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darksky.org
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%&LaTeX
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\bibitem{Cabrera-Cruz_etal2019} Cabrera-Cruz, S. A., Smolinsky, J. A., McCarthy, K. P., \& Buler, J. J. (2019). Urban areas affect flight altitudes of nocturnally migrating birds. \textit{J Anim Ecol}, \textit{in press}.
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\bibitem{Adamczyk-Garbowska2001} Adamczyk-Garbowska M., \textit{The Role of Polish Language and Literature in Bashevis{\textquoteright}s Fiction}, [w:] \textit{The Hidden Isaac Bashevis Singer}, red.~Wolitz, Seth L., Texas, 2001, s. 134--47
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Yevgeny Aksenov \\
Ed Blockley \\
Matthieu Chevallier \\
Danny Feltham \\
Thierry Fichefet \\
Gilles Garric \\
Paul Holland \\
Dorotea Iovino \\
Gurvan Madec \\
Fran\c cois Massonnet \\
Jeff Ridley \\
Cl\'ement Rousset \\
David Salas \\
David Schroeder \\
Steffen Tietsche \\
Martin Vancoppenolle
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\node (P) [block] {Planung}; & \\
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\title{Masthead}
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\cleardoublepage
EDITORS
Harvey Ovshinsky
Peter Werbe
EDITORIAL ASSISTANT
Cathy West
CIRCULATION
Tommye Wiese
NEWS EDITOR
Alan Gotkin
MUSIC EDITOR
John Sinclair
DISTRIBUTION
Eric Watkins
ADVERTISING
Gunnar Lewis
ART DIRECTOR
Ed Bania
CALENDAR
Resa
STAFF
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The FIFTH ESTATE is published on the 1\textsuperscript{st} and 3\textsuperscript{rd} Thursday of each month by the Fifth Estate Newspaper, Inc., 1107 W. Warren, Detroit, Michigan 48201. Subscription rate by third class mail is \$3 for one year; \$5 for two years. Canadian is \$3.50; all other foreign are \$5 per year. Phone 831–6800. The FIFTH ESTATE is a member of the Liberation News Service LNS and the Underground Press Syndicate (UPS).
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\strut
\vfill
\begin{center}
Fifth Estate Collective
Masthead
\bigskip
\href{https://www.fifthestate.org/archive/57-july-4-18-1968/masthead}{\texttt{https://www.fifthestate.org/archive/57-july-4-18-1968/masthead}}
Fifth Estate \#57, July 4–18, 1968
\bigskip
\textbf{fifthestate.anarchistlibraries.net}
\end{center}
% end final page with colophon
\end{document}
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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\hrulefill\vskip1em
\sffamily\Large\textbf{OTÁZKY Z TEORIE ELEKTROMAGNETICKÉHO
POLE}\\Letní semestr 2003/2004 \vskip2pt
\tiny poslední úprava \today
\vskip1em\hrulefill
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Síla současně působící na elektrický náboj v elektrickém a
magnetickém poli (Lorentzova síla)}
\crlf \mat{\vec{F_m}=Q\left[\E+(\vec{v}\times\B)\right]}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Klasifikace látek z hlediska polohy, směru a závislosti na
$\E$ a $\HH$.}
\begin{itemize}
\item Homogenní $\times$ nehomogenní -- ve všech bodech stejné
$\times$ různé vlastnosti;
\item izotropní $\times$ anizotropní -- ve všech směrech stejné
$\times$ různé vlastnosti;
\item lineární $\times$ nelineární -- platí $\times$ neplatí princip superpozice;
\item stacionární $\times$ nestacionární -- má časově stálé $\times$
proměnlivé vlastnosti.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Klasifikace elektromagnetických jevů -- typy polí, jejich
zdroje}
\crlf Rozlišujeme tato pole: \emph{elektrostatické pole} (zdrojem jsou
nepohyblivé náboje), \emph{stacionární} proudové pole (zdrojem je
stejnosměrný proud), \emph{magnetostatické} pole,
\emph{kvazistacionární} (nízkofrekvenční proudy) a
\emph{nestacionární} elektromagnetické pole.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Coulombův zákon, orientace vektorů}
\crlf Mezi dvěma bodovými náboji působí Coulombova síla, která
závisí přímo úměrně na velikosti nábojů a nepřímo na kvadrátu
jejich vzdálenosti. Síla působí ve směru spojnice nábojů, náboje
stejného znaménka se odpuzují. Konstanta $\eps_0$ je permitivita
vakua.
\mat{\vec{F}=\frac1{4\pi \eps_0}\frac{Q_1 Q_2}{r^2}\vec{r_0}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je a kdy lze použít princip superpozice}
\crlf \emph{Síla působící na libovolný náboj je rovna vektorovému
součtu Coulombových sil od ostatních nábojů.} Princip superpozice
vyjadřuje způsob řešení elektrických polí sečtením složek jednotivých
zdrojů -- stejně jako složky vektorů (vyřešených samostatně) a tím
získat výsledné řešení celé soutavy. Princip superpozice lze použít
pouze v lineárním prostředí.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Definice intenzity elektrického pole}
\crlf Intenzita elektrického pole je rovna síle působící na
jednotkový kladný náboj: \mat{\E=\lim_{q\rightarrow 0
}\frac{\vec{F}}{q}\qquad
\E=\frac{\vec{F}}{Q}=\frac{1}{4\pi\eps_0}\frac{Q}{r^2}{\vec{r_0}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Rovnice siločáry $\E$ obecně a zvláště v kartézských
souřadnicích}
\crlf Siločára je myšlená orientovaná křivka, po níž by se pohyboval
v~elektrickém poli uvolněný kladný náboj. Každým bodem prostoru
prochází pouze jediná siločára. Siločáry se nikde neprotínají.
V~elektrostatickém poli siločára začíná na kladném a končí na záporném
náboji. Říkáme, že kladný náboj je zřídlem pole, záporný náboj je
norou. Rovnice siločáry v kartézských souřadnicích:
\mat{\frac{E_x}{\dev x}=\frac{E_y}{\dev y}=\frac{E_z}{\dev z}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Vyjádření vektorového pole $\E$ pomocí skalárního pole
potenciálu $\fii$}
\crlf Gradient skalárního potenciálu $\fii$ je vektor určující směr a
velikost největšího růstu potenciálu (proti směru $\vec E$)
\mat{\E=-\grad\fii=-\nabla\fii=- \left( {{\partial \varphi}\over{\partial x}}\vec x_0 +
{{\partial \varphi}\over{\partial y}}\vec y_0 +
{{\partial \varphi}\over{\partial z}}\vec z_0 \right) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Skalární potenciál $\fii$ buzený elektrickým nábojem $Q$}
\crlf Potenciál bodu pole je roven energii, potřebné k~přenesení
jednotkového kladného náboje vnějšími silami (tedy proti silám pole)
z~bodu vztažného do bodu uvažovaného. Velikost konstanty $K$ závisí na
poloze vztažného místa (místa nulového potenciálu). Vztažný bod se
často pokládá do nekonečna. Definujeme ho vztahem:
\mat{\fii=-\int\E\cdot\dev\vec{r}+K=-\frac{Q}{4\pi
\eps_0}\int\frac{\dev r}{r^2}+K=\frac{Q}{4\pi \eps_0r}+K}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{$\E$ a $\fii$ buzené daným rozložením hustoty náboje $\rho$}
\crlf Potenciální pole vyvolané obecnými zdroji (náboji s~daným
prostorovým uspořádáním) můžeme vyjádřit jako sumu (integrál)
příspěvků jednotlivých elementárních nábojů, na které zdroje
rozdělíme. Tomu se říká Greenovo fundamentální řešení.
\mat{\E=\frac1{4\pi \eps_0}\iiint_V\frac{\rho\dev
V}{r^2}\vec{r_0}\quad\quad
\fii=\frac1{4\pi \eps_0}\iiint_V\frac{\rho\dev V}r + K }
\mat{\vec E = {1 \over {4\pi\varepsilon}} \iint_S{{{\sigma_0 \dev S}
\over r^2}} \qquad \varphi = {1 \over {4\pi\varepsilon}}
\iint_S{{{\sigma_0 \dev S} \over r} + K}}
\mat{\vec E = {1 \over {4\pi\varepsilon}} \int_l {{\tau_0\dev l} \over r^2} \qquad
\varphi = {1 \over {4\pi\varepsilon}} \int_l {{\tau_0\dev l} \over r} + K}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je napětí a jak souvisí s $\E$ a $\fii$}
\crlf Napětí je rozdíl potenciálů mezi dvěma body
elektrostatického pole:
\mat{U_{AB}=\fii_A-\fii_B=\int_B^A\E\cdot\dev\vec{l}=
-\int_A^B\E\cdot\dev\vec{l}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Jaký je charakter elektrostatického pole, rovnice, které ho
popisují}
\crlf Elektrostatické pole je zřídlové (zřídla jsou náboje) a
konzervativní (tzn., že cirkulace vektoru intenzity v~elektrostatickém
poli po uzavřené křivce je nulová). Konzervativní pole se také často
nazývají nevírové či potenciální. Použitím Stokesovy věty poté
dostaneme vztah pro diferenciální podobu vztahu. Jednoznačně je určeno
rovnicemi:\mat{\divrg\E=\frac{\rho}{\eps_0}\qquad \Sint E=\frac{Q_0}{\eps} \qquad \olint\E=0 \qquad\rot\E=\nabla\times\E=\left |
\begin{matrix} \vec i & \vec j & \vec k~\cr
{\partial \over {\partial x}} &
{\partial \over {\partial y}} &
{\partial \over {\partial z}} \cr
E_x & E_y & E_z \cr \end{matrix} \right | = \vec 0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Laplaceova a Poissonova rovnice pro elektrický skalární
potenciál}
\crlf Tyto rovnice získáme tak, že do vztahu $\divrg\E={\rho_0/\eps}$
dosadíme vztah $\E= -\grad\fii$.
\mat{\divrg\E=-\divrg\grad\E=-\nabla^2\fii=\frac{\rho_0}{\eps}}
Takto dostaneme \emph{Poissonovu rovnici}:
\mat{\Delta\varphi = \nabla^2\fii =
\pacc{\fii}{x}+\pacc{\fii}{y}+\pacc{\fii}{z} = \frac{\rho_0}{\eps}}
V oblasti beze zdrojů (tj. vně zdrojů) se Poissonova rovnice redukuje
na \emph{rovnici Laplaceovu}:\mat{\Delta\fii=0}
Laplaceovy nebo Poissononovy diferenciální rovnice je výhodné použít
při výpočtu polí, jejichž elektrody jsou ekvipotenciálními plochami o
známém potenciálu. K úplnému řešení je nutné znát také okrajové
podmínky, ty mohou být dvojího druhu:
\begin{itemize}
\item Dirichletova --- na hranici řešené oblasti je znám
potenciál $\varphi$,
\item Neumannova --- na hranici oblasti je známá
normálová složka intenzity pole $E_n$.
\end{itemize}
Funkce $\varphi$ vyhovující těmto rovnicím a okrajovým podmínkám se nazývají harmonické.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Gaussova věta v elektrostatickém poli a definice elektrického
toku}
\crlf Elektrický tok je definován takto:
\mat{\Psi=\Sint{E}=\sint{E_n}\,.} Tok intenzity uzavřenou plochou
je úměrný volnému náboji $Q$ uzavřenému v této ploše (Gaussova věta):
\mat{\oSint{E}=\frac{Q}{\eps_0}\,.}
Necháme-li objem uzavřený plochou limitovat k~nule a dělíme-li obě
strany rovnice tímto objemem, dostaneme diferenciální tvar Gaussovy
věty. Výraz $\divrg\E$ má význam vydatnosti zdroje toku:
\mat{\divrg\E = \lim_{\Delta V\to 0} {\frac{\oSint{E}}{\Delta V}} =
\lim_{\Delta V\to 0} {{Q_0}\over{\eps \Delta V}} =
{\rho_0 \over \eps} }
\mat{\divrg\E = \nabla\cdot\E = \pspd{E_x}{x} + \pspd{E_y}{y}
+ \pspd{E_z}{z} }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Čemu se rovná $\E$ na povrchu vodiče}
\crlf Při vložení tělesa do vnějšího pole se na jeho povrchu vytvoří takové
rozložení náboje, aby byla intenzita pole uvnitř tělesa nulová. Tomuto
jevu se říká elektrostatická indukce. A náboji, který se na povrchu
nashromáždí se říká indukovaný náboj.
Protože uvnitř vodiče je intenzita pole nulová, je také tok vektoru
intenzity $\vec E$ z~libovolné uzavřené plochy uvnitř vodiče nulový.
To je možné pouze tehdy, pokud plocha neobklopuje žádný náboj.
Z~prvního vztahu pro $\vec E=0$ (viz 8.) vyplývá, že vodivé těleso je
ve statickém poli vždy ekvipotenciální plochou.
Protože je povrch vodivého tělesa ekvipotenciálou, musí k~němu být
siločáry kolmé. To znamená, že intenzita elektrického pole má na
povrchu vodiče směr normály. Velikost intenzity určíme snadno pomocí
Gaussovy věty. Náboj je na povrchu rozložen s~plošnou hustotou
$\sigma_0$. Nyní protneme povrch vodivého tělesa elementární válcovou
plochou kolmou k~povrchu, kterou uzavřeme dnem (slupkou tělesa) a
víkem o~plochách $\dev S$. Uvedená plocha uzavírá náboj $\dev Q_0=\sigma_0
\dev S.$ Tok vektroru $\vec E$ bude vycházet pouze víkem, vně vodivého
tělesa. Proto je:
\mat{\oSint{E}=E_n\dev S=\frac{\dev Q}{\eps_0}=\frac{\sigma\dev S
}{\eps_0}\ \Rightarrow\ E_n=\frac{\sigma}{\eps_0}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{$\E$ osamocené neomezené nabité rovinné vodivé folie}
\crlf Intenzitu určíme pomocí Gaussovy věty~-- rovinu protneme
elementárním válcem průřezu $\dev S$,
kolmým na rovinu, který z~obou stran uzavřeme. Vektor
intenzity $\E$ je kolmý k rovině a směřuje na obě strany roviny.
\mat{E\cdot2\,\dev S= \frac{\sigma_0 \dev S}{\eps} \Rightarrow E=\frac{\sigma_0}{2\eps}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{$\E$ a $\fii$ nabité vodivé koule}
\crlf Vně koule (podle Gaussovy věty) je pole stejné jako od bodového náboje:
\mat{E\cdot4\pi r^2=\frac{Q}{\eps} \Rightarrow E=\frac{Q}{4\pi\eps
r^2}\,,\quad \fii=-\int E \dev r =
-\frac{Q}{4\pi\eps}\int\frac{1}{r^2}\dev r=\frac{Q}{4\pi\eps
r}+K\quad(r>R)\,.} Uvnitř koule je intenzita nulová, potenciál je
konstantní a je roven potenciálu na povrchu koule:
\mat{E=0\,,\quad \fii=\frac{Q}{4\pi\eps R}+K\quad(r<R)\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{$\E$ a $\fii$ dlouhého nabitého válcového vodiče}
\crlf Náboj je rozložen na přímce s~liniovou hustotou $\tau$. Přímku
obklopíme souosým válcem. Tok vektoru $\E$ prochází pouze pláštěm
válce a je na něj kolmý. Z~Gaussovy věty platí: \mat{E\cdot2\pi rl
=\frac{\tau l}{\eps} \Rightarrow E=\frac{\tau}{2\pi \eps
r}\,,\quad \fii=-\int E \dev r=-\frac{\tau}{2\pi \eps}\int\frac{\dev
r}{r}= -\frac{\tau}{2\pi \eps}\ln r+K\quad(r>R)\,.} Uvnitř vodiče je
intenzita nulová, potenciál je konstantní a je roven potenciálu na
povrchu: \mat{E=0\,,\quad \fii=-\frac{\tau}{2\pi \eps}\ln R
+K\quad(r<R)\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Energie elektrostatického pole buzeného $\rho$, $\sigma$ nebo
soustavou elektrod}
\crlf \mat{W=\frac12\sum_{i=1}^N Q_i\fii_i\quad\quad
W=\frac12\int_V\rho\fii\dev V\quad\quad
W=\frac12\int_S\sigma\fii\dev S}
Pro energii pole buzeného $\rho$ (spojité rozložení nábojů
v~prostoru), nahradíme v~předchozím vztahu náboj $Q$ vyjádřením
$\rho\dev V$
a sumu integrálem. Analogicky bychom dostali i vztah pro energii pole
buzeného $\sigma$. \mat{ W = \frac12 \vint{\rho\fii}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Energie elektrostatického pole vyjádřena pomocí vektorů pole}
\crlf \mat{W=\frac12\vint{\D\E}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je elektrický dipól, definice dipólového momentu,
orientace použitých vektorů}
\crlf Elektrický dipól je soustava dvou blízkých nábojů stejné
velikosti a opačného znaménka, jejichž vzdálenost je vzhledem ke
vzdálenosti pozorování zanedbatelná. Dipólový moment je roven součinu
náboje $Q$ a vzdálenosti mezi náboji: $\vec{p}=Q\vec{d}$. Vektor
$\vec{p}$ má směr od záporného náboje ke kladnému.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Dipólový moment soustavy elektrických momentů}
Vložením dipólu do homogenního pole se dipól snaží natáčet tak, aby
směry vektorů $\E$ a $\vec p$ splynuly. Dipól umístěný v poli
bodového náboje (obdobně tak v obecném nehomogenním poli) bude natáčen
a také vtahován do směru rostoucí intenzity pole. Kvadrupól jsou dva
antiparalelní elementární dipóly, jeho potenciál klesá s třetí
mocninou. Složitější seskupení dipólů se nazývá multipól.
\ignore{
$$ \varphi = {1 \over 4\pi\varepsilon}{{p \cos{\vartheta}}\over r^2} \qquad
\vec E = {p \over {4\pi\varepsilon r^3}} \left( 2\cos{\vartheta}\,\vec r_0 + \sin{\vartheta}\,\vec \vartheta_0 \right) $$
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Definice kapacity, význam všech použitých symbolů}
\crlf Kapacita je koeficient lineární závislosti mezi nábojem a
potenciálem, charekterizuje množství náboje $Q$ přeneseného
vynaložením určité práce $U$. Je definována jako
\mat{C=\frac{Q}{\Delta\fii}=\frac{Q}{U}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Řešení Laplaceovy rovnice pro elektrický skalární potenciál v
rovině ($xy$) kartézských souřadnic}
\crlf Hledáme analytické řešení rovnice
\mat{\pacc{\fii}{x}+\pacc{\fii}{y}=0} ve tvaru $\fii(x,y)=X(x)Y(y)$.
Pro jednoduché případy polí lze k řešení Laplaceovy rovnice použít
metody přímé integrace (pokud $\fii$ závisí pouze na jedné proměnné),
separace proměnných, konečných diferencí nebo superrelaxační metodu.
Úpravou Laplaceovy rovnice a pomocí separace proměnných dostaneme
rovnici: \mat{\frac1X\acc{X}{x}+\frac1Y\acc{Y}{y}=0\,.} Jednotlivé
členy jsou na sobě nezávislé, proto můžeme rovnici rozdělit:
\mat{\frac1X\acc{X}{x}=k_x^2,\quad \frac1Y\acc{Y}{y}=k_y^2,\quad
k_x^2+k_y^2=0\,.} Dostali jsme dvě obyčejné diferenciální rovnice,
které už umíme snadno vyřešit.
\ignore{
$$ \varphi = X(x) Y(y)
= A \left\{{\matrix{\sin \cr \cos \cr}\over{\matrix{\sinh \cr \cosh \cr}}}\right\}kx
\left\{{\matrix{\sin \cr \cos \cr}\over{\matrix{\sinh \cr \cosh \cr}}}\right\}ky $$
$$ \varphi = \sum_k \bigl( A_k \sin{kx} + B_k \cos{kx}\bigr) \bigl( C_k \sinh{kx} + D_k \cosh{kx}\bigr)+ Exy + Fx + Gy + H $$
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Gaussova věta v elektrostatickém poli, kdy ji lze použít k
výpočtu $\E$}
\crlf \mat{\oSint{E}=\frac{q}{\eps_o}\,.} Gaussova věta udává silový
tok, procházející uzavřenou plochou. Určuje tzv. zřídlovost
elekstostatického pole. Gaussovu větu můžeme s~výhodou použít
k~výpočtu elektrických polí symetricky uspořádaných nábojů. Podmínkou
je, že se nám podaří najít takovou plochu, uzavřenou kolem náboje
(nábojů), která splňuje tyto podmínky:
\begin{itemize}
\item plocha prochází bodem v~němž intenzitu pole hledáme,
\item intenzita $\E$ má pouze normálovou složku $E_n$ k~nalezené ploše,
\item složka $E_n$ je na této ploše konstantní. Na její části může být nulová.
\end{itemize}
Jako např. prostorový náboj se sférickou, plošnou či lineární symetrií.
Tyto podmínky jsou splněny jen v~těch případech, kdy se dá vhodnou
volbou souřadnic vyjádřit intenzita pole jako funkce jedné proměnné
(rovina, válec, koule). Tohoto postupu lze použít také u~složitějších
polí, u~nichž nejsou splněny uvedené předpoklady. Ale jen tehdy, pokud
se dají pokládat za superpozice několika polí, u~nichž všech tyto
předpoklady splněny jsou.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Jakou úlohu má konformní zobrazení při výpočtu
elektrostatického pole a jak se používá}
\crlf Tato metoda se používá k řešení polí složitějších tvarů. Princip
řešení Laplaceovy rovnice metodou konformního zobrazení~--- hledané
pole, které máme řešit, je dvourozměrné nehomogenní pole nabitých
elektrod obecného tvaru. Toto pole geometricky transformujeme na pole
homogenní, ve kterém řešení snadno nalezneme a výsledek transformujeme
zpátky do původního pole. Transformace, která zachovává vlastnosti
potenciálního pole musí být zprostředkovaná tzv. analytickou funkcí
komplexní proměnné, tj. funkcí která vyhovuje Cauchyovým-Riemannovým
podmínkám. Tato funkce se však obtížně hledá, ale existují jejich
obsáhlé slovníky.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Schema výpočtu elektrického potenciálu ve vnitřním uzlu 2D
čtvercové homogenní sítě superrelaxační metodou konečných
diferencí}
\crlf Při výpočtu potenciálu postupujeme takto: nejdříve všem
vnitřním uzlům sítě přiřadíme libovolnou počáteční hodnotu. Potom
v uspořádaném sledu projdeme všechny vnitřní uzly a v každém
určíme chybu podle vztahu:
\mat{R=\frac14\Big(\fii(x,y+h)+\fii(x,y-h)+\fii(x+h,y)+\fii(x-h,y)\Big)-\fii(x,y)\,.}
Hodnotu potenciálu v aktuálním uzlu nahradíme novou hodnotou:
\mat{\fii_n(x,y)=\fii_s(x,y)+\alpha R\,,} kde $\alpha$ je
relaxační koeficient. Aby metoda konvergovala, musí být
$1\leq\alpha\leq2$. Tento postup opakujeme do té doby, až se nové
hodnoty v žádném uzlu nebudou lišit o více než stanovenou
odchylku.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je a jak je definována elektrická polarizace}
\crlf Vektor polarizace je definován jako objemová hustota dipólových
momentů. Nebo také jako množství vázaného náboje, který se při
polarizaci přesune, vztažené na jednotkovou plochu.
\mat{\vec{P}=\lim_{\Delta V\rightarrow 0}\frac{\sum_i
\vec{p}_i }{\Delta V}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Jak souvisí elektrická polarizace s prostorovým a plošným
vázaným elektrickýn nábojem}
\crlf Velikost vektoru polarizace je rovna plošné hustotě vázaného
náboje, který se při polarizaci přesune: $P_n=-\sigma_v$. Platí:
\mat{|P_n|=\sigma_v\quad\quad \divrg\vec{P}=-\rho_v}
\ignore{
Do předešlého vztahu doplníme za $\vec p = \sigma_v dS \Delta l \vec p_0$ a
dostaneme: $$ \vec P = {\vec p \over{\Delta V}}
= {{\sigma_v dS\Delta l \vec p_0} \over{dS\Delta l}}
= \sigma_v \, \vec p_0 $$
$$ \div\!\vec P = -\rho_v$$
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Definice elektrické indukce, jak souvisí s elektrickou
susceptibilitou a permitivitou v lineárních prostředích, co je
zdrojem $\E$, $\D$, $\vec{P}$}
\crlf Vektor elektrické indukce $\D$ je definován vztahem
$\D=\eps_0\E+\vec{P}$. Platí:
\mat{\D=\eps_0\E+\vec{P}=\eps_0\E+\eps_0\chi\E=\eps_0(1+\chi)\E=\eps_0\eps_r\E=\eps\E\,.}
Bezrozměrná konstanta $\chi$ $(\chi>0)$ se nazývá \emph{elektrická
susceptibilita}.
Zdrojem toku vektoru $\E$ jsou volné i vázané
náboje:
\mat{\oSint{E}=\frac{Q}{\eps_0}=\frac{Q_0+Q_v}{\eps_0}=\frac{Q_0}{\eps_r\eps_0}=\frac{Q_0}{\eps}}
Zdrojem toku $\D$ jsou volné náboje:
\mat{\oSint{D}=Q_0}
Zdrojem toku vektoru $\vec{P}$ jsou vázané náboje:
\mat{\oSint{P}=-Q_v}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Gaussova věta elektrostatického pole v dielektriku a definice
indukčního toku}
\crlf Gaussova věta pro dielektrikum: \mat{\oSint{D}=Q_0\quad \divrg\D
= \rho_0} Elektrický indukční tok $\Psi$ uzavřenou plochou je roven
velikosti volného náboje, který je v~ploše uzavřen. Zdrojem toku
vektoru $\vec D$ je pouze volný náboj. \mat{\Psi=\oSint{D}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Podmínky pro tečné složky pole $\E$, $\D$ na rozhraní dvou
dielektrik, vyznačit orientaci normály k rozhraní}
\crlf Zvolíme integrační dráhu $l$ tak, aby při $h \to 0$ dráha
obepínala úsek rozhraní o~délce $\dev l$, pak ze vztahu $\oLint{E}=0$
vyplývá vztah pro tečné složky:
\mat{E_{1t}=E_{2t}\quad\quad\frac{D_{1t}}{\eps_1}=\frac{D_{2t}}{\eps_2}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Podmínky pro normálové složky pole $\E$, $\D$ na rozhraní
dvou dielektrik, vyznačit orientaci normály k rozhraní}
\crlf Abychom určili vztahy pro normálové složky vektorů pole na
rozhraní dvou prostředí, budeme uvažovat elementární váleček o~výšce
$h \to 0$, který uzavírá část rozhraní tak, že jeho dno a víko leží na
různých jeho stranách. Uvažovaná plocha obepíná (uzavírá) volný náboj
s hustotou $\sigma_0$. Tok vektoru pláštěm válce je nulový. Platí:
\mat{\oSint{D}=Q_0 \Rightarrow D_{1n}-D_{2n}=\sigma_0\,.}
Odtud plyne, že normálové složky elektrické indukce nejsou spojité a
jejich rozdíl je roven plošné hustotě náboje v~uvažovaném bodě
rozhraní. Pokud na rozhraní není přítomen volný náboj, tak jak to mezi
dielektriky bývá, pak platí:\mat{E_{1n}\eps_1=E_{2n}\eps_2\quad\quad
D_{1n}=D_{2n}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Celková kapacita kapacitorů řazených seriově a paralelně}
\crlf Kapacita kapacitorů řazených \emph{sériově} (na všech kapacitorech
je stejně veliký náboj, celkové napětí jednotlivých kapacitorů:
$U=U_1 + U_2 + \dots + U_n$ a z definice kapacity $C=Q/U$ plyne
$\frac{Q}{U} = \frac{Q}{U_1} + \frac{Q}{U_2} + \dots + \frac{ Q}{U_n}$.):
\mat{\frac1C=\frac1C_1+\frac1C_2+\ldots+\frac1C_n\,.}
Kapacita kapacitorů řazených \emph{paralelně} (na všech kapacitorech
je stejné napětí, celkový náboj je dán součtem včech kapacitorů $Q =
Q_1 + Q_2 + \dots + Q_n$, po dosazení $Q=CU$ a vynásobení $1/U$):
\mat{C=C_1+C_2+\ldots+C_n\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Energie nabitého kapacitoru}
\crlf \mat{W=\int_0^Q \frac{q}{C}\dev
q=\frac{Q^2}2\frac1C=\frac12CU^2=\frac12QU}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Definice elektrického proudu ve stacionárním proudovém poli}
\crlf Proud je množství náboje procházející plochou za jednotku
času: \mat{\Delta I=\frac{\Delta q_c}{\Delta
t}=Nq\vec{v}\Delta\vec{S}\,,} kde $q_c$ je celkový náboj
procházející plochou, $N$ je koncentrace náboje, $q$ je
elementární náboj a $\vec{v}$ je rychlost pohybu nábojů. Výraz
$Nq$ vyjadřuje objemovou hustotu náboje: $Nq=\rho$.
\crlf Proudová hustota je množství náboje procházející jednotkovou
plochou za jednotku času: \mat{\jj=Nq\vec{v}=\rho \vec{v} \qquad
I=\Sint{J}=\sint{J_n}\,.}
Za kladný směr proudu považujeme směr pohybu kladných nábojů.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Rovnice kontinuity stacionárního proudu v integrálním a
diferenciálním tvaru}
\crlf Ve stacionárním proudovém poli platí, že proud, který vtéká do
libovolného objemu z něho musí ve stejné velikosti vytékat (1.
Kirchhoffův zákon). V tomto poli je tedy tok vektoru proudové hustoty
uzavřenou plochou nulový. \mat{\oSint{J}=0\quad\quad \divrg\jj=0\,.}
Rovnici $\oSint{J}=0$ můžeme přepsat do tvaru $\sum_j I_j=0$, což je
známý Kirchoffův zákon.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Jaký je charakter stacionárního proudového pole, rovnice
které ho popisují}
\crlf Stacionární proudové pole je nezřídlové~-- to znamená, že
proudové čáry jsou vždy uzavřené křivky. Popisují ho tyto rovnice:
\mat{\oSint{J}=0\,, \quad\quad \oLint{E}=U_e\,.} Vně zdrojů je
stacionární proudové pole nevírové: \mat{\oLint{E}=0\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Ohmův zákon v diferenciálním a integrálním tvaru}
\crlf Ze vztahů $v = -bE$ a $\rho_e = -Ne$, dosazených do $J = \rho v$, dostaneme
$ J = NebE,$ což je Ohmův zákon ve tvaru:
\mat{\jj=\sigma\E \quad\mathrm{diferenciálním}, \qquad
\frac{\Lint{E}}{\Sint{J}}=\frac{U}{I}=R\quad
\mathrm{~integrálním}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Podmínky pro tečné a normálové složky $\jj$ na rozhraní dvou
vodivých prostředí, vyznačit orientaci normály k rozhraní}
\crlf Z rovnice kontinuity plyne podmínka pro normálové složky:
\mat{J_{n1}=J_{n2}\,.} Pro oblast mimo zdrojů platí pro tečné
složky $\E$ to samé jako v elektrostatickém poli; z Ohmova zákona
plyne podmínka
\mat{\frac{J_{t1}}{\sigma_1}=\frac{J_{t2}}{\sigma_2}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Definice elektromotorického napětí a jeho vztah ke svorkovému
napětí zdroje}
\crlf Elektromotorické napětí je práce rozdělujících sil
nutných k přenesení kladného jednotkového náboje od záporné elektrody
ke kladné. Je definováno vztahem \mat{U_{e21}=\int_1^2\E_r\dev\vec{l}\,.}
Svorkové napětí je rovno elektromotorickému napětí zmenšenému o
úbytek na vnitřním odporu zdroje: \mat{U_{12}=U_{e21}-R_iI\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Jouleovy ztráty v proudovém stacionárním poli}
\crlf Elementární práce $\Delta A$ vykonaná polem při přenesení náboje
$Q$ na vzdálenost $\Delta l$ je rovna $\Delta A = Q E\Delta l =
\rho\dev V E \Delta l$, z tohoto vztahu odvodíme vztah pro výkon
$\Delta P = \Delta{}A/\Delta{}t = \rho\dev V E v = JE \dev V, $ potom
také platí, že objemová hustota výkonu stacionárního proudu je
\mat{\spd{P}{V}=\E\jj=\sigma E^2=\frac{J^2}{\sigma}}
%$$ {dP \over dV} = \vec E \cdot \vec J = \sigma E^2 \left[ {W \over m^3} \right] $$
Pro výkon spotřebovaný v objemu $V$ je
\mat{ P = \iiint_V{\E\jj\dev V} = JS \int E\dev l = UI = RI^2\quad}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Definice odporu vodiče, celkový odpor rezistorů řazených
seriově a paralelně}
\crlf Odpor vodiče: \mat{R=\int_0^l\frac{\dev
l}{\sigma S}=\frac{U}{I}=\frac{El}{JS}=\frac{El}{\sigma ES}=\frac{l}{\sigma
S}=\frac1G\,.} Odpor rezistorů řazených sériově:
\mat{R=\sum_{i=1}^n R_i = R_1+R_2+\ldots+R_n\,.} Odpor rezistorů řazených paralelně:
\mat{G=\sum_{i=1}^n G_i=G_1+G_2\ldots+G_n=\frac1R=\frac1R_1+\frac1R_2+\ldots+\frac1R_n\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Biotův-Savartův zákon, nakreslete orientaci vektorů}
\crlf Biotův-Savartův zákon udává velikost magnetické indukce
způsobené vodičem délky $l$ protékaným proudem $i$ v bodě $P$ ve
vzdálenosti $r$ od tohoto vodiče.
\mat{\B=\frac{\mu_0}{4\pi}\oint\frac{i\dev\vec{l}\times
\vec{r_0}}{r^2}\,.}
Směr vektoru $\B$ určíme \emph{pravidlem pravé ruky}~--- palec
pravé ruky položíme do směru proudu a ohnuté prsty ukazují směr
indukce. Biotův-Savartův zákon spolu s principem superpozice umožňuje
zformulovat vztah pro výpočet magnetického pole zdrojů s libovolnou
geometrií. Vzhah $i\dev\vec l$ můžeme nahradit kterýmkoliv ekvivalentním
výrazem $\dev Q\vec v$ nebo $\dev V \vec J$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je magnetická indukce a její definice pomocí pohybujícího
se náboje $Q$, $\jj$, $\vec K$ a $I$}
\crlf Magnetická indukce $\B$ je vektorová veličina popisující
silové působení magnetického pole na pohybující se náboj. Je
definována pomocí vztahu pro Lorentzovu sílu: \mat{\dev\vec F=\dev
Q(\vec v\times\B)=I(\dev\vec l\times\B)=\dev
S(\vec K\times\B)=\dev V(\jj\times\B)\,.}
Její směr odpovídá směru stočení severního pólu magnetky v okolí
proudového vodiče.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Definice magnetického toku}
\crlf Magnetický tok je tok vektoru magnetické indukce plochou
$S$: \mat{\Phi=\Sint{B}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Jaký je charakter magnetostatického pole, rovnice které ho
popisují}
\crlf Magnetostatické pole je nezřídlové a vírové. Je určeno
vztahy (integrální tvar):
\mat{\oLint{B}=\mu_0I \quad \quad
\oSint{B}=0\,,} diferenciální tvar:
\mat{\rot{\B}=\mu_0\jj\quad\quad\divrg\B=0\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Definice vektorového potenciálu, podmínka jednoznačnosti $\B$
určené pomocí $\vec{A}$}
\crlf Vektorový magnetický potenciál je definován vztahem:
$\B=\rot\vec{A}$. Aby bylo pole definováno jednoznačně, položíme
divergenci potenciálu rovnu nule: $\divrg\vec{A}=0$ (Coulombova
podmínka).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Poissonova a Laplaceova rovnice pro vektorový potenciál a
její obecné řešení v integrálním tvaru}
\crlf Dosazením do Ampérova zákona ($\rot\B=\mu_0\jj$) získáme
Poissonovu rovnici: \mat{\Delta\vec{A}=-\mu_0\jj\,.} Pro oblasti
vně zdrojů platí Laplaceova rovnice: \mat{\Delta\vec{A}=0\,.}
Partikulární řešení Poissonovy rovnice má tvar (Greenovo řešení):
\mat{\vec{A}=\frac{\mu_0}{4\pi}\iiint_V\frac{\jj}{r}\dev
V+\vec{K}\qquad \vec{A}=\frac{\mu_0I}{4\pi}\oint\frac{\dev\vec l}{r}+\vec{K}\,,} kde $\vec{K}$ je vektorová konstanta.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Ampérův zákon a kdy ho lze použít k výpočtu $\B$ nebo $\HH$}
\crlf Ampérův zákon říká, že integrál $\oint \B \dev\vec l$ po
libovolné křivce se rovná celkovému proudu, který křivka obepíná
násobenému $\mu_0$ (ve vakuu). Mimo jiné určuje, že magnetické pole je
pole vírové. Ampérův zákon můžeme s výhodou použít k výpočtu
magnetických polí s vyšší symetrií zdrojů (dlouhý přímý vodič, koule,
válec, pole uvnitř toroidní cívky, velmi dlouhého solenoidu).
\mat{\oLint{B}=\mu_0\sum I \qquad \oLint{H}=\sum I_0} Diferenciální
tvar dostaneme použitím Stokesovy věty.
\mat{\rot\B=\mu_0\jj\qquad\rot\vec H=\jj\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Magnetický tok vyjádřený pomocí vektorového potenciálu}
\crlf Magnetický tok $\Phi$ určíme z vektorového potenciálu $\vec A$ pomocí
Stokesovy věty:
\mat{\Phi=\Sint{B}=\sint{\rot\vec{A}}=\oLint{A}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Definice magnetického skalárního potenciálu, kdy ho lze použít}
\crlf Skalární magnetický potenciál je definován takto:
\mat{\B=-\grad\fii_m\mathrm{~nebo~}\B=-\mu_0\,\grad\fii_m.} Lze ho zavést jen v oblastech bez proudu
($\rot\B=0$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Co je magnetický dipól, definice dipólového momentu,
orientace použitých vektorů}
\crlf Magnetický dipól je magnetické pole kruhové smyčky o poloměru
$a,$ jehož velikost se vůči vzdálenosti pozorování $h$ jeví jako
zanedbatelná, tj. platí $a \ll h.$ V tomto poli definujeme moment
magnetického dipólu $\vec m$ jako vektor, jehož směr je určen směrem oběhu
proudu ve smyčce pravidlem pravé ruky a jehož velikost je dána
následujícím vztahem. Proudům, které tyto dipóly způsobují, říkáme
vázané proudy a značíme je $I_v.$ \mat{\vec m = I_v \dev\vec S = I_v\pi a^2
\vec z_0} %[Am^2]
Symbol $\dev \vec{S}$ je orientovaná plocha proudové smyčky.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Statická definice vlastní a vzájemné indukčnosti}
\crlf Uvažujeme-li lineární prostředí, je i závislost magnetického
toku cívkou na proudu, který ho vyvolal, lineární a platí:
$\Phi_c=N\Phi=LI$. Konstanta úměrnosti $L$ se nazývá indukčnost
(vlastní) a je definována vztahem \mat{L=\frac{\Phi_c}{I}\,.}
Mějme dvě cívky ($N_1,I_1,N_2,I_2)$. První cívka vybudí tok
$\Phi_1$ a část tohoto toku $\Phi_{12}$ projde i druhou cívkou.
Mezi proudem $I_1$ a magnetickým tokem $\Phi_{12}$ je lineární
závislost. Konstanta úměrnosti této závislosti se nazývá vzájemná
indukčnost: \mat{M_{12}=\frac{\Phi_{12c}}{I_1}\,.} V lineárních
prostředích platí $ \Phi_{12c} = M_{12}I_1 = M_{21}I_2 $.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Co je a jak je definována magnetizace}
\crlf Vektor magnetizace $\vec{M}$ vyjadřuje míru uspořádanosti
magnetických momentů. Je definován jako objemová hustota
magnetických momentů: \mat{\vec{M}=\lim_{\Delta V\rightarrow
0}\frac{\sum_i\vec{m}_i}{\Delta V}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Jak souvisí magnetizace s prostorovými a plošnými vázanými
proudy}
\crlf Buď $V$ objem, ve kterém je vektor magnetizace konstantní a
výsledný magnetický moment elementárních smyček je $\vec{m_c}$.
Účinek těchto smyček můžeme nahradit ekvivalentní proudovou
vrstvou s vázaným proudem $I_v$. Velikost vektoru magnetizace je
rovna délkové hustotě tohoto vázaného proudu ($j_v$ je hustota
plošného vázaného proudu): \mat{M=\frac{I_v}h=j_v \qquad
I_v=\int_A^B\vec M \dev\vec l}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Definice intenzity magnetického pole, jak souvisí s
magnetickou susceptibilitou a permeabilitou v lineárním prostředí}
\crlf Podle Ampérova zákona platí vztah $ \Lint{B} = \mu_0 (I_0 +
I_v)$. Jehož úpravou dostaneme:
\mat{\olint{\left(\frac{\B}{\mu_0}-\vec M\right)}}
Intenzita magnetického pole $\vec H$ je definována vztahem:
\mat{\HH=\frac{\B}{\mu_0}-\vec{M}\,.} V lineárním a izotropním prostředí je
$\vec{M}=\chi_m\HH$, kde $\chi_m$ je magnetická susceptibilita a
platí:
\mat{\B=\mu_0(\HH+\vec{M})=\mu_0(1+\chi_m)\HH=\mu_0\mu_r\HH=\mu\HH\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Podmínky pro normálové složky pole $\B$, $\HH$ na rozhraní
dvou magnetik, vyznačit orientaci normály k rozhraní}
\crlf Vztah mezi normálovými složkami odvodíme ze vztahu $ \oSint{B} =
0.$ Určíme tok vektoru $\vec B$ povrchem nízkého válečku, jehož výška
$h$ limituje k nule tak, aby váleček stále obsahoval rozhraní. Odtud:
\mat{B_{n1}=B_{n2}\quad\quad \mu_1H_{n1}=\mu_2H_{n2}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Podmínky pro tečné složky pole $\B$, $\HH$ na rozhraní dvou
magnetik, vyznačit orientaci normály k rozhraní}
\crlf Vztah tečných složek odvodíme z Ampérova zákona $ \oLint{H}= I
.$ Cirkulaci vektoru $\vec H$ provedeme po dráze $c$, kde $h$ necháme
limitovat k nule tak, že dráha bude stále obepínat rozhraní. Obepnutý
proud proto musí být nulový.
\mat{H_{t1}=H_{t2}\quad\quad
\frac{B_{t1}}{\mu_1}=\frac{B_{t2}}{\mu_2}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\kkk{Energie magnetostatického pole buzeného $\vec{J}$, $\vec{K}$}
%ZKONTROLOVAT!!
\crlf \mat{W_m=\frac12\sint{\olint{\jj\vec A}}=\frac12\iiint_V\jj\vec
A \dev V=\frac12\sint{\vec K\vec A}%=\frac12\iiint_V\jj\vec
%A \dev V+\frac12\sint{\vec K\vec A}=\frac12}
}
%*** \mat{\dev W=i\dev\Phi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Energie magnetostatického pole vyjádřena pomocí vektorů pole}
\crlf \mat{W=\frac12\vint{\B\HH}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Energie nahromaděná v induktoru, energetická definice
indukčnosti}
\crlf \mat{W=\int_0^\Phi i\dev\Phi=
\int_0^\Phi \frac{\Phi}{L}\dev\Phi=
\frac{\Phi^2}{2}L =
\frac12 \Phi I = \frac12LI^2}
% $$ F = {dW \over dl} = {1 \over 2} I^2 {dL(x) \over dx} $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Energie soustavy induktorů}
\crlf Mějme dvě cívky. První cívkou prochází magnetický tok
$\Phi_A$, druhou $\Phi_B$. Energie pole obou cívek je dána vztahem
\mat{W=\frac12\Phi_AI_1+\frac12\Phi_BI_2=\frac12L_1I_1^2+\frac12L_2I_2^2\pm
MI_1I_2\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Jaké síly působí mezi dvěma paralelními vodiči protékanými
stejným proudem ve stejném směru a opačnými směry}
\crlf Mezi dvěma rovnoběžnými vodiči se vzájemnou vzdáleností $a$
působí síla: \mat{\frac{F}{l}=BI=\frac{\mu_0I^2}{2\pi a}\,.}
Vodiče protékané proudy o stejném směru se přitahují. Na základě
tohoto vztahu je definována jednotka proudu: proud o velikosti 1 A
vyvolá mezi dvěma nekonečně dlouhými vodiči vzdálenými od sebe 1 m
silové působení o velikosti $2\cdot 10^{-7}$~N na 1~m délky vodiče.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Hopkinsonův zákon a definice reluktance}
\crlf Hoplinsonův zákon vyjadřuje přímou úměrnost mezi magnetickým
tokem a magnetomotorickým napětím: \mat{U_m=NI=\Phi R_m\,.}
Reluktance (magnetický odpor) je definována vztahem
\mat{R_m=\int_0^l\frac{\dev l}{\mu S}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Vyjádření vlastní a vzájemné indukčnosti cívek pomocí
reluktance}
\crlf Magnetický tok cívkou je určen vztahem $\Phi = NI/R_m$,
což po dosazení do vztahu statické induk\-čnosti dá:
\mat{L=\frac{\Phi_c}{I}=\frac{N\Phi}{I}=\frac{N^2I}{R_mI}=\frac{N^2}{R_m}\quad\quad
M_{12}=M_{21}=\frac{N_1N_2}{R_m}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Faradayův indukční zákon}
\crlf \mat{U_e=-\spd{\Phi_c}{t}=
\oLint{E}=-\spd{}{t}\Sint{B}\quad\quad\rot\E=-\pspd{\B}{t}}
Velikost elektromotorického napětí $U_e$ indukovaného ve vodivé
smyčce je rovna rychlosti změny magnetického indukčního toku
$\Phi_c$ procházející touto smyčkou.
% Je-li v oblasti indukovaného elektrického pole vodič,
% bude jím protékat proud. Orientace indukovaného napětí je taková,
% aby vzniklý proud bránil změnám magnetického pole.
% To je známé univerzální Lenzovo pravidlo.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Dynamická definice vlastní a vzájemné indukčnosti}
\crlf Jestliže cívkou protéká časově proměnný proud $i(t)$, pak bude
magnetický tok $\Phi(t)$ také čásově proměnný. Po dozazení do
definice statické indukčnosti $\Phi(t)=Li(t)$ a do Faradayova
indukčního zákona bude platit: \mat{u=L\spd{i}{t}\quad\quad
u_2=M\spd{i_1}{t}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Jak transformuje ideální transformátor $u$, $i$, $R$}
\crlf \mat{\frac{i_1}{i_2}=\frac{N_2}{N_1}\quad\quad
\frac{u_1}{u_2}=\frac{N_1}{N_2}\quad\quad}
Je-li na sekundárním vinutí zátěž $R_z$ jeví se tato zátěž
z pohledu primárního vinutí jako efektivní zátěž o velikosti
\mat{R_{1ef}=\frac{u_1}{i_1}=\Big(\frac{N_1}{N_2}\Big)^2R_z.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Zápis okamžité hodnoty $\E$ pomocí fázoru, který definujte}
\crlf Fázor je obecně komplexní veličina nezávisející na čase.
Vyjádříme jej vztahem \mat{\fav E(x,y,z)=\faz E\vec e_0=E_m e^{\rm
j\fii }\vec e_0\,.} Máme-li veličinu měnící se s časem podle
vztahu $\E(x,y,z,t)=\E(x,y,z)\sin\omega t$, vyjádříme ji pomocí
fázoru vektoru \mat{\E(x,y,z,t)=\hbox{Im\,}\{\fav E(x,y,z)e^{{\rm
j}\omega t}\}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Časová střední hodnota energie elektrického a magnetického
pole zapsaná pomocí fázorů}
\crlf\mat{S_{stř}=\frac{1}{2}\Real{ \fav E \times \fav{H}{}^{*} }=\frac12E_mH_m\cos\varphi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Jaké je rozložení $\B$ v harmonicky podélně magnetovaném
feromagnetickém plechu}
\crlf Na povrchu plechu bude indukce maximální $\faz{B}_m$.
Tloušťka plechu je $2a$. \mat{\faz{B}=\faz{B}_m\frac{\cos kx}{\cos
ka}=\faz{B}_m\frac{\cosh (1+\im)\beta x}{\cosh(1+\im) \beta a}}
Na povrchu plechu je indukce maximální, směrem k rovině středního
řezu klesá, uprostřed plechu je minimální.
