}, 46(2):253--281, 1909. \bibitem{ernst2000history} Thomas Ernst. \newblock {\em The history of q-calculus and a new method}. -\newblock Citeseer, 2000. +\newblock {Citeseer}, 2000. \bibitem{ernst2008different} Thomas Ernst. -\newblock The different tongues of q-calculus. -\newblock {\em Proceedings of the Estonian Academy of Sciences}, 57(2), 2008. +\newblock {The different tongues of q-calculus}. +\newblock {\em {Proceedings of the Estonian Academy of Sciences}}, 57(2), 2008. \bibitem{kac2001quantum} Victor Kac and Pokman Cheung. \newblock {\em Quantum calculus}. -\newblock Springer Science \& Business Media, 2001. +\newblock {Springer Science \& Business Media}, 2001. \bibitem{aldwoah2011power} -Khaled~A Aldwoah, Agnieszka~B Malinowska, and Delfim~FM Torres. +{Aldwoah, Khaled A. and Malinowska, Agnieszka B. and Torres, Delfim F.M.} \newblock The power quantum calculus and variational problems. -\newblock {\em arXiv preprint arXiv:1107.0344}, 2011. +\newblock {\em {arXiv preprint arXiv:1107.0344}}, 2011. \newblock \url {https://arxiv.org/abs/1107.0344}. \bibitem{kolosov2016link} -Petro Kolosov. +{Kolosov, Petro}. \newblock {On the link between Binomial Theorem and Discrete Convolution of Polynomials}. -\newblock {\em arXiv preprint arXiv:1603.02468}, 2016. +\newblock {\em {arXiv preprint arXiv:1603.02468}}, 2016. \newblock \url {https://arxiv.org/abs/1603.02468}. -\bibitem{WeissteinBernoulli} -Eric~W Weisstein. -\newblock {"Bernoulli Number." From MathWorld -- A Wolfram Web Resource.} -\newblock \url {http://mathworld.wolfram.com/BernoulliNumber.html}. - \bibitem{hilger1988ein} -S~Hilger. +S.~Hilger. \newblock {\em {E {\i} n Ma {\ss} kettenkalk {\"u} l m {\i} t Anwendung auf Zentrumsmann {\i} gfalt {\i} gke {\i} ten}}. -\newblock PhD thesis, Ph. D. Thesis, Universt {\"a} t W {\"u} rzburg, 1988. +\newblock PhD thesis, {Ph. D. Thesis, Universt {\"a} t W {\"u} rzburg}, 1988. \bibitem{bayour2017truly} -Benaoumeur Bayour, Ahmed Hammoudi, and Delfim~FM Torres. -\newblock A truly conformable calculus on time scales. -\newblock {\em arXiv preprint arXiv:1705.08928}, 2017. +{Bayour, Benaoumeur and Hammoudi, Ahmed and Torres, Delfim F.M.} +\newblock {A truly conformable calculus on time scales}. +\newblock {\em {arXiv preprint arXiv:1705.08928}}, 2017. \newblock \url {https://arxiv.org/abs/1705.08928}. \bibitem{benkhettou2016conformable} -Nadia Benkhettou, Salima Hassani, and Delfim~FM Torres. +{Benkhettou, Nadia and Hassani, Salima and Torres, Delfim F.M.} \newblock A conformable fractional calculus on arbitrary time scales. -\newblock {\em Journal of King Saud University-Science}, 28(1):93--98, 2016. +\newblock {\em {Journal of King Saud University-Science}}, 28(1):93--98, 2016. \bibitem{caputo2009time} -M~Cristina Caputo. -\newblock Time scales: from nabla calculus to delta calculus and vice versa via - duality. -\newblock {\em arXiv preprint arXiv:0910.0085}, 2009. +{Caputo, M. Cristina}. +\newblock {Time scales: from nabla calculus to delta calculus and vice versa + via duality}. +\newblock {\em {arXiv preprint arXiv:0910.0085}}, 2009. \bibitem{martins2009calculus} -Nat {\'a}~lia Martins and Delfim~FM Torres. -\newblock Calculus of variations on time scales with nabla derivatives. -\newblock {\em Nonlinear Analysis: Theory, Methods \& Applications}, +{Martins, Natalia and Torres, Delfim F.M.} +\newblock {Calculus of variations on time scales with nabla derivatives}. +\newblock {\em {Nonlinear Analysis: Theory, Methods \& Applications}}, 71(12):e763--e773, 2009. \bibitem{kolosov2022mathematica} diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index f978ed65fd823ac2e6881afff2316ee60dde8eb5..0457748656d2a091d9f6fa1db14286def19dcd73 100644 GIT binary patch delta 56301 zcmZU)WmFtZyEV+t#F1&81SC%F6I5}W}BCwK@XKyY^m5+t}oaCf(l-1m9D zb)NJ7>FKKKt7`9E*Q#Azz2BOi+kG%%lI1WZDt*z58;Zo=^_@Q4WvpbCmt2P#E;E(t|%skbmu{dx&ww 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F.M.}}, + journal = {{arXiv preprint arXiv:1107.0344}}, note = {\url {https://arxiv.org/abs/1107.0344}}, year = {2011} } @@ -14,31 +14,31 @@ @article{aldwoah2011power @book{Bohner2001DynamicEO, title = {{Dynamic equations on time scales: An introduction with applications}}, - author = {Bohner, Martin and Peterson, Allan}, + author = {{Bohner, Martin and Peterson, Allan}}, year = {2001}, - publisher = {Springer Science \& Business Media}, + publisher = {{Springer Science \& Business Media}}, note = {\url {https://web.mst.edu/~bohner/sample.pdf}} } % jackson q-derivative @article{jackson_1909, - title = {XI.—on q-functions and a certain difference operator}, - author = {Jackson, Frederick H}, - journal = {Earth and Environmental Science Transactions of the Royal Society of Edinburgh}, + title = {{XI.—on q-functions and a certain difference operator}}, + author = {{Jackson, Frederick H.}}, + journal = {{Earth and Environmental Science Transactions of the Royal Society of Edinburgh}}, volume = {46}, number = {2}, pages = {253--281}, year = {1909}, - publisher = {Royal Society of Edinburgh Scotland Foundation} + publisher = {{Royal Society of Edinburgh Scotland Foundation}} } % my work @article{kolosov2016link, title = {{On the link between Binomial Theorem and Discrete Convolution of Polynomials}}, - author = {Kolosov, Petro}, - journal = {arXiv preprint arXiv:1603.02468}, + author = {{Kolosov, Petro}}, + journal = {{arXiv preprint arXiv:1603.02468}}, note = {\url {https://arxiv.org/abs/1603.02468}}, year = {2016} } @@ -47,18 +47,18 @@ @article{kolosov2016link @phdthesis{hilger1988ein, title = {{E {\i} n Ma {\ss} kettenkalk {\"u} l m {\i} t Anwendung auf Zentrumsmann {\i} gfalt {\i} gke {\i} ten}}, - author = {Hilger, S}, + author = {Hilger, S.}, year = {1988}, - school = {Ph. D. Thesis, Universt {\"a} t W {\"u} rzburg} + school = {{Ph. D. Thesis, Universt {\"a} t W {\"u} rzburg}} } % partial time scale derivatives % http://web.mst.edu/~bohner/papers/pdots.pdf @article{bohner2004partial, - title = {Partial differentiation on time scales}, - author = {Bohner, Martin and Guseinov, Gusein Sh}, - journal = {Dynamic systems and applications}, + title = {{Partial differentiation on time scales}}, + author = {{Bohner, Martin and Guseinov, Gusein Sh.