Merge pull request #4 from kolosovpetro/MATH-81

kolosovpetro · web-flow · commit 9e8d96dde495 · 2024-10-19T01:27:28.000+02:00
MATH-81
diff --git a/CHANGELOG.md b/CHANGELOG.md
@@ -5,6 +5,15 @@ 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).
 
+## [1.0.3] - 2024-10-18
+
+## Changed
+
+- Add build scripts
+- Add mathematica program
+- Add table for seventh powers
+- Move figures to separate folder
+
 ## [1.0.2] - 2024-09-25
 
 ## Changed
diff --git a/mathematica/Coefficients_A_as_solution_system_of_linear_equations.txt b/mathematica/Coefficients_A_as_solution_system_of_linear_equations.txt
@@ -0,0 +1,35 @@
+(* Power 0^0 should equal 1 by definition *)
+Unprotect[Power];
+Power[0 | 0., 0 | 0.] = 1;
+Protect[Power];
+(* This Mathematica code was created as supplementary file to arXiv:1603.02468 *)
+(* The code produces the solutions of system (2.4), page 8, for m=0,...,11. Date: 06/05/2018 *)
+(* Original file is placed at https://kolosovpetro.github.io/mathematica_codes/solutions_system_2_4.txt *)
+ClearAll[M];
+ClearAll[A];
+ClearAll[T];
+Off[Solve::svars];
+M[m_, n_, k_] := Sum[A[m, j]*(n - k)^j*k^j, {j, 0, m}];
+T[m_] := Solve[{
+    Sum[M[m, 1, k], {k, 0, 0}] == 1^(2 m + 1),
+    Sum[M[m, 2, k], {k, 0, 1}] == 2^(2 m + 1),
+    Sum[M[m, 3, k], {k, 0, 2}] == 3^(2 m + 1),
+    Sum[M[m, 4, k], {k, 0, 3}] == 4^(2 m + 1),
+    Sum[M[m, 5, k], {k, 0, 4}] == 5^(2 m + 1),
+    Sum[M[m, 6, k], {k, 0, 5}] == 6^(2 m + 1),
+    Sum[M[m, 7, k], {k, 0, 6}] == 7^(2 m + 1),
+    Sum[M[m, 8, k], {k, 0, 7}] == 8^(2 m + 1),
+    Sum[M[m, 9, k], {k, 0, 8}] == 9^(2 m + 1),
+    Sum[M[m, 10, k], {k, 0, 9}] == 10^(2 m + 1),
+    Sum[M[m, 11, k], {k, 0, 10}] == 11^(2 m + 1),
+    Sum[M[m, 12, k], {k, 0, 11}] == 12^(2 m + 1)},
+   {A[m, 0], A[m, 1], A[m, 2], A[m, 3], A[m, 4], A[m, 5], A[m, 6], 
+    A[m, 7], A[m, 8], A[m, 9], A[m, 10], A[m, 11], A[m, 12]}];
+
+(* Print a list of solutions as triangle, solutions for m=0,1,...,5, five initial rows of sequence A302971 *)
+
+Column[Table[T[m], {m, 0, 11}], Center]
+
+(* We can reproduce a particular set of solution for integer m <= 11 *)
+(* By execution T[m] in Mathematica *)
+(* If it's necessary to produce terms for m>11, one has to add the sums to definition of T[m_] solving it for A[m+1,t],(m+1)^(2m+1);A[m+2,t],(m+2)^(2m+1);....;A[m+k,t],(m+k)^(2m+1). *)
diff --git a/out/PolynomialIdentityInvolvingBTandFaulhaber.blg b/out/PolynomialIdentityInvolvingBTandFaulhaber.blg
@@ -1,13 +1,13 @@
 This is BibTeX, Version 0.99d
 Capacity: max_strings=200000, hash_size=200000, hash_prime=170003
-The top-level auxiliary file: PolynomialIdentityInvolvingBTandFaulhaber.aux
+The top-level auxiliary file: E:\RiderProjects\04_LATEX\PolynomialIdentityInvolvingBTandFaulhaber\out/PolynomialIdentityInvolvingBTandFaulhaber.aux
 Reallocating 'name_of_file' (item size: 1) to 6 items.
 The style file: unsrt.bst
 Reallocating 'name_of_file' (item size: 1) to 42 items.