\iii{Jaké je rozložení $\jj$ ve vodiči kruhového průřezu
protékaném harmonicky se měnícím proudem $I$}
\crlf Nejsilnější je pole na vnějším povrchu, povrchový jev je tím
výraznější, čím je $f$ harmonického proudu vyšší (až v extrémním
případě se feromagnetikum chová jako \uv{trubka}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Impedance vodiče při výrazném elektrickém povrchovém jevu,
frekvenční závislost}
\crlf
\mat{Z=\frac{E}{I}=\frac{1+\im}{\sigma\delta}=(1+\im)\sqrt{\frac{\omega\mu}{2\sigma}}}
Poznámka: $\delta=1/\alpha$ -- hloubka vniku,
$\alpha=\sqrt{\omega\mu\sigma/2}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Rovnice kontinuity pro volné náboje a proudy v
nestacionárním poli, diferenciální a integrální tvar}
\crlf Je-li nějaký objem zdrojem toku vektoru proudové hustoty $\jj$,
musí v něm existovat volný náboj $Q$ a jeho hustota $\rho$ se musí v
čase měnit. \mat{\oSint{J}=-\spd{Q}{t}\quad\quad
\divrg\jj=-\pspd{\rho}{t}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Rovnice kontinuity pro polarizační proud a vázané náboje v
nestacionárním poli}
\crlf \mat{\divrg \jj_p=-\pspd{\rho_v}t\ \Rightarrow\
\divrg\jj_p=\pspd{}t(\divrg\vec P)\ \Rightarrow\
\jj_p=\pspd{\vec{P}}t}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{První Maxwellova rovnice v nestacionárním poli v
diferenciálním tvaru pro hmotné prostředí a obecnou časovou
závislost, význam všech použitých symbolů}
\crlf \mat{\rot\HH=\jj+\jj_p=\jj+\pspd{\D}{t}\,,} kde $\jj$ je
plošná hustota volného (kondukčního a konvekčního) proudu a
$\jj_p=\partial\D/\partial t$ je hustota polarizačního (posuvného) proudu.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Druhá, třetí, čtvrtá Maxwellova rovnice v nestacionárním poli
pro hmotné prostředí a obecnou časovou závislost, význam všech
použitých symbolů}
\crlf \mat{\rot\E=-\pspd{\B}{t}\quad\quad \divrg\D=\rho \quad\quad
\divrg\B=0 }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Čtyři Maxwellovy rovnice v integrálním tvaru v nestacionárním
poli, obecná časová závislost}
\crlf \emph{Ampérův-Mawellův zákon} vyjadřuje souvislost mezi cirkulací
magnetické indukce $\B$ podél uzavřené orientované křivky a časovou
změnou toku elektrické intenzity $\oSint{E}$ plochou ohraničenou touto křivkou a
celkovm proudem procházejícím touto plochou. \mat{\oLint{H}=I+I_p=I+\spd{\Psi}{t}}
\emph{Faradayův zákon} vyjadřuje souvislost mezi cirkulací intenzity
elektrického pole $\E$ podél uzavřené orientované křivky a časovou
změnou indukčního magnetického toku $\Phi = \oSint{B} $ plochou ohraničenou touto
křivkou \mat{\oLint{E}=-\spd{\Phi}{t}}
\emph{Gaussův zákon pro elektrostatické
pole} vyjadřuje souvislost mezi
tokem intentzity elektrického pole $\E$ uzavřenou plochou a celkovým
elektrickým nábojem uvnitř této plochy. \mat{\oSint{D}=Q}
\emph{Gaussův
zákon pro magnetické pole} vyjadřuje poznatek, že tok magnetiké indukce
$\B$ libovolnou uzavřenou plochou je roven nule (tj. neexistuje
magnetický náboj) \mat{\oSint{B}=0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Maxwellovy rovnice v diferenciálním tvaru pro harmonicky
proměnné nestacionární pole}
\crlf
\mat{\rot\fav{H}=\fav{J}+\io\fav{D}=\sigma\fav{E}+\io\eps\fav{E}=(\sigma+\io\eps)\fav{E}}
\mat{\rot\fav{E}=-\io\fav{B}=-\io\mu\fav{H}}
\mat{\divrg\fav{D}=\divrg\eps\fav{E}=\rho}
\mat{\divrg\fav{B}=\divrg\mu\fav{H}=0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Maxwellovy rovnice v integrálním tvaru pro harmonicky
proměnné nestacionární pole}
\crlf \mat{\oint_l\fav{H}\dev\vec{l}=I+\io\Psi}
\mat{\oint_l\fav{E}\dev\vec{l}=-\Phi}
\mat{\oiint_S\fav{D}\dev\vec{S}=Q}
\mat{\oiint_S\fav{B}\dev\vec{S}=0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Podmínky na rozhraní dvou prostředí v nestacionárním poli pro
tečné složky $\E$, $\HH$}
\crlf Podmínky na rozhraní pro nestacionární pole jsou dány
rovnicemi: \mat{\hbox{Rot }\HH=\vec K, \quad \hbox{Rot }\E=0,
\quad \hbox{Div }\D=\sigma, \quad \hbox{Div }\B=0\,,} kde
$\vec{K}$ je hustota plošného proudu a $\sigma$ je plošná
hustota volného náboje. Pro tečné složky platí:
\mat{H_{1t}-H_{2t}=K, \quad E_{1t}=E_{2t}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Podmínky na rozhraní dvou prostředí v nestacionárním poli pro
normálové složky $\E$, $\HH$}
\crlf \mat{\eps_1E_{1n}-\eps_2E_{2n}=\sigma,\quad
\mu_1H_{1n}=\mu_2H_{2n}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Energetická bilance elektromagnetického pole, obecná časová
závislost, fyzikální význam jednotlivých členů}
\crlf Energetická bilance (Poyntingův teorém) v diferenciálním a
integrálním tvaru: \mat{-\divrg\left(\E\times\HH\right)=\E\jj+\pspd w
t\quad\quad
-\oiint_S\left(\E\times\HH\right)\dev\vec{S}=\iiint_V\E\jj\dev V+\pspd
W t} Levá strana udává energii, která za jednotku času teče plochou
$S$ do uvažovaného objemu; první člen na pravé straně vyjadřuje
pohlcenou energii za jednotku času v objemu (Jouleovy ztráty), druhý
člen je výkon zvyšující akumulovanou energii. Pro jednoduchost:
energie, která se z daného objemu ztratí, se změní na teplo nebo se
vyzáří.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Poyntingův vektor, definice a zápis pomocí vektorů}
\crlf Poyntingův vektor vyjadřuje okamžitou hodnotu plošné hustoty
výkonu. Směr vektoru $\vec{S}$, který je kolmý na $\E$ a $\HH$, udává směr toku energie. Je definován jako
\mat{\vec{S}=\E\times\HH.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Energetická bilance činného výkonu}
\crlf \mat{2P_s=\osint{\Real{\fav E\times\fav{H}{}^*}}=\vint{\sigma E_m^2}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Energetická bilance jalového výkonu}
\crlf
\mat{2Q_s=\osint{\Imag{\fav{E}\times\fav{H}{}^{*}}}=\omega\vint{\left(\eps
E_m^2-\mu H_m^2\right)}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Vlnová rovnice pro $\E$ nebo $\HH$ v obecném prostředí mimo
oblast zdrojů, obecná časová závislost}
\crlf \mat{\Delta\E-\mu\sigma\pspd{\E}{t}-\mu\eps\pacc{\E}{t}=0 \qquad
\Delta\HH-\mu\sigma\pspd{\HH}{t}-\mu\eps\pacc{\HH}{t}=0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jjj{Vlnová rovnice pro $\E$ nebo $\HH$ v obecném prostředí mimo
oblast zdrojů, harmonické časové změny pole, zápis pomocí fázorů}
\crlf
\mat{\Delta\fav{E}-\io\mu\sigma\fav{E}+\omega^2\mu\eps\fav{E}=
\Delta\fav{E}-\io\mu(\io\eps+\sigma)\fav{E}=\Delta\fav{E}+k^2\fav{E}=0}
\mat{k=\sqrt{\io\mu(\io\eps+\sigma)}=\beta-\im\alpha}%\jedn{m^-1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Zápis $\E$ harmonicky proměnné postupné rovinné vlny v
obecném prostředí a zápis její okamžité hodnoty, význam všech
použitých symbolů}
\crlf Pro rovinnou vlnu šířící se ve směru osy $z$ má řešení tvar
\mat{\fav E(z)=\fav K_1e^{\im kz}+\fav K_2e^{-\im kz}\,,}
kde $\fav K_1$ a $\fav K_2$ jsou konstanty. Pro vlnu šířící se v
kladném směru osy $z$ je \mat{\fav E(z)= \fav E_0 e^{-\im
kz}=E_{0m}e^{\im\fii_0}e^{-\im kz},
\quad \quad E(z,t)=E_{0m}e^{-\alpha z}\sin(\omega t-\beta
z+\fii_0)\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Jaká je plocha konstantní amplitudy a plocha konstantní fáze
u rovinné elektromagnetické vlny, co je uniformní a neuniformní
vlna}
\crlf Ploše vlny (geometrickému místu) s konstantní fází říkáme
vlnoplocha. Uniformní vlna je vlnoplocha, která má navíc konstantní i
amplitudu. Neuniformní vlna je vlnoplocha s proměnnou amlitudou.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je a jak je definována fázová rychlost}
\crlf Fázová rychlost $v_f$ je rychlost pohybu vlnoplochy (místa
konstantní fáze). Je definována vztahem $v_f=\omega/\beta$. V
bezeztrátovém prostředí je
\mat{v_f=\frac{1}{\sqrt{\mu\eps}}=\frac{c_0}{\sqrt{\mu_r\eps_r}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je a jak je definována skupinová rychlost}
\crlf Skupinová (grupová) rychlost je rychlost pohybu maxima
energie vlny. \mat{v_g=\spd\omega\beta=\spd{(\beta
v_f)}\beta=v_f+\beta\spd{v_f}\beta}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Nakreslete orientaci $\E$, $\HH$, $k$ u rovinné vlny, jaký je
vztah těchto tří vektorů, jaký typ vlny je rovinná vlna}
\crlf Uvažujme vlnu šířící se ve směru osy $z$ a souřadnou
soustavu orientovanou tak, aby intenzita $\E$ měla jen složku ve
směru osy $x$ $(E_x)$. Vektory $\E$, $\HH$ a $\vec{z_0}$, kde
$\vec{z_0}$ je směr šíření vlny, tvoří pravotočivý ortogonální
systém. Mezi intenzitami platí vztah:
\mat{\faz{H}_y=\frac{k}{\omega\mu}\faz{E}_x\ \Rightarrow\
\faz{E}_x= \faz{Z}\faz{H}_y\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je a jak je definována charakteristická impedance v
obecném prostředí}
\crlf Vlnová impedance prostředí je pro neohraničené prostředí
definována vztahem
\mat{\faz{Z}=\frac{\omega\mu}k=\sqrt\frac{\io\mu}{\io\eps+\sigma}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Čemu se rovná $k$, $v_f$ a $Z$ v ideálním dielektriku}
\crlf V ideálním dielektriku je $\sigma=0$ a proto nedochází ke
ztrátám. Měrný útlum je nulový a konstanta šíření reálná:
\mat{k=\beta=\omega\sqrt{\mu\eps},\quad \alpha=0, \quad
v_f=\frac1{\sqrt{\mu\eps}},\quad
Z=\sqrt\frac\mu\eps \quad(\omega\eps\gg\sigma),\quad
Z=\sqrt{\frac{\mu_0}{\eps_0}}=120\pi\,\Omega~\mathrm{(ve~vakuu)}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Čemu se rovná $k$, $v_f$ a $Z$ v dobrém vodiči}
\crlf V dobrém vodiči je $\omega\eps\ll\sigma$. Konstantu šíření
můžeme aproximovat $k^2=-\io\mu\sigma$:
\mat{k=\sqrt{-\io\mu\sigma},\quad
\beta=\alpha=\sqrt\frac{\omega\mu\sigma}2,\quad
v_f=\sqrt{\frac{2\omega}{\mu\sigma}},\quad
Z=(1+\im)\sqrt\frac{\omega\mu}{2\sigma}=\sqrt\frac{\omega\mu}\sigma
e^{\im \pi/4} \quad(\omega\eps\ll\sigma)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je a jak je definována hloubka vniku}
\crlf Ekvivalentní hloubka vniku je vzdálenost, kterou musí vlna
urazit, aby její amplituda klesla na $e^{-1}$ násobek (37 \%)
původní hodnoty. Je definována jako převrácená hodnota měrného
útlumu:
\mat{\delta=\frac1\alpha}
%=\sqrt{\frac2{\omega\mu\sigma}}=\sqrt{\frac1{\pi\mu\sigma}}\cdot\sqrt{\frac1f}\,.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Činný výkon přenášený rovinnou vlnou plochou $1\ \metr^2$
obecně a zvláště ve vakuu}
\crlf Činný výkon procházející jednotkovou plochou je roven
střední hodnotě Poyntingova vektoru:
\mat{\vec
S_{stř}=\frac12\Real{\fav{E}\times\fav{H}{}^*}=\frac12E_mH_m\cos\fii\vec{z_0}=
\frac12\frac{E^2_m}{|Z|}\cos\fii\vec{z_0}=\frac12|Z|H^2_m\cos\fii\vec{z_0}\,.}
Ve ztrátovém prostředí závisí amplitudy intenzit na souřadnici
$z$: \mat{\E_m(z)=E_m(0)e^{-\alpha z},\quad H_m(z)=H_m(0)e^{-\alpha
z }\ \Rightarrow\ \vec{S}_{stř}(z)=\vec{S}_{stř}(0)e^{-2\alpha z}\,.} Ve
vakuu je $\fii=0$ a $Z_0=376,73=120\pi\ \Omega$:
\mat{\vec{S}_{stř}=\frac{E^2_m}{240\pi}\vec z_0 = 60\pi H^2_m\vec z_0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Co je a jaké jsou typy polarizace elektromagnetické vlny}
\crlf Je to způsob pohybu koncového bodu vektoru $\E$ v příčné
rovině. Druhy polarizace jsou~--- lineárně polarizovatelná (vertikální,
horizontální), kruhově
polarizovatelná (levotočivě, pravotočivě), elipticky polarizovatelná.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Za jakých podmínek dvě lineárně polarizované vlny vytvoří
vlnu lineárně, kruhově a elipticky polarizovanou}
\crlf Superpozicí dvou lineárně polarizovaných vln vznikne
lineárně polarizovaná vlna tehdy, budou-li vektory intenzit (např.
$\E$) rovnoběžné nebo ve fázi. Kruhově polarizovaná vlna vznikne
superpozicí dvou vln stejné ampitudy s navzájem kolmými vektory
$\E$ a s fázovým posunem $\pi/2$. V ostatních případech vznikne
elipticky polarizovaná vlna.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iii{Telegrafní rovnice pro harmonicky v čase proměnné $u$ nebo
$i$ v případě dvouvodičového vedení, na kterém se šíří vlna TEM,
význam všech použitých symbolů}
\crlf Základní rovnice pro $u$ a $i$:
\mat{-\pspd{u}{x}=Ri+L\pspd{i}{t}\quad\quad
-\pspd{i}{x}=Gu+C\pspd{u}{t}} Vlnová rovnice pro napětí:
\mat{\pacc{u}{x}=LC\pacc{u}{t}+(LG+RC)\pspd{u}{t}+RGu} Vlnová
rovnice pro proud má stejný tvar. $R$ je odpor, $L$ indukčnost,
$C$ kapacita a $G$ svod vedení na jednotku délky. Vlnová rovnice
pro napětí pro harmonický ustálený stav má tvar
\mat{\pacc{\faz{U}}{x}=(R+\io L)(G+\io
C)\faz{U}=\faz{Z_l}\faz{Y_q}\faz{U} \Rightarrow
\pacc{\faz{U}}{x}+k^2\faz{U}=0\,.} Konstanta $k=\alpha-\im\beta$
je konstanta šíření (vlnové číslo), $\alpha$ je fázová konstanta
(měrný posun) a $\beta$ je měrný útlum. $\faz{Z_l}$ je podélná
impedance a $\faz{Y_q}$ je příčná admitance vedení na jednotku
délky.
\iii{Zápis řešení telegrafní rovnice pomocí fázorů, význam všech
použitých symbolů}
\crlf Mějme vedení délky $l$ s charakteristickou (vlnovou)
impedancí $\faz{Z}_0$. Označme $s$ vzdálenost od konce vedení a
$\faz{U}_2$, $\faz{I}_2$ poměry na konci vedení. Pak platí:
\mat{\faz{U}(s)=\faz{U}_2\cos ks+\faz{Z}_0\faz{I}_2\sin ks }
\mat{\faz{I}(s)=\frac{\faz{U}_2}{\faz{Z}_0}\sin ks+\faz{I}_2\cos
ks }
\iii{Impedance vedení s rozprostřenými parametry v závislosti na
poloze}
\crlf Označme $\faz Z_s$ impedanci na konci vedení: $\faz Z_s=\faz
U_2/\faz I_2$. Vztah pro impedanci je \mat{\faz{Z}(s)=
\frac{\faz{U}(s)}{\faz{I}(s)}=\frac{\faz{U}_2\cos
ks+\faz{Z}_0\faz{I}_2\sin ks}{\frac{\faz{U}_2}{\faz{Z}_0}\sin
ks+\faz{I}_2\cos ks}=\faz Z_0\frac{\faz Z_s +\faz Z_0\tg ks}{\faz
Z_0 + \faz Z_s \tg ks} \,.}
\iii{Charakteristická impedance vedení s vlnou TEM u reálného a
bezeztrátového vedení}
\crlf Charakteristická (vlnová) impedance se spočítá podle vztahu:
\mat{\faz Z_0=\sqrt\frac{\faz Z_l}{\faz Y_q}=\sqrt\frac{R+\io
L}{G+\io C}\,,} kde $\faz Z_l$ je podélná impedance a $\faz Y_q$
příčná admitance na jednotku délky vedení. U bezeztrátového vedení
je odpor $R$ a svod $G$ nulový a platí: \mat{Z_0=\sqrt\frac LC\,.}
\iii{Vstupní impedance reálného a bezeztrátového vedení s vlnou
TEM délky $l$ a zakončeného impedancí $Z_s$}
\crlf \mat{\faz Z_v=\faz Z(l)=\faz Z_0\frac{\faz Z_s +\faz Z_0\tg
kl}{\faz Z_0 + \faz Z_s \tg kl}}
\iii{Vstupní impedance bezeztrátového vedení s vlnou TEM délky $l$
na konci zkratovaného a otevřeného}
% Nejsme si jisti zda tady je alpha ci beta :)
\crlf \mat{Z_k=\sqrt\frac LC\, \hbox{tg }\beta l\quad \quad
Z_p=\sqrt\frac LC\, \hbox{cotg }\beta l }
\end{enumerate}
\newpage
{\center
{\sffamily\Large\textbf{Abecední seznam otázek}}\vskip.5em
}
\typeout{}
\typeout{+---------------------------------+}
\typeout{| Po prvnim zpracovanim dokumentu}
\typeout{| je nutne v externim programu }
\typeout{| setridit soubor seznam.tex, }
\typeout{| prizemz se musi brat ohled na }
\typeout{| spravne zarazeni polozek, ktere}
\typeout{| zacinaji matematickym rezimem. }
\typeout{|}
\typeout{| sort seznam.tex > sorted.tex}
\typeout{|}
\typeout{+---------------------------------+}
\typeout{}
{\small
\noindent\input{sorted.tex}
}
\end{document}
|
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|
utexas.edu
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crawl-data/CC-MAIN-2019-39/segments/1568514573465.18/warc/CC-MAIN-20190919081032-20190919103032-00541.warc.gz
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\documentclass[12pt]{report} % The documentclass must be ``report''.
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\author{Scott Austen Robinson} % Required
\address{603 Davis St.\\ Austin, Texas 78701} % Required
\title{The Solar Energy Consumer Agent Decision (SECAD) Model:
Addressing complexity through GIS-integrated agent-based modeling}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Enter names of the supervisor and co-supervisor(s), if any,
% of your dissertation committee. Put one name per line with
% the name in square brackets. The name on the last line, however,
% must be in curly braces.
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% \supervisor
% {Supervisor's Name}
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\supervisor
{Varun Rai}
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% dissertation committee. Put one name per line with the name
% in square brackets. The name on the last line, however, must
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% NOTE: The Office of Graduate Studies may restrict you to a total
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\committeemembers
[Eugenio Arima]
{Jay Zarnikau}
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% Graduation year, in the form as `\graduationyear{2001}'.
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\copyrightpage % Produces the copyright page.
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% NOTE: In a doctoral dissertation, the Committee Certification page
% (with signatures) is BEFORE the Title page.
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% 20 Mar 2002 cwm
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\begin{dedication}
\index{Dedication@\emph{Dedication}}%
This work is dedicated to my parents, Will Robinson and Maria Katherman for showing me the value of curiosity.
\end{dedication}
\begin{acknowledgments} % Optional
\index{Acknowledgments@\emph{Acknowledgments}}%
I wish to acknowledge the hard work of others in the Energy Systems Transformation Group that made this work possible. Without years worth of survey and program data we would not be in a position to run simulations at this scale and level of detail. I would like to acknowledge the tireless support and guidance of Dr. Varun Rai, whose vision and understanding of the underlying issues was irreplaceable. I would like to thank the National Renewable Energy Laboratory and the United States Department of Energy for their support and funding of this research, and Austin Energy, Frontier Associates, the City of Austin, and Travis County for providing data. Finally I would like to acknowledge the Texas Advanced Computing Center for proving the computing power that made city-wide agent simulation possible.
\end{acknowledgments}
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\index{Abstract}%
\indent
This thesis presents a step-by-step implementation of the Solar Energy Consumer Agent Decision (SECAD) model: an empirically-grounded multi-agent model of residential solar photovoltaic (PV) adoption with an integrated geospatial topology. Solar PV diffusion is a complex system with geographic heterogeneity, uncertain information, high financial risk, and important social interaction and feedback effects between consumers. A key limitation for agent-based models in human socio-technical systems is the integration of empirical patterns in the model structure, initialization, and validation efforts. This limitation is addressed though highly granular and interlocking data-streams from the geographic, social network, financial, demographic, and decision-making process of real households in the study. The fitted and validation model is used to simulate implementation of potential policies to inform decision-makers: i) Targeted informational dissemination campaigns, ii) Tiered rebates, and iii) Alternative rebate schedules. Informational campaigns can increase cumulative installations by as much as 12\%, but vary greatly in their effectiveness based on which agents are targeted. Simulations indicated suggest that the wealth inequality between solar adopters and non adopters could be reduced by 22.6\% on average through a slightly increased rebate amount for low-wealth households. Tiered rebates can allow the utility more control over the locational diffusion patterns with regard to load pockets--a \$0.25 higher offering increased the percentage of adopters in the target area from less than 1\% to over 10\%. Relative to flatter rebate schedules, sharply decreasing schedules are effective in terms of motivating adoption but inefficient in small markets. It is our hope that this work will provide a working example for other agent-based models of human socio-technical systems as well as provide insight into the likely outcomes of novel policy-levers such as those described above.
\tableofcontents % Table of Contents will be automatically
% generated and placed here.
\listoftables % List of Tables and List of Figures will be placed
\listoffigures % here, if applicable.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Actual text starts here. %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Including external files for each chapter makes this document simpler,
% makes each chapter simpler, and allows for generating test documents
% with as few as zero chapters (by commenting out the include statements).
% This allows quicker processing by the Makediss command file in case you
% are not working on a specific, long and slow to compile chapter. You
% can even change the chapter order by merely interchanging the order
% of the include statements (something I found helpful in my own
% dissertation).
%
\include{solarABM_ch_Introduction}
\include{solarABM_ch_Principles}
\include{solarABM_ch_Agents}
\include{solarABM_ch_Methods}
\include{solarABM_ch_Initialization}
\include{solarABM_ch_Results}
\include{solarABM_ch_Scenarios}
\include{solarABM_ch_Discussion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Appendix/Appendices %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% If you have only one appendix, use the command \appendix instead
% of \appendices.
%
\appendices
\index{Appendices@\emph{Appendices}}%
\include{solarABM_ch_Appendix}
\include{solarABM_ch_Sensitivity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Generate the index. %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% NOTE: For master's theses and reports, NOTHING is permitted to %
% come between the bibliography and the vita. This section %
% to generate the index (if used) MUST be moved to before %
% the bibliography section. %
% %
\printindex% % Include the index here. Comment out this line %
% % with a percent sign if you do not want an index. %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Generate the bibliography. %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% NOTE: For master's theses and reports, NOTHING is permitted to %
% come between the bibliography and the vita. The command %
% to generate the index (if used) MUST be moved to before %
% this section. %
% %
%\nocite{*} % This command causes all items in the %
% bibliographic database to be added to %
% the bibliography, even if they are not %
% explicitly cited in the text. %
% %
\bibliographystyle{plainnat} %
\bibliography{Solar_ReferencesUpdate} %
\index{Bibliography@\emph{Bibliography}}% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Vita page. %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{vita}
%Scott Austen Robinson was born in Denver, Colorado, in 1987, the son of
%Dr. Willard S. Robinson and Maria J. Katherman. He was raised on a ranch outside Douglas, Wyoming. He received the Bachelor
%of Arts degree in Political Science from Middlebury College
%in 2009, conducting research in political forecasting and the nexus of energy and foreign policy. He went on to coach collegiate Crew, work for the Department of Energy in Washington D.C., and for Senator John Barasso (R-WY) in Casper, Wyoming. In 2011 he joined the University of Texas at Austin to complete the dual degree program in Energy and Earth Resources at the Jackson School of Geoscience, and Public Affairs at the Lyndon B. Johnson School. In 2012 he joined the Energy Systems Transformation Group under Dr. Varun Rai.
%\end{vita}
\end{document}
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| 97,296,857 | 663 |
$\mathop{\cos\/}\nolimits\psi=\frac{\mathop{\cos\/}\nolimits\theta\mathop{\cos%
\/}\nolimits\phi-(\mathop{\sin\/}\nolimits\theta\mathop{\sin\/}\nolimits\phi)%
\Delta(\theta)\Delta(\phi)}{1-k^{2}{\mathop{\sin\/}\nolimits^{2}}\theta{%
\mathop{\sin\/}\nolimits^{2}}\phi},$
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| 104,831,021 | 660 |
\[|x|^{1/2}{\mathop{M\/}\nolimits^{2}}\!\left(x\right)=\tfrac{1}{2}\xi\left({%
\mathop{J_{1/3}\/}\nolimits^{2}}\!\left(\xi\right)+{\mathop{Y_{1/3}\/}%
\nolimits^{2}}\!\left(\xi\right)\right),\]
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| 101,738,194 | 646 |
\[e^{y}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!}\mathop{U\/}\nolimits\!\left(a,b+n,%
x\right),\]
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http://bumble.sourceforge.net/books/javascript/javascript-book.txt.x.tex
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sourceforge.net
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| 45,611,728 | 2,926 |
% -------------------------------------------
% latex generated by: booktolatex.cgi
% from source file : ../htdocs/books/javascript/javascript-book.txt
% on: 11 December 2017, 9:14am
% querystring: books/javascript/javascript-book.txt
% document-root: /var/www/html
% script-name: /cgi-bin/booktolatex.cgi
% Server-name: bumble.sourceforge.net
% Sed-script: booktolatex.sed
% -------------------------------------------
\documentclass[a4paper,12pt]{article}
\usepackage[margin=0.4cm,noheadfoot]{geometry}
\usepackage{color} %% to use colours, use "xcolor" for more
\usepackage{multicol} %% for multiple columns
\usepackage{keystroke} %% for keyboard key images
\usepackage[toc]{multitoc} %% for multi column table of contents
\usepackage{tocloft} %% to customize the table of contents
\setcounter{tocdepth}{2} %% only display 2 levels in the contents
\setlength{\cftbeforesecskip}{0cm} %% make the toc more compact
\usepackage{listings} %% for nice code listings
%\lstset{language={},
\lstset{language=,
%% define special comment delimiters '##(' and ')'
moredelim=[s][\color{grey}\itshape\footnotesize\ttfamily]{~(}{)},
basicstyle=\ttfamily, %% fixed pitch font
xleftmargin=1cm, %% margin on the left outside the frames
breaklines=true, %% break long code lines
breakatwhitespace=false, %% break long code lines anywhere
breakindent=10pt, %% reduce the indent from 20pt to 10
postbreak=\mbox{{\color{blue}\small$\Rightarrow$\space}}, %% mark with arrow
showstringspaces=false, %% dont show spaces within strings
framerule=5pt, %% thickness of the frames
rulecolor=\color{lightgrey}, frame=l} %% source code settings
\usepackage{graphicx} %% to include images
\usepackage{fancybox} %% boxes with rounded corners
\usepackage{wrapfig} %% flow text around tables, images
\usepackage{tabularx} %% change width of tables
\usepackage[table]{xcolor} %% alternate row colour tables
\usepackage{booktabs} %% for heavier rules in tables
\usepackage[small,compact]{titlesec} %% sections more compact, less space
\usepackage{enumitem} %% more compact and better lists
\setlist{noitemsep} %% reduce list item spacing
\usepackage{hyperref} %% make urls into hyperlinks
\hypersetup{ %% add "pdftex," if only pdf output is required
colorlinks=true, %% set up the colours for the hyperlinks
linkcolor=black, %% internal document links black
urlcolor=black, %% url links black
filecolor=red,
citecolor=red,
bookmarks=true, pdfpagemode=UseOutlines}
% define some colours to use
\definecolor{lightgrey}{gray}{0.70}
\definecolor{grey}{gray}{0.30}
\titleformat{\section}[frame] %% titlesec: create framed section headings
{\normalfont}
{\filleft \footnotesize \enspace Section \thesection\enspace\enspace}
{3pt} {\bfseries\itshape\filright}
\title{The javascript language}
\author{}
\date{27 October 2011, 6:33pm}
\setlength{\parindent}{0pt}
% \setlength{\parskip}{1ex}
% label lists with stars
\renewcommand{\labelitemi}{$\star$}
\begin{document}
\centerline{\Large \bf The javascript language} \medskip
\begin{center}
{\huge ``}\textit{}{\huge ''}
\textsc{}
\end{center}
% -----------------------------------
% the toc should be 2 columns because of the \multitoc package
\tableofcontents
\begin{multicols}{2}
\begin{lstlisting}
I learnt javascript before the advent of ajax and was thinking about writing
another vim in javascript. A truly strange project, but hence this book.
VARIABLES
* clear a variable
>> lastinsert = '';
ARRAYS
* define a new array
>> var a = new Array();
* get an array of all <a> html elements
>> var d = document.getElementsByTagName('A');
* set a array element to be an anonymous function
>> this.movements['E'] = function(){this.endOfWord('E');}
* set an array to a method
* this.movements['H'] = this.topOfScreen;
STRING VARIABLES
* get the length of a string
>> var i = s.length;
* a function to check if a string contain a space character
>> function hasSpace(s) { return (s.indexOf(' ') == -1) ? false : true; }
SUBSTITUTING AND REPLACING IN STRINGS
* replace a pattern in a string and assign the result
>> var t = s.replace(/[.?!,:]*$/,"");
MATCHING PATTERNS
* check if a string matches a pattern
>> if (s.match(/^[ \r\n\t]*<.*>[ \r\n\t]*$/)) return true
IF STATEMENT
>> if (z == -1) { x++; }
TERNARY OPERATOR
>> var d = (z > 2) ? -1 : 1;
FUNCTIONS
* define a function called 'f'
>> function name( f ) { return f; }
OBJECTS
It is possible to program in an object oriented style
with javascript
* define a constructor
\begin{lstlisting}
function Editor(s)
{
this.message = s;
}
\end{lstlisting}
* define a new class "Editor" with fields and methods
\begin{lstlisting}
Editor.prototype =
{
text: new Array(),
movements: new Object(),
nextChar: function()
{
return this.text[this.cursorRow].charAt(this.cursorCol);
},
...
};
\end{lstlisting}
\end{document}
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\documentclass[11pt,b4j,landscape,twocolumn]{jsarticle}
% jsarticle.cls 2019年4月現在別途DL不要 http://oku.edu.mie-u.ac.jp/~okumura/jsclasses/
% b4tate.sty 2019年4月現在不要にした http://unilab.gbb60166.jp/tex/b4tate.zip
% furikana.sty http://xymtex.my.coocan.jp/fujitas2/texlatex/index.html#tategumi
\usepackage{ascmac,furikana,utf,bm,epic,eepic}
\usepackage[dvipdfmx]{graphicx,color}
%\usepackage[dvipdfmx,pdfpagemode=FullScreen,setpagesize=false]{hyperref}
%\usepackage{tikz}
%\usetikzlibrary{calc}
%\usetikzlibrary{calc,intersections,through,backgrounds,positioning}
\textwidth=315mm \textheight=220mm \columnseprule=0pt
\topmargin=-25mm \oddsidemargin=-15mm
\newcommand{\Slash}[1]{\ooalign{\hfil\kern-2pt／\hfil\crcr$#1$}}
%\newcommand{\Slash}[1]{\ooalign{#1\llap}}
\newenvironment{nidan}{\begin{minipage}[t]{.225\textwidth}}{\end{minipage}}
\def\nidangumi#1#2{\begin{nidan}{#1}\end{nidan}\quad%
\begin{nidan}{#2}\end{nidan}}
% ■■■ 0より大きいときはdvipdfmxで印刷用PDFを作ることを想定（答えはプリント下部に出力される）
% 0 のときはdvipdfmxでデジタル教材用PDFを作ることを想定（答えは問題のすぐ横に赤で出力され、後でOneNoteで黒く塗りつぶし）
\def\make_pdf_with_insatsu{1}
\newcounter{toi} \newcounter{subtoi}[toi]
\newcounter{subsubtoi}[subtoi]
\def\toi{\refstepcounter{toi}\par % 問
\makebox[2zw][r]{\Large\ajKaku{\value{toi}}\hspace{0.5zw}}%
\hangindent=2zw}
\newcounter{subtoihyoji}
\def\subtoi{\refstepcounter{subtoi}\par % 小問
\setcounter{subtoihyoji}{\value{subtoi}}
\ifnum\thesubtoi>20
\addtocounter{subtoihyoji}{9873}
\else
\addtocounter{subtoihyoji}{8070}
\fi
\makebox[3.4zw][r]{\CID{\thesubtoihyoji}\hspace{1zw}}%
\hangindent=3.4zw}
\newcounter{subsubtoihyoji}
\def\subsubtoi{\refstepcounter{subsubtoi}\par % 小小問
\setcounter{subsubtoihyoji}{\value{subsubtoi}}
\addtocounter{subsubtoihyoji}{7554}
\makebox[4.2zw][r]{\CID{\thesubsubtoihyoji}\hspace{1zw}}%
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\ifnum\make_pdf_with_insatsu>0
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\else
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\fi
\everymath{\displaystyle}
\pagestyle{empty}
\begin{document}
%
% ===========================以下本文===============================
%
\def\namae{%
{\Large\bf 数学A \quad 授業プリント \# 10}\hfill
年\hskip8ex 組\hskip8ex 号 \hskip5ex
\hfill
\underline{氏名\rule[-2ex]{0cm}{6ex}\hskip30ex}}
\namae
\vskip-2ex
{\large\bf ■ 組合せ}
\medskip
\toi 次の問いに答えなさい。
\nidangumi{\subtoi 8人の中から委員長・副委員長・書記の\\
\hspace*{2.6zw}3人を選ぶ選び方は何通りありますか。\\\kotae{336{\bf 通り}} }
{\subtoi 8人の中から図書委員3名を選ぶ選び方\\
\hspace*{2.4zw}は何通りありますか。\kotae{56{\bf 通り}} }\vfill
\begin{center}
\begin{minipage}[t]{0.4\textwidth}
\begin{shadebox}
\vspace*{0.5ex}
\hspace*{1zw}$8$つの中から$3$つを選ぶ選び方が何通りありますかを
\( {}_8 \mbox{C}_3 \)という記号で表す。この記号は次のように計算をする
\vspace*{1zh}
\hskip10zw \( {}_8 \mbox{C}_3 = %
%{7から始めて数字を1ずつ減らしながら3個の数字を
%かけ算 \over 3から1までの数字をかけ算} =
{8 \times 7 \times 6 \over 3 \times 2 \times 1} =
{8 \times 7 \times \Slash{6}^{\displaystyle\Slash{3}^{\displaystyle1}} \over %
{}_{\displaystyle1}\Slash{3} \times \Slash{2}_{\displaystyle1} \times 1} = 8 \times 7 = 56\)
%\Slash{2}_{\displaystyle1}
\vspace*{0.5ex}
\end{shadebox}
\end{minipage}
\end{center}
\fbox{その他の例} \mbox{}\hskip7zw \( {}_9 \mbox{C}_2 = %
{9 \times 8 \over 2 \times 1} =
{9 \times \Slash{8}^{\displaystyle4} \over {}_{\displaystyle1}\Slash{2} \times 1} = 9 \times 4=36\)
\vskip1zh
\mbox{}\hskip12zw \( {}_6 \mbox{C}_4 = %
{6 \times 5 \times 4 \times 3 \over 4 \times 3 \times 2 \times 1} =
{{}^{\displaystyle3}\Slash{6} \times 5 \times \Slash{4}^{\displaystyle1} \times \Slash{3}^{\displaystyle1} \over %
{}_{\displaystyle1}\Slash{4} \times \Slash{3}_{\displaystyle1} \times \Slash{2}_{\displaystyle1} \times 1} =
3 \times 5=15\)
\vskip1zh
\toi 次の値を計算しなさい。
\nidangumi{\subtoi \( {}_6 \mbox{C}_3 \) \kotae{20} }
{\subtoi \( {}_8 \mbox{C}_4 \) \kotae{70} }\vfill
\nidangumi{\subtoi \( {}_5 \mbox{C}_4 \) \kotae{5} }
{\subtoi \( {}_9 \mbox{C}_7 \) \kotae{36} }\vfill
\nidangumi{\subtoi \( {}_5 \mbox{C}_5 \) \kotae{1} }
{}\vfill
\ifnum\make_pdf_with_insatsu>0
\hfill\rotatebox{180}{{\normalsize\bf プリント\#10}\scriptsize
{\normalsize\ajKaku{4}}~\( 120通り \)\
{\normalsize\ajKaku{5}}~\( 200通り \)\
{\normalsize\ajKaku{6}}~\( 18通り \)\
{\normalsize\ajKaku{7}}~\( 60通り \)\
{\normalsize\ajKaku{8}}~\( 2520通り \)\
{\normalsize\ajKaku{9}}~\( 1260通り \)\
}
\fi
\newpage
\toi 次の場合の数を\( {}_n \mbox{C}_r \)の記号を使って表し，何通りあるのかを
計算しなさい。
\subtoi 40人のクラスから代表を2名選ぶ選び方 \kotae{{}_{40} \mbox{C}_2=780{\bf 通り}}\vfill
\subtoi テストの問題が7題ある。この中から4題を選択するとき，
問題の選び方 \kotae{{}_7 \mbox{C}_4=35{\bf 通り}}\vfill
\subtoi 9種類のおかずがある食堂で3種類を選ぶ選び方 \kotae{{}_9 \mbox{C}_3=84{\bf 通り}}\vfill
\subtoi 8チームが総当たりのリーグ戦で試合をするとき，
すべての試合数 \kotae{{}_8 \mbox{C}_2=28{\bf 試合}}\vfill
\subtoi 次のように平面上に8個の点がある。
これらの点を頂点とする三角形の数\kotae{{}_8 \mbox{C}_3=56{\bf 個}}
\unitlength=1mm
\hspace*{30zw}\begin{picture}(30,30)(0,0)
\path(0,0)(0,30)(30,30)(30,0)(0,0)
\put(15,25){\circle*{1}}
\put(15,28){\makebox(0,0){\scriptsize A}}
\put(8,21){\circle*{1}}
\put(7,24){\makebox(0,0){\scriptsize B}}
\put(5,15){\circle*{1}}
\put(2,15){\makebox(0,0){\scriptsize C}}
\put(8,8){\circle*{1}}
\put(7,5){\makebox(0,0){\scriptsize D}}
\put(18,7){\circle*{1}}
\put(18,4){\makebox(0,0){\scriptsize E}}
\put(24,11){\circle*{1}}
\put(25,8){\makebox(0,0){\scriptsize F}}
\put(26,15){\circle*{1}}
\put(28,15){\makebox(0,0){\scriptsize G}}
\put(23,22){\circle*{1}}
\put(25,24){\makebox(0,0){\scriptsize H}}
\end{picture}
\subtoi 13人の部員の中から11人の選手を選ぶ選び方 \kotae{{}_{13} \mbox{C}_{11}=78{\bf 通り}}\vfill
\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\toi 男子6人，女子4人の中から男子3人，女子2人を選ぶ選び方
全部で何通りありますか。\kotae{120{\bf 通り}} \vfill
\toi 男子6人，女子5人の中から男子3人，女子3人を選ぶ選び方は
全部で何通りありますか。\kotae{200{\bf 通り}} \vfill
\toi 次の図の中に平行四辺形は何個あるか。\kotae{18{\bf 通り}}
%%%%%1つの平行四辺形は平行線を2本ずつ選んで出来ることを使って求めなさい。
\unitlength=1mm
\hspace*{1zw}\begin{picture}(45,25)(0,0)
\path(6,0)(12,22)
\path(12,0)(18,22)
\path(24,0)(30,22)
\path(33,0)(39,22)
\path(3,4)(39,4)
\path(4,9)(41,9)
\path(8,18)(45,18)
\end{picture}
\vskip2zh
\toi 次の図の中に平行四辺形は何個あるか。\kotae{60{\bf 通り}}
\unitlength=0.8mm
\hspace*{1zw}\begin{picture}(45,25)(0,0)
\path(6,0)(12,22)
\path(12,0)(18,22)
\path(16,0)(22,22)
\path(24,0)(30,22)
\path(33,0)(39,22)
\path(3,4)(39,4)
\path(4,9)(41,9)
\path(6,15)(43,15)
\path(8,18)(45,18)
\end{picture}
\vskip2zh
\toi 10人を5人，3人，2人に分ける分け方は全部で何通りありますか。\kotae{2520{\bf 通り}}\vfill
\toi 9人を4人，3人，2人に分ける分け方は全部で何通りありますか。\kotae{1260{\bf 通り}}\vfill
\ifnum\make_pdf_with_insatsu>0
\hfill\rotatebox{180}{
{\normalsize\bf プリント\#10}\scriptsize
{\normalsize\ajKaku{1}}~\ajKakko{1}\( 336通り \)\ \ajKakko{2}\( 56通り \)\
{\normalsize\ajKaku{2}}~\ajKakko{1}\( 20 \)\ \ajKakko{2}\( 70 \)\ \ajKakko{3}\( 5 \)\ \ajKakko{4}\( 36 \)\ \ajKakko{5}\( 1 \)\
{\normalsize\ajKaku{3}}~\ajKakko{1}\( 780通り \)\ \ajKakko{2}\( 35通り \)\ \ajKakko{3}\( 84通り \)\
\ajKakko{4}\( 28試合 \)\ \ajKakko{5}\( 56個 \)\ \ajKakko{6}\( 78通り \)\
}
\fi
\end{document}
■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
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| 98,787,199 | 6,015 |
\documentclass[12pt,dvips,twosided]{article}
\usepackage{palatino,a4,graphicx,color,pifont,html}
\def\xfm{\emph{X File Explorer}}
\begin{document}
\pagestyle{empty}
\setlength{\parskip}{.15in}
\vspace*{1cm}
\begin{center}
\Huge
Introduction for New Users\\
\vspace{1cm}
\Large
Cambridge University Engineering Department\\
Teaching Workstations\\
\vspace{0.5cm}
\large Roberto Cipolla, Richard Prager, Tim Love\\
July, 2005\\
\vspace{0.5cm}
\end{center}
\noindent\hbox{\vtop{\kern0pt\hsize=10cm \noindent This document describes how to log on to the teaching system and perform
simple file manipulation using \xfm{}. Common tasks
like printing, editing, getting help, accessing the World Wide Web and using
e-mail are also addressed. An enhanced version of this document is
online in the help system}
\vtop{\kern0pt\hbox{\hspace{5mm}}}
\vtop{\kern0pt\hbox{\includegraphics[width=4cm]{myxfe.eps}}}}
\setlength{\parskip}{0mm}
{\small
\tableofcontents
}
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\setlength{\parskip}{0in}
\newpage
\section{Getting Started}
Here we describe how to log on and perform simple file
manipulation in the windowing environment. Using this simple environment
you will be able to use the \texttt{Unix} operating system
in a straightforward way.
This year there are 2 options when you login. This document describes
the
\textit{This Machine} set-up. The \textit{Server} set-up is described in a
separate
document.
\setlength{\parskip}{.15in}
To start with, the computer will present you with a login window.
If the screen is completely black then move the mouse slightly; this
will turn off the screen saver.
\begin{dinglist}{43}
\item Make sure that \textit{This Machine} is highlighted.
\item Type your username (your \textit{CRS ID}) and press
the \texttt{Return} key. If you don't
know your user identifier it can be found on the notice boards near the
door to the main computer room. If in doubt ask a demonstrator or one
of the operators in the machine room at the west end of the DPO.
\item Type in your CUED password (\textit{not} your hermes password).
First year students should
have chosen a password during registration at the start of term. Other
students should use their old password. When you type your
password it will not be echoed to the screen. If you cannot remember
your password, or it does not work, ask one of the operators. \texttt{Unix} is
\textit{case sensitive}, so make sure that ``\texttt{Caps Lock}'' is not set.
If there are system announcements a yellow window will appear in the
bottom-right corner of the screen. Read the messages then press the
\verb|Return| key while the mouse pointer is in the yellow window.
Once you have logged in successfully a text window will appear and
you will get a prompt something like:\newline
\texttt{tw400/ab123:}
\end{dinglist}
If you are doing a
practical you should now consult your main practical handout to see if
you are required to use a special command (like \texttt{`start camclay'} or
\texttt{`start 1AComputing'}).
\setlength{\parskip}{0in}
\section{Icons \& files}
\subsection*{Your Home Folder}
Next, we are going to start \xfm{}. Click on the
\includegraphics{start.eps} button at the bottom-left of the screen and choose
\texttt{X File Explorer}. The icons you initially see in this
\emph{Xfe}
window represent files or
collections of files in the part of the file system that belongs to you.
\setlength{\parskip}{0in}
\subsection*{Creating a New File}
To create a new file, choose the \texttt{New File} option from the
\texttt{File} menu. Call the file \verb|sports|.
\setlength{\parskip}{.15in}
When you come to name files yourself try to use only
letters and digits --- no spaces or punctuation characters
(except for full stops). If you are used to using PCs it is also worth
noting that Unix distinguishes the case of characters in file names.
The file \texttt{Sports} is not the same file as \texttt{sports}.
\setlength{\parskip}{0in}
\subsection*{Copying Files}
\noindent\hbox{\vtop{\kern0pt\hsize=6cm \noindent To copy the \texttt{sports} file hold the
\texttt{Ctrl} button down while dragging the file elsewhere in the window.
You will be prompted to give the new file a name. The name starts with the folder's name. Append it with \texttt{/activities}
(note the '/' character) and click on the \fbox{Accept} button. }
\vtop{\kern0pt\hbox{\hspace{5mm}}}
\vtop{\kern0pt\hbox{\includegraphics[width=8cm]{copy.eps}}}}
\setlength{\parskip}{0in}
\subsection*{Deleting Files}
Make another copy of \texttt{sports} called \texttt{doomed}. We will use
this file to practice deleting! First select the \texttt{doomed} file
by clicking on it. Then use the \texttt{Delete} menu item in the
\texttt{File} menu. A new box will appear. Click on the
\fbox{OK} option and the file will
disappear from your directory.
\setlength{\parskip}{0in}
\section{Folders}
\subsection*{Creating a New Folder}
Imagine
that the \texttt{sports} file contains a list of the sports you like playing
and
the \texttt{activities} file contains a list of all the other leisure
activities you enjoy. They are rather out of place among all the other
files in your home directory which are currently mostly special user
configuration files.
\setlength{\parskip}{.15in}
It would be nice if you could create a separate place to put
\texttt{sports} and \texttt{activities}. This is what a folder is for.
Folders (\textit{e.g.} your home folder) can contain files \textbf{and
other sub-folders}.
Choose the \texttt{New Folder} option from the
\texttt{File} menu. Call the folder \verb|clump|.
Note how the icon of this folder looks different to ordinary file icons.
Double click on this new icon to open it out. You will see a folder just
like your home folder, only almost empty.
There will be one icon present, called \texttt{`..'} (pronounced `dot dot'). This refers to the folder's parent, in this case
your home folder.
The underlying structure is that all the folders are arranged in a tree
and each one contains icons telling it:
\begin{itemize}
\item
The parent folder (they only have one parent).
This is the \texttt{`..'} icon.
\item
Which (if any) folders are its children.
\item
Which (if any) files it contains.
%\item
%There is also the \texttt{`.'} icon referring back to the directory itself.
\end{itemize}
Note that the full name of the folder you're now looking at is
shown near the top of the window.
Now double-click on the \texttt{`..'} icon. This will open out your home
folder again because that is \texttt{clump}'s parent.
\setlength{\parskip}{0in}
\subsection*{Moving \& Copying Between Folders}
Now we'll try moving the \texttt{sports} files into the \texttt{clump}
directory. If you drag the \texttt{sports} icon onto the \texttt{clump} icon
and click on the \texttt{Accept} button when
asked to confirm, your file will have moved into the folder. If you
want to \textit{copy} the file, leaving the original intact, then drag
while pressing down the \texttt{Ctrl} key.
You may need to use the \texttt{Refresh} option of the \texttt{File}
menu to update the window.
\setlength{\parskip}{0in}
\section{Window management}
When you have lots of windows you need to be able to manage them.
\begin{itemize}
\item \textit{Moving} - you can drag a window around using its title
bar.
\item \textit{Resizing} - you can drag a window's corner to resize it.
\item \textit{Iconify} - if you click on the \fbox{\_} near the
top right hand corner of the folder window, it will tumble
down into the task-bar at the bottom of the screen. The window can
be restored by clicking on its name in the task-bar
\item \textit{Full Screen} - to fill the screen with the window,
click on the \fbox{ } near the
top right hand corner of the directory window.
\item \textit{Kill} - clicking on the \fbox{x} box kills the window.
Using this risks losing your unsaved work. Try to use
the program's \texttt{Exit} option instead.
\item \textit{The Screens Panel} - The Screens box at the bottom-right
shows
9 'virtual desktops'. All your windows
so far have appeared on one virtual desktop (top-left), but by clicking on another
virtual desktop you can start with an uncluttered screen. This facility helps
keep your workspace tidy. You can move windows from one
virtual desktop to another by dragging the mini-window using the middle button.
\end{itemize}
Note that you might have to click on a window before the text that you
type goes into it.
\setlength{\parskip}{0in}
\section{The Emacs Editor}
\subsection*{Starting the Editor}
Create a file called \texttt{pomodoro} in your home directory.
\setlength{\parskip}{.15in}
Now let's put something in the file. Double-click with the left
mouse button on the \texttt{pomodoro} icon. When offered
\texttt{emacs},
accept it. In Emacs you will see the majority of the screen is used to display the
file you are editing. Below this is a bar with something like
\begin{verbatim}
--:-- pomodoro (Fundamental)--L1--ALL-------
\end{verbatim}
Beneath this there is a single line (the mini-buffer) for messages and editor commands.
Along the top of the window is a menu-bar and icons.
Type in several lines
of text (pressing the \texttt{Return} key at the end of each line).
The arrow keys, and Delete key will work though the NumPad keys won't
work until you press the \texttt{Num Lock} key.
There is an online tutorial for advanced emacs
users which is started from emacs' \texttt{Help} menu.
It is \textbf{not} recommended that you do this now.
Further details can be found in the \emph{Unix from the Command Line} handout.
\setlength{\parskip}{0in}
\subsection*{Emacs Menus and Scrollbars}
Across the top of the emacs window is a series of menus. The \texttt{File}
menu contains items that let you load and save files. Select the
\texttt{Open File...} option.
A prompt will appear at the bottom of the window. If
you were to type the name of a file here you would be able to edit that
file. We won't do this just now so go to the \texttt{Minibuf} menu
and select \texttt{Quit} to cancel the operation.
\setlength{\parskip}{.15in}
Go back to the \texttt{File} menu and select the \texttt{Save Buffer} item.
This saves the file you are editing to disk.
You must save files to disk before
you try to perform any operations on them outside the editor.
We'll try searching the text now.
Place the text cursor at the top of the text in
\texttt{pomodoro} and pull down the \texttt{Edit} menu so that you can select the
\texttt{Search} item from the\texttt{Search} submenu. A prompt will appear. Type in a
string you can see at some point in the file and press the \texttt{Return}
key. The editor will find the string and place the text cursor at the end of it.
Make sure \texttt{pomodoro} is at least
five lines long and pull down the \texttt{Edit} menu. From the
\texttt{Go To} sub-menu pick the \texttt{Goto Line} item. When the
prompt appears specify \texttt{5} and press the \texttt{Return}
key. The cursor should move to the
fifth line of the file. Note that the line number is shown in the
status line near the bottom of the window -- \texttt{L5}.
To copy some text, drag the mouse cursor over it using the left mouse
button. Then click with the middle mouse button where you want the
text to be copied. This facility also works between
most kinds of text windows.
To delete a block of text, click once with the left mouse button at
one end, and twice with the right mouse button at the other (or
choose the \texttt{Cut} item in the \texttt{Edit} menu). If you want to
move the text elsewhere, just click on the location with the middle mouse
button.
If you want to move quickly around a document that is too
big to fit on one page, you can use the scrollbar on the left of the window
or the
\texttt{Page Up} and \texttt{Page Down} on the keyboard by the arrow keys.
If you made an editing mistake you can use the \texttt{Undo} option in the
\texttt{Edit} menu repeatedly until you get back to the way you were.
The \texttt{Exit Emacs} item in the \texttt{Files} menu makes the editor offer to
save any files you have
changed during the current editing session, and then kills the editor
program. A backup copy of each file you change is kept by the editor.
These are the files with names ending in the character \verb|`~'|. Don't
worry if you cannot always see them in your directory window, you will
only need to use them if things go badly wrong.
Many of these actions can be performed using the icons under the
menu-bar. Experiment to find out what they do.
\includegraphics{bar.eps}
\setlength{\parskip}{.15in}
\section{Applications and logging off}
If you click on \includegraphics{start.eps} at the bottom-left of the
screen you can explore what programs we have. In particular there's
\noindent\textbf{Mozilla-Firefox} - a web browser. You can use it for webmail too. \emph{You will need to become familiar with the WWW}.
