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https://www.apmep.fr/IMG/tex/Corrige_brevet_Polynesie_juin_2012.tex
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\documentclass[10pt]{article}
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\hypersetup{%
pdfauthor = {APMEP},
pdfsubject = {Corrigé du brevet des collèges},
pdftitle = {Polynésie juin 2012},
allbordercolors = white}
\usepackage[frenchb]{babel}
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\begin{document}
\setlength\parindent{0mm}
\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small Corrigé du brevet des collèges}
\lfoot{\small{Polynésie}}
\rfoot{\small{juin 2012}}
\renewcommand \footrulewidth{.2pt}
\pagestyle{fancy}
\thispagestyle{empty}
\begin{center}
{\Large \textbf{\decofourleft~Corrigé du brevet des collèges Polynésie juin 2012~\decofourright}}
\bigskip
\textbf{Durée : 2 heures} \end{center}
\bigskip
\textbf{ACTIVITÉS NUMÉRIQUES \hfill 12 points}
\bigskip
\textbf{Exercice 1}
\medskip
%Pour chaque ligne du tableau ci-dessous, choisir et entourer la bonne réponse parmi les trois proposées. Aucune justification n'est demandée.
%
%\medskip
%
%\begin{tabularx}{\linewidth}{|m{4.8cm}|*{3}{>{\centering \arraybackslash}X|}}\hline
%L' inverse de 1 est &$- 1$ &1 &2 \\ \hline
%\rule[-3mm]{0mm}{8mm}$\dfrac{2+3}{4\times 7}$ s'écrit aussi :& $(2+3)\div (4 \times 7)$& $2+3 \div 7(4\times7)$& $2 + 3 \div 4 \times 7$\\ \hline
%\rule[-3mm]{0mm}{8mm}$2 + \dfrac{2}{3}\times \dfrac{1}{4}$ est égal à:& $\dfrac{13}{6}$& $\dfrac{4}{12}$&
%$\dfrac{5}{7}$\\ \hline
%Si $x = - 4$ alors $x + 4 + (x + 4 )(2x - 5)$ est égal à :& $-4$& $- 1$& $0$\\ \hline
%\end{tabularx}
\begin{enumerate}
\item 1.
\item $2+3 \div 7(4\times7)$
\item $2 + \dfrac{2}{3}\times \dfrac{1}{4} = 2 + \dfrac{1}{6} = \dfrac{13}{6}$.
\item $- 4 + 4 +(- 4 + 4)(- 8 - 5) = 0 + 0 \times (- 13) = 0$.
\end{enumerate}
\medskip
\vspace{0,5cm}
\textbf{Exercice 2}
\medskip
%L'entreprise \og Punu Pua Toro\fg{} vend des boîtes de corned-beef.
%
%Ces dernières sont de forme cylindrique de 12~cm de diamètre et de 5~cm de hauteur.
%
%Elles sont rangées dans un carton de 84~cm de long, 60~cm de large et 5~cm de hauteur de façon à ce qu'elles se calent les unes contre les autres.
%
%\medskip
\begin{enumerate}
\item %Combien de boîtes peut-on ranger au maximum dans un carton ?
On peut mettre $\dfrac{84}{12} = 7$ boîtes dans la longueur, $\dfrac{60}{12} = 5$ dans la largeur et 1 dans la hauteur, soit $7\times 5 \times 1 = 35$ dans un carton.
\item %Calcule le PGCD de $84$ et $60$.
Par l’algorithme d’Euclide :
$84 = 60 \times 1 + 24$ ;
$60 = 24 \times 2 + 12$ ;
$24 = 12 \times 2 + 0$.
Le PGCD de 84 et 60 est donc 12.
\item %L’entreprise peut-elle ranger dans ce carton des boîtes cylindriques de plus grand diamètre de façon à ce qu'elles se calent les unes contre les autres ? Justifie ta réponse.
La réponse est non, puisque ce diamètre doit être un diviseur de 84 et de 60 et que le plus grand a été trouvé : 12 (cm).
\end{enumerate}
\vspace{0,5cm}
\textbf{Exercice 3}
\medskip
%L'hôtel \og la ora na \fg accueille 125 touristes :
%
%\setlength\parindent{8mm}
%\begin{itemize}
%\item $55$ néo-calédoniens dont $12$ parlent également anglais.
%\item $45$ américains parlant uniquement l'anglais.
%\item Le reste étant des polynésiens dont $8$ parlent également anglais.
%\end{itemize}
%\setlength\parindent{0mm}
%
%\textbf{Les néo-calédoniens et les polynésiens parlent tous le français.}
%
%\medskip
Òn peut faire un tableau :
\begin{center}
\begin{tabularx}{\linewidth}{|*{4}{>{\centering \arraybackslash}X|}}\hline
&Anglophones&Non anglophones&Total\\ \hline
Néo Calédo.&12&43&55\\ \hline
Américains&45&0&45\\ \hline
Polynésiens&8&17&25\\ \hline
Total&65&60&125\\ \hline
\end{tabularx}
\end{center}
\begin{enumerate}
\item %Si je choisis un touriste pris au hasard dans l'hôtel, quelle est la probabilité des évènements suivants:
\begin{enumerate}
\item %Évènement A : \og Le touriste est un américain \fg
La probabilité de A est égale à $\dfrac{45}{125} = \dfrac{9}{25} = \dfrac{36}{100} = 0,36$.
\item %Évènement B : \og Le touriste est un polynésien ne parlant pas anglais \fg
Il y a $125 - (55 + 45) = 125 - 100 = 25$ polynésiens dont 8 parlent anglais, donc $25 - 8 = 17$ polynésiens ne parlant pas anglais.
La probabilité de B est égale à $\dfrac{17}{125} = 0,136 = 13,6\,\%$.
\item %Évènement C : \og Le touriste parle anglais \fg
Parlent l’anglais 12 néo-calédoniens, 45 américains et 8 polynésiens soit en tout 65 touristes.
La probabilité de C est égale à $\dfrac{65}{125} = \dfrac{13}{25} = \dfrac{52}{100} = 0,52 = 52\,\%.$
\end{enumerate}
\item %Si j'aborde un touriste dans cet hôtel, ai-je plus de chance de me faire comprendre en parlant en anglais ou en français ? Justifie ta réponse. (\emph{Toute trace de recherche, même incomplète sera prise en compte dans l'évaluation})
Parlent le français : $55$ néo-calédoniens et $25$ polynésiens soit en tout 80 touristes. Parlent l'anglais : $12 + 45 + 8 = 65$ touristes. Il y a donc plus de chances de se faire comprendre en français qu’en anglais.
\end{enumerate}
\vspace{0,5cm}
\textbf{ACTIVITÉS GÉOMÉTRIQUES \hfill 12 points}
\bigskip
\textbf{Exercice 1}
\medskip
%Teva vient de construire lui-même sa pirogue.
%
%\medskip
%
%\psset{unit=1cm,linewidth=0.8mm}
%\begin{center}
%\begin{pspicture}(-2,0)(6,6)
%%\psgrid
%\psline(-1,3.5)(6,3.5) \psline(-1,5.5)(6,5.5)
%\psline(1.5,5.5)(3,0)\psline(4.5,5.5)(3,0)
%\uput[ul](1.5,5.5){K}\uput[ur](4.5,5.5){L}
%\uput[dl](2,3.5){I}\uput[dr](4,3.5){J}
%\uput[l](3,0){O}\uput[u](5.5,5.5){balancier} \uput[u](5.5,3.5){balancier}
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%\end{pspicture}
%\end{center}
\begin{enumerate}
\item %Pour vérifier que les deux bras du balancier sont parallèles entre eux, il place sur ceux-ci deux bois rectilignes schématisés sur le dessin ci-dessus par les segments [OK] et [OL] avec I $\in$ [OK] et J $\in$ [OL].
%La mesure des longueurs OI, OJ, OK et OL donne les résultats suivants :
%
%\[\text{OI} = 1,5~\text{m}\quad \text{OJ} = 1,65~\text{m} \quad \text{OK} = 2~\text{m} \quad \text{OL} = 2,2~\text{m}.\]
%
%Les deux bras sont-ils parallèles ? Justifie ta réponse.
On a $\dfrac{\text{OI}}{\text{OK}} = \dfrac{1,5}{2} = 0,75$ et $\dfrac{\text{OJ}}{\text{OL}} = \dfrac{1,65}{2,2} = \dfrac{11 \times 0,15}{11 \times 0,2} = \dfrac{0,15}{0,2} = 0,75$.
D’après la réciproque du théorème de Thalès, les droites (IJ) et (KL) sont parallèles.
\item %Pour vérifier que la pièce [AB] est perpendiculaire au balancier il mesure les longueurs AB, AC et CB et obtient:
%\[\text{AB} = 15~\text{cm} \quad \text{AC} = 25~\text{cm} \quad \text{CB} = 20~\text{cm}\]
%
%Peut-il affirmer que la pièce [AB] est perpendiculaire au balancier ? Justifie ta réponse.
On a AC$^2 = 25^2 = 625$ ;
AB$^2 + \text{BC}^2 = 15^2 + 20^2 = 225 + 400 = 625$.
On a donc $\text{AC}^2 = \text{AB}^2 + \text{BC}^2
$, donc d’après la réciproque du théorème de Pythagore, le triangle ABC est rectangle en B.. La pièce [AB] est perpendiculaire au balancier.
\end{enumerate}
\vspace{0,5cm}
\textbf{Exercice 2}
\medskip
\begin{enumerate}
\item ~
%Trace le cercle $\mathcal{C}$ de centre O et de diamètre [AB] tel que AB = 8~cm.
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(-4.5,-4.5)(4.5,4.5)
%\psgrid
\pscircle(0,0){4}
\psarc(-4,0){8}{0}{18}
\psline(-4,0)(3.6,2.48)
\psline(3.22,2.35)(4,0)(-4,0)
\psline(0,0)(3.22,2.35)
\uput[l](-4,0){A}\uput[r](4,0){B}
\uput[u](3.22,2.35){M}\uput[d](0,0){O}
\psline(3.22,2.35)(3.22,-2.35)\uput[dr](4;-36){N}
\psline(0;0)(4;108)\uput[ul](4;108){R}
\psline(0;0)(4;180)\uput[dl](4;252){S}
\psline(0;0)(4;252)
\psline(0;0)(4;-36)
\pspolygon[linecolor=blue](4;-36)(4;36)(4;108)(4;180)(4;252)
\end{pspicture}
\end{center}
\item %Place un point M appartenant à $\mathcal{C}$ tel que $\widehat{\text{BOM}} = 36$\,\degres.
Voir la figure.
\item %Calcule la mesure de l'angle inscrit $\widehat{\text{MAB}}$ qui intercepte le petit arc de cercle $\widearc{\text{MB}}$.
On sait que la mesure de l’angle au centre est le double de la mesure de l’angle inscrit qui intercepte le même arc.
Donc $\widehat{\text{MAB}} = 18\degres$.
\item %À l'aide des données de l'énoncé, laquelle de ces propositions te permet de montrer que AMB est un triangle rectangle en M : (Recopie sur ta copie la bonne proposition)
%\medskip
%
%\textbf{Proposition 1 :}
%
%Si dans le triangle AME on a AB$^2$= AM$^2$ + BM$^2$ alors AME est un triangle rectangle en M.
%
%\textbf{Proposition 2 :}
%
%Si le triangle AMB est inscrit dans le cercle $\mathcal{C}$ dont l'un des diamètres est [AB] alors AMB est un triangle rectangle en M
%
%\textbf{Proposition 3 :}
%
%Si O est le milieu de [AB] alors AMB est un triangle rectangle d'hypoténuse [AB].
C’est la proposition 2.
\item %Calcule la longueur AM et arrondis le résultat au dixième.
Dans le triangle ABM rectangle en M, on a $\cos \widehat{\text{MAB}} = \dfrac{\text{AM}}{\text{AB}}$.
Donc AM $ = \text{AB}\cos \widehat{\text{MAB}} = 8 \cos 18 \approx 7,608$ soit 7,6 au dixième près.
\item %Trace le symétrique N de M par rapport à [AB].
Voir la figure.
\item %Place les points R et S de façon à ce que NMRAS soit un pentagone régulier.
Puisque le pentagone est régulier, l’angle au centre $\widehat{\text{NOM}} = \dfrac{360}{5} = 72 = 2 \times 36$\degres.
Les autres angles ayant la même mesure il suffit de reporter l’arc NM à partir du point M sur le cercle ou de construire des angles au centre de 72\degres.
\end{enumerate}
\vspace{0,5cm}
\textbf{PROBLÈME \hfill 12 points}
\bigskip
\textbf{PREMIÈRE PARTIE}
\medskip
%Taraina dirige une école de danse pour adolescents. Elle a relevé dans un tableau l'âge de ses élèves ainsi que la fréquence des âges.
%
%\medskip
\begin{enumerate}
\item Complète \textbf{sur cette feuille} le tableau suivant :
\medskip
\begin{tabularx}{\linewidth}{|c|*{6}{>{\centering \arraybackslash}X|}}\hline
Âge des élèves &12 &13 &14 &15 &16 &TOTAL\\ \hline
Nombre d'élèves &5 &2 &4 &5 &4 & 20\\ \hline
Fréquence en \,\%& 25 & 10 & 20 &25 &20 &100\\ \hline
\end{tabularx}
\medskip
\item Complète le diagramme en barres des effectifs à l'aide du tableau précédent.
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(-0.5,-0.75)(7,7)
\psgrid[gridlabels=0pt,subgriddiv=1,griddots=5](0,0)(7,7)
\psframe[fillstyle=solid,fillcolor=lightgray](1,5)
\psframe[fillstyle=solid,fillcolor=lightgray](1,0)(2,2)
\psframe[fillstyle=solid,fillcolor=lightgray](2,0)(3,4)
\psframe[fillstyle=solid,fillcolor=lightgray](3,0)(4,5)
\psframe[fillstyle=solid,fillcolor=lightgray](4,0)(5,4)
\psaxes[linewidth=1.5pt,Ox=12,Dy=10]{->}(0,0)(7,7)
\uput[d](6.2,-0.35){Âge (ans)}\uput[r](0,6.5){Effectif}
\end{pspicture}
\end{center}
\item %Quelle est dans cette école la fréquence d'élèves ayant 14 ans ?
20\,\%.
\item %Quel est le nombre d'élèves âgés de 14 ans ou moins ?
Il y a $5 + 2 + 4 = 11$ élèves ayant au plus 14 ans.
\item %Taraina a calculé que l'âge moyen de ses élèves est légèrement supérieur à 14 ans, or pour inscrire son groupe au Heiva dans la catégorie \og Adolescents \fg, l'âge moyen du groupe doit être inférieur ou égal à 14 ans.
%Pour régler ce problème, elle a la possibilité d'accepter dans sa troupe de danse un nouvel élève, soit de 13 ans, soit de 15 ans.
\begin{enumerate}
\item %Lequel va-t-elle choisir ? Pourquoi ? (Toute trace de recherche sera valorisée.)
En prenant l’élève de 15 la moyenne d’âge va augmenter ; en prenant l’élève de 13 ans cette moyenne va baisser.
\item %Montre que l'âge moyen de sa nouvelle troupe est maintenant de 14 ans.
La nouvelle moyenne est égale à :
$\dfrac{5 \times 12 + 3 \times 13 + 4 \times 14 + 5 \times 15 + 4 \times 16}{5 + 3 + 4 + 5 + 4} = \dfrac{294}{21} = 14$~(ans).
\end{enumerate}
\end{enumerate}
\textbf{DEUXIEME PARTIE}
\medskip
Taraina veut inscrire ses 21 élèves aux festivités du Heiva. Deux tarifs lui sont proposés :
Tarif Individuel : 500~F par danseur inscrit.
Tarif Groupe : Paiement d'un forfait de \np{4000}~F pour le groupe puis $300$~F par danseur inscrit.
\medskip
\begin{enumerate}
\item Complète le tableau suivant :
\medskip
\begin{tabularx}{\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline
Nombre d'inscriptions &0 &10 &25\\ \hline
Prix au tarif Individuel en F &0 &\np{5000} &\np{12500}\\ \hline
Prix au tarif Groupe en F &0 &\np{7000} &\np{11500} \\ \hline
\end{tabularx}
\medskip
\item %Soit $x$ le nombre d'inscriptions.
%Le prix $I(x)$ à payer si l'on choisit le tarif individuel en fonction de $x$ est $I(x) = 500x$.
%Exprimer en fonction de $x$, le prix $G(x)$ à payer si l'on choisit le tarif Groupe.
$G(x) = \np{4000} + 300x$.
\item ~
%Dans le repère ci-dessous construire la représentation graphique des deux fonctions $x \longmapsto 500x$ et $x \longmapsto 300x+ \np{4000}$.
\begin{center}
\psset{xunit=0.3cm,yunit=0.001cm}
\begin{pspicture}(-2,-1000)(35,15000)
\multido{\n=0+1}{36}{\psline[linecolor=orange,linewidth=0.2pt](\n,0)(\n,14000)}
\multido{\n=0+500}{29}{\psline[linecolor=orange,linewidth=0.2pt](0,\n)(35,\n)}
\psaxes[linewidth=1.5pt,Dx=2,Dy=1000]{->}(0,0)(35,14000)
\psaxes[linewidth=1.5pt,Dx=2,Dy=1000](0,0)(35,14000)
\uput[dl](0,0){O}
\uput[u](32,0){\footnotesize Nombre d'élèves}
\uput[r](0,14500){\footnotesize Prix à payer}
\psplot{0}{28}{x 500 mul}
\psplot{0}{33.35}{x 300 mul 4000 add}
\psset{arrowsize=3pt 5}
\psline[linestyle=dashed,ArrowInside=->](21,0)(21,10300)(0,10300)
\end{pspicture}
\end{center}
\item %Graphiquement, quel est le tarif le plus avantageux pour l'inscription des $21$ élèves ?
C’est le tarif Groupe qui est le plus avantageux.
%Laisser apparaître les tracés utiles sur le graphique.
\item %Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi ?
Graphiquement il semble que le tarif est identique pour $x = 20$.
Par le calcul :
$I(x) = G(x)$ si $500x = 300x + \np{4000}$ soit $200x = \np{4000}$ et enfin $x = 20$.
%Justifie ta réponse par le calcul.
\end{enumerate}
\end{document}
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\[\mathop{{{}_{2}\phi_{2}}\/}\nolimits\!\left({a,q/a\atop-q,b};q,-b\right)=\frac%
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\def\ds{\displaystyle}
\def\d{\partial}
\begin{document}
\centerline{\large \bf Title of Report}
\vspace{.1truein}
\def\thefootnote{\arabic{footnote}}
\begin{center}
Author 1\footnote{Department, University},
Author 2\footnote{Department, University},
Author 3\footnote{Department, University},
Author 4\footnote{Department, University},
Author 5\footnote{Department, University}
\end{center}
%\vspace{.1truein}
\begin{center}
Faculty Mentors: Mentor 1\footnote{Company},
Mentor 2\footnote{University}
\end{center}
\vspace{.3truein}
\centerline{\bf Abstract}
\begin{itemize}
\item Summarize the results presented in the report, and the contributions
of your research.
\item Readers should not have to look at the rest of the paper in order to
understand the abstract.
\item Keep it short and to the point.
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\begin{itemize}
\item Describe the problem you are trying to solve, the approach
you took, and summarize your contribution and results.
\item Review the history of this problem, and existing literature.
\item Give an outline of the rest of the paper.
\end{itemize}
\section{The Problem}
\begin{itemize}
\item Give a precise technical description of your problem.
\item State and justify all your assumptions.
\item Define notation.
\item Describe your data, how you collected them, their properties,
and whether you did
anything to them (removed noise, filled in missing data,
applied normalizations).
\end{itemize}
\section{The Approach}
\begin{itemize}
\item Present and justify your approach for solving the problem.
\item Explain the advantages of your approach over existing ones.
\item Tell a story.
Don't just say: ``I did this, then I did this, and at last I did this''.
\end{itemize}
\section{Computational Experiments}
Give enough details so that readers can duplicate your experiments.
\begin{itemize}
\item Describe the precise purpose of the experiments, and what they
are supposed to show.
\item Describe and justify your test data, and any assumptions you made to
simplify the problem.
\item Describe the software you used, and the
parameter values you selected.
\item
For every figure, describe the meaning and units of the coordinate axes,
and what is being plotted.
\item Describe the conclusions you can draw from your experiments
\end{itemize}
\section{Summary and Future Work}
\begin{itemize}
\item Briefly summarize your contributions, and their possible
impact on the field (but don't just repeat the abstract or introduction).
\item Identify the limitations of your approach.
\item Suggest improvements for future work.
\item Outline open problems.
\end{itemize}
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\section*{An interrupted plot}
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pdftitle={Literaturverzeichnis}}
\begin{document}
Liste erstellt am 2020-06-27
% \begin{thebibliography}{XXnna}
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% \footnotesize
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%
\bibitem[Ant¢n 2020]{An20}~\\*
Susan C.\ Ant¢n,
\newblock \textsl{All who wander are not lost}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/science/s368-0034-Anton.pdf}%
{science \textbf{368} (2020), 34--35}.
\par
{\footnotesize\setlength{\parindent}{1em}
New hominin cranial fossils highlight the early exploits of Homo
erectus.\par
Scientists have hypothesized that Homo as a genus relied more heavily
on technological extraction of food resources (meat, marrow, and
plants) and was behaviorally more flexible than Paranthropus or
Australopithecus. Even in light of the diversity among Homo species,
H. erectus seems to be the beginning of something new. In the ÷7-%
million-year history of the human lineage, H. erectus was the first
species to leave the African continent (see the figure). In fact,
almost as soon as they arose, H. erectus appeared outside of Africa at
the site of Dmanisi in the Republic of Georgia. Over the next nearly 2
million years, H. erectus occupied a variety of different habitats and
contexts before going extinct well after 0.5 million years ago on
presentday Java.\par
\par}
%
\bibitem[Bae 2017]{Ba17}~\\*
Christopher J.\ Bae, Katerina Douka \& Michael D.\ Petraglia,
\newblock \textsl{On the origin of modern humans},
\textit{Asian perspectives}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/science/s358-1269-Bae.pdf}%
{science \textbf{358} (2017), 1269}.
\par
{\footnotesize\setlength{\parindent}{1em}
The traditional ``out of Africa'' model, which posits a dispersal of
modern Homo sapiens across Eurasia as a single wave at ÷60,000 years
ago and the subsequent replacement of all indigenous populations, is
in need of revision. Recent discoveries from archaeology, hominin
paleontology, geochronology, genetics, and paleoenvironmental studies
have contributed to a better understanding of the Late Pleistocene
record in Asia. Important findings highlighted here include growing
evidence for multiple dispersals predating 60,000 years ago in regions
such as southern and eastern Asia. Modern humans moving into Asia met
Neandertals, Denisovans, mid-Pleistocene Homo, and possibly H.
floresiensis, with some degree of interbreeding occurring.These early
human dispersals,which left at least some genetic traces in modern
populations, indicate that later replacements were not wholesale.\par
\par}
%
\bibitem[Bliege Bird 2020]{Bl20}~\\*
Rebecca Bliege Bird, Chloe McGuire, Douglas W.\ Bird, Michael H.\ Price, David Zeanah \& Dale G.\ Nimmo,
\newblock \textsl{Fire mosaics and habitat choice in nomadic foragers}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/PNAS/pnas117-12904-BliegeBird.pdf}%
{PNAS \textbf{117} (2020), 12904--12914}.
\par
{\footnotesize\setlength{\parindent}{1em}
pnas117-12904-Supplement.pdf\par
In the mid-1950s Western Desert of Australia, Aboriginal populations
were in decline as families left for ration depots, cattle stations,
and mission settlements. In the context of reduced population density,
an ideal free-distribution model predicts landscape use should
contract to the most productive habitats, and people should avoid
areas that show more signs of extensive prior use. However, ecological
or social facilitation due to Allee effects (positive density
dependence) would predict that the intensity of past habitat use
should correlate positively with habitat use. We analyzed fire
footprints and fire mosaics from the accumulation of several years of
landscape use visible on a 35,300-km2 mosaic of aerial photographs
covering much of contemporary Indigenous Martu Native Title Lands
imaged between May and August 1953. Structural equation modeling
revealed that, consistent with an Allee ideal free distribution, there
was a positive relationship between the extent of fire mosaics and the
intensity of recent use, and this was consistent across habitats
regardless of their quality. Fire mosaics build up in regions with low
cost of access to water, high intrinsic food availability, and good
access to trade opportunities; these mosaics (constrained by water
access during the winter) then draw people back in subsequent years or
seasons, largely independent of intrinsic habitat quality. Our results
suggest that the positive feedback effects of landscape burning can
substantially change the way people value landscapes, affecting
mobility and settlement by increasing sedentism and local population
density.\par
{\sffamily Keywords:} ideal free distribution | positive density
dependence | niche construction | historical ecology | hunter-gatherer
mobility\par
{\sffamily Significance:} Models of human habitat choice and landscape
use assume that people have negative effects on resource availability,
which causes them to avoid regions that are already occupied or that
show signs of extensive past use in favor of regions of higher
quality. We show that when people engage in activities that increase
resource productivity, like burning, there is the potential for these
improvements to change habitat preferences in favor of places that
have been previously modified and occupied by people. This process
changes the way we think about intensification (and the origins of
broad-spectrum economies), which may arise not from the negative
effects of people on resources, but from the positive (and often
unintentional) feedbacks between people and their environments.\par
\par}
%
\bibitem[Braun 2019]{Br19}~\\*
David R.\ Braun et al.,
\newblock \textsl{Earliest known Oldowan artifacts at >2.58 Ma from Ledi-Geraru, Ethiopia, highlight early technological diversity}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/PNAS/pnas116-11712-Braun.pdf}%
{PNAS \textbf{116} (2019), 11712--11717}.
\par
{\footnotesize\setlength{\parindent}{1em}
pnas116-11712-Supplement.pdf\par
David R.\ Braun, Vera Aldeias, Will Archer, J.\ Ramon Arrowsmith,
Niguss Baraki, Christopher J.\ Campisano, Alan L.\ Deino, Erin N.\
DiMaggio, Guillaume Dupont-Nivet, Blade Engda, David A.\ Feary,
Dominique I.\ Garello, Zenash Kerfelew, Shannon P.\ McPherron, David
B.\ Patterson, Jonathan S.\ Reeves, Jessica C.\ Thompson \& Kaye E.\
Reed\par
The manufacture of flaked stone artifacts represents a major milestone
in the technology of the human lineage. Although the earliest
production of primitive stone tools, predating the genus Homo and
emphasizing percussive activities, has been reported at 3.3 million
years ago (Ma) from Lomekwi, Kenya, the systematic production of
sharp-edged stone tools is unknown before the 2.58--2.55 Ma Oldowan
assemblages from Gona, Ethiopia. The organized production of Oldowan
stone artifacts is part of a suite of characteristics that is often
associated with the adaptive grade shift linked to the genus Homo.
Recent discoveries from Ledi-Geraru (LG), Ethiopia, place the first
occurrence of Homo ÷250 thousand years earlier than the Oldowan at
Gona. Here, we describe a substantial assemblage of systematically
flaked stone tools excavated in situ from a stratigraphically
constrained context [Bokol Dora 1, (BD 1) hereafter] at LG bracketed
between 2.61 and 2.58 Ma. Although perhaps more primitive in some
respects, quantitative analysis suggests the BD 1 assemblage fits more
closely with the variability previously described for the Oldowan than
with the earlier Lomekwian or with stone tools produced by modern
nonhuman primates. These differences suggest that hominin technology
is distinctly different from generalized tool use that may be a shared
feature of much of the primate lineage. The BD 1 assemblage, near the
origin of our genus, provides a link between behavioral
adaptations---in the form of flaked stone artifacts---and the biological
evolution of our ancestors.\par
{\sffamily Keywords:} Oldowan | stone tools | Homo | cultural
evolution | paleoanthropology\par
{\sffamily Significance:} Humans are distinguished from all other
primates by their reliance on tool use. When this uniquely human
feature began is debated. Evidence of tool use in human ancestors now
extends almost 3.3 Ma and becomes prevalent only after 2.6 Ma with the
Oldowan. Here, we report a new Oldowan locality (BD 1) that dates
prior to 2.6 Ma. These earliest Oldowan tools are distinctive from the
3.3 Ma assemblage and from materials that modern nonhuman primates
produce. So, although tool production and use represent a generalized
trait of many primates, including human ancestors, the production of
Oldowan stone artifacts appears to mark a systematic shift in tool
manufacture that occurs at a time of major environmental changes.\par
\par}
%
\bibitem[Connah 2001]{Co01}~\\*
Graham Connah,
\newblock \textsl{African civilizations},
\textit{An archaeological perspective}.
\newblock \href{http://axel.berger-odenthal.de/Scans/meins/Connah~African-civilizations/}%
{(Cambridge \up{2}2001)}.
%
\bibitem[Connah 2004]{Co04}~\\*
Graham Connah,
\newblock \textsl{Forgotten Africa},
\textit{An introduction to its archaeology}.
\newblock \href{http://axel.berger-odenthal.de/Scans/meins/Connah~Forgotten-Africa/}%
{(Abingdon 2004)}.
%
\bibitem[Diamond 1987]{Di87}~\\*
Jared Diamond,
\newblock \textsl{The Worst Mistake in the History of the Human Race}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/Discover1987.5-064-Diamond.pdf}%
{Discover \textbf{1987}, \romannumeral5\relax , 64--66}.
\par
{\footnotesize\setlength{\parindent}{1em}
At this point it's instructive to recall the common complaint that
archaeology is a luxury, concerned with the remote past, and
offering no lessons for the present. Archaeologists studying the
rise of farming have reconstructed a crucial stage at which we made
the worst mistake in human history. Forced to choose between
limiting population or trying to increase food production, we chose
the latter and ended up with starvation, warfare, and tyranny.\par
\par}
%
\bibitem[Dom¡nguez-Rodrigo 2009]{Do09}~\\*
M.\ Dom¡nguez-Rodrigo et al.,
\newblock \textsl{Unraveling hominin behavior at another anthropogenic site from Olduvai Gorge (Tanzania)},
\textit{New archaeological and taphonomic research at BK, Upper Bed~II}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journ-JZ/JHumEvo057-0260-Dominguez-Rodrigo.pdf}%
{Journal of Human Evolution \textbf{57} (2009), 260--283}.
\par
{\footnotesize\setlength{\parindent}{1em}
New archaeological excavations and research at BK, Upper Bed II
(Olduvai Gorge, Tanzania) have yielded a rich and unbiased collection
of fossil bones. These new excavations show that BK is a stratified
deposit formed in a riverine setting close to an alluvial plain. The
present taphonomic study reveals the secondlargest collection of
hominin-modified bones from Olduvai, with abundant cut marks found on
most of the anatomical areas preserved. Meat and marrow exploitation
is reconstructed using the taphonomic signatures left on the bones by
hominins. Highly cut-marked long limb shafts, especially those of
upper limb bones, suggest that hominins at BK were actively engaged in
acquiring small and middle-sized animals using strategies other than
passive scavenging. The exploitation of large-sized game (Pelorovis)
by Lower Pleistocene hominins, as suggested by previous researchers,
is supported by the present study.\par
M.\ Dom¡nguez-Rodrigo, A.\ Mabulla, H.\,T.\ Bunn, R.\ Barba, F.\ Diez-%
Mart¡n, C.\,P.\ Egeland, E.\ Esp¡lez, A.\ Egeland, J.\ Yravedra \& P.\
S nchez\par
{\sffamily Keywords:} Olduvai Gorge | Meat-eating | Cut marks |
Percussion marks | Taphonomy | Lower Pleistocene archaeology | Hunting
| Scavenging\par
\par}
%
\bibitem[Dom¡nguez-Rodrigo 2010]{Do10}~\\*
Manuel Dom¡nguez-Rodrigo, Travis Rayne Pickering \& Henry T. Bunn,
\newblock \textsl{Configurational approach to identifying the earliest hominin butchers}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/PNAS/pnas107-20929-Dominguez-Rodrigo.pdf}%
{PNAS \textbf{107} (2010), 20929--20934}.
\par
{\footnotesize\setlength{\parindent}{1em}
pnas107-20929-Comment.pdf, pnas107-20929-Reply.pdf\par
The announcement of two approximately 3.4-million-y-old purportedly
butchered fossil bones from the Dikika paleoanthropological research
area (Lower Awash Valley, Ethiopia) could profoundly alter our
understanding of human evolution. Butchering damage on the Dikika
bones would imply that tool-assisted meat-eating began approximately
800,000 y before previously thought, based on butchered bones from
2.6- to 2.5-million-y-old sites at the Ethiopian Gona and Bouri
localities. Further, the only hominin currently known from Dikika at
approximately 3.4 Ma is Australopithecus afarensis, a temporally and
geographically widespread species unassociated previously with any
archaeological evidence of butchering. Our taphonomic configurational
approach to assess the claims of A. afarensis butchery at Dikika
suggests the claims of unexpectedly early butchering at the site are
not warranted. The Dikika research group focused its analysis on the
morphology of the marks in question but failed to demonstrate, through
recovery of similarly marked in situ fossils, the exact provenience of
the published fossils, and failed to note occurrences of random striae
on the cortices of the published fossils (incurred through incidental
movement of the defleshed specimens across and/or within their abrasive
encasing sediments). The occurrence of such random striae (sometimes
called collectively ``trampling'' damage) on the two fossils provide the
configurational context for rejection of the claimed butchery marks.
The earliest best evidence for hominin butchery thus remains at 2.6 to
2.5 Ma, presumably associated with more derived species than A.
afarensis.\par
taphonomy | cut marks | hammerstone percussion | abrasion |
equifinality\par
\par}
%
\bibitem[Dom¡nguez-Rodrigo 2012]{Do12}~\\*
Manuel Dom¡nguez-Rodrigo, Travis Rayne Pickering \& Henry T. Bunn,
\newblock \textsl{Experimental study of cut marks made with rocks unmodified by human flaking }
\textit{and its bearing on claims of ÷3.4-million-year-old butchery evidence from Dikika, Ethiopia}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/JAS/JAS039-0205-Dominguez-Rodrigo.pdf}%
{Journal of Archaeological Science \textbf{39} (2012), 205--214}.
\par
{\footnotesize\setlength{\parindent}{1em}
In order to assess further the recent claims of w3.4 Ma butchery marks
on two fossil bones from the site of Dikika (Ethiopia), we broadened
the actualistic-interpretive zooarchaeological framework by conducting
butchery experiments that utilized nave butchers and rocks unmodified
by human flaking to deflesh chicken and sheep long limb bones. It is
claimed that the purported Dikika cut marks present their unexpectedly
atypical morphologies because they were produced by early hominins
utilizing just such rocks. The composition of the cut mark sample
produced in our experiments is quite dissimilar to the sample of
linear bone surface modifications preserved on the Dikika fossils. This
finding substantiates and expands our earlier conclusion
that---considering the morphologies and patterns of the Dikika bone
surface modifications and the inferred coarse-grained depositional
context of the fossils on which they occur---the Dikika bone damage was
caused incidentally by the movement of the fossils on and/or within
their depositional substrate(s), and not by early hominin butchery.
Thus, contrary to initial claims, the Dikika evidence does not warrant
a major shift in our understanding of early hominin behavioral
evolution with regard to carcass foraging and meat-eating.\par
Keywords: Early hominin carcass foraging | Taphonomy | Cut marks |
Random striae | Trampling\par
\par}
%
\bibitem[Eco 1977]{Ec77}~\\*
Umberto Eco,
\newblock \textsl{How to Write a Thesis}.
\newblock \href{http://axel.berger-odenthal.de/PDF/monographs/Eco~How-Write-Thesis.pdf}%
{(Cambridge 2015)}.
\par
{\footnotesize\setlength{\parindent}{1em}
Eco was aware of this predicament. As a university professor, he knew
that the majority of students in Italian universities seldom attended
classes, that very few of them would continue to write and do
research, and that the degree they eventually earned would not
necessarily improve their social conditions. It would have been easy
to call for the system to be reformed so as not to require a thesis
from students illequipped to write one, and for whom the benefit of
spending several months working on a thesis might be difficult to
justify in cold economic terms.\par
But Eco did not believe that education belonged to an elite, or that
it should lower its standards in including the non-elite. He
understood that the writing of a thesis forced many students outside
of their cultural comfort zone, and that if the shock was too sudden
or strong, they would give up. For him, it was about tailoring the
challenge to students' needs and capabilities, but without giving up
thoroughness, complexity, and rigor. If students' interests and
ambitions could be met, while the limits of their sense of security
were stretched, education would be achieved.\par
\par}
%
\bibitem[Evans 2014]{Ev14}~\\*
Adrian Anthony Evans,
\newblock \textsl{On the importance of blind testing in archaeological science},
\textit{The example from lithic functional studies}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/JAS/JAS048-0005-Evans.pdf}%
{Journal of Archaeological Science \textbf{48} (2014), 5--14}.
\par
{\footnotesize\setlength{\parindent}{1em}
Blind-testing is an important tool that should be used by all
analytical fields as an approach for validating method. Several fields
do this well outside of archaeological science. It is unfortunate that
many applied methods do not have a strong underpinning built on, what
should be considered necessary, blind-testing. Historically lithic
microwear analysis has been subjected to such testing, the results of
which stirred considerable debate. However, putting this aside, it is
argued here that the tests have not been adequately exploited. Too
much attention has been focused on basic results and the implications
of those rather than using the tests as a powerful tool to improve the
method. Here the tests are revisited and reviewed in a new light. This
approach is used to highlight specific areas of methodological weakness
that can be targeted by developmental research. It illustrates the
value in having a large dataset of consistently designed blind-tests
in method evaluation and suggests that fields such as lithic microwear
analysis would greatly benefit from such testing. Opportunity is also
taken to discuss recent developments in quantitative methods within
lithic functional studies and how such techniques might integrate with
current practices.\par
{\sffamily Keywords:} Blind-tests | Quantification | Method improvement
| Lithic microwear | Functional analysis\par
\par}
%
\bibitem[Fauvelle 2013a]{Fa13a}~\\*
Franois-Xavier Fauvelle,
\newblock \textsl{Le rhinocros d'or},
\textit{Histoires du Moyen \^Age africain}.
\newblock folio histoire 239
(Malesherbes 2017).
%
\bibitem[Fauvelle 2013b]{Fa13b}~\\*
Franois-Xavier Fauvelle,
\newblock \textsl{Das Goldene Rhinozeros},
\textit{Afrika im Mittelalter}.
\newblock \href{http://axel.berger-odenthal.de/Scans/USB-Kln/44A2885-Fauvelle~Rhinozeros/}%
{(Mnchen 2017)}.
\newblock Originaltitel: Le Rhinoceros d'or -- Histoires du Moyen \^Age africain.
%
\bibitem[Groucutt 2015]{Gr15}~\\*
Huw S.\ Groucutt et al.,
\newblock \textsl{Rethinking the Dispersal of Homo sapiens out of Africa}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/EvolAnth24-149-Groucutt.pdf}%
{Evolutionary Anthropology \textbf{24} (2015), 149--164}.
\par
{\footnotesize\setlength{\parindent}{1em}
Huw S. Groucutt, Michael D. Petraglia, Geoff Bailey, Eleanor M. L.
Scerri, Ash Parton, Laine Clark-Balzan, Richard P. Jennings, Laura
Lewis, James Blinkhorn, Nick A. Drake, Paul S. Breeze, Robyn H.
Inglis, Maud H. Devs, Matthew Meredith-Williams, Nicole Boivin, Mark
G. Thomas, and Aylwyn Scally\par
Current fossil, genetic, and archeological data indicate that Homo
sapiens originated in Africa in the late Middle Pleistocene. By the
end of the Late Pleistocene, our species was distributed across every
continent except Antarctica, setting the foundations for the
subsequent demographic and cultural changes of the Holocene. The
intervening processes remain intensely debated and a key theme in
hominin evolutionary studies. We review archeological, fossil,
environmental, and genetic data to evaluate the current state of
knowledge on the dispersal of Homo sapiens out of Africa. The emerging
picture of the dispersal process suggests dynamic behavioral
variability, complex interactions between populations, and an
intricate genetic and cultural legacy. This evolutionary and
historical complexity challenges simple narratives and suggests that
hybrid models and the testing of explicit hypotheses are required to
understand the expansion of Homo sapiens into Eurasia.\par
\par}
%
\bibitem[Grn 1991]{Gr91}~\\*
R.\ Grn \& C.\,B.\ Stringer,
\newblock \textsl{Electron Spin Resonance Dating and the Evolution of Modern Humans}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/Archaeometry33-153-Grun.pdf}%
{Archaeometry \textbf{33} (1991), 153--199}.
\par
{\footnotesize\setlength{\parindent}{1em}
An ESR signal used for dating should have the following properties.\par
\noindent\textbf{(i)} A zeroing effect deletes all previously stored
ESR intensity in the sample at the event which is to be dated.\par
\noindent\textbf{(ii)} The signal intensity grows in proportion to the
dose received.\par
\noindent\textbf{(iii)} The signals must have a stability which is at
least one order of magnitude higher than the age of the sample.\par
\noindent\textbf{(iv)} The number of traps is constant or changes in a
predictable manner. Recrystallization, crystal growth or phase
transitions must not have occurred.\par
\noindent\textbf{(v)} The ESR signal is not influenced by sample
preparation (grinding, exposure to laboratory light) or anomalous
fading.\par
\par}
%
\bibitem[Harmand 2015]{Ha15}~\\*
Sonia Harmand et al.,
\newblock \textsl{3.3-million-year-old stone tools from Lomekwi~3, West Turkana, Kenya}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/nature/n521-0310-Harmand.pdf}%
{nature \textbf{521} (2015), 310--315}.
\par
{\footnotesize\setlength{\parindent}{1em}
n521-0310-Supplement.pdf\par
Sonia Harmand, Jason E.\ Lewis, Craig S.\ Feibel, Christopher J.\
Lepre, Sandrine Prat, Arnaud Lenoble, Xavier Bos, Rhonda L.Quinn,
Michel Brenet, Adrian Arroyo,Nicholas Taylor, Sophie Clment,
GuillaumeDaver, Jean-Philip Brugal, Louise Leakey, Richard A.\
Mortlock, James D.\ Wright, Sammy Lokorodi, Christopher Kirwa, Dennis
V.\ Kent \& Hlne Roche\par
Human evolutionary scholars have long supposed that the earliest stone
tools were made by the genusHomo and that this technological
development was directly linked to climate change and the spread of
savannah grasslands. New fieldwork in West Turkana, Kenya, has
identified evidence of much earlier hominin technological behaviour.
We report the discovery of Lomekwi 3, a 3.3-million-year-old
archaeological site where in situ stone artefacts occur in spatio-%
temporal association with Pliocene hominin fossils in a wooded
palaeoenvironment. The Lomekwi 3 knappers, with a developing
understanding of stone's fracture properties, combined core reduction
with battering activities. Given the implications of the Lomekwi 3
assemblage for models aiming to converge environmental change, hominin
evolution and technological origins, we propose for it the name
`Lomekwian', which predates the Oldowan by 700,000 years and marks a
new beginning to the known archaeological record.\par
\par}
%
\bibitem[Hertel 2001]{He01}~\\*
Peter Hertel,
\newblock \textsl{Projekt Diplomarbeit},
\textit{Schreibwerkstatt}.
\newblock \href{http://axel.berger-odenthal.de/PDF/monographs/Anleitung-TeX-p24!.pdf}%
{(Osnabrck 2001)}.
\newblock <\url{http://www.informatik.hs-furtwangen.de/~hanne/LATEX-DA-sws.pdf}> (2017-04-16).
\par
{\footnotesize\setlength{\parindent}{1em}
Wir halten fest: Jedes Dokument, mit dem man sich wegen der
Diplomarbeit beschftigt, ist sofort in der Literaturdatenbank zu
vermerken. Auch dann, wenn Sie noch gar nicht wissen knnen, ob das
Schriftstck zitiert werden soll, oder an welcher Stelle.\par
\par}
%
\bibitem[Hillman 1990a]{Hi90a}~\\*
Gordon C.\ Hillman \& M.\ Stuart Davies,
\newblock \textsl{Measured Domestication Rates in Wild Wheats and Barley Under Primitive Cultivation},
\textit{and Their Archaeological Implications}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/JWoPrehist04-157-Hillman.pdf}%
{Journal of World Prehistory \textbf{4} (1990), 157--222}.
\par
{\footnotesize\setlength{\parindent}{1em}
Man's (or, more probably, Woman's) first cereal crops were sown from
seed gathered from wild stands, and it was in the course of
cultivation that domestication occurred. Experiments in the
measurement of domestication rates indicate that in wild-type crops of
einkorn, emmer, and barley under primitive systems of husbandry: (a)
domestication will occur only if they are harvested when partially or
nearly ripe, using specific harvesting methods; (b) exposure to
shifting cultivation may sometimes have been required; and (c) under
these conditions, the crops could become completely domesticated
within 200 years, and perhaps only 20-30 years, without any conscious
selection. This paper (a) considers possible delays in the start of
domestication due to early crops of wild-type cereals lacking
domestic-types mutants; (b) examines the husbandry practices necessary
for these mutants to enjoy any selective advantage; (c) considers the
state of ripeness at harvest necessary for the crops to respond to
these selective pressures; (d) outlines field measurements of the
selective intensities arising from analogous husbandry practices
applied experimentally to living wild-type crops; (e) summarizes a
mathematical model which incorporates the measured selective
intensities and other key variables and which describes the rate of
increase in domestic-type mutants in early populations of wild-type
cereals under specific combinations of primitive husbandry practices;
(f) considers why very early cultivators should have used those
husbandry methods which, we suggest, led unconsciously to the
domestication of wild wheats and barley; and (g) considers whether
these events are likely to leave archaeologically recognizable traces.\par
{\sffamily Keywords:} domestication rate; agricultural origins;
einkorn wheat; emmer wheat; selection pressures.\par
\par}
%
\bibitem[Hillman 1990b]{Hi90b}~\\*
Gordon C.\ Hillman \& M.\ Stuart Davies,
\newblock \textsl{Domestication rates in wild-type wheats and barley under primitive cultivation}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/BioJLinnSoc039-039-Hillman.pdf}%
{Biological Journal of the Linnean Society \textbf{39} (1990), 39--78}.
\par
{\footnotesize\setlength{\parindent}{1em}
Man's first cereal crops were sown from seed gathered from wild
stands, and it was in the course of cultivation that domestication
occurred. This paper prcsents thr preliminary rrsults of an
experimcntal approach to thc measurement of domestication rate in
crops of wild-type einkorn wheat exposed to primitive systems of
husbandry. The results indicate that in wild-type crops of einkorn,
emmer and barley (a) domestication will have occurred only if they
were harvested in a partially ripe (or near-ripe) state using specific
harvesting methods; (b) exposure to shifting cultivation may also have
been required in somr cases; and (c) given these requirements, the
crops could have become completely domcsticatrd within two centuries,
and maybr in as littlr as 20-30 years without any form of conscious
selection.\par
This paper (1) considers the possible lrngth of delays in the start of
domestication due to early crops of wild-type cereals lacking
domestic-type mutants; (2) examines the combination of primitive
husbandry practices that would have been necessary for any selective
advantage to have been unconsciously conferred on these mutants; (3)
considers the state of ripeness (at harvest) necessary for crops to be
able to respond to these selective prcssures; (4) outlines field
measurements of the selective intensities (selection coefficients)
which arise when analogous husbandry practices are applied
experimentally to living wild-type crops; (5) summarizes the essential
features of a mathrmatical model which inrorporatcs thcsr mcasurrments
of selection coefficients and other key variables, and which describes
the rate of inrreasc in domestic-type mutants that would have occurred
in early populations of wild-type cereals under specific combinations
of primitivc husbandry practices; (6) considers why very early
cultivators should have used that combination of husbandry methods
which, we suggest, unconsciously brought about the domestication of
wild wheats and barley; and (7) concludes by considering whether these
events arc likely to havc left recognizable tracrs in archaeological
remains.\par
{\sffamily Keywords:} Domestication rate -- agricultural origins --
einkorn wheat -- emmer wheat -- barley -- selection pressures --
archaeobotany.\par
\par}
%
\bibitem[Jesse 2010]{Je10}~\\*
Friederike Jesse,
\newblock \textsl{Early Pottery in Northern Africa},
\textit{An Overview}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journ-JZ/JAfrArch08-219-Jesse.pdf}%
{Journal of African Archaeology \textbf{8} (2010), 219--238}.
\par
{\footnotesize\setlength{\parindent}{1em}
The emergence of pottery is a compelling issue for archaeologists. In
Africa , pottery appeared in what now the southern part of the Sahara
and the Sahel different localities and in different contexts in the
10th millennium bp. This paper aims to give an overview the available
data concerning early pottery in Northern Africa. The radiocarbon
evidence is considered as well as technological features of the
pottery ; the decoration and the site context. The areas of the
earliest appearance of pottery in Northern Africa were uninhabited
during hyperarid phase at the end of the Pleistocene. Intriguing
questions are therefore the origin of the Early Holocene occupants and
of their knowledge of potting and of course the role of early pottery
in the prehistoric groups.\par
{\sffamily Keywords:} Northern Africa | pottery | Early Holocene |
Wavy Line\par
\par}
%
\bibitem[Kaplan 2000]{Ka00}~\\*
David Kaplan,
\newblock \textsl{The Darker Side of the ``Original Affluent Society''}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/JAnthRes56-301-Kaplan.pdf}%
{Journal of Anthropological Research \textbf{56} (2000), 301--324}.
\par
{\footnotesize\setlength{\parindent}{1em}
Hunter-gatherers emerged from the ``Man the Hunter'' conference in 1966
as the ``original affluent society''. The main features of this thesis
now seem to be widely accepted by anthropologists, despite the strong
reservations expressed by certain specialists in foraging societies
concerning the data advanced to support the claim. This essay brings
together a portion of the data and argumentation in the literature
that raise a number of questions about hunter-gatherer affluence.
Three topics are addressed: How ``hard'' do foragers work? How well-fed
are members of foraging societies? And what do we mean by ``work'',
``leisure'', and ``affluence'' in the context of foraging societies?
Finally, this essay offers some thoughts about why, given the
reservations and critical observations expressed by anthropologists
who work with foragers, the thesis seems to have been enthusiastically
embraced by most anthropologists.\par
\par}
%
\bibitem[Larson 2007]{La07}~\\*
Greger Larson et al.,
\newblock \textsl{Ancient DNA, pig domestication, and the spread of the Neolithic into Europe}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/PNAS/pnas104-15276-Larson.pdf}%
{PNAS \textbf{104} (2007), 15276--15281}.
\par
{\footnotesize\setlength{\parindent}{1em}
pnas104-15276-Supplement.zip\par
Greger Larson, Umberto Albarella, Keith Dobney, Peter Rowley-Conwy,
Jrg Schibler, Anne Tresset, Jean-Denis Vigne, Ceiridwen J. Edwards,
Angela Schlumbaum, Alexandru Dinu, Adrian B\u{a}l\u{a}sescu, Gaynor
Dolman, Antonio Tagliacozzo, Ninna Manaseryan, Preston Miracle, Louise
Van Wijngaarden-Bakker, Marco Masseti, Daniel G. Bradley and Alan
Cooper\par
The Neolithic Revolution began 11,000 years ago in the Near East and
preceded a westward migration into Europe of distinctive cultural
groups and their agricultural economies, including domesticated
animals and plants. Despite decades of research, no consensus has
emerged about the extent of admixture between the indigenous and
exotic populations or the degree to which the appearance of specific
components of the ``Neolithic cultural package'' in Europe reflects
truly independent development. Here, through the use of mitochondrial
DNA from 323 modern and 221 ancient pig specimens sampled across
western Eurasia, we demonstrate that domestic pigs of Near Eastern
ancestry were definitely introduced into Europe during the Neolithic
(potentially along two separate routes), reaching the Paris Basin by
at least the early 4th millennium B.C. Local European wild boar were
also domesticated by this time, possibly as a direct consequence of
the introduction of Near Eastern domestic pigs. Once domesticated,
European pigs rapidly replaced the introduced domestic pigs of Near
Eastern origin throughout Europe. Domestic pigs formed a key component
of the Neolithic Revolution, and this detailed genetic record of their
origins reveals a complex set of interactions and processes during the
spread of early farmers into Europe.\par
\par}
%
\bibitem[Lessing 1982]{Le82}~\\*
Doris Lessing,
\newblock \textsl{The Making of the Representative for Planet~8}.
\newblock (St Albans 1983).
%
\bibitem[McPherron 2010]{Ph10}~\\*
Shannon P. McPherron et al.,
\newblock \textsl{Evidence for stone-tool-assisted consumption of animal tissues before 3.39 million years ago at Dikika, Ethiopia}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/nature/n466-0857-McPherron.pdf}%
{nature \textbf{466} (2010), 857--860}.
\par
{\footnotesize\setlength{\parindent}{1em}
n466-0857-Supplement.pdf, see: pnas107-20929-Dominguez-Rodrigo.pdf,
n467-0341-Dempsey.pdf, JArchSci39-0205-Dominguez-Rodrigo.pdf\par
Shannon P. McPherron, Zeresenay Alemseged, Curtis W. Marean, Jonathan
G. Wynn, Denn Reed, Denis Geraads, Ren Bobe \& Hamdallah A. Barat\par
The oldest direct evidence of stone tool manufacture comes from Gona
(Ethiopia) and dates to between 2.6 and 2.5 million years (Myr) ago1.
At the nearby Bouri site several cut-marked bones also show stone tool
use approximately 2.5 Myr ago2. Here we report stone-tool-inflicted
marks on bones found during recent survey work in Dikika, Ethiopia, a
research area close to Gona and Bouri. On the basis of low-power
microscopic and environmental scanning electron microscope
observations, these bones show unambiguous stone-tool cut marks for
flesh removal and percussion marks for marrow access. The bones derive
from the Sidi Hakoma Member of the Hadar Formation. Established
40Ar-39Ar dates on the tuffs that bracket this member constrain the
finds to between 3.42 and 3.24 Myrago, and stratigraphic scaling
between these units and other geological evidence indicate that they
are older than 3.39 Myr ago. Our discovery extends by approximately
800,000 years the antiquity of stone tools and of stone-tool-assisted
consumption of ungulates by hominins; furthermore, this behaviour can
now be attributed to Australopithecus afarensis.\par
\par}
%
\bibitem[Merritt 2019]{Me19}~\\*
Stephen R.\ Merritt, Michael C.\ Pante, Trevor L.\ Keevil, Jackson K.\ Njau \& Robert J.\ Blumenschine,
\newblock \textsl{Don't cry over spilled ink},
\textit{Missing context prevents replication and creates the Rorschach effect in bone surface modification studies}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/JAS/JAS102-0071-Merritt.pdf}%
{Journal of Archaeological Science \textbf{102} (2019), 71--79}.
\par
{\footnotesize\setlength{\parindent}{1em}
JAS086-0014-Dominguez-Rodrigo.pdf, JAS102-0080-Dominguez-Rodrigo.pdf\par
The scientific replicability crisis has recently focused on bone
surface modification (BSM) analysis, which underlies zooarchaeological
and anthropological conclusions about the ecology and evolution of
tool-assisted carcass consumption behavior. We review a recent blind
test of inter-analyst correspondence in morphometric analysis of
experimentally generated butchery marks that advocates algorithmic
methods for diagnosing and measuring BSM in an effort to standardize
methodology and minimize inter-analyst error (Dom¡nguez-Rodrigo et
al., 2017. Use and abuse of cut mark analyses: The Rorschach effect.
Journal of Archaeological Science, 86, 14--23.
http://doi.org/10.1016/j.jas.2017.08.001). This study overstates
concern about the inaccuracy of BSM measurement and interpretation,
concluding that BSM analysis is a subjective, non-scientific endeavor.
Based on a minimally described sample of cut marks, it measures
variables that involve inherent inaccuracy and subjectivity and
overlooks how the contexts of experimental sample generation --
particularly the difference between immanent and configurational
processes -- differentially affect cut mark morphometrics. We illustrate
this discussion with experimental taphonomic examples focused on
analytical context including sample construction and control over
factors that affect cut mark cross-sectional size. Our analysis
suggests the relationship between tool attributes and cut mark
morphology is not generalizable to all experimental and archaeological
butchery contexts. We show that our experimental samples capture
metric variability observed in archaeological cut marks, but that
intentionally incised marks and realistic defleshing marks differ in
width and depth. Further, when controlling for factors that impact cut
mark size including animal size class, tool type, butcher experience,
and density across bone portions, overlapping cut mark widths and
depths produced by phonolite and ignimbrite flakes lead to poor
classification of marks according to causal flake material, which casts
doubt on the ability to discriminate cut marks made by different
materials. We build datasets that include diverse experimental
contexts and suggest that meta-analysis can disentangle how multiple
configurational factors contribute to cut mark morphometric attributes.
Ultimately, progress in BSM analysis rests on inter-analyst
replicability, which must be preceded by clear discussion of all parts
of the inferential loop -- from the design of experiments that generate
actualistic analogues, to their use in supporting archaeological
arguments. Otherwise, problematic expert knowledge traditions may mask
arguments from authority in sophisticated methodological language and
underreported experimental context.\par
{\sffamily Keywords:} Experimental taphonomy | Cut marks | Butchery |
Generality | Realism | Context | Expert knowledge\par
\par}
%
\bibitem[Mitchell 2013]{Mi13}~\\*
\textsc{Peter Mitchell \& Paul Lane} (Hrsg.),
\newblock \textsl{The Oxford Handbook of African Archaeology}.
\newblock \href{http://axel.berger-odenthal.de/Scans/Aegyptologie/Mitchell~Oxf-Handb-African-Arch/}%
{(Oxford 2013)}.
%
\bibitem[O'Connell 2020]{Co20}~\\*
James F.\ O'Connell,
\newblock \textsl{Aboriginal fires modify an ideal free distribution}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/PNAS/pnas117-13873-OConnell.pdf}%
{PNAS \textbf{117} (2020), 13873--13875}.
\par
{\footnotesize\setlength{\parindent}{1em}
Bliege Bird et al.'s recent results challenge this picture by
suggesting less rapid resource depletion associated with initial burns
around early occupied sites, allowing more time to gain the benefits
of resource regeneration. A 10-km daily foraging radius around a
permanent water source contains thousands of potential 3-ha burn
locations. It would take a group of 50 people setting 10 fires a week
over several 6-mo burning seasons to cover a significant fraction of
that catchment, long enough for burned areas to begin to display the
enhanced foraging returns associated with a serial recovery process.
This would reduce the incentive to move away from the improving patch
and the social networks it supports. Understanding constraints like
this should aid the development of increasingly realistic,
archaeologically testable models of the pattern and pace of
continental colonization and its ecological consequences.\par
\par}
%
\bibitem[Osborne 2008]{Os08}~\\*
Anne H.\ Osborne, Derek Vance, Eelco J.\ Rohling, Nick Barton, Mike Rogerson \& Nuri Fello,
\newblock \textsl{A humid corridor across the Sahara for the migration of early modern humans out of Africa 120,000 years ago}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/PNAS/pnas105-16444-Osborne.pdf}%
{PNAS \textbf{105} (2008), 16444--16447}.
\par
{\footnotesize\setlength{\parindent}{1em}
It is widely accepted that modern humans originated in subSaharan
Africa ÷150--200 thousand years ago (ka), but their route of dispersal
across the currently hyperarid Sahara remains controversial. Given
that the first modern humans north of the Sahara are found in the
Levant ÷120--90 ka, northward dispersal likely occurred during a humid
episode in the Sahara within Marine Isotope Stage (MIS) 5e (130--117
ka). The obvious dispersal route, the Nile, may be ruled out by
notable differences between archaeological finds in the Nile Valley
and the Levant at the critical time. Further west, space-born radar
images reveal networks of---now buried---fossil river channels that
extend across the desert to the Mediterranean coast, which represent
alternative dispersal corridors. These corridors would explain
scattered findings at desert oases of Middle Stone Age Aterian lithic
industries with bifacial and tanged points that can be linked with
industries further to the east and as far north as the Mediterranean
coast. Here we present geochemical data that demonstrate that water in
these fossil systems derived from the south during wet episodes in
general, and penetrated all of the way to the Mediterranean during MIS
5e in particular. This proves the existence of an uninterrupted
freshwater corridor across a currently hyperarid region of the Sahara
at a key time for early modern human migrations to the north and out
of Africa.\par
Middle Stone Age | Eemian | neodymium | paleochannel | sapropel\par
\par}
%
\bibitem[Phillipson 1993]{Ph93}~\\*
David W. Phillipson,
\newblock \textsl{African Archaeology}.
\newblock \href{http://axel.berger-odenthal.de/Scans/meins/Phillipson~African-Archaeology/}%
{(Cambridge \up{2}1993)}.
%
\bibitem[Phillipson 2005]{Ph05}~\\*
David W. Phillipson,
\newblock \textsl{African Archaeology}.
\newblock (Cambridge \up{3}2005).
%
\bibitem[Proffitt 2014]{Pr14}~\\*
Tomos Proffitt \& Ignacio de la Torre,
\newblock \textsl{The effect of raw material on inter-analyst variation and analyst accuracy for lithic analysis},
\textit{A case study from Olduvai Gorge}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/JAS/JAS045-0270-Proffitt.pdf}%
{Journal of Archaeological Science \textbf{45} (2014), 270--283}.
\par
{\footnotesize\setlength{\parindent}{1em}
This study aims to understand what effect, in terms of inter-analysis
variation and analyst accuracy, different raw material types have on
modern technological analyses of lithic assemblages. This is done
through a series of blind analysis tests undertaken on experimentally
derived assemblages of cores and flakes. Novelties of our approach
include the introduction of refit studies as a method to assess
analyst's accuracy, and the use of statistical tests specifically
designed to address inter-analyst variability, common in other
disciplines but rarely used in Archaeology. The experimental
assemblages were produced from raw materials collected at Olduvai
Gorge, an archaeological sequence that has been a source for studies
of early human technology for several decades, and where re-analyses
of the same assemblages have usually offered different
interpretations. The results of the blind analyses are compared to the
true technological values obtained through full refit analysis of the
experimental material, and suggest that there is a significant
difference in terms of inter-analyst variability as well as accuracy
related to different raw materials. Our paper highlights the
interpretative problems posed by difficult-to-analyse raw materials
such as quartzite, and stresses subjectivity present in stone-tool
technological studies, which may contribute to explain differences in
the interpretation of Early Stone Age lithic assemblages.\par
{\sffamily Keywords:} Lithic technology | Olduvai Gorge | Blind tests
| Inter-analyst variability | Analyst accuracy | Refit analysis\par
\par}
%
\bibitem[Reichholf 1990]{Re90}~\\*
Josef H. Reichholf,
\newblock \textsl{Das Rtsel der Menschwerdung},
\textit{Die Entstehung des Menschen im Wechselspiel der Natur}.
\newblock (Mnchen \up{6}2004).
%
\bibitem[Richerson 2005]{Ri05}~\\*
Peter Richerson \& Robert Boyd,
\newblock \textsl{Not by genes alone},
\textit{How culture transformed human evolution}.
\newblock (Chicago 2005).
\par
{\footnotesize\setlength{\parindent}{1em}
Humans are a striking anomaly in the natural world. While we are
similar to other mammals in many ways, our behavior sets us apart. Our
unparalleled ability to adapt has allowed us to occupy virtually every
habitat on earth, and our societies are larger, more complex, and more
cooperative than any other mammal's. In ``Not by Genes Alone'', Peter J.
Richerson and Robert Boyd argue that only a Darwinian theory of
cultural evolution can explain these unique characteristics.\par
``Not by Genes Alone'' offers a radical interpretation of human
evolution, arguing that our ecological dominance and our singular
social systems stem from a psychology uniquely adapted to create
complex culture. Richerson and Boyd consider culture to be essential
to human adaptation, as much a part of human biology as bipedal
locomotion. Drawing on work in the fields of anthropology, political
science, sociology, and economics -- and building their case with such
fascinating examples as kayaks, clever knots, and yams that require
twelve men to carry them -- Richerson and Boyd convincingly demonstrate
that culture and biology are inextricably linked.\par
In abandoning the nature-versus-nurture debate as fundamentally
misconceived, ``Not by Genes Alone'' is a truly original and
groundbreaking theory of the role of culture in evolution and a book
to be reckoned with for generations to come.\par
\par}
%
\bibitem[Richter 2018]{Ri18}~\\*
Jrgen Richter,
\newblock \textsl{Altsteinzeit},
\textit{Der Weg der frhen Menschen von Afrika bis in die Mitte Europas}.
\newblock \href{http://axel.berger-odenthal.de/Scans/meins/Richter~Altsteinzeit/}%
{(Stuttgart 2018)}.
%
\bibitem[Riemer 2007]{Ri07}~\\*
Heiko Riemer,
\newblock \textsl{When hunters started herding},
\textit{Pastro-foragers and the complexity of Holocene economic change in the Western Desert of Egypt}.
\newblock In: \textsc{Michael Bollig, Olaf Bubenzer, Ralf Vogelsang \& Hans-Peter Wotzka} (Hrsg.),
\newblock \textsl{Aridity, Change and Conflict in Africa},
\textit{Proceedings of an International ACACIA Conference held at Knigswinter, Germany October 1--3, 2003}.
\newblock Colloquium Africanum 2
\href{http://axel.berger-odenthal.de/Scans/Ethnologie/AF351.2-Bollig~Aridity-Change-Conflict/105-Riemer.pdf}%
{(Kln 2007), 105--144}.
\par
{\footnotesize\setlength{\parindent}{1em}
Despite the debate on early Holocene large bovids from the Nabta-%
Kiseiba region, faunal data from archaeological sites in the Eastern
Sahara speak for an introduction and rapid spread of domestic cattle,
goat and sheep around 6000 calBC within a highly mobile hunter-%
gatherer context. However, wild animals and hunting equipment are the
major components of archaeological sites from the 6th millennium.
Diversity in relief and water accessibility, and the seasonal
influence of winter and summer rains formed the individual conditions
of subsistence in which herding played only a minor role. It was not
before the onset of deterioration of the Eastern Sahara, around 5000
calBC, and the following population agglomeration in the Nile Valley
that herding and plant cultivation became dominant in the predynastic
economies which can truly be labelled as the earliest Neolithic in
Egypt.\par
{\sffamily Keywords:} Pastro-foragers | domesticated animals | hunting
| herding | arrow heads | economic change | Holocene | Neolithic |
Egypt\par
\par}
%
\bibitem[Sage 1995]{Sa95}~\\*
Rowan F.\ Sage,
\newblock \textsl{Was low atmospheric CO2 during the Pleistocene a limiting factor for the origin of agriculture?}
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/GlobalChangeBiology01-0093-Sage.pdf}%
{Global Change Biology \textbf{1} (1995), 93--106}.
\par
{\footnotesize\setlength{\parindent}{1em}
Agriculture originated independently in many distinct regions at
approximately the same time in human history. This synchrony in
agricultural origins indicates that a global factor may have
controlled the timing of the transition from foraging to foodproducing
economies. The global factor may have been a rise in atmospheric CO2
from below 200 to near 270 æmol mol-1 which occurred between 15,000
and 12,000 years ago. Atmospheric CO2 directly affects photosynthesis
and plant productivity, with the largest proportional responses
occurring below the current level of 350 æmol mol-1 In the late
Pleistocene, CO2 levels near 200 æmol mol-1 may have been too low to
support the level of productivity required for successful
establishment of agriculture. Recent studies demonstrate that
atmospheric CO2 increase from 200 to 270 æmol mol-1 stimulates
photosynthesis and biomass productivity of C3 plants by 25\,\% to 50\,\%,
and greatly increases the performance of C3 plants relative to weedy
C4 competitors. Rising CO2 also stimulates biological nitrogen
fixation and enhances the capacity of plants to obtain limiting
resources such as water and mineral nutrients. These results indicate
that increases in productivity following the late Pleistocene rise in
CO2 may have been substantial enough to have affected human
subsistence patterns in ways that promoted the development of
agriculture. Increasing CO2 may have simply removed a productivity
barrier to successful domestication and cultivation of plants. Through
effects on ecosystem productivity, rising CO2 may also have been a
catalyst for agricultural origins by promoting population growth,
sedentism, and novel social relationships that in turn led to
domestication and cultivation of preferred plant resources.\par
{\sffamily Keywords:} origin of agriculture, CO2 enrichment, crop
domestication, global change, neolithic transition, photosynthesis\par
\par}
%
\bibitem[Sahlins 1966]{Sa66}~\\*
Marshall Sahlins,
\newblock \textsl{The Original Affluent Society}.
\newblock (Online 1966).
\newblock <\url{http://delong.typepad.com/files/original-affluent-society.pdf}> (2020-06-22).
\par
{\footnotesize\setlength{\parindent}{1em}
Hunter-gatherers consume less energy per capita per year than any
other group of human beings. Yet when you come to examine it the
original affluent society was none other than the hunter's -- in which
all the people's material wants were easily satisfied. To accept that
hunters are affluent is therefore to recognise that the present human
condition of man slaving to bridge the gap between his unlimited wants
and his insufficient means is a tragedy of modern times.\par
\par}
%
\bibitem[Schmidt 2015]{Sc15}~\\*
Christoph Schmidt, Karin Kindermann, Philip van Peer \& Olaf Bubenzer,
\newblock \textsl{Multi-emission luminescence dating of heated chert from the Middle Stone Age sequence at Sodmein Cave }
\textit{(Red Sea Mountains, Egypt)}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/JAS/JAS063-0094-Schmidt.pdf}%
{Journal of Archaeological Science \textbf{63} (2015), 94--103}.
\par
{\footnotesize\setlength{\parindent}{1em}
Sodmein Cave in Egypt is one of the rare archaeological sites in
north-eastern Africa conserving human occupation remains of a period
most relevant for the `Out of Africa II' hypothesis. This underlines
the need for establishing a chronological framework for the more than
4 m of stratified sediments ranging from the Middle Stone Age (MSA) to
the Neolithic. The lowest layer J hosts huge fireplaces, from which we
report luminescence ages of heated chert fragments unearthed from
different depths. The `multiemission' dating approach -- using both the
blue and red TL of each specimen as well as the OSL emission of one
sample -- allowed identifying the most reliable ages. Samples yield
ages between <121\,ñ\,15 ka (maximum age) and 87\,ñ\,9 ka. These data
evidence human presence at the site during MIS 5. For integrating
Sodmein Cave into the actual discussion of the dispersal patterns of
modern humans and to identify potential connections with other sites
in the Nile Valley and in the Middle East, a sound and reliable
chronology is indispensable.\par
{\sffamily Keywords:} Luminescence dating | Burnt chert | Burnt flint |
Egypt | Out of Africa II | Middle Stone Age\par
\par}
%
\bibitem[Sealy 2010]{Se10}~\\*
Judith Sealy,
\newblock \textsl{Isotopic Evidence for the Antiquity of Cattle-Based Pastoralism in Southernmost Africa}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journ-JZ/JAfrArch08-065-Sealy.pdf}%
{Journal of African Archaeology \textbf{8} (2010), 65--81}.
\par
{\footnotesize\setlength{\parindent}{1em}
Pastoralist Khoekhoe people in southern Africa are well known from
17th and 18th century records from the Cape, and from later descendent
communities. The Cape Khoekhoen kept large herds of sheep and cattle,
which constituted wealth and provided the dairy products that formed
dietary staples. The origins and development of this way of life
remain contentious. This paper addresses the issue by means of stable
carbon and nitrogen isotope analyses of 160 adult human skeletons from
the coastal forelands of southernmost Africa. Prior to 2000 bp,
hunter-gatherers ate varying mixes of marine and terrestrial foods,
but terrestrial C4 grasses (and animals grazing on them) were of
relatively minor importance. Sheep (and probably cattle) first
appeared in archaeological sites around 2000 bp, but whatever their
role in peoples `diets, there was no significant shift in the isotope
ratios of human skeletons in the first millennium AD. From the early
second millennium AD, people began to eat significantly more C4 based
foods, probably in the form of animal products (dairy and meat) from
animals grazing on C4 grasses. I argue that the most likely reason is
that domestic stock -- especially cattle -- became more important in
peoples ` diets at this time . There is evidence for a new style of
burial, in which the body was interred in a seated, flexed position,
and the grave capped with stones. Thus, although living sites remain
elusive, important elements of the historically documented Khoekhoe
way of life can be identified for the first time in the early second
millennium AD. This evidence also shows that a cattle-based economy
emerged centuries before Europeans seeking animals to slaughter
increased the demand for stock.\par
{\sffamily Keywords:} Later Stone Age | Khoe | diet | domestic stock | herd\par
\par}
%
\bibitem[Wrangham 1999]{Wr99}~\\*
Richard W. Wrangham, James Holland Jones, Greg Laden, David Pilbeam \& NancyLou Conklin-Brittain,
\newblock \textsl{The Raw and the Stolen},
\textit{Cooking and the Ecology of Human Origins}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/CurrAnth40-567-Wrangham.pdf}%
{Current Anthropology \textbf{40} (1999), 567--594}.
\par
{\footnotesize\setlength{\parindent}{1em}
Cooking is a human universal that must have had widespread effects on
the nutrition, ecology, and social relationships of the species that
invented it. The location and timing of its origins are unknown, but
it should have left strong signals in the fossil record. We suggest
that such signals are detectable at ca. 1.9 million years ago in the
reduced digestive effort (e.g., smaller teeth) and increased supply of
food energy (e.g., larger female body mass) of early Homo erectus. The
adoption of cooking required delay of the consumption of food while it
was accumulated and/ or brought to a processing area, and
accumulations of food were valuable and stealable. Dominant (e.g.,
larger) individuals (typically male) were therefore able to scrounge
from subordinate (e.g., smaller) individuals (typically female)
instead of relying on their own foraging efforts. Because female
fitness is limited by access to resources (particularly energetic
resources), this dynamic would have favored females able to minimize
losses to theft. To do so, we suggest, females formed protective
relationships with male co-defenders. Males would have varied in their
ability or willingness to engage effectively in this relationship, so
females would have competed for the best food guards, partly by
extending their period of sexual attractiveness. This would have
increased the numbers of matings per pregnancy, reducing the intensity
of male intrasexual competition. Consequently, there was reduced
selection for males to be relatively large. This scenario is supported
by the fossil record, which indicates that the relative body size of
males fell only once in hominid evolution, around the time when H.
erectus evolved. Therefore we suggest that cooking was responsible for
the evolution of the unusual human social system in which pair bonds
are embedded within multifemale, multimale communities and supported
by strong mutual and frequently conflicting sexual interest.\par
\par}
%
\bibitem[Wynn 1981]{Wy81}~\\*
Thomas Wynn,
\newblock \textsl{The Intelligence of Oldowan Hominids}.
\newblock
\href{http://axel.berger-odenthal.de/PDF/journals/JHumEvo010-0529-Wynn.pdf}%
{Journal of Human Evolution \textbf{10} (1981), 529--541}.
\par
{\footnotesize\setlength{\parindent}{1em}
This article uses Piagetian genetic epistemology to evaluate the
intelligence of Oldowan hominids. From the analysis of the geometry of
two-million-year-old artifacts from Olduvai Gorge it is concluded that
the hominids who made the tools possessed pre-operational
intelligence. Pre-operational intelligence employs such organizational
features as trial-and-error and control of single variables but lacks
such important modern features as true classification and pre-%
correction of errors. Pre-operational intelligence is also typical of
modern pongids. This implies that Oldowan hominids were not remarkably
intelligent by hominoid standards and that evolution of intelligence
was not significant in human evolution until after about 1.6 million
years ago, at which time it became an important factor in the rapid
increase in reliance on culture.\par
\par}
%
\par
\end{thebibliography}
\end{document}
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https://melusine.eu.org/syracuse/B/BaseBrevet/dta/a2001/geoplane/2001exo09.tex?enregistrement=ok
|
eu.org
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CC-MAIN-2021-21
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application/x-tex
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application/x-tex
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crawl-data/CC-MAIN-2021-21/segments/1620243988774.18/warc/CC-MAIN-20210506235514-20210507025514-00598.warc.gz
| 311,962,718 | 1,100 |
%@Titre: Réunion -- 2001
\par\compo{1}{reunion2001}{1}{Le triangle ci-contre représente un triangle $EST$, isocèle en $E$.\\ $[TH]$ est la hauteur issue de $T$.\\{\em Il n'est pas demandé de reproduire la figure.}\\ On sait que:
\begin{itemize}
\item $ES=ET=12$~cm (les dimensions ne sont pas respectées sur la figure);
\item l'aire du triangle $EST$ est de 42~cm$^2$.
\end{itemize}
}
\begin{myenumerate}
\item Prouver que $TH=7$~cm.
\item Calcule l'angle $\widehat{TES}$ (on donnera sa valeur arrondie au degré près).
\item En déduire une valeur approchée de l'angle $\widehat{EST}$.
\end{myenumerate}
|
https://defelement.com/img/element-serendipity_Hdiv-quadrilateral-1-0.tex
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defelement.com
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CC-MAIN-2022-21
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application/x-tex
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text/x-matlab
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crawl-data/CC-MAIN-2022-21/segments/1652662522309.14/warc/CC-MAIN-20220518183254-20220518213254-00346.warc.gz
| 253,345,198 | 5,342 |
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|
http://www.orsj.or.jp/nc2012s/sample/sample-jis.tex
|
orsj.or.jp
|
CC-MAIN-2021-43
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application/x-tex
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text/x-matlab
|
crawl-data/CC-MAIN-2021-43/segments/1634323585516.51/warc/CC-MAIN-20211022145907-20211022175907-00085.warc.gz
| 132,713,926 | 4,126 |
%
% アブストラクトのサンプルファイル (JIS)
%
% prepared by Keisuke Inakawa (inakawa＠nanzan-u.ac.jp) and
% Mihiro Sasaki (mihiro＠nanzan-u.ac.jp)
%
\documentclass[twoside,twocolumn,11pt]{jarticle} % 2段組の場合
%\documentclass[11pt]{jarticle} % 1段組の場合
\usepackage{latexsym,amssymb}
\usepackage[dvips]{graphicx} % epsファイルを使う場合
\usepackage{orsabs-jis}
%%%%%%%%%% Title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{アブストラクト書式のサンプル}
\author{\begin{tabular}{lll@{}ll}
0101XXX3 & 凹凸大学 & *&学会太郎 & GAKKAI Tarou \\
0209XXX5 & 凸凹企画 & &中部花子 & NAKABE Hanako
\end{tabular}}
\date{}
\begin{document}
\maketitle
%%%%%%%%%% ここから本文%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{はじめに}
%このファイルは，2006年OR学会秋季研究発表会で発表を希望される方が
このファイルは，20XX年OR学会\{春，秋\}季研究発表会で発表を希望される方が
\LaTeXe \cite{okumura,otobe97}を用いてアブストラクトを執筆する
際の原稿作成例です．
このファイルとスタイルファイル(orsabs-jis.sty)をお使いいただけば，
余白等の設定をしていただかなくても，すでに設定済みです．
\LaTeXe をご使用になる際は，出来るだけこのファイルを
利用して原稿を作成することをお勧めいたします．
原稿の構成については特に決まりはありません
ので，自由な形式で原稿作成していただいて結構です．
ただし，アブストラクト集作成上問題が発生する可能性がありますので，
余白の変更は避けてください．
各ページには，ヘッダを差し込む都合上，特に上マージンの変更は厳禁です．
\section{発表者の方へ}
\newcommand{\TM}{$^{\circledR}$}
%日本オペレーションズ・リサーチ学会２００６年秋季研究発表会に発表を希望される方は，以下の手順に従ってアブストラクトの原稿を作成してください．
日本オペレーションズ・リサーチ学会20XX年\{春，秋\}季研究発表会に発表を希望される方は，以下の手順に従ってアブストラクトの原稿を作成してください．
Acrobat\TM Distiller\TM の初期設定はレターサイズになっています．A4 サイズの PDF を作成する場合，{\bf 初期設定のままではできません}ので\underline{PDF の作成方法} をご参照下さい．
\section{アブストラクトの書き方}
アブストラクト集は著者の原稿をそのままフォトコピーして，Ｂ５版にオフセット印刷します．形式が不備の場合は印刷ができない場合がございますので，アブストラクト作成の際には以下の注意書きをお読みいただくよう，お願いいたします．
\begin{itemize}
\item アブストラクト原稿は発表１件につき２ページです．Ａ４版用紙で印刷してください．各ページの余白は上下30mm，左右20mmとしてください．余白部分には統一したヘッダーとして書名，フッターとしてページ番号が挿入されますので，必ず，空白のままにしておいてください．縮小印刷されますので，フォントサイズは（本文，図表とも）9pt．以上でお願いします．アブストラクト集はモノクロ印刷されます．
\item 発表題目，発表者氏名・所属は規定の位置に書いてください．
\begin{enumerate}
\item 発表題目は１枚目の最上段に本文より大きめのフォントを使い書いてください．
\item １行空けてその下に，発表者の「会員番号」，「所属」，「氏名」，「ローマ字読み」を書いてください．ローマ字読みは姓，名の順，姓はすべて大文字，名は頭文字だけ大文字，としてください．連名の場合は同じ形式で全員の氏名を書き，登壇者の姓の前に＊印を付けてください．
\end{enumerate}
\item 図・表・写真などは縮小されても識別できるように，また，モノクロ印刷しても識別できるように，投稿する前にあらかじめテスト印刷して仕上がりを確かめてください．
\\ アブストラクトはAdobe社のAcrobat
\footnote{Acrobat\TM, Reader\TM, Adobe\TM, Distiller\TM は，Adobe システムズ社の商標登録です．}
などでPDF形式に変換してください．Acrobat Readerでは変換できません．
%変換したアブストラクト原稿にはabst\_name.pdfという名前をつけてください．(nameの部分は発表者等の名前（姓のみ）のローマ字読みを入れてください．例：abst\_nakabe.pdf)
\item PDF形式に変換する際は，フォントをすべてインクルードするようにして下さい．
\item PDF形式に変換したファイルをAdobe 社のAcrobat Readerで印刷し，読めることを確認して下さい．フォントの文字化けが生じる可能性がありますので，できれば，環境の違うパソコンでも仕上がりを確かめてください．
（お願い
ワープロで作成したファイルはサイズが大きくなりがちで，ファイル転送の際にトラブルが予想されます．できるだけ上記５，６，７に従ってPDFファイルをご用意ください）
\end{itemize}
\begin{description}
%\item[注意１] アブストラクトのファイル名は「abst\_name.pdf」としてください．name の部分は発表者等の名前（姓のみ）のローマ字読みを入れてください．（例：abst\_nakabe.pdf）
\item[注意] 図表・写真が多い場合はファイルサイズが大きくなり，途中で転送を拒否される場合もありますので，あらかじめご確認ください．
\end{description}
\subsection{\TeX の余白の設定}
内容を記述する前に余白の設定をおこなう TeX での余白設定は一通りではないが，たとえばプリアンブルに以下；
\begin{verbatim}
%%% 上下 3cm, 左右 2cm の余白を定義 %%%
\paperwidth 597pt
\paperheight 845pt
\hoffset -14.0pt
\voffset 14.5pt
\oddsidemargin 0.0pt
\evensidemargin 0.0pt
\topmargin 0.0pt
\headheight 0.0pt
\headsep 0.0pt
\textheight 671.0pt
\textwidth 480.5pt
\marginparsep 0.0pt
\marginparwidth 0.0pt
\footskip 0.0pt
\end{verbatim}
を記述し，これ以外で長さに関する設定を行わなければ， 上下3cm，左右2cm の余白が確保できる． 簡単におこなう場合は，スタイルファイル： orsabs-jis.sty を用いるのがよい．スタイルファイルは，
\begin{verbatim}
\documentclass[twoside,twocolumn,11pt]
{jarticle}
\usepackage[dvips]{graphicx}
\usepackage{latexsym}
\usepackage{orsabs}
\end{verbatim}
などとして読み込む．
%詳しくは，サンプルファイル ( sample.tex , orsabs.sty ) を参考にしてください． サンプルファイルは，リンク上で [右クリック] - [対象をファイルに保存] するとダウンロードできます．
\subsection{WORD の余白設定}
内容を記述する前に余白の設定をおこなう．
\begin{itemize}
\item ~[ファイル/ ページ設定] をクリックし，ページ設定ウィンドウを表示させる．
\item ~[余白] タブをクリックし，余白セクションで以下：
\begin{verbatim}
上 30 mm 下 30 mm
左 20 mm 右 20 mm
とじしろ 0 mm
\end{verbatim}
と設定する．
\item ~[OK] をクリックし，ページ設定ウィンドウを閉じる．
\end{itemize}
\section{PDF の作成方法 }
これらの設定を正しくおこなっても，PDF に変換するとき余白が変更される場合があります．これは，正しく Acrobat Distiller を設定していないため起こるようです．
Acrobat Distiller の初期設定は レターサイズ であり，A4 サイズではありません．これに気づかず，そのまま A4 サイズの PDF を作成すると， 内容が上にずれるような感じを受けます．
Acrobat Distiller の設定をした覚えのない人は，( [ PDF の作成 ] \underline{Acrobat Distiller を用いて作成}) を参照し，正しく Acrobat Distiller を設定してください．
\section{PDF の印刷方法 }
\begin{enumerate}
\item~ PDF を起動し，メニューの[ファイル/印刷]で印刷ウィンドウを開く.
\item~ [ページの拡大/縮小] を "なし" にする． [自動回転と中央配置] にチェックがついている場合は，チェックをはずす．
\\ これらが付いたままになっていると，余白がずれたり， 内容が全体的に縮小されたりします． 必ずこの設定を確認してください．
\item~ [ OK ] (または [印刷]) をクリックして印刷する． 印刷後は，余白の設定などが正しく印刷されていることを確認する．
\end{enumerate}
\section{おわりに}
このファイルに関するご質問等がありましたら，作成者までお問い合わせ下さい．
%%%%%%%%% ここから参考文献 %%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{9}
\bibitem{okumura}
奥村晴彦: [改訂版] \LaTeXe 美文書作成入門,
技術評論社 (2000).
\bibitem{otobe97}
乙部厳己: p\LaTeXe for WINDOWS Another Manual,
ソフトバンク(1997).
\end{thebibliography}
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\begin{center}
\vskip 1cm{\LARGE\bf Extremely Abundant Numbers
\vskip .1in
and the Riemann Hypothesis
}
\vskip 1cm
\large
Sadegh Nazardonyavi and Semyon Yakubovich\\
Departamento de Matem\'{a}tica\\
Faculdade de Ci\^{e}ncias\\
Universidade do Porto\\
4169-007 Porto\\
Portugal\\
\href{mailto:[email protected]}{\tt [email protected]}\\
\href{mailto:[email protected]}{\tt [email protected]}\\
\end{center}
\vskip .2 in
\begin{abstract}
Robin's theorem states that the Riemann hypothesis is equivalent to the
inequality $\sigma(n)<e^\gamma n\log\log n$ for all $n>5040$, where
$\sigma(n)$ is the sum of divisors of $n$ and $\gamma$ is Euler's
constant. It is natural to seek the first integer, if
it exists, that violates this inequality. We introduce the sequence of
extremely abundant numbers, a subsequence of superabundant numbers,
where one might look for this first violating integer. The Riemann
hypothesis is true if and only if there are infinitely many extremely
abundant numbers. These numbers have some connection to the colossally
abundant numbers. We show the fragility of the Riemann hypothesis with
respect to the terms of some supersets of extremely abundant numbers.
\end{abstract}
\section{Introduction}\label{sec-introduction}
There are several statements equivalent to the famous Riemann hypothesis (RH) \cite{Bor}. Some of them are related to the asymptotic behavior of arithmetic functions. In particular, the well-known Robin theorem (inequality, criterion, etc.) deals with the upper bound of the sum-of-divisors function $\sigma$, which is defined by $\sigma(n):=\sum_{d|n}\, d$. Robin \cite[Theorem 1]{Rob} established an elegant connection between RH and the sum of divisors of $n$ by proving that the RH is true if and only if
\begin{equation}\label{Rob1}
\frac{\sigma(n)}{n\log\log n}<e^\gamma,\qquad\text{for all}\quad n>5040,
\end{equation}
where $\gamma$ is Euler's constant.
Throughout the paper, as in Robin \cite{Rob}, we define the function $f$ by setting
\begin{equation}\label{f(n)}
f(n)=\frac{\sigma(n)}{n\log\log n}.
\end{equation}
Gronwall, in his study of the asymptotic maximal
size for the sum of divisors of $n$ \cite{GR},
found that the order of $\sigma(n)$ is always ``very nearly $n$''
\cite[Theorem 323]{HW}. More precisely, he proved the following theorem.
\begin{theorem}[Gronwall] Let $f$ be defined as in (\ref{f(n)}). Then
\begin{equation}
\limsup_{n\rightarrow\infty}f(n)=e^\gamma.
\end{equation}
\end{theorem}
Let us call a positive integer $n$ \cite{Al, Ram.N}
\begin{itemize}
\item[(i)] \textit{colossally abundant}, if for some $\varepsilon>0$,
\begin{align}\label{CA-defn}
\frac{\sigma(n)}{n^{1+\varepsilon}}\geq\frac{\sigma(m)}{m^{1+\varepsilon}},\qquad(m<n)\qquad\text{and}\qquad
\frac{\sigma(n)}{n^{1+\varepsilon}}>\frac{\sigma(m)}{m^{1+\varepsilon}},\qquad (m>n);
\end{align}
\item[(ii)] \textit{generalized superior highly composite}, if there is a positive number $\varepsilon$ such that
$$
\frac{\sigma_{-s}(n)}{n^\varepsilon}\geq\frac{\sigma_{-s}(m)}{m^\varepsilon},\qquad(m<n)\qquad\text{and}\qquad \frac{\sigma_{-s}(n)}{n^\varepsilon}>\frac{\sigma_{-s}(m)}{m^\varepsilon}, \qquad(m>n),
$$
where $\sigma_{-s}(n)=\sum_{d\mid n}d^{-s}$. The parameter $s$ is assumed to be positive in \cite{Ram.N}. In the case $s=1$, (ii) becomes (i).
\end{itemize}
Ramanujan initiated the study of these classes of numbers in an unpublished part of his 1915 work on highly composite numbers (\cite{Ram.h,Ram.N} and
\cite[pp.\ 78--129, 338--339]{Ram.c}). More precisely, he defined rather general classes of these numbers. For instance, he defined generalized highly composite numbers, containing as a subset superabundant numbers \cite[Section 59]{Ram.h}, and he introduced the generalized superior highly composite numbers, including as a particular case colossally abundant numbers. For more details we refer the reader to \cite{Al, Erd, Ram.N}.
We denote by CA the set of all colossally abundant numbers. We also use CA as an abbreviation for the term ``colossally abundant''. Ramanujan \cite{Ram.N} proved that if $n$ is a generalized superior highly composite number, i.e., a CA number, then under the RH we have
\begin{align*}
\liminf_{n\rightarrow\infty}\left(\frac{\sigma(n)}{n}-e^{\gamma}\log\log n\right)\sqrt{\log n}\geq& -e^\gamma(2\sqrt{2}+\gamma-\log4\pi)\approx-1.558,\\[7pt]
\limsup_{n\rightarrow\infty}\left(\frac{\sigma(n)}{n}-e^{\gamma}\log\log n\right)\sqrt{\log n}\leq& -e^\gamma(2\sqrt{2}-4-\gamma+\log4\pi)\approx-1.393.
\end{align*}
Robin \cite{Rob} also established (independent of the RH) the following inequality
\begin{equation}\label{Rob2}
f(n)\leq e^\gamma+\frac{0.648214}{(\log\log n)^2}\ ,\qquad (n\geq3),
\end{equation}
where $0.648214\approx(\frac{7}{3}-e^\gamma\log\log12)\log\log12$ and the left-hand side of (\ref{Rob2}) attains its maximum at $n=12$. In the same spirit, Lagarias \cite{Lag} proved the equivalence of the RH to the problem
$$
\sigma(n)\leq e^{H_n}\log H_n+H_n,\qquad(n\geq1),
$$
where $H_n:=\sum_{j=1}^n 1/j$ is the $n$th harmonic number.
Investigating upper and lower bounds for arithmetic functions, Landau \cite[pp.\ 216--219]{Land} obtained the following limits:
$$
\liminf_{n\rightarrow\infty}\frac{\varphi(n)\log\log n}{n}=e^{-\gamma}\ ,\qquad \limsup_{n\rightarrow\infty}\frac{\varphi(n)}{n}=1,
$$
where $\varphi(n)$ is the Euler totient function, which is defined as the number of positive integers not exceeding $n$ that are relatively prime to $n$. It can also be expressed as a product extended over the distinct prime divisors of $n$ \cite[Theorem 2.4]{Apostol} by
$$
\varphi(n)=n\prod_{p\mid n}\left(1-\frac1p\right).
$$
Nicolas \cite{Nic, Nic2} proved that if the RH is true, then we have for all $k\geq2$
\begin{equation}\label{nicolas-ineq}
\frac{N_k}{\varphi(N_k)\log\log N_k}>e^\gamma,
\end{equation}
where $N_k=\prod_{j=1}^k p_j$ and $p_j$ is the $j$th prime. On the other hand, if the RH is false, then for infinitely many $k$, inequality (\ref{nicolas-ineq}) is true and for infinitely many $k$, inequality (\ref{nicolas-ineq}) is false.
Compared to numbers $N_k$ which are the smallest integers that maximize $n/\varphi(n)$, there are integers which play this role for $\sigma(n)/n$ and they are called superabundant numbers. A positive integer $n$ is said to be \textit{superabundant} \cite{Al, Ram.N} if
\begin{equation}\label{sa}
\frac{\sigma(n)}{n}>\frac{\sigma(m)}{m}\qquad\mbox{for all }\ m<n.
\end{equation}
We will use the symbol SA to denote the set of superabundant numbers and also as an abbreviation for the term ``superabundant''.
Briggs \cite{Briggs} described a computational study of the successive maxima of the relative sum-of-divisors function $\sigma(n)/n$. He also studied the density of these numbers. W\'{o}jtowicz \cite{Wojtowicz} showed that the values of the function $f$ defined in~ (\ref{f(n)}) are close to 0 on a set of asymptotic density 1. Another study on Robin's inequality~ (\ref{Rob1}) is due to Choie et al.\ \cite{Choie}. They have shown that the RH holds true if and only if every natural number divisible by a fifth power greater than 1 satisfies Robin's inequality (\ref{Rob1}).
Akbary and Friggstad \cite{Ak} established the following interesting theorem which enables us to limit our attention to a narrow sequence of positive integers, in order to find a probable counterexample to inequality (\ref{Rob1}).
\begin{theorem}[{\cite[Theorem 3]{Ak}}]
If there is any counterexample to Robin's inequality (\ref{Rob1}), then the least such counterexample is a superabundant number.
\end{theorem}
Unfortunately, to our knowledge, there is no known algorithm (except the formula (\ref{sa}) in the definition) to produce SA numbers. Alaoglu and Erd\H{o}s \cite{Al} proved that
$$
Q(x)> c\,\frac{\log x\log\log x}{(\log\log\log x)^2},
$$
where $Q(x)$ denotes the number of superabundant numbers not exceeding $x$. Later, Erd\H{o}s and Nicolas \cite{Erd} demonstrated a stronger result that for every $\delta<5/48$ we have
$$
Q(x)>(\log x)^{1+\delta},\qquad(x>x_0).
$$
As a natural question in this direction, it is interesting to introduce and study a set of positive integers to which the first probable violation of inequality (\ref{Rob1}) belongs. Following this aim, we introduce the sequence of \emph{extremely abundant numbers}. We will establish another criterion equivalent to the RH by proving that the RH is true if and only if there are infinitely many extremely abundant numbers. Also, we give a connection between extremely abundant numbers and CA numbers. Moreover, we present approximate formula for the prime factorization of (sufficiently large) extremely abundant numbers.
Finally, we establish the fragility of the Riemann hypothesis with
respect to the terms
of certain subsets of superabundant numbers which are quite close to
the set of extremely abundant numbers.
Before stating the main definition and results of this paper,
we mention recent work of Caveney et al.\ \cite{Caveney}. They defined a positive integer $n$ as an \textit{extraordinary} number if $n$ is composite and $f(n)\geq f(kn)$ for all
$$
k\in \mathbb{N}\cup\{1/p:\ p\ \text{is a prime factor of}\ n\}.
$$
Under these conditions they showed that the smallest extraordinary number is $n = 4$. Then they proved that the RH is true if and only if 4 is the only extraordinary number. For more properties of these numbers and comparisons with SA and CA numbers, we refer the reader to \cite{Caveney2}.
\section{Extremely abundant numbers}
We define a new sequence of positive integers related to the RH. Our
primary contribution and motivation of this definition are Theorems
\ref{ext} and \ref{rh}.
Let us now state the main definition of this paper.
\begin{definition}\label{extn}
A positive integer $n$ is \textit{extremely abundant} if either $n=10080$, or $n>10080$ and
\begin{equation}\label{extn1}
\frac{\sigma(n)}{n\log\log n}>\frac{\sigma(m)}{m\log\log m},\qquad \text{for all }\ 10080\leq m<n.
\end{equation}
\end{definition}
Here 10080 has been chosen as the smallest SA number greater than 5040. In Table \ref{tabl} we list the first 20 extremely abundant numbers. To find them, we used a list of SA numbers (see~ Proposition \ref{XA-subset-SA}) provided by Kilminster \cite{Kil} and Noe \cite{Noe}.
\begin{remark}
If we choose (instead of 10080) $n_1$ such that $2520<n_1\leq5040$, and define $n$ to be extremely abundant if either $n=n_1$, or $n>n_1$ and
\begin{equation*}
\frac{\sigma(n)}{n\log\log n}>\frac{\sigma(m)}{m\log\log m},\qquad \text{for all }\ n_1\leq m<n,
\end{equation*}
then we have a finite number of elements $n\leq5040$ that satisfy the above inequality. Using inequality (\ref{Rob2}), we have
$$
f(n)<e^\gamma+\frac{0.648214}{(\log\log n)^2}<f(5040),\qquad\text{for some } s_{10308}<n\leq s_{10309},
$$
where $s_k$ denotes the $k$th SA number listed in \cite{Noe}. Checking by computer for values $n$ between $5040$ and $s_{10309}$, we derive a finite set with maximum $5040$. Similarly, one can easily check for $n_1<2520$, and get only sets with finite number of elements.
\end{remark}
Let XA denote the set of all extremely abundant numbers. (We also use XA as an abbreviation for the term ``extremely abundant''.) Clearly, $XA\neq CA$ (see Table \ref{tabl}). Indeed, we shall prove that infinitely many elements of CA are not in XA and that, if RH holds, then infinitely many elements of XA are in CA. As an elementary result from the definition of XA numbers we have the following proposition.
\begin{proposition}\label{XA-subset-SA}
The inclusion $XA\subset SA$ holds.
\end{proposition}
\begin{proof}
First, $10080\in SA$. Next, if $n>10080$ and $n\in XA$, then for $10080\leq m<n$ we have
$$
\frac{\sigma(n)}{n}=f(n)\log\log n>f(m)\log\log m=\frac{\sigma(m)}{m}.
$$
In particular, for $m=10080$ we get
$$
\frac{\sigma(n)}{n}>\frac{\sigma(10080)}{10080}.
$$
So for $m<10080$, we have
$$
\frac{\sigma(n)}{n}>\frac{\sigma(10080)}{10080}>\frac{\sigma(m)}{m}
$$
since $10080\in SA$. Therefore, $n$ belongs to SA.
\end{proof}
Next, motivating our construction of XA numbers,
we will establish the first interesting result of the paper.
\begin{theorem}\label{ext}
If there is any counterexample to Robin's inequality (\ref{Rob1}), then the least one is an XA number.
\end{theorem}
\begin{proof}
By doing some computer calculations we observe that there is no counterexample to Robin's inequality (\ref{Rob1}) for $5040<n\leq10080$. Now let $n>10080$ be the least counterexample to inequality (\ref{Rob1}). For $m$ satisfying $10080\leq m<n$ we have
$$
f(m)<e^\gamma\leq f(n).
$$
Therefore $n$ is an XA number.
\end{proof}
As we mentioned in Section \ref{sec-introduction} we will prove an equivalent criterion to the RH for which the proof is based on Robin's inequality (\ref{Rob1}) and Gronwall's theorem. Let $\#A$ denote the cardinality of the set $A$. The second stimulus result is the following theorem. This result also has its own interest that will be discussed in Section \ref{sec-delicacy-of-RH}.
\begin{theorem}\label{rh}
The RH is true if and only if $\# XA=\infty$.
\end{theorem}
\begin{proof}
\emph{Sufficiency.} Assume that RH is not true. Then by Theorem \ref{ext} we have $f(m)\geq e^\gamma$ for some $m\geq10080$. From Gronwall's theorem, we
know that $M=\sup_{n\geq10080}f(n)$ is finite and there exists $n_0$ such that $f(n_0)=M\geq e^\gamma$ (if $M=e^\gamma$ then set $n_0=m$). An integer $n>n_0$ satisfies $f(n)\leq M=f(n_0)$ and $n$ can not be in XA, so $\#XA\leq n_0$.
\emph{Necessity.} \ On the other hand, if RH is true, then Robin's inequality (\ref{Rob1}) holds. If $\# XA$ is finite, then let $m$ be its largest element. For every $n > m$ the inequality $f(n)\le f(m)$ holds and therefore
$$
\limsup_{n\rightarrow\infty} f(n)\leq f(m)<e^\gamma,
$$
which is a contradiction to Gronwall's theorem.
\end{proof}
There are some primes which cannot be the largest prime factor of any XA number. For example, referring to Table \ref{tabl}, there is no XA number with the largest prime factor $p(n)=149$. Do there exist infinitely many such primes?
\section{Auxiliary lemmas and inequalities}
Chebyshev's functions $\vartheta(x)$ and $\psi(x)$ are defined by
$$
\vartheta(x)=\sum_{p\leq x}\log p,\qquad\psi(x)=\sum_{p^m\leq x}\log p=\sum_{p\leq x}\left\lfloor\frac{\log x}{\log p}\right\rfloor\log p,
$$
where $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$. It is known that the prime number theorem (PNT) (\cite[Theorem 434]{HW} and \cite[Theorem 3, 12]{Ingham}) is equivalent to
\begin{equation}\label{psi(x)-x}
\psi(x)\sim x.
\end{equation}
In his m\'{e}moir, Chebyshev proved the following lemma that we call
Chebyshev's result.
\begin{lemma}[{\cite[p.\ 379]{Chebyshev}}]
For all $x>1$
\begin{align*}
\vartheta(x)&<\frac65Ax-Ax^{\frac12}+\frac{5}{4\log6}\log^2x+\frac52\log x+2\\
\vartheta(x)&>Ax-\frac{12}{5}Ax^{\frac12}-\frac{5}{8\log6}\log^2x-\frac{15}{4}\log x-3,
\end{align*}
where
$$
A=\log\frac{2^\frac12 3^\frac13 5^\frac15}{30^\frac{1}{30}}\approx0.92129202.
$$
\end{lemma}
We will use the following corollary in the proof of Theorem \ref{p<log(n)-prop}.
\begin{corollary}\label{chebyshev's-result}
We have
$$
\vartheta(x)>\frac x3,\qquad(x\geq3).
$$
\end{corollary}
The next lemma here provides Littlewood's result for oscillation of Chebyshev's $\vartheta$ function.
\begin{lemma}[{\cite[Lemma 4]{Caveney2}}]\label{lemma-4,caveney2}
There exists a constant $c > 0$ such that for infinitely many primes $p$ we have
\begin{equation}\label{theta(p)-p<cp1/2logloglog(p)}
\vartheta(p)<p-c\sqrt{p}\log\log\log p,
\end{equation}
and for infinitely many other primes $p$ we have
$$
\vartheta(p)>p+c\sqrt{p}\log\log\log p.
$$
\end{lemma}
In what follows, we shall frequently use the following elementary inequalities:
\begin{equation}\label{log(1+t)}
\frac{t}{1+t}<\log(1+t)< t,\qquad (t>0),
\end{equation}
and
\begin{equation}\label{log(1+t)2}
\frac{2t}{2+t}<\log (1+t),\qquad(t>0).
\end{equation}
\section{Some properties of SA, CA and XA numbers}
We divide this section into three subsections, for which we shall exhibit several properties of SA, CA and XA numbers, respectively. We denote by $p(n)$ the largest prime factor of $n$ or, when there is no ambiguity, simply by $p$.
\subsection{SA Numbers}
\begin{proposition}\label{an+1<=2an}
Let $n<n'$ be two consecutive SA numbers. Then
$$
\frac{n'}{n}\leq2.
$$
\end{proposition}
\begin{proof}
Let $n=2^{k_2}\cdots p$. We compare $n$ with $2n$. In fact,
$$
\frac{\sigma(2n)/(2n)}{\sigma(n)/n}=\frac{2^{k_2+2}-1}{2^{k_2+2}-2}>1.
$$
Hence $n'\leq2n$.
\end{proof}
\begin{corollary}
For any positive real number $x\geq1$ there exists at least one SA number $n$ such that $x\leq n<2x$.
\end{corollary}
Alaoglu and Erd\H{o}s \cite{Al} have shown that if $n=2^{k_2}\cdot3^{k_3}\cdots p^{k_p}$ is an SA number, then $k_2\geq k_3\geq\cdots \geq k_p$ and $k_p=1$, except for $n=4,\ 36$.
\begin{proposition}[{\cite[Theorem 2]{Al}}]
Let $n\in SA$, and let $q<r$ be prime factors of $n$ with corresponding exponents $k_q$ and $k_r$. Set
$$
\beta:= \left\lfloor\frac{k_q \log q}{\log r}\right\rfloor.
$$
Then $k_r$ has one of the three values: $\beta-1$, $\beta+1$, $\beta$.
\end{proposition}
As we observe, the above proposition determines the exponent of each prime factor of an SA number with error of at most 1 in terms of a smaller prime factor of that number. In the next lemma we prove a relation between the lower bound of an exponent of a prime factor of $n$ and its largest prime factor $p$.
\begin{lemma}\label{log(p)/log(q)<kq}
Let $n\in SA$, and let $q$ be a prime factor of $n$. Then
$$
\left\lfloor\frac{\log p}{\log q}\right\rfloor\leq k_q.
$$
\end{lemma}
\begin{proof}
If $q = p\;(=p(n))$, then the result is trivial. Let $q < p$ and $k_q=k$. Suppose that $k\leq~\lfloor\log p/\log q\rfloor-1$. Then
\begin{equation}\label{qk<p}
q^{k+1}<p.
\end{equation}
Now we compare values of $\sigma(\nu)/\nu$, taking $\nu=n$ and $\nu=m=nq^{k+1}/p$. Since $\sigma(\nu)/\nu$ is multiplicative, we restrict our attention to different factors. But $n$ is an SA number and~ $m<n$. Hence
$$
1<\frac{\sigma(n)/n}{\sigma(m)/m}=\frac{q^{2k+2}-q^{k+1}}{q^{2k+2}-1}\left(1+\frac{1}{p}\right)=\frac{1}{1+1/q^{k+1}}\left(1+\frac{1}{p}\right).
$$
Consequently, $p<q^{k+1}$, which contradicts inequality (\ref{qk<p}).
\end{proof}
\begin{proposition}[{\cite[Theorem 5]{Al}}]
Let $n\in SA$. If $k_q=k$ and $q<(\log p)^{\alpha}$, where $\alpha$ is a constant, then
\begin{equation}\label{log-q(k+1)}
\log\frac{q^{k+1}-1}{q^{k+1}-q}>\frac{\log q}{p\log p}\left(1+O\left(\frac{(\log\log p)^2}{\log p\log q}\right)\right),
\end{equation}
\begin{equation}\label{log-q(k+2)}
\log\frac{q^{k+2}-1}{q^{k+2}-q}<\frac{\log q}{p\log p}\left(1+O\left(\frac{(\log\log p)^2}{\log p\log q}\right)\right).
\end{equation}
\end{proposition}
\begin{lemma}\label{cplogp<2k<c'plogp}
Let $n\in SA$, and let $q$ be a fixed prime factor of $n$. Then there exist two positive constants $c$ and $c\,'$ (depending on $q$) such that
$$
c\,p\frac{\log p}{\log q}<q^{k_q}<c\,'p\frac{\log p}{\log q}.
$$
\end{lemma}
\begin{proof}
By inequality (\ref{log(1+t)})
\begin{align*}
\log\frac{q^{k+1}-1}{q^{k+1}-q}=\log\left(1+\frac{q-1}{q^{k+1}-q}\right)<\frac{q-1}{q^{k+1}-q}\leq\frac{1}{q^k}
\end{align*}
and (\ref{log-q(k+1)}), there exists a $c\,'>0$ such that
\begin{equation*}
q^k<c\,'\,\frac{p\log p}{\log q}.
\end{equation*}
On the other hand, again by inequality (\ref{log(1+t)})
\begin{align*}
\log\frac{q^{k+2}-1}{q^{k+2}-q}=\log\left(1+\frac{q-1}{q^{k+2}-q}\right)>\frac{q-1}{q^{k+2}-1}>\frac{1}{2q^{k+1}}
\end{align*}
and (\ref{log-q(k+2)}), there exists a $c>0$ such that
$$
q^k>c\,\frac{p\log p}{\log q}.
$$
\end{proof}
\begin{corollary}
Let $n=2^{k}\cdots p$ be an SA number. Then for $n$ sufficiently large we have
$$
\left\lfloor\frac{k\log2}{\log p}\right\rfloor=1.
$$
\end{corollary}
We showed \cite{Naz} that for sufficiently large $n\in SA$
\begin{equation}\label{log(n)<p(1+.5/log p)}
\log n<p\left(1+\frac{1}{2\log p}\right).
\end{equation}
Our computation on the list of SA numbers in \cite{Noe} suggests that a weaker inequality
\begin{equation}\label{log(n)<p(1+2/3log(p))}
\log n<p\left(1+\frac{2}{3\log p}\right)
\end{equation}
holds for all $n\geq s_{365}$. The product of exponent of a prime factor and the logarithm of the corresponding prime factor of an SA number can be
controlled, on average, by the logarithm of the largest prime factor of that number. More precisely,
\begin{proposition}[{\cite[Theorem 7]{Al}}]\label{p=log}
If $n\in SA$, then
$$
p(n)\sim\log n.
$$
\end{proposition}
The next proposition gives a lower bound of $n\in SA$ in terms of Chebyshev's $\psi$ function compared to the above-mentioned asymptotic relation.
\begin{proposition}
Let $n\in SA$. Then
\begin{equation*}
\psi(p(n))\leq\log n.
\end{equation*}
Moreover,
\begin{equation}\label{psilog}
\lim_{\substack{n\rightarrow\infty\\n\in SA}}\frac{\psi(p(n))}{\log n}=1.
\end{equation}
\end{proposition}
\begin{proof} In fact, by Lemma \ref{log(p)/log(q)<kq}
$$
\psi(p(n))=\sum_{q\leq p(n)}\left\lfloor\frac{\log p(n)}{\log q}\right\rfloor\log q\leq\sum_{q\leq p(n)}k_q\log q=\log n.
$$
To prove (\ref{psilog}) we appeal to (the equivalent of) the PNT (\ref{psi(x)-x}) and Proposition \ref{p=log}.
\end{proof}
\subsection{CA Numbers}
By the definition of CA numbers (\ref{CA-defn}) it is easily seen that $CA\subset SA$. Here
we give a concise description of the algorithm (essentially borrowed from \cite{Caveney2, Erd, Rob}) to produce CA numbers. For more details on this introduction we refer the reader to \cite{Al, Caveney2, Erd, Rob}.
Let $F$ be defined by
\begin{equation*}\label{F(x,k)}
F(x,k)=\frac{\log(1+1/(x+\cdots+x^k))}{\log x}.
\end{equation*}
For $\varepsilon>0$ we define $x_1=x_1(\varepsilon)$ to be the only number such that
\begin{equation}\label{x(epsilon)}
F(x_1,1)=\varepsilon,
\end{equation}
and $x_k=x_k(\varepsilon)$ (for $k>1$) to be the only number such that
$$
F(x_k,k)=\varepsilon.
$$
Let
$$
E_p=\{F(p,\alpha):\ \alpha\geq1\},\ \ p\ \text{is a prime}
$$
and
$$
E=\bigcup_{p}E_p=\{\varepsilon_1,\,\varepsilon_2,\,\ldots\}.
$$
If $\varepsilon\notin E$, then the function $\sigma(n)/n^{1+\varepsilon}$ attains its maximum at a single point $N_\varepsilon$ whose prime decomposition is
$$
N_\varepsilon=\prod p^{\alpha_p(\varepsilon)},\qquad \alpha_p(\varepsilon)=\left\lfloor\frac{\log\frac{p^{1+\varepsilon}-1}{p^\varepsilon-1}}{\log p}\right\rfloor-1,
$$
or if one prefers
$$
\alpha_p(\varepsilon)=
\begin{cases}
k, & \text{if $x_{k+1}<p<x_k$, $k\geq1$;} \\
0, & \text{if $p>x=x_1$.}
\end{cases}
$$
If $\varepsilon\in E$, then at most two $x_k$'s are prime. Hence, there are either two or four CA numbers of parameter $\varepsilon$, defined by
\begin{equation}\label{CA-N=prod-two-or-four}
N_\varepsilon=\prod_{k=1}^K\prod_{\substack{p<x_k\\\text{or}\\p\leq x_k}}p,
\end{equation}
where $K$ is the largest integer such that $x_K\geq2$. In particular, if $N$ is the largest CA number of parameter $\varepsilon$, then
\begin{equation}\label{p(N)=p}
F(p,1)=\varepsilon\Rightarrow p(N)=p,
\end{equation}
where $p(N)$ is the largest prime factor of $N$. Therefore for any $\varepsilon$, formula (\ref{CA-N=prod-two-or-four}) gives all possible values of a CA number $N$ of parameter $\varepsilon$ \cite{Caveney2}.
Robin \cite[Proposition 1]{Rob} proved that the maximum order of the function $f$ defined in~ (\ref{f(n)}) is attained by CA numbers. More precisely, if $3\leq N< n< N'$, where $N$ and $N'$ are two successive CA numbers, then
\begin{equation*}\label{f(n)<f(N),f(N')}
f(n)\le \max\{f(N ), f(N\,')\}.
\end{equation*}
In the next proposition we improve the above inequality to a strict one.
\begin{proposition}\label{Robin-prop}
Let $3\leq N< n< N'$, where $N$ and $N'$ are two successive CA numbers. Then
\begin{equation}\label{f(n)<max{f(N),f(N')}}
f(n)<\max\{f(N), f(N')\}.
\end{equation}
\end{proposition}
\begin{proof}
In fact, due to the strict convexity of the function $t\mapsto \varepsilon t-\log\log t$,
Robin's proof extends to the strict inequality (\ref{f(n)<max{f(N),f(N')}}).
\end{proof}
Proposition \ref{Robin-prop} shows that if there is a counterexample to inequality (\ref{Rob1}), then there exists at least one CA number that violates it.
\begin{lemma}\label{N<n<N'-ca-xa}
Let $N<N'$ be two consecutive CA numbers. If there exists an XA number $n>10080$ satisfying $N<n<N'$, then $N'$ is also an XA.
\end{lemma}
\begin{proof}
Let us set
$$
B=\{m\in XA:\ N<m<N'\}.
$$
By assumption $n\in XA$, we have $B\neq\emptyset$. Let $n'=\max B$. Since $n'\in XA$ and $n'>N$, it follows that $f(n')>f(N)$. From inequality (\ref{f(n)<max{f(N),f(N')}}) we must have $f(n')<f(N')$. Hence~ $N'\in~ XA$.
\end{proof}
\begin{remark}\label{N<n<N'-ca-xa-remark}
If $n=10080$, then we have $N=5040$ and $N'=55440$, and
$$
f(N)\approx1.7909,\qquad f(n)\approx1.7558,\qquad f(N')\approx1.7512.
$$
Hence, $f(N')<f(n)<f(N)$ and inequality (\ref{f(n)<max{f(N),f(N')}}) is satisfied with
$$
f(n)<f(N)=\max\{f(N),\ f(N')\}.
$$
Therefore $N'\notin XA$. The point here is that $n=10080$ is the initial XA number, so that it misses the property (\ref{extn1}) of the definition of XA numbers which is used in the proof of Lemma~ \ref{N<n<N'-ca-xa}.
\end{remark}
\begin{theorem}\label{RH->CA-are-XA}
If RH holds, then there exist infinitely many CA numbers that are also XA.
\end{theorem}
\begin{proof}
If RH holds, then $\#XA=\infty$ by Theorem \ref{rh}. Let $n\in XA$. Since $\#CA=\infty$ \cite{Al, Erd}, there exist two successive CA numbers $N,\ N'$ such that $N<n\leq N'$. If~ $N'=n$ then it is already in XA, otherwise $N'$ belongs to XA via Lemma \ref{N<n<N'-ca-xa}.
\end{proof}
It can be seen that there exist infinitely many CA numbers $N$ for which the largest prime factor $p\ (=p(N))$ is greater than $\log N$. For this purpose we use the following lemma.
\begin{lemma}[{\cite[Lemma 3]{Caveney2}}]\label{log(N)<theta+cx1/2}
Let $N$ be a CA number of parameter $\varepsilon$ with
$$
\varepsilon<F(2, 1) = \log(3/2)/ \log 2
$$
and define $x=x(\varepsilon)$ by (\ref{x(epsilon)}). Then
\begin{itemize}
\item[{\upshape (i)}] for some constant $c >0$
$$
\log N\leq\vartheta(x)+c\sqrt{x}.
$$
\item[{\upshape (ii)}] Moreover, if $N$ is the largest CA number of parameter $\varepsilon$, then
$$
\vartheta(x)\leq\log N\leq\vartheta(x)+c\sqrt{x}.
$$
\end{itemize}
\end{lemma}
\begin{theorem}\label{log(N)<p(N)}
There are infinitely many CA numbers $N_\varepsilon$ such that $\log N_\varepsilon<p(N_\varepsilon)$.
\end{theorem}
\begin{proof}
Let $p$ be sufficiently large satisfying the inequality (\ref{theta(p)-p<cp1/2logloglog(p)}), and let $N_\varepsilon$ be the largest CA number of parameter
$$
\varepsilon=F(p,1).
$$
Then from (\ref{p(N)=p}) it follows that $p(N_\varepsilon) = p$. By part (ii) of Lemma \ref{log(N)<theta+cx1/2} we have
\begin{align*}
\log N_\varepsilon-\vartheta(p)<c\sqrt{p},\qquad(\text{for some}\ c>0).
\end{align*}
On the other hand, by Lemma \ref{lemma-4,caveney2} there exists a constant $c' > 0$ such that
$$
\vartheta(p)-p<-c'\sqrt{p}\log\log\log p,\qquad(c'>0).
$$
Hence
$$
\log N_\varepsilon-p<\{c-c'\log\log\log p\}\sqrt{p}<0,
$$
which is the desired conclusion.
\end{proof}
\subsection{XA Numbers}
We return to XA numbers and present some of their properties. We begin by the first interesting property of the XA numbers whose proof is essentially an application of the definition of XA numbers.
\begin{theorem}\label{p<log(n)-prop}
Let $n\in XA$. Then
$$
p(n)<\log n.
$$
\end{theorem}
\begin{proof}
For $n=10080$ we have
$$
p(10080)=7<9.218<\log(10080).
$$
Let $n>10080$ be an XA number and $m=n/p(n)$. Then $m>10080$, since for all primes $p$ we have $\vartheta(p)>p/3$ (Corollary \ref{chebyshev's-result}). Thus for $n\in SA$ we have
$$
\log n\geq\psi(p(n))\geq\vartheta(p(n))>\frac13p(n)
$$
and $m=n/p(n)>n/(3\log n)>10080$ if $n\geq400,000$. For $n<400,000$ we can check by computation. Hence by Definition \ref{extn} we obtain
$$
1+\frac1{p(n)}=\frac{\sigma(n)/n}{\sigma(m)/m}>\frac{\log\log n}{\log\log m}.
$$
Therefore,
$$
\frac{1}{p(n)}>\frac{\log\log n}{\log\log m}-1=\frac{\log(1+\log {p(n)}/\log m)}{\log\log m}.
$$
Using inequality (\ref{log(1+t)}) we have
$$
\frac{1}{p(n)}>\frac{\log {p(n)}}{\log n\log\log m}>\frac{\log {p(n)}}{\log n\log\log n}\Rightarrow {p(n)}<\log n.
$$
\end{proof}
We mention here a similar result proved by Choie et al.\ \cite{Choie}.
\begin{proposition}[{\cite[Lemma 6.1]{Choie}}]
Let $t\geq2$ be fixed. Suppose that there exists a $t$-free integer exceeding $5040$ that does not satisfy Robin's inequality (\ref{Rob1}). Let $n$ be the smallest such integer. Then $p(n)<\log n$.
\end{proposition}
In Theorem \ref{RH->CA-are-XA} we showed that if RH holds, then there exist infinitely many CA numbers that are also XA. Next theorem is a conclusion of Theorems \ref{log(N)<p(N)} and \ref{p<log(n)-prop} which is independent of the RH.
\begin{theorem}
There exist infinitely many CA numbers that are not XA.
\end{theorem}
We conclude this subsection with a result describing the structure of (sufficiently large) XA numbers. More precisely, the next theorem will determine the exponents of the prime factors of a (sufficiently large) XA number with an error at most 1.
\begin{theorem}
Let $n=2^{k_2}\cdots q^{k_q}\cdots p\in XA$. Set
$$
\alpha_q(p)=\left\lfloor\log_q\left(1+(q-1)\frac{p\log p}{q\log q}\right)\right\rfloor.
$$
Then for sufficiently large $n\in XA$ we have $|k_q-\alpha_q(p)|\leq1$.
\end{theorem}
\begin{proof}
Assume that $k_q=k$ and $k-\alpha_q(p)\geq2$. Then we have
\begin{equation}\label{q-to-k>}
q^{k}\geq q^{\alpha_q(p)+2}> q\left(1+(q-1)\frac{p\log p}{q\log q}\right).
\end{equation}
Now let us compare $f(n)$ with $f(m)$, where $m=n/q$. Since $n\in XA$ we must have
$$
\frac{\sigma(n)/n}{\sigma(m)/m}=\frac{q^{k+1}-1}{q^{k+1}-q}>\frac{\log\log n}{\log\log m},
$$
or using inequality (\ref{log(1+t)}),
\begin{equation}\label{q-to-k<}
q^k<1+(q-1)\frac{\log n\log\log m}{q\log q}.
\end{equation}
Comparison of (\ref{q-to-k>}) and (\ref{q-to-k<}) gives that
$$
\log n\log\log m-q p\log p>q\log q.
$$
This contradicts inequality (\ref{log(n)<p(1+.5/log p)}). Now we assume that $k-\alpha_q(p)\leq-2$. Then
\begin{equation}\label{q-to-k+2/q-1<plogp/qlogq}
\frac{q^{k+2}-1}{q-1}\leq\frac{p\log p}{q\log q}.
\end{equation}
Choose $m=nq/p$. Comparing $f(n)$ with $f(m)$ we have
$$
\frac{\sigma(n)/n}{\sigma(m)/m}=\left(1-\frac{q-1}{q^{k+2}-1}\right)\left(1+\frac1p\right)>\frac{\log\log n}{\log\log m}>1+\frac{\log p/q}{\log n\log\log m}.
$$
or simply by (\ref{q-to-k+2/q-1<plogp/qlogq})
\begin{equation*}
-\frac{q\log q}{\log p}\left(1+\frac1p\right)>\frac{p\log p/q}{\log n\log\log m}-1.
\end{equation*}
Hence by (\ref{log(n)<p(1+.5/log p)}) we have
$$
-\frac{q\log q}{\log p}\left(1+\frac1p\right)>\frac{\log p/q}{(\log p)\left(1+\frac{1}{2\log p}\right)\left(1+\frac{1}{2\log^2 p}\right)}-1,
$$
which is false for all $2\leq q\leq p$.
\end{proof}
\begin{remark}
Note that if inequality (\ref{log(n)<p(1+2/3log(p))}) holds for XA numbers $n\geq s_{365}$, then by performing computations for two smaller values of XA numbers, i.e., $s_{20}$ and $s_{356}$ (cf.\ Table \ref{tabl}) we see that the above theorem holds true for all $n\in XA$.
\end{remark}
\section{Fragility of the RH and certain supersets of XA numbers}\label{sec-delicacy-of-RH}
In Theorem \ref{rh} we proved that under the RH the cardinality of the set of XA numbers is infinite. Here we present some interesting theorems which demonstrate the fragility of the RH showing the infinitude of some supersets of XA numbers independent of the RH. These sets are defined by inequalities quite close to that in (\ref{extn1}). The basic inequalities used here to define these sets are (\ref{log(1+t)}) and (\ref{log(1+t)2}).
\begin{lemma}\label{(s(n)/n)/(s(m)/m)>1+log(n/m)/(log(n)loglog(m))}
If $m\geq3$, then there exists $n>m$ such that
\begin{equation*}
\frac{\sigma(n)/n}{\sigma(m)/m}>1+\frac{\log n/m}{\log n\log\log m}.
\end{equation*}
\end{lemma}
\begin{proof}
Let $m\geq3$. Then by inequality (\ref{Rob2})
\begin{equation}\label{3.41}
\frac{\sigma(m)}{m}\leq\left(e^\gamma+\frac{0.648214}{(\log\log m)^2}\right)\log\log m.
\end{equation}
Since for $m'>m$
$$
\frac{\log\log m}{\log\log m'}\left(1+\frac{\log m'/m}{\log m'\log\log m}\right)<1
$$
and the left-hand side is decreasing with respect to $m'$ and tends to zero as $m'\rightarrow\infty$, then for some $m'>m$ we have
\begin{equation}\label{3.42}
\frac{\log\log m}{\log\log m'}\left(1+\frac{\log m'/m}{\log m'\log\log m}\right)\left(e^\gamma+\frac{0.648214}{(\log\log m)^2}\right)=e^\gamma-\varepsilon,
\end{equation}
where $\varepsilon>0$ is arbitrarily small and fixed. Hence by Gronwall's theorem there exists $n\geq m'$ such that
\begin{align*}
\frac{\sigma(n)}{n}&>(e^\gamma-\varepsilon)\log\log n\\
&=\frac{\log\log m}{\log\log m'}\left(1+\frac{\log m'/m}{\log m'\log\log m}\right)\left(e^\gamma+\frac{0.648214}{(\log\log m)^2}\right)\log\log n\\
&\geq\left(1+\frac{\log n/m}{\log n\log\log m}\right)\frac{\sigma(m)}{m},
\end{align*}
where the last inequality holds by (\ref{3.41}) and (\ref{3.42}).
\end{proof}
\begin{definition}
Let $n_1=10080$, and let $n_{k+1}$ be the least integer greater than $n_k$ such that
$$
\frac{\sigma(n_{k+1})/n_{k+1}}{\sigma(n_k)/n_k}>1+\frac{\log n_{k+1}/n_k}{\log n_{k+1}\log\log n_k},\qquad(k=1,\,2,\,\ldots).
$$
We define $X'$ to be the set of all $n_1$, $n_2$, $n_3$, \ldots.
\end{definition}
One can easily show that
\begin{equation}\label{3.43}
XA\subset X'\subset SA.
\end{equation}
Our first result towards the fragility of the RH is the following theorem.
\begin{theorem}\label{3.57}
The set $X'$ has an infinite number of elements.
\end{theorem}
\begin{proof}
If the RH is true, then the cardinality of $X'$ is infinite by (\ref{3.43}). If RH is not true, then by Theorem \ref{rh} there exists $m_0\geq10080$ such that
$$
\frac{\sigma(m_0)/m_0}{\sigma(m)/m}\geq\frac{\log\log m_0}{\log\log m},\qquad\text{for all}\ m\geq10080.
$$
By Lemma \ref{(s(n)/n)/(s(m)/m)>1+log(n/m)/(log(n)loglog(m))} there exists $m'>m_0$ such that $m'$ satisfies
$$
\frac{\sigma(m')/m'}{\sigma(m_0)/m_0}>1+\frac{\log m'/m_0}{\log m'\log\log m_0}.
$$
Let $n$ be the least number greater than $m_0$ for which
$$
\frac{\sigma(n)/n}{\sigma(m_0)/m_0}>1+\frac{\log n/m_0}{\log n\log\log m_0}.
$$
Hence $n\in X'$.
\end{proof}
The following lemma can be proved in the same manner as Lemma \ref{(s(n)/n)/(s(m)/m)>1+log(n/m)/(log(n)loglog(m))}.
\begin{lemma}
If $m\geq3$, then there exists $n>m$ such that
\begin{equation*}
\frac{\sigma(n)/n}{\sigma(m)/m}>1+\frac{2\log n/m}{(\log m+\log n)\log\log m}.
\end{equation*}
\end{lemma}
We continue our approach towards the fragility of the RH via (the stronger) inequality (\ref{log(1+t)2}) defining a smaller superset of XA numbers as follows.
\begin{definition}
Let $n_1=10080$, and let $n_{k+1}$ be the least integer greater than $n_k$, such that
$$
\frac{\sigma(n_{k+1})/n_{k+1}}{\sigma(n_k)/n_k}>1+\frac{2\log n_{k+1}/n_k}{(\log n_k+\log n_{k+1})\log\log n_k},\qquad(k=1,\,2,\,\ldots).
$$
We define $X''$ to be the set of all $n_1$, $n_2$, $n_3$, \ldots.
\end{definition}
By elementary inequality (\ref{log(1+t)2}) and
$$
\frac{t}{1+t}<\frac{2t}{2+t},\qquad(t>0)
$$
one can easily show the inclusion $XA\subset X''\subset X'$. The following theorem is a refinement of
Theorem \ref{3.57} with a similar proof.
\begin{theorem}
The set $X''$ has an infinite number of elements.
\end{theorem}
We calculated the number of elements in $XA$, $X'$ and $X''$ up to the $300,000$th element of SA in \cite{Noe}. Note that
$$
\#XA=9240,\qquad\#X'=9535,\qquad\#X''=9279
$$
and
$$
\#(X'-XA)=295,\qquad\#(X''-XA)=39.
$$
It might be interesting to look at the list of elements of $X''-XA$ up to $s_{300,000}$:
\begin{align*}
X''-XA&=\{s_{55}, s_{62}, s_{91}, s_{106}, s_{116}, s_{127}, s_{128}, s_{137}, s_{138}, s_{149}, s_{181}, s_{196}, s_{212}, s_{219},\\
&\hspace{22pt}s_{224}, s_{231}, s_{232}, s_{246}, s_{247}, s_{259}, s_{260}, s_{263}, s_{272}, s_{273}, s_{276}, s_{288}, s_{294},\\
&\hspace{22pt}s_{299}, s_{305}, s_{311}, s_{317}, s_{330}, s_{340}, s_{341}, s_{343}, s_{354}, s_{65343}, s_{271143}, s_{271151}\}
\end{align*}
Note that the second XA number is $s_{356}$ (see Table \ref{tabl}) and
only three out of 39 elements in the set $X''-XA$ up to $s_{300,000}$,
namely $s_{65343}$, $s_{271143}$ and $s_{271151}$, are greater than $s_{356}$.
\section{Numerical experiments}
We present here some numerical results, mainly for the set of XA
numbers (sorted in increasing order) up to 13770th element, which is
less than $C_1:=s_{500,000}$, based on the list provided by Noe~\cite{Noe}.
\begin{property}
Let $n=2^{k_2}\cdots q^{k_q}\cdots r^{k_r}\cdots p\in XA$, where $2\leq q<r\leq p$. Then for $10080<n\leq C_1$
\begin{itemize}
\item[{\upshape (i)}] $\log n<q^{k_q+1}$,
\item[{\upshape (ii)}] $r^{k_r}<q^{k_q+1}<r^{k_r+2}$,
\item[{\upshape (iii)}] $q^{k_q}<k_qp$,
\item[{\upshape (iv)}] $q^{k_q}\log q<\log n\log\log n<q^{k_q+2}$.
\end{itemize}
\end{property}
\begin{property}
Let $n=2^{k_2}\cdots x_k^k\cdots p\in XA$, where $2<x_k<p$ is the greatest prime factor of exponent $k$. Then
$$
\sqrt{p}<x_2<\sqrt{2p},\qquad\text{for}\ \ 10080<n\leq C_1.
$$
\end{property}
\begin{property}
Let $n=2^{k_2}\cdots q^{k_q}\cdots p$ and $n'=2^{k'_2}\cdots q^{k'_q}\cdots p'$ be two consecutive XA numbers. Then for $10080<n<n'\leq C_1$
$$
|k_q-k'_q|\leq1,\qquad \mbox{for all}\ \ 2\leq q\leq p'.
$$
\end{property}
\begin{property}\label{d-p}
If $m$, $n$ are XA numbers, then for $10080\leq m<n\leq C_1$
\begin{itemize}
\item[{\upshape (i)}] $p(m)\leq p(n)$,
\item[{\upshape (ii)}] $d(m)\leq d(n)$.
\end{itemize}
\end{property}
\begin{remark}
We note that Property \ref{d-p} is not true for SA numbers. For instance,
$$
s_{47}=(19\sharp)(3\sharp)^22,\qquad s_{48}=(17\sharp)(5\sharp)(3\sharp)2^3,
$$
$$
p(s_{47})=19>17=p(s_{48}).
$$
and
$$
s_{173}=(59\sharp)(7\sharp)(5\sharp)(3\sharp)^22^3,\qquad s_{174}=(61\sharp)(7\sharp)(3\sharp)^22^2,
$$
$$
d(s_{173})=5308416>5160960=d(s_{174}).
$$
\end{remark}
\begin{property}
If $n$, $n'\in XA$ are consecutive, then for $10080\leq n<n'<C_1$
$$
\frac{n'}{n}>1+c\,\frac{(\log\log n)^2}{\log n},\qquad(0<c\leq4),
$$
$$
\frac{n'}{n}>1+c\,\frac{(\log\log n)^2}{\sqrt{\log n}},\qquad(0<c\leq0.195).
$$
\end{property}
\begin{property}
If $n$, $n'\in XA$ are consecutive, then for $10080\leq n<n'<C_1$
$$
\frac{f(n')}{f(n)}<1+\frac{1}{p\,'},
$$
where $p\,'$ is the largest prime factor of $n'$.
\end{property}
We have checked the following properties up to $C_2=s_{250,000}$ and up to 8150th element of XA numbers which is less than $C_2$.
\begin{property}
If $n$, $n'\in SA$ are consecutive, then
$$
\frac{\sigma(n')/n'}{\sigma(n)/n}<1+\frac{1}{p\,'},\qquad(n'<C_2),
$$
where $p\,'$ is the largest prime factor of $n'$.
\end{property}
The number of distinct prime factors of a number $n$ is denoted by $\omega(n)$ (see \cite{Sandor}).
\begin{property}
If $n$, $n'\in XA$ are consecutive, then for $10080\leq n<n'<C_2$, then
$$
\frac{n}{\omega(n)}\leq\frac{n'}{\omega(n')}.
$$
\end{property}
The comparison of the sets CA and XA is given. We calculated them up to
$C=s_{1,000,000}$ from the list of SA in \cite{Noe}.
\begin{align*}
\#&\{n\in XA: n<C\}=24875,\\ \#&\{n\in CA: n<C\}=21187,\\ \#&\{n\in
CA\cap XA: n<C\}=20468,\\ \#&\{n\in XA\setminus CA: n<C\}=4407,\\
\#&\{n\in CA\setminus XA: n<C\}=719, \end{align*} We conclude this
paper with another remark on choosing the first element of XA as
10080.
\begin{remark}
If we replace $n_1=10080$, the initial number in the definition of XA numbers, by $n_1=665280$, we do not need to pose the condition (i.e., $n>n_1$) in Lemma \ref{N<n<N'-ca-xa}. Indeed, if we choose the initial number $n_1=665280$, then $N=55440<665280<720720=N'$, where in this case $N'$ is also an XA number. Therefore we do not need Remark \ref{N<n<N'-ca-xa-remark}. Moreover, if we choose $n_1=665280$, then there are 37 more XA numbers.
\end{remark}
\begin{table}[!h]
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
& $n$ &Type& $f(n)$ & $p(n)$ & $\log n$ & $k_2$\\
\hline
1&$\ \quad s_{20}=(7\sharp)(3\sharp)2^3=10080$ & & $1.75581$ & $7$ & $9.21831$ & $5$\\[3pt]
2&$s_{356}=(113\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^3$&c & $1.75718$ & $113$ & $126.444$ & $8$\\
3&$s_{368}=(127\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^3$&c & $1.75737$ & $127$ & $131.288$ & $8$\\
4&$s_{380}=(131\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^3$&c & $1.75764$ & $131$ & $136.163$ & $8$\\
5&$s_{394}=(137\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^3$&c & $1.75778$ & $137$ & $141.083$ & $8$\\
6&$s_{408}=(139\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^3$&c & $1.75821$ & $139$ & $146.018$ & $8$\\
7&$s_{409}=(139\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^4$&c & $1.75826$ & $139$ & $146.711$ & $9$\\
8&$s_{438}=(151\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^3$& & $1.75831$ & $151$ & $156.039$ & $8$\\
9&$s_{440}=(151\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^4$&c & $1.75849$ & $151$ & $156.732$ & $9$\\
10&$s_{444}=(151\sharp)(13\sharp)(7\sharp)(3\sharp)^2 2^4$&c & $1.75860$ & $151$ & $158.678$ & $9$\\
11&$s_{455}=(157\sharp)(13\sharp)(5\sharp)(3\sharp)^2 2^4$& & $1.75864$ & $157$ & $161.788$ & $9$\\
12&$s_{458}=(157\sharp)(13\sharp)(7\sharp)(3\sharp)^2 2^3$& & $1.75866$ & $157$ & $163.041$ & $8$\\
13&$s_{459}=(157\sharp)(13\sharp)(7\sharp)(3\sharp)^2 2^4$ &c & $1.75892$ & $157$ & $163.734$ & $9$ \\
14&$s_{476}=(163\sharp)(13\sharp)(7\sharp)(3\sharp)^2 2^4$ &c & $1.75914$ & $163$ & $168.828$ & $9$ \\
15&$s_{486}=(163\sharp)(17\sharp)(7\sharp)(3\sharp)^2 2^4$ & & $1.75918$ & $163$ & $171.661$ & $9$ \\
16&$s_{493}=(167\sharp)(13\sharp)(7\sharp)(3\sharp)^2 2^4$ &c & $1.75943$ & $167$ & $173.946$ & $9$ \\
17&$s_{502}=(167\sharp)(17\sharp)(7\sharp)(3\sharp)^2 2^4$ &c & $1.75966$ & $167$ & $176.779$ & $9$ \\
18&$s_{519}=(173\sharp)(17\sharp)(7\sharp)(3\sharp)^2 2^4$ &c & $1.76006$ & $173$ & $181.933$ & $9$ \\
19&$s_{537}=(179\sharp)(17\sharp)(7\sharp)(3\sharp)^2 2^4$ &c & $1.76038$ & $179$ & $187.120$ & $9$ \\
20&$s_{555}=(181\sharp)(17\sharp)(7\sharp)(3\sharp)^2 2^4$ &c & $1.76089$ & $181$ & $192.318$ & $9$ \\
\hline
\end{tabular}
\centering
\caption{First 20 extremely abundant numbers ($p_k\sharp:=\prod_{j=1}^k p_j$ and c represents a colossally abundant number).}\label{tabl}
\end{table}
\section{Acknowledgments}
We thank J. C. Lagarias, C. Calderon, J. Stopple, and M. Wolf for useful
discussion and sending us some relevant references. Our sincere thanks
to J.-L. Nicolas for careful reading of the manuscript, helpful
comments, and worthwhile suggestions which improved the
presentation of the paper. The work of the first author is supported by
the Calouste Gulbenkian Foundation, under Ph.D.\ grant number
CB/C02/2009/32. Research partially funded by the European Regional
Development Fund through the programme COMPETE and by the Portuguese
Government through the FCT under the project PEst-C/MAT/UI0144/2011.
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T. M. Apostol,
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Springer-Verlag, 1976.
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\emph{The Riemann Hypothesis: A
Resource for the Afficionado and Virtuoso Alike},
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K. Briggs,
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G. Caveney, J.-L. Nicolas, and J. Sondow,
Robin's theorem, primes, and a new elementary reformulation and the Riemann hypothesis,
\emph{Integers} \textbf{11} (2011), 753--763.
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G. Caveney, J.-L. Nicolas, and J. Sondow,
On SA, CA, and GA numbers,
\emph{Ramanujan J.} \textbf{29} (2012), 359--384.
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M\'{e}moir sur les nombres premiers,
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\emph{J. Th\'{e}or. Nombres Bordeaux} \textbf{19} (2007), 357--372.
\bibitem{Erd}
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R\'{e}partition des nombres superabondants,
\emph{Bull. Soc. Math. France} \textbf{103} (1975), 65--90.
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T. H. Gronwall,
Some asymptotic expressions in the theory of numbers,
\emph{Trans. Amer. Math. Soc.} \textbf{14} (1913), 113--122.
\bibitem{HW}
G. H. Hardy and E. M. Wright,
\emph{An Introduction to the Theory of Numbers},
Sixth ed., Oxford University Press, Oxford, 2008.
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A. E. Ingham,
\emph{The Distribution of Prime Numbers}, With a foreword by R. C. Vaughan,
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\bibitem{Kil}
D. Kilminster,
Table of $n$, $a(n)$ for $n$=0..2000, \newline
\url{http://oeis.org/A004394/b004394.txt}.
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J. C. Lagarias,
An elementary problem equivalent to the Riemann hypothesis,
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\bibitem{Land}
E. Landau,
\emph{Handbuch der Lehre von der Verteilung der Primzahlen},
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S. Nazardonyavi and S. Yakubovich,
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\emph{Acta Arith.} \textbf{155} (2012), 311--321.
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First 1,000,000 superabundant numbers,
\url{http://oeis.org/A004394}.
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S. Ramanujan,
Highly composite numbers,
\emph{Proc. Lond. Math. Soc.} \textbf{14} (1915), 347--407.
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S. Ramanujan,
\emph{Collected Papers}, Chelsea, 1962.
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\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2010 \emph{Mathematics Subject Classification}:
Primary 11A25; Secondary 11N37, 11Y70, 11K31.
\noindent \emph{Keywords: }
extremely abundant number,
superabundant number,
colossally abundant number, Robin's theorem, Chebyshev's function.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A004394},
\seqnum{A004490}, and
\seqnum{A217867}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received April 1 2013;
revised versions received August 16 2013; January 15 2014.
Published in {\it Journal of Integer Sequences}, February 7 2014.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in
\end{document}
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\documentclass[12pt,reqno]{article}
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\begin{document}
\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
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\begin{center}
\vskip 1cm{\LARGE\bf Coprime and Prime Labelings of Graphs}
\vskip 1cm
\begin{minipage}[t]{0.5\textwidth}
\begin{center}
Adam H. Berliner \\
Department of Mathematics, Statistics, and Computer Science \\
St.~Olaf College \\
Northfield, MN 55057 \\
USA\\
\href{mailto:[email protected]}{\tt [email protected]} \\
\ \\
Jonelle Hook \\
Department of Mathematics and Computer Science\\
Mount St.~Mary's University \\
Emmitsburg, MD 21727 \\
USA\\
\href{mailto:[email protected]}{\tt [email protected]}\\
\ \\
Aba Mbirika \\
Department of Mathematics\\
University of Wisconsin-Eau Claire \\
Eau Claire, WI 54702 \\
USA\\
\href{mailto:[email protected]}{\tt [email protected]}\\
\end{center}
\end{minipage}
%second column
\begin{minipage}[t]{0.4\textwidth}
\begin{center}
Nathaniel Dean \\
Department of Mathematics\\
Texas State University \\
San Marcos, TX 78666 \\
USA\\
\href{mailto:[email protected]}{\tt [email protected]}\\
\ \\
\ \\
Alison Marr \\
Department of Mathematics and Computer Science\\
Southwestern University \\
Georgetown, TX 78626 \\
USA\\
\href{mailto:[email protected]}{\tt [email protected]}\\
\ \\
Cayla D. McBee \\
Department of Mathematics and Computer Science\\
Providence College \\
Providence, RI 02918 \\
USA\\
\href{mailto:[email protected]}{\tt [email protected]}
\end{center}
\end{minipage}
\end{center}
\vskip .2 in
\begin{abstract}
A coprime labeling of a simple graph of order $n$ is a labeling in which adjacent vertices are given relatively prime labels, and a graph is prime if the labels used can be taken to be the first $n$ positive integers. In this paper, we consider when ladder graphs are prime and when the corresponding labeling may be done in a cyclic manner around the vertices of the ladder. Furthermore, we discuss coprime labelings for complete bipartite graphs.
\end{abstract}
\section{Introduction}\label{sec:Intro}
Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$,
where $n = |V|$ is the number of vertices of $G$. A \textit{coprime
labeling} of $G$ is a labeling of the vertices of $G$ with distinct
integers from the set $\{1, 2, \ldots, k\}$, for some $k \geq n$, in
such a way that the labels of any two adjacent vertices are relatively
prime. We then define $\pr(G)$ to be the minimum value of $k$ for
which $G$ has a coprime labeling. The corresponding labeling of $G$ is
called a \textit{minimal coprime labeling} of $G$.
If $\pr(G)=n$, then a corresponding minimal coprime labeling of $G$ is called a \textit{prime labeling} of $G$ and we call $G$ \textit{prime}. Though these definitions are more common, some of the literature uses the term \textit{coprime} to mean what we refer to as \textit{prime} in this paper (cf.~\cite{erdos-sarkozy96}). In our setting it does not make sense to refer to a graph as {\em coprime}, since all graphs have a coprime labeling (for example, use the first $n$ prime integers as the labels).
Much work has been done on various types of labeling problems, including coprime and prime graphs (see \cite{Gallian} for a detailed survey). For nearly 35 years, Entringer's conjecture that all trees are prime has remained unsolved. Some progress towards the result has been made. In 1994, Fu and Huang proved trees with 15 or fewer vertices are prime \cite{Fu-Huang94}. Pikhurko improved the result for trees of up to 50 vertices \cite{Pikhurko07} in 2007. In 2011, Haxell, Pikhurko, and Taraz \cite{HPT11} proved Entringer's conjecture for trees of sufficiently large order. Specifically, it is known that paths, stars, caterpillars, complete binary trees, and spiders are prime.
Many other classes of graphs have been studied as well, several of which are constructed from trees. If we let $P_n$ denote the path on $n$ vertices, then the Cartesian product $P_n\times P_m$, where $m\leq n$, is called a {\it grid graph}. Some results about prime labelings of grid graphs can be found in \cite{SPS06, Kanetkar}. If $m=2$, then the graph is called a {\it ladder}. Several results are known about ladders. For example, if $n$ and $k$ are prime, then $P_{n} \times P_{2}$, $P_{n+1} \times P_{2}$, $P_{n+k} \times P_{2}$, $P_{3n} \times P_{2}$, and $P_{n+2} \times P_{2}$ are prime \cite{VSN, SPS06, SPS07}. Ladders $P_{n} \times P_{2}$, $P_{n+1} \times P_{2}$, and $P_{n+2} \times P_{2}$ have also been shown to be prime when $2n+1$ is prime \cite{VSN,SPS07,Varkey}. In \cite{Varkey}, it is conjectured that all ladders are prime. While we cite the papers \cite{VSN,Varkey} here, we note that both of them contain some errors and incomplete proofs.
In Section~\ref{sec:ladders}, we further consider prime labelings for ladders. Moreover, we consider instances when a prime labeling exists where the labels occur in numerical order around the vertices of the ladder.
In Section~\ref{sec:bipartite}, we consider complete bipartite graphs $K_{m,n}$. In the case $m=n$, it is clear that $K_{n,n}$ is not prime when $n>2$. Thus, we focus on minimal coprime labelings. In the more general case of $m<n$, for each $m$ we give a sufficiently large lower bound value of $n$ for which $K_{m,n}$ is prime. Specifically, we give all values of $n$ for which $K_{m,n}$ is prime for $m\leq 13$.
In Section~\ref{sec:future_work}, we conclude with some possible directions for future work.
\section{Ladders}\label{sec:ladders}
In this section, we give labelings of ladders that are mainly constructed in a cyclic manner. As mentioned above, several classes of ladders have been shown to be prime. We reproduce some of the results of \cite{VSN} and \cite{SPS07}, but our constructions are arguably more elegant and complete than those given.
\begin{theorem}\label{thm:Deans_idea}
If $n+1$ is prime, then $P_n \times P_2$ has a prime labeling. Moreover, this prime labeling can be realized with top row labels from left to right, $1, 2, \ldots, n$, and bottom row labels from left to right, $n+2, n+3, \ldots, 2n, n+1$.
\end{theorem}
\begin{proof}
Consider the graph $P_n \times P_2$ where $n+1$ is prime. We claim that the following vertex labeling gives a prime labeling:
\begin{center}
\begin{tikzpicture}[xscale=1.5]
\draw (0,0) -- (1.5,0);
\draw (0,1) -- (1.5,1);
\draw (2.5,0) -- (4,0);
\draw (2.5,1) -- (4,1);
\foreach \x in {0,1,3,4}
\draw (\x,0) -- (\x,1);
\node at (2,0) {$\cdots$};
\node at (2,1) {$\cdots$};
\foreach \x in {0,1,3,4}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,3,4}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {1};
\node [above] at (1,1) {2};
\node [above] at (3,1) {$n-1$};
\node [above] at (4,1) {$n$};
\node [below] at (0,0) {$n+2$};
\node [below] at (1,0) {$n+3$};
\node [below] at (3,0) {$2n$};
\node [below] at (4,0) {$n+1$};
\end{tikzpicture}
\end{center}
Since $\gcd(k,k+1) = 1$, it suffices to check only the vertex labels arising from the endpoints of the following $n$ particular edges:
\begin{itemize}
\item the horizontal edge connecting vertex labels $2n$ and $n+1$, and
\item the first $n-1$ vertical edges going from left to right.
\end{itemize}
Since $n+1$ is prime and $2n < 2(n+1)$, then $n+1$ cannot divide $2n$. Hence, $\gcd(2n, n+1)=1$ as desired. Observe that each of the $n-1$ vertical edges under consideration have vertex labels $a$ and $(n+1)+a$ for $1 \leq a \leq n-1$. It follows that $\gcd((n+1)+a,a) = \gcd(n+1,a) = 1$. Thus, the graph $P_n \times P_2$ is prime whenever $n+1$ is prime.
\end{proof}
The remaining theorems involve consecutive cyclic prime labelings of ladders.
Let $P_n \times P_2$ be the ladder with vertices $v_1, v_2, \dots, v_n$ and $u_1, u_2, \dots, u_n$, where $v_i$ is adjacent to $u_i$ for $1\leq i\leq n$, $v_i$ is adjacent to $v_{i+1}$ for $1\leq i\leq n-1$, and $u_i$ is adjacent to $u_{i+1}$ for $1\leq i\leq n-1$. When drawing a ladder graph, we may assume without loss of generality that $v_1$ denotes the top left vertex of the graph.
\begin{definition}\label{def:ccpl}
A \textit{consecutive cyclic prime labeling} of a ladder $P_n \times P_2$ is a prime labeling in which the labels on the vertices wrap around the ladder in a consecutive way. In particular, if the label 1 is placed on vertex $v_i$, then 2 will be placed on $v_{i+1}$, $n-i+1$ will be placed on $v_n$, $n-i+2$ on $u_n$, $2n-i+1$ on $u_1$, $2n-i+2$ on $v_1$, and $2n$ will be placed on vertex $v_{i-1}$. A similar definition holds if 1 is placed on $u_i$.
\end{definition}
The reverse direction of the following theorem is stated and proved in \cite{VSN, Varkey}, but we include our own proof here for completeness.
\begin{theorem}\label{iff_theorem}
$P_n \times P_2$ has a consecutive cyclic prime labeling with the value $1$ assigned to vertex $v_1$ if and only if $2n + 1$ is prime.
\end{theorem}
\begin{proof}
We prove the forward implication by contradiction. Let $p>1$ be a divisor of $2n+1$. The following consecutive cyclic labeling of $P_n \times P_2$ with the value of 1 assigned to the top left vertex of the graph is not a prime labeling. The pair of vertices labeled $p$ and $2n-(p-1)=2n+1 -p$ are not relatively prime since $p\ |\ 2n+1$.
\begin{center}
\begin{tikzpicture}[xscale=1.5]
\draw (0,0) -- (2.5,0);
\draw (0,1) -- (2.5,1);
\draw (3.5,0) -- (4,0);
\draw (3.5,1) -- (4,1);
\draw (4,0) -- (4.5,0);
\draw (4,1) -- (4.5,1);
\foreach \x in {0,1,2,4}
\draw (\x,0) -- (\x,1);
\node at (3,0) {$\cdots$};
\node at (3,1) {$\cdots$};
\foreach \x in {0,1,2,4}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2,4}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node at (5,0) {$\cdots$};
\node at (5,1) {$\cdots$};
\node [above] at (0,1) {1};
\node [above] at (1,1) {2};
\node [above] at (2,1) {3};
\node [above] at (4,1) {$p$};
\node [below] at (0,0) {$2n$};
\node [below] at (1,0) {$2n-1$};
\node [below] at (2,0) {$2n-2$};
\node [below] at (4,0) {$2n-(p-1)$};
\end{tikzpicture}
\end{center}
Conversely, consider the graph $P_n \times P_2$ where $2n+1$ is prime. We claim that the following vertex labeling gives a consecutive cyclic prime labeling:
\begin{center}
\begin{tikzpicture}[xscale=1.5]
\draw (0,0) -- (1.5,0);
\draw (0,1) -- (1.5,1);
\draw (2.5,0) -- (4,0);
\draw (2.5,1) -- (4,1);
\foreach \x in {0,1,3,4}
\draw (\x,0) -- (\x,1);
\node at (2,0) {$\cdots$};
\node at (2,1) {$\cdots$};
\foreach \x in {0,1,3,4}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,3,4}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {1};
\node [above] at (1,1) {2};
\node [above] at (3,1) {$n-1$};
\node [above] at (4,1) {$n$};
\node [below] at (0,0) {$2n$};
\node [below] at (1,0) {$2n-1$};
\node [below] at (3,0) {$n+2$};
\node [below] at (4,0) {$n+1$};
\end{tikzpicture}
\end{center}
Since $\gcd(k,k+1) = 1$, it suffices to check only the vertex labels arising from the endpoints of the first $n-1$ vertical edges going from left to right. Observe that each of the $n-1$ vertical edges under consideration have vertex labels $a$ and $(2n+1)-a$ for $1 \leq a \leq n-1$. We conclude that
$$\gcd(a,(2n+1)-a) = \gcd(a,2n+1) = 1.$$
Thus, the graph $P_n \times P_2$ has a consecutive cyclic prime labeling whenever $2n+1$ is prime.
\end{proof}
When drawing a ladder, $n$ columns are formed consisting of a vertex from the first path, a vertex from the second path, and the edge between them. When we place labels on the vertices, we create $n$ column sums which are just the sum of the label on vertex $u_i$ and $v_i$ for $1\leq i\leq n$. When constructing a consecutive cyclic labeling, without loss of generality we place a value of 1 somewhere in the top row of the ladder and increase each next vertex label by one in a clockwise direction. Thus, there will be a value $k$ directly below the 1 depending on where the 1 is placed in the top row. This creates column sums of $k+1$ for columns to the right of 1 (and including the column with a 1) and column sums of $2n+k+1$ for columns to the left of the 1.
\begin{theorem}\label{pcolsumsmod2n_theorem}
For every consecutive cyclic labeling of $P_n\times P_2$, the column sums are congruent to $k+1$ modulo $2n$, where $k$ is the label on the vertex directly below the vertex with label 1. If $k+1$ is not prime, then the consecutive cyclic labeling is not a prime labeling.
\end{theorem}
\begin{proof}
Consider a ladder graph with the following consecutive cyclic labeling.
\begin{center}
\begin{tikzpicture}[xscale=1.2]
\draw (0.5,0) -- (3.5,0);
\draw (0.5,1) -- (3.5,1);
\node at (0,0) {$\cdots$};
\node at (0,1) {$\cdots$};
\node at (4,0) {$\cdots$};
\node at (4,1) {$\cdots$};
\foreach \x in {1,2,3}
\draw (\x,0) -- (\x,1);
\foreach \x in {1,2,3}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {1,2,3}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (1,1) {$2n$};
\node [above] at (2,1) {1};
\node [above] at (3,1) {2};
\node [below] at (1,0) {$k+1$};
\node [below] at (2,0) {$k$};
\node [below] at (3,0) {$k-1$};
\end{tikzpicture}
\end{center}
First, recall that each column sum in a consecutive cyclic labeling will be either $k+1$ or $2n+k+1$.
If $k+1$ is prime, then all the column sums are congruent to a prime modulo $2n$. So $k+1$ must be composite (which is an odd composite since $k$ must be even). Then $q\ |\ k+1$ for some prime $q$. Note that $k+1=q\cdot s > 2q$ which implies that $q < \dfrac{k+1}{2}$. In the above consecutive cyclic labeling, the labels $q$ and $k+1-q$ are (vertically) adjacent. However, gcd$(q,\ k+1-q)=q$ and so the consecutive cyclic labeling is not a prime labeling.
\end{proof}
The converse of Theorem \ref{pcolsumsmod2n_theorem} does not hold. If $k+1$ is prime, then it does not guarantee there exists a consecutive cyclic prime labeling, as the following example illustrates.
\begin{example}
Below is a consecutive cyclic labeling of $P_5\times P_2$ where $k+1$ equals $5$, but the labeling is not prime.
\begin{center}
\begin{tikzpicture}[xscale=1.2]
\draw (0,0) -- (4,0);
\draw (0,1) -- (4,1);
\foreach \x in {0,3,4}
\draw (\x,0) -- (\x,1);
\draw[dashed] (1,0)--(1,1);
\draw[dashed] (2,0)--(2,1);
\foreach \x in {0,1,2,3,4}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2,3,4}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {8};
\node [above] at (1,1) {9};
\node [above] at (2,1) {10};
\node [above] at (3,1) {1};
\node [above] at (4,1) {2};
\node [below] at (0,0) {7};
\node [below] at (1,0) {6};
\node [below] at (2,0) {5};
\node [below] at (3,0) {4};
\node [below] at (4,0) {3};
\end{tikzpicture}
\end{center}
\end{example}
\begin{lemma}\label{primecolsums} Consider a consecutive cyclic labeling of $P_n\times P_2$ and let $k$ be the label on the vertex directly below the vertex with label 1. If the column sums $k+1$ and $2n+k+1$ are prime, then the labeling is a consecutive cyclic prime labeling of $P_n \times P_2$.
\end{lemma}
\begin{proof}
Consider the graph $P_n \times P_2$ with a consecutive cyclic labeling and prime column sums $k+1$ and $2n+k+1$. To show this labeling is prime we must verify that the vertex labels arising from the endpoints of the vertical edges are relatively prime. Letting $k+1=p$ results in vertical pairs of labels $(a, p-a)$ for $1 \leq a \leq \frac{1}{2}(p-1)$ and $(2n-b, p+b)$ for $0 \leq b \leq n-\frac{1}{2}(p-1)$.
First we claim that gcd($a$, $p-a$)=1. Suppose gcd($a$, $p-a$)=$d$ for $d$ a positive integer. This implies $d\ |\ (p-a+a)$ and so $d\ |\ p$. Therefore, $d=1$ or $p$, but, $d \neq p$ since $d \leq a < p$. Thus, gcd($a$, $p-a$)=1.
Next we claim that gcd($2n-b$, $p+b$)=1. Suppose gcd($2n-b$, $p+b$)=$d$ for $d$ a positive integer. Then $d\ |\ ((2n-b) + (p+b))$ which is equivalent to $d\ |\ (2n+p)$. Since we assumed $2n+p$ is prime, $d=1$ or $2n+p$. However, $d \neq 2n+p$ since $d \leq 2n-b < 2n+p$. Therefore, gcd($2n-b$, $p+b$)=1.
Given the cyclic labeling of the graph, all horizontal edges connect labels that are relatively prime. Thus we can conclude that if the column sums $k+1$ and $2n+k+1$ are prime, then the labeling is a consecutive cyclic prime labeling.
\end{proof}
The converse of Lemma~\ref{primecolsums} is not true, as is shown in the following example.
\begin{example}
The labeling below is a consecutive cyclic prime labeling of $P_4\times P_2$ with column sums 7 and 15.
\begin{center}
\begin{tikzpicture}[xscale=1.2]
\draw (0,0) -- (3,0);
\draw (0,1) -- (3,1);
\foreach \x in {0,1,2,3}
\draw (\x,0) -- (\x,1);
\foreach \x in {0,1,2,3}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2,3}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {8};
\node [above] at (1,1) {1};
\node [above] at (2,1) {2};
\node [above] at (3,1) {3};
\node [below] at (0,0) {7};
\node [below] at (1,0) {6};
\node [below] at (2,0) {5};
\node [below] at (3,0) {4};
\end{tikzpicture}
\end{center}
\end{example}
\begin{theorem}\label{big_theorem}
If $2n+p$ is prime where $p$ is a prime less than $2n+1$, then $P_n \times P_2$ has a consecutive cyclic prime labeling. Moreover, this labeling can be realized by assigning $1$ to the vertex in the location $\frac{1}{2}(p-1)-1$ places from the top right vertex.
\end{theorem}
\begin{proof}
Let $2n+p$ be a prime where $p$ is prime less than $2n+1$. We claim that the following gives a consecutive cyclic labeling of $P_n \times P_2$:
\begin{center}
\begin{tikzpicture}[xscale=1.2]
\draw (0,0) -- (1.5,0);
\draw (0,1) -- (1.5,1);
\draw (2.5,0) -- (6.5,0);
\draw (2.5,1) -- (6.5,1);
\draw (7.5,0) -- (9,0);
\draw (7.5,1) -- (9,1);
\foreach \x in {0,1,3,4,5,6,8,9}
\draw (\x,0) -- (\x,1);
\node at (2,0) {$\cdots$};
\node at (2,1) {$\cdots$};
\node at (7,0) {$\cdots$};
\node at (7,1) {$\cdots$};
\node [right] at (3.95,.5) {$e_L$};
\node [right] at (4.95,.5) {$e_R$};
\foreach \x in {0,1,3,4,5,6,8,9}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,3,4,5,6,8,9}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {$L_T$};
\node [above] at (1,1) {$L_T+1$};
\node [above] at (3,1) {$2n-1$};
\node [above] at (4,1) {$2n$};
\node [above] at (5,1) {$1$};
\node [above] at (6,1) {$2$};
\node [above] at (8,1) {$R_T-1$};
\node [above] at (9,1) {$R_T$};
\node [below] at (0,0) {$L_B$};
\node [below] at (1,0) {$L_B-1$};
\node [below] at (3,0) {$p+1$};
\node [below] at (4,0) {$p$};
\node [below] at (5,0) {$p-1$};
\node [below] at (6,0) {$p-2$};
\node [below] at (8,0) {$R_B+1$};
\node [below] at (9,0) {$R_B$};
\end{tikzpicture}
\end{center}
where the four corner values are
\begin{align*}
L_T &= \frac{1}{2}(p+1)+n & \hspace{.5in} R_T &= \frac{1}{2}(p-1)\\
L_B &= \frac{1}{2}(p-1)+n & \hspace{.5in} R_B &= \frac{1}{2}(p-1)+1.
\end{align*}
Since $\gcd(k,k+1)=1$, it suffices to check only the vertex labels arising from the endpoints of the first $n-1$ vertical edges going from left to right. For the vertical edges right of (and including) $e_R$, the column sum is $p$. For the vertical edges left of (and including) $e_L$, the column sum is $2n+p$. Since both of these column sums are prime, Lemma~\ref{primecolsums} implies the ladder has a consecutive cyclic prime labeling.
\end{proof}
It is of interest to know if there exists a prime number of the form $2n+p$ where $n$ is an integer and $p = 1$ or $p$ is a prime less than $2n+1$. If such a prime exists, then by Theorem~\ref{iff_theorem} and Theorem~\ref{big_theorem} we may conclude that $\pr(P_n \times P_2) = 2n$ for all $n$ and also that every ladder has a consecutive cyclic prime labeling that we may easily construct.
Unfortunately, determining whether such a prime exists is a difficult problem related to Polignac's conjecture. Polignac's conjecture, first stated by Alphonse de Polignac in 1849~\cite{Polignac}, states that for any positive even integer $n$, there are infinitely many prime gaps of size $n$. If $n=2$, the conjecture is equivalent to the twin prime conjecture. In looking for cyclic prime labelings of ladder graphs, we are interested in finding pairs of primes that differ by the even number $2n$ where the smaller prime is less than $2n+1$. Although Polignac's conjecture guarantees the existence of pairs of primes whose difference is $2n$, the result remains as yet to be proved. Also, Polignac's conjecture does not address our additional constraint that the smaller prime be less than $2n+1$. In summary, we have the following observation:
\begin{observation} If every even integer $2n$ can be written in the form $2n=q-p$ where $q$ is a prime and $p$ is either 1 or a prime less than $2n+1$, then all ladder graphs are prime. \end{observation}
The labelings in the following example illustrate our results thus far.
\begin{example}
Consider $P_{10} \times P_2$. Theorem~\ref{iff_theorem} does not apply because $2n+1 = 21$ is not prime. Consequently, assigning 1 to the top left vertex does not yield a consecutive cyclic prime labeling.
The only primes $p$ for which $2n+p$ is prime and $p<2n+1$ are $p = 3,11,17$. Thus, Theorem~\ref{big_theorem} holds and we have the following consecutive cyclic prime labelings:
\begin{center}
\begin{tikzpicture}[xscale=1.2]
\draw (0,0) -- (9,0); \draw (0,1) -- (9,1); \foreach \x in {0,...,9} \draw (\x,0) -- (\x,1);
\node [state2] at (9,.5) {\textcolor{red}{\text{\small$p=3$}}};
\node [above] at (0,1) {12}; \node [below] at (0,0) {11};
\node [above] at (1,1) {13}; \node [below] at (1,0) {10};
\node [above] at (2,1) {14}; \node [below] at (2,0) {9};
\node [above] at (3,1) {15}; \node [below] at (3,0) {8};
\node [above] at (4,1) {16}; \node [below] at (4,0) {7};
\node [above] at (5,1) {17}; \node [below] at (5,0) {6};
\node [above] at (6,1) {18}; \node [below] at (6,0) {5};
\node [above] at (7,1) {19}; \node [below] at (7,0) {4};
\node [above] at (8,1) {20}; \node [below] at (8,0) {3};
\node [above] at (9,1) {1}; \node [below] at (9,0) {2};
\foreach \x in {0,...,9} \shade[ball color=blue] (\x,0) circle (.5ex); \foreach \x in {0,...,9} \shade[ball color=blue] (\x,1) circle (.5ex);
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[xscale=1.2]
\draw (0,0) -- (9,0); \draw (0,1) -- (9,1); \foreach \x in {0,...,9} \draw (\x,0) -- (\x,1);
\node [state2] at (5,.5) {\textcolor{red}{\text{\small$p=11$}}};
\node [above] at (0,1) {16}; \node [below] at (0,0) {15};
\node [above] at (1,1) {17}; \node [below] at (1,0) {14};
\node [above] at (2,1) {18}; \node [below] at (2,0) {13};
\node [above] at (3,1) {19}; \node [below] at (3,0) {12};
\node [above] at (4,1) {20}; \node [below] at (4,0) {11};
\node [above] at (5,1) {1}; \node [below] at (5,0) {10};
\node [above] at (6,1) {2}; \node [below] at (6,0) {9};
\node [above] at (7,1) {3}; \node [below] at (7,0) {8};
\node [above] at (8,1) {4}; \node [below] at (8,0) {7};
\node [above] at (9,1) {5}; \node [below] at (9,0) {6};
\foreach \x in {0,...,9} \shade[ball color=blue] (\x,0) circle (.5ex); \foreach \x in {0,...,9} \shade[ball color=blue] (\x,1) circle (.5ex);
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[xscale=1.2]
\draw (0,0) -- (9,0); \draw (0,1) -- (9,1); \foreach \x in {0,...,9} \draw (\x,0) -- (\x,1);
\node [state2] at (2,.5) {\textcolor{red}{\text{\small$p=17$}}};
\node [above] at (0,1) {19}; \node [below] at (0,0) {18};
\node [above] at (1,1) {20}; \node [below] at (1,0) {17};
\node [above] at (2,1) {1}; \node [below] at (2,0) {16};
\node [above] at (3,1) {2}; \node [below] at (3,0) {15};
\node [above] at (4,1) {3}; \node [below] at (4,0) {14};
\node [above] at (5,1) {4}; \node [below] at (5,0) {13};
\node [above] at (6,1) {5}; \node [below] at (6,0) {12};
\node [above] at (7,1) {6}; \node [below] at (7,0) {11};
\node [above] at (8,1) {7}; \node [below] at (8,0) {10};
\node [above] at (9,1) {8}; \node [below] at (9,0) {9};
\foreach \x in {0,...,9} \shade[ball color=blue] (\x,0) circle (.5ex); \foreach \x in {0,...,9} \shade[ball color=blue] (\x,1) circle (.5ex);
\end{tikzpicture}.
\end{center}
In each graph, we highlight the value of $p$ where the location of the label 1 is determined by Theorem~\ref{big_theorem}. For $p=19$, we observe that $2n+p = 39$ is not prime, yet the following labeling shows that assigning 1 to the prescribed vertex gives a successful labeling.
\begin{center}
\begin{tikzpicture}[xscale=1.2]
\draw (0,0) -- (9,0); \draw (0,1) -- (9,1); \foreach \x in {0,...,9} \draw (\x,0) -- (\x,1);
\node [state2] at (1,.5) {\textcolor{red}{\text{\small$p=19$}}};
\node [above] at (0,1) {20}; \node [below] at (0,0) {19};
\node [above] at (1,1) {1}; \node [below] at (1,0) {18};
\node [above] at (2,1) {2}; \node [below] at (2,0) {17};
\node [above] at (3,1) {3}; \node [below] at (3,0) {16};
\node [above] at (4,1) {4}; \node [below] at (4,0) {15};
\node [above] at (5,1) {5}; \node [below] at (5,0) {14};
\node [above] at (6,1) {6}; \node [below] at (6,0) {13};
\node [above] at (7,1) {7}; \node [below] at (7,0) {12};
\node [above] at (8,1) {8}; \node [below] at (8,0) {11};
\node [above] at (9,1) {9}; \node [below] at (9,0) {10};
\foreach \x in {0,...,9} \shade[ball color=blue] (\x,0) circle (.5ex); \foreach \x in {0,...,9} \shade[ball color=blue] (\x,1) circle (.5ex);
\end{tikzpicture}
\end{center}
An exhaustive check shows that assigning 1 to any other vertex in the top row fails to yield a consecutive cyclic prime labeling.
\end{example}
The previous example proves that the converse of Theorem~\ref{big_theorem} does not hold. That is, there are primes $p$ for which $2n+p$ is not prime, yet the prescribed labeling is successful.
\section{Complete bipartite graphs}\label{sec:bipartite}
In this section, we look at prime labelings and coprime labelings of complete bipartite graphs. We first examine minimal coprime labelings of $K_{n,n}$ for $n>2$. Then, we consider complete bipartite graphs $K_{m,n}$ with $m<n$, which have prime labelings for sufficiently large $n$ (depending on the value of $m$).
\subsection{Minimal coprime labelings of \texorpdfstring{$K_{n,n}$}{Kn,n}}
It is straightforward to see that the complete bipartite graph $K_{n,n}$ has no prime labeling for $n>2$. Hence, the best that one can do is to find the minimal value $\pr(K_{n,n}) > 2n$ such that $2n$ distinct labels chosen from the set $\{ 1, 2, \ldots, \pr(K_{n,n}) \}$ allows a coprime labeling of $K_{n,n}$. We use the term \textit{minimal coprime labeling} to denote the latter assignment of labels on a graph $G$.
\begin{example}
The minimal coprime labelings of $K_{3,3}$ and $K_{4,4}$ below show that $\pr(K_{3,3})=7$ and $\pr(K_{4,4})=9$.
\begin{center}
\begin{tikzpicture}[xscale=1.5]
\foreach \x in {0,1,2}
\foreach \w in {0,1,2}
\draw (\x,1) -- (\w,0);
\foreach \x in {0,1,2}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {1};
\node [above] at (1,1) {3};
\node [above] at (2,1) {5};
\node [below] at (0,0) {2};
\node [below] at (1,0) {4};
\node [below] at (2,0) {7};
\end{tikzpicture}
\hspace{.75in}
\begin{tikzpicture}[xscale=1.5]
\foreach \x in {0,1,2,3}
\foreach \w in {0,1,2,3}
\draw (\x,1) -- (\w,0);
\foreach \x in {0,1,2,3}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2,3}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {1};
\node [above] at (1,1) {3};
\node [above] at (2,1) {5};
\node [above] at (3,1) {9};
\node [below] at (0,0) {2};
\node [below] at (1,0) {4};
\node [below] at (2,0) {7};
\node [below] at (3,0) {8};
\end{tikzpicture}
\end{center}
\end{example}
Using an exhaustive computer check, we give the following values for $\pr(K_{n,n})$.
\begin{center}
\begin{tabular}{|c||ccccccccccccc|}
\hline
$n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline
$\pr(K_{n,n})$ & 2 & 4 & 7 & 9 & 11 & 15 & 17 & 21 & 23 & 27 & 29 & 32 & 37 \\
\hline
\end{tabular}
\end{center}
This sequence~\seqnum{A213273} appears in the
{\it On-Line Encyclopedia of Integer Sequences} (OEIS) \cite{Sloane}. By further computer check, Alois Heinz extended the sequence to $n=23$, thus adding the following values for $\pr(K_{n,n})$:
\begin{center}
\begin{tabular}{|c||cccccccccc|}
\hline
$n$ & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ \hline
$\pr(K_{n,n})$ & 40 & 43 & 46 & 49 & 53 & 57 & 61 & 63 & 67 & 71\\
\hline
\end{tabular}
\end{center}
\begin{observation}
By analyzing the minimal coprime labelings constructed for $K_{n,n}$, the following facts are readily verified for $n\leq 13$:
\begin{itemize}
\item There exists a minimal coprime labeling with the labels 1 and 2 in separate partite sets.
\item All primes up to $\pr(K_{n,n})$ are used in a minimal coprime labeling.
\item There are never more than $\frac{n}{2}$ primes in a minimal coprime labeling of $K_{\frac{n}{2},\frac{n}{2}}$.
\item A small set, $\mathfrak{P}$, of \textit{carefully chosen} primes determine the minimal coprime labeling of $K_{n,n}$ by labeling one of the two partite sets of vertices with numbers from a set of products of powers of these primes (see Definition~\ref{popop}).
\end{itemize}
\end{observation}
\begin{definition}\label{popop}
Let $\mathfrak{P} = \{p_1, \ldots, p_j\}$ be a set of primes. From $\mathfrak{P}$, we build the set of the first $n$ integers (larger than 1) of the form $p_1^{k_1} p_2^{k_2} \cdots p_j^{k_j}$ such that $k_i\geq 0$ for all $i$. Since this set is constructed by taking \textit{products of powers of primes}, we denote this set of $n$ elements as $\popop(\mathfrak{P},n)$.
\end{definition}
For example, for $\mathfrak{P}=\{2,3\}$,
$$\popop(\mathfrak{P},9) = \{ 2, 3, 4, 6, 8, 9, 12, 16, 18 \}.$$
\begin{conjecture}\label{Knn_conjecture}
For $K_{n,n}$, there exists a set of prime numbers $\{p_1, \ldots, p_j\}$ such that this set determines the values in the label sets of the two partite sets of vertices, giving $K_{n,n}$ a minimal coprime labeling.
\end{conjecture}
We believe the conjecture is true if we do the following. Consider a \textit{carefully}\footnote{Currently, how to \textit{carefully} choose a prime set is not clear. For example, $\popop(\{2,3\},n)$ allows a coprime labeling of $K_{n,n}$ for $n=3,8,13$, but fails for every other $n \leq 13$. Whereas, $\popop(\{2,7,11,13\},n)$ allows a coprime labeling of $K_{n,n}$ for $n=3,4,10,11,12$, but fails for every other $n \leq13$.} chosen set of small primes, $\mathfrak{P}=\{p_1, \ldots, p_j\}$. Let $A=\popop(\mathfrak{P},n)$ and $B$ contain the $n$ smallest positive integers which are relatively prime to all the elements of $A$. We claim that the sets $A$ and $B$, when used to label the partite sets, yield a coprime labeling of $K_{n,n}$. Moreover, we claim $\pr(K_{n,n}) = \max\{x \; | \; x \in A \cup B \}$, and hence this is a minimal coprime labeling.
\begin{example}
Via exhaustive search in \textsl{Mathematica}, we verified that there is a unique minimal coprime labeling of $K_{12,12}$ and $\pr(K_{12,12}) = 32$. If Conjecture~\ref{Knn_conjecture} is correct, then this calculation could have been done by the following method. If we let $\mathfrak{P} = \{ 2, 7, 11, 13 \}$, then $\popop(\mathfrak{P},12)$ is the following:
$$A = \{ 2, 4, 7, 8, 11, 13, 14, 16, 22, 26, 28, 32 \}.$$
Hence, the set of the $n$ smallest positive integers all relatively prime to the elements of $A$ is:
$$B = \{ 1, 3, 5, 9, 15, 17, 19, 23, 25, 27, 29, 31 \}.$$
These two sets are exactly the sets in which the exhaustive computer check unveiled as the unique minimal coprime labeling of $K_{12,12}$. Observe that the largest element in $A \cup B$ is 32, and indeed $\pr(K_{12,12}) = 32$.
\end{example}
Again using \textsl{Mathematica}, we observe that there exists a unique
minimal coprime labeling of $K_{n,n}$ for $n = 1,2,5,9,11,12$. On the
other hand, for $n = 3,4,6,7,8,10,$ there are a variety of different
minimal coprime labelings. For example, $K_{8,8}$ has 5 different
coprime labelings while $K_{10,10}$ has 9 different coprime labelings.
Thus it is natural to ask if there is a way to determine the values of
$n$ for which $K_{n,n}$ has a unique coprime labeling. By further
computation, Alois Heinz found the number of minimal coprime labelings
of $K_{n,n}$ for $n\leq 23$, published as sequence \seqnum{A213806} in
the OEIS~\cite{Sloane}.
\subsection{Prime labelings of \texorpdfstring{$K_{m,n}$}{Km,n}}
Although there exist no prime labelings for $K_{n,n}$ when $n>2$, there are prime labelings for $K_{m,n}$ when $m$ is fixed and $n$ is sufficiently large, depending on the value of $m$. In 1990, Fu and Huang \cite{Fu-Huang94} proposed a necessary and sufficient condition for the graph $K_{m,n}$ to be prime. Letting $P(t,v)$ be the set of all primes $x$ such that $t < x \leq v$, they prove the following proposition.
\begin{proposition} Let $m,n$ be positive integers where $m<n$. Then
$K_{m,n}$ is prime if and only if $m \leq \left|P\left(\frac{m+n}{2},m+n\right)\right|+1$.
\end{proposition}
We provide alternate proofs for specific cases of small values of $m$ and then in full generality. First, we must introduce some helpful notation and definitions. Combining the notation of \cite{Fu-Huang94} with ours, we denote the set of labels for each partite set of vertices in a prime labeling of $K_{m,n}$ as $A_{m,n}$ and $B_{m,n}$. For $K_{3,4}$, for example,
\begin{center}
\begin{tikzpicture}[baseline=2.5ex,xscale=1]
\foreach \x in {0.5,1.5,2.5}
\foreach \w in {0,1,2,3}
\draw (\x,1) -- (\w,0);
\foreach \x in {0,1,2,3}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0.5,1.5,2.5}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0.5,1) {1};
\node [above] at (1.5,1) {5};
\node [above] at (2.5,1) {7};
\node [below] at (0,0) {2};
\node [below] at (1,0) {3};
\node [below] at (2,0) {4};
\node [below] at (3,0) {6};
\end{tikzpicture}
\hspace{.1in} corresponds to \hspace{.1in}
$\begin{cases}
A_{3,4} = \{1,5,7\}, \mbox{ and}\\
B_{3,4} = \{2,3,4,6\}.
\end{cases}$
\end{center}
If we let $\pi(x)$ denote the number of primes less than or equal to $x$, then the {\it $n^{\text{th}}$ Ramanujan prime} is the least integer $R_n$ for which $\pi(x) - \pi(\frac{x}{2}) \geq n$ holds for all $x \geq R_n$. The first few Ramanujan primes as given in sequence \seqnum{A104272} in the OEIS~\cite{Sloane} are as follows:
\begin{center}
\begin{tabular}{|c||ccccc|}
\hline
$n$ & 1 & 2 & 3 & 4 & 5 \\ \hline
$R_n$ & 2 & 11 & 17 & 29 & 41 \\
\hline
\end{tabular}
\end{center}
\begin{remark}
In his attempt to give a new proof of Bertrand's postulate in 1919, Ramanujan published a proof in which he not only proves the famous postulate, but also generalizes it to infinitely many cases. This answered the question, ``From which $x$ onward will there be at least $n$ primes lying between $\frac{x}{2}$ and $x$?''~\cite{ramanujan1919}.
\end{remark}
\begin{theorem}\label{main_bipartite_result}
$K_{m,n}$ is prime if $n \geq R_{m-1}-m$.
\end{theorem}
\begin{proof}
There are at least $m-1$ primes in the interval $(\frac{m+n}{2}, m+n]$ for $n\geq R_{m-1}-m$. If we denote the first $m-1$ of these primes by $p_1, p_2, \ldots , p_{m-1}$, then the sets
\begin{align*}
A_{m,n} &= \{1, p_1, p_2, \ldots, p_{m-1}\}\\
B_{m,n} &= \{1, \ldots, m+n\} \backslash A_{m,n}
\end{align*}
give a prime labeling of $K_{m,n}$ for $n \geq R_{m-1}-m$.
\end{proof}
\begin{remark}
Note that Theorem \ref{main_bipartite_result} provides a sufficiently large value of $n$ for which $K_{m,n}$ is always prime. It is interesting to note, however, that for certain smaller values of $n$, the graph $K_{m,n}$ still has a prime labeling. These cases are summarized below for $3 \leq m \leq 13$.
\end{remark}
\begin{center}
\begin{tabular}{|c||l|l|}
\hline
$K_{m,n}$ & Prime (small $n$-cases) & Prime by Theorem \ref{main_bipartite_result}\\ \hline
$K_{3,n}$ & $n = $ 4, 5, 6 & $n \geq $ 8 \\ \hline
$K_{4,n}$ & $n = $ 9 & $n \geq $ 13 \\ \hline
$K_{5,n}$ & $n = $ 14, 15, 16, 18, 19, 20 & $n \geq $ 24 \\ \hline
$K_{6,n}$ & $n = $ 25, 26, 27, 31 & $n \geq $ 35 \\ \hline
$K_{7,n}$ & $n = $ 36, 37, 38 & $n \geq $ 40 \\ \hline
$K_{8,n}$ & $n = $ 45, 46, 47, 48, 49 & $n \geq $ 51 \\ \hline
$K_{9,n}$ & $n = $ 52 & $n \geq $ 58 \\ \hline
$K_{10,n}$ & & $n \geq $ 61 \\ \hline
$K_{11,n}$ & $n = $ 62, 68, 69, 70, 72, 73, 74, & $n \geq $ 86 \\
& 78, 79, 80, 81, 82 & \\
\hline
$K_{12,n}$ & & $n \geq $ 89 \\ \hline
$K_{13,n}$ & $n = $ 90, 91, 92 & $n \geq $ 94 \\
\hline
\end{tabular}
\end{center}
\bigskip
We conclude this subsection by applying Theorem~\ref{main_bipartite_result} to find the exact prime labelings for all graphs $K_{m,n}$ when $m = $ 3, 4, 5, or 6. Moreover, we give prime labelings for $K_{m,n}$ for the values of $n$ smaller than $R_{m-1}-m$ for which a prime labeling is possible.
\begin{proposition}
$K_{3,n}$ is prime if $n =4, 5, 6$ or $n \geq 8$.
\end{proposition}
\begin{proof}
Since $R_2=11$, there are at least two primes $p_1, p_2$ in the interval $(\frac{n+3}{2}, n+3]$ for $n \geq 8$. Hence, the sets $A_{3,n} = \{1, p_1, p_2\} \mbox{ and } B_{3,n} = \{1, \ldots, n+3\} \backslash A_{3,n}$ give a prime labeling of $K_{3,n}$ for $n \geq 8$.
If $n = $ 4, 5, or 6, then $A_{3,n} = \{1, 5, 7\}$ and the sets $B_{3,n}= \{2, 3, 4, 6\}, \{2, 3, 4, 6, 8\}$, and $\{2, 3, 4, 6, 8, 9\}$, respectively, give the following prime labelings of $K_{3,4}$, $K_{3,5}$, and $K_{3,6}$:
\begin{center}
\begin{tikzpicture}[xscale=1]
\foreach \x in {0.5,1.5,2.5}
\foreach \w in {0,1,2,3}
\draw (\x,1) -- (\w,0);
\foreach \x in {0,1,2,3}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0.5,1.5,2.5}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0.5,1) {1};
\node [above] at (1.5,1) {5};
\node [above] at (2.5,1) {7};
\node [below] at (0,0) {2};
\node [below] at (1,0) {3};
\node [below] at (2,0) {4};
\node [below] at (3,0) {6};
\end{tikzpicture}
\hspace{.35in}
\begin{tikzpicture}[xscale=1]
\foreach \x in {1,2,3}
\foreach \w in {0,1,2,3,4}
\draw (\x,1) -- (\w,0);
\foreach \x in {0,1,2,3,4}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {1,2,3}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (1,1) {1};
\node [above] at (2,1) {5};
\node [above] at (3,1) {7};
\node [below] at (0,0) {2};
\node [below] at (1,0) {3};
\node [below] at (2,0) {4};
\node [below] at (3,0) {6};
\node [below] at (4,0) {8};
\end{tikzpicture}
\hspace{.35in}
\begin{tikzpicture}[xscale=1]
\foreach \x in {1.5,2.5,3.5}
\foreach \w in {0,1,2,3,4,5}
\draw (\x,1) -- (\w,0);
\foreach \x in {0,1,2,3,4,5}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {1.5,2.5,3.5}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (1.5,1) {1};
\node [above] at (2.5,1) {5};
\node [above] at (3.5,1) {7};
\node [below] at (0,0) {2};
\node [below] at (1,0) {3};
\node [below] at (2,0) {4};
\node [below] at (3,0) {6};
\node [below] at (4,0) {8};
\node [below] at (5,0) {9};
\end{tikzpicture}
\end{center}
\end{proof}
\begin{proposition}
$K_{4,n}$ is prime if $n=9$ or $n \geq 13$.
\end{proposition}
\begin{proof}
Since $R_3 =17$, there are at least three primes $p_1, p_2, p_3$ in the interval $(\frac{n+4}{2}, n+4]$ for $n \geq 13$. Hence, the sets $A_{4,n} = \{1, p_1, p_2, p_3\} \mbox{ and } B_{4,n} = \{1, \ldots, n+4\} \backslash A_{4,n}$ give a prime labeling of $K_{4,n}$ for $n \geq 13$.
If $n=9$, then the sets $A_{4,9} = \{1, 7, 11, 13\}$ and $B_{4,9} = \{2, 3, 4, 5, 6, 8, 9, 10, 12\}$ give a prime labeling of $K_{4,9}$.
\end{proof}
\begin{proposition}
$K_{5,n}$ is prime if $n=14, 15, 16, 18, 19, 20$ or $n \geq 24$.
\end{proposition}
\begin{proof}
Since $R_4 =29$, there are at least four primes $p_1, p_2, p_3, p_4$ in the interval $(\frac{n+5}{2}, n+5]$ for $n \geq 24$. Hence, the sets $A_{5,n} = \{1, p_1, p_2, p_3, p_4\} \mbox{ and } B_{5,n} = \{1, \ldots, n+5\} \backslash A_{5,n}$ give a prime labeling of $K_{5,n}$ for $n \geq 24$.
If $n = $ 14, 15, or 16, then choose $A_{5,n} = \{1, 11, 13, 17, 19\}$. If $n = $ 18, 19, or 20, then choose $A_{5,n} = \{1, 13, 17, 19, 23\}$. In each case, $B_{5,n}= \{1, \ldots, n+5\} \backslash A_{5,n}$ gives a prime labeling of $K_{5,n}$.
\end{proof}
\begin{proposition}
$K_{6, n}$ is prime if $n=25, 26, 27, 31$ or $n \geq 35$.
\end{proposition}
\begin{proof}
Since $R_5 =41$, there are at least five primes $p_1, p_2, p_3, p_4, p_5$ in the interval $(\frac{n+6}{2}, n+6]$ for $n \geq 35$. Hence, the sets $A_{6,n} = \{1, p_1, p_2, p_3, p_4, p_5\} \mbox{ and } B_{6,n} = \{1, \ldots, n+6\} \backslash A_{6,n}$ give a prime labeling of $K_{6,n}$ for $n \geq 35$.
If $n = $ 25, 26, or 27, then choose $A_{6,n} = \{1, 17, 19, 23, 29, 31\}$. If $n=31$, then choose $A_{6,31} = \{1, 19, 23, 29, 31, 37\}$. In each case, $B_{6,n}=\{1, \ldots, n+6 \} \backslash A_{6,n}$ gives a prime labeling of $K_{6,n}$.
\end{proof}
\begin{question}
Is there any predictability as to the values of $n$ smaller than $R_{m-1}-m$ for which $K_{m,n}$ has a prime labeling? It is interesting to note that when $m=10$ or $m=12$, there are no such values of $n$.
\end{question}
\section{Future work}\label{sec:future_work}
Throughout the paper we have mentioned some questions and conjectures for further research. We conclude here with a few additional open questions.
\begin{question} Is it possible to write every even integer $2n$ in the form $2n=q-p$ where $q$ is prime and $p$ is either 1 or a prime less than $2n+1$? \end{question}
An affirmative answer to this interesting number theory question implies that all ladders are prime and, in fact, have a consecutive cyclic prime labeling.
\begin{question}
Does there exist an inductive method to get a minimal coprime labeling of the graph $K_{n+1,n+1}$ from a minimal coprime labeling of the graph $K_{n,n}$?
\end{question}
We have observed that minimal coprime labelings of $K_{n,n}$ are sometimes properly contained as subgraphs of a minimal coprime labeling of $K_{n+1,n+1}$. For example, we have the following labeling of $K_{6,6}$:
\begin{center}
\begin{tikzpicture}[xscale=1]
\foreach \x in {0,1,2,3,4,5}
\foreach \w in {0,1,2,3,4,5}
\draw (\x,1) -- (\w,0);
\foreach \x in {0,1,2,3,4,5}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2,3,4,5}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {1};
\node [above] at (1,1) {3};
\node [above] at (2,1) {5};
\node [above] at (3,1) {9};
\node [above] at (4,1) {11};
\node [above] at (5,1) {15};
\node [below] at (0,0) {2};
\node [below] at (1,0) {4};
\node [below] at (2,0) {7};
\node [below] at (3,0) {8};
\node [below] at (4,0) {13};
\node [below] at (5,0) {14};
\end{tikzpicture}
\end{center}
Observe that the above graph is properly contained in the following minimal coprime labeling of $K_{7,7}$:
\begin{center}
\begin{tikzpicture}[xscale=1]
\foreach \x in {0,1,2,3,4,5,6}
\foreach \w in {0,1,2,3,4,5,6}
\draw (\x,1) -- (\w,0);
\foreach \x in {0,1,2,3,4,5,6}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2,3,4,5,6}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {1};
\node [above] at (1,1) {3};
\node [above] at (2,1) {5};
\node [above] at (3,1) {9};
\node [above] at (4,1) {11};
\node [above] at (5,1) {15};
\node [above] at (6,1) {17};
\node [below] at (0,0) {2};
\node [below] at (1,0) {4};
\node [below] at (2,0) {7};
\node [below] at (3,0) {8};
\node [below] at (4,0) {13};
\node [below] at (5,0) {14};
\node [below] at (6,0) {16};
\end{tikzpicture}
\end{center}
Also we see that the sets $\popop(\{2,7,13\},6)$ and $\popop(\{2,7,13\},7)$ give the bottom row labelings for $K_{6,6}$ and $K_{7,7}$, respectively. We observed that in all the examples which we looked at for which we go from a minimal coprime labeling of $K_{n,n}$ to $K_{n+1,n+1}$ in which the above phenomena above arose, the prime numbers in the $\popop$ sets coincided.
Secondly, another reason to believe that there may exist an inductive method to get from $K_{n,n}$ to $K_{n+1,n+1}$ is the observation that we can always switch a prime number in the top row with one from the bottom row and still have a coprime labeling if and only if there are no multiples of the particular two prime numbers within the set of the other labels in the graph. For example, we can get from $K_{5,5}$ to $K_{6,6}$ by simply switching the primes 5 and 7 in $K_{5,5}$. The cost of such a switch is that the label 10 in $K_{5,5}$ cannot also be a label in $K_{6,6}$ since both the values 2 and 5 exist in $K_{6,6}$ as a consequence of the switch. So we replace 10 with the smallest available prime, namely 13. Below we illustrate this transition from $K_{5,5}$ (on the left) to $K_{6,6}$ (on the right):
\begin{center}
\begin{tikzpicture}[xscale=.75]
\draw[<->,line width=.5mm, dotted] (2,.1) -- (2,.9);
\foreach \x in {0,1,2,3,4}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2,3,4}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {1};
\node [above] at (1,1) {3};
\node [state, above] at (2,1) {\textbf{7}};
\node [above] at (3,1) {9};
\node [above] at (4,1) {11};
\node [below] at (0,0) {2};
\node [below] at (1,0) {4};
\node [state, below] at (2,0) {\textbf{5}};
\node [below] at (3,0) {8};
\node [below] at (4,0) {10};
\end{tikzpicture}
\hspace{.7in}
\begin{tikzpicture}[xscale=.75]
\foreach \x in {0,1,2,3,4,5}
\shade[ball color=blue] (\x,0) circle (.5ex);
\foreach \x in {0,1,2,3,4,5}
\shade[ball color=blue] (\x,1) circle (.5ex);
\node [above] at (0,1) {1};
\node [above] at (1,1) {3};
\node [state, above] at (2,1) {\textbf{5}};
\node [above] at (3,1) {9};
\node [above] at (4,1) {11};
\node [above] at (5,1) {15};
\node [below] at (0,0) {2};
\node [below] at (1,0) {4};
\node [state, below] at (2,0) {\textbf{7}};
\node [below] at (3,0) {8};
\node [state, below] at (4,0) {\textbf{13}};
\node [below] at (5,0) {14};
\end{tikzpicture}
\end{center}
\section{Acknowledgments}
The authors thank the AIM (American Institute of Mathematics) and NSF
(National Science Foundation) for funding the REUF (Research
Experiences for Undergraduate Faculty) program. Furthermore, we
appreciate the hospitality of ICERM (Institute for Computational and
Experimental Research in Mathematics), who hosted the REUF program in
summer 2012 where the collaborators met and the work on this paper was
initiated.
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H.-L.~Fu and K.-C.~Huang.
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P.~Haxell, O.~Pikhurko, and A.~Taraz.
\newblock Primality of trees.
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J.~A.~Gallian.
\newblock A dynamic survey of graph labeling.
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A.~Kanetkar.
\newblock Prime labeling of grids.
\newblock {\em AKCE J. Graphs. Combin.} \textbf{6} (2009), 135--142.
\bibitem{VSN}
T.~Nicholas, S.~Somasundaram, and V.~Vilfred.
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O.~Pikhurko.
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A.~de Polignac.
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\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05C78; Secondary 11A05.
\noindent \emph{Keywords: } Coprime labeling, prime labeling, prime
graph, consecutive cyclic prime labeling, bipartite graph, ladder
graph, Ramanujan prime.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A104272},
\seqnum{A213273}, and
\seqnum{A213806}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received March 12 2016;
revised versions received June 7 2016.
Published in {\it Journal of Integer Sequences}, June 13 2016.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in
\end{document}
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https://www.bortzmeyer.org/files/vie-privee-bfm-limoges.tex
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\documentclass{beamer}
%\documentclass[handout]{beamer}
%\usepackage{pgfpages}
%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
\usetheme{Antibes}
\usepackage[french]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{bortzmeyer-utils}
\title{Vie privée et Internet : derrière les derniers scandales (\insertframenumber/\inserttotalframenumber)}
\author{Stéphane Bortzmeyer\\\texttt{[email protected]}}
\date{9 mars 2019}
\setlength{\parskip}{15pt plus 10pt minus 10pt}
\newcommand{\of}{.6}
\begin{document}
\maketitle
\begin{frame}
\frametitle{Définition}
\begin{quote}
La vie privée (du latin privatus - séparé de, dépourvu de) est la capacité, pour une personne ou pour un groupe de personnes, de s'isoler afin de protéger ses intérêts. Les limites de la vie privée ainsi que ce qui est considéré comme privé diffèrent selon les groupes, les cultures et les individus, selon les coutumes et les traditions bien qu'il existe toujours un certain tronc commun.
\end{quote}
(Wikipédia « Vie privée »)
\end{frame}
\begin{frame}
\frametitle{Proclamation}
\begin{quote}
Nul ne sera l’objet d’immixtions arbitraires dans sa vie privée, sa famille, son domicile ou sa correspondance, ni d’atteintes à son honneur et à sa réputation. Toute personne a droit à la protection de la loi contre de telles immixtions ou de telles atteintes.
\end{quote} (Déclaration Universelle des Droits Humains, article 12)
\end{frame}
\begin{frame}
\frametitle{Sur l'Internet}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->Comme vous le savez, il y a régulièrement des fuites de
données suite à des piratages ou failles de sécurité\only<2>{
(British Airways, Orange,
Ariane, Aadhaar, Dow Jones, Google+…)}
\item<3->Mais aussi des utilisations scandaleuses\only<3->{
(Cambridge Analytica…)}
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{Les réactions}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->À chaque scandale, on s'indigne…
\item<3->… puis on oublie jusqu'au suivant,
\item<4->Et on tend à considérer chaque fuite comme un cas isolé,
lié à la méchanceté de certains acteurs, ou à leur incompétence.
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{Mais c'est plus compliqué que cela}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->La sécurité, c'est compliqué\only<2>{ : l'attaquant peut
frapper où il veut, le défenseur doit tout protéger}, % ParisWeb https://www.paris-web.fr/2018/conferences/votre-base-de-donnees-avec-des-informations-personnelles-sera-piratee.php
\item<3->Tôt ou tard, il y aura un piratage, % JDLL https://www.bortzmeyer.org/vie-privee-rgpd.html
\item<4->Et les scandales comme Cambridge Analytica ne sont pas
un accident, ou un accès de malhonnêteté ponctuel de Facebook,
\item<5->Ces scandales sont une conséquence naturelle d'un
certain modèle d'affaires.
\item<6->Et le danger n'est pas seulement le piratage par un
tiers, c'est aussi la mauvaise utilisation.
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{La sécurité n'est jamais parfaite}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->Méfiez-vous quand un Monsieur Sérieux dit « ne vous
inquiétez-pas, c'est parfaitement sécurisé »,
\item<3->La sécurité informatique est un domaine complexe, que
peu de gens maitrisent,
\item<4->La sécurité n'est pas juste une affaire de compétence,
et d'outils coûteux : elle dépend de bonnes pratiques suivies
\emph{tout le temps}.
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{Anonymat ?}
% La définition courante n'est pas la définition scientifique
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->Méfiez-vous deux fois plus quand le Monsieur Sérieux dit
« les données sont anonymisées »,
\item<3->La réalité : avec le numérique, tout est trivialement
copié, archivé, traité, et on peut retrouver plein de choses,
\item<4->Le vrai anonymat est \emph{très} difficile à atteindre
sur l'Internet,
\item<5->Dans le meilleur des cas, on a du pseudonymat\only<5->{
(une identité stable, mais qui n'est pas celle gérée par l'État)}.
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{Le modèle d'affaires du Web commercial, c'est la surveillance}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->Si un service est gratuit, demandez-vous comment il
est financé,
\item<3->La publicité est un gros demandeur de données
personnelles (l'État en est un autre).
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{Il n'y a pas que Facebook qui collecte des données}
\includegraphics[scale=0.31]{sncf.png}
Et ces données sont gardées pendant… 13 ans après le dernier achat !
% « La plupart / majorité des données sont conservées trois ans ou
% moins à compter de votre date d’achat ou de dernière activité,
% puis archivées pour une durée de dix ans pour des motifs
% strictement légaux (obligations fiscales...). » https://www.oui.sncf/espaceclient/creer-compte
\end{frame}
\begin{frame}
\frametitle{Pisteurs dans les pages Web}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->Beaucoup de webmestres mettent Google Analytics ou le
bouton Like de Facebook dans leurs pages,
\item<3->Ce faisant, ils transmettent à Google ou Facebook les
données de leurs visiteurs.
\item<4->Il ne s'agit pas que des entreprises
commerciales. France Télévisions a Google Analytics, % https://www.france.tv/
\item<5->L'écrasante majorité des webmestres se moque de la vie privée.
\end{itemize}
\end{overlayarea}
\end{frame}
% Le bandeau cookie : c'est le pistage, le problème, pas le bandeau.
\begin{frame}
\frametitle{Données envoyées aux USA}
\includegraphics[scale=0.23]{trackography.png} % https://trackography.org
\end{frame}
\begin{frame}
\frametitle{Pisteurs dans les applications sur l'ordiphone}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->La plupart des applications signalent vos activités à
l'éditeur mais aussi à Google, Facebook et d'autres,
\item<3->Rapports Exodus Privacy. Exemple : l'Express a
21 pisteurs, dont Google et Facebook. % https://reports.exodus-privacy.eu.org/en/reports/2134/
\item<4->Comme pour les webmestres, très peu de réaction des développeurs.
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{L'Express vous piste}
\includegraphics[scale=0.24]{exodus-privacy.png} % https://reports.exodus-privacy.eu.org/en/reports/2134/
\end{frame}
\begin{frame}
\frametitle{Il n'y a pas que le Web}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->« Objets connectés » : à qui votre brosse à dents, votre
télévision, votre voiture, la poupée Barbie de votre enfant
envoie des données ? % https://www.bbc.com/news/technology-31502898
\item<3->Micros dans la rue qui envoient tous les bruits à la
police (Saint-Étienne).
\end{itemize}
\end{overlayarea}
\end{frame}
% Visages des passagers sur les voies de covoiturage https://www.nextinpact.com/news/107680-voies-covoiturage-et-detecteurs-passagers-senat-reclame-plus-garanties.htm
% Les objections : le ``sacrifice volontaire'', le ``je n'ai rien à
% cacher'', le petit village où tout le monde sait tout.
\begin{frame}
\frametitle{Les solutions}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->Minimisation des données,
\item<3->Solutions individuelles et solutions collectives.
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{Minimisation des données}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->Pourquoi récolter nom et prénom ?
\item<3->Pourquoi récolter la date de naissance ?
\item<4->Les données sont une drogue, il faut se désintoxiquer,
\item<5->Si la base est piratée, et que les données personnelles
fuient, c'est parce que quelqu'un avait décidé de récolter ces
données,
\item<6->Le collecteur des données est responsable.
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{Solutions individuelles}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->D'abord, le bloqueur de
publicités. Indispensable. Exemple : uBlock Origin. % https://github.com/gorhill/uBlock#ublock-origin
\item<3->Bloque également les pisteurs.
\item<4->Autre exemple : Pi-hole, pour protéger tout le
réseau local. % https://pi-hole.net/
\item<5->Minimiser soi-même ce qu'on envoie (quand c'est
possible).
\item<6->Et brouiller les données le reste du temps (« Obfuscation », de Finn Brunton et Helen Nissenbaum, chez C\&F Éditions)
\end{itemize}
\end{overlayarea}
\end{frame}
\begin{frame}
\frametitle{uBlock Origin vous protège}
\includegraphics[scale=0.40]{ublock-origin.png}
\end{frame}
\begin{frame}
\frametitle{Solutions collectives}
\begin{overlayarea}{\textwidth}{\of\textheight}
\begin{itemize}
\item<2->RGPD (Réglement [européen] Général de Protection des
Données.
\begin{enumerate}
\item Minimisation (déjà dans la loi Informatique et
Libertés),
\item Consentement (parfois), justification de la récolte (déjà dans la loi Informatique et
Libertés),
\item Droit d'accès (déjà dans la loi Informatique et
Libertés),
\item Non-territorialité,
\item Notification des failles,
\item Protection de la vie privée dès la conception,
\item Responsabilité du donneur d'ordres et du sous-traitant.
\end{enumerate}
% Exposé JDLL https://www.bortzmeyer.org/vie-privee-rgpd.html
\item<3->L'État est parfois incohérent : RGPD d'un côté, lois
de surveillance de l'autre.
\end{itemize}
\end{overlayarea}
\end{frame}
\end{document}
|
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\input zb-basic
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\iteman{ZMATH 1988b.01666}
\itemau{Henderson, P.B. (State Univ.of New York, Stony Brook (USA). Dept.of Computer Science); Ferguson, D.L. (State Univ.of New York, Stony Brook (USA). Dept.of Technology and Society)}
\itemti{A conceptual approach to algorithmic problem solving. Ein begrifflicher Zugang zum algorithmischen Problemloesen.}
\itemso{National educational computing conference (NECC '86). Proceedings. Siebte Jahrestagung ueber Computer im Bildungswesen (NECC '86). Vortraege. Editor(s): Ryan, W.C. (Swarthmore Coll., PA (USA)) Oregon Univ., Eugene (USA). International Council on Computers in Education 1986. p. 243-249 of 381 p. Copy held by FIZ Karlsruhe. Conference: 7. Annual National Educational Computing Conference (NECC '86), San Diego, CA (USA), 4-6 Jun 1986 [ISBN 0-924667-35-4]}
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\itemli{}
\end
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\bib{1986/hildebrand-rankin}
\yr 1986
\mr 87f:11066
\by Adolf Hildebrand
\by G\'erald Tenenbaum
\paper On integers free of large prime factors
\jour Transactions of the American Mathematical Society
\issn 0002--9947
\vol 296
\pages 265--290
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\[\sinh z=2\!I_{1}\left(z\right)+2\!I_{3}\left(z\right)+2\!I_{5}\left(z\right)+\dots.\]
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\bibitem{An_etal2018} An, P., Kovalyuk, V., Golikov, A., Zubkova, E., Ferrari, S., Korneev, A., et al. (2018). Experimental optimisation of O-ring resonator Q-factor for on-chip spontaneous four wave mixing. In \textit{J. Phys.: Conf. Ser.} (Vol. 1124, 051047).
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\documentclass[10pt]{article}
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\rhead{\textbf{A. P. M. E. P.}}
\lhead{\small Baccalauréat STI Génie électronique, électrotechnique et optique}
\lfoot{\small{Nouvelle--Calédonie}}
\rfoot{\small{novembre 2011}}
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\begin{center}
{\Large \textbf{\decofourleft~Corrigé du baccalauréat STI Nouvelle--Calédonie~\decofourright\\novembre 2011\\Génie électronique, électrotechnique et optique}}
\end{center}
\vspace{0,25cm}
\textbf{\textsc{Exercice 1} \hfill 4,5 points}
\medskip
%Dans une urne, on dispose de cinq boules indiscernables au toucher : trois boules vertes numérotées de 1 à 3, qu'on notera $V_{1},\: V_{2}$ et $V_{3}$, et deux boules rouges numérotées 0 et 5, qu'on notera $R_{0}$ et $R_{5}$.
%
%Un jeu consiste à tirer successivement deux boules au hasard. Avant de tirer la deuxième boule, on remet dans l'urne la boule obtenue au premier tirage.
%
%\begin{itemize}
%\item Si les deux boules tirées sont de la même couleur, le joueur ne reçoit rien.
%\item Si les deux boules tirées sont de couleurs différentes, le joueur remporte le montant en euros égal au nombre formé en prenant le chiffre de la boule verte pour les dizaines et celui de la boule rouge pour les unités.
%\end{itemize}
%
%Par exemple, le tirage du couple $\left(R_{5}~;~V_{3}\right)$ rapporte 35 euros, alors que le tirage du couple $\left(V_{1}~;~V_{2}\right)$ ne rapporte rien.
\begin{enumerate}
\item
\begin{enumerate}
\item ~%Recopier et compléter le tableau suivant en indiquant les différents montants :
\medskip
\begin{tabularx}{\linewidth}{|m{3cm}|*{5}{>{\centering \arraybackslash}X|}}\hline
\backslashbox{1\up{re} boule}{2\up{e} boule}& $V_{1}$ &$V_{2}$ &$V_{3}$ &$R_{0}$ &$R_{5}$\\ \hline
$V_{1}$&0&0&0&10&15\\ \hline
$V_{2}$&0&0&0&20&25\\ \hline
$V_{3}$&0&0&0&30&35\\ \hline
$R_{0}$&10&20&30&0&0 \\ \hline
$R_{5}$&15&25& 35&0&0 \\ \hline
\end{tabularx}
\medskip
\item %Déterminer la probabilité de chacun des évènements suivants :
%A : \og le joueur ne reçoit rien \fg{} ;
Dans 13 cas sur 25 le joueur ne reçoit rien : la probabilité est donc égale à $\dfrac{13}{25} = \dfrac{52}{100} = 0,52$.
%B : \og le joueur remporte un montant supérieur ou égal à 20 euros \fg.
Le joueur reçoit plus de 20 euros dans 6 cas sur 25, d'où une probabilité de $\dfrac{6}{25} = \dfrac{24}{100} = 0,24$.
\end{enumerate}
\item %On suppose dans cette question que le joueur mise 15 euros. On note $X$ la variable aléatoire qui, à chaque tirage, associe le gain du joueur, c'est-à-dire la différence entre ce qu'il reçoit et ce qu'il mise, $X$ étant négative en cas de perte.
\begin{enumerate}
\item %Donner la loi de probabilité de la variable aléatoire $X$.
On construit le tableau des \og gains \fg{} en retranchant 15 de chaque cas du tableau précédent :
\medskip
\begin{tabularx}{\linewidth}{|m{3cm}|*{5}{>{\centering \arraybackslash}X|}}\hline
\backslashbox{1\up{re} boule}{2\up{e} boule}& $V_{1}$ &$V_{2}$ &$V_{3}$ &$R_{0}$ &$R_{5}$\\ \hline
$V_{1}$&$-15$&$-15$&$-15$&$-5$&0\\ \hline
$V_{2}$&$-15$&$-15$&$-15$&5&10\\ \hline
$V_{3}$&$-15$&$-15$&$-15$&15&20\\ \hline
$R_{0}$&$-5$&5&15&$-15$&$-15$ \\ \hline
$R_{5}$&0&10& 20&$-15$&$-15$ \\ \hline
\end{tabularx}
\medskip
On en déduit le tableau de la loi de probabilité de la variable aléatoire $X$ :
\medskip
\begin{tabularx}{\linewidth}{|m{1.75cm}|*{7}{>{\centering \arraybackslash}X|}}\hline
$x_{i}$&$- 15$&$- 5$&0&5&10&15&20\\ \hline
\rule[-3mm]{0mm}{9mm}$p\left(X = x_{i}\right)$&$\dfrac{13}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$\\ \hline
\end{tabularx}
\medskip
\item %Calculer l'espérance de $X$.
On a E$(X) = - 15 \times \dfrac{13}{25} - 5 \times \dfrac{2}{25} + 0 \times \dfrac{2}{25} + 5 \times \dfrac{2}{25} + 10\times \dfrac{2}{25} + 15 \times \dfrac{2}{25} + 20\times \dfrac{2}{25} = \dfrac{- 195 - 10 + 0 + 10 + 20 + 30 + 40}{25} = - \dfrac{105}{25} = - \dfrac{21}{5} = - 4,20$~(\euro).
Sur un grand nombre de parties la perte moyenne pour un joueur est égale à 4,20~\euro.
\item %Un jeu est dit équitable lorsque l'espérance du gain est nulle. Quelle devrait être la mise de départ pour que ce jeu soit équitable ?
En remplaçant 15 par un mise de $m$ euros on obtient une nouvelle loi de probabilité :
\medskip
\begin{footnotesize}
\begin{tabularx}{\linewidth}{|m{1.5cm}|*{8}{>{\centering \arraybackslash}X|}}\hline
$x_{i}$&$- m$&$10 - m$&$10 - m$&$15 - m$&$20 - m$&$25 - m$&$30 - m$&$35 - m$\\ \hline
\rule[-3mm]{0mm}{9mm}$p\left(X = x_{i}\right)$&$\dfrac{13}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$&$\dfrac{2}{25}$\\ \hline
\end{tabularx}
\end{footnotesize}
\medskip
On a alors :
E$(X) = - m \times \dfrac{13}{25} + (10 - m)\times \dfrac{2}{25} + (15 - m)\times \dfrac{2}{25} + (20 - m)\times \dfrac{2}{25} + (25 - m)\times \dfrac{2}{25} + (30 - m)\times \dfrac{2}{25} + (35 - m)\times \dfrac{2}{25} =$
$ \dfrac{-13m + 20 - 2m + 30 - 2m + 40 - 2m + 50 - 2m + 60 - 2m + 70 - 2m}{25} = \dfrac{270 - 25m}{25}$
Le leu est équitable si l'espérance est nulle :
E$(X) = 0 \iff \dfrac{270 - 25m}{25} = 0 \iff 270 - 25m = 0 \iff 270 = 25m \iff 54 = 5m \iff m \dfrac{54}{5} = \dfrac{108}{10} = 10,80$~\euro.
Le jeu est équitable si la mise est égale à 10,80~\euro.
\end{enumerate}
\end{enumerate}
\vspace{0,5cm}
\textbf{\textsc{Exercice 2} \hfill 5,5 points}
\medskip
%Soit i le nombre complexe de module 1 et d'argument $\dfrac{\pi}{2}$.
\begin{enumerate}
\item %Pour tout nombre complexe z, on pose :
%\[P(z) = z^3 + 5z^2 + 10z + 12.\]
\begin{enumerate}
\item %Calculer $P(- 3)$, puis déterminer les réels $b$ et $c$ tels que
$P(- 3) = (- 3)^3 + 5 \times (- 3)^2 + 10 \times (- 3) + 12 = - 27 + 45 - 30 + 12 = 57 - 57 = 0$.
$- 3$ est donc une racine de $P$. Il existe donc des complexes $b$ et $c$ tels que :
$P(z) = (z + 3)\left(z^2 + bz + c\right) = z^3 + bz^2 + cz + 3z^2 + 3bz + 3c = z^3 + (b + 3)z^2 + (3b + c)z + 3c$.
En identifiant avec l'énoncé, on obtient le système :
$\left\{\begin{array}{l c l}
b + 3&=&5\\
3b + c&=&10\\
3c&=&12
\end{array}\right. \iff \left\{\begin{array}{l c l}
b + 3&=&5\\
3b + c&=&10\\
c&=&4
\end{array}\right. \iff \left\{\begin{array}{l c l}
b + 3&=&5\\
3b &=&6\\
c&=&4
\end{array}\right. \iff \left\{\begin{array}{l c l}
b + 3&=&5\\
b &=&2\\
c&=&4
\end{array}\right.$
%$P(z) = (z + 3)\left(z^2 + bz + c\right)$.
Finalement : $P(z) = (z + 3)\left(z^2 + 2z + 4\right)$.
\item %Résoudre dans l'ensemble des nombres complexes l'équation : $P(z) = 0$.
$P(z) = 0 \iff (z + 3)\left(z^2 + 2z + 4\right) = 0 \iff \left\{\begin{array}{l c l}
z + 3&=&0\\
z^2 + 2z + 4&=&0
\end{array}\right. \iff \left\{\begin{array}{l c l}
z &=&- 3\\
z^2 + 2z + 4&=&0
\end{array}\right.$
Résolution de l'équation du second degré : $\Delta = 4 - 4 \times 4 = - 12 = \left(2\text{i}\sqrt{3}\right)^2 < 0$ : il y a donc deux racines complexes :
$z_{2} = \dfrac{- 2 + 2\text{i}\sqrt{3}}{2} = - 1 + \text{i}\sqrt{3}$ et $z_{3} = - 1 - \text{i}\sqrt{3}$.
Finalement les solutions sont :
\[z_{1} = - 3 ;\quad z_{2} = - 1 + \text{i}\sqrt{3} ;\quad z_{3} = - 1 - \text{i}\sqrt{3}.\]
\end{enumerate}
\item %Dans le plan muni d'un repère orthonormal \Ouv, on considère les points A, B et C d'affixes respectives :
%\[z_{\text{A}} = -3 \quad ; \quad z_{\text{B}} = - 1 + \text{i}\sqrt{3} \quad ;\quad z_{\text{C}} = - 1 - \text{i}\sqrt{3}.\]
\begin{enumerate}
\item %Sur la figure donnée en annexe, on a tracé deux triangles. L'un est le triangle ABC, l'autre sera noté $T$. Placer les points A, B et C sur cette figure.
Voir la figure
\item %On considère la translation $t$ de vecteur $4\vect{u}$. Construire les points A$'$, B$'$ et C$'$, images respectives des points A, B et C par la translation $t$.
Voir la figure
\item %On admet que le triangle $T$ est l'image du triangle A$'$B$'$C$'$ par une rotation $r$ de centre O. Quel est l'angle de cette rotation ? (aucune justification n' est demandée).
L'angle vaut 90 degrés.
\item %On appelle S le sommet du triangle $T$ ayant une abscisse strictement négative.
%Placer le point S sur la figure donnée en annexe. Quelle conjecture peut-on faire concernant les points O, B et S ?
Il semble que les points O, B et S sont alignés.
\end{enumerate}
\item
\begin{enumerate}
\item %Calculer l'affixe $z_{\text{B}'}$ du point B$'$ puis écrire $z_{\text{B}'}$ sous forme exponentielle.
On a $z_{\text{B}'} = z_{\text{B}} + 4 = - 1 + \text{i}\sqrt{3} + 4 = 3 + \text{i}\sqrt{3}$.
D'où $\left|z_{\text{B}'} \right|^2 = 3^2 + \left(\sqrt{3} \right)^2 = 9 + 3 = 12$ et $\left|z_{\text{B}'} \right| = \sqrt{12} = 2\sqrt{3}$.
En factorisant ce module on a :
$z_{\text{B}'} = 2\sqrt{3}\left(\dfrac{\sqrt{3}}{2} + \text{i}\dfrac{1}{2}\right)$.
Or $\cos \frac{\pi}{6} = \dfrac{\sqrt{3}}{2}$ et $\sin \frac{\pi}{6} = \dfrac{1}{2}$, donc
$z_{\text{B}'} = 2\sqrt{3}\left(\cos \frac{\pi}{6} + \text{i}\sin \frac{\pi}{6}\right) = 2\sqrt{3}\text{e}^{\frac{\pi}{6}}$.
\item %On admet que le point S est l'image du point B$'$ par la rotation $r$. En déduire l'écriture exponentielle de l'affixe $z_{\text{S}}$ du point S.
On admet que S est l'image du point B$'$ par la rotation $r$ de centre O et d'angle $+ \frac{\pi}{2}$. On a donc :
$z_{\text{S}} = \text{e}^{\frac{\text{i}\pi}{2}}z_{\text{B}'} = 2\sqrt{3}\text{e}^{\frac{\pi}{6} + \frac{\pi}{2}} = 2\sqrt{3}\text{e}^{\frac{4\pi}{6}} = 2\sqrt{3}\text{e}^{\frac{2\pi}{3}}$.
\item %Écrire $z_{\text{B}}$ sous forme exponentielle. Démontrer alors la conjecture faite à la question 2. d.
$z_{\text{B}} = - 1 + \text{i}\sqrt{3}$. Donc $\left|z_{\text{B}} \right|^2 = 1 + 3 = 4 \Rightarrow \left|z_{\text{B}} \right| = 2$.
En factorisant ce module :
$z_{\text{B}} = 2 \left(- \dfrac{1}{2} + \text{i}\dfrac{\sqrt{3}}{2}\right)$.
Or $\cos \frac{2\pi}{3} = - \frac{1}{2}$ et $\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}$, donc
$z_{\text{B}} = 2 \left(\cos \frac{2\pi}{3} + \text{i}\sin \frac{2\pi}{3}\right) = 2\text{e}^{\frac{2\pi}{3}}$.
Conclusion : B et S ont des affixes de même argument : ils sont alignés avec O.
\end{enumerate}
\end{enumerate}
\vspace{0,5cm}
\textbf{\textsc{Problème} \hfill 10 points}
\medskip
%Dans tout le problème, on note I l'intervalle de $\R$ défini par I $= ]0~;~ +\infty[$.
%
%\medskip
\textbf{Partie A}
\medskip
%Soit $g$ la fonction définie sur l'intervalle I par
%
%\[g(x) = x^2 - 2\ln x + 2.\]
\begin{enumerate}
\item %Pour tout réel $x$ de l'intervalle I, déterminer $g^{\prime}(x)$ puis étudier le signe de $g^{\prime}(x)$ sur l'intervalle I.
$g$ somme de fonctions dérivables sur I est dérivable sur cet intervalle et :
$g^{\prime}(x) = 2x - \dfrac{2}{x} = \dfrac{2x^2 - 2}{x} = \dfrac{2}{x}\left(x^2 - 1\right)$.
Comme $x > 0 \Rightarrow \dfrac{1}{x} > 0$ : le signe de $g^{\prime}(x)$ est donc celui du trinôme $x^2 - 1$. Celui-ci est positif sauf entre les racines $- 1$ et 1, donc ici :
$g^{\prime}(x) < 0$ sur $]0~;~1[$ ;
$g^{\prime}(x) > 0$ sur $]1~;~+ \infty[$ ;
$g^{\prime}(1) = 0$.
\item %Dresser le tableau des variations de la fonction $g$ sur l'intervalle I (les limites ne sont pas demandées).
De la question précédente résulte que :
$g$ est décroissante sur $]0~;~1[$ ;
$g$ est croissante sur $]1~;~+ \infty[$ ;
$g(1) = 1 + 2 = 3$ est le minimum de $g$ sur I.
\medskip
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\item %En déduire que, pour tout réel $x$ de l'intervalle I, on a $g(x) > 0$.
On constante que sur I, $g(x) \geqslant 3 > 0$ : la fonction est donc strictement positive sur I.
\end{enumerate}
\medskip
\textbf{Partie B}
\medskip
%Soit $f$ la fonction définie sur l'intervalle I par:
%
%\[f(x) = \dfrac{\ln x}{x} + \dfrac{1}{2}x - 1.\]
%
%On note $\mathcal{C}$ sa courbe représentative dans le repère orthonormal \Oij{} d'unité graphique 4~centimètres.
\begin{enumerate}
\item
\begin{enumerate}
\item %Déterminer la limite de la fonction $f$ en $0$.
De $\displaystyle\lim_{x \to 0} \ln x = - \infty$ et $\displaystyle\lim_{x \to 0} x = 0_{+}$ on déduit que $\displaystyle\lim_{x \to 0} \dfrac{\ln x}{x} = - \infty$ et comme $\displaystyle\lim_{x \to 0} \dfrac{1}{2}x - 1 = - 1$, par somme de limites, on conclut que :
$\displaystyle\lim_{x \to 0} f(x) = - \infty$.
\item %En déduire l'existence d'une droite asymptote à la courbe $\mathcal{C}$, notée $D$, dont on précisera une équation.
Géométriquement le résultat précédent signifie que la droite d'équation $x = 0$ est asymptote verticale à $\mathcal{C}$ au voisinage de zéro.
\end{enumerate}
\item
\begin{enumerate}
\item %Déterminer la limite de la fonction $f$ en $+ \infty$.
On sait que $\displaystyle\lim_{x \to + \infty} \dfrac{\ln x}{x} = 0$ et que $\displaystyle\lim_{x \to + \infty} \dfrac{1}{2}x = + \infty$, d'où par somme de limites :
$\displaystyle\lim_{x \to + \infty} f(x) = + \infty$.
\item %Montrer que la droite $\Delta$ d'équation $y = \dfrac{1}{2}x - 1$ est asymptote à la courbe $\mathcal{C}$ au voisinage de $+ \infty$.
Soit $d$ la fonction définie sur I par : $d(x) = f(x) - \left(\dfrac{1}{2}x - 1 \right) = \dfrac{\ln x}{x}$.
Comme $\displaystyle\lim_{x \to + \infty} d(x) = \displaystyle\lim_{x \to + \infty}\dfrac{\ln x}{x} = 0$, ceci montre que la droite $\Delta$ d'équation $y = \dfrac{1}{2}x - 1$ est asymptote à la courbe $\mathcal{C}$ au voisinage de $+ \infty$.
\item %Préciser la position relative de la courbe $\mathcal{C}$ et de la droite $\Delta$.
On a $d(x) > 0 \iff \dfrac{\ln x}{x} > 0 \iff \ln x > 0 \iff x > 1$ : ceci signifie que la courbe $\mathcal{C}$ est au dessus de la droite $(\Delta)$ sur l'intervalle $]1~;~+ \infty[$.
De même on a $d(x) < 0 \iff x < 1$ : ceci signifie que la courbe $\mathcal{C}$ est au dessus de la droite $(\Delta)$ sur l'intervalle $]0~;~1[$.
\end{enumerate}
\item
\begin{enumerate}
\item %Montrer que, pour tout réel $x,\: f^{\prime}(x) = \dfrac{g(x)}{2x^2}$.
La fonction $f$ somme de fonctions dérivables sur I est dérivable sur I et :
$f^{\prime}(x) = \dfrac{\frac{1}{x}\times x - 1 \times \ln x}{x^2} + \dfrac{1}{2} = \dfrac{1 - \ln x}{x^2} + \dfrac{1}{2} = \dfrac{2 - 2\ln x + x^2}{2x^2} = \dfrac{g(x)}{2x^2}$.
\item %En déduire le tableau complet des variations de la fonction $f$ sur l'intervalle I.
Comme $2x^2 > 0$ sur I, le signe de $f^{\prime}(x)$ est donc celui de $g(x)$ qui a été trouvé à la question A 3. Donc $g(x) > 0$ sur I entra\^{\i}ne $f^{\prime}(x) > 0$ : la fonction $f$ est donc croissante sur I.
D'où le tableau de variation :
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\end{enumerate}
\item
\begin{enumerate}
\item %Calculer les images de $1$ et de $2$ par la fonction $f$.
$f(1) = 0 + \dfrac{1}{2} - 1 = - \dfrac{1}{2}$ ;
$f(2) = \dfrac{\ln 2}{2} + \dfrac{1}{2} \times 2 - 1 = \dfrac{\ln 2}{2} \approx 0,345$.
\item %Montrer que l'équation $f(x) = 0$ admet une unique solution notée $\alpha$ dans l'intervalle [1~;~2].
Sur l'intervalle [1~;~2], la fonction $f$ est strictement croissante de $- \dfrac{1}{2} < 0$ à $\dfrac{\ln 2}{2} > 0$ : il existe donc un unique réel $\alpha \in ]1~;~2[$ tel que $f(\alpha) = 0$
\item %Donner une valeur approchée de $\alpha$ à $10^{-2}$ près.
La calculatrice donne :
$f(1,4) \approx -0,06$ et $f(1,5) \approx 0,02$, donc $1,4 < \alpha < 1,5$ ;
$f(1,47) \approx - 0,003$ et $f(1,48) \approx 0,005$, donc $1,47 < \alpha < 1,48$.
\end{enumerate}
\item %Établir une équation de la tangente $T$ à la courbe $\mathcal{C}$ au point d'abscisse $1$.
$f(1) = - \dfrac{1}{2}$ et $f^{\prime}(1) = \dfrac{2 - 2\ln 1 + 1^2}{2\times 1^2} = \dfrac{3}{2}$.
$M(x~;~y) \in T \iff y - f(1) = f^{\prime}(1) (x - 1) \iff y + \dfrac{1}{2} = \dfrac{3}{2}(x - 1) \iff y = \dfrac{3}{2}x - 2$.
\item %Tracer les droites $D,\: ,\Delta$ et $T$ puis la courbe $\mathcal{C}$.
Voir à la fin
\end{enumerate}
\medskip
\textbf{Partie C}
\medskip
\begin{enumerate}
\item %Soit $h$ la fonction définie sur l'intervalle I par
%\[h(x) = \dfrac{1}{2}(\ln x)^2.\]
%Calculer $h^{\prime}(x)$.
$h$ produit de fonctions dérivables sur I est dérivable sur I et
$h^{\prime}(x) = \dfrac{1}{2} \times 2 \times \ln x \times \dfrac{1}{x} = \dfrac{\ln x}{x}$.
\item %En déduire que $\displaystyle\int_{2}^{\text{e}} \dfrac{\ln x}{x}\:\text{d}x = \dfrac{1}{2}\left[1 - (\ln 2)^2\right]$.
On vient de démontrer que $h$ est une primitive de $\dfrac{\ln x}{x}$ sur I donc en particulier sur l'intervalle $[2~;~\text{e}]$ et
$\displaystyle\int_{2}^{\text{e}} \dfrac{\ln x}{x}\:\text{d}x = \left[h(x)\right]_{2}^{\text{e}} = h(\text{e}) - h(0) = \dfrac{1}{2}(\ln \text{e})^2 - \left(\dfrac{1}{2}(\ln 2)^2 \right) = \dfrac{1}{2} - \dfrac{1}{2}(\ln 2)^2 = \dfrac{1}{2}\left[1 - (\ln 2)^2\right]$.
\item %On considère l'aire $\mathcal{A}$ de la partie du plan délimitée par la courbe $\mathcal{C}$, son asymptote $\Delta$ et les droites d'équation $x = 2$ et $x = \text{e}$.
%Déduire de la question précédente une valeur approchée de $\mathcal{A}$ en centimètres carrés à $10^{-2}$ près.
On a vu que pour $x > 1$, la courbe $\mathcal{C}$ est au dessus de son asymptote $\Delta$. On a donc l'aire $\mathcal{A}$ qui est égale (en unités d'aire) à l'intégrale de la différence de $f(x)$ et de $\dfrac{1}{2}x - 1$.
$\mathcal{A} = \displaystyle\int_{2}{\text{e}} \dfrac{\ln x}{x} + \dfrac{1}{2}x - 1- \left(\dfrac{1}{2}x - 1\right) = \displaystyle\int_{2}^{\text{e}} \dfrac{\ln x}{x}$, soit d'après la question précédente :
$\mathcal{A} = h(\text{e}) - h(2) = \dfrac{1}{2}\left[1 - (\ln 2)^2\right]$ unités d'aire.
Or une unité d'aire vaut $4 \times 4 = 16$~cm$^2$
Donc $\mathcal{A} = 16 \times \dfrac{1}{2}\left[1 - (\ln 2)^2\right] = 8\left[1 - (\ln 2)^2\right]
\approx 4,16$~cm$^2$.
\end{enumerate}
\newpage
\begin{center}
\textbf{\large Annexe : à rendre avec la copie }
\vspace{2cm}
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\uput[dl](0,0){O}\uput[d](0.5,0){$\vect{u}$} \uput[l](0,0.5){$\vect{v}$}
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\newpage
\begin{center}
\textbf{\large Figure du problème }
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\psplot{0}{3}{0.5 x mul 1 sub}
\uput[u](2.8,0.78){$\mathcal{C}$}\uput[d](2.6,1.9){$D$}\uput[d](2.8,0.4){$\Delta$}
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crawl-data/CC-MAIN-2020-29/segments/1593657146247.90/warc/CC-MAIN-20200713162746-20200713192746-00565.warc.gz
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<em>\href{http://grecjawogniu.info/}{Grecja w ogniu}: objavljujemo dio izvještaja \href{http://rieas.gr/images/mares14.pdf}{"Odnosi između ekstremista iz Središnje i Istočne Europe s Grčkom"}
<\Slash{}em>
\textbf{IZVJEŠTAJ}
Ekstremističko se nasilje smatra ozbiljnom prijetnjom modernoj europskoj i globalnoj sigurnosti, nadasve kada je povezano s terorizmom i s drugim gerilskim strategijama. ekstremisti iz različitih zemalja i regija nalaze se danas u bliskom kontaktu i međusobno se osnažuju. Jedan bitan dio današnjeg međunarodnog ekstremizma sastoji se od razmjene strategija i taktičkih elemenata. Pojava ekstremizma u jednoj zemlji ili u jednoj regiji povezana je s razvojem ekstremizma u drugim zemljama ili regijama. Navedeni proces postaje tipičan kada govorimo o utjecaju grčkog ekstremizma na ekstremizme u Središnjoj i Istočnoj Europi. Ovaj članak ispituje kampanje i povezanost unutar ekstremističkih sredina ekstremne ljevice, ekstremne desnice i islamskih ekstremista.
\textbf{EKSTREMIZAM EKSTREMNE LJEVICE}
Suradnja između grčke ljevice s ljevicom Središnje i Istočne Europe je vrlo stara i ima svoje korijene u dvadesetom stoljeću kada su komunistički režimi (nadasve u Čehoslovačkoj) pomagali ljevičarskim emigrantima koji su bježali iz Grčke tokom građanskog rata u četrdesetim i pedesetim godinama. Ta tradicionalna grčka dijaspora nije politički aktivna, osim poneki "ekstremni" komunist. Grčka komunistička stranka (KKE) i njena ideologija predstavljaju model dogmatskoj komunističkoj stranci, nadasve u Češkoj i Slovačkoj. Međutim, KKE ne podržava terorističko nasilje i gerilu današnjice, nego održava tradicionalnu marksističko-lenjinističku politiku masovne revolucije. Njene izjave prevedene su na jezike Središnje Europe (Komunistická strana Řecka 2012).
Mladi ekstremisti ekstremne ljevice iz Grčke i Središnje i Istočne Europe započeli su intenzivnu suradnju tokom borbenog anti-global pokreta na prijelazu u XXI stoljeće. Black Block i drugi militantni aktivisti susreli su se na različitim masovnim prosvjedima, koji su bili popraćeni neredima. Grčki aktivisti posjetili su Prag tokom prosvjeda protiv Samita Svjetske Banke\Slash{}Međunarodnog Monetarnog Fonda u septembru 2000., i tokom jednog prosvjeda protiv samita NATO-a u 2002.
Militantni aktivisti središnjo-istočne Europe bili su u Ateni i u drugim grčkim gradovima (pogotovo u Solunu), na europskom Social Forumu u 2006. Od tada su ostali stalnom kontaktu s grčkom zajednicom (Mezinárodní revoluční organizace 2010). Kada je 2008. počela takozvana "Grčka pobuna" ekstremisti istočne Europe pokrenuli su snažnu kampanju podrške u svojoj zemlji. Pokrenuti su web sajtovi posvećeni specifično situaciji u Grčkoj, pogotovo poljski web sajt "Grčka u plamenu" ("Grecja w ogniu"). Na poljskom jeziku objavljene su izjave borbenih terorističkih grčkih grupa, uključujući i Συνωμοσία των Πυρήνων της Φωτιάς ("Zavjeru Vatrenih Ćelija") i njenih saveznika (Ćelija Nicola i Alfredo\Slash{}FAI 2013.). Grčka kriza je (osim sirijske) ujedno i jedna od glavnih tema češkog komunističkog i radikalnog web sajta “Třídní válka” (“Klasna borba”) (Bastl, Mareš, Smolík, Vejvodová 2011: 52).
Pojedini ekstremisti ekstremne ljevice iz Središnje i Istočne Europe uključeni su i u "borbeni turizam" u Grčkoj, dok drugi su jednostavno tamo zaposleni, i pridružuju se lokalnim anarhističkim strukturama, uključujući i grupe koje koriste nasilje. Napoznatiji slučaj takvog tipa je slučaj Andrewa Mazurka, nazvan "posljednjim zatvorenikom Grčke pobune". 2008. osuđen je u Grčkoj na 7 godina zatvora zbog sudjelovanja u sukobima na prosvjedima. U decembru 2012. deportiran je u poljski zatvor da bi odslužio ostatak presude. U Poljskoj, Slovačkoj i u Češkoj organizirani su prosvjedi i kampanje zbog tog premještaja u Poljsku. (Mareš, Výborný 2013: 30)
U Središnjoj i Istočnoj Europi izvedene su i borbene operacije povezane s ekstremizmom ekstremne ljevice u Grčkoj, ta su djela počinjena nadasve u Republici Češkoj. U aprilu 2010. kolektiv "Angry Brigade" (“Rozhněvaná brigade”) napao je molotovljevim koktelima grčku ambasadu u Pragu. Napad je bio posvećen grčkom anarhističkom zatvoreniku Yiannisu Dimitrakisu. U 2010.-2011. u Češkoj i Slovačkoj uništeno je pregršt rampi na autocestama, a barem jedan od napada koji je počinila grupa imenom "Revolucionarna Borba" (“Revoluční boj”) nadahnut je Grčkom. Napad je posvećen grčkom anarhistu Lambrosu Foundasu (Mareš, Výborný 2013: 31-32).
Posljednji godina je u Središnjoj i Istočnoj Europi može zamijećen utjecaj direktnih akcija koje su počinili grčki i talijanski ekstremisti, i teroristi ekstremne ljevice. Primijećena je i podrška antifašističkoj borbi grčke ljevice, nadasve protiv Zlatne Zore. Što predstavlja bitan element suradnje između ekstremista ekstremne ljevice Grčke i Središnje i Istočne Europe.
\emph{[svaka ispravka prijevoda je dobro došla, nap.prev.]}
Izvor:\href{http://grecjawogniu.info/?p=23701\&utm\_source=feedburner\&utm\_medium=feed\&utm\_campaign=Feed\%3A\%2Bgrecjawogniu\%2B\%28Grecja\%2Bw\%2BOgniu\%29}{Grecja w ogniu}
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ČEŠKA: IZVJEŠTAJ ČEŠKE REPUBLIKE O SURADNJI MEĐU ANARHISTIMA
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\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage[6in,cmex12]{amspapermacs}
%\makeatletter
%\let\@@hold=\@tempa
\usepackage{pb-diagram} %commutative diagrams
\usepackage{lamsarrow} %commutative diagrams
\usepackage{pb-lams} %commutative diagrams
%\let\@tempa=\@@hold
%\makeatother
\newcommand{\cft}{Class Field Theory}
\newcommand{\bs}{\bigbreak}
\newcommand{\ms}{\medbreak}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\F}{{\mathbb F}_p}
\newcommand{\llegendre}[2] {\displaystyle{\left(\frac{#1}{#2}\right)}}
\newcommand{\legendre}[2] {\left(\frac{#1}{#2}\right)}
\newcommand{\R}{{\mathbb R}}
\newcommand{\K}{\tilde K}
\renewcommand{\O}{{\mathcal O}}
\renewcommand{\P}{{\mathfrak p}}
\newcommand{\PP}{{\mathfrak P}}
\newcommand{\M}{{\mathfrak M}}
\newcommand{\N}{N_{L/K}}
\newcommand{\f}{{\mathfrak f}_{L/K}}
\begin{document}
%topmatter
\title [An Overview of Class Field Theory]{An Overview of Class Field Theory}
\author {Thomas R. Shemanske}
\address{Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755}
\email {[email protected]}
\date{\today}
%endtopmatter
\maketitle
\section[intro]{{\bf Introduction}} In these notes, we try to give a reasonably
simple exposition on the question of what is \cft. We strive more for an intuitive
discussion rather than complete accuracy on all points. A great deal of what
follows has been lifted without proper reference from the two very informative
papers by Garbanati and Wyman
\cite{Garbanati},\cite{Wyman}. The questions which we shall pose and try to
answer in the next section are:
\begin{enumerate}
\item What is \cft?
\item What are the goals of \cft?
\item What are the main results of \cft\ over $\Q$?
\end{enumerate}
\section[Origins]{\bf{The Origins of \cft}} In examining the work of Abel,
Kronecker (1821 -- 1891) observed that certain abelian extensions of imaginary
quadratic number fields are generated by the adjunction of special values of
automorphic functions arising from elliptic curves. For example, if $K$ is an
imaginary quadratic number field and ${\mathfrak A} = {\mathbb Z}\omega_1 +
{\mathbb Z}\omega_2$ is an ideal of $K$ with Im$(\omega_1/\omega_2) > 0$, then
$K(j(\omega_1/\omega_2))$ is an abelian extension of $K$, where $j$ is the
modular function.
Kronecker wondered whether all abelian extensions of $K$ could be obtained in
this manner (Kronecker's Jugendtraum). This leads to the question of ``finding"
all abelian extensions of number fields. Kronecker conjectured and Weber (1842
-- 1913) proved:
\begin{nnthm} [Kronecker--Weber (1886--1887)] \label{thm:KronWeber}
Every abelian extension of
$\Q$ is contained in a cyclotomic extension of $\Q$.
\end{nnthm}
To Kronecker and Weber, \cft\ was the task of finding all abelian extensions,
and of finding a generalization of Dirichlet's theorem on primes in arithmetic
progressions which is valid in number fields.
Hilbert saw that \cft\ is much more --- that it is the theory of abelian
extensions. In his famous address to the ICM in Paris in 1900, Hilbert posed
numerous questions two of which are the focus of the endeavors in \cft.
\pagebreak[3]
\begin{Ventry}{Hilbert's 12th:}
\item[Hilbert's 9th:] To develop the most general reciprocity law in an
arbitrary number field, generalizing Gauss' law of quadratic reciprocity.
\begin{Ventry}{---}
\item[---] For abelian extensions, this is the Artin reciprocity law
\item[---] For non-abelian extensions, the question is still open and one
cannot expect an answer similar to the one in the abelian case. In particular,
``congruence conditions will not suffice".
\end{Ventry}
\item[Hilbert's 12th:] Generalize Kronecker's Jugendtraum.
\end{Ventry}
\subsection{\bf What is a reciprocity law?} Let $f \in \Z[X]$ be monic and
irreducible, and let $K_f$ be the splitting field of $f$ over $\Q$. Then
$K_f/\Q$ is a finite Galois extension. Let $p \in \Z$ be a prime and
$\F = \Z/p\Z$. Reducing $f$ mod $p$ gives a polynomial $f_p \in \F[X]$.
If $f_p$ factors into distinct linear factors over $\F$ then we say that $f$
{\it splits completely modulo $p$}.
Define $Spl(f) = \{\,p\in \Z \mid f \text{ splits completely modulo }
p\,\}$. With finite exceptions, $Spl(f) = \{\,p\in \Z \mid p \text{ splits
completely in } K_f\,\}$ via the Dedekind-Kummer theorem (see \S4).
By a reciprocity law, we intend a means by which to describe the factorization of
$f_p$ as a function of $p$, or somewhat less demanding, a ``rule" which
determines which primes belong to $Spl(f)$.
First, why is this of interest? In response, we have the Inclusion theorem:
\begin{nnthm} [Inclusion Theorem] Let $f$, $g$ be irreducible polynomials in
$\Z[X]$ with splitting fields $K_f$, $K_g$ respectively. Then
$K_f \supset K_g$ if and only if $Spl(f) \subset^* Spl(g)$.
\end{nnthm}
\noindent Here $\subset^*$ means with finitely many exceptions.
Thus $K_f = K_g$ if and only if $Spl(f) =^* Spl(g)$, that is the set $Spl(f)$
captures the Galois extension.
\begin{proof} $(\Rightarrow)$ This direction is straightforward. If $p \in
Spl(f)$ then $e(K_f/\Q) = 1$ and $f(K_f/\Q)=1$. Since $K_f \supset K_g$ and $e$
and $f$ are multiplicative in towers, we have $e(K_g/\Q) = 1$ and
$f(K_g/\Q)=1$, and hence $p \in Spl(g)$.
$(\Leftarrow)$ This direction follows from the Tchebotarev density
theorem.
\end{proof}
We give an example.
\ms
\noindent{\bf Example.} Let $p$ be a prime $p\equiv 1\pmod4$, $f(X) = X^2 -
p$, and $g(X) = X^p -1$. Then $K_f = \Q(\sqrt p)$ and $K_g = \Q(\zeta_p)$
where $\zeta_p$ is a primitive $p$-th root of unity. Since $p\equiv 1\pmod4$,
we have $K_f \subset K_g$. We must show that $Spl(f) \supset^* Spl(g)$. It is
well-known that a prime $q \in Spl(g)$ (i.e., $q$ splits completely in
$\Q(\zeta_p)$) iff $q \equiv 1\pmod p$ and $q \in Spl(f)$ (i.e., $q$ splits
completely in $\Q(\sqrt p)$) iff $\legendre q p = 1$ (via the
Dedekind-Kummer theorem). Clearly any prime $q$ satisfying $q \equiv 1\pmod p$
satisfies $\legendre q p = 1$, hence $Spl(f) \supset Spl(g)$.
\bs
Another theorem of great importance is the
\begin{nnthm} [Abelian Polynomial Theorem] The set $Spl(f)$ can be described
by congruences with respect to a modulus depending only on $f$ ($K_f$) if and
only if $K_f$ is an abelian extension of $\Q$.
\end{nnthm}
\noindent The Artin reciprocity law is a precise version of ($\Leftarrow$), and
($\Rightarrow$) says that ``congruence conditions" will not suffice to
characterize a reciprocity law for non-abelian extensions.
\ms
\noindent {\bf Examples}:
\begin{enumerate}
\item Let $p \in \Z$ be an odd prime, and
consider the quadratic polynomial $f(X) = X^2 - q$ where $q$ is an odd prime.
Then modulo $p$, three things can happen:
%\renewcommand{\theenumi}{\alph{enumi}}
%\renewcommand{\labelenumi}{(\theenumi)}
\begin{enumerate}
\item $f_p(X) = l(X)^2$, linear $l(X)$
\item $f_p(X) = l_1(X)l_2(X)$ distinct linear factors ($f$ splits
completely modulo $p$)
\item $f_p$ is irreducible.
\end{enumerate}
\noindent (a) occurs iff $x^2 \equiv q\pmod p$ has one solution iff $p = q$.
\noindent (b) occurs iff $x^2 \equiv q\pmod p$ has two solutions iff
$\legendre{q}{p} = +1$.
\noindent (c) occurs iff $x^2 \equiv q\pmod p$ has no solutions iff
$\legendre{q}{p} = -1$.
To determine for which $p$ the congruence $x^2 \equiv q\pmod p$
is solvable, is apriori an infinite problem. On the other hand, it is one from
which the traditional form of quadratic reciprocity rescues us.
Suppose $q = 17$ in the above example. Then $\legendre{17}{p} = +1$
iff
$\legendre{p}{17} = +1$ iff $p \equiv 1,2,4,8,9,13,15,16\pmod {17}$.
Thus $p \in Spl(x^2 - 17)$ iff (with finite exceptions)
$p \equiv 1,2,4,8,9,13,15,16\pmod {17}$.
\item Next consider the cyclotomic polynomials, $\Phi_n$. Let
$\zeta$ be a primitive $n^{th}$ root of unity and $\Phi_n$ the irreducible
polynomial of
$\zeta$ over $\Q$. We know that the degree of $\Phi_n$ is $\phi(n)$ and that
$x^n - 1 = \prod_{d \mid n} \Phi_d$.
To describe $Spl(\Phi_n)$ we need to answer which primes split completely in
$K_{\Phi_n} = \Q(\zeta)$. If $p \nmid n$ then for any prime of
$K_{\Phi_n}$ lying above $p$, we know $e = 1 $ and $fg = \phi(n)$. Moreover,
$f$ is determined by the relation that it is the smallest positive integer such
that $p^f \equiv 1\pmod n$. Thus $p$ splits completely in $K_{\Phi_n}$ iff
$f=1$, hence $p \in Spl(\Phi_n)$ iff (wfe) $p \equiv 1\pmod n$, characterizing
$Spl(\Phi_n)$ by congruence conditions.
\end{enumerate}
Let us loosely define the {\it arithmetic of a number field $K$} to be the study
of the ideals of $K$ and the quotient rings determined by the ideals of $K$ as
well as the study of the ideal class group and groups isomorphic to subgroups
or quotient groups of the ideal class group.
\ms
\noindent {\bf Goals of \cft}:
\begin{enumerate}
\item Describe all finite abelian extensions of $K$ in terms of the
arithmetic of $K$.
\item Canonically realize $Gal(L/K)$ in terms of the arithmetic of $K$
whenever $Gal(L/K)$ is abelian.
\item Describe the decomposition of a prime ideal from $K$ to $L$ in terms
of the arithmetic of $K$ whenever $L/K$ is abelian (i.e., provide a reciprocity
law).
\end{enumerate}
\subsection{\bf Summary of \cft\ over $\Q$.}
\noindent Notation: $\Q_m = \Q(e^{2\pi i/m})$. We may assume that $m \not\equiv
2(4)$. For if $m\equiv 2\pmod4$ with $m = 2m_0$, then we easily observe that
$-e^{2\pi i/{m_0}}$ is a primitive $m$th root of unity, and hence that
$\Q_m = \Q_{m_0}$. Over
$\Q$, the Kronecker-Weber Theorem motiviates the following definition:
\begin{nndefinition}
Let $L/\Q$ be a finite abelian extension. A positive integer $m$ is called a
{\it defining modulus} or an {\it admissible modulus of $L$} if $L \subset
\Q_m$. Such an $m$ exists by the Kronecker-Weber theorem. The {\it conductor} of
$L$, ${\mathfrak f}_L$, is the smallest admissible modulus of $L$.
\end{nndefinition}
\noindent {\bf Examples:}
\begin{enumerate}
\item $L = \Q_m$. Then ${\mathfrak f}_L = m$, since $\Q_m \subset \Q_n$ implies
that $\Q_m = \Q_m \cap \Q_n = \Q_{(m,n)}$ implies that $m \mid n$.
\item Let $L$ be the maximal real subfield of $\Q_m$. Then $L = \Q(\zeta +
\zeta^{-1})$ where $\zeta = e^{2 \pi i /m}$ (it is the fixed
field of complex conjugation). Note that if $m = 3,4$, then $L = \Q$.
For $m \ge 5$, ${\mathfrak f}_L = m$. For $m = 3,4$, ${\mathfrak f}_L = 1$.
\item $L = \Q(\sqrt d)$, $d$ square-free integer, $|d| > 1$.
Then $${\mathfrak f}_L = |\text{disc}(L)| =
\begin{cases}
|d|& \text{if $d\equiv 1\pmod 4$}\\
|4d|& \text{if $d\equiv 2,3\pmod 4$}.
\end{cases}$$
\end{enumerate}
To gain some feeling for why the last example holds, recall that if $L = \Q_p$
($p$ an odd prime), then disc$(L) = (-1)^{\frac{p-1}2} p^{p-2}$ is the square
of an integer in
$\mathcal O_L$, thus
$$\Q\Big(\sqrt{(-1)^{\frac{p-1}2} p}\Big)
\subset
\Q_p.$$
It follows that for a prime $p$
$$\Q(\sqrt p) \subset
\begin{cases}
\Q_p&\text{if $p \equiv 1\pmod 4$}\\
\Q_{4p}& \text{if $p \equiv 3 \pmod 4$}\\
\Q_8&\text{if $p = 2$}.
\end{cases}$$
\noindent Moreover, if $d = \pm 2^\nu p_1 p_2 \cdots p_r$ is squarefree, then
$\Q(\sqrt d) \subset
\Q(\sqrt{2^\nu},\sqrt{p_1},\sqrt{p_2},\ldots,\sqrt{p_r}) \subset
\Q(\zeta_{4\cdot2^\nu})\Q(\zeta_{p_1},\zeta_{p_2},\ldots,\zeta_{p_r},\zeta_{4})
= \Q(\zeta_{4d})$.
\begin{nnthm} Let $L/\Q$ be a finite abelian extension, and $m$ an
admissible modulus of $L$. Then ${\mathfrak f}_L \mid m$.
\end{nnthm}
\begin{proof} $L \subset \Q_m \cap \Q_{{\mathfrak f}_L} = \Q_{({\mathfrak
f}_L,m)}$ which implies ${\mathfrak f}_L \mid m$.
\end{proof}
\bs
Let $L$ be an abelian extension of $\Q$, and let $m$ be an admissible modulus of
$L$. Then $L \subset \Q_m$. Let $a \in \Z$ with $(a,m) =
1$, and denote by $\llegendre{L}{a}$ the Artin symbol, the
automorphism of $L$ obtained by restricting to the field $L$ the automorphism of
$\Q_m$ determined by ($\zeta
\mapsto \zeta^a$). Then the Artin map is the homomorphism
$$\llegendre{L}{*}:(\Z/m\Z)^\times \to Gal(L/\Q).$$
\noindent The Artin map is onto since every automorphism of $L$ extends to one of
$\Q_m$ which has the above form. Denote the kernel of $\llegendre{L}{*}$
by $I_{L,m}$. Identifying $(\Z/m\Z)^\times$ with $Gal(\Q_m/\Q)$, we see that
$I_{L,m}$ is identified with $Gal(\Q_m/L)$, so under the Galois correspondence (see
diagram below),
$L$ is the fixed field of the subgroup $I_{L,m}$ of $(\Z/m\Z)^\times$.
$$
\begin{diagram}
\node{\Q_m}\arrow{e}\arrow{s,-}
\node{\{1\}}\arrow{w}\arrow{s,-}\\
\node{L}\arrow{e}\arrow{s,-}
\node{I_{L,m}}\arrow{w}\arrow{s,-}\\
\node{\Q}\arrow{e}
\node{(\Z/m\Z)^\times}\arrow{w}
\end{diagram}
$$
\noindent This information is summarized in the
\begin{nnthm} [Artin Reciprocity] Let $L/\Q$ be a finite abelian
extension with defining modulus $m$. Then the following sequence is exact:
$$1 \rightarrow I_{L,m} \hookrightarrow (\Z/m\Z)^\times \rightarrow Gal(L/\Q)
\rightarrow 1.$$
\end{nnthm}
\noindent Thus, the Artin map induces an isomorphism between $Gal(L/\Q)$ and
$(\Z/m\Z)^\times/I_{L,m}$ thus canonically realizing $Gal(L/\Q)$ in terms of the
arithmetic of $\Q$. In particular, this
says that every abelian extension is given in terms of the arithmetic of $\Q$,
and so realizes one of the primary goals of \cft.
\bs
As a special case, if $L$ is a quadratic extension of $\Q$ contained in
$\Q_m$, then $Gal(L/\Q)$ is isomorphic to $\{\pm 1\}$, and identifying the
isomorphic groups, the Artin map essentially can be defined by $\legendre L,a =
\legendre a m$. To make clearer what we mean, we examine some typical cases in
the examples below.
\ms
\noindent {\bf Examples}:
\begin{enumerate}
\item Let $p$ be an prime $p \equiv 1\pmod4$. Then $\Q(\sqrt p) \subset \Q_p$.
If $L = \Q(\sqrt p)$, then since $[L:\Q] = 2$, $I_{L,p}$ is a subgroup of
index two in $(\Z/p\Z)^\times$. Since $(\Z/p\Z)^\times$ is cyclic, there is a
unique such subgroup, namely the squares (or quadratic residues) mod $p$.
For any $a$ prime to $p$, $\legendre L a = \pm1$, and
$I_{L,p}$ is the kernel of $\legendre L *$. With $I_{L,p}$ identified as the
group of squares mod $p$ and $Gal(L/\Q)$ identified with $\{\pm1\}$, it is clear
that $\legendre L a = \legendre a p$.
\item A considerably more complicated example is $L = \Q(\sqrt 7)$.
Clearly the conductor of $L$ is 28, so take $m = 28$ in the setup above.
Here we will see that $\legendre L a$ is almost $\legendre a {28}$.
The only real difficulty in interpreting the quadratic residue symbol
$\legendre a 2$, so we digress for a moment.
Recall that $\legendre a 2$ is defined by
$$\legendre a 2 =
\begin{cases}
1&\text{if $a \equiv 1\pmod 8$}\\
-1&\text{if $a \equiv 5\pmod 8$}\\
0&\text{otherwise}.
\end{cases}$$
In particular, if $a$ is squarefree and $p$ is any prime, then $\legendre a p$ is
1, $-1$ or 0 depending upon whether $p$ splits, is inert, or ramifies in $\Q(\sqrt
a)$. The difficulty we encounter is that if $a \equiv 3\pmod 4$, then
$\legendre a 4 = \legendre a 2^2 \ne \legendre {a^2} 2$, the first expression
equalling zero, while the last equals 1.
To continue, let $\zeta_m = e^{2\pi i/m}$, and consider the tower of fields below.
$$
\begin{diagram}
\node[2]{\Q(\zeta_{28})}
\arrow{sw,-} \arrow{s,-} \arrow{se,-}\\
\node{\Q(\zeta_4)}\arrow{s,=}
\node{\Q(\sqrt {-1},\sqrt {-7})}\arrow{sw,-}\arrow{s,-}\arrow{se,-}
\node{\Q(\zeta_7)}\arrow{s,-}\\
\node{\Q(\sqrt{-1})}\arrow{se,-}
\node{\Q(\sqrt 7)}\arrow{s,-}
\node{\Q(\sqrt{-7})}\arrow{sw,-}\\
\node[2]{\Q}
\end{diagram}
$$
By the Galois correspondence, there is a corresponding lattice of groups.
$$
\begin{diagram}
\node[2]{H(\zeta_{28})}
\arrow{sw,-} \arrow{s,-} \arrow{se,-}\\
\node{H(\zeta_4)}\arrow{s,=}
\node{H(\sqrt {-1},\sqrt {-7})}\arrow{sw,-}\arrow{s,-}\arrow{se,-}
\node{H(\zeta_7)}\arrow{s,-}\\
\node{H(\sqrt{-1})}\arrow{se,-}
\node{H(\sqrt 7)}\arrow{s,-}
\node{H(\sqrt{-7})}\arrow{sw,-}\\
\node[2]{H(1)}
\end{diagram}
$$
Here we set the notation by putting $H(1) = (\Z/28\Z)^\times$ (and
$H(\zeta_{28}) = \{1\}$). Then for example,
$H(\sqrt{-7})$ is the subgroup of $(\Z/28\Z)^\times$
corresponding to $Gal(\Q(\zeta_{28})/\Q(\sqrt{-7}))$.
Our purpose is to calculate $I_{L,{28}}$ where $L = \Q(\sqrt 7)$, and to compare
the values of $\legendre L a$ with those of $\legendre a {28}$. The subgroup
$I_{L,{28}}$ will simply be $H(\sqrt 7)$.
If we consider the tower $\Q \subset \Q(\sqrt{-7}) \subset \Q(\zeta_7)$, then
as a subgroup of $(\Z/7\Z)^\times$, $\Q(\sqrt{-7})$ corresponds to the subgroup
of quadratic residues mod 7 (as in example 1), that is to $\{1, 2, 4\}$. Modulo
28 (i.e. $a \equiv 1,2,4\pmod 7$ and $a\equiv 1,3\pmod4$), this yields
$H(\sqrt{-7}) = \{1, 9, 11, 15, 25, 23\}\subset (\Z/28\Z)^\times$.
The tower $\Q \subset \Q(\sqrt{-1}) = \Q(\zeta_4)$ is degenerate yielding the
trivial subgroup of $(\Z/4\Z)^\times$ corresponding to $\Q(\sqrt{-1})$, or
$\{a | a \equiv 1\pmod4\}$. Modulo 28 (i.e., $a \equiv 1\pmod4$ and
$a \not\equiv 0\pmod7$), this yields
$H(\sqrt{-1}) = \{1, 5, 9, 13, 17, 25\}$.
As $\Q(\sqrt{-1},\sqrt{-7})$ is the compositum of $\Q(\sqrt{-1})$ and
$\Q(\sqrt{-7})$, Galois theory tells us that
$H(\sqrt{-1},\sqrt{-7}) = H(\sqrt{-1}) \cap H(\sqrt{-7}) = \{ 1, 9, 25\}$.
For the record, we note that $\legendre a {28} = +1$ if and only if
$(a,28) = 1$ and $\legendre a 7 = \legendre a 4$, which is true if and only if
$a \equiv 1,2,4\pmod7$ and $a\equiv 1\pmod4$. Note that since
$\legendre a 4 = \legendre a 2^2$, $\legendre a 4$ is never equal to $-1$.
Thus $\{a | \legendre a {28} = 1\} = \{ 1, 9, 25\}$, and is not equal to
$H(\sqrt 7)=I_{L,{28}}$ which has order 6. It is now a trivial matter to deduce
that $H(\sqrt 7)= \{1, 3, 9, 19, 25, 27\}$.
\item To handle more general examples like $L = \Q(\sqrt{\pm35})$, we need only
consider one of the two tower of fields below and use the techniques of the
preceding examples.
If $L = \Q(\sqrt{-35})$, we consider the tower
$$
\begin{diagram}
\node[2]{\Q(\zeta_{35})}
\arrow{sw,-} \arrow{s,-} \arrow{se,-}\\
\node{\Q(\zeta_7)}\arrow{s,-}
\node{\Q(\sqrt {-7},\sqrt {5})}\arrow{sw,-}\arrow{s,-}\arrow{se,-}
\node{\Q(\zeta_5)}\arrow{s,-}\\
\node{\Q(\sqrt{-7})}\arrow{se,-}
\node{\Q(\sqrt{-35})}\arrow{s,-}
\node{\Q(\sqrt{5})}\arrow{sw,-}\\
\node[2]{\Q}
\end{diagram}
$$
\noindent whereas if $L = \Q(\sqrt{35})$, we consider the tower
$$
\begin{diagram}
\node[2]{\Q(\zeta_{140})}
\arrow{sw,-} \arrow{s,-} \arrow{se,-}\\
\node{\Q(\zeta_{28})}\arrow{s,-}
\node{\Q(\sqrt {7},\sqrt {5})}\arrow{sw,-}\arrow{s,-}\arrow{se,-}
\node{\Q(\zeta_5)}\arrow{s,-}\\
\node{\Q(\sqrt{7})}\arrow{se,-}
\node{\Q(\sqrt{35})}\arrow{s,-}
\node{\Q(\sqrt{5})}\arrow{sw,-}\\
\node[2]{\Q}
\end{diagram}
$$
\noindent and proceed as in the previous examples.
\end{enumerate}
\bs
To continue our investigation of class fields, we have the following theorem
which gives information about the conductor of an abelian extension.
\begin{nnthm} [Conductor--Ramification Theorem] If $L$ is a finite
abelian extension of $\Q$, then a prime $p$ of $\Q$ ramifies in $L$ if and only
if $p \mid {\mathfrak f}_L$.
\end{nnthm}
\begin{nncor} If $L \neq \Q$ is a finite abelian extension of $\Q$, then at
least one prime $p$ ramifies in $L$.
\end{nncor}
\begin{proof} Since $L \neq \Q$, $L \nsubseteq \Q_1 = \Q$, hence ${\mathfrak f}_L
> 1$, and so is divisible by at least one prime.
\end{proof}
\noindent For contrast, we have the result of Minkowski that a prime $p$ of $\Q$
ramifies in a number field $L$ if and only of $p \mid \text{disc}(L)$. This
says that for abelian extensions, there should be a connection between the
conductor and the discriminant (see the conductor-discriminant formula below).
\begin{nnthm}[Decomposition Theorem] Let $m$ be a defining modulus
of $L$. If $p \nmid m$ (in particular $p$ is unramified) then the order of
$pI_{L,m}$ in $(\Z/m\Z)^\times/I_{L,m}$ is $f$, the residue class degree.
\end{nnthm}
\noindent
Notice that this generalizes the theorem about the decomposition of primes in
cyclotomic fields. If we choose $L = \Q_m$, then $I_{L,m} = 1$, and we are
reduced to talking about the order of $p$ in $(\Z/m\Z)^\times$.
\noindent Let $m = {\mathfrak f}_L$ in the above theorem. Since $efg = [L:\Q]$,
\begin{align*}
p \in Spl(L/\Q) &\Leftrightarrow e=1,\ f=1\\
&\Leftrightarrow p \nmid {\mathfrak f}_L,\ p\in I_{L,{\mathfrak f}_L}
\end{align*}
the first condition because $e = 1$ and the second because $f = 1 $ via the
Decomposition theorem.
If $I_{L,{\mathfrak f}_L} = \{a_1, \ldots a_s\}$ with $a_i \in \Z$ and
$(a_i,{\mathfrak f}_L) = 1$, then $p \in Spl(L/\Q) \Leftrightarrow
p \equiv a_i\pmod {{\mathfrak f}_L}$ for some $i$. This acomplishes the goal of
describing $Spl(L/\Q)$ in terms of congruence conditions, and hence the
decomposition of primes in terms of the arithmetic of $\Q$.
\subsection{\bf Duality}
Let $X_m$ denote the character group of $(\Z/m\Z)^\times$. That is
$\chi \in X_m$ implies that $\chi:(\Z/m\Z)^\times \to {\mathbb C}^\times$ is a
homomorphism.
\begin{nndefinition} We say that $d$ is a {\it defining modulus} for $\chi$ if $a
\equiv 1\pmod d$ implies that $\chi(a) = 1$. The {\it conductor} of $\chi$,
denoted
${\mathfrak f}_\chi$, is the smallest defining modulus for $\chi$.
\end{nndefinition}
\noindent If $m$ is a defining modulus for a finite abelian extension $L$, let
$$X_{L,m} = \{\chi \in X_m | \chi(h) = 1 \hbox{ for all } h \in I_{L,m}\}$$
\noindent Recall that $I_{L,m}$ is the subgroup of
$ (\Z/m\Z)^\times \cong Gal(\Q_m/\Q) $ corresponding to the subfield $L
\subset \Q_m$ via the Galois correspondence. That is, $I_{L,m} \cong
Gal(\Q_m/L)$, and from duality we see that
$$X_{L,m}\cong Gal(\Q_m/L)^\perp \cong
\widehat{Gal(\Q_m/\Q)/Gal(\Q_m/L)} \cong \widehat{Gal(L/\Q)}.$$
\noindent Finally, we have the
\begin{nnthm}[Conductor--Discriminant Formula] Let $m$ be an
admissible modulus for a finite abelian extension $L$ of $\Q$. Then
$${\mathfrak f}_L = \text{lcm}\{{\mathfrak f}_\chi | \chi \in X_{L,m}\}$$
and
$$|\text{disc}(L)| = \prod_{\chi \in X_{L,m}}{\mathfrak f}_\chi.$$
\end{nnthm}
\noindent In particular, ${\mathfrak f}_L \mid \text{disc}(L)$, and so we always
have the tower of fields:
$$\Q \subset L \subset \Q_{{\mathfrak f}_L} \subset \Q_{|\text{disc}(L)|}.$$
\section{{\bf Global \cft}}
In order to generalize \cft\ to ground fields other than $\Q$, several issues
need to be addressed:
\begin{enumerate}
\item The Kronecker-Weber theorem is valid only for ground field $\Q$, so we need
a new notion of admissible modulus (a very deep theorem).
\item We need to handle all the infinite primes.
\item We need a generalized notion of congruence.
\item With what shall we replace $(\Z/m\Z)^\times$ and $\Q_m$?
\end{enumerate}
\bs
Let $\M$ be a modulus and let $\M_0$ denote its finite part. For a number
field $K$, let $I_K^\M$ denote the group of fractional ideals of $K$ relatively
prime to $\M_0$. Let
$$K_{\M,1} = \{\,\alpha \in K^\times \mid
\alpha \equiv 1\ (\bmod\,^* \M)\,\}.$$
\noindent Recall that $\alpha \equiv 1\ (\bmod\,^* \M)$ means that
\begin{align*}
\text{ord}_\P(\alpha -1) &\ge \text{ord}_\P(\M_0) \quad \text{ for all }\ \P \mid
\M_0\quad \text{ and}\\
\alpha &> 0 \text{ at each real prime dividing $\M$}
\end{align*}
Let $R_\M = \{ \alpha\O_K \mid \alpha \in K_{\M,1}\,\}$. $R_\M$ is called the
{\it ray mod $\M$}. Let $C_\M = I_K^\M/R_\M$, the {\it ray class group}.
Special cases are familiar. If $\M = 1$, then the ray class group $C_1$ is just
the ideal class group of the field $K$. If $K = \Q$ and $\M = mp_\infty$, where
$m$ is a positive integer, then $C_\M \cong (\Z/m\Z)^\times$.
Let $L/K$ be a Galois extension and let $\M$ be a $K$-modulus. Define
$I_L^\M = I_L^{\M_0\O_L}$ and
\begin{align*}
L_{\M,1} = \{\,\alpha \in L^\times \mid &
\alpha \equiv 1\ (\bmod\,^* \M_0\O_L)\\
&\text{and where $\alpha > 0$ at each real prime of $L$}\\
&\text{dividing a real prime occuring in $\M$}\,\}.
\end{align*}
\noindent Finally let $R_{L,\M} = \{ \alpha\O_L \mid \alpha \in L_{\M,1}\,\}$, and
$C_{L,\M} = I_L^\M/R_{L,\M}$.
Recall that the norm of an ideal relative to a Galois extension $L/K$ is
defined as follows: If $\P$ is a prime of $K$ and $\PP$ is a prime of $L$
lying above $\P$ with inertial degree $f$, then we define the norm of $\PP$ to
be
$\N(\PP) = \P^f$. We extend the definition of the norm to the group of
fractional ideals by multiplicativity. Note that when $K = \Q$,
$\N(\PP) = \P^f = p^f\Z$ for the prime $p\Z = \PP\cap \Z$, while the absolute
norm of $\PP$ is equal the cardinality of the residue class field $\O_L/\PP$
which is $p^f$, so this definition provides a natural generalization of the
absolute norm.
One can show that $\N(R_{L,\M}) \subset R_\M$, and so the definition of the
norm can be extended to $C_{L,\M}$ by defining
$\N({\mathfrak A}R_{L,\M}) = \N({\mathfrak A}) R_\M$. Put
$$I_{L/K,\M} = \N(C_{L,\M}) < C_\M.$$
\noindent For example, if $K=\Q$ and $L \subset \Q_m$ (i.e. $m$ is an admissible
modulus of $L$), then we have the diagram:
$$
\begin{diagram}
\node{\Q_m}\arrow{e}\arrow{s,-}
\node{\{1\}}\arrow{w}\arrow{s,-}\\
\node{L}\arrow{e}\arrow{s,-}
\node{I_{L,m}}\arrow{w}\arrow{s,-}\\
\node{\Q}\arrow{e}
\node{(\Z/m\Z)^\times}\arrow{w}
\end{diagram}
$$
If we let $\M = mp_\infty$, then it can be shown that
$I_{L/K,\M} \cong I_{L,m}$. Notice also that $C_\M \cong (\Z/m\Z)^\times$ and
that $[C_\M : I_{L/\Q,\M}] = [L:\Q]$ by the Galois correspondence. Generalizing
this fact, we have the deep theorem:
\begin{nnthm} Let $\M$ be a $K$-modulus and $L/K$ an abelian extension
of number fields. Then there exists a unique $K$-modulus $\f$ such that
$[C_\M : I_{L/K,\M}] = [L:K]$ iff $\f \mid \M$.
\end{nnthm}
\noindent The unique modulus $\f$ is called the {\it conductor} of $L/K$ and any
$K$-modulus divisible by $\f$ is called an {\it admissible} modulus of $L/K$.
This is
not a very intuitive theorem because we don't have something natural like the
Kronecker-Weber theorem with which to define the conductor. The theorem is
proved in two steps. The first inequality to be established was that $[C_\M :
I_{L/K,\M}] \le [L:K]$ for all moduli $\M$. This was done by Weber (1897-8).
It is now known as the ``second inequality". In 1920, Tagaki showed that $[C_\M
: I_{L/K,\M}] \ge [L:K]$ for some modulus $\M$, now known as the ``first
inequality".
\noindent One can also show that
$${\mathfrak f}_{L/\Q} =
\begin{cases}
(f_L)& \text{if $L \subset \R$}\\ (f_L)p_\infty&\text{ if $L \not\subset \R$}
\end{cases}$$
Now we need an analog of the cyclotomic fields and the Kronecker-Weber
theorem.
\begin{nnthm}[Existence] Given a $K$-modulus $\M$ and a subgroup
$I_\M$ of the ray class group $C_\M$, there exists a unique abelian
extension $L/K$ such that
\begin{enumerate}
\item $\M$ is an admissible modulus of $L/K$
\item $I_{L/K,\M} = \N(C_{L,\M}) = I_\M$\qquad or
\item The kernel of the Artin map $I_K^\M \to Gal(L/K)$ is $H_\M$ where
$I_\M = H_\M/R_\M.$
\end{enumerate}
\end{nnthm}
\noindent $L$ is called the {\it class field} of the subgroup $I_\M$. When $I_\M=
R_\M$, the trivial subgroup, the class field $L$ is called the {\it ray class
field} and is denoted $K(R_\M)$. If $K=\Q$ and $\M = mp_\infty$, then $K(R_\M) =
\Q_m$, that is the cyclotomic fields are the ray class fields for the moduli
$\M = mp_\infty$. The ray class field for the modulus $\M = m$ is the maximal
real subfield of $\Q_m$.
We have two more important theorems:
\begin{nnthm}Given an abelian extension $L/K$, there exists a
$K$-modulus $\M$ so that $L \subset K(R_\M)$.
\end{nnthm}
As a consequence, we recover the Kronecker-Weber
theorem.
\begin{nnthm}Given an abelian extension $L/K$, the conductor $\f$ is the
``smallest" $K$-modulus $\M$ such that $L \subset K(R_\M)$. Moreover, $\M$ is
an admissible modulus of $L/K$ iff $L \subset K(R_\M)$.
\end{nnthm}
Thus it follows that every abelian extension of $K$ is a subfield of a ray class
field for $K$. We have classified the abelian extensions, but we have not
constructed them. More later.
We have the following generalization of Dirichlet's theorem on primes in
arithmetic progressions.
\begin{nnthm} Let $I_\M$ be a subgroup of the ray class group $C_\M$.
Then $I_\M = H_\M/R_\M$ where $R_\M \subset H_\M \subset I_K^\M$. Then there
are an infinite number of primes in each coset of $I_K^\M/H_\M$. In fact, the
primes in the coset have density $1/[I_K^\M\,:\,H_\M]$.
\end{nnthm}
\noindent If $K = \Q$ and $\M = mp_\infty$, then $C_\M \cong (\Z/m\Z)^\times$.
If we choose $H_\M = R_\M$, then $I_K^\M/H_\M = C_\M \cong (\Z/m\Z)^\times$,
and we have recovered the Dirichlet theorem over $\Q$.
It can be shown that if $\M$ is an admissible modulus for an abelian extension of
number fields $L/K$, then the Artin map, $\llegendre{L/K} *$, is trivial on
the ray mod $\M$, $R_\M$, and so the definition of the Artin map can be
extended to the ray class group, $C_\M$.
In 1927, Artin proved
\begin{nnthm}[Artin Reciprocity] Let $L/K$ be an abelian
extension of number fields, and let $\M$ be an admissible modulus of $L/K$.
Then the following sequence is exact:
$$1 \rightarrow I_{L/K,\M}= \N(C_{L/K,\M})
\hookrightarrow C_\M \rightarrow Gal(L/K) \rightarrow 1.$$
\end{nnthm}
\begin{nncor}Let $\M$ be an admissible modulus of the abelian
extension $L/K$. Then $L \subset K(R_\M)$, $Gal(K(R_\M)/K) \cong C_\M$ and
$Gal(K(R_\M)/L) \cong I_{L/K,\M}$.
\end{nncor}
\begin{nndefinition} The Hilbert class field of a number field $K$ is the
ray class field $K(R_1)$, and will be denoted $\K$.
\end{nndefinition}
From above we see that $Gal(\K/K) \cong C_1$, the ideal class group of $K$.
Thus much work is done in trying to understand subfields of $\K$ to help
understand the structure of the ideal class group.
\begin{nnthm}
A $K$-prime $\P$ ramifies in $L$ iff $\P\mid\f$.
\end{nnthm}
\begin{nnthm}
The Hilbert class field $\K$ is the maximal
abelian unramified extension of $K$.
\end{nnthm}
\begin{proof} Since $\K = K(R_1)$, ${\mathfrak f}_{\K/K} = (1)$. If $L$ is an
unramified extension of $K$, then $\f=(1)$ by the above theorem. Since $(1)$
is an admissible modulus for $L/K$, we have $L \subset K(R_1) = \K$.
\end{proof}
\begin{nnthm}
Each fractional ideal of $K$ becomes principal in $\K$.
\end{nnthm}
\noindent This does not say that $\K$ has class number one. Instead it suggests
the ``class tower problem". Let $K_0 = K$ and $K_i = \tilde K_{i-1}$ for
$i\ge1$. Does there exist a $j$ such that $K_j = K_{j-1}$? This would imply
that the class number of $K_{j-1}$ equals 1. Golod and Shafarevich (1964)
showed that any imaginary quadratic field $\Q(\sqrt{-d})$ where $d$ is a
positive integer divisible by at least six primes has an infinite class field
tower.
\section{{\bf Equivalence of the reciprocity laws.}}
We consider the case of a prime $p$, $p \equiv 1 \ (\bmod\, 4)$, and $q$ an odd
prime, $q \ne p$.
Gauss' law says that $\legendre p q = \legendre q p$.
Wyman asks for a rule which describes the primes $q$ which split completely in
$\Q(\sqrt p)$.
Artin says that
$$1 \rightarrow H \hookrightarrow I_\Q^{pp_\infty} \rightarrow
Gal(\Q(\sqrt p)/\Q) \rightarrow 1$$
\noindent for an appropriately defined subgroup $H$ is exact where the map
$I_\Q^{pp_\infty} \rightarrow Gal(\Q(\sqrt p)/\Q)$ is the Artin map.
\noindent Consider the diagram:
$$
\begin{diagram}
\node{L=\Q_p} \arrow{s,-}\arrow{e}
\node{\{1\}}\arrow{s,-}\arrow{w}\\
\node{K= \Q(\sqrt p)}\arrow{s,-}\arrow{e}
\node{Gal(\Q_p/\Q(\sqrt p))}\arrow{s,-}\arrow{w}\\
\node{\Q}\arrow{e}\node{Gal(\Q_p/\Q)}\arrow{w}
\end{diagram}
$$
The Artin map which we need to consider is $\llegendre {K/\Q} * $.
However, from the properties of the Frobenius automorphism, we know that
$\llegendre {K/\Q} * = \llegendre {L/\Q} * \bigg|_K$, so we compute
$\llegendre {L/\Q} *$ instead.
\begin{nnlem} Let $m > 0$, $q$ a prime with $q\nmid m$, and ${\mathcal Q}$ a
prime of $\Q_m$ lying above $q$. Then the $m$-th roots of unity are distinct
modulo ${\mathcal Q}$.
\end{nnlem}
\begin{proof} Let $\zeta_m$ be a primitve $m$-th root of unity.
Then
$$ X^m -1 = \prod_{j=0}^{m-1} (X - \zeta_m^j)$$
\noindent implies
$$ X^{m-1} + \cdots + X + 1 = \prod_{j=1}^{m-1} (X - \zeta_m^j)$$
\noindent and hence
$$ m = \prod_{j=1}^{m-1} (1 - \zeta_m^j).$$
If $\zeta_m^j \equiv \zeta_m^k \ (\bmod\, {\mathcal Q})$, then
$(1-\zeta_m^l) \equiv 0 \ (\bmod\, {\mathcal Q})$ for some $l$, hence
$(m,{\mathcal Q})
\ne 1$. Since ${\mathcal Q}$ is prime, we have ${\mathcal Q}\mid m\O$ and hence
$m\O \subset {\mathcal Q}$. Thus $m \in {\mathcal Q}\cap \Z = q\Z$ which implies
$q\mid m$, a contradiction.
\end{proof}
\noindent Now let $\sigma:I_\Q^{mp_\infty} \to Gal(\Q_m/\Q)$ be the Artin map, and
denote by $\sigma_a$ the automorphism $\sigma(a) \in Gal(\Q_m/\Q)$
characterized by $\sigma_a(\zeta_m) = \zeta_m^a$. $\sigma_q$
is the element of the Galois group, $Gal(\Q_m/\Q)$, which satisfies
$\sigma_q(x) \equiv x^q \ (\bmod\, {\mathcal Q})$ for all $x \in \Z[\zeta_m]$. In
particular, $\sigma_q(\zeta_m) \equiv \zeta_m^q \ (\bmod\, {\mathcal Q})$, and
since every automorphism of $Gal(\Q_m/\Q)$ is characterized by $\tau(\zeta_m) =
\zeta_m^a$ for some $a$, we have by the lemma that
$\sigma_q(\zeta_m) = \zeta_m^q$.
\begin{nnthm} The following are equivalent:
\begin{enumerate}
\item $\legendre p q = 1$.
\item $\displaystyle{X^2 - X + \frac{1-p}4 \equiv 0 \ (\bmod\, q)}$ is
solvable.
\item $q$ splits in $\displaystyle{\Q(\sqrt p) =
\Q\Big(\frac{1+\sqrt p}{2}\Big)}$.
\item $\legendre {K/\Q}q = 1$.
\item $\legendre q p = 1$.
\end{enumerate}
\end{nnthm}
\begin{proof} The equivalence of (1) and (2) can be seen directly. If
$\legendre p q = 1$, then $p \equiv \alpha^2 \ (\bmod\, q)$ for some
$\alpha \in \Z$. $\displaystyle{X\equiv \frac{1+\alpha}{2} \ (\bmod\, q)}$
solves
$\displaystyle{X^2 - X + \frac{1-p}{4} \equiv 0 \ (\bmod\, q)}$. Conversely,
if
$\displaystyle{X^2 - X + \frac{1-p}{4} \equiv (X-\alpha)(X-\beta) \ (\bmod\, q)}$,
then
$(\alpha - \beta)^2 \equiv p \ (\bmod\, q)$, hence $\legendre p q = 1$.
Note that this is really pretty obvious if we think of
$\alpha,\beta \equiv \frac{1 \pm \sqrt p} {2}$, the real roots of the
quadratic.
The equivalence of (2) and (3) is a consequence of the Dedekind-Kummer theorem.
\begin{nnthm}[Dedekind-Kummer] Let $A$ be a Dedekind domain with
quotient field $K$, let $E/K$ be a finite separable extension, and let $B$ be
the integral closure of $A$ in $E$. Suppose that $B = A[\alpha]$ for some
$\alpha \in E$ and let $f(X)$ be the irreducible polynomial for $\alpha$ over
$K$. Let $\P$ be a prime ideal of $A$. Let $\overline{f(X)}$ denote the
reduction of $f(X)$ modulo $\P$. Suppose
$$\overline{f(X)} = \overline{P_1(X)}^{e_1}\cdots\overline{P_g(X)}^{e_g}$$
is the factorization of $f(X)$ modulo $\P$ into powers of distinct monic
irreducible polynomials in $(A/\P)[X]$. Let $P_i(X) \in A[X]$ be a monic
polynomial in $A[X]$ which reduces mod $\P$ to $\overline{P_i(X)}$. Let
$\PP_i$ be the ideal of $B$ generated by $\P$ and $P_i(\alpha)$. Then $\PP_i$
is a prime ideal of $B$ lying above $\P$, $e_i$ is the ramification index, the
$\PP_i$'s are distinct, and
$$\P B = \PP_1^{e_1}\cdots\PP_g^{e_g}$$
is the factorization of $\P$ in $B$.
\end{nnthm}
\noindent We merely note that the roots of $X^2 - X + \frac{1-p}{4}$ are
$\frac{1\pm\sqrt p}{ 2}$ which generate the ring of integers of $\Q(\sqrt
p)$. For the converse, observe that if $X^2 - X + \frac{1-p}{4}$ was
irreducible mod $q$, then
$q$ would be inert in $\Q(\sqrt p)$.
The equivalence of (3) and (4) is an elementary property of the Frobenius
automorphism.
The equivalence of (4) and (5) is where the fun is. Recall that for any
integer $a$ not divisible by $p$, we denote by $\sigma_a$ the automorphism
$\llegendre {L/\Q} a$ of $Gal(\Q_p/\Q)$. From above, we know that
$\sigma_a(\zeta_p) = \zeta_p^a$, and from elementary properties of the Frobenius
that $\displaystyle{\sigma_a|_K = \llegendre{K/\Q} a}$.
The map $a \leftrightarrow \sigma_a$ gives the isomorphism between
$(\Z/p\Z)^\times$ and $Gal(\Q_p/\Q)$. Consider the diagram modified from above:
$$
\begin{diagram}
\node{L=\Q_p} \arrow{s,-}\arrow{e}
\node{\{1\}}\arrow{s,-}\arrow{w}\\
\node{K= \Q(\sqrt p)}\arrow{s,-}\arrow{e}
\node{\{\text{squares}\}}\arrow{s,-}\arrow{w}\\
\node{\Q}\arrow{e}\node{(\Z/p\Z)^\times}\arrow{w}
\end{diagram}
$$
\noindent By the Galois correspondence,
$$Gal(\Q(\sqrt p)/\Q) \cong Gal(\Q_p/\Q)/Gal(\Q_p/\Q(\sqrt p))
\cong (\Z/p\Z)^\times/\{\text{squares}\}$$
\noindent Now (4) is true if and only if
$\sigma_q|_K = 1$ in $Gal(\Q(\sqrt p)/\Q)$, hence if and only if
$\sigma_q \in Gal(\Q_p/\Q(\sqrt p))$, hence
under the correspondence above if and only if $q \in \{\text{squares}\}$ if
and only if (5).
\end{proof}
\section{{\bf Examples of Hilbert Class Fields}}
\begin{enumerate}
\item $K = \Q$. Then $\tilde K = \Q$ since any proper extension
of
$\Q$ is ramified (as a consequence of Minkowski's bound on the discriminant).
\item $K = \Q(\sqrt{-15})$. Then $\tilde K = \Q(\sqrt{-3},\sqrt 5)$.
To see this we need to do a little work. Let $L = \Q(\sqrt{-3},\sqrt 5)$ and
consider the tower of fields:
$$
\begin{diagram}
\node[2]{\Q(\sqrt{-3},\sqrt 5)}
\arrow{sw,-} \arrow{s,-} \arrow{se,-}\\
\node{\Q(\sqrt{-15})}\arrow{se,-}
\node{\Q(\sqrt 5)}\arrow{s,-}
\node{\Q(\sqrt{-3})}\arrow{sw,-}\\
\node[2]{\Q}
\end{diagram}
$$
First, we show that $L/K$ is an unramified extension of fields. Consider the
infinite primes first. Since both primes of $K$ are complex, there can be no
ramification from $K$ to $L$ at the infinite places. Observe that
$\Delta_{K/\Q} = -15$ hence 3 and 5 are the only primes which ramifiy in $K$.
It is then clear that 3 and 5 ramify in $L$. Moreover, these are the only
finite primes $p$ of $\Q$ which ramify in $L$, since if $p \ne
3,5$ is prime, then (by checking discriminants) $p$ is unramified in both
$\Q(\sqrt{-3})$ and $\Q(\sqrt 5)$, and hence in the compositum $L$. Thus the
only primes which can ramify from $K$ to $L$ are the primes in $K$ lying above 3
and 5.
Consider a prime $\PP$ of $L$ lying above 3. Note that since $L/\Q$ is Galois
(it is the compositum of Galois extensions), it doesn't really matter which
prime $\PP$ we choose. Let $\P = \PP \cap K$, and $\P' = \PP \cap \Q(\sqrt
5)$. We know that
$$e(\PP/3) = e(\PP/\P)e(\P/3) = e(\PP/\P')e(\P'/3)$$
\noindent and that $e(\P'/3) = 1$, $e(\P/3) = 2$, and that
$e(\PP/\P') \le [L\,:\,\Q(\sqrt 5)] = 2$. This implies that
$e(\PP/\P) = 1$. This together with an analogous argument for the prime 5
shows us that $L/K$ is an unramified (necessarily abelian) extension. Thus
$L \subset \tilde K$.
Out of the study of Dirichlet $L$-series come various analytic formulae for the
class number of number fields (see Borevich and Shafarevich for example).
The significance is that $Gal(\tilde K/K)$ is isomorphic to the ideal class
group of $K$, and hence $[\tilde K\,:\,K] = h_K$.
If $K = \Q(\sqrt
{-d})$ with $d >2$ and the conductor ${\mathfrak f}_K$ of $K$ (in the old sense -- ignoring
the infinite prime) is odd, then
$$h_K = \frac{1}{2 - \legendre 2 d}\cdot
\sum_{\substack{0<a<{\mathfrak f}_K/2\\(a,{\mathfrak f}_K) = 1}}
\legendre a d.$$
\noindent Here $\llegendre 2 n = (-1)^{\frac{n^2-1}{8}}$.
Recall that $\Delta_{K/\Q} = -15$, ${\mathfrak f}_K \mid \Delta_{K/\Q}$ and ${\mathfrak f}_K$ is
divisible by every finite prime of $\Q$ which ramifies in $K$. Thus it is
immediate that ${\mathfrak f}_K = 15$. It is now trivial to check that
$$h_K = \frac{1 }{2 - (1)}\cdot \bigg[\legendre 1 {15} + \legendre 2 {15} +
\legendre 4 {15} + \legendre 7 {15}\bigg] = 2.$$
\noindent Thus $K \subset L \subset \tilde K$ and $[\tilde K\,:\,K] = 2$ and hence
$\tilde K = L = \Q(\sqrt {-3},\sqrt 5)$, as claimed.
\end{enumerate}
% References
\begin{thebibliography}{99}
\bibitem{Garbanati}
D. Garbanati, Class Field Theory Summarized,
{\em Rocky Mountain Journal of Mathematics}, {\bf 11} Number 2, (1981),
195--225.
\bibitem{Wyman} B. F. Wyman, What Is a Reciprocity Law?,
{\em American Mathematical Monthly}, {\bf 79}, (1972), 571--586.
\end{thebibliography}
\end{document}
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\title{Jailed Residents Describe Experiences}
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Sunday night a bunch of us were over at a friend’s house. We didn’t have room to stay there so we thought we’d try to make it back to another guy’s apartment. We were almost home when five cop cars pulled up with guns sticking out of all the windows and stopped us.
We were in two cars. The cops that came over to our car stuck shot guns in our faces and made us get out. They handcuffed our hands behind our backs. The handcuffs were fastened very tightly just at the wrist joint so that today, Thursday, our hands are still numb.
They lined us up against the brick wall of a house and started questioning us, searching us and banging our heads against the wall. There were three of us, two white guys and one black guy. They found some empty cartridges in the black guy’s pocket that he had picked up off the street—because he had the cartridges they thought he must have a gun, too.
They searched the car and couldn’t find one. One cop stuck the barrel of his shotgun in the guy’s throat, cocked it, and told him if he didn’t tell where the gun was hidden they were going to kill him. At the same time they were kicking him and hitting him across the head and back with black jacks.
Meanwhile they were working us over. They lifted our hands up behind our backs as far as they could with the handcuffs and said “Come on you dirty cock suckers, where the hell are you hiding that gun.” They kicked us in the ass and the balls and screamed “Where are the guns, you dirty cock suckers?”
Later my friend who was in the other car told me they stopped his car, cocked a pistol and stuck it in his mouth, and told him to get out of the car. Another one grabbed his hands and handcuffed him. He threw him to the ground and five cops started stomping on him.
He tried to bury his face in the street and they just about peeled the back of his head with their boots. They put him in the back seat of the cop car with two of his friends and told them to put their heads between their knees. While they were driving one cop sitting in the front seat beat them on the backs of their heads with a flashlight yelling at them, “Try to lick your dick.”
We were put into a different cop car, one of a whole caravan going to the Vernor precinct station. On the way they wanted to stop and pick up looters but were afraid of getting shot.
In the car they repeatedly hit us on the head, the back of the neck and ribs with a black jack and jabbed us in the ribs with a shotgun. They were screaming that they were going to put us into a cell with a bunch of black guys and tell them we were setting fire to black people’s homes and let them work us over.
At the station they photographed us and took information. They let us keep our wallets except one guy who reached to take his wallet back was hit with the butt of a pistol and never got his wallet back.
They lined us up and marched us down the hall to a cell. The black guy had been separated from us. As they shoved us into the cell one cop stood and hit us in the face with his fist. They locked six of us in one cell about eight feet long, six and a half feet high and five feet wide. It had a sink, a toilet that kept overflowing, and a bench running along one wall.
Every once in a while they would bring in more prisoners. Almost all of them were brought in for curfew violations and almost all of them had been beaten. People in some cells had no toilets or drinking water. Sometimes police would take them out of the block to get a drink or use a toilet and sometimes they just told them to get fucked.
Monday night they opened up the cells because they were so crowded and let people roam around in the cell block. I talked to one guy who had been picked up Saturday night on a drunk charge and still hadn’t had anything to eat.
Tuesday night they brought fifteen sandwiches on a tray to the cell block. The block was made with twenty-four cells, each for one person and there were now over one hundred people in the block. Those near the door grabbed the food and the cops told us they would be back in a while with more.
People formed a line in front of the window and waited for hours until some of them started fainting. For a while the cops refused to do anything about them. People started screaming and making a lot of noise until finally the cops came and moved them somewhere else.
Then a turnkey came to the door and said “Ten of you come with me.” He took us to the bullpen which is a fairly small room with a sink, a toilet and a bench. There were thirty three of us. Two turnkeys agreed to buy candy bars for us but one took a commission.
I talked to one of the black guys with us in the bullpen. He said “We were in the Packer store on Trumbull and Grand River. Man, there was everybody in there, hillbillies and soul brothers and everybody just takin’ all the shit they could get there ‘hands on and everybody was saying ‘this mother fuckin’ Packer store done robbed everybody for so long we just gonna clean the store out.
“People was pushin’ away whole grocery carts full of food and then this one cop car drives their cop car sittin’ out on the street. Man, people inside just kept right on lootin’. They started throwin’ cans at them cops.”
When I asked him why he thought people were rioting he said, “Man, peoples is workin’ their ass off on their jobs and ain’t makin’ shit. And if they bought the house they’re livin’ in with their life savings the taxes is so high. And then they want to tear it down for some expressway or university or some thing. People is tired of bein screwed over by everything. Seems a lotta white people don’t like it neither.”
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anon.
Jailed Residents Describe Experiences
\bigskip
\href{https://www.fifthestate.org/archive/35-august-1-15-1967/jailed-residents-describe-experiences}{\texttt{https://www.fifthestate.org/archive/35-august-1-15-1967/jailed-residents-describe-experiences}}
Fifth Estate \#35, August 1–15, 1967
\bigskip
\textbf{fifthestate.anarchistlibraries.net}
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#
# ChangeLog for doc/theses/fangren_yu_COOP_F20/Report.tex
#
# Generated by Trac 1.2.1
# Nov 27, 2022, 10:13:58 AM
Tue, 19 Jan 2021 18:31:06 GMT Fangren Yu <f37yu@…> [7722242]
* doc/theses/fangren_yu_COOP_F20/Report.tex (modified)
add acknowledgement section, fix a few format issues
Tue, 19 Jan 2021 14:01:55 GMT Peter A. Buhr <pabuhr@…> [4b1c8da]
* doc/bibliography/pl.bib (modified)
* doc/theses/fangren_yu_COOP_F20/Report.tex (modified)
proofread Fangren's co-op report, add necessary references
Fri, 15 Jan 2021 20:01:11 GMT Fangren Yu <f37yu@…> [2d63023]
* doc/theses/fangren_yu_COOP_F20/Report.tex (modified)
add abstract and introduction section
Wed, 13 Jan 2021 18:03:49 GMT Peter A. Buhr <pabuhr@…> [101cc3a]
* doc/theses/fangren_yu_COOP_F20/Report.tex (modified)
* doc/theses/fangren_yu_COOP_S20/Report.tex (modified)
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Wed, 13 Jan 2021 16:36:57 GMT Peter A. Buhr <pabuhr@…> [91571e5]
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* doc/theses/fangren_yu_COOP_F20/Report.tex (added)
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* doc/theses/fangren_yu_COOP_S20/figures/DeepNodeSharing.fig.bak (deleted)
remove unnecessary files, add Fangren Yu F20 report "Optimization of ...
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%aTpuscrit Denis Vergès
%Relecture Ftançois Hache
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\usepackage{fancyhdr}
\usepackage[dvips]{hyperref}
\hypersetup{%
pdfauthor = {APMEP},
pdfsubject = {Baccalauréat STMG },
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\lhead{\small Baccalauréat Sciences et Technologies du Management et de la Gestion (STMG)}
\lfoot{\small{Métropole-La Réunion}}
\rfoot{\small{janvier 2020}}
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\begin{center}{\Large \decofourleft~\textbf{Baccalauréat STMG Métropole-La Réunion~e3c Auto A ~\decofourright\\[5pt] janvier 2020 }}
\end{center}
\vspace{0,7cm}
\textbf{ÉPREUVE DE MATHÉMATIQUES - Séries technologiques}
\medskip
%Auto A
\begin{center}\textbf{PARTIE 1}\end{center}
\medskip
\textbf{ Automatismes \quad Sans calculatrice \quad Durée: 20 minutes}
\medskip
Pour chaque question, indiquer la réponse dans la case correspondante.
Aucune justification n'est demandée.
\begin{center}
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\begin{tabularx}{\linewidth}{|c|p{8cm}|X|}\cline{2-3}
\multicolumn{1}{c}{}&\multicolumn{1}{|c|}{Énoncé} &\multicolumn{1}{c|}{Réponse}\\ \hline
\textbf{1.}&Résoudre dans $\R$ l'équation $3x - 5 = 7$.&\\ \hline
\textbf{2.}&Une veste coûte $80$ \euro. On obtient une remise de 20\,\% sur son
prix.
Quel est le montant de la remise ?&\\ \hline
\textbf{3.}&Le chiffre d'affaires d'une entreprise pour l'année 2019 est de \np{10000}~\euro. Le chef d'entreprise prévoit une diminution de 5\,\% de ce chiffre d'affaires en 2020.
Calculer le chiffre d'affaires prévisible pour 2020.&\\ \hline
\textbf{4.}&Développer et réduire l'expression $(x - 3)^2$.&\\ \hline
\textbf{5.}&Quel est le signe de la fonction affine $f$ définie par
$f(x) = - 2x + 8$ lorsque $x > 4$ ?&\\ \hline
\textbf{6.}&Exprimer sous la forme d'une puissance de $2$ :
$\dfrac{2^{10}}{2 \times 2^3}$.\rule[-3mm]{0mm}{8mm}&\\ \hline
\textbf{7.}&Déterminer la valeur de l'entier positif $n$ tel que:
$10^n < \np{2019} < 10^{n+1}$.&\\ \hline
\textbf{8.}&Soit $f$ la fonction définie par $f(x) = 3x^2 +1$.
Calculer l'image de 2 par $f$.&\\ \hline
\textbf{9.}& Peut-on dire que la droite d'équation $y = 3x - 1$ passe par le point de coordonnées (2~;~1) ?
Répondre par \og oui\fg ou \og non \fg.&\\ \hline
\textbf{10.}&On considère la fonction $f$ représentée par la courbe ci-dessous:
\begin{center}
\psset{unit=5mm}
\begin{pspicture*}(-5.5,-2.5)(2.5,5.5)
\psgrid[gridlabels=0pt,subgriddiv=1,gridcolor=lightgray]
\psaxes[linewidth=1.25pt,labelFontSize=\scriptscriptstyle](0,0)(-5.5,-2.5)(2.5,5.5)
\psplot[plotpoints=2000,linewidth=1.25pt,linecolor=blue]{-5}{2}{x 4 add x 1 add mul 2 div}
\uput[dr](0,0){O}
\end{pspicture*}
\end{center}
Avec la précision permise par le graphique, lire l'image de $- 1$ par $f$.&\\ \hline
\end{tabularx}
\end{center}
\newpage
\textbf{PARTIE II}
\medskip
\textbf{Calculatrice autorisée selon la réglementation en vigueur}
\smallskip
\textbf{Cette partie est composée de trois exercices indépendants}
\bigskip
\textbf{EXERCICE 2 \hfill 5 points}
\medskip
Au cours de l'année 2019, Adam est embauché par une entreprise qui lui propose un salaire
mensuel net de \np{1500}~\euro.
Son employeur lui annonce que son salaire mensuel net augmentera de $50$~\euro{} au 1\up{er} janvier de
chaque année suivante.
On note $u$ la suite qui modélise le salaire mensuel net d'Adam au cours de l'année $2019 + n$. Ainsi, $u(0) = \np{1500}$ et $u(1) = \np{1550}$.
\medskip
\begin{enumerate}
\item Calculer le salaire mensuel net d'Adam en 2021.
\item Établir une relation entre $u(n + 1)$ et $u(n)$ et préciser la nature de la suite $u$.
\item Quel est le sens de variation de la suite $u$ ? Justifier la réponse.
\end{enumerate}
Au cours de l'année 2019, Alice est embauchée par une entreprise qui lui propose un salaire mensuel net de \np{1400}~\euro.
Son employeur lui annonce que son salaire mensuel net augmentera de 4\,\% au 1\up{er} janvier de chaque année suivante.
On note $v$ la suite qui modélise le salaire mensuel net d'Alice au cours de l'année $2019 +n$.
\begin{enumerate}[resume]
\item Quelle est la nature de la suite $v$ ?
\item A partir de quelle année le salaire mensuel net d'Alice dépassera-t-il pour la première
fois le salaire mensuel net d'Adam ?
\end{enumerate}
\bigskip
\textbf{EXERCICE 3 \hfill 5 points}
\medskip
On considère la fonction $f$ définie sur l'intervalle $[-2~;~4]$.
Sa courbe représentative est une parabole que l'on note $\mathcal{C}$.
\begin{center}
\psset{unit=1.2cm}
\begin{pspicture*}(-2.5,-4.9)(4.5,4.5)
\psgrid[gridlabels=0pt,subgriddiv=1,gridwidth=0.2pt]
\psaxes[linewidth=1.25pt]{->}(0,0)(-2.5,-5)(4.5,4.5)
\psplot[plotpoints=2000,linewidth=1.25pt,linecolor=blue]{-2}{4}{x 1 add x 3 sub mul }
\pscurve*(0,-3)(0.5,-3.4)(1.,-2.6)(1.5,-3.7)(2,-2.8)(2.5,-3.2)(2.25,-3.3)(1.5,-4.2)(1,-4.5)(0.8,-4.2)(0.5,-4.5)(0,-3)
\uput[r](-1.5,2.5){\blue $\mathcal{C}$}
\end{pspicture*}
Une tâche d'encre masque une partie de la courbe $\mathcal{C}$.
\end{center}
\medskip
\begin{enumerate}
\item Lire sur le graphique l'image de $-1$ et celle de 3 par $f$.
\item Résoudre par lecture graphique sur l'intervalle $[-2~;~4]$, l'inéquation $f(x) \leqslant 0$.
\item On admet que l'expression de la fonction $f$ est de la forme
\[f(x) = \left(x - x_1\right)\left(x - x_2\right)\: \text{avec }\: x_1< x_2.\]
Préciser les valeurs respectives de $x_1$ et de $x_2$.
\item Le sommet de la parabole n'apparait pas sur le dessin. Retrouver ses coordonnées en détaillant le raisonnement.
\item Dresser le tableau des variations de la fonction $f$ sur l'intervalle $[-2~;~4]$. On admettra que $f(-2) = f(4)= 5$.
\end{enumerate}
\bigskip
\textbf{EXERCICE 4 \hfill 5 points}
\medskip
Un match de rugby entre deux équipes A et 8 se déroule dans un stade accueillant \np{75000} spectateurs.
Parmi les spectateurs:
\setlength\parindent{1cm}
\begin{itemize}[label=$\bullet~~$]
\item \np{52500} sont des supporters de l'équipe A ;
\item \np{32250} sont licenciés à la Fédération française de rugby (FFR) ;
\item \np{13125} supporters de l'équipe A sont licenciés à la FFR.
\end{itemize}
\setlength\parindent{0cm}
\medskip
\begin{enumerate}
\item Recopier et compléter le tableau croisé d'effectifs avec les données fournies dans l'énoncé.
\begin{center}
\begin{tabularx}{\linewidth}{|*{4}{>{\centering \arraybackslash}X|}}\cline{2-4}
\multicolumn{1}{c|}{~}&Licenciés à la FFR&Non licenciés à la FFR&Total\\ \hline
Supporters de l'équipe A&&&\\ \hline
Supporters de l'équipe 8&&&\\ \hline
Total&&&\np{75000}\rule[-12pt]{0pt}{30pt}\\ \hline
\end{tabularx}
\end{center}
\item On interroge au hasard un spectateur du match. On considère les évènements suivants:
$A$ : \og le spectateur est un supporter de l'équipe A\fg
$B$ : \og le spectateur est un supporter de l'équipe B \fg
$L$ : \og le spectateur est licencié à la FFR \fg.
Pour tout évènement $E$, on note $P(E)$ sa probabilité. Les probabilités seront données sous forme décimale.
\begin{enumerate}
\item Calculer $P(B)$.
\item Décrire l'évènement $ A \cap L$.
\item Calculer $P(A \cap L)$.
\end{enumerate}
\item On interroge au hasard un spectateur du match. C'est un supporter de l'équipe B. Calculer la probabilité qu'il soit licencié à la FFR.
\end{enumerate}
\end{document}
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http://dlmf.nist.gov/18.14.E2.tex
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nist.gov
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crawl-data/CC-MAIN-2016-50/segments/1480698542414.34/warc/CC-MAIN-20161202170902-00065-ip-10-31-129-80.ec2.internal.warc.gz
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\[|\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x\right)|\leq|\mathop{P^{(%
\alpha,\beta)}_{n}\/}\nolimits\!\left(-1\right)|=\frac{\left(\beta+1\right)_{n%
}}{n!},\]
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https://www.dickimaw-books.com/latex/admin/html/exercises/invoice-isodoc-sql.tex
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% This is an example document accompanying "LaTeX for Administrative Work"
% See www.dickimaw-books.com/latex/admin/ for the book details,
% including licence.
%
% This requires two datatooltk calls.
% Arara v4.0 directives for datatooltk:
% arara: datatooltk: { output: order.dbtex,
% arara: --> sqldb: samples,
% arara: --> sqluser: sampleuser,
% arara: --> sql: "SELECT people.surname, people.forenames,
% arara: --> people.title, people.address1, people.address2,
% arara: --> people.town, people.county, people.postcode,
% arara: --> countries.name AS countryname, ordergroups.discount,
% arara: --> ordergroups.postage, SUM(books.price * orders.quantity)
% arara: --> AS subtotal,
% arara: --> (SUM(books.price*orders.quantity)+ordergroups.postage-ordergroups.discount)
% arara: --> AS total
% arara: --> FROM books, orders, ordergroups, people, countries
% arara: --> WHERE orders.groupid = 2 AND ordergroups.id=orders.groupid
% arara: --> AND orders.bookid = books.id
% arara: --> AND people.id = ordergroups.customerid
% arara: --> AND countries.code = people.country"}
% arara: datatooltk: { output: orderlist.dbtex,
% arara: --> sqldb: samples,
% arara: --> sqluser: sampleuser,
% arara: --> sql: "SELECT books.title AS booktitle, books.author,
% arara: --> books.format, books.price, orders.quantity,
% arara: --> (books.price * orders.quantity) AS subtotal
% arara: --> FROM books, orders
% arara: --> WHERE orders.groupid = 2 AND orders.bookid = books.id"}
%
% Arara directive to build document:
% arara: pdflatex
% arara: pdflatex
\documentclass{isodoc}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{datatool}
\DTLloaddbtex{\thisorder}{order.dbtex}
\DTLloaddbtex{\orderlist}{orderlist.dbtex}
\begin{document}
\DTLassign{\thisorder}{1}{%
\Title=title,%
\Forenames=forenames,%
\Surname=surname,%
\AddressI=address1,%
\AddressII=address2,%
\Town=town,%
\County=county,%
\Postcode=postcode,%
\CountryName=countryname,%
\OrderDiscount=discount,%
\Postage=postage,%
\SubTotal=subtotal,%
\Total=total%
}
\invoice
[
to={\DTLifnullorempty{\Title}{}{\Title\ }%
\Forenames\ \Surname\\%
\AddressI\\%
\DTLifnullorempty{\AddressII}{}{\AddressII\\}%
\Town\\%
\DTLifnullorempty{\County}{}{\County\\}%
\Postcode\\%
\CountryName},
currency={\pounds}
]
{
\itable
{
\DTLforeach*{\orderlist}%
{%
\BookTitle=booktitle,%
\BookAuthor=author,%
\BookFormat=format,%
\BookPrice=price,%
\OrderQuantity=quantity,%
\ThisSubTotal=subtotal%
}%
{%
\iitem{\OrderQuantity\ $\times$ \BookTitle\newline
by \BookAuthor\ (\BookFormat) @ \BookPrice\ each}{\ThisSubTotal}%
}%
\itotal[Subtotal]{\SubTotal}
\iitem{Postage and Packaging}{\Postage}%
\iitem{Promotional Discount}{$-\OrderDiscount$}%
\itotal{\Total}
}
}
\end{document}
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\documentclass[a4paper]{article}
\def\npart {III}
\def\nterm {Easter}
\def\nyear {2017}
\def\nlecturer {N.\ S.\ Manton and D.\ Stuart}
\def\ncourse {Classical and Quantum Solitons}
\def\nisofficial {}
\input{header}
\usepackage[compat=1.1.0]{tikz-feynman}
\tikzfeynmanset{/tikzfeynman/momentum/arrow shorten = 0.3}
\tikzfeynmanset{/tikzfeynman/warn luatex = false}
\begin{document}
\maketitle
{\small
\setlength{\parindent}{0em}
\setlength{\parskip}{1em}
Solitons are solutions of classical field equations with particle-like properties. They are localised in space, have finite energy and are stable against decay into radiation. The stability usually has a topological explanation. After quantisation, they give rise to new particle states in the underlying quantum field theory that are not seen in perturbation theory. We will focus mainly on kink solitons in one space dimension, on gauge theory vortices in two dimensions, and on Skyrmions in three dimensions.
\subsubsection*{Pre-requisites}
This course assumes you have taken Quantum Field Theory and Symmetries, Fields and Particles. The small amount of topology that is needed will be developed during the course.
\subsubsection*{Reference}
N.\ Manton and P.\ Sutcliffe, \emph{Topological Solitons}, CUP, 2004
}
\tableofcontents
\setcounter{section}{-1}
\section{Introduction}
Given a classical field theory, if we want to ``quantize'' it, then we find the vacuum of the theory, and then do perturbation theory around this vacuum. If there are multiple vacua, then what we did was that we arbitrarily picked a vacuum, and then expanded around that vacuum.
However, these field theories with multiple vacua often contain \emph{soliton} solutions. These are localized, smooth solutions of the classical field equations, and they ``connect multiple vacuums''. To quantize these solitons solutions, we fix such a soliton, and use it as the ``background''. We then do perturbation theory around these solutions, but this is rather tricky to do. Thus, in a lot of the course, we will just look at the classical part of the theory.
Recall that when quantizing our field theories in perturbation theory, we obtain particles in the quantum theory, despite the classical theory being completely about fields. It turns out solitons also behave like particles, and they are a \emph{new} type of particles. These are non-perturbative phenomena. If we want to do the quantum field theory properly, we have to include these solitons in the quantum field theory. In general this is hard, and so we are not going to develop this a lot.
What does it mean to say that solitons are like particles? In relativistic field theories, we find these solitons have a classical energy. We define the ``mass'' $M$ of the soliton to be the energy in the ``rest frame''. Since this is relativistic, we can do a Lorentz boost, and we obtain a moving soliton. Then we obtain an energy-momentum relation of the form
\[
E^2 - \mathbf{P} \cdot \mathbf{P} = M^2.
\]
This is a Lorentz-invariant property of the soliton. Together with the fact that the soliton is localized, this is a justification for thinking of them as particles.
These particles differ from the particles of perturbative quantum fields, as they have rather different properties. Interesting solitons have a \emph{topological} character different from the classical vacuum. Thus, at least naively, they cannot be thought of perturbatively.
There are also non-relativistic solitons, but they usually don't have interpretations as particles. These appear, for example, as defects in solids. We will not be interested in these much.
What kinds of theories have solitons? To obtain solitons, we definitely need a non-linear field structure and/or non-linear equations. Thus, free field theories with quadratic Lagrangians such as Maxwell theory do not have solitons. We need interaction terms.
Note that in QFT, we dealt with interactions using the interaction picture. We split the Hamiltonian into a ``free field'' part, which we solve exactly, and the ``interaction'' part. However, to quantize solitons, we need to solve the full interacting field equations \emph{exactly}.
Having interactions is not enough for solitons to appear. To obtain solitons, we also need some non-trivial vacuum topology. In other words, we need more than one vacuum. This usually comes from symmetry breaking, and often gauge symmetries are involved.
In this course, we will focus on three types of solitons.
\begin{itemize}
\item In one (space) dimension, we have kinks. We will spend $4$ lectures on this.
\item In two dimensions, we have vortices. We will spend $6$ lectures on this.
\item In three dimensions, there are monopoles and Skyrmions. We will only study Skyrmions, and will spend $6$ lectures on these.
\end{itemize}
These examples are all relativistic. Non-relativistic solitons include \emph{domain walls}, which occur in ferromagnets, and two-dimensional ``baby'' Skyrmions, which are seen in exotic magnets, but we will not study these.
In general, solitons appear in all sorts of different actual, physical scenarios such as in condensed matter physics, optical fibers, superconductors and exotic magnets. ``Cosmic strings'' have also been studied. Since we are mathematicians, we probably will not put much focus on these actual applications. However, we can talk a bit more about Skyrmions.
Skyrmions are solitons in an \emph{effective field theory} of interacting pions, which are thought to be the most important hadrons because they are the lightest. This happens in spite of the lack of a gauge symmetry. While pions have no baryon number, the associated solitons have a topological charge identified with baryon number. This baryon number is conserved for topological reasons.
Note that in QCD, baryon number is conserved because the quark number is conserved. Experiments tried extremely hard to find proton decay, which would be a process that involves baryon number change, but we cannot find such examples. We have very high experimental certainty that baryon number is conserved. And if baryon number is topological, then this is a very good reason for the conservation of baryon number.
Skyrmions give a model of low-energy interactions of baryons. This leads to an (approximate) theory of nucleons (proton and neutron) and larger nuclei, which are bound states of any number of protons and neutrons.
For these ideas to work out well, we need to eventually do quantization. For example, Skyrmions by themselves do not have spin. We need to quantize the theory before spins come out. Also, Skyrmions cannot distinguish between protons and neutrons. These differences only come up after we quantize.
\section{\tph{$\phi^4$}{phi4}{ϕ<sup>4</sup>} kinks}
\subsection{Kink solutions}
In this section, we are going to study \term{$\phi^4$ kinks}\index{kink}\index{kink!$\phi^4$}. These occur in $1 + 1$ dimensions, and involve a single scalar field $\phi(x, t)$. In higher dimensions, we often need many fields to obtain solitons, but in the case of 1 dimension, we can get away with a single field.
In general, the \term{Lagrangian density} of such a scalar field theory is of the form
\[
\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - U(\phi)
\]
for some potential $U(\phi)$ polynomial in $\phi$. Note that in $1 + 1$ dimensions, any such theory is renormalizable. Here we will choose the Minkowski metric to be
\[
\eta^{\mu\nu} =
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix},
\]
with $\mu, \nu = 0, 1$. Then the \term{Lagrangian} is given by
\[
L = \int_{-\infty}^\infty \mathcal{L}\;\d x = \int_{-\infty}^\infty \left(\frac{1}{2} \partial_\mu \phi \partial^\mu \phi - U(\phi)\right)\;\d x,
\]
and the \term{action} is
\[
S[\phi] = \int L\;\d t = \int \mathcal{L} \;\d x\;\d t.
\]
There is a non-linearity in the field equations due to a potential $U(\phi)$ with \emph{multiple vacua}. We need multiple vacua to obtain a soliton. The kink stability comes from the \emph{topology}. It is very simple here, and just comes from counting the discrete, distinct vacua.
As usual, we will write
\[
\dot{\phi} = \frac{\partial \phi}{\partial t},\quad \phi' = \frac{\partial \phi}{\partial x}.
\]
Often it is convenient to (non-relativistically) split the Lagrangian as
\[
L = T - V,
\]
where
\[
T = \int \frac{1}{2} \dot{\phi}^2\;\d x,\quad V = \int \left(\frac{1}{2} \phi'^2 + U(\phi)\right)\;\d x.
\]
In higher dimensions, we separate out $\partial_\mu \phi$ into $\dot{\phi}$ and $\nabla \phi$.
The classical field equation comes from the condition that $S[\phi]$ is stationary under variations of $\phi$. By a standard manipulation, the field equation turns out to be
\[
\partial_\mu \partial^\mu \phi + \frac{\d U}{\d \phi} = 0.
\]
This is an example of a Klein--Gordon type of field equation, but is non-linear if $U$ is not quadratic. It is known as the \emph{non-linear Klein--Gordon equation}\index{Klein--Gordon equation!non-linear}.
We are interested in a soliton that is a static solution. For a static field, the time derivatives can be dropped, and this equation becomes
\[
\frac{\d^2 \phi}{\partial x^2} = \frac{\d U}{\d \phi}.
\]
Of course, the important part is the choice of $U$! In $\phi^4$ theory, we choose
\[
U(\phi) = \frac{1}{2} (1 - \phi^2)^2.
\]
This is mathematically the simplest version, because we set all coupling constants to $1$.
The importance of this $U$ is that it has two minima:
\begin{center}
\begin{tikzpicture}
\draw [->](-3, -1) -- (3, -1) node [right] {$\phi$};
\draw [->] (0, -1.5) -- (0, 3) node [above] {$U(\phi)$};
\draw [mblue, thick, domain=-2.4:2.4, samples=50] plot [smooth] (\x, {4 * ((\x/2)^4 - (\x/2)^2)});
\node at (-1.414, -1) [below] {$-1$};
\node at (1.414, -1) [below] {$1$};
\end{tikzpicture}
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The two classical vacua are
\[
\phi(x) \equiv 1,\quad \phi(x) \equiv -1.
\]
This is, of course, not the only possible choice. We can, for example, include some parameters and set
\[
U(\phi) = \lambda(m^2 - \phi^2)^2.
\]
If we are more adventurous, we can talk about a $\phi^6$ theory with
\[
U(\phi) = \lambda \phi^2 (m^2 - \phi^2)^2.
\]
In this case, we have $3$ minima, instead of $2$. Even braver people can choose
\[
U(\phi) = 1 - \cos \phi.
\]
This has \emph{infinitely} many minima. The field equation involves a $\sin \phi$ term, and hence this theory is called the \term{sine-Gordon theory} (a pun on the name Klein--Gordon, of course).
The sine-Gordon theory is a special case. While it seems like the most complicated potential so far, it is actually \emph{integrable}\index{integrability}. This implies we can find explicit exact solutions involving multiple, interacting solitons in a rather easy way. However, integrable systems is a topic for another course, namely IID Integrable Systems.
For now, we will focus on our simplistic $\phi^4$ theory. As mentioned, there are two vacuum field configurations, both of zero energy. We will in general use the term ``\term{field configuration}'' to refer to fields at a given time that are not necessarily solutions to the classical field equation, but in this case, the vacua are indeed solutions.
If we wanted to quantize this $\phi^4$ theory, then we have to pick one of the vacua and do perturbation theory around it. This is known as \term{spontaneous symmetry breaking}. Of course, by symmetry, we obtain the same quantum theory regardless of which vacuum we expand around.
However, as we mentioned, when we want to study solitons, we have to involve \emph{both} vacua. We want to consider solutions that ``connect'' these two vacua. In other words, we are looking for solutions that look like
\begin{center}
\begin{tikzpicture}
\draw [->] (-3, 0) -- (3, 0) node [right] {$x$};
\draw [->] (0, -2) -- (0, 2) node [above] {$\phi$};
\draw [mblue, thick, domain=-3:3] plot [smooth] (\x, {tanh(1.5*(\x + 1))});
\draw [dashed] (-3, 1) -- (3, 1);
\draw [dashed] (-3, -1) -- (3, -1);
\node [circ] at (-1, 0) {};
\node [anchor = north west] at (-1, 0) {$a$};
\end{tikzpicture}
\end{center}
This is known as a \term{kink solution}.
To actually find such a solution, we need the full field equation, given by
\[
\frac{\d^2 \phi}{\d x^2} = -2 (1 - \phi^2) \phi.
\]
Instead of solving this directly, we will find the kink solutions by considering the energy, since this method generalizes better.
We will work with a general potential $U$ with minimum value $0$. From Noether's theorem, we obtain a conserved energy
\[
E = \int \left(\frac{1}{2} \dot{\phi}^2 + \frac{1}{2} \phi'^2 + U(\phi)\right)\;\d x.
\]
For a static field, we drop the $\dot{\phi}^2$ term. Then this is just the $V$ appearing in the Lagrangian. By definition, the field equation tells us the field is a stationary point of this energy. To find the kink solution, we will in fact find a \emph{minimum} of the energy.
Of course, the global minimum is attained when we have a vacuum field, in which case $E = 0$. However, this is the global minimum only if we don't impose any boundary conditions. In our case, the kinks satisfy the boundary conditions ``$\phi(\infty) = 1$'', ``$\phi(-\infty) = -1$'' (interpreted in terms of limits, of course). The kinks will minimize energy subject to these boundary conditions.
These boundary conditions are important, because they are ``topological''. Eventually, we will want to understand the dynamics of solitons, so we will want to consider fields that evolve with time. From physical considerations, for any fixed $t$, the field $\phi(x, t)$ must satisfy $\phi(x, t) \to \text{vacuum}$ as $x \to \pm \infty$, or else the field will have infinite energy. However, the vacuum of our potential $U$ is discrete. Thus, if $\phi$ is to evolve continuously with time, the boundary conditions must not evolve with time! At least, this is what we expect classically. Who knows what weird tunnelling can happen in quantum field theory.
So from now on, we fix some boundary conditions $\phi(\infty)$ and $\phi(-\infty)$, and focus on fields that satisfy these boundary conditions. The trick is to write the potential in the form
\[
U (\phi) = \frac{1}{2} \left(\frac{\d W (\phi)}{\d \phi}\right)^2.
\]
If $U$ is non-negative, then we can always find $W$ in principle --- we take the square root and then integrate it. However, in practice, this is useful only if we can find a simple form for $W$. Let's assume we've done that. Then we can write
\begin{align*}
E &= \frac{1}{2} \int \left(\phi'^2 + \left(\frac{\d W}{\d \phi}\right)^2 \right)\;\d x\\
&= \frac{1}{2} \int \left(\phi' \mp \frac{\d W}{\d \phi}\right)^2\;\d x \pm \int \frac{\d W}{\d \phi} \frac{\d \phi}{\d x} \;\d x\\
&= \frac{1}{2} \int \left(\phi' \mp \frac{\d W}{\d \phi}\right)^2\;\d x \pm \int \d W\\
&= \frac{1}{2} \int \left(\phi' \mp \frac{\d W}{\d \phi}\right)^2\;\d x \pm (W(\phi(\infty)) - W(\phi(-\infty))).
\end{align*}
The second term depends purely on the boundary conditions, which we have fixed. Thus, we can minimize energy if we can make the first term vanish! Note that when completing the square, the choice of the signs is arbitrary. However, if we want to set the first term to be $0$, the second term had better be non-negative, since the energy itself is non-negative! Hence, we will pick the sign such that the second term is $\geq 0$, and then the energy is minimized when
\[
\phi' = \pm \frac{\d W}{\d \phi}.
\]
In this case, we have
\[
E = \pm (W(\infty) - W(-\infty)).
\]
These are known as the \term{Bogomolny equation} and the \term{Bogomolny energy bound}. Note that if we picked the other sign, then we cannot solve the differential equation $\phi' = \pm \frac{\d W}{\d \phi}$, because we know the energy must be non-negative.
For the $\phi^4$ kink, we have
\[
\frac{\d W}{\d \phi} = 1 - \phi^2.
\]
So we pick
\[
W = \phi - \frac{1}{3} \phi^3.
\]
So when $\phi = \pm 1$, we have $W = \pm \frac{2}{3}$. We need to choose the $+$ sign, and then we know the energy (and hence mass) of the kink is
\[
E \equiv M = \frac{4}{3}.
\]
We now solve for $\phi$. The equation we have is
\[
\phi' = 1 - \phi^2.
\]
Rearranging gives
\[
\frac{1}{1 - \phi^2} \d \phi = \d x,
\]
which we can integrate to give
\[
\phi (x) = \tanh (x - a).
\]
This $a$ is an arbitrary constant of integration, labelling the intersection of the graph of $\phi$ with the $x$-axis. We think of this as the ``\emph{location}'' of the kink.
Note that there is not a unique solution, which is not unexpected by translation invariance. Instead, the solutions are labeled by a parameter $a$. This is known as a \term{modulus} of the solution. In general, there can be multiple moduli, and the space of all possible values of the moduli of static solitons is known as the \term{moduli space}. In the case of a kink, the moduli space is just $\R$.
Is this solution stable? We obtained this kink solution by minimizing the energy within this topological class of solutions (i.e.\ among all solutions with the prescribed boundary conditions). Since a field cannot change the boundary conditions during evolution, it follows that the kink must be stable.
Are there other soliton solutions to the field equations? The solutions are determined by the boundary conditions. Thus, we can classify all soliton solutions by counting all possible combinations of the boundary conditions. We have, of course, two vacuum solutions $\phi \equiv 1$ and $\phi \equiv -1$. There is also an \term{anti-kink}\index{kink!anti-} solution obtained by inverting the kink:
\[
\phi(x) = - \tanh (x - b).
\]
This also has energy $\frac{4}{3}$.
\subsection{Dynamic kink}
We now want to look at kinks that move. Given what we have done so far, this is trivial. Our theory is Lorentz invariant, so we simply apply a Lorentz boost. Then we obtain a field
\[
\phi(x, t) = \tanh \gamma (x - vt),
\]
where, as usual
\[
\gamma = (1 - v^2)^{-1/2}.
\]
But this isn't all. Notice that for small $v$, we can approximate the solution simply by
\[
\phi(x, t) = \tanh (x - vt).
\]
This looks like a kink solution with a modulus that varies with time slowly. This is known as the \term{adiabatic} point of view.
More generally, let's consider a ``moving kink'' field
\[
\phi(x, t) = \tanh (x - a(t))
\]
for some function $a(t)$. In general, this is not a solution to the field equation, but if $\dot{a}$ is small, then it is ``approximately a solution''.
We can now explicitly compute that
\[
\dot{\phi} = - \frac{\d a}{\d t} \phi'.
\]
Let's consider fields of this type, and look at the Lagrangian of the field theory. The kinetic term is given by
\[
T = \int \frac{1}{2} \dot{\phi}^2\;\d x = \frac{1}{2} \left(\frac{\d a}{\d t}\right)^2 \int \phi'^2 \;\d x = \frac{1}{2} M \left(\frac{\d a}{\d t}\right)^2.
\]
To derive this result, we had to perform the integral $\int \phi'^2 \;\d x$, and if we do that horrible integral, we will find a value that happens to be equal to $M = \frac{4}{3}$. Of course, this is not a coincidence. We can derive this result from Lorentz invariance to see that the result of integration is manifestly $M$.
The remaining part of the Lagrangian is less interesting. Since it does not involve taking time derivatives, the time variation of $a$ is not seen by it, and we simply have a constant
\[
V = \frac{4}{3}.
\]
Then the original field Lagrangian becomes a particle Lagrangian
\[
L = \frac{1}{2}M \dot{a}^2 - \frac{4}{3}.
\]
Note that when we first formulated the field theory, the action principle required us to find a field that extremizes the action \emph{among all fields}. However, what we are doing now is to restrict to the set of kink solutions only, and then when we solve the variational problem arising from this Lagrangian, we are extremizing the action among fields of the form $\tanh (x - a(t))$. We can think of this as motion in a ``valley'' in the field configuration space. In general, these solutions will not also extremize the action among all fields. However, as we said, it will do so ``approximately'' if $\dot{a}$ is small.
We can obtain an effective equation of motion
\[
M \ddot{a} = 0,
\]
which is an equation of motion for the variable $a(t)$ \emph{in the moduli space}.
Of course, the solution is just given by
\[
a(t) = vt + \mathrm{const},
\]
where $v$ is an arbitrary constant, which we interpret as the velocity. In this formulation, we do not have any restrictions on $v$, because we took the ``non-relativistic approximation''. This approximation breaks down when $v$ is large.
There is a geometric interpretation to this. We can view the equation of motion $M\ddot{a} = 0$ as the \emph{geodesic equation} in the moduli space $\R$, and we can think of the coefficient $M$ as specifying a Riemannian metric on the moduli space. In this case, the metric is (a scalar multiple of) the usual Euclidean metric $(\d a)^2$.
This seems like a complicated way of describing such a simple system, but this picture generalizes to higher-dimensional systems and allows us to analyze multi-soliton dynamics, in particular, the dynamics of vortices and monopoles.
%We can find the dynamics of $a(t)$ from the Lagrangian of the field theory. Thus, we are reducing the ``field dynamics'' to the ``particle dynamics''. We have
%\[
% T = \int \frac{1}{2} \dot{\phi}^2\;\d x = \frac{1}{2} \left(\frac{\d a}{\d t}\right)^2 \int \phi'^2 \;\d x = \frac{1}{2} M \left(\frac{\d a}{\d t}\right)^2.
%\]
%The factor of $M$ just comes from doing the (tricky) integral explicitly, but we can also work it out from more general principles to make it manifestly $M$, and this is known as Derrick's theorem.
%
%Thus, the kink behaves like a particle with mass $M$! How about the potential energy? The potential energy is \emph{not} time-dependent. We simply integrate some polynomial of $\phi$ over all $x$, and the shift by $a$ does not make a difference. In this case, we have
%\[
% V = \frac{4}{3}.
%\]
%So we've reduced the field Lagrangian to a particle Lagrangian
%\[
% L = \frac{1}{2}M \dot{a}^2 - \frac{4}{3}.
%\]
%We can think of this as motion in a \term{valley} in the field configuration space. We are drifting in the energy minima in the field configuration space.
%
%This method is powerful, and applies to multi-soliton dynamics in high dimensions. From this, we obtain an effective equation of motion in moduli space
%\[
% M \ddot{a} = 0.
%\]
%This has \emph{geometric} interpretation as geometric motion in the moduli space. The moduli space is just the real line $\R$ with its standard metric. We can think of the coefficient $M$ as a Riemannian metric, which happens to be constant (as a function of $a$) in this case.
%
%Of course, the solution is
%\[
% a(t) = vt + \mathrm{const},
%\]
%where $v$ is an arbitrary constant, namely velocity.
%
%In this approximation, $v$ can be anything, and the approximation does not see the speed of light. However, as we plug this back into the actual field equation, we see that the approximation breaks down when $v$ is large.
%
%This motion in moduli space is not exact, but is accurate in the non-relativistic approximation.
%
%This was all rather trivial in our case of kinks. However, it is also important and allows us to analyze the motion of several solitons in higher dimension.
%
%Note that here we started with a Lagrangian that is quadratic in time derivatives of the field. When we pass on to solitons, we have a term that is quadratic in the time derivative in the moduli, and the coefficients provide a Riemannian geometry on the moduli space.
We might ask ourselves if there are multi-kinks in our theory. There aren't in the $\phi^4$ theory, because we saw that the solutions are classified by the boundary conditions, and we have already enumerated all the possible boundary conditions. In more complicated theories like sine-Gordon theory, multiple kinks are possible.
However, while we cannot have two kinks in $\phi^4$ theory, we can have a kink followed by an anti-kink, or more of these pairs. This actually lies in the ``vacuum sector'' of the theory, but it still looks like it's made up of kinks and anti-kinks, and it is interesting to study these.
\subsection{Soliton interactions}
We now want to study interactions between kinks and anti-kinks, and see how they cause each other to move. So far, we were able to label the position of the particle by its ``center'' $a$, and thus we can sensibly talk about how this center moves. However, this center is well-defined only in the very special case of a pure kink or anti-kink, where we can use symmetry to identify the center. If there is some perturbation, or if we have a kink and an anti-kink, it is less clear what should be considered the center.
Fortunately, we can still talk about the momentum of the field, even if we don't have a well-defined center. Indeed, since our theory has translation invariance, Noether's theorem gives us a conserved charge which is interpreted as the momentum.
Recall that for a single scalar field in $1 + 1$ dimensions, the Lagrangian density can be written in the form
\[
\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - U(\phi).
\]
Applying Noether's theorem, to the translation symmetry, we obtain the \term{energy-momentum tensor}
\[
T^\mu_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\nu \phi - \delta^\mu_\nu \mathcal{L} = \partial^\mu \phi \partial_\nu \phi - \delta^\mu_\nu \mathcal{L}.
\]
Fixing a time and integrating over all space, we obtain the conserved energy and conserved momentum. These are
\begin{align*}
E &= \int_{-\infty}^\infty T^0\!_0 \;\d x = \int_{-\infty}^\infty \left(\frac{1}{2}\dot{\phi}^2 + \frac{1}{2} \phi'^2 + U(\phi)\right)\;\d x,\\
P &= - \int_{-\infty}^\infty T^0_1 \;\d x = - \int_{-\infty}^\infty \dot{\phi} \phi' \;\d x.
\end{align*}
We now focus on our moving kink in the adiabatic approximation of the $\phi^4$ theory. Then the field is given by
\[
\phi = \tanh (x - a(t)).
\]
Doing another horrible integral, we find that the momentum is just
\[
P = M \dot{a}.
\]
This is just as we would expect for a particle with mass $M$!
Now suppose what we have is instead a kink-antikink configuration
\begin{center}
\begin{tikzpicture}
\draw [->] (-3, 0) -- (3, 0) node [right] {$x$};
\draw [->] (0, -2) -- (0, 2) node [above] {$\phi$};
\draw [mblue, thick] plot [smooth] coordinates {(-3.0,-0.99933) (-2.9,-0.99851) (-2.8,-0.99668) (-2.7,-0.99263) (-2.6,-0.98367) (-2.5,-0.96403) (-2.4,-0.92167) (-2.3,-0.83365) (-2.2,-0.66404) (-2.1,-0.37995) (-2.0,-0.00000) (-1.9,0.37995) (-1.8,0.66404) (-1.7,0.83365) (-1.6,0.92167) (-1.5,0.96403) (-1.4,0.98367) (-1.3,0.99263) (-1.2,0.99668) (-1.1,0.99851) (-1.0,0.99933) (-0.9,0.99970) (-0.8,0.99986) (-0.7,0.99994) (-0.6,0.99997) (-0.5,0.99999) (-0.4,0.99999) (-0.3,1.00000) (-0.2,1.00000) (-0.1,1.00000) (0.0,1.00000) (0.1,1.00000) (0.2,1.00000) (0.3,1.00000) (0.4,0.99999) (0.5,0.99999) (0.6,0.99997) (0.7,0.99994) (0.8,0.99986) (0.9,0.99970) (1.0,0.99933) (1.1,0.99851) (1.2,0.99668) (1.3,0.99263) (1.4,0.98367) (1.5,0.96403) (1.6,0.92167) (1.7,0.83365) (1.8,0.66404) (1.9,0.37995) (2.0,-0.00000) (2.1,-0.37995) (2.2,-0.66404) (2.3,-0.83365) (2.4,-0.92167) (2.5,-0.96403) (2.6,-0.98367) (2.7,-0.99263) (2.8,-0.99668) (2.9,-0.99851) (3.0,-0.99933)};
% map (\x -> (showFFloat (Just 1) x "", showFFloat (Just 5) (tanh (4 * (x + 2)) - tanh (4 * (x - 2)) - 1) "")) [-3,-2.9..3]
\draw [dashed] (-3, 1) -- (3, 1);
\draw [dashed] (-3, -1) -- (3, -1);
\node [circ] at (-2, 0) {};
\node [anchor = north west] at (-2, 0) {$-a$};
\node [circ] at (2, 0) {};
\node [anchor = north east] at (2, 0) {$a$};
\end{tikzpicture}
\end{center}
Here we have to make the crucial assumption that our kinks are well-separated. Matters get a lot worse when they get close to each other, and it is difficult to learn anything about them analytically. However, by making appropriate approximations, we can understand well-separated kink-antikink configurations.
When the kink and anti-kink are far away, we first pick a point $b$ lying in-between the kink and the anti-kink:
\begin{center}
\begin{tikzpicture}
\draw [->] (-3, 0) -- (3, 0) node [right] {$x$};
\draw [->] (0, -2) -- (0, 2) node [above] {$\phi$};
\draw [mblue, thick] plot [smooth] coordinates {(-3.0,-0.99933) (-2.9,-0.99851) (-2.8,-0.99668) (-2.7,-0.99263) (-2.6,-0.98367) (-2.5,-0.96403) (-2.4,-0.92167) (-2.3,-0.83365) (-2.2,-0.66404) (-2.1,-0.37995) (-2.0,-0.00000) (-1.9,0.37995) (-1.8,0.66404) (-1.7,0.83365) (-1.6,0.92167) (-1.5,0.96403) (-1.4,0.98367) (-1.3,0.99263) (-1.2,0.99668) (-1.1,0.99851) (-1.0,0.99933) (-0.9,0.99970) (-0.8,0.99986) (-0.7,0.99994) (-0.6,0.99997) (-0.5,0.99999) (-0.4,0.99999) (-0.3,1.00000) (-0.2,1.00000) (-0.1,1.00000) (0.0,1.00000) (0.1,1.00000) (0.2,1.00000) (0.3,1.00000) (0.4,0.99999) (0.5,0.99999) (0.6,0.99997) (0.7,0.99994) (0.8,0.99986) (0.9,0.99970) (1.0,0.99933) (1.1,0.99851) (1.2,0.99668) (1.3,0.99263) (1.4,0.98367) (1.5,0.96403) (1.6,0.92167) (1.7,0.83365) (1.8,0.66404) (1.9,0.37995) (2.0,-0.00000) (2.1,-0.37995) (2.2,-0.66404) (2.3,-0.83365) (2.4,-0.92167) (2.5,-0.96403) (2.6,-0.98367) (2.7,-0.99263) (2.8,-0.99668) (2.9,-0.99851) (3.0,-0.99933)};
\draw [dashed] (-3, 1) -- (3, 1);
\draw [dashed] (-3, -1) -- (3, -1);
\node [circ] at (-2, 0) {};
\node [anchor = north west] at (-2, 0) {$-a$};
\node [circ] at (2, 0) {};
\node [anchor = north east] at (2, 0) {$a$};
\draw [dashed] (-0.3, -2) -- (-0.3, 2);
\node [circ] at (-0.3, 0) {};
\node [anchor = north east] at (-0.3, 0) {$b$};
\end{tikzpicture}
\end{center}
The choice of $b$ is arbitrary, but we should choose it so that it is far away from both kinks. We will later see that, at least to first order, the result of our computations does not depend on which $b$ we choose. We will declare that the parts to the left of $b$ belongs to the kink, and the parts to the right of $b$ belong to the anti-kink. Then by integrating the energy-momentum tensor in these two regions, we can obtain the momentum of the kink and the anti-kink separately.
We will focus on the kink only. Its momentum is given by\index{kink!forces}
\[
P = - \int_{-\infty}^b T^0_1 \;\d x = - \int_{-\infty}^b \dot{\phi} \phi' \;\d x.
\]
Since $T^\mu_\nu$ is conserved, we know $\partial_\mu T^\mu\!_\nu = 0$. So we find
\[
\frac{\partial}{\partial t} T^0\!_1 + \frac{\partial}{\partial x}T^1\!_1 = 0.
\]
By Newton's second law, the force $F$ on the kink is given by the rate of change of the momentum:
\begin{align*}
F &= \frac{\d}{\d t} P \\
&= -\int_{-\infty}^b \frac{\partial}{\partial t} T^0\!_1\;\d x \\
&= \int_{-\infty}^b \frac{\partial}{\partial x} T^1\!_1\;\d x\\
&= \left.T^1\!_1\right|_b\\
&= \left(-\frac{1}{2} \dot{\phi}^2 - \frac{1}{2} \phi'^2 + U\right)_b.
\end{align*}
Note that there is no contribution at the $-\infty$ end because it is vacuum and $T^1\!_1$ vanishes.
But we want to actually work out what this is. To do so, we need to be more precise about what our initial configuration is. In this theory, we can obtain it just by adding a kink to an anti-kink. The obvious guess is that it should be
\[
\phi(x) \overset{?}{=} \tanh(x + a) - \tanh(x - a),
\]
but this has the wrong boundary condition. It vanishes on both the left and the right. So we actually want to subtract $1$, and obtain
\[
\phi(x) = \tanh(x + a) - \tanh(x - a) - 1 \equiv \phi_1 + \phi_2 - 1.
\]
Note that since our equation of motion is not linear, this is in general not a genuine solution! However, it is approximately a solution, because the kink and anti-kink are well-separated. However, there is no hope that this will be anywhere near a solution when the kink and anti-kink are close together!
Before we move on to compute $\dot{\phi}$ and $\phi'$ explicitly and plugging numbers in, we first make some simplifications and approximations. First, we restrict our attention to fields that are initially at rest. So we have $\dot{\phi} = 0$ at $t = 0$. Of course, the force will cause the kinks to move, but we shall, for now, ignore what happens when they start moving.
That gets rid of one term. Next, we notice that we only care about the expression when evaluated at $b$. Here we have $\phi_2 - 1 \approx 0$. So we can try to expand the expression to first order in $\phi_2 - 1$ (and hence $\phi_2'$), and this gives
\[
F = \left(-\frac{1}{2}\phi_1'^2 + U(\phi_1) - \phi_1' \phi_2' + (\phi_2 - 1) \frac{\d U}{\d \phi}(\phi_1)\right)_b.
\]
We have a zeroth order term $-\frac{1}{2} \phi_1'^2 + U(\phi_1)$. We claim that this must vanish. One way to see this is that this term corresponds to the force when there is no anti-kink $\phi_2$. Since the kink does not exert a force on itself, this must vanish!
Analytically, we can deduce this from the Bogomolny equation, which says for any kink solution $\phi$, we have
\[
\phi' = \frac{\d W}{\d \phi}.
\]
It then follows that
\[
\frac{1}{2} \phi'^2 = \frac{1}{2} \left(\frac{\d W}{\d \phi}\right)^2 = U(\phi).
\]
Alternatively, we can just compute it directly! In any case, convince yourself that it indeed vanishes in your favorite way, and then move on.
Finally, we note that the field equations tell us
\[
\frac{\d U}{\d \phi}(\phi_1) = \phi_1''.
\]
So we can write the force as
\[
F = \big(-\phi_1' \phi_2' + (\phi_2 - 1) \phi_1''\big)_b.
\]
That's about all the simplifications we can make without getting our hands dirty. We might think we should plug in the $\tanh$ terms and compute, but that is \emph{too} dirty. Instead, we use asymptotic expressions of kinks and anti-kinks far from their centers. Using the definition of $\tanh$, we have
\[
\phi_1 = \tanh(x + a) = \frac{1 - e^{-2(x + a)}}{1 + e^{-2(x + a)}} \approx 1 - 2e^{-2(x + a)}.
\]
This is valid for $x \gg -a$, i.e.\ to the right of the kink. The constant factor of $2$ in front of the exponential is called the \term{amplitude} of the tail. We will later see that the $2$ appearing in the exponent has the interpretation of the mass of the field $\phi$.
For $\phi_2$, take the approximation that $x \ll a$. Then
\[
\phi_2 - 1= -\tanh(x - a) - 1 \approx -2 e^{2(x - a)}.
\]
We assume that our $b$ satisfies both of these conditions. These are obviously easy to differentiate once or twice. Doing this, we obtain
\[
-\phi_1' \phi_2' = (-4e^{-2(x + a)})(-4 e^{2(x - a)}) = 16 e^{-4a}.
\]
Note that this is independent of $x$. In the formula, the $x$ will turn into a $b$, and we see that this part of the force is independent of $b$. Similarly, the other term is
\[
(\phi_2 - 1) \phi_1'' = (-2e^{2(x - a)}) (-8 e^{-2(x + a)}) = 16 e^{-4a}.
\]
Therefore we find
\[
F = 32 e^{-4a},
\]
and as promised, this is independent of the precise position of the cutoff $b$ we chose.
We can write this in a slightly more physical form. Our initial configuration was symmetric around the $y$-axis, but in reality, only the separation matters. We write the separation of the pair as $s = 2a$. Then we have
\[
F = 32 e^{-2s}.
\]
What is the interpretation of the factor of $2$? Recall that our potential was given by
\[
U(\phi) = \frac{1}{2}(1 - \phi^2)^2.
\]
We can do perturbation theory around one of the vacua, say $\phi = 1$. Thus, we set $\phi = 1 + \eta$, and then expanding gives us
\[
U(\eta) \approx \frac{1}{2} (-2\eta)^2 = \frac{1}{2}m^2 \eta^2,
\]
where $m = 2$. This is the same ``2'' that goes into the exponent in the force.
What about the constant factor of $32$? Recall that when we expanded the kink solution, we saw that the amplitude $A$ of the tail was $A = 2$. It turns out if we re-did our theory and put back the different possible parameters, we will find that the force is given by
\[
F = 2 m^2 A^2 e^{-ms}.
\]
This is an interesting and important phenomenon. The mass $m$ was the \emph{perturbative} mass of the field. It is something we obtain by perturbation theory. However, the same mass appears in the force between the solitons, which are non-perturbative phenomenon!
This is perhaps not too surprising. After all, when we tried to understand the soliton interactions, we took the approximation that $\phi_1$ and $\phi_2$ are close to $1$ at $b$. Thus, we are in some sense perturbing around the vacuum $\phi \equiv 1$.
%The perturbative field theory has a meson of mass $2 \hbar$. You might have not noticed this $\hbar$ when doing quantum field theory, because we set $\hbar = 1$. But this makes sense, because in free field theory, we decompose the quantum field into a lot of harmonic oscillators, and the energy of the harmonic oscillator was $\hbar\omega\left(n + \frac{1}{2}\right)$. So the $\hbar$ should be here.
%
%However, in our soliton, we do \emph{not} have an $\hbar$. That was not a mistake. We have only been working classically.
%
%The soliton has mass $M = \frac{4}{3}$, and this is much larger than the meson mass. Note that one should choose $\hbar$ to be small for perturbation theory to be justified. However, soliton is non-perturbative and has larger mass than the meson).
We can interpret the force between the kink and anti-kink diagrammatically. From the quantum field theory point of view, we can think of this force as being due to meson exchange, and we can try to invent a Feynman diagram calculus that involves solitons and mesons. This is a bit controversial, but at least heuristically, we can introduce new propagators representing solitons, using double lines, and draw the interaction as
\begin{center}
\begin{tikzpicture}
\begin{feynman}
\vertex (i);
\vertex [right=of i] (m);
\vertex [above right=of m] (f1);
\vertex [below right=of m] (f2) {$\bar{K}$};
\vertex [above left=of i] (s1);
\vertex [below left=of i] (s2) {$K$};
\diagram*{
(i) -- [scalar] (m), % label ``meson''
(f2) -- [double distance=1.5pt] (m) -- [double distance=1.5pt] (f1), % make these double lines
(s1) -- [double distance=1.5pt] (i) -- [double distance=1.5pt] (s2),
};
\end{feynman}
\end{tikzpicture}
\end{center}
%This is an effective diagram leading to a Yukawa force in $1 + 1$ dimensions, which decays exponentially with separation $s$.
%
%The amplitude of the tail of the of the soliton kink is $A = 2$. The factor of $32$ in the force ultimately comes from the mass being $m = 2$. More generally, one can show that if we put in parameters into our theory, we have
%\[
% F = 2 m^2 A^2 e^{-ms}.
%\]
So what happens to this soliton? The force we derived was positive. So the kink is made to move to the right. By symmetry, we will expect the anti-kink to move towards the left. They will collide!
What happens when they collide? All our analysis so far assumed the kinks were well-separated, so everything breaks down. We can only understand this phenomenon numerically. After doing some numerical simulations, we see that there are two regimes:
\begin{itemize}
\item If the kinks are moving slowly, then they will annihilate into \emph{meson radiation}.
\item If the kinks are moving very quickly, then they often bounce off each other.
\end{itemize}
%We can now study the time-dependence. The kinks will accelerate due to standard Newtonian dynamics, and they will move towards each other. However, the full dynamics of the kink-antikink pair is complicated. When they hit each other, they annihilate, and this happens in the regime where the separation is large. Where does the energy go when they annihilate? The answer is that they annihilate into meson radiation, which we can discover by doing numerical simulations.
%
%Annihilation is what happens if they are initially at rest, or slowly moving. However, at high speed, they happen to bounce of each other (of course, there is still some energy loss to radiation). These are very complicated, and we understand this mostly through numerical simulations.
\subsection{Quantization of kink motion}
We now briefly talk about how to quantize kinks. The most naive way of doing so is pretty straightforward. We use the moduli space approximation, and then we have a very simple kink Lagrangian.
\[
L = \frac{1}{2} M \dot{a}^2.
\]
This is just a free particle moving in $\R$ with mass $M$. This $a$ is known as the \term{collective coordinate} of the kink. Quantizing a free particle is very straightforward. It is just IB Quantum Mechanics. For completeness, we will briefly outline this procedure.
We first put the system in Hamiltonian form. The conjugate momentum to $a$ is given by
\[
P = M \dot{a}.
\]
Then the Hamiltonian is given by
\[
H = P \dot{a} - L = \frac{1}{2M} P^2.
\]
Then to quantize, we replace $P$ by the operator $-i\hbar \frac{\partial}{\partial a}$. In this case, the quantum Hamiltonian is given by
\[
H = - \frac{\hbar^2}{2M} \frac{\partial^2}{\partial a^2}.
\]
A wavefunction is a function of $a$ and $t$, and this is just ordinary QM for a single particle.
As usual, the stationary states are given by
\[
\psi(a) = e^{i\kappa a},
\]
and the momentum and energy (eigenvalues) are
\[
P = \hbar \kappa,\quad H = E = \frac{\hbar^2 \kappa^2}{2M} = \frac{P^2}{2M}.
\]
Is this actually ``correct''? Morally speaking, we really should quantize the complete $1 + 1$ dimensional field theory. What would this look like?
In normal quantum field theory, we consider perturbations around a vacuum solution, say $\phi \equiv 1$, and we obtain mesons. Here if we want to quantize the kink solution, we should consider field oscillations around the kink. Then the solution contains both a kink and a meson. These mesons give rise to quantum corrections to the kink mass $M$.
Should we be worried about these quantum corrections? Unsurprisingly, it turns out these quantum corrections are of the order of the meson mass. So we should not be worried when the meson mass is small.