 Database file #1: PolynomialIdentityInvolvingBTandFaulhaber.bib
 You've used 14 entries,
             1791 wiz_defined-function locations,
-            520 strings with 6325 characters,
+            521 strings with 6510 characters,
 and the built_in function-call counts, 2062 in all, are:
 = -- 178
 > -- 63
diff --git a/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf b/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf
diff --git a/src/Build-Latex.ps1 b/src/Build-Latex.ps1
@@ -0,0 +1,59 @@
+param (
+    [string]$OutputDirectory = "../out"
+)
+
+Write-Output "Setting working directory to PSScriptRoot: $PSScriptRoot"
+Set-Location -Path $PSScriptRoot
+
+Write-Output "=============================================================================="
+
+Write-Output "Determining the root latex file to build ..."
+$LatexFileName = (Get-ChildItem -Filter "*.tex" | Select-Object -First 1).BaseName
+Write-Output "Latex file: $LatexFileName"
+
+$OutputDirectoryAbsPath = (Resolve-Path $OutputDirectory).Path
+$LatexFileAbsPath = (Resolve-Path "$LatexFileName.tex").Path
+$AuxDirectoryAbsPath = $OutputDirectoryAbsPath
+
+Write-Output "=============================================================================="
+
+Write-Output "Compiling $LatexFileName first time... "
+Write-Output "Latex file: $LatexFileAbsPath"
+Write-Output "Output directory: $OutputDirectoryAbsPath"
+Write-Output "Aux directory: $AuxDirectoryAbsPath"
+pdflatex -file-line-error -interaction="nonstopmode" -synctex="1" -output-format="pdf" `
+    -output-directory="$OutputDirectoryAbsPath" `
+    -aux-directory="$AuxDirectoryAbsPath" $LatexFileAbsPath
+
+Write-Output "=============================================================================="
+
+Write-Output "Compiling bibtex ... "
+$WorkingDirectoryForBibtex = "$OutputDirectoryAbsPath/$LatexFileName"
+Write-Output "Working directory for bibtex: $WorkingDirectoryForBibtex"
+bibtex "$WorkingDirectoryForBibtex"
+
+Write-Output "=============================================================================="
+
+Write-Output "Compiling $LatexFileName second time... "
+Write-Output "Latex file: $LatexFileAbsPath"
+Write-Output "Output directory: $OutputDirectoryAbsPath"
+Write-Output "Aux directory: $AuxDirectoryAbsPath"
+pdflatex -file-line-error -interaction="nonstopmode" -synctex="1" -output-format="pdf" `
+    -output-directory="$OutputDirectoryAbsPath" `
+    -aux-directory="$AuxDirectoryAbsPath" $LatexFileAbsPath
+
+Write-Output "=============================================================================="
+
+Write-Output "Compiling $LatexFileName third time... "
+Write-Output "Latex file: $LatexFileAbsPath"
+Write-Output "Output directory: $OutputDirectoryAbsPath"
+Write-Output "Aux directory: $AuxDirectoryAbsPath"
+pdflatex -file-line-error -interaction="nonstopmode" -synctex="1" -output-format="pdf" `
+    -output-directory="$OutputDirectoryAbsPath" `
+    -aux-directory="$AuxDirectoryAbsPath" $LatexFileAbsPath
+
+Write-Output "=============================================================================="
+
+Write-Output "Build complete."
+Write-Output "Changing Powershell Directory... "
+cd ..
diff --git a/src/Rename-Tex-Files.ps1 b/src/Rename-Tex-Files.ps1
@@ -0,0 +1,49 @@
+Write-Output "Setting working directory to PSScriptRoot: $PSScriptRoot"
+Set-Location -Path $PSScriptRoot
+
+Write-Output "=============================================================================="
+
+Write-Output "Determining the root latex file to build ..."
+$LatexFileName = (Get-ChildItem -Filter "*.tex" | Select-Object -First 1).BaseName
+Write-Output "Latex file: $LatexFileName"
+
+Write-Output "=============================================================================="
+
+Write-Output "Determining the root bibtex file ..."
+$BibtexFileName = (Get-ChildItem -Filter "*.bib" | Select-Object -First 1).BaseName
+Write-Output "Bibtex file: $BibtexFileName"
+
+Write-Output "=============================================================================="
+
+Write-Output "Determining New name for TeX and BibTeX files ..."
+Write-Output "New name is the root of Git repository ..."
+$GitRootDirectory = (Get-Item $PSScriptRoot).Parent.Name
+Write-Output "New name for TeX and BibTeX files: $GitRootDirectory"
+
+Write-Output "=============================================================================="
+
+Write-Output "Renaming TeX root file ..."