\noindent\hbox{\vtop{\kern0pt\hsize=6cm \noindent\textbf{Help} - a short-cut to CUED's help-on-the-web and list of
\htmladdnormallink{\textit{Frequently Asked
Questions}}{http://www-h.eng.cam.ac.uk/help/tpl/cuedfaq.html}. \emph{We expect you to become familiar with the help system}}
\vtop{\kern0pt\hbox{\hspace{5mm}}}
\vtop{\kern0pt\hbox{\includegraphics[width=8cm]{help.eps}}}}
\noindent\hbox{\vtop{\kern0pt\hsize=6cm \noindent\textbf{Programs/CUED/Fast
Feedback} - our Fast Feedback Facility. Your chance
to offer instant, anonymous feedback on teaching, computing, etc.}
\vtop{\kern0pt\hbox{\hspace{5mm}}}
\vtop{\kern0pt\hbox{\includegraphics[width=8cm]{1bfff.eps}}}}
\noindent\textbf{Programs/CUED/Survey} - our under-graduate user survey.
\noindent\textbf{XTerminal} - provides a terminal window on the screen. This
can be used to execute Unix shell commands. These are
covered in the \textit{Unix from the command line} document.
\subsection*{Logging off}
To \textbf{Log Out}, use the \texttt{Log Out} item in the
\texttt{start}
menu.
\end{document}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay987}%
$\displaystyle \stensor C$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay991}%
$\displaystyle E ({\svector v}{\svector v}^\intercal ).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline993}%
$ C_{ij}= C_{ji}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline995}%
$ \svector c$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay998}%
$\displaystyle c_i$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1002}%
$\displaystyle E (v_0 v_i ) , ~$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1003}%
$\displaystyle \mbox {for}~ i{=}1,n, ~\mbox{or}$%
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\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1005}%
$\displaystyle \svector c$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1009}%
$\displaystyle E (v_0 {\svector v}).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1012}%
$ \sigma_E^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1015}%
$\displaystyle \sigma_E^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1019}%
$\displaystyle E [ ( v_0- \sum_{i=1}^nw_iv_i)^2 ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1023}%
$\displaystyle E(v_0v_0) - 2 \sum_{i=1}^nE ( v_iv_0) w_i+ \sum_{i=1}^n\sum_{j=1}^nE (v_iv_j) w_iw_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1027}%
$\displaystyle \sigma_v^2- 2 \sum_{i=1}^nc_iw_i+ \sum_{i=1}^n\sum_{j=1}^nC_{ij}w_iw_j.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1034}%
$\displaystyle {\frac{\partial \sigma_E^2}{\partial w_i}} = - 2 c_i+ 2 \sum_{j=1}^nC_{ij}w_j= 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1036}%
$\displaystyle \Rightarrow$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1038}%
$\displaystyle \sum_{j=1}^nC_{ij}w_j= c_i$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1042}%
$\displaystyle \stensor C\cdot {\svector w}= \svector c$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1046}%
$\displaystyle {\svector w}= \stensor C^{-1}\cdot \svector c.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1052}%
$ \stensor C^{-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1061}%
$\displaystyle \sigma_v^2- 2 \sum_{i=1}^nc_iw_i+ \sum_{i=1}^nc_iw_i$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1065}%
$\displaystyle \sigma_v^2- \sum_{i=1}^nc_iw_i$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1069}%
$\displaystyle \sigma_v^2- \svector c^\intercal \cdot {\svector w}.$%
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\lthtmlcheckvsize\clearpage}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1073}%
$\displaystyle C_{ij}\equiv
E [v_i v_j ] \equiv
E [v ({\svector x}_i) v({\svector x}_j) ] \equiv C ( {\svector x}_i - {\svector x}_j )$%
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\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1075}%
$\displaystyle ~$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1076}%
$\displaystyle ~ i{=}0,n~$%
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\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1077}%
$\displaystyle ~ j{=}0,n.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1088}%
$\displaystyle \sum_{i=1}^nw_i\cdot v({\svector x}_i)~$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1090}%
$\displaystyle w_i$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1094}%
$\displaystyle \|{\svector x}_i- {\svector x}_0\|^{-2} / \sum_{j=1}^n\|{\svector x}_j- {\svector x}_0\|^{-2} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1097}%
$\displaystyle \stensor C({\svector x}_i- {\svector x}_j)$%
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\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1099}%
$\displaystyle \propto$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1101}%
$\displaystyle \|{\svector x}_i- {\svector x}_j\|^{-2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1103}%
$\displaystyle \Rightarrow ~~ \sigma_v^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1105}%
$\displaystyle \approx$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1107}%
$\displaystyle \infty$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1109}%
$\displaystyle \Rightarrow ~~ \stensor C$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1113}%
$\displaystyle \stensor I$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1115}%
$\displaystyle \Rightarrow ~~ {\svector w}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1119}%
$\displaystyle \svector c.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1121}%
$ \Delta x$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1128}%
$\displaystyle \min(\Delta x^{-2}, \|{\svector x}_i- {\svector x}_j\| ^{-2} ) .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1133}%
$ x$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1135}%
$ v_2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1137}%
$ v_1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1141}%
$ \|x_1- x_0\| = 1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1143}%
$ \|x_2- x_0\| = 2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1145}%
$ {{\svector w}^\intercal}= [ 0.8 , 0.2 ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1147}%
$ \Delta x= 0.5$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1149}%
$ \|x_2 - x_1\| = 3$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1151}%
$ {{\svector w}^\intercal}= [ 0.82 , 0.18 ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1153}%
$ \|x_2 - x_1\| = 1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1165}%
$ {{\svector w}^\intercal}= [ 1 ,0 ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1169}%
$\displaystyle \gamma_v ({\svector x}_i, {\svector x}_j)$%
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\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1173}%
$\displaystyle \frac{1}{2} E \{ [v ({\svector x}_i) - v({\svector x}_j) ]^2 \} = \sigma_v^2- C_{ij}.$%
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\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1176}%
$\displaystyle \gamma_v ({\svector x}_i- {\svector x}_j) = \sigma_v^2- C( {\svector x}_i - {\svector x}_j )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1178}%
$ {\svector h}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1181}%
$\displaystyle {\svector h}_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1185}%
$\displaystyle {\svector x}_i- {\svector x}_j;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1187}%
$\displaystyle \gamma_v ({\svector h}_{ij})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1191}%
$\displaystyle \sigma_v^2- C( {\svector h}_{ij}).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1194}%
$\displaystyle \gamma ({\svector h}) = C ({\svector 0} ) - C ({\svector h}).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1197}%
$\displaystyle C( {\svector \infty})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1202}%
$\displaystyle \Rightarrow ~~ \gamma( {\svector \infty})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1206}%
$\displaystyle C({\svector 0})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1208}%
$\displaystyle \Rightarrow ~~ C ({\svector h})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1212}%
$\displaystyle \gamma ({\svector \infty}) - \gamma({\svector h}).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1215}%
$\displaystyle \gamma_{\exp} ({\svector h})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1219}%
$\displaystyle 1 - \exp( - 3 \| {\svector h}\| / a), ~~$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1221}%
$\displaystyle C_{\exp} ({\svector h})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1225}%
$\displaystyle \exp( - 3 \| {\svector h}\| / a).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1227}%
$ a$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1232}%
$ {\svector m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1234}%
$ {\svector d}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1236}%
$ {\stensor F}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1238}%
$ {\svector n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1241}%
$\displaystyle {\svector d}= {\stensor F}{\svector m}+ {\svector n}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1244}%
$\displaystyle S ({\svector m}) = ({\svector d}- {\stensor F}{\svector m})^{\intercal} {\stensor C}_n^{-1}({\svector d}- {\stensor F}{\svector m}) + {\svector m}^{\intercal} {\stensor C}_m^{-1}{\svector m}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1246}%
$ {\stensor C}_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1250}%
$ {\stensor C}_n$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1252}%
$ {\stensor C}_d$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1255}%
$\displaystyle {\stensor C}_d= {\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor C}_n.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1259}%
$ \hat{{\svector m}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1262}%
$\displaystyle \frac{\partial S}{\partial {\svector m}^{\intercal}} (\hat{{\svector m}})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1266}%
$\displaystyle - {\stensor F}^\intercal {\stensor C}_n^{-1}( {\svector d}- {\stensor F}\hat{{\svector m}}) + {\stensor C}_m^{-1}\hat{{\svector m}}= 0 ;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1268}%
$\displaystyle \Rightarrow ~~ \hat{{\svector m}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1272}%
$\displaystyle ( {\stensor F}^\intercal {\stensor C}_n^{-1}{\stensor F}+ {\stensor C}_m^{-1}) ^ {-1} {\stensor F}^\intercal {\stensor C}_n^{-1}{\svector d};$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1278}%
$\displaystyle {\stensor C}_m{\stensor F}^\intercal ( {\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor C}_n)^{-1} {\svector d}~=~ {\stensor C}_m{\stensor F}^\intercal {\stensor C}_d^{-1}{\svector d}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1281}%
$\displaystyle ( {\stensor F}^\intercal {\stensor C}_n^{-1}{\stensor F}+ {\stensor C}_m^{-1}) ^ {-1} {\stensor F}^\intercal {\stensor C}_n^{-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1285}%
$\displaystyle {\stensor C}_m{\stensor F}^\intercal ( {\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor C}_n)^{-1} ;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1287}%
$\displaystyle \Rightarrow ~~ {\stensor F}^\intercal {\stensor C}_n^{-1}( {\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor C}_n)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1291}%
$\displaystyle ( {\stensor F}^\intercal {\stensor C}_n^{-1}{\stensor F}+ {\stensor C}_m^{-1}) {\stensor C}_m{\stensor F}^\intercal$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1295}%
$\displaystyle {\stensor F}^\intercal {\stensor C}_n^{-1}{\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor F}^\intercal .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1309}%
$ {\stensor F}^\intercal $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1315}%
$ {\stensor F}{\stensor F}^\intercal = \stensor I$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1317}%
$ {\stensor C}_n= \epsilon \stensor I; \epsilon \rightarrow 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1319}%
$ \hat{{\svector m}}\rightarrow {\stensor F}^\intercal {\svector d}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1324}%
$\displaystyle ~~
{\stensor C}_n$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1328}%
$\displaystyle 0 ;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1330}%
$\displaystyle \Rightarrow ~~ {\stensor C}_d$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1334}%
$\displaystyle {\stensor F}{\stensor C}_m{\stensor F}^\intercal$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1337}%
$\displaystyle \hat{{\svector m}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1341}%
$\displaystyle ({\stensor C}_m{\stensor F}^\intercal ) ( {\stensor F}{\stensor C}_m{\stensor F}^\intercal )^{-1} {\svector d}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1345}%
$\displaystyle ({\stensor F}{\stensor C}_m)^{\intercal} {\stensor C}_d^{-1}{\svector d}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1349}%
$\displaystyle \stensor W^{\intercal}{\svector d};$%
\lthtmlindisplaymathZ
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\title{Abad de Santillán, Diego (1897–1983)}
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\par
Diego Abad de Santillán (a pseudonym of Baudillo Sinesio García Fernández, sometimes also spelled Hernández) was an anarchist intellectual influential both in the Spanish and Argentine movements.
Abad de Santillán was born in Spain, but his family moved to Argentina when he was a boy. He worked various jobs in Argentina and returned to Spain to continue his education, earning a doctorate in Madrid. There, he was active in the 1917 general strike, for which he faced a prison term. Amnestied a year later, he returned to Argentina, where he became a leading figure in the anarchosyndicalist Argentine Regional Work Federation (Fed- eración Obrera Regional Argentina, FORA) as editor of its journal, \emph{La Protesta}. In 1922, as a delegate of the FORA, he attended the founding congress and the 1924 congress of the International Workingmen’s Association (IWMA) in Berlin, where he remained briefly to study medicine. In Germany, he became acquainted with a number of important anarchists, including Rudolf Rocker, Voline, Nestor Makhno, Alexander Schapiro, Emma Goldman, and Alexander Berkman, as well as Max Nettlau, whose work he translated into Spanish and published. Abad de Santillán went to Mexico in 1925 to aid the Mexican anarchosyndicalist General Federation of Workers (Confederación General de Traba- jadores, CGT) and then back to Argentina, where he participated in the international solidarity campaign in defense of Sacco and Vanzetti. He wrote a history of anarchism in Argentina in 1930 and was sentenced to death for sedition that same year, but instead was briefly exiled to Spain, after which he returned to Argentina for two years.
It was in Spain once again, however, that Abad de Santillán was to play a crucial role during the rise and fall of the Republic. He joined the anarchosyndicalist Confederación Nacional del Trabajo (National Labor Con- federation, CNT), where he became an active opponent of reformism, and became the peninsular secretary of the Federación Anar- quista Ibérica (Iberian Anarchist Federation, FAI), as well as editing \emph{Solidaridad Obrera} (Workers’ Solidarity), \emph{Tiempos Nuevos} (New Times), and \emph{Tierra y Libertad} (Land and Free- dom), among other publications. During the Spanish revolution and civil war, he served as the FAI’s delegate to the Anti-Fascist Mili- tias Committee in Catalonia, then became Minister of the Economy in the Catalan regional government. Critical of the policies of Spanish socialist premier Juan Negrín and of the role of the Spanish Communist Party, Abad de Santillán published \emph{El organismo económico de la revolución} (published in English as \emph{After the Revolution}), a brief but ambitious statement of the anarchosyndicalist program for a self-managed economy.
With the defeat of the Republic, he went into exile to France, later returning to Argentina via Santo Domingo and Chile. There he resumed writing scholarly works on the history of anarchism, including \emph{Why Did We Lose the War} (\emph{¿Por qué perdimos la guerra?}, 1978), which was made into a film by his son Luis Galindo in 1978. He was less active in politics, though he did work against the Perón dictatorship. He contributed to numerous dictionaries and encyclopedias and his writings had an enormous impact. He defended the policy of collaboration during the civil war and became increasingly reformist, coming to regard the state as a greater threat than capitalism, which he viewed as a necessary stage of social progress. In 1977, he returned to Spain, dying in Barcelona in 1983.
\hairline
\textbf{REFERENCES AND SUGGESTED READINGS}
\begin{amusebiblio}
Abad de Santillán, D. (1937) \emph{After the Revolution: Economic Reconstruction in Spain Today}. New York: Greenberg. (Translation of \emph{El organismo económico de la revolución: cómo vivios y cómo podríamos vivir en España}, 1936.).
Abad de Santillán, D. (1977) \emph{Memorias, 1897–1936}. Barcelona: Planeta.
Galindo, Luís (dir.) (1978) \emph{Why Did We Lose the War?} (\emph{¿Por qué perdimos la guerra?}). Available at www.brightcove.tv\Slash{}title.jsp?title=236034466\&channel=219646953.
Iñiguez, M. (2001) García Fernández, Baudillo Sinesio. In \emph{Esbozo de una enciclopedia histórica del anarquismo español.} Madrid: Fundación de Estudios Libertarios Anselmo Lorenzo, pp. 252–3.
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Andrew H. Lee
Abad de Santillán, Diego (1897–1983)
2009
\bigskip
Retrieved on 21\textsuperscript{st} November 2021 from \href{https://onlinelibrary.wiley.com/doi/10.1002/9781405198073.wbierp1656}{onlinelibrary.wiley.com}
Published in \emph{The International Encyclopedia of Revolution and Protest.}
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\hline
%\textbf{Studiengang} & 3BaIP\\ \hline
\textbf{Modulbezeichnung}
(eng.) % Algorithms and Data Structures
& \textbf{Algorithmen und Datenstrukturen}
(Algorithms and Data Structures)
\\ \hline
\textbf{Semester} & 3\\ \hline
\textbf{ECTS-Punkte (Dauer)} & 5 (1 Semester)\\ \hline
\textbf{Art} & Pflichtfach\\ \hline
%\textbf{Sprache(n)} & !!!Modulsprache!!!\\ \hline
%\textbf{ECTS-Punkte} & 5\\ \hline
%\textbf{Studentische Arbeitsbelastung} & 60,90\\ \hline
\textbf{Studentische Arbeitsbelastung} & 60 h Kontaktzeit + 90 h Selbststudium\\ \hline
\textbf{Voraussetzungen (laut BPO)} & \\ \hline
\textbf{Empf.\ Voraussetzungen} & Java 1 oder Programmieren 1\\ \hline
\textbf{Verwendbarkeit} & BaIP\\ \hline
\textbf{Prüfungsform und -dauer} & Klausur 1,5 h oder mündliche Prüfung\\ \hline
\textbf{Lehr- und Lernmethoden} & Vorlesung, Praktikum\\ \hline
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\textbf{Qualifikationsziele}\newline
Die Studierenden kennen häufig verwendete Algorithmen mit ihren dazu
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die Wiederverwendbarkeit der Implementierungen für unterschiedliche
Grunddatentypen gelegt.
}
\\ \hline
\multicolumn{2}{|p{0.97\textwidth}|}{
\textbf{Literatur}\newline
Heun, V.: Grundlegende Algorithmen, Vieweg, 2000.
Sedgewick, R.: Algorithmen in Java, 3. überarbeitete Auflage,
Pearson Studium, 2003.
}
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N. Streekmann & Algorithmen und Datenstrukturen & 2\\ \hline
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\begin{center}Math 2173 Spring 2021 Recitation Handout 2\end{center}
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Group Member 1: \underline{\hspace{140mm}}
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Group Member 2: \underline{\hspace{140mm}}
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Group Member 3: \underline{\hspace{140mm}}
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Group Member 4: \underline{\hspace{140mm}}
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Below is a checklist of instructions to follow when completing this assignment. Failure to follow these directions will result in penalty on your final score and/or in some problems not being graded. If multiple directions are not followed, then it is also possible that the assignment will not be accepted for any credit at all. Please contact your TA or make a post on the discussion boards for this course if you have any questions about this assignment or these directions.
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Sohail Farhangi: [email protected], \quad Pan Yan: [email protected], \quad Yilong Zhang: [email protected]
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& This assignment must be completed by a group of 3, 4, 5, or 6 members. \\
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& This assignment is to be uploaded to gradescope as a pdf file no later than 11:59 PM EST on Sunday, January 24.\\
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& The assignment will be uploaded by 1 group member, and that group member will be responsible for manually entering the names of all other collaborators into gradescope.\\
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& This assignment must be completed using this template. You may either print this template to write on it and then scan it (pages ordered correctly) into a pdf file, or you may write directly on the template using programs such as notability.\\
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& If you need more space than what is given to solve a given problem, then you will find blank pages provided at the end of this template. At the end of each problem section of this assignment you will find a space in which to indicate on what page your work is continued in case you used additional pages to complete your solution. You must provide the page number on which your work is continued in the alloted space, or write 'N/A' incase you did not use any additional pages.\\
\hline
& On the additional pages, you will also find space in which to indicate which problem the page is being used for, and if the page is used then that space must also be filled.\\
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& To complete this handout, you may use your textbook, class notes, discussions with your TA and group members, and any resources that are available on Carmen. You should not receive any help from the MSLC or people outside of your group when solving these problems. You may discuss these problems on the Carmen discussion boards, but you should not provide your entire solution when answering a such question, you should only give a hint or a helpful idea.\\
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\huge
{\bf \textcolor{red}{(Ungraded Optional Problem)} Example 13.8.8:} Find the point(s) on the plane $x+2y+z = 2$ closest to the point $P(2,0,4)$.
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{\bf Problem 13.8.55 (10 points):} Find the point on the plane $x+y+z = 4$ nearest the point $P(0,3,6)$. Remember to justify why your answer is a global minimum and not just a local minimum.
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{\bf Problem 13.8.41 (7 points):} Show that the second derivative test is inconclusive when applied to the function $f(x,y) = x^4y^2$ at the point $(0,0)$. Show that $f(x,y)$ has a local minimum at $(0,0)$ by direct analysis.
{\textit Hint: The product of 2 negative numbers is positive.}
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Here, we generalize the martix Riccati differential equation to the coupled matrix Riccati
differential equation. Using Schauder�??s fixed point theorem, the existence of at least one
periodic solution of the coupled matrix Riccati equation with $n \times n$ matrix coefficients is proved.
Finally, two numerical examples are presented.
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https://palm.muk.uni-hannover.de/trac/log/palm/trunk/DOC/misc/prandtl_layer.tex?format=changelog&rev=1771&limit=100&mode=stop_on_copy
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#
# ChangeLog for palm/trunk/DOC/misc/prandtl_layer.tex
#
# Generated by Trac 1.0.5
# Jun 30, 2022 11:15:15 AM
Wed, 06 Aug 2008 16:25:09 GMT letzel [187]
* palm/trunk/DOC/misc/plant_canopy_model.pdf (modified)
* palm/trunk/DOC/misc/plant_canopy_model.tex (added)
* palm/trunk/DOC/misc/prandtl_layer.pdf (added)
* palm/trunk/DOC/misc/prandtl_layer.tex (added)
* palm/trunk/SOURCE/CURRENT_MODIFICATIONS (modified)
* palm/trunk/SOURCE/prandtl_fluxes.f90 (modified)
* palm/trunk/SOURCE/production_e.f90 (modified)
* palm/trunk/SOURCE/wall_fluxes.f90 (modified)
- new: descriptions of plant canopy model and prandtl layer ...
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https://www.apmep.fr/IMG/tex/ES_Antilles_sept_2015.tex
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apmep.fr
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CC-MAIN-2021-49
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\documentclass[10pt]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{fourier}
\usepackage[scaled=0.875]{helvet}
\renewcommand{\ttdefault}{lmtt}
\usepackage{amsmath,amssymb,makeidx}
\usepackage[normalem]{ulem}
\usepackage{fancybox}
\usepackage{tabularx}
\usepackage{ulem}
\usepackage{textcomp}
\newcommand{\euro}{\eurologo{}}
\usepackage{pstricks,pst-plot,pst-tree}
% Tapuscrit Denis Vergès
% Sujet Académie de la Guyane
\usepackage[left=3.5cm, right=3.5cm, top=3cm, bottom=3cm]{geometry}
\newcommand{\vect}[1]{\overrightarrow{\,\mathstrut#1\,}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
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\renewcommand{\labelenumi}{\textbf{\theenumi.}}
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\def\Oij{$\left(\text{O},~\vect{\imath},~\vect{\jmath}\right)$}
\def\Oijk{$\left(\text{O},~\vect{\imath},~\vect{\jmath},~\vect{k}\right)$}
\def\Ouv{$\left(\text{O},~\vect{u},~\vect{v}\right)$}
\setlength{\voffset}{-1,5cm}
\usepackage{fancyhdr}
\usepackage[dvips]{hyperref}
\hypersetup{%
pdfauthor = {APMEP},
pdfsubject = {Terminale ES},
pdftitle = {Antilles-Guyane 11 septembre 2015},
allbordercolors = white,
pdfstartview=FitH}
\usepackage[frenchb]{babel}
\usepackage[np]{numprint}
\begin{document}
\setlength\parindent{0mm}
\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small Baccalauréat ES}
\lfoot{\small{Antilles--Guyane}}
\rfoot{\small septembre 2015}
\pagestyle{fancy}
\thispagestyle{empty}
\begin{center}\textbf{Durée : 3 heures}
\vspace{0,5cm}
{\Large \textbf{\decofourleft~Baccalauréat ES Antilles--Guyane septembre 2015~\decofourright}}
\end{center}
\vspace{0,5cm}
\textbf{\textsc{Exercice 1}\hfill 5 points}
\textbf{Commun à tous les candidats}
\medskip
\emph{Pour chacune des questions suivantes, une seule des quatre réponses proposées est exacte. Aucune justification n'est demandée. Une bonne réponse rapporte un point. Une mauvaise réponse, plusieurs réponses ou l'absence de réponse ne rapportent, ni n'enlèvent aucun point.}
\medskip
\textbf{Indiquer sur la copie le numéro de la question et la réponse choisie.}
\medskip
\begin{enumerate}
\item Soit la fonction $f$ définie sur [1~;~100] par $f(x) = 200 \ln x + 10x$,\: $f’(x)$ désigne la fonction dérivée de $f$. On a :\index{dérivée}
\medskip
\begin{tabularx}{\linewidth}{*{4}{X}}
\textbf{a.~~} $f'(x) = 200 + \dfrac{1}{x}$&\textbf{b.~~} $f'(x) = \dfrac{200}{x} + 10$&\textbf{c.~~} $f'(x) = 200 + 10x$&\textbf{d.~~}$f'(x) = \dfrac{200}{x} + 10x$ \\
\end{tabularx}
\item On note $L$ une primitive sur $]0~;~+\infty[$ de la fonction ln. Cette fonction $L$ est :
\textbf{a.~~} croissante puis décroissante
\textbf{b.~~} décroissante sur $[0~;~+\infty[$
\textbf{c.~~} croissante sur $[0~;~+\infty[$
\textbf{d.~~} décroissante puis croissante
\item La fonction $g$ définie sur $]0~;~+\infty[$ par $g(x) = x - \ln x$ est : \index{fonction convexe}
\textbf{a.~~} convexe sur $]0~;~+\infty[$
\textbf{b.~~} concave sur $]0~;~+\infty[$
\textbf{c.~~} ni convexe ni concave sur $]0~;~+\infty[$
\textbf{d.~~} change de convexité sur $]0~;~+\infty[$
\item On a représenté ci-dessous la courbe représentative d'une fonction $h$ définie et dérivable sur $[0~;~+\infty[$ ainsi que sa tangente au point A d'abscisse 2. Par lecture graphique, on peut conjecturer que : \index{lecture graphique}
\medskip
\parbox{0.35\linewidth}{
\textbf{a.~~} $h'(2) = 2$
\textbf{b.~~} $h'(2) = \dfrac{1}{2}$
\textbf{c.~~} $h'(2) = 0$
\textbf{d.~~} $h'(2) = 1$}\hfill
\parbox{0.6\linewidth}{\psset{unit=1cm}
\begin{pspicture*}(-1.5,-2.15)(6.1,3.2)
\psgrid[gridlabels=0pt,subgriddiv=1,griddots=8]
\psaxes[linewidth=1pt](0,0)(-1.5,-2.5)(6,3.2)
\psaxes[linewidth=1pt](0,0)(6,3.2)
\psaxes[linewidth=1.5pt]{->}(0,0)(1,1)
\psplot[plotpoints=3000,linewidth=1.25pt,linecolor=blue]{0.1}{6}{x ln 1 add 2 ln sub}
\psplot{-1}{6}{0.5 x mul }
\uput[ul](2,1){A}
\end{pspicture*}
}
\item La variable aléatoire $X$ suit une loi normale d'espérance $\mu = 0$ et d'écart type $\sigma$ inconnu mais on sait que $P( -10 < X < 10) = 0,8$. On peut en déduire: \index{loi normale}
\textbf{a.~~} $P(X < 10) = 0,1$
\textbf{b.~~} $P(X < 10) = 0,2$
\textbf{c.~~} $P(X < 10) = 0,5$
\textbf{d.~~} $P(X < 10) = 0,9 $
\end{enumerate}
\vspace{0,5cm}
\textbf{\textsc{Exercice 2}\hfill 5 points}
\textbf{Candidats ES n'ayant pas suivi l'enseignement de spécialité et candidats L }
\medskip
Un supermarché dispose d'un stock de pommes. On sait que 40\,\% des pommes proviennent d'un fournisseur A et le reste d'un fournisseur B.
Il a été constaté que 85\,\% des pommes provenant du fournisseur A sont commercialisables. La proportion de pommes commercialisables est de 95\,\% pour le fournisseur B.
Le responsable des achats prend au hasard une pomme dans le stock. On considère les évènements suivants :
$A$ : \og La pomme provient du fournisseur A \fg.
$B$ : \og La pomme provient du fournisseur B \fg.
$C$ : \og La pomme est commercialisable \fg.\index{probabilités}
\bigskip
\textbf{PARTIE A}
\medskip
\begin{enumerate}
\item Construire un arbre pondéré traduisant cette situation. \index{arbre}
\item Montrer que la probabilité que la pomme ne soit pas commercialisable est 0,09.
\item La pomme choisie est non commercialisable. Le responsable des achats estime qu'il y a deux fois plus de chance qu'elle provienne du fournisseur A que du fournisseur B. A-t-il raison ?
\end{enumerate}
Pour les parties B et C, on admet que la proportion de pommes non commercialisables est $0,09$ et, quand nécessaire, on arrondira les résultats au millième.
\bigskip
\textbf{PARTIE B}
\medskip
On prend au hasard 15 pommes dans le stock. Le stock est suffisamment important pour qu'on puisse assimiler ce prélèvement à un tirage aléatoire avec remise. \index{loi binomiale}
\medskip
\begin{enumerate}
\item Quelle est la probabilité que les 15 pommes soient toutes commercialisables ?
\item Quelle est la probabilité qu'au moins 14 pommes soient commercialisables ?
\end{enumerate}
\bigskip
\textbf{PARTIE C}
\medskip
Le responsable des achats prélève dans le stock un échantillon de 200 pommes. Il s'aperçoit que 22 pommes sont non commercialisables.
Est-ce conforme à ce qu'il pouvait attendre?
\vspace{0,5cm}
\textbf{\textsc{Exercice 2}\hfill 5 points}
\textbf{Candidats ayant suivi l'enseignement de spécialité}
\medskip
\parbox{0.5\linewidth}{Un cycliste désire visiter plusieurs villages notés A, B, C, D, E, F et G reliés entre eux par un réseau de pistes cyclables.
Le graphe ci-contre schématise son plan ; les arêtes représentent les pistes cyclables et les distances sont en kilomètre. }
\hfill
\parbox{0.47\linewidth}{\psset{unit=0.8cm}
\begin{pspicture}(-1,0)(8,6)
%\psgrid
\psdots(0.2,1.5)(0.2,5.5)(2,5.5)(7,5.5)(2,1.5)(7,1.5)(5.5,0.2)%FABCGDE
\uput[ul](0.2,5.5){A}\uput[u](2,5.5){B}\uput[u](7,5.5){C}
\uput[r](7,1.5){D}\uput[u](5.5,0.2){E}\uput[dl](0.2,1.5){F}
\uput[dl](2,1.5){G}
\pspolygon(0.2,1.5)(0.2,5.5)(7,5.5)(7,1.5)(5.5,0.2)(2,1.5)(0.2,1.5)%FACDEGF
\pspolygon(2,1.5)(2,5.5)(7,1.5)%GBD
\uput[u](1.1,5.5){15} \uput[u](4.5,5.5){21} \uput[r](7,3.5){20}
\uput[u](4.5,1.5){17} \uput[u](1.1,1.5){20} \uput[l](0.2,3.5){30}
\uput[ur](4.5,3.5){25} \uput[dl](3.75,0.85){15} \uput[dr](6.25,0.85){10}
\uput[r](2,3.5){10}
\end{pspicture}}
\medskip
\textbf{Partie A}
Pour faire son parcours, le cycliste décide qu'il procèdera selon l'algorithme ci-dessous :\index{algorithme}
\begin{center}
\begin{tabularx}{.85\linewidth}{|m{1cm}|X|}\hline
ligne 1&Marquer sur le plan tous les villages comme non \og visités\fg \\ \hline
ligne 2& Choisir un village de départ \\ \hline
ligne 3&Visiter le village et le marquer \og visité\fg\\ \hline
ligne 4&Rouler vers le village le plus proche\\ \hline
ligne 5& Tant que le village où il arrive n'est pas un village déjà visité \\ \hline
ligne 6&\hspace{0.2cm}|visiter le village et le marquer \og visité\fg\\
ligne 7 &\hspace{0.2cm}|rouler vers le village le plus proche sans revenir en arrière\\ \hline
ligne 8& Fin Tant que\\ \hline
ligne 9&afficher la liste des villages visités \\ \hline
\end{tabularx}
\end{center}
\smallskip
\begin{enumerate}
\item Quelle propriété du graphe permet à la ligne 4 d'être toujours exécutable ? \index{graphe}
\item En partant du village noté G, quelle sera la liste des villages visités ?
\item Existe-t-il un village de départ qui permette, en suivant cet algorithme, de visiter tous les villages ?
\item Le cycliste abandonne l'idée de suivre l'algorithme. Il souhaite maintenant, partant d'un village, y revenir après avoir emprunté toutes les pistes cyclables une et une seule fois. Cela sera-t-il possible ? \index{graphe complet}
\end{enumerate}
\bigskip
\textbf{Partie B}
\medskip
\begin{enumerate}
\item Écrire la matrice $M$ de transition de ce graphe (dans l'ordre $A, B, C, \ldots ,G$).
\item On donne la matrice $M^4$ :\index{matrice}
\[M^4 =\bordermatrix{
&A &B &C &D &E &F &G\cr
A&10 &5 &9 &11 &4 &1 &16\cr
B&5 &30 &12 &23 &18 &16 &16\cr
C&9 &12 &12 &14 &9 &4 &18\cr
D&11 & 23 &14 &28 &14 &11 &23\cr
E&4 &18 &9 &14 &12 &9 &12\cr
F&1 &16 &4 &11 &9 &10 &5\cr
G&16 &16 &18 &23 &12 &5 &30\cr
}\]
Interpréter le terme en gras, ligne A, colonne F (valant 1 ) dans le contexte de l'exercice.
\end{enumerate}
\vspace{0,5cm}
\textbf{\textsc{Exercice 3}\hfill 4 points}
\textbf{Commun à tous les candidats}
\medskip
Un couple fait un placement au taux annuel de 2\,\% dont les intérêts sont capitalisés tous les ans. Son objectif est de constituer un capital de \np{18000}~euros.
Le couple a placé le montant de \np{1000}~euros à l'ouverture le 1\up{er} janvier 2010 puis, tous les ans à chaque 1\up{er} janvier, verse \np{2400}~euros.\index{suite}
\medskip
\begin{enumerate}
\item Déterminer le capital présent sur le compte le 1\up{er} janvier 2011 après le versement annuel.
\item On veut déterminer la somme présente sur le compte après un certain nombre d'années.
On donne ci-dessous trois algorithmes :
%\hspace*{-1.2cm}
{\small \begin{tabularx}{\linewidth}{|X|m{0.1mm}|X|m{0.1mm}|X|}\cline{1-1}\cline{3-3}\cline{5-5}
\textbf{Variables :}&& \textbf{Variables :} &&\textbf{Variables :} \\
$U$ est un nombre réel&&$U$ est un nombre réel&&$U$ est un nombre réel\\
$i$ et $N$ sont des nombres entiers&&$i$ et $N$ sont des nombres entiers&&$i$ et $N$ sont des nombres entiers\\
\textbf{Entrée}&&\textbf{Entrée}&&\textbf{Entrée}\\
Saisir une valeur pour $N$&&Saisir une valeur pour $N$&&Saisir une valeur pour $N$\\
\textbf{Début traitement}&&\textbf{Début traitement}&&\textbf{Début traitement}\\
Affecter \np{1000} à $U$&&Pour $i$ de 1 à $N$ faire&&Affecter \np{1000} à $U$\\
Pour $i$ de 1 à $N$ faire&&\hspace{0.2cm}\begin{tabular}{|l}Affecter \np{1000} à $U$\\
Affecter $1,02\times U + \np{2400}$ à $U$
\end{tabular}&&Pour $i$ de 1 à $N$ faire\\
| Affecter $1,02 \times U + \np{2400}$ à $U$&&&&\begin{tabular}{|l}
Affecter $1,02 \times U + \np{2400}$ à $U$\\
Affecter $N+1$ à $N$\end{tabular}\\
Fin Pour&&Fin Pour&&Fin Pour\\
Afficher $U$&&Afficher $U$&&Afficher $U$ \\
\textbf{Fin traitement}&&\textbf{Fin traitement}&&\textbf{Fin traitement}\\ \cline{1-1}\cline{3-3}\cline{5-5}
\end{tabularx}}
\begin{tabularx}{1.15\linewidth}{Xm{0.4mm}Xm{0.4mm}X}
\textbf{algorithme 1} &&\textbf{algorithme 2}&&\textbf{algorithme 3}\\
\end{tabularx}
\medskip
\begin{enumerate}
\item Pour la valeur 5 de $N$ saisie dans l'algorithme 1, recopier puis compléter, en le prolongeant avec autant de colonnes que nécessaire, le tableau ci-dessous (arrondir les valeurs calculées au centième).
\begin{center}
\begin{tabular}{|l|c|c|p{2cm}}\hline
valeur de $i$& xxx& 1& \ldots \\ \hline
valeur de $U$ & \np{1000}&&\ldots \\ \hline
\end{tabular}
\end{center}
\item Pour la valeur 5 de $N$ saisie, quel affichage obtient-on en sortie de cet algorithme ?
Comment s'interprète cet affichage ?
\item En quoi les algorithmes 2 et 3 ne fournissent pas la réponse attendue?
\end{enumerate}
\item À partir de la naissance de son premier enfant en 2016, le couple décide de ne pas effectuer le versement du premier janvier 2017 et de cesser les versements annuels tout en laissant le capital sur ce compte rémunéré à 2\,\%.
Au premier janvier de quelle année l'objectif de \np{18000}~euros est-il atteint ?
\end{enumerate}
\vspace{0,5cm}
\textbf{\textsc{Exercice 4}\hfill 6 points}
\textbf{Commun à tous les candidats}
\medskip
L'évolution de la population d'une station balnéaire pour l'été 2015 a été modélisée par une fonction $f$, définie sur l'intervalle [0~;~70], dont la courbe représentative est donnée ci-dessous.
\medskip
\parbox{0.4\linewidth}{
Lorsque $x$ est le nombre de jours écoulés après le 1\up{er} juillet, $f(x)$ désigne la population en milliers d'habitants.
Ainsi $x = 30$ correspond au 31 juillet et $f(30)$ représente la population qu'il est prévu d'accueillir le 31 juillet.
On estime qu'un habitant utilisera chaque jour entre 45 et 55~litres d'eau par jour.}\hfill
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\begin{pspicture}(-4,-1)(75,11)
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\psaxes[linewidth=1.25pt,Dx=10,Dy=2]{->}(0,0)(0,0)(75,11)
\psaxes[linewidth=1.25pt,Dx=10,Dy=2](0,0)(75,11)
\psplot[plotpoints=3000,linewidth=1.25pt,linecolor=blue]{0}{70}{0.2 x mul 2.71828 0.025 x mul 1 sub exp div 2 add}
\uput[u](60,0){nombre de jours}
\uput[r](0,10.5){milliers d’habitants}
\end{pspicture}
}
\medskip
\textbf{Partie A } \emph{Dans cette partie, les réponses sont à fournir par lecture graphique}\index{lecture graphique}
\medskip
\begin{enumerate}
\item
\begin{enumerate}
\item Estimer le nombre maximal d'habitants présents dans la station balnéaire selon ce modèle durant l'été 2015 et préciser à quelle date ce maximum serait atteint.
\item La commune est en capacité de fournir \np{600000}~litres d'eau par jour, est-ce suffisant ?
\end{enumerate}
\item Estimer le nombre de jours durant lesquels le nombre d'habitants de la station balnéaire devrait rester supérieur à 80\,\% du nombre maximal prévu.
\end{enumerate}
\bigskip
\textbf{Partie B}
\medskip
On admet que la fonction $f$ est définie sur l'intervalle [0~;~70] par
\[f(x) = 2 + 0,2x\text{e}^{-0,025x+1}.\]
\smallskip
\begin{enumerate}
\item Calculer $f(9)$ puis vérifier que la consommation d'eau le 10 juillet serait, selon ce modèle, au plus de \np{324890}~litres.
\item
\begin{enumerate}
\item Démontrer que $f'(x) = (0,2 - 0,005 x)\text{e}^{-0,025x+1}$ où $f'$ est la fonction dérivée de $f$.\index{dérivée}
\item Étudier le signe de $f'(x)$ sur l'intervalle [0~;~70].
\item En déduire la date de la consommation d'eau maximale.
\end{enumerate}
\end{enumerate}
\bigskip
\textbf{Partie C}
\medskip
On note $g$ la fonction définie sur l'intervalle [0~;~70] par
\[g(x) = 55 f(x) = 110 + 11x\text{e}^{-0,025x+1}.\]
Lorsque $x$ est le nombre de jours écoulés après le 1\up{er} juillet, $g(x)$ représente alors la consommation maximale d'eau prévue ce jour là et exprimée en m$^3$.
Soit la fonction $G$ définie sur l'intervalle [0~;~70] par
\[G(x) = 110x - (440x + \np{17600})\text{e}^{-0,025x+1}.\]
On admet que la fonction $G$ est une primitive de la fonction $g$.
La somme $S = g(10) + g(11) + g(12) + \cdots + g(20)$ représente la consommation maximale d'eau du 10\up{e} au 20\up{e} jour exprimée en m$^3$.
\medskip
\begin{enumerate}
\item En l'illustrant sur la courbe $\mathcal{C}_g$ de l’\textbf{annexe} à rendre avec la copie, donner une interprétation graphique en termes d'aires de la somme $S$.\index{aire et intégrale}
\item En déduire une valeur approximative de cette quantité d'eau consommée du 10\up{e} au 20\up{e} jour.
\end{enumerate}
\newpage
\begin{center}
\textbf{ANNEXE}
\bigskip
\textbf{Annexe à l'exercice 4 à rendre avec la copie}
\bigskip
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\psaxes[linewidth=1.25pt,Dy=50]{->}(0,0)(-0.5,-12)(24,550)
\psaxes[linewidth=1.25pt,Dy=50](0,0)(24,550)
\uput[r](0,525){consommation $\left(\text{m}^3\right)$}
\uput[u](21.5,0){nombre de jours}
\psplot[plotpoints=3000,linewidth=1.25pt,linecolor=blue]{0}{24}{0.2 x mul 2.71828 0.025 x mul 1 sub exp div 2 add 55 mul}
\uput[u](8,305){\blue $\mathcal{C}_g$}
\end{pspicture}
\end{center}
\end{document}
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https://git.rockbox.org/cgit/rockbox.git/plain/manual/platform/ipodvideo.tex?id=f45982abcd8e1b44971ca0359128443ddb4693ac
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rockbox.org
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| 284,807,750 | 1,517 |
% $Id$ %
\def\UseOption{ipodvideo}
\edef\UseOption{\UseOption,HAVE_RB_BL_ON_DISK}
\edef\UseOption{\UseOption,HAVE_LCD_BITMAP}
\edef\UseOption{\UseOption,CONFIG_RTC}
\edef\UseOption{\UseOption,SWCODEC}
\edef\UseOption{\UseOption,IPOD_4G_PAD}
\edef\UseOption{\UseOption,HAVE_LCD_COLOR}
\edef\UseOption{\UseOption,ipod}
\newcommand{\playerman}{Ipod}
\newcommand{\playertype}{Video}
\newcommand{\playerlongtype}{\playertype}
\newcommand{\genericimg}{320x240x16}
\newcommand{\specimg}{ipodvideo}
%Used to name the player, e.g. ...to the \dap
\newcommand{\dap}{player}
%For use when referring to the player. E.g. the \daps\ capacity ...
\newcommand{\daps}{player's}
\newcommand{\firmwarefilename}{\fname{rockbox.ipod}}
\newcommand{\screenshotsize}{8cm}
\newcommand{\ButtonLeft}{\btnfnt{Prev}}
\newcommand{\ButtonRight}{\btnfnt{Next}}
\newcommand{\ButtonMenu}{\btnfnt{Menu}}
\newcommand{\ButtonPlay}{\btnfnt{Play}}
\newcommand{\ButtonSelect}{\btnfnt{Select}}
\newcommand{\ButtonScrollFwd}{\btnfnt{Scroll Fwd}}
\newcommand{\ButtonScrollBack}{\btnfnt{Scroll Back}}
%Button actions, standard context
\newcommand{\ActionStdPrev}{\ButtonScrollBack}
\newcommand{\ActionStdPrevRepeat}{Long \ButtonScrollBack}
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\newcommand{\ActionPsExit}{\ButtonSelect}
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https://unilab.gbb60166.jp/prekou/tex/m1-parabo-jiku.tex
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gbb60166.jp
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CC-MAIN-2020-40
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application/x-tex
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text/x-matlab
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crawl-data/CC-MAIN-2020-40/segments/1600402101163.62/warc/CC-MAIN-20200930013009-20200930043009-00182.warc.gz
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%
% [email protected] https://unilab.gbb60166.jp/prekou/prekou.htm
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% aspectratio= は 1610, 169, 149, 54, 32 の中から選べる（省略時は 43）
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%\def\pgfsysdriver{pgfsys-dvips.def}%デフォルト
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\usepackage{tikz}%（これで、pgfとpgfforが読み込まれます。）
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% フォントはお好みで
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% ■ 以前は{\bf }とかしてましたが \seriesdefault で一気に
% 変更出来ることがわかりました。2017/3/3
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% ↑ 数式モードで式の前後に入るスペースを制御したかったが、
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% カラーユニバーサルデザインを調べたつもりだがあまり自信がありません
% http://www.fukushihoken.metro.tokyo.jp/kiban/machizukuri/kanren/color.files/colorudguideline.pdf
% http://jfly.iam.u-tokyo.ac.jp/colorset/
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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軸は\( x=\textcolor{CUDCyan}{\alt<3->{-2}{\hooklen{-2}{\text{★}}}} \)だ！}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{\small 軸\( x\!=\!\textcolor{CUDCyan}{-2} \)で
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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【\ \((\textcolor{CUDRed}{-1},\textcolor{CUDBlue}{-3})\)を通る\ 】とは\\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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【\ \((\textcolor{CUDRed}{-4},\textcolor{CUDBlue}{0})\)を通る\ 】とは\\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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https://theanarchistlibrary.org/library/adeline-champney-what-is-worth-while.tex
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theanarchistlibrary.org
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\title{What Is Worth While?}
\date{1911}
\author{Adeline Champney}
\subtitle{A Study of Conduct, from the Viewpoint of the Man Awake}
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\par
When we were little we were taught to mind. It used to be the fashion to teach children to mind. Obedience was the \emph{sine qua non} of childhood. A child with a will of its own was marked for special discipline at the hands — often, literally at the hands — of the alarmed parent. A will of its own was a dangerous possession and must be broken at all costs. So the little will was broken; the costs were too often handed down, even unto the third and fourth generation.
On the whole we learned to mind; learned it so well that most of us have minded ever since, becoming devout Christians and exemplary citizens; following the beaten path, thinking the time worn thoughts, moulding our lives after the antique pattern esteemed by our ancestors. To be “good” was to do as we were told — “ours not to make reply, ours not to reason why” — ours to conform to the adult life around us, and to cause as little inconvenience as possible. This was the ideal of juvenile “goodness,” and to be “good” was the most important thing in life. If it did not so appear to our childish minds, it was made so, very much so. Not only were we inflicted with punishments and enticed with rewards, but to offset the human tendency to concealment which naturally followed such treatment, we were assured that God was watching us, and that not merely every act but indeed every thought was “under the law” and subject to the everlasting wrath of the Almighty, “who slumbers not nor sleeps.” With the sacred ten commandments, the laws of the land, personified by the brass-buttoned policeman, and the arbitrary say-so of parents and teachers and other adults too numerous to mention, our little lives were bounded on the north, south, east and west by Authority, and in the sky above lowered the Awful Presence.
This it was to be a child. I am afraid it has not altogether changed to-day. The home, intrenched in its ancient fastnesses, is slow to feel the influences of progressive tendencies. Fortunately, persons feel and respond to these tendencies before their institutions, individuals in advance of groups. Fortunately, too, we are not all “good” children, or we should all remain on our knees at the feet of Authority, murmuring with submissive lip, “Thy will, not mine, be done.”
As the child grows, he gradually becomes aware of certain principles to which all are expected to conform. If he has been “well trained,” by the time he enters upon his teens he has the habit of obedience, fixed as a trait of character. The persistent “Why?” of his normal mental activity has been silenced. He has become beautifully “teachable” and very satisfactorily tractable. The period of youth is one of the inculcation of principles, social ideals, which have come to be held inviolable, and by which the future conduct of his life is to be gauged when he shall assume direction of his own affairs. Life now grows more complex. Obedience was simple; so very simple, so very easy, that many prefer to abrogate all private judgment, to avoid all perplexities, and to remain always good and obedient children. Hence religion survives — religion, which fosters irresponsibility and automatic morality.
These social ideals — remember I am setting aside peculiarities of time and place, and dealing with averages, the great civilized human averages — these social ideals may be broadly stated as: Honesty, Respectability, Prosperity. On these hang all the essentials of conduct. Failing in these, the individual becomes, more or less according to the measure of his deviation, an undesirable.
These standards of conduct, accepted by religious and irreligious alike, are presented to the youth as things sacred in themselves, not to be questioned. One who should ask: “Why should I be honest?” would be suspected of moral degeneracy. It is true they tell us that honesty is the best policy, but that is given us rather as an assuaging circumstance than as a motive. \emph{Of course} one must be honest. One must be honest for honesty’s own sake. Money-honest, that is. In a society where Science and Religion walk hand-in-hand one will hardly look for scrupulousness as to intellectual honesty; nor will one expect to find insistence on emotional and social honesty in a society which worships Respectability. For the greatest of these is Respectability, and respectable one must be though the heavens fall.
Close upon Respectability follows Prosperity. He who fails to get on in the world arouses suspicion, but he who prospers glows with justification. However, the element of opportunity being recognized as a factor in business success, and moreover the good Lord having peculiar ways of chastening his children, some measure of social forgiveness may be meted out to him of small means, but the pillars of the Church and the bulwarks of society are honest, well-thought-of, and well-to-do.
The worship of this blessed trinity is called Duty. By the unpremeditated and involuntary act of being born we are supposed to have incurred a three-fold obligation: our duty to God, our duty to man, and our duty to ourselves — named in the order of their importance. Preacher, teacher, poet and sage alike speak to us of Duty. The world’s literature is full of beautiful tributes to Duty, and stirring exhortations of Conscience — a spiritual faculty the function of which is to admonish us of Duty. Conscience is the voice of God in the soul, say the religious. The nonreligious who have dethroned God and set Right in his place will tell us that conscience is man’s innate sense of right and wrong — a newer edition, revised, of the God-explanation. Of course that settles it, settles it about as well, or ill, as the God-explanation usually settles problems. It is not essential that such an explanation be logical, that it be scientific, that it be consistent; it is not essential even that it should explain. So long as repeating it gives one that superior sanctified air, it will stand through the ages, to be fought for and lived for and died for.
As is the history of the individual, so has been the history of the human race. Human knowledge passes through three stages of development: the Supernatural, the Metaphysical, the Scientific, and the science of human conduct follows it. We find primitive man ruled by fear; worshipping power and mystery; easily coming under the authority of a priesthood which claims to interpret for him the unknown. This is the childish age of Bugaboos and Authority, which is succeeded by the Metaphysical period; the worship of entities, ideals, principles; things to be valued in and for themselves. To this age belongs the reign of Conscience, which especially characterizes our own day. And as our knowledge and understanding of the material universe passes from the realms of mystery into the region of exact knowledge, so must the conduct of life take on the scientific method, and, leaving the darkness of tradition and the fogs of metaphysics, become truly rationalized. As yet it lingers on the borderline between the Supernatural and the Metaphysical. The Scientific Era has not dawned.
In the life of each man and woman sooner or later there comes an awakening. I am inclined to think it comes to all, but very many go to sleep again. The stupor of years of acquiescence, the apathy bred of the habit of conformity, overcomes them. And there are many who count the cost and shut their eyes again. It takes a certain sturdy strength to cross the current, to steer for unknown seas.
But some there are who do not shrink when they come face to face with life, and unto these comes experience and knowledge and insight; and through these comes all the progress of the world. Awakened by some crisis, public or private; or cramped into wakefulness by the pressure of antique traditions or institutions; shocked awake, it may be, by contradictions between scientific and conventional standards; or perhaps stirred by some echo from the unanswered “Why?” of their childhood; they boldly challenge the world. “Why are you here?” they demand of every institution. “What have you to offer me?” they ask of Life itself. And to such there is no rest and no peace until they are answered. The Man Awake recognizes nothing which he may not analyze, nothing which he may not weigh in the balance. Though one by one his cherished idols fall and crumble, he must apply the tests of truth.