}}, + journal = {{Dynamic systems and applications}}, volume = {13}, number = {3-4}, pages = {351--379}, @@ -68,8 +68,8 @@ @article{bohner2004partial @article{ahlbrandt2002partial, title = {Partial differential equations on time scales}, - author = {Ahlbrandt, Calvin D and Morian, Christina}, - journal = {Journal of Computational and Applied Mathematics}, + author = {Ahlbrandt, Calvin D. and Morian, Christina}, + journal = {{Journal of Computational and Applied Mathematics}}, volume = {141}, number = {1-2}, pages = {35--55}, @@ -89,22 +89,10 @@ @article{JACKSON2006391 doi = "https://doi.org/10.1016/j.cam.2005.02.011", url = "http://www.sciencedirect.com/science/article/pii/S0377042705001019", author = "B. Jackson", - note = {\url {https://doi.org/10.1016/j.cam.2005.02.011}}, - keywords = "Time scale, Partial dynamic equation, Laplace transform, Multivariable calculus, Partial derivative, Iterated integral", - abstract = "In this work, we generalize existing ideas of the univariate case of the time scales calculus to the bivariate case. Formal definitions of partial derivatives and iterated integrals are offered, and bivariate partial differential operators are examined. In particular, solutions of the homogeneous and nonhomogeneous heat and wave operators are found when initial distributions given are in terms of elementary functions by means of the generalized Laplace Transform for the time scale setting. Finally, the so-termed mixed time scale setting is discussed. Examples are given and solutions are provided in tabular form." + note = {\url {https://doi.org/10.1016/j.cam.2005.02.011}} } -% bernoulli numbers - -@MISC{WeissteinBernoulli, - author = {Weisstein, Eric W}, - title = {{"Bernoulli Number." From MathWorld -- A Wolfram Web Resource.}}, - howpublished = {\url {http://mathworld.wolfram.com/BernoulliNumber.html}} -} - -% Repository reference - @article{kolosov2022mathematica, author = {Petro Kolosov}, title = {{Supplementary Mathematica Programs}}, @@ -112,66 +100,56 @@ @article{kolosov2022mathematica year = {2020} } -% cited of me, inser in order to replace convolution - @article{bayour2017truly, - title = {A truly conformable calculus on time scales}, - author = {Bayour, Benaoumeur and Hammoudi, Ahmed and Torres, Delfim FM}, - journal = {arXiv preprint arXiv:1705.08928}, + title = {{A truly conformable calculus on time scales}}, + author = {{Bayour, Benaoumeur and Hammoudi, Ahmed and Torres, Delfim F.M.}}, + journal = {{arXiv preprint arXiv:1705.08928}}, note = {\url {https://arxiv.org/abs/1705.08928}}, year = {2017} } -% reference to extend ref counts - @article{benkhettou2016conformable, title = {A conformable fractional calculus on arbitrary time scales}, - author = {Benkhettou, Nadia and Hassani, Salima and Torres, Delfim FM}, - journal = {Journal of King Saud University-Science}, + author = {{Benkhettou, Nadia and Hassani, Salima and Torres, Delfim F.M.}}, + journal = {{Journal of King Saud University-Science}}, volume = {28}, number = {1}, pages = {93--98}, year = {2016}, - publisher = {Elsevier} + publisher = {{Elsevier}} } -% reference to extend ref counts - @article{caputo2009time, - title = {Time scales: from nabla calculus to delta calculus and vice versa via duality}, - author = {Caputo, M Cristina}, - journal = {arXiv preprint arXiv:0910.0085}, + title = {{Time scales: from nabla calculus to delta calculus and vice versa via duality}}, + author = {{Caputo, M. Cristina}}, + journal = {{arXiv preprint arXiv:0910.0085}}, year = {2009} } -% reference to extend ref counts - @article{martins2009calculus, - title = {Calculus of variations on time scales with nabla derivatives}, - author = {Martins, Nat {\'a} lia and Torres, Delfim FM}, - journal = {Nonlinear Analysis: Theory, Methods \& Applications}, + title = {{Calculus of variations on time scales with nabla derivatives}}, + author = {{Martins, Natalia and Torres, Delfim F.M.}}, + journal = {{Nonlinear Analysis: Theory, Methods \& Applications}}, volume = {71}, number = {12}, pages = {e763--e773}, year = {2009}, - publisher = {Elsevier} + publisher = {{Elsevier}} } -% reference to extend ref counts - @book{ernst2000history, title = {The history of q-calculus and a new method}, author = {Ernst, Thomas}, year = {2000}, - publisher = {Citeseer} + publisher = {{Citeseer}} } % q-calculus reference @article{ernst2008different, - title = {The different tongues of q-calculus.}, + title = {{The different tongues of q-calculus}}, author = {Ernst, Thomas}, - journal = {Proceedings of the Estonian Academy of Sciences}, + journal = {{Proceedings of the Estonian Academy of Sciences}}, volume = {57}, number = {2}, year = {2008} @@ -183,7 +161,5 @@ @book{kac2001quantum title = {Quantum calculus}, author = {Kac, Victor and Cheung, Pokman}, year = {2001}, - publisher = {Springer Science \& Business Media} + publisher = {{Springer Science \& Business Media}} } - -% Macaulay method \ No newline at end of file From 473efdbd2ba8e3f8885bb08839e01a158919218d Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Mon, 16 Sep 2024 21:36:43 +0200 Subject: [PATCH 16/22] update bibliography --- out/AStudyOnDynamicEquations.bbl | 3 +-- out/AStudyOnDynamicEquations.pdf | Bin 379631 -> 379641 bytes src/AStudyOnDynamicEquations.bib | 2 +- 3 files changed, 2 insertions(+), 3 deletions(-) diff --git a/out/AStudyOnDynamicEquations.bbl b/out/AStudyOnDynamicEquations.bbl index 201906b..c2023df 100644 --- a/out/AStudyOnDynamicEquations.bbl +++ b/out/AStudyOnDynamicEquations.bbl @@ -56,8 +56,7 @@ Victor Kac and Pokman Cheung. \bibitem{kolosov2016link} {Kolosov, Petro}. -\newblock {On the link between Binomial Theorem and Discrete Convolution of - Polynomials}. +\newblock {On the link between binomial theorem and discrete convolution}. \newblock {\em {arXiv preprint arXiv:1603.02468}}, 2016. \newblock \url {https://arxiv.org/abs/1603.02468}. diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index 0457748656d2a091d9f6fa1db14286def19dcd73..7b85c82ebf48e8cb9d549f806d613b0078e2c178 100644 GIT binary patch delta 5470 zcmai02~>{j7ViJ`Nrg%(xtu0VRK9uWrZgJkkdBfloHA83$cg&JiIj+=K1V`^GBl|E ziY66Fj;TTpM}tTlDVjKv`+n|S=U(q;t=n3yxAwpPXTST|&))CuO49j|q;tWfNy>4& zuN7h@%kr2L(ngOdzms#<+}YhzT_WB3-eg~A>h2kZn&$KJj!y7VPWEVbY`wDgmn%bk zoi2_CW)W^@v&v%i?~dwg`%~vro!