+Write-Output "Old name: $LatexFileName.tex"
+Rename-Item -Path "$LatexFileName.tex" -NewName "$GitRootDirectory.tex"
+Write-Output "New name: $GitRootDirectory.tex"
+
+Write-Output "=============================================================================="
+
+Write-Output "Renaming BibTeX root file ..."
+Write-Output "Old name: $BibtexFileName.bib"
+Rename-Item -Path "$BibtexFileName.bib" -NewName "$GitRootDirectory.bib"
+Write-Output "New name: $GitRootDirectory.bib"
+
+Write-Output "=============================================================================="
+
+Write-Output "Updating TeX root file with new BibTeX file name ..."
+$LatexFileContent = Get-Content -Path "$GitRootDirectory.tex"
+Write-Output "Old BibTeX file name: $BibtexFileName"
+Write-Output "New BibTeX file name: $GitRootDirectory"
+$LatexFileContent = $LatexFileContent -replace "$BibtexFileName", "$GitRootDirectory"
+$LatexFileContent| Set-Content "$GitRootDirectory.tex"
+
+Write-Output "=============================================================================="
+
+Write-Output "Changing Powershell Directory... "
+cd ..
diff --git a/src/auxiliary/oeis_row_sums_cubes.tex b/src/auxiliary/oeis_row_sums_cubes.tex
diff --git a/src/figures/01_fig_finite_differences_cubes.tex b/src/figures/01_fig_finite_differences_cubes.tex
@@ -14,5 +14,5 @@
             7   & 343   &               &                 &
         \end{tabular}
     \end{center}
-    \caption{Table of finite differences of the polynomial $n^3$.} \label{tab:table}
-\end{table}
+    \caption{Table of finite differences of the polynomial $n^3$.} \label{tab:differneces-of-cubes}
+\end{table}
diff --git a/src/figures/02_fig_triangle_row_sums_give_cubes.tex b/src/figures/02_fig_triangle_row_sums_give_cubes.tex
@@ -0,0 +1,18 @@
+\begin{table}[H]
+    \setlength\extrarowheight{-6pt}
+    \begin{tabular}{c|cccccccc}
+        $n/k$ & 0 & 1  & 2  & 3  & 4  & 5  & 6  & 7 \\
+        \hline
+        0     & 1 &    &    &    &    &    &    &   \\
+        1     & 1 & 1  &    &    &    &    &    &   \\
+        2     & 1 & 7  & 1  &    &    &    &    &   \\
+        3     & 1 & 13 & 13 & 1  &    &    &    &   \\
+        4     & 1 & 19 & 25 & 19 & 1  &    &    &   \\
+        5     & 1 & 25 & 37 & 37 & 25 & 1  &    &   \\
+        6     & 1 & 31 & 49 & 55 & 49 & 31 & 1  &   \\
+        7     & 1 & 37 & 61 & 73 & 73 & 61 & 37 & 1
+    \end{tabular}
+    \caption{Values of $6k(n-k) + 1$.
+    See the OEIS entry: \href{https://oeis.org/A287326}{\texttt{A287326}}~\cite{kolosov2017third}.}
+    \label{tab:triangle_row_sums_give_cubes}
+\end{table}
diff --git a/src/figures/03_fig_triangle_row_sums_give_fifth_power.tex b/src/figures/03_fig_triangle_row_sums_give_fifth_power.tex
@@ -0,0 +1,18 @@
+\begin{table}[H]
+    \setlength\extrarowheight{-6pt}
+    \begin{tabular}{c|cccccccc}
+        $n/k$ & 0 & 1    & 2    & 3    & 4    & 5    & 6    & 7 \\
+        \hline
+        0     & 1 &      &      &      &      &      &      &   \\
+        1     & 1 & 1    &      &      &      &      &      &   \\
+        2     & 1 & 31   & 1    &      &      &      &      &   \\
+        3     & 1 & 121  & 121  & 1    &      &      &      &   \\
+        4     & 1 & 271  & 481  & 271  & 1    &      &      &   \\
+        5     & 1 & 481  & 1081 & 1081 & 481  & 1    &      &   \\
+        6     & 1 & 751  & 1921 & 2431 & 1921 & 751  & 1    &   \\
+        7     & 1 & 1081 & 3001 & 4321 & 4321 & 3001 & 1081 & 1
+    \end{tabular}
+    \caption{Values of $30k^2(n-k)^2 + 1$.