With the downfall of the God-idol I shall not here concern myself. It is the simplest, the easiest liberation. When one bears the torch of Reason and uses the compass of Science, all roads lead to Freedom. Many have made this journey, but many have stopped here and lain down again and slept. I concern myself with the Man Awake who sees his liberation but begun; for the God-influence does not perish with the belief in God. God is dead, but worship survives, and it is not God but worship which stultifies man’s growth. The Supernatural passes into the Metaphysical — and the Man Awake still questions. The conduct of life, no longer a matter of the relation of man to occult powers, becomes a relation of man to exalted imaginings and deified principles. While our knowledge and use of our material environment is far advanced into the scientific stage of development, our understanding of and our attitude toward our social environment is still in the Metaphysical stage. We have a science of things, but not as yet a science of men. There are many cobwebs to be swept away before the conduct of life takes on the scientific form. Any ideal which becomes an object of worship, which in and for itself compels observance; any principle, obedience to which is forced upon men, either by violence, by legal enactment, or by the coercion of public opinion, becomes a fetish. The air is full of such. This is an age of mental and emotional fetishism. Chief among these and including most of them — all, indeed, which approach universality-stands Duty. From the cradle to the grave one is admonished of Duty. From the lips of parents and teachers, from preachers and judges and kings, from friend and foe alike, comes the magic word. Come joy or come sorrow, in life or unto death, one must follow Duty; and no man knows whence it comes nor why, and few \emph{can} follow it, but each man says to every other, “Do thy Duty.” Duty, not to be denied, not to be questioned, but potent to guide and to govern a world of men I Of this fetish, then, the Man Awake demands credentials. He has outgrown the theological traditions of his fathers, he has gained a new viewpoint whence everything must be judged anew. He sets about revising his standards. It may be months, it may be years, before he makes the full readjustment, but what matters it? He is free, and growing, and that is very nearly the whole of life — to be free and to grow.
When God vanishes from the skies he takes a great many things with him, some of which are not commonly recognized as pertaining to the God-idea. Not only does his departure into the limbo of past superstition remove the authority of bibles and churches and temples, and the divine authorities of priests and rulers, but it also removes all ultimate authorities whatever, and takes the sanctity from all principles of conduct. The departure of God places man face to face with the material universe, and men face to face with each other. With the abolition of the law-giver all laws disappear. The term “laws of nature” shows how our very language is so tinctured with the teleological conception that we have difficulty in choosing exact terms for our knowledge.
The so-called “laws of nature” are merely the undeviating principles in accordance with which the universe of substance in motion continues its unceasing and eternal change. Forms appear and disappear, phenomena come and go, but in all the universe is found neither beginning nor end, neither first nor last; neither good nor evil, right nor wrong, virtue nor sin, justice nor injustice. To none of these terms is there any absolute meaning whatever. All are man-made distinctions, varying with time and place, differing among races and among individuals. To the history of the human race, then, the Man Awake must go in his search for the meaning of Duty. For development proceeds ever from the simple to the complex, and the basis of sane thinking is found in the study of development. To gain an adequate comprehension of anything one must understand its development. And nothing will so aid in clearing away superstition and traditional prejudice in matters social and ethical as a survey of human history; not merely recorded history but that great story of the prehistoric man which science resurrects for us. What does this history say to us of Duty? Just this: bereft of all theological and metaphysicial sanctities all the human institutions which have demanded obedience from men are seen to rest ultimately on the power to impose themselves on individuals. Religion, government, all property privilege, the marriage institution — all originated in force, and are maintained by force. Back of every “duty” stands a club. Does one “owe” anything to compulsion? Can a “duty” be imposed on one, without one’s own consent? Brought into this world by no act of one’s own, does one inherit the obligations assumed by one’s ancestors, much less those forced upon them? The sole justification of every authority is its power to enforce obedience; and therein lies the justification of every rebellion. Whatever obedience may be exacted, whatever allegiance may be voluntarily rendered, there is no obligation whatsoever. Duty is but a metaphysical cobweb. It has no foundation in fact. “But conscience? Surely I cannot deny the admonition of conscience!” Have you studied the conscience of a savage? Have you made a comparative analysis of conscience among varying peoples and at various periods of history? Have you ever observed the conscience of a very little child? The dictates of conscience are purely and simply a matter of education. Conscience itself is neither more nor less than one’s satisfaction in himself. A clear conscience is the pleasurable sense of self-approval; guilty conscience, the disquieting sense of self-censure. This is the reality of conscience; the grounds for the satisfaction or dissatisfaction lie in our beliefs and principles, and are, largely, a product of our social heredity. They may be well or ill founded. One has only to review the many deeds that have been done “for conscience sake” to perceive how utterly unreliable it is as a “moral” guide. Of the fetish, Duty, with Conscience as its private watchman, investigation leaves not one shred. It follows the gods, the heavens and the hells, and all the spooks that infest intellectual darkness. Not so with conscience as a profound sense of self-judgment. That is an attribute of the mind which is of inestimable value. To the Man Awake it becomes a veritable court of last appeal. There is no greater honor to win than the approval of our own souls. There is no greater faith to keep than faith with ourselves.
There is an idea prevalent among the religious that if once the religious and moral restraints were removed, men would fly off at a tangent, fling open all the hitherto forbidden doors, and plunge into a carnival of crime. If they should do so, what would be to blame, their new-found freedom or their former training? Have all the ages of religion and morality produced no moral sense? The alarmists indict their own institutions! Occasionally one hears of preachers’ sons who “go wrong” — sometimes it is the preacher himself! Sometimes there are children who have been brought up in the sternest and strictest of homes, who, on coming of age, plunge into dissipation, perhaps ruining health and even life. But does any thinking person blame their coming of age? Is it not plain that their religious training has not given them moral stamina, or a rational view of life? That it has weakened their resistage by the constant suggestion of weakness and dependence, and given them only an arbitrary rule of conduct and not a vital purpose in life? Believing themselves “vile worms of the dust,” they act the part!
No. The Man Awake is not going off at a tangent. The conduct of life, now that he no longer gets it ready-made, has become of vastly greater importance to him. It is his own concern, now; he will ask himself as never before — “What is really worth while?” And the answer must be a personal one. Not that out of his inner consciousness he will dig up a set of rules and precepts unrelated to the thought and feeling of the world about him. Not every man is called to blaze a new trail. But he will make sure, when he takes the road, that it leads in his direction, and that he is not merely following in the footsteps of his grandsires. Nor is it needful that he travel alone. He may go hand-in-hand with a comrade, he may join himself to a company, he may even follow a leader; but the comrades must be of his own choosing, related in thought and purpose, and not mere accidents of the wayside; and he will see to it that he is driven by no compulsion save the impulse of his own nature.
Let it not be thought that I disparage ideals. It is not the Ideal but the deification of it that stultifies growth. The leaders of men are always idealists; all the periods of great moral and social uplift have been periods of idealism. If there be any exclusively human characteristic, essentially distinguishing the man from his fellow-animals, it is this power to frame ideal conceptions, to picture better things and to strive toward them. Many of the finest types of manhood which society has produced have been men of vision as well as of insight, ardent dreamers of dreams, with the daring to follow their dreams. These have been strong men, men of striking personality, of resolute self-determination, these idealists. When a man loses himself, when he becomes subservient to an ideal; when he no longer possesses it, but deifies it so that it takes possession of him, then he is no longer a man but a shadow; and his ideal, a spook.
Out of the past have come down to us many maxims and precepts, most of which are so permeated with theology or so befogged with metaphysics as to render them utterly worthless in a modern world. The Man Awake does not despise the Wisdom of the Ages, but there is also a Folly of the Ages, and he reserves the right to make his own selection. He accepts no maxims on say-so, even though the say-so be a repetition of twice ten thousand years. These shreds of old wisdom make an interesting study, revealing, as they do, the stuff of which human conduct has been woven, the woof of the fabric of social custom and usage. But to-day they are mostly rags, rags.
Among them there is one which seems to have an immortal life. It is found in many lands and many tongues, varying but slightly in form; and so general and unquestioned is its acceptance as an efficient guide to social conduct that even an iconoclast hesitates to lay violent hands on the Golden Rule. But we recognize no exemptions; nothing escapes the test. “Whatsoever ye would that men should do unto you, do ye even so to them” might be good sense in a world where all men were alike, possessed of identical needs, desires and tastes. If anyone thinks it applicable in a world of individualities, let him try it out in his daily living. If he attempts to apply it literally, he will speedily discover the arrogance of the assumption that other men are like himself, that what pleases him will be acceptable to them. If he endeavors to disregard the letter but carry out the spirit of it, he will soon be engulfed in the fathomless task of determining what others, actuated by the Golden Rule, would do unto him with a view of having him do so even unto them! And at the best it is not so practical as the familiar “Put yourself in his place.” Good suggestions, both of them, but as adequate rules of conduct, such as the Golden Rule is on every hand assumed to be — childish, utterly childish! In the negative form attributed to Confucius it becomes less fraught with danger and discord. “Do not unto others as you would not that they should do unto you.” Where others are involved, to refrain from action has this advantage: at the worst one becomes guilty of neglect, but never of aggression. But the moment one begins to “do things” unto others, he is on dangerous ground. The Golden Rule, lauded as a social panacea, makes a really pretty plaything for babies, but is more innocuous when written in Chinese!
Another idol must be shattered in the course of this inquiry, the ideal of self-sacrifice. Grim and grisly rise the phantoms of its antecedents: living animals torn asunder, human blood poured out, on the altars of the gods; self-tortures, flagellations, loathsome mortifications of the flesh in the cells and hovels of monks and saints — a gruesome crew! Life and love and treasure offered up to please and placate Deity; and the crowning sacrifice of Deity himself in the person of his son to satisfy his own wrath and save a sinning but well-beloved and eternally damned people! It is doubtless this sacrificial atonement of the ancient churches which has passed into the metaphysical concept of self-sacrifice as a laudable and beautiful thing, a holy and righteous thing, a kind of sublimated duty. Self-mutilations, mortifications of the flesh, are not all in the past. The religious frenzy of the old-time saint is rare, and we call it by its right name now. But in its more subtle form sacrifice unto sanctification is not uncommon among high-strung nervous temperaments. No one can estimate the injury to health, the distortions of mind and character, and that among the finer, more highly developed types of men and women, particularly women. No one can know the loss to society of strong sane womanhood and motherhood, from this sacrifice. Moreover, the strong give place to the weak, the efficient spend their strength in ministering to the inefficient, youth sterilizes itself in the service of age, the fit waste themselves to preserve the unfit, until, viewing the social misery of it, one could almost welcome the restraining hand of a stern but wholesome paganism.
For, mark you, for all this sacrifice the world is scarcely the kinder. Indeed, as Oscar Wilde so keenly says, “It takes a thoroughly selfish age like our own to deify self-sacrifice.” “Living for others,” we say, but deliver us from the arrogance, the insufferable despotism of many of those who insist on living for us. I have seen whole families tyrannized over, kept uncomfortable for years, even disrupted, by one member whose whole purpose in life was to “live for” that family. “Living for others,” we say, and we thrill with admiration; but when one really lives for others, what happens? A spoiled life on the one hand, and spoiled character on the other. Who does not know the unselfish, self-forgetful, overworked mother and the utterly selfish, inefficient children? Self-sacrifice is an abnormality, a demoralizing thing. It is not only an injury to self, it is an insult to its object. Who of us has not felt this? Have you never been made the object of a sacrifice? Have you felt “properly” grateful for it? In spite of your appreciation of the kindness of intent, have you not found yourself half-conscious of a sort of sneaking resentment? Have you not forced yourself to be demonstrative and thankful, when you were secretly inclined to go away and sulk? Yet you did not wish to be ungrateful. Ungrateful! “Ingratitude is the independence of the soul.” The object of a sacrifice, like the object of charity, is placed in a position of weakness, of inefficiency and dependence, and every sturdy soul resents this to the core.
On the other hand, have you not been thrilled into grateful responsiveness upon being made the object of some spontaneous act of affection and thoughtfulness of some expression of the real self of that other? It may have cost nothing, it may have been a real pleasure to the other — and that is precisely why you valued it. It was a genuine tribute to some excellence in you which attracted it. It is ever the spontaneous things that count. It does not always seem fair that the utmost endeavor of one person should count for less than the spontaneous, uncalculated action of another; but it does. We appreciate the effort, but it is spontaneity which attracts us and gives us joy. Being is more beautiful than acting; play is more beautiful than work. It is only when work \emph{is} play that it is beautiful, when the worker enjoys it and puts himself into it. Nothing is beautiful which does not give joy, and all effort that does not tend toward joy is wasted.
We often seem to forget that man is an emotional creature as well as a reasoning being. But in truth our feelings are the important things in life, not our ideas. It is our feelings which impel us to action; our thoughts merely restrain. Even our judgments ultimately rest on feeling. Prof. James puts it in this way: “Our judgments concerning the worth of things, big or little, depend on the feelings the things arouse in us. Where we judge a thing to be precious in consequence of the \emph{idea} we frame of it, this is only because the idea is itself associated already with a feeling. If we were radically feelingless, and if ideas were the only things which our minds could entertain, we should close all our likes and dislikes at a stroke, and be unable to point to any one situation or experience in life more valuable or significant than any other.”
In this alleged reign of reason we are apt to overlook this fact. It is frequently remarked how thin is the veneer which civilization has laid upon the primal savage. When a serious crisis arrives, the veneer cracks and the savage appears. And the whole effort of civilization seems to be, not to develop and improve the savage, but to thicken the veneer. Surely society would be more secure if the savage were not veneered at all.
The whole structure of society must rest either on conflict or on confidence, and confidence is not born of veneer. Any system of education which relies upon the imposition of ideas rather than the development of individualities must result in a hypocrisy which is none the less demoralizing for being well-intentioned; a hypocrisy which destroys confidence, understanding, comradeship and social stability. For the foundation of social stability is the co-operation of spontaneously acting individuals. Restraint is the essence of our governments, and largely the aim of our education, but restraint is not power but the denial of power. Expression is the vital thing, expression of feeling; and the function of restraint is intellectual, the preservation of balance. Reason is normally the handmaid of feeling, developed by the endeavor to fulfill our desires. To discount our emotional life and attempt to live by intellect alone is to dehumanize ourselves just as surely as to abdicate reason and live from impulse alone is to brutalize ourselves. The well-developed individual is he whose impulses and desires are so well-balanced and harmonized that he secures the greatest amount of spontaneous self-satisfaction with the least friction; and the road to this is self-discipline, that self-discipline the true function of which is the freeing of our impulses, and their co-ordination into efficiency and power.
The conduct of life is a matter of valuations, and since our valuations are dependent upon our feelings rather than upon our reason, there must always be a wide variation between the valuations of individuals. Hence it is impossible to be dogmatic, and to limit the activities and the affiliations of the Man Awake. Living is not a matter of conformity but of personality. There are many Men Awake, and while they may travel together for a time, they must part company somewhere, for each man must live his own life. Even the closest are separated by an impassable gulf, and “in the hour of our bitterest need, we are ever alone.” This isolation of individuals in the human race, a species in which each member is more utterly dependent upon his fellows than in any other, is one of the most remarkable of paradoxes. Indeed, self-reliance is an eminent social virtue, but self-limitation is a pitiful individual weakness. This distinction can hardly be too strongly emphasized. The finest type of human development is strongly self-centered, but the self-limited individual is deficient in essential humanity, for man is a social being, not merely a gregarious animal. He does not merely hunt in packs like the wolves, nor herd together for protection like weaker animals; but before man was possible a species of social creatures had appeared, who, living together, sharing in weal and woe, and especially through close association in play, developed a community of feeling which taught them speech and thought and made them the ancestors of the civilizations. One never understands what it is to be human, one never realizes his own individuality until he has gone back across the ages to study his origin, and followed the long, long journey upward. From that hour with the primitive human-like folk, he comes closer in touch with the heart of humanity, feels the great genetic forces which inhere in the race, thrills to the urge and the uplift of human progress. The glory of human joy and the bitterness of human misery press upon him, enter his soul and become one with him. He has thought of himself as belonging to the human race; now he suddenly feels that the human race belongs to him; he has found himself in humanity and humanity in himself. There is no need to talk to him of human brotherhood; he has come closer than brotherhood. The “greatest good of the greatest number sounds like empty words to the sound of his own heart throbs.
Can anyone come close to the origin and history of his kind, and yet feel satisfied? Is he not poor with the poverty of the poorest, and lonely with the desolation of the outcast? So long as some must be cold and hungry and wretched, are there not tears in all his joy, and thorns in all his luxury? Does he not feel with Ernest Crosby —
\begin{quote}
Bitter to eat is the bread that was made by slaves.\forcelinebreak
In the fair white loaf I can taste their sweat and tears.\forcelinebreak
My clothes strangle and oppress me; they burn into my flesh, for I have not justly earned them, and how are they clad that made them?\forcelinebreak
My tapestried walls and inlaid floors chill me and hem me in like the damp stones of a prison house, for I ask why the builders and weavers of them are not living there in my stead.\forcelinebreak
Alas! I am eating of the fruit of the forbidden tree, the tree of others’ labor!”
\end{quote}
Can anyone find humanity and find himself and not become a revolutionist? I cannot. I declare that greater than custom and convention, greater than the laws of the land, greater than schools and philosophies, is the need of human joy. I declare that it is my business to increase it. With Traubel I say —
\begin{quote}
Now I am at last relentless,\forcelinebreak
I declare that the social order is to be superseded by another social order.\forcelinebreak
I know the quality of your folly when you go about the streets looking in the dust of noisy oratory for the complete state.\forcelinebreak
I know very well that when the complete state appears it will be because you bring it to others, not because others bring it to you.\forcelinebreak
And I know that you will bring it, not as a burden upon your back, but as something unscrolled within.
\end{quote}
For who is society but myself and yourself and all selves? And what is human joy but my joy and your joy and the joy of each? And every joy of mine and every joy of yours and every joy that you or I can bring to any, all are so much added joy in the world. For how shall humanity rejoice while you and I are sad?
They tell us much of the social nature of the individual, but they forget to tell us of the individual nature of society. But I tell you that society is myself and yourself and every other self. Shall I serve society by spelling it with a capital? Shall I serve society by lying prostrate before it? Shall I serve society by waiting for it to push me forward? Society does nothing, it is I who do things. It is true that without society I can do nothing, but it is as true that without me — without every individual me — society can do nothing. Let us have done with the worship of society, for at the last there are but men and women, selves, separate and distinct but interdependent. And society progresses only as these progress. And society is great and good and prosperous and happy only with the greatness and goodness and prosperity and happiness of these men and women.
The most and the least which society demands of us is that we be ourselves. We speak of the race-ideals, but the race-ideals are of value to me only as I make them mine, my very own; as I follow them, love them and live them for myself. Then, only, does my living them become of value to my greater self, the social whole. The man in whose being a race-ideal becomes, as it were, focused, becomes from that moment a veritable savior, a leader and maker of history and social destiny; and he becomes this just in the measure of the independence of his thought and action. It is often remarked that great men are the product of their time, expressions of the mass of society; but the significance of this may be easily misconstrued. These men represent the whole by emerging from it; the measure of their greatness, aye, the measure of their service, is the completeness with which they rise above the mass of their fellows. The men who have spoken out the inarticulate desires of the masses, who have become the voice of a great human cry, the right arm of a great human purpose unto action, have been men whose individuality was of the sturdiest and sternest; men who first and foremost have thought their own thoughts and lived their own lives, even unto condemnation and disinheritment at the hands of the very people whose saviors they were. The will of the people is interpreted, is put into action, is brought to fruition, by those individuals of the people who come out from among the people with the fearless and invincible determination — “My will be done!”
We cannot all be saviors, but the impulses which these men personify and concentrate into action are the discontents, the yearnings, the purposes of individuals, and no mystic emanation of the mass as a mass. And as time passes there are more and more individuals and smaller and smaller inarticulate “masses.” The day of the inert mass is passing; the day of the individual is about to dawn, and you and I are either helping or hindering.
I come to you to-day with the question, “What is Worth While?” and I answer it boldly — “Myself!” My own life! And all I demand for myself I accord to you, gladly and with a comrade-word of good cheer — Freedom to live it to the full.
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The Anarchist Library
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Adeline Champney
What Is Worth While?
A Study of Conduct, from the Viewpoint of the Man Awake
1911
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Retrieved on 2 July 2011 from \href{http://en.wikisource.org/wiki/What\_Is\_Worth\_While\%3F}{en.wikisource.org}
(Read before the Cleveland Free Thought Society. Feb. 20, 1910)\forcelinebreak Mother Earth Publishing Association\forcelinebreak 55 West 28\textsuperscript{th} Street, New York
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\begin{center} {Kutat\'asi jelent\'es } \end{center}
\item Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Boz\'o L.$^{3}$: {\it Characterization of regional atmospheric aerosols over Hungary by PIXE elemental analysis.} Applied research on air pollution using nuclear-related analytical techniques. Report on the First Research Co-operation Meeting, Vienna, Austria, 30 March - 2 April, 1993. IAEA, Vienna. (NAHRES-19){ \bf 0} (1994)117-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Boz\'o L.$^{3}$: {\it Characterization of regional atmospheric aerosols over Hungary by PIXE elemental analysis.} Applied Research on Air Pollution Using Nuclear-Related Analytical Techniques. Report on the Second Research Co-ordination Meeting. Menai, Australia, 27-31 March 1995. Report NAHRES-26. Vienna, IAEA{ \bf 0} (1995)9-X./ 0.000$^{ 0}$
\\ {Hivatkoz\'asok:}
\setcounter{hiv} { 0}
\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Parr R. M.$^{4}$,...: IAEA Bulletin {\bf 38} (1996)16.
\item Kert\'esz Zs.$^{1}$,...: PhD \'ertekez\'es. Debreceni Egyetem {\bf 0} (2000)1.
\end{list}
\begin{center} {Konf. abstract, poszter, el\H oad\'as} \end{center}
\item Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Beam transport system with asymmetrical quadrupol lenses.} Rossendorf, Zentralinstitut f\"ur Kernphysik, DDR. 1964. okt.{ \bf 10} (1964)1-X./ 0.000$^{ 0}$
\item Angeli I.$^{2}$$^{+}$, B\'odizs D.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Sz\'amit\'asok \'es m\'er\'esek Van de Graaff gener\'atorok sug\'arv\'edelm\'ere.} I. Nukle\'aris G\'ep\'eszeti Konferencia. Budapest, 1970. nov. 18.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Papp I.$^{2}$$^{+}$: {\it Az ATOMKI-ben \'ep\"ul\"o 5 Mv-os Van de Graaff nyom\'asgener\'ator.} I. Nukle\'aris G\'ep\'eszeti Konferencia. Budapest, 1970. nov. 18.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Papp I.$^{2}$$^{+}$: {\it Az ATOMKI 1 Mv-os Van de Graaff nyom\'asgener\'atora.} I. Nukle\'aris G\'ep\'eszeti Konferencia. Budapest, 1970. nov. 18.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Die Entwicklung eines 5 Mev Van de Graaff Generators in Debrecen. Szemin\'ariumi el\"oad\'as.} Zentralinstitut f\"ur Kernforschung. Rossendorf, DDR, 1970. dec. 3.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$
\item Yavor Sh. Ya.$^{4}$, Koltay E.$^{2}$$^{+}$, Ovsyanikova L. P.$^{4}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Elementy s sil'noi fokusirovkoi dlya sistem transportirovki puchka.} T\"olt\"ott R\'eszecske Gyorsitok II. \"Osszsz\"ovets\'egi Konferenci\'aja. Moszkva, 1970. Nov. 18.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$;N
\item Koltay E.$^{2}$$^{+}$: {\it K\"ozbens\H omag \'elettartam\'anak m\'er\'ese blocking effektus \'utj\'an.} Szimp\'ozium a Van de Graaff gener\'atorok alkalmaz\'as\'ar\'ol, MTA Atommag Kutat\'o Int\'ezete, Debrecen, 1972. m\'aj. 29-31.{ \bf 1} (1972)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Electrostatic design and acceleration tubes in the 5 MV Van de Graaff generator of the ATOMKI.} International Conference on the Technology of Electrostatic Accelerators, Daresbury, 4-7 May, 1973.{ \bf 0} (1973)0-X./ 0.000$^{ 0}$
\item Dang Huy Uyen.$^{2}$$^{+}$, F\'enyes T.$^{1}$$^{+}$, Guly\'as J.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$: {\it Investigation of the 45Sc/p,p'gamma/45Sc reaction and 45Ti- 45Sc decay. (Abstr.: Bp., KFKI, 1975, p.15).} International Symposium on Nuclear Structure, Balatonf\"ured, Sept. 1-6, 1975.{ \bf 0} (1975)0-X./ 0.000$^{ 0}$
\item Dang Huy Uyen.$^{2}$$^{+}$, F\'enyes T.$^{1}$$^{+}$, Guly\'as J.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$: {\it Skhema urovnei 45Sc.} XIV. Soveshchanie po Yadernoi Spektroskopi i Teorii Yadra. Ob"edinennyi Institut Yadernykh Issledovanii, Dubna, 15-19. Iyulya. 1975 G.{ \bf 0} (1975)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it The Ep=2567 MeV resonance in 9Be/p,alfa2/6Li studied through the Doppler broadening of gamma2 line. (Abstr.: p.35).} International Symposium on Nuclear Structure, Balatonf\"ured, Sept. 1-6, 1975.{ \bf 0} (1975)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Opregelenie uglovykh raspredelenii alfa-chastits metodom dopplerovskogo rasshireniya gamma-luchei.} XXV. Soveshchaniya po yadernoi spektroskopii i strukture atomnogo yadra, Leningrad, 28-31. yanvarya, 1975. Abstract, Programma i tezisy dokladov, p. 503.{ \bf 0} (1975)0-X./ 0.000$^{ 0}$
\item Somogyi Gy.$^{2}$$^{+}$, Hunyadi I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it On the detection of low-energy 4He, 12C, 14N, 16O ions in PC foils and its use in nuclear reaction measurements.} IX-th International Conference on Solid State Nuclear Track Detectors, and Meeting of the Working Group on Space Biophysics of the Council of Europe, Neuherberg/M\"unchen, September 30 - October 6, 1976.{ \bf 1} (1976)1-X./ 0.000$^{ 0}$
\item Zolnai L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Hunyadi I.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$: {\it Vozbuzhdenie sostoyaniya 28Si, nablyudaemye v reaktsii 27Al(p,alfa)24Mg pri energii protonov Ep=1540-2220 keV.} XXVI Soveshchanie po Yadernoi Spektroskopii i Strukture Atomnogo Yadra, Baku, 3-6, Fevralya 1976. g.{ \bf 1} (1976)1-X./ 0.000$^{ 0}$
\item Berecz I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Papp I.$^{2}$$^{+}$, Szalay S.$^{2}$$^{+}$, Dzmuran R.$^{4}$: {\it Construction of the stack insulator and acceleration tube elements used in the 5 MV Van de Graaff generator of ATOMKI.} IInd International Conference on Electrostatic Accelerator Technology, Strasbourg, May 26, 1977. (Abstr.: F-4). { \bf 0} (1977)0-X./ 0.000$^{ 0}$;N
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Bornemisza-Pauspertl P.$^{2}$$^{+}$: {\it Modified extraction geometry in a radiofrequency ion source.} IInd International Conference on Electrostatic Accelerator Technology, Strasbourg, May 17, 1977. (Abstr.) { \bf 1} (1977)1-X./ 0.000$^{ 0}$
\item F\'enyes T.$^{1}$$^{+}$, Guly\'as J.$^{2}$$^{+}$, Kib\'edi T.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Gamma-rays from the 100Mo(p,ngamma)100Tc reaction.} International Symposium on Nuclear Reaction Models, Balatonf\"ured, June 27 - July 1, 1977.{ \bf 0} (1977)0-X./ 0.000$^{ 0}$
\item F\'enyes T.$^{1}$$^{+}$, Guly\'as J.$^{2}$$^{+}$, Kib\'edi T.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Study of the 98Mo(p,ngamma)98Tc reaction gamma-rays.} International Symposium on Nuclear Reaction Models, Balatonf\"ured, June 27 - July 1, 1977.{ \bf 0} (1977)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Cseh J.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it Izuchenie vozbuzhdennykh sostoyanii 28Si v reaktsiyakh (alfa,alfa) i (alfa,gamma). (Tezisy Dokladov, Leningrad, Nauka, 1977, p.184).} XXVII.Soveshchaniya po Yadernoi Spektroskopii i Strukture Atomnogo Yadra, Tashkent, 22-25 marta 1977 g.{ \bf 0} (1977)0-X./ 0.000$^{ 0}$
\item Antony M. S.$^{4}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Pintye \'E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, V\'egh L.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Gammaspektroszk\'opiai vizsg\'alatok a Van de Graaff gener\'atoron.} IV. Magyar Magfizikus Tal\'alkoz\'o, Nyiregyh\'aza, 1978. j\'unius 19-23.{ \bf 0} (1978)0-X./ 0.000$^{ 0}$;N
\item \'Arvay Z.$^{2}$$^{+}$, F\'enyes T.$^{1}$$^{+}$, Guly\'as J.$^{2}$$^{+}$, Kib\'edi T.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Krasznahorkay A. J.$^{1}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Nov\'ak D.$^{2}$$^{+}$: {\it Gamma and conversion electron spectra of the 97Mo(p,n-gamma-e)97Tc reaction. (OIYaI Report, D6-11574, 1978, str.101).} 15. Soveshchanie po Jadrnoi Spektroskopii i Teorii Jadra. Dubna, 1978{ \bf 0} (1978)0-X./ 0.000$^{ 0}$
\item Ber\'enyi D.$^{2}$$^{+}$, Mukayama T. ld. 576.$^{4}$, Sarkadi L.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Relative intensities and energy shifts of K x-rays produced in heavy charged-particle collisions. (Abstracts of contributed papers. Ed. E. Anderson, et al. Physics Institute of the Latvian SSR, Academy of Science, 1(1978)377).} VIth International Conference on Atomic Physics, Riga, August 17-22, 1978.{ \bf 0} (1978)0-X./ 0.000$^{ 0}$;N
\item Cseh J.$^{1}$$^{+}$, Hunyadi I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, V\'egh L.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it 28Si resonance states excited by nuclear reactions with p and alfa-particles.} Europhysics Study Conference on the Structure of Lighter Nuclei. Hvar (Yugoslavia), May 8-13,1978.{ \bf 1} (1978)1-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Hunyadi I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$, V\'egh L.$^{2}$$^{+}$: {\it A 28Si mag vizsg\'alata.} IV. Magyar Magfizikus Tal\'alkoz\'o, Nyiregyh\'aza, 1978. j\'unius 19-23.{ \bf 1} (1978)1-X./ 0.000$^{ 0}$
\item Somorjai E.$^{2}$$^{+}$, Cseh J.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Zolnai L.$^{2}$$^{+}$, Decowski P.$^{4}$: {\it Rugalmas sz\'or\'as 19F magon. Tervek direkt-befog\'asi vizsg\'alatokra.} IV. Magyar Magfizikus Tal\'alkoz\'o, Nyiregyh\'aza, 1978. j\'unius 19-23.{ \bf 1} (1978)1-X./ 0.000$^{ 0}$;N
\item V\'egh L.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Cseh J.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it Izuchenie vozbuzhdennykh sostoyanii 23Na v reaktsii 19F(alfa,alfa)19F.(Abstr.p.93).} XV.soveshchanie po Yadernoi Spektroskopii i Teorii Yadra, Dubna, 4-7 Iyulya 1978 G. (Annotatsii Dokladov). Dubna, 1978., Ob"edinennyi Institut Yadernykh Issledovannii.{ \bf 0} (1978)0-X./ 0.000$^{ 0}$
\item Antony M. S.$^{4}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Sistematicheskoe issledovanie vremeni zhizni nekotorykh sostoyanii yadra 14N v reaktsii 13C(p,alfa)14N.} XXIX. Soveshchaniya Po Yadernoi Spektroskopii I Strukture Atomnogo Yadra, Riga, 27-30 Marta 1979 G.{ \bf 0} (1979)0-X./ 0.000$^{ 0}$;N
\item Ber\'enyi D.$^{2}$$^{+}$, Mukoyama T.$^{2}$$^{+}$, Sarkadi L.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it K$\beta$/K$\alpha$ intensity ratios and energy shifts of K$\beta$ rays produced by N-ion impact. (Abstracts of Invited and Contributed Papers. Tokyo, 1979, The Societyfor Atomic Collision Research, p.16).} The VIth International Seminar on Ion-Atom Collisions, Tokai-mura, Ibaraki, Japan, September 6-7, 1979.{ \bf 0} (1979)0-X./ 0.000$^{ 0}$
\item Guly\'as J.$^{2}$$^{+}$, Dombr\'adi Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Krasznahorkay A. J.$^{1}$$^{+}$, F\'enyes T.$^{1}$$^{+}$: {\it Urovni 94Nb, vozbuzhdaemye v reaktsii 94Zr(p,n gamma)94Nb. (Tezisy Dokladov, Leningrad, Nauka, 1979, str. 579).} XXIX. Soveshchaniya po Jadernoj Spektroskopii i Strukture Atomnogo Jadra. Riga, 1979. { \bf 0} (1979)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$, G\"od\'eny S.$^{3}$: {\it PIXE analysis on biological samples.} IInd Working Meeting On Radiation Interaction. Leipzig, September 22-26, 1980.{ \bf 0} (1980)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it A 28Si niv\'oinak vizsg\'alata alfa bomb\'az\'assal.} V.Magyar Magfizikus Tal\'alkoz\'o, Budapest, 1980. j\'ulius 7.{ \bf 0} (1980)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Ionno opticheskie svoistva uskoritel'nykh trubok so spiral'nym polem.} VII. Vsesoyuznoe Soveshchanie Po Uskoritelyam Zaryazhennykh Chastits. Dubna, 14 Oktyabra 1980 G.{ \bf 0} (1980)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Pintye \'E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Opredelenie uglovogo raspredeleniya p1-gruppy reaktsii 10B(alfa,p1-gamma1)13C v intervale 2.9<=E<=3.1 MeV na osnove dopplerovskogo ushireniya gamma-linii. (Tezisy Dokladov, Leningrad, Nauka, 1980, p.622).} XXX. Soveshchaniya po Yadernoi Spektroskopii i Strukture Atomnogo Yadra. Leningrad, 18-21 marta 1980 g. { \bf 0} (1980)0-X./ 0.000$^{ 0}$
\item K\"ov\'er \'A.$^{1}$$^{+}$, Ricz S.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Ber\'enyi D.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, V\'egh J.$^{2}$$^{+}$: {\it Energy distribution of electrons ejected from H2, He by H+, H2+, He (o,8 MeV/nucl.) (Abstr.: Ed.: B.Cobic. Beograd, Boris Kidric Institute of Nuclear Sciences, 1(1980)64 ).} Xth Summer School And Symposium On Physics Of Ionized Gases (SPIG-80), Dubrovnik, August 28, 1980.{ \bf 0} (1980)0-X./ 0.000$^{ 0}$
\item Zolnai L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Cseh J.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it Izuchenie vozbuzhdennykh sostoyanii 23Na v reaktsiyakh (alfa,alfa), (alfa,p) i (alfa,gamma).} XXX. Soveshchaniya po Yadernoi Spektroskopii i Strukture Atomnogo Yadra. Leningrad, 18-21 marta 1980 g. { \bf 0} (1980)0-X./ 0.000$^{ 0}$
\item Abdel-Hady M.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Dve modeli analiza formy dopplerovski ushirennykh gamma linii. (Tezisy Dokladov, Leningrad, Nauka, 1981, str.669).} XXXI. Soveshchanie po Yadernoi Spektroskopii i Strukture Atomnogo Yadra, Samarkand, 14-17 aprelya, 1981 g.{ \bf 0} (1981)0-X./ 0.000$^{ 0}$
\item Csikai Gy.$^{1}$$^{+}$, Sztaricskai T.$^{3}$, Berkes I.$^{3}$, Szegedi S.$^{3}$, Koltay E.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Boh\'atka S.$^{2}$$^{+}$: {\it The intense neutron generator concept in Debrecen.} XIth Symposium on the Interaction of Fast Neutron with Nuclei. Rathen, December 3, 1981.{ \bf 1} (1981)1-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Bremsstrahlung characteristics and particle trajectories in inclined field tubes of alternating or spiraling transverse fields. (Abstracts of contributed papers. Session VI. p.103).} 3rd International Conference On Electrostatic Accelerator Technology, Oak Ridge National Laboratory. Oak Ridge, April 13-16, 1981{ \bf 0} (1981)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it R\'eszecske-induk\'alt r\"ontgenemissz\'o (PIXE) analitikai alkalmaz\'asai az ATOMKI-ban.} 25. Magyar Szink\'epelemz\H o V\'andorgyűl\'es, Sopron, 1982. junius 14-18. Kivonat.{ \bf 0} (1982)264-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it A PIXE m\'odszer interdiszciplin\'aris alkalmaz\'asai az ATOMKI-ban.} VI. Magyar Magfizikus Tal\'alkoz\'o, Hajd\'ub\"osz\"orm\'eny, 1982. j\'unius 21-25.{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Papp Z.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Proton gerjeszt\'eses r\"ontgenemisszi\'o (PIXE) analitikai alkalmaz\'asai.} XII.Magyar Elektronmikroszk\'opos \'es Mikroanalizis Konferencia. Eger, 1982. M\'arcius 29-31. Kivonat.{ \bf 0} (1982)74-X./ 0.000$^{ 0}$
\item G\'acsi Z.$^{1}$$^{+}$, Guly\'as J.$^{2}$$^{+}$, Kib\'edi T.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Krasznahorkay A. J.$^{1}$$^{+}$, F\'enyes T.$^{1}$$^{+}$: {\it Multipolnoszti perehodov iz reakcii 76Ge/p,n gamma/76As.} XXXII Szovescsanije po jadernoj szpektroszkopii i sztrukture atomnogo jadra, 16-18 Marta 1982 g,Kiev{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item G\'acsi Z.$^{1}$$^{+}$, Guly\'as J.$^{2}$$^{+}$, Kib\'edi T.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Krasznahorkay A. J.$^{1}$$^{+}$, F\'enyes T.$^{1}$$^{+}$: {\it Vozbuzhdennye sostojaniya 76As. (Tezisy Dokladov, Leningrad, Nauka, str.557).} XXXII. Soveshchaniya po Yadernoj Spektroskopii i Strukture Atomnogo Yadra, Kiev, 1982{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item G\"od\'eny S.$^{3}$, Jeney F.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it PIXE analizis a sz\"ul\'eszetben.} XII.Magyar Elektronmikroszk\'opos \'es Mikroanalizis Konferencia. Eger, 1982. M\'arcius 29-31. Kivonat.{ \bf 0} (1982)27-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Ion mikronyal\'abok a mikroanal\' \i tik\'aban.} XII.Magyar Elektronmikroszk\'opos \'es Mikroanalizis Konferencia. Eger, 1982. M\'arcius 29-31. Kivonat.{ \bf 0} (1982)41-X./ 0.000$^{ 0}$
\item L\'aszl\'o S.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$: {\it L\'egk\"ori aeroszol mint\'ak PIXE analizise.} XII.Magyar Elektronmikroszk\'opos \'es Mikroanalizis Konferencia. Eger, 1982. M\'arcius 29-31. Kivonat.{ \bf 0} (1982)45-X./ 0.000$^{ 0}$
\item Pintye \'E.$^{2}$$^{+}$, D\'ezsi Z.$^{2}$$^{+}$, Milt\'enyi L.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Kobalt ter\'api\'at k\"ovet\"o v\'er\"osszet\'etel v\'altoz\'as viszg\'alata PIXE m\'odszerrel.} XII.Magyar Elektronmikroszk\'opos \'es Mikroanalizis Konferencia. Eger, 1982. M\'arcius 29-31. Kivonat.{ \bf 0} (1982)60-X./ 0.000$^{ 0}$
\item Somorjai E.$^{2}$$^{+}$, Cseh J.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Magszerkezeti kutat\'asok az ATOMKI elektrosztatikus gyorsit\'o oszt\'aly\'an.} VI. Magyar Magfizikus Tal\'alkoz\'o, Hajd\'ub\"osz\"orm\'eny, 1982. j\'unius 21-25.{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item Bartha L.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, F\'orizs I.$^{2}$$^{+}$: {\it T\"omegkalibr\'al\'as, h\'att\'er- \'es holtid\"o probl\'em\'ak a PIXE m\'odszern\'el. (Abstr.: Szerk. Szilv\'assy Zn\'e. Veszpr\'em, Vegyipari Egyetem Jegyzetsokszoros\' \i t\'oja, 1983, VVE-703/83 p.279).} 26. Magyar Szink\'epelemz\H o V\'andorgyűl\'es, Kecskem\'et, 1983. Oktober 18-21.{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, G\"od\'eny S.$^{3}$: {\it PIXE-analysis of human red blood cell and blood plasma samples taken from pregnant women.} IIIrd Meeting on Nuclear Analytical Methods, Dresden,GDR, 11-15 April,1983.{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Resonances in alpha radiative capture of fluorine.} Europhysics Study Conference on Frontiers of Gamma- and X-ray Spectroscopy.Crete, Greece, 27 June - 1 July, 1983. /Poster/{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item G\"od\'eny S.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Mikro- \'es makroelem vizsg\'alatok terhess\'eg alatt PIXE-vel. (\"Osszefoglal\'as. p.420).} Magyar N\"oorvos T\'arsas\'ag 22. Nagygyűl\'ese, Szeged, 1984. Szeptember 5-8.{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item Hunyadi I.$^{2}$$^{+}$, Kiss A.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Charge state distribution of light heavy ions accelerated in a single ended Van de Graaff accelerator.} VIth Tandem Conference, Chester College, Chester, UK, 18-22 April, 1983{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item Zolnai L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Cseh J.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it Analiz rezonansov v sisteme 19F+alfa (Tezisy dokladov).} XXXIII Soveshchaniya po yadernoi spektroskopii i strukture atomnogo yadra, Moskva, 19-21 aprelya, 1983. Leningrad, 1983, Nauka.{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Pintye \'E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Cluster studies based on low-energy 4He-beam.} IVth International Conference on Clustering Aspects of Nuclear Structure and Nuclear Reactions, 23-27 July, 1984, Chester. Eds. J S Lilley, M A Nagarajan. Daresbury, 1984, Daresbury Laboratory.{ \bf 0} (1984)75-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Pintye \'E.$^{2}$$^{+}$: {\it Line-shape analysis of Doppler-broadened gamma-lines for deriving the angular distribution of the light reaction product.} International Symposium on In-beam Nuclear Spectroscopy, May 14-18, 1984, Debrecen{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Investigation of resonance states in elastic scattering and radiative capture processes.} International Symposium on In-beam Nuclear Spectroscopy, May 14-18, 1984, Debrecen{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item K\"ov\'er \'A.$^{1}$$^{+}$, Szmola E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Hock G.$^{2}$$^{+}$, Ber\'enyi D.$^{2}$$^{+}$, Guly\'as L.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Burkhard M.$^{4}$, Groeneveld K. O.$^{4}$: {\it A study of the electron spectra around 0 for simple collision systems in the 0.8-2.4 MeV impact energy region. .} IInd Workshop On High-energy Ion-atom Collision Processes, August 27-28, 1984, Debrecen.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$;N
\item M\'esz\'aros \'A.$^{3}$, Haszpra L.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Trace element concentrations in atmospheric aerosol over Hungary.} XIth International Conference on Atmospheric Aerosols, Condensation and Ice Nuclei, 3-8 September 1984, Budapest Orsz\'agos Meteorol\'ogiai Szolg\'alat.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\\ {Hivatkoz\'asok:}
\setcounter{hiv} { 0}
\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Heintzenberg I.$^{4}$,...: Id\H oj\'ar\'as {\bf 89} (1985)813.
\end{list}
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$, Nyilas I.$^{2}$$^{+}$: {\it Neutral particles in the beam of a single ended Van de Graaff accelerator,.} Fourth Internat. Conf. on Electrostatic Accelerator Technology and Associated Boosters, April 15-19, 1985, Buenos Aires, Argentina{ \bf 0} (1985)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Pankotai M.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Protongerjeszt\'eses r\"ontgenemisszi\'os analitika kert\'eszeti mint\'ak vizsg\'alat\'ara.} 13. V\'andorgyűl\'es "Fizika-biofizika a Mez\H ogazdas\'agban", Magyar Biofizikai T\'arsas\'ag, MTESZ Debrecen, 1985. Jul. 3-5.{ \bf 0} (1985)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Pankotai M.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it The concentration of trace and macroelements in horticultural samples measured by PIXE method.} 16th Annual Meeting of ESNA(European Society of Nuclear Methods in Agriculture), Warsaw, Sept. 9-13, 1985{ \bf 0} (1985)0-X./ 0.000$^{ 0}$
\\ {Hivatkoz\'asok:}
\setcounter{hiv} { 0}
\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Giese W.$^{4}$,...: XVI.ESNA PROC. WARSAW, 1985 {\bf 0} (1985)57.
\end{list}
\item J\'ozsa M.$^{2}$$^{+}$, Keinonen J.$^{4}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it Highly excited levels of 40Ar from the reaction 36S/alfa,gamma/40Ar.} ELSA-85. Symp. on Electromagnetic Properties of High Spin States. Stockholm,Sveden, May 28-31, 1985. AFI (Research Institute of Physics){ \bf 0} (1985)0-X./ 0.000$^{ 0}$;N
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, J\'ozsa M.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it PIGE-PIXE m\'odszerek kombin\'alt alkalmaz\'asai.} 8. Magyar Magfizikus Tal\'alkoz\'o. Debrecen, 1986. j\'unius 30 - j\'ulius 3.{ \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Haszpra L.$^{3}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, M\'esz\'aros \'A.$^{3}$, M\'esz\'aros E.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Elemental concentrations and regional signatures in atmospheric aerosols over Hungary.} Environmental Physics - Atmospheric Aerosols, Adriatico Research Conference, Trieste, ICTP(Intern.Centre for Theor.Phys.), July 22-25, 1986 { \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, J\'ozsa M.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it On the combined application of PIXE and PIGE methods for elemental analysis of human blood samples.} International Symposium on Microchemical Techniques, Antwerp, August 25-29,1986. Book of Abstr. Antwerp, 1986, Univ. of Antwerp. { \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it PIXE metodikai fejleszt\'esek \'es alkalmaz\'asok.} VIII.Magyar Magfizikus Tal\'alkoz\'o, Debrecen, 1986, Jul.3.{ \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, J\'ozsa M.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Investigation on nuclear levels with low energy 4He beams.} 9th Oaxtepec Symposium on Nuclear Physics, Oaxtepec, Mexico, January 2-4, 1986{ \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item J\'ozsa M.$^{2}$$^{+}$, Keinonen J.$^{4}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it A 36S/alfa,gamma/40Ar reakci\'o.} 8. Magyar Magfizikus Tal\'alkoz\'o. Debrecen, 1986. j\'unius 30 - j\'ulius 3.{ \bf 0} (1986)0-X./ 0.000$^{ 0}$;N
\item Kisp\'eter J.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Trace element concentration in milk powder analysed by PIXE-method (Abstr. p. 78).} Annual Meeting of ESNA(European Society of Nuclear Methods in Agriculture) XVII, Hannover, Sept. 14-19, 1986{ \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Pankotai M.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it A t\"olt\"ottr\'eszecsk\'ekkel keltett r\"ontgensug\'arz\'as analitikai felhaszn\'al\'asa a z\"olds\'egtermeszt\'esben (Abstr.: Szerk.: Blaske Zn\'e. Veszpr\'em, NEVIKI, p. 254).} A Mez\"ogazdas\'ag Kemiz\'al\'asa, Keszthely, 1986, j\'unius 26. { \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, J\'ozsa M.$^{2}$$^{+}$, Keinonen J.$^{4}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it 40Ar nucleus studied by the reaction 36S(alpha,gamma)40Ar (abstracts and programme booklet, p.175).} 6th International Symposium on Capture Gamma-Ray Spectroscopy, Leuven /Belgium/, Aug.31 - Sept.4, 1987{ \bf 0} (1987)0-X./ 0.000$^{ 0}$;N
\\ {Hivatkoz\'asok:}
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\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Van der Leun C.$^{4}$,...: $^{\rm SCI}$Journal of Physics G. Nuclear Physics, Supplement {\bf 14} (1988)109.