Meson-kink scattering can also be studied in the full quantum theory. To first approximation, since the kink is heavy, mesons are reflected or transmitted with some probabilities, while the momentum of the kink is unchanged. But when we work to higher orders, then of course the kink will move as a result. This is all rather complicated.
For more details, see Rajaraman's \emph{Solitons and Instantons}, or Weinberg's \emph{Classical Solutions in Quantum Field Theory}.
The thing that is really hard to understand in the quantum field theory is kink-antikink pair production. This happens in meson collisions when the mesons are very fast, and the theory is highly relativistic. What we have done so far is perturbative and makes the non-relativistic approximation to get the adiabatic picture. It is \emph{very} difficult to understand the highly relativistic regime.
\subsection{Sine-Gordon kinks}
We end the section by briefly talking about kinks in a different theory, namely the \term{sine-Gordon theory}. In this theory, kinks are often known as \emph{solitons}\index{sine-Gordon soliton} instead.
The sine-Gordon theory is given by the potential
\[
U(\phi) = 1 - \cos \phi.
\]
Again, we suppress coupling constants, but it is possible to add them back.
The potential looks like
\begin{center}
\begin{tikzpicture}
\draw [->](-3, 0) -- (3, 0) node [right] {$\phi$};
\draw [->] (0, -0.5) -- (0, 2) node [above] {$U(\phi)$};
\draw [mblue, thick] (0, 0) cos (0.3, 0.7) sin (0.6, 1.4) cos (0.9, 0.7) sin (1.2, 0) cos (1.5, 0.7) sin (1.8, 1.4) cos (2.1, 0.7) sin (2.4, 0) cos (2.7, 0.7) sin (3, 1.4);;
\draw [mblue, thick, xscale=-1] (0, 0) cos (0.3, 0.7) sin (0.6, 1.4) cos (0.9, 0.7) sin (1.2, 0) cos (1.5, 0.7) sin (1.8, 1.4) cos (2.1, 0.7) sin (2.4, 0) cos (2.7, 0.7) sin (3, 1.4);;
\node at (1.2, 0) [below] {$2\pi$};
\node at (2.4, 0) [below] {$4\pi$};
\node at (-1.2, 0) [below] {$2\pi$};
\node at (-2.4, 0) [below] {$4\pi$};
\end{tikzpicture}
\end{center}
Now there are \emph{infinitely many} distinct vacua. In this case, we find we need to pick $W$ such that
\[
\frac{\d W}{\d \phi} = 2 \sin \frac{1}{2}\phi.
\]
\subsubsection*{Static sine-Gordon kinks}
To find the static kinks in the sine-Gordon theory, we again look at the Bogomolny equation. We have to solve
\[
\frac{\d \phi}{\d x} = 2 \sin \frac{1}{2}\phi.
\]
This can be solved. This involves integrating a $\csc$, and ultimately gives us a solution
\[
\phi(x) = 4 \tan^{-1} e^{x - a}.
\]
We can check that this solution interpolates between $0$ and $2\pi$.
\begin{center}
\begin{tikzpicture}
\draw [->] (-4, 0) -- (4, 0) node [right] {$x$};
\draw [->] (0, -0.5) -- (0, 2.5) node [above] {$\phi$};
\draw [mblue, thick] plot [smooth] coordinates {(-4.0,0.00000) (-3.9,0.00000) (-3.8,0.00000) (-3.7,0.00001) (-3.6,0.00001) (-3.5,0.00001) (-3.4,0.00001) (-3.3,0.00002) (-3.2,0.00002) (-3.1,0.00003) (-3.0,0.00005) (-2.9,0.00006) (-2.8,0.00008) (-2.7,0.00011) (-2.6,0.00015) (-2.5,0.00020) (-2.4,0.00027) (-2.3,0.00037) (-2.2,0.00050) (-2.1,0.00068) (-2.0,0.00091) (-1.9,0.00123) (-1.8,0.00166) (-1.7,0.00224) (-1.6,0.00303) (-1.5,0.00409) (-1.4,0.00552) (-1.3,0.00745) (-1.2,0.01005) (-1.1,0.01357) (-1.0,0.01831) (-0.9,0.02472) (-0.8,0.03336) (-0.7,0.04502) (-0.6,0.06074) (-0.5,0.08190) (-0.4,0.11035) (-0.3,0.14847) (-0.2,0.19922) (-0.1,0.26607) (0.0,0.35251) (0.1,0.46091) (0.2,0.59053) (0.3,0.73548) (0.4,0.88474) (0.5,1.02559) (0.6,1.14850) (0.7,1.24946) (0.8,1.32902) (0.9,1.39011) (1.0,1.43628) (1.1,1.47087) (1.2,1.49666) (1.3,1.51583) (1.4,1.53006) (1.5,1.54061) (1.6,1.54843) (1.7,1.55423) (1.8,1.55852) (1.9,1.56170) (2.0,1.56406) (2.1,1.56580) (2.2,1.56710) (2.3,1.56806) (2.4,1.56877) (2.5,1.56929) (2.6,1.56968) (2.7,1.56997) (2.8,1.57019) (2.9,1.57034) (3.0,1.57046) (3.1,1.57055) (3.2,1.57061) (3.3,1.57066) (3.4,1.57070) (3.5,1.57072) (3.6,1.57074) (3.7,1.57076) (3.8,1.57077) (3.9,1.57077) (4.0,1.57078)};
% map (\x -> (showFFloat (Just 1) x "", showFFloat (Just 5) (atan ( exp (3*x - 1))) "")) [-4,-3.9..4]
\draw [dashed] (-4, 1.57078) -- (4, 1.57078);
\node [left] at (-4, 0) {$0$};
\node [right] at (4, 1.57078) {$2\pi$};
\node [circ] at (0.33, 0.785398) {};
\draw [dashed] (0.33, 0.785398) -- (0.33, 0) node [below] {$a$};
\end{tikzpicture}
\end{center}
Unlike the $\phi^4$ theory, dynamical multi-kink solutions exist here and can be derived \emph{exactly}. One of the earlier ways to do so was via B\"acklund transforms, but that was very complicated. People later invented better methods, but they are still not very straightforward. Nevertheless, it can be done. Ultimately, this is due to the sine-Gordon equation being \emph{integrable}. For more details, refer to the IID Integrable Systems course.
\begin{eg}
There is a two-kink solution
\[
\phi (x, t) = 4 \tan^{-1} \left(\frac{v \sinh \gamma x}{ \cosh \gamma vt}\right),
\]
where, as usual, we have
\[
\gamma = (1 - v^2)^{-1/2}.
\]
For $v = 0.01$, this looks like
\begin{center}
\begin{tikzpicture}
\draw [->] (-4, 0) -- (4, 0) node [right] {$x$};
\draw [->] (0, -2.5) -- (0, 2.5) node [above] {$\phi$};
\draw [mblue, thick] plot [smooth] coordinates {(-4.0,-1.56957) (-3.9,-1.56914) (-3.8,-1.56856) (-3.7,-1.56778) (-3.6,-1.56672) (-3.5,-1.56529) (-3.4,-1.56337) (-3.3,-1.56077) (-3.2,-1.55726) (-3.1,-1.55252) (-3.0,-1.54613) (-2.9,-1.53751) (-2.8,-1.52587) (-2.7,-1.51019) (-2.6,-1.48906) (-2.5,-1.46067) (-2.4,-1.42263) (-2.3,-1.37197) (-2.2,-1.30524) (-2.1,-1.21893) (-2.0,-1.11067) (-1.9,-0.98117) (-1.8,-0.83628) (-1.7,-0.68699) (-1.6,-0.54603) (-1.5,-0.42296) (-1.4,-0.32183) (-1.3,-0.24211) (-1.2,-0.18089) (-1.1,-0.13458) (-1.0,-0.09986) (-0.9,-0.07394) (-0.8,-0.05461) (-0.7,-0.04020) (-0.6,-0.02942) (-0.5,-0.02129) (-0.4,-0.01509) (-0.3,-0.01027) (-0.2,-0.00637) (-0.1,-0.00305) (0.0,0.00000) (0.1,0.00305) (0.2,0.00637) (0.3,0.01027) (0.4,0.01509) (0.5,0.02129) (0.6,0.02942) (0.7,0.04020) (0.8,0.05461) (0.9,0.07394) (1.0,0.09986) (1.1,0.13458) (1.2,0.18089) (1.3,0.24211) (1.4,0.32183) (1.5,0.42296) (1.6,0.54603) (1.7,0.68699) (1.8,0.83628) (1.9,0.98117) (2.0,1.11067) (2.1,1.21893) (2.2,1.30524) (2.3,1.37197) (2.4,1.42263) (2.5,1.46067) (2.6,1.48906) (2.7,1.51019) (2.8,1.52587) (2.9,1.53751) (3.0,1.54613) (3.1,1.55252) (3.2,1.55726) (3.3,1.56077) (3.4,1.56337) (3.5,1.56529) (3.6,1.56672) (3.7,1.56778) (3.8,1.56856) (3.9,1.56914) (4.0,1.56957)};
\draw [mred, thick] plot [smooth] coordinates {(-4.0,-1.53726) (-3.9,-1.52555) (-3.8,-1.50975) (-3.7,-1.48847) (-3.6,-1.45988) (-3.5,-1.42157) (-3.4,-1.37057) (-3.3,-1.30340) (-3.2,-1.21659) (-3.1,-1.10779) (-3.0,-0.97784) (-2.9,-0.83269) (-2.8,-0.68347) (-2.7,-0.54286) (-2.6,-0.42031) (-2.5,-0.31974) (-2.4,-0.24053) (-2.3,-0.17975) (-2.2,-0.13381) (-2.1,-0.09939) (-2.0,-0.07374) (-1.9,-0.05467) (-1.8,-0.04052) (-1.7,-0.03002) (-1.6,-0.02224) (-1.5,-0.01648) (-1.4,-0.01221) (-1.3,-0.00904) (-1.2,-0.00670) (-1.1,-0.00496) (-1.0,-0.00367) (-0.9,-0.00271) (-0.8,-0.00200) (-0.7,-0.00147) (-0.6,-0.00108) (-0.5,-0.00078) (-0.4,-0.00055) (-0.3,-0.00038) (-0.2,-0.00023) (-0.1,-0.00011) (0.0,0.00000) (0.1,0.00011) (0.2,0.00023) (0.3,0.00038) (0.4,0.00055) (0.5,0.00078) (0.6,0.00108) (0.7,0.00147) (0.8,0.00200) (0.9,0.00271) (1.0,0.00367) (1.1,0.00496) (1.2,0.00670) (1.3,0.00904) (1.4,0.01221) (1.5,0.01648) (1.6,0.02224) (1.7,0.03002) (1.8,0.04052) (1.9,0.05467) (2.0,0.07374) (2.1,0.09939) (2.2,0.13381) (2.3,0.17975) (2.4,0.24053) (2.5,0.31974) (2.6,0.42031) (2.7,0.54286) (2.8,0.68347) (2.9,0.83269) (3.0,0.97784) (3.1,1.10779) (3.2,1.21659) (3.3,1.30340) (3.4,1.37057) (3.5,1.42157) (3.6,1.45988) (3.7,1.48847) (3.8,1.50975) (3.9,1.52555) (4.0,1.53726)};
% let v = 0.01 in let g = sqrt (1 / (1 - v * v)) in let t = 400 in map (\x -> (showFFloat (Just 1) x "", showFFloat (Just 5) (atan (v * sinh (g * x * 3) / cosh (v * g * t))) "")) [-4,-3.9..4]
\draw [dashed] (-4, 1.57078) -- (4, 1.57078);
\draw [dashed] (-4, -1.57078) -- (4, -1.57078);
\node [right] at (4, 1.57078) {$2\pi$};
\node [left] at (-4, -1.57078) {$-2\pi$};
\node [mblue, left] at (1.73, 0.785) {\small$t = 0$};
\node [mred, right] at (2.85, 0.785) {\small$t = \pm400$};
\end{tikzpicture}
\end{center}
Note that since $\phi(x, t) = \phi(x, -t)$, we see that this solution involves two solitons at first approaching each other, and then later bouncing off. Thus, the two kinks \emph{repel} each other. When we did kinks in $\phi^4$ theory, we saw that a kink and an anti-kink attracted, but here there are two kinks, which is qualitatively different.
We can again compute the force just like the $\phi^4$ theory, but alternatively, since we have a full, exact solution, we can work it out directly from the solution! The answers, fortunately, agree. If we do the computations, we find that the point of closest approach is $\sim 2 \log \left(\frac{2}{v}\right)$ if $v$ is small.
\end{eg}
There are some important comments to make. In the sine-Gordon theory, we can have very complicated interactions between kinks and anti-kinks, and these can connect vastly different vacua. However, \emph{static} solutions must join $2n\pi$ and $2(n \pm 1)\pi$ for some $n$, because if we want to join vacua further apart, we will have more than one kink, and they necessarily interact.
If we have multiple kinks and anti-kinks, then each of these things can have their own velocity, and we might expect some very complicated interaction between them, such as annihilation and pair production. But remarkably, the interactions are \emph{not} complicated. If we try to do numerical simulations, or use the exact solutions, we see that we do not have energy loss due to ``radiation''. Instead, the solitons remain very well-structured and retain their identities. This, again, is due to the theory being integrable.
\subsubsection*{Topology of the sine-Gordon equation}
There are also a lot of interesting things we can talk about without going into details about what the solutions look like.
The important realization is that our potential is periodic in $\phi$. For the sine-Gordon theory, it is much better to think of this as a field modulo $2\pi$, i.e.\ as a function
\[
\phi: \R \to S^1.
\]
In this language, the boundary condition is that $\phi(x) = 0 \bmod 2\pi$ as $x \to \pm \infty$. Thus, instead of thinking of the kink as joining two vacua, we can think of it as ``winding around the circle'' instead.
We can go further. Since the boundary conditions of $\phi$ are now the same on two sides, we can join the ends of the domain $\R$ together, and we can think of $\phi$ as a map
\[
\phi: S^1 \to S^1
\]
instead. This is a \term{compactification} of space.
Topologically, such maps are classified by their \term{winding number}, or the \term{degree}, which we denote \term{$Q$}. This is a topological (homotopy) invariant of a map, and is preserved under continuous deformations of the field. Thus, it is preserved under time evolution of the field.
Intuitively, the winding number is just how many times we go around the circle. There are multiple (equivalent) ways of making this precise.
The first way, which is the naive way, is purely topological. We simply have to go back to the first picture, where we regard $\phi$ as a real value. Suppose the boundary values are
\[
\phi(-\infty) = 2 n_- \pi,\quad \phi(\infty) = 2 n_+ \pi.
\]
Then we set the winding number to be $Q = n_+ - n_-$.
Topologically, we are using the fact that $\R$ is the \term{universal covering space}\index{covering space} of the circle, and thus we are really looking at the induced map on the fundamental group of the circle.
\begin{eg}
As we saw, a single kink has $Q = 1$.
\begin{center}
\begin{tikzpicture}
\draw [->] (-4, 0) -- (4, 0) node [right] {$x$};
\draw [->] (0, -0.5) -- (0, 2.5) node [above] {$\phi$};
\draw [mblue, thick] plot [smooth] coordinates {(-4.0,0.00000) (-3.9,0.00000) (-3.8,0.00000) (-3.7,0.00001) (-3.6,0.00001) (-3.5,0.00001) (-3.4,0.00001) (-3.3,0.00002) (-3.2,0.00002) (-3.1,0.00003) (-3.0,0.00005) (-2.9,0.00006) (-2.8,0.00008) (-2.7,0.00011) (-2.6,0.00015) (-2.5,0.00020) (-2.4,0.00027) (-2.3,0.00037) (-2.2,0.00050) (-2.1,0.00068) (-2.0,0.00091) (-1.9,0.00123) (-1.8,0.00166) (-1.7,0.00224) (-1.6,0.00303) (-1.5,0.00409) (-1.4,0.00552) (-1.3,0.00745) (-1.2,0.01005) (-1.1,0.01357) (-1.0,0.01831) (-0.9,0.02472) (-0.8,0.03336) (-0.7,0.04502) (-0.6,0.06074) (-0.5,0.08190) (-0.4,0.11035) (-0.3,0.14847) (-0.2,0.19922) (-0.1,0.26607) (0.0,0.35251) (0.1,0.46091) (0.2,0.59053) (0.3,0.73548) (0.4,0.88474) (0.5,1.02559) (0.6,1.14850) (0.7,1.24946) (0.8,1.32902) (0.9,1.39011) (1.0,1.43628) (1.1,1.47087) (1.2,1.49666) (1.3,1.51583) (1.4,1.53006) (1.5,1.54061) (1.6,1.54843) (1.7,1.55423) (1.8,1.55852) (1.9,1.56170) (2.0,1.56406) (2.1,1.56580) (2.2,1.56710) (2.3,1.56806) (2.4,1.56877) (2.5,1.56929) (2.6,1.56968) (2.7,1.56997) (2.8,1.57019) (2.9,1.57034) (3.0,1.57046) (3.1,1.57055) (3.2,1.57061) (3.3,1.57066) (3.4,1.57070) (3.5,1.57072) (3.6,1.57074) (3.7,1.57076) (3.8,1.57077) (3.9,1.57077) (4.0,1.57078)};
% map (\x -> (showFFloat (Just 1) x "", showFFloat (Just 5) (atan ( exp (3*x - 1))) "")) [-4,-3.9..4]
\draw [dashed] (-4, 1.57078) -- (4, 1.57078);
\node [left] at (-4, 0) {$0$};
\node [right] at (4, 1.57078) {$2\pi$};
\end{tikzpicture}
\end{center}
\end{eg}
Thus, we can think of the $Q$ as the \emph{net soliton number}.
But this construction we presented is rather specific to maps from $S^1$ to $S^1$. We want something more general that can be used for more complicated systems. We can do this in a more ``physics'' way. We note that there is a \emph{topological} current
\[
j^\mu = \frac{1}{2\pi} \varepsilon^{\mu\nu} \partial_\nu \phi,
\]
where $\varepsilon^{\mu\nu}$ is the anti-symmetric tensor in $1 + 1$ dimensions, chosen so that $\varepsilon^{01} = 1$.
In components, this is just
\[
j^\mu = \frac{1}{2\pi} (\partial_x \phi, - \partial_t \phi).
\]
This is conserved because of the symmetry of mixed partial derivatives, so that
\[
\partial_\mu j^\mu = \frac{1}{2\pi} \varepsilon^{\mu\nu} \partial_\mu \partial_\nu \phi = 0.
\]
As usual, a current induces a conserved charge
\[
Q = \int_{-\infty}^\infty j^0 \;\d x = \frac{1}{2\pi} \int_{-\infty}^\infty \partial_x \phi \;\d x = \frac{1}{2\pi} (\phi(\infty) - \phi(-\infty)) = n_+ - n_-,
\]
which is the formula we had earlier.
Note that all these properties do not depend on $\phi$ satisfying any field equations! It is completely topological.
Finally, there is also a differential geometry way of defining $Q$. We note that the target space $S^1$ has a normalized volume form $\omega$ so that
\[
\int_{S^1} \omega = 1.
\]
For example, we can take
\[
\omega = \frac{1}{2\pi}\;\d \phi.
\]
Now, given a mapping $\phi: \R \to S^1$, we can pull back the volume form to obtain
\[
\phi^* \omega = \frac{1}{2\pi} \frac{\d \phi}{\d x} \;\d x.
\]
We can then define the degree of the map to be
\[
Q = \int \phi^* \omega = \frac{1}{2\pi}\int_{-\infty}^\infty \frac{\d \phi}{\d x}\;\d x.
\]
This is exactly the same as the formula we obtained using the current!
Note that even though the volume form is normalized on $S^1$ and has integral $1$, the integral when pulled back is not $1$. We can imagine this as saying if we wind around the circle $n$ times, then after pulling back, we would have pulled back $n$ ``copies'' of the volume form, and so the integral will be $n$ times that of the integral on $S^1$.
We saw that these three definitions gave the same result, and different definitions have different benefits. For example, in the last two formulations, it is not \emph{a priori} clear that the winding number has to be an integer, while this is clear in the first formulation.
\section{Vortices}
We are now going to start studying \emph{vortices}. These are topological solitons in two space dimensions. While we mostly studied $\phi^4$ kinks last time, what we are going to do here is more similar to the sine-Gordon theory than the $\phi^4$ theory, as it is largely topological in nature.
A lot of the computations we perform in the section are much cleaner when presented using the language of differential forms. However, we shall try our best to provide alternative versions in coordinates for the easily terrified.
\subsection{Topological background}
\subsubsection*{Sine-Gordon kinks}
We now review what we just did for sine-Gordon kinks, and then try to develop some analogous ideas in higher dimension. The sine-Gordon equation is given by
\[
\frac{\partial^2 \theta}{\partial t^2} - \frac{\partial^2 \theta}{\partial x^2} + \sin \theta = 0.
\]
We want to choose boundary conditions so that the energy has a chance to be finite. The first part is, of course, to figure out what the energy is. The energy-momentum conservation equation given by Noether's theorem is
\[
\partial_t \left(\frac{\theta_t^2 + \theta_x^2}{2} + (1 - \cos \theta)\right) + \partial_x (-\theta_t \theta_x) = \partial_\mu P^\mu = 0.
\]
The energy we will be considering is thus
\[
E = \int_{\R} P^0 \;\d x = \int_\R \left(\frac{\theta_t^2 + \theta_x^2}{2} + (1 - \cos \theta)\right)\;\d x.
\]
Thus, to obtain finite energy, we will want $\theta(x) \to 2n_{\pm} \pi$ for some integers $n_{\pm}$ as $x \to \pm \infty$. What is the significance of this $n_{\pm}$?
\begin{eg}
Consider the basic kink
\[
\theta_K(x) = 4\tan^{-1} e^x.
\]
Picking the standard branch of $\tan^{-1}$, this kink looks like
\begin{center}
\begin{tikzpicture}
\draw [->] (-4, 0) -- (4, 0) node [right] {$x$};
\draw [->] (0, -0.5) -- (0, 2.5) node [above] {$\phi$};
\draw [mblue, thick] plot [smooth] coordinates {(-4.0,0.00001) (-3.9,0.00001) (-3.8,0.00001) (-3.7,0.00002) (-3.6,0.00002) (-3.5,0.00003) (-3.4,0.00004) (-3.3,0.00005) (-3.2,0.00007) (-3.1,0.00009) (-3.0,0.00012) (-2.9,0.00017) (-2.8,0.00022) (-2.7,0.00030) (-2.6,0.00041) (-2.5,0.00055) (-2.4,0.00075) (-2.3,0.00101) (-2.2,0.00136) (-2.1,0.00184) (-2.0,0.00248) (-1.9,0.00335) (-1.8,0.00452) (-1.7,0.00610) (-1.6,0.00823) (-1.5,0.01111) (-1.4,0.01499) (-1.3,0.02024) (-1.2,0.02732) (-1.1,0.03687) (-1.0,0.04975) (-0.9,0.06710) (-0.8,0.09047) (-0.7,0.12185) (-0.6,0.16382) (-0.5,0.21953) (-0.4,0.29255) (-0.3,0.38616) (-0.2,0.50193) (-0.1,0.63760) (0.0,0.78540) (0.1,0.93320) (0.2,1.06887) (0.3,1.18464) (0.4,1.27824) (0.5,1.35126) (0.6,1.40698) (0.7,1.44895) (0.8,1.48033) (0.9,1.50369) (1.0,1.52105) (1.1,1.53393) (1.2,1.54348) (1.3,1.55056) (1.4,1.55580) (1.5,1.55969) (1.6,1.56257) (1.7,1.56470) (1.8,1.56628) (1.9,1.56745) (2.0,1.56832) (2.1,1.56896) (2.2,1.56944) (2.3,1.56979) (2.4,1.57005) (2.5,1.57024) (2.6,1.57039) (2.7,1.57049) (2.8,1.57057) (2.9,1.57063) (3.0,1.57067) (3.1,1.57070) (3.2,1.57073) (3.3,1.57075) (3.4,1.57076) (3.5,1.57077) (3.6,1.57078) (3.7,1.57078) (3.8,1.57079) (3.9,1.57079) (4.0,1.57079)};
% map (\x -> (showFFloat (Just 1) x "", showFFloat (Just 5) (atan ( exp (3*x - 1))) "")) [-4,-3.9..4]
\draw [dashed] (-4, 1.57078) -- (4, 1.57078);
\node [left] at (-4, 0) {$0$};
\node [right] at (4, 1.57078) {$2\pi$};
\end{tikzpicture}
\end{center}
This goes from $\theta = 0$ to $\theta = 2\pi$.
\end{eg}
To better understand this, we can think of $\theta$ as an angular variable, i.e.\ we identify $\theta \sim \theta + 2n\pi$ for any $n \in \Z$. This is a sensible thing because the energy density and the equation etc.\ are unchanged when we shift everything by $2n\pi$. Thus, $\theta$ is not taking values in $\R$, but in $\R/2\pi\Z \cong S^1$.
Thus, for each fixed $t$, our field $\theta$ is a map
\[
\theta: \R \to S^1.
\]
The number $Q = n_+ - n_-$ equals the number of times $\theta$ covers the circle $S^1$ on going from $x = -\infty$ to $x = +\infty$. This is the winding number, which is interpreted as the topological charge.
As we previously discussed, we can express this topological charge as the integral of some current. We can write
\[
Q = \frac{1}{2\pi} \int_{\theta(\R)} \d \theta = \frac{1}{2\pi} \int_{-\infty}^\infty \frac{\d \theta}{\d x}\;\d x.
\]
Note that this formula automatically takes into account the orientation. This is the form that will lead to generalization in higher dimensions.
This function $\frac{\d \theta}{\d x}$ appearing in the integral has the interpretation as a topological charge density. Note that there is a topological conservation law
\[
\partial_\mu j^\mu = \frac{\partial j^0}{\partial t} + \frac{\partial j^1}{\partial x} = 0,
\]
where
\[
j^0 = \theta_x,\quad j^1 = -\theta_t.
\]
This conservation law is not a consequence of the field equations, but merely a mathematical identity, namely the commutation of partial derivatives.
\subsubsection*{Two dimensions}
For the sine-Gordon kink, the target space was a circle $S^1$. Now, we are concerned with the \term{unit disk}\index{$D$}
\[
D = \{(x^1, x^2): |\mathbf{x}|^2 < 1\} \subseteq \R^2.
\]
We will then consider fields
\[
\Phi: D \to D.
\]
In the case of a sine-Gordon kink, we still cared about moving solitons. However, here we will mostly work with static solutions, and study fields at a fixed time. Thus, there is no time variable appearing.
Using the canonical isomorphism $\R^2 \cong \C$, we can think of the target space as the unit disk in the complex plane, and write the field as
\[
\Phi = \Phi^1 + i \Phi^2.
\]
However, we will usually view the $D$ in the domain as a real space instead.
We will impose some boundary conditions. We pick any function $\chi: S^1 \to \R$, and consider
\[
g = e^{i\chi}: S^1 \to S^1 = \partial D \subseteq D.
\]
Here $g$ is a genuine function, and has to be single-valued. So $\chi$ must be single-valued modulo $2\pi$. We then require
\[
\Phi_{\partial D} = g = e^{i \chi}.
\]
In particular, $\Phi$ must send the boundary to the boundary.
Now the target space $D$ has a canonical choice of measure $\d \Phi^1 \wedge \d \Phi^2$. Then we can expect the new topological charge to be given by
\[
Q = \frac{1}{\pi} \int_D \d \Phi^1 \wedge \d \Phi^2 = \frac{1}{\pi} \int_D \det
\begin{pmatrix}
\frac{\partial \Phi^1}{\partial x^1} & \frac{\partial \Phi^1}{\partial x^2}\\
\frac{\partial \Phi^2}{\partial x^1} & \frac{\partial \Phi^2}{\partial x^2}
\end{pmatrix}\;\d x^1 \wedge \d x^2.
\]
Thus, the charge density is given by
\[
j^0 = \frac{1}{2} \varepsilon_{ab} \varepsilon_{ij} \frac{\partial \Phi^a}{\partial x^i} \frac{\partial \Phi^b}{\partial x^j}.
\]
Crucially, it turns out this charge density is a total derivative, i.e.\ we have
\[
j^0 = \frac{\partial V^i}{\partial x^i}
\]
for some function $V$. It is not immediately obvious this is the case. However, we can in fact pick
\[
V^i = \frac{1}{2} \varepsilon_{ab}\varepsilon^{ij} \Phi^a \frac{\partial \Phi^b}{\partial x^j}.
\]
To see this actually works, we need to use the anti-symmetry of $\varepsilon^{ij}$ and observe that
\[
\varepsilon^{ij} \frac{\partial^2 \Phi^b}{\partial x^i \partial x^j} = 0.
\]
Equivalently, using the language of differential forms, we view the charge density $j^0$ as the $2$-form
\[
j^0 = \d \Phi^1 \wedge \d \Phi^2 = \d (\Phi^1\;\d \Phi^2) = \frac{1}{2} \d (\Phi^1 \;\d \Phi^2 - \Phi^2 \;\d \Phi^1).
\]
By the divergence theorem, we find that
\[
Q = \frac{1}{2\pi}\oint_{\partial D} \Phi^1 \;\d \Phi^2 - \Phi^2 \;\d \Phi^1.
\]
We then use that on the boundary,
\[
\Phi^1 = \cos \chi,\quad \Phi^2 = \sin \chi,
\]
so
\[
Q = \frac{1}{2\pi} \oint_{\partial D} (\cos^2 \chi + \sin^2 \chi) \;\d \chi = \frac{1}{2\pi} \oint_{\partial D} \;\d \chi = N.
\]
Thus, the charge is just the winding number of $g$!
%But what actually is the degree telling us? Let's fix a boundary condition $\Phi|_{\partial D} = g = e^{i\chi}$, where $\chi: \partial D \cong S^1 \to \R$ is a real function. Since $g$ is now a map from the boundary circle to the boundary circle. So $g$ itself has got a winding number,
%\[
% \frac{1}{2\pi} \int_0^{2\pi} \frac{\d \chi}{\d \theta} \;\d \theta.
%\]
%Let's relate the degree of $\Phi$ to the winding number of $g$. To do so, we will need the fact that the Jacobian determinant is a total derivative, so that we can apply Green's identity. Thus, we want to write $j^0$ as
%\[
% j^0 = \frac{\partial V^i}{\partial x^i}
%\]
%for some $V^i$. It might not be immediately obvious how we can pick such a $V^i$, but notice that
%\[
% \varepsilon_{ij} \frac{\partial^2 \Phi^b}{\partial x^i \partial x^j} = 0
%\]
%by anti-symmetry. So we can in fact pick
%\[
% V^i = \frac{1}{2} \varepsilon_{ab}\varepsilon_{ij} \Phi^a \frac{\partial \Phi^b}{\partial x^j}.
%\]
%Then in this case, we have
%\[
% \int_D j^0 \;\d x = \int \left(\frac{\partial V^1}{\partial x^1} + \frac{\partial V^2}{\partial x^2}\right)\;\d x^1 \;\d x^2 = \oint_{\partial D} V^1 \;\d x^2 - V^2 \;\d x^1.
%\]
%On the boundary, we know $\Phi^1 = \cos \chi$ and $\Phi^2 = \sin \chi$. Substituting that in, we can calculate $V^1$ and $V^2$ on the boundary, and then work out the integral to be
%\[
% \int_D j\;\d x = \frac{1}{2} \oint_D \;\d \chi = \pi N,
%\]
%where $N$ is the winding number of $g: S^1 \to S^1$.
%
%This is a general phenomenon. Often, the appearance of anti-symmetric combinations of derivatives will allow us to reduce complicated quantities into total derivatives. Then this will allow us to relate things that \emph{a priori} depends on the value of $\Phi$ everywhere into something that depends only on the boundary values.
%The next interesting thing to notice is that the charge $Q$ is invariant under scaling of the domain. If we scale $x$ by $R$, then $\d x^1 \wedge \d x^2$ gets scaled up by $R^2$, but the Jacobian gets scaled down by $R^2$. Thus, an interesting thing to do is to take take $R \to \infty$. Then the domain becomes all of $\R^2$.
Now notice that our derivation didn't really depend on our domain being $D$. It could have been any region bounded by a simple closed curve in $\R^2$. In particular, we can take it to be a disk $D_R$ of arbitrary radius $R$.
What we are \emph{actually} interested in is a field
\[
\Phi: \R^2 \to D.
\]
We then impose asymptotic boundary conditions
\[
\Phi \sim g = e^{i\chi}
\]
as $|x| \to \infty$. We can still define the charge or degree by
\[
Q = \frac{1}{\pi} \int_{\R^2} j^0 \;\d x^1 \;\d x^2 = \frac{1}{\pi} \lim_{R \to \infty}\int_{D_R} j^0 \;\d x^1 \;\d x^2
\]
This is then again the winding number of $g$.
%So our field is now a mapping $\Phi: \R^2 \to D$. We then set the asymptotic boundary condition $\Phi \sim g = e^{i \chi}$ as $|x| \to \infty$. For these maps, we have the same formula for degree, namely
%\[
% Q = \frac{1}{\pi} \int_{\R^2} \frac{1}{2} \varepsilon_{ab} \varepsilon_{ij} \partial_i \Phi^a \partial_j \Phi^b \;\d x^1 \;\d x^2 = \frac{1}{2\pi} \lim_{R\to \infty} \oint_{|x| = R} \d \chi \in \Z.
%\]
%This is going to be our basic topological quantity for the vortices. Its existence, and being non-zero, is what gives rise to a stable vortex.
It is convenient to rewrite this in terms of an inner product. $\R^2$ itself has an inner product, and under the identification $\R^2 \cong \C$, the inner product can be written as
\[
(a, b) = \frac{\bar{a} b + a \bar{b}}{2}.
\]
Use of this expression allows calculations to be done efficiently if one makes use of the fact that for real numbers $a$ and $b$, we have
\[
(a, b) = (ai, bi) = ab, \quad (ai, b) = (a, bi) = 0.
\]
In particular, we can evaluate
\[
(i \Phi, \d \Phi) = (i \Phi^1 - \Phi^2, \d \Phi^1 + i \d \Phi^2) = \Phi^1\;\d \Phi^2 - \Phi^2\;\d \Phi^1.
\]
This is just (twice) the current $V$ we found earlier! So we can write our charge as
\[
Q = \frac{1}{2\pi} \lim_{R \to \infty} \oint_{|x| = R} (i \Phi, \d \Phi).
\]
We will refer to $(i \Phi, \d \Phi)$ as the current, and the corresponding charge density is $j^0 = \frac{1}{2} \d (i \Phi, \d \Phi)$.
%We can rewrite this in terms of complex numbers. Instead of viewing a $\Phi$ as a function valued in $D \subseteq \R^2$, we view it as a function $\Phi: D \to \C$. We already implicitly did so when we wrote the boundary condition as $e^{i \chi}$. In terms of complex numbers, we can write the standard inner product on $\R^2$ as
%\[
% (a, b) = \frac{\bar{a} b + a \bar{b}}{2}.
%\]
%Then we can write the integral for the charge as
%\[
% \oint_{\partial D}\;\d \chi = \oint_{\partial D} (i\Phi, \d \Phi).
%\]
%This formulation is sometimes more convenient for calculations.
This current is actually a familiar object from quantum mechanics: recall that for the Schr\"odinger's equation
\[
i \frac{\partial \Phi}{\partial t} = - \frac{1}{2m} \Delta \Phi + V(x) \Phi,\quad \Delta = \nabla^2.
\]
the probability $\int |\Phi|^s \;\d x$ is conserved. The differential form of the probability conservation law is
\[
\frac{1}{2} \partial_t (\Phi, \Phi) + \frac{1}{2m} \nabla \cdot (i \Phi, \nabla \Phi) = 0.
\]
What appears in the flux term here is just the topological current!
%The final comment, which we will not actually use, is that if we have a holomorphic function $f$, then consider the integral
%\[
% \frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z) - w}\;\d z.
%\]
%Using the residue theorem, we can see that this is the number of times $f$ takes value $w$, counted with multiplicity, within the area bounded by $C$. This is the \term{local degree} of the mapping. Indeed,
%\[
% \Res\left(\frac{f'(z)}{f(z) - w}, z_i\right) = n_i,
%\]
%where $f(z) - w = (z - z_i)^{n_i'} h(z)$, with $h(z_i) \not= 0$.
%
%But we can think about this in a different way. The integrand is an exact differential
%\[
% \frac{1}{2\pi i} [\log (f(z) - w)],
%\]
%and if we think about this carefully, this the change in argument of $f(z) - w$ around the curve.
\subsection{Global \texorpdfstring{$U(1)$}{U(1)} Ginzburg--Landau vortices}
We now put the theory into use. We are going to study \emph{Ginzburg--Landau vortices}\index{Ginzburg--Landau!vortex}. Our previous discussion involved a function taking values in the unit disk $D$. We will not impose such a restriction on our vortices. However, we will later see that any solution must take values in $D$.
The potential energy of the Ginzburg--Landau theory is given by
\[
V(\Phi) = \frac{1}{2} \int_{\R^2} \left((\nabla \Phi, \nabla \Phi) + \frac{\lambda}{4} (1 - (\Phi, \Phi))^2 \right)\;\d x^1 \;\d x^2.
\]
where $\lambda > 0$ is some constant.
Note that the inner product is invariant under phase rotation, i.e.
\[
(e^{i\chi} a, e^{i\chi}b) = (a, b)
\]
for $\chi \in \R$. So in particular, the potential satisfies
\[
V(e^{i\chi}\Phi) = V(\Phi).
\]
Thus, our theory has a \term{global $\U(1)$ symmetry}\index{$\U(1)$ symmetry!global}.
The Euler--Lagrange equation of this theory says
\[
- \Delta \Phi = \frac{\lambda}{2}(1 - |\Phi|^2) \Phi.
\]
This is the \emph{ungauged Ginzburg--Landau equation}\index{Ginzburg--Landau!ungauged}.
To justify the fact that our $\Phi$ takes values in $D$, we use the following lemma:
\begin{lemma}
Assume $\Phi$ is a smooth solution of the ungauged Ginzburg--Landau equation in some domain. Then at any interior maximum point $x_*$ of $|\Phi|$, we have $|\Phi(x_*)| \leq 1$.
\end{lemma}
\begin{proof}
Consider the function
\[
W(x) = 1 - |\Phi(x)|^2.
\]
Then we want to show that $W \geq 0$ when $W$ is minimized. We note that if $W$ is at a minimum, then the Hessian matrix must have non-negative eigenvalues. So, taking the trace, we must have $\Delta W(x_*) \geq 0$. Now we can compute $\Delta W$ directly. We have
\begin{align*}
\nabla W &= -2 (\Phi, \nabla \Phi)\\
\Delta W &= \nabla^2 W \\
&= - 2(\Phi, \Delta \Phi) - 2(\nabla \Phi, \nabla \Phi)\\
&= \lambda |\Phi|^2 W - 2 |\nabla \Phi|^2.
\end{align*}
Thus, we can rearrange this to say
\[
2 |\nabla \Phi|^2 + \Delta W = \lambda |\Phi|^2 W.
\]
But clearly $2 |\nabla \Phi|^2 \geq 0$ everywhere, and we showed that $\Delta W(x_*) \geq 0$. So we must have $W(x_*) \geq 0$.
\end{proof}
By itself, this doesn't force $|\Phi| \in [0, 1]$, since we could imagine $|\Phi|$ having no maximum. However, if we prescribe boundary conditions such that $|\Phi| = 1$ on the boundary, then this would indeed imply that $|\Phi| \leq 1$ everywhere. Often, we can think of $\Phi$ as some ``complex order parameter'', in which case the condition $|\Phi| \leq 1$ is very natural.
The objects we are interested in are \emph{vortices}.
\begin{defi}[Ginzburg--Landau vortex]\index{Ginzburg--Landau!vortex}
A global \emph{Ginzburg--Landau vortex} of charge $N > 0$ is a (smooth) solution of the ungauged Ginzburg--Landau equation of the form
\[
\Phi = f_N(r) e^{iN\theta}
\]
in polar coordinates $(r, \theta)$. Moreover, we require that $f_N(r) \to 1$ as $r \to \infty$.
\end{defi}
Note that for $\Phi$ to be a smooth solution, we must have $f_N(0) = 0$. In fact, a bit more analysis shows that we must have $f_N = O(r^N)$ as $r \to 0$. Solutions for $f_N$ do exist, and they look roughly like this:
\begin{center}
\begin{tikzpicture}
\draw [->] (-0.5, 0) -- (5, 0) node [right] {$r$};
\draw [->] (0, -0.5) -- (0, 3) node [above] {$f_N$};
\draw [mblue, thick, domain=0:5] plot [smooth] (\x, {2 * \x^4 / (1 + \x^4)});
\draw [dashed] (-1, 2) -- (5, 2);
\end{tikzpicture}
\end{center}
In the case of $N = 1$, we can visualize the field $\Phi$ as a vector field on $\C$. Then it looks like
\begin{center}
\begin{tikzpicture}
\draw [->] (-3, 0) -- (3, 0);
\draw [->] (0, -3) -- (0, 3);
\foreach \t in {0,30,60,90,120,150,180,210,240,270,300,330,360} {
\begin{scope}[rotate=\t]
\foreach \x in {0.5, 1, 1.5, 2, 2.5} {
\pgfmathsetmacro\arlen{0.5 * (\x^4 / (1 + \x^4))^2}
\draw [-latex'] (\x, 0) -- +(\arlen, 0);
}
\end{scope}
}
\end{tikzpicture}
\end{center}
This is known as a \emph{$2$-dimensional hedgehog}.
For general $N$, it might be more instructive to look at how the current looks like. Recall that the current is defined by $(i\Phi, \d \Phi)$. We can write this more explicitly as
\begin{align*}
(i\Phi, \d \Phi) &= (if_N e^{iN\theta}, (\d f_N) e^{iN\theta} + i f_N N\;\d \theta e^{iN\theta})\\
&= (if_N,\d f_N + i f_N N\;\d \theta).
\end{align*}
We note that $if_N$ and $\d f_N$ are orthogonal, while $i f_N$ and $i f_N N \d \theta$ are parallel. So the final result is
\[
(i\Phi, \d \Phi) = f_N^2 N\;\d \theta.
\]
So the current just looks this:
\begin{center}
\begin{tikzpicture}
\draw [->] (-3, 0) -- (3, 0);
\draw [->] (0, -3) -- (0, 3);
\foreach \t in {0,30,60,90,120,150,180,210,240,270,300,330,360} {
\begin{scope}[rotate=\t]
\foreach \x in {0.5, 1, 1.5, 2, 2.5} {
\pgfmathsetmacro\arlen{0.5 * (\x^4 / (1 + \x^4))^2}
\draw [-latex'] (\x, 0) -- +(0, \arlen);
}
\end{scope}
}
\end{tikzpicture}
\end{center}
As $|x| \to \infty$, we have $f_N \to 1$. So the winding number is given as before, and we can compute the winding number of this system to be
\[
\frac{1}{2\pi} \lim_{R \to \infty} \oint (i \Phi, \d \Phi) = \frac{1}{2\pi} \lim_{R \to \infty} \oint f_N^2 N \;\d \theta = N.
\]
The winding number of these systems is a discrete quantity, and can make the vortex stable.
This theory looks good so far. However, it turns out this model has a problem --- the energy is infinite! We can expand out $V(f_N e^{iN \theta})$, and see it is a sum of a few non-negative terms. We will focus on the $\frac{\partial}{\partial \theta}$ term. We obtain
\begin{align*}
V(f_N e^{iN\theta}) &\geq \int \frac{1}{r^2} \left|\frac{\partial \Phi}{\partial \theta}\right|^2 r\;\d r\;\d \theta\\
&= N^2 \int \frac{1}{r^2} f_N^2 r\;\d r\;\d \theta \\
&= 2\pi N^2 \int_0^\infty \frac{f_N^2}{r}\;\d r.
\end{align*}
Since $f_N \to 1$ as $r \to \infty$, we see that the integral diverges logarithmically.
This is undesirable physically. To understand heuristically why this occurs, decompose $\d \Phi$ into two components --- a mode parallel to $\Phi$ and a mode perpendicular to $\Phi$. For a vortex solution these correspond to the radial and angular modes respectively. We will argue that for fluctuations the parallel mode is massive, while the perpendicular mode is massless. Now given that we just saw that the energy divergence of the vortex arises from the angular part of the energy, we see that it is the massless mode that leads to problems. We will see below that in gauge theories, the Higgs mechanism serves to make all modes massive, thus allowing for finite energy vortices.
We can see the difference between massless and massive modes very explicitly in a different setting, corresponding to Yukawa mesons. Consider the equation
\[
-\Delta u + M^2 u = f.
\]
Working in three dimensions, the solution is given by
\[
u(x) = \frac{1}{4\pi} \int \frac{e^{-M|x - y|}}{|x - y|} f(y)\;\d y.
\]
Thus, the Green's function is
\[
G(x) = \frac{e^{-M|x|}}{4\pi|x|}.
\]
If the system is massless, i.e.\ $M = 0$, then this decays as $\frac{1}{|x|}$. However, if the system is massive with $M > 0$, then this decays exponentially as $|x| \to \infty$. In the nonlinear setting the exponential decay which is characteristic of massive fundamental particles can help to ensure decay of the energy density at a rate fast enough to ensure finite energy of the solution.
So how do we figure out the massive and massless modes? We do not have a genuine decomposition of $\Phi$ itself into ``parallel'' and ``perpendicular'' modes, because what is parallel and what is perpendicular depends on the local value of $\Phi$.
Thus, to make sense of this, we have to consider small fluctuations around a fixed configuration $\Phi$. We suppose $\Phi$ is a solution to the field equations. Then $\frac{\delta V}{\delta \Phi} = 0$. Thus, for small variations $\Phi \mapsto \Phi + \varepsilon \varphi$, we have
\[
V(\Phi + \varepsilon \varphi) = V(\Phi) + \varepsilon^2 \int \left(|\nabla \varphi|^2 + \lambda (\varphi, \Phi)^2 - \frac{2\lambda}{4}(1 - |\Phi|^2)|\varphi|^2\right) \;\d x + O(\varepsilon^3).
\]
Ultimately, we are interested in the asymptotic behaviour $|x| \to \infty$, in which case $1 - |\Phi|^2 \to 0$. Moreover, $|\Phi| \to 1$ implies $(\varphi, \Phi)$ becomes a projection along the direction of $\Phi$. Then the quadratic part of the potential energy density for fluctuations becomes approximately
\[
|\nabla \varphi|^2 + \lambda |\varphi^{\mathrm{parallel}}|^2
\]
for large $x$. Thus, for $\lambda > 0$, the parallel mode is massive, with corresponding ``Yukawa'' mass parameter $M = \sqrt{\lambda}$, while the perpendicular mode is massless. The presence of the massless mode is liable to produce a soliton with slow algebraic decay at spatial infinity, and hence infinite total energy. This is all slightly heuristic, but is a good way to think about the issues. When we study vortices that are gauged, i.e.\ coupled to the electromagnetic field, we will see that the Higgs mechanism renders all components massive, and this problem does not arise.
%
%The divergence of energy arises because if we consider fluctuations $\Phi + \varepsilon \varphi$, and look at the energy
%\[
% V(\Phi + \varepsilon \varphi) = V(\Phi) + \varepsilon \int \frac{\delta V}{\delta \Phi} (\varphi) + \varepsilon^2 \int |\Delta \varphi|^2 + \text{something}.
%\]
%The first order term vanishes because $V$ solves the equations of motion.
%
%We look carefully at what the ``something'' is. We have
%\[
% (1 - (\Phi + \varepsilon \varphi, \Phi + \varepsilon \varphi))^2 = (1 - |\Phi|^2 - 2 \varepsilon (\varphi, \Phi) - \varepsilon ^2 |\varphi|^2)^2.
%\]
%The order $\varepsilon^2$ part is given by
%\[
% 4 \varepsilon^2 (\varphi, \Phi)^2 - 2 \varepsilon^2 (1 - |\phi|^2)|\varphi|^2
%\]
%So we find
%\[
% V(\Phi + \varepsilon \varphi) = V(\Phi) + \varepsilon^2 \int \left(|\nabla \varphi|^2 + \lambda (\varphi, \Phi)^2 - \frac{2\lambda}{4}(1 - |\Phi|^2)|\varphi|^2\right) \;\d x + O(\varepsilon^3).
%\]
%We now remember something about Green's functions, and the difference between massless and massive fields. We consider the ordinary Poisson equation
%\[
% - \Delta u = f.
%\]
%We can write down the solution to this in terms of Green's functions. In three dimensions, this is
%\[
% u(x) = \frac{1}{4\pi} \int \frac{f(y)}{|x - y|}\;\d y.
%\]
%This Green's function $\frac{1}{4\pi|x|}$ decays very slowly with $x$, and this is a sign that we have a massless field.
%
%If we change our integral equation instead to
%\[
% (-\Delta u + M^2 u) = f,
%\]
%then we find that
%\[
% u(x) + \frac{1}{4\pi} \int_{\R^3} \frac{e^{-M|x - y|}}{|x - y|} f(y) \;\d y.
%\]
%The Green's function is now exponentially decaying with $|x|$ if $M > 0$. This is characteristic for the behaviour of static Green's functions for massive fields.
%
%We now look at our change in $V$ above. We look at the quadratic fluctuation energy as $|x| \to \infty$. For $x$ very large, we have $(1 - |\Phi|^2) \to 0$ by the boundary conditions. Then we are left with the first two terms
%\[
% |\nabla \varphi|^2 + \lambda (\Phi, \varphi)^2.
%\]
%What does this mean? $\Phi$ and $\varphi$ are complex numbers, and in the second term, we are essentially projecting in the direction to $\Phi$. We write
%\[
% \varphi = \varphi^{||} + \varphi^{\perp}.
%\]
%Then we have
%\[
% \varphi^{||} \frac{(\Phi, \varphi)\Phi}{|\Phi|^2}.
%\]
%Thus, heuristically, this term is
%\[
% |\nabla \varphi^{||}|^2 + \lambda |\varphi^{||}|^2 + |\Delta \varphi^\perp|^2.
%\]
%So the parallel part behaves like a massive particle with quick decay, and the perpendicular part behaves like a massless particle with slow decay.
\subsection{Abelian Higgs/Gauged Ginzburg--Landau vortices}
We now consider a theory where the complex scalar field $\Phi$ is coupled to a magnetic field. This is a $\U(1)$ gauge theory, with gauge potential given by a smooth real $1$-form
\[
A = A_1 \;\d x^1 + A_2 \;\d x^2.
\]
The coupling between $\Phi$ and $A$ is given by \term{minimal coupling}: this is enacted by introduction of the \term{covariant derivative}\index{$\D$}\index{$\D_A$}
\[
\D \Phi = \D_A \Phi = \d \Phi - i A \Phi = \sum_{j = 1}^2 \D_j \Phi \;\d x^j.
\]
To proceed, it is convenient to have a list of identities involving the covariant derivative.
\begin{prop}
If $f$ is a smooth real-valued function, and $\Phi$ and $\Psi$ are smooth complex scalar fields, then
\begin{align*}
\D (f \Phi) &= (\d f) \; \Phi + f\;\D \Phi,\\
\d (\Phi, \Psi) &= (\D \Phi, \Psi) + (\Phi, \D \Psi).
\end{align*}
(Here $(\ph)$ is the real inner product defined above.) In coordinates, these take the form
\begin{align*}
\D_j (f \Phi) &= (\partial_j f)\;\Phi + f\; \D_j \Phi\\
\partial_j (\Phi, \Psi) &= (\D_j \Phi, \Psi) + (\Phi, \D_j \Psi).
\end{align*}
\end{prop}
The proofs just involve writing all terms out. The first rule is a version of the Leibniz rule, while the second, called unitarity, is analogous to the fact that if $V, W$ are smooth vector fields on a Riemannian manifold, then
\[
\partial_k g(V, W) = g(\nabla_k V, W) + g(V, \nabla_k W)
\]
for the Levi-Civita connection $\nabla$ associated to a Riemannian metric $g$.
The curvature term is given by the magnetic field.
\begin{defi}[Magnetic field/curvature]\index{magnetic field}\index{curvature}
The \emph{magnetic field}, or \emph{curvature} is given by
\[
B = \partial_1 A_2 - \partial_2 A_1.
\]
We can alternatively think of it as the 2-form
\[
F = \d A = B \;\d x^1 \wedge\d x^2.
\]
\end{defi}
The formulation in terms of differential forms is convenient for computations, because we don't have to constrain ourselves to working in Cartesian coordinates --- for example, polar coordinates may be more convenient.
\begin{prop}
If $\Phi$ is a smooth scalar field, then
\[
(\D_1 \D_2 - \D_2 \D_1) \Phi = - i B \Phi.
\]
\end{prop}
The proof is again a direct computation. Alternatively, we can express this without coordinates. We can extend $\D$ to act on $p$-forms by letting $A$ act as $A \wedge$. Then this result says
\begin{prop}
\[
\D \D \Phi = -i F \Phi.
\]
\end{prop}
\begin{proof}
\begin{align*}
\D \D \Phi &= (\d - iA) (\d \Phi - iA\Phi)\\
&= \d^2 \Phi - i \d (A \Phi) - iA\;\d \Phi - A \wedge A\;\Phi\\
&= -i \d (A \Phi) - i A \;\d \Phi\\
&= -i \d A\; \Phi + i A \;\d \Phi - i A\;\d \Phi\\
&= -i (\d A)\; \Phi\\
&= -i F \Phi.\qedhere
\end{align*}
\end{proof}
The point of introducing the covariant derivative is that we can turn the global $\U(1)$ invariance into a local one. Previously, we had a global $\U(1)$ symmetry, where our field is unchanged when we replace $\Phi \mapsto \Phi e^{i \chi}$ for some constant $\chi \in \R$. With the covariant derivative, we can promote this to a \emph{gauge} symmetry.
Consider the simultaneous \term{gauge transformations}
\begin{align*}
\Phi(x) &\mapsto e^{i\chi(x)} \Phi(x)\\
A(x) &\mapsto A(x) + \d \chi(x).
\end{align*}
Then the covariant derivative of $\Phi$ transforms as
\begin{align*}
(\d - iA) \Phi &\mapsto (\d - i (A + \d \chi)) (\Phi e^{i \chi}) \\
&= (\d \Phi + i \Phi \d \chi - i (A + \d \chi) \Phi) e^{i\chi} \\
&= e^{i \chi} (\d - iA) \Phi.
\end{align*}
Similarly, the magnetic field is also invariant under gauge transformations.
As a consequence, we can write down energy functionals that are invariant under these gauge transformations. In particular, we have (using the real inner product defined above)
\[
(\D \Phi, \D \Phi) \mapsto (e^{i\chi} \D \Phi, e^{i\chi} \D \Phi) = (\D \Phi, \D \Phi).
\]
So we can now write down the \emph{gauged Ginzburg--Landau energy}\index{Ginzburg--Landau!gauged}
\[
V_\lambda(A, \Phi) = \frac{1}{2} \int_{\R^2} \left(B^2 + |\D \Phi|^2 + \frac{\lambda}{4} (1 - |\Phi|^2)^2\right)\;\d^2 x.
\]
This is then manifestly gauge invariant.
%where
%\[
% B^2 = \partial_1 A_2 - \partial_2 A_1,
%\]
%which can be shown to be gauge-invariant.
%
%Thus, $V(\lambda)$ is invariant under gauge transformations. In other words,
%\[
% V_\lambda(A + \d \chi, \Phi e^{i\chi}) = V_\lambda (A, \Phi).
%\]
%To proceed, it is convenient to have a list of identities about the covariant derivative.
%\begin{prop}
% If $f(x) \in \R$, then
% \[
% (\nabla_A)_j (f \Phi) = (\partial_j f) \Phi + f (\nabla_A)_j \Phi.
% \]
% If $\Phi$ and $\Psi$ are complex scalar fields, then
% \[
% \partial_j (\Phi, \Psi) = (\D_j \Phi, \Psi) + (\Phi, \D_j \Psi).
% \]
%\end{prop}
%The proofs just involve writing all terms out. The second rule is analogous to fact that
%\[
% \partial_k g(V, W) = g(\nabla_k V, W) + g(V, \nabla_k W)
%\]
%for the Levi-Civita connection.
%
%\begin{prop}
% We have
% \[
% -((\nabla_A)_1 (\nabla_A)_2 - (\nabla_A)_2 (\nabla_A)_1) \Phi = - i B \phi,
% \]
% where
% \[
% B = \partial_1 A_2 - \partial_2 A_1.
% \]
% In geometry, this \term{$B$} is known as the \term{curvature}, and in electromagnetism, this is the \term{magnetic field}.
%\end{prop}
%
%The proof is again writing it out.
%
%Our theory now has gauge invariance. In the ungauged Ginzburg--Landau theory, there was an invariance under a $\U(1)$ action $\Phi \mapsto \Phi e^{i\chi}$ for $\chi \in \R$ constant. This is \emph{global} gauge invariance, because it is constant everywhere.
%
%This invariance is now localized to
%\[
% \Phi(x) \mapsto \Phi(x) e^{i \chi(x)},
%\]
%where $\chi(x) \in \R$ is a smooth function, as long as our $A$ transforms accordingly as
%\[
% A \mapsto A + \d \chi.
%\]
%Then the covariant derivative changes as
%\[
% \D \Phi \mapsto (\D \Phi)e^{i \chi}.
%\]
%Finally, we check that the magnetic field is invariant,
%\[
% B \mapsto B.
%\]
%therefore the abelian Higgs energy functional
%\[
% V_\lambda (A, \Phi) = \frac{1}{2} \int \left(B^2 + |\D \Phi|^2 + \frac{\lambda}{4} (1 - |\Phi|^2)\right)\;\d^2 x
%\]
%satisfies
%\[
% V_\lambda(A + \d \chi, \Phi e^{i \chi}) = V_\lambda (A, \Phi)
%\]
%for all $\chi \in C^1(\R^2)$. So we now have an infinite degeneracy for all solutions.
As before, the equations of motion are given by the Euler--Lagrange equations. Varying $\Phi$, we obtain
\[
- (\D_1^2 + \D_2^2) \Phi - \frac{\lambda}{2} (1 - |\Phi|^2)\Phi = 0.
\]
This is just like the previous vortex equation in the ungauged case, but since we have the covariant derivative, this is now coupled to the gauge potential $A$. The equations of motion satisfied by $A$ are
\begin{align*}
\partial_2 B &= (i \Phi, \D_1 \Phi)\\
-\partial_1 B &= (i \Phi, \D_2 \Phi).
\end{align*}
These are similar to one of Maxwell's equation --- the one relating the curl of the magnetic field to the current.
It is again an exercise to derive these. We refer to the complete system as the gauged Ginzburg--Landau, or Abelian Higgs equations. In deriving them, it is helpful to use the previous identities such as
\[
\partial_j (\Phi, \Psi) = (\D_j \Phi, \Psi) + (\Phi, \D_j \Psi).
\]
So we get the integration by parts formula
\[
\int_{\R^2} (\D_j \Phi, \Psi)\;\d^2 x = - \int (\Phi, \D_j \Psi)\;\d^2 x
\]
under suitable boundary conditions.
%Recall that in the ungauged Ginzburg--Landau theory, we proved that we always had $|\Phi| \leq 1$. This is again true. The same proof works, with $\nabla$ replaced with $\D$. (It is literally the same proof. Just copy-and-paste, find-and-replace.)
\begin{lemma}
Assume $\Phi$ is a smooth solution of the gauged Ginzburg--Landau equation in some domain. Then at any interior maximum point $x_*$ of $|\Phi|$, we have $|\Phi(x_*)| \leq 1$.
\end{lemma}
\begin{proof}
Consider the function
\[
W(x) = 1 - |\Phi(x)|^2.
\]
Then we want to show that $W \geq 0$ when $W$ is minimized. We note that if $W$ is at a minimum, then the Hessian matrix must have non-negative eigenvalues. So, taking the trace, we must have $\Delta W(x_*) \geq 0$. Now we can compute $\Delta W$ directly. We have
\begin{align*}
\partial_j W &= -2 (\Phi, \D_j \Phi)\\
\Delta W &= \partial_j \partial_j W \\
&= - 2(\Phi, \D_j \D_j \Phi) - 2(\D_j \Phi, \D_j \Phi)\\
&= \lambda |\Phi|^2 W - 2 |\nabla \Phi|^2.
\end{align*}
Thus, we can rearrange this to say
\[
2 |\nabla \Phi|^2 + \Delta W = \lambda |\Phi|^2 W.
\]
But clearly $2 |\nabla \Phi|^2 \geq 0$ everywhere, and we showed that $\Delta W(x_*) \geq 0$. So we must have $W(x_*) \geq 0$.
\end{proof}
As before, this suggests we interpret $|\Phi|$ as an order parameter. This model was first used to describe superconductors. The matter can either be in a ``normal'' phase or a superconducting phase, and $|\Phi|$ measures how much we are in the superconducting phase.
Thus, in our model, far away from the vortices, we have $|\Phi| \approx 1$, and so we are mostly in the superconducting phase. The vortices represent a breakdown of the superconductivity. At the core of the vortices, we have $|\Phi| = 0$, and we are left with completely normal matter. Usually, this happens when we have a strong magnetic field. In general, a magnetic field cannot penetrate the superconductor (the ``Meissner effect''), but if it is strong enough, it will cause such breakdown in the superconductivity along vortex ``tubes''.
\subsubsection*{Radial vortices}
Similar to the ungauged case, for $\lambda > 0$, there exist vortex solutions of the form
\begin{align*}
\Phi &= f_N(r) e^{iN\theta}\\
A &= N \alpha_N(r) \;\d \theta.
\end{align*}
The boundary conditions are $f_N, \alpha_N \to 1$ as $r \to \infty$ and $f_N, \alpha_N\to 0$ as $r \to 0$.
Let's say a few words about why these are sensible boundary conditions from the point of view of energy. We want
\[
\lambda \int_{\R^2} (1 - |\Phi|^2)^2 < \infty,
\]
and this is possible only for $f_N \to 1$ as $r \to \infty$. What is less obvious is that we also need $\alpha_N \to 1$. We note that we have
\[
\D_\theta \Phi = \frac{\partial \Phi}{\partial \theta} - i A_\theta \Phi = (iN f_N - iN \alpha_N f_N) e^{iN\theta}.
\]
We want this to approach $0$ as $r \to \infty$. Since $f_N \to 1$, we also need $\alpha_N \to 1$.
The boundary conditions at $0$ can be justified as before, so that the functions are regular at $0$.
\subsubsection*{Topological charge and magnetic flux}
Let's calculate the topological charge. We have (assuming sufficiently rapid approach to the asymptotic values as $r \to \infty$)
\begin{align*}
Q &= \frac{1}{\pi} \int_{\R^2} j^0(\Phi)\;\d^2 x \\
&= \lim_{R \to \infty} \frac{1}{2\pi} \oint_{|x| = R} (i \Phi, \d \Phi) \\
&= \lim_{R \to \infty} \frac{1}{2\pi} \oint (if_N e^{iN\theta}, iN f_N e^{iN\theta})\;\d \theta\\
&= \frac{1}{2\pi} \cdot N \lim_{R \to \infty} \int_0^{2\pi} f_N^2 \;\d \theta\\
&= N.
\end{align*}
Previously, we understood $N$ as the ``winding number'', and it measures how ``twisted'' our field was. However, we shall see shortly that there is an alternative interpretation of this $N$. Previously, in the sine-Gordon theory, we could think of $N$ as the number of kinks present. Similarly, here we can think of this $N$-vortex as a superposition of $N$ vortices at the origin. In the case of $\lambda = 1$, we will see that there are static solutions involving multiple vortices placed at different points in space.
We can compute the magnetic field and total flux as well. It is convenient to use the $\d A$ definition, as we are not working in Cartesian coordinates. We have
\[
\d A = N \alpha_N'(r) \;\d r \wedge \d \theta = \frac{1}{r} N \alpha_N'\;\d x^1 \wedge \d x^2.
\]
Thus it follows that
\[
B = \partial_1 A_2 - \partial_2 A_1 = \frac{N}{r} \alpha_N'.
\]
Working slightly more generally, we assume given a smooth finite energy configuration $A, \Phi$, and suppose in addition that $|\Phi|^2 \to 1$ and $r|A|$ is bounded as $r \to \infty$, and also that $\D_\theta \Phi = o(r^{-1})$. Then we find that
\[
(\d \Phi)_\theta = i A_\theta \Phi + o(r^{-1}).
\]
Then if we integrate around a circular contour, since only the angular part contributes, we obtain
\[
\oint_{|x| = R} (i \Phi, \d \Phi) = \oint_{|x| = R} (i \Phi, i A_\theta \Phi)\;\d \theta + o(1) = \oint_{|x| = R} A + o(1).
\]
Note that here we are explicitly viewing $A$ as a differential form so that we can integrate it. We can then note that $|x| = R$ is the boundary of the disk $|x| \leq R$. So we can apply Stokes' theorem and obtain
\[
\oint_{|x| = R} (i \Phi, \d \Phi) = \int_{|x| \leq R} \d A = \int_{|x| \leq R} B\;\d^2 x.
\]
%
%Now suppose that the covariant derivative
%\[
% \D_\theta \Phi = (\nabla_A)_\theta \Phi = \left(\frac{\partial \Phi}{\partial \theta} - A_\theta \Phi\right) = (iN f_n - iN \alpha_N f_N) e^{iN\theta}
%\]
%is rapidly decreasing as $r \to \infty$, then we know
%\[
% A_\theta - \left(i \Phi, \frac{\partial \Phi}{\partial \theta}\right) \to 0.
%\]
%quickly as $r \to \infty$. Therefore we obtain
%\[
% \oint_{|x| = R} (i \Phi, \d \Phi) = \oint_{|x| = R} \left(i \Phi, \frac{\partial \Phi}{\partial \theta}\right)\;\d \theta = \oint_{|x| = R} A_\theta \;\d \theta + o(1).
%\]
%We now apply Stokes theorem. We note that $A$ only has a $\theta$ direction. So we have
%\[
% A_\theta \;\d \theta = A_1 \;\d x^1 + A_2\;\d x^2.
%\]
%Thus we obtain
%\[
% \oint_{|x| = R} A_1 \;\d x^1 + A_2\;\d x^2 = \int_{|x| \leq R} (\partial_1 A_2 - \partial_2 A_1)\;\d x^1 \wedge \d x^2 = \int_{|x| \leq R} B\;\d x^1 \;\d x^2.
%\]
Now we let $R \to \infty$ to obtain
\[
\lim_{R \to \infty} \int_{x \leq R} B\;\d^2 x = \lim_{R \to \infty} \oint_{|x| = R} A = \lim_{R \to \infty} \oint_{|x| = R} (i \Phi, \d \Phi) = 2 \pi Q.
\]
%If we analyze our derivation of this, it didn't exactly require that we had a radial vortex. It only required that
%\[
% |\D \Phi| \leq \frac{\text{constant}}{r^{1 + \varepsilon}}
%\]
%for some $\varepsilon > 0$.
Physically, what this tells us then is that there is a relation between the topological winding number and the magnetic flux. This is a common property of topological gauge theories. In mathematics, this is already well known --- it is the fact that we can compute characteristic classes of vector bundles by integrating the curvature, as discovered by Chern.
\subsubsection*{Behaviour of vortices as $r \to 0$}
We saw earlier that for reasons of regularity, it was necessary that $f_N, \alpha_N \to 0$ as $r \to 0$.
But actually, we must have $f_N \sim r^N$ as $r \to 0$. This has as a consequence that $\Phi \sim (r e^{i\theta})^N = z^N$. So the local appearance of the vortex is the zero of an analytic function with multiplicity $N$.
To see this, we need to compute that the Euler--Lagrange equations are
\[
-f_N''(r) - \frac{1}{r} f_N'(r) + \frac{(N - \alpha_N)^2}{r^2} f_N = \frac{\lambda}{2} f_N(1 - f_N^2).
\]
With the boundary condition that $f_N$ and $\alpha_N$ vanish at $r = 0$, we can approximate this locally as
\[
- f_N'' - \frac{1}{r} f_N' + \frac{N^2}{r^2} f_N \approx 0
\]
since we have a $\frac{1}{r^2}$ on the left hand side. The approximating equation is homogeneous, and the solutions are just
\[
f_N = r^{\pm N}.
\]
So for regularity, we want the one that $\to 0$, so $f_N \sim r^N$ as $r \to 0$.
\subsection{Bogomolny/self-dual vortices and Taubes' theorem}
As mentioned, we can think of the radial vortex solution as a collection of $N$ vortices all superposed at the origin. Is it possible to have separated vortices all over the plane? Naively, we would expect that the vortices exert forces on each other, and so we don't get a static solution. However, it turns out that in the $\lambda = 1$ case, there do exist static solutions corresponding to vortices at arbitrary locations on the plane.
This is not obvious, and the proof requires some serious analysis. We will not do the analysis, which requires use of Sobolev spaces and PDE theory. However, we will do all the non-hard-analysis part. In particular, % we will obtain Bogomolny bounds as we did in the sine-Gordon case, and reduce the problem to finding solutions of a single scalar PDE, which can be understood with tools from calculus of variations and elliptic PDE.
Recall that for the sine-Gordon kinks, we needed to solve
\[
\theta'' = \sin \theta,
\]
with boundary conditions $\theta(x) \to 0$ or $2\pi$ as $x \to \pm \infty$. The only solutions we found were
\[
\theta_K (x - X)
\]
for any $X \in \R$. This $X$ is interpreted as the location of the kink. So the moduli space of solutions is $\mathcal{M} = \R$.
We shall get a similar but more interesting description for the $\lambda = 1$ vortices. This time, the moduli space will be $\C^N$, given by $N$ complex parameters describing the solutions.
\begin{thm}[Taubes' theorem]
For $\lambda = 1$, the space of (gauge equivalence classes of) solutions of the Euler--Lagrange equations $\delta V_1 = 0$ with winding number $N$ is $\mathcal{M} \cong \C^N$.
To be precise, given $N \in \N$ and an unordered set of points $\{Z_1, \cdots, Z_N\}$, there exists a smooth solution $A(x; Z_1, \cdots, Z_N)$ and $\Phi(x; Z_1, \cdots, Z_n)$ which solves the Euler-Lagrange equations $\delta V_1 = 0$, and also the so-called \term{Bogomolny equations}
\[
\D_1 \Phi + i \D_2 \Phi = 0, \quad B = \frac{1}{2} (1 - |\Phi|^2).
\]
Moreover, $\Phi$ has exactly $N$ zeroes $Z_1, \cdots, Z_N$ counted with multiplicity, where (using the complex coordinates $z = x_1 + ix_2$)
\[
\Phi(x; Z_1, \cdots, Z_N) \sim c_j (z - Z_j)^{n_j}
\]
as $z \to Z_j$, where $n_j = |\{k: Z_k = Z_j\}|$ is the multiplicity and $c_j$ is a nonzero complex number.
This is the unique such solution up to gauge equivalence. Furthermore,
\[
V_1(A(\ph, Z_1, \cdots, Z_N), \Phi(\ph; Z_1, \cdots, Z_N)) = \pi N\tag{$*$}
\]
and
\[
\frac{1}{2\pi} \int_{\R^2} B\;\d^2 x = N = \text{winding number}.
\]
Finally, this gives all finite energy solutions of the gauged Ginzburg--Landau equations.
\end{thm}
Note that it is not immediately clear from our description that the moduli space is $\C^N$. It looks more like $\C^N$ quotiented out by the action of the permutation group $S_N$. However, the resulting quotient is still isomorphic to $\C^N$. (However, it is important for various purposes to remember this quotient structure, and to use holomorphic coordinates which are invariant under the action of the permutation group --- the elementary symmetric polynomials in $\{Z_1, \ldots, Z_n\}$.
There is a lot to be said about this theorem. The equation $(*)$ tells us the energy is just a function of the number of particles, and does not depend on where they are. This means there is no force between the vortices. In situations like this, it is said that the Bogomolny bound is saturated. The final statement suggests that the topology is what is driving the existence of the vortices, as we have already seen. The reader will find it useful to work out the corresponding result in the case of negative winding number (in which case the holomorphicity condition becomes anti-holomorphicity, and the sign of the magnetic field is reversed in the Bogomolny equations).
Note that the Euler--Lagrange equations themselves are second-order equations. However, the Bogomolny equations are \emph{first order}. In general, this is a signature that suggests that interesting mathematical structures are present.
We'll discuss three crucial ingredients in this theorem, but we will not complete the proof, which involves more analysis than is a pre-requisite for this course. The proof can be found in Chapter 3 of Jaffe and Taubes's \emph{Vortices and Monopoles}.
\subsubsection*{Holomorphic structure}
When there are Bogomolny equations, there is often some complex analysis lurking behind. We can explicitly write the first Bogomolny equation as
\[
\D_1 \Phi + i \D_2 \Phi = \frac{\partial \Phi}{\partial x^1} + i\frac{\partial \Phi}{\partial x^2} - i(A_1 + i A_2) \Phi = 0.
\]
Recall that in complex analysis, holomorphic functions can be characterized as complex-valued functions which are continuously differentiable (in the real sense) and also satisfy the Cauchy--Riemann equations
\[
\frac{\partial f}{\partial \bar{z}} = \frac{1}{2} \left(\frac{\partial f}{\partial x^1} + i \frac{\partial f}{\partial x^2}\right) = 0.
\]
So we think of the first Bogomolny equation as the covariant Cauchy--Riemann equations. It is possible to convert this into the standard Cauchy--Riemann equations to deduce the local behaviour at $\Phi$.
To do so, we write
\[
\Phi = e^{\omega} f.
\]
Then
\begin{align*}
\frac{\partial f}{\partial \bar{z}} &= e^{-\omega} \left(\frac{\partial \Phi}{\partial \bar{z}} - \frac{\partial \omega}{\partial \bar{z}} \Phi\right)\\
&= e^{-\omega} \left(\frac{i(A_i + i A_2)}{2} - \frac{\partial \omega}{\partial \bar{z}}\right)\Phi.
\end{align*}
This is equal to $0$ if $\omega$ satisfies
\[
\frac{\partial \omega}{\partial \bar{z}} = i \frac{A_1 + iA_2}{2}.
\]
So the question is --- can we solve this? It turns out we can always solve this equation, and there is an explicit formula for the solution. In general, if $\beta$ is smooth, then the equation
\[
\frac{\partial w}{\partial \bar{z}} = \beta,
\]
has a smooth solution in the disc $\{z: |z| < r\}$, given by
\[
\omega(z, \bar{z}) = \frac{1}{2\pi i} \int_{|w| < r} \frac{\beta(w)}{w - z}\;\d w \wedge \d \bar{w}.
\]
A proof can be found in the book \emph{Griffiths and Harris} on algebraic geometry, on page 5. So we can write
\[
\Phi = e^{\omega} f
\]
where $f$ is holomorphic. Since $e^{\omega}$ is never zero, we can apply all our knowledge of holomorphic functions to $f$, and deduce that $\Phi$ has isolated zeroes, where $\Phi \sim (z - Z_j)^{n_j}$ for some integer power $n_j$.
\subsubsection*{The Bogomolny equations}
We'll now show that $(A, \Phi)$ satisfies $V_1(A, \Phi) = \pi N \geq 0$ iff it satisfies the Bogomolny equations, i.e.
\[
\D_1 \Phi + i \D_2 \Phi = 0,\quad B = \frac{1}{2} (1 - |\Phi|^2).
\]
We first consider the simpler case of the sine-Gordon equation. As in the $\phi^4$ kinks, to find soliton solutions, we write the energy as
\begin{align*}
E &= \int_{-\infty}^\infty \left(\frac{1}{2} \theta_x^2 + (1 - \cos \theta)\right) \;\d x \\
&= \frac{1}{2} \int_{-\infty}^\infty \left(\theta_x^2 + 4 \sin^2 \frac{\theta}{2}\right) \;\d x\\
&= \frac{1}{2} \int_{-\infty}^\infty \left(\left(\theta_x - 2 \sin \frac{\theta}{2}\right)^2 + 4 \theta_x \sin \frac{\theta}{2}\right)\;\d x\\
&= \frac{1}{2} \int_{-\infty}^\infty \left(\theta_x - 2 \sin \frac{\theta}{2}\right)^2\;\d x + \int_{-\infty}^\infty \frac{\partial}{\partial x}\left(-4 \cos \frac{\theta}{2}\right)\;\d x\\
&= \frac{1}{2}\int_{-\infty}^\infty \left(\theta_x - 2 \sin \frac{\theta}{2}\right)^2\;\d x + \left(-4\cos \frac{\theta(+\infty)}{2} + 4 \cos \frac{\theta(-\infty)}{2}\right).
\end{align*}
We then use the kink asymptotic boundary conditions to obtain, say, $\theta(+\infty) = 2\pi$ and $\theta(-\infty) = 0$. So the boundary terms gives $8$. Thus, subject to these boundary conditions, we can write the sine-Gordon energy as
\[
E = \frac{1}{2}\int_{-\infty}^\infty \left(\theta_x - 2 \sin \frac{\theta}{2}\right)^2\;\d x + 8.
\]
Thus, if we try to minimize the energy, then we know the minimum is at least $8$, and if we could solve the first-order equation $\theta_x = 2 \sin \frac{\theta}{2}$, then the minimum would be exactly $8$. The solution we found does satisfy this first-order equation. Moreover, the solutions are all of the form
\[
\theta(x) = \theta_K(x - X),\quad \theta_K(x) = 4 \arctan e^x.
\]
Thus, we have shown that the minimum energy is $8$, and the minimizers are all of this form, parameterized by $X \in \R$.
We want to do something similar for the Ginzburg--Landau theory. In order to make use of the discussion above of the winding number $N$, we will make the same standing assumptions as used in that discussion, but it is possible to generalize the conclusion of the following result to arbitrary finite energy configurations with an appropriate formulation of the winding number.
\begin{lemma}
We have
\[
V_1(A, \Phi) = \frac{1}{2} \int_{\R^2} \left(\left(B - \frac{1}{2}(1 - |\Phi|^2)\right)^2 + 4 |\bar{\partial}_A \Phi|^2\right)\;\d^2 x + \pi N,
\]
where
\[
\bar{\partial}_A \Phi = \frac{1}{2} (\D_1 \Phi + i \D_2 \Phi).
\]
\end{lemma}
It is clear that the desired result follows from this.
\begin{proof}
We complete the square and obtain
\[
V_1(A, \Phi) = \frac{1}{2} \int \left(\left(B - \frac{1}{2}(1 - |\Phi|^2)\right)^2 + B(1 - |\Phi|^2) + |\D_1\Phi|^2 + |\D_2 \Phi|^2\right)\;\d^2 x.
\]
We now dissect the terms one by one. We first use the definition of $B \;\d x^1 \wedge \d x^2= \d A$ and integration by parts to obtain
\[
\int_{\R^2} (1 - |\Phi|^2) \;\d A = - \int_{\R^2} \d (1 - |\Phi|^2) \wedge A = 2 \int_{\R^2} (\Phi, \D \Phi) \wedge A.
\]
Alternatively, we can explicitly write
\begin{align*}
\int_{\R^2} B(1 - |\Phi|^2) \;\d^2 x &= \int_{\R^2} (\partial_1 A_2 - \partial_2 A_1) (1 - |\Phi|^2)\;\d ^2x\\
&= \int_{\R^2} (A_2 \partial_1 |\Phi|^2 - A_1 \partial_2 |\Phi|^2) \;\d^2 x \\
&= 2\int_{\R^2} A_2 (\Phi, \D_1 \Phi) - A_1 (\Phi, \D_2 \Phi).
\end{align*}
Ultimately, we want to obtain something that looks like $|\bar{\partial}_A \Phi|^2$. We can write this out as
\[
(\D_1 \Phi + i \D_2 \Phi, \D_1 \Phi + i \D_2 \Phi) = |\D_1 \Phi|^2 + |\D_2 \Phi|^2 + 2(\D_1 \Phi, i \D_2 \Phi).
\]
We note that $i\Phi$ and $\Phi$ are always orthogonal, and $A_i$ is always a real coefficient. So we can write
\begin{align*}
(\D_1 \Phi, i \D_2 \Phi) &= (\partial_1 \Phi - i A_1 \Phi, i \partial_2 \Phi + A_2 \Phi) \\
&= (\partial_1 \Phi, i \partial _2 \Phi) + A_2 (\Phi, \partial_1 \Phi) - A_1 (\Phi, \partial_2 \Phi).
\end{align*}
We now use again the fact that $(\Phi, i\Phi) = 0$ to replace the usual derivatives with the covariant derivatives. So we have
\[
(\D_1 \Phi, i \D_2 \Phi) = (\partial_1 \Phi, i \partial_2 \Phi) + A_2 (\Phi, \D_1 \Phi) - A_1 (\Phi, \D_2 \Phi).
\]
This tells us we have
\[
\int \left(B(1 - |\Phi|^2) + |\D_1 \Phi|^2 + |\D_2 \Phi|^2\right)\;\d^2 x = \int \left(4 |\bar{\partial}_A \Phi|^2 + 2 (\partial_1 \Phi, i \partial_2 \Phi)\right)\;\d^2 x.
\]
It then remains to show that $(\partial_1 \Phi, i \partial_2 \Phi) = j^0(\Phi)$. But we just write
% This is just like what we had above! So we get
% \begin{align*}
% V_1(A, \Phi) &= \frac{1}{2} \int \left(\left(B - \frac{1}{2}(1 - |\Phi|^2)\right)^2 + |\D_1 \Phi|^2 + |\D_2 \Phi|^2 + 2 (\D_1 \Phi, i \D_2 \Phi) - 2 (\partial_1 \Phi, i \partial_2 \Phi)\right) \;\d^2 x\\
% &= \frac{1}{2} \int \left(\left(B - \frac{1}{2} (1 - |\Phi|^2)\right)^2 + 4 |\bar{\partial}_A \Phi|^2 + 2 j^0(\Phi)\right)\;\d^2 x,
% \end{align*}
% since
\begin{align*}
(\partial_1 \Phi_1 + i \partial_1 \Phi_2, - \partial_2 \Phi_2 + i \partial_2 \Phi_1) &= - (\partial_1 \Phi_1, \partial_2 \Phi_2) + (\partial_1 \Phi_2, \partial_2 \Phi_1)\\
&= -j^0(\Phi)\\
&= - \det
\begin{pmatrix}
\partial_1 \Phi_1 & \partial_2 \Phi_1\\
\partial_1 \Phi_2 & \partial_2 \Phi_2
\end{pmatrix}
\end{align*}
Then we are done.
\end{proof}
\begin{cor}
For any $(A, \Phi)$ with winding number $N$, we always have $V_1(A, \Phi) \geq \pi N$, and those $(A, \Phi)$ that achieve this bound are exactly those that satisfy
\[
\bar{\partial}_A \Phi = 0,\quad B = \frac{1}{2} (1 - |\Phi|^2).
\]
\end{cor}
\subsubsection*{Reduction to scalar equation}
The remaining part of Taubes' theorem is to prove the existence of solutions to these equations, and that they are classified by $N$ unordered complex numbers. This is the main analytic content of the theorem.
To do so, we reduce the two Bogomolny equations into a scalar equation. Note that we have
\[
\D_1 \Phi + i \D_2 \Phi = (\partial_1 \Phi + i \partial_2 \Phi) - i (A_1 + i A_2) \Phi = 0.
\]
So we can write
\[
A_1 + i A_2 = - i (\partial_1 + i \partial_2) \log \Phi.
\]
Thus, once we've got $\Phi$, we can get $A_1$ and $A_2$.
The next step is to use gauge invariance. Under gauge invariance, we can fix the phase of $\Phi$ to anything we want. We write
\[
\Phi = e^{\frac{1}{2} (u + i \theta)}.
\]
Then $|\Phi|^2 = e^u$.
We might think we can get rid of $\theta$ completely. However, this is not quite true, since the argument is not well-defined at a zero of $\Phi$, and in general we cannot get rid of $\theta$ by a smooth gauge transformation. But since
\[
\Phi \sim c_j (z - Z_j)^{n_j}
\]
near $Z_j$, we expect we can make $\theta$ look like
\[
\theta = 2 \sum_{j = 1}^N \arg (z - Z_j).
\]
We will assume we can indeed do so. Then we have
\[
A_1 = \frac{1}{2}(\partial_2 u + \partial_1 \theta),\quad A_2 = - \frac{1}{2} (\partial_1 u - \partial_2 \theta).
\]
We have now solved for $A$ using the first Bogomolny equation. We then use this to work out $B$ and obtain a scalar equation for $u$ by the second Bogomolny equation.
\begin{thm}
In the above situation, the Bogomolny equation $B = \frac{1}{2} (1 - |\Phi|^2)$ is equivalent to the scalar equation for $u$
\[
-\Delta u + (e^u - 1) = -4\pi \sum_{j = 1}^N \delta_{Z_j}.
\]
This is known as \term{Taubes' equation}.
\end{thm}
\begin{proof}
We have
\begin{align*}
B &= \partial_1 A_2 - \partial_2 A_1 \\
&= -\frac{1}{2} \partial_1^2 u - \frac{1}{2} \partial_2^2 u + \frac{1}{2} (\partial_1 \partial_2 - \partial_2 \partial_1) \theta\\
&= - \frac{1}{2} \Delta u + \frac{1}{2} (\partial_1 \partial_2 - \partial_2 \partial_1) \theta.
\end{align*}
We might think the second term vanishes identically, but that is not true. Our $\theta$ has some singularities, and so that expression is not going to vanish at the singularities. The precise statement is that $(\partial_1 \partial_2 - \partial_2 \partial_1) \theta$ is a distribution supported at the points $Z_j$.
To figure out what it is, we have to integrate:
\begin{align*}
\int_{|z - Z_j| \leq \varepsilon} (\partial_1 \partial_2 - \partial_2 \partial_1) \theta \;\d^2 x &= \int_{|z - Z_j| = \varepsilon} \partial_1 \theta \;\d x^1 + \partial_2 \theta \;\d x^2 \\
&= \oint_{|z - Z_j| = \varepsilon} \;\d \theta = 4\pi n_j,
\end{align*}
where $n_j$ is the multiplicity of the zero. Thus, we deduce that
\[
(\partial_1 \partial_2 - \partial_2 \partial_1) \theta = 2\pi \sum \delta_{Z_j}.
\]
But then we are done!
\end{proof}
We can think of this $u$ as a non-linear combination of fundamental solutions to the Laplacian. Near the $\delta$ functions, the $e^u - 1$ term doesn't contribute much, and the solution looks like the familiar fundamental solutions to the Laplacian with logarithmic singularities. However, far away from the singularities, $e^u - 1$ forces $u$ to tend to $0$, instead of growing to infinity.
Taubes proved that this equation has a unique solution, which is smooth on $\R^2 \setminus \{Z_j\}$, with logarithmic singularities at $Z_j$, and such that $u \to 0$ as $|z| \to \infty$. Also, $u < 0$.
It is an exercise to check that the Bogomolny equations imply the second-order Euler--Lagrange equations.
For example, differentiating the second Bogomolny equation and using the first gives
\[
\partial_1 B = - (\Phi, \D_1 \Phi) = (\Phi, i \D_2 \Phi).
\]
We can similarly do this for the sine-Gordon theory.
\subsection{Physics of vortices}
Recall we began with the ungauged Ginzburg--Landau theory, and realized the solitons didn't have finite energy. We then added a gauge field, and the problem went away --- we argued the coupling to the gauge field ``gave mass'' to the transverse component, thus allowing the existence of finite energy soliton solutions. In the book of Jaffe and Taubes there are results on the exponential decay of gauge invariant combinations of the fields which are another expression of this effect --- the Higgs mechanism. However, there is a useful and complementary way of understanding how gauge fields assist in stabilizing finite energy configurations against collapse, and this doesn't require \emph{any} detailed information about the theory at all --- only scaling. We now consider this technique, which is known either as the Derrick or the Pohozaev argument.
Suppose we work in $d$ space dimensions. Then a general scalar field $\Phi: \R^d \to \R^\ell$ has energy functional given by
\[
\int_{\R^d} \left(\frac{1}{2} |\nabla \Phi|^2 + U(\Phi)\right)\;\d^d x.
\]
for some $U$. In the following we consider smooth finite energy configurations for which the energy is stationary. To be precise, we require that the energy is stationary with respect to variations induced by rescaling of space (as is made explicit in the proof); we just refer to these configurations as stationary points.
\begin{thm}[Derrick's scaling argument]\index{Derrick's scaling argument}
Consider a field theory in $d$-dimensions with energy functional
\[
E[\Phi] = \int_{\R^d}\left(\frac{1}{2} |\nabla \Phi|^2 + U(\Phi)\right)\;\d^d x = T + W,
\]
with $T$ the integral of the gradient term and $W$ the integral of the term involving $U$.
\begin{itemize}
\item If $d = 1$, then any stationary point must satisfy
\[
T = W.
\]
\item If $d = 2$, then all stationary points satisfy $W = 0$.
\item If $d \geq 3$, then all stationary points have $T = W = 0$, i.e.\ $\Phi$ is constant.
\end{itemize}
\end{thm}
This forbids the existence of solitons in high dimensions for this type of energy functional.
\begin{proof}
Suppose $\Phi$ were such a stationary point. Then for any variation $\Phi_\lambda$ of $\Phi$ such that $\Phi = \Phi_1$, we have
\[
\left.\frac{\d}{\d \lambda}\right|_{\lambda = 1}E[\Phi_\lambda] = 0.
\]
Consider the particular variation given by
\[
\Phi_\lambda(\mathbf{x}) = \Phi(\lambda \mathbf{x}).
\]
Then we have
\[
W[\Phi_\lambda] = \int_{\R^d} U(\Phi_\lambda(\mathbf{x})) \;\d^d x = \lambda^{-d} \int_{\R^d} U(\Phi(\lambda \mathbf{x}))\;\d^d (\lambda x) = \lambda^{-d} W[\Phi].
\]
On the other hand, since $T$ contains two derivatives, scaling the derivatives as well gives us
\[
T[\Phi_\lambda] = \lambda^{2 -d} T[\Phi].
\]
Thus, we find
\[
E[\Phi_\lambda] = \lambda^{2 - d}T[\Phi] + \lambda^{-d} W[\Phi].
\]
Differentiating and setting $\lambda = 1$, we see that we need
\[
(2 - d) T[\Phi] - d W[\Phi] = 0.
\]
Then the results in different dimensions follow.
\end{proof}
%We can look at vortices for general $\lambda > 0$. We shall begin by some high-level argument involving \term{Derrick's scaling argument}. For scalar fields $\phi: \R^d \to \R^\ell$, suppose we have an energy functional
%\[
% \int_{\R^d} \left(\frac{1}{2} |\Delta \phi|^2 + U(\phi)\right)\;\d^d x.
%\]
%We assume that the potential $U \geq 0$, and that there exists a finite-energy smooth equilibrium $\phi$. In other words, if $\phi_\lambda$ is a smooth family of field configurations with $\phi_\lambda |_{\lambda = 1} = \phi$, then
%\[
% \left.\frac{\d}{\d \lambda} V(\phi_\lambda) \right|_{\lambda = 1} = 0.
%\]
%We now want to ask if these things exist.
%
%We choose our variation by
%\[
% \phi_\lambda(x) = \phi(\lambda x).
%\]
%We can evaluate
%\begin{align*}
% V(\phi_\lambda) &= \int_{\R^d} \left(\frac{1}{2} \lambda^2|(\Delta \phi)(\lambda x)|^2 + U(\phi_\lambda)\right)\;\d^d x\\
% &= \int_{\R^d} \left(\frac{1}{2} \lambda^2 |\Delta \phi(y)|^2 + U(\phi(y))\right)\;\lambda^{-d} \;\d^d y,
%\end{align*}
%where we substituted in $y = \lambda x$. We write
%\[
% T = \int_{\R^d} \frac{1}{2} |\Delta \phi|^2 \;\d^d x,\quad W = \int_{\R^d} U(\phi)\;\d^d x.
%\]
%Then we find that
%\[
% V(\phi_\lambda) = \lambda^{2 - d} T + \lambda^{-d} W.
%\]
%Here $T$ and $W$ are constants completely determined by the soliton solution, and is in particular independent of $\lambda$. So we find
%\[
% \left.\frac{\d}{\d \lambda} (V(\phi_\lambda))\right|_{\lambda = 1} = (2 - d) T - d W = 0.
%\]
%Now if we want interesting, non-vacuum solutions, then $T$ and $W$ are non-zero.
%
%If $d = 1$, then there is no problem. We just get that we must have
%\[
% T = W.
%\]
%However, if $d \geq 3$, then we immediately get an impossibility, as both coefficients are negative. So this happens only if $T = W = 0$. So $\phi$ is constant.
%
%If $d = 2$, then this is a bit border-line. Then the kinetic energy term doesn't matter. Then we get a solution only if $W = 0$. There are some interesting solitons satisfying this, if the vacuum states are degenerate enough.
The $d = 2$ case is rather interesting. We can still get interesting soliton theories if we have sufficiently large space of classical vacua $\{\Phi: W(\Phi) = 0\}$.
\begin{eg}
In $d = 2$, we can take $\ell = 3$ and
\[
W(\phi) = (1 - |\phi|^2)^2.
\]
Then the set $W = 0$ is given by the unit sphere $S^2 \subseteq \R^3$. With $\phi$ constrained to this $2$-sphere, this is a \term{$\sigma$-model}, and there is a large class of such maps $\phi(x)$ which minimize the energy (for a fixed topology) --- in fact they are just rational functions when stereographic projection is used to introduce complex coordinates.
\end{eg}
Derrick's scaling argument is not only a mathematical trick. We can also interpret it physically. Increasing $\lambda$ corresponds to ``collapsing'' down the field. Then we see that in $d \geq 3$, both the gradient and potential terms favour collapsing of the field. However, in $d = 1$, the gradient term wants the field to expand, and the potential term wants the field to collapse. If these two energies agree, then these forces perfectly balance, and one can hope that stationary solitons exist.
We will eventually want to work with theories in higher dimensions, and Derrick's scaling argument shows that for scalar theories with energy functionals as above this isn't going to be successful for three or more dimensions, and places strong restrictions in two dimensions. There are different ways to get around Derrick's argument. In the Skyrme model, which we are going to study in the next chapter, there are no gauge fields, but instead we will have some higher powers of derivative terms. In particular, by introducing fourth powers of derivatives in the energy density, we will have a term that scales as $\lambda^{4 - d}$, and this acts to stabilize scalar field solitons in three dimensions.
Now let's see how gauge theory provides a way around Derrick's argument without having to depart from employing only energy densities which are quadratic in the derivatives (as is highly desirable for quantization). To understand this, we need to know how gauge fields transform under spatial rescaling.
One way to figure this out is to insist that the covariant derivative $\D_j \Phi_\lambda$ must scale as a whole in the same way that ordinary derivatives scale in scalar field theory. Since
\[
\partial_j \Phi_\lambda = \lambda (\partial_j \Phi)(\lambda x),
\]
we would want $A_j$ to scale as
\[
(A_j)_\lambda(x) = \lambda A_j(\lambda x).
\]
We can also see this more geometrically. Consider the function
\begin{align*}
\chi_\lambda: \R^d &\to \R^d\\
x &\mapsto \lambda x.
\end{align*}
Then our previous transformations were just given by pulling back along $\chi_\lambda$. Since $A$ is a $1$-form, it pulls back as
\[
\chi_\lambda^* (A_j \;\d x^j) = \lambda A_j (\lambda x) \;\d x^j.
\]
Then since $B = \d A$, the curvature must scale as
\[
B_\lambda(x) = \lambda^2 B(\lambda x).
\]
Thus, we can obtain a gauged version of Derrick's scaling argument.
Since we don't want to develop gauge theory in higher dimensions, we will restrict our attention to the Ginzburg--Landau model. Since we already used the letter $\lambda$, we will denote the scaling parameter by $\mu$. We have
\begin{align*}
V_\lambda(A_\mu, \Phi_\mu) &= \frac{1}{2} \int \left(\mu^4 B(\mu x)^2 + \mu^2 |\D \Phi(\mu x)|^2 + \frac{\lambda}{4} (1 - |\Phi(\mu x)|^2)^2\right) \frac{1}{\mu^2}\;\d^2 (\mu x)\\
&= \frac{1}{2} \int \left(\mu^2 B^2(y) + |\D \Phi(y)|^2 + \frac{\lambda}{4 \mu^2} (1 - |\Phi(y)|^2)^2\right)\;\d^2 y.
\end{align*}
Again, the gradient term is scale invariant, but the magnetic field term counteracts the potential term. We can find the derivative to be
\[
\left.\frac{\d}{\d \mu}\right|_{\mu = 1} V_\lambda(A_\mu, \Phi_\mu) = \int \left(B^2 - \frac{\lambda}{4}(1 - |\Phi|^2)^2\right)\;\d^2 y.
\]
Thus, for a soliton, we must have
\[
\int B^2\;\d^2 x = \frac{\lambda}{4} \int_{\R^2} (1 - |\Phi|^2)^2 \;\d^2 x.
\]
Such solutions exist, and we see that this is because they are stabilized by the magnetic field energy in the sense that a collapse of the configuration induced by rescaling would be resisted by the increase of magnetic energy which such a collapse would produce. Note that in the case $\lambda = 1$, this equation is just the integral form of one of the Bogomolny equations!
\subsubsection*{Scattering of vortices}
Derrick's scaling argument suggests that vortices can exist if $\lambda > 0$. However, as we previously discussed, for $\lambda \not= 1$, there are forces between vortices in general, and we don't get static, separated vortices. By doing numerical simulations, we find that when $\lambda < 1$, the vortices attract. When $\lambda > 1$, the vortices repel. Thus, when $\lambda > 1$, the symmetric vortices with $N > 1$ are unstable, as they want to break up into multiple single vortices.
We can talk a bit more about the $\lambda = 1$ case, where we have static multi-vortices. For example, for $N = 2$, the solutions are parametrized by pairs of points in $\C$, up to equivalence
\[
(Z_1, Z_2) \sim (Z_2, Z_1).
\]
We said the moduli space is $\C^2$, and this is indeed true. However, $Z_1$ and $Z_2$ are not good coordinates for this space. Instead, good coordinates on the moduli space $\mathcal{M} = \mathcal{M}_2$ are given by some functions symmetric in $Z_1$ and $Z_2$. One valid choice is
\[
Q = Z_1 + Z_2,\quad P = Z_1 Z_2.
\]
In general, for vortex number $N$, we should use the \term{elementary symmetric polynomials} in $Z_1, \cdots, Z_N$ as our coordinates.
Now suppose we set our vortices in motion. For convenience, we fix the center of mass so that $Q(t) = 0$. We can then write $P$ as
\[
P = - Z_1^2.
\]
If we do some simulations, we find that in a head-on collision, after they collide, the vortices scatter off at $90^\circ$. This is the \term{$90^\circ$ scattering phenomenon}, and holds for other $\lambda$ as well.
In terms of our coordinates, $Z_1^2$ is smoothly evolving from a negative to a positive value, going through $0$. This corresponds to $Z_1 \mapsto \pm i Z_1$, $Z_2 = - Z_1$. Note that at the point when they collide, we lose track of which vortex is which.
Similar to the $\phi^4$ kinks, we can obtain effective Lagrangians for the dynamics of these vortices. However, this is much more complicated.
\subsection{Jackiw--Pi vortices}
So far, we have been thinking about electromagnetism, using the abelian Higgs model. There is a different system that is useful in condensed matter physics. We look at vortices in Chern--Simons--Higgs theory. This has a different Lagrangian which is not Lorentz invariant --- instead of having the Maxwell curvature term, we have the Chern--Simons Lagrangian term. We again work in two space dimensions, with the Lagrangian density given by
\[
\mathcal{L} = \frac{\kappa}{4} \varepsilon^{\mu\nu\lambda} A_\mu F_{\nu\lambda} - (i \Phi, \D_0 \Phi) - \frac{1}{2} | \D \Phi|^2 + \frac{1}{2\kappa} |\Phi|^4,
\]
where $\kappa$ is a constant,
\[
F_{\nu \lambda} = \partial_\nu A_\lambda - \partial_\lambda A_\nu
\]
is the electromagnetic field and, as before,
\[
\D_0 \Phi = \frac{\partial \Phi}{\partial t} - i A_0 \Phi.
\]
The first term is the \term{Chern--Simons term}, while the rest is the Schr\"odinger Lagrangian density with a $|\Phi|^4$ potential term.
Varying with respect to $\Phi$, the Euler--Lagrange equation gives the Schr\"odinger equation
\[
i D_0 \Phi + \frac{1}{2} \D_j^2 \Phi + \frac{1}{\kappa} |\Phi|^2 \Phi = 0.
\]
If we take the variation with respect to $A_0$ instead, then we have
\[
\kappa B + |\Phi|^2 = 0.
\]
This is unusual --- it looks more like a constraint than an evolution equation, and is a characteristic feature of Chern--Simons theories.
The other equations give conditions like
\begin{align*}
\partial_1 A_0 &= \partial_0 A_1 + \frac{1}{\kappa} (i \Phi, \D_2 \Phi)\\
\partial_2 A_0 &= \partial_0 A_2 - \frac{1}{\kappa} (i \Phi, \D_1 \Phi).
\end{align*}
This is peculiar compared to Maxwell theory --- the equations relate the current directly to the electromagnetic field, rather than its derivative.
For static solutions, we need
\[
\partial_i A_0 = \frac{1}{\kappa} \varepsilon_{ij} (i \Phi, \D_j \Phi).
\]
To solve this, we assume the field again satisfies the covariant anti-holomorphicity condition
\[
D_j \Phi = i \varepsilon_{jk} \D_k \Phi = 0.
\]
Then we can write
\[
\partial_1 A_0 = +\frac{1}{\kappa} (i \Phi, \D_2 \Phi) = - \frac{1}{\kappa} \partial_1 \frac{|\Phi|^2}{2},
\]
and similarly for the derivative in the second coordinate direction. We can then integrate these to obtain
\[
A_0 = -\frac{|\Phi|^2}{ 2 \kappa}.
\]
We can then look at the other two equations, and see how we can solve those. For static fields, the Schr\"odinger equation becomes
\[
A_0 \Phi + \frac{1}{2} \D_j^2 \Phi + \frac{1}{\kappa} |\Phi|^2 \Phi = 0.
\]
Substituting in $A_0$, we obtain
\[
\D_j^2 \Phi = -\frac{|\Phi|^2}{\kappa}
\]
Let's then see if this makes sense. We need to see whether this can be consistent with the holomorphicity condition. The answer is yes, if we have the equation $\kappa B + |\Phi|^2 = 0$. We calculate
\[
\D_j^2 \Phi = \D_1^2 \Phi + \D_2^2 \Phi = i (\D_1 \D_2 - \D_2 \D_1) \Phi = + B \Phi = -\frac{1}{\kappa} |\Phi|^2 \Phi,
\]
exactly what we wanted.
So the conclusion (check as an exercise) is that we can generate vortex solutions by solving
\begin{align*}
\D_j \Phi - i \varepsilon_{jk} \D_k \Phi &= 0\\
\kappa B + |\Phi|^2 &= 0.
\end{align*}
As in Taubes' theorem there is a reduction to a scalar equation, which is in this case solvable explicitly:
\[
\Delta \log |\Phi| = -\frac{1}{\kappa} |\Phi|^2.
\]
Setting $\rho = |\Phi|^2$, this becomes Liouville's equation
\[
\Delta \log \rho = -\frac{2}{\kappa}\rho,
\]
which can in fact be solved in terms of rational functions --- see for example Chapter 5 of the book \emph{Solitons in Field Theory and Nonlinear Analysis} by Y. Yang. (As in Taubes' theorem, there is a corresponding statement providing solutions via holomorphic rather than anti-holomorphic conditions.)
\section{Skyrmions}
We now move on to one dimension higher, and study \emph{Skyrmions}. In recent years, there has been a lot of interest in what people call ``Skyrmions'', but what they are studying is a 2-dimensional variant of the original idea of Skyrmions. These occur in certain exotic magnetic systems. But instead, we are going to study the original Skyrmions as discovered by Skyrme, which have applications to nuclear physics.
With details to be filled in soon, hadronic physics exhibits (approximate) spontaneously broken \emph{chiral symmetry} $\frac{\SU(2)_L \times \SU(2)_R}{\Z_2} \cong \SO(4)$, where the unbroken group is (diagonal) $\SO(3)$ isospin, and the elements of $\SO(3)$ are $(g, g) \in \SU(2) \times \SU(2)$.
This symmetry is captured in various effective field theories of pions (which are the approximate Goldstone bosons) and heavier mesons. It is also a feature of QCD with very light $u$ and $d$ quarks.
The special feature of Skyrmion theory is that we describe nucleons as solitons in the effective field theory. Skyrme's original idea was that nucleons and bigger nuclei can be modelled by classical approximations to some ``condensates'' of pion fields. To explain the conservation of baryon number, the classical field equations have soliton solutions (Skyrmions) with an integer topological charge. This topological charge is then identified with what is known, physically, as the baryon number.
\subsection{Skyrme field and its topology}
Before we begin talking about the Skyrme field, we first discuss the symmetry group this theory enjoys. Before symmetry breaking, our theory has a symmetry group
\[
\frac{\SU(2) \times \SU(2)}{\{\pm (\mathbf{1}, \mathbf{1})\}} = \frac{\SU(2) \times \SU(2)}{\Z_2}.
\]
This might look like a rather odd symmetry group to work with. We can begin by understanding the $\SU(2) \times \SU(2)$ part of the symmetry group. This group acts naturally on $\SU(2)$ again, by
\[
(A, B) \cdot U = AUB^{-1}.
\]
However, we notice that the pair $(-\mathbf{1}, -\mathbf{1}) \in \SU(2) \times \SU(2)$ acts trivially. So the true symmetry group is the quotient by the subgroup generated by this element. One can check that after this quotienting, the action is faithful.
In the Skyrme model, the field will be valued in $\SU(2)$. It is convenient to introduce coordinates for our Skyrme field\index{Skyrme field}. As usual, we let $\boldsymbol\tau$ be the Pauli matrices, and $\mathbf{1}$ be the unit matrix. Then we can write the Skyrme field as
\[
U(\mathbf{x}, t) = \sigma(\mathbf{x}, t) \mathbf{1} + i \boldsymbol\pi(x, t) \cdot \boldsymbol\tau.
\]
However, the values of $\sigma$ and $\boldsymbol\pi$ cannot be arbitrary. For $U$ to actually lie in $\SU(2)$, we need the coefficients to satisfy
\[
\sigma, \pi_i \in \R,\quad \sigma^2 + \boldsymbol\pi \cdot \boldsymbol\pi = 1.
\]
This is a \emph{non-linear} constraint, and defines what known as a non-linear \term{$\sigma$-model}\index{non-linear $\alpha$-model}.
From this constraint, we see that geometrically, we can identify $\SU(2)$ with $S^3$. We can also see this directly, by writing
\[
\SU(2) =\left\{
\begin{pmatrix}
\alpha & \beta\\
-\bar{\beta} & \bar{\alpha}
\end{pmatrix} \in M_2(\C): |\alpha|^2 + |\beta|^2 = 1\right\},
\]
and this gives a natural embedding of $\SU(2)$ into $\C^2 \cong \R^4$ as the unit sphere, by sending the matrix to $(\alpha, \beta)$.
One can check that the action we wrote down acts by isometries of the induced metric on $S^3$. Thus, we obtain an inclusion
\[
\frac{\SU(2) \times \SU(2)}{\Z_2} \hookrightarrow \SO(4),
\]
which happens to be a surjection.
Our theory will undergo spontaneous symmetry breaking, and the canonical choice of vacuum will be $U = \mathbf{1}$. Equivalently, this is when $\sigma = 1$ and $\boldsymbol\pi = \mathbf{0}$. We see that the stabilizer of $\mathbf{1}$ is given by the diagonal
\[
\Delta : \SU(2) \rightarrow \frac{\SU(2) \times \SU(2)}{\Z_2},
\]
since $A\mathbf{1}B^{-1} = \mathbf{1}$ if and only if $A = B$.
Geometrically, if we view $\frac{\SU(2) \times \SU(2)}{\Z_2} \cong \SO(4)$, then it is obvious that the stabilizer of $\mathbf{1}$ is the copy of $\SO(3) \subseteq \SO(4)$ that fixes the $\mathbf{1}$ axis. Indeed, the image of the diagonal $\Delta$ is $\SU(2)/\{\pm \mathbf{1}\} \cong \SO(3)$.
%Note that $\SU(2) \times \SU(2)$ acts naturally on $\SU(2)$ itself, by left and right multiplication. Since we want to capture this symmetry, we will construct the Skyrme fields as $\SU(2)$-valued matrices.
%
%We let $\boldsymbol\tau$ be the Pauli matrices. Then we can write the Skyrme field as
%\[
% U(\mathbf{x}, t) = \sigma(\mathbf{x}, t) \mathbf{1}_2 + i \boldsymbol\pi(x, t) \cdot \boldsymbol\tau.
%\]
%For this to lie in $\SU(2)$, we need the coefficients to satisfy
%\[
% \sigma, \pi_i \in \R,\quad \sigma^2 + \boldsymbol\pi \cdot \boldsymbol\pi = 1.
%\]
%This is a \emph{non-linear} constraint. This condition is a sum of four squares. So geometrically, we are talking about the unit three sphere $S^3 \subseteq \R^4$. Indeed, $\SU(2)$ is isomorphic to the $3$-sphere. This is known as a \term{$\sigma$-model}.
%
%Since we want a $\SU(2) \times \SU(2)$ symmetry, our Lagrangian has to be invariant under left and right multiplication of $\SU(2)$. Under the identification of $\SU(2) \cong \S^3$ geometrically, this is just the usual $\SO(4)$ action on $S^3$. If we want this $\SO(4)$ to be our isometry group, then there is a natural Riemannian metric to put on the sphere, namely the metric induced by $\R^4$.
Note that in our theory, for any choice of $\boldsymbol\pi$, there are at most two possible choices of $\sigma$. Thus, despite there being four variables, there are only three degrees of freedom. Geometrically, this is saying that $\SU(2) \cong S^3$ is a three-dimensional manifold.
This has some physical significance. We are using the $\boldsymbol\pi$ fields to model pions. We have seen and observed pions a lot. We know they exist. However, as far as we can tell, there is no ``$\sigma$ meson'', and this can be explained by the fact that $\sigma$ isn't a genuine degree of freedom in our theory.
Let's now try to build a Lagrangian for our field $U$. We will want to introduce derivative terms. From a mathematical point of view, the quantity $\partial_\mu U$ isn't a very nice thing to work with. It is a quantity that lives in the tangent space $T_U \SU(2)$, and in particular, the space it lives in depends on the value of $U$.
What we want to do is to pull this back to $T_\mathbf{1} \SU(2) = \su(2)$. To do so, we multiply by $U^{-1}$. We write
\[
R_\mu = (\partial_\mu U)U^{-1},
\]
which is known as the \term{right current}. For practical, computational purposes, it is convenient to note that
\[
U^{-1} = \sigma \mathbf{1} - i \boldsymbol\pi \cdot \boldsymbol\tau.
\]
Using the $(+, -, -, -)$ metric signature, we can now write the Skyrme Lagrangian density as
\[
\mathcal{L} =-\frac{F_{\pi}^2 }{16} \Tr \left( R_\mu R^\mu\right) +\frac{1}{32e^2}\Tr\left([R_\mu, R_\nu][R^\mu, R^\nu]\right) - \frac{1}{8} F_\pi^2 m_\pi^2 \Tr(\mathbf{1} - U).
\]
The three terms are referred to as the $\sigma$-model term, Skyrme term and pion mass term respectively.
The first term is the obvious kinetic term. The second term might seem a bit mysterious, but we \emph{must} have it (or some variant of it). By Derrick's scaling argument, we cannot have solitons if we just have the first term. We need a higher multiple of the derivative term to make solitons feasible.
There are really only two possible terms with four derivatives. The alternative is to have the square of the first term. However, Skyrme rejected that object, because that Lagrangian will have four time derivatives. From a classical point of view, this is nasty, because to specify the initial configuration, not only do we need the initial field condition, but also its first three derivatives. This doesn't happen in our theory, because the commutator vanishes when $\mu = \nu$. The pieces of the Skyrme term are thus at most quadratic in time derivatives.
Now note that the first two terms have an exact chiral symmetry, i.e.\ they are invariant under the $\SO(4)$ action previously described. In the absence of the final term, this symmetry would be spontaneously broken by a choice of vacuum. As described before, there is a conventional choice $U = \mathbf{1}$. After this spontaneous symmetry breaking, we are left with an isospin $\SU(2)$ symmetry\index{isospin symmetry}. This isospin symmetry rotates the $\boldsymbol\pi$ fields among themselves.
The role of the extra term is that now the vacuum \emph{has} to be the identity matrix. The symmetry is now \emph{explicitly broken}. This might not be immediately obvious, but this is because the pion mass term is linear in $\sigma$ and is minimized when $\sigma = 1$. Note that this theory is still invariant under the isospin $\SU(2)$ symmetry. Since the isospin symmetry is not broken, all pions have the same mass. In reality, the pion masses are very close, but not exactly equal, and we can attribute the difference in mass as due to electromagnetic effects. In terms of the $\boldsymbol\pi$ fields we defined, the physical pions are given by
\[
\pi^{\pm} = \pi_1 \pm i \pi_2,\quad \pi^0 = \pi_3.
\]
It is convenient to draw the target space $\SU(2)$ as
\begin{center}
\begin{tikzpicture}[scale=1.5]
\draw circle [radius=1];
\node [circ] at (0, -1) {};
\node [below] at (0, -1) {$\sigma = 1$};
\node [circ] at (0, 1) {};
\node [above] at (0, 1) {$\sigma = -1$};
\draw [->] (2.5, -0.7) -- (2.5, 0.7) node [pos=0.5, right, align=left] {Potential\\ energy};
\draw [dashed] (1, 0) arc(0:180:1 and 0.3);
\draw (1, 0) arc(0:-180:1 and 0.3);
\node [right, align=left] at (1, 0) {$\sigma = 0$\\ $|\boldsymbol\pi| = 1$};
\node at (-0.707, 0.707) [anchor = south east] {$S^3 \cong \SU(2)$};
\end{tikzpicture}
\end{center}
$\sigma = -1$ is the \emph{anti-vacuum}. We will see that in the core of the Skyrmion, $\sigma$ will take value $\sigma = -1$.
In the Skyrme Lagrangian, we have three free parameters. This is rather few for an effective field theory, but we can reduce the number further by picking appropriate coefficients. We introduce an energy unit $\frac{F_\pi}{4e}$ and length unit $\frac{2}{eF_\pi}$. Setting these to $1$, there is one parameter left, which is the dimensionless pion mass. In these units, we have
\[
L = \int \left(-\frac{1}{2}\Tr(R_\mu R^\mu) + \frac{1}{16} \Tr([R_\mu, R_\nu][R^\mu, R^\nu]) - m^2 \Tr(\mathbf{1} - U)\right)\;\d^3 x.
\]
In this notation, we have
\[
m = \frac{2m_\pi}{e F_\pi}.
\]
In general, we will think of $m$ as being ``small''.
Let's see what happens if we in fact put $m = 0$. In this case, the lack of mass term means we no longer have the boundary condition that $U \to 1$ at infinity. Hence, we need to manually impose this condition.
Deriving the Euler--Lagrange equations is slightly messy, since we have to vary $U$ while staying inside the group $\SU(2)$. Thus, we vary $U$ multiplicatively,
\[
U \mapsto U (\mathbf{1} + \varepsilon V)
\]
for some $V \in \su(2)$. We then have to figure out how $R$ varies, do some differentiation, and then the Euler--Lagrange equations turn out to be
\[
\partial_\mu \left(R^\mu + \frac{1}{4} [R^\nu, [R_\nu, R^\mu]]\right) = 0.
\]
For static fields, the energy is given by
\[
E = \int \left(-\frac{1}{2} \Tr(R_i R_i) - \frac{1}{16} \Tr([R_i, R_j] [R_i, R_j])\right)\;\d^3 x \equiv E_2 + E_4.
\]
where we sum $i$ and $j$ from $1$ to $3$. This is a sum of two terms --- the first is quadratic in derivatives while the second is quartic.
Note that the trace functional on $\su(2)$ is negative definite. So the energy is actually positive.
We can again run Derrick's theorem.
\begin{thm}[Derrick's theorem]
We have $E_2 = E_4$ for any finite-energy static solution for $m = 0$ Skyrmions.
\end{thm}
\begin{proof}
Suppose $U(\mathbf{x})$ minimizes $E = E_2 + E_4$. We rescale this solution, and define
\[
\tilde{U}(\mathbf{x}) = U(\lambda \mathbf{x}).
\]
Since $U$ is a solution, the energy is stationary with respect to $\lambda$ at $\lambda = 1$.
We can take the derivative of this and obtain
\[
\partial_i \tilde{U}(\mathbf{x}) = \lambda \tilde{U}(\lambda \mathbf{x}).
\]
Therefore we find
\[
\tilde{R}_i(\mathbf{x}) = \lambda R_i(\lambda \mathbf{x}),
\]
and therefore
\begin{align*}
\tilde{E}_2 &= - \frac{1}{2} \int \Tr(\tilde{R}_i \tilde{R}_i) \;\d^3 x\\
&= - \frac{1}{2} \lambda^2 \int \Tr(R_i R_i)(\lambda \mathbf{x})\;\d^3 x\\
&= - \frac{1}{2} \frac{1}{\lambda} \int \Tr(R_i R_i) (\lambda \mathbf{x})\;\d^3 (\lambda x)\\
&= \frac{1}{\lambda} E_2.
\end{align*}
Similarly,
\[
\tilde{E}_4 = \lambda E_4.
\]
So we find that
\[
\tilde{E} = \frac{1}{\lambda} E_2 + \lambda E_4.
\]
But this function has to have a minimum at $\lambda = 1$. So the derivative with respect to $\lambda$ must vanish at $1$, requiring
\[
0 = \frac{\d \tilde{E}}{\d \lambda} = - \frac{1}{\lambda^2}E_2 + E_4 = 0
\]
at $\lambda = 1$. Thus we must have $E_4 = E_2$.
\end{proof}
We see that we must have a four-derivative term in order to stabilize the soliton. If we have the mass term as well, then the argument is slightly more complicated, and we get some more complicated relation between the energies.
% Note that the Skyrmions, which we will find later, have a definite scale size of order $1$.
\subsubsection*{Baryon number}
Recall that our field is a function $U: \R^3 \to \SU(2)$. Since we have a boundary condition $U = \mathbf{1}$ at infinity, we can imagine compactifying $\R^3$ into $S^3$, where the point at infinity is sent to $\mathbf{1}$.
On the other hand, we know that $\SU(2)$ is isomorphic to $S^3$. Geometrically, we think of the space and $\SU(2)$ as ``different'' $S^3$. We should think of $\SU(2)$ as the ``unit sphere'', and write it as $S^3_1$. However, we can think of the compactification of $\R^3$ as a sphere with ``infinite radius'', so we denote it as $S^3_\infty$. Of course, topologically, they are the same.
So the field is a map
\[
U: S^3_\infty \to S^3_1.
\]
This map has a degree. There are many ways we can think about the degree. For example, we can think of this as the homotopy class of this map, which is an element of $\pi_3(S^3) \cong \Z$. Equivalently, we can think of it as the map induced on the homology or cohomology of $S^3$.
While $U$ evolves with time, because the degree is a discrete quantity, it has to be independent of time (alternatively, the degree of the map is homotopy invariant).
There is an explicit integral formula for the degree. We will not derive this, but it is
\[
B = - \frac{1}{24\pi^2} \int \varepsilon_{ijk} \Tr(R_i R_j R_k) \;\d^3 x.
\]
The factor of $2\pi^2$ comes from the volume of the three sphere, and there is also a factor of $6$ coming from how we anti-symmetrize. We identify $B$ with the conserved, physical baryon number.
If we were to derive this, then we would have to pull back a normalized volume form from $S_1^3$ and then integrate over all space. In this formula, we chose to use the $\SO(4)$-invariant volume form, but in general, we can pull back any normalized volume form.
Locally, near $\sigma = 1$, this volume form is actually
\[
\frac{1}{2\pi^2}\;\d \pi^1 \wedge\d \pi^2 \wedge\d \pi^3.
\]
\subsubsection*{Faddeev--Bogomolny bound}
There is a nice result analogous to the Bogomolny energy bound we saw for kinks and vortices, known as the \term{Faddeev--Bogomolny bound}. We can write the static energy as
\[
E = \int - \frac{1}{2} \Tr\left(\left(R_i \mp \frac{1}{4} \varepsilon_{ijk}[R_j, R_k]\right)^2\right)\;\d^3 x \pm 12 \pi^2 B.
\]
This bound is true for both sign choices. However, to get the strongest result, we should choose the sign such that $\pm 12 \pi^2 B > 0$. Then we find
\[
E \geq 12 \pi^2 |B|.
\]
By symmetry, it suffices to consider the case $B > 0$, which is what we will do.
The Bogomolny equation for $B > 0$ should be
\[
R_i - \frac{1}{4} \varepsilon_{ijk} [R_j, R_k] = 0.
\]
However, it turns out this equation has \emph{no} non-vacuum solution.
Roughly, the argument goes as follows --- by careful inspection, for this to vanish, whenever $R_i$ is non-zero, the three vectors $R_1, R_2, R_3$ must form an orthonormal frame in $\su(2)$. So $U$ must be an isometry. But this isn't possible, because the spheres have ``different radii''.
Therefore, true Skyrmions with $B > 0$ satisfy
\[
E > 12 \pi^2 B.
\]
We get a lower bound, but the actual energy is always greater than this lower bound. It is quite interesting to look at the energies of true solutions numerically, and their energy is indeed bigger.
\subsection{Skyrmion solutions}
The simplest Skyrmion solution has baryon number $B = 1$. We will continue to set $m = 0$.
\subsubsection*{$B = 1$ hedgehog Skyrmion}
Consider the spherically symmetric function
\[
U(\mathbf{x}) = \cos f(r) \mathbf{1} + i \sin f(r) \hat{\mathbf{x}} \cdot \boldsymbol\tau.
\]
This is manifestly in $\SU(2)$, because $\cos^2 f + \sin^2 f = 1$. This is known as a hedgehog, because the unit pion field is $\hat{\mathbf{x}}$, which points radially outwards. We need some boundary conditions. We need $U \to \mathbf{1}$ at $\infty$. On the other hand, we will see that we need $U \to -\mathbf{1}$ at the origin to get baryon number $1$. So $f \to \pi$ as $r \to 0$, and $f \to 0$ as $r \to \infty$. So it looks roughly like this:
\begin{center}
\begin{tikzpicture}
\draw [->] (0, 0) -- (5, 0) node [right] {$r$};
\draw [->] (0, 0) -- (0, 3) node [above] {$f$};
\draw [mblue, thick] (0, 2) .. controls (1, 1) and (3, 0.2) .. (5, 0.2);
\node [circ] at (0, 2) {};
\node [left] at (0, 2) {$\pi$};
\end{tikzpicture}
\end{center}
After some hard work, we find that the energy is given by
\[
E = 4\pi \int_0^\infty \left(f'^2 + \frac{2 \sin^2 f}{r^2}(1 + f'^2) + \frac{\sin^4 f}{r^4}\right) r^2\;\d r.
\]
From this, we can obtain a second-order ordinary differential equation in $f$, which is not simple. Solutions have to be found numerically. This is a sad truth about Skyrmions. Even in the simplest $B = 1, m = 0$ case, we don't have an analytic expression for what $f$ looks like. Numerically, the energy is given by
\[
E = 1.232 \times 12\pi^2.
\]
To compute the baryon number of this solution, we plug our solution into the formula, and obtain
\[
B = - \frac{1}{2\pi^2} \int_0^\infty \frac{\sin^2 f}{r^2} \frac{\d f}{\d r} \cdot 4\pi r^2 \;\d r.
\]
We can interpret $\frac{\d f}{\d r}$ as the radial contribution to $B$, while there are two factors of $\frac{\sin f}{r}$ coming from the angular contribution due to the $i \sin f(r) \hat{\mathbf{x}} \cdot \boldsymbol\tau$ term.
But this integral is just an exact differential. It simplifies to
\[
B = \frac{1}{\pi} \int_0^\pi 2 \sin^2 f \;\d f.
\]
Note that we have lost a sign, because of the change of limits. We can integrate this directly, and just get
\[
B = \frac{1}{\pi}\left(f - \frac{1}{2} \sin 2f\right)^\pi_0 = 1,
\]
as promised.
Intuitively, we see that in this integral, $f$ goes from $0$ to $\pi$, and we can think of this as the field $U$ wrapping around the sphere $S^3$ once.
\subsubsection*{More hedgehogs}
We can consider a more general hedgehog with the same ansatz, but with the boundary conditions
\[
f(0) = n\pi,\quad f(\infty) = 0.
\]
In this case, the same computations gives us $B = n$. So in principle, this gives a Skyrmion of any baryon number. The solutions do exist. However, it turns out they have extremely high energy, and are nowhere near minimizing the energy. In fact, the energy increases much faster than $n$ itself, because the Skyrmion ``onion'' structure highly distorts each $B = 1$ Skyrmion. Unsurprisingly, these solutions are unstable.
This is not what we want in hadronic physics, where we expect the energy to scale approximately linearly with $n$. In fact, since baryons attract, we expect the solution for $B = n$ to have less energy than $n$ times the $B = 1$ energy.
We can easily get energies approximately $n$ times the $B = 1$ energy simply by having very separated baryons, and then since they attract, when they move towards each other, we get even lower energies.
% Finding Skyrmions is hard. Up to today, we only have explicit Skyrmions up to $B \approx 25$, and we don't know how the picture looks like in general.
\subsubsection*{A better strategy --- rational map approximation}
So far, we have been looking at solutions that depend very simply on angle. This means, all the ``winding'' happens in the radial direction. In fact, it is a better idea to wind more in the \emph{angular} direction.
In the case of the $B = 1$ hedgehog, the field looks roughly like this:
\begin{center}
\begin{tikzpicture}
\fill [morange, opacity=0.25] (0, -2) rectangle (4, 2);
\foreach \r in {0.5, 1, 1.5} {
\fill [morange, opacity=0.3] (2, 0) circle [radius=\r];
}
\foreach \r in {0.5, 1, 1.5} {
\draw [opacity=0.9](2, 0) circle [radius=\r];
}
\draw [thick] (0, -2) rectangle (4, 2);
\node [right] at (4, 1.8) {$\infty$};
\draw (9, 0) circle [radius=2];
\fill [morange, opacity=0.25] (9, 0) circle [radius=2];
\foreach \y in {-1.3, -0.2, 0.9} {
\pgfmathsetmacro\len{sqrt(4 - (\y)^2)};
\pgfmathsetmacro\ht{\len * 0.1};
\pgfmathsetmacro\x{\len + 9};
\pgfmathsetmacro\sang{270 - acos(-\y/2)};
\pgfmathsetmacro\eang{270 + acos(-\y/2)};
\fill [morange, opacity=0.05] (\x, \y) arc(0:180:{\len} and \ht) arc(\sang:\eang:2);
\fill [morange, opacity=0.25] (\x, \y) arc(0:-180:{\len} and \ht) arc(\sang:\eang:2);
}
\foreach \y in {-1.3, -0.2, 0.9} {
\pgfmathsetmacro\len{sqrt(4 - (\y)^2)};
\pgfmathsetmacro\ht{\len * 0.1};
\pgfmathsetmacro\x{\len + 9};
\draw [dashed, opacity=0.9] (\x, \y) arc(0:180:{\len} and \ht);
\draw [opacity=0.9] (\x, \y) arc(0:-180:{\len} and \ht);
}
\node [circ] at (2, 0) {};
\draw [opacity=0.5] (2, 0) edge [bend right, ->] (9, -2);
\draw [opacity=0.5] (2.5, 0) edge [bend right=20, ->] (8.8, -1.44);
\draw [opacity=0.5] (3, 0) edge [bend right=10, ->] (8.6, -0.39);
\draw [opacity=0.5] (3.5, 0) edge [bend right=5, ->] (8.6, 0.75);
\draw [opacity=0.5] (4, 1.4) edge [bend left=10, ->] (9, 2);
\node [circ] at (9, 2) {};
\node [above] at (9, 2) {$\sigma = 1$};
\node [circ] at (9, -2) {};
\node [below] at (9, -2) {$\sigma = -1$};
\end{tikzpicture}
\end{center}
In our $B > 1$ spherically symmetric hedgehogs, we wrapped radially around the sphere many times, and it turns out that was not a good idea.
Better is to use a similar radial configuration as the $B = 1$ hedgehog, but introduce more angular twists. We can think of the above $B = 1$ solution as follows --- we slice up our domain $\R^3$ (or rather, $S^3$ since we include the point at infinity) into $2$-spheres of constant radius, say
\[
S^3 = \bigcup_{r \in [0, \infty]} S_r^2.
\]
On the other hand, we can also slice up the $S^3$ in the codomain into constant $\sigma$ levels, which are also $2$-spheres:
\[
S^3 = \bigcup_{\sigma} S_\sigma^2.
\]
Then the function $f(r)$ we had tells us we should map the $2$-sphere $S_r^2$ into the two sphere $S_{\cos f(r)}^2$. What we did, essentially, was that we chose to map $S_r^2$ to $S_{\cos f(r)}^2$ via the ``identity map''. This gave a spherically symmetric hedgehog solution.
But we don't have to do this! Pick any function $R: S^2 \to S^2$. Then we can construct the map $\Sigma_{\cos f} R$ that sends $S_r^2$ to $S_{\cos f(r)}^2$ via the map $R$. For simplicity, we will use the same $R$ for all $r$. If we did this, then we obtain a non-trivial map $\Sigma_{\cos f} R: S^3 \to S^3$.
Since $R$ itself is a map from a sphere to a sphere (but one dimension lower), $R$ also has a degree. It turns out this degree is the same as the degree of the induced map $\Sigma_{\cos f} R$! So to produce higher baryon number hedgehogs, we simply have to find maps $R: S^2 \to S^2$ of higher degree.
Fortunately, this is easier than maps between $3$-spheres, because a $2$-sphere is a Riemann surface. We can then use complex coordinates to work on $2$-spheres. By complex analysis, any complex holomorphic map between $2$-spheres is given by a \term{rational map}.
Pick any rational function $R_k(z)$ of degree $k$. This is a map $S^2 \to S^2$. We use coordinates $r, z$, where $r \in \R^+$ and $z \in \C_\infty = \C \cup \{\infty\} \cong S^2$. Then we can look at generalized hedgehogs
\[
U(\mathbf{x}) = \cos f(r) \mathbf{1} + i \sin f(r) \hat{\mathbf{n}}_{R_k(z)} \cdot \boldsymbol\tau,
\]
with $f(0) = \pi$, $f(\infty) = 0$, and $\hat{\mathbf{n}}_{R_k}$ is the normalized pion field $\hat{\boldsymbol\pi}$, given by the unit vector obtained from $R_k(z)$ if we view $S^2$ as a subset of $\R^3$ in the usual way. Explicitly,
\[
\hat{\mathbf{n}}_R = \frac{1}{1 + |R|^2} (\bar{R} + R, i (\bar{R} - R), 1 - |R|^2).
\]
This construction is in some sense a separation of variables, where we separate the radial and angular dependence of the field.
Note that even if we find a minimum among this class of fields, it is not a true minimum energy Skyrmion. However, it gets quite close, and is much better than our previous attempt.
There is quite a lot of freedom in this construction, since we are free to pick $f(r)$, as well as the rational function $R_k(z)$. The geometric degree $k$ of $R_k(z)$ we care about is the same as the algebraic degree\index{degree!algebraic}\index{degree!rational function} of $R_k(z)$. Precisely, if we write
\[
R_k(z) = \frac{p_k(z)}{q_k(z)},
\]
where $p_k$ and $q_k$ are coprime, then the algebraic degree of $R_k$ is the maximum of the degrees of $p_k$ and $q_k$ as polynomials. Since there are finitely many coefficients for $p_k$ and $q_k$, this is a finite-dimensional problem, which is much easier than solving for arbitrary functions. We will talk more about the degree later.
Numerically, we find that minimal energy fields are obtained with
\begin{align*}
R_1(z) &= z & R_2(z) &= z^2\\
R_3(z) &= \frac{\sqrt{3}i z^2 - 1}{z^3 - \sqrt{3}i z} & R_4(z) &= \frac{z^4 + 2\sqrt{3} i z^2 + 1}{z^4 - 2\sqrt{3} i z^2 + 1}.
\end{align*}
The true minimal energy Skyrmions have also been found numerically, and are very similar to the optimal rational map fields. In fact, the search for the true minima often starts from a rational map field.
\begin{center}
\includegraphics[width=\textwidth]{images/B1-8-Skyrmions.pdf}
{Constant energy density surfaces of Skyrmions up to baryon number 8 \\(for $m = 0$), by R.\ A.\ Battye and P.\ M.\ Sutcliffe}
\end{center}
We observe that
\begin{itemize}
\item for $n = 1$, we recover the hedgehog solution.
\item for $n = 2$, our solution has an axial symmetry.
\item for $n = 3$, the solution might seem rather strange, but it is in fact the unique solution in degree $3$ with \emph{tetrahedral} symmetry.
\item for $n = 4$, the solution has cubic symmetry.
\end{itemize}
In each case, these are the unique rational maps with such high symmetry. It turns out, even though these are not the exact Skyrmion solutions, the exact solutions enjoy the same symmetries.
The function $f$ can be found numerically, and depends on $B$.
Geometrically, what we are doing is that we are viewing $S^3$ as the \emph{suspension} $\Sigma S^2$, and our construction of $\Sigma_{\cos f} R$ from $R$ is just the suspension of maps. The fact that degree is preserved by suspension can be viewed as an example of the fact that homology is \emph{stable}.
\subsubsection*{More on rational maps}
Why does the algebraic degree of $R_k$ agree with the geometric degree? One way of characterizing the geometric degree is by counting pre-images. Consider a generic point $c$ in the target $2$-sphere, and consider the equation
\[
R_k(z) = \frac{p_k(z)}{q_k(z)} = c
\]
for a generic $c$. We can rearrange this to say
\[
p_k - c q_k = 0.
\]
For a \emph{generic} $c$, the $z^k$ terms do not cancel, so this is a polynomial equation of degree $k$. Also, generically, this equation doesn't have repeated roots, so has exactly $k$ solutions. So the number of points in the pre-image of a generic $c$ is $k$.
Because $R_k$ is a holomorphic map, it is automatically orientation preserving (and in fact conformal). So each of these $k$ points contributes $+1$ to the degree.
In the pictures above, we saw that the Skyrmions have some ``hollow polyhedral'' structures. How can we understand these?
The holes turn out to be zeroes of the baryon density, and are where energy density is small. At the center of the holes, the angular derivatives of the Skyrme field $U$ are zero, but the radial derivative is not. % This explains both these points. Using expressions for the baryon density - baryon density is product of two things while energy is sum of squares?
We can find these holes precisely in the rational map approximation. This allows us to find the symmetry of the system. They occur where the derivative $\frac{\d R_k}{\d z} = 0$. Since $R = \frac{p}{q}$, we can rewrite this requirement as
\[
W(z) = p'(z) q(z) - q'(z) p(z) = 0.
\]
$W(z)$ is known as the \term{Wronskian}.
A quick algebraic manipulation shows that $W$ has degree at most $2k - 2$, and generically, it is indeed $2k - 2$. This degree is the number of holes in the Skyrmion.
We can look at our examples and look at the pictures to see this is indeed the case.
\begin{eg}
For
\[
R_4(z) = \frac{z^4 + 2\sqrt{3} i z^2 + 1}{z^4 - 2\sqrt{3} i z^2 + 1},
\]
the Wronskian is
\[
W(z) = (4z^3 + 4 \sqrt{3}i z)(z^4 - 2\sqrt{3} i z^2 + 1) - (z^4 + 2 \sqrt{3} i z^2 + 1) (4z^3 - 4 \sqrt{3} i z).
\]
The highest degree $z^7$ terms cancel. But there isn't any $z^6$ term anywhere either. Thus $W$ turns out to be a degree $5$ polynomial. It is given by
\[
W(z) = - 8 \sqrt{3}i (z^5 - z).
\]
We can easily list the roots --- they are $z = 0, 1, i, -1, -i$.
Generically, we expect there to be $6$ roots. It turns out the Wronksian has a zero at $\infty$ as well. To see this more rigorously, we can rotate the Riemann sphere a bit by a M\"obius map, and then see there are $6$ finite roots. Looking back at our previous picture, there are indeed $6$ holes when $B = 4$.
\end{eg}
% How about energy? We can plot the energies of the energies of the Skyrmions as the baryon number increases:
% insert plot
It turns out although the rational map approximation is a good way to find solutions for $B$ up to $\approx 20$, they are hollow with $U = -1$ at the center. This is not a good model for larger nuclei, especially when we introduce non-zero pion mass.
For $m \approx 1$, there are better, less hollow Skyrmions when $B \geq 8$.
\subsection{Other Skyrmion structures}
There are other ways of trying to get Skyrmion solutions.
\subsubsection*{Product Ansatz}
Suppose $U_1(\mathbf{x})$ and $U_2(\mathbf{x})$ are Skyrmions of baryon numbers $B_1$ and $B_2$. Since the target space is a group $\SU(2)$, we can take the product
\[
U(\mathbf{x}) = U_1(\mathbf{x}) U_2(\mathbf{x}).
\]
Then the baryon number is $B = B_1 + B_2$. To see this, we can consider the product when $U_1$ and $U_2$ are well-separated, i.e.\ consider $U(\mathbf{x} - \mathbf{a}) U_2(\mathbf{x})$ with $|\mathbf{a}|$ large. Then we can see the baryon number easily because the baryons are well-separated. We can then vary $\mathbf{a}$ continuously to $0$, and $B$ doesn't change as we make this continuous deformation. So we are done. Alternatively, this follows from an Eckmann--Hilton argument.
This can help us find Skyrmions with baryon number $B$ starting with $B$ well-separated $B = 1$ hedgehogs. Of course, this will not be energy-minimizing, but we can numerically improve the field by letting the separation vary.
It turns out this is not a good way to find Skyrmions. In general, it doesn't give good approximations to the Skyrmion solutions. They tend to lack the desired symmetry, and this boils down to the problem that the product is not commutative, i.e.\ $U_1 U_2 \not= U_2 U_1$. Thus, we cannot expect to be able to approximate symmetric things with a product ansatz.
The product ansatz can also be used for several $B = 4$ subunits to construct configurations with baryon number $4n$ for $n \in \Z$. For example, the following is a $B = 31$ Skyrmion:
\begin{center}
\includegraphics[clip, width=0.3\textwidth, trim=4cm 4cm 4cm 4cm]{images/B-31-Skyrmion.pdf}
B = 31 Skyrmion by P.\ H.\ C.\ Lau and N.\ S.\ Manton
\end{center}
This is obtained by putting eight $B = 4$ Skyrmions side by side, and then cutting off a corner.
This strategy tends to work quite well. With this idea, we can in fact find Skyrmion solutions with baryon number infinity! We can form an infinite cubic crystal out of $B = 4$ subunits. For $m = 0$, the energy per baryon is $\approx 1.038 \times 12 \pi^2$. This is a very close to the lower bound!
We can also do other interesting things. In the picture below, on the left, we have a usual $B = 7$ Skyrmion. On the right, we have deformed the Skyrmion into what looks like a $B = 4$ Skyrmion and a $B = 3$ Skyrmion. This is a cluster structure, and it turns out this deformation doesn't cost a lot of energy. This two-cluster system can be used as a model of the lithium-7 nucleus.
\begin{center}
\includegraphics[clip, width=0.4\textwidth]{images/B-7-Cluster.pdf}
B = 7 Skyrmions by C.\ J.\ Halcrow
\end{center}
\subsection{Asymptotic field and forces for \texorpdfstring{$B = 1$}{B = 1} hedgehogs}
We now consider what happens when we put different $B = 1$ hedgehogs next to each other. To understand this, we look at the profile function $f$, for $m = 0$. For large $r$, this has the asymptotic form
\[
f(r) \sim \frac{C}{r^2}.
\]
To obtain this, we linearize the differential equation for $f$ and see how it behaves as $r \to \infty$ and $f \to 0$. The linearized equation doesn't determine the coefficient $C$, but the full equation and boundary condition at $r = 0$ does. This has to be worked out numerically, and we find that $C \approx 2.16$.
Thus, as $ \sigma \sim 1$, we find $\boldsymbol\pi \sim C \frac{\mathbf{x}}{r^3}$. So the $B = 1$ hedgehog asymptotically looks like three pion dipoles. Each pion field itself has an axis, but because we have three of them, the whole solution is spherically symmetric.
We can roughly sketch the Skyrmion as
\begin{center}
\begin{tikzpicture}
\draw [mblue] (-0.949, 0) -- (0.949, 0) node [pos=-0.1] {$-$} node [pos=1.1] {$+$};
\draw [mgreen] (0, -0.949) -- (0, 0.949) node [pos=-0.1] {$-$} node [pos=1.1] {$+$};
\draw [mred] (-0.9, -0.3) -- (0.9,0.3) node [pos=-0.1] {$-$} node [pos=1.1] {$+$};
\end{tikzpicture}
\end{center}
Note that unlike in electromagnetism, scalar dipoles attract if oppositely oriented. This is because the fields have low gradient. So the lowest energy arrangement of two $B = 1$ Skyrmions while they are separated is
\begin{center}
\begin{tikzpicture}
\draw [mblue] (-0.949, 0) -- (0.949, 0) node [pos=-0.1] {$-$} node [pos=1.1] {$+$};
\draw [mgreen] (0, -0.949) -- (0, 0.949) node [pos=-0.1] {$-$} node [pos=1.1] {$+$};
\draw [mred] (-0.9, -0.3) -- (0.9,0.3) node [pos=-0.1] {$-$} node [pos=1.1] {$+$};
\begin{scope}[shift={(5, 0)}]
\draw [mblue] (-0.949, 0) -- (0.949, 0) node [pos=-0.1] {$+$} node [pos=1.1] {$-$};
\draw [mgreen] (0, -0.949) -- (0, 0.949) node [pos=-0.1] {$-$} node [pos=1.1] {$+$};
\draw [mred] (-0.9, -0.3) -- (0.9,0.3) node [pos=-0.1] {$+$} node [pos=1.1] {$-$};
\end{scope}
\end{tikzpicture}
\end{center}
The right-hand Skyrmion is rotated by $180^\circ$ about a line perpendicular to the line separating the Skyrmions.
These two Skyrmions attract! So two Skyrmions in this ``attractive channel'' can merge to form the $B = 2$ torus, which is the true minimal energy solution.
\begin{center}
\begin{tikzpicture}
\draw [mgreen] (0, -0.3) -- (0, 1.5) node [above] {$+$};
\draw [mgreen] (0, -1.5) node [below] {$-$} -- (0, -1.1);
\draw [mgreen, opacity=0.3] (0, -0.3) -- (0, -1.1);
\draw (0,0) ellipse (2 and 1.12);
\path[rounded corners=24pt] (-.9,0)--(0,.6)--(.9,0) (-.9,0)--(0,-.56)--(.9,0);
\draw[rounded corners=28pt] (-1.1,.1)--(0,-.6)--(1.1,.1);
\draw[rounded corners=24pt] (-.9,0)--(0,.6)--(.9,0);
\node [mblue] at (0, 0.7) {$\boldsymbol+$};
\node [mblue] at (0, -0.7) {$\boldsymbol+$};
\node [mblue] at (1.5, 0) {$\boldsymbol-$};
\node [mblue] at (-1.5, 0) {$\boldsymbol-$};
\node [mred] at (0.9, 0.5) {$\boldsymbol-$};
\node [mred] at (-0.9, -0.5) {$\boldsymbol-$};
\node [mred] at (0.9, -0.5) {$\boldsymbol+$};
\node [mred] at (-0.9, 0.5) {$\boldsymbol+$};
\end{tikzpicture}
\end{center}
The blue and the red fields have no net dipole, even before they merge. There is only a quadrupole. However, the field has a strong net green dipole. The whole field has toroidal symmetry, and these symmetries are important if we want to think about quantum states and the possible spins these kinds of Skyrmions could have.
For $B = 4$ fields, we can begin with the arrangement
\begin{center}
\begin{tikzpicture}
\draw [gray, opacity=0.5] (0, 0) rectangle (2, 2);
\draw [gray, opacity=0.5] (2, 0) -- (2.9, 0.3) -- (2.9, 2.3) -- (2, 2);
\draw [gray, opacity=0.5] (0, 2) -- (0.9, 2.3) -- (2.9, 2.3);
\draw [dashed, gray, opacity=0.5] (0, 0) -- (0.9, 0.3) -- (2.9, 0.3);
\draw [dashed, gray, opacity=0.5] (0.9, 0.3) -- (0.9, 2.3);
% \draw [mblue] (-0.2, 0) node [left] {\tiny$-$} -- (0.2, 0) node [right] {\tiny$+$};
% \draw [mgreen] (0, -0.2) node [below] {\tiny$-$} -- (0, 0.2) node [above] {\tiny$+$};
% \draw [mred] (-0.15, -0.05) node [left] {\tiny$-$} -- (0.15,0.05) node [right] {\tiny$+$};
%
% \begin{scope}[shift={(0.9, 2.3)}]
% \draw [mblue] (-0.2, 0) node [left] {\tiny$-$} -- (0.2, 0) node [right] {\tiny$+$};
% \draw [mgreen] (0, -0.2) node [below] {\tiny$+$} -- (0, 0.2) node [above] {\tiny$-$};
% \draw [mred] (-0.15, -0.05) node [left] {\tiny$+$} -- (0.15,0.05) node [right] {\tiny$-$};
% \end{scope}
%
% \begin{scope}[shift={(2, 2)}]
% \draw [mblue] (-0.2, 0) node [left] {\tiny$+$} -- (0.2, 0) node [right] {\tiny$-$};
% \draw [mgreen] (0, -0.2) node [below] {\tiny$+$} -- (0, 0.2) node [above] {\tiny$-$};
% \draw [mred] (-0.15, -0.05) node [left] {\tiny$-$} -- (0.15,0.05) node [right] {\tiny$+$};
% \end{scope}
%
% \begin{scope}[shift={(2.9, 0.3)}]
% \draw [mblue] (-0.2, 0) node [left] {\tiny$+$} -- (0.2, 0) node [right] {\tiny$-$};
% \draw [mgreen] (0, -0.2) node [below] {\tiny$-$} -- (0, 0.2) node [above] {\tiny$+$};
% \draw [mred] (-0.15, -0.07) node [left] {\tiny$+$} -- (0.15,0.05) node [right] {\tiny$-$};
% \end{scope}
\begin{scope}[scale=0.3]
\draw [mblue] (-0.949, 0) -- (0.949, 0) node [pos=-0.2] {\tiny$-$} node [pos=1.2] {\tiny$+$};
\draw [mgreen] (0, -0.949) -- (0, 0.949) node [pos=-0.2] {\tiny$-$} node [pos=1.2] {\tiny$+$};
\draw [mred] (-0.9, -0.3) -- (0.9,0.3) node [pos=-0.2] {\tiny$-$} node [pos=1.2] {\tiny$+$};
\end{scope}
\begin{scope}[shift={(0.9, 2.3)}, scale=0.3]
\draw [mblue] (-0.949, 0) -- (0.949, 0) node [pos=-0.2] {\tiny$-$} node [pos=1.2] {\tiny$+$};
\draw [mgreen] (0, -0.949) -- (0, 0.949) node [pos=-0.2] {\tiny$+$} node [pos=1.2] {\tiny$-$};
\draw [mred] (-0.9, -0.3) -- (0.9,0.3) node [pos=-0.2] {\tiny$+$} node [pos=1.2] {\tiny$-$};
\end{scope}
\begin{scope}[shift={(2, 2)}, scale=0.3]
\draw [mblue] (-0.949, 0) -- (0.949, 0) node [pos=-0.2] {\tiny$+$} node [pos=1.2] {\tiny$-$};
\draw [mgreen] (0, -0.949) -- (0, 0.949) node [pos=-0.2] {\tiny$+$} node [pos=1.2] {\tiny$-$};
\draw [mred] (-0.9, -0.3) -- (0.9,0.3) node [pos=-0.2] {\tiny$-$} node [pos=1.2] {\tiny$+$};
\end{scope}
\begin{scope}[shift={(2.9, 0.3)}, scale=0.3]
\draw [mblue] (-0.949, 0) -- (0.949, 0) node [pos=-0.2] {\tiny$+$} node [pos=1.2] {\tiny$-$};
\draw [mgreen] (0, -0.949) -- (0, 0.949) node [pos=-0.2] {\tiny$-$} node [pos=1.2] {\tiny$+$};
\draw [mred] (-0.9, -0.3) -- (0.9,0.3) node [pos=-0.2] {\tiny$+$} node [pos=1.2] {\tiny$-$};
\end{scope}
\end{tikzpicture}
\end{center}
To obtain the orientations, we begin with the bottom-left, and then obtain the others by rotating along the axis perpendicular to the faces of the cube shown here.
This configuration only has a tetrahedral structure. However, the Skyrmions can merge to form a cubic $B = 4$ Skyrmion.
\subsection{Fermionic quantization of the \texorpdfstring{$B = 1$}{B = 1} hedgehog}
We begin by quantizing the $B = 1$ Skyrmion. The naive way to do this is to view the Skyrmion as a rigid body, and quantize it. However, if we do this, then we will find that the result must have integer spin. However, this is not what we want, since we want Skyrmions to model protons and nucleons, which have half-integer spin. In other words, the naive quantization makes the Skyrmion bosonic.
In general, if we want to take the Skyrme model as a low energy effective field theory of QCD with an odd number of colours, then we must require the $B = 1$ Skyrmion to be in a fermionic quantum state, with half-integer spin.
As a field theory, the configuration space for a baryon number $B$ Skyrme field is $\Maps_B(\R^3 \to \SU(2))$, with appropriate boundary conditions. These are all topologically the same as $\Maps_0(\R^3 \to \SU(2))$, because if we fix a single element $U_0 \in \Maps_B(\R^3 \to \SU(2))$, then multiplication by $U_0$ gives us a homeomorphism between the two spaces. Since we imposed the vacuum boundary condition, this space is also the same as $\Maps_0(S^3 \to S^3)$.
This space is \emph{not} simply connected. In fact, it has a first homotopy group of
\[
\pi_1(\Maps_0(S^3 \to S^3)) = \pi_1(\Omega^3 S^3) = \pi_4(S^3) = \Z_2.
\]
Thus, $\Maps_0(S^3 \to S^3)$ has a universal double cover. In our theory, the wavefunctions on $\Maps(S^3 \to S^3)$ are not single-valued, but are well-defined functions on the double cover. The wavefunction $\Psi$ changes sign after going around a non-contractible loop in the configuration space. It can be shown that this is not just a choice, but required in a low-energy version of QCD.
This has some basic consequences:
\begin{enumerate}
\item If we rotate a $1$-Skyrmion by $2\pi$, then $\Psi$ changes sign.
\item $\Psi$ also changes sign when one exchanges two 1-Skyrmions (without rotating them in the process). This was shown by Finkelstein and Rubinstein.
This links spin with statistics. If we quantized Skyrmions as bosons, then both (i) and (ii) do not happen. Thus, in this theory, we obtain the spin-statistics theorem \emph{from topology}.
\item In general, if $B$ is odd, then rotation by $2\pi$ is a non-contractible loop, while if $B$ is even, then it is contractible. Thus, spin is half-integer if $B$ is odd, and integer if $B$ is even.
\item There is another feature of the Skyrme model. So far, our rotations are spatial rotations. We can also rotate the value of the pion field, i.e.\ rotate the target $3$-sphere. This is \term{isospin rotation}\index{isospin}. This behaves similarly to above. Thus, isospin is half-integer if $B$ is odd, and integer if $B$ is even.
\end{enumerate}
\subsection{Rigid body quantization}
We now make the \emph{rigid body approximation}. We allow the Skyrmion to translate, rotate and isorotate rigidly, but do not allow it to deform. This is a low-energy approximation, and we have reduced the infinite dimensional space of configurations to a finite-dimensional space. The group acting is
\[
(\mathrm{translation}) \times (\mathrm{rotation}) \times (\mathrm{isorotation}).
\]
The translation part is trivial. The Skyrmion just gets the ability to move, and thus gains momentum. So we are going to ignore it. The remaining group acting is $\SO(3) \times \SO(3)$. This is a bit subtle. We have $\pi_1(\SO(3)) = \Z_2$, so we would expect $\pi_1(\SO(3) \times \SO(3)) = (\Z_2)^2$. However, in the full theory, we only have a single $\Z_2$. So we need to identify a loop in the first $\SO(3)$ with a loop in the second $\SO(3)$.
Our wavefunction is thus a function on some cover of $\SO(3) \times \SO(3)$. While $\SO(3) \times \SO(3)$ is the symmetry group of the full theory, for a particular classical Skyrmion solution, the orbit is smaller than the whole group. This is because the Skyrmion often is invariant under some subgroup of $\SO(3) \times \SO(3)$. For example, the $B = 3$ solution has a tetrahedral symmetry. Then we require our wavefunction to be invariant under this group up to a sign.
For a single $\SO(3)$, we have a rigid-body wavefunction $\bket{J\; L_3\; J_3}$, where $J$ is the spin, $L_3$ is the third component of spin relative to body axes, and $J_3$ is the third component of spin relative to space axes. Since we have two copies of $\SO(3)$, our wavefunction can be represented by
\[
\bket{J\; L_3\; J_3} \otimes \bket{I\; K_3\; I_3}.
\]
$I_3$ is the isospin we are physically familiar with. The values of $J_3$ and $I_3$ are not constrained, i.e.\ they take all the standard $(2J + 1)(2I + 1)$ values. Thus, we are going to suppress these labels.
However, the symmetry of the Skyrmion places constraints on the body projections, and not all values of $J$ and $I$ are allowed.
\begin{eg}
For the $B = 1$ hedgehog, there is a lot of symmetry. Given an axis $\hat{\mathbf{n}}$ and an angle $\alpha$, we know that classically, if we rotate and isorotate by the same axis and angle, then the wavefunction is unchanged. So we must have
\[
e^{i \alpha \hat{\mathbf{n}} \cdot \mathbf{L}} e^{i \alpha \hat{\mathbf{n}} \cdot \mathbf{K}} \bket{\Psi} = \bket{\Psi}.
\]
It follows, by considering $\alpha$ small, and all $\hat{\mathbf{n}}$, that
\[
(\mathbf{L} + \mathbf{K}) \bket{\Psi} = 0.
\]
So the ``grand spin'' $\mathbf{L} + \mathbf{K}$ must vanish. So $\mathbf{L} \cdot \mathbf{L} = \mathbf{K} \cdot \mathbf{K}$. Recall that $\mathbf{L} \cdot \mathbf{L} = \mathbf{J} \cdot \mathbf{J}$ and $\mathbf{K} \cdot \mathbf{K} = \mathbf{I} \cdot \mathbf{I}$. So it follows that we must have $J = I$.
Since $1$ is an odd number, for any axis $\hat{\mathbf{n}}$, we must also have
\[
e^{i 2\pi \hat{\mathbf{n}} \cdot \mathbf{L}} \bket{\Psi} = - \bket{\Psi}.
\]
So $I$ and $J$ must be half-integer.
Thus, the allowed states are
\[
J = I = \frac{n}{2}
\]
for some $n \in 1 + 2\Z$. If we work out the formula for the energy in terms of the spin, we see that it increases with $J$. It turns out the system is highly unstable if $n \geq 5$, and so $\frac{1}{2}$ and $\frac{3}{2}$ are the only physically allowed values.
The $J = I = \frac{1}{2}$ states corresponds to $p$ and $n$, with spin $\frac{1}{2}$. The $J = I = \frac{3}{2}$ correspond to the $\Delta^{++}, \Delta^+, \Delta^0$ and $\Delta^-$ baryons, with spin $\frac{3}{2}$.
\end{eg}
\begin{eg}
If $B = 2$, then we have a toroidal symmetry. This still has one continuous $\SO(2)$ symmetry. Our first constraint becomes
\[
e^{i \alpha L_3} e^{i 2\alpha J_3} \bket{\Psi} = \bket{\Psi}.
\]
Note that we have $2 \alpha$ instead of $\alpha$. If we look at our previous picture, we see that when we rotate the space around by $2\pi$, the pion field rotates by $4\pi$.
There is another discrete symmetry, given by turning the torus upside down. This gives
\[
e^{i \pi L_1} e^{i \pi K_1} \bket{\Psi} = -\bket{\Psi}.
\]
The sign is not obvious from what we've said so far, but it is correct. Since we have an even baryon number, the allowed states have integer spin and isospin.
States with isospin $0$ are most interesting, and have lowest energy. Then the $K$ operators act trivially, and we have
\[
e^{i \alpha L_3} \bket{\Psi} = \bket{\Psi},\quad e^{i \pi L_1} \bket{\Psi} =- \bket{\Psi}.
\]
We have reduced the problem to one involving only body-fixed spin projection, because the first equation tells us
\[
L_3 \bket{\Psi} = 0.
\]
Thus the allowed states are $\bket{J, L_3 = 0}$.
The second constraint requires $J$ to be odd. So the lowest energy states with zero isospin are $\bket{1, 0}$ and $\bket{3, 0}$. In particular, there are no spin $0$ states. The state $\bket{1, 0}$ represents the deuteron. This is a spin $1$, isospin $0$ bound state of $p$ and $n$. This is a success.
The $\bket{3, 0}$ states have too high energy to be stable, but there is some evidence for a spin $3$ dibaryon resonance that decays into two $\Delta$'s.
There is also a $2$-nucleon resonance with $I = 1$ and $J = 0$, but this is not a bound state. This is also allowed by the Skyrme model.
\end{eg}
\begin{eg}
For $B = 4$, we have cubic symmetry. The symmetry group is rather complicated, and imposes various constraints on our theory. But these constraints always involve $+$ signs on the right-hand side. The lowest allowed state is $\bket{0, 0} \otimes \bket{0, 0}$, which agrees with the $\alpha$-particle. The next $I = 0$ state is
\[
\left(\bket{4, 4} + \sqrt{\frac{14}{5}} \bket{4, 0} + \bket{4, -4}\right) \otimes \bket{0, 0}
\]
involving a combination of $L_3$ values. This is an excited state with spin $4$, and would have rather high energy. Unfortunately, we haven't seen this experimentally. It is, however, a success of the Skyrme model that there are no $J = 1, 2, 3$ states with isospin $0$.
\end{eg}
\printindex
\end{document}
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\title{Against the Gendered Nightmare}
\date{2014}
\author{baedan}
\subtitle{Fragments On Domestication}
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In the past several years, the question of gender has been taken up again and again by the anarchist milieu. And still few attempts amount to much more than a rehashing of old ideas. Most positions on gender remain within the constraints of one or more of the ideologies that have failed us already, mainly Marxist feminism, a watered down eco-feminism, or some sort of liberal “queer anarchism.” Present in all of these are the same problems we’ve howled against already: identity politics, representation, gender essentialism, reformism, and reproductive futurism. While we have no interest in offering another ideology in this discourse, we imagine that an escape route could be charted by asking the question that few will ask; by setting a course straight to the secret center of gendered life which all the ideological answers take for granted. We are speaking, of course, about Civilization itself.
Such a path of inquiry is not one easily travelled. At every step of the way, stories are obscured and falsified by credentialed deceivers and revolutionary careerists. Those ideas presented as Science are separated from Myth only in that their authors claim to abolish mythology. Anthropology, Psychoanalysis, History, Economics—each faces us as another edifice built to hide a vital secret. At every step, we find more questions than answers. And yet this shadowy journey feels all the more necessary at the present moment. At the same time as technological Civilization is undergoing a renewed assault on the very experience of living beings, the horrors of gendered life continue to be inextricable from that assault. Rape, imprisonment, bashings, separations, dysmorphia, displacement, the labors of sexuality, and all the anxieties of techniques of the self—these daily miseries and plagues are only outpaced by the false solutions which strive to foreclose any possibility of escape; queer economies, cybernetic communities, legal reforms, prescription drugs, abstraction, academia, the utopias of activist soothsayers, and the diffusion of countless subcultures and niche identities—so many apparatuses of capture.
The first issue of \emph{Bædan} features a rather involved exegesis of Lee Edelman’s book \emph{No Future}. In it, we attempted to read Edelman against himself; to elaborate his critique of progress and futurity outside of its academic trappings and beyond the limitations of its form. To do so, we explored the traditions of queer revolt to which Edelman’s theory is indebted, particularly the thought of Guy Hocquenghem. Exploring Hocquenghem still proves particularly exciting, because his writing represents some of the earliest queer theory which explicitly rejects Civilization—as well as the families, economies, metaphysics, sexualities and genders which compose it—while also imagining a queer desire which is Civilization’s undoing. That exploration lead us to explore the bodily and spiritual underpinnings of Civilization: domestication, or “the process of the victory of our fathers over our lives; the way in which the social order laid down by the dead continues to haunt the living\dots{} the residue of accumulated memories, culture and relationships which have been transmitted to us through the linear progression of time and the fantasy of the Child\dots{} this investment of the horrors of the past into our present lives which ensures the perpetuation of civilization.”\footnote{“Queers Gone Wild,” \emph{bædan} vol. one, 2012.} Our present inquiry begins here.
To explore the conflict of the wildness of queer desire against domestication is to take aim at an enemy who confronts us from the beginning of Time itself. While our efforts in the first issue of this journal were a refusal of the teleology which situated an end to gender at the conclusion of a linear progression of time, we’ll now address the questions of origins which hint toward an outside at the other end of this line. As we’ve denied ourselves the future, we now turn against the past. In this, we abandon any pretensions of certainty or claims to truth. Instead we have only the experiences of those who revolt against the gendered existent, as well as the stories of those whose revolt we’ve inherited. In the spirit of this revolt, we offer these fragments against gender and domestication.
\section{I}
Domestication, the integration of living beings into the civilized order, must also be the integration of life into the dualism and separation which we experience as gender. The concept is thrown about in a variety of contexts and under various names, and yet very few have attempted to thoroughly define it. It is used colloquially to discuss the vast gulf which exists between wild creatures and those tamed and clawless ones whose existence has been reduced to economic necessities. It is linguistically tied to the realm of the Domestic, and by extension to the Economic through the management of the home, \emph{oikonomia}. It is the violence implied in the concept of \emph{primitive accumulation}, the first (but also the originary) tearing of a being away from its self and its subsequent imprisonment in class society. It is further implied in all the theories of \emph{subjectification}, the construction of all the identities and roles which populate the social order. Being so central to the world we inhabit and the subjects we have become, the concept warrants a more precise and consistent definition.
In our previous engagement with domestication, we primarily looked at the writings of Jacques Camatte. He comes to his theory of domestication through an exploration of the ways that Capital empties, transforms and colonizes human beings; in his words, Capital’s anthropomorphism. Capital dissects and analyzes the human being, ruptures the mind from the body, and reconstructs the human as a willful subject of the social order. The consequence of this rupturing and suturing of life is the recuperation of the vast range of humanist means of resistance; communities become communities of capital, and individuals become little more than consumers. Separation evolves into an image of wholeness which replaces the unity it abolished. Domestication, which limits the possibilities of what we can become, promises a future without limits, because it ties our future to an undead and all-devouring system. We are evacuated of our desires and instincts, and the vacuous space left within us is filled with all the representations of what was taken. Instead of a vast multitude of potentials and ways of relating to the world, our lives are reduced to a microcosm of the linear progression of society. Domestication does more than enslave us to the social order’s future, it creates willful slaves. As individual living beings are reduced to spectators and functions of dead things, the non-living itself becomes autonomous. All the scientific disciplines, the linguists of this autonomous non-living thing, proclaim alongside the fascists: long live death! These disciples of Capital use their methodology to prove that this is the way things always were, they naturalize Capital and demonstrate its inevitability. We are split and dominated in the same way as physicists split and dominate the atom; managed in the same way cyberneticians manage their networks and feedback loops; as above, so below. Thus for Camatte, Capital conquers our imagination both with regard to our future, and also our past.
\begin{quote}
Capital has reduced nature and human beings to a state of domestication. The imagination and the libido have been enclosed as surely as the forests, oceans, and common lands.
The process of domestication is sometimes brought about violently, as happens with primitive accumulation; more often it proceeds insidiously because revolutionaries continue to think according to assumptions which are implicit in capital and the development of productive forces, and all of them share in exalting the one divinity, science. Hence domestication and repressive consciousness have left our minds fossilized more or less to the point of senility; our actions have become rigidified and our thoughts stereotyped. We have been the soulless frozen masses fixated on the post, believing all the time that we were gazing ahead into the future.
\end{quote}
This moment of Camatte’s thought is interesting because it marks his personal shift away from Marxism and toward a critique of civilization (a shift which would be significant for a whole generation of anti-civilization thinkers). Unfortunately though, it is precisely its situation in that shift (an obsession with one particular mode of production) which creates the limit of his definition of domestication. For him, the autonomous non-life which domesticates life is Capital, and he situates this process in a specific moment of capitalism where Capital “escapes” and forms its own community. This is tied up in his esoteric, (and in its own way, exegetical) reading of Marx. He locates domestication at the point at which capitalism has developed into a representation and is thrown into crisis. He calls Capital an endpoint of the processes of democratization, individuation, and massification. He speaks of these processes as presuppositions to Capital which may go as far back as the Greek \emph{Polis} and its representational break of humans from the rest of wild life, and to the “domination of men over women.” And so if we can locate Capital at the endpoint of this ancient chain of separations, how can domestication (separation itself) begin with Capital? Moreover, if gendered domination predates domestication by millennia, how can his version of domestication account for the separation and colonization of life for which gender is a euphemism? His origin myth fails at the point where it begins. His story is not enough for us, because we know this colonization of our very existence did not begin in the last century, or even the one before it. We can still hear the distant cries of those who’ve resisted since long before. Clearly, we must leave Camatte behind if we want to comprehend domestication in its totality.
\section{II}
Camatte’s critique of domestication is most clearly articulated in his essay \emph{The Wandering of Humanity}, which was first published in English in 1975 by Black and Red of Detroit. At the time, the press was run by Lorraine Perlman and her husband Fredy. They self-published the text in a beautiful pamphlet after Fredy completed its first English translation. In reading Perlman’s own writing, the influence of the text is readily apparent. Perlman himself would go on to incorporate these ideas into a scathing critique of Civilization which still inspires much of the anti-civilization perspective within the anarchist milieu. His efforts would largely be motivated by seizing upon the precise limit we’ve identified in Camatte’s story: that of origins.
In her biography of Fredy, \emph{Having Little, Being Much,} Lorraine narrates the way that he spent the following seven years almost single-mindedly focused on exploring the history of the domesticating monster. In particular he spent those years tearing through accounts of the European colonization of the North America, and the domestication process which they unleashed upon all of the living inhabitants of this continent. He stole from Hobbes in naming this monster Leviathan, and undertook the monumental task of telling the tale of those who’ve resisted it. He self-published his findings in 1983 in a wonderful and tragic book, released among friends at a party at his and Lorraine’s house in Detroit. The book was titled \emph{Against His-Story, Against Leviathan!}
Asserting that “resistance is the only human component of the entire His-story,” Fredy suspended his in-depth study of resistance to Leviathanic incursions in the woodlands around the Great Lakes to examine the “barbarians” and untamed tribes who, in earlier times, unequivocally refused the bondage of civilization. Where His-story exults in civic and military achievements, calling them Progress, Fredy’s story views each consolidation of state power as an encroachment on the human community. He addresses the reader as one individual speaking to another and makes no claim to follow scholarly rules: “I take it for granted that resistance is the natural human response to dehumanization and, therefore, does not have to be explained or justified.” The resistance story follows the chronology of Leviathan’s destructive march, but avoids using His-storians’ conventions of dating the events. This, as well as the poetic visionary language, gives the work an epic quality.
Fredy begins his narrative by attempting to isolate the way that other available ideological positions fail to grasp the enemy in its fullness. His method is instructive in that he points to how each ideology is too narrow, and can only offer incredibly superficial solutions to the problem of domestication. In the first chapter, he writes:
\begin{quote}
Marxists point at the Capitalist mode of production, sometimes only at the Capitalist class. Anarchists point at the State. Camatte points at Capital. New Ranters point at Technology or Civilization or both\dots{}
The Marxists see only the mote in the enemy’s eye. They supplant their villain with a hero, the Anti-capitalist mode of production, the Revolutionary Establishment. They fail to see that their hero is the very same “shape with lion body and the head of a man, a gaze blank and pitiless as the sun.” They fail to see that the Anti-capitalist mode of production wants only to outrun its brother in wrecking the Biosphere.
Anarchists are as varied as Mankind. There are governmental and commercial Anarchists as well as a few for hire. Some Anarchists differ from Marxists only in being less informed. They would supplant the state with a network of computer centers, factories and mines coordinated “by the workers themselves” or by an Anarchist union. They would not call this arrangement a State. The name-change would exorcize the beast.
Camatte, the New Ranters and Turner treat the villains of the Marxists and Anarchists as mere attributes of the real protagonist. Camatte gives the monster a body; he names the monster Capital, borrowing the term from Marx but giving it a new content. He promises to describe the monster’s origin and trajectory but has not yet done so\dots{}
\end{quote}
The problems that he draws out about Anarchist and Marxist politics resonate as much today as they did in 1983, and those who’ve drawn other conclusions largely have Fredy to thank for helping to rejuvenate an anarchy without an attachment to industrialism, technology or other fetishes of production. It is from this last point, the failure of Camatte to sketch the origin and trajectory of the monster, that he sketches his own. He draws on the writings of Frederick Turner to articulate the spirit of the monster, but criticizes Turner for his inability to speak of the monster’s body; the cadaverous body which tears apart wild things and incorporates them into itself. Fredy’s narrative strikes out against this body.
Fredy’s project is an important one, because it pushes the critique of domestication beyond the comfortable answers. He interrogates the beast’s machinations before late capitalism, before the colonization of the ‘new world,’ before the rise of capitalism itself. What he accomplished was to write a story about the rise of every Civilization since the first in Sumeria, and thus also of Civilization itself. Significantly, he told this tale while indicting the historians, anthropologists and economists who justify the rise of Leviathan. Instead he told the story from the perspective of those who resisted domestication at every juncture. This is one of the many stylistic and ethical reasons that make the book so genuinely beautiful to read. Whereas I can’t in good faith recommend that one reads the tedious works of Edelman or Camatte, I’d happily gift \emph{Against His-Story} to any of my dearest friends. This is also the reason that it doesn’t make a great deal of sense to attempt a comprehensive paraphrasing. Trying to capture the magic of Fredy’s storytelling would be difficult, if not impossible. Rather I’d suggest that anyone who wants to experience the depth and weight of the book’s critique should simply read it themselves. That being said, we’ll identify a few themes within the story which will help us in our own. These understandings will be useful in moving further with an exploration of Domestication.
In no particular order, some useful themes about domestication which emerge through the text:
\begin{itemize}
\item\relax
\emph{The language of the domesticated always serves to hide widely accepted lies, if barely.} Clearly only those outside of the monster are free, and yet the civilized will use this word to describe themselves. Even the dictionary contains this contradiction: it describes ‘freedom’ as belonging to ‘citizens,’ yet then says that something is free if not constrained by anything other than its own being. There isn’t any way to reconcile this contradiction. Wild birds and trees and insects which are only determined by their own potential and wishes are free. Citizens are constrained by an infinity of un-freedom. The domesticated will refer to those humans who are still free as ‘barbarians’ or ‘savages,’ and yet these terms designate those very people as legitimate prey for the most barbaric atrocities at the hands of the ‘civilized.’ This meaninglessness and deception inherent to language is true of almost every word that the domesticated will use to describe themselves: that which destroys communities is named a Community, that which has a thirst for human blood beyond any reason is called Humanism and Reason. This is important when faced with the writings of those who aim, through words, to justify domestication.
\item\relax
\emph{Leviathan takes the form of artificial life; it has no life of its own, and thus can only function by capturing living beings within itself.} Following Hobbes, Leviathan (or Commonwealth or State or Civitas) is an artificial man. A blond, masculine, crowned man bearing a sword and a scepter. This artificial man is composed of countless faceless human beings, tasked with moving the springs and wheels and levers which make the artificial beast move. Hobbes, in turn, would see these individual human beings as nothing more than a composite of strings and wheels and springs. Fredy imagines that the beast might not be an artificial human but rather a giant worm, not a living worm but a carcass of a worm, a monstrous cadaver, its body consisting of numerous segments, its skin pimpled with spears and wheels and other technological implements. He knows from his own experience that the entire carcass is brought to artificial life by the motions of the human beings trapped inside\dots{} who operate the springs and wheel\dots{} Human beings regress while the worm progresses. The worm’s greatest accomplishment is to remake the people within it into individual mechanized units. These human machines are ultimately replaced by entirely automated machines, more amenable to existence within the labor camps of leviathan. This is a haunting proposition because it implicates us as complicit in the machinery of our own nightmare: both as the living force which animates the monster, but also as having internalized that animation.
\item\relax
\emph{Leviathan constitutes itself through institutions of domestication; these institutions are impersonal and immortal.} Immortality is found among no living creature on the earth. In being immortal, these institutions are a part of death, and death cannot die. Workers, prisoners and soldiers die; and yet factories, prisons and armies live on. As civilization grows, the domain of death grows while the individuals living within it die. No resistance movement has yet been able to deal with this contradiction. Monasteries were an early innovation in these immortal institutions. In these establishments, which are nothing but early schools, human beings are systematically broken, the way horses or oxen are broken, to bear weights and pull loads. They are separated from their own humanity, from all natural activities and sequences, and taught to perform artificial activities and identify with Leviathanic sequences. They become disciplined springs and wheels engaged in a routine that has no relation to human desires or natural cycles. The clock will be invented by monastic beings because the clock is nothing but a miniature monastery whose springs and wheels are made of metal instead of flesh and blood. No amount of institutional reform has exorcised this monstrous aspect of institutions.
\item\relax
\emph{Domesticated humans are defined by their adornment with masks over their faces and armor over their bodies.} These masks and armors are the ways in which the individual internalizes the constraint of Leviathan and acclimates themselves to life within it. These are necessary for surviving the everyday domination and humiliation which is life in this society. They protect individuals from their own emotions, perception and estrangement from being. The armor wraps around the individual and invades their body just as all ecstatic life and freedom is evacuated from the body, save for a potential. All that’s left is the armor. This can also be understood as the formation of civilized identities.
\item\relax
\emph{Domestication is perpetuated through a civilized spirituality which emphasizes dominion over all living things, but more importantly, self-management and self-domination.} All monotheistic religions hold in common that man must have dominion over the fish and foul and all living things. The Catholic church in particular has enforced this decree by declaring war against all living things; the same living things which constitute the autonomy and independence of free people. The church innovated upon this doctrine through the concept of sin. In response to sin, people are compelled to do to themselves what God does to all living things and what the nobles do to the peasants. They turn violence against their own urges and desires, above all the desire for freedom and escape. The war against all life continues as a war against one’s self. No previous leviathan had so thoroughly degraded its human contents. Not only do humans domesticated into the Christian civilization suffer, they suffer a self inflicted violence at their own hands and from their own minds. They enforce a slow tortuous murder upon themselves. This war on the self would be externalized as the Holy War which the Church would later wage against infidels, both domestic and abroad. Such conquest is democratized through the decree that every man should be an emperor in his own home: peasants and nobility alike are joined in this frenzy of violence and control over their subjects. At this point, even the most secular civilized society has been entrenched in this self-constraint for so many generations that such a spiritual form of domination appears also as secular and natural.
\item\relax
\emph{While some Leviathans could be seen as worms, others appear more as octopuses carrying out a pillaging of the earth more intense and widespread than ever before; this expansion is necessary to Leviathans’ survival, but no living being willfully submits to accumulation into these monsters.} Economists and Historians will describe a natural material dialectic by which people willfully enter these beasts, because of their supposedly superior amenities. And yet at every turn, violence must be used to force people to accept these amenities. There is no ‘demand’ until people have been broken from the wild world and from their own abilities to care for themselves. European clothes are only worn by those who have lost their own. These communities of free peoples are attacked by an unprecedented chemical and biological warfare which exists nowhere outside of Civilization itself. All that exists outside of Civilization is viewed as raw materials to be accumulated. This outside is often constructed through a racialized and gendered categories. This accumulation does not happen at the hands of economists, but by lynch mobs, militaries, armies, and all the rest of Leviathan’s police. The genocide carried out by Europeans against native peoples and animals and land bases on the American continents amounts to the most unprecedented of these accumulations. Through the activity of grave diggers (known as archeologists), even the dead become commodities. All of this violence is necessary for Leviathan’s growth, the dead commodities become the seeds of the next wave of accumulation.
\item\relax
\emph{Those whose communities have long since been defeated will carry the banner of their lost community in an attempt to regain that lost freedom by battling an imagined enemy.} The civilized humans wear the mask of something they no longer are or never were, all in an attempt to hide what they’ve become. It amounts to a frenzied rush away from ones self. Christianity, the Reformation, Marxism and Naziism are but a few examples of movements which begin by projecting an image of rejecting the industrial hell, but in fact only reproduce industrial civilization. In fact, most new Leviathans begin as resistance movements.
\item\relax
\emph{“By undergoing what will be called Industrial and Technological Revolutions, the Great Artifice breaches all walls, storms victoriously through every natural and human barrier, increasing its velocity at every turn. But by the time the beast really gets going like a winged rodent out of Inferno, its own soothsayers will be saying an object which approaches the speed of light loses its body and turns to smoke. Such object’s victories are, in the long run Pyrrhic.”} Civilization is marked by over-extension, rapid growth, and a movement toward infinity. This movement is ultimately self-destructive, producing contradictions and break-downs which threaten the machine itself. All of history is littered with the carnage and wreckage of this hubris. This is a complex point about decomposition which warrants more attention. We will return to it later.
\end{itemize}
These points barely scratch the surface of eloquent argumentation in \emph{Against His-Story, Against Leviathan!,} but they are worth drawing out because they help us to understand and elucidate a functioning definition of Domestication beginning with the first Civilizations. Deception, capture, domination, accumulation, annihilation, decline; we will see these themes repeating in all the stories which follow our inquiry.
\section{III}
In the years since Fredy published \emph{Against His-Story, Against Leviathan!}, the topic of domestication has been taken up by a whole range of anti-civilization anarchists and projects. In most of the writings emerging from this milieu, domestication is nearly tautological with civilization. (Civilization is understood as the web of power between the institutions, ideologies, and physical apparatuses which perform domestication and control; while Domestication is understood as the process by which living beings are trapped within the network that is Civilization.) This tautology is instructive, as it points to the autonomous existence of a monster which has the sole purpose of perpetuating itself by bringing all life inside. Fredy would call such a monster a world-destroyer. While different tendencies of anti-civilization thought tend to understand domestication from different angles,\footnote{Primitivists seek to understand domestication at its origins, with particular attention to the cultures it destroyed. Insurrectionaries tend to explore strategies against the institutions of domestication in the present. Others emphasize the metaphysical and spiritual implications of domestication. Queer and feminist anti-civilization perspectives focus on domestication as the origins of patriarchy.} it remains central to the thought and practice of those who believe civilization must be destroyed.
Contemporary anti-civilization writers (many anonymous or pseudonymous) have elaborated the critique of domestication into daily life, indicting countless small operations which serve to domesticate life.
\begin{quote}
Domestication is the process that civilization uses to indoctrinate and control life according to its logic. These time-tested mechanisms of subordination include: taming, breeding, genetically modifying, schooling, caging, intimidating, coercing, extorting, promising, governing, enslaving, terrorizing, murdering\dots{}the list goes on to include almost every civilized social interaction. Their movement and effects can be examined and felt throughout society, enforced through various institutions, rituals, and customs.\footnote{“An Introduction to Anti-Civilization Anarchist Thought and Practice” by the \emph{Green Anarchy} collective.}
\end{quote}
Others have devoted their explorations to the conditions and events which lead to the establishment of agriculture and symbolic thought ten thousand years ago, trying to force the far past to give up its secrets. From this perspective, that originary moment of domestication inaugurated millennia of war, slavery, ecological destruction, and the annihilation of free creatures.
All of these elaborations are useful in that they explain what domestication means in various instances and phenomena, but it is still rare to find a concise and functioning definition of what it means all together. If we need to do so, we could say rather simply that \emph{domestication is capture.} Further, it is the capture of living beings by a dead thing, and the integration of those beings into all the roles and institutions which comprise the dead thing. Furthermore it is all the practices which force those beings to spiritually accede to their capture. And lastly it is the discourse and ideology which justifies that capture. This capture is unending, and the dead thing can only continue its immortal reign if it continues to bring new living beings and commodities within itself.
\section{First Mythos: Enkidu and Shamhat}
\emph{Fredy begins his account of the first civilization emerging in Sumeria. He describes the rise of the first king, the Lugal, and from it all subsequent worm monsters. Sumeria is interesting to our inquiry because it is the birth of civilization, but also of the written word. From this ancient civilization, the oldest written story, that of the Sumerian king Gilgamesh, was etched into tablets of lapis lazuli. As its hero, Gilgamesh is responsible for instituting the ultimate domination of the Sumerian Leviathan over the wild world. He does this because he:}
\begin{verse}
\dots{}leaves no son to his father \\{}
Day and night \\{}
endlessly \\{}
Gilgamesh \\{}
The shepherd of Uruk \\{}
The shepherd of the people \\{}
Leaves no daughter to her mother \\{}
No Warrior’s daughter \\{}
~~~~no young man’s spouse \\{}
No bride to her groom
\end{verse}
\emph{In his endless mobilization of human beings, Gilgamesh built a human machinery which waged war against the wild earth. In response to Gilgamesh and his imposition of order, the Gods created an equal who could oppose him. His name was}
\begin{verse}
Enkidu \\{}
~~~~~~~~Primeval \\{}
~~~~in the wild \\{}
Born of silence \\{}
~~~~knit by Ninurta \\{}
~~~~~~~~war \\{}
His body covered with hair \\{}
On his head as on a woman’s \\{}
~~~~thick as Nissaba \\{}
~~~~~~~~grain \\{}
Knowing neither people nor place \\{}
Dressed as Sakkan commands \\{}
~~~~as the god of animals commands \\{}
~~~~~~~~as animals do \\{}
He fed on the grass with gazelles \\{}
He drank at springs with animals \\{}
Satisfied his thirst with the herd
\end{verse}
\emph{But the hunters and shepherds were angry and terrified of Enkidu, who sabotaged their traps and released their animals. They went to Gilgamesh and asked for his help. He devised a plan involving Shamhat, one of the sacred prostitutes of the temple. He said:}
\begin{verse}
“Go \\{}
Take Shamhat with you \\{}
When the beast comes to the spring \\{}
Let her strip off her clothing \\{}
~~~~~~~~reveal her charms \\{}
He will see her and approach \\{}
And the beasts will reject him” \\{}
And so Shamhat and the hunter set out in search of Enkidu. The hunter said: \\{}
~~~~“Shamhat \\{}
Open your arms \\{}
Open your legs \\{}
~~~~let him take your charms \\{}
Don’t be afraid \\{}
~~~~Take his breath away \\{}
He will see you and approach \\{}
Open your clothes \\{}
~~~~Let him lie upon you \\{}
Do a woman’s work for the man \\{}
Caress and embrace him \\{}
As he embraces you \\{}
And the beasts will reject him”
Shamhat opened her clothes \\{}
Opened her legs \\{}
He saw her charms \\{}
She was not afraid \\{}
And he lay down on her \\{}
She did a woman’s work for the man \\{}
Six days \\{}
~~~~seven nights \\{}
Enkidu coupled with Shamhat \\{}
~~~~breathless \\{}
When he had satisfied his desire \\{}
He faced the wilderness \\{}
~~~~The gazelles shunned him and moved away \\{}
~Exhausted \\{}
~~~~Enkidu’s legs would not move \\{}
As the beasts moved away \\{}
He could not run as he had before \\{}
But he had reason and broad understanding \\{}
He turned and sat at Shamhat’s feet \\{}
Looked at her face \\{}
~~~~as she looked at his \\{}
He listened to her speak \\{}
“You are handsome \\{}
~~~~Enkidu \\{}
~~~~~~~~like a god \\{}
Why wander the wild \\{}
~~~~with the beasts? \\{}
Come \\{}
~~~~let me lead you to Uruk-the-Sheepfold \\{}
To the temple \\{}
~~~~home of Anu and Ishtar”
\end{verse}
\emph{Enkidu agreed, but for the possibility of challenging the mighty Gilgamesh, but Shamhat convinced him otherwise. Gilgamesh had already dreamt of Enkidu’s coming, and the king would take the wild one as a dearest friend, would treat him as a wife. He would domesticate Enkidu.}
\begin{verse}
~Shamhat disrobed and dressed him \\{}
~~~~in one of her robes\dots{} \\{}
The shepherds set bread and beer before him \\{}
Suckled on the milk \\{}
~~~~of the wild \\{}
Enkidu looked \\{}
~~~~squinted \\{}
~~~~stared \\{}
He knew nothing \\{}
~~~~of food
~Shamhat spoke to Enkidu: \\{}
“Eat the bread \\{}
~~~~staff of life \\{}
Drink the beer \\{}
~~~~destiny of the land”
~Enkidu ate of the bread until sated \\{}
He drank of the beer until sated \\{}
~~~~~~~~~~~~seven mugs \\{}
He became a manifestation \\{}
~~~~dressed in robes \\{}
A warrior \\{}
~~~~who took up his weapons \\{}
~~~~~~~~to fight lions \\{}
the shepherds rested at night \\{}
Enkidu fought off wolves \\{}
~~~~~~~~and lions \\{}
The elder shepherds slept \\{}
Enkidu stayed \\{}
~~~~awake.
\end{verse}
\emph{The story of Enkidu and Shamhat is a story of domestication from within the mythology of the first civilization. It shows of the taming of Enkidu through the imposition of sex roles, the wearing of clothes, the drinking of alcohol, and his separation from the wild beasts. Shamhat is a sacred prostitute of the Sumerian temples, a spiritual practitioner of the oldest profession. She serves the goddess Ishtar through the rite of} hieros gamos\emph{, the sacred marriage between the king and the goddess of the city. Ishtar is the goddess of nature, yes, but of nature within the city.} Heiros gamos, \emph{the sacred prostitution, is a ritualistic submission of nature to the power of the king; the bringing of the wild within the walls of the city. In this way, the nature goddess was also the goddess of arts of civilization. These arts included the practices of government and religion, war and peace, crafts, profession, eating, drinking, clothing, bodily adornments, art, music, sex and prostitution. Theirs are the arts of living applicable to every aspect of civilized life. The goddess rules nature within the city, so her} ars vivendi \emph{are the rules of civilization, of domestication. And so it was through these rules that Shamhat, a priestess of Ishtar, made Enkidu into a man. After he is torn from his world, Enkidu becomes a virile and bloodthirsty destroyer of the wild. The imposition of gender unleashes a continuum of separation which endlessly separates the city from the forest, humanity from the rest of wild life, and splits humans into genders.}
\emph{Contemporary readings will of course illustrate a degree of misogyny around Shamhat, implying that women tamed the wild men. But this is incorrect and only reveals how deeply seated gendered domination is to civilization. Enkidu is domesticated by all the} ars vivendi \emph{which define life in the first civilization; by} women’s work \emph{and} men’s work\emph{. Enkidu is} made a man \emph{through these domesticating laws; he is civilized by gender itself.}
\section{IV}
It could be said that perhaps no tendency has taken the question of gender further than primitivism. We say this, because the primitivists view the question through the lens offered by a critique of domestication. While there are obviously heinous examples of masculinist and misogynist theories and individuals within anti-civilization thought, the most lucid and careful writers have always located the rise of patriarchy at the very beginning of civilization. For many (Fredy Perlman and John Zerzan to name just two), Patriarchy emerges alongside domestication and the two are practically synonymous. We can even see small fragments of this perspective in Camatte’s later writing, \emph{Echoes of the Past}, for example. It is also acknowledged in the 2009 editorial statement of \emph{BLOODLUST: a feminist journal against civilization.} The editors articulate that their desire to publish the journal was a result of what felt like a superficial treatment of the critique of gender, and yet they still celebrate that the anti-civilization tendency is one of the few that consistently indicts Patriarchy as a central enemy. While sadly the journal only released one issue, the task of fleshing out the anti-civilization critique of Patriarchy seems like a step toward understanding domestication’s centrality to gender itself.
The primitivist perspective on gender is problematic for reasons we’ll elaborate later, but for a moment we’ll suspend our criticism so as to fairly lay out the argument. Whatever its flaws, this perspective on the rise of patriarchy is useful because it situates the emergence of gendered domination with civilization itself. In doing so, it refuses any ideology which fails to do so. By constantly demonstrating that such misery is older than most other institutions and systems of domination, it equips us with the necessary pessimism to respond to those who assure us that gendered violence will disappear after their specific reform or revolution.
Camatte (and consequently those who are influenced by his writing) is indebted, with regard to his fleeting thoughts on gender, to a French writer named Françoise d’Eaubonne. D’Eaubonne is credited as the person who coined the term \emph{eco-feminism} in her 1974 book, \emph{Feminism or Death.} More interestingly, she was also one of the cofounders of the organization \emph{Front Homosexuel d’Action Revolutionnaire} (FHAR), the same militant gay liberation group which Guy Hocquenghem joined and which shaped his later perspectives. It makes sense then, that two anti-civilization theories of gender would emerge from the same action and discussions; d’Eaubonne’s \emph{eco-feminism,} and Hocquenghem’s \emph{homosexual desire.} It is a tragic detriment to our inquiry that almost nothing of d’Eaubonne’s writing is translated into English. Most Anglophone primitivists and eco-feminists have only been exposed to her ideas though secondary sources (Camatte among them). We’ll cite an excerpt from \emph{Feminism or Death} as it is unlikely that most readers would have access to the text:
\begin{quote}
Practically everybody knows that today the two most immediate threats to survival are overpopulation and the destruction of our resources; fewer recognize the complete responsibility of the male System, in so far as it is male (and not capitalist or socialist) in these two dangers; but even fewer still have discovered that each of the two threats is the logical outcome of one of the two parallel discoveries which gave men their power over fifty centuries ago: their ability to plant the seed in the earth as in women, and their participation in the act of reproduction.
Up until then the male believed [women were] impregnated by the gods. From the moment he discovered at once his two capacities as farmer and procreator, he instituted what Lederer calls ‘the great reversal’ to his own advantage. Having taken possession of the land, thus of productivity (later of industry) and of woman’s body (thus of reproduction), it was natural that the overexploitation of both of these would end in this threatening and parallel menace: overpopulation, surplus births, and destruction of the environment, surplus production.
The only change capable of saving the world today is that of the ‘great reversal’ of male power which is represented, after agricultural overproductivity, by this mortal industrial expansion. Not ‘matriarchy,’ to be sure, nor ‘power-to-the-women,’ but destruction of power by women. And finally, the end of the tunnel: a world to be reborn (and no longer ‘protected’ as is still believed by the first wave of timid ecologists)\dots{}
Therefore, with a society at last in the feminine gender, meaning non-power (and not power-to-the-women), it would be proved that no other human group could have brought about the ecological revolution; because none other was so directly concerned at all levels. And the two sources of wealth which up until now have benefited only the male would once again become the expression of life and no longer the elaboration of death; and human beings would finally be treated first as persons, and not above all else as male or female.
And the planet in the feminine gender would become green again for all.
\end{quote}
While simplistic and essentialist, this line of argument stands out for its singular elaboration of the intrinsic connection between agricultural production and human reproduction. We’ll look at others who’ve expanded on this theory, but we would be hard pressed to find anything in the primitivist canon that deviates too far from this straightforward position. All of it will center the role of man as the husband to his wife and the practitioner of agriculture and animal husbandry. The argument is useful because it is an articulation of the way domestication captures both those humans assigned female and also a vast diversity of non-human life.
One can clearly see the echoes of this in a primer\footnote{\emph{Ibid}.} written by the \emph{Green Anarchy} collective:
\begin{quote}
Toward the beginning in the shift to civilization, an early product of domestication is patriarchy: the formalization of male domination and the development of institutions which reinforce it. By creating false gender distinctions and divisions between men and women, civilization, again, creates an “other” that can be objectified, controlled, dominated, utilized, and commodified. This runs parallel to the domestication of plants for agriculture and animals for herding, in general dynamics, and also in specifics like the control of reproduction. As in other realms of social stratification, roles are assigned to women in order to establish a very rigid and predictable order, beneficial to hierarchy. Woman come to be seen as property, no different then the crops in the field or the sheep in the pasture. Ownership and absolute control, whether of land, plants, animals, slaves, children, or women, is part of the established dynamic of civilization. Patriarchy demands the subjugation of the feminine and the usurpation of nature, propelling us toward total annihilation. It defines power, control and dominion over wildness, freedom, and life. Patriarchal conditioning dictates all of our interactions; with ourselves, our sexuality, our relationships to each other, and our relationship to nature. It severely limits the spectrum of possible experience. The interconnected relationship between the logic of civilization and patriarchy is undeniable; for thousands of years they have shaped the human experience on every level, from the institutional to the personal, while they have devoured life. To be against civilization, one must be against patriarchy; and to question patriarchy, it seems, one must also put civilization into question.
\end{quote}
Fredy Perlman expands on this premise in a few ways. Firstly, he consistently centers rape and the weaponization of the phallus as methods intrinsic to domestication. He connects the phallic towers at the center of early Leviathans to the weapons used by their armies. For him these institutions and apparatuses function to naturalize an unnatural form of domination and power, to subject women to men and to pretend that this arrangement is the natural order of things. At times he describes Leviathanic men as ‘women haters.’ Secondly, he believes His-story to be the process by which the men who control Leviathan narrate their own conquests and achievements. For him His-story is specific to civilized culture and only emerges as a violent annihilation both of a pre-existing matriarchy, but also through the deification of an image of militaristic, Leviathanic men as opposed to former nature goddesses. For him, the earth itself is feminine; a mother who gives birth to all life. By contrast, Leviathan gives birth to nothing but death, and as such, despises the mother Earth. In the following fragments we’ll criticize much of this theory, but it is worth acknowledging that it is rare to find another theory of His-story (especially one written by a man) which locates patriarchy as absolutely inseparable from civilization.
John Zerzan expands upon the theory from a different angle. He primarily concerns himself with studying the work of over a dozen anthropologists (all of them women) who analyze the role of women in social arrangements before domestication. Many of these anthropologists were part of the shift in Anthropology referred to as the shift from “man the hunter” to “woman the gatherer.” Based on their research, he argues that the vast majority of sustenance in most non-civilized societies was provided by gatherers, who tended to be women. He argues that as a consequence, women had significantly more social power and autonomy, because they were not reliant on patriarchal agricultural arrangements for survival. He also follows other anthropologists in claiming that hierarchies around gender were rare among American indigenous tribes, specifically noting the absence of fetishes for virginity and chastity, expectations of monogamy for women, or male control over reproduction. He argues that the sexual division of labor, imposed by domestication, was the first form of the division of labor which constitutes contemporary civilization. He also criticizes the shift from communal tribal relationships of sharing to the privatized and gendered existence of the family-form, arguing that the family is neither inevitable nor universal in human communities. Zerzan argues that the shift toward domestication is marked by the emergence of specialized labor roles, the limiting of women’s labor to reproductive efforts, and the strengthening of kinship bonds above all else. For him, the presence of a gendered division of labor by the time of the earliest recorded symbolic art indicates that it is this division which gave rise to all others. He refuses to believe that these phenomena are coincidence, instead pointing toward a causal relationship between the rise of gendered existence and that of domestication. Both are shifts away from non-separated, non-hierarchical life. He says: “nothing in nature explains the sexual division of labor, nor such institutions as marriage, conjugality or paternal filiation. All are imposed on women by constraint, all are therefore facts of civilization which must be explained, not used as explanations.” His explanation for these shifts involves both the ways that agricultural life immiserated the women it captured, but also that the introduction of patriarchy was a key strategy of colonial civilizers and missionaries around the world. He argues that any attempts to destroy civilization must also be an attempted return to “the wholeness of original genderless existence.”
Much of the primitivist perspective on gender doesn’t sit well from a queer perspective, significantly the emphasis on gender essentialism and the lack of substantive critique of compulsory heterosexuality, to say nothing of the role of Anthropology. And yet still there is something which resonates in the theory. Perhaps the appeal of the primitivist answer is that it implicates literally everything about this world in the horror of gender: the food we eat, the cities we live in, the language we speak, our families, our fetishes—all of it interwoven into the fabric of gendered existence. The implication, then, is that any break from gender would require a break from literally all the assurances and comforts which maintain our capture in it. Even more powerful, is a fiery insistence that our gendered existence is not inevitable nor laid out in the stars. Primitivism could be understood as an attempt to give words and evidence to a visceral experience of not-belonging in this world, to the feeling in our bones and muscles which cries out against the gendering of our lives and possibilities. Primitivism asserts an outside and makes claims to certainty regarding the nature of that outside. We’ll dispense with them on the point of certainty; but the outside itself calls to us.
\section{V}
One of the most lucid points that Fredy Perlman makes in \emph{Against His-Story, Against Leviathan!} is his critique of Anthropology. He often speaks of anthropologists and archeologists as “grave robbers,” whose intention is to enforce their own story about human existence while erasing all other stories. He pays particular attention to the efforts of anthropologists to describe the role of work in primitive societies. Many anthropologists, sympathetic to primitive societies, will claim that the people in those societies worked significantly less than domesticated people. They call them Hunters or Gatherers. They will speak of the four hours a day that are devoted to work. Fredy critiques this position by claiming that it is the operation of the managers of work camps to naturalize work into all other human and animal existence. Yes, primitive people worked less, but because they did not work at all.
\begin{quote}
Modern anthropologists who carry Gulag in their brains reduce such human communities to the motions that look most like work, and give the name Gatherers to people who pick and sometimes store their favorite foods. A bank clerk would call such communities Savings Banks! The [workers] on a coffee plantation in Guatemala are Gatherers, and the anthropologist is a Savings Bank. Their free ancestors had more important things to do.
The !Kung people miraculously survived as a community of free human beings into our own exterminating age. R.E. Leakey observed them in their lush African forest homeland. They cultivated nothing except themselves. They made themselves what they wished to be. They were not determined by anything beyond their own being—not by alarm clocks, not by debts, not by orders from superiors. They feasted and celebrated and played, full-time, except when they slept. They shared everything with their communities: food, experiences, visions, songs. Great personal satisfaction, deep inner joy, came from the sharing.
(In today’s world, wolves still experience the joys that come from sharing. Maybe that’s why governments pay bounties to the killers of wolves.)
\end{quote}
The assertion is simple, but profound: those who live in a world of work can only understand the activity of others as work. Work is a historically determined institution, and yet our civilized metaphysics operates to naturalize this institution; to obscure the violence of our domestication into it. The implications of this operation is all the more sinister, as we live in a world where more and more non-waged activities are subsumed into the world of work. In a sense, domestication functions as a linear enforcement of the world of work, colonizing our past as it does our future.
\begin{quote}
S. Diamond observed other free human beings who survived into our age, also in Africa. He could see that they did no work, but he couldn’t quite bring himself to say it in English. Instead, he said they made no distinction between work and play. Does Diamond mean that the activity of the free people can be seen as work one moment, as play another, depending on how the anthropologist feels? Does he mean that they didn’t know if their activity was work or play? Does he mean we, you and I, Diamond’s armored contemporaries, cannot distinguish their work from their play?
If the !Kung visited our offices and factories, they might think we’re playing. Why else would we be there?
I think Diamond meant to say something more profound. A time-and-motion engineer watching a bear near a berry patch would not know when to punch his clock. Does the bear start working when he walks to the berry patch, when he picks the berry, when he opens his jaws? If the engineer has half a brain he might say the bear makes no distinction between work and play. If the engineer has an imagination he might say that the bear experiences joy from the moment the berries turn deep red, and that none of the bear’s motions are work.
\end{quote}
If we are to attempt to imagine that none of the bear’s (or our distance ancestors, for that matter) activity is work, then we are forced to abandon to scientific disciplines which aim to make claims to certainty about what vanquished peoples’ activities were like. This is an important break from a primitivist orthodoxy which prioritizes the use of anthropological methods. It is understandable why one would want to make such claims as to the precise nature of an \emph{outside} or a \emph{before} civilization. We would assert, however, that such claims aren’t simply wrong (by virtue of their entrenchment in the scientific worldview) but that they are unnecessary to our critique. We do not need to be able to claim with certainty that our ancestors “worked less” in order to refuse the world of work that captures us. That we can point to the world of work as a historically determined institution of domination which emerged with domestication and continues to immiserate our lives is reason enough that world should burn.
This is a different orientation to the \emph{outside}. There is surely comfort and peace of mind in believing the scientific answers about what is outside. There is also a dignity and certainty which comes from believing that utopia once existed on the face of the earth. But what is left to us if we abandon these certainties? What remains is the a mystery and a chaos which evades any rationalist attempt to capture and put it to use. This unknown is precisely that which drives those who speak with certainty crazy. It is the dark and magical world of mystery which all the violence of the scientific operation aims to annihilate. Our proposal is simple: instead of deceiving ourselves about the unknown with this or that Positive Evidence, the unknown itself is something to celebrate. Rather than a primitivist return to an outside that is supposedly mapped into our biology; we’ll pursue an escape into an outside which is at the same time a mystery and an uncertainty. Should we fight less to escape if we don’t know what the outside looks like? One needs only look at the world which presents itself as all too certain to know the answer.
\section{VI}
In considering this provocation in the context of our inquiry into gender and domestication, a glaring contradiction emerges: why is Fredy’s willful embrace of the unknown (with regard to work) not likewise applied to gender? It takes very little effort to extend the critique of anthropological certainty into the gendered world. We could easily parallel it in saying: \emph{Anthropologists, sympathetic to primitive societies, will view the relationships between Men and Women as more fair and desirable in these societies than in civilized societies. They are wrong in that there is no relationship between Men and Women. They live in a world of gender, and so they can only perceive the varied and ineffable existences of others as conforming to those categories. An anthropologist with half a brain will say that these gender relationships are less rigid and dominating than the ones we experience; an anthropologist with an imagination would say that these are not gender relationships in the way we understand them at all.}
This critique can very easily be applied to almost all primitivist writings on gender. Perlman and d’Eaubonne are obviously implicated in this type of essentialism regarding the roles that women and men played in primitive cultures. The archetype of woman as the nurturing and pro-creative center of the universe is clearly as historically constructed by the division of labor, and yet it is all the more sinister because it operates as if natural. While Zerzan’s theory of gender is more overt in mobilizing anthropology, it opens space against essentialism by identifying gender as a socially constructed institution sutured on top of a natural sexual difference. This still warrants critique, however. One of the most worthwhile understandings offered by queer theory is the provocation that the sex\Slash{}gender dichotomy referred to by feminists over the last several decades is not two systems, but actually one. Sex as a binary is no more natural than gender. It is the historical and retrospective arrangement into two categories of a vast range of organs, hormones, gestures, dispositions, body shapes, sexual capacities, etc. The efforts on the part of transgender liberationists are relevant to this shift, as they demonstrate that there is no determinacy or cohesion between any particular arrangement of the above characteristics, but rather that the arrangement of them into categories is always a coercive attack on an individual. The recent struggles of intersex people goes further to clearly undermine the certainty which naturalizes binary sex. The quiet scientific and medical mutilation and reshaping of untold infants to fit into binary sex demonstrates that it is no more natural than binary gender. This institutional capture into one or another sex is just the newest form of what is an ancient regime of diet, medicine, labor, bondage, religion and taboo which functions to shape and exaggerate two sexes out of the vast infinity of possibilities contained by the human body. Sex and Gender are the same his-storical operation of categorization and separation, they are simply different articulations.
It is not uncommon for primitivist thinkers and anthropologists to have a critique of heteronormativity, pointing to evidence of widespread homosexual practices in tribal societies before their colonization. Others will also point to the existence of ‘third genders’ in certain tribes. These stories are relevant in that they undermine the naturalized view of heteronormativity (and with it reproductive futurism), but as long as they function scientifically, they still maintain the stability of gender (even third genders). They point to a more favorable gender arrangement, but lack the imagination to understand that people may have had relationships to one’s body and sexuality outside of the gendered cages which have been built around us. Furthermore, the tendency to universalize these conclusions is a tendency of Leviathan; homogeneity is intrinsic to the domestication process.
If we follow the analogous critique of work, we must come to a place where we can say that we do not know for certain what gendered existence was like before civilization. And yet this revelation in no way alters our certainty that gender \emph{as we know it} begins with civilization. If we invoke an orientation to an outside of civilized gender, then we are actually invoking another mystery, an ineffable which evades definition and capture. What would it mean to participate in life or death struggle against gender without knowing what existed before it? This would mean pursuing an outside which presents itself to us as shadows and chaos. It would mean fighting for the wild, without recourse to the natural. As we’ve intoned before: \emph{though we forego the privilege of naturalness, we are not deterred, for we ally ourselves instead with the chaos and blackness from which Nature itself spills forth.}\footnote{Susan Stryker, “My Words to Victor Frankenstein above the Village of Chamounix: Performing Transgender Rage,” \emph{GLQ: A Journal of Lesbian and Gay Studies,} issue 1 volume 3, 1994.} What we’ve elsewhere called queer desire is a tendency toward this primordial chaos. The task is to live it.
\section{VII}
Having unveiled this contradiction within primitivism, we are left wondering how this blindspot has remained for so long.
One of the beautiful aspects of the primitivist critique is that is provides a lens through which to explore every relation and institution that is naturalized in Leviathanic thought. Within the primitivist canon, one can readily find incisive attacks against the family, race, psychiatry, agriculture, the division of labor, specialization, militarism and countless other dimensions of civilized existence. Primitivists are perhaps at their most imaginative and insightful when they explore a world outside the more deeply embedded abstractions of Leviathanic culture: symbolic thought, numbers, art, language, even nature. Several texts even offer dreamlike attempts to imagine how free people have conceived of \emph{different shapes to time itself.}
How then, has this critical onslaught missed a relation so obvious and entrenched into our being? Those who claim that Civilization inaugurated gender disparity, still maintain the naturalness of those genders. Even those (like Zerzan) who call gender into question, still hold to a natural dualism which is perverted by domestication. That this dualism is considered natural by those who would otherwise refuse any other dualism (human\Slash{}animal, mind\Slash{}body, etc.) as a civilized constraint is not proof of its naturalism. Rather it is proof of how deeply entrenched it is in the process of domestication—so deep that we can scarcely imagine a world before it. Zerzan, to his credit, says the divide (which varies in its form, but not its essence) is the most deeply seated dualism; giving rise to the subject\Slash{}object and mind\Slash{}body splits in turn. He calls it a “categorization\dots{} that may be the single cultural form of greatest significance.” It introduces and legitimizes all other dominations. This line of argument is echoed by Witch Hazel in \emph{BLOODLUST}, who writes that the construction and devaluation of the feminine archetype is a parallel to the mind\Slash{}body split and enables the turn toward domestication and Civilized conquest. This central underpinning of Civilization already divines, without knowing it, the enmity between Civilization and queer desire articulated by Guy Hocquenghem and others; the way that queer desire reveals what is common between the family and the automobile and every other civilized apparatus. This lens allows us to see that in gender, more than anywhere else, the enemy has projected itself throughout time in order to preclude our dreams of an outside. As Fredy narrates this dynamic of projection:
\begin{quote}
The strait that separates us from the other shore has been widening for three hundred generations, and whatever was cannibalized from the other shore is no longer a vestige of their activity but an excretion of ours: it’s shit. Reduced to blank slates by school, we cannot know what it was to grow up heirs to thousands of generations of vision, insight, experience. We cannot know what it was to learn to hear the plants grow, and to feel the growth\dots{}
It becomes very important for the last Leviathan to deny the existence of an outside. The beast’s voices have to project Leviathanic traits into pre-Leviathanic past, into nature, even into the unknown universe.
The post-Hobbesian artificial beast becomes conscious of itself as Leviathan and not as Temple or Heavenly Empire or Vicarate of Christ, and it simultaneously begins to suspect its own frailty, its impermanence. The beast knows itself to be a machine, and it knows that machines break down, decompose, and may even destroy themselves. A frantic search for perpetual motion machines yields no assurance to counter the suspicions, and the beast has no choice but to project itself into realms or beings which are not machines.
\end{quote}
A telling story is that of the interaction between colonizing French Jesuits and the indigenous Montagnais-Naskapi in 17\textsuperscript{th} century Canada, as recounted by Eleanor Leacock, a feminist anthropologist cited by both Zerzan and Silvia Federici. She describes how it became necessary for the Jesuits to ‘civilize’ the Montagnais-Naskapi in order to ensure they’d be disciplined trading partners. This endeavor started with the introduction of hierarchical gender roles.
\begin{quote}
As often happened when Europeans came in contact with native American populations, the French were impressed by Montagnais-Naskapi generosity, their sense of cooperation and indifference to status, but they were scandalized by their ‘lack of morals;’ they saw that the Naskapi had no conception of private property, of authority, of male superiority, and they even refused to punish their children. The Jesuits decided to change all that, setting out to teach the Indians the basic elements of civilization, convinced that this was necessary to turn them into reliable trade partners. In this spirit they first taught them that ‘man is the master,’ that ‘in France women do not rule their husbands,’ and that courting at night, divorce at either partner’s desire, and sexual freedom for both spouses, before or after marriage, had to be forbidden.
\end{quote}
The Jesuits succeeded in convincing the newly appointed chiefs of the tribe to implement male authority over the women. Several Naskapi women fled such novel and offensive constraint, causing men (at the encouragement of the Jesuits) to chase after them and threaten to beat and\Slash{}or imprison them for their disobedience. One Jesuit missionary’s journal proudly includes an account of the incident:
\begin{quote}
Such acts of justice cause no surprise in France, because it is usual there to proceed in that manner. But among these people\dots{}where everyone considers himself from birth as free as the wild animals that roam in their great forests\dots{}it is a marvel, or rather a miracle, to see a peremptory command obeyed, or any act of severity or justice performed.
\end{quote}
Another interesting story is recounted in a brief segment from the journal \emph{Species Traitor} about homosexuality outside of civilization. The segment has the humility to acknowledge that while we can indict universalized homophobia as being unique to modern society, we can know very little about the vast and divergent sexual practices of the majority of cultures that have walked the earth. The segment goes on to cite an example of two anthropologists living among the Huaorani people in the Amazon region of what is now Ecuador. The two anthropologists witnessed two Huaorani men in an intimate embrace. When the Huaorani men saw that they were being watched, one quietly whispered to the other \emph{kowudi}, after which they looked embarrassed at the anthropologists and walked away. \emph{Kowudi} means \emph{outsiders}.
Both of these stories succinctly illustrate the truly partisan role played by those who operate under some notion of objectivity or neutrality. The journals of countless missionaries, explorers and anthropologists show that their accounts are tainted by their civilized attitudes toward gender and sexuality, but also that one of their primary operations is to force those attitudes upon the people they study. In \emph{Witchcraft and the Gay Counterculture,} Arthur Evans points to several of these, including a rather humorous example of the Greek historian Diodorus Siculus’ disgust at the behavior of Celtic men in the first century BC:
\begin{quote}
Although they have good-looking women, they pay very little attention to them, but are really crazy about having sex with men. They are accustomed to sleep on the ground on animal skins and roll around with male bed-mates on both sides. Heedless of their own dignity, they abandon without a qualm the bloom of their bodies to others. And the most incredible thing is that they don’t think this is shameful.
\end{quote}
All of this points to the great flaw of anthropology in regard to the question of gender. As the existence and universality of gendered categories is taken for granted, their accounts (and often their actions) will always function to enact a violence upon a wild range of human experience, severing it from its whole context and recounting that experience as an amputated and gendered one. This isn’t to say that we shouldn’t read these stories. Instead it instructs us on \emph{how} to read them. If we can glean any useful direction from them, it is by reading these scientists as we would read any other enemy; critically, and with attention to the secrets hidden between the lines. And even when we can distill this or that, we still only have one story, from one culture, in one moment. To universalize these stories as representations and truths about all of humanity, as is often done by primitivist anthropology, is to falsify our understanding and erase an infinity of other possibilities and stories of people beyond civilization’s snares. It is a reverence for this infinity which sets our inquiry apart from a scientific one. Science, after all, is also one myth among many. It is different only in that it refuses all stories but its own.
Some interpret these stories to mean that Patriarchy is one of the first pillars of civilization to emerge from domestication. Others glean that the gender division is the first duality, which makes domestication possible. Both versions draw circles around a third possibility:
Gender \emph{is} domestication.
The two supposedly distinct phenomena appear as mutually constituting because they are one and the same phenomenon. Earlier we said that domestication is the capture of living things by something non-living. It is also the process where capture is internalized by living beings who are then shaped into pre-determined roles. The non-living thing is immortal and continues long after its captives are dead, and that it is constantly accumulating new lives in order to reproduce itself. Gender is precisely this non-living institution which tears individuals away from themselves and reconstitutes them as a pre-determined role. Gender would be an empty husk if it wasn’t for its constant capture of new bodies; bodies which in turn give it life. Isn’t the first incursion of Civilization into the life of a wild newborn always to proclaim its gender? It is the first separation which gives rise to all others. Gender is the cipher through which Leviathan categorizes and understands each and every one of the beings trapped in its entrails. A whole destiny of experience is inscribed on our bodies from it.
We should also remember that we previous identified a theme where domesticated people invoke the image of those they are not and never were to justify their own machinations and violence. In gender, we see all the ways that the gender binary is naturalized as sex and projected into pre-history as a way of explaining and rationalizing (essentializing) all of these experiences of violence. We are told those assigned female are meant to be mothers, and therefore it is in their nature to endure pain, to be caretakers, to submit to external authority. Those assigned male are virile hunters and warriors, violence and rape are supposedly intrinsic to their nature. Homosexuals are aberrations in nature, and thus they are fated for exile in their short, brutal and diseased lives. Every mask of the natural is only ever a lie told by Leviathan to justify its own activity.
An understanding of \emph{gender as domestication} is supported by the inquiries of a handful of anti-colonial theorists of gender such as María Lugones, Andrea Smith and Oyèrónkẹ́ Oyěwùmí. Smith, for example, horrifyingly illustrates the use of sexual violence as strategy of Leviathan’s conquest of the Americas.\footnote{Andrea Smith, \emph{Conquest: Sexual Violence and the American Indian Genocide}, 2005.} More so, she argues that colonialism is itself structured by sexual violence. Lugones, as another example, argues that gender itself is violently introduced by colonial civilization.\footnote{María Lugones, \emph{Heterosexualism and the Colonial\Slash{}Modern Gender System}, 2007.} She says it is consistently and contemporarily used to destroy peoples, cosmologies and communities in order to form the building ground of the ‘civilized West.’ She argues that the colonial system produces different racialized genders, but more importantly institutes gender itself as a way of organizing relations, knowledges and cosmic understanding. This is useful because it refuses a universal or natural understanding of Patriarchy that lacks a critique of racial and heteronormative colonialism. Instead, her argument helps us to describe the gender as something that spreads, consumes and destroys. She describes this process as the Colonial\Slash{}Modern Gender System. This system entails the naturalization of the sexual binary, the demonization of a racial and hermaphroditic other, and the violent eradication of everything outside civilization: third genders, homosexuality, gynocentric knowledges and non-gendered existence, etc. Oyèrónkẹ́ Oyěwùmí in \emph{The Invention of Women} describes how gender was not an organizing principle in Yoruba society prior to colonization. She says that patriarchy only emerges when Yoruba society is “translated into english to fit the western pattern of body reasoning.” She locates the dominance of civilization’s gender system in its \emph{documentation} and \emph{interpretation} of the world. “Researchers always find gender when they look for it.”
Within colonialism, new subject categories were created by western Civilization and were racialized and engendered as the foundation of the new colonial state. This creation process is composed of several operations: the introduction and entrenchment of gender roles, the imposition of Male gods, the formation of Patriarchal colonial government, the displacement of people from their traditional means of subsistence and the violent institution of the Family. These operations serve as a revision which recasts and genders tribal life and spirituality. This engendering does more than create the victimized category of women, but also constructs men as collaborators in domestication. Lugones cites the British strategy of bringing indigenous men to English schools where they would be instructed in the ways of civilized gender. These men would work within the colonial state to deprive women of their previous power to declare war, bear arms and determine their own relationships. She also cites the Spanish strategy of criminalizing sodomy among colonized populations, intertwining it with racialized hatred of the Moors and other ‘primitive’ people.
These theorists employ stories and examples of ‘third genders’ not as a literal description of a three gendered system, but instead as a place holder for the infinite range of bodily possibility which exists outside the colonial system. They argue that domestication has to be imposed as gender in order to disintegrate all the communal and free relationships, rituals and overlapping means of survival. And as the civilized ideal of racial gender is naturalized, everything outside of itself is fair game for capture, domination and reshaping. Colonialism itself is often described through the racial and sexual metaphor of the white male explorer uncovering and pillaging the dark female continents, forcing her to submit and planting the seed of civilization.
From this perspective, we can recognize all the incidents of gendered and racial violence in our lives as repetitions of this first capture. Sex work, abusive relationships, body dysmorphia, marriage, sexual abuse, familial constraint, date rape, gang rape, queer bashing, psychiatry, electroshock therapy, eating disorders, domestic labor, unwanted pregnancy, fetishization, emotional labor, street harassment, pornography: each instance is a moment where we are torn from ourselves, taken by another, captured and determined as a brutal repetition of the primary rupture which denied us a life lived by and for ourselves. In this schema, the assimilation and medicalization of queer and transgendered people can be understood as a re-capture of rebellious bodies. Police murder and racist vigilantism can likewise be understood as functions of this capture.
It is worth noting here that to understand \emph{gender as domestication} is crucially different from understanding \emph{patriarchy as a consequence of domestication}, in that the former is a break from the trap of essentialism. None of the above is limited to one subject of the gendered world. Rape, for example, is not solely the experience of women (as is often claimed by various regurgitations of second wave feminism), but is a disgustingly widespread experience among people of all genders. The assertion that any form of gender violence is the exclusive property of one category of people would be laughable if it weren’t for the litany of horrors which serve to disprove it. More sinisterly, these type of essentialist assertions obscure and shame those experience an entire range of very real experiences of gender violence.
Situating gender as domestication is a way to understand gender violence outside of an essentialist and white framework. Without this understanding, all theories which attribute some natural dimension to sex\Slash{}gender (from eco-feminist to Marxist feminist) are structurally unable to account for the violence, capture, and exclusion experienced by anyone who deviates from the gender binary or the heterosexual matrix. These ideologies will expand to pay lip-service to queer and transpeople, but they never alter the structure of their theory. This amounts to little more than the liberal politics of inclusion. If, however, we understand gender as something which captures us, rather than something natural to us (or extracted from our biological existence), we can begin to analyze all the methods of domination experienced by queer or transgender people. Brutality and exclusion come to be recognized as the policing methods by which individuals remain captured; assimilation and exploitation represent a more sophisticated capture. From here I can see the line which binds together the boys who called me \emph{faggot} as a teenager and the gay men who would pay me for sex a few years later. Everything about the refusal of gender follows from this. The criticism of identity, assimilation, medicalization or any technique of the self becomes meaningful once it is placed in this continuum.
\section{VIII}
We’ve said there are some stories which can be stolen from anthropology that might help us in our understanding of gender as domestication. One such story is told by Gayle Rubin in her essay \emph{The Traffic in Women} (not to be confused with the Emma Goldman piece by the same name). This piece is one of the many examples of feminist anthropology which influenced Zerzan and other primitivist writers in their theory of gender. We chose to critically engage with Rubin’s piece for a few reasons. Firstly, within her work, there is a shift from feminist anthropology to queer theory; this feels analogous to shifts within our inquiry. Secondly, she conceives of her own writing as a practice of exegesis, of reading others against themselves to draw conclusions which are opposed to the author’s intentions. Specifically, she heretically reads Levi-Strauss and Freud, (apologists and technicians of gender) for the ways their theories can be subverted. This practice aligns interestingly with our abuse of a whole range of texts. And lastly, she defines her own project as being an attempt to understand the origins of ‘the domestication of women.’ While our own inquiry is more thorough than to be interested in only the domestication of one gendered subject, we cannot help but feel intrigued by a theory of gender that directly interrogates domestication.
In her text, she aims to find the ‘systemic social apparatus’ which transforms ‘females as raw material’ and ‘fashions domesticated women as a product.’ Rubin contends that this apparatus is significant because it dominates the lives ‘of women, of sexual minorities, and of certain aspects of human personalities within individuals.’ She calls this apparatus the \emph{sex\Slash{}gender system} and she believes that both anthropology and psychoanalysis inadvertently describe mechanisms by which this system constructs domesticated gender out of the occurrence of biological sex. It is unfortunate that Rubin advocates the sex\Slash{}gender dichotomy that we’ve critiqued above, but this oversight doesn’t prevent us from being able to use her study. After all, even without a conception of naturalized sex, we are still interested in understanding the social apparatus which transforms wild beings into domesticated gendered products.
Interestingly enough, she begins her exploration of this apparatus by first outlining the failure of Marxist feminism to account for it. She wrote \emph{Traffic} at a time when Marxist feminists such as Selma James, Mariarosa Dalla Costa, and Silvia Federici were articulating a theory of ‘reproductive labor’ and specifically the labor performed by housewives as being the root of women’s oppression and exploitation. This theory stemmed from a desire on the part of these women to locate a theory of gendered oppression that was a concomitant of the capitalist mode of production.
\begin{quote}
Food must be cooked, clothes cleaned, beds made, wood chopped. Housework is therefore a key element in the process of the reproduction of the laborer from whom surplus value is taken. Since it is usually women who do housework, it has been observed that it is through the reproduction of labor power that women are articulated into the surplus-value nexus which is the sine qua non of capitalism. It can be further argued that since no wage is paid for housework, the labor of women in the home contributes to the ultimate quantity of surplus value realized by the capitalist. But to explain women’s usefulness to capitalism is one thing. To argue that this usefulness explains the genesis of the oppression of women is quite another. It is precisely at this point that the analysis of capitalism ceases to explain very much about women and the oppression of women.
\end{quote}
This limit—the conflation of the exploitation of subjects by capitalism with evidence that capitalism is the origin of those subjects—is a flaw of all self-proclaimed ‘scientific’ disciplines which aim to generalize one story into a materialist theory that locates economics as the cause of all woes. Following from this, she identifies a wide range of non-capitalist cultures which are vehemently patriarchal, including pre-capitalist feudal Europe. She then details several practices of gender domination (foot binding, chastity belts, and other fetishized indignities) which cannot be accounted for by a Marxist analysis of the reproduction of labor power. She argues that at most, Marxist Feminism can explain the way capitalism seized upon and tinkered with already existing forces of social control. ‘The analysis of the reproduction of labor power does not even explain why it is usually women rather than men who do domestic work in the home.’ She argues that economics cannot account for the \emph{moral element} which determines that a wife is among the commodities needed by a man, that only men can talk to God, and that women are the ones who perform domestic labor. To her, this \emph{moral element} is the massive and unexplored terrain from which gendered violence emerges and that it is the basis of the femininity and masculinity that capitalism later inherited. It is into this element that she’ll direct the rest of her study. She concludes her critique of Marxist feminism by illustrating the silliness of reducing the vastness of the sex\Slash{}gender system to being simply ‘the reproductive’ sphere. For her, there is far too much excess in that system to be solely the reproductive aspect of industrial production. Not to mention that it is also productive in its own way: producing gendered subjects, for example. The origins of gender domination, she claims, must be located outside the ‘mode of production.’
Her attempt to find this outside is to first look at the writings of Levi-Strauss in his explorations of early kinship structures. His writing places gender and sexuality at the center of these structures; he develops a theory that links their essence to the exchange of women between men of various social groups. In doing so, Rubin believes he has sketched an implicit theory of gendered oppression. He primarily comes to this conclusion after studying the role of gift exchange in pre-state arrangements. He finds that the exchange of gifts was the first measure taken in the long road toward the development of ‘civil society’ and the state. For him, marriage is one of the most significant forms of gift exchange, with women themselves being the gifts given from one man to another. From here, he analyzes the incest taboo as a means of policing and enforcing this exchange of women as gifts. The taboo is less about preventing endogamous sexual relations, and much more about obliging the exchange of sisters and daughters into exogamous relations; it is an early expression of commodity society. The exchange of human beings is more powerful than other gifts because it is not simply an arrangement of reciprocity, but one of kinship. This results in a more long-lasting and expansive relationship which orders all other types of exchange through the established kinship network.
\begin{quote}
The marriage ceremonies recorded in the ethnographic literature are moments in a ceaseless and ordered procession in which women, children, shells, words, cattle, names, fish, ancestors, whale’s teeth, pigs, yams, spells, dances, mats and so on, pass from hand to hand, leaving as their tracks the ties that bind. Kinship is organization, and organization gives power.
\end{quote}
\emph{Organization}, then, is an original structure of power between those who exchange others. This difference between the exchanged and the exchangers is a primary split in the system we’ll call gender. For Rubin, the split is between men as organizers, and women as conduits to organization; men as exchange partners and women as gifts. The circulation of women provides the mystical powers of kinship to the men who exchange them; the men benefit from the subsequent social organization. The vast permutations of gendered \emph{organization} today will not deviate from this unending exchange of bodies. \emph{Women are given in marriage, taken in battle, exchanged for favors, sent as tribute, traded, bought, and sold. Far from being confined to the “primitive world,” these practices seem only to become more pronounced and commercialized in more “civilized” societies.} Rubin finds this concept useful because it locates gender’s emergence in social structures, rather than in biology. Further, it understands gender domination to be more rooted in the exchange of bodies than in the exchange of merchandise. Here, gender is not explained away as a function of reproduction, but is production itself. It is an entire system where individual bodies are produced as gendered subjects and exchanged in the production of kinship structures. This system does not just exchange women, but ancestry, lineage names, social power, children. The inauguration of gender violence emerges from this system within which sex and gender are organized; the economic exploitation of this or that gender is secondary to this.
This story is relevant to the larger one we’re trying to weave because it features gender as inextricably bound to a monster which is Rubin euphemistically calls social organization. We would call the monster domestication, and from this story we can determine a lot about its character and tendencies. Rubin of course, in typical academic fashion, shies away from the totality of these conclusions. She says that, since Levi-Strauss located this exchange as the beginning of the culture of civilization (“his analysis implies that the world-historical defeat of women occurred with the origin of culture, and is a prerequisite of culture”), holding to a firm interpretation of the theory would also imply that her “feminist task” would require the destruction of that culture. This destruction remains unthinkable in her system of thought. Again, we’ll choose to go where others will not. That an argument points to a necessary destruction of everything is precisely why we’d follow it.
The second story that Gayle Rubin recites is one more common: psychoanalysis and its Oedipus complex. Rubin correctly berates psychoanalysis for its tendency to become more than a theory of the mechanisms which reproduces gender and sexuality; she argues it has largely become one of those mechanisms. She follows that a revolt against the mechanisms of gender must then also be a critique of psychoanalysis. This critique isn’t new for us; Hocquenghem’s queer refusal of civilization is predicated on this very refusal of psychoanalysis. Rubin looks at the same concepts as Hocquenghem in an attempt to flesh out her theory of gender’s emergence. Primarily, she concerns herself with how psychoanalysis can hint toward the way children are forced into the categories of boys and girls. Her exegesis of psychoanalysis mostly centers around Lacan, who views his efforts as an attempt to identify the traces left in the individual’s psyche by their conscription into kinship structures, as well as the transformation of their sexuality as they are integrated into civilized culture. For Rubin this is a nice complement to Levi-Strauss; whereas the she had already examined the exchange of individuals within a gender system, she now turns to the interior realities of those exchanged. She begins from Oedipus:
\begin{quote}
Oedipal crisis occurs when a child learns of the sexual rules embedded in the terms for family and relatives. The crisis begins when the child comprehends the system and his or her place in it; Before the Oedipal phase, the sexuality of the child is\dots{} unstructured. Each child contains all the sexual possibilities available to human expression. But in any given society, only some of these possibilities will be expressed, while others will be constrained. Upon leaving the Oedipal phase, the child’s libido and gender identity have been organized in conformity with the rules of the culture which is domesticating it\dots{}
Oedipal complex is an apparatus for the production of sexual personality. Societies will inculcate in their young the character traits appropriate to carry on the business of society\dots{} such as the transformation of the working class into good industrial workers. Just as the social forms of labor demand certain kinds of personality, the social forms of sex and gender demand certain kinds of people. In the most general terms, the Oedipal complex is a machine which fashions the appropriate forms of sexual individuals.
\end{quote}
Psychoanalysis largely concerns itself with how a child can properly adapt to this machine. Rubin would say that the machine needs to be changed. We’ll assert that the machine must be destroyed. Rubin details how the machine functions along with an equally familiar concept, the phallus. She emphasizes that rather than being a biological object, the phallus is primarily a symbol of belonging to a gendered social order. The father possesses it, and so he can exchange it for a woman; if a boy behaves and is properly domesticated, he can one day have the phallus too. The girl is denied it, and thus has nothing with which to bargain for it. The phallus is transmitted through particularly gendered bodies and rests upon others. In the same way as the kinship system detailed by Levi-Strauss gives certain people the ability to exchange others as a commodity, the phallus is the mystical dimension of belonging which is traded for these bodies in turn. For Rubin, these systems cohere into a mutually reinforcing dynamic where women are dispossessed of their very being, and are possessed and exchanged by men. The linkage of these men through their exchange of the woman and phallus creates the social bonds upon which organized civilization is based.
Rubin emphasizes that any part of the body can be a site of active or passive eroticism. But by imbuing certain categories of similar anatomy with the social power of the phallus, domestication concentrates erotic power in certain geographies, tearing all other possibilities away from gendered individuals. Psychoanalysis argues that those gendered as girls are forced to accept their position within a gendered order where they’ve been separated from their access to the phallus, or to socially recognized eroticism. Traditional psychoanalysts describe this as the formation of feminine personality. Rubin breaks from them in describing it instead as a socialized enforcement of psychic brutality which forces young children to internalize a logic of submission. The normative interpretation is that one learns to accept this submission and take pleasure from it. Here the scientists of psychoanalysis allow for the triumphant return of biological essentialism—linking the pain of penetration and child birth to a now rationalized internalization of submission. Rubin will argue that this theory normatively functions to naturalize and justify the gender order, and must be attacked for this function. She proposes a more subversive reading of it as a diagnostic of exactly how this machine functions. Our reading of it should elucidate how that machine can be irreparably sabotaged.
For Rubin, a subversive reading of these two stories begins to unveil aspects of the gender system which would otherwise remain hidden. She calls them preliminary charts of the social machinery. Others today would call it a study of apparatuses. In these charts, she reads a system that is so intractable and monumental that it cannot be exorcised through miniscule reforms. For her, the neat congruity between the two stories indicates that the ancient methods of capture and exchange are still at work in the present. She calls these methods domestication. She argues that domestication will always happen and that the wild profusion of sexual possibilities in the human body will always be tamed. And so she rather cynically argues for a ‘feminist revolution’ to seize this machinery and use it to ‘liberate human personality from the straightjacket of gender.’ We don’t have any hope that this machinery will ever be destroyed on a global scale, but this does not mean that we believe in seizing it for our own use. (Just as we are not interested in seizing state power or the means of production). Our anarchy is the destruction of these machines and our escape from them. Fredy Perlman argued that Leviathan is a dead thing which only has an artificial life when living things inhabit it as captives. If we say that gender is domestication, then Leviathan is one and the same as the gendered machinery described above. Seizing the machinery will only continue the nightmare that is gender: we have to find an escape route.
Rubin argues that these disciplines, psychoanalysis and anthropology function as the most sophisticated rationalization of the sex\Slash{}gender system. We can see this as parallel to the argument made earlier regarding anthropological documentation\Slash{}enforcement of heteronormativity. Surveillance is always a function of policing. Those sciences which aim to analyze the world become blueprints for how the world might be structured to fit their vision of it. We believe that this is true of science in general; later we’ll contend that the same holds for the science of historical materialism. And so just as we must develop an antagonistic reading of anthropological stories, we must also develop a reading of these maps. In them we aren’t looking for how to maintain or even alter the machines. We are reading them as a prisoner might study the stolen blueprints of a prison; as an enemy operation, seeking the points at which they fail. These blueprints are of absolutely no interest for us, save for the image of the world we aim to leave; and even still, these images are two dimensional, bare lines, inscrutable symbols.
The map presented to us is not the one drawn by Marxist feminism. Economics form a dimension of our entrapment, but it is not the end all and be all of gender. The terrain is sexual, psychological, ancestral, familial, technological and moral. It may be economic and political too, but not in any privileged sense. The gender system approaches a totality of all the ways we are captured and the ways in which we internalize that position. Rubin even suggests that the state-form itself may have emerged from this shadowy web of phallic kinship. If we cannot understand and combat gender as a totality, we will never be able to break the curse of the ancient fathers.
While we disagree with Rubin on several of her (mostly political and feminist) conclusions, and are rather bored by her form and obsession with the writings of men of science, we have to appreciate her for her line of inquiry. We can draw on her both in terms of her practice of heretical reading, but also for her unwillingness to accept the simple answers. By problematizing both the conceptions of gender as natural and also as economic, she offers a way of avoiding the pitfalls of an eco-feminist or Marxist-Feminist theory. Her approach is one that is worthwhile if our intention is to locate gender at the moment of domestication; no more and no less.
Perhaps most usefully her two stories correspond to what we might identify as a twofold nature of domestication: bodily and spiritual. On the one hand, domestication takes the form of the capture and exchange of bodies within a social order. On the other, it involves the spiritual taming of those individuals; the internalization of a spirit of submission. These are not two isolated phenomena, but are mutually constituting elements of a self-reproducing dynamic of gender. Form and content. After all, a spiritual linkage is the result of the exchange of body-commodities, just as the Oedipal logic of submission accompanies the entrapment within a particular arrangement of the body. Each assault and constraint upon the body fosters the development of a docile spiritual disposition. Each alienation and dispossession from some dimension of our bodily existence leads to an analogous fragmenting of our psyche. The dualities of sex and gender can be understood as bodily form and spiritual content of the domestication process. The symbolic re-ordering of the body (as in the Phallus) has an accompanying fetish. All the victim subjectivities follow directly from this capture of the body. Equally so, our spiritual complicity with the gendered Leviathan drives us to exchange bodies in pursuit of some mythical belonging. This interplay leads to the creation of the gendered body and the domesticated spirit. This is elsewhere called identity formation. The dualities of sex and gender can be understood as bodily form and spiritual content of the domestication process.
We must take the understanding further than Rubin, by conceptualizing the duality of race as intrinsic to this bodily and spiritual dynamic. In the same way that gender splits bodies and marks them for circulation, race further elaborates this separation. Those captured as black women, for example, were circulated within the slave system and marked as hyper-sexual, perverse, and strong; justifying their rape, hard labor and forced reproduction. The children they produced were taken from them and circulated, while they themselves were forced to wet nurse the white children of their masters. The racist figures of the mammy and the sexually aggressive woman were (and still are) put to use to justify the circulation and domination of the bodies of black women.
We obviously must also take Rubin’s account to task for the latent essentialism within it. While she herself mimes some critique of them, she ends up importing far too much of a conception of naturalized gender from the men she reads. It is up to us to locate this dynamic of bodily and spiritual domestication as being the foundation of all gendered violence, and not simply of the violence against women. We’ve already said that no gendered violence belongs to any one category, but it bears repeating. This dynamic is at much at play in the systematic abuse of young boys by priests as it is in the gang rape in military barracks and fraternities, as it is in and sex slavery in prisons. The circulation of bodies is obvious in these extreme instances, but it is also more subtle: in advertising and pornography (gay and straight), in dating (of the monogamous or polyamorous varieties), in sex work and service work, in the technophilic ways we cruise, and in the ways we learn. It is present in the ‘my’ which always corresponds to boyfriend, wife, daughter, partner. It is what remains unspoken in initiatory rites of secret orders of husbands, rapists and jailers. All of it—from the most abominable to the most minute—is the unending dynamic of bodily capture, spiritual submission, and circulation.
\section{IX}
\begin{quote}
While the ecstasy of the former living community languishes within the Temple and suffers a slow and painful death, the human beings outside the Temple’s precincts but inside the State’s lose their inner ecstasy. The spirit shrivels up inside them. They become nearly empty shells. We’ve seen that this happens even in Leviathans that set out, at least initially, to resist such a shrinkage.
As the generations pass, the individuals within the cadaver’s entrails, the operators of the great worm’s segments, become increasingly like the springs and wheels they operate, so much so that sometime later they will appear as nothing but springs and wheels. They never become altogether reduced to automata; Hobbes and his successors will regret this.
People never become altogether empty shells. A glimmer of life remains in the faceless\dots{} who seem more like springs and wheels than like human beings. They are potential human beings. They are, after all, the living beings responsible for the cadaver’s coming to life, they are the ones who reproduce, wean and move the Leviathan. Its life is but a borrowed life; it neither breathes nor breeds; it is not even a living parasite; it is an excretion and they are the ones who excrete it.
The compulsive and compulsory reproduction of the cadaver’s life is the subject of more than one essay. Why do people do it? This is the great mystery of civilized life.
It is not enough to say that people are constrained. The first captured may do it only because they are physically constrained, but physical constraint no longer explains why their children stick to their levers. It’s not that constraint vanishes. It doesn’t. Labor is always forced labor. But something else happens, something that supplements the physical constraint.
At first the imposed task is taken on as a burden. The newly captured one knows that he is not a ditch-repairman, he knows that he is a free Canaanite filled to the brim with ecstatic life, for he still feels the spirits of the Levantine mountains and forests throbbing inside him. The ditch-fixing is something he takes on to keep from being slaughtered; it is something he merely wears, like a heavy armor or an ugly mask. He knows he will throw off the armor as soon as the manager’s back is turned.
But the tragedy of it is that the longer he wears the armor, the less able he is to remove it. The armor sticks to his body. The mask becomes glued to his face. Attempts to remove the mask become increasingly painful, for the skin tends to come off with it. There’s still a human face below the mask, just as there’s still a potentially free body below the armor, but merely airing them takes almost superhuman effort.
And as if all this weren’t bad enough, something starts to happen to the individual’s inner life, his ecstasy. This starts to dry up. Just as the former community’s living spirits shriveled and died when they were confined to the Temple, so the individual’s spirit shrivels and dies inside the armor. His spirit can breathe in a closed jar no better than the god could. It suffocates. And as the Life inside him shrivels it leaves a growing vacuum. The yawning abyss is filled as quickly as it empties, but not by ecstasy, not by living spirits. The empty space is filled with springs and wheels, with dead things, with Leviathan’s substance.\footnote{\emph{Against His-Story, Against Leviathan!}.}
\end{quote}
\section{X}
We’ve discussed domestication as a process that ensnares us within a monster and infests our very being with the monster’s essence. We continue to endeavor to name this monster gender. Fredy Perlman called it Leviathan, but he also had a name for its spirit: His-story. If domestication integrates us into the form of Leviathan, then it enchants us with His-story. So we turn to this enchantment:
\begin{quote}
His-story is a chronicle of the deeds of the men at the phallus-helm of Leviathan, and in its largest sense it is the “biography” of what Hobbes will call the Artificial Man. There are as many His-stories as there are Leviathans.
But His-story tends to become singular for the same reason that Sumer and now the whole Fertile Crescent becomes singular. The Leviathan is a cannibal. It eats its contemporaries as well as its predecessors. It loves a plurality of Leviathans as little as it loves Earth. Its enemy is everything outside of itself.
His-story is born with Ur, with the first Leviathan. Before or outside of the first Leviathan there is no His-story.
The free individuals of a community without a State did not have a His-story, by definition: they were not encompassed by the immortal carcass that is the subject of His-story. Such a community was a plurality of individuals, a gathering of freedoms. The individuals had biographies, and they were the ones who were interesting. But the community as such did not have a “biography,” a His-story.
Yet the Leviathan does have a biography, an artificial one. “The King is dead; Long Live the King!” Generations die, but Ur lives on. Within the Leviathan, an interesting biography is a privilege conferred on very few or on only one; the rest have dull biographies, as similar to each other as the Egyptian copies of once beautiful originals. What is interesting now is the Leviathan’s story, at least to His scribes and His-storians.
To others, as Macbeth will know, the Leviathan’s story, like its ruler’s, is “a tale told by an idiot, full of sound and fury, signifying nothing.” The ruler is killed by an invader or a usurper and his great deeds die with him. The immortal worm’s story ends when it is swallowed by another immortal. The story of the swallowings is the subject of World His-story, which by its very name already prefigures a single Leviathan which holds all Earth in its Entrails.
\end{quote}
A friend, writing in the nihilist journal \emph{Attentat}\footnote{Anonymous, “History as Decomposition” in \emph{Attentat, the journal of the nihilist position}, 2013.}, takes this to mean that Leviathan is constantly decomposing and that its biographers are trained not to see this decomposition. Instead, historians and intellectuals engineer stories to explain the movement of the beast through time. This is often called History, but can also Progress, Destiny, etc. The writer in \emph{Attentat} says that this subtle contention in Fredy’s thinking entirely breaks from any linear (either progressive or regressive) view of history, arguing instead that history is
\begin{quote}
a process of increasing complication, destructiveness, falling-apart of previous epochs (along with their attitudes, ideas, practices, and so on)\dots{} The very phenomenon of history (as His-Story), its possible unity as narrative and idea, is peculiarly undergirded by this process, which is itself a fragile hanging together of fragments of fragments, endlessly shattering, strangely recombining, giving most observers the sense of ‘delay.’
\end{quote}
In the first issue of this journal, we explored this sense of delay as the perpetual displacement of a future utopia promised to us by the soothsayers of Historical analysis. \emph{It gets better} if only we are patient enough to wait. Most accounts of history are simple variants of this impetus to wait—for the material conditions, for heavenly ascent, for the messiah, for any number of ways to describe the wholeness which awaits us at the end of this or that dialectic. Camatte called this delay the \emph{wandering of humanity} away from its course. We’ll follow our nihilist friend in giving up on this understanding of delay and looking instead to decomposition. This sense of delay cannot be trapped in any periodization (however technical or refined), but rather is descriptive of the whole of time consumed within history. This is the same reason that apocalyptic visions have also always defined the endpoint of Leviathan’s conception of itself. History is the narration of perpetual decomposition.
\emph{Attentat} argues that such a conception of history would mean an awareness of the unique character of events, but without locating them in any temporal logic (\emph{order, progress, explanation, justification}). We interpret this as a collection of stories which hint toward the beast’s tendencies, but never ascertain its totality. Taken as a whole, these stories do not offer a cohesive metanarrative, only fragmenting.
\begin{quote}
The negative or destructive side of history is for some of us more or less all that history has been or done. In the strict sense, nothing is being worked on or built up in or through history. The places, people, and events in past time that we enjoy or claim, appreciate or appropriate, must be creatively reidentified as non-historical, extra-historial, or anti-historical currents.
\end{quote}
Any attempt to systematize the episodic explosions of revolt only rationalizes its defeat, reducing it to just another triumph in the perpetual motion of the decomposing beast.
\begin{quote}
In sum, the perspective that says that decomposition is the logic of His-Story elucidates two things. First, that we were right to deny Progress; second, that we are not believers in its opposite, an inverted Regression away from a golden age. As I imagine it, a principal characteristic of whatever preceded His-Story (civilization, etc) would be its neutrality, its stony silence at the level of metanarrative. Rather than Progress or Regression we could describe historical decomposition as the accelerating complication of events. This acceleration is violent and dangerous. Here and there an eddy may form in which things either slow down or temporarily stabilize in the form of an improvement. What we can say with some certainty is that as historical time elapses, things get more complicated; and these complications so outrun their antecedents that the attempt to explain retroactively becomes ever more confusing.
Situationally, we may be getting some purchase for the moment, an angle, a perspective. But what Debord perhaps could not admit, what Perlman perhaps understood, is that decomposition had always been there in our explanation, our diagnosis, and the actions they are said to justify; and that His-Story is decomposition’s double movement: as Civilization unravels, it narrates its unraveling. The dead thing, Leviathan, organized life, builds itself up as armor in and around it (which would include machines and a certain stiffening of postures and gestures, and concurrently thinking and action, in human bodies). But the dead thing remains dead, and it breaks down. It functions by breaking down. It creates ever more complex organizations (analyses of behavior) that then decompose, i.e. break down.
\end{quote}
If the question of his-story is always already the gender question, then this perspective is crucial to our inquiry because the dead thing in question is gender—the ordering of life, the stiffening of our gestures. But gender has no life of its own. It destroys everything before it, then breaks down, it decays, and its decomposing parts are reorganized again. We are split in half, body and soul are recomposed into a gendered unity which itself decays, we rebel and then this rebellion is identified, split once more. It is this interplay of decomposition and recomposition that concerns us. \emph{What is this re-capture of life other than domestication all over again?} Where do we locate \emph{gender as domestication} if we can see decomposition and recomposition everywhere?
The theories we’ve critiqued have all been attempts to tell an origin story—to historically place gender. But gender cannot be situated at any point along a linear narrative: it is our very inscription into the line. Some theorists of gender will become obsessed with this task: universalizing and totalizing what is really incidence. The outside to gender is not situated at either end of this line, (nor within any neat periodization) but rather where the line breaks apart. If we decide to listen to the self-narration of this breaking apart, then it is because we might hear something within it (maybe a background noise, or a meaningful pause) which shows us where the decomposition can be hastened, where we might sneak out, or ways that others have attempted to evade being recomposed. This is how we can situate our perspective against his-story, Leviathan, gender, et al.
\section{XI}
In the last few years, there have been several attempts within the anarchist milieu to historicize gender. These attempts have largely focused on readings of two books about the same time period: \emph{Caliban and the Witch} by Silvia Federici, and \emph{Witchcraft and the Gay Counterculture} by Arthur Evans. \emph{Caliban} represents a very thorough analysis of the mechanics of gender during the imposition of capitalism, specifically exploring the European colonialism as well as witch hunts in western Europe as a case of the accumulation of women’s bodies and labor. \emph{Witchcraft} narrates the same story, but from a different perspective.
While \emph{Caliban} is worth reading for its wealth of information, its structure is largely problematic. Federici holds to an essentialist view of gender; she wants to tell the story of capitalism’s relationship to \emph{women}, a category she firmly defends. She dismisses all challenges to the naturalization of the gender binary with little more than an assertion of its correctness. Her tautology (\emph{that the category of women is valid because it is a valid category}) is all the more absurd in that she conflates the experiences of women in one part of the world, during one time period, as being the basis of the gendered reality for women all over the globe, at all subsequent times. Consequently, her work wholly ignores the gendered violence against bodies which do not fit within her neat categories. The vast persecution of faggots during the Inquisition and witch hunt, to name one example, is afforded little more than a scarce mention in her book.
To her credit, she does challenge the orthodox Marxist interpretations of History: she claims that the rise of capitalism cannot be seen as progressive if looked at from the perspective of gender, but also that there is no linear transition to capitalism—only a series of violent episodes of capture and reversal. And yet still her perspective remains all too limited by her own autonomous variant of Marxism. For her, all of the atrocities of the witch hunts are to be explained by analyzing the economic necessities of the capitalist mode of production. More specifically, these atrocities are necessitated by the requirement that women perform reproductive labor within the newly forged proletariat. This could be read as a useful movement away from the absurd notion (held by Federici and her contemporaries) that contemporary gender violence can be uniquely and primarily explained in the domestic labor of European women in the last century. Yet still her cathexis upon economics feels like an attempt to project the same notion into the more distant past. We’ve already discussed the limitations of this approach with regard to gender; the re-orientation toward an earlier period doesn’t change these limits. The text feels all the more limited for the fact that she makes maybe two mentions of the existence of gendered violence before this period and offers no explanations as to how that violence came about. This leaves us with that same poverty of naturalized gender.
A central theme of her work is \emph{primitive accumulation}; the first accumulation of a population by Leviathan. She sees this as a transition in her own teleology. However the beast against which her subjects revolt is not born of this or any fixed period, it is constantly decomposing and being born anew. Its mode of capture in the form of gender is not predicated on its mode of production; it is firstly a bodily and spiritual operation upon which an economic mode is sutured. Her story begins amidst a revolt precisely because its subjects are rebelling against their earlier capture. The following round of accumulation, consequently, cannot be \emph{the first}. It is also worth noting that in her exhaustive narrative of the history of the witch hunts, she remains dismissive if not silent about the role of magic itself. This amounts to a purely Materialist reading which cannot account for the spiritual dimensions of domestication as capture. Federici’s tale is one story about an intensification of the process we call gender. She may be wrong in situating that story within a specific periodization, and in her account of why the events played out, but we’re willing to sift through to glean what we may from it. Our instinct is that she may well be correct to pay particular attention to these events, but only on the chance that those rebels burnt at the stake may reveal some occult secrets regarding their own conflict against Leviathanic gender.
Arthur Evans’ book is more interesting in that it diverges from Federici’s on these exact points. Where she asserts an essential Woman, he specifically explores the witch hunts as an attempt to destroy a whole range of sexually deviant and gender variant people. Where Federici limits her critique to the rise of capitalism, Evans indicts all of western civilization in his. Where Federici is indifferent to the practices and beliefs of her story’s victims, Evans tries to listen and perceive what arcane revelations they might offer in a violent and anarchic war against gendered civilization. He also weaves a critique of History throughout his text; indicting (as Fredy Perlman does elsewhere) historians for their complicity in the aggrandizing of Leviathan and the erasure of those it has tried to destroy. Most provocatively, he carves out space for myth within his narrative. And yet still he doesn’t go far enough. Instead of an anti-history, he counters with \emph{Gay History}, as if history’s only problem was its homophobia. As with Federici’s naturalization of the category women, we must also flinch at Evans’ uncritical deployment of some universal \emph{Gay People} into which all the divergent and unique heretics fit. This categorical construction is the exact recomposition alluded to in the \emph{Attentat} piece; the swallowing whole of so many decomposed fragments by a reincarnation of gender. A queer critique must sidestep this trap.
So why read these books? What remains of them if we strip away the grand metanarratives about the movement of abstractions like History; or if we refuse to impose our contemporary subject categories back through time? The remainder is a collection of stories. And these stories differ from his-story, in that they are about the exploits and adventures of individuals, not the machinery which holds them captive. Stories interest us also because they do not seize upon this or that time, but enchant the teller and the listener into active participation. The story is the primary method of the magical practices of oral culture. His-story is the Socratic Ideal of these stories, the One story which cannibalizes all the others. Critical histories like \emph{Caliban} and \emph{Witchcraft} (or any ‘people’s history’) only serve to integrate these tales into the all-consuming one. This becomes a game of abstraction; how a collection of trial statements, handbooks on inquisition, heretical documents and biographies of accused witches become \emph{The Accumulation of Women Within Capitalist Mode of Production} or \emph{Gay History} or \emph{The Old Religion}. Interestingly enough, it has been argued by some (such as David Abram) that
\begin{quote}
the burning alive of tens of thousands of women (most of the herbalists and midwives from peasant backgrounds) as “witches” during the sixteenth and seventeenth centuries may usefully be understood as the attempted, and nearly successful, extermination of the last orally preserved traditions of Europe—the last traditions rooted in the direct, participatory experience of plants, animals and elements—in order to clear the way for the dominion of alphabetic reason over a world increasingly construed as a passive and mechanical set of objects.
\end{quote}
It is not surprising that as a consequence most accounts of this period suffer from the tragedy it imposed upon our conceptions of ourselves and of time. To truly read against His-story is to read with attention to the stories themselves, without an attempt to systematically or universally place them.
In a world that lacks the abstract ideals, directionality and universal moralism of His-storical thought, stories are useful in that they tell us discreet lessons that might assist us in our day to day conflicts. Only when we stop trying to decipher the Truth of His-story, can we actually notice the subtle web of meanings and messages hidden between the stories at our disposal. Here are a few we’ve noticed:
\begin{itemize}
\item\relax
Most of the stories about the imposition of gender are also stories about the creation of institutions and the flight of individuals from them. At times called enclosure or industrialism, these institutions tend to separate us from the vast experiences of life. Once we could find our own food, make our own clothes, discover our own sexual practices, heal ourselves and commune directly with the wild spirits. Now all of these experiences are mediated through farms, schools, churches, and hospitals. The institutionalization of the world could be understood as the material armors of the spiritual poverty imposed through domestication. This institutionalization is always violent. The ascendency of institutional medicine, for example, emerges out of the ashes of herbalists burnt alive by witch hunters. Gender, is constantly re-defined and re-inscribed through these institutions. Foremost among them is the Family. The enclosure of forests and fields corresponds to an enclosure of peoples’ means of care and survival into this private familial unit. The family becomes the primary unit for enforcing private property, enforcement of discipline, and policing of sexuality.
\item\relax
While Leviathan attempts to swallow the entire world, devouring any divergence, it inadvertently brings the outside within. Christianity made this law: \emph{Thou shall have no other gods before me}. The Nazis attempted to perfect this racist science as \emph{Gleichschaltung.} But the elimination of wild diversity is never total. The newly internalized divergence often re-emerges in the form of a heresy. This constant rupture of hegemony often seems like a widespread decomposition of the unity of this or that institution. Regarding gender, this heresy is elsewhere called the queer. Leviathan will, from time to time, deploy a specialized force of police to put down these heresies; these are called inquisitions. The holy war comes home, the war against the outside is turned inward. Little will be known of the doctrines and practices of these heretic sects, for the inquisition’s method is also His-storical: it aims to annihilate their stories as much as their bodies. These inquisitions, whatever century they occur, will each emerge as a more advanced and innovative laboratory of torture and subjection; the most perverse in the recorded history of state repression. No expense will be spared in eliminating these internal colonies.
\item\relax
Active resistance to Leviathan often takes on an ecstatic character. Fredy Perlman will refer to the great dances spreading like fire throughout leagues of deserters. Inquisitors and witch hunters will be haunted by the image of nighttime orgies and \emph{sabbats}. Elsewhere we’ve written that queer desire is the locus point of the dread of an entire social order’s self-annihilation. The most beautiful moments of insurgency are immanent to a decomposition of gendered and sexual roles. Ecstasy, from \emph{ekstasis}, is to be outside one’s self. To flee from domestication is also to flee from the selves (in both their bodily and spiritual dimensions) to which we’ve been constrained. To be outside these selves is the initial break. These breaks are often couched in the language of their times: as animism, or renewal of long vanquished deities, the apocalypse as an immanent lived reality. What is consistent is the emphasis on direct and immediate joy. These eruptions of revolt are not limited to this or that historical period, but are universal throughout His-story. They happen in cities, in the countryside, amidst the peasantry, and in labor camps.
\item\relax
The repression of this ecstatic revolt will always include a sexual dimension. This repression aims to reinscribe the body and spirit of the resisters into their domestic selves. The use of sexual violence as a repressive tactic or the almost universal conflation of criminal charges against homosexuality, heresy and witchcraft help to illustrate this.\footnote{In the handbooks of inquisitors, homosexuality and witchcraft are virtually indistinguishable. From the 1619 \emph{Discours des Sorciers}: “You may well suppose that every kind of obscenity is practiced there, yea, even those abominations for which Heaven poured down fire and brimstone on Sodom and Gomorrah are quite common in these assemblies.” The \emph{Theologia Moralis}, published a few years later, explained that sodomy was a sort of gateway drug to witchcraft.} Many witch hunters implied or explicitly accused witches of having sexual relations with their animal \emph{familiars}, continuing the Christian tradition of separating humanity out of the rest of the living world, while marking \emph{the beastly} as worthy of domination. Nudity, hallucinogens and unkempt hair all become sensual crimes of the body. Collective forms of sexuality and sociality are criminalized in order to maximize productive time. Rape is consistently used as a tactic of domination by conquering armies, torture by inquisitors, and division amidst rebel populations. The state, at various moments, institutionalizes and subsumes prostitution, both as a pressure valve against revolt, but also as a cure for deviant sexual practices. Non-reproductive sexualities are annihilated both for the challenge they pose to the emergent heterosexual matrix, but also for the conspiracies and escape plots implied in these relations. Indigenous resisters are always denounced by missionaries as lacking morality regarding sexuality and gender; this immorality is mobilized in expansive fantasies of colonialists and pioneers. The bodies of colonized resisters are marked for rape and execution. These operations lay the groundwork for the genocidal endeavors of witch hunts and holocausts. As we are alienated from the world, we are alienated from our bodies.
\end{itemize}
In order to pre-empt this type of escape from ourselves, Leviathan must institute ever more complex Subjects for its constituents. These subjects are the end result of a litany of techniques aimed at mechanizing, disciplining , emotionally manipulating and controlling the human body. The reduction of certain bodies to baby-factories is a prime example, but also the scientific diagnostics of various sexual deviants or the disciplinary control of gender variant people. Those who willfully or instinctually resist these techniques must be classified as Other. This othering is often composed of racializing and gendering processes. Against these Others, no violence is excessive. The Other, whether Witch or Terrorist or Drapetomaniac or Faggot or whatever, is the legitimate recipient of all sorts of brutalities designed to either assimilate or annihilate the deviancy. These crimes become \emph{crimen exceptum}.
Once Leviathan has constructed its institutions and the corresponding machine-like bodies, its primary project becomes the movement of these tendencies toward infinity. All of our efforts to critique the \emph{The Child} in the previous issue of this journal are in response to this project of uninhibited growth. Those who practice any form of resistance to this project must therefore be the Other worthy of annihilation. The Child functions as the fantastic future of the parent’s race. Any decline in the (civilized) population will be seen as a threat to the state, which in turn will ramp up the techniques of sexual repression described above. Workers and Slaves will be encouraged to produce more workers and slaves. In these moments, the sexual and abortive dimensions of heresy and witchcraft will come to the forefront of the inquisition trials. It is not a coincidence that witches and queer heretics were executed for having allegedly sacrificed children to the Devil. The demonization of birth control can also be understood through this lens. This fanatical desire to increase population lead even the most misogynist religious and state leaders to proclaim that women’s sole virtue was their natural capacity for childbirth. As Martin Luther said: “whatever their weaknesses, women possess one virtue that cancels them all: they have a womb and they can give birth.”
Rationalism, Reason, Enlightenment (or any other lie told by Leviathan about itself) never lead to the abolition of these genocidal and bloodthirsty practices. Rather, these ideologies only lead to the institutionalization and increased technological sophistication of violence. These ideologies end up serving as justification for brutality against the irrational Other. There is no linear progress out of this brutality. While the good subjects are may be encouraged to infinitely reproduce, the actual children of the racial or colonized Other will often by slaughtered with impunity. Even while promoting the ideology of the Child, the state is constantly and discreetly acting to impose a scientific campaign of eugenics, extermination and forced sterilization upon those it deems to be a racial outside.
These are only a few of an infinity of lessons we might extract from any constellation of stories—lessons which have as much relevance today as they would in centuries past. Rather than a narrative about Domestication as an Idea, we have a fragmentary and esoteric set of tales that each describes what domestication looks like in a particular moment. More excitingly they also describe how people chose to rebel against this process. To tell ourselves these stories is to connect to the individuals and moments which have attempted an escape from the nightmare of His-story. This connection becomes most meaningful when the stories enchant our own being and are given body through our own experiences. These stories only matter insofar as they produce a visceral understanding of flight from this ancient protocol of separation and capture. This is the dimension that must always be centered in a newfound reading of His-story as decomposition. Decomposition isn’t only a force of nature or accident; it is primarily the willful refusal of Leviathan by individuals and groups. Leviathan breaks down when those who maintain its springs and wheels refuse to do so—when they flee to the mountains, sing, dance and practice ecstatic ritual; when they scream, loot and burn; when they rip out the armor, tear off the mask and burn the beast to the ground.
If these stories illustrate instances of domestication, they also illustrate the imposition of gender. The inherent decomposition which afflicts gender is what we call the queer; not this or that historically constituted subject category, but all the divergent bodily and spiritual expressions which escape their roles. In the first issue of the journal, we said that this was a queerness understood negatively. As rebellion\Slash{}decomposition is intrinsic to stories about domestication, so is the outpouring of queer desire.
For this reason, dogmatists (particularly of the Marxist variety), have accused us of being \emph{ahistorical} and \emph{idealist}. To the former, we have no rebuttal. We’d happily find ourselves outside of the Story of mass rapists, kings and industrialists. We certainly won’t cling to any of the Identities offered within it, nor trust any of the prescriptions laid out by its Scholars. Even worse would be to be \emph{organized} by such a prescription of history. When our friends in \emph{Attentat} described the recomposition and further decomposition which follows any decay of history, we read this as the Organization which follows moments of rupture, and the predictable falling-apart of all such political organizations. If we follow Rubin to say that all Organization is predicated on the exchange of gendered bodies, then we must also recognize inevitable rebellion of bodies against political organization. Radical or Feminist organizations are not exempt from this decomposition; it is routinely referred to as burnout or infighting, though we could understand it as an instinctual refusal to be captured and mobilized by this or that Organization.
After all, the tendency of queerness against his-story has always been the ecstasy of life lived outside of time; without concern for whether the \emph{time is right}, for \emph{the material conditions} or for \emph{the Children}. Queerness must always emerge as \emph{out of its time, deviant, irrational.}
To the latter charge, we can only shrug. The Socratic trick of Ideas doesn’t really concern us. We’ll leave the universals and the big stories to the His-storians. We’ll concern ourselves instead with the beautiful moments of heresy and revolt—the lived experiences, bodily practices and spiritual intensity—which hint toward our own.
\begin{quote}
The resistance is the only human component of the entire His-story. All the rest is Leviathanic progress.
\end{quote}
\section{Second Mythos: Lilith and Eve}
\emph{In the patriarchal mythology of Judeo-Christian civilization, Adam was the first man, and God gave him a wife. Most know about Eve, his second wife. Fewer tell of his first; Lilith. Lilith differed in that she refused to be subservient to Adam. She wouldn’t lay beneath him in the missionary position, and so she was expelled from Eden. Upon her expulsion, she became a demon, a succubus who travelled through night and through time, breeding with other demons and unleashing evil spirits. It is said that at night she still tempts women to leave their husbands, turns men into faggots, encourages all manner of non-reproductive sexuality, and even steals and eats children.}
\emph{God the father couldn’t make the same mistake twice, and so he fashioned Adam’s second wife, Eve, out of one of Adam’s own ribs, ensuring her obedience. And still she disobeyed, she ate the fruit from the forbidden tree of knowledge and both she and her husband were banished from Eden. Some, such as Walter Benjamin, will view this as the expulsion of humanity from primitive communism. All the subsequent stories of The Holy Book of this religion is largely a lament of civilized life. Its first chapter narrates the fall, and the following chapters tell of the miseries within and exodus from various civilizations.}
\emph{But what was this forbidden knowledge? What was the original sin? A certain heresy tells that the forbidden knowledge was the realization that a certain type of sex leads to reproduction. Once Adam and Eve knew this, they couldn’t unlearn it. From here, all of their activities were tied to an emerging symbolic order of domination. Whereas before they had simply indulged in utopia without a future, now their actions had consequences. From this knowledge stems the invention of the role of the Father, as well as the knowledge necessary for agriculture, and even the first form of the rational thought which would later become Science. Patriarchy, Civilization, Reproductive Futurism. All of it stems from this abominable discovery.}
\emph{The church’s misogynists will blame Eve for this discovery and expulsion, but as we well know, it is the fathers, herders, husbands, inquisitors and witch hunters who put these arcane secrets to use in the mechanization of the body. These same woman-haters will sentence countless women and faggots to burn for having fallen under the influence of the rebel demon Lilith.}
\emph{If we cannot unlearn these secrets, what would it mean to destroy the machinery which dominates us through them? Can we recall Lilith and fly with her at night?}
\section{XII}
Of all these stories, there is one which occurs consistently in almost any worthwhile history of gender: the splitting of the mind from the body. Various accounts will attribute this split to different times and places, but its centrality and power are beyond question. Anti-civilization critiques will often locate this as a primary emergence of dualism in the world (Zerzan will say it stems immediately from the dualism of gender), whereas Federici will find it in the machinations of the witch hunts; Evans in the rise of industrialism. Again, the precise origins interest us less than its repeated and unending operation. Wherever it started, the split widens and continues to tear us away from ourselves.
It is intuitive that such a split would be necessary in order to acclimate wild beings into those beings fit for labor in the world of work. If one is solely reliant on their own sensual perception of the world—the relation of their body to the bodies of other animals, plants and humans—then that bodily awareness is precisely what must be destroyed for the workers to be born. The disciplining of the body is the precondition of industrial existence.
This disciplining of the body can be understood as an internalization of the warfare occurring outside of it. The battleground of social control becomes the body itself, the site of an eternal conflict between Reason and Passion; Enlightenment and Darkness.
\begin{quote}
On the one side, there are the ‘forces of Reason’: parsimony, prudence, sense of responsibility, self-control. On the other, the ‘low instincts of the Body’: lewdness, idleness, systematic dissipation of one’s vital energies. The battle is fought on many front because Reason must be vigilant against the attacks of the carnal self, and prevent ‘the wisdom of the flesh’ (in Luther’s words) from corrupting the powers of the mind. In the extreme case, the person becomes a terrain for a war of all against all.’\footnote{\emph{Caliban and the Witch}.}
\end{quote}
Others will call this Civil War, we will say it is part and parcel of the capture of the body in domestication. The body is a microcosm for this phenomena.
The commodification of bodies and of their capacities leads to an estrangement from self; a disassociation from the majority of one’s activity and experience. The body is reified and reduced to an object. This separation and objectification of the body reaches arrives at its own self-realization through Cartesian philosophy. Hobbes will enact a related attack upon the body in reducing it to the functioning of a machine. In later times, this mechanized view will reach a new apex through the theory of genetics. More esoteric theorists of genetics will argue that body is a machine-vessel for sentient and \emph{selfish genes} which deploy said bodies in an effort to eternally perpetuate themselves. The philosophical mechanization of the body becomes so total that it is projected back through history and into our very biology. In a strange paradox, science revives God as the ultimate refutation of free will: genetics. Genetic manipulation and nanotechnological methods of surveillance and control are only the most contemporary manifestations of this archaic split.
But the projection of this invention onto the physical world is not done philosophically, it is done through bodily violence. The torture chambers of witch hunters, Nazi doctors and vivisectors are also the laboratories for the emergence of the mechanized body. This is also, of course, the violence of gendered domestication, as gender is that first dualism and remains the primary operation upon the body. The body is continuously dissected so as to identify and naturalize the biological differences which supposedly justify the entirety of the gendered world. The sex\Slash{}gender dichotomy, but also the dichotomies of race are neatly mapped over the body\Slash{}mind, and corresponds to an unending set of disciplinary measures and techniques of the self designed to maintain binary conformity. Black and feminine bodies are imagined as indocile and in need of disciplining, while white masculine bodies are believed to be rational and tame. Bodies viewed with any innate connection to animality can then justifiably be exposed to hard labor, sexual violence, and extermination.
Personally, any inquiry into the split between the mind and body yields a crazy diffusion of revelations. I immediately think of the experience of motion sickness as a worthwhile example. As an instinctual response to feeling motion without consciously perceiving it, this nausea is a helpful defense mechanism against the inadvertent consumption of various poisons. Outside of industrialism, this phenomenon is only experienced on the off chance that someone eats a hallucinogen. Yet in a world like our own, where we are constantly disassociating from the movement of our bodies, this nausea becomes universal. The repetitive motion injuries from my performance of service work (where the quick movement of the wrists and knees corresponds more to the needs of a Point-of-Sale system or bag of groceries than to any other agency) is another reminder of a nearly total disconnect of my perception from the actual movement of my body. The split widens through our acclimation to this constant pain and dizziness; the further severance of perception functions as a tragic survival strategy.
Regarding gender, the split is all the more blatant. As a teenager, my own experience of dysphoria and body dysmorphia led to the self-enactment of a whole range of disciplinary measure and torture in the form of anorexia. This was an experience I shared with the vast majority of my friends who grew up as girls and queers. These techniques of self-control reappear in the context of sex work. In order to more profitably sell our sexual labor, we are constantly project the Ideal of gender upon our bodies; mutilating them and reducing them to objects of our own mechanization. More than just physiology, this domination concerns itself with gestures, grooming, communication, sexual propensity. In the actual experience of sex work, the split widens again. While some horrifying John is touching me, my mind struggles to be anywhere but my own body. I think about the capital, about my bank account, what I’ll have for dinner; \emph{anything besides what is actually occurring to my body.} I’ve experienced this flight from the body in countless other moments; while being arrested, while being sexually assaulted, while drunk. Even the experience of walking through the hallways of a high school can tear us from ourselves: how should I carry myself today so as not to face the predictable violence of a queer basher?
The story of the mind\Slash{}body split gives us a helpful tool in understanding the complexity and nuance of the contention that domestication is the capture and engendering of our bodies. Where Fredy Perlman saw springs and wheels filling the armor encased body, we can read this as the re-ordering of the living body through its conflict with the rational mind. The fantasy of Biological Sex, of Race, and all other supposedly natural categories correspond to this same logic of severance of bodies from each other and the mind from the body. Taxonomies of the body consistently serve to rationalize, systematize and place the varied happenstances of the body into a Leviathanic structure. This mechanistic theory of biology attempts to lay down our destiny.
\section{XIII}
Most theories of the split between mind and body miss a concomitant, yet unique, split: the material from the spiritual. The separation and obscuring of the spiritual dimension of gendered existence leaves us with a tragic inability to express or even really comprehend the implications of these operations of capture. To ignore the spiritual dimension of domestication leaves us with only half the story; with a crass, mechanistic materialism that can only offer us crass, mechanistic solutions.
If the human body and not the steam engine, and not even the clock, was the first machine developed by capitalism, then what is remains of all the capacities of the body which cannot be efficiently put to use or rationalized by this technological innovation? The imposition of a Cartesian Master\Slash{}slave dynamic between the mind\Slash{}body also means the generalization of that dynamic toward all of the forms and capacities of life which once enchanted the body’s sensual connection to the wild world. Our being was inscribed into a soulless world and a machine-body.
Francis Bacon lamented that \emph{magic kills industry}. And this is precisely because the continued relation of human beings to their magical capacities was also their capacity to find meaning and sustenance outside of the world of work and industry. Magical and spiritual beliefs were dangerous simply because their refusal of linear, empty time itself was a source of insubordination. In order for Leviathan to achieve its restructuring of the body, it had to first divorce the body of its participation in a cosmology of power and spirit. The perceived wildness of the witches had to be crushed alongside the wildness of the world. Leviathan alone would possess the ability to alter, enchant and deploy the body. This control over the body certainly happens in a largely metaphysical operation, yet it obscures itself and pretends toward the Natural and Objective. Perhaps the most sinister aspect of the spiritual decimation which mechanizes the body is that it denies the existence of spirit at all.
\begin{quote}
The mechanization of the body is so constitutive of the individual that, at least in industrialized countries, giving space to the belief in occult forces does not jeopardize the regularity of social behavior. Astrology too can be allowed to return, with the certainty that even the most devoted consumer of astral charts will automatically consult the watch before going to work.\footnote{\emph{Ibid}.}
\end{quote}
This mechanization was achieved through the twofold operation of denying the spiritual existence while also destroying the rebel body. Hobbes: “As for witches, I think not that their witchcraft is any real power; but yet they are justly punished, for the false belief they have that they can do such mischief, joined with their purpose to do it if they can.” Fredy Perlman and Arthur Evans will both criticize historians of the witch hunt for reiterating this same domesticated analyses—justifying the massacres of the witch hunts by projecting the mechanistic understanding of the body through time and into the ‘natural’ world.
\begin{quote}
The stakes on which witches and other practitioners of magic died, and the chambers in which their tortures were executed, were a laboratory in which much social discipline was sedimented, and much knowledge about the body was gained. Here those irrationalities were eliminated that stood in the way of the transformation of the individual and social body into a set of predictable and controllable mechanisms. And it was here again that the scientific use of torture was born\dots{}
\end{quote}
\begin{quote}
This battle, significantly occurring at the foot of the gallows, demonstrates both the violence that presided over the scientific rationalization of the world, and the clash of two opposite concepts of the body, two opposite investments in it. On one side, we have a concept of the body that sees it endowed with powers even after death; the corpse does not inspire repulsion, and is not treated as something rotten or irreducibly alien. On the other, the body is seen as dead even when still alive, insofar as it is conceived as a mechanical device, to be taken apart just like any machine. [\dots{}] The course of scientific rationalization was intimately connected to the attempt by the state to impose its control over an unwilling workforce.
\end{quote}
Feral Faun put things another way in “The Quest for the Spiritual”:
\begin{quote}
This civilized, technological, commodity culture in which we live is a wasteland. For most people, most of the time, life is dull and empty, lacking vibrancy, adventure, passion and ecstasy. It’s no surprise that many people search beyond the realm of their normal daily existence for something more. It is in this light that we need to understand the quest for the spiritual\dots{}
\end{quote}
\begin{quote}
I discovered that this dualism [between the material and the spiritual] was common to all religions with the possible exceptions of some forms of Taoism and Buddhism. I also discovered something quite insidious about the flesh\Slash{}spirit dichotomy. Religion proclaims the realm of spirit to be the realm of freedom, of creativity, of beauty, of ecstasy, of joy, of wonder, of life itself. In contrast, the realm of matter is the realm of dead mechanical activity, of grossness, of work, of slavery, of suffering, of sorrow. The earth, the creatures on it, even our own bodies were impediments to our spiritual growth, or at best, tools to be exploited. What a perfect ideological justification for the exploitative activities of civilization\dots{} as exploitation immiserated the lives of people, the ecstatic joy of wild existence and of the flesh unrepressed became fainter and fainter memories until at last they seemed to be not of this world at all. This world was the world of travail (from the Latin root word which gives all the Romance languages their word for work) and sorrow. Joy and ecstasy had to be of another realm—the realm of spirit. Early religion is wildly orgiastic, clearly reflecting the lost way of life for which people longed. But by separating this wild abandon into the realm of spirit, which is in reality just a realm of abstract ideas with no concrete existence, religion made itself the handmaiden of civilized, domesticated culture\dots{}
\end{quote}
This transformation of the body into predictable and controllable operations is absolutely central to the naturalization of the category of sex. The uterus becomes a machine—controlled by the state and doctors—for the production of new bodies. The incomprehensible diversity of the human body becomes reduced to a simplistic and quantitative relation between various chemicals and hormones. Certain shapes are deemed healthy while others abnormal and in need of surgical intervention. The binary of the so-called sex organs is almost achieved through this ongoing mutilation. Certain ratios of the distribution of fat, hair, bone structure and other occurrences come to be immutable proof of the eternal existence of the social prison of sex. In order for this prison to be totalizing, our conception of ourselves must be debased to these material operations. The engendering of humanity into the rational sexual body required the destruction of magic precisely because a magical view of the world holds that it is animated, unpredictable and that there is an occult force in plants, animals, stones, the stars and ourselves. Within this animist worldview, our individual capacities are not limited to the supposed biological destiny of sex; instead we can create, destroy, love, and take pleasure in an infinity of situations. This anarchic, molecular diffusion of powers throughout the world is antithetical to a gendered and social order which aims at capturing and dominating all life. \emph{The world had to be disenchanted to be dominated.}
Here is science born. The disenchanted world can now be explained through rational, objective inquiry. And yet it is a meaningful contradiction that this new science did not mean an end to what it would have seen as an irrational persecution of witches. Instead, mechanistic philosophers celebrated the witch hunts as the advancement of the rational worldview. Francis Bacon, one of the early high priests of science, is explicit in taking methods of scientific inquiry directly out of torture chambers of the inquisition. For science, the whole world becomes analogous to a witch: a body to be interrogated, tortured, raped and unveiled. Far from relegated to this particular period, we can see repeating over and over again in Nazi death camps, the medical experimentation on prisoners, the vivisection of animals, etc. Scientific rationalism is not some progressive intervention against brutality, it is simply the universalization of that brutality against all the wild world, against the body and against the spirit. This scientific approach to the world becomes all the more terrifying when it is taken up by revolutionaries. The bourgeois revolutions fought in the name of Reason and Justice, ended up carving those abstractions into the flesh of individuals through the Guillotine, committees of public safety and health, and other implements of systemic terror. This terror took on a new dimension in the communist revolutions which followed.
We’ll have to say, along with the editors of \emph{Green Anarchy} that the scientific understanding of the world is the culmination of the segmentation of reality which first occurs in gender and in domestication:
\begin{quote}
Science is not neutral. It is loaded with motives and assumptions that come out of, and reinforce, the catastrophe of dissociation, disempowerment, and consuming deadness that we call “civilization.” Science assumes detachment. This is built into the very word “observation.” To “observe” something is to perceive it while distancing oneself emotionally and physically, to have a one-way channel of “information” moving from the observed thing to the “self,” which is defined as not a part of that thing. This death-based or mechanistic view is a religion, the dominant religion of our time. The method of science deals only with the quantitative. It does not admit values or emotions, or the way the air smells when it’s starting to rain; or if it deals with these things, it does so by transforming them into numbers, by turning oneness with the smell of the rain into abstract preoccupation with the chemical formula for ozone, turning the way it makes you feel into the intellectual idea that emotions are only an illusion of firing neurons. Number itself is not truth but a chosen style of thinking. We have chosen a habit of mind that focuses our attention into a world removed from reality, where nothing has quality or awareness or a life of its own. We have chosen to transform the living into the dead. Careful-thinking scientists will admit that what they study is a narrow simulation of the complex real world, but few of them notice that this narrow focus is self-feeding, that it has built technological, economic, and political systems that are all working together, which suck our reality in on itself. As narrow as the world of numbers is, scientific method does not even permit all numbers; only those numbers which are reproducible, predictable, and the same for all observers. Of course reality itself is not reproducible or predictable or the same for all observers. But neither are fantasy worlds derived from reality.
\end{quote}
Science doesn’t stop at pulling us into a dream world; it goes one step further and makes this dream world a nightmare whose contents are selected for predictability and controllability and uniformity. All surprise and sensuality are vanquished. Because of science, states of consciousness that cannot be reliably disposed are classified as insane, or at best “non-ordinary,” and excluded. Anomalous experience, anomalous ideas, and anomalous people are cast off or destroyed like imperfectly-shaped machine components. Science is only a manifestation and locking in of an urge for control that we’ve had at least since we started farming fields and fencing animals instead of surfing the less predictable (but more abundant) world of reality, or “nature.” And from that time to now, this urge has driven every decision about what counts as “progress,” up to and including the genetic restructuring of life.
\section{XIV}
A critique of science now poses a tremendous problem for most theories of resistance. So many of the old means of resistance (especially those which are predicated on science and industrialism) have only reaffirmed this ordering of the world. The blindspot of this resistance is specifically that we ourselves have been domesticated in a biological dimension, in the capture of our bodies and the denial of our spirits. It wouldn’t be enough to destroy all the computer infrastructure in the world, so long as we hold an unspoken view of ourselves as primitive computers. Any attempts to deploy science in the pursuit of liberation can only deepen the tragedy of separation and control which is the very essence of domestication.
This can perhaps be more easily realized in Marxism than in any other system of thought in the last century. Fredy Perlman’s text \emph{The Continuing Appeal of Nationalism} is brutal on this point:
\begin{quote}
Marx had a significant blind spot; most of his disciples, and many militants who were not his disciples, built their platforms on that blind spot. Marx was an enthusiastic supporter of the bourgeoisie’s struggle for liberation from feudal bonds—who was not an enthusiast in those days? He, who observed that the ruling ideas of an epoch were the ideas of the ruling class, shared many of the ideas of the newly empowered middle class. He was an enthusiast of the Enlightenment, of rationalism, of material progress. It was Marx who insightfully pointed out that every time a worker reproduced his labor power, ever minute he devoted to his assigned task, he enlarged the material and social apparatus that dehumanized him. Yet the same Marx was an enthusiast for the application of science to production.
\end{quote}
But this progress had to contend, at every juncture, with the decomposition which accompanied all Leviathanic organization. In order to do this, Leviathan has consistently needed new populations from which it could squeeze surplus. At times, the capture\Slash{}domestication of these populations was achieved through colonialism, whereas at others it was to be found in domestic colonies (of Jews, witches, faggots, Muslims, heretics, etc.) This process of primitive accumulation
\begin{quote}
is responsible for the takeoffs, the windfalls and the great leaps forward. [\dots{}] new injections of preliminary capital are the only known cure to the crises. Without an ongoing primitive accumulation of capital, the production process would stop; each crisis would tend to become permanent.
\end{quote}
\begin{quote}
Genocide, the rationally calculated extermination of human populations designated as legitimate prey, has not been an aberration in an otherwise peaceful march of progress. This is why national armed forces were indispensable to the wielders of capital. These forces did not only protect the owners of capital from the insurrectionary wrath of their own exploited wage workers. These forces also captured the holy grail, the magic lantern, the preliminary capital, by battering the gates of resisting or unresisting outsiders, by looting, deporting and murdering\dots{}
\end{quote}
\begin{quote}
Human communities as variegated in their ways and beliefs as birds are in feathers were invaded, despoiled and at last exterminated beyond imagination’s grasp. The clothes and artifacts of the vanished communities were gathered up as trophies and displayed in museums as additional traces of the march of progress; the extinct beliefs and ways became the curiosities of yet another of the invaders’ many sciences. The expropriated fields, forests, and animals were garnered as bonanzas, as preliminary capital, as the precondition for the production process that was to turn the fields into farms, the trees into lumber, the animals into hats, the minerals into munitions, the human survivors into cheap labor. Genocide was, and still is, the precondition, the cornerstone and groundwork of the military-industrial complexes, of the processed environments of the world of offices and parking lots.\footnote{\emph{The Continuing Appeal of Nationalism}.}
\end{quote}
Perlman goes on to follow this blindspot—the capture, genocide, and exploitation necessitated by industrialization—through the thought of the vast majority of revolutionaries since Marx; anarchists, socialists and Leninists alike. All of them glorify industrialism as key within the progressive movement of history. For Fredy, the most innovative and horrifying consequence of this blindspot can be seen in the Bolshevik revolution and the thought of Lenin.
\begin{quote}
Lenin was a Russian bourgeois who cursed the weakness and ineptitude of the Russian bourgeoisie. An enthusiast for capitalist development, an ardent admirer of American-style progress, he did not make common cause with those he cursed, but rather with their enemies, with the anti-capitalist disciples of Marx. He availed himself of Marx’s blind spot to transform Marx’s critique of the capitalist production process into a manual for developing capital, a ‘how-to-do-it’ guide. Marx’s studies of exploitation and immiseration became food for the famished, a cornucopia, a virtual horn of plenty\dots{}
Russian countryfolk could not be mobilized in terms of their Russianness or orthodoxy or whiteness, but they could be, and were, mobilized in terms of their exploitation, their oppression, their ages of suffering under the despotism of the Tsars. Oppression and exploitation became welding materials. The long sufferings under the Tsars\dots{} were used to organize people into fighting units, into embryos of the national army and the national police.
The presentation of the dictator and of the Party’s central committee as a dictatorship of the liberated proletariat seemed to be something new, but even this was new only in the words that were used. This was something as old as the Pharaohs and Lugals of ancient Egypt and Mesopotamia, who had been chosen by the god to lead the people, who had embodied the people in their dialogues with the god. This was a tried and tested gimmick of the rulers. Even if the ancient precedents were temporarily forgotten, a more recent precedent had been provided by the French Committee of Public Health, which had presented itself as the embodiment of the nation’s general will\dots{}
The goal of the dictator of the proletariat was still American-style progress, capitalist development, electrification, rapid mass transportation, science, the processing of the natural environment. The goal was the capitalism that the weak and inept Russian bourgeoisie had failed to develop\dots{}
Lenin did not live long enough to demonstrate his virtuosity as general manager of Russian capital, but his successor Stalin amply demonstrated the powers of the founder’s machine. The first step was the primitive accumulation of capital. If Marx had not been very clear about this, Preobrazhensky had been very clear. Preobrazhensky was jailed, but his description of the tried and tested methods of procuring preliminary capital was applied to vast Russia. The preliminary capital of English, American, Belgian and other capitalists had come from plundered overseas colonies. Russia had no overseas colonies. This lack was no obstacle. The entire Russian countryside was transformed into a colony.
The peasants were not the only colonials. The former ruling class had already been thoroughly expropriated of all its wealth and property, but yet other sources of preliminary capital were found. With the totality of state power concentrated in their hands, the dictators soon discovered that they could manufacture sources of primitive accumulation. Successful entrepreneurs, dissatisfied workers and peasants, militants of competing organizations, even disillusioned Party members, could be designated as counter-revolutionaries, rounded up, expropriated and shipped off to labor camps. All the deportations, mass executions and expropriations of earlier colonizers were reenacted in Russia.
By [this] time, all the methods of procuring preliminary capital had been tried and tested, and could be scientifically applied.
\end{quote}
Perlman will contend that this innovative method of capture will later inspire the likes of Hitler, Mussolini and Mao, most of whom will dispense of the rhetoric of the Bolsheviks, but maintain the boiled-down scientific essentials of the method. And since the revolution which first implemented this method failed in its rhetorical aim of liberating humanity from wage labor, this too was dispensed of as an embarrassment. Instead, the progress of the techno-industrial state is itself the justification. The primitive accumulation needed for the ascendence of later totalitarian states would be found in the internal enemies of the Parties. Domestication no longer needs to justify itself through anything other than its own scientific method. And science itself would invent methods that earlier genocidal colonialists could only have dreamed of; Eugenics, Gas Chambers, Laboratories. These industrializers will each imagine a triumphant reduction of the entire Eurasian continent to a site of resources to be domesticated and accumulated. Western Rationalists will attempt to explain these mass murderers as irrational, and yet would see people like George Washington and Thomas Jefferson as perfectly reasonable leaders,\emph{even though these men envisioned and began to enact the conquest of a vast continent, the deportation and extermination of the continent’s population, at a time when such a project was much less feasible.}
What is consistent in all of these situations is a deeply seated belief in human progress through the expansion of industrial civilization. Modern day Marxists will say that these applications of Marx’s theory were incorrect and that they were deviant or revisionist. But isn’t this horror the consequence of every attempt to impose any theory on a mass industrial scale?
\begin{quote}
Applied scientists used the discovery [of the atom] to split the atom’s nucleus, to produce weapons which can split every atom’s nucleus; nationalists used the poetry to split and fuse human populations, to mobilize genocidal armies, to perpetuate new holocausts.
The pure scientists, [nationalist] poets and researchers consider themselves innocent of the devastated countrysides and charred bodies\dots{} every minute devoted to the capitalist production process, every thought contributed to the industrial system, further enlarges a power that is inimical to nature, to culture, to life. Applied science is not something alien; it is an integral part of the capitalist production process.
\end{quote}
What becomes clear is that any attempt to flesh out a scientific theory of domination (whatever the intentions of the theorists) becomes put to work by domination itself as a blueprint. This could be understood as the \emph{de\Slash{}recomposition} of history. More significantly it ties into the critique articulated above of other Scientific disciplines: Anthropology and Psychoanalysis. The pure theories of Anthropologists, Psychoanalysts and Marxists always tend to become new means of domestication: universities, asylums and work camps. Camatte is at his most lucid when critiquing the role of theory:
\begin{quote}
Theory, like consciousness, demands objectification to such an extent that even an individual who rejects political rackets can elevate theory to the status of a racket. In a subject posing as revolutionary, theory is a despotism: everyone should recognize this. After the domination of the body by the mind for more than two millennia, it is obvious that theory is still a manifestation of this domination.
\end{quote}
For this reason, it is all the more important that we dispense with scientific certainty and methodology in our inquiry into gender. Otherwise, the solutions will continue to be more of the same: cyber-feminism, the virtual flight from the body, automated reproduction, a flight which is “illusory, a forgetting of the whole train and logic of oppressive institutions that make up patriarchy. The dis-embodied high-tech future can only be more of the same destructive course.”\footnote{John Zerzan, \emph{Patriarchy, Civilization, and the Origins of Gender}.} In the same way that the mind\Slash{}body split assures us that idealist solutions to gender will always fail \emph{(Queering the economy! Queering the State!)} so too does the material\Slash{}spiritual split guarantee us that the blind spot of industrialism will continue its course of annihilation and control.
To return momentarily to Feral Faun:
\begin{quote}
Materialism still accepts the matter\Slash{}spirit dichotomy—but then proclaims that spirit does not exist. Thus, freedom, creativity, beauty, ecstasy, life as something more than mere mechanical existence are utterly eradicated from the world. Mechanistic materialism is the ideology of religion updated to fit the needs of industrial capitalism. For industrial capitalism requires not only a deadened, dispirited earth, but deadened, dispirited human beings who can be made into cogs in a vast machine.
\end{quote}
\section{XV}
Throughout the body of this text we’ve been weaving together a critique of the scientific view of gender, as well as resistance practices which remain rooted in this domestication. We’ll now turn explicitly toward one of the most prominent of these ideologies regarding gender: Marxist Feminism (or its contemporary euphemism, \emph{Materialist Feminism}). This ideology largely emerged in the seventies as an attempt to synthesize the critique of capitalism with the critique of Patriarchy. Gayle Rubin’s inquiry, which we’ve detailed above, was largely a critique of the limitations of the Marxist perspective. Queer theory and black feminism and transfeminism also emerged largely in reaction to the inability of this theory to account for the majority of gender violence experienced by a whole range of subjects excluded from the scientific sample. The theories of contemporary Marxist feminists haven’t deviated all that far from their roots, but the questions posed decades ago remain largely unanswered.
These interventions are relevant to our own critique, but we begin from a different place. Because it is materialist, Materialist Feminism ignores the spiritual dimensions of gender, and as a consequence has not been able to ascertain or critique gender as domestication. Because of its prioritization of the Historical and Economic it offers very little regarding the experience of the individual bodies ensnared or excluded by these Leviathanic abstractions.
In the seventies, Rubin and others said that the primary limitation of Marxist feminism was its conception of origins. For them, the exploitation and domination of women was based in the separation and gendering of the spheres of productive and reproductive labor. Rubin contended that the domination of women originated outside this separation, but also that both the sex\Slash{}gender system and the economic system had their own modes of production and reproduction (the sex\Slash{}gender system is \emph{productive} of gender and sexual identities themselves, while there is also unquantifiable \emph{reproduction} of the economic system that happens ways irreducible to domestic labor). Already then it was sloppy to reduce the two systems as being simply the productive and reproductive spheres of the capitalist mode. For her, the origins of gender are far more archaic, emerging at the beginning of civilization itself. While obviously feminist anthropologists will win against Marxist feminists on the origins debate any day, our inquiry takes us outside this theoretical pissing contest. Rubin’s perspective isn’t interesting to us because its evidence is older (after all, the anthropological method is as rooted in the failures of science as the historical economic one). Instead, we’re interested in the way her text contributes to the elaboration of gender and domestication as being one and the same process with both bodily and spiritual operations.
We’ll contend that in order to plot an escape from a system which holds us captive, let alone to strike out against the beast itself, we must understand not only where it comes from, but more importantly how it operates in the present. Marxist feminism feels inadequate in both these regards. To locate a theory of domination in the performance of \emph{domestic labor} without starting from a critique of domestication will always amount to a partial story; a description of specific moments (or fantasies) in specific times and places, but will miss the discreet enemy function which ties it to all the other moments of gender. More sophisticated iterations of Marxist feminism will say that gender is obviously older than capital, but that capital takes up and consumes all pre-existing social relations, therefore exploiting gender along with all the others. And while it is true that there is a dimension of the unique in every moment, and that genders within capitalism are different than within other modes of production; this does not prove that the essence of gendered domination has changed all that much. Rather, the gender-form emerges from millennia before and stays consistent in its twofold bodily\Slash{}spiritual assault on human existence.
The moments of the accumulation of domestic labor (in the witch hunts, or within Fordism) are two worthwhile stories about how gender has taken its contemporary form, but they remain two stories among many. To over-prioritize these moments of economic exploitation is to silence and undervalue the countless stories which do not fit inside the neat narrative. It is popular for thoughtful Marxists to assert that the State may be far older than capital, but that their inseparable interweaving has completely transformed and reconstituted the state; and the two forms must be destroyed together. And yet all attempted Marxist revolutions have only ever reproduced the state, precisely because the form is more ancient and thoroughly colonizes our being. In this same way, a simple assertion that gender and capital have become terribly intertwined and must be destroyed together is not a theory of how that will happen or even much of an analysis of how this came to be. Just as a focus on the state as part in parcel of capital will in practice function as a blindspot, so too will this situation of gender. We’re reminded of the laughable moments in the last decade where various communist parties had to make a complete reversal of their positions on queer people, without ever altering the structure of their understanding. The effort to expand and adapt the ideology (to account for categories it previously ignored) consistently feels like the same politics of liberal inclusion sutured on top of vulgar Marxism. Yes, gender is exploited by capital and the two are largely indistinguishable and inseparable in the present, but this is not sufficient. Just as a refusal of the state-form would require an understanding of its emergence and function up until the present (without vulgarly systematizing it within capital) so to does gender require such an inquiry. If we want to destroy it, we cannot limit our canon to those moments which fit neatly into a story about capital. We’ll also need to draw upon the archaic origins of gender and the voices and biographies of those who attempted to burn it out of themselves.
The Marxist feminist perspective will always fail on the discussion of origins, because even those who critique the social construction of gender will affirm a naturalized view of sex. For them, socialized gender is a corruption of the biological realities of males and females of the species with regard to reproduction. We’ve already discussed how this split is itself domestication and that it is Leviathan’s function to universalize and naturalize its machinery into the wild. If Marxist feminism has refused this naturalization of sex, we have scarcely seen it. Even those who go as far as to problematize essentialist gender, will still default to discussing a transhistorical ‘men and women’ within all their complex formulations.
Even if we only explored gender in the present, we would still find the story of domestic labor inadequate to the task. The narrative situates the Family as the primary site of the exploitation of women’s reproductive labor, labor which is necessary for the continued function of the capitalist mode of production. It is true that the Family does serve this purpose, but to stop our critique here is to be limited by a mechanistic and materialist view. We’ve already explored a theory that the Family is a structure which emerges out of the exchange of the bodies of others as commodities, and that it is imbued with a mystical power through the enactment of ancient rituals regarding sexuality and kinship. The consolidations of these mystical kinship structures were the basis of more complex human social relations including Leviathan and the State. A specific power of inclusion enchants those who participate in these Families, for they become the inheritors of millennia of lineage and tasked with the transmission of that heritage into the future (we’ve discussed this previously in the symbol of the Child). Fascism fetishizes these bonds, but so too do most political traditions. The Marxist analysis of the Family will tell us that this structure emerges out of the specific economic conditions of capitalism, but this is empirically untrue. Capital has shaped the Family in unique ways, but the bonds which animate and give power to the Family (bonds of kinship, transmission, ancestry, sexuality and reproductive futurism) stream through His-story and constitute an inheritance of millennia of control and domination. To take seriously the task of destroying this unit, we must comprehend it in its totality—in its economic function, sure,but also for its imprisonment and shaping of both the body and the spirit. Why does the family hold such a intrinsic place in all domesticated culture? Why do people form them? Why do they remain in them? Why do some actually claim to love and enjoy their abusive positions within them? Why does it remain the shadowy realm of open secrets and quiet little violences? These questions cannot be answered through economics alone.
A Marxist attempt would answer that women remain in the family because they are denied the wage, and men because they need the free reproductive labor, but this answer feels paltry compared to the enormity of the questions posed. How could this or that arrangement of the wage relationship be the glue which holds together the most formative social relation within civilization? It isn’t. We’ve said already that science is a narrow view of the world which reduces the diversity of reality into the shape of its view. This tendency is all too clear in the scientific interpretation of the family. This view is far too narrow to account for most people’s experiences of gender and violence, but even too narrow to describe most people’s families. Black, brown and indigenous feminists have consistently critiqued the Marxist formulation as being a primarily white understanding which has little to no application to their lives. The formulation even excludes many white families, especially those which are very poor. My Mom, for example, worked two jobs in a factory and a nursing home to support us when I was a child. Her mother still works at the same diner where she has worked for decades. And yet the content of my family retains its domestic character. We’ve followed Fredy and \emph{Attentat} in asserting that history is the decomposition of Leviathanic forms. So too is the family constantly decomposing and rising anew from the ashes. At this point, so many ‘new normal’ familial arrangements exist, none of which are accounted for in the simplistic binarist understanding of gender. How does a Marxist view account for this prolonged moment of the Family’s decomposition?
A queer position contends that the family is a site of our exploitation, yes, but also has been a consistent operation of torture, constraint and domination which vastly outpaces the needs of domestic labor. For others, we often find the family also as a site of exclusion, specifically at the moments when we rebel against it. The Marxist worldview has nothing to say about either our mutilation within the family-form or our expulsion from it. Further, it derides our individual and collective revolt against this form as \emph{ahistorical} and \emph{idealist}. We are acting too soon or without the right conditions; but these rationalistic approaches have only ever affirmed the family (even if critiquing its role economically). Our revolt will never be comprehensible from within it.
Even for the proponents of this theory, it explains very little about their own lives. In the seventies, the situation already was based on a group of women objectively studying an Other. In the present, we have academics studying the ideas of academics who studied this Other body of women (and then calling it historical materialism). I think of those feminist professors whose liberation comes through hiring a housekeeper.
Our inquiry begins firstly from our own lives, and then follows the lines along which we can locate our own struggles within and against gender in the struggles of others. Outside of this, all inquiry feels meaningless and empty. In my own life and experiences, Marxism’s formulations around the split between reproductive and productive labor is incredibly superficial in addressing gender violence. It doesn’t explain why old men pay to have sex with me or to watch videos of my sexual labor. It doesn’t account for what investment people outside my family would have in policing my sexuality and gender expression. It doesn’t explain why rape and sexual violence happens to those of us who don’t have the biological capacity to give birth. It definitely doesn’t account for the prevalence of date rape drugs at queer bars and parties, or for our murder at the hands of bashers and police. While I won’t preclude that possibility that such an accounting could happen someday, we’ve seen no efforts in this direction. A refusal of Materialism isn’t an affirmation of some sort of queer Idealism, rather it is an attempt to explore what has been cut out and discarded by both of these worldviews, the body and the spirit. These experiences require a bodily and spiritual exploration, one which takes seriously the simultaneous question of domestication. Such an exploration seems entirely necessary if we want to comprehend the vast range of gender violence (both the exclusionary and imprisoning violences against queers and gender variant people, and also the more mundane daily exploitations in the family), and to recognize them as one operation.
That the theory of Marxist feminism is flawed is only the beginning of the problem. As with any other theory, its applications will always be haunted by the blind spots within it. We’ve already shown that pure sciences tend to produce horrifying results. The application of this theory, of course, is Organization. Often the organization is so banal and reformist as to not warrant exploration (\emph{Wages for Housework!}, for example). Other false solutions (mechanized reproduction or self-managed housework) have thankfully not been put into practice on any notable scale.
Another application of Marxist feminism is separatism. It is worth focusing on because of the specific tragedy that its history shows. The Separatist project begins from an awareness of the dynamic we’ve also illustrated in organization (for all organization to be constituted through the exchange circulation of gendered bodies), but strives to self-manage this circulation. Women must be organized into this or that group or party, where other \emph{more conscious} women will help to structure their thought and activity. The exclusion of certain genders from the separatist group has never exorcised the demonic quality of organization itself. Beyond this, it has actually taken on a particularly sinister dimension through its willful and vitriolic exclusion of transgendered women and others. Marxist feminist activists were instrumental in the formation of state policies of excluding these women from state services, from activist groups, from shelters. These feminists served as the frontline of the formation of transmisogynist policies in countless political and cultural institutions. As with all scientific theories of domination, this variant of feminism has historically helped to materialize the exclusion of those who cannot fit within its theoretical constructs. Contemporary Marxist feminists will contend that since they are avowedly not transmisogynists, they do not have to answer for this tradition. And yet the theoretical underpinning of this attitude amongst their foremothers has not been changed in any meaningful way. Inclusion of a few references to transwomen at best, repetition of the past at worst. If the tendency is going to substantially break from this history, it would require a thorough analysis which is very far from happening. How can a purely materialist conception of gender explain the choice of individuals to risk their lives, freedom, and wellbeing in order to live openly as a gender other than what they were assigned at birth? It can’t, obviously, unless it explores the interplay of the spiritual and also bodily operations of gender. We have very little faith in the emergence of a categorical theory of gender which does not become an apparatus for policing those categories. This policing is accompanied by the age old problem of politics: that of representation. Claims to be \emph{The Women} or \emph{The Feminists} or even \emph{The Queers} will always tell one tale of gender, at the exclusion of so many others. Those who draw these lines will always draw them through the bodies of others.
One recent answer to these critiques has been the introduction of the concept \emph{not-men}\footnote{In \emph{LIES: A Journal of Materialist Feminism}, as well as other recent publications and debates from within the Marxist Feminist milieu.}. Most attempts at defining this category are extremely clumsy. At times it is used to mean not-cismen, or to explicitly say that faggots are not welcome at certain meetings. At others it simply means women plus trans people. Some feminists have even said that the category at times includes ‘emasculated men of color.’ Usually it is just postmodern shorthand for women. As with any other categories, it only functions if it has a firm border, and this border will always be policed. At every step of the way, it is ceaselessly problematic. The least problematic definitions of it (such as the one in “Undoing Sex”\footnote{In “Undoing Sex,” published in \emph{LIES}, C.E. writes: “Effectively, the not-man cannot speak, cannot be represented with total accuracy, as it is defined through lack and absence. Still, it is a point in a relationship which is constitutive of gendered class, and discussion of it is necessary for any understanding of what it is to be a woman, man, transgender, or queer. Not-man is a means of addressing the problem of patriarchy—the way in which maleness and male subjectivity produces, appropriates, and exploits a condition of silence, death, and lack—while hopefully avoiding the presupposition of a coherent feminist or female subject. Not-maleness is constitutive of gender’s class reality—forms of womanhood and manhood exist only in relation to it—but it is irreducible to one or several classes.”}) are so vague as to not have any practical application. And it is always in the practical applications that these theories enact their violences. The prospect of a political body of largely cisgendered women determining which genderqueer or transfeminine individuals are \emph{not-men} enough to participate in their groups is quite nauseating. This categorical policing mirrors all the others. Meet the new binary, same as the old binary. A way out of this dilemma may be to start from experience rather than identity. To seek out conspirators based on a shared experience of a range of gender violence. Some proponents of not-men have defined it similarly (‘those who are raped,’ ‘those who do caring labor’) but none of these experiences are limited by identity, and to accept a phenomenological or experiential framework would dispense with the utility of the category at all. If the concept is either problematic or useless then why has there been so much fancy footwork put into an attempt to save the concept? What we’re really seeing is a desperate attempt to save binary categories, in a world where they’ve long been decomposing.
\section{XVI}
There is a trend within communist thought which aspires to transcend the limitations illuminated in the various attacks on Marxism: \emph{communization}. While it is beyond the scope of these fragments to explore and critique this textual body in its entirety, we will engage with it because its recent proponents have taken on the question of gender. Most of the writings of American communizers dealing with gender has been influenced by the French group Theorie Communiste. TC posits that in addition to the contradiction of labor and capital, there is a second contradiction between men and women. For them, these two contradictions intersect in the present to form the central dynamic of capitalist society. In this way, TC is similar to Gayle Rubin; imagining two distinct systems of production and of gender which become interwoven. While it is laughable to reduce the dynamic of the present to being two contradictions, we are also not interested in \emph{any} quantifiable arrangement of binary contradictions. Domestication is an infinitely complex and diffuse splitting of life; it introduces countless contradictions which cannot be summarized as any one, two, or five systems. We’ll break from both of them in asserting that there is never a period where these systems are distinct, but rather that they’ve always been examples of the fracturing of domestication.
However contrived TC’s theory of gender feels, it seems worthwhile to explore the ideas of those who’ve drawn inspiration from them. As the cutting edge of Marxist thought on gender, it is here that we’ll look to see if we can find a common critique of domestication. Specifically we’ll briefly look at three texts: “Communization and the Abolition of Gender” by Maya Andrea Gonzalez, “The Gender Distinction in Communization Theory” by P. Valentine from \emph{LIES} journal, and “The Logic of Gender” in the third issue of the journal \emph{Endnotes}.
Gonzalez’s critical reading of TC is interesting for a few reasons. Primarily, she critiques TC for having sutured their theory of gender on top of the already existing theory of the Capitalist Mode of Production, thus dispensing of the historical specificity of gender at the point where they intersect. She criticizes their fetishistic focus on the role of unpaid domestic labor performed by women and says that their domination is tied up in the way class society accumulates their capacity to give birth. This interests us firstly because of its shift outside the more vulgar Marxist understanding, but also because it relates to our critique of reproductive futurism laid out previously. The fantasy of the Child remains the primary structure of the shaping of the social order, and as such has to be indicted as central to the gendered matrix. We are also excited by her attempts to denaturalize both the categories of sex and gender.
\begin{quote}
Not all human beings fit into the categories of male and female. The point is not to use the language of biology to ground a theory of naturalized sexuality, as distinct from socialized gender. Nature, which is without distinction, becomes integrated into a social structure—which takes natural averages and turns them into behavioral norms. Not all ‘women’ bear children; maybe some ‘men’ do. That does not make them any less beholden to society’s strictures, including at the level of their very bodies, which are sometimes altered at birth to ensure conformity with sexual norms.
\end{quote}
This denaturalization fits nicely with a conception of gender as domestication, precisely because it is the domestication process which integrates the wild proliferation of bodies into social structure. The social structure which takes ‘natural averages’ and turns them into police mechanisms is the oldest social structure, the emergent kinship structures which give rise to the first leviathans. To the text’s credit, it situates this policing and categorical construction at the very beginning of class society. Gonzalez’s writing on this point is almost entirely unique in a terrain of thought which otherwise holds sex, if not gender, to be essential. We smile on this point, but have to remind ourselves why this shift feels necessary. To situate gender as domestication is crucial for us, only if our task is also to break genders hold over our lives.
Gonzalez calls for the abolition of gender, and does so through theorizing communization as its overcoming:
\begin{quote}
Since the revolution as communization must abolish all divisions within social life, it must also abolish gender relations—not because gender is inconvenient or objectionable, but because it is part of the totality of relations that daily reproduce the capitalist mode of production. Gender, too, is constitutive of capital’s central contradiction, and so gender must be torn asunder in the process of the revolution. We cannot wait until after the revolution for the gender question to be solved. Its relevance to our existence will not be transformed slowly—whether through planned obsolescence or playful deconstruction, whether as the equality of gender identities or their proliferation into a multitude of differences. On the contrary, in order to be revolution at all, communization must destroy gender in its very course, inaugurating relations between individuals defined in their singularity.
\end{quote}
While we have a great deal of skepticism about this type of total revolution, there is much common ground here: the desire to inaugurate relations between individuals in their singularity, to abolish gender and not simply proliferate it, and to destroy gender alongside our destruction of all the rest. Our disappointment then is precisely at the point where this line of inquiry stops. Gonzalez’s work in this piece amounts to an elaboration of why \emph{this would have to happen}, but remains almost entirely silent on how, when or by whom. In this sense, her text has a problem which is consistent in communization theory. As with most other arguments around communization, it remains stuck as a sort of aspirational tautology. \emph{Communization destroys capital; capital is gender; communization destroys gender; if the revolution does not destroy gender then it is not communization.} The moments in the text which hint toward what this destruction would look like are just a reiteration of the tautology.
\begin{quote}
That overcoming is only the revolution as communization, which destroys gender and all other divisions that come between us.
\end{quote}
We want to read this aspiration as a beginning of a struggle against domestication, but we have not seen this line continue. Gonzalez is correct in articulating the necessary destruction of gender \emph{in course}, but has yet to give a shape to the \emph{course} itself. It is notable that she points to a ‘loosening of the straight jacket of the heterosexual matrix’ but says that queer theory cannot account for this. We’ll argue that this loosening is not a phenomenon deterministically bound to the unfolding of demographics and economics, but rather is the willful activity of many who have attempted to give their own shape to the course of the matrix’s destruction. The materialist historical account of gender is precisely why we feel disappointed by the prescriptions of communization: the possibility of a willful revolt against the straight jacket of gender remains absent.
P. Valentine’s piece begins by reading both the work of TC and Maya Andrea Gonzalez. She affirms much of the same contention, saying that communization theory is uniquely on the brink of being able to offer a theory of gender and capital as a single system. Beyond this, for her, communization is \emph{a demand for the abolition of fundamental material elements of the reproduction of gender.} She, like Gonzalez, critiques TC for their suturing of gender on top of the capitalist mode of production, and strives to find the ‘real material ground’ of the production of gender difference. She contends that this will be the basis for a ‘non-idealist’ theory of the abolition of gender. At best it is funny that she searches for this \emph{material ground} in the \emph{theoretical demand} of esoteric communism. At worst, this attempt to create ‘non-idealist’ content feels eerily complicit in the typical Historical operation of justifying the extermination of those rebels whose escape attempts are not easily rationalized within these material contexts. For Valentine, this ‘real material ground’ is located in the separation of productive and reproductive spheres, but also in the realm of childbirth. To her credit, she explicitly says that neither of these phenomena account for the emergence of the gender distinction, but she has no other theory on this regard.
\begin{quote}
Further, and more fundamentally, how does this appropriation of women, on whatever basis (childbearing or no) begin? In other words, what is the origin of the gender distinction and how is it reproduced? These questions are outside the scope of this article, but we do believe that the answers both involve gendered physical violence and sexual violence.
\end{quote}
What does it mean to assert the necessity of finding the material ground for the emergence of gender, and then to refuse to do so? The material ground is based in sexual violence, but this violence is a tool of domestication’s exchanging of bodies and enforcing of spiritual submission. This dead-end in communization seems like a willful refusal to follow the inquiry to where it should take us. Valentine actually interjected into a panel discussion with Silvia Federici in Oakland when another speaker was beginning to discuss this very question of gender and civilization by mocking ‘what is civilization, even?’ She may not want to let that discussion happen, but it is precisely the discussion we are interested in. Civilization is the archaic monstrosity which produces itself through this very sexual violence and gendering operation Valentine alludes to. It is the holy grail of ‘material ground’ that Marxist feminists search for but can never find. Valentine is unique in situating sexual violence as the basis of the accumulation of women’s labor (and not simply a consequence of accumulation, as almost all other Marxists would say), but still cannot speak about when and why this violence emerges.
She says that “understanding sexual violence as a structuring element of gender also helps us to understand how patriarchy reproduces itself upon and through gay and queer men, trans people, gender nonconforming people and bodies, and children of any gender\dots{}” but she gives absolutely zero content to this ‘understanding.’ She says “that communization opens avenues toward new and more rigorous theories of gender oppression that are able to link the exploitation and oppression of women with violence and oppression based on heteronormativity and cisnormativity.” She can cite that this violence exists, but does not begin to traverse the avenue that is supposedly opened by communization theory. The only heavy lifting she does on gender violence is explicitly limited to ‘violence against women.’ This feels like the same lip service and politics of inclusion we’ve derided already.
This is a noticeable trend in the essay: Valentine identifies limits within other communizer thought, and offers platitudes about how these limits must be overcome, but does little to start the process of that overcoming. This is true of the questions of origins, sexual violence, the gender violence experienced by queer and transgender people, and the violence imposed upon children. She does the same with race, identifying it as a limit to communization thought, but ending there. This strategy appears as a tragic repetition of the academic worldview, but also as the hard limit to the usefulness of communization in our own inquiry. We aren’t interested in academics’ self-congratulatory pontification on how they should start considering our experiences: \emph{we want a way out.}
At the time of writing, the most recent contribution to the gender and communization discourse appears in the third issue of the journal \emph{Endnotes} under the title “The Logic of Gender.” Were we to wager a hope that this piece would flesh out some of the limits set in the first two texts, we would be sorely disappointed. If anything, this piece takes a hard turn away from the questions of origins, sexual violence and the means of destruction. Instead, \emph{Endnotes} is explicit in being only interested in those forms of gender specific to the capitalist mode of production. Ironically, their definition of those forms centers on the trading of bodies as gendered commodities, a process which Camatte, Rubin, Perlman and countless others have identified long before the capitalist mode of production. The piece limits its focus to the contemporary split between two spheres of labor central the capitalist production. Elsewhere defined as public\Slash{}private, productive\Slash{}reproductive, or waged\Slash{}unwaged, \emph{Endnotes} devote most of their intellectual labor to defining more precise, specific and sophisticated terms for these spheres. What they settle on are humorously long-winded \emph{directly market-mediated sphere} (DMM) and the \emph{indirectly market-mediated sphere} (IMM). True to form, they go on and sketch a periodization of these spheres beginning with the primitive accumulation of the 16\textsuperscript{th} and 17\textsuperscript{th} centuries, jumping forward to Fordism, dwelling for a moment on the seventies and concluding with the present Crisis. We could accept this as an interesting constellation of stories, if it wasn’t for the insistence by the storytellers that this is empirical, material His-story—the one story which consumes all others. This His-story is noticeably thin for people who pride themselves on their erudite and meticulous historical analysis; to say nothing of its fixation on those exact same periods on which previous Marxist accounts of gender fixated. This new formulation of DMM and IMM spheres is maybe the most vulgar of all the Marxist formulations we’ve explored so far.
And yet there is one moment of the text which we may find useful. The piece specifically denaturalizes gender and sex (with the help of queer theory) and says that groups of individuals are \emph{anchored} into these binary spheres—spheres which are constantly changing which maintaining the universal binary structure itself. It describes the naturalization of sex and gender as moments of this \emph{anchoring}, and claims that this process happens over and over again, reimposing and reproducing gender. They criticize a formulation for self-managed reproductive labor (put forward by Federici) as just another dreadful reimposition of gender. We’d agree with this, but are interested in locating the other moments of reimposition. If we are to be generous, this process of \emph{anchoring} and \emph{reimposition} of gender could be understood as a euphemism for what we call \emph{domestication}. Sadly the text explores this no further.
In keeping to the motifs of communization theory, the author(s) will allude to more limits that they do not actually explore. In what is essentially a footnote to an addendum, they say that their theory is predicated on taking for granted mechanisms such as the institution of marriage, the availability or not of contraceptives, the enforcement of heteronormativity, the shame around non-reproductive sex acts, etc. These moments which cannot be systematized within their rigorous system are noteworthy in that they amount to a vast and unquantifiable sphere of gendered activity. It is through these untheorized mechanisms that the \emph{anchoring} of gender occurs. If we want to theorize the abolition of gender, we need to depart from the Marxist cathexis upon the spheres of labor, and look also at those mechanisms which naturalize, capture and \emph{anchor} individuals into them.
The piece concludes by repeating another motif of communization theory, an assertion that this or that movement of history \emph{now} makes it possible for us to recognize this or that aspect of identity as an external constraint. Specifically they say that “the process of denaturalization creates the possibility of gender appearing as an external constraint. This is not to say that the constraint of gender is less powerful than before, but that it can now be seen as a constraint, that is, as something outside oneself that it is possible to abolish.” This assertion inadvertently serves the naturalization process through the unfounded implication that gender has not been seen as an external constraint up until this point. Gender is of course something outside of ourselves which imprisons us, but this has been realized from its most primal origin; this realization has been the continuous source of the revolt which tends toward its decomposition. The faggot heretics, witches, and gay rioters show us that domesticated gender has always been experienced as an external constraint. This is exactly why it must be constantly re-naturalized and re-imposed.
The \emph{Endnotes} piece ends in the same way as the others, in asserting the need for a communization theory that can explain how gender will be abolished, without even beginning to conceive of how that abolition will occur. In this way, communization can only be experienced as having a tragically messianic character, as something we must wait for and never something in which we participate. It is a scientific study, constrained like all other theories which stake a claim to certainty and truth. If it has an application outside of this purely academic framework, it remains to be shown. The assertion (that gender and Capital will be overcome together) is merely rhetorical if gender is only understood in its capitalist permutations. If the assertion is to have any content, we must understand the gendered world that Capital inherited as well as the contemporary operations which cannot be explained by Marxist formulations.
\section{XVII}
The preceding fragments point to what we should now state clearly: domestication did not happen to us 10,000 years ago, nor in the 16\textsuperscript{th} and 17\textsuperscript{th} century, and certainly not in the rise of Fordism. Domestication is constantly happening. There is no singular origin to gender as domestication. It is done to us everyday in countless diffuse and often invisible ways. It is a rhythm that is imposed upon our lives; escape and capture, decomposition and recomposition. If gender\Slash{}domestication is active in all the origin stories, but also in every moment of the present, then we need a tool to explain how this happens, and what mechanisms enforce this rhythm. The method of storytelling is one such tool, enchanting us with occurrences not bound in any particular temporality.
Foucault, through Agamben and later Tiqqun, gives us another tool in the concept of the apparatus. An apparatus is a network of relationships between a heterogenous set of discourses, institutions, architectural forms, regulatory decisions, laws, administrative measures, scientific statements, philosophical, moral and philanthropic propositions.
\begin{quote}
It is a heterogenous set that includes virtually anything, linguistic and nonlinguistic, under the same heading: discourses, institutions, buildings, laws, police measures, philosophical propositions, and so on. The apparatus itself is the network that is established between these elements.
\end{quote}
Apparatuses are the pure enforcement of governance and the formation of subjectivities. They include anything useful in governing, controlling and orienting human behavior. In this sense, the system of gender can be understood as a network between all these mechanisms which produce gendered subjects in order to control and orient our very being.
To quote Agamben:
\begin{quote}
I wish to propose to you nothing less than a general and massive partitioning of beings into two large groups or classes: on the one hand, living beings (or substances), and on the other, apparatuses in which living beings are incessantly captured. On one side, then, to return to the terminology of the theologians, lies the ontology of creatures, and on the other side, the \emph{oikonomia} of apparatuses that seek to govern and guide them toward the good.
Further expanding the already large class of Foucauldian apparatuses, I shall call an apparatus literally anything that has in some way the capacity to capture, orient, determine, intercept, model, control, or secure the gestures, behaviors, opinions, or discourses of living beings. Not only, therefore, prisons, madhouses, the panopticon, schools, confession, factories, disciplines, juridical measures, and so forth (whose connection with power is in a certain sense evident), but also the pen, writing, literature, philosophy, agriculture, cigarettes, navigation, computers, cellular telephones and—why not—language itself\dots{}
To recapitulate, we have then two great classes: living beings (or substances) and apparatuses. And, between these two, as a third class, subjects. I call a subject that which results from the relation and, so to speak, from the relentless fight between living beings and apparatuses. Naturally, the substances and the subjects, as in ancient metaphysics, seem to overlap, but not completely. In this sense, for example, the same individual, the same substance, can be the place of multiple processes of subjectification: the user of cellular phones, the web surfer, the writer of stories, the tango aficionado, the anti-globalization activist, and so on and so forth. The boundless growth of apparatuses in our time corresponds to the equally extreme proliferation in processes of subjectification.
\end{quote}
In this description, we cannot help but read a process by which wild life is captured by a dead thing, and is mutilated into a gendered subject. This theory of apparatuses gives us a helpful way to conceive of domestication without origins, of domestication in the present. It also allows us to indict all the emergent non-normative and innovative subjects as new machines of capture along with the old.
\begin{quote}
All of this means that the strategy that we must adopt in our hand-to-hand combat with apparatuses cannot be a simple one. This is because what we are dealing with here is the liberation of that which remains captured and separated by means of apparatuses\dots{}
\end{quote}
Our hand-to-hand conflict with gender must then be conceived of as that same effort to liberate the living remainder from the subjectivities created by the network of dead things. From this perspective, an insurrection against gender begins as an exploration of all the engendering apparatuses which function in our daily lives to reorient and re-anchor our being into these subjects. Equally so, we must also explore those apparatuses which produce racial subjects which are inseparable form gendered ones. What are the machines that hold us hostage? How do they breakdown? How can we evade them? How can we destroy them? A thorough detailing of these infinite enemies is a monumental task, but it is one which we must undertake if an insurgent break from gender is to be possible. We have already indicted several, but we will need to be even more imaginative and aware if we are to indict all those ones that seem neutral if we are to permanently shatter the spectacle of naturalized gender and escape into an ungendered unknown.
Following from this understanding, we can realize that it requires that we have recourse to another: to explore domestication without origins, we need to give a \emph{different shape to Time itself}. Such a new shape will mean dispensing with the concept of the \emph{primitive} as some natural antecedent to an inevitable teleological rise of civilization. Such a concept will always bear the naturalized image of civilization itself into pre-history, obscuring the brutal conquest which those images entail. Instead we need a shape to time which recognizes domestication as a process which is constantly capturing life outside itself; erasing the stories and cosmologies of anything beyond its control.
\section{XVIII}
\begin{quote}
I’ve been using the present tense. Ur is Now. It is not exotic at all. It is our world\dots{}
An individual intimately familiar with the daily rapacity may remain unmoved by critics of the rapacity. She or he must make a choice, she must decide to turn against the authorities and to join the circle of resisters. Such a decision disrupts a person’s whole life, and it needs to be motivated by very good reasons. The good reasons are expressed in the language of the time, not in the language of some future time. A revelation or a visitation is a very good reason. The revelation might come in a dream, or in a vision, or in what we will call a complete mental breakdown. Before this experience, everything was noise and nothing had meaning. After the experience, everything is clear. Now the individual wonders why others are so blind. She might become impatient with the others and leave them to their blindness, or she might decide to return to the others to help them see.
All this is very understandable, very human, and it has been taking place in human communities for a long time. But such sudden disruptions of individual lives are also disruptions of Leviathanic existence. After such experiences, an individual abandons the sequence of meaningless intervals of Leviathanic Time and recovers some of the rhythms of communities in the state of nature\dots{}
The paradox will be problematic to people trapped in linear, Leviathanic time. [Others] knew linear time as well as rhythmic time, and they also knew that what mattered, what was humanly important, did not take place in linear time\dots{} Rhythmic events were the subjects of songs, of dances, of the frequent ceremonies and festivals. [Historical events] will be considered ‘facts’ and “raw data” by the Leviathanized because the linear progression of such events constitutes Leviathanic time, namely His-story. The Leviathanized will remember only fragments of the sole events they consider worth remembering because the memory of such events will not be lodged in living human beings but on stone tablets, on paper, and eventually in machines\dots{}
If tragedy repeat[s], then the event was not linear but rhythmic, and it was already known. Rhythms were grasped with symbols and expressed with music. Musical knowledge was knowledge of the important, the deep, the living. The music of myth expressed the symphony of rhythms that constituted the Cosmos.
In Eurasia, Leviathan destroyed communities and encased human beings in its entrails. Linear His-story replaced the rhythmic cycles of life. Music gave way to the March of Time\dots{}
These very words, written words, are inventions of the Lugal’s scribes. They cannot convey dream time\dots{}
The Renegades from Civilization are notorious. They shed masks. They shed whole armors. They separate from previously indispensable amenities and experience a shedding of an insupportable burden. Mere contact with a community of free human beings gives them insights no Leviathanic education can provide. Nurturing contact stimulates dreams and ultimately even visions. The Renegade is possessed, transformed, humanized. Psyche-manipulators aware of Civilization’s discontents will try to induce such transformations within Leviathan’s entrails, but their most vaunted successes will be miserable failures. Civilization does not nurture humanity\dots{}
The invasion is a silencing of music, a flattening of rhythm; it is a linearization of time, a destruction of the myths and ways that will later be called Culture, a war against communities that nurture freedom, vision and life\dots{}
The resistance persists from generation to generation, in the face of plagues, poisons and explosives. The story of that resistance has been repeatedly and powerfully told. It is a story that does not show Leviathan to be as natural to human beings as hives are to bees. It is a story that shows Leviathan to be an aberration which cannot be imposed, by wile or by force, on human beings who retain the slightest link with community, even a link as tenuous as the remembrance of a Dream Time\dots{}
It is a good time for people to let go of its sanity, its masks and armors, and go mad, for they are already being ejected from its pretty \emph{polis}. In ancient Anatolia people danced on the earth-covered ruins of the Hittite Leviathan and built their lodges with stones which contained the records of the vanished empire’s great deeds.
The cycle has come round again. America is where Anatolia was. It is a place where human beings, just to stay alive, have to jump, to dance, and by dancing revive the rhythms, recover cyclical time. An-archic and pantheistic dancers no longer sense the artifice and its His-story as All but as merely one cycle, one long night, a stormy night that left Earth wounded, but a night that ends, as all nights end, when the sun rises.\footnote{\emph{Against His-Story, Against Leviathan!}.}
\end{quote}
\section{XIX}
We must pause here and ask a question which is implicit in all the ideological understandings of gender; \emph{has there been or will there be a world without gender?}
The nihilist task is to say \emph{no}. As a consequence of a rhythmic shape to time, we cannot rely on any answers which would assert with any certainty that a world without gender ever existed. As a further consequence, we cannot put faith in any utopian vision of a world without gender to come. Whatever is said by the soothsayers of feminism and queer theory, utopia does not approach. We’ve explored countless visions of how such a utopia might emerge, but each feels as unlikely as the last. The eco-feminist matriarchy never existed as a universal, and if it did it is hopelessly lost. The techno-industrial fantasies of mechanical reproduction and automated reproductive labor are simply an intensification of the nightmare. The abolition of gender awaited by the communizers has yet to reveal its shape or really even a hint of its coming. The democratic diffusion of gender in queer subculture amounts to an ever more insidious and diffuse recomposition of gender.
\emph{Against His-Story, Against Leviathan!} can be read as a biographical account of the failures of those who resist the Leviathan. After all, the decomposing or abandoned segments of the monster can always be reconfigured and re-animated. Individuals and communities of resisters will die, but the components and apparatuses of the machine can always be revived to re-capture life anew. Living beings are inferior in this respect. Death is on the side of the machines. The stories of those who’ve escaped are often lost to us. And we ourselves are often so mutilated by the machine that we may not be able to hear anyway. The masks and armors are often to deeply intertwined with our being to tear off, and when we can, we are left wounded.
\begin{quote}
This has tragic consequences for those who at last succeed in disencumbering themselves of the heavy carcass. They cannot return to the old communities, for these have been destroyed by generations of plundering, kidnapping and murdering Civilizations. People cannot resume; they have to start over again. We should not assume that the ways, what we will call Culture, nurtured and cultivated over thousands of generations, can be regenerated overnight.
The messianic stories have lost much of their power.
\end{quote}
It is hard to imagine that any collapse or revolution of divine intervention could truly burn this archaic constraint out of us.
\begin{quote}
All the sweat and labor expended hourly in the beast’s entrails presupposes the beast’s perpetual existence. The notion of a Progress that culminates in a final collapse is Christian but not Leviathanic. The notion is of a piece with Christianity’s commitment to the absurd, and is not altogether absurd if life is considered a vale of tears. But for Leviathan such a notion is contradictory, and Leviathan is an eminently logical entity.
Leviathanic existence, a vale of tears to Christians and outsiders, is to Leviathan a paved highway, and Progress along this highway cannot lead to an Apocalypse but only to more Progress.
Leviathanic self-consciousness expresses itself in the currents of thought known as Enlightenment, Illuminism, Masonry, Marxism, plus a few others. These currents supply the all-swallowing beast with a language suitable to its last days.
\end{quote}
Yet remarkably, we never see in \emph{Against His-Story} an argument to accede to our capture and constraint. Rather, we see a celebration of all the moments of resistance which start in the lives of the resisters themselves. To give up on hope for a world without gender is not to accept defeat. Rather it unchains us from the old traps of Politics and Ideology and allows us to begin again, shifting the scope from all of His-story to our own lives. It allows us to begin again from ourselves, our bodies and our spirits.
\section{XX}
If there was no pre-existing and definite world without gender, then we cannot conceive of our struggle as being for a return to some pre-gendered whole. Rather we must conceive of our escape as the flight of domesticated beings into the wild. Not primitive or prelapsarian beings, we must become feral beings. We can understand queerness similarly. We aren’t naïve enough to project a positive or essential queerness into the unknown before civilization. Instead, we conceive of our queerness negatively, as escape, refusal and failure of gender. What we pursue then, is a \emph{feral queerness} which bucks against all the apparatuses of constraint and subjection; a feral queer which appears as out of time, irrational, inappropriate and wild. We won’t find this in anthropology, history, economics or psychoanalysis. Instead we’ll employ magic, heresy, myth and exegesis.
Those examples we have explored previously take for granted that such a \emph{feral queerness} must emerge through the struggle of the body against its capture. This is largely self evident in the modes of riot, evasion and rebel sexuality which comprise our queer stories. What is more subtle, and requires some elaboration, is that the struggle against domestication must also occur in its spiritual dimension. As the body must flee the machines which capture it, the spirit must expel the machines which colonize it. \emph{We must do violence with ourselves.} To embark on this lifelong endeavor, we’ll have to chart a course against the multiplicity of apparatuses which compose this gendered prison.
Fredy Perlman will speak of this task as the fire which burns against the darkness. A fire which can burn off the mask, burn out the armor and burn Leviathan to the ground.
\begin{quote}
The last communities do a ghost dance, and the ghosts of the last communities will continue to dance within the entrails of the artificial beast. The council-fires of the never-defeated communities are not extinguished by the genocidal invaders, just as the light of Ahura Mazda was not extinguished by rulers who claimed it shone on them. The fire is eclipsed by something dark, but it continues to burn, and its flames shoot out where they are least expected.
\end{quote}
This fire is largely ineffable, and attempts to enshrine it in words often amount to yet another apparatus of capture. We cannot scientifically articulate this fire, as it has to be found in each individual if they are to participate in any personal or collective desertion of the beast. The fire which burns against gender is precisely that inexpressible moment of queerness which lashes out against any capture in language. We cannot comprehend the fire, but we can try to illustrate its contours.
We must reclaim the mystery, passion, intensity and depth of feeling which has been alienated from us and enshrined in religion. We must pursue the spiritual ecstasy which religion cohered in order to abolish. We must pursue the unity and joy which gender has always precluded and imitated. More specifically, we must refuse the binary which relegates these pursuits to some spiritual realm separate from our corporeality. Revolt must take form and content which do not deny and separate the body and spirit. As the fire burns out the mechanistic parts of the self, it must also burn the tethers which maintain our capture.
We’ll briefly return to Feral Faun to quote:
\begin{quote}
The revolutionary project must certainly include the end of religion—but not in the form of a simplistic acceptance of mechanistic materialism. Rather, we must seek to awaken our senses to the fullness of life that is the material world. We must oppose both religion and mechanistic materialism with a vibrant, passionate, living materialism. We must storm the citadel of religion and reclaim the freedom, the creativity, the passion and the wonder that religion has stolen from our earth and our lives. In order to do this we will have to understand what needs and desires religion speaks to and how it fails to fulfill them. I have attempted to express some of my own explorations so that we can carry on the project of creating ourselves as free, wild beings. The project of transforming the world into a realm of sensual joy and pleasure by destroying the civilization that has stolen the fullness of life from us.
\end{quote}
A \emph{feral queerness} may appear as a wildness, as an effort to embody the chaos of the world, while refusing the ordering that is always imposed upon that chaos. It might appear as an orgiastic dance against constraint, or a frenzied tearing off of the masks and armors. It may appear as the rediscovery of all the potentials—sexual, animistic, relational, magical—which have been stunted by domestication. It will seem emotional, cathartic, irrational, but healing.
But it may also appear more quietly as a withdrawal. Sometimes it is easier to discreetly flee the beast. People are constantly plotting escapes and they often succeed. The stories of renegades, maroons, vagabonds and defectors illustrate another form of Leviathan’s decomposition. Rather than proclaiming some new gendered identity, a \emph{feral queerness} might not be visible at all. It may hide, flee, and make a home for itself in the shrouds of mystery outside leviathan’s purview. In a world which calls us to self-identify, we must make a home in anonymity.
Any possible escape from gendered constraint will likely involve both the explosive and clandestine tactics, but also methods which make these forms indistinguishable. When I don the black mask, I participate in the unfolding of a riot, but also withdraw from the apparatuses which would locate and identify me in this or that gender. I obscure my facial features, hair, body—anything which could be engendered; revealing instead my violence. The State, Media, and feminist Left endlessly insist that the violence belongs to men alone; this insistence itself forms another apparatus to capture and engender. My violence, taken from me by so many representations and politics of victimhood, returns and emanates from the inside outward. The black mask forms the fabric which stitches together the refusals of internal submission and external representation. Above all else, the following attacks destroy the barriers and separations within and without. I become a microcosms of the chaos around me, suspending the regulatory practices of identity.
A \emph{feral queerness} must extend this effect to the whole of life. Whatever its form, it must take aim at life itself.
To quote Fredy one last time:
\begin{quote}
I’m impatient to end the story of the artificial beast with human entrails. In a different work I will tell some of the details of the resistance to Americanization on the part of some of the world’s last communities. I cannot tell all, either there or here, because the struggle against His-story, against Leviathan, is synonymous with Life; it is part of the Biosphere’s self-defense against the monster rending her asunder. And the struggle is by no means over; it goes on as long as the beast is animated by living beings.
\end{quote}
To cultivate the fire means to be able to start from oneself and strike out alone. Undeniably a spreading of the wildfire would require the interweaving of one’s personal rebellion with others, but the fire cannot be imposed from the outside. It requires an overcoming of the fear of autonomy, a dependence imposed by domestication. One must oppose their life to the Leviathanic organization of a society which is death appearing as life. Refusal, evasion, attack—all of it flows from that internal fire, or it does not flow at all. We must burn gender out of ourselves before we can help cultivate the fire in others. In the first issue of this journal, we discussed the concept of \emph{jouissance}, the supersession of pleasure and pain, of duality. It is in this break with duality that we can also break with binary gender.
There are several examples we can look to of individuals and small groups fleeing or rebelling against the constraints of gender. In this context we can read the self-organization for survival by street queens of Street Transvestite Action Revolutionaries as an attempt to withdraw from the subjectivizing apparatus of sexual labor, as well as an attempt to cultivate a queer and rebel spirituality. Within prison society, we can see a wide range of stories of queer and gender-variant people revolting against the constraints of gender imposed on their bodies. Men Against Sexism waged an armed struggle against the machinery of rape culture, while the present struggle of Gender Anarky in the California prison system illustrates a clear example of a transgender anarkists waging a spiritual and bodily struggle against civilization from within the hellish intersection of so many apparatuses of gendering and control. In her text “Aspects of Insurrectionary Anarky,” Amazon of Gender Anarky writes:
\begin{quote}
The absence of spiritual awareness in one’s life contributes to fear of consequences. Worse, it leaves a vacuum in the person that gets filled with the debris of the world, clogging them up, stunting their insight. The debris of material possessions, selfishness, uncaring, ignorance, greed, envy, egotism, fear. It is a tragedy because people so afflicted cannot open up to the world around them and draw from it beneficially when their sensibilities are so shut down and distracted, cannot live full lives but live lesser, half lives\dots{} We believe in the spirit. It is an aspect of our insurrection\dots{} Being separated from nature separates us from spiritual awareness and impedes our balance, the totality of our inner self, which is needed to understand and relate to the external world around us: nature and people, the animals, the plant life, the weather and seasons, the suns, planets, moons\dots{} In this there is a direct relationship between anarkist insurrection, which fights for autonomy and the earth, and spirituality.
\end{quote}
Another inspiring example of a revolt against gender from within prisons walls is the communique released by Olga Ekonomidou, imprisoned member of the Conspiracy Cells of Fire in Greece. Olga refused the capture of her body through the apparatus of full body search:
\begin{quote}
In this moment I am writing these few lines from inside isolation; 30 days of solitary confinement is the price I pay for my refusal to sell out my dignity and obey the humiliation of a full body search, which would last 5 minutes. I remain unrepentant in my decision. I won’t give away even a second of compromise to prison guards. I will not exchange my refusals and choices with the ‘warmth’ of a standard cell and the ‘liberty’ of yard time among the general prison population. I’m not looking to become another normal statistic of an inmate who cringes before the prison service, who serves ‘quietly’ her sentence, who trips into hallucinations induced by wacko-pills, who forces herself as an ‘older rank’ on new-coming prisoners. I remain friend, comrade and human with all women and men who keep the fire burning inside them. With those women and men who choose the dangerous paths of wolves instead of sheep pastures. When it comes to all of us, anarchists of praxis, imprisonment is never enough ‘punishment.’ For this, disciplinary penalties, transfers and solitary confinements are due to come down. Isolation is a prison within the prison. You remain 24 hours a day locked up in a cage with a bunk bed, an in-cell toilet and the vigilant eye of a closed-circuit camera. Inside here, your only girlfriends are your thoughts and memories. Inside here, the days and hours are eliminated, lost, dying, pushing slowly each other\dots{}
But these 30 days of solitary confinement I was not left alone. I had some odd and charming visitors by my side that passed secretly and ‘smuggled’ their way into my cell, breaking the isolation. 30 days of solitary confinement and I go on, but the she-wolf inside me doesn’t sleep, doesn’t give consent, doesn’t forgive\dots{}
\end{quote}
Lastly, we have to mention a woman in Juarez, Mexico who goes by the name of Diana the Huntress. The border town of Juarez is notorious for what some have called an ongoing \emph{femicide}, a mass murder and disappearance of countless women. In September of 2013, Diana struck out against this apparatus of capture, shooting two rapist bus drivers. She released a communique claiming responsibility for the murder, indicting those drivers as part of the rape machinery of the city, but also announcing a refusal on her part to play the role of a victim subject.
In these diffuse stories we see moments, fragments, of the burning spiritual clarity which strikes out, through explosive violence or quiet refusal, against gender and domestication.
\section{Third Mythos: Diana}
\emph{Many today praise the greatness of the Roman Empire, the} Res Publica\emph{, the Public Thing, a civilization which recognized and hated itself as such. This self-hatred turned outwards, conquering and destroying everything outside its walls. Countless books have been devoted to the greatness of Rome, to its war engines and death machines—at times to death itself—but Rome’s greatness is posthumous. Among those trapped in its entrails, few loved it; many tried daily to destroy it. Hating what they’d become, many conspired to set fire to Rome.}
\emph{In ancient Rome, some people worshipped a more ancient deity—one who reminded them of a time before: Diana the Huntress. Though associated with the Greek goddess Artemis, she independently emerges from the long forgotten past of the time before either empire. The Romans revered her as the goddess of the moon, animals, and the wild hunt. One of her more well known exploits involves a hunter named Acteon, who inadvertently stumbled upon her bathing in a forest pool. When she realized that Acteon was watching her, she refused to be captured by his gaze. She turned him into a deer, and his own hunting dogs slaughtered him. The domesticated beasts slayed their master; the hunter became the hunted.}
\emph{For this act of wildness and refusal, Diana gained notoriety. A millennia later, she would still be worshipped as the queen of the witches all throughout southern Europe. They danced to her in} sabbats, \emph{and orgiastic rites; they flew with her beneath the stars; they celebrated her as a connection to all that was wild and indomitable. Witch hunters of the Holy Inquisition saw her as the Devil and tortured the accused into confessing their devotion to her. The punishment was death. And yet the sadistic technologies of the inquisitors and the fire of the stake were not enough to eliminate her cult. To this day,} streghas \emph{still venerate her when the moon is full, and when they strike down their enemies. Through her we might invoke the rhythms of the moon, the insight of the animals, a refusal of the techniques of surveillance and subjection, a feral becoming, death to our captors.}
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The Anarchist Library
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baedan
Against the Gendered Nightmare
Fragments On Domestication
2014
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authors’ manuscript, baedan — a queer journal of heresy — issue two
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\textbf{theanarchistlibrary.org}
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\ctr{\twelveb Asymptotics of Young Diagrams and Hook Numbers }
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\ctr{{\bf Amitai Regev}\footnote{$^*$}{Work partially supported by N.S.F.
Grant No.DMS-94-01197.}}
\inst
\par
\ctr{and}
\par
\ctr{\DM}
\ctr{The Pennsylvania State University}
\ctr{University Park, PA 16802,\ \ U.S.A. }
\ske
\ctr{{\bf Anatoly Vershik}\footnote{$^\dagger$}{Partially supported by
Grant INTAS 94-3420 and Russian Fund 96-01-00676 }}
\par
\ctr{St. Petersburg branch of the Mathematics Institute}
\ctr{of the Russian Academy of Science }
\ctr{ Fontanka 27}
\ctr{St. Petersburg, 191011 Russia}
\par
\ctr{and}
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\ctr{The Institute for Advanced Studies of the Hebrew University }
\ctr{Givat Ram }
\ctr{Jerusalem, Israel}
%
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\centerline{Submitted: August 22, 1997; Accepted: September 21, 1997}
\vskip 1.2cm
{\bf Abstract:} \ \ Asymptotic calculations are applied to study
the degrees of certain sequences of characters of symmetric groups.
Starting with a given partition $\mu$, we deduce several skew diagrams
which are related to $\mu$. To each such skew diagram there corresponds
the product of its hook numbers. By asymptotic methods we obtain some
unexpected arithmetic properties between these products. The authors do
not know "finite", nonasymptotic proofs of these results. The problem
appeared in the study of the hook formula for various kinds of Young
diagrams. The proofs are based on properties of shifted Schur functions,
due to Okounkov and Olshanski. The theory of these functions arose from
the asymptotic theory of Vershik and Kerov of the representations of the
symmetric groups.
\endpage
\baselineskip 18pt
%
{\bf \S1.\ \ Introduction and the main results}
\par
Asymptotic calculations are applied to study the degrees of certain
sequences of \ch s of symmetric groups $S_n,\ n\ra\nty$. We obtain
some unexpected arithmetic properties of the set of the hook numbers
for some special families of (fixed) skew-Young diagrams (Theorem 1.2).
The problem appeared in the study of the hook formula for various kinds
of Young diagrams. The proof of 1.2
is based on the properties of shifted Schur
functions \(Ok.Ol\) which appeared in the asymptotic theory of the
\rept\ of the symmetric groups in \(Ver.Ker\). The authors do not know
a ``finite" proof of the theorem.
\par
Given a partition $\mu$, we describe in 1.1 - a construction of certain
skew diagrams which are derived from $\mu$: these are $SQ(\mu),\ SR(\mu)
,\ SR(\mu\pr),\ R$ and $D_\mu$ below. Next, one fills these skew diagrams
with their \corr ing hook numbers \(Mac, page 10\). Theorem 1.2, which is
the main result here, gives some divisibility properties of
the products of these hook numbers.
\par
We remark again that even though the statement of theorem 1.2 has
nothing to do with asymptotics, its proof does use asymptotic methods.
It should be interesting to find an ``asymptotic free" proof of theorem
1.2.
\par
We start with
\par
{\bf 1.1:\ \ A Construction:} \ \ Given a partition (= diagram) $\mu$,
let $D^*_\mu$ denote the double reflection of $\mu$. For example, if
$\mu=(4,2,1)$ then
%
$$ D_\mu=\matrix{x & x & x & x \cr x & x & & \cr x & & & \cr}\ \qquad \
and \ \qquad
D^*_\mu = \matrix{& & & x \cr & & x & x \cr x & x & x & x \cr}. $$
\ske\noindent
Recall that $\mu\pr_1=\ell(\mu)$ is the number of nonzero parts of
$\mu$. Complete $D_\mu^*$ to the $\mu_1\times\mu\pr_1$
rectangle $R(\mu)$, then draw $D^*_\mu$ on top and on the left of $R$.
Finally, erase the first $D^*_\mu$. Denote the resulting skew diagram by
$SQ(\mu)$. For example, with $\mu=(4,2,1)$ we get\vskip 0.5truecm
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$\lft{#}$ & \lft{#} \qquad& $\lft{#}$ \hquad &
$\lft{#}$ \hquad & $\lft{#}$ \hquad & $\lft{#}$ \hquad &
$\lft{#}$ \hquad & $\lft{#}$ \hquad & $\lft{#}$ \hquad &
$\lft{#}$ & $\lft{#}$ & $\lft{#}$
\hquad & $\lft{#}$\hquad & $\lft{#}$\hquad & $\lft{#}$\cr
%
&& && && && & A_2\cr
&& && && && && \searrow &&x\cr
&&&&&&&&&&& x & x \cr
&& && && && & x & x & x & x \cr
\noalign{\vskip -6pt}
\noalign{\hskip 194.7pt{\hbox to 78pt{\hl}}}
\noalign{\vskip -5pt}
SQ(4,2,1)&=&&& A_1 &&&&\vl\cr
\noalign{\vskip -2pt}
&&&&& \searrow &&x&\vl & x & x & x\cr
\noalign{\vskip -2pt}
& & &&& & x & x & \vl & x & x \cr
\noalign{\vskip -6pt}
& & & &&& & & \vl & & &\nwarrow\cr
\noalign{\vskip -12pt}
&&& & x & x & x & x & \vl\cr
\noalign{\vskip -12pt}
& &&& & & & & \vl &&&& A\cr
}} $$}\vskip 0.5truecm
%
We subdivide $SQ(\mu)$ into the three areas $A,\ A_1$ and $A_2$: $A=R-D^*
_\mu,\ A_1$ is the $D^*_\mu$ on the left of $R$ and $A_2$ is the $D^*_
\mu$ on top of $R$. Denote $SR(\mu)=A_1\cup A$, the ``shifted rectangle".
\par
Clearly, $\v A\cup A_1\v=\v A\cup A_2\v=\v R\v,\ \v A_1\v=\v A_2\v=\v
\mu\v$, so $\v SQ(\mu)\v=\v R\v+\v\mu\v$. Now, fill $SQ(\mu),SR(\mu),\
R$ and $\mu$ with their hook numbers. For example, when $\mu=(4,2,1)$
\ske
%
\hbox{\baselineskip 12pt
\def\vl{\vrule height 10pt depth 8pt} %20 & 10
\def\hl{\leaders \hrule height 3pt depth -2.5pt\hfill}
%
$$ \vbox{\halign{\hskip 50pt
$\lft{#}$ & \lft{#} \qquad& $\lft{#}$ \hquad &
$\lft{#}$ \hquad & $\lft{#}$ \hquad & $\lft{#}$ \hquad &
$\lft{#}$ \hquad & $\lft{#}$ \hquad & $\lft{#}$ \hquad &
$\lft{#}$ & $\lft{#}$ & $\lft{#}$
\hquad & $\lft{#}$\hquad & $\lft{#}$\hquad & $\lft{#}$\cr
%
&& && && && && && && 3 \cr
&& && && && && && 4 && 2 \cr
&& && && && & 6 && 5 & 3 && 1 \cr
\noalign{\vskip -6pt}
\noalign{\hskip 180.7pt{\hbox to 70pt{\hl}}}
\noalign{\vskip -5pt}
SQ(4,2,1)& : && & && &&\vl \cr
\noalign{\vskip -2pt}
&& && & && 6 &\vl & 4 && 3 & 1\cr
\noalign{\vskip -2pt}
& & && & & 5 & 4 & \vl & 2 && 1 \cr
\noalign{\vskip -6pt}
& & & &&& & & \vl & & &\cr
\noalign{\vskip -12pt}
&&& & 4 & 3 & 2 & 1 & \vl\cr
\noalign{\vskip -12pt}
& &&& & & & & \vl &&&& \cr
}} $$}\vskip 0.5truecm
%
$$ SR(4,2,1):\hskip 2cm \matrix{& & & 6 & 4 & 3 & 1 \cr
& & 5 & 4 & 2 & 1 \cr 4 & 3 & 2 & 1 & & & \cr} $$
%
\ske
$$ R(4,2,1):\hskip 2cm \matrix{6 && 5 && 4 && 3 \cr 5 && 4 && 3 && 2 \cr
4 && 3 && 2 && 1 \cr} $$
\ske
and
$$ (4,2,1):\hskip 2cm \matrix{6 && 4 && 2 && 1 \cr 3 && 1 && && \cr
1 && && && \cr} $$
\ske
Thus, for example, $\prod_{x\in(4,2,1)}h(x)=1^3\cdot 2\cdot 3\cdot 4
\cdot 6=144$.
\par
Note that the hook numbers in $SR(\mu)$ are the same as those in the
area $A_1\cup A$ of $SQ(\mu)$.
\par
As usual, $\mu\pr_1=\ell(\mu)$ is the number of nonzero parts of $\mu$.
Recall that $s_\mu(x_1,x_2,\cdots)$ is the \corr ing Schur function,
and $s_\mu\underbrace{(1,\cdots,1)}_{\mu\pr_1}$ is the number of (semi%
-standard, i.e. rows weakly and column strictly increasing) tableaux of
shape $\mu$, filled with elements from $\{1,2,\cdots,\mu\pr_1\}$ \(Mac\).
Similarly for $s_{\mu\pr}\underbrace{(1,\cdots,1)}_{\mu_1}$.
\par
{\bf 1.2 \ \ Theorem:} \ \ Let $\mu$ be a partition. With the above
construction of $SQ(\mu)=A\cup A_1\cup A_2$ and $R$, we have
$$ \left(\prod_{x\in R}\ h(x)\right)\raise 15pt\hbox{${\exline}$}
\left(\prod_{x\in A_1\cup A}\ h(x)\right)=s_\mu (\underbrace{1,\cdots,
1}_{\mu\pr_1}). \ \ \leqno(1) $$
%
\Ip, $\prod_{x\in A_1\cup
A}h(x)$ divides $\prod_{x\in R}h(x)$. \(Note that $A\cup A_1\st SQ(\mu)$,
and for $x\in A_1\cup A,\ h(x)$ is the \corr ing hook number in $x\in
SQ(\mu)$\).
%
\parno
(1')\ \ Similarly,
$$ \left(\prod_{x\in R}\ h(x)\right)\raise 15pt\hbox{${\exline}$}
\left(\prod_{x\in A_2\cup A}\
h(x)\right)=s_{\mu\pr}(\underbrace{1,\cdots,1}_{\mu_1}). $$
\par
$$\prod_{x\in SQ(\mu)}\ h(x)=\left(\prod_{x\in R}\ h(x)\right)\cdot\left(
\prod_{x\in\mu}\ h(x)\right).\ \leqno(2) $$
\par
We conjecture that a statement much stronger than 1.2.2 holds, namely:
the two multisets \bk
$\{h(x)\mid x\in SQ(\mu)\}$ and $\{h(x)\mid x\in R\}\cup\{h(x)\mid
x\in\mu\}$ are equal.
\par
Theorem 1.2.1 is an obvious consequence of the following ``asymptotic"
theorem.
\par
{\bf 1.3.\ \ Theorem:} \ \ Let $\mu=(\mu_1,\cdots,\mu_k)$, be a
partition. Let $n=k\ell$, \bk
$\mu_1\leq\ell\ra\nty$, and denote $\l=\l(\ell)
=(\ell^k)$. Then
$$ \lim_{\ell\ra\nty}\ {d_{\l/\mu} \over d_\l} = \left({1\over k}
\right)^{\v\mu\v}\cdot s_\mu(\underbrace{1,\cdots,1}_k) \ \leqno(a) $$
and
$$\lim_{\ell\ra\nty}\ {d_{\l/\mu}\over d_\l}= \left({1
\over k}\right)^{\v\mu\v}\cdot\left(\prod_{x\in R(\mu_1,\mu\pr_1)}h(x)
\right)\raise 15pt\hbox{${\exline}$}\left(\prod_{x\in A_1\cup A}
\ h(x)\right). \ \leqno(b) $$
\parno
Theorem 1.2.1' follows from 1.2.1 by conjugation.