_&4Lw&95Kln!IA}0?S`0MI z&GM!D-jw~m+3@r`?ca{wt}*P7I^CGEE~07g%ICMDCdg{iiUw^yI;1#lzSYtBRH;8% 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of Polynomials}}, + title = {{On the link between binomial theorem and discrete convolution}}, author = {{Kolosov, Petro}}, journal = {{arXiv preprint arXiv:1603.02468}}, note = {\url {https://arxiv.org/abs/1603.02468}}, From 610a561b470afcd83b4500540ae9b0251f57b4d5 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Tue, 17 Sep 2024 15:41:12 +0200 Subject: [PATCH 17/22] update notation --- out/AStudyOnDynamicEquations.pdf | Bin 379641 -> 378738 bytes src/AStudyOnDynamicEquations.tex | 1 + .../conclusion-and-future-research.tex | 4 +- src/sections/proof-of-main-theorem.tex | 18 ++--- src/sections/time-scale-nq.tex | 48 +++++++------ .../time-scale-pure-quantum-power.tex | 62 +++++++++-------- src/sections/time-scale-qn.tex | 62 +++++++++-------- src/sections/time-scale-r.tex | 63 +++++++++--------- 8 files changed, 138 insertions(+), 120 deletions(-) diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index 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+\usepackage{derivative} \newcommand \anglePower [2]{\langle #1 \rangle \sp{#2}} \newcommand \bernoulli [2][B] {{#1}\sb{#2}} diff --git a/src/sections/conclusion-and-future-research.tex b/src/sections/conclusion-and-future-research.tex index d764c9c..ad9aeef 100644 --- a/src/sections/conclusion-and-future-research.tex +++ b/src/sections/conclusion-and-future-research.tex @@ -5,10 +5,10 @@ \[ \pTsDerivative{\polynomialP{m}{b}{x}}{x} + \pTsDerivative{\polynomialP{m}{b}{x}}{b} - = \alpha_m(x,b) (x^{2m+1})^{\Delta}, + = \alpha_m(x,b) (x^{2m+1})^{\Delta} \] 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 $\mathbb{R}^q \times \mathbb{R}^q$. \ No newline at end of file +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 index 695e5c7..ef1b5ce 100644 --- a/src/sections/proof-of-main-theorem.tex +++ b/src/sections/proof-of-main-theorem.tex @@ -1,10 +1,10 @@ -By [Lemma 3.1~\cite{kolosov2016link}], for every $x\in\mathbb{R}, \; m\in\mathbb{N}$ it is true that +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}} +\subsection*{Proof of theorem~\ref{main_theorem}} Let be $x,b\in \Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 \colonequals \{t = (x,b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2\}$. @@ -15,7 +15,7 @@ \subsection*{Proof of theorem ~\ref{main_theorem}} (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}, + \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 $x$ @@ -53,18 +53,18 @@ \subsection*{Proof of theorem ~\ref{main_theorem}} \end{align*} We can collapse the terms $(\sigma(x) - x), \; (\sigma(b) - b)$ in above expressions, as $b\to x$. Therefore, -\[ +\begin{align*} \frac{\polynomialP{m}{\sigma(x)}{\sigma(x)} - \polynomialP{m}{x}{x}}{\sigma(x) - x} - = \polynomialP{m}{b}{x}^{\Delta}_{x} (x, \sigma(x)) - + \polynomialP{m}{b}{x}^{\Delta}_{b} (x, x) -\] + = \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} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t} + ^{\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-nq.tex b/src/sections/time-scale-nq.tex index 01683fa..3d97a5c 100644 --- a/src/sections/time-scale-nq.tex +++ b/src/sections/time-scale-nq.tex @@ -4,11 +4,11 @@ Let be a two-dimensional time scale $\Lambda^2 = {\mathbb{R}}^{q} \times {\mathbb{R}}^{q} \colonequals \{t=(x,b) \colon \; b\in {\mathbb{R}}^{q}, \; x\in{\mathbb{R}}^{q} \}$. For every $t\in {\mathbb{R}}^{q}, \; x,b\in\Lambda^2 = {\mathbb{R}}^{q} \times {\mathbb{R}}^{q}, \; m\in\mathbb{N}$ - \[ + \begin{align*} \qpowerDerivative{t^{2m+1}} - = \pTsDerivative{\polynomialP{m}{b}{x}}{x}(t, \sigma(t)) - + \pTsDerivative{\polynomialP{m}{b}{x}}{b}(t, t), - \] + = \pTsDerivative{\polynomialP{m}{b}{x}}{x}(m, \sigma(t), t) + + \pTsDerivative{\polynomialP{m}{b}{x}}{b}(m, t, t) + \end{align*} where the forward jump operator is defined as $\sigma(t) = t^q, \; q > 1$. \end{cor} \begin{examp} @@ -17,18 +17,21 @@ and let $m=1$, then \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 3 b - 2 b^2 + 3 b^q - 2 b^{2 q} - 2 b^{1 + q} - 3 x + 3 b x + 3 b^q x \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} = -3 t^q + 3 t^{2 q} \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} = t^2 + 3 t^q - 2 t^{2 q} + t^{1 + q} + &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 3 b - 2 b^2 + 3 b^q - 2 b^{2 q} - 2 b^{1 + q} - 3 x + 3 b x + 3 b^q x + \end{align*} + Evaluating in points yields + \begin{align*} + &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (m, \sigma(t), t) = -3 t^q + 3 t^{2 q} \\ + &\pTsDerivative{\polynomialP{1}{b}{x}}{b} (m, t, t) = t^2 + 3 t^q - 2 t^{2 q} + t^{1 + q} \end{align*} Summing up previously obtained partial time-scale derivatives, we get $q-$power derivative of odd polynomial - $t^{2m+1}, \; t\in {\mathbb{R}}^{q}, \; x,b\in\Lambda^2 = {\mathbb{R}}^{q} \times {\mathbb{R}}^{q}, \; m\in\mathbb{N}$ - \[ + $x^{3}$ evaluated in point $t \in {\mathbb{R}}^{q}$ + \begin{align*} \qpowerDerivative{t^{3}} - = \pTsDerivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} + = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (1, \sigma(t), t) + + \pTsDerivative{\polynomialP{1}{b}{x}}{b} (1, t, t) = t^2 + t^{2 q} + t^{1 + q}. - \] + \end{align*} \end{examp} \begin{examp} \label{time_scale_nq_example_2} @@ -44,28 +47,29 @@ &- 15 b^{3 q} x + 30 b^{1 + q} x - 15 b^{2 + q} x - 15 b^{1 + 2 q} x + 5 x^2 - 15 b x^2 + 10 b^2 x^2 \\ &- 15 b^q x^2 + 10 b^{2 q} x^2 + 10 b^{1 + q} x^2 \end{align*} + Evaluation in points yields \begin{align*} - &\pTsDerivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} + &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (m, \sigma(t), t) = -10 t^{2 q} + 15 t^{3q} - 5 t^{4q} + 5 t^{1+q} - 15 t^{1+2q} + 10 t^{1 + 3 q} \\ - &\pTsDerivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} + &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (m, t, t) = t^4 + 10 t^{2 q} - 15 t^{3 q} + 6 t^{4 q} - 5 t^{1 + q} + t^{3 + q} + 15 t^{1 + 2 q} + t^{2 + 2 q} - 9 t^{1 + 3 q} \end{align*} Summing up previously obtained partial time-scale derivatives, we get $q-$power derivative of odd polynomial $t^{2m+1}, \; t\in {\mathbb{R}}^{q}, \; x,b\in\Lambda^2 = {\mathbb{R}}^{q} \times {\mathbb{R}}^{q}, \; m\in\mathbb{N}$ - \[ + \begin{align*} \qpowerDerivative{t^{5}} - = \pTsDerivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} + = \pTsDerivative{\polynomialP{2}{b}{x}}{x} (m, \sigma(t), t) + + \pTsDerivative{\polynomialP{2}{b}{x}}{b} (m, t, t) = t^4 + t^{4 q} + t^{3 + q} + t^{2 + 2 q} + t^{1 + 3 q}. - \] + \end{align*} \end{examp} Another polynomial identity, that is exponential sum holds \begin{cor} \label{time_scale_nq_corollary_1} For every $t\in {\mathbb{R}}^{q}, \; x,b\in\Lambda^2 = {\mathbb{R}}^{q} \times {\mathbb{R}}^{q}, \; t\in\mathbb{R}, \; m\in\mathbb{N}$ - \[ - \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t} + \begin{align*} + \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = \sum_{k=0}^{2m} t^k - \] -\end{cor} \ No newline at end of file + \end{align*} +\end{cor} diff --git a/src/sections/time-scale-pure-quantum-power.tex b/src/sections/time-scale-pure-quantum-power.tex index 41d8de1..142460c 100644 --- a/src/sections/time-scale-pure-quantum-power.tex +++ b/src/sections/time-scale-pure-quantum-power.tex @@ -1,8 +1,8 @@ In this subsection we discuss a pure quantum power time scale $q^{\mathbb{R}^j}$ provided by Aldwoah, Malinowska and Torres in~\cite{aldwoah2011power}, among with the $q-$power derivative operator $\nqderivative{f(t)}$ defined by -\[ +\begin{align*} \nqderivative{f(t)} = \frac{f(qt^n) - f(t)}{qt^n - t}, -\] +\end{align*} where $n$ is odd positive integer and $0 < q < 1$. \begin{cor} (Quantum power derivative~\cite{aldwoah2011power}.) @@ -11,9 +11,9 @@ \colonequals \{t=(x,b) \colon \; b \in q^{\mathbb{R}^j}, \; x \in q^{\mathbb{R}^j} \}$. For every $t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; m\in\mathbb{N}$ \[ - \nqderivative{t^{2m+1}} - = \pTsDerivative{\polynomialP{m}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t}, + \nqderivative{x^{2m+1}} (t) + = \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) + + \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) \] where $\sigma(t) = qt^n, \; \sigma(t) > t$. \end{cor} @@ -23,38 +23,43 @@ and let $m=1$, then \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 3 b - 2 b^2 + 3 b^j q - 2 b^{1 + j} q - 2 b^{2 j} q^2 - 3 x + 3 b x + 3 b^j q x \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{x}(t, \sigma(t)) = -3 q t^j + 3 q^2 t^{2 j} \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(t, t) = t^2 + 3 q t^j - 2 q^2 t^{2 j} + q t^{1 + j} + &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 3 b - 2 b^2 + 3 b^j q - 2 b^{1 + j} q - 2 b^{2 j} q^2 - 3 x + 3 b x + 3 b^j q x +% &\pTsDerivative{\polynomialP{1}{b}{x}}{x}(t, \sigma(t)) = -3 q t^j + 3 q^2 t^{2 j} \\ +% &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(t, t) = t^2 + 3 q t^j - 2 q^2 t^{2 j} + q t^{1 + j} \end{align*} - Summing up previously obtained partial time-scale derivatives, we get $q-$power derivative of odd polynomial + Evaluating in points yields + \begin{align*} + &\pTsDerivative{\polynomialP{1}{b}{x}}{x}(1, \sigma(t), t) = -3 q t^j + 3 q^2 t^{2 j} \\ + &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(1, t, t) = t^2 + 3 q t^j - 2 q^2 t^{2 j} + q t^{1 + j} + \end{align*} + Summing up previously obtained partial timescale derivatives, we get $q-$power derivative of odd polynomial $t^{2m+1}, \; t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; m\in\mathbb{N}$ - \[ + \begin{align*} \nqderivative{t^{3}} - = \pTsDerivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} + = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (1, \sigma(t), t) + + \pTsDerivative{\polynomialP{1}{b}{x}}{b} (1, t, t) = t^2 + q^2 t^{2 j} + q t^{1 + j}. - \] + \end{align*} \end{examp} Another polynomial identity, that is exponential sum holds \begin{cor} \label{time_scale_pure_quantum_power_corollary_1} For every $t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; t\in\mathbb{R}, \; m\in\mathbb{N}$ - \[ - \lim \limits_{j \to 0} \lim \limits_{q \to 1} \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t} + \begin{align*} + \lim \limits_{j \to 0} \lim \limits_{q \to 1} \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = \sum_{k=0}^{2m} t^k - \] + \end{align*} \end{cor} An identity in even polynomials holds too \begin{cor} \label{time_scale_pure_quantum_power_corollary_2} For every $t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; t\in\mathbb{R}, \; m\in\mathbb{N}$ - \[ - \lim \limits_{j \to 0} \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t} + \begin{align*} + \lim \limits_{j \to 0} \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = t^{2m} - \] + \end{align*} \end{cor} \begin{examp} @@ -69,19 +74,20 @@ &+ 30 b^{1 + j} q x - 15 b^{2 + j} q x + 30 b^{2 j} q^2 x - 15 b^{1 + 2 j} q^2 x - 15 b^{3 j} q^3 x + 5 x^2 - 15 b x^2 \\ &+ 10 b^2 x^2 - 15 b^j q x^2 + 10 b^{1 + j} q x^2 + 10 b^{2 j} q^2 x^2 \end{align*} + Evaluation in points yields \begin{align*} - &\pTsDerivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} + &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (2, \sigma(t), t) = -10 q^2 t^{2 j} + 15 q^3 t^{3 j} - 5 q^4 t^{4 j} + 5 q t^{1 + j} - 15 q^2 t^{1 + 2 j} + 10 q^3 t^{1 + 3 j} \\ - &\pTsDerivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} + &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (2, t, t) = t^4 + 10 q^2 t^{2 j} - 15 q^3 t^{3 j} + 6 q^4 t^{4 j} - 5 q t^{1 + j} + q t^{3 + j} + 15 q^2 t^{1 + 2 j} + q^2 t^{2 + 2 j} - 9 q^3 t^{1 + 3 j} \end{align*} - Summing up previously obtained partial time-scale derivatives, we $q-$power derivative of odd polynomial + Summing up previously obtained partial timescale derivatives, we $q-$power derivative of odd polynomial $t^{2m+1}, \; t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; m\in\mathbb{N}$ - \[ + \begin{align*} \nqderivative{t^{5}} - = \pTsDerivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} - = t^4 + q^4 t^{4 j} + q t^{3 + j} + q^2 t^{2 + 2 j} + q^3 t^{1 + 3 j}. - \] -\end{examp} \ No newline at end of file + = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (2, \sigma(t), t) + + \pTsDerivative{\polynomialP{1}{b}{x}}{b} (2, t, t) + = t^4 + q^4 t^{4 j} + q t^{3 + j} + q^2 t^{2 + 2 j} + q^3 t^{1 + 3 j} + \end{align*} +\end{examp} diff --git a/src/sections/time-scale-qn.tex b/src/sections/time-scale-qn.tex index 3a08a98..df858c4 100644 --- a/src/sections/time-scale-qn.tex +++ b/src/sections/time-scale-qn.tex @@ -3,12 +3,12 @@ (Q-derivative~\cite{jackson_1909}.) Let be a two-dimensional time scale $\Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}} \colonequals \{t=(x,b) \colon \; x\in q^{\mathbb{R}}, \; b \in q^{\mathbb{R}} \}$. - For every $t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}, \; m\in\mathbb{N}$ - \[ - \qderivative{t^{2m+1}} - = \pTsDerivative{\polynomialP{m}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t}, - \] + For every $t\in q^{\mathbb{R}}$ and $x,b\in \Lambda^2$ + \begin{align*} + \qderivative{x^{2m+1}}(x) + = \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) + + \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) + \end{align*} where $\sigma(t) = qt, \; q > 1$. \end{cor} \begin{examp} @@ -17,31 +17,34 @@ and let $m=1$, then \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 3 b - 2 b^2 + 3 b q - 2 b^2 q - 2 b^2 q^2 - 3 x + 3 b x + 3 b q x \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} = -3 q t + 3 q^2 t^2 \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} = 3 q t + t^2 + q t^2 - 2 q^2 t^2 + &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 3 b - 2 b^2 + 3 b q - 2 b^2 q - 2 b^2 q^2 - 3 x + 3 b x + 3 b q x \end{align*} - Summing up previously obtained partial time-scale derivatives, we get $q-$derivative of odd polynomial - $t^{2m+1}, \; t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}, \; m\in\mathbb{N}$ - \[ - \qderivative{t^{3}} - = \pTsDerivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} + Evaluating in points yields + \begin{align*} + &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (m, \sigma(t), t) = -3 q t + 3 q^2 t^2 \\ + &\pTsDerivative{\polynomialP{1}{b}{x}}{b} (m, t, t) = 3 q t + t^2 + q t^2 - 2 q^2 t^2 + \end{align*} + Summing up previously obtained partial time-scale derivatives, we get the $q$-derivative of odd polynomial + $x^{2m+1}$ evaluated in point $t\in q^{\mathbb{R}}$ + \begin{align*} + \qderivative{x^{3}}(x) + = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (m, \sigma(t), t) + + \pTsDerivative{\polynomialP{1}{b}{x}}{b} (m, t, t) = t^2 + q t^2 + q^2 t^2. - \] + \end{align*} \end{examp} For every $t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}$ the following polynomial identity holds as $q$ tends to zero -\[ - \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} = t^2 -\] +\begin{align*} + \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{1}{b}{x}}{b} (1, t, t) = t^2 +\end{align*} However, it would be generalized as follows \begin{cor} \label{time_scale_qn_corollary_1} For every $t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}, \; m\in\mathbb{N}$ - \[ - \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t} = t^{2m}. - \] + \begin{align*} + \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = t^{2m}. + \end{align*} \end{cor} \begin{examp} \label{time_scale_qn_example_2} @@ -56,18 +59,19 @@ - 15 b^3 q x + 30 b^2 q^2 x \\ &- 15 b^3 q^2 x - 15 b^3 q^3 x + 5 x^2 - 15 b x^2 + 10 b^2 x^2 - 15 b q x^2 + 10 b^2 q x^2 + 10 b^2 q^2 x^2 \end{align*} + Evaluating in points yields \begin{align*} - &\pTsDerivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} + &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (2, \sigma(t), t) = 5 q t^2 - 10 q^2 t^2 - 15 q^2 t^3 + 15 q^3 t^3 + 10 q^3 t^4 - 5 q^4 t^4 \\ - &\pTsDerivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} + &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (2, t, t) = -5 q t^2 + 10 q^2 t^2 + 15 q^2 t^3 - 15 q^3 t^3 + t^4 + q t^4 + q^2 t^4 - 9 q^3 t^4 + 6 q^4 t^4 \end{align*} Summing up previously obtained partial time-scale derivatives, we get the $q-$derivative of odd polynomial $t^{2m+1}, \; t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}, \; m\in\mathbb{N}$ - \[ + \begin{align*} \qderivative{t^{5}} - = \pTsDerivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pTsDerivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} + = \pTsDerivative{\polynomialP{2}{b}{x}}{x} (2, \sigma(t), t) + + \pTsDerivative{\polynomialP{2}{b}{x}}{b} (2, t, t) = t^4 + q t^4 + q^2 t^4 + q^3 t^4 + q^4 t^4. - \] -\end{examp} \ No newline at end of file + \end{align*} +\end{examp} diff --git a/src/sections/time-scale-r.tex b/src/sections/time-scale-r.tex index ab8d0d3..9a0af0d 100644 --- a/src/sections/time-scale-r.tex +++ b/src/sections/time-scale-r.tex @@ -3,50 +3,53 @@ (Classical derivative.) Let be a two-dimensional time scale $\Lambda^2 = \mathbb{R} \times \mathbb{R} \colonequals \{t=(x, b) \colon \; x\in \mathbb{R}, \; b\in\mathbb{R} \}$. - For every $t\in\mathbb{R}, \; x,b\in \Lambda^2 = \mathbb{R} \times \mathbb{R}, \; m\in\mathbb{N}$ - \[ - \derivative{t^{2m+1}}{t} - = \pderivative{\polynomialP{m}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pderivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t}, - \] + For every $t\in\mathbb{R}$ and $x,b\in \Lambda^2$ + \begin{align*} + \odv{x^{2m+1}}{x} (t) + = \pdv{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) + + \pdv{\polynomialP{m}{b}{x}}{b} (m, t, t) + \end{align*} where $\sigma(t) = t + \Delta t, \; \Delta t \to 0.$ \end{cor} \begin{examp} \label{time_scale_r_example_1} Let be $t\in\mathbb{R}, \; x,b\in \Lambda^2 = \mathbb{R} \times \mathbb{R}, \; m\in\mathbb{N}$ and let $m=1$, then \begin{align*} - \pderivative{\polynomialP{1}{b}{x}}{x} &= -3 b + 3 b^2 \\ - \pderivative{\polynomialP{1}{b}{x}}{b} &= 6 b - 6 b^2 - 3 x + 6 b x + \pdv{\polynomialP{1}{b}{x}}{x} &= -3 b + 3 b^2 \\ + \pdv{\polynomialP{1}{b}{x}}{b} &= 6 b - 6 b^2 - 3 x + 6 b x \end{align*} + Evaluating in points yields \begin{align*} - &\pderivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} = -3t + 3t^2 \\ - &\pderivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} = 3t + &\pdv{\polynomialP{1}{b}{x}}{x} (1, \sigma(t), t) = -3t + 3t^2 \\ + &\pdv{\polynomialP{1}{b}{x}}{b} (1, t, t) = 3t \end{align*} - Summing up previously obtained partial time-scale derivatives, we get classical derivative of odd polynomial - $t^{2m+1}, t\in\mathbb{R}, \; x\in \Lambda^2 = \mathbb{R} \times \mathbb{R}, \; m\in\mathbb{N}$ - \[ - \derivative{t^3}{t} - = \pderivative{\polynomialP{1}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pderivative{\polynomialP{1}{b}{x}}{b} \bigg |_{x = t, \; b = t} + Summing up previously obtained partial timescale derivatives, we get an ordinary derivative of odd polynomial + $x^{3}$ evaluated in point $t \in \mathbb{R}$. + \begin{align*} + \odv{x^3}{x} (t) + = \pdv{\polynomialP{1}{b}{x}}{x} (1, \sigma(t), t) + + \pdv{\polynomialP{1}{b}{x}}{b} (1, t, t) = 3t^2. - \] + \end{align*} \end{examp} \begin{examp} \label{time_scale_r_example_2} Let be $t\in\mathbb{R}, \; x,b\in \Lambda^2 = \mathbb{R} \times \mathbb{R}, \; m\in\mathbb{N}$ and let $m=2$, then \begin{align*} - &\pderivative{\polynomialP{2}{b}{x}}{x} = -15 b^2 + 30 b^3 - 15 b^4 + 10 b x - 30 b^2 x + 20 b^3 x, \\ - &\pderivative{\polynomialP{2}{b}{x}}{b} = 30 b^2 - 60 b^3 + 30 b^4 - 30 b x + 90 b^2 x - 60 b^3 x + 5 x^2 - 30 b x^2 - + 30 b^2 x^2 \\ - &\pderivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} = -5 t^2 + 5 t^4 \\ - &\pderivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} = 5 t^2 + &\pdv{\polynomialP{2}{b}{x}}{x} = -15 b^2 + 30 b^3 - 15 b^4 + 10 b x - 30 b^2 x + 20 b^3 x, \\ + &\pdv{\polynomialP{2}{b}{x}}{b} = 30 b^2 - 60 b^3 + 30 b^4 - 30 b x + 90 b^2 x - 60 b^3 x + 5 x^2 - 30 b x^2 + 30 b^2 x^2 \end{align*} - Summing up previously obtained