\end{list}
\item G\"od\'eny S.$^{3}$, Jeney F.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Determination of magnesium and other elements during pregnancy.} Bayreuther Gesprach \"Uber Magnesium Und Schwangerschaft, Warnensteinach, 22-24 May 1987{ \bf 0} (1987)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Papp I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, F\'elszerfalvi J.$^{3}$, Nyilas I.$^{2}$$^{+}$: {\it On the performance of a straight acceleration tube with axial gradient modulation /abstracts p.19./.} VIIth Tandem Conference, Berlin, April 6-10, 1987.{ \bf 0} (1987)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Particle induced X-ray emission: basic principles, instrumentation and interdisciplinary applications.} NATO Advanced Study Institute "X-ray Spectroscopy in Atomic and Solid State Physics" Vimeiro (Portug\'alia), 30 August - 12 September 1987{ \bf 0} (1987)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Trace elements in atmospheric aerosols over Hungary.} Twelfth International Conference on Atmospheric Aerosols and Nucleation. Held at the University of Vienna, Austria, Aug. 22-7,1988 { \bf 0} (1988)0-X./ 0.000$^{ 0}$
\item J\'ozsa M.$^{2}$$^{+}$, Cseh J.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Keinonen J.$^{4}$: {\it Rezonancia gamma-\'atmenetek vizsg\'alata a 36S(alfa,gamma)40Ar reakcioban.} IX. Magyar Magfizikus Tal\'alkoz\'o, Visegr\'ad, 1988. okt. 20-23{ \bf 0} (1988)0-X./ 0.000$^{ 0}$;N
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Light-heavy ions and molecular beams from a radio-frequency ion source.} 5th International Conference on Electrostatic Accelerators and Associated Boosters, Strasbourg-Heidelberg, 24-30 May, 1989{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Pintye \'E.$^{2}$$^{+}$, Groska E.$^{3}$, Kiss A.$^{3}$: {\it The effect of radiotherapeutic irradiation on the elemental concentration of iron in human erythrocytes and blood plasma (programme and abstracts Vrije Universiteit, Amsterdam, p.15).} 5Th International Conference on Particle Induced X-ray Emission and Analytical Application, Amsterdam, The Netherlands, 21-25 Aug.,1989{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Boz\'o L.$^{3}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$: {\it An evaluation of elemental concentrations in atmospheric aerosols over Hungary: regional signatures and long-range transport modelling (programme and abstracts, Vrije Universiteit, Amsterdam, p.14).} 5Th International Conference on Particle Induced X-ray Emission and Analytical Application, Amsterdam, The Netherlands, 21-25 Aug.,1989{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Boz\'o L.$^{3}$, Koltay E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Elemental composition of aerosol particles under background conditions in Hungary.} Aerosols and Background Pollution, Galway, Ireland, 13-15 June 1989{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$: {\it L\'egk\"ori aeroszolok PIXE analizis\'enek k\"ornyezetkutat\'asi lehet\"os\'egei (abstr.p.25).} I.Magyar Aeroszol Konferencia, Veszpr\'em, 1989.Okt.5-6{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Somorjai E.$^{2}$$^{+}$, J\'ozsa M.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Keinonen J.$^{4}$, Tikkanen P.$^{4}$: {\it The structure of 40Ar studied by 36S(alpha, gamma)40Ar.} Symposium on "Use of Cyclotron", Department of Physics, University of Jyvaskyla, Jyvaskyla, Finland, May 25-26,1989{ \bf 0} (1989)0-X./ 0.000$^{ 0}$;N
\item Biswas S. K.$^{4}$, F\"ul\"op Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it Application of PIGE method for the determination of sodium and boron in aerosol samples.} International Conference on Application of Nuclear Techniques, Crete, Greece, 24-30 June, 1990{ \bf 0} (1990)0-X./ 0.000$^{ 0}$;N
\item Biswas S. K.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, El-Ghawi U. M.$^{4}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it On the applicability of PIXE microanalysis at 100 keV bombarding energies.} International Conference on Application of Nuclear Techniques, Crete, Greece, 24-30 June, 1990{ \bf 0} (1990)0-X./ 0.000$^{ 0}$;N
\item Gesztelyi T.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, F\"ul\"op Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Late Roman glass sealings and their measurement by X- and gamma-rays.} European Workshop "Technology and Analysis of Ancient Gemstones". Ravello, Italy. 9-12 Dec., 1990.{ \bf 0} (1990)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, F\"ul\"op Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Keinonen J.$^{4}$, Tikkanen P.$^{4}$: {\it \'Atmeneti er\H oss\'egek meghat\'aroz\'asa a 25Mg-25Al t\"uk\"ormagokban (abstr.p.8).} X. Magyar Magfizikus Tal\'alkoz\'o, S\'arospatak, 1990. augusztus 21-24{ \bf 0} (1990)0-X./ 0.000$^{ 0}$;N
\item Amemiya S.$^{4}$, Katoh T.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Biswas S. K.$^{4}$: {\it Short-range transport of aerosols emitted by a point source of mixed character in complex terrain (abstract booklet, p.133. 1991) .} 2nd European Conference on Accelerators in Applied Research and Technology. Frankfurt/Main, Germany. 3-7 Sept.,1991.{ \bf 0} (1991)0-X./ 0.000$^{ 0}$;N
\item Amemiya S.$^{4}$, Katoh T.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Varga M.$^{3}$: {\it Vertical concentration profiles of fine and coarse aerosol particles collected over a suburban sampling site in the outskirs of Budapest, Hungary (abstract booklet p.132,1991).} 2nd European Conference on Accelerators in Applied Research and Technology. Frankfurt/Main, Germany. 3-7 Sept.,1991.{ \bf 0} (1991)0-X./ 0.000$^{ 0}$;N
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, F\'elszerfalvi J.$^{3}$: {\it Straight acceleration tube of axial gradient modulation using dish-shaped electrodes.} 6th International Conference on Electrostatic Accelerators and Associated Boosters. Padova, Italy, 1-5 June,1992.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Aeroszol analizis PIXE-m\'odszerrel Debrecenben gy\"ujt\"ott mint\'akon.} 2. Magyar Aeroszol Konferencia. Debrecen, 1992. december 10-11.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Apportionment of atmospheric aerosol collected over Hungary to sources by target transformation factor analysis (Progr.and Abstr. p.29).} 6th International Conference on Particle Induced X-Ray Emission and its Analytical Applications. Tokyo, Japan. July 20-24, 1992.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it L\'egk\"ori aeroszolok PIXE analizise (abstr.: p.8).} XI.Magyar Magfizikus Tal\'alkoz\'o. Szeged, Magyarorsz\'ag. 1992. aug. 17-19.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item F\"ul\"op Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Keinonen J.$^{4}$, Tikkanen P.$^{4}$: {\it Spectroscopy of 38Ar via the 34S(alpha, gamma) reaction. (Abstr.: Verhandlungen der Deutschen Physikalischen Gesellschaft, 1/1992 p.40).} Spring Meeting of Nuclear Physics Sections. Salzburg, Austria, 24-28 Febr.,1992{ \bf 0} (1992)0-X./ 0.000$^{ 0}$;N
\item F\"ul\"op Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Keinonen J.$^{4}$, Tikkanen P.$^{4}$: {\it A 38Ar rezonancia\'allapotainak vizsg\'alata (Abstr.:p.l4).} XI.Magyar Magfizikus Tal\'alkoz\'o. Szeged, Magyarorsz\'ag. 1992. aug. 17-19.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$;N
\item Katoh T.$^{4}$, Amemiya S.$^{4}$, Tsurita Y.$^{4}$, Masuda H.$^{4}$, Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$: {\it PIXE analysis of atmospheric aerosol collected over Hungary and Japan.} 6th International Conference on Particle Induced X-Ray Emission and its Analytical Applications. Tokyo, Japan. July 20-24, 1992.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$;N
\item Katoh T.$^{4}$, Amemiya S.$^{4}$, Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Biswas S. K.$^{4}$: {\it Short-range transport of aerosols emitted by a point source of mixed character as P\'alh\'aza, Hungary.} International Symposium on "Bio-PIXE". Sendai, Japan. 16-18 July, 1992.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$;N
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Amemiya S.$^{4}$, Katoh T.$^{4}$: {\it Pontforr\'as \'altal kibocs\'atott aeroszol r\"ovidt\'av\'u transzportja komplex terepen.} 2. Magyar Aeroszol Konferencia. Debrecen, 1992. december 10-11.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$;N
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Aerosol sampling and analysis by PIXE in the Institute of Nuclear Research, Debrecen.} International Symposium on Environmental Contamination in Central and Eastern Europe. Budapest, Hungary. 12-16 Oct.,1992.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item Moln\'ar \'A.$^{3}$, M\'esz\'aros E.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it A l\'egk\"ori aeroszol r\'eszecsk\'ek elemi \"osszet\'etel\'enek alakul\'asa v\'arosi \'es a h\'att\'erleveg\H oben.} 2. Magyar Aeroszol Konferencia. Debrecen, 1992. december 10-11.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item Szab\'o Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$: {\it Aeroszol adatok f\H okomponens analizise.} 2. Magyar Aeroszol Konferencia. Debrecen, 1992. december 10-11.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, F\"ul\"op Zs.$^{1}$$^{+}$, Gesztelyi T.$^{3}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it The PIXE-PIGE method for the classification of late Roman glass sealings (Abstract book 14.4).} Eleventh International Conference on Ion Beam Analysis. Balatonf\"ured, Hungary, July 5-9,1993.{ \bf 0} (1993)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it The application of PIXE and PIGE techniques in the analytics of atmospheric aerosols (Abstract Book, 2.1).} Eleventh International Conference on Ion Beam Analysis. Balatonf\"ured, Hungary, July 5-9,1993.{ \bf 0} (1993)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Boz\'o L.$^{3}$: {\it Particle characterization at rural, suburban and urban aerosol sampling sites in Hungary (Abstracts... Eds.:J.L.Duggan, I.L.Margan, Univ. of Texas, Dept.of Physics, 1994. p.134).} 13th International Conference on the Application of Accelerators in Research and Industry. Denton, Texas, USA, 7-10 Nov., 1994.{ \bf 0} (1994)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it Characterization of aerosols with other IBA techniques: PIGE and RBS.} Seminario Aplicaci\'on de las T\'ecnicas Analiticas con Haces Iones (PIXE, PIGE, etc.) para la Caracterizaci\'on de Aerosoles Ambientales. Madrid, Spain, 28-29.Nov.,1994.{ \bf 0} (1994)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Elemental composition of urban aerosol collected in Debrecen, Hungary (Programme and Abstracts Book p.67).} VII. International Conference on Particle-induced X-Ray Emission and Its Analytical Application. Padua, Italy, 26-30 May, 1995.{ \bf 0} (1995)0-X./ 0.000$^{ 0}$
\item Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Vad K.$^{1}$$^{+}$: {\it Nuclear microscopy at Debrecen (Abstr.: B.1 p.22).} Accelerator Based Materials Science. Kocovce, Slovakia, 6-9 Nov., 1995{ \bf 0} (1995)0-X./ 0.000$^{ 0}$
\item Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Sz\"o\H or Gy.$^{3}$: {\it The Debrecen scanning proton microprobe facility and its applications to geological samples (Book of Abstracts DO-05).} Twelfth International Conference on Ion Beam Analysis. Arizona State University. Tempe, Arizona, 22-26 May, 1995.{ \bf 0} (1995)0-X./ 0.000$^{ 0}$
\item Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it The new Atomki scanning proton microprobe.} VII. International Conference on Particle-induced X-Ray Emission and Its Analytical Application. Padua, Italy, 26-30 May, 1995.{ \bf 0} (1995)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Gesztelyi T.$^{3}$, Elekes Z.$^{1}$$^{+}$, Rajta I.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Investigation of classical ring-stones and their imitations.} 5th International Conference on Non-destructive Testing, Microanalytical Methods and Environmental Evaluation for Study and Conservation of Works of Art. Budapest, Hungary, 24-28 Sept.,1996{ \bf 0} (1996)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Debrecenben \'es a Hortob\'agyi Nemzeti Parkban gy\"ujt\"ott aeroszol mint\'ak PIXE anal\' \i zise (Az aeroszolok \'es dinamik\'aja, hat\'asa \'es vizsg\'alati m\'odszerei . Szerk.: Czitrowszky A. Bp., MTA KFKI SzFKI p.41).} III.Magyar Aeroszol Konferencia. Budapest., Magyarorsz\'ag, 1996.nov.14-15{ \bf 0} (1996)0-0./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Elemental composition of rural aerosol collected in Hortob\'agy National Park, Hungary (Abstr.:Journal of Aerosol Science Supplement, 27 1996 p.S91).} The 1996 European Aerosol Conference. Delft, The Netherlands, 9-12 Sept.,1996{ \bf 0} (1996)0-0./ 0.000$^{ 0}$
\\ {Hivatkoz\'asok:}
\setcounter{hiv} { 0}
\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Wongphatarakul V.$^{4}$,...: $^{\rm SCI}$Environmental Science and Technology {\bf 32} (1998)3926.
\end{list}
\item Koltay E.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Morales J. R.$^{4}$: {\it Proton mikroszonda alkalmaz\'asa vulk\'ani eredet\H u egyedi aeroszol szemcs\'ek elemanal\' \i zis\'ere (Az aeroszolok dinamik\'aja, hat\'asa \'es vizsg\'alati m\'odszerei. Szerk.:Czitrowszky A. Bp., MTA KFKI SzFKI p. 48).} III.Magyar Aeroszol Konferencia. Budapest., Magyarorsz\'ag, 1996.nov.14-15{ \bf 0} (1996)0-0./ 0.000$^{ 0}$;N
\item Rajta I.$^{1}$$^{+}$, Ontalba M. A.$^{4}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Study of white lead paint layers by the Debrecen nuclear microprobe (Abstr. Q-04 p.116).} 5th International Conference on Nuclear Microprobe Technology and Applications. Santa Fe, New Mexico, USA, 10-15 Nov.,1996{ \bf 0} (1996)0-0./ 0.000$^{ 0}$;N
\item Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Nuclear microscopy at Debrecen.} EMAS'96. 2nd Regional Workshop on Electron Probe Microanalysis of Today - Practical Aspects. Balatonf\"ured, 19-22 May, 1996{ \bf 0} (1996)0-X./ 0.000$^{ 0}$
\item Sundararajan R.$^{4}$, Pet\H o G.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Guczi L.$^{3}$: {\it Investigation of Pt-C and Co-C surfaces from non hydrocarbon sources: Photoemission studies on the nature of interaction between transition metal and carbon (Book of abstr.: P-5, p.77).} Nanoscale Modification of Surfaces and Thin Films: Physical and Chemical Aspects (IUVSTA Workshop) Balatonf\"oldv\'ar, Hungary, 16-19 Sept.,1996{ \bf 0} (1996)0-0./ 0.000$^{ 0}$;N
\item Rajta I.$^{1}$$^{+}$, Grime G. W.$^{4}$, Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Characterization of single aerosol particles from Debrecen region, Hungary using microPIXE.} 13th IBA Conference. Lisboa, Portugal, 27 July - 1 Aug.,1997{ \bf 0} (1997)0-0./ 0.000$^{ 0}$;N
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Tar K.$^{3}$: {\it Composition of urban and rural aerosol samples collected in the Great Hungarian Plain.} 8th International Conference on PIXE and its Analytical Applications. Lund, Sweden, June 14-18, 1998{ \bf 0} (1998)0-0./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Tar K.$^{3}$: {\it K\'etfokozat\'u mintavev\H ovel gy\H ujt\"ott h\'att\'er- \'es v\'arosiaerosol elem\"osszet\'etel\'enek m\'er\'ese PIXE m\'odszerrel. (Abstr.: p.5).} IV. Magyar Aeroszol Konferencia. Veszpr\'em, 1998. okt\'ober 1-2.{ \bf 0} (1998)0-0./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Uzonyi I.$^{1}$$^{+}$: {\it The study of spherules and aerosols. What is common in the two research fields?.} IGCP384 Annual Meeting. Geological Institute of Hungary, Budapest, Hungary. 28-29 Sept., 1998{ \bf 0} (1998)0-0./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Morales J. R.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Characterization of individual aerosol particles from the eruption of Lonquimay volcano in Chile.} 8th International Conference on PIXE and its Analytical Applications. Lund, Sweden, June 14-18, 1998{ \bf 0} (1998)0-0./ 0.000$^{ 0}$;N
\item Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Preliminary study of elemental mass size distribution of urban aerosol collected in Debrecen, Hungary.} 1999 European Aerosol Conference. Prague, Czech Republic, 6-10 Sept., 1999{ \bf 0} (1999)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Tar K.$^{3}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Hossz\'ut\'av\'u tendenci\'ak \'es meteorol\'ogiai hat\'asok megfigyel\'ese v\'arosi \'es vid\'eki aeroszolban, PIXE m\'er\'esek adatai alapj\'an.} Meteorol\'ogiai Tudom\'anyos Napok 2000. A leveg\H ok\"ornyezet monitoringja, \'allapot\'anak \'ert\'ekel\'ese \'es szab\'alyoz\'asa. Budapest, 2000. november 23-24{ \bf 0} (2000)0-0./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Uzonyi I.$^{1}$$^{+}$: {\it V\'arosi \'es barlangi aeroszolok elem\"osszet\'etel\'enek m\'ereteloszl\'asa szerinti vizsg\'alata egyedi szemcse anal\' \i zissel kieg\'esz\' \i tve.} 5. Magyar Aeroszol Konferencia. Szeged, 2000. okt\'ober 5-6.{ \bf 0} (2000)0-0./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it A l\'egk\"ori k\"ornyezet aeroszol terhel\'ese.} 43. Orsz\'agos K\"oz\'episkolai Fizikatan\'ari Ank\'et \'es Eszk\"ozki\'all\' \i t\'as. Keszthely, 2000. \'aprilis 16-20.{ \bf 0} (2000)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Atommagfizika: utak, c\'elok, hat\'asok.} MTA Fizikai Tudom\'anyok Oszt\'alya: A Fizika Magyarorsz\'agon: m\'ult, jelen, j\"ov\H o. c\' \i m\H u tudom\'anyos \"ul\'es. Budapest, 2000. m\'ajus 18. A tudom\'anyos gondolkod\'as Magyarorsz\'agon c. k\"ozgy\H ul\'es rendezv\'enysorozat{ \bf 0} (2000)0-X./ 0.000$^{ 0}$
\item Makra L.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Sutikno.$^{4}$: {\it L\'egk\"ori aeroszol r\'eszecsk\'ek PIXE anal\' \i zise Indon\'ezi\'aban.} 5. Magyar Aeroszol Konferencia. Szeged, 2000. okt\'ober 5-6.{ \bf 0} (2000)0-0./ 0.000$^{ 0}$;N
\item Makra L.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Yongchong Chen.$^{4}$: {\it PIXE analysis of atmospheric aerosol particles in North-Western China (Abstr.: p. 102).} 7th International Conference on Atmospheric Sciences and Applications to Air Quality and Workshop on Air Quality Modelling Challanges. Taipei, Taiwan, 31 Oct. - 3 Nov., 2000{ \bf 0} (2000)0-X./ 0.000$^{ 0}$;N
\item Rajta I.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Morales J. R.$^{4}$: {\it Az 1989-es Lonquimay-i vulk\'ankit\"or\'esb\H ol sz\'armaz\'o egyedi aeroszol szemcs\'ek elemi \"osszet\'eteleinek klaszter anal\' \i zise.} 5. Magyar Aeroszol Konferencia. Szeged, 2000. okt\'ober 5-6.{ \bf 0} (2000)0-0./ 0.000$^{ 0}$;N
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it A l\'egk\"ori aeroszol elem\"osszet\'etel\'enek meghat\'aroz\'asa protoninduk\'alt r\"ontgenemisszi\'os (PIXE) m\'odszerrel (Abstr.: p. 81-82).} PORANAL 2001. 8. Szemcsem\'eret-analitikai, K\"ornyezetv\'edelmi \'es Portechnol\'ogiai Szimp\'ozium. Eger, 2001. szeptember 13-14.{ \bf 0} (2001)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Rajta I.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Uzonyi I.$^{1}$$^{+}$, Boz\'o L.$^{3}$, Makra L.$^{3}$, Morales J. R.$^{4}$, Yongchong Chen.$^{4}$, Tar K.$^{3}$: {\it PIXE analysis of atmospheric aerosol as a tool in environmental research.} Hungarian Science Days in Finland, 28-30 May, 2001: Symposia on Application of Particle Accelerators in Life Environmental and Material Sciences. University of Jyvaskyla, Jyvaskyla, Finland, 29 May, 2001{ \bf 0} (2001)0-X./ 0.000$^{ 0}$;N
\item Makra L.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Yongchong Chen.$^{4}$: {\it Enrichment of desert soil elements in Takla Makan dust aerosol.} 9th International Conference on Particle-Induced X-ray Emission and its Analitical Applications. (PIXE 2001). Guelph, Canada, June 8-12, 2001{ \bf 0} (2001)0-X./ 0.000$^{ 0}$;N
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Boz\'o L.$^{3}$: {\it Terjed\'esi p\'aly\'ak \'es elemi nyomjelz\H ok magyarorsz\'agi aeroszolban 1991 \'es 2000 k\"oz\"ott megfigyelt szaharai epiz\'odok idej\'en.} 6. Magyar Aeroszol Konferencia. Debrecen, 2002. okt\'ober 10-11.{ \bf 0} (2002)0-0./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$: {\it Saharan dust episodes observed by PIXE analyses in Hungarian atmosphere.} 10th International Conference on Particle induced X-ray Emission and its Analytical Applications. Ljubljana - Portoroz, Slovenia, 4-8 June, 2004{ \bf 0} (2004)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Variation of the elemental concentrations of PM10/PM2.5 aerosol collected in Debrecen, Hungary, for the last seven years.} European Aerosol Conference. EAC 2004. Budapest, Hungary, 6-10 Sept., 2004{ \bf 0} (2004)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Bartha L.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Elekes Z.$^{1}$$^{+}$, F\"ul\"op Zs.$^{1}$$^{+}$, Gy\"urky Gy.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Simon A.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Sz\' \i ki G. \'A.$^{2}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Uzonyi I.$^{1}$$^{+}$: {\it The Electrostatic Accelerator Laboratory in Atomki, Debrecen.} IAEA Technical Meeting on Accelerator-based Physics for Sustaining the Flow of Technology and Skills iThemba LABS. Cape Town, South Africa, 7-10 Nov., 2005{ \bf 0} (2005)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Debreceni, v\'arosi PM10 \'es PM2.5 aeroszol koncentr\'aci\'oj\'anak \'es elem\"osszet\'etel\'enek v\'altoz\'as\'aban megfigyelt tendenci\'ak (Abstr.: pp. 49-50).} 8. Magyar Aeroszol Konferencia. MAero2006. Si\'ofok-Szabadif\"urd‹, 2006. m\'ajus 25-26.{ \bf 0} (2006)0-X./ 0.000$^{ 0}$
\item M\'esz\'aros E.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Veres M.$^{3}$, et al.$^{3}$: {\it Similarities and differences in mass and chemical composition of atmospheric fine aerosol at selected sites in Hungary.} Workshop on Similarities and Differences in Airborne Particulate Matter, Exposure and Health Effects Over Europe. Vienna, Austria, Austrian Academy of Sciences, 3-5 April, 2006{ \bf 0} (2006)0-X./ 0.000$^{ 0}$
\item Dobos E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Comparison of Debrecen fine fraction aerosol data with others collected in a European collaboration.} 11th International Conference on Particle Induced X-ray Emission and its Analytical Applications. Puebla, Mexico, 25-29 May, 2007{ \bf 0} (2007)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Historical background for the use of Ion Beam Analysis in Atomki, Debrecen (Abstr.: p. 19).} 11th International Conference on Nuclear Microprobe Technology and Applications. 3rd International Workshop on Proton Beam Writing. Debrecen, Hungary, 20-25 July, 2008{ \bf 0} (2008)0-X./ 0.000$^{ 0}$
\item Szoboszlai Z.$^{1}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Ion beam microanalysis of individual aerosol particles originating from Saharan dust episodes observed in Debrecen, Hungary.} 11th International Conference on Nuclear Microprobe Technology and Applications. 3rd International Workshop on Proton Beam Writing. Debrecen, Hungary, 20-25 July, 2008{ \bf 0} (2008)0-X./ 0.000$^{ 0}$
\item Dobos E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Debreceni PM2.5 \'es finom-frakci\'os elemi koncentr\'aci\'ok eur\'opai sk\'al\'an (Abstr.: pp. 70-70).} 9. Magyar Aeroszol Konferencia. Balatonf\"ured, 2009. \'aprilis 27-28.{ \bf 0} (2009)0-X./ 0.000$^{ 0}$
\item Szoboszlai Z.$^{1}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Egyedi aeroszol r\'eszecsk\'ek anal\' \i zise a debreceni p\'aszt\'az\'o ion mikroszond\'an (Abstr.: pp. 44-45).} 9. Magyar Aeroszol Konferencia. Balatonf\"ured, 2009. \'aprilis 27-28.{ \bf 0} (2009)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Furu E.$^{1}$$^{+}$, Dobos E.$^{2}$$^{+}$, Angyal A.$^{1}$$^{+}$, Szoboszlai Z.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Long-term tendencies and seasonal variation of atmospheric aerosol and its elemental components in Debrecen, Hungary, between 1996 and 2009.} 12th International Conference on Particle Induced X-Ray Emission and Its Analytical Applications. Guilford, UK, 27 June - 2 July, 2010{ \bf 0} (2010)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Furu E.$^{1}$$^{+}$, Dobos E.$^{2}$$^{+}$, Angyal A.$^{1}$$^{+}$, Szoboszlai Z.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Variation of fine and coarse particulate matter and its elemental components in a Central European city from 1996 to 2009.} 2010 International Aerosol Conference. Helsinki, Finland, 29 Aug. - 3 Sept., 2010{ \bf 0} (2010)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Furu E.$^{1}$$^{+}$, Dobos E.$^{2}$$^{+}$, Angyal A.$^{1}$$^{+}$, Szoboszlai Z.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Characterization of atmospheric aerosol in east-Hungary between 1996 and 2009.} International Conference: Natural and Artifical Ecosystems in the Somes, Cris, Mures, Tisa River Basin. Arad, Romania, 7-8 May, 2010{ \bf 0} (2010)0-X./ 0.000$^{ 0}$
\item Szoboszlai Z.$^{1}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Saharan dust episodes observed in East Hungarian atmosphere.} International Conference: Natural and Artifical Ecosystems in the Somes, Cris, Mures, Tisa River Basin. Arad, Romania, 7-8 May, 2010{ \bf 0} (2010)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Dobos E.$^{2}$$^{+}$, Szoboszlai Z.$^{1}$$^{+}$, Angyal A.$^{1}$$^{+}$, Furu E.$^{1}$$^{+}$, T\"or\"ok Zs.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it A debreceni aeroszol adatb\'azis: Mit tudunk kiolvasni a sz\'amokb\'ol?.} 10. Magyar Aeroszol Konferencia. Galyatet\H o, 2011. okt\'ober 20-21.{ \bf 0} (2011)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, T\"or\"ok Zs.$^{1}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Szoboszlai Z.$^{1}$$^{+}$, Angyal A.$^{1}$$^{+}$, Furu E.$^{1}$$^{+}$, Dobos E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Elemental fingerprints of atmospheric aerosol sources and transport in Debrecen, Hungary.} 4T Conference: Turbulence, Transfer, Transport, and Transformation: Interactions Among Environmental Systems. Budapest, Hungary, 24-25 May, 2012{ \bf 0} (2012)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Angyal A.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Furu E.$^{1}$$^{+}$, Major I.$^{1}$$^{+}$, Moln\'ar M.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Szoboszlai Z.$^{1}$$^{+}$, T\"or\"ok Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Long and short term tendencies and variation of PM2.5 and PM10 concentration, composition and sources in an Eastern European city.} 9th Croation Scientific and Professional Conference. Air Protection 2015. Porec, Croatia, 8-12 Sept., 2015{ \bf 0} (2015)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Angyal A.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Furu E.$^{1}$$^{+}$, Major I.$^{1}$$^{+}$, Moln\'ar M.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Szoboszlai Z.$^{1}$$^{+}$, T\"or\"ok Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it 25 years of atmospheric aerosol research at Atomki, Debrecen, Hungary.} 9th Croation Scientific and Professional Conference. Air Protection 2015. Porec, Croatia, 8-12 Sept., 2015{ \bf 0} (2015)0-X./ 0.000$^{ 0}$
\begin{center} {Egy\'eb abstract, poszter, el\H oad\'as} \end{center}
\item Koltay E.$^{2}$$^{+}$: {\it The Application of a Van de Graaff generator to an investigation on the Be9/d,n/B10 nuclear reaction.} A "Fakulta Tech. A Jad. Fyziky Cvut"-ban, Pr\'aga, 1961. okt. 26.{ \bf 10} (1961)26-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Neutrontermel\"o magfolyamatok vizsg\'alata k\"onny\"u magokon.} Fizikus V\'andorgyűl\'es, Debrecen, 1962. aug.{ \bf 8} (1962)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Measurements on the excitation function of the Be9 (d,n) B10 nuclear reaction.} A 3. rossendorf-krakkoi magreakciok \'es magspektroszkopiai szemin\'arium, Krakko, 1963. dec. { \bf 12} (1963)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Besz\'amol\'o az UITF Van de Graaff \'es tandem laboratoriumaiban v\'egzett kis\'erleti vizsg\'alataimrol.} A KFKI Magfizikai I. Oszt\'alya szemin\'ariuma, Bp. 1963. jul.{ \bf 7} (1963)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Hansen O.$^{4}$, Madsen B.$^{4}$: {\it New spin assignments for the excited states of Na23.} A Koppenh\'agai Egyetem Elm\'eleti Fizikai Int\'ezet (UITF) Experimental Group meetingje, Koppenh\'aga, 1963. \'apr.{ \bf 4} (1963)0-X./ 0.000$^{ 0}$;N
\item Koltay E.$^{2}$$^{+}$: {\it A Koppenh\'agai UITF elektrosztatikus gyorsit\'oir\'ol az \'uj gener\'atorszerkeszt\'esi elvek t\"ukr\'eben.} KFKI Gyorsito \"Uzeme. Bp. 1963. okt.{ \bf 10} (1963)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Gyarmati B.$^{2}$$^{+}$: {\it Nekotor\"ue rezultat\"u raszcseta uszkoriteln\"uh trubok sz naklonn\"umi poljami.} Seminar po elektrostaticheskim generatoram Dubna, 1966. febr. { \bf 1} (1966)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Recent results in the Atomki connected with the electron optics of inclined field acceleration tubes.} Szemin\'ariumi el\"oad\'as a Szovjetunio Tudom\'anyos Akad\'emi\'aja A. I. Ioffe Fizikai-technikai Int\'ezet\'enek "t\"olt\"ott r\'eszecsk\'ek optik\'aja" csoportj\'aban, Leningr\'ad, 1968. m\'aj. 24.{ \bf 1} (1968)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Recent results in the ATOMKI connected with field distribution in asymmetric quadrupol lenses.} Szemin\'ariumi el\"oad\'as a Szovjetunio Tudom\'anyos Akad\'emi\'aja A. I. Ioffe Fizikai-technikai Int\'ezet\'enek "t\"olt\"ott r\'eszecsk\'ek optik\'aja" csoportj\'aban, Leningr\'ad, 1968. m\'aj. 27.{ \bf 1} (1968)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Five-electrode lenses for high voltage accelerator use.} Szemin\'ariumi el\"oad\'as a Szovjet\'uni\'o Tudom\'anyos Akad\'emi\'aja "A.F. Joffe" Fizikai Technikai Int\'ezet\'enek "t\"olt\"ott r\'eszecsk\'ek optik\'aja" csoportj\'aban. (Leningr\'ad), 1969. m\'aj.{ \bf 1} (1969)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it The effects of asymmetrization on the effective length of quadrupol lenses.} Szemin\'ariumi el\"oad\'as a Szovjet\'uni\'o Tudom\'anyos Akad\'emi\'aja "A.F. Joffe" Fizikai Technikai Int\'ezet\'enek "t\"olt\"ott r\'eszecsk\'ek optik\'aja" csoportj\'aban. (Leningr\'ad), 1969. jun.{ \bf 1} (1969)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Activities on the accelerator programma in the ATOMKI.} Szemin\'ariumi el\H oad\'as az elektrofizikai berendez\'esek "Jefremov" Tudom\'anyos Kutat\'oint\'ezet\'enek "Elektrosztatikus Gyors\' \i t\'oberendez\'esek" Csoportj\'aban. (Leningr\'ad), 1969. Jun.{ \bf 1} (1969)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it The accelerator project of the ATOMKI.} Szemin\'ariumi el\"oad\'as az obnyinszki Fizikai Energetikai Int\'ezet elektrosztatikus gyorsit\'o laborat\'orium\'aban. Obnyinszk, 1969. okt.{ \bf 1} (1969)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Magfizikai m\'er\"om\'odszerek, sug\'arz\'asok el\"o\'allit\'asa, detekt\'al\'asa.} Orsz\'agos k\"oz\'episkolai fizikatan\'ari ank\'et. Debrecen, 1970. \'apr. 10.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Effective length of multipole components in strong focusing lenses.} Szemin\'ariumi el\"oad\'as a Szovjet\'uni\'o Tudom\'anyos Akad\'emi\'aja "A.F. Joffe" Fizikai Technikai Int\'ezet\'enek "t\"olt\"ott r\'eszecsk\'ek optik\'aja" csoportj\'aban (Leningr\'ad) 1970. jun. 2.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Razrabotka uskoritelya tipa Van de Graafa.} XII. Soveshaniya komiteta po yadernoi fizike pri uchenom sovete po fizike nizkikh energii oiyai. Budapest, 1970. m\'ajus 11-17.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Recent investigations on several physical problems connected with electro-static accelerators.} Daresbury Nuclear Physics Laboratory, seminar. Warrington, United Kingdom, 1971. m\'aj. 28.{ \bf 1} (1971)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it The accelerator project of the ATOMKI.} Reading University, Applied physics section, seminar. Reading, United Kingdom, 1971. m\'aj. 25.{ \bf 1} (1971)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Report on the present status of the accelerator laboratory in the ATOMKI.} Cracow-Rossendorf Seminar, Zentralinstitut f\"ur Kernforschung, Rossendorf, /NDK/, 1973. m\'arc. 22.{ \bf 1} (1973)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Review of the experimental facilities of the Institute of Nuclear Physics in Debrecen. Seminar.} Institut "Rudjer Boskovic", Zagreb, june 25. 1974.{ \bf 1} (1974)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Elektrostaticheskii uskoritel' v institute ATOMKI.} Ob'edinennyi Institut Yadernykh Issledovannii, Laboratoriya Neitronnoi Fiziki, Dubna, 22. mai 1974.{ \bf 1} (1974)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Fizicheskoe issledovanie na novom uskoritely v ATOMKI.} Ob'edinennyi Institut Yadernykh Issledovannii, Laboratoriya Neitronnoi Fiziki, Dubna, 22. mai 1974.{ \bf 1} (1974)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it On going research and its apparative base on the Van de Graaff accelerator.} A Nemzetk\"ozi Atomenergia \"Ugyn\"oks\'eg \'altal szervezett tanulm\'anyi csoport r\'esztvev\"oi sz\'am\'ara. Magyar Tudom\'anyos Akad\'emia Atommag Kutato Int\'ezete, Debrecen, 1976. m\'ajus 20.{ \bf 0} (1976)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Design and construction of the 5 MV Van de Graaff generator and its experimental channels.} A Nemzetk\"ozi Atomenergia \"Ugyn\"oks\'eg \'altal szervezett tanulm\'anyi csoport r\'esztvev\"oi sz\'am\'ara. Magyar Tudom\'anyos Akad\'emia Atommag Kutato Int\'ezete, Debrecen, 1976. m\'ajus 20.{ \bf 1} (1976)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Investigation at the 5 MeV Van de Graaff accelerator of ATOMKI (Institute of Nuclear Research of the Hungarian Academy of Sciences) in Debrecen.} Seminar, Angewandte Kernphysik, Karl Marx Universit„t, Sektion Physik, Leipzig (DDR), September 27, 1976. { \bf 1} (1976)1-X./ 0.000$^{ 0}$
\item Ber\'enyi D.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Seif El-Nasr S. A. H.$^{2}$$^{+}$, Sarkadi L.$^{1}$$^{+}$, V\'egh J.$^{2}$$^{+}$: {\it Zn \"otv\"oz\"oelem m\'elys\'egi koncentr\'aci\'o-eloszl\'as\'anak vizsg\'alata Al minta eset\'en proton-induk\'alt r\"ontgenemisszi\'os m\'odszerrel.} "A magfizikai m\'odszerek alkalmaz\'asa az iparban" Tal\'alkoz\'o ELFT (E\"otv\"os Lor\'and Fizikai T\'arsulat) rendez\'es\'eben. Kecskem\'et, 1977. augusztus 14-16.{ \bf 0} (1977)0-X./ 0.000$^{ 0}$
\item Ber\'enyi D.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Seif El-Nasr S. A. H.$^{2}$$^{+}$, Sarkadi L.$^{1}$$^{+}$, V\'egh J.$^{2}$$^{+}$: {\it M\'elys\'egi analizis protoninduk\'alt r\"ontgenemisszi\'oval. (Hazai tutat\'oint\'ezetekben kifejlesztett \'uj fizikai m\'er\'esi m\'odszerek \'es m\'er\H oeszk\"oz\"ok. Szerk.: Groma G. \'es m\'asok. Bp., METESz. 1980, p.97).} E\"otv\"os Lor\'and Fizikai T\'arsulat 17. V\'andorgyűl\'ese, Eger, 1977. aug. 23-26. { \bf 0} (1977)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it On research work based on the 5 MV Van de Graaff accelerator of ATOMKI (Institute of Nuclear Research of the Hungarian Academy of Sciences).} Seminar. INstitut f\"ur Radiumforschung und Kernphysik der \"Osterreichischenakademie der Wissenschaften, Wien, Sept. 21, 1977.{ \bf 1} (1977)1-X./ 0.000$^{ 0}$
\item Bornemisza-Pauspertl P.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, T\"or\"ok I.$^{2}$$^{+}$, Uray I.$^{2}$$^{+}$: {\it Mikroanalitikai vizsg\'alatok lehet\"os\'ege gyorsitott r\'eszecskenyal\'ab rugalmas sz\'or\'as\'an\'al.} MTA Debreceni Akad\'emiai Bizotts\'ag\'anak Matematikai Fizikai Szakbizotts\'aga \'es a MTESZ Szolnok-megyei Szervezet \'altal rendezett "A fizika helye \'es szerepe a k\"ornyezetv\'edelemben" c. kerekasztal megbesz\'el\'es, Szolnok, 1978. okt\'ober 23.{ \bf 1} (1978)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it A m\'elyen rugalmatlan sz\'or\'as \'es a kett\"os mag rendszer.} Magfizikai \"Oszi Iskola, M\'atraf\"ured, 1978. Oktober 9-14.{ \bf 1} (1978)1-X./ 0.000$^{ 0}$
\item K\"ov\'er \'A.$^{1}$$^{+}$, Ber\'enyi D.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Mukoyama T.$^{2}$$^{+}$, Seif El-Nasr S. A. H.$^{2}$$^{+}$, Ricz S.$^{1}$$^{+}$, Sarkadi L.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, V\'egh J.$^{2}$$^{+}$: {\it Nukle\'aris atomfizikai eredm\'enyek \'es tervek.} IV. Magyar Magfizikus Tal\'alkoz\'o, Nyiregyh\'aza, 1978. j\'unius 19-23.{ \bf 0} (1978)0-X./ 0.000$^{ 0}$
\item G\"od\'eny S.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it A nyomelem vizsg\'alatok jelent\"os\'ege a sz\"ul\'eszetben. A protongerjeszt\'eses r\"ontgenanalitika (PIXE) lehet\"os\'egei.} Tudom\'anyos \"Ul\'es. A Debreceni Orvostudom\'anyi Egyetem \'es a Magyar Tudom\'anyos Akad\'emia Atommagkutat\'o Int\'ezete rendez\'es\'eben, Debrecen, 1979. november 30.{ \bf 1} (1979)1-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it DSA measurement of level lifetimes in 14N via the 13C(p,gamma)14C reaction and some other results in nuclear spectroscopy at the Van de Graaff accelerator laboratory of ATOMKI.} University of Helsinki, Helsinki, Finnorsz\'ag, 1979.M\'ajus 28.{ \bf 0} (1979)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Fejl\"od\'esi ir\'anyok a kis\'erleti magfizik\'aban.} A Magyar Tudom\'anyos Akad\'emia Matematikai \'es Fizikai Tudom\'anyok Oszt\'aly\'anak 1979. \'Evi K\"ozgyűl\'ese keret\'eben tartott tudom\'anyos \"ul\'es, Budapest, 1979. m\'ajus 8.{ \bf 1} (1979)1-X./ 0.000$^{ 0}$
\item Kiss L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, M\'esz\'aros \'A.$^{3}$, L\'aszl\'o S.$^{2}$$^{+}$, G\"od\'eny S.$^{3}$: {\it Proton induced X-ray emission as a tool for analyzing biological and atmospherical samples.} XIth Annual Meeting of European Society of Nuclear Methods in Agriculture, Debrecen, August 25-30, 1980.{ \bf 1} (1980)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Design study on the acceleration tube and beam transport system of a high current neutron generator.} IAEA (International Atomic Energy Agency) Consultants Meeting On Neutron Source Properties. Debrecen, March 20, 1980.{ \bf 1} (1980)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Basic physics and applied investigations on the Van de Graaff beams at the Atomki.} SEMINAR LECTURE. JAGELLONIAN UNIVERSITY, INSTITUTE OF PHISICS. KRAKOW, 14 MAY 1980.{ \bf 1} (1980)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Kis\'erleti magfizikai munk\'ak \"osszefoglal\'asa.} V. Magyar Magfizikus Tal\'alkoz\'o. Budapest, 1980. j\'ulius 7-11.{ \bf 1} (1980)1-X./ 0.000$^{ 0}$
\item M\'esz\'aros \'A.$^{3}$, L\'aszl\'o S.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Preliminari results of atmospheric trace elements study by PIXE analysis in Hungary.} XIth Annual Meeting of European Society of Nuclear Methods in Agriculture, Debrecen, August 25-30, 1980.{ \bf 1} (1980)1-X./ 0.000$^{ 0}$
\\ {Hivatkoz\'asok:}
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\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Diaz de Cuhna K.$^{4}$,...: $^{\rm SCI}$Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms {\bf 187} (2002)401.
\end{list}
\item Bacs\'o J.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Sarkadi L.$^{1}$$^{+}$: {\it A haj endog\'en \'es exog\'en kalcium tartalm\'ar\'ol.} "Nyomelemz\'es" c. ank\'et. A G\'epipari Tudom\'anyos Egyes\"ulet Anyagvizsg\'al\'o Szakoszt\'aly\'anak, A Magyar K\'emikusok Egyes\"ulet\'enek \'es az Orsz\'agos Magyar B\'any\'aszati \'es Koh\'aszati Egyes\"uletnek Szink\'epelemz\"o Szakbiz. k\"oz\"os rendez\'es\'eben. J\'aszber\'eny, 1981. m\'ajus 1 { \bf 0} (1981)0-X./ 0.000$^{ 0}$
\item Ber\'enyi D.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, K\"ov\'er \'A.$^{1}$$^{+}$, Ricz S.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, V\'egh J.$^{2}$$^{+}$, Groeneveld K. O.$^{4}$, Schader J.$^{4}$: {\it Spectra of electrons from the collision of simple sysytem.} International Seminar On High-energy Ion-atom Collision Processes. Debrecen, March 17-19, 1981.{ \bf 1} (1981)1-X./ 0.000$^{ 0}$;N
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Papp Z.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it V\'ekony \'es vastag c\'elt\'argyon v\'egzett protongerjeszt\'eses r\"ontgenanalitikai vizsg\'alatok m\'er\'esi \'es adatfeldolgoz\'asi m\'odszerei.} "Nyomelemz\'es" c. ank\'et. A G\'epipari Tudom\'anyos Egyes\"ulet Anyagvizsg\'al\'o Szakoszt\'aly\'anak, A Magyar K\'emikusok Egyes\"ulet\'enek \'es az Orsz\'agos Magyar B\'any\'aszati \'es Koh\'aszati Egyes\"uletnek Szink\'epelemz\"o Szakbiz. k\"oz\"os rendez\'es\'eben. J\'aszber\'eny, 1981. m\'ajus 1 { \bf 1} (1981)1-X./ 0.000$^{ 0}$
\item G\"od\'eny S.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Papp Z.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Terhesekt\"ol levett v\'er makro- \'es mikroelem \"osszet\'etel\'enek vizsg\'alata PIXE m\'odszerrel.} Debreceni Orvostudom\'anyi Egyetem - Atomki (Magyar Tudom\'anyos Akad\'emia Atommagkutat\'o Int\'ezete) tudom\'anyos \"ul\'es. Debrecen, 1981. december 11.{ \bf 1} (1981)1-X./ 0.000$^{ 0}$
\item Hunyadi I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somogyi Gy.$^{2}$$^{+}$, B\'onis K.$^{3}$: {\it Radiography using SSNTD to analyse environmental aerosol sample.} XIIth International Seminar on Autoradiography. Hark\'any, May 26-29, 1981.{ \bf 0} (1981)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Optical behaviour and bremsstrahlung characteristics of a spiraling field acceleration tube.} IV. Seminar \"Uber Betriebserfahrungen an Elektrostatischen Beschleunigern. Rossendorf, September 9, 1981.{ \bf 0} (1981)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Biol\'ogiai mint\'ak nyomelem anal\' \i zise t\"olt\"ott t\'eszecske nyal\'abok felhaszn\'al\'as\'aval.} "A magyar tudom\'anyos \'elet kapcsolatai a Dubnai Egyes\' \i tett Atommagkutat\'o Int\'ezettel" t\'emak\"or\H u tudom\'anyos \"ul\'es a Dubnai Egyes\' \i tett Atomkutat\'o Int\'ezet megalakul\'as\'anak 25 \'eves \'evfordul\'oja alkalm\'ab\'ol. Budapest, 1981, \'aprilis 14.{ \bf 1} (1981)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it R\'eszecskegyors\' \i t\'os anal\' \i tika (PIXE).} "Modern kis\'erleti modszerek a fizik\'aban." XVI. Fizikus Di\'akk\"ori Ny\'ari Iskola. Nyiregyh\'aza, 1981. augusztus 21-29.{ \bf 1} (1981)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Trace analysis of biological samples PIXE method.} Days of Hungarian science. Symposium in Trace Element Analsys Using Nuclear,Analytical Methods Based on Small Accelerators, for Biological and Industrial Applications. Abo Akademi. Turku, August 31, 1981.{ \bf 1} (1981)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Report on the 5 MeV na de Graaff accelerator and its applications in nuclear physics and in interdisciplinary investigations.} IAEA (International Atomic Energy Agency) Study Tour on the Utilization of Low Energy Accelerators. Debrecen MAY 7, 1981.{ \bf 1} (1981)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Development of accelerators in Debrecen.} Rossendorf-Krakow-Kiev-Rez Seminar School. Biesenthal, March 16-21, 1981.{ \bf 1} (1981)1-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$: {\it Protongerjeszt\'eses r\"ontgenanalitika (PIXE) nyomelemek meghat\'aroz\'as\'ahoz.} A Magyar Tudom\'anyos Akad\'emia \'es a Mez\"ogazdas\'agi \'es \'Elelmez\'es\"ugyi Miniszt\'erium K\"oz\"os Mikroelem Munkabizotts\'agi \"Ul\'ese, Debrecen, 1982.November 9.{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it A PIXE m\'odszer lehet\"os\'egei, m\'er\'esek l\'egk\"ori \'es biol\'ogiai mint\'akon.} Mikroanalitikai Ank\'et, Budapest, Technika H\'aza, 1982. Dec.8.{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item D\'ezsi Z.$^{2}$$^{+}$, Milt\'enyi L.$^{3}$, Pintye \'E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Elem\"osszet\'etel PIXE m\'er\'ese human v\'ermint\'aban, kobalt radioter\'api\'as kezel\'es sor\'an.} A Debreceni Orvostudom\'anyi Egyetem \'es a Magyar Tudom\'anyos Akad\'emia Atommag Kutat\'o Int\'ezete VI. \'evi Egy\"uttes Tudom\'anyos \"Ul\'ese. Debrecen,1982.november 26.{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item F\'elszerfalvi J.$^{3}$, Kiss A.$^{3}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Thek Gy.$^{3}$: {\it Van de Graaff generator fekezesi sugarzasi terenek vizsgalata Tl detektorokkal.} 82 Sug\'arv\'edelmi Tov\'abbk\'epz\"o Tanfolyam, Balatonkenese, 1982. \'Aprilis 14-16. Kivonat{ \bf 0} (1982)12-X./ 0.000$^{ 0}$
\item Szab\'o Gy.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$: {\it Proton-induk\'alt r\"ontgenemisszi\'o analitikai alkalmaz\'asai a l\'egk\"ori \'es biol\'ogiai anyagok vizsg\'alat\'aban.} Hazai Kutat\'ohelyeken Kifejlesztett Fel\"uletanalitikai Elj\'ar\'asok \'es Alkalmaz\'asi Lehet\"os\'egeik a Szil\'ardtestek \'es Biol\'ogiai Anyagok Vizsg\'alat\'aban c.Ank\'et a M\'er\'estechnikai \'es Aut-i Tud. E. Elektromikroszk\'opos Szako. Rend. Bp.1982. Dec.{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, G\"od\'eny S.$^{3}$, Pintye \'E.$^{2}$$^{+}$: {\it Elemental analysis of biological materials related to medical research (PIXE activities in ATOMKI, Debrecen, Hungary).} Seminar Talk. Atomic Energy Centre, Dhaka, Bangladesh, 27 January,1983.{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Pintye \'E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Experiments on nuclear spectroscopy of light nuclei with the beam of a Van de Graaff accelerator.} Seminar talk, Department of Physics, University Of Jyvaskyla, Finland, 27 September, 1983.{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Investigation of the resonances in 19F+alpha and 24Mg+alpha processes below 5 MeV bombarding energy.} Seminar talk, Department of Physics. Abo Academy, Turku, Finland, 6 September, 1983.{ \bf 0} (1983)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, M\'esz\'aros \'A.$^{3}$: {\it V\'ekony mint\'ak PIXE-analizise.} "A R\"ontgenszink\'epelemz\'es \'es Interdiszciplin\'aris Alkalmaz\'asai." c. ank\'et, Debrecen, 1984. november 13-15.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it On the methodology of PIXE microanalytical measurements.} Department of Nuclear Engineering, Kyoto University, December 11,1984. Kyoto.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it PIXE activities in ATOMKI.} Seminar Talk, Institute for Nuclear Study, University of Tokyo, November 27, 1984.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it PIXE activities in ATOMKI.} Seminar Talk, National Laboratory for High Energy Physics, December 17, 1984, Monbusho.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Cseh J.$^{1}$$^{+}$, Fodor I.$^{3}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Papp T.$^{1}$$^{+}$, Pintye \'E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Some clustering aspects of resonant reactions.} Physics Department, University of Jyvaskyla. Seminar talk. Oct.19,1984.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Accelerator based microanalytical investigations in ATOMKI.} Institute of Physical and Chemical Research (Riken), 28 November, 1984, Wako-shi, Saitama.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it The application of PIXE and PIGE microanalytical methods in the ATOMKI.} The Cyclotron and Radioisotope Center, Tohoku University, December 5, 1984, Sendai.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Accelerator based nuclear analytical methods: PIGE and RBS.} International Training Course on Energy Dispersive X-ray Analysis, Organized by the IAEA /International Atomic Energy Agency/ at Jozef Stefan Institute, November, 1984, Ljubljana{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Comments on the application of PIXE in investigations related to Atmospheric physics.} International Training Course on Energy Dispersive X-ray Analysis, Organized by the IAEA /International Atomic Energy Agency/ at Jozef Stefan Institute, November, 1984, Ljubljana{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Comparison of X-ray analysis with proton excited gamma measurements and Rutherford backscattering.} International Training Course on Research and Application of Energy Dispersive X-ray Fluorescence Analysis. Organized by tye IAEA. National University of Singapore. May, 1984. Singapore{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Exotic methods II.: Proton microbeams.} International Training Course on Energy Dispersive X-ray Analysis, Organized by the IAEA /International Atomic Energy Agency/ at Jozef Stefan Institute, November, 1984, Ljubljana{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Physical basis and instrumentation for PIXE.} International Training Course on Research and Application of Energy Dispersive X-ray Fluorescence Analysis. Organized by tye IAEA. National University of Singapore. May, 1984. Singapore{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it PIXE method: basis effects and instrumentation.} International Training Course on Energy Dispersive X-ray Analysis, Organized by the IAEA /International Atomic Energy Agency/ at Jozef Stefan Institute, November, 1984, Ljubljana{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it PIXE mehtod in air pollution studies.} International Training Course on Research and Application of Energy Dispersive X-ray Fluorescence Analysis. Organized by tye IAEA. National University of Singapore. May, 1984. Singapore{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it On the physical processes underlying PIXE and PIGE analytical methods.} The Research Center of Ion Beam Technology. Hosei University. November 29, 1984, Tokyo.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Szab\'o Gy.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'esz\'aros \'A.$^{3}$, L\'aszl\'o S.$^{2}$$^{+}$: {\it Proton-induk\'alt r\"ontgen-emisszi\'os m\'odszer (PIXE) alkalmaz\'asa l\'egk\"ori aeroszolok vizsg\'alat\'ara.} Az MTA 1984. \'evi K\"ozgy\H ul\'es\'ehez Kapcsol\'od\'o "Fizikai M\'odszerek az Emberi K\"ornyezet Kutat\'as\'aban \'es V\'edelm\'eben" c. Tudom\'anyos \"ul\'esszak, Budapest, 1984.M\'ajus 30.{ \bf 0} (1984)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Sampling and PIXE analysis of atmospheric aerosols.} Institute of Atomic Energy. May 14-16, 1985, Beijing { \bf 0} (1985)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it PIXE - physcial basis and methodics.} Institute of Atomic Energy. May 14-16, 1985, Beijing { \bf 0} (1985)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Electron optics and analytical capabilities of proton microbeam devices.} Institute of Atomic Energy. May 14-16, 1985, Beijing { \bf 0} (1985)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it E xperiences and research on accelerator physics in the electrostatic accelerator department of Atomki.} Institute of Atomic Energy. May 14-16, 1985, Beijing { \bf 0} (1985)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Potentialities of the PIXE method.} National Laboratory of Engineering and Industrial Technology. June 25,1985,Sacavem, Portugal { \bf 0} (1985)0-X./ 0.000$^{ 0}$
\item Szab\'o Gy.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Auswertung der PIXE-Spektren.} Institut F\"ur Kernphysik, Aug 20, 1985, Frankfurt{ \bf 0} (1985)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it A protongerjeszt\'eses r\"ontgenemisszi\'o (PIXE) mint analitikai m\'odszer (Abstr.: Szerk.: Gyulai J., M\'ath\'e Gy. 28(1986)88 ).} Anyagvizsg\'alati m\'odszerek \'es m\H uszerek. A KFKI, az Izot\'op Int\'ezet \'es az ATOMKI laboratoriumaiban. Szemin\'arium. Budapest, 1986. \'apr. 26, Debrecen,1986. m\'aj.20. { \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Different applications of the proton induced X-ray emission (PIXE) technique.} Department of Physics, Faculty of Sciences, University of Khartoum,March 12,1986{ \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Analytical applications of PIXE method in (ATOMKI) the Institute of Nuclear Research.} Institute of Atomic Energy, Beijing, Oct. 4, 1986 { \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Haszpra L.$^{3}$, Koltay E.$^{2}$$^{+}$, L\'aszl\'o S.$^{2}$$^{+}$, M\'esz\'aros \'A.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Elemental concentrations and regional signatures in atmospheric aerosol samples over Hungary.} Instituto Nacional de Investigationes Nucleares, Mexico, Jan.16, 1986{ \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Nukle\'aris m\'odszerek a mikroelem anal\' \i tik\'aban.} DOTE/Debreceni Orvostudom\'anyi Egyetem/ - Atomki/Magyar Tudom\'anyos Akad\'emia Atommag Kutat\'o Int\'ezete/, X. Egy\"uttes Tudom\'anyos \"Ul\'ese, Debrecen, 1986. dec. 16{ \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Physics and technology of electrostatic accelerators.} Instituto de Fisica, Universidad Nacional Autonoma, Mexico. Jan. 6, 1986{ \bf 0} (1986)0-X./ 0.000$^{ 0}$
\item Ber\'enyi D.$^{2}$$^{+}$, Bacs\'o J.$^{2}$$^{+}$, Bond\'ar E.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kun K.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Uzonyi I.$^{1}$$^{+}$: {\it A hajduszoboszl\'oi gy\'ogyviz nyomelemtartalm\'anak vizsg\'alata modern nukle\'aris analitikai m\'odszerekkel (Abstr.: p. 37).} Magyar Balneol\'ogiai Egyes\"ulet Tudom\'anyos \"Ul\'ese, Hajduszoboszl\'o, 1987, M\'aj.29-30{ \bf 0} (1987)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'at\'e Z.$^{1}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Pintye \'E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$, Cseh J.$^{1}$$^{+}$: {\it Resonant reactions with low energy 4He beam.} Instituto de Fisica, Universidad Nacional Autonoma de Mexico, January 16, 21, 1987{ \bf 0} (1987)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it A PIXE anal\' \i tikai m\'odszer fizikai alapjai \'es anal\' \i tikai jellemz\H oi.} Magyar K\'emikusok Egyes\"ulete Csepeli Szervezete, szakmai tov\'abbk\'epz\'es.Anyagvizsg\'al\'o \'es G\'epipari Min\H os\'egellen\H orz\H o Int\'ezet, Budapest, 1987, november 23.{ \bf 0} (1987)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it Applications of PIXE analytical method in the Van de Graaff laboratory of ATOMKI.} Szemin\'ariumi el\"oad\'as. Ustav Jadern\'e Fysiky Csav, Rez, Czechoslovakia, 12.10.1988{ \bf 0} (1988)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it PIXE and PIGE research and their interdisciplinary applications in ATOMKI.} Laboratoire de Recherche des Mus\'ees de France. Paris, 6 December, 1988.{ \bf 0} (1988)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$: {\it PIXE measurements on rural aerosol samples in Hungary; regional signatures and long range transport modelling.} Atomic Energy Society of Japan, Nagoya University. Nagoya, Japan, 1989 Sept.22{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Facilities and scientific activities at the accelerator development department of Atomki.} Van de Graaff Laboratory, Faculty of Engineering Nagoya University. Nagoya, Japan, Japan, 1989,Sept.20.{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Moln\'ar \'A.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Elemental composition of aerosol particles in background air.} Austro-Hungarian Seminar on Atmospheric Chemistry. Bp., KLFI, 1989.Febr.27.{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$: {\it An evaluation of elemental concentrations in atmospheric aerosols over Hungary : regional signatures and long-range transport modelling.} Cyric Cyclotron and Radioisotope Center, Tohoku University. Sendai, Japan, 1990.Okt.24{ \bf 0} (1990)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it ELEMENTAL ANALYSIS OF ATMOSPHERIC AEROSOLS.} Department of Nuclear Engineering. Faculty of Engineering. Nagoya University, Nagoya, Japan. 1990.Okt.19.{ \bf 0} (1990)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Katoh T.$^{4}$, Amemiya S.$^{4}$: {\it Investigations in the physics of atmospheric aerosols in ATOMKI and joint activities with Nagoya University.} Department of Material Science and Engineering Waseda University, Tokyo, Japan, 1990.Okt.29.{ \bf 0} (1990)0-X./ 0.000$^{ 0}$;N
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Katoh T.$^{4}$, Amemiya S.$^{4}$: {\it Investigations in the physics of atmospheric aerosols in ATOMKI and joint activities with Nagoya University.} Toyota Central Research and Development Laboratories, Inc. Nagoya, Japan, 1990.Nov.2.{ \bf 0} (1990)0-X./ 0.000$^{ 0}$;N
\item Amemiya S.$^{4}$, Katoh T.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Varga M.$^{3}$: {\it PIXE studies on horizontal and vertical distributions of atmospheric aerosols.} Department of Nuclear Enginerring, Faculty of Engineering, Nagoya University. Nagoya, Japan. 1991.nov.1.{ \bf 0} (1991)0-X./ 0.000$^{ 0}$;N
\item Kiss \'A. Z.$^{2}$$^{+}$, F\"ul\"op Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Gesztelyi T.$^{3}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Analysis of ancient glass sealing by the PIXE-PIGE method.} University of Helsinki, Accelerator Laboratory, Helsinki,Finnorsz\'ag.1991.jun.5.{ \bf 0} (1991)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Appartionment of atmospheric aerosol collected over Hungary to sources by target transformation factor analysis.} Nagoya University, Nagoya, Japan, 1992.j\'ul.28.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it A l\'egk\"ori aeroszol elemi \"osszet\'etele.} Az MTA Agr\'artudom\'anyok Oszt\'alya \'es F\"oldtudom\'anyok Oszt\'alya "A mikroelemek k\"orforgalma a k\"ul\"onb\"oz\H o geoszf\'er\'akban" c. nyilv\'anos \H ul\'ese, Budapest, Magyarorsz\'ag. 1992. m\'ajus 6.{ \bf 0} (1992)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Boz\'o L.$^{3}$: {\it Characterization of regional atmospheric aerosols over Hungary by PIXE elemental analysis.} First Research Co-ordination Meeting on Applied Research on Air Pollution Using Nuclear-Related Analytical Techniques, IAEA, Vienna, Austria. 30 March - 2 Apr.,1993.{ \bf 0} (1993)0-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Boz\'o L.$^{3}$: {\it PIXE, PIGE and their capabilities for source apportionment.} Seminario Aplicaci\'on de las T\'ecnicas Analiticas con Haces Iones (PIXE, PIGE, etc.) para la Caracterizaci\'on de Aerosoles Ambientales. Madrid, Spain, 28-29.Nov.,1994.{ \bf 0} (1994)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it P\'aszt\'az\'o proton-mikroszonda az Atomki-ban (El\H oad\'asok \"osszefoglal\'oi: p.43).} A Magyar Tudom\'anyos Akad\'emia Szabolcs-Szatm\'ar-Bereg Megyei Tudom\'anyos Test\"ulete 1995. \'evi (4.) k\"ozgy\H ul\'essel egybek\"ot\"ott tudom\'anyos \H ul\'ese. Nyiregyh\'aza, Magyarorsz\'ag, 1995. szeptember 29-30.{ \bf 0} (1995)0-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it The new Debrecen scanning proton microprobe facility.} Accelerator Laboratory, University of Helsinki. Helsinki, Finland, June 20, 1995{ \bf 0} (1995)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Boz\'o L.$^{3}$: {\it Characterization of regional atmospheric aerosols over Hungary by PIXE elemental analysis.} Second Research Co-0rdination Meeting on Applied Research on Air Pollution Using Nuclear-Related Analytical Techniques, IAEA. Menai, Australia, 27-31 March, 1995{ \bf 0} (1995)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Boz\'o L.$^{3}$: {\it Characterization of regional atmospheric aerosols over Hungary and qualification of individual volcanic particulates by PIXE elemental analysis.} 3rd Research Co-ordination Meeting on Applied Research on Air Pollution Using Nuclear-related Analytical Techniques. Kingston, Jamaica, 24 Oct.,1996{ \bf 0} (1996)0-0./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Uzonyi I.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it A short introduction to PIXE (Particle Induced X-Ray Emission) method.} IGCP 384 T5 Working Group Meeting "Cosmic Spherules and Aerosols". Debrecen, Hungary, 25-27 Febr.,1997{ \bf 0} (1997)0-0./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$: {\it PIXE investigation on atmospheric aerosol samples in Debrecen.} Institute Jozef Stefan. Ljubljana, Slovenia. 17 Sept., 1998{ \bf 0} (1998)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Nagy L\'aszl\'o egyetemi \'evei. 1950-1954.} Hajdu-Bihar Megyei Nevel\'est\"ort\'eneti Egyes\"ulet Dr. Nagy L\'aszl\'o Eml\'ek\"ul\'ese. Debrecen, 1999. okt\'ober 7.{ \bf 0} (1999)0-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Introduction of the ATOMKI PIXE Group in the field of atmospheric aerosol research.} RIMES Meeting. Genova, Italy, 23-24 Jan., 2003{ \bf 0} (2003)0-X./ 0.000$^{ 0}$
\item Szoboszlai Z.$^{1}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Ion beam microanalysis of individual aerosol particles originating from Saharan dust episodes observed in Debrecen, Hungary.} European Research Course on Atmosphere. ERCA 2009. Grenoble, France, 12 Jan. - 13 Feb., 2009{ \bf 0} (2009)0-X./ 0.000$^{ 0}$
\begin{center} {Ismeretterjeszt\H o k\"ozlem\'eny } \end{center}
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Modellkis\'erletek elektronp\'aly\'ak szeml\'eltet\'es\'ere.} Fizikai Szemle{ \bf 14} (1964)295-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it V.M. Kelman, Sz.Ja. Javor: Elektronoptika. K\"onyvismertet\'es.} Fizikai Szemle{ \bf 17} (1967)320-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it K\"onyvismertet\'es. "Nagy Gy.A. - Szil\'agyi M.: Bevezet\'es a t\'ert\"olt\'esoptika elm\'elet\'ebe. Bp., 1967, Akad\'emiai K.".} Fizikai Szemle{ \bf 19} (1969)349-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Magfizikai m\'er\"om\'odszerek. Sug\'arz\'asok el\"o\'allit\'asa, detekt\'al\'asa.} A XIII. Orsz. K\"oz\'episk. fizikatan\'ari ank\'et \'es eszk\"ozki\'allit\'as. Szerk. F\'elszerfalvy J. stb. Debrecen, 1970. METESZ, ELFT Debreceni csoportja.{ \bf 1} (1970)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it \"Ot \'eves az ATOMKI \"ot milli\'o voltos r\'eszecskegyors\' \i t\'oja.} Hajd\'u-Bihari Napl\'o, dec. 30. 308. sz\'am.{ \bf 33} (1976)8-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Az Atommag Kutat\'o Int\'ezet elektrosztatikus gyorsit\'oja.} \'Elet \'es Tudom\'any{ \bf 32} (1977)351-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it \'uj fizikai t\'argy\'u k\"onyvsorozatot indit az Akad\'emiai Kiad\'o.} Fizikai Szemle{ \bf 33} (1983)157-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Az atommagfizikai k\' \i s\'erleti kutat\'asok helyzete Magyarorsz\'agon.} Magyar Tudom\'any{ \bf 91} (1984)266-X./ 0.000$^{ 0}$
\\ {Hivatkoz\'asok:}
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\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Vlachy J.$^{4}$,...: PHYSICS IN EUROPE A BIBLIOGRAPHY SUPPLEMENT. 1984. CZECHOSLOVAK JOURNAL OF PHYSICS "B" {\bf 34} (1984)721.