\par
Theorem 1.2.2 is a consequence of the following ``asymptotic" theorem
\par
{\bf 1.4. \ \ Theorem:}\ \ Let $\mu$ be a fixed partition. Let $\mu_1
\leq\ell\ra\nty$, \bk
$\mu\pr_1\leq m\ra\nty,\ n=\ell m$ and $\l=\l(\ell,m)=
(\ell^m)$. Then
$$ \lim_{\ell,m\ra\nty}\ {d_{\l/\mu}\over d_\l}=\
{1\over \prod_{x\in\mu}\ h(x)}\ . \ \leqno(a) $$
%
$$ \lim_{\ell,m\ra\nty}\ {d_{\l/\mu}\over d_\l}=\
\left(\prod_{x\in R}\ h(x)\right)\raise 15pt\hbox{${\exline}$}
\left(\prod_{x\in SQ(\mu)}\ h(x)\right)\ . \ \leqno(b) $$
\par
In this note we apply the following main tools:
\parno
a)\ \ The theory of symmetric functions \(Mac\). \Ip, we apply the hook
formula
$$ d_\l=\ {\v\l\v!\over\prod_{x\in\l}\ h(x)} $$
and I.3, Example 4, page 45 in \(Mac\).
\parno
b) \ \ The Okounkov-Olshanski \(Ok.Ol\) theory of ``shifted symmetric
functions". \Ip, we apply formula (0.14) of \(Ok.Ol\): \parno
Let $\mu\vdash k,\ \l\vdash n,\ k\leq n,\ \mu\st\l$, then
$$ {d_{\l/\mu}\over d_\l}\ =\ {s^*_\mu(\l)\over n(n-1)\cdots(n-k+1)}
\ . $$
\parno
Here $s^*_\mu(x)$ is the ``shifted Schur function" \(Ok.Ol\);\ \ one of
its key properties is that \bk
$s^*_\mu(x)=s_\mu(x)+$ lower terms, where
$s_\mu(x)$ is the ordinary Schur function.
\par
We remark that the paper \(Ok.Ol\) was influenced by the work of
Vershik and Kerov on the asymptotic theory of the \rep ations of the
symmetric groups. See for example \(Ver.Ker\), in which the \ch s of
the infinite symmetric group are found from limits involving ordinary
Schur functions. See also the introduction of \(Ok.Ol\).
%
\ske
{\bf \S2.}\ \ Here we prove theorem 1.3 which, as noted before, implies
1.2.1 (and 1.2.1').
\par
{\bf 2.1. \ \ The proof of theorem 1.3.}
$$ {d_{\l(\ell)/\mu}\over d_{\l(\ell)}}\ =\ {s^*_\mu(\l_1(\ell),\cdots,
\l_k(\ell))\over n(n-1)\cdots(n-\v\mu\v+1)}\ , $$
where $n=\v\l\v=k\ell$. Since $\ell\ra\nty,\ n(n-1)\cdots
(n-\v\mu\v+1)\simeq (k\ell)^{\v\mu\v}$. Also,
$$ s^*_\mu(\l)=s_\mu(\l)+(lower\ terms\ in\ \ n), $$
hence
$$ s^*_\mu(\l)\simeq s_\mu(\l)=s_\mu\underbrace{(\ell,\cdots,\ell)}_k
. $$
%
Recall that for two sequences $a_n$, $b_n$ of real numbers,
$a_n \simeq b_n$ means that $\lim_{n\rightarrow\infty}
{a_n\over b_n} = 1$.\par
%
Since $s_\mu(x)$ is \hog\ of degree $\v\mu\v$,
$$ s_\mu(\l)=\ell^{\v\mu\v}\cdot s_\mu(\underbrace{1,\cdots,1}_k)
\ . $$
The proof now follows easily.
\hfill\QED\par
%
\ske
{\bf 2.2. \ \ The proof of theorem 1.3.b:} \ \ Since $\l$ is a
rectangle, hence $d_{\l/\mu}=d_\eta$, where $\eta$
is the double reflection of $\l/\mu$. Denote by $\tilde\mu=D^*_
\mu$ the double reflection of $\mu$. Thus
\ske
%
%REGFIG.TWO
%
\midinsert
\SetLabels
(.05*.46)$\eta$\\
(.85*.35)$D^*_\mu$\\
(.51*.81)$\lscr$\\
\endSetLabels
%\ShowGrid
\centerline{\AffixLabels{%
\epsfysize=1.9in\epsfbox{regfig2.eps}
}}
\endinsert
%
To calculate $d_\l$ and $d_\eta$ by the hook formula, fill $\l=\l(\ell)$
and $\eta$ with their respective hook numbers. In both, examine the
$i^{th}$ row from the bottom - with their respective hook numbers.
Divide $\eta$ into $B_1$ and $B_2$ as follows:
%
%REGFIG3
%
\midinsert
\SetLabels
(.68*.48)$B_1$\\
(.27*.48)$B_2$\\
(.86*.43)$D^*_\mu$\\
(.50*.85)$\lscr$\\
%
(.79*.77)%
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
\hbox to 35pt{\upbracefill}
\hbox to 35pt{\hfill$\mu_1$\hfill}}} $$\\
%
(.68*.15)
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
\hbox to 35pt{\upbracefill}
\hbox to 35pt{\hfill$\mu_1$\hfill}}} $$\\
%
\endSetLabels
%\ShowGrid
%\centerline
\hskip 41pt{\AffixLabels{%
\epsfysize=2.2in\epsfbox{regfig3.eps}
}}
\endinsert
%
\parno
Notice that $B_1=SR(\mu)$ of 1.1. Note also that the hook numbers in
$B_1$ are those in $SR(\mu)$, and they are \ind\ of $\ell$.
\par
Examine the hook numbers in $B_2$. In the $i^{th}$ row (from bottom),
these are \bk
$\mu_1+i,\ \mu_1+i+1,\cdots,\ell+i-1-\mu_i$, consecutive integers.
\par
We also divide $\l(\ell)$ into two rectangles:
%
%REGFIG.FOUR
%
\midinsert
\SetLabels
%
(.86*.03)%
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
\hbox to 35pt{\upbracefill}
\hbox to 35pt{\hfill$\mu_1$\hfill}}} $$\\
%
(.03*.4)$\lambda(\lscr)$\\
(.87*.4)$R_1$\\
(.51*.81)$\lscr$\\
(.49*.4)$R_2$\\
\endSetLabels
%\ShowGrid
\centerline{\AffixLabels{%
\epsfysize=1.9in\epsfbox{regfig4.eps}
}}
\endinsert
%
%
\parno
Again, the hook numbers in $R_1$ are \ind\ of $\ell$, and those in the
$i^{th}$ row (from bottom) of $R_2$ are $\mu_1+i,\mu_1+i+1,\cdots,\ell+i
-1$, again consecutive integers.
\par
By the ``hook" formula, the left hand side of 1.3.b is
$$ {d_{\l(\ell)/\mu}\over d_{\l(\ell)}}\ =\ {d_\eta\over d_{\l(\ell)}}\
=\ \left\({(n-\v\mu\v)!\over \prod_{x\in\eta}h(x)}\right\)
\raise 15pt\hbox{${\exline}$} \left\(
{n!\over\prod_{x\in\l(\ell)}\ h(x)}\right\) $$
$$ =\ {(n-\v\mu\v)!\over n!}\ \cdot\ \left\({\prod_{x\in\l(\ell)}h(x)
\over \prod_{x\in\eta}h(x)}\right\) $$
%
where $n=k\ell$. Since $\ell\ra\nty$,
$$ {(n-\v\mu\v)!\over n!}\ \simeq\ \left({1\over n}\right)
^{\v\mu\v}=\ \left({1\over k\ell}\right)^{\v\mu\v} . $$
\parno
Now
$${\prod_{x\in\l(\ell)}h(x)\over\prod_{x\in\eta}h(x)}\ = \ \left\({\prod
_{x\in R_1}h(x)\over \prod_{x\in B_1}h(x)}\right\)\cdot\left\({\prod_{
x\in R_2}h(x)\over \prod_{x\in B_2}h(x)}\right\)=\a\cdot\b. $$
\parno
Note that the right hand side of 1.3.b is $({1\over k})^{\v\mu\v}\cdot\a$.
\par
We calculate $\b$:
$$ \prod_{x\in R_2}h(x)=\prod^{\mu_1\pr}_{i=1}\((\mu_1+i)(\mu_1+i+1)\cdots
(\ell+i-1)\), $$
$$ \prod_{x\in B_2}h(x)=\prod^{\mu\pr_1}_{i=1}\((\mu_1+i)(\mu_1+i+1)\cdots
(\ell+i-1-\mu_i)\), $$
thus
$$ \b=\prod^{\mu\pr_1}_{i=1}\((\ell+i-\mu_i)(\ell+i-\mu_i+1)\cdots(\ell+
i-1)\)\simeq\ell^{\v\mu\v}, $$
(since $\ell\ra\nty$).
\par
Hence,
$$ \lim_{\ell\ra\nty}\ {d_{\l(\ell)/\mu}\over d_{\l(\ell)}}\ =\
\left({1\over k}\right)^{\v\mu\v}\cdot\a $$
and the proof is complete.
\hfill\QED\par
\ske
%
{\bf \S3.}\ \ Here we prove theorem 1.4 which, as noted before, implies
theorem 1.2.2.
\par
{\bf 3.1. \ \ The proof of 1.4.a:} \ \ Let $\l=\l(\ell,m)=(\ell^m),\
\ell, m\ra\nty$. We show first that $s^*_\mu(\l)\simeq s_\mu(\l)$, as
follows:\ \ By \(Ok.Ol.(0.9)\),
$$ \eqalign{e^*_r(\l) & =\sum_{i\leq i_1<\cdots < i_r\leq m}\ (\ell+r-1)
(\ell+r-2)\cdots\ell = \cr
& = (\ell+r-1)(\ell+r-2)\cdots\ell\cdot{m\choose r}\simeq{\ell^r m^r
\over r!} \ . \cr} $$
\parno
Similarly, $e_r(\l)\simeq {\ell^r m^r\over r!}$ .
\par
Let $\emt$ be given as in \(Ok.Ol.\S13\). By \(Ok.Ol.(13.8)\) it easily
follows that for any $u$ and $r$,
$$ \emt^{-u}e^*_r(\l)\simeq e^*_r(\l)\simeq e_r(\l). $$
\parno
Applying the Jacobi Trudi formulas for $s_\mu(\l)$ |(Mac. I, (3.5),
page 41\) and for $s^*_\mu(\l)$ \(Ok.Ol.(13.10)\), it clearly follows
that $s^*_\mu(\l)\simeq s_\mu(\l)$. Now in 2.1, here
$$ {d_{\l(\ell,m)/\mu}\over d_{\l(\ell,m)}}\ = \ {s^*_\mu(\l_1(\ell,m),
\cdots,\l_{m+k}(\ell,m))\over n(n-1)\cdots (n-\v\mu\v+1)} $$
where
$$ n= \ell m.$$
Here
$$ s^*_\mu(\l(\ell,m))\simeq s_\mu(\l(\ell,m)) = \ell^{\v\mu\v}
s_\mu (\underbrace{1,\cdots,1}_m). $$
Thus
%
$$ {d_{\l(\ell,m)/\mu}\over d_{\l(\ell,m)}}\ \simeq\ \left({1\over n}
\right)^{\v\mu\v}\cdot s_\mu (\underbrace{1,\cdots,1}_m) =
\left({1\over m}\right)^{\v\mu\v}\cdot\prod_{x\in\mu}\ {m+c(x)
\over h(x)}\ , $$
%
(\(Mac, pg. 45, Ex 4\))
%
where $c(x)$ is the content of $x\in\mu$. Since $m\ra\nty,\ \ m+c(x)
\simeq m$ for all $x\in\mu$, and the proof follows.
\hfill\QED\par
%
\ske
{\bf 3.2.\ \ The proof of 1.4.b:} \ \ Choose $\ell,m$ large so that $\mu
\st\l(\ell,m)$. Let $\eta$ be the double reflection of $\l(\ell,m)/\mu$,
so $d_{\l(\ell,m)/\mu}=d_\eta$,
then calculate $d_\eta$ by the hook formula.
To analyze the hook numbers in $\eta$, we subdivide $\eta$ into the
areas $A_{1,\eta},\cdots,A_{4,\eta}$ as shown below:
%
%REGFIG.SIX
%
\midinsert
\SetLabels
(-.02*.43)$\eta:$\\
(.05*.425)$m$\\
(.41*.71)$\lscr$\\
(.35*.48)$A_{1,\eta}$\\
(.7*.48)$A_{3,\eta}$\\
(.35*.22)$A_{2,\eta}$\\
(.82*.33)$A_{4,\eta}$\\
(.8*.1)$D^*_\mu$\\
(.78*.215)$\big\}\mu^\prime_1$\\
%
(.685*.135)%
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
\hbox to 5pt{\upbracefill}
\hbox to 5pt{\hfill$\mu_1$\hfill}}} $$\\
%
\endSetLabels
%\ShowGrid
%\centerline
\hskip 25pt{\AffixLabels{%
\epsfysize=2.7in\epsfbox{regfig6.eps}
}}
\endinsert
%
\parno
i.e., $D^*_\mu$ is drawn at the bottom-right of the $\ell\times m$
rectangle. We then follow 1.1 and construct $A_{4,\eta}=SQ(\mu)$. Now
$A_{1,\eta}$ is the $(\ell-\mu_1)\times(m-\mu\pr_1)$ rectangle, and this
determines $A_{2,\eta}$ and $A_{3,\eta}$.
\par
We also split the $\ell\times m$ rectangle $\l=\l(\ell,m)$ accordingly:
%
%REGFIG.SEVEN
%
\midinsert
\SetLabels
%
(.38*.58)$A_{1,\lambda(\lscr,m)}$\\
(.38*.23)$A_{2,\lambda(\lscr,m)}$\\
(.89*.55)$A_{3,\lambda(\lscr,m)}$\\
(.89*.08)$A_{4,\lambda(\lscr,m)}$\\
(.83*.22)$\bigg\}\mu^\prime_1$\\
%
(.75*.12)%
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
\hbox to 25pt{\upbracefill}
\hbox to 25pt{\hfill$\mu_1$\hfill}}} $$\\
%
\endSetLabels
%\ShowGrid
\centerline{\AffixLabels{%
\epsfysize=1.9in\epsfbox{regfig7.eps}
}}
\endinsert
%
%
\vskip 1cm
Since $\l(\ell,m)\vdash\ell m$ and $\eta\vdash\ell m-\v\mu\v$,
$$ {d_\eta\over d_{\l(\ell,m)}}\simeq\left({1\over \ell m}\right)^{\v\mu
\v}\cdot{\prod_{x\in\l(\ell,m)} h_{\l(\ell,m)}(x)\over \prod_{x\in\eta}
h_\eta(x)}. $$
\par
Now, $h_{\l(\ell,m)}(x)=h_\eta(x)$ for $x\in A_{1,\eta}=A_{1,\l
(\ell,m)}$. As in 2.3
$$ {\prod_{x\in A_{2,\l(\ell,m)}}h_{\l(\ell,m)}(x)\over\prod_{x\in
A_{2,\eta}}h_\eta(x)}\ \simeq\ \ell^{\v\mu\v}. $$
\parno
Similarly (or, by conjugation),
$$ {\prod_{x\in A_{3,\l}}h_{\l(\ell,m)}(x)\over\prod_{x\in A_{3,\eta}}
h_\eta(x)} \ = \ m^{\v\mu\v}\ . $$
\par
After cancellations we have
$$ {d_\eta\over d_\l}\simeq{\prod_{x\in A_{4,\l}}h_{\l(\ell,m)}(x)\over
\prod_{x\in A_{4,\eta}}h_\eta(x)}\ = \ {\prod_{x\in R(\mu_1,\mu\pr_1)}
h(x)\over\prod_{x\in SQ(\mu)}h(x)} $$
%
and the proof is complete.
\hfill\QED\par
\ske
%
\baselineskip 12pt
\ctr{\bf References}\bigskip
%
\offset{40pt}
{\(Ok.Ol\)}Okounkov A. and Olshanski G., Shifted Schur functions,\ \
preprint.\par
%
\offset{40pt}
{\(Mac\)}Macdonald I.G., Symmetric functions and Hall \po s, \ \
Oxford University Press, 2nd edition 1995.\par
%
\offset{45pt}
{\(Ver.Ker\)}Vershik A.M. and Kerov, S.V., Asymptotic Theory of \ch s
of the symmetric group, Funct. Anal. Appl. 15 (1981) 246-255.
%
\bk\bk
email addresses: [email protected], [email protected]
\bye
|
http://manabukano.brilliant-future.net/lecture/kisojoho/020607sample1.tex
|
brilliant-future.net
|
CC-MAIN-2022-40
|
application/x-tex
|
application/x-tex
|
crawl-data/CC-MAIN-2022-40/segments/1664030337971.74/warc/CC-MAIN-20221007045521-20221007075521-00084.warc.gz
| 30,960,058 | 1,718 |
\documentclass{jarticle}
\setlength{\topmargin}{-2cm}
\setlength{\oddsidemargin}{0cm}
\setlength{\evensidemargin}{0cm}
\setlength{\textwidth}{16.0cm}
\setlength{\textheight}{25.0cm}
%
\title{\LaTeX の演習(基礎編)}
\author{加納学（学籍番号：0123456789）}
%
\begin{document}
\maketitle
\LaTeX による文書例です．
\section{文字}
%
{\tiny 非常に小さい文字です．}
{\small 小さい文字です．}
普通の文字です．
{\large 大きい文字です．}
{\LARGE かなり大きい文字です．}
{\Huge 非常に大きい文字です}
和文には，明朝体と{\bf ゴシック体}があります．
英文にも，normal, {\it italic}, {\bf bold}などがあります．
\section{数式}
%
はじめの式を与えます．
\begin{eqnarray}
y=\frac{ax^2+bx+c}{\alpha x+\beta}
\label{eqn:y}
\end{eqnarray}
式(\ref{eqn:y})を次式に代入します．
\begin{eqnarray}
z &=& \int^\infty_0 y dx \\
&=& \int^\infty_0 \frac{ax^2+bx+c}{\alpha x+\beta} dx
\label{eqn:z}
\end{eqnarray}
このように，式番号は自動的に付けられ，その番号を参照することもできます．
もちろん，文章中に$\Sigma_{i=1}^n\int^\infty_0 y_i(x)dx$と複雑な数式を書くこともできます．
行列も簡単に書くことができます．
\begin{eqnarray}
\left[
\begin{array}{c}
x_1(t) \\
x_2(t) \\
\end{array} \right] &=& \left[
\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{array} \right] \, \left[
\begin{array}{c}
x_1(t-1) \\
x_2(t-1) \\
\end{array} \right] + \left[
\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
\end{array} \right] \, \left[
\begin{array}{c}
u_1(t-1) \\
u_2(t-1) \\
u_3(t-1) \\
\end{array} \right] \\
y(t) &=& \left[
\begin{array}{cc}
c_1 & c_2 \\
\end{array} \right] \, \left[
\begin{array}{c}
x_1(t) \\
x_2(t) \\
\end{array} \right]
\end{eqnarray}
\section{箇条書き}
%
まずは、番号付きの箇条書きの例です。
\begin{enumerate}
\item これが１番目
\begin{enumerate}
\item これが１番目の１番目
\item これが１番目の２番目
\end{enumerate}
\item これが２番目
\item これが３番目
\end{enumerate}
続いて、番号無しの箇条書きの例です。
\begin{itemize}
\item これが１番目
\begin{itemize}
\item これが１番目の１番目
\item これが１番目の２番目
\end{itemize}
\item これが２番目
\item これが３番目
\end{itemize}
%
\section{表の作成}
%
ここでは，簡単な時間割を作成します．表番号が自動的に付けられます．
%
\begin{table}[htb]
\begin{center}
\caption{時間割}
\label{tab:schedule}
\begin{tabular}{c|cccccc}
\hline
& 月 & 火 & 水 & 木 & 金 & 土 \\ \hline
１ & 英語 & & & & & 休 \\
２ & & 数学 & & & & \\
３ & & & 理科 & & & \\
４ & & & & 社会 & & \\
５ & & & & & 国語 & \\ \hline
\end{tabular}
\end{center}
\end{table}
%
\end{document}
|
https://www.ttp.kit.edu/Preprints/ttp/ttp99/ttp99-01/ttp99-01.tex
|
kit.edu
|
CC-MAIN-2019-35
|
text/x-tex
|
text/x-matlab
|
crawl-data/CC-MAIN-2019-35/segments/1566027317274.5/warc/CC-MAIN-20190822151657-20190822173657-00350.warc.gz
| 1,023,967,964 | 19,138 |
%Title: Expansion Techniques in Massive Quark Production: Results and Applications
%Author: J.H. Kuehn
%Published: <EM>Radiative corrections: Application of quantum field theory to phenomenology</EM>, Proceedings of the 4th international symposium, Barcelona, 1998, edited by Joan Sola (World Scientific, Singapore, 1999), 202-222.
%hep-ph/9901330
\documentstyle[sprocl,epsf]{article}
%\usepackage{epsf,sprocl}
%---------------------------------------------------------------------------
%\textwidth=15.5cm
%\textheight=23cm
%\topmargin=-2.5cm
%\oddsidemargin0.5cm
%\parindent0cm
%\parskip.2cm
%---------------------------------------------------------------------------
%
% from sprocl.tex:
\font\eightrm=cmr8
\arraycolsep1.5pt
%----------------------------------------------------------------------
\newcommand{\dd}{{\rm d}}
\newcommand{\hmp}{{hard mass procedure}}
\newcommand{\lmp}{{large momentum procedure}}
\newcommand{\bld}[1]{\boldmath{$#1$}}
\newcommand{\order}[1]{{\cal O}(#1)}
\renewcommand{\Re}{{\rm Re}}
\renewcommand{\Im}{{\rm Im}}
\tabcolsep=0em
%----------------------------------------------------------------------
\begin{document}
\title{Expansion Techniques in Massive Quark Production:\\ Results and
Applications}
\author{J.H. K\"uhn}
\address{Institut f\"{u}r Theoretische Teilchenphysik,\\
Universit\"{a}t Karlsruhe,
D--76128 Karlsruhe, Germany}
\date{}
\maketitle
\abstracts{
Recent progress in the calculation of multi-loop, multi-scale diagrams
is reviewed. Expansion techniques combined with new developments in
Computer algebra allow to evaluate the $R$ ratio for massive quarks
up to order $\alpha_s^2$ and, partly, even $\alpha_s^3$. Similar
techniques can be applied to Higgs- or $Z$-boson decays and mixed QCD
and electroweak interactions.
}
\section{Introduction\label{sec::chap1}}
During the past years amazing progress has been made in the experimental
tests of the Standard Model of particle physics. Its electroweak sector
has been scrutinized at LEP, mainly in $Z$ decays and $W$ pair
production, and at the TEVATRON, mainly through the precise measurements
of the $W$ boson and top quark masses. Perturbative QCD has been tested
at electron positron colliders at lower energies as well as in high
energy experiments at LEP, at the TEVATRON through proton-antiproton
collisions, and through lepton-nucleon scattering in particular at HERA.
The large statistics collected in these experiments in conjunction with
the small systematic uncertainty allows to test the theoretical
predictions with high precision, requiring the inclusion of quantum
corrections at least in one-loop approximation, occasionally two or even
three-loop calculations are required. This applies on one hand to
perturbative QCD, for electroweak observables on the other hand either
purely weak two-loop corrections should be included or so-called
``mixed'' QCD and electroweak effects up to two or even three loops.
Considerable progress has been made on the theoretical side which
matches these requirements. Two-loop amplitudes, in particular those
contributing to two-point functions can often be evaluated in closed
form or through straightforward numerical integrals --- even with arbitrary
mass assignments. Three-loop amplitudes, however, are at present only
accessible in a few selected cases. These are, most notably, two-point
functions with massless internal propagators only and vacuum
diagrams with massless and massive propagators --- however, of identical
mass throughout. Other cases of interest are ``on-shell amplitudes''
required for mass renormalization and certain types of integrals
relevant for fermion pair production at threshold.
Despite this seemingly limited set of diagrams a large variety of
problems can be solved with the help of expansion techniques. Frequently
one finds a hierarchy of mass and/or energy scales which suggests to
perform a Taylor series of the integrand, with the resulting integrals
being considerably simpler. However, the integrals are not necessarily
analytic in the expansion parameter, and more refined techniques,
denoted hard mass, large
momentum~\cite{methods,reviews} or threshold
expansion~\cite{BenSmi98,CzaMel98,BenSigSmi98} are required. These can be
formulated in a diagrammatic manner and allow to reduce a given
amplitude into an (infinite) sum of products of simpler ``master''
amplitudes which are known in analytical form. In practice often the
first few terms provide already a sufficiently accurate numerical
answer. In this approach, the demands on computation grow rapidly. Not
only the Dirac algebra and the evaluation of integrals for individual
diagrams has to be performed by computer algebra, with the appearance of
hundreds if not thousands of diagrams also the generation of the basic
diagrams, the application of the hard mass or large momentum procedure,
the transformation of diagrams into algebraic expressions and the
overall book keeping has to be performed automatically. (For a recent
review see~\cite{HarSte:review}.)
In the following talk several characteristic examples will be given,
partly taken from QCD, partly from electroweak interactions. The next
chapter will be concerned with the $\order{\alpha_s^2}$ evaluation of
the cross section for electron positron annihilation into massive
quarks. Information on the vacuum polarization function at small and
large $q^2/(4m^2)$ as obtained via expansion techniques is combined with
the two dominant terms close to threshold. Subsequently a variable
transformation is applied as suggested by the analyticity structure of
$\Pi(q^2)$ in the cut complex plane. The numerical results will be
contrasted with those deduced from the threshold expansion. Techniques,
results and limitations of the large momentum procedure will be
presented in chapter~\ref{sec::chap3}. This includes in particular the
calculation of the vacuum polarization of order $\alpha_s^2$, expanded
in $m^2/q^2$ up to fairly high power. The strategy for an evaluation of
the absorptive part in order $\alpha_s^3 m^2/q^2$ and $\alpha_s^3
(m^2/q^2)^2$ and recent results are given in chapter~\ref{sec::chap4}.
Results of relevance for electroweak measurements are discussed in
chapter~\ref{sec::chap5}. This includes the top quark contribution to
the $\rho$ parameter in three-loop approximation, mixed corrections to
the $Z$ decay rate and ``theory driven'' results for the running QED
coupling at scale $M_Z$. Chapter~\ref{sec::chap6} contains a brief
summary and the conclusions.
\section{Three-Loop Heavy-Quark Vacuum Polarization\label{sec::chap2}}
%
The measurement of the total cross section for electron positron
annihilation into hadrons allows for a unique test of perturbative QCD.
The decay rate $\Gamma(Z \to \mbox{hadrons})$ provides one of the most
precise determinations of the strong coupling constant $\alpha_s$. In
the high energy limit the quark masses can often be neglected. In this
approximation QCD corrections to $R \equiv \sigma(e^+ e^- \to
\mbox{hadrons})/ \sigma(e^+ e^- \to \mu^+ \mu^-)$ have been
calculated~\cite{CheKatTka79DinSap79CelGon80,GorKatLar91}
up to order $\alpha_s^3$.
For precision measurements the dominant mass corrections must be
included through an expansion in $m^2/s$. Terms up to order $\alpha_s^3
m^2/s$ (see~\cite{CheKue90}) and $\alpha_s^2 m^4/s^2$ (see~\cite{CheKue94})
and recently~\cite{rhdiss} even up to $\alpha_s^3 m^4/s^2$ are available
at present, providing an acceptable approximation from the high energy
region down to intermediate energy values. For a number of
measurements, however, the information on the complete mass dependence
is desirable. This includes charm and bottom meson production above the
resonance region, say above $4.5$~GeV and $12$~GeV, respectively, and,
of course, top quark production at a future electron positron collider.
%
To order $\alpha_s$ this calculation was performed by
K\"all\'en and Sabry in the context of QED a long time ago~\cite{KaeSab55}.
With measurements of ever increasing precision, predictions
in order $\alpha_s^2$ are needed for a reliable
comparison between theory and experiment. Furthermore,
when one tries to apply the ${\cal O}(\alpha)$ result
to QCD, with its running coupling
constant, the choice of scale becomes important.
In fact, the distinction between the two intrinsically different
scales, the relative momentum versus the center of mass
energy, is crucial for a stable numerical prediction.
This information can be obtained from a full calculation
to order $\alpha_s^2$ only.
Such a calculation then allows to predict the cross section
in the complete energy region where perturbative QCD can be applied
--- from close to threshold up to high energies. It is then only the
region very close to threshold, where the fixed order result remains
inadequate and Coulomb resummation becomes important.
%
In~\cite{CheKueSte96} results for the cross section were calculated in
order $\alpha_s^2$. They were obtained from the vacuum polarization
$\Pi(q^2)$ which was calculated up to three loops. The imaginary part
of the ``fermionic contribution'' --- derived from diagrams with a
massless quark loop inserted in the gluon propagator --- had been
calculated earlier in~\cite{HoaKueTeu95}. In this latter case all
integrals could be performed to the end and the result was expressed in
terms of polylogarithms. In~\cite{CheKueSte96} the calculation was
extended to the full set of diagrams relevant for QCD. Instead of
trying to perform the integrals analytically, information of
$\Pi(q^2)$ from the large $q^2$ behaviour, the expansion around $q^2=0$
and from threshold was incorporated.
\nopagebreak
%
\subsection[]{Outline of the Calculation~\cite{CheKueSte96}}
%
The different behaviour at threshold makes it necessary to decompose
$\Pi$ according to its colour structure. It is convenient to
write:
\begin{eqnarray}
\Pi(q^2) &=& \Pi^{(0)}(q^2)
+ \frac{\alpha_s(\mu^2)}{\pi}C_F\Pi^{(1)}(q^2)
+ \left(\frac{\alpha_s(\mu^2)}{\pi}\right)^2\Pi^{(2)}(q^2)
+ \cdots,
\\
\Pi^{(2)} &=&
C_F^2 \Pi_{\mbox{\scriptsize\it A}}^{(2)}
+ C_A C_F \Pi_{\mbox{\scriptsize\it NA}}^{(2)}
+ C_F T n_l \Pi_{\mbox{\scriptsize\it l}}^{(2)}
+ C_F T \Pi_{\mbox{\scriptsize\it F}}^{(2)}.
\end{eqnarray}
The same notation is adopted to the
physical observable $R(s)$ which is related to $\Pi(q^2)$ by
\begin{eqnarray}
R(s) &=& 12\pi\, \mbox{Im}\Pi(q^2=s+i\epsilon).
\end{eqnarray}
The contributions from diagrams with $n_l$ light
or one massive
internal fermion loop are denoted
by $C_F T n_l\Pi_l^{(2)}$ and $C_F T \Pi_F^{(2)}$, respectively.
The purely gluonic corrections
are proportional to $C_F^2$ or $C_A C_F$ where the former are the only
contributions in an Abelian theory and the latter are characteristic for
the non-Abelian aspects of QCD.
All steps described below have been performed separately for
the first three contributions to $\Pi^{(2)}$.
In fact, new information is only
obtained for $\Pi_{\mbox{\scriptsize\it A}}^{(2)}$
and $\Pi_{\mbox{\scriptsize\it NA}}^{(2)}$ since
$\mbox{Im}\Pi_{\mbox{\scriptsize\it l}}^{(2)}$
is already known analytically~\cite{HoaKueTeu95}.
The contribution from a four-particle cut
with threshold at $4m$ is given
in terms of a two dimensional integral~\cite{HoaKueTeu95,chkst98}
which can be solved
easily numerically,
so $\Pi_F^{(2)}$ will not be treated.
Let us now discuss the behaviour of $\Pi(q^2)$ in the three different
kinematical regions and the approximation method.
{\it Analysis of the high $q^2$ behaviour:}
%
The high energy behaviour of $\Pi$ provides important
constraints on the complete answer.
In the limit of small $m^2/q^2$ the constant term and the one
proportional to $m^2/q^2$ (modulated by powers of $\ln(\mu^2/q^2$) have been
calculated a long time ago~\cite{GorKatLar86}. The results for terms up to order $(m^2/q^2)^4$
are described in chapter~\ref{sec::chap3}, provide an important cross
check, however, they are not used for the moment.
{\it Threshold behaviour:}
General arguments based on the influence of Coulomb exchange close to
threshold, combined with the information on the perturbative QCD
potential and the running of $\alpha_s$ dictate the singularities
and the structure of the leading cuts close to threshold, that
is for small $v=\sqrt{1-4m^2/s}$.
The $C_F^2$ term
is directly related to the QED result with internal photon lines only.
The leading $1/v$
singularity and the constant term of $R_A$
can be predicted from the nonrelativistic
Greens function for the Coulomb potential
and the ${\cal O}(\alpha_s)$ calculation.
The next-to-leading
term is determined by the combination of one-loop
results again with the Coulomb singularities~\cite{BarGatKoeKun75}. One finds
\begin{eqnarray}
R_{\mbox{\scriptsize\it A}}^{(2)} &=&
3\left(\frac{\pi^4}{8v} - 3\pi^2 + \ldots\right).
\label{Ra}
\end{eqnarray}
The contributions $\sim C_A C_F$ and $\sim C_F T n_l$ can be treated
in parallel.
For these colour structures the perturbative QCD potential~\cite{Fis77}
\begin{eqnarray}
V_{\mbox{\scriptsize QCD}}(\vec{q}\,^2) &=&
-4\pi C_F\frac{\alpha_V(\vec{q}\,^2)}{\vec{q}\,^2},
\\
\alpha_V(\vec{q}\,^2) &=& \alpha_s(\mu^2)\Bigg[
1 + \frac{\alpha_s(\mu^2)}{4\pi}\bigg(
\left(\frac{11}{3}C_A
-\frac{4}{3}T n_l\right)
\left(-\ln\frac{\vec{q}\,^2}{\mu^2}+\frac{5}{3}\right)
-\frac{8}{3}C_A \bigg)
\label{alphav}
\Bigg]
\nonumber
\end{eqnarray}
will become important.
The leading $C_AC_F$ and $C_F T n_l$ term in $R$ is proportional
to $\ln v$ and is responsible for the evolution of the
coupling constant close to threshold. Also the constant term can
be predicted by the observation, that the leading term in
order $\alpha_s$ is induced by the potential.
The ${\cal O}(\alpha_s)$ result
\begin{equation}
R=3\frac{v(3-v^2)}{2}
\left(1
+ C_F\frac{\pi^2(1+v^2)}{2v}\frac{\alpha_s}{\pi}+\ldots
\right)
\end{equation}
is employed to predict the logarithmic and constant $C_FC_A$ and
$C_FTn_l$ terms of ${\cal O}(\alpha_s^2)$ by replacing $\alpha_s$
by $\alpha_V(4\vec{p}\,^2=v^2 s)$ as given in Eq.~(\ref{alphav}).
This implies the following threshold behaviour:
\begin{eqnarray}
R_{\it NA}^{(2)}&=&3\frac{\pi^2}{3}(3-v^2)(1+v^2)
\left(
-\frac{11}{16}\ln\frac{v^2 s}{\mu^2}
+\frac{31}{48}
+\ldots \right),
\label{Rna}
\\
R_l^{(2)}&=&3\frac{\pi^2}{3}(3-v^2)(1+v^2)
\left(
\frac{1}{4}\ln\frac{v^2 s}{\mu^2}
-\frac{5}{12}
+\ldots \right).
\label{Rnl}
\end{eqnarray}
This ansatz suggested in~\cite{CheKueSte96} can be verified for the $C_F
T n_l$ term where the result is known in analytical form~\cite{HoaKueTeu95} and it has also been confirmed for the NA term where
the leading terms for small $v$ have been derived recently (see below).
Extending the ansatz from the behaviour of the imaginary part close to
the branching point into the complex plane allows to predict the leading
terms of $\Pi(q^2)$ $\sim \ln v$ and $\sim \ln^2 v$.
{\it Behaviour at $q^2=0$:}
%
Important information is contained in the Taylor series of $\Pi(q^2)$
around zero. The calculation of the first seven nontrivial terms is
based on the evaluation of three-loop tadpole integrals with
the help of the algebraic program {\tt MATAD} written in {\tt FORM}~\cite{VerFORM}
which
performs the traces,
calculates the derivatives with respect to the external momenta.
It reduces the large number of different
integrals to one master integral and a few simple ones
through an elaborate set of
recursion relations based on the integration-by-parts method~\cite{CheTka81,Bro92}.
The result can be written in the form:
\begin{eqnarray}
\Pi^{(2)} &=&
\frac{3}{16\pi^2}
\sum_{n>0} C_{n}^{(2)} \left(\frac{q^2}{4m^2}\right)^n,
\end{eqnarray}
where the first seven moments are listed in~\cite{CheKueSte96}.
{\it Conformal mapping and Pad\'e approximation:}
%
The vacuum polarization function
$\Pi^{(2)}$ is analytic in the complex plane
cut from $q^2=4m^2$ to $+\infty$. The Taylor series in $q^2$ is
thus convergent in the domain $|q^2|<4m^2$ only. The
conformal mapping
which corresponds to the variable
transformation ($z=q^2/(4m^2)$)
\begin{eqnarray}
\omega = \frac{1-\sqrt{1-z}}{1+\sqrt{1-z}},
\qquad
z = \frac{4\omega}{(1+\omega)^2},
\label{omega}
\end{eqnarray}
transforms the cut complex $z$ plane into the
interior of the
unit circle. The special points
$z=0,1,-\infty$ correspond to $\omega=0,1,-1$, respectively
(Fig.~\ref{fig::trafo}).
\begin{figure}[ht]
\begin{center}
\begin{tabular}{c}
\epsfxsize=7cm
\leavevmode
\epsffile[80 280 540 520]{trafo.ps}
\end{tabular}
\caption{\label{fig::trafo} Transformation between the
$z$ and $\omega$ plane }
\end{center}
\end{figure}
The upper (lower) part of the cut is mapped onto the upper (lower)
perimeter of the circle.
The Taylor series in $\omega$ thus converges in the interior of the
unit circle. To obtain predictions for
$\Pi(q^2)$ at the boundary it has been suggested~\cite{FleTar94,BroFleTar93}
to use
the Pad\'e approximation
which converges towards $\Pi(q^2)$ even on the perimeter.
To improve the accuracy
the singular threshold behaviour
and the large $q^2$ behaviour
is incorporated into simple analytical functions
which are removed
from $\Pi^{(2)}$ before the Pad\'e approximation is
performed.
The quality of this
procedure can be tested by comparing the prediction with
the known result for $\mbox{Im}\Pi_{\mbox{\scriptsize\it l}}^{(2)}$.
The logarithmic singularities at threshold and large $q^2$
are removed by subtraction, the $1/v$ singularity, which is present
for the $C_F^2$ terms only, by multiplication with $v$ as
suggested in~\cite{BaiBro95}.
The imaginary part of the remainder which is actually
approximated by the Pad\'e method is thus smooth in the
entire circle, numerically small and vanishes at
$\omega=1$ and $\omega=-1$.
\tabcolsep=.4em
\begin{figure}[ht]
\leavevmode
\begin{center}
\begin{tabular}{cc}
\epsfxsize=5.5cm
\epsfysize=4cm
\epsffile[110 330 460 520]{pilocf21.ps}
&
\epsfxsize=5.5cm
\epsfysize=4cm
\epsffile[110 330 460 520]{pilocacf1.ps}
\end{tabular}
\caption{\label{fighigh}Complete results (full line) are
compared to the threshold
approximations and the high energy
approximations including the $m^2/s$
(dash-dotted) and the $m^4/s^2$ (dashed) terms
($x=2m/\protect\sqrt{s}$).
}
\end{center}
\end{figure}
%
\begin{figure}[ht]
\leavevmode
\begin{center}
\begin{tabular}{cc}
\epsfxsize=5.5cm
\epsfysize=4cm
\epsffile[110 330 460 520]{pilocf23.ps}
&
\epsfxsize=5.5cm
\epsfysize=4cm
\epsffile[110 330 460 520]{pilocacf3.ps}
\end{tabular}
\caption{\label{figthr}The
threshold behaviour of the remainder $\delta R$
for three different Pad\'e approximants is shown.
(The singular
and constant parts around threshold are subtracted.)
}
\end{center}
\end{figure}
\tabcolsep=0em
\subsection[]{Results}
After performing the Pad\'e approximation for the smooth remainder
with $\omega$ as natural variable, the transformation (\ref{omega})
is inverted and the full vacuum polarization function reconstructed
by reintroducing the threshold and high energy terms. This
procedure provides real and imaginary parts of $\Pi^{(2)}$.
Subsequently only the absorptive part of $\Pi^{(2)}$ (multiplied by
$12\pi$) will be presented.
In the following it will be useful to plot the results as functions of
$x=2m/\sqrt{s}$ and alternatively of $v=\sqrt{1-4m^2/s}$. The first choice
is particularly useful for investigations of the high energy region, the
second one for energies close to threshold. Characteristic values of
$x$, $v$ and $\sqrt{s}$ for charm ($m_c=1.6$~GeV), bottom ($m_b=4.7$~GeV)
and top ($m_t = 175$~GeV) quarks are listed in Table~\ref{tab::1} for
easy comparisons.
%
\tabcolsep=.19em
\begin{table}
\begin{tabular}{|l|r|r|r|r|r|r|r|r|r|r|r|r|}
\hline
x & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 0.95 & 0.97 & 0.98 \\
\hline
v & 0.995 & 0.980 & 0.954 & 0.917 & 0.866 & 0.800 & 0.714 & 0.600 & 0.436 &
0.312 & 0.243 & 0.199\\
\hline
$\sqrt{s_c}$
& 32 & 16 & 10.7 & 8.0 & 6.4 & 5.3 & 4.6 & 4.0 & --- &--- & --- & --- \\
\hline
$\sqrt{s_b}$
& 94 & 47 & 31.3 & 23.5 & 18.8 & 15.7 & 13.4 & 11.8 & 10.4 & --- & --- &
--- \\
\hline
$\sqrt{s_t}$
& 3500 & 1750 & 1167 & 875 & 700 & 583 & 500 & 438 & 389 & 368 & 361 &
357\\
\hline
\end{tabular}
\caption{\label{tab::1} Conversion between $x$ and $v$ and values for
$\sqrt{s}$ in GeV for charm, bottom and top production.}
\end{table}
\tabcolsep=0em
%
Energy values where a perturbative treatment is evidently inapplicable
are denoted by dashes.
In Fig.~\ref{fighigh} the complete results are shown
for $\mu^2=m^2$ with
$R_{\mbox{\scriptsize\it A}}^{(2)}$ and $R_{\mbox{\scriptsize\it NA}}^{(2)}$
displayed separately.
The solid line represents the full correction. The threshold
approximation is given by the dashed curve. In the
high energy region the corrections containing the $m^2/s$ terms and
the quartic approximations are included.
It should be stressed that the latter are not incorporated into the
construction of $R^{(2)}$ but they are evidently very well
reproduced by the method presented here.
This will be investigated in more detail in
chapter~\ref{sec::chap3}. There it will be demonstrated that the high
energy expansion (including sufficiently many terms) and the Pad\'e
result agree remarkably well between $x=0$ and $x=0.7$ which covers most
of the perturbative region for charmed and bottom quarks.
The results which include high
moments up to $C_6$, $C_7$ or even $C_8$ are remarkably stable down to
very small values of $v$.
Different Pad\'e approximations of the same degree
and approximants with a
reduced number of parameters give rise to practically identical
predictions, which could hardly be distinguished in
Fig.~\ref{fighigh}. Minor variations are observed close
to threshold, {\it after} subtracting the singular and constant parts.
The remainder $\delta R$ for up to ten different Pad\'e
approximants is shown in
Fig.~\ref{figthr}. In~\cite{CheKueSte96} it was demonstrated that
there is a perfect agreement for $R^{(2)}_l$;
$R_{\it NA}^{(2)}$ seems to converge to the solid line
($[4/4], [5/3]$ and $[3/5]$)
when more moments from small $q^2$ are included. The dashed lines are from
the $[3/3], [4/2], [2/4]$ and $[3/4]$, the dotted
ones from lower order Pad\'e approximants.
The dash-dotted curve is the [4/3] Pad\'e approximant and has a pole very
close to $\omega=1 (1.07\ldots)$.
For the Abelian part a classification of the different results
can be seen: the dashed lines are $[4/2]$ and $[2/4]$,
the solid ones $[3/2], [2/3], [5/3], [3/5]$ and $[5/4]$
Pad\'e approximants.
Recently additional terms from the high energy expansion have been
injected in the Pad\'e approximation~\cite{rhdiss,CheHarSte98}.
%
\begin{figure}[tf]
\begin{center}
\begin{tabular}{cc}
\small (a) & \small (b) \\[-2ex]
\leavevmode
\epsfxsize=6cm
\epsffile[110 270 480 560]{padera81n.ps }
&
\epsfxsize=6cm
\epsffile[110 270 480 560]{paderna80n.ps }
\end{tabular}
\caption[]{\label{fig::pademn}\sloppy
Variation of the prediction for $R_A^{(2)}$ and $R_{NA}^{(2)}$ by
including an increasing number of terms from the high energy
expansion. The singular and constant pieces at $v=0$ have been
subtracted. (From~\cite{rhdiss}.) }
\end{center}
\end{figure}
%
The results, shown in Fig.~\ref{fig::pademn} confirm those from~\cite{CheKueSte96} and demonstrate the stability of the Pad\'e
approximation. It must be stressed that a safe estimate of the
remaining uncertainty in $R_A$ and $R_{NA}$ amounts to less than $0.02$
for $v$ above $0.1$ and is around $0.05$ in the region $v\approx 0.03$.
This region is, however, entirely dominated by the singular and constant
terms with values around 50 and higher. The perturbative predictions for
$R$ are therefore under excellent control.
It goes without saying that the function $\Pi(q^2)$ constructed this way
and evaluated e.g.\ in the Euclidean region could be a useful tool for
other investigations of interest, for example for sum rules involving
massive quarks.
Quite some effort has been invested in the analytic evaluation
of $R_A$ and $R_{NA}$ close to threshold. As stated above, the singular
and constant terms (Eqs.~(\ref{Ra}), (\ref{Rna}) and (\ref{Rnl})) were
derived through general considerations~\cite{BarGatKoeKun75,CheKueSte96}.
To evaluate the $v\ln v$ and the $v$
terms, however, elaborate analytical calculations of the two-loop form
factor were used~\cite{hoang,CzaMel98} with the results
\begin{eqnarray}
R_A &=& 3\bigg\{ {\pi^4\over 8v} - 3\,\pi^2
+v\,\bigg(-{\pi^4\over 24}
\nonumber\\&&
+{3\over 2}\,
\big({\pi^4\over 6}
+ \pi^2\,
(-{35\over 18} - {2\over 3}\,\ln v
+ {4\over 3}\,\ln 2) +
{39\over 4} - \zeta(3)\big)\bigg) \bigg\}
\\
R_{NA} &=& 3\bigg\{\pi^2\,\left({31\over 48} - {11\over 8}\,\ln 2v\right)
\nonumber\\&&
+{3\over 2}\,v\,\bigg(\pi^2\,({179\over 72} - \ln v
- {8\over 3}\,\ln 2)
- {151\over 36} - {13\over 2}\,\zeta(3)\bigg)\bigg\}
\end{eqnarray}
While the result for $R_A$ was still obtained in the framework of
``classical'' QED calculations, $R_{NA}$ was calculated using a
convenient technique which formalized the expansion for small
$v$ (see~\cite{BenSmi98,CzaMel98,BenSigSmi98}).
Pad\'e and small $v$ results are compared in
Fig.~\ref{fig::rvlnv}, again subtracting first the singular and constant
pieces. The slopes for very small $v$ predicted by the two approaches
are again in nice agreement, giving further credibility to the Pad\'e
result. However, it is also clear from this comparison that the small
$v$ expansion alone cannot lead to a reliable prediction over a larger
energy range. This is explicitely demonstrated for the case of $R_l$
where the analytic result is available. Terms at least of order $v^3$
are needed for a stable prediction.
A compilation of theoretical results can be found in~\cite{chkst98}
where the prediction for massive quark production is compared with the
measurements over a wide energy region (Fig.~\ref{fig::rtot}).
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\leavevmode
\epsfxsize=6cm
\epsffile[100 280 460 560]{rs.ps}
\end{tabular}
\caption[]{\label{fig::rtot}
$R(s)$ plotted against $\protect\sqrt{s}$. The scale $\mu^2=s$ has
been adopted. The dashed curves correspond to the values
$M_c=1.8$~GeV, $M_b=5.0$~GeV and $\alpha_s(M_Z^2)=0.121$, whereas for
the solid curves $M_c=1.4$~GeV, $M_b=4.4$~GeV and
$\alpha_s(M_Z^2)=0.115$ is used. The dotted lines show a recent
compilation of the available experimental data. The central curves
correspond to the mean values, upper and lower curves indicate the
combined statistical and systematical errors. (From~\cite{chkst98}.)
}
\end{center}
\end{figure}
Up to this point only the vector current correlator has been discussed.
However, the techniques described above have also been applied to other
currents~\cite{CheKueSte96:2}: the axial vector, relevant e.g.\ for top
production through the virtual $Z$ boson, as well as scalar and
pseudoscalar currents describing for example the decay of Higgs bosons
into massive quarks. Recently also the singlet piece of top quark
production through the axial current was evaluated~\cite{HarSte97},
completing thus the $\order{\alpha_s^2}$ prediction of massive quark
production.
\section{Large Momentum Expansions\label{sec::chap3}}
%
An alternative route towards an efficient evaluation of the polarization
function is based on the large momentum
expansion.
In this case the
polarization function $\Pi(q^2)$ is expanded in powers of $m^2/q^2$,
multiplied by logarithms of $m^2/q^2$, with the power of the logarithms,
however, limited by the number of loops under consideration. Technically
the expansion is given by a series of products of massless propagator
and massive tadpole integrals.
%Examples for the one-, two-, and
%three-loop case are depicted in Fig.~\ref{fig::lmp}.
\tabcolsep=0em
\begin{figure}
\begin{center}
\begin{tabular}{lll}
\leavevmode
\epsfxsize=4.0cm
\epsffile[110 265 465 560]{ravlnv.ps} &
\epsfxsize=4.0cm
\epsffile[110 265 465 560]{rnavlnv.ps} &
\epsfxsize=4.0cm
\epsffile[110 265 465 560]{rlb.ps}
\end{tabular}
\caption[]{\label{fig::rvlnv}\sloppy\rm
Comparison of the small $v$ expansion (solid curves) of
$\delta R_A$, $\delta R_{NA}$ with the full result (solid and dotted
curves) after subtraction of the singular and constant pieces. The
difference between dotted and solid curve indicate the remaining
uncertainty of the semianalytical result. For $\delta R_l$
successive approximations of the small $v$ expansion (dashed
curves) and exact result (solid) are shown. }
\end{center}
\end{figure}
%\begin{figure}
%\leavevmode
%\begin{center}
%\begin{tabular}{c}
%\epsfxsize=6cm
%\epsffile[170 340 505 390]{figlmp1l.ps}\\[1em]
%\epsfxsize=8cm
%\epsffile[140 315 590 405]{figlmp2l.ps}\\[1em]
%\epsfxsize=8cm
%\epsffile[150 265 525 480]{figlmp3l.ps}
%\end{tabular}
%\caption[]{\label{fig::lmp}\sloppy\rm
% Large momentum procedure for the one-, two- and three-loop case.
% }
%\end{center}
%\end{figure}
In principle the full information on the analytic function is contained
in this series. The structure of the integrals is simplified by moving
from two- to one-scale integrals. However, an enormous proliferation of
the number of diagrams and the amount of algebraic calculation is
observed, requiring the development of programs which implement the
diagrammatic expansion, and translate the resulting diagrams into input
files for other programs which in turn are suited for the algebraic
evaluation of individual diagrams. One example for such a
``superprogram'' is {\tt GEFICOM}~\cite{geficom} which uses
{\tt QGRAF}~\cite{qgraf} for the generation of diagrams,
{\tt LMP}~\cite{rhdiss} or {\tt EXP}~\cite{Sei:dipl} for the diagrammatic
expansion through the hard mass or large momentum procedure, and {\tt
MATAD}~\cite{Stediss} and {\tt MINCER}~\cite{mincer2} for the
evaluation of diagrams. Even nested expansions with a hierarchy of
several scales are possible in this framework. (A more detailed
description of the status of algebraic programs can be found in~\cite{HarSte:review}.) After performing the expansion in $m^2/q^2$ up
to a given power, one may directly take the absorptive part and thus
predict $\Pi(q^2)$ in the high energy region. The comparison with the
Pad\'e result (discussed in chapter~\ref{sec::chap2}) shows excellent
agreement in the region of $x$ between zero and $0.7$ and thus down to
fairly low energies (Fig.~\ref{rvx.ps}).
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\leavevmode
\epsfxsize=5.5cm
\epsfysize=3.5cm
\epsffile[110 265 465 560]{ravx.ps}&
\epsfxsize=5.5cm
\epsfysize=3.5cm
\epsffile[110 265 465 560]{rnavx.ps}
\end{tabular}
\caption[]{\label{rvx.ps}\sloppy\rm
The Abelian contribution $R_A^{(2)}$ and the non-Abelian piece
$R_{\it NA}^{(2)}$. Wide dots: no mass terms; dashed lines:
including successively more mass terms $(m^2/s)^n$ up to $n=5$;
solid line: including mass terms up to $(m^2/s)^6$; narrow dots:
semi-analytical result. The scale $\mu^2 = m^2$ has been adopted.
(From~\cite{CheHarKueSte97}.)
}
\end{center}
\end{figure}
One may even suspect that, given sufficiently many terms of the
absorptive part alone, an approximation of arbitrary precision can be
achieved. This is indeed true for the two-loop result and the ``double
bubble'' contribution with massless quarks in the internal loop, which
are available in analytical form and thus can be used as ``toy models''.
However, in both examples, the only threshold is located at $\sqrt{s} =
2m$ and convergence down to this point is naturally expected. In
contrast, the other diagrams have four-particle cuts at $\sqrt{s} = 4m$,
suggesting convergence only above this point. This has been confirmed
by a detailed study~\cite{CheHarKueSte97} of the double bubble
contribution with equal masses in both fermion loops, where inclusion of
an increasing number of terms does not lead to an improvement beyond
$x\approx 0.5$. Given both real and imaginary parts, this problem could
be and has been circumvented as discussed in chapter~\ref{sec::chap2}.
However, below we will be interested in the situation where the
absorptive piece only is available.
Nevertheless, the relative smallness of the four-particle
contribution and the slow opening of the phase space reduce this effect
and a fairly good approximation of $R_A$ and $R_{NA}$ is achieved even
for $x=0.7$, thus covering most of the interesting energy range (cf.\
Table~\ref{tab::1}). The same technique has also been used for the
axial, the scalar and the pseudoscalar current~\cite{HarSte97,HarSte98}.
(This motivates the step to $\alpha_s^3m^4$ in chapter~\ref{sec::chap4}.)
The agreement of the two-loop result described in
chapter~\ref{sec::chap2} with the threshold and the large momentum
expansion in the respective ranges of validity demonstrate that the
perturbative NLO prediction of $R(s)$ for massive quarks is under full
control.
\section{Quartic Mass Terms in NNLO form Operator Product Expansion
\label{sec::chap4}}
%
Fig.~\ref{rvx.ps} suggests that the first three terms provide an
excellent approximation at $x=0.5$ and are quite acceptable even at
$x=0.7$. Using this line of reasoning, a possible route for a NNLO
prediction (order $\alpha_s^3$) of $R(s)$, including quark mass effects,
is at hand. The massless result~\cite{GorKatLar91} and the $m^2/s$ terms~\cite{CheKue90} have been obtained a long time ago. The strategy used in~\cite{CheKue94} allowed to predict the $\alpha_s^2 m^4/s^2$ terms in
$R(s)$ by evaluating two-loop~(!) tadpoles and massless propagators
only. Additional ingredients are the operator product expansion and the
renormalization group equations, plus certain anomalous dimensions.
Using this method and algebraic programs it is thus possible to obtain
the $\alpha_s^3 m^4/s^2$ terms from three-loop tadpoles and massless
propagators~\cite{rhdiss,chkprep}. Let me briefly describe this method:
In a first step the OPE is applied to the time ordered product of two
currents
\begin{eqnarray}
\int \dd x e^{iqx} T\big(j_\mu(x)j_\nu(0)\big) &=& (q_\mu q_\nu -
g_{\mu\nu}q^2)\bigg\{ A(q^2,\mu^2,\alpha_s){\bf 1} +
\label{eq::tprod}
\\&&
B(q^2,\mu^2,\alpha_s) {\bar m^2\over q^2}(\mu^2) + \sum_{n=1}^6 {1\over
q^4} C_n(q^2,\mu^2,\alpha_s){\cal O}_n\bigg\}\,.
\nonumber
\end{eqnarray}
Only three of the six operators ${\cal O}_n$ with dimension four are
gauge invariant and contribute to physical matrix elements:
\begin{equation}
{\cal O}_1 = G^2, \qquad {\cal O}_2 = m\bar q q, \qquad {\cal O}_6 =
\bar m^4(\mu^2)\,,
\end{equation}
the others are required for the proper construction to the coefficient
functions $C_n$.
For the NNLO calculation, $C_1$, $C_2$ and $C_6$ are required up to
${\cal O}(\alpha_s^2)$, ${\cal O}(\alpha_s^3)$ and ${\cal
O}(\alpha_s^2)$ respectively. To obtain these coefficient functions,
only massless propagator type integrals, at most of three loop, are
needed.
To calculate the vacuum matrix elements of ${\cal O}_1$ and ${\cal
O}_2$, massive tadpole integrals -- at most of three loop -- are
needed. In a last step one uses renormalization group invariance of the
dimension four part of Eq.~(\ref{eq::tprod}). Employing the anomalous
dimension matrix~\cite{CheSpi88} of ${\cal O}_{1,2,6}$ one finally obtains the
coefficients of the terms $\alpha_s^3 \bar m^4 \ln^n q^2/\mu^2$ with
$n=1,2,3$. Only these terms contribute to the absorptive part and one
finally arrives at~\cite{GorKatLar91,CheKue90,rhdiss,chkprep}
\begin{eqnarray}
R^v(s) &=& 3\,\bigg\{1 - 6\,x^2 + {\alpha_s\over
\pi}\,\bigg[1 + 12\,x - 22\,x^2\bigg]
+ \left({\alpha_s\over \pi}\right)^2\,\bigg[1.40923
\nonumber\\&&\mbox{}
+ 104.833\,x +
x^2\,(139.014 - 4.83333\,l_x)\,
\bigg]
+ \left({\alpha_s\over \pi}\right)^3\,\bigg[-12.7671
\nonumber\\&&\mbox{}
+ 541.753\,x
% \nonumber\\&&\mbox{\hspace{2em}}
+ x^2\,(3523.81 - 158.311\,l_x +
9.66667\,l_x^2)\,\bigg]\bigg\}\,,
\end{eqnarray}
with $x\equiv \bar m^2(s)/s$, $l_x \equiv \ln (\bar m^2(s)/s)$ and $n_f
= 5$.
\section{Expansion techniques and electroweak interactions\label{sec::chap5}}
%
Electroweak observables are frequently affected by the interplay between
strong and electroweak interactions. Important examples are the hadronic
contributions to the running of the QED coupling from the Thompson limit
to $M_Z$, QCD effects on the $\rho$ parameter and related quantities,
and ``mixed'' vertex corrections affecting for example the $Z$ decay
rate. Quark mass effects and their perturbative treatment are important
for $\alpha_{\rm QED}(M_Z)$. A detailed discussion of the last topic is
beyond the scope of this paper and can be found in~\cite{chkst98,KueSte98}. Let us present some aspects of the two
remaining items.
{\it Gauge boson self-energies, the mass of the top quark and QCD:}
%
The indirect determination of the top quark through quantum corrections
prior to its observation in hadronic collisions can be considered one of
the triumphs of the Standard Model. The experimental precision of the
key observables, the masses of the top quark and the $W$ boson together
with the weak
mixing angle as determined by asymmetry measurements has increased
during the past years and this process will continue in the foreseeable
future. In order to control the influence of the top quark at an
adequate level the inclusion of QCD corrections in the top and bottom
induced self energies is mandatory. The dominant terms are characterized
by the $\rho$ parameter which, in lowest order, is given by~\cite{veltman}
\begin{equation}
\Delta\rho = 3 \sqrt{2} \frac{G_F m_t^2}{16 \pi^2}.
\end{equation}
In view of the large difference between pole and running mass at scale
$m_t$
\begin{equation}
m_t(pole)=\bar m_t(m_t) \left(1+\frac{4}{3}\frac{\alpha_s}{\pi}+\cdots\right)
\end{equation}
inclusion of two and even three-loop QCD corrections to $\Delta\rho$ is
mandatory.
Analytic results are available in two-loop approximation not only for
the leading term~\cite{djouadi1} in $\Delta\rho$ but also for all self
energies, with arbitrary top and bottom masses~\cite{Kniehl}. The
resulting shift in the prediction for $M_W$ for fixed $G_F$,
$\alpha_{\rm QED}$, $M_Z$ and $m_t=175$~GeV amounts to 68 MeV, well
comparable to the present precision and significantly larger than the
anticipated accuracy of roughly 30 MeV. To arrive at a precise
prediction for the central value and to control the theoretical
uncertainty, three-loop QCD contributions to $\Delta\rho$ as well as to
$\Delta r$ are required. The $\rho$ parameter~\cite{rhoparam} can again
be expressed through diagrams with vanishing external momentum (vacuum
or tadpole diagrams), the remaining quantities involve two point
functions at non-vanishing external momentum and can be
obtained~\cite{deltar} through an expansion in the small mass ratio
$M_Z^2/m_t^2$. The leading three-loop term corresponds to a shift of
$-10.9$~MeV. The first three terms, amounting to $-13.7$~MeV, are adequate
for a prediction with an accuracy better than 1 MeV (Table~\ref{tab2}).
Conversely, this combined shift is equivalent to a reduction of the
effective top mass by about 1.6 GeV.
\tabcolsep=.2em
\begin{table}[th]
%\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|l||r|r|r|}
\hline
$\delta M_W$ in MeV & $\alpha_s^0$ & $\alpha_s^1$ & $\alpha_s^2$ \\
\hline
\hline
$M_t^2$ & 611.9 & -61.3 & -10.9 \\
const. & 136.6 & -6.0 & -2.6 \\
$1/M_t^2$ & -9.0 & -1.0 & -0.2 \\
\hline
\end{tabular}
\end{center}
\caption[]{\label{tab2} The change in $M_W$ separated according
to powers in $\alpha_s$ and $M_t$ in the on-shell
scheme. (From~\cite{deltar}.)}
\end{table}
\tabcolsep=0em
To exploit the precision expected from a future linear collider which
will pin down $m_t$ to better than 200 or perhaps even 100 MeV, the
inclusion of $\alpha_s^2$ terms is evidently mandatory.
Similar considerations~\cite{deltar} are valid for the effective
weak mixing angle which can be deduced from the left right asymmetry or
the forward backward asymmetry in a straightforward way.
An important issue in this connection is the size of uncertainties,
arising either from uncalculated higher orders, or from the
``parametric'' uncertainties in $\alpha_s$ and $m_t$. Shifts of
$\delta\alpha_s = 0.003$ and $\delta m_t = 5$~GeV lead to changes
in $M_W$ of $-2.4$~MeV and 35~MeV respectively. The uncertainty from
uncalculated higher orders is
completely negligible. A $W$-mass determination with a
precision in the 10~MeV range should therefore be accompanied by a top
mass measurement with a precision around or better than 1~GeV.
{\it Mixed QCD and electroweak vertex corrections:}
%
As stated above, gauge boson self energies, in particular those induced
by fermion loops, give rise to the dominant radiative corrections for
electroweak precision observables. Nevertheless, for a complete
treatment of ${\cal O}(\alpha\alpha_s)$ the inclusion of irreducible
vertex corrections is necessary. These have to be distinguished from the
reducible ones which are easily incorporated, if the electroweak result,
including one-loop weak terms, is simply multiplied by the QCD
correction factor $(1+\alpha_s/\pi + \cdots)$.
The physics and the techniques of calculation are markedly different for
vertices leading to light ($u$, $d$, $s$ and $c$) quark pairs on one
hand~\cite{CzarK} and for decays into $b\bar b$ on the other
hand~\cite{fjrt}, a consequence of the presence of top quarks with their
enhanced contribution to the vertex proportional $m_t^2/M_W^2$. The
irreducible one-loop vertex diagrams are shown in Fig.~\ref{fig::2l}.
\begin{figure}[t]
\leavevmode
\centering
\epsfxsize=8cm
\epsffile[105 558 503 717]{zbbdias.ps}
\caption[]{Diagrams contributing to $\delta\Gamma^W_b$
in ${\cal O}(\alpha)$. Thin lines correspond to bottom quarks, thick
lines to top quarks, dotted lines to Goldstone bosons and inner wavy
lines represent $W$ bosons.
\label{fig::2l}
}
\end{figure}
To obtain all one-particle irreducible vertex diagrams in two-loop
approximation , these have to be dressed with gluons in all conceivable
ways. The resulting amplitudes are first studied for arbitrary $q^2$ by
considering expansions in the ratio $x_Z=q^2/M_Z^2$ and $x_W=
q^2/m_W^2$ (or for some diagrams, in $1/x_{Z,W}$). Even for the limiting
values $x_Z=1$ and $x_W= M_Z^2/M_W^2 $ the exact results are well
approximated, once sufficiently many terms are included. Using $\alpha_s
=0.12$, $\alpha=1/129$, $\sin\theta_W = 0.223$ $M_Z = 91.19$~GeV, it is
found~\cite{CzarK} that the net effect of the nonfactorizable corrections
is
\begin{eqnarray}
\lefteqn{\Gamma^{\mbox{\small (2-loop EW/QCD)}} - {\alpha_s\over \pi}
\Gamma^{\mbox{\small (1-loop EW)}}}
\nonumber \\
&& \qquad\qquad =
\left\{
\begin{array}{l}
-1.13(4)\times 10^{-4} \mbox{ GeV}\quad \mbox{for $Z\to \bar uu$}
\\
-1.60(6)\times 10^{-4} \mbox{ GeV}\quad \mbox{for $Z\to \bar dd$}
\end{array}
\right.
\end{eqnarray}
The total change in the partial width $\Gamma(Z\to hadrons)$ is
obtained by summing over 2 down-type and 2 up-type quarks:
\begin{eqnarray}
\Delta \Gamma(Z\to u,d,s,c)= -0.55(3) \mbox{ MeV}
\end{eqnarray}
which translates into the change of the strong coupling constant
determined at LEP 1 equal to
\begin{eqnarray}
\Delta\alpha_s =
-\pi{\Delta \Gamma(Z\to hadrons)\over \Gamma(Z\to hadrons) }
=\pi{0.55\over 1741} \approx 0.001
\end{eqnarray}
This shift is somewhat smaller but still of the same order of
magnitude as the
experimental accuracy and should
to be taken into account in the final analysis of LEP 1 data.
Electroweak parameters extracted from $Z$ decays are not affected
by this correction.
The two-loop corrections to the $Zb\bar b$ vertex are dominated by terms
quadratic in $m_t$. The QCD corrections to these have been evaluated
already some time ago~\cite{fjrt}. The sub-leading terms $\propto \ln
m_t^2$ were obtained in~\cite{KwiSte95} and are of comparable magnitude,
thus indicating relatively slow convergence of the series. The complete
evaluation is thus mandatory and has been recently been
performed~\cite{HarSeiSte97}. In contrast to the previous case one is
confronted with three different scales, the masses of top, the $W$ and
the $Z$ boson.
Using the \hmp\ for $m_t^2\gg M_Z^2,M_W^2$, one may factor out the
$m_t$--dependence. However, for a part of the diagrams one still is left
with two-scale and even three-scale integrals involving $M_Z^2$ and
$M_W^2$ and $\xi_W M_W^2$, where $\xi_W$ is the electroweak gauge
parameter which has been kept in~\cite{HarSeiSte97}. Although they
appear to be only one-loop integrals, their exact evaluation up to
${\cal O}(\epsilon)$ is inconvenient. Instead, the
results~\cite{HarSeiSte97} were obtained by applying the \hmp\ to these
kinds of diagrams once more, this time using $\xi_W M_W^2, M_W^2\gg
M_Z^2$. This seemingly unrealistic choice of scales can be well
justified: It is not possible for an expansion to distinguish the
inequality $M_W^2\gg M_Z^2$ from $4M_W^2\gg M_Z^2$ or $(m_t+M_W)^2\gg
M_Z^2$, the latter ones being perfectly alright. The only matter is to
perform the expansion on the appropriate side of all thresholds, and
here one is concerned with thresholds at $2M_W$ and at $m_t+M_W$.
Therefore, the choice $M_W^2\gg M_Z^2$ is to be understood purely in
this technical sense. Graphically this continued expansion looks as
follows:
\begin{center}
\leavevmode
\epsfxsize=10cm
\epsffile[145 630 550 675]{figzbb.ps}
\end{center}
%
where only those terms are displayed which are relevant in the
discussion above and all others contributing to the \hmp\ are merged
into the ellipse. The thick plain line is the top quark, the thick wavy one
a Goldstone boson with mass squared $\xi_W M_W^2$, for example. The thin
plain lines are $b$--quarks, the inner thin wavy lines are $W$--bosons,
the outer ones $Z$-bosons. The spring-line is a gluon. The mass
hierarchy is assumed to be $m_t^2\gg \xi_W M_W^2 \gg M_W^2 \gg
M_Z^2$. The freedom in choosing the magnitude of $\xi_W$ provides a
welcome check of the routines and the results.
The outcome of this procedure is a nested series: The coefficients of
the $M_W/m_t$--expansion are in turn series in $M_Z/M_W$. Note that in
contrast to the decay into $u,d,s,c$ there is no threshold at $M_W$
which makes an additional expansion in $M_W/M_Z$ unnecessary.
In view of this calculation the procedure of successive application of
the \hmp\ resp.\ the \lmp\ has been implemented in a Fortran 90 program
named {\tt EXP}~\cite{Sei:dipl}. Therefore, given an arbitrary
hierarchy of mass scales, the computation of a three-loop two-point
function can now be done fully automatically. Even more, the link to the
Feynman diagram generator {\tt QGRAF}~\cite{qgraf} in a common
environment called {\tt GEFICOM}~\cite{geficom} allows to obtain the
result of a whole physical process without any human interference except
for specification of the process and final renormalization.
Finally, the result for the $W$--induced corrections to the $Z$--decay
rate $\delta\Gamma^W(Z\to b\bar b)$ is conveniently presented in the
form of the renormalization scheme independent difference to the decay
rate into $d\bar d$. Inserting the on-shell top mass $m_t = 175$~GeV,
the $Z$--mass $M_Z=91.19$~GeV and $\sin^2\theta_W = 0.223$
gives~\cite{HarSeiSte97}
\begin{eqnarray}
&&\delta\Gamma^W(Z\to b\bar b) - \delta\Gamma^W(Z\to d\bar d) =
\Gamma^0 {1\over \sin^2\theta_W} {\alpha\over \pi}
\bigg\{ - 0.50 + (0.71 -0.48)
\nonumber\\&&\mbox{\hspace{0em}}
+ (0.08 - 0.29) + (-0.01 - 0.07) + (-0.007 - 0.006)
%
%
+ {\alpha_s\over \pi} \bigg[ 1.16
+ (1.21
\nonumber\\&&\mbox{\hspace{0em}}
- 0.49)
+ (0.30 - 0.65)
%
+ (0.02 - 0.21 + 0.01) + (-0.01 - 0.04 + 0.004) \bigg] \bigg\}
%
%
\nonumber\\&&\mbox{\hspace{0em}}
=\Gamma^0 {1\over \sin^2\theta_W} {\alpha\over \pi} \bigg\{- 0.50 - 0.07 +
{\alpha_s\over \pi} \bigg[ 1.16 + 0.13 \bigg]\bigg\}\,,
\label{eqgamnum}
\end{eqnarray}
where the factor $\Gamma^0\alpha/(\pi\sin^2\theta_W)$ with $\Gamma^0 =
M_Z \alpha/(4 \sin^2\theta_W \cos^2\theta_W)$ has been pulled out for
convenience. The numbers after the first equality sign correspond to
successively increasing orders in $1/m_t^2$, where the brackets collect
the corresponding constant, $\log m_t$ and, if present, $\log^2
m_t$--terms. The numbers after the second equality sign represent the
leading $m_t^2$--term and the sum of the sub-leading ones. The ${\cal
O}(\alpha)$ and ${\cal O}(\alpha\alpha_s)$--results are displayed
separately. Comparison of this expansion of the one-loop terms to the
exact result~\cite{AkhBarRie86} shows agreement up to $0.01\%$
which gives quite some confidence in the $\alpha\alpha_s$--contribution.
One can see that although the $m_t^2$--, $m_t^0$-- and $m_t^0\log
m_t$--terms are of the same order of magnitude, the final result is
surprisingly well represented by the $m_t^2$--term, since the sub-leading
terms largely cancel among each other.
The uncertainty from uncalculated higher order QCD terms is far below
the foreseeable experimental precision, and the parametric uncertainty
from $\alpha_s$ and $m_t$ dominates. An important lesson to be learned
from the $Z b\bar b$ vertex in one- and two-loop approximation concerns
the interplay between ``dominant'' and ``sub-leading'' pieces: whenever
leading and sub-leading terms are of comparable magnitude, inclusion of
a sizeable number of terms in the expansion is required. The estimate of
the final result or of theoretical uncertainties based on the first two
terms of the series may lead to a wrong or misleading result.
\section{Summary\label{sec::chap6}}
Expansion techniques for Feynman amplitudes combined with sophisticated
computer algebra programs have lead to remarkable progress in multiloop
calculations during the past years. Problems with different mass and
energy scales can be treated with nested series. Powerful computers
allow to evaluate many terms in these expansions, and smallness of the
expansion parameter is thus no longer required. These techniques have
been successfully applied to purely hadronic as well as to electroweak
observables.
\vspace{1em}
{\bf Acknowledgments:} I would like to thank Joan Sola for organizing
this pleasant and very successful conference. The material presented in
this review has been developed in enjoyable and fruitful collaborations
with K.~Chetyrkin, A.~Czarnecki, R.~Harlander, Th.~Seidensticker and
M.~Steinhauser. The paper would never have been completed without the
\TeX nical help of R.~Harlander. Work supported by {\it
DFG-Forschergruppe ``Quantenfeldtheorie, Computeralgebra und
Monte-Carlo-Simulationen''} (DFG Contract KU~502/6-1) and BMBF
Contract 056~KA~93~P6 at the University of Karlsruhe.
\input{journals}
\sloppy
\raggedright
\begin{thebibliography}{99}
\input{bs_ref}
\end{thebibliography}
\end{document}
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\section*{Tests}
\begin{itemize}
\item Test that with externalization, no picture is generated multiple times.
\item Test that there are no warnings with and without using externalization.
\item Test with \texttt{pdflatex} and \texttt{latex} and do not forget to use \texttt{dvips}.
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\input{testRectangle.tikz}%
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\item Use includegraphics without file ending\\%
\includegraphics{testRectangle}%
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\includegraphics[width=\linewidth]{linewidth}%
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\item Use pgfplots without optional parameter\\%
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% \item Use includegraphics with scaling only a node results in an error\\%
% \includegraphics[width=\linewidth]{testNode.tikz}%
\ifpdf
\item Use includegraphics with jpg\\%
\includegraphics{example-grid-100x100bp.jpg}%
\item Use includegraphics with pdf\\%
\includegraphics{example-grid-100x100bp.pdf}%
\item Use includegraphics with png\\%
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\item Use includegraphics with pdf and scaling\\%
\includegraphics[width=0.3\linewidth]{example-grid-100x100bp.pdf}%
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\includegraphics[width=\linewidth]{example-grid-100x100bp.jpg}%
\item Input a 2D pgfplots\\
\input{testgraphic2D.tikz}%
\item Use includegraphics with a two dimensional plot\\%
\includegraphics{testgraphic2D.tikz}%
\item Use includegraphics with a scaled two dimensional plot with line width and an axis ratio of 1\\%
\includegraphics[width=\linewidth,axisratio=1]{testgraphic2D.tikz}%
\item Use includegraphics with a scaled two dimensional plot with given height and an axis ratio of 0.5\\%
\includegraphics[height=\linewidth,axisratio=0.5]{testgraphic2D.tikz}%
\item Use includegraphics with a scaled two dimensional plot with given height and an axis ratio of 0.5 and temporarily deactivated externalization\\%
\tikzexternaldisable
\includegraphics[height=\linewidth,axisratio=0.5]{testgraphic2D.tikz}%
\tikzexternalenable
\item Use includegraphics with a scaled two dimensional plot with given height and an axis ratio of 0.5 again\\%
\includegraphics[height=\linewidth,axisratio=0.5]{testgraphic2D.tikz}%
\item Use includegraphics with a scaled two dimensional plot with line width and a default axis ratio\\%
\includegraphics[width=\linewidth]{testgraphic2D.tikz}%
\item Input a two dimensional plot with a tight frame with width \newlength{\mylen}\settowidth{\mylen}{\frame{\input{testgraphic2D.tikz}}}\the\mylen\\%
\frame{\input{testgraphic2D.tikz}}
\item Use a two dimensional plot with a tight frame with width \settowidth{\mylen}{\frame{\includegraphics{testgraphic2D.tikz}}}\the\mylen\\%
\frame{\includegraphics{testgraphic2D.tikz}}
\else
\item Use includegraphics with eps (this is not found within dvi, even without tikzscale)\\%
\includegraphics{example-grid-100x100bp.eps}%
\item Use includegraphics with eps and full path (this is not shown, even without tikzscale)\\%
\includegraphics{/usr/share/texlive/texmf-dist/tex/latex/mwe/example-grid-100x100bp.eps}%
\fi
\item Use includegraphics with a histogram of a normal distribution\\%
\includegraphics[width=\linewidth,height=0.5\linewidth]{histogramNormal}%
\item {Use \texttt{\textbackslash graphicspath} with superfluous space\graphicspath{{somefolder} } (only defined locally for the current item)
\includegraphics{testRectangle.tikz}}%
\item Use a tikz-3Dplot, which is known to have a different size after externalization compared to the measurements without externalization and is thus rebuilt every time if the counter-measurements are not successful.\\%
\includegraphics[width=\linewidth]{3Dplot}
\end{itemize}
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\usepackage[utf8]{inputenc}
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\begin{document}
\title{20.430 PS2\_3}
\author{Josh Peters}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\sloppy
\section*{{[}3a{]}}
{\label{514602}}
Net number of oxygen molecules, N, crossing the shell over an arbitrary
time interval is
Assumptions: (1) Diffusion is constant (2) No flux at the edge (3) Fixed
media concentration (4) Only radial diffusion (5) Well-mixed~
(i)~\(2\pi rL\cdot{\vec{N}_{O_2}\mid_r-2\pi rL\cdot\vec{N}}_{O_2}\mid_{r+\Delta r}-2\pi rL\Delta r\cdot R=\frac{\partial c}{\partial t}\cdot2\pi r\Delta rL\)
divide through and multiply by 1/r
(ii)~\(\)\(\frac{1}{r}\frac{\left(r\cdot\vec{N}_{O_2}\mid_r-r\cdot\vec{N}_{O_2}\mid_{r+\Delta r}\right)}{\Delta r}-R=\frac{\partial c}{\partial t}\)
so as shell thickness approaches 0,~\(\Delta r\rightarrow0\),
(iii)~\(\frac{\partial c}{\partial t}=\frac{1}{r}\frac{\partial\left(r\cdot\vec{N_{o_2}}\right)}{\partial r^{ }}-R\), which is Fick's 2nd Law with Reaction, R =
k\textsubscript{1}*c(O\textsubscript{2})
(iv)~\(\frac{\partial c}{\partial t}=\frac{D}{r}\frac{\partial}{\partial r}\left(r\cdot\frac{\partial c}{\partial r}\right)-k_1c\)
\section*{}
{\label{514602}}
\section*{{[}3b{]}}
{\label{514602}}
Boundary and initial conditions~include:
\(c\left(r=0,\ t\right)\ =\ c_0\); assuming media flow brings concentration and
consistent oxygen concentration of c\textsubscript{0}
\(\frac{\partial c}{\partial t}\left(r=L,\ t\right)\ =\ 0\); assuming the concentration at the wall boundary does
not change because of impermeability
\(c\left(0<r<L,\ t=0\right)=\ 0\); assuming the concentration of oxygen within the
tissue before media flow is 0
Depending on the time scale of diffusion and reaction, I would expect
the change in concentration over time within the tissue would be a mix
of competing diffusion and degradation. Visually, this would look like a
mixture of linear and exponential decay. If the degradation is much
faster, the concentration would decrease in a fashion similar to
1st-order exponential decay. If diffusion is much faster, then the
concentration would eventually reach a steady state less than or equal
to the initial concentration since the right boundary is impermeable.~
Recasting boundary/initial conditions in Cartesian coordinates results
in by assuming 1D
\(c\left(x=0\right)\ =\ c_0\); assuming media flow brings concentration and
consistent oxygen concentration of c\textsubscript{0}
\(\frac{\partial c}{\partial t}\left(x=L\right)\ =\ 0\); assuming the concentration at the wall boundary does
not change because of impermeabilit
(i) with Fick's 2nd Law with Reaction:~\(\frac{\partial c\left(x,t\right)}{\partial t}=D\frac{\partial^2c\left(x,t\right)}{\partial x^2}-k_1c\)
This equation was solved numerically using MATLAB R2017a (The Mathworks,
Inc.). Code is attached.
\section*{{[}3c{]}}
{\label{514602}}
In this system, the Damk\selectlanguage{ngerman}öhler number (Da) is described by the expression
\(Da\ =\frac{L^2}{D}\cdot k_O\), where L is the tissue thickness, D is the effective
diffusion through the tissue, and k\textsubscript{O}~is the oxygen
consumption rate constant. To plot the behavior of the system at Da =
100, 1, and 0.01, L, D, and k\textsubscript{O~}were set to the following
values.
\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/100/Da100}
\caption{{Da = 100 shows a reaction-dominated time scale with most oxygen being
consumed shortly into the tissue. The profile shows fast 1st order decay
of the oxygen.
{\label{236148}}{\label{236148}}{\label{236148}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/1/Da1}
\caption{{Da = 1 shows mixed effects from the diffusion and reaction components of
the system -- slow decay which allows oxygen to permeate throughout the
system at varying concentrations.
{\label{964313}}{\label{964313}}{\label{964313}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/001/Da001}
\caption{{Da = 0.01 shows diffusion-dominated time scale where oxygen can reach a
saturated concentration over time since degradation is much slower than
diffusion.~
{\label{179987}}%
}}
\end{center}
\end{figure}
\par\null
In this system, Da is determined the length of the diffusing axis, the
degradation constant, and the diffusion. The number relates the
transport timescale to the reaction timescale. At high Da values, the
reaction dominates the particle kinetics, whereas at lower Da numbers,
diffusion dominates. Degradation and diffusion would most likely be
fixed within the tissue of interest in the system. Thus, consideration
must be put in the thickness of tissues and affecting the diffusion into
the system used in the system to ensure the tissues are properly
oxygenated for survival. Perhaps changing flow or increasing initial
oxygen concentration could increase diffusion into the system, but this
would not affect the Da number, only the initial and boundary conditions
of the system.
\section*{{[}3d{]}}
{\label{514602}}
If cells become hypoxic at {[}O\textsubscript{2}{]}\textsubscript{0}/4 =
0.0325 mM, then the maximum tissue thickness that can be obtained in
micrometers would be the large L value where~\(\left[O_2\right]_{t\rightarrow\infty}\ \ge\ 32.5\ \mu M\). ~The
estimated maximum tissue thickness is 184.52 uM.~
\par\null\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/3d/18452um}
\caption{{Numerically solved PDEs with a length of 184.52 uM and literature values
for diffusion, oxygen consumption, and blood oxygen concentration.
{\label{873758}}%
}}
\end{center}
\end{figure}
Comparing Fig. 4 with the previous figures shows that the physiological
constants show a mixed profile of diffusion- and reaction-time scales.
With these dimensions and constants, neither the degradation of oxygen
nor the diffusion of oxygen from the channel dominates the oxygen
concentration profile.
\section*{{[}3e{]}}
{\label{514602}}
In reality, the boundary conditions would not be entirely true,
especially in the length and diameter of the channel are relatively
similar. This would suggest that the concentration of oxygen within the
channel would not be a constant initial concentration and that diffusion
in 3D would be important to investigate since the diffusive area would
be much larger than the length of the channel. The inconsistency in the
cell layer makes sense, because diffusion is not happening the one
direction towards the outer channel wall. Oxygen will diffuse both up
and down the channel within the tissue and in and out of the tissue
between the channel and cell layers. These changes would create dramatic
gradients within the cells which could change the Da number for a small
area of tissue (dA). These factors, various diffusion directions
resulting in varying local concentrations within the tissue which would
increase/decrease the degradation rate, would create inconsistency in
tissue channel morphology.
\par\null
\selectlanguage{english}
\FloatBarrier
\end{document}
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\fancyfoot[R]{jeu. mars 31 12:56:43 CEST 2016}
\lfoot{\Large \textit{Uubu.fr}}
\hypersetup{pdfinfo={
Title={entr},
Author={Sylvain Girod},
Creator={Bash script from uubu.fr's format version 1},
Producer={Bash script and PDFLaTeX},
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\begin{center}
{\Huge entr }
\end{center}
\vspace{1cm}
\begin{flushleft}
{\large Lancer des commandes arbitraires quand des fichiers changent}
\vspace{10mm}
\hspace{1mm} Une liste de fichiers fournis sur l'entrée standard et un programe est exécuté quand ils changent
\vspace{5mm} \\
\vspace{1cm}{\huge OPTIONS }
\
\begin{description} \normalsize
\item[\hspace{1mm} -c]{Exécute /usr/bin/clear avant d'invoquer le programme spécifié}
\item[\hspace{1mm} -d]{Suit les répertoires des fichiers réguliers fournis en entrée et qui si un nouveau fichier est ajouté.}
\item[\hspace{1mm} -p]{Retarde la première exécution du programme jusqu'à ce qu'un fichier soit modifié}
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\end{description}
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\vspace{5mm} \\
\vspace{1cm}{\huge Exemples }
\
\hspace{1mm} Reconstruire un projet si le fichier source change, limitant la sortie aux 20 premières lignes \\
\hspace{1mm} \textbf{find src/ | entr sh -c 'make | head -n 20'} \\
\hspace{1mm} Lancer et recharger automatiquement un serveur node.js \\
\hspace{1mm} \textbf{ls *.js | entr -r node app.js} \\
\hspace{1mm} Effacer l'écran et lancer une requête après que le script SQL soit mis à jours: \\
\hspace{1mm} \textbf{echo my.sql | entr -p psql -f /\_} \\
\hspace{1mm} Reconstruire le projet si un fichier sources est modifié ou ajouté dans le répertoire src/ \\
\hspace{1mm} \textbf{while sleep 1; do ls src/*.rb | entr -d rake; done} \\
\end{flushleft}
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\documentclass{article}
\usepackage[affil-it]{authblk}
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\begin{document}
\title{Photometric Science Alerts From Gaia}
\author{Heather Campbell}
\affil{Affiliation not available}
\date{\today}
\maketitle
\section{What is a Photometric Science Alert?}
A photometric science alert is the appearance of a new source, or a change in flux, which suggests we could learn something from prompt ground-based follow-up. This does not include: periodic variable stars (these sources may be better left to the end of the mission) and moving objects (however, astrometric microlensing would be an exception). The science alerts will be made public, within one to two days of Gaia detection, most of this time is due to downloading the data from the satellite.
\section{Potential Triggers}
Potential triggers for the the Gaia science alerts are objects of scientific interest which would benefit from fast ground based follow-up, as just discussed. Some examples of sources which maybe potential triggers include supernovae, super-luminous supernovae, tidal disruption events, cataclysmic variables, outbursts and eclipses from young stellar objects, X-ray binaries, microlensing events and other theoretical or unexpected phenomena. Figure~\ref{fig:triggers} shows some of these potential triggers and the area of pars space they occupy for their brightness as a function of duration.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig-triggers/potential-triggers}
\caption{{This shows the amplitude and duration of a range of potential triggers for the Gaia science alerts. \label{fig:triggers}%
}}
\end{center}
\end{figure}
\section{Gaia as a Transient Search Machine}
Gaia is comparable to other transient search machines, such as the Catalina Sky Survey and the Palomar Transient Factory, as shown in Table~\ref{tab:transient_machine}, which covers similar areas each day and similar limiting magnitude. The disadvantage of the Gaia survey is that the average cadence is only $\sim$30days whereas transient surveys usually have a cadence of approximately 3 to 5 days. However, there is also a shorter cadence of 106.5 mins from the two mirrors in the satellite, also sometimes a 253.5 mins cadence, and sometimes 3 or more observations are thus obtained (when close to the 45 degrees ecliptic latitude zones for example). This 106.5 mins cadence is a huge advantage and means that changes in brightness should be detected quickly. Also, Gaia will cover the whole sky (including the Galactic plane), which is a significant survey area increase over other transient searches. The Gaia transient alerts will also have high spatial resolution with precise photometry (1$\%$ at G=19) and milliarcsecond astrometry (down to $\sim$20mag), lowres spectra for all objects brighter than $\sim$19mag and colours for fainter objects (see \citet{Jordi:2010} for details of the photometry and lowres spectra).
\begin{center}
\begin{tabular}{c|c|c|c}
Patient & Gaia & Catalina Sky Survey & Palomar Transient Factory\\
\hline
deg2 day-1 & $\sim$1230 & 1500 & 1000\\
Avg Cadence & $\sim$30 days & 14 days & 5 days \\
Limiting mag & 20 (21?) & 19.5 & 21 \\
fsky & all sky & 0.6 & 0.2\\
\end{tabular}
\caption{{\label{tab:transient_machine} Predicted numbers for the Gaia transient search compared to some ongoing surveys.}}
\end{center}
\section{Time line}
The Gaia satellite was launched on the 19th December 2013, and has now successfully been placed into orbit around the second Lagrange point. Over the next few months the telescope will undergo system shake-down and ESA commissioning (Figure~\ref{fig:timeline}). It is planned that in June the Gaia satellite will spend a month scanning the Ecliptic Poles internally verifying the data, and learning how to identify large amplitude variable stars (potential contaminants of the Gaia Science Alerts stream).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/timeline/timeline}
\caption{{\label{fig:timeline} Current timeline for Gaia operations and data accumulation.%
}}
\end{center}
\end{figure}
Then in July Gaia will switch to nominal scanning and history of the whole sky will begin to be accumulated. In Figure~\ref{fig:scanning} we show the expected coverage of Gaia by the end of July and then the end of September 2014. This will give some history of each patch of sky in the Gaia passbands and allow detection of transient objects. We propose to begin Gaia Alerts Spectroscopic Follow-up in the last weeks of August and the first week of September.
\section{Scanning law}
The Gaia satellite consist of two telescopes, which are projected onto one focal plane. The time between the two fields of view being observed is 106.5 mins and then the time between subsequent scans is 6 hours. After these initial observations the field will be revisited every $\sim$10-30 days. Over the full mission each patch of sky will be measured, on average, approximately 70 times. The densest coverage is at 45 degrees to the ecliptic plane and this region is covered with approximately 200 epochs.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/scanning/scan-coverage}
\caption{{\label{fig:scanning} By 30 days 11.6$\%$ of the sky has been observed at least 3 times by Gaia. By 90 days, 52.03$\%$ of the sky has been observed at least 3 times by Gaia.%
}}
\end{center}
\end{figure}
\section{SNa discovery rates}
Simulations, \citet{Belokurov_et_al_2003} and updated by \citet{icella_Cappellaro_Turatto_2012}, predict Gaia will see $\sim$6000 SNe down to G=19 (3/day), and twice this to G=20. One SN per day will be brighter than 18th magnitude (see Fig~\ref{fig:SN_rate}). For cataclysmic variables (CVs) the rate will be approximately similar, and Breedt, (priv. comm) predict Gaia should find 1000 new CVs. \citet{Blagorodnova_in_prep_2014} predict that Gaia will find of order 20 Tidal disruption event's (TDE's) per year. Young stellar objects outbursts will be less common and Gaia will probably only find a few per year.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/SN-rate/SN-detections}
\caption{{\label{fig:SN_rate} Predicted SN detections with Gaia as a function of G-band magnitude.%
}}
\end{center}
\end{figure}
\section{Alert Publication}
Alerts are expected to be discovered and published to the world within $\sim$24-48 hours of observation by the satellite. The Alert Stream will go live once Gaia has mapped at least 10$\%$ of the sky, a minimum of 3 times, which takes approximately one month (see Fig~\ref{fig:scanning}). Once the Gaia alert stream is fully operational all alerts will be made publicly available, and thus accessible for use by the community in their dedicated followup campaigns (see Section~\ref{followup}). During the commissioning, initialisation and early operations phase of Gaia (January - August 2014) - there will be systematic validation of the Gaia alerts, whereby the operational system will be assessed before going `Live'. The science alerts will be available to the community in web-based and email-based formats and will be produced in Virtual Observatory Event (VOEvent) - machine-readable format.
Each Alert package will consist of: coordinates, magnitudes, light curves, spectra, colours, proper motions, parallaxes (when available), astrophysical parameters (pars) (when available), features (random forest classifier see Section~\ref{class}), classifier probabilities, cross match results.
\section{Classification}
\label{class}
Gaia is predicted to detect 44 million transits per day,which is $\sim$150 - 800 GByte/day of data. Within this huge volume of data we expect 100s -1000s of potential interesting astrophysical triggers per day (real variables/moving objects). This precludes visual classification of a rich data stream and thus automated methods which are fast, repeatable and tuneable are essential. The Gaia alerts classification pipeline uses random forest classification. The random forest will use all the information available, and its features will include: light curve photometry (gradient, amplitude, historic rms, magnitude, signal-to-noise ratio, transit rms), lowres spectra (flux v lambda, colours, spectral shape coefficients, spectral type), auxiliary information (neighbour star, shape pars, motion pars, coords, crowding, calibration offset, correlations, QC pars) and crossmatch environment (near known star mags, near known variable class, near galaxy, near galaxy redshift and circumnuclear).
To build up a sufficient sample of classification labels in order to train the random forest classifier (e.g. \citet{Ofek_Cenko_Butler_et_al__2012}) we aim to observe $\sim$500s homogenous high-quality spectra in the first year of the mission, spread across each broad class of transient phenomena (active galactic nuclei, core collapse SN, TDE, SN, Novae, CV and variable stars).
The light curve classification utilises the flux gradient of the transient object. The Gaia observations with 106.5 mins cadence are used to indicate the type of object. The lowers (BP/RP) spectra provide far more information to aid classification \citet{Blagorodnova_in_prep_2014} and provide robust class for most objects, at $>$19mag, when the classifier is fully trained on representative data. In addition, the transient object will be cross matched with archival catalogues, for example, Sloan Digital Sky Survey (SDSS), Two Micron All Sky Survey (2MASS), HST and Visible and Infrared Survey Telescope for Astronomy (VISTA). This will help remove known variable star contaminates and provide environmental information for the transient events, e.g. is there a host galaxy associated with the source and if so what is the type and magnitude.
\section{Follow up}
\label{followup}
We are also co-ordinating a large program of photometric follow-up to improve the light curve sampling of Gaia transients. 47 x (7cm-2m) telescopes are listed as currently active (http://bit.ly/1aHNXzy) and 13 observatories are already doing tests (http://bit.ly/17ViW7s). All make use of our photometric calibration server (a tool developed to maximise the usefulness of the photometric followup data) to place the disparate data onto the same system (Wyrzykowski et al. 2013 $ATEL\#5245$). Additionally, Las Cumbres Observatory Global Telescope Network (LCOGT) are expected to play a key role in the follow-up especially of $\mu$lensing and young star transients. %Other proposals
We point out the strong synergies with external facilities operating at different wavelengths. We will be able to confirm and characterise e.g. Low Frequency Radio Array (LOFAR) transients, and we may also trigger prompt {\it SWIFT} follow-up for particularly interesting events.
There is also a large educational (mostly utilising the Faulkes telescopes) and amateur involvement planned in the followup of these transient events, to assist in compiling light curves and increase the public evolvement and interest.
We need a large sample of well-exposed (S/N$\sim$20$-$50), medium-dispersion (R$\sim$500$-$1000) spectra, over a wide range of classes and magnitudes, to build classification training sets, in order for our (Random Forest) machine learning algorithms (discussed in Section~\ref{class}) to perform well for the Gaia spectra for the remainder of the mission. Therefore we aim to obtain 1.5-4m telescope time to build this training set. It is important to invest time at the beginning of the Gaia mission to understand and characterise the transients that will be discovered with Gaia, so that we can optimise the process, and ensure that the rest of the mission is as productive as possible. We also intend to archive and release our spectroscopic classifications promptly after processing each night's observing.
\section{Summary}
The alert stream is non-proprietary and will be (some of) the first data from Gaia Summer 2014. We have planned an extensive follow-up program for classifying large numbers of transients: e.g. 10,000 SNe Ia over the whole sky. The alerts will be published one to two days after the event was initially detected (most of this time is due to the time taken for the data to be down linked from the satellite and processed). The alerts will be preliminarily classified using random forest classifiers based on the Gaia photometry and lowres spectra with additional cross match information from existing surveys. These classifications should improve after the first few months of ground based followup and retraining of the Bayesian classifiers. The alerts will be published in the VO format. For more information visit: http://www.ast.cam.ac.uk/ioa/wikis/gsawgwiki.
\section{Acknowledgements}
Material used in this work has been provided by the Coordination Unit 5 (CU5) of the Gaia Data Processing and Analysis Consortium (DPAC). They are gratefully acknowledged for their contribution.
HCC, STH, GG, NAW, LW are members of the Gaia Data Processing and Analysis Consortium (DPAC) and this work has been supported by the UK Space Agency. NB has been supported by the The Gaia Research for European Astronomy Training (GREAT-ITN) network, funded through the European Union Seventh Framework Programme ([FP7/2007-2013] under grant agreement n$\,^{\circ}$ 264895.
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\title{Mailings for members}
\begin{article}
\vspace*{-0.5\baselineskip}
The committee has been asked, by a member, for a mailing list of
members of the group. We find, on consulting the Data Protection
Registrar, that we may not legally provide such a list, and we are
investigating what is needed so that we should be able to supply such
lists in future. It seems clear that members must be offered an
opt-out.
In the interim, we are offering members of the group the opportunity
of promoting occasional mailings. We have offered such facilities, on
an \emph{ad hoc} basis, in the past: we have carried fliers for books,
and on one occasion for the \emph{Scientific Word} system. Our
mailing mechanisms are not sophisticated, and significant insertions
may attract an extra charge, as will an insertion that increases the
mail cost.
Members who wish to take advantage of this service should contact the
membership secretary, in the first instance.
\end{article}
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\title{Degradation or evolution? Assessing ecological security network for a
rapid urbanization region in Eastern China}
\author[1]{De Zhou}%
\author[2]{Zhulu Lin}%
\author[1]{Siming Ma}%
\author[1]{Jialing Qi}%
\author[1]{Tingting Yan}%
\affil[1]{Zhejiang Gongshang University}%
\affil[2]{North Dakota State University}%
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Rapid urbanization leads to fragmentation of large land patches,
islandization of ecological landscape, and destruction of ecological
security network. As a basic guarantee of life, a sound ecological
security network promotes connectivity between ecological sources,
improves ecological security patterns, and mitigates the degradation of
an ecological system. The objective of this study was to improve a
framework for assessing the ecological security network. We demonstrated
the application of the proposed framework through a case study of the
urban agglomeration around Hangzhou Bay (UAHB), a rapid urbanization
region in Eastern China's Zhejiang Province. We improved the
identification method of ecological sources by integrating the
evaluations of ecosystem services value and ecological sensitivity,
while we screened ecological sources by using the rank-size rule and the
natural breaks method. Based on the screened ecological sources, the
ecological corridors were reconstructed and optimized for the UAHB
region. Results from this study showed that the structure and function
of the ecological security network were strongly influenced by human
activities and urban sprawl. The ecological security network has
deteriorated locally in eastern coastal areas of UAHB during the past 20
years with strong spatial variability in ecological security patterns.
To maintain a well-protected and sustainable ecological quality, we
proposed a set of 5 measures to improve the ecological security pattern
and the sustainable development of the ecological system in Eastern
China.%
\end{abstract}%
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\begin{center}
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\begin{center}
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\end{center}
\end{figure}\selectlanguage{english}
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\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/FIGURE-5-Identification-of-ecological-sources-in-the-UAHB-in-1995-and-2015/FIGURE-5-Identification-of-ecological-sources-in-the-UAHB-in-1995-and-2015}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/FIGURE-6-Ecological-corridors-system-in-the-UAHB-in-1995-and-2015/FIGURE-6-Ecological-corridors-system-in-the-UAHB-in-1995-and-2015}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/FIGURE-7-Assessment-of-ecological-sources-based-on-shape-index-and-degree-of-ecological-sources/FIGURE-7-Assessment-of-ecological-sources-based-on-shape-index-and-degree-of-ecological-sources}
\end{center}
\end{figure}\selectlanguage{english}
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\begin{center}
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\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
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\end{center}
\end{figure}
\selectlanguage{english}
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\title{The Rattle Of The Thompson Gun}
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The guerrilla struggle against Francoism actually arose in the days following the army revolt against the Spanish Republic on 18 July 1936. In areas which fell immediately to the mutinous army, a bloody repression was promptly set in motion and this obliged many anti-fascists to take to the hills to save their skins. This was repeated over nearly three years of civil war as areas were conquered, one after another, by the Francoist army and it extended to virtually the entirety of the Peninsula after the Republican troops surrendered in the Centre-Levante zone on 31 March 1939.
Very little has been written about the scale of the armed struggle against Franco following the civil war. It was and still is known to few. A thick blanket of silence has been drawn over the fighters, for a variety of reasons. According to Franco’s personal friend Civil Guard Lieutenant-General Camilo Alonso Vega — who was in charge of the anti-guerrilla campaign for twelve years — banditry (the term the Francoists always used to describe the guerrilla activity) was of “great significance” in Spain, in that it “disrupted communications, demoralised folk, wrecked our economy, shattered our unity and discredited us in the eyes of the outside world”.
Only days before those words were uttered General Franco himself had excused the blanket silence imposed on reports of armed opposition and the efforts mounted to stop it, when he had stated that “the Civil Guard’s sacrifices in the years following the Second World War were made selflessly and in silence, because, for political and security reasons it was inappropriate to publicise the locations, the clashes, casualty figures or names of those who fell in performance of their duty, in a heroic and unspoken sacrifice.”
This cover-up has continued right up until our own day. In a Spanish Television (TVE) programme entitled Guerrilla Warfare and broadcast in 1984, General Manuel Prieto Lopez cynically referred to the anti-francoist fighters as bandits and killers. Not that this should come as any surprise — during the period described as the political transition to democracy (November 1975 to October 1982) all political forces, high financiers, industrialists, the military and church authorities decided that references to the past were inappropriate and that the protracted blood-letting of the Franco era should be consigned to oblivion. That consensus holds firm in 1996, and historians eager to lift that veil run up against insurmountable obstacles when they try to examine State, Civil Guard or Police archives.
We have no reliable break-down of the overall figures for guerrillas or for the casualties sustained by or inflicted upon the security forces and Army. If we are to have some grasp of what this unequal struggle against the Dictatorship was like, our only option is to turn to figures made public in 1968 — a one-off it seems — according to which the Civil Guard sustained 628 casualties (258 deaths) between 1943 and 1952: some 5,548 bandits were wiped out in 2,000 skirmishes, many of which amounted to full-scale battles. The figures for this eradication are as follows: killed — 2,166, captured or surrendered — 3,382, arrested as liaisons, accessories or for aiding and abetting — 19,407. An embarrassed silence shrouds the earlier years between 1939 and 1942, when units from the regular army, the Foreign Legion and the Regulars, with artillery support attempted to wipe out the guerillas. The aforesaid figures given for Civil Guard casualties at the guerrillas’ hands can be discounted. If we compare the lists of deceased Civil Guards during these years where no cause of death is listed, with peace-time death-rates, we find a surplus of deaths which are (assuming they were the results of illness or accident) inexplicable and arrive at what is unquestionably a figure closer to the truth: some 1,000 deaths on active service.
The escalation of guerrilla activity began in 1943, when the widespread belief that the Third Reich had victory in its grasp was starting to fade, following the bloody rout of the German Army’s elite divisions at Stalingrad. As the tide of the Second World War turned, the anti-Franco guerrillas, as might have been expected, bounced back in terms of morale and dynamism, and from 1944 onwards flourished to a considerable extent. Its heyday was in 1946–1947. After that, partly as a consequence of international policy which sought a rapprochement with Franco, a decline set in that ended with the demise of guerrilla activity in 1952. In Barcelona, Madrid, Valencia and other cities, urban guerrilla activity persisted for a decade or so longer.
After 1944, guerrillas operating inside Spain received considerable reinforcements from their exiled countrymen who had played an active part in the liberation of France. These were well-trained and experienced men equipped with up-to-date weaponry and easy to use high explosive substances such as plastique. Most of them were drawn from France and a smaller number from across the seas in North Africa. Communist leaders charged with politicising guerrilla activity came in from the Americas via Lisbon and Vigo. The Communists who took it for granted that the war-cry of “Taking Spain back!” would be the signal for a general popular uprising against the Franco regime made a great song and dance about this comparatively massive aid.
Some 3,000 guerrillas organised in France with the very same weaponry they had used in their fight against the Nazis, mounted two main attacks across the Pyrenees in 1944. The first incursion was into Navarre on 3 and 7 October: the second came via Catalonia, the object being to establish a bridge-head in the Vall d’Aran and install a provisional Republican government. It was also taken for granted that, confronted by such a fait accompli, the Allies would be prompted to step in to bring down Franco. These incursions were easily repulsed — having been heralded in advance — for the Spanish government had taken all appropriate measures. Even so, there were lots of guerrillas who refused to return to their bases and opted instead to infiltrate into the interior in small groups. There they reinforced existing guerrilla bands and set up new ones where none existed.
The weapons they brought in were a lot more effective and better suited to guerrilla fighting. The most commonplace weapon was the British Sten gun, or the German M.P. 38. Both were rapid-fire weapons and used 9mm ammunition which was the most plentiful sort. American weapons like the Colt pistol flooded in, as did (in lesser numbers) Thompson sub-machine guns, a heavier but highly effective weapon. One burst of Thompson gunfire in the hills was reminiscent of an artillery salvo. The fighters entering Spain also brought with them a tried and tested morale forged in victories scored against the Nazis and in the staunch belief that Franco could not survive the downfall of Adolf Hitler and Benito Mussolini. They also had organisational experience behind them and solid ideological convictions, anarchist, socialist or communist, qualities that would quickly transform the guerrilla phenomenon as they afforded increased cohesiveness to countless scattered guerrilla bands.
The main areas of guerrilla activity were those whose geographical features made defence and survival most likely ie: mountains ranges and areas which provided adequate cover. For example in Andalusia there were guerrilla bands aplenty, some of them over 100-strong. In Asturias, the guerrillas displayed tremendous enterprise, not unconnected with a deep-rooted political consciousness: the revolution by the Asturias miners in October 1934 had not been all that long ago. In many areas, guerrilla activity was intermittent and random as guerrilla bands moved around for a number of reasons, such as the encroachments of counter-insurgency forces.
The style and nature of the guerrilla struggle varied with the terrain and the resources of the individuals and groups involved. Activities included the bombing of strategic objectives, attendats (political assasinations),the movement of arms, the protection of individuals and groups involved in underground political activity; bank robberies and forgery to fund the struggle and destabilise the economy; as well as some more spectacular actions: rescue missions to free captured comrades, open fire-fights with fascist forces; and even an attempt to bomb Franco from the air! (three men in a light aircraft came within a hair’s breadth of dropping incendiary and fragmentation bombs on the General and his Aides during a Regatta in 1948).
An example that sums up the mentality and spirit of the guerrilla movement of the time is provided by a small team of Anarchist guerrillas, led by the veteran fighter Francisco Sabate Llopart (El Quico). On their return to Spain after the end of the Second World War one of their first missions was the ‘expropriation’ of money and valuables in a series of aggravated robberies of local big-businessmen. On completion of ‘business’, those ‘visited’ would be left a note like the following one, left at the home of a wealthy big-store owner, Manuel Garriga:
“We are not robbers, we are libertarian resistance fighters. What we have just taken will help in a small way to feed the orphaned and starving children of those anti-fascists who you and your kind have shot. We are people who have never and will never beg for what is ours. So long as we have the strength to do so we shall fight for for the freedom of the Spanish working class. As for you, Garriga, although you are a murderer and a thief, we have spared you, because we as libertarians appreciate the value of human life, something which you never have, nor are likely to, understand.”
A small example of how, despite the loss of the war, and despite the ruthlessness of the fascist repression, those involved in the resistance still managed to maintain their politics, their humanity, and their self-respect.
The armed opposition to Franco was no longer a serious problem after 1949 and, as we have said, it petered out around 1952. Aside from the severe blows dealt by the Civil Guard and the Army, the absence of a logistical system capable of keeping the fighters equipped, and, above all else, the fact that the opposition political parties had chosen to gamble upon diplomacy as a substitute for weapons, made it impossible for the resistance’s offensive activity to continue.
Another highly significant element in the winding-up of the guerrilla struggle was the arrival on the scene in 1947 of superbly trained and schooled security force personnel in the shape of “counter-guerrilla bands”, dressed and armed in the guerrillas’ own style and sowing confusion and terror on their home ground. These “counter-gangs” even carried out savage killings that were ascribed to the guerrillas proper, the aim being to bring them into disrepute and strip them of popular support. Then again, the infiltration of police plants into the guerrilla bands was extraordinarily effective and made it possible to dismantle some of the more important groupings.
In Asturias, in 1948, around 30 socialist guerrillas boarded a French fishing smack which had arrived specifically to collect them and deliver them to St Jean de Luz in France. In Levante, the last remaining guerrillas in the area, around two dozen survivors, made it out to France in 1952. In Andalusia, a few bands survived until the end of 1952, but their leaders — like the anarcho-syndicalist, Bernabe Lopez Calle (1889–1949) — had already perished in combat. A few managed to escape to Gibraltar or North Africa, but, for the most part, they were wiped out in armed clashes: others were executed by the \emph{garrote vil} (death by strangulation) or firing squads: those who escaped that fate served prison terms sometimes in excess of 20 years.
In 1953, the United States signed a military and economic assistance treaty with Franco. Two years later, Franco’s Spain was welcomed into the United Nations. However, even though all was lost, a few die-hards refused to give up the fight: in Cantabria, the last two guerrillas, Juan Fernandez Ayala (Juanin) and Franciscxo Bedoya Gutierrez (El Bedoya) met their deaths in April and in December of 1957 respectively. In Catalonia, Ramon Vila Capdevila (Caraquemada), the last anarchist guerrilla, was gunned down by the Civil Guard in August 1963. But the honour of being the last guerrilla has to go to Jose Castro Veiga (El Piloto) who died, without ever having laid down his arms, in the province of Lugo (Galicia), March 1965.
There are a number of reasons for the failure of the Guerrilla campaign against Franco, and although open guerrilla warfare had all but ended in the 50’s, the movement against Franco continued, as did underground political activity, until the regime’s eventual collapse. What the guerrillas had wanted to acheive was open insurrection against Franco. What they show us today, through their ambition and their sacrifice, is that the brutal repression of the progressive working class after the Civil War did not go unchallenged. The full story of the guerrilla struggle, as Tellez states in this article, is still being uncovered. All we can do today is salute the men and women of the resistance who gave their lives, not only in the defence of their class, but for a future where the social structures that create the Francos, are buried along with them.
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The Anarchist Library
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Anti-Copyright
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Anonymous
The Rattle Of The Thompson Gun
\bigskip
Retrieved on January 1, 2005 from \href{http://www.cat.org.au/aprop/rattle.txt}{www.cat.org.au}
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\textbf{theanarchistlibrary.org}
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% This is a LaTeX style file to prepare Nancay Radio Telescope proposals.
% Originally, used for IRAM 30m and interferometer proposals.
% [R. Lucas original version] [J.M. Winters 13 July 2007]
% [J.M. Martin 31 July 2007] [P. Colom 20-21 Aug. 2007]
% [P. Colom 30 Aug. 2007, last modification 11 April 2018 ]
% This should be edited for each deadline ...
%
\def \deadline {14 May 2018}
%
%
\def \period {01 July 2018 --- 31 December 2018}
%
%=====================================================================
% WORKING FILE FOR NANCAY NRT COMITE DES PROGRAMMES PROPOSAL .sty FILE
%
% ATTENTION : after these lignes, add %Modif + initials
% for each MODIFICATIONS. After modification :
% keep file name, but modify VERSION (letter a, b, c, etc...),
% or change the DATE in file name.
%
% HISTORY :
% - 30/07 JMM first updates, these instructions.
% - 20-21/08 PC: modifications for NRT
% - 24/08 PC: pico replaced by nrt (not in picountry !)
% - 27/08 JMM/PC : \rasortedtable added
% - 30/08 PC: \nsour added
% - May/2015 PC: two KP button instead of one
%
%
%=====================================================================
%
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%
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\textwidth 178mm
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%
% Some commands taken from A&A and elsewhere...
%
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%Mod PC 200807
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%
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%
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% environment to the match A&A layout of References.
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{REFERENCES}{REFERENCES}}\list
% The \arabic{enumi} command has been removed because we don't use
% numbered citations in astronomical journals.
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%
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{\usecounter{fign}\setlength{\leftmargin}{0mm}
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\def \endcaptions {\end{list}\normalsize}
%
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% no page headings or numbers whatsoever
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%
% Specific commands
%
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%
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%
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% pico replaced by nrt, picoveleta replaced by nancay
%\newif\if@pico\@picotrue
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%
%\newif\if@pooledobserving \@pooledobservingfalse\def\pooledobserving{\@pooledobservingtrue}
%
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%
%
% at Nan\c{c}ay ?
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%
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% PC 210807 (solar, extragal, galactic, continuum, lines, pulsar, other)
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\newif\if@detection \@detectionfalse\def\detection{\@detectiontrue}
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%
% new section "author" allows two PIs (July 2003) -------------------------
% option "remark" to add a remark concerning the project, the pi, or whatever
%
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\toks@ii{#2}\edef#1{\the\toks@\the\toks@ii}}
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\long\def\@ifxundefined#1{\@ifx{\undefined#1}}
\def\@boolean#1#2{
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\def\@boole@def#1#{\@boolean{#1}} \@boole@def\@ifx#1{\ifx#1}
\long\def\boolean@true#1#2{#1}
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\global\let\@coauthor\@empty
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\def\coauthor#1#2#3{\appdef\@coauthor{#1 (#2 -- #3); }}
%
% end section "author" -------------------------
%
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\def\sourcelist#1{\gdef\@sourcelist{#1}}
\def\epoch#1{\gdef\@epoch{#1}}
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%
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%
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%
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{\large \bf PROPOSAL FOR NAN\c{C}AY 200x35 m$^2$ TELESCOPE}\\
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%
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\begin{minipage}[t]{100mm}
\begin{raggedright}
Mr. Patrick Thivrier, Observatoire de Paris\\
E-mail: {\bf [email protected] {\em at} obspm.fr}\\
Station de Radioastronomie, Nanc{c}ay, FRANCE \\
Fax: +33 1 4507 7939 \/ Phone : +33 2 4851 8632 \\
\end{raggedright}
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%
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Present at Nan{\ccedil}ay ? \,\chec@k{\if@atnancay} \hskip4mm
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Stokes parameter \,\chec@k{\if@stokes} \hskip4mm
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%
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https://ctan.math.washington.edu/tex-archive/info/examples/tlc2/3-4-29.ltx
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washington.edu
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%%
%% The LaTeX Companion, 2ed (second printing August 2004)
%%
%% Example 3-4-29 on page 167.
%%
%% Copyright (C) 2004 Frank Mittelbach, Michel Goossens,
%% Johannes Braams, David Carlisle, and Chris Rowley
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
\documentclass{ttctexa}
\pagestyle{empty}
\setcounter{page}{6}
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\begin{document}
\Verb[fontfamily=courier]+\realdanger{|emph<arg>}+ \\
\verbx[fontfamily=courier]+\realdanger{|emph<arg>}+
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http://cerco.cs.unibo.it/export/1449/Deliverables/D4.2-4.3/reports/D4-3.tex
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unibo.it
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| 22,856,310 | 11,457 |
\documentclass[11pt, epsf, a4wide]{article}
\usepackage{../../style/cerco}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[english]{babel}
\usepackage{graphicx}
\usepackage[utf8x]{inputenc}
\usepackage{listings}
\usepackage{lscape}
\usepackage{stmaryrd}
\usepackage{threeparttable}
\usepackage{url}
\title{
INFORMATION AND COMMUNICATION TECHNOLOGIES\\
(ICT)\\
PROGRAMME\\
\vspace*{1cm}Project FP7-ICT-2009-C-243881 \cerco{}}
\lstdefinelanguage{matita-ocaml}
{keywords={definition,coercion,lemma,theorem,remark,inductive,record,qed,let,let,in,rec,match,return,with,Type,try},
morekeywords={[2]whd,normalize,elim,cases,destruct},
morekeywords={[3]type,of},
mathescape=true,
}
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keywordstyle=\color{red}\bfseries,
keywordstyle=[2]\color{blue},
keywordstyle=[3]\color{blue}\bfseries,
commentstyle=\color{green},
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showspaces=false,showstringspaces=false}
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\DeclareUnicodeCharacter{8797}{:=}
\DeclareUnicodeCharacter{10746}{++}
\DeclareUnicodeCharacter{9001}{\ensuremath{\langle}}
\DeclareUnicodeCharacter{9002}{\ensuremath{\rangle}}
\date{}
\author{}
\begin{document}
\thispagestyle{empty}
\vspace*{-1cm}
\begin{center}
\includegraphics[width=0.6\textwidth]{../../style/cerco_logo.png}
\end{center}
\begin{minipage}{\textwidth}
\maketitle
\end{minipage}
\vspace*{0.5cm}
\begin{center}
\begin{LARGE}
\textbf{
Report n. D4.3\\
Formal semantics of intermediate languages
}
\end{LARGE}
\end{center}
\vspace*{2cm}
\begin{center}
\begin{large}
Version 1.0
\end{large}
\end{center}
\vspace*{0.5cm}
\begin{center}
\begin{large}
Main Authors:\\
Dominic P. Mulligan and Claudio Sacerdoti Coen
\end{large}
\end{center}
\vspace*{\fill}
\noindent
Project Acronym: \cerco{}\\
Project full title: Certified Complexity\\
Proposal/Contract no.: FP7-ICT-2009-C-243881 \cerco{}\\
\clearpage
\pagestyle{myheadings}
\markright{\cerco{}, FP7-ICT-2009-C-243881}
\newpage
\vspace*{7cm}
\paragraph{Abstract}
We describe the encoding in the Calculus of Constructions of the semantics of the CerCo compiler's backend intermediate languages.
The CerCo backend consists of five distinct languages: RTL, RTLntl, ERTL, LTL and LIN.
We describe a process of heavy abstraction of the intermediate languages and their semantics.
We hope that this process will ease the burden of Deliverable D4.4, the proof of correctness for the compiler.
\newpage
\tableofcontents
\newpage
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
% SECTION. %
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
\section{Task}
\label{sect.task}
The Grant Agreement states that Task T4.3, entitled `Formal semantics of intermediate languages' has associated Deliverable D4.3, consisting of the following:
\begin{quotation}
Executable Formal Semantics of back-end intermediate languages: This prototype is the formal counterpart of deliverable D2.1 for the back end side of the compiler and validates it.
\end{quotation}
This report details our implementation of this deliverable.
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
% SECTION. %
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
\subsection{Connections with other deliverables}
\label{subsect.connections.with.other.deliverables}
Deliverable D4.3 enjoys a close relationship with three other deliverables, namely deliverables D2.2, D4.3 and D4.4.
Deliverable D2.2, the O'Caml implementation of a cost preserving compiler for a large subset of the C programming language, is the basis upon which we have implemented the current deliverable.
In particular, the architecture of the compiler, its intermediate languages and their semantics, and the overall implementation of the Matita encodings has been taken from the O'Caml compiler.
Any variations from the O'Caml design are due to bugs identified in the prototype compiler during the Matita implementation, our identification of code that can be abstracted and made generic, or our use of Matita's much stronger type system to enforce invariants through the use of dependent types.
Deliverable D4.2 can be seen as a `sister' deliverable to the deliverable reported on herein.
In particular, where this deliverable reports on the encoding in the Calculus of Constructions of the backend semantics, D4.2 is the encoding in the Calculus of Constructions of the mutual translations of those languages.
As a result, a substantial amount of Matita code is shared between the two deliverables.
Deliverable D4.4, the backend correctness proofs, is the immediate successor of this deliverable.
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
% SECTION. %
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
\section{The backend intermediate languages' semantics in Matita}
\label{sect.backend.intermediate.languages.semantics.matita}
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
% SECTION. %
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
\subsection{Abstracting related languages}
\label{subsect.abstracting.related.languages}
As mentioned in the report for Deliverable D4.2, a systematic process of abstraction, over the O'Caml code, has taken place in the Matita encoding.
In particular, we have merged many of the syntaxes of the intermediate languages (i.e. RTL, ERTL, LTL and LIN) into a single `joint' syntax, which is parameterised by various types.
Equivalent intermediate languages to those present in the O'Caml code can be recovered by specialising this joint structure.
As mentioned in the report for Deliverable D4.2, there are a number of advantages that this process of abstraction brings, from code reuse to allowing us to get a clearer view of the intermediate languages and their structure.
However, the semantics of the intermediate languages allow us to concretely demonstrate this improvement in clarity, by noting that the semantics of the LTL and the semantics of the LIN languages are identical.
In particular, the semantics of both LTL and LIN are implemented in exactly the same way.
The only difference between the two languages is how the next instruction to be interpreted is fetched.
In LTL, this involves looking up in a graph, whereas in LTL, this involves fetching from a list of instructions.
As a result, we see that the semantics of LIN and LTL are both instances of a single, more general language that is parametric in how the next instruction is fetched.
Furthermore, any prospective proof that the semantics of LTL and LIN are identical is now almost trivial, saving a deal of work in Deliverable D4.4.
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
% SECTION. %
%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
\subsection{Type parameters, and their purpose}
\label{subsect.type.parameters.their.purpose}
We mentioned in the Deliverable D4.2 report that all joint languages are parameterised by a number of types, which are later specialised to each distinct intermediate language.
As this parameterisation process is also dependent on designs decisions in the language semantics, we have so far held off summarising the role of each parameter.
We begin the abstraction process with the \texttt{params\_\_} record.
This holds the types of the representations of the different register varieties in the intermediate languages:
\begin{lstlisting}
record params__: Type[1] :=
{
acc_a_reg: Type[0];
acc_b_reg: Type[0];
dpl_reg: Type[0];
dph_reg: Type[0];
pair_reg: Type[0];
generic_reg: Type[0];
call_args: Type[0];
call_dest: Type[0];
extend_statements: Type[0]
}.
\end{lstlisting}
We summarise what these types mean, and how they are used in both the semantics and the translation process:
\begin{center}
\begin{tabular*}{\textwidth}{p{4cm}p{11cm}}
Type & Explanation \\
\hline
\texttt{acc\_a\_reg} & The type of the accumulator A register. In some languages this is implemented as the hardware accumulator, whereas in others this is a pseudoregister.\\
\texttt{acc\_b\_reg} & Similar to the accumulator A field, but for the processor's auxilliary accumulator, B. \\
\texttt{dpl\_reg} & The type of the representation of the low eight bit register of the MCS-51's single 16 bit register, DPL. Can be either a pseudoregister or the hardware DPL register. \\
\texttt{dph\_reg} & Similar to the DPL register but for the eight high bits of the 16-bit register. \\
\texttt{pair\_reg} & Various different `move' instructions have been merged into a single move instruction in the joint language. A value can either be moved to or from the accumulator in some languages, or moved to and from an arbitrary pseudoregister in others. This type encodes how we should move data around the registers and accumulators. \\
\texttt{generic\_reg} & The representation of generic registers (i.e. those that are not devoted to a specific task). \\
\texttt{call\_args} & The actual arguments passed to a function. For some languages this is simply the number of arguments passed to the function. \\
\texttt{call\_dest} & The destination of the function call. \\
\texttt{extend\_statements} & Instructions that are specific to a particular intermediate language, and which cannot be abstracted into the joint language.
\end{tabular*}
\end{center}
As mentioned in the report for Deliverable D4.2, the record \texttt{params\_\_} is enough to be able to specify the instructions of the joint languages:
\begin{lstlisting}
inductive joint_instruction (p: params__) (globals: list ident): Type[0] :=
| COMMENT: String $\rightarrow$ joint_instruction p globals
| COST_LABEL: costlabel $\rightarrow$ joint_instruction p globals
...
| OP1: Op1 $\rightarrow$ acc_a_reg p $\rightarrow$ acc_a_reg p $\rightarrow$ joint_instruction p globals
| COND: acc_a_reg p $\rightarrow$ label $\rightarrow$ joint_instruction p globals
...
\end{lstlisting}
Here, we see that the instruction \texttt{OP1} (a unary operation on the accumulator A) can be given quite a specific type, through the use of the \texttt{params\_\_} data structure.
Joint statements can be split into two subclasses: those who simply pass the flow of control onto their successor statement, and those that jump to a potentially remote location in the program.
Naturally, as some intermediate languages are graph based, and others linearised, the passing act of passing control on to the `successor' instruction can either be the act of following a graph edge in a control flow graph, or incrementing an index into a list.
We make a distinction between instructions that pass control onto their immediate successors, and those that jump elsewhere in the program, through the use of \texttt{succ}, denoting the immediate successor of the current instruction, in the \texttt{params\_} record described below.
\begin{lstlisting}
record params_: Type[1] :=
{
pars__ :> params__;
succ: Type[0]
}.
\end{lstlisting}
The type \texttt{succ} corresponds to labels, in the case of control flow graph based languages, or is instantiated to the unit type for the linearised language, LIN.
Using \texttt{param\_} we can define statements of the joint language:
\begin{lstlisting}
inductive joint_statement (p:params_) (globals: list ident): Type[0] :=
| sequential: joint_instruction p globals $\rightarrow$ succ p $\rightarrow$ joint_statement p globals
| GOTO: label $\rightarrow$ joint_statement p globals
| RETURN: joint_statement p globals.
\end{lstlisting}
Note that in the joint language, instructions are `linear', in that they have an immediate successor.
Statements, on the other hand, consist of either a linear instruction, or a \texttt{GOTO} or \texttt{RETURN} statement, both of which can jump to an arbitrary place in the program. The conditional jump instruction COND is `linear', since it
has an immediate successor, but it also takes an arbitrary location (a label)
to jump to.
For the semantics, we need further parametererised types.
In particular, we parameterise the result and parameter type of an internal function call in \texttt{params0}:
\begin{lstlisting}
record params0: Type[1] :=
{
pars__' :> params__;
resultT: Type[0];
paramsT: Type[0]
}.
\end{lstlisting}
Here, \texttt{resultT} and \texttt{resultT} typically are the (pseudo)registers that store the parameters and result of a function.
We further extend \texttt{params0} with a type for local variables in internal function calls:
\begin{lstlisting}
record params1 : Type[1] :=
{
pars0 :> params0;
localsT: Type[0]
}.
\end{lstlisting}
Again, we expand our parameters with types corresponding to the code representation (either a control flow graph or a list of statements).
Further, we hypothesise a generic method for looking up the next instruction in the graph, called \texttt{lookup}.
Note that \texttt{lookup} may fail, and returns an \texttt{option} type:
\begin{lstlisting}
record params (globals: list ident): Type[1] :=
{
succ_ : Type[0];
pars1 :> params1;
codeT : Type[0];
lookup: codeT $\rightarrow$ label $\rightarrow$ option (joint_statement (mk_params_ pars1 succ_) globals)
}.
\end{lstlisting}
We now have what we need to define internal functions for the joint language.
The first two `universe' fields are only used in the compilation process, for generating fresh names, and do not affect the semantics.
The rest of the fields affect both compilation and semantics.
In particular, we have a description of the result, parameters and the local variables of a function.
Note also that we have lifted the hypothesised \texttt{lookup} function from \texttt{params} into a dependent sigma type, which combines a label (the entry and exit points of the control flow graph or list) combined with a proof that the label is in the graph structure:
\begin{lstlisting}
record joint_internal_function (globals: list ident) (p:params globals) : Type[0] :=
{
joint_if_luniverse: universe LabelTag;
joint_if_runiverse: universe RegisterTag;
joint_if_result : resultT p;
joint_if_params : paramsT p;
joint_if_locals : localsT p;
joint_if_stacksize: nat;
joint_if_code : codeT ... p;
joint_if_entry : $\Sigma$l: label. lookup ... joint_if_code l $\neq$ None ?;
joint_if_exit : $\Sigma$l: label. lookup ... joint_if_code l $\neq$ None ?
}.
\end{lstlisting}
Naturally, a question arises as to why we have chosen to split up the parameterisation into so many intermediate records, each slightly extending earlier ones.
The reason is because some intermediate languages share a host of parameters, and only differ on some others.
For instance, in instantiating the ERTL language, certain parameters are shared with RTL, whilst others are ERTL specific:
\begin{lstlisting}
...
definition ertl_params__: params__ :=
mk_params__ register register register register (move_registers $\times$ move_registers)
register nat unit ertl_statement_extension.
...
definition ertl_params1: params1 := rtl_ertl_params1 ertl_params0.
definition ertl_params: ∀globals. params globals := rtl_ertl_params ertl_params0.
...
definition ertl_statement := joint_statement ertl_params_.
definition ertl_internal_function :=
$\lambda$globals.joint_internal_function ... (ertl_params globals).
\end{lstlisting}
Here, \texttt{rtl\_ertl\_params1} are the common parameters of the ERTL and RTL languages:
\begin{lstlisting}
definition rtl_ertl_params1 := $\lambda$pars0. mk_params1 pars0 (list register).
\end{lstlisting}
The record \texttt{more\_sem\_params} bundles together functions that store and retrieve values in various forms of register:
\begin{lstlisting}
record more_sem_params (p:params_): Type[1] :=
{
framesT: Type[0];
empty_framesT: framesT;
regsT: Type[0];
empty_regsT: regsT;
call_args_for_main: call_args p;
call_dest_for_main: call_dest p;
succ_pc: succ p $\rightarrow$ address $\rightarrow$ res address;
greg_store_: generic_reg p $\rightarrow$ beval $\rightarrow$ regsT $\rightarrow$ res regsT;
greg_retrieve_: regsT $\rightarrow$ generic_reg p $\rightarrow$ res beval;
acca_store_: acc_a_reg p $\rightarrow$ beval $\rightarrow$ regsT $\rightarrow$ res regsT;
acca_retrieve_: regsT $\rightarrow$ acc_a_reg p $\rightarrow$ res beval;
...
dpl_store_: dpl_reg p $\rightarrow$ beval $\rightarrow$ regsT $\rightarrow$ res regsT;
dpl_retrieve_: regsT $\rightarrow$ dpl_reg p $\rightarrow$ res beval;
...
pair_reg_move_: regsT $\rightarrow$ pair_reg p $\rightarrow$ res regsT;
pointer_of_label: label $\rightarrow$ $\Sigma$p:pointer. ptype p = Code
}.
\end{lstlisting}
Here, the fields \texttt{empty\_framesT}, \texttt{empty\_regsT}, \texttt{call\_args\_for\_main} and \texttt{call\_dest\_for\_main} are used for state initialisation.
The field \texttt{succ\_pc} takes an address, and a `successor' label, and returns the address of the instruction immediately succeeding the one at hand.
The fields \texttt{greg\_store\_} and \texttt{greg\_retrieve\_} store and retrieve values from a generic register, respectively.
Similarly, \texttt{pair\_reg\_move} implements the generic move instruction of the joint language.
Here \texttt{framesT} is the type of stack frames, with \texttt{empty\_framesT} an empty stack frame.
The two hypothesised values \texttt{call\_args\_for\_main} and \texttt{call\_dest\_for\_main} deal with problems with the \texttt{main} function of the program, and how it is handled.
In particular, we need to know when the \texttt{main} function has finished executing.
But this is complicated, in C, by the fact that the \texttt{main} function is explicitly allowed to be recursive (disallowed in C++).
Therefore, to understand whether the exiting \texttt{main} function is really exiting, or just recursively calling itself, we need to remember the address to which \texttt{main} will return control once the initial call to \texttt{main} has finished executing.
This is done with \texttt{call\_dest\_for\_main}, whereas \texttt{call\_args\_for\_main} holds the \texttt{main} function's arguments.
We extend \texttt{more\_sem\_params} with yet more parameters via \texttt{more\_sem\_params2}:
\begin{lstlisting}
record more_sem_params2 (globals: list ident) (p: params globals) : Type[1] :=
{
more_sparams1 :> more_sem_params p;
fetch_statement:
genv ... p $\rightarrow$ state (mk_sem_params ... more_sparams1) $\rightarrow$
res (joint_statement (mk_sem_params ... more_sparams1) globals);
...
save_frame:
address $\rightarrow$ nat $\rightarrow$ paramsT ... p $\rightarrow$ call_args p $\rightarrow$ call_dest p $\rightarrow$
state (mk_sem_params ... more_sparams1) $\rightarrow$
res (state (mk_sem_params ... more_sparams1));
pop_frame:
genv globals p $\rightarrow$ state (mk_sem_params ... more_sparams1) $\rightarrow$
res ((state (mk_sem_params ... more_sparams1)));
...
set_result:
list val $\rightarrow$ state (mk_sem_params ... more_sparams1) $\rightarrow$
res (state (mk_sem_params ... more_sparams1));
exec_extended:
genv globals p $\rightarrow$ extend_statements (mk_sem_params ... more_sparams1) $\rightarrow$
succ p $\rightarrow$ state (mk_sem_params ... more_sparams1) $\rightarrow$
IO io_out io_in (trace $\times$ (state (mk_sem_params ... more_sparams1)))
}.
\end{lstlisting}
Here, \texttt{fetch\_statement} fetches the next statement to be executed.
The fields \texttt{save\_frame} and \texttt{pop\_frame} manipulate stack frames.
In particular, \texttt{save\_frame} creates a new stack frame on the top of the stack, saving the destination and parameters of a function, and returning an updated state.
The field \texttt{pop\_frame} destructively pops a stack frame from the stack, returning an updated state.
Further, \texttt{set\_result} saves the result of the function computation, and \texttt{exec\_extended} is a function that executes the extended statements, peculiar to each individual intermediate language.
We bundle \texttt{params} and \texttt{sem\_params} together into a single record.
This will be used in the function \texttt{eval\_statement} which executes a single statement of the joint language:
\begin{lstlisting}
record sem_params2 (globals: list ident): Type[1] :=
{
p2 :> params globals;
more_sparams2 :> more_sem_params2 globals p2
}.
\end{lstlisting}
\noindent
The \texttt{state} record holds the current state of the interpreter:
\begin{lstlisting}
record state (p: sem_params): Type[0] :=
{
st_frms: framesT ? p;
pc: address;
sp: pointer;
isp: pointer;
carry: beval;
regs: regsT ? p;
m: bemem
}.
\end{lstlisting}
Here \texttt{st\_frms} represent stack frames, \texttt{pc} the program counter, \texttt{sp} the stack pointer, \texttt{isp} the internal stack pointer, \texttt{carry} the carry flag, \texttt{regs} the registers (hardware and pseudoregisters) and \texttt{m} external RAM.
Note that we have two stack pointers, as we have two stacks: the physical stack of the MCS-51 microprocessor, and an emulated stack in external RAM.
The MCS-51's own stack is minuscule, therefore it is usual to emulate a much larger, more useful stack in external RAM.
We require two stack pointers as the MCS-51's \texttt{PUSH} and \texttt{POP} instructions manipulate the physical stack, and not the emulated one.
We use the function \texttt{eval\_statement} to evaluate a single joint statement:
\begin{lstlisting}
definition eval_statement:
∀globals: list ident.∀p:sem_params2 globals.
genv globals p $\rightarrow$ state p $\rightarrow$ IO io_out io_in (trace $\times$ (state p)) :=
...
\end{lstlisting}
We examine the type of this function.
Note that it returns a monadic action, \texttt{IO}, denoting that it may have an IO \emph{side effect}, where the program reads or writes to some external device or memory address.
Monads and their use are further discussed in Subsection~\ref{subsect.use.of.monads}.
Further, the function returns a new state, updated by the single step of execution of the program.
Finally, a \emph{trace} is also returned, which records externally observable `events', such as the calling of external functions and the emission of cost labels.
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% SECTION. %
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\subsection{Use of monads}
\label{subsect.use.of.monads}
Monads are a categorical notion that have recently gained an amount of traction in functional programming circles.
In particular, it was noted by Moggi that monads could be used to sequence \emph{effectful} computations in a pure manner.
Here, `effectful computations' cover a lot of ground, from writing to files, generating fresh names, or updating an ambient notion of state.
A monad can be characterised by the following:
\begin{itemize}
\item
A data type, $M$.
For instance, the \texttt{option} type in O'Caml or Matita.
\item
A way to `inject' or `lift' pure values into this data type (usually called \texttt{return}).
We call this function \texttt{return} and say that it must have type $\alpha \rightarrow M \alpha$, where $M$ is the name of the monad.
In our example, the `lifting' function for the \texttt{option} monad can be implemented as:
\begin{lstlisting}
let return x = Some x
\end{lstlisting}
\item
A way to `sequence' monadic functions together, to form another monadic function, usually called \texttt{bind}.
Bind has type $M \alpha \rightarrow (\alpha \rightarrow M \beta) \rightarrow M \beta$.
We can see that bind `unpacks' a monadic value, applies a function after unpacking, and `repacks' the new value in the monad.
In our example, the sequencing function for the \texttt{option} monad can be implemented as:
\begin{lstlisting}
let bind o f =
match o with
None -> None
Some s -> f s
\end{lstlisting}
\item
A series of algebraic laws that relate \texttt{return} and \texttt{bind}, ensuring that the sequencing operation `does the right thing' by retaining the order of effects.
These \emph{monad laws} should also be useful in reasoning about monadic computations in the proof of correctness of the compiler.
\end{itemize}
In the semantics of both front and backend intermediate languages, we make use of monads.
This monadic infrastructure is shared between the frontend and backend languages.
In particular, an `IO' monad, signalling the emission of a cost label, or the calling of an external function, is heavily used in the semantics of the intermediate languages.
Here, the monad's sequencing operation ensures that cost label emissions and function calls are maintained in the correct order.
We have already seen how the \texttt{eval\_statement} function of the joint language is monadic, with type:
\begin{lstlisting}
definition eval_statement:
∀globals: list ident.∀p:sem_params2 globals.
genv globals p $\rightarrow$ state p $\rightarrow$ IO io_out io_in (trace $\times$ (state p)) :=
...
\end{lstlisting}
If we examine the body of \texttt{eval\_statement}, we may also see how the monad sequences effects.
For instance, in the case for the \texttt{LOAD} statement, we have the following:
\begin{lstlisting}
definition eval_statement:
∀globals: list ident. ∀p:sem_params2 globals.
genv globals p $\rightarrow$ state p $\rightarrow$ IO io_out io_in (trace $\times$ (state p)) :=
$\lambda$globals, p, ge, st.
...
match s with
| LOAD dst addrl addrh ⇒
! vaddrh $\leftarrow$ dph_retrieve ... st addrh;
! vaddrl $\leftarrow$ dpl_retrieve ... st addrl;
! vaddr $\leftarrow$ pointer_of_address $\langle$vaddrl,vaddrh$\rangle$;
! v $\leftarrow$ opt_to_res ... (msg FailedLoad) (beloadv (m ... st) vaddr);
! st $\leftarrow$ acca_store p ... dst v st;
! st $\leftarrow$ next ... l st ;
ret ? $\langle$E0, st$\rangle$
\end{lstlisting}
Here, we employ a certain degree of syntactic sugaring.
The syntax
\begin{lstlisting}
...
! vaddrh $\leftarrow$ dph_retrieve ... st addrh;
! vaddrl $\leftarrow$ dpl_retrieve ... st addrl;
...
\end{lstlisting}
is sugaring for the \texttt{IO} monad's binding operation.
We can expand this sugaring to the following much more verbose code:
\begin{lstlisting}
...
bind (dph_retrieve ... st addrh) ($\lambda$vaddrh. bind (dpl_retrieve ... st addrl)
($\lambda$vaddrl. ...))
\end{lstlisting}
Note also that the function \texttt{ret} is implementing the `lifting', or return function of the \texttt{IO} monad.
We believe the sugaring for the monadic bind operation makes the program much more readable, and therefore easier to reason about.
In particular, note that the functions \texttt{dph\_retrieve}, \texttt{pointer\_of\_address}, \texttt{acca\_store} and \texttt{next} are all monadic.
Note, however, that inside this monadic code, there is also another monad hiding.
The \texttt{res} monad signals failure, along with an error message.
The monad's sequencing operation ensures the order of error messages does not get rearranged.
The function \texttt{opt\_to\_res} lifts an option type into this monad, with an error message to be used in case of failure.
The \texttt{res} monad is then coerced into the \texttt{IO} monad, ensuring the whole code snippet typechecks.
\subsection{Memory models}
\label{subsect.memory.models}
Currently, the semantics of the front and backend intermediate languages are built around two distinct memory models.
The frontend languages reuse the CompCert memory model, whereas the backend languages employ a bespoke model tailored to their needs.
This split between the memory models is reflective of the fact that the front and backend language have different requirements from their memory models, to a certain extent.
In particular, the CompCert memory model places quite heavy restrictions on where in memory one can read from.
To read a value in this memory model, you must supply an address, complete with a number of `chunks' to read following that address.
The read is only successful if you attempt to read at a genuine `value boundary', and read the appropriate number of memory chunks for that value.
As a result, with the CompCert memory model you are unable to read the third byte of a 32-bit integer value directly from memory, for instance.
This has some consequences for the CompCert compiler, namely an inability to write a \texttt{memcpy} routine.
However, the CerCo memory model operates differently, as we need to move data `piecemeal' between stacks in the backend of the compiler.
As a result, the bespoke memory model allows one to read data at any memory location, not just on value boundaries.
This has the advantage that we can successfully write a \texttt{memcpy} routine with the CerCo compiler (remembering that \texttt{memcpy} is nothing more than `read a byte, copy a byte' repeated in a loop), an advantage over CompCert.
Right now, the two memory models are interfaced during the translation from RTLabs to RTL.
It is an open question whether we will unify the two memory models, using only the backend, bespoke memory model throughout the compiler, as the CompCert memory model seems to work fine for the frontend, where such byte-by-byte copying is not needed.
However, should we decide to port the frontend to the new memory model, it has been written in such an abstract way that doing so would be relatively straightforward.
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\section{Future work}
\label{sect.future.work}
A few small axioms remain to be closed.
These relate to fetching the next instruction to be interpreted from the control flow graph, or linearised representation, of the language.
Closing these axioms should not be a problem.
Most things related to external function calls are currently axiomatised.
This is due to there being a difficulty with how stackframes are handled with external function calls.
We leave this for further work, due to there being no pressing need to implement this feature at the present time.
There is also, as mentioned, an open problem as to whether the frontend languages should use the same memory model as the backend languages, as opposed to reusing the CompCert memory model.
Should this decision be taken, this will likely be straightforward but potentially time consuming.
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\section{Code listing}
\label{sect.code.listing}
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\subsection{Listing of files}
\label{subsect.listing.files}
Semantics specific files (files relating to language translations ommitted).
Syntax specific files are presented in Table~\ref{table.syntax}.
Here, the O'Caml column denotes the O'Caml source file(s) in the prototype compiler's implementation that corresponds to the Matita script in question.
The ratios are the linecounts of the Matita file divided by the line counts of the corresponding O'Caml file.
These are computed with \texttt{wc -l}, a standard Unix tool.
Individual file's ratios are an over approximation, due to the fact that it's hard to relate an individual O'Caml file to the abstracted Matita code that has been spread across multiple files.
The ratio between total Matita code lines and total O'Caml code lines is more reflective of the compressed and abstracted state of the Matita translation.
Semantics specific files are presented in Table~\ref{table.semantics}.
\begin{landscape}
\begin{table}
\begin{threeparttable}
\begin{tabular}{llrlrl}
Description & Matita & Lines & O'Caml & Lines & Ratio \\
\hline
Abstracted syntax for backend languages & \texttt{joint/Joint.ma} & 173 & N/A & N/A & N/A \\
The syntax of RTLabs & \texttt{RTLabs/syntax.ma} & 73 & \texttt{RTLabs/RTLabs.mli} & 113 & 0.65 \\
The syntax of RTL & \texttt{RTL/RTL.ma} & 49 & \texttt{RTL/RTL.mli} & 120 & 1.85\tnote{a} \\
The syntax of ERTL & \texttt{ERTL/ERTL.ma} & 25 & \texttt{ERTL/ERTL.mli} & 191 & 1.04\tnote{a} \\
The syntax of the abstracted combined LTL and LIN language & \texttt{LIN/joint\_LTL\_LIN.ma} & 10 & N/A & N/A & N/A \\
The specialisation of the above file to the syntax of LTL & \texttt{LTL/LTL.ma} & 10 & \texttt{LTL/LTL.mli} & 104 & 1.86\tnote{b} \\
The specialisation of the above file to the syntax of LIN & \texttt{LIN/LIN.ma} & 17 & \texttt{LIN/LIN.mli} & 88 & 2.27\tnote{b} \\
\end{tabular}
\begin{tablenotes}
\item[a] After inlining of \texttt{joint/Joint.ma}.
\item[b] After inlining of \texttt{joint/Joint\_LTL\_LIN.ma} and \texttt{joint/Joint.ma}.\\
\begin{tabular}{ll}
Total lines of Matita code for the above files:& 347 \\
Total lines of O'Caml code for the above files:& 616 \\
Ratio of total lines:& 0.56
\end{tabular}
\end{tablenotes}
\end{threeparttable}
\caption{Syntax specific files in the intermediate language semantics}
\label{table.syntax}
\end{table}
\end{landscape}
\begin{landscape}
\begin{table}
\begin{threeparttable}
\begin{tabular}{llrlrl}
Description & Matita & Lines & O'Caml & Lines & Ratio \\
\hline
Semantics of the abstracted languages & \texttt{joint/semantics.ma} & 64 & N/A & N/A & N/A \\
Generic utilities used in semantics `joint' languages & \texttt{joint/SemanticUtils.ma} & 77 & N/A & N/A & N/A \\
Semantics of RTLabs & \texttt{RTLabs/semantics.ma} & 223 & \texttt{RTLabs/RTLabsInterpret.ml} & 355 & 0.63 \\
Semantics of RTL & \texttt{RTL/semantics.ma} & 121 & \texttt{RTL/RTLInterpret.ml} & 324 & 1.88\tnote{a} \\
Semantics of ERTL & \texttt{ERTL/semantics.ma} & 125 & \texttt{ERTL/ERTLInterpret.ml} & 504 & 1.22\tnote{a} \\
Semantics of the joint LTL-LIN language & \texttt{LIN/joint\_LTL\_LIN\_semantics.ma} & 64 & N/A & N/A & N/A \\
Semantics of LTL & \texttt{LTL/semantics.ma} & 6 & \texttt{LTL/LTLInterpret.ml} & 416 & 1.25\tnote{b} \\
Semantics of LIN & \texttt{LIN/semantics.ma} & 22 & \texttt{LIN/LINInterpret.ml} & 379 & 1.52\tnote{b}
\end{tabular}
\begin{tablenotes}
\item{a} Includes \texttt{joint/semantics.ma} and \texttt{joint/SemanticUtils.ma}.
\item{b} Includes \texttt{joint/semantics.ma}, \texttt{joint/SemanticUtils.ma} and \texttt{joint/joint\_LTL\_LIN\_semantics.ma}. \\
\begin{tabular}{ll}
Total lines of Matita code for the above files:& 1125 \\
Total lines of O'Caml code for the above files:& 1978 \\
Ration of total lines:& 0.57
\end{tabular}
\end{tablenotes}
\end{threeparttable}
\caption{Semantics specific files in the intermediate language semantics}
\label{table.semantics}
\end{table}
\end{landscape}
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\subsection{Listing of important functions and axioms}
\label{subsect.listing.important.functions.axioms}
We list some important functions and axioms in the backend semantics:
\paragraph{From RTLabs/semantics.ma}
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{make\_initial\_state} & Build an initial state \\
\texttt{eval\_statement} & Evaluate a single RTLabs statement \\
\texttt{is\_final} & Check whether a state is in a `final' configuration \\
\texttt{RTLabs\_exec} & Execute an RTLabs program
\end{tabular*}
\end{center}
\paragraph{From RTL/semantics.ma}
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{rtl\_exec\_extended} & Execute a single step of the RTL language's extended instructions \\
\texttt{rtl\_fullexec} & Execute an RTL program
\end{tabular*}
\end{center}
\paragraph{From ERTL/semantics.ma}
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{ertl\_exec\_extended} & Execute a single step of the ERTL language's extended instructions \\
\texttt{ertl\_fullexec} & Execute an ERTL program
\end{tabular*}
\end{center}
\paragraph{From LTL/semantics.ma}
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{ltl\_fullexec} & Execute an LTL program
\end{tabular*}
\end{center}
\paragraph{From LIN/semantics.ma}
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{lin\_fullexec} & Execute a LIN program
\end{tabular*}
\end{center}
\paragraph{From LIN/joint\_LTL\_LIN\_semantics.ma}
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{ltl\_lin\_exec\_extended} & Execute a single step of the joint LTL-LIN language's extended instructions \\
\texttt{ltl\_lin\_fullexec} & Execute a joint LTL-LIN language program
\end{tabular*}
\end{center}
\paragraph{From joint/semantics.ma}
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{eval\_statement} & Evaluate a single joint language statement \\
\texttt{is\_final} & Check whether a state is in a `final' configuration \\
\texttt{joint\_fullexec} & Execute a joint language program
\end{tabular*}
\end{center}
\paragraph{From joint/SemanticUtils.ma}
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{graph\_fetch\_statement} & Fetch a statement from a control flow graph
\end{tabular*}
\end{center}
\end{document}
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https://www.emis.de/journals/EJC/Volume_14/Abstracts/v14i1n19.abs.tex
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emis.de
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\magnification=1200
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\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
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\nopagenumbers
\noindent
%
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{\bf Peter Hamburger, Penny Haxell and Alexandr Kostochka}
%
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\medskip
\noindent
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{\bf On Directed Triangles in Digraphs}
%
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\vskip 5mm
\noindent
%
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Using a recent result of Chudnovsky, Seymour, and Sullivan, we
slightly improve two bounds related to the Caccetta-Haggkvist
Conjecture. Namely, we show that if $\alpha\geq 0.35312$, then each
$n$-vertex digraph $D$ with minimum outdegree at least $\alpha n$ has
a directed $3$-cycle. If $\beta\geq 0.34564$, then every $n$-vertex
digraph $D$ in which the outdegree and the indegree of each vertex is
at least $\beta n$ has a directed $3$-cycle.
\bye
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https://theanarchistlibrary.org/library/freedom-press-london-freedom-discussion-meetings-oct-1888.tex
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theanarchistlibrary.org
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\title{Freedom Discussion Meetings [Oct, 1888]}
\date{October, 1888}
\author{Freedom Press (London)}
\subtitle{}
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pdftitle={Freedom Discussion Meetings [Oct, 1888]},%
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{\usekomafont{title}{\huge Freedom Discussion Meetings [Oct, 1888]\par}}%
\vskip 1em
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{\usekomafont{author}{Freedom Press (London)\par}}%
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{\usekomafont{date}{October, 1888\par}}%
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\par
\bigskip
The meeting on September 14 was opened by Comrade Marsh with a paper on "Work and Social Utility," the substance of which will be found in another column. There was no direct opposition to the opener's contention that a share in work of social utility, such as providing food, clothing, shelter, etc, ought to be taken by every able-bodied person, and that such work, if fairly shared by all members of the community, would not fall so heavily on any individual as to prevent him or her from exercising special artistic or intellectual capacities at least as fully and as beneficially as they are exercised to-day, when brain and hand labor are almost entirely divided and brain workers are considered as a superior class.
\bigskip
Comrade Kropotkin said that whilst several collectivist schools consider it necessary to make a distinction between different kinds of work, according to the skill required, length of apprenticeship, agreeableness or disagreeableness, and so on, Communist Anarchists are all agreed that no such distinction must be made. We deny the necessity for a special class of brain workers and refuse to recognize an aristocracy in this or any other direction.
We have heard something from Comrade Marsh, he continued, as to the disadvantage to art of class legislation. The same is true with regard to science. Take Medicine. The functions of the doctor and the nurse are now separated. The doctor only comes and looks at the patient once or twice a day and then goes away again; whereas the nurse is continually with the sick person, watching his symptoms, attending to his food and all his needs. It is evident that if each nurse had received a high medical education and each doctor had to perform the functions of the nurse, progress in Medicine would not have been so slow as it has been. In fact, we are now learning more and more that the science of medicine is the science of hygiene, and the art of healing is the art of sanitation. And as nurses have become better educated, it is in their province that the greatest improvements have been initiated.
The President of the British Association spoke of industry helping science, and science industry ; but the help cannot be really effectual as long as brain and hand work are isolated from one another. Formerly scientists were themselves handicraftsmen and themselves inventors. The great astronomer Galileo made his own telescopes, and now we see that the great inventions of to-day, telegraphs, sewing machines telephones, electric lighting, and so on, spring not from the professional scientists, but from practical men like Morse or Edison and the many unknown handicraftsmen who work with them. In fact the numberless inventions of uneducated working people show that work with one's own hand is the great stimulus to inventive genius. For instance, one of the most marvelous machines in existence, that used at Nottingham in lace-making, was originally invented by a drunkard to get money for more gin after he had broken the bottle over his wife's head ; and it has been perfected by the ingenuity of three generations of workmen.
In the interest of science itself it is desirable that scientist and hand worker be one and the same person ; and this is no unattainable state of things even in our present society. At Moscow there is a great college where the students learn pure mathematics and practical mechanics side by side, and the experience there shows that at twenty years of age the young man who can construct a steam-engine with his own hands is able to pass a stiff examination in the higher branches of mathematics as successfully as the youth who has done nothing but brain work all his life.
But granted that this is true for average men and women, -hall we under such a system of mixed brain and band work have great geniuses like Darwin? Darwin's \emph{whole life} was spent in laborious experiment arid research. Yes; but why was this necessary? Thirty years before Darwin's great book on the "Origin of Species" was published, when he as a young man returned from his voyage in the Beagle, be had already framed the hypothesis which has cast such a flood of light on modern thought. What was needed, was to collect facts to prove or disprove it. To verify his hypothesis he had to spend \emph{thirty} years in collecting materials, because he was forced to work almost single-handed. But suppose we had all received a good scientific education and Darwin had been able to make appeal to a wide circle of intelligent and accurately trained minds to help him, then all that information could have been collected in five years.
You see how much to the advantage of scientific progress it would be that the immediately necessary work of the world should be shared by all, so that all should use both hands and brains, and all enjoy a certain amount of leisure.
Is it \emph{practical}, to spend our time in discussing the best lines for the organization of labor after the Social Revolution?
We are nearly all agreed that the time is approaching when there will be again wide-spread popular movements, such as those which have occurred in the past. In the past these movements have usually ended in a change of rulers, and in the people expressing their desires to the new government with more or less earnestness and intelligence, proportionate to those desires having been thought out with more or less clearness beforehand. The people have never yet been so thoroughly convinced of what they needed as to venture to act directly for themselves for any length of time. Now we are, trying to prepare for the next great popular movement by leading as many men and women as possible to think out clearly what they want and make their minds ready to do it for themselves as soon as the chance occurs. Past revolutions have done so little because the workers were prepared to change so little. In the Commune of Paris there was nothing to prevent the workmen from taking possession of the houses and factories, if they had wished it; but before the outbreak occurred their leaders had always told them that it was not practical to think about expropriation and Socialism. The practical thing was to discuss the separation of church and state, the reduction of rents, the evils of night work in bakeries, and such like comparatively trivial matters; so when the chance to act came, it was these small palliatives and no radical changes that the working people of Paris sought and obtained.
The practical people are those who try intelligently to understand what is likely to happen and who prepare for it.
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The Anarchist Library
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
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\end{center}
\strut
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\begin{center}
Freedom Press (London)
Freedom Discussion Meetings [Oct, 1888]
October, 1888
\bigskip
Freedom: A Journal of Anarchist Socialism, Vol. 3, No. 25, online source \href{http://www.revoltlib.com/?id=3004}{RevoltLib.com}, retrieved on April 12, 2020.
\bigskip
\textbf{theanarchistlibrary.org}
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\documentclass[titlepage,letterpaper]{article}
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\author{Matthias Blume \\
Research Institute for Mathematical Sciences \\
Kyoto University}
\title{{\bf CM}\\
The SML/NJ Compilation and Library Manager \\
{\it\small (for SML/NJ version \smlmj.\smlmn~and later)} \\
User Manual}
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\begin{document}
\bibliographystyle{alpha}
\maketitle
\pagebreak
\tableofcontents
\pagebreak
\section{Introduction}
This manual describes a new implementation of CM, the ``Compilation
and Library Manager'' for Standard ML of New Jersey (SML/NJ). Like its
previous version, CM is in charge of managing separate compilation and
facilitates access to stable libraries.
Programming projects that use CM are typically composed of separate
{\em libraries}. Libraries are collections of ML compilation units
and themselves can be internally sub-structured using CM's notion of
{\em groups}. Using libraries and groups, programs can be viewed as a
{\em hierarchy of modules}. The organization of large projects tends
to benefit from this approach~\cite{blume:appel:cm99}.
CM uses {\em cutoff} techniques~\cite{tichy94} to minimize
recompilation work and provides automatic dependency analysis to free
the programmer from having to specify a detailed module dependency
graph by hand~\cite{blume:depend99}.
This new version of CM emphasizes {\em working with libraries}. This
contrasts with the previous implementation where the focus was on
compilation management while libraries were added as an afterthought.
Beginning now, CM takes a very library-centric view of the world. In
fact, the implementation of SML/NJ itself has been restructured to
conform to this approach.
\section{The CM model}
A CM library is a (possibly empty) collection of ML source files and
may also contain references to other libraries. Each library comes
with an explicit export interface which lists all toplevel-defined
symbols of the library that shall be exported to its clients. A
library is described by the contents of its {\em description
file}.\footnote{The description file may also contain references to
input files for {\em tools} like {\tt ml-lex} or {\tt ml-yacc} that
produce ML source files. See section~\ref{sec:tools}.}
\noindent Example:
\begin{verbatim}
Library
signature BAR
structure Foo
is
bar.sig
foo.sml
helper.sml
$/basis.cm (* or just $basis.cm *)
\end{verbatim}
This library exports two definitions, one for a structure named {\tt
Foo} and one for a signature named {\tt BAR}. The specification for
such exports appear between the keywords {\tt Library} and {\tt is}.
The {\em members} of the library are specified after the keyword {\tt
is}. Here we have three ML source files ({\tt bar.sig}, {\tt
foo.sml}, and {\tt helper.sml}) as well as a reference to one external
library ({\tt \$/basis.cm}). The entry {\tt \$/basis.cm} typically denotes
the description file for the {\it Standard ML Basis
Library}~\cite{reppy99:basis}; most programs will want to list it in
their own description file(s).
\subsection{Library descriptions}
Members of a library do not have to be listed in any particular order
since CM will automatically calculate the dependency graph. Some
minor restrictions on the source language are necessary to make this
work:
\begin{enumerate}
\item All top-level definitions must be {\em module} definitions
(structures, signatures, functors, or functor signatures). In other
words, there can be no top-level type-, value-, or infix-definitions.
\item For a given symbol, there can be at most one ML source file per
library (or---more correctly---one file per library component; see
Section~\ref{sec:groups}) that defines the symbol at top level.
\item If more than one sub-library or sub-group is exporting the same
symbol, then the definition (i.e., the ML source file that actually
defines the symbol) must be identical in all cases.
\label{rule:diamond}
\item The use of ML's {\bf open} construct is not permitted at the top
level of ML files compiled by CM. (The use is still ok at the
interactive top level.)
\end{enumerate}
Note that these rules do not require the exports of sub-groups or
sub-libraries to be distinct from the exports of ML source files in
the current library or group. If an ML source file re-defines an
imported name, then the disambiguating rule is that the definition
from the ML source takes precedence over the definition imported from
the group or library.
Rule~\ref{rule:diamond} may come as a bit of a surprise considering
that each ML source file can be a member of at most one group or
library (see section~\ref{sec:multioccur}). However, it is indeed
possible for two libraries to export the ``same'' definition provided
they both import that definition from a third library. For example,
let us assume that {\tt a.cm} exports a structure {\tt X} which was
defined in {\tt x.sml}---one of {\tt a.cm}'s members. Now, if both
{\tt b.cm} and {\tt c.cm} re-export that same structure {\tt X} after
importing it from {\tt a.cm}, it is legal for a fourth library {\tt
d.cm} to import from both {\tt b.cm} and {\tt c.cm}.
The full syntax for library description files also includes provisions
for a simple ``conditional compilation'' facility (see
Section~\ref{sec:preproc}), for access control (see
Section~\ref{sec:access}), and it accepts ML-style nestable comments
delimited by \verb|(*| and \verb|*)|.
\subsection{Name visibility}
In general, all definitions exported from members (i.e., ML source
files, subgroups and sublibraries) of a library are visible in all ML
source files of that library. The source code in those source files
can refer to them directly without further qualification. Here,
``exported'' means either a top-level definition within an ML source
file or a definition listed in a sublibrary's export list.
If a library is structured into library components using {\em groups}
(see Section~\ref{sec:groups}), then---as far as name visibility is
concerned---each component (group) is treated like a separate library.
Cyclic dependencies among libraries, library components, or ML source
files within a library are detected and flagged as errors.
\subsection{Groups}
\label{sec:groups}
CM's group model eliminates a whole class of potential naming problems
by providing control over name spaces for program linkage. The group
model in full generality sometimes requires bindings to be renamed at
the time of import. As has been described
separately~\cite{blume:appel:cm99}, in the case of ML this can also be
achieved using ``administative'' libaries, which is why CM can get
away with not providing more direct support for renaming.
However, under CM, the term ``library'' does not only mean namespace
management (as it would from the point of view of the pure group
model) but also refers to actual file system objects (e.g., CM
description files and stable library files). It would be inconvenient
if name resolution problems would result in a proliferation of
additional library files. Therefore, CM also provides the notion of
groups (or: ``library components''). Name resolution for groups works
like name resolution for entire libraries, but grouping is entirely
internal to each library.
When a library is {\em stabilized} (via {\tt CM.stabilize} -- see
Section~\ref{sec:api}), the entire library is compiled to a single
file (hence groups do not result in separate stable files).
During development, each group has its own description file which will
be referred to by the surrounding library or by other groups of that
library. The syntax of group description files is the same as that of
library description files with the following exceptions:
\begin{itemize}
\item The initial keyword {\tt Library} is replaced with {\tt Group}.
It is followed by the name of the surrounding library's description
file in parentheses.
\item The export list can be left empty, in which case CM will provide
a default export list: all exports from ML source files plus all
exports from subcomponents of the component. (Note that this does not
include the exports of other libraries.)
\item There are some small restrictions on access control
specifications (see Section~\ref{sec:access}).
\end{itemize}
As an example, let us assume that
{\tt foo-utils.cm} contains the following text:
%note: emacs gets temporarily confused by the single dollar
\begin{verbatim}
Group (foo-lib.cm)
is
set-util.sml
map-util.sml
$/basis.cm
\end{verbatim}
This description defines group {\tt foo-utils.cm} to have the
following properties:
\begin{itemize}
\item it is a component of library {\tt foo-lib.cm} (meaning that only
foo-lib.cm itself or other groups thereof may list {\tt foo-utils.cm} as one
of their members)
\item {\tt set-utils.sml} and {\tt map-util.sml} are ML source files
belonging to this component
\item exports from the Standard Basis Library are available when
compiling these ML source files
\item since the export list has been left blank, the only (implicitly
specified) exports of this component are the top-level definitions in
its ML source files
\end{itemize}
With this, the library description file {\tt foo-lib.cm} could list
{\tt foo-utils.cm} as one of its members:
\begin{verbatim}
Library
signature FOO
structure Foo
is
foo.sig
foo.sml
foo-utils.cm
$/basis.cm
\end{verbatim}
%note: emacs should be sufficiently un-confused again by now
No harm is done if {\tt foo-lib.cm} does not actually mention {\tt
foo-utils.cm}. In this case it could be that\linebreak {\tt
foo-utils.cm} is mentioned indirectly via a chain of other components
of {\tt foo-lib.cm}. The other possibility is that it is not
mentioned at all (in which case CM would never know about it, so it
cannot complain).
\subsection{Multiple occurences of the same member}
\label{sec:multioccur}
The following rules apply to multiple occurences of the same ML source
file, the same library, or the same group within a program:
\begin{itemize}
\item Within the same description file, each member can be specified
at most once.
\item Libraries can be referred to freely from as many other groups or
libraries as the programmer desires.
\item A group cannot be used from outside the uniquely defined library
(as specified in its description file) of which it is a component.
However, within that library it can be referred to from arbitrarily
many other groups.
\item The same ML source file cannot appear more than once. If an ML
source file is to be referred to by multiple clients, it must first be
``wrapped'' into a library (or---if all references are from within the
same library---a group).
\end{itemize}
\subsection{Top-level groups}
Mainly to facilitate some superficial backward-compatibility, CM also
allows groups to appear at top level, i.e., outside of any library.
Such groups must omit the parenthetical library specification and then
cannot also be used within libraries. One could think of the top level
itself as a ``virtual unnamed library'' whose components are these
top-level groups.
\section{Naming objects in the file system}
\subsection{Motivation}
File naming has been an area notorious for its problems and was the
cause of most of the gripes from CM's users. With this in mind, CM
now takes a different approach to file name resolution.
The main difficulty lies in the fact that files or even whole
directories may move after CM has already partially (but not fully)
processed them. For example, this happens when the {\em autoloader}
(see Section~\ref{sec:autoload}) has been invoked and the session
(including CM's internal state) is then frozen (i.e., saved to a file)
via {\tt SMLofNJ.exportML}. The new CM is now able to resume such a
session even when operating in a different environment, perhaps on a
different machine with different file system mounted, or a different
location of the SML/NJ installation.
To make this possible, CM provides a configurable mechanism for
locating file system objects. Moreover, it invokes this mechanism
always as late as possible and is prepared to re-invoke it after the
configuration changes.
\subsection{Basic rules}
\label{sec:basicrules}
CM uses its own ``standard'' syntax for pathnames which for the most
part happens to be the same as the one used by most Unix-like systems:
path name components are separated by ``{\bf /}'', paths beginning
with ``{\bf /}'' are considered {\em absolute} while other paths are
{\em relative}. There is an important third form of standard paths:
{\em anchored} paths. Anchored paths always start with ``{\bf \$}''.
Since this standard syntax does not cover system-specific aspects such
as volume names, it is also possible to revert to ``native'' syntax by
enclosing the name in double-quotes. Of course, description files
that use path names in native syntax are not portable across operating
systems.
\begin{description}
\item[Absolute pathnames] are resolved in the usual manner
specific to the operating system. However, it is advisable to avoid
absolute pathnames because they are certain to ``break'' if the
corresponding file moves to a different location.
\item[Relative pathnames that occur in some CM description file] whose
name is {\it path}{\tt /}{\it file}{\tt .cm} will be resolved relative
to {\it path}, i.e., relative to the directory that contains the
description file.
\item[Relative pathnames that have been entered interactively,] for
example as an argument to one of CM's interface functions,
will be resolved in the OS-specific manner, i.e., relative to the
current working directory. However, CM will internally represent the
name in such a way that it remembers the corresponding working
directory. Should the working directory change during an ongoing CM
session while there still is a reference to the name, then CM will
switch its mode of operation and prepend the path of the original
working directory. As a result, two names specified using identical
strings but at different times when different working directories were
in effect will be kept distinct and continue to refer to the file
system locations that they referred to when they were first seen.
\item[Anchored paths] consist of an anchor name (of non-zero length)
and a non-empty list of additional arcs. The name is enclosed by
the path's leading {\bf \$} on the left and the path's first {\bf /}
on the right. The list of arcs follows the first {\bf /}. As with
all standard paths, the arcs themselves are also separated by {\bf /}.
An error is signalled if the anchor name is not known to CM.
If $a$ is a know anchor name currently bound to some directory name
$d$, then the standard path {\tt \$}$a${\tt /}$p$ (where $p$ is a list
of arcs) refers to $d${\tt /}$p$. The frequently occuring case where
$a$ coincides with the first arc of $p$ can be abbreviated as {\tt
\$/}$p$.
\end{description}
\subsection{Anchor environments}
\label{sec:anchor:env}
Anchor names are resolved in the {\em anchor environment} that is in
effect at the time the anchor is read.
The basis for all anchor environments is the {\em root environment}.
Conceptually, the root environments is a fixed mapping that binds
every possible anchor to a mutable location. The location can store a
native directory name or can be marked ``undefined''. Most locations
initially start out undefined. The contents of each location is
configurable (see Section~\ref{sec:anchor:config}).
At the time a CM description file $a${\tt .cm} refers to another
library's or library component's description file $b${\tt .cm}, it can
augment the current anchor environment with new bindings. The new
bindings are in effect while $b${\tt .cm} (including any description
files {\it it}\/ mentions!) is being processed. If a new binding
binds an anchor name that was already bound in the current
environment\footnote{which is technically always the case given our
explanation of the root environment}, then the old binding is being
hidden. The effect is scoping for anchor names.
Using CM's {\em tool parameter} mechanism (see
Section~\ref{sec:toolparam}), a new binding is specified as a pair of
anchor name and anchor value. The value has the form of another path
name (standard or native). Example:
\begin{verbatim}
a.cm (bind:(anchor:lib value:$mystuff/a-lib)
bind:(anchor:support value:$lib)
bind:(anchor:utils value:/home/bob/stuff/ML/utils))
\end{verbatim}
As shown in this example, it is perfectly legal for the specification
of the value to involve the use of another anchor. That anchor will
be resolved in the original anchor environment. Thus, a path anchored
at {\tt \$lib} in {\tt a.cm} will be resolved using the binding for
{\tt \$mystuff} that is currently in effect. The point here is that a
re-configuration of the root environment that affects {\tt \$mystuff}
now also affects how {\tt \$lib} is resolved as it occurs within {\tt
a.cm}.
The list of {\tt bind:}-directives is processed ``in parallel,'' which
means that {\tt \$support} is {\em not} being bound to\linebreak {\tt
\$mystuff/a-lib/asupport} but will refer to the original meaning of
{\tt \$lib}.
The example also demonstrates that {\tt value:}-paths can be single
anchors. In other words, the restriction that there has to be at least
one arc after the anchor does not apply here.
\subsection{Anchor configuration}
\label{sec:anchor:config}
Anchor configuration is concerned with the values that are stored in
the root anchor environment. At startup time, the root environment is
initialized by reading two configuration files: an
installation-specific one and a user-specific one. After that, the
contents of root locations can be maintained using CM's interface
functions {\tt CM.Anchor.anchor} and {\tt CM.Anchor.reset} (see
Section~\ref{sec:api}).
The default location of the installation-specific configuration file
is {\tt /usr/lib/smlnj-pathconfig}. However, normally this default
gets replaced (via an environment variable named {\tt
CM\_PATHCONFIG\_DEFAULT}) at installation time by a path pointing to
wherever the installation actually puts the configuration file.
The user can specify a new location at startup time using the
environment variable {\tt CM\_PATHCONFIG}.
The default location of the user-specific configuration file is {\tt
.smlnj-pathconfig} in the user's home directory (which must be given
by the {\tt HOME} environment variable). At startup time, this
default can be overridden by a fixed location which must be given as
the value of the environment variable {\tt CM\_LOCAL\_PATHCONFIG}.
The syntax of all configuration files is identical. Lines are
processed from top to bottom. White space divides lines into tokens.
\begin{itemize}
\item A line with exactly two tokens associates an anchor (the first
token) with a directory in native syntax (the second token). Neither
anchor nor directory name may contain white space and the anchor
should not contain a {\bf /}. If the directory name is a relative
name, then it will be expanded by prepending the name of the directory
that contains the configuration file.
\item A line containing exactly one token that is the name of an
anchor cancels any existing association of that anchor with a
directory.
\item A line with a single token that consists of a single minus sign
{\bf -} cancels all existing anchors. This typically makes sense only
at the beginning of the user-specific configuration file and
erases any settings that were made by the installation-specific
configuration file.
\item Lines with no token (i.e., empty lines) will be silently ignored.
\item Any other line is considered malformed and will cause a warning
but will otherwise be ignored.
\end{itemize}
\section{Using CM}
\subsection{Structure CM}
\label{sec:api}
Functions that control CM's operation are accessible as members of a
structure named {\tt CM}. This structure itself is exported from a
library called {\tt \$smlnj/cm/full.cm} (or, alternatively, {\tt
\$smlnj/cm.cm}). This library is pre-registered for auto-loading at
the interactive top level.
Other libraries can exploit CM's functionality simply by putting a
{\tt \$smlnj/cm/full.cm} entry into their own description file.
Section~\ref{sec:dynlink} shows one interesting use of this feature.
Here is a description of all members:
\subsubsection*{Compiling}
Two main activities when using CM are to compile ML source code and to
build stable libraries:
\begin{verbatim}
val recomp : string -> bool
val stabilize : bool -> string -> bool
\end{verbatim}
{\tt CM.recomp} takes the name of a program's ``root'' description
file and compiles or recompiles all ML source files that are necessary
to provide definitions for the root library's export list. ({\em
Note:} The difference to {\tt CM.make} is that no linking takes
place.)
{\tt CM.stabilize} takes a boolean flag and then the name of a library
and {\em stabilizes} this library. A library is stabilized by writing
all information pertaining to it, including all of its library
components (i.e., subgroups), into a single file. Sublibraries do not
become part of the stabilized library; CM records stub entries for them.
When a stabilized library is used in other programs, all members of
the library are guaranteed to be up-to-date; no dependency analysis
work and no recompilation work will be necessary. If the boolean flag
is {\tt false}, then all sublibraries of the library must already be
stable. If the flag is {\tt true}, then CM will recursively stabilize
all libraries reachable from the given root.
After a library has been stabilized it can be used even if none of its
original sources---including the description file---are present.
The boolean result of {\tt CM.recomp} and {\tt CM.stabilize} indicates
success or failure of the operation ({\tt true} = success).
\subsubsection*{Linking}
In SML/NJ, linking means executing top-level code (i.e., module
creation and initialization code) of each compilation unit. The
resulting bindings can then be registered at the interactive top
level.
\begin{verbatim}
val make : string -> bool
val autoload : string -> bool
\end{verbatim}
{\tt CM.make} first acts like {\tt CM.recomp}. If the
(re-)compilation is successful, then it proceeds by linking all
modules that require linking. Provided there are no link-time errors,
it finally introduces new bindings at top level.
During the course of the same {\tt CM.make}, the code of each
compilation module that is reachable from the root will be executed at
most once. Code in units that are marked as {\it private} (see
Section~\ref{sec:sharing}) will be executed exactly once. Code in
other units will be executed only if the unit has been recompiled
since it was executed last time or if it depends on another
compilation unit whose code has been executed since.
In effect, different invocations of {\tt CM.make} (and {\tt
CM.autoload}) will share dynamic state created at link time as much as
possible unless the compilation units in question have been explicitly
marked private.
{\tt CM.autoload} acts like {\tt CM.make}, only ``lazily''. See
Section~\ref{sec:autoload} for more information.
As before, the result of {\tt CM.make} indicates success or failure of
the operation. The result of {\tt CM.autoload} indicates success or
failure of the {\em registration}. (It does not know yet whether
loading will actually succeed.)
\subsubsection*{Registers}
Several internal registers control the operation of CM. A register of
type $T$ is accessible via a variable of type $T$ {\tt controller},
i.e., a pair of {\tt get} and {\tt set} functions.\footnote{The type
constructor {\tt controller} is defined as part of {\tt structure
CM}.} Any invocation of the corresponding {\tt get} function reads
the current value of the register. An invocation of the {\tt set}
function replaces the current value with the argument given to {\tt
set}.
Controllers are members of {\tt CM.Control}, a sub-structure of
structure {\tt CM}.
\begin{verbatim}
type 'a controller = { get: unit -> 'a, set: 'a -> unit }
structure Control : sig
val verbose : bool controller
val debug : bool controller
val keep_going : bool controller
val parse_caching : int controller
val warn_obsolete : bool controller
val conserve_memory : bool controller
end
\end{verbatim}
{\tt CM.Control.verbose} can be used to turn off CM's progress
messages. The default is {\em true} and can be overriden at startup
time by the environment variable {\tt CM\_VERBOSE}.
In the case of a compile-time error {\tt CM.Contol.keep\_going}
instructs the {\tt CM.recomp} phase to continue working on parts of
the dependency graph that are not related to the error. (This does
not work for outright syntax errors because a correct parse is needed
before CM can construct the dependency graph.) The default is {\em
false}, meaning ``quit on first error'', and can be overriden at
startup by the environment variable {\tt CM\_KEEP\_GOING}.
{\tt CM.Control.parse\_caching} sets a limit on how many parse trees
are cached in main memory. In certain cases CM must parse source
files in order to be able to calculate the dependency graph. Later,
the same files may need to be compiled, in which case an existing
parse tree saves the time to parse the file again. Keeping parse
trees can be expensive in terms of memory usage. Moreover, CM makes
special efforts to avoid re-parsing files in the first place unless
they have actually been modified. Therefore, it may not make much
sense to set this value very high. The default is {\em 100} and can
be overriden at startup time by the environment variable {\tt
CM\_PARSE\_CACHING}.
This version of CM uses an ML-inspired syntax for expressions in its
conditional compilation subsystem (see Section~\ref{sec:preproc}).
However, for the time being it will accept most of the original
C-inspired expressions but produces a warning for each occurrence of
an old-style operator. {\tt CM.Control.warn\_obsolete} can be used to
turn these warnings off. The default is {\em true}, meaning ``warnings
are issued'', and can be overriden at startup time by the environment
variable {\tt CM\_WARN\_OBSOLETE}.
{\tt CM.Control.debug} can be used to turn on debug mode. This
currently has the effect of dumping a trace of the master-slave
protocol for parallel and distributed compilation (see
Section~\ref{sec:parmake}) to TextIO.stdOut. The default is {\em
false} and can be overriden at startup time by the environment
variable {\tt CM\_DEBUG}.
Using {\tt CM.Control.conserve\_memory}, CM can be told to be slightly
more conservative with its use of main memory at the expense of
occasionally incurring additional input from stable library files.
This does not save very much and, therefore, is normally turned off.
The default ({\em false}) can be overridden at startup by the
environment variable {\tt CM\_CONSERVE\_MEMORY}.
\subsubsection*{Path anchors}
Structure {\tt CM} also provides functions to explicitly manipulate
the path anchor configuration. These functions are members of
structure {\tt CM.Anchor}.
\begin{verbatim}
structure Anchor : sig
val anchor : string -> string option controller
val reset : unit -> unit
end
\end{verbatim}
{\tt CM.Anchor.anchor} returns a pair of {\tt get} and {\tt set}
functions that can be used to query and modify the status of the named
anchor. Note that the {\tt get}-{\tt set}-pair operates over type
{\tt string option}; a value of {\tt NONE} means that the anchor is
currently not bound (or, in the case of {\tt set}, that it is being
cancelled). The (optional) string given to {\tt set} must be a
directory name in native syntax ({\em without} trailing arc separator,
e.g., {\bf /} in Unix). If it is specified as a relative path name,
then it will be expanded by prepending the name of the current working
directory.
{\tt CM.Anchor.reset} erases the entire existing path configuration.
After a call of this function has completed, all root environment
locations are marked as being ``undefined''.
\subsubsection*{Setting CM variables}
CM variables are used by the conditional compilation system (see
Section~\ref{sec:cmvars}). Some of these variables are predefined,
but the user can add new ones and alter or remove those that already
exist.
\begin{verbatim}
val symval : string -> int option controller
\end{verbatim}
Function {\tt CM.symval} returns a {\tt get}-{\tt set}-pair for the
symbol whose name string was specified as the argument. Note that the
{\tt get}-{\tt set}-pair operates over type {\tt int option}; a value
of {\tt NONE} means that the variable is not defined.
\noindent Examples:
\begin{verbatim}
#get (CM.symval "X") (); (* query value of X *)
#set (CM.symval "Y") (SOME 1); (* set Y to 1 *)
#set (CM.symval "Z") NONE; (* remove definition for Z *)
\end{verbatim}
Some care is necessary as {\tt CM.symval} does not check whether the
syntax of the argument string is valid. (However, the worst thing
that could happen is that a variable defined via {\tt CM.symval} is
not accessible\footnote{from within CM's description files} because
there is no legal syntax to name it.)
\subsubsection*{Library registry}
\label{sec:libreg}
To be able to share associated data structures such as symbol tables
and dependency graphs, CM maintains an internal registry of all stable
libraries that it has encountered during an ongoing interactive
session. The {\tt CM.Library} sub-structure of structure {\tt CM}
provides access to this registry.
\begin{verbatim}
structure Library : sig
type lib
val known : unit -> lib list
val descr : lib -> string
val osstring : lib -> string
val dismiss : lib -> unit
val unshare : lib -> unit
end
\end{verbatim}
{\tt CM.Library.known}, when called, produces a list of currently
known stable libraries. Each such library is represented by an
element of the abstract data type {\tt CM.Library.lib}.
{\tt CM.Library.descr} extracts a string describing the location of
the CM description file associated with the given library. The syntax
of this string is the same as that being used by CM's master-slave
protocol (see section~\ref{sec:pathencode}).
{\tt CM.Library.osstring} produces a string denoting the given
library's description file using the underlying operating system's
native pathname syntax. In other words, the result of a call to {\tt
CM.Library.osstring} is suitable as an argument to {\tt
TextIO.openIn}.
{\tt CM.Library.dismiss} is used to remove a stable library from CM's
internal registry. Although removing a library from the registry may
recover considerable amounts of main memory, doing so also eliminates
any chance of sharing the associated data structures with later
references to the same library. Therefore, it is not always in the
interest of memory-conscious users to use this feature.
While dependency graphs and symbol tables need to be reloaded when a
previously dismissed library is referenced again, the sharing of
link-time state created by this library is {\em not} affected.
(Link-time state is independently maintained in a separate data
structure. See the discussion of {\tt CM.unshare} below.)
{\tt CM.Library.unshare} is used to remove a stable library from CM's
internal registry, and---at the same time---to inhibit future sharing
with its existing link-time state. Any future references to this
library will see newly created state (which will then be properly
shared again). ({\bf Warning:} {\it This feature is not the preferred
way of creating unshared state; use functors for that. However, it
can come in handy when two different (and perhaps incompatible)
versions of the same library are supposed to coexist---especially if
one of the two versions is used by SML/NJ itself. Normally, only
programmers working on SML/NJ's compiler are expected to be using this
facility.})
\subsubsection*{Internal state}
For CM to work correctly, it must maintain an up-to-date picture of
the state of the surrounding world (as far as that state affects CM's
operation). Most of the time, this happens automatically and should be
transparent to the user. However, occasionally it may become
necessary to intervene expliticly.
Access to CM's internal state is facilitated by members of the {\tt
CM.State} structure.
\begin{verbatim}
structure State : sig
val pending : unit -> string list
val synchronize : unit -> unit
val reset : unit -> unit
end
\end{verbatim}
{\tt CM.State.pending} produces a list of strings, each string naming
one of the symbols that are currently registered (i.e., ``virtually
bound'') but not yet resolved by the autoloading mechanism.
{\tt CM.State.synchronize} updates tables internal to CM to reflect
changes in the file system. In particular, this will be necessary
when the association of file names to ``file IDs'' (in Unix: inode
numbers) changes during an ongoing session. In practice, the need for
this tends to be rare.
{\tt CM.State.reset} completely erases all internal state in CM. To
do this is not very advisable since it will also break the association
with pre-loaded libraries. It may be a useful tool for determining
the amount of space taken up by the internal state, though.
\subsubsection*{Compile servers}
On Unix-like systems, CM supports parallel compilation. For computers
connected using a LAN, this can be extended to distributed compilation
using a network file system and the operating system's ``rsh''
facility. For a detailed discussion, see Section~\ref{sec:parmake}.
Sub-structure {\tt CM.Server} provides access to and manipulation of
compile servers. Each attached server is represented by a value of
type {\tt CM.Server.server}.
\begin{verbatim}
structure Server : sig
type server
val start : { name: string,
cmd: string * string list,
pathtrans: (string -> string) option,
pref: int } -> server option
val stop : server -> unit
val kill : server -> unit
val name : server -> string
end
\end{verbatim}
CM is put into ``parallel'' mode by attaching at least one compile
server. Compile servers are attached using invocations of {\tt
CM.Server.start}. The function takes the name of the server (as an
arbitrary but unique string) ({\tt name}), the Unix command used to
start the server in a form suitable as an argument to {\tt
Unix.execute} ({\tt cmd}), an optional ``path transformation
function'' for converting local path names to remote pathnames ({\tt
pathtrans}), and a numeric ``preference'' value that is used to choose
servers at times when more than one is idle ({\tt pref}). The
optional result is the handle representing the successfully attached
server.
An existing server can be shut down and detached using {\tt
CM.Server.stop} or {\tt CM.Server.kill}. The argument in either case
must be the result of an earlier call to {\tt CM.Server.start}.
Function {\tt CM.Server.stop} uses CM's master-slave protocol to
instruct the server to shut down gracefully. Only if this fails it
may become necessary to use {\tt CM.Server.kill}, which will send a
Unix TERM signal to destroy the server.
Given a server handle, function {\tt CM.Server.name} returns the
string that was originally given to the call of\linebreak {\tt
CM.Server.start} used to created the server.
\subsubsection*{Plug-ins}
As an alternative to {\tt CM.make} or {\tt CM.autoload}, where the
main purpose is to subsequently be able to access the library from
interactively entered code, one can instruct CM to load libraries
``for effect''.
\begin{verbatim}
val load_plugin : string -> bool
\end{verbatim}
Function {\tt CM.load\_plugin} acts exactly like {\tt CM.make} except
that even in the case of success no new symbols will be bound in the
interactive top-level environment. That means that link-time
side-effects will be visible, but none of the exported definitions
become available. This mechanism can be used for ``plug-in'' modules:
a core library provides hooks where additional functionality can be
registered later via side-effects; extensions to this core are
implemented as additional libraries which, when loaded, register
themselves with those hooks. By using {\tt CM.load\_plugin} instead
of {\tt CM.make}, one can avoid polluting the interactive top-level
environment with spurious exports of the extension module.
CM itself uses plug-in modules in its member-class subsystem (see
section~\ref{sec:classes}). This makes it possible to add new classes
and tools very easily without having to reconfigure or recompile CM,
not to mention modify its source code.
\subsubsection*{Building stand-alone programs}
CM can be used to build stand-alone programs. In fact SML/NJ
itself---including CM---is an example of this. (The interactive
system cannot rely on an existing compilation manager when starting
up.)
A stand-alone program is constructed by the runtime system from
existing binfiles or members of existing stable libraries. CM must
prepare those binfiles or libraries together with a list that
describes them to the runtime system.
\begin{verbatim}
val mk_standalone : bool option -> string -> string list option
\end{verbatim}
Depending on the optional boolean argument, function {\tt
CM.mk\_standalone} first acts like either {\tt CM.recomp} or {\tt
CM.stabilize}. {\tt NONE} means {\tt CM.recomp}, and {\tt (SOME $r$)}
means {\tt CM.stabilize $r$}. After recompilation (or stabilization)
is successful, {\tt CM.mk\_standalone} constructs a topologically
sorted list of strings that, when written to a file, can be passed to the
runtime system in order to perform stand-alone linkage of the given
program. Upon failure, {\tt CM.mk\_standalone} returns {\tt NONE}.
\paragraph {\bf ml-build}: The programmer should normally have no need
to invoke {\tt CM.mk\_standalone} directly. Instead, SML/NJ provides
a command {\tt ml-build} which does all the work. To be able to use
{\tt ml-build}, one must implement a library exporting a structure
that has some function suitable to be an argument to {\tt
SMLofNJ.exportFn}. Suppose the library is called {\tt myproglib.cm}, the
structure is called {\tt MyProg}, and the function is called {\tt
MyProg.main}. If one wishes to produce a heap image file {\tt myprog}
one simply has to invoke the following command:
\begin{verbatim}
ml-build myproglib.cm MyProg.main myprog
\end{verbatim}
\subsubsection*{Finding all sources}
The {\tt CM.sources} function can be used to find the names of all
source files that a given library depends on. It returns the names of
all files involved with the exception of skeleton files and binfiles
(see Section~\ref{sec:files}). Stable libraries are represented by
their library file; their description file or consitutent members are
{\em not} listed.
Normally, the function reports actual file names as used for accessing
the file system. For (stable) library files this behavior can be
inconvenient because these names depend on architecture and operating
system. For this reason, {\tt CM.sources} accepts an optional pair of
strings that then will be used in place of the architecture- and
OS-specific part of these names.
\begin{verbatim}
val sources :
{ arch: string, os: string } option ->
string ->
{ file: string, class: string, derived: bool } list option
\end{verbatim}
In case there was some error analyzing the specified library or group,
{\tt CM.sources} returns {\tt NONE}. Otherwise the result is a list
of records, each carrying a file name, the corresponding class, and
information about whether or not the source was created by some tool.
Examples:
\begin{description}
\item[generating ``make'' dependencies:]
To generate dependency information usable by Unix' {\tt make} command,
one would be interested in all files that were not derived by some
tool application. Moreover, one would probably like to use shell
variables instead of concrete architecture- and OS-names:
\begin{verbatim}
Option.map (List.filter (not o #derived))
(CM.sources (SOME { arch = "$ARCH", os = "$OPSYS" })
"foo.cm");
\end{verbatim}
\item[finding all {\tt noweb} sources:]
To find all {\tt noweb} sources (see Section~\ref{sec:builtin-tools}),
e.g., to be able to run the document preparation program {\tt noweave}
on them, one can simply look for entries of the {\tt noweb} class.
Here, one would probably want to include derived sources:
\begin{verbatim}
Option.map (List.filter (fn x => #class x = "noweb"))
(CM.sources NONE "foo.cm");
\end{verbatim}
\end{description}
\subsection{The autoloader}
\label{sec:autoload}
From the user's point of view, a call to {\tt CM.autoload} acts very
much like the corresponding call to {\tt CM.make} because the same
bindings that {\tt CM.make} would introduce into the top-level
enviroment are also introduced by {\tt CM.autoload}. However, most
work will be deferred until some code that is entered later refers to
one or more of these bindings. Only then will CM go and perform just
the minimal work necessary to provide the actual definitions.
The autoloader plays a central role for the interactive system.
Unlike in earlier versions, it cannot be turned off since it provides
many of the standard pre-defined top-level bindings.
The autoloader is a convenient mechanism for virtually ``loading'' an
entire library without incurring an undue increase in memory
consumption for library modules that are not actually being used.
\subsection{Sharing of state}
\label{sec:sharing}
Whenever it is legal to do so, CM lets multiple invocations of {\tt
CM.make} or {\tt CM.autoload} share dynamic state created by link-time
effects. Of course, sharing is not possible (and hence not ``legal'')
if the compilation unit in question has recently been recompiled or
depends on another compilation unit whose code has recently been
re-executed. The programmer can explicitly mark certain ML files as
{\em shared}, in which case CM will issue a warning whenever the
unit's code has to be re-executed.
State created by compilation units marked as {\em private} is never
shared across multiple calls to {\tt CM.make} or {\tt CM.autoload}.
To understand this behavior it is useful to introduce the notion of a
{\em traversal}. A traversal is the process of traversing the
dependency graph on behalf of {\tt CM.make} or {\tt CM.autoload}.
Several traversals can be executed interleaved with each other because
a {\tt CM.autoload} traversal normally stays suspended and is
performed incrementally driven by input from the interactive top level
loop.
As far as sharing is concerned, the rule is that during one traversal
each compilation unit will be executed at most once. This means that
the same ``program'' will not see multiple instantiations of the same
compilation unit (where ``program'' refers to the code managed by one
call to {\tt CM.make} or {\tt CM.autoload}). Each compilation unit
will be linked at most once during a traversal and private state
will not be confused with private state of other traversals that might
be active at the same time.
% Need a good example here.
\subsubsection*{Sharing annotations}
ML source files in CM description files can be specified as being {\em
private} or {\em shared}. This is done by adding a {\em tool
parameter} specification for the file in the library- or group
description file (see Section~\ref{sec:classes}). To mark an ML file
as {\em private}, follow the file name with the word {\tt private} in
parentheses. For {\em shared} ML files, replace {\tt private} with
{\tt shared}.
An ML source file that is not annotated will typically be treated as
{\em shared} unless it statically depends on some other {\em private}
source. It is an error, checked by CM, for a {\em shared} source to
depend on a {\em private} source.
\subsubsection*{Sharing with the interactive system}
The SML/NJ interactive system, which includes the compiler, is itself
created by linking modules from various libraries. Some of these
libraries can also be used in user programs. Examples are the
Standard ML Basis Library {\tt \$/basis.cm}, the SML/NJ library {\tt
\$/smlnj-lib.cm}, and the ML-Yacc library {\tt \$/ml-yacc-lib.cm}.
If a module from a library is used by both the interactive system and
a user program running under control of the interactive system, then
CM will let them share code and dynamic state. Moreover, the affected
portion of the library will never have to be ``relinked''.
\section{Version numbers}
\label{sec:versions}
A CM library can carry a version number. Version numbers are
specified in parentheses after the keyword {\tt Library} as non-empty
dot-separated sequences of non-negative integers. Example:
\begin{verbatim}
Library (1.4.1.4.2.1.3.5)
structure Sqrt2
is
sqrt2.sml
\end{verbatim}
\subsection{How versions are compared}
Version numbers are compared lexicographically, dot-separated
component by dot-separated component, from left to right. The
components themselves are compared numerically.
\subsection{Version checking}
An importing library or library component can specify which version of
the imported library it would like to see. See the discussion is
section~\ref{sec:toolparam} for how this is done. Where a version
number is requested, an error is signalled if one of the following is
true:
\begin{itemize}
\item the imported library does not carry a version number
\item the imported library's version number is smaller than the
one requested
\item the imported library's version number has a first component
(known as the ``major'' version number) that is greater than the one
requested
\end{itemize}
A warning (but no error) is issued if the imported library has the
same major version but the version as a whole is greater than the one
requested.
Note: {\it Version numbers should be incremented on every change to a
library. The major version number should be increased on every change
that is not backward-compatible.}
\section{Member classes and tools}
\label{sec:classes}
Most members of groups and libraries are either plain ML files or
other description files. However, it is possible to incorporate other
types of files---as long as their contents can in some way be expanded
into ML code or CM descriptions. The expansion is carried out by CM's
{\it tools} facility.
CM maintains an internal registry of {\em classes} and associated {\em
rules}. Each class represents the set of source files that its
corresponding rule is applicable to. For example, the class {\tt
mlyacc} is responsible for files that contain input for the parser
generator ML-Yacc~\cite{tarditi90:yacc}. The rule for {\tt mlyacc}
takes care of expanding an ML-Yacc specifications {\tt foo.grm} by
invoking the auxiliary program {\tt ml-yacc}. The resulting ML files
{\tt foo.grm.sig} and {\tt foo.grm.sml} are then used as if their
names had directly been specified in place of {\tt foo.grm}.
CM knows a small number of built-in classes. In many situations these
classes will be sufficient, but in more complicated cases it may be
worthwhile to add a new class. Since class rules are programmed in
ML, adding a class is not as simple a matter as writing a rule for
{\sc Unix}' {\tt make} program~\cite{feldman79}. Of course,
using ML has also advantages because it keeps CM extremely flexible in
what rules can do. Moreover, it is not necessary to learn yet another
``little language'' in order to be able to program CM's tool facility.
When looking at the member of a description file, CM determines which
tool to use by looking at clues like the file name suffix. However,
it is also possible to specify the class of a member explicitly. For
this, the member name is followed by a colon {\bf :} and the name of
the member class. All class names are case-insensitive.
In addition to genuine tool classes, there are two member classes
that refer to facilities internal to CM: {\tt sml} is the class of
ordinary ML source files and {\tt cm} is the class of CM library or
group description files.
CM automatically classifies files with a {\tt .sml} suffix, a {\tt
.sig} suffix, or a {\tt .fun} suffix as ML-source, file names ending
in {\tt .cm} as CM descriptions.\footnote{Suffixes that are not known
and for which no plugin module can be found are treated as ML source
code. However, as new tools are added there is no guarantee that
this behavior will be preserved in future versions of CM.}
\subsection{Tool parameters}
\label{sec:toolparam}
In many cases the name of the member that caused a rule to be invoked
is the only input to that rule. However, rules can be written in such
a way that they take additional parameters. Those parameters, if
present, must be specified in the CM description file between
parentheses following the name of the member and the optional member
class.
CM's core mechanism parses these tool options and breaks them up into
a list of items, where each item is either a filename (i.e., {\em
looks} like a filename) or a named list of sub-options. However, CM
itself does not interpret the result but passes it on to the tool's
rule function. It is in each rule's own responsibility to assign
meaning to its options.
The {\tt sml} class accepts one parameter which must be either the
word {\tt shared} or the word {\tt private}. (Technically, the
strings {\tt private} and {\tt shared} fall under the {\em filename}
category from above, but the tool ignores that aspect and uses the
name directly.) If {\tt shared} is specified, then dynamic state
created by the compilation unit at link-time must be shared across
invocations of {\tt CM.make} or {\tt CM.autoload}. As explained
earlier (Section~\ref{sec:sharing}), the {\tt private} annotation
means that dynamic state cannot be shared across such calls to {\tt
CM.make} or {\tt CM.autoload}.
The {\tt cm} class understands two kinds of parameters. The first is
a named parameter labeled by the string {\tt version}. It must have
the format of a version number. CM will interpret this as a version
request, thereby insuring that the imported library is not too old or
too new. (See section~\ref{sec:versions} for more on this topic.)
All named sub-option lists (for any class) are specified by a name
string followed by a colon {\bf :} and a parenthesized list of other
tool options. If the list contains precisely one element, the
parentheses may be omitted. Example:
\begin{verbatim}
euler.cm (version:2.71828)
pi.cm (version:3.14159)
\end{verbatim}
Normally, CM looks for stable library files in directory
{\tt CM/}{\it arch}{\tt -}{\it os} (see section~\ref{sec:files}).
However, if an explicit version has been requested, it will first try
directory {\tt CM/}{\it version}{\tt /}{\it arch}{\tt -}{\it os}
before looking at the default location. This way it is possible to
keep several versions of the same library in the file system.
However, CM normally does {\em not} permit the simultaneous use of
multiple versions of the same library in one session. The
disambiguating rule is that the version that gets loaded first
``wins''; subsequent attempts to load different versions result in
warnings or errors. (See the discussion of {\tt CM.unshare} in
section~\ref{sec:libreg} for how to to circumvent this restriction.)
The second kind of parameter understood by {\tt cm} is a named
parameter labeled by the string {\tt bind} (see
Section~\ref{sec:anchor:env}). It can occur arbitrarily many times
and each occurence must be a suboption-list of the form {\tt
(anchor:$a$ value:$v$)}. The set of {\tt bind}-parameters augments
the current anchor environment to form the environment that is used
while processing the contents of the named CM description file.
\subsection{Built-in tools}
\label{sec:builtin-tools}
\subsubsection*{The ML-Yacc tool}
The ML-Yacc tool is responsible for files that are input to the
ML-Yacc parser generator. Its class name is {\tt mlyacc}. Recognized
file name suffixes are {\tt .grm} and {\tt .y}. For a source file
$f$, the tool produces two targets $f${\tt .sig} and $f${\tt .sml},
both of which are always treated as ML source files. Parameters are
passed on without change to the $f${\tt .sml} file but not to the
$f${\tt .sig} file. This means that the parameter can either be the
word {\tt private} or the word {\tt shared}, and that this sharing
annotation will apply to the $f${\tt .sml} file.
The tool invokes the {\tt ml-yacc} command if the targets are
``outdated''. A target is outdated if it is missing or older than the
source. Unless anchored using the path anchor mechanism (see
Section~\ref{sec:anchor:env}), the command {\tt ml-yacc} will be located
using the operating system's path search mechanism (e.g., the {\tt
\$PATH} environment variable).
\subsubsection*{ML-Lex}
The ML-Lex tool governs files that are input to the ML-Lex lexical
analyzer generator~\cite{appel89:lex}. Its class name is {\tt mllex}.
Recognized file name suffixes are {\tt .lex} and {\tt .l}. For a
source file $f$, the tool produces one targets $f${\tt .sml} which
will always be treated as ML source code. Tool parameters are passed
on without change to that file.
The tool invokes the {\tt ml-lex} command if the target is outdated
(just like in the case of ML-Yacc). Unless anchored using the path
anchor mechanism (see Section~\ref{sec:anchor:env}), the command {\tt
ml-lex} will be located using the operating system's path search
mechanism (e.g., the {\tt \$PATH} environment variable).
\subsubsection*{ML-Burg}
The ML-Burg tool deals with files that are input to the ML-Burg
code-generater generator~\cite{mlburg93}. Its class name is {\tt
mlburg}. The only recognized file name suffix is {\tt .burg}. For a
source file $f${\tt .burg}, the tool produces one targets $f${\tt
.sml} which will always be treated as ML source code. Any tool
parameters are passed on without change to the target.
The tool invokes the {\tt ml-burg} command if the target is outdated.
Unless anchored using the path anchor mechanism (see
Section~\ref{sec:anchor:env}), the command {\tt ml-lex} will be located
using the operating system's path search mechanism (e.g., the {\tt
\$PATH} environment variable).
\subsubsection*{Shell}
The Shell tool can be used to specify arbitrary shell commands to be
invoked on behalf of a given file. The name of the class is {\tt
shell}. There are no recognized file name suffixes. This means that
in order to use the shell tool one must always specify the {\tt shell}
member class explicitly.
The rule for the {\tt shell} class relies on tool parameters. The
parameter list must be given in parentheses and follow the {\tt shell}
class specification.
Consider the following example:
\begin{verbatim}
foo.pp : shell (target:foo.sml options:(shared)
/lib/cpp -P -Dbar=baz %s %t)
\end{verbatim}
This member specification says that file {\tt foo.sml} can be obtained
from {\tt foo.pp} by running it through the C preprocessor {\tt cpp}.
The fact that the target file is given as a tool parameter implies
that the member itself is the source. The named parameter {\tt
options} lists the tool parameters to be used for that target. (In the
example, the parentheses around {\tt shared} are optional because it
is the only element of the list.) The command line itself is given by
the remaining non-keyword parameters. Here, a single {\bf \%s} is
replaced by the source file name, and a single {\bf \%t} is replaced
by the target file name; any other string beginning with {\bf \%} is
shortened by its first character.
In the specification one can swap the positions of source and target
(i.e., let the member name be the target) by using a {\tt source}
parameter:
\begin{verbatim}
foo.sml : shell (source:foo.pp options:shared
/lib/cpp -P -Dbar=baz %s %t)
\end{verbatim}
Exactly one of the {\tt source} and {\tt target} parameters must be
specified; the other one is taken to be the member name itself. The
target class can be given by writing a {\tt class} parameter whose
single sub-option must be the desired class name.
The usual distinction between native and standard filename syntax
applies to any given {\tt source} or {\tt target} parameters.
For example, if one were working on a Win32 system and the target file
is supposed to be in the root directory on volume {\tt D:},
then one must use native syntax to write it. One way of doing this
would be:
\begin{verbatim}
"D:\\foo.sml" : shell (source : foo.pp options : shared
cpp -P -Dbar=baz %s %t)
\end{verbatim}
\noindent As a result, {\tt foo.sml} is interpreted using native
syntax while {\tt foo.pp} uses standard conventions (although in this
case it does not make a difference). Had we used the {\tt target}
version from above, one would have to write:
\begin{verbatim}
foo.pp : shell (target : "D:\\foo.sml" options : shared
cpp -P -Dbar=baz %s %t)
\end{verbatim}
The shell tool invokes its command whenever the target is outdated
with respect to the source.
\subsubsection*{Make}
The Make tool (class {\tt make}) can (almost) be seen as a specialized
version of the Shell tool. It has no source and one target (the
member itself) which is always considered outdated. As with the Shell
tool, it is possible to specify target class and parameters using the
{\tt class} and {\tt options} keyword parameters.
The tool invokes the shell command {\tt make} on the target. Unless
anchored using the path anchor mechanism~\ref{sec:anchor:env}, the
command will be located using the operating system's path search
mechanism (e.g., the {\tt \$PATH} environment variable).
Any parameters other than the {\tt class} and {\tt options}
specifications must be plain strings and are given as additional
command line arguments to {\tt make}. The target name is always the
last command line argument.
Example:
\begin{verbatim}
bar-grm : make (class:mlyacc -f bar-grm.mk)
\end{verbatim}
Here, file {\tt bar-grm} is generated (and kept up-to-date) by
invoking the command:
\begin{verbatim}
make -f bar-grm.mk bar-grm
\end{verbatim}
\noindent The target file is then treated as input for {\tt ml-yacc}.
Cascading Shell- and Make-tools is easily possible. Here is an
example that first uses Make to build {\tt bar.pp} and then filters
the contens of {\tt bar.pp} through the C preprocessor to arrive at
{\tt bar.sml}:
\begin{verbatim}
bar.pp : make (class:shell
options:(target:bar.sml cpp -Dbar=baz %s %t)
-f bar-pp.mk)
\end{verbatim}
\subsubsection*{Noweb}
The {\tt noweb} class handles sources written for Ramsey's {\it noweb}
literate programming facility~\cite{ramsey:simplified}. Files ending
with suffix {\tt .nw} are automatically recognized as belonging to
this class.
The list of targets that are to be extracted from a noweb file must be
specified using tool options. A target can then have a variety of its
own options. Each target is specified by a separate tool option
labelled {\tt target}. The option usually has the form of a
sub-option list. Recognized sub-options are:
\begin{description}
\item[name] the name of the target
\item[root] the (optional) root tag for the target (given to the {\tt
-R} command line switch for the {\tt notangle} command); if {\tt root}
is missing, {\tt name} is used instead
\item[class] the (optional) class of the target
\item[options] (optional) options for the tool that handles the
target's class
\item[lineformat] a string that will be passed to the {\tt -L} command
line option of {\tt notangle}
\item[cpif] an optional boolean value (the word {\tt true} or {\tt
false}); if set to {\tt true} then the target will not be overwritten
if its contents would be unchanged\footnote{The tradeoff is between
running {\tt notangle} too often or recompiling the result too
often.}; default: {\tt true}
\end{description}
Example:
\begin{verbatim}
project.nw (target:(name:main.sml options:(private) cpif:false)
target:(name:grammar class:mlyacc)
target:(name:parse.sml))
\end{verbatim}
In place of the sub-option list there can be a single string option
which will be used for {\tt name} or even an unnamed parameter (i.e.,
without the {\tt target} label). If no targets are specified, the
tool will assume two default targets by stripping the {\tt .nw}
suffix (if present) from the source name and adding {\tt .sig} as well
as {\tt .sml}.
The following four examples are all equivalent:
\begin{verbatim}
foo.nw (target:(name:foo.sig) target:(name:foo.sml))
foo.nw (target:foo.sig target:foo.sml)
foo.nw (foo.sig foo.sml)
foo.nw
\end{verbatim}
If {\tt lineformat} is missing, then a default based on the target
class is used. Currently only the {\tt sml} and {\tt cm} classes are
known to CM; other classes can be added or removed by using the {\tt
NowebTool.lineNumbering} controller function exported from library
{\tt \$/noweb-tool.cm}:
\begin{verbatim}
val lineNumbering: string -> { get: unit -> string option,
set: string option -> unit }
\end{verbatim}
\section{Conditional compilation}
\label{sec:preproc}
In its description files, CM offers a simple conditional compilation
facility inspired by the preprocessor for the C language~\cite{k&r2}.
However, it is not really a {\it pre}-processor, and the syntax of the
controlling expressions is borrowed from SML.
Sequences of members can be guarded by {\tt \#if}-{\tt \#endif}
brackets with optional {\tt \#elif} and {\tt \#else} lines in between.
The same guarding syntax can also be used to conditionalize the export
list. {\tt \#if}-, {\tt \#elif}-, {\tt \#else}-, and {\tt
\#endif}-lines must start in the first column and always
extend to the end of the current line. {\tt \#if} and {\tt \#elif}
must be followed by a boolean expression.
Boolean expressions can be formed by comparing arithmetic expressions
(using operators {\tt <}, {\tt <=}, {\tt =}, {\tt >=}, {\tt >}, or
{\tt <>}), by logically combining two other boolean expressions (using
operators {\tt andalso}, {\tt orelse}, {\tt =}, or {\tt <>}, by
querying the existence of a CM symbol definition, or by querying the
existence of an exported ML definition.
Arithmetic expressions can be numbers or references to CM symbols, or
can be formed from other arithmetic expressions using operators {\tt
+}, {\tt -} (subtraction), \verb|*|, {\tt div}, {\tt mod}, or $\tilde{~}$
(unary minus). All arithmetic is done on signed integers.
Any expression (arithmetic or boolean) can be surrounded by
parentheses to enforce precedence.
\subsection{CM variables}
\label{sec:cmvars}
CM provides a number of ``variables'' (names that stand for certain
integers). These variables may appear in expressions of the
conditional-compilation facility. The exact set of variables provided
depends on SML/NJ version number, machine architecture, and
operating system. A reference to a CM variable is considered an
arithmetic expression. If the variable is not defined, then it
evaluates to 0. The expression {\tt defined}($v$) is a boolean
expression that yields true if and only if $v$ is a defined CM
variable.
The names of CM variables are formed starting with a letter followed
by zero or more occurences of letters, decimal digits, apostrophes, or
underscores.
The following variables will be defined and bound to 1:
\begin{itemize}
\item depending on the operating system: {\tt OPSYS\_UNIX}, {\tt
OPSYS\_WIN32}, {\tt OPSYS\_MACOS}, {\tt OPSYS\_OS2}, or {\tt
OPSYS\_BEOS}
\item depending on processor architecture: {\tt ARCH\_SPARC}, {\tt
ARCH\_ALPHA32}, {\tt ARCH\_MIPS}, {\tt ARCH\_X86}, {\tt ARCH\_HPPA},
{\tt ARCH\_RS6000}, or {\tt ARCH\_PPC}
\item depending on the processor's endianness: {\tt BIG\_ENDIAN} or
{\tt LITTLE\_ENDIAN}
\item depending on the native word size of the implementation: {\tt
SIZE\_32} or {\tt SIZE\_64}
\item the symbol {\tt NEW\_CM}
\end{itemize}
Furthermore, the symbol {\tt SMLNJ\_VERSION} will be bound to the
major version number of SML/NJ (i.e., the number before the first dot)
and {\tt SMLNJ\_MINOR\_VERSION} will be bound to the system's minor
version number (i.e., the number after the first dot).
Using the {\tt CM.symval} interface one can define additional
variables or modify existing ones.
\subsection{Querying exported definitions}
An expression of the form {\tt defined}($n$ $s$), where $s$ is an ML
symbol and $n$ is an ML namespace specifier, is a boolean expression
that yields true if and only if any member included before this test
exports a definition under this name. Therefore, order among members
matters after all (but it remains unrelated to the problem of
determining static dependencies)! The namespace specifier must be one
of: {\tt structure}, {\tt signature}, {\tt functor}, or {\tt funsig}.
If the query takes place in the ``exports'' section of a description
file, then it yields true if {\em any} of the included members exports
the named symbol.
\noindent Example:
\begin{verbatim}
Library
structure Foo
#if defined(structure Bar)
structure Bar
#endif
is
#if SMLNJ_VERSION > 110
new-foo.sml
#else
old-foo.sml
#endif
#if defined(structure Bar)
bar-client.sml
#else
no-bar-so-far.sml
#endif
\end{verbatim}
Here, the file {\tt bar-client.sml} gets included if {\tt
SMLNJ\_VERSION} is greater than 110 and {\tt new-foo.sml} exports a
structure {\tt Bar} {\em or} if {\tt SMLNJ\_VERSION <= 110} and {\tt
old-foo.sml} exports structure {\tt Bar}. Otherwise\linebreak {\tt
no-bar-so-far.sml} gets included instead. In addition, the export of
structure {\tt Bar} is guarded by its own existence. (Structure {\tt
Bar} could also be defined by {\tt no-bar-so-far.sml} in
which case it would get exported regardless of the outcome of the
other {\tt defined} test.)
\subsection{Explicit errors}
A pseudo-member of the form {\tt \#error $\ldots$}, which---like other
{\tt \#}-items---starts in the first column and extends to the end of
the line, causes an explicit error message to be printed unless it
gets excluded by the conditional compilation logic. The error message
is given by the remainder of the line after the word {\tt error}.
\section{Access control}
\label{sec:access}
The basic idea behind CM's access control is the following: In their
description files, groups and libraries can specify a list of
{\em privileges} that the client must have in order to be able to use them.
Privileges at this level are just names (strings) and must be written
in front of the initial keyword {\tt Library} or {\tt Group}. If one
group or library imports from another group or library, then
privileges (or rather: privilege requirements) are being inherited.
In effect, to be able to use a program, one must have all privileges
for all its libraries, sub-libraries and library components,
components of sub-libraries, and so on.
Of course, this alone would not yet be satisfactory. The main service
of the access control system is that it can let a client use an
``unsafe'' library ``safely''. For example, a library {\tt LSafe.cm}
could ``wrap'' all the unsafe operations in {\tt LUnsafe.cm} with
enough error checking that they become safe. Therefore, a user of
{\tt LSafe.cm} should not also be required to possess the privileges
that would be required if one were to use {\tt LUnsafe.cm} directly.
In CM's access control model it is possible for a library to ``wrap''
privileges. If a privilege $P$ has been wrapped, then the user of the
library does not need to have privilege $P$ even though the library is
using another library that requires privilege $P$. In essence, the
library acts as a ``proxy'' who provides the necessary credentials for
privilege $P$ to the sub-library.
Of course, not everybody can be allowed to establish a library with
such a ``wrapped'' privilege $P$. The programmer who does that should at
least herself have privilege P (but perhaps better, she should have
{\em permission to wrap $P$}---a stronger requirement).
In CM, wrapping a privilege is done by specifying the name of that
privilege within parenthesis. The wrapping becomes effective once the
library gets stabilized via {\tt CM.stabilize}. The (not yet
implemented) enforcement mechanism must ensure that anyone who
stabilizes a library that wraps $P$ has permission to wrap $P$.
Note that privileges cannot be wrapped at the level of CM groups.
Access control is a new feature. At the moment, only the basic
mechanisms are implemented, but there is no enforcement. In other
words, everybody is assumed to have every possible privilege. CM
merely reports which privileges ``would have been required''.
\section{The pervasive environment}
The {\em pervasive environment} can be thought of as a compilation
unit that all compilation units implicitly depend upon. The pervasive
enviroment exports all non-modular bindings (types, values, infix
operators, overloaded symbols) that are mandated by the specification
for the Standard ML Basis Library~\cite{reppy99:basis}. (All other
bindings of the Basis Library are exported by {\tt \$/basis.cm} which is
a genuine CM library.)
The pervasive environment is the only place where CM conveys
non-modular bindings from one compilation unit to another, and its
definition is fixed.
\section{Files}
\label{sec:files}
CM uses three kinds of files to store derived information during and
between sessions:
\begin{enumerate}
\item {\it Skeleton files} are used to store a highly abbreviated
version of each ML source file's abstract syntax tree---just barely
sufficient to drive CM's dependency analysis. Skeleton files are much
smaller and easier to read than actual ML source code. Therefore, the
existence of valid skeleton files makes CM a lot faster because
usually most parsing operations can be avoided that way.
\item {\it Binfiles} are the SML/NJ equivalent of object files. They
contain executable code and a symbol table for the associated ML
source file.
\item {\it Library files} (sometimes called: {\em stablefiles}) contain
dependency graph, executable code, and symbol tables for an entire CM
library including all of its components (groups). Other libraries
used by a stable library are not included in full. Instead,
references to those libraries are recorded using their (preferably
anchored) pathnames.
\end{enumerate}
Normally, all these files are stored in a subdirectory of directory
{\tt CM}. {\tt CM} itself is a subdirectory of the directory where the
original ML source file or---in the case of library files---the
original CM description file is located.
Skeleton files are machine- and operating system-independent.
Therefore, they are always placed into the same directory {\tt
CM/SKEL}. Parsing (for the purpose of dependency analysis) will be
done only once even if the same file system is accessible from
machines of different type.
Binfiles and library files contain executable code and other
information that is potentially system- and architecture-dependent.
Therefore, they are stored under {\tt CM/}{\it arch}{\tt -}{\it os}
where {\it arch} is a string indicating the type of the current
CPU architecture and {\it os} a string denoting the current operating
system type.
Library files are a bit of an exception in the sense that they do not
require any source files or any other derived files of the same
library to exist. As a consequence, the location of such a library
file is best described as being relative to ``the location of the
original CM description file if that description file still existed''.
(Of course, nothing precludes the CM description file from actually
existing, but in the presence of a corresponding library file CM will
not take any notice.)
{\em Note:} As discussed in section~\ref{sec:toolparam}, CM sometimes
looks for library files in
{\tt CM/}{\it version}{\tt /}{\it arch}{\tt -}{\it os}.
However, library files are never {\em created} there by CM. If
several versions of the same library are to be provided, an
administrator must arrange the directory hierarchy accordingly ``by
hand''.
\subsection{Time stamps}
For skeleton files and binfiles, CM uses file system time stamps
(i.e., modification time) to determine whether a file has become
outdated. The rule is that in order to be considered ``up-to-date''
the time stamp on skeleton file and binfile has to be exactly the
same\footnote{CM explicitly sets the time stamp to be the same.} as
the one on the ML source file. This guarantees that all changes to a
source will be noticed.\footnote{except for the pathological case where
two different versions of the same source file have exactly the same
time stamp}
CM also uses time stamps to decide whether tools such as ML-Yacc or
ML-Lex need to be run (see Section~\ref{sec:tools}). However, the
difference is that a file is considered outdated if it is older than
its source. Some care on the programmers side is necessary since this
scheme does not allow CM to detect the situation where a source file
gets replaced by an older version of itself.
\section{Tools}
\label{sec:tools}
CM's tool set is extensible: new tools can be added by writing a few
lines of ML code. The necessary hooks for this are provided by a
structure {\tt Tools} which is exported by the {\tt \$smlnj/cm/tools.cm}
library.
If the tool is implemented as a ``typical'' shell command, then all
that needs to be done is a single call to:
\begin{verbatim}
Tools.registerStdShellCmdTool
\end{verbatim}
For example, suppose you have made a
new, improved version of ML-Yacc (``New-ML-Yacc'') and want to
register it under a class called {\tt nmlyacc}. Here is what you
write:
\begin{verbatim}
val _ = Tools.registerStdShellCmdTool
{ tool = "New-ML-Yacc",
class = "nmlyacc",
suffixes = ["ngrm", "ny"],
cmdStdPath = "new-ml-yacc",
template = NONE,
extensionStyle =
Tools.EXTEND [("sig", SOME "sml", fn _ => NONE),
("sml", SOME "sml", fn x => x)],
dflopts = [] }
\end{verbatim}
\begin{sloppy}
This code can either be packaged as a CM library or entered at the
interactive top level after loading the {\tt \$smlnj/cm/ tools.cm}
library via {\tt CM.make} or {\tt CM.load\_plugin}. ({\tt
CM.autoload} is not enough because of its lazy nature which prevents
the required side-effects to occur.)
\end{sloppy}
In our example, the shell command name for our tool is {\tt
new-ml-yacc}. When looking for this command in the file system, CM
first tries to treat it as a path anchor (see
section~\ref{sec:anchor:env}). For example, suppose {\tt new-ml-yacc} is
mapped to {\tt /bin}. In this case the command to be
invoked would be {\tt /bin/new-ml-yacc}. If path anchor resolution
fails, then the command name will be used as-is. Normally this
causes the shell's path search mechanism to be used as a fallback.
{\tt Tools.registerStdShellCmdTool} creates the class and installs the
tool for it. The arguments must be specified as follows:
\begin{description}
\item[tool] a descriptive name of the tool (used in error messages);
type: {\tt string}
\item[class] the name of the class; the string must not contain
upper-case letters; type: {\tt string}
\item[suffixes] a list of file name suffixes that let CM automatically
recognize files of the class; type: {\tt string list}
\item[cmdStdPath] the command string from above; type: {\tt string}
\item[template] an optional string that describes how the command line
is to be constructed from pieces; \\
The string is taken verbatim except for embedded \% format specifiers:
\begin{description}\setlength{\itemsep}{0pt}
\item[\%c] the command name (i.e., the elaboration of {\tt cmdStdPath})
\item[\%s] the source file name in native pathname syntax
\item[\%$n$t] the $n$-th target file in native pathname syntax; \\
($n$ is specified as a decimal number, counting starts at $1$, and
each target file name is constructed from the corresponding {\tt
extensionStyle} entry; if $n$ is $0$ (or missing), then all
targets---separated by single spaces---are inserted;
if $n$ is not in the range between $0$ and the number of available
targets, then {\bf \%$n$t} expands into itself)
\item[\%$n$o] the $n$-th tool parameter; \\
(named sub-option parameters are ignored;
$n$ is specified as a decimal number, counting starts at $1$;
if $n$ is $0$ (or missing), then all options---separated by single
spaces---are inserted;
if $n$ is not in the range between $0$ and the number of available
options, then {\bf \%$n$o} expands into itself)
\item[\%$x$] the character $x$ (where $x$ is neither {\bf c}, nor
{\bf s}, {\bf t}, or {\bf o})
\end{description}
If no template string is given, then it defaults to {\tt "\%c \%s"}.
\item[extensionStyle] a specification of how the names of files
generated by the tool relate to the name of the tool input file;
type: {\tt Tools.extensionStyle}. \\
Currently, there are two possible cases:
\begin{enumerate}
\item ``{\tt Tools.EXTEND} $l$'' says that if the tool source file is
{\it file} then for each suffix {\it sfx} in {\tt (map \#1 $l$)} there
will be one tool output file named {\it file}{\tt .}{\it sfx}. The
list $l$ consists of triplets where the first component specifies the
suffix string, the second component optionally specifies the
member class name of the corresponding derived file, and the
third component is a function to calculate tool options for the
target from those of the source. (Argument and result type of these
functions is {\tt Tools.toolopts option}.)
\item ``{\tt Tools.REPLACE }$(l_1, l_2)$'' specifies that given the
base name {\it base} there will be one tool output file {\it base}{\tt
.}{\it sfx} for each suffix {\it sfx} in {\tt (map \#1 $l_2$)}. Here,
{\it base} is determined by the following rule: If the name of the
tool input file has a suffix that occurs in $l_1$, then {\it base} is
the name without that suffix. Otherwise the whole file name is taken
as {\it base} (just like in the case of {\tt Tools.EXTEND}). As with
{\tt Tools.EXTEND}, the second components of the elements of $l_2$ can
optionally specify the member class name of the corresponding derived
file, and the third component maps source options to target options.
\end{enumerate}
\item[dflopts] a list of tool options which is used for
substituting {\bf \%$n$o} fields in {\tt template} (see above) if no
options were specified. (Note that the value of {\tt dflopts} is never
passed to the option mappers in {\tt Tools.EXTEND} or {\tt
Tools. REPLACE}.) Type: {\tt Tools.toolopts}.
\end{description}
Less common kinds of rules can also be defined using the generic
interface {\tt Tools.registerClass}.
\subsection{Plug-in Tools}
If CM comes across a member class name $c$ that it does not know
about, then it tries to load a plugin module named {\tt \$}$c${\tt
-tool.cm} or {\tt ./}$c${\tt -tool.cm}. If it sees a file whose name
ends in suffix $s$ for which no explicit member class has been
specified in the CM description file and for which automatic
member classification fails, then it tries to load a plugin module
named {\tt \$}$s${\tt -ext.cm} or {\tt ./}$s${\tt -ext.cm}. The
so-loaded module can then register the required tool which enables CM
to successfully deal with the previously unknown member.
This mechanism makes it possible for new tools to be added by simply
placing appropriately-named plug-in libraries in such a way that CM
can find them. This can be done in one of two ways:
\begin{enumerate}
\item For general-purpose tools that are installed in some central
place, corresponding tool description files {\tt \$}$c${\tt -tool.cm}
and {\tt \$}$s${\tt -ext.cm} should be registered using the path
anchor mechanism. If this is done, actual description files can be
placed in arbitrary locations.
\item For special-purpose tools that are part of a specific program
and for which there is no need for central installation, one should
simply put the tool description files into the same directory as the
one that contains their ``client'' description file.
\end{enumerate}
\section{Parallel and distributed compilation}
\label{sec:parmake}
To speed up recompilation of large projects with many ML source files,
CM can exploit parallelism that is inherent in the dependency graph.
Currently, the only kind of operating system for which this is
implemented is Unix ({\tt OPSYS\_UNIX}), where separate processes are
used. From there, one can distribute the work across a network of
machines by taking advantage of the network file system and the
``rsh'' facility.
To perform parallel compilations, one must attach ``compile servers''
to CM. This is done using function\linebreak {\tt CM.Server.start}
with the following signature:
\begin{verbatim}
structure Server : sig
type server
val start : { name: string,
cmd: string * string list,
pathtrans: (string -> string) option,
pref: int } -> server option
end
\end{verbatim}
Here, {\tt name} is a string uniquely identifying the server and {\tt
cmd} is a value suitable as argument to {\tt Unix.execute}.
The program to be specified by {\tt cmd} should be another instance of
CM---running in ``slave mode''. To start CM in slave mode, start {\tt
sml} with a single command-line argument of {\tt @CMslave}. For
example, if you have installed in /path/to/smlnj/bin/sml, then a
server process on the local machine could be started by
\begin{verbatim}
CM.Server.start { name = "A", pathtrans = NONE, pref = 0,
cmd = ("/path/to/smlnj/bin/sml",
["@CMslave"]) };
\end{verbatim}
To run a process on a remote machine, e.g., ``thatmachine'', as
compute server, one can use ``rsh''.\footnote{On certain systems it
may be necessary to wrap {\tt rsh} into a script that protects rsh
from interrupt signals.} Unfortunately, at the moment it
is necessary to specify the full path to ``rsh'' because {\tt
Unix.execute} (and therefore {\tt CM.Server.start}) does not perform
a {\tt PATH} search. The remote machine
must share the file system with the local machine, for example via NFS.
\begin{verbatim}
CM.Server.start { name = "thatmachine",
pathtrans = NONE, pref = 0,
cmd = ("/usr/ucb/rsh",
["thatmachine",
"/path/to/smlnj/bin/sml",
"@CMslave"]) };
\end{verbatim}
You can start as many servers as you want, but they all must have
different names. If you attach any servers at all, then you should
attach at least two (unless you want to attach one that runs on a
machine vastly more powerful than your local one). Local servers make
sense on multi-CPU machines: start as many servers as there are CPUs.
Parallel make is most effective on multiprocessor machines because
network latencies can have a severely limiting effect on what can be
gained in the distributed case.
(Be careful, though. Since there is no memory-sharing to speak of
between separate instances of {\tt sml}, you should be sure to check
that your machine has enough main memory.)
If servers on machines of different power are attached, one can give
some preference to faster ones by setting the {\tt pref} value higher.
(But since the {\tt pref} value is consulted only in the rare case
that more than one server is idle, this will rarely lead to vastly
better throughput.) All attached servers must use the same
architecture-OS combination as the controlling machine.
In parallel mode, the master process itself normally does not compile
anything. Therefore, if you want to utilize the master's CPU for
compilation, you should start a compile server on the same machine
that the master runs on (even if it is a uniprocessor machine).
The {\tt pathtrans} argument is used when connecting to a machine with
a different file-system layout. For local servers, it can safely be
left at {\tt NONE}. The ``path transformation'' function is used to
translate local path names to their remote counterparts. This can be
a bit tricky to get right, especially if the machines use automounters
or similar devices. The {\tt pathtrans} functions consumes and
produces names in CM's internal ``protocol encoding'' (see
Section~\ref{sec:pathencode}).
Once servers have been attached, one can invoke functions like
{\tt CM.recomp}, {\tt CM.make}, and {\tt CM.stabilize}. They should
work the way the always do, but during compilation they will take
advantage of parallelism.
When CM is interrupted using Control-C (or such), one will sometimes
experience a certain delay if servers are currently attached and busy.
This is because the interrupt-handling code will wait for the servers
to finish what they are currently doing and bring them back to an
``idle'' state first.
\subsection{Pathname protocol encoding}
\label{sec:pathencode}
The master-slave protocol encodes pathnames in the following way:
A pathname consists of {\bf /}-separated arcs (like Unix patnames).
The first arc can be interpreted relative to the current working directory,
relative to the root of the file system, relative to the root of a
volume (on systems that support separate volumes), or relative to a
directory that corresponds to a pathname anchor. The first character
of the pathname is used to distinguish between these cases.
\begin{itemize}
\item If the name starts with {\bf ./}, then the name is relative to
the working directory.
\item If the name starts with {\bf /}, then the name is relative to
the file system root.
\item If the name starts with {\bf \%}, then the substring between this
{\bf \%} and the first {\bf /} is used as the name of a volume and the
remaining arcs are interpreted relative to the root of that volume.
\item If the name starts with {\bf \$}, then the substring between
this {\bf \$} and the first {\bf /} must be the name of a pathname
anchor. The remaining arcs are interpreted relative to the directory
that (on the slave side) is denoted by the anchor.
\item Any other name is interpreted relative to the current working
directory.
\end{itemize}
\subsection{Parallel bootstrap compilation}
The bootstrap compiler\footnote{otherwise not mentioned in this
document} with its main function {\tt CMB.make} and the corresponding
cross-compilation variants of the bootstrap compiler will also use any
attached compile servers. If one intends to exclusively use the
bootstrap compiler, one can even attach servers that run on machines
with different architecture or operating system.
Since the master-slave protocol is fairly simple, it cannot handle
complicated scenarios such as the one necessary for compiling the
``init group'' (i.e., the small set of files necessary for setting up
the ``pervasive'' environment) during {\tt CMB.make}. Therefore, this
will always be done locally by the master process.
\section{Example: Dynamic linking}
\label{sec:dynlink}
Autoloading is convenient and avoids wasted memory for modules that
should be available at the interactive prompt but have not actually
been used so far. However, sometimes one wants to be even more
aggressive and save the space needed for a function until---at
runtime---that function is actually being dynamically invoked.
CM does not provide immediate support for this kind of {\em dynamic
linking}, but it is quite simple to achieve the effect by carefully
arranging some helper libraries and associated stub code.
Consider the following module:
\begin{verbatim}
structure F = struct
fun f (x: int): int =
G.g x + H.h (2 * x + 1)
end
\end{verbatim}
Let us further assume that the implementations of structures {\tt G}
and {\tt H} are rather large so that it would be worthwhile to avoid
loading the code for {\tt G} and {\tt H} until {\tt F.f} is called
with some actual argument. Of course, if {\tt F} were bigger, then we
also want to avoid loading {\tt F} itself.
To achieve this goal, we first define a {\em hook} module which will
be the place where the actual implementation of our function will be
registered once it has been loaded. This hook module is then wrapped
into a hook library. Thus, we have {\tt f-hook.cm}:
\begin{verbatim}
Library
structure F_Hook
is
f-hook.sml
\end{verbatim}
and {\tt f-hook.sml}:
\begin{verbatim}
structure F_Hook = struct
local
fun placeholder (i: int) : int =
raise Fail "F_Hook.f: unitinialized"
val r = ref placeholder
in
fun init f = r := f
fun f x = !r x
end
end
\end{verbatim}
The hook module provides a reference cell into which a function of
type equal to {\tt F.f} can be installed. Here we have chosen to hide
the actual reference cell behind a {\bf local} construct. Accessor
functions are provided to install something into the hook
({\tt init}) and to invoke the so-installed value ({\tt f}).
With this preparation we can write the implementation module {\tt f-impl.sml}
in such a way that not only does it provide the actual
code but also installs itself into the hook:
\begin{verbatim}
structure F_Impl = struct
local
fun f (x: int): int =
G.g x + H.h (2 * x + 1)
in
val _ = F_Hook.init f
end
end
\end{verbatim}
\noindent The implementation module is wrapped into its implementation
library {\tt f-impl.cm}:
\begin{verbatim}
Library
structure F_Impl
is
f-impl.sml
f-hook.cm
g.cm (* imports G *)
h.cm (* imports H *)
\end{verbatim}
\noindent Note that {\tt f-impl.cm} must mention {\tt f-hook.cm} for
{\tt f-impl.sml} to be able to access structure {\tt F\_Hook}.
Finally, we replace the original contents of {\tt f.sml} with a stub
module that defines structure {\tt F}:
\begin{verbatim}
structure F = struct
local
val initialized = ref false
in
fun f x =
(if !initialized then ()
else if CM.make "f-impl.cm" then initialized := true
else raise Fail "dynamic linkage for F.f failed";
F_Hook.f x)
end
end
\end{verbatim}
\noindent The trick here is to explicitly invoke {\tt CM.make} the
first time {\tt F.f} is called. This will then cause {\tt f-impl.cm}
(and therefore {\tt g.cm} and also {\tt h.cm}) to be loaded and the
``real'' implementation of {\tt F.f} to be registered with the hook
module from where it will then be available to this and future calls
of {\tt F.f}.
For the new {\tt f.sml} to be compiled successfully it must be placed
into a library {\tt f.cm} that mentions {\tt f-hook.cm} and {\tt
\$smlnj/cm/full.cm}. As we have seen, {\tt f-hook.cm} exports {\tt
F\_Hook.f} and {\tt \$smlnj/cm/full.cm} is needed because it exports
{\tt CM.make}:
\begin{verbatim}
Library
structure F
is
f.sml
f-hook.cm
$smlnj/cm.cm (* or $smlnj/cm/full.cm *)
\end{verbatim}
\noindent{\bf Beware!} This solution makes use of {\tt \$smlnj/cm.cm}
which in turn requires the SML/NJ compiler to be present. Therefore,
is worthwhile only for really large program modules where the benefits
of their absence are not outweighed be the need for the compiler.
\section{Some history}
Although its programming model is more general, CM's implementation is
closely tied to the Standard ML programming language~\cite{milner97}
and its SML/NJ implementation~\cite{appel91:sml}.
The current version is preceded by several other compilation managers.
Of those, the most recent went by the same name
``CM''~\cite{blume95:cm}, while earlier ones were known as IRM ({\it
Incremental Recompilation Manager})~\cite{harper94:irm} and SC (for
{\it Separate Compilation})~\cite{harper-lee-pfenning-rollins-CM}. CM
owes many ideas to SC and IRM.
Separate compilation in the SML/NJ system heavily relies on mechanisms
for converting static environments (i.e., the compiler's symbol
tables) into linear byte stream suitable for storage on
disks~\cite{appel94:sepcomp}. However, unlike all its predecessors,
the current implementation of CM is integrated into the main compiler
and no longer relies on the {\em Visible Compiler} interface.
\pagebreak
\appendix
\section{CM description file syntax}
\subsection{Lexical Analysis}
The CM parser employs a context-sensitive scanner. In many cases this
avoids the need for ``escape characters'' or other lexical devices
that would make writing description files cumbersome. On the other
hand, it increases the complexity of both documentation and implementation.
The scanner skips all nestable SML-style comments (enclosed with {\bf
(*} and {\bf *)}).
Lines starting with {\bf \#line} may list up to three fields separated
by white space. The first field is taken as a line number and the
last field (if more than one field is present) as a file name. The
optional third (middle) field specifies a column number. A line of
this form resets the scanner's idea about the name of the file that it
is currently processing and about the current position within that
file. If no file is specified, the default is the current file. If
no column is specified, the default is the first column of the
(specified) line. This feature is meant for program-generators or
tools such as {\tt noweb} but is not intended for direct use by
programmers.
The following lexical classes are recognized:
\begin{description}
\item[Namespace specifiers:] {\bf structure}, {\bf signature},
{\bf functor}, or {\bf funsig}. These keywords are recognized
everywhere.
\item[CM keywords:] {\bf group}, {\bf Group}, {\bf GROUP}, {\bf
library}, {\bf Library}, {\bf LIBRARY}, {\bf is}, {\bf IS}. These
keywords are recognized everywhere except within ``preprocessor''
lines (lines starting with {\bf \#}) or following one of the namespace
specifiers.
\item[Preprocessor control keywords:] {\bf \#if}, {\bf \#elif}, {\bf
\#else}, {\bf \#endif}, {\bf \#error}. These keywords are recognized
only at the beginning of the line and indicate the start of a
``preprocessor'' line. The initial {\bf \#} character may be
separated from the rest of the token by white space (but not by comments).
\item[Preprocessor operator keywords:] {\bf defined}, {\bf div}, {\bf
mod}, {\bf andalso}, {\bf orelse}, {\bf not}. These keywords are
recognized only when they occur within ``preprocessor'' lines. Even
within such lines, they are not recognized as keywords when they
directly follow a namespace specifier---in which case they are
considered SML identifiers.
\item[SML identifiers (\nt{mlid}):] Recognized SML identifiers
include all legal identifiers as defined by the SML language
definition. (CM also recognizes some tokens as SML identifiers that
are really keywords according to the SML language definiten. However,
this can never cause problems in practice.) SML identifiers are
recognized only when they directly follow one of the namespace
specifiers.
\item[CM identifiers (\nt{cmid}):] CM identifiers have the same form
as those ML identifiers that are made up solely of letters, decimal
digits, apostrophes, and underscores. CM identifiers are recognized when they
occur within ``preprocessor'' lines, but not when they directly follow
some namespace specifier.
\item[Numbers (\nt{number}):] Numbers are non-empty sequences of
decimal digits. Numbers are recognized only within ``preprocessor''
lines.
\item[Preprocessor operators:] The following unary and binary operators are
recognized when they occur within ``preprocessor'' lines: {\tt +},
{\tt -}, {\tt *}, {\tt /}, {\tt \%}, {\tt <>}, {\tt !=}, {\tt <=},
{\tt <}, {\tt >=}, {\tt >}, {\tt ==}, {\tt =}, $\tilde{~}$, {\tt
\&\&}, {\tt ||}, {\tt !}. Of these, the following (``C-style'')
operators are considered obsolete and trigger a warning
message\footnote{The use of {\tt -} as a unary minus also triggers
this warning.} as long as {\tt CM.Control.warn\_obsolete} is set to
{\tt true}: {\tt /}, {\tt \%}, {\tt !=}, {\tt ==}, {\tt \&\&}, {\tt
||}, {\tt !}.
\item[Standard path names (\nt{stdpn}):] Any non-empty sequence of
upper- and lower-case letters, decimal digits, and characters drawn
from {\tt '\_.;,!\%\&\$+/<=>?@$\tilde{~}$|\#*-\verb|^|} that occurs
outside of ``preprocessor'' lines and is neither a namespace specifier
nor a CM keyword will be recognized as a stardard path name. Strings
that lexically constitute standard path names are usually---but not
always---interpreted as file names. Sometimes they are simply taken as
literal strings. When they act as file names, they will be
interpreted according to CM's {\em standard syntax} (see
Section~\ref{sec:basicrules}). (Member class names, names of
privileges, and many tool optios are also specified as standard path
names even though in these cases no actual file is being named.)
\item[Native path names (\nt{ntvpn}):] A token that has the form of an
SML string is considered a native path name. The same rules as in SML
regarding escape characters apply. Like their ``standard''
counterparts, native path names are not always used to actually name
files, but when they are, they use the native file name syntax of the
underlying operating system.
\item[Punctuation:] A colon {\bf :} is recognized as a token
everywhere except within ``preprocessor'' lines. Parentheses {\bf ()}
are recognized everywhere.
\end{description}
\subsection{EBNF for preprocessor expressions}
\noindent{\em Lexical conventions:}\/ Syntax definitions use {\em
Extended Backus-Naur Form} (EBNF). This means that vertical bars
\vb separate two or more alternatives, curly braces \{\} indicate
zero or more copies of what they enclose (``Kleene-closure''), and
square brackets $[]$ specify zero or one instances of their enclosed
contents. Round parentheses () are used for grouping. Non-terminal
symbols appear in \nt{this}\/ typeface; terminal symbols are
\tl{underlined}.
\noindent The following set of rules defines the syntax for CM's
preprocessor expressions (\nt{ppexp}):
\begin{tabular}{rcl}
\nt{aatom} &\ar& \nt{number} \vb \nt{cmid} \vb \tl{(} \nt{asum} \tl{)} \vb (\ttl{$\tilde{~}$} \vb \ttl{-}) \nt{aatom} \\
\nt{aprod} &\ar& \{\nt{aatom} (\ttl{*} \vb \tl{div} \vb \tl{mod}) \vb \ttl{/} \vb \ttl{\%} \} \nt{aatom} \\
\nt{asum} &\ar& \{\nt{aprod} (\ttl{+} \vb \ttl{-})\} \nt{aprod} \\
\\
\nt{ns} &\ar& \tl{structure} \vb \tl{signature} \vb \tl{functor} \vb \tl{funsig} \\
\nt{mlsym} &\ar& \nt{ns} \nt{mlid} \\
\nt{query} &\ar& \tl{defined} \tl{(} \nt{cmid} \tl{)} \vb \tl{defined} \tl{(} \nt{mlsym} \tl{)} \\
\\
\nt{acmp} &\ar& \nt{asum} (\ttl{<} \vb \ttl{<=} \vb \ttl{>} \vb \ttl{>=} \vb \ttl{=} \vb \ttl{==} \vb \ttl{<>} \vb \ttl{!=}) \nt{asum} \\
\\
\nt{batom} &\ar& \nt{query} \vb \nt{acmp} \vb (\tl{not} \vb \ttl{!}) \nt{batom} \vb \tl{(} \nt{bdisj} \tl{)} \\
\nt{bcmp} &\ar& \nt{batom} [(\ttl{=} \vb \ttl{==} \vb \ttl{<>} \vb \ttl{!=}) \nt{batom}] \\
\nt{bconj} &\ar& \{\nt{bcmp} (\tl{andalso} \vb \ttl{\&\&})\} \nt{bcmp} \\
\nt{bdisj} &\ar& \{\nt{bconj} (\tl{orelse} \vb \ttl{||})\} \nt{bconj} \\
\\
\nt{ppexp} &\ar& \nt{bdisj}
\end{tabular}
\subsection{EBNF for export lists}
The following set of rules defines the syntax for export lists (\nt{elst}):
\begin{tabular}{rcl}
\nt{guardedexport} &\ar& \{ \nt{export} \} (\tl{\#endif} \vb
\tl{\#else} \{ \nt{export} \} \tl{\#endif} \vb \tl{\#elif} \nt{ppexp}
\nt{guardedexports}) \\
\nt{restline} &\ar& rest of current line up to next newline character \\
\nt{export} &\ar& \nt{mlsym} \vb \tl{\#if} \nt{ppexp}
\nt{guardedexports} \vb \tl{\#error} \nt{restline} \\
\nt{elst} &\ar& \nt{export} \{ \nt{export} \} \\
\end{tabular}
\subsection{EBNF for tool options}
The following set of rules defines the syntax for tool options
(\nt{toolopts}):
\begin{tabular}{rcl}
\nt{pathname} &\ar& \nt{stdpn} \vb \nt{ntvpn} \\
\nt{toolopts} &\ar& \{ \nt{pathname} [\tl{:} (\tl{(} \nt{toolopts} \tl{)} \vb \nt{pathname})] \}
\end{tabular}
\subsection{EBNF for member lists}
The following set of rules defines the syntax for member lists (\nt{members}):
\begin{tabular}{rcl}
\nt{class} &\ar& \nt{stdpn} \\
\nt{member} &\ar& \nt{pathname} [\tl{:} \nt{class}] [\tl{(} \nt{toolopts} \tl{)}] \\
\nt{guardedmembers} &\ar& \nt{members} (\tl{\#endif} \vb \tl{\#else} \nt{members} \tl{\#endif} \vb \tl{\#elif} \nt{ppexp} \nt{guardedmembers}) \\
\nt{members} &\ar& \{ (\nt{member} \vb \tl{\#if} \nt{ppexp}
\nt{guardedmembers} \vb \tl{\#error} \nt{restline}) \}
\end{tabular}
\subsection{EBNF for library descriptions}
The following set of rules defines the syntax for library descriptions
(\nt{library}). Notice that although the syntax used for \nt{version}
is the same as that for \nt{stdpn}, actual version strings will
undergo further analysis according to the rules given in
section~\ref{sec:versions}:
\begin{tabular}{rcl}
\nt{libkw} &\ar& \tl{library} \vb \tl{Library} \vb \tl{LIBRARY} \\
\nt{version} &\ar& \nt{stdpn} \\
\nt{privilege} &\ar& \nt{stdpn} \\
\nt{lprivspec} &\ar& \{ \nt{privilege} \vb \tl{(} \{ \nt{privilege} \} \tl{)} \} \\
\nt{library} &\ar& [\nt{lprivspec}] \nt{libkw} [\tl{(} \nt{version} \tl{)}] \nt{elst} (\tl{is} \vb \tl{IS}) \nt{members}
\end{tabular}
\subsection{EBNF for library component descriptions (group descriptions)}
The main differences between group- and library-syntax can be
summarized as follows:
\begin{itemize}\setlength{\itemsep}{0pt}
\item Groups use keyword \tl{group} instead of \tl{library}.
\item Groups may have an empty export list.
\item Groups cannot wrap privileges, i.e., names of privileges (in
front of the \tl{group} keyword) never appear within parentheses.
\item Groups have no version.
\item Groups have an optional owner.
\end{itemize}
\noindent The following set of rules defines the syntax for library
component (group) descriptions (\nt{group}):
\begin{tabular}{rcl}
\nt{groupkw} &\ar& \tl{group} \vb \tl{Group} \vb \tl{GROUP} \\
\nt{owner} &\ar& \nt{pathname} \\
\nt{gprivspec} &\ar& \{ \nt{privilege} \} \\
\nt{group} &\ar& [\nt{gprivspec}] \nt{groupkw} [\tl{(} \nt{owner} \tl{)}] [\nt{elst}] (\tl{is} \vb \tl{IS}) \nt{members}
\end{tabular}
\section{Full signature of {\tt structure CM}}
Structure {\tt CM} serves as the compilation manager's user interface
and also constitutes the major part of the API. The structure is the
(only) export of library {\tt \$smlnj/cm.cm}. The standard
installation procedure of SML/NJ registers this library for
autoloading at the interactive top level.
\begin{verbatim}
signature CM = sig
val autoload : string -> bool
val make : string -> bool
val recomp : string -> bool
val stabilize : bool -> string -> bool
type 'a controller = { get : unit -> 'a, set : 'a -> unit }
structure Anchor : sig
val anchor : string -> string option controller
val reset : unit -> unit
end
structure Control : sig
val keep_going : bool controller
val verbose : bool controller
val parse_caching : int controller
val warn_obsolete : bool controller
val debug : bool controller
val conserve_memory : bool controller
end
structure Library : sig
type lib
val known : unit -> lib list
val descr : lib -> string
val osstring : lib -> string
val dismiss : lib -> unit
val unshare : lib -> unit
end
structure State : sig
val synchronize : unit -> unit
val reset : unit -> unit
val pending : unit -> string list
end
structure Server : sig
type server
val start : { cmd : string * string list,
name : string,
pathtrans : (string -> string) option,
pref : int } -> server option
val stop : server -> unit
val kill : server -> unit
val name : server -> string
end
val sources :
{ arch: string, os: string } option ->
string -> { file: string, class: string, derived: bool } list option
val symval : string -> int option controller
val load_plugin : string -> bool
val mk_standalone : bool option -> string -> string list option
end
structure CM : CM
\end{verbatim}
\section{Listing of all pre-defined CM identifiers}
\begin{center}
\begin{tabular}{l||c|c|c|c|c|c|c}
& Alpha32 & HP-PA & PowerPC & PowerPC & Sparc & IA32 & IA32 \\
& Unix & Unix & MACOS & Unix & Unix & Unix & Win32 \\
\hline \hline
{\tt ARCH\_ALPHA32} & 1 & & & & & & \\
{\tt ARCH\_HPPA} & & 1 & & & & & \\
{\tt ARCH\_PPC} & & & 1 & 1 & & & \\
{\tt ARCH\_SPARC} & & & & & 1 & & \\
{\tt ARCH\_X86} & & & & & & 1 & 1 \\
{\tt OPSYS\_UNIX} & 1 & 1 & & 1 & 1 & 1 & \\
{\tt OPSYS\_MACOS} & & & 1 & & & & \\
{\tt OPSYS\_WIN32} & & & & & & & 1 \\
{\tt BIG\_ENDIAN} & & & & & 1 & & \\
{\tt LITTLE\_ENDIAN} & 1 & 1 & 1 & 1 & & 1 & 1 \\
{\tt SIZE\_32} & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
{\tt NEW\_CM} & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
{\tt SMLNJ\_VERSION} & \smlmj & \smlmj & \smlmj & \smlmj & \smlmj & \smlmj & \smlmj \\
{\tt SMLNJ\_MINOR\_VERSION} & \smlmn & \smlmn & \smlmn & \smlmn & \smlmn & \smlmn & \smlmn
\end{tabular}
\end{center}
\section{Listing of all CM-specific environment variables}
Most control parameters that affect CM's operation can be adjusted
using environment variables $v_s$ at startup time, i.e, when the {\tt
sml} command is invoked. Each such parameter has a default setting.
Default settings are determined at bootstrap time, i.e., the time when
the heap image for SML/NJ's interactive system is
built.\footnote{Normally this is the same as installation time, but
for SML/NJ compiler there is also a {\tt makeml} script for the
purpose of bootstrapping.} At bootstrap time, it is possible to
adjust defaults by using a different set of environment variables
$v_b$. If neither $v_s$ nor $v_b$ were set, a hard-wired fallback
value will be used.
The rule for constructing (the names of) $v_s$ and $v_b$ is the
following: For each adjustable parameter $x$ there is a {\em name
stem}. If the stem for $x$ is $s$, then $v_s = \mbox{\tt CM\_}s$ and
$v_b = v_s\mbox{\tt \_DEFAULT}$.
Since the normal installation procedure for SML/NJ sets some of the
$v_b$ variables at bootstrap time, there are two columns with default
values in the following table. The value labeled {\em fallback} is
the one that would have been used had there been no environment
variable at bootrap time, the one labeled {\em default} is the one the
user will actually see.
To save space, the table lists the stem but not the names for its
associated (longer) $v_s$ and $v_b$. For example, since the the table
shows {\tt VERBOSE} in the row for {\tt CM.Control.verbose}, CM's
per-session verbosity can be adjusted using {\tt CM\_VERBOSE} and the
boot-time default can be set using {\tt CM\_VERBOSE\_DEFAULT}.
\begin{center}
\begin{small}
\begin{tabular}{@{}l||c|c|c|c|p{1.5in}@{}}
{\tt CM.Control.}$c$ & stem & type & fallback & default & default's meaning \\
\hline \hline
{\tt verbose} & {\tt VERBOSE} & {\tt bool} & {\tt true} & same & issue
progess messages \\
{\tt debug} & {\tt DEBUG} & {\tt bool} & {\tt false} & same & do not
issue debug messages \\
{\tt keep\_going} & {\tt KEEP\_GOING} & {\tt bool} & {\tt false} &
same & quit on first error \\
(none) & {\tt PATHCONFIG} & {\tt string} & see below & see below &
standard library directory of SML/NJ installation \\
{\tt parse\_caching} & {\tt PARSE\_CACHING} & {\tt int} & {\tt 100} &
same & at most 100 parse trees will be cached in main memory \\
(none) & {\tt LOCAL\_PATHCONFIG} & {\tt string} & see below & same &
user-specific path configuration file \\
{\tt warn\_obsolete} & {\tt WARN\_OBSOLETE} & {\tt bool} & {\tt true}
& same & issue warnings about obsolete C-style operators in
description files \\
{\tt conserve\_memory} & {\tt CONSERVE\_MEMORY} & {\tt bool} & {\tt
false} & same & avoid repeated I/O operations by keeping certain
information in main memory
\end{tabular}
\end{small}
\end{center}
The fallback for {\tt PATHCONFIG} is {\tt /usr/lib/smlnj-pathconfig},
but the standard installation overrides this and uses {\tt
\$INSTALLDIR/lib/pathconfig} (where {\tt \$INSTALLDIR} is the SML/NJ
installation directory) instead.
The default for the ``local'' path configuration file is {\tt
.smlnj-pathconfig}. This file is located in the user's home directory
(given by the environment variable {\tt \$HOME}).
\section{Listing of all class names and their tools}
\begin{center}
\begin{tabular}{c|l|c|l}
class & file contents & tool & file name suffixes \\
\hline\hline
sml & ML source code & built-in & {\tt .sig}, {\tt .sml}, {\tt .fun} \\
cm & CM description file & built-in & {\tt .cm} \\
mlyacc & ML-Yacc grammar & ml-yacc & {\tt .grm}, {\tt .y} \\
mllex & ML-Lex specification & ml-lex & {\tt .lex}, {\tt .l} \\
mlburg & ML-Burg specification & ml-burg & {\tt .burg} \\
noweb & literate program & noweb & {\tt .nw} \\
make & makefile & make & \\
shell & arbitrary & shell command &
\end{tabular}
\end{center}
\section{Available libraries}
Compiler and interactive system of SML/NJ consist of several hundred
individual compilation units. Like modules of application programs,
these compilation units are also organized using CM libraries.
Some of the libraries that make up SML/NJ are actually the same ones
that application programmers are likely to use, others exist for
organizational purposes only. There are ``plugin'' libraries---mainly
for the CM ``tools'' subsystem---that will be automatically loaded on
demand, and libraries such as {\tt \$smlnj/cmb.cm} can be used to
obtain access to functionality that by default is not present.
\subsection{Libraries for general programming}
Libraries listed in the following table provide a broad palette of
general-purpose programming tools\footnote{Recall that anchored paths
of the form {\tt \$$/x[/\cdots]$} act as an abbreviation for {\tt
\$$x/x[/\cdots]$}.}:
\begin{center}
\begin{tabular}{p{2.3in}||p{2.8in}|c|c}
name & description & installed & loaded \\
\hline\hline
{\tt \$/basis.cm} & Standard Basis Library & always & auto \\
\hline\hline
{\tt \$/ml-yacc-lib.cm} & ML-Yacc library & always & no \\
\hline\hline
{\tt \$/smlnj-lib.cm} & SML/NJ general-purpose utility library &
always & no \\
\hline
{\tt \$/unix-lib.cm} & SML/NJ Unix programming utility library &
optional & no \\
\hline
{\tt \$/inet-lib.cm} & SML/NJ internet programming utility library &
optional & no \\
\hline
{\tt \$/regexp-lib.cm} & SML/NJ regular expression library & optional
& no \\
\hline
{\tt \$/reactive-lib.cm} & SML/NJ reactive programming library &
optional & no \\
\hline
{\tt \$/pp-lib.cm} & SML/NJ pretty-printing library & always & no \\
\hline
{\tt \$/html-lib.cm} & SML/NJ HTML handling library & always & no
\end{tabular}
\end{center}
\subsection{Libraries for controlling SML/NJ's operation}
The following table lists those libraries that provide access to the
so-called {\em visible compiler} infrastructure and to the compilation
manager API.
\begin{center}
\begin{tabular}{p{2.3in}||p{2.5in}|c|c}
name & description & installed & loaded \\
\hline\hline
{\tt \$smlnj/compiler.cm} \newline
{\tt \$smlnj/compiler/current.cm} & visible compiler for current
architecture & always & auto \\
\hline\hline
{\tt \$smlnj/cm.cm} \newline
{\tt \$smlnj/cm/full.cm} & compilation manager & always & auto \\
\hline
{\tt \$smlnj/cm/tools.cm} & API for extending CM with new tools &
always & no \\
\hline\hline
{\tt \$/mllex-tool.cm} & plugin library for class {\tt mllex} & always
& on demand \\
\hline
{\tt \$/lex-ext.cm} & plugin library for extension {\tt .lex} & always
& on demand \\
\hline
{\tt \$/mlyacc-tool.cm} & plugin library for class {\tt mlyacc} &
always & on demand \\
\hline
{\tt \$/grm-ext.cm} & plugin library for extension {\tt .grm} & always
& on demand \\
\hline
{\tt \$/mlburg-tool.cm} & plugin library for class {\tt mlburg} &
always & on demand \\
\hline
{\tt \$/burg-ext.cm} & plugin library for extension {\tt .burg} &
always & on demand \\
\hline
{\tt \$/noweb-tool.cm} & plugin library for class {\tt noweb} & always
& on demand \\
\hline
{\tt \$/nw-ext.cm} & plugin library for extension {\tt .nw} & always &
on demand \\
\hline
{\tt \$/make-tool.cm} & plugin library for class {\tt make} & always &
on demand \\
\hline
{\tt \$/shell-tool.cm} & plugin library for class {\tt shell} & always
& on demand \\
\end{tabular}
\end{center}
\subsection{Libraries for SML/NJ compiler hackers}
The following table lists libraries that provide access to the SML/NJ
{\em bootstrap compiler}. The bootstrap compiler is a derivative of
the compilation manager. In addition to being able to recompile
SML/NJ for the ``host'' system there are also cross-compilers that
can target all of SML/NJ's supported platforms.
\begin{center}
\begin{tabular}{p{2.3in}||p{2.8in}|c|c}
name & description & installed & loaded \\
\hline\hline
{\tt \$smlnj/cmb.cm} \newline
{\tt \$smlnj/cmb/current.cm} & bootstrap compiler for current
architecture and OS & always & no \\
\hline\hline
{\tt \$smlnj/cmb/alpha32-unix.cm} & bootstrap compiler for Alpha/Unix
systems & always & no \\
\hline
{\tt \$smlnj/cmb/hppa-unix.cm} & bootstrap compiler for HP-PA/Unix
systems & always & no \\
\hline
{\tt \$smlnj/cmb/ppc-macos.cm} & bootstrap compiler for PowerPC/Unix
systems & always & no \\
\hline
{\tt \$smlnj/cmb/ppc-unix.cm} & bootstrap compiler for PowerPC/MacOS
systems & always & no \\
\hline
{\tt \$smlnj/cmb/sparc-unix.cm} & bootstrap compiler for Sparc/Unix
systems & always & no \\
\hline
{\tt \$smlnj/cmb/x86-unix.cm} & bootstrap compiler for IA32/Unix
systems & always & no \\
\hline
{\tt \$smlnj/cmb/x86-win32.cm} & bootstrap compiler for IA32/Win32
systems & always & no \\
\hline\hline
{\tt \$smlnj/compiler/alpha32.cm} & visible compiler for
Alpha-specific cross-compiler & always & no \\
\hline
{\tt \$smlnj/compiler/hppa.cm} & visible compiler for
HP-PA-specific cross-compiler & always & no \\
\hline
{\tt \$smlnj/compiler/ppc.cm} & visible compiler for
PowerPC-specific cross-compiler & always & no \\
\hline
{\tt \$smlnj/compiler/sparc.cm} & visible compiler for
Sparc-specific cross-compiler & always & no \\
\hline
{\tt \$smlnj/compiler/x86.cm} & visible compiler for
IA32-specific cross-compiler & always & no \\
\hline
{\tt \$smlnj/compiler/all.cm} & visible compilers for all
architecture-specific cross-compilers and all cross-compilation
bootstrap compilers & always & no \\
\end{tabular}
\end{center}
\subsection{Internal libraries}
For completeness, here is the list of other libraries that are part of
SML/NJ's implementation:
\begin{center}
\begin{tabular}{p{2.8in}||p{2.3in}|c|c}
name & description & installed & loaded \\
\hline\hline
{\tt \$MLRISC/Lib.cm} & utility library for MLRISC backend & always &
no \\
\hline
{\tt \$MLRISC/Control.cm} & control facilities for MLRISC backend &
always & no \\
\hline
{\tt \$MLRISC/MLRISC.cm} & architecture-neutral core of MLRISC backend
& always & no \\
\hline
{\tt \$MLRISC/ALPHA.cm} & Alpha-specific MLRISC backend & always & no \\
\hline
{\tt \$MLRISC/HPPA.cm} & HP-PA-specific MLRISC backend & always & no \\
\hline
{\tt \$MLRISC/PPC.cm} & PowerPC-specific MLRISC backend & always & no \\
\hline
{\tt \$MLRISC/SPARC.cm} & Sparc-specific MLRISC backend & always & no \\
\hline
{\tt \$MLRISC/IA32.cm} & IA32-specific MLRISC backend & always & no \\
\hline\hline
{\tt \$/comp-lib.cm} & utility library for compiler & always & no \\
\hline
{\tt \$smlnj/viscomp/core.cm} & architecture-neutral core of compiler
& always & no \\
\hline
{\tt \$smlnj/viscomp/alpha32.cm} & Alpha-specific part of compiler &
always & no \\
\hline
{\tt \$smlnj/viscomp/hppa.cm} & HP-PA-specific part of compiler &
always & no \\
\hline
{\tt \$smlnj/viscomp/ppc.cm} & PowerPC-specific part of compiler &
always & no \\
\hline
{\tt \$smlnj/viscomp/sparc.cm} & Sparc-specific part of compiler &
always & no \\
\hline
{\tt \$smlnj/viscomp/x86.cm} & IA32-specific part of compiler & always
& no \\
\hline \hline
{\tt \$smlnj/init/init.cmi} & initial ``glue''; implementation of
pervasive environment & always & no \\
\hline \hline
{\tt \$smlnj/internal/cm-lib.cm} & implementation of compilation
manager (not yet specialized to specific backends) & always & no \\
\hline
{\tt \$smlnj/internal/host-compiler-0.cm} & selection of host-specific
visible compiler and specialization of compilation manager & always &
no \\
\hline
{\tt \$smlnj/internal/intsys.cm} & root library implementing the
interactive system and glueing all the other parts together & always &
no
\end{tabular}
\end{center}
\pagebreak
\bibliography{blume,appel,ml}
\end{document}
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\documentclass{article}
\usepackage{graphicx}
\usepackage{makeidx}
\makeindex
\begin{document}
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{Table of Contents}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
% This is an example of generating a table of contents, and list of
% figures and tables.
\tableofcontents
\listoffigures
\listoftables
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{Index Example}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\noindent This is an example of generating an index.
\\
\\
As you learned in the first article, in LaTeX \index{LaTeX} you can manage your
bibliographic citations using BibTeX\index{BibTeX}. The index is
printed on the next page.
\printindex
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{BibTeX Example}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\noindent This is an example of BibTeX.
\\
\\
Here I cite Wilson \cite{wilson-ltxx}, then Horspool
\cite{horspool-analysis}. Notice that source.bib contains three
entries. LaTeX only displays the cited references. To cite the other
reference, you can use the \\nocite command.
\bibliographystyle{plain}
\bibliography{examples}
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{Footnote Example}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
This is an example of the LaTeX footnote command.\footnote{Place the text for
the footnote here.}
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{Table Example}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\noindent Three examples of simple tables.
\begin{table}[h]
\centering
\caption{Simple Table - Top 5 Hockey Players of All Time}
\begin{tabular}{|l|c|r|}
\hline
Name & Number & Main Team \\
\hline
Bobby Orr & 4 & Boston \\
Wayne Gretzky & 99 & Edmonton \\
Mario Lemieux & 66 & Pittsburgh \\
Gordie Howe & 9 & Detroit \\
Maurice Richard & 9 & Montreal \\
\hline
\end{tabular}
\end{table}
\begin{table}[h]
\caption{Simple Table with Double Lines - Top 5 Hockey Players of All Time}
\centering
\begin{tabular}{|l||c||r|}
\hline
\textbf{Name} & \textbf{Number} & \textbf{Main Team} \\
\hline \hline
Bobby Orr & 4 & Boston \\
Wayne Gretzky & 99 & Edmonton \\
Mario Lemieux & 66 & Pittsburgh \\
Gordie Howe & 9 & Detroit \\
Maurice Richard & 9 & Montreal \\
\hline
\end{tabular}
\end{table}
\begin{table}[h]
\caption{Extended Table - Top 5 Hockey Players of All Time}
\centering
\begin{tabular*}{6in}{|l@{\extracolsep{\fill}}cr|}
\hline
Name & Number & Main Team \\
\hline
Bobby Orr & 4 & Boston \\
Wayne Gretzky & 99 & Edmonton \\
Mario Lemieux & 66 & Pittsburgh \\
Gordie Howe & 9 & Detroit \\
Maurice Richard & 9 & Montreal \\
\hline
\end{tabular*}
\end{table}
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{Figures Example}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\noindent This is an example of a figure.
\begin{figure}[h]
\centering
\caption{Mississippi John Hurt}
\includegraphics{mjh.jpg}
\end{figure}
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{Graphics Example}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\noindent This is an example of inserting a EPS file into LaTeX
document. The LaTeX file is processed with latex.
\\
% \includegraphics{mjh.eps}
% This is an example of inserting a JPG file into a LaTeX
% document. The LaTeX file is processed with pdflatex.
\includegraphics{mjh.jpg}
Mississippi John Hurt, the great Delta Blues musician.
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{Math Example}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\noindent This is an example of embedding math in a document.
\\
\\
The time-points $x_{i}$ to $x_{j}$ are embedded in the main text.
\\
\\
\noindent This is an example of math enclosed in the math command.
\\
\\
\begin{math}
(l_{1} \leq x_{j} - x_{i} \leq u_{i}) \vee ... \vee (l_{n} \leq x_{j} - x_{i} \leq u_{n})
\end{math}
\pagebreak
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\section{List Example}
%=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
\noindent This is an example of the description list.
\begin{description}
\item[Item 1] This is the first item in the description list
\item[Item 2] This is the second item in the description list
\end{description}
\noindent This is an example of the enumerate list.
\begin{enumerate}
\item This is the first item in the enumerate list
\item This is the second item in the enumerate list
\end{enumerate}
\noindent This is an example of the itemize list.
\begin{itemize}
\item This is the first item in the itemize list
\item This is the second item in the itemize list
\end{itemize}
\noindent This is an example of a nested list.
\begin{enumerate}
\item This is the first item in the nested list
\begin{itemize}
\item Item 1 of the sub list
\begin{itemize}
\item Item 1 of the sub/sub list
\end{itemize}
\end{itemize}
\item This is the second item in the nested list
\end{enumerate}
\end{document}
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%---------------------------------------------------------
% Sample LaTex template for preparing your 1-page FOMMS abstract
% You shouldn't need to mess with the commands below for most LaTex
% distributions.
% You will need to add the title, authors, author affiliations, the
% main abstract body and and referenences. It should be obvious how to
% do this by looking at the file.
% Problems? Send an email to the fomms.org mailing list
%---------------------------------------------------------
\documentclass [12pt]{article}
% Packages - these should be all you need
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[superscript]{cite}
\usepackage[T1]{fontenc}
\usepackage{ae,aecompl}
% Page Format - should not need to modify
\topmargin=0pt
\oddsidemargin=0pt
\evensidemargin=0pt
\parindent=0pt
\textheight=220truemm
\textwidth=170truemm
% Definitions - don't modify
\makeatletter
\renewenvironment{samepage}%
{\begin{center}\begin{minipage}{\textwidth}}%
{\end{minipage}\end{center}\vspace{2mm}}%
\newskip\abstlineskip%
\setlength\abstlineskip{15pt plus 10pt minus 5pt}%
\def\abstinput#1{%
\setlength\parskip{0pt}
\setcounter{equation}{0} % reset "equation"
\def\theequation{\arabic{equation}}
\input{#1}%
\vspace{\abstlineskip}%
}%
\makeatother
%----------------------------------------------------
\begin{document}
% ---------------- Begin document --------------------------------
\begin{samepage}
\begin{center}
\Large \textbf{
% Please add the title of your abstract here (14 pt boldface)%
My FOMMS Abstract Title
}
% ---------------- Authors section -------------------------------
\vspace{1cm}
\normalsize
% % Add author names. Please make sure the presenter's name is
% % underlined. Use superscripts to list affiliations
Joseph Student$^1$, Jane Postdoc$^2$, Robert Undergrad$^1$ and \underline{Mary Professor}$^3$
\vspace{5mm}
% %------------- Affiliations section-----------------------------
% Author affiliations go here in 12 pt italics.
% Try to use line breaks \\ to balance the affiliations
\em{
$^1$ Department of Chemical Engineering, Name of University, \\
Name of City, Name of State, Zip Code, Country. \\[2mm]
$^2$Research Division, Name of Company, Name of
City, Name of State, Zip Code, Country.\\[2mm]
$^3$ Department of Chemistry, Name of University, \\
Name of City, Name of State, Zip Code, Country.
}
\vspace{5mm}
\end{center}
\end{samepage}
\nopagebreak
\normalsize
% -----------------Abstract section-----------------------------------
\centerline{{\bf Abstract}} % Don't change this line
% Please type in abstract below. 1 page limit please!
The abstract should be no longer than one page. Please provide
background information, key findings, and a brief summary of the
results. If you would like to cite references in your abstract, please
cite references in the following way \cite{myref1}. A reference to a
book is \cite{myref2}. You may also add a figure but this is not
required.
% Here is where you list the references. Please use this
% format. Include all authors and titles. You can go as small as 10 pt
% type if you need to, but 12 pt is preferred.
% -----------------References section-----------------------------------
\begin{thebibliography}{99}
\bibitem{myref1}
J. M. ~Smith, Y. T.~Lee and J.~Black, ``Article Title'', {\em Journal Name},
{\bf year}, volume, page numbers.
\bibitem{myref2}
T. R.~Roosevelt and G. ~Marshall, {\em Book title}, publisher, address, year.
\end{thebibliography}
\end{document}
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\documentclass[11pt]{article}
\usepackage{jeffe,handout,graphicx}
\newtheorem{theorem}{Theorem}
% ==========================================================
\begin{document}
\begin{center}
\LARGE
\textbf{Algorithms and Theoretical Computer Science}
\\
\textbf{Ph.D. Qualifying Examination}
\\
\textbf{Spring 2006}
\bigskip
\bigskip
%
%\begin{tabular}{|p{2.5in}|p{2.5in}|}
%\hline
%\multicolumn{2}{|l|}{Name:}
%\\\hline
%Net ID: & Alias:
%\\\hline
%\end{tabular}
%\vfil
\hrule
\normalsize
\begin{itemize}
\item
The exam consists of eight written questions, four in the morning and
four in the afternoon. You will have three hours for each group of four
questions. Please start your answers to each numbered question on a new
sheet of paper.
\item
All else being equal, it is better to solve some of the problems
completely than to get partial credit on every problem.
\end{itemize}
\hrule
%
%\vfil
%\includegraphics[height=4in]{dino-qual.pdf}
\vfil
\LARGE
\begin{tabular}{|c||c|c|c|c||c|c|c|c||c|}
\hline
\# & ~1~ & ~2~ & ~3~ & ~4~ & ~1~ & ~2~ & ~3~ & ~4~ & ~$\Sum$~ \\ \hline\hline
Score & & & & & & & & & \\ \hline
Grader & & & & & & & & & \\ \hline
\end{tabular}
\end{center}
% ----------------------------------------------------------------------
\headers{Algorithms and Theory Qual}{}{Spring 2006}
\section*{Algorithms and Data Structures}
\begin{enumerate}
\parindent2em
% ----------------------
\item % SH
Let $G$ be an undirected complete graph with edge weights that satisfy the triangle inequality. For any integer $r$, an \emph{$r$-tour} of $G$ is a simple cycle $\pi$ such that every vertex in $G$ is within distance $r$ of some vertex in $\pi$ (where distances are computed according to the edge weights).
\begin{itemize}
\item{[5 pts]}
Prove that for any fixed integer $r$, computing the shortest $r$-tour of a given graph is NP-Complete.
\item{[10 pts]}
Show that if we know the \emph{vertices} of the shortest $r$-tour, then we can compute an $r$-tour that is at most twice as long as the shortest $r$-tour.
\item{[35 pts]}
Describe a polynomial-time algorithm that outputs an $\alpha r$-tour of $G$, of length at most $\beta\cdot\textsc{Opt}_r$, where $\textsc{Opt}_r$ is the length of the optimal $r$-tour of $G$ and $\alpha$ and $\beta$ are constants. (You should try to make $\alpha$ and $\beta$ as small as possible, but any constant values are interesting.)
\end{itemize}
% ----------------------
\bigskip\vfil
\item % JE
Suppose you are given a set of $n$ axis-aligned ellipses in the plane, all centered at the origin. Describe and analyze an efficient algorithm to compute the union of these ellipses. The output should be a combinatorial description of the boundary of the union. Don't worry about the underlying algebraic details---assume that you can compute any useful function of a constant number of ellipses in constant time. (But please tell us what those useful functions are!)
\begin{center}
\includegraphics[width=5in]{ellipses}
\end{center}
%% ----------------------
%\bigskip
%\item % SH
%Let $L$ be a multiset of $m$ lines in the plane. The \emph{crossing distance} between two points $p$ and~$q$ in the plane is the number of lines in $L$ that intersect the closed segment $\overline{pq}$.
%\begin{enumerate}
%\item
%An \emph{arrangement vertex} is the intersection point of two lines in $L$. Prove, that for any point $p$ in the plane and any $\delta>8$, there are $\Omega(\delta^2)$ arrangement vertices within crossing distance $\delta$ of $p$.
%\item
%let $P$ be a set of $n$ points in the plane. Prove that there are two points $u,v \in P$ whose crossing distance (with respect to $L$) is at most $O(m /\sqrt{n})$.
%\item
%Prove that
%\[
% \prod_{i=1}^n \left(1 + \frac{1}{\sqrt{i}}\right) \leq \exp( c \sqrt{n})
%\]
%for some constant $c$.
%\item
%Consider the following algorithm for computing a spanning tree of $P$. Starting with $T=\varnothing$, repeat the following process until $P$ contains only one point:
%\begin{itemize}
%\item
%Find the pair of points $u,v\in P$ whose crossing distance (with respect to $L$) is minimized.
%\item
%Add the segment $\overline{uv}$ to $T$.
%\item
%Remove either $u$ or $v$ from $P$. (It doesn't matter which.)
%\item
%Duplicate the lines in $L$ that intersect $\overline{uv}$ and add them back to $L$. (Alternatively, just double the weight of all these lines.)
%\end{itemize}
%Prove that if $m = O(n^2)$, then no line of $L$ crosses more than $O( \sqrt{n} )$ edges of the spanning tree $T$ computed by this algorithm.
%\end{enumerate}
% ----------------------
\bigskip\vfil
\item % SH
Let $\pi(n)$ denote the number of prime numbers smaller than $n$.
\begin{enumerate}
\item
Show that the product of all primes $p$ with $m < p \leq 2m$ is at most $\binom{2m}{m}$.
\item
Using part (a), prove that $\pi(n) =O(n /\log n)$.
\item
Let $p$ be a prime number, and let $m$ and $k$ be natural numbers. Prove that if $p^k$ divides $\binom{2m}{m}$ then $p^k \leq 2m$.
\item
Using part (c), prove that $\pi(n) = \Omega( n / \log n)$.
\end{enumerate}
% ----------------------
\newpage
\item % JE
\begin{enumerate}
\item{[10 pts]}
In a \emph{rooted ordered binary tree}, each node has a right child, a left child, both, or neither. Sketch an efficient algorithm to compute, given two rooted ordered binary trees $A$ and~$B$, the smallest rooted ordered binary tree that contains both $A$ and $B$ as rooted ordered subtrees.
\item{[10 pts]}
In a \emph{rooted binary tree}, each node has zero, one, or two children. Sketch an efficient algorithm to compute, given two rooted binary trees $A$ and~$B$, the smallest rooted binary tree that contains both $A$ and $B$ as rooted subtrees.
\item{[20 pts]}
In a \emph{rooted tree}, each node has zero or more children. Sketch an efficient algorithm to compute, given two rooted trees $A$ and $B$, the smallest rooted tree that contains both $A$ and $B$ as rooted subtrees.
\item{[10 pts]}
A \emph{free tree} is a connected acyclic undirected graph. Sketch an efficient algorithm to compute, given two free trees $A$ and $B$, the smallest free tree that contains both $A$ and~$B$ as subtrees.
\end{enumerate}
\noindent
In each problem, `smallest' means `with the fewest nodes'. In each part, you may use the preceding parts as subroutines. You don't need to describe each algorithm in complete detail; just give a high-level overview and a convincing argument that its running time is polynomial.
%
%% ----------------------
%\item %MP
%\newcommand{\col}[1]{\ensuremath{{\mathrm{col}}(#1)}}
%\newcommand{\minent}[1]{\ensuremath{{\mathrm H}_\infty(#1)}}
%\newcommand{\from}[1]{\ensuremath{\stackrel{#1}{\leftarrow}}}
%\newcommand{\sdiff}[2]{\ensuremath{\Delta(#1,#2)}}
%\newcommand{\hrep}{\ensuremath{\langle h \rangle}}
%\newcommand{\defeq}{\ensuremath{\mathrel{\mathop :}=}}
%\newcommand{\pr}{\ensuremath{\mathrm{Pr}}}
%\def\X{\mathcal{X}}
%In this problem we shall see how a ``pair-wise independent'' hash family can
%be used to ``smooth out'' the randomness in an a probability distribution.
%Roughly, we shall prove the following: If a random hash function from the
%hash family is applied to an input drawn from a distribution which is not
%close to uniform, but has some amount of randomness, then the output
%distribution will be close to uniform and contains almost all the randomness
%used in the process.
%First we define a few standard notions about probability distributions and prove a few useful relations.
%\begin{itemize}
%\item
%A \emph{probability distribution} over a set $\X$ is a function $P:\X\to[0,1]$ such that
%\[
% \sum_{x\in \X} P(x) = 1.
%\]
%\item
%The \emph{collision probability} of a distribution $P$ over $\X$ is
%\[
% \col{P}
% \defeq \pr_{x\from{P}\X, y\from{P}\X}[x=y]
% = \sum_{x\in \X} (P(x))^2.
%\]
%\item
%The \emph{min-entropy} of $P$ is $\minent{P} \defeq -\log_2\left(\max_{x\in \X} P(x)\right)$.
%\item
%The \emph{statistical difference} between two distributions $P$ and $Q$
%(both over $\X$) is
%\[
% \sdiff P Q \defeq \frac12 \sum_{x\in\X} \abs{P(x)-Q(x)}.
%\]
%\end{itemize}
%\begin{enumerate}
%\parindent 1.5em
%\item
%\textbf{Prove that} $\col P \le 2^{-\minent P}$ for any probability distribution $P$.
%\medskip
%\item
%Let $U_m$ denote the uniform distribution over the set $\set{0,1}^m$.
%\textbf{Prove that} for any probability distribution $P$ over $\set{0,1}^m$,
%\[
% \sdiff P {U_m} \le \frac12 \sqrt{2^m \col P - 1}.
%\]
%\Hint{Use the Cauchy-Schwarz inequality to relate $\sum_x \abs{a_x}$ and
%$\sqrt{\sum_x a_x^2}$.}
%\medskip
%\item
%Now consider hash functions from $n$ bits to $\ell$ bits. A collection $\X$ of $D$ such hash functions is said to be a \emph{pairwise independent family} if for every possible pair of \emph{distinct} inputs $x_1,x_2 \in \set{0,1}^n$ and every possible pair of outputs $y_1,y_2\in\set{0,1}^\ell$, the number of hash functions $h\in\X$ such that $h(x_1)=y_1$ and $h(x_2)=y_2$ is exactly $D/\ell^2$.
%Suppose we are given a pairwise independent family $\X$ of $D=2^d$ hash functions. Any hash function in $\X$ can be specified by $d$ bits; let $\hrep$ denote the $d$-bit representation of $h$. Define $f_h:\set{0,1}^n\times\set{0,1}^d\to\set{0,1}^{d+\ell}$ as $f(x,\hrep)=(\hrep,h(x))$.
%Consider a probability distribution $P$ over $\set{0,1}^n$. Let $Z=f(P,U_d)$. That is, let $Z$ be the distribution over $\set{0,1}^m$ obtained as output of $f$, when the input is drawn from $P\times U_d$. Thus $Z(\hrep,y) = 2^{-d}\sum_{x: h(x)=y} P(x)$.
%\textbf{Prove that} $\col Z = 2^{-d} \left(\col P + (1-\col P)2^{-\ell}\right)$.
%\end{enumerate}
%These three claims imply that if $P$ is a distribution over $\set{0,1}^n$ with min-entropy at least $\ell + 2\log\frac1{\epsilon}$, then $\sdiff {f(P,U_d)}{U_m} < \epsilon/2$.
%
%\paragraph{Solution.}
%\newcommand{\expect}{\ensuremath{\mathbf{E}}}
%\begin{enumerate}
%\item
%$\sum_x (P(x))^2 \le (\max_x P(x))(\sum_x P(x)) = \max_x P(x)$.
%\item
%\begin{align*}
% \sdiff P {U_m}
% &\defeq
% \frac12 \sum_x \abs{P(x) - 2^{-m}}
% &\text{[definition of $\Delta$]}
%\\ &\le
% \frac12 \sqrt{2^m \left(\sum_x (P(x) - 2^{-m})^2\right)}
% &\text{[Cauchy-Schwarz $\ne$]}
%\\ &=
% \frac12 \sqrt{2^m \left(\sum_x (P(x))^2 - 2^{-m}\right)}
% &\text{[algebra]}
%\\ &=
% \frac12 \sqrt{2^m \col Z - 1}
% &\text{[definition of col]}
%\end{align*}
%\item
%\begin{align*}
% \col Z
% &=
% \col{U_d}\cdot \expect_{\hrep\from{U_d}\set{0,1}^d}
% \left[ \pr_{x,y\from{P}\set{0,1}^n}[h(x)=h(y)] \right]
%\\ & =
% \col{U_d}\cdot \expect_{\hrep\from{U_d}\set{0,1}^d}
% \left[ \col P +
% (1-\col P)\,\pr_{x,y\from{P}\set{0,1}^n}[h(x)=h(y)\mid x\ne y]
% \right]
%\\ & = 2^{-d}
% \left(\col P +
% (1-\col P)\,
% \expect_{\hrep\from{U_d}\{0,1\}^d}
% \left[\pr_{x,y\from{P}\{0,1\}^n}[h(x)=h(y)\mid x\ne y]\right]
% \right)
%\\ & =
% 2^{-d} (\col P + (1-\col P)2^{-\ell})
%\end{align*}
%\end{enumerate}
\end{enumerate}
% =====================
\newpage
\section*{Formal Languages and Complexity Theory}
\begin{enumerate}
%
%% ----------------------
%\item % LP
%(a) Prove that logspace reductions are closed under composition.
%(b) Use this to prove that if $A$ is $P$-Complete, and $A$ is in $L$ (= \textsc{Logspace}), then $L = P$.
% ----------------------
\bigskip
\item % MV
Suppose $L \in \mathsf{DSPACE}(o(\log\log n))$. Prove that there is an integer $n_0$ such that $L$ can in fact be recognized in $\mathsf{DSPACE}(\log\log n_0)$; in other words, show that $L$ can be recognized in deterministic \emph{constant} space.
\hidesolutions
\begin{solution}
Consider $L$ recognized by $M$ in $o(\log\log n)$
space. We know that $M$ has $2^{o(\log\log n)}$ configurations, and
therefore runs in $o(\log n)$ time. For a position $i$ on the input
tape, the \emph{crossing sequence} at $i$ is the sequence
$aC_0C_1\ldots C_k$ where $a$ is symbol at position $i$, and $C_j$ is
the configuration of $M$ when it reads $a$ for the $j$th time.
Observe that the number of crossing sequences is at most
$(2^{o(\log\log n)})^{(2^{o(\log \log n)})} = 2^{2^{o(\log\log n)}} =
o(n)$. Thus there is an integer $n_0$, such that for all $n \geq n_0$,
the number of crossing sequences is less than $n/3$. We claim that
$L \in \mathsf{DPSACE}(\log\log n_0) = \mathsf{DSPACE}(1)$.
Suppose $x$ is the shortest string longer than $n_0$ on which the
space requirement is more than $\log\log n_0$. Since $\abs{x} > n_0$, we
know that $x$ can be written as $\alpha a\beta a\gamma a\delta$ such that
the $a$'s in the decomposition have the same crossing sequences.
Now, from the definition of crossing sequences, $\alpha a\gamma a
\delta$ and $\alpha a\beta a\delta$ have the same crossing sequences
in $\alpha$, $\beta$, $\gamma$, and $\delta$. Thus, at least one of them
has the same space requirements as $x$ (depending on whether
the maximum space is needed in $\alpha$, $\beta$, $\gamma$, or $\delta$).
So we have a shorter string that requires more than $\log\log n_0$
space, which contradicts minimality at $x$.
\end{solution}
% ----------------------
\bigskip
\item % LP
Tree automata are a generalization of finite state automata, where the inputs are labeled binary trees instead of strings. More precisely, the input is a finite, oriented binary tree where every node is either a \emph{leaf} with no children, or an \emph{internal node} with a left child, a right child, and a \emph{label} from some finite alphabet $\Sigma$.
Formally, a \emph{tree automaton} is a $5$-tuple $(Q, \Sigma, \delta, q_0, F)$, where just as for standard automata, $Q$ is a finite set of states, $F \subseteq Q$ is the set of accepting states, $q_0$ is the initial state, and $\Sigma$ is the input alphabet (in this case, the legal labels of interior nodes of input trees). Finally, $\delta: Q \times Q \times \Sigma \rightarrow Q$ is the transition function, which associates a state with each node of the tree inductively as follows.
\begin{itemize}
\item
The state associated with each leaf is $q_0$.
\item
The state associated with any internal node $v$ is $\delta(q_L, q_R, a)$, where $q_L$ is the state associated with $v$'s left child, $q_R$ is the state associated with $v$'s right child, and $a$ is the label of $v$.
\end{itemize}
A tree is \emph{accepted} if the state associated with the root is an accepting state.
\begin{enumerate}
\item
State and prove a ``pumping lemma" for tree automata that generalizes the standard pumping lemma for finite state automata.
\item
Using your pumping lemma as part of the proof, exhibit a family of trees that is not accepted by any tree automaton.
\end{enumerate}
% ----------------------
%\bigskip
%\item % MV
%Recall that a \emph{deterministic context free language} (DCFL) is a language $L$ that can be recognized by a deterministic pushdown automata by final state. Show that the language $L = \set{a^mb^nc^k \mid \text{either $m \neq n$ or $n \neq k$}}$ is not a DCFL.
%
%{\bf Solution:} Observe that the class of DCFLs (unlike CFLs) is
%closed under complementation. Thus, if $L$ is DCFL then $\bar{L}$ is
%also DCFL. Consider $L' = \bar{L} \cap a^*b^*c^*$; since $L'$ is an
%intersection of a CFL and a regular language $L'$ is a CFL. But $L' =
%\{a^nb^nc^n\: |\: n \geq 0\}$ which can easily shown to be not a CFL
%(by a pumping lemma argument), and hence we get a contradiction. $L$
%is not a DCFL.
% ----------------------
\bigskip
\item % MV
Recall that a \emph{linear context-free grammar} is a context-free grammar where the right-hand side of every production has at most one non-terminal. A language $L$ is said to be a \emph{linear context-free language} if there is a linear CFG $G$ such that $L = L(G)$. Show that for any linear CFL $L$ and any regular language $R$, the language $L \cap R$ is a linear CFL.
% {\bf Solution:} Let $G = (V, \Sigma, P, S)$ be the linear grammar
% generating $L$ and let $D = (Q, \Sigma, q_0, \delta, F)$ be the DFA
% reconizing $R$. Construct a grammar $G'$ for $L \cap R$ whose
% variables are of the form $[pAq]$, where $A \in V$ and $p,q \in Q$.
% The intuition is that $[pAq]$ will generate strings $w$ iff $w$ is
% generated by $A$, and the DFA $D$ goes from $p$ to $q$ on $w$.
% Defining the rules for grammar $G'$ is straightforward based on this
% intuition.
% ----------------------
\bigskip
\item % LP
The \emph{leaf complexity} of a Boolean function $f:\set{0,1}^n\to\set{0,1}$, denoted
$LC(f)$, is defined as the fewest number of leaves in any formula over the gate set $\{\lor, \land, \lnot\}$ that computes $f$.
\begin{enumerate}
\item
Let $\oplus_n$ denote the parity function on $n$ input bits. Show that $LC(\oplus_n) \leq n^2$. You may assume that $n$ is a power of two.
\item
A \emph{formal complexity measure} is a function $FC : \left(\set{0,1}^n\to\set{0,1}\right)\to\Natural$ that maps $n$-ary Boolean functions to natural numbers and that has the following properties:
\begin{itemize}
\item \emph{Atomicity:} $FC(x_i)$ = 1 for any index $i$.
\item \emph{Symmetry:} $FC(f) = FC(\neg f)$ for any function $f$.
\item \emph{Subadditivity:} $FC(f \lor g) \leq FC(f) + FC(g)$ for any functions $f$ and $g$.
\end{itemize}
Prove that $FC(f) \leq LC(f)$ for any formal complexity measure $FC$ and any $n$-ary Boolean function $f$.
\end{enumerate}
\end{enumerate}
\end{document}
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\[\int_{0}^{1}\mathop{B_{n}\/}\nolimits\!\left(t\right)\mathop{B_{m}\/}\nolimits%
\!\left(t\right)dt=\frac{(-1)^{n-1}m!n!}{(m+n)!}\mathop{B_{m+n}\/}\nolimits.\]
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\[\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits%
\theta\right)=\pi^{{1/2}}2^{{\mu}}(\mathop{\sin\/}\nolimits\theta)^{{\mu}}\*%
\sum_{{k=0}}^{{\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+k+1%
\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+k+\frac{3}{2}\right)}\frac{%
\left(\mu+\frac{1}{2}\right)_{{k}}}{k!}\*\mathop{\cos\/}\nolimits\!\left((\nu+%
\mu+2k+1)\theta\right).\]
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%%
%% The LaTeX Companion, 2ed (second printing August 2004)
%%
%% Example 3-3-25 on page 147.
%%
%% Copyright (C) 2004 Frank Mittelbach, Michel Goossens,
%% Johannes Braams, David Carlisle, and Chris Rowley
%%
%% It may be distributed and/or modified under the conditions
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\bibitem[{Ciancamerla et~al.}(2003){Ciancamerla, Minichino, Serro \& Tronci}]{Ciancamerla_etal2003} Ester Ciancamerla, Michele Minichino, Stefano Serro, and Enrico Tronci. "Automatic Timeliness Verification of a Public Mobile Network." In \textit{22nd International Conference on Computer Safety, Reliability, and Security (SAFECOMP)}, edited by S. Anderson, M. Felici and B. Littlewood, 35--48. Lecture Notes in Computer Science 2788. Edinburgh, UK: Springer, 2003. ISSN: 978-3-540-20126-7. DOI: 10.1007/978-3-540-39878-3_4.
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\begin{document}
%
% ===========================以下本文===============================
%
\def\namae{%
{\Large\gtfamily\bfseries 数学A \quad 授業プリント \# 22}\hfill
年\hskip8ex 組\hskip8ex 号 \hskip5ex
\hfill
\underline{氏名\rule[-2ex]{0cm}{6ex}\hskip30ex}}
\namae
\vskip-2ex
{\large\gtfamily\bfseries ■ 確率（独立試行）}
\toi A,B 2人の野球選手が打席でヒットを打つ確率は
それぞれ \( {\,1\, \over 3}, {\,1\, \over 4} \) である
とき，次の確率を求めなさい。
\nidangumi{\subtoi 2人ともヒットを打つ \kotae{{1\over12}} }
{\subtoi 2人ともヒットを打たない \kotae{{1\over2}} }\vfill
\nidangumi{\subtoi どちらか1人がヒットを打つ \kotae{{5\over12}} }
{}\vfill
\toi A,B,Cの3人が簿記検定試験に合格する確率は，
それぞれ\( {\,3\, \over 8}, {\,2\, \over 5}, {\,1\, \over 3} \)で
あるという。このとき，次の確率を求めなさい。
\ajKakko{1} Aが不合格になる \kotae{{5\over8}} \hfill
\ajKakko{2} Bが不合格になる \kotae{{3\over5}} \hfill
\ajKakko{3} Cが不合格になる \kotae{{2\over3}} \hfill\mbox{}\vfill
■ これらのことを使って次の問いに答えよ。
\ajKakko{4} 3人とも合格する \kotae{{1\over20}} \hfill
\ajKakko{5} 3人とも合格しない \kotae{{1\over4}} \hfill
\ajKakko{6} 少なくとも1人は合格する \kotae{{3\over4}} \hfill\mbox{}\vfill
\nidangumi{\ajKakko{7} 1人だけが合格する \kotae{{53\over120}} }{\ajKakko{8} 2人が合格する \kotae{{31\over120}} }\vfill
\newpage
\toi A,B,Cの3人が簿記検定試験に合格する確率は，
それぞれ\( {\,2\, \over 5}, {\,3\, \over 4}, {\,1\, \over 3} \)で
あるという。このとき，次の確率を求めなさい。
\ajKakko{1} 3人とも合格する \kotae{{1\over10}} \hfill
\ajKakko{2} 3人とも合格しない \kotae{{1\over10}} \hfill
\ajKakko{3} 少なくとも1人は合格する \kotae{{9\over10}} \hfill\mbox{}\vfill
\nidangumi{\ajKakko{4} 1人だけが合格する \kotae{{5\over12}} }{\ajKakko{5} 2人が合格する \kotae{{23\over60}} }\vfill
\ifnum\make_pdf_with_insatsu>0
\rotatebox{0}{{\normalsize\gtfamily\bfseries 数学プリント\#22}\scriptsize
{\normalsize\ajKaku{1}}~\ajKakko{1}\( {1\over12} \)\ \ajKakko{2}\( {1\over2} \)\ \ajKakko{3}\( {5\over12} \)\
{\normalsize\ajKaku{2}}~\ajKakko{1}\( {5\over8} \)\ \ajKakko{2}\( {3\over5} \)\ \ajKakko{3}\( {2\over3} \)\ \ajKakko{4}\( {1\over20} \) \
\ajKakko{5}\( {1\over4} \) \ \ajKakko{6}\( {3\over4} \) \ \ajKakko{7}\( {53\over120} \) \ \ajKakko{8}\( {31\over120} \)
%{\normalsize\ajKaku{3}}~\ajKakko{1}\( {1\over10} \) \ \ajKakko{2}\( {1\over10} \) \ \ajKakko{3}\( {9\over10} \) \
%\ajKakko{4}\( {5\over12} \) \ \ajKakko{5}\( {23\over60} \)
%{\normalsize\ajKaku{4}}~\ajKakko{1}\( {20\over243} \) \ \ajKakko{2}\( {13\over729} \)
%{\normalsize\ajKaku{5}}~\ajKakko{1}\( {1\over32} \) \ \ajKakko{2}\( {31\over32} \) \
%{\normalsize\ajKaku{6}}~\( {15\over64} \)
}
\rotatebox{0}{\phantom{\normalsize\gtfamily\bfseries 数学プリント\#22}\scriptsize
%{\normalsize\ajKaku{1}}~\ajKakko{1}\( {1\over12} \)\ \ajKakko{2}\( {1\over2} \)\ \ajKakko{3}\( {5\over12} \)\
%{\normalsize\ajKaku{2}}~\ajKakko{1}\( {5\over8} \)\ \ajKakko{2}\( {3\over5} \)\ \ajKakko{3}\( {2\over3} \)\ \ajKakko{4}\( {1\over20} \) \
%\ajKakko{5}\( {1\over4} \) \ \ajKakko{6}\( {3\over4} \) \ \ajKakko{7}\( {53\over120} \) \ \ajKakko{8}\( {31\over120} \)
{\normalsize\ajKaku{3}}~\ajKakko{1}\( {1\over10} \) \ \ajKakko{2}\( {1\over10} \) \ \ajKakko{3}\( {9\over10} \) \
\ajKakko{4}\( {5\over12} \) \ \ajKakko{5}\( {23\over60} \)
%{\normalsize\ajKaku{4}}~\ajKakko{1}\( {20\over243} \) \ \ajKakko{2}\( {13\over729} \)
%{\normalsize\ajKaku{5}}~\ajKakko{1}\( {1\over32} \) \ \ajKakko{2}\( {31\over32} \) \
%{\normalsize\ajKaku{6}}~\( {15\over64} \)
}
\fi
\end{document}
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\bibitem{Fotios+Yao2018} Fotios, S., \& Yao, Q. (2018). The association between correlated colour temperature and scotopic/photopic ratio. \textit{Lighting Research \& Technology}, \textit{in press}.
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\bibitem{Wrobel2002} Wr{\'o}bel Miros{\l}aw S, \textit{Kim jest antropoktonos w J 8,44?}, "Roczniki Teologiczne", t. 4 (2002), s. 77--92.
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%% Example 5-4-2 on page 224.
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%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
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%% See http://www.latex-project.org/lppl.txt for details.
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\include{preamble}
\setcounter{try}{25}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\title{}
\author{Zajj Daugherty}
\date{\today}
\usepackage{multicol}
\begin{document}
\begin{flushright}
SOLUTIONS\\
Homework 6\\
Due 3/20/19
\end{flushright}
%-----------------% HANDOUT 10 %-----------------%
\begin{try}~
\begin{enumerate}[(a)]
\item Consider strings of length 10 consisting of 1's, 2's, and/or 3's.
\begin{enumerate}[(i)]
\item How many of these are there (with no additional restrictions)?
\Ans{$3^{10}$ (three choices for each digit)}
\item How many of these are there that contain exactly three 1's, two 2's, and five 3's?
\begin{ans}
We're counting the anagrams of $1112233333$:
$$\frac{10!}{3!2!5!} = \binom{10}{3}\binom{10-3}{2}\binom{10-3-5}{5}.$$
\end{ans}
\end{enumerate}
\item How many anagrams are there of MISSISSIPPI?
\begin{ans}
There are 4 S's, 4 I's, 2 P's, and 1 M, therefore, there are
$$\frac{11!}{4!4!2!1!} = \binom{11}{4}\binom{11-4}{4}\binom{11-4-4}{2}\binom{11-4-4-2}{1}$$
anagrams.
\end{ans}
\item Suppose you've got eight varieties of doughnuts to choose from at a doughnuts shop.
\begin{enumerate}[(i)]
\item How many ways can you select 6 doughnuts?
\begin{ans}
We're putting 6 indistinguishable objects (our choices) into 8 distinguishable boxes (the varieties of doughnuts):
$$\binom{6+(8-1)}{6}.$$
(This is a ``stars and bars'' problem.)
\end{ans}
\item How many ways can you select a dozen (12) doughnuts?
\begin{ans}
We're putting 12 indistinguishable objects (our choices) into 8 distinguishable boxes (the varieties of doughnuts):
$$\binom{12+(8-1)}{12}.$$
(This is a ``stars and bars'' problem.)
\end{ans}
\item How many ways can you select a dozen doughnuts with at least one of each kind?\\
{[Hint: if there's at least one of each kind, then how many choices are you really making?]}
\begin{ans}Since 8 choices have already been made, we're left with putting 4 indistinguishable objects (our choices) into 8 distinguishable boxes (the varieties of doughnuts):
$$\binom{4+8-1}{4}.$$
(This is still a ``stars and bars'' problem, but accounting for choices already determined.)
\end{ans}
\end{enumerate}
\item How many different combinations of pennies, nickels, dimes, quarters, and half dollars can a jar contain if it has 20 coins in it?
\begin{ans}We're putting 200 indistinguishable objects (instances of coins) into 5 distinguishable boxes (the varieties of coins):
$$\binom{20+5-1}{20}.$$
(This is a ``stars and bars'' problem.)
\end{ans}
\item Counting solutions.
\begin{enumerate}[(i)]
\item How many solutions are there to the equation
$x_1 + x_2 + x_3 = 10,$
where $x_1, x_2$, and $x_3$ are nonnegative integers?
\begin{ans}We're putting 10 indistinguishable objects (one unit at a time) into 3 distinguishable boxes (the value of the variables):
$$\binom{10+3-1}{10}.$$
(This is a ``stars and bars'' problem.)\end{ans}
\item How many solutions are there to the equation
$x_1 + x_2 + x_3 = 10,$
where $x_1, x_2$, and $x_3$ are strictly positive integers?\\
{[Hint: See problem (c)(iii)]}
\begin{ans}Since 3 ``units'' have already been assigned (one to $x_1$, one to $x_2$, and one to $x_3$), we're left with putting 7 indistinguishable objects (our units) into 3 distinguishable boxes (the variables):
$$\binom{7+3-1}{7}.$$
(This is still a ``stars and bars'' problem, but accounting for choices already determined.)
\end{ans}
\item How many solutions are there to the equation
$x_1 + x_2 + x_3 \leq 10,$
where $x_1, x_2$, and $x_3$ are nonnegative integers?\quad
{[}Hint: Use an extra variable $x_4$ such that $x_1 + x_2 + x_3 + x_4 = 10${]}
\begin{ans}
The nonnegative integer solutions to $x_1 + x_2 + x_3 \leq 10$ is the same as the nonnegative integer solutions to $x_1 + x_2 + x_3 + x_4 = 10$ (where $x_4 = 10 -(x_1 + x_2 + x_3)$). So, similarly to the previous part, there are
$$\binom{10+4-1}{10} \qquad \text{ solutions. }$$
(This is still a ``stars and bars'' problem.)
\end{ans}
\end{enumerate}
\end{enumerate}
\end{try}
\pagebreak
\begin{try}
~
\begin{enumerate}[(a)]
\item List the partitions of 6, both as box diagrams and as sequences.
\begin{ans}
\begin{align*}
\PART{6} & \qquad (6)\\
\PART{5,1} & \qquad (5,1)\\
\PART{4,2}& \qquad (4,2)\\
\PART{4,1,1}& \qquad (4,1,1)\\
\PART{3,3}& \qquad (3,3)\\
\PART{3,2,1}& \qquad (3,2,1)\\
\PART{3,1,1}& \qquad (3,1,1)\\
\PART{2,2,2}& \qquad (2,2,2)\\
\PART{2,2,1,1}& \qquad (2,2,1,1)\\
\PART{2,1,1,1,1}& \qquad (2,1,1,1,1)\\
\PART{1,1,1,1,1,1}& \qquad (1,1,1,1,1,1)
\end{align*}
\end{ans}
\item How many ways are there to distribute 6 identical cookies into 6 identical lunch boxes, possibly leaving some empty?
\begin{ans}
This is the number of partitions of 6 with at most 6 parts, of which there are 12 (see part (a)).
\end{ans}
\item How many ways are there to distribute 6 identical snack bars into 4 identical lunch boxes, possibly leaving some empty?
\begin{ans}
This is the number of partitions of 6 with at most 4 parts, of which there are 9 (see part (a)).
\end{ans}
\item How many ways are there to distribute 4 identical apples into 6 identical lunch boxes, possibly leaving some empty?
\begin{ans}
This is the number of partitions of 4 with at most 6 parts, which is the same as the number of partitions with at most 4 parts, of which there are 5:
$$\PART{4}, \quad \PART{3,1}, \quad \PART{2,2} \quad \PART{2,1,1}, \quad \text{ and } \quad \PART{1,1,1,1}.$$
\end{ans}
\end{enumerate}
\end{try}
\pagebreak
\begin{try}~
\begin{enumerate}[(a)]
\item Basic counting:
\begin{enumerate}[(a)]
\item[(i)] How many ways are there to distribute 5 distinguishable objects into 3 distinguishable boxes, possibly leaving some empty?
\Ans{$3^5$}
\medskip
\item[(ii)] How many ways are there to distribute 5 indistinguishable objects into 3 distinguishable boxes, possibly leaving some empty?
\Ans{$\binom{5+3-1}{5}$}
\medskip
\item[(iii)] How many ways are there to distribute 5 distinguishable objects into 3 indistinguishable boxes, possibly leaving some empty?
\begin{ans}
There are
$$S(5,3) + S(5,2) + S(5,1)$$
ways, where $S(n,j)$ is the Stirling number of the second kind.
To compute this number explicitly, you can either use the formula
$$S(n,j) = \frac{1}{j!} \sum_{\ell=0}^{j-1} (-1)^\ell \binom{j}{\ell}(j - \ell)^n;$$
or you can count directly by cases as follows.
\smallskip
\noindent \underline{$ S(5,1) = 1$}: There is one way to put all 5 things into one box.
\smallskip
\noindent \underline{$ S(5,2) = 5+\binom{5}{2} = 15$}: If we split 5 things into two boxes then that split either looks like
$$\{a,b,c,d\}, \{e\} \qquad \text{ or } \qquad \{a,b,c\}, \{d,e\}.$$
In the first case, there are 5 ways to do this (5 ways to choose $e$); in the second case, there are $\binom{5}{2} = 10$ ways to choose $d$ and $e$.
\smallskip
\noindent \underline{$ S(5,3) = \binom{5}{2} + \frac{1}{2} \binom{5}{2} \binom{3}{2} = 25$}: If we split 5 things into three boxes then that split either looks like
$$\{a,b,c\}, \{d\}, \{e\} \qquad \text{ or } \qquad \{a,b\}, \{c,d\}, \{e\}.$$
In the first case, there are $\binom{5}{2}$ to choose $d$ and $e$ (the order doesn't matter since the boxes are indistinguishable--all we care about is that $d$ and $e$ get their own box, and the rest have to share a box). In the second case, there are $\frac{1}{2} \binom{5}{2} \binom{3}{2}$ ways--pick $a$ and $b$, then pick $c$ and $d$, and then divide by the permutations of the first two sets (again since I can't tell the difference between the boxes).
\smallskip
So
$$S(5,3) + S(5,2) + S(5,1) = 25 + 15 + 1.$$
\end{ans}
\item[(iv)] How many ways are there to distribute 5 indistinguishable objects into 3 indistinguishable boxes, possibly leaving some empty?
\begin{ans}
The ways to do this are in bijection with integer partitions of 5 with at most 3 parts, so there are
$$p_3(5) = \left|\left\{ \PART{5}, \PART{4,1}, \PART{3,2}, \PART{3,1,1}, \PART{2,2,2} \right\}\right| = 5$$
ways
\end{ans}
\pagebreak
\item[(v)] How many ways are there to distribute 6 distinguishable objects into 4 indistinguishable boxes, possibly leaving some empty?
\begin{ans}
$$S(6,4) + S(6,3) + S(6,2) + S(6,1)$$
\end{ans}
\item[(vi)] How many ways are there to distribute 6 distinguishable objects into 4 indistinguishable boxes so that each of the boxes contains at least one object?
\Ans{$S(6,4)$}
\end{enumerate}
\bigskip
\item How many ways are there to pack 8 identical DVDs into 5 indistinguishable boxes? How many ways to do this task so that each box contains
at least one DVD?
\begin{ans}
In general,
$$S(8,5) + S(8,4)+ S(8,3)+ S(8,2) + S(8,1);$$
but only $S(8,5)$ if each box contains at least one DVD.
\end{ans}
\item How many ways are there to distribute 5 balls into 7 boxes if
\begin{enumerate}[(i)]
\item both the balls and boxes are labeled? \Ans{$7^5$}
\medskip
\item the balls are labeled, but the boxes are unlabeled?
\begin{ans}
There are
$$S(5,7) + S(5,6) + S(5,5) + S(5,4) + S(5,3) + S(5,2) + S(5,1) = 0 + 0 + 1 + S(5,4) + S(5,3) + S(5,2) + S(5,1)$$
ways.
Note that $S(5,7) = S(5,6) = 0$ since there is no way to leave no box empty when there are more boxes than balls.
\end{ans}
\item the balls are unlabeled, but the boxes are labeled? \Ans{$\binom{5+7-1}{5}$}
\item both the balls and boxes are unlabeled?
\begin{ans}
$$p_7(5) = \left|\left\{\PART{5}, \PART{4,1}, \PART{3,2}, \PART{3,1,1}, \PART{2,2,2}, \PART{2,2,1,1}, \PART{2,1,1,1,1}, \PART{1,1,1,1,1,1}\right\}\right|$$
\end{ans}
\end{enumerate}
\item Repeat parts (i)--(iv) of part (c), adding the condition that each bucket can have at \emph{most} one ball in it.
\begin{enumerate}[(i)]
\item both the balls and boxes are labeled:
\begin{ans}
There are $7*6*5*4*3$ ways (pick 5 boxes from 7 in order without replacement).
\end{ans}
\pagebreak
\item the balls are labeled, but the boxes are unlabeled:
\begin{ans}
Each ball has to go into a separate box, but we can't tell the difference between the buckets. So there's one way.
\end{ans}
\item the balls are unlabeled, but the boxes are labeled:
\begin{ans}
Each ball has to go into a separate box, and we can't tell the difference between the balls, but we can tell which buckets have been filled. So there are $\binom{7}{5}$ ways.
\end{ans}
\item both the balls and boxes are unlabeled:
\begin{ans}
Each ball has to go into a separate box, and we can't tell the difference between the balls or the buckets. So there's one way.
\end{ans}
\end{enumerate}
\end{enumerate}
\end{try}
%--------- Handout 11 ---------------%
\begin{try}
Draw a tree-diagram that tells you how many ways to form the following results, and count the possible outcomes.
\begin{enumerate}[(a)]
\item Strings of 1's and 0's of length-four with three consecutive 0's.
\begin{ans}
$$
\begin{tikzpicture}[yscale=-1]
\node[bV] (e) at (0,0) {};
\node[bV](0) at (-1,1) {};
\node[bV](1) at (1,1) {};
\node[bV](00) at (-1,2) {};
\node[bV](000) at (-1,3) {};
\node[bV](0000) at (-1.5,4) {};
\node[bV](0001) at (-.5,4) {};
\node[bV](10) at (1,2) {};
\node[bV](100) at (1,3) {};
\node[bV](1000) at (1,4) {};
\draw (e) to node[above left] {$0$} (0);
\draw (e) to node[above right] {$1$} (1);
\draw (0) to node[left] {$0$} (00);
\draw (00) to node[left] {$0$} (000);
\draw (000) to node[above left] {$0$} (0000);
\draw (000) to node[above right] {$1$} (0001);
\draw (1) to node[above right] {$0$} (10) to node[above right] {$0$} (100)
to node[above right] {$0$} (1000) ;
\foreach \x in {0000, 0001, 1000}{\node[right, rotate=-90] at (\x) {$\x$};}
\end{tikzpicture}$$
\end{ans}
\pagebreak
\item Subsets of the set $\{3,7,9,11,24\}$ whose elements sum to less than 28.
\begin{ans}
$$
\begin{tikzpicture}
\node[bV] (r) at (0,0){};
\node[bV] (1) at (-4,-1){};
\node[bV] (0) at (6,-1){};
\node[bV] (10) at (-4,-2){};
\node[bV] (01) at (3,-2){};
\node[bV] (00) at (9,-2){};
\node[bV] (100) at (-4,-3){};
\node[bV] (011) at (1.5,-3){};
\node[bV] (010) at (4.5,-3){};
\node[bV] (001) at (7.5,-3){};
\node[bV] (000) at (10.5,-3){};
\node[bV] (1000) at (-4,-4){};
\node[bV] (0111) at (.75,-4){};
\node[bV] (0110) at (2.25,-4){};
\node[bV] (0101) at (3.75,-4){};
\node[bV] (0100) at (5.25,-4){};
\node[bV] (0011) at (6.75,-4){};
\node[bV] (0010) at (8.25,-4){};
\node[bV] (0001) at (9.75,-4){};
\node[bV] (0000) at (11.25,-4){};
\node[bV] (10001) at (-5,-5.5){};
\node[bV] (10000) at (-3,-5.5){};
\node[bV] (01110) at (.75,-5.5){};
\node[bV] (01101) at (1.9,-5.5){};
\node[bV] (01100) at (2.6,-5.5){};
\node[bV] (01011) at (3.4,-5.5){};
\node[bV] (01010) at (4.1,-5.5){};
\node[bV] (01001) at (4.9,-5.5){};
\node[bV] (01000) at (5.6,-5.5){};
\node[bV] (00111) at (6.4,-5.5){};
\node[bV] (00110) at (7.1,-5.5){};
\node[bV] (00101) at (7.9,-5.5){};
\node[bV] (00100) at (8.6,-5.5){};
\node[bV] (00011) at (9.4,-5.5){};
\node[bV] (00010) at (10.1,-5.5){};
\node[bV] (00001) at (10.9,-5.5){};
\node[bV] (00000) at (11.6,-5.5){};
\draw (r) to node[sloped, above] {24 in} (1);
\draw (1) to node[right] {11 out} (10);
\draw (10) to node[right] {9 out} (100);
\draw (100) to node[right] {7 out} (1000);
%
\draw (r) to node[sloped, above] {24 out} (0);
\draw (0) to node[sloped, above] {11 in} (01);
\draw (0) to node[sloped, above] {11 out} (00);
\draw (01) to node[sloped, above] {9 in} (011);
\draw (01) to node[sloped, above] {9 out} (010);
\draw (00) to node[sloped, above] {9 in} (001);
\draw (00) to node[sloped, above] {9 out} (000);
\foreach \x in {011,010,001,000}
{\draw (\x) to node[sloped, above] {7 in} (\x1);
\draw (\x) to node[sloped, above] {7 out} (\x0);}
\foreach \x in {1000,0110,0101,0100,0011,0010,0001,0000}
{\draw (\x) to node[sloped, above] {3 in} (\x1);
\draw (\x) to node[sloped, above] {3 out} (\x0);}
\draw (0111) to node[sloped, above] {3 out} (01110);
\node[right, rotate=-90] at (10001) {$\{24, 3\}$};
\node[right, rotate=-90] at (10000) {$\{24\}$};
\node[right, rotate=-90] at (01110) {$\{11, 9, 7\}$};
\node[right, rotate=-90] at (01101) {$\{11,9,3\}$};
\node[right, rotate=-90] at (01011) {$\{11,7,3\}$};
\node[right, rotate=-90] at (01100) {$\{11,9\}$};
\node[right, rotate=-90] at (01010) {$\{11,7\}$};
\node[right, rotate=-90] at (01001) {$\{11,3\}$};
\node[right, rotate=-90] at (01000) {$\{11\}$};
\node[right, rotate=-90] at (00111) {$\{9,7,3\}$};
\node[right, rotate=-90] at (00110) {$\{9,7\}$};
\node[right, rotate=-90] at (00101) {$\{9,3\}$};
\node[right, rotate=-90] at (00100) {$\{9\}$};
\node[right, rotate=-90] at (00011) {$\{7,3\}$};
\node[right, rotate=-90] at (00001) {$\{3\}$};
\node[right, rotate=-90] at (00010) {$\{7\}$};
\node[right, rotate=-90] at (00000) {$\emptyset$};
\end{tikzpicture}$$
\end{ans}
\end{enumerate}
$\quad$\hfill\emph{To check your answers: (a) 3; (b) 17.}
\end{try}
\begin{try}~
\begin{enumerate}[(a)]
\item Permutations.
\begin{enumerate}[(i)]
\item Find a recurrence relation and initial conditions for the number of permutations of a set with $n$ elements.
\begin{ans}
For each permutation of $n-1$, insert an $n$. There are $n$ ways to do this (insert at the beginning, after the first term, after the second term, \dots, after the $n-1$ term):
$$a_n = na_{n-1}.$$
This needs one initial condition. There is one permutation of one thing, so
$$a_1 = 1.$$
\end{ans}
\item Check your recurrence relation by iteratively calculating the first 5 terms of your sequence, and using the known closed formula for counting permutations.
\begin{ans}
We know that there are $n!$ permutations of $n$ elements.
\begin{align*}
a_2 &= 2a_1 = 2*1 = 2!& \checkmark\\
a_3 &= 3a_2 = 3*2*1 = 3!& \checkmark\\
a_4 &= 4a_3 = 4*3*2*1 = 4!& \checkmark\\
a_5 &= 5a_4 = 5*4*3*2*1 = 5!& \checkmark
\end{align*}
\end{ans}
\end{enumerate}
\pagebreak
\item Bit strings.
\begin{enumerate}[(i)]
\item Find a recurrence relation and initial conditions for the number of bit strings of length $n$ that contain a pair of consecutive $0$s.
\begin{ans}
For every good bit string (a bit string containing at least one pair of consecutive 0's) or length $n$, removing the last bit leave either a good or a bad bit string of length $n-1$. For those that leave a good bit sting, either the last digit is a 1 or a 0, so there are
$$a_{n-1}*2 \qquad \text{ of these.}$$
For those that leave a bad bit string, this means that the $n-1$ bit has to be a 0 and the $n-2$ bit has to be a 1. The rest of the bits are free. Thus there are
$$\text{(total $n-3$ strings) $-$ (number of good $n-3$ strings)} = 2^{n-3} - a_{n-3} \qquad \text{ of these.}$$
So
$$a_n = 2 a_{n-1} + 2^{n-3} - a_{n-3.}$$
This requires three initial conditions. There are no good bit strings of length 1; there is 1 good string of length 2 ($00$), and there are three good strings of length 3 ($000$, $100$, and $001$). So
$$a_1 = 0, \quad a_2 = 1, \quad a_3 = 3.$$
\end{ans}
\item Check your answer for $n=4$ by iteratively using your recurrence relation, and then by listing the possibilities.
\begin{ans}
Decision tree:
$$\begin{tikzpicture}
\node[bV] (root) at (0,0){};
\node[bV] (0) at (-4,-1){};
\node[bV] (1) at (4,-1){};
\node[bV] (00) at (-6,-2){};
\node[bV] (01) at (-2,-2){};
\node[bV] (10) at (2,-2){};
\node[bV] (11) at (6,-2){};
\node[bV] (000) at (-7,-3){};
\node[bV] (001) at (-5,-3){};
\node[bV] (010) at (-2,-3){};
\node[bV] (100) at (2,-3){};
\node[bV] (110) at (6,-3){};
\node[bV] (0000) at (-7.5,-4){};
\node[bV] (0001) at (-6.5,-4){};
\node[bV] (0010) at (-5.5,-4){};
\node[bV] (0011) at (-4.5,-4){};
\node[bV] (0100) at (-2,-4){};
\node[bV] (1000) at (1.5,-4){};
\node[bV] (1001) at (2.5,-4){};
\node[bV] (1100) at (6,-4){};
\foreach \x in {0,1,00,000,001,100}{\draw (\x) to node[above left]{\tiny 0} (\x0);}
\foreach \x in {0,1,00,000,001,100}{\draw (\x) to node[above right]{\tiny 1} (\x1);}
\draw (root) to node[above left]{\tiny 0} (0);
\draw (root) to node[above right]{\tiny 1} (1);
\draw (01) to node[left]{\tiny 0} (010);
\draw (010) to node[left]{\tiny 0} (0100);
\draw (10) to node[left]{\tiny 0} (100);
\draw (110) to node[left]{\tiny 0} (1100);
\draw (11) to node[left]{\tiny 0} (110);
\end{tikzpicture}$$
This shows that there are 8 good 4-strings. Alternatively,
$$a_4 = 2a_3 + 2^1 - a_1 = 2*3 + 2 - 0 = 8 \qquad \checkmark$$
\end{ans}
\end{enumerate}
\pagebreak
\item Climbing stairs.
\begin{enumerate}[(i)]
\item
Find a recurrence relation and initial conditions for the number of ways to climb $n$ stairs if the person climbing the stairs can take one stair or two stairs at a time.
\begin{ans}
Consider the last step taken. This is either two stairs or one. If it's two, there are $a_{n-2}$ ways to lead up to this; if it's one, there are $a_{n-1}$ ways to lead up to this. So
$$a_n = a_{n-1} + a_{n-2}.$$
If there's only one stair, there's one way to climb it. If there are two stairs, the person can take them one at a time, or both at once. So
$$a_1 = 1 \qquad \text{ and } \qquad a_2 = 2.$$
\end{ans}
\item Check your answer for $n=4$ by iteratively using your recurrence relation, and by counting the number of these sequences by hand using a decision tree.
\begin{ans}
$$\begin{tikzpicture}
\node[bV] (root) at (0,0){};
\node[bV] (2) at (-4,-1){};
\node[bV] (1) at (4,-1){};
\node[bV] (22) at (-6,-2){};
\node[bV] (21) at (-2,-2){};
\node[bV] (12) at (2,-2){};
\node[bV] (11) at (6,-2){};
\node[bV] (211) at (-2,-3){};
\node[bV] (121) at (2,-3){};
\node[bV] (112) at (4,-3){};
\node[bV] (111) at (8,-3){};
\node[bV] (1111) at (8,-4){};
\foreach \x in {2,1,11}{\draw (\x) to node[above left]{\tiny 2} (\x2);}
\foreach \x in {2,1,11}{\draw (\x) to node[above right]{\tiny 1} (\x1);}
\draw (root) to node[above left]{\tiny 2} (2);
\draw (root) to node[above right]{\tiny 1} (1);
\foreach \x in {21,12,111}{\draw (\x) to node[left]{\tiny 1} (\x1);}
\end{tikzpicture}$$
This shows that there are 5 ways to climb 4 stairs. Alternatively,
\begin{align*}
a_3 &= a_2 + a_1 = 2+1 = 3\\
a_4&= a_3 + a_2 = 3+ 1 = 5 & \checkmark
\end{align*}
\end{ans}
\item Calculate the number of ways to climb 8 stairs in this way.
\begin{ans}
Continuing from before:
\begin{align*}
a_5 &= a_4 + a_3 = 5+3 = 8\
a_6 &= a_5 + a_4 = 8+5 = 13\\
a_7 &= a_6 + a_5 = 13+8 = 21\\
a_8 &= a_7 + a_6 = 21+13 = 34.
\end{align*}
\end{ans}
\end{enumerate}
\pagebreak
\item Tiling boards.
\begin{enumerate}[(i)]
\item Find a recurrence relation and initial conditions for the number of ways to completely cover a $2 \times n$ checkerboard with $1 \times 2$ dominoes. For example, if $n=3$, one solution is
$$
\begin{tikzpicture}[scale=.8]
\node at (1.5,2.5) {$2 \times 3$ checkerboard:};
\foreach \x/\y in {0/0, 1/1, 2/0}{
\filldraw[black!60] (\x,\y) rectangle (\x+1,\y+1);
}
\foreach \x in {0,1,2,3}{\draw (\x,0) to (\x,2);}
\foreach \x in {0,1,2}{\draw (0,\x) to (3,\x);}
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=.8]
\node at (1.5,2.5) {covered with 3 dominoes:};
\foreach \x/\y in {0/0, 1/1, 2/0}{
\filldraw[black!60] (\x,\y) rectangle (\x+1,\y+1);
}
\foreach \x in {0,1,2,3}{\draw (\x,0) to (\x,2);}
\foreach \x in {0,1,2}{\draw (0,\x) to (3,\x);}
\draw[fill=black!20] (.1,.1) rectangle (.9,1.9);
\draw[fill=black!20] (1.1,.1) rectangle (2.9,.9);
\draw[fill=black!20] (1.1,1.1) rectangle (2.9,1.9);\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=.8]
\node at (1.5,2.5) {shorthand for same solution:};
%\foreach \x/\y in {0/0, 1/1, 2/0}{
%\filldraw[black!60] (\x,\y) rectangle (\x+1,\y+1);
%}
\foreach \x in {0,1,2,3}{\draw (\x,0) to (\x,2);}
\foreach \x in {0,1,2}{\draw (0,\x) to (3,\x);}
\draw[line width=5pt] (.5,.5) rectangle (.5,1.5);
\draw[line width=5pt] (1.5,.5) rectangle (2.5,.5);
\draw[line width=5pt] (1.5,1.5) rectangle (2.5,1.5);\end{tikzpicture}
$$
{[}Hint: Consider separately the coverings where the position in the top right corner of the checkerboard is covered by a domino positioned horizontally and where it is covered by a domino positioned vertically.{]}
\begin{ans}
If the top right corner is covered by a vertical domino, then the remainder of the board is a tiling of a $2 \times (n-1)$ board, of which there are $a_{n-1}$ ways. If the top right corner is covered by a horizontal domino, then the bottom right corner is tiled by a horizontal domino. The rest of the board is a $2 \times (n-2)$ board, of which there are $a_{n-2}$ ways to do this. So
$$a_n = a_{n-1} + a_{n-2}.$$
There's one way to tile a $2 \times 1$ board, and two ways to tile a $2 \times 2$ board, so
$$a_1 = 1 \qquad \text{ and } \qquad a_2 = 2.$$
\end{ans}
\item Check your answer for $n=4 $ by iteratively using your recurrence relation, and by counting the number of these sequences by hand.
\begin{ans}
The $2 \times 4$ tilings:
$$\begin{tikzpicture}[scale=.5]
\foreach \x in {0,1,2,3,4}{\draw (\x,0) to (\x,2);}
\foreach \x in {0,1,2}{\draw (0,\x) to (4,\x);}
\draw[line width=5pt] (.5,.5) rectangle (.5,1.5);
\draw[line width=5pt] (1.5,.5) rectangle (1.5,1.5);
\draw[line width=5pt] (2.5,.5) rectangle (2.5,1.5);
\draw[line width=5pt] (3.5,.5) rectangle (3.5,1.5);\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=.5]
\foreach \x in {0,1,2,3,4}{\draw (\x,0) to (\x,2);}
\foreach \x in {0,1,2}{\draw (0,\x) to (4,\x);}
\draw[line width=5pt] (.5,.5) rectangle (.5,1.5);
\draw[line width=5pt] (1.5,.5) rectangle (1.5,1.5);
\draw[line width=5pt] (2.5,.5) rectangle (3.5,.5);
\draw[line width=5pt] (2.5,1.5) rectangle (3.5,1.5);\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=.5]
\foreach \x in {0,1,2,3,4}{\draw (\x,0) to (\x,2);}
\foreach \x in {0,1,2}{\draw (0,\x) to (4,\x);}
\draw[line width=5pt] (.5,.5) rectangle (.5,1.5);
\draw[line width=5pt] (1.5,.5) rectangle (2.5,.5);
\draw[line width=5pt] (1.5,1.5) rectangle (2.5,1.5);
\draw[line width=5pt] (3.5,.5) rectangle (3.5,1.5);\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=.5]
\foreach \x in {0,1,2,3,4}{\draw (\x,0) to (\x,2);}
\foreach \x in {0,1,2}{\draw (0,\x) to (4,\x);}
\draw[line width=5pt] (2.5,.5) rectangle (2.5,1.5);
\draw[line width=5pt] (3.5,.5) rectangle (3.5,1.5);
\draw[line width=5pt] (.5,.5) rectangle (1.5,.5);
\draw[line width=5pt] (.5,1.5) rectangle (1.5,1.5);\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=.5]
\foreach \x in {0,1,2,3,4}{\draw (\x,0) to (\x,2);}
\foreach \x in {0,1,2}{\draw (0,\x) to (4,\x);}
\draw[line width=5pt] (2.5,.5) rectangle (3.5,.5);
\draw[line width=5pt] (2.5,1.5) rectangle (3.5,1.5);
\draw[line width=5pt] (.5,.5) rectangle (1.5,.5);
\draw[line width=5pt] (.5,1.5) rectangle (1.5,1.5);\end{tikzpicture}
$$
Alternatively,
$$a_3= 2+1 = 3, \quad \text{ so } \quad a_4 = 3+2 = 5. \qquad \checkmark$$
\end{ans}
\item How many ways are there to completely cover a $2 \times 6$ checkerboard with $1 \times 2$ dominoes?
\begin{ans}
Continuing from above,
$$a_5= 5+3 = 8, \quad \text{ so } \quad a_6 = 8+5 = 13.$$
\end{ans}
\end{enumerate}
\pagebreak
\item Increasing sequences
\begin{enumerate}[(i)]
\item Find a recurrence relation for the number of strictly increasing sequences of positive integers that have $1$ as their first term and $n$ as their last term, where $n$ is a positive integer. That is, sequences $a_1$, $a_2$, \dots, $a_k$, where $a_1=1$, $a_k=n$, and $a_j<a_{j+1}$ for $j= 1,2,\dots,k - 1$.
\begin{ans}
Let $H_n$ be the number of these sequences. Each sequence ending in $n$ either has $n-1$ in it or it doesn't. By removing the last term from a sequence that has $n-1$ in it, you're left with an increasing sequence starting at 1 and ending at $n-1$, of which there are $H_{n-1}$. If the sequence doesn't have an $n-1$ in it, then replacing $n$ with $n-1$ leaves an increasing sequence starting at 1 and ending at $n-1$, of which there are $H_{n-1}$. So
$$H_n = 2 H_{n-1}.$$
There is one sequence starting at 1 and ending at 2, so $H_2 = 1$ (it doesn't make sense to start with $H_2$).
\end{ans}
\item Check your answer for $n=4 $ by iteratively using your recurrence relation, and by counting the number of these sequences by hand using a decision tree.
\begin{ans}
$$\begin{tikzpicture}
\node[bV, label=above:{\tiny 1}] (1) at (0,0){};
\node[bV] (2) at (-4,-1){};
\node[bV] (3) at (0,-1){};
\node[bV] (4) at (4,-1){};
\node[bV] (23) at (-6,-2){};
\node[bV] (234) at (-6,-3){};
\node[bV] (24) at (-4,-2){};
\node[bV] (34) at (0,-2){};
\draw (1) to node[above left]{\tiny 2} (2);
\draw (1) to node[left]{\tiny 3} (3);
\draw (1) to node[above right]{\tiny 4} (4);
\draw (2) to node[above left]{\tiny 3} (23);
\draw (2) to node[ left]{\tiny 4} (24);
\draw (3) to node[ left]{\tiny 4} (34);
\draw (23) to node[ left]{\tiny 4} (234);
\end{tikzpicture}$$
This says there are four such sequences. Alternatively,
$$a_3 = 2a_2 = 2, \quad \text{ and so } \quad a_4 = 2a_3 = 2*2=4 \quad \checkmark.$$
\end{ans}
\item Explain why there are infinitely many such sequences if we replace ``strictly increasing'' with ``weakly increasing'' in part (i), i.e.\ turn ``$<$'' into ``$\leq$''.
\begin{ans}
This will include sequences like $\underbrace{1,1,\dots, 1}_{\text{ arbitrarily many 1's}},n$
\end{ans}
\end{enumerate}
\end{enumerate}
\end{try}
\end{document}
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http://ida.darksky.org/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%2C%20area%20FROM%20refs%20WHERE%20serial%20%3D%202819%20ORDER%20BY%20created_date%20DESC%2C%20created_time%20DESC%2C%20modified_date%20DESC%2C%20modified_time%20DESC%2C%20serial%20DESC&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=5&rowOffset=&wrapResults=1&citeOrder=creation-date&citeStyle=APA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg=
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%&LaTeX
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{Orlando_etal2020} Orlando, L., Ortega, L., \& Defeo, O. (2020). Urbanization effects on sandy beach macrofauna along an estuarine gradient. \textit{Ecological Indicators}, \textit{111}, in press.
\end{thebibliography}
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\[\mathrm{Si}\left(a,z\right)=\Gamma\left(a\right)\sin\left(\tfrac{1}{2}\pi a%
\right)-\mathrm{si}\left(a,z\right),\]
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\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\begin{document}
\parindent=0pt
{\bf Question}
Find the work done in moving a particle in the force field $${\bf
F} = 3x^2 {\bf i} + (2xz - y){\bf j} + z{\bf k}$$
\begin{description}
\item[(a)] along the straight line from (0,0,0) to (2,1,3),
\item[(b)] along the space curve $x = 2t^2, \, y = t, \, z = 4t^2 - t,$
from $t = 0 $ to $ t = 1.$
\item[(c)] Is the work done independent of the path? Explain.
\end{description}
\vspace{.25in}
{\bf Answer}
\begin{description}
\item[(a)]
Straight line $x = 2t$, $y = t$, $z = 3t$ $0 \leq t \leq 1$
\begin{eqnarray*} {\rm Work\ done} & = & \int {\bf F} \cdot d{\bf r} \\ & = &
\int_0^1 {\bf F} \cdot \frac{d {\bf r}}{dt} dt \\ & = & \int_0^1
[3(2t)^2 {\bf i} + (2.2t.3t - t){\bf j} + 3t{\bf k}] \cdot [2{\bf
i} + {\bf j} + 3{\bf k}] dt \\ & = & \int_0^1 [24t^2 + 12t^2 - t +
9t] dt \\ & = & \left[8t^3 + 4t^3 \frac{1}{2}t^2 +
\frac{9}{2}t^2\right]_0^1
\\ & = & 8 + 4 + 4 \\ & = & 16 \end{eqnarray*}
\item[(b)]
\begin{eqnarray*} {\rm Work\ done} & = & \int {\bf F} \cdot d{\bf r} \\ & = &
\int_0^1 {\bf F} \cdot \frac{d {\bf r}}{dt} dt \\ & = & \int_0^1
\left[3((2t)^2)^2 {\bf i} + (2.2t^2(4t^2 - t).(- t)){\bf j} +
(4t^2 - t){\bf k}\right]\\ & &\hspace{.3in} \cdot [4t{\bf i} + {\bf j} +
(8t-1){\bf k}] dt
\\ & = & \int_0^1 \left[48t^5 + 16 t^4 - 4t^3 - t + (4t^t - t)(8t -1)\right] dt \\
& = & \left[8t^6 + \frac{16}{5}t^5 - t^4 - \frac{1}{2}t^2 +
\frac{1}{2}(4t^2 - t)^2\right]_0^1 \\ & = & 8 + \frac{16}{5} - 1 -
\frac{1}{2} + \frac{9}{2} \\ & = & 14\frac{1}{5} \end{eqnarray*}
\item[(c)]
No. The force is not conservative.
\end{description}
\end{document}
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\title{Dangerous Fallacies}
\date{January 1896}
\author{Saverio Merlino}
\subtitle{}
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\par
Anarchists, whether individualistic or communistic, and even some Social Democrats, are fond of speaking of the “absolute sovereignty of the individual,” and they claim for each individual “free access to the means of production.” “Let everybody do whatever he likes,” they say, and the implication is that society will then be organized to perfection, or rather that it will do without organization, individuals will agree or disagree, groups will cooperate spontaneously, without any coercive power, without any settled plan, and without any permanent individual initiative. Every man will go to his work, will choose his own accord or be allowed the occupation most congenial to his own aptitudes, and yet that will happen to be the very sort of work society at that moment is peculiarly in need of. Each individual will likewise consume what he may take a fancy to, consulting but his own pleasure, and yet he will not waste the resources of society—he will not destroy the means for further production, nor appropriate to his secondary needs that which is essential to the subsistence of his fellow men. And it is also said, that in spite of the complications of social relations—of individual interests, in spite of the variety of needs, capabilities, climates, customs, civilizations, etc., no man would try and get the best of his neighbor, each would act in a true spirit of solidarity, and no conflict of any kind would arise, but perfect order and harmony would prevail. And it is sometimes assumed that science would suggest to each individual the right function to perform in society, would prescribe his food, measure his volume of air, light, etc., and would indicate the best purpose to which might be turned each parcel of the soil and each stock of commodities. Indeed, each individual would carry in his head the whole plan of social economy, and, wonderful enough, the plan of each would exactly coincide with those of the hundreds of millions of his fellow men. And ultimately there would be such an abundance of all the good things of this world—each region, perhaps each group if not each individual, would supply all necessary requirements, that even exchanges would not be any longer requisite.
Such things have been said and repeated with an insistence and a good faith worthy of a better cause. No doubt many a great truth underlies such paradoxes—truths which it is all-important to bring home to the people. For instance, it should be known that human society is not even now altogether led by the weak threads called laws, rules, and punishments, handed down by cunning and rapacious men to suit their own interests. There are other forces at play besides police and tribunals—besides rent, profit, and interest. There are ignored or suppressed energies in the masses of the people, the powerful spring of common interests, the manifest advantages of cooperation, and lastly, but not least, the sentiment of solidarity; and these may grow by education and constant practice to become part and parcel of human nature.
But, this admission having been made, we must look the practical difficulties of a social re-organization square in the face, and admit that society is much more complicated than it appears to some people to be. We have to discard the notion of the “perfect individual,” which is at the bottom of many of the views just referred to.
We must also, however unwillingly, refuse to believe that science can provide us with an incontrovertible ready-made solution of the problem of the organization of labor and distribution of the produce. Science may perhaps one day give us the data for such a solution, or rather for a variety of solutions, the number of possible combinations being infinite, but the practical solution must be found out by man in each particular case.
We must also dismiss the supposition of such an abundant supply of the various commodities being at once obtained that men shall have more than they require for the actual satisfaction of their needs. Of course if such an abundant supply of commodities were the immediate result of new social surroundings things might proceed smoothly enough under almost any system. Men’s needs, however, are not a fixed quantity—they admit of indefinite expansion. The production of superfluous commodities is not likely to occur, but as soon as there be enough of a certain commodity other commodities will be produced and the standard or life will be raised.
There is but one argument left in favor of the views which I am criticizing—that the individual will exercise discretion in his choice of labor, and in his choice of consumption—that he will not shirk work, nor take more than his legitimate share of the common stock—that labor will be a pleasure and consumption will be a matter of indifference to him.
Speaking however of the immediate future we must expect there will still be people who, by education, tradition, and instinct, will be willing to live at other people’s expense. It will suffice that a few such people set an example: many more will follow.
But let us waive this objection, and suppose a society composed of the very best men. How could the individual know what particular labor his fellow-men expect of him at any time? How could he know what commodities he might consume without injury to them? How could each group know what raw materials it might receive of other groups? How could it be prevented that one or many groups, severally or jointly, took advantage either of the more favorable situation of their land, factory, mine, or railway, of a new invention, the opening of a road, or even of their own greater industry, skill, or thrift, in order to dictate harsh terms to other groups or individuals, accumulate wealth, and ultimately become a menace to the liberty and well being of the people?
These problems admit of no solution so long as we take our stand on the principle of liberty or the will or the needs of the individual, and leave social interests—(by which I men the permanent interest of a community, in the continuity of its existence over and above the monetary or apparent interests of the individual)—to chance arrangements of individuals.
What the real Anarchist-Socialist solution of these problems might be I will try to explain in another article.
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The Anarchist Library
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Anti-Copyright
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\begin{center}
Saverio Merlino
Dangerous Fallacies
January 1896
\bigskip
https:\Slash{}\Slash{}www.libertarian-labyrinth.org\Slash{}progress-reports\Slash{}saverio-merlino-dangerous-fallacies-1896\Slash{}
S. Merlino, “Dangerous Fallacies,” \emph{Liberty} (London) 3 no. 1 (January, 1896): 2-3.
\bigskip
\textbf{theanarchistlibrary.org}
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\[\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{K_{{\nu}}\/}\nolimits\!%
\left(z\right)\sim\frac{1}{2z}\left(1-\frac{1}{2}\frac{\mu-1}{(2z)^{2}}+\frac{%
1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2z)^{4}}-\cdots\right),\]
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\documentclass{article}
\usepackage{axiom}
\setlength{\textwidth}{400pt}
\begin{document}
\title{\$SPAD/src/input rich1b.input}
\author{Albert Rich and Timothy Daly}
\maketitle
\begin{abstract}
There are:
\begin{itemize}
\item 578 integrals in this file.
\item 578 supplied "optimal results".
\item 467 matching answers. (436 first matches, 31 second matches)
\item 31 cases where Axiom supplied 2 results.
\item 99 cases that Axiom failed to integrate.
\item 28 that contain expressions Axiom does not recognize.
\end{itemize}
There are two classes of failures. Axiom claims
\begin{verbatim}
integrate: implementation incomplete (non-algebraic residues)
\end{verbatim}
but will generate a correct integral if it is re-tried a few times.
Some of the results will only simplify to zero if you assume that
square roots only use the positive branch. Axiom does not like to
make this assumption so it will not simplify to zero such things as:
\[\sqrt(3)\sqrt(7)\sqrt(21)-21\]
\end{abstract}
\eject
\tableofcontents
\eject
\section{Integrands of the form $x^m (a+b x)^n$}
\subsection{Integrands of the form $x^m$}
\subsubsection{Integrands of the form $a$}
\begin{chunk}{*}
)set break resume
)sys rm -f rich1b.output
)spool rich1b.output
)set message test on
)set message auto off
)clear all
--S 1 of 2952
t0000:=x^(5/3)*(a+b*x)
--R
--R
--R 2 3+-+2
--R (1) (b x + a x)\|x
--R Type: Expression(Integer)
--E 1
--S 2 of 2952
r0000:=3/8*a*x^(8/3)+3/11*b*x^(11/3)
--R
--R
--R 3 2 3+-+2
--R (24b x + 33a x )\|x
--R (2) ----------------------
--R 88
--R Type: Expression(Integer)
--E 2
--S 3 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 2 3+-+2
--R (24b x + 33a x )\|x
--R (3) ----------------------
--R 88
--R Type: Union(Expression(Integer),...)
--E 3
--S 4 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 4
--S 5 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 5
)clear all
--S 6 of 2952
t0000:=x^(4/3)*(a+b*x)
--R
--R
--R 2 3+-+
--R (1) (b x + a x)\|x
--R Type: Expression(Integer)
--E 6
--S 7 of 2952
r0000:=3/7*a*x^(7/3)+3/10*b*x^(10/3)
--R
--R
--R 3 2 3+-+
--R (21b x + 30a x )\|x
--R (2) ---------------------
--R 70
--R Type: Expression(Integer)
--E 7
--S 8 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 2 3+-+
--R (21b x + 30a x )\|x
--R (3) ---------------------
--R 70
--R Type: Union(Expression(Integer),...)
--E 8
--S 9 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 9
--S 10 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 10
)clear all
--S 11 of 2952
t0000:=x^(2/3)*(a+b*x)
--R
--R
--R 3+-+2
--R (1) (b x + a)\|x
--R Type: Expression(Integer)
--E 11
--S 12 of 2952
r0000:=3/5*a*x^(5/3)+3/8*b*x^(8/3)
--R
--R
--R 2 3+-+2
--R (15b x + 24a x)\|x
--R (2) ---------------------
--R 40
--R Type: Expression(Integer)
--E 12
--S 13 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 3+-+2
--R (15b x + 24a x)\|x
--R (3) ---------------------
--R 40
--R Type: Union(Expression(Integer),...)
--E 13
--S 14 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 14
--S 15 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 15
)clear all
--S 16 of 2952
t0000:=x^(1/3)*(a+b*x)
--R
--R
--R 3+-+
--R (1) (b x + a)\|x
--R Type: Expression(Integer)
--E 16
--S 17 of 2952
r0000:=3/4*a*x^(4/3)+3/7*b*x^(7/3)
--R
--R
--R 2 3+-+
--R (12b x + 21a x)\|x
--R (2) --------------------
--R 28
--R Type: Expression(Integer)
--E 17
--S 18 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 3+-+
--R (12b x + 21a x)\|x
--R (3) --------------------
--R 28
--R Type: Union(Expression(Integer),...)
--E 18
--S 19 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 19
--S 20 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 20
)clear all
--S 21 of 2952
t0000:=(a+b*x)/x^(1/3)
--R
--R
--R b x + a
--R (1) -------
--R 3+-+
--R \|x
--R Type: Expression(Integer)
--E 21
--S 22 of 2952
r0000:=3/2*a*x^(2/3)+3/5*b*x^(5/3)
--R
--R
--R 3+-+2
--R (6b x + 15a)\|x
--R (2) -----------------
--R 10
--R Type: Expression(Integer)
--E 22
--S 23 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3+-+2
--R (6b x + 15a)\|x
--R (3) -----------------
--R 10
--R Type: Union(Expression(Integer),...)
--E 23
--S 24 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 24
--S 25 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 25
)clear all
--S 26 of 2952
t0000:=(a+b*x)/x^(2/3)
--R
--R
--R b x + a
--R (1) -------
--R 3+-+2
--R \|x
--R Type: Expression(Integer)
--E 26
--S 27 of 2952
r0000:=3*a*x^(1/3)+3/4*b*x^(4/3)
--R
--R
--R 3+-+
--R (3b x + 12a)\|x
--R (2) ----------------
--R 4
--R Type: Expression(Integer)
--E 27
--S 28 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3+-+
--R (3b x + 12a)\|x
--R (3) ----------------
--R 4
--R Type: Union(Expression(Integer),...)
--E 28
--S 29 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 29
--S 30 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 30
)clear all
--S 31 of 2952
t0000:=(a+b*x)/x^(4/3)
--R
--R
--R b x + a
--R (1) -------
--R 3+-+
--R x\|x
--R Type: Expression(Integer)
--E 31
--S 32 of 2952
r0000:=-3*a/x^(1/3)+3/2*b*x^(2/3)
--R
--R
--R 3b x - 6a
--R (2) ---------
--R 3+-+
--R 2\|x
--R Type: Expression(Integer)
--E 32
--S 33 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3b x - 6a
--R (3) ---------
--R 3+-+
--R 2\|x
--R Type: Union(Expression(Integer),...)
--E 33
--S 34 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 34
--S 35 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 35
)clear all
--S 36 of 2952
t0000:=(a+b*x)/x^(5/3)
--R
--R
--R b x + a
--R (1) -------
--R 3+-+2
--R x \|x
--R Type: Expression(Integer)
--E 36
--S 37 of 2952
r0000:=-3/2*a/x^(2/3)+3*b*x^(1/3)
--R
--R
--R 6b x - 3a
--R (2) ---------
--R 3+-+2
--R 2\|x
--R Type: Expression(Integer)
--E 37
--S 38 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 6b x - 3a
--R (3) ---------
--R 3+-+2
--R 2\|x
--R Type: Union(Expression(Integer),...)
--E 38
--S 39 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 39
--S 40 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 40
)clear all
--S 41 of 2952
t0000:=x^(5/3)*(a+b*x)^2
--R
--R
--R 2 3 2 2 3+-+2
--R (1) (b x + 2a b x + a x)\|x
--R Type: Expression(Integer)
--E 41
--S 42 of 2952
r0000:=3/8*a^2*x^(8/3)+6/11*a*b*x^(11/3)+3/14*b^2*x^(14/3)
--R
--R
--R 2 4 3 2 2 3+-+2
--R (132b x + 336a b x + 231a x )\|x
--R (2) ------------------------------------
--R 616
--R Type: Expression(Integer)
--E 42
--S 43 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 4 3 2 2 3+-+2
--R (132b x + 336a b x + 231a x )\|x
--R (3) ------------------------------------
--R 616
--R Type: Union(Expression(Integer),...)
--E 43
--S 44 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 44
--S 45 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 45
)clear all
--S 46 of 2952
t0000:=x^(4/3)*(a+b*x)^2
--R
--R
--R 2 3 2 2 3+-+
--R (1) (b x + 2a b x + a x)\|x
--R Type: Expression(Integer)
--E 46
--S 47 of 2952
r0000:=3/7*a^2*x^(7/3)+3/5*a*b*x^(10/3)+3/13*b^2*x^(13/3)
--R
--R
--R 2 4 3 2 2 3+-+
--R (105b x + 273a b x + 195a x )\|x
--R (2) -----------------------------------
--R 455
--R Type: Expression(Integer)
--E 47
--S 48 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 4 3 2 2 3+-+
--R (105b x + 273a b x + 195a x )\|x
--R (3) -----------------------------------
--R 455
--R Type: Union(Expression(Integer),...)
--E 48
--S 49 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 49
--S 50 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 50
)clear all
--S 51 of 2952
t0000:=x^(2/3)*(a+b*x)^2
--R
--R
--R 2 2 2 3+-+2
--R (1) (b x + 2a b x + a )\|x
--R Type: Expression(Integer)
--E 51
--S 52 of 2952
r0000:=3/5*a^2*x^(5/3)+3/4*a*b*x^(8/3)+3/11*b^2*x^(11/3)
--R
--R
--R 2 3 2 2 3+-+2
--R (60b x + 165a b x + 132a x)\|x
--R (2) ----------------------------------
--R 220
--R Type: Expression(Integer)
--E 52
--S 53 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 3 2 2 3+-+2
--R (60b x + 165a b x + 132a x)\|x
--R (3) ----------------------------------
--R 220
--R Type: Union(Expression(Integer),...)
--E 53
--S 54 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 54
--S 55 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 55
)clear all
--S 56 of 2952
t0000:=x^(1/3)*(a+b*x)^2
--R
--R
--R 2 2 2 3+-+
--R (1) (b x + 2a b x + a )\|x
--R Type: Expression(Integer)
--E 56
--S 57 of 2952
r0000:=3/4*a^2*x^(4/3)+6/7*a*b*x^(7/3)+3/10*b^2*x^(10/3)
--R
--R
--R 2 3 2 2 3+-+
--R (42b x + 120a b x + 105a x)\|x
--R (2) ---------------------------------
--R 140
--R Type: Expression(Integer)
--E 57
--S 58 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 3 2 2 3+-+
--R (42b x + 120a b x + 105a x)\|x
--R (3) ---------------------------------
--R 140
--R Type: Union(Expression(Integer),...)
--E 58
--S 59 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 59
--S 60 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 60
)clear all
--S 61 of 2952
t0000:=(a+b*x)^2/x^(1/3)
--R
--R
--R 2 2 2
--R b x + 2a b x + a
--R (1) ------------------
--R 3+-+
--R \|x
--R Type: Expression(Integer)
--E 61
--S 62 of 2952
r0000:=3/2*a^2*x^(2/3)+6/5*a*b*x^(5/3)+3/8*b^2*x^(8/3)
--R
--R
--R 2 2 2 3+-+2
--R (15b x + 48a b x + 60a )\|x
--R (2) ------------------------------
--R 40
--R Type: Expression(Integer)
--E 62
--S 63 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 2 2 3+-+2
--R (15b x + 48a b x + 60a )\|x
--R (3) ------------------------------
--R 40
--R Type: Union(Expression(Integer),...)