partial time-scale derivatives, we get classical derivative of an odd polynomial - $t^{2m+1}, \; t\in\mathbb{R}, \; x\in \Lambda^2 = \mathbb{R} \times \mathbb{R}, \; m\in\mathbb{N}$ - \[ - \derivative{t^5}{t} - = \pderivative{\polynomialP{2}{b}{x}}{x} \bigg |_{x = t, \; b = \sigma(t)} - + \pderivative{\polynomialP{2}{b}{x}}{b} \bigg |_{x = t, \; b = t} + Evaluation in points yields + \begin{align*} + &\pdv{\polynomialP{2}{b}{x}}{x} (2, \sigma(t), t) = -5 t^2 + 5 t^4 \\ + &\pdv{\polynomialP{2}{b}{x}}{b} (2, \sigma(t), t) = 5 t^2 + \end{align*} + Summing up previously obtained partial timescale derivatives, we get classical derivative of an odd polynomial + $x^5$ evaluated in point $t\in\mathbb{R}$ + \begin{align*} + \odv{t^5}{t} + = \pdv{\polynomialP{2}{b}{x}}{x} (2, \sigma(t), t) + + \pdv{\polynomialP{2}{b}{x}}{b} (2, \sigma(t), t) = 5t^4. - \] -\end{examp} \ No newline at end of file + \end{align*} +\end{examp} From a8c510156754a9a5557adea71f79e49d0fd8aeca Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Tue, 17 Sep 2024 15:45:21 +0200 Subject: [PATCH 18/22] work in progress --- out/AStudyOnDynamicEquations.pdf | Bin 378738 -> 378738 bytes src/sections/proof-of-main-theorem.tex | 4 +++- 2 files changed, 3 insertions(+), 1 deletion(-) diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index 045f5f892289b8977292b57c49ff6d37bd813ff1..2abdab829456c659037024f66f421631ee9d7679 100644 GIT binary patch delta 133 zcmey=C;q8VyrG4$g{g(Pg=GtC>~bbU!|CzMSrviw_NL{m6XX+I&0S0!O-x~bapqv`R>Srviw_NL{m6XX-j%pIN04J-^DoeZ3v z%*>5k%q&eUos68^%nS_7%`FV=6l@48iB+((<0>vmEGnreN=@T3Ha9RcHRDoMb@g}S F0st0)BZL3| diff --git a/src/sections/proof-of-main-theorem.tex b/src/sections/proof-of-main-theorem.tex index ef1b5ce..dc7dab2 100644 --- a/src/sections/proof-of-main-theorem.tex +++ b/src/sections/proof-of-main-theorem.tex @@ -1,10 +1,12 @@ +%! 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}} +\subsection{Proof of theorem~\ref{main_theorem}} +\label{subsec:proof-of-theorem} Let be $x,b\in \Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 \colonequals \{t = (x,b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2\}$. From f3b962cabcfee21e3a98a60f0c21dcfbc6cb01fd Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Tue, 17 Sep 2024 15:49:16 +0200 Subject: [PATCH 19/22] work in progress --- src/sections/proof-of-main-theorem.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/sections/proof-of-main-theorem.tex b/src/sections/proof-of-main-theorem.tex index dc7dab2..b4c259b 100644 --- a/src/sections/proof-of-main-theorem.tex +++ b/src/sections/proof-of-main-theorem.tex @@ -11,7 +11,7 @@ \subsection{Proof of theorem~\ref{main_theorem}} $x,b\in \Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 \colonequals \{t = (x,b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2\}$. Let be $\mathbb{T}_1 = \mathbb{T}_2$. -Assume that time-scale derivative $(x^{2m+1})^{\Delta}$ is +Assume that timescale derivative $(x^{2m+1})^{\Delta}$ is \begin{equation} \label{eq:proof1} (x^{2m+1})^{\Delta} From db8c1e91c218d576245d6f0029aeba249a108238 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Tue, 17 Sep 2024 15:55:18 +0200 Subject: [PATCH 20/22] update notations --- src/sections/time-scale-z.tex | 24 +++++++++--------------- 1 file changed, 9 insertions(+), 15 deletions(-) diff --git a/src/sections/time-scale-z.tex b/src/sections/time-scale-z.tex index d820437..1eb98aa 100644 --- a/src/sections/time-scale-z.tex +++ b/src/sections/time-scale-z.tex @@ -5,7 +5,7 @@ $\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$. For every $t\in\mathbb{Z}$ and $x,b\in \Lambda^2$ \begin{align*} - \finiteDifference{x^{2m+1}} (t) + \finiteDifference{x^{+1}} (t) = \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) + \pTsDerivative{\polynomialP{m}{b}{x}}{b}(m, \sigma(t), t) \end{align*} @@ -17,22 +17,18 @@ \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 1 - 6 b^2 + 6 b x -% &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (t, \sigma(t)) = 3 t + 3 t^2 \\ -% &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(t,t) = 1 \end{align*} Evaluating in points yields \begin{align*} -% &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ -% &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 1 - 6 b^2 + 6 b x \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (\sigma(t), t) = 3 t + 3 t^2 \\ - &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(t,t) = 1 + &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (1, \sigma(t), t) = 3 t + 3 t^2 \\ + &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(1, t,t) = 1 \end{align*} Summing up previously obtained partial timescale derivatives, we get ordinary finite difference of odd-powered polynomial $x^{3}$ evaluated in point $ t\in\mathbb{Z}, \; x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$ \begin{align*} \finiteDifference{x^{3}}(t) - = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (\sigma(t), t) - + \pTsDerivative{\polynomialP{1}{b}{x}}{b}(t,t) + = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (1, \sigma(t), t) + + \pTsDerivative{\polynomialP{1}{b}{x}}{b}(1,t,t) = 3 t + 3 t^2 + 1 \end{align*} \end{examp} @@ -43,20 +39,18 @@ \begin{align*} &\pTsDerivative{\polynomialP{2}{b}{x}}{x} = 5 b - 30 b^2 + 40 b^3 - 15 b^4 + 10 b x - 30 b^2 x + 20 b^3 x \\ &\pTsDerivative{\polynomialP{2}{b}{x}}{b} = 1 + 30 b^4 - 60 b^3 x + 30 b^2 x^2 -% &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (t, \sigma(t)) = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ -% &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (t, t) = 1 \end{align*} Evaluating in points yields \begin{align*} - &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (\sigma(t), t) = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ - &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (t, t) = 1 + &\pTsDerivative{\polynomialP{2}{b}{x}}{x} (1, \sigma(t), t) = 5 t + 10 t^2 + 10 t^3 + 5 t^4 \\ + &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (1, t, t) = 1 \end{align*} Summing up previously obtained partial timescale derivatives, we get time ordinary finite difference of odd-powered polynomial $x^{2m+1}$ and $t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ \begin{align*} \finiteDifference{x^{5}} (t) - = \pTsDerivative{\polynomialP{2}{b}{x}}{x} (t, \sigma(t)) - + \pTsDerivative{\polynomialP{2}{b}{x}}{b} (t, t) + = \pTsDerivative{\polynomialP{2}{b}{x}}{x} (1, t, \sigma(t)) + + \pTsDerivative{\polynomialP{2}{b}{x}}{b} (1, t, t) = 1 + 5 t + 10 t^2 + 10 t^3 + 5 t^4 \end{align*} \end{examp} From 59bbc51fc675bf60346a3a1203a1c8abc8003572 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Wed, 18 Sep 2024 16:41:17 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zz&-X58Ul0+*jfPE5deW;EeK=}G!