\end{list}
\item Koltay E.$^{2}$$^{+}$: {\it Gyors\' \i t\'ofejleszt\'es \'es magreakci\'o-kutat\'as.} Fizikai Szemle{ \bf 35} (1985)2-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Tudom\'anyos \"ul\'es Szalay S\'andor akad\'emikus 75. sz\"ulet\'esnapja alkalm\'ab\'ol.} Fizikai Szemle{ \bf 35} (1985)438-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Van-e el\'eg p\'enz a kutat\'asra?.} Hajd\'u-Bihari Napl\'o{ \bf 43} (1986)3-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Kov\'ach \'A.$^{2}$$^{+}$: {\it Szalay S\'andor 1909-1987.} Fizikai Szemle{ \bf 38} (1988)42-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Profile of foreign visitors to NIRS: Ede Koltay (Jap\'anul).} Hoshasen Kagaku (Radiological Sciences){ \bf 32} (1989)364-X./ 0.000$^{ 0}$
\item Makra L.$^{3}$, S\"umeghy Z.$^{3}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Porszennyez\H od\'es - Sivatagi s\'o.} \'Elet \'es Tudom\'any{ \bf 55} (2000)1235-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Eml\'ekez\'es a 100 \'eves sz\"uletett Szalay S\'andor akad\'emikusra.} Debreceni Szemle { \bf 18} (2010)1:56-58./ 0.000$^{ 0}$
\begin{center} {Ismeretterjeszt\H o el\H oad\'as } \end{center}
\item Koltay E.$^{2}$$^{+}$: {\it Atomfizika a laboratoriumban \'es az iskol\'aban.} A Besze Nagy J\'anos gimn\'azium fizikus k\"or\'eben, Gy\"ongy\"os, 1975. okt\'ober 23.{ \bf 1} (1975)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Uj \'ori\'asok az elektrosztatikus gyorsit\'ok vil\'ag\'aban.} Kossuth Lajos Tudom\'anyegyetem fizikus klubja. Debrecen, 1975. m\'arcius 4.{ \bf 1} (1975)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it K\'epek a debreceni fizika multj\'ab\'ol \'es jelen\'eb\"ol. megnyito el\"oad\'as.} "\"Oszi Fizikusnapok". A K\"olcsey Ferenc M\H uvel\H od\'esi K\"ozpont, az Atomki (Magyar Tudom\'anyos Akad\'emia AtommagKutat\'o Int\'ezete) \'es a TIT (Tudom\'anyos Ismeretterjeszt\H o T\'arsulat) rendez\'es\'eben, Debrecen, 1979. november 12-15.{ \bf 1} (1979)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it A nagyenergi\'aj\'u fizika gyors\' \i t\'o berendez\'esei.} Az E\"otv\"os Lor\'and Fizikai T\'arsulat \H oszi Magfizikai Iskol\'aja. Balatonf\"oldv\'ar, 1980. okt\'ober 30.{ \bf 1} (1980)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Ujdons\'agok a nagyenergi\'aj\'u gyors\' \i t\'ok vit\'aj\'aban.} Kossuth Lajos Tudom\'anyegyetem Term\'eszettudom\'anyi Kar\'anak Fizikus Szakhete. Debrecen, 1980. \'Aprilis 22.{ \bf 1} (1980)1-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Nagyenergiaj\'u gyors\' \i t\'oberendez\'esek technol\'ogiai alkalmaz\'asai.} E\"otv\"os Lor\'and Fizikai T\'arsulat Soproni Csoportja rendez\'es\'eben. Sopron, 1982. Janu\'ar 18.{ \bf 0} (1982)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Gyarmati B.$^{2}$$^{+}$, Kov\'ach \'A.$^{2}$$^{+}$: {\it A "debreceni atomfizikai iskola" multja, jelene \'es j\"ov\"oje.} E\"otv\"os Lor\'and Fizikai T\'arsulat Sopron V\'arosi Csoportja, 1989.Febru\'ar 13.{ \bf 0} (1989)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Mir\H ol besz\'el egy porszem?.} Ember - K\"ornyezet - Fizika. 16. Fizikusnapok. Debrecen, Magyarorsz\'ag, 1995. m\'arcius 6-11.{ \bf 0} (1995)0-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$: {\it Aeroszolok \'es expressz terjed\'es\"uk a l\'egk\"orben.} 23. Fizikusnapok - A F\"old, amelyen \'el\"unk. Debrecen, Atomki, 2002. m\'arcius 4-8.{ \bf 0} (2002)0-X./ 0.000$^{ 0}$
\begin{center} {\'Evk\"onyv } \end{center}
\item Cseh J.$^{1}$$^{+}$, Hunyadi I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$, V\'egh L.$^{2}$$^{+}$: {\it Resonance levels of 28Si excited with low energy protons and alpha particles.} INDC(HUN)-16/g Progress Report Nuclear Data Programme In Hungary 1977{ \bf 0} (1978)1-X./ 0.000$^{ 0}$
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Papp I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it The operation of the Van de Graaff accelerator in 1986.} Atomki Annual Report 1986{ \bf 0} (1987)53-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, J\'ozsa M.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Zolnai L.$^{2}$$^{+}$: {\it The analytical application of charged particle induced X-ray and gamma-ray emission (PIXE, PIGE).} Atomki Annual Report 1986{ \bf 0} (1987)43-X./ 0.000$^{ 0}$
\item J\'ozsa M.$^{2}$$^{+}$, Keinonen J.$^{4}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nyak\'o B. M.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it 40Ar levels at 9 to 10 MeV studied by the reaction 36S/alpha,gamma/40Ar.} Atomki Annual Report 1986{ \bf 0} (1987)8-X./ 0.000$^{ 0}$;N
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Papp I.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it The operation of the Van de Graaff accelerators in 1987.} Atomki Annual Report 1987{ \bf 0} (1988)88-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it The analytical application of charged particle induced X-ray emission method.} Atomki Annual Report 1987{ \bf 0} (1988)70-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it New data for 12C/p,gamma/13N radiative capture process: note on an anomaly in thick target measurements.} Atomki Annual Report 1987{ \bf 0} (1988)33-X./ 0.000$^{ 0}$
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it The operation of the Van de Graaff accelerators in 1988.} Atomki Annual Report 1988{ \bf 0} (1989)123-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it The analitical application of charged particle induced X-ray emission method.} Atomki Annual Report 1988{ \bf 0} (1989)101-X./ 0.000$^{ 0}$
\item Kiss \'A. Z.$^{2}$$^{+}$, Tikkanen P.$^{4}$, Keinonen J.$^{4}$, Koltay E.$^{2}$$^{+}$, Pintye \'E.$^{2}$$^{+}$: {\it DSA lifetimes in 29P from the reaction 28Si(p,gamma)29P.} Atomki Annual Report 1988{ \bf 0} (1989)1-X./ 0.000$^{ 0}$;N
\\ {Hivatkoz\'asok:}
\setcounter{hiv} { 0}
\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Endt P. M.$^{4}$,...: $^{\rm SCI}$Nuclear Physics A {\bf 521} (1991)1.
\end{list}
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Activities at the Van de Graaff accelerator laboratory.} Atomki Annual Report 1989{ \bf 0} (1990)87-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it The analytical application of charged particle induced X-ray emission method.} Atomki Annual Report 1989{ \bf 0} (1990)84-X./ 0.000$^{ 0}$
\item Keinonen J.$^{4}$, Kuronen A.$^{4}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Pintye \'E.$^{2}$$^{+}$, Wildenthal B H.$^{4}$: {\it Short lifetimes in 29Si-29P for test of shell-model wave functions.} University of Helsinki, Department of Physics, Annual Report 1989{ \bf 0} (1990)25-X./ 0.000$^{ 0}$;N
\item Amemiya S.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Katoh T.$^{4}$, Koltay E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Varga M.$^{3}$: {\it Vertical concentration profiles of fine and coarse aerosol particles collected over an urban sampling site in the outskirts of Budapest.} Atomki Annual Report 1990{ \bf 0} (1991)77-X./ 0.000$^{ 0}$;N
\item Amemiya S.$^{4}$, Biswas S. K.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Katoh T.$^{4}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Short-range transport of aerosols emitted by the P\'alh\'aza pearlite plant, Hungary.} Atomki Annual Report 1990{ \bf 0} (1991)79-X./ 0.000$^{ 0}$;N
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Activities at the Van de Graaff accelerator laboratory.} Atomki Annual Report 1990{ \bf 0} (1991)128-X./ 0.000$^{ 0}$
\item Biswas S. K.$^{4}$, F\"ul\"op Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$: {\it Application of PIGE method for the determination of sodium and boron in aerosol samples.} Atomki Annual Report 1990{ \bf 0} (1991)82-X./ 0.000$^{ 0}$;N
\item Biswas S. K.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, El-Ghawi U. M.$^{4}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it On the applicability of PIXE microanalysis at 100 keV bombarding energies.} Atomki Annual Report 1990{ \bf 0} (1991)110-X./ 0.000$^{ 0}$;N
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, F\"ul\"op Zs.$^{1}$$^{+}$, Gesztelyi T.$^{3}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Classification of late Roman glass sealings by the PIXE-PIGE method.} Atomki Annual Report 1990{ \bf 0} (1991)81-X./ 0.000$^{ 0}$
\item Amemiya S.$^{4}$, Biswas S. K.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Katoh T.$^{4}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Short-range transport of aerosols emitted by a point source of mixed character in complex terrain.} Atomki Annual Report 1991{ \bf 0} (1992)112-X./ 0.000$^{ 0}$;N
\item Amemiya S.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Katoh T.$^{4}$, Koltay E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Varga M.$^{3}$: {\it Dry deposition velocities of atmospheric aerosol particles from vertical concentration profiles measured by PIXE method.} Atomki Annual Report 1991{ \bf 0} (1992)114-X./ 0.000$^{ 0}$;N
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Activities at the Van de Graaff accelerator laboratory.} Atomki Annual Report 1991{ \bf 0} (1992)161-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, F\"ul\"op Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Combined applications of the PIXE and PIGE analytical methods.} Atomki Annual Report 1991{ \bf 0} (1992)103-X./ 0.000$^{ 0}$
\item F\"ul\"op Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Keinonen J.$^{4}$, Tikkanen P.$^{4}$: {\it Spectroscopy of 38Ar via the 34S(alpha, gamma) reaction.} Atomki Annual Report 1991{ \bf 0} (1992)4-X./ 0.000$^{ 0}$;N
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Activities at the Van de Graaff accelerator laboratory.} Atomki Annual Report 1992{ \bf 0} (1993)138-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it PIXE data set on atmospheric aerosol evaluated by wind sector and target transformation factor analyses.} Atomki Annual Report 1992{ \bf 0} (1993)104-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Aerosol sampling and analysis by PIXE.} Atomki Annual Report 1992{ \bf 0} (1993)107-X./ 0.000$^{ 0}$
\item F\"ul\"op Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Keinonen J.$^{4}$, Tikkanen P.$^{4}$: {\it New results on the resonance states of 38Ar.} Atomki Annual Report 1992{ \bf 0} (1993)11-X./ 0.000$^{ 0}$;N
\item Ali A. E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Medve F.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Bacs\'o J.$^{2}$$^{+}$: {\it Trace element concentrations in atmospheric aerosol particles at three different sites of Debrecen city.} Atomki Annual Report 1993{ \bf 0} (1994)114-X./ 0.000$^{ 0}$
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Activities at the Van de Graaff accelerator laboratory.} Atomki Annual Report 1993{ \bf 0} (1994)166-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Aerosol sampling and analysis by PIXE.} Atomki Annual Report 1993{ \bf 0} (1994)120-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Boz\'o L.$^{3}$, Koltay E.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Szab\'o Gy.$^{2}$$^{+}$: {\it A search for Saharan dust intrusion in atmospheric aerosol over Hungary.} Atomki Annual Report 1993{ \bf 0} (1994)121-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, M\'esz\'aros E.$^{3}$, Moln\'ar \'A.$^{3}$, Boz\'o L.$^{3}$: {\it Characterization of regional atmospheric aerosol over Hungary by PIXE elemental analysis.} Atomki Annual Report 1993{ \bf 0} (1994)118-X./ 0.000$^{ 0}$
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Activities at the Van de Graaff accelerator laboratory.} Atomki Annual Report 1994{ \bf 0} (1995)99-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Aerosol sampling and analysis by PIXE.} Atomki Annual Report 1994{ \bf 0} (1995)76-X./ 0.000$^{ 0}$
\item Rajta I.$^{1}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it The new scanning proton microprobe facility.} Atomki Annual Report 1994{ \bf 0} (1995)91-X./ 0.000$^{ 0}$
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Activities at the Van de Graaff accelerator laboratory.} Atomki Annual Report 1995{ \bf 0} (1996)106-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Aerosol sampling and analysis by PIXE.} Atomki Annual Report 1995{ \bf 0} (1996)78-X./ 0.000$^{ 0}$
\item Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, M\'orik Gy.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Activities at the scanning proton microprobe.} Atomki Annual Report 1995{ \bf 0} (1996)105-X./ 0.000$^{ 0}$
\item Bartha L.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Activities at the Van de Graaff accelerator laboratory.} Atomki Annual Report 1996{ \bf 0} (1997)89-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Aerosol sampling and analysis by PIXE.} Atomki Annual Report 1996{ \bf 0} (1997)68-X./ 0.000$^{ 0}$
\item Elekes Z.$^{1}$$^{+}$, Ontalba M. A.$^{4}$, Rajta I.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Gesztelyi T.$^{3}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Applications of ion beam analysis in archaeometry.} Atomki Annual Report 1996{ \bf 0} (1997)58-X./ 0.000$^{ 0}$;N
\item Koltay E.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Morales J. R.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Micro-PIXE characterization of volcanic individual aerosol particles from a Chilean eruption.} Atomki Annual Report 1996{ \bf 0} (1997)66-X./ 0.000$^{ 0}$;N
\item Rajta I.$^{1}$$^{+}$, Simon A.$^{1}$$^{+}$, Uzonyi I.$^{1}$$^{+}$, Ontalba M. A.$^{4}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Bartha L.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$: {\it Activities at the scanning proton microprobe.} Atomki Annual Report 1996{ \bf 0} (1997)86-X./ 0.000$^{ 0}$;N
\\ {Hivatkoz\'asok:}
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\begin{list}%
{\arabic{hiv}.}{\usecounter{hiv}\setlength{\rightmargin}{\leftmargin}}
\item Bujdos\'o E.$^{2}$,...: $^{\rm SCI}$Journal of Radioanalytical and Nuclear Chemistry {\bf 246} (2000)467.
\end{list}
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Tar K.$^{3}$: {\it Seasonal variation of the composition of urban and rural atmospheric aerosol.} Atomki Annual Report 1997{ \bf 0} (1998)67-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Tar K.$^{3}$: {\it Dependence of the composition of urban atmospheric aerosol on wind direction.} Atomki Annual Report 1997{ \bf 0} (1998)68-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Composition of coarse and fine mode aerosol samples collected in rural and urban sites.} Atomki Annual Report 1998{ \bf 0} (1999)60-X./ 0.000$^{ 0}$
\item Elekes Z.$^{1}$$^{+}$, Uzonyi I.$^{1}$$^{+}$, Bartha L.$^{2}$$^{+}$, Nagy A.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Gy\"urky Gy.$^{1}$$^{+}$, Somorjai E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Installation of a new PGT Si(Li) detector and analytical application of a new clover-Ge-BGO detector system at the scanning nuclear microprobe of Atomki.} Atomki Annual Report 1998{ \bf 0} (1999)44-X./ 0.000$^{ 0}$
\item Kert\'esz Zs.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Preliminary study of elemental mass size distribution of urban aerosol collected in Debrecen.} Atomki Annual Report 1999{ \bf 0} (2000)62-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Tar K.$^{3}$: {\it Long range tendencies and meteorological effects in aerosol concentrations.} Atomki Annual Report 2000{ \bf 0} (2001)56-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Boz\'o L.$^{3}$: {\it Aerosol index, air mass trajectories and elemental characteristics during Saharan dust episodes in the Hungarian atmosphere between 1991 and 2000.} Atomki Annual Report 2002{ \bf 0} (2003)69-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Changes in the elemental concentration of PM10 and PM2.5 aerosol during the last seven years.} Atomki Annual Report 2004{ \bf 0} (2005)64-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Elemental concentrations of PM10 and PM2.5 aerosol in Debrecen.} Atomki Annual Report 2005{ \bf 0} (2006)58-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Variation of the PM10, PM2.5 and BC particulate masses of urban aerosol in Debrecen.} Atomki Annual Report 2006{ \bf 0} (2007)57-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Koltay E.$^{2}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Variation of the elemental concentrations of urban aerosol in Debrecen.} Atomki Annual Report 2006{ \bf 0} (2007)58-X./ 0.000$^{ 0}$
\item Koltay E.$^{2}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Dobos E.$^{2}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$: {\it Comparison of Debrecen fine fraction aerosol data with others collected in some European collaboration.} Atomki Annual Report 2006{ \bf 0} (2007)59-X./ 0.000$^{ 0}$
\item Borb\'ely-Kiss I.$^{2}$$^{+}$, Husz\'ank R.$^{1}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Kiss \'A. Z.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$, Rajta I.$^{1}$$^{+}$, Simon A.$^{1}$$^{+}$, Szab\'o Gy.$^{2}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Szilasi S. Z.$^{1}$$^{+}$, Szoboszlai Z.$^{1}$$^{+}$, Uzonyi I.$^{1}$$^{+}$: {\it Laboratory of Ion Beam Applications at ATOMKI.} Atomki Annual Report 2008{ \bf 0} (2009)1-14./ 0.000$^{ 0}$
\item Szoboszlai Z.$^{1}$$^{+}$, Kert\'esz Zs.$^{1}$$^{+}$, Szikszai Z.$^{1}$$^{+}$, Borb\'ely-Kiss I.$^{2}$$^{+}$, Koltay E.$^{2}$$^{+}$: {\it Ion micro-beam analysis of single aerosol particles originating from Saharan dust episodes observed in Debrecen, Hungary.} Atomki Annual Report 2008{ \bf 0} (2009)56-X./ 0.000$^{ 0}$
\end{list}
\end{document}
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https://genkuroki.github.io/documents/20160501StirlingFormula/20160501StirlingFormula-0.18.tex
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\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.
様々な確率分布についての付録(\secref{sec:dists})を追加した.
Maxwell-Boltzmann則の導出も追加した(\secref{sec:MB1}).
2016年5月14日Ver.0.13(46頁): 細かい計算ミスを訂正し, MB則の解説を補充した.
2016年5月15～18日Ver.0.14(50頁): ギャンブルに関する逆正弦法則(\fnref{fn:arcsin}),
Wignerの半円則(\fnref{fn:Wigner}),
$\sin^2$ 型分布が佐藤・Tate予想に登場すること(\fnref{fn:Sato-Tate})
のコメントを追加した.
二項分布と第一種ベータ分布の関係(\secref{sec:Bin-Beta})と
Poisson分布とガンマ分布の関係(\secref{sec:Poisson-Gamma})の
簡単で大雑把な解説を追加した.
Stirlingの公式のよりシンプルな証明の筋道の解説(\secref{sec:pconv-2})を追加した.
細かな誤りを訂正した.
2016年5月23日Ver.0.15(57頁): 簡単なTauber型定理とその応用に関する\secref{sec:Tauber}
を追加した. 応用例はWallisの公式と $x-x^2+x^4-x^8+x^{16}-\cdots$ の $x\nearrow 1$
での漸近挙動の2つ. Wallisの公式型の漸近挙動からどのようにして逆正弦分布が
出て来るかも解説してある.
2016年5月24日Ver.0.16(61頁): \theoremref{theorem:Tauber-Laplace}の証明が
あまりにも雑だったので, Stone-Weierstrassの多項式近似定理から得られる
\lemmaref{lemma:SW}を追加して, 詳しく書き直した.
大きく書き直した直後なので誤りが残っているものと思われる.
他にも細かな訂正と追加をした.
2016年5月24日Ver.0.17(61頁): 細かな誤りの訂正.
2016年5月25日Ver.0.18(67頁): Taylorの定理の証明の仕方に関する\secref{sec:Taylor}
を追加した(iterated integralsによる線形常微分方程式の解法(\secref{sec:LODE})を含む).
}
\\[\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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\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誤差が $0.1\%$ 程度と相当に小さい:
\[
\frac{\sqrt{2\pi}}{e}\left(1+\frac{1}{12}\right) = 0.9989\cdots \approx 1.
\]
このようにStirlingの公式は階乗の近似公式として非常に優秀である%
\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
\[
n!
=\left[\left(n + \frac{1}{12n-\frac{1}{10n+\cdots}}\right)\frac{1}{e}\right]^n\sqrt{2\pi n}
=n^n e^{-n} \sqrt{2\pi n}
\left(1+\frac{1}{12n^2}+\frac{1}{1440n^4}+\cdots \right)^n
\]
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{Fourier反転公式を用いない方法}
\label{sec:pconv-2}
ガンマ函数の定義より,
\[
n! = \Gamma(n+1)=\int_0^\infty e^{-x} x^n\,dx.
\]
積分変数を $x=n+\sqrt{n}\,y=n(1+y/\sqrt{n})$ によって $y$ に変換すると,
\[
n! = n^n e^{-n}\sqrt{n} \int_{-\sqrt{n}}^\infty e^{-\sqrt{n}\,y}\left(1+\frac{y}{\sqrt{n}}\right)^n\,dy.
\]
ゆえに
\[
c_n = \frac{n!}{n^n e^{-n}\sqrt{n}},
\qquad
h_n(y) =
\begin{cases}
e^{-\sqrt{n}\,y}(1+y/\sqrt{n})^n & (y>\sqrt{n}), \\
0 & (y\leqq -\sqrt{n}).
\end{cases}
\]
とおくと, $c_n=\int_{-\infty}^\infty h_n(y)\,dy$ となる.
$\log h_n(y)$ の $y=0$ における Taylor 展開によって
$\log h_n(y) = -y^2/2 + o(1)$ ($n\to\infty$)
となることがわかるので, $\lim_{n\to\infty} h_n(y)=e^{-y^2/2}$ となることがわかる.
さらに
\[
\lim_{n\to\infty}\int_{-\infty}^\infty h_n(y)\,dy
=\int_{-\infty}^\infty e^{-y^2/2}\,dy
%\tag{$\$$}
\]
という積分と極限の順序の交換を示すことができれば%
\footnote{$y\geqq 0$ で $h_n(y)\leqq h_1(y)=e^{-y}(1+y)$ が,
$y\leqq 0$ で $h_n(y)\leqq e^{-y^2/2}$ が成立しているので,
Lebesgueの収束定理を使えば容易に示すことができる.
Lebesgueの収束定理を使わなくても, $|y|\leqq M$ で $h_n$ が
一様収束することを用いて示すこともできる.},
$\lim_{n\to\infty}c_n=\sqrt{2\pi}$ が得られる. すなわちStirlingの公式
\[
\lim_{n\to\infty} \frac{n!}{n^n e^{-n} \sqrt{2\pi n}}=1
\]
が得られる. この筋道であればFourier解析の知識は必要ではなくなる.
%積分と極限の順序交換をLebesgueの収束定理で示すためには
%\[
%0\leqq h_n(y)\leqq
%\begin{cases}
%e^{-y}(1+y) & (y\geqq 0), \\
%e^{-y^2/2} & (y\leqq 0).
%\end{cases}
%\]
%を示せば十分である($\phi(y)$ は可積分函数).
%$y>-\sqrt{n}$ とし, $l_n(y)=\log h_n(y)$ を微分すると,
%\begin{align*}
%&
%l'_n(y)
%=\frac{\sqrt{n}}{1+y/\sqrt{n}}-\sqrt{n}
%=\frac{-y}{1+y/\sqrt{n}},
%\\ &
%l''_n(y)=\frac{-1}{(1+y/\sqrt{n})^2}<0,
%\\ &
%%\qquad
%l'''_n(y)=\frac{2/\sqrt{n}}{(1+y/\sqrt{n})^3}>0,
%\\ &
%l_n(0)=0, \qquad\;
%l'_n(0)=0, \qquad\;
%l''_n(1)=-1.
%\end{align*}
%Taylorの定理より, 各 $y>-\sqrt{n}$ ごとにある $0<\theta<1$ が存在して,
%\[
%l_n(y) = -\frac{y^2}{2} + Ay^3, \qquad
%A = \frac{1}{3!}l'''_n(\theta y) =
%\frac{1}{3\sqrt{n}(1+\theta y/\sqrt{n})^3} > 0.
%\]
%これより $\lim_{n\to\infty}l_n(y)=-y^2/2$.
%ゆえに $\lim_{n\to\infty}h_n(y)=e^{-y^2/2}$ となることがわかる.
%
%$y\leqq 0$ のとき,
%$Ay^3\leqq 0$ なので $l_n(y)\leqq e^{-y^2/2}$ となるので,
%$h_n(y)\leqq e^{-y^2/2}$.
%
%$y\geqq 0$ と仮定し, $l_1(y)=\log(e^{-y}(1+y))$ と $l_n(y)$ ($n\geqq 1$)を比較しよう.
%まず $l_1(0)=l_n(0)$ である.
%そして $l'_1(y)=-y/(1+y)$, $l'_n(y)=-y/(1+y/\sqrt{n})$
%の分母を比較すると,
%$\sqrt{n}\geqq 1$ より $1+y\geqq 1+y/\sqrt{n}$ なので,
%$l_1'(y)\geqq l'_n(y)$ ($y\geqq 0$) となる.
%ゆえに, $y\geqq 0$ のとき $l_1(y)\geqq l_n(y)$ となる.
%すなわち $h_n(y)\leqq h_1(y)=e^{-y}(1+y)$ となる.
%
%これで示すべきことが示された.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
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}
以上では確率分布の「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 $f$ に収束する可積分函数列であり,
ある可積分函数 $\varphi\geqq 0$ ですべての $n$ について $|f_n|\leqq\varphi$ を
満たすものが存在するとき,
$f_n$ の収束先の $f$ も可積分函数になり,
積分 $\int_\R f_n(x)\,dx$ は $n\to\infty$ で
$\int_\R f(x)\,dx$ に収束する.
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
\[
\lim_{N\to\infty}s_N(f)(0)
=\lim_{N\to\infty}
\frac{1}{\pi}\int_0^\delta \sin(Ny)\frac{2e^{-y^2/2}}{y}\,dy
=e^{-0^2/2}=1.
\]
ゆえに
\[
\lim_{N\to\infty}
\left(
\int_0^\delta \frac{\sin(Ny)}{y}\,dy
+ \int_0^\delta \sin(Ny)\frac{e^{-y^2/2}-1}{y}\,dy
\right)
=\frac{\pi}{2}.
\]
左辺の後者の積分はRiemann-Lebesgueの定理より $N\to\infty$ で $0$ 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 $a>0$ に対する $ax$ で置換し,
両辺を $\pm 1$ 倍することによって
\[
\lim_{R\to\infty}\int_0^R \frac{\sin(\pm ax)}{x}\,dx = \pm\frac{\pi}{2}
\qquad (a>0,\ \text{復号同順}).
\]
すなわち次が成立している:
\[
\lim_{R\to\infty}\int_0^R \frac{\sin(ax)}{x}\,dx
=
\begin{cases}
\pi/2 & (a>0), \\
0 & (a=0), \\
-\pi/2 & (a<0).
\end{cases}
\]
Dirichlet積分の公式はこの形で使われることが多い.
$\R$ 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 $f$ が点 $x$ 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 $f$ が微分可能な点 $x$ において,
$\lim_{N\to\infty} s_N(f)(x)=f(x)$ 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}
\begin{example}
$a>1$ に対して函数 $f_a(x)$ を次のように定める:
\[
f_a(x)=
\begin{cases}
1/(2a) & (-a<x<a), \\
1/(4a) & (x=\pm a), \\
0 & (x<-a\ \text{または}\ a<x).
\end{cases}
\]
このとき
\[
\widehat{f_a}(p)
=\frac{1}{2a}\int_{-a}^a e^{-ipx}\,dx
=\frac{e^{-iap}-e^{iap}}{-2iap}=\frac{\sin(ap)}{ap}.
\]
これは偶函数である.
ゆえにFourier変換の $N$ 部分和は次のようになる:
\begin{align*}
s_N(f_a)(x)
&= \frac{1}{2\pi}\int_{-N}^N e^{ixp}\frac{\sin(ap)}{ap}\,dp
%\\ &
%= \frac{1}{2\pi a}\int_{-N}^N \cos(xp)\frac{\sin(ap)}{p}\,dp
= \frac{2}{2\pi a}\int_0^N \cos(xp)\frac{\sin(ap)}{p}\,dp
\\ &
= \frac{1}{2\pi a}\int_0^N \frac{\sin((a+x)p)+\sin((a-x)p)}{p}\,dp
\end{align*}
これの $N\to\infty$ での極限は2つのDirichlet積分の和の $2\pi a$ 分の $1$ になる.
1つ目のDirichlet積分は
$x>-a$ のとき $\pi/2$ になり, $x=-a$ のとき $0$ になり, $x<-a$ のとき $-\pi/2$ になり,
2つ目のDirichlet積分は
$x<a$ のとき $\pi/2$ になり, $x=a$ のとき $0$ なり, $x>a$ のとき $-\pi/2$ になる.
それらの和は $-a<x<a$ のとき $\pi$ になり, $x=\pm a$ のとき $\pi/2$ になり,
$x<-a$ または $a<x$ のとき $0$ になる. ゆえに
\[
\lim_{N\to\infty} s_N(f_a)=f_a(x)
\]
となることがわかる.
\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,
\[
\pi\cot(\pi t)
=\frac{1}{t} + \sum_{n=1}^\infty\frac{2t}{t^2-n^2}
=\frac{1}{t} + \sum_{n=1}^\infty\left(\frac{1}{t-n}+\frac{1}{t+n}\right)
\]
を得る%
\footnote{$\coth z=-i\cot(-iz)$ より,
\(\displaystyle
\coth(\pi t)=-i\pi\cot(-\pi i t)
=\frac{1}{t} + \sum_{n=1}^\infty\frac{2t}{t^2+n^2}.
\)}.
$\sin(\pi t)$ 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, 次が得られる%
\footnote{$\sinh z=i\sin(-iz)$ より,
\(\displaystyle
\sinh(\pi s)=\pi s \prod_{n=1}^\infty\left(1+\frac{s^2}{n^2}\right).
\)
}
\[
\sin(\pi s)
=\pi s \prod_{n=1}^\infty\left( 1-\frac{s^2}{n^2} \right).
\]
$\sin$ 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,
\[
1=\sin\left(\frac{\pi}{2}\right)
=\frac{\pi}{2}\prod_{n=1}^\infty\left(1-\frac{1}{(2n)^2}\right)
=\frac{\pi}{2}\prod_{n=1}^\infty\frac{(2n-1)(2n+1)}{2n\cdot 2n}.
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Tauber型定理を用いたWallisの公式の証明}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{付録: 様々な確率分布について}
\label{sec:dists}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{正規分布}
\label{sec:normal}
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}
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}
\label{sec:Gamma}
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:
\[
f_{\alpha,\tau}(x)\,dx
=\frac{e^{-x/\tau}x^{\alpha-1}}{\Gamma(\alpha)\tau^\alpha}\,dx
=\frac{e^{-x/\tau}(x/\tau)^\alpha}{\Gamma(\alpha)}\frac{dx}{x}
\qquad (x>0).
\]
平均は $x=\alpha\tau$, 分散は $\alpha\tau^2$ であり,
$\alpha\geqq 0$ のとき最頻値は $x=(\alpha-1)\tau$ になる.
\paragraph{再生性}
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}
\label{sec:Beta2nd}
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.
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}
\label{sec:Beta1st}
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 $x=\alpha/(\alpha+\beta)$,
分散は $(\alpha\beta)/((\alpha+\beta)^2(\alpha+\beta+1))$ になり,
$\alpha,\beta>1$ のとき最頻値は $x=(\alpha-1)/(\alpha+\beta-2)$ 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.
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
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Maxwell-Boltzmann則 (1)}
\label{sec:MB1}
$X_i$ 達は独立な標準正規分布であるとし, $R_n=\sqrt{X_1^2+\cdots+X_n^2}$,
$Z^{(n)}_i=X_i/R_n$ とおく.
このとき $(Z^{(n)}_1,\ldots,Z^{(n)}_n)$ 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 $Z^{(n)}_i$ 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-3)/2}\,dz
= B\left( \frac{1}{2},\frac{n-1}{2} \right)
= 2^{n-2} B\left(\frac{n-1}{2},\frac{n-1}{2}\right)
\end{align*}
になる. 以下, これを示そう.
$n-2$ 次元単位球面 $S^{n-2}=\{\,(x_2,\ldots,x_n)\mid x_2^2+\cdots+x_n^2=1\,\}$
の面積要素を $d\omega'$ と書き,
$r'=\sqrt{x_2^2+\cdots+x_n^2}$ と置き,
$x_2,\ldots,x_n$ から $r'$ と $n-2$ 次元単位球面上の座標の組に変数変換すると,
半径 $r'$ の $n-2$ 次元球面の面積は $r'^{n-2}$ に比例するので,
\[
dx_1\wedge dx_2\wedge\cdots\wedge dx_n
=r'^{n-2}dx_1\wedge dr'\wedge d\omega'.
\]
さらに, $r'$ から $r=\sqrt{x_1^2+\cdots+x_n^2}$ に変数変換すると,
$r'=\sqrt{r^2-x_1^2}$, $\d r'/\d r=r/r'$ なので,
\[
dx_1\wedge dx_2\wedge\cdots\wedge dx_n
=r(r^2-x_1^2)^{(n-3)/2}\,dx_1\wedge dr\wedge d\omega'.
\]
最後に $x_1$ から $z=x_1/r$ に変数変換すると,
\[
dx_1\wedge dx_2\wedge\cdots\wedge dx_n
=r^{n-1}(1-z^2)^{(n-3)/2}\,dz\wedge dr\wedge d\omega'.
\]
したがって, $\R^n$ 上の球対称な確率密度函数 $\rho(r)$ に対して,
\[
\int_{\R^n} g(z)\rho(r)\,dx_1\cdots dx_n
=
\int_{-1}^1 g(z)(1-z^2)^{(n-3)/2}\,dz
\int_0^\infty r^{n-1}\rho(r)\,dr
\int_{S^{n-2}}d\omega'.
\]
後ろの2つの積分の積を $c_n^{-1}$ と書くと,
\[
c_n=\int_{-1}^1 (1-z^2)^{(n-3)/2}\,dz
\]
$c_n$ を2通りの方法で計算しよう.
1つ目は $z=t^{1/2}$, $dz=t^{-1/2}\,dt/2$ と変数変換する方法である:
\[
c_n
=2\int_0^1 (1-z^2)^{(n-3)/2}\,dz
=\int_0^1 t^{-1/2}(1-t)^{(n-3)/2}\,dt
=B\left(\frac{1}{2},\frac{n-1}{2}\right).
\]
2つ目は $(1-z^2)=(1+z)(1-z)$ と因数分解し,
$z=2t-1$, $dz=2\,dt$ と変数変換する方法である:
\[
c_n
=\int_0^1 2^{(n-3)/2}t^{(n-3)/2}2^{(n-3)/2}(1-t)^{(n-3)/2}2\,dt
=2^{n-2}B\left(\frac{n-1}{2},\frac{n-1}{2}\right).
\]
これで示すべきことがすべて示された.
副産物として, ガンマ函数の duplication formula も
得られていることを注意しておこう.
$(n-1)/2$ を任意の正の実数 $s$ に置き換えても
$c_n$ の二通りの表示は成立している:
\[
\int_{-1}^1 (1-z^2)^{s-1}\,dz
=B(1/2,s)
=2^{2s-1}B(s,s).
\]
ベータ函数にガンマ函数を代入すると
\[
\frac{\Gamma(1/2)\Gamma(s)}{\Gamma(s+1/2)}
=\frac{2^{2s-1}\Gamma(s)^2}{\Gamma(2s)}.
\]
すなわち, $\Gamma(1/2)=\sqrt{\pi}$ より,
\[
\Gamma(2s) = \frac{2^{2s-1}}{\sqrt{\pi}}\Gamma(s)\Gamma(s+1/2).
\]
この公式は (Legendre's) duplication formula と呼ばれている%
\footnote{Legendre's duplication formula は
任意の正の整数 $n$ に対する次の Gauss's multiplication theorem に一般化される:
\[
\Gamma(ns)=\frac{n^{ns-1/2}}{(2\pi)^{(n-1)/2}}
\Gamma(s)\Gamma(s+1/n)\Gamma(s+2/n)\cdots\Gamma(s+(n-1)/n).
\]
たとえば \(
\Gamma(3s)=3^{3s-1/2}\Gamma(s)\Gamma(s+1/3)\Gamma(s+2/3)/(2\pi)
\).
}.
$Z^{(n)}_i$ の確率密度函数の例%
\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.}:
\begin{itemize}
\item $g_2(z)\,dz = \dfrac{1}{\pi}\dfrac{dz}{\sqrt{1-z^2}}$ \;\;\;\quad ($-1<z<1$).
\qquad 平均 $0$, 分散 $1/2$.
\\[\smallskipamount]
$z=\sin\theta$ を代入すると,
$\dfrac{1}{\pi}\,d\theta$ ($-\pi/2\leqq\theta\leqq\pi/2$) と一様分布になる(当たり前).
ゆえに累積分布函数は $1/2+\theta/\pi=1/2+(\arcsin z)/\pi$ ($-1\leqq z\leqq 1$) になる.
逆正弦函数が出て来るのでこの分布は{\bf 逆正弦分布}と呼ばれる%
\footnote{\label{fn:arcsin}%
ギャンブルをやり続けるとき,
トータルで勝ち越している状態の時間の長さの総和から
負け越している状態の時間の長さの総和を引いた結果の確率分布は
適当に規格化すると逆正弦分布に近付くことが知られている.
これは{\bf 逆正弦法則}と呼ばれている.
逆正弦分布の確率密度函数は両端に近付くほど大きくなり,
真ん中の $0$ 付近は小さくなる.
ゆえに, 逆正弦法則は勝ち越している時間と負け越している時間の差の絶対値
は $0$ 付近に留まらずに大きくなる傾向が強いということを意味している.
ギャンブル好きならばこの事実を経験的によく知っているはずである.
単なる偶然で, 勝ち続けたり, 負け続けたりすることの方が多い.
}.
\item $g_3(z)\,dz = \dfrac{1}{2}\,dz$ \;\;\qquad\qquad ($-1\leqq z\leqq 1$).
\qquad 平均 $0$, 分散 $1/3$.
\\[\smallskipamount]
2次元球面上の一様分布の原点を通る直線上への射影は一様分布になる.
\item $g_4(z)\,dz = \dfrac{2}{\pi}\sqrt{1-z^2}\,dz$ \quad ($-1\leqq z\leqq 1$).
\qquad 平均 $0$, 分散 $1/4$.
\\[\smallskipamount]
この分布は{\bf 半円分布}と呼ばれる%
\footnote{\label{fn:Wigner}%
半円分布は行列模型における固有値の分布密度に関する
{\bf Wignerの半円則}に現われる.
$N$ 次実対称行列に値を持つ確率変数 $M$ の確率密度函数
は $\prod_i e^{-M_{ii}^2/2}dM_{ii} \prod_{i<j}e^{-M_{ij}^2/2}dM_{ij}$
に比例していると仮定し, ランダムな実対称行列 $M$ の固有値の確率分布を
考える. そのとき, スケール変換によって分散が $1/4$ になるように規格化すると,
その確率分布は $N\to\infty$ で分散 $1/4$ の半円分布に収束するというのが
Wignerの半円則である.
半円分布は量子中心極限定理における収束先として現われる典型的な確率分布である.
たとえば,
\href{http://www.math.is.tohoku.ac.jp/~obata/student/graduate/file/2013-Meijo-QP-Graph.pdf}
{尾畑伸明, 量子確率論とその応用, 無限次元解析特論(名城大学, 2013.10)}に解説がある.
}.