--E 63
--S 64 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 64
--S 65 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 65
)clear all
--S 66 of 2952
t0000:=(a+b*x)^2/x^(2/3)
--R
--R
--R 2 2 2
--R b x + 2a b x + a
--R (1) ------------------
--R 3+-+2
--R \|x
--R Type: Expression(Integer)
--E 66
--S 67 of 2952
r0000:=3*a^2*x^(1/3)+3/2*a*b*x^(4/3)+3/7*b^2*x^(7/3)
--R
--R
--R 2 2 2 3+-+
--R (6b x + 21a b x + 42a )\|x
--R (2) ----------------------------
--R 14
--R Type: Expression(Integer)
--E 67
--S 68 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 2 2 3+-+
--R (6b x + 21a b x + 42a )\|x
--R (3) ----------------------------
--R 14
--R Type: Union(Expression(Integer),...)
--E 68
--S 69 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 69
--S 70 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 70
)clear all
--S 71 of 2952
t0000:=(a+b*x)^2/x^(4/3)
--R
--R
--R 2 2 2
--R b x + 2a b x + a
--R (1) ------------------
--R 3+-+
--R x\|x
--R Type: Expression(Integer)
--E 71
--S 72 of 2952
r0000:=-3*a^2/x^(1/3)+3*a*b*x^(2/3)+3/5*b^2*x^(5/3)
--R
--R
--R 2 2 2
--R 3b x + 15a b x - 15a
--R (2) ----------------------
--R 3+-+
--R 5\|x
--R Type: Expression(Integer)
--E 72
--S 73 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 2 2
--R 3b x + 15a b x - 15a
--R (3) ----------------------
--R 3+-+
--R 5\|x
--R Type: Union(Expression(Integer),...)
--E 73
--S 74 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 74
--S 75 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 75
)clear all
--S 76 of 2952
t0000:=(a+b*x)^2/x^(5/3)
--R
--R
--R 2 2 2
--R b x + 2a b x + a
--R (1) ------------------
--R 3+-+2
--R x \|x
--R Type: Expression(Integer)
--E 76
--S 77 of 2952
r0000:=-3/2*a^2/x^(2/3)+6*a*b*x^(1/3)+3/4*b^2*x^(4/3)
--R
--R
--R 2 2 2
--R 3b x + 24a b x - 6a
--R (2) ---------------------
--R 3+-+2
--R 4\|x
--R Type: Expression(Integer)
--E 77
--S 78 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 2 2
--R 3b x + 24a b x - 6a
--R (3) ---------------------
--R 3+-+2
--R 4\|x
--R Type: Union(Expression(Integer),...)
--E 78
--S 79 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 79
--S 80 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 80
)clear all
--S 81 of 2952
t0000:=x^(5/3)*(a+b*x)^3
--R
--R
--R 3 4 2 3 2 2 3 3+-+2
--R (1) (b x + 3a b x + 3a b x + a x)\|x
--R Type: Expression(Integer)
--E 81
--S 82 of 2952
r0000:=3/8*a^3*x^(8/3)+9/11*a^2*b*x^(11/3)+_
9/14*a*b^2*x^(14/3)+3/17*b^3*x^(17/3)
--R
--R
--R 3 5 2 4 2 3 3 2 3+-+2
--R (1848b x + 6732a b x + 8568a b x + 3927a x )\|x
--R (2) ----------------------------------------------------
--R 10472
--R Type: Expression(Integer)
--E 82
--S 83 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 5 2 4 2 3 3 2 3+-+2
--R (1848b x + 6732a b x + 8568a b x + 3927a x )\|x
--R (3) ----------------------------------------------------
--R 10472
--R Type: Union(Expression(Integer),...)
--E 83
--S 84 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 84
--S 85 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 85
)clear all
--S 86 of 2952
t0000:=x^(4/3)*(a+b*x)^3
--R
--R
--R 3 4 2 3 2 2 3 3+-+
--R (1) (b x + 3a b x + 3a b x + a x)\|x
--R Type: Expression(Integer)
--E 86
--S 87 of 2952
r0000:=3/7*a^3*x^(7/3)+9/10*a^2*b*x^(10/3)+9/13*a*b^2*x^(13/3)+_
3/16*b^3*x^(16/3)
--R
--R
--R 3 5 2 4 2 3 3 2 3+-+
--R (1365b x + 5040a b x + 6552a b x + 3120a x )\|x
--R (2) ---------------------------------------------------
--R 7280
--R Type: Expression(Integer)
--E 87
--S 88 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 5 2 4 2 3 3 2 3+-+
--R (1365b x + 5040a b x + 6552a b x + 3120a x )\|x
--R (3) ---------------------------------------------------
--R 7280
--R Type: Union(Expression(Integer),...)
--E 88
--S 89 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 89
--S 90 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 90
)clear all
--S 91 of 2952
t0000:=x^(2/3)*(a+b*x)^3
--R
--R
--R 3 3 2 2 2 3 3+-+2
--R (1) (b x + 3a b x + 3a b x + a )\|x
--R Type: Expression(Integer)
--E 91
--S 92 of 2952
r0000:=3/5*a^3*x^(5/3)+9/8*a^2*b*x^(8/3)+9/11*a*b^2*x^(11/3)+_
3/14*b^3*x^(14/3)
--R
--R
--R 3 4 2 3 2 2 3 3+-+2
--R (660b x + 2520a b x + 3465a b x + 1848a x)\|x
--R (2) --------------------------------------------------
--R 3080
--R Type: Expression(Integer)
--E 92
--S 93 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 4 2 3 2 2 3 3+-+2
--R (660b x + 2520a b x + 3465a b x + 1848a x)\|x
--R (3) --------------------------------------------------
--R 3080
--R Type: Union(Expression(Integer),...)
--E 93
--S 94 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 94
--S 95 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 95
)clear all
--S 96 of 2952
t0000:=x^(1/3)*(a+b*x)^3
--R
--R
--R 3 3 2 2 2 3 3+-+
--R (1) (b x + 3a b x + 3a b x + a )\|x
--R Type: Expression(Integer)
--E 96
--S 97 of 2952
r0000:=3/4*a^3*x^(4/3)+9/7*a^2*b*x^(7/3)+9/10*a*b^2*x^(10/3)+_
3/13*b^3*x^(13/3)
--R
--R
--R 3 4 2 3 2 2 3 3+-+
--R (420b x + 1638a b x + 2340a b x + 1365a x)\|x
--R (2) -------------------------------------------------
--R 1820
--R Type: Expression(Integer)
--E 97
--S 98 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 4 2 3 2 2 3 3+-+
--R (420b x + 1638a b x + 2340a b x + 1365a x)\|x
--R (3) -------------------------------------------------
--R 1820
--R Type: Union(Expression(Integer),...)
--E 98
--S 99 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 99
--S 100 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 100
)clear all
--S 101 of 2952
t0000:=(a+b*x)^3/x^(1/3)
--R
--R
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R (1) ----------------------------
--R 3+-+
--R \|x
--R Type: Expression(Integer)
--E 101
--S 102 of 2952
r0000:=3/2*a^3*x^(2/3)+9/5*a^2*b*x^(5/3)+9/8*a*b^2*x^(8/3)+3/11*b^3*x^(11/3)
--R
--R
--R 3 3 2 2 2 3 3+-+2
--R (120b x + 495a b x + 792a b x + 660a )\|x
--R (2) ---------------------------------------------
--R 440
--R Type: Expression(Integer)
--E 102
--S 103 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 3 2 2 2 3 3+-+2
--R (120b x + 495a b x + 792a b x + 660a )\|x
--R (3) ---------------------------------------------
--R 440
--R Type: Union(Expression(Integer),...)
--E 103
--S 104 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 104
--S 105 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 105
)clear all
--S 106 of 2952
t0000:=(a+b*x)^3/x^(2/3)
--R
--R
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R (1) ----------------------------
--R 3+-+2
--R \|x
--R Type: Expression(Integer)
--E 106
--S 107 of 2952
r0000:=3*a^3*x^(1/3)+9/4*a^2*b*x^(4/3)+9/7*a*b^2*x^(7/3)+3/10*b^3*x^(10/3)
--R
--R
--R 3 3 2 2 2 3 3+-+
--R (42b x + 180a b x + 315a b x + 420a )\|x
--R (2) -------------------------------------------
--R 140
--R Type: Expression(Integer)
--E 107
--S 108 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 3 2 2 2 3 3+-+
--R (42b x + 180a b x + 315a b x + 420a )\|x
--R (3) -------------------------------------------
--R 140
--R Type: Union(Expression(Integer),...)
--E 108
--S 109 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 109
--S 110 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 110
)clear all
--S 111 of 2952
t0000:=(a+b*x)^3/x^(4/3)
--R
--R
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R (1) ----------------------------
--R 3+-+
--R x\|x
--R Type: Expression(Integer)
--E 111
--S 112 of 2952
r0000:=-3*a^3/x^(1/3)+9/2*a^2*b*x^(2/3)+9/5*a*b^2*x^(5/3)+3/8*b^3*x^(8/3)
--R
--R
--R 3 3 2 2 2 3
--R 15b x + 72a b x + 180a b x - 120a
--R (2) ------------------------------------
--R 3+-+
--R 40\|x
--R Type: Expression(Integer)
--E 112
--S 113 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 3 2 2 2 3
--R 15b x + 72a b x + 180a b x - 120a
--R (3) ------------------------------------
--R 3+-+
--R 40\|x
--R Type: Union(Expression(Integer),...)
--E 113
--S 114 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 114
--S 115 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 115
)clear all
--S 116 of 2952
t0000:=(a+b*x)^3/x^(5/3)
--R
--R
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R (1) ----------------------------
--R 3+-+2
--R x \|x
--R Type: Expression(Integer)
--E 116
--S 117 of 2952
r0000:=-3/2*a^3/x^(2/3)+9*a^2*b*x^(1/3)+9/4*a*b^2*x^(4/3)+3/7*b^3*x^(7/3)
--R
--R
--R 3 3 2 2 2 3
--R 12b x + 63a b x + 252a b x - 42a
--R (2) -----------------------------------
--R 3+-+2
--R 28\|x
--R Type: Expression(Integer)
--E 117
--S 118 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 3 3 2 2 2 3
--R 12b x + 63a b x + 252a b x - 42a
--R (3) -----------------------------------
--R 3+-+2
--R 28\|x
--R Type: Union(Expression(Integer),...)
--E 118
--S 119 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 119
--S 120 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 120
)clear all
--S 121 of 2952
t0000:=x^(5/3)/(a+b*x)
--R
--R
--R 3+-+2
--R x \|x
--R (1) -------
--R b x + a
--R Type: Expression(Integer)
--E 121
--S 122 of 2952
r0000:=-3/2*a*x^(2/3)/b^2+3/5*x^(5/3)/b-a^(5/3)*log(a^(1/3)+_
b^(1/3)*x^(1/3))/b^(8/3)+1/2*a^(5/3)*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/b^(8/3)-_
a^(5/3)*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))*sqrt(3)/b^(8/3)
--R
--R
--R (2)
--R 3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R 5a \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+2 3+-+3+-+ 3+-+ +-+3+-+2 2\|b \|x - \|a
--R - 10a \|a log(\|b \|x + \|a ) + 10a\|3 \|a atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R 3+-+2 3+-+2
--R (6b x - 15a)\|b \|x
--R /
--R 2 3+-+2
--R 10b \|b
--R Type: Expression(Integer)
--E 122
--S 123 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +----+ +----+2 +----+
--R | 2 | 2 | 2
--R | a 3+-+2 | a 3+-+ | a
--R - 5a |- -- log(a \|x - b |- -- \|x - a |- --)
--R 3| 2 3| 2 3| 2
--R \| b \| b \| b
--R +
--R +----+ +----+2
--R | 2 | 2
--R | a 3+-+ | a
--R 10a |- -- log(a\|x + b |- -- )
--R 3| 2 3| 2
--R \| b \| b
--R +
--R +----+2
--R | 2
--R 3+-+ | a
--R +----+ 2a\|x - b |- --
--R | 2 3| 2
--R +-+ | a \| b 3+-+2
--R - 10a\|3 |- -- atan(------------------- + (6b x - 15a)\|x
--R 3| 2 +----+2
--R \| b | 2
--R +-+ | a
--R b\|3 |- --
--R 3| 2
--R \| b
--R /
--R 2
--R 10b
--R Type: Union(Expression(Integer),...)
--E 123
--S 124 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - a \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +----+ +----+2 +----+
--R | 2 | 2 | 2
--R | a 3+-+2 3+-+2 | a 3+-+ | a
--R - a |- -- \|b log(a \|x - b |- -- \|x - a |- --)
--R 3| 2 3| 2 3| 2
--R \| b \| b \| b
--R +
--R +----+ +----+2
--R | 2 | 2
--R 3+-+2 3+-+3+-+ 3+-+ | a 3+-+2 3+-+ | a
--R 2a \|a log(\|b \|x + \|a ) + 2a |- -- \|b log(a\|x + b |- -- )
--R 3| 2 3| 2
--R \| b \| b
--R +
--R 3+-+3+-+ 3+-+
--R +-+3+-+2 2\|b \|x - \|a
--R - 2a\|3 \|a atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +----+2
--R | 2
--R 3+-+ | a
--R +----+ 2a\|x - b |- --
--R | 2 3| 2
--R +-+ | a 3+-+2 \| b
--R - 2a\|3 |- -- \|b atan(-------------------)
--R 3| 2 +----+2
--R \| b | 2
--R +-+ | a
--R b\|3 |- --
--R 3| 2
--R \| b
--R /
--R 2 3+-+2
--R 2b \|b
--R Type: Expression(Integer)
--E 124
--S 125 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 125
)clear all
--S 126 of 2952
t0000:=x^(4/3)/(a+b*x)
--R
--R
--R 3+-+
--R x\|x
--R (1) -------
--R b x + a
--R Type: Expression(Integer)
--E 126
--S 127 of 2952
r0000:=-3*a*x^(1/3)/b^2+3/4*x^(4/3)/b+a^(4/3)*log(a^(1/3)+_
b^(1/3)*x^(1/3))/b^(7/3)-1/2*a^(4/3)*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/b^(7/3)-_
a^(4/3)*atan((a^(1/3)-_
2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/b^(7/3)
--R
--R
--R (2)
--R 3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - 2a\|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+ 3+-+3+-+ 3+-+ +-+3+-+ 2\|b \|x - \|a
--R 4a\|a log(\|b \|x + \|a ) + 4a\|3 \|a atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R 3+-+3+-+
--R (3b x - 12a)\|b \|x
--R /
--R 2 3+-+
--R 4b \|b
--R Type: Expression(Integer)
--E 127
--S 128 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +-+ +-+ +-+2 +-+ +-+
--R |a 3+-+2 |a 3+-+ |a |a 3+-+ |a
--R - 2a 3|- log(\|x - 3|- \|x + 3|- + 4a 3|- log(\|x + 3|- )
--R \|b \|b \|b \|b \|b
--R +
--R +-+
--R 3+-+ |a
--R +-+ 2\|x - 3|-
--R +-+ |a \|b 3+-+
--R 4a\|3 3|- atan(------------ + (3b x - 12a)\|x
--R \|b +-+
--R +-+ |a
--R \|3 3|-
--R \|b
--R /
--R 2
--R 4b
--R Type: Union(Expression(Integer),...)
--E 128
--S 129 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R a\|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +-+ +-+ +-+2
--R |a 3+-+ 3+-+2 |a 3+-+ |a 3+-+ 3+-+3+-+ 3+-+
--R - a 3|- \|b log(\|x - 3|- \|x + 3|- - 2a\|a log(\|b \|x + \|a )
--R \|b \|b \|b
--R +
--R +-+ +-+ 3+-+3+-+ 3+-+
--R |a 3+-+ 3+-+ |a +-+3+-+ 2\|b \|x - \|a
--R 2a 3|- \|b log(\|x + 3|- - 2a\|3 \|a atan(----------------)
--R \|b \|b +-+3+-+
--R \|3 \|a
--R +
--R +-+
--R 3+-+ |a
--R +-+ 2\|x - 3|-
--R +-+ |a 3+-+ \|b
--R 2a\|3 3|- \|b atan(------------)
--R \|b +-+
--R +-+ |a
--R \|3 3|-
--R \|b
--R /
--R 2 3+-+
--R 2b \|b
--R Type: Expression(Integer)
--E 129
--S 130 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 130
)clear all
--S 131 of 2952
t0000:=x^(2/3)/(a+b*x)
--R
--R
--R 3+-+2
--R \|x
--R (1) -------
--R b x + a
--R Type: Expression(Integer)
--E 131
--S 132 of 2952
r0000:=3/2*x^(2/3)/b+a^(2/3)*log(a^(1/3)+b^(1/3)*x^(1/3))/b^(5/3)-_
1/2*a^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+_
b^(2/3)*x^(2/3))/b^(5/3)+a^(2/3)*atan((a^(1/3)-_
2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/b^(5/3)
--R
--R
--R (2)
--R 3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+2 3+-+3+-+ 3+-+ +-+3+-+2 2\|b \|x - \|a 3+-+2 3+-+2
--R 2\|a log(\|b \|x + \|a ) - 2\|3 \|a atan(----------------) + 3\|b \|x
--R +-+3+-+
--R \|3 \|a
--R /
--R 3+-+2
--R 2b \|b
--R Type: Expression(Integer)
--E 132
--S 133 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +--+ +--+2 +--+ +--+ +--+2
--R | 2 | 2 | 2 | 2 | 2
--R |a 3+-+2 |a 3+-+ |a |a 3+-+ |a
--R - |-- log(a \|x - b |-- \|x + a |--) + 2 |-- log(a\|x + b |-- )
--R 3| 2 3| 2 3| 2 3| 2 3| 2
--R \|b \|b \|b \|b \|b
--R +
--R +--+2
--R | 2
--R 3+-+ |a
--R +--+ 2a\|x - b |--
--R | 2 3| 2
--R +-+ |a \|b 3+-+2
--R - 2\|3 |-- atan(----------------- + 3\|x
--R 3| 2 +--+2
--R \|b | 2
--R +-+ |a
--R b\|3 |--
--R 3| 2
--R \|b
--R /
--R 2b
--R Type: Union(Expression(Integer),...)
--E 133
--S 134 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +--+ +--+2 +--+
--R | 2 | 2 | 2
--R |a 3+-+2 3+-+2 |a 3+-+ |a
--R - |-- \|b log(a \|x - b |-- \|x + a |--)
--R 3| 2 3| 2 3| 2
--R \|b \|b \|b
--R +
--R +--+ +--+2
--R | 2 | 2
--R 3+-+2 3+-+3+-+ 3+-+ |a 3+-+2 3+-+ |a
--R - 2\|a log(\|b \|x + \|a ) + 2 |-- \|b log(a\|x + b |-- )
--R 3| 2 3| 2
--R \|b \|b
--R +
--R +--+2
--R | 2
--R 3+-+ |a
--R +--+ 2a\|x - b |--
--R 3+-+3+-+ 3+-+ | 2 3| 2
--R +-+3+-+2 2\|b \|x - \|a +-+ |a 3+-+2 \|b
--R 2\|3 \|a atan(----------------) - 2\|3 |-- \|b atan(-----------------)
--R +-+3+-+ 3| 2 +--+2
--R \|3 \|a \|b | 2
--R +-+ |a
--R b\|3 |--
--R 3| 2
--R \|b
--R /
--R 3+-+2
--R 2b \|b
--R Type: Expression(Integer)
--E 134
--S 135 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 135
)clear all
--S 136 of 2952
t0000:=x^(1/3)/(a+b*x)
--R
--R
--R 3+-+
--R \|x
--R (1) -------
--R b x + a
--R Type: Expression(Integer)
--E 136
--S 137 of 2952
r0000:=3*x^(1/3)/b-a^(1/3)*log(a^(1/3)+b^(1/3)*x^(1/3))/b^(4/3)+_
1/2*a^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+_
b^(2/3)*x^(2/3))/b^(4/3)+a^(1/3)*atan((a^(1/3)-_
2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/b^(4/3)
--R
--R
--R (2)
--R 3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2 3+-+ 3+-+3+-+ 3+-+
--R \|a log(\|b \|x - \|a \|b \|x + \|a ) - 2\|a log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R +-+3+-+ 2\|b \|x - \|a 3+-+3+-+
--R - 2\|3 \|a atan(----------------) + 6\|b \|x
--R +-+3+-+
--R \|3 \|a
--R /
--R 3+-+
--R 2b\|b
--R Type: Expression(Integer)
--E 137
--S 138 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +---+ +---+ +---+2 +---+ +---+
--R | a 3+-+2 | a 3+-+ | a | a 3+-+ | a
--R - 3|- - log(\|x + 3|- - \|x + 3|- - ) + 2 3|- - log(\|x - 3|- - )
--R \| b \| b \| b \| b \| b
--R +
--R +---+
--R 3+-+ | a
--R +---+ 2\|x + 3|- -
--R +-+ | a \| b 3+-+
--R - 2\|3 3|- - atan(-------------- + 6\|x
--R \| b +---+
--R +-+ | a
--R \|3 3|- -
--R \| b
--R /
--R 2b
--R Type: Union(Expression(Integer),...)
--E 138
--S 139 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +---+ +---+ +---+2
--R | a 3+-+ 3+-+2 | a 3+-+ | a 3+-+ 3+-+3+-+ 3+-+
--R - 3|- - \|b log(\|x + 3|- - \|x + 3|- - ) + 2\|a log(\|b \|x + \|a )
--R \| b \| b \| b
--R +
--R +---+ +---+ 3+-+3+-+ 3+-+
--R | a 3+-+ 3+-+ | a +-+3+-+ 2\|b \|x - \|a
--R 2 3|- - \|b log(\|x - 3|- - + 2\|3 \|a atan(----------------)
--R \| b \| b +-+3+-+
--R \|3 \|a
--R +
--R +---+
--R 3+-+ | a
--R +---+ 2\|x + 3|- -
--R +-+ | a 3+-+ \| b
--R - 2\|3 3|- - \|b atan(--------------)
--R \| b +---+
--R +-+ | a
--R \|3 3|- -
--R \| b
--R /
--R 3+-+
--R 2b\|b
--R Type: Expression(Integer)
--E 139
--S 140 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 140
)clear all
--S 141 of 2952
t0000:=1/(x^(1/3)*(a+b*x))
--R
--R
--R 1
--R (1) -------------
--R 3+-+
--R (b x + a)\|x
--R Type: Expression(Integer)
--E 141
--S 142 of 2952
r0000:=-log(a^(1/3)+b^(1/3)*x^(1/3))/(a^(1/3)*b^(2/3))+_
1/2*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/_
(a^(1/3)*b^(2/3))-atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))*sqrt(3)/(a^(1/3)*b^(2/3))
--R
--R
--R (2)
--R 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2 3+-+3+-+ 3+-+
--R log(\|b \|x - \|a \|b \|x + \|a ) - 2log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R +-+ 2\|b \|x - \|a
--R 2\|3 atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R /
--R 3+-+3+-+2
--R 2\|a \|b
--R Type: Expression(Integer)
--E 142
--S 143 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +------+ +------+2
--R 3| 2 3+-+2 3| 2 3+-+
--R - log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+2
--R +------+2 3| 2 3+-+
--R 3| 2 3+-+ +-+ 2\|- a b \|x - a b
--R 2log(\|- a b \|x + a b) - 2\|3 atan(---------------------)
--R +-+
--R a b\|3
--R /
--R +------+
--R 3| 2
--R 2\|- a b
--R Type: Union(Expression(Integer),...)
--E 143
--S 144 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +------+
--R 3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|- a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +------+ +------+2
--R 3+-+3+-+2 3| 2 3+-+2 3| 2 3+-+
--R - \|a \|b log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+ +------+2
--R 3| 2 3+-+3+-+ 3+-+ 3+-+3+-+2 3| 2 3+-+
--R 2\|- a b log(\|b \|x + \|a ) + 2\|a \|b log(\|- a b \|x + a b)
--R +
--R +------+ 3+-+3+-+ 3+-+
--R +-+3| 2 2\|b \|x - \|a
--R - 2\|3 \|- a b atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +------+2
--R 3| 2 3+-+
--R +-+3+-+3+-+2 2\|- a b \|x - a b
--R - 2\|3 \|a \|b atan(---------------------)
--R +-+
--R a b\|3
--R /
--R +------+
--R 3| 2 3+-+3+-+2
--R 2\|- a b \|a \|b
--R Type: Expression(Integer)
--E 144
--S 145 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 145
)clear all
--S 146 of 2952
t0000:=1/(x^(2/3)*(a+b*x))
--R
--R
--R 1
--R (1) --------------
--R 3+-+2
--R (b x + a)\|x
--R Type: Expression(Integer)
--E 146
--S 147 of 2952
r0000:=log(a^(1/3)+b^(1/3)*x^(1/3))/(a^(2/3)*b^(1/3))-_
1/2*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/_
(a^(2/3)*b^(1/3))-atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))*sqrt(3)/(a^(2/3)*b^(1/3))
--R
--R
--R (2)
--R 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2 3+-+3+-+ 3+-+
--R - log(\|b \|x - \|a \|b \|x + \|a ) + 2log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R +-+ 2\|b \|x - \|a
--R 2\|3 atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R /
--R 3+-+2 3+-+
--R 2\|a \|b
--R Type: Expression(Integer)
--E 147
--S 148 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +---+2 +---+ +---+
--R 3| 2 3+-+2 3| 2 3+-+ 2 3| 2 3+-+
--R - log(\|a b \|x - a\|a b \|x + a ) + 2log(\|a b \|x + a)
--R +
--R +---+
--R 3| 2 3+-+
--R +-+ 2\|a b \|x - a
--R 2\|3 atan(---------------)
--R +-+
--R a\|3
--R /
--R +---+
--R 3| 2
--R 2\|a b
--R Type: Union(Expression(Integer),...)
--E 148
--S 149 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +---+2 +---+
--R 3+-+2 3+-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R - \|a \|b log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R 3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R \|a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +---+ +---+
--R 3+-+2 3+-+ 3| 2 3+-+ 3| 2 3+-+3+-+ 3+-+
--R 2\|a \|b log(\|a b \|x + a - 2\|a b log(\|b \|x + \|a )
--R +
--R +---+
--R 3| 2 3+-+ +---+ 3+-+3+-+ 3+-+
--R +-+3+-+2 3+-+ 2\|a b \|x - a +-+3| 2 2\|b \|x - \|a
--R 2\|3 \|a \|b atan(--------------- - 2\|3 \|a b atan(----------------)
--R +-+ +-+3+-+
--R a\|3 \|3 \|a
--R /
--R +---+
--R 3+-+2 3+-+3| 2
--R 2\|a \|b \|a b
--R Type: Expression(Integer)
--E 149
--S 150 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 150
)clear all
--S 151 of 2952
t0000:=1/(x^(4/3)*(a+b*x))
--R
--R
--R 1
--R (1) ----------------
--R 2 3+-+
--R (b x + a x)\|x
--R Type: Expression(Integer)
--E 151
--S 152 of 2952
r0000:=(-3)/(a*x^(1/3))+b^(1/3)*log(a^(1/3)+b^(1/3)*x^(1/3))/a^(4/3)-_
1/2*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+_
b^(2/3)*x^(2/3))/a^(4/3)+b^(1/3)*atan((a^(1/3)-_
2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/a^(4/3)
--R
--R
--R (2)
--R 3+-+3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|b \|x log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+3+-+ 3+-+3+-+ 3+-+ +-+3+-+3+-+ 2\|b \|x - \|a 3+-+
--R 2\|b \|x log(\|b \|x + \|a ) - 2\|3 \|b \|x atan(----------------) - 6\|a
--R +-+3+-+
--R \|3 \|a
--R /
--R 3+-+3+-+
--R 2a\|a \|x
--R Type: Expression(Integer)
--E 152
--S 153 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +-+ +-+2 +-+
--R |b 3+-+ 3+-+2 |b 3+-+ |b
--R - 3|- \|x log(b \|x - a 3|- \|x + a 3|-)
--R \|a \|a \|a
--R +
--R +-+2
--R 3+-+ |b
--R +-+ +-+2 +-+ 2b\|x - a 3|-
--R |b 3+-+ 3+-+ |b +-+ |b 3+-+ \|a
--R 2 3|- \|x log(b\|x + a 3|- - 2\|3 3|- \|x atan(---------------- - 6
--R \|a \|a \|a +-+2
--R +-+ |b
--R a\|3 3|-
--R \|a
--R /
--R 3+-+
--R 2a\|x
--R Type: Union(Expression(Integer),...)
--E 153
--S 154 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R \|b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +-+ +-+2 +-+
--R 3+-+ |b 3+-+2 |b 3+-+ |b 3+-+ 3+-+3+-+ 3+-+
--R - \|a 3|- log(b \|x - a 3|- \|x + a 3|- - 2\|b log(\|b \|x + \|a )
--R \|a \|a \|a
--R +
--R +-+ +-+2 3+-+3+-+ 3+-+
--R 3+-+ |b 3+-+ |b +-+3+-+ 2\|b \|x - \|a
--R 2\|a 3|- log(b\|x + a 3|- + 2\|3 \|b atan(----------------)
--R \|a \|a +-+3+-+
--R \|3 \|a
--R +
--R +-+2
--R 3+-+ |b
--R +-+ 2b\|x - a 3|-
--R +-+3+-+ |b \|a
--R - 2\|3 \|a 3|- atan(----------------)
--R \|a +-+2
--R +-+ |b
--R a\|3 3|-
--R \|a
--R /
--R 3+-+
--R 2a\|a
--R Type: Expression(Integer)
--E 154
--S 155 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 155
)clear all
--S 156 of 2952
t0000:=1/(x^(5/3)*(a+b*x))
--R
--R
--R 1
--R (1) -----------------
--R 2 3+-+2
--R (b x + a x)\|x
--R Type: Expression(Integer)
--E 156
--S 157 of 2952
r0000:=(-3/2)/(a*x^(2/3))-b^(2/3)*log(a^(1/3)+_
b^(1/3)*x^(1/3))/a^(5/3)+1/2*b^(2/3)*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/a^(5/3)+_
b^(2/3)*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))*_
sqrt(3)/a^(5/3)
--R
--R
--R (2)
--R 3+-+2 3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R \|b \|x log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+2 3+-+2 3+-+3+-+ 3+-+ +-+3+-+2 3+-+2 2\|b \|x - \|a
--R - 2\|b \|x log(\|b \|x + \|a - 2\|3 \|b \|x atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R 3+-+2
--R - 3\|a
--R /
--R 3+-+2 3+-+2
--R 2a \|a \|x
--R Type: Expression(Integer)
--E 157
--S 158 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +----+ +----+ +----+2
--R | 2 | 2 | 2
--R | b 3+-+2 2 3+-+2 | b 3+-+ 2 | b
--R - |- -- \|x log(b \|x + a b |- -- \|x + a |- -- )
--R 3| 2 3| 2 3| 2
--R \| a \| a \| a
--R +
--R +----+ +----+
--R | 2 | 2
--R | b 3+-+2 3+-+ | b
--R 2 |- -- \|x log(b\|x - a |- --)
--R 3| 2 3| 2
--R \| a \| a
--R +
--R +----+
--R | 2
--R 3+-+ | b
--R +----+ 2b\|x + a |- --
--R | 2 3| 2
--R +-+ | b 3+-+2 \| a
--R - 2\|3 |- -- \|x atan(----------------- - 3
--R 3| 2 +----+
--R \| a | 2
--R +-+ | b
--R a\|3 |- --
--R 3| 2
--R \| a
--R /
--R 3+-+2
--R 2a \|x
--R Type: Union(Expression(Integer),...)
--E 158
--S 159 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +----+ +----+ +----+2
--R | 2 | 2 | 2
--R | b 3+-+2 2 3+-+2 | b 3+-+ 2 | b
--R - |- -- \|a log(b \|x + a b |- -- \|x + a |- -- )
--R 3| 2 3| 2 3| 2
--R \| a \| a \| a
--R +
--R +----+ +----+
--R | 2 | 2
--R 3+-+2 3+-+3+-+ 3+-+ | b 3+-+2 3+-+ | b
--R 2\|b log(\|b \|x + \|a ) + 2 |- -- \|a log(b\|x - a |- --)
--R 3| 2 3| 2
--R \| a \| a
--R +
--R +----+
--R | 2
--R 3+-+ | b
--R +----+ 2b\|x + a |- --
--R 3+-+3+-+ 3+-+ | 2 3| 2
--R +-+3+-+2 2\|b \|x - \|a +-+ | b 3+-+2 \| a
--R 2\|3 \|b atan(----------------) - 2\|3 |- -- \|a atan(-----------------)
--R +-+3+-+ 3| 2 +----+
--R \|3 \|a \| a | 2
--R +-+ | b
--R a\|3 |- --
--R 3| 2
--R \| a
--R /
--R 3+-+2
--R 2a \|a
--R Type: Expression(Integer)
--E 159
--S 160 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 160
)clear all
--S 161 of 2952
t0000:=x^(5/3)/(a+b*x)^2
--R
--R
--R 3+-+2
--R x \|x
--R (1) ------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Expression(Integer)
--E 161
--S 162 of 2952
r0000:=5/2*x^(2/3)/b^2-x^(5/3)/(b*(a+b*x))+5/3*a^(2/3)*log(a^(1/3)+_
b^(1/3)*x^(1/3))/b^(8/3)-5/6*a^(2/3)*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/b^(8/3)+_
5*a^(2/3)*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/(a^(1/3)*_
sqrt(3)))/(b^(8/3)*sqrt(3))
--R
--R
--R (2)
--R +-+3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (- 5b x - 5a)\|3 \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +-+3+-+2 3+-+3+-+ 3+-+
--R (10b x + 10a)\|3 \|a log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+2 2\|b \|x - \|a +-+3+-+2 3+-+2
--R (- 30b x - 30a)\|a atan(----------------) + (9b x + 15a)\|3 \|b \|x
--R +-+3+-+
--R \|3 \|a
--R /
--R 3 2 +-+3+-+2
--R (6b x + 6a b )\|3 \|b
--R Type: Expression(Integer)
--E 162
--S 163 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +--+ +--+2 +--+
--R | 2 | 2 | 2
--R +-+ |a 3+-+2 |a 3+-+ |a
--R (- 5b x - 5a)\|3 |-- log(a \|x - b |-- \|x + a |--)
--R 3| 2 3| 2 3| 2
--R \|b \|b \|b
--R +
--R +--+ +--+2
--R | 2 | 2
--R +-+ |a 3+-+ |a
--R (10b x + 10a)\|3 |-- log(a\|x + b |-- )
--R 3| 2 3| 2
--R \|b \|b
--R +
--R +--+2
--R | 2
--R +-+3+-+ +-+ |a
--R +--+ 2a\|3 \|x - b\|3 |--
--R | 2 3| 2
--R |a \|b +-+3+-+2
--R (- 30b x - 30a) |-- atan(------------------------ + (9b x + 15a)\|3 \|x
--R 3| 2 +--+2
--R \|b | 2
--R |a
--R 3b |--
--R 3| 2
--R \|b
--R /
--R 3 2 +-+
--R (6b x + 6a b )\|3
--R Type: Union(Expression(Integer),...)
--E 163
--S 164 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +-+3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R 5\|3 \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +--+ +--+2 +--+
--R | 2 | 2 | 2
--R +-+ |a 3+-+2 3+-+2 |a 3+-+ |a
--R - 5\|3 |-- \|b log(a \|x - b |-- \|x + a |--)
--R 3| 2 3| 2 3| 2
--R \|b \|b \|b
--R +
--R +--+ +--+2
--R | 2 | 2
--R +-+3+-+2 3+-+3+-+ 3+-+ +-+ |a 3+-+2 3+-+ |a
--R - 10\|3 \|a log(\|b \|x + \|a ) + 10\|3 |-- \|b log(a\|x + b |-- )
--R 3| 2 3| 2
--R \|b \|b
--R +
--R +--+2
--R | 2
--R +-+3+-+ +-+ |a
--R +--+ 2a\|3 \|x - b\|3 |--
--R 3+-+3+-+ 3+-+ | 2 3| 2
--R 3+-+2 2\|b \|x - \|a |a 3+-+2 \|b
--R 30\|a atan(----------------) - 30 |-- \|b atan(------------------------)
--R +-+3+-+ 3| 2 +--+2
--R \|3 \|a \|b | 2
--R |a
--R 3b |--
--R 3| 2
--R \|b
--R /
--R 2 +-+3+-+2
--R 6b \|3 \|b
--R Type: Expression(Integer)
--E 164
--S 165 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 165
)clear all
--S 166 of 2952
t0000:=x^(4/3)/(a+b*x)^2
--R
--R
--R 3+-+
--R x\|x
--R (1) ------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Expression(Integer)
--E 166
--S 167 of 2952
r0000:=4*x^(1/3)/b^2-x^(4/3)/(b*(a+b*x))-4/3*a^(1/3)*log(a^(1/3)+_
b^(1/3)*x^(1/3))/b^(7/3)+2/3*a^(1/3)*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/b^(7/3)+_
4*a^(1/3)*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))/_
(b^(7/3)*sqrt(3))
--R
--R
--R (2)
--R +-+3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (2b x + 2a)\|3 \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +-+3+-+ 3+-+3+-+ 3+-+
--R (- 4b x - 4a)\|3 \|a log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+ 2\|b \|x - \|a +-+3+-+3+-+
--R (- 12b x - 12a)\|a atan(----------------) + (9b x + 12a)\|3 \|b \|x
--R +-+3+-+
--R \|3 \|a
--R /
--R 3 2 +-+3+-+
--R (3b x + 3a b )\|3 \|b
--R Type: Expression(Integer)
--E 167
--S 168 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +---+ +---+ +---+2
--R +-+ | a 3+-+2 | a 3+-+ | a
--R (- 2b x - 2a)\|3 3|- - log(\|x + 3|- - \|x + 3|- - )
--R \| b \| b \| b
--R +
--R +---+ +---+
--R +-+ | a 3+-+ | a
--R (4b x + 4a)\|3 3|- - log(\|x - 3|- - )
--R \| b \| b
--R +
--R +---+
--R +-+3+-+ +-+ | a
--R +---+ 2\|3 \|x + \|3 3|- -
--R | a \| b +-+3+-+
--R (- 12b x - 12a) 3|- - atan(---------------------- + (9b x + 12a)\|3 \|x
--R \| b +---+
--R | a
--R 3 3|- -
--R \| b
--R /
--R 3 2 +-+
--R (3b x + 3a b )\|3
--R Type: Union(Expression(Integer),...)
--E 168
--S 169 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +-+3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - 2\|3 \|a log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +---+ +---+ +---+2
--R +-+ | a 3+-+ 3+-+2 | a 3+-+ | a
--R - 2\|3 3|- - \|b log(\|x + 3|- - \|x + 3|- - )
--R \| b \| b \| b
--R +
--R +---+ +---+
--R +-+3+-+ 3+-+3+-+ 3+-+ +-+ | a 3+-+ 3+-+ | a
--R 4\|3 \|a log(\|b \|x + \|a ) + 4\|3 3|- - \|b log(\|x - 3|- - )
--R \| b \| b
--R +
--R +---+
--R +-+3+-+ +-+ | a
--R 3+-+3+-+ 3+-+ +---+ 2\|3 \|x + \|3 3|- -
--R 3+-+ 2\|b \|x - \|a | a 3+-+ \| b
--R 12\|a atan(----------------) - 12 3|- - \|b atan(----------------------)
--R +-+3+-+ \| b +---+
--R \|3 \|a | a
--R 3 3|- -
--R \| b
--R /
--R 2 +-+3+-+
--R 3b \|3 \|b
--R Type: Expression(Integer)
--E 169
--S 170 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 170
)clear all
--S 171 of 2952
t0000:=x^(2/3)/(a+b*x)^2
--R
--R
--R 3+-+2
--R \|x
--R (1) ------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Expression(Integer)
--E 171
--S 172 of 2952
r0000:=-x^(2/3)/(b*(a+b*x))-2/3*log(a^(1/3)+_
b^(1/3)*x^(1/3))/(a^(1/3)*b^(5/3))+1/3*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/(a^(1/3)*b^(5/3))-_
2*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))/_
(a^(1/3)*b^(5/3)*sqrt(3))
--R
--R
--R (2)
--R +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (b x + a)\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R +-+ 3+-+3+-+ 3+-+ 2\|b \|x - \|a
--R (- 2b x - 2a)\|3 log(\|b \|x + \|a ) + (6b x + 6a)atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+3+-+2 3+-+2
--R - 3\|3 \|a \|b \|x
--R /
--R 2 +-+3+-+3+-+2
--R (3b x + 3a b)\|3 \|a \|b
--R Type: Expression(Integer)
--E 172
--S 173 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +------+ +------+2
--R +-+ 3| 2 3+-+2 3| 2 3+-+
--R (- b x - a)\|3 log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+2
--R +-+ 3| 2 3+-+
--R (2b x + 2a)\|3 log(\|- a b \|x + a b)
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+ +------+
--R 2\|3 \|- a b \|x - a b\|3 +-+3| 2 3+-+2
--R (- 6b x - 6a)atan(-----------------------------) - 3\|3 \|- a b \|x
--R 3a b
--R /
--R +------+
--R 2 +-+3| 2
--R (3b x + 3a b)\|3 \|- a b
--R Type: Union(Expression(Integer),...)
--E 173
--S 174 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +------+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|3 \|- a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +------+ +------+2
--R +-+3+-+3+-+2 3| 2 3+-+2 3| 2 3+-+
--R - \|3 \|a \|b log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+
--R +-+3| 2 3+-+3+-+ 3+-+
--R 2\|3 \|- a b log(\|b \|x + \|a )
--R +
--R +------+2
--R +-+3+-+3+-+2 3| 2 3+-+
--R 2\|3 \|a \|b log(\|- a b \|x + a b)
--R +
--R +------+ 3+-+3+-+ 3+-+
--R 3| 2 2\|b \|x - \|a
--R - 6\|- a b atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+
--R 3+-+3+-+2 2\|3 \|- a b \|x - a b\|3
--R - 6\|a \|b atan(-----------------------------)
--R 3a b
--R /
--R +------+
--R +-+3| 2 3+-+3+-+2
--R 3b\|3 \|- a b \|a \|b
--R Type: Expression(Integer)
--E 174
--S 175 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 175
)clear all
--S 176 of 2952
t0000:=x^(1/3)/(a+b*x)^2
--R
--R
--R 3+-+
--R \|x
--R (1) ------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Expression(Integer)
--E 176
--S 177 of 2952
r0000:=-x^(1/3)/(b*(a+b*x))+1/3*log(a^(1/3)+_
b^(1/3)*x^(1/3))/(a^(2/3)*b^(4/3))-1/6*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/(a^(2/3)*b^(4/3))-_
atan((a^(1/3)-2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))/_
(a^(2/3)*b^(4/3)*sqrt(3))
--R
--R
--R (2)
--R +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (- b x - a)\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R +-+ 3+-+3+-+ 3+-+ 2\|b \|x - \|a
--R (2b x + 2a)\|3 log(\|b \|x + \|a ) + (6b x + 6a)atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+2 3+-+3+-+
--R - 6\|3 \|a \|b \|x
--R /
--R 2 +-+3+-+2 3+-+
--R (6b x + 6a b)\|3 \|a \|b
--R Type: Expression(Integer)
--E 177
--S 178 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +---+2 +---+
--R +-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R (- b x - a)\|3 log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R +-+ 3| 2 3+-+
--R (2b x + 2a)\|3 log(\|a b \|x + a)
--R +
--R +---+
--R +-+3| 2 3+-+ +-+ +---+
--R 2\|3 \|a b \|x - a\|3 +-+3| 2 3+-+
--R (6b x + 6a)atan(-----------------------) - 6\|3 \|a b \|x
--R 3a
--R /
--R +---+
--R 2 +-+3| 2
--R (6b x + 6a b)\|3 \|a b
--R Type: Union(Expression(Integer),...)
--E 178
--S 179 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +---+2 +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R - \|3 \|a \|b log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R \|3 \|a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +---+ +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+ +-+3| 2 3+-+3+-+ 3+-+
--R 2\|3 \|a \|b log(\|a b \|x + a - 2\|3 \|a b log(\|b \|x + \|a )
--R +
--R +---+
--R +-+3| 2 3+-+ +-+ +---+ 3+-+3+-+ 3+-+
--R 3+-+2 3+-+ 2\|3 \|a b \|x - a\|3 3| 2 2\|b \|x - \|a
--R 6\|a \|b atan(----------------------- - 6\|a b atan(----------------)
--R 3a +-+3+-+
--R \|3 \|a
--R /
--R +---+
--R +-+3+-+2 3+-+3| 2
--R 6b\|3 \|a \|b \|a b
--R Type: Expression(Integer)
--E 179
--S 180 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 180
)clear all
--S 181 of 2952
t0000:=1/(x^(1/3)*(a+b*x)^2)
--R
--R
--R 1
--R (1) ------------------------
--R 2 2 2 3+-+
--R (b x + 2a b x + a )\|x
--R Type: Expression(Integer)
--E 181
--S 182 of 2952
r0000:=x^(2/3)/(a*(a+b*x))-1/3*log(a^(1/3)+_
b^(1/3)*x^(1/3))/(a^(4/3)*b^(2/3))+1/6*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/(a^(4/3)*b^(2/3))-_
atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(2/3)*sqrt(3))
--R
--R
--R (2)
--R +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (b x + a)\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R +-+ 3+-+3+-+ 3+-+ 2\|b \|x - \|a
--R (- 2b x - 2a)\|3 log(\|b \|x + \|a ) + (6b x + 6a)atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+3+-+2 3+-+2
--R 6\|3 \|a \|b \|x
--R /
--R 2 +-+3+-+3+-+2
--R (6a b x + 6a )\|3 \|a \|b
--R Type: Expression(Integer)
--E 182
--S 183 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +------+ +------+2
--R +-+ 3| 2 3+-+2 3| 2 3+-+
--R (- b x - a)\|3 log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+2
--R +-+ 3| 2 3+-+
--R (2b x + 2a)\|3 log(\|- a b \|x + a b)
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+ +------+
--R 2\|3 \|- a b \|x - a b\|3 +-+3| 2 3+-+2
--R (- 6b x - 6a)atan(-----------------------------) + 6\|3 \|- a b \|x
--R 3a b
--R /
--R +------+
--R 2 +-+3| 2
--R (6a b x + 6a )\|3 \|- a b
--R Type: Union(Expression(Integer),...)
--E 183
--S 184 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +------+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|3 \|- a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +------+ +------+2
--R +-+3+-+3+-+2 3| 2 3+-+2 3| 2 3+-+
--R - \|3 \|a \|b log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+
--R +-+3| 2 3+-+3+-+ 3+-+
--R 2\|3 \|- a b log(\|b \|x + \|a )
--R +
--R +------+2
--R +-+3+-+3+-+2 3| 2 3+-+
--R 2\|3 \|a \|b log(\|- a b \|x + a b)
--R +
--R +------+ 3+-+3+-+ 3+-+
--R 3| 2 2\|b \|x - \|a
--R - 6\|- a b atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+
--R 3+-+3+-+2 2\|3 \|- a b \|x - a b\|3
--R - 6\|a \|b atan(-----------------------------)
--R 3a b
--R /
--R +------+
--R +-+3| 2 3+-+3+-+2
--R 6a\|3 \|- a b \|a \|b
--R Type: Expression(Integer)
--E 184
--S 185 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 185
)clear all
--S 186 of 2952
t0000:=1/(x^(2/3)*(a+b*x)^2)
--R
--R
--R 1
--R (1) -------------------------
--R 2 2 2 3+-+2
--R (b x + 2a b x + a )\|x
--R Type: Expression(Integer)
--E 186
--S 187 of 2952
r0000:=x^(1/3)/(a*(a+b*x))+2/3*log(a^(1/3)+b^(1/3)*x^(1/3))/_
(a^(5/3)*b^(1/3))-1/3*log(a^(2/3)-a^(1/3)*b^(1/3)*_
x^(1/3)+b^(2/3)*x^(2/3))/(a^(5/3)*b^(1/3))-2*atan((a^(1/3)-_
2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(1/3)*sqrt(3))
--R
--R
--R (2)
--R +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (- b x - a)\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R +-+ 3+-+3+-+ 3+-+ 2\|b \|x - \|a
--R (2b x + 2a)\|3 log(\|b \|x + \|a ) + (6b x + 6a)atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+2 3+-+3+-+
--R 3\|3 \|a \|b \|x
--R /
--R 2 +-+3+-+2 3+-+
--R (3a b x + 3a )\|3 \|a \|b
--R Type: Expression(Integer)
--E 187
--S 188 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +---+2 +---+
--R +-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R (- b x - a)\|3 log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R +-+ 3| 2 3+-+
--R (2b x + 2a)\|3 log(\|a b \|x + a)
--R +
--R +---+
--R +-+3| 2 3+-+ +-+ +---+
--R 2\|3 \|a b \|x - a\|3 +-+3| 2 3+-+
--R (6b x + 6a)atan(-----------------------) + 3\|3 \|a b \|x
--R 3a
--R /
--R +---+
--R 2 +-+3| 2
--R (3a b x + 3a )\|3 \|a b
--R Type: Union(Expression(Integer),...)
--E 188
--S 189 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +---+2 +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R - \|3 \|a \|b log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R \|3 \|a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +---+ +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+ +-+3| 2 3+-+3+-+ 3+-+
--R 2\|3 \|a \|b log(\|a b \|x + a - 2\|3 \|a b log(\|b \|x + \|a )
--R +
--R +---+
--R +-+3| 2 3+-+ +-+ +---+ 3+-+3+-+ 3+-+
--R 3+-+2 3+-+ 2\|3 \|a b \|x - a\|3 3| 2 2\|b \|x - \|a
--R 6\|a \|b atan(----------------------- - 6\|a b atan(----------------)
--R 3a +-+3+-+
--R \|3 \|a
--R /
--R +---+
--R +-+3+-+2 3+-+3| 2
--R 3a\|3 \|a \|b \|a b
--R Type: Expression(Integer)
--E 189
--S 190 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 190
)clear all
--S 191 of 2952
t0000:=1/(x^(4/3)*(a+b*x)^2)
--R
--R
--R 1
--R (1) --------------------------
--R 2 3 2 2 3+-+
--R (b x + 2a b x + a x)\|x
--R Type: Expression(Integer)
--E 191
--S 192 of 2952
r0000:=(-4)/(a^2*x^(1/3))+1/(a*x^(1/3)*(a+b*x))+_
4/3*b^(1/3)*log(a^(1/3)+b^(1/3)*x^(1/3))/a^(7/3)-_
2/3*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+_
b^(2/3)*x^(2/3))/a^(7/3)+4*b^(1/3)*atan((a^(1/3)-_
2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))/(a^(7/3)*sqrt(3))
--R
--R
--R (2)
--R +-+3+-+3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (- 2b x - 2a)\|3 \|b \|x log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +-+3+-+3+-+ 3+-+3+-+ 3+-+
--R (4b x + 4a)\|3 \|b \|x log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+3+-+ 2\|b \|x - \|a +-+3+-+
--R (- 12b x - 12a)\|b \|x atan(----------------) + (- 12b x - 9a)\|3 \|a
--R +-+3+-+
--R \|3 \|a
--R /
--R 2 3 +-+3+-+3+-+
--R (3a b x + 3a )\|3 \|a \|x
--R Type: Expression(Integer)
--E 192
--S 193 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +-+ +-+2 +-+
--R +-+ |b 3+-+ 3+-+2 |b 3+-+ |b
--R (- 2b x - 2a)\|3 3|- \|x log(b \|x - a 3|- \|x + a 3|-)
--R \|a \|a \|a
--R +
--R +-+ +-+2
--R +-+ |b 3+-+ 3+-+ |b
--R (4b x + 4a)\|3 3|- \|x log(b\|x + a 3|- )
--R \|a \|a
--R +
--R +-+2
--R +-+3+-+ +-+ |b
--R +-+ 2b\|3 \|x - a\|3 3|-
--R |b 3+-+ \|a +-+
--R (- 12b x - 12a) 3|- \|x atan(----------------------- + (- 12b x - 9a)\|3
--R \|a +-+2
--R |b
--R 3a 3|-
--R \|a
--R /
--R 2 3 +-+3+-+
--R (3a b x + 3a )\|3 \|x
--R Type: Union(Expression(Integer),...)
--E 193
--S 194 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +-+3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R 2\|3 \|b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +-+ +-+2 +-+
--R +-+3+-+ |b 3+-+2 |b 3+-+ |b
--R - 2\|3 \|a 3|- log(b \|x - a 3|- \|x + a 3|-)
--R \|a \|a \|a
--R +
--R +-+ +-+2
--R +-+3+-+ 3+-+3+-+ 3+-+ +-+3+-+ |b 3+-+ |b
--R - 4\|3 \|b log(\|b \|x + \|a ) + 4\|3 \|a 3|- log(b\|x + a 3|- )
--R \|a \|a
--R +
--R +-+2
--R +-+3+-+ +-+ |b
--R 3+-+3+-+ 3+-+ +-+ 2b\|3 \|x - a\|3 3|-
--R 3+-+ 2\|b \|x - \|a 3+-+ |b \|a
--R 12\|b atan(----------------) - 12\|a 3|- atan(-----------------------)
--R +-+3+-+ \|a +-+2
--R \|3 \|a |b
--R 3a 3|-
--R \|a
--R /
--R 2 +-+3+-+
--R 3a \|3 \|a
--R Type: Expression(Integer)
--E 194
--S 195 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 195
)clear all
--S 196 of 2952
t0000:=1/(x^(5/3)*(a+b*x)^2)
--R
--R
--R 1
--R (1) ---------------------------
--R 2 3 2 2 3+-+2
--R (b x + 2a b x + a x)\|x
--R Type: Expression(Integer)
--E 196
--S 197 of 2952
r0000:=(-5/2)/(a^2*x^(2/3))+1/(a*x^(2/3)*(a+b*x))-5/3*b^(2/3)*log(a^(1/3)+_
b^(1/3)*x^(1/3))/a^(8/3)+5/6*b^(2/3)*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/a^(8/3)+_
5*b^(2/3)*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/(a^(1/3)*sqrt(3)))/_
(a^(8/3)*sqrt(3))
--R
--R
--R (2)
--R +-+3+-+2 3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (5b x + 5a)\|3 \|b \|x log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +-+3+-+2 3+-+2 3+-+3+-+ 3+-+
--R (- 10b x - 10a)\|3 \|b \|x log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+2 3+-+2 2\|b \|x - \|a +-+3+-+2
--R (- 30b x - 30a)\|b \|x atan(---------------- + (- 15b x - 9a)\|3 \|a
--R +-+3+-+
--R \|3 \|a
--R /
--R 2 3 +-+3+-+2 3+-+2
--R (6a b x + 6a )\|3 \|a \|x
--R Type: Expression(Integer)
--E 197
--S 198 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +----+ +----+ +----+2
--R | 2 | 2 | 2
--R +-+ | b 3+-+2 2 3+-+2 | b 3+-+ 2 | b
--R (- 5b x - 5a)\|3 |- -- \|x log(b \|x + a b |- -- \|x + a |- -- )
--R 3| 2 3| 2 3| 2
--R \| a \| a \| a
--R +
--R +----+ +----+
--R | 2 | 2
--R +-+ | b 3+-+2 3+-+ | b
--R (10b x + 10a)\|3 |- -- \|x log(b\|x - a |- --)
--R 3| 2 3| 2
--R \| a \| a
--R +
--R +----+
--R | 2
--R +-+3+-+ +-+ | b
--R +----+ 2b\|3 \|x + a\|3 |- --
--R | 2 3| 2
--R | b 3+-+2 \| a
--R (- 30b x - 30a) |- -- \|x atan(-------------------------)
--R 3| 2 +----+
--R \| a | 2
--R | b
--R 3a |- --
--R 3| 2
--R \| a
--R +
--R +-+
--R (- 15b x - 9a)\|3
--R /
--R 2 3 +-+3+-+2
--R (6a b x + 6a )\|3 \|x
--R Type: Union(Expression(Integer),...)
--E 198
--S 199 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +-+3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - 5\|3 \|b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +----+ +----+ +----+2
--R | 2 | 2 | 2
--R +-+ | b 3+-+2 2 3+-+2 | b 3+-+ 2 | b
--R - 5\|3 |- -- \|a log(b \|x + a b |- -- \|x + a |- -- )
--R 3| 2 3| 2 3| 2
--R \| a \| a \| a
--R +
--R +----+ +----+
--R | 2 | 2
--R +-+3+-+2 3+-+3+-+ 3+-+ +-+ | b 3+-+2 3+-+ | b
--R 10\|3 \|b log(\|b \|x + \|a ) + 10\|3 |- -- \|a log(b\|x - a |- --)
--R 3| 2 3| 2
--R \| a \| a
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+2 2\|b \|x - \|a
--R 30\|b atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +----+
--R | 2
--R +-+3+-+ +-+ | b
--R +----+ 2b\|3 \|x + a\|3 |- --
--R | 2 3| 2
--R | b 3+-+2 \| a
--R - 30 |- -- \|a atan(-------------------------)
--R 3| 2 +----+
--R \| a | 2
--R | b
--R 3a |- --
--R 3| 2
--R \| a
--R /
--R 2 +-+3+-+2
--R 6a \|3 \|a
--R Type: Expression(Integer)
--E 199
--S 200 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 200
)clear all
--S 201 of 2952
t0000:=x^(5/3)/(a+b*x)^3
--R
--R
--R 3+-+2
--R x \|x
--R (1) ----------------------------
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Expression(Integer)
--E 201
--S 202 of 2952
r0000:=-1/2*x^(5/3)/(b*(a+b*x)^2)-5/6*x^(2/3)/(b^2*(a+b*x))-_
5/9*log(a^(1/3)+b^(1/3)*x^(1/3))/(a^(1/3)*b^(8/3))+_
5/18*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/_
(a^(1/3)*b^(8/3))-5/3*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(8/3)*sqrt(3))
--R
--R
--R (2)
--R 2 2 2 +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (5b x + 10a b x + 5a )\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 2 2 2 +-+ 3+-+3+-+ 3+-+
--R (- 10b x - 20a b x - 10a )\|3 log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 2 2 2 2\|b \|x - \|a
--R (30b x + 60a b x + 30a )atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+3+-+2 3+-+2
--R (- 24b x - 15a)\|3 \|a \|b \|x
--R /
--R 4 2 3 2 2 +-+3+-+3+-+2
--R (18b x + 36a b x + 18a b )\|3 \|a \|b
--R Type: Expression(Integer)
--E 202
--S 203 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +------+ +------+2
--R 2 2 2 +-+ 3| 2 3+-+2 3| 2 3+-+
--R (- 5b x - 10a b x - 5a )\|3 log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+2
--R 2 2 2 +-+ 3| 2 3+-+
--R (10b x + 20a b x + 10a )\|3 log(\|- a b \|x + a b)
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+
--R 2 2 2 2\|3 \|- a b \|x - a b\|3
--R (- 30b x - 60a b x - 30a )atan(-----------------------------)
--R 3a b
--R +
--R +------+
--R +-+3| 2 3+-+2
--R (- 24b x - 15a)\|3 \|- a b \|x
--R /
--R +------+
--R 4 2 3 2 2 +-+3| 2
--R (18b x + 36a b x + 18a b )\|3 \|- a b
--R Type: Union(Expression(Integer),...)
--E 203
--S 204 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +------+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - 5\|3 \|- a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +------+ +------+2
--R +-+3+-+3+-+2 3| 2 3+-+2 3| 2 3+-+
--R - 5\|3 \|a \|b log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+
--R +-+3| 2 3+-+3+-+ 3+-+
--R 10\|3 \|- a b log(\|b \|x + \|a )
--R +
--R +------+2
--R +-+3+-+3+-+2 3| 2 3+-+
--R 10\|3 \|a \|b log(\|- a b \|x + a b)
--R +
--R +------+ 3+-+3+-+ 3+-+
--R 3| 2 2\|b \|x - \|a
--R - 30\|- a b atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+
--R 3+-+3+-+2 2\|3 \|- a b \|x - a b\|3
--R - 30\|a \|b atan(-----------------------------)
--R 3a b
--R /
--R +------+
--R 2 +-+3| 2 3+-+3+-+2
--R 18b \|3 \|- a b \|a \|b
--R Type: Expression(Integer)
--E 204
--S 205 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 205
)clear all
--S 206 of 2952
t0000:=x^(4/3)/(a+b*x)^3
--R
--R
--R 3+-+
--R x\|x
--R (1) ----------------------------
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Expression(Integer)
--E 206
--S 207 of 2952
r0000:=-1/2*x^(4/3)/(b*(a+b*x)^2)-2/3*x^(1/3)/(b^2*(a+b*x))+_
2/9*log(a^(1/3)+b^(1/3)*x^(1/3))/(a^(2/3)*b^(7/3))-_
1/9*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/_
(a^(2/3)*b^(7/3))-2/3*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(7/3)*sqrt(3))
--R
--R
--R (2)
--R 2 2 2 +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (- 2b x - 4a b x - 2a )\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 2 2 2 +-+ 3+-+3+-+ 3+-+
--R (4b x + 8a b x + 4a )\|3 log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 2 2 2 2\|b \|x - \|a
--R (12b x + 24a b x + 12a )atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+2 3+-+3+-+
--R (- 21b x - 12a)\|3 \|a \|b \|x
--R /
--R 4 2 3 2 2 +-+3+-+2 3+-+
--R (18b x + 36a b x + 18a b )\|3 \|a \|b
--R Type: Expression(Integer)
--E 207
--S 208 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +---+2 +---+
--R 2 2 2 +-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R (- 2b x - 4a b x - 2a )\|3 log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R 2 2 2 +-+ 3| 2 3+-+
--R (4b x + 8a b x + 4a )\|3 log(\|a b \|x + a)
--R +
--R +---+
--R +-+3| 2 3+-+ +-+
--R 2 2 2 2\|3 \|a b \|x - a\|3
--R (12b x + 24a b x + 12a )atan(-----------------------)
--R 3a
--R +
--R +---+
--R +-+3| 2 3+-+
--R (- 21b x - 12a)\|3 \|a b \|x
--R /
--R +---+
--R 4 2 3 2 2 +-+3| 2
--R (18b x + 36a b x + 18a b )\|3 \|a b
--R Type: Union(Expression(Integer),...)
--E 208
--S 209 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +---+2 +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R - \|3 \|a \|b log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R \|3 \|a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +---+ +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+ +-+3| 2 3+-+3+-+ 3+-+
--R 2\|3 \|a \|b log(\|a b \|x + a - 2\|3 \|a b log(\|b \|x + \|a )
--R +
--R +---+
--R +-+3| 2 3+-+ +-+ +---+ 3+-+3+-+ 3+-+
--R 3+-+2 3+-+ 2\|3 \|a b \|x - a\|3 3| 2 2\|b \|x - \|a
--R 6\|a \|b atan(----------------------- - 6\|a b atan(----------------)
--R 3a +-+3+-+
--R \|3 \|a
--R /
--R +---+
--R 2 +-+3+-+2 3+-+3| 2
--R 9b \|3 \|a \|b \|a b
--R Type: Expression(Integer)
--E 209
--S 210 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 210
)clear all
--S 211 of 2952
t0000:=x^(2/3)/(a+b*x)^3
--R
--R
--R 3+-+2
--R \|x
--R (1) ----------------------------
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Expression(Integer)
--E 211
--S 212 of 2952
r0000:=-1/2*x^(2/3)/(b*(a+b*x)^2)+1/3*x^(2/3)/(a*b*(a+b*x))-_
1/9*log(a^(1/3)+b^(1/3)*x^(1/3))/(a^(4/3)*b^(5/3))+_
1/18*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/_
(a^(4/3)*b^(5/3))-1/3*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(5/3)*sqrt(3))
--R
--R
--R (2)
--R 2 2 2 +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (b x + 2a b x + a )\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 2 2 2 +-+ 3+-+3+-+ 3+-+
--R (- 2b x - 4a b x - 2a )\|3 log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 2 2 2 2\|b \|x - \|a
--R (6b x + 12a b x + 6a )atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+3+-+2 3+-+2
--R (6b x - 3a)\|3 \|a \|b \|x
--R /
--R 3 2 2 2 3 +-+3+-+3+-+2
--R (18a b x + 36a b x + 18a b)\|3 \|a \|b
--R Type: Expression(Integer)
--E 212
--S 213 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +------+ +------+2
--R 2 2 2 +-+ 3| 2 3+-+2 3| 2 3+-+
--R (- b x - 2a b x - a )\|3 log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+2
--R 2 2 2 +-+ 3| 2 3+-+
--R (2b x + 4a b x + 2a )\|3 log(\|- a b \|x + a b)
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+
--R 2 2 2 2\|3 \|- a b \|x - a b\|3
--R (- 6b x - 12a b x - 6a )atan(-----------------------------)
--R 3a b
--R +
--R +------+
--R +-+3| 2 3+-+2
--R (6b x - 3a)\|3 \|- a b \|x
--R /
--R +------+
--R 3 2 2 2 3 +-+3| 2
--R (18a b x + 36a b x + 18a b)\|3 \|- a b
--R Type: Union(Expression(Integer),...)
--E 213
--S 214 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +------+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|3 \|- a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +------+ +------+2
--R +-+3+-+3+-+2 3| 2 3+-+2 3| 2 3+-+
--R - \|3 \|a \|b log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+
--R +-+3| 2 3+-+3+-+ 3+-+
--R 2\|3 \|- a b log(\|b \|x + \|a )
--R +
--R +------+2
--R +-+3+-+3+-+2 3| 2 3+-+
--R 2\|3 \|a \|b log(\|- a b \|x + a b)
--R +
--R +------+ 3+-+3+-+ 3+-+
--R 3| 2 2\|b \|x - \|a
--R - 6\|- a b atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+
--R 3+-+3+-+2 2\|3 \|- a b \|x - a b\|3
--R - 6\|a \|b atan(-----------------------------)
--R 3a b
--R /
--R +------+
--R +-+3| 2 3+-+3+-+2
--R 18a b\|3 \|- a b \|a \|b
--R Type: Expression(Integer)
--E 214
--S 215 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 215
)clear all
--S 216 of 2952
t0000:=x^(1/3)/(a+b*x)^3
--R
--R
--R 3+-+
--R \|x
--R (1) ----------------------------
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Expression(Integer)
--E 216
--S 217 of 2952
r0000:=-1/2*x^(1/3)/(b*(a+b*x)^2)+1/6*x^(1/3)/(a*b*(a+b*x))+_
1/9*log(a^(1/3)+b^(1/3)*x^(1/3))/(a^(5/3)*b^(4/3))-_
1/18*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/_
(a^(5/3)*b^(4/3))-1/3*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(4/3)*sqrt(3))
--R
--R
--R (2)
--R 2 2 2 +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (- b x - 2a b x - a )\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 2 2 2 +-+ 3+-+3+-+ 3+-+
--R (2b x + 4a b x + 2a )\|3 log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 2 2 2 2\|b \|x - \|a
--R (6b x + 12a b x + 6a )atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+2 3+-+3+-+
--R (3b x - 6a)\|3 \|a \|b \|x
--R /
--R 3 2 2 2 3 +-+3+-+2 3+-+
--R (18a b x + 36a b x + 18a b)\|3 \|a \|b
--R Type: Expression(Integer)
--E 217
--S 218 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +---+2 +---+
--R 2 2 2 +-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R (- b x - 2a b x - a )\|3 log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R 2 2 2 +-+ 3| 2 3+-+
--R (2b x + 4a b x + 2a )\|3 log(\|a b \|x + a)
--R +
--R +---+
--R +-+3| 2 3+-+ +-+
--R 2 2 2 2\|3 \|a b \|x - a\|3
--R (6b x + 12a b x + 6a )atan(-----------------------)
--R 3a
--R +
--R +---+
--R +-+3| 2 3+-+
--R (3b x - 6a)\|3 \|a b \|x
--R /
--R +---+
--R 3 2 2 2 3 +-+3| 2
--R (18a b x + 36a b x + 18a b)\|3 \|a b
--R Type: Union(Expression(Integer),...)
--E 218
--S 219 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +---+2 +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R - \|3 \|a \|b log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R \|3 \|a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +---+ +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+ +-+3| 2 3+-+3+-+ 3+-+
--R 2\|3 \|a \|b log(\|a b \|x + a - 2\|3 \|a b log(\|b \|x + \|a )
--R +
--R +---+
--R +-+3| 2 3+-+ +-+ +---+ 3+-+3+-+ 3+-+
--R 3+-+2 3+-+ 2\|3 \|a b \|x - a\|3 3| 2 2\|b \|x - \|a
--R 6\|a \|b atan(----------------------- - 6\|a b atan(----------------)
--R 3a +-+3+-+
--R \|3 \|a
--R /
--R +---+
--R +-+3+-+2 3+-+3| 2
--R 18a b\|3 \|a \|b \|a b
--R Type: Expression(Integer)
--E 219
--S 220 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 220
)clear all
--S 221 of 2952
t0000:=1/(x^(1/3)*(a+b*x)^3)
--R
--R
--R 1
--R (1) ----------------------------------
--R 3 3 2 2 2 3 3+-+
--R (b x + 3a b x + 3a b x + a )\|x
--R Type: Expression(Integer)
--E 221
--S 222 of 2952
r0000:=1/2*x^(2/3)/(a*(a+b*x)^2)+2/3*x^(2/3)/(a^2*(a+b*x))-_
2/9*log(a^(1/3)+b^(1/3)*x^(1/3))/(a^(7/3)*b^(2/3))+_
1/9*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/_
(a^(7/3)*b^(2/3))-2/3*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(2/3)*sqrt(3))
--R
--R
--R (2)
--R 2 2 2 +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (2b x + 4a b x + 2a )\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 2 2 2 +-+ 3+-+3+-+ 3+-+
--R (- 4b x - 8a b x - 4a )\|3 log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 2 2 2 2\|b \|x - \|a
--R (12b x + 24a b x + 12a )atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+3+-+2 3+-+2
--R (12b x + 21a)\|3 \|a \|b \|x
--R /
--R 2 2 2 3 4 +-+3+-+3+-+2
--R (18a b x + 36a b x + 18a )\|3 \|a \|b
--R Type: Expression(Integer)
--E 222
--S 223 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +------+ +------+2
--R 2 2 2 +-+ 3| 2 3+-+2 3| 2 3+-+
--R (- 2b x - 4a b x - 2a )\|3 log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+2
--R 2 2 2 +-+ 3| 2 3+-+
--R (4b x + 8a b x + 4a )\|3 log(\|- a b \|x + a b)
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+
--R 2 2 2 2\|3 \|- a b \|x - a b\|3
--R (- 12b x - 24a b x - 12a )atan(-----------------------------)
--R 3a b
--R +
--R +------+
--R +-+3| 2 3+-+2
--R (12b x + 21a)\|3 \|- a b \|x
--R /
--R +------+
--R 2 2 2 3 4 +-+3| 2
--R (18a b x + 36a b x + 18a )\|3 \|- a b
--R Type: Union(Expression(Integer),...)
--E 223
--S 224 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +------+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - \|3 \|- a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +------+ +------+2
--R +-+3+-+3+-+2 3| 2 3+-+2 3| 2 3+-+
--R - \|3 \|a \|b log(b\|- a b \|x + \|- a b \|x - a b)
--R +
--R +------+
--R +-+3| 2 3+-+3+-+ 3+-+
--R 2\|3 \|- a b log(\|b \|x + \|a )
--R +
--R +------+2
--R +-+3+-+3+-+2 3| 2 3+-+
--R 2\|3 \|a \|b log(\|- a b \|x + a b)
--R +
--R +------+ 3+-+3+-+ 3+-+
--R 3| 2 2\|b \|x - \|a
--R - 6\|- a b atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +------+2
--R +-+3| 2 3+-+ +-+
--R 3+-+3+-+2 2\|3 \|- a b \|x - a b\|3
--R - 6\|a \|b atan(-----------------------------)
--R 3a b
--R /
--R +------+
--R 2 +-+3| 2 3+-+3+-+2
--R 9a \|3 \|- a b \|a \|b
--R Type: Expression(Integer)
--E 224
--S 225 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 225
)clear all
--S 226 of 2952
t0000:=1/(x^(2/3)*(a+b*x)^3)
--R
--R
--R 1
--R (1) -----------------------------------
--R 3 3 2 2 2 3 3+-+2
--R (b x + 3a b x + 3a b x + a )\|x
--R Type: Expression(Integer)
--E 226
--S 227 of 2952
r0000:=1/2*x^(1/3)/(a*(a+b*x)^2)+5/6*x^(1/3)/(a^2*(a+b*x))+_
5/9*log(a^(1/3)+b^(1/3)*x^(1/3))/(a^(8/3)*b^(1/3))-_
5/18*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/_
(a^(8/3)*b^(1/3))-5/3*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(1/3)*sqrt(3))
--R
--R
--R (2)
--R 2 2 2 +-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R (- 5b x - 10a b x - 5a )\|3 log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 2 2 2 +-+ 3+-+3+-+ 3+-+
--R (10b x + 20a b x + 10a )\|3 log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 2 2 2 2\|b \|x - \|a
--R (30b x + 60a b x + 30a )atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +-+3+-+2 3+-+3+-+
--R (15b x + 24a)\|3 \|a \|b \|x
--R /
--R 2 2 2 3 4 +-+3+-+2 3+-+
--R (18a b x + 36a b x + 18a )\|3 \|a \|b
--R Type: Expression(Integer)
--E 227
--S 228 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +---+2 +---+
--R 2 2 2 +-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R (- 5b x - 10a b x - 5a )\|3 log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R 2 2 2 +-+ 3| 2 3+-+
--R (10b x + 20a b x + 10a )\|3 log(\|a b \|x + a)
--R +
--R +---+
--R +-+3| 2 3+-+ +-+
--R 2 2 2 2\|3 \|a b \|x - a\|3
--R (30b x + 60a b x + 30a )atan(-----------------------)
--R 3a
--R +
--R +---+
--R +-+3| 2 3+-+
--R (15b x + 24a)\|3 \|a b \|x
--R /
--R +---+
--R 2 2 2 3 4 +-+3| 2
--R (18a b x + 36a b x + 18a )\|3 \|a b
--R Type: Union(Expression(Integer),...)
--E 228
--S 229 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +---+2 +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+2 3| 2 3+-+ 2
--R - 5\|3 \|a \|b log(\|a b \|x - a\|a b \|x + a )
--R +
--R +---+
--R +-+3| 2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R 5\|3 \|a b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +---+ +---+
--R +-+3+-+2 3+-+ 3| 2 3+-+ +-+3| 2 3+-+3+-+ 3+-+
--R 10\|3 \|a \|b log(\|a b \|x + a - 10\|3 \|a b log(\|b \|x + \|a )
--R +
--R +---+
--R +-+3| 2 3+-+ +-+ +---+ 3+-+3+-+ 3+-+
--R 3+-+2 3+-+ 2\|3 \|a b \|x - a\|3 3| 2 2\|b \|x - \|a
--R 30\|a \|b atan(----------------------- - 30\|a b atan(----------------)
--R 3a +-+3+-+
--R \|3 \|a
--R /
--R +---+
--R 2 +-+3+-+2 3+-+3| 2
--R 18a \|3 \|a \|b \|a b
--R Type: Expression(Integer)
--E 229
--S 230 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 230
)clear all
--S 231 of 2952
t0000:=1/(x^(4/3)*(a+b*x)^3)
--R
--R
--R 1
--R (1) ------------------------------------
--R 3 4 2 3 2 2 3 3+-+
--R (b x + 3a b x + 3a b x + a x)\|x
--R Type: Expression(Integer)
--E 231
--S 232 of 2952
r0000:=(-14/3)/(a^3*x^(1/3))+1/2/(a*x^(1/3)*(a+b*x)^2)+_
7/6/(a^2*x^(1/3)*(a+b*x))+14/9*b^(1/3)*log(a^(1/3)+_
b^(1/3)*x^(1/3))/a^(10/3)-7/9*b^(1/3)*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/a^(10/3)+_
14/3*b^(1/3)*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(10/3)*sqrt(3))
--R
--R
--R (2)
--R 2 2 2 +-+3+-+3+-+
--R (- 14b x - 28a b x - 14a )\|3 \|b \|x
--R *
--R 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 2 2 2 +-+3+-+3+-+ 3+-+3+-+ 3+-+
--R (28b x + 56a b x + 28a )\|3 \|b \|x log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 2 2 2 3+-+3+-+ 2\|b \|x - \|a
--R (- 84b x - 168a b x - 84a )\|b \|x atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R 2 2 2 +-+3+-+
--R (- 84b x - 147a b x - 54a )\|3 \|a
--R /
--R 3 2 2 4 5 +-+3+-+3+-+
--R (18a b x + 36a b x + 18a )\|3 \|a \|x
--R Type: Expression(Integer)
--E 232
--S 233 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +-+ +-+2 +-+
--R 2 2 2 +-+ |b 3+-+ 3+-+2 |b 3+-+ |b
--R (- 14b x - 28a b x - 14a )\|3 3|- \|x log(b \|x - a 3|- \|x + a 3|-)
--R \|a \|a \|a
--R +
--R +-+ +-+2
--R 2 2 2 +-+ |b 3+-+ 3+-+ |b
--R (28b x + 56a b x + 28a )\|3 3|- \|x log(b\|x + a 3|- )
--R \|a \|a
--R +
--R +-+2
--R +-+3+-+ +-+ |b
--R +-+ 2b\|3 \|x - a\|3 3|-
--R 2 2 2 |b 3+-+ \|a
--R (- 84b x - 168a b x - 84a ) 3|- \|x atan(-----------------------)
--R \|a +-+2
--R |b
--R 3a 3|-
--R \|a
--R +
--R 2 2 2 +-+
--R (- 84b x - 147a b x - 54a )\|3
--R /
--R 3 2 2 4 5 +-+3+-+
--R (18a b x + 36a b x + 18a )\|3 \|x
--R Type: Union(Expression(Integer),...)
--E 233
--S 234 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +-+3+-+ 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R 7\|3 \|b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +-+ +-+2 +-+
--R +-+3+-+ |b 3+-+2 |b 3+-+ |b
--R - 7\|3 \|a 3|- log(b \|x - a 3|- \|x + a 3|-)
--R \|a \|a \|a
--R +
--R +-+ +-+2
--R +-+3+-+ 3+-+3+-+ 3+-+ +-+3+-+ |b 3+-+ |b
--R - 14\|3 \|b log(\|b \|x + \|a ) + 14\|3 \|a 3|- log(b\|x + a 3|- )
--R \|a \|a
--R +
--R +-+2
--R +-+3+-+ +-+ |b
--R 3+-+3+-+ 3+-+ +-+ 2b\|3 \|x - a\|3 3|-
--R 3+-+ 2\|b \|x - \|a 3+-+ |b \|a
--R 42\|b atan(----------------) - 42\|a 3|- atan(-----------------------)
--R +-+3+-+ \|a +-+2
--R \|3 \|a |b
--R 3a 3|-
--R \|a
--R /
--R 3 +-+3+-+
--R 9a \|3 \|a
--R Type: Expression(Integer)
--E 234
--S 235 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 235
)clear all
--S 236 of 2952
t0000:=1/(x^(5/3)*(a+b*x)^3)
--R
--R
--R 1
--R (1) -------------------------------------
--R 3 4 2 3 2 2 3 3+-+2
--R (b x + 3a b x + 3a b x + a x)\|x
--R Type: Expression(Integer)
--E 236
--S 237 of 2952
r0000:=(-10/3)/(a^3*x^(2/3))+1/2/(a*x^(2/3)*(a+b*x)^2)+_
4/3/(a^2*x^(2/3)*(a+b*x))-20/9*b^(2/3)*log(a^(1/3)+_
b^(1/3)*x^(1/3))/a^(11/3)+10/9*b^(2/3)*log(a^(2/3)-_
a^(1/3)*b^(1/3)*x^(1/3)+b^(2/3)*x^(2/3))/a^(11/3)+_
20/3*b^(2/3)*atan((a^(1/3)-2*b^(1/3)*x^(1/3))/_
(a^(1/3)*sqrt(3)))/(a^(11/3)*sqrt(3))
--R
--R
--R (2)
--R 2 2 2 +-+3+-+2 3+-+2
--R (20b x + 40a b x + 20a )\|3 \|b \|x
--R *
--R 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R 2 2 2 +-+3+-+2 3+-+2 3+-+3+-+ 3+-+
--R (- 40b x - 80a b x - 40a )\|3 \|b \|x log(\|b \|x + \|a )
--R +
--R 3+-+3+-+ 3+-+
--R 2 2 2 3+-+2 3+-+2 2\|b \|x - \|a
--R (- 120b x - 240a b x - 120a )\|b \|x atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R 2 2 2 +-+3+-+2
--R (- 60b x - 96a b x - 27a )\|3 \|a
--R /
--R 3 2 2 4 5 +-+3+-+2 3+-+2
--R (18a b x + 36a b x + 18a )\|3 \|a \|x
--R Type: Expression(Integer)
--E 237
--S 238 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R +----+
--R | 2
--R 2 2 2 +-+ | b 3+-+2
--R (- 20b x - 40a b x - 20a )\|3 |- -- \|x
--R 3| 2
--R \| a
--R *
--R +----+ +----+2
--R | 2 | 2
--R 2 3+-+2 | b 3+-+ 2 | b
--R log(b \|x + a b |- -- \|x + a |- -- )
--R 3| 2 3| 2
--R \| a \| a
--R +
--R +----+ +----+
--R | 2 | 2
--R 2 2 2 +-+ | b 3+-+2 3+-+ | b
--R (40b x + 80a b x + 40a )\|3 |- -- \|x log(b\|x - a |- --)
--R 3| 2 3| 2
--R \| a \| a
--R +
--R +----+
--R | 2
--R +-+3+-+ +-+ | b
--R +----+ 2b\|3 \|x + a\|3 |- --
--R | 2 3| 2
--R 2 2 2 | b 3+-+2 \| a
--R (- 120b x - 240a b x - 120a ) |- -- \|x atan(-------------------------)
--R 3| 2 +----+
--R \| a | 2
--R | b
--R 3a |- --
--R 3| 2
--R \| a
--R +
--R 2 2 2 +-+
--R (- 60b x - 96a b x - 27a )\|3
--R /
--R 3 2 2 4 5 +-+3+-+2
--R (18a b x + 36a b x + 18a )\|3 \|x
--R Type: Union(Expression(Integer),...)
--E 238
--S 239 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R +-+3+-+2 3+-+2 3+-+2 3+-+3+-+3+-+ 3+-+2
--R - 10\|3 \|b log(\|b \|x - \|a \|b \|x + \|a )
--R +
--R +----+ +----+ +----+2
--R | 2 | 2 | 2
--R +-+ | b 3+-+2 2 3+-+2 | b 3+-+ 2 | b
--R - 10\|3 |- -- \|a log(b \|x + a b |- -- \|x + a |- -- )
--R 3| 2 3| 2 3| 2
--R \| a \| a \| a
--R +
--R +----+ +----+
--R | 2 | 2
--R +-+3+-+2 3+-+3+-+ 3+-+ +-+ | b 3+-+2 3+-+ | b
--R 20\|3 \|b log(\|b \|x + \|a ) + 20\|3 |- -- \|a log(b\|x - a |- --)
--R 3| 2 3| 2
--R \| a \| a
--R +
--R 3+-+3+-+ 3+-+
--R 3+-+2 2\|b \|x - \|a
--R 60\|b atan(----------------)
--R +-+3+-+
--R \|3 \|a
--R +
--R +----+
--R | 2
--R +-+3+-+ +-+ | b
--R +----+ 2b\|3 \|x + a\|3 |- --
--R | 2 3| 2
--R | b 3+-+2 \| a
--R - 60 |- -- \|a atan(-------------------------)
--R 3| 2 +----+
--R \| a | 2
--R | b
--R 3a |- --
--R 3| 2
--R \| a
--R /
--R 3 +-+3+-+2
--R 9a \|3 \|a
--R Type: Expression(Integer)
--E 239
--S 240 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 240
)clear all
--S 241 of 2952
t0000:=(1-x)^(1/4)/(1+x)
--R
--R
--R 4+-------+
--R \|- x + 1
--R (1) ----------
--R x + 1
--R Type: Expression(Integer)
--E 241
--S 242 of 2952
r0000:=4*(1-x)^(1/4)-2*2^(1/4)*atan((1-x)^(1/4)/2^(1/4))-_
2*2^(1/4)*atanh((1-x)^(1/4)/2^(1/4))
--R
--R
--R 4+-------+ 4+-------+
--R 4+-+ \|- x + 1 4+-+ \|- x + 1 4+-------+
--R (2) - 2\|2 atanh(----------) - 2\|2 atan(----------) + 4\|- x + 1
--R 4+-+ 4+-+
--R \|2 \|2
--R Type: Expression(Integer)
--E 242
--S 243 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 4+-+ 4+-------+ 4+-+ 4+-+ 4+-------+ 4+-+
--R - \|2 log(\|- x + 1 + \|2 ) + \|2 log(\|- x + 1 - \|2 )
--R +
--R 4+-+
--R 4+-+ \|2 4+-------+
--R 2\|2 atan(----------) + 4\|- x + 1
--R 4+-------+
--R \|- x + 1
--R Type: Union(Expression(Integer),...)
--E 243
--S 244 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 4+-+ 4+-------+ 4+-+ 4+-+ 4+-------+ 4+-+
--R - \|2 log(\|- x + 1 + \|2 ) + \|2 log(\|- x + 1 - \|2 )
--R +
--R 4+-------+ 4+-------+ 4+-+
--R 4+-+ \|- x + 1 4+-+ \|- x + 1 4+-+ \|2
--R 2\|2 atanh(----------) + 2\|2 atan(----------) + 2\|2 atan(----------)
--R 4+-+ 4+-+ 4+-------+
--R \|2 \|2 \|- x + 1
--R Type: Expression(Integer)
--E 244
--S 245 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 245
)clear all
--S 246 of 2952
t0000:=x^3*(a+b*x)^n
--R
--R
--R 3 n
--R (1) x (b x + a)
--R Type: Expression(Integer)
--E 246
--S 247 of 2952
r0000:=-a^3*(a+b*x)^(1+n)/(b^4*(1+n))+3*a^2*(a+b*x)^(2+n)/(b^4*(2+n))-_
3*a*(a+b*x)^(3+n)/(b^4*(3+n))+(a+b*x)^(4+n)/(b^4*(4+n))
--R
--R
--R (2)
--R 3 2 n + 4
--R (n + 6n + 11n + 6)(b x + a)
--R +
--R 3 2 n + 3
--R (- 3a n - 21a n - 42a n - 24a)(b x + a)
--R +
--R 2 3 2 2 2 2 n + 2
--R (3a n + 24a n + 57a n + 36a )(b x + a)
--R +
--R 3 3 3 2 3 3 n + 1
--R (- a n - 9a n - 26a n - 24a )(b x + a)
--R /
--R 4 4 4 3 4 2 4 4
--R b n + 10b n + 35b n + 50b n + 24b
--R Type: Expression(Integer)
--E 247
--S 248 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 4 3 4 2 4 4 4 3 3 3 2 3 3
--R (b n + 6b n + 11b n + 6b )x + (a b n + 3a b n + 2a b n)x
--R +
--R 2 2 2 2 2 2 3 4
--R (- 3a b n - 3a b n)x + 6a b n x - 6a
--R *
--R n log(b x + a)
--R %e
--R /
--R 4 4 4 3 4 2 4 4
--R b n + 10b n + 35b n + 50b n + 24b
--R Type: Union(Expression(Integer),...)
--E 248
--S 249 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 4 3 4 2 4 4 4 3 3 3 2 3 3
--R (b n + 6b n + 11b n + 6b )x + (a b n + 3a b n + 2a b n)x
--R +
--R 2 2 2 2 2 2 3 4
--R (- 3a b n - 3a b n)x + 6a b n x - 6a
--R *
--R n log(b x + a)
--R %e
--R +
--R 3 2 n + 4
--R (- n - 6n - 11n - 6)(b x + a)
--R +
--R 3 2 n + 3
--R (3a n + 21a n + 42a n + 24a)(b x + a)
--R +
--R 2 3 2 2 2 2 n + 2
--R (- 3a n - 24a n - 57a n - 36a )(b x + a)
--R +
--R 3 3 3 2 3 3 n + 1
--R (a n + 9a n + 26a n + 24a )(b x + a)
--R /
--R 4 4 4 3 4 2 4 4
--R b n + 10b n + 35b n + 50b n + 24b
--R Type: Expression(Integer)
--E 249
--S 250 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 250
)clear all
--S 251 of 2952
t0000:=x^2*(a+b*x)^n
--R
--R
--R 2 n
--R (1) x (b x + a)
--R Type: Expression(Integer)
--E 251
--S 252 of 2952
r0000:=a^2*(a+b*x)^(1+n)/(b^3*(1+n))-2*a*(a+b*x)^(2+n)/(b^3*(2+n))+_
(a+b*x)^(3+n)/(b^3*(3+n))
--R
--R
--R (2)
--R 2 n + 3 2 n + 2
--R (n + 3n + 2)(b x + a) + (- 2a n - 8a n - 6a)(b x + a)
--R +
--R 2 2 2 2 n + 1
--R (a n + 5a n + 6a )(b x + a)
--R /
--R 3 3 3 2 3 3
--R b n + 6b n + 11b n + 6b
--R Type: Expression(Integer)
--E 252
--S 253 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 2 3 3 3 2 2 2 2 2 3 n log(b x + a)
--R ((b n + 3b n + 2b )x + (a b n + a b n)x - 2a b n x + 2a )%e
--R -----------------------------------------------------------------------------
--R 3 3 3 2 3 3
--R b n + 6b n + 11b n + 6b
--R Type: Union(Expression(Integer),...)
--E 253
--S 254 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 3 2 3 3 3 2 2 2 2 2 3
--R ((b n + 3b n + 2b )x + (a b n + a b n)x - 2a b n x + 2a )
--R *
--R n log(b x + a)
--R %e
--R +
--R 2 n + 3 2 n + 2
--R (- n - 3n - 2)(b x + a) + (2a n + 8a n + 6a)(b x + a)
--R +
--R 2 2 2 2 n + 1
--R (- a n - 5a n - 6a )(b x + a)
--R /
--R 3 3 3 2 3 3
--R b n + 6b n + 11b n + 6b
--R Type: Expression(Integer)
--E 254
--S 255 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 255
)clear all
--S 256 of 2952
t0000:=x*(a+b*x)^n
--R
--R
--R n
--R (1) x (b x + a)
--R Type: Expression(Integer)
--E 256
--S 257 of 2952
r0000:=-a*(a+b*x)^(1+n)/(b^2*(1+n))+(a+b*x)^(2+n)/(b^2*(2+n))
--R
--R
--R n + 2 n + 1
--R (n + 1)(b x + a) + (- a n - 2a)(b x + a)
--R (2) --------------------------------------------------
--R 2 2 2 2
--R b n + 3b n + 2b
--R Type: Expression(Integer)
--E 257
--S 258 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 2 2 2 n log(b x + a)
--R ((b n + b )x + a b n x - a )%e
--R (3) ---------------------------------------------
--R 2 2 2 2
--R b n + 3b n + 2b
--R Type: Union(Expression(Integer),...)
--E 258
--S 259 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 2 2 2 2 n log(b x + a) n + 2
--R ((b n + b )x + a b n x - a )%e + (- n - 1)(b x + a)
--R +
--R n + 1
--R (a n + 2a)(b x + a)
--R /
--R 2 2 2 2
--R b n + 3b n + 2b
--R Type: Expression(Integer)
--E 259
--S 260 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 260
)clear all
--S 261 of 2952
t0000:=(a+b*x)^n
--R
--R
--R n
--R (1) (b x + a)
--R Type: Expression(Integer)
--E 261
--S 262 of 2952
r0000:=(a+b*x)^(1+n)/(b*(1+n))
--R
--R
--R n + 1
--R (b x + a)
--R (2) --------------
--R b n + b
--R Type: Expression(Integer)
--E 262
--S 263 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R n log(b x + a)
--R (b x + a)%e
--R (3) -------------------------
--R b n + b
--R Type: Union(Expression(Integer),...)
--E 263
--S 264 of 2952
m0000:=a0000 - r0000
--R
--R
--R n log(b x + a) n + 1
--R (b x + a)%e - (b x + a)
--R (4) ------------------------------------------
--R b n + b
--R Type: Expression(Integer)
--E 264
--S 265 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 265
)clear all
--S 266 of 2952
t0000:=(a+b*x)^n/x
--R
--R
--R n
--R (b x + a)
--R (1) ----------
--R x
--R Type: Expression(Integer)
--E 266
--S 267 of 2952
--r0000:=-(a+b*x)^(1+n)*hypergeometric(1,1+n,2+n,(a+b*x)/a)/(a*(1+n))
--E 267
--S 268 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n
--R ++ (%BJ b + a)
--R (2) | ------------ d%BJ
--R ++ %BJ
--R Type: Union(Expression(Integer),...)
--E 268
--S 269 of 2952
--m0000:=a0000 - r0000
--E 269
--S 270 of 2952
--d0000:=D(m0000,x)
--E 270
)clear all
--S 271 of 2952
t0000:=(a+b*x)^n/x^2
--R
--R
--R n
--R (b x + a)
--R (1) ----------
--R 2
--R x
--R Type: Expression(Integer)
--E 271
--S 272 of 2952
--r0000:=-(a+b*x)^(1+n)/(a*x)-b*n*(a+b*x)^(1+n)*_
-- hypergeometric(1,1+n,2+n,(a+b*x)/a)/(a^2*(1+n))
--E 272
--S 273 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n
--R ++ (%BJ b + a)
--R (2) | ------------ d%BJ
--R ++ 2
--R %BJ
--R Type: Union(Expression(Integer),...)
--E 273
--S 274 of 2952
--m0000:=a0000 - r0000
--E 274
--S 275 of 2952
--d0000:=D(m0000,x)
--E 275
)clear all
--S 276 of 2952
t0000:=(a+b*x)^n/x^3
--R
--R
--R n
--R (b x + a)
--R (1) ----------
--R 3
--R x
--R Type: Expression(Integer)
--E 276
--S 277 of 2952
--r0000:=-1/2*(a+b*x)^(1+n)/(a*x^2)+1/2*b*(1-n)*(a+b*x)^(1+n)/(a^2*x)+_
-- 1/2*b^2*(1-n)*n*(a+b*x)^(1+n)*_
-- hypergeometric(1,1+n,2+n,(a+b*x)/a)/(a^3*(1+n))
--E 277
--S 278 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n
--R ++ (%BJ b + a)
--R (2) | ------------ d%BJ
--R ++ 3
--R %BJ
--R Type: Union(Expression(Integer),...)
--E 278
--S 279 of 2952
--m0000:=a0000 - r0000
--E 279
--S 280 of 2952
--d0000:=D(m0000,x)
--E 280
)clear all
--S 281 of 2952
t0000:=x^(-4+n)/(a+b*x)^n
--R
--R
--R n - 4
--R x
--R (1) ----------
--R n
--R (b x + a)
--R Type: Expression(Integer)
--E 281
--S 282 of 2952
r0000:=-x^(-3+n)*(a+b*x)^(1-n)/(a*(3-n))+_
2*b*x^(-2+n)*(a+b*x)^(1-n)/(a^2*(2-n)*(3-n))-_
2*b^2*x^(-1+n)*(a+b*x)^(1-n)/(a^3*(1-n)*(2-n)*(3-n))
--R
--R
--R (2)
--R 2 n - 1 n - 2 2 2 2 2 n - 3
--R (2b x + (2a b n - 2a b)x + (a n - 3a n + 2a )x )
--R *
--R - n + 1
--R (b x + a)
--R /
--R 3 3 3 2 3 3
--R a n - 6a n + 11a n - 6a
--R Type: Expression(Integer)
--E 282
--S 283 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 4 2 3 2 2 2 2 3 2 3 3
--R (2b x + 2a b n x + (a b n - a b n)x + (a n - 3a n + 2a )x)
--R *
--R (n - 4)log(x)
--R %e
--R /
--R 3 3 3 2 3 3 n log(b x + a)
--R (a n - 6a n + 11a n - 6a )%e
--R Type: Union(Expression(Integer),...)
--E 283
--S 284 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 2 n - 1 n - 2 2 2 2 2 n - 3
--R (- 2b x + (- 2a b n + 2a b)x + (- a n + 3a n - 2a )x )
--R *
--R - n + 1 n log(b x + a)
--R (b x + a) %e
--R +
--R 3 4 2 3 2 2 2 2 3 2 3 3
--R (2b x + 2a b n x + (a b n - a b n)x + (a n - 3a n + 2a )x)
--R *
--R (n - 4)log(x)
--R %e
--R /
--R 3 3 3 2 3 3 n log(b x + a)
--R (a n - 6a n + 11a n - 6a )%e
--R Type: Expression(Integer)
--E 284
--S 285 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 285
)clear all
--S 286 of 2952
t0000:=x^(-3+n)/(a+b*x)^n
--R
--R
--R n - 3
--R x
--R (1) ----------
--R n
--R (b x + a)
--R Type: Expression(Integer)
--E 286
--S 287 of 2952
r0000:=-x^(-2+n)*(a+b*x)^(1-n)/(a*(2-n))+_
b*x^(-1+n)*(a+b*x)^(1-n)/(a^2*(1-n)*(2-n))
--R
--R
--R n - 1 n - 2 - n + 1
--R (b x + (a n - a)x )(b x + a)
--R (2) --------------------------------------------
--R 2 2 2 2
--R a n - 3a n + 2a
--R Type: Expression(Integer)
--E 287
--S 288 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 3 2 2 2 (n - 3)log(x)
--R (b x + a b n x + (a n - a )x)%e
--R (3) ----------------------------------------------
--R 2 2 2 2 n log(b x + a)
--R (a n - 3a n + 2a )%e
--R Type: Union(Expression(Integer),...)
--E 288
--S 289 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R n - 1 n - 2 - n + 1 n log(b x + a)
--R (- b x + (- a n + a)x )(b x + a) %e
--R +
--R 2 3 2 2 2 (n - 3)log(x)
--R (b x + a b n x + (a n - a )x)%e
--R /
--R 2 2 2 2 n log(b x + a)
--R (a n - 3a n + 2a )%e
--R Type: Expression(Integer)
--E 289
--S 290 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 290
)clear all
--S 291 of 2952
t0000:=x^(-2+n)/(a+b*x)^n
--R
--R
--R n - 2
--R x
--R (1) ----------
--R n
--R (b x + a)
--R Type: Expression(Integer)
--E 291
--S 292 of 2952
r0000:=-x^(-1+n)*(a+b*x)^(1-n)/(a*(1-n))
--R
--R
--R n - 1 - n + 1
--R x (b x + a)
--R (2) ----------------------
--R a n - a
--R Type: Expression(Integer)
--E 292
--S 293 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 (n - 2)log(x)
--R (b x + a x)%e
--R (3) ---------------------------
--R n log(b x + a)
--R (a n - a)%e
--R Type: Union(Expression(Integer),...)
--E 293
--S 294 of 2952
m0000:=a0000 - r0000
--R
--R
--R n - 1 - n + 1 n log(b x + a) 2 (n - 2)log(x)
--R - x (b x + a) %e + (b x + a x)%e
--R (4) ----------------------------------------------------------------------
--R n log(b x + a)
--R (a n - a)%e
--R Type: Expression(Integer)
--E 294
--S 295 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 295
)clear all
--S 296 of 2952
t0000:=x^(-1+n)/(a+b*x)^n
--R
--R
--R n - 1
--R x
--R (1) ----------
--R n
--R (b x + a)
--R Type: Expression(Integer)
--E 296
--S 297 of 2952
--r0000:=x^n*(a+b*x)^(1-n)/(a*n)-x^n*(a+b*x)^(1-n)*_
-- hypergeometric(1-n,-n,2-n,(a+b*x)/a)/(a*(1-n)*n*(-b*x/a)^n)
--E 297
--S 298 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n - 1
--R ++ %BJ
--R (2) | ------------ d%BJ
--R ++ n
--R (%BJ b + a)
--R Type: Union(Expression(Integer),...)
--E 298
--S 299 of 2952
--m0000:=a0000 - r0000
--E 299
--S 300 of 2952
--d0000:=D(m0000,x)
--E 300
)clear all
--S 301 of 2952
t0000:=x^n/(a+b*x)^n
--R
--R
--R n
--R x
--R (1) ----------
--R n
--R (b x + a)
--R Type: Expression(Integer)
--E 301
--S 302 of 2952
--r0000:=x^n*(a+b*x)^(1-n)*_
-- hypergeometric(1-n,-n,2-n,(a+b*x)/a)/(b*(1-n)*(-b*x/a)^n)
--E 302
--S 303 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n
--R ++ %BJ
--R (2) | ------------ d%BJ
--R ++ n
--R (%BJ b + a)
--R Type: Union(Expression(Integer),...)
--E 303
--S 304 of 2952
--m0000:=a0000 - r0000
--E 304
--S 305 of 2952
--d0000:=D(m0000,x)
--E 305
)clear all
--S 306 of 2952
t0000:=x^(1+n)/(a+b*x)^n
--R
--R
--R n + 1
--R x
--R (1) ----------
--R n
--R (b x + a)
--R Type: Expression(Integer)
--E 306
--S 307 of 2952
--r0000:=1/2*x^(1+n)*(a+b*x)^(1-n)/b-1/2*a*(1+n)*x^n*(a+b*x)^(1-n)*_
-- hypergeometric(1-n,-n,2-n,(a+b*x)/a)/(b^2*(1-n)*(-b*x/a)^n)
--E 307
--S 308 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n + 1
--R ++ %BJ
--R (2) | ------------ d%BJ
--R ++ n
--R (%BJ b + a)
--R Type: Union(Expression(Integer),...)
--E 308
--S 309 of 2952
--m0000:=a0000 - r0000
--E 309
--S 310 of 2952
--d0000:=D(m0000,x)
--E 310
)clear all
--S 311 of 2952
t0000:=x^(2+n)/(a+b*x)^n
--R
--R
--R n + 2
--R x
--R (1) ----------
--R n
--R (b x + a)
--R Type: Expression(Integer)
--E 311
--S 312 of 2952
--r0000:=-1/6*a*(2+n)*x^(1+n)*(a+b*x)^(1-n)/b^2+_
-- 1/3*x^(2+n)*(a+b*x)^(1-n)/b+1/6*a^2*(1+n)*(2+n)*x^n*(a+b*x)^(1-n)*_
-- hypergeometric(1-n,-n,2-n,(a+b*x)/a)/(b^3*(1-n)*(-b*x/a)^n)
--E 312
--S 313 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n + 2
--R ++ %BJ
--R (2) | ------------ d%BJ
--R ++ n
--R (%BJ b + a)
--R Type: Union(Expression(Integer),...)
--E 313
--S 314 of 2952
--m0000:=a0000 - r0000
--E 314
--S 315 of 2952
--d0000:=D(m0000,x)
--E 315
)clear all
--S 316 of 2952
t0000:=x^(3/2)*(a+b*x)^n
--R
--R
--R n +-+
--R (1) x (b x + a) \|x
--R Type: Expression(Integer)
--E 316
--S 317 of 2952
--r0000:=2*x^(3/2)*(a+b*x)^(1+n)/(b*(5+2*n))-_
-- 6*a*(a+b*x)^(1+n)*sqrt(x)/(b^2*(3+2*n)*(5+2*n))+_
-- 3*a^2*(a+b*x)^(1+n)*_
-- hypergeometric(1/2,1+n,2+n,(a+b*x)/a)*sqrt(-b*x/a)/_
-- (b^3*(1+n)*(3+2*n)*(5+2*n)*sqrt(x))
--E 317
--S 318 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x
--R ++ n +---+
--R (2) | %BJ (%BJ b + a) \|%BJ d%BJ
--R ++
--R Type: Union(Expression(Integer),...)
--E 318
--S 319 of 2952
--m0000:=a0000 - r0000
--E 319
--S 320 of 2952
--d0000:=D(m0000,x)
--E 320
)clear all
--S 321 of 2952
t0000:=x^(1/2)*(a+b*x)^n
--R
--R
--R n +-+
--R (1) (b x + a) \|x
--R Type: Expression(Integer)
--E 321
--S 322 of 2952
--r0000:=2*(a+b*x)^(1+n)*sqrt(x)/(b*(3+2*n))-_
-- a*(a+b*x)^(1+n)*hypergeometric(1/2,1+n,2+n,(a+b*x)/a)*_
-- sqrt(-b*x/a)/(b^2*(1+n)*(3+2*n)*sqrt(x))
--E 322
--S 323 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x
--R ++ n +---+
--R (2) | (%BJ b + a) \|%BJ d%BJ
--R ++
--R Type: Union(Expression(Integer),...)
--E 323
--S 324 of 2952
--m0000:=a0000 - r0000
--E 324
--S 325 of 2952
--d0000:=D(m0000,x)
--E 325
)clear all
--S 326 of 2952
t0000:=(a+b*x)^n/x^(1/2)
--R
--R
--R n
--R (b x + a)
--R (1) ----------
--R +-+
--R \|x
--R Type: Expression(Integer)
--E 326
--S 327 of 2952
--r0000:=(a+b*x)^(1+n)*_
-- hypergeometric(1/2,1+n,2+n,(a+b*x)/a)*sqrt(-b*x/a)/(b*(1+n)*sqrt(x))
--E 327
--S 328 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n
--R ++ (%BJ b + a)
--R (2) | ------------ d%BJ
--R ++ +---+
--R \|%BJ
--R Type: Union(Expression(Integer),...)
--E 328
--S 329 of 2952
--m0000:=a0000 - r0000
--E 329
--S 330 of 2952
--d0000:=D(m0000,x)
--E 330
)clear all
--S 331 of 2952
t0000:=(a+b*x)^n/x^(3/2)
--R
--R
--R n
--R (b x + a)
--R (1) ----------
--R +-+
--R x\|x
--R Type: Expression(Integer)
--E 331
--S 332 of 2952
--r0000:=-2*(a+b*x)^(1+n)/(a*sqrt(x))+(1+2*n)*(a+b*x)^(1+n)*_
-- hypergeometric(1/2,1+n,2+n,(a+b*x)/a)*sqrt(-b*x/a)/(a*(1+n)*sqrt(x))
--E 332
--S 333 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n
--R ++ (%BJ b + a)
--R (2) | ------------ d%BJ
--R ++ +---+
--R %BJ\|%BJ
--R Type: Union(Expression(Integer),...)
--E 333
--S 334 of 2952
--m0000:=a0000 - r0000
--E 334
--S 335 of 2952
--d0000:=D(m0000,x)
--E 335
)clear all
--S 336 of 2952
t0000:=(a+b*x)^n/x^(5/2)
--R
--R
--R n
--R (b x + a)
--R (1) ----------
--R 2 +-+
--R x \|x
--R Type: Expression(Integer)
--E 336
--S 337 of 2952
--r0000:=-2/3*(a+b*x)^(1+n)/(a*x^(3/2))+_
-- 2/3*b*(1-2*n)*(a+b*x)^(1+n)/(a^2*sqrt(x))-_
-- 1/3*b*(1-2*n)*(1+2*n)*(a+b*x)^(1+n)*_
-- hypergeometric(1/2,1+n,2+n,(a+b*x)/a)*sqrt(-b*x/a)/(a^2*(1+n)*sqrt(x))
--E 337
--S 338 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R x n
--R ++ (%BJ b + a)
--R (2) | ------------ d%BJ
--R ++ 2 +---+
--R %BJ \|%BJ
--R Type: Union(Expression(Integer),...)
--E 338
--S 339 of 2952
--m0000:=a0000 - r0000
--E 339
--S 340 of 2952
--d0000:=D(m0000,x)
--E 340
)clear all
--S 341 of 2952
t0000:=x^(-1+n)*(a+b*x)^(-1-n)
--R
--R
--R n - 1 - n - 1
--R (1) x (b x + a)
--R Type: Expression(Integer)
--E 341
--S 342 of 2952
r0000:=x^n/(a*n*(a+b*x)^n)
--R
--R
--R n
--R x
--R (2) --------------
--R n
--R a n (b x + a)
--R Type: Expression(Integer)
--E 342
--S 343 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 (- n - 1)log(b x + a) (n - 1)log(x)
--R (b x + a x)%e %e
--R (3) --------------------------------------------------
--R a n
--R Type: Union(Expression(Integer),...)
--E 343
--S 344 of 2952
m0000:=a0000 - r0000
--R
--R
--R 2 n (- n - 1)log(b x + a) (n - 1)log(x) n
--R (b x + a x)(b x + a) %e %e - x
--R (4) -----------------------------------------------------------------
--R n
--R a n (b x + a)
--R Type: Expression(Integer)
--E 344
--S 345 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 345
)clear all
--S 346 of 2952
t0000:=x^(-3-n)*(a+b*x)^n
--R
--R
--R - n - 3 n
--R (1) x (b x + a)
--R Type: Expression(Integer)
--E 346
--S 347 of 2952
r0000:=-x^(-2-n)*(a+b*x)^(1+n)/(a*(2+n))+_
b*x^(-1-n)*(a+b*x)^(1+n)/(a^2*(1+n)*(2+n))
--R
--R
--R - n - 1 - n - 2 n + 1
--R (b x + (- a n - a)x )(b x + a)
--R (2) ------------------------------------------------
--R 2 2 2 2
--R a n + 3a n + 2a
--R Type: Expression(Integer)
--E 347
--S 348 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 3 2 2 2 (- n - 3)log(x) n log(b x + a)
--R (b x - a b n x + (- a n - a )x)%e %e
--R (3) ------------------------------------------------------------------
--R 2 2 2 2
--R a n + 3a n + 2a
--R Type: Union(Expression(Integer),...)
--E 348
--S 349 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 2 3 2 2 2 (- n - 3)log(x) n log(b x + a)
--R (b x - a b n x + (- a n - a )x)%e %e
--R +
--R - n - 1 - n - 2 n + 1
--R (- b x + (a n + a)x )(b x + a)
--R /
--R 2 2 2 2
--R a n + 3a n + 2a
--R Type: Expression(Integer)
--E 349
--S 350 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 350
)clear all
--S 351 of 2952
t0000:=x^(2*n-3*(1+n))*(a+b*x)^n
--R
--R
--R - n - 3 n
--R (1) x (b x + a)
--R Type: Expression(Integer)
--E 351
--S 352 of 2952
r0000:=-x^(-2-n)*(a+b*x)^(1+n)/(a*(2+n))+_
b*x^(-1-n)*(a+b*x)^(1+n)/(a^2*(1+n)*(2+n))
--R
--R
--R - n - 1 - n - 2 n + 1
--R (b x + (- a n - a)x )(b x + a)
--R (2) ------------------------------------------------
--R 2 2 2 2
--R a n + 3a n + 2a
--R Type: Expression(Integer)
--E 352
--S 353 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 3 2 2 2 (- n - 3)log(x) n log(b x + a)
--R (b x - a b n x + (- a n - a )x)%e %e
--R (3) ------------------------------------------------------------------
--R 2 2 2 2
--R a n + 3a n + 2a
--R Type: Union(Expression(Integer),...)
--E 353
--S 354 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 2 3 2 2 2 (- n - 3)log(x) n log(b x + a)
--R (b x - a b n x + (- a n - a )x)%e %e
--R +
--R - n - 1 - n - 2 n + 1
--R (- b x + (a n + a)x )(b x + a)
--R /
--R 2 2 2 2
--R a n + 3a n + 2a
--R Type: Expression(Integer)
--E 354
--S 355 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 355
)clear all
--S 356 of 2952
t0000:=1/(e*(c+d*x)+sqrt(-a))
--R
--R
--R 1
--R (1) --------------------
--R +---+
--R \|- a + d e x + c e
--R Type: Expression(Integer)
--E 356
--S 357 of 2952
r0000:=log(e*(c+d*x)+sqrt(-a))/(d*e)
--R
--R
--R +---+
--R log(\|- a + d e x + c e)
--R (2) -------------------------
--R d e
--R Type: Expression(Integer)
--E 357
--S 358 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R +---+
--R log(\|- a + d e x + c e)
--R (3) -------------------------
--R d e
--R Type: Union(Expression(Integer),...)
--E 358
--S 359 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 359
--S 360 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 360
)clear all
--S 361 of 2952
t0000:=(c+d*(a+b*x))^(5/2)
--R
--R
--R 2 2 2 2 2 2 2 +---------------+
--R (1) (b d x + (2a b d + 2b c d)x + a d + 2a c d + c )\|b d x + a d + c
--R Type: Expression(Integer)
--E 361
--S 362 of 2952
r0000:=2/7*(c+d*(a+b*x))^(7/2)/(b*d)
--R
--R
--R (2)
--R 3 3 3 2 3 2 2 2 2 3 2 2
--R 2b d x + (6a b d + 6b c d )x + (6a b d + 12a b c d + 6b c d)x
--R +
--R 3 3 2 2 2 3
--R 2a d + 6a c d + 6a c d + 2c
--R *
--R +---------------+
--R \|b d x + a d + c
--R /
--R 7b d
--R Type: Expression(Integer)
--E 362
--S 363 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 3 3 2 3 2 2 2 2 3 2 2
--R 2b d x + (6a b d + 6b c d )x + (6a b d + 12a b c d + 6b c d)x
--R +
--R 3 3 2 2 2 3
--R 2a d + 6a c d + 6a c d + 2c
--R *
--R +---------------+
--R \|b d x + a d + c
--R /
--R 7b d
--R Type: Union(Expression(Integer),...)
--E 363
--S 364 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 364
--S 365 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 365
)clear all
--S 366 of 2952
t0000:=(c+d*(a+b*x))^(3/2)
--R
--R
--R +---------------+
--R (1) (b d x + a d + c)\|b d x + a d + c
--R Type: Expression(Integer)
--E 366
--S 367 of 2952
r0000:=2/5*(c+d*(a+b*x))^(5/2)/(b*d)
--R
--R
--R (2)
--R 2 2 2 2 2 2 2 +---------------+
--R (2b d x + (4a b d + 4b c d)x + 2a d + 4a c d + 2c )\|b d x + a d + c
--R ------------------------------------------------------------------------
--R 5b d
--R Type: Expression(Integer)
--E 367
--S 368 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 2 2 2 2 2 2 +---------------+
--R (2b d x + (4a b d + 4b c d)x + 2a d + 4a c d + 2c )\|b d x + a d + c
--R ------------------------------------------------------------------------
--R 5b d
--R Type: Union(Expression(Integer),...)
--E 368
--S 369 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 369
--S 370 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 370
)clear all
--S 371 of 2952
t0000:=(c+d*(a+b*x))^(1/2)
--R
--R
--R +---------------+
--R (1) \|b d x + a d + c
--R Type: Expression(Integer)
--E 371
--S 372 of 2952
r0000:=2/3*(c+d*(a+b*x))^(3/2)/(b*d)
--R
--R
--R +---------------+
--R (2b d x + 2a d + 2c)\|b d x + a d + c
--R (2) --------------------------------------
--R 3b d
--R Type: Expression(Integer)
--E 372
--S 373 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R +---------------+
--R (2b d x + 2a d + 2c)\|b d x + a d + c
--R (3) --------------------------------------
--R 3b d
--R Type: Union(Expression(Integer),...)
--E 373
--S 374 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 374
--S 375 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 375
)clear all
--S 376 of 2952
t0000:=1/(c+d*(a+b*x))^(1/2)
--R
--R
--R 1
--R (1) ------------------
--R +---------------+
--R \|b d x + a d + c
--R Type: Expression(Integer)
--E 376
--S 377 of 2952
r0000:=2*sqrt(c+d*(a+b*x))/(b*d)
--R
--R
--R +---------------+
--R 2\|b d x + a d + c
--R (2) -------------------
--R b d
--R Type: Expression(Integer)
--E 377
--S 378 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R +---------------+
--R 2\|b d x + a d + c
--R (3) -------------------
--R b d
--R Type: Union(Expression(Integer),...)
--E 378
--S 379 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 379
--S 380 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 380
)clear all
--S 381 of 2952
t0000:=1/(c+d*(a+b*x))^(3/2)
--R
--R
--R 1
--R (1) -----------------------------------
--R +---------------+
--R (b d x + a d + c)\|b d x + a d + c
--R Type: Expression(Integer)
--E 381
--S 382 of 2952
r0000:=(-2)/(b*d*sqrt(c+d*(a+b*x)))
--R
--R
--R 2
--R (2) - ---------------------
--R +---------------+
--R b d\|b d x + a d + c
--R Type: Expression(Integer)
--E 382
--S 383 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2
--R (3) - ---------------------
--R +---------------+
--R b d\|b d x + a d + c
--R Type: Union(Expression(Integer),...)
--E 383
--S 384 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 384
--S 385 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 385
)clear all
--S 386 of 2952
t0000:=1/(c+d*(a+b*x))^(5/2)
--R
--R
--R 1
--R (1) ---------------------------------------------------------------------
--R 2 2 2 2 2 2 2 +---------------+
--R (b d x + (2a b d + 2b c d)x + a d + 2a c d + c )\|b d x + a d + c
--R Type: Expression(Integer)
--E 386
--S 387 of 2952
r0000:=(-2/3)/(b*d*(c+d*(a+b*x))^(3/2))
--R
--R
--R 2
--R (2) - ---------------------------------------------
--R 2 2 2 +---------------+
--R (3b d x + 3a b d + 3b c d)\|b d x + a d + c
--R Type: Expression(Integer)
--E 387
--S 388 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2
--R (3) - ---------------------------------------------
--R 2 2 2 +---------------+
--R (3b d x + 3a b d + 3b c d)\|b d x + a d + c
--R Type: Union(Expression(Integer),...)
--E 388
--S 389 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 389
--S 390 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 390
)clear all
--S 391 of 2952
t0000:=(a+b*x)^m*(c+d*x)
--R
--R
--R m
--R (1) (d x + c)(b x + a)
--R Type: Expression(Integer)
--E 391
--S 392 of 2952
r0000:=(b*c-a*d)*(a+b*x)^(1+m)/(b^2*(1+m))+d*(a+b*x)^(2+m)/(b^2*(2+m))
--R
--R
--R m + 2 m + 1
--R (d m + d)(b x + a) + ((- a d + b c)m - 2a d + 2b c)(b x + a)
--R (2) ----------------------------------------------------------------------
--R 2 2 2 2
--R b m + 3b m + 2b
--R Type: Expression(Integer)
--E 392
--S 393 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 2 2 2 2 2
--R ((b d m + b d)x + ((a b d + b c)m + 2b c)x + a b c m - a d + 2a b c)
--R *
--R m log(b x + a)
--R %e
--R /
--R 2 2 2 2
--R b m + 3b m + 2b
--R Type: Union(Expression(Integer),...)
--E 393
--S 394 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 2 2 2 2 2 2
--R ((b d m + b d)x + ((a b d + b c)m + 2b c)x + a b c m - a d + 2a b c)
--R *
--R m log(b x + a)
--R %e
--R +
--R m + 2 m + 1
--R (- d m - d)(b x + a) + ((a d - b c)m + 2a d - 2b c)(b x + a)
--R /
--R 2 2 2 2
--R b m + 3b m + 2b
--R Type: Expression(Integer)
--E 394
--S 395 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 395
)clear all
--S 396 of 2952
t0000:=(a+b*x)^4*(c+d*x)
--R
--R
--R (1)
--R 4 5 3 4 4 2 2 3 3 3 2 2 2
--R b d x + (4a b d + b c)x + (6a b d + 4a b c)x + (4a b d + 6a b c)x
--R +
--R 4 3 4
--R (a d + 4a b c)x + a c
--R Type: Polynomial(Integer)
--E 396
--S 397 of 2952
r0000:=1/5*(b*c-a*d)*(a+b*x)^5/b^2+1/6*d*(a+b*x)^6/b^2
--R
--R
--R (2)
--R 1 6 6 4 5 1 6 5 3 2 4 5 4
--R - b d x + (- a b d + - b c)x + (- a b d + a b c)x
--R 6 5 5 2
--R +
--R 4 3 3 2 4 3 1 4 2 3 3 2 4 2 1 6 1 5
--R (- a b d + 2a b c)x + (- a b d + 2a b c)x + a b c x - -- a d + - a b c
--R 3 2 30 5
--R /
--R 2
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 397
--S 398 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 4 6 4 3 1 4 5 3 2 2 3 4 4 3 2 2 3
--R - b d x + (- a b d + - b c)x + (- a b d + a b c)x + (- a b d + 2a b c)x
--R 6 5 5 2 3
--R +
--R 1 4 3 2 4
--R (- a d + 2a b c)x + a c x
--R 2
--R Type: Polynomial(Fraction(Integer))
--E 398
--S 399 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 6 1 5
--R -- a d - - a b c
--R 30 5
--R (4) ----------------
--R 2
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 399
--S 400 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 400
)clear all
--S 401 of 2952
t0000:=(a+b*x)^3*(c+d*x)
--R
--R
--R (1)
--R 3 4 2 3 3 2 2 2 3 2 3
--R b d x + (3a b d + b c)x + (3a b d + 3a b c)x + (a d + 3a b c)x + a c
--R Type: Polynomial(Integer)
--E 401
--S 402 of 2952
r0000:=1/4*(b*c-a*d)*(a+b*x)^4/b^2+1/5*d*(a+b*x)^5/b^2
--R
--R
--R (2)
--R 1 5 5 3 4 1 5 4 2 3 4 3
--R - b d x + (- a b d + - b c)x + (a b d + a b c)x
--R 5 4 4
--R +
--R 1 3 2 3 2 3 2 3 2 1 5 1 4
--R (- a b d + - a b c)x + a b c x - -- a d + - a b c
--R 2 2 20 4
--R /
--R 2
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 402
--S 403 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 3 5 3 2 1 3 4 2 2 3 1 3 3 2 2
--R - b d x + (- a b d + - b c)x + (a b d + a b c)x + (- a d + - a b c)x
--R 5 4 4 2 2
--R +
--R 3
--R a c x
--R Type: Polynomial(Fraction(Integer))
--E 403
--S 404 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 5 1 4
--R -- a d - - a b c
--R 20 4
--R (4) ----------------
--R 2
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 404
--S 405 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 405
)clear all
--S 406 of 2952
t0000:=(a+b*x)^2*(c+d*x)
--R
--R
--R 2 3 2 2 2 2
--R (1) b d x + (2a b d + b c)x + (a d + 2a b c)x + a c
--R Type: Polynomial(Integer)
--E 406
--S 407 of 2952
r0000:=1/3*(b*c-a*d)*(a+b*x)^3/b^2+1/4*d*(a+b*x)^4/b^2
--R
--R
--R (2)
--R 1 4 4 2 3 1 4 3 1 2 2 3 2 2 2 1 4
--R - b d x + (- a b d + - b c)x + (- a b d + a b c)x + a b c x - -- a d
--R 4 3 3 2 12
--R +
--R 1 3
--R - a b c
--R 3
--R /
--R 2
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 407
--S 408 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 1 2 4 2 1 2 3 1 2 2 2
--R (3) - b d x + (- a b d + - b c)x + (- a d + a b c)x + a c x
--R 4 3 3 2
--R Type: Polynomial(Fraction(Integer))
--E 408
--S 409 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 4 1 3
--R -- a d - - a b c
--R 12 3
--R (4) ----------------
--R 2
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 409
--S 410 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 410
)clear all
--S 411 of 2952
t0000:=(a+b*x)*(c+d*x)
--R
--R
--R 2
--R (1) b d x + (a d + b c)x + a c
--R Type: Polynomial(Integer)
--E 411
--S 412 of 2952
r0000:=a*c*x+1/2*(b*c+a*d)*x^2+1/3*b*d*x^3
--R
--R
--R 1 3 1 1 2
--R (2) - b d x + (- a d + - b c)x + a c x
--R 3 2 2
--R Type: Polynomial(Fraction(Integer))
--E 412
--S 413 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 1 3 1 1 2
--R (3) - b d x + (- a d + - b c)x + a c x
--R 3 2 2
--R Type: Polynomial(Fraction(Integer))
--E 413
--S 414 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Polynomial(Fraction(Integer))
--E 414
--S 415 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Polynomial(Fraction(Integer))
--E 415
)clear all
--S 416 of 2952
t0000:=c+d*x
--R
--R
--R (1) d x + c
--R Type: Polynomial(Integer)
--E 416
--S 417 of 2952
r0000:=c*x+1/2*d*x^2
--R
--R
--R 1 2
--R (2) - d x + c x
--R 2
--R Type: Polynomial(Fraction(Integer))
--E 417
--S 418 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 1 2
--R (3) - d x + c x
--R 2
--R Type: Polynomial(Fraction(Integer))
--E 418
--S 419 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Polynomial(Fraction(Integer))
--E 419
--S 420 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Polynomial(Fraction(Integer))
--E 420
)clear all
--S 421 of 2952
t0000:=(c+d*x)/(a+b*x)
--R
--R
--R d x + c
--R (1) -------
--R b x + a
--R Type: Fraction(Polynomial(Integer))
--E 421
--S 422 of 2952
r0000:=d*x/b+(b*c-a*d)*log(a+b*x)/b^2
--R
--R
--R (- a d + b c)log(b x + a) + b d x
--R (2) ---------------------------------
--R 2
--R b
--R Type: Expression(Integer)
--E 422
--S 423 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (- a d + b c)log(b x + a) + b d x
--R (3) ---------------------------------
--R 2
--R b
--R Type: Union(Expression(Integer),...)
--E 423
--S 424 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 424
--S 425 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 425
)clear all
--S 426 of 2952
t0000:=(c+d*x)/(a+b*x)^2
--R
--R
--R d x + c
--R (1) ------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 426
--S 427 of 2952
r0000:=(-b*c+a*d)/(b^2*(a+b*x))+d*log(a+b*x)/b^2
--R
--R
--R (b d x + a d)log(b x + a) + a d - b c
--R (2) -------------------------------------
--R 3 2
--R b x + a b
--R Type: Expression(Integer)
--E 427
--S 428 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (b d x + a d)log(b x + a) + a d - b c
--R (3) -------------------------------------
--R 3 2
--R b x + a b
--R Type: Union(Expression(Integer),...)
--E 428
--S 429 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 429
--S 430 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 430
)clear all
--S 431 of 2952
t0000:=(c+d*x)/(a+b*x)^3
--R
--R
--R d x + c
--R (1) ----------------------------
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 431
--S 432 of 2952
r0000:=-1/2*(c+d*x)^2/((b*c-a*d)*(a+b*x)^2)
--R
--R
--R 1 2 2 1 2
--R - d x + c d x + - c
--R 2 2
--R (2) --------------------------------------------------
--R 2 3 2 2 2 3 2
--R (a b d - b c)x + (2a b d - 2a b c)x + a d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 432
--S 433 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R - 2b d x - a d - b c
--R (3) ----------------------
--R 4 2 3 2 2
--R 2b x + 4a b x + 2a b
--R Type: Union(Expression(Integer),...)
--E 433
--S 434 of 2952
m0000:=a0000 - r0000
--R
--R
--R 2
--R d
--R (4) - -------------
--R 2 3
--R 2a b d - 2b c
--R Type: Expression(Integer)
--E 434
--S 435 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 435
)clear all
--S 436 of 2952
t0000:=(c+d*x)/(a+b*x)^4
--R
--R
--R d x + c
--R (1) --------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 436
--S 437 of 2952
r0000:=1/3*(-b*c+a*d)/(b^2*(a+b*x)^3)-1/2*d/(b^2*(a+b*x)^2)
--R
--R
--R 1 1 1
--R - - b d x - - a d - - b c
--R 2 6 3
--R (2) ------------------------------
--R 5 3 4 2 2 3 3 2
--R b x + 3a b x + 3a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 437
--S 438 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R - 3b d x - a d - 2b c
--R (3) ----------------------------------
--R 5 3 4 2 2 3 3 2
--R 6b x + 18a b x + 18a b x + 6a b
--R Type: Union(Expression(Integer),...)
--E 438
--S 439 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 439
--S 440 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 440
)clear all
--S 441 of 2952
t0000:=(c+d*x)/(a+b*x)^5
--R
--R
--R d x + c
--R (1) --------------------------------------------------
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R b x + 5a b x + 10a b x + 10a b x + 5a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 441
--S 442 of 2952
r0000:=1/4*(-b*c+a*d)/(b^2*(a+b*x)^4)-1/3*d/(b^2*(a+b*x)^3)
--R
--R
--R 1 1 1
--R - - b d x - -- a d - - b c
--R 3 12 4
--R (2) ----------------------------------------
--R 6 4 5 3 2 4 2 3 3 4 2
--R b x + 4a b x + 6a b x + 4a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 442
--S 443 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R - 4b d x - a d - 3b c
--R (3) -----------------------------------------------
--R 6 4 5 3 2 4 2 3 3 4 2
--R 12b x + 48a b x + 72a b x + 48a b x + 12a b
--R Type: Union(Expression(Integer),...)
--E 443
--S 444 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 444
--S 445 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 445
)clear all
--S 446 of 2952
t0000:=(a+b*x)^m*(c+d*x)^2
--R
--R
--R 2 2 2 m
--R (1) (d x + 2c d x + c )(b x + a)
--R Type: Expression(Integer)
--E 446
--S 447 of 2952
r0000:=(b*c-a*d)^2*(a+b*x)^(1+m)/(b^3*(1+m))+_
2*d*(b*c-a*d)*(a+b*x)^(2+m)/(b^3*(2+m))+d^2*(a+b*x)^(3+m)/(b^3*(3+m))
--R
--R
--R (2)
--R 2 2 2 2 m + 3
--R (d m + 3d m + 2d )(b x + a)
--R +
--R 2 2 2 2
--R ((- 2a d + 2b c d)m + (- 8a d + 8b c d)m - 6a d + 6b c d)
--R *
--R m + 2
--R (b x + a)
--R +
--R 2 2 2 2 2 2 2 2 2 2 2
--R (a d - 2a b c d + b c )m + (5a d - 10a b c d + 5b c )m + 6a d
--R +
--R 2 2
--R - 12a b c d + 6b c
--R *
--R m + 1
--R (b x + a)
--R /
--R 3 3 3 2 3 3
--R b m + 6b m + 11b m + 6b
--R Type: Expression(Integer)
--E 447
--S 448 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 2 2 3 2 3 2 3
--R (b d m + 3b d m + 2b d )x
--R +
--R 2 2 3 2 2 2 3 3 2
--R ((a b d + 2b c d)m + (a b d + 8b c d)m + 6b c d)x
--R +
--R 2 3 2 2 2 2 2 3 2 3 2
--R ((2a b c d + b c )m + (- 2a b d + 6a b c d + 5b c )m + 6b c )x
--R +
--R 2 2 2 2 2 2 3 2 2 2 2
--R a b c m + (- 2a b c d + 5a b c )m + 2a d - 6a b c d + 6a b c
--R *
--R m log(b x + a)
--R %e
--R /
--R 3 3 3 2 3 3
--R b m + 6b m + 11b m + 6b
--R Type: Union(Expression(Integer),...)
--E 448
--S 449 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 3 2 2 3 2 3 2 3
--R (b d m + 3b d m + 2b d )x
--R +
--R 2 2 3 2 2 2 3 3 2
--R ((a b d + 2b c d)m + (a b d + 8b c d)m + 6b c d)x
--R +
--R 2 3 2 2 2 2 2 3 2 3 2
--R ((2a b c d + b c )m + (- 2a b d + 6a b c d + 5b c )m + 6b c )x
--R +
--R 2 2 2 2 2 2 3 2 2 2 2
--R a b c m + (- 2a b c d + 5a b c )m + 2a d - 6a b c d + 6a b c
--R *
--R m log(b x + a)
--R %e
--R +
--R 2 2 2 2 m + 3
--R (- d m - 3d m - 2d )(b x + a)
--R +
--R 2 2 2 2 m + 2
--R ((2a d - 2b c d)m + (8a d - 8b c d)m + 6a d - 6b c d)(b x + a)
--R +
--R 2 2 2 2 2 2 2 2 2 2 2
--R (- a d + 2a b c d - b c )m + (- 5a d + 10a b c d - 5b c )m - 6a d
--R +
--R 2 2
--R 12a b c d - 6b c
--R *
--R m + 1
--R (b x + a)
--R /
--R 3 3 3 2 3 3
--R b m + 6b m + 11b m + 6b
--R Type: Expression(Integer)
--E 449
--S 450 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 450
)clear all
--S 451 of 2952
t0000:=(a+b*x)^4*(c+d*x)^2
--R
--R
--R (1)
--R 4 2 6 3 2 4 5 2 2 2 3 4 2 4
--R b d x + (4a b d + 2b c d)x + (6a b d + 8a b c d + b c )x
--R +
--R 3 2 2 2 3 2 3 4 2 3 2 2 2 2
--R (4a b d + 12a b c d + 4a b c )x + (a d + 8a b c d + 6a b c )x
--R +
--R 4 3 2 4 2
--R (2a c d + 4a b c )x + a c
--R Type: Polynomial(Integer)
--E 451
--S 452 of 2952
r0000:=1/5*(b*c-a*d)^2*(a+b*x)^5/b^3+_
1/3*d*(b*c-a*d)*(a+b*x)^6/b^3+1/7*d^2*(a+b*x)^7/b^3
--R
--R
--R (2)
--R 1 7 2 7 2 6 2 1 7 6 6 2 5 2 8 6 1 7 2 5
--R - b d x + (- a b d + - b c d)x + (- a b d + - a b c d + - b c )x
--R 7 3 3 5 5 5
--R +
--R 3 4 2 2 5 6 2 4 1 4 3 2 8 3 4 2 5 2 3
--R (a b d + 3a b c d + a b c )x + (- a b d + - a b c d + 2a b c )x
--R 3 3
--R +
--R 4 3 3 4 2 2 4 3 2 1 7 2 1 6 1 5 2 2
--R (a b c d + 2a b c )x + a b c x + --- a d - -- a b c d + - a b c
--R 105 15 5
--R /
--R 3
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 452
--S 453 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 4 2 7 2 3 2 1 4 6 6 2 2 2 8 3 1 4 2 5
--R - b d x + (- a b d + - b c d)x + (- a b d + - a b c d + - b c )x
--R 7 3 3 5 5 5
--R +
--R 3 2 2 2 3 2 4 1 4 2 8 3 2 2 2 3
--R (a b d + 3a b c d + a b c )x + (- a d + - a b c d + 2a b c )x
--R 3 3
--R +
--R 4 3 2 2 4 2
--R (a c d + 2a b c )x + a c x
--R Type: Polynomial(Fraction(Integer))
--E 453
--S 454 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 7 2 1 6 1 5 2 2
--R - --- a d + -- a b c d - - a b c
--R 105 15 5
--R (4) ----------------------------------
--R 3
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 454
--S 455 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 455
)clear all
--S 456 of 2952
t0000:=(a+b*x)^3*(c+d*x)^2
--R
--R
--R (1)
--R 3 2 5 2 2 3 4 2 2 2 3 2 3
--R b d x + (3a b d + 2b c d)x + (3a b d + 6a b c d + b c )x
--R +
--R 3 2 2 2 2 2 3 2 2 3 2
--R (a d + 6a b c d + 3a b c )x + (2a c d + 3a b c )x + a c
--R Type: Polynomial(Integer)
--E 456
--S 457 of 2952
r0000:=1/4*(b*c-a*d)^2*(a+b*x)^4/b^3+2/5*d*(b*c-a*d)*(a+b*x)^5/b^3+_
1/6*d^2*(a+b*x)^6/b^3
--R
--R
--R (2)
--R 1 6 2 6 3 5 2 2 6 5 3 2 4 2 3 5 1 6 2 4
--R - b d x + (- a b d + - b c d)x + (- a b d + - a b c d + - b c )x
--R 6 5 5 4 2 4
--R +
--R 1 3 3 2 2 4 5 2 3 3 3 3 2 4 2 2 3 3 2
--R (- a b d + 2a b c d + a b c )x + (a b c d + - a b c )x + a b c x
--R 3 2
--R +
--R 1 6 2 1 5 1 4 2 2
--R -- a d - -- a b c d + - a b c
--R 60 10 4
--R /
--R 3
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 457
--S 458 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 3 2 6 3 2 2 2 3 5 3 2 2 3 2 1 3 2 4
--R - b d x + (- a b d + - b c d)x + (- a b d + - a b c d + - b c )x
--R 6 5 5 4 2 4
--R +
--R 1 3 2 2 2 2 3 3 3 2 2 2 3 2
--R (- a d + 2a b c d + a b c )x + (a c d + - a b c )x + a c x
--R 3 2
--R Type: Polynomial(Fraction(Integer))
--E 458
--S 459 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 6 2 1 5 1 4 2 2
--R - -- a d + -- a b c d - - a b c
--R 60 10 4
--R (4) ---------------------------------
--R 3
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 459
--S 460 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 460
)clear all
--S 461 of 2952
t0000:=(a+b*x)^2*(c+d*x)^2
--R
--R
--R (1)
--R 2 2 4 2 2 3 2 2 2 2 2
--R b d x + (2a b d + 2b c d)x + (a d + 4a b c d + b c )x
--R +
--R 2 2 2 2
--R (2a c d + 2a b c )x + a c
--R Type: Polynomial(Integer)
--E 461
--S 462 of 2952
r0000:=1/3*(b*c-a*d)^2*(a+b*x)^3/b^3+1/2*d*(b*c-a*d)*(a+b*x)^4/b^3+_
1/5*d^2*(a+b*x)^5/b^3
--R
--R
--R (2)
--R 1 5 2 5 1 4 2 1 5 4 1 2 3 2 4 4 1 5 2 3
--R - b d x + (- a b d + - b c d)x + (- a b d + - a b c d + - b c )x
--R 5 2 2 3 3 3
--R +
--R 2 3 4 2 2 2 3 2 1 5 2 1 4 1 3 2 2
--R (a b c d + a b c )x + a b c x + -- a d - - a b c d + - a b c
--R 30 6 3
--R /
--R 3
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 462
--S 463 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 2 2 5 1 2 1 2 4 1 2 2 4 1 2 2 3
--R - b d x + (- a b d + - b c d)x + (- a d + - a b c d + - b c )x
--R 5 2 2 3 3 3
--R +
--R 2 2 2 2 2
--R (a c d + a b c )x + a c x
--R Type: Polynomial(Fraction(Integer))
--E 463
--S 464 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 5 2 1 4 1 3 2 2
--R - -- a d + - a b c d - - a b c
--R 30 6 3
--R (4) --------------------------------
--R 3
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 464
--S 465 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 465
)clear all
--S 466 of 2952
t0000:=(a+b*x)*(c+d*x)^2
--R
--R
--R 2 3 2 2 2 2
--R (1) b d x + (a d + 2b c d)x + (2a c d + b c )x + a c
--R Type: Polynomial(Integer)
--E 466
--S 467 of 2952
r0000:=-1/3*(b*c-a*d)*(c+d*x)^3/d^2+1/4*b*(c+d*x)^4/d^2
--R
--R
--R (2)
--R 1 4 4 1 4 2 3 3 3 1 2 2 2 2 2
--R - b d x + (- a d + - b c d )x + (a c d + - b c d )x + a c d x
--R 4 3 3 2
--R +
--R 1 3 1 4
--R - a c d - -- b c
--R 3 12
--R /
--R 2
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 467
--S 468 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 1 2 4 1 2 2 3 1 2 2 2
--R (3) - b d x + (- a d + - b c d)x + (a c d + - b c )x + a c x
--R 4 3 3 2
--R Type: Polynomial(Fraction(Integer))
--E 468
--S 469 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 3 1 4
--R - - a c d + -- b c
--R 3 12
--R (4) -------------------
--R 2
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 469
--S 470 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 470
)clear all
--S 471 of 2952
t0000:=(c+d*x)^2
--R
--R
--R 2 2 2
--R (1) d x + 2c d x + c
--R Type: Polynomial(Integer)
--E 471
--S 472 of 2952
r0000:=1/3*(c+d*x)^3/d
--R
--R
--R 1 3 3 2 2 2 1 3
--R - d x + c d x + c d x + - c
--R 3 3
--R (2) ------------------------------
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 472
--S 473 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 1 2 3 2 2
--R (3) - d x + c d x + c x
--R 3
--R Type: Polynomial(Fraction(Integer))
--E 473
--S 474 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 3
--R - c
--R 3
--R (4) - ----
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 474
--S 475 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 475
)clear all
--S 476 of 2952
t0000:=(c+d*x)^2/(a+b*x)
--R
--R
--R 2 2 2
--R d x + 2c d x + c
--R (1) ------------------
--R b x + a
--R Type: Fraction(Polynomial(Integer))
--E 476
--S 477 of 2952
r0000:=d*(b*c-a*d)*x/b^2+1/2*(c+d*x)^2/b+(b*c-a*d)^2*log(a+b*x)/b^3
--R
--R
--R (2)
--R 2 2 2 2 2 2 2 2 2
--R (2a d - 4a b c d + 2b c )log(b x + a) + b d x + (- 2a b d + 4b c d)x
--R +
--R 2 2
--R b c
--R /
--R 3
--R 2b
--R Type: Expression(Integer)
--E 477
--S 478 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 2 2 2 2 2 2 2 2
--R (2a d - 4a b c d + 2b c )log(b x + a) + b d x + (- 2a b d + 4b c d)x
--R -----------------------------------------------------------------------
--R 3
--R 2b
--R Type: Union(Expression(Integer),...)
--E 478
--S 479 of 2952
m0000:=a0000 - r0000
--R
--R
--R 2
--R c
--R (4) - --
--R 2b
--R Type: Expression(Integer)
--E 479
--S 480 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 480
)clear all
--S 481 of 2952
t0000:=(c+d*x)^2/(a+b*x)^2
--R
--R
--R 2 2 2
--R d x + 2c d x + c
--R (1) ------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 481
--S 482 of 2952
r0000:=d^2*x/b^2-(b*c-a*d)^2/(b^3*(a+b*x))+2*d*(b*c-a*d)*log(a+b*x)/b^3
--R
--R
--R (2)
--R 2 2 2 2 2 2 2 2
--R ((- 2a b d + 2b c d)x - 2a d + 2a b c d)log(b x + a) + b d x + a b d x
--R +
--R 2 2 2 2
--R - a d + 2a b c d - b c
--R /
--R 4 3
--R b x + a b
--R Type: Expression(Integer)
--E 482
--S 483 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 2 2 2 2 2 2 2
--R ((- 2a b d + 2b c d)x - 2a d + 2a b c d)log(b x + a) + b d x + a b d x
--R +
--R 2 2 2 2
--R - a d + 2a b c d - b c
--R /
--R 4 3
--R b x + a b
--R Type: Union(Expression(Integer),...)
--E 483
--S 484 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 484
--S 485 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 485
)clear all
--S 486 of 2952
t0000:=(c+d*x)^2/(a+b*x)^3
--R
--R
--R 2 2 2
--R d x + 2c d x + c
--R (1) ----------------------------
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 486
--S 487 of 2952
r0000:=-1/2*(b*c-a*d)^2/(b^3*(a+b*x)^2)-2*d*(b*c-a*d)/(b^3*(a+b*x))+_
d^2*log(a+b*x)/b^3
--R
--R
--R (2)
--R 2 2 2 2 2 2 2 2 2 2
--R (2b d x + 4a b d x + 2a d )log(b x + a) + (4a b d - 4b c d)x + 3a d
--R +
--R 2 2
--R - 2a b c d - b c
--R /
--R 5 2 4 2 3
--R 2b x + 4a b x + 2a b
--R Type: Expression(Integer)
--E 487
--S 488 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 2 2 2 2 2 2 2 2 2
--R (2b d x + 4a b d x + 2a d )log(b x + a) + (4a b d - 4b c d)x + 3a d
--R +
--R 2 2
--R - 2a b c d - b c
--R /
--R 5 2 4 2 3
--R 2b x + 4a b x + 2a b
--R Type: Union(Expression(Integer),...)
--E 488
--S 489 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 489
--S 490 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 490
)clear all
--S 491 of 2952
t0000:=(c+d*x)^2/(a+b*x)^4
--R
--R
--R 2 2 2
--R d x + 2c d x + c
--R (1) --------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 491
--S 492 of 2952
r0000:=-1/3*(c+d*x)^3/((b*c-a*d)*(a+b*x)^3)
--R
--R
--R (2)
--R 1 3 3 2 2 2 1 3
--R - d x + c d x + c d x + - c
--R 3 3
--R ------------------------------------------------------------------------
--R 3 4 3 2 2 3 2 3 2 2 4 3
--R (a b d - b c)x + (3a b d - 3a b c)x + (3a b d - 3a b c)x + a d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 492
--S 493 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 2 2 2 2 2 2 2 2
--R - 3b d x + (- 3a b d - 3b c d)x - a d - a b c d - b c
--R (3) ---------------------------------------------------------
--R 6 3 5 2 2 4 3 3
--R 3b x + 9a b x + 9a b x + 3a b
--R Type: Union(Expression(Integer),...)
--E 493
--S 494 of 2952
m0000:=a0000 - r0000
--R
--R
--R 3
--R d
--R (4) - -------------
--R 3 4
--R 3a b d - 3b c
--R Type: Expression(Integer)
--E 494
--S 495 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 495
)clear all
--S 496 of 2952
t0000:=(c+d*x)^2/(a+b*x)^5
--R
--R
--R 2 2 2
--R d x + 2c d x + c
--R (1) --------------------------------------------------
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R b x + 5a b x + 10a b x + 10a b x + 5a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 496
--S 497 of 2952
r0000:=-1/4*(b*c-a*d)^2/(b^3*(a+b*x)^4)-2/3*d*(b*c-a*d)/(b^3*(a+b*x)^3)-_
1/2*d^2/(b^3*(a+b*x)^2)
--R
--R
--R 1 2 2 2 1 2 2 2 1 2 2 1 1 2 2
--R - - b d x + (- - a b d - - b c d)x - -- a d - - a b c d - - b c
--R 2 3 3 12 6 4
--R (2) -------------------------------------------------------------------
--R 7 4 6 3 2 5 2 3 4 4 3
--R b x + 4a b x + 6a b x + 4a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 497
--S 498 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 2 2 2 2 2 2 2 2
--R - 6b d x + (- 4a b d - 8b c d)x - a d - 2a b c d - 3b c
--R (3) -----------------------------------------------------------
--R 7 4 6 3 2 5 2 3 4 4 3
--R 12b x + 48a b x + 72a b x + 48a b x + 12a b
--R Type: Union(Expression(Integer),...)
--E 498
--S 499 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 499
--S 500 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 500
)clear all
--S 501 of 2952
t0000:=(c+d*x)^2/(a+b*x)^6
--R
--R
--R 2 2 2
--R d x + 2c d x + c
--R (1) -------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 501
--S 502 of 2952
r0000:=-1/5*(b*c-a*d)^2/(b^3*(a+b*x)^5)-1/2*d*(b*c-a*d)/(b^3*(a+b*x)^4)-_
1/3*d^2/(b^3*(a+b*x)^3)
--R
--R
--R 1 2 2 2 1 2 1 2 1 2 2 1 1 2 2
--R - - b d x + (- - a b d - - b c d)x - -- a d - -- a b c d - - b c
--R 3 6 2 30 10 5
--R (2) --------------------------------------------------------------------
--R 8 5 7 4 2 6 3 3 5 2 4 4 5 3
--R b x + 5a b x + 10a b x + 10a b x + 5a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 502
--S 503 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 2 2 2 2 2 2 2 2 2
--R - 10b d x + (- 5a b d - 15b c d)x - a d - 3a b c d - 6b c
--R (3) --------------------------------------------------------------
--R 8 5 7 4 2 6 3 3 5 2 4 4 5 3
--R 30b x + 150a b x + 300a b x + 300a b x + 150a b x + 30a b
--R Type: Union(Expression(Integer),...)
--E 503
--S 504 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 504
--S 505 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 505
)clear all
--S 506 of 2952
t0000:=(c+d*x)^2/(a+b*x)^7
--R
--R
--R (1)
--R 2 2 2
--R d x + 2c d x + c
--R ------------------------------------------------------------------------
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R b x + 7a b x + 21a b x + 35a b x + 35a b x + 21a b x + 7a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 506
--S 507 of 2952
r0000:=-1/6*(b*c-a*d)^2/(b^3*(a+b*x)^6)-2/5*d*(b*c-a*d)/(b^3*(a+b*x)^5)-_
1/4*d^2/(b^3*(a+b*x)^4)
--R
--R
--R 1 2 2 2 1 2 2 2 1 2 2 1 1 2 2
--R - - b d x + (- -- a b d - - b c d)x - -- a d - -- a b c d - - b c
--R 4 10 5 60 15 6
--R (2) ---------------------------------------------------------------------
--R 9 6 8 5 2 7 4 3 6 3 4 5 2 5 4 6 3
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 507
--S 508 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 2 2 2 2 2 2 2 2
--R - 15b d x + (- 6a b d - 24b c d)x - a d - 4a b c d - 10b c
--R ---------------------------------------------------------------------------
--R 9 6 8 5 2 7 4 3 6 3 4 5 2 5 4 6 3
--R 60b x + 360a b x + 900a b x + 1200a b x + 900a b x + 360a b x + 60a b
--R Type: Union(Expression(Integer),...)
--E 508
--S 509 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 509
--S 510 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 510
)clear all
--S 511 of 2952
t0000:=(a+b*x)^m*(c+d*x)^3
--R
--R
--R 3 3 2 2 2 3 m
--R (1) (d x + 3c d x + 3c d x + c )(b x + a)
--R Type: Expression(Integer)
--E 511
--S 512 of 2952
r0000:=(b*c-a*d)^3*(a+b*x)^(1+m)/(b^4*(1+m))+_
3*d*(b*c-a*d)^2*(a+b*x)^(2+m)/(b^4*(2+m))+_
3*d^2*(b*c-a*d)*(a+b*x)^(3+m)/(b^4*(3+m))+_
d^3*(a+b*x)^(4+m)/(b^4*(4+m))
--R
--R
--R (2)
--R 3 3 3 2 3 3 m + 4
--R (d m + 6d m + 11d m + 6d )(b x + a)
--R +
--R 3 2 3 3 2 2
--R (- 3a d + 3b c d )m + (- 21a d + 21b c d )m
--R +
--R 3 2 3 2
--R (- 42a d + 42b c d )m - 24a d + 24b c d
--R *
--R m + 3
--R (b x + a)
--R +
--R 2 3 2 2 2 3 2 3 2 2 2 2
--R (3a d - 6a b c d + 3b c d)m + (24a d - 48a b c d + 24b c d)m
--R +
--R 2 3 2 2 2 2 3 2 2 2
--R (57a d - 114a b c d + 57b c d)m + 36a d - 72a b c d + 36b c d
--R *
--R m + 2
--R (b x + a)
--R +
--R 3 3 2 2 2 2 3 3 3
--R (- a d + 3a b c d - 3a b c d + b c )m
--R +
--R 3 3 2 2 2 2 3 3 2
--R (- 9a d + 27a b c d - 27a b c d + 9b c )m
--R +
--R 3 3 2 2 2 2 3 3 3 3 2 2
--R (- 26a d + 78a b c d - 78a b c d + 26b c )m - 24a d + 72a b c d
--R +
--R 2 2 3 3
--R - 72a b c d + 24b c
--R *
--R m + 1
--R (b x + a)
--R /
--R 4 4 4 3 4 2 4 4
--R b m + 10b m + 35b m + 50b m + 24b
--R Type: Expression(Integer)
--E 512
--S 513 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 4 3 3 4 3 2 4 3 4 3 4
--R (b d m + 6b d m + 11b d m + 6b d )x
--R +
--R 3 3 4 2 3 3 3 4 2 2
--R (a b d + 3b c d )m + (3a b d + 21b c d )m
--R +
--R 3 3 4 2 4 2
--R (2a b d + 42b c d )m + 24b c d
--R *
--R 3
--R x
--R +
--R 3 2 4 2 3 2 2 3 3 2 4 2 2
--R (3a b c d + 3b c d)m + (- 3a b d + 15a b c d + 24b c d)m
--R +
--R 2 2 3 3 2 4 2 4 2
--R (- 3a b d + 12a b c d + 57b c d)m + 36b c d
--R *
--R 2
--R x
--R +
--R 3 2 4 3 3 2 2 2 3 2 4 3 2
--R (3a b c d + b c )m + (- 6a b c d + 21a b c d + 9b c )m
--R +
--R 3 3 2 2 2 3 2 4 3 4 3
--R (6a b d - 24a b c d + 36a b c d + 26b c )m + 24b c
--R *
--R x
--R +
--R 3 3 3 2 2 2 3 3 2
--R a b c m + (- 3a b c d + 9a b c )m
--R +
--R 3 2 2 2 2 3 3 4 3 3 2 2 2 2
--R (6a b c d - 21a b c d + 26a b c )m - 6a d + 24a b c d - 36a b c d
--R +
--R 3 3
--R 24a b c
--R *
--R m log(b x + a)
--R %e
--R /
--R 4 4 4 3 4 2 4 4
--R b m + 10b m + 35b m + 50b m + 24b
--R Type: Union(Expression(Integer),...)
--E 513
--S 514 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 4 3 3 4 3 2 4 3 4 3 4
--R (b d m + 6b d m + 11b d m + 6b d )x
--R +
--R 3 3 4 2 3 3 3 4 2 2
--R (a b d + 3b c d )m + (3a b d + 21b c d )m
--R +
--R 3 3 4 2 4 2
--R (2a b d + 42b c d )m + 24b c d
--R *
--R 3
--R x
--R +
--R 3 2 4 2 3 2 2 3 3 2 4 2 2
--R (3a b c d + 3b c d)m + (- 3a b d + 15a b c d + 24b c d)m
--R +
--R 2 2 3 3 2 4 2 4 2
--R (- 3a b d + 12a b c d + 57b c d)m + 36b c d
--R *
--R 2
--R x
--R +
--R 3 2 4 3 3 2 2 2 3 2 4 3 2
--R (3a b c d + b c )m + (- 6a b c d + 21a b c d + 9b c )m
--R +
--R 3 3 2 2 2 3 2 4 3 4 3
--R (6a b d - 24a b c d + 36a b c d + 26b c )m + 24b c
--R *
--R x
--R +
--R 3 3 3 2 2 2 3 3 2
--R a b c m + (- 3a b c d + 9a b c )m
--R +
--R 3 2 2 2 2 3 3 4 3 3 2 2 2 2
--R (6a b c d - 21a b c d + 26a b c )m - 6a d + 24a b c d - 36a b c d
--R +
--R 3 3
--R 24a b c
--R *
--R m log(b x + a)
--R %e
--R +
--R 3 3 3 2 3 3 m + 4
--R (- d m - 6d m - 11d m - 6d )(b x + a)
--R +
--R 3 2 3 3 2 2 3 2
--R (3a d - 3b c d )m + (21a d - 21b c d )m + (42a d - 42b c d )m
--R +
--R 3 2
--R 24a d - 24b c d
--R *
--R m + 3
--R (b x + a)
--R +
--R 2 3 2 2 2 3
--R (- 3a d + 6a b c d - 3b c d)m
--R +
--R 2 3 2 2 2 2
--R (- 24a d + 48a b c d - 24b c d)m
--R +
--R 2 3 2 2 2 2 3 2 2 2
--R (- 57a d + 114a b c d - 57b c d)m - 36a d + 72a b c d - 36b c d
--R *
--R m + 2
--R (b x + a)
--R +
--R 3 3 2 2 2 2 3 3 3
--R (a d - 3a b c d + 3a b c d - b c )m
--R +
--R 3 3 2 2 2 2 3 3 2
--R (9a d - 27a b c d + 27a b c d - 9b c )m
--R +
--R 3 3 2 2 2 2 3 3 3 3 2 2
--R (26a d - 78a b c d + 78a b c d - 26b c )m + 24a d - 72a b c d
--R +
--R 2 2 3 3
--R 72a b c d - 24b c
--R *
--R m + 1
--R (b x + a)
--R /
--R 4 4 4 3 4 2 4 4
--R b m + 10b m + 35b m + 50b m + 24b
--R Type: Expression(Integer)
--E 514
--S 515 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 515
)clear all
--S 516 of 2952
t0000:=(a+b*x)^5*(c+d*x)^3
--R
--R
--R (1)
--R 5 3 8 4 3 5 2 7 2 3 3 4 2 5 2 6
--R b d x + (5a b d + 3b c d )x + (10a b d + 15a b c d + 3b c d)x
--R +
--R 3 2 3 2 3 2 4 2 5 3 5
--R (10a b d + 30a b c d + 15a b c d + b c )x
--R +
--R 4 3 3 2 2 2 3 2 4 3 4
--R (5a b d + 30a b c d + 30a b c d + 5a b c )x
--R +
--R 5 3 4 2 3 2 2 2 3 3 3
--R (a d + 15a b c d + 30a b c d + 10a b c )x
--R +
--R 5 2 4 2 3 2 3 2 5 2 4 3 5 3
--R (3a c d + 15a b c d + 10a b c )x + (3a c d + 5a b c )x + a c
--R Type: Polynomial(Integer)
--E 516
--S 517 of 2952
r0000:=1/6*(b*c-a*d)^3*(a+b*x)^6/b^4+3/7*d*(b*c-a*d)^2*(a+b*x)^7/b^4+_
3/8*d^2*(b*c-a*d)*(a+b*x)^8/b^4+1/9*d^3*(a+b*x)^9/b^4
--R
--R
--R (2)
--R 1 9 3 9 5 8 3 3 9 2 8
--R - b d x + (- a b d + - b c d )x
--R 9 8 8
--R +
--R 10 2 7 3 15 8 2 3 9 2 7
--R (-- a b d + -- a b c d + - b c d)x
--R 7 7 7
--R +
--R 5 3 6 3 2 7 2 5 8 2 1 9 3 6
--R (- a b d + 5a b c d + - a b c d + - b c )x
--R 3 2 6
--R +
--R 4 5 3 3 6 2 2 7 2 8 3 5
--R (a b d + 6a b c d + 6a b c d + a b c )x
--R +
--R 1 5 4 3 15 4 5 2 15 3 6 2 5 2 7 3 4
--R (- a b d + -- a b c d + -- a b c d + - a b c )x
--R 4 4 2 2
--R +
--R 5 4 2 4 5 2 10 3 6 3 3 3 5 4 2 5 4 5 3 2 5 4 3
--R (a b c d + 5a b c d + -- a b c )x + (- a b c d + - a b c )x + a b c x
--R 3 2 2
--R +
--R 1 9 3 1 8 2 1 7 2 2 1 6 3 3
--R - --- a d + -- a b c d - -- a b c d + - a b c
--R 504 56 14 6
--R /
--R 4
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 517
--S 518 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 5 3 9 5 4 3 3 5 2 8 10 2 3 3 15 4 2 3 5 2 7
--R - b d x + (- a b d + - b c d )x + (-- a b d + -- a b c d + - b c d)x
--R 9 8 8 7 7 7
--R +
--R 5 3 2 3 2 3 2 5 4 2 1 5 3 6
--R (- a b d + 5a b c d + - a b c d + - b c )x
--R 3 2 6
--R +
--R 4 3 3 2 2 2 3 2 4 3 5
--R (a b d + 6a b c d + 6a b c d + a b c )x
--R +
--R 1 5 3 15 4 2 15 3 2 2 5 2 3 3 4
--R (- a d + -- a b c d + -- a b c d + - a b c )x
--R 4 4 2 2
--R +
--R 5 2 4 2 10 3 2 3 3 3 5 2 5 4 3 2 5 3
--R (a c d + 5a b c d + -- a b c )x + (- a c d + - a b c )x + a c x
--R 3 2 2
--R Type: Polynomial(Fraction(Integer))
--E 518
--S 519 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 9 3 1 8 2 1 7 2 2 1 6 3 3
--R --- a d - -- a b c d + -- a b c d - - a b c
--R 504 56 14 6
--R (4) ----------------------------------------------
--R 4
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 519
--S 520 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 520
)clear all
--S 521 of 2952
t0000:=(a+b*x)^4*(c+d*x)^3
--R
--R
--R (1)
--R 4 3 7 3 3 4 2 6 2 2 3 3 2 4 2 5
--R b d x + (4a b d + 3b c d )x + (6a b d + 12a b c d + 3b c d)x
--R +
--R 3 3 2 2 2 3 2 4 3 4
--R (4a b d + 18a b c d + 12a b c d + b c )x
--R +
--R 4 3 3 2 2 2 2 3 3 3
--R (a d + 12a b c d + 18a b c d + 4a b c )x
--R +
--R 4 2 3 2 2 2 3 2 4 2 3 3 4 3
--R (3a c d + 12a b c d + 6a b c )x + (3a c d + 4a b c )x + a c
--R Type: Polynomial(Integer)
--E 521
--S 522 of 2952
r0000:=1/5*(b*c-a*d)^3*(a+b*x)^5/b^4+1/2*d*(b*c-a*d)^2*(a+b*x)^6/b^4+_
3/7*d^2*(b*c-a*d)*(a+b*x)^7/b^4+1/8*d^3*(a+b*x)^8/b^4
--R
--R
--R (2)
--R 1 8 3 8 4 7 3 3 8 2 7 2 6 3 7 2 1 8 2 6
--R - b d x + (- a b d + - b c d )x + (a b d + 2a b c d + - b c d)x
--R 8 7 7 2
--R +
--R 4 3 5 3 18 2 6 2 12 7 2 1 8 3 5
--R (- a b d + -- a b c d + -- a b c d + - b c )x
--R 5 5 5 5
--R +
--R 1 4 4 3 3 5 2 9 2 6 2 7 3 4
--R (- a b d + 3a b c d + - a b c d + a b c )x
--R 4 2
--R +
--R 4 4 2 3 5 2 2 6 3 3 3 4 4 2 3 5 3 2 4 4 3
--R (a b c d + 4a b c d + 2a b c )x + (- a b c d + 2a b c )x + a b c x
--R 2
--R +
--R 1 8 3 1 7 2 1 6 2 2 1 5 3 3
--R - --- a d + -- a b c d - -- a b c d + - a b c
--R 280 35 10 5
--R /
--R 4
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 522
--S 523 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 4 3 8 4 3 3 3 4 2 7 2 2 3 3 2 1 4 2 6
--R - b d x + (- a b d + - b c d )x + (a b d + 2a b c d + - b c d)x
--R 8 7 7 2
--R +
--R 4 3 3 18 2 2 2 12 3 2 1 4 3 5
--R (- a b d + -- a b c d + -- a b c d + - b c )x
--R 5 5 5 5
--R +
--R 1 4 3 3 2 9 2 2 2 3 3 4
--R (- a d + 3a b c d + - a b c d + a b c )x
--R 4 2
--R +
--R 4 2 3 2 2 2 3 3 3 4 2 3 3 2 4 3
--R (a c d + 4a b c d + 2a b c )x + (- a c d + 2a b c )x + a c x
--R 2
--R Type: Polynomial(Fraction(Integer))
--E 523
--S 524 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 8 3 1 7 2 1 6 2 2 1 5 3 3
--R --- a d - -- a b c d + -- a b c d - - a b c
--R 280 35 10 5
--R (4) ----------------------------------------------
--R 4
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 524
--S 525 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 525
)clear all
--S 526 of 2952
t0000:=(a+b*x)^3*(c+d*x)^3
--R
--R
--R (1)
--R 3 3 6 2 3 3 2 5 2 3 2 2 3 2 4
--R b d x + (3a b d + 3b c d )x + (3a b d + 9a b c d + 3b c d)x
--R +
--R 3 3 2 2 2 2 3 3 3 3 2 2 2 2 3 2
--R (a d + 9a b c d + 9a b c d + b c )x + (3a c d + 9a b c d + 3a b c )x
--R +
--R 3 2 2 3 3 3
--R (3a c d + 3a b c )x + a c
--R Type: Polynomial(Integer)
--E 526
--S 527 of 2952
r0000:=1/4*(b*c-a*d)^3*(a+b*x)^4/b^4+3/5*d*(b*c-a*d)^2*(a+b*x)^5/b^4+_
1/2*d^2*(b*c-a*d)*(a+b*x)^6/b^4+1/7*d^3*(a+b*x)^7/b^4
--R
--R
--R (2)
--R 1 7 3 7 1 6 3 1 7 2 6 3 2 5 3 9 6 2 3 7 2 5
--R - b d x + (- a b d + - b c d )x + (- a b d + - a b c d + - b c d)x
--R 7 2 2 5 5 5
--R +
--R 1 3 4 3 9 2 5 2 9 6 2 1 7 3 4
--R (- a b d + - a b c d + - a b c d + - b c )x
--R 4 4 4 4
--R +
--R 3 4 2 2 5 2 6 3 3 3 3 4 2 3 2 5 3 2 3 4 3
--R (a b c d + 3a b c d + a b c )x + (- a b c d + - a b c )x + a b c x
--R 2 2
--R +
--R 1 7 3 1 6 2 3 5 2 2 1 4 3 3
--R - --- a d + -- a b c d - -- a b c d + - a b c
--R 140 20 20 4
--R /
--R 4
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 527
--S 528 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 3 3 7 1 2 3 1 3 2 6 3 2 3 9 2 2 3 3 2 5
--R - b d x + (- a b d + - b c d )x + (- a b d + - a b c d + - b c d)x
--R 7 2 2 5 5 5
--R +
--R 1 3 3 9 2 2 9 2 2 1 3 3 4
--R (- a d + - a b c d + - a b c d + - b c )x
--R 4 4 4 4
--R +
--R 3 2 2 2 2 3 3 3 3 2 3 2 3 2 3 3
--R (a c d + 3a b c d + a b c )x + (- a c d + - a b c )x + a c x
--R 2 2
--R Type: Polynomial(Fraction(Integer))
--E 528
--S 529 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 7 3 1 6 2 3 5 2 2 1 4 3 3
--R --- a d - -- a b c d + -- a b c d - - a b c
--R 140 20 20 4
--R (4) ----------------------------------------------
--R 4
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 529
--S 530 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 530
)clear all
--S 531 of 2952
t0000:=(a+b*x)^2*(c+d*x)^3
--R
--R
--R (1)
--R 2 3 5 3 2 2 4 2 3 2 2 2 3
--R b d x + (2a b d + 3b c d )x + (a d + 6a b c d + 3b c d)x
--R +
--R 2 2 2 2 3 2 2 2 3 2 3
--R (3a c d + 6a b c d + b c )x + (3a c d + 2a b c )x + a c
--R Type: Polynomial(Integer)
--E 531
--S 532 of 2952
r0000:=1/4*(b*c-a*d)^2*(c+d*x)^4/d^3-2/5*b*(b*c-a*d)*(c+d*x)^5/d^3+_
1/6*b^2*(c+d*x)^6/d^3
--R
--R
--R (2)
--R 1 2 6 6 2 6 3 2 5 5 1 2 6 3 5 3 2 2 4 4
--R - b d x + (- a b d + - b c d )x + (- a d + - a b c d + - b c d )x
--R 6 5 5 4 2 4
--R +
--R 2 5 2 4 1 2 3 3 3 3 2 2 4 3 3 2 2 3 3
--R (a c d + 2a b c d + - b c d )x + (- a c d + a b c d )x + a c d x
--R 3 2
--R +
--R 1 2 4 2 1 5 1 2 6
--R - a c d - -- a b c d + -- b c
--R 4 10 60
--R /
--R 3
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 532
--S 533 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 2 3 6 2 3 3 2 2 5 1 2 3 3 2 3 2 2 4
--R - b d x + (- a b d + - b c d )x + (- a d + - a b c d + - b c d)x
--R 6 5 5 4 2 4
--R +
--R 2 2 2 1 2 3 3 3 2 2 3 2 2 3
--R (a c d + 2a b c d + - b c )x + (- a c d + a b c )x + a c x
--R 3 2
--R Type: Polynomial(Fraction(Integer))
--E 533
--S 534 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 2 4 2 1 5 1 2 6
--R - - a c d + -- a b c d - -- b c
--R 4 10 60
--R (4) ---------------------------------
--R 3
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 534
--S 535 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 535
)clear all
--S 536 of 2952
t0000:=(a+b*x)*(c+d*x)^3
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3 3
--R b d x + (a d + 3b c d )x + (3a c d + 3b c d)x + (3a c d + b c )x + a c
--R Type: Polynomial(Integer)
--E 536
--S 537 of 2952
r0000:=-1/4*(b*c-a*d)*(c+d*x)^4/d^2+1/5*b*(c+d*x)^5/d^2
--R
--R
--R (2)
--R 1 5 5 1 5 3 4 4 4 2 3 3
--R - b d x + (- a d + - b c d )x + (a c d + b c d )x
--R 5 4 4
--R +
--R 3 2 3 1 3 2 2 3 2 1 4 1 5
--R (- a c d + - b c d )x + a c d x + - a c d - -- b c
--R 2 2 4 20
--R /
--R 2
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 537
--S 538 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 3 5 1 3 3 2 4 2 2 3
--R - b d x + (- a d + - b c d )x + (a c d + b c d)x
--R 5 4 4
--R +
--R 3 2 1 3 2 3
--R (- a c d + - b c )x + a c x
--R 2 2
--R Type: Polynomial(Fraction(Integer))
--E 538
--S 539 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 4 1 5
--R - - a c d + -- b c
--R 4 20
--R (4) -------------------
--R 2
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 539
--S 540 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 540
)clear all
--S 541 of 2952
t0000:=(c+d*x)^3
--R
--R
--R 3 3 2 2 2 3
--R (1) d x + 3c d x + 3c d x + c
--R Type: Polynomial(Integer)
--E 541
--S 542 of 2952
r0000:=1/4*(c+d*x)^4/d
--R
--R
--R 1 4 4 3 3 3 2 2 2 3 1 4
--R - d x + c d x + - c d x + c d x + - c
--R 4 2 4
--R (2) -----------------------------------------
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 542
--S 543 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R 1 3 4 2 3 3 2 2 3
--R (3) - d x + c d x + - c d x + c x
--R 4 2
--R Type: Polynomial(Fraction(Integer))
--E 543
--S 544 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 4
--R - c
--R 4
--R (4) - ----
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 544
--S 545 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 545
)clear all
--S 546 of 2952
t0000:=(c+d*x)^3/(a+b*x)
--R
--R
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R (1) ----------------------------
--R b x + a
--R Type: Fraction(Polynomial(Integer))
--E 546
--S 547 of 2952
r0000:=d*(b*c-a*d)^2*x/b^3+1/2*(b*c-a*d)*(c+d*x)^2/b^2+1/3*(c+d*x)^3/b+_
(b*c-a*d)^3*log(a+b*x)/b^4
--R
--R
--R (2)
--R 3 3 2 2 2 2 3 3 3 3 3
--R (- 6a d + 18a b c d - 18a b c d + 6b c )log(b x + a) + 2b d x
--R +
--R 2 3 3 2 2 2 3 2 2 3 2 2 2
--R (- 3a b d + 9b c d )x + (6a b d - 18a b c d + 18b c d)x - 3a b c d
--R +
--R 3 3
--R 5b c
--R /
--R 4
--R 6b
--R Type: Expression(Integer)
--E 547
--S 548 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 3 2 2 2 2 3 3 3 3 3
--R (- 6a d + 18a b c d - 18a b c d + 6b c )log(b x + a) + 2b d x
--R +
--R 2 3 3 2 2 2 3 2 2 3 2
--R (- 3a b d + 9b c d )x + (6a b d - 18a b c d + 18b c d)x
--R /
--R 4
--R 6b
--R Type: Union(Expression(Integer),...)
--E 548
--S 549 of 2952
m0000:=a0000 - r0000
--R
--R
--R 2 3
--R 3a c d - 5b c
--R (4) --------------
--R 2
--R 6b
--R Type: Expression(Integer)
--E 549
--S 550 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 550
)clear all
--S 551 of 2952
t0000:=(c+d*x)^3/(a+b*x)^2
--R
--R
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R (1) ----------------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 551
--S 552 of 2952
r0000:=d^2*(3*b*c-2*a*d)*x/b^3+1/2*d^3*x^2/b^2-(b*c-a*d)^3/(b^4*(a+b*x))+_
3*d*(b*c-a*d)^2*log(a+b*x)/b^4
--R
--R
--R (2)
--R 2 3 2 2 3 2 3 3 2 2 2 2
--R ((6a b d - 12a b c d + 6b c d)x + 6a d - 12a b c d + 6a b c d)
--R *
--R log(b x + a)
--R +
--R 3 3 3 2 3 3 2 2 2 3 2 2 3 3
--R b d x + (- 3a b d + 6b c d )x + (- 4a b d + 6a b c d )x + 2a d
--R +
--R 2 2 2 2 3 3
--R - 6a b c d + 6a b c d - 2b c
--R /
--R 5 4
--R 2b x + 2a b
--R Type: Expression(Integer)
--E 552
--S 553 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 3 2 2 3 2 3 3 2 2 2 2
--R ((6a b d - 12a b c d + 6b c d)x + 6a d - 12a b c d + 6a b c d)
--R *
--R log(b x + a)
--R +
--R 3 3 3 2 3 3 2 2 2 3 2 2 3 3
--R b d x + (- 3a b d + 6b c d )x + (- 4a b d + 6a b c d )x + 2a d
--R +
--R 2 2 2 2 3 3
--R - 6a b c d + 6a b c d - 2b c
--R /
--R 5 4
--R 2b x + 2a b
--R Type: Union(Expression(Integer),...)
--E 553
--S 554 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 554
--S 555 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 555
)clear all
--S 556 of 2952
t0000:=(c+d*x)^3/(a+b*x)^3
--R
--R
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R (1) ----------------------------
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 556
--S 557 of 2952
r0000:=d^3*x/b^3-1/2*(b*c-a*d)^3/(b^4*(a+b*x)^2)-_
3*d*(b*c-a*d)^2/(b^4*(a+b*x))+3*d^2*(b*c-a*d)*log(a+b*x)/b^4
--R
--R
--R (2)
--R 2 3 3 2 2 2 3 2 2 3 3
--R (- 6a b d + 6b c d )x + (- 12a b d + 12a b c d )x - 6a d
--R +
--R 2 2
--R 6a b c d
--R *
--R log(b x + a)
--R +
--R 3 3 3 2 3 2 2 3 2 2 3 2 3 3
--R 2b d x + 4a b d x + (- 4a b d + 12a b c d - 6b c d)x - 5a d
--R +
--R 2 2 2 2 3 3
--R 9a b c d - 3a b c d - b c
--R /
--R 6 2 5 2 4
--R 2b x + 4a b x + 2a b
--R Type: Expression(Integer)
--E 557
--S 558 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 3 3 2 2 2 3 2 2 3 3
--R (- 6a b d + 6b c d )x + (- 12a b d + 12a b c d )x - 6a d
--R +
--R 2 2
--R 6a b c d
--R *
--R log(b x + a)
--R +
--R 3 3 3 2 3 2 2 3 2 2 3 2 3 3
--R 2b d x + 4a b d x + (- 4a b d + 12a b c d - 6b c d)x - 5a d
--R +
--R 2 2 2 2 3 3
--R 9a b c d - 3a b c d - b c
--R /
--R 6 2 5 2 4
--R 2b x + 4a b x + 2a b
--R Type: Union(Expression(Integer),...)
--E 558
--S 559 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 559
--S 560 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 560
)clear all
--S 561 of 2952
t0000:=(c+d*x)^3/(a+b*x)^4
--R
--R
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R (1) --------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 561
--S 562 of 2952
r0000:=-1/3*(b*c-a*d)^3/(b^4*(a+b*x)^3)-3/2*d*(b*c-a*d)^2/(b^4*(a+b*x)^2)-_
3*d^2*(b*c-a*d)/(b^4*(a+b*x))+d^3*log(a+b*x)/b^4
--R
--R
--R (2)
--R 3 3 3 2 3 2 2 3 3 3
--R (6b d x + 18a b d x + 18a b d x + 6a d )log(b x + a)
--R +
--R 2 3 3 2 2 2 3 2 2 3 2 3 3
--R (18a b d - 18b c d )x + (27a b d - 18a b c d - 9b c d)x + 11a d
--R +
--R 2 2 2 2 3 3
--R - 6a b c d - 3a b c d - 2b c
--R /
--R 7 3 6 2 2 5 3 4
--R 6b x + 18a b x + 18a b x + 6a b
--R Type: Expression(Integer)
--E 562
--S 563 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 3 3 2 3 2 2 3 3 3
--R (6b d x + 18a b d x + 18a b d x + 6a d )log(b x + a)
--R +
--R 2 3 3 2 2 2 3 2 2 3 2 3 3
--R (18a b d - 18b c d )x + (27a b d - 18a b c d - 9b c d)x + 11a d
--R +
--R 2 2 2 2 3 3
--R - 6a b c d - 3a b c d - 2b c
--R /
--R 7 3 6 2 2 5 3 4
--R 6b x + 18a b x + 18a b x + 6a b
--R Type: Union(Expression(Integer),...)
--E 563
--S 564 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 564
--S 565 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 565
)clear all
--S 566 of 2952
t0000:=(c+d*x)^3/(a+b*x)^5
--R
--R
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R (1) --------------------------------------------------
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R b x + 5a b x + 10a b x + 10a b x + 5a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 566
--S 567 of 2952
r0000:=-1/4*(c+d*x)^4/((b*c-a*d)*(a+b*x)^4)
--R
--R
--R (2)
--R 1 4 4 3 3 3 2 2 2 3 1 4
--R - d x + c d x + - c d x + c d x + - c
--R 4 2 4
--R /
--R 4 5 4 2 3 4 3 3 2 2 3 2
--R (a b d - b c)x + (4a b d - 4a b c)x + (6a b d - 6a b c)x
--R +
--R 4 3 2 5 4
--R (4a b d - 4a b c)x + a d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 567
--S 568 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 3 3 2 3 3 2 2 2 3 2 2 3 2
--R - 4b d x + (- 6a b d - 6b c d )x + (- 4a b d - 4a b c d - 4b c d)x
--R +
--R 3 3 2 2 2 2 3 3
--R - a d - a b c d - a b c d - b c
--R /
--R 8 4 7 3 2 6 2 3 5 4 4
--R 4b x + 16a b x + 24a b x + 16a b x + 4a b
--R Type: Union(Expression(Integer),...)
--E 568
--S 569 of 2952
m0000:=a0000 - r0000
--R
--R
--R 4
--R d
--R (4) - -------------
--R 4 5
--R 4a b d - 4b c
--R Type: Expression(Integer)
--E 569
--S 570 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 570
)clear all
--S 571 of 2952
t0000:=(c+d*x)^3/(a+b*x)^6
--R
--R
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R (1) -------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 571
--S 572 of 2952
r0000:=-1/5*(c+d*x)^4/((b*c-a*d)*(a+b*x)^5)+_
1/20*d*(c+d*x)^4/((b*c-a*d)^2*(a+b*x)^4)
--R
--R
--R (2)
--R 1 5 5 1 5 4 4 1 2 3 3 3 2 3 3 2 2
--R -- b d x + - a d x + (a c d - - b c d )x + (- a c d - b c d )x
--R 20 4 2 2
--R +
--R 3 2 3 4 1 4 1 5
--R (a c d - - b c d)x + - a c d - - b c
--R 4 4 5
--R /
--R 2 5 2 6 7 2 5 3 4 2 2 5 6 2 4
--R (a b d - 2a b c d + b c )x + (5a b d - 10a b c d + 5a b c )x
--R +
--R 4 3 2 3 4 2 5 2 3 5 2 2 4 3 3 4 2 2
--R (10a b d - 20a b c d + 10a b c )x + (10a b d - 20a b c d + 10a b c )x
--R +
--R 6 2 5 2 4 3 2 7 2 6 5 2 2
--R (5a b d - 10a b c d + 5a b c )x + a d - 2a b c d + a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 572
--S 573 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 3 3 2 3 3 2 2
--R - 10b d x + (- 10a b d - 20b c d )x
--R +
--R 2 3 2 2 3 2 3 3 2 2 2 2 3 3
--R (- 5a b d - 10a b c d - 15b c d)x - a d - 2a b c d - 3a b c d - 4b c
--R /
--R 9 5 8 4 2 7 3 3 6 2 4 5 5 4
--R 20b x + 100a b x + 200a b x + 200a b x + 100a b x + 20a b
--R Type: Union(Expression(Integer),...)
--E 573
--S 574 of 2952
m0000:=a0000 - r0000
--R
--R
--R 5
--R d
--R (4) - -----------------------------
--R 2 4 2 5 6 2
--R 20a b d - 40a b c d + 20b c
--R Type: Expression(Integer)
--E 574
--S 575 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 575
)clear all
--S 576 of 2952
t0000:=(c+d*x)^3/(a+b*x)^7
--R
--R
--R (1)
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R ------------------------------------------------------------------------
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R b x + 7a b x + 21a b x + 35a b x + 35a b x + 21a b x + 7a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 576
--S 577 of 2952
r0000:=-1/6*(b*c-a*d)^3/(b^4*(a+b*x)^6)-3/5*d*(b*c-a*d)^2/(b^4*(a+b*x)^5)-_
3/4*d^2*(b*c-a*d)/(b^4*(a+b*x)^4)-1/3*d^3/(b^4*(a+b*x)^3)
--R
--R
--R (2)
--R 1 3 3 3 1 2 3 3 3 2 2
--R - - b d x + (- - a b d - - b c d )x
--R 3 4 4
--R +
--R 1 2 3 3 2 2 3 3 2 1 3 3 1 2 2
--R (- -- a b d - -- a b c d - - b c d)x - -- a d - -- a b c d
--R 10 10 5 60 20
--R +
--R 1 2 2 1 3 3
--R - -- a b c d - - b c
--R 10 6
--R /
--R 10 6 9 5 2 8 4 3 7 3 4 6 2 5 5 6 4
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 577
--S 578 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 3 3 2 3 3 2 2
--R - 20b d x + (- 15a b d - 45b c d )x
--R +
--R 2 3 2 2 3 2 3 3 2 2 2 2 3 3
--R (- 6a b d - 18a b c d - 36b c d)x - a d - 3a b c d - 6a b c d - 10b c
--R /
--R 10 6 9 5 2 8 4 3 7 3 4 6 2 5 5
--R 60b x + 360a b x + 900a b x + 1200a b x + 900a b x + 360a b x
--R +
--R 6 4
--R 60a b
--R Type: Union(Expression(Integer),...)
--E 578
--S 579 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 579
--S 580 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 580
)clear all
--S 581 of 2952
t0000:=(c+d*x)^3/(a+b*x)^8
--R
--R
--R (1)
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R /
--R 8 8 7 7 2 6 6 3 5 5 4 4 4 5 3 3 6 2 2
--R b x + 8a b x + 28a b x + 56a b x + 70a b x + 56a b x + 28a b x
--R +
--R 7 8
--R 8a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 581
--S 582 of 2952
r0000:=-1/7*(b*c-a*d)^3/(b^4*(a+b*x)^7)-1/2*d*(b*c-a*d)^2/(b^4*(a+b*x)^6)-_
3/5*d^2*(b*c-a*d)/(b^4*(a+b*x)^5)-1/4*d^3/(b^4*(a+b*x)^4)
--R
--R
--R (2)
--R 1 3 3 3 3 2 3 3 3 2 2
--R - - b d x + (- -- a b d - - b c d )x
--R 4 20 5
--R +
--R 1 2 3 1 2 2 1 3 2 1 3 3 1 2 2
--R (- -- a b d - - a b c d - - b c d)x - --- a d - -- a b c d
--R 20 5 2 140 35
--R +
--R 1 2 2 1 3 3
--R - -- a b c d - - b c
--R 14 7
--R /
--R 11 7 10 6 2 9 5 3 8 4 4 7 3 5 6 2 6 5
--R b x + 7a b x + 21a b x + 35a b x + 35a b x + 21a b x + 7a b x
--R +
--R 7 4
--R a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 582
--S 583 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 3 3 2 3 3 2 2
--R - 35b d x + (- 21a b d - 84b c d )x
--R +
--R 2 3 2 2 3 2 3 3 2 2 2 2 3 3
--R (- 7a b d - 28a b c d - 70b c d)x - a d - 4a b c d - 10a b c d - 20b c
--R /
--R 11 7 10 6 2 9 5 3 8 4 4 7 3 5 6 2
--R 140b x + 980a b x + 2940a b x + 4900a b x + 4900a b x + 2940a b x
--R +
--R 6 5 7 4
--R 980a b x + 140a b
--R Type: Union(Expression(Integer),...)
--E 583
--S 584 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 584
--S 585 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 585
)clear all
--S 586 of 2952
t0000:=(c+d*x)^3/(a+b*x)^9
--R
--R
--R (1)
--R 3 3 2 2 2 3
--R d x + 3c d x + 3c d x + c
--R /
--R 9 9 8 8 2 7 7 3 6 6 4 5 5 5 4 4 6 3 3
--R b x + 9a b x + 36a b x + 84a b x + 126a b x + 126a b x + 84a b x
--R +
--R 7 2 2 8 9
--R 36a b x + 9a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 586
--S 587 of 2952
r0000:=-1/8*(b*c-a*d)^3/(b^4*(a+b*x)^8)-3/7*d*(b*c-a*d)^2/(b^4*(a+b*x)^7)-_
1/2*d^2*(b*c-a*d)/(b^4*(a+b*x)^6)-1/5*d^3/(b^4*(a+b*x)^5)
--R
--R
--R (2)
--R 1 3 3 3 1 2 3 1 3 2 2
--R - - b d x + (- -- a b d - - b c d )x
--R 5 10 2
--R +
--R 1 2 3 1 2 2 3 3 2 1 3 3 1 2 2
--R (- -- a b d - - a b c d - - b c d)x - --- a d - -- a b c d
--R 35 7 7 280 56
--R +
--R 3 2 2 1 3 3
--R - -- a b c d - - b c
--R 56 8
--R /
--R 12 8 11 7 2 10 6 3 9 5 4 8 4 5 7 3 6 6 2
--R b x + 8a b x + 28a b x + 56a b x + 70a b x + 56a b x + 28a b x
--R +
--R 7 5 8 4
--R 8a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 587
--S 588 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 3 3 2 3 3 2 2
--R - 56b d x + (- 28a b d - 140b c d )x
--R +
--R 2 3 2 2 3 2 3 3 2 2 2 2
--R (- 8a b d - 40a b c d - 120b c d)x - a d - 5a b c d - 15a b c d
--R +
--R 3 3
--R - 35b c
--R /
--R 12 8 11 7 2 10 6 3 9 5 4 8 4
--R 280b x + 2240a b x + 7840a b x + 15680a b x + 19600a b x
--R +
--R 5 7 3 6 6 2 7 5 8 4
--R 15680a b x + 7840a b x + 2240a b x + 280a b
--R Type: Union(Expression(Integer),...)
--E 588
--S 589 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 589
--S 590 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 590
)clear all
--S 591 of 2952
t0000:=(a+b*x)^m*(c+d*x)^7
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R (d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c )
--R *
--R m
--R (b x + a)
--R Type: Expression(Integer)
--E 591
--S 592 of 2952
r0000:=(b*c-a*d)^7*(a+b*x)^(1+m)/(b^8*(1+m))+_
7*d*(b*c-a*d)^6*(a+b*x)^(2+m)/(b^8*(2+m))+_
21*d^2*(b*c-a*d)^5*(a+b*x)^(3+m)/(b^8*(3+m))+_
35*d^3*(b*c-a*d)^4*(a+b*x)^(4+m)/(b^8*(4+m))+_
35*d^4*(b*c-a*d)^3*(a+b*x)^(5+m)/(b^8*(5+m))+_
21*d^5*(b*c-a*d)^2*(a+b*x)^(6+m)/(b^8*(6+m))+_
7*d^6*(b*c-a*d)*(a+b*x)^(7+m)/(b^8*(7+m))+_
d^7*(a+b*x)^(8+m)/(b^8*(8+m))
--R
--R
--R (2)
--R 7 7 7 6 7 5 7 4 7 3 7 2 7
--R d m + 28d m + 322d m + 1960d m + 6769d m + 13132d m + 13068d m
--R +
--R 7
--R 5040d
--R *
--R m + 8
--R (b x + a)
--R +
--R 7 6 7 7 6 6
--R (- 7a d + 7b c d )m + (- 203a d + 203b c d )m
--R +
--R 7 6 5 7 6 4
--R (- 2401a d + 2401b c d )m + (- 14945a d + 14945b c d )m
--R +
--R 7 6 3 7 6 2
--R (- 52528a d + 52528b c d )m + (- 103292a d + 103292b c d )m
--R +
--R 7 6 7 6
--R (- 103824a d + 103824b c d )m - 40320a d + 40320b c d
--R *
--R m + 7
--R (b x + a)
--R +
--R 2 7 6 2 2 5 7
--R (21a d - 42a b c d + 21b c d )m
--R +
--R 2 7 6 2 2 5 6
--R (630a d - 1260a b c d + 630b c d )m
--R +
--R 2 7 6 2 2 5 5
--R (7686a d - 15372a b c d + 7686b c d )m
--R +
--R 2 7 6 2 2 5 4
--R (49140a d - 98280a b c d + 49140b c d )m
--R +
--R 2 7 6 2 2 5 3
--R (176589a d - 353178a b c d + 176589b c d )m
--R +
--R 2 7 6 2 2 5 2
--R (353430a d - 706860a b c d + 353430b c d )m
--R +
--R 2 7 6 2 2 5 2 7
--R (360024a d - 720048a b c d + 360024b c d )m + 141120a d
--R +
--R 6 2 2 5
--R - 282240a b c d + 141120b c d
--R *
--R m + 6
--R (b x + a)
--R +
--R 3 7 2 6 2 2 5 3 3 4 7
--R (- 35a d + 105a b c d - 105a b c d + 35b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 6
--R (- 1085a d + 3255a b c d - 3255a b c d + 1085b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 5
--R (- 13685a d + 41055a b c d - 41055a b c d + 13685b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 4
--R (- 90335a d + 271005a b c d - 271005a b c d + 90335b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 3
--R (- 334040a d + 1002120a b c d - 1002120a b c d + 334040b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 2
--R (- 684740a d + 2054220a b c d - 2054220a b c d + 684740b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4
--R (- 710640a d + 2131920a b c d - 2131920a b c d + 710640b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4
--R - 282240a d + 846720a b c d - 846720a b c d + 282240b c d
--R *
--R m + 5
--R (b x + a)
--R +
--R 4 7 3 6 2 2 2 5 3 3 4 4 4 3 7
--R (35a d - 140a b c d + 210a b c d - 140a b c d + 35b c d )m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R 1120a d - 4480a b c d + 6720a b c d - 4480a b c d
--R +
--R 4 4 3
--R 1120b c d
--R *
--R 6
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R 14630a d - 58520a b c d + 87780a b c d - 58520a b c d
--R +
--R 4 4 3
--R 14630b c d
--R *
--R 5
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R 100240a d - 400960a b c d + 601440a b c d - 400960a b c d
--R +
--R 4 4 3
--R 100240b c d
--R *
--R 4
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R 384755a d - 1539020a b c d + 2308530a b c d - 1539020a b c d
--R +
--R 4 4 3
--R 384755b c d
--R *
--R 3
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R 815920a d - 3263680a b c d + 4895520a b c d - 3263680a b c d
--R +
--R 4 4 3
--R 815920b c d
--R *
--R 2
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R 870660a d - 3482640a b c d + 5223960a b c d - 3482640a b c d
--R +
--R 4 4 3
--R 870660b c d
--R *
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R 352800a d - 1411200a b c d + 2116800a b c d - 1411200a b c d
--R +
--R 4 4 3
--R 352800b c d
--R *
--R m + 4
--R (b x + a)
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4 4 4 3
--R - 21a d + 105a b c d - 210a b c d + 210a b c d - 105a b c d
--R +
--R 5 5 2
--R 21b c d
--R *
--R 7
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R - 693a d + 3465a b c d - 6930a b c d + 6930a b c d
--R +
--R 4 4 3 5 5 2
--R - 3465a b c d + 693b c d
--R *
--R 6
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R - 9387a d + 46935a b c d - 93870a b c d + 93870a b c d
--R +
--R 4 4 3 5 5 2
--R - 46935a b c d + 9387b c d
--R *
--R 5
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R - 67095a d + 335475a b c d - 670950a b c d + 670950a b c d
--R +
--R 4 4 3 5 5 2
--R - 335475a b c d + 67095b c d
--R *
--R 4
--R m
--R +
--R 5 7 4 6 3 2 2 5
--R - 270144a d + 1350720a b c d - 2701440a b c d
--R +
--R 2 3 3 4 4 4 3 5 5 2
--R 2701440a b c d - 1350720a b c d + 270144b c d
--R *
--R 3
--R m
--R +
--R 5 7 4 6 3 2 2 5
--R - 602532a d + 3012660a b c d - 6025320a b c d
--R +
--R 2 3 3 4 4 4 3 5 5 2
--R 6025320a b c d - 3012660a b c d + 602532b c d
--R *
--R 2
--R m
--R +
--R 5 7 4 6 3 2 2 5
--R - 673008a d + 3365040a b c d - 6730080a b c d
--R +
--R 2 3 3 4 4 4 3 5 5 2
--R 6730080a b c d - 3365040a b c d + 673008b c d
--R *
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R - 282240a d + 1411200a b c d - 2822400a b c d + 2822400a b c d
--R +
--R 4 4 3 5 5 2
--R - 1411200a b c d + 282240b c d
--R *
--R m + 3
--R (b x + a)
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4 2 4 4 3
--R 7a d - 42a b c d + 105a b c d - 140a b c d + 105a b c d
--R +
--R 5 5 2 6 6
--R - 42a b c d + 7b c d
--R *
--R 7
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R 238a d - 1428a b c d + 3570a b c d - 4760a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R 3570a b c d - 1428a b c d + 238b c d
--R *
--R 6
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R 3346a d - 20076a b c d + 50190a b c d - 66920a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R 50190a b c d - 20076a b c d + 3346b c d
--R *
--R 5
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R 25060a d - 150360a b c d + 375900a b c d - 501200a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R 375900a b c d - 150360a b c d + 25060b c d
--R *
--R 4
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R 107023a d - 642138a b c d + 1605345a b c d - 2140460a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R 1605345a b c d - 642138a b c d + 107023b c d
--R *
--R 3
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R 256942a d - 1541652a b c d + 3854130a b c d - 5138840a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R 3854130a b c d - 1541652a b c d + 256942b c d
--R *
--R 2
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R 312984a d - 1877904a b c d + 4694760a b c d - 6259680a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R 4694760a b c d - 1877904a b c d + 312984b c d
--R *
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R 141120a d - 846720a b c d + 2116800a b c d - 2822400a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R 2116800a b c d - 846720a b c d + 141120b c d
--R *
--R m + 2
--R (b x + a)
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - a d + 7a b c d - 21a b c d + 35a b c d - 35a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R 21a b c d - 7a b c d + b c
--R *
--R 7
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 35a d + 245a b c d - 735a b c d + 1225a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 1225a b c d + 735a b c d - 245a b c d + 35b c
--R *
--R 6
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 511a d + 3577a b c d - 10731a b c d + 17885a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 17885a b c d + 10731a b c d - 3577a b c d + 511b c
--R *
--R 5
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 4025a d + 28175a b c d - 84525a b c d + 140875a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 140875a b c d + 84525a b c d - 28175a b c d + 4025b c
--R *
--R 4
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 18424a d + 128968a b c d - 386904a b c d + 644840a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 644840a b c d + 386904a b c d - 128968a b c d + 18424b c
--R *
--R 3
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 48860a d + 342020a b c d - 1026060a b c d + 1710100a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 1710100a b c d + 1026060a b c d - 342020a b c d + 48860b c
--R *
--R 2
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 69264a d + 484848a b c d - 1454544a b c d + 2424240a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 2424240a b c d + 1454544a b c d - 484848a b c d + 69264b c
--R *
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 40320a d + 282240a b c d - 846720a b c d + 1411200a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 1411200a b c d + 846720a b c d - 282240a b c d + 40320b c
--R *
--R m + 1
--R (b x + a)
--R /
--R 8 8 8 7 8 6 8 5 8 4 8 3 8 2
--R b m + 36b m + 546b m + 4536b m + 22449b m + 67284b m + 118124b m
--R +
--R 8 8
--R 109584b m + 40320b
--R Type: Expression(Integer)
--E 592
--S 593 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 8 7 7 8 7 6 8 7 5 8 7 4 8 7 3
--R b d m + 28b d m + 322b d m + 1960b d m + 6769b d m
--R +
--R 8 7 2 8 7 8 7
--R 13132b d m + 13068b d m + 5040b d
--R *
--R 8
--R x
--R +
--R 7 7 8 6 7 7 7 8 6 6
--R (a b d + 7b c d )m + (21a b d + 203b c d )m
--R +
--R 7 7 8 6 5 7 7 8 6 4
--R (175a b d + 2401b c d )m + (735a b d + 14945b c d )m
--R +
--R 7 7 8 6 3 7 7 8 6 2
--R (1624a b d + 52528b c d )m + (1764a b d + 103292b c d )m
--R +
--R 7 7 8 6 8 6
--R (720a b d + 103824b c d )m + 40320b c d
--R *
--R 7
--R x
--R +
--R 7 6 8 2 5 7 2 6 7 7 6 8 2 5 6
--R (7a b c d + 21b c d )m + (- 7a b d + 161a b c d + 630b c d )m
--R +
--R 2 6 7 7 6 8 2 5 5
--R (- 105a b d + 1435a b c d + 7686b c d )m
--R +
--R 2 6 7 7 6 8 2 5 4
--R (- 595a b d + 6335a b c d + 49140b c d )m
--R +
--R 2 6 7 7 6 8 2 5 3
--R (- 1575a b d + 14518a b c d + 176589b c d )m
--R +
--R 2 6 7 7 6 8 2 5 2
--R (- 1918a b d + 16184a b c d + 353430b c d )m
--R +
--R 2 6 7 7 6 8 2 5 8 2 5
--R (- 840a b d + 6720a b c d + 360024b c d )m + 141120b c d
--R *
--R 6
--R x
--R +
--R 7 2 5 8 3 4 7
--R (21a b c d + 35b c d )m
--R +
--R 2 6 6 7 2 5 8 3 4 6
--R (- 42a b c d + 525a b c d + 1085b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 5
--R (42a b d - 756a b c d + 5061a b c d + 13685b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 4
--R (420a b d - 4830a b c d + 23835a b c d + 90335b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 3
--R (1470a b d - 13860a b c d + 57414a b c d + 334040b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 2
--R (2100a b d - 17808a b c d + 66360a b c d + 684740b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4
--R (1008a b d - 8064a b c d + 28224a b c d + 710640b c d )m
--R +
--R 8 3 4
--R 282240b c d
--R *
--R 5
--R x
--R +
--R 7 3 4 8 4 3 7
--R (35a b c d + 35b c d )m
--R +
--R 2 6 2 5 7 3 4 8 4 3 6
--R (- 105a b c d + 945a b c d + 1120b c d )m
--R +
--R 3 5 6 2 6 2 5 7 3 4 8 4 3 5
--R (210a b c d - 2205a b c d + 9905a b c d + 14630b c d )m
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4
--R - 210a b d + 2940a b c d - 16485a b c d + 50715a b c d
--R +
--R 8 4 3
--R 100240b c d
--R *
--R 4
--R m
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4
--R - 1260a b d + 12390a b c d - 53235a b c d + 131180a b c d
--R +
--R 8 4 3
--R 384755b c d
--R *
--R 3
--R m
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4
--R - 2310a b d + 19740a b c d - 74130a b c d + 160020a b c d
--R +
--R 8 4 3
--R 815920b c d
--R *
--R 2
--R m
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4
--R - 1260a b d + 10080a b c d - 35280a b c d + 70560a b c d
--R +
--R 8 4 3
--R 870660b c d
--R *
--R m
--R +
--R 8 4 3
--R 352800b c d
--R *
--R 4
--R x
--R +
--R 7 4 3 8 5 2 7
--R (35a b c d + 21b c d )m
--R +
--R 2 6 3 4 7 4 3 8 5 2 6
--R (- 140a b c d + 1015a b c d + 693b c d )m
--R +
--R 3 5 2 5 2 6 3 4 7 4 3 8 5 2 5
--R (420a b c d - 3360a b c d + 11585a b c d + 9387b c d )m
--R +
--R 4 4 6 3 5 2 5 2 6 3 4 7 4 3
--R - 840a b c d + 7560a b c d - 29540a b c d + 65485a b c d
--R +
--R 8 5 2
--R 67095b c d
--R *
--R 4
--R m
--R +
--R 5 3 7 4 4 6 3 5 2 5 2 6 3 4
--R 840a b d - 9240a b c d + 43260a b c d - 114240a b c d
--R +
--R 7 4 3 8 5 2
--R 188300a b c d + 270144b c d
--R *
--R 3
--R m
--R +
--R 5 3 7 4 4 6 3 5 2 5 2 6 3 4
--R 2520a b d - 21840a b c d + 83160a b c d - 182000a b c d
--R +
--R 7 4 3 8 5 2
--R 251020a b c d + 602532b c d
--R *
--R 2
--R m
--R +
--R 5 3 7 4 4 6 3 5 2 5 2 6 3 4
--R 1680a b d - 13440a b c d + 47040a b c d - 94080a b c d
--R +
--R 7 4 3 8 5 2
--R 117600a b c d + 673008b c d
--R *
--R m
--R +
--R 8 5 2
--R 282240b c d
--R *
--R 3
--R x
--R +
--R 7 5 2 8 6 7
--R (21a b c d + 7b c d)m
--R +
--R 2 6 4 3 7 5 2 8 6 6
--R (- 105a b c d + 651a b c d + 238b c d)m
--R +
--R 3 5 3 4 2 6 4 3 7 5 2 8 6 5
--R (420a b c d - 2835a b c d + 8085a b c d + 3346b c d)m
--R +
--R 4 4 2 5 3 5 3 4 2 6 4 3 7 5 2
--R - 1260a b c d + 9240a b c d - 29085a b c d + 50925a b c d
--R +
--R 8 6
--R 25060b c d
--R *
--R 4
--R m
--R +
--R 5 3 6 4 4 2 5 3 5 3 4 2 6 4 3
--R 2520a b c d - 20160a b c d + 70140a b c d - 138285a b c d
--R +
--R 7 5 2 8 6
--R 168294a b c d + 107023b c d
--R *
--R 3
--R m
--R +
--R 6 2 7 5 3 6 4 4 2 5 3 5 3 4
--R - 2520a b d + 22680a b c d - 89460a b c d + 202440a b c d
--R +
--R 2 6 4 3 7 5 2 8 6
--R - 288330a b c d + 265944a b c d + 256942b c d
--R *
--R 2
--R m
--R +
--R 6 2 7 5 3 6 4 4 2 5 3 5 3 4
--R - 2520a b d + 20160a b c d - 70560a b c d + 141120a b c d
--R +
--R 2 6 4 3 7 5 2 8 6
--R - 176400a b c d + 141120a b c d + 312984b c d
--R *
--R m
--R +
--R 8 6
--R 141120b c d
--R *
--R 2
--R x
--R +
--R 7 6 8 7 7 2 6 5 2 7 6 8 7 6
--R (7a b c d + b c )m + (- 42a b c d + 231a b c d + 35b c )m
--R +
--R 3 5 4 3 2 6 5 2 7 6 8 7 5
--R (210a b c d - 1260a b c d + 3115a b c d + 511b c )m
--R +
--R 4 4 3 4 3 5 4 3 2 6 5 2 7 6
--R - 840a b c d + 5460a b c d - 14910a b c d + 21945a b c d
--R +
--R 8 7
--R 4025b c
--R *
--R 4
--R m
--R +
--R 5 3 2 5 4 4 3 4 3 5 4 3 2 6 5 2
--R 2520a b c d - 17640a b c d + 52710a b c d - 86940a b c d
--R +
--R 7 6 8 7
--R 85078a b c d + 18424b c
--R *
--R 3
--R m
--R +
--R 6 2 6 5 3 2 5 4 4 3 4
--R - 5040a b c d + 37800a b c d - 122640a b c d
--R +
--R 3 5 4 3 2 6 5 2 7 6 8 7
--R 223860a b c d - 249648a b c d + 171864a b c d + 48860b c
--R *
--R 2
--R m
--R +
--R 7 7 6 2 6 5 3 2 5 4 4 3 4
--R 5040a b d - 40320a b c d + 141120a b c d - 282240a b c d
--R +
--R 3 5 4 3 2 6 5 2 7 6 8 7
--R 352800a b c d - 282240a b c d + 141120a b c d + 69264b c
--R *
--R m
--R +
--R 8 7
--R 40320b c
--R *
--R x
--R +
--R 7 7 7 2 6 6 7 7 6
--R a b c m + (- 7a b c d + 35a b c )m
--R +
--R 3 5 5 2 2 6 6 7 7 5
--R (42a b c d - 231a b c d + 511a b c )m
--R +
--R 4 4 4 3 3 5 5 2 2 6 6 7 7 4
--R (- 210a b c d + 1260a b c d - 3115a b c d + 4025a b c )m
--R +
--R 5 3 3 4 4 4 4 3 3 5 5 2 2 6 6
--R 840a b c d - 5460a b c d + 14910a b c d - 21945a b c d
--R +
--R 7 7
--R 18424a b c
--R *
--R 3
--R m
--R +
--R 6 2 2 5 5 3 3 4 4 4 4 3 3 5 5 2
--R - 2520a b c d + 17640a b c d - 52710a b c d + 86940a b c d
--R +
--R 2 6 6 7 7
--R - 85078a b c d + 48860a b c
--R *
--R 2
--R m
--R +
--R 7 6 6 2 2 5 5 3 3 4 4 4 4 3
--R 5040a b c d - 37800a b c d + 122640a b c d - 223860a b c d
--R +
--R 3 5 5 2 2 6 6 7 7
--R 249648a b c d - 171864a b c d + 69264a b c
--R *
--R m
--R +
--R 8 7 7 6 6 2 2 5 5 3 3 4
--R - 5040a d + 40320a b c d - 141120a b c d + 282240a b c d
--R +
--R 4 4 4 3 3 5 5 2 2 6 6 7 7
--R - 352800a b c d + 282240a b c d - 141120a b c d + 40320a b c
--R *
--R m log(b x + a)
--R %e
--R /
--R 8 8 8 7 8 6 8 5 8 4 8 3 8 2
--R b m + 36b m + 546b m + 4536b m + 22449b m + 67284b m + 118124b m
--R +
--R 8 8
--R 109584b m + 40320b
--R Type: Union(Expression(Integer),...)
--E 593
--S 594 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 8 7 7 8 7 6 8 7 5 8 7 4 8 7 3
--R b d m + 28b d m + 322b d m + 1960b d m + 6769b d m
--R +
--R 8 7 2 8 7 8 7
--R 13132b d m + 13068b d m + 5040b d
--R *
--R 8
--R x
--R +
--R 7 7 8 6 7 7 7 8 6 6
--R (a b d + 7b c d )m + (21a b d + 203b c d )m
--R +
--R 7 7 8 6 5 7 7 8 6 4
--R (175a b d + 2401b c d )m + (735a b d + 14945b c d )m
--R +
--R 7 7 8 6 3 7 7 8 6 2
--R (1624a b d + 52528b c d )m + (1764a b d + 103292b c d )m
--R +
--R 7 7 8 6 8 6
--R (720a b d + 103824b c d )m + 40320b c d
--R *
--R 7
--R x
--R +
--R 7 6 8 2 5 7
--R (7a b c d + 21b c d )m
--R +
--R 2 6 7 7 6 8 2 5 6
--R (- 7a b d + 161a b c d + 630b c d )m
--R +
--R 2 6 7 7 6 8 2 5 5
--R (- 105a b d + 1435a b c d + 7686b c d )m
--R +
--R 2 6 7 7 6 8 2 5 4
--R (- 595a b d + 6335a b c d + 49140b c d )m
--R +
--R 2 6 7 7 6 8 2 5 3
--R (- 1575a b d + 14518a b c d + 176589b c d )m
--R +
--R 2 6 7 7 6 8 2 5 2
--R (- 1918a b d + 16184a b c d + 353430b c d )m
--R +
--R 2 6 7 7 6 8 2 5 8 2 5
--R (- 840a b d + 6720a b c d + 360024b c d )m + 141120b c d
--R *
--R 6
--R x
--R +
--R 7 2 5 8 3 4 7
--R (21a b c d + 35b c d )m
--R +
--R 2 6 6 7 2 5 8 3 4 6
--R (- 42a b c d + 525a b c d + 1085b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 5
--R (42a b d - 756a b c d + 5061a b c d + 13685b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 4
--R (420a b d - 4830a b c d + 23835a b c d + 90335b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 3
--R (1470a b d - 13860a b c d + 57414a b c d + 334040b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 2
--R (2100a b d - 17808a b c d + 66360a b c d + 684740b c d )m
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4
--R (1008a b d - 8064a b c d + 28224a b c d + 710640b c d )m
--R +
--R 8 3 4
--R 282240b c d
--R *
--R 5
--R x
--R +
--R 7 3 4 8 4 3 7
--R (35a b c d + 35b c d )m
--R +
--R 2 6 2 5 7 3 4 8 4 3 6
--R (- 105a b c d + 945a b c d + 1120b c d )m
--R +
--R 3 5 6 2 6 2 5 7 3 4 8 4 3 5
--R (210a b c d - 2205a b c d + 9905a b c d + 14630b c d )m
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4
--R - 210a b d + 2940a b c d - 16485a b c d + 50715a b c d
--R +
--R 8 4 3
--R 100240b c d
--R *
--R 4
--R m
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4
--R - 1260a b d + 12390a b c d - 53235a b c d + 131180a b c d
--R +
--R 8 4 3
--R 384755b c d
--R *
--R 3
--R m
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4
--R - 2310a b d + 19740a b c d - 74130a b c d + 160020a b c d
--R +
--R 8 4 3
--R 815920b c d
--R *
--R 2
--R m
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4
--R - 1260a b d + 10080a b c d - 35280a b c d + 70560a b c d
--R +
--R 8 4 3
--R 870660b c d
--R *
--R m
--R +
--R 8 4 3
--R 352800b c d
--R *
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--R x
--R +
--R 7 4 3 8 5 2 7
--R (35a b c d + 21b c d )m
--R +
--R 2 6 3 4 7 4 3 8 5 2 6
--R (- 140a b c d + 1015a b c d + 693b c d )m
--R +
--R 3 5 2 5 2 6 3 4 7 4 3 8 5 2 5
--R (420a b c d - 3360a b c d + 11585a b c d + 9387b c d )m
--R +
--R 4 4 6 3 5 2 5 2 6 3 4 7 4 3
--R - 840a b c d + 7560a b c d - 29540a b c d + 65485a b c d
--R +
--R 8 5 2
--R 67095b c d
--R *
--R 4
--R m
--R +
--R 5 3 7 4 4 6 3 5 2 5 2 6 3 4
--R 840a b d - 9240a b c d + 43260a b c d - 114240a b c d
--R +
--R 7 4 3 8 5 2
--R 188300a b c d + 270144b c d
--R *
--R 3
--R m
--R +
--R 5 3 7 4 4 6 3 5 2 5 2 6 3 4
--R 2520a b d - 21840a b c d + 83160a b c d - 182000a b c d
--R +
--R 7 4 3 8 5 2
--R 251020a b c d + 602532b c d
--R *
--R 2
--R m
--R +
--R 5 3 7 4 4 6 3 5 2 5 2 6 3 4
--R 1680a b d - 13440a b c d + 47040a b c d - 94080a b c d
--R +
--R 7 4 3 8 5 2
--R 117600a b c d + 673008b c d
--R *
--R m
--R +
--R 8 5 2
--R 282240b c d
--R *
--R 3
--R x
--R +
--R 7 5 2 8 6 7
--R (21a b c d + 7b c d)m
--R +
--R 2 6 4 3 7 5 2 8 6 6
--R (- 105a b c d + 651a b c d + 238b c d)m
--R +
--R 3 5 3 4 2 6 4 3 7 5 2 8 6 5
--R (420a b c d - 2835a b c d + 8085a b c d + 3346b c d)m
--R +
--R 4 4 2 5 3 5 3 4 2 6 4 3 7 5 2
--R - 1260a b c d + 9240a b c d - 29085a b c d + 50925a b c d
--R +
--R 8 6
--R 25060b c d
--R *
--R 4
--R m
--R +
--R 5 3 6 4 4 2 5 3 5 3 4 2 6 4 3
--R 2520a b c d - 20160a b c d + 70140a b c d - 138285a b c d
--R +
--R 7 5 2 8 6
--R 168294a b c d + 107023b c d
--R *
--R 3
--R m
--R +
--R 6 2 7 5 3 6 4 4 2 5 3 5 3 4
--R - 2520a b d + 22680a b c d - 89460a b c d + 202440a b c d
--R +
--R 2 6 4 3 7 5 2 8 6
--R - 288330a b c d + 265944a b c d + 256942b c d
--R *
--R 2
--R m
--R +
--R 6 2 7 5 3 6 4 4 2 5 3 5 3 4
--R - 2520a b d + 20160a b c d - 70560a b c d + 141120a b c d
--R +
--R 2 6 4 3 7 5 2 8 6
--R - 176400a b c d + 141120a b c d + 312984b c d
--R *
--R m
--R +
--R 8 6
--R 141120b c d
--R *
--R 2
--R x
--R +
--R 7 6 8 7 7 2 6 5 2 7 6 8 7 6
--R (7a b c d + b c )m + (- 42a b c d + 231a b c d + 35b c )m
--R +
--R 3 5 4 3 2 6 5 2 7 6 8 7 5
--R (210a b c d - 1260a b c d + 3115a b c d + 511b c )m
--R +
--R 4 4 3 4 3 5 4 3 2 6 5 2 7 6
--R - 840a b c d + 5460a b c d - 14910a b c d + 21945a b c d
--R +
--R 8 7
--R 4025b c
--R *
--R 4
--R m
--R +
--R 5 3 2 5 4 4 3 4 3 5 4 3 2 6 5 2
--R 2520a b c d - 17640a b c d + 52710a b c d - 86940a b c d
--R +
--R 7 6 8 7
--R 85078a b c d + 18424b c
--R *
--R 3
--R m
--R +
--R 6 2 6 5 3 2 5 4 4 3 4
--R - 5040a b c d + 37800a b c d - 122640a b c d
--R +
--R 3 5 4 3 2 6 5 2 7 6 8 7
--R 223860a b c d - 249648a b c d + 171864a b c d + 48860b c
--R *
--R 2
--R m
--R +
--R 7 7 6 2 6 5 3 2 5 4 4 3 4
--R 5040a b d - 40320a b c d + 141120a b c d - 282240a b c d
--R +
--R 3 5 4 3 2 6 5 2 7 6 8 7
--R 352800a b c d - 282240a b c d + 141120a b c d + 69264b c
--R *
--R m
--R +
--R 8 7
--R 40320b c
--R *
--R x
--R +
--R 7 7 7 2 6 6 7 7 6
--R a b c m + (- 7a b c d + 35a b c )m
--R +
--R 3 5 5 2 2 6 6 7 7 5
--R (42a b c d - 231a b c d + 511a b c )m
--R +
--R 4 4 4 3 3 5 5 2 2 6 6 7 7 4
--R (- 210a b c d + 1260a b c d - 3115a b c d + 4025a b c )m
--R +
--R 5 3 3 4 4 4 4 3 3 5 5 2 2 6 6
--R 840a b c d - 5460a b c d + 14910a b c d - 21945a b c d
--R +
--R 7 7
--R 18424a b c
--R *
--R 3
--R m
--R +
--R 6 2 2 5 5 3 3 4 4 4 4 3 3 5 5 2
--R - 2520a b c d + 17640a b c d - 52710a b c d + 86940a b c d
--R +
--R 2 6 6 7 7
--R - 85078a b c d + 48860a b c
--R *
--R 2
--R m
--R +
--R 7 6 6 2 2 5 5 3 3 4 4 4 4 3
--R 5040a b c d - 37800a b c d + 122640a b c d - 223860a b c d
--R +
--R 3 5 5 2 2 6 6 7 7
--R 249648a b c d - 171864a b c d + 69264a b c
--R *
--R m
--R +
--R 8 7 7 6 6 2 2 5 5 3 3 4
--R - 5040a d + 40320a b c d - 141120a b c d + 282240a b c d
--R +
--R 4 4 4 3 3 5 5 2 2 6 6 7 7
--R - 352800a b c d + 282240a b c d - 141120a b c d + 40320a b c
--R *
--R m log(b x + a)
--R %e
--R +
--R 7 7 7 6 7 5 7 4 7 3 7 2
--R - d m - 28d m - 322d m - 1960d m - 6769d m - 13132d m
--R +
--R 7 7
--R - 13068d m - 5040d
--R *
--R m + 8
--R (b x + a)
--R +
--R 7 6 7 7 6 6
--R (7a d - 7b c d )m + (203a d - 203b c d )m
--R +
--R 7 6 5 7 6 4
--R (2401a d - 2401b c d )m + (14945a d - 14945b c d )m
--R +
--R 7 6 3 7 6 2
--R (52528a d - 52528b c d )m + (103292a d - 103292b c d )m
--R +
--R 7 6 7 6
--R (103824a d - 103824b c d )m + 40320a d - 40320b c d
--R *
--R m + 7
--R (b x + a)
--R +
--R 2 7 6 2 2 5 7
--R (- 21a d + 42a b c d - 21b c d )m
--R +
--R 2 7 6 2 2 5 6
--R (- 630a d + 1260a b c d - 630b c d )m
--R +
--R 2 7 6 2 2 5 5
--R (- 7686a d + 15372a b c d - 7686b c d )m
--R +
--R 2 7 6 2 2 5 4
--R (- 49140a d + 98280a b c d - 49140b c d )m
--R +
--R 2 7 6 2 2 5 3
--R (- 176589a d + 353178a b c d - 176589b c d )m
--R +
--R 2 7 6 2 2 5 2
--R (- 353430a d + 706860a b c d - 353430b c d )m
--R +
--R 2 7 6 2 2 5 2 7
--R (- 360024a d + 720048a b c d - 360024b c d )m - 141120a d
--R +
--R 6 2 2 5
--R 282240a b c d - 141120b c d
--R *
--R m + 6
--R (b x + a)
--R +
--R 3 7 2 6 2 2 5 3 3 4 7
--R (35a d - 105a b c d + 105a b c d - 35b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 6
--R (1085a d - 3255a b c d + 3255a b c d - 1085b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 5
--R (13685a d - 41055a b c d + 41055a b c d - 13685b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 4
--R (90335a d - 271005a b c d + 271005a b c d - 90335b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 3
--R (334040a d - 1002120a b c d + 1002120a b c d - 334040b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4 2
--R (684740a d - 2054220a b c d + 2054220a b c d - 684740b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4
--R (710640a d - 2131920a b c d + 2131920a b c d - 710640b c d )m
--R +
--R 3 7 2 6 2 2 5 3 3 4
--R 282240a d - 846720a b c d + 846720a b c d - 282240b c d
--R *
--R m + 5
--R (b x + a)
--R +
--R 4 7 3 6 2 2 2 5 3 3 4 4 4 3 7
--R (- 35a d + 140a b c d - 210a b c d + 140a b c d - 35b c d )m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R - 1120a d + 4480a b c d - 6720a b c d + 4480a b c d
--R +
--R 4 4 3
--R - 1120b c d
--R *
--R 6
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R - 14630a d + 58520a b c d - 87780a b c d + 58520a b c d
--R +
--R 4 4 3
--R - 14630b c d
--R *
--R 5
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R - 100240a d + 400960a b c d - 601440a b c d + 400960a b c d
--R +
--R 4 4 3
--R - 100240b c d
--R *
--R 4
--R m
--R +
--R 4 7 3 6 2 2 2 5
--R - 384755a d + 1539020a b c d - 2308530a b c d
--R +
--R 3 3 4 4 4 3
--R 1539020a b c d - 384755b c d
--R *
--R 3
--R m
--R +
--R 4 7 3 6 2 2 2 5
--R - 815920a d + 3263680a b c d - 4895520a b c d
--R +
--R 3 3 4 4 4 3
--R 3263680a b c d - 815920b c d
--R *
--R 2
--R m
--R +
--R 4 7 3 6 2 2 2 5
--R - 870660a d + 3482640a b c d - 5223960a b c d
--R +
--R 3 3 4 4 4 3
--R 3482640a b c d - 870660b c d
--R *
--R m
--R +
--R 4 7 3 6 2 2 2 5 3 3 4
--R - 352800a d + 1411200a b c d - 2116800a b c d + 1411200a b c d
--R +
--R 4 4 3
--R - 352800b c d
--R *
--R m + 4
--R (b x + a)
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4 4 4 3
--R 21a d - 105a b c d + 210a b c d - 210a b c d + 105a b c d
--R +
--R 5 5 2
--R - 21b c d
--R *
--R 7
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R 693a d - 3465a b c d + 6930a b c d - 6930a b c d
--R +
--R 4 4 3 5 5 2
--R 3465a b c d - 693b c d
--R *
--R 6
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R 9387a d - 46935a b c d + 93870a b c d - 93870a b c d
--R +
--R 4 4 3 5 5 2
--R 46935a b c d - 9387b c d
--R *
--R 5
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R 67095a d - 335475a b c d + 670950a b c d - 670950a b c d
--R +
--R 4 4 3 5 5 2
--R 335475a b c d - 67095b c d
--R *
--R 4
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R 270144a d - 1350720a b c d + 2701440a b c d - 2701440a b c d
--R +
--R 4 4 3 5 5 2
--R 1350720a b c d - 270144b c d
--R *
--R 3
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R 602532a d - 3012660a b c d + 6025320a b c d - 6025320a b c d
--R +
--R 4 4 3 5 5 2
--R 3012660a b c d - 602532b c d
--R *
--R 2
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R 673008a d - 3365040a b c d + 6730080a b c d - 6730080a b c d
--R +
--R 4 4 3 5 5 2
--R 3365040a b c d - 673008b c d
--R *
--R m
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R 282240a d - 1411200a b c d + 2822400a b c d - 2822400a b c d
--R +
--R 4 4 3 5 5 2
--R 1411200a b c d - 282240b c d
--R *
--R m + 3
--R (b x + a)
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4 2 4 4 3
--R - 7a d + 42a b c d - 105a b c d + 140a b c d - 105a b c d
--R +
--R 5 5 2 6 6
--R 42a b c d - 7b c d
--R *
--R 7
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R - 238a d + 1428a b c d - 3570a b c d + 4760a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R - 3570a b c d + 1428a b c d - 238b c d
--R *
--R 6
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R - 3346a d + 20076a b c d - 50190a b c d + 66920a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R - 50190a b c d + 20076a b c d - 3346b c d
--R *
--R 5
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R - 25060a d + 150360a b c d - 375900a b c d + 501200a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R - 375900a b c d + 150360a b c d - 25060b c d
--R *
--R 4
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R - 107023a d + 642138a b c d - 1605345a b c d + 2140460a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R - 1605345a b c d + 642138a b c d - 107023b c d
--R *
--R 3
--R m
--R +
--R 6 7 5 6 4 2 2 5
--R - 256942a d + 1541652a b c d - 3854130a b c d
--R +
--R 3 3 3 4 2 4 4 3 5 5 2 6 6
--R 5138840a b c d - 3854130a b c d + 1541652a b c d - 256942b c d
--R *
--R 2
--R m
--R +
--R 6 7 5 6 4 2 2 5
--R - 312984a d + 1877904a b c d - 4694760a b c d
--R +
--R 3 3 3 4 2 4 4 3 5 5 2 6 6
--R 6259680a b c d - 4694760a b c d + 1877904a b c d - 312984b c d
--R *
--R m
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4
--R - 141120a d + 846720a b c d - 2116800a b c d + 2822400a b c d
--R +
--R 2 4 4 3 5 5 2 6 6
--R - 2116800a b c d + 846720a b c d - 141120b c d
--R *
--R m + 2
--R (b x + a)
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R a d - 7a b c d + 21a b c d - 35a b c d + 35a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 21a b c d + 7a b c d - b c
--R *
--R 7
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 35a d - 245a b c d + 735a b c d - 1225a b c d + 1225a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 735a b c d + 245a b c d - 35b c
--R *
--R 6
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R 511a d - 3577a b c d + 10731a b c d - 17885a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R 17885a b c d - 10731a b c d + 3577a b c d - 511b c
--R *
--R 5
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R 4025a d - 28175a b c d + 84525a b c d - 140875a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R 140875a b c d - 84525a b c d + 28175a b c d - 4025b c
--R *
--R 4
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R 18424a d - 128968a b c d + 386904a b c d - 644840a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R 644840a b c d - 386904a b c d + 128968a b c d - 18424b c
--R *
--R 3
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R 48860a d - 342020a b c d + 1026060a b c d - 1710100a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R 1710100a b c d - 1026060a b c d + 342020a b c d - 48860b c
--R *
--R 2
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R 69264a d - 484848a b c d + 1454544a b c d - 2424240a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R 2424240a b c d - 1454544a b c d + 484848a b c d - 69264b c
--R *
--R m
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R 40320a d - 282240a b c d + 846720a b c d - 1411200a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R 1411200a b c d - 846720a b c d + 282240a b c d - 40320b c
--R *
--R m + 1
--R (b x + a)
--R /
--R 8 8 8 7 8 6 8 5 8 4 8 3 8 2
--R b m + 36b m + 546b m + 4536b m + 22449b m + 67284b m + 118124b m
--R +
--R 8 8
--R 109584b m + 40320b
--R Type: Expression(Integer)
--E 594
--S 595 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 595
)clear all
--S 596 of 2952
t0000:=(a+b*x)^9*(c+d*x)^7
--R
--R
--R (1)
--R 9 7 16 8 7 9 6 15 2 7 7 8 6 9 2 5 14
--R b d x + (9a b d + 7b c d )x + (36a b d + 63a b c d + 21b c d )x
--R +
--R 3 6 7 2 7 6 8 2 5 9 3 4 13
--R (84a b d + 252a b c d + 189a b c d + 35b c d )x
--R +
--R 4 5 7 3 6 6 2 7 2 5 8 3 4 9 4 3 12
--R (126a b d + 588a b c d + 756a b c d + 315a b c d + 35b c d )x
--R +
--R 5 4 7 4 5 6 3 6 2 5 2 7 3 4 8 4 3
--R 126a b d + 882a b c d + 1764a b c d + 1260a b c d + 315a b c d
--R +
--R 9 5 2
--R 21b c d
--R *
--R 11
--R x
--R +
--R 6 3 7 5 4 6 4 5 2 5 3 6 3 4 2 7 4 3
--R 84a b d + 882a b c d + 2646a b c d + 2940a b c d + 1260a b c d
--R +
--R 8 5 2 9 6
--R 189a b c d + 7b c d
--R *
--R 10
--R x
--R +
--R 7 2 7 6 3 6 5 4 2 5 4 5 3 4 3 6 4 3
--R 36a b d + 588a b c d + 2646a b c d + 4410a b c d + 2940a b c d
--R +
--R 2 7 5 2 8 6 9 7
--R 756a b c d + 63a b c d + b c
--R *
--R 9
--R x
--R +
--R 8 7 7 2 6 6 3 2 5 5 4 3 4 4 5 4 3
--R 9a b d + 252a b c d + 1764a b c d + 4410a b c d + 4410a b c d
--R +
--R 3 6 5 2 2 7 6 8 7
--R 1764a b c d + 252a b c d + 9a b c
--R *
--R 8
--R x
--R +
--R 9 7 8 6 7 2 2 5 6 3 3 4 5 4 4 3
--R a d + 63a b c d + 756a b c d + 2940a b c d + 4410a b c d
--R +
--R 4 5 5 2 3 6 6 2 7 7
--R 2646a b c d + 588a b c d + 36a b c
--R *
--R 7
--R x
--R +
--R 9 6 8 2 5 7 2 3 4 6 3 4 3 5 4 5 2
--R 7a c d + 189a b c d + 1260a b c d + 2940a b c d + 2646a b c d
--R +
--R 4 5 6 3 6 7
--R 882a b c d + 84a b c
--R *
--R 6
--R x
--R +
--R 9 2 5 8 3 4 7 2 4 3 6 3 5 2 5 4 6
--R 21a c d + 315a b c d + 1260a b c d + 1764a b c d + 882a b c d
--R +
--R 4 5 7
--R 126a b c
--R *
--R 5
--R x
--R +
--R 9 3 4 8 4 3 7 2 5 2 6 3 6 5 4 7 4
--R (35a c d + 315a b c d + 756a b c d + 588a b c d + 126a b c )x
--R +
--R 9 4 3 8 5 2 7 2 6 6 3 7 3
--R (35a c d + 189a b c d + 252a b c d + 84a b c )x
--R +
--R 9 5 2 8 6 7 2 7 2 9 6 8 7 9 7
--R (21a c d + 63a b c d + 36a b c )x + (7a c d + 9a b c )x + a c
--R Type: Polynomial(Integer)
--E 596
--S 597 of 2952
r0000:=1/10*(b*c-a*d)^7*(a+b*x)^10/b^8+7/11*d*(b*c-a*d)^6*(a+b*x)^11/b^8+_
7/4*d^2*(b*c-a*d)^5*(a+b*x)^12/b^8+_
35/13*d^3*(b*c-a*d)^4*(a+b*x)^13/b^8+_
5/2*d^4*(b*c-a*d)^3*(a+b*x)^14/b^8+_
7/5*d^5*(b*c-a*d)^2*(a+b*x)^15/b^8+_
7/16*d^6*(b*c-a*d)*(a+b*x)^16/b^8+1/17*d^7*(a+b*x)^17/b^8
--R
--R
--R (2)
--R 1 17 7 17 9 16 7 7 17 6 16
--R -- b d x + (-- a b d + -- b c d )x
--R 17 16 16
--R +
--R 12 2 15 7 21 16 6 7 17 2 5 15
--R (-- a b d + -- a b c d + - b c d )x
--R 5 5 5
--R +
--R 3 14 7 2 15 6 27 16 2 5 5 17 3 4 14
--R (6a b d + 18a b c d + -- a b c d + - b c d )x
--R 2 2
--R +
--R 126 4 13 7 588 3 14 6 756 2 15 2 5 315 16 3 4
--R --- a b d + --- a b c d + --- a b c d + --- a b c d
--R 13 13 13 13
--R +
--R 35 17 4 3
--R -- b c d
--R 13
--R *
--R 13
--R x
--R +
--R 21 5 12 7 147 4 13 6 3 14 2 5 2 15 3 4
--R -- a b d + --- a b c d + 147a b c d + 105a b c d
--R 2 2
--R +
--R 105 16 4 3 7 17 5 2
--R --- a b c d + - b c d
--R 4 4
--R *
--R 12
--R x
--R +
--R 84 6 11 7 882 5 12 6 2646 4 13 2 5 2940 3 14 3 4
--R -- a b d + --- a b c d + ---- a b c d + ---- a b c d
--R 11 11 11 11
--R +
--R 1260 2 15 4 3 189 16 5 2 7 17 6
--R ---- a b c d + --- a b c d + -- b c d
--R 11 11 11
--R *
--R 11
--R x
--R +
--R 18 7 10 7 294 6 11 6 1323 5 12 2 5 4 13 3 4
--R -- a b d + --- a b c d + ---- a b c d + 441a b c d
--R 5 5 5
--R +
--R 3 14 4 3 378 2 15 5 2 63 16 6 1 17 7
--R 294a b c d + --- a b c d + -- a b c d + -- b c
--R 5 10 10
--R *
--R 10
--R x
--R +
--R 8 9 7 7 10 6 6 11 2 5 5 12 3 4 4 13 4 3
--R a b d + 28a b c d + 196a b c d + 490a b c d + 490a b c d
--R +
--R 3 14 5 2 2 15 6 16 7
--R 196a b c d + 28a b c d + a b c
--R *
--R 9
--R x
--R +
--R 1 9 8 7 63 8 9 6 189 7 10 2 5 735 6 11 3 4
--R - a b d + -- a b c d + --- a b c d + --- a b c d
--R 8 8 2 2
--R +
--R 2205 5 12 4 3 1323 4 13 5 2 147 3 14 6 9 2 15 7
--R ---- a b c d + ---- a b c d + --- a b c d + - a b c
--R 4 4 2 2
--R *
--R 8
--R x
--R +
--R 9 8 6 8 9 2 5 7 10 3 4 6 11 4 3 5 12 5 2
--R a b c d + 27a b c d + 180a b c d + 420a b c d + 378a b c d
--R +
--R 4 13 6 3 14 7
--R 126a b c d + 12a b c
--R *
--R 7
--R x
--R +
--R 7 9 8 2 5 105 8 9 3 4 7 10 4 3 6 11 5 2 5 12 6
--R - a b c d + --- a b c d + 210a b c d + 294a b c d + 147a b c d
--R 2 2
--R +
--R 4 13 7
--R 21a b c
--R *
--R 6
--R x
--R +
--R 9 8 3 4 8 9 4 3 756 7 10 5 2 588 6 11 6 126 5 12 7 5
--R (7a b c d + 63a b c d + --- a b c d + --- a b c d + --- a b c )x
--R 5 5 5
--R +
--R 35 9 8 4 3 189 8 9 5 2 7 10 6 6 11 7 4
--R (-- a b c d + --- a b c d + 63a b c d + 21a b c )x
--R 4 4
--R +
--R 9 8 5 2 8 9 6 7 10 7 3 7 9 8 6 9 8 9 7 2
--R (7a b c d + 21a b c d + 12a b c )x + (- a b c d + - a b c )x
--R 2 2
--R +
--R 9 8 7 1 17 7 1 16 6 1 15 2 2 5 1 14 3 3 4
--R a b c x - ------ a d + ----- a b c d - ---- a b c d + --- a b c d
--R 194480 11440 1430 286
--R +
--R 7 13 4 4 3 7 12 5 5 2 7 11 6 6 1 10 7 7
--R - --- a b c d + --- a b c d - --- a b c d + -- a b c
--R 572 220 110 10
--R /
--R 8
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 597
--S 598 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 9 7 17 9 8 7 7 9 6 16
--R -- b d x + (-- a b d + -- b c d )x
--R 17 16 16
--R +
--R 12 2 7 7 21 8 6 7 9 2 5 15
--R (-- a b d + -- a b c d + - b c d )x
--R 5 5 5
--R +
--R 3 6 7 2 7 6 27 8 2 5 5 9 3 4 14
--R (6a b d + 18a b c d + -- a b c d + - b c d )x
--R 2 2
--R +
--R 126 4 5 7 588 3 6 6 756 2 7 2 5 315 8 3 4 35 9 4 3 13
--R (--- a b d + --- a b c d + --- a b c d + --- a b c d + -- b c d )x
--R 13 13 13 13 13
--R +
--R 21 5 4 7 147 4 5 6 3 6 2 5 2 7 3 4 105 8 4 3
--R -- a b d + --- a b c d + 147a b c d + 105a b c d + --- a b c d
--R 2 2 4
--R +
--R 7 9 5 2
--R - b c d
--R 4
--R *
--R 12
--R x
--R +
--R 84 6 3 7 882 5 4 6 2646 4 5 2 5 2940 3 6 3 4
--R -- a b d + --- a b c d + ---- a b c d + ---- a b c d
--R 11 11 11 11
--R +
--R 1260 2 7 4 3 189 8 5 2 7 9 6
--R ---- a b c d + --- a b c d + -- b c d
--R 11 11 11
--R *
--R 11
--R x
--R +
--R 18 7 2 7 294 6 3 6 1323 5 4 2 5 4 5 3 4 3 6 4 3
--R -- a b d + --- a b c d + ---- a b c d + 441a b c d + 294a b c d
--R 5 5 5
--R +
--R 378 2 7 5 2 63 8 6 1 9 7
--R --- a b c d + -- a b c d + -- b c
--R 5 10 10
--R *
--R 10
--R x
--R +
--R 8 7 7 2 6 6 3 2 5 5 4 3 4 4 5 4 3
--R a b d + 28a b c d + 196a b c d + 490a b c d + 490a b c d
--R +
--R 3 6 5 2 2 7 6 8 7
--R 196a b c d + 28a b c d + a b c
--R *
--R 9
--R x
--R +
--R 1 9 7 63 8 6 189 7 2 2 5 735 6 3 3 4 2205 5 4 4 3
--R - a d + -- a b c d + --- a b c d + --- a b c d + ---- a b c d
--R 8 8 2 2 4
--R +
--R 1323 4 5 5 2 147 3 6 6 9 2 7 7
--R ---- a b c d + --- a b c d + - a b c
--R 4 2 2
--R *
--R 8
--R x
--R +
--R 9 6 8 2 5 7 2 3 4 6 3 4 3 5 4 5 2
--R a c d + 27a b c d + 180a b c d + 420a b c d + 378a b c d
--R +
--R 4 5 6 3 6 7
--R 126a b c d + 12a b c
--R *
--R 7
--R x
--R +
--R 7 9 2 5 105 8 3 4 7 2 4 3 6 3 5 2 5 4 6
--R - a c d + --- a b c d + 210a b c d + 294a b c d + 147a b c d
--R 2 2
--R +
--R 4 5 7
--R 21a b c
--R *
--R 6
--R x
--R +
--R 9 3 4 8 4 3 756 7 2 5 2 588 6 3 6 126 5 4 7 5
--R (7a c d + 63a b c d + --- a b c d + --- a b c d + --- a b c )x
--R 5 5 5
--R +
--R 35 9 4 3 189 8 5 2 7 2 6 6 3 7 4
--R (-- a c d + --- a b c d + 63a b c d + 21a b c )x
--R 4 4
--R +
--R 9 5 2 8 6 7 2 7 3 7 9 6 9 8 7 2 9 7
--R (7a c d + 21a b c d + 12a b c )x + (- a c d + - a b c )x + a c x
--R 2 2
--R Type: Polynomial(Fraction(Integer))
--E 598
--S 599 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 17 7 1 16 6 1 15 2 2 5 1 14 3 3 4
--R ------ a d - ----- a b c d + ---- a b c d - --- a b c d
--R 194480 11440 1430 286
--R +
--R 7 13 4 4 3 7 12 5 5 2 7 11 6 6 1 10 7 7
--R --- a b c d - --- a b c d + --- a b c d - -- a b c
--R 572 220 110 10
--R /
--R 8
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 599
--S 600 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 600
)clear all
--S 601 of 2952
t0000:=(a+b*x)^8*(c+d*x)^7
--R
--R
--R (1)
--R 8 7 15 7 7 8 6 14 2 6 7 7 6 8 2 5 13
--R b d x + (8a b d + 7b c d )x + (28a b d + 56a b c d + 21b c d )x
--R +
--R 3 5 7 2 6 6 7 2 5 8 3 4 12
--R (56a b d + 196a b c d + 168a b c d + 35b c d )x
--R +
--R 4 4 7 3 5 6 2 6 2 5 7 3 4 8 4 3 11
--R (70a b d + 392a b c d + 588a b c d + 280a b c d + 35b c d )x
--R +
--R 5 3 7 4 4 6 3 5 2 5 2 6 3 4 7 4 3
--R 56a b d + 490a b c d + 1176a b c d + 980a b c d + 280a b c d
--R +
--R 8 5 2
--R 21b c d
--R *
--R 10
--R x
--R +
--R 6 2 7 5 3 6 4 4 2 5 3 5 3 4 2 6 4 3
--R 28a b d + 392a b c d + 1470a b c d + 1960a b c d + 980a b c d
--R +
--R 7 5 2 8 6
--R 168a b c d + 7b c d
--R *
--R 9
--R x
--R +
--R 7 7 6 2 6 5 3 2 5 4 4 3 4 3 5 4 3
--R 8a b d + 196a b c d + 1176a b c d + 2450a b c d + 1960a b c d
--R +
--R 2 6 5 2 7 6 8 7
--R 588a b c d + 56a b c d + b c
--R *
--R 8
--R x
--R +
--R 8 7 7 6 6 2 2 5 5 3 3 4 4 4 4 3
--R a d + 56a b c d + 588a b c d + 1960a b c d + 2450a b c d
--R +
--R 3 5 5 2 2 6 6 7 7
--R 1176a b c d + 196a b c d + 8a b c
--R *
--R 7
--R x
--R +
--R 8 6 7 2 5 6 2 3 4 5 3 4 3 4 4 5 2
--R 7a c d + 168a b c d + 980a b c d + 1960a b c d + 1470a b c d
--R +
--R 3 5 6 2 6 7
--R 392a b c d + 28a b c
--R *
--R 6
--R x
--R +
--R 8 2 5 7 3 4 6 2 4 3 5 3 5 2 4 4 6
--R 21a c d + 280a b c d + 980a b c d + 1176a b c d + 490a b c d
--R +
--R 3 5 7
--R 56a b c
--R *
--R 5
--R x
--R +
--R 8 3 4 7 4 3 6 2 5 2 5 3 6 4 4 7 4
--R (35a c d + 280a b c d + 588a b c d + 392a b c d + 70a b c )x
--R +
--R 8 4 3 7 5 2 6 2 6 5 3 7 3
--R (35a c d + 168a b c d + 196a b c d + 56a b c )x
--R +
--R 8 5 2 7 6 6 2 7 2 8 6 7 7 8 7
--R (21a c d + 56a b c d + 28a b c )x + (7a c d + 8a b c )x + a c
--R Type: Polynomial(Integer)
--E 601
--S 602 of 2952
r0000:=1/9*(b*c-a*d)^7*(a+b*x)^9/b^8+7/10*d*(b*c-a*d)^6*(a+b*x)^10/b^8+_
21/11*d^2*(b*c-a*d)^5*(a+b*x)^11/b^8+_
35/12*d^3*(b*c-a*d)^4*(a+b*x)^12/b^8+_
35/13*d^4*(b*c-a*d)^3*(a+b*x)^13/b^8+_
3/2*d^5*(b*c-a*d)^2*(a+b*x)^14/b^8+_
7/15*d^6*(b*c-a*d)*(a+b*x)^15/b^8+1/16*d^7*(a+b*x)^16/b^8
--R
--R
--R (2)
--R 1 16 7 16 8 15 7 7 16 6 15
--R -- b d x + (-- a b d + -- b c d )x
--R 16 15 15
--R +
--R 2 14 7 15 6 3 16 2 5 14
--R (2a b d + 4a b c d + - b c d )x
--R 2
--R +
--R 56 3 13 7 196 2 14 6 168 15 2 5 35 16 3 4 13
--R (-- a b d + --- a b c d + --- a b c d + -- b c d )x
--R 13 13 13 13
--R +
--R 35 4 12 7 98 3 13 6 2 14 2 5 70 15 3 4 35 16 4 3 12
--R (-- a b d + -- a b c d + 49a b c d + -- a b c d + -- b c d )x
--R 6 3 3 12
--R +
--R 56 5 11 7 490 4 12 6 1176 3 13 2 5 980 2 14 3 4
--R -- a b d + --- a b c d + ---- a b c d + --- a b c d
--R 11 11 11 11
--R +
--R 280 15 4 3 21 16 5 2
--R --- a b c d + -- b c d
--R 11 11
--R *
--R 11
--R x
--R +
--R 14 6 10 7 196 5 11 6 4 12 2 5 3 13 3 4
--R -- a b d + --- a b c d + 147a b c d + 196a b c d
--R 5 5
--R +
--R 2 14 4 3 84 15 5 2 7 16 6
--R 98a b c d + -- a b c d + -- b c d
--R 5 10
--R *
--R 10
--R x
--R +
--R 8 7 9 7 196 6 10 6 392 5 11 2 5 2450 4 12 3 4
--R - a b d + --- a b c d + --- a b c d + ---- a b c d
--R 9 9 3 9
--R +
--R 1960 3 13 4 3 196 2 14 5 2 56 15 6 1 16 7
--R ---- a b c d + --- a b c d + -- a b c d + - b c
--R 9 3 9 9
--R *
--R 9
--R x
--R +
--R 1 8 8 7 7 9 6 147 6 10 2 5 5 11 3 4 1225 4 12 4 3
--R - a b d + 7a b c d + --- a b c d + 245a b c d + ---- a b c d
--R 8 2 4
--R +
--R 3 13 5 2 49 2 14 6 15 7
--R 147a b c d + -- a b c d + a b c
--R 2
--R *
--R 8
--R x
--R +
--R 8 8 6 7 9 2 5 6 10 3 4 5 11 4 3 4 12 5 2
--R a b c d + 24a b c d + 140a b c d + 280a b c d + 210a b c d
--R +
--R 3 13 6 2 14 7
--R 56a b c d + 4a b c
--R *
--R 7
--R x
--R +
--R 7 8 8 2 5 140 7 9 3 4 490 6 10 4 3 5 11 5 2
--R - a b c d + --- a b c d + --- a b c d + 196a b c d
--R 2 3 3
--R +
--R 245 4 12 6 28 3 13 7
--R --- a b c d + -- a b c
--R 3 3
--R *
--R 6
--R x
--R +
--R 8 8 3 4 7 9 4 3 588 6 10 5 2 392 5 11 6 4 12 7 5
--R (7a b c d + 56a b c d + --- a b c d + --- a b c d + 14a b c )x
--R 5 5
--R +
--R 35 8 8 4 3 7 9 5 2 6 10 6 5 11 7 4
--R (-- a b c d + 42a b c d + 49a b c d + 14a b c )x
--R 4
--R +
--R 8 8 5 2 56 7 9 6 28 6 10 7 3 7 8 8 6 7 9 7 2
--R (7a b c d + -- a b c d + -- a b c )x + (- a b c d + 4a b c )x
--R 3 3 2
--R +
--R 8 8 7 1 16 7 1 15 6 1 14 2 2 5 7 13 3 3 4
--R a b c x - ------ a d + ---- a b c d - --- a b c d + ---- a b c d
--R 102960 6435 858 1287
--R +
--R 7 12 4 4 3 7 11 5 5 2 7 10 6 6 1 9 7 7
--R - --- a b c d + --- a b c d - -- a b c d + - a b c
--R 396 165 90 9
--R /
--R 8
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 602
--S 603 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 8 7 16 8 7 7 7 8 6 15
--R -- b d x + (-- a b d + -- b c d )x
--R 16 15 15
--R +
--R 2 6 7 7 6 3 8 2 5 14
--R (2a b d + 4a b c d + - b c d )x
--R 2
--R +
--R 56 3 5 7 196 2 6 6 168 7 2 5 35 8 3 4 13
--R (-- a b d + --- a b c d + --- a b c d + -- b c d )x
--R 13 13 13 13
--R +
--R 35 4 4 7 98 3 5 6 2 6 2 5 70 7 3 4 35 8 4 3 12
--R (-- a b d + -- a b c d + 49a b c d + -- a b c d + -- b c d )x
--R 6 3 3 12
--R +
--R 56 5 3 7 490 4 4 6 1176 3 5 2 5 980 2 6 3 4 280 7 4 3
--R -- a b d + --- a b c d + ---- a b c d + --- a b c d + --- a b c d
--R 11 11 11 11 11
--R +
--R 21 8 5 2
--R -- b c d
--R 11
--R *
--R 11
--R x
--R +
--R 14 6 2 7 196 5 3 6 4 4 2 5 3 5 3 4 2 6 4 3
--R -- a b d + --- a b c d + 147a b c d + 196a b c d + 98a b c d
--R 5 5
--R +
--R 84 7 5 2 7 8 6
--R -- a b c d + -- b c d
--R 5 10
--R *
--R 10
--R x
--R +
--R 8 7 7 196 6 2 6 392 5 3 2 5 2450 4 4 3 4 1960 3 5 4 3
--R - a b d + --- a b c d + --- a b c d + ---- a b c d + ---- a b c d
--R 9 9 3 9 9
--R +
--R 196 2 6 5 2 56 7 6 1 8 7
--R --- a b c d + -- a b c d + - b c
--R 3 9 9
--R *
--R 9
--R x
--R +
--R 1 8 7 7 6 147 6 2 2 5 5 3 3 4 1225 4 4 4 3
--R - a d + 7a b c d + --- a b c d + 245a b c d + ---- a b c d
--R 8 2 4
--R +
--R 3 5 5 2 49 2 6 6 7 7
--R 147a b c d + -- a b c d + a b c
--R 2
--R *
--R 8
--R x
--R +
--R 8 6 7 2 5 6 2 3 4 5 3 4 3 4 4 5 2
--R a c d + 24a b c d + 140a b c d + 280a b c d + 210a b c d
--R +
--R 3 5 6 2 6 7
--R 56a b c d + 4a b c
--R *
--R 7
--R x
--R +
--R 7 8 2 5 140 7 3 4 490 6 2 4 3 5 3 5 2 245 4 4 6
--R - a c d + --- a b c d + --- a b c d + 196a b c d + --- a b c d
--R 2 3 3 3
--R +
--R 28 3 5 7
--R -- a b c
--R 3
--R *
--R 6
--R x
--R +
--R 8 3 4 7 4 3 588 6 2 5 2 392 5 3 6 4 4 7 5
--R (7a c d + 56a b c d + --- a b c d + --- a b c d + 14a b c )x
--R 5 5
--R +
--R 35 8 4 3 7 5 2 6 2 6 5 3 7 4
--R (-- a c d + 42a b c d + 49a b c d + 14a b c )x
--R 4
--R +
--R 8 5 2 56 7 6 28 6 2 7 3 7 8 6 7 7 2 8 7
--R (7a c d + -- a b c d + -- a b c )x + (- a c d + 4a b c )x + a c x
--R 3 3 2
--R Type: Polynomial(Fraction(Integer))
--E 603
--S 604 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 16 7 1 15 6 1 14 2 2 5 7 13 3 3 4
--R ------ a d - ---- a b c d + --- a b c d - ---- a b c d
--R 102960 6435 858 1287
--R +
--R 7 12 4 4 3 7 11 5 5 2 7 10 6 6 1 9 7 7
--R --- a b c d - --- a b c d + -- a b c d - - a b c
--R 396 165 90 9
--R /
--R 8
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 604
--S 605 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 605
)clear all
--S 606 of 2952
t0000:=(a+b*x)^7*(c+d*x)^7
--R
--R
--R (1)
--R 7 7 14 6 7 7 6 13 2 5 7 6 6 7 2 5 12
--R b d x + (7a b d + 7b c d )x + (21a b d + 49a b c d + 21b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 11
--R (35a b d + 147a b c d + 147a b c d + 35b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 10
--R (35a b d + 245a b c d + 441a b c d + 245a b c d + 35b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R 21a b d + 245a b c d + 735a b c d + 735a b c d + 245a b c d
--R +
--R 7 5 2
--R 21b c d
--R *
--R 9
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 7a b d + 147a b c d + 735a b c d + 1225a b c d + 735a b c d
--R +
--R 6 5 2 7 6
--R 147a b c d + 7b c d
--R *
--R 8
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R a d + 49a b c d + 441a b c d + 1225a b c d + 1225a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R 441a b c d + 49a b c d + b c
--R *
--R 7
--R x
--R +
--R 7 6 6 2 5 5 2 3 4 4 3 4 3 3 4 5 2
--R 7a c d + 147a b c d + 735a b c d + 1225a b c d + 735a b c d
--R +
--R 2 5 6 6 7
--R 147a b c d + 7a b c
--R *
--R 6
--R x
--R +
--R 7 2 5 6 3 4 5 2 4 3 4 3 5 2 3 4 6
--R 21a c d + 245a b c d + 735a b c d + 735a b c d + 245a b c d
--R +
--R 2 5 7
--R 21a b c
--R *
--R 5
--R x
--R +
--R 7 3 4 6 4 3 5 2 5 2 4 3 6 3 4 7 4
--R (35a c d + 245a b c d + 441a b c d + 245a b c d + 35a b c )x
--R +
--R 7 4 3 6 5 2 5 2 6 4 3 7 3
--R (35a c d + 147a b c d + 147a b c d + 35a b c )x
--R +
--R 7 5 2 6 6 5 2 7 2 7 6 6 7 7 7
--R (21a c d + 49a b c d + 21a b c )x + (7a c d + 7a b c )x + a c
--R Type: Polynomial(Integer)
--E 606
--S 607 of 2952
r0000:=1/8*(b*c-a*d)^7*(a+b*x)^8/b^8+7/9*d*(b*c-a*d)^6*(a+b*x)^9/b^8+_
21/10*d^2*(b*c-a*d)^5*(a+b*x)^10/b^8+_
35/11*d^3*(b*c-a*d)^4*(a+b*x)^11/b^8+_
35/12*d^4*(b*c-a*d)^3*(a+b*x)^12/b^8+_
21/13*d^5*(b*c-a*d)^2*(a+b*x)^13/b^8+_
1/2*d^6*(b*c-a*d)*(a+b*x)^14/b^8+1/15*d^7*(a+b*x)^15/b^8
--R
--R
--R (2)
--R 1 15 7 15 1 14 7 1 15 6 14
--R -- b d x + (- a b d + - b c d )x
--R 15 2 2
--R +
--R 21 2 13 7 49 14 6 21 15 2 5 13
--R (-- a b d + -- a b c d + -- b c d )x
--R 13 13 13
--R +
--R 35 3 12 7 49 2 13 6 49 14 2 5 35 15 3 4 12
--R (-- a b d + -- a b c d + -- a b c d + -- b c d )x
--R 12 4 4 12
--R +
--R 35 4 11 7 245 3 12 6 441 2 13 2 5 245 14 3 4
--R -- a b d + --- a b c d + --- a b c d + --- a b c d
--R 11 11 11 11
--R +
--R 35 15 4 3
--R -- b c d
--R 11
--R *
--R 11
--R x
--R +
--R 21 5 10 7 49 4 11 6 147 3 12 2 5 147 2 13 3 4
--R -- a b d + -- a b c d + --- a b c d + --- a b c d
--R 10 2 2 2
--R +
--R 49 14 4 3 21 15 5 2
--R -- a b c d + -- b c d
--R 2 10
--R *
--R 10
--R x
--R +
--R 7 6 9 7 49 5 10 6 245 4 11 2 5 1225 3 12 3 4
--R - a b d + -- a b c d + --- a b c d + ---- a b c d
--R 9 3 3 9
--R +
--R 245 2 13 4 3 49 14 5 2 7 15 6
--R --- a b c d + -- a b c d + - b c d
--R 3 3 9
--R *
--R 9
--R x
--R +
--R 1 7 8 7 49 6 9 6 441 5 10 2 5 1225 4 11 3 4
--R - a b d + -- a b c d + --- a b c d + ---- a b c d
--R 8 8 8 8
--R +
--R 1225 3 12 4 3 441 2 13 5 2 49 14 6 1 15 7
--R ---- a b c d + --- a b c d + -- a b c d + - b c
--R 8 8 8 8
--R *
--R 8
--R x
--R +
--R 7 8 6 6 9 2 5 5 10 3 4 4 11 4 3 3 12 5 2
--R a b c d + 21a b c d + 105a b c d + 175a b c d + 105a b c d
--R +
--R 2 13 6 14 7
--R 21a b c d + a b c
--R *
--R 7
--R x
--R +
--R 7 7 8 2 5 245 6 9 3 4 245 5 10 4 3 245 4 11 5 2
--R - a b c d + --- a b c d + --- a b c d + --- a b c d
--R 2 6 2 2
--R +
--R 245 3 12 6 7 2 13 7
--R --- a b c d + - a b c
--R 6 2
--R *
--R 6
--R x
--R +
--R 7 8 3 4 6 9 4 3 441 5 10 5 2 4 11 6 3 12 7 5
--R (7a b c d + 49a b c d + --- a b c d + 49a b c d + 7a b c )x
--R 5
--R +
--R 35 7 8 4 3 147 6 9 5 2 147 5 10 6 35 4 11 7 4
--R (-- a b c d + --- a b c d + --- a b c d + -- a b c )x
--R 4 4 4 4
--R +
--R 7 8 5 2 49 6 9 6 5 10 7 3 7 7 8 6 7 6 9 7 2
--R (7a b c d + -- a b c d + 7a b c )x + (- a b c d + - a b c )x
--R 3 2 2
--R +
--R 7 8 7 1 15 7 1 14 6 7 13 2 2 5 7 12 3 3 4
--R a b c x - ----- a d + ---- a b c d - ---- a b c d + --- a b c d
--R 51480 3432 3432 792
--R +
--R 7 11 4 4 3 7 10 5 5 2 7 9 6 6 1 8 7 7
--R - --- a b c d + --- a b c d - -- a b c d + - a b c
--R 264 120 72 8
--R /
--R 8
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 607
--S 608 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 7 7 15 1 6 7 1 7 6 14
--R -- b d x + (- a b d + - b c d )x
--R 15 2 2
--R +
--R 21 2 5 7 49 6 6 21 7 2 5 13
--R (-- a b d + -- a b c d + -- b c d )x
--R 13 13 13
--R +
--R 35 3 4 7 49 2 5 6 49 6 2 5 35 7 3 4 12
--R (-- a b d + -- a b c d + -- a b c d + -- b c d )x
--R 12 4 4 12
--R +
--R 35 4 3 7 245 3 4 6 441 2 5 2 5 245 6 3 4 35 7 4 3 11
--R (-- a b d + --- a b c d + --- a b c d + --- a b c d + -- b c d )x
--R 11 11 11 11 11
--R +
--R 21 5 2 7 49 4 3 6 147 3 4 2 5 147 2 5 3 4 49 6 4 3
--R -- a b d + -- a b c d + --- a b c d + --- a b c d + -- a b c d
--R 10 2 2 2 2
--R +
--R 21 7 5 2
--R -- b c d
--R 10
--R *
--R 10
--R x
--R +
--R 7 6 7 49 5 2 6 245 4 3 2 5 1225 3 4 3 4 245 2 5 4 3
--R - a b d + -- a b c d + --- a b c d + ---- a b c d + --- a b c d
--R 9 3 3 9 3
--R +
--R 49 6 5 2 7 7 6
--R -- a b c d + - b c d
--R 3 9
--R *
--R 9
--R x
--R +
--R 1 7 7 49 6 6 441 5 2 2 5 1225 4 3 3 4 1225 3 4 4 3
--R - a d + -- a b c d + --- a b c d + ---- a b c d + ---- a b c d
--R 8 8 8 8 8
--R +
--R 441 2 5 5 2 49 6 6 1 7 7
--R --- a b c d + -- a b c d + - b c
--R 8 8 8
--R *
--R 8
--R x
--R +
--R 7 6 6 2 5 5 2 3 4 4 3 4 3 3 4 5 2
--R a c d + 21a b c d + 105a b c d + 175a b c d + 105a b c d
--R +
--R 2 5 6 6 7
--R 21a b c d + a b c
--R *
--R 7
--R x
--R +
--R 7 7 2 5 245 6 3 4 245 5 2 4 3 245 4 3 5 2 245 3 4 6
--R - a c d + --- a b c d + --- a b c d + --- a b c d + --- a b c d
--R 2 6 2 2 6
--R +
--R 7 2 5 7
--R - a b c
--R 2
--R *
--R 6
--R x
--R +
--R 7 3 4 6 4 3 441 5 2 5 2 4 3 6 3 4 7 5
--R (7a c d + 49a b c d + --- a b c d + 49a b c d + 7a b c )x
--R 5
--R +
--R 35 7 4 3 147 6 5 2 147 5 2 6 35 4 3 7 4
--R (-- a c d + --- a b c d + --- a b c d + -- a b c )x
--R 4 4 4 4
--R +
--R 7 5 2 49 6 6 5 2 7 3 7 7 6 7 6 7 2 7 7
--R (7a c d + -- a b c d + 7a b c )x + (- a c d + - a b c )x + a c x
--R 3 2 2
--R Type: Polynomial(Fraction(Integer))
--E 608
--S 609 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 15 7 1 14 6 7 13 2 2 5 7 12 3 3 4
--R ----- a d - ---- a b c d + ---- a b c d - --- a b c d
--R 51480 3432 3432 792
--R +
--R 7 11 4 4 3 7 10 5 5 2 7 9 6 6 1 8 7 7
--R --- a b c d - --- a b c d + -- a b c d - - a b c
--R 264 120 72 8
--R /
--R 8
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 609
--S 610 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 610
)clear all
--S 611 of 2952
t0000:=(a+b*x)^6*(c+d*x)^7
--R
--R
--R (1)
--R 6 7 13 5 7 6 6 12 2 4 7 5 6 6 2 5 11
--R b d x + (6a b d + 7b c d )x + (15a b d + 42a b c d + 21b c d )x
--R +
--R 3 3 7 2 4 6 5 2 5 6 3 4 10
--R (20a b d + 105a b c d + 126a b c d + 35b c d )x
--R +
--R 4 2 7 3 3 6 2 4 2 5 5 3 4 6 4 3 9
--R (15a b d + 140a b c d + 315a b c d + 210a b c d + 35b c d )x
--R +
--R 5 7 4 2 6 3 3 2 5 2 4 3 4 5 4 3
--R 6a b d + 105a b c d + 420a b c d + 525a b c d + 210a b c d
--R +
--R 6 5 2
--R 21b c d
--R *
--R 8
--R x
--R +
--R 6 7 5 6 4 2 2 5 3 3 3 4 2 4 4 3
--R a d + 42a b c d + 315a b c d + 700a b c d + 525a b c d
--R +
--R 5 5 2 6 6
--R 126a b c d + 7b c d
--R *
--R 7
--R x
--R +
--R 6 6 5 2 5 4 2 3 4 3 3 4 3 2 4 5 2
--R 7a c d + 126a b c d + 525a b c d + 700a b c d + 315a b c d
--R +
--R 5 6 6 7
--R 42a b c d + b c
--R *
--R 6
--R x
--R +
--R 6 2 5 5 3 4 4 2 4 3 3 3 5 2 2 4 6
--R 21a c d + 210a b c d + 525a b c d + 420a b c d + 105a b c d
--R +
--R 5 7
--R 6a b c
--R *
--R 5
--R x
--R +
--R 6 3 4 5 4 3 4 2 5 2 3 3 6 2 4 7 4
--R (35a c d + 210a b c d + 315a b c d + 140a b c d + 15a b c )x
--R +
--R 6 4 3 5 5 2 4 2 6 3 3 7 3
--R (35a c d + 126a b c d + 105a b c d + 20a b c )x
--R +
--R 6 5 2 5 6 4 2 7 2 6 6 5 7 6 7
--R (21a c d + 42a b c d + 15a b c )x + (7a c d + 6a b c )x + a c
--R Type: Polynomial(Integer)
--E 611
--S 612 of 2952
r0000:=1/8*(b*c-a*d)^6*(c+d*x)^8/d^7-2/3*b*(b*c-a*d)^5*(c+d*x)^9/d^7+_
3/2*b^2*(b*c-a*d)^4*(c+d*x)^10/d^7-_
20/11*b^3*(b*c-a*d)^3*(c+d*x)^11/d^7+_
5/4*b^4*(b*c-a*d)^2*(c+d*x)^12/d^7-_
6/13*b^5*(b*c-a*d)*(c+d*x)^13/d^7+1/14*b^6*(c+d*x)^14/d^7
--R
--R
--R (2)
--R 1 6 14 14 6 5 14 7 6 13 13
--R -- b d x + (-- a b d + -- b c d )x
--R 14 13 13
--R +
--R 5 2 4 14 7 5 13 7 6 2 12 12
--R (- a b d + - a b c d + - b c d )x
--R 4 2 4
--R +
--R 20 3 3 14 105 2 4 13 126 5 2 12 35 6 3 11 11
--R (-- a b d + --- a b c d + --- a b c d + -- b c d )x
--R 11 11 11 11
--R +
--R 3 4 2 14 3 3 13 63 2 4 2 12 5 3 11 7 6 4 10 10
--R (- a b d + 14a b c d + -- a b c d + 21a b c d + - b c d )x
--R 2 2 2
--R +
--R 2 5 14 35 4 2 13 140 3 3 2 12 175 2 4 3 11
--R - a b d + -- a b c d + --- a b c d + --- a b c d
--R 3 3 3 3
--R +
--R 70 5 4 10 7 6 5 9
--R -- a b c d + - b c d
--R 3 3
--R *
--R 9
--R x
--R +
--R 1 6 14 21 5 13 315 4 2 2 12 175 3 3 3 11
--R - a d + -- a b c d + --- a b c d + --- a b c d
--R 8 4 8 2
--R +
--R 525 2 4 4 10 63 5 5 9 7 6 6 8
--R --- a b c d + -- a b c d + - b c d
--R 8 4 8
--R *
--R 8
--R x
--R +
--R 6 13 5 2 12 4 2 3 11 3 3 4 10 2 4 5 9
--R a c d + 18a b c d + 75a b c d + 100a b c d + 45a b c d
--R +
--R 5 6 8 1 6 7 7
--R 6a b c d + - b c d
--R 7
--R *
--R 7
--R x
--R +
--R 7 6 2 12 5 3 11 175 4 2 4 10 3 3 5 9 35 2 4 6 8
--R - a c d + 35a b c d + --- a b c d + 70a b c d + -- a b c d
--R 2 2 2
--R +
--R 5 7 7
--R a b c d
--R *
--R 6
--R x
--R +
--R 6 3 11 5 4 10 4 2 5 9 3 3 6 8 2 4 7 7 5
--R (7a c d + 42a b c d + 63a b c d + 28a b c d + 3a b c d )x
--R +
--R 35 6 4 10 63 5 5 9 105 4 2 6 8 3 3 7 7 4
--R (-- a c d + -- a b c d + --- a b c d + 5a b c d )x
--R 4 2 4
--R +
--R 6 5 9 5 6 8 4 2 7 7 3 7 6 6 8 5 7 7 2 6 7 7
--R (7a c d + 14a b c d + 5a b c d )x + (- a c d + 3a b c d )x + a c d x
--R 2
--R +
--R 1 6 8 6 1 5 9 5 1 4 2 10 4 1 3 3 11 3 1 2 4 12 2
--R - a c d - -- a b c d + -- a b c d - -- a b c d + --- a b c d
--R 8 12 24 66 264
--R +
--R 1 5 13 1 6 14
--R - ---- a b c d + ----- b c
--R 1716 24024
--R /
--R 7
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 612
--S 613 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 6 7 14 6 5 7 7 6 6 13
--R -- b d x + (-- a b d + -- b c d )x
--R 14 13 13
--R +
--R 5 2 4 7 7 5 6 7 6 2 5 12
--R (- a b d + - a b c d + - b c d )x
--R 4 2 4
--R +
--R 20 3 3 7 105 2 4 6 126 5 2 5 35 6 3 4 11
--R (-- a b d + --- a b c d + --- a b c d + -- b c d )x
--R 11 11 11 11
--R +
--R 3 4 2 7 3 3 6 63 2 4 2 5 5 3 4 7 6 4 3 10
--R (- a b d + 14a b c d + -- a b c d + 21a b c d + - b c d )x
--R 2 2 2
--R +
--R 2 5 7 35 4 2 6 140 3 3 2 5 175 2 4 3 4 70 5 4 3
--R - a b d + -- a b c d + --- a b c d + --- a b c d + -- a b c d
--R 3 3 3 3 3
--R +
--R 7 6 5 2
--R - b c d
--R 3
--R *
--R 9
--R x
--R +
--R 1 6 7 21 5 6 315 4 2 2 5 175 3 3 3 4 525 2 4 4 3
--R - a d + -- a b c d + --- a b c d + --- a b c d + --- a b c d
--R 8 4 8 2 8
--R +
--R 63 5 5 2 7 6 6
--R -- a b c d + - b c d
--R 4 8
--R *
--R 8
--R x
--R +
--R 6 6 5 2 5 4 2 3 4 3 3 4 3 2 4 5 2 5 6
--R a c d + 18a b c d + 75a b c d + 100a b c d + 45a b c d + 6a b c d
--R +
--R 1 6 7
--R - b c
--R 7
--R *
--R 7
--R x
--R +
--R 7 6 2 5 5 3 4 175 4 2 4 3 3 3 5 2 35 2 4 6 5 7 6
--R (- a c d + 35a b c d + --- a b c d + 70a b c d + -- a b c d + a b c )x
--R 2 2 2
--R +
--R 6 3 4 5 4 3 4 2 5 2 3 3 6 2 4 7 5
--R (7a c d + 42a b c d + 63a b c d + 28a b c d + 3a b c )x
--R +
--R 35 6 4 3 63 5 5 2 105 4 2 6 3 3 7 4
--R (-- a c d + -- a b c d + --- a b c d + 5a b c )x
--R 4 2 4
--R +
--R 6 5 2 5 6 4 2 7 3 7 6 6 5 7 2 6 7
--R (7a c d + 14a b c d + 5a b c )x + (- a c d + 3a b c )x + a c x
--R 2
--R Type: Polynomial(Fraction(Integer))
--E 613
--S 614 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 6 8 6 1 5 9 5 1 4 2 10 4 1 3 3 11 3 1 2 4 12 2
--R - - a c d + -- a b c d - -- a b c d + -- a b c d - --- a b c d
--R 8 12 24 66 264
--R +
--R 1 5 13 1 6 14
--R ---- a b c d - ----- b c
--R 1716 24024
--R /
--R 7
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 614
--S 615 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 615
)clear all
--S 616 of 2952
t0000:=(a+b*x)^5*(c+d*x)^7
--R
--R
--R (1)
--R 5 7 12 4 7 5 6 11 2 3 7 4 6 5 2 5 10
--R b d x + (5a b d + 7b c d )x + (10a b d + 35a b c d + 21b c d )x
--R +
--R 3 2 7 2 3 6 4 2 5 5 3 4 9
--R (10a b d + 70a b c d + 105a b c d + 35b c d )x
--R +
--R 4 7 3 2 6 2 3 2 5 4 3 4 5 4 3 8
--R (5a b d + 70a b c d + 210a b c d + 175a b c d + 35b c d )x
--R +
--R 5 7 4 6 3 2 2 5 2 3 3 4 4 4 3 5 5 2 7
--R (a d + 35a b c d + 210a b c d + 350a b c d + 175a b c d + 21b c d )x
--R +
--R 5 6 4 2 5 3 2 3 4 2 3 4 3 4 5 2
--R 7a c d + 105a b c d + 350a b c d + 350a b c d + 105a b c d
--R +
--R 5 6
--R 7b c d
--R *
--R 6
--R x
--R +
--R 5 2 5 4 3 4 3 2 4 3 2 3 5 2 4 6 5 7 5
--R (21a c d + 175a b c d + 350a b c d + 210a b c d + 35a b c d + b c )x
--R +
--R 5 3 4 4 4 3 3 2 5 2 2 3 6 4 7 4
--R (35a c d + 175a b c d + 210a b c d + 70a b c d + 5a b c )x
--R +
--R 5 4 3 4 5 2 3 2 6 2 3 7 3
--R (35a c d + 105a b c d + 70a b c d + 10a b c )x
--R +
--R 5 5 2 4 6 3 2 7 2 5 6 4 7 5 7
--R (21a c d + 35a b c d + 10a b c )x + (7a c d + 5a b c )x + a c
--R Type: Polynomial(Integer)
--E 616
--S 617 of 2952
r0000:=-1/8*(b*c-a*d)^5*(c+d*x)^8/d^6+5/9*b*(b*c-a*d)^4*(c+d*x)^9/d^6-_
b^2*(b*c-a*d)^3*(c+d*x)^10/d^6+10/11*b^3*(b*c-a*d)^2*(c+d*x)^11/d^6-_
5/12*b^4*(b*c-a*d)*(c+d*x)^12/d^6+1/13*b^5*(c+d*x)^13/d^6
--R
--R
--R (2)
--R 1 5 13 13 5 4 13 7 5 12 12
--R -- b d x + (-- a b d + -- b c d )x
--R 13 12 12
--R +
--R 10 2 3 13 35 4 12 21 5 2 11 11
--R (-- a b d + -- a b c d + -- b c d )x
--R 11 11 11
--R +
--R 3 2 13 2 3 12 21 4 2 11 7 5 3 10 10
--R (a b d + 7a b c d + -- a b c d + - b c d )x
--R 2 2
--R +
--R 5 4 13 70 3 2 12 70 2 3 2 11 175 4 3 10 35 5 4 9 9
--R (- a b d + -- a b c d + -- a b c d + --- a b c d + -- b c d )x
--R 9 9 3 9 9
--R +
--R 1 5 13 35 4 12 105 3 2 2 11 175 2 3 3 10 175 4 4 9
--R - a d + -- a b c d + --- a b c d + --- a b c d + --- a b c d
--R 8 8 4 4 8
--R +
--R 21 5 5 8
--R -- b c d
--R 8
--R *
--R 8
--R x
--R +
--R 5 12 4 2 11 3 2 3 10 2 3 4 9 4 5 8
--R a c d + 15a b c d + 50a b c d + 50a b c d + 15a b c d
--R +
--R 5 6 7
--R b c d
--R *
--R 7
--R x
--R +
--R 7 5 2 11 175 4 3 10 175 3 2 4 9 2 3 5 8 35 4 6 7
--R - a c d + --- a b c d + --- a b c d + 35a b c d + -- a b c d
--R 2 6 3 6
--R +
--R 1 5 7 6
--R - b c d
--R 6
--R *
--R 6
--R x
--R +
--R 5 3 10 4 4 9 3 2 5 8 2 3 6 7 4 7 6 5
--R (7a c d + 35a b c d + 42a b c d + 14a b c d + a b c d )x
--R +
--R 35 5 4 9 105 4 5 8 35 3 2 6 7 5 2 3 7 6 4
--R (-- a c d + --- a b c d + -- a b c d + - a b c d )x
--R 4 4 2 2
--R +
--R 5 5 8 35 4 6 7 10 3 2 7 6 3 7 5 6 7 5 4 7 6 2
--R (7a c d + -- a b c d + -- a b c d )x + (- a c d + - a b c d )x
--R 3 3 2 2
--R +
--R 5 7 6 1 5 8 5 5 4 9 4 1 3 2 10 3 1 2 3 11 2
--R a c d x + - a c d - -- a b c d + -- a b c d - --- a b c d
--R 8 72 36 132
--R +
--R 1 4 12 1 5 13
--R --- a b c d - ----- b c
--R 792 10296
--R /
--R 6
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 617
--S 618 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 5 7 13 5 4 7 7 5 6 12
--R -- b d x + (-- a b d + -- b c d )x
--R 13 12 12
--R +
--R 10 2 3 7 35 4 6 21 5 2 5 11
--R (-- a b d + -- a b c d + -- b c d )x
--R 11 11 11
--R +
--R 3 2 7 2 3 6 21 4 2 5 7 5 3 4 10
--R (a b d + 7a b c d + -- a b c d + - b c d )x
--R 2 2
--R +
--R 5 4 7 70 3 2 6 70 2 3 2 5 175 4 3 4 35 5 4 3 9
--R (- a b d + -- a b c d + -- a b c d + --- a b c d + -- b c d )x
--R 9 9 3 9 9
--R +
--R 1 5 7 35 4 6 105 3 2 2 5 175 2 3 3 4 175 4 4 3
--R - a d + -- a b c d + --- a b c d + --- a b c d + --- a b c d
--R 8 8 4 4 8
--R +
--R 21 5 5 2
--R -- b c d
--R 8
--R *
--R 8
--R x
--R +
--R 5 6 4 2 5 3 2 3 4 2 3 4 3 4 5 2 5 6 7
--R (a c d + 15a b c d + 50a b c d + 50a b c d + 15a b c d + b c d)x
--R +
--R 7 5 2 5 175 4 3 4 175 3 2 4 3 2 3 5 2 35 4 6
--R - a c d + --- a b c d + --- a b c d + 35a b c d + -- a b c d
--R 2 6 3 6
--R +
--R 1 5 7
--R - b c
--R 6
--R *
--R 6
--R x
--R +
--R 5 3 4 4 4 3 3 2 5 2 2 3 6 4 7 5
--R (7a c d + 35a b c d + 42a b c d + 14a b c d + a b c )x
--R +
--R 35 5 4 3 105 4 5 2 35 3 2 6 5 2 3 7 4
--R (-- a c d + --- a b c d + -- a b c d + - a b c )x
--R 4 4 2 2
--R +
--R 5 5 2 35 4 6 10 3 2 7 3 7 5 6 5 4 7 2 5 7
--R (7a c d + -- a b c d + -- a b c )x + (- a c d + - a b c )x + a c x
--R 3 3 2 2
--R Type: Polynomial(Fraction(Integer))
--E 618
--S 619 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 5 8 5 5 4 9 4 1 3 2 10 3 1 2 3 11 2 1 4 12
--R - - a c d + -- a b c d - -- a b c d + --- a b c d - --- a b c d
--R 8 72 36 132 792
--R +
--R 1 5 13
--R ----- b c
--R 10296
--R /
--R 6
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 619
--S 620 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 620
)clear all
--S 621 of 2952
t0000:=(a+b*x)^4*(c+d*x)^7
--R
--R
--R (1)
--R 4 7 11 3 7 4 6 10 2 2 7 3 6 4 2 5 9
--R b d x + (4a b d + 7b c d )x + (6a b d + 28a b c d + 21b c d )x
--R +
--R 3 7 2 2 6 3 2 5 4 3 4 8
--R (4a b d + 42a b c d + 84a b c d + 35b c d )x
--R +
--R 4 7 3 6 2 2 2 5 3 3 4 4 4 3 7
--R (a d + 28a b c d + 126a b c d + 140a b c d + 35b c d )x
--R +
--R 4 6 3 2 5 2 2 3 4 3 4 3 4 5 2 6
--R (7a c d + 84a b c d + 210a b c d + 140a b c d + 21b c d )x
--R +
--R 4 2 5 3 3 4 2 2 4 3 3 5 2 4 6 5
--R (21a c d + 140a b c d + 210a b c d + 84a b c d + 7b c d)x
--R +
--R 4 3 4 3 4 3 2 2 5 2 3 6 4 7 4
--R (35a c d + 140a b c d + 126a b c d + 28a b c d + b c )x
--R +
--R 4 4 3 3 5 2 2 2 6 3 7 3
--R (35a c d + 84a b c d + 42a b c d + 4a b c )x
--R +
--R 4 5 2 3 6 2 2 7 2 4 6 3 7 4 7
--R (21a c d + 28a b c d + 6a b c )x + (7a c d + 4a b c )x + a c
--R Type: Polynomial(Integer)
--E 621
--S 622 of 2952
r0000:=1/8*(b*c-a*d)^4*(c+d*x)^8/d^5-4/9*b*(b*c-a*d)^3*(c+d*x)^9/d^5+_
3/5*b^2*(b*c-a*d)^2*(c+d*x)^10/d^5-4/11*b^3*(b*c-a*d)*(c+d*x)^11/d^5+_
1/12*b^4*(c+d*x)^12/d^5
--R
--R
--R (2)
--R 1 4 12 12 4 3 12 7 4 11 11
--R -- b d x + (-- a b d + -- b c d )x
--R 12 11 11
--R +
--R 3 2 2 12 14 3 11 21 4 2 10 10
--R (- a b d + -- a b c d + -- b c d )x
--R 5 5 10
--R +
--R 4 3 12 14 2 2 11 28 3 2 10 35 4 3 9 9
--R (- a b d + -- a b c d + -- a b c d + -- b c d )x
--R 9 3 3 9
--R +
--R 1 4 12 7 3 11 63 2 2 2 10 35 3 3 9 35 4 4 8 8
--R (- a d + - a b c d + -- a b c d + -- a b c d + -- b c d )x
--R 8 2 4 2 8
--R +
--R 4 11 3 2 10 2 2 3 9 3 4 8 4 5 7 7
--R (a c d + 12a b c d + 30a b c d + 20a b c d + 3b c d )x
--R +
--R 7 4 2 10 70 3 3 9 2 2 4 8 3 5 7 7 4 6 6 6
--R (- a c d + -- a b c d + 35a b c d + 14a b c d + - b c d )x
--R 2 3 6
--R +
--R 4 3 9 3 4 8 126 2 2 5 7 28 3 6 6 1 4 7 5 5
--R (7a c d + 28a b c d + --- a b c d + -- a b c d + - b c d )x
--R 5 5 5
--R +
--R 35 4 4 8 3 5 7 21 2 2 6 6 3 7 5 4
--R (-- a c d + 21a b c d + -- a b c d + a b c d )x
--R 4 2
--R +
--R 4 5 7 28 3 6 6 2 2 7 5 3 7 4 6 6 3 7 5 2
--R (7a c d + -- a b c d + 2a b c d )x + (- a c d + 2a b c d )x
--R 3 2
--R +
--R 4 7 5 1 4 8 4 1 3 9 3 1 2 2 10 2 1 3 11 1 4 12
--R a c d x + - a c d - -- a b c d + -- a b c d - --- a b c d + ---- b c
--R 8 18 60 330 3960
--R /
--R 5
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 622
--S 623 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 4 7 12 4 3 7 7 4 6 11
--R -- b d x + (-- a b d + -- b c d )x
--R 12 11 11
--R +
--R 3 2 2 7 14 3 6 21 4 2 5 10
--R (- a b d + -- a b c d + -- b c d )x
--R 5 5 10
--R +
--R 4 3 7 14 2 2 6 28 3 2 5 35 4 3 4 9
--R (- a b d + -- a b c d + -- a b c d + -- b c d )x
--R 9 3 3 9
--R +
--R 1 4 7 7 3 6 63 2 2 2 5 35 3 3 4 35 4 4 3 8
--R (- a d + - a b c d + -- a b c d + -- a b c d + -- b c d )x
--R 8 2 4 2 8
--R +
--R 4 6 3 2 5 2 2 3 4 3 4 3 4 5 2 7
--R (a c d + 12a b c d + 30a b c d + 20a b c d + 3b c d )x
--R +
--R 7 4 2 5 70 3 3 4 2 2 4 3 3 5 2 7 4 6 6
--R (- a c d + -- a b c d + 35a b c d + 14a b c d + - b c d)x
--R 2 3 6
--R +
--R 4 3 4 3 4 3 126 2 2 5 2 28 3 6 1 4 7 5
--R (7a c d + 28a b c d + --- a b c d + -- a b c d + - b c )x
--R 5 5 5
--R +
--R 35 4 4 3 3 5 2 21 2 2 6 3 7 4
--R (-- a c d + 21a b c d + -- a b c d + a b c )x
--R 4 2
--R +
--R 4 5 2 28 3 6 2 2 7 3 7 4 6 3 7 2 4 7
--R (7a c d + -- a b c d + 2a b c )x + (- a c d + 2a b c )x + a c x
--R 3 2
--R Type: Polynomial(Fraction(Integer))
--E 623
--S 624 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 4 8 4 1 3 9 3 1 2 2 10 2 1 3 11 1 4 12
--R - - a c d + -- a b c d - -- a b c d + --- a b c d - ---- b c
--R 8 18 60 330 3960
--R (4) -------------------------------------------------------------------
--R 5
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 624
--S 625 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 625
)clear all
--S 626 of 2952
t0000:=(a+b*x)^3*(c+d*x)^7
--R
--R
--R (1)
--R 3 7 10 2 7 3 6 9 2 7 2 6 3 2 5 8
--R b d x + (3a b d + 7b c d )x + (3a b d + 21a b c d + 21b c d )x
--R +
--R 3 7 2 6 2 2 5 3 3 4 7
--R (a d + 21a b c d + 63a b c d + 35b c d )x
--R +
--R 3 6 2 2 5 2 3 4 3 4 3 6
--R (7a c d + 63a b c d + 105a b c d + 35b c d )x
--R +
--R 3 2 5 2 3 4 2 4 3 3 5 2 5
--R (21a c d + 105a b c d + 105a b c d + 21b c d )x
--R +
--R 3 3 4 2 4 3 2 5 2 3 6 4
--R (35a c d + 105a b c d + 63a b c d + 7b c d)x
--R +
--R 3 4 3 2 5 2 2 6 3 7 3
--R (35a c d + 63a b c d + 21a b c d + b c )x
--R +
--R 3 5 2 2 6 2 7 2 3 6 2 7 3 7
--R (21a c d + 21a b c d + 3a b c )x + (7a c d + 3a b c )x + a c
--R Type: Polynomial(Integer)
--E 626
--S 627 of 2952
r0000:=-1/8*(b*c-a*d)^3*(c+d*x)^8/d^4+1/3*b*(b*c-a*d)^2*(c+d*x)^9/d^4-_
3/10*b^2*(b*c-a*d)*(c+d*x)^10/d^4+1/11*b^3*(c+d*x)^11/d^4
--R
--R
--R (2)
--R 1 3 11 11 3 2 11 7 3 10 10
--R -- b d x + (-- a b d + -- b c d )x
--R 11 10 10
--R +
--R 1 2 11 7 2 10 7 3 2 9 9
--R (- a b d + - a b c d + - b c d )x
--R 3 3 3
--R +
--R 1 3 11 21 2 10 63 2 2 9 35 3 3 8 8
--R (- a d + -- a b c d + -- a b c d + -- b c d )x
--R 8 8 8 8
--R +
--R 3 10 2 2 9 2 3 8 3 4 7 7
--R (a c d + 9a b c d + 15a b c d + 5b c d )x
--R +
--R 7 3 2 9 35 2 3 8 35 2 4 7 7 3 5 6 6
--R (- a c d + -- a b c d + -- a b c d + - b c d )x
--R 2 2 2 2
--R +
--R 3 3 8 2 4 7 63 2 5 6 7 3 6 5 5
--R (7a c d + 21a b c d + -- a b c d + - b c d )x
--R 5 5
--R +
--R 35 3 4 7 63 2 5 6 21 2 6 5 1 3 7 4 4
--R (-- a c d + -- a b c d + -- a b c d + - b c d )x
--R 4 4 4 4
--R +
--R 3 5 6 2 6 5 2 7 4 3 7 3 6 5 3 2 7 4 2 3 7 4
--R (7a c d + 7a b c d + a b c d )x + (- a c d + - a b c d )x + a c d x
--R 2 2
--R +
--R 1 3 8 3 1 2 9 2 1 2 10 1 3 11
--R - a c d - -- a b c d + --- a b c d - ---- b c
--R 8 24 120 1320
--R /
--R 4
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 627
--S 628 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 3 7 11 3 2 7 7 3 6 10
--R -- b d x + (-- a b d + -- b c d )x
--R 11 10 10
--R +
--R 1 2 7 7 2 6 7 3 2 5 9
--R (- a b d + - a b c d + - b c d )x
--R 3 3 3
--R +
--R 1 3 7 21 2 6 63 2 2 5 35 3 3 4 8
--R (- a d + -- a b c d + -- a b c d + -- b c d )x
--R 8 8 8 8
--R +
--R 3 6 2 2 5 2 3 4 3 4 3 7
--R (a c d + 9a b c d + 15a b c d + 5b c d )x
--R +
--R 7 3 2 5 35 2 3 4 35 2 4 3 7 3 5 2 6
--R (- a c d + -- a b c d + -- a b c d + - b c d )x
--R 2 2 2 2
--R +
--R 3 3 4 2 4 3 63 2 5 2 7 3 6 5
--R (7a c d + 21a b c d + -- a b c d + - b c d)x
--R 5 5
--R +
--R 35 3 4 3 63 2 5 2 21 2 6 1 3 7 4
--R (-- a c d + -- a b c d + -- a b c d + - b c )x
--R 4 4 4 4
--R +
--R 3 5 2 2 6 2 7 3 7 3 6 3 2 7 2 3 7
--R (7a c d + 7a b c d + a b c )x + (- a c d + - a b c )x + a c x
--R 2 2
--R Type: Polynomial(Fraction(Integer))
--E 628
--S 629 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 3 8 3 1 2 9 2 1 2 10 1 3 11
--R - - a c d + -- a b c d - --- a b c d + ---- b c
--R 8 24 120 1320
--R (4) ----------------------------------------------------
--R 4
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 629
--S 630 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 630
)clear all
--S 631 of 2952
t0000:=(a+b*x)^2*(c+d*x)^7
--R
--R
--R (1)
--R 2 7 9 7 2 6 8 2 7 6 2 2 5 7
--R b d x + (2a b d + 7b c d )x + (a d + 14a b c d + 21b c d )x
--R +
--R 2 6 2 5 2 3 4 6 2 2 5 3 4 2 4 3 5
--R (7a c d + 42a b c d + 35b c d )x + (21a c d + 70a b c d + 35b c d )x
--R +
--R 2 3 4 4 3 2 5 2 4 2 4 3 5 2 2 6 3
--R (35a c d + 70a b c d + 21b c d )x + (35a c d + 42a b c d + 7b c d)x
--R +
--R 2 5 2 6 2 7 2 2 6 7 2 7
--R (21a c d + 14a b c d + b c )x + (7a c d + 2a b c )x + a c
--R Type: Polynomial(Integer)
--E 631
--S 632 of 2952
r0000:=1/8*(b*c-a*d)^2*(c+d*x)^8/d^3-2/9*b*(b*c-a*d)*(c+d*x)^9/d^3+_
1/10*b^2*(c+d*x)^10/d^3
--R
--R
--R (2)
--R 1 2 10 10 2 10 7 2 9 9
--R -- b d x + (- a b d + - b c d )x
--R 10 9 9
--R +
--R 1 2 10 7 9 21 2 2 8 8 2 9 2 8 2 3 7 7
--R (- a d + - a b c d + -- b c d )x + (a c d + 6a b c d + 5b c d )x
--R 8 4 8
--R +
--R 7 2 2 8 35 3 7 35 2 4 6 6
--R (- a c d + -- a b c d + -- b c d )x
--R 2 3 6
--R +
--R 2 3 7 4 6 21 2 5 5 5
--R (7a c d + 14a b c d + -- b c d )x
--R 5
--R +
--R 35 2 4 6 21 5 5 7 2 6 4 4
--R (-- a c d + -- a b c d + - b c d )x
--R 4 2 4
--R +
--R 2 5 5 14 6 4 1 2 7 3 3 7 2 6 4 7 3 2 2 7 3
--R (7a c d + -- a b c d + - b c d )x + (- a c d + a b c d )x + a c d x
--R 3 3 2
--R +
--R 1 2 8 2 1 9 1 2 10
--R - a c d - -- a b c d + --- b c
--R 8 36 360
--R /
--R 3
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 632
--S 633 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 2 7 10 2 7 7 2 6 9 1 2 7 7 6 21 2 2 5 8
--R -- b d x + (- a b d + - b c d )x + (- a d + - a b c d + -- b c d )x
--R 10 9 9 8 4 8
--R +
--R 2 6 2 5 2 3 4 7 7 2 2 5 35 3 4 35 2 4 3 6
--R (a c d + 6a b c d + 5b c d )x + (- a c d + -- a b c d + -- b c d )x
--R 2 3 6
--R +
--R 2 3 4 4 3 21 2 5 2 5
--R (7a c d + 14a b c d + -- b c d )x
--R 5
--R +
--R 35 2 4 3 21 5 2 7 2 6 4 2 5 2 14 6 1 2 7 3
--R (-- a c d + -- a b c d + - b c d)x + (7a c d + -- a b c d + - b c )x
--R 4 2 4 3 3
--R +
--R 7 2 6 7 2 2 7
--R (- a c d + a b c )x + a c x
--R 2
--R Type: Polynomial(Fraction(Integer))
--E 633
--S 634 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 2 8 2 1 9 1 2 10
--R - - a c d + -- a b c d - --- b c
--R 8 36 360
--R (4) -----------------------------------
--R 3
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 634
--S 635 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 635
)clear all
--S 636 of 2952
t0000:=(a+b*x)*(c+d*x)^7
--R
--R
--R (1)
--R 7 8 7 6 7 6 2 5 6
--R b d x + (a d + 7b c d )x + (7a c d + 21b c d )x
--R +
--R 2 5 3 4 5 3 4 4 3 4 4 3 5 2 3
--R (21a c d + 35b c d )x + (35a c d + 35b c d )x + (35a c d + 21b c d )x
--R +
--R 5 2 6 2 6 7 7
--R (21a c d + 7b c d)x + (7a c d + b c )x + a c
--R Type: Polynomial(Integer)
--E 636
--S 637 of 2952
r0000:=-1/8*(b*c-a*d)*(c+d*x)^8/d^2+1/9*b*(c+d*x)^9/d^2
--R
--R
--R (2)
--R 1 9 9 1 9 7 8 8 8 2 7 7
--R - b d x + (- a d + - b c d )x + (a c d + 3b c d )x
--R 9 8 8
--R +
--R 7 2 7 35 3 6 6 3 6 4 5 5
--R (- a c d + -- b c d )x + (7a c d + 7b c d )x
--R 2 6
--R +
--R 35 4 5 21 5 4 4 5 4 7 6 3 3
--R (-- a c d + -- b c d )x + (7a c d + - b c d )x
--R 4 4 3
--R +
--R 7 6 3 1 7 2 2 7 2 1 8 1 9
--R (- a c d + - b c d )x + a c d x + - a c d - -- b c
--R 2 2 8 72
--R /
--R 2
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 637
--S 638 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 7 9 1 7 7 6 8 6 2 5 7
--R - b d x + (- a d + - b c d )x + (a c d + 3b c d )x
--R 9 8 8
--R +
--R 7 2 5 35 3 4 6 3 4 4 3 5
--R (- a c d + -- b c d )x + (7a c d + 7b c d )x
--R 2 6
--R +
--R 35 4 3 21 5 2 4 5 2 7 6 3 7 6 1 7 2
--R (-- a c d + -- b c d )x + (7a c d + - b c d)x + (- a c d + - b c )x
--R 4 4 3 2 2
--R +
--R 7
--R a c x
--R Type: Polynomial(Fraction(Integer))
--E 638
--S 639 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 8 1 9
--R - - a c d + -- b c
--R 8 72
--R (4) -------------------
--R 2
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 639
--S 640 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 640
)clear all
--S 641 of 2952
t0000:=(c+d*x)^7
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R Type: Polynomial(Integer)
--E 641
--S 642 of 2952
r0000:=1/8*(c+d*x)^8/d
--R
--R
--R (2)
--R 1 8 8 7 7 7 2 6 6 3 5 5 35 4 4 4 5 3 3 7 6 2 2
--R - d x + c d x + - c d x + 7c d x + -- c d x + 7c d x + - c d x
--R 8 2 4 2
--R +
--R 7 1 8
--R c d x + - c
--R 8
--R /
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 642
--S 643 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 7 8 6 7 7 2 5 6 3 4 5 35 4 3 4 5 2 3 7 6 2 7
--R - d x + c d x + - c d x + 7c d x + -- c d x + 7c d x + - c d x + c x
--R 8 2 4 2
--R Type: Polynomial(Fraction(Integer))
--E 643
--S 644 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 8
--R - c
--R 8
--R (4) - ----
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 644
--S 645 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 645
)clear all
--S 646 of 2952
t0000:=(c+d*x)^7/(a+b*x)
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R ------------------------------------------------------------------------
--R b x + a
--R Type: Fraction(Polynomial(Integer))
--E 646
--S 647 of 2952
r0000:=d*(b*c-a*d)^6*x/b^7+1/2*(b*c-a*d)^5*(c+d*x)^2/b^6+_
1/3*(b*c-a*d)^4*(c+d*x)^3/b^5+1/4*(b*c-a*d)^3*(c+d*x)^4/b^4+_
1/5*(b*c-a*d)^2*(c+d*x)^5/b^3+1/6*(b*c-a*d)*(c+d*x)^6/b^2+_
1/7*(c+d*x)^7/b+(b*c-a*d)^7*log(a+b*x)/b^8
--R
--R
--R (2)
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 420a d + 2940a b c d - 8820a b c d + 14700a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 14700a b c d + 8820a b c d - 2940a b c d + 420b c
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 60b d x + (- 70a b d + 490b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (84a b d - 588a b c d + 1764b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 105a b d + 735a b c d - 2205a b c d + 3675b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (140a b d - 980a b c d + 2940a b c d - 4900a b c d + 4900b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 210a b d + 1470a b c d - 4410a b c d + 7350a b c d
--R +
--R 6 4 3 7 5 2
--R - 7350a b c d + 4410b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R 420a b d - 2940a b c d + 8820a b c d - 14700a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R 14700a b c d - 8820a b c d + 2940b c d
--R *
--R x
--R +
--R 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2 6 6
--R - 210a b c d + 1190a b c d - 2765a b c d + 3339a b c d - 2163a b c d
--R +
--R 7 7
--R 669b c
--R /
--R 8
--R 420b
--R Type: Expression(Integer)
--E 647
--S 648 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 420a d + 2940a b c d - 8820a b c d + 14700a b c d
--R +
--R 3 4 4 3 2 5 5 2 6 6 7 7
--R - 14700a b c d + 8820a b c d - 2940a b c d + 420b c
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 60b d x + (- 70a b d + 490b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (84a b d - 588a b c d + 1764b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 105a b d + 735a b c d - 2205a b c d + 3675b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (140a b d - 980a b c d + 2940a b c d - 4900a b c d + 4900b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 210a b d + 1470a b c d - 4410a b c d + 7350a b c d
--R +
--R 6 4 3 7 5 2
--R - 7350a b c d + 4410b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R 420a b d - 2940a b c d + 8820a b c d - 14700a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R 14700a b c d - 8820a b c d + 2940b c d
--R *
--R x
--R /
--R 8
--R 420b
--R Type: Union(Expression(Integer),...)
--E 648
--S 649 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 5 2 5 4 3 4 3 2 4 3 2 3 5 2 4 6
--R 210a c d - 1190a b c d + 2765a b c d - 3339a b c d + 2163a b c d
--R +
--R 5 7
--R - 669b c
--R /
--R 6
--R 420b
--R Type: Expression(Integer)
--E 649
--S 650 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 650
)clear all
--S 651 of 2952
t0000:=(c+d*x)^7/(a+b*x)^2
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R ------------------------------------------------------------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 651
--S 652 of 2952
r0000:=21*d^2*(b*c-a*d)^5*x/b^7-(b*c-a*d)^7/(b^8*(a+b*x))+_
35/2*d^3*(b*c-a*d)^4*(a+b*x)^2/b^8+35/3*d^4*(b*c-a*d)^3*(a+b*x)^3/b^8+_
21/4*d^5*(b*c-a*d)^2*(a+b*x)^4/b^8+7/5*d^6*(b*c-a*d)*(a+b*x)^5/b^8+_
1/6*d^7*(a+b*x)^6/b^8+7*d*(b*c-a*d)^6*log(a+b*x)/b^8
--R
--R
--R (2)
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R 420a b d - 2520a b c d + 6300a b c d - 8400a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R 6300a b c d - 2520a b c d + 420b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 420a d - 2520a b c d + 6300a b c d - 8400a b c d + 6300a b c d
--R +
--R 2 5 5 2 6 6
--R - 2520a b c d + 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 10b d x + (- 14a b d + 84b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (21a b d - 126a b c d + 315b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 35a b d + 210a b c d - 525a b c d + 700b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (70a b d - 420a b c d + 1050a b c d - 1400a b c d + 1050b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 210a b d + 1260a b c d - 3150a b c d + 4200a b c d
--R +
--R 6 4 3 7 5 2
--R - 3150a b c d + 1260b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 231a b d - 546a b c d - 525a b c d + 2800a b c d - 3150a b c d
--R +
--R 6 5 2
--R 1260a b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 651a d - 3066a b c d + 5775a b c d - 5600a b c d + 3150a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 1260a b c d + 420a b c d - 60b c
--R /
--R 9 8
--R 60b x + 60a b
--R Type: Expression(Integer)
--E 652
--S 653 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R 420a b d - 2520a b c d + 6300a b c d - 8400a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R 6300a b c d - 2520a b c d + 420b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 420a d - 2520a b c d + 6300a b c d - 8400a b c d + 6300a b c d
--R +
--R 2 5 5 2 6 6
--R - 2520a b c d + 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 10b d x + (- 14a b d + 84b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (21a b d - 126a b c d + 315b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 35a b d + 210a b c d - 525a b c d + 700b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (70a b d - 420a b c d + 1050a b c d - 1400a b c d + 1050b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 210a b d + 1260a b c d - 3150a b c d + 4200a b c d
--R +
--R 6 4 3 7 5 2
--R - 3150a b c d + 1260b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R - 360a b d + 2100a b c d - 5040a b c d + 6300a b c d
--R +
--R 2 5 4 3 6 5 2
--R - 4200a b c d + 1260a b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 60a d - 420a b c d + 1260a b c d - 2100a b c d + 2100a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 1260a b c d + 420a b c d - 60b c
--R /
--R 9 8
--R 60b x + 60a b
--R Type: Union(Expression(Integer),...)
--E 653
--S 654 of 2952
m0000:=a0000 - r0000
--R
--R
--R 6 7 5 6 4 2 2 5 3 3 3 4 2 4 4 3
--R - 591a d + 2646a b c d - 4515a b c d + 3500a b c d - 1050a b c d
--R (4) ---------------------------------------------------------------------
--R 8
--R 60b
--R Type: Expression(Integer)
--E 654
--S 655 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 655
)clear all
--S 656 of 2952
t0000:=(c+d*x)^7/(a+b*x)^3
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R ------------------------------------------------------------------------
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 656
--S 657 of 2952
r0000:=35*d^3*(b*c-a*d)^4*x/b^7-1/2*(b*c-a*d)^7/(b^8*(a+b*x)^2)-_
7*d*(b*c-a*d)^6/(b^8*(a+b*x))+35/2*d^4*(b*c-a*d)^3*(a+b*x)^2/b^8+_
7*d^5*(b*c-a*d)^2*(a+b*x)^3/b^8+7/4*d^6*(b*c-a*d)*(a+b*x)^4/b^8+_
1/5*d^7*(a+b*x)^5/b^8+21*d^2*(b*c-a*d)^5*log(a+b*x)/b^8
--R
--R
--R (2)
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 420a b d + 2100a b c d - 4200a b c d + 4200a b c d
--R +
--R 6 4 3 7 5 2
--R - 2100a b c d + 420b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R - 840a b d + 4200a b c d - 8400a b c d + 8400a b c d
--R +
--R 2 5 4 3 6 5 2
--R - 4200a b c d + 840a b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - 420a d + 2100a b c d - 4200a b c d + 4200a b c d - 2100a b c d
--R +
--R 2 5 5 2
--R 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 4b d x + (- 7a b d + 35b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (14a b d - 70a b c d + 140b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 35a b d + 175a b c d - 350a b c d + 350b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (140a b d - 700a b c d + 1400a b c d - 1400a b c d + 700b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3 2
--R (259a b d - 1575a b c d + 3500a b c d - 3500a b c d + 1400a b c d )x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R - 322a b d + 1050a b c d - 1400a b c d + 1400a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R - 1400a b c d + 840a b c d - 140b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - 371a d + 1575a b c d - 2800a b c d + 2800a b c d - 1750a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R 630a b c d - 70a b c d - 10b c
--R /
--R 10 2 9 2 8
--R 20b x + 40a b x + 20a b
--R Type: Expression(Integer)
--E 657
--S 658 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 420a b d + 2100a b c d - 4200a b c d + 4200a b c d
--R +
--R 6 4 3 7 5 2
--R - 2100a b c d + 420b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R - 840a b d + 4200a b c d - 8400a b c d + 8400a b c d
--R +
--R 2 5 4 3 6 5 2
--R - 4200a b c d + 840a b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - 420a d + 2100a b c d - 4200a b c d + 4200a b c d - 2100a b c d
--R +
--R 2 5 5 2
--R 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 4b d x + (- 7a b d + 35b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (14a b d - 70a b c d + 140b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 35a b d + 175a b c d - 350a b c d + 350b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (140a b d - 700a b c d + 1400a b c d - 1400a b c d + 700b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3 2
--R (500a b d - 2380a b c d + 4410a b c d - 3850a b c d + 1400a b c d )x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 160a b d - 560a b c d + 420a b c d + 700a b c d - 1400a b c d
--R +
--R 6 5 2 7 6
--R 840a b c d - 140b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - 130a d + 770a b c d - 1890a b c d + 2450a b c d - 1750a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R 630a b c d - 70a b c d - 10b c
--R /
--R 10 2 9 2 8
--R 20b x + 40a b x + 20a b
--R Type: Union(Expression(Integer),...)
--E 658
--S 659 of 2952
m0000:=a0000 - r0000
--R
--R
--R 5 7 4 6 3 2 2 5 2 3 3 4
--R 241a d - 805a b c d + 910a b c d - 350a b c d
--R (4) -------------------------------------------------
--R 8
--R 20b
--R Type: Expression(Integer)
--E 659
--S 660 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 660
)clear all
--S 661 of 2952
t0000:=(c+d*x)^7/(a+b*x)^4
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R ------------------------------------------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 661
--S 662 of 2952
r0000:=35*d^4*(b*c-a*d)^3*x/b^7-1/3*(b*c-a*d)^7/(b^8*(a+b*x)^3)-_
7/2*d*(b*c-a*d)^6/(b^8*(a+b*x)^2)-21*d^2*(b*c-a*d)^5/(b^8*(a+b*x))+_
21/2*d^5*(b*c-a*d)^2*(a+b*x)^2/b^8+7/3*d^6*(b*c-a*d)*(a+b*x)^3/b^8+_
1/4*d^7*(a+b*x)^4/b^8+35*d^3*(b*c-a*d)^4*log(a+b*x)/b^8
--R
--R
--R (2)
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R 420a b d - 1680a b c d + 2520a b c d - 1680a b c d
--R +
--R 7 4 3
--R 420b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R 1260a b d - 5040a b c d + 7560a b c d - 5040a b c d
--R +
--R 6 4 3
--R 1260a b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R 1260a b d - 5040a b c d + 7560a b c d - 5040a b c d
--R +
--R 2 5 4 3
--R 1260a b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 420a d - 1680a b c d + 2520a b c d - 1680a b c d + 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 3b d x + (- 7a b d + 28b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (21a b d - 84a b c d + 126b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 105a b d + 420a b c d - 630a b c d + 420b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 3
--R (- 455a b d + 1820a b c d - 2520a b c d + 1260a b c d )x
--R +
--R 5 2 7 4 3 6 2 5 3 4 6 4 3 7 5 2 2
--R (- 105a b d + 420a b c d - 1260a b c d + 1260a b c d - 252b c d )x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 525a b d - 2100a b c d + 3780a b c d - 3780a b c d + 1890a b c d
--R +
--R 6 5 2 7 6
--R - 252a b c d - 42b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 315a d - 1260a b c d + 2100a b c d - 1820a b c d + 770a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 84a b c d - 14a b c d - 4b c
--R /
--R 11 3 10 2 2 9 3 8
--R 12b x + 36a b x + 36a b x + 12a b
--R Type: Expression(Integer)
--E 662
--S 663 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R 420a b d - 1680a b c d + 2520a b c d - 1680a b c d
--R +
--R 7 4 3
--R 420b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R 1260a b d - 5040a b c d + 7560a b c d - 5040a b c d
--R +
--R 6 4 3
--R 1260a b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R 1260a b d - 5040a b c d + 7560a b c d - 5040a b c d
--R +
--R 2 5 4 3
--R 1260a b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 420a d - 1680a b c d + 2520a b c d - 1680a b c d + 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 3b d x + (- 7a b d + 28b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (21a b d - 84a b c d + 126b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 105a b d + 420a b c d - 630a b c d + 420b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 3
--R (- 556a b d + 2044a b c d - 2646a b c d + 1260a b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 408a b d + 1092a b c d - 378a b c d - 1260a b c d
--R +
--R 6 4 3 7 5 2
--R 1260a b c d - 252b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 222a b d - 1428a b c d + 3402a b c d - 3780a b c d + 1890a b c d
--R +
--R 6 5 2 7 6
--R - 252a b c d - 42b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 214a d - 1036a b c d + 1974a b c d - 1820a b c d + 770a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 84a b c d - 14a b c d - 4b c
--R /
--R 11 3 10 2 2 9 3 8
--R 12b x + 36a b x + 36a b x + 12a b
--R Type: Union(Expression(Integer),...)
--E 663
--S 664 of 2952
m0000:=a0000 - r0000
--R
--R
--R 4 7 3 6 2 2 2 5
--R - 101a d + 224a b c d - 126a b c d
--R (4) -------------------------------------
--R 8
--R 12b
--R Type: Expression(Integer)
--E 664
--S 665 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 665
)clear all
--S 666 of 2952
t0000:=(c+d*x)^7/(a+b*x)^5
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R ------------------------------------------------------------------------
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R b x + 5a b x + 10a b x + 10a b x + 5a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 666
--S 667 of 2952
r0000:=21*d^5*(b*c-a*d)^2*x/b^7-1/4*(b*c-a*d)^7/(b^8*(a+b*x)^4)-_
7/3*d*(b*c-a*d)^6/(b^8*(a+b*x)^3)-21/2*d^2*(b*c-a*d)^5/(b^8*(a+b*x)^2)-_
35*d^3*(b*c-a*d)^4/(b^8*(a+b*x))+7/2*d^6*(b*c-a*d)*(a+b*x)^2/b^8+_
1/3*d^7*(a+b*x)^3/b^8+35*d^4*(b*c-a*d)^3*log(a+b*x)/b^8
--R
--R
--R (2)
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 420a b d + 1260a b c d - 1260a b c d + 420b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 3
--R (- 1680a b d + 5040a b c d - 5040a b c d + 1680a b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 2
--R (- 2520a b d + 7560a b c d - 7560a b c d + 2520a b c d )x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R (- 1680a b d + 5040a b c d - 5040a b c d + 1680a b c d )x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 420a d + 1260a b c d - 1260a b c d + 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 4b d x + (- 14a b d + 42b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (84a b d - 252a b c d + 252b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 4
--R (518a b d - 1386a b c d + 1008a b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (392a b d - 504a b c d - 1008a b c d + 1680a b c d - 420b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 672a b d + 3024a b c d - 5292a b c d + 3780a b c d
--R +
--R 6 4 3 7 5 2
--R - 630a b c d - 126b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R - 1008a b d + 3696a b c d - 5208a b c d + 3080a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R - 420a b c d - 84a b c d - 28b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - 357a d + 1239a b c d - 1617a b c d + 875a b c d - 105a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 21a b c d - 7a b c d - 3b c
--R /
--R 12 4 11 3 2 10 2 3 9 4 8
--R 12b x + 48a b x + 72a b x + 48a b x + 12a b
--R Type: Expression(Integer)
--E 667
--S 668 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 420a b d + 1260a b c d - 1260a b c d + 420b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 3
--R (- 1680a b d + 5040a b c d - 5040a b c d + 1680a b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 2
--R (- 2520a b d + 7560a b c d - 7560a b c d + 2520a b c d )x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R (- 1680a b d + 5040a b c d - 5040a b c d + 1680a b c d )x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4
--R - 420a d + 1260a b c d - 1260a b c d + 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6
--R 4b d x + (- 14a b d + 42b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (84a b d - 252a b c d + 252b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 4
--R (556a b d - 1428a b c d + 1008a b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (544a b d - 672a b c d - 1008a b c d + 1680a b c d - 420b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 444a b d + 2772a b c d - 5292a b c d + 3780a b c d
--R +
--R 6 4 3 7 5 2
--R - 630a b c d - 126b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R - 856a b d + 3528a b c d - 5208a b c d + 3080a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R - 420a b c d - 84a b c d - 28b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - 319a d + 1197a b c d - 1617a b c d + 875a b c d - 105a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 21a b c d - 7a b c d - 3b c
--R /
--R 12 4 11 3 2 10 2 3 9 4 8
--R 12b x + 48a b x + 72a b x + 48a b x + 12a b
--R Type: Union(Expression(Integer),...)
--E 668
--S 669 of 2952
m0000:=a0000 - r0000
--R
--R
--R 3 7 2 6
--R 19a d - 21a b c d
--R (4) -------------------
--R 8
--R 6b
--R Type: Expression(Integer)
--E 669
--S 670 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 670
)clear all
--S 671 of 2952
t0000:=(c+d*x)^7/(a+b*x)^6
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R ------------------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 671
--S 672 of 2952
r0000:=d^6*(7*b*c-6*a*d)*x/b^7+1/2*d^7*x^2/b^6-_
1/5*(b*c-a*d)^7/(b^8*(a+b*x)^5)-7/4*d*(b*c-a*d)^6/(b^8*(a+b*x)^4)-_
7*d^2*(b*c-a*d)^5/(b^8*(a+b*x)^3)-_
35/2*d^3*(b*c-a*d)^4/(b^8*(a+b*x)^2)-_
35*d^4*(b*c-a*d)^3/(b^8*(a+b*x))+21*d^5*(b*c-a*d)^2*log(a+b*x)/b^8
--R
--R
--R (2)
--R 2 5 7 6 6 7 2 5 5
--R (420a b d - 840a b c d + 420b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 4
--R (2100a b d - 4200a b c d + 2100a b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 3
--R (4200a b d - 8400a b c d + 4200a b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2
--R (4200a b d - 8400a b c d + 4200a b c d )x
--R +
--R 6 7 5 2 6 4 3 2 5 7 7 6 6
--R (2100a b d - 4200a b c d + 2100a b c d )x + 420a d - 840a b c d
--R +
--R 5 2 2 5
--R 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6 2 5 7 6 6 5
--R 10b d x + (- 70a b d + 140b c d )x + (- 500a b d + 700a b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 400a b d - 700a b c d + 2100a b c d - 700b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (1300a b d - 5600a b c d + 6300a b c d - 1400a b c d - 350b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R 2700a b d - 8400a b c d + 7700a b c d - 1400a b c d - 350a b c d
--R +
--R 7 5 2
--R - 140b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 1875a b d - 5250a b c d + 4375a b c d - 700a b c d - 175a b c d
--R +
--R 6 5 2 7 6
--R - 70a b c d - 35b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 459a d - 1218a b c d + 959a b c d - 140a b c d - 35a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 14a b c d - 7a b c d - 4b c
--R /
--R 13 5 12 4 2 11 3 3 10 2 4 9 5 8
--R 20b x + 100a b x + 200a b x + 200a b x + 100a b x + 20a b
--R Type: Expression(Integer)
--E 672
--S 673 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 5 7 6 6 7 2 5 5
--R (420a b d - 840a b c d + 420b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 4
--R (2100a b d - 4200a b c d + 2100a b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 3
--R (4200a b d - 8400a b c d + 4200a b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2
--R (4200a b d - 8400a b c d + 4200a b c d )x
--R +
--R 6 7 5 2 6 4 3 2 5 7 7 6 6
--R (2100a b d - 4200a b c d + 2100a b c d )x + 420a d - 840a b c d
--R +
--R 5 2 2 5
--R 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 7 6 6 2 5 7 6 6 5
--R 10b d x + (- 70a b d + 140b c d )x + (- 500a b d + 700a b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 400a b d - 700a b c d + 2100a b c d - 700b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (1300a b d - 5600a b c d + 6300a b c d - 1400a b c d - 350b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R 2700a b d - 8400a b c d + 7700a b c d - 1400a b c d - 350a b c d
--R +
--R 7 5 2
--R - 140b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 1875a b d - 5250a b c d + 4375a b c d - 700a b c d - 175a b c d
--R +
--R 6 5 2 7 6
--R - 70a b c d - 35b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 459a d - 1218a b c d + 959a b c d - 140a b c d - 35a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 14a b c d - 7a b c d - 4b c
--R /
--R 13 5 12 4 2 11 3 3 10 2 4 9 5 8
--R 20b x + 100a b x + 200a b x + 200a b x + 100a b x + 20a b
--R Type: Union(Expression(Integer),...)
--E 673
--S 674 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 674
--S 675 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 675
)clear all
--S 676 of 2952
t0000:=(c+d*x)^7/(a+b*x)^7
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R ------------------------------------------------------------------------
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R b x + 7a b x + 21a b x + 35a b x + 35a b x + 21a b x + 7a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 676
--S 677 of 2952
r0000:=d^7*x/b^7-1/6*(b*c-a*d)^7/(b^8*(a+b*x)^6)-_
7/5*d*(b*c-a*d)^6/(b^8*(a+b*x)^5)-_
21/4*d^2*(b*c-a*d)^5/(b^8*(a+b*x)^4)-_
35/3*d^3*(b*c-a*d)^4/(b^8*(a+b*x)^3)-_
35/2*d^4*(b*c-a*d)^3/(b^8*(a+b*x)^2)-_
21*d^5*(b*c-a*d)^2/(b^8*(a+b*x))+7*d^6*(b*c-a*d)*log(a+b*x)/b^8
--R
--R
--R (2)
--R 6 7 7 6 6 2 5 7 6 6 5
--R (- 420a b d + 420b c d )x + (- 2520a b d + 2520a b c d )x
--R +
--R 3 4 7 2 5 6 4 4 3 7 3 4 6 3
--R (- 6300a b d + 6300a b c d )x + (- 8400a b d + 8400a b c d )x
--R +
--R 5 2 7 4 3 6 2 6 7 5 2 6
--R (- 6300a b d + 6300a b c d )x + (- 2520a b d + 2520a b c d )x
--R +
--R 7 7 6 6
--R - 420a d + 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 6 2 5 7 6 6 7 2 5 5
--R 60b d x + 360a b d x + (- 360a b d + 2520a b c d - 1260b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 4050a b d + 9450a b c d - 3150a b c d - 1050b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R - 8200a b d + 15400a b c d - 4200a b c d - 1400a b c d
--R +
--R 7 4 3
--R - 700b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 7725a b d + 13125a b c d - 3150a b c d - 1050a b c d
--R +
--R 6 4 3 7 5 2
--R - 525a b c d - 315b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R - 3594a b d + 5754a b c d - 1260a b c d - 420a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R - 210a b c d - 126a b c d - 84b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - 669a d + 1029a b c d - 210a b c d - 70a b c d - 35a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 21a b c d - 14a b c d - 10b c
--R /
--R 14 6 13 5 2 12 4 3 11 3 4 10 2 5 9
--R 60b x + 360a b x + 900a b x + 1200a b x + 900a b x + 360a b x
--R +
--R 6 8
--R 60a b
--R Type: Expression(Integer)
--E 677
--S 678 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 6 7 7 6 6 2 5 7 6 6 5
--R (- 420a b d + 420b c d )x + (- 2520a b d + 2520a b c d )x
--R +
--R 3 4 7 2 5 6 4 4 3 7 3 4 6 3
--R (- 6300a b d + 6300a b c d )x + (- 8400a b d + 8400a b c d )x
--R +
--R 5 2 7 4 3 6 2 6 7 5 2 6
--R (- 6300a b d + 6300a b c d )x + (- 2520a b d + 2520a b c d )x
--R +
--R 7 7 6 6
--R - 420a d + 420a b c d
--R *
--R log(b x + a)
--R +
--R 7 7 7 6 7 6 2 5 7 6 6 7 2 5 5
--R 60b d x + 360a b d x + (- 360a b d + 2520a b c d - 1260b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 4050a b d + 9450a b c d - 3150a b c d - 1050b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R - 8200a b d + 15400a b c d - 4200a b c d - 1400a b c d
--R +
--R 7 4 3
--R - 700b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 7725a b d + 13125a b c d - 3150a b c d - 1050a b c d
--R +
--R 6 4 3 7 5 2
--R - 525a b c d - 315b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4
--R - 3594a b d + 5754a b c d - 1260a b c d - 420a b c d
--R +
--R 2 5 4 3 6 5 2 7 6
--R - 210a b c d - 126a b c d - 84b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R - 669a d + 1029a b c d - 210a b c d - 70a b c d - 35a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 21a b c d - 14a b c d - 10b c
--R /
--R 14 6 13 5 2 12 4 3 11 3 4 10 2 5 9
--R 60b x + 360a b x + 900a b x + 1200a b x + 900a b x + 360a b x
--R +
--R 6 8
--R 60a b
--R Type: Union(Expression(Integer),...)
--E 678
--S 679 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 679
--S 680 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 680
)clear all
--S 681 of 2952
t0000:=(c+d*x)^7/(a+b*x)^8
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 8 8 7 7 2 6 6 3 5 5 4 4 4 5 3 3 6 2 2
--R b x + 8a b x + 28a b x + 56a b x + 70a b x + 56a b x + 28a b x
--R +
--R 7 8
--R 8a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 681
--S 682 of 2952
r0000:=-1/7*(b*c-a*d)^7/(b^8*(a+b*x)^7)-7/6*d*(b*c-a*d)^6/(b^8*(a+b*x)^6)-_
21/5*d^2*(b*c-a*d)^5/(b^8*(a+b*x)^5)-_
35/4*d^3*(b*c-a*d)^4/(b^8*(a+b*x)^4)-_
35/3*d^4*(b*c-a*d)^3/(b^8*(a+b*x)^3)-_
21/2*d^5*(b*c-a*d)^2/(b^8*(a+b*x)^2)-_
7*d^6*(b*c-a*d)/(b^8*(a+b*x))+d^7*log(a+b*x)/b^8
--R
--R
--R (2)
--R 7 7 7 6 7 6 2 5 7 5 3 4 7 4
--R 420b d x + 2940a b d x + 8820a b d x + 14700a b d x
--R +
--R 4 3 7 3 5 2 7 2 6 7 7 7
--R 14700a b d x + 8820a b d x + 2940a b d x + 420a d
--R *
--R log(b x + a)
--R +
--R 6 7 7 6 6 2 5 7 6 6 7 2 5 5
--R (2940a b d - 2940b c d )x + (13230a b d - 8820a b c d - 4410b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (26950a b d - 14700a b c d - 7350a b c d - 4900b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R 30625a b d - 14700a b c d - 7350a b c d - 4900a b c d
--R +
--R 7 4 3
--R - 3675b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R 20139a b d - 8820a b c d - 4410a b c d - 2940a b c d
--R +
--R 6 4 3 7 5 2
--R - 2205a b c d - 1764b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 7203a b d - 2940a b c d - 1470a b c d - 980a b c d - 735a b c d
--R +
--R 6 5 2 7 6
--R - 588a b c d - 490b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 1089a d - 420a b c d - 210a b c d - 140a b c d - 105a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 84a b c d - 70a b c d - 60b c
--R /
--R 15 7 14 6 2 13 5 3 12 4 4 11 3
--R 420b x + 2940a b x + 8820a b x + 14700a b x + 14700a b x
--R +
--R 5 10 2 6 9 7 8
--R 8820a b x + 2940a b x + 420a b
--R Type: Expression(Integer)
--E 682
--S 683 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 6 2 5 7 5 3 4 7 4
--R 420b d x + 2940a b d x + 8820a b d x + 14700a b d x
--R +
--R 4 3 7 3 5 2 7 2 6 7 7 7
--R 14700a b d x + 8820a b d x + 2940a b d x + 420a d
--R *
--R log(b x + a)
--R +
--R 6 7 7 6 6 2 5 7 6 6 7 2 5 5
--R (2940a b d - 2940b c d )x + (13230a b d - 8820a b c d - 4410b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (26950a b d - 14700a b c d - 7350a b c d - 4900b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R 30625a b d - 14700a b c d - 7350a b c d - 4900a b c d
--R +
--R 7 4 3
--R - 3675b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R 20139a b d - 8820a b c d - 4410a b c d - 2940a b c d
--R +
--R 6 4 3 7 5 2
--R - 2205a b c d - 1764b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R 7203a b d - 2940a b c d - 1470a b c d - 980a b c d - 735a b c d
--R +
--R 6 5 2 7 6
--R - 588a b c d - 490b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3
--R 1089a d - 420a b c d - 210a b c d - 140a b c d - 105a b c d
--R +
--R 2 5 5 2 6 6 7 7
--R - 84a b c d - 70a b c d - 60b c
--R /
--R 15 7 14 6 2 13 5 3 12 4 4 11 3
--R 420b x + 2940a b x + 8820a b x + 14700a b x + 14700a b x
--R +
--R 5 10 2 6 9 7 8
--R 8820a b x + 2940a b x + 420a b
--R Type: Union(Expression(Integer),...)
--E 683
--S 684 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 684
--S 685 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 685
)clear all
--S 686 of 2952
t0000:=(c+d*x)^7/(a+b*x)^9
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 9 9 8 8 2 7 7 3 6 6 4 5 5 5 4 4 6 3 3
--R b x + 9a b x + 36a b x + 84a b x + 126a b x + 126a b x + 84a b x
--R +
--R 7 2 2 8 9
--R 36a b x + 9a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 686
--S 687 of 2952
r0000:=-1/8*(c+d*x)^8/((b*c-a*d)*(a+b*x)^8)
--R
--R
--R (2)
--R 1 8 8 7 7 7 2 6 6 3 5 5 35 4 4 4 5 3 3 7 6 2 2
--R - d x + c d x + - c d x + 7c d x + -- c d x + 7c d x + - c d x
--R 8 2 4 2
--R +
--R 7 1 8
--R c d x + - c
--R 8
--R /
--R 8 9 8 2 7 8 7 3 6 2 7 6
--R (a b d - b c)x + (8a b d - 8a b c)x + (28a b d - 28a b c)x
--R +
--R 4 5 3 6 5 5 4 4 5 4 6 3 5 4 3
--R (56a b d - 56a b c)x + (70a b d - 70a b c)x + (56a b d - 56a b c)x
--R +
--R 7 2 6 3 2 8 7 2 9 8
--R (28a b d - 28a b c)x + (8a b d - 8a b c)x + a d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 687
--S 688 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 7 6 6
--R - 8b d x + (- 28a b d - 28b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (- 56a b d - 56a b c d - 56b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 70a b d - 70a b c d - 70a b c d - 70b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (- 56a b d - 56a b c d - 56a b c d - 56a b c d - 56b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R - 28a b d - 28a b c d - 28a b c d - 28a b c d - 28a b c d
--R +
--R 7 5 2
--R - 28b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3 6 5 2
--R - 8a b d - 8a b c d - 8a b c d - 8a b c d - 8a b c d - 8a b c d
--R +
--R 7 6
--R - 8b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2 6 6
--R - a d - a b c d - a b c d - a b c d - a b c d - a b c d - a b c d
--R +
--R 7 7
--R - b c
--R /
--R 16 8 15 7 2 14 6 3 13 5 4 12 4 5 11 3
--R 8b x + 64a b x + 224a b x + 448a b x + 560a b x + 448a b x
--R +
--R 6 10 2 7 9 8 8
--R 224a b x + 64a b x + 8a b
--R Type: Union(Expression(Integer),...)
--E 688
--S 689 of 2952
m0000:=a0000 - r0000
--R
--R
--R 8
--R d
--R (4) - -------------
--R 8 9
--R 8a b d - 8b c
--R Type: Expression(Integer)
--E 689
--S 690 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 690
)clear all
--S 691 of 2952
t0000:=(c+d*x)^7/(a+b*x)^10
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R b x + 10a b x + 45a b x + 120a b x + 210a b x + 252a b x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210a b x + 120a b x + 45a b x + 10a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 691
--S 692 of 2952
r0000:=-1/9*(c+d*x)^8/((b*c-a*d)*(a+b*x)^9)+_
1/72*d*(c+d*x)^8/((b*c-a*d)^2*(a+b*x)^8)
--R
--R
--R (2)
--R 1 9 9 1 9 8 8 1 2 7 7 7 2 7 7 3 6 6
--R -- b d x + - a d x + (a c d - - b c d )x + (- a c d - - b c d )x
--R 72 8 2 2 3
--R +
--R 3 6 21 4 5 5 35 4 5 5 4 4
--R (7a c d - -- b c d )x + (-- a c d - 7b c d )x
--R 4 4
--R +
--R 5 4 35 6 3 3 7 6 3 7 2 2 7 2 7 8
--R (7a c d - -- b c d )x + (- a c d - 3b c d )x + (a c d - - b c d)x
--R 6 2 8
--R +
--R 1 8 1 9
--R - a c d - - b c
--R 8 9
--R /
--R 2 9 2 10 11 2 9 3 8 2 2 9 10 2 8
--R (a b d - 2a b c d + b c )x + (9a b d - 18a b c d + 9a b c )x
--R +
--R 4 7 2 3 8 2 9 2 7
--R (36a b d - 72a b c d + 36a b c )x
--R +
--R 5 6 2 4 7 3 8 2 6
--R (84a b d - 168a b c d + 84a b c )x
--R +
--R 6 5 2 5 6 4 7 2 5
--R (126a b d - 252a b c d + 126a b c )x
--R +
--R 7 4 2 6 5 5 6 2 4
--R (126a b d - 252a b c d + 126a b c )x
--R +
--R 8 3 2 7 4 6 5 2 3
--R (84a b d - 168a b c d + 84a b c )x
--R +
--R 9 2 2 8 3 7 4 2 2 10 2 9 2 8 3 2
--R (36a b d - 72a b c d + 36a b c )x + (9a b d - 18a b c d + 9a b c )x
--R +
--R 11 2 10 9 2 2
--R a d - 2a b c d + a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 692
--S 693 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 7 6 6
--R - 36b d x + (- 84a b d - 168b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (- 126a b d - 252a b c d - 378b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 126a b d - 252a b c d - 378a b c d - 504b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (- 84a b d - 168a b c d - 252a b c d - 336a b c d - 420b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R - 36a b d - 72a b c d - 108a b c d - 144a b c d - 180a b c d
--R +
--R 7 5 2
--R - 216b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R - 9a b d - 18a b c d - 27a b c d - 36a b c d - 45a b c d
--R +
--R 6 5 2 7 6
--R - 54a b c d - 63b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2
--R - a d - 2a b c d - 3a b c d - 4a b c d - 5a b c d - 6a b c d
--R +
--R 6 6 7 7
--R - 7a b c d - 8b c
--R /
--R 17 9 16 8 2 15 7 3 14 6 4 13 5
--R 72b x + 648a b x + 2592a b x + 6048a b x + 9072a b x
--R +
--R 5 12 4 6 11 3 7 10 2 8 9 9 8
--R 9072a b x + 6048a b x + 2592a b x + 648a b x + 72a b
--R Type: Union(Expression(Integer),...)
--E 693
--S 694 of 2952
m0000:=a0000 - r0000
--R
--R
--R 9
--R d
--R (4) - -------------------------------
--R 2 8 2 9 10 2
--R 72a b d - 144a b c d + 72b c
--R Type: Expression(Integer)
--E 694
--S 695 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 695
)clear all
--S 696 of 2952
t0000:=(c+d*x)^7/(a+b*x)^11
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 11 11 10 10 2 9 9 3 8 8 4 7 7 5 6 6
--R b x + 11a b x + 55a b x + 165a b x + 330a b x + 462a b x
--R +
--R 6 5 5 7 4 4 8 3 3 9 2 2 10 11
--R 462a b x + 330a b x + 165a b x + 55a b x + 11a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 696
--S 697 of 2952
r0000:=-1/10*(c+d*x)^8/((b*c-a*d)*(a+b*x)^10)+_
1/45*d*(c+d*x)^8/((b*c-a*d)^2*(a+b*x)^9)-_
1/360*d^2*(c+d*x)^8/((b*c-a*d)^3*(a+b*x)^8)
--R
--R
--R (2)
--R 1 2 10 10 1 10 9 1 2 10 8
--R --- b d x + -- a b d x + - a d x
--R 360 36 8
--R +
--R 2 9 2 8 1 2 3 7 7 7 2 2 8 14 3 7 7 2 4 6 6
--R (a c d - a b c d + - b c d )x + (- a c d - -- a b c d + - b c d )x
--R 3 2 3 4
--R +
--R 2 3 7 21 4 6 21 2 5 5 5
--R (7a c d - -- a b c d + -- b c d )x
--R 2 5
--R +
--R 35 2 4 6 5 5 35 2 6 4 4
--R (-- a c d - 14a b c d + -- b c d )x
--R 4 6
--R +
--R 2 5 5 35 6 4 2 7 3 3
--R (7a c d - -- a b c d + 5b c d )x
--R 3
--R +
--R 7 2 6 4 7 3 21 2 8 2 2 2 7 3 7 8 2 7 2 9
--R (- a c d - 6a b c d + -- b c d )x + (a c d - - a b c d + - b c d)x
--R 2 8 4 9
--R +
--R 1 2 8 2 2 9 1 2 10
--R - a c d - - a b c d + -- b c
--R 8 9 10
--R /
--R 3 10 3 2 11 2 12 2 13 3 10
--R (a b d - 3a b c d + 3a b c d - b c )x
--R +
--R 4 9 3 3 10 2 2 11 2 12 3 9
--R (10a b d - 30a b c d + 30a b c d - 10a b c )x
--R +
--R 5 8 3 4 9 2 3 10 2 2 11 3 8
--R (45a b d - 135a b c d + 135a b c d - 45a b c )x
--R +
--R 6 7 3 5 8 2 4 9 2 3 10 3 7
--R (120a b d - 360a b c d + 360a b c d - 120a b c )x
--R +
--R 7 6 3 6 7 2 5 8 2 4 9 3 6
--R (210a b d - 630a b c d + 630a b c d - 210a b c )x
--R +
--R 8 5 3 7 6 2 6 7 2 5 8 3 5
--R (252a b d - 756a b c d + 756a b c d - 252a b c )x
--R +
--R 9 4 3 8 5 2 7 6 2 6 7 3 4
--R (210a b d - 630a b c d + 630a b c d - 210a b c )x
--R +
--R 10 3 3 9 4 2 8 5 2 7 6 3 3
--R (120a b d - 360a b c d + 360a b c d - 120a b c )x
--R +
--R 11 2 3 10 3 2 9 4 2 8 5 3 2
--R (45a b d - 135a b c d + 135a b c d - 45a b c )x
--R +
--R 12 3 11 2 2 10 3 2 9 4 3 13 3 12 2
--R (10a b d - 30a b c d + 30a b c d - 10a b c )x + a d - 3a b c d
--R +
--R 11 2 2 10 3 3
--R 3a b c d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 697
--S 698 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 7 6 6
--R - 120b d x + (- 210a b d - 630b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (- 252a b d - 756a b c d - 1512b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 210a b d - 630a b c d - 1260a b c d - 2100b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (- 120a b d - 360a b c d - 720a b c d - 1200a b c d - 1800b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R - 45a b d - 135a b c d - 270a b c d - 450a b c d - 675a b c d
--R +
--R 7 5 2
--R - 945b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R - 10a b d - 30a b c d - 60a b c d - 100a b c d - 150a b c d
--R +
--R 6 5 2 7 6
--R - 210a b c d - 280b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2
--R - a d - 3a b c d - 6a b c d - 10a b c d - 15a b c d - 21a b c d
--R +
--R 6 6 7 7
--R - 28a b c d - 36b c
--R /
--R 18 10 17 9 2 16 8 3 15 7 4 14 6
--R 360b x + 3600a b x + 16200a b x + 43200a b x + 75600a b x
--R +
--R 5 13 5 6 12 4 7 11 3 8 10 2 9 9
--R 90720a b x + 75600a b x + 43200a b x + 16200a b x + 3600a b x
--R +
--R 10 8
--R 360a b
--R Type: Union(Expression(Integer),...)
--E 698
--S 699 of 2952
m0000:=a0000 - r0000
--R
--R
--R 10
--R d
--R (4) - --------------------------------------------------
--R 3 8 3 2 9 2 10 2 11 3
--R 360a b d - 1080a b c d + 1080a b c d - 360b c
--R Type: Expression(Integer)
--E 699
--S 700 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 700
)clear all
--S 701 of 2952
t0000:=(c+d*x)^7/(a+b*x)^12
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 12 12 11 11 2 10 10 3 9 9 4 8 8 5 7 7
--R b x + 12a b x + 66a b x + 220a b x + 495a b x + 792a b x
--R +
--R 6 6 6 7 5 5 8 4 4 9 3 3 10 2 2 11 12
--R 924a b x + 792a b x + 495a b x + 220a b x + 66a b x + 12a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 701
--S 702 of 2952
r0000:=-1/11*(c+d*x)^8/((b*c-a*d)*(a+b*x)^11)+_
3/110*d*(c+d*x)^8/((b*c-a*d)^2*(a+b*x)^10)-_
1/165*d^2*(c+d*x)^8/((b*c-a*d)^3*(a+b*x)^9)+_
1/1320*d^3*(c+d*x)^8/((b*c-a*d)^4*(a+b*x)^8)
--R
--R
--R (2)
--R 1 3 11 11 1 2 11 10 1 2 11 9 1 3 11 8
--R ---- b d x + --- a b d x + -- a b d x + - a d x
--R 1320 120 24 8
--R +
--R 3 10 3 2 2 9 2 3 8 1 3 4 7 7
--R (a c d - - a b c d + a b c d - - b c d )x
--R 2 4
--R +
--R 7 3 2 9 2 3 8 21 2 4 7 7 3 5 6 6
--R (- a c d - 7a b c d + -- a b c d - - b c d )x
--R 2 4 5
--R +
--R 3 3 8 63 2 4 7 63 2 5 6 7 3 6 5 5
--R (7a c d - -- a b c d + -- a b c d - - b c d )x
--R 4 5 2
--R +
--R 35 3 4 7 2 5 6 35 2 6 5 3 7 4 4
--R (-- a c d - 21a b c d + -- a b c d - 5b c d )x
--R 4 2
--R +
--R 3 5 6 35 2 6 5 2 7 4 35 3 8 3 3
--R (7a c d - -- a b c d + 15a b c d - -- b c d )x
--R 2 8
--R +
--R 7 3 6 5 2 7 4 63 2 8 3 7 3 9 2 2
--R (- a c d - 9a b c d + -- a b c d - - b c d )x
--R 2 8 3
--R +
--R 3 7 4 21 2 8 3 7 2 9 2 7 3 10 1 3 8 3 1 2 9 2
--R (a c d - -- a b c d + - a b c d - -- b c d)x + - a c d - - a b c d
--R 8 3 10 8 3
--R +
--R 3 2 10 1 3 11
--R -- a b c d - -- b c
--R 10 11
--R /
--R 4 11 4 3 12 3 2 13 2 2 14 3 15 4 11
--R (a b d - 4a b c d + 6a b c d - 4a b c d + b c )x
--R +
--R 5 10 4 4 11 3 3 12 2 2 2 13 3 14 4 10
--R (11a b d - 44a b c d + 66a b c d - 44a b c d + 11a b c )x
--R +
--R 6 9 4 5 10 3 4 11 2 2 3 12 3 2 13 4 9
--R (55a b d - 220a b c d + 330a b c d - 220a b c d + 55a b c )x
--R +
--R 7 8 4 6 9 3 5 10 2 2 4 11 3 3 12 4 8
--R (165a b d - 660a b c d + 990a b c d - 660a b c d + 165a b c )x
--R +
--R 8 7 4 7 8 3 6 9 2 2 5 10 3 4 11 4 7
--R (330a b d - 1320a b c d + 1980a b c d - 1320a b c d + 330a b c )x
--R +
--R 9 6 4 8 7 3 7 8 2 2 6 9 3 5 10 4 6
--R (462a b d - 1848a b c d + 2772a b c d - 1848a b c d + 462a b c )x
--R +
--R 10 5 4 9 6 3 8 7 2 2 7 8 3 6 9 4 5
--R (462a b d - 1848a b c d + 2772a b c d - 1848a b c d + 462a b c )x
--R +
--R 11 4 4 10 5 3 9 6 2 2 8 7 3 7 8 4 4
--R (330a b d - 1320a b c d + 1980a b c d - 1320a b c d + 330a b c )x
--R +
--R 12 3 4 11 4 3 10 5 2 2 9 6 3 8 7 4 3
--R (165a b d - 660a b c d + 990a b c d - 660a b c d + 165a b c )x
--R +
--R 13 2 4 12 3 3 11 4 2 2 10 5 3 9 6 4 2
--R (55a b d - 220a b c d + 330a b c d - 220a b c d + 55a b c )x
--R +
--R 14 4 13 2 3 12 3 2 2 11 4 3 10 5 4 15 4
--R (11a b d - 44a b c d + 66a b c d - 44a b c d + 11a b c )x + a d
--R +
--R 14 3 13 2 2 2 12 3 3 11 4 4
--R - 4a b c d + 6a b c d - 4a b c d + a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 702
--S 703 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 7 6 6
--R - 330b d x + (- 462a b d - 1848b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (- 462a b d - 1848a b c d - 4620b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 330a b d - 1320a b c d - 3300a b c d - 6600b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4 7 4 3 3
--R (- 165a b d - 660a b c d - 1650a b c d - 3300a b c d - 5775b c d )x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R - 55a b d - 220a b c d - 550a b c d - 1100a b c d - 1925a b c d
--R +
--R 7 5 2
--R - 3080b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R - 11a b d - 44a b c d - 110a b c d - 220a b c d - 385a b c d
--R +
--R 6 5 2 7 6
--R - 616a b c d - 924b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2
--R - a d - 4a b c d - 10a b c d - 20a b c d - 35a b c d - 56a b c d
--R +
--R 6 6 7 7
--R - 84a b c d - 120b c
--R /
--R 19 11 18 10 2 17 9 3 16 8 4 15 7
--R 1320b x + 14520a b x + 72600a b x + 217800a b x + 435600a b x
--R +
--R 5 14 6 6 13 5 7 12 4 8 11 3
--R 609840a b x + 609840a b x + 435600a b x + 217800a b x
--R +
--R 9 10 2 10 9 11 8
--R 72600a b x + 14520a b x + 1320a b
--R Type: Union(Expression(Integer),...)
--E 703
--S 704 of 2952
m0000:=a0000 - r0000
--R
--R
--R 11
--R d
--R (4) - --------------------------------------------------------------------
--R 4 8 4 3 9 3 2 10 2 2 11 3 12 4
--R 1320a b d - 5280a b c d + 7920a b c d - 5280a b c d + 1320b c
--R Type: Expression(Integer)
--E 704
--S 705 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 705
)clear all
--S 706 of 2952
t0000:=(c+d*x)^7/(a+b*x)^13
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 13 13 12 12 2 11 11 3 10 10 4 9 9 5 8 8
--R b x + 13a b x + 78a b x + 286a b x + 715a b x + 1287a b x
--R +
--R 6 7 7 7 6 6 8 5 5 9 4 4 10 3 3 11 2 2
--R 1716a b x + 1716a b x + 1287a b x + 715a b x + 286a b x + 78a b x
--R +
--R 12 13
--R 13a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 706
--S 707 of 2952
r0000:=-1/12*(c+d*x)^8/((b*c-a*d)*(a+b*x)^12)+_
1/33*d*(c+d*x)^8/((b*c-a*d)^2*(a+b*x)^11)-_
1/110*d^2*(c+d*x)^8/((b*c-a*d)^3*(a+b*x)^10)+_
1/495*d^3*(c+d*x)^8/((b*c-a*d)^4*(a+b*x)^9)-_
1/3960*d^4*(c+d*x)^8/((b*c-a*d)^5*(a+b*x)^8)
--R
--R
--R (2)
--R 1 4 12 12 1 3 12 11 1 2 2 12 10 1 3 12 9 1 4 12 8
--R ---- b d x + --- a b d x + -- a b d x + -- a b d x + - a d x
--R 3960 330 60 18 8
--R +
--R 4 11 3 2 10 2 2 3 9 3 4 8 1 4 5 7 7
--R (a c d - 2a b c d + 2a b c d - a b c d + - b c d )x
--R 5
--R +
--R 7 4 2 10 28 3 3 9 21 2 2 4 8 28 3 5 7 7 4 6 6 6
--R (- a c d - -- a b c d + -- a b c d - -- a b c d + - b c d )x
--R 2 3 2 5 6
--R +
--R 4 3 9 3 4 8 126 2 2 5 7 3 6 6 4 7 5 5
--R (7a c d - 21a b c d + --- a b c d - 14a b c d + 3b c d )x
--R 5
--R +
--R 35 4 4 8 3 5 7 2 2 6 6 3 7 5 35 4 8 4 4
--R (-- a c d - 28a b c d + 35a b c d - 20a b c d + -- b c d )x
--R 4 8
--R +
--R 4 5 7 70 3 6 6 2 2 7 5 35 3 8 4 35 4 9 3 3
--R (7a c d - -- a b c d + 30a b c d - -- a b c d + -- b c d )x
--R 3 2 9
--R +
--R 7 4 6 6 3 7 5 63 2 2 8 4 28 3 9 3 21 4 10 2 2
--R (- a c d - 12a b c d + -- a b c d - -- a b c d + -- b c d )x
--R 2 4 3 10
--R +
--R 4 7 5 7 3 8 4 14 2 2 9 3 14 3 10 2 7 4 11
--R (a c d - - a b c d + -- a b c d - -- a b c d + -- b c d)x
--R 2 3 5 11
--R +
--R 1 4 8 4 4 3 9 3 3 2 2 10 2 4 3 11 1 4 12
--R - a c d - - a b c d + - a b c d - -- a b c d + -- b c
--R 8 9 5 11 12
--R /
--R 5 12 5 4 13 4 3 14 2 3 2 15 3 2 16 4 17 5 12
--R (a b d - 5a b c d + 10a b c d - 10a b c d + 5a b c d - b c )x
--R +
--R 6 11 5 5 12 4 4 13 2 3 3 14 3 2 2 15 4
--R 12a b d - 60a b c d + 120a b c d - 120a b c d + 60a b c d
--R +
--R 16 5
--R - 12a b c
--R *
--R 11
--R x
--R +
--R 7 10 5 6 11 4 5 12 2 3 4 13 3 2 3 14 4
--R 66a b d - 330a b c d + 660a b c d - 660a b c d + 330a b c d
--R +
--R 2 15 5
--R - 66a b c
--R *
--R 10
--R x
--R +
--R 8 9 5 7 10 4 6 11 2 3 5 12 3 2
--R 220a b d - 1100a b c d + 2200a b c d - 2200a b c d
--R +
--R 4 13 4 3 14 5
--R 1100a b c d - 220a b c
--R *
--R 9
--R x
--R +
--R 9 8 5 8 9 4 7 10 2 3 6 11 3 2
--R 495a b d - 2475a b c d + 4950a b c d - 4950a b c d
--R +
--R 5 12 4 4 13 5
--R 2475a b c d - 495a b c
--R *
--R 8
--R x
--R +
--R 10 7 5 9 8 4 8 9 2 3 7 10 3 2
--R 792a b d - 3960a b c d + 7920a b c d - 7920a b c d
--R +
--R 6 11 4 5 12 5
--R 3960a b c d - 792a b c
--R *
--R 7
--R x
--R +
--R 11 6 5 10 7 4 9 8 2 3 8 9 3 2
--R 924a b d - 4620a b c d + 9240a b c d - 9240a b c d
--R +
--R 7 10 4 6 11 5
--R 4620a b c d - 924a b c
--R *
--R 6
--R x
--R +
--R 12 5 5 11 6 4 10 7 2 3 9 8 3 2
--R 792a b d - 3960a b c d + 7920a b c d - 7920a b c d
--R +
--R 8 9 4 7 10 5
--R 3960a b c d - 792a b c
--R *
--R 5
--R x
--R +
--R 13 4 5 12 5 4 11 6 2 3 10 7 3 2
--R 495a b d - 2475a b c d + 4950a b c d - 4950a b c d
--R +
--R 9 8 4 8 9 5
--R 2475a b c d - 495a b c
--R *
--R 4
--R x
--R +
--R 14 3 5 13 4 4 12 5 2 3 11 6 3 2
--R 220a b d - 1100a b c d + 2200a b c d - 2200a b c d
--R +
--R 10 7 4 9 8 5
--R 1100a b c d - 220a b c
--R *
--R 3
--R x
--R +
--R 15 2 5 14 3 4 13 4 2 3 12 5 3 2 11 6 4
--R 66a b d - 330a b c d + 660a b c d - 660a b c d + 330a b c d
--R +
--R 10 7 5
--R - 66a b c
--R *
--R 2
--R x
--R +
--R 16 5 15 2 4 14 3 2 3 13 4 3 2 12 5 4
--R 12a b d - 60a b c d + 120a b c d - 120a b c d + 60a b c d
--R +
--R 11 6 5
--R - 12a b c
--R *
--R x
--R +
--R 17 5 16 4 15 2 2 3 14 3 3 2 13 4 4 12 5 5
--R a d - 5a b c d + 10a b c d - 10a b c d + 5a b c d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 707
--S 708 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 7 6 6
--R - 792b d x + (- 924a b d - 4620b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (- 792a b d - 3960a b c d - 11880b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 495a b d - 2475a b c d - 7425a b c d - 17325b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R - 220a b d - 1100a b c d - 3300a b c d - 7700a b c d
--R +
--R 7 4 3
--R - 15400b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R - 66a b d - 330a b c d - 990a b c d - 2310a b c d - 4620a b c d
--R +
--R 7 5 2
--R - 8316b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R - 12a b d - 60a b c d - 180a b c d - 420a b c d - 840a b c d
--R +
--R 6 5 2 7 6
--R - 1512a b c d - 2520b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2
--R - a d - 5a b c d - 15a b c d - 35a b c d - 70a b c d - 126a b c d
--R +
--R 6 6 7 7
--R - 210a b c d - 330b c
--R /
--R 20 12 19 11 2 18 10 3 17 9
--R 3960b x + 47520a b x + 261360a b x + 871200a b x
--R +
--R 4 16 8 5 15 7 6 14 6 7 13 5
--R 1960200a b x + 3136320a b x + 3659040a b x + 3136320a b x
--R +
--R 8 12 4 9 11 3 10 10 2 11 9 12 8
--R 1960200a b x + 871200a b x + 261360a b x + 47520a b x + 3960a b
--R Type: Union(Expression(Integer),...)