+    See the OEIS entry \href{https://oeis.org/A300656}{\texttt{A300656}}~\cite{kolosov2018fifth}.}
+    \label{tab:row-sums-gives-fifth-power}
+\end{table}
diff --git a/src/figures/04_fig_triangle_row_sums_give_seventh_power.tex b/src/figures/04_fig_triangle_row_sums_give_seventh_power.tex
@@ -0,0 +1,18 @@
+\begin{table}[H]
+    \setlength\extrarowheight{-6pt}
+    \begin{tabular}{c|cccccccc}
+        $n/k$ & 0 & 1     & 2      & 3      & 4      & 5      & 6     & 7 \\
+        \hline
+        0     & 1 &       &        &        &        &        &       &   \\
+        1     & 1 & 1     &        &        &        &        &       &   \\
+        2     & 1 & 127   & 1      &        &        &        &       &   \\
+        3     & 1 & 1093  & 1093   & 1      &        &        &       &   \\
+        4     & 1 & 3739  & 8905   & 3739   & 1      &        &       &   \\
+        5     & 1 & 8905  & 30157  & 30157  & 8905   & 1      &       &   \\
+        6     & 1 & 17431 & 71569  & 101935 & 71569  & 17431  & 1     &   \\
+        7     & 1 & 30157 & 139861 & 241753 & 241753 & 139861 & 30157 & 1
+    \end{tabular}
+    \caption{Values of $140 k^3 (n-k)^3 - 14k(n-k) + 1$.
+    See the OEIS entry \href{https://oeis.org/A300785}{\texttt{A300785}}~\cite{kolosov2018seventh}.}
+    \label{tab:row-sums-gives-seventh-power}
+\end{table}
diff --git a/src/figures/05_fig_coefficients_a.tex b/src/figures/05_fig_coefficients_a.tex
@@ -14,6 +14,6 @@
             7     & 1 & -450054 & 491400 & -60060 & 0   & 0    & 0     & 51480
         \end{tabular}
     \end{center}
-    \caption{Coefficients $\coeffA{m}{r}$. See the OEIS entries (two links to references).}
+    \caption{Coefficients $\coeffA{m}{r}$. See OEIS sequences~\cite{kolosov2018numerator,kolosov2018denominator}.}
     \label{tab:table_of_coefficients_a}
-\end{table}
+\end{table}
diff --git a/src/sections/01_introduction/introduction.tex b/src/sections/01_introduction/introduction.tex
@@ -1,22 +1,5 @@
 Considering the table of forward finite differences of the polynomial $n^3$
-\begin{table}[H]
-    \begin{center}
-        \setlength\extrarowheight{-6pt}
-        \begin{tabular}{c|cccc}
-            $n$ & $n^3$ & $\Delta(n^3)$ & $\Delta^2(n^3)$ & $\Delta^3(n^3)$ \\
-            \hline
-            0   & 0     & 1             & 6               & 6               \\
-            1   & 1     & 7             & 12              & 6               \\
-            2   & 8     & 19            & 18              & 6               \\
-            3   & 27    & 37            & 24              & 6               \\
-            4   & 64    & 61            & 30              & 6               \\
-            5   & 125   & 91            & 36              &                 \\
-            6   & 216   & 127           &                 &                 \\
-            7   & 343   &               &                 &
-        \end{tabular}
-    \end{center}
-    \caption{Table of finite differences of the polynomial $n^3$.} \label{tab:table}
-\end{table}
+\input{figures/01_fig_finite_differences_cubes}
 We can easily observe that finite differences
 \footnote{\input{sections/01_introduction/footnote}}
 of the polynomial $n^3$ may be expressed according
diff --git a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example1.tex b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example1.tex
@@ -35,22 +35,5 @@
     \end{equation*}
     It is also clearly seen why the above identity is true evaluating the terms $6k(n-k) + 1$ over $0 \leq k \leq n$ as
     the following table shows
-    \begin{table}[H]
-        \setlength\extrarowheight{-6pt}
-        \begin{tabular}{c|cccccccc}
-            $n/k$ & 0 & 1  & 2  & 3  & 4  & 5  & 6  & 7 \\
-            \hline
-            0     & 1 &    &    &    &    &    &    &   \\
-            1     & 1 & 1  &    &    &    &    &    &   \\
-            2     & 1 & 7  & 1  &    &    &    &    &   \\
-            3     & 1 & 13 & 13 & 1  &    &    &    &   \\
-            4     & 1 & 19 & 25 & 19 & 1  &    &    &   \\
-            5     & 1 & 25 & 37 & 37 & 25 & 1  &    &   \\
-            6     & 1 & 31 & 49 & 55 & 49 & 31 & 1  &   \\
-            7     & 1 & 37 & 61 & 73 & 73 & 61 & 37 & 1
-        \end{tabular}
-        \caption{Values of $6k(n-k) + 1$.