\\
$z=-\cos\theta$ を代入すると, $\sin^2$ 型分布
$\dfrac{2}{\pi}\sin^2\theta\,d\theta$ ($0\leqq\theta\leqq\pi$) になる%
\footnote{\label{fn:Sato-Tate}%
{\bf 佐藤・Tate予想}にこの型の分布が登場する.
佐藤・Tate予想とは
「有理数体上定義された虚数乗法を持たない楕円曲線の
素数位数 $p$ の有限体上での有理点の個数
から $p+1$ を引いて $2\sqrt{p}$ で割って得られる数値の分布が $\sin^2$ 型分布になる」
という内容の1960年代に独立に発見された予想である.
\href{http://www.math.ias.edu/~rtaylor/}{現在では完全に解決されている}らしい.
\href{http://www.kurims.kyoto-u.ac.jp/~gokun/R=T.html}
{Ｒ＝Ｔの最近の発展についての勉強会(2008)}の報告集にまとまった解説がある.
佐藤幹夫氏の側がどのように「佐藤 $\sin^2$ 予想」を発見したかについては,
\href{http://www2.tsuda.ac.jp/suukeiken/math/suugakushi/sympo16/16_8nanba.pdf}
{難波莞爾, Dedekind η 関数と佐藤 sin2-予想, 第16回数学史シンポジウム,
津田塾大学 (2005)}に詳しい. 当時まだ大学院生だった難波莞爾さんが
コンピューターで遊んでいることを佐藤先生らにビアガーデンで話したときについて
「少し意味のある計算をやってみませんか、ということになった。
それで、楕円母数形式、志村・谷山…などの概念や文字列と遭遇することになった
のである」と書いてある. その「少し意味のある計算」の積み重ねによって
「佐藤 $\sin^2$ 予想」が発見された.
$SU(2)$ 上の一様分布(Haar測度)から誘導される $SU(2)$ の共役類全体の
空間上の分布は $\sin^2$ 型分布になる. その理由は以下の通り.
$A\in SU(2)$ の共役類は $-1\leqq\operatorname{tr}(A)/2\leqq 1$
で一意に特徴付けられる. (一般に $GL_r(\C)$ のコンパクトLie部分群の
元の共役類はその特性多項式(すなわち固有値たち)で一意に特徴づけられる.)
$A\in SU(2)$ に $\operatorname{tr}(A)/2$ を
対応させる写像は, $SU=S^3\subset\R^4$ という同一視のもとで,
$S^3$ から $\R^4$ の $1$ 次元部分空間への射影に一致している.
このことから $SU(2)$ 上の一様分布がその共役類全体の空間上に
誘導する分布は確率密度函数は $\sin^2$ 型分布になることがわかる.
佐藤・Tate予想は「有理数体上の虚数乗法を持たない楕円曲線から
各素数 $p$ ごとに得られる $SU(2)$ の共役類達が
$3$ 次元球面 $S^3=SU(2)$ 上の一様分布から誘導される分布にしたがっている」
という話であるとみなせる.
}.
\end{itemize}
$n\geqq 4$ のとき $g_n(z)$ はグラフが釣鐘型の函数になる.
平均はどれも $0$ で分散は以下で示すように $1/n$ になる.
$Z^{(n)}_i$ 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, $Z^{(n)}_i$ の分散が $1/n$ になることを示せる:
\[
c_n^{-1}\int_{-1}^1 z^2(1-z^2)^{(n-3)/2}\,dz
=c_n^{-1}(c_n-c_{n+2})
=1-\frac{c_n}{c_{n+1}}
=\frac{1}{n}.
\]
ここで $z^2$ に $1-(1-z^2)$ を代入する計算を行った.
$Y^{(n)}_i=\sqrt{n}\,Z^{(n)}_i$ は平均 $0$, 分散 $1$ の確率変数になり, その確率密度函数は
\[
g_n\left(\frac{y}{\sqrt{n}}\right)\frac{dy}{\sqrt{n}}
=\frac{1}{\sqrt{n}\,c_n} \left(1-\frac{y^2}{n}\right)^{(n-3)/2}\,dy
\]
になる. $n\to\infty$ 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, $Y^{(n)}_i$ 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}
\[
\lim_{n\to\infty}
\frac{1}{\sqrt{n}}g_n\left(\frac{y}{\sqrt{n}}\right)
=\lim_{n\to\infty}
%\frac{\left(1-\dfrac{y^2}{n}\right)^{(n-3)/2}}{\sqrt{n}\,2^{n-2}B(\frac{n-1}{2},\frac{n-1}{2})}
\frac{(1-y^2/n)^{(n-3)/2}}{\sqrt{n}\,2^{n-2}B(\frac{n-1}{2},\frac{n-1}{2})}
=\frac{e^{-y^2/2}}{\sqrt{2\pi}}.
\]
以上をまとめると, 実数 $y$ の有界連続函数 $g(y)$ について,
\[
C_n^{-1}\int_{\sqrt{n}\,S^{n-1}} g(y_i) \,d\omega_n
\longrightarrow
\int_\R g(y)\frac{e^{-y^2/2}}{\sqrt{2\pi}}\,dy
\qquad (n\to\infty).
\]
ここで, $\sqrt{n}\,S^{n-1}=\{\,(y_1,\ldots,y_n)\in\R^n\mid y_1^2+\cdots+y_n^2=n \,\}$
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 Maxwell-Boltzmann則}としてよく知られている.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$n-1$ 次元球面上の一様分布とMaxwell-Boltzmann則 (2)}
\label{sec:MB2}
前節では半径 $\sqrt{n}$ の $n-1$ 次元球面上の一様分布
の $x_i$ 軸への射影の極限が標準正規分布になることを証明した.
同様の方法で,
半径 $\sqrt{n}$ の $n-1$ 次元球面上の一様分布の
$m$ 次元部分空間への射影が $m$ 次元の標準正規分布に収束することも示せる.
以下でその筋道を簡単に説明しておく.
前節の記号をそのまま引き継ぐ.
$n-m-1$ 次元単位球面 $S^{n-m-1}=\{\,(x_{m+1},\ldots,x_n)\mid x_{m+1}^2+\cdots+x_n^2=1\,\}$
の面積要素を $d\omega'$ と書き,
$r'=\sqrt{x_{m+1}^2+\cdots+x_n^2}$ と置き,
$x_{m+1},\ldots,x_n$ から $r'$ と $n-m-1$ 次元単位球面上の座標の組に変数変換すると
\[
dx_1\wedge\cdots\wedge dx_n
=r'^{n-m-1}dx_1\wedge\cdots\wedge dx_m\wedge dr'\wedge d\omega'.
\]
さらに, $r'$ から $r=\sqrt{x_1^2+\cdots+x_n^2}$ に変数変換すると,
$r'=\sqrt{r^2-x_1^2-\cdots-x_m^2}$ かつ $\d r'/\d r=r/r'^{-1}$ なので,
\[
dx_1\wedge\cdots\wedge dx_n
=r(r^2-x_1^2)^{(n-m-2)/2}\,dx_1\wedge\cdots\wedge dx_m\wedge dr\wedge d\omega'.
\]
最後に $x_i$ ($i=1,\ldots,m$) から $z_i=x_1/r$ ($i=1,\ldots,m$) に変数変換すると,
\[
dx_1\wedge\cdots\wedge dx_n
=r^{n-1}(1-z_1^2-\cdots-z_m^2)^{(n-m-2)/2}\,dz\wedge dr\wedge d\omega'.
\]
したがって, 球対称な確率密度函数 $\rho(r)$ に対して,
\begin{align*}
&
\int_{\R^n} g(z_1,\ldots,z_m)\rho(r)\,dx_1\cdots dx_n
\\ & \qquad
={c^{(n)}_{m}}^{-1}
\int_{z_1^2+\cdots+z_m^2<1}
g(z_1,\ldots,z_m)(1-z_1^2-\cdots-z_m^2)^{(n-m-2)/2}\,dz_1\cdots dz_m.
\tag{$*$}
\end{align*}
ここで
\[
{c^{(n)}_m}^{-1}
=
\int_0^\infty r^{n-1}\rho(r)\,dr\,
\int_{S^{n-m-1}}d\omega'
\]
である. もっとも極端な場合として $m=0$ の場合を考えると $c^{(n)}_0=1$ と
なる. このことより, $\rho(r)=e^{-r^2/2}/(2\pi)^{n/2}$ とすることによって,
$n-1$ 次元単位球面の面積は
\[
\int_{S^{n-1}}d\omega
=(2\pi)^{n/2}\left(\int_0^\infty r^{n-1} e^{-r^2/2}\,dr \right)^{-1}
=\frac{2^{n/2}\pi^{n/2}}{2^{n/2-1}\Gamma(n/2)}
=\frac{2\pi^{n/2}}{\Gamma(n/2)}
=\frac{n\pi^{n/2}}{\Gamma(n/2+1)}
\]
と計算される($d\omega$ は $n-1$ 次元単位球面 $S^{n-1}$ の面積要素).
次の公式を使った:
\[
\int_0^\infty r^{s-1}e^{-r^2/2}\,dr
=\int_0^\infty e^{-t}(2t)^{(s-2)/2}\,dt
=2^{s/2-1}\Gamma(s/2).
\]
積分変数を $r^2/2=t$, $r\,dr=dt$, $r^{s-1}\,dr=r^{s-2}\,r\,dr$ と変換
すればこの公式が得られる.
以上より, $\int_0^\infty r^{n-1}\rho(r)\,dr$ は常に $n-1$ 次元単位球面の
面積の逆数になることもわかる. したがって,
\[
c^{(n)}_m
=\frac{\int_{S^{n-1}}d\omega}{\int_{S^{n-m-1}}d\omega'}
=\frac{(\text{$n-1$ 次元単位球面の面積})}{(\text{$n-m-1$ 次元単位球面の面積})}.
\]
これが定数 $c^{(n)}_m$ の幾何学的意味である.
定数 $c^{(n)}_{m}$ は以下のように計算される%
\footnote{$n=m+2$ のとき $c_m^{(m+2)}=\pi^{m/2}/\Gamma(m/2+1)$
は $m$ 次元単位球体の体積に等しい.}:
\begin{align*}
c^{(n)}_{m}
&=
\int_{z_1^2+\cdots+z_m^2<1}
(1-z_1^2-\cdots-z_m^2)^{(n-m-2)/2}\,dz_1\cdots dz_m.
\\ &
= \int_{t_i>0,\, \sum_{i=1}^m t_i<1}
t_1^{-1/2}\cdots t_m^{-1/2}
(1-t_1-\cdots-t_m)^{(n-m-2)/2}\,dt_1\cdots dt_m
\\ &
= \frac{\Gamma(1/2)^m\Gamma((n-m)/2)}{\Gamma(n/2)}.
\end{align*}
2つ目の等号で $z_i=\sqrt{t_i}$ と変数変換し,
最後の等号で次の公式を使った: $p_i>0$ に対して,
\begin{align*}
&
\frac{\Gamma(p_1)\cdots\Gamma(p_{m+1})}{\Gamma(p_1+\cdots+p_{m+1})}
%\\ &
=
\int_{t_i>0,\, \sum_{i=1}^m t_i<1}
t_1^{p_1-1}\cdots t_m^{p_m-1}
(1-t_1-\cdots-t_m)^{p_{m+1}-1}\,dt_1\cdots dt_m.
\end{align*}
証明の方法はガンマ函数とベータ函数の関係とまったく同様である.
もしくは右辺を $B(p_1,\ldots,p_{m+1})$ と書くと,
\[
B(p_1,\ldots,p_{m+1})
=B(p_1,\ldots,p_{m-1},p_m+p_{m+1})B(p_m,p_{m+1})
\tag{B}
\]
が成立することから, 帰納法で証明することもできる.
実際, $t_m=(1-t_1-\cdots-t_{m-1})u$ によって $t_m$ から $u$ に変数変換すると
\begin{align*}
&
B(p_1,\ldots,p_m,p_{m+1})
\\ & \qquad
=
\int_{t_i>0,\;\sum_{i=1}^{m-1}t_i<1}
dt_1\cdots dt_{m-1}
\int_{-1}^1 du
\\ & \qquad\qquad
t_1^{p_1-1}\cdots t_{m-1}^{p_{m-1}-1}
(1-t_1-\cdots-t_{m-1})^{p_m+p_{m+1}-1}
u^{p_m-1}(1-u)^{p_{m+1}-1}.
\end{align*}
これより上の公式(B)が成立することがわかる.
公式($*$)より, ベクトル値確率変数 $(Z^{(n)}_1,\ldots,Z^{(n)}_m)$ の
確率密度函数は
\[
g_n(z_1,\ldots,z_m)\,dz_1\cdots dz_m
=
{c^{(n)}_{m}}^{-1}(1-z_1^2-\cdots-z_m^2)^{(n-m-2)/2}\,dz_1\cdots dz_m
\]
である.
これより, $\sigma>0$ に対して, \(
(Y^{(n)}_1,\ldots,Y^{(n)}_m)
=\sqrt{n}\,\sigma\,(Z^{(n)}_1,\ldots,Z^{(n)}_m)
\) の確率密度函数は
\[
\left(1-\frac{1}{n\sigma^2}\sum_{i=1}^m y_i^2 \right)^{(n-m-2)/2}\,dy_1\cdots dy_m
\]
の定数倍になる%
\footnote{$Y^{(n)}_i$ たちは独立ではないことに注意せよ.}. そして,
\[
\lim_{n\to\infty}
\left(1-\frac{1}{n\sigma^2}\sum_{i=1}^m y_i^2 \right)^{\frac{n-m-2}{2}}
=\exp\left( -\frac{1}{2\sigma^2} \sum_{i=1}^m y_i^2 \right)
\]
なので $(Y^{(n)}_1,\ldots,Y^{(n)}_m)$ は $n\to\infty$ で
$m$ 次元の正規分布にしたがうベクトル値確率変数に収束する%
\footnote{$Y^{(n)}_i$ 達は有限な $n$ で独立ではないが,
$n\to\infty$ の極限で独立な標準正規分布に収束する.}.
すなわち,
\begin{align*}
&
\frac{1}{C_n(\sqrt{n}\,\sigma)}
\int_{\sqrt{n}\,\sigma\,S^{n-1}} g(y_1,\ldots,y_m) \,d\omega_n
\\ &
\longrightarrow
\frac{1}{(2\sigma^2)^{m/2}}
\int_{\R^m} g(y_1,\ldots,y_m)
\exp\left( -\frac{1}{2\sigma^2}\sum_{i=1}^my_i^2 \right)\,dy_1\cdots dy_m.
\end{align*}
ここで, \(
\sqrt{n}\,\sigma\,S^{n-1}
=\{\,(y_1,\ldots,y_n)\in\R^n\mid y_1^2+\cdots+y_n^2=n\sigma^2 \,\}
\) は半径 $\sqrt{n}\,\sigma$ の $n-1$ 次元球面であり,
$C_n(\sqrt{n}\,\sigma)$ 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.
これは物理的には{\bf Maxwell-Boltzmann則}としてよく知られており,
分散 $\sigma^2$ は絶対温度のBoltzmann定数倍 $kT$ だと解釈される.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{二項分布と第一種ベータ分布}
\label{sec:Bin-Beta}
$0<p<1$ とする.
$n$ は非負の整数であるとする.
離散型確率変数 $B_{p,n}$ がパラメーター $n$ と $p$ の二項分布にしたがうとは
\[
P(B_{p,n}=k) = \binom{n}{k}p^k(1-p)^{n-k}
\qquad (k=0,1,2,\ldots,n)
\]
が成立することであると定める. 平均と分散はそれぞれ $np$ と $np(1-p)$ になり,
特性函数は $E[e^{itB_{p,n}}=(pe^{it}+q)^n$ となる.
二項分布はパラメーター $n$ に関して再生性を持つ.
ゆえに中心極限定理より, $p$ を一定のまま $n$ を大きくすると,
$(B_{p,n}-np)/\sqrt{np(1-p)}$ は標準正規分布にしたがう確率変数で近似される.
二項分布と第一種ベータ分布の関係は以下の通り.
$\Gamma(s+1)=s!$, $\binom{s}{t}=s!/(t!(s-t)!)$ と書くことにすると
\[
\frac{1}{B(\alpha,\beta)}
=\frac{(\alpha+\beta-1)!}{(\alpha-1)!(\beta-1)!}
=(\alpha+\beta-1)\binom{\alpha+\beta-2}{\alpha-1}
\]
なので, パラメーター $\alpha,\beta>0$ を持つ第一種ベータ分布の確率密度函数は
\[
f_{\alpha,\beta}(p)\,dp
=(\alpha+\beta-1)\binom{\alpha+\beta-2}{\alpha-1} p^{\alpha-1}(1-p)^{\beta-1}\,dp
\qquad (0<p<1)
\]
と表される.
平均は $\alpha/(\alpha+\beta)$,
分散は $(\alpha\beta)/((\alpha+\beta)^2(\alpha+\beta+1))$ になり,
$\alpha,\beta>1$ のとき最頻値は $p=(\alpha-1)/(\alpha+\beta-2)$ になるのであった.
ゆえに $\alpha+\beta-2=n$, $\alpha-1=k$ のとき,
第一種ベータ分布の確率密度函数は
\[
f_{k+1,n-k+1}(p)\,dp
=(n+1)\binom{n}{k} p^k(1-p)^{n-k}\,dp
\qquad (0<p<1)
\]
となり, 平均値は $p=(k+1)/(n+2)$, 分散は $((k+1)(n-k+1))/((n+2)^2(n+2))$,
最頻値は $p=k/n$ になる%
\footnote{$k\sim np$ ($n\to\infty$, $p$ は一定) ならば, $n\to\infty$ で
平均値と最頻値は $p$ に収束し, 分散は $0$ に収束する.}.
以上の結果から, AがBと $n$ 回対戦して $k$ 回勝ったとき,
AがBに勝つ確率はパラメーターが $\alpha=k+1$, $\beta=n-k+1$ の
第一種ベータ分布にしたがっているとみなすと便利なことがわかる%
\footnote{共役事前分布の話.}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Poisson分布とガンマ分布}
\label{sec:Poisson-Gamma}
離散型確率変数 $N_{\lambda T}$ が
パラメーター $\lambda T>0$ のPoisson分布にしたがうとは
\[
P(N_{\lambda T}=k) = \frac{e^{-\lambda T}(\lambda T)^k}{k!}
\qquad (k=0,1,2,3,\ldots)
\]
が成立することであると定める.
平均と分散はどちらも $\lambda T$ になる.
$T$ は測定する時間の長さを,
$\lambda$ は単位時間あたりにまれな事象が起こる回数の期待値を意味している.
特性函数は $E[e^{itN_{\lambda T}}]=e^{\lambda T(e^{it}-1)}$ となる.
Poisson分布は $\lambda T$ について再生性を持つ.
ゆえに中心極限定理より, $\lambda T$ を大きくすると,
$(N_{\lambda T}-\lambda T)/\sqrt{\lambda T}$ は標準正規分布にしたがう確率変数で近似される.
Poisson分布とガンマ分布の関係は以下の通り.
次の確率密度函数で定義される確率分布を
shape $\alpha=k+1>0$, scale $\tau=1/T$ のガンマ分布と呼ぶのであった:
\[
f_{k+1,1/T}(\lambda)\,d\lambda
=\frac{e^{-\lambda T}(\lambda T)^k}{k!}\,d\lambda
\qquad (\lambda>0).
\]
平均は $\lambda=(k+1)/T$, 分散は $(k+1)/T^2$ になり,
最頻値は $\lambda=k/T$ になる%
\footnote{$k\sim\lambda T$ ($T\to\infty$, $\lambda$ は一定)
ならば $T\to\infty$ で平均と最頻値は $\lambda$
に収束し, 分散は $0$ に収束する.}.
このことから, $T$ 単位時間の観測でまれな事象が $k$ 回起こったならば,
単位時間あたりにまれな事象が起こる回数の平均値 $\lambda$ の推定値が
shape $\alpha=k+1$, scale $\tau=1/T$ のガンマ分布にしたがっていると
みなすことが十分に合理的なことがわかる.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{基本的な数学用語の大雑把な説明}
確率変数にその期待値(平均)を
対応させる汎函数 $E[\ \ ]$ は以下を満たしている%
\footnote{確率空間 $(\Omega,\mathcal{F},\mu)$ 上の可測函数 $X$ を確率変数と呼ぶ.
可積分函数 $X$ に $\int_\Omega X(x)\,\mu(dx)$ を対応させる汎函数
を期待値汎函数と呼び $E[\ \ ]$ と表わす.}:
\begin{itemize}
\item $E[\alpha X+\beta Y]=\alpha E[X]+\beta E[Y]$ (線形性).
\item $f\geqq 0$ ならば $E[f(X)]\geqq 0$ (単調性).
\item $E[1]=1$ (規格化条件).
\end{itemize}
たったこれだけの性質だけからかなりのことが言える.
確率変数 $X$ の平均値(期待値)が存在するとは $E\bigl[|X|\bigr]<\infty$ となることである.
そのとき $\mu_X=E[X]$ を $X$ の平均値もしくは期待値と呼ぶ.
$X$ の平均値 $\mu_X$ が存在するとき,
$(X-\mu_X)^2$ の平均値を $X$ の分散と呼び,
$\sigma_X^2$ と表わし, 分散の平方根 $\sigma_X$ を標準偏差と呼ぶ.
分散と標準偏差は無限大になることがありえる.
もしも $E\bigl[|X|^r\bigr]<\infty$ ならば $X$ の $r$ 次のモーメントが存在する
と言い, $E[X^r]$ を $X$ の $r$ 次のモーメントと言う.
$X$ の $1$ 次のモーメントは $X$ の平均 $\mu_X=E[X]$ であり,
$2$ 次のモーメントについて $E[X^2]=\sigma_X^2+\mu_X^2$
なので $\sigma_X^2=E[X^2]-E[X]^2$ となる.
確率変数 $X$ に対して $\varphi_X(t)=E[e^{itX}]$ を $X$ の特性函数と呼ぶ.
特性函数は $t$ について一様連続函数になる.
特性函数が等しい確率変数は確率分布を持つ%
\footnote{確率変数とは確率空間 $(\Omega,\mathcal{F},\mu)$ 上
の実数値可測函数 $X:\Omega\to\R$ のことである.
$\R$ のBorel部分集合 $A$ に対して $\mu_X(A)=\mu(X^{-1}(A))$ と定めることによって,
$\R$ 上の確率測度 $\mu_X$ が定まる. $\mu_X$ を確率変数 $X$ の確率分布と呼ぶ.
もしも $\mu_X$ がLebesgue測度の函数 $f(x)$ 倍と表示されるとき,
$f(x)$ を確率変数 $X$ の確率密度函数と呼ぶ.
$\R$ 上の可測函数 $g(x)$ に対して $X$ と $g$ の合成を $g(X)$ と書く.
$g(X)$ も確率変数になる.
$g(x)$ が有界連続関数のとき,
$g(X)$ の期待値は $E[g(X)]=\int_\R g(x)\,\mu_X(dx)$ と表わされる.
$X$ の確率密度函数 $f(x)$ が存在するならば $E[g(X)]=\int_\R g(x)f(x)\,dx$.
}.
確率変数 $X$, $Y$ が同じ確率分布を持つとき, $X\sim Y$ と書くことにする.
$X$ の $r$ 次以下のモーメントがすべて存在するとき,
特性函数 $\varphi_X(t)$ は $t=0$ で $r$ 回微分可能になり,
$\varphi_X^{(k)}(0)=E[X^k]$ ($k=0,1,\ldots,r$) となる.
$X$ と $Y$ は平均値と有限の分散を持つ確率変数であるとする.
このとき Cauchy-Schwarz の不等式より,
$E[|(X-\mu_X)(Y-\mu_Y)]\leqq\sigma_X\sigma_Y$ となるので,
$\sigma_{XY}=E[(X-\mu_X)(Y-\mu_Y)]$ がwell-definedになり,
$\bigl|E[(X-\mu_X)(Y-\mu_Y)]\bigr|\leqq\sigma_X\sigma_Y$ となる.
$\sigma_{XY}$ を $X$ と $Y$ の共分散と呼ぶ.
$\rho_{XY}=\sigma_{XY}/(\sigma_X\sigma_Y)$ を$X$ と $Y$ の相関係数と呼ぶ.
相関係数の絶対値は $1$ 以下になる.
共分散は線形代数での「ベクトルの内積」に対応し,
相関係数は「ベクトルのあいだの角度を $\theta$ と書くときの $\cos\theta$」
に対応している.
確率変数 $X$ を平均が $0$ になるように値を平行移動した $X-\mu_X$
はベクトルの類似物であり, $E(X-\mu_X)(Y-\mu_Y)]$ が内積の類似物であることを
理解できれば, 線形代数学で学んだことがすべて役に立つ.
確率変数たち $X_i$ が独立であるとは,
$i_1,\ldots,i_r$ が互いに異なるとき,
\[
E[f_1(X_{i_1})\cdots f_r(X_{i_r})]
=
E[f_1(X_{i_1})]\cdots E[f_r(X_{i_r})]
\]
が成立することである($f_k$ たちは有界な連続函数).
$X$ と $Y$ が独立ならば $X$ と $Y$ の共分散と相関係数は $0$ になるが,
逆は成立しない.
$D_\alpha$ はパラメーター $\alpha>0$ を持つ確率変数であるとし,
$X\sim D_\alpha$, $Y\sim D_\beta$ であり, $X,Y$ は独立であるとする.
このとき, もしも $X+Y\sim D_{\alpha+\beta}$ が成立するとき,
$D_\alpha$ の確率分布は再生性を持つと言う.
確率変数 $X_1,\ldots,X_r$ が独立であるとき,
$\varphi_{X_1+\cdots+X_r}=\prod_{i=k}^r\varphi_{X_k}$ が成立する.
ゆえに, $\varphi_{D_\alpha}=\phi^\alpha$ が成立することと,
$D_\alpha$ の確率分布は再生性を持つことは同値である.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{付録: 簡単なTauber型定理とその応用}
\label{sec:Tauber}
\subsection{不定積分型}
\begin{theorem}
$f(t)$ は $t>0$ で定義された正値単調減少%
\footnote{$x\leqq x'$ ならば $f(x)\geqq f(x')$ が成立することを「単調減少」と呼んでいる.
字義通りに解釈できるようにするためには「非増加函数」と呼ぶべきかもしれないが,
慣習に合わせてこのように呼んでいる.}%
函数であるとし, $\alpha,a>0$ であるとする. このとき
\[
\int_0^x f(t)\,dt \sim a x^\alpha \qquad (x\to\infty)
\]
ならば%
\footnote{$F(x)\sim G(x)$ ($x\to\infty$) は
$\lim_{x\to\infty}(F(x)/G(x))=1$ を意味する},
\[
f(x) \sim a \alpha x^{\alpha-1} \qquad (x\to\infty)
\]
が成立する. (この結論の式は前提の式の両辺を形式的に $x$ で微分した形をしている.)
\end{theorem}
\begin{proof}
$f$ が単調減少函数であることより, 任意の $c>1$ に対して,
\[
\frac{\int_0^{cx} f(t)\,dt-\int_0^x f(t)\,dt}{cx-x}
\leqq
f(x)
\leqq
\frac{\int_0^x f(t)\,dt-\int_0^{c^{-1}x} f(t)\,dt}{x-c^{-1}x}.
\]
これの全体を $ax^{\alpha-1}$ で割ると,
\[
\frac{\dfrac{\int_0^{cx} f(t)\,dt}{ax^\alpha}-\dfrac{\int_0^x f(t)\,dt}{ax^\alpha}}{c-1}
\leqq
\frac{f(x)}{ax^{\alpha-1}}
\leqq
\frac{\dfrac{\int_0^x f(t)\,dt}{ax^{\alpha}}-\dfrac{\int_0^{c^{-1}x} f(t)\,dt}{ax^\alpha}}{1-c^{-1}}.
\]
ゆえに $x\to\infty$ とすることによって%
\footnote{$\int_0^{cx}f(t)\,dt\sim ac^\alpha x^\alpha$ ($x\to\infty$) を用いる.},
\[
\frac{c^\alpha-1}{c-1}
\leqq
\liminf_{x\to\infty}\frac{f(x)}{ax^{\alpha-1}}
\leqq
\limsup_{x\to\infty}\frac{f(x)}{ax^{\alpha-1}}
\leqq
\frac{1-c^{-\alpha}}{1-c^{-1}}.
\]
さらに $c\searrow 1$ とすることによって
\[
\alpha
\leqq
\liminf_{x\to\infty}\frac{f(x)}{ax^{\alpha-1}}
\leqq
\limsup_{x\to\infty}\frac{f(x)}{ax^{\alpha-1}}
\leqq
\alpha.
\]
を得る. ゆえに
\[
\lim_{x\to\infty}\frac{f(x)}{ax^{\alpha-1}}=\alpha, \qquad
\text{つまり} \quad f(x)\sim a\alpha x^{\alpha-1} \quad(x\to\infty).
\]
これで示すべきことが示された.
\qed
\end{proof}
正値単調減少数列 $a_n$ に対して $f(n)=a_n$ を満たす正値単調減少函数 $f(t)$ を
適切に定めることによって次の結果が得られる.
\begin{cor}
$a_1,a_2,a_3,\ldots$ は正値単調減少数列であるとし, $a,\alpha>0$ であるとする.
このとき
\[
\sum_{k=1}^n a_k \sim a n^\alpha \qquad (n\to\infty)
\]
ならば
\[
a_n \sim a\alpha n^{\alpha-1} \qquad (n\to\infty)
\]
が成立する.
\qed
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Laplace変換型}
Stone-Weierstrassの多項式近似定理%
\footnote{閉区間上の任意の連続函数が多項式函数で一様近似されるという定理.}%
を用いてまず次を示そう.
\begin{lemma}
\label{lemma:SW}
$\phi(y)$ は閉区間 $[0,1]$ 上の非負値可積分函数であるとし,
$g(y)$ は閉区間 $[0,1]$ 上の函数で一点 $c\in(0,1)$ でのみ
不連続で他の点では連続であるものであるとし,
極限 $g(c\pm 0)=\lim_{\eps\searrow 0}g(c\pm\eps)$ が存在すると仮定する.
このとき, 任意の $\eps>0$ に対して, 多項式函数 $P(y)$, $Q(y)$ で
\begin{align*}
&
P(y)\leqq g(y)\leqq Q(y) \quad (0\leqq y\leqq 1),
\\ &
\int_0^1 g(y)\phi(y)\,dy-\eps
\leqq
\int_0^1 P(y)\phi(y)\,dy
\leqq
\int_0^1 g(y)\phi(y)\,dy,
\\ &
\int_0^1 g(y)\phi(y)\,dy
\leqq
\int_0^1 Q(y)\phi(y)\,dy
\leqq
\int_0^1 g(y)\phi(y)\,dy+\eps
\end{align*}
を満たすものが存在する.
\end{lemma}
\begin{proof}
条件を満たす多項式函数 $Q(y)$ の存在のみを示せばよい.
($g(y)$ の代わりに $-g(y)$ を考えれば $P(y)$ の存在も示される.)
さらに $g(c-0)\leqq g(c+0)$ と仮定してよい.
($g(c-0)\geqq g(c+0)$ ならば $g(y)$ の代わりに $g(1-y)$ を考えればよい.)
$\phi(y)$ は非負値可積分函数なので $N=\int_0^1|\phi(y)|\,dy=\int_0^1\phi(y)\,dy$
とおくと, $N<\infty$ となる.
$g(y)$ は $[0,1]$ 上有界なので, ある $M>0$ で $|g(y)|\leqq M$
($0\leqq y\leqq 1$) をみたすものが取れる.
任意に $\eps>0$ を取る.
$c$ 未満の $\delta>0$ に対して,
$g(y)$ を近似する連続函数 $g_\delta(y)$ を次のように定める:
\[
g_\delta(y)=
\begin{cases}
g(y) & (0\leqq y\leqq c-\delta), \\
\max\{a(y-c)+g(c+0), g(y) \} & (c-\delta\leqq y\leqq c), \\
g(y) & (c\leqq y\leqq 1).
\end{cases}
\]
ここで $a=(g(c+0)-g(c-\delta))/\delta$ であり,
$a(y-c)+g(c+0)=a(y-(c-\delta))+g(c-\delta)$ であることに注意せよ.
定義より
\[
-M\leqq g(y)\leqq g_\delta(y)\leqq M \qquad (0\leqq y\leqq 1)
\]
となっている.
$|g_\delta(y)\phi(y)|\leqq M|\phi(y)|$ ($0\leqq y\leqq 1$) かつ
$\lim_{\delta\searrow 0}g_\delta(y)\phi(y)=g(y)\phi(y)$ ($y\ne c$) なので
Lebesgueの収束定理より,
\[
\lim_{\delta\searrow 0}\int_0^1 g_\delta(y)\phi(y)\,dy = \int_0^1 g(y)\phi(y)\,dy.
\]
このことを使って, $\delta>0$ を十分小さくして
\[
\int_0^1 g(y)\phi(y)\,dy
\leqq
\int_0^1 g_\delta(y)\phi(y)\,dy
\leqq
\int_0^1 g(y)\phi(y)\,dy + \frac{\eps}{3}
\]
となるようにしておく.
Stone-Weierstrassの多項式近似定理より, ある多項式函数 $Q(y)$ で
\[
\left|Q(y)-g_\delta(y)-\frac{\eps}{3N}\right|\leqq\frac{\eps}{3N} \qquad (0\leqq y\leqq 1)
\]
を満たすものが存在する.
このとき $g(y)\leqq g_\delta(y)\leqq Q(y)$ ($0\leqq y\leqq 1$) が成立しており,
\begin{align*}
&
\int_0^1 Q(y)\phi(y)\,dy
\\ &
\leqq
\int_0^1 \left|Q(y)-g_\delta(y)-\frac{\eps}{3N}\right|\phi(y)\,dy
+\int_0^1g_\delta(y)\phi(y)\,dy + \int_0^1 \frac{\eps}{3N}\phi(y)\,dy
\\ &
\leqq
\frac{\eps}{3N}\int_0^1\phi(y)\,dy
+\int_0^1g(y)\phi(y)\,dy + \frac{\eps}{3}
+\frac{\eps}{3N}\int_0^1\phi(y)\,dy
\\ &
=\frac{\eps}{3N}N+\int_0^1g(y)\phi(y)\,dy+\frac{\eps}{3}+\frac{\eps}{3N}N
\\ &
=\int_0^1g(y)\phi(y)\,dy+\eps.
\end{align*}
これで示すべきことが示された.
\qed
\end{proof}
\begin{theorem}
\label{theorem:Tauber-Laplace}
$f(t)$ は $t>0$ で定義された非負値可測函数であるとし,
$a,\alpha>0$ であると仮定する. このとき
\[
\int_0^\infty e^{-xt}f(t)\,dt \sim \frac{a}{x^\alpha}
\qquad (x\searrow 0)
\]
ならば
\[
\int_0^{1/x} f(t)\,dt \sim \frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha+1)}
\qquad (x\searrow 0).
\]
が成立する. (ガンマ函数が出て来る理由は以下の証明を見ればわかる.)
\end{theorem}
\begin{proof}
$F(x)=\int_0^\infty e^{-xt}f(t)\,dt$ とおくと,
仮定 $F(x)\sim a/x^\alpha$ ($x\searrow 0$) より, $k=0,1,2,\ldots$ に対して,
\begin{align*}
F((k+1)x)&=
\int_0^\infty e^{-xt}\left(e^{-xt}\right)^k f(t)\,dt
\\ &
\sim \frac{a}{(k+1)^\alpha x^\alpha}
=\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha)}
\int_0^\infty e^{-t}\left(e^{-t}\right)^k t^{\alpha-1}\,dt
\qquad(x\searrow 0).
\end{align*}
ここで次の公式を使った:
\[
\frac{1}{c^\alpha} = \frac{1}{\Gamma(\alpha)}\int_0^\infty e^{-ct} t^{\alpha-1}\,dt
\qquad (c>0).
\]
したがって任意の多項式函数 $p(y)$ について
\[
\int_0^\infty e^{-xt}p(e^{-xt})f(t)\,dt
\sim
\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha)}\int_0^\infty e^{-t}p(e^{-t})t^{\alpha-1}\,dt
\qquad(x\searrow 0).
\]
閉区間 $[0,1]$ 上の可積分函数 $\phi(y)$ を
\[
\phi(y)=(-\log y)^{\alpha-1}\quad (0<y\leqq 1), \qquad
\phi(0)=0
\]
と定め%
\footnote{$\int_0^1\phi(y)\,dy=\int_0^1(-\log y)^{\alpha-1}\,dy$
で $y=e^{-t}$ とおくと,
$\int_0^1\phi(y)\,dy=\int_0^\infty e^{-t} t^{\alpha-1}\,dt=\Gamma(\alpha)$
となる. このことから $\phi(y)=(-\log y)^{\alpha-1}$ は $\alpha>0$ のとき $[0,1]$ で
可積分であることがわかる.
}, %
$y=e^{-1}$ にのみ不連続点を持つ $[0,1]$ 上の函数 $g(y)$ を
\[
g(y) =
\begin{cases}
0 & (0\leqq y<e^{-1}) \\
y^{-1} & (e^{-1}\leqq y\leqq 1)
\end{cases}
\]
と定める.
\lemmaref{lemma:SW}より, 任意の $\eps>0$ に対して,
ある多項式函数 $P(y),Q(y)$ で
\begin{align*}
&
P(y)\leqq g(y)\leqq Q(y) \quad (0\leqq y\leqq 1),
\\ &
\int_0^1 g(y)\phi(y)\,dy-\eps
\leqq
\int_0^1 P(y)\phi(y)\,dy
\leqq
\int_0^1 g(y)\phi(y)\,dy,
\\ &
\int_0^1 g(y)\phi(y)\,dy
\leqq
\int_0^1 Q(y)\phi(y)\,dy
\leqq
\int_0^1 g(y)\phi(y)\,dy+\eps
\end{align*}
を満たすものが存在する. このとき $y=e^{-t}$ とおくと,
\begin{align*}
&
\int_0^\infty e^{-t}g(e^{-t})t^{\alpha-1}\,dy-\eps
\leqq
\int_0^\infty e^{-t}P(e^{-t})t^{\alpha-1}\,dy
\leqq
\int_0^\infty e^{-t}g(e^{-t})t^{\alpha-1}\,dy,
\\ &
\int_0^\infty e^{-t}g(e^{-t})t^{\alpha-1}\,dy
\leqq
\int_0^\infty e^{-t}Q(e^{-t})t^{\alpha-1}\,dy
\leqq
\int_0^\infty e^{-t}g(e^{-t})t^{\alpha-1}\,dy+\eps.
\end{align*}
一方, $f(t)\geqq 0$ であることより%
\footnote{ここで $f(t)$ の非負性を使っている.},
\[
\int_0^\infty e^{-xt}P(e^{-xt})f(t)\,dt
\leqq
\int_0^\infty e^{-xt}g(e^{-xt})f(t)\,dt
\leqq
\int_0^\infty e^{-xt}Q(e^{-xt})f(t)\,dt
\]
なので, これの全体を $a/(x^\alpha\Gamma(\alpha+1))$ で割って,
$x\searrow 0$ の極限を取ると,
\begin{align*}
&
\int_0^\infty e^{-t}P(e^{-t})t^{\alpha-1}\,dt
\leqq
\liminf_{x\searrow 0}
\frac{x^\alpha\Gamma(\alpha+1)}{a}
\int_0^\infty e^{-xt}g(e^{-xt})f(t)\,dt
\\ & \qquad
\leqq
\limsup_{x\searrow 0}
\frac{x^\alpha\Gamma(\alpha+1)}{a}
\int_0^\infty e^{-xt}g(e^{-xt})f(t)\,dt
\leqq
\int_0^\infty e^{-t}Q(e^{-t})t^{\alpha-1}\,dt
\end{align*}
以上の2つの段落の結果を合わせ,
$\eps>0$ をいくらでも小さくできることに注意すれば
次が成立することがわかる:
\[
\lim_{x\searrow 0}
\frac{x^\alpha\Gamma(\alpha+1)}{a}
\int_0^\infty e^{-xt}g(e^{-xt})f(t)\,dt
=
\int_0^\infty e^{-t}g(e^{-t})t^{\alpha-1}\,dt.
\]
すなわち次が得られた%
\footnote{以上のStone-Weierstrassの多項式近似定理を
使う鮮やかな方法はJovan Karamataによる.}:
\[
\int_0^\infty e^{-xt}g(e^{-xt})f(t)\,dt
\sim
\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha)}\int_0^\infty e^{-t}g(e^{-t})t^{\alpha-1}\,dt
\qquad(x\searrow 0).
\]
$e^{-1} \leqq e^{-xt}$ と $t\leqq 1/x$ は同値であり,
$t\leqq 1/x$ のとき $e^{-xt}g(e^{-xt})=1$ となり,
$t>1/x$ のとき $g(e^{-xt})=0$ なので,
すぐ上の式は次のように書き直される:
\[
\int_0^{1/x} f(t)\,dt
\sim
\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha)}\int_0^1 t^{\alpha-1}\,dt
=\frac{a}{x^\alpha}\frac{1}{\alpha\Gamma(\alpha)}
=\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha+1)}
\qquad (x\searrow 0).
\]
これで示すべきことがすべて示された.
\qed
\end{proof}
上の定理と前節の訂正を合わせることによって次の結果が得られる.
\begin{cor}
\label{cor:Tauber-cor-Laplace-all}
$f(t)$ は $t>0$ で定義された正値単調減少函数であるとし,
$\alpha,a>0$ であるとする. このとき
\[
\int_0^\infty e^{-xt} f(t)\,dt \sim \frac{a}{x^\alpha} \qquad (x\searrow 0)
\]
ならば,
\[
\int_0^t f(t')\,dt' \sim \frac{at^\alpha}{\Gamma(\alpha+1)},
\qquad
f(t) \sim \frac{a t^{\alpha-1}}{\Gamma(\alpha)}
\qquad (x\to\infty)
\]
が成立する. \qed
\end{cor}
数列 $a_n$ に対して $f(n)=a_n$ を満たす函数 $f(t)$
を適切に定義することによって近似したり,
Stieltjes積分版の定理を証明し直したり,
さらに $y=e^{-x}$ と置いて $x\searrow 0$ の極限
を $y\nearrow 1$ の極限に書き直すことによって,
もしくは直接証明し直すことによって以下の結果が得られる.
($1-e^{-x}\sim x$ ($x\searrow 0$) であることに注意せよ.)
\begin{cor}
\label{cor:Tauber-cor-power-series}
$a_0,a_1,a_2,\ldots$ は非負値数列であるとし, $\alpha,a>0$ であるとする.
このとき
\[
\lim_{y\nearrow 1}(1-y)^\alpha\sum_{n=0}^\infty a_n y^n = a
\]
ならば
\[
\sum_{k=0}^n a_k \sim \frac{an^\alpha}{\Gamma(\alpha+1)}
\qquad (n\to\infty)
\]
が成立する. (ガンマ函数が出て来る理由は以下の証明を見ればわかる.)
\qed
\end{cor}
\begin{proof}
直接証明し直しておこう.
$x>0$ とし, $y=e^{-x}$ とおくと, $1-y\sim x$ ($x\searrow 0$) なので,
\[
F(x):=
\sum_{n=0}^\infty e^{-nx} a_n \sim \frac{a}{x^\alpha}
\qquad (x\searrow 0).
\]
ゆえに任意の $k=0,1,2,\ldots$ に対して,
\begin{align*}
F((k+1)x)
&=\sum_{n=0}^\infty e^{-nx}(e^{-nx})^k a_n
\\ &
\sim
\frac{a}{(k+1)^\alpha x^\alpha}
=\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha)}
\int_0^\infty e^{-t}\left(e^{-t}\right)^k t^{\alpha-1}\,dt
\qquad(x\searrow 0).
\end{align*}
ここで次の公式を使った:
\[
\frac{1}{c^\alpha} = \frac{1}{\Gamma(\alpha)}\int_0^\infty e^{-ct} t^{\alpha-1}\,dt
\qquad (c>0).
\]
したがって, 任意の多項式函数 $p(y)$ について
\[
\sum_{n=0}^\infty e^{-nx}p(e^{-nx})a_n
\sim
\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha)}\int_0^\infty e^{-t}p(e^{-t})t^{\alpha-1}\,dt
\qquad(x\searrow 0).
\]
多項式函数で函数
\[
g(y) =
\begin{cases}
0 & (0\leqq y<e^{-1}), \\
y^{-1} & (e^{-1}\leqq y\leqq1)
\end{cases}
\]
を近似することによって次が得られる%
\footnote{実際には\lemmaref{lemma:SW}を用いた注意深い議論が必要になる.
その議論の詳細を見ないと, どうして $a_n\geqq 0$ と仮定しているか, よくわからないだろう.
議論の詳細については\theoremref{theorem:Tauber-Laplace}の証明を参照せよ.
この方法は Jovan Karamata による.}:
\[
\sum_{x=0}^\infty e^{-nx}g(e^{-nx})a_n
\sim
\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha)}\int_0^\infty e^{-t}g(e^{-t})t^{\alpha-1}\,dt
\qquad(x\searrow 0).
\]
$e^{-1} \leqq e^{-nx}$ と $n\leqq 1/x$ は同値であり,
$n\leqq 1/x$ のとき $e^{-nx}g(e^{-xx})=1$ なので,
すぐ上の式は次のように書き直される:
\[
\sum_{0\leqq n\leqq{1/x}} a_n
\sim
\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha)}\int_0^1 t^{\alpha-1}\,dt
=\frac{a}{x^\alpha}\frac{1}{\alpha\Gamma(\alpha)}
=\frac{a}{x^\alpha}\frac{1}{\Gamma(\alpha+1)}
\qquad (x\searrow 0).
\]
右辺で $x=1/n$ とおき, 左辺の和を $k=0,1,\ldots,n$ の和に書き直すと,
\[
\sum_{k=0}^n a_k
\sim \frac{a n^\alpha}{\Gamma(\alpha+1)}
\qquad (n\to\infty).
\]
これで示すべきことが示された.
\qed
\end{proof}
\begin{cor}
\label{cor:Tauber-cor-power-series-all}
$a_0,a_1,a_2,\ldots$ は正値単調減少数列であるとし, $\alpha,a>0$ であるとする.
このとき
\[
\lim_{y\nearrow 1}(1-y)^\alpha\sum_{n=0}^\infty a_n y^n = a
\]
ならば
\[
\sum_{k=0}^n a_k \sim \frac{an^\alpha}{\Gamma(\alpha+1)}, \qquad
a_n \sim \frac{an^{\alpha-1}}{\Gamma(\alpha)}
\qquad (n\to\infty)
\]
が成立する. \qed
\end{cor}
\theoremref{theorem:Tauber-Laplace}のStieltjes積分版は次の通り.
\begin{theorem}
$\varphi(t)$ は $\varphi(t)=0$ ($t<0$) を満たす右連続単調増加函数であるとし,
$a,\alpha>0$ であるとする. このとき
\[
F(x):=\int_{-0}^\infty e^{-xt} d\varphi(t) \sim \frac{a}{x^\alpha} \qquad (x\searrow 0)
\]
ならば
\[
\varphi(t) \sim \frac{at^\alpha}{\Gamma(\alpha+1)}
\qquad (t\to\infty)
\]
が成立する. \qed
\end{theorem}
この定理の特別な場合として,
もしくは\theoremref{theorem:Tauber-Laplace}の証明と
完全に同様の筋道をたどることによって次の結果が得られる.
\begin{cor}
$\lambda_n\geqq 0$ は単調増加数列であるとし, $a,\alpha>0$ であるとする. このとき
\[
\sum_{n=1}^\infty e^{-\lambda_n x} \sim \frac{a}{x^\alpha} \qquad (x\searrow 0)
\]
ならば
\[
\#\{\, n \mid \lambda_n\leqq t \,\} \sim \frac{at^\alpha}{\Gamma(\alpha+1)}
\qquad (t\to\infty)
\]
が成立する. さらに $t=\lambda_n$ の場合を考えることによって
\[
\lambda_n \sim \left(\frac{\Gamma(\alpha+1)}{a}\right)^{1/\alpha}n^{1/\alpha}
\qquad (n\to\infty)
\]
も得られる. \qed
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Wallisの公式と逆正弦分布}
べき級数展開 $(1-y)^{-1/2}=\sum_{n=0}^\infty a_n y^n$ ($|y|<1$) で
数列 $a_n$ を定めると
\begin{align*}
a_n
&
=(-1)^n\binom{-1/2}{n}
=\frac{\frac{1}{2}(\frac{1}{2}+1)\cdots(\frac{1}{2}+n-1)}{n!}
\\ &
=\frac{1\cdot 3\cdots(2n-1)}{2^n n!}
=\frac{(2n)!}{2^{2n}(n!)^2}
=\frac{1}{2^{2n}}\binom{2n}{n}>0.
\end{align*}
4つ目の等号で分子分母に $2\cdot 4\cdots(2n)=2^n n!$ をかけた.
さらに
\[
a_{n+1}=\frac{n+1/2}{n+1}a_n<a_n.
\]
ゆえに $a_n$ は正値単調減少数列である.
$(1-y)^{1/2}\sum_{n=0}^\infty a_n y^n=1$
なので\corref{cor:Tauber-cor-power-series-all}を適用することによって次が得られる:
\[
a_n=\frac{1}{2^{2n}}\binom{2n}{n}
\sim \frac{n^{1/2-1}}{\Gamma(1/2)}
=\frac{1}{\sqrt{\pi n}}
\qquad (n\to\infty).
\]
この公式は{\bf Wallisの公式}と呼ばれている.
逆正弦分布が出て来る一つのパターンについて説明しよう.
そのために正値数列 $a_n$ はWallisの公式型の漸近挙動
$a_n\sim 1/\sqrt{\pi n}$ ($n\to\infty$) を満たしていると仮定し,
$p_{n,k}=a_k a_{n-k}$ とおく.
このとき, 仮定より $\min\{k,n-k\}\to\infty$ において
\[
p_{n,k} \sim \frac{1}{\pi}\frac{1}{\sqrt{k(n-k)}}
= \frac{1}{\pi}\frac{1}{\sqrt{\frac{k}{n}\left(1-\frac{k}{n}\right)}}\frac{1}{n}
\]
となるので, $0\leqq a<b\leqq1$ に対して,
\begin{align*}
\lim_{n\to\infty}
\sum_{a\leqq k/n\leqq b} p_{n,k}
=\lim_{n\to\infty}
\sum_{a\leqq k/n\leqq b}
\frac{1}{\pi}\frac{1}{\sqrt{\frac{k}{n}\left(1-\frac{k}{n}\right)}}\frac{1}{n}
=\frac{1}{\pi}\int_a^b\frac{dx}{\sqrt{x(1-x)}}.
\end{align*}
確率密度函数 $\pi^{-1}(x(1-x))^{-1/2}\,dx$ が定める確率分布は逆正弦分布と呼ばれている.
そのように呼ばれる理由はその累積確率分布函数が次のように逆正弦函数で表わされるからである:
\[
\frac{1}{\pi}\int_0^x \frac{dt}{\sqrt{t(1-t)}}
=\frac{2}{\pi}\int_0^{\sqrt{x}}\frac{dy}{\sqrt{1-y^2}}
= \frac{2}{\pi}\arcsin\sqrt{x}.
\]
この確率分布は中心が $(x,y)=(1/2,0)$ で半径が $1/2$ の円周上の一様分布を $x$ 軸上に
射影したものに等しい.
このようにWallisの公式型の漸近挙動の仮定から逆正弦分布が出て来る.
そしてWallisの公式型の漸近挙動はTauber型の定理
(\corref{cor:Tauber-cor-power-series-all})から出て来る.
一般的な1次元ランダムウォークに関する逆正弦法則はそのような方針で証明される.