--E 708
--S 709 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R -
--R 12
--R d
--R /
--R 5 8 5 4 9 4 3 10 2 3 2 11 3 2
--R 3960a b d - 19800a b c d + 39600a b c d - 39600a b c d
--R +
--R 12 4 13 5
--R 19800a b c d - 3960b c
--R Type: Expression(Integer)
--E 709
--S 710 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 710
)clear all
--S 711 of 2952
t0000:=(c+d*x)^7/(a+b*x)^14
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 14 14 13 13 2 12 12 3 11 11 4 10 10
--R b x + 14a b x + 91a b x + 364a b x + 1001a b x
--R +
--R 5 9 9 6 8 8 7 7 7 8 6 6 9 5 5
--R 2002a b x + 3003a b x + 3432a b x + 3003a b x + 2002a b x
--R +
--R 10 4 4 11 3 3 12 2 2 13 14
--R 1001a b x + 364a b x + 91a b x + 14a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 711
--S 712 of 2952
r0000:=-1/13*(b*c-a*d)^7/(b^8*(a+b*x)^13)-_
7/12*d*(b*c-a*d)^6/(b^8*(a+b*x)^12)-_
21/11*d^2*(b*c-a*d)^5/(b^8*(a+b*x)^11)-_
7/2*d^3*(b*c-a*d)^4/(b^8*(a+b*x)^10)-_
35/9*d^4*(b*c-a*d)^3/(b^8*(a+b*x)^9)-_
21/8*d^5*(b*c-a*d)^2/(b^8*(a+b*x)^8)-_
d^6*(b*c-a*d)/(b^8*(a+b*x)^7)-1/6*d^7/(b^8*(a+b*x)^6)
--R
--R
--R (2)
--R 1 7 7 7 1 6 7 7 6 6
--R - - b d x + (- - a b d - b c d )x
--R 6 6
--R +
--R 1 2 5 7 3 6 6 21 7 2 5 5
--R (- - a b d - - a b c d - -- b c d )x
--R 8 4 8
--R +
--R 5 3 4 7 5 2 5 6 35 6 2 5 35 7 3 4 4
--R (- -- a b d - -- a b c d - -- a b c d - -- b c d )x
--R 72 12 24 9
--R +
--R 1 4 3 7 1 3 4 6 7 2 5 2 5 14 6 3 4 7 7 4 3 3
--R (- -- a b d - - a b c d - -- a b c d - -- a b c d - - b c d )x
--R 36 6 12 9 2
--R +
--R 1 5 2 7 1 4 3 6 7 3 4 2 5 14 2 5 3 4 21 6 4 3
--R - --- a b d - -- a b c d - -- a b c d - -- a b c d - -- a b c d
--R 132 22 44 33 22
--R +
--R 21 7 5 2
--R - -- b c d
--R 11
--R *
--R 2
--R x
--R +
--R 1 6 7 1 5 2 6 7 4 3 2 5 7 3 4 3 4
--R - --- a b d - --- a b c d - --- a b c d - -- a b c d
--R 792 132 264 99
--R +
--R 7 2 5 4 3 7 6 5 2 7 7 6
--R - -- a b c d - -- a b c d - -- b c d
--R 44 22 12
--R *
--R x
--R +
--R 1 7 7 1 6 6 7 5 2 2 5 7 4 3 3 4
--R - ----- a d - ---- a b c d - ---- a b c d - ---- a b c d
--R 10296 1716 3432 1287
--R +
--R 7 3 4 4 3 7 2 5 5 2 7 6 6 1 7 7
--R - --- a b c d - --- a b c d - --- a b c d - -- b c
--R 572 286 156 13
--R /
--R 21 13 20 12 2 19 11 3 18 10 4 17 9 5 16 8
--R b x + 13a b x + 78a b x + 286a b x + 715a b x + 1287a b x
--R +
--R 6 15 7 7 14 6 8 13 5 9 12 4 10 11 3
--R 1716a b x + 1716a b x + 1287a b x + 715a b x + 286a b x
--R +
--R 11 10 2 12 9 13 8
--R 78a b x + 13a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 712
--S 713 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 7 6 6
--R - 1716b d x + (- 1716a b d - 10296b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (- 1287a b d - 7722a b c d - 27027b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 715a b d - 4290a b c d - 15015a b c d - 40040b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R - 286a b d - 1716a b c d - 6006a b c d - 16016a b c d
--R +
--R 7 4 3
--R - 36036b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4 6 4 3
--R - 78a b d - 468a b c d - 1638a b c d - 4368a b c d - 9828a b c d
--R +
--R 7 5 2
--R - 19656b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R - 13a b d - 78a b c d - 273a b c d - 728a b c d - 1638a b c d
--R +
--R 6 5 2 7 6
--R - 3276a b c d - 6006b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2
--R - a d - 6a b c d - 21a b c d - 56a b c d - 126a b c d - 252a b c d
--R +
--R 6 6 7 7
--R - 462a b c d - 792b c
--R /
--R 21 13 20 12 2 19 11 3 18 10
--R 10296b x + 133848a b x + 803088a b x + 2944656a b x
--R +
--R 4 17 9 5 16 8 6 15 7 7 14 6
--R 7361640a b x + 13250952a b x + 17667936a b x + 17667936a b x
--R +
--R 8 13 5 9 12 4 10 11 3 11 10 2
--R 13250952a b x + 7361640a b x + 2944656a b x + 803088a b x
--R +
--R 12 9 13 8
--R 133848a b x + 10296a b
--R Type: Union(Expression(Integer),...)
--E 713
--S 714 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 714
--S 715 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 715
)clear all
--S 716 of 2952
t0000:=(c+d*x)^7/(a+b*x)^15
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 15 15 14 14 2 13 13 3 12 12 4 11 11
--R b x + 15a b x + 105a b x + 455a b x + 1365a b x
--R +
--R 5 10 10 6 9 9 7 8 8 8 7 7 9 6 6
--R 3003a b x + 5005a b x + 6435a b x + 6435a b x + 5005a b x
--R +
--R 10 5 5 11 4 4 12 3 3 13 2 2 14 15
--R 3003a b x + 1365a b x + 455a b x + 105a b x + 15a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 716
--S 717 of 2952
r0000:=-1/14*(b*c-a*d)^7/(b^8*(a+b*x)^14)-_
7/13*d*(b*c-a*d)^6/(b^8*(a+b*x)^13)-_
7/4*d^2*(b*c-a*d)^5/(b^8*(a+b*x)^12)-_
35/11*d^3*(b*c-a*d)^4/(b^8*(a+b*x)^11)-_
7/2*d^4*(b*c-a*d)^3/(b^8*(a+b*x)^10)-_
7/3*d^5*(b*c-a*d)^2/(b^8*(a+b*x)^9)-_
7/8*d^6*(b*c-a*d)/(b^8*(a+b*x)^8)-1/7*d^7/(b^8*(a+b*x)^7)
--R
--R
--R (2)
--R 1 7 7 7 1 6 7 7 7 6 6
--R - - b d x + (- - a b d - - b c d )x
--R 7 8 8
--R +
--R 1 2 5 7 7 6 6 7 7 2 5 5
--R (- -- a b d - -- a b c d - - b c d )x
--R 12 12 3
--R +
--R 1 3 4 7 7 2 5 6 7 6 2 5 7 7 3 4 4
--R (- -- a b d - -- a b c d - - a b c d - - b c d )x
--R 24 24 6 2
--R +
--R 1 4 3 7 7 3 4 6 14 2 5 2 5 14 6 3 4 35 7 4 3 3
--R (- -- a b d - -- a b c d - -- a b c d - -- a b c d - -- b c d )x
--R 66 66 33 11 11
--R +
--R 1 5 2 7 7 4 3 6 7 3 4 2 5 7 2 5 3 4 35 6 4 3
--R - --- a b d - --- a b c d - -- a b c d - -- a b c d - -- a b c d
--R 264 264 66 22 44
--R +
--R 7 7 5 2
--R - - b c d
--R 4
--R *
--R 2
--R x
--R +
--R 1 6 7 7 5 2 6 7 4 3 2 5 7 3 4 3 4
--R - ---- a b d - ---- a b c d - --- a b c d - --- a b c d
--R 1716 1716 429 143
--R +
--R 35 2 5 4 3 7 6 5 2 7 7 6
--R - --- a b c d - -- a b c d - -- b c d
--R 286 26 13
--R *
--R x
--R +
--R 1 7 7 1 6 6 1 5 2 2 5 1 4 3 3 4 5 3 4 4 3
--R - ----- a d - ---- a b c d - --- a b c d - --- a b c d - --- a b c d
--R 24024 3432 858 286 572
--R +
--R 1 2 5 5 2 1 6 6 1 7 7
--R - -- a b c d - -- a b c d - -- b c
--R 52 26 14
--R /
--R 22 14 21 13 2 20 12 3 19 11 4 18 10
--R b x + 14a b x + 91a b x + 364a b x + 1001a b x
--R +
--R 5 17 9 6 16 8 7 15 7 8 14 6 9 13 5
--R 2002a b x + 3003a b x + 3432a b x + 3003a b x + 2002a b x
--R +
--R 10 12 4 11 11 3 12 10 2 13 9 14 8
--R 1001a b x + 364a b x + 91a b x + 14a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 717
--S 718 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 7 6 6
--R - 3432b d x + (- 3003a b d - 21021b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (- 2002a b d - 14014a b c d - 56056b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 1001a b d - 7007a b c d - 28028a b c d - 84084b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R - 364a b d - 2548a b c d - 10192a b c d - 30576a b c d
--R +
--R 7 4 3
--R - 76440b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 91a b d - 637a b c d - 2548a b c d - 7644a b c d
--R +
--R 6 4 3 7 5 2
--R - 19110a b c d - 42042b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R - 14a b d - 98a b c d - 392a b c d - 1176a b c d - 2940a b c d
--R +
--R 6 5 2 7 6
--R - 6468a b c d - 12936b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2
--R - a d - 7a b c d - 28a b c d - 84a b c d - 210a b c d - 462a b c d
--R +
--R 6 6 7 7
--R - 924a b c d - 1716b c
--R /
--R 22 14 21 13 2 20 12 3 19 11
--R 24024b x + 336336a b x + 2186184a b x + 8744736a b x
--R +
--R 4 18 10 5 17 9 6 16 8 7 15 7
--R 24048024a b x + 48096048a b x + 72144072a b x + 82450368a b x
--R +
--R 8 14 6 9 13 5 10 12 4 11 11 3
--R 72144072a b x + 48096048a b x + 24048024a b x + 8744736a b x
--R +
--R 12 10 2 13 9 14 8
--R 2186184a b x + 336336a b x + 24024a b
--R Type: Union(Expression(Integer),...)
--E 718
--S 719 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 719
--S 720 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 720
)clear all
--S 721 of 2952
t0000:=(c+d*x)^7/(a+b*x)^16
--R
--R
--R (1)
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R d x + 7c d x + 21c d x + 35c d x + 35c d x + 21c d x + 7c d x + c
--R /
--R 16 16 15 15 2 14 14 3 13 13 4 12 12
--R b x + 16a b x + 120a b x + 560a b x + 1820a b x
--R +
--R 5 11 11 6 10 10 7 9 9 8 8 8 9 7 7
--R 4368a b x + 8008a b x + 11440a b x + 12870a b x + 11440a b x
--R +
--R 10 6 6 11 5 5 12 4 4 13 3 3 14 2 2
--R 8008a b x + 4368a b x + 1820a b x + 560a b x + 120a b x
--R +
--R 15 16
--R 16a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 721
--S 722 of 2952
r0000:=-1/15*(b*c-a*d)^7/(b^8*(a+b*x)^15)-_
1/2*d*(b*c-a*d)^6/(b^8*(a+b*x)^14)-_
21/13*d^2*(b*c-a*d)^5/(b^8*(a+b*x)^13)-_
35/12*d^3*(b*c-a*d)^4/(b^8*(a+b*x)^12)-_
35/11*d^4*(b*c-a*d)^3/(b^8*(a+b*x)^11)-_
21/10*d^5*(b*c-a*d)^2/(b^8*(a+b*x)^10)-_
7/9*d^6*(b*c-a*d)/(b^8*(a+b*x)^9)-1/8*d^7/(b^8*(a+b*x)^8)
--R
--R
--R (2)
--R 1 7 7 7 7 6 7 7 7 6 6
--R - - b d x + (- -- a b d - - b c d )x
--R 8 72 9
--R +
--R 7 2 5 7 7 6 6 21 7 2 5 5
--R (- --- a b d - -- a b c d - -- b c d )x
--R 120 15 10
--R +
--R 7 3 4 7 7 2 5 6 21 6 2 5 35 7 3 4 4
--R (- --- a b d - -- a b c d - -- a b c d - -- b c d )x
--R 264 33 22 11
--R +
--R 7 4 3 7 7 3 4 6 7 2 5 2 5 35 6 3 4 35 7 4 3 3
--R (- --- a b d - -- a b c d - -- a b c d - -- a b c d - -- b c d )x
--R 792 99 22 33 12
--R +
--R 7 5 2 7 7 4 3 6 21 3 4 2 5 35 2 5 3 4
--R - ---- a b d - --- a b c d - --- a b c d - --- a b c d
--R 3432 429 286 143
--R +
--R 35 6 4 3 21 7 5 2
--R - -- a b c d - -- b c d
--R 52 13
--R *
--R 2
--R x
--R +
--R 1 6 7 1 5 2 6 3 4 3 2 5 5 3 4 3 4
--R - ---- a b d - --- a b c d - --- a b c d - --- a b c d
--R 3432 429 286 143
--R +
--R 5 2 5 4 3 3 6 5 2 1 7 6
--R - -- a b c d - -- a b c d - - b c d
--R 52 13 2
--R *
--R x
--R +
--R 1 7 7 1 6 6 1 5 2 2 5 1 4 3 3 4
--R - ----- a d - ---- a b c d - ---- a b c d - --- a b c d
--R 51480 6435 1430 429
--R +
--R 1 3 4 4 3 1 2 5 5 2 1 6 6 1 7 7
--R - --- a b c d - -- a b c d - -- a b c d - -- b c
--R 156 65 30 15
--R /
--R 23 15 22 14 2 21 13 3 20 12 4 19 11
--R b x + 15a b x + 105a b x + 455a b x + 1365a b x
--R +
--R 5 18 10 6 17 9 7 16 8 8 15 7 9 14 6
--R 3003a b x + 5005a b x + 6435a b x + 6435a b x + 5005a b x
--R +
--R 10 13 5 11 12 4 12 11 3 13 10 2 14 9 15 8
--R 3003a b x + 1365a b x + 455a b x + 105a b x + 15a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 722
--S 723 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 7 7 6 7 7 6 6
--R - 6435b d x + (- 5005a b d - 40040b c d )x
--R +
--R 2 5 7 6 6 7 2 5 5
--R (- 3003a b d - 24024a b c d - 108108b c d )x
--R +
--R 3 4 7 2 5 6 6 2 5 7 3 4 4
--R (- 1365a b d - 10920a b c d - 49140a b c d - 163800b c d )x
--R +
--R 4 3 7 3 4 6 2 5 2 5 6 3 4
--R - 455a b d - 3640a b c d - 16380a b c d - 54600a b c d
--R +
--R 7 4 3
--R - 150150b c d
--R *
--R 3
--R x
--R +
--R 5 2 7 4 3 6 3 4 2 5 2 5 3 4
--R - 105a b d - 840a b c d - 3780a b c d - 12600a b c d
--R +
--R 6 4 3 7 5 2
--R - 34650a b c d - 83160b c d
--R *
--R 2
--R x
--R +
--R 6 7 5 2 6 4 3 2 5 3 4 3 4 2 5 4 3
--R - 15a b d - 120a b c d - 540a b c d - 1800a b c d - 4950a b c d
--R +
--R 6 5 2 7 6
--R - 11880a b c d - 25740b c d
--R *
--R x
--R +
--R 7 7 6 6 5 2 2 5 4 3 3 4 3 4 4 3 2 5 5 2
--R - a d - 8a b c d - 36a b c d - 120a b c d - 330a b c d - 792a b c d
--R +
--R 6 6 7 7
--R - 1716a b c d - 3432b c
--R /
--R 23 15 22 14 2 21 13 3 20 12
--R 51480b x + 772200a b x + 5405400a b x + 23423400a b x
--R +
--R 4 19 11 5 18 10 6 17 9
--R 70270200a b x + 154594440a b x + 257657400a b x
--R +
--R 7 16 8 8 15 7 9 14 6
--R 331273800a b x + 331273800a b x + 257657400a b x
--R +
--R 10 13 5 11 12 4 12 11 3 13 10 2
--R 154594440a b x + 70270200a b x + 23423400a b x + 5405400a b x
--R +
--R 14 9 15 8
--R 772200a b x + 51480a b
--R Type: Union(Expression(Integer),...)
--E 723
--S 724 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 724
--S 725 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 725
)clear all
--S 726 of 2952
t0000:=(a+b*x)^m*(c+d*x)^10
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R *
--R m
--R (b x + a)
--R Type: Expression(Integer)
--E 726
--S 727 of 2952
r0000:=(b*c-a*d)^10*(a+b*x)^(1+m)/(b^11*(1+m))+_
10*d*(b*c-a*d)^9*(a+b*x)^(2+m)/(b^11*(2+m))+_
45*d^2*(b*c-a*d)^8*(a+b*x)^(3+m)/(b^11*(3+m))+_
120*d^3*(b*c-a*d)^7*(a+b*x)^(4+m)/(b^11*(4+m))+_
210*d^4*(b*c-a*d)^6*(a+b*x)^(5+m)/(b^11*(5+m))+_
252*d^5*(b*c-a*d)^5*(a+b*x)^(6+m)/(b^11*(6+m))+_
210*d^6*(b*c-a*d)^4*(a+b*x)^(7+m)/(b^11*(7+m))+_
120*d^7*(b*c-a*d)^3*(a+b*x)^(8+m)/(b^11*(8+m))+_
45*d^8*(b*c-a*d)^2*(a+b*x)^(9+m)/(b^11*(9+m))+_
10*d^9*(b*c-a*d)*(a+b*x)^(10+m)/(b^11*(10+m))+_
d^10*(a+b*x)^(11+m)/(b^11*(11+m))
--R
--R
--R (2)
--R 10 10 10 9 10 8 10 7 10 6 10 5
--R d m + 55d m + 1320d m + 18150d m + 157773d m + 902055d m
--R +
--R 10 4 10 3 10 2 10 10
--R 3416930d m + 8409500d m + 12753576d m + 10628640d m + 3628800d
--R *
--R m + 11
--R (b x + a)
--R +
--R 10 9 10 10 9 9
--R (- 10a d + 10b c d )m + (- 560a d + 560b c d )m
--R +
--R 10 9 8 10 9 7
--R (- 13650a d + 13650b c d )m + (- 190200a d + 190200b c d )m
--R +
--R 10 9 6
--R (- 1672230a d + 1672230b c d )m
--R +
--R 10 9 5
--R (- 9653280a d + 9653280b c d )m
--R +
--R 10 9 4
--R (- 36862550a d + 36862550b c d )m
--R +
--R 10 9 3
--R (- 91331800a d + 91331800b c d )m
--R +
--R 10 9 2
--R (- 139262760a d + 139262760b c d )m
--R +
--R 10 9 10 9
--R (- 116552160a d + 116552160b c d )m - 39916800a d + 39916800b c d
--R *
--R m + 10
--R (b x + a)
--R +
--R 2 10 9 2 2 8 10
--R (45a d - 90a b c d + 45b c d )m
--R +
--R 2 10 9 2 2 8 9
--R (2565a d - 5130a b c d + 2565b c d )m
--R +
--R 2 10 9 2 2 8 8
--R (63540a d - 127080a b c d + 63540b c d )m
--R +
--R 2 10 9 2 2 8 7
--R (898290a d - 1796580a b c d + 898290b c d )m
--R +
--R 2 10 9 2 2 8 6
--R (7999425a d - 15998850a b c d + 7999425b c d )m
--R +
--R 2 10 9 2 2 8 5
--R (46695285a d - 93390570a b c d + 46695285b c d )m
--R +
--R 2 10 9 2 2 8 4
--R (180021510a d - 360043020a b c d + 180021510b c d )m
--R +
--R 2 10 9 2 2 8 3
--R (449614260a d - 899228520a b c d + 449614260b c d )m
--R +
--R 2 10 9 2 2 8 2
--R (690085080a d - 1380170160a b c d + 690085080b c d )m
--R +
--R 2 10 9 2 2 8
--R (580543200a d - 1161086400a b c d + 580543200b c d )m
--R +
--R 2 10 9 2 2 8
--R 199584000a d - 399168000a b c d + 199584000b c d
--R *
--R m + 9
--R (b x + a)
--R +
--R 3 10 2 9 2 2 8 3 3 7 10
--R (- 120a d + 360a b c d - 360a b c d + 120b c d )m
--R +
--R 3 10 2 9 2 2 8 3 3 7 9
--R (- 6960a d + 20880a b c d - 20880a b c d + 6960b c d )m
--R +
--R 3 10 2 9 2 2 8 3 3 7 8
--R (- 175320a d + 525960a b c d - 525960a b c d + 175320b c d )m
--R +
--R 3 10 2 9 2 2 8
--R - 2517840a d + 7553520a b c d - 7553520a b c d
--R +
--R 3 3 7
--R 2517840b c d
--R *
--R 7
--R m
--R +
--R 3 10 2 9 2 2 8
--R - 22748040a d + 68244120a b c d - 68244120a b c d
--R +
--R 3 3 7
--R 22748040b c d
--R *
--R 6
--R m
--R +
--R 3 10 2 9 2 2 8
--R - 134522640a d + 403567920a b c d - 403567920a b c d
--R +
--R 3 3 7
--R 134522640b c d
--R *
--R 5
--R m
--R +
--R 3 10 2 9 2 2 8
--R - 524563080a d + 1573689240a b c d - 1573689240a b c d
--R +
--R 3 3 7
--R 524563080b c d
--R *
--R 4
--R m
--R +
--R 3 10 2 9 2 2 8
--R - 1322982960a d + 3968948880a b c d - 3968948880a b c d
--R +
--R 3 3 7
--R 1322982960b c d
--R *
--R 3
--R m
--R +
--R 3 10 2 9 2 2 8
--R - 2047105440a d + 6141316320a b c d - 6141316320a b c d
--R +
--R 3 3 7
--R 2047105440b c d
--R *
--R 2
--R m
--R +
--R 3 10 2 9 2 2 8
--R - 1733313600a d + 5199940800a b c d - 5199940800a b c d
--R +
--R 3 3 7
--R 1733313600b c d
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--E 727
--S 728 of 2952
a0000:=integrate(t0000,x)
--R
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--R - 237106800a b c d + 554122800a b c d - 911111040a b c d
--R +
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--R 1663200a b d - 19202400a b c d + 101001600a b c d
--R +
--R 4 7 3 7 3 8 4 6 2 9 5 5
--R - 319636800a b c d + 676695600a b c d - 1007249040a b c d
--R +
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--R *
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--R 907200a b d - 9979200a b c d + 49896000a b c d
--R +
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--R - 149688000a b c d + 299376000a b c d - 419126400a b c d
--R +
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--R +
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--R (- 840a b c d + 7080a b c d + 2835b c d )m
--R +
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--R (5040a b c d - 45360a b c d + 180000a b c d + 78120b c d )m
--R +
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--R - 25200a b c d + 241920a b c d - 1033200a b c d
--R +
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--R +
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--R - 302400a b c d + 3326400a b c d - 16506000a b c d
--R +
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--R +
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--R +
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--R - 127890000a b c d + 275531760a b c d - 413738640a b c d
--R +
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--R - 604800a b d + 8467200a b c d - 52920000a b c d
--R +
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--R 196257600a b c d - 482428800a b c d + 829241280a b c d
--R +
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--R - 1022028000a b c d + 908571360a b c d + 1003011660b c d
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--R - 1814400a b d + 21168000a b c d - 112492800a b c d
--R +
--R 5 6 3 7 4 7 4 6 3 8 5 5
--R 359654400a b c d - 769204800a b c d + 1156720320a b c d
--R +
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--R - 1249859520a b c d + 972590400a b c d + 1727578440b c d
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--R - 1209600a b d + 13305600a b c d - 66528000a b c d
--R +
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--R 199584000a b c d - 399168000a b c d + 558835200a b c d
--R +
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--R - 558835200a b c d + 399168000a b c d + 1608573600b c d
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--R +
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--R (- 360a b c d + 2745a b c d + 640b c d)m
--R +
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--R (2520a b c d - 20520a b c d + 72630a b c d + 17970b c d)m
--R +
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--R +
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--R - 6728400a b c d + 10195605a b c d + 2992710b c d
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--R +
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--R +
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--R 907200a b c d - 9374400a b c d + 43621200a b c d
--R +
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--R +
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--R - 1814400a b c d + 19958400a b c d - 99489600a b c d
--R +
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--R 296427600a b c d - 585884880a b c d + 805210560a b c d
--R +
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--R - 783102240a b c d + 536489820a b c d + 274727240b c d
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--R 1814400a b d - 21772800a b c d + 118843200a b c d
--R +
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--R - 389793600a b c d + 854431200a b c d - 1315954080a b c d
--R +
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--R +
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--R 1814400a b d - 19958400a b c d + 99792000a b c d
--R +
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--R - 299376000a b c d + 598752000a b c d - 838252800a b c d
--R +
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--R +
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--R +
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--R (- 90a b c d + 630a b c d + 65b c )m
--R +
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--R (720a b c d - 5400a b c d + 17340a b c d + 1860b c )m
--R +
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--R - 5040a b c d + 40320a b c d - 139860a b c d
--R +
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--R +
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--R - 151200a b c d + 1360800a b c d - 5418000a b c d
--R +
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--R +
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--R +
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--R - 60404400a b c d + 96803280a b c d - 103987800a b c d
--R +
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--R - 1814400a b c d + 18144000a b c d - 81496800a b c d
--R +
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--R +
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--R +
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--R - 511358400a b c d + 955402560a b c d - 1234820160a b c d
--R +
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--R +
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--R +
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--R +
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--R - 1676505600a b c d + 1197504000a b c d - 598752000a b c d
--R +
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--R +
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--R +
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--R (- 720a b c d + 5400a b c d - 17340a b c d + 30810a b c )m
--R +
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--R 5040a b c d - 40320a b c d + 139860a b c d - 273420a b c d
--R +
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--R +
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--R - 1121515200a b c d + 710445600a b c d - 303343200a b c d
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--R +
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--E 728
--S 729 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
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--R +
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--R +
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--R +
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--R +
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--R +
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--R +
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--R (- 360a b d + 9420a b c d + 63540b c d )m
--R +
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--R (- 5460a b d + 105420a b c d + 898290b c d )m
--R +
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--R (- 45360a b d + 723450a b c d + 7999425b c d )m
--R +
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--R (- 224490a b d + 3142230a b c d + 46695285b c d )m
--R +
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--R (- 672840a b d + 8582480a b c d + 180021510b c d )m
--R +
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--R (- 1181240a b d + 14089480a b c d + 449614260b c d )m
--R +
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--R (- 1095840a b d + 12457440a b c d + 690085080b c d )m
--R +
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--R (- 403200a b d + 4435200a b c d + 580543200b c d )m
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--R (90a b d - 3510a b c d + 45900a b c d + 175320b c d )m
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--R +
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--R +
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--R +
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--R +
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--R (- 360a b c d + 6120a b c d + 12390b c d )m
--R +
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--R (720a b c d - 15120a b c d + 132480a b c d + 317520b c d )m
--R +
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--R - 720a b d + 23040a b c d - 261360a b c d
--R +
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--R 1590480a b c d + 4638060b c d
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--R - 15120a b d + 292320a b c d - 2419200a b c d
--R +
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--R 11614680a b c d + 42592410b c d
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--R - 126000a b d + 1915200a b c d - 13071240a b c d
--R +
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--R 53219880a b c d + 255740310b c d
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--R - 529200a b d + 6990480a b c d - 42018480a b c d
--R +
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--R 152023920a b c d + 1011120180b c d
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--R - 1169280a b d + 14132160a b c d - 77905440a b c d
--R +
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--R 258815520a b c d + 2581262040b c d
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--R - 1270080a b d + 14489280a b c d - 75297600a b c d
--R +
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--R 235396800a b c d + 4035361680b c d
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--R +
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--R - 518400a b d + 5702400a b c d - 28512000a b c d
--R +
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--R 85536000a b c d + 3445243200b c d
--R *
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--R 1197504000b c d
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--R +
--R 10 4 6 11 5 5 10
--R (210a b c d + 252b c d )m
--R +
--R 2 9 3 7 10 4 6 11 5 5 9
--R (- 840a b c d + 11130a b c d + 15120b c d )m
--R +
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--R 2520a b c d - 37800a b c d + 250740a b c d
--R +
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--R *
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--R +
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--R - 5040a b c d + 90720a b c d - 700560a b c d
--R +
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--R 3133620a b c d + 5866560b c d
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--R +
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--R 5040a b d - 131040a b c d + 1285200a b c d
--R +
--R 2 9 3 7 10 4 6 11 5 5
--R - 6930000a b c d + 23790690a b c d + 54871236b c d
--R *
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--R +
--R 5 6 10 4 7 9 3 8 2 8
--R 75600a b d - 1260000a b c d + 9223200a b c d
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--R +
--R 9 10 8 9 7 2 2 8 6 3 3 7
--R 17970a d - 161730a b c d + 646920a b c d - 1509480a b c d
--R +
--R 5 4 4 6 4 5 5 5 3 6 6 4
--R 2264220a b c d - 2264220a b c d + 1509480a b c d
--R +
--R 2 7 7 3 8 8 2 9 9
--R - 646920a b c d + 161730a b c d - 17970b c d
--R *
--R 8
--R m
--R +
--R 9 10 8 9 7 2 2 8
--R 290760a d - 2616840a b c d + 10467360a b c d
--R +
--R 6 3 3 7 5 4 4 6 4 5 5 5
--R - 24423840a b c d + 36635760a b c d - 36635760a b c d
--R +
--R 3 6 6 4 2 7 7 3 8 8 2 9 9
--R 24423840a b c d - 10467360a b c d + 2616840a b c d - 290760b c d
--R *
--R 7
--R m
--R +
--R 9 10 8 9 7 2 2 8
--R 2992710a d - 26934390a b c d + 107737560a b c d
--R +
--R 6 3 3 7 5 4 4 6 4 5 5 5
--R - 251387640a b c d + 377081460a b c d - 377081460a b c d
--R +
--R 3 6 6 4 2 7 7 3 8 8 2
--R 251387640a b c d - 107737560a b c d + 26934390a b c d
--R +
--R 9 9
--R - 2992710b c d
--R *
--R 6
--R m
--R +
--R 9 10 8 9 7 2 2 8
--R 20390160a d - 183511440a b c d + 734045760a b c d
--R +
--R 6 3 3 7 5 4 4 6 4 5 5 5
--R - 1712773440a b c d + 2569160160a b c d - 2569160160a b c d
--R +
--R 3 6 6 4 2 7 7 3 8 8 2
--R 1712773440a b c d - 734045760a b c d + 183511440a b c d
--R +
--R 9 9
--R - 20390160b c d
--R *
--R 5
--R m
--R +
--R 9 10 8 9 7 2 2 8
--R 92615030a d - 833535270a b c d + 3334141080a b c d
--R +
--R 6 3 3 7 5 4 4 6 4 5 5 5
--R - 7779662520a b c d + 11669493780a b c d - 11669493780a b c d
--R +
--R 3 6 6 4 2 7 7 3 8 8 2
--R 7779662520a b c d - 3334141080a b c d + 833535270a b c d
--R +
--R 9 9
--R - 92615030b c d
--R *
--R 4
--R m
--R +
--R 9 10 8 9 7 2 2 8
--R 274727240a d - 2472545160a b c d + 9890180640a b c d
--R +
--R 6 3 3 7 5 4 4 6 4 5 5 5
--R - 23077088160a b c d + 34615632240a b c d - 34615632240a b c d
--R +
--R 3 6 6 4 2 7 7 3 8 8 2
--R 23077088160a b c d - 9890180640a b c d + 2472545160a b c d
--R +
--R 9 9
--R - 274727240b c d
--R *
--R 3
--R m
--R +
--R 9 10 8 9 7 2 2 8
--R 503126280a d - 4528136520a b c d + 18112546080a b c d
--R +
--R 6 3 3 7 5 4 4 6 4 5 5 5
--R - 42262607520a b c d + 63393911280a b c d - 63393911280a b c d
--R +
--R 3 6 6 4 2 7 7 3 8 8 2
--R 42262607520a b c d - 18112546080a b c d + 4528136520a b c d
--R +
--R 9 9
--R - 503126280b c d
--R *
--R 2
--R m
--R +
--R 9 10 8 9 7 2 2 8
--R 502927200a d - 4526344800a b c d + 18105379200a b c d
--R +
--R 6 3 3 7 5 4 4 6 4 5 5 5
--R - 42245884800a b c d + 63368827200a b c d - 63368827200a b c d
--R +
--R 3 6 6 4 2 7 7 3 8 8 2
--R 42245884800a b c d - 18105379200a b c d + 4526344800a b c d
--R +
--R 9 9
--R - 502927200b c d
--R *
--R m
--R +
--R 9 10 8 9 7 2 2 8
--R 199584000a d - 1796256000a b c d + 7185024000a b c d
--R +
--R 6 3 3 7 5 4 4 6 4 5 5 5
--R - 16765056000a b c d + 25147584000a b c d - 25147584000a b c d
--R +
--R 3 6 6 4 2 7 7 3 8 8 2
--R 16765056000a b c d - 7185024000a b c d + 1796256000a b c d
--R +
--R 9 9
--R - 199584000b c d
--R *
--R m + 2
--R (b x + a)
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - a d + 10a b c d - 45a b c d + 120a b c d - 210a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9
--R 252a b c d - 210a b c d + 120a b c d - 45a b c d + 10a b c d
--R +
--R 10 10
--R - b c
--R *
--R 10
--R m
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 65a d + 650a b c d - 2925a b c d + 7800a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 13650a b c d + 16380a b c d - 13650a b c d + 7800a b c d
--R +
--R 2 8 8 2 9 9 10 10
--R - 2925a b c d + 650a b c d - 65b c
--R *
--R 9
--R m
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 1860a d + 18600a b c d - 83700a b c d + 223200a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4
--R - 390600a b c d + 468720a b c d - 390600a b c d
--R +
--R 3 7 7 3 2 8 8 2 9 9 10 10
--R 223200a b c d - 83700a b c d + 18600a b c d - 1860b c
--R *
--R 8
--R m
--R +
--R 10 10 9 9 8 2 2 8
--R - 30810a d + 308100a b c d - 1386450a b c d
--R +
--R 7 3 3 7 6 4 4 6 5 5 5 5
--R 3697200a b c d - 6470100a b c d + 7764120a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2
--R - 6470100a b c d + 3697200a b c d - 1386450a b c d
--R +
--R 9 9 10 10
--R 308100a b c d - 30810b c
--R *
--R 7
--R m
--R +
--R 10 10 9 9 8 2 2 8
--R - 326613a d + 3266130a b c d - 14697585a b c d
--R +
--R 7 3 3 7 6 4 4 6 5 5 5 5
--R 39193560a b c d - 68588730a b c d + 82306476a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2
--R - 68588730a b c d + 39193560a b c d - 14697585a b c d
--R +
--R 9 9 10 10
--R 3266130a b c d - 326613b c
--R *
--R 6
--R m
--R +
--R 10 10 9 9 8 2 2 8
--R - 2310945a d + 23109450a b c d - 103992525a b c d
--R +
--R 7 3 3 7 6 4 4 6 5 5 5 5
--R 277313400a b c d - 485298450a b c d + 582358140a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2
--R - 485298450a b c d + 277313400a b c d - 103992525a b c d
--R +
--R 9 9 10 10
--R 23109450a b c d - 2310945b c
--R *
--R 5
--R m
--R +
--R 10 10 9 9 8 2 2 8
--R - 11028590a d + 110285900a b c d - 496286550a b c d
--R +
--R 7 3 3 7 6 4 4 6 5 5 5 5
--R 1323430800a b c d - 2316003900a b c d + 2779204680a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2
--R - 2316003900a b c d + 1323430800a b c d - 496286550a b c d
--R +
--R 9 9 10 10
--R 110285900a b c d - 11028590b c
--R *
--R 4
--R m
--R +
--R 10 10 9 9 8 2 2 8
--R - 34967140a d + 349671400a b c d - 1573521300a b c d
--R +
--R 7 3 3 7 6 4 4 6 5 5 5 5
--R 4196056800a b c d - 7343099400a b c d + 8811719280a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2
--R - 7343099400a b c d + 4196056800a b c d - 1573521300a b c d
--R +
--R 9 9 10 10
--R 349671400a b c d - 34967140b c
--R *
--R 3
--R m
--R +
--R 10 10 9 9 8 2 2 8
--R - 70290936a d + 702909360a b c d - 3163092120a b c d
--R +
--R 7 3 3 7 6 4 4 6 5 5 5 5
--R 8434912320a b c d - 14761096560a b c d + 17713315872a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2
--R - 14761096560a b c d + 8434912320a b c d - 3163092120a b c d
--R +
--R 9 9 10 10
--R 702909360a b c d - 70290936b c
--R *
--R 2
--R m
--R +
--R 10 10 9 9 8 2 2 8
--R - 80627040a d + 806270400a b c d - 3628216800a b c d
--R +
--R 7 3 3 7 6 4 4 6 5 5 5 5
--R 9675244800a b c d - 16931678400a b c d + 20318014080a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2
--R - 16931678400a b c d + 9675244800a b c d - 3628216800a b c d
--R +
--R 9 9 10 10
--R 806270400a b c d - 80627040b c
--R *
--R m
--R +
--R 10 10 9 9 8 2 2 8
--R - 39916800a d + 399168000a b c d - 1796256000a b c d
--R +
--R 7 3 3 7 6 4 4 6 5 5 5 5
--R 4790016000a b c d - 8382528000a b c d + 10059033600a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2
--R - 8382528000a b c d + 4790016000a b c d - 1796256000a b c d
--R +
--R 9 9 10 10
--R 399168000a b c d - 39916800b c
--R *
--R m + 1
--R (b x + a)
--R /
--R 11 11 11 10 11 9 11 8 11 7 11 6
--R b m + 66b m + 1925b m + 32670b m + 357423b m + 2637558b m
--R +
--R 11 5 11 4 11 3 11 2
--R 13339535b m + 45995730b m + 105258076b m + 150917976b m
--R +
--R 11 11
--R 120543840b m + 39916800b
--R Type: Expression(Integer)
--E 729
--S 730 of 2952
d0000:=normalize m0000
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 730
)clear all
--S 731 of 2952
t0000:=(a+b*x)^12*(c+d*x)^10
--R
--R
--R (1)
--R 12 10 22 11 10 12 9 21
--R b d x + (12a b d + 10b c d )x
--R +
--R 2 10 10 11 9 12 2 8 20
--R (66a b d + 120a b c d + 45b c d )x
--R +
--R 3 9 10 2 10 9 11 2 8 12 3 7 19
--R (220a b d + 660a b c d + 540a b c d + 120b c d )x
--R +
--R 4 8 10 3 9 9 2 10 2 8 11 3 7 12 4 6 18
--R (495a b d + 2200a b c d + 2970a b c d + 1440a b c d + 210b c d )x
--R +
--R 5 7 10 4 8 9 3 9 2 8 2 10 3 7
--R 792a b d + 4950a b c d + 9900a b c d + 7920a b c d
--R +
--R 11 4 6 12 5 5
--R 2520a b c d + 252b c d
--R *
--R 17
--R x
--R +
--R 6 6 10 5 7 9 4 8 2 8 3 9 3 7
--R 924a b d + 7920a b c d + 22275a b c d + 26400a b c d
--R +
--R 2 10 4 6 11 5 5 12 6 4
--R 13860a b c d + 3024a b c d + 210b c d
--R *
--R 16
--R x
--R +
--R 7 5 10 6 6 9 5 7 2 8 4 8 3 7
--R 792a b d + 9240a b c d + 35640a b c d + 59400a b c d
--R +
--R 3 9 4 6 2 10 5 5 11 6 4 12 7 3
--R 46200a b c d + 16632a b c d + 2520a b c d + 120b c d
--R *
--R 15
--R x
--R +
--R 8 4 10 7 5 9 6 6 2 8 5 7 3 7
--R 495a b d + 7920a b c d + 41580a b c d + 95040a b c d
--R +
--R 4 8 4 6 3 9 5 5 2 10 6 4 11 7 3
--R 103950a b c d + 55440a b c d + 13860a b c d + 1440a b c d
--R +
--R 12 8 2
--R 45b c d
--R *
--R 14
--R x
--R +
--R 9 3 10 8 4 9 7 5 2 8 6 6 3 7
--R 220a b d + 4950a b c d + 35640a b c d + 110880a b c d
--R +
--R 5 7 4 6 4 8 5 5 3 9 6 4 2 10 7 3
--R 166320a b c d + 124740a b c d + 46200a b c d + 7920a b c d
--R +
--R 11 8 2 12 9
--R 540a b c d + 10b c d
--R *
--R 13
--R x
--R +
--R 10 2 10 9 3 9 8 4 2 8 7 5 3 7
--R 66a b d + 2200a b c d + 22275a b c d + 95040a b c d
--R +
--R 6 6 4 6 5 7 5 5 4 8 6 4 3 9 7 3
--R 194040a b c d + 199584a b c d + 103950a b c d + 26400a b c d
--R +
--R 2 10 8 2 11 9 12 10
--R 2970a b c d + 120a b c d + b c
--R *
--R 12
--R x
--R +
--R 11 10 10 2 9 9 3 2 8 8 4 3 7
--R 12a b d + 660a b c d + 9900a b c d + 59400a b c d
--R +
--R 7 5 4 6 6 6 5 5 5 7 6 4 4 8 7 3
--R 166320a b c d + 232848a b c d + 166320a b c d + 59400a b c d
--R +
--R 3 9 8 2 2 10 9 11 10
--R 9900a b c d + 660a b c d + 12a b c
--R *
--R 11
--R x
--R +
--R 12 10 11 9 10 2 2 8 9 3 3 7 8 4 4 6
--R a d + 120a b c d + 2970a b c d + 26400a b c d + 103950a b c d
--R +
--R 7 5 5 5 6 6 6 4 5 7 7 3 4 8 8 2
--R 199584a b c d + 194040a b c d + 95040a b c d + 22275a b c d
--R +
--R 3 9 9 2 10 10
--R 2200a b c d + 66a b c
--R *
--R 10
--R x
--R +
--R 12 9 11 2 8 10 2 3 7 9 3 4 6
--R 10a c d + 540a b c d + 7920a b c d + 46200a b c d
--R +
--R 8 4 5 5 7 5 6 4 6 6 7 3 5 7 8 2
--R 124740a b c d + 166320a b c d + 110880a b c d + 35640a b c d
--R +
--R 4 8 9 3 9 10
--R 4950a b c d + 220a b c
--R *
--R 9
--R x
--R +
--R 12 2 8 11 3 7 10 2 4 6 9 3 5 5
--R 45a c d + 1440a b c d + 13860a b c d + 55440a b c d
--R +
--R 8 4 6 4 7 5 7 3 6 6 8 2 5 7 9 4 8 10
--R 103950a b c d + 95040a b c d + 41580a b c d + 7920a b c d + 495a b c
--R *
--R 8
--R x
--R +
--R 12 3 7 11 4 6 10 2 5 5 9 3 6 4
--R 120a c d + 2520a b c d + 16632a b c d + 46200a b c d
--R +
--R 8 4 7 3 7 5 8 2 6 6 9 5 7 10
--R 59400a b c d + 35640a b c d + 9240a b c d + 792a b c
--R *
--R 7
--R x
--R +
--R 12 4 6 11 5 5 10 2 6 4 9 3 7 3
--R 210a c d + 3024a b c d + 13860a b c d + 26400a b c d
--R +
--R 8 4 8 2 7 5 9 6 6 10
--R 22275a b c d + 7920a b c d + 924a b c
--R *
--R 6
--R x
--R +
--R 12 5 5 11 6 4 10 2 7 3 9 3 8 2 8 4 9
--R 252a c d + 2520a b c d + 7920a b c d + 9900a b c d + 4950a b c d
--R +
--R 7 5 10
--R 792a b c
--R *
--R 5
--R x
--R +
--R 12 6 4 11 7 3 10 2 8 2 9 3 9 8 4 10 4
--R (210a c d + 1440a b c d + 2970a b c d + 2200a b c d + 495a b c )x
--R +
--R 12 7 3 11 8 2 10 2 9 9 3 10 3
--R (120a c d + 540a b c d + 660a b c d + 220a b c )x
--R +
--R 12 8 2 11 9 10 2 10 2 12 9 11 10 12 10
--R (45a c d + 120a b c d + 66a b c )x + (10a c d + 12a b c )x + a c
--R Type: Polynomial(Integer)
--E 731
--S 732 of 2952
r0000:=1/13*(b*c-a*d)^10*(a+b*x)^13/b^11+_
5/7*d*(b*c-a*d)^9*(a+b*x)^14/b^11+_
3*d^2*(b*c-a*d)^8*(a+b*x)^15/b^11+_
15/2*d^3*(b*c-a*d)^7*(a+b*x)^16/b^11+_
210/17*d^4*(b*c-a*d)^6*(a+b*x)^17/b^11+_
14*d^5*(b*c-a*d)^5*(a+b*x)^18/b^11+_
210/19*d^6*(b*c-a*d)^4*(a+b*x)^19/b^11+_
6*d^7*(b*c-a*d)^3*(a+b*x)^20/b^11+_
15/7*d^8*(b*c-a*d)^2*(a+b*x)^21/b^11+_
5/11*d^9*(b*c-a*d)*(a+b*x)^22/b^11+1/23*d^10*(a+b*x)^23/b^11
--R
--R
--R (2)
--R 1 23 10 23 6 22 10 5 23 9 22
--R -- b d x + (-- a b d + -- b c d )x
--R 23 11 11
--R +
--R 22 2 21 10 40 22 9 15 23 2 8 21
--R (-- a b d + -- a b c d + -- b c d )x
--R 7 7 7
--R +
--R 3 20 10 2 21 9 22 2 8 23 3 7 20
--R (11a b d + 33a b c d + 27a b c d + 6b c d )x
--R +
--R 495 4 19 10 2200 3 20 9 2970 2 21 2 8 1440 22 3 7
--R --- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 19 19 19 19
--R +
--R 210 23 4 6
--R --- b c d
--R 19
--R *
--R 19
--R x
--R +
--R 5 18 10 4 19 9 3 20 2 8 2 21 3 7
--R 44a b d + 275a b c d + 550a b c d + 440a b c d
--R +
--R 22 4 6 23 5 5
--R 140a b c d + 14b c d
--R *
--R 18
--R x
--R +
--R 924 6 17 10 7920 5 18 9 22275 4 19 2 8 26400 3 20 3 7
--R --- a b d + ---- a b c d + ----- a b c d + ----- a b c d
--R 17 17 17 17
--R +
--R 13860 2 21 4 6 3024 22 5 5 210 23 6 4
--R ----- a b c d + ---- a b c d + --- b c d
--R 17 17 17
--R *
--R 17
--R x
--R +
--R 99 7 16 10 1155 6 17 9 4455 5 18 2 8 7425 4 19 3 7
--R -- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 2 2 2 2
--R +
--R 5775 3 20 4 6 2079 2 21 5 5 315 22 6 4 15 23 7 3
--R ---- a b c d + ---- a b c d + --- a b c d + -- b c d
--R 2 2 2 2
--R *
--R 16
--R x
--R +
--R 8 15 10 7 16 9 6 17 2 8 5 18 3 7
--R 33a b d + 528a b c d + 2772a b c d + 6336a b c d
--R +
--R 4 19 4 6 3 20 5 5 2 21 6 4 22 7 3 23 8 2
--R 6930a b c d + 3696a b c d + 924a b c d + 96a b c d + 3b c d
--R *
--R 15
--R x
--R +
--R 110 9 14 10 2475 8 15 9 17820 7 16 2 8 6 17 3 7
--R --- a b d + ---- a b c d + ----- a b c d + 7920a b c d
--R 7 7 7
--R +
--R 5 18 4 6 4 19 5 5 3 20 6 4 3960 2 21 7 3
--R 11880a b c d + 8910a b c d + 3300a b c d + ---- a b c d
--R 7
--R +
--R 270 22 8 2 5 23 9
--R --- a b c d + - b c d
--R 7 7
--R *
--R 14
--R x
--R +
--R 66 10 13 10 2200 9 14 9 22275 8 15 2 8 95040 7 16 3 7
--R -- a b d + ---- a b c d + ----- a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 194040 6 17 4 6 199584 5 18 5 5 103950 4 19 6 4
--R ------ a b c d + ------ a b c d + ------ a b c d
--R 13 13 13
--R +
--R 26400 3 20 7 3 2970 2 21 8 2 120 22 9 1 23 10
--R ----- a b c d + ---- a b c d + --- a b c d + -- b c
--R 13 13 13 13
--R *
--R 13
--R x
--R +
--R 11 12 10 10 13 9 9 14 2 8 8 15 3 7
--R a b d + 55a b c d + 825a b c d + 4950a b c d
--R +
--R 7 16 4 6 6 17 5 5 5 18 6 4 4 19 7 3
--R 13860a b c d + 19404a b c d + 13860a b c d + 4950a b c d
--R +
--R 3 20 8 2 2 21 9 22 10
--R 825a b c d + 55a b c d + a b c
--R *
--R 12
--R x
--R +
--R 1 12 11 10 120 11 12 9 10 13 2 8 9 14 3 7
--R -- a b d + --- a b c d + 270a b c d + 2400a b c d
--R 11 11
--R +
--R 8 15 4 6 7 16 5 5 6 17 6 4 5 18 7 3
--R 9450a b c d + 18144a b c d + 17640a b c d + 8640a b c d
--R +
--R 4 19 8 2 3 20 9 2 21 10
--R 2025a b c d + 200a b c d + 6a b c
--R *
--R 11
--R x
--R +
--R 12 11 9 11 12 2 8 10 13 3 7 9 14 4 6
--R a b c d + 54a b c d + 792a b c d + 4620a b c d
--R +
--R 8 15 5 5 7 16 6 4 6 17 7 3 5 18 8 2
--R 12474a b c d + 16632a b c d + 11088a b c d + 3564a b c d
--R +
--R 4 19 9 3 20 10
--R 495a b c d + 22a b c
--R *
--R 10
--R x
--R +
--R 12 11 2 8 11 12 3 7 10 13 4 6 9 14 5 5
--R 5a b c d + 160a b c d + 1540a b c d + 6160a b c d
--R +
--R 8 15 6 4 7 16 7 3 6 17 8 2 5 18 9
--R 11550a b c d + 10560a b c d + 4620a b c d + 880a b c d
--R +
--R 4 19 10
--R 55a b c
--R *
--R 9
--R x
--R +
--R 12 11 3 7 11 12 4 6 10 13 5 5 9 14 6 4
--R 15a b c d + 315a b c d + 2079a b c d + 5775a b c d
--R +
--R 8 15 7 3 7 16 8 2 6 17 9 5 18 10
--R 7425a b c d + 4455a b c d + 1155a b c d + 99a b c
--R *
--R 8
--R x
--R +
--R 12 11 4 6 11 12 5 5 10 13 6 4 26400 9 14 7 3
--R 30a b c d + 432a b c d + 1980a b c d + ----- a b c d
--R 7
--R +
--R 22275 8 15 8 2 7920 7 16 9 6 17 10
--R ----- a b c d + ---- a b c d + 132a b c
--R 7 7
--R *
--R 7
--R x
--R +
--R 12 11 5 5 11 12 6 4 10 13 7 3 9 14 8 2
--R 42a b c d + 420a b c d + 1320a b c d + 1650a b c d
--R +
--R 8 15 9 7 16 10
--R 825a b c d + 132a b c
--R *
--R 6
--R x
--R +
--R 12 11 6 4 11 12 7 3 10 13 8 2 9 14 9
--R 42a b c d + 288a b c d + 594a b c d + 440a b c d
--R +
--R 8 15 10
--R 99a b c
--R *
--R 5
--R x
--R +
--R 12 11 7 3 11 12 8 2 10 13 9 9 14 10 4
--R (30a b c d + 135a b c d + 165a b c d + 55a b c )x
--R +
--R 12 11 8 2 11 12 9 10 13 10 3
--R (15a b c d + 40a b c d + 22a b c )x
--R +
--R 12 11 9 11 12 10 2 12 11 10 1 23 10
--R (5a b c d + 6a b c )x + a b c x + -------- a d
--R 14872858
--R +
--R 1 22 9 1 21 2 2 8 1 20 3 3 7 5 19 4 4 6
--R - ------ a b c d + ----- a b c d - ---- a b c d + ---- a b c d
--R 646646 58786 8398 8398
--R +
--R 1 18 5 5 5 3 17 6 6 4 3 16 7 7 3 3 15 8 8 2
--R - --- a b c d + --- a b c d - --- a b c d + -- a b c d
--R 442 442 182 91
--R +
--R 5 14 9 9 1 13 10 10
--R - -- a b c d + -- a b c
--R 91 13
--R /
--R 11
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 732
--S 733 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 12 10 23 6 11 10 5 12 9 22
--R -- b d x + (-- a b d + -- b c d )x
--R 23 11 11
--R +
--R 22 2 10 10 40 11 9 15 12 2 8 21
--R (-- a b d + -- a b c d + -- b c d )x
--R 7 7 7
--R +
--R 3 9 10 2 10 9 11 2 8 12 3 7 20
--R (11a b d + 33a b c d + 27a b c d + 6b c d )x
--R +
--R 495 4 8 10 2200 3 9 9 2970 2 10 2 8 1440 11 3 7
--R --- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 19 19 19 19
--R +
--R 210 12 4 6
--R --- b c d
--R 19
--R *
--R 19
--R x
--R +
--R 5 7 10 4 8 9 3 9 2 8 2 10 3 7 11 4 6
--R 44a b d + 275a b c d + 550a b c d + 440a b c d + 140a b c d
--R +
--R 12 5 5
--R 14b c d
--R *
--R 18
--R x
--R +
--R 924 6 6 10 7920 5 7 9 22275 4 8 2 8 26400 3 9 3 7
--R --- a b d + ---- a b c d + ----- a b c d + ----- a b c d
--R 17 17 17 17
--R +
--R 13860 2 10 4 6 3024 11 5 5 210 12 6 4
--R ----- a b c d + ---- a b c d + --- b c d
--R 17 17 17
--R *
--R 17
--R x
--R +
--R 99 7 5 10 1155 6 6 9 4455 5 7 2 8 7425 4 8 3 7
--R -- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 2 2 2 2
--R +
--R 5775 3 9 4 6 2079 2 10 5 5 315 11 6 4 15 12 7 3
--R ---- a b c d + ---- a b c d + --- a b c d + -- b c d
--R 2 2 2 2
--R *
--R 16
--R x
--R +
--R 8 4 10 7 5 9 6 6 2 8 5 7 3 7 4 8 4 6
--R 33a b d + 528a b c d + 2772a b c d + 6336a b c d + 6930a b c d
--R +
--R 3 9 5 5 2 10 6 4 11 7 3 12 8 2
--R 3696a b c d + 924a b c d + 96a b c d + 3b c d
--R *
--R 15
--R x
--R +
--R 110 9 3 10 2475 8 4 9 17820 7 5 2 8 6 6 3 7
--R --- a b d + ---- a b c d + ----- a b c d + 7920a b c d
--R 7 7 7
--R +
--R 5 7 4 6 4 8 5 5 3 9 6 4 3960 2 10 7 3
--R 11880a b c d + 8910a b c d + 3300a b c d + ---- a b c d
--R 7
--R +
--R 270 11 8 2 5 12 9
--R --- a b c d + - b c d
--R 7 7
--R *
--R 14
--R x
--R +
--R 66 10 2 10 2200 9 3 9 22275 8 4 2 8 95040 7 5 3 7
--R -- a b d + ---- a b c d + ----- a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 194040 6 6 4 6 199584 5 7 5 5 103950 4 8 6 4 26400 3 9 7 3
--R ------ a b c d + ------ a b c d + ------ a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 2970 2 10 8 2 120 11 9 1 12 10
--R ---- a b c d + --- a b c d + -- b c
--R 13 13 13
--R *
--R 13
--R x
--R +
--R 11 10 10 2 9 9 3 2 8 8 4 3 7 7 5 4 6
--R a b d + 55a b c d + 825a b c d + 4950a b c d + 13860a b c d
--R +
--R 6 6 5 5 5 7 6 4 4 8 7 3 3 9 8 2 2 10 9
--R 19404a b c d + 13860a b c d + 4950a b c d + 825a b c d + 55a b c d
--R +
--R 11 10
--R a b c
--R *
--R 12
--R x
--R +
--R 1 12 10 120 11 9 10 2 2 8 9 3 3 7 8 4 4 6
--R -- a d + --- a b c d + 270a b c d + 2400a b c d + 9450a b c d
--R 11 11
--R +
--R 7 5 5 5 6 6 6 4 5 7 7 3 4 8 8 2
--R 18144a b c d + 17640a b c d + 8640a b c d + 2025a b c d
--R +
--R 3 9 9 2 10 10
--R 200a b c d + 6a b c
--R *
--R 11
--R x
--R +
--R 12 9 11 2 8 10 2 3 7 9 3 4 6 8 4 5 5
--R a c d + 54a b c d + 792a b c d + 4620a b c d + 12474a b c d
--R +
--R 7 5 6 4 6 6 7 3 5 7 8 2 4 8 9 3 9 10
--R 16632a b c d + 11088a b c d + 3564a b c d + 495a b c d + 22a b c
--R *
--R 10
--R x
--R +
--R 12 2 8 11 3 7 10 2 4 6 9 3 5 5 8 4 6 4
--R 5a c d + 160a b c d + 1540a b c d + 6160a b c d + 11550a b c d
--R +
--R 7 5 7 3 6 6 8 2 5 7 9 4 8 10
--R 10560a b c d + 4620a b c d + 880a b c d + 55a b c
--R *
--R 9
--R x
--R +
--R 12 3 7 11 4 6 10 2 5 5 9 3 6 4 8 4 7 3
--R 15a c d + 315a b c d + 2079a b c d + 5775a b c d + 7425a b c d
--R +
--R 7 5 8 2 6 6 9 5 7 10
--R 4455a b c d + 1155a b c d + 99a b c
--R *
--R 8
--R x
--R +
--R 12 4 6 11 5 5 10 2 6 4 26400 9 3 7 3
--R 30a c d + 432a b c d + 1980a b c d + ----- a b c d
--R 7
--R +
--R 22275 8 4 8 2 7920 7 5 9 6 6 10
--R ----- a b c d + ---- a b c d + 132a b c
--R 7 7
--R *
--R 7
--R x
--R +
--R 12 5 5 11 6 4 10 2 7 3 9 3 8 2 8 4 9
--R 42a c d + 420a b c d + 1320a b c d + 1650a b c d + 825a b c d
--R +
--R 7 5 10
--R 132a b c
--R *
--R 6
--R x
--R +
--R 12 6 4 11 7 3 10 2 8 2 9 3 9 8 4 10 5
--R (42a c d + 288a b c d + 594a b c d + 440a b c d + 99a b c )x
--R +
--R 12 7 3 11 8 2 10 2 9 9 3 10 4
--R (30a c d + 135a b c d + 165a b c d + 55a b c )x
--R +
--R 12 8 2 11 9 10 2 10 3 12 9 11 10 2 12 10
--R (15a c d + 40a b c d + 22a b c )x + (5a c d + 6a b c )x + a c x
--R Type: Polynomial(Fraction(Integer))
--E 733
--S 734 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 23 10 1 22 9 1 21 2 2 8 1 20 3 3 7
--R - -------- a d + ------ a b c d - ----- a b c d + ---- a b c d
--R 14872858 646646 58786 8398
--R +
--R 5 19 4 4 6 1 18 5 5 5 3 17 6 6 4 3 16 7 7 3
--R - ---- a b c d + --- a b c d - --- a b c d + --- a b c d
--R 8398 442 442 182
--R +
--R 3 15 8 8 2 5 14 9 9 1 13 10 10
--R - -- a b c d + -- a b c d - -- a b c
--R 91 91 13
--R /
--R 11
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 734
--S 735 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 735
)clear all
--S 736 of 2952
t0000:=(a+b*x)^11*(c+d*x)^10
--R
--R
--R (1)
--R 11 10 21 10 10 11 9 20
--R b d x + (11a b d + 10b c d )x
--R +
--R 2 9 10 10 9 11 2 8 19
--R (55a b d + 110a b c d + 45b c d )x
--R +
--R 3 8 10 2 9 9 10 2 8 11 3 7 18
--R (165a b d + 550a b c d + 495a b c d + 120b c d )x
--R +
--R 4 7 10 3 8 9 2 9 2 8 10 3 7 11 4 6 17
--R (330a b d + 1650a b c d + 2475a b c d + 1320a b c d + 210b c d )x
--R +
--R 5 6 10 4 7 9 3 8 2 8 2 9 3 7 10 4 6
--R 462a b d + 3300a b c d + 7425a b c d + 6600a b c d + 2310a b c d
--R +
--R 11 5 5
--R 252b c d
--R *
--R 16
--R x
--R +
--R 6 5 10 5 6 9 4 7 2 8 3 8 3 7
--R 462a b d + 4620a b c d + 14850a b c d + 19800a b c d
--R +
--R 2 9 4 6 10 5 5 11 6 4
--R 11550a b c d + 2772a b c d + 210b c d
--R *
--R 15
--R x
--R +
--R 7 4 10 6 5 9 5 6 2 8 4 7 3 7
--R 330a b d + 4620a b c d + 20790a b c d + 39600a b c d
--R +
--R 3 8 4 6 2 9 5 5 10 6 4 11 7 3
--R 34650a b c d + 13860a b c d + 2310a b c d + 120b c d
--R *
--R 14
--R x
--R +
--R 8 3 10 7 4 9 6 5 2 8 5 6 3 7
--R 165a b d + 3300a b c d + 20790a b c d + 55440a b c d
--R +
--R 4 7 4 6 3 8 5 5 2 9 6 4 10 7 3 11 8 2
--R 69300a b c d + 41580a b c d + 11550a b c d + 1320a b c d + 45b c d
--R *
--R 13
--R x
--R +
--R 9 2 10 8 3 9 7 4 2 8 6 5 3 7
--R 55a b d + 1650a b c d + 14850a b c d + 55440a b c d
--R +
--R 5 6 4 6 4 7 5 5 3 8 6 4 2 9 7 3
--R 97020a b c d + 83160a b c d + 34650a b c d + 6600a b c d
--R +
--R 10 8 2 11 9
--R 495a b c d + 10b c d
--R *
--R 12
--R x
--R +
--R 10 10 9 2 9 8 3 2 8 7 4 3 7 6 5 4 6
--R 11a b d + 550a b c d + 7425a b c d + 39600a b c d + 97020a b c d
--R +
--R 5 6 5 5 4 7 6 4 3 8 7 3 2 9 8 2
--R 116424a b c d + 69300a b c d + 19800a b c d + 2475a b c d
--R +
--R 10 9 11 10
--R 110a b c d + b c
--R *
--R 11
--R x
--R +
--R 11 10 10 9 9 2 2 8 8 3 3 7 7 4 4 6
--R a d + 110a b c d + 2475a b c d + 19800a b c d + 69300a b c d
--R +
--R 6 5 5 5 5 6 6 4 4 7 7 3 3 8 8 2
--R 116424a b c d + 97020a b c d + 39600a b c d + 7425a b c d
--R +
--R 2 9 9 10 10
--R 550a b c d + 11a b c
--R *
--R 10
--R x
--R +
--R 11 9 10 2 8 9 2 3 7 8 3 4 6 7 4 5 5
--R 10a c d + 495a b c d + 6600a b c d + 34650a b c d + 83160a b c d
--R +
--R 6 5 6 4 5 6 7 3 4 7 8 2 3 8 9 2 9 10
--R 97020a b c d + 55440a b c d + 14850a b c d + 1650a b c d + 55a b c
--R *
--R 9
--R x
--R +
--R 11 2 8 10 3 7 9 2 4 6 8 3 5 5
--R 45a c d + 1320a b c d + 11550a b c d + 41580a b c d
--R +
--R 7 4 6 4 6 5 7 3 5 6 8 2 4 7 9 3 8 10
--R 69300a b c d + 55440a b c d + 20790a b c d + 3300a b c d + 165a b c
--R *
--R 8
--R x
--R +
--R 11 3 7 10 4 6 9 2 5 5 8 3 6 4
--R 120a c d + 2310a b c d + 13860a b c d + 34650a b c d
--R +
--R 7 4 7 3 6 5 8 2 5 6 9 4 7 10
--R 39600a b c d + 20790a b c d + 4620a b c d + 330a b c
--R *
--R 7
--R x
--R +
--R 11 4 6 10 5 5 9 2 6 4 8 3 7 3
--R 210a c d + 2772a b c d + 11550a b c d + 19800a b c d
--R +
--R 7 4 8 2 6 5 9 5 6 10
--R 14850a b c d + 4620a b c d + 462a b c
--R *
--R 6
--R x
--R +
--R 11 5 5 10 6 4 9 2 7 3 8 3 8 2 7 4 9
--R 252a c d + 2310a b c d + 6600a b c d + 7425a b c d + 3300a b c d
--R +
--R 6 5 10
--R 462a b c
--R *
--R 5
--R x
--R +
--R 11 6 4 10 7 3 9 2 8 2 8 3 9 7 4 10 4
--R (210a c d + 1320a b c d + 2475a b c d + 1650a b c d + 330a b c )x
--R +
--R 11 7 3 10 8 2 9 2 9 8 3 10 3
--R (120a c d + 495a b c d + 550a b c d + 165a b c )x
--R +
--R 11 8 2 10 9 9 2 10 2 11 9 10 10 11 10
--R (45a c d + 110a b c d + 55a b c )x + (10a c d + 11a b c )x + a c
--R Type: Polynomial(Integer)
--E 736
--S 737 of 2952
r0000:=1/12*(b*c-a*d)^10*(a+b*x)^12/b^11+_
10/13*d*(b*c-a*d)^9*(a+b*x)^13/b^11+_
45/14*d^2*(b*c-a*d)^8*(a+b*x)^14/b^11+_
8*d^3*(b*c-a*d)^7*(a+b*x)^15/b^11+_
105/8*d^4*(b*c-a*d)^6*(a+b*x)^16/b^11+_
252/17*d^5*(b*c-a*d)^5*(a+b*x)^17/b^11+_
35/3*d^6*(b*c-a*d)^4*(a+b*x)^18/b^11+_
120/19*d^7*(b*c-a*d)^3*(a+b*x)^19/b^11+_
9/4*d^8*(b*c-a*d)^2*(a+b*x)^20/b^11+_
10/21*d^9*(b*c-a*d)*(a+b*x)^21/b^11+1/22*d^10*(a+b*x)^22/b^11
--R
--R
--R (2)
--R 1 22 10 22 11 21 10 10 22 9 21
--R -- b d x + (-- a b d + -- b c d )x
--R 22 21 21
--R +
--R 11 2 20 10 11 21 9 9 22 2 8 20
--R (-- a b d + -- a b c d + - b c d )x
--R 4 2 4
--R +
--R 165 3 19 10 550 2 20 9 495 21 2 8 120 22 3 7 19
--R (--- a b d + --- a b c d + --- a b c d + --- b c d )x
--R 19 19 19 19
--R +
--R 55 4 18 10 275 3 19 9 275 2 20 2 8 220 21 3 7
--R -- a b d + --- a b c d + --- a b c d + --- a b c d
--R 3 3 2 3
--R +
--R 35 22 4 6
--R -- b c d
--R 3
--R *
--R 18
--R x
--R +
--R 462 5 17 10 3300 4 18 9 7425 3 19 2 8 6600 2 20 3 7
--R --- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 17 17 17 17
--R +
--R 2310 21 4 6 252 22 5 5
--R ---- a b c d + --- b c d
--R 17 17
--R *
--R 17
--R x
--R +
--R 231 6 16 10 1155 5 17 9 7425 4 18 2 8 2475 3 19 3 7
--R --- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 8 4 8 2
--R +
--R 5775 2 20 4 6 693 21 5 5 105 22 6 4
--R ---- a b c d + --- a b c d + --- b c d
--R 8 4 8
--R *
--R 16
--R x
--R +
--R 7 15 10 6 16 9 5 17 2 8 4 18 3 7
--R 22a b d + 308a b c d + 1386a b c d + 2640a b c d
--R +
--R 3 19 4 6 2 20 5 5 21 6 4 22 7 3
--R 2310a b c d + 924a b c d + 154a b c d + 8b c d
--R *
--R 15
--R x
--R +
--R 165 8 14 10 1650 7 15 9 6 16 2 8 5 17 3 7
--R --- a b d + ---- a b c d + 1485a b c d + 3960a b c d
--R 14 7
--R +
--R 4 18 4 6 3 19 5 5 2 20 6 4 660 21 7 3
--R 4950a b c d + 2970a b c d + 825a b c d + --- a b c d
--R 7
--R +
--R 45 22 8 2
--R -- b c d
--R 14
--R *
--R 14
--R x
--R +
--R 55 9 13 10 1650 8 14 9 14850 7 15 2 8 55440 6 16 3 7
--R -- a b d + ---- a b c d + ----- a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 97020 5 17 4 6 83160 4 18 5 5 34650 3 19 6 4 6600 2 20 7 3
--R ----- a b c d + ----- a b c d + ----- a b c d + ---- a b c d
--R 13 13 13 13
--R +
--R 495 21 8 2 10 22 9
--R --- a b c d + -- b c d
--R 13 13
--R *
--R 13
--R x
--R +
--R 11 10 12 10 275 9 13 9 2475 8 14 2 8 7 15 3 7
--R -- a b d + --- a b c d + ---- a b c d + 3300a b c d
--R 12 6 4
--R +
--R 6 16 4 6 5 17 5 5 4 18 6 4 3 19 7 3
--R 8085a b c d + 9702a b c d + 5775a b c d + 1650a b c d
--R +
--R 825 2 20 8 2 55 21 9 1 22 10
--R --- a b c d + -- a b c d + -- b c
--R 4 6 12
--R *
--R 12
--R x
--R +
--R 1 11 11 10 10 12 9 9 13 2 8 8 14 3 7
--R -- a b d + 10a b c d + 225a b c d + 1800a b c d
--R 11
--R +
--R 7 15 4 6 6 16 5 5 5 17 6 4 4 18 7 3
--R 6300a b c d + 10584a b c d + 8820a b c d + 3600a b c d
--R +
--R 3 19 8 2 2 20 9 21 10
--R 675a b c d + 50a b c d + a b c
--R *
--R 11
--R x
--R +
--R 11 11 9 99 10 12 2 8 9 13 3 7 8 14 4 6
--R a b c d + -- a b c d + 660a b c d + 3465a b c d
--R 2
--R +
--R 7 15 5 5 6 16 6 4 5 17 7 3 4 18 8 2
--R 8316a b c d + 9702a b c d + 5544a b c d + 1485a b c d
--R +
--R 3 19 9 11 2 20 10
--R 165a b c d + -- a b c
--R 2
--R *
--R 10
--R x
--R +
--R 11 11 2 8 440 10 12 3 7 3850 9 13 4 6 8 14 5 5
--R 5a b c d + --- a b c d + ---- a b c d + 4620a b c d
--R 3 3
--R +
--R 7 15 6 4 6 16 7 3 5 17 8 2 1100 4 18 9
--R 7700a b c d + 6160a b c d + 2310a b c d + ---- a b c d
--R 3
--R +
--R 55 3 19 10
--R -- a b c
--R 3
--R *
--R 9
--R x
--R +
--R 11 11 3 7 1155 10 12 4 6 3465 9 13 5 5 17325 8 14 6 4
--R 15a b c d + ---- a b c d + ---- a b c d + ----- a b c d
--R 4 2 4
--R +
--R 7 15 7 3 10395 6 16 8 2 1155 5 17 9 165 4 18 10
--R 4950a b c d + ----- a b c d + ---- a b c d + --- a b c
--R 4 2 4
--R *
--R 8
--R x
--R +
--R 11 11 4 6 10 12 5 5 9 13 6 4 19800 8 14 7 3
--R 30a b c d + 396a b c d + 1650a b c d + ----- a b c d
--R 7
--R +
--R 14850 7 15 8 2 6 16 9 5 17 10
--R ----- a b c d + 660a b c d + 66a b c
--R 7
--R *
--R 7
--R x
--R +
--R 11 11 5 5 10 12 6 4 9 13 7 3 2475 8 14 8 2
--R 42a b c d + 385a b c d + 1100a b c d + ---- a b c d
--R 2
--R +
--R 7 15 9 6 16 10
--R 550a b c d + 77a b c
--R *
--R 6
--R x
--R +
--R 11 11 6 4 10 12 7 3 9 13 8 2 8 14 9
--R 42a b c d + 264a b c d + 495a b c d + 330a b c d
--R +
--R 7 15 10
--R 66a b c
--R *
--R 5
--R x
--R +
--R 11 11 7 3 495 10 12 8 2 275 9 13 9 165 8 14 10 4
--R (30a b c d + --- a b c d + --- a b c d + --- a b c )x
--R 4 2 4
--R +
--R 11 11 8 2 110 10 12 9 55 9 13 10 3
--R (15a b c d + --- a b c d + -- a b c )x
--R 3 3
--R +
--R 11 11 9 11 10 12 10 2 11 11 10 1 22 10
--R (5a b c d + -- a b c )x + a b c x + ------- a d
--R 2 7759752
--R +
--R 1 21 9 1 20 2 2 8 5 19 3 3 7 5 18 4 4 6
--R - ------ a b c d + ----- a b c d - ----- a b c d + ---- a b c d
--R 352716 33592 25194 5304
--R +
--R 3 17 5 5 5 1 16 6 6 4 2 15 7 7 3 15 14 8 8 2
--R - --- a b c d + --- a b c d - -- a b c d + --- a b c d
--R 884 104 91 364
--R +
--R 5 13 9 9 1 12 10 10
--R - -- a b c d + -- a b c
--R 78 12
--R /
--R 11
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 737
--S 738 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 11 10 22 11 10 10 10 11 9 21
--R -- b d x + (-- a b d + -- b c d )x
--R 22 21 21
--R +
--R 11 2 9 10 11 10 9 9 11 2 8 20
--R (-- a b d + -- a b c d + - b c d )x
--R 4 2 4
--R +
--R 165 3 8 10 550 2 9 9 495 10 2 8 120 11 3 7 19
--R (--- a b d + --- a b c d + --- a b c d + --- b c d )x
--R 19 19 19 19
--R +
--R 55 4 7 10 275 3 8 9 275 2 9 2 8 220 10 3 7 35 11 4 6 18
--R (-- a b d + --- a b c d + --- a b c d + --- a b c d + -- b c d )x
--R 3 3 2 3 3
--R +
--R 462 5 6 10 3300 4 7 9 7425 3 8 2 8 6600 2 9 3 7
--R --- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 17 17 17 17
--R +
--R 2310 10 4 6 252 11 5 5
--R ---- a b c d + --- b c d
--R 17 17
--R *
--R 17
--R x
--R +
--R 231 6 5 10 1155 5 6 9 7425 4 7 2 8 2475 3 8 3 7
--R --- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 8 4 8 2
--R +
--R 5775 2 9 4 6 693 10 5 5 105 11 6 4
--R ---- a b c d + --- a b c d + --- b c d
--R 8 4 8
--R *
--R 16
--R x
--R +
--R 7 4 10 6 5 9 5 6 2 8 4 7 3 7 3 8 4 6
--R 22a b d + 308a b c d + 1386a b c d + 2640a b c d + 2310a b c d
--R +
--R 2 9 5 5 10 6 4 11 7 3
--R 924a b c d + 154a b c d + 8b c d
--R *
--R 15
--R x
--R +
--R 165 8 3 10 1650 7 4 9 6 5 2 8 5 6 3 7
--R --- a b d + ---- a b c d + 1485a b c d + 3960a b c d
--R 14 7
--R +
--R 4 7 4 6 3 8 5 5 2 9 6 4 660 10 7 3 45 11 8 2
--R 4950a b c d + 2970a b c d + 825a b c d + --- a b c d + -- b c d
--R 7 14
--R *
--R 14
--R x
--R +
--R 55 9 2 10 1650 8 3 9 14850 7 4 2 8 55440 6 5 3 7
--R -- a b d + ---- a b c d + ----- a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 97020 5 6 4 6 83160 4 7 5 5 34650 3 8 6 4 6600 2 9 7 3
--R ----- a b c d + ----- a b c d + ----- a b c d + ---- a b c d
--R 13 13 13 13
--R +
--R 495 10 8 2 10 11 9
--R --- a b c d + -- b c d
--R 13 13
--R *
--R 13
--R x
--R +
--R 11 10 10 275 9 2 9 2475 8 3 2 8 7 4 3 7
--R -- a b d + --- a b c d + ---- a b c d + 3300a b c d
--R 12 6 4
--R +
--R 6 5 4 6 5 6 5 5 4 7 6 4 3 8 7 3
--R 8085a b c d + 9702a b c d + 5775a b c d + 1650a b c d
--R +
--R 825 2 9 8 2 55 10 9 1 11 10
--R --- a b c d + -- a b c d + -- b c
--R 4 6 12
--R *
--R 12
--R x
--R +
--R 1 11 10 10 9 9 2 2 8 8 3 3 7 7 4 4 6
--R -- a d + 10a b c d + 225a b c d + 1800a b c d + 6300a b c d
--R 11
--R +
--R 6 5 5 5 5 6 6 4 4 7 7 3 3 8 8 2 2 9 9
--R 10584a b c d + 8820a b c d + 3600a b c d + 675a b c d + 50a b c d
--R +
--R 10 10
--R a b c
--R *
--R 11
--R x
--R +
--R 11 9 99 10 2 8 9 2 3 7 8 3 4 6 7 4 5 5
--R a c d + -- a b c d + 660a b c d + 3465a b c d + 8316a b c d
--R 2
--R +
--R 6 5 6 4 5 6 7 3 4 7 8 2 3 8 9 11 2 9 10
--R 9702a b c d + 5544a b c d + 1485a b c d + 165a b c d + -- a b c
--R 2
--R *
--R 10
--R x
--R +
--R 11 2 8 440 10 3 7 3850 9 2 4 6 8 3 5 5 7 4 6 4
--R 5a c d + --- a b c d + ---- a b c d + 4620a b c d + 7700a b c d
--R 3 3
--R +
--R 6 5 7 3 5 6 8 2 1100 4 7 9 55 3 8 10
--R 6160a b c d + 2310a b c d + ---- a b c d + -- a b c
--R 3 3
--R *
--R 9
--R x
--R +
--R 11 3 7 1155 10 4 6 3465 9 2 5 5 17325 8 3 6 4
--R 15a c d + ---- a b c d + ---- a b c d + ----- a b c d
--R 4 2 4
--R +
--R 7 4 7 3 10395 6 5 8 2 1155 5 6 9 165 4 7 10
--R 4950a b c d + ----- a b c d + ---- a b c d + --- a b c
--R 4 2 4
--R *
--R 8
--R x
--R +
--R 11 4 6 10 5 5 9 2 6 4 19800 8 3 7 3
--R 30a c d + 396a b c d + 1650a b c d + ----- a b c d
--R 7
--R +
--R 14850 7 4 8 2 6 5 9 5 6 10
--R ----- a b c d + 660a b c d + 66a b c
--R 7
--R *
--R 7
--R x
--R +
--R 11 5 5 10 6 4 9 2 7 3 2475 8 3 8 2 7 4 9
--R 42a c d + 385a b c d + 1100a b c d + ---- a b c d + 550a b c d
--R 2
--R +
--R 6 5 10
--R 77a b c
--R *
--R 6
--R x
--R +
--R 11 6 4 10 7 3 9 2 8 2 8 3 9 7 4 10 5
--R (42a c d + 264a b c d + 495a b c d + 330a b c d + 66a b c )x
--R +
--R 11 7 3 495 10 8 2 275 9 2 9 165 8 3 10 4
--R (30a c d + --- a b c d + --- a b c d + --- a b c )x
--R 4 2 4
--R +
--R 11 8 2 110 10 9 55 9 2 10 3 11 9 11 10 10 2
--R (15a c d + --- a b c d + -- a b c )x + (5a c d + -- a b c )x
--R 3 3 2
--R +
--R 11 10
--R a c x
--R Type: Polynomial(Fraction(Integer))
--E 738
--S 739 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 22 10 1 21 9 1 20 2 2 8 5 19 3 3 7
--R - ------- a d + ------ a b c d - ----- a b c d + ----- a b c d
--R 7759752 352716 33592 25194
--R +
--R 5 18 4 4 6 3 17 5 5 5 1 16 6 6 4 2 15 7 7 3
--R - ---- a b c d + --- a b c d - --- a b c d + -- a b c d
--R 5304 884 104 91
--R +
--R 15 14 8 8 2 5 13 9 9 1 12 10 10
--R - --- a b c d + -- a b c d - -- a b c
--R 364 78 12
--R /
--R 11
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 739
--S 740 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 740
)clear all
--S 741 of 2952
t0000:=(a+b*x)^10*(c+d*x)^10
--R
--R
--R (1)
--R 10 10 20 9 10 10 9 19
--R b d x + (10a b d + 10b c d )x
--R +
--R 2 8 10 9 9 10 2 8 18
--R (45a b d + 100a b c d + 45b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 17
--R (120a b d + 450a b c d + 450a b c d + 120b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 16
--R (210a b d + 1200a b c d + 2025a b c d + 1200a b c d + 210b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7 9 4 6
--R 252a b d + 2100a b c d + 5400a b c d + 5400a b c d + 2100a b c d
--R +
--R 10 5 5
--R 252b c d
--R *
--R 15
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7 2 8 4 6
--R 210a b d + 2520a b c d + 9450a b c d + 14400a b c d + 9450a b c d
--R +
--R 9 5 5 10 6 4
--R 2520a b c d + 210b c d
--R *
--R 14
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 120a b d + 2100a b c d + 11340a b c d + 25200a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R 25200a b c d + 11340a b c d + 2100a b c d + 120b c d
--R *
--R 13
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7 4 6 4 6
--R 45a b d + 1200a b c d + 9450a b c d + 30240a b c d + 44100a b c d
--R +
--R 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R 30240a b c d + 9450a b c d + 1200a b c d + 45b c d
--R *
--R 12
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R 10a b d + 450a b c d + 5400a b c d + 25200a b c d + 52920a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2 10 9
--R 52920a b c d + 25200a b c d + 5400a b c d + 450a b c d + 10b c d
--R *
--R 11
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R a d + 100a b c d + 2025a b c d + 14400a b c d + 44100a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2
--R 63504a b c d + 44100a b c d + 14400a b c d + 2025a b c d
--R +
--R 9 9 10 10
--R 100a b c d + b c
--R *
--R 10
--R x
--R +
--R 10 9 9 2 8 8 2 3 7 7 3 4 6 6 4 5 5
--R 10a c d + 450a b c d + 5400a b c d + 25200a b c d + 52920a b c d
--R +
--R 5 5 6 4 4 6 7 3 3 7 8 2 2 8 9 9 10
--R 52920a b c d + 25200a b c d + 5400a b c d + 450a b c d + 10a b c
--R *
--R 9
--R x
--R +
--R 10 2 8 9 3 7 8 2 4 6 7 3 5 5 6 4 6 4
--R 45a c d + 1200a b c d + 9450a b c d + 30240a b c d + 44100a b c d
--R +
--R 5 5 7 3 4 6 8 2 3 7 9 2 8 10
--R 30240a b c d + 9450a b c d + 1200a b c d + 45a b c
--R *
--R 8
--R x
--R +
--R 10 3 7 9 4 6 8 2 5 5 7 3 6 4
--R 120a c d + 2100a b c d + 11340a b c d + 25200a b c d
--R +
--R 6 4 7 3 5 5 8 2 4 6 9 3 7 10
--R 25200a b c d + 11340a b c d + 2100a b c d + 120a b c
--R *
--R 7
--R x
--R +
--R 10 4 6 9 5 5 8 2 6 4 7 3 7 3 6 4 8 2
--R 210a c d + 2520a b c d + 9450a b c d + 14400a b c d + 9450a b c d
--R +
--R 5 5 9 4 6 10
--R 2520a b c d + 210a b c
--R *
--R 6
--R x
--R +
--R 10 5 5 9 6 4 8 2 7 3 7 3 8 2 6 4 9
--R 252a c d + 2100a b c d + 5400a b c d + 5400a b c d + 2100a b c d
--R +
--R 5 5 10
--R 252a b c
--R *
--R 5
--R x
--R +
--R 10 6 4 9 7 3 8 2 8 2 7 3 9 6 4 10 4
--R (210a c d + 1200a b c d + 2025a b c d + 1200a b c d + 210a b c )x
--R +
--R 10 7 3 9 8 2 8 2 9 7 3 10 3
--R (120a c d + 450a b c d + 450a b c d + 120a b c )x
--R +
--R 10 8 2 9 9 8 2 10 2 10 9 9 10 10 10
--R (45a c d + 100a b c d + 45a b c )x + (10a c d + 10a b c )x + a c
--R Type: Polynomial(Integer)
--E 741
--S 742 of 2952
r0000:=1/11*(b*c-a*d)^10*(a+b*x)^11/b^11+5/6*d*(b*c-a*d)^9*(a+b*x)^12/b^11+_
45/13*d^2*(b*c-a*d)^8*(a+b*x)^13/b^11+_
60/7*d^3*(b*c-a*d)^7*(a+b*x)^14/b^11+_
14*d^4*(b*c-a*d)^6*(a+b*x)^15/b^11+_
63/4*d^5*(b*c-a*d)^5*(a+b*x)^16/b^11+_
210/17*d^6*(b*c-a*d)^4*(a+b*x)^17/b^11+_
20/3*d^7*(b*c-a*d)^3*(a+b*x)^18/b^11+_
45/19*d^8*(b*c-a*d)^2*(a+b*x)^19/b^11+_
1/2*d^9*(b*c-a*d)*(a+b*x)^20/b^11+1/21*d^10*(a+b*x)^21/b^11
--R
--R
--R (2)
--R 1 21 10 21 1 20 10 1 21 9 20
--R -- b d x + (- a b d + - b c d )x
--R 21 2 2
--R +
--R 45 2 19 10 100 20 9 45 21 2 8 19
--R (-- a b d + --- a b c d + -- b c d )x
--R 19 19 19
--R +
--R 20 3 18 10 2 19 9 20 2 8 20 21 3 7 18
--R (-- a b d + 25a b c d + 25a b c d + -- b c d )x
--R 3 3
--R +
--R 210 4 17 10 1200 3 18 9 2025 2 19 2 8 1200 20 3 7
--R --- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 17 17 17 17
--R +
--R 210 21 4 6
--R --- b c d
--R 17
--R *
--R 17
--R x
--R +
--R 63 5 16 10 525 4 17 9 675 3 18 2 8 675 2 19 3 7
--R -- a b d + --- a b c d + --- a b c d + --- a b c d
--R 4 4 2 2
--R +
--R 525 20 4 6 63 21 5 5
--R --- a b c d + -- b c d
--R 4 4
--R *
--R 16
--R x
--R +
--R 6 15 10 5 16 9 4 17 2 8 3 18 3 7
--R 14a b d + 168a b c d + 630a b c d + 960a b c d
--R +
--R 2 19 4 6 20 5 5 21 6 4
--R 630a b c d + 168a b c d + 14b c d
--R *
--R 15
--R x
--R +
--R 60 7 14 10 6 15 9 5 16 2 8 4 17 3 7
--R -- a b d + 150a b c d + 810a b c d + 1800a b c d
--R 7
--R +
--R 3 18 4 6 2 19 5 5 20 6 4 60 21 7 3
--R 1800a b c d + 810a b c d + 150a b c d + -- b c d
--R 7
--R *
--R 14
--R x
--R +
--R 45 8 13 10 1200 7 14 9 9450 6 15 2 8 30240 5 16 3 7
--R -- a b d + ---- a b c d + ---- a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 44100 4 17 4 6 30240 3 18 5 5 9450 2 19 6 4 1200 20 7 3
--R ----- a b c d + ----- a b c d + ---- a b c d + ---- a b c d
--R 13 13 13 13
--R +
--R 45 21 8 2
--R -- b c d
--R 13
--R *
--R 13
--R x
--R +
--R 5 9 12 10 75 8 13 9 7 14 2 8 6 15 3 7
--R - a b d + -- a b c d + 450a b c d + 2100a b c d
--R 6 2
--R +
--R 5 16 4 6 4 17 5 5 3 18 6 4 2 19 7 3
--R 4410a b c d + 4410a b c d + 2100a b c d + 450a b c d
--R +
--R 75 20 8 2 5 21 9
--R -- a b c d + - b c d
--R 2 6
--R *
--R 12
--R x
--R +
--R 1 10 11 10 100 9 12 9 2025 8 13 2 8 14400 7 14 3 7
--R -- a b d + --- a b c d + ---- a b c d + ----- a b c d
--R 11 11 11 11
--R +
--R 44100 6 15 4 6 63504 5 16 5 5 44100 4 17 6 4 14400 3 18 7 3
--R ----- a b c d + ----- a b c d + ----- a b c d + ----- a b c d
--R 11 11 11 11
--R +
--R 2025 2 19 8 2 100 20 9 1 21 10
--R ---- a b c d + --- a b c d + -- b c
--R 11 11 11
--R *
--R 11
--R x
--R +
--R 10 11 9 9 12 2 8 8 13 3 7 7 14 4 6
--R a b c d + 45a b c d + 540a b c d + 2520a b c d
--R +
--R 6 15 5 5 5 16 6 4 4 17 7 3 3 18 8 2
--R 5292a b c d + 5292a b c d + 2520a b c d + 540a b c d
--R +
--R 2 19 9 20 10
--R 45a b c d + a b c
--R *
--R 10
--R x
--R +
--R 10 11 2 8 400 9 12 3 7 8 13 4 6 7 14 5 5
--R 5a b c d + --- a b c d + 1050a b c d + 3360a b c d
--R 3
--R +
--R 6 15 6 4 5 16 7 3 4 17 8 2 400 3 18 9
--R 4900a b c d + 3360a b c d + 1050a b c d + --- a b c d
--R 3
--R +
--R 2 19 10
--R 5a b c
--R *
--R 9
--R x
--R +
--R 10 11 3 7 525 9 12 4 6 2835 8 13 5 5 7 14 6 4
--R 15a b c d + --- a b c d + ---- a b c d + 3150a b c d
--R 2 2
--R +
--R 6 15 7 3 2835 5 16 8 2 525 4 17 9 3 18 10
--R 3150a b c d + ---- a b c d + --- a b c d + 15a b c
--R 2 2
--R *
--R 8
--R x
--R +
--R 10 11 4 6 9 12 5 5 8 13 6 4 14400 7 14 7 3
--R 30a b c d + 360a b c d + 1350a b c d + ----- a b c d
--R 7
--R +
--R 6 15 8 2 5 16 9 4 17 10
--R 1350a b c d + 360a b c d + 30a b c
--R *
--R 7
--R x
--R +
--R 10 11 5 5 9 12 6 4 8 13 7 3 7 14 8 2
--R 42a b c d + 350a b c d + 900a b c d + 900a b c d
--R +
--R 6 15 9 5 16 10
--R 350a b c d + 42a b c
--R *
--R 6
--R x
--R +
--R 10 11 6 4 9 12 7 3 8 13 8 2 7 14 9 6 15 10 5
--R (42a b c d + 240a b c d + 405a b c d + 240a b c d + 42a b c )x
--R +
--R 10 11 7 3 225 9 12 8 2 225 8 13 9 7 14 10 4
--R (30a b c d + --- a b c d + --- a b c d + 30a b c )x
--R 2 2
--R +
--R 10 11 8 2 100 9 12 9 8 13 10 3 10 11 9 9 12 10 2
--R (15a b c d + --- a b c d + 15a b c )x + (5a b c d + 5a b c )x
--R 3
--R +
--R 10 11 10 1 21 10 1 20 9 5 19 2 2 8
--R a b c x + ------- a d - ------ a b c d + ----- a b c d
--R 3879876 184756 92378
--R +
--R 5 18 3 3 7 15 17 4 4 6 3 16 5 5 5 2 15 6 6 4
--R - ----- a b c d + ---- a b c d - --- a b c d + --- a b c d
--R 14586 9724 572 143
--R +
--R 30 14 7 7 3 15 13 8 8 2 5 12 9 9 1 11 10 10
--R - ---- a b c d + --- a b c d - -- a b c d + -- a b c
--R 1001 286 66 11
--R /
--R 11
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 742
--S 743 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 10 10 21 1 9 10 1 10 9 20
--R -- b d x + (- a b d + - b c d )x
--R 21 2 2
--R +
--R 45 2 8 10 100 9 9 45 10 2 8 19
--R (-- a b d + --- a b c d + -- b c d )x
--R 19 19 19
--R +
--R 20 3 7 10 2 8 9 9 2 8 20 10 3 7 18
--R (-- a b d + 25a b c d + 25a b c d + -- b c d )x
--R 3 3
--R +
--R 210 4 6 10 1200 3 7 9 2025 2 8 2 8 1200 9 3 7
--R --- a b d + ---- a b c d + ---- a b c d + ---- a b c d
--R 17 17 17 17
--R +
--R 210 10 4 6
--R --- b c d
--R 17
--R *
--R 17
--R x
--R +
--R 63 5 5 10 525 4 6 9 675 3 7 2 8 675 2 8 3 7 525 9 4 6
--R -- a b d + --- a b c d + --- a b c d + --- a b c d + --- a b c d
--R 4 4 2 2 4
--R +
--R 63 10 5 5
--R -- b c d
--R 4
--R *
--R 16
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7 2 8 4 6
--R 14a b d + 168a b c d + 630a b c d + 960a b c d + 630a b c d
--R +
--R 9 5 5 10 6 4
--R 168a b c d + 14b c d
--R *
--R 15
--R x
--R +
--R 60 7 3 10 6 4 9 5 5 2 8 4 6 3 7 3 7 4 6
--R -- a b d + 150a b c d + 810a b c d + 1800a b c d + 1800a b c d
--R 7
--R +
--R 2 8 5 5 9 6 4 60 10 7 3
--R 810a b c d + 150a b c d + -- b c d
--R 7
--R *
--R 14
--R x
--R +
--R 45 8 2 10 1200 7 3 9 9450 6 4 2 8 30240 5 5 3 7
--R -- a b d + ---- a b c d + ---- a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 44100 4 6 4 6 30240 3 7 5 5 9450 2 8 6 4 1200 9 7 3
--R ----- a b c d + ----- a b c d + ---- a b c d + ---- a b c d
--R 13 13 13 13
--R +
--R 45 10 8 2
--R -- b c d
--R 13
--R *
--R 13
--R x
--R +
--R 5 9 10 75 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - a b d + -- a b c d + 450a b c d + 2100a b c d + 4410a b c d
--R 6 2
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 75 9 8 2 5 10 9
--R 4410a b c d + 2100a b c d + 450a b c d + -- a b c d + - b c d
--R 2 6
--R *
--R 12
--R x
--R +
--R 1 10 10 100 9 9 2025 8 2 2 8 14400 7 3 3 7
--R -- a d + --- a b c d + ---- a b c d + ----- a b c d
--R 11 11 11 11
--R +
--R 44100 6 4 4 6 63504 5 5 5 5 44100 4 6 6 4 14400 3 7 7 3
--R ----- a b c d + ----- a b c d + ----- a b c d + ----- a b c d
--R 11 11 11 11
--R +
--R 2025 2 8 8 2 100 9 9 1 10 10
--R ---- a b c d + --- a b c d + -- b c
--R 11 11 11
--R *
--R 11
--R x
--R +
--R 10 9 9 2 8 8 2 3 7 7 3 4 6 6 4 5 5
--R a c d + 45a b c d + 540a b c d + 2520a b c d + 5292a b c d
--R +
--R 5 5 6 4 4 6 7 3 3 7 8 2 2 8 9 9 10
--R 5292a b c d + 2520a b c d + 540a b c d + 45a b c d + a b c
--R *
--R 10
--R x
--R +
--R 10 2 8 400 9 3 7 8 2 4 6 7 3 5 5 6 4 6 4
--R 5a c d + --- a b c d + 1050a b c d + 3360a b c d + 4900a b c d
--R 3
--R +
--R 5 5 7 3 4 6 8 2 400 3 7 9 2 8 10
--R 3360a b c d + 1050a b c d + --- a b c d + 5a b c
--R 3
--R *
--R 9
--R x
--R +
--R 10 3 7 525 9 4 6 2835 8 2 5 5 7 3 6 4 6 4 7 3
--R 15a c d + --- a b c d + ---- a b c d + 3150a b c d + 3150a b c d
--R 2 2
--R +
--R 2835 5 5 8 2 525 4 6 9 3 7 10
--R ---- a b c d + --- a b c d + 15a b c
--R 2 2
--R *
--R 8
--R x
--R +
--R 10 4 6 9 5 5 8 2 6 4 14400 7 3 7 3 6 4 8 2
--R 30a c d + 360a b c d + 1350a b c d + ----- a b c d + 1350a b c d
--R 7
--R +
--R 5 5 9 4 6 10
--R 360a b c d + 30a b c
--R *
--R 7
--R x
--R +
--R 10 5 5 9 6 4 8 2 7 3 7 3 8 2 6 4 9
--R 42a c d + 350a b c d + 900a b c d + 900a b c d + 350a b c d
--R +
--R 5 5 10
--R 42a b c
--R *
--R 6
--R x
--R +
--R 10 6 4 9 7 3 8 2 8 2 7 3 9 6 4 10 5
--R (42a c d + 240a b c d + 405a b c d + 240a b c d + 42a b c )x
--R +
--R 10 7 3 225 9 8 2 225 8 2 9 7 3 10 4
--R (30a c d + --- a b c d + --- a b c d + 30a b c )x
--R 2 2
--R +
--R 10 8 2 100 9 9 8 2 10 3 10 9 9 10 2 10 10
--R (15a c d + --- a b c d + 15a b c )x + (5a c d + 5a b c )x + a c x
--R 3
--R Type: Polynomial(Fraction(Integer))
--E 743
--S 744 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 21 10 1 20 9 5 19 2 2 8 5 18 3 3 7
--R - ------- a d + ------ a b c d - ----- a b c d + ----- a b c d
--R 3879876 184756 92378 14586
--R +
--R 15 17 4 4 6 3 16 5 5 5 2 15 6 6 4 30 14 7 7 3
--R - ---- a b c d + --- a b c d - --- a b c d + ---- a b c d
--R 9724 572 143 1001
--R +
--R 15 13 8 8 2 5 12 9 9 1 11 10 10
--R - --- a b c d + -- a b c d - -- a b c
--R 286 66 11
--R /
--R 11
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 744
--S 745 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 745
)clear all
--S 746 of 2952
t0000:=(a+b*x)^9*(c+d*x)^10
--R
--R
--R (1)
--R 9 10 19 8 10 9 9 18
--R b d x + (9a b d + 10b c d )x
--R +
--R 2 7 10 8 9 9 2 8 17
--R (36a b d + 90a b c d + 45b c d )x
--R +
--R 3 6 10 2 7 9 8 2 8 9 3 7 16
--R (84a b d + 360a b c d + 405a b c d + 120b c d )x
--R +
--R 4 5 10 3 6 9 2 7 2 8 8 3 7 9 4 6 15
--R (126a b d + 840a b c d + 1620a b c d + 1080a b c d + 210b c d )x
--R +
--R 5 4 10 4 5 9 3 6 2 8 2 7 3 7 8 4 6
--R 126a b d + 1260a b c d + 3780a b c d + 4320a b c d + 1890a b c d
--R +
--R 9 5 5
--R 252b c d
--R *
--R 14
--R x
--R +
--R 6 3 10 5 4 9 4 5 2 8 3 6 3 7 2 7 4 6
--R 84a b d + 1260a b c d + 5670a b c d + 10080a b c d + 7560a b c d
--R +
--R 8 5 5 9 6 4
--R 2268a b c d + 210b c d
--R *
--R 13
--R x
--R +
--R 7 2 10 6 3 9 5 4 2 8 4 5 3 7 3 6 4 6
--R 36a b d + 840a b c d + 5670a b c d + 15120a b c d + 17640a b c d
--R +
--R 2 7 5 5 8 6 4 9 7 3
--R 9072a b c d + 1890a b c d + 120b c d
--R *
--R 12
--R x
--R +
--R 8 10 7 2 9 6 3 2 8 5 4 3 7 4 5 4 6
--R 9a b d + 360a b c d + 3780a b c d + 15120a b c d + 26460a b c d
--R +
--R 3 6 5 5 2 7 6 4 8 7 3 9 8 2
--R 21168a b c d + 7560a b c d + 1080a b c d + 45b c d
--R *
--R 11
--R x
--R +
--R 9 10 8 9 7 2 2 8 6 3 3 7 5 4 4 6
--R a d + 90a b c d + 1620a b c d + 10080a b c d + 26460a b c d
--R +
--R 4 5 5 5 3 6 6 4 2 7 7 3 8 8 2 9 9
--R 31752a b c d + 17640a b c d + 4320a b c d + 405a b c d + 10b c d
--R *
--R 10
--R x
--R +
--R 9 9 8 2 8 7 2 3 7 6 3 4 6 5 4 5 5
--R 10a c d + 405a b c d + 4320a b c d + 17640a b c d + 31752a b c d
--R +
--R 4 5 6 4 3 6 7 3 2 7 8 2 8 9 9 10
--R 26460a b c d + 10080a b c d + 1620a b c d + 90a b c d + b c
--R *
--R 9
--R x
--R +
--R 9 2 8 8 3 7 7 2 4 6 6 3 5 5 5 4 6 4
--R 45a c d + 1080a b c d + 7560a b c d + 21168a b c d + 26460a b c d
--R +
--R 4 5 7 3 3 6 8 2 2 7 9 8 10
--R 15120a b c d + 3780a b c d + 360a b c d + 9a b c
--R *
--R 8
--R x
--R +
--R 9 3 7 8 4 6 7 2 5 5 6 3 6 4 5 4 7 3
--R 120a c d + 1890a b c d + 9072a b c d + 17640a b c d + 15120a b c d
--R +
--R 4 5 8 2 3 6 9 2 7 10
--R 5670a b c d + 840a b c d + 36a b c
--R *
--R 7
--R x
--R +
--R 9 4 6 8 5 5 7 2 6 4 6 3 7 3 5 4 8 2
--R 210a c d + 2268a b c d + 7560a b c d + 10080a b c d + 5670a b c d
--R +
--R 4 5 9 3 6 10
--R 1260a b c d + 84a b c
--R *
--R 6
--R x
--R +
--R 9 5 5 8 6 4 7 2 7 3 6 3 8 2 5 4 9
--R 252a c d + 1890a b c d + 4320a b c d + 3780a b c d + 1260a b c d
--R +
--R 4 5 10
--R 126a b c
--R *
--R 5
--R x
--R +
--R 9 6 4 8 7 3 7 2 8 2 6 3 9 5 4 10 4
--R (210a c d + 1080a b c d + 1620a b c d + 840a b c d + 126a b c )x
--R +
--R 9 7 3 8 8 2 7 2 9 6 3 10 3
--R (120a c d + 405a b c d + 360a b c d + 84a b c )x
--R +
--R 9 8 2 8 9 7 2 10 2 9 9 8 10 9 10
--R (45a c d + 90a b c d + 36a b c )x + (10a c d + 9a b c )x + a c
--R Type: Polynomial(Integer)
--E 746
--S 747 of 2952
r0000:=-1/11*(b*c-a*d)^9*(c+d*x)^11/d^10+3/4*b*(b*c-a*d)^8*(c+d*x)^12/d^10-_
36/13*b^2*(b*c-a*d)^7*(c+d*x)^13/d^10+_
6*b^3*(b*c-a*d)^6*(c+d*x)^14/d^10-_
42/5*b^4*(b*c-a*d)^5*(c+d*x)^15/d^10+_
63/8*b^5*(b*c-a*d)^4*(c+d*x)^16/d^10-_
84/17*b^6*(b*c-a*d)^3*(c+d*x)^17/d^10+_
2*b^7*(b*c-a*d)^2*(c+d*x)^18/d^10-_
9/19*b^8*(b*c-a*d)*(c+d*x)^19/d^10+1/20*b^9*(c+d*x)^20/d^10
--R
--R
--R (2)
--R 1 9 20 20 9 8 20 10 9 19 19
--R -- b d x + (-- a b d + -- b c d )x
--R 20 19 19
--R +
--R 2 7 20 8 19 5 9 2 18 18
--R (2a b d + 5a b c d + - b c d )x
--R 2
--R +
--R 84 3 6 20 360 2 7 19 405 8 2 18 120 9 3 17 17
--R (-- a b d + --- a b c d + --- a b c d + --- b c d )x
--R 17 17 17 17
--R +
--R 63 4 5 20 105 3 6 19 405 2 7 2 18 135 8 3 17
--R -- a b d + --- a b c d + --- a b c d + --- a b c d
--R 8 2 4 2
--R +
--R 105 9 4 16
--R --- b c d
--R 8
--R *
--R 16
--R x
--R +
--R 42 5 4 20 4 5 19 3 6 2 18 2 7 3 17 8 4 16
--R -- a b d + 84a b c d + 252a b c d + 288a b c d + 126a b c d
--R 5
--R +
--R 84 9 5 15
--R -- b c d
--R 5
--R *
--R 15
--R x
--R +
--R 6 3 20 5 4 19 4 5 2 18 3 6 3 17 2 7 4 16
--R 6a b d + 90a b c d + 405a b c d + 720a b c d + 540a b c d
--R +
--R 8 5 15 9 6 14
--R 162a b c d + 15b c d
--R *
--R 14
--R x
--R +
--R 36 7 2 20 840 6 3 19 5670 5 4 2 18 15120 4 5 3 17
--R -- a b d + --- a b c d + ---- a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 17640 3 6 4 16 9072 2 7 5 15 1890 8 6 14 120 9 7 13
--R ----- a b c d + ---- a b c d + ---- a b c d + --- b c d
--R 13 13 13 13
--R *
--R 13
--R x
--R +
--R 3 8 20 7 2 19 6 3 2 18 5 4 3 17
--R - a b d + 30a b c d + 315a b c d + 1260a b c d
--R 4
--R +
--R 4 5 4 16 3 6 5 15 2 7 6 14 8 7 13 15 9 8 12
--R 2205a b c d + 1764a b c d + 630a b c d + 90a b c d + -- b c d
--R 4
--R *
--R 12
--R x
--R +
--R 1 9 20 90 8 19 1620 7 2 2 18 10080 6 3 3 17
--R -- a d + -- a b c d + ---- a b c d + ----- a b c d
--R 11 11 11 11
--R +
--R 26460 5 4 4 16 31752 4 5 5 15 17640 3 6 6 14 4320 2 7 7 13
--R ----- a b c d + ----- a b c d + ----- a b c d + ---- a b c d
--R 11 11 11 11
--R +
--R 405 8 8 12 10 9 9 11
--R --- a b c d + -- b c d
--R 11 11
--R *
--R 11
--R x
--R +
--R 9 19 81 8 2 18 7 2 3 17 6 3 4 16
--R a c d + -- a b c d + 432a b c d + 1764a b c d
--R 2
--R +
--R 15876 5 4 5 15 4 5 6 14 3 6 7 13 2 7 8 12
--R ----- a b c d + 2646a b c d + 1008a b c d + 162a b c d
--R 5
--R +
--R 8 9 11 1 9 10 10
--R 9a b c d + -- b c d
--R 10
--R *
--R 10
--R x
--R +
--R 9 2 18 8 3 17 7 2 4 16 6 3 5 15
--R 5a c d + 120a b c d + 840a b c d + 2352a b c d
--R +
--R 5 4 6 14 4 5 7 13 3 6 8 12 2 7 9 11 8 10 10
--R 2940a b c d + 1680a b c d + 420a b c d + 40a b c d + a b c d
--R *
--R 9
--R x
--R +
--R 9 3 17 945 8 4 16 7 2 5 15 6 3 6 14
--R 15a c d + --- a b c d + 1134a b c d + 2205a b c d
--R 4
--R +
--R 5 4 7 13 2835 4 5 8 12 3 6 9 11 9 2 7 10 10
--R 1890a b c d + ---- a b c d + 105a b c d + - a b c d
--R 4 2
--R *
--R 8
--R x
--R +
--R 9 4 16 8 5 15 7 2 6 14 6 3 7 13
--R 30a c d + 324a b c d + 1080a b c d + 1440a b c d
--R +
--R 5 4 8 12 4 5 9 11 3 6 10 10
--R 810a b c d + 180a b c d + 12a b c d
--R *
--R 7
--R x
--R +
--R 9 5 15 8 6 14 7 2 7 13 6 3 8 12 5 4 9 11
--R 42a c d + 315a b c d + 720a b c d + 630a b c d + 210a b c d
--R +
--R 4 5 10 10
--R 21a b c d
--R *
--R 6
--R x
--R +
--R 9 6 14 8 7 13 7 2 8 12 6 3 9 11
--R 42a c d + 216a b c d + 324a b c d + 168a b c d
--R +
--R 126 5 4 10 10
--R --- a b c d
--R 5
--R *
--R 5
--R x
--R +
--R 9 7 13 405 8 8 12 7 2 9 11 6 3 10 10 4
--R (30a c d + --- a b c d + 90a b c d + 21a b c d )x
--R 4
--R +
--R 9 8 12 8 9 11 7 2 10 10 3 9 9 11 9 8 10 10 2
--R (15a c d + 30a b c d + 12a b c d )x + (5a c d + - a b c d )x
--R 2
--R +
--R 9 10 10 1 9 11 9 3 8 12 8 6 7 2 13 7 3 6 3 14 6
--R a c d x + -- a c d - -- a b c d + --- a b c d - --- a b c d
--R 11 44 143 143
--R +
--R 6 5 4 15 5 3 4 5 16 4 3 3 6 17 3 1 2 7 18 2
--R --- a b c d - ---- a b c d + ---- a b c d - ---- a b c d
--R 715 1144 4862 9724
--R +
--R 1 8 19 1 9 20
--R ----- a b c d - ------- b c
--R 92378 1847560
--R /
--R 10
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 747
--S 748 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 9 10 20 9 8 10 10 9 9 19
--R -- b d x + (-- a b d + -- b c d )x
--R 20 19 19
--R +
--R 2 7 10 8 9 5 9 2 8 18
--R (2a b d + 5a b c d + - b c d )x
--R 2
--R +
--R 84 3 6 10 360 2 7 9 405 8 2 8 120 9 3 7 17
--R (-- a b d + --- a b c d + --- a b c d + --- b c d )x
--R 17 17 17 17
--R +
--R 63 4 5 10 105 3 6 9 405 2 7 2 8 135 8 3 7 105 9 4 6 16
--R (-- a b d + --- a b c d + --- a b c d + --- a b c d + --- b c d )x
--R 8 2 4 2 8
--R +
--R 42 5 4 10 4 5 9 3 6 2 8 2 7 3 7 8 4 6
--R -- a b d + 84a b c d + 252a b c d + 288a b c d + 126a b c d
--R 5
--R +
--R 84 9 5 5
--R -- b c d
--R 5
--R *
--R 15
--R x
--R +
--R 6 3 10 5 4 9 4 5 2 8 3 6 3 7 2 7 4 6
--R 6a b d + 90a b c d + 405a b c d + 720a b c d + 540a b c d
--R +
--R 8 5 5 9 6 4
--R 162a b c d + 15b c d
--R *
--R 14
--R x
--R +
--R 36 7 2 10 840 6 3 9 5670 5 4 2 8 15120 4 5 3 7
--R -- a b d + --- a b c d + ---- a b c d + ----- a b c d
--R 13 13 13 13
--R +
--R 17640 3 6 4 6 9072 2 7 5 5 1890 8 6 4 120 9 7 3
--R ----- a b c d + ---- a b c d + ---- a b c d + --- b c d
--R 13 13 13 13
--R *
--R 13
--R x
--R +
--R 3 8 10 7 2 9 6 3 2 8 5 4 3 7 4 5 4 6
--R - a b d + 30a b c d + 315a b c d + 1260a b c d + 2205a b c d
--R 4
--R +
--R 3 6 5 5 2 7 6 4 8 7 3 15 9 8 2
--R 1764a b c d + 630a b c d + 90a b c d + -- b c d
--R 4
--R *
--R 12
--R x
--R +
--R 1 9 10 90 8 9 1620 7 2 2 8 10080 6 3 3 7
--R -- a d + -- a b c d + ---- a b c d + ----- a b c d
--R 11 11 11 11
--R +
--R 26460 5 4 4 6 31752 4 5 5 5 17640 3 6 6 4 4320 2 7 7 3
--R ----- a b c d + ----- a b c d + ----- a b c d + ---- a b c d
--R 11 11 11 11
--R +
--R 405 8 8 2 10 9 9
--R --- a b c d + -- b c d
--R 11 11
--R *
--R 11
--R x
--R +
--R 9 9 81 8 2 8 7 2 3 7 6 3 4 6 15876 5 4 5 5
--R a c d + -- a b c d + 432a b c d + 1764a b c d + ----- a b c d
--R 2 5
--R +
--R 4 5 6 4 3 6 7 3 2 7 8 2 8 9 1 9 10
--R 2646a b c d + 1008a b c d + 162a b c d + 9a b c d + -- b c
--R 10
--R *
--R 10
--R x
--R +
--R 9 2 8 8 3 7 7 2 4 6 6 3 5 5 5 4 6 4
--R 5a c d + 120a b c d + 840a b c d + 2352a b c d + 2940a b c d
--R +
--R 4 5 7 3 3 6 8 2 2 7 9 8 10
--R 1680a b c d + 420a b c d + 40a b c d + a b c
--R *
--R 9
--R x
--R +
--R 9 3 7 945 8 4 6 7 2 5 5 6 3 6 4 5 4 7 3
--R 15a c d + --- a b c d + 1134a b c d + 2205a b c d + 1890a b c d
--R 4
--R +
--R 2835 4 5 8 2 3 6 9 9 2 7 10
--R ---- a b c d + 105a b c d + - a b c
--R 4 2
--R *
--R 8
--R x
--R +
--R 9 4 6 8 5 5 7 2 6 4 6 3 7 3 5 4 8 2
--R 30a c d + 324a b c d + 1080a b c d + 1440a b c d + 810a b c d
--R +
--R 4 5 9 3 6 10
--R 180a b c d + 12a b c
--R *
--R 7
--R x
--R +
--R 9 5 5 8 6 4 7 2 7 3 6 3 8 2 5 4 9
--R 42a c d + 315a b c d + 720a b c d + 630a b c d + 210a b c d
--R +
--R 4 5 10
--R 21a b c
--R *
--R 6
--R x
--R +
--R 9 6 4 8 7 3 7 2 8 2 6 3 9 126 5 4 10 5
--R (42a c d + 216a b c d + 324a b c d + 168a b c d + --- a b c )x
--R 5
--R +
--R 9 7 3 405 8 8 2 7 2 9 6 3 10 4
--R (30a c d + --- a b c d + 90a b c d + 21a b c )x
--R 4
--R +
--R 9 8 2 8 9 7 2 10 3 9 9 9 8 10 2 9 10
--R (15a c d + 30a b c d + 12a b c )x + (5a c d + - a b c )x + a c x
--R 2
--R Type: Polynomial(Fraction(Integer))
--E 748
--S 749 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 9 11 9 3 8 12 8 6 7 2 13 7 3 6 3 14 6
--R - -- a c d + -- a b c d - --- a b c d + --- a b c d
--R 11 44 143 143
--R +
--R 6 5 4 15 5 3 4 5 16 4 3 3 6 17 3 1 2 7 18 2
--R - --- a b c d + ---- a b c d - ---- a b c d + ---- a b c d
--R 715 1144 4862 9724
--R +
--R 1 8 19 1 9 20
--R - ----- a b c d + ------- b c
--R 92378 1847560
--R /
--R 10
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 749
--S 750 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 750
)clear all
--S 751 of 2952
t0000:=(a+b*x)^8*(c+d*x)^10
--R
--R
--R (1)
--R 8 10 18 7 10 8 9 17
--R b d x + (8a b d + 10b c d )x
--R +
--R 2 6 10 7 9 8 2 8 16
--R (28a b d + 80a b c d + 45b c d )x
--R +
--R 3 5 10 2 6 9 7 2 8 8 3 7 15
--R (56a b d + 280a b c d + 360a b c d + 120b c d )x
--R +
--R 4 4 10 3 5 9 2 6 2 8 7 3 7 8 4 6 14
--R (70a b d + 560a b c d + 1260a b c d + 960a b c d + 210b c d )x
--R +
--R 5 3 10 4 4 9 3 5 2 8 2 6 3 7 7 4 6
--R 56a b d + 700a b c d + 2520a b c d + 3360a b c d + 1680a b c d
--R +
--R 8 5 5
--R 252b c d
--R *
--R 13
--R x
--R +
--R 6 2 10 5 3 9 4 4 2 8 3 5 3 7 2 6 4 6
--R 28a b d + 560a b c d + 3150a b c d + 6720a b c d + 5880a b c d
--R +
--R 7 5 5 8 6 4
--R 2016a b c d + 210b c d
--R *
--R 12
--R x
--R +
--R 7 10 6 2 9 5 3 2 8 4 4 3 7 3 5 4 6
--R 8a b d + 280a b c d + 2520a b c d + 8400a b c d + 11760a b c d
--R +
--R 2 6 5 5 7 6 4 8 7 3
--R 7056a b c d + 1680a b c d + 120b c d
--R *
--R 11
--R x
--R +
--R 8 10 7 9 6 2 2 8 5 3 3 7 4 4 4 6
--R a d + 80a b c d + 1260a b c d + 6720a b c d + 14700a b c d
--R +
--R 3 5 5 5 2 6 6 4 7 7 3 8 8 2
--R 14112a b c d + 5880a b c d + 960a b c d + 45b c d
--R *
--R 10
--R x
--R +
--R 8 9 7 2 8 6 2 3 7 5 3 4 6 4 4 5 5
--R 10a c d + 360a b c d + 3360a b c d + 11760a b c d + 17640a b c d
--R +
--R 3 5 6 4 2 6 7 3 7 8 2 8 9
--R 11760a b c d + 3360a b c d + 360a b c d + 10b c d
--R *
--R 9
--R x
--R +
--R 8 2 8 7 3 7 6 2 4 6 5 3 5 5 4 4 6 4
--R 45a c d + 960a b c d + 5880a b c d + 14112a b c d + 14700a b c d
--R +
--R 3 5 7 3 2 6 8 2 7 9 8 10
--R 6720a b c d + 1260a b c d + 80a b c d + b c
--R *
--R 8
--R x
--R +
--R 8 3 7 7 4 6 6 2 5 5 5 3 6 4 4 4 7 3
--R 120a c d + 1680a b c d + 7056a b c d + 11760a b c d + 8400a b c d
--R +
--R 3 5 8 2 2 6 9 7 10
--R 2520a b c d + 280a b c d + 8a b c
--R *
--R 7
--R x
--R +
--R 8 4 6 7 5 5 6 2 6 4 5 3 7 3 4 4 8 2
--R 210a c d + 2016a b c d + 5880a b c d + 6720a b c d + 3150a b c d
--R +
--R 3 5 9 2 6 10
--R 560a b c d + 28a b c
--R *
--R 6
--R x
--R +
--R 8 5 5 7 6 4 6 2 7 3 5 3 8 2 4 4 9
--R 252a c d + 1680a b c d + 3360a b c d + 2520a b c d + 700a b c d
--R +
--R 3 5 10
--R 56a b c
--R *
--R 5
--R x
--R +
--R 8 6 4 7 7 3 6 2 8 2 5 3 9 4 4 10 4
--R (210a c d + 960a b c d + 1260a b c d + 560a b c d + 70a b c )x
--R +
--R 8 7 3 7 8 2 6 2 9 5 3 10 3
--R (120a c d + 360a b c d + 280a b c d + 56a b c )x
--R +
--R 8 8 2 7 9 6 2 10 2 8 9 7 10 8 10
--R (45a c d + 80a b c d + 28a b c )x + (10a c d + 8a b c )x + a c
--R Type: Polynomial(Integer)
--E 751
--S 752 of 2952
r0000:=1/11*(b*c-a*d)^8*(c+d*x)^11/d^9-2/3*b*(b*c-a*d)^7*(c+d*x)^12/d^9+_
28/13*b^2*(b*c-a*d)^6*(c+d*x)^13/d^9-4*b^3*(b*c-a*d)^5*(c+d*x)^14/d^9+_
14/3*b^4*(b*c-a*d)^4*(c+d*x)^15/d^9-7/2*b^5*(b*c-a*d)^3*(c+d*x)^16/d^9+_
28/17*b^6*(b*c-a*d)^2*(c+d*x)^17/d^9-4/9*b^7*(b*c-a*d)*(c+d*x)^18/d^9+_
1/19*b^8*(c+d*x)^19/d^9
--R
--R
--R (2)
--R 1 8 19 19 4 7 19 5 8 18 18
--R -- b d x + (- a b d + - b c d )x
--R 19 9 9
--R +
--R 28 2 6 19 80 7 18 45 8 2 17 17
--R (-- a b d + -- a b c d + -- b c d )x
--R 17 17 17
--R +
--R 7 3 5 19 35 2 6 18 45 7 2 17 15 8 3 16 16
--R (- a b d + -- a b c d + -- a b c d + -- b c d )x
--R 2 2 2 2
--R +
--R 14 4 4 19 112 3 5 18 2 6 2 17 7 3 16 8 4 15 15
--R (-- a b d + --- a b c d + 84a b c d + 64a b c d + 14b c d )x
--R 3 3
--R +
--R 5 3 19 4 4 18 3 5 2 17 2 6 3 16 7 4 15
--R 4a b d + 50a b c d + 180a b c d + 240a b c d + 120a b c d
--R +
--R 8 5 14
--R 18b c d
--R *
--R 14
--R x
--R +
--R 28 6 2 19 560 5 3 18 3150 4 4 2 17 6720 3 5 3 16
--R -- a b d + --- a b c d + ---- a b c d + ---- a b c d
--R 13 13 13 13
--R +
--R 5880 2 6 4 15 2016 7 5 14 210 8 6 13
--R ---- a b c d + ---- a b c d + --- b c d
--R 13 13 13
--R *
--R 13
--R x
--R +
--R 2 7 19 70 6 2 18 5 3 2 17 4 4 3 16 3 5 4 15
--R - a b d + -- a b c d + 210a b c d + 700a b c d + 980a b c d
--R 3 3
--R +
--R 2 6 5 14 7 6 13 8 7 12
--R 588a b c d + 140a b c d + 10b c d
--R *
--R 12
--R x
--R +
--R 1 8 19 80 7 18 1260 6 2 2 17 6720 5 3 3 16
--R -- a d + -- a b c d + ---- a b c d + ---- a b c d
--R 11 11 11 11
--R +
--R 14700 4 4 4 15 14112 3 5 5 14 5880 2 6 6 13 960 7 7 12
--R ----- a b c d + ----- a b c d + ---- a b c d + --- a b c d
--R 11 11 11 11
--R +
--R 45 8 8 11
--R -- b c d
--R 11
--R *
--R 11
--R x
--R +
--R 8 18 7 2 17 6 2 3 16 5 3 4 15 4 4 5 14
--R a c d + 36a b c d + 336a b c d + 1176a b c d + 1764a b c d
--R +
--R 3 5 6 13 2 6 7 12 7 8 11 8 9 10
--R 1176a b c d + 336a b c d + 36a b c d + b c d
--R *
--R 10
--R x
--R +
--R 8 2 17 320 7 3 16 1960 6 2 4 15 5 3 5 14
--R 5a c d + --- a b c d + ---- a b c d + 1568a b c d
--R 3 3
--R +
--R 4900 4 4 6 13 2240 3 5 7 12 2 6 8 11 80 7 9 10
--R ---- a b c d + ---- a b c d + 140a b c d + -- a b c d
--R 3 3 9
--R +
--R 1 8 10 9
--R - b c d
--R 9
--R *
--R 9
--R x
--R +
--R 8 3 16 7 4 15 6 2 5 14 5 3 6 13
--R 15a c d + 210a b c d + 882a b c d + 1470a b c d
--R +
--R 4 4 7 12 3 5 8 11 2 6 9 10 7 10 9
--R 1050a b c d + 315a b c d + 35a b c d + a b c d
--R *
--R 8
--R x
--R +
--R 8 4 15 7 5 14 6 2 6 13 5 3 7 12 4 4 8 11
--R 30a c d + 288a b c d + 840a b c d + 960a b c d + 450a b c d
--R +
--R 3 5 9 10 2 6 10 9
--R 80a b c d + 4a b c d
--R *
--R 7
--R x
--R +
--R 8 5 14 7 6 13 6 2 7 12 5 3 8 11
--R 42a c d + 280a b c d + 560a b c d + 420a b c d
--R +
--R 350 4 4 9 10 28 3 5 10 9
--R --- a b c d + -- a b c d
--R 3 3
--R *
--R 6
--R x
--R +
--R 8 6 13 7 7 12 6 2 8 11 5 3 9 10 4 4 10 9 5
--R (42a c d + 192a b c d + 252a b c d + 112a b c d + 14a b c d )x
--R +
--R 8 7 12 7 8 11 6 2 9 10 5 3 10 9 4
--R (30a c d + 90a b c d + 70a b c d + 14a b c d )x
--R +
--R 8 8 11 80 7 9 10 28 6 2 10 9 3 8 9 10 7 10 9 2
--R (15a c d + -- a b c d + -- a b c d )x + (5a c d + 4a b c d )x
--R 3 3
--R +
--R 8 10 9 1 8 11 8 2 7 12 7 14 6 2 13 6 2 5 3 14 5
--R a c d x + -- a c d - -- a b c d + --- a b c d - --- a b c d
--R 11 33 429 143
--R +
--R 2 4 4 15 4 1 3 5 16 3 1 2 6 17 2 1 7 18
--R --- a b c d - --- a b c d + ---- a b c d - ----- a b c d
--R 429 858 4862 43758
--R +
--R 1 8 19
--R ------ b c
--R 831402
--R /
--R 9
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 752
--S 753 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 8 10 19 4 7 10 5 8 9 18
--R -- b d x + (- a b d + - b c d )x
--R 19 9 9
--R +
--R 28 2 6 10 80 7 9 45 8 2 8 17
--R (-- a b d + -- a b c d + -- b c d )x
--R 17 17 17
--R +
--R 7 3 5 10 35 2 6 9 45 7 2 8 15 8 3 7 16
--R (- a b d + -- a b c d + -- a b c d + -- b c d )x
--R 2 2 2 2
--R +
--R 14 4 4 10 112 3 5 9 2 6 2 8 7 3 7 8 4 6 15
--R (-- a b d + --- a b c d + 84a b c d + 64a b c d + 14b c d )x
--R 3 3
--R +
--R 5 3 10 4 4 9 3 5 2 8 2 6 3 7 7 4 6
--R 4a b d + 50a b c d + 180a b c d + 240a b c d + 120a b c d
--R +
--R 8 5 5
--R 18b c d
--R *
--R 14
--R x
--R +
--R 28 6 2 10 560 5 3 9 3150 4 4 2 8 6720 3 5 3 7
--R -- a b d + --- a b c d + ---- a b c d + ---- a b c d
--R 13 13 13 13
--R +
--R 5880 2 6 4 6 2016 7 5 5 210 8 6 4
--R ---- a b c d + ---- a b c d + --- b c d
--R 13 13 13
--R *
--R 13
--R x
--R +
--R 2 7 10 70 6 2 9 5 3 2 8 4 4 3 7 3 5 4 6
--R - a b d + -- a b c d + 210a b c d + 700a b c d + 980a b c d
--R 3 3
--R +
--R 2 6 5 5 7 6 4 8 7 3
--R 588a b c d + 140a b c d + 10b c d
--R *
--R 12
--R x
--R +
--R 1 8 10 80 7 9 1260 6 2 2 8 6720 5 3 3 7 14700 4 4 4 6
--R -- a d + -- a b c d + ---- a b c d + ---- a b c d + ----- a b c d
--R 11 11 11 11 11
--R +
--R 14112 3 5 5 5 5880 2 6 6 4 960 7 7 3 45 8 8 2
--R ----- a b c d + ---- a b c d + --- a b c d + -- b c d
--R 11 11 11 11
--R *
--R 11
--R x
--R +
--R 8 9 7 2 8 6 2 3 7 5 3 4 6 4 4 5 5
--R a c d + 36a b c d + 336a b c d + 1176a b c d + 1764a b c d
--R +
--R 3 5 6 4 2 6 7 3 7 8 2 8 9
--R 1176a b c d + 336a b c d + 36a b c d + b c d
--R *
--R 10
--R x
--R +
--R 8 2 8 320 7 3 7 1960 6 2 4 6 5 3 5 5 4900 4 4 6 4
--R 5a c d + --- a b c d + ---- a b c d + 1568a b c d + ---- a b c d
--R 3 3 3
--R +
--R 2240 3 5 7 3 2 6 8 2 80 7 9 1 8 10
--R ---- a b c d + 140a b c d + -- a b c d + - b c
--R 3 9 9
--R *
--R 9
--R x
--R +
--R 8 3 7 7 4 6 6 2 5 5 5 3 6 4 4 4 7 3
--R 15a c d + 210a b c d + 882a b c d + 1470a b c d + 1050a b c d
--R +
--R 3 5 8 2 2 6 9 7 10
--R 315a b c d + 35a b c d + a b c
--R *
--R 8
--R x
--R +
--R 8 4 6 7 5 5 6 2 6 4 5 3 7 3 4 4 8 2
--R 30a c d + 288a b c d + 840a b c d + 960a b c d + 450a b c d
--R +
--R 3 5 9 2 6 10
--R 80a b c d + 4a b c
--R *
--R 7
--R x
--R +
--R 8 5 5 7 6 4 6 2 7 3 5 3 8 2 350 4 4 9
--R 42a c d + 280a b c d + 560a b c d + 420a b c d + --- a b c d
--R 3
--R +
--R 28 3 5 10
--R -- a b c
--R 3
--R *
--R 6
--R x
--R +
--R 8 6 4 7 7 3 6 2 8 2 5 3 9 4 4 10 5
--R (42a c d + 192a b c d + 252a b c d + 112a b c d + 14a b c )x
--R +
--R 8 7 3 7 8 2 6 2 9 5 3 10 4
--R (30a c d + 90a b c d + 70a b c d + 14a b c )x
--R +
--R 8 8 2 80 7 9 28 6 2 10 3 8 9 7 10 2 8 10
--R (15a c d + -- a b c d + -- a b c )x + (5a c d + 4a b c )x + a c x
--R 3 3
--R Type: Polynomial(Fraction(Integer))
--E 753
--S 754 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 8 11 8 2 7 12 7 14 6 2 13 6 2 5 3 14 5
--R - -- a c d + -- a b c d - --- a b c d + --- a b c d
--R 11 33 429 143
--R +
--R 2 4 4 15 4 1 3 5 16 3 1 2 6 17 2 1 7 18
--R - --- a b c d + --- a b c d - ---- a b c d + ----- a b c d
--R 429 858 4862 43758
--R +
--R 1 8 19
--R - ------ b c
--R 831402
--R /
--R 9
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 754
--S 755 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 755
)clear all
--S 756 of 2952
t0000:=(a+b*x)^7*(c+d*x)^10
--R
--R
--R (1)
--R 7 10 17 6 10 7 9 16
--R b d x + (7a b d + 10b c d )x
--R +
--R 2 5 10 6 9 7 2 8 15
--R (21a b d + 70a b c d + 45b c d )x
--R +
--R 3 4 10 2 5 9 6 2 8 7 3 7 14
--R (35a b d + 210a b c d + 315a b c d + 120b c d )x
--R +
--R 4 3 10 3 4 9 2 5 2 8 6 3 7 7 4 6 13
--R (35a b d + 350a b c d + 945a b c d + 840a b c d + 210b c d )x
--R +
--R 5 2 10 4 3 9 3 4 2 8 2 5 3 7 6 4 6
--R 21a b d + 350a b c d + 1575a b c d + 2520a b c d + 1470a b c d
--R +
--R 7 5 5
--R 252b c d
--R *
--R 12
--R x
--R +
--R 6 10 5 2 9 4 3 2 8 3 4 3 7 2 5 4 6
--R 7a b d + 210a b c d + 1575a b c d + 4200a b c d + 4410a b c d
--R +
--R 6 5 5 7 6 4
--R 1764a b c d + 210b c d
--R *
--R 11
--R x
--R +
--R 7 10 6 9 5 2 2 8 4 3 3 7 3 4 4 6
--R a d + 70a b c d + 945a b c d + 4200a b c d + 7350a b c d
--R +
--R 2 5 5 5 6 6 4 7 7 3
--R 5292a b c d + 1470a b c d + 120b c d
--R *
--R 10
--R x
--R +
--R 7 9 6 2 8 5 2 3 7 4 3 4 6 3 4 5 5
--R 10a c d + 315a b c d + 2520a b c d + 7350a b c d + 8820a b c d
--R +
--R 2 5 6 4 6 7 3 7 8 2
--R 4410a b c d + 840a b c d + 45b c d
--R *
--R 9
--R x
--R +
--R 7 2 8 6 3 7 5 2 4 6 4 3 5 5 3 4 6 4
--R 45a c d + 840a b c d + 4410a b c d + 8820a b c d + 7350a b c d
--R +
--R 2 5 7 3 6 8 2 7 9
--R 2520a b c d + 315a b c d + 10b c d
--R *
--R 8
--R x
--R +
--R 7 3 7 6 4 6 5 2 5 5 4 3 6 4 3 4 7 3
--R 120a c d + 1470a b c d + 5292a b c d + 7350a b c d + 4200a b c d
--R +
--R 2 5 8 2 6 9 7 10
--R 945a b c d + 70a b c d + b c
--R *
--R 7
--R x
--R +
--R 7 4 6 6 5 5 5 2 6 4 4 3 7 3 3 4 8 2
--R 210a c d + 1764a b c d + 4410a b c d + 4200a b c d + 1575a b c d
--R +
--R 2 5 9 6 10
--R 210a b c d + 7a b c
--R *
--R 6
--R x
--R +
--R 7 5 5 6 6 4 5 2 7 3 4 3 8 2 3 4 9
--R 252a c d + 1470a b c d + 2520a b c d + 1575a b c d + 350a b c d
--R +
--R 2 5 10
--R 21a b c
--R *
--R 5
--R x
--R +
--R 7 6 4 6 7 3 5 2 8 2 4 3 9 3 4 10 4
--R (210a c d + 840a b c d + 945a b c d + 350a b c d + 35a b c )x
--R +
--R 7 7 3 6 8 2 5 2 9 4 3 10 3
--R (120a c d + 315a b c d + 210a b c d + 35a b c )x
--R +
--R 7 8 2 6 9 5 2 10 2 7 9 6 10 7 10
--R (45a c d + 70a b c d + 21a b c )x + (10a c d + 7a b c )x + a c
--R Type: Polynomial(Integer)
--E 756
--S 757 of 2952
r0000:=-1/11*(b*c-a*d)^7*(c+d*x)^11/d^8+7/12*b*(b*c-a*d)^6*(c+d*x)^12/d^8-_
21/13*b^2*(b*c-a*d)^5*(c+d*x)^13/d^8+_
5/2*b^3*(b*c-a*d)^4*(c+d*x)^14/d^8-_
7/3*b^4*(b*c-a*d)^3*(c+d*x)^15/d^8+_
21/16*b^5*(b*c-a*d)^2*(c+d*x)^16/d^8-_
7/17*b^6*(b*c-a*d)*(c+d*x)^17/d^8+1/18*b^7*(c+d*x)^18/d^8
--R
--R
--R (2)
--R 1 7 18 18 7 6 18 10 7 17 17
--R -- b d x + (-- a b d + -- b c d )x
--R 18 17 17
--R +
--R 21 2 5 18 35 6 17 45 7 2 16 16
--R (-- a b d + -- a b c d + -- b c d )x
--R 16 8 16
--R +
--R 7 3 4 18 2 5 17 6 2 16 7 3 15 15
--R (- a b d + 14a b c d + 21a b c d + 8b c d )x
--R 3
--R +
--R 5 4 3 18 3 4 17 135 2 5 2 16 6 3 15 7 4 14 14
--R (- a b d + 25a b c d + --- a b c d + 60a b c d + 15b c d )x
--R 2 2
--R +
--R 21 5 2 18 350 4 3 17 1575 3 4 2 16 2520 2 5 3 15
--R -- a b d + --- a b c d + ---- a b c d + ---- a b c d
--R 13 13 13 13
--R +
--R 1470 6 4 14 252 7 5 13
--R ---- a b c d + --- b c d
--R 13 13
--R *
--R 13
--R x
--R +
--R 7 6 18 35 5 2 17 525 4 3 2 16 3 4 3 15
--R -- a b d + -- a b c d + --- a b c d + 350a b c d
--R 12 2 4
--R +
--R 735 2 5 4 14 6 5 13 35 7 6 12
--R --- a b c d + 147a b c d + -- b c d
--R 2 2
--R *
--R 12
--R x
--R +
--R 1 7 18 70 6 17 945 5 2 2 16 4200 4 3 3 15
--R -- a d + -- a b c d + --- a b c d + ---- a b c d
--R 11 11 11 11
--R +
--R 7350 3 4 4 14 5292 2 5 5 13 1470 6 6 12 120 7 7 11
--R ---- a b c d + ---- a b c d + ---- a b c d + --- b c d
--R 11 11 11 11
--R *
--R 11
--R x
--R +
--R 7 17 63 6 2 16 5 2 3 15 4 3 4 14 3 4 5 13
--R a c d + -- a b c d + 252a b c d + 735a b c d + 882a b c d
--R 2
--R +
--R 2 5 6 12 6 7 11 9 7 8 10
--R 441a b c d + 84a b c d + - b c d
--R 2
--R *
--R 10
--R x
--R +
--R 7 2 16 280 6 3 15 5 2 4 14 4 3 5 13
--R 5a c d + --- a b c d + 490a b c d + 980a b c d
--R 3
--R +
--R 2450 3 4 6 12 2 5 7 11 6 8 10 10 7 9 9
--R ---- a b c d + 280a b c d + 35a b c d + -- b c d
--R 3 9
--R *
--R 9
--R x
--R +
--R 7 3 15 735 6 4 14 1323 5 2 5 13 3675 4 3 6 12
--R 15a c d + --- a b c d + ---- a b c d + ---- a b c d
--R 4 2 4
--R +
--R 3 4 7 11 945 2 5 8 10 35 6 9 9 1 7 10 8
--R 525a b c d + --- a b c d + -- a b c d + - b c d
--R 8 4 8
--R *
--R 8
--R x
--R +
--R 7 4 14 6 5 13 5 2 6 12 4 3 7 11 3 4 8 10
--R 30a c d + 252a b c d + 630a b c d + 600a b c d + 225a b c d
--R +
--R 2 5 9 9 6 10 8
--R 30a b c d + a b c d
--R *
--R 7
--R x
--R +
--R 7 5 13 6 6 12 5 2 7 11 525 4 3 8 10
--R 42a c d + 245a b c d + 420a b c d + --- a b c d
--R 2
--R +
--R 175 3 4 9 9 7 2 5 10 8
--R --- a b c d + - a b c d
--R 3 2
--R *
--R 6
--R x
--R +
--R 7 6 12 6 7 11 5 2 8 10 4 3 9 9 3 4 10 8 5
--R (42a c d + 168a b c d + 189a b c d + 70a b c d + 7a b c d )x
--R +
--R 7 7 11 315 6 8 10 105 5 2 9 9 35 4 3 10 8 4
--R (30a c d + --- a b c d + --- a b c d + -- a b c d )x
--R 4 2 4
--R +
--R 7 8 10 70 6 9 9 5 2 10 8 3 7 9 9 7 6 10 8 2
--R (15a c d + -- a b c d + 7a b c d )x + (5a c d + - a b c d )x
--R 3 2
--R +
--R 7 10 8 1 7 11 7 7 6 12 6 7 5 2 13 5 5 4 3 14 4
--R a c d x + -- a c d - --- a b c d + --- a b c d - --- a b c d
--R 11 132 286 572
--R +
--R 1 3 4 15 3 1 2 5 16 2 1 6 17 1 7 18
--R --- a b c d - ---- a b c d + ----- a b c d - ------ b c
--R 429 2288 19448 350064
--R /
--R 8
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 757
--S 758 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 7 10 18 7 6 10 10 7 9 17
--R -- b d x + (-- a b d + -- b c d )x
--R 18 17 17
--R +
--R 21 2 5 10 35 6 9 45 7 2 8 16
--R (-- a b d + -- a b c d + -- b c d )x
--R 16 8 16
--R +
--R 7 3 4 10 2 5 9 6 2 8 7 3 7 15
--R (- a b d + 14a b c d + 21a b c d + 8b c d )x
--R 3
--R +
--R 5 4 3 10 3 4 9 135 2 5 2 8 6 3 7 7 4 6 14
--R (- a b d + 25a b c d + --- a b c d + 60a b c d + 15b c d )x
--R 2 2
--R +
--R 21 5 2 10 350 4 3 9 1575 3 4 2 8 2520 2 5 3 7
--R -- a b d + --- a b c d + ---- a b c d + ---- a b c d
--R 13 13 13 13
--R +
--R 1470 6 4 6 252 7 5 5
--R ---- a b c d + --- b c d
--R 13 13
--R *
--R 13
--R x
--R +
--R 7 6 10 35 5 2 9 525 4 3 2 8 3 4 3 7 735 2 5 4 6
--R -- a b d + -- a b c d + --- a b c d + 350a b c d + --- a b c d
--R 12 2 4 2
--R +
--R 6 5 5 35 7 6 4
--R 147a b c d + -- b c d
--R 2
--R *
--R 12
--R x
--R +
--R 1 7 10 70 6 9 945 5 2 2 8 4200 4 3 3 7 7350 3 4 4 6
--R -- a d + -- a b c d + --- a b c d + ---- a b c d + ---- a b c d
--R 11 11 11 11 11
--R +
--R 5292 2 5 5 5 1470 6 6 4 120 7 7 3
--R ---- a b c d + ---- a b c d + --- b c d
--R 11 11 11
--R *
--R 11
--R x
--R +
--R 7 9 63 6 2 8 5 2 3 7 4 3 4 6 3 4 5 5
--R a c d + -- a b c d + 252a b c d + 735a b c d + 882a b c d
--R 2
--R +
--R 2 5 6 4 6 7 3 9 7 8 2
--R 441a b c d + 84a b c d + - b c d
--R 2
--R *
--R 10
--R x
--R +
--R 7 2 8 280 6 3 7 5 2 4 6 4 3 5 5 2450 3 4 6 4
--R 5a c d + --- a b c d + 490a b c d + 980a b c d + ---- a b c d
--R 3 3
--R +
--R 2 5 7 3 6 8 2 10 7 9
--R 280a b c d + 35a b c d + -- b c d
--R 9
--R *
--R 9
--R x
--R +
--R 7 3 7 735 6 4 6 1323 5 2 5 5 3675 4 3 6 4 3 4 7 3
--R 15a c d + --- a b c d + ---- a b c d + ---- a b c d + 525a b c d
--R 4 2 4
--R +
--R 945 2 5 8 2 35 6 9 1 7 10
--R --- a b c d + -- a b c d + - b c
--R 8 4 8
--R *
--R 8
--R x
--R +
--R 7 4 6 6 5 5 5 2 6 4 4 3 7 3 3 4 8 2
--R 30a c d + 252a b c d + 630a b c d + 600a b c d + 225a b c d
--R +
--R 2 5 9 6 10
--R 30a b c d + a b c
--R *
--R 7
--R x
--R +
--R 7 5 5 6 6 4 5 2 7 3 525 4 3 8 2 175 3 4 9
--R 42a c d + 245a b c d + 420a b c d + --- a b c d + --- a b c d
--R 2 3
--R +
--R 7 2 5 10
--R - a b c
--R 2
--R *
--R 6
--R x
--R +
--R 7 6 4 6 7 3 5 2 8 2 4 3 9 3 4 10 5
--R (42a c d + 168a b c d + 189a b c d + 70a b c d + 7a b c )x
--R +
--R 7 7 3 315 6 8 2 105 5 2 9 35 4 3 10 4
--R (30a c d + --- a b c d + --- a b c d + -- a b c )x
--R 4 2 4
--R +
--R 7 8 2 70 6 9 5 2 10 3 7 9 7 6 10 2 7 10
--R (15a c d + -- a b c d + 7a b c )x + (5a c d + - a b c )x + a c x
--R 3 2
--R Type: Polynomial(Fraction(Integer))
--E 758
--S 759 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 7 11 7 7 6 12 6 7 5 2 13 5 5 4 3 14 4
--R - -- a c d + --- a b c d - --- a b c d + --- a b c d
--R 11 132 286 572
--R +
--R 1 3 4 15 3 1 2 5 16 2 1 6 17 1 7 18
--R - --- a b c d + ---- a b c d - ----- a b c d + ------ b c
--R 429 2288 19448 350064
--R /
--R 8
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 759
--S 760 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 760
)clear all
--S 761 of 2952
t0000:=(a+b*x)^6*(c+d*x)^10
--R
--R
--R (1)
--R 6 10 16 5 10 6 9 15
--R b d x + (6a b d + 10b c d )x
--R +
--R 2 4 10 5 9 6 2 8 14
--R (15a b d + 60a b c d + 45b c d )x
--R +
--R 3 3 10 2 4 9 5 2 8 6 3 7 13
--R (20a b d + 150a b c d + 270a b c d + 120b c d )x
--R +
--R 4 2 10 3 3 9 2 4 2 8 5 3 7 6 4 6 12
--R (15a b d + 200a b c d + 675a b c d + 720a b c d + 210b c d )x
--R +
--R 5 10 4 2 9 3 3 2 8 2 4 3 7 5 4 6
--R 6a b d + 150a b c d + 900a b c d + 1800a b c d + 1260a b c d
--R +
--R 6 5 5
--R 252b c d
--R *
--R 11
--R x
--R +
--R 6 10 5 9 4 2 2 8 3 3 3 7 2 4 4 6
--R a d + 60a b c d + 675a b c d + 2400a b c d + 3150a b c d
--R +
--R 5 5 5 6 6 4
--R 1512a b c d + 210b c d
--R *
--R 10
--R x
--R +
--R 6 9 5 2 8 4 2 3 7 3 3 4 6 2 4 5 5
--R 10a c d + 270a b c d + 1800a b c d + 4200a b c d + 3780a b c d
--R +
--R 5 6 4 6 7 3
--R 1260a b c d + 120b c d
--R *
--R 9
--R x
--R +
--R 6 2 8 5 3 7 4 2 4 6 3 3 5 5 2 4 6 4
--R 45a c d + 720a b c d + 3150a b c d + 5040a b c d + 3150a b c d
--R +
--R 5 7 3 6 8 2
--R 720a b c d + 45b c d
--R *
--R 8
--R x
--R +
--R 6 3 7 5 4 6 4 2 5 5 3 3 6 4 2 4 7 3
--R 120a c d + 1260a b c d + 3780a b c d + 4200a b c d + 1800a b c d
--R +
--R 5 8 2 6 9
--R 270a b c d + 10b c d
--R *
--R 7
--R x
--R +
--R 6 4 6 5 5 5 4 2 6 4 3 3 7 3 2 4 8 2
--R 210a c d + 1512a b c d + 3150a b c d + 2400a b c d + 675a b c d
--R +
--R 5 9 6 10
--R 60a b c d + b c
--R *
--R 6
--R x
--R +
--R 6 5 5 5 6 4 4 2 7 3 3 3 8 2 2 4 9
--R 252a c d + 1260a b c d + 1800a b c d + 900a b c d + 150a b c d
--R +
--R 5 10
--R 6a b c
--R *
--R 5
--R x
--R +
--R 6 6 4 5 7 3 4 2 8 2 3 3 9 2 4 10 4
--R (210a c d + 720a b c d + 675a b c d + 200a b c d + 15a b c )x
--R +
--R 6 7 3 5 8 2 4 2 9 3 3 10 3
--R (120a c d + 270a b c d + 150a b c d + 20a b c )x
--R +
--R 6 8 2 5 9 4 2 10 2 6 9 5 10 6 10
--R (45a c d + 60a b c d + 15a b c )x + (10a c d + 6a b c )x + a c
--R Type: Polynomial(Integer)
--E 761
--S 762 of 2952
r0000:=1/11*(b*c-a*d)^6*(c+d*x)^11/d^7-1/2*b*(b*c-a*d)^5*(c+d*x)^12/d^7+_
15/13*b^2*(b*c-a*d)^4*(c+d*x)^13/d^7-_
10/7*b^3*(b*c-a*d)^3*(c+d*x)^14/d^7+b^4*(b*c-a*d)^2*(c+d*x)^15/d^7-_
3/8*b^5*(b*c-a*d)*(c+d*x)^16/d^7+1/17*b^6*(c+d*x)^17/d^7
--R
--R
--R (2)
--R 1 6 17 17 3 5 17 5 6 16 16
--R -- b d x + (- a b d + - b c d )x
--R 17 8 8
--R +
--R 2 4 17 5 16 6 2 15 15
--R (a b d + 4a b c d + 3b c d )x
--R +
--R 10 3 3 17 75 2 4 16 135 5 2 15 60 6 3 14 14
--R (-- a b d + -- a b c d + --- a b c d + -- b c d )x
--R 7 7 7 7
--R +
--R 15 4 2 17 200 3 3 16 675 2 4 2 15 720 5 3 14
--R -- a b d + --- a b c d + --- a b c d + --- a b c d
--R 13 13 13 13
--R +
--R 210 6 4 13
--R --- b c d
--R 13
--R *
--R 13
--R x
--R +
--R 1 5 17 25 4 2 16 3 3 2 15 2 4 3 14 5 4 13
--R - a b d + -- a b c d + 75a b c d + 150a b c d + 105a b c d
--R 2 2
--R +
--R 6 5 12
--R 21b c d
--R *
--R 12
--R x
--R +
--R 1 6 17 60 5 16 675 4 2 2 15 2400 3 3 3 14
--R -- a d + -- a b c d + --- a b c d + ---- a b c d
--R 11 11 11 11
--R +
--R 3150 2 4 4 13 1512 5 5 12 210 6 6 11
--R ---- a b c d + ---- a b c d + --- b c d
--R 11 11 11
--R *
--R 11
--R x
--R +
--R 6 16 5 2 15 4 2 3 14 3 3 4 13 2 4 5 12
--R a c d + 27a b c d + 180a b c d + 420a b c d + 378a b c d
--R +
--R 5 6 11 6 7 10
--R 126a b c d + 12b c d
--R *
--R 10
--R x
--R +
--R 6 2 15 5 3 14 4 2 4 13 3 3 5 12 2 4 6 11
--R 5a c d + 80a b c d + 350a b c d + 560a b c d + 350a b c d
--R +
--R 5 7 10 6 8 9
--R 80a b c d + 5b c d
--R *
--R 9
--R x
--R +
--R 6 3 14 315 5 4 13 945 4 2 5 12 3 3 6 11
--R 15a c d + --- a b c d + --- a b c d + 525a b c d
--R 2 2
--R +
--R 2 4 7 10 135 5 8 9 5 6 9 8
--R 225a b c d + --- a b c d + - b c d
--R 4 4
--R *
--R 8
--R x
--R +
--R 6 4 13 5 5 12 4 2 6 11 2400 3 3 7 10
--R 30a c d + 216a b c d + 450a b c d + ---- a b c d
--R 7
--R +
--R 675 2 4 8 9 60 5 9 8 1 6 10 7
--R --- a b c d + -- a b c d + - b c d
--R 7 7 7
--R *
--R 7
--R x
--R +
--R 6 5 12 5 6 11 4 2 7 10 3 3 8 9 2 4 9 8
--R 42a c d + 210a b c d + 300a b c d + 150a b c d + 25a b c d
--R +
--R 5 10 7
--R a b c d
--R *
--R 6
--R x
--R +
--R 6 6 11 5 7 10 4 2 8 9 3 3 9 8 2 4 10 7 5
--R (42a c d + 144a b c d + 135a b c d + 40a b c d + 3a b c d )x
--R +
--R 6 7 10 135 5 8 9 75 4 2 9 8 3 3 10 7 4
--R (30a c d + --- a b c d + -- a b c d + 5a b c d )x
--R 2 2
--R +
--R 6 8 9 5 9 8 4 2 10 7 3 6 9 8 5 10 7 2
--R (15a c d + 20a b c d + 5a b c d )x + (5a c d + 3a b c d )x
--R +
--R 6 10 7 1 6 11 6 1 5 12 5 5 4 2 13 4 5 3 3 14 3
--R a c d x + -- a c d - -- a b c d + --- a b c d - ---- a b c d
--R 11 22 286 1001
--R +
--R 1 2 4 15 2 1 5 16 1 6 17
--R ---- a b c d - ---- a b c d + ------ b c
--R 1001 8008 136136
--R /
--R 7
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 762
--S 763 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 6 10 17 3 5 10 5 6 9 16
--R -- b d x + (- a b d + - b c d )x
--R 17 8 8
--R +
--R 2 4 10 5 9 6 2 8 15
--R (a b d + 4a b c d + 3b c d )x
--R +
--R 10 3 3 10 75 2 4 9 135 5 2 8 60 6 3 7 14
--R (-- a b d + -- a b c d + --- a b c d + -- b c d )x
--R 7 7 7 7
--R +
--R 15 4 2 10 200 3 3 9 675 2 4 2 8 720 5 3 7 210 6 4 6 13
--R (-- a b d + --- a b c d + --- a b c d + --- a b c d + --- b c d )x
--R 13 13 13 13 13
--R +
--R 1 5 10 25 4 2 9 3 3 2 8 2 4 3 7 5 4 6
--R - a b d + -- a b c d + 75a b c d + 150a b c d + 105a b c d
--R 2 2
--R +
--R 6 5 5
--R 21b c d
--R *
--R 12
--R x
--R +
--R 1 6 10 60 5 9 675 4 2 2 8 2400 3 3 3 7 3150 2 4 4 6
--R -- a d + -- a b c d + --- a b c d + ---- a b c d + ---- a b c d
--R 11 11 11 11 11
--R +
--R 1512 5 5 5 210 6 6 4
--R ---- a b c d + --- b c d
--R 11 11
--R *
--R 11
--R x
--R +
--R 6 9 5 2 8 4 2 3 7 3 3 4 6 2 4 5 5
--R a c d + 27a b c d + 180a b c d + 420a b c d + 378a b c d
--R +
--R 5 6 4 6 7 3
--R 126a b c d + 12b c d
--R *
--R 10
--R x
--R +
--R 6 2 8 5 3 7 4 2 4 6 3 3 5 5 2 4 6 4
--R 5a c d + 80a b c d + 350a b c d + 560a b c d + 350a b c d
--R +
--R 5 7 3 6 8 2
--R 80a b c d + 5b c d
--R *
--R 9
--R x
--R +
--R 6 3 7 315 5 4 6 945 4 2 5 5 3 3 6 4 2 4 7 3
--R 15a c d + --- a b c d + --- a b c d + 525a b c d + 225a b c d
--R 2 2
--R +
--R 135 5 8 2 5 6 9
--R --- a b c d + - b c d
--R 4 4
--R *
--R 8
--R x
--R +
--R 6 4 6 5 5 5 4 2 6 4 2400 3 3 7 3 675 2 4 8 2
--R 30a c d + 216a b c d + 450a b c d + ---- a b c d + --- a b c d
--R 7 7
--R +
--R 60 5 9 1 6 10
--R -- a b c d + - b c
--R 7 7
--R *
--R 7
--R x
--R +
--R 6 5 5 5 6 4 4 2 7 3 3 3 8 2 2 4 9
--R 42a c d + 210a b c d + 300a b c d + 150a b c d + 25a b c d
--R +
--R 5 10
--R a b c
--R *
--R 6
--R x
--R +
--R 6 6 4 5 7 3 4 2 8 2 3 3 9 2 4 10 5
--R (42a c d + 144a b c d + 135a b c d + 40a b c d + 3a b c )x
--R +
--R 6 7 3 135 5 8 2 75 4 2 9 3 3 10 4
--R (30a c d + --- a b c d + -- a b c d + 5a b c )x
--R 2 2
--R +
--R 6 8 2 5 9 4 2 10 3 6 9 5 10 2 6 10
--R (15a c d + 20a b c d + 5a b c )x + (5a c d + 3a b c )x + a c x
--R Type: Polynomial(Fraction(Integer))
--E 763
--S 764 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 6 11 6 1 5 12 5 5 4 2 13 4 5 3 3 14 3
--R - -- a c d + -- a b c d - --- a b c d + ---- a b c d
--R 11 22 286 1001
--R +
--R 1 2 4 15 2 1 5 16 1 6 17
--R - ---- a b c d + ---- a b c d - ------ b c
--R 1001 8008 136136
--R /
--R 7
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 764
--S 765 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 765
)clear all
--S 766 of 2952
t0000:=(a+b*x)^5*(c+d*x)^10
--R
--R
--R (1)
--R 5 10 15 4 10 5 9 14
--R b d x + (5a b d + 10b c d )x
--R +
--R 2 3 10 4 9 5 2 8 13
--R (10a b d + 50a b c d + 45b c d )x
--R +
--R 3 2 10 2 3 9 4 2 8 5 3 7 12
--R (10a b d + 100a b c d + 225a b c d + 120b c d )x
--R +
--R 4 10 3 2 9 2 3 2 8 4 3 7 5 4 6 11
--R (5a b d + 100a b c d + 450a b c d + 600a b c d + 210b c d )x
--R +
--R 5 10 4 9 3 2 2 8 2 3 3 7 4 4 6
--R a d + 50a b c d + 450a b c d + 1200a b c d + 1050a b c d
--R +
--R 5 5 5
--R 252b c d
--R *
--R 10
--R x
--R +
--R 5 9 4 2 8 3 2 3 7 2 3 4 6 4 5 5
--R 10a c d + 225a b c d + 1200a b c d + 2100a b c d + 1260a b c d
--R +
--R 5 6 4
--R 210b c d
--R *
--R 9
--R x
--R +
--R 5 2 8 4 3 7 3 2 4 6 2 3 5 5 4 6 4
--R 45a c d + 600a b c d + 2100a b c d + 2520a b c d + 1050a b c d
--R +
--R 5 7 3
--R 120b c d
--R *
--R 8
--R x
--R +
--R 5 3 7 4 4 6 3 2 5 5 2 3 6 4 4 7 3
--R 120a c d + 1050a b c d + 2520a b c d + 2100a b c d + 600a b c d
--R +
--R 5 8 2
--R 45b c d
--R *
--R 7
--R x
--R +
--R 5 4 6 4 5 5 3 2 6 4 2 3 7 3 4 8 2
--R 210a c d + 1260a b c d + 2100a b c d + 1200a b c d + 225a b c d
--R +
--R 5 9
--R 10b c d
--R *
--R 6
--R x
--R +
--R 5 5 5 4 6 4 3 2 7 3 2 3 8 2 4 9
--R 252a c d + 1050a b c d + 1200a b c d + 450a b c d + 50a b c d
--R +
--R 5 10
--R b c
--R *
--R 5
--R x
--R +
--R 5 6 4 4 7 3 3 2 8 2 2 3 9 4 10 4
--R (210a c d + 600a b c d + 450a b c d + 100a b c d + 5a b c )x
--R +
--R 5 7 3 4 8 2 3 2 9 2 3 10 3
--R (120a c d + 225a b c d + 100a b c d + 10a b c )x
--R +
--R 5 8 2 4 9 3 2 10 2 5 9 4 10 5 10
--R (45a c d + 50a b c d + 10a b c )x + (10a c d + 5a b c )x + a c
--R Type: Polynomial(Integer)
--E 766
--S 767 of 2952
r0000:=-1/11*(b*c-a*d)^5*(c+d*x)^11/d^6+5/12*b*(b*c-a*d)^4*(c+d*x)^12/d^6-_
10/13*b^2*(b*c-a*d)^3*(c+d*x)^13/d^6+_
5/7*b^3*(b*c-a*d)^2*(c+d*x)^14/d^6-_
1/3*b^4*(b*c-a*d)*(c+d*x)^15/d^6+1/16*b^5*(c+d*x)^16/d^6
--R
--R
--R (2)
--R 1 5 16 16 1 4 16 2 5 15 15
--R -- b d x + (- a b d + - b c d )x
--R 16 3 3
--R +
--R 5 2 3 16 25 4 15 45 5 2 14 14
--R (- a b d + -- a b c d + -- b c d )x
--R 7 7 14
--R +
--R 10 3 2 16 100 2 3 15 225 4 2 14 120 5 3 13 13
--R (-- a b d + --- a b c d + --- a b c d + --- b c d )x
--R 13 13 13 13
--R +
--R 5 4 16 25 3 2 15 75 2 3 2 14 4 3 13 35 5 4 12 12
--R (-- a b d + -- a b c d + -- a b c d + 50a b c d + -- b c d )x
--R 12 3 2 2
--R +
--R 1 5 16 50 4 15 450 3 2 2 14 1200 2 3 3 13
--R -- a d + -- a b c d + --- a b c d + ---- a b c d
--R 11 11 11 11
--R +
--R 1050 4 4 12 252 5 5 11
--R ---- a b c d + --- b c d
--R 11 11
--R *
--R 11
--R x
--R +
--R 5 15 45 4 2 14 3 2 3 13 2 3 4 12 4 5 11
--R a c d + -- a b c d + 120a b c d + 210a b c d + 126a b c d
--R 2
--R +
--R 5 6 10
--R 21b c d
--R *
--R 10
--R x
--R +
--R 5 2 14 200 4 3 13 700 3 2 4 12 2 3 5 11
--R 5a c d + --- a b c d + --- a b c d + 280a b c d
--R 3 3
--R +
--R 350 4 6 10 40 5 7 9
--R --- a b c d + -- b c d
--R 3 3
--R *
--R 9
--R x
--R +
--R 5 3 13 525 4 4 12 3 2 5 11 525 2 3 6 10 4 7 9
--R 15a c d + --- a b c d + 315a b c d + --- a b c d + 75a b c d
--R 4 2
--R +
--R 45 5 8 8
--R -- b c d
--R 8
--R *
--R 8
--R x
--R +
--R 5 4 12 4 5 11 3 2 6 10 1200 2 3 7 9
--R 30a c d + 180a b c d + 300a b c d + ---- a b c d
--R 7
--R +
--R 225 4 8 8 10 5 9 7
--R --- a b c d + -- b c d
--R 7 7
--R *
--R 7
--R x
--R +
--R 5 5 11 4 6 10 3 2 7 9 2 3 8 8 25 4 9 7
--R 42a c d + 175a b c d + 200a b c d + 75a b c d + -- a b c d
--R 3
--R +
--R 1 5 10 6
--R - b c d
--R 6
--R *
--R 6
--R x
--R +
--R 5 6 10 4 7 9 3 2 8 8 2 3 9 7 4 10 6 5
--R (42a c d + 120a b c d + 90a b c d + 20a b c d + a b c d )x
--R +
--R 5 7 9 225 4 8 8 3 2 9 7 5 2 3 10 6 4
--R (30a c d + --- a b c d + 25a b c d + - a b c d )x
--R 4 2
--R +
--R 5 8 8 50 4 9 7 10 3 2 10 6 3 5 9 7 5 4 10 6 2
--R (15a c d + -- a b c d + -- a b c d )x + (5a c d + - a b c d )x
--R 3 3 2
--R +
--R 5 10 6 1 5 11 5 5 4 12 4 5 3 2 13 3 5 2 3 14 2
--R a c d x + -- a c d - --- a b c d + --- a b c d - ---- a b c d
--R 11 132 429 2002
--R +
--R 1 4 15 1 5 16
--R ---- a b c d - ----- b c
--R 3003 48048
--R /
--R 6
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 767
--S 768 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 5 10 16 1 4 10 2 5 9 15
--R -- b d x + (- a b d + - b c d )x
--R 16 3 3
--R +
--R 5 2 3 10 25 4 9 45 5 2 8 14
--R (- a b d + -- a b c d + -- b c d )x
--R 7 7 14
--R +
--R 10 3 2 10 100 2 3 9 225 4 2 8 120 5 3 7 13
--R (-- a b d + --- a b c d + --- a b c d + --- b c d )x
--R 13 13 13 13
--R +
--R 5 4 10 25 3 2 9 75 2 3 2 8 4 3 7 35 5 4 6 12
--R (-- a b d + -- a b c d + -- a b c d + 50a b c d + -- b c d )x
--R 12 3 2 2
--R +
--R 1 5 10 50 4 9 450 3 2 2 8 1200 2 3 3 7 1050 4 4 6
--R -- a d + -- a b c d + --- a b c d + ---- a b c d + ---- a b c d
--R 11 11 11 11 11
--R +
--R 252 5 5 5
--R --- b c d
--R 11
--R *
--R 11
--R x
--R +
--R 5 9 45 4 2 8 3 2 3 7 2 3 4 6 4 5 5
--R a c d + -- a b c d + 120a b c d + 210a b c d + 126a b c d
--R 2
--R +
--R 5 6 4
--R 21b c d
--R *
--R 10
--R x
--R +
--R 5 2 8 200 4 3 7 700 3 2 4 6 2 3 5 5 350 4 6 4
--R 5a c d + --- a b c d + --- a b c d + 280a b c d + --- a b c d
--R 3 3 3
--R +
--R 40 5 7 3
--R -- b c d
--R 3
--R *
--R 9
--R x
--R +
--R 5 3 7 525 4 4 6 3 2 5 5 525 2 3 6 4 4 7 3
--R 15a c d + --- a b c d + 315a b c d + --- a b c d + 75a b c d
--R 4 2
--R +
--R 45 5 8 2
--R -- b c d
--R 8
--R *
--R 8
--R x
--R +
--R 5 4 6 4 5 5 3 2 6 4 1200 2 3 7 3 225 4 8 2
--R 30a c d + 180a b c d + 300a b c d + ---- a b c d + --- a b c d
--R 7 7
--R +
--R 10 5 9
--R -- b c d
--R 7
--R *
--R 7
--R x
--R +
--R 5 5 5 4 6 4 3 2 7 3 2 3 8 2 25 4 9
--R 42a c d + 175a b c d + 200a b c d + 75a b c d + -- a b c d
--R 3
--R +
--R 1 5 10
--R - b c
--R 6
--R *
--R 6
--R x
--R +
--R 5 6 4 4 7 3 3 2 8 2 2 3 9 4 10 5
--R (42a c d + 120a b c d + 90a b c d + 20a b c d + a b c )x
--R +
--R 5 7 3 225 4 8 2 3 2 9 5 2 3 10 4
--R (30a c d + --- a b c d + 25a b c d + - a b c )x
--R 4 2
--R +
--R 5 8 2 50 4 9 10 3 2 10 3 5 9 5 4 10 2 5 10
--R (15a c d + -- a b c d + -- a b c )x + (5a c d + - a b c )x + a c x
--R 3 3 2
--R Type: Polynomial(Fraction(Integer))
--E 768
--S 769 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 5 11 5 5 4 12 4 5 3 2 13 3 5 2 3 14 2
--R - -- a c d + --- a b c d - --- a b c d + ---- a b c d
--R 11 132 429 2002
--R +
--R 1 4 15 1 5 16
--R - ---- a b c d + ----- b c
--R 3003 48048
--R /
--R 6
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 769
--S 770 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 770
)clear all
--S 771 of 2952
t0000:=(a+b*x)^4*(c+d*x)^10
--R
--R
--R (1)
--R 4 10 14 3 10 4 9 13 2 2 10 3 9 4 2 8 12
--R b d x + (4a b d + 10b c d )x + (6a b d + 40a b c d + 45b c d )x
--R +
--R 3 10 2 2 9 3 2 8 4 3 7 11
--R (4a b d + 60a b c d + 180a b c d + 120b c d )x
--R +
--R 4 10 3 9 2 2 2 8 3 3 7 4 4 6 10
--R (a d + 40a b c d + 270a b c d + 480a b c d + 210b c d )x
--R +
--R 4 9 3 2 8 2 2 3 7 3 4 6 4 5 5 9
--R (10a c d + 180a b c d + 720a b c d + 840a b c d + 252b c d )x
--R +
--R 4 2 8 3 3 7 2 2 4 6 3 5 5 4 6 4 8
--R (45a c d + 480a b c d + 1260a b c d + 1008a b c d + 210b c d )x
--R +
--R 4 3 7 3 4 6 2 2 5 5 3 6 4 4 7 3 7
--R (120a c d + 840a b c d + 1512a b c d + 840a b c d + 120b c d )x
--R +
--R 4 4 6 3 5 5 2 2 6 4 3 7 3 4 8 2 6
--R (210a c d + 1008a b c d + 1260a b c d + 480a b c d + 45b c d )x
--R +
--R 4 5 5 3 6 4 2 2 7 3 3 8 2 4 9 5
--R (252a c d + 840a b c d + 720a b c d + 180a b c d + 10b c d)x
--R +
--R 4 6 4 3 7 3 2 2 8 2 3 9 4 10 4
--R (210a c d + 480a b c d + 270a b c d + 40a b c d + b c )x
--R +
--R 4 7 3 3 8 2 2 2 9 3 10 3
--R (120a c d + 180a b c d + 60a b c d + 4a b c )x
--R +
--R 4 8 2 3 9 2 2 10 2 4 9 3 10 4 10
--R (45a c d + 40a b c d + 6a b c )x + (10a c d + 4a b c )x + a c
--R Type: Polynomial(Integer)
--E 771
--S 772 of 2952
r0000:=1/11*(b*c-a*d)^4*(c+d*x)^11/d^5-1/3*b*(b*c-a*d)^3*(c+d*x)^12/d^5+_
6/13*b^2*(b*c-a*d)^2*(c+d*x)^13/d^5-2/7*b^3*(b*c-a*d)*(c+d*x)^14/d^5+_
1/15*b^4*(c+d*x)^15/d^5
--R
--R
--R (2)
--R 1 4 15 15 2 3 15 5 4 14 14
--R -- b d x + (- a b d + - b c d )x
--R 15 7 7
--R +
--R 6 2 2 15 40 3 14 45 4 2 13 13
--R (-- a b d + -- a b c d + -- b c d )x
--R 13 13 13
--R +
--R 1 3 15 2 2 14 3 2 13 4 3 12 12
--R (- a b d + 5a b c d + 15a b c d + 10b c d )x
--R 3
--R +
--R 1 4 15 40 3 14 270 2 2 2 13 480 3 3 12 210 4 4 11
--R (-- a d + -- a b c d + --- a b c d + --- a b c d + --- b c d )
--R 11 11 11 11 11
--R *
--R 11
--R x
--R +
--R 4 14 3 2 13 2 2 3 12 3 4 11 126 4 5 10 10
--R (a c d + 18a b c d + 72a b c d + 84a b c d + --- b c d )x
--R 5
--R +
--R 4 2 13 160 3 3 12 2 2 4 11 3 5 10 70 4 6 9 9
--R (5a c d + --- a b c d + 140a b c d + 112a b c d + -- b c d )x
--R 3 3
--R +
--R 4 3 12 3 4 11 2 2 5 10 3 6 9 4 7 8 8
--R (15a c d + 105a b c d + 189a b c d + 105a b c d + 15b c d )x
--R +
--R 4 4 11 3 5 10 2 2 6 9 480 3 7 8 45 4 8 7 7
--R (30a c d + 144a b c d + 180a b c d + --- a b c d + -- b c d )x
--R 7 7
--R +
--R 4 5 10 3 6 9 2 2 7 8 3 8 7 5 4 9 6 6
--R (42a c d + 140a b c d + 120a b c d + 30a b c d + - b c d )x
--R 3
--R +
--R 4 6 9 3 7 8 2 2 8 7 3 9 6 1 4 10 5 5
--R (42a c d + 96a b c d + 54a b c d + 8a b c d + - b c d )x
--R 5
--R +
--R 4 7 8 3 8 7 2 2 9 6 3 10 5 4
--R (30a c d + 45a b c d + 15a b c d + a b c d )x
--R +
--R 4 8 7 40 3 9 6 2 2 10 5 3 4 9 6 3 10 5 2
--R (15a c d + -- a b c d + 2a b c d )x + (5a c d + 2a b c d )x
--R 3
--R +
--R 4 10 5 1 4 11 4 1 3 12 3 1 2 2 13 2 1 3 14
--R a c d x + -- a c d - -- a b c d + --- a b c d - ---- a b c d
--R 11 33 143 1001
--R +
--R 1 4 15
--R ----- b c
--R 15015
--R /
--R 5
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 772
--S 773 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 4 10 15 2 3 10 5 4 9 14
--R -- b d x + (- a b d + - b c d )x
--R 15 7 7
--R +
--R 6 2 2 10 40 3 9 45 4 2 8 13
--R (-- a b d + -- a b c d + -- b c d )x
--R 13 13 13
--R +
--R 1 3 10 2 2 9 3 2 8 4 3 7 12
--R (- a b d + 5a b c d + 15a b c d + 10b c d )x
--R 3
--R +
--R 1 4 10 40 3 9 270 2 2 2 8 480 3 3 7 210 4 4 6 11
--R (-- a d + -- a b c d + --- a b c d + --- a b c d + --- b c d )x
--R 11 11 11 11 11
--R +
--R 4 9 3 2 8 2 2 3 7 3 4 6 126 4 5 5 10
--R (a c d + 18a b c d + 72a b c d + 84a b c d + --- b c d )x
--R 5
--R +
--R 4 2 8 160 3 3 7 2 2 4 6 3 5 5 70 4 6 4 9
--R (5a c d + --- a b c d + 140a b c d + 112a b c d + -- b c d )x
--R 3 3
--R +
--R 4 3 7 3 4 6 2 2 5 5 3 6 4 4 7 3 8
--R (15a c d + 105a b c d + 189a b c d + 105a b c d + 15b c d )x
--R +
--R 4 4 6 3 5 5 2 2 6 4 480 3 7 3 45 4 8 2 7
--R (30a c d + 144a b c d + 180a b c d + --- a b c d + -- b c d )x
--R 7 7
--R +
--R 4 5 5 3 6 4 2 2 7 3 3 8 2 5 4 9 6
--R (42a c d + 140a b c d + 120a b c d + 30a b c d + - b c d)x
--R 3
--R +
--R 4 6 4 3 7 3 2 2 8 2 3 9 1 4 10 5
--R (42a c d + 96a b c d + 54a b c d + 8a b c d + - b c )x
--R 5
--R +
--R 4 7 3 3 8 2 2 2 9 3 10 4
--R (30a c d + 45a b c d + 15a b c d + a b c )x
--R +
--R 4 8 2 40 3 9 2 2 10 3 4 9 3 10 2 4 10
--R (15a c d + -- a b c d + 2a b c )x + (5a c d + 2a b c )x + a c x
--R 3
--R Type: Polynomial(Fraction(Integer))
--E 773
--S 774 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 1 4 11 4 1 3 12 3 1 2 2 13 2 1 3 14 1 4 15
--R - -- a c d + -- a b c d - --- a b c d + ---- a b c d - ----- b c
--R 11 33 143 1001 15015
--R -------------------------------------------------------------------------
--R 5
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 774
--S 775 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 775
)clear all
--S 776 of 2952
t0000:=(a+b*x)^3*(c+d*x)^10
--R
--R
--R (1)
--R 3 10 13 2 10 3 9 12 2 10 2 9 3 2 8 11
--R b d x + (3a b d + 10b c d )x + (3a b d + 30a b c d + 45b c d )x
--R +
--R 3 10 2 9 2 2 8 3 3 7 10
--R (a d + 30a b c d + 135a b c d + 120b c d )x
--R +
--R 3 9 2 2 8 2 3 7 3 4 6 9
--R (10a c d + 135a b c d + 360a b c d + 210b c d )x
--R +
--R 3 2 8 2 3 7 2 4 6 3 5 5 8
--R (45a c d + 360a b c d + 630a b c d + 252b c d )x
--R +
--R 3 3 7 2 4 6 2 5 5 3 6 4 7
--R (120a c d + 630a b c d + 756a b c d + 210b c d )x
--R +
--R 3 4 6 2 5 5 2 6 4 3 7 3 6
--R (210a c d + 756a b c d + 630a b c d + 120b c d )x
--R +
--R 3 5 5 2 6 4 2 7 3 3 8 2 5
--R (252a c d + 630a b c d + 360a b c d + 45b c d )x
--R +
--R 3 6 4 2 7 3 2 8 2 3 9 4
--R (210a c d + 360a b c d + 135a b c d + 10b c d)x
--R +
--R 3 7 3 2 8 2 2 9 3 10 3
--R (120a c d + 135a b c d + 30a b c d + b c )x
--R +
--R 3 8 2 2 9 2 10 2 3 9 2 10 3 10
--R (45a c d + 30a b c d + 3a b c )x + (10a c d + 3a b c )x + a c
--R Type: Polynomial(Integer)
--E 776
--S 777 of 2952
r0000:=-1/11*(b*c-a*d)^3*(c+d*x)^11/d^4+_
1/4*b*(b*c-a*d)^2*(c+d*x)^12/d^4-3/13*b^2*(b*c-a*d)*(c+d*x)^13/d^4+_
1/14*b^3*(c+d*x)^14/d^4
--R
--R
--R (2)
--R 1 3 14 14 3 2 14 10 3 13 13
--R -- b d x + (-- a b d + -- b c d )x
--R 14 13 13
--R +
--R 1 2 14 5 2 13 15 3 2 12 12
--R (- a b d + - a b c d + -- b c d )x
--R 4 2 4
--R +
--R 1 3 14 30 2 13 135 2 2 12 120 3 3 11 11
--R (-- a d + -- a b c d + --- a b c d + --- b c d )x
--R 11 11 11 11
--R +
--R 3 13 27 2 2 12 2 3 11 3 4 10 10
--R (a c d + -- a b c d + 36a b c d + 21b c d )x
--R 2
--R +
--R 3 2 12 2 3 11 2 4 10 3 5 9 9
--R (5a c d + 40a b c d + 70a b c d + 28b c d )x
--R +
--R 3 3 11 315 2 4 10 189 2 5 9 105 3 6 8 8
--R (15a c d + --- a b c d + --- a b c d + --- b c d )x
--R 4 2 4
--R +
--R 3 4 10 2 5 9 2 6 8 120 3 7 7 7
--R (30a c d + 108a b c d + 90a b c d + --- b c d )x
--R 7
--R +
--R 3 5 9 2 6 8 2 7 7 15 3 8 6 6
--R (42a c d + 105a b c d + 60a b c d + -- b c d )x
--R 2
--R +
--R 3 6 8 2 7 7 2 8 6 3 9 5 5
--R (42a c d + 72a b c d + 27a b c d + 2b c d )x
--R +
--R 3 7 7 135 2 8 6 15 2 9 5 1 3 10 4 4
--R (30a c d + --- a b c d + -- a b c d + - b c d )x
--R 4 2 4
--R +
--R 3 8 6 2 9 5 2 10 4 3 3 9 5 3 2 10 4 2
--R (15a c d + 10a b c d + a b c d )x + (5a c d + - a b c d )x
--R 2
--R +
--R 3 10 4 1 3 11 3 1 2 12 2 1 2 13 1 3 14
--R a c d x + -- a c d - -- a b c d + --- a b c d - ---- b c
--R 11 44 286 4004
--R /
--R 4
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 777
--S 778 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 3 10 14 3 2 10 10 3 9 13
--R -- b d x + (-- a b d + -- b c d )x
--R 14 13 13
--R +
--R 1 2 10 5 2 9 15 3 2 8 12
--R (- a b d + - a b c d + -- b c d )x
--R 4 2 4
--R +
--R 1 3 10 30 2 9 135 2 2 8 120 3 3 7 11
--R (-- a d + -- a b c d + --- a b c d + --- b c d )x
--R 11 11 11 11
--R +
--R 3 9 27 2 2 8 2 3 7 3 4 6 10
--R (a c d + -- a b c d + 36a b c d + 21b c d )x
--R 2
--R +
--R 3 2 8 2 3 7 2 4 6 3 5 5 9
--R (5a c d + 40a b c d + 70a b c d + 28b c d )x
--R +
--R 3 3 7 315 2 4 6 189 2 5 5 105 3 6 4 8
--R (15a c d + --- a b c d + --- a b c d + --- b c d )x
--R 4 2 4
--R +
--R 3 4 6 2 5 5 2 6 4 120 3 7 3 7
--R (30a c d + 108a b c d + 90a b c d + --- b c d )x
--R 7
--R +
--R 3 5 5 2 6 4 2 7 3 15 3 8 2 6
--R (42a c d + 105a b c d + 60a b c d + -- b c d )x
--R 2
--R +
--R 3 6 4 2 7 3 2 8 2 3 9 5
--R (42a c d + 72a b c d + 27a b c d + 2b c d)x
--R +
--R 3 7 3 135 2 8 2 15 2 9 1 3 10 4
--R (30a c d + --- a b c d + -- a b c d + - b c )x
--R 4 2 4
--R +
--R 3 8 2 2 9 2 10 3 3 9 3 2 10 2 3 10
--R (15a c d + 10a b c d + a b c )x + (5a c d + - a b c )x + a c x
--R 2
--R Type: Polynomial(Fraction(Integer))
--E 778
--S 779 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 3 11 3 1 2 12 2 1 2 13 1 3 14
--R - -- a c d + -- a b c d - --- a b c d + ---- b c
--R 11 44 286 4004
--R (4) -------------------------------------------------------
--R 4
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 779
--S 780 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 780
)clear all
--S 781 of 2952
t0000:=(a+b*x)^2*(c+d*x)^10
--R
--R
--R (1)
--R 2 10 12 10 2 9 11 2 10 9 2 2 8 10
--R b d x + (2a b d + 10b c d )x + (a d + 20a b c d + 45b c d )x
--R +
--R 2 9 2 8 2 3 7 9
--R (10a c d + 90a b c d + 120b c d )x
--R +
--R 2 2 8 3 7 2 4 6 8
--R (45a c d + 240a b c d + 210b c d )x
--R +
--R 2 3 7 4 6 2 5 5 7
--R (120a c d + 420a b c d + 252b c d )x
--R +
--R 2 4 6 5 5 2 6 4 6
--R (210a c d + 504a b c d + 210b c d )x
--R +
--R 2 5 5 6 4 2 7 3 5
--R (252a c d + 420a b c d + 120b c d )x
--R +
--R 2 6 4 7 3 2 8 2 4
--R (210a c d + 240a b c d + 45b c d )x
--R +
--R 2 7 3 8 2 2 9 3 2 8 2 9 2 10 2
--R (120a c d + 90a b c d + 10b c d)x + (45a c d + 20a b c d + b c )x
--R +
--R 2 9 10 2 10
--R (10a c d + 2a b c )x + a c
--R Type: Polynomial(Integer)
--E 781
--S 782 of 2952
r0000:=1/11*(b*c-a*d)^2*(c+d*x)^11/d^3-_
1/6*b*(b*c-a*d)*(c+d*x)^12/d^3+1/13*b^2*(c+d*x)^13/d^3
--R
--R
--R (2)
--R 1 2 13 13 1 13 5 2 12 12
--R -- b d x + (- a b d + - b c d )x
--R 13 6 6
--R +
--R 1 2 13 20 12 45 2 2 11 11
--R (-- a d + -- a b c d + -- b c d )x
--R 11 11 11
--R +
--R 2 12 2 11 2 3 10 10
--R (a c d + 9a b c d + 12b c d )x
--R +
--R 2 2 11 80 3 10 70 2 4 9 9
--R (5a c d + -- a b c d + -- b c d )x
--R 3 3
--R +
--R 2 3 10 105 4 9 63 2 5 8 8
--R (15a c d + --- a b c d + -- b c d )x
--R 2 2
--R +
--R 2 4 9 5 8 2 6 7 7
--R (30a c d + 72a b c d + 30b c d )x
--R +
--R 2 5 8 6 7 2 7 6 6
--R (42a c d + 70a b c d + 20b c d )x
--R +
--R 2 6 7 7 6 2 8 5 5
--R (42a c d + 48a b c d + 9b c d )x
--R +
--R 2 7 6 45 8 5 5 2 9 4 4
--R (30a c d + -- a b c d + - b c d )x
--R 2 2
--R +
--R 2 8 5 20 9 4 1 2 10 3 3 2 9 4 10 3 2
--R (15a c d + -- a b c d + - b c d )x + (5a c d + a b c d )x
--R 3 3
--R +
--R 2 10 3 1 2 11 2 1 12 1 2 13
--R a c d x + -- a c d - -- a b c d + --- b c
--R 11 66 858
--R /
--R 3
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 782
--S 783 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 2 10 13 1 10 5 2 9 12
--R -- b d x + (- a b d + - b c d )x
--R 13 6 6
--R +
--R 1 2 10 20 9 45 2 2 8 11
--R (-- a d + -- a b c d + -- b c d )x
--R 11 11 11
--R +
--R 2 9 2 8 2 3 7 10 2 2 8 80 3 7 70 2 4 6 9
--R (a c d + 9a b c d + 12b c d )x + (5a c d + -- a b c d + -- b c d )x
--R 3 3
--R +
--R 2 3 7 105 4 6 63 2 5 5 8
--R (15a c d + --- a b c d + -- b c d )x
--R 2 2
--R +
--R 2 4 6 5 5 2 6 4 7 2 5 5 6 4 2 7 3 6
--R (30a c d + 72a b c d + 30b c d )x + (42a c d + 70a b c d + 20b c d )x
--R +
--R 2 6 4 7 3 2 8 2 5 2 7 3 45 8 2 5 2 9 4
--R (42a c d + 48a b c d + 9b c d )x + (30a c d + -- a b c d + - b c d)x
--R 2 2
--R +
--R 2 8 2 20 9 1 2 10 3 2 9 10 2 2 10
--R (15a c d + -- a b c d + - b c )x + (5a c d + a b c )x + a c x
--R 3 3
--R Type: Polynomial(Fraction(Integer))
--E 783
--S 784 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 2 11 2 1 12 1 2 13
--R - -- a c d + -- a b c d - --- b c
--R 11 66 858
--R (4) --------------------------------------
--R 3
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 784
--S 785 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 785
)clear all
--S 786 of 2952
t0000:=(a+b*x)*(c+d*x)^10
--R
--R
--R (1)
--R 10 11 10 9 10 9 2 8 9
--R b d x + (a d + 10b c d )x + (10a c d + 45b c d )x
--R +
--R 2 8 3 7 8 3 7 4 6 7
--R (45a c d + 120b c d )x + (120a c d + 210b c d )x
--R +
--R 4 6 5 5 6 5 5 6 4 5
--R (210a c d + 252b c d )x + (252a c d + 210b c d )x
--R +
--R 6 4 7 3 4 7 3 8 2 3
--R (210a c d + 120b c d )x + (120a c d + 45b c d )x
--R +
--R 8 2 9 2 9 10 10
--R (45a c d + 10b c d)x + (10a c d + b c )x + a c
--R Type: Polynomial(Integer)
--E 786
--S 787 of 2952
r0000:=-1/11*(b*c-a*d)*(c+d*x)^11/d^2+1/12*b*(c+d*x)^12/d^2
--R
--R
--R (2)
--R 1 12 12 1 12 10 11 11 11 9 2 10 10
--R -- b d x + (-- a d + -- b c d )x + (a c d + - b c d )x
--R 12 11 11 2
--R +
--R 2 10 40 3 9 9 3 9 105 4 8 8
--R (5a c d + -- b c d )x + (15a c d + --- b c d )x
--R 3 4
--R +
--R 4 8 5 7 7 5 7 6 6 6
--R (30a c d + 36b c d )x + (42a c d + 35b c d )x
--R +
--R 6 6 7 5 5 7 5 45 8 4 4
--R (42a c d + 24b c d )x + (30a c d + -- b c d )x
--R 4
--R +
--R 8 4 10 9 3 3 9 3 1 10 2 2 10 2 1 11
--R (15a c d + -- b c d )x + (5a c d + - b c d )x + a c d x + -- a c d
--R 3 2 11
--R +
--R 1 12
--R - --- b c
--R 132
--R /
--R 2
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 787
--S 788 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 10 12 1 10 10 9 11 9 9 2 8 10
--R -- b d x + (-- a d + -- b c d )x + (a c d + - b c d )x
--R 12 11 11 2
--R +
--R 2 8 40 3 7 9 3 7 105 4 6 8
--R (5a c d + -- b c d )x + (15a c d + --- b c d )x
--R 3 4
--R +
--R 4 6 5 5 7 5 5 6 4 6 6 4 7 3 5
--R (30a c d + 36b c d )x + (42a c d + 35b c d )x + (42a c d + 24b c d )x
--R +
--R 7 3 45 8 2 4 8 2 10 9 3 9 1 10 2
--R (30a c d + -- b c d )x + (15a c d + -- b c d)x + (5a c d + - b c )x
--R 4 3 2
--R +
--R 10
--R a c x
--R Type: Polynomial(Fraction(Integer))
--E 788
--S 789 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 11 1 12
--R - -- a c d + --- b c
--R 11 132
--R (4) -----------------------
--R 2
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 789
--S 790 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 790
)clear all
--S 791 of 2952
t0000:=(c+d*x)^10
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R Type: Polynomial(Integer)
--E 791
--S 792 of 2952
r0000:=1/11*(c+d*x)^11/d
--R
--R
--R (2)
--R 1 11 11 10 10 2 9 9 3 8 8 4 7 7 5 6 6
--R -- d x + c d x + 5c d x + 15c d x + 30c d x + 42c d x
--R 11
--R +
--R 6 5 5 7 4 4 8 3 3 9 2 2 10 1 11
--R 42c d x + 30c d x + 15c d x + 5c d x + c d x + -- c
--R 11
--R /
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 792
--S 793 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 1 10 11 9 10 2 8 9 3 7 8 4 6 7 5 5 6 6 4 5
--R -- d x + c d x + 5c d x + 15c d x + 30c d x + 42c d x + 42c d x
--R 11
--R +
--R 7 3 4 8 2 3 9 2 10
--R 30c d x + 15c d x + 5c d x + c x
--R Type: Polynomial(Fraction(Integer))
--E 793
--S 794 of 2952
m0000:=a0000 - r0000
--R
--R
--R 1 11
--R -- c
--R 11
--R (4) - ------
--R d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 794
--S 795 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 795
)clear all
--S 796 of 2952
t0000:=(c+d*x)^10/(a+b*x)
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R b x + a
--R Type: Fraction(Polynomial(Integer))
--E 796
--S 797 of 2952
r0000:=d*(b*c-a*d)^9*x/b^10+1/2*(b*c-a*d)^8*(c+d*x)^2/b^9+_
1/3*(b*c-a*d)^7*(c+d*x)^3/b^8+1/4*(b*c-a*d)^6*(c+d*x)^4/b^7+_
1/5*(b*c-a*d)^5*(c+d*x)^5/b^6+1/6*(b*c-a*d)^4*(c+d*x)^6/b^5+_
1/7*(b*c-a*d)^3*(c+d*x)^7/b^4+1/8*(b*c-a*d)^2*(c+d*x)^8/b^3+_
1/9*(b*c-a*d)*(c+d*x)^9/b^2+1/10*(c+d*x)^10/b+_
(b*c-a*d)^10*log(a+b*x)/b^11
--R
--R
--R (2)
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2520a d - 25200a b c d + 113400a b c d - 302400a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R 529200a b c d - 635040a b c d + 529200a b c d - 302400a b c d
--R +
--R 2 8 8 2 9 9 10 10
--R 113400a b c d - 25200a b c d + 2520b c
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 252b d x + (- 280a b d + 2800b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (315a b d - 3150a b c d + 14175b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 360a b d + 3600a b c d - 16200a b c d + 43200b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 420a b d - 4200a b c d + 18900a b c d - 50400a b c d
--R +
--R 10 4 6
--R 88200b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 504a b d + 5040a b c d - 22680a b c d + 60480a b c d
--R +
--R 9 4 6 10 5 5
--R - 105840a b c d + 127008b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 630a b d - 6300a b c d + 28350a b c d - 75600a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 132300a b c d - 158760a b c d + 132300b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 840a b d + 8400a b c d - 37800a b c d + 100800a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 176400a b c d + 211680a b c d - 176400a b c d + 100800b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 1260a b d - 12600a b c d + 56700a b c d - 151200a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 264600a b c d - 317520a b c d + 264600a b c d - 151200a b c d
--R +
--R 10 8 2
--R 56700b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 2520a b d + 25200a b c d - 113400a b c d + 302400a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 529200a b c d + 635040a b c d - 529200a b c d + 302400a b c d
--R +
--R 9 8 2 10 9
--R - 113400a b c d + 25200b c d
--R *
--R x
--R +
--R 8 2 2 8 7 3 3 7 6 4 4 6 5 5 5 5
--R 1260a b c d - 10920a b c d + 41790a b c d - 92484a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R 129990a b c d - 119640a b c d + 71325a b c d - 25930a b c d + 4861b c
--R /
--R 11
--R 2520b
--R Type: Expression(Integer)
--E 797
--S 798 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2520a d - 25200a b c d + 113400a b c d - 302400a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R 529200a b c d - 635040a b c d + 529200a b c d - 302400a b c d
--R +
--R 2 8 8 2 9 9 10 10
--R 113400a b c d - 25200a b c d + 2520b c
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 252b d x + (- 280a b d + 2800b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (315a b d - 3150a b c d + 14175b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 360a b d + 3600a b c d - 16200a b c d + 43200b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 420a b d - 4200a b c d + 18900a b c d - 50400a b c d
--R +
--R 10 4 6
--R 88200b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 504a b d + 5040a b c d - 22680a b c d + 60480a b c d
--R +
--R 9 4 6 10 5 5
--R - 105840a b c d + 127008b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 630a b d - 6300a b c d + 28350a b c d - 75600a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 132300a b c d - 158760a b c d + 132300b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 840a b d + 8400a b c d - 37800a b c d + 100800a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 176400a b c d + 211680a b c d - 176400a b c d + 100800b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 1260a b d - 12600a b c d + 56700a b c d - 151200a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 264600a b c d - 317520a b c d + 264600a b c d - 151200a b c d
--R +
--R 10 8 2
--R 56700b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 2520a b d + 25200a b c d - 113400a b c d + 302400a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 529200a b c d + 635040a b c d - 529200a b c d + 302400a b c d
--R +
--R 9 8 2 10 9
--R - 113400a b c d + 25200b c d
--R *
--R x
--R /
--R 11
--R 2520b
--R Type: Union(Expression(Integer),...)
--E 798
--S 799 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 8 2 8 7 3 7 6 2 4 6 5 3 5 5
--R - 1260a c d + 10920a b c d - 41790a b c d + 92484a b c d
--R +
--R 4 4 6 4 3 5 7 3 2 6 8 2 7 9
--R - 129990a b c d + 119640a b c d - 71325a b c d + 25930a b c d
--R +
--R 8 10
--R - 4861b c
--R /
--R 9
--R 2520b
--R Type: Expression(Integer)
--E 799
--S 800 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 800
)clear all
--S 801 of 2952
t0000:=(c+d*x)^10/(a+b*x)^2
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 801
--S 802 of 2952
r0000:=45*d^2*(b*c-a*d)^8*x/b^10-(b*c-a*d)^10/(b^11*(a+b*x))+_
60*d^3*(b*c-a*d)^7*(a+b*x)^2/b^11+70*d^4*(b*c-a*d)^6*(a+b*x)^3/b^11+_
63*d^5*(b*c-a*d)^5*(a+b*x)^4/b^11+42*d^6*(b*c-a*d)^4*(a+b*x)^5/b^11+_
20*d^7*(b*c-a*d)^3*(a+b*x)^6/b^11+45/7*d^8*(b*c-a*d)^2*(a+b*x)^7/b^11+_
5/4*d^9*(b*c-a*d)*(a+b*x)^8/b^11+1/9*d^10*(a+b*x)^9/b^11+_
10*d*(b*c-a*d)^9*log(a+b*x)/b^11
--R
--R
--R (2)
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 2520a b d + 22680a b c d - 90720a b c d + 211680a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4
--R - 317520a b c d + 317520a b c d - 211680a b c d
--R +
--R 2 8 7 3 9 8 2 10 9
--R 90720a b c d - 22680a b c d + 2520b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2520a d + 22680a b c d - 90720a b c d + 211680a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 317520a b c d + 317520a b c d - 211680a b c d + 90720a b c d
--R +
--R 2 8 8 2 9 9
--R - 22680a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 28b d x + (- 35a b d + 315b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (45a b d - 405a b c d + 1620b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 60a b d + 540a b c d - 2160a b c d + 5040b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (84a b d - 756a b c d + 3024a b c d - 7056a b c d + 10584b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 126a b d + 1134a b c d - 4536a b c d + 10584a b c d
--R +
--R 9 4 6 10 5 5
--R - 15876a b c d + 15876b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 210a b d - 1890a b c d + 7560a b c d - 17640a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 26460a b c d - 26460a b c d + 17640b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 420a b d + 3780a b c d - 15120a b c d + 35280a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 52920a b c d + 52920a b c d - 35280a b c d + 15120b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 1260a b d - 11340a b c d + 45360a b c d - 105840a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 158760a b c d - 158760a b c d + 105840a b c d - 45360a b c d
--R +
--R 10 8 2
--R 11340b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 4211a b d + 29079a b c d - 82296a b c d + 116424a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 68796a b c d - 26460a b c d + 70560a b c d - 45360a b c d
--R +
--R 9 8 2
--R 11340a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 6731a d + 51759a b c d - 173016a b c d + 328104a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 386316a b c d + 291060a b c d - 141120a b c d + 45360a b c d
--R +
--R 2 8 8 2 9 9 10 10
--R - 11340a b c d + 2520a b c d - 252b c
--R /
--R 12 11
--R 252b x + 252a b
--R Type: Expression(Integer)
--E 802
--S 803 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 2520a b d + 22680a b c d - 90720a b c d + 211680a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4
--R - 317520a b c d + 317520a b c d - 211680a b c d
--R +
--R 2 8 7 3 9 8 2 10 9
--R 90720a b c d - 22680a b c d + 2520b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2520a d + 22680a b c d - 90720a b c d + 211680a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 317520a b c d + 317520a b c d - 211680a b c d + 90720a b c d
--R +
--R 2 8 8 2 9 9
--R - 22680a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 28b d x + (- 35a b d + 315b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (45a b d - 405a b c d + 1620b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 60a b d + 540a b c d - 2160a b c d + 5040b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (84a b d - 756a b c d + 3024a b c d - 7056a b c d + 10584b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 126a b d + 1134a b c d - 4536a b c d + 10584a b c d
--R +
--R 9 4 6 10 5 5
--R - 15876a b c d + 15876b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 210a b d - 1890a b c d + 7560a b c d - 17640a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 26460a b c d - 26460a b c d + 17640b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 420a b d + 3780a b c d - 15120a b c d + 35280a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 52920a b c d + 52920a b c d - 35280a b c d + 15120b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 1260a b d - 11340a b c d + 45360a b c d - 105840a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 158760a b c d - 158760a b c d + 105840a b c d - 45360a b c d
--R +
--R 10 8 2
--R 11340b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 2268a b d - 20160a b c d + 79380a b c d - 181440a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R 264600a b c d - 254016a b c d + 158760a b c d - 60480a b c d
--R +
--R 9 8 2
--R 11340a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 252a d + 2520a b c d - 11340a b c d + 30240a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 52920a b c d + 63504a b c d - 52920a b c d + 30240a b c d
--R +
--R 2 8 8 2 9 9 10 10
--R - 11340a b c d + 2520a b c d - 252b c
--R /
--R 12 11
--R 252b x + 252a b
--R Type: Union(Expression(Integer),...)
--E 803
--S 804 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 9 10 8 9 7 2 2 8 6 3 3 7
--R 6479a d - 49239a b c d + 161676a b c d - 297864a b c d
--R +
--R 5 4 4 6 4 5 5 5 3 6 6 4 2 7 7 3
--R 333396a b c d - 227556a b c d + 88200a b c d - 15120a b c d
--R /
--R 11
--R 252b
--R Type: Expression(Integer)
--E 804
--S 805 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 805
)clear all
--S 806 of 2952
t0000:=(c+d*x)^10/(a+b*x)^3
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 3 3 2 2 2 3
--R b x + 3a b x + 3a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 806
--S 807 of 2952
r0000:=120*d^3*(b*c-a*d)^7*x/b^10-1/2*(b*c-a*d)^10/(b^11*(a+b*x)^2)-_
10*d*(b*c-a*d)^9/(b^11*(a+b*x))+105*d^4*(b*c-a*d)^6*(a+b*x)^2/b^11+_
84*d^5*(b*c-a*d)^5*(a+b*x)^3/b^11+105/2*d^6*(b*c-a*d)^4*(a+b*x)^4/b^11+_
24*d^7*(b*c-a*d)^3*(a+b*x)^5/b^11+15/2*d^8*(b*c-a*d)^2*(a+b*x)^6/b^11+_
10/7*d^9*(b*c-a*d)*(a+b*x)^7/b^11+1/8*d^10*(a+b*x)^8/b^11+_
45*d^2*(b*c-a*d)^8*log(a+b*x)/b^11
--R
--R
--R (2)
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 2520a b d - 20160a b c d + 70560a b c d - 141120a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 176400a b c d - 141120a b c d + 70560a b c d - 20160a b c d
--R +
--R 10 8 2
--R 2520b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 5040a b d - 40320a b c d + 141120a b c d - 282240a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R 352800a b c d - 282240a b c d + 141120a b c d - 40320a b c d
--R +
--R 9 8 2
--R 5040a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2520a d - 20160a b c d + 70560a b c d - 141120a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R 176400a b c d - 141120a b c d + 70560a b c d - 20160a b c d
--R +
--R 2 8 8 2
--R 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 7b d x + (- 10a b d + 80b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (15a b d - 120a b c d + 420b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 24a b d + 192a b c d - 672a b c d + 1344b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (42a b d - 336a b c d + 1176a b c d - 2352a b c d + 2940b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 84a b d + 672a b c d - 2352a b c d + 4704a b c d
--R +
--R 9 4 6 10 5 5
--R - 5880a b c d + 4704b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 210a b d - 1680a b c d + 5880a b c d - 11760a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 14700a b c d - 11760a b c d + 5880b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 840a b d + 6720a b c d - 23520a b c d + 47040a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 58800a b c d + 47040a b c d - 23520a b c d + 6720b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 129a b d + 5232a b c d - 31752a b c d + 87024a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 132300a b c d + 117600a b c d - 58800a b c d + 13440a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 4782a b d - 29856a b c d + 77616a b c d - 108192a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R 88200a b c d - 47040a b c d + 23520a b c d - 13440a b c d
--R +
--R 9 8 2 10 9
--R 5040a b c d - 560b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 3651a d - 25008a b c d + 74088a b c d - 124656a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R 132300a b c d - 94080a b c d + 47040a b c d - 16800a b c d
--R +
--R 2 8 8 2 9 9 10 10
--R 3780a b c d - 280a b c d - 28b c
--R /
--R 13 2 12 2 11
--R 56b x + 112a b x + 56a b
--R Type: Expression(Integer)
--E 807
--S 808 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 2520a b d - 20160a b c d + 70560a b c d - 141120a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 176400a b c d - 141120a b c d + 70560a b c d - 20160a b c d
--R +
--R 10 8 2
--R 2520b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 5040a b d - 40320a b c d + 141120a b c d - 282240a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R 352800a b c d - 282240a b c d + 141120a b c d - 40320a b c d
--R +
--R 9 8 2
--R 5040a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2520a d - 20160a b c d + 70560a b c d - 141120a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R 176400a b c d - 141120a b c d + 70560a b c d - 20160a b c d
--R +
--R 2 8 8 2
--R 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 7b d x + (- 10a b d + 80b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (15a b d - 120a b c d + 420b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 24a b d + 192a b c d - 672a b c d + 1344b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (42a b d - 336a b c d + 1176a b c d - 2352a b c d + 2940b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 84a b d + 672a b c d - 2352a b c d + 4704a b c d
--R +
--R 9 4 6 10 5 5
--R - 5880a b c d + 4704b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 210a b d - 1680a b c d + 5880a b c d - 11760a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 14700a b c d - 11760a b c d + 5880b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 840a b d + 6720a b c d - 23520a b c d + 47040a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 58800a b c d + 47040a b c d - 23520a b c d + 6720b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 3248a b d + 25480a b c d - 86940a b c d + 168000a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 199920a b c d + 148176a b c d - 64680a b c d + 13440a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 1456a b d + 10640a b c d - 32760a b c d + 53760a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 47040a b c d + 14112a b c d + 11760a b c d - 13440a b c d
--R +
--R 9 8 2 10 9
--R 5040a b c d - 560b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R 532a d - 4760a b c d + 18900a b c d - 43680a b c d + 64680a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2
--R - 63504a b c d + 41160a b c d - 16800a b c d + 3780a b c d
--R +
--R 9 9 10 10
--R - 280a b c d - 28b c
--R /
--R 13 2 12 2 11
--R 56b x + 112a b x + 56a b
--R Type: Union(Expression(Integer),...)
--E 808
--S 809 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 8 10 7 9 6 2 2 8 5 3 3 7
--R - 3119a d + 20248a b c d - 55188a b c d + 80976a b c d
--R +
--R 4 4 4 6 3 5 5 5 2 6 6 4
--R - 67620a b c d + 30576a b c d - 5880a b c d
--R /
--R 11
--R 56b
--R Type: Expression(Integer)
--E 809
--S 810 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 810
)clear all
--S 811 of 2952
t0000:=(c+d*x)^10/(a+b*x)^4
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 811
--S 812 of 2952
r0000:=210*d^4*(b*c-a*d)^6*x/b^10-1/3*(b*c-a*d)^10/(b^11*(a+b*x)^3)-_
5*d*(b*c-a*d)^9/(b^11*(a+b*x)^2)-45*d^2*(b*c-a*d)^8/(b^11*(a+b*x))+_
126*d^5*(b*c-a*d)^5*(a+b*x)^2/b^11+70*d^6*(b*c-a*d)^4*(a+b*x)^3/b^11+_
30*d^7*(b*c-a*d)^3*(a+b*x)^4/b^11+9*d^8*(b*c-a*d)^2*(a+b*x)^5/b^11+_
5/3*d^9*(b*c-a*d)*(a+b*x)^6/b^11+1/7*d^10*(a+b*x)^7/b^11+_
120*d^3*(b*c-a*d)^7*log(a+b*x)/b^11
--R
--R
--R (2)
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 2520a b d + 17640a b c d - 52920a b c d + 88200a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 88200a b c d + 52920a b c d - 17640a b c d + 2520b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 7560a b d + 52920a b c d - 158760a b c d + 264600a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 264600a b c d + 158760a b c d - 52920a b c d + 7560a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 7560a b d + 52920a b c d - 158760a b c d + 264600a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 264600a b c d + 158760a b c d - 52920a b c d + 7560a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2520a d + 17640a b c d - 52920a b c d + 88200a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 88200a b c d + 52920a b c d - 17640a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 3b d x + (- 5a b d + 35b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (9a b d - 63a b c d + 189b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 18a b d + 126a b c d - 378a b c d + 630b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (42a b d - 294a b c d + 882a b c d - 1470a b c d + 1470b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 126a b d + 882a b c d - 2646a b c d + 4410a b c d
--R +
--R 9 4 6 10 5 5
--R - 4410a b c d + 2646b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 630a b d - 4410a b c d + 13230a b c d - 22050a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 22050a b c d - 13230a b c d + 4410b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 2124a b d - 16758a b c d + 54684a b c d - 95550a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4
--R 95550a b c d - 52920a b c d + 13230a b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 1188a b d + 2646a b c d + 5292a b c d - 22050a b c d
--R +
--R 4 6 4 6 2 8 6 4 9 7 3 10 8 2
--R 22050a b c d - 13230a b c d + 7560a b c d - 945b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 4968a b d + 29106a b c d - 74088a b c d + 110250a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 110250a b c d + 79380a b c d - 39690a b c d + 11340a b c d
--R +
--R 9 8 2 10 9
--R - 945a b c d - 105b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2496a d + 15582a b c d - 42336a b c d + 66150a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 66150a b c d + 44100a b c d - 19110a b c d + 4620a b c d
--R +
--R 2 8 8 2 9 9 10 10
--R - 315a b c d - 35a b c d - 7b c
--R /
--R 14 3 13 2 2 12 3 11
--R 21b x + 63a b x + 63a b x + 21a b
--R Type: Expression(Integer)
--E 812
--S 813 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 2520a b d + 17640a b c d - 52920a b c d + 88200a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 88200a b c d + 52920a b c d - 17640a b c d + 2520b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 7560a b d + 52920a b c d - 158760a b c d + 264600a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 264600a b c d + 158760a b c d - 52920a b c d + 7560a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 7560a b d + 52920a b c d - 158760a b c d + 264600a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 264600a b c d + 158760a b c d - 52920a b c d + 7560a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2520a d + 17640a b c d - 52920a b c d + 88200a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 88200a b c d + 52920a b c d - 17640a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 3b d x + (- 5a b d + 35b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (9a b d - 63a b c d + 189b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 18a b d + 126a b c d - 378a b c d + 630b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (42a b d - 294a b c d + 882a b c d - 1470a b c d + 1470b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 126a b d + 882a b c d - 2646a b c d + 4410a b c d
--R +
--R 9 4 6 10 5 5
--R - 4410a b c d + 2646b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 630a b d - 4410a b c d + 13230a b c d - 22050a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 22050a b c d - 13230a b c d + 4410b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 3773a b d - 25655a b c d + 74025a b c d - 116760a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4
--R 107310a b c d - 55566a b c d + 13230a b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 3759a b d - 24045a b c d + 63315a b c d - 85680a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 57330a b c d - 7938a b c d - 13230a b c d + 7560a b c d
--R +
--R 10 8 2
--R - 945b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 21a b d + 2415a b c d - 16065a b c d + 46620a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 74970a b c d + 71442a b c d - 39690a b c d + 11340a b c d
--R +
--R 9 8 2 10 9
--R - 945a b c d - 105b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 847a d + 6685a b c d - 22995a b c d + 44940a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3
--R - 54390a b c d + 41454a b c d - 19110a b c d + 4620a b c d
--R +
--R 2 8 8 2 9 9 10 10
--R - 315a b c d - 35a b c d - 7b c
--R /
--R 14 3 13 2 2 12 3 11
--R 21b x + 63a b x + 63a b x + 21a b
--R Type: Union(Expression(Integer),...)
--E 813
--S 814 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 7 10 6 9 5 2 2 8 4 3 3 7 3 4 4 6
--R 1649a d - 8897a b c d + 19341a b c d - 21210a b c d + 11760a b c d
--R +
--R 2 5 5 5
--R - 2646a b c d
--R /
--R 11
--R 21b
--R Type: Expression(Integer)
--E 814
--S 815 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 815
)clear all
--S 816 of 2952
t0000:=(c+d*x)^10/(a+b*x)^5
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R b x + 5a b x + 10a b x + 10a b x + 5a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 816
--S 817 of 2952
r0000:=252*d^5*(b*c-a*d)^5*x/b^10-1/4*(b*c-a*d)^10/(b^11*(a+b*x)^4)-_
10/3*d*(b*c-a*d)^9/(b^11*(a+b*x)^3)-_
45/2*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^2)-_
120*d^3*(b*c-a*d)^7/(b^11*(a+b*x))+_
105*d^6*(b*c-a*d)^4*(a+b*x)^2/b^11+_
40*d^7*(b*c-a*d)^3*(a+b*x)^3/b^11+_
45/4*d^8*(b*c-a*d)^2*(a+b*x)^4/b^11+_
2*d^9*(b*c-a*d)*(a+b*x)^5/b^11+_
1/6*d^10*(a+b*x)^6/b^11+210*d^4*(b*c-a*d)^6*log(a+b*x)/b^11
--R
--R
--R (2)
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 2520a b d - 15120a b c d + 37800a b c d - 50400a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 37800a b c d - 15120a b c d + 2520b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 10080a b d - 60480a b c d + 151200a b c d - 201600a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4
--R 151200a b c d - 60480a b c d + 10080a b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 15120a b d - 90720a b c d + 226800a b c d - 302400a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4
--R 226800a b c d - 90720a b c d + 15120a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 10080a b d - 60480a b c d + 151200a b c d - 201600a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4
--R 151200a b c d - 60480a b c d + 10080a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2520a d - 15120a b c d + 37800a b c d - 50400a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4
--R 37800a b c d - 15120a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 2b d x + (- 4a b d + 24b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (9a b d - 54a b c d + 135b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 24a b d + 144a b c d - 360a b c d + 480b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (84a b d - 504a b c d + 1260a b c d - 1680a b c d + 1260b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 504a b d + 3024a b c d - 7560a b c d + 10080a b c d
--R +
--R 9 4 6 10 5 5
--R - 7560a b c d + 3024b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 3150a b d + 19404a b c d - 48510a b c d + 62160a b c d
--R +
--R 2 8 4 6 9 5 5
--R - 41580a b c d + 12096a b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 2520a b d + 17136a b c d - 42840a b c d + 47040a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 15120a b c d - 12096a b c d + 10080a b c d - 1440b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 3780a b d - 19656a b c d + 49140a b c d - 80640a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 90720a b c d - 63504a b c d + 22680a b c d - 2160a b c d
--R +
--R 10 8 2
--R - 270b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 5880a b d - 33264a b c d + 83160a b c d - 120960a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R 110880a b c d - 62496a b c d + 18480a b c d - 1440a b c d
--R +
--R 9 8 2 10 9
--R - 180a b c d - 40b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2100a d - 12096a b c d + 30240a b c d - 42840a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2
--R 37170a b c d - 19404a b c d + 5250a b c d - 360a b c d - 45a b c d
--R +
--R 9 9 10 10
--R - 10a b c d - 3b c
--R /
--R 15 4 14 3 2 13 2 3 12 4 11
--R 12b x + 48a b x + 72a b x + 48a b x + 12a b
--R Type: Expression(Integer)
--E 817
--S 818 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 2520a b d - 15120a b c d + 37800a b c d - 50400a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 37800a b c d - 15120a b c d + 2520b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 10080a b d - 60480a b c d + 151200a b c d - 201600a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4
--R 151200a b c d - 60480a b c d + 10080a b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 15120a b d - 90720a b c d + 226800a b c d - 302400a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4
--R 226800a b c d - 90720a b c d + 15120a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 10080a b d - 60480a b c d + 151200a b c d - 201600a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4
--R 151200a b c d - 60480a b c d + 10080a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2520a d - 15120a b c d + 37800a b c d - 50400a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4
--R 37800a b c d - 15120a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 2b d x + (- 4a b d + 24b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (9a b d - 54a b c d + 135b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 24a b d + 144a b c d - 360a b c d + 480b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (84a b d - 504a b c d + 1260a b c d - 1680a b c d + 1260b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 504a b d + 3024a b c d - 7560a b c d + 10080a b c d
--R +
--R 9 4 6 10 5 5
--R - 7560a b c d + 3024b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 4043a b d + 23250a b c d - 54765a b c d + 66720a b c d
--R +
--R 2 8 4 6 9 5 5
--R - 42840a b c d + 12096a b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 6092a b d + 32520a b c d - 67860a b c d + 65280a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 20160a b c d - 12096a b c d + 10080a b c d - 1440b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 1578a b d + 3420a b c d + 11610a b c d - 53280a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R 83160a b c d - 63504a b c d + 22680a b c d - 2160a b c d
--R +
--R 10 8 2
--R - 270b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 2308a b d - 17880a b c d + 58140a b c d - 102720a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R 105840a b c d - 62496a b c d + 18480a b c d - 1440a b c d
--R +
--R 9 8 2 10 9
--R - 180a b c d - 40b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R 1207a d - 8250a b c d + 23985a b c d - 38280a b c d + 35910a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9
--R - 19404a b c d + 5250a b c d - 360a b c d - 45a b c d - 10a b c d
--R +
--R 10 10
--R - 3b c
--R /
--R 15 4 14 3 2 13 2 3 12 4 11
--R 12b x + 48a b x + 72a b x + 48a b x + 12a b
--R Type: Union(Expression(Integer),...)
--E 818
--S 819 of 2952
m0000:=a0000 - r0000
--R
--R
--R 6 10 5 9 4 2 2 8 3 3 3 7 2 4 4 6
--R - 893a d + 3846a b c d - 6255a b c d + 4560a b c d - 1260a b c d
--R (4) ----------------------------------------------------------------------
--R 11
--R 12b
--R Type: Expression(Integer)
--E 819
--S 820 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 820
)clear all
--S 821 of 2952
t0000:=(c+d*x)^10/(a+b*x)^6
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 821
--S 822 of 2952
r0000:=210*d^6*(b*c-a*d)^4*x/b^10-1/5*(b*c-a*d)^10/(b^11*(a+b*x)^5)-_
5/2*d*(b*c-a*d)^9/(b^11*(a+b*x)^4)-_
15*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^3)-_
60*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^2)-_
210*d^4*(b*c-a*d)^6/(b^11*(a+b*x))+_
60*d^7*(b*c-a*d)^3*(a+b*x)^2/b^11+_
15*d^8*(b*c-a*d)^2*(a+b*x)^3/b^11+_
5/2*d^9*(b*c-a*d)*(a+b*x)^4/b^11+1/5*d^10*(a+b*x)^5/b^11+_
252*d^5*(b*c-a*d)^5*log(a+b*x)/b^11
--R
--R
--R (2)
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 2520a b d + 12600a b c d - 25200a b c d + 25200a b c d
--R +
--R 9 4 6 10 5 5
--R - 12600a b c d + 2520b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 12600a b d + 63000a b c d - 126000a b c d + 126000a b c d
--R +
--R 2 8 4 6 9 5 5
--R - 63000a b c d + 12600a b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 25200a b d + 126000a b c d - 252000a b c d + 252000a b c d
--R +
--R 3 7 4 6 2 8 5 5
--R - 126000a b c d + 25200a b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 25200a b d + 126000a b c d - 252000a b c d + 252000a b c d
--R +
--R 4 6 4 6 3 7 5 5
--R - 126000a b c d + 25200a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 12600a b d + 63000a b c d - 126000a b c d + 126000a b c d
--R +
--R 5 5 4 6 4 6 5 5
--R - 63000a b c d + 12600a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2520a d + 12600a b c d - 25200a b c d + 25200a b c d
--R +
--R 6 4 4 6 5 5 5 5
--R - 12600a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 2b d x + (- 5a b d + 25b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (15a b d - 75a b c d + 150b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 60a b d + 300a b c d - 600a b c d + 600b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (420a b d - 2100a b c d + 4200a b c d - 4200a b c d + 2100b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R 3654a b d - 17850a b c d + 33600a b c d - 29400a b c d
--R +
--R 9 4 6
--R 10500a b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 5670a b d - 26250a b c d + 42000a b c d - 21000a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 10500a b c d + 12600a b c d - 2100b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 1260a b d + 10500a b c d - 42000a b c d + 84000a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 84000a b c d + 37800a b c d - 4200a b c d - 600b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 9660a b d + 52500a b c d - 126000a b c d + 168000a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 126000a b c d + 46200a b c d - 4200a b c d - 600a b c d
--R +
--R 10 8 2
--R - 150b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 7980a b d + 42000a b c d - 94500a b c d + 115500a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 78750a b c d + 26250a b c d - 2100a b c d - 300a b c d
--R +
--R 9 8 2 10 9
--R - 75a b c d - 25b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2100a d + 10920a b c d - 23940a b c d + 28140a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2
--R - 18270a b c d + 5754a b c d - 420a b c d - 60a b c d - 15a b c d
--R +
--R 9 9 10 10
--R - 5a b c d - 2b c
--R /
--R 16 5 15 4 2 14 3 3 13 2 4 12 5 11
--R 10b x + 50a b x + 100a b x + 100a b x + 50a b x + 10a b
--R Type: Expression(Integer)
--E 822
--S 823 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 2520a b d + 12600a b c d - 25200a b c d + 25200a b c d
--R +
--R 9 4 6 10 5 5
--R - 12600a b c d + 2520b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 12600a b d + 63000a b c d - 126000a b c d + 126000a b c d
--R +
--R 2 8 4 6 9 5 5
--R - 63000a b c d + 12600a b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 25200a b d + 126000a b c d - 252000a b c d + 252000a b c d
--R +
--R 3 7 4 6 2 8 5 5
--R - 126000a b c d + 25200a b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 25200a b d + 126000a b c d - 252000a b c d + 252000a b c d
--R +
--R 4 6 4 6 3 7 5 5
--R - 126000a b c d + 25200a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 12600a b d + 63000a b c d - 126000a b c d + 126000a b c d
--R +
--R 5 5 4 6 4 6 5 5
--R - 63000a b c d + 12600a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2520a d + 12600a b c d - 25200a b c d + 25200a b c d
--R +
--R 6 4 4 6 5 5 5 5
--R - 12600a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 2b d x + (- 5a b d + 25b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (15a b d - 75a b c d + 150b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 60a b d + 300a b c d - 600a b c d + 600b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (420a b d - 2100a b c d + 4200a b c d - 4200a b c d + 2100b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R 4127a b d - 19375a b c d + 35250a b c d - 30000a b c d
--R +
--R 9 4 6
--R 10500a b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 8035a b d - 33875a b c d + 50250a b c d - 24000a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 10500a b c d + 12600a b c d - 2100b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 3470a b d - 4750a b c d - 25500a b c d + 78000a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 84000a b c d + 37800a b c d - 4200a b c d - 600b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 4930a b d + 37250a b c d - 109500a b c d + 162000a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 126000a b c d + 46200a b c d - 4200a b c d - 600a b c d
--R +
--R 10 8 2
--R - 150b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 5615a b d + 34375a b c d - 86250a b c d + 112500a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 78750a b c d + 26250a b c d - 2100a b c d - 300a b c d
--R +
--R 9 8 2 10 9
--R - 75a b c d - 25b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 1627a d + 9395a b c d - 22290a b c d + 27540a b c d
--R +
--R 6 4 4 6 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2
--R - 18270a b c d + 5754a b c d - 420a b c d - 60a b c d - 15a b c d
--R +
--R 9 9 10 10
--R - 5a b c d - 2b c
--R /
--R 16 5 15 4 2 14 3 3 13 2 4 12 5 11
--R 10b x + 50a b x + 100a b x + 100a b x + 50a b x + 10a b
--R Type: Union(Expression(Integer),...)
--E 823
--S 824 of 2952
m0000:=a0000 - r0000
--R
--R
--R 5 10 4 9 3 2 2 8 2 3 3 7
--R 473a d - 1525a b c d + 1650a b c d - 600a b c d
--R (4) ----------------------------------------------------
--R 11
--R 10b
--R Type: Expression(Integer)
--E 824
--S 825 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 825
)clear all
--S 826 of 2952
t0000:=(c+d*x)^10/(a+b*x)^7
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R b x + 7a b x + 21a b x + 35a b x + 35a b x + 21a b x + 7a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 826
--S 827 of 2952
r0000:=120*d^7*(b*c-a*d)^3*x/b^10-1/6*(b*c-a*d)^10/(b^11*(a+b*x)^6)-_
2*d*(b*c-a*d)^9/(b^11*(a+b*x)^5)-_
45/4*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^4)-_
40*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^3)-_
105*d^4*(b*c-a*d)^6/(b^11*(a+b*x)^2)-_
252*d^5*(b*c-a*d)^5/(b^11*(a+b*x))+_
45/2*d^8*(b*c-a*d)^2*(a+b*x)^2/b^11+_
10/3*d^9*(b*c-a*d)*(a+b*x)^3/b^11+1/4*d^10*(a+b*x)^4/b^11+_
210*d^6*(b*c-a*d)^4*log(a+b*x)/b^11
--R
--R
--R (2)
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 2520a b d - 10080a b c d + 15120a b c d - 10080a b c d
--R +
--R 10 4 6
--R 2520b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R 15120a b d - 60480a b c d + 90720a b c d - 60480a b c d
--R +
--R 9 4 6
--R 15120a b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 37800a b d - 151200a b c d + 226800a b c d - 151200a b c d
--R +
--R 2 8 4 6
--R 37800a b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 50400a b d - 201600a b c d + 302400a b c d - 201600a b c d
--R +
--R 3 7 4 6
--R 50400a b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 37800a b d - 151200a b c d + 226800a b c d - 151200a b c d
--R +
--R 4 6 4 6
--R 37800a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 15120a b d - 60480a b c d + 90720a b c d - 60480a b c d
--R +
--R 5 5 4 6
--R 15120a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2520a d - 10080a b c d + 15120a b c d - 10080a b c d
--R +
--R 6 4 4 6
--R 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 3b d x + (- 10a b d + 40b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (45a b d - 180a b c d + 270b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 360a b d + 1440a b c d - 2160a b c d + 1440b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 6
--R (- 3810a b d + 14160a b c d - 18360a b c d + 8640a b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 7740a b d + 24480a b c d - 19440a b c d - 8640a b c d
--R +
--R 9 4 6 10 5 5
--R 15120a b c d - 3024b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 450a b d - 14400a b c d + 64800a b c d - 97200a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 56700a b c d - 7560a b c d - 1260b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 16200a b d - 86400a b c d + 187200a b c d - 196800a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R 92400a b c d - 10080a b c d - 1680a b c d - 480b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 21600a b d - 102600a b c d + 197100a b c d - 185400a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R 78750a b c d - 7560a b c d - 1260a b c d - 360a b c d - 135b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 11664a b d - 53136a b c d + 96984a b c d - 86256a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R 34524a b c d - 3024a b c d - 504a b c d - 144a b c d - 54a b c d
--R +
--R 10 9
--R - 24b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R 2364a d - 10536a b c d + 18684a b c d - 16056a b c d + 6174a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 504a b c d - 84a b c d - 24a b c d - 9a b c d - 4a b c d - 2b c
--R /
--R 17 6 16 5 2 15 4 3 14 3 4 13 2 5 12
--R 12b x + 72a b x + 180a b x + 240a b x + 180a b x + 72a b x
--R +
--R 6 11
--R 12a b
--R Type: Expression(Integer)
--E 827
--S 828 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 2520a b d - 10080a b c d + 15120a b c d - 10080a b c d
--R +
--R 10 4 6
--R 2520b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R 15120a b d - 60480a b c d + 90720a b c d - 60480a b c d
--R +
--R 9 4 6
--R 15120a b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 37800a b d - 151200a b c d + 226800a b c d - 151200a b c d
--R +
--R 2 8 4 6
--R 37800a b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 50400a b d - 201600a b c d + 302400a b c d - 201600a b c d
--R +
--R 3 7 4 6
--R 50400a b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 37800a b d - 151200a b c d + 226800a b c d - 151200a b c d
--R +
--R 4 6 4 6
--R 37800a b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 15120a b d - 60480a b c d + 90720a b c d - 60480a b c d
--R +
--R 5 5 4 6
--R 15120a b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R 2520a d - 10080a b c d + 15120a b c d - 10080a b c d
--R +
--R 6 4 4 6
--R 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 3b d x + (- 10a b d + 40b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (45a b d - 180a b c d + 270b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 360a b d + 1440a b c d - 2160a b c d + 1440b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 6
--R (- 4043a b d + 14660a b c d - 18630a b c d + 8640a b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 9138a b d + 27480a b c d - 21060a b c d - 8640a b c d
--R +
--R 9 4 6 10 5 5
--R 15120a b c d - 3024b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 3945a b d - 6900a b c d + 60750a b c d - 97200a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R 56700a b c d - 7560a b c d - 1260b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 11540a b d - 76400a b c d + 181800a b c d - 196800a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R 92400a b c d - 10080a b c d - 1680a b c d - 480b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 18105a b d - 95100a b c d + 193050a b c d - 185400a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R 78750a b c d - 7560a b c d - 1260a b c d - 360a b c d - 135b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 10266a b d - 50136a b c d + 95364a b c d - 86256a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R 34524a b c d - 3024a b c d - 504a b c d - 144a b c d - 54a b c d
--R +
--R 10 9
--R - 24b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R 2131a d - 10036a b c d + 18414a b c d - 16056a b c d + 6174a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 504a b c d - 84a b c d - 24a b c d - 9a b c d - 4a b c d - 2b c
--R /
--R 17 6 16 5 2 15 4 3 14 3 4 13 2 5 12
--R 12b x + 72a b x + 180a b x + 240a b x + 180a b x + 72a b x
--R +
--R 6 11
--R 12a b
--R Type: Union(Expression(Integer),...)
--E 828
--S 829 of 2952
m0000:=a0000 - r0000
--R
--R
--R 4 10 3 9 2 2 2 8
--R - 233a d + 500a b c d - 270a b c d
--R (4) --------------------------------------
--R 11
--R 12b
--R Type: Expression(Integer)
--E 829
--S 830 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 830
)clear all
--S 831 of 2952
t0000:=(c+d*x)^10/(a+b*x)^8
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 8 8 7 7 2 6 6 3 5 5 4 4 4 5 3 3 6 2 2
--R b x + 8a b x + 28a b x + 56a b x + 70a b x + 56a b x + 28a b x
--R +
--R 7 8
--R 8a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 831
--S 832 of 2952
r0000:=45*d^8*(b*c-a*d)^2*x/b^10-1/7*(b*c-a*d)^10/(b^11*(a+b*x)^7)-_
5/3*d*(b*c-a*d)^9/(b^11*(a+b*x)^6)-9*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^5)-_
30*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^4)-_
70*d^4*(b*c-a*d)^6/(b^11*(a+b*x)^3)-_
126*d^5*(b*c-a*d)^5/(b^11*(a+b*x)^2)-_
210*d^6*(b*c-a*d)^4/(b^11*(a+b*x))+_
5*d^9*(b*c-a*d)*(a+b*x)^2/b^11+_
1/3*d^10*(a+b*x)^3/b^11+120*d^7*(b*c-a*d)^3*log(a+b*x)/b^11
--R
--R
--R (2)
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 2520a b d + 7560a b c d - 7560a b c d + 2520b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 6
--R (- 17640a b d + 52920a b c d - 52920a b c d + 17640a b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7 5
--R (- 52920a b d + 158760a b c d - 158760a b c d + 52920a b c d )x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7 4
--R (- 88200a b d + 264600a b c d - 264600a b c d + 88200a b c d )x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7 3
--R (- 88200a b d + 264600a b c d - 264600a b c d + 88200a b c d )x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7 2
--R (- 52920a b d + 158760a b c d - 158760a b c d + 52920a b c d )x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R (- 17640a b d + 52920a b c d - 52920a b c d + 17640a b c d )x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2520a d + 7560a b c d - 7560a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 7b d x + (- 35a b d + 105b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (315a b d - 945a b c d + 945b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 7
--R (3675a b d - 9450a b c d + 6615a b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 8085a b d - 13230a b c d - 6615a b c d + 17640a b c d
--R +
--R 10 4 6
--R - 4410b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 2205a b d + 39690a b c d - 99225a b c d + 79380a b c d
--R +
--R 9 4 6 10 5 5
--R - 13230a b c d - 2646b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 33075a b d + 154350a b c d - 253575a b c d + 161700a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 22050a b c d - 4410a b c d - 1470b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 55125a b d + 220500a b c d - 319725a b c d + 183750a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 22050a b c d - 4410a b c d - 1470a b c d - 630b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 43659a b d + 164052a b c d - 223587a b c d + 120834a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R - 13230a b c d - 2646a b c d - 882a b c d - 378a b c d - 189b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 17493a b d + 63504a b c d - 83349a b c d + 43218a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R - 4410a b c d - 882a b c d - 294a b c d - 126a b c d - 63a b c d
--R +
--R 10 9
--R - 35b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - 2859a d + 10152a b c d - 12987a b c d + 6534a b c d - 630a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 126a b c d - 42a b c d - 18a b c d - 9a b c d - 5a b c d - 3b c
--R /
--R 18 7 17 6 2 16 5 3 15 4 4 14 3 5 13 2
--R 21b x + 147a b x + 441a b x + 735a b x + 735a b x + 441a b x
--R +
--R 6 12 7 11
--R 147a b x + 21a b
--R Type: Expression(Integer)
--E 832
--S 833 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 2520a b d + 7560a b c d - 7560a b c d + 2520b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 6
--R (- 17640a b d + 52920a b c d - 52920a b c d + 17640a b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7 5
--R (- 52920a b d + 158760a b c d - 158760a b c d + 52920a b c d )x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7 4
--R (- 88200a b d + 264600a b c d - 264600a b c d + 88200a b c d )x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7 3
--R (- 88200a b d + 264600a b c d - 264600a b c d + 88200a b c d )x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7 2
--R (- 52920a b d + 158760a b c d - 158760a b c d + 52920a b c d )x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R (- 17640a b d + 52920a b c d - 52920a b c d + 17640a b c d )x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7
--R - 2520a d + 7560a b c d - 7560a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 7b d x + (- 35a b d + 105b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (315a b d - 945a b c d + 945b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 7
--R (3773a b d - 9555a b c d + 6615a b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 8771a b d - 13965a b c d - 6615a b c d + 17640a b c d
--R +
--R 10 4 6
--R - 4410b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 147a b d + 37485a b c d - 99225a b c d + 79380a b c d
--R +
--R 9 4 6 10 5 5
--R - 13230a b c d - 2646b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 29645a b d + 150675a b c d - 253575a b c d + 161700a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 22050a b c d - 4410a b c d - 1470b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 51695a b d + 216825a b c d - 319725a b c d + 183750a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 22050a b c d - 4410a b c d - 1470a b c d - 630b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 41601a b d + 161847a b c d - 223587a b c d + 120834a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R - 13230a b c d - 2646a b c d - 882a b c d - 378a b c d - 189b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 16807a b d + 62769a b c d - 83349a b c d + 43218a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R - 4410a b c d - 882a b c d - 294a b c d - 126a b c d - 63a b c d
--R +
--R 10 9
--R - 35b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - 2761a d + 10047a b c d - 12987a b c d + 6534a b c d - 630a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 126a b c d - 42a b c d - 18a b c d - 9a b c d - 5a b c d - 3b c
--R /
--R 18 7 17 6 2 16 5 3 15 4 4 14 3 5 13 2
--R 21b x + 147a b x + 441a b x + 735a b x + 735a b x + 441a b x
--R +
--R 6 12 7 11
--R 147a b x + 21a b
--R Type: Union(Expression(Integer),...)
--E 833
--S 834 of 2952
m0000:=a0000 - r0000
--R
--R
--R 3 10 2 9
--R 14a d - 15a b c d
--R (4) --------------------
--R 11
--R 3b
--R Type: Expression(Integer)
--E 834
--S 835 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 835
)clear all
--S 836 of 2952
t0000:=(c+d*x)^10/(a+b*x)^9
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 9 9 8 8 2 7 7 3 6 6 4 5 5 5 4 4 6 3 3
--R b x + 9a b x + 36a b x + 84a b x + 126a b x + 126a b x + 84a b x
--R +
--R 7 2 2 8 9
--R 36a b x + 9a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 836
--S 837 of 2952
r0000:=d^9*(10*b*c-9*a*d)*x/b^10+1/2*d^10*x^2/b^9-_
1/8*(b*c-a*d)^10/(b^11*(a+b*x)^8)-_
10/7*d*(b*c-a*d)^9/(b^11*(a+b*x)^7)-_
15/2*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^6)-_
24*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^5)-_
105/2*d^4*(b*c-a*d)^6/(b^11*(a+b*x)^4)-_
84*d^5*(b*c-a*d)^5/(b^11*(a+b*x)^3)-_
105*d^6*(b*c-a*d)^4/(b^11*(a+b*x)^2)-_
120*d^7*(b*c-a*d)^3/(b^11*(a+b*x))+45*d^8*(b*c-a*d)^2*log(a+b*x)/b^11
--R
--R
--R (2)
--R 2 8 10 9 9 10 2 8 8
--R (2520a b d - 5040a b c d + 2520b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 7
--R (20160a b d - 40320a b c d + 20160a b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 6
--R (70560a b d - 141120a b c d + 70560a b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 5
--R (141120a b d - 282240a b c d + 141120a b c d )x
--R +
--R 6 4 10 5 5 9 4 6 2 8 4
--R (176400a b d - 352800a b c d + 176400a b c d )x
--R +
--R 7 3 10 6 4 9 5 5 2 8 3
--R (141120a b d - 282240a b c d + 141120a b c d )x
--R +
--R 8 2 10 7 3 9 6 4 2 8 2
--R (70560a b d - 141120a b c d + 70560a b c d )x
--R +
--R 9 10 8 2 9 7 3 2 8 10 10
--R (20160a b d - 40320a b c d + 20160a b c d )x + 2520a d
--R +
--R 9 9 8 2 2 8
--R - 5040a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 28b d x + (- 280a b d + 560b c d )x
--R +
--R 2 8 10 9 9 8
--R (- 3248a b d + 4480a b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 5824a b d - 4480a b c d + 20160a b c d - 6720b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 14896a b d - 86240a b c d + 105840a b c d - 23520a b c d
--R +
--R 10 4 6
--R - 5880b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R 76832a b d - 266560a b c d + 258720a b c d - 47040a b c d
--R +
--R 9 4 6 10 5 5
--R - 11760a b c d - 4704b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 140140a b d - 421400a b c d + 367500a b c d - 58800a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 14700a b c d - 5880a b c d - 2940b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 140336a b d - 393568a b c d + 322224a b c d - 47040a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 11760a b c d - 4704a b c d - 2352a b c d - 1344b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 81928a b d - 220304a b c d + 172872a b c d - 23520a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R - 5880a b c d - 2352a b c d - 1176a b c d - 672a b c d - 420b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 26288a b d - 68704a b c d + 52272a b c d - 6720a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 1680a b c d - 672a b c d - 336a b c d - 192a b c d
--R +
--R 9 8 2 10 9
--R - 120a b c d - 80b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R 3601a d - 9218a b c d + 6849a b c d - 840a b c d - 210a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 84a b c d - 42a b c d - 24a b c d - 15a b c d - 10a b c d - 7b c
--R /
--R 19 8 18 7 2 17 6 3 16 5 4 15 4
--R 56b x + 448a b x + 1568a b x + 3136a b x + 3920a b x
--R +
--R 5 14 3 6 13 2 7 12 8 11
--R 3136a b x + 1568a b x + 448a b x + 56a b
--R Type: Expression(Integer)
--E 837
--S 838 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 2 8 10 9 9 10 2 8 8
--R (2520a b d - 5040a b c d + 2520b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 7
--R (20160a b d - 40320a b c d + 20160a b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 6
--R (70560a b d - 141120a b c d + 70560a b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 5
--R (141120a b d - 282240a b c d + 141120a b c d )x
--R +
--R 6 4 10 5 5 9 4 6 2 8 4
--R (176400a b d - 352800a b c d + 176400a b c d )x
--R +
--R 7 3 10 6 4 9 5 5 2 8 3
--R (141120a b d - 282240a b c d + 141120a b c d )x
--R +
--R 8 2 10 7 3 9 6 4 2 8 2
--R (70560a b d - 141120a b c d + 70560a b c d )x
--R +
--R 9 10 8 2 9 7 3 2 8 10 10
--R (20160a b d - 40320a b c d + 20160a b c d )x + 2520a d
--R +
--R 9 9 8 2 2 8
--R - 5040a b c d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 10 9 9
--R 28b d x + (- 280a b d + 560b c d )x
--R +
--R 2 8 10 9 9 8
--R (- 3248a b d + 4480a b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 5824a b d - 4480a b c d + 20160a b c d - 6720b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 14896a b d - 86240a b c d + 105840a b c d - 23520a b c d
--R +
--R 10 4 6
--R - 5880b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R 76832a b d - 266560a b c d + 258720a b c d - 47040a b c d
--R +
--R 9 4 6 10 5 5
--R - 11760a b c d - 4704b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 140140a b d - 421400a b c d + 367500a b c d - 58800a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 14700a b c d - 5880a b c d - 2940b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 140336a b d - 393568a b c d + 322224a b c d - 47040a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 11760a b c d - 4704a b c d - 2352a b c d - 1344b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 81928a b d - 220304a b c d + 172872a b c d - 23520a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R - 5880a b c d - 2352a b c d - 1176a b c d - 672a b c d - 420b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 26288a b d - 68704a b c d + 52272a b c d - 6720a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 1680a b c d - 672a b c d - 336a b c d - 192a b c d
--R +
--R 9 8 2 10 9
--R - 120a b c d - 80b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R 3601a d - 9218a b c d + 6849a b c d - 840a b c d - 210a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 84a b c d - 42a b c d - 24a b c d - 15a b c d - 10a b c d - 7b c
--R /
--R 19 8 18 7 2 17 6 3 16 5 4 15 4
--R 56b x + 448a b x + 1568a b x + 3136a b x + 3920a b x
--R +
--R 5 14 3 6 13 2 7 12 8 11
--R 3136a b x + 1568a b x + 448a b x + 56a b
--R Type: Union(Expression(Integer),...)
--E 838
--S 839 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 839
--S 840 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 840
)clear all
--S 841 of 2952
t0000:=(c+d*x)^10/(a+b*x)^10
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R b x + 10a b x + 45a b x + 120a b x + 210a b x + 252a b x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210a b x + 120a b x + 45a b x + 10a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 841
--S 842 of 2952
r0000:=d^10*x/b^10-1/9*(b*c-a*d)^10/(b^11*(a+b*x)^9)-_
5/4*d*(b*c-a*d)^9/(b^11*(a+b*x)^8)-_
45/7*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^7)-_
20*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^6)-_
42*d^4*(b*c-a*d)^6/(b^11*(a+b*x)^5)-_
63*d^5*(b*c-a*d)^5/(b^11*(a+b*x)^4)-_
70*d^6*(b*c-a*d)^4/(b^11*(a+b*x)^3)-_
60*d^7*(b*c-a*d)^3/(b^11*(a+b*x)^2)-_
45*d^8*(b*c-a*d)^2/(b^11*(a+b*x))+10*d^9*(b*c-a*d)*log(a+b*x)/b^11
--R
--R
--R (2)
--R 9 10 10 9 9 2 8 10 9 9 8
--R (- 2520a b d + 2520b c d )x + (- 22680a b d + 22680a b c d )x
--R +
--R 3 7 10 2 8 9 7
--R (- 90720a b d + 90720a b c d )x
--R +
--R 4 6 10 3 7 9 6
--R (- 211680a b d + 211680a b c d )x
--R +
--R 5 5 10 4 6 9 5
--R (- 317520a b d + 317520a b c d )x
--R +
--R 6 4 10 5 5 9 4
--R (- 317520a b d + 317520a b c d )x
--R +
--R 7 3 10 6 4 9 3
--R (- 211680a b d + 211680a b c d )x
--R +
--R 8 2 10 7 3 9 2
--R (- 90720a b d + 90720a b c d )x
--R +
--R 9 10 8 2 9 10 10 9 9
--R (- 22680a b d + 22680a b c d )x - 2520a d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 9
--R 252b d x + 2268a b d x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 2268a b d + 22680a b c d - 11340b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 54432a b d + 136080a b c d - 45360a b c d - 15120b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 197568a b d + 388080a b c d - 105840a b c d - 35280a b c d
--R +
--R 10 4 6
--R - 17640b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 375732a b d + 661500a b c d - 158760a b c d - 52920a b c d
--R +
--R 9 4 6 10 5 5
--R - 26460a b c d - 15876b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 439236a b d + 725004a b c d - 158760a b c d - 52920a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 26460a b c d - 15876a b c d - 10584b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 328104a b d + 518616a b c d - 105840a b c d - 35280a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 17640a b c d - 10584a b c d - 7056a b c d - 5040b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 153576a b d + 235224a b c d - 45360a b c d - 15120a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 7560a b c d - 4536a b c d - 3024a b c d - 2160a b c d
--R +
--R 10 8 2
--R - 1620b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 41229a b d + 61641a b c d - 11340a b c d - 3780a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 1890a b c d - 1134a b c d - 756a b c d - 540a b c d
--R +
--R 9 8 2 10 9
--R - 405a b c d - 315b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - 4861a d + 7129a b c d - 1260a b c d - 420a b c d - 210a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 126a b c d - 84a b c d - 60a b c d - 45a b c d - 35a b c d - 28b c
--R /
--R 20 9 19 8 2 18 7 3 17 6 4 16 5
--R 252b x + 2268a b x + 9072a b x + 21168a b x + 31752a b x
--R +
--R 5 15 4 6 14 3 7 13 2 8 12 9 11
--R 31752a b x + 21168a b x + 9072a b x + 2268a b x + 252a b
--R Type: Expression(Integer)
--E 842
--S 843 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 9 10 10 9 9 2 8 10 9 9 8
--R (- 2520a b d + 2520b c d )x + (- 22680a b d + 22680a b c d )x
--R +
--R 3 7 10 2 8 9 7
--R (- 90720a b d + 90720a b c d )x
--R +
--R 4 6 10 3 7 9 6
--R (- 211680a b d + 211680a b c d )x
--R +
--R 5 5 10 4 6 9 5
--R (- 317520a b d + 317520a b c d )x
--R +
--R 6 4 10 5 5 9 4
--R (- 317520a b d + 317520a b c d )x
--R +
--R 7 3 10 6 4 9 3
--R (- 211680a b d + 211680a b c d )x
--R +
--R 8 2 10 7 3 9 2
--R (- 90720a b d + 90720a b c d )x
--R +
--R 9 10 8 2 9 10 10 9 9
--R (- 22680a b d + 22680a b c d )x - 2520a d + 2520a b c d
--R *
--R log(b x + a)
--R +
--R 10 10 10 9 10 9
--R 252b d x + 2268a b d x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 2268a b d + 22680a b c d - 11340b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 54432a b d + 136080a b c d - 45360a b c d - 15120b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 197568a b d + 388080a b c d - 105840a b c d - 35280a b c d
--R +
--R 10 4 6
--R - 17640b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 375732a b d + 661500a b c d - 158760a b c d - 52920a b c d
--R +
--R 9 4 6 10 5 5
--R - 26460a b c d - 15876b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 439236a b d + 725004a b c d - 158760a b c d - 52920a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 26460a b c d - 15876a b c d - 10584b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 328104a b d + 518616a b c d - 105840a b c d - 35280a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 17640a b c d - 10584a b c d - 7056a b c d - 5040b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 153576a b d + 235224a b c d - 45360a b c d - 15120a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 7560a b c d - 4536a b c d - 3024a b c d - 2160a b c d
--R +
--R 10 8 2
--R - 1620b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 41229a b d + 61641a b c d - 11340a b c d - 3780a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 1890a b c d - 1134a b c d - 756a b c d - 540a b c d
--R +
--R 9 8 2 10 9
--R - 405a b c d - 315b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - 4861a d + 7129a b c d - 1260a b c d - 420a b c d - 210a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 126a b c d - 84a b c d - 60a b c d - 45a b c d - 35a b c d - 28b c
--R /
--R 20 9 19 8 2 18 7 3 17 6 4 16 5
--R 252b x + 2268a b x + 9072a b x + 21168a b x + 31752a b x
--R +
--R 5 15 4 6 14 3 7 13 2 8 12 9 11
--R 31752a b x + 21168a b x + 9072a b x + 2268a b x + 252a b
--R Type: Union(Expression(Integer),...)
--E 843
--S 844 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 844
--S 845 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 845
)clear all
--S 846 of 2952
t0000:=(c+d*x)^10/(a+b*x)^11
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 11 11 10 10 2 9 9 3 8 8 4 7 7 5 6 6
--R b x + 11a b x + 55a b x + 165a b x + 330a b x + 462a b x
--R +
--R 6 5 5 7 4 4 8 3 3 9 2 2 10 11
--R 462a b x + 330a b x + 165a b x + 55a b x + 11a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 846
--S 847 of 2952
r0000:=-1/10*(b*c-a*d)^10/(b^11*(a+b*x)^10)-_
10/9*d*(b*c-a*d)^9/(b^11*(a+b*x)^9)-_
45/8*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^8)-_
120/7*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^7)-_
35*d^4*(b*c-a*d)^6/(b^11*(a+b*x)^6)-_
252/5*d^5*(b*c-a*d)^5/(b^11*(a+b*x)^5)-_
105/2*d^6*(b*c-a*d)^4/(b^11*(a+b*x)^4)-_
40*d^7*(b*c-a*d)^3/(b^11*(a+b*x)^3)-_
45/2*d^8*(b*c-a*d)^2/(b^11*(a+b*x)^2)-_
10*d^9*(b*c-a*d)/(b^11*(a+b*x))+d^10*log(a+b*x)/b^11
--R
--R
--R (2)
--R 10 10 10 9 10 9 2 8 10 8 3 7 10 7
--R 2520b d x + 25200a b d x + 113400a b d x + 302400a b d x
--R +
--R 4 6 10 6 5 5 10 5 6 4 10 4 7 3 10 3
--R 529200a b d x + 635040a b d x + 529200a b d x + 302400a b d x
--R +
--R 8 2 10 2 9 10 10 10
--R 113400a b d x + 25200a b d x + 2520a d
--R *
--R log(b x + a)
--R +
--R 9 10 10 9 9
--R (25200a b d - 25200b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (170100a b d - 113400a b c d - 56700b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (554400a b d - 302400a b c d - 151200a b c d - 100800b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 1102500a b d - 529200a b c d - 264600a b c d - 176400a b c d
--R +
--R 10 4 6
--R - 132300b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R 1450008a b d - 635040a b c d - 317520a b c d - 211680a b c d
--R +
--R 9 4 6 10 5 5
--R - 158760a b c d - 127008b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 1296540a b d - 529200a b c d - 264600a b c d - 176400a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 132300a b c d - 105840a b c d - 88200b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 784080a b d - 302400a b c d - 151200a b c d - 100800a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 75600a b c d - 60480a b c d - 50400a b c d - 43200b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 308205a b d - 113400a b c d - 56700a b c d - 37800a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 28350a b c d - 22680a b c d - 18900a b c d - 16200a b c d
--R +
--R 10 8 2
--R - 14175b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 71290a b d - 25200a b c d - 12600a b c d - 8400a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 6300a b c d - 5040a b c d - 4200a b c d - 3600a b c d
--R +
--R 9 8 2 10 9
--R - 3150a b c d - 2800b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R 7381a d - 2520a b c d - 1260a b c d - 840a b c d - 630a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9
--R - 504a b c d - 420a b c d - 360a b c d - 315a b c d - 280a b c d
--R +
--R 10 10
--R - 252b c
--R /
--R 21 10 20 9 2 19 8 3 18 7 4 17 6
--R 2520b x + 25200a b x + 113400a b x + 302400a b x + 529200a b x
--R +
--R 5 16 5 6 15 4 7 14 3 8 13 2
--R 635040a b x + 529200a b x + 302400a b x + 113400a b x
--R +
--R 9 12 10 11
--R 25200a b x + 2520a b
--R Type: Expression(Integer)
--E 847
--S 848 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 9 2 8 10 8 3 7 10 7
--R 2520b d x + 25200a b d x + 113400a b d x + 302400a b d x
--R +
--R 4 6 10 6 5 5 10 5 6 4 10 4 7 3 10 3
--R 529200a b d x + 635040a b d x + 529200a b d x + 302400a b d x
--R +
--R 8 2 10 2 9 10 10 10
--R 113400a b d x + 25200a b d x + 2520a d
--R *
--R log(b x + a)
--R +
--R 9 10 10 9 9
--R (25200a b d - 25200b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (170100a b d - 113400a b c d - 56700b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (554400a b d - 302400a b c d - 151200a b c d - 100800b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R 1102500a b d - 529200a b c d - 264600a b c d - 176400a b c d
--R +
--R 10 4 6
--R - 132300b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R 1450008a b d - 635040a b c d - 317520a b c d - 211680a b c d
--R +
--R 9 4 6 10 5 5
--R - 158760a b c d - 127008b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R 1296540a b d - 529200a b c d - 264600a b c d - 176400a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 132300a b c d - 105840a b c d - 88200b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R 784080a b d - 302400a b c d - 151200a b c d - 100800a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 75600a b c d - 60480a b c d - 50400a b c d - 43200b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R 308205a b d - 113400a b c d - 56700a b c d - 37800a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 28350a b c d - 22680a b c d - 18900a b c d - 16200a b c d
--R +
--R 10 8 2
--R - 14175b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R 71290a b d - 25200a b c d - 12600a b c d - 8400a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 6300a b c d - 5040a b c d - 4200a b c d - 3600a b c d
--R +
--R 9 8 2 10 9
--R - 3150a b c d - 2800b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R 7381a d - 2520a b c d - 1260a b c d - 840a b c d - 630a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9
--R - 504a b c d - 420a b c d - 360a b c d - 315a b c d - 280a b c d
--R +
--R 10 10
--R - 252b c
--R /
--R 21 10 20 9 2 19 8 3 18 7 4 17 6
--R 2520b x + 25200a b x + 113400a b x + 302400a b x + 529200a b x
--R +
--R 5 16 5 6 15 4 7 14 3 8 13 2
--R 635040a b x + 529200a b x + 302400a b x + 113400a b x
--R +
--R 9 12 10 11
--R 25200a b x + 2520a b
--R Type: Union(Expression(Integer),...)
--E 848
--S 849 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 849
--S 850 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 850
)clear all
--S 851 of 2952
t0000:=(c+d*x)^10/(a+b*x)^12
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 12 12 11 11 2 10 10 3 9 9 4 8 8 5 7 7
--R b x + 12a b x + 66a b x + 220a b x + 495a b x + 792a b x
--R +
--R 6 6 6 7 5 5 8 4 4 9 3 3 10 2 2 11 12
--R 924a b x + 792a b x + 495a b x + 220a b x + 66a b x + 12a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 851
--S 852 of 2952
r0000:=-1/11*(c+d*x)^11/((b*c-a*d)*(a+b*x)^11)
--R
--R
--R (2)
--R 1 11 11 10 10 2 9 9 3 8 8 4 7 7 5 6 6
--R -- d x + c d x + 5c d x + 15c d x + 30c d x + 42c d x
--R 11
--R +
--R 6 5 5 7 4 4 8 3 3 9 2 2 10 1 11
--R 42c d x + 30c d x + 15c d x + 5c d x + c d x + -- c
--R 11
--R /
--R 11 12 11 2 10 11 10 3 9 2 10 9
--R (a b d - b c)x + (11a b d - 11a b c)x + (55a b d - 55a b c)x
--R +
--R 4 8 3 9 8 5 7 4 8 7
--R (165a b d - 165a b c)x + (330a b d - 330a b c)x
--R +
--R 6 6 5 7 6 7 5 6 6 5
--R (462a b d - 462a b c)x + (462a b d - 462a b c)x
--R +
--R 8 4 7 5 4 9 3 8 4 3
--R (330a b d - 330a b c)x + (165a b d - 165a b c)x
--R +
--R 10 2 9 3 2 11 10 2 12 11
--R (55a b d - 55a b c)x + (11a b d - 11a b c)x + a d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 852
--S 853 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 11b d x + (- 55a b d - 55b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 165a b d - 165a b c d - 165b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 330a b d - 330a b c d - 330a b c d - 330b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7 10 4 6 6
--R (- 462a b d - 462a b c d - 462a b c d - 462a b c d - 462b c d )x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7 9 4 6
--R - 462a b d - 462a b c d - 462a b c d - 462a b c d - 462a b c d
--R +
--R 10 5 5
--R - 462b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7 2 8 4 6
--R - 330a b d - 330a b c d - 330a b c d - 330a b c d - 330a b c d
--R +
--R 9 5 5 10 6 4
--R - 330a b c d - 330b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7 3 7 4 6
--R - 165a b d - 165a b c d - 165a b c d - 165a b c d - 165a b c d
--R +
--R 2 8 5 5 9 6 4 10 7 3
--R - 165a b c d - 165a b c d - 165b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7 4 6 4 6
--R - 55a b d - 55a b c d - 55a b c d - 55a b c d - 55a b c d
--R +
--R 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R - 55a b c d - 55a b c d - 55a b c d - 55b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 11a b d - 11a b c d - 11a b c d - 11a b c d - 11a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2 10 9
--R - 11a b c d - 11a b c d - 11a b c d - 11a b c d - 11b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6 5 5 5 5
--R - a d - a b c d - a b c d - a b c d - a b c d - a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - a b c d - a b c d - a b c d - a b c d - b c
--R /
--R 22 11 21 10 2 20 9 3 19 8 4 18 7
--R 11b x + 121a b x + 605a b x + 1815a b x + 3630a b x
--R +
--R 5 17 6 6 16 5 7 15 4 8 14 3 9 13 2
--R 5082a b x + 5082a b x + 3630a b x + 1815a b x + 605a b x
--R +
--R 10 12 11 11
--R 121a b x + 11a b
--R Type: Union(Expression(Integer),...)
--E 853
--S 854 of 2952
m0000:=a0000 - r0000
--R
--R
--R 11
--R d
--R (4) - -----------------
--R 11 12
--R 11a b d - 11b c
--R Type: Expression(Integer)
--E 854
--S 855 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 855
)clear all
--S 856 of 2952
t0000:=(c+d*x)^10/(a+b*x)^13
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 13 13 12 12 2 11 11 3 10 10 4 9 9 5 8 8
--R b x + 13a b x + 78a b x + 286a b x + 715a b x + 1287a b x
--R +
--R 6 7 7 7 6 6 8 5 5 9 4 4 10 3 3 11 2 2
--R 1716a b x + 1716a b x + 1287a b x + 715a b x + 286a b x + 78a b x
--R +
--R 12 13
--R 13a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 856
--S 857 of 2952
r0000:=-1/12*(c+d*x)^11/((b*c-a*d)*(a+b*x)^12)+_
1/132*d*(c+d*x)^11/((b*c-a*d)^2*(a+b*x)^11)
--R
--R
--R (2)
--R 1 12 12 1 12 11 11 1 2 10 10
--R --- b d x + -- a d x + (a c d - - b c d )x
--R 132 11 2
--R +
--R 2 10 10 3 9 9 3 9 45 4 8 8
--R (5a c d - -- b c d )x + (15a c d - -- b c d )x
--R 3 4
--R +
--R 4 8 5 7 7 5 7 6 6 6
--R (30a c d - 24b c d )x + (42a c d - 35b c d )x
--R +
--R 6 6 7 5 5 7 5 105 8 4 4
--R (42a c d - 36b c d )x + (30a c d - --- b c d )x
--R 4
--R +
--R 8 4 40 9 3 3 9 3 9 10 2 2
--R (15a c d - -- b c d )x + (5a c d - - b c d )x
--R 3 2
--R +
--R 10 2 10 11 1 11 1 12
--R (a c d - -- b c d)x + -- a c d - -- b c
--R 11 11 12
--R /
--R 2 12 2 13 14 2 12
--R (a b d - 2a b c d + b c )x
--R +
--R 3 11 2 2 12 13 2 11
--R (12a b d - 24a b c d + 12a b c )x
--R +
--R 4 10 2 3 11 2 12 2 10
--R (66a b d - 132a b c d + 66a b c )x
--R +
--R 5 9 2 4 10 3 11 2 9
--R (220a b d - 440a b c d + 220a b c )x
--R +
--R 6 8 2 5 9 4 10 2 8
--R (495a b d - 990a b c d + 495a b c )x
--R +
--R 7 7 2 6 8 5 9 2 7
--R (792a b d - 1584a b c d + 792a b c )x
--R +
--R 8 6 2 7 7 6 8 2 6
--R (924a b d - 1848a b c d + 924a b c )x
--R +
--R 9 5 2 8 6 7 7 2 5
--R (792a b d - 1584a b c d + 792a b c )x
--R +
--R 10 4 2 9 5 8 6 2 4
--R (495a b d - 990a b c d + 495a b c )x
--R +
--R 11 3 2 10 4 9 5 2 3
--R (220a b d - 440a b c d + 220a b c )x
--R +
--R 12 2 2 11 3 10 4 2 2
--R (66a b d - 132a b c d + 66a b c )x
--R +
--R 13 2 12 2 11 3 2 14 2 13 12 2 2
--R (12a b d - 24a b c d + 12a b c )x + a d - 2a b c d + a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 857
--S 858 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 66b d x + (- 220a b d - 440b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 495a b d - 990a b c d - 1485b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 792a b d - 1584a b c d - 2376a b c d - 3168b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 924a b d - 1848a b c d - 2772a b c d - 3696a b c d
--R +
--R 10 4 6
--R - 4620b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 792a b d - 1584a b c d - 2376a b c d - 3168a b c d
--R +
--R 9 4 6 10 5 5
--R - 3960a b c d - 4752b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 495a b d - 990a b c d - 1485a b c d - 1980a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 2475a b c d - 2970a b c d - 3465b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7 3 7 4 6
--R - 220a b d - 440a b c d - 660a b c d - 880a b c d - 1100a b c d
--R +
--R 2 8 5 5 9 6 4 10 7 3
--R - 1320a b c d - 1540a b c d - 1760b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7 4 6 4 6
--R - 66a b d - 132a b c d - 198a b c d - 264a b c d - 330a b c d
--R +
--R 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R - 396a b c d - 462a b c d - 528a b c d - 594b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 12a b d - 24a b c d - 36a b c d - 48a b c d - 60a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2 10 9
--R - 72a b c d - 84a b c d - 96a b c d - 108a b c d - 120b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6 5 5 5 5
--R - a d - 2a b c d - 3a b c d - 4a b c d - 5a b c d - 6a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 7a b c d - 8a b c d - 9a b c d - 10a b c d - 11b c
--R /
--R 23 12 22 11 2 21 10 3 20 9 4 19 8
--R 132b x + 1584a b x + 8712a b x + 29040a b x + 65340a b x
--R +
--R 5 18 7 6 17 6 7 16 5 8 15 4
--R 104544a b x + 121968a b x + 104544a b x + 65340a b x
--R +
--R 9 14 3 10 13 2 11 12 12 11
--R 29040a b x + 8712a b x + 1584a b x + 132a b
--R Type: Union(Expression(Integer),...)
--E 858
--S 859 of 2952
m0000:=a0000 - r0000
--R
--R
--R 12
--R d
--R (4) - -----------------------------------
--R 2 11 2 12 13 2
--R 132a b d - 264a b c d + 132b c
--R Type: Expression(Integer)
--E 859
--S 860 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 860
)clear all
--S 861 of 2952
t0000:=(c+d*x)^10/(a+b*x)^14
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 14 14 13 13 2 12 12 3 11 11 4 10 10
--R b x + 14a b x + 91a b x + 364a b x + 1001a b x
--R +
--R 5 9 9 6 8 8 7 7 7 8 6 6 9 5 5
--R 2002a b x + 3003a b x + 3432a b x + 3003a b x + 2002a b x
--R +
--R 10 4 4 11 3 3 12 2 2 13 14
--R 1001a b x + 364a b x + 91a b x + 14a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 861
--S 862 of 2952
r0000:=-1/13*(c+d*x)^11/((b*c-a*d)*(a+b*x)^13)+_
1/78*d*(c+d*x)^11/((b*c-a*d)^2*(a+b*x)^12)-_
1/858*d^2*(c+d*x)^11/((b*c-a*d)^3*(a+b*x)^11)
--R
--R
--R (2)
--R 1 2 13 13 1 13 12 1 2 13 11
--R --- b d x + -- a b d x + -- a d x
--R 858 66 11
--R +
--R 2 12 2 11 1 2 3 10 10
--R (a c d - a b c d + - b c d )x
--R 3
--R +
--R 2 2 11 20 3 10 5 2 4 9 9
--R (5a c d - -- a b c d + - b c d )x
--R 3 2
--R +
--R 2 3 10 45 4 9 2 5 8 8
--R (15a c d - -- a b c d + 9b c d )x
--R 2
--R +
--R 2 4 9 5 8 2 6 7 7
--R (30a c d - 48a b c d + 20b c d )x
--R +
--R 2 5 8 6 7 2 7 6 6
--R (42a c d - 70a b c d + 30b c d )x
--R +
--R 2 6 7 7 6 63 2 8 5 5
--R (42a c d - 72a b c d + -- b c d )x
--R 2
--R +
--R 2 7 6 105 8 5 70 2 9 4 4
--R (30a c d - --- a b c d + -- b c d )x
--R 2 3
--R +
--R 2 8 5 80 9 4 2 10 3 3
--R (15a c d - -- a b c d + 12b c d )x
--R 3
--R +
--R 2 9 4 10 3 45 2 11 2 2
--R (5a c d - 9a b c d + -- b c d )x
--R 11
--R +
--R 2 10 3 20 11 2 5 2 12 1 2 11 2 1 12 1 2 13
--R (a c d - -- a b c d + - b c d)x + -- a c d - - a b c d + -- b c
--R 11 6 11 6 13
--R /
--R 3 13 3 2 14 2 15 2 16 3 13
--R (a b d - 3a b c d + 3a b c d - b c )x
--R +
--R 4 12 3 3 13 2 2 14 2 15 3 12
--R (13a b d - 39a b c d + 39a b c d - 13a b c )x
--R +
--R 5 11 3 4 12 2 3 13 2 2 14 3 11
--R (78a b d - 234a b c d + 234a b c d - 78a b c )x
--R +
--R 6 10 3 5 11 2 4 12 2 3 13 3 10
--R (286a b d - 858a b c d + 858a b c d - 286a b c )x
--R +
--R 7 9 3 6 10 2 5 11 2 4 12 3 9
--R (715a b d - 2145a b c d + 2145a b c d - 715a b c )x
--R +
--R 8 8 3 7 9 2 6 10 2 5 11 3 8
--R (1287a b d - 3861a b c d + 3861a b c d - 1287a b c )x
--R +
--R 9 7 3 8 8 2 7 9 2 6 10 3 7
--R (1716a b d - 5148a b c d + 5148a b c d - 1716a b c )x
--R +
--R 10 6 3 9 7 2 8 8 2 7 9 3 6
--R (1716a b d - 5148a b c d + 5148a b c d - 1716a b c )x
--R +
--R 11 5 3 10 6 2 9 7 2 8 8 3 5
--R (1287a b d - 3861a b c d + 3861a b c d - 1287a b c )x
--R +
--R 12 4 3 11 5 2 10 6 2 9 7 3 4
--R (715a b d - 2145a b c d + 2145a b c d - 715a b c )x
--R +
--R 13 3 3 12 4 2 11 5 2 10 6 3 3
--R (286a b d - 858a b c d + 858a b c d - 286a b c )x
--R +
--R 14 2 3 13 3 2 12 4 2 11 5 3 2
--R (78a b d - 234a b c d + 234a b c d - 78a b c )x
--R +
--R 15 3 14 2 2 13 3 2 12 4 3 16 3 15 2
--R (13a b d - 39a b c d + 39a b c d - 13a b c )x + a d - 3a b c d
--R +
--R 14 2 2 13 3 3
--R 3a b c d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 862
--S 863 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 286b d x + (- 715a b d - 2145b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 1287a b d - 3861a b c d - 7722b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 1716a b d - 5148a b c d - 10296a b c d - 17160b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 1716a b d - 5148a b c d - 10296a b c d - 17160a b c d
--R +
--R 10 4 6
--R - 25740b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 1287a b d - 3861a b c d - 7722a b c d - 12870a b c d
--R +
--R 9 4 6 10 5 5
--R - 19305a b c d - 27027b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 715a b d - 2145a b c d - 4290a b c d - 7150a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 10725a b c d - 15015a b c d - 20020b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 286a b d - 858a b c d - 1716a b c d - 2860a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 4290a b c d - 6006a b c d - 8008a b c d - 10296b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7 4 6 4 6
--R - 78a b d - 234a b c d - 468a b c d - 780a b c d - 1170a b c d
--R +
--R 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R - 1638a b c d - 2184a b c d - 2808a b c d - 3510b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 13a b d - 39a b c d - 78a b c d - 130a b c d - 195a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2 10 9
--R - 273a b c d - 364a b c d - 468a b c d - 585a b c d - 715b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6 5 5 5 5
--R - a d - 3a b c d - 6a b c d - 10a b c d - 15a b c d - 21a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 28a b c d - 36a b c d - 45a b c d - 55a b c d - 66b c
--R /
--R 24 13 23 12 2 22 11 3 21 10
--R 858b x + 11154a b x + 66924a b x + 245388a b x
--R +
--R 4 20 9 5 19 8 6 18 7 7 17 6
--R 613470a b x + 1104246a b x + 1472328a b x + 1472328a b x
--R +
--R 8 16 5 9 15 4 10 14 3 11 13 2
--R 1104246a b x + 613470a b x + 245388a b x + 66924a b x
--R +
--R 12 12 13 11
--R 11154a b x + 858a b
--R Type: Union(Expression(Integer),...)
--E 863
--S 864 of 2952
m0000:=a0000 - r0000
--R
--R
--R 13
--R d
--R (4) - ----------------------------------------------------
--R 3 11 3 2 12 2 13 2 14 3
--R 858a b d - 2574a b c d + 2574a b c d - 858b c
--R Type: Expression(Integer)
--E 864
--S 865 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 865
)clear all
--S 866 of 2952
t0000:=(c+d*x)^10/(a+b*x)^15
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 15 15 14 14 2 13 13 3 12 12 4 11 11
--R b x + 15a b x + 105a b x + 455a b x + 1365a b x
--R +
--R 5 10 10 6 9 9 7 8 8 8 7 7 9 6 6
--R 3003a b x + 5005a b x + 6435a b x + 6435a b x + 5005a b x
--R +
--R 10 5 5 11 4 4 12 3 3 13 2 2 14 15
--R 3003a b x + 1365a b x + 455a b x + 105a b x + 15a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 866
--S 867 of 2952
r0000:=-1/14*(c+d*x)^11/((b*c-a*d)*(a+b*x)^14)+_
3/182*d*(c+d*x)^11/((b*c-a*d)^2*(a+b*x)^13)-_
1/364*d^2*(c+d*x)^11/((b*c-a*d)^3*(a+b*x)^12)+_
1/4004*d^3*(c+d*x)^11/((b*c-a*d)^4*(a+b*x)^11)
--R
--R
--R (2)
--R 1 3 14 14 1 2 14 13 1 2 14 12 1 3 14 11
--R ---- b d x + --- a b d x + -- a b d x + -- a d x
--R 4004 286 44 11
--R +
--R 3 13 3 2 2 12 2 3 11 1 3 4 10 10
--R (a c d - - a b c d + a b c d - - b c d )x
--R 2 4
--R +
--R 3 2 12 2 3 11 15 2 4 10 3 5 9 9
--R (5a c d - 10a b c d + -- a b c d - 2b c d )x
--R 2
--R +
--R 3 3 11 135 2 4 10 2 5 9 15 3 6 8 8
--R (15a c d - --- a b c d + 27a b c d - -- b c d )x
--R 4 2
--R +
--R 3 4 10 2 5 9 2 6 8 120 3 7 7 7
--R (30a c d - 72a b c d + 60a b c d - --- b c d )x
--R 7
--R +
--R 3 5 9 2 6 8 2 7 7 105 3 8 6 6
--R (42a c d - 105a b c d + 90a b c d - --- b c d )x
--R 4
--R +
--R 3 6 8 2 7 7 189 2 8 6 3 9 5 5
--R (42a c d - 108a b c d + --- a b c d - 28b c d )x
--R 2
--R +
--R 3 7 7 315 2 8 6 2 9 5 3 10 4 4
--R (30a c d - --- a b c d + 70a b c d - 21b c d )x
--R 4
--R +
--R 3 8 6 2 9 5 2 10 4 120 3 11 3 3
--R (15a c d - 40a b c d + 36a b c d - --- b c d )x
--R 11
--R +
--R 3 9 5 27 2 10 4 135 2 11 3 15 3 12 2 2
--R (5a c d - -- a b c d + --- a b c d - -- b c d )x
--R 2 11 4
--R +
--R 3 10 4 30 2 11 3 5 2 12 2 10 3 13 1 3 11 3
--R (a c d - -- a b c d + - a b c d - -- b c d)x + -- a c d
--R 11 2 13 11
--R +
--R 1 2 12 2 3 2 13 1 3 14
--R - - a b c d + -- a b c d - -- b c
--R 4 13 14
--R /
--R 4 14 4 3 15 3 2 16 2 2 17 3 18 4 14
--R (a b d - 4a b c d + 6a b c d - 4a b c d + b c )x
--R +
--R 5 13 4 4 14 3 3 15 2 2 2 16 3 17 4 13
--R (14a b d - 56a b c d + 84a b c d - 56a b c d + 14a b c )x
--R +
--R 6 12 4 5 13 3 4 14 2 2 3 15 3 2 16 4 12
--R (91a b d - 364a b c d + 546a b c d - 364a b c d + 91a b c )x
--R +
--R 7 11 4 6 12 3 5 13 2 2 4 14 3
--R 364a b d - 1456a b c d + 2184a b c d - 1456a b c d
--R +
--R 3 15 4
--R 364a b c
--R *
--R 11
--R x
--R +
--R 8 10 4 7 11 3 6 12 2 2 5 13 3
--R 1001a b d - 4004a b c d + 6006a b c d - 4004a b c d
--R +
--R 4 14 4
--R 1001a b c
--R *
--R 10
--R x
--R +
--R 9 9 4 8 10 3 7 11 2 2 6 12 3
--R 2002a b d - 8008a b c d + 12012a b c d - 8008a b c d
--R +
--R 5 13 4
--R 2002a b c
--R *
--R 9
--R x
--R +
--R 10 8 4 9 9 3 8 10 2 2 7 11 3
--R 3003a b d - 12012a b c d + 18018a b c d - 12012a b c d
--R +
--R 6 12 4
--R 3003a b c
--R *
--R 8
--R x
--R +
--R 11 7 4 10 8 3 9 9 2 2 8 10 3
--R 3432a b d - 13728a b c d + 20592a b c d - 13728a b c d
--R +
--R 7 11 4
--R 3432a b c
--R *
--R 7
--R x
--R +
--R 12 6 4 11 7 3 10 8 2 2 9 9 3
--R 3003a b d - 12012a b c d + 18018a b c d - 12012a b c d
--R +
--R 8 10 4
--R 3003a b c
--R *
--R 6
--R x
--R +
--R 13 5 4 12 6 3 11 7 2 2 10 8 3
--R 2002a b d - 8008a b c d + 12012a b c d - 8008a b c d
--R +
--R 9 9 4
--R 2002a b c
--R *
--R 5
--R x
--R +
--R 14 4 4 13 5 3 12 6 2 2 11 7 3
--R 1001a b d - 4004a b c d + 6006a b c d - 4004a b c d
--R +
--R 10 8 4
--R 1001a b c
--R *
--R 4
--R x
--R +
--R 15 3 4 14 4 3 13 5 2 2 12 6 3
--R 364a b d - 1456a b c d + 2184a b c d - 1456a b c d
--R +
--R 11 7 4
--R 364a b c
--R *
--R 3
--R x
--R +
--R 16 2 4 15 3 3 14 4 2 2 13 5 3 12 6 4 2
--R (91a b d - 364a b c d + 546a b c d - 364a b c d + 91a b c )x
--R +
--R 17 4 16 2 3 15 3 2 2 14 4 3 13 5 4 18 4
--R (14a b d - 56a b c d + 84a b c d - 56a b c d + 14a b c )x + a d
--R +
--R 17 3 16 2 2 2 15 3 3 14 4 4
--R - 4a b c d + 6a b c d - 4a b c d + a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 867
--S 868 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 1001b d x + (- 2002a b d - 8008b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 3003a b d - 12012a b c d - 30030b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 3432a b d - 13728a b c d - 34320a b c d - 68640b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 3003a b d - 12012a b c d - 30030a b c d - 60060a b c d
--R +
--R 10 4 6
--R - 105105b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 2002a b d - 8008a b c d - 20020a b c d - 40040a b c d
--R +
--R 9 4 6 10 5 5
--R - 70070a b c d - 112112b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 1001a b d - 4004a b c d - 10010a b c d - 20020a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 35035a b c d - 56056a b c d - 84084b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 364a b d - 1456a b c d - 3640a b c d - 7280a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 12740a b c d - 20384a b c d - 30576a b c d - 43680b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7 4 6 4 6
--R - 91a b d - 364a b c d - 910a b c d - 1820a b c d - 3185a b c d
--R +
--R 3 7 5 5 2 8 6 4 9 7 3 10 8 2
--R - 5096a b c d - 7644a b c d - 10920a b c d - 15015b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 14a b d - 56a b c d - 140a b c d - 280a b c d - 490a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2 10 9
--R - 784a b c d - 1176a b c d - 1680a b c d - 2310a b c d - 3080b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6 5 5 5 5
--R - a d - 4a b c d - 10a b c d - 20a b c d - 35a b c d - 56a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 84a b c d - 120a b c d - 165a b c d - 220a b c d - 286b c
--R /
--R 25 14 24 13 2 23 12 3 22 11
--R 4004b x + 56056a b x + 364364a b x + 1457456a b x
--R +
--R 4 21 10 5 20 9 6 19 8 7 18 7
--R 4008004a b x + 8016008a b x + 12024012a b x + 13741728a b x
--R +
--R 8 17 6 9 16 5 10 15 4 11 14 3
--R 12024012a b x + 8016008a b x + 4008004a b x + 1457456a b x
--R +
--R 12 13 2 13 12 14 11
--R 364364a b x + 56056a b x + 4004a b
--R Type: Union(Expression(Integer),...)
--E 868
--S 869 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R 14
--R d
--R - -------------------------------------------------------------------------
--R 4 11 4 3 12 3 2 13 2 2 14 3 15 4
--R 4004a b d - 16016a b c d + 24024a b c d - 16016a b c d + 4004b c
--R Type: Expression(Integer)
--E 869
--S 870 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 870
)clear all
--S 871 of 2952
t0000:=(c+d*x)^10/(a+b*x)^16
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 16 16 15 15 2 14 14 3 13 13 4 12 12
--R b x + 16a b x + 120a b x + 560a b x + 1820a b x
--R +
--R 5 11 11 6 10 10 7 9 9 8 8 8 9 7 7
--R 4368a b x + 8008a b x + 11440a b x + 12870a b x + 11440a b x
--R +
--R 10 6 6 11 5 5 12 4 4 13 3 3 14 2 2
--R 8008a b x + 4368a b x + 1820a b x + 560a b x + 120a b x
--R +
--R 15 16
--R 16a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 871
--S 872 of 2952
r0000:=-1/15*(c+d*x)^11/((b*c-a*d)*(a+b*x)^15)+_
2/105*d*(c+d*x)^11/((b*c-a*d)^2*(a+b*x)^14)-_
2/455*d^2*(c+d*x)^11/((b*c-a*d)^3*(a+b*x)^13)+_
1/1365*d^3*(c+d*x)^11/((b*c-a*d)^4*(a+b*x)^12)-_
1/15015*d^4*(c+d*x)^11/((b*c-a*d)^5*(a+b*x)^11)
--R
--R
--R (2)
--R 1 4 15 15 1 3 15 14 1 2 2 15 13 1 3 15 12
--R ----- b d x + ---- a b d x + --- a b d x + -- a b d x
--R 15015 1001 143 33
--R +
--R 1 4 15 11
--R -- a d x
--R 11
--R +
--R 4 14 3 2 13 2 2 3 12 3 4 11 1 4 5 10 10
--R (a c d - 2a b c d + 2a b c d - a b c d + - b c d )x
--R 5
--R +
--R 4 2 13 40 3 3 12 2 2 4 11 3 5 10 5 4 6 9 9
--R (5a c d - -- a b c d + 15a b c d - 8a b c d + - b c d )x
--R 3 3
--R +
--R 4 3 12 3 4 11 2 2 5 10 3 6 9 45 4 7 8 8
--R (15a c d - 45a b c d + 54a b c d - 30a b c d + -- b c d )x
--R 7
--R +
--R 4 4 11 3 5 10 2 2 6 9 480 3 7 8 4 8 7 7
--R (30a c d - 96a b c d + 120a b c d - --- a b c d + 15b c d )x
--R 7
--R +
--R 4 5 10 3 6 9 2 2 7 8 3 8 7 70 4 9 6 6
--R (42a c d - 140a b c d + 180a b c d - 105a b c d + -- b c d )x
--R 3
--R +
--R 4 6 9 3 7 8 2 2 8 7 3 9 6 126 4 10 5 5
--R (42a c d - 144a b c d + 189a b c d - 112a b c d + --- b c d )x
--R 5
--R +
--R 4 7 8 3 8 7 2 2 9 6 3 10 5 210 4 11 4 4
--R (30a c d - 105a b c d + 140a b c d - 84a b c d + --- b c d )x
--R 11
--R +
--R 4 8 7 160 3 9 6 2 2 10 5 480 3 11 4 4 12 3 3
--R (15a c d - --- a b c d + 72a b c d - --- a b c d + 10b c d )x
--R 3 11
--R +
--R 4 9 6 3 10 5 270 2 2 11 4 3 12 3 45 4 13 2 2
--R (5a c d - 18a b c d + --- a b c d - 15a b c d + -- b c d )x
--R 11 13
--R +
--R 4 10 5 40 3 11 4 2 2 12 3 40 3 13 2 5 4 14
--R (a c d - -- a b c d + 5a b c d - -- a b c d + - b c d)x
--R 11 13 7
--R +
--R 1 4 11 4 1 3 12 3 6 2 2 13 2 2 3 14 1 4 15
--R -- a c d - - a b c d + -- a b c d - - a b c d + -- b c
--R 11 3 13 7 15
--R /
--R 5 15 5 4 16 4 3 17 2 3 2 18 3 2 19 4 20 5 15
--R (a b d - 5a b c d + 10a b c d - 10a b c d + 5a b c d - b c )x
--R +
--R 6 14 5 5 15 4 4 16 2 3 3 17 3 2 2 18 4
--R 15a b d - 75a b c d + 150a b c d - 150a b c d + 75a b c d
--R +
--R 19 5
--R - 15a b c
--R *
--R 14
--R x
--R +
--R 7 13 5 6 14 4 5 15 2 3 4 16 3 2
--R 105a b d - 525a b c d + 1050a b c d - 1050a b c d
--R +
--R 3 17 4 2 18 5
--R 525a b c d - 105a b c
--R *
--R 13
--R x
--R +
--R 8 12 5 7 13 4 6 14 2 3 5 15 3 2
--R 455a b d - 2275a b c d + 4550a b c d - 4550a b c d
--R +
--R 4 16 4 3 17 5
--R 2275a b c d - 455a b c
--R *
--R 12
--R x
--R +
--R 9 11 5 8 12 4 7 13 2 3 6 14 3 2
--R 1365a b d - 6825a b c d + 13650a b c d - 13650a b c d
--R +
--R 5 15 4 4 16 5
--R 6825a b c d - 1365a b c
--R *
--R 11
--R x
--R +
--R 10 10 5 9 11 4 8 12 2 3 7 13 3 2
--R 3003a b d - 15015a b c d + 30030a b c d - 30030a b c d
--R +
--R 6 14 4 5 15 5
--R 15015a b c d - 3003a b c
--R *
--R 10
--R x
--R +
--R 11 9 5 10 10 4 9 11 2 3 8 12 3 2
--R 5005a b d - 25025a b c d + 50050a b c d - 50050a b c d
--R +
--R 7 13 4 6 14 5
--R 25025a b c d - 5005a b c
--R *
--R 9
--R x
--R +
--R 12 8 5 11 9 4 10 10 2 3 9 11 3 2
--R 6435a b d - 32175a b c d + 64350a b c d - 64350a b c d
--R +
--R 8 12 4 7 13 5
--R 32175a b c d - 6435a b c
--R *
--R 8
--R x
--R +
--R 13 7 5 12 8 4 11 9 2 3 10 10 3 2
--R 6435a b d - 32175a b c d + 64350a b c d - 64350a b c d
--R +
--R 9 11 4 8 12 5
--R 32175a b c d - 6435a b c
--R *
--R 7
--R x
--R +
--R 14 6 5 13 7 4 12 8 2 3 11 9 3 2
--R 5005a b d - 25025a b c d + 50050a b c d - 50050a b c d
--R +
--R 10 10 4 9 11 5
--R 25025a b c d - 5005a b c
--R *
--R 6
--R x
--R +
--R 15 5 5 14 6 4 13 7 2 3 12 8 3 2
--R 3003a b d - 15015a b c d + 30030a b c d - 30030a b c d
--R +
--R 11 9 4 10 10 5
--R 15015a b c d - 3003a b c
--R *
--R 5
--R x
--R +
--R 16 4 5 15 5 4 14 6 2 3 13 7 3 2
--R 1365a b d - 6825a b c d + 13650a b c d - 13650a b c d
--R +
--R 12 8 4 11 9 5
--R 6825a b c d - 1365a b c
--R *
--R 4
--R x
--R +
--R 17 3 5 16 4 4 15 5 2 3 14 6 3 2
--R 455a b d - 2275a b c d + 4550a b c d - 4550a b c d
--R +
--R 13 7 4 12 8 5
--R 2275a b c d - 455a b c
--R *
--R 3
--R x
--R +
--R 18 2 5 17 3 4 16 4 2 3 15 5 3 2
--R 105a b d - 525a b c d + 1050a b c d - 1050a b c d
--R +
--R 14 6 4 13 7 5
--R 525a b c d - 105a b c
--R *
--R 2
--R x
--R +
--R 19 5 18 2 4 17 3 2 3 16 4 3 2 15 5 4
--R 15a b d - 75a b c d + 150a b c d - 150a b c d + 75a b c d
--R +
--R 14 6 5
--R - 15a b c
--R *
--R x
--R +
--R 20 5 19 4 18 2 2 3 17 3 3 2 16 4 4 15 5 5
--R a d - 5a b c d + 10a b c d - 10a b c d + 5a b c d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 872
--S 873 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 3003b d x + (- 5005a b d - 25025b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 6435a b d - 32175a b c d - 96525b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 6435a b d - 32175a b c d - 96525a b c d - 225225b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 5005a b d - 25025a b c d - 75075a b c d - 175175a b c d
--R +
--R 10 4 6
--R - 350350b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 3003a b d - 15015a b c d - 45045a b c d - 105105a b c d
--R +
--R 9 4 6 10 5 5
--R - 210210a b c d - 378378b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 1365a b d - 6825a b c d - 20475a b c d - 47775a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 95550a b c d - 171990a b c d - 286650b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 455a b d - 2275a b c d - 6825a b c d - 15925a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 31850a b c d - 57330a b c d - 95550a b c d - 150150b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 105a b d - 525a b c d - 1575a b c d - 3675a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 7350a b c d - 13230a b c d - 22050a b c d - 34650a b c d
--R +
--R 10 8 2
--R - 51975b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 15a b d - 75a b c d - 225a b c d - 525a b c d - 1050a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R - 1890a b c d - 3150a b c d - 4950a b c d - 7425a b c d
--R +
--R 10 9
--R - 10725b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6 5 5 5 5
--R - a d - 5a b c d - 15a b c d - 35a b c d - 70a b c d - 126a b c d
--R +
--R 4 6 6 4 3 7 7 3 2 8 8 2 9 9 10 10
--R - 210a b c d - 330a b c d - 495a b c d - 715a b c d - 1001b c
--R /
--R 26 15 25 14 2 24 13 3 23 12
--R 15015b x + 225225a b x + 1576575a b x + 6831825a b x
--R +
--R 4 22 11 5 21 10 6 20 9 7 19 8
--R 20495475a b x + 45090045a b x + 75150075a b x + 96621525a b x
--R +
--R 8 18 7 9 17 6 10 16 5 11 15 4
--R 96621525a b x + 75150075a b x + 45090045a b x + 20495475a b x
--R +
--R 12 14 3 13 13 2 14 12 15 11
--R 6831825a b x + 1576575a b x + 225225a b x + 15015a b
--R Type: Union(Expression(Integer),...)
--E 873
--S 874 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R -
--R 15
--R d
--R /
--R 5 11 5 4 12 4 3 13 2 3 2 14 3 2
--R 15015a b d - 75075a b c d + 150150a b c d - 150150a b c d
--R +
--R 15 4 16 5
--R 75075a b c d - 15015b c
--R Type: Expression(Integer)
--E 874
--S 875 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 875
)clear all
--S 876 of 2952
t0000:=(c+d*x)^10/(a+b*x)^17
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 17 17 16 16 2 15 15 3 14 14 4 13 13
--R b x + 17a b x + 136a b x + 680a b x + 2380a b x
--R +
--R 5 12 12 6 11 11 7 10 10 8 9 9 9 8 8
--R 6188a b x + 12376a b x + 19448a b x + 24310a b x + 24310a b x
--R +
--R 10 7 7 11 6 6 12 5 5 13 4 4 14 3 3
--R 19448a b x + 12376a b x + 6188a b x + 2380a b x + 680a b x
--R +
--R 15 2 2 16 17
--R 136a b x + 17a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 876
--S 877 of 2952
r0000:=-1/16*(c+d*x)^11/((b*c-a*d)*(a+b*x)^16)+_
1/48*d*(c+d*x)^11/((b*c-a*d)^2*(a+b*x)^15)-_
1/168*d^2*(c+d*x)^11/((b*c-a*d)^3*(a+b*x)^14)+_
1/728*d^3*(c+d*x)^11/((b*c-a*d)^4*(a+b*x)^13)-_
1/4368*d^4*(c+d*x)^11/((b*c-a*d)^5*(a+b*x)^12)+_
1/48048*d^5*(c+d*x)^11/((b*c-a*d)^6*(a+b*x)^11)
--R
--R
--R (2)
--R 1 5 16 16 1 4 16 15 5 2 3 16 14 5 3 2 16 13
--R ----- b d x + ---- a b d x + ---- a b d x + --- a b d x
--R 48048 3003 2002 429
--R +
--R 5 4 16 12 1 5 16 11
--R --- a b d x + -- a d x
--R 132 11
--R +
--R 5 15 5 4 2 14 10 3 2 3 13 5 2 3 4 12 4 5 11
--R a c d - - a b c d + -- a b c d - - a b c d + a b c d
--R 2 3 2
--R +
--R 1 5 6 10
--R - - b c d
--R 6
--R *
--R 10
--R x
--R +
--R 5 2 14 50 4 3 13 3 2 4 12 2 3 5 11 25 4 6 10
--R 5a c d - -- a b c d + 25a b c d - 20a b c d + -- a b c d
--R 3 3
--R +
--R 10 5 7 9
--R - -- b c d
--R 7
--R *
--R 9
--R x
--R +
--R 5 3 13 225 4 4 12 3 2 5 11 2 3 6 10 225 4 7 9
--R 15a c d - --- a b c d + 90a b c d - 75a b c d + --- a b c d
--R 4 7
--R +
--R 45 5 8 8
--R - -- b c d
--R 8
--R *
--R 8
--R x
--R +
--R 5 4 12 4 5 11 3 2 6 10 1200 2 3 7 9 4 8 8
--R 30a c d - 120a b c d + 200a b c d - ---- a b c d + 75a b c d
--R 7
--R +
--R 40 5 9 7
--R - -- b c d
--R 3
--R *
--R 7
--R x
--R +
--R 5 5 11 4 6 10 3 2 7 9 525 2 3 8 8 350 4 9 7
--R 42a c d - 175a b c d + 300a b c d - --- a b c d + --- a b c d
--R 2 3
--R +
--R 5 10 6
--R - 21b c d
--R *
--R 6
--R x
--R +
--R 5 6 10 4 7 9 3 2 8 8 2 3 9 7 4 10 6
--R 42a c d - 180a b c d + 315a b c d - 280a b c d + 126a b c d
--R +
--R 252 5 11 5
--R - --- b c d
--R 11
--R *
--R 5
--R x
--R +
--R 5 7 9 525 4 8 8 700 3 2 9 7 2 3 10 6
--R 30a c d - --- a b c d + --- a b c d - 210a b c d
--R 4 3
--R +
--R 1050 4 11 5 35 5 12 4
--R ---- a b c d - -- b c d
--R 11 2
--R *
--R 4
--R x
--R +
--R 5 8 8 200 4 9 7 3 2 10 6 1200 2 3 11 5 4 12 4
--R 15a c d - --- a b c d + 120a b c d - ---- a b c d + 50a b c d
--R 3 11
--R +
--R 120 5 13 3
--R - --- b c d
--R 13
--R *
--R 3
--R x
--R +
--R 5 9 7 45 4 10 6 450 3 2 11 5 75 2 3 12 4 225 4 13 3
--R 5a c d - -- a b c d + --- a b c d - -- a b c d + --- a b c d
--R 2 11 2 13
--R +
--R 45 5 14 2
--R - -- b c d
--R 14
--R *
--R 2
--R x
--R +
--R 5 10 6 50 4 11 5 25 3 2 12 4 100 2 3 13 3 25 4 14 2
--R a c d - -- a b c d + -- a b c d - --- a b c d + -- a b c d
--R 11 3 13 7
--R +
--R 2 5 15
--R - - b c d
--R 3
--R *
--R x
--R +
--R 1 5 11 5 5 4 12 4 10 3 2 13 3 5 2 3 14 2 1 4 15
--R -- a c d - -- a b c d + -- a b c d - - a b c d + - a b c d
--R 11 12 13 7 3
--R +
--R 1 5 16
--R - -- b c
--R 16
--R /
--R 6 16 6 5 17 5 4 18 2 4 3 19 3 3 2 20 4 2
--R a b d - 6a b c d + 15a b c d - 20a b c d + 15a b c d
--R +
--R 21 5 22 6
--R - 6a b c d + b c
--R *
--R 16
--R x
--R +
--R 7 15 6 6 16 5 5 17 2 4 4 18 3 3 3 19 4 2
--R 16a b d - 96a b c d + 240a b c d - 320a b c d + 240a b c d
--R +
--R 2 20 5 21 6
--R - 96a b c d + 16a b c
--R *
--R 15
--R x
--R +
--R 8 14 6 7 15 5 6 16 2 4 5 17 3 3
--R 120a b d - 720a b c d + 1800a b c d - 2400a b c d
--R +
--R 4 18 4 2 3 19 5 2 20 6
--R 1800a b c d - 720a b c d + 120a b c
--R *
--R 14
--R x
--R +
--R 9 13 6 8 14 5 7 15 2 4 6 16 3 3
--R 560a b d - 3360a b c d + 8400a b c d - 11200a b c d
--R +
--R 5 17 4 2 4 18 5 3 19 6
--R 8400a b c d - 3360a b c d + 560a b c
--R *
--R 13
--R x
--R +
--R 10 12 6 9 13 5 8 14 2 4 7 15 3 3
--R 1820a b d - 10920a b c d + 27300a b c d - 36400a b c d
--R +
--R 6 16 4 2 5 17 5 4 18 6
--R 27300a b c d - 10920a b c d + 1820a b c
--R *
--R 12
--R x
--R +
--R 11 11 6 10 12 5 9 13 2 4 8 14 3 3
--R 4368a b d - 26208a b c d + 65520a b c d - 87360a b c d
--R +
--R 7 15 4 2 6 16 5 5 17 6
--R 65520a b c d - 26208a b c d + 4368a b c
--R *
--R 11
--R x
--R +
--R 12 10 6 11 11 5 10 12 2 4 9 13 3 3
--R 8008a b d - 48048a b c d + 120120a b c d - 160160a b c d
--R +
--R 8 14 4 2 7 15 5 6 16 6
--R 120120a b c d - 48048a b c d + 8008a b c
--R *
--R 10
--R x
--R +
--R 13 9 6 12 10 5 11 11 2 4 10 12 3 3
--R 11440a b d - 68640a b c d + 171600a b c d - 228800a b c d
--R +
--R 9 13 4 2 8 14 5 7 15 6
--R 171600a b c d - 68640a b c d + 11440a b c
--R *
--R 9
--R x
--R +
--R 14 8 6 13 9 5 12 10 2 4 11 11 3 3
--R 12870a b d - 77220a b c d + 193050a b c d - 257400a b c d
--R +
--R 10 12 4 2 9 13 5 8 14 6
--R 193050a b c d - 77220a b c d + 12870a b c
--R *
--R 8
--R x
--R +
--R 15 7 6 14 8 5 13 9 2 4 12 10 3 3
--R 11440a b d - 68640a b c d + 171600a b c d - 228800a b c d
--R +
--R 11 11 4 2 10 12 5 9 13 6
--R 171600a b c d - 68640a b c d + 11440a b c
--R *
--R 7
--R x
--R +
--R 16 6 6 15 7 5 14 8 2 4 13 9 3 3
--R 8008a b d - 48048a b c d + 120120a b c d - 160160a b c d
--R +
--R 12 10 4 2 11 11 5 10 12 6
--R 120120a b c d - 48048a b c d + 8008a b c
--R *
--R 6
--R x
--R +
--R 17 5 6 16 6 5 15 7 2 4 14 8 3 3
--R 4368a b d - 26208a b c d + 65520a b c d - 87360a b c d
--R +
--R 13 9 4 2 12 10 5 11 11 6
--R 65520a b c d - 26208a b c d + 4368a b c
--R *
--R 5
--R x
--R +
--R 18 4 6 17 5 5 16 6 2 4 15 7 3 3
--R 1820a b d - 10920a b c d + 27300a b c d - 36400a b c d
--R +
--R 14 8 4 2 13 9 5 12 10 6
--R 27300a b c d - 10920a b c d + 1820a b c
--R *
--R 4
--R x
--R +
--R 19 3 6 18 4 5 17 5 2 4 16 6 3 3
--R 560a b d - 3360a b c d + 8400a b c d - 11200a b c d
--R +
--R 15 7 4 2 14 8 5 13 9 6
--R 8400a b c d - 3360a b c d + 560a b c
--R *
--R 3
--R x
--R +
--R 20 2 6 19 3 5 18 4 2 4 17 5 3 3
--R 120a b d - 720a b c d + 1800a b c d - 2400a b c d
--R +
--R 16 6 4 2 15 7 5 14 8 6
--R 1800a b c d - 720a b c d + 120a b c
--R *
--R 2
--R x
--R +
--R 21 6 20 2 5 19 3 2 4 18 4 3 3 17 5 4 2
--R 16a b d - 96a b c d + 240a b c d - 320a b c d + 240a b c d
--R +
--R 16 6 5 15 7 6
--R - 96a b c d + 16a b c
--R *
--R x
--R +
--R 22 6 21 5 20 2 2 4 19 3 3 3 18 4 4 2 17 5 5
--R a d - 6a b c d + 15a b c d - 20a b c d + 15a b c d - 6a b c d
--R +
--R 16 6 6
--R a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 877
--S 878 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 8008b d x + (- 11440a b d - 68640b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 12870a b d - 77220a b c d - 270270b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 11440a b d - 68640a b c d - 240240a b c d - 640640b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 8008a b d - 48048a b c d - 168168a b c d - 448448a b c d
--R +
--R 10 4 6
--R - 1009008b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 4368a b d - 26208a b c d - 91728a b c d - 244608a b c d
--R +
--R 9 4 6 10 5 5
--R - 550368a b c d - 1100736b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 1820a b d - 10920a b c d - 38220a b c d - 101920a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 229320a b c d - 458640a b c d - 840840b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 560a b d - 3360a b c d - 11760a b c d - 31360a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 70560a b c d - 141120a b c d - 258720a b c d - 443520b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 120a b d - 720a b c d - 2520a b c d - 6720a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 15120a b c d - 30240a b c d - 55440a b c d - 95040a b c d
--R +
--R 10 8 2
--R - 154440b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 16a b d - 96a b c d - 336a b c d - 896a b c d - 2016a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R - 4032a b c d - 7392a b c d - 12672a b c d - 20592a b c d
--R +
--R 10 9
--R - 32032b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - a d - 6a b c d - 21a b c d - 56a b c d - 126a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9
--R - 252a b c d - 462a b c d - 792a b c d - 1287a b c d - 2002a b c d
--R +
--R 10 10
--R - 3003b c
--R /
--R 27 16 26 15 2 25 14 3 24 13
--R 48048b x + 768768a b x + 5765760a b x + 26906880a b x
--R +
--R 4 23 12 5 22 11 6 21 10
--R 87447360a b x + 209873664a b x + 384768384a b x
--R +
--R 7 20 9 8 19 8 9 18 7
--R 549669120a b x + 618377760a b x + 549669120a b x
--R +
--R 10 17 6 11 16 5 12 15 4
--R 384768384a b x + 209873664a b x + 87447360a b x
--R +
--R 13 14 3 14 13 2 15 12 16 11
--R 26906880a b x + 5765760a b x + 768768a b x + 48048a b
--R Type: Union(Expression(Integer),...)
--E 878
--S 879 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R -
--R 16
--R d
--R /
--R 6 11 6 5 12 5 4 13 2 4 3 14 3 3
--R 48048a b d - 288288a b c d + 720720a b c d - 960960a b c d
--R +
--R 2 15 4 2 16 5 17 6
--R 720720a b c d - 288288a b c d + 48048b c
--R Type: Expression(Integer)
--E 879
--S 880 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 880
)clear all
--S 881 of 2952
t0000:=(c+d*x)^10/(a+b*x)^18
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 18 18 17 17 2 16 16 3 15 15 4 14 14
--R b x + 18a b x + 153a b x + 816a b x + 3060a b x
--R +
--R 5 13 13 6 12 12 7 11 11 8 10 10
--R 8568a b x + 18564a b x + 31824a b x + 43758a b x
--R +
--R 9 9 9 10 8 8 11 7 7 12 6 6 13 5 5
--R 48620a b x + 43758a b x + 31824a b x + 18564a b x + 8568a b x
--R +
--R 14 4 4 15 3 3 16 2 2 17 18
--R 3060a b x + 816a b x + 153a b x + 18a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 881
--S 882 of 2952
r0000:=-1/17*(c+d*x)^11/((b*c-a*d)*(a+b*x)^17)+_
3/136*d*(c+d*x)^11/((b*c-a*d)^2*(a+b*x)^16)-_
1/136*d^2*(c+d*x)^11/((b*c-a*d)^3*(a+b*x)^15)+_
1/476*d^3*(c+d*x)^11/((b*c-a*d)^4*(a+b*x)^14)-_
3/6188*d^4*(c+d*x)^11/((b*c-a*d)^5*(a+b*x)^13)+_
1/12376*d^5*(c+d*x)^11/((b*c-a*d)^6*(a+b*x)^12)-_
1/136136*d^6*(c+d*x)^11/((b*c-a*d)^7*(a+b*x)^11)
--R
--R
--R (2)
--R 1 6 17 17 1 5 17 16 1 2 4 17 15 5 3 3 17 14
--R ------ b d x + ---- a b d x + ---- a b d x + ---- a b d x
--R 136136 8008 1001 1001
--R +
--R 5 4 2 17 13 1 5 17 12 1 6 17 11
--R --- a b d x + -- a b d x + -- a d x
--R 286 22 11
--R +
--R 6 16 5 2 15 4 2 3 14 3 3 4 13 2 4 5 12
--R a c d - 3a b c d + 5a b c d - 5a b c d + 3a b c d
--R +
--R 5 6 11 1 6 7 10
--R - a b c d + - b c d
--R 7
--R *
--R 10
--R x
--R +
--R 6 2 15 5 3 14 75 4 2 4 13 3 3 5 12 2 4 6 11
--R 5a c d - 20a b c d + -- a b c d - 40a b c d + 25a b c d
--R 2
--R +
--R 60 5 7 10 5 6 8 9
--R - -- a b c d + - b c d
--R 7 4
--R *
--R 9
--R x
--R +
--R 6 3 14 135 5 4 13 4 2 5 12 3 3 6 11
--R 15a c d - --- a b c d + 135a b c d - 150a b c d
--R 2
--R +
--R 675 2 4 7 10 135 5 8 9 6 9 8
--R --- a b c d - --- a b c d + 5b c d
--R 7 4
--R *
--R 8
--R x
--R +
--R 6 4 13 5 5 12 4 2 6 11 2400 3 3 7 10
--R 30a c d - 144a b c d + 300a b c d - ---- a b c d
--R 7
--R +
--R 2 4 8 9 5 9 8 6 10 7
--R 225a b c d - 80a b c d + 12b c d
--R *
--R 7
--R x
--R +
--R 6 5 12 5 6 11 4 2 7 10 3 3 8 9 2 4 9 8
--R 42a c d - 210a b c d + 450a b c d - 525a b c d + 350a b c d
--R +
--R 5 10 7 210 6 11 6
--R - 126a b c d + --- b c d
--R 11
--R *
--R 6
--R x
--R +
--R 6 6 11 5 7 10 945 4 2 8 9 3 3 9 8 2 4 10 7
--R 42a c d - 216a b c d + --- a b c d - 560a b c d + 378a b c d
--R 2
--R +
--R 1512 5 11 6 6 12 5
--R - ---- a b c d + 21b c d
--R 11
--R *
--R 5
--R x
--R +
--R 6 7 10 315 5 8 9 4 2 9 8 3 3 10 7
--R 30a c d - --- a b c d + 350a b c d - 420a b c d
--R 2
--R +
--R 3150 2 4 11 6 5 12 5 210 6 13 4
--R ---- a b c d - 105a b c d + --- b c d
--R 11 13
--R *
--R 4
--R x
--R +
--R 6 8 9 5 9 8 4 2 10 7 2400 3 3 11 6 2 4 12 5
--R 15a c d - 80a b c d + 180a b c d - ---- a b c d + 150a b c d
--R 11
--R +
--R 720 5 13 4 60 6 14 3
--R - --- a b c d + -- b c d
--R 13 7
--R *
--R 3
--R x
--R +
--R 6 9 8 5 10 7 675 4 2 11 6 3 3 12 5 675 2 4 13 4
--R 5a c d - 27a b c d + --- a b c d - 75a b c d + --- a b c d
--R 11 13
--R +
--R 135 5 14 3 6 15 2
--R - --- a b c d + 3b c d
--R 7
--R *
--R 2
--R x
--R +
--R 6 10 7 60 5 11 6 25 4 2 12 5 200 3 3 13 4 75 2 4 14 3
--R a c d - -- a b c d + -- a b c d - --- a b c d + -- a b c d
--R 11 2 13 7
--R +
--R 5 15 2 5 6 16
--R - 4a b c d + - b c d
--R 8
--R *
--R x
--R +
--R 1 6 11 6 1 5 12 5 15 4 2 13 4 10 3 3 14 3 2 4 15 2
--R -- a c d - - a b c d + -- a b c d - -- a b c d + a b c d
--R 11 2 13 7
--R +
--R 3 5 16 1 6 17
--R - - a b c d + -- b c
--R 8 17
--R /
--R 7 17 7 6 18 6 5 19 2 5 4 20 3 4 3 21 4 3
--R a b d - 7a b c d + 21a b c d - 35a b c d + 35a b c d
--R +
--R 2 22 5 2 23 6 24 7
--R - 21a b c d + 7a b c d - b c
--R *
--R 17
--R x
--R +
--R 8 16 7 7 17 6 6 18 2 5 5 19 3 4 4 20 4 3
--R 17a b d - 119a b c d + 357a b c d - 595a b c d + 595a b c d
--R +
--R 3 21 5 2 2 22 6 23 7
--R - 357a b c d + 119a b c d - 17a b c
--R *
--R 16
--R x
--R +
--R 9 15 7 8 16 6 7 17 2 5 6 18 3 4
--R 136a b d - 952a b c d + 2856a b c d - 4760a b c d
--R +
--R 5 19 4 3 4 20 5 2 3 21 6 2 22 7
--R 4760a b c d - 2856a b c d + 952a b c d - 136a b c
--R *
--R 15
--R x
--R +
--R 10 14 7 9 15 6 8 16 2 5 7 17 3 4
--R 680a b d - 4760a b c d + 14280a b c d - 23800a b c d
--R +
--R 6 18 4 3 5 19 5 2 4 20 6 3 21 7
--R 23800a b c d - 14280a b c d + 4760a b c d - 680a b c
--R *
--R 14
--R x
--R +
--R 11 13 7 10 14 6 9 15 2 5 8 16 3 4
--R 2380a b d - 16660a b c d + 49980a b c d - 83300a b c d
--R +
--R 7 17 4 3 6 18 5 2 5 19 6 4 20 7
--R 83300a b c d - 49980a b c d + 16660a b c d - 2380a b c
--R *
--R 13
--R x
--R +
--R 12 12 7 11 13 6 10 14 2 5 9 15 3 4
--R 6188a b d - 43316a b c d + 129948a b c d - 216580a b c d
--R +
--R 8 16 4 3 7 17 5 2 6 18 6 5 19 7
--R 216580a b c d - 129948a b c d + 43316a b c d - 6188a b c
--R *
--R 12
--R x
--R +
--R 13 11 7 12 12 6 11 13 2 5 10 14 3 4
--R 12376a b d - 86632a b c d + 259896a b c d - 433160a b c d
--R +
--R 9 15 4 3 8 16 5 2 7 17 6 6 18 7
--R 433160a b c d - 259896a b c d + 86632a b c d - 12376a b c
--R *
--R 11
--R x
--R +
--R 14 10 7 13 11 6 12 12 2 5
--R 19448a b d - 136136a b c d + 408408a b c d
--R +
--R 11 13 3 4 10 14 4 3 9 15 5 2
--R - 680680a b c d + 680680a b c d - 408408a b c d
--R +
--R 8 16 6 7 17 7
--R 136136a b c d - 19448a b c
--R *
--R 10
--R x
--R +
--R 15 9 7 14 10 6 13 11 2 5 12 12 3 4
--R 24310a b d - 170170a b c d + 510510a b c d - 850850a b c d
--R +
--R 11 13 4 3 10 14 5 2 9 15 6 8 16 7
--R 850850a b c d - 510510a b c d + 170170a b c d - 24310a b c
--R *
--R 9
--R x
--R +
--R 16 8 7 15 9 6 14 10 2 5 13 11 3 4
--R 24310a b d - 170170a b c d + 510510a b c d - 850850a b c d
--R +
--R 12 12 4 3 11 13 5 2 10 14 6 9 15 7
--R 850850a b c d - 510510a b c d + 170170a b c d - 24310a b c
--R *
--R 8
--R x
--R +
--R 17 7 7 16 8 6 15 9 2 5 14 10 3 4
--R 19448a b d - 136136a b c d + 408408a b c d - 680680a b c d
--R +
--R 13 11 4 3 12 12 5 2 11 13 6 10 14 7
--R 680680a b c d - 408408a b c d + 136136a b c d - 19448a b c
--R *
--R 7
--R x
--R +
--R 18 6 7 17 7 6 16 8 2 5 15 9 3 4
--R 12376a b d - 86632a b c d + 259896a b c d - 433160a b c d
--R +
--R 14 10 4 3 13 11 5 2 12 12 6 11 13 7
--R 433160a b c d - 259896a b c d + 86632a b c d - 12376a b c
--R *
--R 6
--R x
--R +
--R 19 5 7 18 6 6 17 7 2 5 16 8 3 4
--R 6188a b d - 43316a b c d + 129948a b c d - 216580a b c d
--R +
--R 15 9 4 3 14 10 5 2 13 11 6 12 12 7
--R 216580a b c d - 129948a b c d + 43316a b c d - 6188a b c
--R *
--R 5
--R x
--R +
--R 20 4 7 19 5 6 18 6 2 5 17 7 3 4
--R 2380a b d - 16660a b c d + 49980a b c d - 83300a b c d
--R +
--R 16 8 4 3 15 9 5 2 14 10 6 13 11 7
--R 83300a b c d - 49980a b c d + 16660a b c d - 2380a b c
--R *
--R 4
--R x
--R +
--R 21 3 7 20 4 6 19 5 2 5 18 6 3 4
--R 680a b d - 4760a b c d + 14280a b c d - 23800a b c d
--R +
--R 17 7 4 3 16 8 5 2 15 9 6 14 10 7
--R 23800a b c d - 14280a b c d + 4760a b c d - 680a b c
--R *
--R 3
--R x
--R +
--R 22 2 7 21 3 6 20 4 2 5 19 5 3 4
--R 136a b d - 952a b c d + 2856a b c d - 4760a b c d
--R +
--R 18 6 4 3 17 7 5 2 16 8 6 15 9 7
--R 4760a b c d - 2856a b c d + 952a b c d - 136a b c
--R *
--R 2
--R x
--R +
--R 23 7 22 2 6 21 3 2 5 20 4 3 4 19 5 4 3
--R 17a b d - 119a b c d + 357a b c d - 595a b c d + 595a b c d
--R +
--R 18 6 5 2 17 7 6 16 8 7
--R - 357a b c d + 119a b c d - 17a b c
--R *
--R x
--R +
--R 24 7 23 6 22 2 2 5 21 3 3 4 20 4 4 3
--R a d - 7a b c d + 21a b c d - 35a b c d + 35a b c d
--R +
--R 19 5 5 2 18 6 6 17 7 7
--R - 21a b c d + 7a b c d - a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 882
--S 883 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 19448b d x + (- 24310a b d - 170170b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 24310a b d - 170170a b c d - 680680b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 19448a b d - 136136a b c d - 544544a b c d - 1633632b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 12376a b d - 86632a b c d - 346528a b c d - 1039584a b c d
--R +
--R 10 4 6
--R - 2598960b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 6188a b d - 43316a b c d - 173264a b c d - 519792a b c d
--R +
--R 9 4 6 10 5 5
--R - 1299480a b c d - 2858856b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 2380a b d - 16660a b c d - 66640a b c d - 199920a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 499800a b c d - 1099560a b c d - 2199120b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 680a b d - 4760a b c d - 19040a b c d - 57120a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 142800a b c d - 314160a b c d - 628320a b c d - 1166880b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 136a b d - 952a b c d - 3808a b c d - 11424a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 28560a b c d - 62832a b c d - 125664a b c d - 233376a b c d
--R +
--R 10 8 2
--R - 408408b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 17a b d - 119a b c d - 476a b c d - 1428a b c d - 3570a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R - 7854a b c d - 15708a b c d - 29172a b c d - 51051a b c d
--R +
--R 10 9
--R - 85085b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - a d - 7a b c d - 28a b c d - 84a b c d - 210a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9
--R - 462a b c d - 924a b c d - 1716a b c d - 3003a b c d - 5005a b c d
--R +
--R 10 10
--R - 8008b c
--R /
--R 28 17 27 16 2 26 15 3 25 14
--R 136136b x + 2314312a b x + 18514496a b x + 92572480a b x
--R +
--R 4 24 13 5 23 12 6 22 11
--R 324003680a b x + 842409568a b x + 1684819136a b x
--R +
--R 7 21 10 8 20 9 9 19 8
--R 2647572928a b x + 3309466160a b x + 3309466160a b x
--R +
--R 10 18 7 11 17 6 12 16 5
--R 2647572928a b x + 1684819136a b x + 842409568a b x
--R +
--R 13 15 4 14 14 3 15 13 2 16 12
--R 324003680a b x + 92572480a b x + 18514496a b x + 2314312a b x
--R +
--R 17 11
--R 136136a b
--R Type: Union(Expression(Integer),...)
--E 883
--S 884 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R -
--R 17
--R d
--R /
--R 7 11 7 6 12 6 5 13 2 5 4 14 3 4
--R 136136a b d - 952952a b c d + 2858856a b c d - 4764760a b c d
--R +
--R 3 15 4 3 2 16 5 2 17 6 18 7
--R 4764760a b c d - 2858856a b c d + 952952a b c d - 136136b c
--R Type: Expression(Integer)
--E 884
--S 885 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 885
)clear all
--S 886 of 2952
t0000:=(c+d*x)^10/(a+b*x)^19
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 19 19 18 18 2 17 17 3 16 16 4 15 15
--R b x + 19a b x + 171a b x + 969a b x + 3876a b x
--R +
--R 5 14 14 6 13 13 7 12 12 8 11 11
--R 11628a b x + 27132a b x + 50388a b x + 75582a b x
--R +
--R 9 10 10 10 9 9 11 8 8 12 7 7 13 6 6
--R 92378a b x + 92378a b x + 75582a b x + 50388a b x + 27132a b x
--R +
--R 14 5 5 15 4 4 16 3 3 17 2 2 18 19
--R 11628a b x + 3876a b x + 969a b x + 171a b x + 19a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 886
--S 887 of 2952
r0000:=-1/18*(c+d*x)^11/((b*c-a*d)*(a+b*x)^18)+_
7/306*d*(c+d*x)^11/((b*c-a*d)^2*(a+b*x)^17)-_
7/816*d^2*(c+d*x)^11/((b*c-a*d)^3*(a+b*x)^16)+_
7/2448*d^3*(c+d*x)^11/((b*c-a*d)^4*(a+b*x)^15)-_
1/1224*d^4*(c+d*x)^11/((b*c-a*d)^5*(a+b*x)^14)+_
1/5304*d^5*(c+d*x)^11/((b*c-a*d)^6*(a+b*x)^13)-_
1/31824*d^6*(c+d*x)^11/((b*c-a*d)^7*(a+b*x)^12)+_
1/350064*d^7*(c+d*x)^11/((b*c-a*d)^8*(a+b*x)^11)
--R
--R
--R (2)
--R 1 7 18 18 1 6 18 17 1 2 5 18 16 1 3 4 18 15
--R ------ b d x + ----- a b d x + ---- a b d x + --- a b d x
--R 350064 19448 2288 429
--R +
--R 5 4 3 18 14 7 5 2 18 13 7 6 18 12 1 7 18 11
--R --- a b d x + --- a b d x + --- a b d x + -- a d x
--R 572 286 132 11
--R +
--R 7 17 7 6 2 16 5 2 3 15 35 4 3 4 14 3 4 5 13
--R a c d - - a b c d + 7a b c d - -- a b c d + 7a b c d
--R 2 4
--R +
--R 7 2 5 6 12 6 7 11 1 7 8 10
--R - - a b c d + a b c d - - b c d
--R 2 8
--R *
--R 10
--R x
--R +
--R 7 2 16 70 6 3 15 105 5 2 4 14 4 3 5 13 175 3 4 6 12
--R 5a c d - -- a b c d + --- a b c d - 70a b c d + --- a b c d
--R 3 2 3
--R +
--R 2 5 7 11 35 6 8 10 10 7 9 9
--R - 30a b c d + -- a b c d - -- b c d
--R 4 9
--R *
--R 9
--R x
--R +
--R 7 3 15 315 6 4 14 5 2 5 13 525 4 3 6 12
--R 15a c d - --- a b c d + 189a b c d - --- a b c d
--R 4 2
--R +
--R 3 4 7 11 945 2 5 8 10 6 9 9 9 7 10 8
--R 225a b c d - --- a b c d + 35a b c d - - b c d
--R 8 2
--R *
--R 8
--R x
--R +
--R 7 4 14 6 5 13 5 2 6 12 4 3 7 11 3 4 8 10
--R 30a c d - 168a b c d + 420a b c d - 600a b c d + 525a b c d
--R +
--R 2 5 9 9 6 10 8 120 7 11 7
--R - 280a b c d + 84a b c d - --- b c d
--R 11
--R *
--R 7
--R x
--R +
--R 7 5 13 6 6 12 5 2 7 11 3675 4 3 8 10
--R 42a c d - 245a b c d + 630a b c d - ---- a b c d
--R 4
--R +
--R 2450 3 4 9 9 2 5 10 8 1470 6 11 7 35 7 12 6
--R ---- a b c d - 441a b c d + ---- a b c d - -- b c d
--R 3 11 2
--R *
--R 6
--R x
--R +
--R 7 6 12 6 7 11 1323 5 2 8 10 4 3 9 9
--R 42a c d - 252a b c d + ---- a b c d - 980a b c d
--R 2
--R +
--R 3 4 10 8 5292 2 5 11 7 6 12 6 252 7 13 5
--R 882a b c d - ---- a b c d + 147a b c d - --- b c d
--R 11 13
--R *
--R 5
--R x
--R +
--R 7 7 11 735 6 8 10 5 2 9 9 4 3 10 8
--R 30a c d - --- a b c d + 490a b c d - 735a b c d
--R 4
--R +
--R 7350 3 4 11 7 735 2 5 12 6 1470 6 13 5 7 14 4
--R ---- a b c d - --- a b c d + ---- a b c d - 15b c d
--R 11 2 13
--R *
--R 4
--R x
--R +
--R 7 8 10 280 6 9 9 5 2 10 8 4200 4 3 11 7
--R 15a c d - --- a b c d + 252a b c d - ---- a b c d
--R 3 11
--R +
--R 3 4 12 6 2520 2 5 13 5 6 14 4 7 15 3
--R 350a b c d - ---- a b c d + 60a b c d - 8b c d
--R 13
--R *
--R 3
--R x
--R +
--R 7 9 9 63 6 10 8 945 5 2 11 7 525 4 3 12 6
--R 5a c d - -- a b c d + --- a b c d - --- a b c d
--R 2 11 4
--R +
--R 1575 3 4 13 5 135 2 5 14 4 6 15 3 45 7 16 2
--R ---- a b c d - --- a b c d + 21a b c d - -- b c d
--R 13 2 16
--R *
--R 2
--R x
--R +
--R 7 10 8 70 6 11 7 35 5 2 12 6 350 4 3 13 5 3 4 14 4
--R a c d - -- a b c d + -- a b c d - --- a b c d + 25a b c d
--R 11 2 13
--R +
--R 2 5 15 3 35 6 16 2 10 7 17
--R - 14a b c d + -- a b c d - -- b c d
--R 8 17
--R *
--R x
--R +
--R 1 7 11 7 7 6 12 6 21 5 2 13 5 5 4 3 14 4 7 3 4 15 3
--R -- a c d - -- a b c d + -- a b c d - - a b c d + - a b c d
--R 11 12 13 2 3
--R +
--R 21 2 5 16 2 7 6 17 1 7 18
--R - -- a b c d + -- a b c d - -- b c
--R 16 17 18
--R /
--R 8 18 8 7 19 7 6 20 2 6 5 21 3 5 4 22 4 4
--R a b d - 8a b c d + 28a b c d - 56a b c d + 70a b c d
--R +
--R 3 23 5 3 2 24 6 2 25 7 26 8
--R - 56a b c d + 28a b c d - 8a b c d + b c
--R *
--R 18
--R x
--R +
--R 9 17 8 8 18 7 7 19 2 6 6 20 3 5
--R 18a b d - 144a b c d + 504a b c d - 1008a b c d
--R +
--R 5 21 4 4 4 22 5 3 3 23 6 2 2 24 7 25 8
--R 1260a b c d - 1008a b c d + 504a b c d - 144a b c d + 18a b c
--R *
--R 17
--R x
--R +
--R 10 16 8 9 17 7 8 18 2 6 7 19 3 5
--R 153a b d - 1224a b c d + 4284a b c d - 8568a b c d
--R +
--R 6 20 4 4 5 21 5 3 4 22 6 2 3 23 7
--R 10710a b c d - 8568a b c d + 4284a b c d - 1224a b c d
--R +
--R 2 24 8
--R 153a b c
--R *
--R 16
--R x
--R +
--R 11 15 8 10 16 7 9 17 2 6 8 18 3 5
--R 816a b d - 6528a b c d + 22848a b c d - 45696a b c d
--R +
--R 7 19 4 4 6 20 5 3 5 21 6 2 4 22 7
--R 57120a b c d - 45696a b c d + 22848a b c d - 6528a b c d
--R +
--R 3 23 8
--R 816a b c
--R *
--R 15
--R x
--R +
--R 12 14 8 11 15 7 10 16 2 6 9 17 3 5
--R 3060a b d - 24480a b c d + 85680a b c d - 171360a b c d
--R +
--R 8 18 4 4 7 19 5 3 6 20 6 2 5 21 7
--R 214200a b c d - 171360a b c d + 85680a b c d - 24480a b c d
--R +
--R 4 22 8
--R 3060a b c
--R *
--R 14
--R x
--R +
--R 13 13 8 12 14 7 11 15 2 6 10 16 3 5
--R 8568a b d - 68544a b c d + 239904a b c d - 479808a b c d
--R +
--R 9 17 4 4 8 18 5 3 7 19 6 2 6 20 7
--R 599760a b c d - 479808a b c d + 239904a b c d - 68544a b c d
--R +
--R 5 21 8
--R 8568a b c
--R *
--R 13
--R x
--R +
--R 14 12 8 13 13 7 12 14 2 6
--R 18564a b d - 148512a b c d + 519792a b c d
--R +
--R 11 15 3 5 10 16 4 4 9 17 5 3
--R - 1039584a b c d + 1299480a b c d - 1039584a b c d
--R +
--R 8 18 6 2 7 19 7 6 20 8
--R 519792a b c d - 148512a b c d + 18564a b c
--R *
--R 12
--R x
--R +
--R 15 11 8 14 12 7 13 13 2 6
--R 31824a b d - 254592a b c d + 891072a b c d
--R +
--R 12 14 3 5 11 15 4 4 10 16 5 3
--R - 1782144a b c d + 2227680a b c d - 1782144a b c d
--R +
--R 9 17 6 2 8 18 7 7 19 8
--R 891072a b c d - 254592a b c d + 31824a b c
--R *
--R 11
--R x
--R +
--R 16 10 8 15 11 7 14 12 2 6
--R 43758a b d - 350064a b c d + 1225224a b c d
--R +
--R 13 13 3 5 12 14 4 4 11 15 5 3
--R - 2450448a b c d + 3063060a b c d - 2450448a b c d
--R +
--R 10 16 6 2 9 17 7 8 18 8
--R 1225224a b c d - 350064a b c d + 43758a b c
--R *
--R 10
--R x
--R +
--R 17 9 8 16 10 7 15 11 2 6
--R 48620a b d - 388960a b c d + 1361360a b c d
--R +
--R 14 12 3 5 13 13 4 4 12 14 5 3
--R - 2722720a b c d + 3403400a b c d - 2722720a b c d
--R +
--R 11 15 6 2 10 16 7 9 17 8
--R 1361360a b c d - 388960a b c d + 48620a b c
--R *
--R 9
--R x
--R +
--R 18 8 8 17 9 7 16 10 2 6
--R 43758a b d - 350064a b c d + 1225224a b c d
--R +
--R 15 11 3 5 14 12 4 4 13 13 5 3
--R - 2450448a b c d + 3063060a b c d - 2450448a b c d
--R +
--R 12 14 6 2 11 15 7 10 16 8
--R 1225224a b c d - 350064a b c d + 43758a b c
--R *
--R 8
--R x
--R +
--R 19 7 8 18 8 7 17 9 2 6 16 10 3 5
--R 31824a b d - 254592a b c d + 891072a b c d - 1782144a b c d
--R +
--R 15 11 4 4 14 12 5 3 13 13 6 2
--R 2227680a b c d - 1782144a b c d + 891072a b c d
--R +
--R 12 14 7 11 15 8
--R - 254592a b c d + 31824a b c
--R *
--R 7
--R x
--R +
--R 20 6 8 19 7 7 18 8 2 6 17 9 3 5
--R 18564a b d - 148512a b c d + 519792a b c d - 1039584a b c d
--R +
--R 16 10 4 4 15 11 5 3 14 12 6 2
--R 1299480a b c d - 1039584a b c d + 519792a b c d
--R +
--R 13 13 7 12 14 8
--R - 148512a b c d + 18564a b c
--R *
--R 6
--R x
--R +
--R 21 5 8 20 6 7 19 7 2 6 18 8 3 5
--R 8568a b d - 68544a b c d + 239904a b c d - 479808a b c d
--R +
--R 17 9 4 4 16 10 5 3 15 11 6 2
--R 599760a b c d - 479808a b c d + 239904a b c d
--R +
--R 14 12 7 13 13 8
--R - 68544a b c d + 8568a b c
--R *
--R 5
--R x
--R +
--R 22 4 8 21 5 7 20 6 2 6 19 7 3 5
--R 3060a b d - 24480a b c d + 85680a b c d - 171360a b c d
--R +
--R 18 8 4 4 17 9 5 3 16 10 6 2 15 11 7
--R 214200a b c d - 171360a b c d + 85680a b c d - 24480a b c d
--R +
--R 14 12 8
--R 3060a b c
--R *
--R 4
--R x
--R +
--R 23 3 8 22 4 7 21 5 2 6 20 6 3 5
--R 816a b d - 6528a b c d + 22848a b c d - 45696a b c d
--R +
--R 19 7 4 4 18 8 5 3 17 9 6 2 16 10 7
--R 57120a b c d - 45696a b c d + 22848a b c d - 6528a b c d
--R +
--R 15 11 8
--R 816a b c
--R *
--R 3
--R x
--R +
--R 24 2 8 23 3 7 22 4 2 6 21 5 3 5
--R 153a b d - 1224a b c d + 4284a b c d - 8568a b c d
--R +
--R 20 6 4 4 19 7 5 3 18 8 6 2 17 9 7
--R 10710a b c d - 8568a b c d + 4284a b c d - 1224a b c d
--R +
--R 16 10 8
--R 153a b c
--R *
--R 2
--R x
--R +
--R 25 8 24 2 7 23 3 2 6 22 4 3 5
--R 18a b d - 144a b c d + 504a b c d - 1008a b c d
--R +
--R 21 5 4 4 20 6 5 3 19 7 6 2 18 8 7 17 9 8
--R 1260a b c d - 1008a b c d + 504a b c d - 144a b c d + 18a b c
--R *
--R x
--R +
--R 26 8 25 7 24 2 2 6 23 3 3 5 22 4 4 4
--R a d - 8a b c d + 28a b c d - 56a b c d + 70a b c d
--R +
--R 21 5 5 3 20 6 6 2 19 7 7 18 8 8
--R - 56a b c d + 28a b c d - 8a b c d + a b c
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 887
--S 888 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 43758b d x + (- 48620a b d - 388960b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 43758a b d - 350064a b c d - 1575288b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 31824a b d - 254592a b c d - 1145664a b c d - 3818880b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 18564a b d - 148512a b c d - 668304a b c d - 2227680a b c d
--R +
--R 10 4 6
--R - 6126120b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 8568a b d - 68544a b c d - 308448a b c d - 1028160a b c d
--R +
--R 9 4 6 10 5 5
--R - 2827440a b c d - 6785856b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 3060a b d - 24480a b c d - 110160a b c d - 367200a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 1009800a b c d - 2423520a b c d - 5250960b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 816a b d - 6528a b c d - 29376a b c d - 97920a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 269280a b c d - 646272a b c d - 1400256a b c d - 2800512b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 153a b d - 1224a b c d - 5508a b c d - 18360a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 50490a b c d - 121176a b c d - 262548a b c d - 525096a b c d
--R +
--R 10 8 2
--R - 984555b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 18a b d - 144a b c d - 648a b c d - 2160a b c d - 5940a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R - 14256a b c d - 30888a b c d - 61776a b c d - 115830a b c d
--R +
--R 10 9
--R - 205920b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - a d - 8a b c d - 36a b c d - 120a b c d - 330a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2 9 9
--R - 792a b c d - 1716a b c d - 3432a b c d - 6435a b c d - 11440a b c d
--R +
--R 10 10
--R - 19448b c
--R /
--R 29 18 28 17 2 27 16 3 26 15
--R 350064b x + 6301152a b x + 53559792a b x + 285652224a b x
--R +
--R 4 25 14 5 24 13 6 23 12
--R 1071195840a b x + 2999348352a b x + 6498588096a b x
--R +
--R 7 22 11 8 21 10 9 20 9
--R 11140436736a b x + 15318100512a b x + 17020111680a b x
--R +
--R 10 19 8 11 18 7 12 17 6
--R 15318100512a b x + 11140436736a b x + 6498588096a b x
--R +
--R 13 16 5 14 15 4 15 14 3
--R 2999348352a b x + 1071195840a b x + 285652224a b x
--R +
--R 16 13 2 17 12 18 11
--R 53559792a b x + 6301152a b x + 350064a b
--R Type: Union(Expression(Integer),...)
--E 888
--S 889 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4)
--R -
--R 18
--R d
--R /
--R 8 11 8 7 12 7 6 13 2 6
--R 350064a b d - 2800512a b c d + 9801792a b c d
--R +
--R 5 14 3 5 4 15 4 4 3 16 5 3
--R - 19603584a b c d + 24504480a b c d - 19603584a b c d
--R +
--R 2 17 6 2 18 7 19 8
--R 9801792a b c d - 2800512a b c d + 350064b c
--R Type: Expression(Integer)
--E 889
--S 890 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 890
)clear all
--S 891 of 2952
t0000:=(c+d*x)^10/(a+b*x)^20
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 20 20 19 19 2 18 18 3 17 17 4 16 16
--R b x + 20a b x + 190a b x + 1140a b x + 4845a b x
--R +
--R 5 15 15 6 14 14 7 13 13 8 12 12
--R 15504a b x + 38760a b x + 77520a b x + 125970a b x
--R +
--R 9 11 11 10 10 10 11 9 9 12 8 8
--R 167960a b x + 184756a b x + 167960a b x + 125970a b x
--R +
--R 13 7 7 14 6 6 15 5 5 16 4 4 17 3 3
--R 77520a b x + 38760a b x + 15504a b x + 4845a b x + 1140a b x
--R +
--R 18 2 2 19 20
--R 190a b x + 20a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 891
--S 892 of 2952
r0000:=-1/19*(b*c-a*d)^10/(b^11*(a+b*x)^19)-_
5/9*d*(b*c-a*d)^9/(b^11*(a+b*x)^18)-_
45/17*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^17)-_
15/2*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^16)-_
14*d^4*(b*c-a*d)^6/(b^11*(a+b*x)^15)-_
18*d^5*(b*c-a*d)^5/(b^11*(a+b*x)^14)-_
210/13*d^6*(b*c-a*d)^4/(b^11*(a+b*x)^13)-_
10*d^7*(b*c-a*d)^3/(b^11*(a+b*x)^12)-_
45/11*d^8*(b*c-a*d)^2/(b^11*(a+b*x)^11)-_
d^9*(b*c-a*d)/(b^11*(a+b*x)^10)-1/9*d^10/(b^11*(a+b*x)^9)
--R
--R
--R (2)
--R 1 10 10 10 1 9 10 10 9 9
--R - - b d x + (- - a b d - b c d )x
--R 9 9
--R +
--R 1 2 8 10 9 9 9 45 10 2 8 8
--R (- -- a b d - -- a b c d - -- b c d )x
--R 11 11 11
--R +
--R 2 3 7 10 6 2 8 9 30 9 2 8 10 3 7 7
--R (- -- a b d - -- a b c d - -- a b c d - 10b c d )x
--R 33 11 11
--R +
--R 14 4 6 10 42 3 7 9 210 2 8 2 8 70 9 3 7
--R - --- a b d - --- a b c d - --- a b c d - -- a b c d
--R 429 143 143 13
--R +
--R 210 10 4 6
--R - --- b c d
--R 13
--R *
--R 6
--R x
--R +
--R 2 5 5 10 18 4 6 9 90 3 7 2 8 30 2 8 3 7
--R - --- a b d - --- a b c d - --- a b c d - -- a b c d
--R 143 143 143 13
--R +
--R 90 9 4 6 10 5 5
--R - -- a b c d - 18b c d
--R 13
--R *
--R 5
--R x
--R +
--R 2 6 4 10 6 5 5 9 30 4 6 2 8 10 3 7 3 7
--R - --- a b d - --- a b c d - --- a b c d - -- a b c d
--R 429 143 143 13
--R +
--R 30 2 8 4 6 9 5 5 10 6 4
--R - -- a b c d - 6a b c d - 14b c d
--R 13
--R *
--R 4
--R x
--R +
--R 1 7 3 10 3 6 4 9 15 5 5 2 8 5 4 6 3 7
--R - --- a b d - --- a b c d - --- a b c d - -- a b c d
--R 858 286 286 26
--R +
--R 15 3 7 4 6 3 2 8 5 5 7 9 6 4 15 10 7 3
--R - -- a b c d - - a b c d - - a b c d - -- b c d
--R 26 2 2 2
--R *
--R 3
--R x
--R +
--R 1 8 2 10 9 7 3 9 45 6 4 2 8 15 5 5 3 7
--R - ---- a b d - ---- a b c d - ---- a b c d - --- a b c d
--R 4862 4862 4862 442
--R +
--R 45 4 6 4 6 9 3 7 5 5 21 2 8 6 4 45 9 7 3 45 10 8 2
--R - --- a b c d - -- a b c d - -- a b c d - -- a b c d - -- b c d
--R 442 34 34 34 17
--R *
--R 2
--R x
--R +
--R 1 9 10 1 8 2 9 5 7 3 2 8 5 6 4 3 7
--R - ----- a b d - ---- a b c d - ---- a b c d - ---- a b c d
--R 43758 4862 4862 1326
--R +
--R 5 5 5 4 6 1 4 6 5 5 7 3 7 6 4 5 2 8 7 3
--R - --- a b c d - -- a b c d - --- a b c d - -- a b c d
--R 442 34 102 34
--R +
--R 5 9 8 2 5 10 9
--R - -- a b c d - - b c d
--R 17 9
--R *
--R x
--R +
--R 1 10 10 1 9 9 5 8 2 2 8 5 7 3 3 7
--R - ------ a d - ----- a b c d - ----- a b c d - ----- a b c d
--R 831402 92378 92378 25194
--R +
--R 5 6 4 4 6 1 5 5 5 5 7 4 6 6 4 5 3 7 7 3
--R - ---- a b c d - --- a b c d - ---- a b c d - --- a b c d
--R 8398 646 1938 646
--R +
--R 5 2 8 8 2 5 9 9 1 10 10
--R - --- a b c d - --- a b c d - -- b c
--R 323 171 19
--R /
--R 30 19 29 18 2 28 17 3 27 16 4 26 15
--R b x + 19a b x + 171a b x + 969a b x + 3876a b x
--R +
--R 5 25 14 6 24 13 7 23 12 8 22 11
--R 11628a b x + 27132a b x + 50388a b x + 75582a b x
--R +
--R 9 21 10 10 20 9 11 19 8 12 18 7
--R 92378a b x + 92378a b x + 75582a b x + 50388a b x
--R +
--R 13 17 6 14 16 5 15 15 4 16 14 3 17 13 2
--R 27132a b x + 11628a b x + 3876a b x + 969a b x + 171a b x
--R +
--R 18 12 19 11
--R 19a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 892
--S 893 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 92378b d x + (- 92378a b d - 831402b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 75582a b d - 680238a b c d - 3401190b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 50388a b d - 453492a b c d - 2267460a b c d - 8314020b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 27132a b d - 244188a b c d - 1220940a b c d - 4476780a b c d
--R +
--R 10 4 6
--R - 13430340b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 11628a b d - 104652a b c d - 523260a b c d - 1918620a b c d
--R +
--R 9 4 6 10 5 5
--R - 5755860a b c d - 14965236b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 3876a b d - 34884a b c d - 174420a b c d - 639540a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 1918620a b c d - 4988412a b c d - 11639628b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 969a b d - 8721a b c d - 43605a b c d - 159885a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 479655a b c d - 1247103a b c d - 2909907a b c d - 6235515b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 171a b d - 1539a b c d - 7695a b c d - 28215a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 84645a b c d - 220077a b c d - 513513a b c d - 1100385a b c d
--R +
--R 10 8 2
--R - 2200770b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7 5 5 4 6
--R - 19a b d - 171a b c d - 855a b c d - 3135a b c d - 9405a b c d
--R +
--R 4 6 5 5 3 7 6 4 2 8 7 3 9 8 2
--R - 24453a b c d - 57057a b c d - 122265a b c d - 244530a b c d
--R +
--R 10 9
--R - 461890b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - a d - 9a b c d - 45a b c d - 165a b c d - 495a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2
--R - 1287a b c d - 3003a b c d - 6435a b c d - 12870a b c d
--R +
--R 9 9 10 10
--R - 24310a b c d - 43758b c
--R /
--R 30 19 29 18 2 28 17 3 27 16
--R 831402b x + 15796638a b x + 142169742a b x + 805628538a b x
--R +
--R 4 26 15 5 25 14 6 24 13
--R 3222514152a b x + 9667542456a b x + 22557599064a b x
--R +
--R 7 23 12 8 22 11 9 21 10
--R 41892683976a b x + 62839025964a b x + 76803253956a b x
--R +
--R 10 20 9 11 19 8 12 18 7
--R 76803253956a b x + 62839025964a b x + 41892683976a b x
--R +
--R 13 17 6 14 16 5 15 15 4
--R 22557599064a b x + 9667542456a b x + 3222514152a b x
--R +
--R 16 14 3 17 13 2 18 12 19 11
--R 805628538a b x + 142169742a b x + 15796638a b x + 831402a b
--R Type: Union(Expression(Integer),...)
--E 893
--S 894 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 894
--S 895 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 895
)clear all
--S 896 of 2952
t0000:=(c+d*x)^10/(a+b*x)^21
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 21 21 20 20 2 19 19 3 18 18 4 17 17
--R b x + 21a b x + 210a b x + 1330a b x + 5985a b x
--R +
--R 5 16 16 6 15 15 7 14 14 8 13 13
--R 20349a b x + 54264a b x + 116280a b x + 203490a b x
--R +
--R 9 12 12 10 11 11 11 10 10 12 9 9
--R 293930a b x + 352716a b x + 352716a b x + 293930a b x
--R +
--R 13 8 8 14 7 7 15 6 6 16 5 5 17 4 4
--R 203490a b x + 116280a b x + 54264a b x + 20349a b x + 5985a b x
--R +
--R 18 3 3 19 2 2 20 21
--R 1330a b x + 210a b x + 21a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 896
--S 897 of 2952
r0000:=-1/20*(b*c-a*d)^10/(b^11*(a+b*x)^20)-_
10/19*d*(b*c-a*d)^9/(b^11*(a+b*x)^19)-_
5/2*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^18)-_
120/17*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^17)-_
105/8*d^4*(b*c-a*d)^6/(b^11*(a+b*x)^16)-_
84/5*d^5*(b*c-a*d)^5/(b^11*(a+b*x)^15)-_
15*d^6*(b*c-a*d)^4/(b^11*(a+b*x)^14)-_
120/13*d^7*(b*c-a*d)^3/(b^11*(a+b*x)^13)-_
15/4*d^8*(b*c-a*d)^2/(b^11*(a+b*x)^12)-_
10/11*d^9*(b*c-a*d)/(b^11*(a+b*x)^11)-1/10*d^10/(b^11*(a+b*x)^10)
--R
--R
--R (2)
--R 1 10 10 10 1 9 10 10 10 9 9
--R - -- b d x + (- -- a b d - -- b c d )x
--R 10 11 11
--R +
--R 3 2 8 10 15 9 9 15 10 2 8 8
--R (- -- a b d - -- a b c d - -- b c d )x
--R 44 22 4
--R +
--R 6 3 7 10 60 2 8 9 30 9 2 8 120 10 3 7 7
--R (- --- a b d - --- a b c d - -- a b c d - --- b c d )x
--R 143 143 13 13
--R +
--R 3 4 6 10 30 3 7 9 15 2 8 2 8 60 9 3 7 10 4 6 6
--R (- --- a b d - --- a b c d - -- a b c d - -- a b c d - 15b c d )x
--R 143 143 13 13
--R +
--R 6 5 5 10 12 4 6 9 6 3 7 2 8 24 2 8 3 7 9 4 6
--R - --- a b d - --- a b c d - -- a b c d - -- a b c d - 6a b c d
--R 715 143 13 13
--R +
--R 84 10 5 5
--R - -- b c d
--R 5
--R *
--R 5
--R x
--R +
--R 3 6 4 10 15 5 5 9 15 4 6 2 8 15 3 7 3 7
--R - ---- a b d - --- a b c d - --- a b c d - -- a b c d
--R 1144 572 104 26
--R +
--R 15 2 8 4 6 21 9 5 5 105 10 6 4
--R - -- a b c d - -- a b c d - --- b c d
--R 8 4 8
--R *
--R 4
--R x
--R +
--R 3 7 3 10 15 6 4 9 15 5 5 2 8 30 4 6 3 7
--R - ---- a b d - ---- a b c d - --- a b c d - --- a b c d
--R 4862 2431 442 221
--R +
--R 15 3 7 4 6 21 2 8 5 5 105 9 6 4 120 10 7 3
--R - -- a b c d - -- a b c d - --- a b c d - --- b c d
--R 34 17 34 17
--R *
--R 3
--R x
--R +
--R 1 8 2 10 5 7 3 9 5 6 4 2 8 5 5 5 3 7
--R - ---- a b d - ---- a b c d - --- a b c d - --- a b c d
--R 9724 4862 884 221
--R +
--R 5 4 6 4 6 7 3 7 5 5 35 2 8 6 4 20 9 7 3 5 10 8 2
--R - -- a b c d - -- a b c d - -- a b c d - -- a b c d - - b c d
--R 68 34 68 17 2
--R *
--R 2
--R x
--R +
--R 1 9 10 5 8 2 9 5 7 3 2 8 10 6 4 3 7
--R - ----- a b d - ----- a b c d - ---- a b c d - ---- a b c d
--R 92378 46189 8398 4199
--R +
--R 5 5 5 4 6 7 4 6 5 5 35 3 7 6 4 40 2 8 7 3
--R - --- a b c d - --- a b c d - --- a b c d - --- a b c d
--R 646 323 646 323
--R +
--R 5 9 8 2 10 10 9
--R - -- a b c d - -- b c d
--R 19 19
--R *
--R x
--R +
--R 1 10 10 1 9 9 1 8 2 2 8 1 7 3 3 7
--R - ------- a d - ------ a b c d - ----- a b c d - ---- a b c d
--R 1847560 184756 33592 8398
--R +
--R 1 6 4 4 6 7 5 5 5 5 7 4 6 6 4 2 3 7 7 3
--R - ---- a b c d - ---- a b c d - ---- a b c d - --- a b c d
--R 2584 6460 2584 323
--R +
--R 1 2 8 8 2 1 9 9 1 10 10
--R - -- a b c d - -- a b c d - -- b c
--R 76 38 20
--R /
--R 31 20 30 19 2 29 18 3 28 17 4 27 16
--R b x + 20a b x + 190a b x + 1140a b x + 4845a b x
--R +
--R 5 26 15 6 25 14 7 24 13 8 23 12
--R 15504a b x + 38760a b x + 77520a b x + 125970a b x
--R +
--R 9 22 11 10 21 10 11 20 9 12 19 8
--R 167960a b x + 184756a b x + 167960a b x + 125970a b x
--R +
--R 13 18 7 14 17 6 15 16 5 16 15 4
--R 77520a b x + 38760a b x + 15504a b x + 4845a b x
--R +
--R 17 14 3 18 13 2 19 12 20 11
--R 1140a b x + 190a b x + 20a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 897
--S 898 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 184756b d x + (- 167960a b d - 1679600b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 125970a b d - 1259700a b c d - 6928350b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 77520a b d - 775200a b c d - 4263600a b c d - 17054400b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 38760a b d - 387600a b c d - 2131800a b c d - 8527200a b c d
--R +
--R 10 4 6
--R - 27713400b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 15504a b d - 155040a b c d - 852720a b c d - 3410880a b c d
--R +
--R 9 4 6 10 5 5
--R - 11085360a b c d - 31039008b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 4845a b d - 48450a b c d - 266475a b c d - 1065900a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 3464175a b c d - 9699690a b c d - 24249225b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 1140a b d - 11400a b c d - 62700a b c d - 250800a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4 10 7 3
--R - 815100a b c d - 2282280a b c d - 5705700a b c d - 13041600b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 190a b d - 1900a b c d - 10450a b c d - 41800a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 135850a b c d - 380380a b c d - 950950a b c d - 2173600a b c d
--R +
--R 10 8 2
--R - 4618900b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 20a b d - 200a b c d - 1100a b c d - 4400a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 14300a b c d - 40040a b c d - 100100a b c d - 228800a b c d
--R +
--R 9 8 2 10 9
--R - 486200a b c d - 972400b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - a d - 10a b c d - 55a b c d - 220a b c d - 715a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2
--R - 2002a b c d - 5005a b c d - 11440a b c d - 24310a b c d
--R +
--R 9 9 10 10
--R - 48620a b c d - 92378b c
--R /
--R 31 20 30 19 2 29 18 3 28 17
--R 1847560b x + 36951200a b x + 351036400a b x + 2106218400a b x
--R +
--R 4 27 16 5 26 15 6 25 14
--R 8951428200a b x + 28644570240a b x + 71611425600a b x
--R +
--R 7 24 13 8 23 12 9 22 11
--R 143222851200a b x + 232737133200a b x + 310316177600a b x
--R +
--R 10 21 10 11 20 9 12 19 8
--R 341347795360a b x + 310316177600a b x + 232737133200a b x
--R +
--R 13 18 7 14 17 6 15 16 5
--R 143222851200a b x + 71611425600a b x + 28644570240a b x
--R +
--R 16 15 4 17 14 3 18 13 2
--R 8951428200a b x + 2106218400a b x + 351036400a b x
--R +
--R 19 12 20 11
--R 36951200a b x + 1847560a b
--R Type: Union(Expression(Integer),...)