}RPkT5W$!Qz9X{+`VJ7yki;5*Q<4F#^+9Y_$SC zGz1%&12n{+LujBJ1A4`7ZGm1J`ey|M^bep4;CpQT-1zrQ8{5;N5G0^PEItSlCocfa zGthTn@c}b;Z0`pK2{14Q%V_{2&cGR%`y#NLI$$~p$Hs|J7#NB(*!bP=U}+VYzWy^5 zf&cCOP}uJ{!IByVfI_UA4-A|^pm@B(->m~yhy#;eG*$x%OnQMR#o+^T8a54pSU3uY zgHhN67%2R=4gVwNe_aO{A%d}?KG4j9u>%OeslYf6>x^(DP+4(kKm=ja(2zezSn+3d zL>F-;G(e$7Vb?*xhr!s`384Ltx4VE^i!C1j3~>|)?#A$u&~(g_(8y+INAb+1~}SBzyMT)rBwtBh2z>l ziT|@n`_CR|+{3{^C_$W)0B`Q$s1Oc;q2eXliEc~DB4uE=VQ?8az)mv2U t$. \end{cor} \begin{examp} \label{time_scale_pure_quantum_power_example_1} - Let be $t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; m\in\mathbb{N}$ - and let $m=1$, then + Let be $t\in q^{\mathbb{R}^j}, \; (x,b) \in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}$, let $m=1$ then \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 3 b - 2 b^2 + 3 b^j q - 2 b^{1 + j} q - 2 b^{2 j} q^2 - 3 x + 3 b x + 3 b^j q x -% &\pTsDerivative{\polynomialP{1}{b}{x}}{x}(t, \sigma(t)) = -3 q t^j + 3 q^2 t^{2 j} \\ -% &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(t, t) = t^2 + 3 q t^j - 2 q^2 t^{2 j} + q t^{1 + j} \end{align*} Evaluating in points yields \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x}(1, \sigma(t), t) = -3 q t^j + 3 q^2 t^{2 j} \\ &\pTsDerivative{\polynomialP{1}{b}{x}}{b}(1, t, t) = t^2 + 3 q t^j - 2 q^2 t^{2 j} + q t^{1 + j} \end{align*} - Summing up previously obtained partial timescale derivatives, we get $q-$power derivative of odd polynomial - $t^{2m+1}, \; t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; m\in\mathbb{N}$ + Summing up previously obtained partial timescale derivatives, we get $q-$power derivative of odd-powered polynomial + $x^{3}$ evaluated in point $t\in q^{\mathbb{R}^j}$ \begin{align*} - \nqderivative{t^{3}} + \nqderivative{x^{3}}(t) = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (1, \sigma(t), t) + \pTsDerivative{\polynomialP{1}{b}{x}}{b} (1, t, t) = t^2 + q^2 t^{2 j} + q t^{1 + j}. @@ -45,7 +42,7 @@ Another polynomial identity, that is exponential sum holds \begin{cor} \label{time_scale_pure_quantum_power_corollary_1} - For every $t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; t\in\mathbb{R}, \; m\in\mathbb{N}$ + For every $t\in q^{\mathbb{R}^j}, \; (x,b) \in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; t\in\mathbb{R}$ \begin{align*} \lim \limits_{j \to 0} \lim \limits_{q \to 1} \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = \sum_{k=0}^{2m} t^k @@ -55,7 +52,7 @@ An identity in even polynomials holds too \begin{cor} \label{time_scale_pure_quantum_power_corollary_2} - For every $t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; t\in\mathbb{R}, \; m\in\mathbb{N}$ + For every $t\in q^{\mathbb{R}^j}, \; (x,b) \in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}$ \begin{align*} \lim \limits_{j \to 0} \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = t^{2m} @@ -64,7 +61,7 @@ \begin{examp} \label{time_scale_pure_quantum_power_example_2} - Let be $t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; m\in\mathbb{N}$ and let $m=2$, then + Let be $t\in q^{\mathbb{R}^j}, \; (x,b) \in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}$, let $m=2$ then \begin{align*} \pTsDerivative{\polynomialP{2}{b}{x}}{x} &= -15 b^2 + 30 b^3 - 15 b^4 + 5 b x - 15 b^2 x + 10 b^3 x + 5 b q x^j - 15 b^2 q x^j + 10 b^3 q x^j \\ @@ -83,9 +80,9 @@ + q^2 t^{2 + 2 j} - 9 q^3 t^{1 + 3 j} \end{align*} Summing up previously obtained partial timescale derivatives, we $q-$power derivative of odd polynomial - $t^{2m+1}, \; t\in q^{\mathbb{R}^j}, \; x,b\in\Lambda^2 = q^{\mathbb{R}^j} \times q^{\mathbb{R}^j}, \; m\in\mathbb{N}$ + $x^{5}$ evaluated in point $t\in q^{\mathbb{R}^j}$ \begin{align*} - \nqderivative{t^{5}} + \nqderivative{x^{5}} (t) = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (2, \sigma(t), t) + \pTsDerivative{\polynomialP{1}{b}{x}}{b} (2, t, t) = t^4 + q^4 t^{4 j} + q t^{3 + j} + q^2 t^{2 + 2 j} + q^3 t^{1 + 3 j} diff --git a/src/sections/time-scale-qn.tex b/src/sections/time-scale-qn.tex index df858c4..09f6c2b 100644 --- a/src/sections/time-scale-qn.tex +++ b/src/sections/time-scale-qn.tex @@ -3,9 +3,9 @@ (Q-derivative~\cite{jackson_1909}.) Let be a two-dimensional time scale $\Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}} \colonequals \{t=(x,b) \colon \; x\in q^{\mathbb{R}}, \; b \in q^{\mathbb{R}} \}$. - For every $t\in q^{\mathbb{R}}$ and $x,b\in \Lambda^2$ + For every $t\in q^{\mathbb{R}}$ and $(x,b)\in \Lambda^2$ \begin{align*} - \qderivative{x^{2m+1}}(x) + \qderivative{x^{2m+1}}(t) = \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) + \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) \end{align*} @@ -13,8 +13,7 @@ \end{cor} \begin{examp} \label{time_scale_qn_example_1} - Let be $t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}, \; m\in\mathbb{N}$ - and let $m=1$, then + Let be $t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}$, let $m=1$ then \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 3 b - 2 b^2 + 3 b q - 2 b^2 q - 2 b^2 q^2 - 3 x + 3 b x + 3 b q x @@ -24,16 +23,16 @@ &\pTsDerivative{\polynomialP{1}{b}{x}}{x} (m, \sigma(t), t) = -3 q t + 3 q^2 t^2 \\ &\pTsDerivative{\polynomialP{1}{b}{x}}{b} (m, t, t) = 3 q t + t^2 + q t^2 - 2 q^2 t^2 \end{align*} - Summing up previously obtained partial time-scale derivatives, we get the $q$-derivative of odd polynomial - $x^{2m+1}$ evaluated in point $t\in q^{\mathbb{R}}$ + Summing up previously obtained partial time-scale derivatives, we get the $q$-derivative of odd-powered polynomial + $x^{3}$ evaluated in point $t\in q^{\mathbb{R}}$ \begin{align*} - \qderivative{x^{3}}(x) + \qderivative{x^{3}}(t) = \pTsDerivative{\polynomialP{1}{b}{x}}{x} (m, \sigma(t), t) + \pTsDerivative{\polynomialP{1}{b}{x}}{b} (m, t, t) = t^2 + q t^2 + q^2 t^2. \end{align*} \end{examp} -For every $t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}$ +For every $t\in q^{\mathbb{R}}, \; (x,b) \in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}$ the following polynomial identity holds as $q$ tends to zero \begin{align*} \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{1}{b}{x}}{b} (1, t, t) = t^2 @@ -41,14 +40,14 @@ However, it would be generalized as follows \begin{cor} \label{time_scale_qn_corollary_1} - For every $t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}, \; m\in\mathbb{N}$ + For every $t\in q^{\mathbb{R}}, \; (x,b) \in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}$ \begin{align*} \lim \limits_{q \to 0} \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = t^{2m}. \end{align*} \end{cor} \begin{examp} \label{time_scale_qn_example_2} - Let be $t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}, \; m\in\mathbb{N}$ and let $m=2$, then + Let be $t\in q^{\mathbb{R}}, \; (x,b) \in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}$, let $m=2$ then \begin{align*} \pTsDerivative{\polynomialP{2}{b}{x}}{x} &= -15 b^2 + 30 b^3 - 15 b^4 + 5 b x - 15 b^2 x + 10 b^3 x + 5 b q x - 15 b^2 q x + 10 b^3 q x \\ @@ -67,7 +66,7 @@ = -5 q t^2 + 10 q^2 t^2 + 15 q^2 t^3 - 15 q^3 t^3 + t^4 + q t^4 + q^2 t^4 - 9 q^3 t^4 + 6 q^4 t^4 \end{align*} Summing up previously obtained partial time-scale derivatives, we get the $q-$derivative of odd polynomial - $t^{2m+1}, \; t\in q^{\mathbb{R}}, \; x,b\in \Lambda^2 = q^{\mathbb{R}} \times q^{\mathbb{R}}, \; m\in\mathbb{N}$ + $x^{5}$ evaluated in point $t\in q^{\mathbb{R}}$ \begin{align*} \qderivative{t^{5}} = \pTsDerivative{\polynomialP{2}{b}{x}}{x} (2, \sigma(t), t) diff --git a/src/sections/time-scale-r.tex b/src/sections/time-scale-r.tex index 9a0af0d..6e58313 100644 --- a/src/sections/time-scale-r.tex +++ b/src/sections/time-scale-r.tex @@ -1,9 +1,9 @@ \begin{cor} \label{derivative_case} (Classical derivative.) - Let be a two-dimensional time scale + Let be a two-dimensional timescale $\Lambda^2 = \mathbb{R} \times \mathbb{R} \colonequals \{t=(x, b) \colon \; x\in \mathbb{R}, \; b\in\mathbb{R} \}$. - For every $t\in\mathbb{R}$ and $x,b\in \Lambda^2$ + For every $t\in\mathbb{R}$ and $(x,b) \in \Lambda^2$ \begin{align*} \odv{x^{2m+1}}{x} (t) = \pdv{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) @@ -13,7 +13,7 @@ \end{cor} \begin{examp} \label{time_scale_r_example_1} - Let be $t\in\mathbb{R}, \; x,b\in \Lambda^2 = \mathbb{R} \times \mathbb{R}, \; m\in\mathbb{N}$ and let $m=1$, then + Let be $t\in\mathbb{R}, \; (x,b) \in \Lambda^2 = \mathbb{R} \times \mathbb{R}$, let $m=1$ then \begin{align*} \pdv{\polynomialP{1}{b}{x}}{x} &= -3 b + 3 b^2 \\ \pdv{\polynomialP{1}{b}{x}}{b} &= 6 b - 6 b^2 - 3 x + 6 b x @@ -34,7 +34,7 @@ \end{examp} \begin{examp} \label{time_scale_r_example_2} - Let be $t\in\mathbb{R}, \; x,b\in \Lambda^2 = \mathbb{R} \times \mathbb{R}, \; m\in\mathbb{N}$ and let $m=2$, then + Let be $t\in\mathbb{R}, \; (x,b) \in \Lambda^2 = \mathbb{R} \times \mathbb{R}$, let $m=2$ then \begin{align*} &\pdv{\polynomialP{2}{b}{x}}{x} = -15 b^2 + 30 b^3 - 15 b^4 + 10 b x - 30 b^2 x + 20 b^3 x, \\ &\pdv{\polynomialP{2}{b}{x}}{b} = 30 b^2 - 60 b^3 + 30 b^4 - 30 b x + 90 b^2 x - 60 b^3 x + 5 x^2 - 30 b x^2 + 30 b^2 x^2 @@ -47,7 +47,7 @@ Summing up previously obtained partial timescale derivatives, we get classical derivative of an odd polynomial $x^5$ evaluated in point $t\in\mathbb{R}$ \begin{align*} - \odv{t^5}{t} + \odv{x^5}{x} (t) = \pdv{\polynomialP{2}{b}{x}}{x} (2, \sigma(t), t) + \pdv{\polynomialP{2}{b}{x}}{b} (2, \sigma(t), t) = 5t^4. diff --git a/src/sections/time-scale-z.tex b/src/sections/time-scale-z.tex index 1eb98aa..9d32223 100644 --- a/src/sections/time-scale-z.tex +++ b/src/sections/time-scale-z.tex @@ -1,11 +1,11 @@ \begin{cor} \label{finite_difference_case} - (Finite difference.) + (Divided difference.) Let be a two-dimensional timescale $\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$. For every $t\in\mathbb{Z}$ and $x,b\in \Lambda^2$ \begin{align*} - \finiteDifference{x^{+1}} (t) + \frac{\finiteDifference{x^{2m+1}}}{\Delta x} (t) = \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) + \pTsDerivative{\polynomialP{m}{b}{x}}{b}(m, \sigma(t), t) \end{align*} @@ -13,7 +13,7 @@ \end{cor} \begin{examp} \label{time_scale_z_example_1} - Let be $t \in \mathbb{Z}, \; x,b \in \Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ and let $m=1$, then + Let be $t \in \mathbb{Z}, \; x,b \in \Lambda^2 = \mathbb{Z} \times \mathbb{Z}$, let $m=1$ then \begin{align*} &\pTsDerivative{\polynomialP{1}{b}{x}}{x} = -3 b + 3 b^2 \\ &\pTsDerivative{\polynomialP{1}{b}{x}}{b} = 1 - 6 b^2 + 6 b x @@ -34,8 +34,7 @@ \end{examp} \begin{examp} \label{time_scale_z_example_2} - Let be $t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ - and let $m=2$, then + Let be $t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$, let $m=2$ then \begin{align*} &\pTsDerivative{\polynomialP{2}{b}{x}}{x} = 5 b - 30 b^2 + 40 b^3 - 15 b^4 + 10 b x - 30 b^2 x + 20 b^3 x \\ &\pTsDerivative{\polynomialP{2}{b}{x}}{b} = 1 + 30 b^4 - 60 b^3 x + 30 b^2 x^2 @@ -46,7 +45,7 @@ &\pTsDerivative{\polynomialP{2}{b}{x}}{b} (1, t, t) = 1 \end{align*} Summing up previously obtained partial timescale derivatives, we get time ordinary finite difference of odd-powered - polynomial $x^{2m+1}$ and $t\in\mathbb{Z}, \;x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ + polynomial $x^{5}$ and $t\in\mathbb{Z}, \; (x,b) \in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}$ \begin{align*} \finiteDifference{x^{5}} (t) = \pTsDerivative{\polynomialP{2}{b}{x}}{x} (1, t, \sigma(t)) @@ -56,7 +55,7 @@ \end{examp} \begin{cor} \label{time_scale_z_corollary_1} - For every $t\in\mathbb{Z}, \; x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ + For every $t\in\mathbb{Z}, \; (x,b) \in \Lambda^2 = \mathbb{Z} \times \mathbb{Z}$ \begin{align*} \pTsDerivative{\polynomialP{m}{b}{x}}{x} (m, \sigma(t), t) = \sum_{r=1}^{2m} \binom{2m+1}{r} t^{r} @@ -64,7 +63,7 @@ \end{cor} \begin{cor} \label{time_scale_z_corollary_2} - For every $t\in\mathbb{Z}, \; x,b\in\Lambda^2 = \mathbb{Z} \times \mathbb{Z}, \; m\in\mathbb{N}$ + For every $t\in\mathbb{Z}, \; (x,b) \in \Lambda^2 = \mathbb{Z} \times \mathbb{Z}$ \begin{align*} \pTsDerivative{\polynomialP{m}{b}{x}}{b} (m, t, t) = 1 \end{align*} From 8210962a409c6f12ca2409d5c7ae9fe139c87445 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Wed, 18 Sep 2024 16:49:38 +0200 Subject: [PATCH 22/22] updated --- out/AStudyOnDynamicEquations.pdf | Bin 378847 -> 377533 bytes src/AStudyOnDynamicEquations.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf index 145bd507e17d65dc5f1626dfff340175f064c468..7f41a61211c50b2169a877626802417f3ab39e0b 100644 GIT binary patch delta 62308 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partial dynamic equation on time scales involving derivatives of polynomials} \author[Petro Kolosov]{Petro Kolosov} \email{kolosovp94@gmail.com}