-        See the OEIS entry: \href{https://oeis.org/A287326}{\texttt{A287326}}~\cite{kolosov2017third}.}
-        \label{tab:table-row-sums-gives-cubes}
-    \end{table}
+    \input{figures/02_fig_triangle_row_sums_give_cubes}
 \end{examp}
diff --git a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example2.tex b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example2.tex
@@ -40,22 +40,5 @@
     It is also clearly seen
     why the above identity is true evaluating the terms $30k^2(n-k)^2 + 1$ over $0 \leq k \leq n$ as
     the following table shows
-    \begin{table}[H]
-        \setlength\extrarowheight{-6pt}
-        \begin{tabular}{c|cccccccc}
-            $n/k$ & 0 & 1    & 2    & 3    & 4    & 5    & 6    & 7 \\
-            \hline
-            0     & 1 &      &      &      &      &      &      &   \\
-            1     & 1 & 1    &      &      &      &      &      &   \\
-            2     & 1 & 31   & 1    &      &      &      &      &   \\
-            3     & 1 & 121  & 121  & 1    &      &      &      &   \\
-            4     & 1 & 271  & 481  & 271  & 1    &      &      &   \\
-            5     & 1 & 481  & 1081 & 1081 & 481  & 1    &      &   \\
-            6     & 1 & 751  & 1921 & 2431 & 1921 & 751  & 1    &   \\
-            7     & 1 & 1081 & 3001 & 4321 & 4321 & 3001 & 1081 & 1
-        \end{tabular}
-        \caption{Values of $30k^2(n-k)^2 + 1$.
-        See the OEIS entry \href{https://oeis.org/A300656}{\texttt{A300656}}~\cite{kolosov2018fifth}}
-        \label{tab:row-sums-gives-fifth-power}
-    \end{table}
+    \input{figures/03_fig_triangle_row_sums_give_fifth_power}
 \end{examp}
diff --git a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example3.tex b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example3.tex
@@ -52,5 +52,6 @@
     \end{equation*}
     It is also clearly seen
     why the above identity is true evaluating the terms $140 k^3 (n-k)^3 - 14k(n-k) + 1$ over $0 \leq k \leq n$ as
-    the OEIS sequence \href{https://oeis.org/A300785}{\textit{A300785}}~\cite{kolosov2018seventh} shows.
+    the OEIS sequence \href{https://oeis.org/A300785}{\textit{A300785}}~\cite{kolosov2018seventh} shows
+    \input{figures/04_fig_triangle_row_sums_give_seventh_power}
 \end{examp}
diff --git a/src/sections/03_finding_a_recurrence_relation/finding_a_recurrence_relation.tex b/src/sections/03_finding_a_recurrence_relation/finding_a_recurrence_relation.tex
@@ -124,25 +124,7 @@
 where $\bernoulli{t}$ are Bernoulli numbers~\cite{bateman1953higher}.
 It is assumed that $\bernoulli{1}=\frac{1}{2}$.
 For example,
-\begin{table}[H]
-    \begin{center}
-        \setlength\extrarowheight{-6pt}
-        \begin{tabular}{c|cccccccc}
-            $m/r$ & 0 & 1       & 2      & 3      & 4   & 5    & 6     & 7 \\ [3px]
-            \hline
-            0     & 1 &         &        &        &     &      &       &       \\
-            1     & 1 & 6       &        &        &     &      &       &       \\
-            2     & 1 & 0       & 30     &        &     &      &       &       \\
-            3     & 1 & -14     & 0      & 140    &     &      &       &       \\
-            4     & 1 & -120    & 0      & 0      & 630 &      &       &       \\
-            5     & 1 & -1386   & 660    & 0      & 0   & 2772 &       &       \\
-            6     & 1 & -21840  & 18018  & 0      & 0   & 0    & 12012 &       \\
-            7     & 1 & -450054 & 491400 & -60060 & 0   & 0    & 0     & 51480
-        \end{tabular}
-    \end{center}
-    \caption{Coefficients $\coeffA{m}{r}$.}
-    \label{tab:table_of_coefficients_a}
-\end{table}
+\input{figures/05_fig_coefficients_a}
 The coefficients $\coeffA{m}{r}$ are also registered in the OEIS~\cite{kolosov2018numerator,kolosov2018denominator}.
 It is as well interesting to notice that row sums of the $\coeffA{m}{r}$ give powers of $2$
 \begin{equation*}