--E 898
--S 899 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 899
--S 900 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 900
)clear all
--S 901 of 2952
t0000:=(c+d*x)^10/(a+b*x)^22
--R
--R
--R (1)
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R d x + 10c d x + 45c d x + 120c d x + 210c d x + 252c d x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210c d x + 120c d x + 45c d x + 10c d x + c
--R /
--R 22 22 21 21 2 20 20 3 19 19 4 18 18
--R b x + 22a b x + 231a b x + 1540a b x + 7315a b x
--R +
--R 5 17 17 6 16 16 7 15 15 8 14 14
--R 26334a b x + 74613a b x + 170544a b x + 319770a b x
--R +
--R 9 13 13 10 12 12 11 11 11 12 10 10
--R 497420a b x + 646646a b x + 705432a b x + 646646a b x
--R +
--R 13 9 9 14 8 8 15 7 7 16 6 6
--R 497420a b x + 319770a b x + 170544a b x + 74613a b x
--R +
--R 17 5 5 18 4 4 19 3 3 20 2 2 21 22
--R 26334a b x + 7315a b x + 1540a b x + 231a b x + 22a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 901
--S 902 of 2952
r0000:=-1/21*(b*c-a*d)^10/(b^11*(a+b*x)^21)-_
1/2*d*(b*c-a*d)^9/(b^11*(a+b*x)^20)-_
45/19*d^2*(b*c-a*d)^8/(b^11*(a+b*x)^19)-_
20/3*d^3*(b*c-a*d)^7/(b^11*(a+b*x)^18)-_
210/17*d^4*(b*c-a*d)^6/(b^11*(a+b*x)^17)-_
63/4*d^5*(b*c-a*d)^5/(b^11*(a+b*x)^16)-_
14*d^6*(b*c-a*d)^4/(b^11*(a+b*x)^15)-_
60/7*d^7*(b*c-a*d)^3/(b^11*(a+b*x)^14)-_
45/13*d^8*(b*c-a*d)^2/(b^11*(a+b*x)^13)-_
5/6*d^9*(b*c-a*d)/(b^11*(a+b*x)^12)-1/11*d^10/(b^11*(a+b*x)^11)
--R
--R
--R (2)
--R 1 10 10 10 5 9 10 5 10 9 9
--R - -- b d x + (- -- a b d - - b c d )x
--R 11 66 6
--R +
--R 15 2 8 10 15 9 9 45 10 2 8 8
--R (- --- a b d - -- a b c d - -- b c d )x
--R 286 26 13
--R +
--R 30 3 7 10 30 2 8 9 180 9 2 8 60 10 3 7 7
--R (- ---- a b d - -- a b c d - --- a b c d - -- b c d )x
--R 1001 91 91 7
--R +
--R 2 4 6 10 2 3 7 9 12 2 8 2 8 9 3 7 10 4 6 6
--R (- --- a b d - -- a b c d - -- a b c d - 4a b c d - 14b c d )x
--R 143 13 13
--R +
--R 3 5 5 10 3 4 6 9 9 3 7 2 8 3 2 8 3 7 21 9 4 6
--R - --- a b d - -- a b c d - -- a b c d - - a b c d - -- a b c d
--R 572 52 26 2 4
--R +
--R 63 10 5 5
--R - -- b c d
--R 4
--R *
--R 5
--R x
--R +
--R 15 6 4 10 15 5 5 9 45 4 6 2 8 15 3 7 3 7
--R - ---- a b d - --- a b c d - --- a b c d - -- a b c d
--R 9724 884 442 34
--R +
--R 105 2 8 4 6 315 9 5 5 210 10 6 4
--R - --- a b c d - --- a b c d - --- b c d
--R 68 68 17
--R *
--R 4
--R x
--R +
--R 5 7 3 10 5 6 4 9 5 5 5 2 8 5 4 6 3 7
--R - ----- a b d - ---- a b c d - --- a b c d - -- a b c d
--R 14586 1326 221 51
--R +
--R 35 3 7 4 6 35 2 8 5 5 140 9 6 4 20 10 7 3
--R - --- a b c d - -- a b c d - --- a b c d - -- b c d
--R 102 34 51 3
--R *
--R 3
--R x
--R +
--R 5 8 2 10 5 7 3 9 15 6 4 2 8 5 5 5 3 7
--R - ----- a b d - ---- a b c d - ---- a b c d - --- a b c d
--R 92378 8398 4199 323
--R +
--R 35 4 6 4 6 105 3 7 5 5 140 2 8 6 4 20 9 7 3 45 10 8 2
--R - --- a b c d - --- a b c d - --- a b c d - -- a b c d - -- b c d
--R 646 646 323 19 19
--R *
--R 2
--R x
--R +
--R 1 9 10 1 8 2 9 3 7 3 2 8 1 6 4 3 7
--R - ------ a b d - ----- a b c d - ---- a b c d - --- a b c d
--R 184756 16796 8398 646
--R +
--R 7 5 5 4 6 21 4 6 5 5 14 3 7 6 4 2 2 8 7 3
--R - ---- a b c d - ---- a b c d - --- a b c d - -- a b c d
--R 1292 1292 323 19
--R +
--R 9 9 8 2 1 10 9
--R - -- a b c d - - b c d
--R 38 2
--R *
--R x
--R +
--R 1 10 10 1 9 9 1 8 2 2 8 1 7 3 3 7
--R - ------- a d - ------ a b c d - ----- a b c d - ----- a b c d
--R 3879876 352716 58786 13566
--R +
--R 1 6 4 4 6 1 5 5 5 5 2 4 6 6 4 2 3 7 7 3
--R - ---- a b c d - ---- a b c d - --- a b c d - --- a b c d
--R 3876 1292 969 399
--R +
--R 3 2 8 8 2 1 9 9 1 10 10
--R - --- a b c d - -- a b c d - -- b c
--R 266 42 21
--R /
--R 32 21 31 20 2 30 19 3 29 18 4 28 17
--R b x + 21a b x + 210a b x + 1330a b x + 5985a b x
--R +
--R 5 27 16 6 26 15 7 25 14 8 24 13
--R 20349a b x + 54264a b x + 116280a b x + 203490a b x
--R +
--R 9 23 12 10 22 11 11 21 10 12 20 9
--R 293930a b x + 352716a b x + 352716a b x + 293930a b x
--R +
--R 13 19 8 14 18 7 15 17 6 16 16 5
--R 203490a b x + 116280a b x + 54264a b x + 20349a b x
--R +
--R 17 15 4 18 14 3 19 13 2 20 12 21 11
--R 5985a b x + 1330a b x + 210a b x + 21a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 902
--S 903 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 10 10 10 9 10 10 9 9
--R - 352716b d x + (- 293930a b d - 3233230b c d )x
--R +
--R 2 8 10 9 9 10 2 8 8
--R (- 203490a b d - 2238390a b c d - 13430340b c d )x
--R +
--R 3 7 10 2 8 9 9 2 8 10 3 7 7
--R (- 116280a b d - 1279080a b c d - 7674480a b c d - 33256080b c d )x
--R +
--R 4 6 10 3 7 9 2 8 2 8 9 3 7
--R - 54264a b d - 596904a b c d - 3581424a b c d - 15519504a b c d
--R +
--R 10 4 6
--R - 54318264b c d
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 7 2 8 2 8 3 7
--R - 20349a b d - 223839a b c d - 1343034a b c d - 5819814a b c d
--R +
--R 9 4 6 10 5 5
--R - 20369349a b c d - 61108047b c d
--R *
--R 5
--R x
--R +
--R 6 4 10 5 5 9 4 6 2 8 3 7 3 7
--R - 5985a b d - 65835a b c d - 395010a b c d - 1711710a b c d
--R +
--R 2 8 4 6 9 5 5 10 6 4
--R - 5990985a b c d - 17972955a b c d - 47927880b c d
--R *
--R 4
--R x
--R +
--R 7 3 10 6 4 9 5 5 2 8 4 6 3 7
--R - 1330a b d - 14630a b c d - 87780a b c d - 380380a b c d
--R +
--R 3 7 4 6 2 8 5 5 9 6 4
--R - 1331330a b c d - 3993990a b c d - 10650640a b c d
--R +
--R 10 7 3
--R - 25865840b c d
--R *
--R 3
--R x
--R +
--R 8 2 10 7 3 9 6 4 2 8 5 5 3 7
--R - 210a b d - 2310a b c d - 13860a b c d - 60060a b c d
--R +
--R 4 6 4 6 3 7 5 5 2 8 6 4 9 7 3
--R - 210210a b c d - 630630a b c d - 1681680a b c d - 4084080a b c d
--R +
--R 10 8 2
--R - 9189180b c d
--R *
--R 2
--R x
--R +
--R 9 10 8 2 9 7 3 2 8 6 4 3 7
--R - 21a b d - 231a b c d - 1386a b c d - 6006a b c d
--R +
--R 5 5 4 6 4 6 5 5 3 7 6 4 2 8 7 3
--R - 21021a b c d - 63063a b c d - 168168a b c d - 408408a b c d
--R +
--R 9 8 2 10 9
--R - 918918a b c d - 1939938b c d
--R *
--R x
--R +
--R 10 10 9 9 8 2 2 8 7 3 3 7 6 4 4 6
--R - a d - 11a b c d - 66a b c d - 286a b c d - 1001a b c d
--R +
--R 5 5 5 5 4 6 6 4 3 7 7 3 2 8 8 2
--R - 3003a b c d - 8008a b c d - 19448a b c d - 43758a b c d
--R +
--R 9 9 10 10
--R - 92378a b c d - 184756b c
--R /
--R 32 21 31 20 2 30 19 3 29 18
--R 3879876b x + 81477396a b x + 814773960a b x + 5160235080a b x
--R +
--R 4 28 17 5 27 16 6 26 15
--R 23221057860a b x + 78951596724a b x + 210537591264a b x
--R +
--R 7 25 14 8 24 13 9 23 12
--R 451151981280a b x + 789515967240a b x + 1140411952680a b x
--R +
--R 10 22 11 11 21 10 12 20 9
--R 1368494343216a b x + 1368494343216a b x + 1140411952680a b x
--R +
--R 13 19 8 14 18 7 15 17 6
--R 789515967240a b x + 451151981280a b x + 210537591264a b x
--R +
--R 16 16 5 17 15 4 18 14 3
--R 78951596724a b x + 23221057860a b x + 5160235080a b x
--R +
--R 19 13 2 20 12 21 11
--R 814773960a b x + 81477396a b x + 3879876a b
--R Type: Union(Expression(Integer),...)
--E 903
--S 904 of 2952
m0000:=a0000 - r0000
--R
--R
--R (4) 0
--R Type: Expression(Integer)
--E 904
--S 905 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 905
)clear all
--S 906 of 2952
t0000:=(a+b*x)^m/(c+d*x)
--R
--R
--R m
--R (b x + a)
--R (1) ----------
--R d x + c
--R Type: Expression(Integer)
--E 906
--S 907 of 2952
--r0000:=(a+b*x)^(1+m)*_
-- hypergeometric(1,1+m,2+m,-d*(a+b*x)/(b*c-a*d))/((b*c-a*d)*(1+m))
--E 907
--S 908 of 2952
--a0000:=integrate(t0000,x)
--E 908
--S 909 of 2952
--m0000:=a0000 - r0000
--E 909
--S 910 of 2952
--d0000:=D(m0000,x)
--E 910
)clear all
--S 911 of 2952
t0000:=(a+b*x)^5/(c+d*x)
--R
--R
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R b x + 5a b x + 10a b x + 10a b x + 5a b x + a
--R (1) --------------------------------------------------
--R d x + c
--R Type: Fraction(Polynomial(Integer))
--E 911
--S 912 of 2952
r0000:=b*(b*c-a*d)^4*x/d^5-1/2*(b*c-a*d)^3*(a+b*x)^2/d^4+_
1/3*(b*c-a*d)^2*(a+b*x)^3/d^3-1/4*(b*c-a*d)*(a+b*x)^4/d^2+_
1/5*(a+b*x)^5/d-(b*c-a*d)^5*log(c+d*x)/d^6
--R
--R
--R (2)
--R 5 5 4 4 3 2 2 3 2 3 3 2 4 4
--R 60a d - 300a b c d + 600a b c d - 600a b c d + 300a b c d
--R +
--R 5 5
--R - 60b c
--R *
--R log(d x + c)
--R +
--R 5 5 5 4 5 5 4 4
--R 12b d x + (75a b d - 15b c d )x
--R +
--R 2 3 5 4 4 5 2 3 3
--R (200a b d - 100a b c d + 20b c d )x
--R +
--R 3 2 5 2 3 4 4 2 3 5 3 2 2
--R (300a b d - 300a b c d + 150a b c d - 30b c d )x
--R +
--R 4 5 3 2 4 2 3 2 3 4 3 2 5 4 5 5
--R (300a b d - 600a b c d + 600a b c d - 300a b c d + 60b c d)x + 77a d
--R +
--R 4 4 3 2 2 3 2 3 3 2
--R - 145a b c d + 110a b c d - 30a b c d
--R /
--R 6
--R 60d
--R Type: Expression(Integer)
--E 912
--S 913 of 2952
a0000:=integrate(t0000,x)
--R
--R
--R (3)
--R 5 5 4 4 3 2 2 3 2 3 3 2 4 4
--R 60a d - 300a b c d + 600a b c d - 600a b c d + 300a b c d
--R +
--R 5 5
--R - 60b c
--R *
--R log(d x + c)
--R +
--R 5 5 5 4 5 5 4 4
--R 12b d x + (75a b d - 15b c d )x
--R +
--R 2 3 5 4 4 5 2 3 3
--R (200a b d - 100a b c d + 20b c d )x
--R +
--R 3 2 5 2 3 4 4 2 3 5 3 2 2
--R (300a b d - 300a b c d + 150a b c d - 30b c d )x
--R +
--R 4 5 3 2 4 2 3 2 3 4 3 2 5 4
--R (300a b d - 600a b c d + 600a b c d - 300a b c d + 60b c d)x
--R /
--R 6
--R 60d
--R Type: Union(Expression(Integer),...)
--E 913
--S 914 of 2952
m0000:=a0000 - r0000
--R
--R
--R 5 3 4 2 3 2 2 2 3 3
--R - 77a d + 145a b c d - 110a b c d + 30a b c
--R (4) ----------------------------------------------
--R 4
--R 60d
--R Type: Expression(Integer)
--E 914
--S 915 of 2952
d0000:=D(m0000,x)
--R
--R
--R (5) 0
--R Type: Expression(Integer)
--E 915
)clear all
--S 916 of 2952
t0000:=(a+b*x)^4/(c+d*x)
--R
--R
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R (1) --------------------------------------
--R d x + c
--R Type: Fraction(Polynomial(Integer))
--E 916
--S 917 of 2952
r0000:=-b*(b*c-a*d)^3*x/d^4+1/2*(b*c-a*d)^2*(a+b*x)^2/d^3-_
1/3*(b*c-a*d)*(a+b*x)^3/d^2+1/4*(a+b*x)^4/d+(b*c-a*d)^4*log(c+d*x)/d^5
--R
--R
--R (2)
--R 4 4 3 3 2 2 2 2 3 3 4 4
--R (12a d - 48a b c d + 72a b c d - 48a b c d + 12b c )log(d x + c)
--R +
--R 4 4 4 3 4 4 3 3 2 2 4 3 3 4 2 2 2
--R 3b d x + (16a b d - 4b c d )x + (36a b d - 24a b c d + 6b c d )x
--R +
--R 3 4 2 2 3 3 2 2 4 3 4 4 3 3
--R (48a b d - 72a b c d + 48a b c d - 12b c d)x + 13a d - 16a b c d
--R +
--R 2 2 2 2
--R 6a b c d
|
https://share.cocalc.com/share/5d54f9d642cd3ef1affd88397ab0db616c17e5e0/www/tables/serremodpq/serremodpq.aux?viewer=raw
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Star& P(hours) &Puls.&$B$&$V$&$B$--$V$& obs. & obs. & obs. & obs. \cr
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[Fe/H] &P(day)&Mode&$B_0$&$V_0$&($B$--$V$)$_0$&pred.&pred.&pred.&pred.\cr
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Draco-V157 &22.476 &F& 19.24& 18.90& 0.34& 14.55& 14.33&--0.43&--0.65\cr
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Sculptor-V119 &27.6 &F& 19.03& - & - & 14.39& - &--0.48& - \cr
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0.02 &"0.061 & & - & - & & "0.05& - & & \cr
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Leo~II-V203 &31.75 &F& 20.55& - & - & 14.21& - &--0.56& - \cr
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Sculptor-V26=B &32.30 &F& 18.96& 18.75& 0.21& 14.32& 14.16&--0.55&--0.72\cr
--1.7 &"1.346 & & 18.88& 18.69& 0.19& 14.12& 13.88&--0.75&--1.00\cr
0.02 &"0.129 & & "1.2"& "0.8"& & "0.20& "0.28& & \cr
\noalign{\vskip 4pt}
Leo~II-V27 &37.18 &F& 20.24& - & - & 13.90& - &--0.87& - \cr
--1.4 &"1.549 & & 20.20& - & - & 13.93& - &--0.84& - \cr
0.01 &"0.190 & & "1.64& - & &--0.03& - & & \cr
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Draco-V194 &38.1631 &F& 18.53& 18.13& 0.40& 13.84& 13.56&--1.14&--1.42\cr
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% \multispan{10}{The numbers in this table are from Run~109. \hfil }\cr
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}$$
\bye
\end
SMC-V58 (G75) &10.226 &H& 18.50& - & - & & - & & - \cr
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SMC-V15 (G75) &12.6 &F& 19.35& - & - &15.24& - & "0.76& - \cr
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--1.4 &"0.58232 & & 18.63& - & - &14.57& - &--0.20& - \cr
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SMC-V82 (G75) &15.3 &H& 18.53& - & - &14.42& - &--0.35& - \cr
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"0.06$\pm$0.03 &--0.194 & & "1.0"& - & &--0.02& - & & \cr
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SMC-V67 (G75) &18.46 &H& 18.48& - & - &14.37& - &--0.40& - \cr
--1.4 &"0.76923 & & 18.32& - & - &14.18& - &--0.58& - \cr
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SMC-V29 (G75) &34.3 &F& 18.28& - & - & 14.17& - &--0.60& - \cr
--1.4 &"1.42957 & & 18.12& - & - & 14.02& - &--0.75& - \cr
"0.06$\pm$0.03 &"0.155 & & "1.0"& - & & "0.15& - & & \cr
\noalign{\vskip 4pt}
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https://git.rockbox.org/cgit/rockbox.git/plain/manual/appendix/wps_tags.tex?id=8ac46f844f437a31f72d73a41a6d3852d8b96143
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% $Id$ %
\chapter{\label{ref:wps_tags}Theme Tags}
Themeing is discussed in detail in section \reference{ref:ConfiguringtheWPS},
what follows is a list of the available tags.
\note{The ``bar-type tags'' (such as \%pb, \%pv, \%bl etc.) can be further
themed -- see \reference{ref:bar_tags}.}
\section{Status Bar}
\begin{tagmap}
\config{\%we} & Display Status Bar\\
\config{\%wd} & Hide Status Bar\\
\config{\%wi} & Display the inbuilt Status Bar in the current viewport\\
\end{tagmap}
These tags override the player setting for the display of the status bar.
They must be noted on their own line (which will not be shown in the WPS).
\section{Hardware Capabilities}
\begin{tagmap}
\config{\%cc} & Check for presence of a real time clock, returns ``c''
when used unconditionally\\
\config{\%tp} & Does this target have a radio?\\
\config{\%Tp} & Indicates that the target has a touchscreen\\
\end{tagmap}
With the above tags it is possible to find out about the presence of certain
hardware and make the theme adapt to it. This can be very useful for designing
a theme that works on multiple targets with differing hardware capabilities, e.g.
targets that do and do not have a clock. When used conditionally, the ``true''
branch is completely ignored if it does not apply.
Example:
\config{\%?cc<\%cH:\%cM|No clock detected>}
\section{Information from the track tags}
\begin{tagmap}
\config{\%ia} & Artist\\
\config{\%ic} & Composer\\
\config{\%iA} & Album Artist\\
\config{\%id} & Album Name\\
\config{\%iG} & Grouping\\
\config{\%ig} & Genre Name\\
\config{\%in} & Track Number\\
\config{\%it} & Track Title\\
\config{\%iC} & Comment\\
\config{\%iv} & ID3 version (1.0, 1.1, 2.2, 2.3, 2.4, or empty if not an ID3 tag)\\
\config{\%iy} & Year\\
\config{\%ik} & Disc Number\\
\end{tagmap}
Remember that this information is not always available, so use the
conditionals to show alternate information in preference to assuming.
These tags, when written with a capital ``I'' (e.g. \config{\%Ia} or \config{\%Ic}),
show the information for the next song to be played.
\section{Viewports}
\begin{tagmap}
\nopt{lcd_non-mono}{%
\config{\%V(x,y,[width],\tabnlindent[height],[font])}
& See section \ref{ref:Viewports}\\}
\nopt{lcd_color}{\opt{lcd_non-mono}{%
\config{\%V(x,y,[width],\tabnlindent[height],[font])}\newline
\config{\%Vf([fgshade])}\newline
\config{\%Vb([bgshade])}
& See section \ref{ref:Viewports}\\}}
\opt{lcd_color}{%
\config{\%V(x,y,[width],\tabnlindent[height],[font])}\newline
\config{\%Vf([fgcolour])}\newline
\config{\%Vb([bgcolour])}\newline
\config{\%Vg(start,end \tabnlindent[,text])}
& See section \ref{ref:Viewports}\\}
\opt{lcd_non-mono}{%
\config{\%Vs(mode[,param])}
& See section \ref{ref:Viewports}\\}
\config{\%Vl('identifier',\newline\dots)} & Preloads a viewport for later
display. `identifier' is a single lowercase letter (a-z) and the `\dots'
parameters use the same logic as the \%V tag explained above.\\
\config{\%Vd('identifier')} & Display the `identifier' viewport. E.g.
\config{\%?C<\%Vd(a)|\%Vd(b)>}
will show viewport `a' if album art is found, and `b' if it isn't.\\
\config{\%Vi('label',\dots)} &
Declare a Custom UI Viewport. The `\dots' parameters use the same logic as
the \config{\%V} tag explained above. See section \ref{ref:Viewports}.\\
\config{\%VI('label')} & Set the Info Viewport to use the viewport called
label, as declared with the previous tag.\\
\config{\%VB} & Draw this viewport on the backdrop layer.\\
\end{tagmap}
\section{Additional Fonts}
\begin{tagmap}
\config{\%Fl('id',filename)} & See section \ref{ref:multifont}.\\
\end{tagmap}
\section{Misc Coloring Tags}
\begin{tagmap}
\config{\%dr(x,y,width,height,[color1,color2])} & Color a rectangle. \\
\end{tagmap}
width and height can be ``$-$'' to fill the viewport. If no color is
specified the viewports foreground color will be used. If two
colors are specified it will do a gradient fill.
\section{Power Related Information}
\begin{tagmap}
\config{\%bl} & Numeric battery level in percents.
Can also be used in a conditional:
\config{\%?bl<-1|0|1|2|\ldots|N>},
where the value $-1$ is used when the battery level isn't
known (it usually is). The value $N$ is only used when the
battery level is exactly 100 percent.
An image can also be used, the proportion of the image
shown corresponds to the battery level:
\config{\%bl(x,y,[width],[height],image.bmp)}\\
\config{\%bv} & The battery level in volts\\
\config{\%bt} & Estimated battery time left\\
\config{\%bp} & ``p'' if the charger is connected (only on targets
that can charge batteries)\\
\config{\%bc} & ``c'' if the unit is currently charging the battery (only on
targets that have software charge control or monitoring)\\
\config{\%bs} & Remaining time of the sleep timer (if it is set)\\
\end{tagmap}
\section{Information about the file}
\begin{tagmap}
\config{\%fb} & File Bitrate (in kbps)\\
\config{\%fc} & File Codec (e.g. ``MP3'' or ``FLAC'').
This tag can also be used in a conditional tag:
\config{\%?fc<mp1|mp2|mp3|aiff|wav|ogg|\newline
flac|mpcsv7|a52|wavpack|alac|aac|shn|sid|adx|nsf|\newline
speex|spc|ape|wma|wmpapro|mod|sap|realaudiocook|\newline
realaudioaac|realaudioac3|realaudioatrac3|cmc|\newline
cm3|cmr|cms|dmc|dlt|mpt|mpd|rmt|tmc|tm8|tm2|\newline
omaatrac3|smaf|au|vox|wave64|tta|wmavoice|mpcsv8|\newline
aache|ay|gbs|hes|sgc|vgm|kss|unknown>}.
The codec order is as shown above.\\
\config{\%ff} & File Frequency (in Hz)\\
\config{\%fk} & File Frequency (in kHz)\\
\config{\%fm} & File Name\\
\config{\%fn} & File Name (without extension)\\
\config{\%fp} & File Path\\
\config{\%fs} & File Size (in Kilobytes)\\
\config{\%fv} & ``(avg)'' if variable bit rate or empty string if constant bit rate\\
\config{\%d(N)} & N-th segment from the end of the file's directory
(N can be 1, 2, 3, \dots)\\
\end{tagmap}
Example for the \config{\%d(N)} commands: If the file is
``\fname{/Rock/Kent/Isola/11 - 747.mp3}'', \config{\%d(1)} is ``\fname{Isola}'',
\config{\%d(2)} is ``\fname{Kent}'' and \config{\%d(3)} is ``\fname{Rock}''.
These tags, when written with the first letter capitalized (e.g. \config{\%Fn}
or \config{\%D(2)}), produce the information for the next file to be played.
\section{Playlist/Song Info}
\begin{tagmap}
\config{\%pb} & Progress Bar.
\opt{lcd_bitmap}{
This will replace the entire line with a progress bar.
You can set the position, width and height of the progressbar
(in pixels) and load a custom image for it:
\config{\%pb(x,y,[width],[height],image.bmp)}} \\
\config{\%px} & Percentage played in song\\
\config{\%pc} & Current time in song\\
\config{\%pe} & Total number of playlist entries\\
\config{\%pm} & Peak Meter. The entire line is used as %
volume peak meter.\\%
\config{\%pL} & Peak meter for the left channel. Can be used as a value, %
a conditional tag or a bar tag.\\
\config{\%pR} & Peak meter for the right channel. Can be used as a value, %
a conditional tag or a bar tag.\\
\config{\%pn} & Playlist name (without path or extension)\\
\config{\%pp} & Playlist position\\
\config{\%pr} & Remaining time in song\\
\config{\%ps} & ``s'' if shuffle mode is enabled\\
\config{\%pt} & Total track time\\
\config{\%pv} & Current volume (in dB). Can also be used in a conditional:
\config{\%?pv<Mute|\ldots|0 dB|Above 0 dB>}\newline
Mute is 0\% volume, \ldots is the values between Mute and max, 0 dB is max volume, and Above 0 dB is amplified volume
\opt{lcd_bitmap}{This can also be used like \%pb to provide a continuous scale:
\config{\%pv(x,y,[width],[height],image.bmp)}} \\
\config{\%pS} & Track is starting. An optional number gives how many seconds
the tag remains true for after the start of the track. The default is
10 seconds if no number is specified.
\config{\%?pS(7)<in the first 7 seconds of track|in
the rest of the track>}\\
\config{\%pE} & Track is ending. An optional number gives how many seconds
before the end of the track the tag becomes true. The default is
10 seconds if no number is specified.
\config{\%?pE(7)<in the last 7 seconds of track|in
the rest of the track>}\\
\config{\%Sp} & Current playback pitch\\
\end{tagmap}
\section{Playlist Viewer}
\begin{tagmap}
\config{\%Vp(start,code to render)} & Display the playlist viewer in
the current viewport.\\
\end{tagmap}
\begin{itemize}
\item `start' is the offset relative to the currently playing track for the
playlist to display from (0 the current track, 1 is the next track, etc.).
\item `code to render' is a line of skin code which will be displayed for
each line in the viewer. All text tags are supported (including conditionals
and sublines)
\end{itemize}
The entire viewport will be used, so don't expect other tags in the same
viewport to work well. Supported tags are \%pp, all tags starting with \%i,
most tags starting with \%f, \%pt and \%s.\\
Example: \config{\%Vp(1,\%pp - \%it,\%pp - \%fn)} -- Display the playlist
position, then either the track title (from the tags) or
the filename. The viewer will display as many tracks as will fit in the
viewport.
\section{Runtime Database}
\begin{tagmap}
\config{\%rp} & Song playcount\\
\config{\%rr} & Song rating (0-10). This tag can also be used in a conditional tag: %
\config{\%?rr<0|1|2|3|4|5|6|7|8|9|10>}\\
\config{\%ra} & Autoscore for the song\\
\end{tagmap}
\opt{swcodec}{
\section{Sound (DSP) settings}
\begin{tagmap}
\config{\%Sp} & Current playback pitch \\
\opt{swcodec}{
\config{\%xf} & Crossfade setting, in the order: Off, Auto Skip, Man Skip,
Shuffle, Shuffle and Man Skip, Always\\
\config{\%rg} & ReplayGain value in use (x.y~dB). If used as a conditional,
Replaygain type in use: \config{\%?rg<Off|Track%
|Album|TrackShuffle|AlbumShuffle%
|No tag>}\\
}
\end{tagmap}
}
\section{Hold}
\begin{tagmap}
\config{\%mh} & ``h'' if the main unit keys are locked\\
\opt{remote_button_hold}{%
\config{\%mr} & ``r'' if the remote keys are locked\\
}
\end{tagmap}
\section{Virtual LED}
\begin{tagmap}
\config{\%lh} & ``h'' if the \disk\ is accessed\\
\end{tagmap}
\section{Repeat Mode}
\begin{tagmap}
\config{\%mm} & Repeat mode, 0-4, in the order: Off, All, One, Shuffle, A-B\\
\end{tagmap}
Example: \config{\%?mm<Off|All|One|Shuffle|A-B>}
\section{Playback Mode}
\begin{tagmap}
\config{\%mp} & Play status, 0-4, in the order: Stop, Play, Pause,
Fast Forward, Rewind, Recording, Recording paused, FM Radio playing,
FM Radio muted\\
\end{tagmap}
Example: \config{\%?mp<Stop|Play|Pause|Ffwd|Rew|Rec|Rec pause|FM|FM pause>}
\section{Current Screen}
\begin{tagmap}
\config{\%cs} & The current screen, 1-20, in the order shown below\\
\end{tagmap}
\begin{table}
\begin{rbtabular}{.75\textwidth}{lX}%
{\textbf{Number} & \textbf{Screen}}{}{}
1 & Menus \\
2 & WPS \\
3 & Recording screen \\
4 & FM Radio screen \\
5 & Current Playlist screen \\
6 & Settings menus \\
7 & File browser \\
8 & Database browser \\
9 & Plugin browser \\
10 & Quickscreen \\
11 & Pitchscreen \\
12 & Setting chooser \\
13 & Playlist Catalogue Viewer \\
14 & Plugin \\
15 & Context menu \\
16 & System Info screen \\
17 & Time and Date Screen \\
18 & Bookmark browser \\
19 & Shortcuts menu \\
20 & Track Info screen \\
\end{rbtabular}
\end{table}
The tag can also be used as the switch in a conditional tag. For players without
certain capabilities (e.g. no FM radio) some values will never be returned.
Examples:
\config{You are in the \%?cs<Main menu|WPS|Recording screen|FM Radio screen>}
\config{\%?if(\%cs, =, 2)<This is the WPS>}
\section{List Title (\fname{.sbs} only)}
\begin{tagmap}
\config{\%Lt} & Title text. Should be used in a conditional so that non-list
screens don't show a title when they shouldn't\\
\config{\%Li} & Title icon. This uses the same order as custom icons (see
\wikilink{CustomIcons} in the wiki) except that here \config{0} is ``no
icon''\\
\end{tagmap}
This tag can be used to give custom formatting to list titles.
Define a viewport with the font and formatting desired, and then use
\config{\%?Lt<\%Lt>} to display the title within the
viewport. If \config{\%Lt} is present anywhere in the \fname{.sbs}, then the
\config{\%Vi} viewport will not show the title.
\section{Changing Volume}
\begin{tagmap}
\config{\%mv(t)} & ``v'' if the volume is being changed\\
\end{tagmap}
The tag produces the letter ``v'' while the volume is being changed and some
amount of time after that, i.e. after the volume button has been released. The
optional parameter \config{t} specifies that amount of time, in seconds. If it
is not specified, 1 second is assumed.
The tag can be used as the switch in a conditional tag to display different things
depending on whether the volume is being changed. It can produce neat effects
when used with conditional viewports.
Example: \config{\%?mv(2.5)<Volume changing|\%pv>}
The example above will display the text ``Volume changing'' if the volume is
being changed and 2.5 seconds after the volume button has been released. After
that, it will display the volume value.
\section{Settings}
\begin{tagmap}
\config{\%St(<setting\tabnlindent{}name>)} & The value of the Rockbox
setting with the specified name. See \reference{ref:config_file_options}
for the list of the available settings.\\
\config{\%St(...)} & Draw a bar using from the setting.
See \reference{ref:bar_tags} for details.\\
\end{tagmap}
Examples:
\begin{enumerate}
\item As a simple tag: \config{\%St(skip length)}
\item As a conditional: \config{\%?St(eq enabled)<Eq is enabled|Eq is disabled>}
\end{enumerate}
\opt{lcd_bitmap}{
\section{\label{ref:wps_images}Images}
\begin{tagmap}
\config{\%X(filename.bmp)}
& Load and set a backdrop image for the WPS.
This image must be exactly the same size as your LCD.\\
}%
\config{\%x(n,filename[,x,y])}
& Load and display an image\newline
\config{n}: image ID for later referencing in \config{\%xd}\newline
\config{filename}: file name relative to \fname{/.rockbox/} and including ``.bmp''\newline
\config{x}: x coordinate (defaults to 0 if both x and y are not specified)\newline
\config{y}: y coordinate. (defaults to 0 if both x and y are not specified)\\
\config{\%xl(n,filename,[x,y],\tabnlindent[nimages])}
& Preload an image for later display (useful for when your images are displayed conditionally).\newline
\config{n}: image ID for later referencing in \config{\%xd}\newline
\config{filename}: file name relative to \fname{/.rockbox/} and including ``.bmp''\newline
If the filename is ``\_\_list\_icons\_\_'' the list icon bitmap will be used instead\newline
\config{x}: x coordinate (defaults to 0 if both x and y are not specified)\newline
\config{y}: y coordinate. (defaults to 0 if both x and y are not specified)\newline
\config{nimages}: (optional) number of sub-images (tiled vertically, of the same height)
contained in the bitmap. Default is 1.\\
\config{\%xd(n[i] [,tag] [,offset])} & Display a preloaded image.
\config{n}: image ID as it was specified in \config{\%x} or \config{\%xl}\newline
\config{i}: (optional) number of the sub-image to display (a-z for 1-26 and A-Z for 27-52).
(ignored when \config{tag} is used). Only useable if the ID is a single letter.
By default the first (i.e. top most) sub-image will be used.\newline
\config{tag}: (optional) Another tag to calculate the subimage from e.g \config{\%xd(A, \%mh)} would
use the first subimage when \config{\%mh} is on and the second when it is off\newline
\config{offset}: (optional) Add this number to the value from the \config{tag} when
chosing the subimage (may be negative)\\
\config{\%x9(n)}
& Display an image as a 9-patch bitmap covering the entire viewport.\newline
9-patch images are bitmaps split into 9 segments where the four corners
are unscaled, the four middle sections are scaled along one axis and the middle
section is scaled on both axis.\newline
\config{n}: image ID\\
\end{tagmap}
Examples:
\begin{enumerate}
\item Load and display the image \fname{/.rockbox/bg.bmp} with ID ``a'' at 37, 109:\\
\config{\%x(a,bg.bmp,37,109)}
\item Load a bitmap strip containing 5 volume icon images (all the same size)
with image ID ``M'', and then reference the individual sub-images in a conditional:\\
\config{\%xl(M,volume.bmp,134,153,5)}\\
\config{\%?pv<\%xd(Ma)|\%xd(Mb)|\%xd(Mc)|%
\%xd(Md)|\%xd(Me)>}
\end{enumerate}
\note{
\begin{itemize}
\item The images must be in BMP format
\item The image tag must be on its own line
\item The ID is case sensitive
\item The size of the LCD screen for each \dap{} varies. See table below
for appropriate sizes of each device. The x and y coordinates must
respect each of the \daps{} limits.
\end{itemize}
}
\opt{albumart}{
\subsection{How to display the album art}
Once the album art files are present on your \dap, they can be displayed as
follows.
\begin{tagmap}
\config{\%Cl(x,y,[maxwidth],\tabnlindent[maxheight],\tabnlindent{}hor\_align,\tabnlindent{}vert\_align)}
& Define the settings for album art\newline
\config{x}: x coordinate\newline
\config{y}: y coordinate\newline
\config{maxwidth}: Maximum height\newline
\config{maxheight}: Maximum width\newline
\config{hor\_align}: Horizontal alignment, enter as `l', `c' or `r' for
left, centre or right. Centre is default\newline
\config{vert\_align}: Vertical alignment, enter as `t', `c' or `b' for
top, centre or bottom. Centre is default\\
\config{\%Cd} & Display the album art as configured. \\
\config{\%C} & Use in a conditional to determine if an image is available. \\
\end{tagmap}
The picture will be rescaled, preserving aspect ratio to fit the given
\config{maxwidth} and \config{maxheight}. If the aspect ratio doesn't match the
configured values, the picture will be placed according to the alignment flags.
Examples:
\begin{enumerate}
\item Load albumart at position 20,40 and display it without resizing:\\
\config{\%Cl(20,40,,)}
\item Load albumart at position 0,20 and resize it to be at most 100$\times$100
pixels. If the image isn't square, align it to the bottom-right
corner:\\
\config{\%Cl(0,20,100,100,r,b)}
\end{enumerate}
For general information where to put album art see \reference{ref:album_art}.
}
\opt{radio}{
\section{FM Radio}
\begin{tagmap}
\config{\%tt} & Is the tuner tuned?\\
\config{\%tm} & Scan or preset mode? Scan is ``true'', preset is ``false''.\\
\config{\%ts} & Is the station in stereo?\\
\config{\%ta} & Minimum frequency (region specific) in MHz.\\
\config{\%tb} & Maximum frequency (region specific) in MHz.\\
\config{\%tf} & Current frequency in MHz.\\
\config{\%Ti} & Current preset id, i.e. 1-based number of the preset
within the presets list (usable in playlist viewer).\\
\config{\%Tn} & Current preset name (usable in playlist viewer).\\
\config{\%Tf} & Current preset frequency (usable in playlist viewer).\\
\config{\%Tc} & Preset count, i.e. the number of stations in the current
preset list.\\
\config{\%tx} & Is RDS available?\\
\config{\%ty} & RDS name.\\
\config{\%tz} & RDS text.\\
\config{\%tr} & Signal strength (RSSI). Can be used in a conditional or as
a bar.\\
\end{tagmap}
It is also possible to show ``Radio Art'' which can be used to display images
associated with presets. The tags are exactly the same as for album art,
described above. Images need to be placed in \fname{/.rockbox/fmpresets/},
and must have the same name as the preset. They need to be in either
\fname{.bmp} or \fname{.jpg} format, and the radio must be in preset mode
and tuned to a preset (and not recording) in order for them to be shown.
}
\section{Alignment and language direction}
\begin{tagmap}
\config{\%al} & Align the text left\\
\config{\%aL} & Align the text left, or to the right if RTL language is in use\\
\config{\%ac} & Centre the text\\
\config{\%ar} & Align the text right\\
\config{\%aR} & Align the text right, or to the left if RTL language is in use\\
\config{\%ax} & The next tag should follow the set language direction. When
prepended to a viewport declaration, the viewport will
be horizontally mirrored if the user language is set to
a RTL language. Currently the \%Cl, \%V and \%Vl tags
support this.\\
\config{\%Sr} & Use as a conditional to define options for left to right, or
right to left languages. \%?Sr<RTL|LTR>\\
\end{tagmap}
All alignment tags may be present in one line, but they need to be in the
order left -- centre -- right. If the aligned texts overlap, they are merged.
Example: \config{\%ax\%V(\dots)}
\section{Conditional Tags}
\begin{tagmap}
\config{\%?xx<true|false>}
& If / Else: Evaluate for true or false case \\
\config{\%?xx<alt1|alt2|\tabnlindent{}alt3|\dots|else>}
& Enumerations: Evaluate for first / second / third / \dots / last condition \\
\config{\%if(tag, operator, operand, [option count])}
& Allows very simple comparisons with other tags.\newline
\config{tag}: the tag to check against.\newline
\config{operator}: the comparison to perform -- possible options are =, !=,
>, >=, <, <=\newline
\config{operand}: either a second tag, a number, or text.\newline
\config{[option count]}: optional parameter used to select which parameter
of a tag to use when the tag has multiple options, e.g. \%?pv<a|b|c|d>\\
\config{\%and(tag1, tag2, ..., tagN)}\newline
& Logical ``and'' operator. Will be evaluate to true if all the tag parameters are true.\\
\config{\%or(tag1, tag2, ..., tagN)}\newline
& Logical ``or'' operator. Will be evaluate to true if any of the tag parameters are true.\\
\end{tagmap}
Examples of the \%if tag:\\
\config{\%?if(\%pv, >=, 0)<Clipping possible|Volume OK>} will display ``Clipping
possible'' if the volume is higher than or equal to 0 dB, ``Volume OK'' if it
is lower.\\
\config{\%?if(\%ia, =, \%Ia)<same artist>} -- this artist and the next artist
are the same.\\
\note{When performing a comparison against a string tag such as \%ia, only = and
!= work, and the comparison is not case sensitive.}
\section{Subline Tags}
\begin{tagmap}
\config{\%t(time)}
& Set the subline display cycle time (\%t(5) or \%t(3.4) formats) \\
\config{;}
& Split items on a line into separate sublines \\
\end{tagmap}
Allows grouping of several items (sublines) onto one line, with the
display cycling round the defined sublines. See
\reference{ref:AlternatingSublines} for details.
\opt{rtc}{
\section{Time and Date}
\begin{tagmap}
\config{\%cd} & Day of month from 01 to 31\\
\config{\%ce} & Zero padded day of month from 1 to 31\\
\config{\%cf} & A conditional for 12/24 hour format.\newline
\config{\%?cf<24 hour stuff|12 hour stuff>}\\
\config{\%cH} & Zero padded hour from 00 to 23 (24 hour format)\\
\config{\%ck} & Hour from 0 to 23 (24 hour format)\\
\config{\%cI} & Zero padded hour from 01 to 12 (am/pm format)\\
\config{\%cl} & Hour from 1 to 12 (am/pm format)\\
\config{\%cm} & Month from 01 to 12\\
\config{\%cM} & Minutes\\
\config{\%cS} & Seconds\\
\config{\%cy} & 2-digit year\\
\config{\%cY} & 4-digit year\\
\config{\%cP} & Capital AM/PM\\
\config{\%cp} & Lowercase am/pm\\
\config{\%ca} & Weekday name\\
\config{\%cb} & Month name\\
\config{\%cu} & Day of week from 1 to 7, 1 is Monday\\
\config{\%cw} & Day of week from 0 to 6, 0 is Sunday\\
\end{tagmap}
}
\section{Text Translation}
\begin{tagmap}
\config{\%Sx(English)}
& Display the translation of ``English'' in the current language\\
\end{tagmap}
\begin{itemize}
\item ``English'' must be a phrase used in the language file.
\item It should match the \config{Source:} line in the language file.
\end{itemize}
\note{checkwps cannot verify that the string is correct, so please check on
either the simulator or on target.}
\opt{touchscreen}{
\section{Touchscreen Areas}
\begin{tagmap}
\config{\%T(x,y,width,\tabnlindent{}height, action, [options])}
& Invoke the action specified when the user presses in the defined
touchscreen area.\\
\end{tagmap}
Possible actions are:
\begin{description}
\item[none] -- Do nothing.
\item[play] -- Play/pause playback.
\item[stop] -- Stop playback and exit the WPS.
\item[prev] -- Previous track/item.
\item[next] -- Next track/item.
\item[wps\_prev] -- Previous track.
\item[wps\_next] -- Next track.
\item[ffwd] -- Seek forwards in the track.
\item[rwd] -- Seek backwards in the track.
\item[progressbar] -- Seek to the appropriate position in the track based on the touch.
\item[shuffle] -- Toggle shuffle mode.
\item[repmode] -- Cycle through the repeat modes.
\item[volume] -- Set the volume to the appropriate level based on the touch.
\item[voldown] -- Decrease the volume by one step.
\item[volup] -- Increase the volume by one step.
\item[mute] -- Un/Mute playback.
\item[createbookmark] -- Create a bookmark in the currently-playing track.
\item[hotkey] -- Performs the action assigned to the hotkey (see Hotkeys section).
\item[menu] -- Go to the main menu.
\item[browse] -- Go back to the file browser or database.
\item[resumeplayback] -- Go back to the last music screen (WPS or radio screen).
\item[quickscreen] -- Go to the quickscreen.
\item[contextmenu] -- Open the context menu.
\item[playlist] -- Go to the playlist viewer.
\item[listbookmarks] -- List the bookmarks for the currently-playing directory or playlist.
\item[trackinfo] -- Open the track info viewer.
\item[pitch] -- Open the pitchscreen.
\item[setting\_inc] -- Increment the subsequently specified setting (e.g
\config{\%T(0,0,32,32, setting\_inc, volume)} increases the volume by one step).
\item[setting\_dec] -- Decrement the subsequently specified setting (e.g
\config{\%T(0,0,32,32, setting\_dec, volume)} decreases the volume by one step).
\item[setting\_set] -- Set the subsequently specified setting to a specific value (e.g
\config{\%T(0,0,32,32, setting\_set, volume, 0)} sets the volume to 0).
\item[lock] -- Soft locks the touchscreen. All touch areas are disabled except for
areas with the lock action or ones that have the allow\_while\_locked option (see below).
\end{description}
Any (or muliple) of the following options can be used after the action is specified
\subsection{Options}
\begin{description}
\item[repeat\_press] -- This area will trigger mulitple times when held (i.e for seeking)
\item[long\_press] -- This area will trigger once after it is held for a long press
\item[reverse\_bar] -- Reverse the bars touch direction (i.e seek right to left)
\item[allow\_while\_locked] -- Allows the area to be pressable when the
skin is locked by the lock touch action
\end{description}
\section{Last Touchscreen Press}
\begin{tagmap}
\config{\%Tl} & Indicates that the touchscreen is pressed.\\
\end{tagmap}
This tag can be used to display text or images or a viewport when the
touchscreen is pressed (like an On Screen Display). If you put a number
straight after \%Tl it will be used as a timeout in seconds
(e.g \%Tl(2.5) will give a 2.5 second timeout) between the touchscreen press
being released and the tag going false. If no number is specified it will
use a 1 second timeout. It can also be used as a conditional, and can be
used with conditional viewports.
}
\section{\label{ref:bar_tags}Bar Tags}
Some tags can be used to display a bar which draws according to the value of
the tag. To use these tags like a bar you need to use the following parameters
(\%XX should be replaced with the actual tag).
\opt{touchscreen}{
Volume and progress bars automatically create touch regions the same size
as the bar (slightly larger actually). This can be disabled with the
\config{notouch} option.
}
\begin{tagmap}
\config{\%XX(x, y, width, height, [options])}
& Draw the specified tag as a bar\newline
\config{x}: x co-ordinate at which to start drawing the bar.\newline
\config{y}: y co-ordinate at which to start drawing the bar (- to make the
bar appear on the line of the tag, as if it was a text tag) .\newline
\config{width}: width of the bar (- for the full viewport width).\newline
\config{height}: height of the bar (- to set to the font height for
horizontal bars and to the viewport height for vertical bars).\newline
\config{options}: any of the options set out below.\\
\end{tagmap}
\subsection{Options}
\begin{description}
\item[image] -- the next option is either the filename or image label to
use for the fill image.
\item[horizontal] -- force the bar to be drawn horizontally.
\item[vertical] -- force the bar to be drawn vertically.
\item[invert] -- invert the draw direction (i.e. right to left, or top to
bottom).
\item[slider] -- draw a preloaded image over the top of the bar so that
the centre of the image matches the current position. This must be
followed by the label of the desired image.
\item[backdrop] -- draw a preloaded image under the bar. The full
image will be displayed and must be the same size as the bar.
This must be followed by the label of the desired image.
\item[nofill] -- don't draw the bar, only its frame (for use with the
``slider'' option).
\item[noborder] -- don't draw the border for image-less bars, instead maximise
the filling over the specified area. This doesn't work for bars which
specify an image.
\item[nobar] -- don't draw the bar or its frame (for use with the
``slider'' option).
\opt{touchscreen}{
\item[notouch] -- don't create the touchregion for progress/volume bars.
}
\item[setting] -- Specify the setting name to draw the bar from (bar must be
\%St type), the next param is the settings config name.
\end{description}
Example: \config{\%pb(0,0,-,-,-,nofill, slider, slider\_image, invert)} -- draw
a horizontal progressbar which doesn't fill and draws the image
``slider\_image'' which moves right to left.
\note{If the slider option is used, the bar will be shrunk so that the slider
fits inside the specified width and height. Example: A 100px bar image with a
16px slider image needs the bar to be 116px wide, and should be offset 8px
left of the backdrop image to align correctly.}
\section{Other Tags}
\begin{tagmap}
\config{\%ss(start, length, tag [,number]} & Get a substring from another tag.\\
\end{tagmap}
Use this tag to get a substring from another tag.
\begin{description}
\item[start] -- first character to take (0 being the start of the string, negative means from the end of the string)
\item[length] -- length of the substring to return (- for the rest of the string)
\item[tag] -- tag to get
\item[number] -- OPTIONAL. if this is present it will assume the
substring is a number so it can be used with conditionals. (i.e \config{\%cM}).
0 is the first conditional option
\end{description}
\begin{tagmap}
\config{\%(} & The character `('\\
\config{\%)} & The character `)'\\
\config{\%,} & The character `,'\\
\config{\%\%} & The character `\%'\\
\config{\%<} & The character `<'\\
\config{\%|} & The character `|'\\
\config{\%>} & The character `>'\\
\config{\%;} & The character `;'\\
\config{\%\#} & The character `\#'\\
\config{\%s} & Indicate that the line should scroll. Can occur
anywhere in a line (given that the text is
displayed; see conditionals above). You can specify
up to ten scrolling lines. Scrolling lines can not
contain dynamic content such as timers, peak meters
or progress bars.\\
\end{tagmap}
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\section*{Conference Proceedings}
\subsection*{2009}
Kinshuk, C., M., Graf, S., \& Yang, G. (2009). Adaptivity and Personalization in Mobile Learning. In \textit{Proceedings of the American Educational Research Association Annual Meeting (AERA 2009)}. San Diego, CA , USA.: Technology, Instruction, Cognition and Learning Special Interest Group, Session on The Future of Adaptive Tutoring and Personalized Instruction,.
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\title{W(i)ara od Kościoła}
\date{16 października 2020}
\author{Kamil Kupiński}
\subtitle{}
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Oficjalnie według statystyk prawie wszyscy w tym kraju są katolikami nawet na lewicy. Możemy oczywiście krzyczeć i tupać, że Kościół jest zły, księża to pedofile i do tego kradną, ale potem przyjdzie się zmierzyć z rzeczywistością i wyborcami. Pora by lewica stanęła w jednym szeregu z Kościołem!
\section{Słowo wyjaśnienia}
No dobra, trochę przesadziłem. Obiecuję, że się to powtórzy jeszcze nie raz, ale jakoś trzeba Was zachęcić do kliknięcia. O co zatem chodzi w tym \emph{clickbaitowym} tytule? Na pewno nie o to, żeby wybielać złe uczynki polskiego Kościoła Rzymsko-Katolickiego i zachęcać do zawierania jakiegoś wyraźnego sojuszu, bo aż tak cyniczny, to ja nie jestem.
Jakbyśmy źle nie myśleli o Kościele, czego byśmy sobie nie wyobrażali o nim i jakich byśmy klątw na nich nie rzucali, to jedna rzecz jest niezaprzeczalna – nawet pomimo pandemii Kościół w Polsce to jest siła. Kościół nie ogranicza się do świątyń i klasztorów. Przedstawiciele tego związku wyznaniowego mają w swoim posiadaniu majątki ziemskie, szkoły, uniwersytety, biznesy, media głównego nurtu i regionalne i pewnie jeszcze parę rzeczy, o których nie pamiętam albo nie wiem. Poza tym jest też sporo podobnych instytucji, których nie mają na własność, ale mają w nich wpływy, które przy okazji rozciągają się też na ludzi władzy (każdej władzy).
Jak już powiedzieliśmy sobie na głos, z czym mamy do czynienia, to łatwiej Wam przyjdzie zrozumienie mojego stanowiska. Będziemy tu rozmawiać o grze o władzę, a ta jest podła, okrutna, cyniczna i ogólnie nie lubi zasad. Na naszej planszy nie znajdują się jednak same partie polityczne. Jednym z graczy jest właśnie polski Kościół Katolicki, który, choć sam bezpośrednio wygrać nie może, to ma niebagatelny wpływ na końcowy wynik i jeśli nie jest możliwe, żeby był naszym sojusznikiem, to jest wysoce porządane, żeby zachował wobec nas neutralność. Powiem Wam to wprost – pójście na czołówkę z Kościołem w Polsce to gwarancja przegranej. Właściwie każda władza od trzydziestu lat zginała tu nogi przed biskupami, włączając w to SLD, które tylko na czas kampanii wypuszczało Joannę Senyszyn, żeby podłapać głosy antyklerykałów. Czy to oznacza, że dla władzy lewicowi politycy też powinni uklęknąć?
Bynajmniej. Trzeba do tego podejść ze sprytem.
\section{Co ludzie myślą?}
Sam Kościół jest oczywiście potężną organizacją, ale jest to tylko połowa jego siły. Drugą połowę stanowią jego wyznawcy. No może „wyznawcy” to za mocne słowo, biorąc pod uwagę to, jak Polki i Polacy podchodzą do religii, ale jest ich mnóstwo. Oficjalna statystyka, według której prawie wszyscy jesteśmy katolikami, informuje jedynie o stanie formalnym, ale mówi też jak bardzo jesteśmy zdeterminowani, żeby zdławić wpływy Kościoła. Zacznijmy jednak od tego, czym jest Kościół dla ludzi.
Dla jednych to przyzwyczajenie, dla innych konieczność ze względu na presję społeczną, a dla innych spełnieniem potrzeb duchowych lub wspólnotowych. To ostatnie jest szczególnie ważne dla osób starszych w czasach kapitalizmu i silnego indywidualizmu, zwłaszcza gdy Kościół jest jedyną organizacją wychodzącą do nich z ofertą. Niezależnie jednak od tych motywacji, prawie każdy przypadek jest podszyty, chociaż tą minimalną ilością wiary, która trzyma ich przy Kościele. Poza tym dla nich jest to coś naturalnego, oczywistego. Chodzili, chodzą i będą chodzić do kościoła, bo przeważnie nic złego dla nich z tego nie wynika.
Postawię śmiałą tezę, bardzo odważną, być może dla niektórych nawet zbyt skrajną, że stosunek większości ludzi do Kościoła warunkują nie hierarchowie, a księża z ich parafii, których mają na co dzień. A gdyby się zdarzyło, że wszyscy lokalni księża irytowali? Zawsze zostaje radio albo telewizja i jakoś uda się utrzymać łączność z Panem Bogiem. W każdym razie ludzie mają dość szeroki wybór sposobów uczestniczenia, zanim postanowią być wierzącymi niepraktykującymi (o apostazji nawet nie marząc). Proces laicyzacji społeczeństwa co prawda trwa, ale w obecnych warunkach samoistnie będzie trwał jeszcze długie lata, zanim Kościół w odczuwalny sposób utraci na znaczeniu, o ile w ogóle to jest możliwe bez ingerencji.
Wiemy już jak szeroką bazę odbiorców posiada Kościół, ale co nam to daje? Po pierwsze możemy uzmysłowić sobie, jak dużo ludzi\dots{} hmm\dots{} Powiedzieć, że ma poglądy konserwatywne, to chyba trochę za dużo. Powiedzmy, że dużo ludzi ma w sobie pewną dozę konserwatyzmu. Niby wszyscy to wiedzą i jest to raczej truizm, ale mimo to sporo lewicowców zdaje się to wypierać i idzie w zaparte, nie próbując dopasować przekazu (nie programu) do wyborców, których jest więcej niż te niemożliwe 3 do 5 procent. Tutaj Was pewnie zaskoczę, ale nie namawiam po raz kolejny do walki o elektorat PiSu. Ludzie, których tu opisuje są tak samo wyborcami PiSu, PO, PSLu, a nawet SLD. Wbrew pozorom to nie społeczeństwo jest konserwatywne, bo partie są konserwatywne, ale jest na odwrót. Oczywiście ta czy inna partia, docierając do kolejnych wyborców może ich bardziej wpychać w ramiona konserwatyzmu, ale jest to efekt sprzężenia zwrotnego. Potem partie chcą być jeszcze bardziej konserwatywne i kółko się nakręca (albo zamienia w cyrk, patrzcie na PO).
Kolejną rzeczą, jaką możemy i musimy sobie uświadomić jest to, w jakiej skali i jak sprawnie trafia przekaz Kościoła do ludzi. Podpowiadam – w dużej skali i bardzo sprawnie. Przy obecnej zdolności lewicy w docieraniu do ludzi i bez własnych mediów walka ze ściemą serwowaną przez Kościół jest daremna. Nie do przecenienia jest tu efekt kotwiczenia i przewaga tworzenia pierwszej wersji historii. Przy takim rozdaniu kart ludzie lewicy zawsze będą skazani na odkłamywanie, co samo w sobie jest trudne, a przy małej liczebności i bez adekwatnych środków niemożliwym staje się skuteczne działanie. Czy zatem da się coś zrobić w takich warunkach?
\section{Graj muzyko}
Wielu zapewne nie spodoba się (zaskoczenie) to, co za chwilę przedstawię. Ja sam nie wiem, czy to ma w ogóle szanse powodzenia, ale mam silne przeczucie, że obecna strategia (albo właściwie jej brak) nie daje szans na uzyskanie oczekiwanych rezultatów.
W związku z tym poniżej powstała lista zmian, jakie według mnie musiałyby zajść, żeby to Kościół został wyrolowany z parkietu w tym piekielnym tańcu:
\section{1. Ludzie Kościoła to też\dots{} ludzie}
Dobra, wiem, znowu truizm, ale problem polega na tym, że rzadko kiedy widzę taką świadomość u polityków, aktywistów czy przeciętnych użytkowników mediów społecznościowych, którzy lokują się po lewej stronie. Póki co widzę, że większość obsiada pozycje walczących z organizacją Zło. Nie jest to w sumie zbyt dalekie od prawdy, ale wbrew wyobrażeniom wiecznych wojowniczek i wojowników ludzie Kościoła nie postrzegają się tak. Ludzie ogólnie są dobrzy albo przynajmniej wydaje im się, że tacy są i myślą, że postępują słusznie. Niestety mamy dzisiaj tak spolaryzowane społeczeństwo, że nawet szczera próba dyskusji jest odbierana jak atak, a ponieważ każdy myśli, że jest dobry, to ci drudzy automatycznie stają się źli i potem przychodzi jeszcze jakiś buc, żeby podgrzać atmosferę. W związku z tym wszystkim myślę, że trzymanie w głowie tego truizmu z punktu pierwszego znacznie ułatwiłoby nam tworzenie czegoś konstruktywnego.
\section{2. Zwijamy front}
Systemowe krycie pedofilii, kradzieże, wyłudzenia, homofobia, rasizm, faworyzowanie przez państwo i zapewne wiele innych rzeczy, które możemy znaleźć w Kościele, a które to sprawiają, że nas krew zalewa i najchętniej przejechalibyśmy go walcem. Byłoby miło, ale w tej grze to Kościół jest nawet nie tyle walcem ile armią walców. Co prawda – nigdy tego nie widziałem, ale gdyby mała grupa ludzi ot, tak o spróbowała zatrzymać rząd walców, to walce najpewniej pojechałyby dalej. Co innego gdyby zaczęli przemykać dołami, rozpraszać walce, kopać większe doły i eliminować walce pojedynczo lub napuszczać je na siebie.
Dla wielu z Was będzie to rozczarowanie, ale nowa strategia wymagałaby schowania do szuflady pomysłów z zerwaniem konkordatu, usunięciem Funduszu Kościelnego czy nawet odzyskaniem majątku rozdanego w wyniku działań Komisji Likwidacyjnej, nie wspominając już o tych prawdziwie radykalnych. Jak bardzo słuszne by one jednak nie były i jak bardzo motywowane chęcią niesienia sprawiedliwości, to potraktowane będą jako wypowiedzenie otwartej wojny i spotkają się z adekwatną odpowiedzią. Powiem Wam nawet, że te pomysły, mimo iż słuszne, to są w pewien sposób niesprawiedliwe, ale o tym przeczytacie trochę dalej. Póki co zajmijmy się czymś bardziej przyziemnym.
Kojarzycie, gdy jakiś działacz trzeciej kategorii, piąty asystent trzeciego asystenta czwartorzędnego polityka niebędącego nawet w radzie krajowej partii powie coś głupiego, a potem prawica oczekuje przeprosin od całego zarządu, a, wy mówicie, że to jest głupie i najwyżej ta osoba powinna przeprosić? No to macie rację, ale to działa też w drugą stronę, gdy jakiś proboszcz z parafii w Piździszewie Dolnym też powie coś głupiego, to powinni się tym zająć najwyżej lokalni działacze, a wywieranie presji na wysokie władzę Kościoła, pompowanie całej aferki do wyższej rangi i oskarżanie całego Kościoła o odpowiedzialność za to jest równie głupie. Oczywiście podkoloryzowałem oba przypadki, ale tryb działania jest raczej znajomy. Mało tego. Powiem Wam, że ograniczenie „działań wojennych” jest wskazane nawet wtedy, gdy mówi to ktoś w randze arcybiskupa. Wszystkim pewnie przyszedł teraz na myśl znany i kochany arcybiskup Jędraszewski.
Jakby ktoś jakimś cudem nie pamiętał, to ten od tęczowej zarazy. Tak, nawet tu nie ma atakować całego Kościoła. Takie zdania jak „Arcybiskup Jędraszewski tymi słowami przynosi hańbę nie tylko najwyższej władzy Kościoła, ale też tysiącom księży codziennie niosącym wśród swoich parafian Dobrą Nowinę. Arcybiskup powinien przeprosić nie tylko adresatów swojej wypowiedzi, ale też swoje siostry i braci w wierze.” jednocześnie potępiają zachowanie Jędraszewskiego, nie rozpętują wojny z całym Kościołem, a także podsycają wewnętrzne podziały. Atak powinien być precyzyjny i niezauważalny, żeby nikt nie mógł przeprowadzić odwetu.
\section{3. Pozytywne wibracje}
Ponownie wbrew wyobrażeniom osób antyklerykalnych nie wszystko, co robi Kościół, jest złe. Żeby daleko nie szukać, wspomnę tylko o tym jak nie tylko nie przyłączył się do rasistowskiej nagonki w trakcie kryzysu migracyjnego, ale też ją potępił, zachęcał do pomocy uchodźcom i nawet realnie im pomagał. Wtedy zetknąłem się z reakcją w stylu „no, w końcu zrobili coś dobrego” i była to reakcja błędna. Co prawda poparto słuszne działania, ale jednocześnie ustawiono się na wrogiej pozycji. Zamiast tego można by przyjąć bardziej przyjazny ton i nie tylko (nie)szczerze pogratulować, ale też zadeklarować wsparcie choćby symboliczne i podkreślić, jak wiele nas łączy. Wiem, że w praktyce jest to umacnianie pozycji Kościoła, ale w dłuższej perspektywie opłaci się, żeby zyskać przewagę i uderzyć wtedy, gdy będzie miało to szanse powodzenia. Nie jest to jednak jedyny sposób na tworzenie pozytywnych więzi z Kościołem.
Jedną z głupszych rzeczy, z jaką spotkałem się wśród młodej lewicy w internecie było naśmiewanie się ze śmierci księdza Jerzego Popiełuszki. Człowiek ten nie zrobił nic, żeby zasłużyć sobie na drwiny czy potępienie. Ba! On się właśnie świetnie nadaje do wzięcia na sztandary przez lewicę nie tylko wierzącą. Wszak stał on w jednym szeregu z robotnikami i został za to zamordowany przez reżim. Właśnie brakuje mi w lewicy grzebania w historii i przyszywania sobie postaci, jak np. trochę mniej dziś znanego księdza Stanisława Brzóski, generała powstania styczniowego działającego w powiecie łukowskim, który został powieszony ze swoim adiutantem Franciszkiem Wilczyńskim jako ostatni utrzymujący się powstańcy. Albo w ogóle sięgnąć głębiej i powiązać postępowość dzisiejszej lewicy z tym co robił Kościół zakładając pierwsze szkoły (tak, wiem, że tylko dla chłopców, ale przypominam, że to nie był XXI wiek i nie mamy na celu podsycać tu podziałów). Chcemy czy nie Kościół jest nierozerwalnie związany z naszym krajem, raz lepiej się zachowywał, raz gorzej, ale my musimy to zaakceptować i sobie z tym poradzić.
\section{Fałszywe obietnice}
Myślicie teraz pewnie o tym, co obiecamy Kościołowi, żeby finalnie wbić mu sztylet w plecy? Przepraszam Was, bo to ja Wam złożyłem fałszywą obietnicę w tym tekście, tworząc miraż pokonania Kościoła. Niestety uważam, że nie da się tego zrobić. Przynajmniej nie za naszego życia lub bez rozlewu krwi. Strategia, którą Wam przedstawiłem, ma inny cel, ale najpierw powiem Wam, co powinna dać nam wymiernie.
Przede wszystkim chciałbym, żeby Kościół przestał widzieć wroga w lewicy i zrezygnował z ataków na jej idee. Poza tym byłoby to wytrącenie oręże przeciwnikowi. Bo jak tu mówić o wrogu, skoro deklaruje się jako przyjaciel? A nawet jeśli poszczególni księża nie rezygnowaliby z wrogiej postawy, to nawet opóźniony przekaz lewicy tworzyłby wśród wiernych dysonans poznawczy. Wykazanie przyjaznych intencji przy zachowaniu zdrowego dystansu po prostu mogłoby przekonać część konserwatywnych wyborców.
Wspomniałem też wcześniej o pewnych niesprawiedliwościach wobec Kościoła w związku z pewnymi pomysłami. Warto pamiętać, że mimo wszelkich animozji i konfliktów ludzie Kościoła to tak jak my obywatelki i obywatele Polski i należy im się sprawiedliwe traktowanie. Nie jest winą księdza mieszkającego i służącego całe życie we wsi na wschodniej Lubelszczyźnie, że ustanowiono Fundusz Kościelny, dzięki któremu będzie miał emeryturę, mimo iż nigdy normalnie nie pracował. Tak samo nie jest winą sióstr zakonnych, których wtedy nawet nie było w zakonie, gdy 20 lat wcześniej poprzednia przeorysza za sprawą Komisji Likwidacyjnej nabyła ziemię i postawiła tam nowy budynek klasztorny. Gdyby teraz siostry te usłyszały, że mają się wynosić i szukać nowego miejsca zamieszkania, to byłoby zbliżone do sytuacji, w której mieszkańcy kamienicy są wydalani po reprywatyzacji. Tak samo gdy przychodzimy pracować do nowej firmy, raczej nie mamy świadomości o jej machlojkach, jesteśmy przekonani, że będziemy działać zgodnie z prawem, nasza szefowa bądź szef jest uczciwą osobą, a my poniesiemy odpowiedzialność tylko za to, w czym weźmiemy świadomy udział, gdyby było inaczej.
Trzeba tu też spojrzeć z bardziej praktycznej strony. Nawet jeśli przyjąć, że reformy te nie tylko niosłyby sprawiedliwość dziejową, ale też uznalibyśmy, że są one sprawiedliwe mimo wszystko, to w Kościele wytworzyłoby się poczucie głębokiej niesprawiedliwości. W dłuższej perspektywie przerodziłoby się to w chęć odwetu, co skutkowałoby dążeniem do zmiany władzy i wykorzystaniem pierwszej nadążającej się okazji, żeby odzyskać to co utracone.
Wracając zatem do początkowej tezy tej części tekstu, że Kościoła nie da się pokonać w jednej chwili, odpowiedzmy sobie na pytanie, czy można coś w ogóle zrobić. Tak samo jak wiemy, że nie da się z dnia na dzień pozbyć korporacji, to chcemy na początek uczynić je czymś lepszym. Można by zaproponować Kościołowi (po dojściu do władzy) szereg reform, które zapewniłyby stopniowe przejście w nowy sposób współistnienia, majątek niestety odkupić, konkordat spokojnie renegocjować punkt po punkcie, a to wszystko pod hasłem przywrócenia zdrowych relacji z Kościołem i zaufania do niego (jakby kiedykolwiek były, no ale wiecie sami). Walki frontalnej nie wygramy, ale świat możemy uczynić lepszym niż jest.
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Anarcho-Biblioteka
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Dobry pieróg to wywrotowy pieróg
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Kamil Kupiński
W(i)ara od Kościoła
16 października 2020
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https:\Slash{}\Slash{}megafon.lhub.pl\Slash{}opinie\Slash{}wiara-od-kosciola\Slash{}
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\textbf{pl.anarchistlibraries.net}
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\begin{document}
\begin{slide}
\begin{center} \bf \" Uber die lokale Struktur endlicher Gruppen
Ulrich Meierfrankenfeld
Perspektiven in Algebra und Geometrie
Erlangen
29 April 2003 \end{center}
\end{slide}
\begin{slide}
Gegeben eine endliche Gruppe $G$.
Wie kann man $G$ identifizieren?
Eine Antwort: Anhand der echten Untergruppen.
Wie findet man echten Untergruppen?
Satz von Sylow: Sei $p^k$ eine Primzahlpotenz, die $|G|$ teilt. Dann
gibt es eine Untergruppe der Ordung $p^k$ in $G$.
Weitere Untergruppen?
Sei $P$ eine $p$-Unterguppe von $G$ und
$$L=N_G(P)=\{g\in P\mid g^{-1}Pg=P\}$$
Dann hei\ss t $L$ eine $p$-{\bf lokale Untergruppe} von $G$.
\end{slide}
\begin{slide}
{\bf Beispiel}: Sei $G=GL_n(\F)$, $\F$ ein endlicher K\" orper. Au\ss erdem
sei $t\in G$ eine Involution, d.h. $|t|=2$. Dann ist
$$N_G(\<t\>)=C_G(t)=\{g\in G\mid gt=tg\}.$$
Zwei F\"alle: $\chr \F\neq 2$ und $\chr \F = 2$.\bigskip
\begin{center}\bf 1. Fall: $\chr \F \neq 2$.\end{center}
Jordansche Normalform von $t$:
$$\begin{pmatrix} I_k & 0 \\ 0 & -I_m\end{pmatrix}$$
$$C_G(t)=\{ \begin{pmatrix} A & 0 \\ 0 & B\end{pmatrix}\}$$
$ A\in
GL_k(\F),B\in GL_m(\F)$
$$C_G(t)\cong GL_k(\F)\times GL_m(\F)$$
\end{slide}
\begin{note}
$$\begin{pmatrix} I_k & 0 \\ 0 & -I_m\end{pmatrix}\cdot
\begin{pmatrix} A & C \\ D & B\end{pmatrix}=\begin{pmatrix} A & C
\\ -D &- B\end{pmatrix}$$
$$ \begin{pmatrix} A & C \\ D & B\end{pmatrix}\cdot\begin{pmatrix} I_k
& 0 \\ 0 & -I_m\end{pmatrix}=\begin{pmatrix} A & -C \\ D &-
B\end{pmatrix}$$
$$C=-C\quad,\quad D=-D$$
$$\chr \F\neq 2$$
$$C=0\quad,\quad D=0$$
\end{note}
\begin{slide}
\begin{center}\bf 2. Fall: $\chr \F = 2$.\end{center}
Jordansche Normalform von $t$:
$$\begin{pmatrix} I_k & 0 & 0 \\ 0 & I_m & 0\\ I_k & O&
I_k\end{pmatrix}\}$$
$$C_G(t)=\{ \begin{pmatrix} A & 0 &0 \\ \star & B& 0 \\ \star &\star
& A\end{pmatrix}\}$$
$A\in
GL_k(\F),B\in GL_m(\F)$
Sei $Q= \{ \begin{pmatrix} I_k & 0 &0 \\ \star & I_m& 0 \\ \star &\star
& I_k\end{pmatrix}\}$
Dann ist $Q$ ein Normalteiler von $C_G(t)$, $Q$ ist eine $2$-Gruppe und
$$C_G(T)/Q\cong GL_k(\F)\times GL_m(\F)$$
\end{slide}
\begin{slide}
In ersten Fall des vorherigen Beispiels nennt man $t$ halbeinfach, im
zweiten Fall unipotent. Gruppentheoretisch l\"a\ss t sich das wie folgt
definieren:
Sei $p$ eine Primzahl und $P$ eine $p$-Untergruppe von $G$. Setze
$L=F^*(C_G(P))$.
Ist $L$ eine $p$-Gruppe, so hei\ss t $P$ unipotent in $G$.
Ist $F(L)=Z(L)$, so hei\ss t $P$ halbeinfach in $G$.
\end{slide}
\begin{note}
Das letztere is genau dann der Fall falls es Untergruppen $L_0,L_1,\ldots,L_m$ in $ L$ mit
\bl[i]
\li a $L=L_0L_1\ldots L_m$
\li b $[L_i,L_j]=1$ f\"ur alle $0\leq i< j\leq m$.
\li c $L_0$ is abelsch.
\li d $L_i$ is quasieinfach, $1\leq i\leq m$.
\el
$K$ quasieinfach: $K$ ist perfekt und $K/Z(K)$ is einfach.
\end{note}
\begin{slide}
\begin{center}\bf Inzidenzgeometrien\end{center}
Sie $I$ eine (endliche) Menge. Eine Inzidenzgeometrie \"uber $I$
besteht aus einer Menge
$$\ca G,$$
einer Funktion
$$\tau: \ca G\to I$$
(genannt Typ)
und einer symmetrischen Relation
$$\sim$$ (genannt Inzidenz), so dass zwei Element von gleichen Typ nie
inzident sind.
{\bf Beispiel:}
Sei $V$ ein endlich dimensionaler Vektorraum, $\ca G$ die Menge der echte
Unterr\"aume, $\tau=\dim$ und $A\sim B$ falls $A< B$ oder $B<A$.
$\ca G$ ist dann also der zu $V$ geh\"orige projektive Raum $Proj(V)$.
\end{slide}
\begin{slide}
\begin{center}{\bf Die unipotente Geometrie}\end{center}
Sei $p$ eine Primzahl, so dass $G$ unipotente $p$-Untergruppen
besitzt. Eine Borelgruppe von $G$ ist der Normalizator einer
maximalen unipotenten Untergruppe von $G$.
Eine Untergruppe $L$ von $G$ hei\ss t parabolisch falls
\bl[i]
\li 1 $L=N_G(U)$ f\"ur eine unipotente Untergruppe $U$ von $G$.
\li 2 $L$ enth\"alt eine Borelgruppe von $G$.
\el
Zwei parabolische Untergruppen
hei\ss en inzident, falls ihr Durchschnitt wieder parabolisch
ist.
Die unipotente Geometrie von $G$ ist die Menge der maximalen
parabolischen Untergruppen zusammen mit der obigen Inzidenzrelation.
\end{slide}
\begin{note}
Die maximalen unipotenten
$p$-Untergruppen sind gerade die $p$-Sylowgruppen von $G$ sind.
\end{note}
\begin{slide}
{\bf Beispiel:} Sei wieder $G=GL_\F(V)$, wobei $\F$ eine endlicher
K\"orper and $V$ ein endlich dimensionaler Vektorraum \"uber $V$ ist.
Sei $p=\chr \F$. Jede $p$-Untergruppe von $G$ ist dann unipotent. Sei $U$ eine
unipotente Untergruppe. Dann ist $W:=C_V(U)\neq 0$ und $N_G(U)\leq
N_G(W)$.
Damit sind die maximalen parabolischen Untergruppen gerade die Normalizatoren
der echten Unterr\"aume von $V$. Zwei maximale Parabolische sind
inzident genau dann, wenn die entsprechenden Unter\"aume ineinander
enthalten sind.
Also ist die unipotente Geometrie von $\GL_\F(V)$ genau der zu $V$
geh\"orige projektive
Raum.
\end{slide}
\begin{slide}
\begin{center}\bf Halbeinfache Geometrien \end{center}
Wir betrachten halbeinfache Geometrien nur am Beispiel
$GL_\F(V)$.
Sei $p$ eine Primzahl, so dass $\F$ eine $p$-te Einheitswurzel
enth\"alt. Sei $t$ ein Element der Ordnung $p$ in $G$. Dann ist
$$V= V_1\oplus V_2\oplus \ldots \oplus V_m$$
die direkte Summe der Eigenr\"aume von $t$ und
$$C_G(t)=GL_{\F}(V_1)\times GL_{\F}(V_2)\times \ldots GL_{\F}(V_m).$$
Die maximalen Objekte der halbeinfachen Geometrie entsprechen in
diesem Beispiel den Paaren
$(V_1,V_2)$ von Unterr\"aumen mit $V=V_1\oplus V_2$. Und $(V_1,V_2)$ ist inzident
mit $(W_1,W_2)$ falls $V_1$ mit $W_1$, und $V_2$ mit $W_2$ inzident ist.
\end{slide}
\begin{slide}
Ein etwas allgemeineres {\bf Beispiel:}
Sei $G$ eine Gruppe von Lie-Typ definiert \"uber einen endlichen
K\"orper $\F$.
1. Sei $p=\chr \F$. Dann ist die unipotente Geometrie von $G$ gerade
das zu $G$ geh\"orende Geb\"aude.
2. Sei $p$ ein Primteiler von $|\F|-1$. Dann besteht die halbeinfache
Geometrie von $G$ aus all den Paaren $(a,b)$, wobei $a$ und $b$
gegen\"uberliegende Elemente des Geb\"audes sind.
\end{slide}
\begin{slide}\small{
Gegeben eine Inzidengeometrie $\ca G$. Wie kann man $\ca
G$ idenifizieren?
Fahne in $\ca G$: eine Teilmenge $\ca F$, so dass je zwei verschiedene
Elemente von $\ca F$ inzident sind.
$\tau(\ca F)$ nennt man den Typ $\ca F$. Eine Fahne kann hoechstens sowiele
Elemente enthalten wie es Typen gibt.
Residium einer Fahne $\ca F$ vom Typ $J$:
$$Res_\ca F=\{ a\in \ca G\setminus \ca F\mid \ca F\cup \{a\} \text{
ist eine Fahne}\}. $$
$Res_\ca F$ ist eine Inzidenzgeometrie \"uber $I\setminus J$.
Diagramm von $\ca F$:
$$(Res_a\mid a\in \ca F)\text{ (modulo Isomorphismen})$$
Diagramm von $\ca G$: Diagramm einer maximalen Fahne (nicht notwendig
eindeutig)
}
\end{slide}
\begin{slide}
{\bf Beispiel:} projektiver Raum
Maximale Fahne: ($n+1=\dim V$)
$$0< V_1< V_2<\ldots V_n< V.$$
$$Res_W=Proj(W)\cup Proj(V/W).$$
\medskip
Diagramm einer projektiven Ebene:
$$\node\arc\node$$
Diagramm eines projektiven Raumes:
$$\an$$
\end{slide}
\begin{slide}
Oft ist eine Geometrie durch ihr Diagramm bestimmt.
{\bf Satz:} Zwei spherische Geb\"aude vom Rank mindestens 4 und
gleichem Diagramm sind isomorph.
\end{slide}
\begin{slide}
Wie kann man das Diagramm der parabolische Geometrie bestimmen?
Sei $B$ eine Borelgruppe von $G$.
Definiere
$$\ca F=\{M\mid B\leq M, M \text{ maximale Parabolische}\}$$
Dann ist $\ca F$ eine maximale Fahne der Parabolische Geometrie.
Sei $M\in \ca F$ und $L$ eine Parabolische inzident mit $L$. Dann ist
$M\cap L$ eine Parabolische von $M$. Damit gilt $O_p(M)\leq M\cap L$
und $M\cap L/O_p(M)$ ist (im allgemeinen) eine Parabolische von
$M\cap L/O_p(M)$. Also
$$Res_M$$ $$\leftrightarrow$$$$ \text{parabolische Geometrie von} M/O_p(M)$$
Also ist Kenntnis des Diagrammes mehr oder weniger \"aquivalent zur
Kenntnis von $M/O_p(M)$ f\"ur alle maximalen parabolischen Untergruppen $M$
von $G$.
\end{slide}
\begin{slide} Kurze Zusammenfassung:
Wie kann man eine gegebene endlichen Gruppe identifizieren:
Schritt 1: Definiere eine gegeignete Geometrie auf der die Gruppe operiert.
Schritt 2: Bestimme das Diagramm der Geometrie.
Schritt 3: Identifiziere die Geometrie anhand des Diagrammes.\bs
\end{slide}
\begin{slide}
Schritt 2 wird fast immer rein gruppentheoretisch durchgef\"uhrt.
{\bf Offene Frage:} Gibt es geometrische Methoden, die bei der
Bestimmung der Diagramme geeigneter Geometrien der endlichen
einfachen Gruppen hilfreich seien k\"onnten?
\end{slide}
\end{document}
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% given that we said footinclude=false, this should be safe
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\title{Rationale}
\date{}
\author{Marco Pessotto}
\subtitle{Why another wiki engine?}
% https://groups.google.com/d/topic/comp.text.tex/6fYmcVMbSbQ/discussion
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\section{Markup and formats}
Apparently, the world is full of wiki engines, some of them good or
very good. So the question is if this particular engine was really
needed.
The problem AmuseWiki wants to address is, anyway, very practical. I
needed and wanted a decent range of output format. Not just a PDF
output (MediaWiki does that as well, for example), but a nice, good,
readable PDF. Now, given that the procedure needs to be completely
automated, the perfection is hardly reached, but we can do better than
simply render an HTML page and stuff the output in a PDF container.
TeX has been around more than 30 years by now. So the idea was of
course to use that.
Then I needed a markup, possibly an existing one. \href{http://daringfireball.net/projects/markdown/syntax}{Markdown} was
considered, of course, but then discarded, because the original
specification didn't support footnotes, explicitly permitted random
inline HTML, and some other questionable (in my very humble opinion)
design choices. Creating another markdown dialect would have bring me
to square zero.
Given that I'm a Emacs user, I encountered \href{http://mwolson.org/projects/EmacsMuse.html}{Emacs Muse} some years ago,
and I truly liked the syntax. By now the project is more or less
stalled, but the elisp code still works, is distributed in Debian, has
a nice manual, and provided a first reference implementation for the
output.
The other alternative would have been the \href{http://orgmode.org}{org-mode} markup, but, beside
to be very large and complicated by lots of plugins that people would
expect to work, it have some (again) questionable markup elements for
things often used as quotations.
All considered, the muse markup was small, compact and expressive
enough for my needs. The bottom line is: every lightweight markup
needs something like a couple of minutes to be learned, so better
choose one I like, not the one everyone uses, and have a \href{http://www.amusewiki.org/library/manual}{manual} for
that.
In turn, this could have been another questionable design choice, but
that's the upside of being the author of some software: you write it
your way.
The markup implementation was developed in Perl and is available on
\href{https://metacpan.org/pod/Text::Amuse}{CPAN} as Text::Amuse. I added the "A" prefix because the markup is not
1:1 compatible with the original one.
Once the markup was able to produce HTML and TeX code, I needed
something to create the imposed versions of the PDF, i.e., a PDF file
which can be printed, folded and clipped to create booklets. During
the past years, I tried various different solutions, which worked (to
some extent), but weren't really satisfying. This resulted in
\href{https://metacpan.org/release/PDF-Imposition}{PDF::Imposition}, which so far appears to work well and is extensible
enough to accomodate any future need (being Perl code, and not an
hack).
I truly hate to read on screen. For the same reason, an EPUB output
would have been nice as well, and given that the HTML was already
there, a couple of CPAN modules were put together (notably Template
Toolkit and EBook::EPUB), and a module to wrap the pieces together on
the command line was created: \href{https://metacpan.org/release/Text-Amuse-Compile}{Text::Amuse::Compile}. This code provides
also a command line utility to generate the formats on the command
line.
Last but not the least, experience showed that people are used to type
the character \texttt{"} and have it rendered as \texttt{“} depending on the position
and the language used. Same goes for the dash and other typographical
elements. Also, there was the need for some code capable (to some
extend) to take some HTML code and convert it into the markup. For
this I wrote the code that later became \href{https://metacpan.org/pod/Text::Amuse::Preprocessor}{Text::Amuse::Preprocessor}.
The modules above provide a way to work locally on the texts without
needing any access to the internet (they install the command line
scripts as well).
\section{The web front-end}
Sometimes writing Amusewiki itself felt like reinventing the wheel for
the n\textsuperscript{th} time, so, while for the core modules I kept the
dependencies at the minimum, for the web pieces I used many available
modules on CPAN everywhere was possible.
The application itself is built on DBIx::Class, Catalyst and Template
Toolkit. The almost “standard” tools for web things written in Perl
out there. I considered Dancer as well, but given that the previous
incarnation of amusewiki used Dancer (and I had the feeling it was
already becoming too messy), I chose to jump on the catalyst train,
and I don't have any regret because it's really \emph{elegant}.
A note about the database. I'm not a database fan. Actually, I don't
like them at all for managing texts. I think that the various
db-driven CMS out there are just doing the wrong thing, because
they're coupling the texts with the application.
For Amusewiki, being that it's archive- and library-oriented, I wanted
the texts stored in a directory tree under revision control. Given
that Git now is ubiquitous, choosing it was the easy part. Also, the
revision history and the possibility to have the archive distributed
and decentralized was just too good to be ignored.
To sum-up: the markup is decoupled by everything and can work
stand-alone on a single file or a whole tree. The archive is decoupled
by everything else and is just a git archive which follows some simple
directory-naming conventions.
The web application is then in charge to extract the relevant
information from the texts, store it into the database, keep them
up-to-date, and use it (hating databases doesn't mean I don't
recognize their usefulness when properly used -- it's just using them
as text archives that it's so bad).
DBIx::Class was handy to provide another layer of abstraction between
the web application itself and the archive, so the code could be
shared among the Catalyst part, and the back-end part (which compiles
the texts, updates the database and the git archive).
For the layout, which I admit was kind of hacked together by a
programmer (myself), not a designer, I went straight for bootstrap and
jquery and some other useful javascript libraries (many are used
around). It's not the best out there, but I feel that definitively I
could have done much worse.
% begin final page
\clearpage
% new page for the colophon
\thispagestyle{empty}
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\bigskip
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\strut
\vfill
\begin{center}
Marco Pessotto
Rationale
Why another wiki engine?
\bigskip
\bigskip
\textbf{amusewiki.org}
\end{center}
% end final page with colophon
\end{document}
% No format ID passed.
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\begin{center}
{\bf \large Chercheur Boogie}
\vspace*{1ex}
{\bf Texte~:} Nik\aa
{\bf Inspiration~:} Stevie Ray Vaughan, ZZ TOP
{\bf Musique, chant, guitare, etc., etc.~:} Mich'L
\end{center}
\vfill
\begin{verse}
D\'eja tout petit, je m'disais\\
J'peux pas rester comm'\c{c}a\\
Les fesses par terre, le bec en l'air, dis !\\[1ex]
{\bf Refrain :}\\
Faut qu'\c{c}a marche,
faut qu'\c{c}a marche,
faut qu'\c{c}a marche !\\
Jusqu'\`a minuit, je m'disais\\
Bon sang\\
Faut qu'\c{c}a marche\\
Jusqu'\`a minuit, je m'disais\\
Faut qu'\c{c}a marche,
faut qu'\c{c}a marche\\
faut qu'\c{c}a marche,
faut qu'\c{c}a marche ...\\[1ex]
Plus tard \`a l'\'ecole, on m'disait\\
Faut pas rester comm'\c{c}a\\
J'craignais l'pire, fallait lire, dis !\\
{\bf Refrain}\\
Et au temps des filles, qui m'disait\\
Faut pas r\^ever comm'\c{c}a\\
Sans l'bachot, pas d'boulot, dites ?\\
{\bf Refrain}\\
Puis c'est BelleIsa qui m'disait\\
J'peux pas rester comm'\c{c}a\\
Je voudrais bien de beaux gamins, dis ?\\
{\bf Refrain}\\
Bernard, Philippe, ils m'disaient\\
Toujours des choses comme \c{c}a\\
Alors la science ? Est-ce qu'elle avance, dis ?\\
{\bf Refrain}\\
Avec l'HDR, d\'esormais\\
Ca va changer, c'est moi\\
Qui maint'nant aux \'etudiants dis:\\[1ex]
Faut qu'\c{c}a marche ! Faut qu'\c{c}a marche !\\
Jusqu'\`a minuit, j'leur dirais\\
Bon sang\\
Faut qu'\c{c}a marche ! Faut qu'\c{c}a marche !\\
Faut qu'\c{c}a marche ! Faut qu'\c{c}a marche !\\
\end{verse}
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#
# ChangeLog for doc/user/user.tex
#
# Generated by Trac 1.2.1
# Jan 29, 2022, 12:38:53 AM
Tue, 14 Jun 2016 16:22:45 GMT Peter A. Buhr <pabuhr@…> [45576af9]
* doc/LaTeXmacros/common.tex (modified)
* doc/user/Cdecl.fig (modified)
* doc/user/user.tex (modified)
update user manual and global latex macros
Sat, 04 Jun 2016 16:34:24 GMT Peter A. Buhr <pabuhr@…> [e229c22]
* .gitignore (modified)
* doc/LaTeXmacros/common.tex (modified)
* doc/bibliography/cfa.bib (modified)
* doc/refrat/.gitignore (added)
* doc/refrat/refrat.tex (modified)
* doc/user/.gitignore (added)
* doc/user/user.tex (modified)
change gitignore with respect to latex-generated files, small updates ...
Tue, 03 May 2016 13:30:55 GMT Peter A. Buhr <pabuhr@…> [0638c44]
* doc/LaTeXmacros/common.tex (modified)
* doc/refrat/refrat.tex (modified)
* doc/user/user.tex (modified)
* src/examples/fstream_test.c (modified)
* src/examples/sum.c (modified)
more formatting changes to documents, update I/O for examples
Sat, 30 Apr 2016 18:05:06 GMT Peter A. Buhr <pabuhr@…> [e945826]
* doc/LaTeXmacros/common.tex (modified)
* doc/refrat/refrat.tex (modified)
* doc/user/user.tex (modified)
* src/examples/io.c (modified)
* src/libcfa/iostream.c (modified)
formatting in iostream.c, and change escape sequences in
documentation
Thu, 28 Apr 2016 18:48:14 GMT Peter A. Buhr <pabuhr@…> [8bc4ef8]
* doc/LaTeXmacros/common.tex (modified)
* doc/user/user.tex (modified)
* src/libcfa/fstream (modified)
* src/libcfa/stdlib.c (modified)
small updates missing from last commit
Thu, 28 Apr 2016 02:52:23 GMT Peter A. Buhr <pabuhr@…> [b72bad4f]
* doc/user/user.tex (modified)
* src/examples/io.c (modified)
* src/libcfa/fstream.c (modified)
* src/libcfa/iostream.c (modified)
* src/libcfa/stdlib (modified)
add math library to user manual, fix sepDisable bug in iostream
Thu, 21 Apr 2016 12:24:07 GMT Peter A. Buhr <pabuhr@…> [6e991d6]
* doc/user/Cdecl.fig (added)
* doc/user/user.tex (modified)
* src/driver/cfa.cc (modified)
* src/examples/swap.c (modified)
* src/libcfa/Makefile.am (modified)
* src/libcfa/Makefile.in (modified)
* src/libcfa/math (added)
* src/libcfa/math.c (added)
* src/libcfa/rational.c (modified)
* src/libcfa/stdlib (modified)
* src/libcfa/stdlib.c (modified)
add -fgnu89-inline flag to compile, cleanup swap example I/O, stdlib ...
Wed, 20 Apr 2016 01:36:29 GMT Peter A. Buhr <pabuhr@…> [6b6597c]
* doc/user/Makefile (modified)
* doc/user/user.tex (modified)
* src/examples/io.c (modified)
* src/libcfa/fstream (modified)
* src/libcfa/stdlib (modified)
* src/libcfa/stdlib.c (modified)
user manual updates, extend I/O test, fix memset in stdlib, ...
Tue, 12 Apr 2016 22:28:37 GMT Peter A. Buhr <pabuhr@…> [b52d900]
* doc/LaTeXmacros/common.tex (modified)
* doc/refrat/refrat.tex (modified)
* doc/user/user.tex (modified)
extend user manual, update latex macros, and update reference manual
Sat, 09 Apr 2016 14:04:50 GMT Peter A. Buhr <pabuhr@…> [e55ca05]
* doc/LaTeXmacros/common.tex (added)
* doc/LaTeXmacros/indexstyle (moved)
* doc/bibliography/cfa.bib (added)
* doc/refrat/Makefile (modified)
* doc/refrat/refrat.tex (modified)
* doc/user/Makefile (modified)
* doc/user/indexstyle (deleted)
* doc/user/user.bib (deleted)
* doc/user/user.tex (modified)
fix bibliography for manuals, refactor common LaTeX macros
Thu, 07 Apr 2016 02:08:32 GMT Peter A. Buhr <pabuhr@…> [53ba273]
* doc/refrat/refrat.tex (modified)
* doc/user/user.tex (modified)
* src/driver/cfa.cc (modified)
* src/examples/io.c (added)
* src/examples/io.data (moved)
* src/examples/rational.c (added)
* src/examples/rational.cc (added)
* src/examples/read.c (deleted)
* src/libcfa/Makefile.am (modified)
* src/libcfa/Makefile.in (modified)
* src/libcfa/fstream (modified)
* src/libcfa/fstream.c (modified)
* src/libcfa/iostream (modified)
* src/libcfa/iostream.c (modified)
* src/libcfa/limits (modified)
* src/libcfa/limits.c (added)
* src/libcfa/rational (added)
* src/libcfa/rational.c (added)
* src/libcfa/stdlib (modified)
* src/libcfa/stdlib.c (modified)
switch from std=c99 to std=gnu99, update latex macros, refrat and ...
Wed, 23 Mar 2016 01:40:14 GMT Peter A. Buhr <pabuhr@…> [6c91065]
* doc/user/indexstyle (added)
* doc/user/user.bib (added)
* doc/user/user.tex (added)
start user manual
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\textwidth .72\paperwidth % WIDTH OF TEXT, REMAINS .28% FOR MARGINS
\if@proc
\addtolength\textwidth{.02\paperwidth}
\fi
\setlength\@tempdima{.76\paperheight} % HEIGHT OF TEXT WITH FOOTNOTES
\divide\@tempdima\baselineskip % EXCLUDING HEAD (AND FOOT)
\@tempcnta=\@tempdima % rounded to integer (#lines-1)
\setlength\textheight{\@tempcnta\baselineskip}
\addtolength\textheight{\topskip}
\voffset -1in
\topmargin .05\paperheight % FROM TOP OF PAGE TO TOP OF HEADING (0=1inch)
\headheight .02\paperheight % HEIGHT OF HEADING BOX.
\headsep .03\paperheight % VERT. SPACE BETWEEN HEAD AND TEXT.
\footskip .07\paperheight % FROM END OF TEX TO BASE OF FOOTER. (40pt)
\marginparsep 9\p@ % HOR. SPACE BETWEEN TEXT AND MARGINAL NOTE
\marginparpush 6\p@ % MIN. VERT. SPACE BETWEEN MARGINAL NOTES
\hoffset -1in % TO ADJUST WITH PAPER:
\if@twoside % IF TWO-SIDED:
\oddsidemargin .13\paperwidth % LEFT MARGIN FOR ODD PAGES (10)
\evensidemargin .15\paperwidth % LEFT MARGIN FOR EVEN PAGES (30)
\marginparwidth .10\paperwidth % TEXTWIDTH OF MARGINALNOTES
\reversemarginpar % BECAUSE OF TITLEPAGE.
\else % IF ONE-SIDED:
\oddsidemargin .14\paperwidth % LEFT MARGIN FOR ODD PAGES (20)
\evensidemargin .14\paperwidth % LEFT MARGIN FOR EVEN PAGES (20)
\marginparwidth .11\paperwidth % TEXTWIDTH OF MARGINALNOTES
\fi %
\if@proc
\addtolength\oddsidemargin{-.01\paperwidth}
\addtolength\evensidemargin{-.01\paperwidth}
\fi
%%%%%%%%%%%%%%%%%%%%%%% HYPER (AND DRAFT) STUFF %%%%%%%%%%%%%%%%%%%%%%%
%\let\textref\@gobble
%\if@hyper % IF NOHYPER WE DO LESS DAMAGE AS POSSIBLE.***
\bgroup\catcode`\#=12\gdef\hash{#}\egroup % DEFINED \hash=#.
%\fi
\def\H@tilde{\string~}
\newcommand{\href}[2]{\bgroup\let~\H@tilde
\if@hyper\noexpand\special{html:<a href="#1">}\fi
{#2}\egroup\if@hyper\special{html:</a>}\fi}
\newcommand{\name}[1]{\if@hyper\noexpand\special% NAME HAS NO TAGGED TEXT HERE.
{html:<a name="#1">}\special{html:</a>}\fi}
\newcommand{\base}[1]{\if@hyper\bgroup\let~\H@tilde
\noexpand\special % BASE HAS NO TAGGED TEXT.
{html:<base href="#1">}\egroup\fi}
\newcommand{\textref}[2]{\vrule height \z@ width \z@\href{\hash ref-#1}{#2}}
%------------------------------------------------------------------%
% SECTION TAGS and THEIR REFERENCE IN TOC \d@t will be '.' for sections only
\if@hyper
\renewcommand{\@seccntformat}[1]{\name{sec\csname the#1\endcsname}%
\csname the#1\endcsname\d@t\hspace{1ex}}
\renewcommand{\numberline}[1]{\hb@xt@\@tempdima{\href
{\hash\hyp@typ#1}{#1}\d@t\hfil}}
\else
\renewcommand{\@seccntformat}[1]{\csname the#1\endcsname\d@t\hspace{1ex}}
\renewcommand{\numberline}[1]{\hb@xt@\@tempdima{#1\d@t\hfil}}
\fi
%------------------------------------------------------------------%
% CITATION TAGS ARE TOGETHER WITH BIB STUFF.
% CITATION HREF
\def\@citex[#1]#2{% % UNFORTUNATELY REDEFINED!!!!
\let\@citea\@empty
\@cite{\@for\@citeb:=#2\do
{\@citea\def\@citea{,\penalty\@m\ }%
\edef\@citeb{\expandafter\@firstofone\@citeb}%
\if@filesw\immediate\write\@auxout{\string\citation{\@citeb}}\fi
\@ifundefined{b@\@citeb}{\mbox{\reset@font\bfseries ?}%
\G@refundefinedtrue
\@latex@warning
{Citation `\@citeb' on page \thepage \space undefined}}%
{\edef\tmp@bn{\csname b@\@citeb\endcsname}%
\hbox{\href{\hash bib\tmp@bn}{\tmp@bn}}}}}{#1}% % **HYPER**
\if@draft\norm@note{CIT: }{#2}\fi % **DRAFT**
}
%-----------------------------------------------------------------%
% LABEL, REF AND PAGEREF, ~ COMPATIBLE:
\let\old@label\label % OLD LATEX COMMAND.
\renewcommand{\label}[1]{\name{ref-#1}% % LABEL IN HYPER TAG.
\old@label{#1}% %
\if@draft% % DRAFT:
\ifmmode\math@note{#1}%
\else\norm@note{}{LAB: #1}\fi%
\fi%
}
\if@hyper
\let\old@ref\ref \let\old@pageref\pageref % OLD LATEX COMMANDS.
\renewcommand{\ref}[1]{\@ifundefined{r@#1}{}{\href{\hash ref-#1}}%
{\old@ref{#1}}} % LABEL INSIDE HREF.
\renewcommand{\pageref}[1]{\@ifundefined{r@#1}{}{% % UNDEFINED => NO HREF
\edef\tmp@ref{\noexpand\@secondoftwo\csname r@#1\endcsname}%
\href{\hash pag\tmp@ref}}{\old@pageref{#1}}}
\fi
%------------------------------------------------------------------%
%%%%%%%%%%%%%%%%%%%%%%%%%% END HYPER-STUFF %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%% JHEP HEADINGS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\ps@JHEP{%
\def\@oddfoot{\reset@font\hfil\thepage\hfil}
\if@draft\edef\cur@opt{\@curroptions}\fi
\def\@oddhead{\name{pag\thepage}\hfil\if@draft\copy\drft@box\fi}
\if@proc
\def\@oddhead{\name{pag\thepage}
\smash{\if@draft\raise 2.5em\rlap{\copy\drft@box}\fi
\vbox{\hsize=\textwidth\noindent
\copy\conf@box\hfill
\copy\@firstauthorbox\hskip 0.5em {\it et al}.\vskip.17em \hrule}}
}%
\fi
\if@twoside\let\@evenhead\@oddhead\let\@evenfoot\@oddfoot\fi
\let\@mkboth\@gobbletwo
\let\sectionmark\@gobble
\let\subsectionmark\@gobble
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TITLE PAGE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\maketitle{\JHEP@ignol{\maketitle}}
\newcommand\auto@maketitle{\begingroup
\renewcommand\thefootnote{\@fnsymbol\c@footnote}%
\def\@makefnmark{\rlap{\@textsuperscript{\normalfont\@thefnmark}}}%
\long\def\@makefntext##1{\parindent 1em\noindent
\hb@[email protected]{\hss\@textsuperscript{\normalfont\@thefnmark}}##1}%
\let\footnote\thanks%
\newpage %
\if@proc % IF PROCEEDINGS:
\global\let\@nodocument\relax % LaTeX FOULED:)
\twocolumn[\@maketitle]% % TWOCOLUMN AFTER REAL TITLE
\@thanks % TITLE THANKS IN 1st COLUMN
\let\@evenfoot\@empty % NO PAGENO.
\let\@oddfoot\@empty
\setcounter{page}{1}%
\thispagestyle{empty}% % PROCS HAVE NO PAGENUMBER
\else
\@maketitle % CALL ACTUAL FORMATTING
\@thanks % TITLE THANKS
\let\@evenfoot\@empty % NO PAGENO.
\let\@oddfoot\@empty
\setcounter{page}\z@
\newpage % FINE PAGINA
\fi
\endgroup
\setcounter{footnote}{0}%
\if@todotoc % IF THERE IS A SECTION
\noindent\rule\textwidth{.1pt} % THEN MAKE TOC...
\vskip 2em \@plus 2ex \@minus 2ex
\tableofcontents
\vskip 2em \@plus 2ex \@minus 2ex
\noindent\rule\textwidth{.1pt}
\vskip 2em \@plus 2ex \@minus 2ex
\fi
\if@preprint\else
\gdef\tableofcontents{\JHEP@igno{\tableofcontents\space is automatic}}
\fi
\global\let\thanks\@gobble
\setbox\@tmpbox=\vbox{\rm\@author}% % FOR PROCS RESET HEADERS
% \global\let\maketitle\relax % NO LONGER EXISTS.
\global\let\@maketitle\relax %
\global\let\@thanks\@empty %
\global\let\@author\@empty % KILL ALL
\global\let\@date\@empty %
\global\let\@title\@empty %
\global\let\@abstract\@empty %
\global\let\title\relax %
\global\let\author\relax %
\global\let\date\relax %
\global\let\and\relax %
\global\let\email\@gobble
\global\let\received\relax
\global\let\accepted\relax
\global\let\keywords\relax
\global\let\endkeywords\relax
}
%%%%%%%%%%%%%%%%%%%%%%%%%% ACTUAL TITLEPAGE %%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\@maketitle{% % PAGE IS FLUSHED LEFT
\begin{flushleft}% % PAY ATTENTION TO \par'S
\if@preprint %%% PREPRINT HEADER:
\vskip-7em %
\underline{\tiny Preprint typeset %
in JHEP style. - %
\if@hyper{HYPER VERSION} %
\else{PAPER VERSION}\fi } %
\normalsize\hfill% % *** MAYBE PUT IN A ZERO BOX
\begin{tabular}[t]{r}\@preprint\end{tabular}% % PREPRINT NUMBERS
\else %
\if@proc %%% PROCEEDINGS: LOGO ETC..
\vskip-4.48em\hfill\copy\conf@box
\vskip-.7\baselineskip\logo\hrulefill
\vskip-.4\baselineskip\hskip32\p@% % *** IDEM BOXIZE
{\tiny PROCEEDINGS}
\else
% %%% PUBLISHED: LOGO ETC..
\vskip-7em\hfill\unhbox\rece@box\unhbox\acce@box
\vskip-.7\baselineskip\logo\hrulefill
\vskip-.5\baselineskip\hskip23\p@% % *** IDEM BOXIZE
{\tiny \if@hyper{HYPER VERSION}\else{PAPER VERSION}\fi}
\fi\fi\null
\vskip 1.5em plus .4fil % V. SPACE BEFORE TITLE
{\LARGE \sffamily % TITLE: large sans-serif bf
\bfseries %
% \if@hyper\else\huge\fi % IF PAPER: no longer HUGE.
\@title\par} %
\vskip .6em plus .06fil minus .5ex %
\rule\textwidth{\if@proc\else1.5\fi\p@} % RULE for PROC = 1pt else 1.5pt.
\vskip 1em plus .06fil minus .6ex % (symmetric is 1em)
{\normalsize \bfseries \sffamily %
\@author \par} % AUTHORS\\ADDRESSES
\vskip 1em plus 0.05fil minus 1ex %
\parbox\textwidth{\unhbox\abstract@box} % ABSTRACT IF PRESENT (SHOULD)
\vskip 1em plus 1em minus 1ex %
\par %
\@keywords % KEYWORDS IF PRESENT
\par
\vskip\baselineskip
\dedic@box % DEDICATION, IF PRESENT
\end{flushleft}% % FINISHED.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%% STANDARD STUFF %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{secnumdepth}{3}
\newcounter {part}
\newcounter {section}
\newcounter {subsection}[section]
\newcounter {subsubsection}[subsection]
\newcounter {paragraph}[subsubsection]
\newcounter {subparagraph}[paragraph]
\renewcommand\thepart {\@Roman\c@part}
\renewcommand\thesection {\@arabic\c@section}
\renewcommand\thesubsection {\thesection.\@arabic\c@subsection}
\renewcommand\thesubsubsection{\thesubsection.\@arabic\c@subsubsection}
\renewcommand\theparagraph {\thesubsubsection.\@arabic\c@paragraph}
\renewcommand\thesubparagraph {\theparagraph.\@arabic\c@subparagraph}
%%%%%%%%%%%%%%%%%%%%%%%%%%% SECTION FORMATS %%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand\secstyle{\bfseries}
\def\ts@flag{\let\d@t.% % Sections get ``.''
\immediate\write\@auxout % IF THERE IS A SECTION
{\string\global\string\@todotoctrue}} % THEN SIGNAL IN AUX.
\newcommand\part{\par
\addvspace{4ex}%
\@afterindentfalse
\secdef\@part\@spart}
\def\@part[#1]#2{\ts@flag
\ifnum \c@secnumdepth >\m@ne
\refstepcounter{part}% % DAMN LATEX !!! BELOW,
\addcontentsline{toc}{part}{\string\href % UNFORTUNATELY, NO NUMBERLINE:
{\string\hash\space part\the\c@part}%
{\thepart}\hspace{1em}#1}%
\else
\addcontentsline{toc}{part}{#1}%
\fi
{\parindent \z@ \raggedright
\interlinepenalty \@M
\normalfont
\ifnum \c@secnumdepth >\m@ne
\name{part\the\c@part}\Large\bfseries \partname~\thepart
\par\nobreak
\fi
\huge \bfseries #2%
\markboth{}{}\par}%
\nobreak
\vskip 3ex
\@afterheading\let\d@t\relax}
\def\@spart#1{%
{\parindent \z@ \raggedright
\interlinepenalty \@M
\normalfont
\huge \bfseries #1\par}%
\nobreak
\vskip 3ex
\@afterheading}
\newcommand\section{\@startsection{section}{1}{\z@}%
{-3.5ex \@plus -1.3ex \@minus -.7ex}%
{2.3ex \@plus.4ex \@minus .4ex}%
{\normalfont\large\secstyle}}
\newcommand\subsection{\@startsection{subsection}{2}{\z@}%
{-2.3ex\@plus -1ex \@minus -.5ex}%
{1.2ex \@plus .3ex \@minus .3ex}%
{\normalfont\normalsize\secstyle}}
\newcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
{-2.3ex\@plus -1ex \@minus -.5ex}%
{1ex \@plus .2ex \@minus .2ex}%
{\normalfont\normalsize\secstyle}}
\newcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
{1.75ex \@plus1ex \@minus.2ex}%
{-1em}%
{\normalfont\normalsize\bfseries}}
\newcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
{1.75ex \@plus1ex \@minus .2ex}%
{-1em}%
{\normalfont\normalsize\bfseries}}
% ---- turnaround for sections with figures ------ %
\def\bef@sec{\iffigprocessing\JHEP@warnl{Floating figure \the\ffigcount\space
and startsection colliding}\fi
\iftabprocessing\JHEP@warnl{Floating table \the\ftabcount\space
and startsection colliding}\fi
\edef\bef@everypar{\the\everypar}} % HACK FOR FLOATS AND \everypar
\def\aft@sec{\let\d@t\relax % RESTORE: NO DOTTED SUBSECS..
\edef\tmp@everypar{\the\everypar}% % DELETED CLUBPENALTY, BELOW.
\everypar\expandafter{\bef@everypar{\tmp@everypar}}}% RESTORE.
\let\old@sec\section %
\let\old@ssec\subsection %
\let\old@sssec\subsubsection % OLD SECTIONING COMMANDS.
\let\old@par\paragraph %
\let\old@spar\subparagraph %
% ------- new forms ------- %
\renewcommand{\section}{\secdef\JHEP@sec\JHEP@secs}
\renewcommand{\subsection}{\secdef\JHEP@ssec\JHEP@ssecs}
\renewcommand{\subsubsection}{\secdef\JHEP@sssec\JHEP@sssecs}
\renewcommand{\paragraph}{\secdef\JHEP@par\JHEP@pars}
\renewcommand{\subparagraph}{\secdef\JHEP@spar\JHEP@spars}
% ---- unstarred forms ---- %
\def\JHEP@sec[#1]#2{\ts@flag\bef@sec\old@sec[#1]{#2}\aft@sec}
\def\JHEP@ssec[#1]#2{\bef@sec\old@ssec[#1]{#2}\aft@sec}
\def\JHEP@sssec[#1]#2{\bef@sec\old@sssec[#1]{#2}\aft@sec}
\def\JHEP@par[#1]#2{\bef@sec\old@par[#1]{#2}\aft@sec}
\def\JHEP@spar[#1]#2{\bef@sec\old@spar[#1]{#2}\aft@sec}
% ----- starred forms ----- %
\def\JHEP@secs#1{\bef@sec\old@sec*{#1}\aft@sec}
\def\JHEP@ssecs#1{\bef@sec\old@ssec*{#1}\aft@sec}
\def\JHEP@sssecs#1{\bef@sec\old@sssec*{#1}\aft@sec}
\def\JHEP@pars#1{\bef@sec\old@par*{#1}\aft@sec}
\def\JHEP@spars#1{\bef@sec\old@spar*{#1}\aft@sec}
% ----- end hack for sections with floats --------%
\if@twocolumn
\setlength\leftmargini {2em}
\else
\setlength\leftmargini {2.5em}
\fi
\leftmargin \leftmargini
\setlength\leftmarginii {2.2em}
\setlength\leftmarginiii {1.87em}
\setlength\leftmarginiv {1.7em}
\setlength\leftmarginv {1em}
\setlength\leftmarginvi {1em}
\setlength \labelsep {.5em}
\setlength \labelwidth{\leftmargini}
\addtolength\labelwidth{-\labelsep}
\@beginparpenalty -\@lowpenalty
\@endparpenalty -\@lowpenalty
\@itempenalty -\@lowpenalty
\renewcommand\theenumi{\@arabic\c@enumi}
\renewcommand\theenumii{\@alph\c@enumii}
\renewcommand\theenumiii{\@roman\c@enumiii}
\renewcommand\theenumiv{\@Alph\c@enumiv}
\newcommand\labelenumi{\theenumi.}
\newcommand\labelenumii{(\theenumii)}
\newcommand\labelenumiii{\theenumiii.}
\newcommand\labelenumiv{\theenumiv.}
\renewcommand\p@enumii{\theenumi}
\renewcommand\p@enumiii{\theenumi(\theenumii)}
\renewcommand\p@enumiv{\p@enumiii\theenumiii}
\newcommand\labelitemi{$\m@th\bullet$}
\newcommand\labelitemii{\normalfont\bfseries --}
\newcommand\labelitemiii{$\m@th\ast$}
\newcommand\labelitemiv{$\m@th\cdot$}
\newenvironment{description}
{\list{}{\labelwidth\z@ \itemindent-\leftmargin
\let\makelabel\descriptionlabel}}
{\endlist}
\newcommand*\descriptionlabel[1]{\hspace\labelsep
\normalfont\bfseries #1}
%%%%%%%%%%%%%%%%%%%%%%%%%% AUTHORS/ADDRESSES %%%%%%%%%%%%%%%%%%%%%%%%%%
% *** After proceedings, the authors go in a hbox. => \break does not work!!
\newtoks\prev@t
\newtoks\cur@t
\newbox\@firstauthorbox
\renewcommand\@author{% % FIRST TIME \\=>ADDR.
\def\\{\egroup %
\copy\@firstauthorbox\par % CLOSE & COPY HBOX
\vskip.6em\@plus.02fil\@minus.3ex% GLUE UNDER AUTHOR
\hskip1em% % ADDRESS INDENT
\vbox\bgroup\hsize=.9\textwidth %
\let\\\par\small\it\raggedright}%% STYLE FOR ADDRESSES
}
\renewcommand\author[1]{%
\if@proc\if@author\@PROCerr\fi\fi % PROCS ONLY 1 AUTHOR!
\global\@authortrue
\prev@t=\expandafter{\@author}% % TWO TOKEN LISTS.
\cur@t={\global\setbox\@firstauthorbox %
\hbox\bgroup #1\egroup\par % ACTUAL AUTH.\\ADD.
\vskip.6em\@plus.03fil\@minus.2ex}% % Glue!
\long\xdef\@author{\the\prev@t\the\cur@t}%
} % STORE ALL IN \@AUTHOR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROCEEDINGS %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newbox\conf@box
\newcommand\conference[1]{\global\setbox\conf@box\hbox{%
\itshape #1}\@conftrue}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RECEIVED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newbox\rece@box
\newcommand\received[1]{\global\setbox\rece@box\hbox{\small
{\scshape\receivedname} \itshape #1, }\@recetrue}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PUBLISHED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newbox\acce@box
\newcommand\accepted[1]{\global\setbox\acce@box\hbox{\small
{\scshape\acceptedname} \itshape #1}\@accetrue}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEDICATED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\gdef\dedic@box{\relax}
\newcommand\dedicated[1]{\gdef\dedic@box{\vbox{\small\it\raggedleft #1}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%% ACKNOWELEDGMENTS %%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand\acknowledgments{\section*{\acknowlname}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EMAIL %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand\email[1]{{\tt\href{mailto:#1}{#1}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PREPRINT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\@preprint{\relax}
\newcommand\preprint[1]{\long\gdef\@preprint{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newbox\abstract@box% % HBOX FOR WHOLE ABSTRACT
\newcommand{\abstract}[1]% % ABSTR. IN NORMALSIZE
{\global\setbox\abstract@box=\hbox{\noindent{\scshape
\abstractname}\ \ignorespaces #1}\global\@abstracttrue}
%\newenvironment{abstract}% % ABSTR. ENV. IN NORMALSIZE
% {\global\setbox\abstract@box=\hbox\bgroup\noindent{\scshape
% \abstractname}\ \ignorespaces}
% {\egroup\global\@abstracttrue}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KEYWORDS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\@keywords{\relax}% % CS FOR WHOLE KEYWORDS
\def\keywords{\bgroup\gdef\@comma{,}\catcode`\ =\active\catcode`,=\active
\@@keywords}
\begingroup
\catcode`\ =\active\catcode`,=\active\global
\def\@@keywords#1{\gdef\@keywords{\noindent{\scshape\keywordsname}
\bgroup\def, {+}\def {_}% %ATTENTION: NO EXPANSION.
\href{http://jhep.sissa.it/stdsearch?keywords=#1}%
{\let,\@comma\let \ #1}.
\egroup}\egroup\global\@keywordstrue}%
\endgroup
%%%%%%%%%%%%%%%%%%%%%%%%%% OTHER ENVIRONMENTS %%%%%%%%%%%%%%%%%%%%%%%%%
\newenvironment{verse}
{\let\\\@centercr
\list{}{\itemsep \z@
\itemindent -1.5em%
\listparindent\itemindent
\rightmargin \leftmargin
\advance\leftmargin 1.5em}%
\item\relax}
{\endlist}
\newenvironment{quotation}
{\list{}{\listparindent 1.5em%
\itemindent \listparindent
\rightmargin \leftmargin
\parsep \z@ \@plus\p@}%
\item\relax}
{\endlist}
\newenvironment{quote}
{\list{}{\rightmargin\leftmargin}%
\item\relax}
{\endlist}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% APPENDIX %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand\appendix{\par
\setcounter{section}{0}%
\setcounter{subsection}{0}%
\renewcommand\thesection{\@Alph\c@section}}
%%%%%%%%%%%%%%%%%%%%%%%%%%% SOME MORE LENGHTS %%%%%%%%%%%%%%%%%%%%%%%%%
\setlength\arraycolsep{2\p@} % TO MATCH eqnarrays<->equations
\setlength\tabcolsep{6\p@}
\setlength\arrayrulewidth{.4\p@}
\setlength\doublerulesep{2\p@}
\setlength\tabbingsep{\labelsep}
\skip\@mpfootins = \skip\footins
\setlength\fboxsep{3\p@}
\setlength\fboxrule{.4\p@}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% THE EQUATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand\theequation{\ifnum\c@section=0\else\thesection.\fi
\@arabic\c@equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIGURES, TABLES %%%%%%%%%%%%%%%%%%%%%%%%%%
\newcounter{figure}
\renewcommand\thefigure{\@arabic\c@figure}
\def\fps@figure{Htbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{\figurename~\thefigure}
\newenvironment{figure}
{\@ENVwarn{\FIGURE{...}}\@float{figure}}
{\end@float}
\newenvironment{figure*}
{\@ENVwarn{\FIGURE{...}}\@dblfloat{figure}}
{\end@dblfloat}
\newcounter{table}
\renewcommand\thetable{\@arabic\c@table}
\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{\tablename~\thetable}
\newenvironment{table}
{\@ENVwarn{\TABLE{...}}\@float{table}}
{\end@float}
\newenvironment{table*}
{\@ENVwarn{\TABLE{...}}\@dblfloat{table}}
{\end@dblfloat}
%%-----------------------------------------------------------------%%
\newcommand{\FIGURE}[2][v]{\begin{floatingfigure}[#1]#2
\end{floatingfigure}}
\newcommand{\TABLE}[2][v]{\begin{floatingtable}[#1]{#2}
\end{floatingtable}}
\newcommand{\EPSFIGURE}[3][v]{\begin{floatingfigure}[#1]\epsfig{file=#2}
\caption{#3}\end{floatingfigure}}
\newcommand{\TABULAR}[4][v]{\begin{floatingtable}[#1]{\begin{tabular}{#2}
#3\end{tabular}}\caption{#4}
\end{floatingtable}}
%Check below, and put automatic width ***
\newcommand{\DOUBLEFIGURE}[5][ht]{\@dblfloat{figure}[#1]\centerline{%
\parbox{.45\textwidth}{\centerline{\epsfig{file=#2}}}~~~~
\parbox{.45\textwidth}{\centerline{\epsfig{file=#3}}}}
\centerline{\parbox[t]{.45\textwidth}{\caption{#4}}~~~~
\parbox[t]{.45\textwidth}{\caption{#5}}}\end@dblfloat}
\newcommand{\DOUBLETABLE}[5][ht]{\@dblfloat{table}[#1]\centerline{%
\parbox{.45\textwidth}{\centerline{#2}}~~~~
\parbox{.45\textwidth}{\centerline{#3}}}
\centerline{\parbox[t]{.45\textwidth}{\caption{#4}}~~~~
\parbox[t]{.45\textwidth}{\caption{#5}}}\end@dblfloat}
%%%%%%---------------- FROM FLOATFLT PACKAGE ------------------%%%%%%%%
%% Original file `floatflt.sty', modified by F.Nesti <[email protected]> to:
%% be inserted in JHEP.cls;
%% correct float placement when shifting past pages;
%% correct incompatibility with \marginpar;
%% correct incompatibility with \@startsection;
%% correct persisting indentation in following pars;
%% let float also at beginning of pages;
%% avoid the widht specification.
%%
%% Original was 1994-1996 by Mats Dahlgren <[email protected]>.
%%
%%%%%%%%\NeedsTeXFormat{LaTeX2e}[1994/06/01] %LEFT THIS STUFF TO
%%%%%%%%\ProvidesPackage{floatflt}[1996/02/27 v. 1.3] %RECONVERT TO PACKAGE.
\newcounter{OptionTest}
\if@twoside
\setcounter{OptionTest}{0}
\else % POS: 2side=p/1side=r
\setcounter{OptionTest}{1}
\fi
%%%%%%%%\DeclareOption{rflt}{\setcounter{OptionTest}{1}}
%%%%%%%%\DeclareOption{lflt}{\setcounter{OptionTest}{2}} % CHOSEN.
%%%%%%%%\DeclareOption{vflt}{\setcounter{OptionTest}{0}}
%%%%%%%%\DeclareOption*{\OptionNotUsed}
%%%%%%%%\ProcessOptions
\newbox\@tmpbox
\newbox\figbox
\newbox\tabbox
\newbox\pagebox
\newcount\ffigcount
\newcount\ftabcount
\newcount\hangcount
\newcount\nosuccesstryfig
\newcount\nosuccesstrytab
\newdimen\figgutter \figgutter=1truepc
\newdimen\tabgutter \tabgutter=1truepc
\newdimen\fl@wd
\newdimen\fl@ht
%\newdimen\fl@gut
\newdimen\htdone \htdone=\z@
\newdimen\pageht
\newdimen\startpageht
\newdimen\floatfltwidth
\newdimen\fltitemwidth
\newif\iftryingfig \tryingfigfalse
\newif\iftryingtab \tryingtabfalse
\newif\ifdoingfig \doingfigfalse
\newif\ifdoingtab \doingtabfalse
\newif\iffigprocessing \figprocessingfalse
\newif\iftabprocessing \tabprocessingfalse
\newif\ifpageafterfig \pageafterfigfalse
\newif\ifpageaftertab \pageaftertabfalse
\newif\ifoddpages
\newif\ifoutput
\newtoks\outputpretest
\def\@captype{}
%%---------------- ORRIBLE HACKS, SORRY -------------------------%%
\let\old@marginpar\marginpar %
\renewcommand\marginpar[1]{{\outputpretest={\outputtrue}% REDEF \marginpar.
\old@marginpar{#1}}} %
\def\g@addto#1#2{{\toks@\expandafter{#1#2}%
\xdef#1{\the\toks@}}}%
\def\postpone@captions{% %
% \global\let\tmp@label\label % REDEF \caption AND
\global\let\tmp@caption\caption % \label, LATER.
\global\let\later@capt\relax % UN PO' PESANTI
\gdef\later@label{} % MA D'ALTRONDE...
\renewcommand\caption[1]{\gdef\later@capt{\tmp@caption{##1}}}% LOCAL!!
\renewcommand\label[1]{\name{ref-##1}% % TAG WHERE CALLED, AT LEAST,
\if@draft\norm@note{}{LAB: ##1}\fi% & DRAFTNOTE.
\g@addto\later@label{\old@label{##1}}}%
}
\AtEndDocument{\iftryingfig\JHEP@warnl{Floating figure \the\ffigcount\space
remains undone}\fi
\iftryingtab\JHEP@warnl{Floating table \the\ftabcount\space
remains undone}\fi}
%%----------------- FIGURE ---------------------------------------%%
\newenvironment{floatingfigure}[1][v]%
{\@tfor \@tempa :=#1\do {\xdef\@fside{\@tempa}}%
\global\advance\ffigcount by 1%
\iffigprocessing {\count0=\ffigcount\advance\count0 by -1%
\JHEP@warnl{Floating figures \the\count0\space\space and
\the\ffigcount\space colliding}}\fi
\iftabprocessing \JHEP@warnl{Floating table \the\ftabcount\space and
floating figure \the\ffigcount\space colliding}\fi
\postpone@captions
\global\setbox\@tmpbox=\hbox\bgroup% begin of figbox %HBOX FOR AUTO WIDTH!
}
{%
\egroup% % CALCULATE WIDTH (NO CAPTION FOR NOW)
\global\setlength{\floatfltwidth}{\the\wd\@tmpbox}%
% New behavior: wd < .6 columnwidth => floatflt.
% wd < columnwidth => float{figure}
% wd > columnwidth => dblfloat{figure}
\ifdim\floatfltwidth<.6\columnwidth% % IF NOT TOO WIDE OK:
\global\setbox\figbox=\vbox{\hsize=\floatfltwidth
\def\@captype{figure}%
\noindent\unhbox\@tmpbox
\later@capt\later@label}%
\global\figprocessingtrue
\global\everypar={\tryfig\oldeverypar}% must be set globally!
\figinsert\par
\else% % ELSE NORMAL LATEX FIGURE, SIGH.
% \JHEP@mess{Floating figure \the\ffigcount\space is
% wide becomes a LaTeX float}%
\if\@fside v \def\@fside{ht}\fi
\ifdim\floatfltwidth<\columnwidth
\def\@tempa{\@float{figure}[}\let\@tempb\end@float
\else
\def\@tempa{\@dblfloat{figure}[}\let\@tempb\end@dblfloat
\fi
\expandafter\@tempa\@fside]\center\unhbox\@tmpbox\later@capt\later@label\@tempb
\fi
}
%%------------ TABLE ----------------------------------------------%%
\newenvironment{floatingtable}[1][v]%
{\@tfor \@tempa :=#1\do {\global\edef\@tside{\@tempa}}%
\global\advance\ftabcount by 1%
\iftabprocessing {\count0=\ftabcount\advance\count0 by -1%
\JHEP@warnl{Floating tables \the\count0\space
\space and \the\ftabcount \space colliding}}\fi
\iffigprocessing \JHEP@warnl{Floating figure \the\ffigcount\space and
floating table \the\ftabcount\space colliding}\fi
\postpone@captions
\global\setbox\@tmpbox=\hbox\bgroup% begin of tabbox, ACTUALLY AN HBOX.
}
{%
\egroup% end of \tabbox, % ACTUALLY \@tmpbox
\global\setlength{\floatfltwidth}{\the\wd\@tmpbox}%
\ifdim\floatfltwidth<.6\columnwidth% % IF NOT TOO WIDE OK:
\global\setbox\tabbox=\vbox{\hsize=\floatfltwidth
\def\@captype{table}%
\noindent\unhbox\@tmpbox
\later@capt\later@label}%
\global\tabprocessingtrue
\global\everypar={\trytab\oldeverypar}%
\tabinsert\par%
\else% % ELSE NORMAL LATEX TABLE.
% \JHEP@mess{Floating table \the\ftabcount\space is
% wide becomes a LaTeX float}%
\ifdim\floatfltwidth<1.1\columnwidth
\if\@tside v \def\@tside{ht}\fi
\def\@tempa{\@float{table}[}\let\@tempb\end@float
\else
\if\@tside v \def\@tside{t}\fi
\def\@tempa{\@dblfloat{table}[}\let\@tempb\end@dblfloat
\fi
\expandafter\@tempa\@tside]% % ACTUALLY CALL THE LATEX FLOAT
\center\unhbox\@tmpbox\later@capt\later@label
\@tempb
\fi%
}
%----------- prepare for tries ----------------------------%
\def\figinsert{%
\global\nosuccesstryfig=0%
\global\outputpretest={\do@test}%
\global\tryingfigtrue \global\doingfigfalse%
\global\pageafterfigfalse}%
\def\tabinsert{%
\global\nosuccesstrytab=0%
\global\outputpretest={\do@test}%
\global\tryingtabtrue \global\doingtabfalse%
\global\pageaftertabfalse}%
%----------- init -----------------------------------------%
\AtBeginDocument{%
\edef\oldoutput{\the\output}
\output={\the\outputpretest\ifoutput\oldoutput\fi}%
\outputpretest={\outputtrue}%
\edef\oldeverypar{\the\everypar}% PERCHE' NESSUNO LO RIMETTE PIU' A POSTO?***
}
%---- tests during \outputpretest -------------------------%
\def\do@test{%
\ifnum\outputpenalty=-10005
\setbox\pagebox=\vbox{\unvbox255}%
\global\pageht=\ht\pagebox
\global\outputfalse
\unvbox\pagebox
\dimen0=\pageht\advance\dimen0 by 2 \baselineskip
\ifdim\dimen0>\vsize\pagebreak[4]\global\pageht\z@\fi
\else
\global\outputtrue
\ifnum\outputpenalty>-\@Mi %ie \marginpar's give penalty.
\ifdoingtab\global\pageaftertabtrue\fi
\ifdoingfig\global\pageafterfigtrue\fi
\fi
\fi}%
%-------- check for side ----------------------------------%
\def\chk@side#1{% DEFINED \fl@sid, BUT IF CHOSEN A DEFAULT, ELIMINATE ALL ***
{\edef\fl@sid{#1}%
\if\fl@sid r\global\oddpagestrue\fi
\if\fl@sid l\global\oddpagesfalse\fi
\if\fl@sid p%
\ifodd\c@page\global\oddpagesfalse
\else\global\oddpagestrue\fi
\fi
\if\fl@sid v%
\ifnum\theOptionTest=0%
\ifodd\c@page\global\oddpagesfalse
\else\global\oddpagestrue\fi
\else
\ifodd\theOptionTest\global\oddpagestrue
\else\global\oddpagesfalse\fi
\fi
\fi
}}%
%-------- get point in page -------------------------------%
\def\get@pageht{{\everypar={\relax}\setbox0=\lastbox
\parindent=\wd0 \parskip=\z@ \par
\penalty-10005 \leavevmode}}%
%-------- try at everypar ---------------------------------%
\def\tryfig{%
\global\fl@ht\ht\figbox
\global\fl@wd\wd\figbox
\global\let\fl@gut\figgutter
\iftryingfig
\get@pageht
\dimen0=\vsize
\advance\dimen0 by -\pageht
\advance\dimen0 by -2\baselineskip
\ifdim\dimen0>\fl@ht
\chk@side\@fside
\dimen0=0.3\baselineskip
\vrule depth \dimen0 width \z@%height 1.5\baselineskip
\vadjust{\kern -\dimen0
\vtop to \dimen0{%
\baselineskip=\dimen0
\vss \vbox to 1ex{%
\ifoddpages
\hb@xt@\hsize{\hss\copy\figbox}%
\else% leftsetting
\hb@xt@\hsize{\copy\figbox\hss}%
\fi
\vss}\null}}%
\global\tryingfigfalse
\global\doingfigtrue
\global\startpageht=\pageht
\global\htdone=\z@
\dohang
\ifnum\nosuccesstryfig>0%
\JHEP@mess{Flt. fig. \the\ffigcount\space set on page \the\count0,
shifted \the\nosuccesstryfig\space par(s) forward}%
% \else
% \JHEP@mess{Floating figure \the\ffigcount\space
% set on page \the\count0}%
\fi
\else
\global\advance\nosuccesstryfig by 1
\fi
\else% % IF NOT TRYING
\ifdoingfig
\get@pageht
\global\htdone=\pageht
\global\advance\htdone by -\startpageht
\ifpageafterfig
\global\doingfigfalse% IN CASE A PAGEBREAK JUST BELOW?
\else
\dimen0=\fl@ht
\advance\dimen0 by .85\baselineskip % .85 ENOUGH: WE ARE AT NEW PAR
% \typeout{FLOAT HT: \the\dimen0, HTDONE: \the\htdone}
\ifdim\htdone<\dimen0%
\dohang
\else
\global\doingfigfalse
\fi
\fi
\ifdoingfig\else\global\figprocessingfalse\fi% IF NO MORE DOING.
\else
\global\outputpretest={\outputtrue}% NOT RESTORE EVPAR IF NOT DOING?
\fi% % END IF DOING
\fi% % END IF TRYING
}
\def\trytab{%
\global\fl@ht\ht\tabbox
\global\fl@wd\wd\tabbox
\global\let\fl@gut\tabgutter
\iftryingtab
\get@pageht
\dimen0=\vsize
\advance\dimen0 by -\pageht
\advance\dimen0 by -2\baselineskip
\ifdim\dimen0>\fl@ht
\chk@side\@tside
\dimen0=0.3\baselineskip
\vrule depth \dimen0 width \z@
\vadjust{\kern -\dimen0
\vtop to \dimen0{%
\baselineskip=\dimen0
\vss \vbox to 1ex{%
\ifoddpages
\hb@xt@\hsize{\hss\copy\tabbox}%
\else% leftsetting
\hb@xt@\hsize{\copy\tabbox\hss}%
\fi
\vss}\null}}%
\global\tryingtabfalse
\global\doingtabtrue
\global\startpageht=\pageht
\global\htdone=\z@
\dohang
\ifnum\nosuccesstrytab>0%
\JHEP@mess{Flt. tab. \the\ftabcount\space set on page \the\count0,
shifted \the\nosuccesstrytab\space par(s) forward}%
% \else
% \JHEP@mess{Floating table \the\ftabcount\space
% set on page \the\count0}%
\fi
\else
\global\advance\nosuccesstrytab by 1
\fi
\else
\ifdoingtab
\get@pageht
\global\htdone=\pageht
\global\advance\htdone by -\startpageht
\ifpageaftertab
\global\doingtabfalse
\else
\dimen0=\fl@ht
\advance\dimen0 by .85\baselineskip % .85 ENOUGH: WE ARE AT NEW PAR
\ifdim\htdone<\dimen0%
\dohang
\else
\global\doingtabfalse
\fi
\fi
\ifdoingtab\relax\else\global\tabprocessingfalse\fi
\else
\global\outputpretest={\outputtrue}%
\fi
\fi
}
%----- hanging lord -----------------------------%
\def\dohang{%
\dimen0=\fl@ht
\advance\dimen0 by -\htdone
\advance\dimen0 by 1.999\baselineskip % (16/12/98) WAS 1.49 BUT
\hangcount=\dimen0 % TEX ROUNDS ON STRICT INTEGERS
\divide\hangcount by \baselineskip % IN THIS DIVISION!
% \typeout{HANGCOUNT: \the\hangcount.}
% \ifnum\hangcount > 0 % IF ZERO ALSO INDENT MUST BE NULL. (7/6/97)
\dimen0=\fl@wd
\advance\dimen0 by \fl@gut
\ifoddpages
\global\hangindent=-\dimen0% placing right
\else
\global\hangindent=\dimen0% placing left
\fi
\global\hangafter=-\hangcount
% \fi
}
%----------------------- TWO NEWITEMS ---------------------%
\newcommand{\fltitem}[2][\z@]{\setlength{\fltitemwidth}{\linewidth}%
\addtolength{\fltitemwidth}{-\floatfltwidth}%
\addtolength{\fltitemwidth}{-0.5em}%
\item \parbox[t]{\fltitemwidth}{#2}\\[#1]}
\newcommand{\fltditem}[3][\z@]{\setlength{\fltitemwidth}{\linewidth}%
\addtolength{\fltitemwidth}{-\floatfltwidth}%
\addtolength{\fltitemwidth}{-0.5em}%
\item[#2] \parbox[t]{\fltitemwidth}{#3}\\[#1]}
%%%%%%\endinput
%%
%% End of ex-file `floatflt.sty'.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CAPTIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newlength\abovecaptionskip
\newlength\belowcaptionskip
\setlength\abovecaptionskip{8\p@}
\setlength\belowcaptionskip{0\p@}
\long\def\@makecaption#1#2{%
\if@hyper{\edef\@pippo{the\@captype}
\name{\@captype\expandafter\csname\@pippo\endcsname}}\fi%
\vskip\abovecaptionskip
{\let\label\@gobble% % FN 10.2.97 REMOVED LATEX BUG:
\let\index\@gobble% % LARGE CAPTIONS PROCESS LABEL
\let\glossary\@gobble% % TWO TIMES.
\sbox\@tempboxa{\small {\bfseries #1:} #2}% %
\global\dimen0\wd\@tempboxa}% %
\ifdim \dimen0 >\hsize
\small {\bfseries #1:} #2\par
\else
\global\@minipagefalse \sbox\@tempboxa{\small {\bfseries #1:} #2}%
\hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
\fi
\vskip\belowcaptionskip}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FONTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
\DeclareRobustCommand*\cal{\@fontswitch\relax\mathcal}
\DeclareRobustCommand*\mit{\@fontswitch\relax\mathnormal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TOC %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand\tocsecs{}
\newcommand\@pnumwidth{1.55em}
\newcommand\@tocrmarg{2.55em}
\newcommand\@dotsep{4.5}
\setcounter{tocdepth}{3}
\newcommand\tableofcontents{%
\section*{\contentsname\label{-TOC-}}
% \@mkboth{%
% \MakeUppercase\contentsname}{\MakeUppercase\contentsname}%
\@starttoc{toc}%
\gdef\tableofcontents{\JHEP@igno{\tableofcontents\space already done}}}
\newcommand*\l@part[2]{\def\hyp@typ{part}%
\ifnum \c@tocdepth >-2\relax
\addpenalty\@secpenalty
\addvspace{2.25em \@plus\p@}%
\begingroup
\setlength\@tempdima{3em}%
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
{\leavevmode
\large \bfseries \tocsecs#1\hfil \hb@xt@\@pnumwidth{\hss
\href{\hash pag#2}{#2}}}\par
\nobreak
\if@compatibility
\global\@nobreaktrue
\everypar{\global\@nobreakfalse\everypar{}}%
\fi
\endgroup
\fi}
\newcommand*\l@section[2]{\def\hyp@typ{sec}%
\ifnum \c@tocdepth >\z@
\addpenalty\@secpenalty
\addvspace{1.0em \@plus\p@}%
\setlength\@tempdima{1.8em}% %WIDTH OF SECT. N.
\begingroup
\let\d@t.%
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode \bfseries
\advance\leftskip\@tempdima
\hskip -\leftskip
\tocsecs#1\nobreak\hfil %HREF HERE WOULD GET ALSO THE TITLE(...)
\nobreak\hb@xt@\@pnumwidth{\hss \href{\hash pag#2}{#2}}\par
\endgroup
\fi}
\newcommand*\l@subsection{\def\hyp@typ{sec}\@tocline{2}{1.8em}{2.3em}}
\newcommand*\l@subsubsection{\def\hyp@typ{sec}\@tocline{3}{4.1em}{3.1em}}
\newcommand*\l@paragraph{\def\hyp@typ{sec}\@tocline{4}{7.2em}{4.3em}}
\newcommand*\l@subparagraph{\def\hyp@typ{sec}\@tocline{5}{10.5em}{5em}}
\if@hyper
\let\old@dtl\@dottedtocline
\def\@dottedtocline#1#2#3#4#5{\old@dtl{#1}{#2}{#3}{#4}{%
\href{\hash pag#5}{#5}}}
\fi
\def\@tocline#1#2#3#4#5{%
\ifnum #1>\c@tocdepth \else
\vskip \z@ \@plus.2\p@
{\leftskip #2\relax \rightskip \@tocrmarg \parfillskip -\rightskip
\parindent #2\relax\@afterindenttrue
\interlinepenalty\@M
\leavevmode
\@tempdima #3\relax
\advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
{#4}\nobreak
%\leaders\hbox{$\m@th
% \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
% mu$}
\hfill
\nobreak
\hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor \href{\hash pag#5}{#5}}%
\par}%
\fi}
%%%%%%%%%%%%%%%%%%%%%% LIST OF FIGURES AND TABLES %%%%%%%%%%%%%%%%%%%%%
\if@draft % ONLY IF DRAFT! ...
\newcommand\listoffigures{%
\section*{\listfigurename
\@mkboth{\MakeUppercase\listfigurename}%
{\MakeUppercase\listfigurename}}%
\@starttoc{lof}%
}
\newcommand*\l@figure{\def\hyp@typ{figure}\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand\listoftables{%
\section*{\listtablename
\@mkboth{%
\MakeUppercase\listtablename}{\MakeUppercase\listtablename}}%
\@starttoc{lot}%
}
\newcommand*\l@table{\def\hyp@typ{table}\@dottedtocline{1}{1.5em}{2.3em}}
\else
\newcommand\listoffigures{\JHEP@ignol{List of figures (draft mode only)}}
\newcommand\listoftables{\JHEP@ignol{List of tables (draft mode only)}}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%%%%%%%%%%***
\newdimen\bibindent
\setlength\bibindent{1.5em}
\newenvironment{thebibliography}[1]% %UNFORTUNATELY MODIFIED..
{\bgroup\small\section*{\refname
\@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}}%
\list{\name{bib\@arabic\c@enumiv}% HOPE!
\@biblabel{\@arabic\c@enumiv}}%
{\settowidth\labelwidth{\@biblabel{#1}}%
\leftmargin\labelwidth
\advance\leftmargin\labelsep
\@openbib@code
\usecounter{enumiv}%
\let\p@enumiv\@empty
\renewcommand\theenumiv{\@arabic\c@enumiv}}%
\sloppy\clubpenalty4000\widowpenalty4000%
\sfcode`\.\@m}
{\def\@noitemerr
{\@latex@warning{Empty `thebibliography' environment}}%
\endlist\egroup}
\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em}
\if@draft
\let\old@bbt\@bibitem\let\old@lbbt\@lbibitem% LOOK THE DISASTER HERE BELOW.
\def\@lbibitem[#1]#2{\old@lbbt[#1]{#2}\reversemarginpar{\sf\bfseries\small#2}}
\def\@bibitem#1{\old@bbt{#1}\reversemarginpar{\sf\bfseries\small#1}}
\fi
\let\@openbib@code\@empty
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JOURNALS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\@spires#1{\href{http://www-spires.slac.stanford.edu/spires/find/hep/www?j=#1}}
% %PITY THAT target IS NOT IMPLEMENTED.
\catcode`\%=12
\newcommand\adp[3] {\@spires{ADPHA%2C#1%2C#3}
{{\it Adv.\ Phys.\ }{\bf #1} (#2) #3}}
\newcommand\ap[3] {\@spires{APNYA%2C#1%2C#3}
{{\it Ann.\ Phys.\ (NY) }{\bf #1} (#2) #3}}
\newcommand\app[3] {\@spires{APHYE%2C#1%2C#3}
{{\it Astropart.\ Phys.\ }{\bf #1} (#2) #3}}
\newcommand\appol[3] {\@spires{APPOL%2C#1%2C#3}
{{\it Acta Phys.\ Polon.\ }{\bf #1} (#2) #3}}
\newcommand\arnps[3] {\@spires{ARNUA%2C#1%2C#3}
{{\it Ann.\ Rev.\ Nucl.\ Part.\ Sci.\ }{\bf #1} (#2) #3}}
\newcommand\atmp[3] {\@spires{00203%2C#1%2C#3}
{{\it Adv.\ Theor.\ Math.\ Phys.\ }{\bf #1} (#2) #3}}
\newcommand\cpc[3] {\@spires{CPHCB%2C#1%2C#3}
{{\it Comput.\ Phys.\ Commun.\ }{\bf #1} (#2) #3}}
\newcommand\cmp[3] {\@spires{CMPHA%2C#1%2C#3}
{{\it Comm.\ Math.\ Phys.\ }{\bf #1} (#2) #3}}
\newcommand\dmj[3] {\@spires{DUMJA%2C#1%2C#3}
{{\it Duke Math.\ J. }{\bf #1} (#2) #3}}
\newcommand\epjc[3] {\@spires{EPHJA%2CC#1%2C#3}
{{\it Eur.\ Phys.\ J. }{\bf C #1} (#2) #3}}
\newcommand\jmp[3] {\@spires{JMAPA%2C#1%2C#3}
{{\it J.\ Math.\ Phys.\ }{\bf #1} (#2) #3}}
\newcommand\jgp[3] {\@spires{JGPHE%2C#1%2C#3}
{{\it J.\ Geom.\ Phys.\ }{\bf #1} (#2) #3}}
\newcommand\jphg[3] {\@spires{JPAGB%2CG#1%2C#3}
{{\it J. Phys.\ }{\bf G #1} (#2) #3}}
\newcommand\cqg[3] {\@spires{CQGRD%2C#1%2C#3}
{{\it Class.\ and Quant.\ Grav.\ }{\bf #1} (#2) #3}}
\newcommand\hpa[3] {\@spires{HPACA%2C#1%2C#3}
{{\it Helv.\ Phys.\ Acta }{\bf #1} (#2) #3}}
\newcommand\jhep[3] {\href{http://jhep.sissa.it/stdsearch?paper=#1%28#2%29#3}
{{\it J. High Energy Phys.\ }{\bf #1} (#2) #3}}
\newcommand\lmp[3] {\@spires{LMPHD%2CA#1%2C#3}
{{\it Lett.\ Math.\ Phys.\ }{\bf #1} (#2) #3}}
\newcommand\npa[3] {\@spires{NUPHA%2CA#1%2C#3}
{{\it Nucl.\ Phys.\ }{\bf A #1} (#2) #3}}
\newcommand\npb[3] {\@spires{NUPHA%2CB#1%2C#3}
{{\it Nucl.\ Phys.\ }{\bf B #1} (#2) #3}}
\newcommand\npps[3] {\@spires{NUPHZ%2C#1%2C#3}
{{\it Nucl.\ Phys.\ }{\bf #1} {\it(Proc.\ Suppl.)} (#2) #3}}
\newcommand\pla[3] {\@spires{PHLTA%2CA#1%2C#3}
{{\it Phys.\ Lett.\ }{\bf A #1} (#2) #3}}
\newcommand\plb[3] {\@spires{PHLTA%2CB#1%2C#3}
{{\it Phys.\ Lett.\ }{\bf B #1} (#2) #3}}
\newcommand\ppnp[3] {\@spires{PPNPD%2C#1%2C#3}
{{\it Prog.\ Part.\ Nucl.\ Phys.\ }{\bf #1} (#2) #3}}
\newcommand\pr[3] {\@spires{PHRVA%2C#1%2C#3}
{{\it Phys.\ Rev.\ }{\bf #1} (#2) #3}}
\newcommand\pra[3] {\@spires{PHRVA%2CA#1%2C#3}
{{\it Phys.\ Rev.\ }{\bf A #1} (#2) #3}}
\newcommand\prb[3] {\@spires{PHRVA%2CB#1%2C#3}
{{\it Phys.\ Rev.\ }{\bf B #1} (#2) #3}}
\newcommand\prc[3] {\@spires{PHRVA%2CC#1%2C#3}
{{\it Phys.\ Rev.\ }{\bf C #1} (#2) #3}}
\newcommand\prd[3] {\@spires{PHRVA%2CD#1%2C#3}
{{\it Phys.\ Rev.\ }{\bf D #1} (#2) #3}}
\newcommand\pre[3] {\@spires{PHRVA%2CE#1%2C#3}
{{\it Phys.\ Rev.\ }{\bf E #1} (#2) #3}}
\newcommand\prep[3] {\@spires{PRPLC%2C#1%2C#3}
{{\it Phys.\ Rep.\ }{\bf #1} (#2) #3}}
\newcommand\prl[3] {\@spires{PRLTA%2C#1%2C#3}
{{\it Phys.\ Rev.\ Lett.\ }{\bf #1} (#2) #3}}
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cheatography.com
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\documentclass[10pt,a4paper]{article}
% Packages
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%\usepackage{opensans} % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem} % For underlining links
% Most of the following are not required for the majority
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\vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Mercurial workflow Cheat Sheet}}}} \\
\normalsize{by \textcolor{DarkBackground}{alexo\_o} via \textcolor{DarkBackground}{\uline{cheatography.com/20190/cs/3031/}}}
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\vspace{-2pt}alexo\_o \\
\uline{cheatography.com/alexo-o} \\
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\mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}} \\
\vspace{-2pt}Published 11th December, 2014.\\
Updated 10th May, 2016.\\
Page {\thepage} of \pageref{LastPage}.
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\vfill
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\vspace{-5pt}
%\includegraphics[width=48px,height=48px]{dave.jpeg}
Measure your website readability!\\
www.readability-score.com
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\end{multicols}}
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% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
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\begin{multicols*}{3}
\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Hotfix branches}} \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{May branch off from: \newline % Row Count 1 (+ 1)
`default` \newline % Row Count 2 (+ 1)
Must merge back into: \newline % Row Count 3 (+ 1)
`dev` and `default` \newline % Row Count 4 (+ 1)
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\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\mymulticolumn{1}{x{5.377cm}}{Creating the hotfix branch: \newline `\textgreater{} hg up default` \newline `\textgreater{} hg branch hotfix-1.2.1` \newline `\textgreater{} ./bump-version.sh 1.2.1` \newline `\textgreater{} hg commit -m "Bumped version number to 1.2.1"` \newline `\textgreater{} ./bugfix` \newline `\textgreater{} hg commit -m "Fixed severe production problem"` \newline \newline Finishing a hotfix branch: \newline `\textgreater{} hg up default` \newline `\textgreater{} hg merge hotfix-1.2.1` \newline `\textgreater{} hg tag 1.2.1` \newline `\textgreater{} hg com` \newline `\textgreater{} hg up dev` \newline `\textgreater{} hg merge hotfix-1.2.1` \newline `\textgreater{} hg com` \newline `\textgreater{} hg up hotfix-1.2.1` \newline `\textgreater{} hg com -m "closing branch" -{}-close-branch`} \tn
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\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{May branch off from: \newline % Row Count 1 (+ 1)
`dev` \newline % Row Count 2 (+ 1)
Must merge back into: \newline % Row Count 3 (+ 1)
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Branch naming convention: \newline % Row Count 5 (+ 1)
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\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Creating a feature branch: \newline `\textgreater{} hg up dev` \newline `\textgreater{} hg branch myfeature` \newline \newline Incorporating a finished feature on develop: \newline `\textgreater{} hg up dev` \newline `\textgreater{} hg pull` \newline `\textgreater{} hg merge myfeature` \newline `\textgreater{} hg com -m "Merge myfeature"` \newline `\textgreater{} hg up myfeature` \newline `\textgreater{} hg com -m "closing branch" -{}-close-branch` \newline `\textgreater{} hg up dev` \newline `\textgreater{} hg push` \newline \newline \newline \newline \newline \newline -} \tn
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}
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\SetRowColor{DarkBackground}
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\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{May branch off from: \newline % Row Count 1 (+ 1)
`dev` \newline % Row Count 2 (+ 1)
Must merge back into: \newline % Row Count 3 (+ 1)
`dev and default` \newline % Row Count 4 (+ 1)
Branch naming convention: \newline % Row Count 5 (+ 1)
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\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Creating a release branch: \newline `\textgreater{} hg up dev` \newline `\textgreater{} hg branch release-1.2` \newline `\textgreater{} ./bump-version.sh 1.2` \newline `\textgreater{} hg commit -m "Bumped version number to 1.2"` \newline \newline Finishing a release branch: \newline `\textgreater{} hg up default` \newline `\textgreater{} hg merge release-1.2` \newline `\textgreater{} hg tag 1.2` \newline `\textgreater{} hg com` \newline `\textgreater{} hg up dev` \newline `\textgreater{} hg merge release-1.2` \newline `\textgreater{} hg com` \newline `\textgreater{} hg up release-1.2` \newline `\textgreater{} hg com -m "closing branch" -{}-close-branch` \newline \newline \newline \newline \newline \newline \newline \newline \newline -} \tn
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}
% That's all folks
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https://ctan.math.washington.edu/tex-archive/info/examples/lgc2/D-11-7.ltx
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washington.edu
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%%
%% It may be distributed and/or modified under the conditions
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https://kohnlehome.de/datenbanken/kardinalitaet.tex
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kohnlehome.de
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Kardinalität - Multiplizität
\end{center}
\paragraph{Kardinalität (Cardinality):}
Anzahl der Elemente einer Menge
\paragraph{Multiplizität (Multiplicity):}
Bereich min .. max
\section*{Kardinalität}
\begin{tabular}{c | c | c | c}
UML & Chen-Notation & Modifizierte Chen-Notation (MC) & Bedeutung\\ \hline \hline
0..1 &1 & c (can) & 0 oder 1\\
1..1 & - & 1 & genau 1 \\ \hline
0..* & m, n & mc & beliebig viele \\
1..* & - & m (must, multiple) & mindestens 1
\end{tabular}
\section*{Assoziation - Relationship}
\begin{tabular}{l | c | c | c | c}
& Chen-Notation & MC & UML & Relationale Datenbank\\ \hline \hline
One-To-One & 1:1 & c - c & 0..1 - 0..1 & PK - FK \\
& & c - 1 & 0..1 - 1..1 & \\
& & 1 - 1 & 1..1 - 1..1 & \\ \hline
One-To-Many & 1:n & c - mc & 0..1 - 0..* & PK - FK \\
& & 1 - mc & 1..1 - 0..* & \\
& & c - m & 0..1 - 1..* & \\
& & 1 - m & 1..1 - 1..* & \\ \hline
Many-To-Many & m:n & mc - mc & 0..* - 0..* & Beziehungstabelle \\
& & m - mc & 1..* - 0..* & \\
& & m - m & 1..* - 1..* & \\ \hline
\end{tabular}
\end{document}
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There is a sizable and systematic discrepancy between experimental data on the $b\overline{b}$ production in $p\overline{p}$, $\gamma p$ and $\gamma \gamma$ collisions and the existing theoretical calculations within perturbative QCD. The uncertainty is coming from renormalization and factorization scale dependence of finite order perturbative calculations of the total cross section of $b\overline{b}$ production in such collisions and will be discussed for $p\overline{p}$ collision in detail. If we employ the approach of "Complete RG-improvement(CORGI)", in which one should perform a resummation to all-orders of renormalization and factorization group-predictable terms at each order of perturbation theory, then the scales dependence will be avoided and the mentioned discrepency is reduced significantly.
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\[\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{%
-1}{6}}}{\sqrt{\pi}(48)^{\ifrac{1}{6}}\mathop{\Gamma\/}\nolimits\!\left(\frac{%
5}{6}\right)}\int_{0}^{\infty}e^{-t}t^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}%
\right)^{-\ifrac{1}{6}}\mathrm{d}t,\]
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\section{Links}
%% NOTE: this should re restructured...
\begin{itemize}
\item Forward secrecy mit debian wheezy: postfix, dovecot, nginx: \url{https://www.incertum.net/archives/72-Forward-Secrecy-mit-Debianwheezy-postfix,-dovecot,-nginx.html}
\item SSL cipher settings: \url{http://www.skytale.net/blog/archives/22-SSL-cipher-setting.html}
\item Perfect Forward Secrecy mit Apple Mail: \url{http://www.kuketz-blog.de/perfect-forward-secrecy-mit-apple-mail/}
\item Perfect Forward Secrecy (PFS) f\"ur Postfix und Dovecot: \url{https://www.heinlein-support.de/blog/security/perfect-forward-secrecy-pfs-fur-postfix-und-dovecot/#more-1085}
\item Elliptic curves and their implementation (04 Dec 2010): \url{https://www.imperialviolet.org/2010/12/04/ecc.html}
\item A (relatively easy to understand) primer on elliptic curve cryptography: \url{http://arstechnica.com/security/2013/10/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography}
\item Duraconf (Jake Applebaum's github): \url{https://github.com/ioerror/duraconf}
\item Attacks on SSL a comprehensive study of BEAST, CRIME, TIME, BREACH, LUCKY 13 \& RC4 Biases: \url{https://www.isecpartners.com/media/106031/ssl_attacks_survey.pdf}
\item EFF How to deploy HTTPS correctly: \url{https://www.eff.org/https-everywhere/deploying-https}
\item Bruce Almighty: Schneier preaches security to Linux faithful (on not recommending to use Blowfish anymore in favour of Twofish): \url{https://www.computerworld.com.au/article/46254/bruce_almighty_schneier_preaches_security_linux_faithful/?pp=3}
\end{itemize}
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https://svn.geocomp.uq.edu.au/escript/trunk/doc/user/pyvisi.tex?r1=879&r2=882&pathrev=999&sortby=date&view=patch
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--- trunk/doc/user/pyvisi.tex 2006/10/25 03:56:34 879
+++ trunk/doc/user/pyvisi.tex 2006/10/27 08:15:33 882
@@ -89,10 +89,6 @@
\end{classdesc}
The following are some of the methods available:
-\begin{methoddesc}[Camera]{setClippingRange}{near_clipping, far_clipping}
-Set the near and far clipping plane of the camera.
-\end{methoddesc}
-
\begin{methoddesc}[Camera]{setFocalPoint}{position}
Set the focal point of the camera.
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@@ -133,6 +129,10 @@
View the right side of the rendered object.
\end{methoddesc}
+\begin{methoddesc}[Camera]{isometricView}{}
+View the isometric side of the rendered object.
+\end{methoddesc}
+
The following is a sample code using the \Camera class.
\fig{fig:camera.1} shows the corresponding output.
\verbatiminput{../examples/drivercamera.py}
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% -*- coding: utf-8 -*-
% !TEX program = xelatex
% 直接包含 A4 试卷的 PDF 文件，生成双栏的 A3 试卷
\documentclass[a3input]{jnuexam}
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%\includepdf[pages=-,nup=2x1,offset=0 0,delta=0 0,frame]{exam-b}
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\input booklet
\hskip1pt\vadjust{\vskip2.5in}\par
\pageno=2
\footline={\hfill\folio\hfill}
{
\font\rm=cmssi9 \rm
\baselineskip=9pt
\obeylines
\parindent=14pt
\narrower
\parskip=0pt
From this hour I ordain myself loos'd of limits and
\quad imaginary lines
Going where I list, my own master total and absolute,
Listening to others, considering well what they say,
Pausing, searching, receiving, contemplating,
Gently, but with undeniable will, divesting myself
\quad of the holds that would hold me.
}
\line{\hfill -- Walt Whitman}
\smallskip
Welcome to Without Borders Anarchist Conference and Festival '89! ¨
This is the fourth such North American conference in as many¨
years. In July of 1986 some anarchists in Chicago called a¨
national conference to commemorate the 100th anniversary of the¨
Haymarket Affair. Over 300 people came together for three days to¨
explore anarchist ideas, history and practice with the goal of¨
establishing an on going network of North American anarchists. ¨
The gathering was so successful in creating personal and¨
political connections between those attending that they have¨
become yearly events with conferences in Minneapolis in 1987 and¨
Toronto in 1988. As the number of folks attending has grown (over¨
900 in Toronto), the work and play of weaving a North American¨
network/movement has continued.
The title of this years meeting, ``Without Borders Anarchist¨
Conference and Festival 89'' reflects the desire of many¨
attending last year's gathering that we begin to reach out to¨
anarchists around the world. It also expresses our commitment to¨
struggle against all forms of ``borders'' that keep us from¨
living as free, spontaneous and empowered people in harmony with¨
each other and the earth. The conference is a context for serious¨
discussion and organizing around ideas and issues of importance¨
to the anarchist community. The festival highlights and¨
celebrates the vitality of contemporary anarchism as embodied in¨
the works of poets, musicians, artists and performers whose art¨
draws on and expresses the spirit and vision of anarchism.
Anarchists in the Bay Area began meeting last August to work out¨
the logistics and explore our visions for the Con/Fest. ¨
Philadelphia comrades hosted a national planning meeting in¨
January. At that meeting, consensus was reached that two days of¨
the conference would be devoted to workshops dealing with the¨
issues of racism and sexism. One day would be set aside for the¨
networks to meet. Please see the schedule for these days.
Many people have spent countless hours to make this gathering a¨
reality. Putting on one of these events is a full time job (yech)¨
for 20 people. A heartfelt thanks to all who have made Without¨
Borders Con/Fest happen.
Anarchism, the belief and practice that people are capable of¨
directing and living their lives without the imposition of¨
external authority, is one of the greatest visions of humanity. ¨
With its emphasis on people acting responsibly; practicing¨
individual and collective direct action; fighting injustice; and¨
creating a freer society, it shines the brightest light in the¨
tunnel of contemporary daily life, which constantly threatens to¨
entomb the world.
vive l'anarchie!
\line{\hfil -- Without Borders}
\bigskip
\centerline{\hl Donations to the Con/Fest}
\smallskip
Without Borders has cost approximately \$15,000 to put on. Figure¨
that to be about \$20 per person and please donate accordingly. ¨
If you can afford to, give more to help cover those who don't¨
have the \$20.
Thanks to all who have sent in donations over the past year. By¨
the end of the gathering we plan to have as complete an¨
accounting as possible of all funds received and spent. The¨
disbursement of any remaining funds will be decided at a meeting¨
on the last day of the Con/Fest.
\bigskip
\centerline{\hl The Daily Newsletter}
\smallskip
There's going to be a newsletter, hopefully generated before the¨
start of each day, that will cover last minute changes, additions¨
and general info on the gathering. In addition, it will carry¨
article, discussions, arguments and visions, written by you, the¨
con/fest participants. We need your help, skills and input; ask¨
at the registration table.
\bigskip
\centerline{\hl This Conference and the Media}
\smallskip
The final Without Borders general planning meeting consensed,¨
reiterating previous decisions, that commercial mass-media (TV,¨
radio, print, etc.) would not be allowed to use or bring¨
recording devices of any kind into the building(s) in which the¨
conference is to be held.
In addition, ``friendly'' media (to be determined on a
case-by-case basis) is also asked not to use recording devices¨
without making their presence known at the morning assembly, as¨
well as in the workshops themselves.
There will be leaflets available to the general public that will¨
give basic information about the conference; these will also¨
serve a ``media-kit''.
In general, be considerate of others if you plan to record any¨
sessions or general goings-on at the conference; ask people if¨
they mind being taped or photographed, and so on.
Many people will be taping and recording their sessions for later¨
use; if you have a problem with this you might want to make¨
arrangements not to be there (or to get a copy of the tape for¨
yourself!)
\bigskip\goodbreak
\centerline{\hl Fashion Advice for}
\centerline{\hl Your Visit to San Francisco}
No matter how warm, beautiful and sunny it may be when you get up¨
in the morning (or afternoon) remember, DON'T GO OUT WITHOUT A¨
JACKET! Puss print is fine as long as it's fake. Sweaters,¨
however, just won't cut it. San Francisco is notorious for many¨
things, especially sudden shifts in the weather. One minute you¨
can be basking in the hot sun, the next drowning in fog with 50¨
degree temperatures. If you don't heed this sensible fashion¨
advice, don't say we didn't warn you.
\vskip\parskip
\line{\hfil -- The Fashion Committee}
\bye
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\title{Anarcho-syndicalism in the 20th Century}
\date{Monday, September 28th 2009}
\author{Vadim Damier}
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\chapter{Translator's introduction}
In the first decade of the 21st century many labour unions and labour federations worldwide celebrated their 100th anniversaries. This was an occasion for reflecting on the past century of working class history. Mainstream labour organizations typically understand their own histories as never-ending struggles for better working conditions and a higher standard of living for their members – as the wresting of piecemeal concessions from capitalists and the State.
But there is another current of the labour movement which aims somewhat higher. The anarcho-syndicalists set as their goal no less than seizing control of society from Capitalists and the State and instituting worker self-management in the spheres of production, distribution, and consumption.
The standard work in English on anarcho-syndicalism has long been a translation of [URL=\Slash{}tags\Slash{}rudolf-rocker] Rudolf Rocker’s slim book on the subject,1 written over 70 years ago by a key figure in the movement. Since Rocker’s book was written, there have been many limited studies of the movement but nothing much in the way of an attempt to grasp the movement as a whole or cover the entire sweep of its history.
Anarcho-syndicalism has always been a global movement embraced by many different cultures and indeed modes of production. Its appearance in so many different settings has created a daunting task for historians who would do justice to its scope and diversity. The source materials are found in many different languages and in widely scattered archives which have not always been accessible. The Russian historian V. Damier, author of a monumental history of the anarcho- syndicalist International in the 1920’s - 1930’s,2 has tackled this task with great skill and the mastery of an enormous variety of material. Even in this brief survey of the history of the movement, he has had to refer to archival sources since the secondary literature is inadequate on many vital aspects of the movement.
Anarcho-syndicalism in the 20th Century was first published in Moscow in 2000. For the English edition the author has provided additional material: an historiographic essay, more in-depth coverage of the Spanish Revolution, an update on contemporary Russia, etc. As a result, the English edition is at least twice as long as the original Russian book.
Although addressed primarily to a Russian readership by someone active in the Russian anarcho-syndicalist movement, it is hoped that with this English edition the book will find the global audience it deserves.
To assist the reader in tracking down references, where a footnote refers to previous documentation (by means of “op. cit.” or some other device), the number X of the previous note is given in brackets “(nX).”
I would like to thank Vadim Damier for his generous assistance in preparing this edition; also Gail Silvius for expert editorial work.
<em>Malcolm Archibald
September 2009<\Slash{}em>
\begin{enumerate}[1.]
\item\relax
R. Rocker, Anarcho-Syndicalism: Theory and Practice (AK Press, 2004). This work is in print in a number of English editions with slightly different titles, including electronic versions. The work was originally written in German.
\item\relax
Vadim Damier, Забытый Интернационал. Международное анархо- синдикалистское движение между двумя мировыми войнами. [The Forgotten International. The international anarcho-syndicalist movement between the two world wars.]: Vol. 1. От революционного синдикализма к анархо- синдикализму. 1918-1930. [From revolutionary syndicalism to anarchosyndicalism. 1918-1930.] (Moscow, 2006), 904 pp., ill.; Vol. 2. Международный анархо-синдикализм в условиях “Великого кризиса” и наступления фашизма. 1930-1939. [International anarcho-syndicalism faces the “Great Crisis” and fascist aggression. 1930-1939.] (Moscow, 2007), 736 pp., ill.
\end{enumerate}
\chapter{Preface}
Anarcho-syndicalism is a fundamental tendency in the global workers’ movement. It is made up of revolutionary unions of workers (“syndicat” in French means “trade union”), acting to bring about a stateless (anarchist), selfmanaged society.
Anarcho-syndicalism, the only mass variant of the anarchist movement in history, arose and acquired strength during a period of profound social, economic, and political changes – the first decades of the 20th century. In the countries which formed the “centre” of the global industrial-capitalist system, a transition to a developed industrial society was taking place, while on the “periphery” and “semi-periphery” the process of industrialization was still only getting started. The furious pace of social change often caused much suffering for the workers, forcing them to abandon traditional occupations and forms of life and pushing them into factories, frequently under onerous conditions. Former agricultural labourers were uprooted from their accustomed mode of life – conditioned by centuries, while skilled craftsmen experienced anguish when they were forced into narrowly specialized or unskilled work. The workers’ consciousness was scarred by the growing alienation and atomization of the human personality under the conditions of the rise of “mass society.”
The workers’ movement arose, to a significant extent, as an alternative force in relation to the industrial-capitalist system. As the Italian sociologist Marco Revelli has noted, “the modern State from the very beginning counterpoised these two forces to each other, as opposing tendencies.”1 Of course, this opposition could be regarded in different ways, either more radically (as in the case of the English Luddites who resisted the introduction of the factory system), or less radically (in the form of workers’ mutual aid societies, taking upon themselves control of the social sphere). But almost always this “early” workers’ movement was based on the spirit of independence, communal life, and collectivism preserved from the pre-industrial era of artisan workshops, in opposition to factory despotism. The division of labour had still not reached the level of Taylorist fragmentation.
Skilled workers, with a good understanding of their own work and where it fit in the production process, were quite capable of thinking they could control production on their own. On the other hand, the State mechanisms of social integration had not yet achieved sufficient development; rather the social sphere was almost completely controlled by the institutions and organizations of the workers’ movement (associations, syndicates, bourses de travail, etc.), which frequently were regarded as the basis for a possible self-managed alternative.
In the social realities of those times there was undeniably a place for radical tendencies which to some degree aimed at the dismantling, elimination, or radical transformation of the industrial-capitalist system. Although the majority of revolutionary syndicalists and anarchists were by no means immune from certain myths and concepts about the progressiveness of industrialism, still their social goals on the whole were oriented to a rupture with the system and its replacement with a new social structure based on selfmanagement and decision-making by means of agreements arrived at “from the bottom up.” Such views were compatible in many respects with the desires of the working masses in that epoch.
It is impossible to regard anarcho-syndicalism as some kind of insignificant, marginal phenomenona – as the extravagant escapades of “extremist grouplets” or the fantasies of salon intellectuals. This is a global movement which spread to countries as different as Spain and Russia, France and Japan, Argentina and Sweden, Italy and China, Portugal and Germany. It possesses strong, healthy social roots and traditions, and was able to attract hundreds of thousands, indeed millions, of wage workers. Anarcho-syndicalists not only took an active part in the most important social upheavals and conflicts of the 20th century, often leaving their own indelible imprint on these events, but also in many countries they formed the centre of a special, inimitable, working class culture with its own values, norms, customs, and symbols.
The ideas and traditions of anarcho-syndicalism, and the slogans it put forth about workplace and territorial selfmanagement, exerted an influence on many other social movements, including the workers’ councils of Budapest (1956), the student and youth uprisings of 1968, Polish “Solidarity” in 1980-81, the Argentine “popular assemblies,” etc.
Without knowing the history of anarcho-syndicalism, it is impossible to gain a reliable understanding of the history of many countries of the world; it is impossible to grasp in its fullness the course of development and destiny of humanity throughout the last 120 years.
\begin{enumerate}[1.]
\item\relax
M. Revelli, “Der Sozialstaat in den Brennesseln,” Die Aktion (Hamburg), no. 113\Slash{}119 (March 1994), p. 1932.
\end{enumerate}
\chapter{Part 1: Revolutionary Syndicalism}
\section{Chapter 1: From the First International to Revolutionary Syndicalism}
The prehistory of anarcho-syndicalism has its origin in the anti-authoritarian wing of the First International – the Bakuninists and federalists. The First International was created in 1864 and included adherents of various socialist tendencies. In the course of discussions in this international workers’ organization, ideas were formed about labour unions as an instrument of social liberation, about the role of the general strike, about the primacy of economic struggle, about the replacement of organs of the State by organizations of producers, about the self-management of society, and about “direct action,” i.e. the workers acting directly in their own interests and not handing over the job to political parties and leaders. After the split of the International in 1872, these views were upheld by anti-authoritarian anarchists. Their Marxist opponents set about creating social-democratic and socialist parties which engaged in the struggle for political power and the “conquest of the State.”
The rivalry between the two tendencies (anarchist and Marxist) gripped the workers’ movement. It developed unevenly and in different ways in various countries. But by the beginning of the 20th century it seemed the state socialists (social-democrats) had definitely gained the upper hand. Their opponents – the anti-authoritarian socialists (anarchists) – had been driven out of the workers’ movement in the majority of countries. On the one hand, the anarchists themselves had assisted this development at the end of the 19th century by their mistaken tactic of assuming they could bring forth revolution directly by means of symbolic acts of violence, without the necessity for solid, long-term organizing of working class forces. On the other hand, the rapid economic growth of the 1880’s strengthened illusions about the possibility of the peaceful improvement of the situation of the workers within the framework of the industrial-capitalist system.1
Social-democracy originated from the concept that the history of humanity proceeds along an ascending line of progress. Its theoreticians assumed that capitalism by its own development prepares the basis for the future socialist society, a society which in many aspects (technology, industrial and political centralization, division of labour, specialization of productive and social functions) becomes the continuation of capitalist society.2 The fundamental difference between the two social formations was located by the social-democrats in the control of political power: thus it was necessary to wrest power from the capitalists and transfer it to the workers, thereby putting the industrial machine created by capitalism at the service of everyone. In other words, the factory system of organizing production was to be extended to the whole of society. The liberation of the working classes and socialism were understood not as a break with the logic of capitalism and industrialization, but as their consequent development according to their own natural laws.
Towards the beginning of the 20th century the major labour union associations of Europe were controlled by social-democratic parties: the German and Austro-Hungarian Free Trade Unions; a number of French, Dutch, Belgian, and Portuguese workers’ associations; the General Workers’ Union (UGT) of Spain; the federations of trade unions of the Scandinavian countries, Switzerland, etc. The majority of British trade unions endorsed parliamentary socialism and supported the creation of the Labour Party.
The characteristic tactic of the social-democrats in the trade union movement consisted in subjecting the mass workers’ movements to the party line, strengthening the power and influence of the union bureaucracy and its control over the disbursement of union funds, and promoting an orientation towards purely economic struggle while leaving political and social questions entirely to the competence of the party.
Anarchists and other anti-authoritarian socialists retained influence only in the workers’ movements of Spain and Latin America, and also to some extent in workers’ organizations in France, Portugal, and Italy.
However, at the beginning of the 20th century the hegemony of social-democracy was challenged. Dissatisfaction with the parliamentary strategy of the workers’ parties generated not only intra-party left oppositions, but also resistance in the labour union milieu. A new radical current arose – revolutionary syndicalism. This term began to be applied to a labour union movement “which recommended ‘revolutionary direct action’ for the transformation of economic and social conditions of the working masses\dots{} in contrast to parliamentary reformism.”3
Researchers have identified some of the causes of this radicalization of the attitudes and actions of the workers. First of all, it was connected with a change in the position of the workers themselves within the structure of industrial production. Up to the 1890’s and the first decade of the 20th century the organization of industrial production on the whole had not reached a level of specialization which would allow the division of the labour process into separate operations.
Labour in industrial enterprises was still characterized by a certain integrity not unlike the labour of craftsmen, from which factory workers inherited the psychology and ethic of autonomy and independence. They possessed complex production knowledge: in their own area of expertise, in the sphere of organizing their labour, in the distribution of labour-time, etc. All this favoured the formation of ideas among the workers about the possibility of workers’ control of the whole production process, and both production- and social-oriented self-management.4
A systematic revolution in production, beginning at the turn of the century (based on new sources of energy, and the increasing use of electricity and the internal combustion engine) led to changes in the relations of the various branches of industry and the appearance of new ones. The widespread application of technical innovations resulted in advances in production processes and changes in working and living conditions for the workers.5 The working class was more and more concentrated in cities in homogeneous neighbourhoods which strengthened class consciousness and the feeling of solidarity among wage workers. Along with the precipitous rise in the profits of enterprises, almost everywhere stagnation or even a decline in real wages was the rule. Technical and organizational changes in production undermined the professional craft skills of workers. The addition of mechanical and electrical components to machines and operations fragmented the labour process, leading to the downgrading of workers’ skills so they were less able to grasp the labour process in its entirety and correspondingly lost the possibility of controlling it.6 New methods of organizing work and management (direct hiring of all workers, piece-work, the bonus system, models of internal incentives, and the introduction of intra-factory hierarchies) allowed enterprises and administrations to control and intensify production more rigorously, increasing both the workload and the working time of the labour force. All this reinforced the dissatisfaction of the workers, first of all in such branches of industry as manufacturing, mining, and railway transport.
At the same time, there was a growing number of unskilled temporary and seasonal workers in construction, shipping, agriculture, and the oil and gas industry. Their situation was insecure and unstable but they were less dependent on specialized labour and specific employers and liable to act quickly to defend their own rights and interests.
Observers noted a rapid growth in the sense of solidarity among workers. Evidence of this can be seen in the huge strikes of transport workers in Britain, the Netherlands, and France of 1911-1912, which acquired an international character. The mutual support of sailors, stevedores, and surface transport workers brought success to the cause of wage labourers. It was characteristic that workers of different countries effectively used similar methods of mutual aid, such organizing free meals and childcare.7 The strike movement was observed to be growing almost everywhere.
In a number of countries general or “political” strikes took place. The workers were less and less satisfied with the traditional politics of social-democratic workers’ parties and trade unions. Social-democracy rejected the notion of general strikes as “total nonsense.” At a congress of the German Free Trade Unions in Cologne (1905), it was once more affirmed that “the idea of the general strike, which is upheld by the anarchists and other people lacking any experience in the field of economic struggle, is not worth discussing.”8 Even in the case of economic struggle for partial demands, trade unions under the influence of social-democracy were more and more inclined towards reformism and compromises with governments and enterprises, having recourse to strikes only in extreme circumstances. In their organizational setup the reformist unions were orientated towards a centralized operation (for example, in Germany strikes had to be sanctioned by the central industrial union association). In these labour unions a ramified and despotic bureaucracy took form. The model of a large organization with a multilevel structure for decision-making, and the assignment of projects to specially selected professionals, was based on the assumption that the rank-and-file members should have limited power and restricted access to resources. Full-time officers of labour unions were more interested in preserving and strengthening the structure of their organization than in taking part in a struggle the outcome of which was uncertain.9 Frequently union leaders preferred to avoid conducting strikes in order not to risk the money accumulated in their organization’s strike funds. In other cases the leadership of workers’ organizations compelled their members to terminate strikes, as happened, for example, in the course of the struggle of the Berlin metalworkers in December 1911. In this connection, the defeat of strike actions by German wage workers in the metallurgical, ceramic, tobacco, shoe-making, textile, and other branches of industry at the beginning of the second decade of the 20th century led many activists throughout Europe to conclude that the performance of the German model of centralized trade unions had reached a dead end.10 Instead of direct strike action, reformist union leaders preferred to follow the practice of central “wage agreements” between enterprises and unions – agreements which were concluded between the unions and the business owners for specific occupations and territories and bound both sides for the duration of a mutually agreed period of time. Among the workers such actions provoked a growing indignation, since they were often saddled with unfavourable conditions and deprived of their right to have a say in decisions about labour questions which affected them in an important way. “On the whole and on all the most important questions, the central administration enjoys supreme authority\dots{},” according to a brochure published in 1911 by the British Federation of Miners. “They, the leaders, are becoming ‘gentlemen’ and Members of Parliament and, as a result of their powerful positions, they have acquired an impressive social standing\dots{} . What really should be condemned is this politics of conciliation which finds a use for such leaders\dots{} .”11 In the words of the German trade union activist Karl Roche, “Within the workers’ movement itself, supposedly struggling to liquidate all class contradictions\dots{} two classes have formed” – the all-powerful “paid officials” and the applauding, voting “ordinary folk.”12
\begin{enumerate}[1.]
\item\relax
See: A. Castel, De la Premiere Internationale a l’Association Internationale des Travailleurs (Marseille, 1995), pp. 13-15.
\item\relax
See H.-J. Steinberg, “Zukunftsvorstellungen innerhalb der deutschen Sozialdemokratie vor dem 1. Weltkrieg,” Soziale Bewegungen. Jahrbuch 2: Auf dem Wege nach Utopia (Frankfurt a.M. \Slash{} New York, 1985), pp. 48-58.
\item\relax
C. Cornelissen, Uber die theoretische und wirtschaftliche Grundlagen des Syndikalismus,” in Forschungen zur Volkerpsychologie und Soziologie, Bd.2. Partei und Klasse im Lebensprozess der Gesellschaft (Leipzig, 1926), p. 63.
\item\relax
See K. H. Roth (ed.), Die Wiederkehr der Proletaritat. Dokumentation der Debatte (Koln, 1994), p. 271.
\item\relax
See M. Van der Linden and W. Thorpe, “Aufstieg und Niedergang des revolutionaren Syndikalismus,” in 1999. Zeitschrift fur Sozialgeschichte des 20. und 21. Jahrhunderts. 1990, no. 3, p. 15.
\item\relax
See W. Thorpe, ‘The Workers Themselves’, Revolutionary Syndicalism and International Labour, 1913-1923 (Dordrecht \Slash{} Boston \Slash{} London \Slash{} Amsterdam, 1918), p. 24.
\item\relax
C. Cornelissen, “Die neueste Entwicklung des Syndikalismus,” Archiv fur Sozialwissenchaft und Sozialpolitik, Bd.36, (Tubingen, 1913), p. 135.
\item\relax
Cited by N. Luskin-Antonov, Очерки по новейшей истории Германии. 1890-1914 [Essays on the contemporary history of Germany, 1890-1914] (Moscow \Slash{} Leningrad, 1925), p. 321.
\item\relax
See K. Schonhoven, “Lokalismus – Berufsorientierung – Industrieverband: Zur Entwicklung der organisatgorischen Binnenstrukturen der deutschen Gewerkschaften vor 1914,” in W. J. Mommsen and H. G. Husung (eds.), Auf dem Wege zur Massengewerkschaft: die Entwicklung der Gewerkschaften in Deutschland und Grossbritannien 1880-1914 (Stuttgart, 1984), pp. 291, 295.
\item\relax
C. Cornelissen, “Die neueste Entwicklung des Syndikalismus\dots{},” p. 131. (n8)
\item\relax
Cited by C. Cornelissen, “Die neueste Entwicklung\dots{},” p. 128-129. (n8)
\item\relax
K. Roche, Aus dem roten Sumpf oder: Wie es in einem nicht ganz kleinem Zentralverband hergeht (Berlin, 1909); reprint (Hamburg\Slash{}Altona, 1990), p. 4.
\end{enumerate}
\section{Chapter 2: the Rise of the Revolutionary Syndicalist Movement}
The challenge to social-democracy in the workers’ movement, and to everything connected with it – parliamentary orientation, reformism, and the dominance of party and union bureaucracies – first appeared in France. It was here the workers began to work out the tactic of revolutionary syndicalism from below. This line was disseminated initially in the bourses de travail. The first of them was created in 1886 in Paris. Originally these places were labour exchanges for the workforce but they soon began to function as workers’ clubs and cultural-educational centres. From a local type of inter-occupational organization, the bourses were transformed over a period of time into union centres oriented towards the class struggle. In 1892 they were united in a national federation. The bourses de travail carried on active work creating solidarity among workers at the local level, independent of political parties and individual unions which often turned out to be under party influence. The bourses became a unique kind of centre for the self-organization and mutual aid of workers: they helped the unemployed and people seeking work; they also helped the sick and victims of workplace accidents; they created libraries, social museums, and both specialist and generalist courses; and they carried on propaganda for the creation of unions, backing this up in a systematic way by organizing strikes, setting up strike funds, engaging in general agitation, etc.1 A weak point of the bourses de travail was their dependence on financing from municipal governments, which gave rise to constant conflicts between government bureaucrats and worker-activists.
The French socialists – “Guesdists” – did not wield any influence in the bourses de travail movement. The participants in the bourses were mainly rank-and-file union activists, disillusioned by the lack of social and labour legislation of the 1880’s and 1890’s; members of socialist groups (especially the “Alemanists”) opposed to the Socialist Party of Jules Guesdes; and also a certain number of anarchists who worked in the trade unions in such cities as Paris, Rouen, Toulouse, Algiers, etc. The anarchists hoped that, in the event of revolution, the local bourses and the unions would become “associations of producers” – the embryos of a self-managed, libertarian, and stateless society, a transitional stage on the road to “full” anarchism (if the revolution occurred before an anarchist consciousness had taken root among the workers) or the initial stage of libertarian (anarchist) communism – a society without either the State or money. The anarchist Fernand Pelloutier was elected secretary of the Federation of bourses de travail. He was to play an important role in the formation of revolutionary syndicalism.
Within the confines of the French bourses de travail movement a number of the most important principles of revolutionary syndicalism were formulated. Some of them were similar to those proposed by the anti-authoritarian (“Bakuninist”) wing of the First International: independence from political parties, non-participation in political struggles, “direct action” (that is, people standing up directly on behalf of their own interests2), an orientation towards economic struggle in which the workers negotiated directly with business owners for partial improvements in the working conditions of wage-workers, and the preparation of the general strike as the vehicle of social revolution. This similarity can be explained not only by the influence of the anarchists participating in the movement, but also by the practical experience of many French workers of that era.
In 1902 the Federation of bourses de travail joined with another union central – the General Confederation of Labour (CGT) in a unified CGT. The new CGT became the largest workers’ organization in France: in 1912 it included 600,000 of the one million organized wage-workers of the country.3 The leadership of the confederation was in the hands of adherents of revolutionary syndicalism. This ideological stance was supported by the following labour federations: longshoremen, metalworkers, and production workers in the industries manufacturing graphite pencils, jewelry, matches, and hats; workers in the printing, construction, paper-manufacturing, and food industries; workers producing means of transportation; municipal service workers, etc.
But the CGT also included unions which were dominated by reformists: railway workers, bookbinders, textile workers, mechanics, workers in the war industry, musicians, workers in the ceramic industry, gas and electric utility workers, tobacco workers, and teamsters.4 The relation of forces was unstable and could change quickly. However, during the period of active struggle revolutionary syndicalism was also embraced by workers belonging to reformist unions.
The radicalism of the CGT found expression not only in leading strikes, but also in organizing campaigns, especially against militarism and colonialism, as well as for the eighthour day. Starting on May 1 1905, the French union central launched a massive agitation for the purpose of having the workers institute the 8-day hour day starting on May 1, 1906, without prior authorization. Throughout the whole country signs and leaflets were distributed, slogans were posted, meetings were held, and reports presented. “\dots{} within the working class an almost chiliastic mood took root which had the effect of inhibiting those trade unionists who had a grip on reality (in many factories it was possible to read signs like: ‘70 more days – and we shall be free’ or ‘67 more days – and our liberation will begin’). At the same time the bourgeoisie was seized by a collective psychosis. The Great Fear prevailed.”5 The government arrested the leaders of the CGT and brought troops into the cities. During the week before May 1 1906, strikes broke out in many sectors for the 8-hour working day, and on May 1 a general strike took place, in which up to 200,000 workers took part in Paris alone.
There were battles in the streets and at the barricades and a full cessation of economic life in many industrial centres. A multi-month wave of rear-guard strikes wrested a number of concessions from the authorities: a reduction in work time and increase in pay in individual enterprises; the legislated introduction of a day off every week and an abbreviated work day on Saturdays; and a reduction in the intensity of work in construction.
In the following years repression against the CGT increased.
The government frequently used troops against strikers and the soldiers opened fire on workers; street battles erupted. The organization could not endure the excessive strain on its resources. By the end of 1908 the leadership of the CGT had passed into the hands of reformers. Nevertheless, right up to 1914 strong revolutionary moments could be observed in the activities of the confederation: the organization continued its active anti-militarist and anti-war campaigns, its struggle against pension legislation which did not meet the workers’ needs, and against inflation. [19]
From France revolutionary syndicalism spread to other European countries. After the general strike of 1903 the National Secretariat of Labour of the Netherlands, created in 1893, broke with reformist social-democracy and adopted a position of revolutionary syndicalism.
In Italy, starting in 1891, there arose local “houses of labour” similar to the French bourses de travail. The general strike of 1904, general strikes and clashes in the South in 1905, and the general strike of May 1906, in Turin, increased the tendency towards the unification of workers. In 1906 the General Confederation of Labour (CGL) was created; its leadership was captured by socialists and the revolutionary syndicalists headed the opposition. Dissatisfaction of the workers with the reformist politics of the socialist leadership of the CGL grew after it refused to support a strike of railway workers in Milan in 1907 and a regional strike in Parma in 1908. The revolutionary syndicalists, on the other hand, during the period 1908-1911 led large-scale actions of agricultural labourers in Apulia, and metalworkers in Turin and Genoa; strikes against Italian intervention in Africa; strikes of foundry workers in Piombino and on the island of Elba; a strike of bricklayers in Carrara, etc. Gradually the synchronized structures of a revolutionary syndicalist movement were formed. Finally, in 1912, the Italian Syndicalist Union (USI) was created, having a federalist and self-governing internal structure. In 1914 it already counted 124,000 members.6 The revolutionary syndicalists organized the largest actions of the Italian workers, such as the general strike of workers of the marble industry; the general strike of the Milan metalworkers; actions of construction workers, sailors, agricultural labourers, and railway workers; the general strike in solidarity with workers in the furniture manufacturing industry in 1913; and the strikes of bricklayers in Carrara in 1914. In June 1914 anti-militarism protests grew into an insurrection (“Red Week”) above all in the Marche (Ancona) and Emilia Romagna. The USI actively participated in these actions, while the leaders of the CGL sabotaged them in whatever way they could.
In Portugal, where the anarchists had taken an active part in workers’ association from the beginning of the 1890’s, the example of French revolutionary syndicalism aided the majority of organized workers to free themselves from the influence of the socialists. An active strike movement grew, which put the methods of direct action into practice. Already in 1907 several unions, emerging from under the control of reformists, had joined together in the General Federation of Labour. In 1909 the anarchists and revolutionary syndicalists, brushing aside the socialists, convened a congress of trade union and co-operative associations in Lisbon. The participants put forward the demand for the 8-hour work day and agreed on the creation of a confederation of all workers with the goal of “obtaining an increasing influence over the production of essential goods.” In the north of the country in Porto an autonomous General Union of Labour started up in 1911, independent of the Socialist Party. The second syndicalist congress in the same year consolidated its revolutionary syndicalist orientation. In 1910-1912 the country was rocked by a wave of strikes of a radical, insurrectionary character, accompanied by clashes with troops and police and acts of sabotage. In 1912 as a sign of solidarity with the strike of 20,000 agricultural workers of the Evora region, syndicalists declared a general strike. Workers armed themselves and Lisbon literally found itself in the hands of the toilers. The politics of the reformist trade unions helped to supress the revolt to a significant degree. The subsequent repression forced the syndicalists and socialists to seek common ground. At the 1st all-national workers’ congress in Tomar in 1914 representatives of both tendencies were present.
The result was the creation of a single National Workers’ Union (UON) in which each ideological tendency received full independence. However the ideas and practice of revolutionary syndicalism enjoyed increasing influence and at the national convention in 1917 revolutionary syndicalism was officially recognized.7 In Germany and the Scandinavian countries, the sources of both the anarchist and revolutionary syndicalist movements were found among the left activists and trade union opposition within social-democracy itself. The Free Association of German Trade Unions (FVdG), created in 1897 by “localists” (opponents of the formation of bureaucratic, centralized trade union associations), at the beginning of the 1900’s adopted the concept of the general strike and methods of direct action. In 1912 it approved a program put together under the influence of the French CGT. In response the Social Democratic Party of Germany in 1908 prohibited its members from joining the FVdG. In Sweden the “young socialists,” in the course of trade union debates in 1908, spoke out in support of methods of struggle and tactics close to the CGT.8 The defeat of a general strike in the following year strengthened the disenchantment with the line of the social-democratic trade union leadership, and in 1910 delegates from a number of unions announced the creation of a “Central Organization of Swedish Workers” (SAC).9 The organization of syndicalist oppositions also took place in Norway (the Norwegian Syndicalist Union) and in Denmark.
The wave of lockouts in the Scandinavian countries in the summer of 1911 and the compromises agreed to under such conditions by the trade union leadership with business owners, served to promote the spreading of the revolutionary syndicalist movement in Scandinavia.
In the Anglo-Saxon countries revolutionary syndicalism arose in the practice of “industrial unionism,” i.e. organizing workers not on an occupational, but rather on a sectoral or industrial, basis. In contrast to the French and Italian syndicalist unions, “industrial unionism” regarded as its organizational basis the lowest production unit; and at a higher level – the industry association; and finally – “the one big union” of all the workers, regardless of their occupation.
In the U.S.A. in 1905 the Industrial Workers of the World (IWW) was created through the initiative of radical unions. The IWW also became more and more revolutionary syndicalist in character. It was oriented towards direct action, striving to combine actions aimed at improving the situation of workers with the struggle for social revolution and a new society, organized on the basis of unions managing production. In contrast to the official trade unions, the IWW included in its membership unskilled workers, immigrants, and women. In 1906-1916 the IWW participated in a number of the bitterest and most radical strikes in the history of the U.S.A.: a general insurgence by workers of various occupations in Goldfield, Nevada and a strike by sawmill workers in Portland, Oregon (1906-1907); strikes of multi-thousands of textile workers in Skowhegan, Maine (1907) and Lawrence, Massachusetts (1909); a steelworkers’ strike in McKees Rocks, Pennsylvania (1909); and so on. The response to this was repression against the activists of the IWW10
In Australia the organization of the IWW took place as a reaction to the introduction of compulsory state arbitration in labour disputes and the suppression of strikes. Workers’ organizations based on the IWW platform were also created in Great Britain, South Africa, in Russia in 1917, and in Germany after the First World War.
The revolutionary syndicalist movement in Britain arose also under the influence of agitation by the IWW and the newspaper The Syndicalist, published by the worker-activists Tom Mann and Guy Bowman. In 1910 the Industrial Syndicalist Education League (ISEL) was formed. The British syndicalists set out not to create their own separate organizations, but to win over the craft unions. They succeeded in taking control over the key unions of miners and railway workers. In the pre-war years a rapid growth of syndicalism took place in Great Britain. The mass actions organized by syndicalists (the 1911 general strike of seamen which gave rise to the first international movement of solidarity, and the strike of one million coal miners in the spring of 1912) were on a scale which exceeded anything previously known to the world in the way of class conflicts.11 The action by British seamen was supported by their colleagues in Belgium, Holland, and the U.S.A.; by longshoremen; and also by other categories of British transport workers. Significantly, during the miners’ strike decisions were arrived at by a referendum of the workers, and in the course of negotiations with the owners the workers tried to impose clear-cut and binding instructions on their own representatives and, in the spirit of federalism, to observe the autonomy of individual mines and regions. The South Wales Miners’ Federation developed a plan of re-organization in which envisaged the introduction of revolutionary syndicalist principles: the autonomy of lodges as the highest instance of decision-making, the rejection of full-time paid union leaders, the taking control of industry by the workers as a goal, etc.12
At the beginning of the 20th century revolutionary syndicalist tendencies spread to a number of other countries: Belgium (the Union of Syndicates of the Province of Liège from 1910, the Belgium Syndical Confederation from 1913), Switzerland, Russia (it was here, according to some sources, that the term “anarcho-syndicalism” was coined13), Austro- Hungary, the Balkans, Canada (the “One Big Union” which arose in 1919), etc.
\begin{enumerate}[1.]
\item\relax
For details, see: F. Pelloutier, Histoire des bourses du travail: origin – institutions – avenir (Paris, 1978).
\item\relax
“Direct action,” explained Victor Griffuelhes, one of the leading activists of French revolutionary syndicalism, “denotes the actions of the workers themselves, i.e. actions directly carried out by people in their own interests. The worker himself applies his efforts: he personally exerts his influence on the forces which rule over him in order to obtain from them the desired benefit. With the help of direct action the worker himself creates his own struggle; he takes full responsibility for it and does not hand off the matter of his personal liberation to anyone else.” (Cited by: G. Aigte, “Uber die Entwicklung der revolutionaren syndikalistischen Arbeiterbewegung Frankreichs und Deutschlands in der Kriegs- und Nachkriegszeit,” Die Internationale, 1931, no. 4 (Februar), p. 88.
\item\relax
W. Thorpe, “The Workers Themselves\dots{},” p. 26. (n7)
\item\relax
L. Mercier-Vega and V. Griffuelhes, L’Anarcho-syndicalisme et le syndicalisme revolutionnaire (Paris, 1978), p. 14.
\item\relax
direkte aktion, 1993, no. 98 (Mai-Juni), p. 7.
\item\relax
G. Careri, L’Unione Sindacale Italiana tra sindacalismo di base e trasformazione sociale (n. p., 1997), p. 9.
\item\relax
See: Die Internationale,no.5 (Juni 1925), p. 148ff.; C. Da Fonseca, Introduction a l’histoire du mouvement libertaire au Portugal (Lausanne, 1973).
\item\relax
H. Rubner, Freiheit und Brot (Berlin \Slash{} Koln, 994), pp. 23-32.
\item\relax
For details, see: L. K. Persson, Syndikalismen I Sverige 1903-1922 (Stockholm, 1975).
\item\relax
See: L. Adamic, Dynamit: Geschichte des Klassenkampfs in den U.S.A. (1880-1930), 3. Aufl. (Stuttgart, 1985). About the IWW see also: V. Trautman, D. Ettor, et al., Производственный синдикализм (индустриализм) [Industrial Unionism], coll. of articles (Petersburg-Moscow, 1919).
\item\relax
C. Cornelissen, “Die neueste Entwicklung\dots{},” p. 138. (n8)
\item\relax
Ibid., pp. 144-147.
\item\relax
A. Schapiro, preface to P. Besnard, L’Anarcho-syndicalisme et l’anarchisme (Marseille, 1997).
\end{enumerate}
\section{Chapter 3: Revolutionary Syndicalism and Anarchism}
The revolutionary syndicalism of the early 20th century was not born in the heads of theoreticians. It was the practice of the workers’ movement which sought its own doctrine1 – above all, the practice of direct action. What this meant, according to the words of Émile Pouget, one of the leading activists of the French CGT, was that the working class, finding itself in constant conflict with contemporary society, “expects nothing from anyone, any government, or any powers external to themselves, but creates the conditions for its own struggle and draws on its own resources for the means of action.”2 “Direct action varies according to the circumstances,” pointed out Georges Yvetot, one of the leaders of the CGT, “the workers find new methods depending on their occupations, their imaginations, or their initiatives. In principle direct action excludes any concern about legality\dots{}
Direct action consists in forcing the owner to make concessions from considerations of fear or self-interest.”3
Such methods include, in the first place, means of economic struggle which are pointed directly at the counter-agent of the workers in production – the entrepreneur or capitalist (the boycott, individual or group sabotage of production, partial or general strike), and also revolutionary syndicalist propaganda and anti-militarist activity. Political struggle as a task of the organized workers’ movement was rejected. It was assumed that from the economic struggle of workers for their rights and the improvement of their situation within the framework of the existing system would develop a frontal assault on Capital and its State. As a result, capitalism would be overthrown, the system of wage labour eliminated, and the workers, organized in labour unions, would take over control of production. In this sense strikes played a very special role for revolutionary syndicalists: they were viewed not as an end in themselves but as a “revolutionary drill,” as preparation of the workers for the imminent revolution.
The revolutionary syndicalist movement was not able to formulate a coherent ideological doctrine. At the level of theory revolutionary syndicalism remained a complex of ideas from various sources. Very different tendencies contributed to this complex. The Dutch syndicalist Christiaan Cornelissen, one of the first to study the movement, distinguished three groups among the activists of revolutionary syndicalism: the trade unionists, who considered syndicalism “self-sufficient” and distinct from any ideology and occupied radical positions based on their practice of class struggle; the anarchists, who saw in the trade union movement the possibility of moving from agitation to action; and finally, people from the socialist parties and groups who hoped to extricate socialism from the impasse of parliamentarism.4
The anarchists who were working in the trade unions and trying to draw them closer to libertarian positions considered the unions not just as an organ of the struggle of workers for the direct improvement of their situation, but also as the instrument which by way of the General Strike would carry out the social revolution, seize control of the economy, and plan both production and consumption in the interests of the whole of society. In 1909 two prominent French revolutionary syndicalists, Émile Pataud and Émile Pouget, published the programmatic book “How We Shall Make the Revolution.”5 They proceeded from the assumption that the unions in the course of a revolutionary strike would expropriate capitalist property and transform themselves into an association of producers. Each union would occupy itself with carrying out the re-organization of production and disribution in its own area of expertise. The trade unions, with their territorial and industrial federations at all levels (up to and including the national congress and its executive) would become the organs of a new society, making decisions and carrying them out in the sphere of economic and social life: gathering statistics and sharing them, coordinating production and distribution on the basis of these statistics, and ensuring the social processes by which administration takes place from bottom to top. In this scheme groups and associations which are engaged in governing inhabitants on a territorial basis are assigned only a subsidiary role in the organization of life at the local level.
In the designs and elaborations of the revolutionary syndicalists one can discover many basic features of anarchist (libertarian) self-managed alternatives to industrial-capitalist society. However, there are differences on some points of principle. First of all, revolutionary syndicalism is much more favourably disposed towards industrial progress and industrial forms of organization than anarcho-communist doctrine. Anarchism rejected not only capitalism, private property, and the State; but also the centralization of social life and the division and specialization of labour. Anarchist theoreticians did not object to professional associations and other groups based on common interests, but they considered that the free society of the future would be based on self-managed, autonomous, territorial communes, joined together by federations. To industrial centralization with its occupational hierarchy and specialization, and to factory tyranny with its strict division of labour and its cult of production and productivity, the anarchists counterpoised a break with the logic of industrialism: the decentralization and breaking up into smaller units of industry; its re-orientation towards local needs; the integration of industrial and agricultural, intellectual, and physical labour; and the maximum possible self-sufficiency of communes and regions.6 On the contrary, many syndicalists aspired to have an influence on the labour process in existing enterprises, rather than liquidating the system of large-scale centralized industry.
Thus, Cornelissen affirmed that the division of labour has “great advantages” for the wage worker and will contribute to his liberation. In the spirit of the industrial Marxism of the Second International, he declared that the liquidation of capitalist ownership in the means of production by no means implies that all the workers in an enterprise must participate in management. Cornelissen also defended the institution of full-time functionaries – the trade union bureaucracy.7
In other words, a section of the anarchists, those working in the trade unions, tended to consider syndicalism as the anarchism appropriate to the new, industrial century. “I am an anarchist, but anarchy does not interest me,” declared E. Pouget.8
Some of the anarchists in the revolutionary syndicalist movement recognized the divergence between anarchist social doctrine and the model of a hierarchical, centralized production system, administered by the trade unions. However they stressed that such a “syndicalist system,” although not yet dispensing with the State, nevertheless in its subsequent evolution would lead to the “total implementation of communist principles in economic relations” and “to the total disappearance” of the State “as a consequence\dots{} of its superfluousness,” i.e. it would lead to anarchy.9
The theory of anarcho-communism proceeded from the assumption that immediately after the social revolution, which would eliminate private property and the State, society would switch to a communist system of production and distribution according to the principle “from each according to their abilities, to each according to their needs.”
The book by Pataud and Pouget proposed an intermediate, “collectivist” variant, similar to that espoused in those days by the Marxists: communist distribution of goods of prime necessity and distribution “according to labour” (by means of worker’s time sheets) for all remaining goods. And Cornelissen, like the social-democrats, asserted that in the contemporary industrial era with the growth of interdependency in the world economy, self-sufficiency was impossible because both prices as well as the compensation of labour were in the form of money and would remain so in a socialist society, at least until a state of affluence prevailed. [37]
A significant number of Marxists at the end of the 19th – beginning of the 20th centuries, disenchanted with the “senility” of parliamentary socialism and reformism, saw in revolutionary syndicalism the means to envigorate and save socialism. The syndicalist “neo-Marxist” theoreticians (Georges Sorel, Edouard Berth, and Hubert Lagardelle in France; Arturo Labriola and Enrico Leone in Italy; etc.) tried to return to that aspect of Marxist doctrine which critiqued the State and factory discipline and was oriented towards their liquidation. However their ideas about the mobilizing role of violence, about the vanguardist-elitist function of the “revolutionary minority” in contrast to the “democracy of numbers” and, finally, about the myths in which each participant of the movement must believe even if they were not destined to realize them in full measure (such myths were ascribed by Sorel, for example, to the syndicalist concept of the general strike and the Marxist doctrine about “catastrophic revolution”10) – these ideas were antithetical to libertarian views. Nevertheless, the works of these authors received very wide distribution and in many countries became associated with the revolutionary syndicalist movement, exerting a significant influence on its development.
The theoreticians of anarcho-communism ( Petr Kropotkin, Ericco Malatesta, and others) maintained that the roots of social development lie in progress of the ethical concepts of humanity; that capitalism is a regressive system since it undermines the intrinsic social nature of humanity based on mutual aid; and that the division of humanity into warring classes plays a reactionary role, retarding the set11 self-realization of the human personality. From this the anarcho-communists drew their demand for the liquidation of the division of society into classes. The path to this result they saw in the resistance of oppressed social layers, but they emphasized: “The anarchist revolution which we seek is far from being restricted to the interests of one distinct class. Its goal is the complete liberation of the whole of humanity oppressed at the present time in three senses of the word – economic, political, and ethical.”12 On the other hand, revolutionary syndicalism adopted the Marxist concept of the primacy of the economy and the progressive nature of class struggle in social development. It proceeded from the assumptions that the development of industrial capitalism creates the economic and social basis for a free society, and that the struggle of the proletariat for its own class interests necessarily leads to its overthrow of capitalism. These assumptions resulted in the organizational and programmatic views of the revolutionary syndicalists, embodied above all in the “Charter of Amiens” – a document adopted by a congress of the French CGT in Amiens in 1906. Although the Charter represented a compromise between different tendencies present in the French trade union confederation, it exerted a decisive influence on the workers’ movement of many countries, namely as a declaration of the principles of revolutionary syndicalism.
According to this document, the CGT was not based on ideology but on class, embracing all workers, “regardless of any political tendencies,” who acknowledged the necessity of “struggle for the riddance of wage labour and entrepreneurial activity.” The Charter agreed in principle with the class struggle in the economic arena “against any form of exploitation and oppression.” It stated that syndicalism has a dual purpose: to lead the struggle for the immediate improvement of the situation of the working class, and simultaneously to prepare for “complete liberation” by means of “expropriation of the capitalists” in the course of a general strike, so that the trade union (syndicate) would in the future be transformed into a “group for production and redistribution, the basis of social reorganization.” Concerning political parties, ideological tendencies, religious beliefs, etc., it was proposed that workers belonging to a trade union keep their own individual convictions outside of the union in the name of class unity. However, the right of workers to struggle for their own ideas outside the union was recognized.13
Thus, in comparison with anarcho-communism, revolutionary syndicalism represented only a partial, inconsistent, and contradictory rupture with the industrial-capitalist system. Therefore it was not surprising that in anarchist circles the new movement was often regarded critically. It’s true Kropotkin was one of the first to encourage anarchists to work in the trade unions14 and even wrote an introduction to the book by Pataud and Pouget, emphasizing the closeness of the revolutionary syndicalist program to anarchism in the matter of workers’ self-organization and self-management.15
But by no means did all the anarchists perceive revolutionary syndicalism in a sympathetic way. Sharp disputes about the relationship between anarchism and syndicalism flared up at the congress of anarchists in Amsterdam in August 1907, which was convened, not surprisingly, through the efforts of the Dutch syndicalist Cornelissen. The French delegate Pierre Monatte, active in the CGT, stressed the shared positions and reciprocal influences of anarchism and syndicalism, insisting that syndicalism, “as defined by the Amiens congress of 1906,” was self-sufficient. He presented it as a sort of renewal of anarchist goals and “the way the movement and revolution are conceived.” A number of other participants at the congress critiqued the notion of the “self-sufficiency” of syndicalism. Thus, the Czech anarchist K. Vokryzek declared that syndicalism must be only a means, an instrument of anarchist propaganda, but not the goal. Cornelissen argued that anarchists should not support just any kind of syndicalism or any kind of direct action, but only those “which are revolutionary in their aims.” But the most outspoken criticism of Monatte’s position came from the Italian anarchist E. Malatesta. He also spoke in favour of anarchists working in the trade unions, but assigned to the unions, and indeed the workers’ movement as such, the role of one of the means of revolutionary struggle. Malatesta did not deny trade unions could in the future provide “groups which are capable of taking the management of production in their own hands,” however, he considered the main point about unions was that they were created and exist as instruments to defend collective material interests within the framework of existing society. He disputed the idea that solidarity between workers can develop out of common economic class interests, since it was completely possible to satisfy the aspirations of some groups at the expense of others.
But on the other hand, he supposed there was a possibility of “ethical solidarity” of proletarians – based on a common ideal. Malatesta also denied the possibility that the general strike by itself could replace social revolution: a stoppage of work could serve to start a revolution, but could not replace insurrection and expropriation. Finally, he appealed to anarchists to “awaken” the trade unions to the anarchist ideal.
But at the same time he rejected the idea of special, purely revolutionary, trade unions and spoke in favour of single, “absolutely neutral,” workers’ unions.16 However, already at the Amsterdam congress A. Dunois articulated the concept, closely related to future anarcho-syndicalism, of “workers’ anarchism,” which would replace the abstract and purely literary “pure anarchism.”17 The congress created a bureau of the anarchist International which included syndicalists (the Russian Aleksandr Shapiro and the Englishman John Turner), and also the German anarchist Rudolph Rocker, who was sympathetic to syndicalism. However the bureau had ceased its work already by the end of 1911.18
In spite of the criticism of revolutionary syndicalism in anarchist circles, the new current exerted a significant influence on the anarchist workers’ movement in those countries where it had existed since the time of the First International (in Spain), or where it had arisen later (for example, in Latin America).
In Spain the tradition of mass anarchist labour unions could be traced to the Spanish Regional Federation of the First International (1870) and the Federation of the Workers of the Spanish Region (1880’s). In spite of the attempt to recreate the latter organization in 1900, the majority of worker’s organizations essentially acted independently, under conditions of severe state repression. In 1907 the autonomous workers’ societies of Barcelona, which were under the influence of anarchists, created a federation of “Worker’s Solidarity” with the stated goal of replacing the capitalist system with a “workers’ organization, transformed into a social system of labour.” The activity of the federation soon spread to the whole of Catalonia – the most developed industrial region of the country. In 1909 the federation was able to conduct a general strike in Barcelona in protest against the colonial war in Morocco, a strike which was cruelly suppressed by troops (the “Tragic Week”). Analogous organizations began to spring up in other regions. The impetus for the growth of the movement was the example of the French CGT. In October-November 1910 at a congress in Barcelona, a national association of Spanish workers was created – the National Confederation of Labour (CNT). The organizational structure of the CNT was based on the model of the CGT, and the workers’ societies were converted into trade unions (“syndicates”). The resolutions and decisions adopted reflected an attempt at an original synthesis of anarchism and revolutionary syndicalism. Along with points which were close to syndicalist positions (such as the necessity of struggle for partial improvements, the 8-hour day, a fixed minimum wage, the application of methods of direct action, and the general revolutionary strike), the resolutions of the CNT congress contained formulas decisively rejecting politics and parties and which continued the traditions of the anarchist movement. The Spanish anarcho-syndicalists again adopted the slogan of the First International (“The liberation of the workers is the task of the workers themselves”).
They stated that syndicalism is not an end in itself but a means of organizing the revolutionary general strike and attaining “the total liberation of the workers by way of the revolutionary expropriation of the bourgeoisie.” They also announced it was necesssary to propagandize the new “powerful ideas” among the people – the new formulas of radical social renewal, i.e. anarchism. In 1911 the CNT already had 30,000 members. It was able to organize big strikes in Madrid, Bilbao, Seville, Jerez-de-la-Frontera, Málaga, and Tarrasa; a general strike in Zaragoza; a general revolutionary strike against the war in Morocco (autumn 1911); a strike of 100,000 textile workers; a general strike in Valencia (March 1914), etc. In 1911 the CNT was banned and had to go underground until 1914.19
Anarchists in Latin American countries such as Mexico, Cuba, and Brazil worked in the trade union movement. Anarchism reached its highest development in the workers’ movement in Argentina and Uruguay, where groups of adherents of the First International were active already in the 1870’s. Ettore Matei, Errico Malatesta and other well known anarchists took part in the creation of the first workers’ organizations in Argentina. In 1901 a national workers’ federation sprang up (from 1904 it was known as the Argentine Regional Workers’ Federation – FORA). A year after its creation the social-democrats withdrew and, at its 1905 congress, the FORA recommended to its members to propagandize “the economic and philosophical principles of anarcho-communism” among the workers. At the same time the Argentine workers’ organization rejected not only the concept of the “self-sufficiency” of syndicalism, but also the idea of “neutral” trade unions (which was held by the French revolutionary syndicalists, as well as by Malatesta).
The FORA organized many local and general strikes, achieving a reduction in the work day and the improvement of working conditions. For example, general strikes were conducted in solidarity with workers in the sugar industry (Rosario, 1901), and with sales clerks (Buenos Aires, 1902; on a national scale, 1904). There were large strikes of bakery workers in Buenos Aires (1902), and longshoremen (1902 and 1903-1904). Hundreds of thousands of workers took part in national general strikes of solidarity and protest against repressions in 1907, 1909, and 1910. In 1907, on the initiative of the anarchists, a general strike of tenants was organized.
These actions and demonstrations often resulted in violent clashes and street battles with police, and harsh repressions which were answered in turn by protest strikes.20 “One must say that the anarchist movement here – is unlike any other in the world,” wrote the correspondent of a European anarchist newspaper in 1907, “since here almost all the workers are anarchists.”21 In 1916 supporters of “neutral” syndicalism succeeded in splitting the FORA – the more moderate breakaway organization was known as the “FORA of the 9th Congress.”
Under the influence of the FORA the Uruguayan Regional Workers’ Federation (FORU) was formed in 1905. It developed more quietly, experiencing a number of ups and downs. Nevertheless, the Uruguayan worker anarchists were able to lead important strikes of street car conductors, bakers, leather workers, construction workers, transport workers, printers, metalworkers, packing plant workers, etc. as well as several general strikes. It was able to compel the government to introduce the 8-hour working day.22 The Argentine FORA also served as a model for the Regional Workers’ Central of Paraguay, founded in 1916.
Anarchists from the very beginning exerted a fundamental influence on the workers’ movements of such countries as Mexico, Cuba, and Brazil.23 Mexican anarchists were involved in founding the first association of the country’s labour unions – the Great Circle of Mexican Workers (GCOM) in 1870. At the beginning of the 20th century, they carried on a tenacious struggle against the dictatorship of Porfirio Díaz; however, during the revolutionary period 1910-1917 their forces split. A section of the activists led by Ricardo Flores Magón organized an insurgent movement which eventually resulted in the overthrow of the dictatorship.
But this section continued to act against the new regime to obtain the goals of social revolution, “land and freedom.”
The other section took part in creating a syndicalist labour union central – the House of the World Worker (COM) in 1912. Mexican syndicalists formed an alliance with the leaders of the liberal-constitutional wing of the Revolution, counting on receiving from them the possibility of freedom in the workplace, and helped them defeat the revolutionaries of the North led by F. Villa and the insurgent peasants of the South under E. Zapata. But already in 1916 the syndicalists were smashed by the government.
In Cuba, a colony of Spain up until 1898, the anarchist movement developed originally under the influence of the anarchists of the metropolis. Many trade unionists in Cuba at the beginning of the 20th century were under the influence of the anarchists.
In Brazil the anarchists, overshadowing the socialists, achieved predominance in the labour federations of a number of states, and in 1906 by their initiative a national labour union central was formed – the Brazilian Workers’ Confederation (COB). Active strike warfare was carried on in the country.
The anarchist workers’ movement also spread to other countries of Latin America. In Chile the anarchists worked in numerous Resistance Societies of skilled workers and in “Mancomunales” (which were simultaneously trade unions, mutual aid societies, and regional workers’ associations), and organized a number of powerful strikes. However in 1907 the movement received a heavy blow: the government suppressed a strike of 30,000 nitrate workers organized by the anarchists in which as many as 4,000 people were killed.24
In Peru worker-anarchists headed labour unions of bakers, textile workers, dockers, seafarers, casual labourers, etc.
They acted as the initiators of powerful strikes (including a general strike in Callao in 1913, after which the 8-hour day was introduced for a number of occupations), and developed work among indigenous communalists.25 A number of active trade unions were under anarchist influence as well in Boliva, Ecuador, Panama\dots{}
The rapid spread of the revolutionary syndicalist and anarchist workers’ movement throughout the whole world soon led to the first contacts between organizations and attempts to create an international association of radical trade unions. In August 1907, during the anarchist congress in Amsterdam, a meeting of syndicalists was held. In accor- dance with a proposal by the Free Association of German Trade Unions (FVdG), it was decided to start publishing an “International bulletin of the syndicalist movement” in four languages, which would further the development of the contacts between the syndicalist organizations of different countries. The bulletin was published in Paris and its editor was C. Cornelissen. The publication was financed by the syndicalists of the Netherland, Germany, Bohemia, Sweden, and France, and also received support periodically from the American IWW26
Rank-and-file activists in the revolutionary syndicalist organizations of the Netherlands, Germany, and France frequently urged the French CGT to convene an international trade union congress with the participation not only of reformists, but also revolutionary unions. Some of the French revolutionary syndicalists spoke out in favour of giving a higher priority to developing connections with other revolutionary trade union and initiatives; however, the leadership of the CGT declined to do so for the sake of preserving unity in the workers’ movement. The CGT joined a global association of trade unions under the aegis of social-democrats and reformists – the International Secretariat of the National Centers of Trade Unions (ISNTUC). It boycotted the conferences organized by this secretariat in 1905 and 1907 because the German trade unions would not allow the inclusion on the agenda of resolutions about the general strike and antimilitarism, but from 1909 on the CGT participated in the conferences but was unsuccessful in obtaining their transformation into plenipotentiary congresses of delegates. The banding together of the revolutionary syndicalist forces now continued without the participation of the CGT.27
New proposals about international connections were raised at the 6th convention of the IWW (1911) and by the syndicalist trade union associations of Italy, Germany, and the Netherlands. Finally, the responsibility for holding an international meeting was taken upon itself by the British Industrial Syndicalist Education League (ISEL). Participants at the conference were supposed to be “revolutionary workers, organized in independent trade unions” and rejecting political parties: “activists,” not “functionaries.” The preparatory committee called the international syndicalist congress for London in September-October 1913.
Sessions of the congress took place at Holborn Town Hall, London. There were delegates representing the Free Association of German Trade Unions; the Argentine FORA and the syndicalist “Regional Workers’ Confederation of Argentina” (CORA); the Brazilian workers’ confederation; the trade union organizations of Belgium, Cuba, France, Spain, the Netherlands, Britain; the Italian syndicalist union and a number of local trade union organizations of Italy; and the Swedish trade union association SAC which also represented the syndicalists of Norway and Denmark. A representative of the IWW was present as an observer. C. Cornelissen was elected secretary of the congress, and its translator was the Russian anarcho-syndicalist A. Shapiro. Discussed were questions of international collaboration; theory and tactics; anti-militarism and anti-war work; migrant workers, etc.
In the course of the sessions serious differences surfaced between those who, like the Italian delegate Alceste De Ambris, tried to soften the anti-statist and anti-capitalist slant of the proposed resolutions and avoid “splitting the working class” by creating a new trade union International; and adherents of a more consistently revolutionary line. In the end the congress adopted a declaration of principles which included the basic positions of revolutionary syndicalism: “Capitalist slavery and State oppression” were rejected, and the “class struggle” was proclaimed as the inevitable consequence of private property and workers’ solidarity. This document contained appeals for the creation of independent industrial unions on the basis of free association, both for the fight for everyday necessities for the workers, as well as for the overthrow of the capitalist system and the State. It was maintained that workers’ organizations must overcome the divisions brought about by “political and religious differences.”
The declaration expressed the view that trade unions will become organs of the socialization of property and the management of production in the interests of the whole of society. Direct action was recognized the means of struggle. Finally, the congress took a decisive step towards the creating of a new syndicalist International: it called for international solidarity and established an International Syndicalist Information Bureau to coordinate communications and cooperation, make preparations for new congresses, etc. The functions of the Bureau were entrusted to the Netherlands NAS, although De Ambris expressed dissatisfaction with this circumstance and proposed to place it in Paris (effectively under the control of the CGT). The Bureau, composed of Gerrit van Erkel (chair), Thomas Markmann (secretary), A. J. Hooze (treasurer), M. A. van der Hage, and F. Drewes, set to work officially on January 1 1914.
The further unification of worker anarchists and revolutionary syndicalists was prevented by the outbreak several months later of the First World War. The war demonstrated all the contradictions and inconsistencies of the revolutionary.
\begin{enumerate}[1.]
\item\relax
“In the revolutionary syndicalist workers’ movement, more than in other movements, one sees the lively instincts of the [working – V. D.] class, searching about and finding its own way\dots{},” noted in this connection the German researcher of the 1930’s Gerhard Aigte. “That is why this movement did not spring up as a result of some well-defined, polished theory, but arose from the requirements of practical life. The revolutionary syndicalists\dots{} always emphasized that syndicalism – is the workers going about their own business, and not the speculative creation of isolated intellectuals.” (G. Aigte, Die Internationale, 1930, no. 2 (Dezember), p. 45).
\item\relax
E. Pouget, L’Action directe (Marseille, 1997), p. 1.
\item\relax
G. Yvetot, A.B.C. syndicaliste + F. Pelloutier, L’Organisation corporative et l’anarchie (Toulouse, n.d.), p. 33.
\item\relax
C. Cornelissen, “Uber den internationalen Syndikalismus,” Archiv fur Sozialwissenschaft und Sozialpolitik, Bd. XXX (Tubingen, 1910), pp. 153-154.
\item\relax
E. Pataud and E. Pouget, Comment nous ferons revolution (Paris, 1909).
\item\relax
See, for example: P. A. Kropotkin, Fields, factories, and workshops (Montreal: Black Rose Books, 1996).
\item\relax
C. Cornelissen, “Uber den internationalen Syndikalismus\dots{},” pp. 158, 161, 165; C. Cornelissen, “Zur internationalen syndikalistischen Bewegung,” Archiv fur Sozialwissenschaft und Sozialpolitik, Bd. XXXII (Tubingen, 1911), p. 842.
\item\relax
Cited by V. Garcia, Antologia del anarcosindicalismo (Caracas \Slash{} Montady, 1988), p. 17.
\item\relax
M. Rayevsky, Anarcho-syndicalism and critical syndicalism (New York, 1988), p. 17.
\item\relax
See, for example: G. Sorel, Reflections sur la violence (Paris, 1906).
\item\relax
Concerning the economic views of Cornelissen, see: C. Cornelissen, Theorie de la valeur (Paris,1903).
\item\relax
Cited by E. Malatesta, Anarchie (Berlin, 1995), p. 290.
\item\relax
For the text of the “Charter of Amiens” see: H. Dubief (ed.), Le Syndicalisme revolutionnaire, Paris (1969), pp. 95-96.
\item\relax
See: Anarchistes en exil. Corresondence inedite de Pierre Kropotkine a Marie Goldsmith 1897-1917 (Paris, 1995), p. 290.
\item\relax
P. Kropotkin, preface to E. Pataud and E. Pouget, How we shall bring about the revolution (London \Slash{} Winchester (Mass.), 1990); P. Kropotkin, Syndikalismus und Anarchismus (reprint) (Meppen, 1981), p. 16.
\item\relax
For texts of speeches and the corresponding resolutions of the congress, see: Congres Anarchiste tenu a Amsterdam. Aout 1907. Compte-rendu analytique et resume de rapports sur l’etat du mouvement dans le monde entier, Paris (1908).
\item\relax
V. Garcia, Antologia del anarcosindicalismo\dots{}, p.18.
\item\relax
Brandenburgisches Landeshauptarchiv (eh. Staatsarchiv Potsdam). Pr. Br. Rep. 30, Berlin C Polizeiprasidium, Tit.94, Lit.A, Nr. 24: Die Anarchistische Internationale. 1908-1915. (15644), Bl. 14,16.
\item\relax
See: “Prefigurando futuro”: 75° aniversario de la CNT. 1910-1995, (Madrid, 1985), p. 4-8; Congresos anarcosindicalistas en Espana. 1870-1936 (Toulouse\Slash{} Paris, 1977), pp. 35-40; J. Peirats, Les anarchistes espagnols. Revolution de 1936 et luttes de toujours (Toulouse, 1989), pp. 9-13.
\item\relax
See: E. Lopez Arango and D. Abad de Santillan, El anarquismo en el movimniento obrero (Barcelona,1925); A. Lopez, La FORA en el movimiento obrero (Buenos Aires, 1987).
\item\relax
E. Lopez Arango and D. Abad de Santillan, op. cit., pp. 20-21.
\item\relax
F. Pintos, Профсоюзное движение в Уругвае [The Labour Union Movement in Uruguay] (Moscow, 1964); C. Zubillaga and J. Balbis, Historia del movimiento sindical uruguaya (Montevideo, 1984).
\item\relax
See: J. M. Hart, Anarchism and the Mexican Working Class, 1860-1931 (Austin, 1987); F. Fernandez, El Anarquismo en Cuba (Madrid, 2000); S. Dolgoff, The Cuban Revolution: a Critical Perspective (Montreal, 1976); E. Rodriques, Socialismo e sindicalismo no Brasil (Rio de Janeiro, 1969); E. Rodrigues, Pequena historia da imprensa social no Brasil (Florianopolis, 1997).
\item\relax
J. Godio, Historia del movimiento obrero latinoamericano, Vol. 1., Anarquistas y socialistas 1850-1918 (Mexico, 1980); L. Gambone, “The Libertarian Movement in Chile,” Black Flag, 1990, January, No. 196; L. Vitale, Contribucion a una Historia del Anarquismo en America Latina (Santiago, 1998).
\item\relax
El anarcosindicalismo en el Peru (Mexico, 1961).
\item\relax
See: C. Cornelissen, “Uber den internationalen Syndikalismus\dots{},” p. 150 (n31); M. Van der Linden and W. Thorpe (eds.), Revolutionary syndicalism: an international perspective\dots{}, p. 239 (n19).
\item\relax
From this point on in the text, only informative footnotes are included. For footnotes including references, please see the full PDF text.
\end{enumerate}
\section{Chapter 4: Revolutionary Syndicalism during the First World War}
The First World War was a serious test for the internationalist and anti-militarist position proclaimed by the syndicalists. Some of them (Alexander Berkman, Antonio Bernardo, V. García, A. Shapiro, Bill Shatov) together with E. Malatesta and Emma Goldman signed a manifesto against the war, denouncing it as a war of aggression by both sides.
They declared their intention to “incite insurrection and organize revolution.” Others (like Christiaan Cornelissen) supported the position of P. Kropotkin, Jean Grave, Charles Malato, and a number of other prominent anarchists who rallied to the side of the Entente since they considered German imperialism the “greater evil.”
The decline of revolutionary syndicalism in France could be noted even before the war. The progress of industralization brought with it a temporary stabilization in standards of living and some increase in wages; strikes acquired a more peaceful character, and among the workers and labour unions there arose a inclination to solve problems through negotiations. The leaders of the CGT (its general secretary Léon Jouhaux, P. Monatte and others) were compelled more and more to take into account the reality of industrial development. “After 1910 the ideological pretensions of the revolutionary syndicalists and the actual behaviour of workers in the CGT itself began to diverge more and more\dots{} The Amiens compromise, which pointed to the future, had nothing to offer.” The outbreak of the war deepened the crisis of French revolutionary syndicalism. The federal bureau of the CGT did not proclaim a general strike against the war, but issued a call “to defend the nation.” During the war years, representatives of the CGT collaborated in various “mixed commissions” created by the State. At the same time, an antiwar opposition surfaced within the organization in 1915, led by Alphonse Merrheim and P. Monatte, and grouped around the newspaper La Vie ouvriere. During the next year the left revolutionary syndicalists formed a Committee of Syndicalist Defense (CDS) which, despite taking an extreme anti-war position which referred to the “Charter of Amiens,” achieved a large measure of independence from the left socialist opponents of the war. In 1917 the Committee supported strike action by the workers, and spoke out against the worsening of living conditions and the intensification of labour.
In Italy the question of what stance to take regarding the war lead to a split in the USI. The group led by the general secretary A. De Ambris endorsed participation in the war on the grounds that this would facilitate the “revolutionization” of the country (a position which was labelled “revolutionary interventionism”). However this group did not enjoy the support of the majority of members and organizations of the USI. A new general secretary was elected – Armando Borghi.
In 1915 the USI endorsed the idea of a general strike against the war, although lacking the practical possibility of carrying it out. Adherents of “interventionism” were expelled from a number of unions.
The American syndicalists of the IWW launched an active struggle against entry into the war, which provoked furious persecution on the part of the government and nationalists.
In 1915 the well known IWW activist Joe Hill was executed, in 1916 five union members were shot by police in an atmosphere of nationalist hysteria, and in 1917 1,200 members of the IWW were deported to the New Mexico desert in connection with a miners’ strike in Arizona. Meanwhile, the IWW was successful in helping large strikes in Wheatland (California, 1915) and the Mesabi Range (Minnesota, 1916).
In the spring of 1917, job actions and sabotage organized by the IWW inflicted significant losses on branches of industry – woodworking and copper mining – vitally important for the prosecution of war. Between 1916 and 1917 the number of members of the IWW grew from 40,000 to 75,000, and by the end of the summer of 1917 had swollen, according to various sources, to between 125,000 and 250,000.
In Germany the syndicalist movement was virtually paralyzed soon after the start of the war, and the FVdG and its press were banned. In Great Britain as well nothing in the way of active work occurred.
The longer the war continued, the worse the lives of the workers became. In many countries strikes flared up as well as hunger riots. Anarchists and syndicalists took an active part in them. In France in May 1918, a congress of revolutionary syndicalists came out in favour of a general revolutionary strike against the war. In protest demonstrations an especially active role was played by the metalworkers of the Loire and Paris region, resulting in substantial losses to the war industry. The movement was suppressed, activists were dispatched to the front, and the leader of the Committee of Syndicalist Defense Raymond Péricat was convicted of treason against the State.
In Spain (neutral, but economically sucked into the war) in 1916 workers all over the country protested against the rise in the cost of living; the country was paralyzed. The CNT signed a “revolutionary alliance” with the socialist General Workers’ Union (UGT). In May-June 1917 Spain stood on the threshold of revolution. In August a general strike broke out, on a scale unseen up to that time, accompanied by armed struggle. The outbreak was suppressed after a battle lasting many days.
In Portugal protests against increases in the cost of living and the number of unemployed workers constantly developed into acts of resistance which often were spontaneous in character. In September 1914 unrest flared up in Lisbon, and the first fatalities occurred. In the spring of 1915 unemployed workers seized the ministry of agriculture and destroyed it. Riots and mayhem gave way to strikes, organized by the trade unions. By 1917 the revolutionary syndicalists had achieved dominance in the National Workers’ Union (UON), completely overshadowing the socialists.
Regaining their composure after the first shock, the anarchists and revolutionary syndicalists tried to re-establish regular international contacts. In 1915 an international antimilitarist congress was organized in the Spanish region of Galicia. It assembled not only many prominent Spanish working class anarchists (such as Ángel Pestaña, M. Andreu, F. Miranda, L. Bouza, Eusebio Carb\_, Eleuterio Quintanilla, and others), but also delegates from Portugal (notably M. J. de Sousa), France, England, Italy, Brazil, Argentina, and Cuba. At the meeting the question of an international general strike was discussed. The meeting also played an important role in renewing the Spanish CNT. In December 1916 the NAS of neutral Holland called on workers’ organizations of all countries to gather at a world congress of revolutionary syndicalism, but this idea was not carried out until the end of the war.
The inability of workers’ organizations to prevent World War I, the impotence of “neutral” syndicalism, and the increase in revolutionary sentiments among the labouring masses made changes in the syndicalist movement itself all the more urgent. “The Great War swept away neutral syndicalism,” noted A. Shapiro later. To many activists it became clear that syndicalism by itself was insufficient, that it was necessary to combine the self-organized workers’ movement with direct action animated by clear revolutionary ideas.
\chapter{Part 2: Anarcho-syndicalism}
\section{Chapter 5: The Revolutionary Years}
The global revolutionary wave which started in 1917 in Russia gradually enveloped other countries. Anarchists and syndicalists took an active part in events and were frequently found in the front ranks of revolutionary actions.
The general enthusiasm and mass self-organization of the workers imparted a new impulse to the libertarian workers’ movement.
The Russian anarcho-syndicalists in 1917-1918 were grouped around the newspapers Golos truda and Novy golos truda, and in 1918 they held two All-Russian conferences (in August-September and November-December). In 1920 the Russian Confederation of Anarcho-Syndicalists (RKAS) was created. In Ukraine the anarcho-syndicalists took part in creating the Confederation of Anarchists of Ukraine – Nabat, which exerted a substantial influence on the Makhnovist movement.
The libertarians enjoyed appreciable support in the factory committees and independent labour unions. At the end of 1917 and beginning of 1918 they were successful in organizing 25-30 thousand miners of Debaltsevo (in the Donbass) on the basis of the platform of the Industrial Workers of the World (IWW). They were recognized by the miners of Cheremkhovo in Siberia, stevedores and workers in the cement industry in the Kuban and Novorossiysk, railway workers, workers in the perfume industry, and workers in other fields.
In 1918 the anarcho-syndicalists supported bakery workers in Moscow, Kharkov, and Kiev; postal-telegraph workers in Petrograd; river transport workers in the Volga region; etc. Some of these organizations were destroyed by the Whites, others were neutralized by the Bolshevik authorities by means of mergers and outright oppression of activists. As a result, while at the First All-Russia Congress of Trade Unions (1918) the syndicalist and Maximalist delegates represented around 88,000 workers, at the Second Congress (1919) they represented 53,000, and at the Third (1920) 35,000 at most. An attempt by some of the syndicalists to organize a General Confederation of Labour independent of the Bolshevik government was suppressed. By 1922 the unions created by the anarcho-syndicalists had been disbanded, and their publishing operations shut down. The leading activists of the movement were arrested: Vsevolod Volin, Aron Baron, Mark Mrachny, and other anarchists and syndicalists who took part in the Makhnovist movement – in November and December 1920; Grigory Maksimov – in March 1921, etc. After a ten day hunger strike in Tagansk Prison in 1921, and protests by foreign delegations arriving in Moscow in connection with the First Congress of the Profintern, Volin, Maksimov, Mrachny, and several of their comrades were deported from Soviet Russia in January 1922. Another prominent Russian anarcho-syndicalist, Aleksandr Shapiro, was arrested by the Bolshevik authorities after his return from a syndicalist conference in Berlin in the summer of 1922. After numerous protests from abroad he was also deported.
In Germany the anarchists were part of the Council movement; two prominent anarchists (Gustav Landauer and Erich Mühsam) took part in the executive organs of the Bavarian Soviet Republic. The FVdG resumed its activity soon after the November revolution of 1918 and began to publish its newspaper Der Sindikalist. The FVdG “presented itself as the only organizational alternative at the time for those workers disillusioned with the politics of the official parties and identifying with radical unionism.” Considering themselves the left wing of the Council movement, the syndicalists took the position that these organs were not like political parties, but should take the economic functions of management into their own hands. “Workers Councils must have control over all the revenues and expenditures of enterprises, and actively participate in accepting orders and ordering raw materials. In doing so they are acting in the interests not only of the workers but of the whole of society. In the final analysis, the workers become the sole masters of the means of labour, thereby completing their humanity,” emphasized the FVdG newspaper. The German syndicalists were influential in the Workers’ Council at the Thyssen machine-building plant in Mülheim, in the Workers’ and Soldiers’ Council in the same city, played a decisive role in the strike movement in Hamborn, and were represented in the Munich Soviet.
To the extent the Council movement went into decline and was integrated into the system of the Weimar republic (law about Councils of enterprises of February 4, 1920), the FVdG regarded the possibility of the spread and development of Councils within capitalist society as an illusion.
The influence of the syndicalists rose quickly after the armed suppression of a general strike in the Ruhr in April 1919. In December of that year the FVdG was transformed into the Free Workers’ Union of Germany (FAUD); almost 112,000 workers were represented at its founding congress.
This organization called for a general strike to turn back the counterrevolution, but its initiative did not find a response.
In 1919-1920 during the course of radical strikes in the Ruhr, syndicalist methods of direct action were often used. In March 1920 during a general strike against the Kapp putsch, which evolved into armed revolts in a number of regions, branches of the FAUD in many cities led the struggle, despite the cautious stance of the central executive committee of the union which condemned “putschism.” The FAUD took part in Workers’ Councils in Essen, Mülheim, Oberhausen, Duisberg, and Dortmund. In Mülheim and Hamborn Factory Councils followed the advice of the FAUD and took control (“socialized”) the gigantic Thyssen plants. Forty-five percent of the soldiers of the “Red Army of the Ruhr” were members of FAUD. In the Thuringian industrial city of Sömmerda the syndicalists and left communists declared a Soviet republic. Although the movement was harshly suppressed, the popularity of the FAUD in these revolutionary years continued to grow. In 1921 it counted 150,000 members.
In March of that year, despite the negative attitude of the executive committee in Berlin, Thuringian members of the FAUD together with left communists again took part in an armed revolt.
The ebb of the revolutionary wave and government repressions led to a rapid decrease in the membership of the organization. At its congress in 1922, only about 70,000 members were represented. However the FAUD still remained a significant force, especially at the local level (among the miners and metalworkers of the Ruhr and Rhineland, construction workers in Berlin, and workers of Central Germany). In 1923, under conditions of crisis and revolutionary fervour after the occupation of the Ruhr by Franco-Belgian troops, the anarcho-syndicalists supported many strikes and demonstrations by the unemployed, calling for a general strike and social revolution. However the economic catastrophe and mass unemployment undermined the strength of FAUD and its ranks fell to 30,000.
In Italy the revolutionary syndicalist trade union USI already in the summer of 1919, in spite of repression, unleashed a strike movement in La Spezia and a 48-hour general strike in Bologna. The USI endorsed the seizure of factories by the workers. At its third congress in Parma (December 1919), the USI proposed a system of “autonomous and free” Councils “antithetical to the State.” These Councils were seen as organs both for the defense of the workers and for the administration of the future society. The USI supported the initiatives of workers to create Factory Councils and urged that they not be allowed to fall into reformist “degeneration.” In February 1920 metalworkers belonging to the USI seized factories in Sestri-Ponente and neighbouring cities and set up Councils to manage them. In March workers’ unrest spread to Turin, and in April convulsed the whole of Piedmont and Napoli. In Pombino workers organized in the USI rose in revolt to protest the dismissal of 1,500 workers of the Ilva firm and took over the city. The syndicalists were also active as organizers of strikes of farm labourers and anti-militarist demonstrations. In July 1920 the USI called on metalworkers to carry out a wholesale seizure of factories in response to the intransigence of the owners and lockouts. In August – September armed workers created a “Red Guard” which seized around 300 enterprises in Milan; the movement then spread throughout the whole country. Factories were taken over by Councils. However the numerically dominant CGL, controlled by socialists, was content with promises of minimal concessions and, not desiring revolution, put the brakes on the movement, while the USI, with its 500,000 members (several times smaller than the CGL) did not risk continuing the struggle alone. After this the revolutionary wave in Italy went into decline, although in March of the following year the USI was able to conduct a general strike in Milan and a shutdown of the USI-controlled “houses of labour” [labour exchanges] in support of imprisoned members of the organization.
From the winter – spring of 1921 the syndicalists, along with other leftists, became the objects of armed attacks on the part of the fascists, who destroyed the “houses of labour” and interfered with the activities of left-wing trade unionists and parties throughout the whole country. “Faced with attacks by fascist gangs, the USI organized itself on various levels in order to resist the wave of reaction – both by radicalizing the social struggle and by having recourse to arms. In contrast to the indecisiveness of other parties and unions, the USI chose direct action\dots{} In order to put an end to the fascist strategy of systematic attacks in areas where level of antifascist and class struggle was high\dots{}, the USI encouraged the creation of armed volunteer groups of ‘people’s heroes’\dots{} and transformed their main ‘houses of labour’ into small fortresses, capable of withstanding attacks by fascist gangs.” The syndicalists and anarchists responded to the fascist assault with proletarian class action – with strikes – but did not succeed in vanquishing the fascists, who were, for all intents and purposes, supported by the country’s rightwing circles. It’s true the struggle against the “blackshirts” led to an agreement between the Italian trade unions to create an “Alliance of Labour” which, in July 1922, declared a general antifascist strike. In a few cities (Parma, Bari, and others) this developed into an armed revolt. But the reformists also retreated on this occasion. “The fact remains that fascism\dots{} was able to become an irresistible force and, with the support of the tried and tested repressive apparatus of the monarchist State, it was able to sweep aside all obstacles in its path. The equivocal actions of the reformist Left, the sectarianism of the Communist Party, and the military and political unpreparedness of the revolutionary forces hastened the defeat of the workers’ movement.” Several months later (in October 1922) a government came to power headed by the fascist leader B. Mussolini. After the new regime was established, naked repression led to a destruction of all the local sections of the USI, and the mass arrest or emigration of the most energetic members of the organization, which was forced to restrict its activities to the underground.
The revolutionary workers’ movement in Spain grew rapidly. New syndicates of the CNT sprang up everywhere.
By a decision taken at a Catalonian regional congress in July 1918, these syndicates were “integrated” at the local level, i.e. they were industrial rather than craft unions. The CNT already had more than one million members. A national conference of anarchists in November 1918 urged all libertarians to join the CNT. In February 1919 as a sign of solidarity with striking workers at the “La Canadiense” company, the anarchist syndicalists launched a general strike – one of the largest and most successful in the history of the Spanish labour movement. It induced panic among the ruling classes. Even the declaration of martial law did not save the owners. The action ended with the complete triumph of the workers. The centre of the workers’ struggle was Barcelona.
Large-scale events included a struggle against a lockout at the end of 1919, a general strike against repression in November 1920, and a strike of transport workers in 1923. The CNT had already started to collect statistical data which would allow it to run the economy smoothly after the forthcoming social revolution.
Then the ruling classes had recourse to a different tactic: they began to create “yellow” trade unions and terrorist gangs of “pistoleros,” murdering activists of the workers’ movement.
In December 1919 in an atmosphere of revolutionary enthusiasm, a congress of the CNT in Madrid announced as its goal the liquidation of the State and the establishing of libertarian (anarchist) communism, in other words, finally and officially rejecting the concept of “neutral syndicalism” and declaring the correctness of the tradition of the Bakuninist wing of the First International. In response to the unceasing wave of strikes the government unleashed systematic repression. The leading activists of the CNT were arrested, including the members of the Confederation’s executive (in March 1921). The organization was deprived of its leadership and forced to go underground. In the spring of 1923 the prominent working class leaders Salvador Segui and F. Comas were murdered. The anarchists and syndicalists answered counterrevolutionary terror with strikes and armed actions. The stand-off continued until the dictatorship of General Primo de Rivera was installed in September 1923 and independent trade union activity was prohibited.
In Portugal the UON, in which the revolutionary syndicalists now predominated, organized a successful general strike in the Lisbon region in support of construction workers, offering armed resistance to the police and the national guard. The federation of construction workers called for an armed revolt in the course of a new general strike planned for November 1918 which had been announced by the UON.
The failure of this revolt did not discourage the workers. In 1919 protests of workers against the rising cost of living and unemployment continued. In some sectors of the economy there were breakthroughs in gaining the 8-hour workday. A workers’ congress in September 1919 transformed the UON into a united organization of the Portuguese workers – the General Confederation of Labour (CGT). The principles of revolutionary syndicalism were enshrined in its articles.
All the tendencies in the Confederation were in agreement that pure trade unionism was insufficient. The Portuguese CGT included not only trade unions but, starting from 1922, also students and artists, tenants’ associations, consumer cooperatives, and “groups of syndicalist solidarity.” The number of members of the CGT, which reached 120,000 – 150,000 in 1919, had fallen somewhat by 1922 but the organization as before still united the majority of organized workers in the country. However its activities to a significant degree were spontaneous in character. They consisted usually in the organization of a sudden tide of protest which soon ebbed without being channeled into building a strong organization and solidarity between workers (although many strikes were carried out successfully, and in February 1924 the largest workers’ demonstration in Portuguese history took place with more than 100,000 participants).
A rebirth of the revolutionary workers’ movement began also in France. The dampening of the strike movement by the reformist leadership of the French CGT ignited the trade union opposition grouped around P. Monatte and the newspaper Vie ouvriere. This opposition was strengthened at the congress of the CGT in September 1919, and it formed its own coordinating body and it started setting up “Revolutionary Syndicalist Committees” (CRS), trying to establish its influence in individual unions and “bourses de travail.”
It succeeded in consolidating its position in the union of railway workers. At the beginning of 1920 the country was paralyzed by railroad strikes. The revolutionary syndicalists organized a general strike for May 1, which was joined by metalworkers, construction workers, dockers, and miners.
But the hopes this insurgency would grow into revolution were not realized. In September 1921 at a conference of the opposition in Lyon a Central Committee of the CRS was created, headed by P. Monatte. In December 1921 at a congress in Paris the revolutionary syndicalists announced their split from the CGT and in July 1922 at a congress in Saint-Étienne they created the new “Unitarian CGT” (CGTU).
Anarchists and syndicalists were active in the workers’ movements of some other European countries. The membership of SAC in Sweden reached 32,000 workers in 1920, chiefly bricklayers, construction workers, workers in the forestry and paper industries, and metalworkers. Although it remained small in comparison with the social-democratic union movement, it participated in a broad range of post- war strikes. The syndicalist federation of Norway and Danish syndicalists had close connections with SAC. The Netherlands Labour Secretariat (NAS) strengthened its own position during the war years, thanks to its energetic support of the movement against military service and the high cost of living, and engaged in a wave of strikes and protests in the first post-war years. Its membership grew to 49,000 in 1918 but as before it was smaller than unions of a socialdemocratic or clerical persuasion. The failure of strike actions in 1920-1922 led to a shrinkage in the membership (by the autumn of 1922 the NAS was down to 26,000 members) and favoured the intensification of internal disagreements.
In other regions of Europe, despite the presence of a strong anarchist movement (Bulgaria) or a definite syndicalist tendency in the union movement (Great Britain, Czechoslovakia, Belgium), in the postwar years it did not prove possible to create an anarcho-syndicalist union central.
The revolutionary wave which began in Russia, coupled with the postwar economic difficulties, inspired a powerful expansion in working class actions in Argentina, in which the FORA and its member unions played a leading role. The most important of these actions were the general strike in Buenos Aires in January 1919, which was accompanied by battles at the barricades and harsh repressions (“the tragic week”), a general strike in the capital in May 1920, and a strike and revolt of agricultural labourers in Patagonia (1921) which was suppressed by government troops with great cruelty. The Uruguayan FORU in 1917-1921 virtually headed the strike movement in the country, organizing a series of stubborn general and local strikes. In Brazil the anarchists even during the war period were at the epicentre of the movement against militarism and increases in food prices due to profiteering. Massive general strikes took place in 1917 in São Paulo, Santos, and Rio de Janeiro. In the course of the struggle the workers were able to achieve significant concessions and the adoption of labour legislation. In November 1918 the anarchists of Rio de Janeiro rose in revolt, intending to overthrow the government and proclaim a “communist republic.” The uprising was suppressed, and the government smashed the pro-anarchist workers’ federation of the state, which included as many as 150,000 workers.
However the anarchists still maintained their position in the workers’ movement which was confirmed by the outcome of the 3rd congress of the Brazilian workers’ confederation in 1920. The destruction of the anarchist workers’ movement happened only after the army mutinies of 1924.
In Mexico the anarchists criticized the collaboration of working class leaders with government authorities, and also the pro-government policies of the trade union activists headed by Luis Morones, who founded the Mexican Regional Workers’ Confederation (CROM) in May 1918. Anarchist and syndicalist groups convened a congress in 1921 in Mexico; at it the creation of the General Confederation of Workers (CGT) was announced. It was based on the unions of textile workers, streetcar conductors, telephone operators, oil field workers, etc. During the 1920’s the anarcho-syndicalists led the strike struggle of these categories of wage workers. The confederation, which had a membership of about 60,000 workers, endorsed “libertarian communism.” In Chile the anarchists and syndicalists worked in the Federation of Chilean Workers until 1921, making up its extreme left wing, but then the centre of attraction of anarchists became the Chilean section of the Industrial Workers of the World, formed in 1918-1919, which had a membership of over 25,000 members in 1920 – including dockers, seafarers, construction workers, shoemakers, etc. The Chilean IWW took an active part in actions against the high cost of living and shortages in food supplies. It also supported the student movement and was active until the installation of the military dictatorship of Carlos Ibáñez in 1927.
Anarcho-syndicalist union centrals occupied a leading position in the workers’ movement in a number of other countries of Latin America. The regional workers’ centre of Paraguay headed a strike movement, including a strike of electrical workers and a general strike in Ascención in 1923-1924. In Bolivia the Local Labour Federation of La Paz (founded in 1918) and the syndicalist miners’ union launched a desperate strike struggle. Peruvian anarcho-syndicalists (in particular, stevedores, bakers, textile workers, etc.) continued a stubborn struggle for the inauguration of the 8-hour day and against the rising cost of living. In the midst of a wave of general strikes in 1919, which took on a revolutionary character, a Peruvian regional workers’ federation sprang up; the government was compelled to agree with the demanded reduction in the length of the workday. The movement was destroyed by a military dictatorship in the middle 1930’s, and influence in the trade unions shifted to communist party members and national-reformists. In Ecuador under the influence of anarchists a regional federation of workers appeared in 1922. In October – November of the same year, it organized the largest general strike in the history of the country in Guayaquil, in the course of which the city was for a time under the control of the workers. The harsh suppression of the strike dealt the movement a heavy blow from which it recovered somewhat only in the second half of the 1920’s, when the anarcho-syndicalists were able to revive a number of labour unions.
In Cuba the anarchists and syndicalists predominated in the leadership of the Workers’ Federation of Havana (1921) and the National Federation of Workers of Cuba (1925), up to the point when they were destroyed by the dictatorship of J. Machado in 1925-1927. It was this disaster, as the Cuban communists themselves have admitted, which allowed them to establish their control of the workers’ movement of the country.
In the countries of Central America the anarchists and anarcho-syndicalists for a time enjoyed an appreciable influence in organizations of the labour movement, including: the General Confederation of Labour of Costa Rica (1913- 1923), the Workers’ Federation of Panama (1921-1923), the General Labour Union of the Workers of Panama (mid 1920’s), the Regional Federation of Workers of Salvador, the Committee for Trade Union Activity of Guatemala (end of the 1920’s), etc.1
The workers’ movement in Japan became radicalized in a hurry in the first postwar years under the influence of the food riots of 1918 and the wave of strikes of 1919-1921, in the course of which methods of direct action were widely used by the workers. In the most important union central of the country, Yu-Ai-Kai, the influence of the anarchists, revolutionary syndicalists, and adherents of Russian Bolshevism gained strength. At their insistence the congress of Yu-Ai-Kai in 1920 approved the principles of class struggle and direct action; in 1921 the union central was renamed the Japanese Federation of Labour (Sodomei). But already by 1922 a regroupment of forces in the workers’ movement of the country took place. Reformist leaders of the union central and the communists came out in favour of a re-organization of the union movement on a sectoral basis, while the anarchists and the syndicalists who were close to them upheld federalist principles and the autonomy of labour unions. The libertarians left Sodomei, but a number of unions remained under their influence, including the printers, mechanics, metalworkers, electrical workers, and the regional association of unions of Tokyo. [
The association of anarchist unions of Japan was able to impede the repressions after the “great earthquake” of September 1923, in the course of which the leading anarcho- syndicalist Ōsugi Sakae was killed. Only in 1926 did a labour union central appear which approved the principles of anarchist communism – the All-Japanese Libertarian Federation of Trade Unions (Zenkoku Jiren). This federation existed until the mid-1930’s, when it was annihilated by government persecutions.
In China the anarchists were the organizers of the first labour unions of the modern type in Guangzhou in the 1910’s, and also organized the first strikes. At the beginning of the 1920’s the workers’ organizations of this city, being under the influence of anarchists (especially the dockers and service workers), were united in a Workers’ Mutual Aid Society; however, in 1923-1924 it fell apart. In November 1920, on the initiative of anarchists a Society of Workers of the Province of Hunan was formed, uniting the workers of the most varied branches of heavy and light industry. It organized important demonstrations of textile workers, but in January 1922 it was destroyed by the provincial authorities and its leaders executed. In the 1920’s the centre of the anarchist and syndicalist movement shifted to Shanghai, where the anarchists and other non-communist workers’ unions formed a Federation of Labour Unions in March 1924. It participated actively in a strike movement. But in 1927 control of the federation passed into the hands of members of the Guomindang. In 1926 anarchists and anarchosyndicalists formed a Federation of People’s Struggle, which affiliated to the IWA; this organization ceased its existence under conditions of civil war towards the end of the 1920’s.
In the majority of colonial countries of the Far East, where the social struggle was centred on the acquisition of independence for a national state, the anti-statist slogans of the anarchists were not widely disseminated. A group of revolutionary emigrants from India led by M. P. T. Acharya adopted anarcho-syndicalist positions. The group tried to carry on work in Indian labour unions, but its propaganda was suppressed by the British colonial authorities. In Korea and Taiwan the anarchists, strongly influenced by their Japanese comrades, acted in the 1920’s to create a number of unions and underground groups which were soon wiped out. Anarchist unions of Chinese workers were active in the 1920’s in Malaya and in other countries of Southeast Asia.
In the postwar years the activity of the Industrial Workers of the World (IWW) increased – this was a special variety of the syndicalist movement. Like European syndicalists, its members embraced the idea of unions carrying out the revolution and running things themselves, and they applied the tactics of direct action and were critical of parliamentarism and political parties. However they rejected federalism and were in favour of creating “one big union” of all the workers with divisions according to various branches of industry.
Anarchists did not play a decisive role in the unions of the IWW, in fact activists of various leftist Marxists parties were much in evidence. In the U.S.A. members of the IWW suffered greatly from government repression in 1917-1920. Another industrial unionist labour central – the One Big Union (OBU) – arose in 1919 in Canada and headed a powerful general strike in the western part of the country. The North American IWW and OBU did not develop along the lines of anarcho-syndicalism. In Australia and New Zealand, the initial groups of the IWW carried on work in the existing labour unions, trying to encourage them to associate on an industrial basis and adopt the principles of the IWW. They suffered greatly from repression during the First World War, and then many of their leading activists joined communist parties. In South Africa industrial unionists were grouped around the IWW (1910-1914), the International Socialist League (1915-1921), and the Industrial Socialist League (1918-1921). They acted as organizers of major strikes (including a general strike of miners in 1921-1922) and a number of active unions of “whites,” “blacks,” and Indian workers. But after 1921 the majority of the South African unionists joined communist parties.
\begin{enumerate}[1.]
\item\relax
In Columbia and Venezuela the anarcho-syndicalists tendencies began to have an impact only towards the middle and end of the 1920’s.
\end{enumerate}
\section{Chapter 6: From Revolutionary Syndicalism to Anarcho-syndicalism}
The Russian Revolution, it seemed, offered the workers’ movement a revolutionary alternative to social-reformism.
The idea of soviets – not as state organs staffed by party officials but as instruments of non-party self-organization and workers’ self-management of production and of local living arrangements – played an important part in the belief systems of many anarchists and syndicalists.The majority of libertarians were enthralled by events in Russia, seeing in them what they wished to see rather than what was actually transpiring. In the words of Malatesta, they interpreted the dictatorship of the proletariat not as a system of government, but as “a revolutionary action with the help of which the workers would take possession of the land and the means of production, and would attempt to build a society in which there was no place for class, no place for exploitative and oppressive owners. In this case the ‘dictatorship of the proletariat’ would denote the dictatorship of everyone and therefore would not be a dictatorship at all, the same as a government of everyone is no longer a government in the authoritarian, historical, and practical sense of the word,” the old anarchist noted. A section of the libertarians became convinced that the Bolshevik system of “the dictatorship of the proletariat” is some kind of intermediate stage on the road to the anarchist organization of society (the phenomenon of “anarcho-bolshevism”). It was years before the anarchists and syndicalists grasped that behind the “power of the soviets” was hidden a new party-state dictatorship.
The revolutionary syndicalists were faced with the necessity of choosing between anarchism and Bolshevism. The question of the orientation and goals of the movement was central to the process of its unification on a global scale. At the end of 1918 the Dutch and German syndicalists renewed their appeal for the convening of an international congress, but at a conference in February 1919 in Copenhagen, only the Scandinavian delegates were able to be present. Attempts during 1919-1920 to assemble a congress in the Netherlands and Sweden were unsuccessful. Meanwhile the Bolsheviks, along with Communist parties and groups in a number of European countries, announced the creation of the Communist International. To many anarchists and revolutionary syndicalists it seemed that this new international association could be the centre of attraction not only for the left-radical wing of social-democracy but also for libertarians, as a sort of historical compromise between Marx and Bakunin on the basis of revolutionary principles. Announcements about joining the Comintern were made by the French “Committee of Syndicalist Defense” of R. Péricat (renamed the Communist Party in the spring of 1919, and later – the Communist Federation of Soviets), by the Italian USI (in July 1919 and confirmed at a USI congress in December), and even by – “temporarily” in anticipation of the holding of a congress in Spain to organize a “genuine workers’ International” – the Spanish CNT (at a congress in December 1919). A number of prominent leaders of Anglo-Saxon syndicalism joined communist parties: Bill Haywood (American IWW), T. Mann (the leading British revolutionary syndicalist), and others.
There were some anarchists who spoke out early on with a sharp critique of the Bolsheviks and their dictatorship.
Among them were the Italian Luigi Fabbri and the German Rudolph Rocker. Already in 1919 skepticism regarding the Bolsheviks’ break with the centralism of social-democracy was expressed by the Swedish revolutionary syndicalists (SAC). <fn>In 1922 SAC declared that affiliating to the International being created in Moscow was incompatible with the syndicalist principle of independence from political parties (RGASPI: F.532, Op. 7, D. 624, L. 23, 36, 65-66).<\Slash{}fn> But the centre of resistance to the influence of Bolshevism became the German revolutionary trade union association FAUD.
In December 1918 FAUD called for co-operation with revolutionary socialists. Within its organization there were supporters and even members of the Communist Party.
In the spring of 1919 the prevailing view within its ranks was support for a non-party “dictatorship of the proletariat” in the form of Councils, in contrast to parliamentary activity, although it was maintained that socialization could only be carried out by revolutionary unions. In December 1919 at the 12th congress of the FVdG, which morphed into the FAUD, solidarity was expressed with Soviet Russia. But at this same congress R. Rocker took the floor with a report on the principles of syndicalism. His speech and the resulting “Declaration concerning the Principles of Syndicalism” set forth a synthesis of anarchism and revolutionary syndicalism on which the ideology of the anarcho-syndicalist movement was based. An adherent of the anarcho-communism of P. Kropotkin, Rocker combined the traditional goals of anarchism (doing away with the State, private property, and the system of the division of labour; creation of a federation of free communes and a diversified economy aimed at the satisfaction of the real needs of people – the ethical basis of socialism) with ideas developed by the German anarchist G. Landauer about a new culture and the creation of the elements of a future free society without waiting for a general social upheaval. Rocker was convinced the social revolution could not be carried through spontaneously, that it must be prepared still within the framework of existing capitalist society and that the better it was prepared, the less trouble and pain there would be in carrying it through. Following the revolutionary syndicalists, he considered the unions (syndicates) to be the organs and elements of preparation for the revolution. The unions, in Rocker’s opinion, struggling not only for momentary improvements, but also for revolution, are “not a transitory product of capitalist society, but the cells of the future socialist economic organization.”
Rejecting private property as a “monopoly of possessions” and government as a “monopoly of decision-making,” the syndicalists should strive “for collectivization of land, work tools, raw materials, and all social wealth; for the reorganization of the whole of economic life on the basis of libertarian, i.e. stateless, communism, which finds its expression in the slogan: ‘From each according to their abilities, to each according to their needs!’” Rocker criticized not only the bourgeois State, State boundaries, parliamentarism and political parties; but also Bolshevism (party communism) since centralization, preservation of State power, and nationalization (government ownership) of the economy can “lead only to the worst form of exploitation – State capitalism, rather than socialism.” The syndicalists should act not to win political power, but for the eradication of political power generally. As for socialism – in the final analysis this is a question of culture – it cannot be established by any kind of decisions from above. It is only possible in the form of an association of self-managed groups of producers, of workers performing both mental and physical labour. By this means “groups, enterprises, and branches of production” would work as “autonomous members of a general economic organism, which on the basis of mutual and free agreements would systematically carry out production and distribution in the common interest.” As the instruments for such “planning from below” Rocker considered statistics and voluntary agreements. “The organization of enterprises and workshops by economic councils, the organization of the whole of production by industrial and agricultural associations, and the organization of consumption by workers’ exchanges” (i.e. industrial associations of workers at the local level) – he proclaimed.
According to the notion of the German anarcho-syndicalists, in the course of a victorious general strike it was appropriate to carry out the expropriation of private property, enterprises, food stores, real estate, etc. The management of enterprises was to be transferred into the hands of Councils of workers and employees [office workers]; the management of dwellings into the hands of Councils of tenants. Delegates from enterprises and districts would constitute a Commune.
Money and the system of commodity production (for sale) was slated to be abolished: the regulation of consumption (fixed levels in the beginning, later driven by demand) was to be entrusted to “labour exchanges” and tenants’ councils.
The fundamental difference between anarcho-syndicalism and revolutionary syndicalism lay in the fact that syndicalism did not consider direct action to be “selfsufficient” as a means of achieving anarchist communism.
“\dots{} Anarcho-syndicalism exists as the organizational force of the social revolution on a libertarian-communist basis; anarcho-communists must be anarcho-syndicalists in order to organize the revolution, and every anarchist who is able to become a member of a trade union should be a member of the anarcho-syndicalist Confederation of Labour,” the general secretary of the anarcho-syndicalist International, A. Shapiro, declared later.
In spite of the openly anti-Bolshevik orientation of the new doctrine, the German anarcho-syndicalists in the beginning still permitted limited co-operation with Communist Party members. Thus, in January 1921, the executive committee of FAUD stated in a letter to the Central Committee of the United Communist Party of Germany that the syndicalists were agreeable to joint actions under the condition that the participating organizations harmonize their demands in advance (including the 6-hour workday, abolition of piece-work, rejection of weapons production) and their tactics, as well as treating participants as equals. But these conditions were unacceptable to State-communists. In 1921 FAUD announced that membership in political parties was incompatible with being in a syndicalist organization.
However in 1920 the possibility of co-operation in practice was still conceivable. At the invitation the Soviets, revolutionary trade union organizations of various countries sent their own representatives to the 2nd Congress of the Comintern in Moscow in the summer of 1920. The FAUD sent its own delegates – the Australian Paul Freeman and the German Augustin Souchy – with a mandate “to study the economic Soviet system in Russia so we have a clear picture of what’s going on and can evaluate the experience of the Russian comrades for our own country.” Freeman later became a supporter of Bolshevism, while A. Souchy returned from Moscow a fervent opponent. The latter described his impressions of the Russian Revolution in a timely book.
Subjecting to a sharp critique the Bolshevist modus operandi of seizing political power, centralization, and dictatorial state socialism, the German syndicalist made this recommendation: “[the Bolshevik method] should not be followed if a revolution should begin in our own country.”
At the 2nd Congress of the Comintern there were also syndicalist delegates or observers from other countries: Spain (Ángel Pestaña), France (Marcel Verge and Berto Lepti), a delegation of British shop stewards led by John Tanner, and representatives of the IWW. Immediately after the Congress the leading activist of the Italian USI, Armando Borghi, arrived in Moscow. In the course of meetings before the Congress, organized by the Executive Committee of the Comintern, the Bolsheviks proposed to create a new revolutionary International of Trade Unions so that in each country trade unions would have to act under the leadership of the Communist Party affiliated with the Comintern. It was envisaged that this project would also involve acceptance of the dictatorship of the proletariat. A. Pestaña, A. Souchy, and J. Tanner rejected the Bolshevist ideas about the necessity of working in reformist trade unions, the dictatorship of the proletariat, the conquest of political power, and the subordination of unions to communist parties. The Spanish delegate, bound by the decision of the CNT about joining the Comintern, agreed to sign the draft plan, but only after the Bolsheviks promised to exclude from it any mention of the dictatorship of the proletariat and the seizure of political power. However it turned out Pestaña was deceived: the text was published in the original form, but with his signature.
During the Congress itself the same disagreements were on display.
Now the revolutionary unions were faced with the decision whether or not to join to the newly created “Red International of Trade Unions” (Profintern). Declarations about affiliating were made by the British shop stewards and the French revolutionary syndicalists (at a conference in September 1920 in Orléans, accompanying this with an affirmation of loyalty to the Charter of Amiens). In December 1920 in Berlin the long-awaited international syndicalist conference convened with the participation of delegates from the FAUD (Germany, but also representing Czechoslovakia), FORA (Argentina), IWW (U.S.A.), CRS (France), NAS (Netherlands), shop stewards’ and workers’ committees (Britain), and SAC (Sweden). Declarations of support for the conference were made by syndicalists from Norway and Denmark, and by the Portuguese CGT. A delegation from Russian trade unions also arrived and urged the participants to endorse the dictatorship of the proletariat and the Profintern, which they insisted was a structure separate from the Comintern. The Swedish and German delegates took the floor with a critique of Moscow and the persecution of anarchists in Russia; the French representatives showed themselves to be solid supporters of the Bolsheviks; the Dutch delegation was split; and other delegates called for spelling out concrete demands for the form to be taken by an international association of revolutionary unions. These demands, approved by all the delegates with the exception of the Russians and French, became known as the “Berlin Declaration.” According to it, the Profintern would have to base itself on class struggle, aiming at the liquidation of the rule of the capitalist system and the creation of a free communist society. In this connection it was noted that the liberation of the working class must be carried out only with the help of economic means of struggle, and that the regulation of production and distribution must become the task of economic organizations of the proletariat. The complete independence of the trade union International from any political party was emphasized, although co-operation with parties and other political organizations was to be allowed.
All the revolutionary syndicalist organizations of the world were urged to take part in the Moscow congress of the Profitern. An international syndicalist information bureau was created in Amsterdam (its secretary was the Dutchman Bernard Lansink and the other members were R. Rocker from Germany and J. Tanner from Great Britain).
The Bolsheviks, the Western European communist parties loyal to them, and the Moscow organizing committee, tried to persuade the revolutionary syndicalists to take part in the new international trade union association under the aegis of the communists. The chief opposition to this was considered to come from the German FAUD. Thus, the section of the Communist Party of Germany which dealt with trade union work in the mining industry issued a directive to district secretaries and party fractions in the unions, ordering them to “struggle and defeat” this organization.
The communists encouraged breakaways from the FAUD in every way possible. The German anarcho-syndicalists did not send delegates to the Moscow congress. In France, where an internal opposition in the CGT existed, the communists distinguished “three tendencies: (1) anarcho-syndicalists, (2) old syndicalists who wanted to return to the Amiens program of 1906, and (3) communist syndicalists.” Moscow was counting on the third tendency for support and hoped to neutralize the first. Nevertheless the secretary of the Central Committee of the CRS, Pierre Besnard, took a position of opposition to Bolshevism. A group of new leaders of the Spanish CNT (Joaquín Maurín, Andrés Nin, and others) aspired to join with Moscow. They moved to the forefront at a plenum in Barcelona in April 1921 after the arrest of the members of the previous Confederational Committee. “In some sections of our Confederation one finds a certain opposition to joining the Red International of Labour Unions. But it is our firm hope that the CNT will join the Profintern,” they wrote to Moscow.
At the congress of the Profintern held in July 1921, the communists succeeded, thanks to a system of representation which favoured them, in assuring themselves a sizeable majority. All the revolutionary syndicalist organizations which took part in the 1920 Berlin conference sent representatives (with the exception of FAUD). But a motion proposed by Albert Lemoine that the Profintern not be subordinate to the Comintern failed, despite being supported by the French syndicalists, FORA, IWW, NAS, SAC, and the German leftcommunist workers’ unions. Also defeated was a proposal by the CNT, USI, NAS, IWW, FORA, the French and Canadian syndicalists, the Uruguayan regional workers federation, and the German unions opposing work in reformist unions.
After this the oppositionist syndicalists, getting together in Moscow, adopted a “Manifesto of the revolutionary syndicalists of the world” and agreed to create an “Association of revolutionary syndicalists elements of the world.” This association would include the CNT, USI, CSR, IWW, SAC, NAS, FORA, the German workers’ organizations, and unions from Denmark, Norway, Canada, and Uruguay giving a total membership of almost 2.8 million. It was proposed to locate the bureau of the new association in Paris. But the organization was not created at this time.
The Bolsheviks succeeded in sundering the united bloc of the syndicalist opposition. The leadership of the Profintern made a deal with the delegation of the Spanish CNT, promising that the communists would facilitate the merger of the socialist trade unions of the UGT with the CNT. The French delegates held meetings with representatives of the Profintern and agreed to join the Red International, but only on condition that the “Charter of Amiens” was observed, namely that the organizational independence of unions from parties would be preserved. In principle none of the syndicalists objected to belonging to the Profintern as long as a number of conditions were met – and only the FORA repudiated its delegate to the Moscow congress.
The situation began to change in an sense unfavourable for Moscow in connection with the repression against the anarchists and anarcho-syndicalists in Russia and Ukraine (a delegation of foreign syndicalists in Moscow demanded their release) and also because the Bolsheviks continued to insist on the subordination of the unions to the Comintern.
In October 1921 at an international conference of syndicalists from Germany, the Netherlands, Sweden, Czechoslovakia, and from the IWW, which was held in Düsseldorf on the occasion of the 13th congress of FAUD, a resolution was adopted to consider the founding of an International of trade unions abortive. The participants announced themselves in favour of convening a new international congress in Germany on the basis of the Berlin declaration. The preparation for this meeting was entrusted to an international Information Bureau of revolutionary syndicalists which set about putting out the appropriate international bulletin. The Italian USI also answered the call; at its own 4th congress in March 1922 it turned down a proposal by Nicolo Vecci’s group to join the Profintern until questions about the mutual relations of trade unions with the Comintern had been thrashed out at a new congress outside Soviet territory. The members of the Swedish trade union central SAC in a referendum turned down an amendment to their declaration of principles which would have envisaged the possibility of joining the Comintern and forming links with communist parties. The Spanish CNT at a plenum in August 1921 re-affirmed its independence from political parties and policy of organizing the social revolution and libertarian communism. Its newly elected National Committee was composed of anarchists.
In June 1922 at a plenum in Zaragoza the CNT adopted a resolution about withdrawing from the Comintern as a matter of principle and sending delegates to the conference of syndicalists.
Basically, the demands the syndicalists made to the Profintern reduced to the following points: “(1) cancellation of reciprocal representation between the Comintern and the Profintern in order to preserve the independence of the revolutionary union movement; (2) the second congress of the Profintern must be held abroad, in order to avoid the anticipated harmful influence of Russia on the gathering; (3) non-admission of separate delegations from the labour unions of Georgia, Armenia, Ukraine, and similar nations under Russian control; (4) relocation of the residence of the executive committee of the Profintern outside of the Soviet Union; (5) independence of the labour union movement from political parties, i.e. from communist parties, at the national and international levels; (6) denial of the right of representation to revolutionary minorities, which was meant to include communist opposition fractions in labour unions affiliated with the Amsterdam International [the international trade union association controlled by socialdemocracy – V. D.]; (7) voting at international congresses of the Profintern to be conducted on the basis of countries, regardless of the number of members of organizations; (8) restriction of the Profintern to the sphere of international affairs – prohibition of interference in practice and tactics in individual countries.” <fn>SAPMO: Bestand RY1\Slash{}I2\Slash{}708, Aktenband No. 53, Bl. 75-78.<\Slash{}fn>
The search for common ground between the Profintern and the syndicalists was initiated by the French Unitary General Confederation of Labour (CGTU). This organization was formed in 1922 by leftist tendencies which had withdrawn from the CGT. In a letter dated March 8 1922 directed to the Executive Office of the Profintern, its syndicalist leadership demanded the strictest observance of the complete independence of national labour union centrals from communist parties and the Comintern – only in this situation were they ready to join the Profintern. In this connection the CGTU was prepared to allow co-operation with communists within the framework of “coalitions of all the revolutionary forces” by means of specially created “Coordinating Committees.” In forwarding this proposal to Moscow, the Spanish communist Hilari Arlandis urged its acceptance, in order “to disarm the libertarians as quickly as possible” since these ideas enjoyed wide popularity among international syndicalist and even partly among communist circles, especially in Latin countries where the Profintern found itself in an “extremely delicate” situation and there was active anti-Bolshevik agitation by Russian anarchists. “If we don’t put an end to this opposition movement once and for all by making a declaration in favour of the complete independence of the Profintern,” he warned, “We shall be at high risk of never seeing an end to this issue; \dots{} if today the non-negotiable demand of the syndicalist opposition is organizational independence with no strings attached, then tomorrow the libertarians will be raising questions about the dictatorship of the proletariat.”
The leadership of the Profintern suggested on March 10 that the CGTU send two representatives to Moscow for negotiations in order to “prepare the ground for a second congress in the interests of all tendencies which would be a great benefit for our common interests.” But the syndicalists preferred the idea of negotiations with Moscow on a broader scale. A congress of the Italian USI in March 1922 approved a proposal of the CGTU to convene an international conference to discuss the conditions of agreement.
It was originally scheduled for June 16-18 in Paris. In connection with this, the Administrative Commission of the CGTU at a meeting on April 28 rejected an invitation from the Profintern to send French delegates to Moscow.
It informed the General Secretary of the Red International Lozovsky about the decision to convene a “preliminary conference” in Paris, the purpose of which was “to make the differences disappear” which were preventing the syndicalists from affiliating with the Moscow International. The CGTU asked the USI, which was organizing the conference, to relocate it to Berlin in order to make it easier for delegations from Russian labour unions to attend.
On May 19 1922 the leaders of the USI A. Borghi and A. Giovannetti informed the “secretary of the Russian labour union central” that on June 16-18 in the capital of Germany would take place an “international syndicalist conference for the purpose of studying the differences in views existing between the revolutionary syndicalist movement of all countries and the Red International of Labour Unions, and to agree on the formation of a Revolutionary Labour Union International if the differences with the Red International could not be resolved.” The USI reported that invitations had been extended to labour union associations in Italy, France, Germany, Spain, Portugal, and also to the “syndicalist minorities” of various countries. In the instructions given to delegations of the Profintern to the international syndicalist conference, it was stated that discussions, and even concessions, about contentious issues were possible, with the exclusion of three basic questions – about the independence of labour unions from political parties, about the banning of communist fractions in reformist labour unions, and about non-interference in the internal affairs of individual organizations. “We must take a stand for our positions on these three most important questions and on this basis we are prepared to go all the way to an open rupture\dots{},” the instructions went on to say.
The international syndicalist conference convened in Berlin in June 1922 with the participation of delegations from France, Germany, Norway, Spain, and also Russian anarcho-syndicalists and official Russian labour unions, representing the Profintern. The communist fraction in the USI and labour unions which had split from the German FAUD were not allowed to cast deciding votes. This prompted the Soviet delegates to quit the conference. A majority of the delegates were sharply critical of the repression of the anarchists in Soviet Russia. This was the final break between the syndicalists and the communists. And although the French delegates refrained from voting because of internal differences, the remaining delegates resolved to break with the Profintern and create an international congress of revolutionary labour unions. To prepare for this a bureau was set up in Berlin headed by R. Rocker assisted by A. Borghi (USI), A. Pestaña (CNT), Albert Jensen (from the Scandinavian syndicalists), and A. Shapiro (from the Russian anarcho-syndicalists). A declaration of principles was adopted, based on the corresponding declaration of the FAUD.
It rejected political parties, parliamentarism, militarism, nationalism, and centralism. Its positive program included the complete autonomy of economic organizations of both physical and intellectual labour, and direct action with the general strike being its highest expression, the “prelude to the social revolution.” The goal of this revolution would be the reconstruction of economic and social life, the liquidation of all State functions in the life of society, and the creation of a system of libertarian communism. The dictatorship of the proletariat and Bolshevik methods were decisively condemned. In the words of researcher [URL=\Slash{}tags\Slash{}wayne-thorpe] W. Thorpe, the declaration “signified an important advance in syndicalist thought, since it confirmed and made clear what had often only been implied in pre-war European syndicalism.” It enunciated “not simply political neutrality, as expressed in the ‘Charter of Amiens’, but opposition to all political parties, which were regarded as qualitatively different, hostile organizations, inevitably striving to establish their control over labour unions; and also the smashing of the political state\dots{} In short, this document, adopted by the delegates in Berlin, elaborated syndicalist principles.”
In a last-ditch attempt to draw at least part of the revolutionary syndicalists to their side, the leaders of the Comintern and Profintern agreed to do away with reciprocal representation of both “red” Internationals, although they continued to insist on the “leading role” of communists in the labour unions. This concession seemed sufficient to the leadership of the French CGTU, which announced its affiliation with the Profintern; its libertarian minority formed a “Committee of Syndicalist Defense” (CDS). Satisfied with the measures taken by Moscow, a majority of the leadership of the Netherlands NAS took a position opposed to the creation of a new syndicalist International. The remaining revolutionary syndicalist unions endorsed an organizational demarcation between themselves and Bolshevism. Thus, at the congress of the Portuguese CGT in October 1922, 55 locals supported the creation of a new International and only 22 were for joining the Profintern.
At the same time, the rupture between the ascendant European anarcho-syndicalism and both the pre-war syndicalism and Bolshevism seemed inadequate to some of the revolutionary unions. Thus, the Argentine FORA, in its “Memorandum” addressed to the upcoming constitutional congress of the syndicalist International, expressed complete agreement with the proposed organizational system and methods of struggle, and endorsed the social goal of the new international organization – libertarian communism.
However it categorically rejected the notion that labour unions – organs which arose under capitalism in response to capitalist conditions and fulfilled a service as the best means of worker resistance against the State and Capital – would be transformed in the course of the revolution into the basis and ruling organs of the new society. “With the liquidation of the capitalist production system and rule of the State, the syndicalist economic organs will end their historical role as the fundamental weapon in the struggle with the system of exploitation and tyranny. Consequently, these organs must give way to free associations and free federations of free producers and consumers.” FORA took a stand against industrial (sectoral) forms of organization, considering that they imitated Capitalism. Finally, FORA categorically rejected any form of a “united front” with labour unions led by communists.
The final formation of the anarcho-syndicalist International (sometimes also known as the “Berlin International of labour unions”) took place at the constitutional congress which took place illegally in Berlin from December 25 1922 to January 2 1923, punctuated by police raids and arrests.
Represented at it were the Argentine FORA, the Italian USI, the German FAUD, the Chilean division of the IWW, the Swedish union central SAC, the Norwegian syndicalist federation, the Union for syndicalist propaganda of Denmark, the Netherlands NAS, and the Mexican General Confederation of Workers. The delegates of the Spanish CNT were arrested before they reached Berlin. The Portuguese CGT sent a written endorsement. Attending with a deliberative vote were representatives of the left-communist German General Workers Union-Unitary Organization (AAUD-E), the German anarcho-syndicalist youth, the French CDS, the French federation of construction workers, the Federation of Youth of the Seine, delegates of the Russian anarcho-syndicalist emigration, the Czechoslovak Free Workers Union, and representatives of the international syndicalist bureaus created in 1920 and 1922 in the Netherlands and Germany.
Altogether these organizations accounted for roughly two million members. The 14th annual convention of the American IWW declared it did not intend to affiliate with either the Profintern or the syndicalist International, since neither one were suitable for it.
All the delegates, except the representatives of the Netherlands NAS, rejected the “concessions” of the Bolsheviks and participation in the Profintern. The creation of a new, anarcho-syndicalist International was announced. By way of a motion proposed by the Italian Alibrando Giovanetti, as a symbol of continuity the new organization took the historical name of the First International – the “International Workers’ Association” (IWA). The declaration of principles of the IWA (“Principles of revolutionary syndicalism”) in essence repeated the basic positions of the Berlin declaration of June 1922. Elected to the Secretariat of the IWA in Berlin were R. Rocker, A. Souchy, and A. Shapiro.
The records of the congress contain harsh condemnations not only of capitalism and the reformism of social-democracy, but also of the Bolshevist “State socialism.” The delegates accused Bolshevism of suppressing revolution in Russia and creating a new state-capitalist system, in which the workers of the USSR remained exploited as wage workers. “Forcibly destroying with relentless consistency all institutions which arose out of the people’s initiative, namely soviets, co-operatives, etc., in order to subject the masses to a newly created class of commissar-rulers, [Bolshevism] paralyzed the creative activity of the masses and gave birth to a new despotism, stifling any kind of free thought and confining the spiritual life of the country to the banal party mold,” according to the appeal “To Working People of All Countries and Nationalities.” The so-called “dictatorship of the proletariat – a fig leaf for Bolshevist reaction – had proven itself able to stabilize the rule of a new upper stratum over the broad masses of the people and condemn to death revolutionaries of all tendencies, but was incapable of guiding the economic and social life of the country on a new path and carrying out really constructive work in the spirit of socialism.”
As R. Rocker explained later, for anarcho-syndicalists the Bolsheviks were the heirs of “the absolutist trend of thought in socialism,” a special kind of “socialist Jacobins,” i.e. essentially they were revolutionaries who were political rather than social, and bourgeois rather than proletarian.
In spite of this harsh critique of Bolshevism, some syndicalists still believed in the possibility of coming to an arrangement with the Profitern about a “united front” of the revolutionary proletariat. A corresponding draft resolution was introduced at the Berlin congress by the French delegation.
A majority of the other participants did not exhibit any great enthusiasm for this project, but went along with this idea so as not to complicate the situation of the French comrades. The FORA emphatically objected to such a compromise and abstained from voting on the resolution.
The creation of the IWA was officially confirmed at congresses or referenda of its sections. In Europe affiliation to the IWA was speedily approved by the FAUD, USI, SAC, and CNT. At a referendum in Norway the creation of the International was approved unanimously, and in Portugal (October 1924) 104 syndicates declared for the IWA, six for the Profintern. In the Netherlands, the communists and other supporters of the Profintern were able to gain a slight majority in a referendum of syndicates, and IWA members organized a new trade union central – the Netherlands Syndicalist Trade Union Federation (NSV). Also declaring its affiliation to the IWA was the Revolutionary-Syndicalist CGT (CGT-SR), finally splitting from the French CGTU.
During the 1920’s and 1930’s sections and groups of adherents of the IWA also appeared in Austria, Denmark, Belgium, Switzerland, Bulgaria, Poland, and Rumania.
In America, affiliation with the IWA was also confirmed by a congress of the Mexican General Confederation of Labour (CGT) in December 1923. A congress of the FORA, extremely unhappy with the resolution adopted in Berlin about “revolutionary unity,” decided in March 1923 to join the anarcho-syndicalist International conditionally and to hold a referendum on this matter. But then, after the contentious resolution was repealed, the objection against participation in the IWA was removed. Also joining the IWA were anarcho-syndicalists from Uruguay, Brazil, Paraguay, Bolivia, Columbia, Peru, Guatemala, Ecuador, Cuba, Costa Rica, and El Salvador (in May 1929 an American continental association of workers was created as a section of the IWA). Sections also sprang up in Japan and China. In the U.S.A. the Marine Transport Workers Industrial Union of the IWW affiliated with the IWA.
\section{Chapter 7: The World Anarcho-Syndicalist Movement in the 1920’s and 1930’s}
The International Association of Workers was reconstituted at a moment when the global revolutionary wave had already begun to subside. Many of its sections were soon subjected to harsh repression and were crushed. In Italy after the regime of Mussolini took power, the activity of local branches of the USI was paralyzed already by April 1924.
Going underground, the labour federation re-organized and was able to lead a number of significant strikes (miners in Valdarno and on Elba), marble workers in Carrara, and metalworkers).
But by 1927 the USI had finally been destroyed, its leading activists either arrested or forced to emigrate.
In Portugal after the installation of a military dictatorship, the CGT tried to organize a general strike in February 1927. The strike was suppressed, nearly 100 people were killed, many activists were arrested, and the CGT was outlawed. It succeeded in re-organizing its forces underground and re-established a number of unions and branches of the federation. In 1929-1930 the organization had 32 unions with 15,000-20,000 members, and by 1934 it included seven federations. The Portuguese anarcho-syndicalists continued a tenacious struggle against unemployment and the high cost of living, for the 8-hour day, and the right for unions to exist. In January 1934 decrees of the Salazar government about replacing unions with corporations of the fascist type were greeted by the CGT with a “general revolutionary strike” and an uprising. The revolt suffered defeat. The heroic resistance of the Portuguese workers could not avert the destruction of the CGT.
In Argentina the FORA towards the end of the 1920’s had a membership, according to various sources, somewhere between 40,000 and 100,000 and conducted successful general and localized strikes, achieving the implementation of the 6-hour work day. However a military coup in 1930 and the subsequent persecution dealt a heavy blow to the organization, from which it was unable to recover.
In Germany, after the downturn of the revolutionary movement in 1923, the membership of the FAUD began to fall sharply: in 1929 it still had 9,500 members, but under conditions of catastrophic mass unemployment this number decreased to 6,600 in 1931 and 4,300 in 1932. This small organization was no longer able to conduct strikes independently.
It carried on active cultural work and campaigns for the boycotting of elections, and participated in strikes organized by the reformist labour unions in order to impart to them a more radical character. Emphasizing direct action and strikes of solidarity, it tried to oppose the onslaught of Nazism. After Hitler took power, the FAUD continued to resist underground until the second half of the 1930’s.
The headquarters of the IWA in Berlin was seized by the Nazis and the members of the Secretariat barely succeeded in fleeing Germany.
As a result of massive government repression anarchosyndicalist unions were destroyed in Peru, Brazil (after 1930), Columbia, Japan (in the mid 1930’s), Cuba (after 1925-1927), Bulgaria (the Confederation of Labour which appeared at the beginning of the 1930’s had been wiped out by the end of the decade), and the countries of Central America. In Paraguay and Bolivia activities of the anarcho-syndicalist workers’ organizations were banned during the Chaco War (1932- 1935) and subsequently were not able to attain their previous level. The French section was also unable to acquire a mass character. The great crisis of 1929-1933, accompanied by the growth of nationalist and statist sentiments, significantly weakened the movement in the majority of other countries.
In Mexico the leadership of the CGT collaborated with the national-reformist government, accepting the principle of arbitration of labour disputes by the State; the Confederation quit the anarcho-syndicalist International. By the end of the 1930’s legal anarcho-syndicalist trade union associations existed only in Chile (General Confederation of Workers, 1931), Bolivia (Local Federation of La Paz), and Uruguay (FORU); the FORA operated in the underground.
The main stronghold of anarcho-syndicalism remained Spain where, following the fall of the monarchy in 1931, a vigorous growth of the strength and influence of the CNT took place. “From all sides, from Germany, Poland, France, and other countries where there are IWA sections, the Secretariat receives communications about the existing state of mind, which \dots{} it is possible to express in the following form: ‘International fascism has destroyed our revolutionary movement in most countries\dots{} Only in one country do we entertain hope that the social revolution can overcome it [fascist reaction, – V. D.] – in Spain’,” – wrote members of the IWA Secretariat in a message to the CNT in June 1934.
At the first legal congress of the labour federation in 1931, more than 500,000 members were represented and a few years later the number of members exceeded one million.
During the first year and a half of the republic’s existence, 30 general and 3,600 localized strikes were organized, mainly by the CNT. The peasantry, organized by the anarcho-syndicalists, seized land from the estate owners, demanding socialization, on a massive scale. In 1932-1933 a wave of local revolutionary uprisings rolled across the country: members of the CNT seized control of population centres and proclaimed libertarian communism. The authorities were able to suppress the movement only with difficulty. Thousands of people were killed or arrested, but the influence of anarchosyndicalism in Spain continued to grow.
Confronted with aggressive reaction, the anarchosyndicalists had to deal with a series of tactical questions. First of all, an IWA plenum at Innsbruck (December 1923) once and for all condemned the actions of the Bolsheviks, repealed the concessions made to the French syndicalists at the constitutional congress, and rejected the possibility of a united front with the communist parties. The second congress of the IWA (1925) confirmed its negative attitude towards all political parties which were regarded as tools in the struggle for power, rather than for freedom. Any long-term alliance with political parties was impossible, for this would contradict the goals of the IWA. Participants at the congress perceived fascism and Bolshevism as “reaction of a new type,” resorting to naked tyranny and massive repression. The congress expressed the conviction that it was necessary to defend civil and union freedoms as conquests of the workers, but not as part of a democratic system which was liable to be overthrown along with capitalism.
Anarcho-syndicalists should act independently and not make official alliances with anyone else even if, in the course of struggling with fascist and military dictatorships, they happened to “cross paths with other political forces.”
In the struggle with Bolshevism any kind of collaboration with other forces was impermissible. It was noted that the liberal bourgeoisie, when confronted with a threat to their own rule, was always prepared to transfer power to dictators.
Therefore the struggle with dictatorship must not be carried on in such way as to strengthen democracy as a system of government. The best means of struggle with dictatorship, according to a resolution of the congress, is the class struggle of the workers. More or less the same tone was displayed in a resolution adopted at the 4th Congress (1931). The IWA was oriented, in the first instance, to working together with other groups with similar views (anarchist federations and groups, anti-militarists, etc.), but also permitted practical co-operation for concrete goals with other labour unions, supporting strikes and conducting solidarity campaigns. The IWA frequently made approaches to Internationals of socialdemocratic and communist labour unions about mutually organizing boycotts of fascist and dictatorial states and the goods produced in them, and trying to stop the delivery of raw materials from other countries in the case of strikes, etc. At the beginning of the 1930’s the struggle with fascist reaction became even more urgent for anarcho-syndicalists, but they endeavoured in dealing with the problem to adhere to their social-revolutionary line. In the appeal issued by the IWA for May 1 1932 it was said that “in a number of countries in the immediate future the question will arise: revolution or fascism?” [158] In 1933 the anarcho-syndicalist International called for a global boycott of Nazi Germany.
The Spanish and Swedish sections worked out plans to avoid handling German goods and vessels, accompanied by consumer boycotts – this idea was also supported in Holland.
But the French section expressed opposition, fearing such actions could be exploited by Hitlerian propaganda. Repression against the CNT at the end of 1933 finally put an end to these plans. In their attempts to oppose international reaction, the anarcho-syndicalists did not put their faith in social-democrats and communists and boycotted their “antifascist” and “anti-militarist” congresses. After the proposal by the communists about the creation of a “United Front,” the Secretariat of the IWA queried the sections, but ended up sharply rejecting the idea (only the FAUD, already being in emigration, supported the notion of a “united front against fascism”). In May 1934, the Secretariat issued a declaration once more rejecting any possibility of organizing a “united front.” A corresponding resolution, proposed by the French section, was passed at the 5th Congress of the IWA in Paris (1935).
\section{Chapter 8: Ideological-Theoretical Discussions in Anarcho-syndicalism in the 1920’s-1930’s}
In spite of heavy defeats in a majority of countries, the repressions of dictators, and the politics of communists aimed at subverting the anarcho-syndicalist movement, the period of the 1920’s and 1930’s was a time of lively ideological-theoretical discussions among anarchists and anarcho-syndicalists. The participants in these discussions not only put forth a penetrating analysis of contemporary capitalist society, but also described the contours of a social alternative with great insight.
In all the documents and decisions of the IWA there is emphasis on the basis of unity of anarcho-syndicalists: their common goal (libertarian communism, free socialism) and their common principles and methods of struggle (direct action up to and including social revolution). However within this framework there existed significant divergences within the world anarcho-syndicalist movement. “We are well aware that within organizations and, even more so, within an international association of various national organizations, it is impossible to arrange complete harmony,” said R. Rocker at the 2nd Congress of the IWA in Amsterdam in 1925. “On the other hand, we even consider that different opinions on certain questions within one and the same organization can serve a useful purpose by assisting spiritual development and encouraging independent judgement. We have seen this occurring in the IWA.”
The experience of the Russian Revolution and the outbreaks of revolution after World War I had made a deep impression on the views of libertarians about contemporary society and the alternative to it. It was in this period that so-called “anarchist revisionism” developed. In Italy E. Malatesta and [URL=\Slash{}tags\Slash{}camillo-berneri] Camillo Berneri acted as its propagandists.
The former, long known as one of the leading theoreticians of anarcho-communism, while not renouncing his basic ideological principles, now believed as a result of the Russian experience that “for the organization on a broad scale of a communist society one must radically transform the whole of economic life – the means of production, exchange, and consumption – and this can only be done one step at a time.” He believed that during the course of a revolution, anarchists would find themselves in a minority at first and ought not to impose their own ideas and concepts on the whole of society. Revolutions, in his opinion, were liable to lead to the emergence of a pluralistic society, composed of a multitude of communes bound together by communistic, but also commercial, relations.1 Berneri advanced the notion of the coexistence of different economic forms in an anarchist society. “All anarchists are atheists, but I’m an agnostic,” he wrote, “All anarchists are communists, but I’m a liberal, that is, I’m for free competition between co-operative and individual labour and trade.”2
Some anarchists, trying to figure out why the Bolsheviks gained victory in the Russian Revolution, came to the conclusion there was something to be learned from the Bolsheviks in the field of tactics and organization. Thus, the “Platformists” (a group led by [URL=\Slash{}tags\Slash{}nestor-makhno] Nestor Makhno and Petr Arshinov) took a position for the acknowledgement of the principle of class struggle in history, and for the creation of a strong organization of anarchists (in fact – a type of party) which could take part as a unitary force in Soviets and in the trade union movement, and play a leading ideological and constructive role in the revolution. Essentially, the “platformists” allowed for stages in the revolutionary process and the fulfilment of governmental functions by soviets. They maintained that in the productive system of the future society decentralization and integration of labour would be technical questions, subject to needs of a unified economy, rather than questions of principle. In fact they adopted the industrial form of organizing production, proposing only to get rid of private ownership and hand over control of production to Factory Committees. A significant number of anarchists (Vsevolod Volin and other Russian emigrants, E. Malatesta, Sebastien Faure) subjected such positions to criticism, considering them a departure from anti-authoritarian principles and the values of libertarian communism.
Another argument against the immediate implementation of anarchist communism is that the notion of a free commune is in contradiction to “the real spirit and tendencies” of the industrial stage of development of society with its striving for universality and increasing specialization. For example, the well known historian of anarchism, Max Nettlau, criticized the “rural-industrial atomization of humanity in anarcho-communism and declared: “Decentralization \dots{} creates something just the opposite to solidarity and multiplies the sources of friction and stress. Our hopes for improvement are based on building solidarity, in federating larger units, and breaking down local barriers and boundaries, and in the collective control of the natural resources and other forms of wealth of our planet.” At the same time, he assumed that the principles of “collectivism” (distribution according to labour) and monetary compensation for labour were more compatible with the industrial form of organizing production.
The heated discussions and quarrels about the trajectories of social revolution which were carried on in the IWA to some extent served as a continuation of the polemics between anarcho-communists and syndicalists at the beginning of the century. One group were in favour not only of the elimination of Capitalism and the State, but also for the demolition of the industrial system itself with its factory despotism, rigid division of labour, and dehumanizing technology. A second group welcomed industrial-technological progress and hoped to construct a socialist society using it as a base.
Their quarrel was closely connected with the analysis of the latest trends in the development of Capitalism itself – its rationalization of production in its Fordist-Taylorist phase.
This stage of industrial development was accompanied by the introduction of mechanization and conveyor technology on a massive scale, dividing the labour process into a series of operations and severely undermining control on the part of the worker, who lost the sense of the integrity and meaning of their own labour, but in exchange acquired the possibility of mass consumption.
The problems of “capitalist rationalization” were first dealt with at the 3rd Congress of the IWA in Liège in 1928.
The delegates declared themselves in favour of “progress in all fields of endeavour,” but considered its manifestations in the sphere of capitalist production to be negative as far as the workers were concerned. The resolution passed by the Congress appraised the ongoing process as the direct result of a new phase of development of society, which was reflected in the transition from the “old private capitalism” to “contemporary collective capitalism” (trusts, cartels), from untrammeled competition to the exploitation of the whole world by a unified system. It was emphasized that rationalization was being carried out in the interest of capitalists, and its implications for workers involved the undermining of their physical and mental health, along with their subordination to the mechanisms of “industrial slavery.” Rationalization condemned working people to the loss of jobs, unemployment, and, consequently, a worsening of living conditions. The Congress declared that it considered such a transformation of the capitalist economy as a precondition not of socialism, but rather of a future state capitalism. The path to socialism, it was noted in the resolution, is defined not by the constant growth of production, but, in the first instance, by clear thinking and firm will on the part of the people. Socialism is not just an economic problem, it is also cultural and psychological; it assumes people believe in their own capabilities and that work is complex and absorbing – and that all this is incompatible with the ongoing rationalization.
The resolution spoke in favour of decentralization rather than centralization of the economy, for the unity rather than specialization and division of labour, and for the integral formation and development of all the abilities of people. In response to the creation of gigantic national and international structures of capital, the workers should strengthen their own international economic organization, enabling them to struggle for everyday demands as well as for the re-organization of society, for the shortening of the work day to six hours, to resist unemployment, organize international strikes and boycott campaigns, etc.
However such a critical stance towards the process of development of the industrial-capitalist system and the demands for a radical break with it encountered objections from a substantial number of anarcho-syndicalists who, following the Marxists, associated socialism with advances in technology and an increase in the productivity of labour.
They did not consider the new forms of technology and the organization of production as incompatible with socialism.
Such an approach logically entailed the centralization of production and the economy as a whole, and the rejection of the notion of federations of decentralized and largely self-sufficient communes, and therefore rejection of the communist principle of distribution. The old ideas of collectivism were considered much more appropriate for the industrial century. Even Rocker began stating at the end of the 1920’s that, although remaining in principle an adherent of anarchist communism, he considered the collectivist principle “to each the full product of his\Slash{}her labour” to be more realistic in a period of revolutionary transformations and during the first phases of the creation of a new society. He referred to the inevitable economic difficulties accompanying revolution, to the growth of selfish attitudes in contemporary society, and – like the Marxists – he associated the implementation of communist distribution with material “abundance.”
Souchy, debating these problems with Cornelissen, proposed that only “in a pre-industrial society would it be possible, and then only in small communities, to introduce a pure distributive economy. In a contemporary industrial society and with the current interdependence of global economies, from which an individual country cannot withdraw, the exchange of products inevitably determines values. Speaking more precisely, exchange determines prices which in turn determines wages.” The alternative would be to introduce centralized planning, which is contrary to the principles of anarchism. Such a situation, in his opinion, would obtain at least until the epoch of universal abundance.
Lively discussions about the question of industrial development and the nature of the future free society were carried on in the pages of the journal Die Internationale – the de facto organ of the IWA, published by its German section.
If previously FAUD had unequivocally declared itself as the “bearer of communist anarchism” , now many of its leading activists began to oppose the anarcho-communist principle of distribution “according to demand” as a “crazy idea,” calling instead for the study of existing economic categories (Helmut Rüdiger) and adjusting distribution in accordance with the real “productivity” of labour (Gerhard Wartenberg – “Gerhard”). It was even asserted that “rationing” by means of monetary regulation was “fairer” than communist anarchism (Fritz Dettmer). The opinion was expressed that in a “socialist-federative system” there must exist an “industrial interlocking of the productive forces,” a regulated and planned economy, and economic democracy (Fritz Linov). Finally, some found it conceivable that the social functions of the State “should be kept intact” even after revolution (Wartenberg), and a federative system of Councils should be introduced only after a transitional stage, as soon as the revolution managed to put together a “united front” in which the anarcho-syndicalists would be in a minority (Reinhold Busch). On the other hand, a section of the German anarcho-syndicalists continued to insist on the classical anti-industrial principles of anarchocommunism.
Thus, Heinrich Drewes condemned such innovations as “capitalistic thinking” and supported the complete transformation of the existing profit-based economy.
He supported the creation of a non-monetary communist economic system, in which associations of workers would organize planning from below, based on the determination of the people’s real needs. He rejected “gigantomania” and centralization the borrowed from Marxism and was in favour of the re-organization of the economic life based on “agrarianization” as opposed to “industrialization.” In 1932 the leadership of FAUD was almost paralyzed by bitter ideological and theoretical disputes.
The industrialist tendency was strongest in the French section of the IWA – the Revolutionary-Syndicalist CGT. The theoretician and practitioner of French anarcho-syndicalism Pierre Besnard, like many of the syndicalists before the First World War, started from the assumption of the progressiveness of the industrial development of humanity. According to Besnard, technological changes (associated with the production-line, “Fordist-Taylorist” era) opened new, broad perspectives for the social liberation of workers. Workers’ organizations, while carrying on the struggle with capitalism, should arrange their internal structure in imitation of capitalist economic formations, so that immediately after the victorious general revolutionary strike they could take over management of the economy. In other words, the syndicates and their federations emerging within the capitalist structure were destined to become the nervous system of the new society, the organs of economic coordination, planning, etc. The first stage, which Besnard called “libertarian communism,” would involve the preservation of elements of the monetary system and distribution “according to labour.”
Only at the second stage (Besnard named it “free communism”) would it be possible to carry to completion the ideal of a self-managed communist society.
This departure from the principles of anarcho-communism provoked a sharp rejoinder from anarchists in Latin America, above all from those in the Argentine FORA. Its theoreticians set themselves the task of providing a sound basis for their own traditional critique of revolutionary syndicalism (as being semi-Marxist in essence) and European anarcho-syndicalism (as an attempt to synthesize anarchism and revolutionary syndicalism). They raised questions about the conceptions of a syndicalist structure of the postrevolutionary society and about a united class front of the proletariat. Simultaneously they also criticized the notion of “ideological-political” organizations of anarchists separate from the workers’ movement (as proposed by Malatesta, on the one hand, and by the Platformists on the other). FORA countered this by advancing a model of an “anarchist organization of workers,” structured like a syndicate but not limiting itself to strictly economic problems but also taking up issues of solidarity, mutual aid, and anarchist communism.
The theoreticians of the FORA presented a thorough critique of the Marxist-industrial viewpoint on history, contemporary capitalism, and social revolution, one of the first such critiques in the 20th century. Above all, they criticized the theory of linear progress and Marxist historical materialism, affirming (following Kropotkin) that the development of humanity is impelled not just by economic laws, but also by the evolution of ethical concepts and compelling ideas. According, the FORA sharply criticized economic and historical determinism and denied that capitalism and its economic organization were progressive by nature. The theoreticians of the FORA perceived the economic structure of industrial capitalist society (the factory system, sectoral specialization, extreme division of labour, etc.) as an “economic state” – in tandem with the “political state,” i.e. the government. The new, free society should not develop according to the laws of the old society, according to their logic, but represent a decisive, radical break with it. The base of the new society should be the free commune and the free association; their slogan should not be “All Power to the Syndicates!,” but rather “No Power to Anyone!” An anarchist communist system must not under any circumstances be built “within the bowels” of the old social organism, or else it could expect the fate of the Russian Revolution – warned the leading ideologue of the FORA Emilio López Arango. The proletariat was “destined to become the wall which would stem the tide of industrial imperialism. Only by creating ethical values which would enable the proletariat to understand social problems independently from bourgeois civilization would it be possible to arrive at an indestructible basis for an anti-capitalist and anti-Marxist revolution – a revolution which would do away with the regime of large-scale industry and financial, industrial, and commercial trusts.” The purely economic interests of the proletarians within capitalism could be completely fulfilled within the framework of the existing system, mainly at the expense of other proletarians, which was why a united front of the proletariat was an impossibility.
It was important to spread the habits and notions of solidarity and freedom; it was possible to accomplish this in the course of economic direct action, but in doing so the ultimate goal should never be lost sight of. Therefore the anarchist workers’ organization should be not simply “for all the workers,” but, above all for those who share the ideal of anarchist communism.
The most lively debates about tendencies in the development of capitalism and the concomitant changes in the tactics of anarcho-syndicalism unfolded at the 4th Congress of the IWA in Madrid in 1931. This congress took place at the height of the world economic crisis, which the anarchosyndicalist theoreticians understood as a consequence of capitalist rationalization. This rationalization led, on the one hand, to a runaway growth in production but, on the other hand, to a reduction of positions in the workplace and a reduction in the buying power of workers. Two approaches – one industrial and the other anti-industrial, clashed at the congress in a most acrimonious manner. According to Muños Congost, author of historical notes to the publication of materials of the congresses of the IWA, the essence of the discussion reduced to the following. “On one side, the draft of the document about rationalization, prepared by Shapiro and serving as the basis for final editing according to the wishes of the Congress, insisted on the advantages of the new methods of organizing production connected with increasing mechanization. These methods were regarded as fundamental in preparing the consciousness of the working masses, and as the starting point for the future organization of the economic content of the revolution. On the other side, a more anarchist conception was put forward [by Rocker, – V. D.] about the direct responsibility of the producers, who cannot and must not divorce their own productive activities from all the other forms of activity of conscious individuals\dots{}
This approach did not oppose rationalization as such, but rather required a balance between the participation of the individuals in social production and the preservation of their own individuality, their personalities.” Rocker “declared that the revolution must transform the slave conception of labour-as-exploitation, as an obligation sanctified by tradition and the church over many centuries, \dots{} into a different form, more compatible with an harmonious organization of human relationships,” on the basis of the integrity of labour. The German anarcho-syndicalist conceded that technical development can humanize life, but not in a capitalist setting, where human beings exist for production. Long before people began talking about alienation and ecological problems, he noted that the production of goods which are harmful to health is “social suicide.” Working according to the monotonous rhythm of a machine destroys a person’s personality. It follows that people must be placed at the centre of the economy, and production – oriented according to the needs of real consumers. He warned: “If the rationalization of labour is preserved in its present form for another 50 years, any hope for socialism will be lost.”
Basing himself on an industrial analysis of the changes which were occurring, although also not agreeing with Shapiro’s proposal about sanctioning the creation of Factory Councils which would take control over the financial management of enterprises, Besnard proposed a “Plan for Reorganizing International Syndicalism.” Since capitalism was now in the throes of “simultaneously carrying out two rationalizations – economic and social,” the syndicalist movement should “position itself on the same level as its opponent” and carry through a “rationalization on a global scale” on its own. He called for a reorganization of the international organization using a model for industrial unions which would be applied in all countries from bottom to top: union Factory Councils joined together in networks up to the national level, and then affiliation to the corresponding international organs. The various structures must be completely independent of enterprises and the State, being the embryo of the economic system of the future. Their task would be the collection of managerial and technical information, the implementation of workers’ control over enterprises, the relocation of work forces, and the preparation of workers for managing production at all levels, including the international level.
Besnard’s conception was the subject of a sharp attack by the Argentine FORA, which went much farther than Rocker in its critique of rationalization. One of the Argentine delegates declared at the congress: “Not only political fascism, but also capitalist industrialism is the most dangerous form of tyranny. Comrades are assuming the economic question alone has decisive significance. However the capitalist apparatus, if it remains as is, even in our hands will never be an instrument for the liberation of humanity, a humanity crushed by a gigantic mechanism. The economic crisis has triggered an enormous growth in machines and rationalization, and this growth is by no means limited to urban industry but has also spread to the rural economy. This is a universal crisis which can only be resolved through social revolution.” Consequently, the Latin American delegates at the congress rejected the plan proposed by the French syndicalists to reorganize the international anarcho-syndicalist movement as a global structure of industrial syndicates, capable of taking over the existing system of industrial production in the case of revolution. “Industrialization is not necessary,” they asserted, “People lived without it for thousands of years; happy lives and well-being do not depend on industrialization.” “It must not be assumed that the impending revolution will decide everything once and for all. The next revolution will not be the last. In the revolutionary upheaval all preparations will be thrown overboard, and the revolution will create for itself its own forms of living.” According to one of the Argentine delegates, the French syndicalists “have committed an error in trying to mechanize the IWA. One should not think exclusively about production, but more about people; the main problem is not the organization of the economic system, but the propagation of anarchist ideology.” He spoke against rationalization, since “the people don’t exist for society, but society for the people” and called for “a pure syndicalism: a return to nature, to agriculture, to communes. Only by following these principles can we surmount production for the market and switch to a system of free distribution.”
The objections of the FORA to the plans of Besnard were supported also by the Uruguayan FORU.
The theoreticians of the Japanese labour federation Zenkoku Jiren criticized syndicalist industrialism even more severely than the Latin American worker-anarchists. Their conception of anarchist revolution, which they expounded in detail, implied a cardinal break with the logic of industrial capitalism. The current system, they said, was based on the division of labour and the consequent hierarchy; this division and its attendant mechanization deprived the workers of any responsibility and required coordinating and administrative authorities which were incompatible with the principles of libertarian communism. Therefore the structure of the future free society could not be compatible with the existing authoritarian and capitalist structure. The new society must surmount industrialism with its soul-destroying division of labour and base itself on a different conception of the interrelation of production and consumption, but with the emphasis on consumption. The fundamental unit of this new society must be the self-sufficient, autonomous commune, uniting industry and agriculture.
The Japanese anarchists acknowledged the class struggle as an historical fact, but refused to see in it the basis for libertarian revolution which, in their opinion, would emerge not from the contradictions of capitalism and not from the material interests of classes, but rather from the desire of humanity for freedom and the liquidation of classes generally.
Since “class struggle and revolution are different things,” “it was a great mistake to claim\dots{}, that revolution takes place by means of class struggle,” emphasized the Japanese theoretician of anarchism Hatta Shûzô.
Zenkoku Jiren rejected traditional syndicalism, seeing in it elements of the reproduction of the industrial-capitalist model. The continuation of the division of society into groups according to occupation, the preservation of the factory system and centralization, and the organization of society throughout on the basis of professional and industrial unions, would perpetuate the division of labour and the hierarchy of management. “Syndicalism,” wrote Hatta, “will adopt the capitalist means of production, and will also preserve the system of big factories, and first and foremost it will also retain the division of labour and the mode of economic organization which go together with capitalist means of production.” The structure of the syndicates grows out of the capitalist means of production and creates an organization which serves as a mirror image of industrial-capitalist structures. If the capitalist bosses are simply removed and the mines handed over to the miners, the foundries to the foundry workers, etc., then the contradictions between different branches of production and the inequality between individual groups of workers will be preserved. Consequently some kind of arbitrage or organ for resolving disputes between different sectors and groups is required. This creates a real danger of regenerating classes and leads to the appearance of a new state or government in the form of a union bureaucracy. The Japanese anarchists also considered totally wrong any plans of organizing a new society on the basis of a system of Workers’ Councils. Because they originated in production, such councils also reproduced the capitalist division of labour. Moreover, they would also inevitably be power bases and would discriminate against those who did not take part directly in the production of material wealth or who worked in “secondary” branches of the economy. “No matter how the councils were oriented economically,” emphasized Hatta, “it remains clear that their creation would always be accompanied by the emergence of authoritarian rule.”
Thus a choice was posed: the commune or the industrial union? industrial rationalization or integration of decentralized industrial and rural economies? The majority of the sections of the IWA occupied an intermediate position between these extreme positions. The 1931 congress decided to submit the question about “international re-organization” to a referendum of the sections. In 1935 the regular IWA congress in Paris, meeting at a time when the Latin American organizations had been shattered by government terror, approved the proposal of the French Revolutionary-Syndicalist CGT. But this decision about re-organizing the IWA was not in fact implemented.
The conceptions of the FORA contained a critique of the alien and destructive character of the industrial-capitalist system which was brilliant for its time – the FORA’s proposals anticipated by half a century the recommendations and prescriptions of the contemporary ecological movement.
Nevertheless their critique had a point of vulnerability – a categorical refusal to elaborate more concrete notions about the future society, how to get to it and how to prepare for it.
According to the thinking of the Argentine theoreticians, to do so would be to infringe on revolutionary spontaneity and the improvisations of the masses themselves. The achievement of socialism was not a matter of technical and organizational preparation, but rather the dissemination of feelings of freedom, equality, and solidarity – insisted the Argentine worker-anarchists. Nevertheless, objected the European anarcho-syndicalists, such an approach provides no protection from authoritarianism, and could be conducive to the appropriation of the gains of the Revolution by some kind of elitist “vanguard.” Thus from the Marxist reluctance to imagine the forms and mechanisms of functioning of a socialist society logically ensued the rule of “scientific socialists” over immature and ignorant masses. At the moment of Revolution these masses already know what they don’t want, but don’t yet have an understanding of what is required for a new, liberated life. Instead they end up with the Enlightenment or Jacobin concept of an “educational dictatorship.” “The Social Revolution must be prepared in detail, in order to be crowned with success. It doesn’t make any sense to wish to improvise everything,” argued the Swedish delegate Albert Jensen at the 4th IWA Congress, “Such a position can be exploited by political demagogues in order to get control of the Revolution, restore political power, and establish a dictatorship.” At this moment special attention was focused on the anarcho-syndicalists of Spain – a country where social revolution was soon to become a reality. That is why the delegates of the CNT at the 4th Congress of the IWA supported Besnard’s proposal.3 “It is necessary to nourish the constructive capabilities of the workers. Capitalism won’t die by itself. Constructive action is more important than barricades,” declared Victor Orobón Fernández. “Destruction by itself is not at all creative. The most important day of the Revolution is the second day, when new construction begins.” He referred to the example of Russia, where “the anarchists fought, while the Bolsheviks started building on their own.” The more people are prepared for revolution, the better they will know what to do after the overthrow and expropriation of Capital and the State, the easier and less painful it will be to carry out the Revolution, and the less danger there will be of usurpation by an avant-garde. The significance of the arguments of the European anarcho-syndicalists lay in their insistence on the insufficiency of just spreading libertarian values and ideas.
They maintained it was necessary to prepare people technically and organizationally so their grasp of production was such that they could take over management of production after the Revolution. “It’s quite indisputable: in order for a certain ideal to triumph, it must be ingrained in the heads of those who will defend it. Insufficient preparation of the people leads to vacillations, always fatal for the matter being defended. That’s why we recognize that before proceeding to the anarchist organization of society, it is quite essential that the people be prepared beforehand,” emphasized V. Márquez Sicilia in the theoretical journal of the Spanish anarchists La Revista blanca. He maintained that, although the Revolution will be violent, the main path to the new society is propaganda: “Victory can only be gained as the result of a general effort which, moreover, will be contingent on the support of a majority of the people. And this combined action, this support of the majority of the people, can be achieved only in the course of a prolonged period of ideological propaganda, but propaganda which is competent, serious, deliberate, and responsible\dots{}” J. Masgomieri, another author of La Revista blanca indicated it was not a matter of an interminable process of waiting until all the people became anarchists: “In order for the anarchist social revolution to become \dots{} an invincible and triumphant force which embraces the whole population, it is first necessary that everyone knows and understands without any kind of intellectual effort the organizational mechanism of the new order of things. And this clear understanding, this material knowledge of the new system, to a much greater degree than abstract and philosophical studies, will give rise to revolutionary consciousness which will become the surest guarantee of development of the Revolution.” The Spanish anarchists categorically rejected the notion advanced by some syndicalists about the difference between an anarchist society and libertarian communism: vague ideas about Anarchy as the simple removal of any sort of restrictions can only give rise to some kind of “sad state of affairs” which amounts to “unconscious sabotage of one’s own ideal and paves the way for the schemes of newly minted politicians.”
In the Spanish CNT there existed tendencies close both to revolutionary syndicalism with its notion of the “syndicalist construction of society,” and to the conception of “libertarian communism.” The debate was ongoing about what to do after the Revolution triumphed by means of a general strike and insurrection. The communitarians, following the anarcho-communist tradition, believed the basis of the future society should be the libertarian commune (“free municipality”), autonomous and self-sufficient to the maximum degree. Correspondingly, they ascribed less significance to problems of economic linkages and the management of coordinated activities between such communes, assuming that any surpluses could be exchanged on an unpaid basis. The industrialists were partial to the revolutionary syndicalist scheme, according to which after the Revolution centralized factory management structures and forms of organization of the economy would be preserved and transferred from private or State control into the hands of the associated syndicates (labour unions). Their strong point was working out solutions to economic problems according to libertarian planning principles. The best known theoreticians of the communitarians were the writer and publicist Federico Urales (editor of the theoretical and literary magazine La Revista blanca) and the physician Isaac Puente. Urales combined Kropotkin’s reasoning with the traditions of the Spanish village communes, which he considered the most suitable base for realizing the collective principles of solidarity.
He maintained that the Revolution would break out after a phase of capitalist crisis, and result in the regeneration of the communal traditions in the free villages. At the same time, Urales and his supporters counted on the presence of revolutionary spontaneity.
Other anarchists considered it essential to formulate ideas about a free society which could provide guidelines for experiments in workers’ insurgency. (Such was the viewpoint of the activists of the Nosotros group, which was behind many of the anarcho-syndicalist uprisings of 1932-1933.) These ideas were popularized by Puente, one of the leaders of the uprisings, in his book The Goal of the CNT – Libertarian Communism. It contained a plan for the creation of a system of libertarian communism in Spain and arguments in favour of its being put into practice. Similar to Urales, Puente followed Kropotkin’s understanding of the social inclinations of humanity. He rejected the idea of a revolutionary or post-revolutionary elite and a transition period.
He believed that the communitarian movement was in tune with the social instincts of mankind. The author proceeded from the assumption that libertarian communism could be established in Spain which would then withstand the capitalist world. Puente conceded that the commune as a popular organ (general assembly of all inhabitants) could exist only in villages and small cities, and that in large population centres its functions would be carried out by the organs of syndicates (associations of producers). But, in the anarchocommunist tradition, he emphasized the voluntary nature and social-economic self-sufficiency of the communes. He was skeptical of “the architects of the new world,” to managerial planning and industrial development. Social wealth, the means of production, and the products produced with the help of these means, would become the property of everyone; each member of society had an obligation to work to the extent of their own powers and in exchange would receive the possibility to satisfy their own needs. Money in any form whatsoever was not required; wealth would be distributed “in proportion to the demands for it.” Finally, the economy of the country “would be the result of coordination between various localities,” which would make arrangements between themselves at the lowest level about combining their efforts at plenums, congresses, and through industrial federations. [193] The book enjoyed a huge popularity in anarchist circles; it was reprinted and widely discussed. One of the main theoreticians of the industrialists was Diego Abad de Santillan, who arrived in Spain from Argentina and renounced the views of the FORA. His work The Economic Organism of the Revolution embraced contemporary industry and emphasized the necessity of planning and economic coordination. He criticized Kropotkin for economic localism and declared free communes an anachronism, a “reactionary utopia.” Abad de Santillan ascribed great significance to free experimentation, allowing for various forms of a future society. But in principle he favoured a comparatively rigid syndicalist structure for the whole of society, similar to the ideas of Besnard. Moreover, like many of the other industrialists, he interpreted libertarian communism as a sort of transitional society on the way to complete anarchy (communism), in which in the beginning a departure from communist principles of distribution (“according to needs”) was permitted.
These theoretical and tactical differences led to splits, the most important of which was the withdrawal from the organization of supporters of a more reformist and pragmatic syndicalist approach, formulated in 1931 in the “Manifesto of the Thirty” (Juan Peiró, Ángel Pestaña, and others). In the middle of the 1930’s it became clear that Spain was on the verge of a social revolution, and that the CNT was faced with the urgent problem of converting the generalized positions of the anarchist “program” into a real plan for the transformation of society on the bases of free communism.
The congress of the CNT in Zaragoza in May 1936 approved a document which was one of the first in history to set out an anarchist program of concrete measures for social revolution – “The Conception of Libertarian Communism.” It combined the ideas and approaches of both currents, but was heavily dependent on the scheme of Puente. Libertarian communism (principle: from each according to their abilities, to each according to their needs within the framework of economic possibility) must be established without any kind of “transition period” immediately after the victory of the social revolution. At the basis of the future free society must lie a dual organization: territorial (free communes and their federations) and industrial (syndicates as association of producers and economic organs of the communes).
The program endorsed decentralized planning from below on the basis of the statistical determination of needs and production possibilities. Money was liable to be abolished and replaced by cards for producers\Slash{}consumers – the only function of such a card was to show that its possessor was actually working. “Once the violent phase of the Revolution is finished, private property, the State, the principle of authority and, consequently, classes, will be abolished\dots{} Wealth will be socialized, organizations of free producers will take the direct management of production and consumption in their own hands. In each locality a Free Commune will be established, which will initiate a new social mechanism. Producers united in labour unions in each industry and profession will freely determine the form of their organizations in their own work places.” It was proposed to entrust the coordination of economic and social life, functions of defense, etc. to communes, syndicates, and their federations. The program emphasized the communist principle of distribution, transformation in relations between the sexes, and education – especially the free development of art and science. The State and permanent army were slated to be abolished and replaced by federations of communes and workers’ militias.
\begin{enumerate}[1.]
\item\relax
E. Malatesta, “Quelques considerations sur le regime de la propriete apres la revolution” in Articles politiques (Paris, 1979), pp. 379-390.
\item\relax
Cited by: P. Adamo, “Anarchismo tra ethos e progetto,” A – Rivista anarchica, 1997, no. 1 (233), Febbraio, p. 36.
\item\relax
This position was by no means shared by all members of the CNT. At the 3\_\_ Congress of CNT in June 1931 a bitter dispute flared up regarding the plan for rebuilding the organization on the basis of industrial unions, as proposed by the syndicalist wing led by Juan Peiro. The anarchists spoke out against this plan. “Supporters of industrial federations have arrived at this position because they have lost faith in \dots{} the goal, and are pinning their hopes on the efficacy of machines,” declared, for example, the prominent anarchist Jose Alberola. “But I say that a machine cannot create vital forces but rather depletes them, and in this sense we are creating a mentality which contradicts everything that speaks to the initiative of the individual\dots{} We need an ideal, and in the final analysis this capitalist machine will sooner or later destroy our ideal.” In the end the draft resolution was adopted by 302,000 votes to 91,000, but in fact was never applied in practice. See: A. Paz, op. cit., pp. 219-222 (n64); J. Peirats, Les anarchistes espagnols\dots{}, pp. 63-64 (n46).
\end{enumerate}
\chapter{Part 3: The Spanish Revolution}
\section{Chapter 9: The Uprising of July 19th 1936}
The uprising was prepared and organized by “committees of defense” which were created in Barcelona’s working class neighbourhoods from members of the CNT, the Federation of Anarchists of Iberia (FAI), and Libertarian Youth. The most active role in the uprising was played by members of one of the anarchist groups – Nosotros (Buenaventura Durruti, Francisco Ascaso, Juan García Oliver, Ricardo Sanz, Aurelio Fernandez, and others), which constituted something like a Central Revolutionary Committee of Defense.
The army mutiny in Barcelona was suppressed. But the workers did not limit themselves to simple clashes with army units. They spontaneously began to carry out the social revolution: they seized enterprises and introduced workers’ selfmanagement; they took supply, transport, and social services into their own hands; they organized a new life. The CNT union of food industry workers opened communal cafeterias where people could eat for free. Even during the fighting, in each working class quarter of the city food committees were organized to arrange the requisition of food products from warehouses and to set up the exchange of manufactured goods for food with the peasantry. Market commerce and the money system were replaced to a significant extent by non-monetary exchange. The food supplies acquired in these exchanges was distributed according to norms established by the committees. Clothing and other consumer goods were distributed through shops and stores. There were instances where workers raided banks and monasteries and burned the money confiscated as a symbol of the hated Capitalism. Items from pawn shops were returned to the people who had been compelled to pawn them. The labour unions (syndicates) confiscated large government and privately-owned buildings and set up their headquarters in them. At the majority of industrial enterprises, in transportation, and in social services, general meetings of worker collectives took place which elected management committees, most of the members of which were representatives of the CNT. Such a seizure of production units by a collective received the name “collectivization.” In several sectors (woodworking in Barcelona, bakeries, railway transport, and others the collectivization of industry went on to the next stage of socialization: the whole production process from start to finish was subject to the self-management of workers, who created the appropriate organs. Within a few days life in Barcelona had already normalized: transport was running, enterprises were working, shops were open, and communications systems were operating. Researchers concur that all the revolutionary measures and the normalization of daily living were, basically, the spontaneous actions of workers belong to the CNT; the corresponding orders had not been issued by some higher committee of the union federation. Initiatives most often came from rank-and-file members of the unions (syndicates) of the CNT or from front-line anarcho-syndicalist and anarchist activists.
“\dots{} the proletariat of Catalonia,” according to Andre Capdevilla, a member of a CNT syndicate of textile workers, “was saturated with anarcho-syndicalist revolutionary propaganda. Over a period of many decades the notion had taken root among the workers that they should make the most of any opportunity to carry out the Revolution. So they acted as soon as the possibility presented itself.”
The Revolution also took hold in other cities (above all, in Catalonia), and also rural areas (in Catalonia, parts of Aragon, Andalusia, and Valencia). In regions with large estates the peasants seized the land from its owners. In many regions they agreed to carry on agricultural work on a group basis – by forming “collectives.” In regions such as Aragon and Andalusia, the anarchists had carried on agitation among the village population over a period of many decades. “In those most backward regions to which they were sent,” according to [URL=\Slash{}tags\Slash{}gaston-leval] Gaston Leval, an eye-witness, participant, and researcher of these events, “our comrades joined in working in the fields and were able to communicate more advanced technical ideas, and teach the children to read. The result was that the Good News [anarchism] penetrated into the socially most backward areas of the countryside.” The German anarchosyndicalist Augustin Souchy told the story of an anarchist from the Aragonese village of Munesa, who worked for a long time in Barcelona, and then went back to his native village and acquainted the peasants with libertarian ideas. Under his influence his fellow-villagers organized a collective – a free commune. “A Spanish edition of Kropotkin’s book The Conquest of Bread lay on the table. In the evenings members of the collective would gather, and one of them would read the book out loud. This was the new Gospel.”
During the first days of the Revolution, new structures of social self-management appeared, spontaneously formed by revolutionary workers and peasants in enterprises, village communes, and urban neighbourhoods. At the base of these structures one always found general meetings (“assemblies”) of the residents or of the labour collective. They elected revolutionary committees, committees or councils of enterprises, councils of soldiers and sailors, etc. to carry out routine, coordinating, technical, and executive functions. The members of the committees acted within a framework where they were obligated to carry out the orders of the assembly which elected them, and could be recalled at any moment. All important decisions of the committees were adopted only in accordance with the wishes of the collective of the commune.
In Barcelona the revolutionary committees, which grew out of the neighbourhood committees of defense of the CNT and “barricade committees,” occupied themselves with street-level organizing – arranging food and other services, and maintaining order. In many villages, immediately after the failure of the military mutiny, the inhabitants removed the local administration, and a revolutionary committee, elected at a general meeting, took over administrative as well as economic functions. Often the revolutionary com- mittees immediately applied themselves to such revolutionary measures as the burning of all documents about private ownership; the confiscation of the land, buldings, crops, and inventory of big landowners; the conversion of churches into storage facilities; the collectivization of land, and the organization of a volunteer militia.
Of course it was not only the anarcho-syndicalists who took part in the formation of popular organs. There were also other workers, mainly rank-and-file members of the other trade union central – the General Union of Workers (UGT) – which was oriented towards the Socialist Party Consequently, the composition of these organs reflected the correlation of forces between the CNT, the UGT, and other forces.
In any case, the power of the State ceased to function over a significant part of the territory of Spain. The central government of the Republic in Madrid was completely discredited by its inability to oppose the military mutiny and lost all its authority. The regional government of Catalonia (the Generalitat) headed by Luis Companys controlled only its own building. Local administrations were either removed or neutralized. The army and police were either disbanded or destroyed. Barcelona was controlled by workers’ militias, primarily anarcho-syndicalist in composition. “\dots{} power was lying in the street, and it was embodied by the people armed,” noted the contemporary researcher Abel Paz. The anarcho-syndicalists, who now enjoyed a dominant influence among the workers of Catalonia, were confronted by a decision about what to do with this power: whether to destroy it, take it into their own hands, or hand it over to others.
\section{Chapter 10: Libertarian Communism or Anti-Fascist Unity?}
Theoretically the relationship of the Spanish anarchosyndicalists to the question of power was determined long before July 1936. The Spanish anarchist (libertarian) movement from its very beginning in the 1870’s preached the simultaneous annihilation of Capitalism and the State by means of social revolution, and the transition to a stateless system – a federation of free communes and workers’ unions. A plan of action in a situation of social revolution had been outlined by the end of 1933, just before a planned uprising against a right-wing government which had just acceded to power. Guidelines for building a new society were enshrined in the Zaragoza Program (“The Conception of Libertarian Communism”) of 1936.
In spite of having a more or less clear idea about what had to be done at the moment of revolution, the Spanish anarcho-syndicalist movement paradoxically was unable to pin down the criteria for determining the “ripeness” of a society for social transformation. In other words: how does one establish if the time is right to start implementing a blueprint for building a new society? The CNT in July 1936 was not able to find an unambiguous answer to this question. “The Conception of Libertarian Communism” talked about the revolutionary character of the epoch as a whole, but was rather vague when it came to the moment of revolution itself. Within the CNT there had long existed a belief that a genuine social revolution would be possible only when the CNT represented an overwhelming majority of the workers in the whole of Spain, or when the CNT had created an all-embracing union structure which was prepared to take over the management of the whole economy in the course of a social revolution. There were radical anarchists in the CNT (the Nosotros group and others who shared its views) who took a different position. They considered that the readiness of the masses for revolution was first and foremost a matter of psychology, and that this readiness would develop under the conditions of an ongoing revolutionary situation. They also did not make much of an effort to theorize and explain the moment of qualitative change. Moreover, the CNT frequently emphasized that in Spain the alternatives were clear: fascism or libertarian communism – and the appropriate response to a fascist putsch was social revolution.1 There was also a lack of clarity concerning relations with the other large union federation – the UGT, which was controlled by the Socialist Party. On the one hand, the anarcho-syndicalists expressed their desire for an “alliance” with the UGT; but on the other hand, at the Zaragoza congress they approved the conditions for such a pact which would require the UGT to repudiate the Socialist Party and adopt a position of social revolution.
All this created uncertainty. That is why at the very moment when events in Barcelona, in practically the whole of Catalonia, and partly in other regions of the country, “gifted” the anarchists with that for which they had struggled and dreamed for decades, they found themselves unprepared to make use of this “gift.”
One must also take note of the fact that the CNT had always harboured reformist tendencies which from time to time took control of the organization. Thus, Pestaña and Piero, who headed the CNT at the end of the 1920’s and the beginning of the 1930’s, supported close contacts with republican political organizations, and in 1931-1932 became the leaders of a reformist group, the “Treintistas.” A significant part of this fraction quit the CNT, but returned to it in 1936.
However, besides the “Treintistas” there remained a substantial number of “pure” syndicalists in the union federation as well as members who were simply pragmatically inclined. To a certain extent, this was a consequence of the contradictory organizational vision of Spanish anarcho-syndicalism, which tried to combine anarchist goals and social ideals with the revolutionary syndicalist principle of trade unions being open “to all workers,” independently of their convictions.
The membership of the CNT were far from being made up entirely of conscious anarchists; this was particularly true of those who had joined during the period of the Republic (from 1931 on). These partisans of a pragmatic approach could be relied upon by those activists and members of the executive organs of the CNT who preferred to avoid risky, “extremist” decisions.
On July 20 1936 the president of the Generalitat, Companys, made contact with the Catalan Regional Committee of the CNT and invited its representatives to a meeting to discuss the situation emerging after the suppression of the “fascist mutiny” of the military. A plenary assembly of delegates of the CNT unions, committees, and FAI groups was convened to analyze this proposal. The opinions of the participants diverged right from the start. Their spectrum extended from the proposal of García Oliver, a member of the Nosotros group, to declare libertarian communism; to the position of Abad de Santillan, who spoke in favour of uniting with other antifascist forces. An intermediate position was maintained by those who, like Manuel Escorza, proposed for the time being a “hands off” policy towards the government of Companys, not making any agreements with him, but setting about carrying out the socialization of the economy and thereby depriving him of any real power. Escorza declared real power was found in the hands of the CNT; consequently, political power could be ignored. The delegation of anarcho-syndicalists from the working class area of Baja Llobregat led by José Xena objected strongly to collaborating with the government, but did not want to support García Oliver and was inclined to support Escorza’s point of view. The debate was turbulent, at times bitter. In the end a decision was arrived at which was provisional in nature: to send an armed delegation to meet with Companys for the purpose of exchanging information.
Receiving the delegation of the CNT and FAI, Companys congratulated the anarchists on their victory and expressed his willingness to resign. But he then tried to convince them they would not be able to manage without traditional political forces. He reminded the libertarians that the battle with fascism was far from won and required a broad coalition of antifascist forces. Companys proposed to form a coalition organ with the participation of the anarcho-syndicalists – a “Committee of Militias” with the mission of organizing the final defeat of the rebels. The anarchist delegation explained it lacked the authority to make an agreement with him, but would transmit his proposal to their own organizations. Without waiting for the agreement of the CNT, Companys issued a declaration about the creation of popular militias and the corresponding chief organ made up of people close to him. The Regional Committee of the CNT, after listening to the reports of García Oliver and Durruti about the meeting, resolved to contact Companys and let him know the CNT could offer provisional support for the creation of such an organ, but that the final decision would have to come from a regional plenum of the Catalan CNT.
At the regional conference (plenum of local organizations) of the Catalan CNT on July 21 1936, the delegation from Baja Llobregat proposed to withdraw from the newlycreated Central Committee of Antifascist Militias (CCMA) and proclaim libertarian communism, as stipulated in the decisions, principles, and ideological goals of the organization.
The Nosotros member García Oliver, speaking for his group, supported the demand from Baja Llobregat. He called for the errors which had been committed to be rectified and for the social revolution to be carried through to the end: the CCMA should be dissolved and libertarian communism established throughout the whole country. Speaking against these proposals were the well known FAI activists Federica Montseny, Abad de Santillan, and the secretary of the Catalan CNT Mariano Vasquez. Montseny urged that events not be forced since, in her opinion, this would lead to the establishment of an anarchist dictatorship which would be in contradiction to the essence of anarchism. She proposed to have recourse to concessions: to take part in the CCMA, and then – after the final defeat of the military mutineers – withdraw from this organ and return to the work of creating an anarchist society. Abad de Santillan pronounced in favour of participation in the “Committee of Militias,” and stressed that global capitalism would not permit libertarian communism in Spain and would have recourse to military intervention. He warned against war on two fronts and called for “deferring” libertarian communism to the future.
Vasquez, speaking at the second session of the plenum, argued that even by not “carrying things through to the end,” the CNT could still rule from the street, depending on its own real strength. Consequently he considered it worthwhile to remain in the CCMA and avoid a dictatorship.
In the course of subsequent discussions, the delegation from Baja Llobregat stood firm on their proposals, and García Oliver attempted to refute the arguments of his opponents. He denied accusations of wanting a “trade unionist” or “anarchist” dictatorship and urged that a decision be made right away so as not to leave a vacuum which could be used by the enemies of the Revolution, as had happened in Russian in 1917. “I am convinced that syndicalism, both in Spain and in the rest of the world, finds itself faced with the act of proclaiming its values openly to humanity and to history,” he insisted. “If we don’t demonstrate that we can build libertarian socialism, the future will belong, just like before, to the sort of politics which came out of the French Revolution – starting with a bunch of political parties and ending with one.” García Oliver also criticized attempts to “sow fear,” emphasizing that the Revolution could deal with interventionists as well as the mutiny. García Oliver repeated his call to declare libertarian communism and “carry things through to the end.”
After everyone had spoken, Abad de Santillan officially stated the alternatives: endorse membership in the CCMA or declare libertarian communism. The question was put to a vote; only the delegation from Baja Llobregat voted for declaring libertarian communism; the rest of the delegates were in favour of “anti-fascist co-operation.” The decision adopted took the view that the Revolution was going through an “antifascist stage,” that libertarian communism was inappropriate, and that at the present time it was necessary to consolidate the “antifascist front which was taking shape in the street.”
What had caused such a major volte-face on the part of the CNT, essentially discarding the program of action which it had adopted just two months before these events?
The decision upheld by the Catalan CNT not to declare libertarian communism and to enter into collaboration with other antifascist forces (socialists, communists, and republicans) was, as many anarcho-syndicalists recognized later, the result of a hasty evaluation of a complex situation.
Victorious only in Catalonia, the libertarians did not feel sure of themselves in other regions of the country. “We agreed to cooperate,” said the CNT’s report to the IWA Congress in 1937, “Why? The Levant [Valencia] was defenseless and vacillating – its barracks were full of putschists. In Madrid our forces were in the minority. Andalusia was in a confused state, with groups of workers, badly armed with hunting rifles, carrying on the struggle in the mountains. The situation in the North remained uncertain, and the rest of Spain was presumably in the hands of the fascists. The enemy was established in Aragon, at the very gates of Catalonia. The real state of our foes was unknown to us – whether on the national or the international level.” The activists of the CNT did not risk taking the path of independent revolutionary action, dreading the prospect of war on three fronts: against the fascists, the government, and possibly foreign interventionists. In other words, the majority of the activists believed it was premature to talk about social revolution on a country-wide scale, while libertarian communism in Catalonia alone was inevitably doomed.
Nevertheless, the real situation of things was far from being as hopeless as it seemed to the Catalan anarchosyndicalists, who were probably still living in the shadow of the defeat of the insurrections of 1932-1933. This time it was not a case of an isolated local outbreak. The socialrevolutionary movement spread throughout Catalonia and parts of Aragon and Valencia, and the way to Andalusia was open. In other words, the economically pivotal industrial and agrarian regions of the Iberian peninsula had fallen into the hands of the revolutionaries. In such a situation it was possible to risk “going to the end.” “In the given case,” wrote the contemporary Spanish anarcho-syndicalist Abel Paz, “we believe the question of power was decided in too much of a hurry, and this haste prevented taking into account “the whole significance of the Revolution,” as the report [of the CNT] made clear. If the proposals of García Oliver had been accepted, then the problem of Revolution would undoubtedly have been cleared up at the grass roots level.” But now the anarcho-syndicalists lost valuable time and conceded the initiative to their enemies.
Finally, there was still one factor which García Oliver mentions casually in his memoirs: the delegates gathered hurriedly, not previously being aware about what they were to discuss. In other words, they adopted a decision at the plenum without having instructions from the unions and other organizations they were representing. This was the first serious violation of federalist procedure within the CNT – a tendency which was to become prevalent subsequently. “The first error,” notes Paz, “was committed already on July 19 and 20, when a group of activists substituted themselves for the members themselves and made decisions for them. From this moment on a gap manifested itself between the base and the upper levels: the base wanted to broaden the Revolution, the superstructure tried to control and limit it\dots{}”
Other members of Nosotros did not speak at the plenum. One of its prominent members, Ricardo Sanz, subsequently recalled: “As a group, we did not exert pressure on the results [of the discussion]. We knew our organization was against dictatorship. And that’s what would have happened if our position had been adopted\dots{} But in any case, we did not try to force a decision, since there was other urgent business:
Companys had agreed that Durruti would lead the militia forces, which must occupy Zaragoza which had fallen into the hands of the enemy\dots{}” In the evening after the conclusion of the CNT plenum, a meeting of Nosotros and its supporters (Marcos Alcon, García Vivancos, Domingo, Joaquín Ascasco, and others) was held. All were agreed it was necessary to move beyond alliances with political parties and form new organs of popular self-government, based on the revolutionary committees and labour unions of the CNT. However differences arose about the time-table for such actions. García Oliver urged the group “to finish the work begun on July 18” by having the forces of the anarchosyndicalist militia occupy the government buildings and key installations of Barcelona. Durruti called this plan “excellent,” but considered the moment “inauspicious” when the mood of the CNT activists was taken into account. He proposed to wait ten days, until the libertarian militia had taken Zaragoza – the capital of Aragon – thereby saving Catalonia itself from a possible economic and political blockade. García Oliver objected, arguing that the capture of the city could wait, but his arguments did not find support.
At the first meeting of the CCMA, the anarcho-syndicalists rejected the plan of Companys, which attempted to reduce the role of the new organ to carrying out military and technical tasks. They insisted on its transformation into an institution for the economic, political, and military administration of Catalonia, so that the functions of Companys as President of the Generalitat would become purely nominal. The CCMA became a semi-governmental, semi-grassroots organ. Besides the anarchists, who held key posts in it, there were also representatives of the UGT, the Catalan left nationalists, Communists (controlled by the Comintern and formed in July into the Unified Socialist Party of Catalonia – closely linked with the Communist Party of Spain), anti-Stalinist Communists from the Workers’ Party of Marxist Unity (POUM), and others. The Committee made decisions on fundamental social-political questions, but at the same time it was impossible to view it as an organ of a purely governmental type since its members were responsible primarily to the committees at the head of their organizations, to which they owed their positions as delegates. So in fact these organizations made decisions, and the CCMA only ratified them. Up until August 10 1936 its official documents were valid only if they bore the imprint of the Catalan Regional Committee of the FAI.
The maintenance of order in Catalonia was carried out by patrols organized by the militias of the various organizations and movements belonging to the CCMA. The most powerful of these was the militia of the CNT. The members of the CNT, the FAI, and FIJL also constituted the basis of those volunteer forces which fought with the insurgents at the front of the unfolding Civil War. On July 24 1936 the first of such columns with a complement of 2,000 led by Durruti set out for Aragon. So it happened that volunteer units, formed by various organizations and movements, were able successfully to oppose the insurgent armed forces for the whole first period of the Civil War and achieve significant successes.
Durruti’s column, which liberated a large part of Aragon from the enemy, was organized on the basis of libertarian principles: all the commanders were elected and lived in the same manner as the rank-and-filers, there was no penal code, and everyone observed voluntary self-discipline. The CNT columns which fought in Aragon were 16,000 strong.
The anarcho-syndicalists rejected the decree concerning mobilization of reservists issued by the central republican government at the beginning of August. However in Catalonia on August 6 1936 the CNT gave consent to partial conscription by the Generalitat and the CCMA, which was already a fundamental departure from principles. Nevertheless, the anarcho-syndicalist militias continued to be based on the principle of voluntary popular armed forces.
\begin{enumerate}[1.]
\item\relax
“Only by carrying through the social revolution is it possible to smash fascism,” wrote, for example, the newspaper of the Catalan CNT Solidaridad Obrera just before July 19 [Solidaridad Obrera, 17.07.1936].
\end{enumerate}
\section{Chapter 11: Under the Pressure of Circumstances}
Thus, the CNT made a principled decision (and one which, as became clear later, had fatal consequences) to renounce “total revolution,” to set aside libertarian communism until victory was gained over the coalition of military, fascist Falangists and Monarchists opposing the Republic. The official position of the anarcho-syndicalists on the question of State power in this period was expressed in the article “The Uselessness of Government,” published in the “Information Bulletin of Propaganda of the CNT-FAI” and in the Catalan CNT’s newspaper Solidaridad Obrera.
This position boiled down to the notion of the necessity of continuing the Revolution in the social-economic sphere, not paying any attention to the State, and preserving the Popular Anti-fascist Front “from below.” In the article it was emphasized that the central and Catalan republican governments had not undertaken any measures to prevent or suppress the mutiny and that their existence was inessential for the antifascist struggle. The anarcho-syndicalists believed the “social struggle” was unfolding throughout the country.
“The coordination of the forces of the Popular Front and the organization of the food supply by means of the simultaneous collectivization of enterprises is vitally important for the achievement of our goals\dots{},” they noted. “However up until now this has been carried out not under the control of the State, but rather in a decentralized, demilitarized fashion,” based on the CNT and UGT labour unions. The existing government is “basically only a weak preserver of the ‘status quo’ in tending to the property rights of international financial interests.” In such a situation a government of the Popular Front was unnecessary and even harmful, since it would either serve as a means of compromise and paralyze the decision-making process with its coalition politics and internal struggles, or prepare the way for a new dictatorship in the form of a “workers’ state.”
The leaders of the CNT and the FAI compromised with the antifascist parties and movement and made concessions to them, justifying this by reference to “developing circumstances,” namely the necessity of victory in the Civil War. They agreed (in order to avoid foreign intervention) not to expropriate enterprises belonging to foreign capital; such enterprises would only be subject to workers’ control.
New organs (revolutionary committees, committees of the antifascist militias, etc.) were now quite often put together not at general meetings, but – like the CCMA – on the basis of agreements between the CNT, UGT, and other organizations. Frequently revolutionary organs existed in parallel to the surviving pre-revolutionary structures at the local level, which sometimes gave rise to sharp conflicts between them.
The anarcho-syndicalist masses paid little attention during the first months to the compromises agreed to “above.” They carried out the social revolution on their own “from below,” impelled by their own libertarian “idée-force.” The scale of self-management by workers during this period of the Spanish Revolution has no equal in history. Thus, in Barcelona 70\% of enterprises were taken from their owners and transferred to the control of the CNT and UGT; in Valencia – 50\%. Collectivization was also widely embraced in the rural economy. A regional plenum of the peasant syndicates of Catalonia, belonging to the CNT, resolved on September 5-7 1936 to collectivize large estates and any land which was being worked with the help of a hired workforce. All expropriated land passed under the control and management of a syndicate and was cultivated directly in the interests of its members, namely “the workers as a whole.” Subsequently in Catalonia, Valencia, and other regions a wide-ranging process evolved of peasants coming together in self-managed collectives. This phenomenon was particularly widespread in the territory of Aragon which had been liberated by the anarcho-syndicalist militias, where such peasant collectives controlled up to 60\% of all the land of the region and transformed themselves essentially into free, self-managed communes in the anarcho-syndicalist spirit.
However very soon the political compromises became an obstacle in the path of the grassroots initiatives. Thus, since libertarian communism had not been proclaimed, the notion of abolishing money and carrying out distribution according to needs had to be renounced. In the cities the circulation of money was fully retained; the most that was accomplished was the introduction in a number of cases of the so-called “family allowance” system, namely equal pay for each worker with a supplement for members of the worker’s family. More typically, there was a significant increase in the wage rates for the lowest paid workers, which reduced the gap between the earnings of different groups of workers. In the villages, at first there were attempts to experiment with unfettered consumption, rationing, introduction of local currency, the “family allowance,” etc. However all these measures were characterized by a lack of coordination. There was an absence of any sort of coordination of the activities of local revolutionary organs; in spite of the anarcho-syndicalist “program,” these organs were not united in a federation, but operated exclusively at the local level.
In their efforts above all to advance beyond “collectivization” (transition stage of management by workers’ collectives) to complete socialization of the economy, the anarcho-syndicalists initiated the creation on August 11 1936 of the Economic Council of Catalonia, which was to carry out the overall coordination and planning of the economy and establishing pricing policy. However this organ also bore the stamp of compromise both as to its make-up (it included members of CNT, UGT, and political parties) and as to the tasks it undertook to carry out. Its goals included such diverse measures as the regulation of production guided by the needs of consumption; the monopoly of external trade; the development of collectivization in industry, commerce, in the rural economy, and in transport; the fostering of cooperation between the peasantry and consumers; job placement for the unemployed; reform of the tax system, etc.
Abad de Santillan, who played a key role in the Economic Council, was convinced this organ would be able to bring about the creation of a new economic system. On the other hand, the radical wing of the anarcho-syndicalists (Durruti and others) feared such a “legalization” of the conquests of the Revolution would only tend to strengthen the power of the Generalitat and could lead to “State Capitalism” or “State Socialism.”
The unstable equilibrium of forces could not be preserved for long. State power – not liquidated by the anarchists – as well as the political parties and social strata which supported them, made use of the breathing space granted them to pass over to an offensive against the Revolution. In the hands of the unabolished State remained powerful levers, above all currency and other financial resources. Collectivized industry lacked raw materials. “The Marxists and Republicans formed a bloc and, possessing money and armaments, they pursued a politics of patronage in relation to their supporters, distributing to them food, weapons, administrative jobs, means of communication and transport\dots{},” it was acknowledged in the report of the CNT to the congress of the anarcho-syndicalist International in 1937. “Catalonia had to organize its own foreign trade, competing abroad with other parts of the country, in order to feed its own citizens and satisfy the needs of the Aragon Front\dots{} The government, taking advantage of our efforts to avoid causing harm to antifascist unity and to not provoke a rupture of official relations with foreign nations, used its privileged diplomatic situation and ruthlessly sabotaged our actions in all fields. [222]
The governments in Madrid and Catalonia began to exert increasing pressure on the anarcho-syndicalists in three directions at once: impeding the supply of weapons and ammunition to the badly armed militias, trying to limit the scope and course of collectivizations in industry and in the rural economy, and attempting to impose the replacement of the militias by the regular army. In September 1936 a massive campaign was begun in the Catalan press directed against “out-of-control” anarchists, who were accused of concealing weapons instead of sending them to the Front (it was the committees of defense which were being targeted here), and also against “utopian experiments” in the economy.
Having embedded itself in the power system, the leadership of the CNT was forced to change itself. It had reconstructed itself in order to conform to the demands of the moment, justifying the mushrooming bureaucratic apparatus by the real requirements of coordinating economic and social life. Taking advantage of the fact that the activist members of the CNT and FAI were either fighting at the Front or completely weighed down with the work of workers’ self-management at the local level, many labour federation officers (members of the national, regional, or district committees; aides to the various union commissions, the Committee of Militias, the Economic Council, etc.) began to take into account the needs and desires of the anarcho-syndicalist masses less and less . The rank-and-file activists simply could not keep track of the endless chain of conferences, plenums, and meetings and look into the matters discussed in detail.
As noted by José Peirats , the historiographer of the CNT, there was essentially a breakdown of the federalist norms of the organization (transformation of the National Committee into a “machine for issuing orders” to individual unions, the convening of plenums by means of announcements from above, the adoption of important decisions by committees at all levels or at meetings of picked activists with subsequent approval at general assemblies). All these practices were in contradiction to the principles of anarcho-syndicalism, corresponding to which initiatives in the organizations ought to advance not “from the top down,” but “from the bottom up,” and committees and commissions were to be convened not to adopt independent decisions on fundamental questions, but to carry out the orders of the “ordinary members” at general assemblies.
Many anarcho-syndicalists spoke out against the nascent bureaucratization of the CNT and against the policy of more and more concessions into it after 490 to the State and political parties on the part of the CNT leaders. Durruti frequently expressed his concern and indignation on this score. The radical wing tried to turn the course of events at the regional plenum of the Catalan CNT at the beginning of August 1936. García Oliver and Durruti demanded an end to the collaboration with political forces, which was causing the Revolution to lose its bearings and depriving it of its strength. They called for further progress in the Revolution. But the majority feared above all civil war in the “antifascist camp.” The course pursued since July 20 remained without significant changes.
A decision was adopted about the necessity of a “revolutionary alliance” with the UGT and the creation of a National Committee of Defense for military-political leadership. The radical minority, noted the historian Paz, submitted this time around, obeying organizational discipline. “The only way out of this impasse would have been to break with ‘the activist’s sense of responsibility’ and, without the consent of their own organization, take the revolutionary problem into the streets. But none of the activists felt capable of doing this\dots{}” In the middle of August the CNT attempted to put into practice the idea of an alliance with the UGT by entering into negotiations with its leader, the socialist Largo Caballero. The possibility was discussed that both union federations could combine to topple the central republican government and replace it with a revolutionary junta of defense. At the last moment Largo Caballero renounced this plan, since he did not want to destroy the legitimacy of the republican government. On September 4 1936, he was appointed prime minister of the Spanish Republic.
Tensions between the anarcho-syndicalists and the antifascist parties and movements continued to grow. In response to the accusation that the anarchists were “hiding weapons,” the “committees of defense” of Barcelona declared that it intended to store weapons “as long as the Revolution has not resolved the problem of political power, and as long as there exist armed forces submitting to the orders of the government in Madrid,” since they considered weapons “the guarantee of our revolutionary conquests.” The newspaper Solidaridad obrera defended the collectives in industry and in the rural economy, and reminded its readers about “the revolutionary character” of the war. In a radio broadcast from the Front, Durruti emphasized that “fascism and capitalism – are one and the same,” and the company committees and the military committee of the “Durruti column” threatened to march on Barcelona if weapons allegedly concealed in the Barcelona barracks of the Communists were not immediately sent to the Front. Eight machine guns, discovered in the office of the Communists in Sabadella, were sent to the front-line soldiers.
\section{Chapter 12: The CNT Enters the Government}
Meanwhile, the logic of “circumstances” induced the leadership of the CNT to take the following step: it began to seek ways to participate in the direction of military-political affairs, hoping this would help to consolidate the revolutionary conquests. On September 15 1936, at a plenum of the regional federations of the CNT, the National Committee was able to get adoption of a resolution about the necessity of a National Council of Defense as a “national organ, empowered to carry out executive functions in the area of military planning, and functions of coordination in the area of political and economic planning.” The Council, headed by Largo Caballero, was to include “delegates” from all three political tendencies (anarcho-syndicalist, Marxist, and republican), and the army and police were to be replaced by popular militias. The economic program of the Council was to include the socialization of banks and church property, estates, big industry, and commerce; the socialized means of production would be handed over to management by syndicates, and provision would be made for the freedom to carry out revolutionary economic experiments.
Similar councils would be formed at the regional and local level. The plenum resolved to submit this draft to the UGT along with a proposal about an alliance. As Peirats justly remarked, such a Council of Defense would have been the government, but under another name. Nevertheless, the “nongovernmental” form of this organ was important to the anarchists. Understanding perfectly the contradictions built into this proposal, Largo Caballero rejected it as violating constitutional principles. However, according to Paz who has made a detailed study of the events of those days, both sides – Largo Caballero and the National Committee of the CNT (headed by a new General Secretary and proponent of the reformist line Horacio Martínez Prieto), had a good grasp of what the other side wanted, and from this moment on carried on interminable haggling during which they had recourse to various kinds of pressure tactics. The trump card of the prime minister was the question about money and weapons for the anarchist militias at the Front, which carried on fighting in the hopes that by taking Zaragoza and Huesca they could compel the CNT committees to put an end to concessions and proclaim libertarian communism.
The volunteer units at the Front were becoming weaker and weaker due to lack of weapons and ammunition. The situation became so critical that Durruti and Abad de Santillan came up with a scheme for an anarchist column to attack the National Bank in Madrid in order to expropriate its resources and use them to purchase weapons. However the frightened members of the National Committee vetoed this. Meanwhile, in Catalonia the Regional Committee of the CNT, under constant pressure from the government of Largo Caballero to put an end to “dual power,” announced its consent to the dissolution of the CCMA; in exchange, three representatives of CNT joined the Generalitat. Thus, for the first time anarcho-syndicalists openly became part of a government organ. Prominent activists of the Catalan CNT such as García Oliver, A. Fernandez, Xena, and Marcos Alcon, gritting their teeth, reconciled themselves to this decision.
The reaction of the rank-and-file activists of the CNT to the continual concessions of the leadership of the Catalan organization was different. Marcus Alcon, one of the key figures of the CNT (first with the glassworkers’ union, then with the union of workers in the entertainment industry), who enjoyed great popularity in Barcelona, recalled that soon after the CCMA was dissolved and the CNT joined the Catalan government, he was confronted by representatives of a commission of Committees of Defense of Barcelona – Daniel Sanchez, Ángel Carbalera, Trapota, and others. They informed him that at a meeting of the Committees of Defense a resolution was passed empowering them to go to the headquarters of the CNT and the FAI and dismiss the Regional Committees of those organizations, which were “stifling the Revolution.” The delegates proposed that Marcos Alcon become the new secretary of the Catalan Regional Committee of the CNT. Alcon was in agreement with the activists in their evaluation of the situation and the concessions which had been made. But he was resolutely against the proposed measures, considering them “irresponsible” and harmful for the organization. With difficulty he persuaded the Committees of Defense to refrain from taking action, urging them instead to “build up their strength in the unions” and, basing themselves on the unions, compel the CNT committees to carry out the will of the members of the organization.
Thus one of the last chances to continue the development of the social revolution in Catalonia was lost.
At this critical juncture a plenum of the regional federations of the CNT was convened on September 28, at which there was an expression of regret in connection with the negative reaction of other unions and political organizations to the proposal about creating a National Council of Defense.
The CNT complained that the exclusion of its representatives from the leadership of the struggle was undermining the authority of that leadership, and once more called upon the UGT to join in a “revolutionary alliance,” threatening to “decline all responsibility” for the consequences in the case of refusal.
The problem of the lack of weaponry, it appeared, made some headway after a meeting of the General Secretary of the anarcho-syndicalist International, Pierre Besnard, and Durruti with Prime Minister Caballero in Madrid on October 1 1936. Durruti warned the Prime Minister that if the government did not allocate sufficient financial resources for the purchases of arms for the CNT-FAI columns, then the front-line soldiers would march on Madrid. After this, the Spanish government agreed to spend 1.6 million pesetas on the purchase of armaments, of which a third would be spent on material earmarked for Catalonia and Aragon. But just a few days later the proposed deal with an armaments firm was cancelled, since the Soviet Union had interfered in the matter, offering its own assistance to the Republican government.1 Aid from the USSR led to a dramatic increase in the influence of the enemies of the anarcho-syndicalists – the Communists of the PCE, who opposed socialist revolution in Spain.
As a counterbalance to the conciliatory course of the leaders of the CNT in Madrid and Catalonia, the front-line and Aragonese anarcho-syndicalists formed their own central. They began to hurl open challenges at their own organization and preferred to create something along the lines of a “rallying point” for the Spanish Revolution. After the return of Durruti from Madrid to the Aragon Front, a regional conference of delegates from the villages and anarcho-syndicalist columns was held on October 6 1936 in Bujaraloz. At this conference a Council for the Defense of Aragaon was formed, composed exclusively of anarchists. It was empowered to coordinate all activities in the military, economic, and social spheres. The Council was made up of sections assigned to various fields of activity and thus it resembled a governmental organ. However the originators of this organ envisaged federalist rather than hierarchical mutual relations between it and the grassroots general assemblies: “The sections will develop a plan which will be presented to the representatives of the organizations and requires their consent. But once approved, it will become generally obligatory and will be carried out in all its aspects.” In citing this document A. Paz notes: “For the first time in the history of society, an entire region initiated revolutionary activity independently of any political parties, having as its exclusive basis the General Assembly, which was declared sovereign. In actual fact, the organization of society which was developed in Aragon is about as close as you can get to libertarian communism.”
The central and Catalan governments did not recognize the Aragonese Councils.2 With the help of Durruti and the soldiers of his column, federations of self-managed villager collectives began to form in the region, which finally took shape at a congress in Caspe in February 1937.
But while the Revolution was in the ascendant in Aragon, in other parts of the Republic its development was slowing down. State power intensified its efforts to control revolutionary spontaneity, and the leadership of the CNT did nothing to prevent this from happening.
On October 9 the Catalan government issued a decree about the dissolution of all local committees and various administrative, cultural, and other organs created after July 20 1936. In their place, the Generalitat instituted new communal councils, the members of which were not elected, but delegated by the movements and parties which were taking part in the regional government. Failure to observe this decree was equated with treason with regard to the State. However in practice many revolutionary committees ignored the decree and were unwilling to give up their power to the new organs. A “dual power” system persisted for several months at the local level, until the revolutionary organs gave up, mainly because of constant pressure from the CNT which appealed to its own members to observe the government decree.
The central government of Largo Caballero issued a whole series of decrees which stipulated the restoration of military discipline, a command hierarchy, codes of punishment for their violation, and also aimed at assimilating the militias into the regular army. On September 30 a decree was issued according to which on October 10 militia detachments of the Central Front were to be converted to regular military units; the conversion was to take place on October 20 on the remaining fronts. On October 21 the government published a decree about the creation of a regular army. The government’s decision ignited a storm of indignation in the anarcho-syndicalist columns and militias. “If we deprive the war of all its revolutionary content, its ideas of social transformation\dots{}, then there is nothing left except a war for independence [of Spain], which \dots{} is no longer \dots{} a revolutionary war for a new society,” was stated in a declaration of internationalist soldiers of the anarchist “Ascaso” column.
The CNT militias in central Spain accused the government of trying to fetter the proletariat with “new chains,” and described the restoration of the army as a “typical tactic of authoritarianism” and the entrenchment of militarism as “an integral part of fascism.” They called the restoration of the army “a return to the past” and threatened the working class would not stand for the loss of that for which it had shed its blood. Durruti himself made it clear in an interview he had no objection to bolstering conscious discipline nor instituting a unified command (referring to the ongoing opposition of the communist columns to attempts at unification), but at the same time he did not intend to observe any military ranks, salutes, drills, or code of punishment. He continued to insist that in a revolutionary war, volunteer corps, made up of people who understood what they were fighting for, were extremely effective. In September – October 1936 soldiers of the anarchist “Iron Column” took part in sensational incidents in Valencia. They withdrew from the Front and made their way to the rear areas, where they demanded the break up and disarming of the State’s reserve formations and the dispatch of their members to the Front. Meanwhile the CNT leadership confirmed its commitment to militias in principle, but tried to get its fighters to comply with the government decision.
The Republican authorities began to ratchet up the pressure on self-management in industry and in the rural economy. The government of Largo Caballero ordered the nationalization of the war industry, placing it under control of the State bureaucracy. She The anarcho-syndicalist Fabregas, becoming minister of the economy in the Generalitat, on October 2 appealed to the workers to refrain from further expropriations of enterprises; his appeal was not heeded, at least in the beginning. However on October 24 in Catalonia a decree was approved which, on the one hand, legalized industrial collectivizations but, on the other hand, exempted small businesses with hired labour and a portion of medium sized businesses. The decree introduced the position of director (elected by the workers’ committee, it’s true) as well as State control over self-managed enterprises, especially in large-scale industry. Here a compromise with the State had already been effected through the direct participation of the leadership of the CNT, which was pursuing a policy of “legalizing the Revolution.” As far as the rural economy was concerned, a decree of October 7 1936, signed by the communist Uribe, minister of agriculture in the Largo Caballero government, recognized as legal only the confiscation of land belonging to estate owners who were considered mutineers. Thus many agrarian collectives which had seized large estates now found themselves outside the law.
In October 1936 H. Prieto, the General Secretary of the CNT, carried on negotiations about the entry of the union federation into the Republican government. He demanded six positions for the CNT, but Largo Caballero would agree to allocate only four to the anarcho-syndicalists. As a precursor to the agreement, on October 25 1936 a pact was signed about unity of action between the Catalan regional organizations of the CNT and the UGT, and also between the FAI and the pro-Soviet Unified Socialist Party of Catalonia (PSUC). This pact stipulated that the collectivization of the economy must be directed and coordinated by the Generalitat. It also specified the municipalization of housing, the introduction of a unified military command, compulsory mobilization into the militias (with the intention of transforming them subsequently into a “people’s army”), the introduction of workers’ control, the nationalization of banks, and the establishment of State control over banking operations. There was special emphasis on the necessity of struggle with “undisciplined groups,” i.e. with independent initiatives from below.
In order to put pressure on the government of Largo Caballero, the leaders of the CNT had recourse to threats.
On October 23 1936 a plenum of the regional CNT federations of Central Spain, Valencia, Aragon, Catalonia, and Andalusia discussed the National Committee’s report about confronting the government “concerning our participation in the leadership of the struggle against fascism and in the structure of the political-economic life of the Revolution.”
The resolution adopted reflected the inconsistency and vacillation of the anarcho-syndicalist activists: for them it was not a matter about the “cost” of taking power (as it was, probably, for H. Prieto himself and a number of the other “leaders”), but rather was about an attempt to alter the correlation of forces in their favour. The resolution represented essentially an ultimatum to the government of the Republic.
The plenum decided to create a commission of representatives of the regional organizations of Valencia, Central Spain, and Catalonia to engage in talks with President M. \_\_\_\_\_, “in order to explain to the crisis-ridden government the necessity \dots{} of having the CNT join it \dots{} under the conditions approved by the plenum of regional organizations of September 15.” The commission was instructed to wait up to 48 hours for an answer. In the case of a negative response, the CNT threatened to undertake “measures of a military character, in order to secure communication between Madrid, Valencia, Aragon, Andalusia, and Catalonia and to control the passage of people and supplies from these regions to Madrid.” To carry out this decision the National Committee was to appoint a National War Council to unify the fronts in Catalonia, Aragon, the Levant, and Andalusia. The CNT, together with the regional committees, proposed to mobilize 100,000 of its members for this Council. The confederation intended “to organize together with all our regional forces an action which would allow us to obtain control over the economy and the coordination of reserves.” At the same time, it was decided “to consult with diplomatic representatives of Russia, in the event this is necessary to achieve the carrying out of the decisions adopted at this plenum.”
The threats of the CNT were a bluff, as Largo Caballero understood perfectly, not to mention the USSR which was supporting his plans. As Abad de Santillan later acknowledged, in an article published in the newspaper Tierra y Libertad, at this time he was already convinced of the necessity of a “disciplined army” for the struggle with fascism and a “transitional State.”
In the final account, an agreement was reached according to which the CNT received four positions in the government with the proviso that it could appoint its own candidates. Their selection was made behind closed doors by H. Prieto himself, without even informing the National Committee. Juan López and Juan Peiró, representatives of the moderate wing of the CNT, were simply told over the phone by Prieto that they were appointed ministers of trade and industry, respectively. The FAI members Montseny and García Oliver had to be persuaded, and for this purpose Prieto travelled to Barcelona. Montseny at first refused to take up a ministerial post, however Prieto and the secretary of the Catalan regional organization of the CNT, Mariano Vasquez, insisted.
Then she asked for 24 hours to think it over and sought the advice of her father – the old anarchist Federico Urales. He told her that this meant “the liquidation of anarchism and the CNT,” but that if the organization demanded it, then, taking account of the circumstances, it was necessary to agree.
When the discussion with Prieto was taken up again, the General Secretary reminded her about her responsibility to the organization, and Montseny gave her consent although, in her own words, it was painful for her to take this step which represented “a break with the whole course of her life.” García Oliver also did not immediately agree to join the government. Up to now he had been considered one of the radicals. He was more swayed by tactical considerations: he did not wish to leave Barcelona where he was playing a key role in organizing the war effort. But in the end he gave in and agreed, although he insisted on the responsibility of the National Committee of the CNT for his action. Although subsequently García Oliver maintained he had only obeyed the decision of his organization, in reality from this moment on he became a fervent partisan of collaboration with political parties and tendencies.
Returning to Madrid, Prieto settled the last details with Largo Caballero. On November 4 1936 rank-and-file members of the CNT and FAI were amazed to learn from the newspapers of the appearance in the Largo Caballero government of four new members from their organizations: minister of justice García Oliver, minister of industry J. Peiró, minister of trade López Sánchez, and minister of public health Montseny. The CNT leadership assured the members of the organization that these ministers would be expressing not their own personal views, but the positions of their organization, the “collective will of the majority of the united toiling masses, previously formulated at general assemblies.”3
This line of argument was in stark contradiction to the antistatist ideals of anarchism, which always considered the State as an instrument of oppression and class rule. In an article it was maintained that “circumstances had altered the essence of the government and the Spanish State”: “The government in the current situation has ceased to be the main instrument of State rule, a force of oppression directed against the working class; just as the State is no longer an organ which divides society into classes. And both the government and the State, now that the CNT has entered into them, are still farther from oppressing the people.” That last thought was entirely compatible with the thesis of supporters of state socialism according to which that it was “merely” necessary to place the State at the service “of the people as a whole” by staffing it with the representatives of the people themselves. “The CNT’s entry into the central government,” announced the article, “is one of the most important events in the political history of our country.” Now “the functions of the State, with the concurrence of workers’ organizations, will be restricted to directing the course of the economic and social life of the country. And the government will only have the task of conducting the war properly and coordinating revolutionary work according to a common plan.” In a manifesto of the CNT National Committee, it was explained that consent to join the government was given in view of “the delicate situation of our military fronts.” The confederation was striving for “the triumph of the Iberian proletarian revolution,” “has never renounced and will never renounce its own tenets,” and remained apolitical; but in view of the serious situation was compelled “to demand a position of responsibility in the government.” The same tone was maintained in a manifesto of the CNT organization of the Central region: “The CNT in no way is renouncing its own program and its own principles. It agreed to enter the government only and exclusively in order to win the war.”
On the day the CNT joined the government, Durruti made an address on the radio. Its text has not been preserved and the versions published in the press, according to the testimony of some witnesses, were subjected to heavy censorship and distorted. Marcos Alcon recalled that Durruti “made them [the responsible figures of the CNT and FAI] tremble with fear, declaring to them in an extraordinarily harsh way that they had not succeeded in stifling the Revolution under the pretext of their insipid antifascism\dots{}” . This was the last speech by the leader of the anarchist radicals. Madrid was on the point of being captured by fascist troops, and the Republican government abandoned the city in a panic on November 6. Giving in to numerous entreaties, Durruti’s column went to the aid of besieged Madrid and, in stubborn battles, helped to save it from falling. However Durruti himself was killed on November 19 1936 under mysterious circumstances. The opponents of concessions and governmental collaboration lost their most outstanding, iconic, and popular with the anarcho-syndicalist masses figure.
\begin{enumerate}[1.]
\item\relax
Details of these negotiations about the purchase of weapons are recounted in the report of the General Secretary to the IWA Congress of 1937, which is preserved in the archives of the International in the International Institute of Social History. See: IISG: IWMA Archive: Nr. 21, Extraordinary Congress, Paris, 1937, Rapport moral par P. Besnard, membre du Secretariat.
\item\relax
The Council of Defense for Aragon received official recognition by the central authorities at the end of December 1936 after the anarchists agreed to include representatives of other tendencies in its make-up.
\item\relax
V. Richards, op. cit., p. 69 (n219). It must be acknowledged that the members of the government from the CNT – FAI were able to carry out a number of transformations. Thus, on the initiative of F. Montseny, a free medical service was introduced throughout the whole Republican zone, new medical clinics were built, abortions were legalized, etc. Garcia Oliver achieved the legalization of “free” marriages, softened the regimen for prisons and concentration camps, etc. (For details, see: A. V. Shubin, Анархо-синдакалисты в испанской гражданской войне 1936-1939 гг. [Anarcho-syndicalists in the Spanish civil war 1936-1939], (Moscow, 1997), pp. 17-18. Nevertheless, these measures had no connection with the anarcho-syndicalists’ own program and did not correspond to their “identity.”
\end{enumerate}
\section{Chapter 13: The CNT in Government - Results and Lessons}
The representatives of the CNT remained in the government until May 1937. The result of this “passage into power” turned out to be catastrophic for Spanish anarchosyndicalism. Its ministers were able neither to bring about an improvement in the military situation, nor stop the assault on the revolutionary conquests. Montseny publicly acknowledged the failure of participation in the government, and López stressed the impossibility of any kind of achievement in a situation where the other economic posts were in the hands of communists and right-wing socialists. The syndicalists were not able to obtain labour union control over “the monopoly of foreign trade” nor the adoption of their proposed drafts of decrees about collectivization in industry and financial assistance to collectives. A government decree of February 22 1937 envisaged the possibility of State control and ownership in industry.
Moreover, the activities of the “comrade-ministers,” as the CNT-FAI members of the government were known in libertarian circles, not only represented a break with the fundamental principles and traditions of the movement, but also caused trouble for the anarchists. Thus, the judicial reforms of García Oliver included not only the awarding of equal rights to women and the abrogation of punishment for crimes committed before July 19 1936, but also eliminated such “libertarian” projects as the organization of “labour camps” for criminals. Some of the decrees he came up with (for example, prison terms of up to 20 years for hiding weapons or explosives) were used against the anarchists themselves in Barcelona after May 1937.
Under the cover of “sharing responsibilities” with the CNT and FAI, the Spanish and Catalan republican authorities were able, during the period when the labour federations were represented in the government, to proceed to carry out counterrevolutionary measures such as liquidation of the popular militias and their complete replacement by the regular army (January 29 1937) – which, as the subsequent course of the war proved, was much less battle-worthy; the dissolution of revolutionary committees and local councils through the whole country, replacing them with appointed organs (January 4 1937);1 and the elimination of workers’ detachments for the maintenance of order in Catalonia (in favour of “disciplined patrols”) (March 1937). The basic problem for the authorities in this period was the disarming of the workers. Efforts to relieve anarcho-syndicalist workers’ organizations of frontier control in April 1937 led to fierce fighting in the Catalan border zone with France. Attacks by communists, right-wing socialists, and republicans on collectivization in the economy became more frequent; violent conflicts erupted between the Spanish Ministry of Agriculture and the workers’ collectives of the orange tree plantations of Valencia, created by the CNT and UGT; between the Catalan Ministry of Food Rationing and the Barcelona union of the CNT which was trying to socialize distribution; etc.
Finally in May 1937 a crisis, provoked by a police attack on the Barcelona telephone exchange (under workers’ control), set off a mass uprising of the city’s anarcho-syndicalist workers: the basic units of self-organization of the workers, just as in July 1936, were the block committees of defense. The anarcho-syndicalist masses succeeded in taking control of a large part of the city and the real possibility arose that the social revolution could become more profound. However the leadership of the CNT and FAI, fearing the collapse of “antifascist unity,” convinced the workers to abandon the barricades. After this the “republican counterrevolution” went on the counterattack: Largo Caballero – the supporter of compromise – was dismissed from the post of Premier, the representatives of the CNT and FAI were removed from their posts in the central and Catalan governments, the Council of Defense of Aragon was dissolved by a government decree in August 1937, and republican troops under the command of a member of the Communist Party, Enrique Lister, destroyed a large part of the rural communes of the region. In the course of the second half of 1937-1938, the government of Juan Negrín approved a number of decrees which dissolved unregistered agrarian collectives, placed the remaining ones under State control, and also (under the pretext of wartime necessity) gradually reduced the sphere of workers’ self-management in industry – to the point where a large part of industry was either nationalized or militarized. Thousands of anarcho-syndicalists were arrested as “undisciplined elements.” The leaders of the CNT and FAI offered virtually no resistance to this assault on the workers’ movement, continuing to proclaim the necessity of “first of all, winning the war with fascism.” But discord was growing in the leadership of these organizations. By and large, while the majority of the leading figures of the Peninsular Committee of the FAI continued to affirm they had not retreated one step from traditional anarcho-syndicalist ideas and would revert to their implementation after the victorious end of the war, at the same time people around the National Committee of the CNT, starting with the general secretary Vasquez and the éminence gris H. Prieto, increased their efforts to review a number of fundamental conceptions of anarcho-syndicalism from the social-democratic perspective of “workers’ democracy” with a “mixed economy.” They favoured the transformation of the FAI into a political party, controlling the CNT. In spite of internal disputes about the scale and extent of concessions to the political authorities, the leading circles of the movement until the end of the Civil War remained hostages to the notions of “antifascist unity” and “the lesser evil.” In April 1938 the CNT again occupied a second tier government post – the Ministry of Education and Public Health.
The whole tactic of “postponing” or “restraining” the social revolution for the sake of victory in the Civil War between the bourgeois-republican and fascist camps turned out to be unfavourable even for the outcome of the war itself.
Events showed it was impossible to win by fighting a normal or even “antifascist” war, by means of a regular army and a militarized State, following all the rules of military expertise.
Only the Spanish workers could defeat Francoism, workers who were full of hope in July 1936 and had, as Durruti said, “a new world in our hearts” while defending their revolutionary conquests. “We knew,” acknowledged D. Abad de Santillan after the defeat, “that our cause could not triumph without winning the war. We sacrificed the Revolution, not understanding that this sacrifice entailed renouncing the real goals of the war.” With nothing to fight for, the masses had already lost their revolutionary enthusiasm. It’s no accident that by the beginning of 1939 desertion from the republican army had reach massive proportions, and there were even cases of fraternization between soldiers of the republican and Francoist troops.
\begin{enumerate}[1.]
\item\relax
In connection with the re-constitution of local organs of power in Aragon, the agrarian collectives of the region passed a resolution at their conference in February 1937 that these organs must not interfere in the economy of the Federation of collectives.
\end{enumerate}
\section{Chapter 14: Notwithstanding “Circumstances”}
Many researchers who belong to the Marxist tendency have attempted to lay all the blame for the defeat of the Spanish Revolution on the anarcho-syndicalist movement, maintaining that the governmental collaboration of leaders of the CNT and FAI was a consequence of anarchist ideology which rejects the taking of political power by the workers.1 However such a viewpoint is untenable. First of all, anarchist conceptions not only repudiate the creation of new, “proletarian” political authorities, but also envisage the liquidation of the old – a process which the leaders of the CNT acted to prevent. In any event, they acted in the way they did not “because of ” libertarian theory, but in spite of it.
Besides, it is incorrect to assert that the anarcho-syndicalist masses of Spain refused to carry out the social revolution only because their “leaders” called on them to put an end to the revolutionary process. The facts show that the hundreds of thousands of rank-and-file members of the CNT and FAI, who played an outstanding role in the organization of workers’ and peasants’ self-management, “did not consider themselves constrained by political maneuvering,” but took action independently at the level of the enterprise, the syndicate, or the commune without waiting for any orders or appeals. Namely, this autonomous creativity “from below” did not depend on the “leaders” and often took place in spite of them, thereby proving the power of the anarchist “ideé-force.”
Although anarcho-syndicalism was, first and foremost, an urban rather than a rural movement, the Revolution in the Spanish village on the whole went further than in the cities, where government pressure and concessions were more effective. Here the associations which were created (collectives) embraced not only the realm of production, but other spheres of life as well. Members of the collectives voluntarily combined their own land with the land seized from the estate owners and often pooled their own financial resources. Each family preserved a small garden exclusively for their own needs. The rights of those who wished to continue to work their land on an individual basis were usually respected, so long as they promised to do so only with their own efforts, without using hired labour. It’s difficult to exclude the possibility of moral pressure on “individuals” by fellow-villagers, but cases of direct physical compulsion were virtually unknown in the saga of Spanish “collectivization.”
The collectives often included all the inhabitants of a village or at least the overwhelming majority of them. In many collectives “family allowances” were introduced.
Monetary wealth was expropriated by revolutionary committees and deposited in banks. Some places issued their own money or coupons. Committees took over distribution, and prices were established collectively and controlled. Collective warehouses and stores were organized; frequently they were accommodated in former churches.
Social transformations were uncoordinated and took various forms. Often this was connected with peculiarities of the structure of land ownership. If in Aragon 80\% of cultivated land belonged to large landowners, then in the Levant (Valencia region) and Catalonia small land-holdings predominated. And although there were a good many anarchists among these small owners, who also began to create collectives, there were greater obstacles in their path in the Levant and Catalonia. In these regions only the lands of the large estate owners were confiscated. But war-induced food shortages prompted the setting up of communal councils to take measures to limit private trade and to promote socialization.
This was followed by the creation of complete collectives, although these did not enjoy as much support from the local population as the Aragon collectives. Some of these collectives were very large and prosperous, but in the majority of them monetary relations were still retained.
In Aragon around 400 to 500 agricultural collectives were formed, in the regions of Valencia – 900, in Castile – 300, in Catalonia – 40, and in Estremadura – 30 collectives.
In Aragon at the end of 1936 and the beginning of 1937 between 300,000 and 400,000 people lived in agrarian collectives belonging to the Federation (until its destruction by republican troops, and then – also by the triumphant Francoists). Here to the maximum extent an anarchist social structure was put into effect – “without proprietors, casiques [local bosses], priests, and exploiters.”
The Aragonese collectives included up to 70\% of the population of the region; approximately 60\% of cultivated land was at their disposal. In February 1937 a congress of collectives in Caspe officially confirmed that persons who wished to farm individually, without joining a group and without using hired labour, had the right to do so, as long as they did not benefit from services provided by the collectives. Such individuals could only retain as much land as they could cultivate by their own efforts.
Handicraft workshops and other types of local industry in the Aragonese villages, as well as shops and institutions of education and culture, were also socialized. In these villages there were strong, ancient, communal traditions, and their preservation made it easier to bring people together in free territorial and economic communities, appropriate for an anarcho-communist society.
Inside the collectives there was an absence of any kind of hierarchy and all members possessed equal rights. The main decision-making body was the regular general meeting of the members, which convened usually once a month. For the on-going coordination of communal and economic life, committees were elected, often based on the former revolutionary committees. Their members – generally delegates from the various sections – did not enjoy any special privileges and did not receive any special reward for this work. All of them, except secretaries and bookkeepers, had to continue their normal work activity. Each adult member of the collective (with the exception of pregnant women) worked. Labour was organized on the basis of self-management. Brigades, composed of from five to ten people, made decisions about all basic work-related questions at meetings held every evening.
Delegates elected at these meetings also carried out functions of coordination and exchange of information with other brigades. In many collectives the principle of rotation of jobs was put into practice, and workers moved from one section to another according to the requirements of the moment. Industrial enterprises were included in the communal structure, which facilitated the integration of industry and the rural economy. Collectives were joined together through regional federations.
The circulation of currency was gradually liquidated. In the first weeks after their creation, many collectives abolished the remuneration of labour and introduced unlimited free consumption of all goods from the common stores. But under conditions of war and shortages, this turned out not to be an easy matter, especially since currency still circulated outside the collectives. In September 1936 the majority of communes converted to the so-called “family allowance” system. Each family in the collective received an equal sum of money (depending on the collective, this was approximately 7 – 10 pesetas for the head of the family, 50\% more for his wife, and 15\% more for each additional member of the family). These allowances were intended only for the purchase of food and objects of consumption and were not to be put into savings. In many communes coupons were introduced in place of the national currency. In others there were cards or tokens. Under war conditions, certain types of food products were rationed almost everywhere, while others (wine, butter, etc.) were available in virtually unlimited supply in many places. Until a final decision about abolishing money “in a third of the 510 villages and towns adopting collectivization in Aragon, money was abolished and goods were available free of charge from the collective’s store upon presentation of a consumer’s booklet,” and “in two thirds a replacement currency was put to use: bonds, coupons, tokens, etc. which were valid only within the confines of the communes issuing them.”
The first occasion in the activity of individual communes in which a certain parochial tendency displayed itself had to do with the initial inequality of collectives: some of them started off being more prosperous, others poorer. As confirmed by an eye-witness – the German syndicalist Souchy – in the beginning some collectives opposed the idea of economic planning under the slogan of “self-sufficiency.”
The complete independence of collectives from one another, and differences in the distribution systems of the communes, made it difficult to coordinate their economic activity. The anarchists – proponents of intensifying the social revolution – applied themselves to solving this problem, including Durruti, who personally campaigned for “collectives.” In February 1937 in the town of Caspe a congress of the Aragon collectives was held with the participation of hundreds of delegates. The participants agreed to step up propaganda on behalf of “collectivization,” to create experimental farms and technical schools, and to organize mutual aid between collectives so that machines and labour power could be shared. The boundaries between settlements were eliminated and limits on communal ownership were also abolished. The federated collectives decided to coordinate exchanges with the external world, creating for this purpose a common stock of products intended for exchange rather than the internal consumption of the communes, and also started the gathering of statistics about possible exchange products. Finally, it was proposed to completely do away with any form of money circulation inside the collectives and their federation and the introduction of a universal consumer booklet (normally upon the presentation of this booklet, items of consumption were given out free of charge). These booklets were to help to establish the real requirements of each of the inhabitants of the region, in order that production could be geared to the concrete needs of people, thereby moving to the anarchocommunist practice of “planning from below.”
The activity of the Aragonese collectives was very successful. Even according to official data, the harvest in the region in 1937 grew by 20\% at a time when there was a decrease in many other areas of the country. In Aragon roads, schools, hospitals, farms, and cultural institutions were built – in many settlements for the first time; the mechanization of labour was also applied. The inhabitants received access to medical services and free, anti-authoritarian education (physicians and teachers became full-fledged members of the collectives). Many collectives did not pay taxes. They preferred to support the Front directly and voluntarily.
Social transformations in the Spanish cities took place in a more uncoordinated fashion. On the one hand, the majority of industrial enterprises were occupied by the workers and passed under their control. On the other hand, the transition from the expropriation of enterprises by unions and collectives to full-scale socialization of industry did not take place, since commodity-money relations had not been done away with and money remained in the hands of the capitalists and the State. According to the eye-witness Gaston Leval, “very often workers in Barcelona and Valencia took over the factory, the workshop, the machines, and the raw materials and, taking advantage of the preservation of the monetary system and normal capitalist commercial relations, they organized production on their own account, selling the products of their labour for their own benefit.”
The pressure to compromise with the government did not allow the workers “to do more, and this distorted everything right from the start. This was \dots{} not real socialization, but a workers’ neo-capitalism, a self-management vacillating between capitalism and socialism, which would not have happened – it should be emphasized – if the Revolution could have been carried out to completion under the direction of our Syndicates.” “We did not organize the economic body which did the planning. We were satisfied to chase the owners out of the factories and set up committees for control. We did not undertake any attempt to establish links between ourselves or coordinate the economy in a practical way. We worked without any plan, really not knowing what we were doing,” admitted Abad de Santillan, who dealt with economic questions in the CNT.
The socialization of distribution was not implemented in the cities, which soon had repercussions. In Barcelona, after the formation of the Catalan Central Committee of Militias, a Central Committee for Food Supply was created which included representatives of various political forces. It organized the supply of provisions for the Front and for hospitals, opened stores, and maintained a network of “people’s cafeterias.” But the system of private commerce was retained and towards the end of the year in Barcelona there such phenomena appeared as a shortage of food items, a speculative rise in prices, and other abuses. Already in December 1936 one syndicate of workers of the distribution sector of the CNT called on the workers of stores and shops to fight against speculation, by keeping a close watch on the owners to make sure they were not selling goods “to the wrong customers,” and also by not allowing arbitrary increases in prices.
Workers continued to receive wages. In a number of cases it was possible to inaugurate the so-called “family allowance,” namely equal pay for each worker with supplements for the members of his family (for example, in Barcelona). But more often matters were limited to reducing gaps in the scale of wages and a significant increase in the rates for the lowest-paid categories.
Nevertheless, in a number of places and branches of industry, syndicalization moved beyond the level of individual enterprises and spread to whole sectors. So-called “groups” of enterprises began to operate in a coordinated way like a single enterprise (in this manner were organized, for example, all the branches of industry in Alcoy; the supply of gas, water, and electricity in Catalonia; the streetcars in Barcelona; in various places – transport and public health facilities).
The anarcho-syndicalist unions strived to continue and deepen the revolutionary transformations, in spite of the war situation and the concessions of the “leaders.” Thus, one syndicate of the woodworking sector stressed that anarchists from the very beginning could realize their own will: “to replace the regime which died on July 19 with another which is more humane and equal – libertarian communism.” In Barcelona and in Catalonia “this transformation has begun.”
However “other organizations exploited the enthusiasm of the members of the CNT and FAI” to divert the “popular trend” in the direction of new defeats. As a result, “instead of proceeding to genuine expropriation, which would have satisfied the widespread desires of the people, the owners were forced to pay wages on a weekly basis and the daily pay increased but the hourly pay decreased – and this at the height of the war!” In enterprises which had already been confiscated, a large number of “parasitical bureaucrats” and control committees made their appearance – which were not involved in production as such. Moreover the collectives which sprang up in industry found themselves in an unequal situation. They tended to resemble co-operatives, trying to compete at their own risk, which gave rise to “two classes: the new rich and ever-present poor.”
The anarcho-syndicalists hoped to wrest economic activity from under the control of the estate. They were convinced “the petty bourgeoisie, represented in the government and similar official bodies,” bureaucrats, functionaries, and “useless agents and middlemen” were incapable of ensuring the normal operation and development of the economy. The unions and their organizations had an obligation “to control the whole of production and manage it.” As, for example, one of the syndicates of the woodworking sector explained, the anarcho-syndicalists recognized the Generalitat’s decree about collectivization, but in practice tried to impute to it a different orientation. “We agreed with the collectivization of all branches of industry, but with a single financial centre, switching to an egalitarian distribution system. We did not agree that some collectives should be rich and others poor\dots{} .”
The syndicates and federations of the CNT actively discussed plans for socialization of the economy. The federation which included the unions of workers of water, gas, and electrical utilities worked out a plan for collectivizing the supply of electrical power. Representatives of the textile federations of the CNT and the UGT, holding a joint meeting, resolved “to go over to full collectivization of the textile sector in Catalonia” and approved a system of self- management for it. The participants of a local plenum of syndicates of the CNT in Barcelona declared the necessity of “implementing the socialization of branches of industry on a nation-wide scale.” They proposed a scheme of organizing self-management at all levels, including councils for factories, sections, and branches as well as an overall Economic Council. Each section of an industrial branch would have to make a complete and detailed study of the situation in its branch and provide the Economic Council with a plan for socialization with a precise data on current capacity and productivity, number of workers, raw materials on hand, markets for sales, and possibilities for economic development. On January 1 1937 a national congress of the transport industry discussed the question of nationalization or socialization of its sector.
In the Levant the regional federation of peasants and the united syndicate of workers in the fruit export business issued an appeal to the peasants growing oranges and other fruits, which constituted one of the basic sources of foreign currency. The existing state of affairs, in which each population centre or syndicate engaged independently in the export business and disposed of the monies earned, and which resulted in rivalries, was termed “unfortunate.”
The syndicates called for the creation of a “central organ” with a common reserve of products and a mutual aid fund, controlled by the peasants themselves. Subsequently the peasant federations of the Levant succeeded in unifying about one half of the production of oranges; up to 70\% of the harvest was routed through its trade organization to the European markets. [281]
In February 1937 a congress of the Catalan CNT approved a plan for re-structuring the industrial syndicates, which would embrace and control the whole cycle of production – from the cultivation of crops or extraction of raw materials to the distribution of the finished products. In Catalonia an economic survey of local syndicates and associations was organized. In this way information was gathered to serve for the creation of “revolutionary economics” with a system of “planning from below.” These statistics included, specifically, data about the geographical location and climate, traditions of the social-revolutionary movement, the economic situation and economic links of the locality, the housing situation, possibilities for the future, etc.
The gradual reversal of the Revolution from 1937 on did not allow plans for wide-scale socialization to be implemented. Under wartime conditions, the government was always more oriented to establishing State control over economic activity or even direct nationalization of industries, especially industries producing essential military goods. Correspondingly, the notion spread among some of the activists of forming a separate syndicalist managerial sector, run by the CNT, with autonomous structures of coordination and planning, to provide overall direction for the industrial federations and economic councils, with its own bank, etc. This concept was approved at the National Economic Congress of the CNT in Valencia in January 1938. In spite of all its suspended and incomplete projects, the significance of the social transformations brought about by the anarcho-syndicalist workers of Spain can scarcely be overestimated. These transformations have no equal in history on such a scale. Anarcho-syndicalism put into practice much of what had been “envisaged at all its congresses: workers’ control of factory and field, the planned development of production, equality in economic relations and in the possibility of adopting constructive decisions\dots{} All this took place outside the framework of the Republican government\dots{}”
In Aragon especially the possibility of implementing libertarian communism was demonstrated in principle.
The retreat of the leaders of the CNT and the FAI from the idea of “total revolution” and their concessions to the governments and parties of the Popular Front provoked bitter resistance and direct insubordination among the rank-and-file anarcho-syndicalists. Information about such happenings are fragmentary, and systematic investigations of organized opposition in the CNT, FAI, and Federation of Libertarian Youth do not exist up to this time. Therefore it is very difficult to gauge the real scale of opposition. Briefly summarizing the scattered information available, it is possible to distinguish three basic forms of such resistance. In the first place, this was resistance on the part of the lowest level unions of the CNT to the politics of nationalization (statification) of economic and social life, and a defense of gains in the area of workers’ self-management. Clashes between the republican authorities, on the one hand, and the unions and “collectives” on the other, were constantly flaring up. At the beginning of 1937 the Minister of Agriculture of Catalonia opposed plans for the socialization of distribution as proposed by the CNT syndicates in Barcelona. A sharp crisis was provoked by the efforts of the government to take over control of the economic activity of the workerp easant collectives of the orange plantations of the Valencia region. The Minister of Commerce Juan López, a member of the CNT, in support of the Minister of Agriculture – the communist Uribe, issued a decree at the beginning of 1937 about government control over the exports of agricultural collectives. However, a number of Valencian co-operatives refused to recognize his decree. The government sent military-police units with artillery and tanks against the strategic villages of Tulluera and Alfara, but the peasants, armed with hunting rifles and two old cannons, offered stubborn resistance. They were supported by the inhabitants of the neighbouring districts of Jativa, Carcagente, Gandia, and Sueca, forming the “Gandia Front.” The peasants of the villages of Catarroja, Liria, Moncada. Paterna, and Burriana formed the “Vilanesa Front.” To the aid of the collectives rush two battalions of the libertarian “Iron Column” and two battalions of the CNT columns, vacating the Teruel – Segorbe sector of the Front. Fighting in the region of Cullera continued for four days, after which the government forces attempted a flanking manoeuvre. After the intervention of the CNT an agreement was reached for a cease-fire and the mutual release of prisoners. The collectives of the Levant retained control over the production and export of oranges. Information exists about the strike launched by the union of workers in the entertainment industry of Barcelona early in 1938 (despite pressure from the leadership of the CNTFAI), in opposition to the introduction of State control of their sector. In the same category it is possible to include the protests of soldiers of the anarchist militias against their militarization and absorption into the regular army. As a result of the resultant crisis, the Catalan Regional Committee of the CNT was compelled to consent to allowing soldiers unwilling to submit to army orders to quit the Front. In the second place, a whole series of anarcho-syndicalist publications appeared which openly and quite severely criticized the “collaborationist” and “concessionist” course of the CNT and FAI committees. These publications denounced the winding down of the Revolution on the pretext of “antifascist unity” and collaboration with the government. The most important of these was the newspaper Ideas, which started coming out on December 29 1936. It was published by the local organizations of the CNT and FAI of Bajo Llobregat, and its editor was Liberto Calejas, formerly director of the Catalan CNT’s organ Solidaridad Obrera, but forced to vacate this post because of disagreements with the progovernment policies of the leadership of the CNT and FAI. Ideas became the centre of attraction of the whole revolutionary opposition inside the anarcho-syndicalist movement.
Among the writers who contributed blistering critiques were such well known anarchists as José Alberola, Felipe Alaiz, José Peirats, Severino Campos, Floreal Ocaña, Francisco Carreño, Jaime Balius, etc. Among the other oppositional anarchist publications it is possible to name Acracia in Lerida (editor – Peirats), Ciudad i Campo in Tortosa, Nosotros in Valencia; and also the organs of the Catalan Libertarian Youth (FIJL) – Ruta and Esfuerzo; and the newspapers of the Friends of Durruti (La Noche, and after May 1937 – El Amigo del Pueblo). All these publications were read with interest by the rank-and-file activists of the anarcho-syndicalist movement and enjoyed their support.
Finally, there also existed opposition groups. Thus, in Valencia some sections of the FAI and Libertarian Youth were grouped around the publication Nosotros which took a strong position against participation in the government.
In the same place in December 1936 manifestos were frequently distributed signed by the Iconoclasta group. They contained harsh criticism of the persons representing the CNT in the government and other organs of the State. It is likely these manifestos received a favourable response from members of the CNT, since the National Committee of the CNT considered it necessary to react in a brusque manner, denouncing its “undisciplined and irresponsible” critics which “do not represent anyone.”
The most important of the regional federations of libertarian youth – the Libertarian Youth of Catalonia – openly took a position against participation in the government, turning away from anarchist ideas, giving in to “circumstances,” and the collaboration of the “leaders.” After taking an active part in the events of May 1937, Libertarian Youth passed over into open opposition, refusing to submit to the decisions of the leadership of the CNT and FAI and concluding an agreement with the youth organizations of the antifascist parties. In response the leaders of the anarcho-syndicalist movement threatened sanctions against the “undisciplined” organ of Libertarian Youth – the newspaper Ruta.
In the spring of 1937 a section of the anarcho-syndicalists, dissatisfied with policies of the committees of the CNT and FAI, along with former soldiers of the militias, created the “Friends of Durruti” group, which included as many as four or five thousand members. They condemned the refusal to proclaim libertarian communism, participation in the government, and collaboration with socialists, communists, and bourgeois republicans. The members of the group also criticized both “orthodox” and reformist notions of anarchism, and called for a further development of anarchist theory and tactics, which would be based on the following fundamental positions: “the free city” (commune), management of the economy by syndicates, creation of a revolutionary committee for the defense of the Revolution, and coordination of the activities of local committees of defense.
But the Friends of Durruti did not become a centre of attraction for other oppositional groups in the anarchist movement, which criticized them for having an inclination for authoritarian methods. These groups, active in the FAI and CNT (Ideas and The Incorrigibles from Baja Llobregat, Los Quijotes del Ideal in Barcelona, Acracia in Lerida, etc.), advocated a return to the traditional principles and ideals of anarcho-syndicalism, resisting plans to transform the organizations into a political party and attempts at unifying and centralizing the libertarian movement. Thus, at the end of 1937 the prominent anarchists Santana Calero, Severino Campos, and Peirats published a brochure on behalf of “the main oppositional current of the conscious part of the libertarian movement.” Accusing the “leaders” of betraying the “ideological principles of anarchism,” violating the “essence of anarchism” in the name of “the demands of circumstances,” and “poisoning the lungs and brain of the body of the CNT-FAI with their stinking abomination of a policy,” they called for deliverance from being “strangled by statification and centralization.”
Like the Friends of Durruti, the supporters of a return to orthodox anarcho-syndicalism did not envisage any field of action for themselves other than the mass libertarian organizations – the CNT and FAI. Working among rank-and-file activists, they tried to alter the official line of the movement by speaking out at plenums and conferences. At the national plenum of regional committees of the CNT, FAI, and FIJL in October 1938, the opponents of “collaboration” tried to give battle one last time to the policy of taking part in government. A delegate of the Catalan “Libertarian Youth” declared: “Trying to insinuate yourself inside the State in order to destroy it, is like sending your wife and sister to a brothel in order to liquidate prostitution,” and Xena, a representative of the Catalan FAI, stormed out of the meeting hall as a sign of protest against the stated possibility of participation of the Federation in politics. However the opposition did not succeed in getting the changes they sought. It remained fragmented and organizationally inchoate. As usual the activists were encumbered with their faith in “their own organization” and any sort of appeal to the masses outside of its framework seemed inconceivable. Moreover, in Spanish anarcho-syndicalism there was no experience of systematic, coordinated fractional struggle, which could have helped the oppositionists to remove the leadership of the CNT and FAI committees.
\begin{enumerate}[1.]
\item\relax
One of the first to make this assertion was the Trotskyist writer F. Morrow in 1938. See: F. Morrow, Revolution and Counter-Revolution in Spain (Atlanta, 1974).
\end{enumerate}
\section{Chapter 15: The Spanish Revolution and World Anarcho-syndicalism}
The international anarcho-syndicalist movement in 1936- 1939 was torn between all out practical solidarity with the Spanish Revolution and criticism of the policies of the leading activists of the CNT. Besnard, the General Secretary of the IWA from 1936, visited revolutionary Spain three times in the autumn of that year and ultimately found a deep departure from the principles of anarcho-syndicalism which he associated with the regression of the Revolution.
He sharply criticized the entry into the government, collaboration with political parties, militarization, the refusal to allow the syndicates to take control of the economy, the refusal to criticize the Stalinist USSR, and the refusal to work on establishing libertarian communism. But at the same time, as shown by the plenums of the International in 1936 and 1937 as well as the Extraordinary Congress of 1937, the IWA did not possess any real possibility of exerting influence on the line being pursued by the CNT. The Secretariat of the International itself was split: its members Helmut Rüdiger and Nemesio Galve differed with P. Besnard and defended the “forced” tactics of the CNT. The anarchist workers’ organizations of Argentina and Uruguay (the FORA and FORU) denounced the Spanish CNT in very strong terms, viewing its policies as the logical result of the errors of revolutionary syndicalism. The French CGT-SR also condemned the CNT. These organizations called on the Spanish comrades to review their decisions and tactics and confirm their adherence to the principles of the IWA.
The “Francophone Anarchist Federation” (FAF), in which the Russian emigrant-anarchist Volin played a prominent role, declared its solidarity with the oppositional tendencies of the Spanish anarchists and anarcho-syndicalists which were struggling against the participation of the CNT in the government and the collaborationist line of its leadership. The FAF addressed itself to “the genuine CNT-FAI,” to those Spanish anarchists who condemned “spinelessness” and “ideological betrayal,” and declared that it considered “as inevitable a split in the ranks of the CNT and FAI themselves, as well as in the entire international anarchist movement.”
Before the Extraordinary Congress of the IWA in 1937 there were even discussions about expelling the CNT from the International.
But the leadership of the CNT was able to paralyze the waves of critics by referring to the “extraordinary circumstances” in which the Spanish Revolution found itself, to the weakness of the anarcho-syndicalist movement in other countries, and the absence elsewhere of revolutionary outbreaks.
It succeeded in obtaining the removal of Besnard from the post of General Secretary of the IWA. Moreover, the CNT leadership demanded changes in the declaration of principles and statutes of the IWA so as to exclude “obsolete” points and add provisions concerning the armed defense of the Revolution and “sweeping autonomy” for the sections, which would allow them to pursue whatever tactical line they considered necessary. The anarcho-syndicalist groups of German emigrants, led by Rüdiger, went even further in this direction. They called for a fundamental revision of the ideas and tactics of anarcho-syndicalism, for a review of the declaration of principles in order to have it register the possibility of collaboration with other antifascist forces, as well as taking an anti-imperialist stance and expressing support for revolutionary wars. Rüdiger spoke in favour of “elastic” tactics and a “clearer conception” which would include the necessity of political activities, “revolutionary” government, collaboration with statist and party organs, the creation of a disciplined “revolutionary army” and apparatus of repression, as well as retention of the bourgeoisie and safeguarding private property. However there was also no unity in the ranks of the critics of the CNT. The Swedish SAC condemned participation in government, but defended the policy of “antifascist co-operation” and also proposed to include in IWA documents a policy about the tactical autonomy of the sections. The French CGT-SR and Besnard sharply denounced “participation in democratic Capitalism,” collaboration with the State, with parties, and with armies, and the rejection of basic principles of anarcho-syndicalism.
But these critics could not offer any clear alternatives and agreed to a certain “modification of tactics,” and the inclusion in the declaration of principles of clauses about the possibility of revolutionary and anti-colonial wars. From another perspective, the Argentine and Uruguayan FORA and FORU took a resolute stance against changing the principles and tactics of the IWA, which were grounded in the struggle with the State and direct action, as well as the rejection of politics and collaboration with political forces. They called for the re-affirmation of opposition to all wars, since wars were inevitably tied to the struggle for power between different groups of capitalists, and for opposing war with revolution.
Finally, the Latin American anarchists made a clear statement that they saw no distinction in principle between fascism and non-class-based antifascism, i.e. the defense of democracy, since either one were “enemies of proletarian liberation.”
This ideological and tactical confusion impeded the work of the IWA and allowed the leaders of the Spanish CNT to obtain approval of their course of action from the international organization. Although the Extraordinary Congress in December 1937 turned down the proposal of the Spanish delegation about holding a meeting of “the three Internationals” and the creation of a permanent committee of representatives of all “three socialist schools” (anarchists, party communists, and social-democrats) for the struggle with fascism and imperialism, the participants adopted a resolution introduced by the CGT-SR which gave the right to the CNT to continue the “experiment” it had started “under its own responsibility.” An appeal to the international association of social-democratic unions (the Amsterdam International) was drafted, with a proposal to organize a global boycott of ships and goods from fascist countries. However the leaders of this International rejected this overture.
Finally, at the 6th Congress in 1938, in the absence of Latin American delegates and representatives of the French CGT-SR, the delegates of Spain, Sweden, and Portugal succeeded, despite the opposition of the Dutch delegates, in revising the charter of the IWA. These alterations envisaged, among other things, the “broad tactical autonomy” of sections and control of the syndicates over workers’ militias during revolutionary periods. The actions of the CGT-SR were officially condemned. The opinions of the FORA and FORU, expressed in written form in the absence of their delegates, were generally not taken into account.
The victory of the leaders of the CNT over their critics in the international arena could change nothing in the general situation and did not help to strengthen their position inside Spain. The war was lost. Early in 1939 the whole territory of the Spanish republic was under the control of the troops of the rebel generals. The bloody regime of terror was firmly established in the country, the CNT was annihilated, and hundreds of thousands of people were forced to flee across the border. Individual armed groups of anarcho-syndicalists continued partisan struggle in Spain until the beginning of the 1960’s.
In emigration, the Spanish anarcho-syndicalist movement found within itself the strength to give a self-critical evaluation of its experience of “participation in government” during the Civil War and to draw the appropriate lessons.
The intercontinental conference of the “Spanish Libertarian Movement (CNT – FAI – Federation of Libertarian Youth), held in April 1947 in Toulouse, considered the “consequences of collaboration in government” “catastrophic” and announced the return to traditional anarchist concepts about the necessity of liquidating State power and its replacement by universal self-management by the workers.
\chapter{Part 4: Decline and Possible Regeneration}
\section{Chapter 16: Anarcho-Syndicalism during the Second World War}
Several months after the defeat in Spain, the Second World War broke out – completely paralyzing the activity of the IWA. The FORA, disturbed by the decisions of the 1938 congress, resolved to “temporarily cease to have relations with the IWA,” until the next congress re-examined these decisions.
The Argentine and Uruguayan anarchists continued to insist the functions of syndicates must cease as soon as revolution took place and, as a consequence, they rejected the notion of syndicalist control over working class militias.
They objected to cooperation with the State and political parties under the pretext of “tactical autonomy,” to the decisions of the 1938 congress about introducing proportional representation of sections at IWA congresses (instead of the previous equality), and to the creation of a special world federation of syndicalist youth.
As far as World War II was concerned, both FORA and FORU confirmed their previous anti-war and anti-militarist position: the war was taking place between different groups of States and capitalists which were fighting for their own rule and privileges. In no way did the war correspond to the interests and hopes of people struggling for freedom and justice. Antifascism, according to the anarchists of Latin America, serves only as a screen for the interests of Capital of one of the groups of warring States. Therefore they called upon workers not to support the war under the banner and pretext of antifascism. Instead they advanced the slogan: “Neither Fascism, nor Antifascism.” Appealing for intensified antiwar and antimilitarist activity, they announced: “The unique solution to the war, in fact to all wars – is the revolutionary union of peoples.”
In Europe itself during the Second World War the anarcho-syndicalists on the whole were too weak to exert themselves as an independent force. In France the CGT-SR, with 6,000 members at the end of the 1930’s, was dissolved, while the syndicalist and anarchist organizations of Poland, the Netherlands, Belgium, Norway, and Denmark were outlawed following the occupation of these countries by the Nazis. The IWA Secretariat was located in Sweden and was deprived of almost all contact with libertarians in the belligerent nations.
The majority of the libertarian organizations at the very beginning of the war took a position which they termed “internationalist,” by analogy with the traditional slogans of revolutionary leftists about the transformation of imperialist war into social revolution. A declaration of the IWA Secretariat pointed out that “the war is the result of the capitalist system,” an “expression of the cruel competition between groups of capitalists for raw materials, colonies, and markets,” and the “struggle of imperialist States to ensure their influence and control over the world and its riches in the interests of their own group of States.” The IWA perceived fascism as “the cruelest form of capitalism” and “Enemy No. 1 of humanity,” but also called upon workers not to trust thedemocracies, since “they are soft on reaction, soft on bloody wars,” and “cannot guarantee peace.” “\dots{} If humanity wants to live a free life and liberate itself from constant wars, it must get rid of Capitalism\dots{},” said the IWA in its declaration.
“The war between nations must be transformed into a war between classes. The international working class must act with all its energy to liquidate Capitalism.” Declarations in the same spirit were issued by anarchist and anarchosyndicalist organizations in France, Sweden, the Netherlands, and Belgium.1 But in reality a significant number of anarchists soon abandoned this position and began to orient themselves towards the struggle with Fascism as “the greatest evil.” Many German anarcho-syndicalists in emigration, using the Swedish syndicalists as a go-between, co-operated with the intelligence services of the Western powers. French anarchists participated in the Résistance. In Poland syndicalists and anarchists called for the “defense of the country” (although “not jointly with the bourgeoisie”), and created their own partisan detachments, which were then merged with the partisan detachments of the socialists in the “Polish People’s Army” and took an active part in the Warsaw Uprising of 1944. In Italy and Bulgaria the anarchists formed their own partisan detachments which engaged in battles with the armed forces of the Fascist regimes. While participating in the creation of underground territorial and workplace organs, the Italian anarchists at the same time tried to preserve their organizational independence from political parties and groups. They took part in the Resistance and assisted in preparing and conducting strikes which were directed not only against the fascists and the German authorities, but also against Italian entrepreneurs.
“Active operations were accompanied by ongoing efforts to work out the appropriate strategy for the current phase of events (the struggle against Nazism-Fascism) which could broaden the situation into a possible revolution,” noted one researcher. “The proposal for a ‘United Front of Working People’\dots{}, addressed to worker activists and rank-and-file members of left-wing parties, was\dots{} part of a project which regarded the original underground organs of the Resistance as elements of a counter-power in the spirit of anarchism and Workers’ Councils. The participation\dots{} of anarchists in Factory Committees must be viewed in this light, rather than as a concession to the democratic program of the liberation struggle as a second Risorgimento.”
We have knowledge about at least one attempt at organizing armed struggle undertaken by anarchists in Ukraine. A former participant in the Makhnovist movement, Osip Tsebry, returned to the country illegally in 1942 and organized a partisan detachment in the Kiev region. In the tradition of its predecessors, it acted against both Germany and the USSR, until it was defeated by German forces in 1943.
In Hungary small groups of anarchist student youth took part in partisan detachments and organized acts of sabotage in Budapest at the end of 1944. Anarchists and anarchosyndicalists of the Netherlands and Belgium put forward a position for a “Third Front,” that is, against both warring sides; they agitated for civil disobedience and the organization of a workers’ movement independent of political parties.
The Spanish anarchists after losing the war with the Francoists remained in a state of disunity, split between supporters of continued collaboration with antifascist forces and those who were favour of a return to traditional anarchist positions and against participation in any kind of coalition with antifascist or republican statist structures. The traditionalists considered the Second World War as a purely inter- Capitalist conflict and proposed that “in the case of open conflict between the French Resistance and the Germans, activists of the Confederation should seek shelter among the civilian population.” Those who advised continuing the alliance with the republican forces called upon Spanish anarchist-emigrants to join the French Resistance. The Spanish libertarians continued an underground struggle on the Iberian peninsula and tried to organize the assassinations of Franco and Hitler.
The French anarchists occupied an internationalist position. A particularly active role was played by a group in Marseille, gathered around Vsevolod Volin and André Arru. It distributed leaflets with an appeal to workers to act not only against German and Italian Fascism, but also against Soviet Stalinism and the democratic Capitalism of the West as well as against the slogan “national liberation,” seen as an attempt to unify the ruling and oppressed classes. The Marseilles group, agitating for social revolution and known under the name “International Revolutionary-syndicalist Federation,” became a centre of attraction for other anarchist groups throughout the whole country. The British anarchists also spoke out against the imperialist war which was being sold as a struggle between fascism and democracy.
They carried on active anti-war agitation, supported the strike movement, and tried to organize Soldiers’ Councils in the British Army.
\begin{enumerate}[1.]
\item\relax
Delo truda – Probyzhdeniye, 1940, no.1, Yanvar – Fevral, pp. 7-12. Characteristically, a “group of Belgium, Spanish, Italian, French, and German anarchists” expressed its disagreement with the fact that the IWA manifesto considered fascism to be “Enemy No. 1.” In their declaration they said: “The enemy today, like yesterday and even more so tomorrow, is our bosses. And our Enemy No. 1 is the State – the Government, its organs of suppression, the official and semi-official institutions which support it, the Army, the Bureaucracy, the Church – all the perpetual accomplices in the oppression of freedom and individuality.” (cited in: Service de presse. AIT., 1939, no.14).
\end{enumerate}
\section{Chapter 17: Anarcho-syndicalism After World War II}
Despite the hopes of the anarchists, World War II did not develop into social revolution; on the contrary, it led to the strengthening of national States and the establishment in Western Europe of a system of social partnership within the framework of “democratic corporatism” – collaboration between government, corporations, and trade unions. In Eastern Europe there were dictatorial regimes led by communist parties.
The East European governments suppressed all attempts to revive the libertarian movement. In Bulgaria in 1944 the Federation of Anarchist-Communists was re-established and in 1946 – a National Confederation of Labour. By 1947 there were 11,000 anarchists in the country (including 1,000 anarcho-syndicalists). But soon the libertarian organizations were banned and broken up, and their leading activists arrested.
In East Germany hundreds of members of anarchist and libertarian-socialist groups were arrested in 1948-1949, and the leader of the movement Willi Jelinek was murdered in prison in March 1952. The Polish syndicalist organizations which sprang up during the war years ceased to function after 1944, and in Hungary the anarchists were completely crushed after the strike of the “Csepel” workers, which was partially under their influence.
The anarcho-syndicalists of Spain and Portugal continued to struggle in the deep underground. The CNT tried to re-establish illegal syndicates while some activists preferred armed struggle with the Franco regime. Heavy repressions prevented the organization from rebuilding and it was set back again and again. The situation was complicated by a split in the CNT after 1946: one part of the organization rejected the mistakes committed during the period of revolution, while the other part insisted on a united front with other anti-Francoist forces; as a result the organization lapsed into a deep crisis. Unity was re-established only in 1960. Under these conditions the main burden of work was placed on the Spanish anarchist emigration in France where in the 1940’s there were no fewer than 30,000 members of the Confederation, issuing various newspapers and journals.
Under the conditions of the Salazar dictatorship the activity of the Portuguese CGT gradually died down; the activity of underground syndicates and issuing of illegal publications came to an end in the 1960’s.
The anarchists and anarcho-syndicalists of South America found themselves under heavy pressure from State power.
In 1946 the Argentine FORA could still come up with 3,000 people for a May 1 demonstration. It offered stubborn resistance to the regime of General J. Perón, organizing, despite restrictions and prohibitions, strikes of bakers and dockers in 1946-1948 and demonstrating against interference by the State in labour conflicts. However in the following years the shutting down of independent labour unions and libertarian publications struck the movement with new blows. The influx of new members into the organization almost stopped, and contact with the new generation of social activists did not come about. The veterans faded away but there was no one to replace them. In neighbouring Uruguay the FORU shrank to small groups. At the beginning of the 1950’s the Chilean CGT and the Local Labour Federation of La Paz in Bolivia ceased to exist: they were forced to join unified national labour union centrals.
In the majority of countries of Western Europe anarchosyndicalists after the war had the possibility of legal activity.
But the revival of the anarchist and anarcho-syndicalist movements on a massive scale did not occur. Only in France for a brief moment did things take off: the National Confederation of Labour (CNT) united several tens of thousands of workers (mainly in Paris, Bordeaux, Marseilles, and Toulouse). But the organization soon began to experience great material difficulties and a dearth of staunch activists. The majority of workers who joined it soon left for other, more moderate labour unions, and the French anarchists regarded anarchosyndicalism as a factor which was splitting the workers’ movement. Soon the French CNT shrank to the scale of small labour union initiatives. The anarchist movement of Italy also took a position for trade union unity and against a special anarcho-syndicalist union movement. The re-organization of the formerly powerful USI was announced only in 1950, but it remained an insignificant organization. The ranks of the Swedish SAC remained relatively numerous, but the numbers also fell from 22,000 in 1945 to 16,000 in 1957.
“The most profound explanation of the disappearance of syndicalism as a mass movement must take into consideration not only transitory factors, such as government repression, but also changes in capitalist society,” justly noted the historians M. Van der Linden and W. Thorpe. First of all, one should note carefully R. Rocker’s warning about the negative influence on working class radicalism of the rationalization of capitalist production. Actually, as researchers have noted, beginning from the 1920’s and really taking off after the Second World War, the automation of production processes, the symbol of which was the introduction of the conveyor belt, favoured the extreme specialization and division of labour into partial operations. The new social type of “mass specialized worker” had no sense of production as a whole and therefore did not press demands to take full control over it. The axis of social contradiction was displaced from the sphere of production with its problems of the content of labour and the independence of the producer to the sphere of distribution of the produced surplus product and consumption. This corresponded to a decline in the radical workers’ movement, which had arisen as an alternative to the industrial-Capitalist system and was oriented to the struggle for control by the workers over production.
Parallel to these developments was the growing tendency towards State interference in the economic and social sphere, which after the Second World War led to the formation of a model of the “Social State” or “Welfare State.” The Keynsian policy of stimulating purchasing power led to an increase in prosperity of workers in the developed Capitalist countries and gave the workers a vested interest in the functioning of the system as a whole and expectations of satisfying their growing consumer needs within the framework of a “social partnership” model.1 The new realities, as researchers have noted, confronted the syndicalist organizations with “only three possibilities, each of which would have disastrous consequences for them.
The movement could: (1) continue to maintain its own principles – in which case it would be subject to inevitable marginalization; (2) completely change course to accommodate themselves to the new conditions – in which case they would have to renounce syndicalist principles; (3) if the first two possibilities were rejected, either dissolve themselves or, what amounts to the same thing, join a non-syndicalist labour union.”
The IWA went the first way, waiting for the moment when conditions for the anarcho-syndicalist movement would become more favourable again, and its ideas would again find resonance in society. Taking up its work anew after the Second World War, it provided a home for Spanish revolutionary-emigrants, small labour unions, and action groups in a number of European and Latin American countries. After the Spanish CNT in exile adopted a decision about a return to the anarchist principles of rejection of collaboration with statist political forces and an orientation to social revolution, it proposed at the 7th Congress of the IWA (1951) to repeal the amendment about “tactical autonomy” introduced in 1938. After a long and animated discussion, accompanied by a split in the International, such a resolution was finally adopted at the 9th Congress (1956). This allowed the FORA to return to the international organization. Delegates at the next , 10th Congress (1958), acting on a motion by the Argentinans, announced that “only those groups can belong to the IWA which recognize as their goal libertarian (anarchist) communism and federalism.” In connection with these ideological discussions, the Swedish SAC and the Dutch Syndicalists left the IWA in 1958.
SAC continued to consider itself a “libertarian-syndicalist” labour union, but in practice it followed the second path – a revision of anarchist principles under the guise of “modernization.”
A strong influence on the ideological views of the “revisionists” was exerted by the German emigrant-syndicalist Rüdiger, who had settled in Sweden at the end of the 1930’s.
Already during the period of the Spanish Revolution he had called for a revision of a number of traditional tenets of anarcho-syndicalism, in essence proposing to renounce the struggle for the establishment of an anarcho-communist society, acknowledge the notion of a “transition period,” etc.
Now Rüdiger proposed to repudiate anarchist “orthodoxy” and instead of liquidating the State, try to reform it. “\dots{} As a result of changes undergone by the State since the time of Proudhon, Bakunin, and Kropotkin, and also Marx and Landauer, one can assert that \emph{the destruction of the State would not only mean the destruction of the apparatus of oppression, but also of a whole complex of social functions which are vitally important. It is impossible to arouse the people for such an action. Under the conditions of social relations today, more than previously we are faced with the question about transforming social functions which are today being carried out by the State into genuinely social functions} . In this struggle one often has recourse to the path of reform.” Rüdiger declared that it followed that one should not wait for “social revolution,” but “should act now inside the existing State and economic structure for the renewal of the (democratic) system of representation,” joining for this purpose in alliances with other political forces and tendencies and even allowing for thepossibility of participation in local elections.
As practical way of getting involved in carrying out functions of the Welfare State and simultaneously increasing the popularity of their labour union central, the “revisionists” in SAC advocated participating in the administration of unemployment insurance funds. Such funds were financed by enterprises and the State, but also by contributions from trade union members. The operation of the fund bureaus was entrusted to the unions. Syndicalists had traditionally fought against State interference in labour questions and refused to participate in organs of social partnership which were subsided by the State. But now the “revisionist” wing of SAC sought to have the union central join in carrying out reforms of the social insurance system.
In the course of an internally organized referendum in 1952, the members of SAC voted to approve a change in their statement of principles and create an unemployment insurance fund run by the syndicalist union central. According to a 1952 declaration, the goal of the syndicalists was stated to be the implementation of “industrial democracy.” Radical means of direct action (such as violent opposition and sabotage of production) were perceived as senseless. SAC proposed to hand over the administration of enterprises to worker collectives and expressed its intention to undertake efforts to “introduce workers’ control in private, municipal, and State enterprises.” As Evert Arvidsson, editor of the trade union central’s press organ Arbetaren, explained, “We have completely renounced the ‘magic wand’ of revolution.” The Swedish syndicalists now considered partial reforms to be “the practical means of influencing development in the desired direction\dots{} . SAC regards the progressive democratization of the economy as its primary task\dots{} . The basic idea consists in gradually transferring economic power from the shareholders to the producers.” In this connection, SAC endorsed the participation of worker representatives in the management of private enterprises. At the same time, Swedish syndicalism renounced the role of alternative to the industrial-capitalist system and occupied a position on the left, oppositionist flank of the Welfare State system.
The creation by SAC of unemployment insurance funds as an element of the “Swedish model of the Welfare State” encouraged the involvement of workers in the syndicalist ranks for a time and slowed the decline of Swedish syndicalism. But on the other hand, the re-orientation of SAC led to a breakdown of relations between the trade union central and the international anarcho-syndicalist movement, which subjected the Swedish syndicalists to harsh criticism for their reformism and collaboration with the State.2
The influence of the anarcho-syndicalist International reached its lowest point in the 1960’s. During this period anarcho-syndicalists were compelled to occupy themselves mainly with theoretical work: the analysis of contemporary social development, the evolution of Capitalism and the State, and the situations in the countries of so-called “actually existing socialism” (which the IWA identified as State Capitalism) and in the developing countries; an assessment of the potential of the co-operative movement, and proposals about the agrarian question and about counteraction to the threat of war. After the global wave of student and worker protests in 1968-69 and the liquidation of the Spanish Francoist regime (1975-77), it was possible to observe a growth in the interest in anarcho-syndicalism in Europe and North America. There was a rebirth of the CNT in Spain and structures of the Italian Syndicalist Union (USI). Anarchosyndicalist groups revived in a number of other countries.
The IWA was busy in these years with an analysis of global problems and new social movements, trying to evaluate them from a social-revolutionary point of view. In the 1980’s the processes of globalization of the economy, transition to neoliberalism, and dismantling of the model of the “Welfare State” throughout the whole world was accompanied by a crisis of the statist left-wing (social-democratic and communist) parties and the trade unions under their influence.
The collapse of communist party regimes in the USSR and East European countries took place, social-democratic parties adopted a number of the tenets of neoliberalism, and labour unions found themselves helpless to prevent real cutbacks in pay for many categories of workers, as well as reductions in social benefits and other gains made by wage workers over the previous several decades. There evolved a process of “precarization” – the introduction of an unstable, unprotected by legally enforceable labour relations, system of casual employment and worsening working conditions, as well as a model of “flexible” organization of working hours which were arranged according to the interests of the enterprise rather than its workers. Anarcho-syndicalists perceived these new developments at the end of the century as a sort of “challenge of the times” to which the “traditional left” was unable to respond. From their point of view, the breakup of the USSR, the collapse of communist party regimes, and the advent of the free market model with its “neoliberal totalitarianism” – all this indicated that “the notion of State control, which was the basis of the politics of both the revolutionary and the social-democratic left, had suffered defeat\dots{}
A fundamental re-thinking was necessary,” to a significant extent a return to the discussions between the libertarian and authoritarian socialists in the First International. “The core of any socialist re-examination must be an alternative to Capitalism\dots{} Capitalism cannot be reformed, it must be abolished. We must learn the most important lesson of the history of the 20th century: there is no State which can guarantee freedom to the workers, quite the opposite.”
In the 1990’s a revival of the world anarcho-syndicalist movement took place. New sections and groups of supporters of the IWA appeared, including ones in Russia, Eastern Europe, and America; after the start Argentine revolution of 2001 a rebirth of the FORA began. Sections in Spain, Italy, and France succeeded in becoming active, although small, labour unions. Now, rather than trying to absorb the whole workers’ movement, they are oriented towards the development and radicalization of self-managed and self-organized workers’ initiatives, independent of reformist unions and parties – initiatives in the course of which all decisions are made at general meetings (assemblies) of workers and methods of direct action are implemented.
At the end of the 20th century and beginning of the 21st century, anarcho-syndicalists of many countries took an active role in social and labour conflicts. The Spanish CNT, with a membership of 10,000, is the most noteworthy in this respect. The toughest strikes in Spain are associated with the CNT.
Thus, in 1985-1986 on the initiative of the members of the CNT, the movement by workers against the planned closing of shipyards in Puerto Real grew into a broad social protest which was accompanied by the occupation of enterprises by workers and mass demonstration by the inhabitants of the city. The leadership of the struggle was not concentrated in trade union committees and other representative organs. All basic decisions were adopted directly by workers at their general meetings. Characteristically, these assemblies of workers took place without the sanction of the bureaucrats of the official unions; the proposals of the CNT were always adopted, despite the attempts of other unions which failed to obtain the adoption of their own resolutions. In such a way it was established that every Thursday the workers would occupy the shipyards and hold general meetings in them.
During the strike general assemblies of the inhabitants of the towns and villages of the region were held on a weekly basis. Anyone who was interested in the goings-on, regardless of whether they worked in the shipyards, could come to these assemblies, vote, and participate in the process of adopting decisions on questions which interested them. At the general meetings decisions were adopted about concrete measures and forms of struggle, as well as about the carrying out of acts of sabotage and direct action.
Shock troops were hurled against the rebellious city. More then 1,000 police were drawn from all corners of the country to Puerto Real in an attempt to halt the revolt. In response, people began to put up barricades on the outskirts of the city, not wishing to allow access to the police. People threw rocks, furniture, any kind of junk from rooftops at police vehicles. They engaged in street battles with the cops. Frequently barricades were set up on the railway, the highway, and a strategic bridge, telephone poles were cut down, etc. The struggle of the workers and other city residents brought them victory.
The new activization of the anarcho-syndicalist strike movement in Spain carried over into the beginning of the 21st century. The CNT organized or supported such actions as the strike of garbage collectors in the Andalusian city of Tomares (it lasted 134 days), “indefinite-term” strikes of railway cleaners and crane operators in Seville, municipal workers in Adra, workers at the “Mercadona” department store near Barcelona (lasting 180 days), protest marches with many thousands of participants against the social-economic policies of the government, etc. The Italian syndicalists of the USI took part in a series of General Strikes, led by “alternative” labour unions (including some anti-militarism strikes)\dots{}
Despite the fact that in Spain, France, and Italy new splits took place with breakaway groups trying to achieve a mass base at the expense of jettisoning a number of anarchosyndicalist principles (rejection of political parties, nonparticipation in organs of social partnership in production, etc.) ,3 the IWA is striving to preserve its traditional role as an alternative to the industrial-capitalist system as a whole. Playing the role of “catalyst” for self-organization, the anarcho-syndicalists hope that as people stand up for their own rights and interests on a day-to-day basis, they will acquire the skills and structures of social self-management.
\begin{enumerate}[1.]
\item\relax
One of the first to analyze this phenomonen was the philosopher Herbert Marcuse, cf.: H. Marcuse, Одномерный человек [One-Dimensional Man] (Moscow, 1994), pp. 38-44.
\item\relax
The new course did not save SAC. The organization failed to find a common language with the 1960’s generation of “youth rebellion.” Changes in the structure of Swedish industry and the crisis in the “Swedish model” at the end of the 20th century inflicted more damage on syndicalism in Sweden. In 2002 only about 7,000 members remained in SAC.
\item\relax
Thus, the General Confederation of Labour (CGT) which united syndicates splitting from the Spanish CNT in 1984; and the French CNT with headquarters on “la rue des Vignoles” in Paris, which in 1995 separated from CNT-AIT France; along with other reformist labour union centrals take part in elections to committees – organs of “social partnership” – formed for the purpose of carrying on negotiations with business owners. Like other “official” unions, the CGT receives subsidies from the State and has full-time officials.
\end{enumerate}
\section{Chapter 18: Anarcho-syndicalism in contemporary Russia}
The panorama of world anarcho-syndicalism at the beginning of 21st century would be incomplete without a brief mention of analogous initiatives in contemporary Russia. The revival of the libertarian movement in the Soviet Union began in the era of perestroika at the end of the 1980’s. However the views of the first activists were often quite muddled, which can be explained to a large extent by the decades of isolation of self-educated oppositionists from the rest of the world. In 1989 the Confederation of Anarcho-Syndicalists (KAS) was formed, which for a short time united almost all the existing libertarian groups with the participation of several hundred activists. But, despite its name, Proudhonist views and notions of “stateless market socialism” predominated in KAS, quite far removed from the world anarcho-syndicalist tradition. Changes in the social-political situation, the break-up of the USSR, and the transition to market capitalism deepened the ideological and tactical contradictions in the organization, and in the beginning of the 1990’s KAS, in essence, disintegrated. Some of its individual members tried to put into practice a model of syndicalist labour unions within the framework of an independent regional union central – the Siberian Confederation of Labour.
The first libertarian group to return to the classical ideas of anarcho-communism and anarcho-syndicalism was the Moscow-based Initiative of Revolutionary Anarchists (IREAN), which sprang up in March 1991. In 1995 its activists, together with representatives of a number of other anarcho-communist groups, created the Confederation of Revolutionary Anarcho-syndicalists (KRAS), which at the 20th Congress of the IWA (1996) was accepted into the anarcho-syndicalist International as its Russian section. KRAS regarded itself as a labour union initiative (profinitsiativa), a transitional stage on the road to creating anarcho-syndicalist labour unions. Its development over the past few years has been an up-and-down process, usually in sync with the general dynamic of social movements and protests in Russia. At various times groups or members of KRAS-IWA have acted in Moscow, Baikalsk, Gomel, Yaroslavl, Rostov-on-Don, St. Petersburg, and other cities; in Moscow it created, besides intersectoral initiatives, also groups of workers in education, science, and techology. An important part of the activities of the Russian anarchosyndicalists continues to be agitational work in the form of holding meetings and publishing (the newspaper Прямое действие [Direct Action], the magazineЛибертарная мысль[Libertarian Thought], brochures, etc.). In Baikal members of KRAS were involved in founding the Industrial Labour Union which, in the middle of the 1990’s, organized a strike in a cellulose-paper complex which was smashed by government repression. Activists of KRAS rendered support and technical assistance to participants of strikes and worker demonstrations: to teachers of the Moscow suburbs (1995), workers at the “Rostselmash” plant in Rostov-on-Don (1998), workers of the Yasnogorsky machine tool plant (1999: a strike directed by a general assembly of workers and accompanied by a plant occupation), imported construction workers in Moscow (1999), workers at the Ford plant in Vsevolozhsk (2007), etc. In rendering assistance to strikers, they have tried to disseminate in the workers’ movement anarchosyndicalist methods of self-organization, direct action, and independence from political parties and the structures of bureaucratic labour unions. The members of KRAS actively carry on anti-militarism agitation, and took part in actions against the war in Chechnya (1994-1996 and from 1999 on) and the Trans-Caucasus (2008), and other anti-war actions; in ecological campaigns, demonstrations against pension “reforms” for seniors (2005), in the movement against ZhKR (Housing and Communal Services Reform) and elite home construction in Moscow (in 2007 until the issue was taken over by political parties), against the rising cost of rail transport, etc.
\chapter{Bibliographic Essay}
The history of anarcho-syndicalism has been little studied. Social historians have been attracted in the first instance to social-democratic and communist trends in the workers’ movement; less frequently they have studied Christian and other “mainstream” trade unions. In the Soviet Union, under the conditions of the ideological monopoly of the CPSU, anarcho-syndicalism was perceived as an ideological enemy with which one must carry on an uncompromising struggle. In the books and brochures of V. Yagov, B. M. Leibzon, V. V. Komin, F. Ya. Polyansky, N. V. Ponomarev, S. N. Kanev, E. M. Kornoukhov, I. S. Rozental, et al, this tendency was considered a variety of “petty-bourgeois revolutionism” (along with Trotskyism and Maoism). These authors acknowledged that anarcho-syndicalism had involved significant masses of workers in various countries and in different periods of time; however, this fact was interpreted as a manifestation of the “weakness” and immaturity of the workers’ movement. The fundamental ideas and viewpoints of anarchists and syndicalists were reduced to a simplistic level or, as often happened – just falsified; the intention of these works did not consist in analyzing the content of the positions being criticized, but rather in exposing “ultra-leftists.” The anarcho-syndicalist International was hardly mentioned, and lumped under the rubric “anarcho-syndicalism” without any distinction were the revolutionary syndicalism of the early 20th century, the syndicalist “neo-Marxists” G. Sorel and A. Labriola, such very different union centrals as the Industrial Workers of the World and the Spanish National Confederation of Labour, and even the “Workers’ Opposition” inside the Bolshevik Party at the beginning of the 1920’s.
To some degree or other problems connected with the revolutionary syndicalist and anarcho-syndicalist movement were touched upon by the authors of studies of the history of specific countries: France (S. N. Gurvich, V. M. Dalin, G. Morozov, R. Sabsovich, and others) , Spain (S. P. Pozharskaya, L. V. Ponomareva, and others) , Italy (Z. P. Yakhimovich), and the states of Latin America (B. I. Koval, and others). In general these works were not devoted particularly to the history of anarchism (as a rare exception one can mention Ye. Yu. Staburova’s investigation of anarchism in China). Without deviating from official conceptions, these historians adduced information and facts which broadened the understanding of revolutionary syndicalism and anarcho-syndicalism as components of the global workers’ movement. Nevertheless, here also one finds the predominance of an ideologized assessment of the role of anarchists and syndicalists and their “influence on the masses.”
The elimination of the ideological monopoly of the CPSU in 1990-1991 and the opening of the archives allowed native historians to study social movements at a higher level. Researchers began to write more objectively about the role of the anarchists. A two-volume collection of documents about the Russian anarchists was published, and works appeared about the anarchists and anarcho-syndicalists in Russia. At the same time, it must be acknowledged that an in-depth study of the role of the anarcho-syndicalists in the Russian Revolution still does not exist.
The study of the international anarcho-syndicalist movement was also initiated. A. V. Shubin published several works which covered the role not only of the anarchists in the Makhnovist movement in Ukraine, but also Spanish anarcho-syndicalism in the period of the Spanish Revolution of the 1930’s and the discussions in the Russian emigration and in the global anarchist movement during the inter-war period. Above all he discussed in detail the social transformations carried out by anarcho-syndicalist workers in Spain and the political practice of the National Confederation of Labour (CNT), and demonstrated the baselessness of many of the myths about anarchism and the accusations directed at the CNT. At the same time, one must regard as unproven his ideas about a transition of anarcho-syndicalism in the 1920’s and 1930’s to a position of “market socialism” and about its “reversion” from Kropotkin to Bakunin.
On the whole, despite significant progress in the study of anarcho-syndicalism, in Russian historiography up to now there have been no investigations devoted to the history of the anarcho-syndicalism International and its sections in a majority of the countries of the world.
Elsewhere a number of works have been published about anarcho-syndicalist organizations and unions in individual countries of the world. The most investigated has been the most powerful movement – the Spanish; indeed the majority of authors were part of it themselves (M. Buenacasa, M. Iñigez, J. Gómez Casas, G. Leval, S. Lorenzo, A. Paz, J. Peirats, and others). Of course, this circumstance has left its imprint on their works: in their pages one finds the continuation of polemics around questions which have long divided the Spanish anarcho-syndicalists, such as the role of the anarchist federation FAI, the struggle with reformism, and tactics in the period of Revolution and Civil War 1936-1939. The study of the Spanish movement has also been taken up by authors far removed from it – A. Balcells, A. Bar, B. Bolloten, J. Brademas, A. Elorza, J. Garner, et al. Historians have been able to show the unique character of syndicalism in Spain, which drew on a tradition which can be traced back directly to the anarchism of the Bakuninist wing of the First International, and formed an original “symbiosis” of both tendencies. Simultaneously the Spanish movement to some extent also felt the influence of French revolutionary syndicalism. In investigations up to the present there exist varying analyses of the activity of the anarchist groups which were formed inside the anarcho-syndicalist unions of Spain: some authors consider them harmful (S. Lorenzo); others – understandable in the light of efforts to oppose reformist and communist tendencies, but useless; and a third group inclined to interpret the actions of at least some of these groups in a positive way (A. Paz, J. Gómez Casas). However, in studies of Spanish anarcho-syndicalism there remain issues and episodes which have been less studied. This applies, in particular, to the battles between supporters and opponents of the Profintern in the CNT, to the development of the “worker anarchism” tendency in the CNT, and the internal struggle in the anarchist movement after the coup of Primo de Rivera in 1923.
One special theme, to which a multitude of books and articles is devoted, is the activity and role of the anarchosyndicalists in the period of the Spanish Revolution and Civil War 1936-1939.
As for other European countries, the greatest interest of researchers has been drawn to French revolutionary syndicalism, frequently regarded as the prototype of all other syndicalist movements. The most important contributions to its study have been made by E. Dolléans, G. Lefranc, J. Maitron, J. Julliard, et al. But still insufficiently studied is the problem of the social base and some concrete moments of the history of the syndicalist movement in France (composition, membership, relationship to social legislation). The least studied aspect remains the activity of the small union central of French anarcho-syndicalists in the inter-war period. In works by German historians since the end of the 1960’s (H. M. Bock, A. Vogel, U. Klan, D. Nelles, H. Rübner, et al.) there is sufficient detail on the founding and development of the Free Association of German Trade Unions (the German section of the anarcho-syndicalist International) and the social organizations connected with it. Comparatively less attention has been devoted to the internal ideological discussions within the ranks of the German movement.
Italian syndicalism has been the subject of investigations by M. Antonioli, C. Venza, E. Falco, G. Careri, et al. The history of anarcho-syndicalism in Portugal is reflected in the workers of the libertarian authors E. Rodrigues, J. Freire, and P. F. Zarcone. Concerning the syndicalist movement in other European countries only investigations limited in scope have been published.
There are a number of monographs and articles about the history of the anarchist workers’ movement in Argentina (E. Bilsky, A. López, S. Marotta, I. Oved, J. Solomonoff, and others). Unfortunately, the emphasis in these works is on the period up to 1920-1921, and the presence of a new surge of working class anarchism in Argentina in the 1920’s is frequently ignored. The ideological-theoretical positions of the FORA, which it defended in the course of debates in the international anarcho-syndicalist movement, also deserve a more substantial analysis.
In Latin America the best studied anarcho-syndicalist movements are those of Chile, Brazil, Mexico, and Cuba. But even here there more than a few neglected moments and details so that the reader, instead of a systematic and thorough picture of the development of organizations, is more often than not presented with sketches describing events with varying degrees of detail. There are also individual works on the history of anarchism and syndicalism in other countries of the region.
The study of Chinese anarchism has been taken up by R. Scalapino, J.-J. Gandini, A. Dirlik, Nohara Shiro, et al.
Unfortunately, the anarcho-syndicalist movement receives significantly less attention in these works; thus, the history of libertarian ideas in China after the mid 1920’s remains basically a “white patch.” The study of Japanese anarchism and syndicalism in the period between the two world wars has received valuable contributions from the European and North American researchers J. Crump, P. Pelletier, S. Large, et al. Works have been published in the Japanese language by Kiyoshi Akiyama, Akinobu Gotô, Ryuji Komatsu, and Yasuyuki Suzuki. The book by Yoshikharu Hashimoto was translated into English; the rest, unfortunately, are inaccessible to the European reader. The history of Korean anarchism is the subject only one substantial, but far from exhaustive, investigation – the work of Ha Ki-Rak.
A special place in the international syndicalist movement is occupied by syndicalism and revolutionary unionism in the English-speaking countries. For a long time the predominant point of view was that the rise of syndicalist tendencies in Great Britain before the First World War was an isolated, temporary episode which did not play an important role in the history of the British workers’ movement. However, in recent decades historians have begun to direct more attention to such phenomena as the ongoing tradition of the struggle for workers’ control, the movement for merger (“amalgamation”) of trade unions, the opposition movements of rank-and- file members, and other examples of the influence of syndicalism. Researchers have come to the conclusion that British syndicalism was not an alien phenomenon, but a natural and appropriate response to the existing historical situation, a manifestation of the drive to overcome shop-level and professional particularism in favour of the community of interests of workers in one or other industries.
To the study of the syndicalist movement in other English- speaking countries (the Industrial Workers of the World and the One Big Union) contributions have been made by such authors as F. Thompson, P. Renshaw, M. Dubofsky, P. Carlson, and M. Hargis (U.S.A.) ; G. Jewel and D. Bercuson (Canada) ; L. van der Walt (South Africa) et al. But the whole story of this “industrial” tendency in syndicalism has not yet been written.
In global historiography a discussion about the historical place and role of anarcho-syndicalism in the workers’ movement is ongoing.
The Marxist tradition is inclined to view it as a product of the “underdevelopment” of the workers’ movement, the evolution of which is understood as a linear-progressive process. Syndicalism and anarcho-syndicalism are associated with economic backwardness, a manifestation of the pre-industrial, “primitive” rebellion of people from a peasant and handicraft milieu (“first generation workers”) who are unable to adjust to the realities of industrial-capitalist society. This phase was completed with the onset of the period of contemporary large-scale industry, mass production, and mass consumption. Anarcho-syndicalism “lingered on” for some time only in “backward” countries where, at the beginning of the 20th century, handicraft or semi-handicraft production still predominated (in France, Italy, Spain, Portugal, Latin America, etc.). The presence of certain customs and traditions supposedly led to “weak” self-discipline and the spread of insurrectionary methods of “direct action,” instead of the practice of collective bargaining between the enterprises and the workers. Correspondingly, the development of large-scale industry was viewed as a factor which led to the spread of Marxist ideas within the working class. As a result, a new type of trade union was established, based not on resolute opposition to enterprises and the contesting of their powers as such, but on negotiations and the pursuit of coordinated efforts to assure the functioning of production.
A contrast to this point of view, based to a significant extent on technical-economic determinism, emphasized in the first instance the particularism of individual countries, differences in culture and mentality, forms and functions of the State, and traditions of class resistance. In connection with this, a thesis was put forward according to which syndicalism and anarcho-syndicalism were perceived as above all “Romance” phenomena, peculiar to Romanic peoples (French, Spaniards, Latin Americans, etc.). It’s interesting that such a position has traditionally been upheld by many syndicalists, as well as a number of social-democratic authors (M. Adler, W. Sombart). Some historians to this day are inclined to make a comparison between the pragmatic (Anglo-Saxon) and social-democratic (continental) tendency in the trade union movement with the Romance-syndicalist tendency, which is characterized as having a lower level of self-discipline, less responsibility in the handling of the members’ dues, and a weakness for radical forms of action.
The majority of researchers nowadays eschew “extreme” points of view and call for the study of various factors and circumstances. The thesis about anarcho-syndicalism as a manifestation of “lack of consciousness” and “backwardness” of workers is not confirmed by the facts. The characterization of anarchism as a utopian, petty-bourgeois movement cannot explain why it enjoyed popularity among significant strata of workers in very different countries of the world. A concrete-historical investigation shows that syndicalism attracted not only skilled and handicraft workers (in the construction and metalworking trades) who were afraid of losing the value of their skill as a result of the introduction of new technologies and methods of organizing labour, but also workers who had received industrial training, and young, unskilled migrant-workers who had been drawn into production as a result of an industrial boom or a restructuring of production for military ends and who were ignored by “traditional” unions.
A number of authors have raised doubts about the legitimacy of the linear conception of the development of the workers’ movement, which associates radical activities and decentralized forms of organization with “backwardness.”
They note that handicraft and communal traditions of the “early” workers’ movement facilitated the formation of attitudes which could lead to and in fact led to more class-conscious, independent activity on the part of the workers. This class-consciousness included such elements as a conception of the social significance of labour, a striving for more independence and responsibility in the production process, and the desire to control the production process and its results.
The thesis about the “Romance” character of anarchosyndicalism as such also denied the facts. Researchers have shown that revolutionary syndicalism and working class anarchism propagated to very different countries and regions of the world – not only to Romanic, but also to Englishspeaking, Germanic, Slavic, and Asiatic. This forces the assumption that at the basis of the given phenomenon there must lie certain common causal factors.
Historians who have attempted a comparative analysis of the syndicalist movement in different countries (P. Schöttler, G. Haupt, L. Peterson, P. Lösche, W. Thorpe, M. van der Linden, and others) tend to interpret it in the context of the general transition from liberal to “organized” capitalism which was characterized by a high degree of State intervention. Radical protest, in their opinion, was directed not so much against the concentration of workers in large enterprises, as against the de-skilling of labour. At the same time they try to take into consideration the appearance of new strata of workers who are not satisfied with the previous relations and forms of organization of the working class, originating in the 19th century. These discontented categories believe that centralized trade unions and the political, parliamentary activities of the socialists are insufficient in themselves to defend their interests and needs. But these historians have failed to show a direct dependency between the scale of enterprises and the spread of syndicalist attitudes. The syndicalist movement pulled together very different strata of workers who rejected the authoritarian structures taking shape in the workplace.
Finally, some authors are inclined to view the rise of the revolutionary syndicalist and anarcho-syndicalist movement in the first decades of the 20th century in the context of the history of the establishment and development of industrialcapitalist civilization itself – as a form of resistance against it and an effort to counterpoise to it a different, alternative model of society, based on self-management and a distinctive working-class culture. The introduction in the 20th century of the “Fordist-Taylorist” model of mass production, based on the division of labour into a series of discrete operations and the severe limiting of initiative, undermined the sense of wholeness of the production process and, consequently, any conception of the possibility of controlling it. This led to, among other things, the collapse of working class radicalism and then a decline in the workers’ movement as such and the “dissolution” of working-class culture.
The decline of mass radicalism in the workers’ movement (including anarcho-syndicalism) facilitated, in the opinion of a number of scholars, the rise of the “Social State” which took shape in the second third of the 20th century; thanks to this political development, the centre of social conflicts shifted from the sphere of production (and the battle for control over it) to the sphere of distribution and consumption. Workers relied more and more on the social and distributive role of the State and were less inclined to concur with the stateless alternative of the anarchists. Looked at from this point of view, the decrease in popularity of anarcho-syndicalism in the second half of the 20th century cannot be seen as “irreversible,” especially in light of the current crises of the “Social State” and the “Fordist model.”
In analyzing the “common” factors favouring the rise of revolutionary syndicalism and anarcho-syndicalism as a global movement in the first decades of the 20th century, historians can not forget the special features of individual countries and regions. These include the forms and models of organization, social basis, ideological tendency, emergent themes and problems, relationship to political parties, and, above all, the focus of labour union or social-cultural work.
On the whole one can say that the international anarchosyndicalist and revolutionary syndicalist movement has been studied in a very uneven manner. Along with a large number of monographs on the history of syndicalism in Memories of Class (London, 1982); et al. a few countries, there are only a few articles or pamphlets dealing with other countries. Of the various themes which have been studied in only a cursory fashion, one can mention ideological discussions, the organizational life of anarchosyndicalist unions and federations, and their international connections and relationships with the anarcho-syndicalist International. Issues concerning the social basis and historical place of anarcho-syndicalism in the history of the workers’ movement continue to be contentious.
Little work has been done on the history of the anarchosyndicalist International – the International Workers’ Association (IWA). Mainly there are some small pamphlets written by members of either the Secretariat of the International or anarcho-syndicalist organizations. In them one finds a demonstration of the origins of the IWA in the First International (at least its anti-authoritarian wing), and the continuity in positions between the two organizations. Much attention is devoted to the confrontation with Bolshevism in the 1920’s, and brief overviews of the congresses of the anarcho-syndicalist International and their resolutions are given. In these condensed outlines there is simply no room for detailed analyses of the course of events and their causes.
There are also some articles of greater scientific value by researchers who are sympathetic to anarchist attitudes. But such works are few in number and only deal with isolated moments in the history of the movement.
The Canadian historian W. Thorpe has made a noteworthy contribution to the history of the creation of the Berlin International. In collaboration with the International Institute of Social History in Amsterdam, he published an article about the London conference of syndicalists in 1913, followed by a fundamental investigation of the international contacts of revolutionary syndicalists before the First World War, their differentiation from Bolshevism, and the processes which led ultimately to the creation of the Berlin International. Thorpe’s work includes a general survey of syndicalism in the world prior to the First World War and an analysis of the discussions among syndicalists about setting up an international strategy. In a convincing manner he describes the dilemma which confronted syndicalism in connection with the attempts of Communist parties to subordinate trade unions to their party line. Finally, Thorpe traces the establishment of an international association of anarcho-syndicalists using materials from their meetings, conferences, and congresses. Unfortunately, his work devotes almost no attention to the internal development and activity of syndicalist organizations in individual countries, their participation in revolutionary events and strikes, and their accomplishments in the elaboration of ideological-theoretical ideas. Moreover, Thorpe makes almost no use of material from Soviet archives and archives of Communist parties.
In attempting to compensate to some extent for these deficiencies, Thorpe and the Dutch historian M. van der Linden published in 1990 the collection Revolutionary Syndicalism: an International Perspective, which was the first attempt to pull together articles about the development of revolutionary syndicalism and anarcho-syndicalism in France, the Netherlands, Germany, Sweden, Great Britain, Spain, Italy, Portugal, Argentina, Mexico, the U.S.A., and Canada.
This collection includes Thorpe’s article: “Syndicalist Internationalism before World War II” with a brief survey of the history of the IWA up to 1939. The obvious value of the book consists in the fact that its editors invited the participation of the leading specialists in the history of syndicalist movements in individual countries. At the same time, the story of the anarcho-syndicalist International is covered in a very general way, and scarcely delves into the concrete moments in its work and activity; the analysis of ideological discussions is virtually absent. The articles on individual countries are relatively brief, and vary substantially in the level with which various aspects are dealt with; in some cases essential moments of the movement are covered in insufficient depth or not even mentioned at all.
Thus it can be said that a general history of the rise of the international anarcho-syndicalist movement – treated as an integral, global phenomenon and taking into account the mutual influence of international and national factors and social-revolutionary processes in individual countries – has yet to be written.
\chapter{Acronyms}
The anarcho-syndicalist International is commonly referred to by one of
several acronyms:
IWA International Workers’ Association (English, used in this book)
AIT Association Internationale des Travailleurs (French)
Asociacion Internacional de los Trabajadores (Spanish)
IAA Internationale Arbeiter-Assoziation (German)
MAT Mezhdunarodnaya Assotsiatsiya Trudyashchikhsya (Russian)
AAUD-E Allgemeine Arbeiter-Union Deutschlands-Einheits-organisation
General Workers Union of Germany – Unitary Organization
CCMA Comite Central de las Milicias Antifascistas de Cataluna
Central Committee of Antifascist Militias of Catalonia
CDS Comite de defense syndicaliste
Committee of Syndicalist Defense
CGL Confederazione Generale del Lavoro
General Confederation of Labor
CGT Confederation generale du travail
General Confederation of Labour
CGT Confederacao Geral do Trabalho
General Confederation of Labour (Portguese)
CGT Confederacion General de Trabajadores
General Confederation of Workers (Mexican)
CGT-SR Confederation generale du travail - syndicaliste revolutionnaire
General Confederation of Labour - Revolutionary Syndicalist
CGTU Confederation generale du travail unitaire - Unitary General Confederation of Labour
CNT Confederacion Nacional del Trabajo
National Confederation of Labour
CNT-AIT Confederation nationale du travail
National Confederation of Labour (French section of AIT)
CNT-f Confederation nationale du travail
National Confederation of Labour (CNT-Vignoles)
COB Confederacao Operaria Brasileira
Brazilian Workers Confederation
COM Casa del Obrero Mundial
House of the World Worker
CORA Confederacion Obrera Regional Argentina
Regional Workers’ Confederation of Argentina
CPSU See “KPSS”
CROM Confederacion Regional Obrera Mexicana
Regional Confederation of Mexican Workers
CSR Comites syndicalistes revolutionnaires
Revolutionary Syndicalist Committees
FAF Federation anarchiste francophone
Francophone Anarchist Federation
FAI Federacion Anarquista Iberica
Iberian Anarchist Federation
FAUD Freie Arbeiter Union Deutschlands
Free Workers‘ Union of Germany
FIJL Federacion Iberica de Juventudes Libertarias
Iberian Federation of Libertarian Youth
FOCH Federacion de Obreros de Chile
Federation of Chilean Workers
FORA Federacion Obrera Regional Argentina
Regional Workers’ Federation of Argentina
FORU Federacion Obrera Regional Uruguayo
Regional Workers’ Federation of Uruguay
FVdG Freie Vereinigung deutscher Gewerkschaften
Free Association of German Trade Unions
GCOM Gran Circulo de Obreros de Mexico
Great Circle of Mexican Workers
ISNTUC International Secretariat of the National Centers of Trade
Unions
IREAN Initsiativa revolyutsionnykh anarkhistov
Initiative of Revolutionary Anarchists
ISEL Industrial Syndicalist Education League
IWW Industrial Workers of the World
KAS Konfederatsiya anarkho-sindikalistov
Confederation of Anarcho-syndicalists
KPSS Kommunisticheskaya Partiya Sovetskogo Soyuza
Communist Party of the Soviet Union
KRAS Konfederatsiya revolyutsionnykh anarkho-sindikalistov
Confederation of Revolutionary Anarcho-Syndicalists
NAS Nationaal Arbeids-Secretariaat
National Labour Secretariat
NSV Nederlands Syndicalistisch Vakverbond
Netherlands Syndicalist Trade Union Federation
OBU One Big Union
PCE Partido Comunista de Espana
Communist Party of Spain
POUM Partido Obrero de Unificacion Marxista
Workers’ Party of Marxist Unification
PSUC Partit Socialista Unificat de Catalunya
Unified Socialist Party of Catalonia
RKAS Rossiyskaya confederatsia anarcho-sindikalistov
Russian Confederation of Anarcho-Syndicalists
RILU Red International of Labour Unions (Profintern)
SAC Sveriges Arbetares Centralorganisation
Central Organisation of the Workers of Sweden
UGT Union General de Trabajadores
General Workers’ Union
UON Uniao Operaria Nacional
National Workers’ Union
USI Unione Sindacale Italiana
Italian Syndicalist Union
VKPD Vereinigte Kommunistische Partei Deutschlands
United Communist Party of Germany
Archives:
SAPMO Stiftung Archive der Parteien und Massenorganisationen der
DDR (Berlin)
IISG International Institute of Social History (Amsterdam)
RGASPI Russian State Archive of Social and Political History (Moscow)
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\begin{center}
The Anarchist Library
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Vadim Damier
Anarcho-syndicalism in the 20th Century
Monday, September 28th 2009
\bigskip
Retrieved on March 22nd, 2011
\emph{English translation © Black Cat Press. Republished with permission.}
\bigskip
\textbf{theanarchistlibrary.org}
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%&LaTeX
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\usepackage[utf8]{inputenc}
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\begin{thebibliography}{1}
\bibitem{Hoyos-Diaz2018} Hoyos-D{\'\i}az, J. M., Villalba-Alem{\'a}n, E., Ramoni-Perazzi, P., \& Mu{\~n}oz-Romo, M. (2018). Impact of artificial lighting on capture success in two species of frugivorous bats (chiroptera: phyllostomidae) in an urban locality from the Venezuelan Andes. \textit{Mastozoologia Neotropical}, \textit{25}(2).
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%%
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\chapter{Transition Plugins}%
\label{cha:transition_plugin}
\todo{wrong border for title's number}
When one edit ends and another edit begins, the default behavior is to have the first edit's output immediately become the output of the second edit when played back. Transitions are a way for the first edit’s output to become the second edit’s output with different variations. The audio and video transitions are listed in the Resources window as figure~\ref{fig:transition}.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.8\linewidth]{images/transition.png}
\caption{Resources window displaying the Video Transitions.}
\label{fig:transition}
\end{figure}
Note the colored bar above the \textit{Shape Wipe} transition.
Transitions may only apply to the matching track type. Transitions under audio transitions can only apply to audio tracks. Transitions under video transitions can only apply to video tracks.
An example usage of a transition follows:
\begin{enumerate}
\item Load a single video file and cut away a section from the center or make a blade cut so that you make two edits out of a single file. Make sure the edit boundary between the two edits is visible on the timeline.
\item Go to the \textit{Resources window} and click on the \texttt{Video transitions} folder. Drag a transition from the transition list onto the second video edit on the timeline. A colored box highlights over where the transition will appear. Releasing over the $2^{nd}$ edit applies the transition between the $1^{st}$ and $2^{nd}$ edit.
\end{enumerate}
Once the transition is in place, it can be edited similarly to a plugin. Move the \textit{pointer} over the transition and \texttt{right click} to bring up the transition menu. The show option brings up specific parameters for the transition in question if any. The \texttt{length} option adjusts the length of the transition in seconds. The \texttt{detach} option removes the transition from the timeline. If the insertion point or the In point is over an edit, the beginning of the edit is covered by the transition.
Dragging and dropping transitions from the Resource window to the Program window can be tedious so there are shortcuts to solve this issue. Once you drag a transition from the Resources window, the \texttt{U} and \texttt{u} keys will paste the same transition. The U key pastes the last video transition and the u key pastes the last audio transition on all the recordable tracks. Another easy way to add the same transition to multiple edits is to get into \texttt{Arrow mode} (Drag and Drop editing), select the edits you would like to add the transition to and use the Video or Audio pulldown to \texttt{Attach transition}. Choose which transition you would like and click the checkmark \texttt{OK}. You can also set your default transition at any time by doing so in the Attach transition popup box – highlight your choice, and then at the bottom, click on the button \texttt{Set Default Transition}. You will see that name appear.
Transitions make two edits overlap for a certain amount of time. Cinelerra does not move edits during transitions. Instead it uses spare frames from the source file to lengthen the first edit enough to make it overlap the second edit for the duration of the transition. The exact point in time when the transition takes effect is the beginning of the second edit. The transition lasts a set amount of time into the second edit. For example, if you set a duration of 1 second for a dissolve transition, it will not start at the last 0.5 second of the first edit and continue 0.5 second into the second edit. In fact, it will start exactly at the beginning of the second edit and last for 1 second into that second edit. On the timeline a colored bar over the transition symbol visually represents the position and the duration of the transition.
The most important consequence of this behavior is that the first asset needs to have enough spare data after the end boundary to fill the transition into the second edit. Spare data duration should be equal or greater than the length of the transition effect set in the Length parameter of the transition popup menu.
If the last frame shown on the timeline is the last frame of the source file, Cinelerra will lengthen the first edit using the last frame only, with the unpleasant result of having the first edit freezing into the transition.
It should be noted that when playing transitions from the timeline to a hardware accelerated video device, the hardware acceleration will usually be turned \textit{off} momentarily during the transition and \textit{on} after the transition, in order to render the transition. Using an un-accelerated video device for the entire timeline normally removes the disturbance.
\section{Audio Transitions}%
\label{sec:audio_transition}
\subsection*{Crossfade}%
\label{sub:crossfade}
Creates a smooth transition from one audio source edit to another. The crossfade has the first source \textit{fade out} while the second \textit{fades in}.
\section{Video Transitions}%
\label{sec:video_transition}
In order to use a transition that you have dragged to the timeline, first right mouse click on the transition icon in the timeline. A menu will pop-up with the following controls:
\begin{description}
\item[Show] Pops up the transition specific menu if available (not available on the dissolve transition).
\item[On] Toggle on/off the transition effect.
\item[Transition length] Set the length in seconds, frames, samples, \textit{H:M:S:frm} or \textit{H:M:S.xxx} for the transition to complete. In addition you can use the mouse wheel to change the length in real time.
\item[Detach] Remove the transition from the timeline.
\end{description}
\subsection*{BandSlide}%
\label{sub:bandslide}
Bands slide across video and you see the image slide.
\subsection*{BandWipe}%
\label{sub:bandwipe}
Bands wipe across the video and you see the mask slides.
\subsection{Flash}%
\label{sub:flash}
The video flashes when transitioning between segments.
\subsection*{IrisSquare}%
\label{sub:irissquare}
Video switches segments via a small rectangular view that gradually grows to full size.
\subsection*{Shape Wipe}%
\label{sub:shape_wipe}
Wipe a specific shape across the video. Currently available shapes are: \textit{burst}, \textit{circle}, \textit{clock}, \textit{heart}, \textit{specks}, \textit{spiral}, \textit{tile2x2h}, and \textit{tile2x2v}.
You can add your own images to the Shape Wipe transition and there are some free ones available to download such as at \url{assistcg.com}.
To include new images in the Shape Wipe Transition, simply copy the \texttt{{shape}.png} file to your location of cinelerra in the subdirectory \texttt{plugins/shapes}.
\subsection*{Slide}%
\label{sub:slide}
Image slides into view; you can set \texttt{Left/Right/In/Out}.
\subsection*{Wipe}%
\label{sub:wipe}
Wipe the image across screen starting left or right.
\subsection*{Zoom}%
\label{sub:zoom}
Zoom out video at $\frac{X}{Y}$ magnification for some seconds.
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%Tapuscrit Jean-Claude Souque
%Corrigé François Hache
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pdfauthor = {APMEP},
pdfsubject = {Baccalauréat STL biotechnologies - Corrigé},
pdftitle = {Métropole - 16 juin 2017 },
allbordercolors = white,
pdfstartview=FitH}
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\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small Corrigé du baccalauréat STL Biotechnologies}
\lfoot{\small{Métropole--La Réunion}}
\rfoot{\small{16 juin 2017}}
\pagestyle{fancy}
\thispagestyle{empty}
\begin{center}{\Large \textbf{\decofourleft~Corrigé du baccalauréat STL biotechnologies Métropole~\decofourright\\[7pt] 16 juin 2017}}
\end{center}
\section*{\textsc{Exercice 1} \hfill 6 points}
%\vspace{0.25cm}
%\emph{Les deux parties de cet exercice peuvent être traitées indépendamment.}
%\medskip
\subsection*{PARTIE A}
%\medskip
Une société souhaite exploiter un nouveau détecteur qui permet de mesurer la désintégration de
noyaux radioactifs. Pour tester ce détecteur, la société l'utilise pour déterminer le nombre de
noyaux radioactifs présents dans un échantillon radioactif à des instants donnés. Voici les
résultats des relevés réalisés au cours des heures qui ont suivi le début du test:
\smallskip
\begin{center}
\begin{tabularx}{\linewidth}{|m{5cm}|*{6}{>{\centering \arraybackslash}X|}}
\hline
Nombre $t_i$ d'heures écoulées depuis le début du test& 0& 2 &4& 6& 8& 10\\\hline
Nombre de noyaux $N_i$ détectés dans l'échantillon (en milliards)&500 &440& 395 &362& 316 &279\\\hline
\end{tabularx}
\end{center}
\smallskip
\begin{enumerate}
\item \begin{enumerate}
\item On complète le tableau ci-dessous en arrondissant les valeurs à $10^{-3}$:
\begin{center}
\begin{tabularx}{\linewidth}{|c|*{7}{>{\centering \arraybackslash}X|}}\hline
$t_i$ &0 &2 &4 &6 &8 &10 \\\hline
$y_i = \ln N_i$ &6,215 &6,087 &5,979 &5,892 &5,756 &5,631\\\hline
\end{tabularx}
\end{center}
\item On représente le nuage de points de coordonnées $(t_i~;~ y_i)$ sur l'\textbf{annexe 1}.
\item Les points obtenus sont presque alignés donc un ajustement affine est envisageable.
\item \label{equa} À l'aide de la calculatrice, on détermine une équation de la droite $\mathcal D$ d'ajustement de $y$ en $t$ par la méthode des moindres carrés sous la forme $y = at + b$, où les coefficients $a$ et $b$ sont arrondis à $10^{-3}$: $y=-0,057t+6,210$.
\item On trace alors la droite $\mathcal D$ sur l'\textbf{annexe 1}.
\end{enumerate}
\item \begin{enumerate}
\item On choisit la droite $\mathcal D$ comme modèle d'ajustement du nuage de points $M_i(t_i~;~y_i )$.
%À l'aide de la question \ref{equa}, montrer alors que, pour tout réel $t$ positif ou nul, le nombre
%de noyaux, en milliards, détectés dans l'échantillon au bout de $t$ heures écoulées depuis
%le début du test, est de la forme: $A\e^{Bt}$ où $A$ (arrondi à l'unité) et $B$ (arrondi au millième)
%sont deux réels à préciser.
$y=-0,057t+6,210 \iff \ln(N)= -0,057t+6,210 \iff N=\e^{-0,057t+6,210} \\[3pt]
\phantom{y=-0,057t+6,210} \iff N=\e^{-0,057t} \times \e^{6,210} \iff N \approx 498\e^{-0,057t}$
\item La loi de désintégration assure que la fonction $f$, qui à tout réel $t$ positif ou nul, associe le
nombre de noyaux, en milliards, présents dans l'échantillon au bout de $t$ heures, est
définie par $f(t) = 500\e^{-0,06t}$ .
%Le test réalisé doit-il conduire la société à exploiter le nouveau détecteur? Pourquoi?
Le test conduit à la fonction qui à $t$ associe $498\e^{-0,057t}$ qui est proche de la fonction $f$: on peut donc utiliser ce dispositif.
\end{enumerate}
\end{enumerate}
%\medskip
\subsection*{PARTIE B}
%\medskip
\def\interA{\cd 0~;~+\infty\cd}
On étudie à présent la fonction $f$ définie sur $\interA$ par
$f(t) = 500\e^{- 0,06t}$.
\smallskip
\begin{enumerate}
\item On admet que : $\displaystyle \lim_{t\to +\infty} \e^{- 0,06t} = 0$ donc $\displaystyle \lim_{t\to +\infty} 500 \e^{- 0,06t} = 0$ ce qui équivaut à $\displaystyle \lim_{t\to +\infty} f(t) = 0$
On en déduit que la courbe représentant la fonction $f$ admet la droite d'équation $y=0$ (l'axe des abscisses) comme asymptote horizontale en $+\infty$.
%Déterminer et interpréter graphiquement la limite de $f$ en $+\infty$.
\item %Calculer $f'(t$) où $f'$ est la fonction dérivée de $f$.
$f'(t) = 500\times (-0,06)\e^{-0,06t} = -30\e^{-0,06t} <0$ sur $\interA$.
\item $f(0)=500$ et $\ds\lim_{t\to +\infty} f(t) \text{\,d}t = 0$;
on déduit le tableau de variations de la fonction $f$ sur $\interA$:
\begin{center}
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\psset{nodesep=3pt,arrowsize=2pt 3} % paramètres
\def\esp{4cm}% pour modifier la largeur du tableau
\def\hauteur{20pt}% mettre au moins 12pt pour augmenter la hauteur
$\begin{array}{|c| *3{c}|}
\hline
t & 0 & \hspace*{\esp} & +\infty \\
% \hline
%f'(x) & & \pmb{-} & \\
\hline
& \Rnode{max}{500} & & \\
f(t) & & & \rule{0pt}{\hauteur} \\
& & & \Rnode{min}{0} \rule{0pt}{\hauteur}
\ncline{->}{max}{min}
%\rput*(-2,0.65){\Rnode{zero}{\red 0}}
%\rput(-2,1.75){\Rnode{alpha}{\red \alpha}}
%\ncline[linestyle=dotted, linecolor=red]{alpha}{zero}
\\
\hline
\end{array}$
}
\end{center}
\item %On rappelle que $f(t)$ est le nombre de noyaux, en milliards, présents dans l'échantillon radioactif $t$ heures après le début du test.
\begin{enumerate}
\item Le nombre de noyaux présents dans l'échantillon 24 heures après le début du test est en milliards
$f(24) \approx \np{118,463879341}$ donc il y a $\np{118463879341}$ noyaux.
\item La moitié des noyaux présents dans l'échantillon au début du test aura disparu quand $t$ sera tel que
$f(t)=250$; on résout cette équation:
$f(t)=250 \iff 500\e^{-0,06t} = 250 \iff \e^{-0,06t} = 0,5 \iff -0,06 t = \ln(0,5) \iff $
$t= -\dfrac{\ln(0,5)}{0,06}$
Or $ -\dfrac{\ln(0,5)}{0,06} \approx 11,6$ donc la moitié des noyaux présents aura disparu au bout de 12 heures.
\end{enumerate}
\end{enumerate}
%\vspace{0,25cm}
\section*{\textsc{Exercice 2} \hfill 6 points}
%\vspace{0.25cm}
On s'intéresse à une modélisation de la concentration d'un médicament, injecté dans le sang d'un patient, en fonction du temps.
À 7 heures du matin, on injecte le médicament au patient. Toutes les heures, on relève la
concentration de médicament dans le sang, exprimée en $\micro \mathrm{g}\cdot \mathrm{mL}^{-1}$. À l'injection, cette concentration est égale à \np[\micro g\cdot mL^{-1}]{3,4}.
Le nuage de points ci-dessous donne la concentration de ce médicament dans le sang en fonction du temps écoulé depuis l'injection.
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%\medskip
\subsection*{PARTIE A}
%\medskip
Dans cette partie, on modélise la concentration de ce médicament par une fonction définie sur l'intervalle [0~;~5].
Parmi les trois modélisations proposées, une seule est correcte. %Laquelle? Justifier.
\textbf{a~~}$f :x\longmapsto 0,6x+3,4$\hfill
\textbf{b~~} $g: x \longmapsto 3,4\e^{ -0,223x}$\hfill
\textbf{c~~} $h: x\longmapsto \dfrac{9}{3+x}$
\begin{list}{\textbullet}{}
\item La fonction $f$ est croissante donc ne peut modéliser la concentration du médicament.
\item $h(0)=3 \neq 3,4$ donc la fonction $h$ ne modélise pas la concentration du médicament.
\item C'est donc la fonction $g$ qui modélise la concentration du médicament.
\end{list}
%\medskip
\subsection*{PARTIE B}
%\medskip
Dans cette partie, on choisit de modéliser la concentration du médicament par une suite, en prenant, pour valeurs des trois premiers termes de la suite, les valeurs données par le graphique placé avant la partie A.
\begin{enumerate}
\item Pour tout entier naturel $n$, on note $C_n$ la concentration, exprimée en $\micro \mathrm{g}\cdot \mathrm{mL}^{-1}$, au bout de $n$ heures, de ce médicament dans le sang.% Une partie de ce médicament est éliminée toutes les heures.
\begin{enumerate}
\item Par lecture du graphique: $C_0=3,400$, $C_1=2,720$ et $C_2=2,176$.
\item %Que peut-on alors conjecturer sur la nature de la suite $\left(C_n\right)$ ? Pourquoi ?
$\dfrac{C_1}{C_0}= \dfrac{2,72}{3,4} = 0,8$ et $\dfrac{C_2}{C_1} = \dfrac{2,176}{2,72\rule[-3pt]{0pt}{0pt}}=0,8$ donc $C_0$, $C_1$ et $C_2$ sont les premiers termes d'une suite géométrique de raison $q=0,8$ et de premier terme $C_0=3,4$.
\end{enumerate}
\end{enumerate}
On admet qu'à chaque heure, la concentration du médicament restante baisse de 20\,\%.
\begin{enumerate}[resume]
\item% Pour tout entier naturel $n$, exprimer $C_n$ en fonction de $n$.
Baisser de 20\,\%, c'est multiplier par $0,8$ donc la suite $(C_n)$ est géométrique de raison $q=0,8$ et de premier terme $C_0=3,4$ donc, pour tout $n$, $C_n=C_0\times q^n = 3,4\times 0,8^n$.
\item% Déterminer alors la limite de la suite $\left(C_n\right)$ lorsque $n$ tend vers l'infini. Interpréter cette limite dans le contexte de l'exercice.
La suite $(C_n)$ est géométrique de raison $0,8$ et $0<0,8<1$ donc la suite $(C_n)$ a pour limite 0 quand $n$ tend vers l'infini.
La concentration du médicament tend vers 0 quand le temps augmente indéfiniment.
\item Soit l'algorithme suivant:
\begin{center}
\fbox{
\begin{tabular}{l}
\textbf{Variables:}\\[5pt]
\hspace*{1.2em}$n$ entier naturel\\
\hspace*{1.2em}$C$ réel\\[5pt]
\textbf{Initialisation :}\\[5pt]
\hspace*{1.2em}Affecter à $n$ la valeur 0\\
\hspace*{1.2em}Affecter à $C$ la valeur 3,4\\[5pt]
\textbf{Traitement:}\\[5pt]
\hspace*{1.2em}Tant que $C$ est supérieur à 1\\
\hspace*{3.2em}Affecter à $n$ la valeur $n+1$\\
\hspace*{3.2em}Affecter à $C$ la valeur $0,8\times C$\\
\hspace*{1.2em}Fin tant que\\[5pt]
\textbf{Sortie:}\\[5pt]
\hspace*{1.2em}Afficher $n$\\[5pt]
\end{tabular}
}
\end{center}
%Quelle valeur affiche l'algorithme ? Interpréter le résultat dans le contexte de cet exercice.
Dans l'algorithme, la variable $C$ correspond à la concentration $C_n$ au bout de $n$ heures.
On fait tourner l'algorithme en arrondissant les résultats au centième:
\begin{center}
\newcommand{\ca}{\centering\arraybackslash}
\begin{tabularx}{0.6\linewidth}{|*8{>{\ca}X|}}
\hline
$n$ & 0 & 1 & 2 & 3 & 4 & 5 & \textcolor{red}{6} \\
\hline
$C_n$ & $3,4$ & $2,72$ & $2,18$ & $1,74$ & $1,39$ & $1,11$ & \textcolor{red}{$0,89$} \\
\hline
\end{tabularx}
\end{center}
Au bout de 6 heures, la concentration du médicament devient inférieure à 1 \micro g.mL$^{-1}$.
\emph{Remarque} - On peut aussi résoudre l'inéquation $C_n \pp 1$ c'est-à-dire $3,4\times 0,8^n \pp 1$.
\item Pour des raisons d'efficacité, le patient reçoit immédiatement une nouvelle injection de médicament dès que, lors d'un relevé à une heure donnée, la concentration $\boldsymbol{c}$ du médicament dans le sang est inférieure ou égale à \np[\micro \mathrm{g}\cdot mL^{-1}]{1}. À la nouvelle injection, la concentration du médicament dans le sang est alors égale à $\boldsymbol{c} + 3,4\, \micro \mathrm{g}\cdot \mathrm{mL}^{-1}$.
\begin{enumerate}
\item Le patient devra-t-il recevoir une deuxième injection dès la 6\ieme{} heure.
\item d'après le tableau de la question précédente, la 6\ieme{} heure avant la deuxième injection le taux de médicament dans le sang est de $0,89$~$\micro$g.mL$^{-1}$; après la deuxième injection, la concentration est de $0,89 + 3,4 = 4,29$ soit en arrondissant: $4,3~\micro$g.mL$^{-1}$
%On arrondira le résultat à \np[\micro g\cdot mL^{-1}]{0.1}.
\item% À quelle heure le patient devra-t-il recevoir une troisième injection?
On continue le processus, en arrondissant au dixième:
\begin{center}
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\hline
$n$ & 6 & 7 & 8 & 9 & 10 & 11 & 12 & \textcolor{red}{13} \\
\hline
{\small Concentration} & $4,3$ & $3,4$ & $2,7$ & $2,2$ & $1,8$ & $1,4$ & $1,1$ & \textcolor{red}{$0,9$} \\
\hline
\end{tabularx}
\end{center}
Il faudra donc procéder à une troisième injection dès la 13\ieme{} heure.
\emph{Remarque} - On peut également résoudre l'inéquation d'inconnue $n$: $4,3\times 0,8^n \pp 1$.
\end{enumerate}
\end{enumerate}
%\vspace{0,25cm}
\section*{\textsc{Exercice 3} \hfill 4 points}
%\vspace{0.25cm}
La Direction de la recherche, des études, de l'évaluation et des statistiques (Drees) affirme qu'en France : 7 adultes sur 10 portent des lunettes.
On prélève au hasard un échantillon de 40 adultes parmi la population française. On assimile ce
prélèvement à un tirage avec remise.
Soit $X$ la variable aléatoire qui, à tout échantillon de ce type, associe le nombre de porteurs de
lunettes dans l'échantillon.
\begin{enumerate}
\item \begin{enumerate}
\item %Montrer que $X$ suit une loi binomiale dont on précisera les paramètres.
Le prélèvement est assimilé à un tirage avec remise, donc la variable aléatoire $X$ qui donne le nombre de porteurs de lunettes suit une loi binomiale de paramètres $n=40$ et $p=\dfrac{7}{10}=0,7$.
\item La probabilité qu'il y ait au moins 30 porteurs de lunettes dans un tel échantillon de 40 adultes est $p5X\pg 30)$; à la calculatrice, on trouve, arrondi à $10^{-3}$, $0,309$.
% On donnera la valeur arrondie à $10^{-3}$.
\end{enumerate}
On admet que la loi binomiale de la variable aléatoire $X$ précédente peut être approchée par une loi normale de paramètres \textmu{} et $\sigma$.
\item On a représenté ci-dessous un diagramme en bâtons et une courbe $\mathcal C$. L'une de ces deux représentations est la représentation de la loi binomiale suivie par $X$; l'autre celle de la loi normale de paramètres \textmu{} et $\sigma$.
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\begin{enumerate}
\item La loi binomiale est associée au diagramme en bâtons car c'est une loi discrète.
La loi normale est continue, elle est donc associée à la courbe $\mathcal C$.
\item La courbe $\mathcal C$ est symétrique par rapport à la droite verticale d'équation $x=\mu$; par lecture graphique on peut dire que $\mu=28$.
\item On affirme que l'écart type $\sigma$ de la loi normale est égal à 8. %Cette affirmation est-elle correcte ? Pourquoi ?
On sait que, si $X$ suit une loi normale de paramètres $\mu$ et $\sigma$, $p(\mu-\sigma \pp X \pp \mu+\sigma) \approx 0,68$, ce qui donnerait avec $\mu=28$ et $\sigma=8$: $p(20\pp X \pp 36)\approx 0,68$.
Il faudrait donc que l'aire du domaine compris entre la courbe, l'axe des abscisses et les droites d'équations $x=20$ et $x=28$ soit égale à $0,68$, sachant que l'aire totale sous la courbe est égale à 1. C'est mainfestement faux donc l'écart-type n'est pas égal à 8.
\end{enumerate}
\item \begin{enumerate}
\item% Déterminer un intervalle de fluctuation asymptotique à 95\,\% de la fréquence des porteurs de lunettes dans un échantillon aléatoire de 40 adultes en France. On arrondira les bornes de l'intervalle à $10^{-3}$.
$n=40 \pg 30$, $np=40\times 0,7=28\pg 5$ et $n(1-p) = 40\times 0,3 = 12 \pg 5$ donc on peut déterminer l'intervalle de fluctuation à 95\,\%:
\smallskip
$I = \left [ p-1,96 \ds\sqrt{\dfrac{p(1-p)}{n}}~;~p+1,96 \ds\sqrt{\dfrac{p(1-p)}{n}}\right ]
= $
$\left [ 0,7-1,96 \ds\sqrt{\dfrac{0,7\times 0,3}{40}}~;~0,7+1,96 \ds\sqrt{\dfrac{0,7\times 0,3}{40}}\right ] \approx \cd 0,558~;~0,842 \cg$
\smallskip
\item Dans un échantillon de 40 adultes en France, on compte 24 porteurs de lunettes, ce qui fait une fréquence de $f=\dfrac{24}{40}=0,6$.
%Déduire de la question précédente si cet échantillon remet en cause l'affirmation de la Drees qui figure au début de l'exercice.
$f\in I$ donc cet échantillon ne remet pas en cause l'affirmation de la Drees.
\end{enumerate}
\end{enumerate}
%\vspace{0,25cm}
\section*{\textsc{Exercice 4} \hfill 4 points}
%\vspace{0.25cm}
\def\interA{\cd 0~;~7\cg}
Soient les fonctions $f$ et $g$ définies sur $\interA$ par
$f(x) = 20x\e^{-x}\qquad \text{ et }\qquad g(x)= 20x^2\e^{-x}$.
On note $C_f$ et $C_g$ les courbes représentatives respectives des fonctions $f$ et $g$ représentées en \textbf{annexe 2}.
\begin{enumerate}
\item On note :
\begin{itemize}
\item $D_1$ l'aire du domaine délimité par la courbe $C_f$, l'axe des abscisses et les droites d'équations $x = 1$ et $x = 3$ ;
\item $D_2$ l'aire du domaine délimité par les courbes $C_g$, $C_f$ et les droites d'équation $x = 3$
et $x = 6$.
\end{itemize}
\begin{enumerate}
\item On hachure les domaines $D_l$ et $D_2$ sur le graphique donné en \textbf{annexe 2}.
\item% Encadrer, par deux entiers consécutifs, les aires, en unités d'aire, des domaines $D_1$ et $D_2$.
%\emph{Il paraît très difficile d'encadrer les domaines $D_1$ et $D_2$ par deux entiers consécutifs en déterminant des polygones intérieurs et extérieurs aux domaines; ou alors il faut faire des découpages \og audacieux \fg{}.}
On encadre la première surface par deux trapèzes de même hauteur 2 et de bases respectives 3 et 7 pour l'intérieur (en vert) et 3 et 8 pour l'extérieur (en rouge), ce qui donne :
\[\dfrac{3 + 7}{2} \times 2 = 10 < D_1 < \dfrac{3 + 8}{2}{2} = 11.\]
%On trouve néanmoins $10<D_1<11$ et $10<D_2<11$.
La deuxième surface est plus délicate à encadrer. En utilisant pour l'intérieur les points de coordonnées (3~;~3) (3~;~9) (4,5~;~4,5) (6~;~1,5) (6~;~0,5) (4,5~;~1) et pour l'extérieur les points de coordonnées (3~;~3) (3~;~9) (4,5~;~4,5) (6~;~1,5) (6~;~0,5) (4,5~;~1) on obtient l'encadrement
\[10,625 < D_2 < 11,5.\]
On trouve donc une aire proche de 11.
\end{enumerate}
\item La commande $Int(f(x),x,a,b)$ d'un logiciel de calcul formel permet de calculer la valeur de l'intégrale $\displaystyle \int_{a}^{b} f(x)\mathrm{d}x$.
On obtient alors les résultats suivants pour quatre intégrales:
\begin{center}
\renewcommand\arraystretch{1.7}
\begin{tabular}{|c| m{7cm}|}
\hlinewd{1mm}
1& $Int(20x\e^{-x} ,x,1,2)$\\\hline
&$40\e^{-1}-60\e^{-2}$\\\hlinewd{1mm}
2&$Int(20x\e^{-x}, x,2,3$)\\\hline
&$60\e^{-2}-80\e^{-3}$\\\hlinewd{1mm}
3& $Int(20x\e^{-x}, x,3,6)$\\
&$80\e^{-3} -140\e^{-6}$\\\hlinewd{1mm}
4& $Int(20x^2\e^{-x} ,x,3,6)$\\
&$340\e^{-3} -\np{1 000}\e^{-6}$\\\hlinewd{1mm}
\end{tabular}
\end{center}
\begin{enumerate}
\item %Déterminer les aires des domaines $D_1$ et $D_2$ en justifiant la réponse. On donnera les valeurs exactes.
La fonction $f$ est positive sur $\cd 1~;~3\cg$ donc $D_1 = \ds\int_{1}^{3} f(x) \d x$:
$D_1 = \ds\int_{1}^{3} f(x) \d x = \ds\int_{1}^{2} f(x) \d x + \ds\int_{2}^{3} f(x) \d x
= \left (40\e^{-1}-60\e^{-2}\right ) + \left (60\e^{-2}-80\e^{-3} \right )\\[5pt]
\phantom{D_1}
= 40\e^{-1}-80\e^{-3}$~U.A.
\smallskip
La fonction $g$ est supérieure à $f$ sur $\cd 3~;~6\cg$ donc $g-f>0$ et donc $D_2 = \ds\int_{3}^{6} \left ( g(x)- f(x)\right ) \d x $:
$D_2 = \ds\int_{3}^{6} \left ( g(x)- f(x)\right ) \d x
= \ds\int_{3}^{6} g(x) \d x - \ds\int_{3}^{6} f(x) \d x
=$
$D_2 = \left (340\e^{-3} -\np{1 000}\e^{-6} \right ) - \left ( 80\e^{-3} -140\e^{-6}\right )
= 340\e^{-3} -\np{1 000}\e^{-6} - 80\e^{-3} +140\e^{-6}
= $
$D_2 =260\e^{-3} - 860\e^{-6}$~U.A.
\item% Comparer les valeurs des deux aires obtenues.
$D_1= 40\e^{-1}-80\e^{-3} \approx 10,73$ et $D_2 = 260\e^{-3} - 860\e^{-6} \approx 10,81$ donc $D_1<D_2$.
\end{enumerate}
\end{enumerate}
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\section*{Annexe 1 \hspace{1em}(exercice 1)}
\textbf{(À rendre avec la copie)}
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\section*{Annexe 2\hspace{1em}(exercice 4)}
\textbf{(À rendre avec la copie)}
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\expandafter\def\csname simpleicon@adobeacrobatreader\endcsname {\simpleiconsmapOne\symbol{28}}
\expandafter\def\csname simpleicon@adobeaftereffects\endcsname {\simpleiconsmapOne\symbol{29}}
\expandafter\def\csname simpleicon@adobeaudition\endcsname {\simpleiconsmapOne\symbol{30}}
\expandafter\def\csname simpleicon@adobecreativecloud\endcsname {\simpleiconsmapOne\symbol{31}}
\expandafter\def\csname simpleicon@adobedreamweaver\endcsname {\simpleiconsmapOne\symbol{32}}
\expandafter\def\csname simpleicon@adobefonts\endcsname {\simpleiconsmapOne\symbol{33}}
\expandafter\def\csname simpleicon@adobeillustrator\endcsname {\simpleiconsmapOne\symbol{34}}
\expandafter\def\csname simpleicon@adobeindesign\endcsname {\simpleiconsmapOne\symbol{35}}
\expandafter\def\csname simpleicon@adobelightroom\endcsname {\simpleiconsmapOne\symbol{36}}
\expandafter\def\csname simpleicon@adobelightroomclassic\endcsname {\simpleiconsmapOne\symbol{37}}
\expandafter\def\csname simpleicon@adobephotoshop\endcsname {\simpleiconsmapOne\symbol{38}}
\expandafter\def\csname simpleicon@adobepremierepro\endcsname {\simpleiconsmapOne\symbol{39}}
\expandafter\def\csname simpleicon@adobexd\endcsname {\simpleiconsmapOne\symbol{40}}
\expandafter\def\csname simpleicon@adonisjs\endcsname {\simpleiconsmapOne\symbol{41}}
\expandafter\def\csname simpleicon@adyen\endcsname {\simpleiconsmapOne\symbol{42}}
\expandafter\def\csname simpleicon@aerlingus\endcsname {\simpleiconsmapOne\symbol{43}}
\expandafter\def\csname simpleicon@aeroflot\endcsname {\simpleiconsmapOne\symbol{44}}
\expandafter\def\csname simpleicon@aeromexico\endcsname {\simpleiconsmapOne\symbol{45}}
\expandafter\def\csname simpleicon@aerospike\endcsname {\simpleiconsmapOne\symbol{46}}
\expandafter\def\csname simpleicon@aew\endcsname {\simpleiconsmapOne\symbol{47}}
\expandafter\def\csname simpleicon@affinity\endcsname {\simpleiconsmapOne\symbol{48}}
\expandafter\def\csname simpleicon@affinitydesigner\endcsname {\simpleiconsmapOne\symbol{49}}
\expandafter\def\csname simpleicon@affinityphoto\endcsname {\simpleiconsmapOne\symbol{50}}
\expandafter\def\csname simpleicon@affinitypublisher\endcsname {\simpleiconsmapOne\symbol{51}}
\expandafter\def\csname simpleicon@aframe\endcsname {\simpleiconsmapOne\symbol{52}}
\expandafter\def\csname simpleicon@agora\endcsname {\simpleiconsmapOne\symbol{53}}
\expandafter\def\csname simpleicon@aib\endcsname {\simpleiconsmapOne\symbol{54}}
\expandafter\def\csname simpleicon@aidungeon\endcsname {\simpleiconsmapOne\symbol{55}}
\expandafter\def\csname simpleicon@aiohttp\endcsname {\simpleiconsmapOne\symbol{56}}
\expandafter\def\csname simpleicon@aiqfome\endcsname {\simpleiconsmapOne\symbol{57}}
\expandafter\def\csname simpleicon@airasia\endcsname {\simpleiconsmapOne\symbol{58}}
\expandafter\def\csname simpleicon@airbnb\endcsname {\simpleiconsmapOne\symbol{59}}
\expandafter\def\csname simpleicon@airbus\endcsname {\simpleiconsmapOne\symbol{60}}
\expandafter\def\csname simpleicon@airbyte\endcsname {\simpleiconsmapOne\symbol{61}}
\expandafter\def\csname simpleicon@aircall\endcsname {\simpleiconsmapOne\symbol{62}}
\expandafter\def\csname simpleicon@aircanada\endcsname {\simpleiconsmapOne\symbol{63}}
\expandafter\def\csname simpleicon@airchina\endcsname {\simpleiconsmapOne\symbol{64}}
\expandafter\def\csname simpleicon@airfrance\endcsname {\simpleiconsmapOne\symbol{65}}
\expandafter\def\csname simpleicon@airplayaudio\endcsname {\simpleiconsmapOne\symbol{66}}
\expandafter\def\csname simpleicon@airplayvideo\endcsname {\simpleiconsmapOne\symbol{67}}
\expandafter\def\csname simpleicon@airtable\endcsname {\simpleiconsmapOne\symbol{68}}
\expandafter\def\csname simpleicon@ajv\endcsname {\simpleiconsmapOne\symbol{69}}
\expandafter\def\csname simpleicon@alacritty\endcsname {\simpleiconsmapOne\symbol{70}}
\expandafter\def\csname simpleicon@albertheijn\endcsname {\simpleiconsmapOne\symbol{71}}
\expandafter\def\csname simpleicon@aldinord\endcsname {\simpleiconsmapOne\symbol{72}}
\expandafter\def\csname simpleicon@aldisud\endcsname {\simpleiconsmapOne\symbol{73}}
\expandafter\def\csname simpleicon@alfaromeo\endcsname {\simpleiconsmapOne\symbol{74}}
\expandafter\def\csname simpleicon@alfred\endcsname {\simpleiconsmapOne\symbol{75}}
\expandafter\def\csname simpleicon@algolia\endcsname {\simpleiconsmapOne\symbol{76}}
\expandafter\def\csname simpleicon@algorand\endcsname {\simpleiconsmapOne\symbol{77}}
\expandafter\def\csname simpleicon@alibabacloud\endcsname {\simpleiconsmapOne\symbol{78}}
\expandafter\def\csname simpleicon@alibabadotcom\endcsname {\simpleiconsmapOne\symbol{79}}
\expandafter\def\csname simpleicon@aliexpress\endcsname {\simpleiconsmapOne\symbol{80}}
\expandafter\def\csname simpleicon@alipay\endcsname {\simpleiconsmapOne\symbol{81}}
\expandafter\def\csname simpleicon@alitalia\endcsname {\simpleiconsmapOne\symbol{82}}
\expandafter\def\csname simpleicon@allegro\endcsname {\simpleiconsmapOne\symbol{83}}
\expandafter\def\csname simpleicon@alliedmodders\endcsname {\simpleiconsmapOne\symbol{84}}
\expandafter\def\csname simpleicon@allocine\endcsname {\simpleiconsmapOne\symbol{85}}
\expandafter\def\csname simpleicon@alltrails\endcsname {\simpleiconsmapOne\symbol{86}}
\expandafter\def\csname simpleicon@alpinedotjs\endcsname {\simpleiconsmapOne\symbol{87}}
\expandafter\def\csname simpleicon@alpinelinux\endcsname {\simpleiconsmapOne\symbol{88}}
\expandafter\def\csname simpleicon@altiumdesigner\endcsname {\simpleiconsmapOne\symbol{89}}
\expandafter\def\csname simpleicon@amazon\endcsname {\simpleiconsmapOne\symbol{90}}
\expandafter\def\csname simpleicon@amazonalexa\endcsname {\simpleiconsmapOne\symbol{91}}
\expandafter\def\csname simpleicon@amazonapigateway\endcsname {\simpleiconsmapOne\symbol{92}}
\expandafter\def\csname simpleicon@amazonaws\endcsname {\simpleiconsmapOne\symbol{93}}
\expandafter\def\csname simpleicon@amazoncloudwatch\endcsname {\simpleiconsmapOne\symbol{94}}
\expandafter\def\csname simpleicon@amazondynamodb\endcsname {\simpleiconsmapOne\symbol{95}}
\expandafter\def\csname simpleicon@amazonec2\endcsname {\simpleiconsmapOne\symbol{96}}
\expandafter\def\csname simpleicon@amazonecs\endcsname {\simpleiconsmapOne\symbol{97}}
\expandafter\def\csname simpleicon@amazoneks\endcsname {\simpleiconsmapOne\symbol{98}}
\expandafter\def\csname simpleicon@amazonfiretv\endcsname {\simpleiconsmapOne\symbol{99}}
\expandafter\def\csname simpleicon@amazonlumberyard\endcsname {\simpleiconsmapOne\symbol{100}}
\expandafter\def\csname simpleicon@amazonpay\endcsname {\simpleiconsmapOne\symbol{101}}
\expandafter\def\csname simpleicon@amazonprime\endcsname {\simpleiconsmapOne\symbol{102}}
\expandafter\def\csname simpleicon@amazonrds\endcsname {\simpleiconsmapOne\symbol{103}}
\expandafter\def\csname simpleicon@amazons3\endcsname {\simpleiconsmapOne\symbol{104}}
\expandafter\def\csname simpleicon@amazonsqs\endcsname {\simpleiconsmapOne\symbol{105}}
\expandafter\def\csname simpleicon@amd\endcsname {\simpleiconsmapOne\symbol{106}}
\expandafter\def\csname simpleicon@americanairlines\endcsname {\simpleiconsmapOne\symbol{107}}
\expandafter\def\csname simpleicon@americanexpress\endcsname {\simpleiconsmapOne\symbol{108}}
\expandafter\def\csname simpleicon@amg\endcsname {\simpleiconsmapOne\symbol{109}}
\expandafter\def\csname simpleicon@amp\endcsname {\simpleiconsmapOne\symbol{110}}
\expandafter\def\csname simpleicon@amul\endcsname {\simpleiconsmapOne\symbol{111}}
\expandafter\def\csname simpleicon@ana\endcsname {\simpleiconsmapOne\symbol{112}}
\expandafter\def\csname simpleicon@anaconda\endcsname {\simpleiconsmapOne\symbol{113}}
\expandafter\def\csname simpleicon@analogue\endcsname {\simpleiconsmapOne\symbol{114}}
\expandafter\def\csname simpleicon@anchor\endcsname {\simpleiconsmapOne\symbol{115}}
\expandafter\def\csname simpleicon@andela\endcsname {\simpleiconsmapOne\symbol{116}}
\expandafter\def\csname simpleicon@android\endcsname {\simpleiconsmapOne\symbol{117}}
\expandafter\def\csname simpleicon@androidauto\endcsname {\simpleiconsmapOne\symbol{118}}
\expandafter\def\csname simpleicon@androidstudio\endcsname {\simpleiconsmapOne\symbol{119}}
\expandafter\def\csname simpleicon@angellist\endcsname {\simpleiconsmapOne\symbol{120}}
\expandafter\def\csname simpleicon@angular\endcsname {\simpleiconsmapOne\symbol{121}}
\expandafter\def\csname simpleicon@angularjs\endcsname {\simpleiconsmapOne\symbol{122}}
\expandafter\def\csname simpleicon@angularuniversal\endcsname {\simpleiconsmapOne\symbol{123}}
\expandafter\def\csname simpleicon@anilist\endcsname {\simpleiconsmapOne\symbol{124}}
\expandafter\def\csname simpleicon@ansible\endcsname {\simpleiconsmapOne\symbol{125}}
\expandafter\def\csname simpleicon@ansys\endcsname {\simpleiconsmapOne\symbol{126}}
\expandafter\def\csname simpleicon@anta\endcsname {\simpleiconsmapOne\symbol{127}}
\expandafter\def\csname simpleicon@antdesign\endcsname {\simpleiconsmapOne\symbol{128}}
\expandafter\def\csname simpleicon@antena3\endcsname {\simpleiconsmapOne\symbol{129}}
\expandafter\def\csname simpleicon@anydesk\endcsname {\simpleiconsmapOne\symbol{130}}
\expandafter\def\csname simpleicon@aol\endcsname {\simpleiconsmapOne\symbol{131}}
\expandafter\def\csname simpleicon@apache\endcsname {\simpleiconsmapOne\symbol{132}}
\expandafter\def\csname simpleicon@apacheairflow\endcsname {\simpleiconsmapOne\symbol{133}}
\expandafter\def\csname simpleicon@apacheant\endcsname {\simpleiconsmapOne\symbol{134}}
\expandafter\def\csname simpleicon@apachecassandra\endcsname {\simpleiconsmapOne\symbol{135}}
\expandafter\def\csname simpleicon@apachecloudstack\endcsname {\simpleiconsmapOne\symbol{136}}
\expandafter\def\csname simpleicon@apachecordova\endcsname {\simpleiconsmapOne\symbol{137}}
\expandafter\def\csname simpleicon@apachecouchdb\endcsname {\simpleiconsmapOne\symbol{138}}
\expandafter\def\csname simpleicon@apachedruid\endcsname {\simpleiconsmapOne\symbol{139}}
\expandafter\def\csname simpleicon@apacheecharts\endcsname {\simpleiconsmapOne\symbol{140}}
\expandafter\def\csname simpleicon@apacheflink\endcsname {\simpleiconsmapOne\symbol{141}}
\expandafter\def\csname simpleicon@apachegroovy\endcsname {\simpleiconsmapOne\symbol{142}}
\expandafter\def\csname simpleicon@apachehadoop\endcsname {\simpleiconsmapOne\symbol{143}}
\expandafter\def\csname simpleicon@apachehive\endcsname {\simpleiconsmapOne\symbol{144}}
\expandafter\def\csname simpleicon@apachejmeter\endcsname {\simpleiconsmapOne\symbol{145}}
\expandafter\def\csname simpleicon@apachekafka\endcsname {\simpleiconsmapOne\symbol{146}}
\expandafter\def\csname simpleicon@apachekylin\endcsname {\simpleiconsmapOne\symbol{147}}
\expandafter\def\csname simpleicon@apachemaven\endcsname {\simpleiconsmapOne\symbol{148}}
\expandafter\def\csname simpleicon@apachenetbeanside\endcsname {\simpleiconsmapOne\symbol{149}}
\expandafter\def\csname simpleicon@apacheopenoffice\endcsname {\simpleiconsmapOne\symbol{150}}
\expandafter\def\csname simpleicon@apachepulsar\endcsname {\simpleiconsmapOne\symbol{151}}
\expandafter\def\csname simpleicon@apacherocketmq\endcsname {\simpleiconsmapOne\symbol{152}}
\expandafter\def\csname simpleicon@apachesolr\endcsname {\simpleiconsmapOne\symbol{153}}
\expandafter\def\csname simpleicon@apachespark\endcsname {\simpleiconsmapOne\symbol{154}}
\expandafter\def\csname simpleicon@apachetomcat\endcsname {\simpleiconsmapOne\symbol{155}}
\expandafter\def\csname simpleicon@aparat\endcsname {\simpleiconsmapOne\symbol{156}}
\expandafter\def\csname simpleicon@apollographql\endcsname {\simpleiconsmapOne\symbol{157}}
\expandafter\def\csname simpleicon@apostrophe\endcsname {\simpleiconsmapOne\symbol{158}}
\expandafter\def\csname simpleicon@appannie\endcsname {\simpleiconsmapOne\symbol{159}}
\expandafter\def\csname simpleicon@appian\endcsname {\simpleiconsmapOne\symbol{160}}
\expandafter\def\csname simpleicon@apple\endcsname {\simpleiconsmapOne\symbol{161}}
\expandafter\def\csname simpleicon@applearcade\endcsname {\simpleiconsmapOne\symbol{162}}
\expandafter\def\csname simpleicon@applemusic\endcsname {\simpleiconsmapOne\symbol{163}}
\expandafter\def\csname simpleicon@applenews\endcsname {\simpleiconsmapOne\symbol{164}}
\expandafter\def\csname simpleicon@applepay\endcsname {\simpleiconsmapOne\symbol{165}}
\expandafter\def\csname simpleicon@applepodcasts\endcsname {\simpleiconsmapOne\symbol{166}}
\expandafter\def\csname simpleicon@appletv\endcsname {\simpleiconsmapOne\symbol{167}}
\expandafter\def\csname simpleicon@appsignal\endcsname {\simpleiconsmapOne\symbol{168}}
\expandafter\def\csname simpleicon@appstore\endcsname {\simpleiconsmapOne\symbol{169}}
\expandafter\def\csname simpleicon@appveyor\endcsname {\simpleiconsmapOne\symbol{170}}
\expandafter\def\csname simpleicon@appwrite\endcsname {\simpleiconsmapOne\symbol{171}}
\expandafter\def\csname simpleicon@aqua\endcsname {\simpleiconsmapOne\symbol{172}}
\expandafter\def\csname simpleicon@aral\endcsname {\simpleiconsmapOne\symbol{173}}
\expandafter\def\csname simpleicon@arangodb\endcsname {\simpleiconsmapOne\symbol{174}}
\expandafter\def\csname simpleicon@archicad\endcsname {\simpleiconsmapOne\symbol{175}}
\expandafter\def\csname simpleicon@archiveofourown\endcsname {\simpleiconsmapOne\symbol{176}}
\expandafter\def\csname simpleicon@archlinux\endcsname {\simpleiconsmapOne\symbol{177}}
\expandafter\def\csname simpleicon@ardour\endcsname {\simpleiconsmapOne\symbol{178}}
\expandafter\def\csname simpleicon@arduino\endcsname {\simpleiconsmapOne\symbol{179}}
\expandafter\def\csname simpleicon@argo\endcsname {\simpleiconsmapOne\symbol{180}}
\expandafter\def\csname simpleicon@argos\endcsname {\simpleiconsmapOne\symbol{181}}
\expandafter\def\csname simpleicon@arkecosystem\endcsname {\simpleiconsmapOne\symbol{182}}
\expandafter\def\csname simpleicon@arlo\endcsname {\simpleiconsmapOne\symbol{183}}
\expandafter\def\csname simpleicon@arm\endcsname {\simpleiconsmapOne\symbol{184}}
\expandafter\def\csname simpleicon@artifacthub\endcsname {\simpleiconsmapOne\symbol{185}}
\expandafter\def\csname simpleicon@artixlinux\endcsname {\simpleiconsmapOne\symbol{186}}
\expandafter\def\csname simpleicon@artstation\endcsname {\simpleiconsmapOne\symbol{187}}
\expandafter\def\csname simpleicon@arxiv\endcsname {\simpleiconsmapOne\symbol{188}}
\expandafter\def\csname simpleicon@asana\endcsname {\simpleiconsmapOne\symbol{189}}
\expandafter\def\csname simpleicon@asciidoctor\endcsname {\simpleiconsmapOne\symbol{190}}
\expandafter\def\csname simpleicon@asciinema\endcsname {\simpleiconsmapOne\symbol{191}}
\expandafter\def\csname simpleicon@asda\endcsname {\simpleiconsmapOne\symbol{192}}
\expandafter\def\csname simpleicon@aseprite\endcsname {\simpleiconsmapOne\symbol{193}}
\expandafter\def\csname simpleicon@askfm\endcsname {\simpleiconsmapOne\symbol{194}}
\expandafter\def\csname simpleicon@askubuntu\endcsname {\simpleiconsmapOne\symbol{195}}
\expandafter\def\csname simpleicon@assemblyscript\endcsname {\simpleiconsmapOne\symbol{196}}
\expandafter\def\csname simpleicon@astonmartin\endcsname {\simpleiconsmapOne\symbol{197}}
\expandafter\def\csname simpleicon@astro\endcsname {\simpleiconsmapOne\symbol{198}}
\expandafter\def\csname simpleicon@asus\endcsname {\simpleiconsmapOne\symbol{199}}
\expandafter\def\csname simpleicon@atandt\endcsname {\simpleiconsmapOne\symbol{200}}
\expandafter\def\csname simpleicon@atari\endcsname {\simpleiconsmapOne\symbol{201}}
\expandafter\def\csname simpleicon@atlassian\endcsname {\simpleiconsmapOne\symbol{202}}
\expandafter\def\csname simpleicon@atom\endcsname {\simpleiconsmapOne\symbol{203}}
\expandafter\def\csname simpleicon@auchan\endcsname {\simpleiconsmapOne\symbol{204}}
\expandafter\def\csname simpleicon@audacity\endcsname {\simpleiconsmapOne\symbol{205}}
\expandafter\def\csname simpleicon@audi\endcsname {\simpleiconsmapOne\symbol{206}}
\expandafter\def\csname simpleicon@audible\endcsname {\simpleiconsmapOne\symbol{207}}
\expandafter\def\csname simpleicon@audioboom\endcsname {\simpleiconsmapOne\symbol{208}}
\expandafter\def\csname simpleicon@audiomack\endcsname {\simpleiconsmapOne\symbol{209}}
\expandafter\def\csname simpleicon@audiotechnica\endcsname {\simpleiconsmapOne\symbol{210}}
\expandafter\def\csname simpleicon@aurelia\endcsname {\simpleiconsmapOne\symbol{211}}
\expandafter\def\csname simpleicon@auth0\endcsname {\simpleiconsmapOne\symbol{212}}
\expandafter\def\csname simpleicon@authy\endcsname {\simpleiconsmapOne\symbol{213}}
\expandafter\def\csname simpleicon@autodesk\endcsname {\simpleiconsmapOne\symbol{214}}
\expandafter\def\csname simpleicon@autohotkey\endcsname {\simpleiconsmapOne\symbol{215}}
\expandafter\def\csname simpleicon@automattic\endcsname {\simpleiconsmapOne\symbol{216}}
\expandafter\def\csname simpleicon@autoprefixer\endcsname {\simpleiconsmapOne\symbol{217}}
\expandafter\def\csname simpleicon@avajs\endcsname {\simpleiconsmapOne\symbol{218}}
\expandafter\def\csname simpleicon@avast\endcsname {\simpleiconsmapOne\symbol{219}}
\expandafter\def\csname simpleicon@awesomelists\endcsname {\simpleiconsmapOne\symbol{220}}
\expandafter\def\csname simpleicon@awesomewm\endcsname {\simpleiconsmapOne\symbol{221}}
\expandafter\def\csname simpleicon@awsamplify\endcsname {\simpleiconsmapOne\symbol{222}}
\expandafter\def\csname simpleicon@awsfargate\endcsname {\simpleiconsmapOne\symbol{223}}
\expandafter\def\csname simpleicon@awslambda\endcsname {\simpleiconsmapOne\symbol{224}}
\expandafter\def\csname simpleicon@axios\endcsname {\simpleiconsmapOne\symbol{225}}
\expandafter\def\csname simpleicon@azureartifacts\endcsname {\simpleiconsmapOne\symbol{226}}
\expandafter\def\csname simpleicon@azuredataexplorer\endcsname {\simpleiconsmapOne\symbol{227}}
\expandafter\def\csname simpleicon@azuredevops\endcsname {\simpleiconsmapOne\symbol{228}}
\expandafter\def\csname simpleicon@azurefunctions\endcsname {\simpleiconsmapOne\symbol{229}}
\expandafter\def\csname simpleicon@azurepipelines\endcsname {\simpleiconsmapOne\symbol{230}}
\expandafter\def\csname simpleicon@babel\endcsname {\simpleiconsmapOne\symbol{231}}
\expandafter\def\csname simpleicon@backbonedotjs\endcsname {\simpleiconsmapOne\symbol{232}}
\expandafter\def\csname simpleicon@backendless\endcsname {\simpleiconsmapOne\symbol{233}}
\expandafter\def\csname simpleicon@badgr\endcsname {\simpleiconsmapOne\symbol{234}}
\expandafter\def\csname simpleicon@badoo\endcsname {\simpleiconsmapOne\symbol{235}}
\expandafter\def\csname simpleicon@baidu\endcsname {\simpleiconsmapOne\symbol{236}}
\expandafter\def\csname simpleicon@bamboo\endcsname {\simpleiconsmapOne\symbol{237}}
\expandafter\def\csname simpleicon@bandcamp\endcsname {\simpleiconsmapOne\symbol{238}}
\expandafter\def\csname simpleicon@bandlab\endcsname {\simpleiconsmapOne\symbol{239}}
\expandafter\def\csname simpleicon@bandrautomation\endcsname {\simpleiconsmapOne\symbol{240}}
\expandafter\def\csname simpleicon@bandsintown\endcsname {\simpleiconsmapOne\symbol{241}}
\expandafter\def\csname simpleicon@bankofamerica\endcsname {\simpleiconsmapOne\symbol{242}}
\expandafter\def\csname simpleicon@barclays\endcsname {\simpleiconsmapOne\symbol{243}}
\expandafter\def\csname simpleicon@baremetrics\endcsname {\simpleiconsmapOne\symbol{244}}
\expandafter\def\csname simpleicon@basecamp\endcsname {\simpleiconsmapOne\symbol{245}}
\expandafter\def\csname simpleicon@bastyon\endcsname {\simpleiconsmapOne\symbol{246}}
\expandafter\def\csname simpleicon@bata\endcsname {\simpleiconsmapOne\symbol{247}}
\expandafter\def\csname simpleicon@bathasu\endcsname {\simpleiconsmapOne\symbol{248}}
\expandafter\def\csname simpleicon@battledotnet\endcsname {\simpleiconsmapOne\symbol{249}}
\expandafter\def\csname simpleicon@bbc\endcsname {\simpleiconsmapOne\symbol{250}}
\expandafter\def\csname simpleicon@bbciplayer\endcsname {\simpleiconsmapOne\symbol{251}}
\expandafter\def\csname simpleicon@beatport\endcsname {\simpleiconsmapOne\symbol{252}}
\expandafter\def\csname simpleicon@beats\endcsname {\simpleiconsmapOne\symbol{253}}
\expandafter\def\csname simpleicon@beatsbydre\endcsname {\simpleiconsmapOne\symbol{254}}
\expandafter\def\csname simpleicon@behance\endcsname {\simpleiconsmapOne\symbol{255}}
\expandafter\def\csname simpleicon@beijingsubway\endcsname {\simpleiconsmapTwo\symbol{0}}
\expandafter\def\csname simpleicon@bem\endcsname {\simpleiconsmapTwo\symbol{1}}
\expandafter\def\csname simpleicon@bentley\endcsname {\simpleiconsmapTwo\symbol{2}}
\expandafter\def\csname simpleicon@betfair\endcsname {\simpleiconsmapTwo\symbol{3}}
\expandafter\def\csname simpleicon@bigbasket\endcsname {\simpleiconsmapTwo\symbol{4}}
\expandafter\def\csname simpleicon@bigbluebutton\endcsname {\simpleiconsmapTwo\symbol{5}}
\expandafter\def\csname simpleicon@bigcartel\endcsname {\simpleiconsmapTwo\symbol{6}}
\expandafter\def\csname simpleicon@bigcommerce\endcsname {\simpleiconsmapTwo\symbol{7}}
\expandafter\def\csname simpleicon@bilibili\endcsname {\simpleiconsmapTwo\symbol{8}}
\expandafter\def\csname simpleicon@binance\endcsname {\simpleiconsmapTwo\symbol{9}}
\expandafter\def\csname simpleicon@biolink\endcsname {\simpleiconsmapTwo\symbol{10}}
\expandafter\def\csname simpleicon@bit\endcsname {\simpleiconsmapTwo\symbol{11}}
\expandafter\def\csname simpleicon@bitbucket\endcsname {\simpleiconsmapTwo\symbol{12}}
\expandafter\def\csname simpleicon@bitcoin\endcsname {\simpleiconsmapTwo\symbol{13}}
\expandafter\def\csname simpleicon@bitcoincash\endcsname {\simpleiconsmapTwo\symbol{14}}
\expandafter\def\csname simpleicon@bitcoinsv\endcsname {\simpleiconsmapTwo\symbol{15}}
\expandafter\def\csname simpleicon@bitdefender\endcsname {\simpleiconsmapTwo\symbol{16}}
\expandafter\def\csname simpleicon@bitly\endcsname {\simpleiconsmapTwo\symbol{17}}
\expandafter\def\csname simpleicon@bitrise\endcsname {\simpleiconsmapTwo\symbol{18}}
\expandafter\def\csname simpleicon@bitwarden\endcsname {\simpleiconsmapTwo\symbol{19}}
\expandafter\def\csname simpleicon@bitwig\endcsname {\simpleiconsmapTwo\symbol{20}}
\expandafter\def\csname simpleicon@blackberry\endcsname {\simpleiconsmapTwo\symbol{21}}
\expandafter\def\csname simpleicon@blazemeter\endcsname {\simpleiconsmapTwo\symbol{22}}
\expandafter\def\csname simpleicon@blazor\endcsname {\simpleiconsmapTwo\symbol{23}}
\expandafter\def\csname simpleicon@blender\endcsname {\simpleiconsmapTwo\symbol{24}}
\expandafter\def\csname simpleicon@blockchaindotcom\endcsname {\simpleiconsmapTwo\symbol{25}}
\expandafter\def\csname simpleicon@blogger\endcsname {\simpleiconsmapTwo\symbol{26}}
\expandafter\def\csname simpleicon@bloglovin\endcsname {\simpleiconsmapTwo\symbol{27}}
\expandafter\def\csname simpleicon@blueprint\endcsname {\simpleiconsmapTwo\symbol{28}}
\expandafter\def\csname simpleicon@bluetooth\endcsname {\simpleiconsmapTwo\symbol{29}}
\expandafter\def\csname simpleicon@bmcsoftware\endcsname {\simpleiconsmapTwo\symbol{30}}
\expandafter\def\csname simpleicon@bmw\endcsname {\simpleiconsmapTwo\symbol{31}}
\expandafter\def\csname simpleicon@boehringeringelheim\endcsname {\simpleiconsmapTwo\symbol{32}}
\expandafter\def\csname simpleicon@boeing\endcsname {\simpleiconsmapTwo\symbol{33}}
\expandafter\def\csname simpleicon@bookalope\endcsname {\simpleiconsmapTwo\symbol{34}}
\expandafter\def\csname simpleicon@bookbub\endcsname {\simpleiconsmapTwo\symbol{35}}
\expandafter\def\csname simpleicon@bookmeter\endcsname {\simpleiconsmapTwo\symbol{36}}
\expandafter\def\csname simpleicon@bookmyshow\endcsname {\simpleiconsmapTwo\symbol{37}}
\expandafter\def\csname simpleicon@bookstack\endcsname {\simpleiconsmapTwo\symbol{38}}
\expandafter\def\csname simpleicon@boost\endcsname {\simpleiconsmapTwo\symbol{39}}
\expandafter\def\csname simpleicon@boots\endcsname {\simpleiconsmapTwo\symbol{40}}
\expandafter\def\csname simpleicon@bootstrap\endcsname {\simpleiconsmapTwo\symbol{41}}
\expandafter\def\csname simpleicon@borgbackup\endcsname {\simpleiconsmapTwo\symbol{42}}
\expandafter\def\csname simpleicon@bosch\endcsname {\simpleiconsmapTwo\symbol{43}}
\expandafter\def\csname simpleicon@bose\endcsname {\simpleiconsmapTwo\symbol{44}}
\expandafter\def\csname simpleicon@boulanger\endcsname {\simpleiconsmapTwo\symbol{45}}
\expandafter\def\csname simpleicon@bower\endcsname {\simpleiconsmapTwo\symbol{46}}
\expandafter\def\csname simpleicon@box\endcsname {\simpleiconsmapTwo\symbol{47}}
\expandafter\def\csname simpleicon@boxysvg\endcsname {\simpleiconsmapTwo\symbol{48}}
\expandafter\def\csname simpleicon@brandfolder\endcsname {\simpleiconsmapTwo\symbol{49}}
\expandafter\def\csname simpleicon@brave\endcsname {\simpleiconsmapTwo\symbol{50}}
\expandafter\def\csname simpleicon@breaker\endcsname {\simpleiconsmapTwo\symbol{51}}
\expandafter\def\csname simpleicon@britishairways\endcsname {\simpleiconsmapTwo\symbol{52}}
\expandafter\def\csname simpleicon@broadcom\endcsname {\simpleiconsmapTwo\symbol{53}}
\expandafter\def\csname simpleicon@bt\endcsname {\simpleiconsmapTwo\symbol{54}}
\expandafter\def\csname simpleicon@buddy\endcsname {\simpleiconsmapTwo\symbol{55}}
\expandafter\def\csname simpleicon@budibase\endcsname {\simpleiconsmapTwo\symbol{56}}
\expandafter\def\csname simpleicon@buefy\endcsname {\simpleiconsmapTwo\symbol{57}}
\expandafter\def\csname simpleicon@buffer\endcsname {\simpleiconsmapTwo\symbol{58}}
\expandafter\def\csname simpleicon@bugatti\endcsname {\simpleiconsmapTwo\symbol{59}}
\expandafter\def\csname simpleicon@bugcrowd\endcsname {\simpleiconsmapTwo\symbol{60}}
\expandafter\def\csname simpleicon@bugsnag\endcsname {\simpleiconsmapTwo\symbol{61}}
\expandafter\def\csname simpleicon@buildkite\endcsname {\simpleiconsmapTwo\symbol{62}}
\expandafter\def\csname simpleicon@bukalapak\endcsname {\simpleiconsmapTwo\symbol{63}}
\expandafter\def\csname simpleicon@bulma\endcsname {\simpleiconsmapTwo\symbol{64}}
\expandafter\def\csname simpleicon@bun\endcsname {\simpleiconsmapTwo\symbol{65}}
\expandafter\def\csname simpleicon@bunq\endcsname {\simpleiconsmapTwo\symbol{66}}
\expandafter\def\csname simpleicon@burgerking\endcsname {\simpleiconsmapTwo\symbol{67}}
\expandafter\def\csname simpleicon@burton\endcsname {\simpleiconsmapTwo\symbol{68}}
\expandafter\def\csname simpleicon@buymeacoffee\endcsname {\simpleiconsmapTwo\symbol{69}}
\expandafter\def\csname simpleicon@buzzfeed\endcsname {\simpleiconsmapTwo\symbol{70}}
\expandafter\def\csname simpleicon@byjus\endcsname {\simpleiconsmapTwo\symbol{71}}
\expandafter\def\csname simpleicon@byte\endcsname {\simpleiconsmapTwo\symbol{72}}
\expandafter\def\csname simpleicon@bytedance\endcsname {\simpleiconsmapTwo\symbol{73}}
\expandafter\def\csname simpleicon@c\endcsname {\simpleiconsmapTwo\symbol{74}}
\expandafter\def\csname simpleicon@cachet\endcsname {\simpleiconsmapTwo\symbol{75}}
\expandafter\def\csname simpleicon@caffeine\endcsname {\simpleiconsmapTwo\symbol{76}}
\expandafter\def\csname simpleicon@cairometro\endcsname {\simpleiconsmapTwo\symbol{77}}
\expandafter\def\csname simpleicon@cakephp\endcsname {\simpleiconsmapTwo\symbol{78}}
\expandafter\def\csname simpleicon@campaignmonitor\endcsname {\simpleiconsmapTwo\symbol{79}}
\expandafter\def\csname simpleicon@canonical\endcsname {\simpleiconsmapTwo\symbol{80}}
\expandafter\def\csname simpleicon@canva\endcsname {\simpleiconsmapTwo\symbol{81}}
\expandafter\def\csname simpleicon@capacitor\endcsname {\simpleiconsmapTwo\symbol{82}}
\expandafter\def\csname simpleicon@carrefour\endcsname {\simpleiconsmapTwo\symbol{83}}
\expandafter\def\csname simpleicon@carthrottle\endcsname {\simpleiconsmapTwo\symbol{84}}
\expandafter\def\csname simpleicon@carto\endcsname {\simpleiconsmapTwo\symbol{85}}
\expandafter\def\csname simpleicon@cashapp\endcsname {\simpleiconsmapTwo\symbol{86}}
\expandafter\def\csname simpleicon@castbox\endcsname {\simpleiconsmapTwo\symbol{87}}
\expandafter\def\csname simpleicon@castorama\endcsname {\simpleiconsmapTwo\symbol{88}}
\expandafter\def\csname simpleicon@castro\endcsname {\simpleiconsmapTwo\symbol{89}}
\expandafter\def\csname simpleicon@caterpillar\endcsname {\simpleiconsmapTwo\symbol{90}}
\expandafter\def\csname simpleicon@cbs\endcsname {\simpleiconsmapTwo\symbol{91}}
\expandafter\def\csname simpleicon@cdprojekt\endcsname {\simpleiconsmapTwo\symbol{92}}
\expandafter\def\csname simpleicon@celery\endcsname {\simpleiconsmapTwo\symbol{93}}
\expandafter\def\csname simpleicon@centos\endcsname {\simpleiconsmapTwo\symbol{94}}
\expandafter\def\csname simpleicon@ceph\endcsname {\simpleiconsmapTwo\symbol{95}}
\expandafter\def\csname simpleicon@cesium\endcsname {\simpleiconsmapTwo\symbol{96}}
\expandafter\def\csname simpleicon@chai\endcsname {\simpleiconsmapTwo\symbol{97}}
\expandafter\def\csname simpleicon@chainlink\endcsname {\simpleiconsmapTwo\symbol{98}}
\expandafter\def\csname simpleicon@chakraui\endcsname {\simpleiconsmapTwo\symbol{99}}
\expandafter\def\csname simpleicon@chartdotjs\endcsname {\simpleiconsmapTwo\symbol{100}}
\expandafter\def\csname simpleicon@chartmogul\endcsname {\simpleiconsmapTwo\symbol{101}}
\expandafter\def\csname simpleicon@chase\endcsname {\simpleiconsmapTwo\symbol{102}}
\expandafter\def\csname simpleicon@chatbot\endcsname {\simpleiconsmapTwo\symbol{103}}
\expandafter\def\csname simpleicon@checkio\endcsname {\simpleiconsmapTwo\symbol{104}}
\expandafter\def\csname simpleicon@checkmarx\endcsname {\simpleiconsmapTwo\symbol{105}}
\expandafter\def\csname simpleicon@chef\endcsname {\simpleiconsmapTwo\symbol{106}}
\expandafter\def\csname simpleicon@chemex\endcsname {\simpleiconsmapTwo\symbol{107}}
\expandafter\def\csname simpleicon@chevrolet\endcsname {\simpleiconsmapTwo\symbol{108}}
\expandafter\def\csname simpleicon@chinaeasternairlines\endcsname {\simpleiconsmapTwo\symbol{109}}
\expandafter\def\csname simpleicon@chinasouthernairlines\endcsname {\simpleiconsmapTwo\symbol{110}}
\expandafter\def\csname simpleicon@chocolatey\endcsname {\simpleiconsmapTwo\symbol{111}}
\expandafter\def\csname simpleicon@chromecast\endcsname {\simpleiconsmapTwo\symbol{112}}
\expandafter\def\csname simpleicon@chrysler\endcsname {\simpleiconsmapTwo\symbol{113}}
\expandafter\def\csname simpleicon@chupachups\endcsname {\simpleiconsmapTwo\symbol{114}}
\expandafter\def\csname simpleicon@cilium\endcsname {\simpleiconsmapTwo\symbol{115}}
\expandafter\def\csname simpleicon@cinema4d\endcsname {\simpleiconsmapTwo\symbol{116}}
\expandafter\def\csname simpleicon@circle\endcsname {\simpleiconsmapTwo\symbol{117}}
\expandafter\def\csname simpleicon@circleci\endcsname {\simpleiconsmapTwo\symbol{118}}
\expandafter\def\csname simpleicon@cirrusci\endcsname {\simpleiconsmapTwo\symbol{119}}
\expandafter\def\csname simpleicon@cisco\endcsname {\simpleiconsmapTwo\symbol{120}}
\expandafter\def\csname simpleicon@citrix\endcsname {\simpleiconsmapTwo\symbol{121}}
\expandafter\def\csname simpleicon@citroen\endcsname {\simpleiconsmapTwo\symbol{122}}
\expandafter\def\csname simpleicon@civicrm\endcsname {\simpleiconsmapTwo\symbol{123}}
\expandafter\def\csname simpleicon@civo\endcsname {\simpleiconsmapTwo\symbol{124}}
\expandafter\def\csname simpleicon@ckeditor4\endcsname {\simpleiconsmapTwo\symbol{125}}
\expandafter\def\csname simpleicon@claris\endcsname {\simpleiconsmapTwo\symbol{126}}
\expandafter\def\csname simpleicon@clickhouse\endcsname {\simpleiconsmapTwo\symbol{127}}
\expandafter\def\csname simpleicon@clickup\endcsname {\simpleiconsmapTwo\symbol{128}}
\expandafter\def\csname simpleicon@clion\endcsname {\simpleiconsmapTwo\symbol{129}}
\expandafter\def\csname simpleicon@cliqz\endcsname {\simpleiconsmapTwo\symbol{130}}
\expandafter\def\csname simpleicon@clockify\endcsname {\simpleiconsmapTwo\symbol{131}}
\expandafter\def\csname simpleicon@clojure\endcsname {\simpleiconsmapTwo\symbol{132}}
\expandafter\def\csname simpleicon@cloud66\endcsname {\simpleiconsmapTwo\symbol{133}}
\expandafter\def\csname simpleicon@cloudbees\endcsname {\simpleiconsmapTwo\symbol{134}}
\expandafter\def\csname simpleicon@cloudcannon\endcsname {\simpleiconsmapTwo\symbol{135}}
\expandafter\def\csname simpleicon@cloudera\endcsname {\simpleiconsmapTwo\symbol{136}}
\expandafter\def\csname simpleicon@cloudflare\endcsname {\simpleiconsmapTwo\symbol{137}}
\expandafter\def\csname simpleicon@cloudflarepages\endcsname {\simpleiconsmapTwo\symbol{138}}
\expandafter\def\csname simpleicon@cloudfoundry\endcsname {\simpleiconsmapTwo\symbol{139}}
\expandafter\def\csname simpleicon@cloudsmith\endcsname {\simpleiconsmapTwo\symbol{140}}
\expandafter\def\csname simpleicon@cloudways\endcsname {\simpleiconsmapTwo\symbol{141}}
\expandafter\def\csname simpleicon@clubhouse\endcsname {\simpleiconsmapTwo\symbol{142}}
\expandafter\def\csname simpleicon@clyp\endcsname {\simpleiconsmapTwo\symbol{143}}
\expandafter\def\csname simpleicon@cmake\endcsname {\simpleiconsmapTwo\symbol{144}}
\expandafter\def\csname simpleicon@cncf\endcsname {\simpleiconsmapTwo\symbol{145}}
\expandafter\def\csname simpleicon@cnn\endcsname {\simpleiconsmapTwo\symbol{146}}
\expandafter\def\csname simpleicon@cockpit\endcsname {\simpleiconsmapTwo\symbol{147}}
\expandafter\def\csname simpleicon@cockroachlabs\endcsname {\simpleiconsmapTwo\symbol{148}}
\expandafter\def\csname simpleicon@cocoapods\endcsname {\simpleiconsmapTwo\symbol{149}}
\expandafter\def\csname simpleicon@cocos\endcsname {\simpleiconsmapTwo\symbol{150}}
\expandafter\def\csname simpleicon@coda\endcsname {\simpleiconsmapTwo\symbol{151}}
\expandafter\def\csname simpleicon@codacy\endcsname {\simpleiconsmapTwo\symbol{152}}
\expandafter\def\csname simpleicon@codeberg\endcsname {\simpleiconsmapTwo\symbol{153}}
\expandafter\def\csname simpleicon@codecademy\endcsname {\simpleiconsmapTwo\symbol{154}}
\expandafter\def\csname simpleicon@codeceptjs\endcsname {\simpleiconsmapTwo\symbol{155}}
\expandafter\def\csname simpleicon@codechef\endcsname {\simpleiconsmapTwo\symbol{156}}
\expandafter\def\csname simpleicon@codeclimate\endcsname {\simpleiconsmapTwo\symbol{157}}
\expandafter\def\csname simpleicon@codecov\endcsname {\simpleiconsmapTwo\symbol{158}}
\expandafter\def\csname simpleicon@codefactor\endcsname {\simpleiconsmapTwo\symbol{159}}
\expandafter\def\csname simpleicon@codeforces\endcsname {\simpleiconsmapTwo\symbol{160}}
\expandafter\def\csname simpleicon@codeigniter\endcsname {\simpleiconsmapTwo\symbol{161}}
\expandafter\def\csname simpleicon@codemagic\endcsname {\simpleiconsmapTwo\symbol{162}}
\expandafter\def\csname simpleicon@codemirror\endcsname {\simpleiconsmapTwo\symbol{163}}
\expandafter\def\csname simpleicon@codenewbie\endcsname {\simpleiconsmapTwo\symbol{164}}
\expandafter\def\csname simpleicon@codepen\endcsname {\simpleiconsmapTwo\symbol{165}}
\expandafter\def\csname simpleicon@codeproject\endcsname {\simpleiconsmapTwo\symbol{166}}
\expandafter\def\csname simpleicon@codereview\endcsname {\simpleiconsmapTwo\symbol{167}}
\expandafter\def\csname simpleicon@codersrank\endcsname {\simpleiconsmapTwo\symbol{168}}
\expandafter\def\csname simpleicon@coderwall\endcsname {\simpleiconsmapTwo\symbol{169}}
\expandafter\def\csname simpleicon@codesandbox\endcsname {\simpleiconsmapTwo\symbol{170}}
\expandafter\def\csname simpleicon@codeship\endcsname {\simpleiconsmapTwo\symbol{171}}
\expandafter\def\csname simpleicon@codewars\endcsname {\simpleiconsmapTwo\symbol{172}}
\expandafter\def\csname simpleicon@codingame\endcsname {\simpleiconsmapTwo\symbol{173}}
\expandafter\def\csname simpleicon@codingninjas\endcsname {\simpleiconsmapTwo\symbol{174}}
\expandafter\def\csname simpleicon@codio\endcsname {\simpleiconsmapTwo\symbol{175}}
\expandafter\def\csname simpleicon@coffeescript\endcsname {\simpleiconsmapTwo\symbol{176}}
\expandafter\def\csname simpleicon@cognizant\endcsname {\simpleiconsmapTwo\symbol{177}}
\expandafter\def\csname simpleicon@coil\endcsname {\simpleiconsmapTwo\symbol{178}}
\expandafter\def\csname simpleicon@coinbase\endcsname {\simpleiconsmapTwo\symbol{179}}
\expandafter\def\csname simpleicon@coinmarketcap\endcsname {\simpleiconsmapTwo\symbol{180}}
\expandafter\def\csname simpleicon@commerzbank\endcsname {\simpleiconsmapTwo\symbol{181}}
\expandafter\def\csname simpleicon@commitlint\endcsname {\simpleiconsmapTwo\symbol{182}}
\expandafter\def\csname simpleicon@commodore\endcsname {\simpleiconsmapTwo\symbol{183}}
\expandafter\def\csname simpleicon@commonworkflowlanguage\endcsname {\simpleiconsmapTwo\symbol{184}}
\expandafter\def\csname simpleicon@composer\endcsname {\simpleiconsmapTwo\symbol{185}}
\expandafter\def\csname simpleicon@comsol\endcsname {\simpleiconsmapTwo\symbol{186}}
\expandafter\def\csname simpleicon@conan\endcsname {\simpleiconsmapTwo\symbol{187}}
\expandafter\def\csname simpleicon@concourse\endcsname {\simpleiconsmapTwo\symbol{188}}
\expandafter\def\csname simpleicon@condaforge\endcsname {\simpleiconsmapTwo\symbol{189}}
\expandafter\def\csname simpleicon@conekta\endcsname {\simpleiconsmapTwo\symbol{190}}
\expandafter\def\csname simpleicon@confluence\endcsname {\simpleiconsmapTwo\symbol{191}}
\expandafter\def\csname simpleicon@construct3\endcsname {\simpleiconsmapTwo\symbol{192}}
\expandafter\def\csname simpleicon@consul\endcsname {\simpleiconsmapTwo\symbol{193}}
\expandafter\def\csname simpleicon@contactlesspayment\endcsname {\simpleiconsmapTwo\symbol{194}}
\expandafter\def\csname simpleicon@containerd\endcsname {\simpleiconsmapTwo\symbol{195}}
\expandafter\def\csname simpleicon@contentful\endcsname {\simpleiconsmapTwo\symbol{196}}
\expandafter\def\csname simpleicon@conventionalcommits\endcsname {\simpleiconsmapTwo\symbol{197}}
\expandafter\def\csname simpleicon@convertio\endcsname {\simpleiconsmapTwo\symbol{198}}
\expandafter\def\csname simpleicon@cookiecutter\endcsname {\simpleiconsmapTwo\symbol{199}}
\expandafter\def\csname simpleicon@coop\endcsname {\simpleiconsmapTwo\symbol{200}}
\expandafter\def\csname simpleicon@cora\endcsname {\simpleiconsmapTwo\symbol{201}}
\expandafter\def\csname simpleicon@coronaengine\endcsname {\simpleiconsmapTwo\symbol{202}}
\expandafter\def\csname simpleicon@coronarenderer\endcsname {\simpleiconsmapTwo\symbol{203}}
\expandafter\def\csname simpleicon@corsair\endcsname {\simpleiconsmapTwo\symbol{204}}
\expandafter\def\csname simpleicon@couchbase\endcsname {\simpleiconsmapTwo\symbol{205}}
\expandafter\def\csname simpleicon@counterstrike\endcsname {\simpleiconsmapTwo\symbol{206}}
\expandafter\def\csname simpleicon@countingworkspro\endcsname {\simpleiconsmapTwo\symbol{207}}
\expandafter\def\csname simpleicon@coursera\endcsname {\simpleiconsmapTwo\symbol{208}}
\expandafter\def\csname simpleicon@coveralls\endcsname {\simpleiconsmapTwo\symbol{209}}
\expandafter\def\csname simpleicon@cpanel\endcsname {\simpleiconsmapTwo\symbol{210}}
\expandafter\def\csname simpleicon@cplusplus\endcsname {\simpleiconsmapTwo\symbol{211}}
\expandafter\def\csname simpleicon@craftcms\endcsname {\simpleiconsmapTwo\symbol{212}}
\expandafter\def\csname simpleicon@cratedb\endcsname {\simpleiconsmapTwo\symbol{213}}
\expandafter\def\csname simpleicon@createreactapp\endcsname {\simpleiconsmapTwo\symbol{214}}
\expandafter\def\csname simpleicon@creativecommons\endcsname {\simpleiconsmapTwo\symbol{215}}
\expandafter\def\csname simpleicon@credly\endcsname {\simpleiconsmapTwo\symbol{216}}
\expandafter\def\csname simpleicon@crehana\endcsname {\simpleiconsmapTwo\symbol{217}}
\expandafter\def\csname simpleicon@criticalrole\endcsname {\simpleiconsmapTwo\symbol{218}}
\expandafter\def\csname simpleicon@crowdin\endcsname {\simpleiconsmapTwo\symbol{219}}
\expandafter\def\csname simpleicon@crowdsource\endcsname {\simpleiconsmapTwo\symbol{220}}
\expandafter\def\csname simpleicon@crunchbase\endcsname {\simpleiconsmapTwo\symbol{221}}
\expandafter\def\csname simpleicon@crunchyroll\endcsname {\simpleiconsmapTwo\symbol{222}}
\expandafter\def\csname simpleicon@cryengine\endcsname {\simpleiconsmapTwo\symbol{223}}
\expandafter\def\csname simpleicon@crystal\endcsname {\simpleiconsmapTwo\symbol{224}}
\expandafter\def\csname simpleicon@csharp\endcsname {\simpleiconsmapTwo\symbol{225}}
\expandafter\def\csname simpleicon@css3\endcsname {\simpleiconsmapTwo\symbol{226}}
\expandafter\def\csname simpleicon@cssmodules\endcsname {\simpleiconsmapTwo\symbol{227}}
\expandafter\def\csname simpleicon@csswizardry\endcsname {\simpleiconsmapTwo\symbol{228}}
\expandafter\def\csname simpleicon@cucumber\endcsname {\simpleiconsmapTwo\symbol{229}}
\expandafter\def\csname simpleicon@curl\endcsname {\simpleiconsmapTwo\symbol{230}}
\expandafter\def\csname simpleicon@curseforge\endcsname {\simpleiconsmapTwo\symbol{231}}
\expandafter\def\csname simpleicon@cycling74\endcsname {\simpleiconsmapTwo\symbol{232}}
\expandafter\def\csname simpleicon@cypress\endcsname {\simpleiconsmapTwo\symbol{233}}
\expandafter\def\csname simpleicon@cytoscapedotjs\endcsname {\simpleiconsmapTwo\symbol{234}}
\expandafter\def\csname simpleicon@d\endcsname {\simpleiconsmapTwo\symbol{235}}
\expandafter\def\csname simpleicon@d3dotjs\endcsname {\simpleiconsmapTwo\symbol{236}}
\expandafter\def\csname simpleicon@dacia\endcsname {\simpleiconsmapTwo\symbol{237}}
\expandafter\def\csname simpleicon@daf\endcsname {\simpleiconsmapTwo\symbol{238}}
\expandafter\def\csname simpleicon@dailymotion\endcsname {\simpleiconsmapTwo\symbol{239}}
\expandafter\def\csname simpleicon@daimler\endcsname {\simpleiconsmapTwo\symbol{240}}
\expandafter\def\csname simpleicon@dapr\endcsname {\simpleiconsmapTwo\symbol{241}}
\expandafter\def\csname simpleicon@darkreader\endcsname {\simpleiconsmapTwo\symbol{242}}
\expandafter\def\csname simpleicon@dart\endcsname {\simpleiconsmapTwo\symbol{243}}
\expandafter\def\csname simpleicon@darty\endcsname {\simpleiconsmapTwo\symbol{244}}
\expandafter\def\csname simpleicon@daserste\endcsname {\simpleiconsmapTwo\symbol{245}}
\expandafter\def\csname simpleicon@dash\endcsname {\simpleiconsmapTwo\symbol{246}}
\expandafter\def\csname simpleicon@dashlane\endcsname {\simpleiconsmapTwo\symbol{247}}
\expandafter\def\csname simpleicon@dask\endcsname {\simpleiconsmapTwo\symbol{248}}
\expandafter\def\csname simpleicon@dassaultsystemes\endcsname {\simpleiconsmapTwo\symbol{249}}
\expandafter\def\csname simpleicon@databricks\endcsname {\simpleiconsmapTwo\symbol{250}}
\expandafter\def\csname simpleicon@datacamp\endcsname {\simpleiconsmapTwo\symbol{251}}
\expandafter\def\csname simpleicon@datadog\endcsname {\simpleiconsmapTwo\symbol{252}}
\expandafter\def\csname simpleicon@datadotai\endcsname {\simpleiconsmapTwo\symbol{253}}
\expandafter\def\csname simpleicon@datagrip\endcsname {\simpleiconsmapTwo\symbol{254}}
\expandafter\def\csname simpleicon@dataiku\endcsname {\simpleiconsmapTwo\symbol{255}}
\expandafter\def\csname simpleicon@datastax\endcsname {\simpleiconsmapThree\symbol{0}}
\expandafter\def\csname simpleicon@dataverse\endcsname {\simpleiconsmapThree\symbol{1}}
\expandafter\def\csname simpleicon@dataversioncontrol\endcsname {\simpleiconsmapThree\symbol{2}}
\expandafter\def\csname simpleicon@datocms\endcsname {\simpleiconsmapThree\symbol{3}}
\expandafter\def\csname simpleicon@datto\endcsname {\simpleiconsmapThree\symbol{4}}
\expandafter\def\csname simpleicon@dazn\endcsname {\simpleiconsmapThree\symbol{5}}
\expandafter\def\csname simpleicon@dblp\endcsname {\simpleiconsmapThree\symbol{6}}
\expandafter\def\csname simpleicon@dbt\endcsname {\simpleiconsmapThree\symbol{7}}
\expandafter\def\csname simpleicon@dcentertainment\endcsname {\simpleiconsmapThree\symbol{8}}
\expandafter\def\csname simpleicon@debian\endcsname {\simpleiconsmapThree\symbol{9}}
\expandafter\def\csname simpleicon@dedge\endcsname {\simpleiconsmapThree\symbol{10}}
\expandafter\def\csname simpleicon@deepin\endcsname {\simpleiconsmapThree\symbol{11}}
\expandafter\def\csname simpleicon@deepnote\endcsname {\simpleiconsmapThree\symbol{12}}
\expandafter\def\csname simpleicon@deezer\endcsname {\simpleiconsmapThree\symbol{13}}
\expandafter\def\csname simpleicon@delicious\endcsname {\simpleiconsmapThree\symbol{14}}
\expandafter\def\csname simpleicon@deliveroo\endcsname {\simpleiconsmapThree\symbol{15}}
\expandafter\def\csname simpleicon@dell\endcsname {\simpleiconsmapThree\symbol{16}}
\expandafter\def\csname simpleicon@delonghi\endcsname {\simpleiconsmapThree\symbol{17}}
\expandafter\def\csname simpleicon@delphi\endcsname {\simpleiconsmapThree\symbol{18}}
\expandafter\def\csname simpleicon@delta\endcsname {\simpleiconsmapThree\symbol{19}}
\expandafter\def\csname simpleicon@deno\endcsname {\simpleiconsmapThree\symbol{20}}
\expandafter\def\csname simpleicon@dependabot\endcsname {\simpleiconsmapThree\symbol{21}}
\expandafter\def\csname simpleicon@derspiegel\endcsname {\simpleiconsmapThree\symbol{22}}
\expandafter\def\csname simpleicon@designernews\endcsname {\simpleiconsmapThree\symbol{23}}
\expandafter\def\csname simpleicon@deutschebahn\endcsname {\simpleiconsmapThree\symbol{24}}
\expandafter\def\csname simpleicon@deutschebank\endcsname {\simpleiconsmapThree\symbol{25}}
\expandafter\def\csname simpleicon@devdotto\endcsname {\simpleiconsmapThree\symbol{26}}
\expandafter\def\csname simpleicon@devexpress\endcsname {\simpleiconsmapThree\symbol{27}}
\expandafter\def\csname simpleicon@deviantart\endcsname {\simpleiconsmapThree\symbol{28}}
\expandafter\def\csname simpleicon@devpost\endcsname {\simpleiconsmapThree\symbol{29}}
\expandafter\def\csname simpleicon@devrant\endcsname {\simpleiconsmapThree\symbol{30}}
\expandafter\def\csname simpleicon@dgraph\endcsname {\simpleiconsmapThree\symbol{31}}
\expandafter\def\csname simpleicon@dhl\endcsname {\simpleiconsmapThree\symbol{32}}
\expandafter\def\csname simpleicon@diagramsdotnet\endcsname {\simpleiconsmapThree\symbol{33}}
\expandafter\def\csname simpleicon@dialogflow\endcsname {\simpleiconsmapThree\symbol{34}}
\expandafter\def\csname simpleicon@diaspora\endcsname {\simpleiconsmapThree\symbol{35}}
\expandafter\def\csname simpleicon@digg\endcsname {\simpleiconsmapThree\symbol{36}}
\expandafter\def\csname simpleicon@digikeyelectronics\endcsname {\simpleiconsmapThree\symbol{37}}
\expandafter\def\csname simpleicon@digitalocean\endcsname {\simpleiconsmapThree\symbol{38}}
\expandafter\def\csname simpleicon@dior\endcsname {\simpleiconsmapThree\symbol{39}}
\expandafter\def\csname simpleicon@directus\endcsname {\simpleiconsmapThree\symbol{40}}
\expandafter\def\csname simpleicon@discogs\endcsname {\simpleiconsmapThree\symbol{41}}
\expandafter\def\csname simpleicon@discord\endcsname {\simpleiconsmapThree\symbol{42}}
\expandafter\def\csname simpleicon@discourse\endcsname {\simpleiconsmapThree\symbol{43}}
\expandafter\def\csname simpleicon@discover\endcsname {\simpleiconsmapThree\symbol{44}}
\expandafter\def\csname simpleicon@disqus\endcsname {\simpleiconsmapThree\symbol{45}}
\expandafter\def\csname simpleicon@disroot\endcsname {\simpleiconsmapThree\symbol{46}}
\expandafter\def\csname simpleicon@django\endcsname {\simpleiconsmapThree\symbol{47}}
\expandafter\def\csname simpleicon@dlib\endcsname {\simpleiconsmapThree\symbol{48}}
\expandafter\def\csname simpleicon@dlna\endcsname {\simpleiconsmapThree\symbol{49}}
\expandafter\def\csname simpleicon@dm\endcsname {\simpleiconsmapThree\symbol{50}}
\expandafter\def\csname simpleicon@docker\endcsname {\simpleiconsmapThree\symbol{51}}
\expandafter\def\csname simpleicon@docsdotrs\endcsname {\simpleiconsmapThree\symbol{52}}
\expandafter\def\csname simpleicon@docusign\endcsname {\simpleiconsmapThree\symbol{53}}
\expandafter\def\csname simpleicon@dogecoin\endcsname {\simpleiconsmapThree\symbol{54}}
\expandafter\def\csname simpleicon@dolby\endcsname {\simpleiconsmapThree\symbol{55}}
\expandafter\def\csname simpleicon@doordash\endcsname {\simpleiconsmapThree\symbol{56}}
\expandafter\def\csname simpleicon@dotenv\endcsname {\simpleiconsmapThree\symbol{57}}
\expandafter\def\csname simpleicon@dotnet\endcsname {\simpleiconsmapThree\symbol{58}}
\expandafter\def\csname simpleicon@douban\endcsname {\simpleiconsmapThree\symbol{59}}
\expandafter\def\csname simpleicon@doubanread\endcsname {\simpleiconsmapThree\symbol{60}}
\expandafter\def\csname simpleicon@dpd\endcsname {\simpleiconsmapThree\symbol{61}}
\expandafter\def\csname simpleicon@dragonframe\endcsname {\simpleiconsmapThree\symbol{62}}
\expandafter\def\csname simpleicon@draugiemdotlv\endcsname {\simpleiconsmapThree\symbol{63}}
\expandafter\def\csname simpleicon@dribbble\endcsname {\simpleiconsmapThree\symbol{64}}
\expandafter\def\csname simpleicon@drone\endcsname {\simpleiconsmapThree\symbol{65}}
\expandafter\def\csname simpleicon@drooble\endcsname {\simpleiconsmapThree\symbol{66}}
\expandafter\def\csname simpleicon@dropbox\endcsname {\simpleiconsmapThree\symbol{67}}
\expandafter\def\csname simpleicon@drupal\endcsname {\simpleiconsmapThree\symbol{68}}
\expandafter\def\csname simpleicon@dsautomobiles\endcsname {\simpleiconsmapThree\symbol{69}}
\expandafter\def\csname simpleicon@dtube\endcsname {\simpleiconsmapThree\symbol{70}}
\expandafter\def\csname simpleicon@ducati\endcsname {\simpleiconsmapThree\symbol{71}}
\expandafter\def\csname simpleicon@duckdb\endcsname {\simpleiconsmapThree\symbol{72}}
\expandafter\def\csname simpleicon@duckduckgo\endcsname {\simpleiconsmapThree\symbol{73}}
\expandafter\def\csname simpleicon@dungeonsanddragons\endcsname {\simpleiconsmapThree\symbol{74}}
\expandafter\def\csname simpleicon@dunked\endcsname {\simpleiconsmapThree\symbol{75}}
\expandafter\def\csname simpleicon@duolingo\endcsname {\simpleiconsmapThree\symbol{76}}
\expandafter\def\csname simpleicon@dvc\endcsname {\simpleiconsmapThree\symbol{77}}
\expandafter\def\csname simpleicon@dwavesystems\endcsname {\simpleiconsmapThree\symbol{78}}
\expandafter\def\csname simpleicon@dwm\endcsname {\simpleiconsmapThree\symbol{79}}
\expandafter\def\csname simpleicon@dynamics365\endcsname {\simpleiconsmapThree\symbol{80}}
\expandafter\def\csname simpleicon@dynatrace\endcsname {\simpleiconsmapThree\symbol{81}}
\expandafter\def\csname simpleicon@e\endcsname {\simpleiconsmapThree\symbol{82}}
\expandafter\def\csname simpleicon@ea\endcsname {\simpleiconsmapThree\symbol{83}}
\expandafter\def\csname simpleicon@eagle\endcsname {\simpleiconsmapThree\symbol{84}}
\expandafter\def\csname simpleicon@easyjet\endcsname {\simpleiconsmapThree\symbol{85}}
\expandafter\def\csname simpleicon@ebay\endcsname {\simpleiconsmapThree\symbol{86}}
\expandafter\def\csname simpleicon@eclipseche\endcsname {\simpleiconsmapThree\symbol{87}}
\expandafter\def\csname simpleicon@eclipseide\endcsname {\simpleiconsmapThree\symbol{88}}
\expandafter\def\csname simpleicon@eclipsejetty\endcsname {\simpleiconsmapThree\symbol{89}}
\expandafter\def\csname simpleicon@eclipsemosquitto\endcsname {\simpleiconsmapThree\symbol{90}}
\expandafter\def\csname simpleicon@eclipsevertdotx\endcsname {\simpleiconsmapThree\symbol{91}}
\expandafter\def\csname simpleicon@edeka\endcsname {\simpleiconsmapThree\symbol{92}}
\expandafter\def\csname simpleicon@editorconfig\endcsname {\simpleiconsmapThree\symbol{93}}
\expandafter\def\csname simpleicon@edotleclerc\endcsname {\simpleiconsmapThree\symbol{94}}
\expandafter\def\csname simpleicon@edx\endcsname {\simpleiconsmapThree\symbol{95}}
\expandafter\def\csname simpleicon@egghead\endcsname {\simpleiconsmapThree\symbol{96}}
\expandafter\def\csname simpleicon@egnyte\endcsname {\simpleiconsmapThree\symbol{97}}
\expandafter\def\csname simpleicon@eightsleep\endcsname {\simpleiconsmapThree\symbol{98}}
\expandafter\def\csname simpleicon@elastic\endcsname {\simpleiconsmapThree\symbol{99}}
\expandafter\def\csname simpleicon@elasticcloud\endcsname {\simpleiconsmapThree\symbol{100}}
\expandafter\def\csname simpleicon@elasticsearch\endcsname {\simpleiconsmapThree\symbol{101}}
\expandafter\def\csname simpleicon@elasticstack\endcsname {\simpleiconsmapThree\symbol{102}}
\expandafter\def\csname simpleicon@electron\endcsname {\simpleiconsmapThree\symbol{103}}
\expandafter\def\csname simpleicon@electronbuilder\endcsname {\simpleiconsmapThree\symbol{104}}
\expandafter\def\csname simpleicon@element\endcsname {\simpleiconsmapThree\symbol{105}}
\expandafter\def\csname simpleicon@elementary\endcsname {\simpleiconsmapThree\symbol{106}}
\expandafter\def\csname simpleicon@elementor\endcsname {\simpleiconsmapThree\symbol{107}}
\expandafter\def\csname simpleicon@eleventy\endcsname {\simpleiconsmapThree\symbol{108}}
\expandafter\def\csname simpleicon@elixir\endcsname {\simpleiconsmapThree\symbol{109}}
\expandafter\def\csname simpleicon@eljueves\endcsname {\simpleiconsmapThree\symbol{110}}
\expandafter\def\csname simpleicon@ello\endcsname {\simpleiconsmapThree\symbol{111}}
\expandafter\def\csname simpleicon@elm\endcsname {\simpleiconsmapThree\symbol{112}}
\expandafter\def\csname simpleicon@elsevier\endcsname {\simpleiconsmapThree\symbol{113}}
\expandafter\def\csname simpleicon@embarcadero\endcsname {\simpleiconsmapThree\symbol{114}}
\expandafter\def\csname simpleicon@emberdotjs\endcsname {\simpleiconsmapThree\symbol{115}}
\expandafter\def\csname simpleicon@emby\endcsname {\simpleiconsmapThree\symbol{116}}
\expandafter\def\csname simpleicon@emirates\endcsname {\simpleiconsmapThree\symbol{117}}
\expandafter\def\csname simpleicon@emlakjet\endcsname {\simpleiconsmapThree\symbol{118}}
\expandafter\def\csname simpleicon@empirekred\endcsname {\simpleiconsmapThree\symbol{119}}
\expandafter\def\csname simpleicon@enpass\endcsname {\simpleiconsmapThree\symbol{120}}
\expandafter\def\csname simpleicon@enterprisedb\endcsname {\simpleiconsmapThree\symbol{121}}
\expandafter\def\csname simpleicon@envato\endcsname {\simpleiconsmapThree\symbol{122}}
\expandafter\def\csname simpleicon@epel\endcsname {\simpleiconsmapThree\symbol{123}}
\expandafter\def\csname simpleicon@epicgames\endcsname {\simpleiconsmapThree\symbol{124}}
\expandafter\def\csname simpleicon@epson\endcsname {\simpleiconsmapThree\symbol{125}}
\expandafter\def\csname simpleicon@equinixmetal\endcsname {\simpleiconsmapThree\symbol{126}}
\expandafter\def\csname simpleicon@erlang\endcsname {\simpleiconsmapThree\symbol{127}}
\expandafter\def\csname simpleicon@esbuild\endcsname {\simpleiconsmapThree\symbol{128}}
\expandafter\def\csname simpleicon@esea\endcsname {\simpleiconsmapThree\symbol{129}}
\expandafter\def\csname simpleicon@eslgaming\endcsname {\simpleiconsmapThree\symbol{130}}
\expandafter\def\csname simpleicon@eslint\endcsname {\simpleiconsmapThree\symbol{131}}
\expandafter\def\csname simpleicon@esphome\endcsname {\simpleiconsmapThree\symbol{132}}
\expandafter\def\csname simpleicon@espressif\endcsname {\simpleiconsmapThree\symbol{133}}
\expandafter\def\csname simpleicon@etcd\endcsname {\simpleiconsmapThree\symbol{134}}
\expandafter\def\csname simpleicon@ethereum\endcsname {\simpleiconsmapThree\symbol{135}}
\expandafter\def\csname simpleicon@ethiopianairlines\endcsname {\simpleiconsmapThree\symbol{136}}
\expandafter\def\csname simpleicon@etihadairways\endcsname {\simpleiconsmapThree\symbol{137}}
\expandafter\def\csname simpleicon@etsy\endcsname {\simpleiconsmapThree\symbol{138}}
\expandafter\def\csname simpleicon@eventbrite\endcsname {\simpleiconsmapThree\symbol{139}}
\expandafter\def\csname simpleicon@eventstore\endcsname {\simpleiconsmapThree\symbol{140}}
\expandafter\def\csname simpleicon@evernote\endcsname {\simpleiconsmapThree\symbol{141}}
\expandafter\def\csname simpleicon@exercism\endcsname {\simpleiconsmapThree\symbol{142}}
\expandafter\def\csname simpleicon@exordo\endcsname {\simpleiconsmapThree\symbol{143}}
\expandafter\def\csname simpleicon@exoscale\endcsname {\simpleiconsmapThree\symbol{144}}
\expandafter\def\csname simpleicon@expensify\endcsname {\simpleiconsmapThree\symbol{145}}
\expandafter\def\csname simpleicon@expertsexchange\endcsname {\simpleiconsmapThree\symbol{146}}
\expandafter\def\csname simpleicon@expo\endcsname {\simpleiconsmapThree\symbol{147}}
\expandafter\def\csname simpleicon@express\endcsname {\simpleiconsmapThree\symbol{148}}
\expandafter\def\csname simpleicon@expressvpn\endcsname {\simpleiconsmapThree\symbol{149}}
\expandafter\def\csname simpleicon@eyeem\endcsname {\simpleiconsmapThree\symbol{150}}
\expandafter\def\csname simpleicon@f1\endcsname {\simpleiconsmapThree\symbol{151}}
\expandafter\def\csname simpleicon@f5\endcsname {\simpleiconsmapThree\symbol{152}}
\expandafter\def\csname simpleicon@facebook\endcsname {\simpleiconsmapThree\symbol{153}}
\expandafter\def\csname simpleicon@facebookgaming\endcsname {\simpleiconsmapThree\symbol{154}}
\expandafter\def\csname simpleicon@facebooklive\endcsname {\simpleiconsmapThree\symbol{155}}
\expandafter\def\csname simpleicon@faceit\endcsname {\simpleiconsmapThree\symbol{156}}
\expandafter\def\csname simpleicon@facepunch\endcsname {\simpleiconsmapThree\symbol{157}}
\expandafter\def\csname simpleicon@falcon\endcsname {\simpleiconsmapThree\symbol{158}}
\expandafter\def\csname simpleicon@fampay\endcsname {\simpleiconsmapThree\symbol{159}}
\expandafter\def\csname simpleicon@fandango\endcsname {\simpleiconsmapThree\symbol{160}}
\expandafter\def\csname simpleicon@fandom\endcsname {\simpleiconsmapThree\symbol{161}}
\expandafter\def\csname simpleicon@farfetch\endcsname {\simpleiconsmapThree\symbol{162}}
\expandafter\def\csname simpleicon@fastapi\endcsname {\simpleiconsmapThree\symbol{163}}
\expandafter\def\csname simpleicon@fastify\endcsname {\simpleiconsmapThree\symbol{164}}
\expandafter\def\csname simpleicon@fastlane\endcsname {\simpleiconsmapThree\symbol{165}}
\expandafter\def\csname simpleicon@fastly\endcsname {\simpleiconsmapThree\symbol{166}}
\expandafter\def\csname simpleicon@fathom\endcsname {\simpleiconsmapThree\symbol{167}}
\expandafter\def\csname simpleicon@fauna\endcsname {\simpleiconsmapThree\symbol{168}}
\expandafter\def\csname simpleicon@favro\endcsname {\simpleiconsmapThree\symbol{169}}
\expandafter\def\csname simpleicon@fdroid\endcsname {\simpleiconsmapThree\symbol{170}}
\expandafter\def\csname simpleicon@feathub\endcsname {\simpleiconsmapThree\symbol{171}}
\expandafter\def\csname simpleicon@fedex\endcsname {\simpleiconsmapThree\symbol{172}}
\expandafter\def\csname simpleicon@fedora\endcsname {\simpleiconsmapThree\symbol{173}}
\expandafter\def\csname simpleicon@feedly\endcsname {\simpleiconsmapThree\symbol{174}}
\expandafter\def\csname simpleicon@ferrari\endcsname {\simpleiconsmapThree\symbol{175}}
\expandafter\def\csname simpleicon@ferrarinv\endcsname {\simpleiconsmapThree\symbol{176}}
\expandafter\def\csname simpleicon@ffmpeg\endcsname {\simpleiconsmapThree\symbol{177}}
\expandafter\def\csname simpleicon@fiat\endcsname {\simpleiconsmapThree\symbol{178}}
\expandafter\def\csname simpleicon@fidoalliance\endcsname {\simpleiconsmapThree\symbol{179}}
\expandafter\def\csname simpleicon@fifa\endcsname {\simpleiconsmapThree\symbol{180}}
\expandafter\def\csname simpleicon@figma\endcsname {\simpleiconsmapThree\symbol{181}}
\expandafter\def\csname simpleicon@figshare\endcsname {\simpleiconsmapThree\symbol{182}}
\expandafter\def\csname simpleicon@fila\endcsname {\simpleiconsmapThree\symbol{183}}
\expandafter\def\csname simpleicon@files\endcsname {\simpleiconsmapThree\symbol{184}}
\expandafter\def\csname simpleicon@filezilla\endcsname {\simpleiconsmapThree\symbol{185}}
\expandafter\def\csname simpleicon@fing\endcsname {\simpleiconsmapThree\symbol{186}}
\expandafter\def\csname simpleicon@firebase\endcsname {\simpleiconsmapThree\symbol{187}}
\expandafter\def\csname simpleicon@firefox\endcsname {\simpleiconsmapThree\symbol{188}}
\expandafter\def\csname simpleicon@firefoxbrowser\endcsname {\simpleiconsmapThree\symbol{189}}
\expandafter\def\csname simpleicon@first\endcsname {\simpleiconsmapThree\symbol{190}}
\expandafter\def\csname simpleicon@fitbit\endcsname {\simpleiconsmapThree\symbol{191}}
\expandafter\def\csname simpleicon@fite\endcsname {\simpleiconsmapThree\symbol{192}}
\expandafter\def\csname simpleicon@fivem\endcsname {\simpleiconsmapThree\symbol{193}}
\expandafter\def\csname simpleicon@fiverr\endcsname {\simpleiconsmapThree\symbol{194}}
\expandafter\def\csname simpleicon@flask\endcsname {\simpleiconsmapThree\symbol{195}}
\expandafter\def\csname simpleicon@flat\endcsname {\simpleiconsmapThree\symbol{196}}
\expandafter\def\csname simpleicon@flathub\endcsname {\simpleiconsmapThree\symbol{197}}
\expandafter\def\csname simpleicon@flatpak\endcsname {\simpleiconsmapThree\symbol{198}}
\expandafter\def\csname simpleicon@flattr\endcsname {\simpleiconsmapThree\symbol{199}}
\expandafter\def\csname simpleicon@flickr\endcsname {\simpleiconsmapThree\symbol{200}}
\expandafter\def\csname simpleicon@flipboard\endcsname {\simpleiconsmapThree\symbol{201}}
\expandafter\def\csname simpleicon@flipkart\endcsname {\simpleiconsmapThree\symbol{202}}
\expandafter\def\csname simpleicon@floatplane\endcsname {\simpleiconsmapThree\symbol{203}}
\expandafter\def\csname simpleicon@flood\endcsname {\simpleiconsmapThree\symbol{204}}
\expandafter\def\csname simpleicon@fluentbit\endcsname {\simpleiconsmapThree\symbol{205}}
\expandafter\def\csname simpleicon@fluentd\endcsname {\simpleiconsmapThree\symbol{206}}
\expandafter\def\csname simpleicon@flutter\endcsname {\simpleiconsmapThree\symbol{207}}
\expandafter\def\csname simpleicon@flyway\endcsname {\simpleiconsmapThree\symbol{208}}
\expandafter\def\csname simpleicon@fmod\endcsname {\simpleiconsmapThree\symbol{209}}
\expandafter\def\csname simpleicon@fnac\endcsname {\simpleiconsmapThree\symbol{210}}
\expandafter\def\csname simpleicon@folium\endcsname {\simpleiconsmapThree\symbol{211}}
\expandafter\def\csname simpleicon@fonoma\endcsname {\simpleiconsmapThree\symbol{212}}
\expandafter\def\csname simpleicon@fontawesome\endcsname {\simpleiconsmapThree\symbol{213}}
\expandafter\def\csname simpleicon@fontbase\endcsname {\simpleiconsmapThree\symbol{214}}
\expandafter\def\csname simpleicon@foodpanda\endcsname {\simpleiconsmapThree\symbol{215}}
\expandafter\def\csname simpleicon@ford\endcsname {\simpleiconsmapThree\symbol{216}}
\expandafter\def\csname simpleicon@forestry\endcsname {\simpleiconsmapThree\symbol{217}}
\expandafter\def\csname simpleicon@formstack\endcsname {\simpleiconsmapThree\symbol{218}}
\expandafter\def\csname simpleicon@fortinet\endcsname {\simpleiconsmapThree\symbol{219}}
\expandafter\def\csname simpleicon@fortran\endcsname {\simpleiconsmapThree\symbol{220}}
\expandafter\def\csname simpleicon@fossa\endcsname {\simpleiconsmapThree\symbol{221}}
\expandafter\def\csname simpleicon@fossilscm\endcsname {\simpleiconsmapThree\symbol{222}}
\expandafter\def\csname simpleicon@foursquare\endcsname {\simpleiconsmapThree\symbol{223}}
\expandafter\def\csname simpleicon@foursquarecityguide\endcsname {\simpleiconsmapThree\symbol{224}}
\expandafter\def\csname simpleicon@foxtel\endcsname {\simpleiconsmapThree\symbol{225}}
\expandafter\def\csname simpleicon@fozzy\endcsname {\simpleiconsmapThree\symbol{226}}
\expandafter\def\csname simpleicon@framer\endcsname {\simpleiconsmapThree\symbol{227}}
\expandafter\def\csname simpleicon@framework7\endcsname {\simpleiconsmapThree\symbol{228}}
\expandafter\def\csname simpleicon@franprix\endcsname {\simpleiconsmapThree\symbol{229}}
\expandafter\def\csname simpleicon@fraunhofergesellschaft\endcsname {\simpleiconsmapThree\symbol{230}}
\expandafter\def\csname simpleicon@freebsd\endcsname {\simpleiconsmapThree\symbol{231}}
\expandafter\def\csname simpleicon@freecodecamp\endcsname {\simpleiconsmapThree\symbol{232}}
\expandafter\def\csname simpleicon@freedesktopdotorg\endcsname {\simpleiconsmapThree\symbol{233}}
\expandafter\def\csname simpleicon@freelancer\endcsname {\simpleiconsmapThree\symbol{234}}
\expandafter\def\csname simpleicon@freenas\endcsname {\simpleiconsmapThree\symbol{235}}
\expandafter\def\csname simpleicon@frontendmentor\endcsname {\simpleiconsmapThree\symbol{236}}
\expandafter\def\csname simpleicon@fsecure\endcsname {\simpleiconsmapThree\symbol{237}}
\expandafter\def\csname simpleicon@fugacloud\endcsname {\simpleiconsmapThree\symbol{238}}
\expandafter\def\csname simpleicon@fujifilm\endcsname {\simpleiconsmapThree\symbol{239}}
\expandafter\def\csname simpleicon@fujitsu\endcsname {\simpleiconsmapThree\symbol{240}}
\expandafter\def\csname simpleicon@funimation\endcsname {\simpleiconsmapThree\symbol{241}}
\expandafter\def\csname simpleicon@furaffinity\endcsname {\simpleiconsmapThree\symbol{242}}
\expandafter\def\csname simpleicon@furrynetwork\endcsname {\simpleiconsmapThree\symbol{243}}
\expandafter\def\csname simpleicon@futurelearn\endcsname {\simpleiconsmapThree\symbol{244}}
\expandafter\def\csname simpleicon@g2\endcsname {\simpleiconsmapThree\symbol{245}}
\expandafter\def\csname simpleicon@g2a\endcsname {\simpleiconsmapThree\symbol{246}}
\expandafter\def\csname simpleicon@gameandwatch\endcsname {\simpleiconsmapThree\symbol{247}}
\expandafter\def\csname simpleicon@gamejolt\endcsname {\simpleiconsmapThree\symbol{248}}
\expandafter\def\csname simpleicon@garmin\endcsname {\simpleiconsmapThree\symbol{249}}
\expandafter\def\csname simpleicon@gatling\endcsname {\simpleiconsmapThree\symbol{250}}
\expandafter\def\csname simpleicon@gatsby\endcsname {\simpleiconsmapThree\symbol{251}}
\expandafter\def\csname simpleicon@geant\endcsname {\simpleiconsmapThree\symbol{252}}
\expandafter\def\csname simpleicon@geeksforgeeks\endcsname {\simpleiconsmapThree\symbol{253}}
\expandafter\def\csname simpleicon@generalelectric\endcsname {\simpleiconsmapThree\symbol{254}}
\expandafter\def\csname simpleicon@generalmotors\endcsname {\simpleiconsmapThree\symbol{255}}
\expandafter\def\csname simpleicon@genius\endcsname {\simpleiconsmapFour\symbol{0}}
\expandafter\def\csname simpleicon@gentoo\endcsname {\simpleiconsmapFour\symbol{1}}
\expandafter\def\csname simpleicon@geocaching\endcsname {\simpleiconsmapFour\symbol{2}}
\expandafter\def\csname simpleicon@gerrit\endcsname {\simpleiconsmapFour\symbol{3}}
\expandafter\def\csname simpleicon@ghost\endcsname {\simpleiconsmapFour\symbol{4}}
\expandafter\def\csname simpleicon@ghostery\endcsname {\simpleiconsmapFour\symbol{5}}
\expandafter\def\csname simpleicon@gimp\endcsname {\simpleiconsmapFour\symbol{6}}
\expandafter\def\csname simpleicon@giphy\endcsname {\simpleiconsmapFour\symbol{7}}
\expandafter\def\csname simpleicon@git\endcsname {\simpleiconsmapFour\symbol{8}}
\expandafter\def\csname simpleicon@gitbook\endcsname {\simpleiconsmapFour\symbol{9}}
\expandafter\def\csname simpleicon@gitea\endcsname {\simpleiconsmapFour\symbol{10}}
\expandafter\def\csname simpleicon@gitee\endcsname {\simpleiconsmapFour\symbol{11}}
\expandafter\def\csname simpleicon@gitextensions\endcsname {\simpleiconsmapFour\symbol{12}}
\expandafter\def\csname simpleicon@github\endcsname {\simpleiconsmapFour\symbol{13}}
\expandafter\def\csname simpleicon@githubactions\endcsname {\simpleiconsmapFour\symbol{14}}
\expandafter\def\csname simpleicon@githubpages\endcsname {\simpleiconsmapFour\symbol{15}}
\expandafter\def\csname simpleicon@githubsponsors\endcsname {\simpleiconsmapFour\symbol{16}}
\expandafter\def\csname simpleicon@gitignoredotio\endcsname {\simpleiconsmapFour\symbol{17}}
\expandafter\def\csname simpleicon@gitkraken\endcsname {\simpleiconsmapFour\symbol{18}}
\expandafter\def\csname simpleicon@gitlab\endcsname {\simpleiconsmapFour\symbol{19}}
\expandafter\def\csname simpleicon@gitlfs\endcsname {\simpleiconsmapFour\symbol{20}}
\expandafter\def\csname simpleicon@gitpod\endcsname {\simpleiconsmapFour\symbol{21}}
\expandafter\def\csname simpleicon@gitter\endcsname {\simpleiconsmapFour\symbol{22}}
\expandafter\def\csname simpleicon@glassdoor\endcsname {\simpleiconsmapFour\symbol{23}}
\expandafter\def\csname simpleicon@glitch\endcsname {\simpleiconsmapFour\symbol{24}}
\expandafter\def\csname simpleicon@globus\endcsname {\simpleiconsmapFour\symbol{25}}
\expandafter\def\csname simpleicon@gmail\endcsname {\simpleiconsmapFour\symbol{26}}
\expandafter\def\csname simpleicon@gnome\endcsname {\simpleiconsmapFour\symbol{27}}
\expandafter\def\csname simpleicon@gnometerminal\endcsname {\simpleiconsmapFour\symbol{28}}
\expandafter\def\csname simpleicon@gnu\endcsname {\simpleiconsmapFour\symbol{29}}
\expandafter\def\csname simpleicon@gnubash\endcsname {\simpleiconsmapFour\symbol{30}}
\expandafter\def\csname simpleicon@gnuemacs\endcsname {\simpleiconsmapFour\symbol{31}}
\expandafter\def\csname simpleicon@gnuicecat\endcsname {\simpleiconsmapFour\symbol{32}}
\expandafter\def\csname simpleicon@gnuprivacyguard\endcsname {\simpleiconsmapFour\symbol{33}}
\expandafter\def\csname simpleicon@gnusocial\endcsname {\simpleiconsmapFour\symbol{34}}
\expandafter\def\csname simpleicon@go\endcsname {\simpleiconsmapFour\symbol{35}}
\expandafter\def\csname simpleicon@gocd\endcsname {\simpleiconsmapFour\symbol{36}}
\expandafter\def\csname simpleicon@godaddy\endcsname {\simpleiconsmapFour\symbol{37}}
\expandafter\def\csname simpleicon@godotengine\endcsname {\simpleiconsmapFour\symbol{38}}
\expandafter\def\csname simpleicon@gofundme\endcsname {\simpleiconsmapFour\symbol{39}}
\expandafter\def\csname simpleicon@gogdotcom\endcsname {\simpleiconsmapFour\symbol{40}}
\expandafter\def\csname simpleicon@goland\endcsname {\simpleiconsmapFour\symbol{41}}
\expandafter\def\csname simpleicon@goldenline\endcsname {\simpleiconsmapFour\symbol{42}}
\expandafter\def\csname simpleicon@goodreads\endcsname {\simpleiconsmapFour\symbol{43}}
\expandafter\def\csname simpleicon@google\endcsname {\simpleiconsmapFour\symbol{44}}
\expandafter\def\csname simpleicon@googleadmob\endcsname {\simpleiconsmapFour\symbol{45}}
\expandafter\def\csname simpleicon@googleads\endcsname {\simpleiconsmapFour\symbol{46}}
\expandafter\def\csname simpleicon@googleadsense\endcsname {\simpleiconsmapFour\symbol{47}}
\expandafter\def\csname simpleicon@googleanalytics\endcsname {\simpleiconsmapFour\symbol{48}}
\expandafter\def\csname simpleicon@googleassistant\endcsname {\simpleiconsmapFour\symbol{49}}
\expandafter\def\csname simpleicon@googlecalendar\endcsname {\simpleiconsmapFour\symbol{50}}
\expandafter\def\csname simpleicon@googlecardboard\endcsname {\simpleiconsmapFour\symbol{51}}
\expandafter\def\csname simpleicon@googlechat\endcsname {\simpleiconsmapFour\symbol{52}}
\expandafter\def\csname simpleicon@googlechrome\endcsname {\simpleiconsmapFour\symbol{53}}
\expandafter\def\csname simpleicon@googleclassroom\endcsname {\simpleiconsmapFour\symbol{54}}
\expandafter\def\csname simpleicon@googlecloud\endcsname {\simpleiconsmapFour\symbol{55}}
\expandafter\def\csname simpleicon@googlecolab\endcsname {\simpleiconsmapFour\symbol{56}}
\expandafter\def\csname simpleicon@googledomains\endcsname {\simpleiconsmapFour\symbol{57}}
\expandafter\def\csname simpleicon@googledrive\endcsname {\simpleiconsmapFour\symbol{58}}
\expandafter\def\csname simpleicon@googleearth\endcsname {\simpleiconsmapFour\symbol{59}}
\expandafter\def\csname simpleicon@googlefit\endcsname {\simpleiconsmapFour\symbol{60}}
\expandafter\def\csname simpleicon@googlefonts\endcsname {\simpleiconsmapFour\symbol{61}}
\expandafter\def\csname simpleicon@googlehangouts\endcsname {\simpleiconsmapFour\symbol{62}}
\expandafter\def\csname simpleicon@googlehome\endcsname {\simpleiconsmapFour\symbol{63}}
\expandafter\def\csname simpleicon@googlekeep\endcsname {\simpleiconsmapFour\symbol{64}}
\expandafter\def\csname simpleicon@googlelens\endcsname {\simpleiconsmapFour\symbol{65}}
\expandafter\def\csname simpleicon@googlemaps\endcsname {\simpleiconsmapFour\symbol{66}}
\expandafter\def\csname simpleicon@googlemarketingplatform\endcsname {\simpleiconsmapFour\symbol{67}}
\expandafter\def\csname simpleicon@googlemeet\endcsname {\simpleiconsmapFour\symbol{68}}
\expandafter\def\csname simpleicon@googlemessages\endcsname {\simpleiconsmapFour\symbol{69}}
\expandafter\def\csname simpleicon@googlemybusiness\endcsname {\simpleiconsmapFour\symbol{70}}
\expandafter\def\csname simpleicon@googlenearby\endcsname {\simpleiconsmapFour\symbol{71}}
\expandafter\def\csname simpleicon@googlenews\endcsname {\simpleiconsmapFour\symbol{72}}
\expandafter\def\csname simpleicon@googleoptimize\endcsname {\simpleiconsmapFour\symbol{73}}
\expandafter\def\csname simpleicon@googlepay\endcsname {\simpleiconsmapFour\symbol{74}}
\expandafter\def\csname simpleicon@googlephotos\endcsname {\simpleiconsmapFour\symbol{75}}
\expandafter\def\csname simpleicon@googleplay\endcsname {\simpleiconsmapFour\symbol{76}}
\expandafter\def\csname simpleicon@googlepodcasts\endcsname {\simpleiconsmapFour\symbol{77}}
\expandafter\def\csname simpleicon@googlescholar\endcsname {\simpleiconsmapFour\symbol{78}}
\expandafter\def\csname simpleicon@googlesearchconsole\endcsname {\simpleiconsmapFour\symbol{79}}
\expandafter\def\csname simpleicon@googlesheets\endcsname {\simpleiconsmapFour\symbol{80}}
\expandafter\def\csname simpleicon@googlestreetview\endcsname {\simpleiconsmapFour\symbol{81}}
\expandafter\def\csname simpleicon@googletagmanager\endcsname {\simpleiconsmapFour\symbol{82}}
\expandafter\def\csname simpleicon@googletranslate\endcsname {\simpleiconsmapFour\symbol{83}}
\expandafter\def\csname simpleicon@gotomeeting\endcsname {\simpleiconsmapFour\symbol{84}}
\expandafter\def\csname simpleicon@grab\endcsname {\simpleiconsmapFour\symbol{85}}
\expandafter\def\csname simpleicon@gradle\endcsname {\simpleiconsmapFour\symbol{86}}
\expandafter\def\csname simpleicon@grafana\endcsname {\simpleiconsmapFour\symbol{87}}
\expandafter\def\csname simpleicon@grammarly\endcsname {\simpleiconsmapFour\symbol{88}}
\expandafter\def\csname simpleicon@grandfrais\endcsname {\simpleiconsmapFour\symbol{89}}
\expandafter\def\csname simpleicon@graphql\endcsname {\simpleiconsmapFour\symbol{90}}
\expandafter\def\csname simpleicon@grav\endcsname {\simpleiconsmapFour\symbol{91}}
\expandafter\def\csname simpleicon@gravatar\endcsname {\simpleiconsmapFour\symbol{92}}
\expandafter\def\csname simpleicon@graylog\endcsname {\simpleiconsmapFour\symbol{93}}
\expandafter\def\csname simpleicon@greensock\endcsname {\simpleiconsmapFour\symbol{94}}
\expandafter\def\csname simpleicon@griddotai\endcsname {\simpleiconsmapFour\symbol{95}}
\expandafter\def\csname simpleicon@gridsome\endcsname {\simpleiconsmapFour\symbol{96}}
\expandafter\def\csname simpleicon@groupme\endcsname {\simpleiconsmapFour\symbol{97}}
\expandafter\def\csname simpleicon@groupon\endcsname {\simpleiconsmapFour\symbol{98}}
\expandafter\def\csname simpleicon@grubhub\endcsname {\simpleiconsmapFour\symbol{99}}
\expandafter\def\csname simpleicon@grunt\endcsname {\simpleiconsmapFour\symbol{100}}
\expandafter\def\csname simpleicon@gtk\endcsname {\simpleiconsmapFour\symbol{101}}
\expandafter\def\csname simpleicon@guangzhoumetro\endcsname {\simpleiconsmapFour\symbol{102}}
\expandafter\def\csname simpleicon@guilded\endcsname {\simpleiconsmapFour\symbol{103}}
\expandafter\def\csname simpleicon@gulp\endcsname {\simpleiconsmapFour\symbol{104}}
\expandafter\def\csname simpleicon@gumroad\endcsname {\simpleiconsmapFour\symbol{105}}
\expandafter\def\csname simpleicon@gumtree\endcsname {\simpleiconsmapFour\symbol{106}}
\expandafter\def\csname simpleicon@gunicorn\endcsname {\simpleiconsmapFour\symbol{107}}
\expandafter\def\csname simpleicon@gurobi\endcsname {\simpleiconsmapFour\symbol{108}}
\expandafter\def\csname simpleicon@gutenberg\endcsname {\simpleiconsmapFour\symbol{109}}
\expandafter\def\csname simpleicon@habr\endcsname {\simpleiconsmapFour\symbol{110}}
\expandafter\def\csname simpleicon@hackaday\endcsname {\simpleiconsmapFour\symbol{111}}
\expandafter\def\csname simpleicon@hackclub\endcsname {\simpleiconsmapFour\symbol{112}}
\expandafter\def\csname simpleicon@hackerearth\endcsname {\simpleiconsmapFour\symbol{113}}
\expandafter\def\csname simpleicon@hackernoon\endcsname {\simpleiconsmapFour\symbol{114}}
\expandafter\def\csname simpleicon@hackerone\endcsname {\simpleiconsmapFour\symbol{115}}
\expandafter\def\csname simpleicon@hackerrank\endcsname {\simpleiconsmapFour\symbol{116}}
\expandafter\def\csname simpleicon@hackster\endcsname {\simpleiconsmapFour\symbol{117}}
\expandafter\def\csname simpleicon@hackthebox\endcsname {\simpleiconsmapFour\symbol{118}}
\expandafter\def\csname simpleicon@handlebarsdotjs\endcsname {\simpleiconsmapFour\symbol{119}}
\expandafter\def\csname simpleicon@handshake\endcsname {\simpleiconsmapFour\symbol{120}}
\expandafter\def\csname simpleicon@handshakeprotocol\endcsname {\simpleiconsmapFour\symbol{121}}
\expandafter\def\csname simpleicon@happycow\endcsname {\simpleiconsmapFour\symbol{122}}
\expandafter\def\csname simpleicon@harbor\endcsname {\simpleiconsmapFour\symbol{123}}
\expandafter\def\csname simpleicon@harmonyos\endcsname {\simpleiconsmapFour\symbol{124}}
\expandafter\def\csname simpleicon@hashnode\endcsname {\simpleiconsmapFour\symbol{125}}
\expandafter\def\csname simpleicon@haskell\endcsname {\simpleiconsmapFour\symbol{126}}
\expandafter\def\csname simpleicon@hasura\endcsname {\simpleiconsmapFour\symbol{127}}
\expandafter\def\csname simpleicon@hatenabookmark\endcsname {\simpleiconsmapFour\symbol{128}}
\expandafter\def\csname simpleicon@haveibeenpwned\endcsname {\simpleiconsmapFour\symbol{129}}
\expandafter\def\csname simpleicon@haxe\endcsname {\simpleiconsmapFour\symbol{130}}
\expandafter\def\csname simpleicon@hbo\endcsname {\simpleiconsmapFour\symbol{131}}
\expandafter\def\csname simpleicon@hcl\endcsname {\simpleiconsmapFour\symbol{132}}
\expandafter\def\csname simpleicon@headlessui\endcsname {\simpleiconsmapFour\symbol{133}}
\expandafter\def\csname simpleicon@headspace\endcsname {\simpleiconsmapFour\symbol{134}}
\expandafter\def\csname simpleicon@hellofresh\endcsname {\simpleiconsmapFour\symbol{135}}
\expandafter\def\csname simpleicon@hellyhansen\endcsname {\simpleiconsmapFour\symbol{136}}
\expandafter\def\csname simpleicon@helm\endcsname {\simpleiconsmapFour\symbol{137}}
\expandafter\def\csname simpleicon@helpdesk\endcsname {\simpleiconsmapFour\symbol{138}}
\expandafter\def\csname simpleicon@helpscout\endcsname {\simpleiconsmapFour\symbol{139}}
\expandafter\def\csname simpleicon@here\endcsname {\simpleiconsmapFour\symbol{140}}
\expandafter\def\csname simpleicon@heroku\endcsname {\simpleiconsmapFour\symbol{141}}
\expandafter\def\csname simpleicon@hetzner\endcsname {\simpleiconsmapFour\symbol{142}}
\expandafter\def\csname simpleicon@hexo\endcsname {\simpleiconsmapFour\symbol{143}}
\expandafter\def\csname simpleicon@hey\endcsname {\simpleiconsmapFour\symbol{144}}
\expandafter\def\csname simpleicon@hibernate\endcsname {\simpleiconsmapFour\symbol{145}}
\expandafter\def\csname simpleicon@hibob\endcsname {\simpleiconsmapFour\symbol{146}}
\expandafter\def\csname simpleicon@hilton\endcsname {\simpleiconsmapFour\symbol{147}}
\expandafter\def\csname simpleicon@hitachi\endcsname {\simpleiconsmapFour\symbol{148}}
\expandafter\def\csname simpleicon@hive\endcsname {\simpleiconsmapFour\symbol{149}}
\expandafter\def\csname simpleicon@hiveblockchain\endcsname {\simpleiconsmapFour\symbol{150}}
\expandafter\def\csname simpleicon@homeadvisor\endcsname {\simpleiconsmapFour\symbol{151}}
\expandafter\def\csname simpleicon@homeassistant\endcsname {\simpleiconsmapFour\symbol{152}}
\expandafter\def\csname simpleicon@homeassistantcommunitystore\endcsname {\simpleiconsmapFour\symbol{153}}
\expandafter\def\csname simpleicon@homebrew\endcsname {\simpleiconsmapFour\symbol{154}}
\expandafter\def\csname simpleicon@homebridge\endcsname {\simpleiconsmapFour\symbol{155}}
\expandafter\def\csname simpleicon@homify\endcsname {\simpleiconsmapFour\symbol{156}}
\expandafter\def\csname simpleicon@honda\endcsname {\simpleiconsmapFour\symbol{157}}
\expandafter\def\csname simpleicon@hootsuite\endcsname {\simpleiconsmapFour\symbol{158}}
\expandafter\def\csname simpleicon@hoppscotch\endcsname {\simpleiconsmapFour\symbol{159}}
\expandafter\def\csname simpleicon@hotelsdotcom\endcsname {\simpleiconsmapFour\symbol{160}}
\expandafter\def\csname simpleicon@hotjar\endcsname {\simpleiconsmapFour\symbol{161}}
\expandafter\def\csname simpleicon@houdini\endcsname {\simpleiconsmapFour\symbol{162}}
\expandafter\def\csname simpleicon@houzz\endcsname {\simpleiconsmapFour\symbol{163}}
\expandafter\def\csname simpleicon@hp\endcsname {\simpleiconsmapFour\symbol{164}}
\expandafter\def\csname simpleicon@html5\endcsname {\simpleiconsmapFour\symbol{165}}
\expandafter\def\csname simpleicon@htmlacademy\endcsname {\simpleiconsmapFour\symbol{166}}
\expandafter\def\csname simpleicon@httpie\endcsname {\simpleiconsmapFour\symbol{167}}
\expandafter\def\csname simpleicon@huawei\endcsname {\simpleiconsmapFour\symbol{168}}
\expandafter\def\csname simpleicon@hubspot\endcsname {\simpleiconsmapFour\symbol{169}}
\expandafter\def\csname simpleicon@hugo\endcsname {\simpleiconsmapFour\symbol{170}}
\expandafter\def\csname simpleicon@hulu\endcsname {\simpleiconsmapFour\symbol{171}}
\expandafter\def\csname simpleicon@humblebundle\endcsname {\simpleiconsmapFour\symbol{172}}
\expandafter\def\csname simpleicon@hungryjacks\endcsname {\simpleiconsmapFour\symbol{173}}
\expandafter\def\csname simpleicon@hurriyetemlak\endcsname {\simpleiconsmapFour\symbol{174}}
\expandafter\def\csname simpleicon@husqvarna\endcsname {\simpleiconsmapFour\symbol{175}}
\expandafter\def\csname simpleicon@hyper\endcsname {\simpleiconsmapFour\symbol{176}}
\expandafter\def\csname simpleicon@hyperledger\endcsname {\simpleiconsmapFour\symbol{177}}
\expandafter\def\csname simpleicon@hypothesis\endcsname {\simpleiconsmapFour\symbol{178}}
\expandafter\def\csname simpleicon@hyundai\endcsname {\simpleiconsmapFour\symbol{179}}
\expandafter\def\csname simpleicon@i18next\endcsname {\simpleiconsmapFour\symbol{180}}
\expandafter\def\csname simpleicon@iata\endcsname {\simpleiconsmapFour\symbol{181}}
\expandafter\def\csname simpleicon@ibeacon\endcsname {\simpleiconsmapFour\symbol{182}}
\expandafter\def\csname simpleicon@ibm\endcsname {\simpleiconsmapFour\symbol{183}}
\expandafter\def\csname simpleicon@ibmcloud\endcsname {\simpleiconsmapFour\symbol{184}}
\expandafter\def\csname simpleicon@ibmwatson\endcsname {\simpleiconsmapFour\symbol{185}}
\expandafter\def\csname simpleicon@iceland\endcsname {\simpleiconsmapFour\symbol{186}}
\expandafter\def\csname simpleicon@icinga\endcsname {\simpleiconsmapFour\symbol{187}}
\expandafter\def\csname simpleicon@icloud\endcsname {\simpleiconsmapFour\symbol{188}}
\expandafter\def\csname simpleicon@icomoon\endcsname {\simpleiconsmapFour\symbol{189}}
\expandafter\def\csname simpleicon@icon\endcsname {\simpleiconsmapFour\symbol{190}}
\expandafter\def\csname simpleicon@iconfinder\endcsname {\simpleiconsmapFour\symbol{191}}
\expandafter\def\csname simpleicon@iconify\endcsname {\simpleiconsmapFour\symbol{192}}
\expandafter\def\csname simpleicon@iconjar\endcsname {\simpleiconsmapFour\symbol{193}}
\expandafter\def\csname simpleicon@icons8\endcsname {\simpleiconsmapFour\symbol{194}}
\expandafter\def\csname simpleicon@icq\endcsname {\simpleiconsmapFour\symbol{195}}
\expandafter\def\csname simpleicon@ieee\endcsname {\simpleiconsmapFour\symbol{196}}
\expandafter\def\csname simpleicon@ifixit\endcsname {\simpleiconsmapFour\symbol{197}}
\expandafter\def\csname simpleicon@ifood\endcsname {\simpleiconsmapFour\symbol{198}}
\expandafter\def\csname simpleicon@ifttt\endcsname {\simpleiconsmapFour\symbol{199}}
\expandafter\def\csname simpleicon@iheartradio\endcsname {\simpleiconsmapFour\symbol{200}}
\expandafter\def\csname simpleicon@ikea\endcsname {\simpleiconsmapFour\symbol{201}}
\expandafter\def\csname simpleicon@imagej\endcsname {\simpleiconsmapFour\symbol{202}}
\expandafter\def\csname simpleicon@imdb\endcsname {\simpleiconsmapFour\symbol{203}}
\expandafter\def\csname simpleicon@imgur\endcsname {\simpleiconsmapFour\symbol{204}}
\expandafter\def\csname simpleicon@immer\endcsname {\simpleiconsmapFour\symbol{205}}
\expandafter\def\csname simpleicon@imou\endcsname {\simpleiconsmapFour\symbol{206}}
\expandafter\def\csname simpleicon@indeed\endcsname {\simpleiconsmapFour\symbol{207}}
\expandafter\def\csname simpleicon@infiniti\endcsname {\simpleiconsmapFour\symbol{208}}
\expandafter\def\csname simpleicon@influxdb\endcsname {\simpleiconsmapFour\symbol{209}}
\expandafter\def\csname simpleicon@informatica\endcsname {\simpleiconsmapFour\symbol{210}}
\expandafter\def\csname simpleicon@infosys\endcsname {\simpleiconsmapFour\symbol{211}}
\expandafter\def\csname simpleicon@ingress\endcsname {\simpleiconsmapFour\symbol{212}}
\expandafter\def\csname simpleicon@inkdrop\endcsname {\simpleiconsmapFour\symbol{213}}
\expandafter\def\csname simpleicon@inkscape\endcsname {\simpleiconsmapFour\symbol{214}}
\expandafter\def\csname simpleicon@insomnia\endcsname {\simpleiconsmapFour\symbol{215}}
\expandafter\def\csname simpleicon@instacart\endcsname {\simpleiconsmapFour\symbol{216}}
\expandafter\def\csname simpleicon@instagram\endcsname {\simpleiconsmapFour\symbol{217}}
\expandafter\def\csname simpleicon@instapaper\endcsname {\simpleiconsmapFour\symbol{218}}
\expandafter\def\csname simpleicon@instatus\endcsname {\simpleiconsmapFour\symbol{219}}
\expandafter\def\csname simpleicon@instructables\endcsname {\simpleiconsmapFour\symbol{220}}
\expandafter\def\csname simpleicon@integromat\endcsname {\simpleiconsmapFour\symbol{221}}
\expandafter\def\csname simpleicon@intel\endcsname {\simpleiconsmapFour\symbol{222}}
\expandafter\def\csname simpleicon@intellijidea\endcsname {\simpleiconsmapFour\symbol{223}}
\expandafter\def\csname simpleicon@interactjs\endcsname {\simpleiconsmapFour\symbol{224}}
\expandafter\def\csname simpleicon@intercom\endcsname {\simpleiconsmapFour\symbol{225}}
\expandafter\def\csname simpleicon@intermarche\endcsname {\simpleiconsmapFour\symbol{226}}
\expandafter\def\csname simpleicon@internetarchive\endcsname {\simpleiconsmapFour\symbol{227}}
\expandafter\def\csname simpleicon@internetexplorer\endcsname {\simpleiconsmapFour\symbol{228}}
\expandafter\def\csname simpleicon@intigriti\endcsname {\simpleiconsmapFour\symbol{229}}
\expandafter\def\csname simpleicon@invision\endcsname {\simpleiconsmapFour\symbol{230}}
\expandafter\def\csname simpleicon@invoiceninja\endcsname {\simpleiconsmapFour\symbol{231}}
\expandafter\def\csname simpleicon@iobroker\endcsname {\simpleiconsmapFour\symbol{232}}
\expandafter\def\csname simpleicon@ionic\endcsname {\simpleiconsmapFour\symbol{233}}
\expandafter\def\csname simpleicon@ionos\endcsname {\simpleiconsmapFour\symbol{234}}
\expandafter\def\csname simpleicon@ios\endcsname {\simpleiconsmapFour\symbol{235}}
\expandafter\def\csname simpleicon@iota\endcsname {\simpleiconsmapFour\symbol{236}}
\expandafter\def\csname simpleicon@ipfs\endcsname {\simpleiconsmapFour\symbol{237}}
\expandafter\def\csname simpleicon@issuu\endcsname {\simpleiconsmapFour\symbol{238}}
\expandafter\def\csname simpleicon@istio\endcsname {\simpleiconsmapFour\symbol{239}}
\expandafter\def\csname simpleicon@itchdotio\endcsname {\simpleiconsmapFour\symbol{240}}
\expandafter\def\csname simpleicon@iterm2\endcsname {\simpleiconsmapFour\symbol{241}}
\expandafter\def\csname simpleicon@itunes\endcsname {\simpleiconsmapFour\symbol{242}}
\expandafter\def\csname simpleicon@iveco\endcsname {\simpleiconsmapFour\symbol{243}}
\expandafter\def\csname simpleicon@jabber\endcsname {\simpleiconsmapFour\symbol{244}}
\expandafter\def\csname simpleicon@jaguar\endcsname {\simpleiconsmapFour\symbol{245}}
\expandafter\def\csname simpleicon@jamboard\endcsname {\simpleiconsmapFour\symbol{246}}
\expandafter\def\csname simpleicon@jameson\endcsname {\simpleiconsmapFour\symbol{247}}
\expandafter\def\csname simpleicon@jamstack\endcsname {\simpleiconsmapFour\symbol{248}}
\expandafter\def\csname simpleicon@jasmine\endcsname {\simpleiconsmapFour\symbol{249}}
\expandafter\def\csname simpleicon@javascript\endcsname {\simpleiconsmapFour\symbol{250}}
\expandafter\def\csname simpleicon@jbl\endcsname {\simpleiconsmapFour\symbol{251}}
\expandafter\def\csname simpleicon@jcb\endcsname {\simpleiconsmapFour\symbol{252}}
\expandafter\def\csname simpleicon@jeep\endcsname {\simpleiconsmapFour\symbol{253}}
\expandafter\def\csname simpleicon@jekyll\endcsname {\simpleiconsmapFour\symbol{254}}
\expandafter\def\csname simpleicon@jellyfin\endcsname {\simpleiconsmapFour\symbol{255}}
\expandafter\def\csname simpleicon@jenkins\endcsname {\simpleiconsmapFive\symbol{0}}
\expandafter\def\csname simpleicon@jenkinsx\endcsname {\simpleiconsmapFive\symbol{1}}
\expandafter\def\csname simpleicon@jest\endcsname {\simpleiconsmapFive\symbol{2}}
\expandafter\def\csname simpleicon@jet\endcsname {\simpleiconsmapFive\symbol{3}}
\expandafter\def\csname simpleicon@jetbrains\endcsname {\simpleiconsmapFive\symbol{4}}
\expandafter\def\csname simpleicon@jetpackcompose\endcsname {\simpleiconsmapFive\symbol{5}}
\expandafter\def\csname simpleicon@jfrog\endcsname {\simpleiconsmapFive\symbol{6}}
\expandafter\def\csname simpleicon@jfrogbintray\endcsname {\simpleiconsmapFive\symbol{7}}
\expandafter\def\csname simpleicon@jinja\endcsname {\simpleiconsmapFive\symbol{8}}
\expandafter\def\csname simpleicon@jira\endcsname {\simpleiconsmapFive\symbol{9}}
\expandafter\def\csname simpleicon@jirasoftware\endcsname {\simpleiconsmapFive\symbol{10}}
\expandafter\def\csname simpleicon@jitsi\endcsname {\simpleiconsmapFive\symbol{11}}
\expandafter\def\csname simpleicon@johndeere\endcsname {\simpleiconsmapFive\symbol{12}}
\expandafter\def\csname simpleicon@joomla\endcsname {\simpleiconsmapFive\symbol{13}}
\expandafter\def\csname simpleicon@joplin\endcsname {\simpleiconsmapFive\symbol{14}}
\expandafter\def\csname simpleicon@jordan\endcsname {\simpleiconsmapFive\symbol{15}}
\expandafter\def\csname simpleicon@jpeg\endcsname {\simpleiconsmapFive\symbol{16}}
\expandafter\def\csname simpleicon@jquery\endcsname {\simpleiconsmapFive\symbol{17}}
\expandafter\def\csname simpleicon@jrgroup\endcsname {\simpleiconsmapFive\symbol{18}}
\expandafter\def\csname simpleicon@jsdelivr\endcsname {\simpleiconsmapFive\symbol{19}}
\expandafter\def\csname simpleicon@jsfiddle\endcsname {\simpleiconsmapFive\symbol{20}}
\expandafter\def\csname simpleicon@json\endcsname {\simpleiconsmapFive\symbol{21}}
\expandafter\def\csname simpleicon@jsonwebtokens\endcsname {\simpleiconsmapFive\symbol{22}}
\expandafter\def\csname simpleicon@jss\endcsname {\simpleiconsmapFive\symbol{23}}
\expandafter\def\csname simpleicon@julia\endcsname {\simpleiconsmapFive\symbol{24}}
\expandafter\def\csname simpleicon@junipernetworks\endcsname {\simpleiconsmapFive\symbol{25}}
\expandafter\def\csname simpleicon@junit5\endcsname {\simpleiconsmapFive\symbol{26}}
\expandafter\def\csname simpleicon@jupyter\endcsname {\simpleiconsmapFive\symbol{27}}
\expandafter\def\csname simpleicon@justeat\endcsname {\simpleiconsmapFive\symbol{28}}
\expandafter\def\csname simpleicon@justgiving\endcsname {\simpleiconsmapFive\symbol{29}}
\expandafter\def\csname simpleicon@k3s\endcsname {\simpleiconsmapFive\symbol{30}}
\expandafter\def\csname simpleicon@k6\endcsname {\simpleiconsmapFive\symbol{31}}
\expandafter\def\csname simpleicon@kaggle\endcsname {\simpleiconsmapFive\symbol{32}}
\expandafter\def\csname simpleicon@kahoot\endcsname {\simpleiconsmapFive\symbol{33}}
\expandafter\def\csname simpleicon@kaios\endcsname {\simpleiconsmapFive\symbol{34}}
\expandafter\def\csname simpleicon@kakao\endcsname {\simpleiconsmapFive\symbol{35}}
\expandafter\def\csname simpleicon@kakaotalk\endcsname {\simpleiconsmapFive\symbol{36}}
\expandafter\def\csname simpleicon@kalilinux\endcsname {\simpleiconsmapFive\symbol{37}}
\expandafter\def\csname simpleicon@kaniko\endcsname {\simpleiconsmapFive\symbol{38}}
\expandafter\def\csname simpleicon@karlsruherverkehrsverbund\endcsname {\simpleiconsmapFive\symbol{39}}
\expandafter\def\csname simpleicon@kasasmart\endcsname {\simpleiconsmapFive\symbol{40}}
\expandafter\def\csname simpleicon@kashflow\endcsname {\simpleiconsmapFive\symbol{41}}
\expandafter\def\csname simpleicon@kaspersky\endcsname {\simpleiconsmapFive\symbol{42}}
\expandafter\def\csname simpleicon@katacoda\endcsname {\simpleiconsmapFive\symbol{43}}
\expandafter\def\csname simpleicon@katana\endcsname {\simpleiconsmapFive\symbol{44}}
\expandafter\def\csname simpleicon@kaufland\endcsname {\simpleiconsmapFive\symbol{45}}
\expandafter\def\csname simpleicon@kde\endcsname {\simpleiconsmapFive\symbol{46}}
\expandafter\def\csname simpleicon@kdenlive\endcsname {\simpleiconsmapFive\symbol{47}}
\expandafter\def\csname simpleicon@keepachangelog\endcsname {\simpleiconsmapFive\symbol{48}}
\expandafter\def\csname simpleicon@keepassxc\endcsname {\simpleiconsmapFive\symbol{49}}
\expandafter\def\csname simpleicon@kentico\endcsname {\simpleiconsmapFive\symbol{50}}
\expandafter\def\csname simpleicon@keras\endcsname {\simpleiconsmapFive\symbol{51}}
\expandafter\def\csname simpleicon@keybase\endcsname {\simpleiconsmapFive\symbol{52}}
\expandafter\def\csname simpleicon@keycdn\endcsname {\simpleiconsmapFive\symbol{53}}
\expandafter\def\csname simpleicon@keystone\endcsname {\simpleiconsmapFive\symbol{54}}
\expandafter\def\csname simpleicon@kfc\endcsname {\simpleiconsmapFive\symbol{55}}
\expandafter\def\csname simpleicon@khanacademy\endcsname {\simpleiconsmapFive\symbol{56}}
\expandafter\def\csname simpleicon@khronosgroup\endcsname {\simpleiconsmapFive\symbol{57}}
\expandafter\def\csname simpleicon@kia\endcsname {\simpleiconsmapFive\symbol{58}}
\expandafter\def\csname simpleicon@kibana\endcsname {\simpleiconsmapFive\symbol{59}}
\expandafter\def\csname simpleicon@kicad\endcsname {\simpleiconsmapFive\symbol{60}}
\expandafter\def\csname simpleicon@kickstarter\endcsname {\simpleiconsmapFive\symbol{61}}
\expandafter\def\csname simpleicon@kik\endcsname {\simpleiconsmapFive\symbol{62}}
\expandafter\def\csname simpleicon@kingstontechnology\endcsname {\simpleiconsmapFive\symbol{63}}
\expandafter\def\csname simpleicon@kinopoisk\endcsname {\simpleiconsmapFive\symbol{64}}
\expandafter\def\csname simpleicon@kirby\endcsname {\simpleiconsmapFive\symbol{65}}
\expandafter\def\csname simpleicon@kitsu\endcsname {\simpleiconsmapFive\symbol{66}}
\expandafter\def\csname simpleicon@klarna\endcsname {\simpleiconsmapFive\symbol{67}}
\expandafter\def\csname simpleicon@klm\endcsname {\simpleiconsmapFive\symbol{68}}
\expandafter\def\csname simpleicon@klook\endcsname {\simpleiconsmapFive\symbol{69}}
\expandafter\def\csname simpleicon@knative\endcsname {\simpleiconsmapFive\symbol{70}}
\expandafter\def\csname simpleicon@knowledgebase\endcsname {\simpleiconsmapFive\symbol{71}}
\expandafter\def\csname simpleicon@known\endcsname {\simpleiconsmapFive\symbol{72}}
\expandafter\def\csname simpleicon@koa\endcsname {\simpleiconsmapFive\symbol{73}}
\expandafter\def\csname simpleicon@koc\endcsname {\simpleiconsmapFive\symbol{74}}
\expandafter\def\csname simpleicon@kodi\endcsname {\simpleiconsmapFive\symbol{75}}
\expandafter\def\csname simpleicon@kofax\endcsname {\simpleiconsmapFive\symbol{76}}
\expandafter\def\csname simpleicon@kofi\endcsname {\simpleiconsmapFive\symbol{77}}
\expandafter\def\csname simpleicon@komoot\endcsname {\simpleiconsmapFive\symbol{78}}
\expandafter\def\csname simpleicon@konami\endcsname {\simpleiconsmapFive\symbol{79}}
\expandafter\def\csname simpleicon@kong\endcsname {\simpleiconsmapFive\symbol{80}}
\expandafter\def\csname simpleicon@kongregate\endcsname {\simpleiconsmapFive\symbol{81}}
\expandafter\def\csname simpleicon@konva\endcsname {\simpleiconsmapFive\symbol{82}}
\expandafter\def\csname simpleicon@kotlin\endcsname {\simpleiconsmapFive\symbol{83}}
\expandafter\def\csname simpleicon@koyeb\endcsname {\simpleiconsmapFive\symbol{84}}
\expandafter\def\csname simpleicon@krita\endcsname {\simpleiconsmapFive\symbol{85}}
\expandafter\def\csname simpleicon@ktm\endcsname {\simpleiconsmapFive\symbol{86}}
\expandafter\def\csname simpleicon@kuaishou\endcsname {\simpleiconsmapFive\symbol{87}}
\expandafter\def\csname simpleicon@kubernetes\endcsname {\simpleiconsmapFive\symbol{88}}
\expandafter\def\csname simpleicon@kubuntu\endcsname {\simpleiconsmapFive\symbol{89}}
\expandafter\def\csname simpleicon@kuma\endcsname {\simpleiconsmapFive\symbol{90}}
\expandafter\def\csname simpleicon@kyocera\endcsname {\simpleiconsmapFive\symbol{91}}
\expandafter\def\csname simpleicon@labview\endcsname {\simpleiconsmapFive\symbol{92}}
\expandafter\def\csname simpleicon@lada\endcsname {\simpleiconsmapFive\symbol{93}}
\expandafter\def\csname simpleicon@lamborghini\endcsname {\simpleiconsmapFive\symbol{94}}
\expandafter\def\csname simpleicon@landrover\endcsname {\simpleiconsmapFive\symbol{95}}
\expandafter\def\csname simpleicon@lapce\endcsname {\simpleiconsmapFive\symbol{96}}
\expandafter\def\csname simpleicon@laragon\endcsname {\simpleiconsmapFive\symbol{97}}
\expandafter\def\csname simpleicon@laravel\endcsname {\simpleiconsmapFive\symbol{98}}
\expandafter\def\csname simpleicon@laravelhorizon\endcsname {\simpleiconsmapFive\symbol{99}}
\expandafter\def\csname simpleicon@laravelnova\endcsname {\simpleiconsmapFive\symbol{100}}
\expandafter\def\csname simpleicon@lastdotfm\endcsname {\simpleiconsmapFive\symbol{101}}
\expandafter\def\csname simpleicon@lastpass\endcsname {\simpleiconsmapFive\symbol{102}}
\expandafter\def\csname simpleicon@latex\endcsname {\simpleiconsmapFive\symbol{103}}
\expandafter\def\csname simpleicon@launchpad\endcsname {\simpleiconsmapFive\symbol{104}}
\expandafter\def\csname simpleicon@lazarus\endcsname {\simpleiconsmapFive\symbol{105}}
\expandafter\def\csname simpleicon@lbry\endcsname {\simpleiconsmapFive\symbol{106}}
\expandafter\def\csname simpleicon@leaderprice\endcsname {\simpleiconsmapFive\symbol{107}}
\expandafter\def\csname simpleicon@leaflet\endcsname {\simpleiconsmapFive\symbol{108}}
\expandafter\def\csname simpleicon@leanpub\endcsname {\simpleiconsmapFive\symbol{109}}
\expandafter\def\csname simpleicon@leetcode\endcsname {\simpleiconsmapFive\symbol{110}}
\expandafter\def\csname simpleicon@legacygames\endcsname {\simpleiconsmapFive\symbol{111}}
\expandafter\def\csname simpleicon@lemmy\endcsname {\simpleiconsmapFive\symbol{112}}
\expandafter\def\csname simpleicon@lenovo\endcsname {\simpleiconsmapFive\symbol{113}}
\expandafter\def\csname simpleicon@lens\endcsname {\simpleiconsmapFive\symbol{114}}
\expandafter\def\csname simpleicon@leroymerlin\endcsname {\simpleiconsmapFive\symbol{115}}
\expandafter\def\csname simpleicon@less\endcsname {\simpleiconsmapFive\symbol{116}}
\expandafter\def\csname simpleicon@letsencrypt\endcsname {\simpleiconsmapFive\symbol{117}}
\expandafter\def\csname simpleicon@letterboxd\endcsname {\simpleiconsmapFive\symbol{118}}
\expandafter\def\csname simpleicon@levelsdotfyi\endcsname {\simpleiconsmapFive\symbol{119}}
\expandafter\def\csname simpleicon@lg\endcsname {\simpleiconsmapFive\symbol{120}}
\expandafter\def\csname simpleicon@lgtm\endcsname {\simpleiconsmapFive\symbol{121}}
\expandafter\def\csname simpleicon@liberapay\endcsname {\simpleiconsmapFive\symbol{122}}
\expandafter\def\csname simpleicon@librariesdotio\endcsname {\simpleiconsmapFive\symbol{123}}
\expandafter\def\csname simpleicon@librarything\endcsname {\simpleiconsmapFive\symbol{124}}
\expandafter\def\csname simpleicon@libreoffice\endcsname {\simpleiconsmapFive\symbol{125}}
\expandafter\def\csname simpleicon@libuv\endcsname {\simpleiconsmapFive\symbol{126}}
\expandafter\def\csname simpleicon@lichess\endcsname {\simpleiconsmapFive\symbol{127}}
\expandafter\def\csname simpleicon@lidl\endcsname {\simpleiconsmapFive\symbol{128}}
\expandafter\def\csname simpleicon@lifx\endcsname {\simpleiconsmapFive\symbol{129}}
\expandafter\def\csname simpleicon@lighthouse\endcsname {\simpleiconsmapFive\symbol{130}}
\expandafter\def\csname simpleicon@line\endcsname {\simpleiconsmapFive\symbol{131}}
\expandafter\def\csname simpleicon@lineageos\endcsname {\simpleiconsmapFive\symbol{132}}
\expandafter\def\csname simpleicon@linear\endcsname {\simpleiconsmapFive\symbol{133}}
\expandafter\def\csname simpleicon@linkedin\endcsname {\simpleiconsmapFive\symbol{134}}
\expandafter\def\csname simpleicon@linkerd\endcsname {\simpleiconsmapFive\symbol{135}}
\expandafter\def\csname simpleicon@linkfire\endcsname {\simpleiconsmapFive\symbol{136}}
\expandafter\def\csname simpleicon@linktree\endcsname {\simpleiconsmapFive\symbol{137}}
\expandafter\def\csname simpleicon@linode\endcsname {\simpleiconsmapFive\symbol{138}}
\expandafter\def\csname simpleicon@linux\endcsname {\simpleiconsmapFive\symbol{139}}
\expandafter\def\csname simpleicon@linuxcontainers\endcsname {\simpleiconsmapFive\symbol{140}}
\expandafter\def\csname simpleicon@linuxfoundation\endcsname {\simpleiconsmapFive\symbol{141}}
\expandafter\def\csname simpleicon@linuxmint\endcsname {\simpleiconsmapFive\symbol{142}}
\expandafter\def\csname simpleicon@lionair\endcsname {\simpleiconsmapFive\symbol{143}}
\expandafter\def\csname simpleicon@liquibase\endcsname {\simpleiconsmapFive\symbol{144}}
\expandafter\def\csname simpleicon@lit\endcsname {\simpleiconsmapFive\symbol{145}}
\expandafter\def\csname simpleicon@litecoin\endcsname {\simpleiconsmapFive\symbol{146}}
\expandafter\def\csname simpleicon@litiengine\endcsname {\simpleiconsmapFive\symbol{147}}
\expandafter\def\csname simpleicon@livechat\endcsname {\simpleiconsmapFive\symbol{148}}
\expandafter\def\csname simpleicon@livejournal\endcsname {\simpleiconsmapFive\symbol{149}}
\expandafter\def\csname simpleicon@livewire\endcsname {\simpleiconsmapFive\symbol{150}}
\expandafter\def\csname simpleicon@llvm\endcsname {\simpleiconsmapFive\symbol{151}}
\expandafter\def\csname simpleicon@lmms\endcsname {\simpleiconsmapFive\symbol{152}}
\expandafter\def\csname simpleicon@lodash\endcsname {\simpleiconsmapFive\symbol{153}}
\expandafter\def\csname simpleicon@logitech\endcsname {\simpleiconsmapFive\symbol{154}}
\expandafter\def\csname simpleicon@logmein\endcsname {\simpleiconsmapFive\symbol{155}}
\expandafter\def\csname simpleicon@logstash\endcsname {\simpleiconsmapFive\symbol{156}}
\expandafter\def\csname simpleicon@looker\endcsname {\simpleiconsmapFive\symbol{157}}
\expandafter\def\csname simpleicon@loom\endcsname {\simpleiconsmapFive\symbol{158}}
\expandafter\def\csname simpleicon@loop\endcsname {\simpleiconsmapFive\symbol{159}}
\expandafter\def\csname simpleicon@loopback\endcsname {\simpleiconsmapFive\symbol{160}}
\expandafter\def\csname simpleicon@lospec\endcsname {\simpleiconsmapFive\symbol{161}}
\expandafter\def\csname simpleicon@lotpolishairlines\endcsname {\simpleiconsmapFive\symbol{162}}
\expandafter\def\csname simpleicon@lua\endcsname {\simpleiconsmapFive\symbol{163}}
\expandafter\def\csname simpleicon@lubuntu\endcsname {\simpleiconsmapFive\symbol{164}}
\expandafter\def\csname simpleicon@ludwig\endcsname {\simpleiconsmapFive\symbol{165}}
\expandafter\def\csname simpleicon@lufthansa\endcsname {\simpleiconsmapFive\symbol{166}}
\expandafter\def\csname simpleicon@lumen\endcsname {\simpleiconsmapFive\symbol{167}}
\expandafter\def\csname simpleicon@lunacy\endcsname {\simpleiconsmapFive\symbol{168}}
\expandafter\def\csname simpleicon@lydia\endcsname {\simpleiconsmapFive\symbol{169}}
\expandafter\def\csname simpleicon@lyft\endcsname {\simpleiconsmapFive\symbol{170}}
\expandafter\def\csname simpleicon@maas\endcsname {\simpleiconsmapFive\symbol{171}}
\expandafter\def\csname simpleicon@macos\endcsname {\simpleiconsmapFive\symbol{172}}
\expandafter\def\csname simpleicon@macys\endcsname {\simpleiconsmapFive\symbol{173}}
\expandafter\def\csname simpleicon@magasinsu\endcsname {\simpleiconsmapFive\symbol{174}}
\expandafter\def\csname simpleicon@magento\endcsname {\simpleiconsmapFive\symbol{175}}
\expandafter\def\csname simpleicon@magisk\endcsname {\simpleiconsmapFive\symbol{176}}
\expandafter\def\csname simpleicon@mailchimp\endcsname {\simpleiconsmapFive\symbol{177}}
\expandafter\def\csname simpleicon@maildotru\endcsname {\simpleiconsmapFive\symbol{178}}
\expandafter\def\csname simpleicon@mailgun\endcsname {\simpleiconsmapFive\symbol{179}}
\expandafter\def\csname simpleicon@majorleaguehacking\endcsname {\simpleiconsmapFive\symbol{180}}
\expandafter\def\csname simpleicon@makerbot\endcsname {\simpleiconsmapFive\symbol{181}}
\expandafter\def\csname simpleicon@mamp\endcsname {\simpleiconsmapFive\symbol{182}}
\expandafter\def\csname simpleicon@man\endcsname {\simpleiconsmapFive\symbol{183}}
\expandafter\def\csname simpleicon@manageiq\endcsname {\simpleiconsmapFive\symbol{184}}
\expandafter\def\csname simpleicon@manjaro\endcsname {\simpleiconsmapFive\symbol{185}}
\expandafter\def\csname simpleicon@mapbox\endcsname {\simpleiconsmapFive\symbol{186}}
\expandafter\def\csname simpleicon@mariadb\endcsname {\simpleiconsmapFive\symbol{187}}
\expandafter\def\csname simpleicon@mariadbfoundation\endcsname {\simpleiconsmapFive\symbol{188}}
\expandafter\def\csname simpleicon@markdown\endcsname {\simpleiconsmapFive\symbol{189}}
\expandafter\def\csname simpleicon@marketo\endcsname {\simpleiconsmapFive\symbol{190}}
\expandafter\def\csname simpleicon@marko\endcsname {\simpleiconsmapFive\symbol{191}}
\expandafter\def\csname simpleicon@marriott\endcsname {\simpleiconsmapFive\symbol{192}}
\expandafter\def\csname simpleicon@maserati\endcsname {\simpleiconsmapFive\symbol{193}}
\expandafter\def\csname simpleicon@mastercard\endcsname {\simpleiconsmapFive\symbol{194}}
\expandafter\def\csname simpleicon@mastercomfig\endcsname {\simpleiconsmapFive\symbol{195}}
\expandafter\def\csname simpleicon@mastodon\endcsname {\simpleiconsmapFive\symbol{196}}
\expandafter\def\csname simpleicon@materialdesign\endcsname {\simpleiconsmapFive\symbol{197}}
\expandafter\def\csname simpleicon@materialdesignicons\endcsname {\simpleiconsmapFive\symbol{198}}
\expandafter\def\csname simpleicon@matomo\endcsname {\simpleiconsmapFive\symbol{199}}
\expandafter\def\csname simpleicon@matrix\endcsname {\simpleiconsmapFive\symbol{200}}
\expandafter\def\csname simpleicon@matterdotjs\endcsname {\simpleiconsmapFive\symbol{201}}
\expandafter\def\csname simpleicon@mattermost\endcsname {\simpleiconsmapFive\symbol{202}}
\expandafter\def\csname simpleicon@matternet\endcsname {\simpleiconsmapFive\symbol{203}}
\expandafter\def\csname simpleicon@max\endcsname {\simpleiconsmapFive\symbol{204}}
\expandafter\def\csname simpleicon@maxplanckgesellschaft\endcsname {\simpleiconsmapFive\symbol{205}}
\expandafter\def\csname simpleicon@maytag\endcsname {\simpleiconsmapFive\symbol{206}}
\expandafter\def\csname simpleicon@mazda\endcsname {\simpleiconsmapFive\symbol{207}}
\expandafter\def\csname simpleicon@mcafee\endcsname {\simpleiconsmapFive\symbol{208}}
\expandafter\def\csname simpleicon@mcdonalds\endcsname {\simpleiconsmapFive\symbol{209}}
\expandafter\def\csname simpleicon@mclaren\endcsname {\simpleiconsmapFive\symbol{210}}
\expandafter\def\csname simpleicon@mdbook\endcsname {\simpleiconsmapFive\symbol{211}}
\expandafter\def\csname simpleicon@mdnwebdocs\endcsname {\simpleiconsmapFive\symbol{212}}
\expandafter\def\csname simpleicon@mdx\endcsname {\simpleiconsmapFive\symbol{213}}
\expandafter\def\csname simpleicon@mediafire\endcsname {\simpleiconsmapFive\symbol{214}}
\expandafter\def\csname simpleicon@mediamarkt\endcsname {\simpleiconsmapFive\symbol{215}}
\expandafter\def\csname simpleicon@mediatek\endcsname {\simpleiconsmapFive\symbol{216}}
\expandafter\def\csname simpleicon@mediatemple\endcsname {\simpleiconsmapFive\symbol{217}}
\expandafter\def\csname simpleicon@medium\endcsname {\simpleiconsmapFive\symbol{218}}
\expandafter\def\csname simpleicon@meetup\endcsname {\simpleiconsmapFive\symbol{219}}
\expandafter\def\csname simpleicon@mega\endcsname {\simpleiconsmapFive\symbol{220}}
\expandafter\def\csname simpleicon@mendeley\endcsname {\simpleiconsmapFive\symbol{221}}
\expandafter\def\csname simpleicon@mercedes\endcsname {\simpleiconsmapFive\symbol{222}}
\expandafter\def\csname simpleicon@merck\endcsname {\simpleiconsmapFive\symbol{223}}
\expandafter\def\csname simpleicon@mercurial\endcsname {\simpleiconsmapFive\symbol{224}}
\expandafter\def\csname simpleicon@messenger\endcsname {\simpleiconsmapFive\symbol{225}}
\expandafter\def\csname simpleicon@meta\endcsname {\simpleiconsmapFive\symbol{226}}
\expandafter\def\csname simpleicon@metabase\endcsname {\simpleiconsmapFive\symbol{227}}
\expandafter\def\csname simpleicon@metafilter\endcsname {\simpleiconsmapFive\symbol{228}}
\expandafter\def\csname simpleicon@meteor\endcsname {\simpleiconsmapFive\symbol{229}}
\expandafter\def\csname simpleicon@metro\endcsname {\simpleiconsmapFive\symbol{230}}
\expandafter\def\csname simpleicon@metrodelaciudaddemexico\endcsname {\simpleiconsmapFive\symbol{231}}
\expandafter\def\csname simpleicon@metrodemadrid\endcsname {\simpleiconsmapFive\symbol{232}}
\expandafter\def\csname simpleicon@metrodeparis\endcsname {\simpleiconsmapFive\symbol{233}}
\expandafter\def\csname simpleicon@mewe\endcsname {\simpleiconsmapFive\symbol{234}}
\expandafter\def\csname simpleicon@microbit\endcsname {\simpleiconsmapFive\symbol{235}}
\expandafter\def\csname simpleicon@microdotblog\endcsname {\simpleiconsmapFive\symbol{236}}
\expandafter\def\csname simpleicon@microgenetics\endcsname {\simpleiconsmapFive\symbol{237}}
\expandafter\def\csname simpleicon@micropython\endcsname {\simpleiconsmapFive\symbol{238}}
\expandafter\def\csname simpleicon@microsoft\endcsname {\simpleiconsmapFive\symbol{239}}
\expandafter\def\csname simpleicon@microsoftacademic\endcsname {\simpleiconsmapFive\symbol{240}}
\expandafter\def\csname simpleicon@microsoftaccess\endcsname {\simpleiconsmapFive\symbol{241}}
\expandafter\def\csname simpleicon@microsoftazure\endcsname {\simpleiconsmapFive\symbol{242}}
\expandafter\def\csname simpleicon@microsoftbing\endcsname {\simpleiconsmapFive\symbol{243}}
\expandafter\def\csname simpleicon@microsoftedge\endcsname {\simpleiconsmapFive\symbol{244}}
\expandafter\def\csname simpleicon@microsoftexcel\endcsname {\simpleiconsmapFive\symbol{245}}
\expandafter\def\csname simpleicon@microsoftexchange\endcsname {\simpleiconsmapFive\symbol{246}}
\expandafter\def\csname simpleicon@microsoftoffice\endcsname {\simpleiconsmapFive\symbol{247}}
\expandafter\def\csname simpleicon@microsoftonedrive\endcsname {\simpleiconsmapFive\symbol{248}}
\expandafter\def\csname simpleicon@microsoftonenote\endcsname {\simpleiconsmapFive\symbol{249}}
\expandafter\def\csname simpleicon@microsoftoutlook\endcsname {\simpleiconsmapFive\symbol{250}}
\expandafter\def\csname simpleicon@microsoftpowerpoint\endcsname {\simpleiconsmapFive\symbol{251}}
\expandafter\def\csname simpleicon@microsoftsharepoint\endcsname {\simpleiconsmapFive\symbol{252}}
\expandafter\def\csname simpleicon@microsoftsqlserver\endcsname {\simpleiconsmapFive\symbol{253}}
\expandafter\def\csname simpleicon@microsoftteams\endcsname {\simpleiconsmapFive\symbol{254}}
\expandafter\def\csname simpleicon@microsofttranslator\endcsname {\simpleiconsmapFive\symbol{255}}
\expandafter\def\csname simpleicon@microsoftvisio\endcsname {\simpleiconsmapSix\symbol{0}}
\expandafter\def\csname simpleicon@microsoftword\endcsname {\simpleiconsmapSix\symbol{1}}
\expandafter\def\csname simpleicon@microstrategy\endcsname {\simpleiconsmapSix\symbol{2}}
\expandafter\def\csname simpleicon@midi\endcsname {\simpleiconsmapSix\symbol{3}}
\expandafter\def\csname simpleicon@minds\endcsname {\simpleiconsmapSix\symbol{4}}
\expandafter\def\csname simpleicon@minecraft\endcsname {\simpleiconsmapSix\symbol{5}}
\expandafter\def\csname simpleicon@minetest\endcsname {\simpleiconsmapSix\symbol{6}}
\expandafter\def\csname simpleicon@mini\endcsname {\simpleiconsmapSix\symbol{7}}
\expandafter\def\csname simpleicon@minutemailer\endcsname {\simpleiconsmapSix\symbol{8}}
\expandafter\def\csname simpleicon@miro\endcsname {\simpleiconsmapSix\symbol{9}}
\expandafter\def\csname simpleicon@mitsubishi\endcsname {\simpleiconsmapSix\symbol{10}}
\expandafter\def\csname simpleicon@mix\endcsname {\simpleiconsmapSix\symbol{11}}
\expandafter\def\csname simpleicon@mixcloud\endcsname {\simpleiconsmapSix\symbol{12}}
\expandafter\def\csname simpleicon@mlb\endcsname {\simpleiconsmapSix\symbol{13}}
\expandafter\def\csname simpleicon@mlflow\endcsname {\simpleiconsmapSix\symbol{14}}
\expandafter\def\csname simpleicon@mobx\endcsname {\simpleiconsmapSix\symbol{15}}
\expandafter\def\csname simpleicon@mobxstatetree\endcsname {\simpleiconsmapSix\symbol{16}}
\expandafter\def\csname simpleicon@mocha\endcsname {\simpleiconsmapSix\symbol{17}}
\expandafter\def\csname simpleicon@modx\endcsname {\simpleiconsmapSix\symbol{18}}
\expandafter\def\csname simpleicon@mojangstudios\endcsname {\simpleiconsmapSix\symbol{19}}
\expandafter\def\csname simpleicon@moleculer\endcsname {\simpleiconsmapSix\symbol{20}}
\expandafter\def\csname simpleicon@momenteo\endcsname {\simpleiconsmapSix\symbol{21}}
\expandafter\def\csname simpleicon@monero\endcsname {\simpleiconsmapSix\symbol{22}}
\expandafter\def\csname simpleicon@moneygram\endcsname {\simpleiconsmapSix\symbol{23}}
\expandafter\def\csname simpleicon@mongodb\endcsname {\simpleiconsmapSix\symbol{24}}
\expandafter\def\csname simpleicon@monkeytie\endcsname {\simpleiconsmapSix\symbol{25}}
\expandafter\def\csname simpleicon@monogames\endcsname {\simpleiconsmapSix\symbol{26}}
\expandafter\def\csname simpleicon@monoprix\endcsname {\simpleiconsmapSix\symbol{27}}
\expandafter\def\csname simpleicon@monster\endcsname {\simpleiconsmapSix\symbol{28}}
\expandafter\def\csname simpleicon@monzo\endcsname {\simpleiconsmapSix\symbol{29}}
\expandafter\def\csname simpleicon@moo\endcsname {\simpleiconsmapSix\symbol{30}}
\expandafter\def\csname simpleicon@morrisons\endcsname {\simpleiconsmapSix\symbol{31}}
\expandafter\def\csname simpleicon@moscowmetro\endcsname {\simpleiconsmapSix\symbol{32}}
\expandafter\def\csname simpleicon@motorola\endcsname {\simpleiconsmapSix\symbol{33}}
\expandafter\def\csname simpleicon@mozilla\endcsname {\simpleiconsmapSix\symbol{34}}
\expandafter\def\csname simpleicon@msi\endcsname {\simpleiconsmapSix\symbol{35}}
\expandafter\def\csname simpleicon@msibusiness\endcsname {\simpleiconsmapSix\symbol{36}}
\expandafter\def\csname simpleicon@mta\endcsname {\simpleiconsmapSix\symbol{37}}
\expandafter\def\csname simpleicon@mtr\endcsname {\simpleiconsmapSix\symbol{38}}
\expandafter\def\csname simpleicon@mui\endcsname {\simpleiconsmapSix\symbol{39}}
\expandafter\def\csname simpleicon@mulesoft\endcsname {\simpleiconsmapSix\symbol{40}}
\expandafter\def\csname simpleicon@muller\endcsname {\simpleiconsmapSix\symbol{41}}
\expandafter\def\csname simpleicon@mumble\endcsname {\simpleiconsmapSix\symbol{42}}
\expandafter\def\csname simpleicon@musescore\endcsname {\simpleiconsmapSix\symbol{43}}
\expandafter\def\csname simpleicon@musicbrainz\endcsname {\simpleiconsmapSix\symbol{44}}
\expandafter\def\csname simpleicon@mxlinux\endcsname {\simpleiconsmapSix\symbol{45}}
\expandafter\def\csname simpleicon@myanimelist\endcsname {\simpleiconsmapSix\symbol{46}}
\expandafter\def\csname simpleicon@myob\endcsname {\simpleiconsmapSix\symbol{47}}
\expandafter\def\csname simpleicon@myspace\endcsname {\simpleiconsmapSix\symbol{48}}
\expandafter\def\csname simpleicon@mysql\endcsname {\simpleiconsmapSix\symbol{49}}
\expandafter\def\csname simpleicon@n26\endcsname {\simpleiconsmapSix\symbol{50}}
\expandafter\def\csname simpleicon@namebase\endcsname {\simpleiconsmapSix\symbol{51}}
\expandafter\def\csname simpleicon@namecheap\endcsname {\simpleiconsmapSix\symbol{52}}
\expandafter\def\csname simpleicon@nano\endcsname {\simpleiconsmapSix\symbol{53}}
\expandafter\def\csname simpleicon@nasa\endcsname {\simpleiconsmapSix\symbol{54}}
\expandafter\def\csname simpleicon@nationalgrid\endcsname {\simpleiconsmapSix\symbol{55}}
\expandafter\def\csname simpleicon@nativescript\endcsname {\simpleiconsmapSix\symbol{56}}
\expandafter\def\csname simpleicon@naver\endcsname {\simpleiconsmapSix\symbol{57}}
\expandafter\def\csname simpleicon@nba\endcsname {\simpleiconsmapSix\symbol{58}}
\expandafter\def\csname simpleicon@nbb\endcsname {\simpleiconsmapSix\symbol{59}}
\expandafter\def\csname simpleicon@ndr\endcsname {\simpleiconsmapSix\symbol{60}}
\expandafter\def\csname simpleicon@nec\endcsname {\simpleiconsmapSix\symbol{61}}
\expandafter\def\csname simpleicon@neo4j\endcsname {\simpleiconsmapSix\symbol{62}}
\expandafter\def\csname simpleicon@neovim\endcsname {\simpleiconsmapSix\symbol{63}}
\expandafter\def\csname simpleicon@nestjs\endcsname {\simpleiconsmapSix\symbol{64}}
\expandafter\def\csname simpleicon@netapp\endcsname {\simpleiconsmapSix\symbol{65}}
\expandafter\def\csname simpleicon@netbsd\endcsname {\simpleiconsmapSix\symbol{66}}
\expandafter\def\csname simpleicon@netflix\endcsname {\simpleiconsmapSix\symbol{67}}
\expandafter\def\csname simpleicon@netlify\endcsname {\simpleiconsmapSix\symbol{68}}
\expandafter\def\csname simpleicon@nette\endcsname {\simpleiconsmapSix\symbol{69}}
\expandafter\def\csname simpleicon@netto\endcsname {\simpleiconsmapSix\symbol{70}}
\expandafter\def\csname simpleicon@neutralinojs\endcsname {\simpleiconsmapSix\symbol{71}}
\expandafter\def\csname simpleicon@newbalance\endcsname {\simpleiconsmapSix\symbol{72}}
\expandafter\def\csname simpleicon@newjapanprowrestling\endcsname {\simpleiconsmapSix\symbol{73}}
\expandafter\def\csname simpleicon@newrelic\endcsname {\simpleiconsmapSix\symbol{74}}
\expandafter\def\csname simpleicon@newyorktimes\endcsname {\simpleiconsmapSix\symbol{75}}
\expandafter\def\csname simpleicon@nextbilliondotai\endcsname {\simpleiconsmapSix\symbol{76}}
\expandafter\def\csname simpleicon@nextcloud\endcsname {\simpleiconsmapSix\symbol{77}}
\expandafter\def\csname simpleicon@nextdoor\endcsname {\simpleiconsmapSix\symbol{78}}
\expandafter\def\csname simpleicon@nextdotjs\endcsname {\simpleiconsmapSix\symbol{79}}
\expandafter\def\csname simpleicon@nfc\endcsname {\simpleiconsmapSix\symbol{80}}
\expandafter\def\csname simpleicon@nginx\endcsname {\simpleiconsmapSix\symbol{81}}
\expandafter\def\csname simpleicon@ngrok\endcsname {\simpleiconsmapSix\symbol{82}}
\expandafter\def\csname simpleicon@niconico\endcsname {\simpleiconsmapSix\symbol{83}}
\expandafter\def\csname simpleicon@nike\endcsname {\simpleiconsmapSix\symbol{84}}
\expandafter\def\csname simpleicon@nim\endcsname {\simpleiconsmapSix\symbol{85}}
\expandafter\def\csname simpleicon@nintendo\endcsname {\simpleiconsmapSix\symbol{86}}
\expandafter\def\csname simpleicon@nintendo3ds\endcsname {\simpleiconsmapSix\symbol{87}}
\expandafter\def\csname simpleicon@nintendogamecube\endcsname {\simpleiconsmapSix\symbol{88}}
\expandafter\def\csname simpleicon@nintendonetwork\endcsname {\simpleiconsmapSix\symbol{89}}
\expandafter\def\csname simpleicon@nintendoswitch\endcsname {\simpleiconsmapSix\symbol{90}}
\expandafter\def\csname simpleicon@nissan\endcsname {\simpleiconsmapSix\symbol{91}}
\expandafter\def\csname simpleicon@nixos\endcsname {\simpleiconsmapSix\symbol{92}}
\expandafter\def\csname simpleicon@nodedotjs\endcsname {\simpleiconsmapSix\symbol{93}}
\expandafter\def\csname simpleicon@nodemon\endcsname {\simpleiconsmapSix\symbol{94}}
\expandafter\def\csname simpleicon@nodered\endcsname {\simpleiconsmapSix\symbol{95}}
\expandafter\def\csname simpleicon@nokia\endcsname {\simpleiconsmapSix\symbol{96}}
\expandafter\def\csname simpleicon@norco\endcsname {\simpleiconsmapSix\symbol{97}}
\expandafter\def\csname simpleicon@nordvpn\endcsname {\simpleiconsmapSix\symbol{98}}
\expandafter\def\csname simpleicon@norwegian\endcsname {\simpleiconsmapSix\symbol{99}}
\expandafter\def\csname simpleicon@notepadplusplus\endcsname {\simpleiconsmapSix\symbol{100}}
\expandafter\def\csname simpleicon@notion\endcsname {\simpleiconsmapSix\symbol{101}}
\expandafter\def\csname simpleicon@notist\endcsname {\simpleiconsmapSix\symbol{102}}
\expandafter\def\csname simpleicon@nounproject\endcsname {\simpleiconsmapSix\symbol{103}}
\expandafter\def\csname simpleicon@now\endcsname {\simpleiconsmapSix\symbol{104}}
\expandafter\def\csname simpleicon@npm\endcsname {\simpleiconsmapSix\symbol{105}}
\expandafter\def\csname simpleicon@nrwl\endcsname {\simpleiconsmapSix\symbol{106}}
\expandafter\def\csname simpleicon@nubank\endcsname {\simpleiconsmapSix\symbol{107}}
\expandafter\def\csname simpleicon@nucleo\endcsname {\simpleiconsmapSix\symbol{108}}
\expandafter\def\csname simpleicon@nuget\endcsname {\simpleiconsmapSix\symbol{109}}
\expandafter\def\csname simpleicon@nuke\endcsname {\simpleiconsmapSix\symbol{110}}
\expandafter\def\csname simpleicon@numba\endcsname {\simpleiconsmapSix\symbol{111}}
\expandafter\def\csname simpleicon@numpy\endcsname {\simpleiconsmapSix\symbol{112}}
\expandafter\def\csname simpleicon@nunjucks\endcsname {\simpleiconsmapSix\symbol{113}}
\expandafter\def\csname simpleicon@nutanix\endcsname {\simpleiconsmapSix\symbol{114}}
\expandafter\def\csname simpleicon@nuxtdotjs\endcsname {\simpleiconsmapSix\symbol{115}}
\expandafter\def\csname simpleicon@nvidia\endcsname {\simpleiconsmapSix\symbol{116}}
\expandafter\def\csname simpleicon@nx\endcsname {\simpleiconsmapSix\symbol{117}}
\expandafter\def\csname simpleicon@nzxt\endcsname {\simpleiconsmapSix\symbol{118}}
\expandafter\def\csname simpleicon@observable\endcsname {\simpleiconsmapSix\symbol{119}}
\expandafter\def\csname simpleicon@obsidian\endcsname {\simpleiconsmapSix\symbol{120}}
\expandafter\def\csname simpleicon@obsstudio\endcsname {\simpleiconsmapSix\symbol{121}}
\expandafter\def\csname simpleicon@ocaml\endcsname {\simpleiconsmapSix\symbol{122}}
\expandafter\def\csname simpleicon@octanerender\endcsname {\simpleiconsmapSix\symbol{123}}
\expandafter\def\csname simpleicon@octave\endcsname {\simpleiconsmapSix\symbol{124}}
\expandafter\def\csname simpleicon@octoprint\endcsname {\simpleiconsmapSix\symbol{125}}
\expandafter\def\csname simpleicon@octopusdeploy\endcsname {\simpleiconsmapSix\symbol{126}}
\expandafter\def\csname simpleicon@oculus\endcsname {\simpleiconsmapSix\symbol{127}}
\expandafter\def\csname simpleicon@odnoklassniki\endcsname {\simpleiconsmapSix\symbol{128}}
\expandafter\def\csname simpleicon@odysee\endcsname {\simpleiconsmapSix\symbol{129}}
\expandafter\def\csname simpleicon@ohdear\endcsname {\simpleiconsmapSix\symbol{130}}
\expandafter\def\csname simpleicon@okcupid\endcsname {\simpleiconsmapSix\symbol{131}}
\expandafter\def\csname simpleicon@okta\endcsname {\simpleiconsmapSix\symbol{132}}
\expandafter\def\csname simpleicon@oneplus\endcsname {\simpleiconsmapSix\symbol{133}}
\expandafter\def\csname simpleicon@onlyfans\endcsname {\simpleiconsmapSix\symbol{134}}
\expandafter\def\csname simpleicon@onlyoffice\endcsname {\simpleiconsmapSix\symbol{135}}
\expandafter\def\csname simpleicon@onnx\endcsname {\simpleiconsmapSix\symbol{136}}
\expandafter\def\csname simpleicon@onstar\endcsname {\simpleiconsmapSix\symbol{137}}
\expandafter\def\csname simpleicon@opel\endcsname {\simpleiconsmapSix\symbol{138}}
\expandafter\def\csname simpleicon@openaccess\endcsname {\simpleiconsmapSix\symbol{139}}
\expandafter\def\csname simpleicon@openai\endcsname {\simpleiconsmapSix\symbol{140}}
\expandafter\def\csname simpleicon@openaigym\endcsname {\simpleiconsmapSix\symbol{141}}
\expandafter\def\csname simpleicon@openapiinitiative\endcsname {\simpleiconsmapSix\symbol{142}}
\expandafter\def\csname simpleicon@openbadges\endcsname {\simpleiconsmapSix\symbol{143}}
\expandafter\def\csname simpleicon@openbsd\endcsname {\simpleiconsmapSix\symbol{144}}
\expandafter\def\csname simpleicon@openbugbounty\endcsname {\simpleiconsmapSix\symbol{145}}
\expandafter\def\csname simpleicon@opencollective\endcsname {\simpleiconsmapSix\symbol{146}}
\expandafter\def\csname simpleicon@opencontainersinitiative\endcsname {\simpleiconsmapSix\symbol{147}}
\expandafter\def\csname simpleicon@opencv\endcsname {\simpleiconsmapSix\symbol{148}}
\expandafter\def\csname simpleicon@openfaas\endcsname {\simpleiconsmapSix\symbol{149}}
\expandafter\def\csname simpleicon@opengl\endcsname {\simpleiconsmapSix\symbol{150}}
\expandafter\def\csname simpleicon@openid\endcsname {\simpleiconsmapSix\symbol{151}}
\expandafter\def\csname simpleicon@openjdk\endcsname {\simpleiconsmapSix\symbol{152}}
\expandafter\def\csname simpleicon@openlayers\endcsname {\simpleiconsmapSix\symbol{153}}
\expandafter\def\csname simpleicon@openmined\endcsname {\simpleiconsmapSix\symbol{154}}
\expandafter\def\csname simpleicon@opennebula\endcsname {\simpleiconsmapSix\symbol{155}}
\expandafter\def\csname simpleicon@openproject\endcsname {\simpleiconsmapSix\symbol{156}}
\expandafter\def\csname simpleicon@opensea\endcsname {\simpleiconsmapSix\symbol{157}}
\expandafter\def\csname simpleicon@opensearch\endcsname {\simpleiconsmapSix\symbol{158}}
\expandafter\def\csname simpleicon@opensourceinitiative\endcsname {\simpleiconsmapSix\symbol{159}}
\expandafter\def\csname simpleicon@openssl\endcsname {\simpleiconsmapSix\symbol{160}}
\expandafter\def\csname simpleicon@openstack\endcsname {\simpleiconsmapSix\symbol{161}}
\expandafter\def\csname simpleicon@openstreetmap\endcsname {\simpleiconsmapSix\symbol{162}}
\expandafter\def\csname simpleicon@opensuse\endcsname {\simpleiconsmapSix\symbol{163}}
\expandafter\def\csname simpleicon@opentelemetry\endcsname {\simpleiconsmapSix\symbol{164}}
\expandafter\def\csname simpleicon@openverse\endcsname {\simpleiconsmapSix\symbol{165}}
\expandafter\def\csname simpleicon@openvpn\endcsname {\simpleiconsmapSix\symbol{166}}
\expandafter\def\csname simpleicon@openwrt\endcsname {\simpleiconsmapSix\symbol{167}}
\expandafter\def\csname simpleicon@openzeppelin\endcsname {\simpleiconsmapSix\symbol{168}}
\expandafter\def\csname simpleicon@openzfs\endcsname {\simpleiconsmapSix\symbol{169}}
\expandafter\def\csname simpleicon@opera\endcsname {\simpleiconsmapSix\symbol{170}}
\expandafter\def\csname simpleicon@opnsense\endcsname {\simpleiconsmapSix\symbol{171}}
\expandafter\def\csname simpleicon@opsgenie\endcsname {\simpleiconsmapSix\symbol{172}}
\expandafter\def\csname simpleicon@opslevel\endcsname {\simpleiconsmapSix\symbol{173}}
\expandafter\def\csname simpleicon@oracle\endcsname {\simpleiconsmapSix\symbol{174}}
\expandafter\def\csname simpleicon@orcid\endcsname {\simpleiconsmapSix\symbol{175}}
\expandafter\def\csname simpleicon@oreilly\endcsname {\simpleiconsmapSix\symbol{176}}
\expandafter\def\csname simpleicon@org\endcsname {\simpleiconsmapSix\symbol{177}}
\expandafter\def\csname simpleicon@origin\endcsname {\simpleiconsmapSix\symbol{178}}
\expandafter\def\csname simpleicon@osano\endcsname {\simpleiconsmapSix\symbol{179}}
\expandafter\def\csname simpleicon@oshkosh\endcsname {\simpleiconsmapSix\symbol{180}}
\expandafter\def\csname simpleicon@osmc\endcsname {\simpleiconsmapSix\symbol{181}}
\expandafter\def\csname simpleicon@osu\endcsname {\simpleiconsmapSix\symbol{182}}
\expandafter\def\csname simpleicon@otto\endcsname {\simpleiconsmapSix\symbol{183}}
\expandafter\def\csname simpleicon@overcast\endcsname {\simpleiconsmapSix\symbol{184}}
\expandafter\def\csname simpleicon@overleaf\endcsname {\simpleiconsmapSix\symbol{185}}
\expandafter\def\csname simpleicon@ovh\endcsname {\simpleiconsmapSix\symbol{186}}
\expandafter\def\csname simpleicon@owasp\endcsname {\simpleiconsmapSix\symbol{187}}
\expandafter\def\csname simpleicon@oxygen\endcsname {\simpleiconsmapSix\symbol{188}}
\expandafter\def\csname simpleicon@oyo\endcsname {\simpleiconsmapSix\symbol{189}}
\expandafter\def\csname simpleicon@p5dotjs\endcsname {\simpleiconsmapSix\symbol{190}}
\expandafter\def\csname simpleicon@packagist\endcsname {\simpleiconsmapSix\symbol{191}}
\expandafter\def\csname simpleicon@packer\endcsname {\simpleiconsmapSix\symbol{192}}
\expandafter\def\csname simpleicon@paddypower\endcsname {\simpleiconsmapSix\symbol{193}}
\expandafter\def\csname simpleicon@pagekit\endcsname {\simpleiconsmapSix\symbol{194}}
\expandafter\def\csname simpleicon@pagerduty\endcsname {\simpleiconsmapSix\symbol{195}}
\expandafter\def\csname simpleicon@pagespeedinsights\endcsname {\simpleiconsmapSix\symbol{196}}
\expandafter\def\csname simpleicon@pagseguro\endcsname {\simpleiconsmapSix\symbol{197}}
\expandafter\def\csname simpleicon@palantir\endcsname {\simpleiconsmapSix\symbol{198}}
\expandafter\def\csname simpleicon@paloaltosoftware\endcsname {\simpleiconsmapSix\symbol{199}}
\expandafter\def\csname simpleicon@pandas\endcsname {\simpleiconsmapSix\symbol{200}}
\expandafter\def\csname simpleicon@pandora\endcsname {\simpleiconsmapSix\symbol{201}}
\expandafter\def\csname simpleicon@pantheon\endcsname {\simpleiconsmapSix\symbol{202}}
\expandafter\def\csname simpleicon@paperspace\endcsname {\simpleiconsmapSix\symbol{203}}
\expandafter\def\csname simpleicon@paritysubstrate\endcsname {\simpleiconsmapSix\symbol{204}}
\expandafter\def\csname simpleicon@parsedotly\endcsname {\simpleiconsmapSix\symbol{205}}
\expandafter\def\csname simpleicon@passport\endcsname {\simpleiconsmapSix\symbol{206}}
\expandafter\def\csname simpleicon@pastebin\endcsname {\simpleiconsmapSix\symbol{207}}
\expandafter\def\csname simpleicon@patreon\endcsname {\simpleiconsmapSix\symbol{208}}
\expandafter\def\csname simpleicon@payoneer\endcsname {\simpleiconsmapSix\symbol{209}}
\expandafter\def\csname simpleicon@paypal\endcsname {\simpleiconsmapSix\symbol{210}}
\expandafter\def\csname simpleicon@paytm\endcsname {\simpleiconsmapSix\symbol{211}}
\expandafter\def\csname simpleicon@pcgamingwiki\endcsname {\simpleiconsmapSix\symbol{212}}
\expandafter\def\csname simpleicon@peakdesign\endcsname {\simpleiconsmapSix\symbol{213}}
\expandafter\def\csname simpleicon@peertube\endcsname {\simpleiconsmapSix\symbol{214}}
\expandafter\def\csname simpleicon@pegasusairlines\endcsname {\simpleiconsmapSix\symbol{215}}
\expandafter\def\csname simpleicon@pelican\endcsname {\simpleiconsmapSix\symbol{216}}
\expandafter\def\csname simpleicon@peloton\endcsname {\simpleiconsmapSix\symbol{217}}
\expandafter\def\csname simpleicon@penny\endcsname {\simpleiconsmapSix\symbol{218}}
\expandafter\def\csname simpleicon@pepsi\endcsname {\simpleiconsmapSix\symbol{219}}
\expandafter\def\csname simpleicon@percy\endcsname {\simpleiconsmapSix\symbol{220}}
\expandafter\def\csname simpleicon@perforce\endcsname {\simpleiconsmapSix\symbol{221}}
\expandafter\def\csname simpleicon@perl\endcsname {\simpleiconsmapSix\symbol{222}}
\expandafter\def\csname simpleicon@personio\endcsname {\simpleiconsmapSix\symbol{223}}
\expandafter\def\csname simpleicon@petsathome\endcsname {\simpleiconsmapSix\symbol{224}}
\expandafter\def\csname simpleicon@peugeot\endcsname {\simpleiconsmapSix\symbol{225}}
\expandafter\def\csname simpleicon@pexels\endcsname {\simpleiconsmapSix\symbol{226}}
\expandafter\def\csname simpleicon@pfsense\endcsname {\simpleiconsmapSix\symbol{227}}
\expandafter\def\csname simpleicon@phabricator\endcsname {\simpleiconsmapSix\symbol{228}}
\expandafter\def\csname simpleicon@philipshue\endcsname {\simpleiconsmapSix\symbol{229}}
\expandafter\def\csname simpleicon@phonepe\endcsname {\simpleiconsmapSix\symbol{230}}
\expandafter\def\csname simpleicon@photobucket\endcsname {\simpleiconsmapSix\symbol{231}}
\expandafter\def\csname simpleicon@photocrowd\endcsname {\simpleiconsmapSix\symbol{232}}
\expandafter\def\csname simpleicon@photopea\endcsname {\simpleiconsmapSix\symbol{233}}
\expandafter\def\csname simpleicon@php\endcsname {\simpleiconsmapSix\symbol{234}}
\expandafter\def\csname simpleicon@phpmyadmin\endcsname {\simpleiconsmapSix\symbol{235}}
\expandafter\def\csname simpleicon@phpstorm\endcsname {\simpleiconsmapSix\symbol{236}}
\expandafter\def\csname simpleicon@picardsurgeles\endcsname {\simpleiconsmapSix\symbol{237}}
\expandafter\def\csname simpleicon@picartodottv\endcsname {\simpleiconsmapSix\symbol{238}}
\expandafter\def\csname simpleicon@picnic\endcsname {\simpleiconsmapSix\symbol{239}}
\expandafter\def\csname simpleicon@picpay\endcsname {\simpleiconsmapSix\symbol{240}}
\expandafter\def\csname simpleicon@pihole\endcsname {\simpleiconsmapSix\symbol{241}}
\expandafter\def\csname simpleicon@pimcore\endcsname {\simpleiconsmapSix\symbol{242}}
\expandafter\def\csname simpleicon@pinboard\endcsname {\simpleiconsmapSix\symbol{243}}
\expandafter\def\csname simpleicon@pingdom\endcsname {\simpleiconsmapSix\symbol{244}}
\expandafter\def\csname simpleicon@pinterest\endcsname {\simpleiconsmapSix\symbol{245}}
\expandafter\def\csname simpleicon@pioneerdj\endcsname {\simpleiconsmapSix\symbol{246}}
\expandafter\def\csname simpleicon@pivotaltracker\endcsname {\simpleiconsmapSix\symbol{247}}
\expandafter\def\csname simpleicon@piwigo\endcsname {\simpleiconsmapSix\symbol{248}}
\expandafter\def\csname simpleicon@pix\endcsname {\simpleiconsmapSix\symbol{249}}
\expandafter\def\csname simpleicon@pixabay\endcsname {\simpleiconsmapSix\symbol{250}}
\expandafter\def\csname simpleicon@pixiv\endcsname {\simpleiconsmapSix\symbol{251}}
\expandafter\def\csname simpleicon@pkgsrc\endcsname {\simpleiconsmapSix\symbol{252}}
\expandafter\def\csname simpleicon@planet\endcsname {\simpleiconsmapSix\symbol{253}}
\expandafter\def\csname simpleicon@planetscale\endcsname {\simpleiconsmapSix\symbol{254}}
\expandafter\def\csname simpleicon@plangrid\endcsname {\simpleiconsmapSix\symbol{255}}
\expandafter\def\csname simpleicon@platformdotsh\endcsname {\simpleiconsmapSeven\symbol{0}}
\expandafter\def\csname simpleicon@platzi\endcsname {\simpleiconsmapSeven\symbol{1}}
\expandafter\def\csname simpleicon@plausibleanalytics\endcsname {\simpleiconsmapSeven\symbol{2}}
\expandafter\def\csname simpleicon@playcanvas\endcsname {\simpleiconsmapSeven\symbol{3}}
\expandafter\def\csname simpleicon@playerdotme\endcsname {\simpleiconsmapSeven\symbol{4}}
\expandafter\def\csname simpleicon@playerfm\endcsname {\simpleiconsmapSeven\symbol{5}}
\expandafter\def\csname simpleicon@playstation\endcsname {\simpleiconsmapSeven\symbol{6}}
\expandafter\def\csname simpleicon@playstation2\endcsname {\simpleiconsmapSeven\symbol{7}}
\expandafter\def\csname simpleicon@playstation3\endcsname {\simpleiconsmapSeven\symbol{8}}
\expandafter\def\csname simpleicon@playstation4\endcsname {\simpleiconsmapSeven\symbol{9}}
\expandafter\def\csname simpleicon@playstation5\endcsname {\simpleiconsmapSeven\symbol{10}}
\expandafter\def\csname simpleicon@playstationvita\endcsname {\simpleiconsmapSeven\symbol{11}}
\expandafter\def\csname simpleicon@playwright\endcsname {\simpleiconsmapSeven\symbol{12}}
\expandafter\def\csname simpleicon@pleroma\endcsname {\simpleiconsmapSeven\symbol{13}}
\expandafter\def\csname simpleicon@plesk\endcsname {\simpleiconsmapSeven\symbol{14}}
\expandafter\def\csname simpleicon@plex\endcsname {\simpleiconsmapSeven\symbol{15}}
\expandafter\def\csname simpleicon@plotly\endcsname {\simpleiconsmapSeven\symbol{16}}
\expandafter\def\csname simpleicon@pluralsight\endcsname {\simpleiconsmapSeven\symbol{17}}
\expandafter\def\csname simpleicon@plurk\endcsname {\simpleiconsmapSeven\symbol{18}}
\expandafter\def\csname simpleicon@pluscodes\endcsname {\simpleiconsmapSeven\symbol{19}}
\expandafter\def\csname simpleicon@pm2\endcsname {\simpleiconsmapSeven\symbol{20}}
\expandafter\def\csname simpleicon@pnpm\endcsname {\simpleiconsmapSeven\symbol{21}}
\expandafter\def\csname simpleicon@pocket\endcsname {\simpleiconsmapSeven\symbol{22}}
\expandafter\def\csname simpleicon@pocketbase\endcsname {\simpleiconsmapSeven\symbol{23}}
\expandafter\def\csname simpleicon@pocketcasts\endcsname {\simpleiconsmapSeven\symbol{24}}
\expandafter\def\csname simpleicon@podcastaddict\endcsname {\simpleiconsmapSeven\symbol{25}}
\expandafter\def\csname simpleicon@podman\endcsname {\simpleiconsmapSeven\symbol{26}}
\expandafter\def\csname simpleicon@poetry\endcsname {\simpleiconsmapSeven\symbol{27}}
\expandafter\def\csname simpleicon@pointy\endcsname {\simpleiconsmapSeven\symbol{28}}
\expandafter\def\csname simpleicon@pokemon\endcsname {\simpleiconsmapSeven\symbol{29}}
\expandafter\def\csname simpleicon@polkadot\endcsname {\simpleiconsmapSeven\symbol{30}}
\expandafter\def\csname simpleicon@poly\endcsname {\simpleiconsmapSeven\symbol{31}}
\expandafter\def\csname simpleicon@polymerproject\endcsname {\simpleiconsmapSeven\symbol{32}}
\expandafter\def\csname simpleicon@polywork\endcsname {\simpleiconsmapSeven\symbol{33}}
\expandafter\def\csname simpleicon@popos\endcsname {\simpleiconsmapSeven\symbol{34}}
\expandafter\def\csname simpleicon@porsche\endcsname {\simpleiconsmapSeven\symbol{35}}
\expandafter\def\csname simpleicon@portainer\endcsname {\simpleiconsmapSeven\symbol{36}}
\expandafter\def\csname simpleicon@postcss\endcsname {\simpleiconsmapSeven\symbol{37}}
\expandafter\def\csname simpleicon@postgresql\endcsname {\simpleiconsmapSeven\symbol{38}}
\expandafter\def\csname simpleicon@postman\endcsname {\simpleiconsmapSeven\symbol{39}}
\expandafter\def\csname simpleicon@postmates\endcsname {\simpleiconsmapSeven\symbol{40}}
\expandafter\def\csname simpleicon@powerapps\endcsname {\simpleiconsmapSeven\symbol{41}}
\expandafter\def\csname simpleicon@powerautomate\endcsname {\simpleiconsmapSeven\symbol{42}}
\expandafter\def\csname simpleicon@powerbi\endcsname {\simpleiconsmapSeven\symbol{43}}
\expandafter\def\csname simpleicon@powerfx\endcsname {\simpleiconsmapSeven\symbol{44}}
\expandafter\def\csname simpleicon@powerpages\endcsname {\simpleiconsmapSeven\symbol{45}}
\expandafter\def\csname simpleicon@powers\endcsname {\simpleiconsmapSeven\symbol{46}}
\expandafter\def\csname simpleicon@powershell\endcsname {\simpleiconsmapSeven\symbol{47}}
\expandafter\def\csname simpleicon@powervirtualagents\endcsname {\simpleiconsmapSeven\symbol{48}}
\expandafter\def\csname simpleicon@prdotco\endcsname {\simpleiconsmapSeven\symbol{49}}
\expandafter\def\csname simpleicon@preact\endcsname {\simpleiconsmapSeven\symbol{50}}
\expandafter\def\csname simpleicon@precommit\endcsname {\simpleiconsmapSeven\symbol{51}}
\expandafter\def\csname simpleicon@premierleague\endcsname {\simpleiconsmapSeven\symbol{52}}
\expandafter\def\csname simpleicon@prestashop\endcsname {\simpleiconsmapSeven\symbol{53}}
\expandafter\def\csname simpleicon@presto\endcsname {\simpleiconsmapSeven\symbol{54}}
\expandafter\def\csname simpleicon@prettier\endcsname {\simpleiconsmapSeven\symbol{55}}
\expandafter\def\csname simpleicon@prezi\endcsname {\simpleiconsmapSeven\symbol{56}}
\expandafter\def\csname simpleicon@prime\endcsname {\simpleiconsmapSeven\symbol{57}}
\expandafter\def\csname simpleicon@primevideo\endcsname {\simpleiconsmapSeven\symbol{58}}
\expandafter\def\csname simpleicon@prisma\endcsname {\simpleiconsmapSeven\symbol{59}}
\expandafter\def\csname simpleicon@prismic\endcsname {\simpleiconsmapSeven\symbol{60}}
\expandafter\def\csname simpleicon@privateinternetaccess\endcsname {\simpleiconsmapSeven\symbol{61}}
\expandafter\def\csname simpleicon@probot\endcsname {\simpleiconsmapSeven\symbol{62}}
\expandafter\def\csname simpleicon@processingfoundation\endcsname {\simpleiconsmapSeven\symbol{63}}
\expandafter\def\csname simpleicon@processwire\endcsname {\simpleiconsmapSeven\symbol{64}}
\expandafter\def\csname simpleicon@producthunt\endcsname {\simpleiconsmapSeven\symbol{65}}
\expandafter\def\csname simpleicon@progate\endcsname {\simpleiconsmapSeven\symbol{66}}
\expandafter\def\csname simpleicon@progress\endcsname {\simpleiconsmapSeven\symbol{67}}
\expandafter\def\csname simpleicon@prometheus\endcsname {\simpleiconsmapSeven\symbol{68}}
\expandafter\def\csname simpleicon@prosieben\endcsname {\simpleiconsmapSeven\symbol{69}}
\expandafter\def\csname simpleicon@protocolsdotio\endcsname {\simpleiconsmapSeven\symbol{70}}
\expandafter\def\csname simpleicon@protodotio\endcsname {\simpleiconsmapSeven\symbol{71}}
\expandafter\def\csname simpleicon@protondb\endcsname {\simpleiconsmapSeven\symbol{72}}
\expandafter\def\csname simpleicon@protonmail\endcsname {\simpleiconsmapSeven\symbol{73}}
\expandafter\def\csname simpleicon@protonvpn\endcsname {\simpleiconsmapSeven\symbol{74}}
\expandafter\def\csname simpleicon@protools\endcsname {\simpleiconsmapSeven\symbol{75}}
\expandafter\def\csname simpleicon@protractor\endcsname {\simpleiconsmapSeven\symbol{76}}
\expandafter\def\csname simpleicon@proxmox\endcsname {\simpleiconsmapSeven\symbol{77}}
\expandafter\def\csname simpleicon@pubg\endcsname {\simpleiconsmapSeven\symbol{78}}
\expandafter\def\csname simpleicon@publons\endcsname {\simpleiconsmapSeven\symbol{79}}
\expandafter\def\csname simpleicon@pubmed\endcsname {\simpleiconsmapSeven\symbol{80}}
\expandafter\def\csname simpleicon@pug\endcsname {\simpleiconsmapSeven\symbol{81}}
\expandafter\def\csname simpleicon@pulumi\endcsname {\simpleiconsmapSeven\symbol{82}}
\expandafter\def\csname simpleicon@puma\endcsname {\simpleiconsmapSeven\symbol{83}}
\expandafter\def\csname simpleicon@puppet\endcsname {\simpleiconsmapSeven\symbol{84}}
\expandafter\def\csname simpleicon@puppeteer\endcsname {\simpleiconsmapSeven\symbol{85}}
\expandafter\def\csname simpleicon@purescript\endcsname {\simpleiconsmapSeven\symbol{86}}
\expandafter\def\csname simpleicon@purgecss\endcsname {\simpleiconsmapSeven\symbol{87}}
\expandafter\def\csname simpleicon@purism\endcsname {\simpleiconsmapSeven\symbol{88}}
\expandafter\def\csname simpleicon@pusher\endcsname {\simpleiconsmapSeven\symbol{89}}
\expandafter\def\csname simpleicon@pwa\endcsname {\simpleiconsmapSeven\symbol{90}}
\expandafter\def\csname simpleicon@pycharm\endcsname {\simpleiconsmapSeven\symbol{91}}
\expandafter\def\csname simpleicon@pyg\endcsname {\simpleiconsmapSeven\symbol{92}}
\expandafter\def\csname simpleicon@pypi\endcsname {\simpleiconsmapSeven\symbol{93}}
\expandafter\def\csname simpleicon@pypy\endcsname {\simpleiconsmapSeven\symbol{94}}
\expandafter\def\csname simpleicon@pyscaffold\endcsname {\simpleiconsmapSeven\symbol{95}}
\expandafter\def\csname simpleicon@pytest\endcsname {\simpleiconsmapSeven\symbol{96}}
\expandafter\def\csname simpleicon@python\endcsname {\simpleiconsmapSeven\symbol{97}}
\expandafter\def\csname simpleicon@pytorch\endcsname {\simpleiconsmapSeven\symbol{98}}
\expandafter\def\csname simpleicon@pytorchlightning\endcsname {\simpleiconsmapSeven\symbol{99}}
\expandafter\def\csname simpleicon@pyup\endcsname {\simpleiconsmapSeven\symbol{100}}
\expandafter\def\csname simpleicon@qantas\endcsname {\simpleiconsmapSeven\symbol{101}}
\expandafter\def\csname simpleicon@qatarairways\endcsname {\simpleiconsmapSeven\symbol{102}}
\expandafter\def\csname simpleicon@qemu\endcsname {\simpleiconsmapSeven\symbol{103}}
\expandafter\def\csname simpleicon@qgis\endcsname {\simpleiconsmapSeven\symbol{104}}
\expandafter\def\csname simpleicon@qi\endcsname {\simpleiconsmapSeven\symbol{105}}
\expandafter\def\csname simpleicon@qiita\endcsname {\simpleiconsmapSeven\symbol{106}}
\expandafter\def\csname simpleicon@qiskit\endcsname {\simpleiconsmapSeven\symbol{107}}
\expandafter\def\csname simpleicon@qiwi\endcsname {\simpleiconsmapSeven\symbol{108}}
\expandafter\def\csname simpleicon@qmk\endcsname {\simpleiconsmapSeven\symbol{109}}
\expandafter\def\csname simpleicon@qt\endcsname {\simpleiconsmapSeven\symbol{110}}
\expandafter\def\csname simpleicon@qualcomm\endcsname {\simpleiconsmapSeven\symbol{111}}
\expandafter\def\csname simpleicon@qualtrics\endcsname {\simpleiconsmapSeven\symbol{112}}
\expandafter\def\csname simpleicon@qualys\endcsname {\simpleiconsmapSeven\symbol{113}}
\expandafter\def\csname simpleicon@quantcast\endcsname {\simpleiconsmapSeven\symbol{114}}
\expandafter\def\csname simpleicon@quantconnect\endcsname {\simpleiconsmapSeven\symbol{115}}
\expandafter\def\csname simpleicon@quarkus\endcsname {\simpleiconsmapSeven\symbol{116}}
\expandafter\def\csname simpleicon@quasar\endcsname {\simpleiconsmapSeven\symbol{117}}
\expandafter\def\csname simpleicon@qubesos\endcsname {\simpleiconsmapSeven\symbol{118}}
\expandafter\def\csname simpleicon@quest\endcsname {\simpleiconsmapSeven\symbol{119}}
\expandafter\def\csname simpleicon@quickbooks\endcsname {\simpleiconsmapSeven\symbol{120}}
\expandafter\def\csname simpleicon@quicklook\endcsname {\simpleiconsmapSeven\symbol{121}}
\expandafter\def\csname simpleicon@quicktime\endcsname {\simpleiconsmapSeven\symbol{122}}
\expandafter\def\csname simpleicon@quip\endcsname {\simpleiconsmapSeven\symbol{123}}
\expandafter\def\csname simpleicon@quora\endcsname {\simpleiconsmapSeven\symbol{124}}
\expandafter\def\csname simpleicon@qwiklabs\endcsname {\simpleiconsmapSeven\symbol{125}}
\expandafter\def\csname simpleicon@qzone\endcsname {\simpleiconsmapSeven\symbol{126}}
\expandafter\def\csname simpleicon@r\endcsname {\simpleiconsmapSeven\symbol{127}}
\expandafter\def\csname simpleicon@r3\endcsname {\simpleiconsmapSeven\symbol{128}}
\expandafter\def\csname simpleicon@rabbitmq\endcsname {\simpleiconsmapSeven\symbol{129}}
\expandafter\def\csname simpleicon@racket\endcsname {\simpleiconsmapSeven\symbol{130}}
\expandafter\def\csname simpleicon@radar\endcsname {\simpleiconsmapSeven\symbol{131}}
\expandafter\def\csname simpleicon@radiopublic\endcsname {\simpleiconsmapSeven\symbol{132}}
\expandafter\def\csname simpleicon@railway\endcsname {\simpleiconsmapSeven\symbol{133}}
\expandafter\def\csname simpleicon@rainmeter\endcsname {\simpleiconsmapSeven\symbol{134}}
\expandafter\def\csname simpleicon@rakuten\endcsname {\simpleiconsmapSeven\symbol{135}}
\expandafter\def\csname simpleicon@ram\endcsname {\simpleiconsmapSeven\symbol{136}}
\expandafter\def\csname simpleicon@rancher\endcsname {\simpleiconsmapSeven\symbol{137}}
\expandafter\def\csname simpleicon@rarible\endcsname {\simpleiconsmapSeven\symbol{138}}
\expandafter\def\csname simpleicon@rasa\endcsname {\simpleiconsmapSeven\symbol{139}}
\expandafter\def\csname simpleicon@raspberrypi\endcsname {\simpleiconsmapSeven\symbol{140}}
\expandafter\def\csname simpleicon@ray\endcsname {\simpleiconsmapSeven\symbol{141}}
\expandafter\def\csname simpleicon@razer\endcsname {\simpleiconsmapSeven\symbol{142}}
\expandafter\def\csname simpleicon@razorpay\endcsname {\simpleiconsmapSeven\symbol{143}}
\expandafter\def\csname simpleicon@react\endcsname {\simpleiconsmapSeven\symbol{144}}
\expandafter\def\csname simpleicon@reacthookform\endcsname {\simpleiconsmapSeven\symbol{145}}
\expandafter\def\csname simpleicon@reactivex\endcsname {\simpleiconsmapSeven\symbol{146}}
\expandafter\def\csname simpleicon@reactos\endcsname {\simpleiconsmapSeven\symbol{147}}
\expandafter\def\csname simpleicon@reactquery\endcsname {\simpleiconsmapSeven\symbol{148}}
\expandafter\def\csname simpleicon@reactrouter\endcsname {\simpleiconsmapSeven\symbol{149}}
\expandafter\def\csname simpleicon@reacttable\endcsname {\simpleiconsmapSeven\symbol{150}}
\expandafter\def\csname simpleicon@readthedocs\endcsname {\simpleiconsmapSeven\symbol{151}}
\expandafter\def\csname simpleicon@realm\endcsname {\simpleiconsmapSeven\symbol{152}}
\expandafter\def\csname simpleicon@reason\endcsname {\simpleiconsmapSeven\symbol{153}}
\expandafter\def\csname simpleicon@reasonstudios\endcsname {\simpleiconsmapSeven\symbol{154}}
\expandafter\def\csname simpleicon@redbubble\endcsname {\simpleiconsmapSeven\symbol{155}}
\expandafter\def\csname simpleicon@reddit\endcsname {\simpleiconsmapSeven\symbol{156}}
\expandafter\def\csname simpleicon@redhat\endcsname {\simpleiconsmapSeven\symbol{157}}
\expandafter\def\csname simpleicon@redhatopenshift\endcsname {\simpleiconsmapSeven\symbol{158}}
\expandafter\def\csname simpleicon@redis\endcsname {\simpleiconsmapSeven\symbol{159}}
\expandafter\def\csname simpleicon@redmine\endcsname {\simpleiconsmapSeven\symbol{160}}
\expandafter\def\csname simpleicon@redux\endcsname {\simpleiconsmapSeven\symbol{161}}
\expandafter\def\csname simpleicon@reduxsaga\endcsname {\simpleiconsmapSeven\symbol{162}}
\expandafter\def\csname simpleicon@redwoodjs\endcsname {\simpleiconsmapSeven\symbol{163}}
\expandafter\def\csname simpleicon@reebok\endcsname {\simpleiconsmapSeven\symbol{164}}
\expandafter\def\csname simpleicon@relianceindustrieslimited\endcsname {\simpleiconsmapSeven\symbol{165}}
\expandafter\def\csname simpleicon@remix\endcsname {\simpleiconsmapSeven\symbol{166}}
\expandafter\def\csname simpleicon@renault\endcsname {\simpleiconsmapSeven\symbol{167}}
\expandafter\def\csname simpleicon@render\endcsname {\simpleiconsmapSeven\symbol{168}}
\expandafter\def\csname simpleicon@renovatebot\endcsname {\simpleiconsmapSeven\symbol{169}}
\expandafter\def\csname simpleicon@renpy\endcsname {\simpleiconsmapSeven\symbol{170}}
\expandafter\def\csname simpleicon@renren\endcsname {\simpleiconsmapSeven\symbol{171}}
\expandafter\def\csname simpleicon@replit\endcsname {\simpleiconsmapSeven\symbol{172}}
\expandafter\def\csname simpleicon@republicofgamers\endcsname {\simpleiconsmapSeven\symbol{173}}
\expandafter\def\csname simpleicon@rescript\endcsname {\simpleiconsmapSeven\symbol{174}}
\expandafter\def\csname simpleicon@rescuetime\endcsname {\simpleiconsmapSeven\symbol{175}}
\expandafter\def\csname simpleicon@researchgate\endcsname {\simpleiconsmapSeven\symbol{176}}
\expandafter\def\csname simpleicon@resurrectionremixos\endcsname {\simpleiconsmapSeven\symbol{177}}
\expandafter\def\csname simpleicon@retroarch\endcsname {\simpleiconsmapSeven\symbol{178}}
\expandafter\def\csname simpleicon@retropie\endcsname {\simpleiconsmapSeven\symbol{179}}
\expandafter\def\csname simpleicon@revealdotjs\endcsname {\simpleiconsmapSeven\symbol{180}}
\expandafter\def\csname simpleicon@reverbnation\endcsname {\simpleiconsmapSeven\symbol{181}}
\expandafter\def\csname simpleicon@revoltdotchat\endcsname {\simpleiconsmapSeven\symbol{182}}
\expandafter\def\csname simpleicon@revolut\endcsname {\simpleiconsmapSeven\symbol{183}}
\expandafter\def\csname simpleicon@revue\endcsname {\simpleiconsmapSeven\symbol{184}}
\expandafter\def\csname simpleicon@rewe\endcsname {\simpleiconsmapSeven\symbol{185}}
\expandafter\def\csname simpleicon@rezgo\endcsname {\simpleiconsmapSeven\symbol{186}}
\expandafter\def\csname simpleicon@rhinoceros\endcsname {\simpleiconsmapSeven\symbol{187}}
\expandafter\def\csname simpleicon@rider\endcsname {\simpleiconsmapSeven\symbol{188}}
\expandafter\def\csname simpleicon@rimacautomobili\endcsname {\simpleiconsmapSeven\symbol{189}}
\expandafter\def\csname simpleicon@ring\endcsname {\simpleiconsmapSeven\symbol{190}}
\expandafter\def\csname simpleicon@riotgames\endcsname {\simpleiconsmapSeven\symbol{191}}
\expandafter\def\csname simpleicon@ripple\endcsname {\simpleiconsmapSeven\symbol{192}}
\expandafter\def\csname simpleicon@riseup\endcsname {\simpleiconsmapSeven\symbol{193}}
\expandafter\def\csname simpleicon@roamresearch\endcsname {\simpleiconsmapSeven\symbol{194}}
\expandafter\def\csname simpleicon@roblox\endcsname {\simpleiconsmapSeven\symbol{195}}
\expandafter\def\csname simpleicon@robotframework\endcsname {\simpleiconsmapSeven\symbol{196}}
\expandafter\def\csname simpleicon@rocketdotchat\endcsname {\simpleiconsmapSeven\symbol{197}}
\expandafter\def\csname simpleicon@rocksdb\endcsname {\simpleiconsmapSeven\symbol{198}}
\expandafter\def\csname simpleicon@rockylinux\endcsname {\simpleiconsmapSeven\symbol{199}}
\expandafter\def\csname simpleicon@roku\endcsname {\simpleiconsmapSeven\symbol{200}}
\expandafter\def\csname simpleicon@rollsroyce\endcsname {\simpleiconsmapSeven\symbol{201}}
\expandafter\def\csname simpleicon@rollupdotjs\endcsname {\simpleiconsmapSeven\symbol{202}}
\expandafter\def\csname simpleicon@rome\endcsname {\simpleiconsmapSeven\symbol{203}}
\expandafter\def\csname simpleicon@roots\endcsname {\simpleiconsmapSeven\symbol{204}}
\expandafter\def\csname simpleicon@rootsbedrock\endcsname {\simpleiconsmapSeven\symbol{205}}
\expandafter\def\csname simpleicon@rootssage\endcsname {\simpleiconsmapSeven\symbol{206}}
\expandafter\def\csname simpleicon@ros\endcsname {\simpleiconsmapSeven\symbol{207}}
\expandafter\def\csname simpleicon@rossmann\endcsname {\simpleiconsmapSeven\symbol{208}}
\expandafter\def\csname simpleicon@rotaryinternational\endcsname {\simpleiconsmapSeven\symbol{209}}
\expandafter\def\csname simpleicon@rottentomatoes\endcsname {\simpleiconsmapSeven\symbol{210}}
\expandafter\def\csname simpleicon@roundcube\endcsname {\simpleiconsmapSeven\symbol{211}}
\expandafter\def\csname simpleicon@rss\endcsname {\simpleiconsmapSeven\symbol{212}}
\expandafter\def\csname simpleicon@rstudio\endcsname {\simpleiconsmapSeven\symbol{213}}
\expandafter\def\csname simpleicon@rte\endcsname {\simpleiconsmapSeven\symbol{214}}
\expandafter\def\csname simpleicon@rtl\endcsname {\simpleiconsmapSeven\symbol{215}}
\expandafter\def\csname simpleicon@rtlzwei\endcsname {\simpleiconsmapSeven\symbol{216}}
\expandafter\def\csname simpleicon@rubocop\endcsname {\simpleiconsmapSeven\symbol{217}}
\expandafter\def\csname simpleicon@ruby\endcsname {\simpleiconsmapSeven\symbol{218}}
\expandafter\def\csname simpleicon@rubygems\endcsname {\simpleiconsmapSeven\symbol{219}}
\expandafter\def\csname simpleicon@rubyonrails\endcsname {\simpleiconsmapSeven\symbol{220}}
\expandafter\def\csname simpleicon@rubysinatra\endcsname {\simpleiconsmapSeven\symbol{221}}
\expandafter\def\csname simpleicon@runkeeper\endcsname {\simpleiconsmapSeven\symbol{222}}
\expandafter\def\csname simpleicon@runkit\endcsname {\simpleiconsmapSeven\symbol{223}}
\expandafter\def\csname simpleicon@rust\endcsname {\simpleiconsmapSeven\symbol{224}}
\expandafter\def\csname simpleicon@rxdb\endcsname {\simpleiconsmapSeven\symbol{225}}
\expandafter\def\csname simpleicon@ryanair\endcsname {\simpleiconsmapSeven\symbol{226}}
\expandafter\def\csname simpleicon@s7airlines\endcsname {\simpleiconsmapSeven\symbol{227}}
\expandafter\def\csname simpleicon@sabanci\endcsname {\simpleiconsmapSeven\symbol{228}}
\expandafter\def\csname simpleicon@safari\endcsname {\simpleiconsmapSeven\symbol{229}}
\expandafter\def\csname simpleicon@sahibinden\endcsname {\simpleiconsmapSeven\symbol{230}}
\expandafter\def\csname simpleicon@sailfishos\endcsname {\simpleiconsmapSeven\symbol{231}}
\expandafter\def\csname simpleicon@salesforce\endcsname {\simpleiconsmapSeven\symbol{232}}
\expandafter\def\csname simpleicon@saltproject\endcsname {\simpleiconsmapSeven\symbol{233}}
\expandafter\def\csname simpleicon@samsung\endcsname {\simpleiconsmapSeven\symbol{234}}
\expandafter\def\csname simpleicon@samsungpay\endcsname {\simpleiconsmapSeven\symbol{235}}
\expandafter\def\csname simpleicon@sandisk\endcsname {\simpleiconsmapSeven\symbol{236}}
\expandafter\def\csname simpleicon@sanfranciscomunicipalrailway\endcsname {\simpleiconsmapSeven\symbol{237}}
\expandafter\def\csname simpleicon@saopaulometro\endcsname {\simpleiconsmapSeven\symbol{238}}
\expandafter\def\csname simpleicon@sap\endcsname {\simpleiconsmapSeven\symbol{239}}
\expandafter\def\csname simpleicon@sass\endcsname {\simpleiconsmapSeven\symbol{240}}
\expandafter\def\csname simpleicon@sat1\endcsname {\simpleiconsmapSeven\symbol{241}}
\expandafter\def\csname simpleicon@saturn\endcsname {\simpleiconsmapSeven\symbol{242}}
\expandafter\def\csname simpleicon@saucelabs\endcsname {\simpleiconsmapSeven\symbol{243}}
\expandafter\def\csname simpleicon@scala\endcsname {\simpleiconsmapSeven\symbol{244}}
\expandafter\def\csname simpleicon@scaleway\endcsname {\simpleiconsmapSeven\symbol{245}}
\expandafter\def\csname simpleicon@scania\endcsname {\simpleiconsmapSeven\symbol{246}}
\expandafter\def\csname simpleicon@schneiderelectric\endcsname {\simpleiconsmapSeven\symbol{247}}
\expandafter\def\csname simpleicon@scikitlearn\endcsname {\simpleiconsmapSeven\symbol{248}}
\expandafter\def\csname simpleicon@scipy\endcsname {\simpleiconsmapSeven\symbol{249}}
\expandafter\def\csname simpleicon@scopus\endcsname {\simpleiconsmapSeven\symbol{250}}
\expandafter\def\csname simpleicon@scpfoundation\endcsname {\simpleiconsmapSeven\symbol{251}}
\expandafter\def\csname simpleicon@scratch\endcsname {\simpleiconsmapSeven\symbol{252}}
\expandafter\def\csname simpleicon@screencastify\endcsname {\simpleiconsmapSeven\symbol{253}}
\expandafter\def\csname simpleicon@scribd\endcsname {\simpleiconsmapSeven\symbol{254}}
\expandafter\def\csname simpleicon@scrimba\endcsname {\simpleiconsmapSeven\symbol{255}}
\expandafter\def\csname simpleicon@scrollreveal\endcsname {\simpleiconsmapEight\symbol{0}}
\expandafter\def\csname simpleicon@scrumalliance\endcsname {\simpleiconsmapEight\symbol{1}}
\expandafter\def\csname simpleicon@scrutinizerci\endcsname {\simpleiconsmapEight\symbol{2}}
\expandafter\def\csname simpleicon@seagate\endcsname {\simpleiconsmapEight\symbol{3}}
\expandafter\def\csname simpleicon@seat\endcsname {\simpleiconsmapEight\symbol{4}}
\expandafter\def\csname simpleicon@securityscorecard\endcsname {\simpleiconsmapEight\symbol{5}}
\expandafter\def\csname simpleicon@sefaria\endcsname {\simpleiconsmapEight\symbol{6}}
\expandafter\def\csname simpleicon@sega\endcsname {\simpleiconsmapEight\symbol{7}}
\expandafter\def\csname simpleicon@selenium\endcsname {\simpleiconsmapEight\symbol{8}}
\expandafter\def\csname simpleicon@sellfy\endcsname {\simpleiconsmapEight\symbol{9}}
\expandafter\def\csname simpleicon@semanticrelease\endcsname {\simpleiconsmapEight\symbol{10}}
\expandafter\def\csname simpleicon@semanticscholar\endcsname {\simpleiconsmapEight\symbol{11}}
\expandafter\def\csname simpleicon@semanticuireact\endcsname {\simpleiconsmapEight\symbol{12}}
\expandafter\def\csname simpleicon@semanticweb\endcsname {\simpleiconsmapEight\symbol{13}}
\expandafter\def\csname simpleicon@semaphoreci\endcsname {\simpleiconsmapEight\symbol{14}}
\expandafter\def\csname simpleicon@semver\endcsname {\simpleiconsmapEight\symbol{15}}
\expandafter\def\csname simpleicon@sencha\endcsname {\simpleiconsmapEight\symbol{16}}
\expandafter\def\csname simpleicon@sennheiser\endcsname {\simpleiconsmapEight\symbol{17}}
\expandafter\def\csname simpleicon@sensu\endcsname {\simpleiconsmapEight\symbol{18}}
\expandafter\def\csname simpleicon@sentry\endcsname {\simpleiconsmapEight\symbol{19}}
\expandafter\def\csname simpleicon@sepa\endcsname {\simpleiconsmapEight\symbol{20}}
\expandafter\def\csname simpleicon@sequelize\endcsname {\simpleiconsmapEight\symbol{21}}
\expandafter\def\csname simpleicon@serverfault\endcsname {\simpleiconsmapEight\symbol{22}}
\expandafter\def\csname simpleicon@serverless\endcsname {\simpleiconsmapEight\symbol{23}}
\expandafter\def\csname simpleicon@sessionize\endcsname {\simpleiconsmapEight\symbol{24}}
\expandafter\def\csname simpleicon@sfml\endcsname {\simpleiconsmapEight\symbol{25}}
\expandafter\def\csname simpleicon@shadow\endcsname {\simpleiconsmapEight\symbol{26}}
\expandafter\def\csname simpleicon@shanghaimetro\endcsname {\simpleiconsmapEight\symbol{27}}
\expandafter\def\csname simpleicon@sharp\endcsname {\simpleiconsmapEight\symbol{28}}
\expandafter\def\csname simpleicon@shazam\endcsname {\simpleiconsmapEight\symbol{29}}
\expandafter\def\csname simpleicon@shell\endcsname {\simpleiconsmapEight\symbol{30}}
\expandafter\def\csname simpleicon@shelly\endcsname {\simpleiconsmapEight\symbol{31}}
\expandafter\def\csname simpleicon@shenzhenmetro\endcsname {\simpleiconsmapEight\symbol{32}}
\expandafter\def\csname simpleicon@shieldsdotio\endcsname {\simpleiconsmapEight\symbol{33}}
\expandafter\def\csname simpleicon@shikimori\endcsname {\simpleiconsmapEight\symbol{34}}
\expandafter\def\csname simpleicon@shopify\endcsname {\simpleiconsmapEight\symbol{35}}
\expandafter\def\csname simpleicon@shopware\endcsname {\simpleiconsmapEight\symbol{36}}
\expandafter\def\csname simpleicon@shotcut\endcsname {\simpleiconsmapEight\symbol{37}}
\expandafter\def\csname simpleicon@showpad\endcsname {\simpleiconsmapEight\symbol{38}}
\expandafter\def\csname simpleicon@showtime\endcsname {\simpleiconsmapEight\symbol{39}}
\expandafter\def\csname simpleicon@shutterstock\endcsname {\simpleiconsmapEight\symbol{40}}
\expandafter\def\csname simpleicon@siemens\endcsname {\simpleiconsmapEight\symbol{41}}
\expandafter\def\csname simpleicon@signal\endcsname {\simpleiconsmapEight\symbol{42}}
\expandafter\def\csname simpleicon@simkl\endcsname {\simpleiconsmapEight\symbol{43}}
\expandafter\def\csname simpleicon@simpleanalytics\endcsname {\simpleiconsmapEight\symbol{44}}
\expandafter\def\csname simpleicon@simpleicons\endcsname {\simpleiconsmapEight\symbol{45}}
\expandafter\def\csname simpleicon@simplenote\endcsname {\simpleiconsmapEight\symbol{46}}
\expandafter\def\csname simpleicon@sinaweibo\endcsname {\simpleiconsmapEight\symbol{47}}
\expandafter\def\csname simpleicon@singlestore\endcsname {\simpleiconsmapEight\symbol{48}}
\expandafter\def\csname simpleicon@sitepoint\endcsname {\simpleiconsmapEight\symbol{49}}
\expandafter\def\csname simpleicon@sketch\endcsname {\simpleiconsmapEight\symbol{50}}
\expandafter\def\csname simpleicon@sketchfab\endcsname {\simpleiconsmapEight\symbol{51}}
\expandafter\def\csname simpleicon@sketchup\endcsname {\simpleiconsmapEight\symbol{52}}
\expandafter\def\csname simpleicon@skillshare\endcsname {\simpleiconsmapEight\symbol{53}}
\expandafter\def\csname simpleicon@skoda\endcsname {\simpleiconsmapEight\symbol{54}}
\expandafter\def\csname simpleicon@sky\endcsname {\simpleiconsmapEight\symbol{55}}
\expandafter\def\csname simpleicon@skynet\endcsname {\simpleiconsmapEight\symbol{56}}
\expandafter\def\csname simpleicon@skypack\endcsname {\simpleiconsmapEight\symbol{57}}
\expandafter\def\csname simpleicon@skype\endcsname {\simpleiconsmapEight\symbol{58}}
\expandafter\def\csname simpleicon@skypeforbusiness\endcsname {\simpleiconsmapEight\symbol{59}}
\expandafter\def\csname simpleicon@slack\endcsname {\simpleiconsmapEight\symbol{60}}
\expandafter\def\csname simpleicon@slackware\endcsname {\simpleiconsmapEight\symbol{61}}
\expandafter\def\csname simpleicon@slashdot\endcsname {\simpleiconsmapEight\symbol{62}}
\expandafter\def\csname simpleicon@slickpic\endcsname {\simpleiconsmapEight\symbol{63}}
\expandafter\def\csname simpleicon@slides\endcsname {\simpleiconsmapEight\symbol{64}}
\expandafter\def\csname simpleicon@slideshare\endcsname {\simpleiconsmapEight\symbol{65}}
\expandafter\def\csname simpleicon@smart\endcsname {\simpleiconsmapEight\symbol{66}}
\expandafter\def\csname simpleicon@smartthings\endcsname {\simpleiconsmapEight\symbol{67}}
\expandafter\def\csname simpleicon@smashdotgg\endcsname {\simpleiconsmapEight\symbol{68}}
\expandafter\def\csname simpleicon@smashingmagazine\endcsname {\simpleiconsmapEight\symbol{69}}
\expandafter\def\csname simpleicon@smrt\endcsname {\simpleiconsmapEight\symbol{70}}
\expandafter\def\csname simpleicon@smugmug\endcsname {\simpleiconsmapEight\symbol{71}}
\expandafter\def\csname simpleicon@snapchat\endcsname {\simpleiconsmapEight\symbol{72}}
\expandafter\def\csname simpleicon@snapcraft\endcsname {\simpleiconsmapEight\symbol{73}}
\expandafter\def\csname simpleicon@snowflake\endcsname {\simpleiconsmapEight\symbol{74}}
\expandafter\def\csname simpleicon@snowpack\endcsname {\simpleiconsmapEight\symbol{75}}
\expandafter\def\csname simpleicon@snyk\endcsname {\simpleiconsmapEight\symbol{76}}
\expandafter\def\csname simpleicon@socialblade\endcsname {\simpleiconsmapEight\symbol{77}}
\expandafter\def\csname simpleicon@society6\endcsname {\simpleiconsmapEight\symbol{78}}
\expandafter\def\csname simpleicon@socketdotio\endcsname {\simpleiconsmapEight\symbol{79}}
\expandafter\def\csname simpleicon@sogou\endcsname {\simpleiconsmapEight\symbol{80}}
\expandafter\def\csname simpleicon@solid\endcsname {\simpleiconsmapEight\symbol{81}}
\expandafter\def\csname simpleicon@solidity\endcsname {\simpleiconsmapEight\symbol{82}}
\expandafter\def\csname simpleicon@sololearn\endcsname {\simpleiconsmapEight\symbol{83}}
\expandafter\def\csname simpleicon@solus\endcsname {\simpleiconsmapEight\symbol{84}}
\expandafter\def\csname simpleicon@sonarcloud\endcsname {\simpleiconsmapEight\symbol{85}}
\expandafter\def\csname simpleicon@sonarlint\endcsname {\simpleiconsmapEight\symbol{86}}
\expandafter\def\csname simpleicon@sonarqube\endcsname {\simpleiconsmapEight\symbol{87}}
\expandafter\def\csname simpleicon@sonarsource\endcsname {\simpleiconsmapEight\symbol{88}}
\expandafter\def\csname simpleicon@songkick\endcsname {\simpleiconsmapEight\symbol{89}}
\expandafter\def\csname simpleicon@songoda\endcsname {\simpleiconsmapEight\symbol{90}}
\expandafter\def\csname simpleicon@sonicwall\endcsname {\simpleiconsmapEight\symbol{91}}
\expandafter\def\csname simpleicon@sonos\endcsname {\simpleiconsmapEight\symbol{92}}
\expandafter\def\csname simpleicon@sony\endcsname {\simpleiconsmapEight\symbol{93}}
\expandafter\def\csname simpleicon@soundcharts\endcsname {\simpleiconsmapEight\symbol{94}}
\expandafter\def\csname simpleicon@soundcloud\endcsname {\simpleiconsmapEight\symbol{95}}
\expandafter\def\csname simpleicon@sourceengine\endcsname {\simpleiconsmapEight\symbol{96}}
\expandafter\def\csname simpleicon@sourceforge\endcsname {\simpleiconsmapEight\symbol{97}}
\expandafter\def\csname simpleicon@sourcegraph\endcsname {\simpleiconsmapEight\symbol{98}}
\expandafter\def\csname simpleicon@sourcetree\endcsname {\simpleiconsmapEight\symbol{99}}
\expandafter\def\csname simpleicon@southwestairlines\endcsname {\simpleiconsmapEight\symbol{100}}
\expandafter\def\csname simpleicon@spacemacs\endcsname {\simpleiconsmapEight\symbol{101}}
\expandafter\def\csname simpleicon@spacex\endcsname {\simpleiconsmapEight\symbol{102}}
\expandafter\def\csname simpleicon@spacy\endcsname {\simpleiconsmapEight\symbol{103}}
\expandafter\def\csname simpleicon@sparkar\endcsname {\simpleiconsmapEight\symbol{104}}
\expandafter\def\csname simpleicon@sparkasse\endcsname {\simpleiconsmapEight\symbol{105}}
\expandafter\def\csname simpleicon@sparkfun\endcsname {\simpleiconsmapEight\symbol{106}}
\expandafter\def\csname simpleicon@sparkpost\endcsname {\simpleiconsmapEight\symbol{107}}
\expandafter\def\csname simpleicon@spdx\endcsname {\simpleiconsmapEight\symbol{108}}
\expandafter\def\csname simpleicon@speakerdeck\endcsname {\simpleiconsmapEight\symbol{109}}
\expandafter\def\csname simpleicon@spectrum\endcsname {\simpleiconsmapEight\symbol{110}}
\expandafter\def\csname simpleicon@speedtest\endcsname {\simpleiconsmapEight\symbol{111}}
\expandafter\def\csname simpleicon@spinnaker\endcsname {\simpleiconsmapEight\symbol{112}}
\expandafter\def\csname simpleicon@spinrilla\endcsname {\simpleiconsmapEight\symbol{113}}
\expandafter\def\csname simpleicon@splunk\endcsname {\simpleiconsmapEight\symbol{114}}
\expandafter\def\csname simpleicon@spond\endcsname {\simpleiconsmapEight\symbol{115}}
\expandafter\def\csname simpleicon@spotify\endcsname {\simpleiconsmapEight\symbol{116}}
\expandafter\def\csname simpleicon@spotlight\endcsname {\simpleiconsmapEight\symbol{117}}
\expandafter\def\csname simpleicon@spreadshirt\endcsname {\simpleiconsmapEight\symbol{118}}
\expandafter\def\csname simpleicon@spreaker\endcsname {\simpleiconsmapEight\symbol{119}}
\expandafter\def\csname simpleicon@spring\endcsname {\simpleiconsmapEight\symbol{120}}
\expandafter\def\csname simpleicon@springcreators\endcsname {\simpleiconsmapEight\symbol{121}}
\expandafter\def\csname simpleicon@springboot\endcsname {\simpleiconsmapEight\symbol{122}}
\expandafter\def\csname simpleicon@springsecurity\endcsname {\simpleiconsmapEight\symbol{123}}
\expandafter\def\csname simpleicon@spyderide\endcsname {\simpleiconsmapEight\symbol{124}}
\expandafter\def\csname simpleicon@sqlite\endcsname {\simpleiconsmapEight\symbol{125}}
\expandafter\def\csname simpleicon@square\endcsname {\simpleiconsmapEight\symbol{126}}
\expandafter\def\csname simpleicon@squareenix\endcsname {\simpleiconsmapEight\symbol{127}}
\expandafter\def\csname simpleicon@squarespace\endcsname {\simpleiconsmapEight\symbol{128}}
\expandafter\def\csname simpleicon@ssrn\endcsname {\simpleiconsmapEight\symbol{129}}
\expandafter\def\csname simpleicon@stackbit\endcsname {\simpleiconsmapEight\symbol{130}}
\expandafter\def\csname simpleicon@stackblitz\endcsname {\simpleiconsmapEight\symbol{131}}
\expandafter\def\csname simpleicon@stackedit\endcsname {\simpleiconsmapEight\symbol{132}}
\expandafter\def\csname simpleicon@stackexchange\endcsname {\simpleiconsmapEight\symbol{133}}
\expandafter\def\csname simpleicon@stackoverflow\endcsname {\simpleiconsmapEight\symbol{134}}
\expandafter\def\csname simpleicon@stackpath\endcsname {\simpleiconsmapEight\symbol{135}}
\expandafter\def\csname simpleicon@stackshare\endcsname {\simpleiconsmapEight\symbol{136}}
\expandafter\def\csname simpleicon@stadia\endcsname {\simpleiconsmapEight\symbol{137}}
\expandafter\def\csname simpleicon@staffbase\endcsname {\simpleiconsmapEight\symbol{138}}
\expandafter\def\csname simpleicon@starbucks\endcsname {\simpleiconsmapEight\symbol{139}}
\expandafter\def\csname simpleicon@stardock\endcsname {\simpleiconsmapEight\symbol{140}}
\expandafter\def\csname simpleicon@starlingbank\endcsname {\simpleiconsmapEight\symbol{141}}
\expandafter\def\csname simpleicon@starship\endcsname {\simpleiconsmapEight\symbol{142}}
\expandafter\def\csname simpleicon@startrek\endcsname {\simpleiconsmapEight\symbol{143}}
\expandafter\def\csname simpleicon@starz\endcsname {\simpleiconsmapEight\symbol{144}}
\expandafter\def\csname simpleicon@statamic\endcsname {\simpleiconsmapEight\symbol{145}}
\expandafter\def\csname simpleicon@statuspage\endcsname {\simpleiconsmapEight\symbol{146}}
\expandafter\def\csname simpleicon@statuspal\endcsname {\simpleiconsmapEight\symbol{147}}
\expandafter\def\csname simpleicon@steam\endcsname {\simpleiconsmapEight\symbol{148}}
\expandafter\def\csname simpleicon@steamdb\endcsname {\simpleiconsmapEight\symbol{149}}
\expandafter\def\csname simpleicon@steamdeck\endcsname {\simpleiconsmapEight\symbol{150}}
\expandafter\def\csname simpleicon@steamworks\endcsname {\simpleiconsmapEight\symbol{151}}
\expandafter\def\csname simpleicon@steelseries\endcsname {\simpleiconsmapEight\symbol{152}}
\expandafter\def\csname simpleicon@steem\endcsname {\simpleiconsmapEight\symbol{153}}
\expandafter\def\csname simpleicon@steemit\endcsname {\simpleiconsmapEight\symbol{154}}
\expandafter\def\csname simpleicon@steinberg\endcsname {\simpleiconsmapEight\symbol{155}}
\expandafter\def\csname simpleicon@stellar\endcsname {\simpleiconsmapEight\symbol{156}}
\expandafter\def\csname simpleicon@stencyl\endcsname {\simpleiconsmapEight\symbol{157}}
\expandafter\def\csname simpleicon@stimulus\endcsname {\simpleiconsmapEight\symbol{158}}
\expandafter\def\csname simpleicon@stitcher\endcsname {\simpleiconsmapEight\symbol{159}}
\expandafter\def\csname simpleicon@stmicroelectronics\endcsname {\simpleiconsmapEight\symbol{160}}
\expandafter\def\csname simpleicon@stopstalk\endcsname {\simpleiconsmapEight\symbol{161}}
\expandafter\def\csname simpleicon@storyblok\endcsname {\simpleiconsmapEight\symbol{162}}
\expandafter\def\csname simpleicon@storybook\endcsname {\simpleiconsmapEight\symbol{163}}
\expandafter\def\csname simpleicon@strapi\endcsname {\simpleiconsmapEight\symbol{164}}
\expandafter\def\csname simpleicon@strava\endcsname {\simpleiconsmapEight\symbol{165}}
\expandafter\def\csname simpleicon@streamlit\endcsname {\simpleiconsmapEight\symbol{166}}
\expandafter\def\csname simpleicon@stripe\endcsname {\simpleiconsmapEight\symbol{167}}
\expandafter\def\csname simpleicon@strongswan\endcsname {\simpleiconsmapEight\symbol{168}}
\expandafter\def\csname simpleicon@stubhub\endcsname {\simpleiconsmapEight\symbol{169}}
\expandafter\def\csname simpleicon@styledcomponents\endcsname {\simpleiconsmapEight\symbol{170}}
\expandafter\def\csname simpleicon@stylelint\endcsname {\simpleiconsmapEight\symbol{171}}
\expandafter\def\csname simpleicon@styleshare\endcsname {\simpleiconsmapEight\symbol{172}}
\expandafter\def\csname simpleicon@stylus\endcsname {\simpleiconsmapEight\symbol{173}}
\expandafter\def\csname simpleicon@subaru\endcsname {\simpleiconsmapEight\symbol{174}}
\expandafter\def\csname simpleicon@sublimetext\endcsname {\simpleiconsmapEight\symbol{175}}
\expandafter\def\csname simpleicon@substack\endcsname {\simpleiconsmapEight\symbol{176}}
\expandafter\def\csname simpleicon@subversion\endcsname {\simpleiconsmapEight\symbol{177}}
\expandafter\def\csname simpleicon@suckless\endcsname {\simpleiconsmapEight\symbol{178}}
\expandafter\def\csname simpleicon@sumologic\endcsname {\simpleiconsmapEight\symbol{179}}
\expandafter\def\csname simpleicon@supabase\endcsname {\simpleiconsmapEight\symbol{180}}
\expandafter\def\csname simpleicon@supermicro\endcsname {\simpleiconsmapEight\symbol{181}}
\expandafter\def\csname simpleicon@superuser\endcsname {\simpleiconsmapEight\symbol{182}}
\expandafter\def\csname simpleicon@surveymonkey\endcsname {\simpleiconsmapEight\symbol{183}}
\expandafter\def\csname simpleicon@suse\endcsname {\simpleiconsmapEight\symbol{184}}
\expandafter\def\csname simpleicon@suzuki\endcsname {\simpleiconsmapEight\symbol{185}}
\expandafter\def\csname simpleicon@svelte\endcsname {\simpleiconsmapEight\symbol{186}}
\expandafter\def\csname simpleicon@svg\endcsname {\simpleiconsmapEight\symbol{187}}
\expandafter\def\csname simpleicon@svgo\endcsname {\simpleiconsmapEight\symbol{188}}
\expandafter\def\csname simpleicon@swagger\endcsname {\simpleiconsmapEight\symbol{189}}
\expandafter\def\csname simpleicon@swarm\endcsname {\simpleiconsmapEight\symbol{190}}
\expandafter\def\csname simpleicon@swc\endcsname {\simpleiconsmapEight\symbol{191}}
\expandafter\def\csname simpleicon@swift\endcsname {\simpleiconsmapEight\symbol{192}}
\expandafter\def\csname simpleicon@swiggy\endcsname {\simpleiconsmapEight\symbol{193}}
\expandafter\def\csname simpleicon@swiper\endcsname {\simpleiconsmapEight\symbol{194}}
\expandafter\def\csname simpleicon@symantec\endcsname {\simpleiconsmapEight\symbol{195}}
\expandafter\def\csname simpleicon@symfony\endcsname {\simpleiconsmapEight\symbol{196}}
\expandafter\def\csname simpleicon@symphony\endcsname {\simpleiconsmapEight\symbol{197}}
\expandafter\def\csname simpleicon@sympy\endcsname {\simpleiconsmapEight\symbol{198}}
\expandafter\def\csname simpleicon@synology\endcsname {\simpleiconsmapEight\symbol{199}}
\expandafter\def\csname simpleicon@tableau\endcsname {\simpleiconsmapEight\symbol{200}}
\expandafter\def\csname simpleicon@tablecheck\endcsname {\simpleiconsmapEight\symbol{201}}
\expandafter\def\csname simpleicon@tacobell\endcsname {\simpleiconsmapEight\symbol{202}}
\expandafter\def\csname simpleicon@tado\endcsname {\simpleiconsmapEight\symbol{203}}
\expandafter\def\csname simpleicon@tails\endcsname {\simpleiconsmapEight\symbol{204}}
\expandafter\def\csname simpleicon@tailwindcss\endcsname {\simpleiconsmapEight\symbol{205}}
\expandafter\def\csname simpleicon@talend\endcsname {\simpleiconsmapEight\symbol{206}}
\expandafter\def\csname simpleicon@talenthouse\endcsname {\simpleiconsmapEight\symbol{207}}
\expandafter\def\csname simpleicon@tampermonkey\endcsname {\simpleiconsmapEight\symbol{208}}
\expandafter\def\csname simpleicon@taobao\endcsname {\simpleiconsmapEight\symbol{209}}
\expandafter\def\csname simpleicon@tapas\endcsname {\simpleiconsmapEight\symbol{210}}
\expandafter\def\csname simpleicon@target\endcsname {\simpleiconsmapEight\symbol{211}}
\expandafter\def\csname simpleicon@task\endcsname {\simpleiconsmapEight\symbol{212}}
\expandafter\def\csname simpleicon@tasmota\endcsname {\simpleiconsmapEight\symbol{213}}
\expandafter\def\csname simpleicon@tata\endcsname {\simpleiconsmapEight\symbol{214}}
\expandafter\def\csname simpleicon@tauri\endcsname {\simpleiconsmapEight\symbol{215}}
\expandafter\def\csname simpleicon@taxbuzz\endcsname {\simpleiconsmapEight\symbol{216}}
\expandafter\def\csname simpleicon@teamcity\endcsname {\simpleiconsmapEight\symbol{217}}
\expandafter\def\csname simpleicon@teamspeak\endcsname {\simpleiconsmapEight\symbol{218}}
\expandafter\def\csname simpleicon@teamviewer\endcsname {\simpleiconsmapEight\symbol{219}}
\expandafter\def\csname simpleicon@ted\endcsname {\simpleiconsmapEight\symbol{220}}
\expandafter\def\csname simpleicon@teespring\endcsname {\simpleiconsmapEight\symbol{221}}
\expandafter\def\csname simpleicon@tekton\endcsname {\simpleiconsmapEight\symbol{222}}
\expandafter\def\csname simpleicon@tele5\endcsname {\simpleiconsmapEight\symbol{223}}
\expandafter\def\csname simpleicon@telegram\endcsname {\simpleiconsmapEight\symbol{224}}
\expandafter\def\csname simpleicon@telegraph\endcsname {\simpleiconsmapEight\symbol{225}}
\expandafter\def\csname simpleicon@temporal\endcsname {\simpleiconsmapEight\symbol{226}}
\expandafter\def\csname simpleicon@tencentqq\endcsname {\simpleiconsmapEight\symbol{227}}
\expandafter\def\csname simpleicon@tensorflow\endcsname {\simpleiconsmapEight\symbol{228}}
\expandafter\def\csname simpleicon@teradata\endcsname {\simpleiconsmapEight\symbol{229}}
\expandafter\def\csname simpleicon@teratail\endcsname {\simpleiconsmapEight\symbol{230}}
\expandafter\def\csname simpleicon@terraform\endcsname {\simpleiconsmapEight\symbol{231}}
\expandafter\def\csname simpleicon@tesco\endcsname {\simpleiconsmapEight\symbol{232}}
\expandafter\def\csname simpleicon@tesla\endcsname {\simpleiconsmapEight\symbol{233}}
\expandafter\def\csname simpleicon@testcafe\endcsname {\simpleiconsmapEight\symbol{234}}
\expandafter\def\csname simpleicon@testin\endcsname {\simpleiconsmapEight\symbol{235}}
\expandafter\def\csname simpleicon@testinglibrary\endcsname {\simpleiconsmapEight\symbol{236}}
\expandafter\def\csname simpleicon@tether\endcsname {\simpleiconsmapEight\symbol{237}}
\expandafter\def\csname simpleicon@textpattern\endcsname {\simpleiconsmapEight\symbol{238}}
\expandafter\def\csname simpleicon@thealgorithms\endcsname {\simpleiconsmapEight\symbol{239}}
\expandafter\def\csname simpleicon@theconversation\endcsname {\simpleiconsmapEight\symbol{240}}
\expandafter\def\csname simpleicon@theirishtimes\endcsname {\simpleiconsmapEight\symbol{241}}
\expandafter\def\csname simpleicon@themighty\endcsname {\simpleiconsmapEight\symbol{242}}
\expandafter\def\csname simpleicon@themodelsresource\endcsname {\simpleiconsmapEight\symbol{243}}
\expandafter\def\csname simpleicon@themoviedatabase\endcsname {\simpleiconsmapEight\symbol{244}}
\expandafter\def\csname simpleicon@thenorthface\endcsname {\simpleiconsmapEight\symbol{245}}
\expandafter\def\csname simpleicon@theregister\endcsname {\simpleiconsmapEight\symbol{246}}
\expandafter\def\csname simpleicon@thesoundsresource\endcsname {\simpleiconsmapEight\symbol{247}}
\expandafter\def\csname simpleicon@thespritersresource\endcsname {\simpleiconsmapEight\symbol{248}}
\expandafter\def\csname simpleicon@thewashingtonpost\endcsname {\simpleiconsmapEight\symbol{249}}
\expandafter\def\csname simpleicon@thingiverse\endcsname {\simpleiconsmapEight\symbol{250}}
\expandafter\def\csname simpleicon@thinkpad\endcsname {\simpleiconsmapEight\symbol{251}}
\expandafter\def\csname simpleicon@threadless\endcsname {\simpleiconsmapEight\symbol{252}}
\expandafter\def\csname simpleicon@threedotjs\endcsname {\simpleiconsmapEight\symbol{253}}
\expandafter\def\csname simpleicon@threema\endcsname {\simpleiconsmapEight\symbol{254}}
\expandafter\def\csname simpleicon@thumbtack\endcsname {\simpleiconsmapEight\symbol{255}}
\expandafter\def\csname simpleicon@thunderbird\endcsname {\simpleiconsmapNine\symbol{0}}
\expandafter\def\csname simpleicon@thymeleaf\endcsname {\simpleiconsmapNine\symbol{1}}
\expandafter\def\csname simpleicon@ticketmaster\endcsname {\simpleiconsmapNine\symbol{2}}
\expandafter\def\csname simpleicon@tidal\endcsname {\simpleiconsmapNine\symbol{3}}
\expandafter\def\csname simpleicon@tide\endcsname {\simpleiconsmapNine\symbol{4}}
\expandafter\def\csname simpleicon@tietoevry\endcsname {\simpleiconsmapNine\symbol{5}}
\expandafter\def\csname simpleicon@tiktok\endcsname {\simpleiconsmapNine\symbol{6}}
\expandafter\def\csname simpleicon@tile\endcsname {\simpleiconsmapNine\symbol{7}}
\expandafter\def\csname simpleicon@timescale\endcsname {\simpleiconsmapNine\symbol{8}}
\expandafter\def\csname simpleicon@tinder\endcsname {\simpleiconsmapNine\symbol{9}}
\expandafter\def\csname simpleicon@tinyletter\endcsname {\simpleiconsmapNine\symbol{10}}
\expandafter\def\csname simpleicon@tistory\endcsname {\simpleiconsmapNine\symbol{11}}
\expandafter\def\csname simpleicon@tmobile\endcsname {\simpleiconsmapNine\symbol{12}}
\expandafter\def\csname simpleicon@tmux\endcsname {\simpleiconsmapNine\symbol{13}}
\expandafter\def\csname simpleicon@todoist\endcsname {\simpleiconsmapNine\symbol{14}}
\expandafter\def\csname simpleicon@toggl\endcsname {\simpleiconsmapNine\symbol{15}}
\expandafter\def\csname simpleicon@tokyometro\endcsname {\simpleiconsmapNine\symbol{16}}
\expandafter\def\csname simpleicon@tomorrowland\endcsname {\simpleiconsmapNine\symbol{17}}
\expandafter\def\csname simpleicon@topcoder\endcsname {\simpleiconsmapNine\symbol{18}}
\expandafter\def\csname simpleicon@toptal\endcsname {\simpleiconsmapNine\symbol{19}}
\expandafter\def\csname simpleicon@torbrowser\endcsname {\simpleiconsmapNine\symbol{20}}
\expandafter\def\csname simpleicon@torproject\endcsname {\simpleiconsmapNine\symbol{21}}
\expandafter\def\csname simpleicon@toshiba\endcsname {\simpleiconsmapNine\symbol{22}}
\expandafter\def\csname simpleicon@toyota\endcsname {\simpleiconsmapNine\symbol{23}}
\expandafter\def\csname simpleicon@tplink\endcsname {\simpleiconsmapNine\symbol{24}}
\expandafter\def\csname simpleicon@tqdm\endcsname {\simpleiconsmapNine\symbol{25}}
\expandafter\def\csname simpleicon@traefikmesh\endcsname {\simpleiconsmapNine\symbol{26}}
\expandafter\def\csname simpleicon@traefikproxy\endcsname {\simpleiconsmapNine\symbol{27}}
\expandafter\def\csname simpleicon@trainerroad\endcsname {\simpleiconsmapNine\symbol{28}}
\expandafter\def\csname simpleicon@trakt\endcsname {\simpleiconsmapNine\symbol{29}}
\expandafter\def\csname simpleicon@transportforireland\endcsname {\simpleiconsmapNine\symbol{30}}
\expandafter\def\csname simpleicon@transportforlondon\endcsname {\simpleiconsmapNine\symbol{31}}
\expandafter\def\csname simpleicon@travisci\endcsname {\simpleiconsmapNine\symbol{32}}
\expandafter\def\csname simpleicon@treehouse\endcsname {\simpleiconsmapNine\symbol{33}}
\expandafter\def\csname simpleicon@trello\endcsname {\simpleiconsmapNine\symbol{34}}
\expandafter\def\csname simpleicon@trendmicro\endcsname {\simpleiconsmapNine\symbol{35}}
\expandafter\def\csname simpleicon@treyarch\endcsname {\simpleiconsmapNine\symbol{36}}
\expandafter\def\csname simpleicon@triller\endcsname {\simpleiconsmapNine\symbol{37}}
\expandafter\def\csname simpleicon@trino\endcsname {\simpleiconsmapNine\symbol{38}}
\expandafter\def\csname simpleicon@tripadvisor\endcsname {\simpleiconsmapNine\symbol{39}}
\expandafter\def\csname simpleicon@tripdotcom\endcsname {\simpleiconsmapNine\symbol{40}}
\expandafter\def\csname simpleicon@trove\endcsname {\simpleiconsmapNine\symbol{41}}
\expandafter\def\csname simpleicon@trpc\endcsname {\simpleiconsmapNine\symbol{42}}
\expandafter\def\csname simpleicon@truenas\endcsname {\simpleiconsmapNine\symbol{43}}
\expandafter\def\csname simpleicon@trulia\endcsname {\simpleiconsmapNine\symbol{44}}
\expandafter\def\csname simpleicon@trustedshops\endcsname {\simpleiconsmapNine\symbol{45}}
\expandafter\def\csname simpleicon@trustpilot\endcsname {\simpleiconsmapNine\symbol{46}}
\expandafter\def\csname simpleicon@tryhackme\endcsname {\simpleiconsmapNine\symbol{47}}
\expandafter\def\csname simpleicon@tryitonline\endcsname {\simpleiconsmapNine\symbol{48}}
\expandafter\def\csname simpleicon@tsnode\endcsname {\simpleiconsmapNine\symbol{49}}
\expandafter\def\csname simpleicon@tubi\endcsname {\simpleiconsmapNine\symbol{50}}
\expandafter\def\csname simpleicon@tui\endcsname {\simpleiconsmapNine\symbol{51}}
\expandafter\def\csname simpleicon@tumblr\endcsname {\simpleiconsmapNine\symbol{52}}
\expandafter\def\csname simpleicon@tunein\endcsname {\simpleiconsmapNine\symbol{53}}
\expandafter\def\csname simpleicon@turborepo\endcsname {\simpleiconsmapNine\symbol{54}}
\expandafter\def\csname simpleicon@turbosquid\endcsname {\simpleiconsmapNine\symbol{55}}
\expandafter\def\csname simpleicon@turkishairlines\endcsname {\simpleiconsmapNine\symbol{56}}
\expandafter\def\csname simpleicon@tutanota\endcsname {\simpleiconsmapNine\symbol{57}}
\expandafter\def\csname simpleicon@tvtime\endcsname {\simpleiconsmapNine\symbol{58}}
\expandafter\def\csname simpleicon@twilio\endcsname {\simpleiconsmapNine\symbol{59}}
\expandafter\def\csname simpleicon@twitch\endcsname {\simpleiconsmapNine\symbol{60}}
\expandafter\def\csname simpleicon@twitter\endcsname {\simpleiconsmapNine\symbol{61}}
\expandafter\def\csname simpleicon@twoo\endcsname {\simpleiconsmapNine\symbol{62}}
\expandafter\def\csname simpleicon@typeform\endcsname {\simpleiconsmapNine\symbol{63}}
\expandafter\def\csname simpleicon@typescript\endcsname {\simpleiconsmapNine\symbol{64}}
\expandafter\def\csname simpleicon@typo3\endcsname {\simpleiconsmapNine\symbol{65}}
\expandafter\def\csname simpleicon@uber\endcsname {\simpleiconsmapNine\symbol{66}}
\expandafter\def\csname simpleicon@ubereats\endcsname {\simpleiconsmapNine\symbol{67}}
\expandafter\def\csname simpleicon@ubiquiti\endcsname {\simpleiconsmapNine\symbol{68}}
\expandafter\def\csname simpleicon@ubisoft\endcsname {\simpleiconsmapNine\symbol{69}}
\expandafter\def\csname simpleicon@ublockorigin\endcsname {\simpleiconsmapNine\symbol{70}}
\expandafter\def\csname simpleicon@ubuntu\endcsname {\simpleiconsmapNine\symbol{71}}
\expandafter\def\csname simpleicon@udacity\endcsname {\simpleiconsmapNine\symbol{72}}
\expandafter\def\csname simpleicon@udemy\endcsname {\simpleiconsmapNine\symbol{73}}
\expandafter\def\csname simpleicon@ufc\endcsname {\simpleiconsmapNine\symbol{74}}
\expandafter\def\csname simpleicon@uikit\endcsname {\simpleiconsmapNine\symbol{75}}
\expandafter\def\csname simpleicon@ulule\endcsname {\simpleiconsmapNine\symbol{76}}
\expandafter\def\csname simpleicon@umbraco\endcsname {\simpleiconsmapNine\symbol{77}}
\expandafter\def\csname simpleicon@unacademy\endcsname {\simpleiconsmapNine\symbol{78}}
\expandafter\def\csname simpleicon@underarmour\endcsname {\simpleiconsmapNine\symbol{79}}
\expandafter\def\csname simpleicon@underscoredotjs\endcsname {\simpleiconsmapNine\symbol{80}}
\expandafter\def\csname simpleicon@undertale\endcsname {\simpleiconsmapNine\symbol{81}}
\expandafter\def\csname simpleicon@unicode\endcsname {\simpleiconsmapNine\symbol{82}}
\expandafter\def\csname simpleicon@unilever\endcsname {\simpleiconsmapNine\symbol{83}}
\expandafter\def\csname simpleicon@unitedairlines\endcsname {\simpleiconsmapNine\symbol{84}}
\expandafter\def\csname simpleicon@unity\endcsname {\simpleiconsmapNine\symbol{85}}
\expandafter\def\csname simpleicon@unlicense\endcsname {\simpleiconsmapNine\symbol{86}}
\expandafter\def\csname simpleicon@unocss\endcsname {\simpleiconsmapNine\symbol{87}}
\expandafter\def\csname simpleicon@unraid\endcsname {\simpleiconsmapNine\symbol{88}}
\expandafter\def\csname simpleicon@unrealengine\endcsname {\simpleiconsmapNine\symbol{89}}
\expandafter\def\csname simpleicon@unsplash\endcsname {\simpleiconsmapNine\symbol{90}}
\expandafter\def\csname simpleicon@untangle\endcsname {\simpleiconsmapNine\symbol{91}}
\expandafter\def\csname simpleicon@untappd\endcsname {\simpleiconsmapNine\symbol{92}}
\expandafter\def\csname simpleicon@upcloud\endcsname {\simpleiconsmapNine\symbol{93}}
\expandafter\def\csname simpleicon@uplabs\endcsname {\simpleiconsmapNine\symbol{94}}
\expandafter\def\csname simpleicon@uploaded\endcsname {\simpleiconsmapNine\symbol{95}}
\expandafter\def\csname simpleicon@ups\endcsname {\simpleiconsmapNine\symbol{96}}
\expandafter\def\csname simpleicon@upstash\endcsname {\simpleiconsmapNine\symbol{97}}
\expandafter\def\csname simpleicon@uptimekuma\endcsname {\simpleiconsmapNine\symbol{98}}
\expandafter\def\csname simpleicon@uptobox\endcsname {\simpleiconsmapNine\symbol{99}}
\expandafter\def\csname simpleicon@upwork\endcsname {\simpleiconsmapNine\symbol{100}}
\expandafter\def\csname simpleicon@usps\endcsname {\simpleiconsmapNine\symbol{101}}
\expandafter\def\csname simpleicon@v\endcsname {\simpleiconsmapNine\symbol{102}}
\expandafter\def\csname simpleicon@v2ex\endcsname {\simpleiconsmapNine\symbol{103}}
\expandafter\def\csname simpleicon@v8\endcsname {\simpleiconsmapNine\symbol{104}}
\expandafter\def\csname simpleicon@vaadin\endcsname {\simpleiconsmapNine\symbol{105}}
\expandafter\def\csname simpleicon@vagrant\endcsname {\simpleiconsmapNine\symbol{106}}
\expandafter\def\csname simpleicon@valorant\endcsname {\simpleiconsmapNine\symbol{107}}
\expandafter\def\csname simpleicon@valve\endcsname {\simpleiconsmapNine\symbol{108}}
\expandafter\def\csname simpleicon@vapor\endcsname {\simpleiconsmapNine\symbol{109}}
\expandafter\def\csname simpleicon@vault\endcsname {\simpleiconsmapNine\symbol{110}}
\expandafter\def\csname simpleicon@vauxhall\endcsname {\simpleiconsmapNine\symbol{111}}
\expandafter\def\csname simpleicon@vbulletin\endcsname {\simpleiconsmapNine\symbol{112}}
\expandafter\def\csname simpleicon@vectorlogozone\endcsname {\simpleiconsmapNine\symbol{113}}
\expandafter\def\csname simpleicon@vectorworks\endcsname {\simpleiconsmapNine\symbol{114}}
\expandafter\def\csname simpleicon@veeam\endcsname {\simpleiconsmapNine\symbol{115}}
\expandafter\def\csname simpleicon@veepee\endcsname {\simpleiconsmapNine\symbol{116}}
\expandafter\def\csname simpleicon@velog\endcsname {\simpleiconsmapNine\symbol{117}}
\expandafter\def\csname simpleicon@venmo\endcsname {\simpleiconsmapNine\symbol{118}}
\expandafter\def\csname simpleicon@vercel\endcsname {\simpleiconsmapNine\symbol{119}}
\expandafter\def\csname simpleicon@verdaccio\endcsname {\simpleiconsmapNine\symbol{120}}
\expandafter\def\csname simpleicon@veritas\endcsname {\simpleiconsmapNine\symbol{121}}
\expandafter\def\csname simpleicon@verizon\endcsname {\simpleiconsmapNine\symbol{122}}
\expandafter\def\csname simpleicon@vexxhost\endcsname {\simpleiconsmapNine\symbol{123}}
\expandafter\def\csname simpleicon@vfairs\endcsname {\simpleiconsmapNine\symbol{124}}
\expandafter\def\csname simpleicon@viadeo\endcsname {\simpleiconsmapNine\symbol{125}}
\expandafter\def\csname simpleicon@viber\endcsname {\simpleiconsmapNine\symbol{126}}
\expandafter\def\csname simpleicon@vim\endcsname {\simpleiconsmapNine\symbol{127}}
\expandafter\def\csname simpleicon@vimeo\endcsname {\simpleiconsmapNine\symbol{128}}
\expandafter\def\csname simpleicon@vimeolivestream\endcsname {\simpleiconsmapNine\symbol{129}}
\expandafter\def\csname simpleicon@virgin\endcsname {\simpleiconsmapNine\symbol{130}}
\expandafter\def\csname simpleicon@virginmedia\endcsname {\simpleiconsmapNine\symbol{131}}
\expandafter\def\csname simpleicon@virtualbox\endcsname {\simpleiconsmapNine\symbol{132}}
\expandafter\def\csname simpleicon@virustotal\endcsname {\simpleiconsmapNine\symbol{133}}
\expandafter\def\csname simpleicon@visa\endcsname {\simpleiconsmapNine\symbol{134}}
\expandafter\def\csname simpleicon@visualstudio\endcsname {\simpleiconsmapNine\symbol{135}}
\expandafter\def\csname simpleicon@visualstudiocode\endcsname {\simpleiconsmapNine\symbol{136}}
\expandafter\def\csname simpleicon@vite\endcsname {\simpleiconsmapNine\symbol{137}}
\expandafter\def\csname simpleicon@vitess\endcsname {\simpleiconsmapNine\symbol{138}}
\expandafter\def\csname simpleicon@vivaldi\endcsname {\simpleiconsmapNine\symbol{139}}
\expandafter\def\csname simpleicon@vivino\endcsname {\simpleiconsmapNine\symbol{140}}
\expandafter\def\csname simpleicon@vk\endcsname {\simpleiconsmapNine\symbol{141}}
\expandafter\def\csname simpleicon@vlcmediaplayer\endcsname {\simpleiconsmapNine\symbol{142}}
\expandafter\def\csname simpleicon@vmware\endcsname {\simpleiconsmapNine\symbol{143}}
\expandafter\def\csname simpleicon@vodafone\endcsname {\simpleiconsmapNine\symbol{144}}
\expandafter\def\csname simpleicon@volkswagen\endcsname {\simpleiconsmapNine\symbol{145}}
\expandafter\def\csname simpleicon@volvo\endcsname {\simpleiconsmapNine\symbol{146}}
\expandafter\def\csname simpleicon@vonage\endcsname {\simpleiconsmapNine\symbol{147}}
\expandafter\def\csname simpleicon@vowpalwabbit\endcsname {\simpleiconsmapNine\symbol{148}}
\expandafter\def\csname simpleicon@vox\endcsname {\simpleiconsmapNine\symbol{149}}
\expandafter\def\csname simpleicon@vsco\endcsname {\simpleiconsmapNine\symbol{150}}
\expandafter\def\csname simpleicon@vtex\endcsname {\simpleiconsmapNine\symbol{151}}
\expandafter\def\csname simpleicon@vuedotjs\endcsname {\simpleiconsmapNine\symbol{152}}
\expandafter\def\csname simpleicon@vuetify\endcsname {\simpleiconsmapNine\symbol{153}}
\expandafter\def\csname simpleicon@vulkan\endcsname {\simpleiconsmapNine\symbol{154}}
\expandafter\def\csname simpleicon@vultr\endcsname {\simpleiconsmapNine\symbol{155}}
\expandafter\def\csname simpleicon@w3c\endcsname {\simpleiconsmapNine\symbol{156}}
\expandafter\def\csname simpleicon@wacom\endcsname {\simpleiconsmapNine\symbol{157}}
\expandafter\def\csname simpleicon@wagtail\endcsname {\simpleiconsmapNine\symbol{158}}
\expandafter\def\csname simpleicon@wakatime\endcsname {\simpleiconsmapNine\symbol{159}}
\expandafter\def\csname simpleicon@walkman\endcsname {\simpleiconsmapNine\symbol{160}}
\expandafter\def\csname simpleicon@wallabag\endcsname {\simpleiconsmapNine\symbol{161}}
\expandafter\def\csname simpleicon@walmart\endcsname {\simpleiconsmapNine\symbol{162}}
\expandafter\def\csname simpleicon@wappalyzer\endcsname {\simpleiconsmapNine\symbol{163}}
\expandafter\def\csname simpleicon@warnerbros\endcsname {\simpleiconsmapNine\symbol{164}}
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https://genkuroki.github.io/documents/20160501StirlingFormula/20160501StirlingFormula-0.12.tex
|
github.io
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crawl-data/CC-MAIN-2019-39/segments/1568514576345.90/warc/CC-MAIN-20190923084859-20190923110859-00088.warc.gz
| 481,090,021 | 34,082 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\TITLE{\bf ガンマ分布の中心極限定理とStirlingの公式}
\def\AUTHOR{黒木玄}
\def\DATE{2016年5月1日作成%
\thanks{%
最新版は下記URLからダウンロードできる.
飽きるまで継続的に更新と訂正を続ける予定である.
2016年5月1日Ver.0.1.
2016年5月2日Ver.0.2: 対数版の易しいStirlingの公式の節を追加した.
2016年5月3日Ver.0.3: 色々追加. 特にFourierの反転公式に関する付録を追加した.
2016年5月4日Ver.0.4: ガウス分布のFourier変換の付録とGauss積分の計算の付録
を追加した.
2016年5月5日Ver.0.5: 誤りの訂正と様々な追加(全17頁).
2016年5月5日Ver.0.6: ファイル名を変更し,
対数版の易しいStirlingの公式の微小な改良の節を追加した(全18頁).
2016年5月6日Ver.0.7: ガンマ函数の正値性と対数凸性と函数等式による特徴付けと
無限乗積展開の証明の節や対数版の易しいStirlingの公式を改良して
通常のStirlingの公式を導くことなどを色々追加した(全24頁).
2016年5月7日Ver.0.8: 正弦函数の無限乗積展開を $\cos(tx)$ の
Fourier級数展開を使って導く方法の解説を追加した(全25頁).
2016年5月8日Ver.0.9: Riemann-Lebesgueの定理の節と
Fourier変換の部分和とFourier級数の部分和の収束に関する解説を追加(全30頁).
2016年5月9日Ver.0.10: 二項分布の中心極限定理の解説を追加(全33頁).
2016年5月12日Ver.0.11: Laplaceの方法による補正項の計算の仕方の解説と
\tableref{table:Stirling}を追加(全37頁).
2016年5月13日Ver.0.12(43頁): 自由度の大きなカイ2乗分布が正規分布で近似できることと
Stirlingの公式が同値であるというコメントを追加した.
正規分布から派生する様々な確率分布についての付録を追加した.
Boltzmann-Maxwell則の導出も追加した(\secref{sec:BM}).
}
\\[\bigskipamount]
{\small
\href{http://www.math.tohoku.ac.jp/~kuroki/LaTeX/20160501StirlingFormula.pdf}
{\tt http://www.math.tohoku.ac.jp/{\textasciitilde}kuroki/LaTeX/20160501StirlingFormula.pdf}
}}
\def\PDFTITLE{Stirlingの公式}
\def\PDFAUTHOR{黒木玄}
\def\PDFSUBJECT{確率論}
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\setcounter{section}{-1} % 最初の節番号を0にする
\pagebreak
\section{はじめに}
{\bf Stirlingの公式}とは
\[
n! \sim n^n e^{-n} \sqrt{2\pi n} \qquad (n\to \infty)
\]
という階乗の近似公式のことである.
ここで $a_n\sim b_n$ ($n\to\infty$)は $\lim_{n\to\infty}(a_n/b_n)=1$ を
意味する. より精密には
\[
n! = n^n e^{-n} \sqrt{2\pi n}\left(1+\frac{1}{12n}+O\left(\frac{1}{n^2}\right)\right) \qquad (n\to \infty)
\]
が成立している%
\footnote{\secref{sec:Laplace}を見よ.}.
このノートではまず最初にガンマ分布に関する中心極限定理からStirlingの公式が
``導出''されることを説明する.
その後は様々な方法でStirlingの公式を導出する.
精密かつ厳密な議論はしない.
このノートの後半の付録群では関連の基礎知識の解説を行なう.
このノートの全体は学生向けのGauss積分入門, ガンマ函数入門, ベータ函数入門,
Fourier解析入門になることを意図して書かれた雑多な解説の寄せ集めである.
前の方の節で後の方の節で説明した結果を使うことが多いので
読者は注意して欲しい.
基本的な方針として易しい話しか扱わないことにする.
\begin{table}[htbp]
\caption{Stirlingの公式による階乗の近似}
\label{table:Stirling}
\centering
\begin{tabular}{|c||c|cc|cc|} \hline
$n$ & $n!$ & $A_n=n^ne^{-n}\sqrt{2\pi n}$ & ($\text{誤差}/n!$) & $A_n(1+1/(12n))$ & ($\text{誤差}/n!$) \\ \hline\hline
$1$ & 1 & $0.92\cdots$ & (7.78\%) & $0.9989\cdots$ & ($0.10\%$) \\ \hline
$3$ & 6 & $5.836\cdots$ & (2.73\%) & $5.998\cdots$ & ($0.028\%$) \\ \hline
$10$ & 3628800 & $3598695.6\cdots$ & (0.83\%) & $3628684.7\cdots$ & ($0.0032\%$) \\ \hline
$30$ & $2.6525\cdots\times10^{32}$ & $2.6451\cdots\times10^{32}$ & (0.28\%) & $2.6525\cdots\times10^{32}$ & ($3.7\times10^{-6}$) \\ \hline
$100$ & $9.3326\cdots\times10^{157}$ & $9.3248\cdots\times10^{157}$ & (0.08\%) & $9.3326\cdots\times10^{157}$ & ($3.4\times10^{-7}$) \\
\hline
\end{tabular}
\end{table}
\tableref{table:Stirling}を見ればわかるように,
$n^n e^{-n}\sqrt{2\pi n}$ による $n!$ の近似の誤差は,
$n=3$ の段階ですでに $3\%$ を切っており,
$n=10$ の段階では $1\%$ を切っている.
さらに $1/(12n)$ で補正すると誤差は劇的に小さくなり,
$n=1$ の段階ですでに近似の精度が $0.1\%$ 程度になる:
\[
\frac{\sqrt{2\pi}}{e}\left(1+\frac{1}{12}\right) = 0.9989\cdots \approx 1.
\]
このようにStirlingの公式は階乗の近似公式として極めて優秀である%
\footnote{\href
{http://www.ebyte.it/library/downloads/2007_MTH_Nemes_GammaFunction.pdf}
{Gerg\"o Nemes, New aymptotic expansion for the $\Gamma(z)$ function, 2007}
に階乗の様々な近似公式の比較がある. たとえば Nemes の公式
\[
n!
=n^n e^{-n} \sqrt{2\pi n}
\left(1+\frac{1}{12n^2}+\frac{1}{1440n^4}+\cdots \right)^n
\]
は極めて優秀な近似公式である.
}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ガンマ分布に関する中心極限定理からの``導出''}
ガンマ分布とは次の確率密度函数で定義される確率分布のことである%
\footnote{ガンマ函数は $s>0$ に対して
$\Gamma(s)=\int_0^\infty e^{-x}x^{s-1}\,dx$ と定義される.
直接の計算によって $\Gamma(1)=1$ を,
部分積分によって $\Gamma(s+1)=s\Gamma(s)$ を示せるので,
$0$ 以上の整数 $n$ について $\Gamma(n+1)=n!$ となる.}:
\[
f_{\alpha,\tau}(x) =
\begin{cases}
\dfrac{e^{-x/\tau}x^{\alpha-1}}{\Gamma(\alpha)\tau^\alpha} & \qquad (x>0), \\
0 & \qquad (x\leqq 0).
\end{cases}
\]
ここで $\alpha,\tau>0$ はガンマ分布を決めるパラメーターである%
\footnote{$\alpha$ は shape parameter と,
$\tau$ は scale parameter と呼ばれているらしい.
ガンマ分布の平均と分散はそれぞれ $\alpha\tau$ と $\alpha\tau^2$ になる.}.
以下簡単のため $\alpha=n>0$, $\tau=1$ の場合のガンマ分布のみを扱うため
に $f_n(x)=f_{n,1}(x)$ とおく:
\[
f_n(x) = \frac{e^{-x} x^{n-1}}{\Gamma(n)} \qquad (x>0).
\]
確率密度函数 $f_n(x)$ で定義される確率変数を $X_n$ と書くことにする.
確率変数 $X_n$ の平均 $\mu_n$ と分散 $\sigma_n^2$ は両方 $n$ になる%
\footnote{確率密度函数 $f(x)$ を持つ確率変数 $X$ に対して,
期待値汎函数が $E[g(X)]=\int_\R g(x)f(x)\,dx$ と定義され,
平均が $\mu=E[X]$ と定義され,
分散が $\sigma^2=E[(X-\mu)^2]=E[X^2]-\mu^2$ と定義される.}:
\begin{align*}
&
\mu_n = E[X_n] = \int_0^\infty x f_n(x)\,dx = \frac{\Gamma(n+1)}{\Gamma(n)}=n,
\\ &
E[X_n^2] = \int_0^\infty x^2 f_n(x)\,dx = \frac{\Gamma(n+2)}{\Gamma(n)}=(n+1)n,
\\ &
\sigma_n^2 = E[X_n^2]-\mu_n^2 = n.
\end{align*}
ゆえに確率変数 $Y_n=(X_n-\mu_n)/\sigma_n=(X_n-n)/\sqrt{n}$ の
平均と分散はそれぞれ $0$ と $1$ になり, その確率密度函数は
\[
\sqrt{n}f_n(\sqrt{n}y+n)
=
\sqrt{n}\frac{e^{-(\sqrt{n}y+n)}(\sqrt{n}y+n)^{n-1}}{\Gamma{n}}
\]
になる%
\footnote{確率変数 $X$ の確率分布函数が $f(x)$ のとき, 確率変数 $Y$ を $Y=(X-a)/b$ と
定めると, $E[g(Y)]=\int_\R g((x-a)/b)f(x)\,dx = \int_\R g(y) b f(by+a)\,dy$ なので,
$Y$ の確率分布函数は $b f(by+a)$ になる.}.
この確率密度函数で $y=0$ とおくと
\[
\sqrt{n}f_n(n)
=
\sqrt{n}\frac{e^{-n}n^{n-1}}{\Gamma(n)}
=
\frac{n^n e^{-n}\sqrt{n}}{\Gamma(n+1)}
\]
となる. $n>0$ が整数のとき $\Gamma(n+1)=n!$ なので,
これが $n\to\infty$ で $1/\sqrt{2\pi}$ に収束することとStirlingの公式の成立は同値になる.
ガンマ分布が再生性を満たしていることより,
中心極限定理を適用できるので,
$\R$ 上の有界連続函数 $\varphi(x)$ に対して, $n\to\infty$ のとき
\[
\int_0^\infty \varphi\left(\frac{x-n}{\sqrt{n}}\right)f_n(x)\,dx
=
\int_0^\infty \varphi(y)\sqrt{n}f_n(\sqrt{n}y+n)\,dy
\longrightarrow
\int_{-\infty}^\infty \varphi(y)\frac{e^{-y^2/2}}{\sqrt{2\pi}}\,dy.
\]
$\varphi(y)$ をデルタ函数 $\delta(y)$ に近付けることによって
(すなわち確率密度函数の $y$ に $0$ を代入することによって),
\[
\sqrt{n}f_n(n)
=
\sqrt{n}\frac{e^{-n}n^{n-1}}{\Gamma(n)}
=
\frac{n^n e^{-n} \sqrt{n}}{\Gamma(n+1)}
\longrightarrow
\frac{1}{\sqrt{2\pi}}
\qquad(n\to\infty)
\]
を得る.
この結果はStirlingの公式の成立を意味する.
以上の``導出''の最後で確率密度函数の $y$ に $0$ を代入するステップ
には論理的にギャップがある.
このギャップを埋めるためには
中心極限定理をブラックボックスとして利用するのではなく,
中心極限定理の特性函数を用いた証明に戻る必要がある.
そのような証明の方針については次の節を見て欲しい.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ガンマ分布の特性函数を用いた表示からの導出}
前節では中心極限定理を便利なブラックボックスとして用いて
Stirlingの公式を``導出''した.
しかし, その``導出''には論理的なギャップがあった.
そのギャップを埋めるためには,
中心極限定理が確率密度函数を特性函数(確率密度函数の逆Fourier変換)の
Fourier変換で表示することによって証明されることを思い出す必要がある.
この節ではガンマ分布の確率密度函数を特性函数のFourier変換で表わす公式を
用いて, 直接的にStirlingの公式を証明する%
\footnote{筆者はこの証明法を
\href
{https://www.math.kyoto-u.ac.jp/~nobuo/pdf/prob/stir.pdf}
{https://www.math.kyoto-u.ac.jp/{\textasciitilde}nobuo/pdf/prob/stir.pdf}
を見て知った.}.
\subsection{Stirlingの公式の証明}
ガンマ分布の確率密度函数 $f_n(x)=e^{-x}x^{n-1}/\Gamma(n)$ ($x>0$)
の特性函数(逆Fourier変換) $F_n(t)$ は次のように計算される%
\footnote{確率分布がパラメーター $n$ について再生性を持つことと
特性函数がある函数の $n$ 乗の形になることは同値である.}:
\[
F_n(t)
=\int_0^\infty e^{itx} f_n(x)\,dx
=\frac{1}{\Gamma(n)}\int_0^\infty e^{-(1-it)x} x^{n-1}\,dx
%=\frac{1}{\Gamma(n)}\frac{\Gamma(n)}{(1-it)^n}
=\frac{1}{(1-it)^n}.
\]
ここで, 実部が正の複素数 $\alpha$ に対して
\[
\frac{1}{\Gamma(n)}\int_0^\infty e^{-\alpha t} t^{n-1}\,dt = \frac{1}{\alpha^n}
\]
となること使った. この公式はCauchyの積分定理を使って示せる%
\footnote{
Cauchyの積分定理を使わなくても示せる.
左辺を $f(\alpha)$ と書くと, $f(1)=1$ でかつ部分積分によって
$f'(\alpha)=-(n/\alpha)f(\alpha)$ となることがわかるので,
その公式が得られる.
正の実数 $\alpha$ に対するこの公式は $t=x/\alpha$ という
置換積分によって容易に証明される.
}.
Fourierの反転公式より%
\footnote{Fourierの反転公式の証明の概略については\secref{sec:Fourier}を参照せよ.},
\[
f_n(x)
=
\frac{e^{-x} x^{n-1}}{\Gamma(n)}
=
\frac{1}{2\pi}\int_{-\infty}^\infty e^{-itx}F_n(t)\,dt
=
\frac{1}{2\pi}\int_{-\infty}^\infty \frac{e^{-itx}}{(1-it)^n}\,dt
\qquad (x>0).
\]
この公式さえ認めてしまえばStirlingの公式の証明は易しい.
この公式より, $t=\sqrt{n}u$ と置換することによって,
\begin{align*}
\sqrt{n}f_n(n)
=
\frac{n^n e^{-n}\sqrt{n}}{\Gamma(n+1)}
=
\frac{\sqrt{n}}{2\pi}
\int_{-\infty}^\infty
\frac{e^{-itn}}{(1-it)^n}
\,dt
=
\frac{1}{2\pi}
\int_{-\infty}^\infty
\frac{e^{-iu\sqrt{n}}}{(1-iu/\sqrt{n})^n}\,du.
\end{align*}
Stirlingの公式を証明するためには,
これが $n\to\infty$ で $1/\sqrt{2\pi}$ に収束することを示せばよい.
そのために被積分函数の対数の様子を調べよう:
\begin{align*}
\log\frac{e^{-iu\sqrt{n}}}{(1-iu/\sqrt{n})^n}
&
=-n\log\left(1-\frac{iu}{\sqrt{n}}\right)-iu\sqrt{n}
\\&
=n\left(\frac{iu}{\sqrt{n}}-\frac{u^2}{2n}+o\left(\frac{1}{n}\right)\right)-iu\sqrt{n}
=-\frac{u^2}{2} + o(1).
\end{align*}
したがって, $n\to\infty$ のとき
\[
\frac{e^{-iu\sqrt{n}}}{(1-iu/\sqrt{n})^n} \longrightarrow e^{-u^2/2}.
\]
これより, $n\to\infty$ のとき
\[
\sqrt{n}f_n(n)
=
\frac{n^n e^{-n}\sqrt{n}}{\Gamma(n+1)}
=
\frac{1}{2\pi}
\int_{-\infty}^\infty
\frac{e^{-iu\sqrt{n}}}{(1-iu/\sqrt{n})^n}\,du
\longrightarrow
\frac{1}{2\pi}
\int_{-\infty}^\infty
e^{-u^2/2}\,du
=
\frac{1}{\sqrt{2\pi}}
\]
となることがわかる%
\footnote{厳密に証明したければ, たとえばLebesgueの収束定理を使えばよい.}.
最後の等号で一般に正の実数 $\alpha$ に対して
\[
\int_{-\infty}^\infty e^{-u^2/\alpha}\,du = \sqrt{\alpha\pi}
\]
となることを用いた%
\footnote{この公式はGauss積分の公式
$\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}$
で $x=u/\sqrt{\alpha}$ と積分変数を変換すれば得られる.
Gauss積分の公式は以下のようにして証明される.
左辺を $I$ とおくと
$I^2=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)}\,dx\,dy$
であり, $I^2$ は $z=e^{-(x^2+y^2)}$ のグラフと平面 $z=0$ で挟まれた
「小山状の領域」の体積だと解釈される.
その小山の高さ $0< z\leqq 1$ における断面積は $-\pi \log z$ に
なるので, その体積は $\int_0^1(-\pi\log z)\,dz=-\pi[z\log z-z]_0^1=\pi$
になる. ゆえに $I=\sqrt{\pi}$.
Gauss積分の公式の不思議なところは円周率が出て来るところであり,
しかもその平方根が出て来るところである.
しかしその二乗が小山の体積であることがわかれば, その高さ $z$ での断面が
円盤の形になることから円周率 $\pi$ が出て来る理由がわかる.
平方根になるのは $I$ そのものを直接計算したのではなく,
$I^2$ の方を計算したからである.
}. %
これでStirlingの公式が証明された.
\subsection{正規化されたガンマ分布の確率密度函数の各点収束}
確率密度函数 $f_n(x)=e^{-x}x^{n-1}$ を持つ確率変数を $X_n$ と書くとき,
$Y_n=(X_n-n)/\sqrt{n}$ の平均と分散はそれぞれ $0$ と $1$ になるので
あった(前節を見よ). $Y_n$ の確率密度函数は
\[
\sqrt{n}f_n(\sqrt{n}y+n)
=\sqrt{n}\frac{e^{-\sqrt{n}y-n}(\sqrt{n}y+n)^{n-1}}{\Gamma(n)}
=\frac{e^{-n}n^{n-1/2}}{\Gamma(n)}
\frac{e^{-\sqrt{n}y}(1+y/\sqrt{n})^n}{1+y/\sqrt{n}}
\]
になる. そして, $n\to\infty$ のとき
\begin{align*}
\log\left(e^{-\sqrt{n}y}\left(1+\frac{y}{\sqrt{n}}\right)^n\right)
&=
n\log\left(1+\frac{y}{\sqrt{n}}\right)-\sqrt{n}y
\\ &
=n\left(\frac{y}{\sqrt{n}}-\frac{y^2}{2n}+o\left(\frac{1}{n}\right)\right)
-\sqrt{n}y
=-\frac{y^2}{2}+o(1)
\end{align*}
なので, $n\to\infty$ で $e^{\sqrt{n}y}(1+y/\sqrt{n})^n\to e^{-y^2/2}$ と
なり, さらに $1+y/\sqrt{n}\to 1$ となる.
ゆえに, 次が成立することと Stirling の公式は同値になる:
\[
\sqrt{n}f_n(\sqrt{n}y+n)
=\sqrt{n}\frac{e^{-\sqrt{n}y-n}(\sqrt{n}y+n)^{n-1}}{\Gamma(n)}
\longrightarrow
\frac{e^{-y^2/2}}{\sqrt{2\pi}}
\qquad (n\to\infty).
\]
すなわち $Y_n$ の確率密度函数が標準正規分布の確率密度函数に各点収束すること
とStirlingの公式は同値である.
ガンマ分布について確率密度函数の各点収束のレベルで中心極限定理が
成立していることと Stirling の公式は同じ深さにある.
$Y_n$ の確率分布函数が標準正規分布の確率密度函数に各点収束することの
直接的証明は $\sqrt{n}f(n)$ の収束の証明と同様に以下のようにして得られる:
\begin{align*}
\sqrt{n}f_n(\sqrt{n}y+n)
&=
\frac{\sqrt{n}}{2\pi}
\int_{-\infty}^\infty
\frac{e^{-it(\sqrt{n}y+n)}}{(1-it)^n}\,dt
=\frac{1}{2\pi}
\int_{-\infty}^\infty
e^{-iuy}\frac{e^{-it\sqrt{n}}}{(1-iu/\sqrt{n})^n}\,dt
\\ &
\longrightarrow
\frac{1}{2\pi}
\int_{-\infty}^\infty e^{-iuy}e^{-u^2/2}\,du
=
\frac{1}{\sqrt{2\pi}}e^{-y^2/2}
\qquad(n\to\infty).
\end{align*}
最後の等号で, Cauchyの積分定理より%
\footnote{複素解析を使わなくても容易に証明される.
たとえば, $e^{-ity}$ のTaylor展開を代入して項別積分を実行しても証明できる.
もしくは, 両辺が $f'(y)=-y f(y)$, $f(0)=\sqrt{2\pi}$ を満たしていることからも
導かれる(左辺が満たしていることは部分積分すればわかる).
Cauchyの積分定理を使えば
形式的に $u+iy$ ($u>0$) を $v>0$ で置き換える
置換積分を実行したのと同じように見える証明が得られる.}
\[
\int_{-\infty}^\infty e^{-iuy}e^{-u^2/2}\,du
=\int_{-\infty}^\infty e^{-(u+iy)^2/2-y^2/2}\,du
=e^{-y^2/2}\int_{-\infty}^\infty e^{-v^2/2}\,dv
=e^{-y^2/2}\sqrt{2\pi}
\]
となることを用いた.
このように,
ガンマ分布の確率密度函数の特性函数のFourier変換による表示を使えば
確率密度函数の各点収束のレベルでの中心極限定理を容易に示すことができ,
その結果は Stirling の公式と同値になっている.
\subsection{自由度が大きなカイ2乗分布が正規分布で近似できることとの関係}
互いに独立な標準正規分布する確率変数 $n$ 個の確率変数 $X_1,\ldots,X_n$
によって $Y_n=X_1^2+\cdots+X_n^2$ と定義された確率変数 $Y_n$ の確率分布を
自由度 $n$ の{\bf カイ2乗分布}と呼ぶ.
自由度 $n$ のカイ2乗分布は
shape が $\alpha=n/2$ で scale が $\tau=2$ のガンマ分布に等しい.
特に自由度 $n$ のカイ2乗分布の確率密度函数は
\[
f_{n/2,2}(y) =
\begin{cases}
\dfrac{e^{-y/2}y^{n/2-1}}{\Gamma(n/2)2^{n/2}} & \qquad (x>0), \\
0 & \qquad (y\leqq 0).
\end{cases}
\]
になり, その平均と分散はそれぞれ $n$ と $2n$ になる. すなわち,
\[
\int_0^\infty g(y) \frac{e^{-y/2}y^{n/2-1}}{\Gamma(n/2)2^{n/2}}\,dy
=\int_{\R^n} g(x_1^2+\cdots+x_n^2) \frac{e^{-(x_1^2+\cdots+x_n^2)/2}}{(2\pi)^{n/2}}\,dx_1\cdots dx_n.
\]
この事実を示すためには, ガンマ分布の再生性より, $n=1$ の場合を示せば十分である.
$n=1$ の場合の計算は本質的にガウス積分と $\Gamma(1/2)$ の関係そのものである.
実際, $x>0$ で $x=\sqrt{y}$ と積分変数を置換することによって
\[
\int_{-\infty}^\infty g(x^2)\frac{e^{-x^2/2}}{\sqrt{2\pi}}\,dx
=2\int_0^\infty g(y) \frac{e^{-y/2}}{\sqrt{2\pi}}\frac{y^{-1/2}}{2}\,dy
=\int_0^\infty g(y)\frac{e^{-y/2}y^{1/2-1}}{\Gamma(1/2)2^{1/2}}\,dy.
\]
最後の等号で $\Gamma(1/2)=\sqrt{\pi}$ を使った.
統計学の世界では, 自由度 $n$ を大きくすると,
カイ2乗分布は平均が $n$ で分散が $2n$ の正規分布にゆっくり近付くことが
よく知られている.
その事実はガンマ分布の中心極限定理そのものである.
そして, 前節で示したように正規化されたガンマ分布の確率密度函数が
標準正規分布に各点収束するという結果とStirlingの公式は同値
(同じ深さの結果)なのであった.
以上をまとめると次のようにも言えることがわかる:
\begin{quote}
自由度 $n$ のカイ2乗分布を変数変換で平均 $0$, 分散 $1$ に正規化するとき,
$n\to\infty$ でその確率密度函数が標準正規分布の確率密度函数に収束する
という統計学においてよく知られている結果はStirlingの公式と同値である.
\end{quote}
要するに統計学をよく知っている人は, Stirlingの公式は
$n\to\infty$ でカイ2乗分布が正規分布に近づくことと同じことを意味していると思ってよい.
\subsection{一般の場合の中心極限定理に関する大雑把な解説}
一般の場合の中心極限定理について大雑把にかつ簡単に解説する.
$X_1,X_2,X_3,\ldots$ は互いに独立で等しい確率分布を持つ確率変数の列であるとする.
さらにそれらは平均 $\mu=E[X_k]$ と分散 $\sigma^2=E[(X_k-\mu)^2]=E[X_k]^2-\mu^2$
を持つと仮定する.
$Y_n=(X_1+\cdots+X_n-n\mu)/\sqrt{n\sigma^2}$ とおくと $Y_n$ の平均と分散は
それぞれ $0$ と $1$ になる.
このとき $n\to\infty$ の極限で $Y_n$ の確率分布が平均 $0$, 分散 $1$ の
標準正規分布に(適切な意味で)収束するというのが中心極限定理である.
記述の簡単のため $X_k$ を $(X_k-\mu)/\sigma$ で置き換えることにする.
このように置き換えても $Y_n$ は変わらない.
このとき $X_k$ の平均と分散はそれぞれ $0$ と $1$ になるので,
$X_k$ の特性函数を $\varphi(t)=E[e^{itX_k}]$ と書くと,
\[
\varphi(t) = 1 - \frac{t^2}{2} + o(t^2).
\]
$Y_n=(X_1+\cdots+X_n)/\sqrt{n}$ とおくと
$Y_n$ の平均と分散もそれぞれ $0$ と $1$ になり,
$Y_n$ の特性函数の極限は次のように計算される:
\begin{align*}
E[e^{itY_n}]
&=\prod_{k=1}^n E[e^{itX_k/\sqrt{n}}]
=\varphi\left(\frac{t}{\sqrt{n}}\right)^n
\\ &
=\left( 1 - \frac{t^2}{2n} + o\left(\frac{1}{n}\right) \right)^n
\longrightarrow e^{-t^2/2}
\qquad (n\to\infty).
\end{align*}
ゆえに, Fourierの反転公式より%
\footnote{$\varphi(t/\sqrt{n})^n$ が可積分ならば
$Y_n$ に関するFourier 反転公式の結果は函数になるが,
可積分でない場合には測度になり, 測度の収束を考えることになる.},
$Y_n$ の確率密度函数%
\footnote{一般には $\R$ 上の確率測度になる.}
$f_n(y)$ は
\[
f_n(y)
= \frac{1}{2\pi}\int_{-\infty}^\infty
e^{-ity}\varphi\left(\frac{t}{\sqrt{n}}\right)^n\,dt
\]
になり, これは $n\to\infty$ で標準正規分布の確率密度函数
\[
\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ity}e^{-t^2/2}\,dt
=\frac{e^{-y^2/2}}{\sqrt{2\pi}}
\]
に収束する\footnote{厳密には適切な意味での収束を考える必要がある.}.
\subsection{二項分布の中心極限定理}
前節では確率分布の「適切な意味での収束」についてほとんど何も説明
しなかった. この節ではその点について二項分布を例に用いて大雑把に説明する%
\footnote{アイデアの説明はするが, 厳密な議論はしない.}.
$X_n$ が二項分布する確率変数のとき, $g(X_n)$ の期待値は
\[
E[g(X_n)] = \sum_{k=0}^n g(k) \binom{n}{k}p^k q^{n-k}
\]
と定義される. ここで $0<p<1$, $q=1-p$ であり, $n$ は正の整数であるとし,
$\binom{n}{k}$ は二項係数を表わす:
\[
\binom{n}{k}
=\frac{n!}{k!(n-k)!}, \qquad
(x+y)^n
=\sum_{k=0}^n \binom{n}{k} x^k y^{n-k}.
\]
$E[g(X_n)]$ を積分の形式で書くためにはデルタ函数(デルタ測度) $\delta(x-a)\,dx$ を
使う必要がある%
\footnote{デルタ函数(デルタ測度) $\delta(x-a)\,dx$ は連続函数 $f(x)$ に対して,
$\int_\R g(x)\delta(x-a)\,dx = g(a)$ によって定義されていると考える.}:
\[
E[g(X_n)] = \int_\R g(x)f_n(x)\,dx,
\quad
f_n(x) = \sum_{k=0}^n\binom{n}{k}p^k q^{n-k}\delta(x-k).
\]
このように, 二項分布の確率密度函数 $f_n(x)$ は
デルタ函数(デルタ測度)を使って表わされると考えられ,
通常の函数ではなく超函数(より正確には測度)になってしまう.
特に確率密度函数の収束を通常の函数の各点収束で考えることは
できなくなる.
そのような場合には確率密度函数の各点収束ではなく,
期待値汎函数 $g\mapsto E[g(X)]$ の収束を考えればよい%
\footnote{この型の収束は{\bf 弱収束}と呼ばれる.}.
具体的な議論では, 一般の函数 $g$ に対する $E[g(X)]$ を扱うのではなく,
ある特別な形の函数 $g$ に関する $E[g(X)]$ を扱い,
その特別な場合の計算から一般の場合を導くというようなことがよく行われる.
その典型例が確率変数 $X$ の特性函数 $\varphi_X(t)=E[e^{itX}]$ を扱うことである.
特性函数は $\R$ 上で常に絶対値が $1$ 以下の一様連続函数になる:
\begin{align*}
&
|\varphi_X(t)|=\left|E[e^{itX}]\right|\leqq E\left[|e^{itX}|\right] = E[1]=1,
\\ &
\sup_{t\in\R}|\varphi_X(t+h)-\varphi(t)|
=\sup_{t\in\R}|E[e^{itX}(e^{ith}-1)]|
\leqq E\left[|e^{ihX}-1|\right]
\longrightarrow 0 \quad (h\to 0).
\end{align*}
最後の $0$ への収束ではLebesgueの収束定理を用いた.
函数 $g(x)$ が
\[
g(x) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{itx} \widehat{g}(t)\,dt
\]
と表わされていたとする%
\footnote{たとえば $g(x)$ が急減少函数であれば
急減少函数 $\widehat{g}(t)$ でこのように $g(x)$ を表示できる.}.
このとき, $E[\ ]$ と積分の順序を交換することによって
\[
E[g(X)]
= \frac{1}{2\pi}\int_{-\infty}^\infty \widehat{g}(t) E[e^{itX}]\,dt
= \frac{1}{2\pi}\int_{-\infty}^\infty \widehat{g}(t) \varphi_X(t)\,dt.
\]
この公式より, 確率変数列 $Y_n$ と確率変数 $Y$ について,
特性函数列 $\varphi_{Y_n}$ が特性函数 $\varphi_Y$ に各点収束していれば,
適切なクラス\footnote{たとえば有界な連続函数の集合.}%
に含まれる任意の函数 $g(y)$ に対して $E[g(Y_n)]$ は $E[g(Y)]$ に
収束することを示せる%
\footnote{実際の証明では,
$g(y)$ が急減少函数であるような扱い易い場合に収束を示し,
その極限として $g(t)$ がより広い函数のクラス(例えば有界連続函数の集合)
に含まれる場合の結果を導く.}.
離散型確率変数を含む一般の場合の中心極限定理はこのような形で定式化される.
\begin{remark*}
確率変数 $Y_n$ の特性函数 $\varphi_{Y_n}$ が函数 $\varphi$ に各点収束していても
収束先の函数 $\varphi$ がある確率変数の特性函数になっていない場合には
確率変数 $Y_n$ は確率変数に収束しない.
特性函数列 $\varphi_{Y_n}$ が原点で連続な函数 $\varphi$ に
各点収束するならば, 特性函数 $\varphi$ を持つ確率変数 $Y$ が存在して,
確率変数列 $Y_n$ が $Y$ に弱収束することが知られている\footnote{Bochnerの定理.}.
\qed
\end{remark*}
二項分布の中心極限定理を示そう.
二項分布の特性函数は
\begin{align*}
\varphi_{X_n}(t)
&=E[e^{itX_n}]
=\sum_{k=0}^n e^{itk}\binom{n}{k}p^kq^{n-k}
\\ &
=\sum_{k=0}^n \binom{n}{k}(pe^{it})^nq^{n-k}
=(pe^{it}+q)^n
\end{align*}
となる. 二項分布の平均と分散はそれぞれ $\mu_n=np$ と $\sigma_n^2=npq$
である. ゆえに確率変数
\[
Y_n=\frac{X_n-\mu_n}{\sigma_n}=\frac{X_n-np}{\sqrt{npq)}}
\]
の平均と分散はそれぞれ $0$ と $1$ になり, その特性函数は
\begin{align*}
\varphi_{Y_n}(t)
&
=E\left[e^{itY_n}\right]
=E\left[e^{-itnp/\sqrt{npq}}e^{itX_n/\sqrt{npq}}\right]
\\ &
=e^{-itnp/\sqrt{npq}}\varphi_{X_n}(t/\sqrt{npq})
%\\ &
=e^{-itnp/\sqrt{npq}}\left( pe^{it/\sqrt{npq}}+q \right)^n
\\ &
=\left( pe^{itq/\sqrt{npq}} + qe^{-itp/\sqrt{npq}} \right)^n
\end{align*}
となる%
\footnote{たとえば $p=q=1/2$ のとき $\varphi_{Y_n}(t)=\left( \cos(t/\sqrt{n}) \right)^n$.}.
$X_n$ の特性函数の公式を経由せずに,
$X_n-np=X_n(p+q)-np=qX_n-p(n-X_n)$ を用いて, 直接的に
\begin{align*}
\varphi_{Y_n}(t)
&
=E\left[e^{itY_n}\right]
=E\left[e^{itqX_n/\sqrt{npq}}e^{-itp(n-X_n)/\sqrt{npq}}\right]
\\ &
=\sum_{k=0}^n e^{itqk/\sqrt{npq}}e^{-itp(n-k)/\sqrt{npq}} \binom{n}{k}p^kq^{n-k}
\\ &
=\sum_{k=0}^n \binom{n}{k}
\left(pe^{itq/\sqrt{npq}}\right)^k \left(qe^{-itp/\sqrt{npq}}\right)^{n-k}
\\ &
=\left( pe^{itq/\sqrt{npq}} + qe^{-itp/\sqrt{npq}} \right)^n
\end{align*}
と計算することもできる. これに
\begin{align*}
&
pe^{itq/\sqrt{npq}}
= p + \frac{itpq}{\sqrt{npq}} - \frac{qt^2}{2n} + O\left(\frac{1}{n\sqrt{n}}\right),
\\ &
qe^{-itp/\sqrt{npq}}
= q - \frac{itpq}{\sqrt{npq}} - \frac{pt^2}{2n} + O\left(\frac{1}{n\sqrt{n}}\right)
\end{align*}
を代入すると
\[
\varphi_{Y_n}(t)=\left(1-\frac{t^2}{2n}+O\left(\frac{1}{n\sqrt{n}}\right)\right)^n
\]
なので
\[
\lim_{n\to\infty}\varphi_{Y_n}(t) = e^{-t^2/2}
\]
一方, 標準正規分布する確率変数 $Y$ の特性函数は
\[
\varphi_Y(t)
= E[e^{itY}]
= \int_{-\infty}^\infty e^{ity} \frac{e^{-y^2/2}}{\sqrt{2\pi}}\,dy
= e^{-t^2/2}.
\]
これより, 適切なクラスに含まれる函数%
\footnote{この場合には有界な連続函数
や $a\leqq y\leqq b$ で値が $1$ にそうでないとき $0$ になる函数など.} %
$g(y)$ について
\[
\lim_{n\to\infty} E[g(Y_n)] = E[g(Y)]
\]
となることを示せる. すなわち
\[
\lim_{n\to\infty}
\sum_{k=0}^n
g\left(\frac{k-np}{\sqrt{npq}}\right)
\binom{n}{k}p^k q^{n-p}
=
\int_{-\infty}^\infty g(y) \frac{e^{-y^2/2}}{2\pi}\,dy.
\]
$g(y)$ が $a\leqq y\leqq b$ のとき値が $1$ になり, そうでないとき $0$
になる函数の場合には
\[
\lim_{n\to\infty}
P\left(a\leqq \frac{X_n-np}{\sqrt{npq}}\leqq b\right)
=
\int_a^b \frac{e^{-y^2/2}}{2\pi}\,dy.
\]
以上が二項分布の確率変数 $X_n$ の中心極限定理である.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Laplaceの方法による導出}
\label{sec:Laplace}
前節までに説明したStirlingの公式の証明は
本質的にガンマ函数(ガンマ分布)がGauss積分(正規分布)で近似されることを
用いた証明だと考えられる.
Gauss積分による近似を{\bf Laplaceの方法}と呼ぶことがある.
\subsection{ガンマ函数のGauss積分による近似を使った導出}
ガンマ函数の値をGauss積分で直接近似することによって
Stirlingの公式を示そう.
$\log(e^{-x}x^n)=n\log x-x$ を $x=n$ でTaylor展開すると
\[
n\log x - x
=n\log n-n
-\frac{(x-n)^2}{2n}
+\frac{(x-n)^3}{3n^2}
-\frac{(x-n)^4}{4n^3}
+\cdots
\]
なので, $n$ が大きなとき $n!=\Gamma(n+1)=\int_0^\infty e^{-x}x^n\,dx$ が
\[
\int_{-\infty}^\infty \exp\left(n\log n-n-\frac{(x-n)^2}{2n}\right)\,dx
=n^n e^{-n} \int_{-\infty}^\infty e^{-(x-n)^2/(2n)}\,dx
=n^n e^{-n} \sqrt{2\pi n}
\]
で近似されることがわかる. ゆえに
\[
n!\sim n^n e^{-n} \sqrt{2\pi n} \qquad (n\to\infty).
\]
この近似の様子をscilabで描くことによって作った画像を
\href{http://twilog.org/genkuroki/date-150709}
{ツイッターの過去ログ}で見ることができる.
無料の数値計算ソフトscilabについては
\href{http://twilog.org/genkuroki/search?word=scilab&ao=a}
{関連のツイート}を参照して欲しい.
以上の証明法ではStirlingの公式中の因子 $n^n e^{-n}$, $\sqrt{2\pi n}$ の
それぞれが $g_n(x)=\log(e^{-x}x^n)=n\log x-x$ の $x=n$ における
Taylor展開の定数項と2次の項に由来していることがわかる.
$3$ 次の項は $\int_{-\infty}^\infty y^3 e^{-y^2/\alpha}\,dy=0$
なので寄与しない.
以上の方法を拡張して第1補正項の $1/(12n)$ まで導出してみよう%
\footnote{%
\href{https://www.jstage.jst.go.jp/article/sugaku1947/31/3/31_3_262/_article/references/-char/ja/}
{一松信, Stirlingの公式の第1剰余項までの初等的証明,
数学 Vol.~31 (1979) No.~3, 262--263}
ではWallisの公式の精密化によって第1補正項を得る方法が解説されている.
第1補正項付きのStirling公式の易しい証明については,
\href{https://www.jstage.jst.go.jp/article/sugaku1947/36/2/36_2_175/_article/references/-char/ja/}
{鍋谷清治, 連続変数に対するStirlingの公式の初等的証明,
数学 Vol.~36 (1984) No.~2, 175--178}
という文献がある. 後者の文献の解説を以下では参考にした.}.
準備. ガウス型積分とガンマ函数の関係は以下の通り:
\begin{align*}
\int_{-\infty}^\infty e^{-x^2/2}x^{2k}\,dx
&=2\int_0^\infty e^{-x^2/2} (x^2)^k \,dx
=2\int_0^\infty e^{-t} (2t)^k \sqrt{2}\frac{t^{-1/2}}{2}\,dt
\\ &
=2^k\sqrt{2}\int_0^\infty e^{-t} t^{k-1/2}\,dt
=2^k\sqrt{2}\Gamma(k+1/2)
\\ &
=2^k\sqrt{2}\frac{1\cdot3\cdots(2k-1)}{2^k}\sqrt{\pi}
=1\cdot3\cdots(2k-1)\sqrt{2\pi}.
\end{align*}
たとえば,
\(
\int_{-\infty}^\infty e^{-x^2/2}\,dx
=\int_{-\infty}^\infty e^{-x^2/2}x^2\,dx
=\sqrt{2\pi}
\),
\[
\qquad
\int_{-\infty}^\infty e^{-x^2/2}x^4\,dx = 3\sqrt{2\pi}, \qquad
\int_{-\infty}^\infty e^{-x^2/2}x^6\,dx = 15\sqrt{2\pi}.
\]
これらの公式を以下で使う.
ガンマ函数の積分表示の積分変数 $x$ に $n(1+x/\sqrt{n})$ を代入すると
\begin{align*}
n!
&=\Gamma(n+1)
=\int_0^\infty e^{-x}x^n\,dx
\\ &
=n^n e^{-n}\sqrt{n}
\int_{-\sqrt{n}}^\infty e^{-\sqrt{n}\,x} \left( 1+\frac{x}{\sqrt{n}} \right)^n \,dx
\\ &
\sim n^n e^{-n}\sqrt{n}
\int_{-1}^1 e^{-\sqrt{n}\,x} \left( 1+\frac{x}{\sqrt{n}} \right)^n \,dx
\qquad (n\to\infty).
\end{align*}
被積分函数の対数を $\phi_n(x)$ と書くと:
\begin{align*}
\phi_n(x)
&=n\log\left(1+\frac{x}{\sqrt{n}}\right)-\sqrt{n}\,x
=-\frac{x^2}{2} + \frac{x^3}{3\sqrt{n}}-\frac{x^4}{4n}+ o\left(\frac{1}{n}\right)
\qquad (n\to\infty).
\end{align*}
最後の $o(1/n)$ の部分は $n$ をかけた後に $n\to\infty$ とすると
$|x|\leqq 1$ で $0$ に一様収束する.
ゆえに $|x|\leqq 1$ において一様に
\begin{align*}
e^{-\sqrt{n}\,x} \left( 1+\frac{x}{\sqrt{n}} \right)^n
&=e^{-x^2/2}
\exp\left( \frac{x^3}{3\sqrt{n}}-\frac{x^4}{4n}+ o\left(\frac{1}{n}\right) \right)
\\ &
=e^{-x^2/2}
\left(
1
+\frac{x^3}{3\sqrt{n}}
-\frac{x^4}{4n}
+\frac{1}{2}\left( \frac{x^3}{3\sqrt{n}} \right)^2
+o\left(\frac{1}{n}\right)
\right)
\\ &
=e^{-x^2/2}
\left(
1
+\frac{x^3}{3\sqrt{n}}
-\frac{x^4}{4n}
+\frac{x^6}{18n}
+o\left( \frac{1}{n} \right)
\right).
\end{align*}
$o(1/n)$ の部分に含まれる $n$ の半整数乗分の $1$ の項の係数
は $x$ について奇函数になることに注意せよ.
奇函数と $e^{-x^2/2}$ の積の $-1\leqq x\leqq 1$ での
積分は消えるので, 上で準備しておいた公式によって次が得られる:
\begin{align*}
\int_{-1}^1
e^{-\sqrt{n}\,x} \left( 1+\frac{x}{\sqrt{n}} \right)^n\,dx
& \sim
\int_{-\infty}^{\infty}
e^{-x^2/2}
\left(
1
-\frac{x^4}{4n}
+\frac{x^6}{18n}
\right)
\,dx
+O\left( \frac{1}{n^2} \right)
\\ &
=
\sqrt{2\pi}
-\frac{3\sqrt{2\pi}}{4n}
+\frac{15\sqrt{2\pi}}{18n}
+O\left( \frac{1}{n^2} \right)
\\ &
=\sqrt{2\pi}\left(1 + \frac{1}{12n} + O\left(\frac{1}{n^2}\right) \right).
\end{align*}
ゆえに
\[
n!
=
n^n e^{-n}\sqrt{2\pi n}
\left(1+\frac{1}{12n}+O\left(\frac{1}{n^2}\right)\right)
\qquad(n\to\infty).
\]
これで第1補正項 $1/(12n)$ が得られた%
\footnote{高次の補正項も同様にして得られる.}
第1補正項 $1/(12n)$ は, $n$ が大きなとき,
$n!$ の $n^n e^{-n}\sqrt{2\pi n}$ による近似の誤差は $n$ が
大きなとき $n!$ の値の $12n$ 分の1程度になることを意味している.
\subsection{ガンマ函数のガンマ函数を用いた近似で補正項を計算する方法}
Laplaceの方法によるStirlingの公式の証明とその一般化に関しては
\href{https://www.cs.elte.hu/blobs/diplomamunkak/msc_mat/2012/nemes_gergo.pdf}
{Gerg\"o Nemes, Asymptotic expansions for integrals, 2012, M.~Sc.~Thesis, 40~pages}
が詳しい. 以下で説明する方法の詳細はこの論文の Example 1.2.1 にある.
そこに書いてある方法を使っても,
Stirlingの公式の補正項 $1/(12n)$ を容易に得ることができる.
次の公式を使うことを考える: 任意の $a>0$ ($a=\infty$ を含む)に対して,
\[
\int_0^a e^{-nt} t^{s-1}\,dt
= \frac{1}{n^s}\int_0^{an} e^{-x} x^{s-1}\,dx
\sim
\frac{\Gamma(s)}{n^s}
\qquad (n\to\infty).
\]
$t=x/n$ と積分変数を置換した. この公式を使えば,
\[
\int_0^a e^{-nt} (\alpha_1 t^{s_1-1} + \alpha_2 t^{s_2-1} + \cdots)\,dt
=
\frac{\alpha_1\Gamma(s_1)}{n^{s_1}} + \frac{\alpha_2\Gamma(s_2)}{n^{s_2}} + \cdots
\qquad (n\to\infty)
\]
のような計算が可能になる.
これを用いてStirlingの公式の最初の補正項 $1/(12n)$を得てみよう.
函数 $f(x)$ を
\[
f(x) = x-\log(1+x) \qquad (x>-1)
\]
と定め, 積分変数を $y=n(1+x)$ と置換することによって,
\begin{align*}
n!
&= \Gamma(n+1)
= \int_0^\infty e^{-y} y^n\,dy
\\ &
= \int_{-1}^\infty e^{-n-nx}n^n(1+x)^n n\,dx
= n^{n+1}e^{-n}\int_{-1}^\infty e^{-nf(x)}\,dx.
\end{align*}
さらに積分を $x>0$ と $x<0$ に分けることによって
\[
\frac{n!}{n^{n+1}e^{-n}}
= \int_0^\infty e^{-nf(x)}\,dx + \int_0^1 e^{-nf(-x)}\,dx.
\]
もしも $f(x)=t$ もしくは $f(-x)=t$ と
積分変数を置換できれば, 積分の形が上で説明した形に
なりそうである.
実際にそれが可能なことを確認しよう.
$f(x)=x-\log(1+x)$ の導函数は
\[
f'(x) = 1 - \frac{1}{1+x} = \frac{x}{1+x}
\]
なので $x>0$ で $f'(x)>0$ となり, $-1<x<0$ で $f'(x)<0$ となる.
$f(x)$ は $x=0$ で最低値 $f(0)=0$ を持ち, $x>0$ で単調増加し,
$x<0$ で単調減少する.
ゆえに $x>0$ と $-1<x<0$ のそれぞれで $t=f(x)$ は逆函数 $x=x(t)$ を持つ.
$x=x(t)$ の原点近くでの振る舞いを調べるために,
\[
x = \alpha t^{1/2} + \beta t + \gamma t^{3/2} + \cdots
\]
とおいて
\[
t = f(x) = x - \log(1+x)
= \frac{x^2}{2} - \frac{x^3}{3} + \frac{x^4}{4} - \cdots
\]
に代入して%
\footnote{$|x|<1$ における
Taylor展開 $\log(1+x)=x-x^2/2+x^3/3-x^4/4+\cdots$ は非常によく使われる.},
$\alpha,\beta,\gamma$ を求めてみよう.
実際に代入すると,
\[
t
= \frac{\alpha^2}{2} t
+ \left( \alpha\beta + \frac{\alpha^3}{3} \right) t^{3/2}
+ \left( \alpha\gamma + \frac{\beta^2}{2} + \alpha^2\beta + \alpha^4 \right) t^2
+ \cdots.
\]
両辺を比較して $\alpha,\beta,\gamma$ を求めると,
\[
\alpha = \sqrt{2}, \qquad
\beta = \frac{2}{3}, \qquad
\gamma = \frac{\sqrt{2}}{18}
\]
を得る. すなわち
\[
x = \sqrt{2}\,t^{1/2} + \frac{2}{3}t + \frac{\sqrt{2}}{18}t^{3/2} + \cdots
\]
とおくと $f(x)=t$ となる.
$x>0$ ではこの表示をそのまま用いる.
$x<0$ では $t^{1/2}$ を $-t^{1/2}$ で置き換え,
さらに $x$ を $-x$ で置き換えた表示を用いる.
すなわち
\[
x = \sqrt{2}\,t^{1/2} - \frac{2}{3}t + \frac{\sqrt{2}}{18}t^{3/2} - \cdots
\]
とおくと $f(-x)=t$ となる. 以上のそれぞれの場合において,
おいて
\[
\frac{dx}{dt}
=
\frac{\sqrt{2}}{2}\,t^{1/2-1}
\pm \frac{2}{3}t^{1-1}
+ \frac{\sqrt{2}}{12}t^{3/2-1}
\pm \cdots
\]
以上の2つの場合で $t$ の整数次の項には $-1$ 倍の違いがある.
準備が整った.
$f(x)=t$ と積分変数を置換することによって, $n\to\infty$ のとき
\begin{align*}
\int_0^\infty e^{-nf(x)}\,dx
&=\int_0^\infty e^{-nt}\frac{dx}{dt}\,dt
\\ &
= \int_0^\infty e^{-nt}
\left(
\frac{\sqrt{2}}{2}\,t^{1/2-1}
+ \frac{2}{3}t^{1-1}
+ \frac{\sqrt{2}}{12}t^{3/2-1}
+ \cdots
\right)
\,dt
\\ &
=
\frac{\sqrt{2}\Gamma(1/2)}{2n^{1/2}}
+\frac{2\Gamma(1)}{3}
+\frac{\sqrt{2}\Gamma(3/2)}{12n^{3/2}}
+\cdots
\\ &
=
\frac{\sqrt{2\pi}}{2n^{1/2}}
+\frac{2}{3n}
+\frac{\sqrt{2\pi}}{24n^{3/2}}
+\cdots
\end{align*}
となる. 最後に $\Gamma(1/2)=\sqrt{\pi}$, $\Gamma(1)=1$,
$\Gamma(3/2)=(1/2)\Gamma(1/2)=\sqrt{\pi}/2$ を使った.
もう一方の積分についても,
$f(-x)=t$ と積分変数を置換することによって同様にして,
$n\to\infty$ のとき
\[
\int_0^1 e^{-nf(-x)}\,dx
=
\frac{\sqrt{2\pi}}{2n^{1/2}}
-\frac{2}{3n}
+\frac{\sqrt{2\pi}}{24n^{3/2}}
-\cdots
\]
となる. 以上の2つを足し合わせると,
$n$ の整数乗分の1の項がすべてキャンセルし,
次が得られる:
\[
\frac{n!}{n^{n+1}e^{-n}}
=
\frac{\sqrt{2\pi}}{n^{1/2}}
+\frac{\sqrt{2\pi}}{12n^{3/2}}
+O\left(\frac{1}{n^{5/2}}\right)
\qquad
(n\to\infty).
\]
これは次のように書き直される:
\[
n!
= n^n e^{-n}\sqrt{2\pi n}
\left(1 + \frac{1}{12n} + O\left(\frac{1}{n^2}\right) \right)
\qquad (n\to\infty).
\]
これで第1の補正項 $1/(12n)$ もLaplaceの方法で求められることがわかった.
第2の補正項以降も同様にして求められる.
\begin{remark*}
以上の計算において ``$+\cdots$'' と書いた部分については注意が必要である.
そのことは以下の計算例を見ればわかる.
\[
\frac{1}{1+t} = 1-t+t^2-t^3+\cdots+(-1)^{k-1}t^{k-1}+(-1)^k\frac{t^k}{1+t}
\]
なので
\begin{align*}
\int_0^\infty \frac{e^{-nt}\,dt}{1+t}
&
=\frac{\Gamma(1)}{n}
-\frac{\Gamma(2)}{n^2}
%+\frac{\Gamma(3)}{n^3}
%-\frac{\Gamma(4)}{n^4}
+\cdots
+(-1)^{k-1}\frac{\Gamma(k)}{n^k}
+(-1)^k\int_0^\infty\frac{e^{-nt}t^k\,dt}{1+t}
\\ &
=\frac{0!}{n}
-\frac{1!}{n^2}
%+\frac{2!}{n^3}
%-\frac{3!}{n^4}
+\cdots
+(-1)^{k-1}\frac{(k-1)!}{n^k}
+(-1)^k\int_0^\infty\frac{e^{-nt}t^k\,dt}{1+t}.
\end{align*}
上の議論ではこのような和の途中から先を ``$+\cdots$'' と略記して来た.
すぐ上の式は正しい公式だが,
\[
\int_0^\infty \frac{e^{-nt}\,dt}{1+t}
=\sum_{k=1}^\infty (-1)^{k-1}\frac{(k-1)!}{n^k}
\]
は通常の意味で正しい公式ではない.
なぜならば右辺はどんなに大きな $n$ に対しても収束しないからである.
``$+\cdots$'' の部分は``無限和''を意味すると解釈するのではなく,
``有限和+剰余項''を意味すると解釈しておかなければいけない.
\qed
\end{remark*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{対数版の易しいStirlingの公式}
Stirlingの公式は次と同値である:
\[
\log n! - (n+1/2)\log n + n \longrightarrow \log\sqrt{2\pi}
\qquad (n\to\infty).
\]
これより, 次の弱い結果が導かれる:
\[
\log n! = n\log n - n + o(n)
\qquad (n\to\infty).
\]
ここで $o(n)$ は $n$ で割った後に $n\to\infty$ と
すると $0$ に収束する量を意味する.
これをこの節では{\bf 対数版の易しい Stirling の公式}と呼ぶことにする.
この公式であれば以下で説明するように初等的に証明することができる%
\footnote{以下の証明を見ればわかるように $o(n)$ の部分は $O(\log n)$
であることも証明できる. ここで $O(\log n)$ は $\log n$ で割った後に
有界になる量を意味している.}.
\subsection{対数版の易しい Stirling の公式の易しい証明}
\label{sec:easy}
単調増加函数 $f(x)$ について $f(k)\leqq\int_k^{k+1} f(x)\,dx\leqq f(k+1)$
が成立しているので, $f(1)\geqq 0$ を満たす単調増加函数 $f(x)$ について,
\[
f(1)+f(2)+\cdots+f(n-1)\leqq \int_1^n f(x)\,dx \leqq f(1)+f(2)+\cdots+f(n).
\]
ゆえに
\[
\int_1^n f(x)\,dx\leqq f(1)+f(2)+\cdots+f(n)\leqq \int_1^n f(x)\,dx + f(n).
\]
これを $f(x)=\log x$ に適用すると
\[
\int_1^n \log x\,dx = [x\log x-x]_1^n = n\log n - n + 1, \qquad
\log 1+\log 2+\cdots+\log n=\log n!
\]
なので
\[
n\log n - n + 1 \leqq \log n! \leqq n\log x - n + 1 + \log n.
\]
すなわち
\[
1 \leqq \log n! - n\log n + n \leqq 1+\log n.
\]
したがって
\[
\log n!=n\log n-n+O(\log n)=n\log n-n+o(n)
\qquad (n\to\infty).
\]
ここで $O(\log n)$ は $\log n$ で割ると有界になるような量を意味している.
\subsection{大学入試問題への応用例}
対数版の易しいStirlingの公式を使うと,
$an$ 個から $bn$ 個取る組み合わせの数(二項係数)の対数は
\begin{align*}
\log\binom{an}{bn}
&=\log(an)! - \log(bn)! -\log((a-b)n)!
\\ &
=an\log a+an\log n - an + o(n)
\\ &
-bn\log b-bn\log n + bn + o(n)
\\ &
-(a-b)n\log(a-b)-(a-b)n\log n + (a-b)n
+o(n)
\\ &
= n\log\frac{a^a}{b^b(a-b)^{a-b}} + o(n).
\end{align*}
となる. ゆえに
\[
\log\binom{an}{bn}^{1/n}
\longrightarrow \log\frac{a^b}{b^b(a-b)^{a-b}}
\qquad (n\to\infty).
\]
すなわち
\[
\lim_{n\to\infty}\binom{an}{bn}^{1/n}
=\lim_{n\to\infty}\left(\frac{(an)!}{(bn)!((a-b)n)!}\right)^{1/n}
=\frac{a^a}{b^b(a-b)^{a-b}}.
\]
要するに $an$ 個から $bn$ 個取る組み合わせの数の $n$ 乗根の $n\to\infty$
での極限は二項係数部分の式の分子分母の $(kn)!$ を $k^k$ で置き換えれば得られる.
この結果を使えば次の
\href{https://www.google.co.jp/search?q=\%E6\%9D\%B1\%E5\%B7\%A5\%E5\%A4\%A7\%E5\%85\%A5\%E8\%A9\%A6\%E5\%95\%8F\%E9\%A1\%8C+1988+\%E6\%95\%B0\%E5\%AD\%A6}
{東工大の1988年の数学の入試問題}を暗算で解くことができる:
\[
\lim_{n\to\infty}\left(\frac{{}_{3n}C_n}{{}_{2n}C_n}\right)^{1/n}\ \text{を求めよ.}
\]
この極限の値は
\[
\frac{3^3/(1^12^2)}{2^2/(1^11^1)}=\frac{3^3}{2^4}=\frac{27}{16}.
\]
入試問題を作った人は, まずStirlingの公式を使うと容易に解ける問題を考え,
その後に高校数学の範囲内でも解けることを確認したのだと思われる.
\begin{remark*}
上で示したことより,
\[
\lim_{n\to\infty}\binom{2n}{n}^{1/n}=\frac{2^2}{1^11^1}=2^2.
\]
これは次を意味している($o(n)$ は $n$ で割ると $n\to\infty$ で $0$ に収束する量):
\[
\binom{2n}{n}=2^{2n} e^{o(n)}
\qquad (n\to\infty).
\]
Wallisの公式(\secref{sec:Wallis})
\[
\binom{2n}{n}\sim\frac{2^{2n}}{\sqrt{\pi n}}
\qquad (n\to\infty)
\]
はその精密化になっている.
\qed
\end{remark*}
\begin{remark*}
\href{http://d.hatena.ne.jp/gould2007/touch/20071127}
{東工大では1968年にも次の問題を出しているようだ}:
\[
\lim_{n\to\infty}\frac{1}{n}\sqrt[n]{{}_{2n}P_n}\ \text{を求めよ.}
\qquad(\text{答えは $2^2 e^{-1}$}.)
\]
この問題も明らかに元ネタはStirlingの公式である. より一般に次を示せる:
\[
\lim_{n\to\infty} \frac{((an)!)^{1/n}}{n^a}
%=\lim_{n\to\infty}\left( (an)! n^{-an} \right)^{1/n}
= a^a e^{-a}.
\]
なぜならば
\begin{align*}
\log\frac{((an)!)^{1/n}}{n^a}
&=
\frac{1}{n}\log(an)!-a\log n
\\ &
=\frac{1}{n}(an\log a + an\log n - an + o(n)) - a\log n
\\ &
=a\log a - a + o(1)
\\ &
=\log(a^a e^{-a})+o(1).
\end{align*}
やはりStirlingの公式を使えば容易に示せる結果を
高校数学の範囲内で解けるように調節して入試問題にしているのだと思われる.
\qed
\end{remark*}
\subsection{対数版の易しいStirlingの公式の改良}
少し工夫すると次を示せる. ある定数 $c$ が存在して,
\[
\log n! = n \log n + \frac{1}{2}\log n - n + c + o(1)
\qquad (n\to\infty).
\]
以下ではこの公式を証明しよう%
\footnote{定数 $c$ が $\log\sqrt{2\pi}$ であることは既知であるが,
Wallisの公式を使えば $e^c=\sqrt{2\pi}$ であることを示せる.}.
\secref{sec:easy}で証明した対数版の易しいStirlingの公式と
上の公式の違いは $(1/2)\log n$ の項と定数項 $c$ を付け加えて
改良しているところである.
それらの項を出すアイデアは次の通り.
$\int_1^n\log x\,dx=[x\log x-x]_1^n=n\log n-n+1$ を $k=1,2,3,\ldots,n-1$ に対する
長方形 $[k-1/2,k+1/2]\times[0,\log k]$ の面積の総和
と長方形 $[n-1/2,n]\times[0,\log n]$ の面積の
和 $\log(n-1)!+(1/2)\log n=\log n!-(1/2)\log n$ で近似すれば,
自然に $(1/2)\log n$ の項が得られる.
さらに, それらの長方形の和集合と
領域 $\{\,(x,y)\mid 1\leqq x\leqq n,\ 0\leqq y\leqq\log x\,\}$
の違いを注意深く分析すれば,
$\int_1^n\log x\,dx$ と長方形の面積の総和の差が $n\to\infty$ で
ある定数に収束することがわかり, 定数項も得られる.
$\log x$ は単調増加函数なので正の実数 $\alpha_k, \beta_k$ を
\[
\alpha_k=\int_k^{k+1/2}\log x\,dx-\frac{1}{2}\log k, \qquad
\beta_k =\frac{1}{2}\log k-\int_{k-1/2}^k\log x\,dx
\]
と定めることができる. このとき,
\begin{align*}
&
\log n! - \frac{1}{2}\log n - \int_1^n \log x\,dx
=
\sum_{k=1}^{n-1}\log k+\frac{1}{2}\log n - \int_1^n \log x\,dx
\\ & \qquad\qquad
= -\alpha_1+\beta_2-\alpha_2+\beta_3-\cdots+\beta_{n-1}-\alpha_{n-1}+\beta_n.
\end{align*}
この交代和が $n\to\infty$ で収束することを示したい.
$\log x$ が上に凸であることより,
数列 $\alpha_1,\beta_2,\alpha_2,\beta_3,\alpha_3,\ldots$ が
単調減少することがわかり,
$\log x$ の導函数が $x\to\infty$ で $0$ に収束することより,
その数列は $0$ に収束することもわかる.
ゆえに上の交代和は $n\to\infty$ で収束する%
\footnote{$0$ 以上の実数で構成された $0$ に収束する単調減少列 $a_n$ が
定める交代級数 $\sum_{k=1}^\infty (-1)^{k-1}a_k$ は収束する.
(絶対収束するとは限らない.)}.
その収束先を $a$ と書き, $c=1+a$ とおくと, $n\to\infty$ のとき
\[
\log n!
= \frac{1}{2}\log n + \int_1^n\log x\,dx + a + o(1)
= n\log n +\frac{1}{2}\log n - n + c + o(1).
\]
$c=\log\sqrt{2\pi}$ であることをWallisの公式(\secref{sec:Wallis})
を使って証明しよう.
$n!=n^{n+1/2}e^{-n}e^ce^{o(1)}$ をWallisの公式
\[
\sqrt{\pi}=\lim_{n\to\infty}\frac{2^{2n}(n!)^2}{(2n)!\sqrt{n}}
\]
に代入すると,
\[
\sqrt{\pi}
=\lim_{n\to\infty}
\frac{2^{2n}n^{2n+1}e^{-2n}e^{2c}}{2^{2n+1/2}n^{2n+1}e^{-2n}e^c}
=\frac{e^c}{\sqrt{2}}.
\]
ゆえに $e^c=\sqrt{2\pi}$ である.
これでWallisの公式を使えば,
対数版の易しいStirlingの公式を改良することによって,
通常のStirlingの公式 $n!\sim n^n e^{-n}\sqrt{2\pi n}$ が
得られることがわかった.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{付録: Fourierの反転公式}
\label{sec:Fourier}
厳密な証明をするつもりはないが,
Fourierの反転公式の証明の概略について説明しよう.
函数 $f(x)$ に対してその逆Fourier変換 $F(p)$ を
\[
F(p) = \int_{-\infty}^\infty e^{ipx} f(x)\,dx
\]
と定める. このとき函数 $f$ について適切な条件を仮定しておくと,
それに応じた適切な意味で
\[
f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx} F(p)\,dp
\]
が成立する. これをFourierの反転公式と呼ぶ.
\subsection{Gauss分布の場合}
$a>0$ であるとし,
\[
f(x)=e^{-x^2/(2a)}
\]
とおき, $F(p)$ はその逆Fourier変換であるとする. このとき
\[
F(p)
=\int_{-\infty}^\infty e^{ipx} e^{-x^2/(2a)}\, dx
=e^{-p^2/(2a^{-1})}\sqrt{2a\pi}
\]
が容易に得られる%
\footnote{Cauchyの積分定理を使う方法,
$e^{ipx}$ のTaylor展開を代入して項別積分する方法,
左辺と右辺が同じ微分方程式を満たしていることを使う方法
など複数の方法で容易に計算可能である.}. %
この公式で $x$, $a$ のそれぞれと $p$, $a^{-1}$ の立場を
交換することによって
\[
\int_{-\infty}^\infty e^{-ipx} e^{-p^2/(2a^{-1})}\, dp
=e^{-x^2/(2a)}\sqrt{2a^{-1}\pi}
\]
が得られる. 以上の2つの結果を合わせると,
\[
f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx} F(p)\,dp
\]
が得られる. すなわち $f(x)=e^{-x^2/(2a)}$ については
Fourierの反転公式が成立している.
一般に $f(x)$ についてFourierの反転公式が成立していれば
$f(x)$ を平行移動して得られる函数 $f(x-\mu)$ についても
Fourierの反転公式が成立していることが容易に示される.
実際, $F(p)$ を $f(x)$ の逆Fourier変換とすると,
$f(x-\mu)$ の逆Fourier変換は
\[
\int_{-\infty}^\infty e^{ipx} f(x-\mu)\,dx
=\int_{-\infty}^\infty e^{ip(x'+\mu)} f(x')\,dx'
=e^{ip\mu}F(p)
\]
になり,
\[
\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}e^{ip\mu}F(p)\,dp
=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ip(x-\mu)}F(p)\,dp
=f(x-\mu).
\]
以上によって, $f(x-\mu)=e^{-(x-\mu)^2/(2a)}$ についても
Fourierの反転公式が成立することがわかった.
逆Fourier変換およびFourier変換の線形性より,
$f(x-\mu)=e^{-(x-\mu)^2/(2a)}$ の形の函数の線形和についても
Fourierの反転公式が成立していることがわかる%
\footnote{``任意の函数''はそのような線形和の``極限''で表わされる.
したがって, Fourierの反転公式の証明の本質的部分はこれで終了している
とみなせる.}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{一般の場合}
\label{sec:generalcase}
$a>0$ に対して函数 $\rho_a(x)$ を
\[
\rho_a(x) = \frac{1}{\sqrt{2\pi a}}e^{-x^2/(2a)}
\]
と定める. これは $\rho_a(x)>0$ と $\int_{-\infty}^\infty \rho_a(x)\,dx=1$
を満たしている. そして前節の結果によって, $\rho_a(x-\mu)$ は
Fourierの反転公式を満たしている.
函数 $f(x)$ に対して函数 $f_a(x)$ を
$\rho_a$ との畳み込み積によって函数 $f_a(x)$ を定める:
\[
f_a(x) = \int_{-\infty}^\infty \rho_a(x-y) f(y)\,dy.
\]
このとき $f_a(x)$ についてはFourierの反転公式が成立している%
\footnote{$f_a(x)$ はFourierの反転公式が成立している函数
$\rho_a(x-\mu)$ の重み $f(\mu)$ での重ね合わせなので,
これはほとんど明らかである.}.
実際, $f_a(x)$ の逆Fourier変換 $F_a(p)$ と書くと,
\begin{align*}
F_a(p)
&= \int_{-\infty}^\infty e^{ipx} f_a(x)\,dx
= \int_{-\infty}^\infty
\left( \int_{-\infty}^\infty e^{ipx} \rho_a(x-y)\,dx \right)
f(y)\,dy
\end{align*}
なので
\begin{align*}
\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}F_a(p)\,dp
&=
\int_{-\infty}^\infty
\left(
\frac{1}{2\pi}
\int_{-\infty}^\infty
e^{-ipx}
\left(
\int_{-\infty}^\infty
e^{ipx'}\rho_a(x'-y)\,dx'
\right)
\,dp
\right)
f(y)\,dy
\\ &
=
\int_{-\infty}^\infty \rho_a(x-y) f(y) \,dy
=
f_a(x).
\end{align*}
2つ目の等号で $\rho_a(x-\mu)$ について
Fourierの反転公式が成立することを使った.
さらに
\[
\int_{-\infty}^\infty e^{ipx} \rho_a(x-y)\,dx
=e^{ipy}e^{-ap^2/2}
\]
なので
\[
F_a(p)=\int_{-\infty}^\infty e^{ipy}e^{-ap^2/2}f(y)\,dy=e^{-ap^2/2}F(p)
\]
となる%
\footnote{これは畳み込み積の逆Fourier変換が
逆Fourier変換の積に等しいことの特殊な場合にすぎない.}.
ゆえに
\[
\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}F_a(p)\,dp
=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}e^{-ap^2/2}F(p)\,dp.
\]
したがって
\[
\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}e^{-ap^2/2}F(p)\,dp
= \int_{-\infty}^\infty \rho_a(x-y)f(y)\,dy
= f_a(x).
\]
もしも $F(p)$ が可積分ならば, Lebesgueの収束定理より, 左辺について
\[
\lim_{a\to 0}\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}e^{-ap^2/2}F(p)\,dp
=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}F(p)\,dp
\]
が言える.
あとは, 函数 $f(x)$ について適切な条件を仮定したとき,
$a\to 0$ のとき函数 $f_a(x)$ が
適切な意味で函数 $f(x)$ に収束することを示せれば,
$f(x)$ 自身が適切な意味でFourierの反転公式を満たすことがわかる%
\footnote{$\rho_a(x)$ の $a\to 0$ での様子のグラフを描けば,
$\rho_a(x)$ がDiracのデルタ函数(超函数)に``収束''しているように
見えることから, これもほとんど明らかだと言える.}.
たとえば, $f$ は有界かつ点 $x$ で連続だと仮定する.
ある $M>0$ が存在して $|f(y)-f(x)|\leqq M$ ($y\in\R$)となる.
任意に $\eps>0$ を取る.
ある $\delta >0$ が存在して
$|y-x|\leqq\delta$ ならば $|f(y)-f(x)|\leqq\eps/2$ となる.
函数 $\rho_a$ の定義より,
$a>0$ を十分小さくすると $\int_{|y-x|>\delta}\rho_a(x-y)\,dy\leqq\eps/(2M)$
となることもわかる. 以上の状況のもとで
\begin{align*}
|f_a(x)-f(x)|
&=
\left|
\int_{-\infty}^\infty \rho_a(x-y)(f(y)-f(x))\,dy
\right|
\\ &
\leqq
\int_{-\infty}^\infty \rho_a(x-y)|f(y)-f(x)|\,dy
\\ &
\leqq
\int_{|y-x|\leqq\delta} \rho_a(x-y)\frac{\eps}{2}\,dy
+
\int_{|y-x|>\delta} \rho_a(x-y)M\,dy
\\ &
\leqq \frac{\eps}{2}+\frac{\eps}{2M}M
=\eps.
\end{align*}
これで $\lim_{a\to 0}f_a(x)=f(x)$ が示された.
筆者は実解析一般については次の教科書をおすすめする.
\begin{center}
\href{http://www.amazon.co.jp/dp/4000054449}{%
猪狩惺,
実解析入門,
岩波書店 (1996), xii+324頁,
定価3,800円}.
\end{center}
筆者は学生時代に猪狩惺先生の授業で
Lebesgue積分論やFourier解析について勉強した.
信じられないほどクリアな講義であり,
数学の分野の中で実解析が最もクリアな分野なのではないかと思えて来るほどだった.
上の教科書が2016年5月3日現在品切れ中であり,
プレミア価格のついた中古本しか手に入らないことはとても残念なことである.
\subsection{Riemann-Lebesgueの定理}
$f(x)$ は $\R$ 上の可積分函数%
\footnote{$\R$ 上の可測函数で $\int_\R |f(x)|\,dx<\infty$ を満たすものを
$\R$ 上の可積分函数と呼ぶ. 可測函数の定義を知らない人は以下のように考えてよい.
区間 $I=[a,b]$ に対して $I$ 上で $1$ になり $I$ の外で $0$ になる
函数を $1_I$ と書く. 数 $\alpha_i$ と区間 $I_i$ たちによって
$\sum_{i=1}^n \alpha_i 1_{I_i}$ と表される函数は{\bf 階段函数}と呼ばれる.
階段函数の全体は和とスカラー倍で閉じており, 自然にベクトル空間をなす.
階段函数 $f=\sum_{i=1}^n \alpha_i 1_{I_i}$, $I_i=[a_i,b_i]$, $a_i<b_i$ の
積分が $\int_\R f(x)\,dx=\sum_{i=1}^n\alpha_i(b_i-a_i)$ と定義することができる.
階段函数列 $f_n(x)$ は
$\int_\R|f_m(x)-f_n(x)|\,dx\to 0$ ($m,n\to\infty$) を満たおり,
ほとんどすべての $x\in\R$ について $f_n(x)$ は収束していると仮定する.
(前者の仮定からほとんどいたる所収束する部分列を取れることを示せる.)
このとき $f(x)=\lim_{n\to\infty}f_n(x)$ で函数 $f(x)$ が定まる
(収束しない $x$ における $f$ の値は任意に決めておく).
このとき \(
\left|\int_\R f_m(x)\,dx - \int_\R f_n(x)\,dx\right|
\leqq \int_\R |f_m(x)-f_n(x)|\,dx
\to 0
\) ($m,n\to\infty$)なので $\int_\R f_n(x)\,dx$ は $n\to\infty$ で
収束する. その収束先の値を $\int_\R f(x)\,dx$ と書く.
このような函数 $f(x)$ を可積分函数と呼んでよい.
さらにそのとき \(
\left|\int_\R |f_m(x)|\,dx - \int_\R |f_n(x)|\,dx\right|
\leqq \int_\R |f_m(x)-f_n(x)|\,dx
\to 0
\) ($m,n\to\infty$)でもあるので,
$\int_\R|f_n(x)|\,dx$ は有限の値に収束し,
$\int_\R|f(x)|\,dx<\infty$ も成立している.
}
であるとする.
このとき, その Fourier変換 $\widehat{f}(p)=\int_{-\infty}^\infty e^{-ipx}f(x)\,dx$ は
連続函数になり, $|p|\to\infty$ で $0$ に収束する. 特に
\[
\lim_{|p|\to\infty} \int_{-\infty}^\infty e^{-ipx}f(x)\,dx=0.
\]
これは{\bf Riemann-Lebesgueの定理}(リーマン・ルベーグの定理)と呼ばれている.
$\hat{f}(p)$ の連続性はLebesgueの収束定理%
\footnote{Lebesgueの収束定理とは次の結果のことである.
$f_n$ はほとんどいたる所収束する可積分函数列であり,
ある可積分函数 $\varphi\geqq 0$ で $|f_n|\leqq\varphi$ を満たすものが
存在するとき, 積分 $\int_\R f_n(x)\,dx$ は $n\to\infty$ で収束する.
この定理は非常に便利なので空気のごとく使われる.}によって示される.
実際, $|e^{ihx}-1||f(x)|\leqq 2|f(x)|$ でかつ $|f(x)|$ は可積分なので,
\[
|\widehat{f}(x+h)-\widehat{f}(x)|
\leqq\int_\R|e^{ihx}-1||f(x)|\,dx
\longrightarrow 0
\qquad(h\to 0).
\]
これで $\hat{f}$ の連続性が示された.
Riemann-Lebesgueの定理の証明は可積分函数が階段函数列で $L^1$ 近似
されることからただちに得られる. 区間 $I=[a,b]$ 上で $1$ になり,
その外で $0$ になる函数を $1_I$ と書くと,
\[
\widehat{1_I}(p)
= \int_a^b e^{-ipx}\,dx
= \frac{e^{-ipb}-e^{-ipa}}{-ip}
\]
なので, $\widehat{1_I}(p)\to 0$ ($|p|\to\infty$).
一般の可積分函数に関する結果はこれよりしたがう.
\subsection{Fourier変換の部分和の収束}
\label{sec:Ftransf-N}
$N>0$ とする.
$\R$ 上の可積分函数 $f$ の
Fourier変換 $\widehat{f}(p)=\int_{-\infty}^\infty e^{-ipy}f(y)\,dx$ に対して,
\[
s_N(f)(x) = \frac{1}{2\pi}\int_{-N}^N e^{ipx} \widehat{f}(p)\,dp
\]
をFourier変換の $N$ 部分和と呼ぶ. $N$ 部分和は次のように変形される:
\begin{align*}
s_N(f)(x)
&=\int_{-\infty}^\infty
\left(\frac{1}{2\pi}\int_{-N}^N e^{ip(x-y)} \,dp\right) f(y)\,dy
\\ &
=\int_{-\infty}^\infty
\frac{e^{iN(x-y)}-e^{e^{-iN(x-y)}}}{2\pi i(x-y)} f(y)\,dy
\\ &
=\int_{-\infty}^\infty
\frac{\sin(N(x-y))}{\pi(x-y)} f(y)\,dy.
\\ &
=\int_0^\infty
\frac{\sin(Ny)}{\pi y} (f(x+y)+f(x-y))\,dy
\\ &
=\frac{1}{\pi}\int_0^\infty
\sin(Ny) \frac{f(x+y)+f(x-y)}{y} \,dy.
\end{align*}
4つ目の等号で $y$ を $x+y$ でおきかえ, $\sin(Ny)/y$ が偶函数であることを
使った.
$\delta>0$ を任意に取る.
$y\geqq \delta$ で $(f(x+y)+f(x-y))/y$ は可積分である.
ゆえに Riemann-Lebesgue の定理より,
\[
\lim_{N\to\infty}
\int_\delta^\infty
\sin(Ny) \frac{f(x+y)+f(x-y)}{y} \,dy
= 0.
\]
したがって $N$ 部分和 $s_N(f)(x)$ が $N\to\infty$ で収束することと,
\[
\frac{1}{\pi}\int_0^\delta
\sin(Ny) \frac{f(x+y)+f(x-y)}{y} \,dy
\]
が $N\to\infty$ で収束することは同値になり,
それらが収束するときそれらの値は一致する.
以上の結果を{\bf Riemannの局所性定理}と呼ぶ.
以上の結果を $f(x)=e^{-x^2/2}$ の場合に適用することによって
{\bf Dirichlet積分}(ディリクレ積分)の公式
\[
\lim_{R\to\infty}\int_0^R \frac{\sin x}{x}\,dx = \frac{\pi}{2}
\]
を証明できる. $f(x)=e^{-x^2/2}$ とおく.
このとき, \secref{sec:Gauss-Fourier}での計算より,
$\widehat{f}(p)=e^{-p^2/2}\sqrt{2\pi}$ でかつ
\[
\lim_{N\to\infty}s_N(f)(x)
= \frac{1}{2\pi}\int_{-\infty}^\infty e^{ipx}\widehat{f}(p)\,dp
= f(x).
\]
ゆえに, Riemannの局所性定理を $x=0$ の場合に適用すると,
任意の $\delta>0$ について
\[
\lim_{N\to\infty}s_N(f)(x)
=\lim_{N\to\infty}
\frac{1}{\pi}\int_0^\delta \sin(Ny)\frac{2e^{-y^2/2}}{y}\,dy
=e^{-0^2/2}=1.
\]
ゆえに
\[
\lim_{N\to\infty}
\left(
\int_0^\delta \frac{\sin(Ny)}{y}\,dy
+ \int_0^\delta \sin(Ny)\frac{e^{-y^2/2}-1}{y}\,dy
\right)
=\frac{\pi}{2}.
\]
左辺の後者の積分の極限はRiemann-Lebesgueの定理より $0$ に収束する.
したがって
\[
\lim_{N\to\infty}\int_0^\delta \frac{\sin(Ny)}{y}\,dy = \frac{\pi}{2}.
\]
さらに $y=x/N$ と積分変数を変換することによって,
\[
\frac{\pi}{2}
=\lim_{N\to\infty}\int_0^{N\delta} \frac{\sin x}{x}\,dx
= \lim_{R\to\infty}\int_0^R \frac{\sin x}{x}\,dx.
\]
このようにDirichlet積分の公式はRiemannの局所性定理とRiemann-Lebesgueの定理
と $e^{-x^2/2}$ のFourier変換の計算から得られる%
\footnote{複素解析を使った証明もある.}.
Dirichlet積分の公式で積分変数 $x$ を $Nx$ で置換することによって
\[
\lim_{R\to\infty}\int_0^R \frac{\sin(Nx)}{x}\,dx = \frac{\pi}{2}.
\]
という公式が得られる.
$\R$ 上の可積分函数 $f$ と $x\in\R$ に対して,
ある $\delta>0$ が存在して
\[
\frac{(f(x+y)+f(x-y))/2-f(x)}{y}
\]
が $0<y<\delta$ で可積分になるならば%
\footnote{この条件は{\bf Diniの条件}と呼ばれている.},
Fourier変換の $N$ 部分和の $x$ における値は $f(x)$ に収束する:
\[
\lim_{N\to\infty} s_N(f)(x)=f(x).
\]
この事実は上で述べたことを合わせると容易に導かれる.
Riemannの局所性定理より, 任意の $\delta>0$ について,
$N\to\infty$ のとき
\[
s_N(f)(x)
=\frac{1}{\pi}\int_0^\delta \sin(Nx)\frac{f(x+y)+f(x-y)}{y}\,dy+o(1).
\]
Dirichlet積分の公式の証明より, $N\to\infty$ のとき
\[
f(x)
= \lim_{N\to\infty} \frac{2}{\pi}\int_0^\delta \frac{\sin(Ny)}{y}\,dy\,f(x)
= \frac{2}{\pi}\int_0^\delta \sin(Ny) \frac{f(x)}{y}\,dy + o(1).
\]
ゆえに
\[
s_N(f)(x)-f(x)
=\frac{2}{\pi} \int_0^\delta
\sin(Ny)\frac{(f(x+y)+f(x-y))/2-f(x)}{y}\,dy
+o(1).
\]
もしも $[(f(x+y)+f(x-y))/2-f(x)]/y$ が $0<y<\delta$ で可積分ならば
Riemann-Lebesgueの定理より, 右辺は $N\to\infty$ で $0$ に収束する.
これで示すべきことが示された.
\begin{example*}
可積分函数 $f$ が $x$ で微分可能ならば,
十分小さな $\delta>0$ について, \\
$[(f(x+y)+f(x-y))/2-f(x)]/y$ は $0<y<\delta$ で有界になる. \\
したがって $\lim_{N\to\infty} s_N(f)(x)=f(x)$ が成立する.
\qed
\end{example*}
\begin{example*}
可積分函数 $f$ の値の点 $x$ における左右からの極限
\[
f(x-0)=\lim_{\eps\searrow 0}f(x-\eps), \qquad
f(x+0)=\lim_{\eps\searrow 0}f(x+\eps)
\]
が存在し, $f(x)=(f(x+0)+f(x-0))/2$ であると仮定する.
さらに点 $x$ における左右の微係数
\[
f'(x-0)=\lim_{\eps\searrow 0}\frac{f(x-\eps)-f(x-0)}{-\eps}, \qquad
f'(x+0)=\lim_{\eps\searrow 0}\frac{f(x+\eps)-f(x+0)}{\eps}
\]
が存在すると仮定する.
このとき, 十分小さな $\delta>0$ について,
\[
\frac{(f(x+y)+f(x-y))/2-f(x)}{y}
=\frac{1}{2}\left[\frac{f(x+y)-f(x+0)}{y}-\frac{f(x-y)-f(x-0)}{-y}\right]
\]
は $0<y<\delta$ で有界になる.
したがって
\[
\lim_{N\to\infty} s_N(f)(x)
=\lim_{N\to\infty}\frac{1}{2\pi}\int_{-N}^N e^{ipx}\widehat{f}(p)\,dp
=f(x)=\frac{f(x+0)+f(x-0)}{2}
\]
となる.
\qed
\end{example*}
\subsection{Fourier級数の部分和の収束}
\label{sec:Fseries-N}
以下, $f$ は $\R$ 上の周期 $2\pi$ を持つ函数であり,
$0\leqq x\leqq 2\pi$ で可積分であると仮定する.
このとき $f$ のFourier係数 $a_n$ ($n\in\Z$) が
\[
a_n = \frac{1}{2\pi}\int_0^{2\pi} e^{-iny}f(y)\,dy
\]
と定義される. 正の整数 $N$ に対して,
次を $f$ のFourier級数の $N$ 部分和と呼ぶ:
\[
s_N(f)(x) = \sum_{n=-N}^N a_n e^{inx}.
\]
$N$ 部分和は次のように変形される:
\begin{align*}
s_N(f)(x)
&=\frac{1}{2\pi}\int_0^{2\pi}
\left(\sum_{n=-N}^N e^{in(x-y)}\right) f(y)\,dy
\\ &
=\frac{1}{2\pi}\int_0^{2\pi}
\frac{e^{i(N+1)(x-y)}-e^{-iN(x-y)}}{e^{i(x-y)}-1} f(y)\,dy
\\ &
=\frac{1}{2\pi}\int_0^{2\pi}
\frac{e^{i(N+1/2)(x-y)}-e^{-i(N+1/2)(x-y)}}{e^{i(x-y)/2}-e^{-i(x-y)/2}} f(y)\,dy
\\ &
=\frac{1}{2\pi}\int_0^{2\pi}
\frac{\sin((N+1/2)(x-y))}{\sin((x-y)/2)} f(y)\,dy
\\ &
=\frac{1}{2\pi}\int_0^{2\pi}
\frac{\sin((N+1/2)y)}{\sin(y/2)}f(x+y)\,dy
\\ &
=\frac{1}{2\pi}\int_0^{\pi}
\frac{\sin((N+1/2)y)}{\sin(y/2)}(f(x+y)+f(x-y))\,dy
\\ &
=\frac{1}{\pi}\int_0^{\pi}
\sin((N+1/2)y)\frac{y/2}{\sin(y/2)}\frac{f(x+y)+f(x-y)}{y}\,dy.
\end{align*}
5つ目の等号で $y$ を $x+y$ で置き換え,
$\sin(\alpha x)/\sin(\beta x)$ が偶函数であることを使い,
さらに6つ目の等号で被積分函数の周期性を使った.
$\lim_{t\to 0}(t/\sin t)=1$ に注意すれば,
\secref{sec:Ftransf-N}とまったく同様にして,
$N$ 部分和の収束に関する類似の結果が得られることがわかる.
Dirichlet積分の公式の代わりに次の公式を使わなければいけない:
\[
\frac{1}{2\pi}\int_0^{2\pi}
\frac{\sin((N+1/2)y)}{\sin(y/2)}\,dy
= s_N(1)(0)=1.
\]
さらに非積分函数の周期性と偶函数性より,
\[
\frac{1}{\pi}\int_0^{\pi}\frac{\sin((N+1/2)y)}{\sin(y/2)}\,dy
= 1.
\]
$s_N(1)(0)=1$ の証明は次の通り:
\[
s_N(1)(0)
=\sum_{n=-N}^N \frac{1}{2\pi}\int_0^{2\pi}e^{-iny}dy
=\sum_{n=-N}^N \delta_{n0}
=1.
\]
$e^{-i0y}=1$ 以外の $e^{-iny}$ の $0$ から $2\pi$ までの積分が
消えることを使った.
上の公式を使うと,
\[
f(x)
=\frac{1}{\pi}\int_0^{\pi}\frac{\sin((N+1/2)y)}{\sin(y/2)}\,dy\,f(x)
=\frac{1}{\pi}\int_0^{\pi}\sin((N+1/2)y)\frac{y/2}{\sin(y/2)}\frac{2f(x)}{y}\,dy.
\]
ゆえに上の $s_N(f)(x)$ の表示より,
\[
s_N(f)(x)-f(x)
=\frac{2}{\pi}\int_0^\pi
\sin((N+1/2)y)
\frac{y/2}{\sin(y/2)}\frac{(f(x+y)+f(x-y))/2-f(x)}{y}\,dy.
\]
右辺の積分の被積分函数の $\sin((N+1/2)y)$ 以外の部分
は $\delta\leqq y<\pi$ で可積分なので
Riemann-Lebesgueの定理より, $\delta>0$ に対して,
\[
\lim_{N\to\infty}
\int_\delta^\pi
\sin((N+1/2)y)
\frac{y/2}{\sin(y/2)}\frac{(f(x+y)+f(x-y))/2-f(x)}{y}\,dy=0.
\]
ゆえに, $N\to\infty$ のとき,
\begin{align*}
&
s_N(f)(x)-f(x)
\\ &
=\frac{2}{\pi}\int_0^\delta
\sin((N+1/2)y)
\frac{y/2}{\sin(y/2)}\frac{(f(x+y)+f(x-y))/2-f(x)}{y}\,dy + o(1).
\end{align*}
ゆえに $0<y<\delta$ で
\[
\frac{(f(x+y)+f(x-y))/2-f(x)}{y}
\]
が可積分ならば $N\to 0$ で $s_N(f)(x)-f(x)$ が $0$ に収束し,
$\lim_{N\to\infty}s_N(f)(x)=f(x)$ が成立することがわかる.
この条件が成立するための簡単な十分条件の例も\secref{sec:Ftransf-N}
と同様である.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{付録: ガウス分布のFourier変換}
\label{sec:Gauss-Fourier}
$t>0$ に対して次の公式が成立している:
\[
\int_{-\infty}^\infty e^{-ipx} \frac{e^{-x^2/(2t)}}{\sqrt{2\pi t}}\,dx
= e^{-tp^2/2}.
\tag{$*$}
\]
この公式が成立していることを複数の方法で示そう.
\subsection{熱方程式を使う方法}
函数 $u=u(t,x)$ を次のように定める:
\[
u(t,x)
= \frac{e^{-x^2/(2t)}}{\sqrt{2\pi t}}.
\]
この函数 $u=u(t,x)$ は熱方程式の基本解になっている:
\[
u_t = \frac{1}{2}u_{xx}, \qquad
\lim_{t\to 0}\int_{-\infty}^\infty f(x) u(t,x)\,dx=f(0).
\]
ここで $f(x)$ は有界な連続函数である.
$u=u(t,x)$ が熱方程式を満たすことは偏微分の計算で容易に示される.
後者の極限の証明は実質的に\secref{sec:generalcase}の終わりに書いてある.
ゆえに, $U(t,p)=\int_{-\infty}^\infty e^{-ipx} u(t,x)\,dx$ とおくと,
\begin{align*}
\frac{\d}{\d t}U(t,p)
&=
\frac{1}{2}
\int_{-\infty}^\infty e^{-ipx} \frac{\d^2 u(t,x)}{\d x^2}\,dx
%\\ &
=
\frac{1}{2}
\int_{-\infty}^\infty \frac{\d^2 e^{-ipx}}{\d x^2} u(t,x)\,dx
%\\ &
=
-\frac{p^2}{2}U(t,p).
\end{align*}
2つ目の等号で部分積分を2回行なった. さらに
\[
\lim_{t\to 0}U(t,p)
=\lim_{t\to 0} \int_{-\infty}^\infty e^{-ipx} u(t,x)\,dx
=e^{-ip0}
=1.
\]
したがって
\[
U(t,p)=e^{-tp^2/2}
\]
となることがわかる. これで公式($*$)が示された.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{両辺が同一の常微分方程式を満たしていることを使う方法}
前節の記号をそのまま使うと,
\begin{align*}
\frac{\d}{\d p}U(t,p)
&=\int_{-\infty}^\infty (-ix)e^{-ipx}u(t,x)\,dx
=it\int_{-\infty}^\infty e^{-ipx}\frac{\d}{\d x}u(t,x)\,dx
\\ &
=-it\int_{-\infty}^\infty \left(\frac{\d}{\d x}e^{-ipx}\right)u(t,x)\,dx
=-it\int_{-\infty}^\infty (-ip)e^{-ipx}u(t,x)\,dx
\\ &
=-tp U(t,p).
\end{align*}
2つ目の等号で $u_x=-(x/t)u$ を使い,
3つ目の等号で部分積分を使った.
さらに
\[
U(t,0)=\int_{-\infty}^\infty u(t,x)\,dx=1
\]
となる. これらより $U(t,p)=e^{-tp^2/2}$ となることがわかる.
この方針であれば $u(t,x)$ が熱方程式の基本解であることを使わずにすむ.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{項別積分で計算する方法}
もしも $t=1$ の場合の公式($*$)
\[
\int_{-\infty}^\infty e^{-ipx} \frac{e^{-x^2/2}}{\sqrt{2\pi}}\,dx = e^{-p^2/2}
\tag{$**$}
\]
が示されたならば,
$x$, $p$ をそれぞれ $x/\sqrt{t}$, $\sqrt{t}\,p$ で
置換することによって一般の $t>0$ に関する公式($*$)が得られる.
ゆえに公式($*$)を示すためには公式($**$)を証明すれば十分である.
さらに $\sin(px)$ は奇函数なので
$\int_{-\infty}^\infty e^{-x^2/2} \sin(px)\,dx=0$ となる.
ゆえに
\[
\int_{-\infty}^\infty e^{-x^2/2}\cos(px)\,dx=e^{-p^2/2}\sqrt{2\pi}
\]
を示せば十分である. 左辺の $\cos(px)$ にそのTaylor-Maclaulin展開を代入
した後に項別積分することによってこの公式を示そう.
準備. まず $\int_{-\infty}^\infty e^{-x^2/2}x^{2n}\,dx$ を計算しよう.
部分積分によって
\begin{align*}
\int_{-\infty}^\infty e^{-x^2/2} x^{2n}\,dx
&=
\int_{-\infty}^\infty \left(-e^{-x^2/2}\right)' x^{2n-1}\,dx
\\ &
=\int_{-\infty}^\infty e^{-x^2/2} (x^{2n-1})'\,dx
=(2n-1)\int_{-\infty}^\infty e^{-x^2/2} x^{2n-2}\,dx.
\end{align*}
ゆえに帰納的に $n=0,1,2,\ldots$ に対して
\[
\int_{-\infty}^\infty e^{-x^2/2} x^{2n}\,dx
=(2n-1)\cdots 5\cdot 3\cdot 1\sqrt{2\pi}
=\frac{(2n)!}{2^n n!}\sqrt{2\pi}.
\]
2つ目の等号は左辺の分子分母に$2n\cdots 4\cdot 2=2^n n!$ を
かけることによって得られる.
上で準備した結果を用いると,
\begin{align*}
&
\int_{-\infty}^\infty e^{-x^2/2}\cos(px)\,dx
=
\int_{-\infty}^\infty e^{-x^2/2}
\sum_{n=0}^\infty (-1)^n\frac{(px)^{2n}}{(2n)!}
\,dx
\\ & \qquad
=
\sum_{n=0}^\infty \frac{(-p^2)^n}{(2n)!}
\int_{-\infty}^\infty e^{-x^2/2}x^{2n}\,dx
%\\ &
=
\sum_{n=0}^\infty \frac{(-p^2/2)^n}{n!}\sqrt{2\pi}
=
e^{-p^2/2}\sqrt{2\pi}.
\end{align*}
これで公式($**$)が示された.
\subsection{Cauchyの積分定理を使う方法}
複素解析を知っている人であれば詳しい説明は必要ないと思うので,
以下の説明では大幅に手抜きをする.
Cauchyの積分定理を使うと実数 $p$ に対して
\[
\int_{-\infty}^\infty e^{-(x+ip)^2/2}\,dx
=\int_{-\infty}^\infty e^{-x^2/2}\,dx
=\sqrt{2\pi}
\]
となることを示せる. ゆえに
\[
\int_{-\infty}^\infty e^{-ipx}e^{-x^2/2}\,dx
=
\int_{-\infty}^\infty e^{-(x+ip)^2/2-p^2/2}\,dx
=
e^{-p^2/2}\int_{-\infty}^\infty e^{-(x+ip)^2/2}\,dx
=
e^{-p^2/2}\sqrt{2\pi}.
\]
これで公式($**$)が示された.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{付録: Gauss積分の計算}
次の公式の様々な証明の仕方を解説しよう:
\[
I:=\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.
\]
この公式の面白いところ(不思議なところ)は円周率の気配が見えない
積分の値が円周率の平方根になっていることである.
実際の証明では
\[
I^2 = \iint_{\R^2} e^{-(x^2+y^2)}\,dx\,dy = \pi
\]
を示すことになる.
\subsection{同一の体積の2通りの積分表示を用いた計算}
$I^2=\iint_{\R^2}e^{-(x^2+y^2)}\,dx\,dy$ は $z=e^{-(x^2+y^2)}$ の
小山状のグラフと平面 $z=0$ に挟まれた部分の体積を表わしている.
その体積は高さ $z$ の断面の面積%
\footnote{$z=e^{-(x^2+y^2)}$, $r^2=x^2+y^2$ とおくと,
$\pi r^2=\pi(-\log z)$ となる.} %
$\pi(-\log z)$ を $0<z\leqq 1$ で
積分することによっても求められる. ゆえに
\[
I^2=\int_0^1 \pi(-\log z)\,dz = -\pi[z\log z-z]_0^1 = \pi.
\]
おそらくこの方法が最も簡明である.
\subsection{極座標変換による計算}
$x=r\cos\theta$, $y=r\sin\theta$ と極座標変換すると,
\[
I^2
=\iint_{\R^2} e^{-(x^2+y^2)}\,dx\,dy
=\int_0^{2\pi}d\theta \int_0^\infty e^{-r^2}r \,dr
=2\pi\left[\frac{e^{-r^2}}{-2}\right]_0^\infty
=\pi.
\]
2つ目の等号で極座標変換のJacobianが $r$ になることを使った.
もしくは
\begin{align*}
dx\wedge dy
=(\cos\theta\,dr-r\sin\theta\,d\theta)\wedge
(\sin\theta\,dr+r\cos\theta\,d\theta)
=r\,dr\wedge d\theta
\end{align*}
なので, $K=\{\,(r,\theta)\mid r>0,\ 0\leqq\theta<2\pi\,\}$ とおくと,
\[
I^2
=\iint_{\R^2} e^{-(x^2+y^2)}\,dx\wedge dy
=\iint_K e^{-r^2}r \,dr\wedge d\theta
=\int_0^{2\pi}d\theta \int_0^\infty e^{-r^2}r \,dr
=\pi.
\]
\subsection{Jacobianを使わずにすむ座標変換による計算}
$y$ から $\theta$ に $y=x\tan\theta$ によって積分変数を変換すると,
\begin{align*}
I^2
&=4
\int_0^\infty
\left(
\int_0^\infty e^{-(x^2+y^2)}\,dy
\right)\,dx
=4
\int_0^\infty
\left(
\int_0^{\pi/2} e^{-x^2\cos^2\theta}\,x\cos^2\theta\,d\theta
\right)\,dx
\\ &
=4
\int_0^{\pi/2}
\left(
\int_0^\infty e^{-x^2\cos^2\theta}\,x\cos^2\theta\,dx
\right)\,d\theta
=4
\int_0^{\pi/2}
\left[
\frac{e^{-x^2\cos^2\theta}}{-2}
\right]_{x=0}^{x=\infty}
\,d\theta
\\ &
=4
\int_0^{\pi/2}\frac{1}{2}\,d\theta
=
\pi.
\end{align*}
3つ目の等号で積分の順序交換を行なった.
\subsection{ガンマ函数とベータ函数の関係を用いた計算}
\label{sec:GaussGamma}
前節ではJacobianが出て来ない1変数の積分の置換積分のみを用いて
Gauss積分を計算する方法を説明した.
それと似たような方法によって,
ガンマ函数とベータ函数の関係式を
1変数の積分の置換積分のみを用いて証明することができて,
その関係式の特別な場合としてGauss積分の値を計算することもできる.
この節の内容は前節の内容の一般化であると考えられる.
統計学でよく使われる確率密度函数の記述にはガンマ函数や
ベータ函数を与える積分の被積分函数が現われる(\secref{sec:dists}).
だから, 統計学に興味がある読者は
Gauss積分の計算の一般化として
ガンマ函数とベータ函数についても学んでおいた方が
効率が良いとも考えられる.
$s,p,q>0$ (もしくは実部が正の複素数 $s,p,q$)に対して,
\[
\Gamma(s)=\int_0^\infty e^{-x}x^{s-1}\,dx
\qquad
B(p,q)=\int_0^1 x^{p-1}(1-x)^{q-1}\,dx
\]
によってガンマ函数 $\Gamma(s)$ とベータ函数 $B(p,q)$ が定義される%
\footnote{他にもたくさんの同値な定義の仕方がある.
以下では解析接続については扱わない.}.
部分積分によって $\Gamma(s+1)=s\Gamma(s)$ であることがわかり,
$\Gamma(1)=1$ なので, 0以上の整数 $n$ に対して $\Gamma(n+1)=n!$ となる.
Gauss積分 $I$ は $\Gamma(1/2)$ に等しい:
\[
I
=2\int_0^\infty e^{-x^2}\,dx
=2\int_0^\infty e^{-t} \frac{t^{-1/2}}{2}\,dt
=\int_0^\infty e^{-t}t^{1/2-1}\,dt
=\Gamma(1/2).
\]
2つ目の等号で $x=\sqrt{t}$ とおいた.
したがって $\Gamma(1/2)^2=\pi$ を証明できれば
Gauss積分が計算できたことになる.
ベータ函数は以下のような複数の表示を持つ:
\begin{align*}
B(p,q)
=2\int_0^{\pi/2} \cos^{2p-1}\theta\,\sin^{2q-1}\theta\,d\theta
=\int_0^\infty \frac{t^{p-1}\,dt}{(1+t)^{p+q}}
=\frac{1}{p}\int_0^\infty \frac{du}{(1+u^{1/p})^{p+q}}.
\end{align*}
$x=\cos^2\theta=t/(1+t)$, $t=u^{1/p}$ と変数変換した.
3つ目の(最後の)表示の $p=1/2$ の場合の被積分函数
が $t$ 分布の確率密度函数の表示で使用され,
2つ目の表示の被積分函数は $F$ 分布の確率密度函数の表示で使用される.
$\Gamma(1/2)$ のGauss積分による表示の被積分函数は
正規分布の確率密度函数の表示で使用され,
ガンマ函数の定義式の被積分函数は $\chi^2$ 分布の被積分函数の表示で使用される.
このようにガンマ函数とベータ函数は実用的によく利用される確率分布を
理解するためには必須の教養になっている(\secref{sec:dists}).
特に最初の表示より $B(1/2,1/2)=\pi$ となることがわかる.
ゆえに, もしも
\[
\Gamma(p)\Gamma(q)=\Gamma(p+q)B(p,q)
\]
が示されたならば, $\Gamma(1/2)^2=B(1/2,1/2)=\pi$ となることがわかる.
したがってGauss積分の計算はガンマ函数とベータ函数のあいだの関係式を
示すことに帰着される.
ガンマ函数とベータ函数のあいだの関係式は1変数の置換積分と
積分順序の交換のみを使って証明可能である.
以下でそのことを簡単に説明しよう.
条件 $A$ に対して, $x,y$ が $A$ をみたすとき値が $1$ になり,
それ以外のときに値が $0$ になる $x,y$ の函数を $1_A(x,y)$ と書くことにすると,
\begin{align*}
\Gamma(p)\Gamma(q)
&=
\int_0^\infty
\left(
\int_0^\infty e^{-(x+y)} x^{p-1} y^{q-1}\,dy
\right)\,dx
\\ &
=
\int_0^\infty
\left(
\int_x^\infty e^{-z} x^{p-1} (z-x)^{q-1}\,dz
\right)\,dx
\\ &
=
\int_0^\infty
\left(
\int_0^\infty 1_{x<z}(x,z) e^{-z} x^{p-1} (z-x)^{q-1}\,dz
\right)\,dx
\\ &
=
\int_0^\infty
\left(
\int_0^\infty 1_{x<z}(x,z) e^{-z} x^{p-1} (z-x)^{q-1}\,dx
\right)\,dz
\\ &
=
\int_0^\infty
\left(
\int_0^z e^{-z} x^{p-1} (z-x)^{q-1}\,dx
\right)\,dz
\\ &
=
\int_0^\infty
\left(
\int_0^1 e^{-z} (zt)^{p-1} (z-zt)^{q-1}z\,dt
\right)\,dz
\\ &
=\int_0^\infty e^{-z}z^{p+q-1}\,dz
\,\int_0^1 t^{p-1}(1-t)^{q-1}\,dt
=\Gamma(p+q)B(p,q).
\end{align*}
2つ目の等号で $y=z-x$ と置換積分し,
4つ目の等号で積分の順序を交換し,
6つ目の等号で $x=zt$ と置換積分した.
\subsection{他の方法}
他の方法については
\href
{http://folk.ntnu.no/oistes/Diverse/gaussian-integral-puzzle.pdf}
{Hirokazu Iwasawa, Gaussian Integral Puzzles,
The Mathematical Intelligencer,
Vol.~31, No.~3, 2009, pp.~38-41}
および
\href
{http://www.math.unl.edu/~sdunbar1/ProbabilityTheory/Lessons/StirlingsFormula/GaussianDensity/gaussiandensity.pdf}
{Steven R.~Dunbar, Evaluation of the Gaussian Density Integral, October 22, 2011}
を参照して欲しい.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{付録: ガンマ函数}
\secref{sec:GaussGamma}でガンマ函数について簡単に解説した.
以下ではそこでは解説できなかったガンマ函数の性質について説明しよう.
\subsection{ガンマ函数と正弦函数の関係式}
\secref{sec:GaussGamma}で示した $\Gamma(1/2)^2=B(1/2,1/2)=\pi$ は
次の有名な公式の特別な場合である:
\[
\Gamma(s)\Gamma(1-s)=B(s,1-s)=\frac{\pi}{\sin(\pi s)}.
\]
この公式にも複数の証明法がある.
1つ目の方法は $\sin z$ と $\Gamma(s)$ の無限乗積展開
\begin{align*}
&
\sin z = z\prod_{n=1}^\infty\left(1-\frac{z^2}{\pi^2 n^2}\right),
\qquad\text{{\it i.e.}}\quad
\frac{\sin(\pi s)}{\pi}=s\prod_{n=1}^\infty\left(1-\frac{s^2}{n^2}\right),
\\ &
\frac{1}{\Gamma(s)}
=\lim_{n\to\infty}\frac{s(s+1)\cdots(s+n)}{n!n^s}
=e^{\gamma s}s\prod_{n=1}^\infty\left[ \left(1+\frac{s}{n}\right)e^{-s/n} \right]
\end{align*}
を使う方法である%
\footnote{$\Gamma(s)\Gamma(1-s)=\pi/\sin(\pi s)$ を先に証明しておいて
(たとえば複素解析を使えば容易に示せる),
ガンマ函数の無限乗積展開から $\sin z$ の無限乗積展開を導出することもできる.}.
ここで $\gamma$ はEuler定数
\[
\gamma=\lim_{n\to\infty}\left(\frac11+\frac12+\cdots+\frac1n-\log n\right)
\]
である. これらの公式を認めると,
\[
\frac{1}{\Gamma(s)\Gamma(1-s)}
=\frac{1}{\Gamma(s)(-s)\Gamma(-s)}
=\frac{s(-s)}{-s}\prod_{n=1}^\infty\left[\left(1+\frac{s}{n}\right)\left(1-\frac{s}{n}\right)\right]
=\frac{\sin(\pi s)}{\pi}.
\]
2つ目の方法は次の定積分を複素解析を用いて計算することである:
\[
\Gamma(s)\Gamma(1-s)=B(s,1-s) = \int_0^\infty \frac{t^{s-1}}{1+t}\,dt.
\]
$0<s<1$ であると仮定し, $0<\eps<1<R$ に対して定まる次の積分経路を $C$ と書く:
まず $\eps$ から $R$ までまっすぐに進む.
次に複素平面上の原点を中心とする半径 $R$ の円周上を反時計回りで1周する.
そして $R$ から $\eps$ までまっすぐに進む.
最後に複素平面上の原点を中心とする半径 $\eps$ の円周上を時計回りで1周する.
このとき $\int_C z^{s-1}\,dz/(1+z)$ は $z^{s-1}\,dz/(1+z)$ の $z=-1$ での留数
の $2\pi i$ 倍に等しい:
\[
\int_C \frac{z^{s-1}\,dz}{1+z} = - 2\pi i e^{\pi i s}.
\]
$\eps\to 0$, $R\to\infty$ の極限を考えることによって $\int_C z^{s-1}\,dz/(1+z)$ は
$\int_0^\infty t^{s-1}\,dt/(1+z)$ からそれ自身の $e^{2\pi i s}$ 倍%
\footnote{$z^s$ の値は原点の周囲を反時計回りに1周すると $e^{2\pi i s}$ 倍になる.}
を引いた結果に等しいこともわかる:
\[
\int_C \frac{z^{s-1}\,dz}{1+z} = (1-e^{2\pi i s})\int_0^\infty\frac{t^{s-1}\,dt}{1+t}.
\]
以上の2つの結果を比較することによって
\[
B(s,1-s)
=\int_0^\infty \frac{t^{s-1}\,dt}{1+t}
=\frac{-2\pi i e^{\pi i s}}{1-e^{2\pi i s}}
=\frac{2\pi i}{e^{\pi i s}-e^{-\pi i s}}
=\frac{\pi}{\sin(\pi s)}.
\]
この積分は $t=u^{1/s}$ とおくことによって
$s^{-1}\int_0^\infty du/(1+u^{1/s})$ に変形できる.
ゆえに, 次の公式も得られたことになる:
\[
B(1+s,1-s)=sB(s,1-s)
=\int_0^\infty \frac{du}{1+u^{1/s}} = \frac{\pi s}{\sin(\pi s)}.
\]
この公式を直接示すこともできる.
$R>1$ であるとし,
複素平面上を原点から $R$ までまっすぐ進み,
次に時計回りに角度 $2\pi s$ だけ回転して $Re^{2\pi is}$ まで進み,
そこから原点までまっすぐに戻る経路を $C$ と書くと,
$\int_C dz/(1+z^{1/s})$ は $dz/(1+z^{1/s})$ の $z=e^{\pi is}$ に
おける留数 $-s e^{\pi is}$ の $2\pi i$ 倍に等しく,
$R\to\infty$ の極限で $\int_C dz/(1+z^{1/s})$
は $\int_0^\infty du/(1+u^{1/s})$ からそれ自身の $e^{2\pi is}$ 倍を引いた
ものに等しい%
\footnote{$z^{1/s}$ は $z$ を $e^{2\pi is}$ 倍しても不変だが,
$dz$ は $e^{2\pi is}$ 倍になる.}. ゆえに
\[
\int_0^\infty \frac{du}{1+u^{1/s}}
=\frac{-2\pi is e^{\pi is}}{1-e^{2\pi is}}
=\frac{2\pi is}{e^{\pi is}-e^{-\pi is}}
=\frac{\pi s}{\sin(\pi s)}.
\]
定積分を計算した結果に円周率倍がよく現われるのは
極の周囲を1周する積分が留数の $2\pi i$ 倍になるからである.
複素解析と初等函数とガンマ函数の解説については,
\href{http://www.amazon.co.jp/dp/4000051717}
{高木貞治『解析概論』(岩波書店)}の第5章(201--267頁)をおすすめする.
複素函数論の一般論だけではなく,
具体的な函数の性質の詳しい解説も含めて67頁におさまっているのは
驚異的だと思う.
\subsection{ガンマ函数の無限乗積展開}
\label{sec:Gamma-prod}
函数 $f(s)$ ($s>0$)は以下の3つの条件を満たしていると仮定する:
\begin{itemize}
\item 正値性: $f(s)>0$ ($s>0$),
\item 函数等式: $f(s+1)=sf(s)$ ($s>0$),
\item 対数凸性: $\log f(s)$ は $s>0$ の下に凸な函数である.
\end{itemize}
この3つの条件を満たす函数は次の表示を持つ:
\[
f(s) = f(1)\lim_{n\to\infty}\frac{n!n^s}{s(s+1)\cdots(s+n)}
\qquad (s>0).
\tag{$*$}
\]
特に $\Gamma(s)$ が上の3つの条件と $\Gamma(1)=1$ を満たしていることから,
{\bf Gaussの公式}
\[
\Gamma(s)=\lim_{n\to\infty}\frac{n!n^s}{s(s+1)\cdots(s+n)}
\]
が成立しており, 上の3つの条件を満たしている函数は $\Gamma(s)$ の定数倍
になることもわかる.
以上で述べたことを証明しよう.
まず, ($*$)の極限の分子分母をひっくり返して得られる極限
\[
\lim_{n\to\infty}\frac{s(s+1)\cdots(s+n)}{n!n^s}
\]
が常に収束することを示そう.
\begin{align*}
&
\frac{s(s+1)\cdots(s+n)}{n!n^s}
\\ &
=
s\left(1+\frac{s}{1}\right)\left(1+\frac{s}{2}\right)\cdots\left(1+\frac{s}{n}\right)
e^{-s\log n}
\\ &
=
s\left(1+\frac{s}{1}\right)e^{-s}\left(1+\frac{s}{2}\right)e^{-\frac{s}{2}}
\cdots\left(1+\frac{s}{n}\right)e^{-\frac{s}{n}}
e^{s\left(1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n\right)}
\end{align*}
$1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n$ は $n\to\infty$ でEuler定数 $\gamma$ に
収束する%
\footnote{$1/x$ は単調減少函数なので,
$1+1/2+\cdots+1/n-\log n\geqq\int_1^{n+1}dx/x-\log n=\log(n+1)-\log n\geqq 0$
でかつ $1/(n+1)\leqq\int_n^{n+1}dx/x=\log(n+1)-\log n$ なので,
$1+1/2+\cdots+1/n-\log n$ は有界かつ単調減少する. ゆえに収束する.}.
ゆえに $\prod_{k=1}^n(1+s/k)e^{-s/k}$ が $n\to\infty$ で収束することを示せばよい.
$z$ の複素正則函数 $(1+z)e^{-z}-1$ は原点 $z=0$ で2位の零点を持つので,
$(1+z)e^{-z}=1+O(z^2)$ ($z\to 0$) となる.
ゆえに $(1+s/k)e^{-s/k}=1+O(s^2/k^2)$ ($k\to\infty$).
これより無限積 $\prod_{k=1}^\infty(1+s/k)e^{-s/k}$ が収束することがわかる.
まとめ:
\[
\lim_{n\to\infty}\frac{s(s+1)\cdots(s+n)}{n!n^s}
=e^{\gamma s}s\prod_{n=1}^\infty\left[ \left(1+\frac{s}{n}\right)e^{-s/n} \right]
\]
は常に収束する%
\footnote{この極限を $1/\Gamma(s)$ の定義とすることもできる.
この方法であれば最初から $1/\Gamma(s)$ が複素平面全体で定義されており,
$\Gamma(s)$ の極が $s=0,-1,-2,\ldots$ のみにあることも自明になる.}.
右辺の無限積が $1/\Gamma(s)$ に等しいという公式を{\bf Weierstrass の公式}と
呼ぶことがある.
この極限の逆数を $F(s)$ と書くと,
\[
F(s+1)
=\lim_{n\to\infty}
\frac{ns}{s+1+n}\frac{n!n^s}{s(s+1)\cdots(s+n)}
=sF(s), \quad
F(1)=\frac{n!\,n}{(n+1)!}=1.
\]
ゆえに目標である($*$)の公式 $f(s)=f(1)F(s)$ ($s>0$) を
示すためには, $0<s<1$ のとき $f(s)=f(1)F(s)$ となることを示せば十分である.
次に, $f(s)$ の正値性と対数凸性を用いて,
2以上の整数 $n$ と $0<s<1$ について,
$f(n+s)$ の大きさを $f(n-1),f(n),f(n+1)$ を用いて上下から評価する不等式
\[
\left(\frac{f(n)}{f(n-1)}\right)^s f(n)
\leqq f(n+s)
\leqq \left(\frac{f(n+1)}{f(n)}\right)^s f(n)
\qquad(0<s<1)
\tag{$\#$}
\]
を示そう. 一般に下に凸な函数 $g(s)$ は $a<b<c$ に対して
\[
\frac{g(b)-g(a)}{b-a}
\leqq \frac{g(c)-g(a)}{c-a}
\leqq \frac{g(c)-g(b)}{c-b}
\]
を満たしている
\footnote{図を描けば直観的に明らかだろう.}.
これの左半分を $g(s)=\log f(s)$, $(a,b,c)=(n,n+s,n+1)$ に
適用すると,
\[
\frac{\log f(n+s)-\log f(n)}{s}\leqq \log f(n)-\log f(n+1).
\]
右半分を $(a,b,c)=(n-1,n,n+s)$ に適用すると,
\[
\log f(n)-\log f(n-1)\leqq\frac{\log f(n+s)-\log f(n)}{s}.
\]
以上の2つの不等式を整理し直すと $f(n+s)$ の評価($\#$)が得られる.
$f(n+s)$ の評価($\#$)に $f$ の函数等式を適用しよう. $f$ の函数等式より
\[
\frac{f(n+1)}{f(n)}=n, \quad
f(s+n)=(s+n-1)\cdots(s+1)sf(s), \quad
f(n)=(n-1)!f(1)
\]
などが成立している.
($\#$)の左半分で $n$ を $n+1$ に置き換えると,
\[
n^s n! f(1)\leqq (n+s)(n-1+s)\cdots s f(s),
\qquad\therefore\quad
\frac{f(0)n!n^s}{s(s+1)\cdots(s+n)}\leqq f(s).
\]
($\#$)の右半分より,
\begin{align*}
f(s)\leqq \frac{f(1)(n-1)!n^s}{s(s+1)\cdots(s+n-1)}
=\frac{n+s}{n}\frac{f(1)n!n^s}{s(s+1)\cdots(s+n)}.
\end{align*}
以上をまとめると
\[
\frac{f(1)n!n^s}{s(s+1)\cdots(s+n)}
\leqq
f(s)
\leqq \frac{n+s}{n}
\frac{f(1)n!n^s}{s(s+1)\cdots(s+n)}.
%\tag{$\&$}
\]
これより, 示したかった($*$)が得られる.
ガンマ函数が3つの条件(正値性, 函数等式, 対数凸性)を満たしていることを
証明しよう. 正値性は定義 $\Gamma(s)=\int_0^\infty e^{-x}x^{s-1}\,dx$
より明らかであり, 函数等式は部分積分によって容易に証明される.
対数凸性を示すためには $g(s)=\log\Gamma(s)$ とおくとき,
$g''(s)\geqq 0$ を示せば十分である.
より一般に次のように定義される函数 $f(s)$ に対して $g(s)=\log f(s)$
とおくと $g''(s)\geqq 0$ となることを示そう:
\[
f(s)=\int_a^b e^{s\phi(x)+\psi(x)}\,dx.
\]
ここで $\phi(x),\psi(x)$ は実数値函数であり,
$s$ に関する積分記号化の微分が可能だと仮定しておく.
$(a,b)=(0,\infty)$, $\phi(x)=\log x$, $\psi(x)=-x-\log x$ の
とき $f(s)=\Gamma(s)$ となる%
\footnote{$(a,b)=(0,1)$, $\psi(x)=\log x$ $\phi(x)=t\log(1-x)$
のとき $f(s)=B(s,t)$ となる.
$B(s,t)$ も $s$ の函数として対数凸になる.
ゆえに $F(s)=\Gamma(s+t)B(s,t)$ も $s$ の函数として対数凸になる.
$F(s+1)=sF(s)$, $F(1)=\Gamma(t)$ なので $F(s)=\Gamma(s)\Gamma(t)$
であることがわかる. このようにガンマ函数の特徴付けによって
ガンマ函数とベータ函数の関係式を証明することもできる.}.
このとき, $g(s)=\log f(s)$ とおくと
\[
g''
=\frac{d}{ds}\frac{f'}{f}
=\frac{ff''-f'^2}{f^2}.
\]
ゆえに $f'^2-ff''\leqq 0$ を示せばよい.
$f(s)$ の定義より,
\begin{align*}
f(s)\lambda^2+2f'(s)\lambda+f''(s)
&
=\int_a^b e^{s\phi(x)+\psi(x)}(\lambda^2+2\phi(x)\lambda+\phi(x)^2)\,dx
\\ &
=\int_a^b e^{s\phi(x)+\psi(x)}(\lambda+\phi(x))^2\,dx
\geqq 0.
\end{align*}
ゆえに $f'^2-ff''\leqq 0$ となる.
特に $\Gamma(s)$ も対数凸である.
これでガンマ函数のGaussの公式と無限乗積展開も証明されたことになる.
補足. 以上で説明したガンマ函数に関するGaussの公式の証明は
ガンマ函数そのものではなく、正値対数凸でガンマ函数と同じ函数等式を
満たす函数に対して証明されたのであった.
積分で定義されたガンマ函数に関するGaussの公式を
以下のようにして直接的に証明することもできる.
函数 $n^s B(s,n+1)$ について,
\[
n^sB(s,n+1)
=\frac{n^s\Gamma(s)\Gamma(n+1)}{\Gamma(s+n+1)}
=\frac{n^s n!}{s(s+1)\cdots(s+n)}
\]
でかつ
\[
n^sB(s,n+1)
=n^s\int_0^1 x^{s-1}(1-x)^n\,dx
=\int_0^n t^{s-1}\left(1-\frac{t}{n}\right)^n\,dt
\]
2つ目の等号で $x=t/n$ とおいた. ゆえに, $n\to\infty$ のとき,
\[
\frac{n^s n!}{s(s+1)\cdots(s+n)}
=\int_0^n t^{s-1}\left(1-\frac{t}{n}\right)^n\,dt
\longrightarrow
\int_0^\infty t^{s-1}e^{-t}\,dt
=\Gamma(s).
\]
最後のステップを別の方法で証明することもできる.
評価($\#$)を $f(s)=\Gamma(s)$ の場合に適用すると,
$0<s<1$ のとき
\[
\Gamma(s+n+1)\sim n^s\Gamma(n+1)
\qquad(n\to\infty).
\]
ガンマ函数の函数等式より, これは任意の $s>0$ で成立している. ゆえに
\[
\frac{n^s n!}{s(s+1)\cdots(s+n)}
=\frac{n^s\Gamma(s)\Gamma(n+1)}{\Gamma(s+n+1)}
\longrightarrow
\Gamma(s)
\qquad(n\to\infty).
\]
このように, ガンマ函数の正値性, 対数凸性, 函数等式による特徴付けを
経由せずに, 直接的にガンマ函数に関するGaussの公式を(したがって無限乗積展開も)
得ることは易しい. 以上によって次の公式も証明されたことになる:
\[
\lim_{n\to\infty}n^s B(s,n+1)=\Gamma(s).
\]
まとめ:
\[
\Gamma(s)
=\lim_{n\to\infty}n^sB(s,n+1)
=\lim_{n\to\infty}\frac{n^s n!}{s(s+1)\cdots(s+n)}
=\frac{1}{e^{\gamma s}s}\prod_{n=1}^\infty\left[\left(1+\frac{s}{n}\right)e^{-s/n}\right]^{-1}.
\]
ここで $\gamma$ はEuler定数である.
\subsection{正弦函数の無限乗積展開}
ガンマ函数の無限乗積展開の応用として $\sin z$ の無限乗積展開を証明しよう.
積分の順序交換を用いて証明されるガンマ函数とベータ函数の関係と
複素解析を用いて証明されるベータ函数と正弦函数の関係より
\[
\Gamma(s)\Gamma(1-s)=B(s,1-s)=\frac{\pi}{\sin(\pi s)}.
\]
一方, ガンマ函数の無限乗積展開より,
\[
\frac{1}{\Gamma(s)\Gamma(1-s)}
=\frac{1}{\Gamma(s)(-s)\Gamma(-s)}
=s\prod_{n=1}^\infty\left(1-\frac{s^2}{n^2}\right).
\]
以上を比較すると,
\[
\sin(\pi s)=\pi s\prod_{n=1}^\infty\left(1-\frac{s^2}{n^2}\right),
\qquad\therefore\quad
\sin z=z\prod_{n=1}^\infty\left(1-\frac{z^2}{\pi^2n^2}\right).
\]
このように, $\sin(\pi s)=\pi/(\Gamma(s)(-s)\Gamma(-s))$ なので
ガンマ函数の無限乗積展開\footnote{直接証明すれば易しい.}から
正弦函数の無限乗積展開が得られるのである.
正弦函数の無限乗積展開を直接示すためには,
$\sin z$ の対数微分 $\cot z$ の部分分数展開
\[
\cot z
= \frac{1}{z}
+ \sum_{n=1}^\infty\left(\frac{1}{z-n\pi}+\frac{1}{z+n\pi}\right)
\]
を複素解析を用いて証明し, 項別に積分すればよい.
詳しくは高木貞治『解析概論』の235頁を見よ.
以下では, 複素解析ではなく,
Fourier級数の理論を使って正弦函数の無限乗積展開を得る方法
を紹介しておこう\footnote{以下では厳密な議論はしないが,
Fourier級数の収束については\secref{sec:Fseries-N}を参照せよ.}.
まず $x$ の函数 $\cos(tx)$ の $-\pi\leqq x\leqq\pi$ での値のFourier級数展開を求め,
そこから $\cot(\pi t)$ の部分分数展開が得られることを示そう%
\footnote{$x$ の偶函数 $\cos(tx)$ の $-\pi\leqq x\leqq\pi$ での値を周期 $2\pi$
で $\R$ 全体に拡張して得られる連続周期函数 $f_t(x)$ のFourier級数を考える.
$\cos(tx)$ の $0\leqq x<2\pi$ での値を周期 $2\pi$ で拡張するのではない
ことに注意せよ.
}.
$e^{itx}$ の Fourier係数は
\begin{align*}
a_n
&= \frac{1}{2\pi}\int_{-\pi}^\pi e^{-inx}e^{itx}\,dx
=\frac{1}{2\pi}\left[ \frac{e^{-inx}e^{itx}}{i(t-n)} \right]_{x=-\pi}^{x=\pi}
\\ &
=\frac{(-1)^n(e^{i\pi t}-e^{-i\pi t})}{2\pi i(t-n)}
=(-1)^n\frac{\sin(\pi t)}{\pi}\frac{1}{t-n}
\end{align*}
なので, $e^{itx}$ のFourier級数展開は
\begin{align*}
e^{itx}
&=\lim_{N\to\infty} \sum_{n=-N}^N a_n e^{inx}
=\frac{\sin(\pi t)}{\pi}
\lim_{N\to\infty} \sum_{n=-N}^N \frac{(-1)^n e^{inx}}{t-n}
\\ &
=\frac{\sin(\pi t)}{\pi} \left[
\frac{1}{t}
+ \sum_{n=1}^\infty (-1)^n
\left(\frac{e^{inx}}{t-n}+\frac{e^{-inx}}{t+n} \right)
\right]
\\ &
=\frac{\sin(\pi t)}{\pi} \left[
\frac{1}{t}
+ \sum_{n=1}^\infty (-1)^n
\left(\frac{2t\cos(nx)}{t^2-n^2}+i\frac{2n\sin(nx)}{t^2-n^2} \right)
\right]
\end{align*}
になる. ゆえに $\cos(tx)$ のFourier級数展開は
\[
\cos(tx)
=\frac{\sin(\pi t)}{\pi}
\left[
\frac{1}{t} + \sum_{n=1}^\infty (-1)^n\frac{2t\cos(nx)}{t^2-n^2}
\right]
\]
になる. したがって,
\[
\pi\cot(tx)
=\frac{\pi\cos(\pi t)}{\sin(\pi t)}
=\frac{1}{t} + \sum_{n=1}^\infty (-1)^n\frac{2t\cos(nx)}{t^2-n^2}
\]
両辺の $x\to\pi$ での極限を取ることによって,
\[
\pi\cot(\pi t)
=\frac{1}{t} + \sum_{n=1}^\infty\frac{2t}{t^2-n^2}
=\frac{1}{t} + \sum_{n=1}^\infty\left(\frac{1}{t-n}+\frac{1}{t+n}\right).
\]
$\sin(\pi t)$ の対数微分は $\pi\cot(\pi t)$ に等しいので,
\[
\frac{d}{dt}\log\frac{\sin(\pi t)}{\pi t}
=\sum_{n=1}^\infty\left(\frac{1}{t-n}+\frac{1}{t+n}\right)
=\sum_{n=1}^\infty\left( \frac{-1/n}{1-t/n} + \frac{1/n}{1+t/n} \right).
\]
両辺を $t=0$ から $t=s$ まで積分すると,
\[
\log\frac{\sin(\pi s)}{\pi s}
=\sum_{n=1}^\infty
\left(\log\left( 1-\frac{s}{n} \right)+\log\left( 1+\frac{s}{n} \right)\right)
=\log\prod_{n=1}^\infty\left( 1-\frac{s^2}{n^2} \right)
\]
したがって
\[
\sin(\pi s)
=\pi s \prod_{n=1}^\infty\left( 1-\frac{s^2}{n^2} \right).
\]
$\sin$ の無限乗積展開とガンマ函数の無限乗積展開の公式を認めて使うことを許せば,
$1/(\Gamma(s)\Gamma(1-s))$ と $\sin(\pi s)$ を比較することによって
\[
\Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin(\pi s)}
\]
を示せる. さらに $\Gamma(p)\Gamma(q)=\Gamma(p+q)B(p,q)$ を
1変数の積分の置換積分と積分の順序交換のみを用いて容易に証明できることを
使えば, 次の公式も得られる:
\[
\frac{\pi}{\sin(\pi s)}
=B(s,1-s)
=\int_0^1x^s(1-x)^{1-s}\,dx
=\int_0^\infty \frac{t^{s-1}\,dt}{1+t}
=\frac{1}{s}\int_0^\infty\frac{du}{1+u^{1/s}}.
\]
これらの公式はどれか一つを証明できれば他も芋づる式に得られるようになっている.
\subsection{Wallisの公式}
\label{sec:Wallis}
次の公式は{\bf Wallisの公式}と呼ばれている:
\[
\lim_{n\to\infty}\frac{2^{2n}(n!)^2}{(2n)!\sqrt{n}}
=\sqrt{\pi},
\qquad
\text{\it i.e.}\quad
\binom{2n}{n}\sim\frac{2^{2n}}{\sqrt{\pi n}}.
\]
Wallisの公式の面白いところは円周率の平方根が
整数の比の極限で表わされているところである.
Wallisの公式はガンマ函数に関するGaussの公式に $s=1/2$ を代入すれば得られる:
\begin{align*}
\sqrt{\pi}&
=\Gamma(1/2)
=\lim_{n\to\infty}\frac{n^{1/2} n!}{(1/2)(1/2+1)\cdots(1/2+n)}
\\ &
=\lim_{n\to\infty}
\frac{2^{n+1}n^{1/2}n!}{1\cdot3\cdots(2n+1)}
=\lim_{n\to\infty}
\frac{2^{n+1}n^{1/2}n!}{1\cdot3\cdots(2n+1)}\frac{2^n n!}{2\cdot4\cdots(2n)}
\\ &
=\lim_{n\to\infty}
\frac{2^{2n+1}n^{1/2}(n!)^2}{(2n+1)!}
=\lim_{n\to\infty}
\frac{2^{2n}(n!)^2}{(2n)!}\frac{2n^{1/2}}{2n+1}
=\lim_{n\to\infty}
\frac{2^{2n}(n!)^2}{(2n)!\sqrt{n}}.
\end{align*}
次の公式も{\bf Wallisの公式}と呼ばれている:
\[
\prod_{n=1}^\infty\frac{2n\cdot 2n}{(2n-1)(2n+1)} = \frac{\pi}{2}.
\]
この公式は次の公式で $s=1/2$ とおけば得られる:
\[
\sin(\pi s)
= \frac{\pi}{\Gamma(s)\Gamma(1-s)}
= \pi s\prod_{n=1}^\infty\left(1-\frac{s^2}{n^2}\right).
\]
実際,
\[
1=\sin\left(\frac{\pi}{2}\right)
=\frac{\pi}{2}\prod_{n=1}^\infty\left(1-\frac{1}{(2n)^2}\right)
=\frac{\pi}{2}\prod_{n=1}^\infty\frac{(2n-1)(2n+1)}{2n\cdot 2n}.
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{document}
\section{付録: 正規分布から派生する様々な確率分布について}
\label{sec:dists}
\subsection{正規分布}
次の確率密度函数で定義される確率分布を
平均 $\mu$, 分散 $\sigma$ の正規分布と呼ぶ:
\[
f_{\mu,\sigma}(x)\,dx
=\frac{e^{-(x-\mu)^2/(2\sigma^2)}}{\sqrt{2\pi \sigma^2}}\,dx.
\]
平均 $0$, 分散 $1$ の正規分布を標準正規分布と呼ぶ.
\paragraph{再生性}
互いに独立な確率変数 $X$, $Y$ がそれぞれ平均 $\mu_X,\mu_Y$, 分散 $\sigma_X^2,\sigma_Y^2$
の正規分布にしたがうとき, $X+Y$ は平均 $\mu_X+\mu_Y$, 分散 $\sigma_X^2+\sigma_Y^2$ の
正規分布にしたがう.
\subsection{ガンマ分布とカイ2乗分布}
次の確率密度函数で定義される確率分布を
shape $\alpha>0$, scale $\tau>0$ のガンマ分布と呼ぶ:
\[
f_{\alpha,\tau}(x)\,dx
=\frac{x^{-x/\tau}x^{\alpha-1}}{\Gamma(\alpha)\tau^\alpha}\,dx
\qquad (x>0).
\]
平均は $\alpha\tau$, 分散は $\alpha\tau^2$ になる.
\paragraph{再生性}
互いに独立な確率変数 $X,Y$ がそれぞれ shape $\alpha_X,\alpha_Y$, scale $\tau,\tau$ の
ガンマ分布にしたがうとき, $X+Y$ は shape $\alpha_X+\alpha_Y$, scale $\tau$ の
ガンマ分布にしたがう.
カイ2乗分布($\chi^2$ 分布)はガンマ分布の特別な場合である.
すなわち, shape $n/2$, scale $2$ のガンマ分布を
自由度 $n$ のカイ2乗分布($\chi^2$ 分布)と呼ぶ.
カイ2乗分布は自由度 $n$ について再生性を持つ.
確率変数 $X_1,\ldots,X_n$ が標準正規分布にしたがうとき,
$Y=X_1^2+\cdots+X_n^2$ は自由度 $n$ のカイ2乗分布にしたがう.
\subsection{第二種ベータ分布と $t$ 分布}
次の確率密度函数で定義される確率分布を
パラメーター $\alpha,\beta>0$ を持つ第二種ベータ分布
(Beta distribution of the second kind もしくは Beta prime distribution)と呼ぶ:
\[
\tf_{\alpha,\beta}(x)\,dx
=\frac{1}{B(\alpha,\beta)}\frac{x^{\alpha-1}}{(1+x)^{\alpha+\beta}}\,dx
\qquad (x>0).
\]
$\beta>1$ ならば平均は $\alpha/(\beta-1)$ になり,
$\beta>2$ ならば分散は $(\alpha(\alpha+\beta-1))/((\beta-2)(\beta-1)^2)$ になる.
第2種ベータ分布の確率密度函数に $x=t^2/\gamma$ ($\gamma>0$) を代入して,
確率分布を $-\infty<t<\infty$ に拡張すると, 確率密度函数は次の形になる:
\[
\tf_{\alpha,\beta}\left(\frac{t^2}{\gamma}\right)\frac{t}{\gamma}\,dt
=\frac{1}{\gamma^\alpha B(\alpha,\beta)}\frac{t^{2\alpha-1}}{(1+t^2/\gamma)^{\alpha+\beta}}\,dt
\]
$n>0$ に対して,
$\alpha=1/2$, $\beta=n/2$, $\gamma=n$ のとき, この確率密度函数で定義される
確率分布を自由度 $n$ の $t$ 分布と呼ぶ.
すなわち, 自由度 $n$ の $t$ 分布とは次の確率密度函数で定義される確率分布のことである:
\[
\tg_n(t)\,dt = c_n\left( 1+\frac{t^2}{n} \right)^{-(n+1)/2}\,dt.
\]
ここで
\[
c_n
=\frac{1}{n^{1/2}B(1/2,n/2)}
=\frac{\Gamma((n+1)/2)}{\sqrt{n\pi}\,\Gamma(n/2)}.
\]
自由度が $n>1$ ならば $t$ 分布は平均 $0$ を持つ.
自由度が $n\leqq 1$ のとき $t$ 分布は平均を持たない.
自由度が $n>2$ ならば $t$ 分布は分散 $n/(n-2)$ を持つ.
自由度を無限大にする極限で $t$ 分布の平均と分散はそれぞれ $0$ と $1$ に収束する.
自由度が $n\leqq 2$ ならば $t$ 分布の分散は無限大になる.
互いに独立な確率変数 $X_1,\dots,X_n$ がどれも
平均 $\mu$, 分散 $\sigma^2$ の正規分布にしたがうとき,
\[
M=\frac{1}{n}\sum_{k=1}^n X_k, \qquad
S^2 = \frac{1}{n-1}\sum_{k=1}^n(X_k-M)^2, \qquad
T = \frac{M-\mu}{S/\sqrt{n}}
\]
とおくと, $T$ は自由度 $n-1$ の $t$ 分布にしたがう.
\begin{remark*}
すぐ上の設定のもとで, $E[S^2]=\sigma^2$ となる.
$S^2$ は不偏分散と呼ばれている.
正規分布の再生性より, $M$ は平均 $\mu$, 分散 $\sigma^2/n$ の正規分布
にしたがう. ゆえに $T$ に類似の確率変数
\[
Z = \frac{M-\mu}{\sigma/\sqrt{n}}
\]
は標準正規分布にしたがう. 上で述べたことは,
分母の $\sigma$ を確率変数 $S$ で置き換えると標準正規分布ではなく,
自由度 $n-1$ の $t$ 分布にしたがうということである.
すでに母分散 $\sigma^2$ がわかっている場合には $Z$ を利用できるが,
母分散がわかっていない場合には $Z$ を利用できない.
そこで母分散 $\sigma^2$ の代わりに不偏分散 $S^2$ を使用すると,
確率分布は正規分布からずれた $t$ 分布になってしまうのである.
\qed
\end{remark*}
\subsection{第一種ベータ分布と $F$ 分布}
次の確率密度函数で定義される確率分布を
パラメーター $\alpha,\beta>0$ を持つ第一種ベータ分布
(Beta distribution of the first kind もしくは単にベータ分布)と呼ぶ:
\[
f_{\alpha,\beta}(x)\,dx
=\frac{1}{B(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1}\,dx
\qquad (0<x<1).
\]
平均は $\alpha/(\alpha+\beta)$,
分散は $(\alpha\beta)/((\alpha+\beta)^2(\alpha+\beta+1))$ になる.
$m,n>0$ とし,
第一種ベータ分布の確率密度函数の $x$ に $mx/(mx+n)$ ($x>0$) を代入すると,
\[
f_{\alpha,\beta}\left( \frac{mx}{mx+n} \right)\frac{mn}{(mx+n)^2}\,dx
=
\frac{1}{B(\alpha,\beta)}
\left( \frac{mx}{mx+n} \right)^\alpha
\left( 1-\frac{mx}{mx+n} \right)^\beta
\frac{dx}{x}
\quad (x>0)
\]
と整理される($1-mx/(mx+n)=b/(mx+n)$ を用いた).
これは $\alpha=m/2$, $\beta=n/2$ のとき次の形になる:
\[
g_{m,n}(x)\,dx
=
\frac{1}{B(m/2,n/2)}
\left( \frac{mx}{mx+n} \right)^{m/2}
\left( 1-\frac{mx}{mx+n} \right)^{n/2}
\frac{dx}{x}
\qquad (x>0).
\]
この確率密度函数で定義される確率分布をパラメーター $m,n$ の $F$ 分布と呼ぶ.
$F$ 分布は $n>2$ のとき平均が $n/(n-2)$ になり,
$n>4$ のとき分散が $(2n^2(m+n-2))/(m(n-2)^2(n-4))$ になる.
$F$ 分布の定義より,
$X$ がパラメーター $m,n$ の $F$ 分布にしたがうならば,
$mX/(mX+n)$ はパラメーター $m/2,n/2$ の第一種ベータ分布にしたがう.
互いに独立な確率変数 $U_1$, $U_2$ が
それぞれ自由度 $d_1$, $d_2$ のカイ2乗分布にしたがうとき,
\[
X = \frac{U_1/d_1}{U_2/d_2}
\]
はパラメーター $d_1,d_2$ の $F$ 分布にしたがう.
すなわち, $X^{(i)}_1,\ldots,X^{(i)}_{d_i}$ ($i=1,2$) がすべて
互いに独立な確率変数であり,
各々の $X^{(i)}_k$ は平均 $0$, 分散 $\sigma_i^2$ の正規分布にしたがうとき,
\[
s_i^2 = \frac{1}{d_1}\sum_{k=1}^{d_i} (X^{(i)}_k)^2, \qquad
X=\frac{s_1^2/\sigma_1^2}{s_2^2/\sigma_2^2}
\]
とおくと, $X$ はパラメーター $d_1,d_2$ の $F$ 分布にしたがう.
第一種ベータ分布の確率密度函数 $f_{\alpha,\beta}(x)\,dx$ の $x$
に $x/(1+x)$ を代入したものは,
第二種ベータ分布の確率密度函数 $\tf_{\alpha,\beta}(x)\,dx$ に一致する.
さらに第二種ベータ分布の確率密度函数に $x=t^2/n$, $\alpha=1/2$, $\beta=n/2$
を代入したものは自由度 $n$ の $t$ 分布の確率密度函数になるのであった.
このことから確率変数 $T$ が自由度 $n$ の $t$ 分布にしたがうとき,
$T^2$ はパラメーター $1,n$ の $F$ 分布にしたがい,
$T^{-2}$ はパラメーター $n,1$ の $F$ 分布にしたがうことがわかる.
この意味で $T$ 分布は本質的に
片方の自由度が $1$ の場合の $F$ 分布であることがわかる.
このことは以下の直接的な計算によっても確かめられる.
$F$ 分布の確率密度函数は次のように書き直される:
\[
g_{m,n}(x)\,dx
=
\frac{(m/n)^{m/2}}{B(m/2,n/2)}
\frac{x^{m/2-1}}{(1+mx/n)^{(m+n)/2}}
\,dx.
\]
$m=1$ を代入すると,
\[
g_{1,n}(x)\,dx
=
\frac{1}{\sqrt{n}\,B(1/2,n/2)}
\frac{x^{-1/2}}{(1+x/n)^{(n+1)/2}}
\,dx.
\]
さらに $x=t^2$ を代入して,
分布を $-\infty<t<\infty$ に拡張したものの確率密度函数は
\[
g_{1,n}(t^2)t\,dt
=
\frac{1}{\sqrt{n}\,B(1/2,n/2)}
\frac{dt}{(1+t^2/n)^{(n+1)/2}}
\]
になる. これは $t$ 分布の確率密度函数 $\tg_n(t)\,dt$ に一致する.
\subsection{$n-1$ 次元球面上の一様分布とBoltzmann-Maxwell則}
\label{sec:BM}
$X_i$ 達は互いに独立な標準正規分布であるとし, $R_n=\sqrt{X_1^2+\cdots+X_n^2}$,
$Z_{ni}=X_i/R_n$ とおく.
このとき $(Z_{n1},\ldots,Z_{nn})$ は $n-1$ 次元単位球面上の一様分布になる%
\footnote{この方法を使えば標準正規正規分布する乱数から
球面上一様分布する乱数が得られる.}.
確率変数 $Z_{ni}$ の確率密度函数は
\begin{align*}
&
g_n(z)\,dz = c_n^{-1} (1-z^2)^{(n-3)/2}\,dz \qquad (-1<z<1),
\\ &
c_n = \int_{-1}^1 (1-z^2)^{(n-1)/2-1}\,dz=2^{n-2} B\left(\frac{n-1}{2},\frac{n-1}{2}\right)
\end{align*}
になる. たとえば
\[
g_2(z)\,dz = \frac{1}{\pi}(1-z^2)^{-1/2}\,dz \qquad(-1<z<1).
\]
$g_3(z)\,dz$ は $-1\leqq z\leqq 1$ における一様分布函数になり,
$n\geqq 4$ のとき $g_n(x)$ はグラフが釣鐘型の函数になる%
\footnote{これらは本質的に第一種ベータ分布の特別な場合である.}.
$Z_{ni}$ の平均は $0$ である.
さらにベータ函数とガンマ函数の関係およびガンマ函数の函数等式
より $c_n/c_{n+2}=(n-1)/n=1-1/n$ となることがわかる.
そのことを使うと, $Z_{ni}$ の分散が $1/n$ になることを示せる:
\[
c_n^{-1}\int_{-1}^1 z^2(1-z^2)^{(n-3)/2}
=c_n^{-1}(c_n-c_{n+2})
=1-\frac{c_n}{c_{n+1}}
=\frac{1}{n}.
\]
ここで $z^2$ に $1-(1-z^2)$ を代入する計算を行った.
$Y_{ni}=\sqrt{n}Z_{ni}$ は平均 $0$, 分散 $1$ の確率変数になり, その確率密度函数は
\[
\frac{1}{\sqrt{n}}g_n\left(\frac{y}{\sqrt{n}}\right)\,dy
=\frac{1}{\sqrt{n}\,c_n} \left(1-\frac{y^2}{n}\right)^{(n-3)/2}\,dy
\]
となる. $n\to\infty$ のとき, $\nu=(n-1)/2$ とおくと,
\begin{align*}
&
\left(1-\frac{y^2}{n}\right)^{(n-3)/2}
=\left(1-\frac{y^2}{n}\right)^{-3/2} \left(1-\frac{y^2/2}{n/2}\right)^{n/2}
\longrightarrow e^{-y^2/2}
\\ &
\sqrt{n}\,c_n
=\sqrt{2\nu+1}\,\,2^{2\nu-1}B(\nu,\nu)
%\\ &
\sim
\sqrt{2\nu}\,2^{2\nu-1}\frac{2}{\nu}\frac{\sqrt{\pi\nu}}{2^{2\nu}}
=\sqrt{2\pi}
\end{align*}
となる%
\footnote{$\sqrt{n}\,c_n=\int_{-1}^1(1-y^2/n)^{(n-3)/2}\,dy$ なので,
前者の \(
\lim_{n\to\infty}(1-y^2/n)^{(n-3)/2}=e^{-y^2/2}
\) から後者の $\lim_{n\to\infty}\sqrt{n}\,c_n=\sqrt{2\pi}$ を導くこともできる.
実際, そうした方が簡単だろう. }. %
途中の計算で Wallis の公式より
\[
B(\nu,\nu)
= \frac{\Gamma(\nu)^2}{\Gamma(2\nu)}
= \frac{2\nu}{\nu^2}\frac{\Gamma(\nu+1)^2}{\Gamma(2\nu+1)}
= \frac{2}{\nu}\binom{2\nu}{\nu}^{-1}
\sim \frac{2}{\nu}\frac{\sqrt{\pi\nu}}{2^{2\nu}}
\]
となることを使った%
\footnote{以上の計算を逆にたどることによって, 逆にWallisの公式を証明することもできる.}.
したがって, $Y_{ni}$ は $n\to\infty$ の極限で標準正規分布にしたがう確率変数に収束する:
%\vspace{-4mm}
\[
\lim_{n\to\infty}
\frac{1}{\sqrt{n}}g_n\left(\frac{y}{\sqrt{n}}\right)
=\lim_{n\to\infty}
\frac{\left(1-\dfrac{y^2}{n}\right)^{(n-3)/2}}{\sqrt{n}\,2^{n-2}B(\frac{n-1}{2},\frac{n-1}{2})}
=\frac{e^{-y^2/2}}{\sqrt{2\pi}}.
\]
以上をまとめると, 実数 $y$ の有界連続函数 $g(y)$ について,
\[
C_n^{-1}\int_{\sqrt{n}\,S^{n-1}} g(y_i) \,d\omega_n
\longrightarrow
\int_\R g(y)\frac{e^{-y^2/2}}{\sqrt{2\pi}}\,dy.
\qquad (n\to\infty).
\]
ただし, $\sqrt{n} S^{n-1}=\{\,(y_1,\ldots,y_n)\in\R^n\mid y_1^2+\cdots+y_n^2=n \,\}$
は半径 $\sqrt{n}$ の $n-1$ 次元球面であり,
$C_n$ はその球面の表面積であり,
$d\omega_n$ はその球面上の面積要素である.
この結果は物理的には {\bf Boltzmann-Maxwell 則}としてよく知られている.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bib{1999/knudsen}
\yr 1999
\isbn 354066226X
\editor Lars Knudsen
\book Fast software encryption: 6th international workshop, FSE'99, Rome, Italy, March 1999: proceedings
\series Lecture Notes in Computer Science
\seriesvol 1636
\publ Springer-Verlag
\publaddr Berlin
\url http://link.springer.de/link/service/series/0558/tocs/t1636.htm
\endref
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\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2016b.00534}
\itemau{Menze, Malte}
\itemti{Three white minus five grey ones. Addition and subtraction of integers with small tiles. (Drei Wei{\ss}e minus f\"unf Graue. Addieren und Subtrahieren von ganzen Zahlen mit Legepl\"attchen.)}
\itemso{Mathematik 5 bis 10 33, 46-47 (2015).}
\itemab
Aus dem Text: Manche meiner Sch\"uler haben auch in h\"oheren Jahrg\"angen noch Schwierigkeiten, wenn zwei der Rechenzeichen ``+" oder ``-" aufeinandertreffen. Ist das Endergebnis nun positiv oder negativ? Ein m\"oglicher Hintergrund ist, dass sie keine Vorstellung davon haben, warum die Rechenregeln gelten, die sie (nur) auswendig gelernt haben. Wieder andere sind irritiert, wenn sie sich mit dem Unterschied zwischen Rechen- und Vorzeichen besch\"aftigen, da die entsprechende Kategorisierung von der Betrachtungsweise abh\"angt: Wird beispielsweise bei der Rechnung ``$5-3$" eine positive 3 subtrahiert oder vielleicht doch eine negative 3 addiert? Deutlich wird die unterschiedliche Perspektive wenn man den Term umschreibt als ``$-3+5$". Um f\"ur diese Fragen ein besseres Verst\"andnis zu erm\"oglichen und zu f\"ordern, verwende ich in einer Unterrichtseinheit gern wei{\ss}e und graue Moosgummipl\"attchen.
\itemrv{~}
\itemcc{F43 U63}
\itemut{negative numbers; integers; arithmetic; teaching aids; manipulative materials; small tiles; addition; subtraction}
\itemli{}
\end
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%%% version = "1.117",
%%% date = "28 December 2019",
%%% time = "10:17:21 MDT",
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%%% address = "University of Utah
%%% Department of Mathematics, 110 LCB
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%%% Kallen--Pauli equation; neutrino; nuclear
%%% beta decay; Pauli algebra; Pauli effect
%%% (humorous); Pauli exclusion principle; Pauli
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%%% Pauli--Fierz operator; Pauli--Jung dialog;
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\title{A Selected Bibliography of Publications by, and about, Wolfgang Pauli}
\author{%
Nelson H. F. Beebe\\
University of Utah\\
Department of Mathematics, 110 LCB\\
155 S 1400 E RM 233\\
Salt Lake City, UT 84112-0090\\
USA\\[\medskipamount]
Tel: +1 801 581 5254\\
FAX: +1 801 581 4148\\[\medskipamount]
E-mail: \protect\[email protected]=,
\protect\[email protected]=,\\
\hphantom{E-mail:\ }
\protect\[email protected]= (Internet)\\
WWW URL: \protect\path=http://www.math.utah.edu/~beebe/=
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\date{28 December 2019 \\
Version 1.117}
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\section*{Title word cross-reference}
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\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2010c.00496}
\itemau{Glaz, Sarah; Liang, Su}
\itemti{Modelling with poetry in an introductory college algebra course and beyond.}
\itemso{J. Math. Arts 3, No. 3, 123-133 (2009).}
\itemab
Summary: The main focus of this article is the pedagogical use of poetry in a college course, Introductory College Algebra and Mathematical Modelling, which was designed to prepare students with weak mathematics background for science courses. We use poetry projects to ease the difficulties students have with the transition between word-problems representing natural phenomena, and the corresponding mathematical models-the equations representing the phenomena. We also use poetry projects to stimulate classroom participation, develop mathematical intuition, enhance number sense and introduce new concepts in an engaging setting. We present two examples of group-projects that we employed in our course, as well as a number of other poems which may be used to construct additional group or individual projects in courses of similar mathematical level. We conclude by providing a brief survey of the pedagogical uses of poetry in the K-14 mathematics classroom, and examples of poetry that may be used to enhance learning experience in calculus classes.
\itemrv{~}
\itemcc{M80 D30 D40}
\itemut{poetry; intermediate algebra; mathematical modelling; Bhaskaracharya; Martin Gardner; calculus}
\itemli{doi:10.1080/17513470903167731}
\end
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| 100,318,977 | 24,565 |
\documentclass[11pt]{jarticle}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\begin{document}
\begin{center}
\underline{\textbf{\LARGE 多変数解析函数に就て}}\\
\medskip
\underline{\Large \rm{\bf{XI}} \bf{\Large --- 擬凸状域と有限正則域,}}\\
\underline{\textbf{\Large 有限正則域に於ける諸定理}}\\
\end{center}
\bigskip
第一次研究の結果を, 第一及び第二基礎的補助定理に依て,
分岐点を内点として持たない有限領域へ拡張します. 但し, 此の度は次の
一聯の問題を考察するに止めます： 擬凸状域は正則域かと云ふこと,
\underline{Cousin} の第一問題, 及び函数の展開
\footnote{結果に就ては, 第10節の定理 $\rm{I}$ 及び第11節の諸定理参照.}.
\underline{Cousin} の第二問題及び積分表示に就ても略々同様と考へます
\footnote{之等は上の三定理の様に不可分の関係にある釈ではなく, それに
此の度の拡張は中間的のものですから, 他の機會に確めようと思ひます.}.
$\ll$ 此の論文に現れる領域もすべて, 分岐点を内点として持たない有限領域です.
それ故, 此の条件は引き続き, 一般には明示しません.$\gg$
\bigskip
\begin{center}
\underline{$\rm{I}$ --- 有限葉正則域に於ける諸定理}
\end{center}
\bigskip
\underline{1} --- 此の章に於ては, \underline{有限葉}正則域に於ける, \underline{Cousin} の第一問題と函数の展開とに
就て述べ, 第二章以後の準備をします. 方法は, 第一基礎的補助定理と, \underline{H. Cartan--P. Thullen} の定理とに依るものであって, 畢竟第一報告のそれと同一です.
\footnote{此の為には, 必ずしも第一基礎的補助定理を要しないのであって,
第八報告の定理 1 があれば充分です（方法に就ては, 前報告, 第1節参照).
然し, 此の方法は分岐点が入って来れば使へなくなります. 所で,
此の第一次拡張（第七及至第十一報告)の目的の一つは, 此の方向の将来の研究を
秩序立てるにあったのです. 本文の方法を撰んだ所以です. 尚, 此の第八報告の
定理 1 に依る方法に依て本章の結果が得られることは, H. Behnke, K. Stein も
屡々指摘して居ます（下記論文参照).
H. Behnke--K. Stein : Approximation analytischer Funktionen in vorgegebenen
Be-\\ reichen des Raumes von $n$ komplexen Ver\"anderlichen, 1939, (Nachrichten
von der Gesellschaft der Wissenchaften zu G\"ottingen).
H. Behnke--K. Stein : Die Konvexit\"at in der Funktionentheorie mehrerer
komplexer Ver\"anderlichen, 1940, (Mitteilungen der Mathematischen
Gesellschaft in Hamburg).
H. Behnke--K. Stein : Die S\"atze von Weierstrass und Mittag--Leffler auf
Riemann-\\ schen Fl\"achen, 1940, (Vierteljahrsschaft der Naturforschenden
Gesellschaft in Z\"urich).}
先づ, （基礎的)補助定理 $\rm{I}$ を場合に適合した形に変へませう.
此の補助定理（第八報告)を今一度申しますと :
\medskip
\underline{補助定理 $\rm{I}$} --- $\ll$ 空間 $(x)$ の単葉有限筒状域 $(X)$ に固有集合体
$\Sigma$ を考へ, $\Sigma$ を含み $(X)$ に含まれる一つの単葉開集合を $V$ とし,
$\Sigma$ は $V$ に於ける正則函数 $f_1 (x), f_2 (x), \ldots, f_p (x)$ の
共通零点の集合として与へられて居ると考へる. $(X^0)$ を $(X^0) \Subset (X)$ なる
筒状域とし, $\Sigma$ の $(X^0)$ 内の部分を $\Sigma_0$ とする. 然るときは,
$\varphi (x)$ を $V$ に於て正則有界な任意の函数とするとき, $V$ に於て
$ | \varphi (x) | < M$ とすれば, $(X^0)$ に於て正則な函数 $\Phi (x)$ を,
$\Sigma_0$の各点で
$$
\Phi (x) \equiv \varphi (x) \quad (\mathrm{mod}.\ f_1, f_2, \ldots, f_p)
$$
であって, $(X^0)$ で
$$
|\Phi (x) | < K M
$$
を充たす様に見出すことが出来る.
ここに $K$ は $\varphi (x)$ に無関係な正数である.$\gg$
\medskip
$R$ を $n$ 複素変数 $x_1, x_2, \ldots, x_n$ の空間に於ける
(分岐点を内点として持たない有限)領域, 又は共通点を持たない様な
可附番個のか様な領域の和とし, $R$ 上に \underline{次の三つの条件を充たす様な
解析的多面体 (点集合) $\Delta$} を考へます：
\underline{$1^\circ, \ \Delta \Subset R$ であること.} (従って $\Delta$ は $R$ の連結成分の
有限個内に含まれて居て, 有界且つ有界葉です.)
\underline{$2^\circ, \ \Delta$ は次の形に定義せられること} :
$$
P \!\in\! R, \ x_i \!\in\! X_i, \ f_j (P) \!\in\! Y_j \ \
(i= 1, 2, \ldots, n; j= 1, 2, \ldots, \nu), \leqno{\quad (\Delta)}
$$
此処に $(x)$ は点 $P$ の座標, $X_i, Y_j$ は平面上の(有限) \underline{単葉}領域,
$f_j (P)$ は $R$ に於ける正則函数（$R$ の各連結成分に於て \underline{一価}正則な
解析函数の意, 以下同様)です.
\underline{$3^\circ$, \ 数値系 $[x_1, x_2, \ldots, x_n, f_1 (P), f_2
(P), \ldots, f_\nu (P)]$ は,
$\Delta$ の相異なる}\\ \underline{点に対して, 必ず相異なること.}
\medskip
複素変数 $y_1, y_2, \ldots, y_\nu$ を導入して空間 $(x, y)$ を考へ, 其処に
筒状域 $(X, Y), \ x_i \!\in\! X_i, \, y_j \!\in\! Y_j \
(i= 1, 2, \ldots, n; j= 1, 2, \ldots, \nu)$ と, 固有集合体,
$$
y_j= f_j (P), \ P \!\in\! \Delta \qquad (j= 1, 2, \ldots, \nu)
\leqno{\quad (\Sigma)}
$$
とを考へます. $\Delta$ 上の $(x)$を座標とする点$P$と, $\Sigma$上
の $[(x, f (P)]$ を座標とする点 $M$ とを
対応せしめます. 条件 $3^\circ$ に依て, $\Delta$ の相異なる二点 $P_1, P_2$ には
常に $\Sigma$ の相異なる二点 $M_1, M_2$ が対応するのですから此の対応は
一対一です. $\Sigma$ の点は總て $(X, Y)$ に含まれ, 其の境点は總て $(X, Y)$ の
境界上にあります. ($f_j (P) \ (j= 1, 2, \ldots, \nu)$ を単に
$\Delta$ に於て正則な函数としますと, 前半は成立しますが, 後半は成立しません.)
$X^{0}_{i}, Y^{0}_{j} \ (i= 1, 2, \ldots, n; j= 1, 2, \ldots, \nu)$ を
$X^{0}_{i} \Subset X_i, \, Y^{0}_{j} \Subset Y_j$ なる平面上の領域とし, 之に応ずる
$\Delta$ を $\Delta_0$ としますと, $\Delta_0 \Subset \Delta$ です. $\Delta_0$ を
$\Delta_0 \Subset \Delta$ なる任意の集合としますと, 若し $P_1, P_2$ が何れも
$\Delta_0$ に屬し, 且つ同じ座標を持つならば, $M_1, M_2$ 間の距離は
$0$ と異なる下端を持ちます.
$\varphi (P)$ を $\Delta$ に於て正則な任意の函数とします. $\Delta$ の点
$P$ と $\Sigma$ の点 $M$ とが上述の対応関係にあるとして,
$$
\varphi (M) = \varphi (P)
$$
に依て, $\Sigma$ 上に於て函数 $\varphi (M)$ を考へます. 上に見ましたことから,
$\Sigma$ を含む或る決った単葉開集合内で定義せられて居て, $\Sigma$ 上で
$\varphi (M)$ となり, 局地的には $(y)$ に無関係である様な, $(x, y)$ に関する
正則函数を考へ得ることが分かります. 故に, 補助定理 $\rm{I}$ に次の形を与へる
ことが出来ます：
\medskip
\underline{補助定理 $\rm{I'}$} --- \underline{上述の状勢に於て, $(X^{0},
Y^{0})$を
$(X^{0}, Y^{0}) \Subset (X, Y)$なる}\\ \underline{筒状域とすれば,\:\:
$\Delta$ に於て正則有界な函数
$\varphi (P)$ \,が与へられた時, \ $(X^{0},$}\\ \underline{$Y^{0})$ \,に於て正則な函数\, $\Phi (x, y)$
\,を求め,\:\: $\Delta$ \,に於て\, $| \varphi (P)| < N$ \,ならば,}\\
\underline{$(X^{0}, Y^{0})$ \,に於て\,
$| \Phi (x, y)| < K N$ \:であって,\:\: $P$ \,の座標を $(x)$ とすれば,}\\
\underline{$(X^{0},
Y^{0})$ 内の $\Sigma$ 上の任意の点 $[x, f (P)]$ に於て, \ 値 $\varphi (P)$
をとる様にす}\\
\underline{ることが出来る. 此処に $K$ は $\varphi (P)$ に無関係な或る正数である.}\vspace{1mm}
\medskip
此の解析的多面体 $\Delta$ と, 正則函数族に関して凸状な有限葉領域との間には,
次の関係があります
\footnote{凸性の定義に就ては, 前報告, 第一節参照.} :
\medskip
\underline{補助定理1} --- \underline{$\mathfrak{D}$ を空間 $(x)$ の正則
域とし, \ $\mathfrak{D}_0$ を\, $\mathfrak{D}$ \,に於ける正則函}\\ \underline{数の全体に関して凸状であって, 有限葉である様な,
$\mathfrak{D}$の部分開集合とす}\\ \underline{る. 然る時は, $E \Subset \mathfrak{D}_0$ なる
任意の集合 $E$ に対して, 解析的多面体 $\Delta$ を,}\\ \underline{$E
\Subset \Delta$ であって, \
$\mathfrak{D}_0$ の或る部分開集合 $R$ に関して, \,上述の三条件を充}\\ \underline{たす様に
選ぶことが出来る. \ \,尚, \,此の際\, $f_j (P) \ (j= 1, 2, \ldots, \nu)$ として,}
\\
\underline{$\mathfrak{D}$ \,に於ける正則函数をとり, \ \,$X_i \ (i= 1, 2, \ldots, n)$ として
円 \ $| x_i| < r$ を,}\\ \underline{$Y_j$ として円 \ $| y_j| < 1$ をとることが出来る.}\vspace{1mm}
\footnote{此の補助定理に就ては, H. Behnke--K. Stein の前掲三論文中,
始めの二つ参照. }
\medskip
\underline{証明}~ $F$ を $\mathfrak{D}_0$ に関して有界な $\mathfrak{D}_0$ の任意の部分集合
としますと, $\mathfrak{D}_0$ は有限葉ですから,
$$
F \Subset \mathfrak{D}_0
$$
であることが容易に分かります. 逆に, $F \Subset \mathfrak{D}_0$ ならば $F$ は
($\mathfrak{D}_0$ が有限葉でなくても) $\mathfrak{D}_0$ に関して有界です.
故に此の際此の二つの概念は一致します.
$\mathfrak{D}$ に於て正則な函数のすべてからなる函数族を $(\mathfrak{F})$ としますと,
$\mathfrak{D}_0$ は $(\mathfrak{F})$ に関して凸状であって, $E \Subset \mathfrak{D}_0$
ですから, 上に見ましたことから, $E \subset \mathfrak{D}^{'}_{0} \Subset \mathfrak{D}_0$
なる開集合 $\mathfrak{D}^{'}_{0}$ を撰び, $\mathfrak{D}^{'}_{0}$ に属しない
$\mathfrak{D}_0$ の任意の点 $P_0$ に対し, $(\mathfrak{F})$ 中に少なくとも一つの函数
$\varphi (P)$ が存在し,
$$
| \varphi (P_0)| > \max | \varphi (E)|
$$
(右辺は $E$ に於ける $| \varphi (P)| $ の上端)となる様にすることが出来ます.
$\mathfrak{D}^{'}_{0}$ の $\mathfrak{D}_0$ に関する最短距離を $\rho$ とし, $r$ を $E$
の任意の点 $P (x)$ が $| x_i| < r \ (i=1, 2, \ldots, n)$ を充たす様な正
数とします. \vspace{1mm} $\mathfrak{D}_0$に関する境界距離が
{\small$\displaystyle\frac {1}{2}$}\,$\rho$ \vspace{1mm} である様な
$\mathfrak{D}_0$ の点の集合を考へ, 其の閉多円筒 $| x_i| \leq 2r$ 上の部分を
$\Gamma$ とします. $\Gamma$ は上に見ましたことから, 閉集合です. $\Gamma$ の
任意の点 $M$ に対し, $M$ を中心とする $\mathfrak{D}$ に属する充分小さな多円筒 $(\gamma)$
と, $(\mathfrak{F})$ の函数 $f (P)$ とが対応し,
$$
\max |f [(\gamma)]| > 1, \quad \max | f (E)| < 1
$$
を充たすこと明白です. 故に, \underline{Borel--Lebesgue} の補助定理に依て, か様な
$(\gamma)$ の有限個を以て $\Gamma$ を被覆することが出来ます. 之等に
対応する函数を $f_1 (P), f_2 (P), \ldots, f_\lambda (P)$ とします.
$R = \mathfrak{D}^{(\frac {\rho}{2})}_{0} \vspace{1mm} \ (\mathfrak{D}_0$ に関する
境界距離が {\small$\displaystyle\frac {\rho}{2}$}\vspace{1mm} より大きい様な $\mathfrak{D}_0$ の点の集合)
として, 次の解析的多面体 $\Delta$ を考へます：
$$
P \!\in\! R, \ | x_i| \!<\! r, \ | f_j (P)| \!<\! 1
\ \ (i\!=\! 1, 2, \ldots, n; j\!=\! 1, 2, \ldots, \lambda) \leqno{\quad (\Delta)}
$$
$E \subset \Delta$ であって, $\Delta \Subset R$ であること明らかです
(補助定理の条件は $E \Subset \Delta$ となって居ますが, 之はどちらでも同じです.)
条件 $3^\circ$ を調べませう. $\mathfrak{D}$ は正則域ですから, $\mathfrak{D}$ を
自身の正則域とする様な函数が存在します. 其の一つを $F (P)$ とします.
然うしますと, 正則域の定義
\footnote{Behnke--Thullen の著書の16頁参照.}
に依て, $\mathfrak{D}$ の相重なる（同じ座標を持つ)二点 $P_1, P_2$ に於ける
$F (P)$ の要素は
必ず違って居ます. 故に, $F (P)$ と $x_i \ (i= 1, 2, \ldots, n)$ に関する其の導函数との中には, $P_1, P_2$ で相異なる値をとるものが必ずあります. 之等の導函数も
亦 $\mathfrak{D}$ で正則であること云ふ迄もありません. $\Delta$ と其の境界とからなる集合を $\overline{\Delta}$ としますと $\Delta \Subset \mathfrak{D}_0$ ですから
$\overline{\Delta}$ は閉集合です. 故に, \underline{Borel--Lebesgue} の補助定理に依て,
$F (P)$ 及び其の導函数中から, 有限個の函数
$$
\varphi_1 (P), \ \varphi_2 (P), \ldots, \ \varphi_\mu (P)
$$
を撰び, $\overline{\Delta}$ の相重なる二点に対して, 数値系
$[\varphi_1 (P), \varphi_2 (P), \ldots, \varphi_\mu (P)]$ が必ず違って
居る様に出来ます. 之等の函数は $\Delta$ に於て有界です.
$$
\max | \varphi_k (\Delta)| < N, \quad f_{\lambda + k} (P) =
\frac {1}{N} \varphi_k (P) \qquad (k= 1, 2, \ldots, \mu)
$$
とし, $P \!\in\! R, \ | x_i| \!<\! r, \ | f_j (P)| \!<\! 1
\ \ (i \!=\! 1, 2, \ldots, n;\,j \!=\! 1, 2, \ldots, \nu;\,\nu \!=\! \lambda + \mu)$
なる三条件を同時に充たす $\mathfrak{D}$ の点の集合を考へますと,
此の集合は $\Delta$ です. $\Delta$ の此の形の表現は条件 $1^\circ, 2^\circ, 3^\circ$ を
すべて充たします. ~ (証明終)
所で正則域は次の性質を持って居ます：
\medskip
\underline{H.\:Cartan--P.\:Thullen\:の第一定理} --- \underline{有限正則域は
其処に於ける正則函}\\ \underline{数の
全体に関して凸状である.}
\medskip
此の定理は, \ \:\underline{H. Cartan--P. Thullen \:の同時解析接続の可能性に関
する}\\ \underline{基礎定理}
\footnote{Behnke--Thullen の著書の第六章, 第1節及び下記論文参照 :
H. Cartan--P. Thullen : Regularit\"ats--und Konvergenzbereiche, 1932,
(Math. Annalen). }
の直接の結果です.
\footnote{か様に, 我々は Cartan--Thullen の基礎定理を使ひます.
所で, 此の定理は, 分岐点や無窮遠の点が入って来れば成立しなくなります.
それで, 将来此の点をどうするかと云ふ問題がある釈ですが, 実際は,
此の論文では, 此の定理はなくてもすむのです. 定理 $\rm{I}$ の脚註参照.
尤も, 此の種の問題は, 他にも無い釈ではありませんが, 上述のものが一
番目立つ様に思ひます. }
\bigskip
\underline{2} --- 函数の展開に就て御話します
\footnote{第一報告, 第4節参照.}.
補助定理 1 の $\Delta$ を考へます. 但し, 終りに追加した諸条件をも
充たすものをとります. $\Delta$ は次の形です：
$$
P \!\in\! R, \ | x_i| \!<\! r, \ | f_j (P)| \!<\! 1 \quad
(i\!=\! 1, 2, \ldots, n; j\!=\! 1, 2, \ldots, \nu) \leqno{\quad (\Delta)}
$$
複素変数 $y_1, y_2, \ldots, y_\nu$ を導入し, 空間 $(x, y)$ に多円筒,
$$
| x_i| < r, \quad | y_j| < 1 \qquad
(i= 1, 2, \ldots, n; j= 1, 2, \ldots, \nu) \leqno{\quad (C)}
$$
と, 固有集合体,
$$
y_j= f_j (P), \quad P \in \Delta \qquad (j= 1, 2, \ldots, \nu)
\leqno{\quad (\Sigma)}
$$
とを考へます. $r_0, \rho_0 $を $r_0 < r, \ \rho_0 < 1$ なる正数とし, $\Delta, (C),
\Sigma$ に於ける $(r, 1)$ を此の $(r_0, \rho_0)$ を以て置き換へて
得るものを夫々 $\Delta_0, (C_0), \Sigma_0$ とします.
$\varphi (P)$ を $\Delta$ に於ける任意の正則函数としますと,
補助定理 $\rm{I'}$ に依て, 多円筒 $(C_0)$ に於て正則な函数 $\Phi (x, y)$ を,
$\Sigma_0$ 上の任意の点 $[x, f (P)]$ に於て値 $\varphi (P)$ をとる様に
作ることが出来ます. 此の $\Phi (x, y)$ を $(C_0)$ に於て,
原点を中心とする Taylor 級数に展開します. 収歛は $(C_0)$ の各点に於て斉一です.
此の展開に $y_j= f_j (P) \ (j= 1, 2, \ldots, \nu)$ を代入しますと,
$\varphi (P)$ の $\Delta_0$ に於ける展開が得られます. 其の各項は $\mathfrak{D}$ に於ける正則函数であって, 収歛は $\Delta_0$ の各点に於て斉一です.
$\mathfrak{D}_0$ は此の $\Delta$ と同じ性質を持つ $\mathfrak{D}_0$ の
部分集合の逓増列の極限ですから, 次の定理が成立します：
\medskip
\underline{定理1} --- \underline{$\mathfrak{D}$ を空間 $(x)$ の正則域とし, $\mathfrak{D}_0$を
$\mathfrak{D}$ に於ける正則函数の全}\\ \underline{体 $(\mathfrak{F})$ に関して凸状であって,
有限葉である様な$\mathfrak{D}$の部分開集合とする.}\\ \underline{然る時は,
$\mathfrak{D}_0$に於ける任意の正則函数は, $\mathfrak{D}_0$の各点に於て
斉一に収歛す}\\ \underline{る様な, $(\mathfrak{F})$ の函数の級数に展開することが出来る.}
\bigskip
\underline{3} --- 次に \underline{Cousin} の第一問題に就て述べませう
\footnote{第一報告の第3節, 及び第5節の定理 $\rm{I}$ の証明参照.}.
補助定理から始めます.
\medskip
\underline{補助定理2} --- \underline{補助定理 $\rm{I'}$ の $\Delta$ をとり, 其の一基点を通る超平面を
$L$,}\\ \underline{$\Delta$ の$L$上の部分を$S$とする. $\Delta_0$ を $\Delta_0
\Subset \Delta$ なる任意の開集合とし, $\Delta_0$}\\ \underline{の$L$の一方の側にある部分を
$\Delta^{'}_{0}$, 他方の側にある部分を $\Delta^{''}_{0}$ とする. 然}\\ \underline{る時は,
$R$上の$S$の近傍に於て正則な函数 $\varphi (P)$ が与へられた時,
$\Delta^{'}_{0}$ に}\\ \underline{於て正則な函数 $\varphi_1 (P)$ と $\Delta^{''}_{0}$
に於て正則な函数 $\varphi_2 (P)$ とを, 何れも $\Delta_0$}\\ \underline{ 内の $S$ の各点に
於て矢張り正則であって, 恒等的に}
$$
\varphi_1 (P) - \varphi_2 (P) = \varphi (P)
$$
\underline{となる様に, 見出すことが出来る.}\vspace{1mm}
\medskip
\underline{証明} ~ $x_1$ を実部と虚部とに分ち, $x_1 = \xi + i \,\eta$
（$i$ は虚単位)とします. $L$ を
$$
\xi = 0 \leqno{\quad (L)}
$$
であると見ませう. $(x)$ に関する一次変換によって常に此の形にすることが
出来ますから, か様に考へて支障ありません. $\Delta$ は次の形でした：
$$
P \!\in\! R, \ x_j \!\in\! X_j, \ f_k (P) \!\in\! Y_k \ \
(j\!=\! 1, 2, \ldots, n; k\!=\! 1, 2, \ldots, \nu). \leqno{\quad (\Delta)}
$$
之に対応して, 上に繰り返しました様に, 空間 $(x, y)$ に筒状域 $(X, Y)$ 及び固有集合体
$\Sigma$ を考へます. $X^{0}_{j}, X^{1}_{j}, Y^{0}_{k}, Y^{1}_{k}$ を
$$
X^{0}_{j} \!\Subset\! X^{1}_{j} \!\Subset\! X_j, \ Y^{0}_{k} \!\Subset\! Y^{1}_{k} \!\Subset\! Y_k
\ \ (j\!=\! 1, 2, \ldots, n; k\!=\! 1, 2, \ldots, \nu)
$$
なる平面上の領域とし, $\Delta$ の $(X, Y)$ を $(X^0, Y^0)$ に依て
置き換へたものを $\Delta_0$ とします. 補助定理の $\Delta_0$ を此の形で
あると看做して支障ありません.
$x_1$ 平面上に於て, 直線 $\xi = 0$ の $X_1$ 内の部分を含み, $X_1$ に
含まれる開集合を $A$ とします. 但し $A$ を充分此の直線に近くとり,
$\Delta$ の $x_1 \in A$ 上の部分に於て $\varphi (P)$ が正則である様にします.
$A_1$ を $A_1 \Subset A$ であって, $X^{1}_{1}$ に関して, $X_1$ に関する $A$ と
同じ関係にある開集合とします.
補助定理 $\rm{I'}$ に依て, 筒状域, $x_1 \in A_1, \, (x, y) \in (X^1, Y^1)$
に於て正則であって, 此の筒状域内の $\Sigma$ 上の任意の点 $[x, f (P)]$ に於て
値 $\varphi (P)$ をとる様な函数 $\Phi (x, y)$ が存在します. $x_1$ 平面の
虚軸上に一つ又は有限個の線分の和（閉集合) $l$ を, $A_1$ に含まれ,
虚軸の $X^{0}_{1}$ 内の部分を含む様にとり, \underline{Cousin の積分}
$$
\Psi (x, y) = \frac {1}{2 \pi i} \int_{l}
\frac {\Phi (t, x_2, \ldots, x_n, y)}{t - x_1}\,dt
$$
を考へます. 但し, $\Delta_0$ の $L$ の左方 $(\xi < 0)$ の部分が
$\Delta^{'}_{0}$ であって, 右方の部分が $\Delta^{''}_{0}$ であるとして,
虚軸の正の方向に積分します. $(X^0, Y^0)$ の $\xi < 0$ の部分を $(C')$,
$\xi > 0$ の部分を $(C'')$ としますと, $\Psi (x, y)$ は $(C')$ 及び
$(C'')$ に於て正則です. $(C')$ に於ける $\Psi$ を $\Psi_1$, $(C'')$ のそれを
$\Psi_2$ として区別しますと, $\Psi_1, \Psi_2$ は何れも $(X^0, Y^0)$ 内の
$\xi = 0$ の各点に於て矢張り正則であって, 其の間に
$$
\Psi_1 (x, y) - \Psi_2 (x, y) = \Phi (x, y)
$$
なる関係があります. 故に
$$
\varphi_1 (P) =\Psi_1 [x, f (P)], \quad \varphi_2 (P) =\Psi_2 [x, f (P)]
$$
$(x)$ は $R$ の点 $P$ の座標, は所求の函数です. （証明終)
\medskip
$\mathfrak{D}$ を空間 $(x)$ の領域とします. $\mathfrak{D}$ の
各点 $P$ に対し, $P$ を中心とする $\mathfrak{D}$ に属する多円筒
$(\gamma)$ と, $(\gamma)$ に於ける有理型函数 $g (P)$ とが対応し,
其の全体が次の同等条件を充たすとしませう : \ $(\gamma_1), (\gamma_2)$ を共通部分 $(\delta)$ を持つ任意の一対の
$(\gamma)$ としますと, 対応する函数 $g_1 (P), g_2 (P)$ は $(\delta)$ に
於て同等, 詳しく云ひますと, $g_1 (P) - g_2 (P)$ は正則であること.
か様にして $\mathfrak{D}$ に於て極が定義せられました. 此の時,
$\mathfrak{D}$ に於て有理型函数 $G (P)$ を, 与へられた極をとる様に,
云ひ換へますと, 各 $(\gamma)$ に於て $g (P)$ と同等である様に作ること,
之が \underline{Cousin の第一問題}です.
$\mathfrak{D}$ を有限葉正則域とします. \underline{Cartan--Thullen} の第一定理に依て,
$\mathfrak{D}$ は $\mathfrak{D}$ に於ける正則函数の全体 $(\mathfrak{F})$ に関して
凸状です. 故に, 補助定理1に於て $\mathfrak{D}$ = $\mathfrak{D}_0$ と
見ることが出来ますから, $\mathfrak{D}$ には此の補助定理に述べた $\Delta$ が
存在します. 但し, 此の度は $X_i, Y_j \ (i= 1, 2, \ldots, n; j= 1, 2,
\ldots, \nu)$ を有界閉領域として, \underline{解析的閉多面体 $\Delta$} をとるのが便利です.
(勿論, $f_j (P)$ は, $(\mathfrak{F})$ の中から撰びます.) か様に, $\mathfrak{D}$
は解析的閉多面体の列,
$$
\Delta_1, \Delta_2, \ldots, \Delta_p, \ldots
$$
の極限です. 此処に $\Delta_p$ は上述の $\Delta$ と同じ性質であって,
$\Delta_{p + 1}$ の内点の集合を $E$ とする時, $\Delta_p \Subset E$ です.
$\Delta_p$ に就て見ます. 前報告の第三節で述べました様に $\Delta_p$ を
$(A)$ に分割します. 但し, 此度は $(A)$ 及び其の基領域 $(\alpha)$ として,
$2 n$ 次元閉立方体を撰びます. 又, $(A)$ 中には不完全な形のものがあっても
よいことにします. $(A)$ を充分小さくして, 任意の $(A)$ が上述の $(\gamma)$ の
適当な一つに対し, $(A) \Subset (\gamma)$ となる様にします. 各 $(A)$ に対し,
か様な $(\gamma)$ の任意の一つを選び, それに附随する $g (P)$ を
此の $(A)$ に附随せしめます.
$(A)_1, (A)_2$ を一つの面（$2 n - 1$ 次元閉立方体)に於て隣接する一対の
$(A)$ としますと, 之等に附随する $g_1 (P), g_2 (P)$ は共通面の近傍
（$\mathfrak{D}$ 上に於ける近傍, 以下同様)に於て同等です. 故に,
上の補助定理2に依て, 和 $(A)_1 \cup (A)_2$ の近傍に於て,
与へられた極をとる有理型函数が存在します. 其の基領域が, 例へば
$$
(\alpha^{(1)}_{j, q}, \alpha^{(2)}, \ldots, \alpha^{(n)}),
$$
$\alpha$ は閉正方形, $q$ 及び $\alpha^{(2)}, \ldots, \alpha^{(n)}$ は
決まったもの, $j$ は任意のもの, なる形の $(A)$ の全体の和に就ても同様です.
此の際, 此の和は勿論連結でなくてもよろしい. 此の推理法を反覆することに依て,
$\Delta_p$ の近傍に於て, 与へられた極をとる有理型函数の存在することが分かります.
其の一つを $G (P)$ とします.
か様にして,
$$
G_1 (P), G_2 (P), \ldots, G_p (P), \ldots,
$$
が得られました.
$$
H (P) = G_{p + 1} (P) - G_p (P)
$$
を見ますと, $H (P)$ は $\Delta_p$ の近傍に於ける正則函数です. 故に,
定理1に依て, $H_p$ を, $\Delta_p$ の近傍に於て斉一に収歛する様な,
$(\mathfrak{F})$ の函数の級数に展開することが出来ます. 与へられた極をとる有理型函数
$G (P)$ が $\mathfrak{D}$ に於て存在することは, 之から直ちに分かります.
(証明法は $\mathfrak{D}$ が単葉筒状域である場合と全く同様です.) か様に, 次の定理が成立します.
\medskip
\underline{定理2} --- \underline{有限葉正則域に於ては, Cousin の第一問題は常に解ける.}
\begin{center}
\bigskip
\underline{$\rm{II}$ --- 主問題}
\end{center}
\bigskip
\underline{4} --- 此の章では, 始めに御話した一聯の問題から抽き出した主要部分を,
第一基礎的補助定理に依て解決します.
\footnote{此の定理を使ふと云ふ点以外は, 本質的には,
第六報告の第一章と同一です.}
先づ, 問題を説明します. $\mathfrak{D}$ を空間 $(x)$ の \underline{有界有限葉}領域とし,
$\mathfrak{D}$ の基領域と交はる, 一つの超平面を考へます. $x_1$ を
実部と虚部に分ち
$$
x_1= \xi + i \,\eta
$$
とします. 簡単の為, 此の超平面を $\xi = 0$ であると看做しませう. $a_1, a_2$ を
$$
a_2 < 0 < a_1
$$
なる実数とし, 超平面 $\xi = a_1, \ \xi = a_2$ も, 何れも $\mathfrak{D}$ の
基領域と交はると見ます. $\mathfrak{D}$ の $\xi < a_1$ なる部分を
$\mathfrak{D_1}, \ \xi > a_2$ なる部分を
$\mathfrak{D}_2$, \ $a_2 < \xi < a_1$ なる部分を $\mathfrak{D}_3$ と
します. \underline{$\mathfrak{D}_1, \mathfrak{D}_2$ の各連結成分は正則域}であると假定
します. 然うしますと, $\mathfrak{D}_3$ に就ても必然同様です.
$f_j (P) \ (j= 1, 2, \ldots, \nu)$ を \underline{$\mathfrak{D}_3$ に於ける正則函数}と
します. 次の様な $\mathfrak{D}$ の部分集合 $E$ を考へます\,: \ $\mathfrak{D}_3$ に属しない $\mathfrak{D}$ の点は總て $E$ に属する.
$\mathfrak{D}_3$ の点は, $\nu$ 個の条件
$$
| f_j (P)| < 1 \qquad (j= 1, 2, \ldots, \nu)
$$
を同時に充たすならば $E$ に属し, 然うでなければ $E$ に属しない.
$E$ は $\xi < a_2$ なる部分及び $\xi > a_1$ なる部分に同時に広がって
居る連結成分を持つと考へ, 其の一つを $\Delta$ としませう.
\underline{此の $\Delta$ に関して, 次の三つの条件が充たされると假定しま
す :}
\medskip
\underline{$1^\circ$, \ $\delta_1$ を $0 < \delta_1 < a_1, \,-a_2$ なる実数とします.
$ | \xi| < \delta_1$ を充たす$\Delta$の}\\ \underline{点 $P (x)$ の集合を $A$ としますと,}
$$
A \Subset \mathfrak{D}.
$$
\underline{$2^\circ$, \ $\delta_2$ を正数, $\varepsilon_0$ を 1 より小さい正数とします.
$p$ を $1, 2, \ldots, \nu$ の任}\\ \underline{意の一つとした時,}
$$
| f_p (P)| \geq 1 - \varepsilon_0
$$
\underline{を充たす $\mathfrak{D}_3$ の点は,}
\begin{center}
$| \xi - a_1 | < \delta_2$ \quad \underline{又は} \quad $| \xi - a_2 | < \delta_2$
\end{center}
\underline{が充たされる部分に存在しないこと.}\vspace{1mm}
\underline{$3^\circ$, 数値系}
$$
[f_1 (P), f_2 (P), \ldots, f_\nu (P)]
$$
\underline{は $A$ の相重なる二点で決して同一とならないこと.}\vspace{2mm}
第二の条件に依て, $\Delta$ は領域です. $\rho_0$ を $1 - \varepsilon_0 <
\rho_0 < 1$ なる実数とし, 次の様な $\Delta$ の部分集合 $\Delta_0$ を
考へます： $\mathfrak{D}_3$ に属しない $\Delta$ の点は總て $\Delta_0$ に
属する. $\mathfrak{D}_3$ に属する $\Delta$ の点は, $\nu$ 個の条件
$$
| f_j (P)| < \rho_0 \qquad (j= 1, 2, \ldots, \nu)
$$
を同時に充たすならば $\Delta_0$ に属し, 然うでなければ, $\Delta_0$ に属しない.
上述の条件 $2^\circ$ に依て $\Delta_0$ は開集合です. $\xi < 0$ なる $\Delta_0$ の
部分を $\Delta^{'}_{0}$ , \ $\xi > 0$ のそれを $\Delta^{''}_{0}$ とします.
此の章の課題は次の通りです.
\medskip
\underline{此の状勢に於て, $A$ に於て正則な函数 $\varphi (P)$ が与へられた時,
\,$\Delta^{'}_{0}$ に於}\\ \underline{て正則な函数 $\varphi_1 (P)$と, $\Delta^{''}_{0}$に
於て正則な函数 $\varphi_2 (P)$とを, 何れも $\xi = 0$}\\ \underline{上の $\Delta_0$ の
各点に於て矢張り正則であって, 恒等的に}
$$
\varphi_1 (P) - \varphi_2 (P) = \varphi (P)
$$
\underline{となる様に, 作ること.}
\bigskip
\underline{5} --- 先づ補助定理 2 の方法を適用して, 此の問題の $\mathfrak{D}_3$ に関する
部分を解決しませう. $y_1, y_2, \ldots, y_\nu$ を複素変数とし,
空間 $(x, y)$ に固有集合体,
$$
y_k= f_k (P), \quad P \in \mathfrak{D}_3 \qquad (k= 1, 2, \ldots, \nu)
\leqno{\quad (\Sigma)}
$$
を考へます. $r,\,r_0$ を $r_0 \!<\! r$ なる正数とし, $r_0$ を充分大きくとり,
有界領域 $\mathfrak{D}$ が, 原点を中心とし $r_0$ を半徑とする多円筒に
含まれる様にします. $\rho$ を $\rho_0 < \rho < 1$ なる実数とし, 円筒
$(C), \ | x_j| < r, \,| y_k| < \rho \ (j= 1, 2, \ldots, n;
k= 1, 2, \ldots, \nu)$ ,
及び $(C_0), \ | x_j| < r_0, \,| y_k| < \rho_0$ を考へます.
$\delta$ を $\delta < \delta_1$ なる正数とし, 集合
$$
P \in A, \quad | \xi| < \delta, \quad | f_k (P)| < \rho
\qquad (k= 1, 2, \ldots, \nu) \leqno{\quad (A')}
$$
を考へます.
$\varphi (P)$ は $A$ に於て正則ですから, 補助定理 $\rm{I'}$ に依て,
$(C)$ と $| \xi| < \delta$ との共通部分に於て正則な函数 $\Phi (x, y)$ を,
$P \in A'$ なる $\Sigma$ の各点 $[x, f (P)]$ に於て値 $\varphi (P)$ となる様に
作ることが出来ます. $x_1$ 平面の虚軸上に線分（連結閉集合) $l$ を円 $| x_1|
< r$ に含まれ, 其の両端が円 $| x_1| < r_0$ 外にある様にとり, \underline{Cousin の積分}
$$
\Psi (x, y) = \frac {1}{2 \pi i} \int_{l}
\frac {\Phi (t, x_2, \ldots, x_n, y_1, \ldots, y_\nu)}{t - x_1} \,dt
$$
を考へます. 但し, 虚軸の正の方向に積分します.
$\Psi (x, y)$ に, $y_k = f_k (P)$ を代入しますと,
$$
\psi (P) = \frac {1}{2 \pi i} \int_{l}
\frac {\Phi [t, x_2, \ldots, x_n, f_1 (P), \ldots, f_\nu (P)]}{t - x_1} \,dt
$$
が得られます. $\psi (P)$ は $\Delta^{'}_{0} \cap \mathfrak{D}_3$ 及び
$\Delta^{''}_{0} \cap \mathfrak{D}_3$ に於て, 夫々正則な函数
$\psi_1 (P), \psi_2 (P)$ を表はします. 之等の函数は何れも $\Delta_0$ 内の
$\xi = 0$ 上の各点で矢張り正則であって, 其の間に $\psi_1 (P) - \psi_2 (P)
= \varphi (P)$ なる関係があります.
此の解の表現を少し変へませう. 平面上に原点を中心とし $\rho_0$ を半徑とする
円周 $\Gamma$ を描きます. \underline{Cauchy} に依て, $| \xi| < \delta, \, | x_j| < r,
\, | y_k| < \rho_0 \ (j= 1, 2, \ldots, n; k= 1, 2, \ldots, \nu)$ に
於て
$$
\Phi (x, y) = \frac {1}{(2 \pi i)^\nu} \int_{\Gamma} \int_{\Gamma}
\ldots \int_{\Gamma}
\frac {\Phi (x_1, \ldots, x_n, u_1, \ldots, u_\nu)}{(u_1 - y_1) \ldots
(u_\nu - y_\nu)} \,d u_1 d u_2 \ldots d u_\nu
$$
です. 積分は勿論, 各$\Gamma$ に就て正の方向に行ひます.
之を次の様に略記しませう：
$$
\Phi (x, y) = \frac {1}{(2 \pi i)^\nu} \int_{(\Gamma)}
\frac {\Phi (x, u)}{(u_1 - y_1) \ldots (u_\nu - y_\nu)} \,du.
$$
此の $\Phi (x, y)$ の積分表示に, $y_k = f_k (P)
\ (k = 1, 2, \ldots, \nu)$ を代入し,
$x_1$ を $t$ と置き換へ, 上に述べた $\psi (P)$ の積分表示に代入します.
$t = u_0$ としますと
$$
\psi (P) = \int_{(l, \Gamma)} \chi (u, P) \Phi (x', u) \,du, \leqno{\quad (1)}
$$
$$
\chi (u, P) = \frac {1}{(2 \pi i)^{\nu + 1}
(u_0 - x_1) [u_1 - f_1 (P)] \ldots [u_\nu - f_\nu (P)]}
$$
となります. 但し, $\Phi (x', u)$ は $\Phi (u_0, x_2, \ldots, x_n, u_1,
\ldots, u_\nu)$ の略記であって, 積分記号に就ては上に説明したと全く同様の
略記法を用ひました. 改めて説明しなくても明瞭と思ひます.
$\Delta_0 \cap \mathfrak{D}_3$ に於て, 上に述べた $\psi (P)$ の
積分表示の代りに此の $(1)$ を使ふことが出来ます.
\bigskip
\underline{6} --- 空間 $(u)$ の筒状閉集合 $(l, \Gamma), \ u_0 \in l, \, u_k
\in \Gamma \ (k = 1, 2, \ldots, \nu)$ を見ますに,
$(l, \Gamma)$ の如何程でも近くに, 之を含む単葉正則域があります. 其の一つを
$V$ とします. 但し, $V$ は充分 $(l, \Gamma)$ に近くとるのですが,
それに就ては其の都度説明することにしませう.
先づ, \ \,\underline{$(V, \mathfrak{D}_1) \ \ ((u) \in V, \ P (x) \in \mathfrak{D}_1)$ に於て
有理型函数 \,$\chi_1 (u, P)$ \,を,}\\ \underline{$(V, \mathfrak{D}_3)$ に於て, (1)式の
$\chi (u, P)$ と同じ極を持ち, 他の部分で極を持たな}\\ \underline{い様に作らう}と云ふのですが,
$(V, \mathfrak{D}_1)$ は有限葉正則域であって, 極の分布に就ては, $\Delta$ に
関する条件 $2^\circ$ に依て, $V$ を充分 $(l, \Gamma)$ に近くとって置けば,
同等条件が充たされますから, 定理2に依て, 此のことは可能です.
$\chi - \chi_1$ は $(V, \mathfrak{D}_3)$ に於て正則です. 所で, \underline{Cartan--Thullen} の
第一定理に依て, $(V, \mathfrak{D}_3)$ は明らかに, $(V, \mathfrak{D}_1)$ に於て
正則な函数の全体に関して凸状です. 故に, 定理1に依て, $\chi - \chi_1$ は,
$(V, \mathfrak{D}_3)$ の各点で斉一に収歛する様な, $(V, \mathfrak{D}_1)$ に於て
正則な函数の級数に展開することが出来ます. それ故, $V$ を $(l, \Gamma)$ に更に
充分近く撰んで置きますと, 正数 $\varepsilon$ に対し, 次の様な函数 $F_1
(u, P)$ が応じます : \ \ \underline{$F_1 (u, P)$ は \,$(V, \mathfrak{D}_1)$ \,に於て正則であって,}\\
\underline{第 4 節で述べた解析的多面体 $A$ に対し, $(V, A)$ に於て}
$$
| \chi - \chi_1 - F_1| < \varepsilon.
$$
$$
K_1 (u, P) = \chi - \chi_1 - F_1
$$
と置きます. $K_1 (u, P)$ は $(V, \mathfrak{D}_3)$ に於ける正則函数であって,
$(V, A)$ に於ては $| K_1| < \varepsilon$ です. $\mathfrak{D}_2$ に対しては,
全く同様にして $K_2 (u, P)$ を作ります. 之等に依て, 積分 (1) を次の様に
変形します：
$$
I_1 (P) = \int_{(l, \Gamma)} [\chi (u, P) - K_1 (u, P)]\Phi (x', u) \,du
\leqno{\quad (2)}
$$
$$
I_2 (P) = \int_{(l, \Gamma)} [\chi (u, P) - K_2 (u, P)]\Phi (x', u) \,du
$$
$(u) \in (l, \Gamma)$ ならば, $\chi - K_1$ は $\chi_1 + F_1$ に等しいのですから,
$P (x)$ に関し $\mathfrak{D}_1$ に於て有理型であって, 特に $\Delta^{'}_{0}$ に
於て正則です. \:故に, \,\underline{$I_1 (P)$ は}\\
\underline{$\Delta^{'}_{0}$ に於て正則} です. 同様に,
\underline{$I_2 (P)$ は $\Delta^{''}_{0}$ に於て正則}です.\vspace{1mm}
解析函数 $I_1 (P), I_2 (P)$ は, $\xi = 0$ 上の $\Delta_0$ の各点に於て,
矢張り正則です. 何となれば, (1) の $\psi (P)$ が此の性質を持って居て,
$K_1, K_2$ は正則函数だからです. $\psi (P)$ の性質から, $I_1 (P), I_2 (P)$ の
間に\underline{次の関係}のあることが分かります：
$$
I_1 (P) - I_2 (P) = \varphi (P) - \int_{(l, \Gamma)}
[K_1 (u, P) - K_2 (u, P)] \Phi (x', u) \,d u. \leqno{\quad (3)}
$$
$$
K (u, P) = K_1 (u, P) - K_2 (u, P)
$$
と書き表します. 此の恒等式を改めて見ますに, $\varphi (P)$ は $P \in A$ に
於ける正則函数, $K$ は $(u) \!\in\! V, \,P \!\in\! \mathfrak{D}_3$ に於ける正則函数で
あって, $\Phi (x, y)$ は $(x, y) \in (C), \ | \xi| < \delta$ \,に於ける正則函数
ですから, \ 右辺は $P (x)$ に関する\\ \underline{$A$ に於ける正則函数}です.
故に, \underline{左辺に就ても其の通り}です.
$$
\varphi_0 (P) = I_1 (P) - I_2 (P)
$$
と置きます.
$\varphi_0, K$ を与へられた函数, $\varphi, \Phi$ を次に説明する様な関係を持つ
一対の 未知函数として, \underline{函数方程式}
$$
\varphi (P) = \int_{(l, \Gamma)} K (u, P) \Phi (x', u) \,d u + \varphi_0 (P)
\leqno{\quad (4)}
$$
を考へます. $\ll$ 此処に, $\Phi (x', u)$ は $\Phi (u_0, x_2, \ldots, x_n, u_1,
\ldots, u_\nu)$ の略記であって $\varphi_0 (P)$ は $A$ に於ける正則函数,
$K (u, P)$ は $(V, \mathfrak{D}_3)$ に於ける正則函数です. $(V, A)$ に於
ては \,$| K (u, P)| < 2\varepsilon$ \,です. \underline{未知函数 $\varphi
(P), \Phi (x, y)$}\\ \underline{に
就ては, (4) 以外に, 次の条件が要求せられて居ます} : \ $\varphi (P)$ は
$P \in A$ に於ける正則函数, $\Phi (x, y)$ は $(x, y) \in (C), \,| \xi| < \delta$ に於ける正則函数であって, $P \in A'$ なる $\Sigma$ の各点
$[x, f (P)]$ に於て, $\Phi (x, y) = \varphi (P)$ となること. $\gg$ \
か様な条件が課せられて居ますから, 此の函数方程式は定積分方程式と
餘り違ひません.
我々は, 正数 $\varepsilon$ を充分小さく撰べば, 此の方程式が必ず解けることを
云はうと云ふのですが, 其の前に, 此のことさへ云へば, それでよいことを確めて
置きませう. 上述の解 $\varphi (P), \Phi (x, y)$ が存在したと假定します.
此の $\Phi (x', u)$ を (2) に代入しますと, かくて得る $I_1 (P)$ は明らかに
$\Delta^{'}_{0}$ に於て正則です. 同様に, $I_2 (P)$ は $\Delta^{''}_{0}$ に
於て正則です. 之等の解析函数は, $\xi= 0$ 上の $\Delta_0$ の各点に於て
矢張り正則であること明かです. 其の間に (3) なる関係のあることも容易に
分ります. (以上は一度行った推理を, 改めて条件を明らかにして, 点検したに
過ぎません.) 故に, 此の $I_1 (P), I_2 (P)$ は第4節に述べた問題の解です.
か様に, 函数方程式 (4) が解けさへすればよろしい. 勿論其の際 $\varepsilon$ を
如何程小さく撰んで置いても支障ありません.
方程式 (4) を解きませう. 解析多面体 $A$ は次の形でした：
$$
P \!\in\! \Delta, \quad | \xi| \!<\! \delta_1, \quad | f_k (P)| \!<\! 1
\qquad (k \!=\! 1, 2, \ldots, \nu) \leqno{\quad (A)}
$$
又, $0 \!<\! \delta \!<\! \delta_1, \ \rho_0 \!<\! \rho \!<\! 1$ なる $(\delta, \rho)$
に依て, $A$ の $(\delta_1, 1)$ を置き換へて得る多面体が $A'$ でした.
$\rho \!<\! \rho' \!<\! 1,\ \delta \!<\! \delta' \!<\! \delta_1$ なる $(\delta', \rho')$ を
撰び $A'$ の $(\delta, \rho)$ を此の $(\delta', \rho')$ に依て置き換へて得る
解析的多面体を $A''$ とします. 之等の間には次の関係があります：
$$
A' \Subset A'' \Subset A.
$$
$\varphi_0 (P)$ は $A$ に於ける正則函数です. 故に $A''$ に於て有界です.
$A''$ に於て
$$
| \varphi_0 (P)| < M_0
$$
とします. \def\ーー{\rule[0.33zh]{2zh}{.03zh}} \underline{補助定理
$\rm{I'}$ に依て}, $(x, y) \!\in\! (C),\
| \xi| \!<\! \delta$ \ーー 今後此の筒状域を $(C')$ に依て表はします \ーー
\ \underline{$(C')$ に於て}正則な函数 $\Phi_0 (x, y)$ を, $P \!\in\! A'$ なる $\Sigma$ 上の各点
$[x, f (P)]$ に於て値 $\varphi_0 (P)$ をとり, \underline{$(C')$ に於て}
$$
| \Phi_0 (x, y)| < N M_0
$$
となる様に作ることが出来ます. 此処に $N$ は $\varphi_0 (P)$
(及び $M_0$, 並びに $\varphi_0 (P)$ が $A$ で正則であると云ふこと)
に無関係な或る正数です. 此の $\Phi_0 (x, y)$ から
$$
\varphi_1 (P) = K (\Phi_0) = \int_{(l, \Gamma)} K (u, P) \Phi_0 (x', u) \,d u
$$
なる操作 $K (\Phi_0)$ に依て, 函数 $\varphi_1 (P)$ を作ります. $(u) \in
(l, \Gamma)$ ならば, $P (x)$ に関して, $K (u, P)$ は $\mathfrak{D}_3$ に於て
正則であって, $\Phi_0 (x', u)$ は $| x_j| < r \ (j = 2, 3, \ldots, n)$, 従って
$(C)$ に於て正則です. 故に, \underline{$\varphi_1 (P)$ は
$\mathfrak{D}_3$ に於}\\
\underline{ける正則函数}です.
次に, $\varphi_1 (P)$ を評価しませう. $(u) \!\in\! (l, \Gamma), \,P \!\in\! A$ ならば,
$| K (u, P)| < 2 \varepsilon$ であって, $(C')$ に於て $| \Phi_0 (x, y)| <
N M_0$ ですから, \underline{$A$ に於て}
$$
| \varphi_1 (P)| < 2 \varepsilon N N_1 M_0, \quad N_1 = 2 r (2 \pi \rho_0)^\nu
$$
です. それで $\varepsilon$ を,
$$
2 \varepsilon N N_1 = \lambda < 1
$$
となる様に, 始めに撰んで置きませう.
\medskip
か様に, \underline{$\varphi_1 (P)$ は $A$ に於て, 従って勿論 $A''$ に於て, 有界な正則函数}
ですから, $\varphi_0 (P)$ に対して, $\Phi_0 (x, y)$ を一つ撰んだと全く同様に,
$\varphi_1 (P)$ に対して, $\Phi_1 (x, y)$ を一つ撰び, $\varphi_2 (P) =
K (\Phi_1)$ なる操作によって $\varphi_2 (P)$ を作ります. 此の方法を反覆して,
$\varphi_p (P), \,\Phi_p (x, y)$ を作り, 函数級数
$$
\varphi_0 (P) + \varphi_1 (P) + \cdots + \varphi_p (P) + \cdots,\vspace{-4mm}
\leqno{\quad (5)}
$$
$$
\Phi_0 (x, y) + \Phi_1 (x, y) + \cdots + \Phi_p (x, y) + \cdots,
\leqno{\quad (6)}
$$
を考へます.
$\varphi_p (P)$ は $\mathfrak{D}_3$ に於て正則, $\Phi_p (x, y)$ は $(C')$ に
於て正則です. $A$ に於て,
$$
| \varphi_p (P)| < \lambda^p M_0 (p > 0),
$$
$(C')$ に於て
$$
| \Phi_p (x, y)| < \lambda^p N M_0
$$
です. 故に, (5) は $A$ に於て, (6) は $(C')$ に於て, 何れも斉一に収歛します.
其の極限を夫々 $\varphi (P), \,\Phi (x, y)$ としますと, $\varphi (P)$ は
$A$ に於て, $\Phi (x, y)$ は $(C')$ に於て, 何れも正則です. $P \in A'$ なる
$\Sigma$ の各点 $[x, f (P)]$ に於て, $\Phi_p (x, y) \ (p = 0, 1, \ldots)$ は値
$\varphi_p (P)$ を採りますから, $\Phi (x, y)$ は値 $\varphi (P)$ をとります.
故に, $\varphi (P), \,\Phi (x', u)$ が, $P \!\in\! A$ に於て, 函数方程式 (4) を
満足することを云へばそれでよろしい. 所で, $P \!\in\! A$ に於て
$$
\varphi_0 \!=\! \varphi_0, \ \varphi_1 \!=\! K (\Phi_0), \ \varphi_2 \!=\! K (\Phi_1), \ldots, \varphi_{p + 1} \!=\! K (\Phi_p), \ldots.
$$
$$
∴ \varphi = K (\Phi) + \varphi_0.
$$
\underline{第4節の問題は, か様に常に解けます.}
\bigskip
\begin{center}
\underline{$\rm{III}$ --- 擬凸状域と正則域, 正則域に於ける諸定理}
\end{center}
\bigskip
\underline{7} --- 課題を離れて暫く補助定理を準備します（7 -- 9節).
第二基礎的補助定理の形を今少し整へることから始めませう.
\medskip
\underline{補助定理 $\rm{II}$} --- \underline{空間 \,$(x)$ \,の分岐点を内点
として持たない有限擬凸状域}\\
\underline{$\mathfrak{D}$ に於ては, \,次の二つの条件を充たす連続実数値函数
$\varphi_0 (P)$ が必ず存在}\\ \underline{する : \ \,$1^\circ$, \,任意の実数
$\alpha$ に対し, \ $\varphi_0 (P) <
\alpha$ を充たす $\mathfrak{D}$ の点の集合を}\\ \underline{$\mathfrak{D}_\alpha$ とすれば,
\ $\mathfrak{D}_\alpha$ は, 若し存在すれば, \,$\mathfrak{D}_\alpha \Subset
\mathfrak{D}$ であること. \:\,$2^\circ$, \ $\mathfrak{D}$ の}\\
\underline{任意の点 \,$P_0$ \,に対し, \ \,其の近傍に於て,
\ 此の点を通り, \ 此の点以外では}\\ \underline{$\varphi_0 (P) > \varphi_0 (P_0)$ の部分のみに
ある様な正則固有面を撰び得ること.}
\medskip
\underline{証明} ~ 前報告に依て, 条件 $1^\circ$ を充たし, 条件 $2^\circ$ を, $\mathfrak{D}$ 内に
集積しない様な例外点の集合を別として, 充たす様な連続\underline{擬凸状}函数ならば,
$\mathfrak{D}$ に於て常に存在することを知って居ます. 其の一つを $\varphi (P)$
とし, 其の例外点の集合を, 実在するとして, $E_0$ としましょう. $\varphi
(P) = \lambda$ 上に $E_0$ の点が存在する様な実数値
$\lambda$ を $\varphi (P)$ の\underline{例外値}と呼びませう. $\alpha$ を任意の実数と
する時, $\varphi (P) < \alpha$ を充たす $\mathfrak{D}$ の点の集合を
$\mathfrak{D}_\alpha$ とします. 条件 $1^\circ$ に依て, $\mathfrak{D}_\alpha
\Subset \mathfrak{D}$ ですから, $\mathfrak{D}_\alpha$ は有界且つ有界葉です.
このことは $\alpha$ を少し大きくしても勿論変りませんから, $\mathfrak{D}_\alpha$ 内に於ける $E_0$ の点は有限個です. 云ふ迄もなく, $\lim_{\alpha \rightarrow
\infty} \mathfrak{D}_\alpha = \mathfrak{D}$ ですから, 例外値は可附番個です.
之を
$$
\lambda_1, \lambda_2, \ldots, \lambda_p, \ldots, \qquad \lambda_p <
\lambda_{p + 1}
$$
としませう.
$\alpha_0$ を\underline{例外値以外}の実数値として, $\mathfrak{D}_{\alpha_0} = \Delta$ と
します. $\Delta$ に於て
$$
\psi (P) = - \log d (P)
$$
を考へます. 此処に $d (P)$ は $\Delta$ に関するユークリッド的境界距離であって,
対数記号は其の実分枝を意味します. $\Delta$ は有界ですから $\psi (P)$ は
連続函数です. $\alpha$ を任意の実数とする時, $\psi (P) < \alpha$ を充たす
$\Delta$ の点の集合を $\Delta_\alpha$ としますと, $\Delta_\alpha \Subset \Delta$
です. か様に, $\psi (P)$ は $\Delta$ に於て条件 $1^\circ$ を充たします. 次に条件
$2^\circ$ を調べませう. $P_0$ を $\Delta$ の任意の点とし, $\psi (P_0) = \beta$
とします. 然うしますと, $P_0$ を中心とし, $e^{- \beta}$ を半徑として,
$\mathfrak{D}$ 上に $2n$ 次元球 $S$ を描きますと, $S \subset \Delta$ であって, 其の境界上に $\varphi (P) = \alpha_0$ を充たす点があります. 其の一つを $M$
とします. $\varphi (P)$ は $\varphi (P) = \alpha_0$ の近傍に於て条件 $2^\circ$ を
充たしますから, $M$ の近傍に於て, $M$ を通り, $M$ 以外では $\varphi (P) >
\alpha_0$ の部分のみを走る様な正則固有面 $\sigma$ があります. 平行移動
$$
x'_{i} = x_i + a_i \qquad (i = 1, 2, \ldots, n) \leqno{\quad (T)}
$$
に依て, $M$ を $P_0$ に, $\sigma$ を $\sigma'$ に移します.
$\sigma'$ は $P_0$ の近傍に於て定義せられて居ます. $\sigma'$ を見ますに,
其の $P_0$ 以外の任意の点を $P'$ としますと, 之に対応する $\sigma$ の点 $P$ は
$\varphi (P) > \alpha_0$ の部分にあって, $P$, $P'$ の（ユークリッド幾何学的)
距離は $e^{- \beta}$ ですから, $P'$ は, $\Delta$ に属するならば, $\psi (P) >
\alpha$ の部分にあります. か様に, \underline{$\psi (P)$ は $\Delta$ に於
て条件 $1^\circ, 2^\circ$ を充たす連続函}\\ \underline{数です.}
\medskip
実数列 $\mu_1, \mu_2,\ldots, \mu_p, \ldots$ を
$$
\mu_1 < \lambda_1, \quad \lambda_p < \mu_{p + 1} < \lambda_{p + 1}
$$
となる様に撰びます. $\alpha_0$ を
$$
\lambda_1 < \alpha_0 < \mu_2
$$
となる様にとり, 之に対して上述の $\psi (P)$ を考へます. $\alpha_0$ を充分
$\lambda_1$ に近く撰びますと, 此の $\psi (P)$ に対しては, 実数 $\beta$ を
$$
\mathfrak{D}_{\mu_1} \Subset \Delta_{\beta} \Subset \mathfrak{D}_{\lambda_1}
$$
となる様に撰ぶことが出来ます. か様にして得た $\psi (P)$ に依て (前報告の終りに
述べたと同様にして), $\varphi (P)$ を補正して $\varphi_1 (P)$ を作ります.
それに就て説明しませう.
$\beta_1, \beta_2$ を上述の $\beta$ と同じ性質を持つ二つの実数とし,
$$
\beta_1 < \beta_2
$$
とします. $\gamma_1, \gamma_2$ を
$$
\lambda_1 < \gamma_1 < \gamma_2 < \alpha_0
$$
なる二つの実数とします. $\mathfrak{D}$ を5つの部分 $\mathfrak{D}_j \quad
(j = 1, 2, \ldots, 5)$ に分かち,
$$
\mathfrak{D}_1 = \Delta_{\beta_1}, \quad
\Delta_{\beta_1} \cup \mathfrak{D}_2 = \Delta_{\beta_2}, \quad
\Delta_{\beta_2} \cup \mathfrak{D}_3 = \mathfrak{D}_{\gamma_1},
$$
$$
\mathfrak{D}_{\gamma_1} \cup \mathfrak{D}_4 = \mathfrak{D}_{\gamma_2}, \quad
\mathfrak{D}_{\gamma_2} \cup \mathfrak{D}_5 = \mathfrak{D}
$$
とします. 実数 $B$ を適当に撰び, 正数 $A$ を充分大きく撰んで,
$$
\Psi (P) = A [\psi (P) - B]
$$
が,\\
\qquad \qquad \ \ $\mathfrak{D}_1$ に対しては, \qquad \qquad \quad $\varphi (P) > \Psi (P)$,
\qquad \quad \ \ $\mathfrak{D}_3$ 及び $\mathfrak{D}_4$ に対しては,
\ \ \ \,$\varphi (P) < \Psi (P)$
\noindent となる様にします. 又, 実数 $B'$ を適当に撰び, 正数 $A'$ を充分大きく撰んで,
$$
\Phi (P) = A' [\psi (P) - B']
$$
が,\\
\qquad \qquad \ \ $\mathfrak{D}_3$ に対しては, \qquad \qquad \quad $\Psi (P) > \Phi (P)$,
\qquad \quad \ \ $\mathfrak{D}^{'}_{5}$ に対しては, \qquad \qquad \quad $\Psi (P) < \Phi (P)$,
\qquad \quad \ \ $\mathfrak{D}_5$ に対しては, $\qquad \qquad \ \ \> \>\varphi (P) < \Phi (P)$
\noindent となる様にします. 此処に $\mathfrak{D}^{'}_{5}$ は $\mathfrak{D}_5$ の
点集合 $\varphi (P) \!=\! \gamma_2$ の近傍の部分です.
$\varphi_1 (P)$ を次の様に定義します :
\qquad \quad \ \ $\mathfrak{D}_1$ に於ては, \qquad \quad $\varphi_1 (P) = \varphi (P)$ .
\qquad \quad \ \ $\mathfrak{D}_2$ に対しては, \qquad $\varphi_1 (P) =
\max [\varphi (P), \Psi (P)]$.
\qquad \quad \ \ $\mathfrak{D}_3$ に於ては, \qquad \quad $\varphi_1 (P) = \Psi (P)$.
\qquad \quad \ \ $\mathfrak{D}_4$ に対しては, \qquad $\varphi_1 (P) =
\max [\Psi (P), \Phi (P)]$.
\qquad \quad \ \ $\mathfrak{D}_5$ に於ては, \qquad \quad $\varphi_1 (P) = \Phi (P)$.
か様に定義せられた $\varphi_1 (P)$ を点檢しませう. $\varphi_1 (P)$ は
$\mathfrak{D}$ に於ける一価実数値函数です. 連続であることも容易に
確められます. $\psi (P)$ は性質 $2^\circ$ を持ち, $\varphi (P)$ は
$\mathfrak{D}$ 内に集積しない様な例外点の集合以外では性質 $2^\circ$ を
持ちますから, $\varphi_1 (P)$ は $\varphi (P)$ と同様です. $\varphi_1 (P)$ の
例外値を調べますと, $\mathfrak{D}_3$ では $\varphi_1 \!=\! \Psi$ であって,
$\mathfrak{D}_5$ では $\varphi_1 \!=\! \Phi$ ですから,
$$
\lambda^{'}_{2}, \,\lambda^{'}_{3}, \ldots, \,\lambda^{'}_{p}, \ldots
$$
です. ここに $\varphi_1 (P) \!=\! \lambda^{'}_{p}$ と $\varphi (P) \!=\!
\lambda_p$ とは同じ点集合です. $\varphi_1 (P)$ を原の $\varphi (P)$ と
比較しますと, $\mathfrak{D}_{\mu_1}$ に於て, $\varphi_1 (P) \!=\! \varphi
(P), \
\mathfrak{D}$ に於て, $\varphi_1 (P) \!\geq\! \varphi (P)$ であることが容易に
分かります. $\varphi_1 \!\geq\! \varphi$ ですから $\varphi_1$ は性質 $1^\circ$ を
持ちます. 此の $\varphi_1 (P)$ は $\varphi (P)$ と大体同じ性質の函数です.
唯, 擬凸状函数であるかどうかと云ふ点が違って居ますが, 上の操作には
$\varphi (P)$ の此の性質は使ひませんでした. それ故, $\varphi (P)$ から
$\varphi_1 (P)$ を作ったと全く同様にして, $\varphi_1 (P)$ から
$\varphi_2 (P)$ を作ることが出来ます. 此の操作を例外値が残って居る限り
繰り返します. か様にして,
$$
\varphi_1 (P), \,\varphi_2 (P), \ldots, \,\varphi_p (P), \ldots
$$
が得られます. $\varphi_p (P) \ (p \!>\! 1)$ の性質の中 $p$ と共に変る部分を
挙げますと, $\varphi_p (P)$ の例外値は
$$
\lambda^{(p)}_{p + 1}, \,\lambda^{(p)}_{p + 2}, \ldots,
\,\lambda^{(p)}_{p + q}, \ldots
$$
\ーー此処に $\varphi_p (P) \!=\! \lambda^{(p)}_{p + q}$ と $\varphi (P) \!=\!
\lambda_{p + q}$ とは同じ点集合です\ーーであって, $\mathfrak{D}_{\mu_p}$
に於て $\varphi_p (P) \!=\! \varphi_{p - 1} (P), \ \mathfrak{D}$ に於て $\varphi_p (P) \!\geq\!
\varphi_{p - 1} (P)$ です ($\mathfrak{D}_5$ に於て $\varphi_1 \!=\! \Phi$ で
あることに注意). か様な $\varphi_p (P)$ を撰ぶことが出来ます.
故に, 此の函数列の極限, 又は列の函数が有限個の場合には, 其の最後のものを
$\varphi_0 (P)$ としますと, $\varphi_0 (P)$ は明らかに所求の函数です. ~ (証明終)
\medskip
か様にして作った $\varphi_0 (P)$ は, 実際は擬凸状函数です.
\footnote{其の為には, $\mathcal{D}_{\alpha_0} \!=\! \Delta$ が擬凸状であれば
よろしい (第九報告, 定理3). 此のことに就ては第9節参照}
\bigskip
\underline{8} --- 第二報告の始めに, 有理整函数に関して外的に凸状な H\"ulle に就て
述べました. 之を少し拡張して, 前節の基礎定理を補ひませう. 但し, 此の度は
便宜上 (内的) 凸性を考へます.
\medskip
\underline{補助定理3} --- \underline{$\mathfrak{D}$を空間$(x)$に於ける有限葉正則域とし,
$E_0$を $E_0 \Subset \mathfrak{D}$}\\ \underline{なる開集合とする. \ 然る時は,
\ \,$1^\circ$, \ $E_0$を
含み, $\mathfrak{D}$に於ける正則函数の全}\\ \underline{体に関して凸状で
ある様な
\ \,$\mathfrak{D}$ の部分開集合中には 最小のものが存在し,}\\ \underline{之を
$H$ とすれば,
$H \Subset \mathfrak{D}$ である. \ \ $2^\circ$, \ 次の条件を充たす固有面片
$\sigma$ は
}\\ \underline{存在しない : \ $\sigma$ は $H$ の界点を通り, \ $H, \,E_0$
及び $E_0$ の境界を
通らず, $\sigma$}\\ \underline{の界点は $H$ 及び其の境界上に無いこと. \ 及
び, $\sigma$ は
次の形に定義せられ}\\ \underline{ること :}
$$
\varphi (P) = 0, \quad P \in V,
$$
\underline{此処に $V$は $V \Subset \mathfrak{D}$ なる領域であって, $\varphi (P)$ は
$\mathfrak{D}$ 上の $V$の近傍に於け}\\ \underline{る正則函数である.}
\footnote{此の $1^\circ$ の部分は, H. Cartan--P.Thullen の K--Konvexe H\"ulle の
存在に関する定理から直ちに出ます. 唯, 原著の証明は同時解析接続の可能性に関する
基礎定理に立脚して居ます. Cartan--Thullen の前掲の論文参照. (尚,
定理 $\rm{I}$ の脚註参照.)}
\medskip
\underline{証明} ~ $1^\circ$, \ 先づ, H\"ulle $H$ の存在を云はうと云ふのですが,
其の為少し準備します.
$\mathfrak{D}$ は \underline{有限葉}ですから, $\mathfrak{D}'$ が $\mathfrak{D}$ に
関して有界な $\mathfrak{D}$ の部分集合であると云ふことと, $\mathfrak{D}'
\Subset \mathfrak{D}$ であると云ふこととは一致します. $\mathfrak{D}$ に於ける
正則函数の全体を $(\mathfrak{F})$ とします. $\mathfrak{D}$ は正則域ですから,
\underline{Cartan--Thullen} の第一定理に依て, $\mathfrak{D}$ は $(\mathfrak{F})$ に関して
凸状です. 故に, 補助定理 1 に依て, $\mathfrak{D} = \mathfrak{D}_0$ と見て,
此の補助定理に述べた解析的多面体 $\Delta$ を作ることが出来ます.
$\Delta$ は次の形です：
$$
P \!\in\! R, \ | x_i| \!<\! r, \ | f_j (P)| \!<\! 1 \ \ (i \!=\! 1, 2, \ldots,
n; j \!=\! 1, 2, \ldots, \nu). \leqno{\quad (\Delta)}
$$
此処に, $f_j (P)$ は $(\mathfrak{F})$ の函数であって, $R$ は $\Delta \Subset R$ なる
$\mathfrak{D}$ の部分開集合です. 尚, $E \Subset \mathfrak{D}$ なる与へられた集合
$E$ に対し, $E \Subset \Delta$ です.
$\rho$ を任意の正数とし, $\mathfrak{D}$ に関するユークリッド的境界距離を
$d (P)$ とする時, $d (P) > \rho$ なる $\mathfrak{D}$ の点の集合を
$\mathfrak{D}_\rho$ とします. (但し, $\rho$ は $\mathfrak{D}_\rho$ が存在する
程小さくとります.) $\mathfrak{D}$ が有限空間 $(x)$ と一致する場合には
$\mathfrak{D}_\rho$ = $\mathfrak{D}$ です. 平行移動,
$$
x'_{i} = x_i + a_i, \quad \Sigma | a_i|^2 \leq \rho^2 \quad
(i = 1, 2, \ldots, n) \leqno{\quad (T)}
$$
に依て, $\mathfrak{D}_\rho$ の点 $P$ を $\mathfrak{D}$ の $P'$ に移します.
$P$ が決れば, $P'$ は一意に決まります. $(\mathfrak{F})$ の函数 $f (P)$ から,
$$
F (P) = f (P')
$$
を作りますと, $F (P)$ は $\mathfrak{D}_\rho$ に於ける正則函数です. $(T)$ を
上述の制限内に於ける任意の平行移動とし, $f (P)$ を $(\mathfrak{F})$ の任意の
函数とした時の $F (P)$ の全体を $(\mathfrak{F}_\rho)$ とします.
$A$ を $A \Subset \mathfrak{D}$ なる開集合とします. $A$ を $(\mathfrak{F})$ に関して凸状と
しますと, \underline{$A_0 \Subset A$ \ なる任意の集合 $A_0$ に対し, \
$A$ \,の任意の界点\, $M$ の
如何程でも}\\ \underline{近くに点\, $P_0$ \,があって, \ $(\mathfrak{F})$ \,の少くとも一つの函数\, $f (P)$ \,が
\ $| f (P_0)| >$ }\\ \underline{$\max |f (A_0)|$ \,なる関係を充たします.}
之を暫く\underline{性質 $(\alpha)$}
と呼びませう. 逆に, $A$ が性質 $(\alpha)$ を持てば, $A$ は \underline{$(\mathfrak{F})$ に関して
凸状}であることを証明しませう. $A \Subset \mathfrak{D}$ ですから, 上に説明した
解析的多面体 $\Delta$ を, $A \Subset \Delta$ となる様に撰ぶことが出来ます.
正数 $\rho$ を充分小さく, $\Delta \subset \mathfrak{D}_\rho$ となる様に
とります. $A$ は, 性質 $(\alpha)$ を持ちますから, $(\mathfrak{F}_\rho)$ に関して
凸状であること明白です. 所で, $(\mathfrak{F}_\rho)$ の任意の函数は $\Delta$ に於て
正則ですから, 定理 1 に依て, $\Delta$ の各点に於て斉一に収歛する様な
$(\mathfrak{F})$ の函数の級数に展開することが出来ます. 故に, $A$ は $(\mathfrak{F})$ に
関して凸状であること明らかです.
扱て, $A$ を $E_0$ を含み $(\mathfrak{F})$ に関して凸状である様な $\mathfrak{D}$ の
任意の部分開集合とします. $A$ の總てに属する $\mathfrak{D}$ の点の集合を考へ,
其の内点の集合を $H$ とします.
$E_0$ は開集合ですから, $E_0 \subset H$ です. 上述の $\Delta$ に於て,
$E = E_0$ ととることが出来ますから, $H \Subset \mathfrak{D}$ です. $H$ は明らかに
性質 $(\alpha)$ を持ちます. 故に, $H$ は $(\mathfrak{F})$ に関して凸状です. か様に,
$H$ は $E_0$ を含み $(\mathfrak{F})$ に関して凸状である様な最小の $\mathfrak{D}$ の
部分開集合であって, $H \Subset \mathfrak{D}$ です.
\medskip
$2^\circ$ , 補助定理に述べた性質の固有面片 $\sigma$ が存在したと假定しませう.
之が矛盾に終ればよろしい. $V \Subset V' \subset \mathfrak{D}$ なる $V'$ に於て,
$\varphi (P)$ が正則であるとします. 正数 $\rho$ を充分小さく撰び, $V'$ に
関するユークリッド的境界距離を $d (P)$ とすれば, $\min d (V) > \rho$ となる様に
します (不等式の左辺は $V$ に於ける $d (P)$ の下端). 平行移動,
$$
x'_{i} = x_i + z_i, \quad \Sigma |z_i|^2 \leq \rho^2 \qquad (i = 1, 2, \ldots, n)
$$
に依て, $V$ の点 $P$ を $V'$ の点 $P'$ に移します. $(z)$ を複素助変数と
看做し,
$$
\psi (P, z) = \varphi (P')
$$
として, 固有面片族 $(\mathfrak{G}), \ \psi (P, z) = 0,\,P \in V$ を考へます. $\rho$ を
充分小さく撰び, $(\mathfrak{G})$ の固有面片の界点が決して $H$ 内に入って
来ない様にします.
$(\mathfrak{G})$ のどの固有面片にもぞくしない様な $H$ の点の集合を $H_0$ とします.
$A_0$ を $A_0 \Subset H_0$ なる開集合とします. 上に見ました様に, $A_0$ を含み
$(\mathfrak{F})$ に関して凸状な $\mathfrak{D}$ の部分開集合中には最小のものがあります.
之を $A$ とします. $H$ は $(\mathfrak{F})$ に関して凸状ですから, 補助定理1に依て,
上に見た $H$ の場合と同様にして, $A \Subset H$ であることが分かります.
$A \subset H_0$ であることを証明しませう.
空間 $(z)$ に原点を中心とし半徑を $\rho$ とする $2n$ 次元球 $S$ を描きます.
空間 $(x, z)$ に於ける $(H, S) \ ((x) \in H, (z) \in S)$ は,
領域 $(x) \in \mathfrak{D}$ に於ける正則函数の全体に関して凸状な,
此の領域の部分開集合です. 故に, 定理2に依て, $(H, S)$ に於て, 有理型函数
$G (P, Z)$ を撰び, $(V, S)$ との共通部分に於て
$$
1 / \psi (P, z)
$$
と同等であって, 他の部分で極を持たない様にすることが出来ます.
(定理 2 は有限葉正則域に於てはと云ふのですが, 実際は此の領域が
補助定理 1 の $\mathfrak{D}_0$ に課せられた条件を充たすことしか使って
居ませんから).
$A$ が $H_0$ に含まれないと假定します. 然うしますと $A$ は開集合ですから,
$H_0$ 外の点を含みます. 故に, $S$ 内に点 $(z^0)$ を適当に撰び $\psi (P, z^0)
= 0$ 上に $A$ の点 $P_0$ がある様に出来ます. $t$ を複素変数として, 函数
$$
G (P, t, z^0)
$$
を考へますと, 此の函数は $P$ が $H$ にあり, $t$ が線分 $(0, 1)$ の近傍にある
時有理型であって, $P = P_0, \,t = 1$ に於て極を持ち, $G (P, 0)$ は $A$ の近傍
($\mathfrak{D}$ 上に於ける)に於て極を持ちません. $t$ が線分 $(0, 1)$ を
1 から 0 に向って描くとき, $G (P, t, z^0)$ が $A$ 又は其の境界上に於て最後に
極を持つ様な点を $t_0$ とします. $G (P, t_0, z^0)$ は $A$ の境界上に於て極を
持ち, $A$ に於て正則でなければなりません. 其の極の一つを $M$ とします.
$M$ に充分近い $A$ の点を $P_1$ としますと, $A_0 \Subset H_0$ であって, $M$ は
不定点ではありませんから,
$$
|G (P_1, t_0, z^0)| > \max |G (A_0, t_0, z^0)|.
$$
所で, 定理1に依て, $G (P, t_0, z^0)$ を, $A$ の各点で斉一に収歛する様な,
$(\mathfrak{F})$ の函数の級数に展開することが出来ます. 之は明らかに $A$ の最小性と
相容れません. 故に, $A \subset H_0$ でなければなりません.
$A_0$ は $A_0 \Subset H_0$ なる任意の開集合ですから, 上の結果は, 開集合 $H_0$ が
性質 $(\alpha)$ を持つことを示します. 故に, $H_0$ は $(\mathfrak{F})$ に関して凸状です.
此の結果は $\rho$ を如何程小さくとっても成立します. 所で, $\rho$ を充分
小さくしますと, $E_0 \subset H_0$ です. 之は, $H$ の最小性と矛盾します. ~ (証明終)
\newpage
\underline{9} --- 補助定理3から, 次の二つの補助定理が容易に得られます.
\medskip
\underline{補助定理4}\,--- \underline{$\Delta$を空間$(x)$の有理整函数
に関して凸状な単葉領域とし,}\\
\underline{$\varphi (x)$ を $\Delta$ の近傍に於ける, 補助定理 $\rm{II}$ に述べた性質
$\,2^\circ$ を持つ様な, 実}\\ \underline{数値連続函数とする. $\alpha$ を任意の実数とする時,
$\varphi (x) < \alpha$ を充たす $\Delta$ の}\\ \underline{点の集合を$\Delta_\alpha$と
すれば, $\Delta_\alpha$は, 若し存在すれば, 有理整函数に関して凸}\\ \underline{状である.}
\medskip
\underline{証明} ~ $\Delta_\alpha$ を含み, 単葉であって, 有理整函数に関して凸状である様な
開集合中には, 補助定理3に依て最小のものがあります. 之を $H$ とします.
明かに $H \subset \Delta$ です. 故に, $H$ の近傍に於て $\varphi (x)$ が
定義せられて居ます. $H$ と其のすべての界点とからなる閉集合を $\overline {H}$
とし, $\varphi (x)$ の $\overline {H}$ 上に於ける最大値を $\beta$ としますと,
$\overline {H}$ 上には $\varphi (x) = \beta$ となる点があります. 其の一つを
$M$ とします. $\varphi (x)$ は性質 $2^\circ$ を持ちますから, $M$ は $H$ の界点で
なければなりません. 更に, 同じ性質に依て, $M$ の近傍に於て, $M$ を通り,
此の点以外では $\overline {H}$ を通らない様な固有面が存在します. 故に,
補助定理 3 に依て, $M$ は $\Delta_\alpha$ の界点でなければなりません. 故に,
$\beta = \alpha$, 従って $H = \Delta_\alpha$ です. 故に $\Delta_\alpha$ は
有理整函数に関して凸状です. （証明終)
\medskip
\underline{補助定理5} --- \underline{$\varphi (P)$を空間$(x)$の領域 $\mathfrak{D}$に於て,
補助定理 $\rm{II}$ に述べた}\\ \underline{性質 $2^\circ$ を持つ実数値連
続函数とし, \ $\Delta$を
$\Delta \Subset \mathfrak{D}$ なる正則域とする. \,若}\\ \underline{し或る実数 $\alpha$ に対して,
\ $\varphi (P) < \alpha$ \ なる $\mathfrak{D}$ の点の集合を $\mathfrak{D}_\alpha$
とする時,}\\ \underline{$\mathfrak{D}_\alpha \Subset \Delta$ ならば, $\mathfrak{D}_\alpha$ は
$\Delta$ に於ける正則函数の全体に関して凸状である.}
\medskip
$\Delta \Subset \mathfrak{D}$ ですから, △は有界葉です. か様に, △は有界葉正則域で
あって, $\mathfrak{D}_\alpha \Subset \Delta$ ですから, 補助定理 3 を適用することが
出来ます. 以下, 上の場合と全く同様です.
\medskip
次に, \underline{H. Cartan--P. Thullen} の定理及び \underline{H. Behnke--K. Stein} の定理を述べます.
\medskip
\underline{H. Cartan--P. Thullen の第二定理} --- \underline{$\mathfrak{D}$ を空間$(x)$の領域とし,
$(\mathfrak{F})$を}\\ \underline{$\mathfrak{D}$ に於て正則なすべての函数
からなる函数族とする. \ 若し,
\ 次の二つの}\\ \underline{条件が充たされるならば, \ $\mathfrak{D}$ は正則域
である : \ \ \ $1^\circ, \ \ \mathfrak{D}_0 \Subset \mathfrak{D}$ なる任
意}\\ \underline{の集合 $\mathfrak{D}_0$ に対し, \ \
$\mathfrak{D}_0 \Subset \mathfrak{D}' \Subset \mathfrak{D}$ \ なる開集合 $\mathfrak{D}'$
が対応し, \ $\mathfrak{D}'$ はその}\\ \underline{任意の界点\, $M$ に対し,
\ \ $(\mathfrak{F})$ 中に少くとも
一つの函数 $f (P)$ が存在して,}\\ \underline{$|f (M)| > \max|f
(\mathfrak{D}_0)|$ \ \ となると
云ふ性質を持つこと. \ \ \ $2^\circ, \ \ \mathfrak{D}$ \,の相異}\\
\underline{なる二点 \ $P_1, \,P_2$ に対し, \
$(\mathfrak{F})$ には少くとも一つの函数 \ $f (P)$ があって,}\\
\underline{$f (P_1) \neq f (P_2)$ となること.}
\footnote{原著者は此の第二定理（及び第一定理)を $K$--凸性に就て述べて
居るのですが, 此処では便宜上, 上の形にしました. 証明法は全く同様であって,
直接的です. (Cartan--Thullen の論文, 前掲, 参照.)}
\medskip
\underline{H. Behnke--K. Stein の補助定理} --- \underline{$\mathfrak{D}$ を空間 $(x)$ の領域とし,}
$$
\mathfrak{D}_1, \mathfrak{D}_2, \ldots, \mathfrak{D}_p, \ldots
$$
\underline{を, \ $\mathfrak{D}_p \Subset \mathfrak{D}_{p + 1}$ であって,
\ $\mathfrak{D}$ を
極限とする様な, $\mathfrak{D}$ の部分開集合の列と}\\ \underline{する. 若
し, \ $1^\circ, \ \mathfrak{D}_p$ が$\mathfrak{D}_{p + 1}$に於ける正則函
数の全体$(\mathfrak{F} _{p + 1})$に関して凸}\\ \underline{状であって, \
$2^\circ, \ \mathfrak{D}_p$ の相異なる
二点$P_1, P_2$に対し, $(\mathfrak{F} _{p + 1})$中に$f(P_1) \neq
$}\\
\underline{$f(P_2)$ \,なる函数 $f(P)$ \,があるならば, \ $\mathfrak{D}_p$ は $\mathfrak{D}$ に於ける
正則函数の全体}\\ \underline{$(\mathfrak{F})$ に関して上述の性質 $1^\circ, \,2^\circ$ を持つ.}
\footnote{H. Behnke--K. Stein : Konvergente Folgen von Regularit\"atsbereichen
und die Meromorphiekonvexit\"at, 1938, (Math. Annalen). }
\medskip
\underline{証明} ~ (\underline{Cartan--Thullen} の第二定理に依て, $\mathfrak{D}_{p + 1}$ は
正則域ですから), 定理1に依て, $\mathfrak{D}_p$ に於ける任意の正則函数
$\varphi (P)$ を, $\mathfrak{D}_p$ の各点に於て斉一に収歛する様な,
$(\mathfrak{F} _{p + 1})$ の函数の級数に展関出来ます. 此のことは勿論, $p + 1, p + 2,
\ldots$ に対しても成立しますから, $\varphi (P)$ を $(\mathfrak{F})$ の函数に依て,
同様に展開出来ます. 故に, $\mathfrak{D}_p$ は明らかに $(\mathfrak{F})$ に関して
性質 $1^\circ, \,2^\circ$ を持ちます. （証明終)
\medskip
\underline{H. Behnke--K. Stein の定理} --- \underline{$\mathfrak{D}$ を空間 $(x)$ の領域とする.
\,若し, $\mathfrak{D}_0$}\\ \underline{$\Subset \mathfrak{D}$ なる任意の集合 $\mathfrak{D}_0$
に対し, $\mathfrak{D}_0 \subset \mathfrak{D}' \Subset \mathfrak{D}$ なる
正則域 $\mathfrak{D}'$ が対応す}\\ \underline{るならば, $\mathfrak{D}$ は正則域である.}
\footnote{15 に同じ.}
\medskip
\underline{証明} ~ $\mathfrak{D}'$ は正則域ですから, \underline{F. Hartogs} に依て擬凸状域です.
故に, 第九報告, 定理 2 の系 2 に依て, $\mathfrak{D}$ は擬凸状域です.
故に, $\mathfrak{D}$ に対して, 補助定理 $\rm{II}$ の函数 $\varphi_0 (P)$ が
存在します. 所で, 補助定理5に依て, $\mathfrak{D}_\alpha \ (\varphi_0 (P) \!<\!
\alpha, \,P \!\in\! \mathfrak{D})$ は $\mathfrak{D}_\alpha \Subset \mathfrak{D}'$ なる
正則域 $\mathfrak{D}'$ に於ける正則函数の全体に関して凸状です. 従って,
$\alpha, \,\beta$ を $\alpha \!<\! \beta$ なる任意の二つの実数としますと,
$\mathfrak{D}_\alpha$ は $\mathfrak{D}_\beta$ に於ける正則函数の全体に関して,
\underline{Behnke--Stein} の補助定理に挙げた二つの条件を充たしますから,
$\mathfrak{D}_\alpha$ は $\mathfrak{D}$ に於ける正則函数の全体に
関しても同様です. 故に, \underline{Cartan--Thullen} の第二定理に依て,
$\mathfrak{D}$ は正則域です. ~ (証明終)
補助定理4, 5の一部分を今少し拡張して置きませう.
\medskip
\underline{補助定理6} --- \underline{$\mathfrak{D}$ を空間 $(x)$ に於ける有限葉正則域とし,
$\varphi (P)$ を $\mathfrak{D}$ に}\\ \underline{於て, 補助定理 $\rm{II}$の性質 $2^\circ$ を
持つ実数値連続函数とする. $\alpha$ を任意の実}\\ \underline{数とする時,
\ $\varphi (P) <
\alpha$ を充たす $\mathfrak{D}$ の点の集合を $\mathfrak{D}_\alpha$ とすれば,
\,($\mathfrak{D}_\alpha$ が}\\ \underline{実在すれば,) $\mathfrak{D}_\alpha$ の各連結成分は
正則域である.}
\footnote{実際は, $\mathcal{D}_\alpha$ は $\mathcal{D}$ に於ける
正則函数の全体に関して凸状です. }
\medskip
\underline{証明} \ $\mathfrak{D}_\alpha$ が実在すると考へます. $\mathfrak{D}$ は
正則域ですから, \underline{F. Hartogs} に依て擬凸状域であって, 従って, 此の
$\mathfrak{D}$ に対して, 補助定理 $\rm{II}$ に述べた実数値函数が存在します.
之を $\psi (P)$ とし, $\beta$ を $\beta < \alpha$ なる実数, $\gamma$ を
任意の実数として, 開集合,
$$
P \in \mathfrak{D}, \quad \varphi (P) < \beta, \quad \psi (P) < \gamma
\leqno{\quad (\mathfrak{D}_{\beta \gamma})}
$$
を考へます. $\mathfrak{D}$ は有限葉正則域であって,
$\mathfrak{D}_{\beta \gamma} \Subset \mathfrak{D}$ ですから,
$E_0 = \mathfrak{D}_{\beta \gamma}$ として, 補助定理 3 を適用することが
出来ます. 以下補助定理 4 の場合と全く同様にして,
$\mathfrak{D}_{\beta \gamma}$ が $\mathfrak{D}$ に於ける正則函数の全体に
関して凸状であることが容易に分かります. 故に, $\mathfrak{D}_{\beta \gamma}
\subset \mathfrak{D}$ ですから \underline{Cartan--Thullen} の第二定理に依て,
$\mathfrak{D}_{\beta \gamma}$ の各連結成分は正則域です. 所で,
$\mathfrak{D}_{\beta \gamma} \Subset \mathfrak{D}_\alpha$ であって,
$\mathfrak{D}_{\beta \gamma}$ は $\mathfrak{D}_\alpha$ の如何程でも
近くにとれますから, \underline{Behnke--Stein} の定理に依て, $\mathfrak{D}_\alpha$ の
各連結成分は正則域です. ~ (証明終)
\footnote{F. Hartogs に依て, 正則域は擬凸状域ですから, 補助定理4から,
此の節の諸定理と第九報告の諸定理とに依て, 擬凸状域の性質が容易に分かりま
す : \
$\ll$ $\varphi (x)$ を $2 n$ 次元球 $S$ の近傍に於ける擬凸状函数とし,
$\varphi (x) < \alpha$ （$\alpha$ は任意の実数)を充たす $S$ の点の集合を
$S_\alpha$ とすれば, $S_\alpha$ は, 若し存在すれば, 擬凸状である.$\gg$ }
\bigskip
\underline{10} --- 課題に帰ります. 先づ擬凸状域が正則域であることを申しませう.
空間 $(x)$ に\underline{有限葉}領域 $\mathfrak{D}$ を考へます. $x_1$ を実部と虚部とに分ち,
$$
x_1 = \xi + i \,\eta
$$
とします. $a_1, a_2$ を
$$
a_2 < 0 < a_1
$$
なる実数とし, $\xi \!<\! a_1$ なる $\mathfrak{D}$ の部分を $\mathfrak{D}_1$,
\
$\xi \!>\! a_2$ なる $\mathfrak{D}$ の部分を $\mathfrak{D}_2, \ \mathfrak{D}_1,
\,\mathfrak{D}_2$ の共通部分を $\mathfrak{D}_3$ とします. $\mathfrak{D}$
には \ $\xi \!<\! a_2$ なる部分及び \ $\xi \!>\! a_1$ なる部分があると考へ, 之等の部分に夫々
一点 $Q_1, \,Q_2$ をとります. \underline{$\mathfrak{D}_1, \,\mathfrak{D}_2$ の各連結成分
は正則域}であると假定します. 然うしますと, $\mathfrak{D}_3$ も必ず同様です.
\underline{F. Hartogs} に依て, 正則域は擬凸状域ですから, $\mathfrak{D}$ は擬凸状域です.
故に, 此の $\mathfrak{D}$ に関して, 補助定理 $\rm{II}$ に述べた実数値函数
$\varphi_0 (P)$ を考へることが出来ます. $\alpha$ を実数として,
$\varphi_0 (P) \!<\! \alpha$ を充たす $\mathfrak{D}$ の点の集合
$\mathfrak{D}_\alpha$ を考へます. $\alpha$ を充分大きくとれば,
$\mathfrak{D}_\alpha$ は定点 $Q_1, \,Q_2$ を同じ一つの連結成分中に含みます.
此の連結成分を $A$ とします. $A$ は有界且つ有界葉です. $\xi \!<\! a_1, \ \xi \!>\! a_2$
及び $a_2 \!<\! \xi \!<\! a_1$ を充たす $A$ の部分を夫々 $A_1, \,A_2, \,A_3$ とします.
補助定理6に依て, $A_1, \,A_2, \,A_3$ の各連結成分は正則域です.
$\xi \!=\! 0$ 上の $A$ の界点の集合を $\Gamma$ とし, $\Gamma$ の任意の点を
$M$ とします. $M$ の近傍に於て定義せられ, $M$ を通り, 此の点以外は
$\mathfrak{D}$ 上の $\sigma$ の近傍に於ける $\varphi_0 (P) \!>\! \alpha$ の
部分のみにある様な固有面片 $\sigma$ があります. $\sigma$ の方程式を
$\psi (P) \!=\! 0$ ($\psi (P)$ は正則函数) とします. $\alpha \!<\! \beta$ なる
$\beta$ を充分 $\alpha$ に近くとりますと, $\mathfrak{D}_\beta \
(\varphi_0 (P) \!<\! \beta)$ 内に於て $\sigma$ は界点を持ちません.
(但し, 其の際若し必要ならば, $\sigma$ の境界の近傍を少し切り捨てます.)
$\mathfrak{D}_\beta$ の $a_2 \!<\! \xi \!<\! a_1$ なる部分を $B$ としますと,
$B$ は有界葉であって, 其の各連結成分は正則域です. 故に, 定理2に依て,
$B$ に於て有理型であって, $\sigma$ に於て極 $1 / \psi (P)$ を持ち,
他に極を持たない様な函数 $G (P)$ が存在します. $G (P)$ は $A_3$ に
於ては正則です. $\Gamma$ の任意の点 $M$ に対し, か様な函数 $G (P)$
が対応します. 又, $A_3$ の各連結成分は正則域です（補助定理 1 の証明法参照).
故に, 正数 $\delta_0, \,\varepsilon_0$ を充分小さく撰べば, 次に述べる
三つの条件を充たす様な正則函数 $f_j (P) \ (j \!=\! 1, 2, \ldots, \nu)$ が
$A_3$ に於て存在することが, 慣用の推理法に依て, 容易に分かります. 条件は :
$1^\circ$, \ $|\xi| \!<\! \delta_0, \, |f_j (P)| \!<\! 1 \ (j \!=\! 1, 2, \ldots, \nu)$
を充たす $A$ の点の集合を $A_0$ とすれば, $A_0 \Subset A$.
$2^\circ$, \ $p$ を $1, 2, \ldots, \nu$ の任意の一つとすれば, $|f_p (P)|
\!\geq\! 1 - \varepsilon_0$ を充たす $\mathfrak{D}_3$ の点は $|\xi - a_1|
\!<\! \delta_0$ の部分にも, $|\xi - a_2| \!<\! \delta_0$ の部分にも存在しないこと.
$3^\circ$, \ 数値系 $[f_1 (P), f_2 (P), \ldots, f_\nu (P)]$ は $A_0$ の相重なる
二点で決して同一とならないこと.
尚, $|f_j (P)| \!<\! 1 \ (j \!=\! 1, 2, \ldots, \nu)$ を充たす $A_3$ の点の集合を
$A_4$ としますと, $A_4$ は如何程でも $A_3$ に近くとれることが分かります.
$\xi \!\leq\! a_2$ 又は $\xi \!\geq\! a_1$ を充たす $A$ の部分と $A_4$ との和を
考へますと, 之は開集合です. $A_4$ が充分 $A_3$ の近くにある様に $f_j (P)
\ (j \!=\! 1, 2, \ldots, \nu)$ を撰び, 此の開集合の連結成分中に,
定点 $Q_1, Q_2$ を同時に含むものがある様にし, 此の連結成分を $\Delta$ と
します. \underline{$\Delta$ は第4節に挙げた諸条件を充たします.}
実数 $\alpha$ を或る値 $\alpha_0$ より大きくさへとれば,
$\mathfrak{D}_\alpha$ の $Q_1, Q_2$ を同時に含む連結成分として $A$ を
考へることが出来ます. $\alpha'$ を $\alpha_0 \!<\! \alpha' \!<\! \alpha$ となる様に
撰び, $\alpha$ に $A$ を対応せしめたと全く同様にして, $\alpha'$ に $A'$ を
対応せしめます. $A' \Subset A$ であること云ふ迄もありません. $A'$ の $\xi \!<\! 0$ の
部分を $A^{'}_{1}$, \ $\xi > 0$ のそれを $A^{'}_{2}$ とします. $\Delta$ は
如何程でも $A$ に近くとれるのですから, 前章の結果から, 直ちに次の結果が
得られます : ~ \underline{開集合, \ $P \!\in\! A, \ \,|\xi| \!<\! \delta_0$ ($\delta_0$ は如何程
小さくてもよい) に於て正則な函数}\\ \underline{$\Phi (P)$ が与へられた時, $A^{'}_{1}$ に
於て正則な函数 $\Phi_1 (P)$ と $A^{'}_{2}$ に於て正則な}\\
\underline{函数 \ $\Phi_2
(P)$とを, \ 何れも \,$\xi \!=\! 0$ 上の $A'$ の各点に於て矢張り正則であっ
}\\ \underline{て, 恒等的に
$\Phi_1 (P) - \Phi_2 (P) = \Phi (P)$ となる様に作ることが出来る.}
\medskip
$A$ に極 $(\wp)$ が与へられたとしませう. 定理2に依て, $A_1$ に於て,
有理型函数 $G_1 (P)$ を, $(\wp)$ を極とする様に作ることが出来ます.
$A_2$ に於ても同様です. 之を $G_2 (P)$ とします. $G_1 (P) - G_2 (P)$ は
$A_3$ に於て正則です. 故に, 上の結果から, 次のことが分かります : \ \underline{$A$ に於て
Cousin の第一問}\\ \underline{題が与へられた時, 之を $A'$ に於て解くことが出来る.}
\medskip
改めて $A$ を考へます. $A$ は $\mathfrak{D}_\alpha \ (\alpha_0 \!<\! \alpha)$ の
$Q_1, Q_2$ を含む連結成分です. $A$ の任意の界点を $M$ とします. $(\gamma)$ を
$\mathfrak{D}$ 上に描かれた, $M$ を中心とする多円筒とします. $(\gamma)$ を
充分小さくとりますと, $(\gamma)$ に於て定義せられた固有面片 $\sigma$ を,
$M$ を通り, 此の点以外は $\varphi_0 (P) \!>\! \alpha$ の部分に存在する様に,
撰ぶことが出来ます. $\sigma$ を
$$
\psi (P) = 0, P \in (\gamma)
$$
とします. $\psi (P)$ は $(\gamma)$ に於ける正則函数です. 若し必要ならば
$(\gamma)$ を少し小さくしますと, $\alpha$ に充分近い $\alpha \!<\! \alpha''$ なる
$\alpha''$ が存在して, 之に対応する $A''$ 内に $\sigma$ の界点はありません.
故に, 上に述べましたことから, $\alpha''$ を更に $\alpha$ に近くとりますと,
$A''$ に於て有理型函数 $G (P)$ を求め, $\sigma$ 上で, 極 $1 / \psi (P)$ を
とり, 他に極を持たない様にすることが出来ます. 此処に $M$ は $A$ の任意の
界点です.
$A$ に関して \underline{Cartan--Thullen} の第二定理の二つの条件を点檢しませう.
$A$ に於ける正則函数の全体を $(\mathfrak{F})$ とします. 上に見ましたことから,
明かに, \ $1^\circ, \ A$ は $(\mathfrak{F})$ に関して凸状です.
$A$ の相重なる任意の一対の点を $P_1, P_2$ とし, 其の共通の基点を \underline{$P$} と
します. \underline{$P$} を一端として, 空間 $(x)$ に半直線 \underline{$L$} を描きます.
$A$ 上に, $P_1$ から出発して半直線を, 其の基点がすべて \underline{$L$} 上にある様に
描きますと, $A$ は有界ですから, 此の半直線は必ず $A$ の境界と交はります.
其の一点を $M_1$ とし, 此の線分 $(P_1, M_1)$ を $L_1$ とします.
同様にして, $P_2$ を一端とする線分 $L_2$ を描きます. 線分 $L_1$ の
長さが $L_2$ の長さを超えないと考へませう. (か様に假定しても勿論一般性は
失はれません.) $M = M_1$ に対応する $G (P)$ を $G_0 (P)$ とします.
$G_0 (P)$ は $A$ に於て正則, $M_1$ 以外の $A$ のすべての界点に於ても矢張り
正則であって, $M_1$ に於ては極を持ちます. 故に, $G_0 (P)$ は $P_1, P_2$ に
於て相異なる要素を持たなければなりません. 従って, \ 2$^\circ$, \ $A$ の相異なる
二点に対し, $(\mathfrak{F})$ 中には之等の二点に於て相異なる値をとる函数が必ずあります.
か様に条件 1$^\circ, 2^\circ$ が充たされますから, \underline{Cartan--Thullen} の第二定理に依て,
$A$ は正則域です. $\mathfrak{D}$ は有限葉領域であって, $A$ は其の如何程でも
近くにとれますから, \underline{Behnke--Stein} の定理に依て,
\underline{$\mathfrak{D}$ は正則域です.}
\medskip
$\mathfrak{D}$ を改めて空間 $(x)$ に於ける \underline{擬凸状域}とします.
此の $\mathfrak{D}$ に関して, 補助定理 $\rm{II}$ に述べた函数 \,$\varphi_0
(P)$\, をとり, $\alpha$ を任意の実数として, \,$\mathfrak{D}_\alpha \ (\varphi_0 (P) \!<\! \alpha)$ を考へます. (但し, $\mathfrak{D}_\alpha$ が
実在する程 $\alpha$ を大きくとります.) 定理2の証明に於てしました様に
(第 3 節, 及び前報告の第 3 節参照), $\mathfrak{D}_\alpha$ を小 $2 n$ 次元
立方体 (開集合) $(A)$ に分ちます. 但し, $(A)$ は必ずしも完全な形とは
限りません. 充分細かく分ちますと, 補助定理 4 に依て, 各 $(A)$ は (完全な形で
あるときは云ふ迄もなく, 然うでない場合に於ても) 有理整函数に関して凸状な
単葉開集合です. 従って, \underline{Cartan--Thullen} の第二定理に依て, 其の各連結成分は
正則域です. 分割を充分細かくすれば, 各 $(B)$ に就ても同様です. ($(B)_0$ は
$(A)_0$ を中心とする $9^n$ 個の $(A)$ と, 夫等の境界の適当な部分とからなる
$2n$ 次元立方体であって, 勿論不完全な形であってもよろしい.) 故に,
上に得た結果によって, $\mathfrak{D}_\alpha$ の各連結成分が正則域であることが,
\underline{Cousin} の第一問題の時と同様にして容易に分かります. 故に, \underline{Behnke--Stein} の
定理に依て, $\mathfrak{D}$ は正則域です.
\medskip
\underline{定理 $\!\rm{I}$} --- \underline{分岐点を内点として持たない有限擬凸状域は正則域である.}
\footnote{以上の所論から Cartan--Thullen の第一定理を抜き去るには
$\ll$ 正則域 $\gg$ と云ふ言葉の代りに $\ll$ 次の二つの条件を充たす領域
$\mathcal{D}$ $\gg$ と云ふ言葉を使へばよろしい. 条件 : \ 1$^\circ$, \
$\mathcal{D}$ に於ける正則函数の全体を $(\mathcal{F})$ とすれば, $\mathcal{D}$ は
$(\mathcal{F})$ に関して凸状であること. \ 2$^\circ$, $\mathcal{D}$ の相異なる
任意の一対の点に対し, 此の二点に於て相異なる値をとる函数が $(\mathcal{F})$ 中に
存在すること. 結果として, 定理 $\rm{I}$ と Cartan--Thullen の第一定理とが
同時に得られます.}
\medskip
此の定理に依て, 或る領域が正則域かと云ふ問題は, 總て其の領域が擬凸状域かと
云ふ問題に還元せられます.
\footnote{第六報告の序言参照. 屡々遭遇する一例を求めますと, 擬凸状域の
\"Uberlargerungs-\\ bereich は矢張り擬凸状域ですから, 例へば, Cartan--Thullen の
第二定理に於て, 第二条件は要りません.}
\bigskip
\underline{11} --- 凸性の定義 (前報告, 第1節) を少し拡張し, 改め
て次の様に定義します.
\medskip
\underline{定義} --- \underline{$\mathfrak{D}$を空間$(x)$の分岐点を内点
として持たない有限領域とし,\,$(\mathfrak{F})$}\\ \underline{を$\mathfrak{D}$に於ける正則函数からなる一つの函数族とする.
$\mathfrak{D}$ が $(\mathfrak{F})$ に関して}\\ \underline{凸状とは, $\mathfrak{D}_0$を
$\mathfrak{D}_0 \Subset \mathfrak{D}$ なる任意の集合とする時, $\mathfrak{D}_0
\subset \mathfrak{D}' \subset \mathfrak{D}$なる$\mathfrak{D}$}\\ \underline{に関して
有界な開集合$\mathfrak{D}'$が対応し,\,$\mathfrak{D}'$に属しない$\mathfrak{D}$の任意の点$P_0$に対}\\ \underline{し,\,$(\mathfrak{F})$中に少なくとも一つの函数
$f (P)$があって, $|f (P_0)| > \max |f (\mathfrak{D}_0)|$}\\
\underline{となることを云ふ.\ $\mathfrak{D}$ が共通点を持たない数個 (有限又は無限) の上述の}\\ \underline{性質を持つ
領域の和である時も同様に名づける.}
\medskip
此の定義は明らかに之迄の定義を含みます. 次の凸性を併せ考へるのが便利です.
\medskip
\underline{定義} --- \underline{上述の状況に於て, \,$\mathfrak{D}$ が $(\mathfrak{F})$ に関して狭義凸状とは,
\,$\mathfrak{D}_0 \Subset \mathfrak{D}$}\\ \underline{なる任意の集合$\mathfrak{D}_0$に対し,
$\mathfrak{D}_0 \subset \mathfrak{D}' \Subset \mathfrak{D}$なる開集合$\mathfrak{D}'$ が対応して上述}\\ \underline{の条件を充たすことを云ふ.}
\medskip
狹義凸状ならば, 明らかに凸状です. $\mathfrak{D}$ が有限葉ならば, 此の新しく
定義した二つの凸性及び前に定義したものは一致します. $\mathfrak{D}$ が
$\mathfrak{D}$ に於ける正則函数の全体に関して凸状 (或は狹義凸状)
であることを, 簡単に, $\mathfrak{D}$ は\underline{正則凸状} (或は
\underline{狹義正則凸状}) で
あると云ひませう.
\footnote{H. Behnke 及び其の一門は狹義凸性を指して凸性と呼んで居ます.
(Behnke--Thullen の著書及び第一節の始めに挙げた H. Behnke--K. Stein の
三つの論文中始めの二つ, 特に, 第二のもの参照.) 尚, 前に一度申しました様に,
正則函数族に関する全域的凸性なる概念は H. Cartan に依て導入せられました.
(第四報告の終り脚註に於ける, H. Cartan の論文参照.)}
正則域は狹義正則凸状であるかと云ふことが前報告以来の懸案でした
\footnote{其の第1節参照. 我々は第一次研究 (第一乃至第六報告) が一通り
済むまでは単葉領域を離れませんでしたが, 其の理由の一つは此処にありました.}.
先づ此の点を調べます.
\medskip
\underline{補助定理7} --- \underline{補助定理$\rm{II}$
(第7節) に於て, $\mathfrak{D}_\alpha$は
$\mathfrak{D}$に於ける正則函数}\\ \underline{の全体に関して凸状である.}
\medskip
\underline{証明} ~ $\mathfrak{D}_\alpha$ は擬凸状域です (補助定理4, \underline{Cartan--Thullen} の
第二定理及び \underline{Hartogs} の定理に依る). 故に, 定理 $\rm{I}$ に依て正則域です.
故に, $\beta$ を $\alpha < \beta$ なる実数としますと, 補助定理 5 に依て,
$\mathfrak{D}_\alpha$ は $\mathfrak{D}_\beta$ に於ける正則函数の全体に関して
凸状です. 故に, \underline{Behnke--Stein} の補助定理に依て, $\mathfrak{D}_\alpha$ は
$\mathfrak{D}$ に於ける正則函数の全体に関して凸状です. ~ (証明終)
\medskip
\underline{定理 $\rm{II}$} --- \underline{有限正則域は狹義正則凸状である.}
\medskip
\underline{証明} ~ $\mathfrak{D}$ を空間 $(x)$ の (有限) 正則域とします. $E$ を,
$E \Subset \mathfrak{D}$ なる任意の点集合とします. 補助定理 $\rm{II}$ に於ける
$\mathfrak{D}_\alpha$ を $E \Subset \mathfrak{D}_\alpha$ となる様にとります.
上の補助定理 7 に依て, $\mathfrak{D}_\alpha$ は $\mathfrak{D}$ に於ける
正則函数の全体 $(\mathfrak{F})$ に関して凸状ですから, 補助定理 1 に依て,
$\mathfrak{D}_0 = \mathfrak{D}_\alpha$ と見て, 解析的多面体 $\Delta$ を,
$$
P \!\in\! R, \ |x_i| \!<\! r, \ |f_j (P)| \!<\! 1 \ \ (i \!=\! 1, 2,\ldots, n; j \!=\! 1,
2,\ldots, \nu) \leqno{\quad (\Delta)}
$$
の形に, $E \Subset \Delta$ となる様に撰ぶことが出来ます. 此処に, $f_j (P)$ は
$(\mathfrak{F})$ の函数, $R$ は $\Delta \Subset R \subset \mathfrak{D}$ なる適当な
開集合です.
$\Delta$ に属しない $\mathfrak{D}$ の任意の点を $P_0$ とします.
此の $P_0$ に対して, $(\mathfrak{F})$ の函数 $f (P)$ を, $|f (P_0)| > \max |f (E)|$
となる様に撰びうることを云へばよろしい. $\Delta$ と同じ性質の $\Delta'$ を,
$\Delta \Subset \Delta', \ P_0 \in \Delta'$ となる様にとります. $\Delta'$ の形を,
$$
P \!\in\! R', \ |x_i| \!<\! r', \ |F_k (P)| \!<\! 1 \ \ (i \!=\! 1, 2,\ldots, n; k \!=\! 1,
2,\ldots, \mu) \leqno{\quad (\Delta')}
$$
とします. 但し, $r \!\leq\! r'$ となる様に撰びます. $\Delta, \,\Delta'$ から
$$
P \!\in\! R', \quad |x_i| \!<\! r, \quad |f_j (P)| \!<\! 1, \quad |F_k (P)| \!<\! 1
\qquad \vspace{-2mm}
\leqno{\quad (\Delta'')}
$$
$$
\qquad \qquad \qquad (i \!=\! 1, 2, \ldots, n; j \!=\! 1, 2, \ldots, \nu; k \!=\! 1, 2, \ldots, \mu)
$$
を作ります. $\Delta$ は明らかに $\Delta''$ の一つ又は数個の連結成分の和です.
$P_0$ が $\Delta''$ に属しないならば, $x_i, \,f_j (P)$ の中には, 所求の条件を
充たすものが必ずあります. $P_0$ が $\Delta''$ に属するならば, $\Delta''$ に
於て, $\Delta$ で 0, 他の部分で 1 となる函数を考へますと, 此の函数は,
$\Delta''$ に於て正則ですから, 定理 1 に依て, $\Delta''$ の各点で斉一に
収歛する様な, $(\mathfrak{F})$ の函数の級数に展開されます. 故に, 此の場合にも
所求の函数は存在します. ~ (証明終)
\medskip
\underline{系} --- \underline{$\mathfrak{D}$ を空間$(x)$の有限正則域とし, $\mathfrak{D}_0$ を
$\mathfrak{D}$ に於ける正則函数の全}\\ \underline{体 $(\mathfrak{F})$ に関して凸状な
$\mathfrak{D}$ の部分開集合とすれば, \ $\mathfrak{D}_0$ は $(\mathfrak{F})$ に関して
狹}\\ \underline{義凸状である.}
\medskip
\underline{証明} ~ $\mathfrak{D}_0$ は $(\mathfrak{F})$ に関して凸状ですから, $E$ を,
$E \Subset \mathfrak{D}_0$ なる任意の集合としますと, $\mathfrak{D}_0$ には
$E \subset \mathfrak{D}' \subset \mathfrak{D}_0$ であって $\mathfrak{D}_0$ に
関して有界な開集合 $\mathfrak{D}'$ があって, 定義に述べた条件を充たします.
他方, 上の定理 $\rm{II}$ に依て, $\mathfrak{D}$ には $E \subset
\mathfrak{D}'' \Subset \mathfrak{D}$ なる $\mathfrak{D}''$ があって,
$\mathfrak{D}$ に関して, 従って勿論 $\mathfrak{D}_0$ に関して同じ条件を
充たします. $\mathfrak{D}' \cap \mathfrak{D}'' = \mathfrak{D}_1$ を考へますと,
$E \subset \mathfrak{D}_1 \subset \mathfrak{D}_0$ であって, 此の条件を
充たします. 所で, $\mathfrak{D}''$ は有界葉であって, $\mathfrak{D}'$ は
$\mathfrak{D}_0$ に関して有界ですから, $\mathfrak{D}_1 \Subset \mathfrak{D}_0$
です. 故に, $\mathfrak{D}_0$ は $(\mathfrak{F})$ に関して狹義凸状です. ~ (証明終)
\medskip
定理 1 と此の系とから, 次の結果が得られます.
\medskip
\underline{定理 $\rm{III}$} --- \underline{$\mathfrak{D}$ を空間 $(x)$ の有限正則域とし,
$\mathfrak{D}_0$ を $\mathfrak{D}$ に於ける正則函}\\ \underline{数の全体 $(\mathfrak{F})$ に
関して凸状な $\mathfrak{D}$ の部分開集合とすれば, \ $\mathfrak{D}_0$ に
於ける任}\\ \underline{意の正則函数は,\,$\mathfrak{D}_0$の各点に於て斉一に収歛する様な,
$(\mathfrak{F})$の函数の級数}\\ \underline{に展開することが出来る.}
\medskip
定理 2 と定理 $\rm{II}, {III}$ とから, 次の結果が得られます.
\medskip
\underline{定理 $\rm{IV}$} --- \underline{有限正則域に於て, Cousin の第一問題は常に解ける.}
\begin{flushright}
(第十一報告終, \ \ 3. 12. 12) \hspace*{2zw}
\end{flushright}
\end{document}
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\hypersetup{%
pdfauthor = {APMEP},
pdfsubject = {Corrigé du brevet des collèges},
pdftitle = {Amérique du Nord juin 2012},
allbordercolors = white}
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\begin{document}
\setlength\parindent{0mm}
\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small Corrigé du brevet des collèges}
\lfoot{\small{Amérique du Nord}}
\rfoot{\small{8 juin 2012}}
\renewcommand \footrulewidth{.2pt}
\pagestyle{fancy}
\thispagestyle{empty}
\begin{center}
\textbf{Durée : 2 heures}
\vspace{0,5cm}
{\Large \textbf{\decofourleft~Corrigé du brevet des collèges Amérique du Nord ~\decofourright\\8 juin 2012}}
\vspace{0,5cm}
L'utilisation d'une calculatrice est autorisée.
\end{center}
\vspace{0,5cm}
\textbf{\textsc{Activités numériques} \hfill 12 points}
\medskip
\textbf{Exercice 1}
\medskip
%Quatre affirmations sont données ci-dessous.
%
%\medskip
\textbf{Affirmation 1 :} $\dfrac{1}{8} = \dfrac{125}{\np{1000}} = 0,125$. Vraie % est un nombre décimal.
\medskip
\textbf{Affirmation 2 :} $72 = 1\times 72 = 2 \times 36 = 3 \times 24$ cela fait déjà 6 diviseurs. Fausse.% a exactement cinq diviseurs.
\medskip
\textbf{Affirmation 3 :} %Si $n$ est un entier, $(n - 1) (n + 1) + 1$ est toujours égal au carré d'un entier.
$(n - 1) (n + 1) + 1 = n^2 - 1 + 1 = n^2$. Vraie
\medskip
\textbf{Affirmation 4 :} %Deux nombres impairs sont toujours premiers entre eux.
Fausse : 3 et 9 sont impairs et ne sont pas premiers entre eux.
%\medskip
%Pour chacune, indiquer si elle est vraie ou fausse en argumentant la réponse.
%
\bigskip
\textbf{Exercice 2}
\medskip
%Deux classes du collège ont répondu à la question suivante :
%
%\og Combien de livres avez-vous empruntés durant les 12 derniers mois ? \fg
%
%Les deux classes ont communiqué les réponses de deux façons différentes :
%
%\begin{tabular}{l l}
%Classe \no 1 :& 1~;~2~;~ 2~;~2~;~2~;~3~;~3~;~3~;~3~;~3~;~3~;~3~;~3~;~6~;~6~;~ 6~;~6~;~6~;~7~;~7~;~7\\
%\end{tabular}
%
%\begin{tabular}{l l r}
%Classe \no 2 :& Effectif total :& 25\\
% &Moyenne : &4\\
% &Étendue : &8\\
% &Médiane : &5\\
%\end{tabular}
%
%\medskip
\begin{enumerate}
\item %Comparer les nombres moyens de livres empruntés dans chaque classe.
Classe 1 : moyenne des livres empruntés $\dfrac{84}{21} = 4$ identique à celle de la classe 2.
\item %Un \og grand lecteur \fg{} est un élève qui a emprunté 5 livres ou plus.
%Quelle classe a le plus de \og grands lecteurs \fg{}?
Il y en a 8 dans la classe 1 et il y en a au moins 12 dans la classe 2.
\item %Dans quelle classe se trouve l'élève ayant emprunté le plus de livres ?
Dans la classe 2 l'étendue est égale à 8 ; même s'il y a un élève qui n'a pas emprunté de livre, il y en a au moins un qui a emprunté au moins 8 livres. C'est donc dans la classe 2 qu'il y a le plus grand lecteur des deux classes.
\end{enumerate}
\bigskip
\textbf{Exercice 3}
\medskip
%Léa observe à midi, au microscope, une cellule de bambou.
%
%Au bout d'une heure, la cellule s'est divisée en deux. On a alors deux cellules.
%
%Au bout de deux heures, ces deux cellules se sont divisées en deux.
%
%Léa note toutes les heures les résultats de son observation.
%
%À quelle heure notera-t-elle, pour la première fois, plus de 200 cellules ?
%
%\emph{Vous laisserez apparentes toutes vos recherches.\\
% Même si le travail n'est pas terminé, il en sera tenu compte dans la notation.}
Chaque heure le nombre de cellules est doublé : on a donc successivement :
$2~;~4~;~8~;~16~;~32~;~64~;~128~;~256~;~\ldots$
C'est donc à la 8\up{e} heure que le nombre dépassera 200.
\vspace{0,5cm}
\textbf{\textsc{Activités géométriques} \hfill 12 points}
\medskip
\textbf{Exercice 1}
\medskip
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%On a modélisé géométriquement un tabouret pliant par les segments [CB] et [AD] pour l'armature métallique et le segment [CD] pour l'assise en toile.
%
%On a CG = DG = 30 cm, AG = BG = 45 cm et AB = 51 cm.
%
%Pour des raisons de confort, l'assise [CD] est parallèle au sol représenté par la droite (AB).
%
%\medskip
%
%Déterminer la longueur CD de l'assise.
%
%\emph{Vous laisserez apparentes toutes vos recherches.
%Même si le travail n'est pas terminé, il en sera tenu compte dans la notation.}
Toutes les conditions d'application du théorème de Thalès sont établies ; on peut donc écrire :
$\dfrac{\text{CG}}{\text{GB}} = \dfrac{\text{DG}}{\text{GA}} = \dfrac{\text{CD}}{\text{AB}}$, soit
$\dfrac{30}{45} = \dfrac{30}{45} = \dfrac{\text{CD}}{51}$.
En simplifiant par 15, on a donc $\dfrac{2}{3} = \dfrac{\text{CD}}{51}$, soit CD $ = 51 \times \dfrac{2}{3} = 34$~(cm).
\bigskip
\textbf{Exercice 2}
\medskip
%\parbox{0.65\linewidth}{On considère un sablier composé de deux cônes identiques
%de même sommet C et dont le rayon de la base est AK = 1,5~cm. Pour le protéger, il est enfermé dans un cylindre de hauteur 6~cm et de même base que les deux cônes.
%\begin{enumerate}
%\item On note V le volume du cylindre et V$_{1}$ le volume du sablier.
%
%Tous les volumes seront exprimés en cm$^3$.
\begin{enumerate}
\item
\begin{enumerate}
\item %Montrer que la valeur exacte du volume V du cylindre est $13,5 \pi$.
Le cylindre a un rayon de 1,5~cm et une hauteur de 6~cm, son volume est :
V $ = \pi \times 1,5^2 \times 6 = \pi \times 2,25 \times 6 = 13,5\pi$.
\item %Montrer que la valeur exacte de V$_{1}$ est $4,5 \pi$.
Les deux cônes sont identiques, ils ont la même hauteur $\dfrac{6}{2} = 3$ et pour rayon de leur base 1,5~cm. Leur volume est donc :
$\dfrac{\pi \times 1,5^2 \times 3}{3} = 2,25\pi$, donc V$_1 = 2\times 2,25\pi = 4,5\pi$.
\item %Quelle fraction du volume du cylindre, le volume du sablier occupe-t-il ?
On a $\dfrac{\text{V}_1}{\text{V}} = \dfrac{4,5\pi}{13,5\pi} = \dfrac{4,5}{13,5} = \dfrac{1}{3}$.
%(On donnera le résultat sous la forme d'une fraction irréductible)
\end{enumerate}
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%
%\emph{Rappel :} La formule du volume du cône est : $\dfrac{\text{aire de la base} \times \text{hauteur}}{3}$
%
%\begin{enumerate}
\item%[\textbf{2.}] On a mis 12 cm$^3$ de sable dans le sablier.
%Sachant que le sable va s'écouler d'un cône à l'autre avec un débit de 240~cm$^3$/h, quel temps sera mesuré par ce sablier ?
On a vu que le volume de la partie haute est égale à $2,25\pi \approx 7,1$~cm$^3$. Il va être très difficile de mettre 12~cm$^3$ dedans...
\emph{Remarque } : si le sable pouvait rentrer il s'écoulerait à la vitesse de 240 cm$^3$ par 60 minutes soit à la vitesse de 4 cm$^3$ par minute. Comme $12 = 3 \times 4$, le sable s'écoulera en 3~min (durée habituelle d'un sablier).
\end{enumerate}
\bigskip
\textbf{Exercice 3}
\medskip
%\parbox{0.55\linewidth}{Construire un carré dont l'aire est égale à la somme des aires des deux carrés représentés ci-contre.
%
%\emph{Vous laisserez apparentes toutes vos
%recherches.\\
%Même si le travail n'est pas terminé, il en sera tenu compte dans la notation}.}
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La mesure chaque côté du petit carré est égale à 2, puisque $2^2 = 4$.
Donc la mesure chaque côté du grand est égale à 4 ; son aire est égale à $4^2 = 16$.
Il faut donc trouver un carré de côté $c$ tel que :
$c^2 = 4 + 16 = 20$ ; on en déduit que $c = \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$.
Pour construire cette longueur il suffit de construire un triangle rectangle dont les côtés mesurent 2 et 4, puisque comme on vient de le voir d'après les théorème de Pythagore :
$2^2 + 4^2 = 20 = \left(2\sqrt{5}\right)^2$.
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\vspace{0,5cm}
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\textbf{\textsc{Problème} \hfill 12 points}
\medskip
%Dans un collège de Caen (Normandie) est organisé un échange avec le Mexique pour les élèves de 3\up{e} qui étudient l'espagnol en seconde langue.
%
%\medskip
%
%\textbf{Partie 1 - L'inscription des élèves}
%
%\medskip
%
%Le tableau ci-dessous permet de déterminer la répartition de la seconde langue étudiée par les 320 élèves de 4\up{e} et de 3\up{e} de ce collège.
%
%\begin{center}
%\begin{tabularx}{0.75\linewidth}{|m{3.5cm}|*{3}{>{\centering \arraybackslash}X|}}\hline
%Seconde langue étudiée &4\up{e} &3\up{e} &Total\\ \hline\hline
%Espagnol &84 & & \\ \hline
%Allemand &22 &24 &\\ \hline
%Italien &62 &50 & \\ \hline
%Total & & &320\\ \hline
%\end{tabularx}
%\end{center}
\begin{enumerate}
\item %Combien d'élèves peuvent être concernés par cet échange ?
Soit $x$ le nombre d'élèves de 3\up{e} faisant de l'espagnol en seconde langue.
Le total des élèves est :
$84 + x + 22 + 24 + 62 + 50 = 320$ ou $x + 242 = 320$ et enfin $x = 78$.
Il y a donc $84 + 78 = 162$ élèves qui peuvent être concernés par cet échange.
\item %24 élèves vont participer à ce voyage.
%Est-il vrai que cela représente plus de 12\,\% des élèves de 3\up{e}?
Il y a en 3\up{e} $78 + 24 + 50 = 152$ élèves.
Or $\dfrac{24}{152} \times 100 = \dfrac{300}{19} \approx 15,8\,\%$ soit effectivement plus de 12\,\%.
\end{enumerate}
\bigskip
\textbf{Partie II - Le financement}
\medskip
%Afin de financer cet échange, deux actions sont mises en {\oe}uvre : un repas mexicain et une tombola.
%
%\medskip
\begin{enumerate}
\item %Le repas mexicain, où chaque participant paye 15~\euro.
%Au menu, on trouve un plat typique du Mexique, le \emph{Chili con carne}.
%
%\medskip
%
%\begin{center}
%\begin{tabularx}{0.75\linewidth}{|*{2}{>{\centering \arraybackslash}X|}}\hline
%\multicolumn{2}{|c|}{\large Recette pour 4 personnes}\\ \hline\hline
%50~g de beurre &500~g de b{\oe}uf haché\\
%2 gros oignons &65~g de concentré de \\
%2 gousses d'ail&tomate \\
%30~cl de bouillon de b{\oe}uf&400~g de haricots rouges\\ \hline
%\end{tabularx}
%\end{center}
%
%\medskip
%
%50 personnes participent à ce repas.
\begin{enumerate}
\item %Donner la quantité de b{\oe}uf haché, de haricots rouges, d'oignons et de concentré de tomate nécessaire.
On a $\dfrac{50}{4} = \dfrac{25}{2} = 12,5$. Il faut donc multiplier toutes les quantités de la recette par 12,5.
B{\oe}uf haché : \np{6250}~g ; haricots rouges \np{5000}~g soit 5~kg ;
Oignons : 25 ; concentré de tomate : 812,5~g.
\item %Les dépenses pour ce repas sont de 261~\euro, quel est le bénéfice ?
On récupère : $50 \times 15 = 750$~\euro. IL y a donc un bénéfice de $750 - 261 = 489$~\euro.
\end{enumerate}
\item %La tombola, où 720~tickets sont vendus au prix de 2~\euro.
%Les lots sont fournis gratuitement par trois magasins qui ont accepté de sponsoriser le projet.
%
%Il y a trois lots à gagner : un lecteur DVD portable, une machine à pain et une mini-chaîne Hifi.
%
%Un élève achète 1 ticket.
\begin{enumerate}
\item %Quelle probabilité a-t-il de gagner l'un des lots ?
La probabilité est égale à $\dfrac{3}{720} = \dfrac{1}{240} \approx \np{0,0042}$ soit 0,4\,\% à peu près.
\item %Quelle probabilité a-t-il de gagner la mini-chaîne Hifi ?
La probabilité est égale à $\dfrac{1}{720} \approx \np{0,0014}$ soit 0,15\,\% à peu près.
\end{enumerate}
\item %Montrer que la somme récupérée par les deux actions est de \np{1929}~\euro.
Avec la tombola on récupère $720 \times 2 = \np{1440}$~\euro.
La somme récupérée au total est égale à : $489 + \np{1440} = \np{1929}$~\euro.
\end{enumerate}
\bigskip
\textbf{Partie II - Le voyage}
\medskip
%Le voyage se décompose en deux parties : le trajet Caen-Paris (256~km) se fait en bus puis le trajet Paris-Mexico (\np{9079}~km) en avion.
%
%\medskip
\begin{enumerate}
\item %Le prix d'un billet d'avion aller-retour coûte 770,30~\euro{} par personne.
%L'argent récolté par le repas mexicain et la tombola permet de réduire équitablement ce prix pour les 24 élèves participants.
%Quelle est la participation demandée par élève pour les billets d'avion ? (arrondir à l'unité)
Les billets d'avion reviennent à $24 \times 770,30 = \np{18487,20}$~\euro.
On déduit les \np{1929}~\euro des deux actions ; reste à payer : $\np{18487,20} - \np{1929} = \np{16658,20}$~\euro.
Il faut donc demander à chaque élève : $\dfrac{\np{16658,20}}{24} \approx 690$~\euro à l'euro près.
\item %Le décollage se fait à 13~h~30. Cependant, les élèves et les accompagnateurs doivent être impérativement à l'aéroport de Paris-Roissy à 11~h~30.
%On estime la vitesse moyenne du bus à 80~km/h.
%Jusqu'à quelle heure peut-il partir de Caen ?
Pour parcourir 256 km à 80 km/h, il faut $\dfrac{256}{80} = 3,2$~h soit 3~h et $0,2 \times 60 = 12$~min.
Il faut donc partir au plus tard à 11 h 30 moins 3 h 12 min soit 8~h 18~min.
\item %L'avion arrive à Mexico à 17~h~24 heure locale. Il faut compter 7 heures de décalage avec la France.
\begin{enumerate}
\item %Quelle est la durée du trajet ?
Quand l'avion arrive il est 17 h 24 + 7 h = 24 h 24 heure française. Soit un trajet de :
24 h 24 moins 13 h 30 = 10~h 54~min.
\item %Quelle est la vitesse moyenne de l'avion ? (arrondir à l'unité)
10 h 54 min correspondent à 600 + 54 = 654~min
La vitesse moyenne de l'avion est $\dfrac{\np{9709}}{654} \approx \np{14,8456}$~(km/min) soit $60 \times \np{14,8456} \approx 891$~(km/h).
\end{enumerate}
\end{enumerate}
\end{document}
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% This issue 90 of the Massive Star Newsletter
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\hfill NEWSLETTER \hfill
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\centerline{\small formely known as {\em the hot star newsletter}}
\centerline{$\star $}
\\[0.9mm] \lightframe{0.5}{2}{1cm}
{No. 90} 2005 Octobre--December
% \\ \mbox{}
\hfill {\small http://www.astroscu.unam.mx/massive\_\_stars}
\\[0.5mm] editor: Philippe Eenens
\hfill {\small http://www.star.ucl.ac.uk/$\sim$hsn/index.html}
\\[0.5mm] [email protected]
% \\ \mbox{}
\hfill {\small ftp://ftp.sron.nl/pub/karelh/{\sc uploads/wrbib/}}
}}
\vspace*{1.5cm}
\centerline{\lightframe{0.7}{5}{6cm}{
\centerline{\Large Contents of this newsletter}}}
% \vspace*{0.5cm}
\begin{quote}
\begin{quote}
\smallskip News \dotfill 1 \\
%\smallskip Working group matters \dotfill 1 \\
%\smallskip Discussion forum \dotfill 888 \\
%\smallskip Commentary \dotfill 888 \\
\smallskip Abstracts of 10 accepted papers \dotfill 2 \\
%\smallskip Abstracts of 99 submitted papers \dotfill 888 \\
\smallskip Abstracts of 8 proceedings papers \dotfill 9 \\
%\smallskip Abstract of 99 dissertation thesis \dotfill 888 \\
\smallskip Jobs \dotfill 12 \\
%\smallskip Meetings \dotfill 888 \\
\end{quote}
\end{quote}
\centerline{\lightframe{0.7}{5}{1.5cm}{
\centerline{\Large News }}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4mm}
\begin{center}{\Large\bf % WRITE THE Title BELOW THIS LINE
Proceedings of the JENAM 2005 Workshop on ``Massive Stars and
High-Energy Emission in OB Associations'' held in Li\`ege (Belgium) on
July 4 - 5, 2005}
\end{center}
\centerline{\bf G.\ Rauw$^1$, Y.\ Naz\'e$^1$, R.\ Blomme$^2$ and E.\
Gosset$^1$
}{\footnotesize
$^1$ Institut d'Astrophysique et de G\'eophysique, Universit\'e de
Li\`ege, 4000 Li\`ege, Belgium
\\ $^2$ Royal Observatory of Belgium, 1180 Brussels, Belgium
}\vspace*{4mm} \\
% WRITE TEXT OF ABSTRACT HERE
In the framework of the Joint European and National Astronomy Meeting
(JENAM) held in Li\`ege (Belgium) from July 4 -- 7, 2005, one of the
mini-symposia was focused on ``Massive Stars and High-Energy Emission in
OB Associations''. This workshop addressed recent developments in
several hot topics related to massive star research. Recent analyses of
the fundamental parameters of these important objects, of their
interactions within binary systems as well as with their surroundings
(including the feedback of massive stars on the formation of other
stars) were presented. Special emphasis was put on the studies of OB
associations and young open clusters with current high-energy space
observatories (XMM-Newton, Chandra and INTEGRAL) which provide not only
information on the massive stars, but also on pre-main sequence stars.
The papers presented during this workshop include\\
{\small
- {\it X-ray and gamma-ray emission from single and binary early-type
stars}
(I.R.\ Stevens)\\
- {\it X-ray survey of Wolf-Rayet stars in the Magellanic Clouds} (M.A.\
Guerrero \& Y.-H.\ Chu)\\
- {\it Parameters of massive stars in the Milky Way and nearby galaxies}
(A.\ Herrero \& F.\ Najarro)\\
- {\it The peculiar Of?p stars HD\,108 and HD\,191612} (Y.\ Naz\'e et
al.)\\
- {\it A new paradigm for the X-rays from O-stars} (A.M.T.\ Pollock \&
A.J.J.\ Raassen)\\
- {\it CN status of a sample of galactic OB supergiants} (M.\ Sarta
Dekovic \& D.\ Kotnik-Karuza)\\
- {\it Observations of non-thermal radio emission in O-type stars} (R.\
Blomme)\\
- {\it Radio emission from colliding-wind binaries: observations and
models} (S.M.\ Dougherty et al.)\\
- {\it The XMM-Newton view of Plaskett's star and its surroundings} (N.\
Linder \& G.\ Rauw)\\
- {\it Non-thermal X-ray and $\gamma$-ray emission from the
colliding-wind binary WR\,140} (J.M.\ Pittard \& S.M.\ Dougherty)\\
- {\it Can single O-stars produce non-thermal radio emission? Or are
they binaries?} (S.\ Van Loo)\\
- {\it Are WC9 Wolf-Rayet stars in colliding-wind binaries?} (P.M.\
Williams et al.)\\
- {\it X-ray analysis of the close binary system FO\,15} (J.F.\ Albacete
Colombo \& G.\ Micela)\\
- {\it Evidence for phase-locked X-ray variations from the CWB
Cyg\,OB2\,\#8a} (M.\ De Becker \& G.\ Rauw)\\
- {\it Preliminary results of an observational campaign aiming at the
study of the binary system LSS\,3074} (E.\ Gosset et al.)\\
- {\it The colliding winds of WR\,146: seeing the works} (E.P.\ O'Connor
et al.)\\
- {\it On the multiplicity of the non-thermal radio emitters 9\,Sgr and
HD\,168112} (G.\ Rauw et al.)\\
- {\it CPD$-41^{\circ}$\,7742: an unusual wind interaction} (H.\ Sana et
al.)\\
- {\it Energetic processes and non-thermal emission of star forming
complexes} (A.M.\ Bykov)\\
- {\it X-raying the super star clusters in the Galactic center} (L.M.\
Oskinova)\\
- {\it XMM-Newton observations of the Cyg\,OB2 association} (G.\ Rauw et
al.)\\
- {\it The young open cluster NGC\,6231: five years of investigations}
(H.\ Sana et al.)\\
- {\it A spectroscopic investigation of the young open cluster IC\,1805}
(M.\ De Becker \& G.\ Rauw)\\
- {\it A survey for $\gamma$-ray emission from OB associations with
INTEGRAL: some preliminary results} (J.-C.\ Leyder \& G.\ Rauw)}
The proceedings of the workshop are now available on the web at the URL
http://www.astro.ulg.ac.be/RPub/Colloques/JENAM/proceedings/proceedings.html
We would like to take this opportunity to thank all the participants in
the workshop.
%------------------------------------------------------------------------
\vspace*{5mm}
\vspace*{5mm}
\centerline{\lightframe{0.7}{5}{4cm}{
\centerline{\Large Accepted Papers}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\begin{center}{\Large\bf
Reduced Wolf-Rayet Line Luminosities at Low Metallicity
}\end{center}
\centerline{\bf Paul A Crowther, L J Hadfield}
Sheffield
New NTT/EMMI spectrophotometry of single WN2-5 stars in the Magellanic Clouds
are presented, from which HeII 4686 line luminosities have been
derived, and compared with observations of other Magellanic Cloud WR
stars. SMC WN3-4 stars possess line luminosities which are a factor of
4 times lower than LMC counterparts, incorporating several binary SMC
WN3-4 stars. Similar results are found for WN5-6 stars, despite
reduced statistics, incorporating observations of single LMC WN5-9
stars. CIV 5808 line luminosities of carbon sequence WR stars in the
SMC and IC1613 (both WO subtypes) are a factor of 3 lower than LMC WC
stars from Mt Stromlo/DBS spectrophotometry, although similar results
are also obtained for the sole LMC WO star. We demonstrate how reduced
line luminosities at low metallicity follow naturally if WR winds are
Z-dependent, as recent results suggest. We apply mass loss-Z scalings
to atmospheric non-LTE models of Milky Way and LMC WR stars to predict
the wind signatures of WR stars in the metal-poor star forming WR
galaxy IZw18. WN HeII 4686 line luminosities are 7-20 times lower than
in Z-rich counterparts of identical bolometric luminosity, whilst WC
CIV 5808 line luminosities are 3-6 times lower. Significant He$^+$ Lyman
continuum fluxes are predicted for Z-poor early-type WR
stars. Consequently, our results suggest the need for larger
population of WR stars in IZw18 than is presently assumed,
particularly for WN stars, potentially posing a severe challenge to
evolutionary models at very low Z. Finally, reduced wind strengths
from WR stars at low Z impacts upon the immediate circumstellar
environment of long duration GRB afterglows, particularly since the
host galaxies of high-redshift GRBs tend to be Z-poor.
Reference: Accepted for A\&A
% \\ Status: Manuscript has been accepted
Weblink: astro-ph/0512183
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
$\alpha$ Eri: Rotational Distortion, Stellar and Circumstellar
Activity
}\end{center}
\centerline{\bf M.M.F. Vinicius$^1$
J. Zorec$^2$
N.V. Leister$^1$
R.S. Levenhagen$^1$}
1 - Instituto de Astronomia, Geof\'{\i}sica e Ci\^{e}ncias Atmosf\'ericas
da Universidade de S\~ao Paulo, CUASO, 05508-900 S\~ao Paulo SP,
Brazil
\\ 2 - Institut d'Astrophysique de Paris, UMR 7095 CNRS-Universit\'e
Pierre \& Marie Curie, 98bis Boulevard Arago, 75014 Paris, France
We explore the geometrical distortion and the stellar and
circumstellar activity of $\alpha$~Eri (HD 10144), the brightest Be star in
the sky. We present a thorough discussion of the fundamental parameters of the
object for an independent determination of its rotational distortion. We used
stellar atmosphere models and evolutionary tracks calculated for fast rotating
early-type stars. If the star is a rigid rotator, its angular velocity rate is
$\Omega/\Omega_c \simeq$ 0.8, so that its rotational distortion is smaller
than the one inferred from recent interferometric measurements. We then
discuss the stellar surface activity using high resolution and high S/N
spectroscopic observations of He\,{\sc i} and Mg\,{\sc ii} lines, which
concern a period of H$\alpha$ line emission decline. The variations in the
He\,{\sc i} lines are interpreted as due to non-radial pulsations. Time series
analysis of variations was performed with the {\sc cleanest} algorithm, which
enabled us to detect the following frequencies: 0.49, 0.76, 1.27 and 1.72 c/d
and pulsation degrees $\ell \sim (3-4)$ for $\nu$ = 0.76 c/d; $\ell \sim
(2-3)$ for $\nu$ = 1.27 c/d and $\ell \sim (3-4)$ for $\nu$ = 1.72 c/d. The
study of the absolute deviation of the He\,{\sc i} $\lambda $6678 \AA\
spectral line revealed mass ejection between 1997 and 1998. We conclude that
the lowest frequency found, $\nu =$ 0.49 c/d, is due to the circumstellar
environment, which is present even at epochs of low emission in the wings of
He\,{\sc i} $\lambda $6678 \AA\ and Mg\,{\sc ii} $\lambda$4481 \AA\ line
profiles, as well as during nearly normal aspects of the H$\alpha$ line.
This suggests that there may be matter around the star affecting some spectral
regions, even though the object displays a B-normal like phase. The long-term
changes of the H$\alpha$ line emission in $\alpha$~Eri are studied. We pay
much attention to the H$\alpha$ line emission at the epoch of interferometric
observations. The H$\alpha$ line emission is modeled and interpreted in terms
of varying structures of the circumstellar disc. We conclude that during the
epoch of interferometric measurements there was enough circumstellar matter
near the star to produce $\lambda 2.2\mu$m flux excess, which could account
for the overestimated stellar equatorial angular diameter. From the study of
the latest B[e] phase transition of $\alpha$~Eri we concluded that
the H$\alpha$ line emission formation regions underwent changes so that: a) the
low H$\alpha$ emission phases are characterized by extended emission zones in
the circumstellar disc and a steep outward matter density decline; b) during
the strong H$\alpha$ emission phases the emitting regions are less extended
and have a constant density distribution. The long-term variations of the
H$\alpha$ line in $\alpha$~Eri seem to have a 14-15 year cyclic B[e] phase
transition. The disc formation time scales, interpreted as the periods during
which the H$\alpha$ line emission increases from zero to its maximum, agree
with the viscous decretion model. On the other hand, the time required for the
disc dissipation ranges from 6 to 12 years which questions the viscous disc
model.
Reference: Astronomy and Astrophysics
% \\ Status: Manuscript has been accepted
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
Collisionless Damping of Fast MHD Waves in Magneto-rotational Winds
}\end{center}
\centerline{\bf
Takeru K. Suzuki(1), Huirong Yan(2,3) Alex Lazarian(2) \& Joseph P. Cassinelli(2)}
1 : Dept. of Physics, Kyoto University, Kyoto 606-8502, Japan
\\ 2 : Dept. of Astronomy, University of Wisconsin, 475 N. Charter St., Madison, WI 53706
\\ 3 : Present Address : CITA, University of Toronto, 60 St. George Street,
Toronto, Ontario, M5S 3H8, Canada
We propose collisionless damping of fast MHD waves as an important mechanism
for the heating and acceleration of winds from rotating stars.
Stellar rotation causes magnetic field lines anchored at the surface
to form a spiral pattern and magneto-rotational winds can be driven.
If the structure is a magnetically dominated, fast MHD waves
generated at the surface can propagate almost radially outward and
cross the field lines. The propagating waves undergo collisionless damping
owing to interactions with particles surfing on magnetic mirrors that are
formed by the waves themselves. The energy damping rate is especially
effective where the angle between the wave propagation and the field
lines becomes moderately large ($\sim 20$ to $80^{\circ}$). The angle tends
naturally to increase into this range because the field in magneto-rotational
winds develops an increasingly large azimuthal component.
The dissipation of the wave energy produces heating and acceleration of the
outflow.
We show using specified wind structures that this damping process can be
important in both solar-type stars and massive stars that have moderately
large rotation rates.
This mechanism can play a role in coronae of young solar-type stars which
are rapidly rotating and show X-ray luminosities much larger than the sun.
The mechanism could also be important for producing the
extended X-ray emitting regions inferred to exist in massive stars of
spectral type middle B and later.
Reference: ApJ, in press
% \\ Status: Manuscript has been accepted
Weblink: http://arxiv.org/abs/astro-ph/0505013
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
On the He II Emission In Eta Carinae and the Origin of Its Spectroscopic Events
}\end{center}
\centerline{\bf
J. C. Martin, K. Davidson, R. M. Humphreys, D. J. Hillier, K. Ishibashi}
School of Physics and Astronomy, University of Minnesota; Department
of Physics and Astronomy, University of Pittsburgh; Center for Space
Research, Massachusetts Institute of Technology
We describe and analyze Hubble Space Telescope (HST) observations of
transient emission near 4680 {\AA} in Eta Car, reported earlier by
Steiner \& Damineli (2004). If, as seems probable, this is He II
$\lambda$4687, then it is a unique clue to Eta Car's 5.5-year
cycle. According to our analysis, several aspects of this feature
support a mass-ejection model of the observed spectroscopic events,
and not an eclipse model. The He II emission appeared in early 2003,
grew to a brief maximum during the 2003.5 spectroscopic event, and
then abruptly disappeared. It did not appear in any other HST spectra
before or after the event. The peak brightness was larger than
previously reported, and is difficult to explain even if one allows
for an uncertainty factor of order 3. The stellar wind must provide a
temporary larger-than-normal energy supply, and we describe a special
form of radiative amplification that may also be needed. These
characteristics are consistent with a class of mass-ejection or
wind-disturbance scenarios, which have implications for the physical
structure and stability of Eta Car.
Reference: Astrophysical Journal
% \\ Status: Manuscript has been accepted
Weblink: http://arxiv.org/abs/astro-ph/0504151
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
Nonlocal Radiative Coupling in Non Monotonic Stellar Winds
}\end{center}
\centerline{\bf A. Feldmeier and R. Nikutta}
Potsdam University, Germany
There is strong observational evidence of shocks and
clumping in radiation-driven stellar winds from hot, luminous
stars. The resulting non monotonic velocity law allows for radiative
coupling between distant locations, which is so far not accounted for
in hydrodynamic wind simulations. In the present paper, we determine
the Sobolev source function and radiative line force in the presence
of radiative coupling in spherically symmetric flows, extending the
geometry-free formalism of Rybicki and Hummer (1978) to the case of
three-point coupling, which can result from, e.g., corotating
interaction regions, wind shocks, or mass overloading. For a simple
model of an overloaded wind, we find that, surprisingly, the flow
decelerates at all radii above a certain height when nonlocal
radiative coupling is accounted for. We discuss whether
radiation-driven winds might in general not be able to re-accelerate
after a non monotonicity has occurred in the velocity law.
Reference: A\&A, in press
% \\ Status: Manuscript has been accepted
Weblink: http://de.arxiv.org/pdf/astro-ph/0510806
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
The Early Star Generations: the Dominant Effect of Rotation on the CNO Yields
}\end{center}
\centerline{\bf Georges Meynet, Sylvia Ekstrom, Andre Maeder}
Geneva Observatory
We examine the role of rotation on the evolution and chemical yields
of very metal--poor stars. The models include the same physics, which
was applied successfully at the solar $Z$ and for the SMC, in
particular, shear diffusion, meridional circulation, horizontal
turbulence, and rotationally enhanced mass loss. Models of very low
$Z$ experience a much stronger internal mixing in all phases than at
solar $Z$. Also, rotating models at very low $Z$, contrary to the
usual considerations, show a large mass loss, which mainly results
from the efficient mixing of the products of the 3$\alpha$ reaction
into the H--burning shell. This mixing allows convective dredge--up to
enrich the stellar surface in heavy elements during the red supergiant
phase, which in turn favours a large loss of mass by stellar winds,
especially as rotation also increases the duration of this phase. On
the whole, the low $Z$ stars may lose about half of their
mass. Massive stars initially rotating at half of their critical
velocity are likely to avoid the pair--instability supernova. The
chemical composition of the rotationally enhanced winds of very low
$Z$ stars show large CNO enhancements by factors of $10^3$ to $10^7$,
together with large excesses of $^{13}$C and $^{17}$O and moderate
amounts of Na and Al. The excesses of primary N are particularly
striking. When these ejecta from the rotationally enhanced winds are
diluted with the supernova ejecta from the corresponding CO cores, we
find [C/Fe], [N/Fe],[O/Fe abundance ratios that are very similar to
those observed in the C--rich, extremely metal--poor stars (CEMP). We
show that rotating AGB stars and rotating massive stars have about the
same effects on the CNO enhancements. Abundances of s-process elements
and the ${12}$C/$^{13}$C ratio could help us to distinguish between
contributions from AGB and massive stars.
Reference: Astronomy and Astrophysics
% \\ Status: Manuscript has been accepted
Weblink: astro-ph/0510560
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
The Massive Eclipsing LMC Wolf-Rayet Binary BAT99-129 1. Orbital
Parameters, Hydrogen Content and Spectroscopic Characteristics
}\end{center}
\centerline{\bf
C. Foellmi (1), A.F.J. Moffat (2), S.V. Marchenko (3)}
1 - ESO (Chile); 2 - U. de Montreal (Canada); 3 - U. Western Kentucky (USA)
BAT99-129 in the LMC is one among a handful of extra-galactic
eclipsing Wolf-Rayet binaries known. We present blue,
medium-resolution, phase-dependent NTT-EMMI spectra of this system
that allow us to separate the spectra of the two components of the
binary and to obtain a reliable orbital solution for both stars. We
assign an O5V spectral type to the companion, and WN3(h)a to the
Wolf-Rayet component. We discuss the spectroscopic characteristics of
the system: luminosity ratio, radii, rotation velocities. We find a
possible oversynchronous rotation velocity for the O
star. Surprisingly, the extracted Wolf-Rayet spectrum clearly shows
the presence of blueshifted absorption lines, similar to what has been
found in all single hot WN stars in the SMC and some in the LMC. We
also discuss the presence of such intrinsic lines in the context of
hydrogen in SMC and LMC Wolf-Rayet stars, WR+O binary evolution and
GRB progenitors. Altogether, BAT99~129 is the extragalactic
counterpart of the well-known Galactic WR binary V444 Cygni.
Reference: Accepted by A\&A
% \\ Status: Manuscript has been accepted
Weblink: http://arxiv.org/abs/astro-ph/0510528
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
Multi-Periodic Photospheric Pulsations and Connected Wind Structures in HD64760
}\end{center}
\centerline{\bf
A.Kaufer(1), O.Stahl(2), R.K.Prinja(3), and D.Witherick(3)}
(1) European Southern Observatory,
(2) Landessternwarte Heidelberg,
(3) University College London
We report on the results of an extended optical spectroscopic
monitoring campaign on the early-type B supergiant HD64760 (B0.5Ib)
designed to probe the deep-seated origin of spatial wind structure in
massive stars. This new study is based on high-resolution echelle
spectra obtained with the FEROS instrument at ESO La Silla. 279
spectra were collected over 10 nights between February 14 and 24,
2003. From the period analysis of the line-profile variability of the
photospheric lines we identify three closely spaced periods around
4.810hrs and a splitting of +/-3\%. The velocity - phase diagrams of
the line-profile variations for the distinct periods reveal
characteristic prograde non-radial pulsation patterns of high order
corresponding to pulsation modes with l and m in the range 6-10. A
detailed modeling of the multi-periodic non-radial pulsations with the
BRUCE and KYLIE pulsation-model codes favors either three modes with
l=-m and l=8,6,8 or m=-6 and l=8,6,10 with the second case maintaining
the closely spaced periods in the co-rotating frame. The pulsation
models predict photometric variations of 0.012-0.020mag consistent
with the non-detection of any of the spectroscopic periods by
photometry. The three pulsation modes have periods clearly shorter
than the characteristic pulsation time scale and show small horizontal
velocity fields and hence are identified as p-modes. The beating of
the three pulsation modes leads to a retrograde beat pattern with two
regions of constructive interference diametrically opposite on the
stellar surface and a beat period of 162.8hrs (6.8days). This beat
pattern is directly observed in the spectroscopic time series of the
photospheric lines. The wind-sensitive lines display features of
enhanced emission, which appear to follow the maxima of the
photospheric beat pattern. The pulsation models predict for the two
regions normalized flux amplitudes of A=+0.33,-0.28, sufficiently
large to raise spiral co-rotating interaction regions. We further
investigate the observed Halpha wind-profile variations with a simple
rotating wind model with wind-density modulations to simulate the
effect of possible streak lines originating from the localized surface
spots created by the NRP beat pattern. It is found that such a simple
scenario can explain the time scales and some but not all
characteristics of the observed Halpha line-profile variations.
Reference: Accepted for A\&A on October 17, 2005.
% \\ Status: Manuscript has been accepted
Weblink: http://arxiv.org/abs/astro-ph/0510511
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
The Discordance of Mass-Loss Estimates for Galactic O-Type Stars
}\end{center}
\centerline{\bf
A. W. Fullerton (1,2), D. L. Massa (3), and R. K. Prinja (4)}
(1) Dept. of Physics \& Astronomy, University of Victoria
\\ (2) Dept. of Physics \& Astronomy, Johns Hopkins University
\\ (3) SGT Inc., NASA's Goddard Space Flight Center
\\ (4) Dept. of Physics \& Astronomy, University College London
We have determined accurate values of the product of the mass-loss rate
and the ion fraction of P$^{4+}$, Mdot q(P$^{4+}$), for a sample of 40 Galactic
O-type stars by fitting stellar-wind profiles to observations of the P V
resonance doublet obtained with FUSE, ORFEUS/BEFS, and Copernicus.
When P$^{4+}$ is the dominant ion in the wind, Mdot q(P$^{4+}$) approximates
the mass-loss rate to within a factor of 2. Theory predicts that P$^{4+}$ is
the dominant ion in the winds of O7-O9.7 stars, though an empirical estimator
suggests that the range from O4-O7 may be more appropriate. However, we find
that the mass-loss rates obtained from P V wind profiles are systematically
smaller than those obtained from fits to Halpha emission profiles or radio
free-free emission by median factors of about 130 (if P$^{4+}$ is dominant
between O7 and O9.7) or about 20 (if P$^{4+}$ is dominant between O4 and O7).
These discordant measurements can be reconciled if the winds of O stars in
the relevant temperature range are strongly clumped on small spatial scales.
We use a simplified two-component model to investigate the volume filling
factors of the denser regions. This clumping implies that mass-loss rates
determined from "density squared" diagnostics have been systematically
over-estimated by factors of 10 or more, at least for a subset of O stars.
Reductions in the mass-loss rates of this size have important implications for
the evolution of massive stars and quantitative estimates of the feedback that
hot-star winds provide to their interstellar environments.
Reference: ApJ, in press
% \\ Status: Manuscript has been accepted
Weblink: http://arxiv.org/abs/astro-ph/0510252
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
Radio Emission Models of Colliding-Wind Binary Systems - Inclusion of IC Cooling
}\end{center}
\centerline{\bf
J.M. Pittard$^1$, S.M. Dougherty$^2$, R.F. Coker$^3$, E. O'Connor$^4$, N.J. Bolingbroke$^5$}
1 - School of Physics and Astronomy, The University of Leeds, Leeds LS2 9JT, UK
\\ 2 - National Research Council of Canada, Herzberg Institute for
Astrophysics, Dominion Radio Astrophysical Observatory, P.O. Box 248,
Penticton, BC, V2A 6J9, Canada
\\ 3 - Los Alamos National Laboratory, X-2 MS T-087, Los Alamos, NM 87545, USA
\\ 4 - Physics Department, University of Prince Edward Island, Charlottetown, PEI, Canada
\\ 5 - Department of Physics and Astronomy, University of Victoria, 3800
Finnerty Rd, Victoria, BC, Canada
Radio emission models of colliding wind binaries (CWBs) have been
discussed by Dougherty et al. (2003). We extend these models by
considering the temporal and spatial evolution of the energy
distribution of relativistic electrons as they advect downstream from
their shock acceleration site. The energy spectrum evolves
significantly due to the strength of inverse-Compton (IC) cooling in
these systems, and a full numerical evaluation of the synchrotron
emission and absorption coefficients is made. We have demonstrated
that the geometry of the WCR and the streamlines of the flow within it
lead to a spatially dependent break frequency in the synchrotron
emission. We therefore do not observe a single, sharp break in the
synchrotron spectrum integrated over the WCR, but rather a steepening
of the synchrotron spectrum towards higher frequencies. We also
observe that emission from the wind-collision region (WCR) may appear
brightest near the shocks, since the impact of IC cooling on the
non-thermal electron distribution is greatest near the contact
discontinuity (CD), and demonstrate that the impact of IC cooling on
the observed radio emission increases significantly with decreasing
binary separation. We study how the synchrotron emission changes in
response to departures from equipartition, and investigate how the
thermal flux from the WCR varies with binary separation. Since the
emission from the WCR is optically thin, we see a substantial fraction
of this emission at certain viewing angles, and we show that the
thermal emission from a CWB can mimic a thermal plus non-thermal
composite spectrum if the thermal emission from the WCR becomes
comparable to that from the unshocked winds. We demonstrate that the
observed synchrotron emission depends upon the viewing angle and the
wind-momentum ratio, and find that the observed synchrotron emission
decreases as the viewing angle moves through the WCR from the WR shock
to the O shock. We obtain a number of insights relevant to models of
closer systems such as WR140. Finally, we apply our new models to the
very wide system WR147. The acceleration of non-thermal electrons
appears to be very efficient in our models of WR147, and we suggest
that the shock structure may be modified by feedback from the
accelerated particles.
Reference: Accepted for A\&A
% \\ Status: Manuscript has been accepted
Weblink: http://arxiv.org/abs/astro-ph/0510283
Comments: 21 pages, 16 figures
Email: [email protected]
\newpage
%\vspace*{5mm} %----------------------------
%\vspace*{5mm} %----------------------------
\centerline{\lightframe{0.7}{5}{3.8cm}{
\centerline{\Large In Proceedings}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\begin{center}{\Large\bf
New Photometric Observations of sigma Ori E
}\end{center}
\centerline{\bf
Mary Oksala \& Rich Townsend}
Bartol Research Institute, University of Delaware
We present new UBVRI observations of the magnetic Bp star sigma Ori
E. The basic features of the star's lightcurve have not changed since
the previous monitoring by Hesser et al. (1977), indicating that the
star's magnetosphere has remained stable over the past three
decades. Interestingly, we find a rotation period that is slightly
longer than in the Hesser et al. (1977) analysis, suggesting possible
spindown of the star.
Reference: To appear in "Active OB Stars: Laboratories for Stellar \&
Circumstellar Physics", ASP Conf. Ser. 2005, S. Stefl, S. P. Owocki \&
A. Okazaki, eds.
% \\ Status: Conference proceedings
Weblink: http://www.star.ucl.ac.uk/$\sim$rhdt/publications/
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
sigma Ori E: The Archetypal Rigidly Rotating Magnetosphere
}\end{center}
\centerline{\bf Rich Townsend}
Bartol Research Institute, University of Delaware
I review the basic concepts of the Rigidly Rotating Magnetosphere
model for the circumstellar plasma distribution around the
helium-strong star sigma Ori E. I demonstrate that the model can
furnish a good fit to the photometric, spectroscopic and magneticv
ariability exhibited by this star, and argue that the variability of
other helium-strong stars may be amenable to a similar interpretation.
Reference: To appear in "Active OB Stars: Laboratories for Stellar \&
Circumstellar Physics", ASP Conf. Ser. 2005, S. Stefl, S. P. Owocki \&
A. Okazaki, eds.
% \\ Status: Conference proceedings
Weblink: http://www.star.ucl.ac.uk/$\sim$rhdt/publications/
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
Kappa-Mechanism Excitation of Retrograde Mixed Modes in B-Type Stars
}\end{center}
\centerline{\bf
Rich Townsend}
Bartol Research Institute, University of Delaware
The stability of retrograde mixed modes in rotating B-type stars is
investigated. It is found that these modes are susceptible to
kappa-mechanism excitation, due to the iron opacity bump at log T ~
5.3. The findings are discussed in the context of the pulsation of SPB
and Be stars.
Reference: To appear in "Active OB Stars: Laboratories for Stellar \&
Circumstellar Physics", ASP Conf. Ser. 2005, S. Stefl, S. P. Owocki \&
A. Okazaki, eds.
% \\ Status: Conference proceedings
Weblink: http://www.star.ucl.ac.uk/$\sim$rhdt/publications/
Email: [email protected]
\newpage
% \vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
Outflowing Disk Winds in B[e] Supergiants
}\end{center}
\centerline{\bf
Michel Cur\'e (1), Diego F. Rial(2) \& Lydia Cidale(3)}
(1) Departamento de Fisica, Facultad de Ciencias,
Universidad de Valparaiso, Chile.
\\ (2) Departamento de Matem\'aticas, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Argentina.
\\ (3) Facultad de Ciencias Astron\'omicas y Geofisicas, Universidad Nacional de La Plata,
Argentina.
The effects of rapid rotation and bi--stability upon the density
contrast between the equatorial and polar directions of a B[e]
supergiant are investigated. Based on a new slow solution for
different high rotational radiation--driven winds and the fact that
bi--stability allows a change in the line--force parameters ($\alpha,
k, \delta$), the equatorial densities are about 10$^2$ - 10$^3$ times
higher than the polar ones. These values are in qualitative agreement
with the observations. This calculation also permits to obtain the
aperture angle of the disk.
Reference: To appear in ``Stars with the B[e] Phenomenon'', ASP
Conf. Ser. 2005, Michaela Kraus \& Anatoly S. Miroshnichenko, eds.
% \\ Status: Conference proceedings
Weblink: astro-ph/0510695
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
X-ray Observations of Binary and Single
Wolf-Rayet Stars with XMM-Newton and Chandra
}\end{center}
\centerline{\bf
S. Skinner$^1$, M. Guedel$^2$, W. Schmutz$^3$
and S. Zhekov$^4$}
1 CASA, Univ. of Colorado, Boulder, CO 80309 USA
\\ 2 Paul Scherrer Inst., CH-5232 Villigen PSI, Switz.
\\ 3 PMOD, Dorfstr. 33, CH-7260 Davos Dorf, Switz.
\\ 4 JILA, Univ. of Colorado, Boulder, CO 80309 USA and Space Res. Inst., Sofia 1000, Bulgaria
We present an overview ofrecent X-ray observations of Wolf-Rayet (WR) stars with
XMM-Newton and Chandra. A new XMM spectrum of
the nearby WN8 + OB binary WR 147 shows hard
heavily absorbed emission, including the Fe
K-alpha line complex, characteristic of colliding wind shock sources. In contrast,
sensitive observations of four of the closest
known single WC (carbon-rich) WR stars have
yielded only non-detections. These results
tentatively suggest that single WC stars are
X-ray quiet. The presence of a companion may
thus be an essential factor in elevating the
X-ray emission of WC + OB stars to detectable
levels.
Reference: To appear in: Close Binaries in the 21st Century - New Opportunities and Challenges
(eds. A. Gimenez, E. Guinan, P. Niarchos,
S. Rucinski), Astrophys. and Space Sci. special issue, 2006.
% \\ Status: Conference proceedings
Weblink: astro-ph/0511137 (after 7 Nov. 2005)
Comments: 4 pages, 2 figures, 1 table
Email: [email protected]
\newpage
% \vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
Massive Star Feedback - From the First Stars to the Present
}\end{center}
\centerline{\bf
Jorick S. Vink}
Keele University, UK
The amount of mass loss is of fundamental importance to the lives and
deaths of very massive stars, their input of chemical elements and
momentum into the interstellar and intergalactic media, as well as
their emitted ionizing radiation. I review mass-loss predictions for
hot massive stars as a function of metal content for groups of OB
stars, Luminous Blue Variables, and Wolf-Rayet stars. Although it is
found that the predicted mass-loss rates drop steeply with decreasing
metal content (Mdot ~ Z$^{0.7-0.85}$), I highlight two pieces of physics
that are often overlooked in cosmologocal studies: (i) mass-loss
predictions for massive stars approaching the Eddington limit, and for
(ii) stars that have enriched their own atmospheres with primary
elements such as carbon. Both of these effects may significantly boost
the mass-loss rates of the first stars -- relevant for the
reionization of the Universe, and a potential pre-enrichment of the
intergalactic medium -- prior to the first supernova explosions.
Reference: Vink, 2006, in: "Stellar Evolution at Low Metallicity:
Mass-Loss, Explosions, Cosmology", eds: H. Lamers, N. Langer,
T. Nugis), ASP Conf Series
% \\ Status: Conference proceedings
Comments: Invited Review in Tartu workshop, Aug 2005.
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
O Star X-ray Line Profiles Explained by Radiation Transfer in Inhomogeneous Stellar Wind
}\end{center}
\centerline{\bf
L.M. Oskinova, A. Feldmeier, W.-R. Hamann}
Universit\"at Potsdam, Astrophysik
It is commonly adopted that X-rays from O stars are produced deep
inside the stellar wind, and transported outwards through the bulk of
the expanding matter which attenuates the radiation and affects the
shape of emission line profiles. The ability of Chandra and XMM-Newton
to resolve these lines spectroscopically provided a stringent test for
the theory of X-ray production. It turned out that none of the
existing models was able to reproduce the observations
consistently. The major caveat of these
models was the underlying assumption of a smooth stellar
wind. Motivated by the various observational evidence that the stellar
winds are in fact structured, we present a 2-D model of a stochastic,
inhomogeneous wind. The X-ray radiative transfer is derived for such
media. It is shown that profiles from a clumped wind differ
drastically from those predicted by
conventional homogeneous models. We review the up-to-date observations
of X-ray line profiles from stellar winds and present line fits
obtained from the inhomogeneous wind model. The necessity to account
for inhomogeneities in calculating the X-ray transport in massive star
winds, including for HMXB is highlighted.
Reference: "The X-ray Universe 2005", ESA, El Escorial, Madrid, Spain, 26 - 30 September 2005
% \\ Status: Conference proceedings
Weblink: astro-ph/0511019
Email: [email protected]
\newpage
% \vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
A Simple Nozzle Analysis of Slow-Acceleration Solutions in 1-D Models
of Rotating Line-Driven Stellar Winds
}\end{center}
\centerline{\bf
Stan Owocki}
Bartol Research Institute, University of Delaware
Newark, DE 19716 USA
For a star rotating at more than about 75\% of the critical rate,
one-dimensional (1-D) models for the equatorial regions of a
line-driven stellar wind show a sudden shift to a slow-acceleration
solution, implying a slower, denser equatorial outflow that might be
associated with the dense disks inferred for sgB[e] stars. To clarify
the nature of this solution shift, I present here a simple analysis of
the 1-D flow equations based on a nozzle analogy for the terms that
constrain the local mass flux. At low rotation rates the nozzle
minimum (or ``throat'') occurs near the stellar surface, allowing a
near-surface transition to a steeply accelerating, supercritical flow
solution. But for rotations above about 75\% of the critical rate,
this {\em local}, inner nozzle minimum exceeds the {\em globa}l
minimum approached asymptotically at large radii, implying that
near-surface supercritical solutions would now have an overloaded mass
loss rate. Maintaining a monotonically positive acceleration is then
only possible if the flow is kept subcritical out to large radii,
where the nozzle function approaches its {\em absolute} minimum. For
fixed line-driving parameters, the associated enhancements in
equatorial density are typically a factor 5-30 relative to the polar
(or nonrotating) wind. However, when gravity darkening and 2-D flow
effects are accounted for, it still seems unlikely that rotationally
modified equatorial wind outflows could account for the very large
densities inferred for the disks around supergiant B[e] stars.
Reference: To appear in ``Stars with the B[e] Phenomenon'', ASP
Conf. Ser. 2005, Michaela Kraus \& Anatoly S. Miroshnichenko, eds.
% \\ Status: Conference proceedings
Weblink: http://www.bartol.udel.edu/$\sim$owocki/preprints/vlieland-rotnoz.pdf
Email: [email protected]
\vspace*{5mm} %----------------------------
\centerline{\lightframe{0.7}{5}{1.5cm}{
\centerline{\Large Jobs }}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\begin{center}{\Large\bf
Graduate Student Research Assistantships at STScI in Massive Star LTSA Project,
Space Telescope Science Institute
}\end{center}
\centerline{\bf
Don Figer {\rm (STScI)}}
The Space Telescope Science Institute (STScI) invites applications
from advanced graduate students to pursue PhD thesis research with
Dr. Don Figer. A stipend of approximately \$22,000 per year (depending
on qualifications) is provided by STScI. Some support for tuition is
also available if required. This position is for the 2005 academic
year, with a likely opportunity for an extension up to three
years. IDL experience is required.
As part of a 5-year funded NASA Long Term Space Astrophysics project,
we seek an energetic graduate student to identify and characterize the
most massive stars in the Galaxy. The following is the abstract for
this project.
Until the Spitzer Space Telescope, there was no wide area survey that
could identify massive stars at all distances in the Galaxy. Indeed,
the sample of known O-stars is woefully incomplete, as it has largely
been generated using optical observations that suffer from the
absorption produced by dust in the disk. We are now at the cusp of a
revolution in massive star research that Spitzer will trigger, and we
propose to capitalize on that opportunity by performing the first
survey of massive stars covering the majority of the Galactic
volume. We will find and measure the physical properties of the most
massive stars in the Galaxy using HST, Spitzer, Chandra, SOFIA, and
ground-based observatories, using a survey technique that probes the
majority of the Galaxy. This program addresses fundamental questions
whose answers are basic requirements for studying many of the most
important topics in Astrophysics: the formation and evolution of the
most massive stars, the effects of massive stars on lower mass
protostellar/protoplanetary systems, gamma-ray burst (GRB)
progenitors, nature of the first stars in the Universe, chemical
enrichment of the interstellar medium, Galactic gas dynamics, star
formation in starbursts and merging galaxies (particularly in the
early Universe). The results of our program will influence the science
programs for future NASA projects, i.e. JWST, SOFIA, SIM, TPF-C, and
TPC-I.
Applicants must have a Bachelor's degree (or equivalent), must have
completed all required graduate course work, and must have been
admitted to the PhD program at their home universities. Enquiries
about this program may be directed to Dr. Don Figer (410-338-4377,
[email protected]).
Applications should include three copies each of the following
material: a signed cover letter, a curriculum vitae, a statement of
research interests, and a letter from the official advisor or
departmental chairperson giving permission to work at STScI. These
should be sent by regular mail to Human Resources, Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore MD,
21218. Applicants should also arrange for their academic transcripts
and three letters of recommendation to be sent to the same address.
Applications and letters of reference received before December 1, 2005
will receive full consideration. The Space Telescope Science Institute
is an affirmative action, equal opportunity employer. Women and
members of minority groups are strongly encouraged to apply.
Attention: Christine Rueter - Reg 479
%Reference: http://members.aas.org/JobReg/JobdetailPage.cfm?JID=22139
% \\ Status: Other
Weblink: http://members.aas.org/JobReg/JobdetailPage.cfm?JID=22139
Email: [email protected]
\vspace*{5mm} %----------------------------
\begin{center}{\Large\bf
Post Doc at STScI in Massive Star LTSA Project
\\Space Telescope Science Institute
}\end{center}
\centerline{\bf
Don Figer {\rm (STScI)}}
The Space Telescope Science Institute (STScI) invites applications
from postdoctoral researchers to pursue research with Dr. Don
Figer. This position is for 1 year with possible extension to 3 years.
As part of a 5-year funded NASA Long Term Space Astrophysics project,
we seek an energetic postdoctoral scholar to identify and characterize
the most massive stars in the Galaxy. The following is the abstract
for this project.
Until the Spitzer Space Telescope, there was no wide area survey that
could identify massive stars at all distances in the Galaxy. Indeed,
the sample of known O-stars is woefully incomplete, as it has largely
been generated using optical observations that suffer from the
absorption produced by dust in the disk. We are now at the cusp of a
revolution in massive star research that Spitzer will trigger, and we
propose to capitalize on that opportunity by performing the first
survey of massive stars covering the majority of the Galactic
volume. We will find and measure the physical properties of the most
massive stars in the Galaxy using HST, Spitzer, Chandra, SOFIA, and
ground-based observatories, using a survey technique that probes the
majority of the Galaxy. This program addresses fundamental questions
whose answers are basic requirements for studying many of the most
important topics in Astrophysics: the formation and evolution of the
most massive stars, the effects of massive stars on lower mass
protostellar/protoplanetary systems, gamma-ray burst (GRB)
progenitors, nature of the first stars in the Universe, chemical
enrichment of the interstellar medium, Galactic gas dynamics, star
formation in starbursts and merging galaxies (particularly in the
early Universe). The results of our program will influence the science
programs for future NASA projects, i.e. JWST, SOFIA, SIM, TPF-C, and
TPC-I.
Applicants must have a PhD. Enquiries about this program may be
directed to Dr. Don Figer (410-338-4377, [email protected]).
Applications should include three copies each of the following
material: a signed cover letter, a curriculum vitae, and a statement
of research interests. Applicants should also arrange for their
academic transcripts and three letters of recommendation to be sent to
the same address.
Applications and letters of reference received before December 1, 2005
will receive full consideration.
The Space Telescope Science Institute is an affirmative action, equal
opportunity employer. Women and members of minority groups are
strongly encouraged to apply.
Attention: Christine Rueter - Req480
% Reference: http://members.aas.org/JobReg/JobdetailPage.cfm?JID=22138
% \\ Status: Other
Weblink: http://members.aas.org/JobReg/JobdetailPage.cfm?JID=22138
Email: [email protected]
% \newpage
% \vspace*{5mm} %----------------------------
% \centerline{\lightframe{0.7}{5}{2cm}{
% \centerline{\Large Theses}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%
% \vspace*{2mm}
%
%
% \vspace*{5mm}
%
%
%
% \centerline{\lightframe{0.7}{5}{2.1cm}{
% \centerline{\Large Meetings }}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \vspace*{2mm}
\end{document} ---------------------------------------------------------
% WHEN SUBMITTING AN ABSTRACT, PLEASE USE THE FOLLOWING TEMPLATE IF POSSIBLE:
\begin{center}{\Large\bf % WRITE THE Title BELOW THIS LINE
}\end{center}
\centerline{\bf A. Author$^1$ and B. Author$^2$
}{\footnotesize
$^1$ Institute One and Address
\\ $^2$ Institute Two and Address
}\vspace*{4mm} \\
% WRITE TEXT OF ABSTRACT HERE
{\bf Accepted by /or/ Submitted to JOURNAL} \\
{\it Preprints from} {\tt YOUR ELECTRONIC ADDRESS} \\
{\it or by anonymous ftp to} {\tt ... } \\
{\it or on the web at} {\tt ... } \\
------------------------------------------------------------------------
------------------------------------------------------------------------
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https://dlmf.nist.gov/27.6.E2.tex
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crawl-data/CC-MAIN-2018-17/segments/1524125946077.4/warc/CC-MAIN-20180423144933-20180423164933-00465.warc.gz
| 600,131,564 | 709 |
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