逆正弦法則とは
「左右対称もしくは期待値が $0$ で有限の分散を持つような
原点から出発する1次元ランダムウォークにおいて,
原点より右側に留まっている時間の総和の割合の分布が
時間無限大の極限で逆正弦分布に収束する」という法則のことである%
\footnote{詳しくは
Frank Spitzer, Principles of Random Walk, Springer GTM~34 (1964)
の第20節を参照せよ.
特にそのpp.~225--227あたりを参照すればこのノートとの関係がわかるはずである.
}.
逆正弦法則の証明にはかなりややこしい計算が必要になる.
確率 $1/2$ で左右に1ステップずつ進む単純なランダムウォークに関する
逆正弦法則の場合でさえ, 組み合わせ論的にややこしい議論が必要になる%
\footnote{単純なランダムウォークの逆正弦法則の証明に興味がある人は例えば
\href{http://web.econ.keio.ac.jp/staff/hattori/srw.pdf}
{服部哲也, 確率論講義付録, 20030525}
もしくは\href
{http://web.econ.keio.ac.jp/staff/hattori/kyoritu.htm}
{服部哲也著『ランダムウォークとくりこみ群』共立出版(2004)}の
第1章を参照して欲しい.
服部哲也さんが書いたものはサービス精神が旺盛でどれも楽しいので,
数楽好きの人にはおすすめできる.}.
ここではその手の面倒な議論には一切触れないことにする.
しかし, 逆正弦法則によれば結果的に以下が成立している:
\begin{align*}
&
\lim_{n\to\infty}
\frac
{\# \left\{\,(x_i)_{i=1}^n\in\{\pm 1\}^n
\,\left|\, na<\#\{\,k\mid x_1+\cdots+x_k>0\,\}<nb \right.\right\}}
{2^n}
=\frac{1}{\pi}\int_a^b \frac{dx}{\sqrt{x(1-x)}},
\\[\medskipamount]
&
\lim_{n\to\infty}
\frac{1}{2^n}\!\!\!\!\!\!
\mathop{\int\cdots\int}%
\limits_{\substack{-1<x_1,\ldots,x_n<1,\\ na<\#\{\,k\mid x_1+\cdots+x_k>0\,\}<nb}}
\!\!\!\!\!\!\!\!dx_1\cdots dx_n
=\frac{1}{\pi}\int_a^b \frac{dx}{\sqrt{x(1-x)}}.
\end{align*}
ギャンブルにたとえれば,
条件 $x_1+\cdots+x_k>0$ は「浮いていること」(トータルで勝っている状態)を意味し,
$\#\{\,k\mid x_1+\cdots+x_k>0\,\}$ は浮いている時間の長さを意味しており,
条件 $na<\#\{\,k\mid x_1+\cdots+x_k>0\,\}<nb$ は
浮いている時間の長さの割合が $a$ より大きく $b$ より小さいことを意味している.
ランダムウォークに関する逆正弦法則は
「浮いている時間の割合」が $n\to\infty$ で逆正弦分布にしたがうこと
を意味している.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\mathbf{x-x^2+x^4-x^8+x^{16}-x^{32}+\cdots}$ で $\mathbf{x\nearrow 1}$ とすると?}
函数 $F(x)$ を
\[
F(x) = x-x^2+x^4-x^8+x^{16}-x^{32}+\cdots=\sum_{k=0}^\infty(-1)^k x^{2^k}
\qquad (|x|<1)
\]
と定める. このとき $x\nearrow 1$ で $F(x)$ は収束するか?
収束するとしたらその収束先の値は何になるか?%
\footnote{筆者はこの問題の存在を
\href{https://www.uam.es/personal_pdi/ciencias/dragan/respub/Duren_Tauberian_Talk_2013-10_UAM.pdf}
{Peter Duren, Sums for Divergent Series: A Tauberian Adventure, 2013-10} (講演スライド)
で学んだ. それによれば G.~H.~Hardy がこの問題を1907年に解いたらしい.
数値計算すれば $x\nearrow 1$ で $F(x)$ の値が $0.5$ の
周囲を\href
{https://twitter.com/genkuroki/status/734774669069287424}
{小さく無限に振動している様子}を見られる.
そして上の講演スライドでは実際に無限に振動していることが証明されている.
}
仮に収束するとしたら, その収束先は $1/2$ でなければいけないことはすぐにわかる.
なせならば,
\[
F(x^2)=x-F(x)
\]
が成立しているからである.
さらに数値計算してみると, $x$ が $1$ の近くで $F(x)$ の値は $0.5$ に
かなり近いこともわかる. たとえば
\[
F(0.99) \simeq 0.494098, \qquad
F(0.999)\simeq 0.500124.
\]
$F(x)$ は $x\nearrow 1$ で $1/2$ に収束するのだろうか？
数列 $a_n$, $s_n$ を
\[
F(x)=\sum_{n=0}^\infty a_n x^n, \qquad
s_n = a_0+a_1+\cdots+a_n
\]
と定める. 一般に
\[
\sum_{n=0}^\infty a_n x^n = (1-x)\sum_{n=0}^\infty s_n x^n
\]
のとき
\[
s_n = a_0+a_1+\cdots+a_n
\]
となることに注意せよ.
上の状況で
\begin{align*}
F(x)
&=x(1-x)+x^4(1-x^4)+x^{16}(1-x^{16})+\cdots
=(1-x)f(x),
\\
f(x)
&=x+x^4(1+x+x^2+x^3)+x^{16}(1+x+\cdots+x^15)+\cdots
\\ &
=x+(x^4+x^5+x^6+x^7)+(x^{16}+x^{17}+\cdots+x^{31})+\cdots.
\end{align*}
となるので, $s_n\geqq 0$ である.
ゆえにもしも
\[
\lim_{x\nearrow 1} F(x) = \lim_{x\nearrow 1}(1-x)\sum_{n=0}^\infty s_n x^n = a
\]
と収束するならば, \corref{cor:Tauber-cor-power-series} ($\alpha=1$) より,
\[
\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^n s_n = a
\]
と収束するはずである. しかし,
\begin{align*}
&
\lim_{k\to\infty}\frac{s_0+s_1+\cdots+s_{2^{2k}-1}}{2^{2k}-1}
=\lim_{k\to\infty}\frac{1+4+16+\cdots+4^{k-1}}{4^k-1}
=\lim_{k\to\infty}\frac{\dfrac{4^k-1}{4-1}}{4^k-1}
=\frac{1}{3},
\\ &
\lim_{k\to\infty}\frac{s_0+s_1+\cdots+s_{2^{2k-1}-1}}{2^{2k-1}-1}
=\lim_{k\to\infty}\frac{1+4+16+\cdots+4^{k-1}}{\frac{1}{2}4^k-1}
=\lim_{k\to\infty}\frac{\dfrac{4^k-1}{4-1}}{\frac{1}{2}4^k-1}
=\frac{2}{3}
\end{align*}
なので, $n^{-1}\sum_{k=0}^n s_n$ は $n\to\infty$ で収束しない.
したがって $x\nearrow 1$ で $F(x)$ も収束しない.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{付録: Taylorの定理に証明の仕方}
\label{sec:Taylor}
「(Taylor展開の途中まで)+(剰余項)」の形式の公式をTaylorの公式と言う.
この節ではTaylorの定理の導出の方針について説明する.
この節の内容は非常に易しい.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{積分剰余項型Taylorの定理}
積分表示された剰余項を持つTaylorの定理は
\[
f(x)=f(a)+\int_a^x f(x_1)\,dx_1
\]
を単純に繰り返し用いることによって証明可能である%
\footnote{部分積分さえ使う必要がない!}.
実際, これに
\[
f'(x_1)=f'(a)+\int_a^x f''(x_2)\,dx_2
\]
を代入すると
\[
f(x)=f(a)+f'(a) \int_a^x dx_1+\int_a^x dx_1\int_a^{x_1} f''(x_2)\,dx_2.
\]
ここで括弧の使用量を減らすために
\[
\int_a^x dx_1\int_a^{x_1} f''(x_2)\,dx_2
=\int_a^x \left(\int_a^{x_1} f''(x_2)\,dx_2 \right)\,dx_1
\]
という書き方を用いた. さらに
\[
f''(x_2)=f''(a)+\int_a^{x_2} f'''(x_3)\,dx_3
\]
を代入すると
\[
f(x)=f(a)+f'(a)\int_a^x\!\!\!dx_1+f''(a)\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2
+\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2\int_a^{x_2}f'''(x_3)\,dx_3.
\]
さらに同じ計算をもう一度繰り返すと
\begin{align*}
&
f(x)=f(a)+f'(a)\int_a^x\!\!\!dx_1+f''(a)\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2
+f'''(a)\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2\int_a^{x_2}\!\!\!dx_3 +R_4
\\ &
R_4 = \int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2\int_a^{x_3}\!\!\!dx_3\int_a^{x_3}f^{(4)}(x_4)\,dx_4.
\end{align*}
以上の計算を続ければ帰納的に次が成立することがわかる:
\begin{align*}
&
f(x)=\sum_{k=0}^{n-1} f^{(k)}(a)\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2\cdots\!\!\int_a^{x_{k-1}}\!\!\!dx_k + R_n,
\\ &
R_n=\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2\cdots\int_a^{x_{n-2}}\!\!\!dx_{n-1}\int_a^{x_{n-1}}f^{(n)}(x_n)\,dx_n.
\end{align*}
$R_n$ を{\bf 剰余項}と呼ぶ.
以上の計算では積分の線形性しか使っていない.
剰余項以外の逐次積分は以下のように順番に(すぐ上の式を使うことによって次々に)容易に計算される:
\begin{align*}
&
\int_a^x\!\!\!dx_1 = x-a,
\\ &
\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2
=\int_a^x (x_1-a)\,dx_1
=\frac{(x-a)^2}{2},
\\ &
\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2\int_a^{x_2}\!\!\!dx_3
=\int_a^x\frac{(x_1-a)^2}{2}\,dx_1
=\frac{(x-a)^3}{3!},
\\ &
\cdots\cdots\cdots
\\ &
\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2\cdots\!\!\int_a^{x_{k-1}}\!\!\!dx_k
=\int_a^x\frac{(x_1-a)^{k-1}}{(k-1)!}\,dx_1
=\frac{(x-a)^k}{k!}.
\end{align*}
$x>a$ のとき, この計算結果は $k$ 次元立方体の体積の $k!$ 分の $1$ を意味している.
\begin{itemize}
\item$k=1$ の場合は線分 $\{\, x_1 \mid a\leqq x_1\leqq x\,\}$ の長さである.
\item $k=2$ の場合は直角二等辺三角形
$\{\,(x_1,x_2)\mid a\leqq x_2\leqq x_1\leqq x\,\}$ の面積
すなわち正方形の面積 $(x-a)^2$ の半分である.
半分になる理由は, $x_2\geqq x_1$ を満たす $(x_1,x_2)$
のみについて積分するからである. 正方形全体の面積を得るためには
$x_1\leqq x_2$ を満たす $(x_1,x_2)$ についても積分しなければいけない.
\item $k=3$ の場合は
$\{\,(x_1,x_2,x_3)\mid a\leqq x_3\leqq x_2\leqq x_1\leqq x\,\}$ の体積であり,
立方体の体積 $(x-a)^3$ の $3!$ 分の1になる.
$3!$ 分の1になる理由は, $x_3\leqq x_2\leqq x_1$ を
満たす $(x_1,x_2,x_3)$ のみについて積分するからである.
立方体全体の体積を得るためにはすべての大きさの順番について
積分しなければいけない. $x_i$ の個数は $3$ 個なので
並べ方の総数は $3!$ 個ある.
\item 一般の $k$ の場合も以上と同様である.
$x_1,x_2,\ldots,x_k$ の並べ方の総数が $k!$ なので
逐次積分の結果は $k$ 次元立方体の体積 $(x-a)^k$ の $k!$ 分の1になる.
\end{itemize}
{\bf 以上によってTaylor展開の各項の分母に階乗が現われる理由も明瞭になった!}
以上のまとめ:
\[
f(x)=\sum_{k=0}^{n-1}f^{(k)}(a)\frac{(x-a)^k}{k!} + R_n, \quad
R_n=\int_a^x\!\!\!dx_1\cdots\int_a^{x_{n-2}}\!\!\!dx_{n-1}\int_a^{x_{n-1}}f^{(n)}(x_n)\,dx_n.
\]
これを積分剰余項型の{\bf Taylorの定理}と呼ぶことにする.
剰余項 $R_n$ が $n\to\infty$ で $0$ に収束するならば
\[
f(x)=\sum_{k=0}^\infty\frac{1}{k!}f^{(k)}(x-a)^k
\]
が成立する. これを函数 $f$ の $x=a$ における{\bf Taylor展開}と呼ぶ.
剰余項の大きさを上から評価するためには次のようにすればよい.
まず $R>0$ を取って, $x$ の動く範囲を $|x-a|\leqq R$ に限定する.
そして, ある $M_n>0$ で $|f^{(n)}(x)|\leqq M_n$ ($|x-a|\leqq R$)を
満たすものを見付ける. そのとき
\begin{align*}
|R_n|
\leqq
M_n \left|\int_a^x\!\!\!dx_1\cdots\int_a^{x_{n-2}}\!\!\!dx_{n-1}\int_a^{x_{n-1}}\!\!\!dx_n\right|
=
\frac{M_n|x-a|^n}{n!}
\leqq
\frac{M_n R^n}{n!}
\tag{R}
\end{align*}
となるので, $M_n R^n/n!\to 0$ ならばTaylor展開が $|x-a|\leqq R$ において $f(x)$ に一様収束する.
剰余項の具体的な形そのものよりも剰余項の大きさの評価式(R)の方がよく使われる.
たとえば $M_n$ 増大速度が $n$ の指数函数程度ならばTaylor展開は収束する($A^nR^n/n!\to 0$).
そのことから, $e^x$, $\cos x$, $\sin x$ のTaylor展開がどのような $a$, $x$ についても
常に収束することが容易に確かめられる:
\[
e^x = \sum_{k=0}^\infty \frac{x^n}{n!}, \qquad
\cos x = \sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!}, \qquad
\sin x = \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)}.
\]
$M_n$ の増大速度が $n!$ と同じ程度な場合には $|x-a|<1$ でTaylor展開は $f(x)$ に収束する.
たとえば, $f(x)=(1+x)^\alpha$ のとき,
\[
f^{(n)}(x)=\alpha(\alpha-1)\cdots(\alpha-n+1)(1+x)^{\alpha-n}
\]
なので, この $f$ の $x=0$ でのTaylor展開は $|x|<1$ で収束することがわかる.
同様にして $f(x)=\log(1+x)$ の $x=0$ でのTaylor展開は $|x|<1$ で収束することがわかる.
\[
(1+x)^\alpha = \sum_{k=0}^\infty\binom{\alpha}{k}x^k, \qquad
\log(1+x) = \sum_{k=1}^\infty(-1)^{k-1}\frac{x^k}{k}.
\]
上の評価式(R)を知っていれば十分なのだが, 剰余項 $R_n$ を逐次積分ではなく,
1回の積分で表示する公式があるのでそれを紹介しておこう.
簡単のため $a\leqq x$ と仮定しよう($a\geqq x$ の場合も同様である).
$R_n$ の逐次積分は $a\leqq x_n\leqq x_{n-1}\leqq\cdots\leqq x_1\leqq x$
にわたる積分であることに注意しながら, $x_n$ による積分を一番外側に出すと,
\begin{align*}
R_n
= \int_a^x f^{(n)}(x_n)\left(\int_{x_n}^x\!\!\!dx_1\int_{x_n}^{x_1}\!\!\!dx_2\cdots\int_{x_n}^{x_{n-2}}\!\!\!dx_{n-1}\right)\,dx_n
= \int_a^x f^{(n)}(x_n)\frac{(x-x_n)^{n-1}}{(n-1)!}\,dx_n.
\end{align*}
2つ目の等号で逐次積分が $(n-1)$ 次元立方体の体積
の $(n-1)!$ 分の1になるという上の方で説明した結果を使った.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{線形常微分方程式の解法}
\label{sec:LODE}
次の線形微分方程式を考える:
\[
\frac{dU(t)}{dt} = A(t)U(t), \qquad U(0)=E.
\]
ここで $A(t)$ は正方行列値連続函数であり,
$U(t)$ は初期値が正方行列の正方行列値函数である.
この微分方程式は次と同値である:
\[
U(t) = E + \int_0^t A(t_1)U(t_1)\,dt_1.
\]
前節を読んだ読者はTaylorの定理を証明した場合と同様に
この式を繰り返し用いれば解が得られそうなことに気付くはずである.
この式で $t,t_1$ のそれぞれを $t_1,t_2$ に置き換えた式を
その式自身に代入すると,
\[
U(t)=E+\int_0^t A(t_1)\,dt_1 + \int_0^t\!\!\!dt_1\int_0^{t_1}A(t_1)A(t_2)U(t_2)\,dt_2.
\]
同じ操作をもう一度繰り返すと
\begin{align*}
&
U(t)=E+\int_0^t A(t_1)\,dt_1
+\int_0^t\!\!\!dt_1\int_0^{t_1} A(t_1)A(t_2)\,dt_2
+R_3,
\\ &
R_3 = \int_0^t\!\!\!dt_1\int_0^{t_1}\!\!\!dt_2\int_0^{t_2}A(t_1)A(t_2)A(t_3)U(t_3)\,dt_3.
\end{align*}
同様に繰り返すと, 帰納的に次が成立していることがわかる:
\begin{align*}
&
U(t)=\sum_{k=0}^{n-1}
\int_0^t\!\!\!dt_1\int_0^{t_1}\!\!\!dt_2\cdots\int_0^{t_{k-1}}
A(t_1)A(t_2)\cdots A(t_k)\,dt_k
+R_n,
\\ &
R_n=\int_0^t\!\!\!dt_1\int_0^{t_1}\!\!\!dt_2\cdots\int_0^{t_{n-1}}
A(t_1)A(t_2)\cdots A(t_n)U(t_n)\,dt_n.
\end{align*}
実はこれの $n\to\infty$ の極限で微分方程式の解が
\[
U(t)=\sum_{k=0}^\infty
\int_0^t\!\!\!dt_1\int_0^{t_1}\!\!\!dt_2\cdots\int_0^{t_{k-1}}
A(t_1)A(t_2)\cdots A(t_k)\,dt_k.
\]
で得られることを示せる.
積分中の $A(t)$ は大きな $t$ の順番に並んでいることに注意せよ.
時間順序積 $T[\ \ ]$ を次のように定める:
\[
T[A(t_1)\ldots A(t_k)]
= A(t_{\sigma(1)})\cdots A(t_{\sigma(k)}),
\quad
t_{\sigma(1)}\geqq\cdots\geqq t_{\sigma(k)},
\quad
\sigma\in S_k.
\]
この記号法のもとで上の公式は次のように書き直される:
\[
U(t)=\sum_{k=0}^\infty\frac{1}{k!}\int_0^t\cdots\int_0^t T[A(t_1)\cdots A(t_k)]\,dt_1\cdots dt_k.
\]
さらに形式的に $T[\ \ ]$ を積分と和の外に出すことを許せばこれは次のように書き直される:
\begin{align*}
U(t)
&=T\left[
\sum_{k=0}^\infty \frac{1}{k!}
\left(\int_0^t A(s)\,ds\right)^k
\right]
=T\left[\exp\int_0^t A(s)\,ds \right]
\end{align*}
この形の公式は物理の教科書などで散見される.
以上で解説した線形常微分方程式の解法は
Picardの逐次代入法の特別な場合である.
すなわち以上の方法は非線形の場合にも適用できる.
このようにTaylorの定理の逐次積分による証明法を知っていれば,
線形常微分方程式の逐次代入法による解法もすぐに思い付くだろうし,
さらにPicardの逐次代入法にまで一般化される.
このような理由からTaylorの定理を逐次積分で証明する方法の紹介は
相当に教育的だと思われる.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{微分剰余項型Taylorの定理}
逐次積分表示された剰余項を持つTaylorの定理を知っているだけで困らないはずなのだが,
多くの文献で剰余項を高階の導函数で表示する公式もよく使われているので
簡単に紹介しておこう.
\paragraph{Cauchyの平均値の定理：}
実函数 $F$, $G$ が微分可能でかつ $F(t)\ne G(a)$ でかつ
速度ベクトル $(F',G')$ が $a$ と $t$ のあいだで決して $0$ にならないならば
$a$ と $t$ のあいだのある実数 $t_1$ で
\[
\frac{G(t)-G(a)}{F(t)-F(a)}=\frac{G'(t_1)}{F'(t_1)}
\]
を満たすものが存在する.
この定理が成立することは $xy$ 平面に $(F(s),G(s))$ の奇跡の曲線を描き,
点 $(F(a),G(a))$ と点 $(F(t),G(t))$ を結ぶ直線と速度ベクトルが平行になる時刻 $t_1$ の存在が
視覚的に読み取れることを確認すれば納得できるだろう.
厳密な証明にはRollの定理を使うが,
個人的には「直観的に明らかな定理」とみなして問題ないと思う.
まず例として $f$ が $4$ 回微分可能な場合を扱おう.
\begin{align*}
&
G(t)=f(t)-f(a)-f'(a)(t-a)-f''(a)\frac{(x-a)^2}{2}-f'''(a)\frac{(x-a)^3}{3!},
\\ &
F(t)=(t-a)^4
\end{align*}
とおく. $F(a)=F'(a)=F''(a)=F'''(a)=0$, $G(a)=G'(a)=G''(a)=G'''(a)=0$
に注意しながら, Cauchyの平均値の定理を次々に適用すると,
$a$ と $t$ のあいだのある実数 $t_1,t_2,t_3,t_4$ で以下を満たすものの
存在が示される:
\begin{align*}
&
\frac
{f(t)-f(a)-f'(a)(t-a)-f''(a)\dfrac{(t-a)^2}{2}-f'''(a)\dfrac{(t-a)^3}{3!}}
{(t-a)^4}
\\
=&\frac
{f'(t_1)-f'(a)-f''(a)(t_1-a)-f'''(a)\dfrac{(t_1-a)^2}{2!}}
{4(t_1-a)^3}
\\
=&\frac
{f''(t_2)-f''(a)-f'''(a)(t_2-a)}
{4\cdot 3(t_2-a)^2}
\\
=&\frac
{f'''(t_3)-f'''(a)}
{4\cdot 3\cdot 2(t_3-a)}
\\
=&\frac
{f^{(4)}(t_4)}
{4!}.
\end{align*}
形式的には分子分母を微分して次の番号の $t_i$ を代入する操作を繰り返しただけである.
以上によって $a$ と $t$ のあいだのある実数 $t_4$ で
\begin{align*}
&
f(t)=f(a)+f'(a)(t-a)+f''(a)\frac{(t-a)^2}{2}+f'''(a)\frac{(t-a)^3}{3!}+R_4,
\\ &
R_4=f^{(4)}(t_4)\frac{(t-a)^4}{4!}
\end{align*}
を満たすものの存在が証明された.
さらに上の計算中のうしろから2番目の等号の右辺の $t\to a$ での極限は
\[
\frac{1}{4!}\lim_{t_3\to a}\frac{f'''(t_3)-f'''(a)}{t_3-a}
=\frac{1}{4!}f^{(4)}(a)
\]
になる. $t_4$ は $a$ と $t$ のあいだにあるので, $t\to a$ で
$t_4\to a$ となることに注意せよ.
これは上の計算の出発点の式の $t\to a$ での極限に等しい:
\[
\lim_{t\to a}
\frac
{f(t)-f(a)-f'(a)(t-a)-f''(a)\dfrac{(t-a)^2}{2}-f'''(a)\dfrac{(t-a)^3}{3!}}
{(t-a)^4}
=
\frac{1}{4!}f^{(4)}(a).
\]
これは次と同値である:
\begin{align*}
f(t)
&=
f(a)+f'(a)(t-a)+f''(a)\dfrac{(x-a)^2}{2}
\\ &
+f'''(a)\dfrac{(x-a)^3}{3!}
+f^{(4)}(a)\frac{(t-a)^4}{4!}+o((t-a)^4).
\end{align*}
これで $4$ 回微分可能な函数に関する微分剰余項型Taylorの定理が証明できた.
以上と全く同様にして, $f$ が $n$ 回微分可能ならば
\begin{align*}
%&
f(t)=
\sum_{k=0}^{n-1} f^{(k)}(a)\frac{(t-a)^k}{k!}
+R_n,
\qquad
%\\ &
R_n = f^{(n)}(t_n)\frac{(t-a)^n}{n!}
\end{align*}
を満たす $a$ と $t$ のあいだの実数 $t_n$ の存在が証明できる.
ゆえに $|t-a|\leqq R$, $|f^{(n)}(s)|\leqq M_n$ ($|s-a|\leqq R$) ならば
\[
|R_n| \leqq \frac{M_n R^n}{n!}
\]
となる. すなわち積分表示された剰余項の場合と同じ形の評価式が得られる.
さらに上と同様にして次も証明される:
\[
f(t) = \sum_{k=0}^n f^{(k)}(a)\frac{(t-a)^k}{k!} + o((t-a)^n).
\]
これらの結果も{\bf Taylorの定理}と呼ばれる.
以上の証明は $n$ 回微分可能性のみを仮定すれば得られる.
$n$ 階の導函数の連続性を仮定する必要はない.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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https://anarhija.info/library/albi-francia-atto-di-creazione-di-f-u-m-i-e-r-17-09-2017-it.tex
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anarhija.info
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\title{Albi [Francia]: Atto di creazione di F.U.M.I.E.R (17\Slash{}09\Slash{}2017)}
\date{}
\author{}
\subtitle{}
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\textbf{Fronte Unito di Movimenti Insurrezionalisiti delle Aree Rurali} [\emph{F.U.M.I.E.R.\Slash{}letamaio ndt}]
17 settembre, Albi, capitale del crimine, perché il crimine paga.
Due membri del FUMIER, che volevano scambiarsi via Whatsapp foto di gattini che ridono, si trovarono improvvisamente senza rete! Avevano già sentito della partecipazione di Orange [\emph{compagnia di telecomunicazioni, ndt}] al mondo capitalista alquanto egemonico e un po’ sull’orlo\dots{} Quindi, questo problema della rete è stato veramente la goccia che ha fatto traboccare il vaso\dots{}
Senza perdere tempo, si mandano un fax, e decidono di ritrovarsi la stessa notte con della diavolina (bio), in piazza Vigan.
Furiosi, partono in fretta in direzione degli uffici di Orange per esporre un reclamo, e là si offre loro una serie di veicoli dei quell’azienda.
Senza perdere tempo, ben cinque di essi presero subito fuoco! In effetti, il segretario generale e il suo vice-sindaco, sempre molto arrabbiato, avevano collocato diversi accendifuoco sotto le ruote anteriori.
Quella notte il capitalismo ha singhiozzato di paura, tremando d’angoscia, davanti all’improvvisa nascita del Fronte Unito di Movimenti Insurrezionalisti delle Aree Rurali!
Il segretario generale, per mano del suo vicesindanco, vi invita a creare, anche voi, delle sezioni locali del LETAMAIO [\emph{FUMIER, ndt}]. Perché siamo tutti e tutte dei LETAMAI, e che il LETAMAIO è dappertutto.
Il L.E.T.A.M.A.I.O.
\emph{P.S.: Eravamo un po’ arrabbiati, ma questo andrà di male in peggio, sappiatelo!}
\emph{\textbf{Segretariato generale}}
\bigskip
\emph{(tradotto da anarhija.info \& guerresociale)}
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Anarhija.info
\strut
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\strut
\vfill
\begin{center}
Albi [Francia]: Atto di creazione di F.U.M.I.E.R (17\Slash{}09\Slash{}2017)
\bigskip
\href{https://attaque.noblogs.org/post/2017/09/19/albi-acte-de-creation-du-f-u-m-i-e-r/}{attaque.noblogs.org}
\bigskip
\textbf{anarhija.info}
\end{center}
% end final page with colophon
\end{document}
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http://alex.laties.info/resume/resume.tex
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laties.info
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\documentclass[a4paper]{article}
\usepackage{verbatim}
\begin{comment}
Why hello there! I see you've chosen the red pill!
Welcome to one of the more painful yet rewarding ways to write a resume!
Hope you enjoy the mess below! -Alex
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\begin{LARGE}
\noindent
Alexander Laties\\
\end{LARGE}
\fbox{\parbox{0.35 \textwidth}{
245 Manhattan Avenue\\
Apt 5B\\
Brooklyn, NY 11211\\
Phone: 267 825 2530\\
Website: http://alex.laties.info/ \\
E-Mail: [email protected] \\
Github: https://github.com/alex-laties
}}
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\noindent
Education\\
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\noindent
Fall 2007 - Spring 2010 \\
University of Pennsylvania School of Engineering \\
Majored in Computer Science.
\begin{itemize}
\item Learned some theory
\item Implemented a CPU, an OS shell, and some sweet buffer overflows
\item Dropped out due to health reasons.
\end{itemize}
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\noindent
General Purpose Software Developer \\
\end{Large}
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I'm general purpose programmer who has worked on building, maintaining, and scaling services. I have no strong language preference. I have reasonable experience with Ruby, Python, and Golang. I have experience with larger IDEs, such as Visual Studio, MonoDevelop, and XCode, but claim no great proficiencies. \\
\par
\noindent
I have worked with teams in an agile style to quickly deliver features and products. I have worked with ticketing systems, such as Jira and Trello, to manage work loads. I have worked with source control systems, like Subversion and Git. I have experience with Test Driven Development through the use of unit tests. \\
\par
\noindent
I have contributed some code to open source projects like Bottlepy, watchwithme, and pyrollbar.\\
\par
\noindent
I like attempting to automate my home in my free time.\\
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Work History
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\noindent
January 2016 - February 2017 \\
Uber Technologies, LLC. \\
Network Engineer \\
\begin{itemize}
\item Worked for the UberATG effort.
\item Implemented and maintained an IPAM system with Collins and phpIPAM.
\item Implemented and maintained a metrics and alerting system for network devices and services using Prometheus and alertmanager.
\item Implemented and maintained a serial port based initial configuration system for Juniper switches with golang.
\item Triaged, diagnosed, and solved mission critical flaws in systems, networks, and deployments.
\end{itemize}
\columnbreak
April 2014 - December 2015 \\
Tumblr, Inc \\
Site Reliability Engineer \\
\begin{itemize}
\item Worked with Ruby and Puppet to maintain internal services.
\item Maintained an HTTP loadbalancing layer implemented with haproxy and nginx.
\item Maintained an HTTP caching layer implemented with varnish.
\item Improved availability of services through anycast with OSPF + bird and quagga.
\item Stress tested the HTTP loadbalancing layer with load generators vegeta, siege, apib, slowloris.pl, and hping3.
\item Improved in-memory credential management across Ruby, Python, and PHP services with a C++ library and wrappers.
\end{itemize}
\columnbreak
March 2012 - October 2013 \\
Voxy, Inc \\
Software Engineer
\begin{itemize}
\item Worked with Django on Python 2.7 to maintain a web application and mobile API.
\item Built prototypes for internal CRMs.
\item Built a new mobile API stack in Bottle.
\item Set up loadbalancing for web and mobile stacks using haproxy and pacemaker.
\item Set up and wrote initial recipes for Chef provisioning service.
\item Also dealt with PostgreSQL, MongoDB, Varnish, Celery, Redis, uwsgi, gunicorn, and nginx.
\end{itemize}
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%
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This are some random comments made by Yossi Gil.
Clearly,
\[
Knuth \ne Knott
\] can be written as $Knott \ne Knuth$ and indeed Knuth and
Knott are not the same person.
\TeX{} by Knuth~\cite{Knuth:84} is claimed to be bug-free.
However, Dr. Knuth never claimed that \TeX is {\em surprise} free!
In fact, the previous sentence had at least three (!) \TeX\ bugs...
The definition of the class NC is attributed by
Cook~\cite{Cook:??} to Pippenger~\cite{Pippenger:??}.
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\title{Shutdown}
\date{}
\author{Saint Just}
\subtitle{Policing the Crisis in Pittsburgh and Boston}
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{\usekomafont{title}{\huge Shutdown\par}}%
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\centering
\includegraphics[keepaspectratio=true,height=0.75\textheight,width=0.50\textwidth]{3-f-390-fall-2013-shutdown-1.png}
\caption[]{\noindent At the 2009 G-20 meeting in Pittsburgh, fully militarized police mobilized to protect the world’s financial ministers, rioted and attacked people indiscriminately.}
\end{wrapfigure}
I have had the unfortunate privilege of experiencing two urban shutdowns in the U.S. In 2009, while living in a quickly gentrifying neighborhood adjacent to the University of Pittsburgh as a graduate student, I experienced a preemptive shutdown of a major city during the G-20 summit meeting of the finance ministers of the top world economies.
I witnessed the first domestic use of a Long Range Acoustic Device (or LRAD), the smell of tear gas wafting into my apartment, and the aftermath of my dumbfounded undergraduate students, ostensibly apolitical, recently gassed and rubber-bulleted, wonder how their adopted city and university officials could collude so callously at their expense.
Less than four years later, now living on the gentrifying border between Roxbury and Jamaica Plain in Boston as a post-doctorate student, I heard police and fire sirens topple out of the tight, winding streets of the Hub during and after the April 15 Boston Marathon bombing, received the email, text-message, television, and radio suggestions (orders?) to “shelter in place,” and during the aftermath listened to liberal public and conservative talk radio hosts out-do themselves in pride of how my recently adopted city had come together to get the bad guy and be, “Boston Strong.”
Outside of professional hockey and football news, Boston and Pittsburgh are not generally compared or contrasted, and they were not in the days following the marathon bombing this spring. Why would they?
During the G-20, Pittsburgh did not suffer a terrorist attack and during the marathon attack and ensuing search, Boston-area residents were not singled-out by law-enforcement because they marched in protests.
But as my experiences glanced off listening to the facts on the evening news about a locked-down city, a resonant chill continued to drift through my ears and eyes. Perhaps Pittsburgh and Boston, both now firmly on the gentrifying side of their 1990s urban renaissances, both invested deeply in meds and eds as well as security and neurotechnology, were not so different in their reactions and responses to policing a crisis.
The assemblage of forces, reactions, and repercussions that accompanied both events are striking. In both cases, major US cities were brought to a standstill, not by terrorists, but by the reaction of local officials, heavily armed police, and the stunned acquiescence and assent of local residents. In Pittsburgh, the shutdown was preemptive in expectation of the demonstration against the representatives of world finance. Many businesses, terrified by media and police hysteria began boarding up their operations for fear of mass unrest from protesters days before the G-20 events began.
In Boston, the shutdown came in the wake of the attack, carried out, again largely voluntarily, by a citywide manhunt by an occupation-sized force of police, and a public that, from all reports, believed in the justness of the search. Both media events happened during a time of increased surveillance and during a time when there has been more regular domestic use of military technologies brought home from our wars in the Middle East. Each occupation--it is hard to call them by any other name--demanded assent on the street during and at the polls after the periods of crisis.
During the Pittsburgh events, my reaction was to march in an unpermitted pre-G-20 march, then run from earsplitting noise emitted by the LRAD, speak out after the fact at a police review board hearing, and then screen, when appropriate, documentaries in my classroom in the following semesters. What struck me at the time, and what I focused on in my scholarly research after the events, was the domestic use of military weapons technology by the state to suppress dissent and the legitimacy such use gained among the population.
However, in the Boston aftermath, it is the acquiescence of residents that strikes me as the most important fact of both events. That Boston residents would so easily stay inside and give their city over to police without a declaration of martial law was deeply troubling. This was reminiscent of the willingness of Pittsburgh businesses to give into police propaganda and abandon their city during the G-20, and citizen failure to demand any serious consequences from city officials after the police rioted in the neighborhoods where many university students lived and studied. What resonates with me here in Boston four years later, is the seeming entrenched cynicism of twelve years of the global war on terror instead of a wider skepticism of the very wisdom of so-called “Homeland Security.”
There must be a caveat here: living for only a few days under vague threat from a police-force-on-steroids is not remotely as intense as the experience of living in Afghanistan or Iraq over the last decade or, for that matter, the experience of living day-in and day-out in many US neighborhoods where the difference between policing and military occupation seem arbitrary and unnervingly thin: the flashing close-captioned television cameras in West Baltimore, the regular use of curfews in Detroit to constrain youth during major events, or the constant police presence in neighborhoods like Homewood or the Hill in Pittsburgh or Roxbury and Dorchester in Boston.
Which is why on the Friday after the bombing, when I was supposed to be staying inside, I got in my car and drove to the Pit Stop Barbeque in Mattapan near Dorchester and ate a plate of ribs. Luckily, the hard-working owners of the delicious establishment had, like me, simply ignored the official memo.
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Saint Just
Shutdown
Policing the Crisis in Pittsburgh and Boston
\bigskip
\href{https://www.fifthestate.org/archive/390-fall-2013/shutdown}{\texttt{https://www.fifthestate.org/archive/390-fall-2013/shutdown}}
Fifth Estate \#390, Fall, 2013
\bigskip
\textbf{fifthestate.anarchistlibraries.net}
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\documentclass[letterpaper, 12pt]{article}
\usepackage{amsmath,amssymb,amsthm,setspace,latexsym}
\usepackage{epsfig,psfrag,graphicx,epstopdf}
\usepackage[usenames,dvipsnames]{color}
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%\usepackage{showkeys}
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\begin{document}
\title{Entropic Chaoticity for the Steady State of a Current Carrying System.}
\author[*]{F. Bonetto}
\author[*]{M. Loss}
\affil[*]{School of Mathematics, Georgia Institute of Technology, Atlanta, GA}
\renewcommand\Authands{ and }
\maketitle
\begin{abstract}
The steady state for a system of $N$ particle under the influence of an external
field and a Gaussian thermostat and colliding with random ``virtual'' scatterers
can be obtained explicitly in the limit of small field. We show the sequence
of steady state distribution, as $N$ varies, forms a chaotic sequence in the
sense that the $k$ particle marginal, in the limit of large $N$, is the $k$-fold
tensor product of the 1 particle marginal. We also show that the
chaoticity properties holds in the stronger form of entropic chaoticity.
\end{abstract}
\section{introduction}
Ever since its introduction by Kac \cite{kac} in 1956, the notion of a chaotic
sequences has become an important concept in studying many body systems. Chaotic
sequences and propagation of chaos are the principal tools for passing from many
body problems to effective equations. The aim of this article is to give yet
another example of the interplay of chaotic sequences and effective equations.
In [BDLR] the authors consider a system consisting of $N$ particles moving in a
2 dimensional billiard and colliding with convex scatterers that form a
dispersing billiard. The particle are subject to an external electric field
$\bE$ and a Gaussian thermostat that keeps the kinetic energy fixed. The
equation of motion between collisions are thus:
\begin{equation}\label{eq2}
\left\{
\begin{array}{l}
\dot\bq_i = \bv_i \qquad\qquad i=1,\ldots,N \\
\dot\bv_i = \bF_i = \bE-\frac{\displaystyle \bE\cdot
\bj}{\displaystyle U}\,\bv_i + \cF_i \end{array}
\right.
\end{equation}
where
\begin{equation} \label{eq3}
\bj(\bV)=\sum_{i=1}^N \bv_i,\qquad U(\bV) = \sum_{i=1}^N |\bv_i|^2
\end{equation}
and $\cF_i$ is the force exerted on the $i$th particle by collisions with the
fixed scatterers. Very little is known about billiards with more than one
particle. In particular there is no existence theorem for the SRB measure of
this system.
The authors introduced a stochastic version of the above model in which the
deterministic collisions are replaced by Poisson distributed collision
processes. More precisely, in the time interval between $t$ and $t+dt$, the
$i$-th
particle as a probability $|\bv_i|^\alpha dt$ of suffering a collision. When a
collision happens, the velocity of the particle is randomly updated, i.e. if the
particle's velocity direction before collision is $\omega=\bv/|\bv|$, after
collision it will be distributed as $K(\omega'\cdot \omega)d\omega'$. The
details of the collision kernel $K$ are largely irrelevant. For what follows
it will be enough to assume that $K(x)>0$ for $x$ in a open set
${\cal U}\in[-1,1]$. We
note that this stochastic process make sense for any
dimension $d$.
The master equation for this process is given by
\begin{eqnarray} \label{dW}
\frac{\partial W(\bQ,\bV,t;\bE)}{\partial t} =
&-&\sum_{i=1}^N\bv_i\frac{\partial W(\bQ,\bV,t;\bE)}{\partial \bq_i}
-\sum_{i=1}^N
\nabla_{\bv_i}\biggl[\Bigl(\bE-\frac{\bE\cdot\bj(\bV)}{U(\bV)}
\bv_i\Bigl)W(\bQ,\bV,t;\bE)
\biggr]\nonumber\\
&+&\sum_{i=1}^N|\bv_i|^\alpha\int_{\cS^{d-1}(1)} K(\omega_i \cdot \omega'_i)
\bigl[W(\bQ,\bV_i',t;\bE)-W(\bQ,\bV,t;\bE)\bigr]\,d\sigma^{d-1}
(\omega')\nonumber\\
\end{eqnarray}
where $\cS^{m}(R)$ is the $m$-dimensional sphere with radius $R$,
$d \sigma^{m}(\cdot)$ is the uniform surfaces measure on $\cS^{m}(R)$ and $\bj =
\bJ/U$ as in \eqref{eq3}. Moreover
\begin{equation}
\bQ = (\bq_1,\ldots,\bq_N)\quad\text{,}\quad
\bV = (\bv_1,\ldots,\bv_i,\ldots,\bv_N)
\quad\text{and}\quad
\bV_i' = (\bv_1,\ldots,\bv_i',\ldots,\bv_N),
\end{equation}
and $\bv_i'=|\bv_i|\omega_i'$ if $\bv_i=|\bv_i|\omega_i$. Note that
the variable $\bQ$ is not part of the dynamics, i.e., if the initial condition
$W(\bQ,\bV,0)$ is independent of $\bQ$ so will be $W(\bQ,\bV,t)$. Moreover,
if $W(\bQ,\bV,0)$ is concentrated on the surface of given
energy $U_0$, that is if
\[
W(\bV,0)=\delta(U(\bV)-U_0)F(\bV,0)
\]
then so will be the solution of .\eqref{dW}:
\[
W(\bV,t;\bE)=\delta(U(\bV)-U_0)F(\bV,t;\bE).
\]
Finally if $F(\bV,0)$ is a symmetric function so is $F(\bV,t;
\bE)$. Thus we will,
henceforth, only consider symmetric spatially homogeneous solutions
concentrated on the surface of energy $U_0=N$.
Recall that the $k$-particle marginal $f^{(k)}_N(\bv_1, \dots, \bv_k)$ of a
distribution $F_N(\bV)$ is defined by the equation
\[
\int_{\cS^{dN-1}(\sqrt{N})} \varphi(\bv_1, \dots, \bv_k) F_N(\bV) d
\sigma^{dN-1}(\bV) =
\int_{\mathbb{R}^{dk}} \varphi(\bv_1, \dots, \bv_k)
f^{(k)}_N(\bv_1, \dots, \bv_k) d \bv_1 \cdots d \bv_k \ ,
\]
where $\varphi(\bv_1, \dots, \bv_k)$ is any bounded continuous function on
$\mathbb{R}^{dk}$.
%$\cS^{m}(R)$ is the $m$-dimensional sphere with radius $R$
%and $d \sigma^{m}(\cdot)$ is the uniform surfaces measure on $\cS^{m}(R)$.
Simple computations show that
\begin{equation} \label{kmarginal}
f^{(k)}_N(\bV_k) = \sqrt{\frac{N}{N - |\bV_k|}}
\int_{\cS^{d(N-k)-1}\left(\sqrt{N - |\bV_k|^2}\right)}
F_N(\bV_k, \bV^k) d \sigma^{d(N-k)-1}(\bV^k) \ ,
\end{equation}
where $\bV_k = (\bv_{1} , \dots, \bv_k)$ and $\bV^k = (\bv_{k+1} , \dots,
\bv_N)$. A sequence of densities
$\{F_N\}_{N=1}^\infty $ form a {\bf chaotic sequence with marginal $f $} if for
any bounded continuous function $\varphi$
\begin{equation}\label{weak}
\lim_{N \to \infty} \int_{\cS^{dN-1}(\sqrt{N})} \varphi(\bV_k)
F_N(\bV) d \sigma^{dN-1}(\bV) =
\int_{\mathbb{R}^{dk}} \varphi(\bV_k)
\prod_{j=1}^k f(\bv_j) d \bv_1 \cdots d \bv_k \
\end{equation}
It was shown in \cite{BCELM} that for finite time $t$ the master
equations \eqref{dW} propagates chaos, i.e., the solution of the master
equation \eqref{dW} forms a chaotic sequence if the initial condition does.
More precisely, for any bounded continuous function on $\mathbb{R}^{dk}$,
$\varphi(\bv_1, \dots, \bv_k)$, we have
\[
\lim_{N \to \infty} \int_{\cS^{dN-1} } \varphi(\bV_k) F(\bV, t;
\bE) d\sigma(\bV) =
\int_{\mathbb{R}^{dk}} \varphi(\bV_k) \prod_{j=1}^k f (\bv_j, t;
\bE) d\bv_1 \cdots d\bv_k
\]
where
\[
f (\bv_1,t;\bE)=\lim_{N\to\infty} f^{(1)}_N(\bv_1,t;\bE) \ .
\]
It was shown in \cite{BCELM} that $ f(\bv_1,t;\bE)$ satisfies
the Boltzmann equation
\begin{equation}\label{BE}
\frac{f(\bv,t;\bE)}{dt}+\frac{\partial}{\partial
\bv}\left[\bigl(\bE-(\bE\cdot\hat\bj(t,E)) \bv\bigl)f
\right]=|\bv|^\alpha\int_{\cS^{d-1}(1)} K(\omega \cdot \omega')
\bigl[f(\bv',t;\bE)-f(\bv,t;\bE)\bigr]\,d\sigma^{d-1}(\omega')
\end{equation}
where $\hat \bj(t,E)$ is given by the self consistent condition
\[
\hat \bj(t,E)=\frac{1}{u}\int \bv f(\bv,t;\bE) d\bv
\]
with
\[
u=\int |\bv|^2 f(\bv,t;\bE) d\bv.
\]
The initial condition is given by $f(\bv) = \lim_{N \to \infty} f^{(1)}_N(\bv,
0)$. It is easy to see that $u$ is independent of time and since we have chosen
$U_0=N$, $u=1$.
Concerning steady states, the situation is far from clear. In \cite{BDLR} and
\cite{BCKLs} it
was shown that a steady state $F_{\rm ss} (\bV;\bE)$ exists for the Kac master
equation \eqref{dW} provided that $\bE \not= 0$. If $\bE = 0$ any density
$F(\bV)$ that depends only on the magnitude of the velocities furnishes a
stationary state. It is, however, an open question whether $F_{\rm ss}
(\bV;\bE)$ tends to a limiting distribution as $\bE \to 0$. Interestingly,
assuming the {\it existence} of the limiting distribution, it can be computed
exactly and it is given by
\begin{equation}\label{exN}
F_{\rm ss}(\bV;0)=\frac{1}{\widetilde{Z}_N}\delta(U(\bV)-N)F(\bV)
\end{equation}
with
\begin{equation} \label{eff}
F_N(\bV)=\frac{1}{\left(\sum_{i=1}^N
|\bv_i|^{2+\alpha}\right)^{\frac{dN-1}{2+\alpha}} }
\end{equation}
where $\widetilde{Z}_N$ is the normalization constant
\begin{equation} \label{normalization}
\widetilde{Z}_N= \int_{\cS^{dN-1}(\sqrt{N})}F_N(\bV)d\sigma^{Nd-1} (\bV) \ .
\end{equation}
For details, the reader should consult \cite{BDLR} and \cite{BCKLs}. Thus, the
electric field
`selects' the right steady state as the electric field tends to zero.
A similar problem exists on the level of the Boltzmann equation. Again it is
possible to show that the steady state $f_{\rm ss}(\bv;\bE)$ for the Boltzmann
equation \eqref{BE} exists and is unique if $E\not=0$. This clearly implies the
existence of a steady state current $\hat \bj_{\rm ss}(E)$. In \cite{BL} it
was shown that, assuming that $f(v)=\lim_{E\to 0} f_{\rm ss}(v,E)$ exists and
that $\hat \bj_{\rm ss}(E)=O(E)$, one has
\begin{equation}\label{exin}
f(\bv)=\frac{\mu^{\frac{d}{2}}}{c}e^{-(\sqrt{\mu} |\bv|)^{2+\alpha}}
\end{equation}
where $c$ and $\mu$ are uniquely determined by normalization and the
condition $u=1$. One easily get
\begin{equation} \label{constants}
\mu=\frac{\Gamma\left(\frac{d+2}{2+\alpha}\right)}{\Gamma\left(\frac{d}{
2+\alpha
} \right)}\qquad\qquad
c=\frac{|\cS^{d-1}(1)|\Gamma\left(\frac{d}{2+\alpha
} \right) }{(2+\alpha)}
\end{equation}
which for $\alpha=1$ and $d=2$ gives
\[
\mu=\frac{\Gamma\left(\frac{4}{3}\right)}{\Gamma\left(\frac{2}{3}\right)}
\approx 0.65948\qquad\qquad
c=\frac{2\pi}{3}\Gamma\left(\frac{2}{3}\right)
\approx 2.83605
\]
For details the reader may consult \cite{BL} where the existence of the small
field limit of the steady state distribution is proved for $d=1$.
It is now natural to ask whether the distribution \eqref{exN} is chaotic with
marginal $f$ given by \eqref{exin}. For the reasons mentioned above, this
cannot be deduced from the previous results on propagation of chaos since these
results do not hold uniform in time.
A more serious impediment is the fact that
the small field limits of the steady states are not known to exist.
As explained before, the limit as $\bE \to 0$
selects a steady state for the Kac master equation as well as for its Boltzmann version.
It is far from clear that the selection mechanism is such as to preserve
chaoticity. In this note
we prove that, nevertheless, the distribution \eqref{exN} is chaotic with
marginal \eqref{exin}.
\medskip
\begin{thm} \label{chaoticss}
Let $f_N(\bv)$ be the one particle marginal of $F_N(\bV)$ i.e., defined by
\begin{equation}\label{NN}
f^{(1)}_N(\bv_1)=\frac{\sqrt{N}}{\sqrt{N-
|\bv_1|^2}} \frac{1}{\widetilde{Z}_N}\int_{\cS^{d{N-1}-1}(\sqrt{N-|\bv_1|^2})}
F_N(\bV)d\sigma(\bV^1)
\end{equation}
and set
\begin{equation}
f(\bv)=\frac{\mu^{\frac{d}{2}}}{c}e^{-(\sqrt{\mu} |\bv|)^{2+\alpha}}
\end{equation}
with the constants given by \eqref{constants}. Then for any bounded continuous function $\varphi(\bv)$
\begin{equation}\label{prove}
\lim_{N\to\infty} \int_{\mathbb{R}^d} \varphi(\bv) f_N^{(1)}(\bv) d \bv =\int_{\mathbb{R}^d} \varphi(\bv) f(\bv) d \bv
\end{equation}
and for every $k$,
the $k$ particle marginal $f^{(k)}_N(\bv_1,\ldots,\bv_k)$ of $F(\bV)$ satisfies
\begin{equation}\label{prove1}
\lim_{N\to\infty}\int_{\mathbb{R}^{kd}} \varphi(\bv_1,\ldots,\bv_k) f^{(k)}_N(\bv_1,\ldots,\bv_k) d \bv_1 \cdots d \bv_k =
\int_{\mathbb{R}^{kd}} \varphi(\bv_1,\ldots,\bv_k) \prod_{i=1}^k f(\bv_k) d \bv_1 \cdots d \bv_k
\end{equation}
where, again, $\varphi$ is any bounded continuous function on $\mathbb{R}^{kd}$. Thus $F_N(\bV)$
from a chaotic sequence with marginal $f$.
% Finally the limit \eqref{prove} and
% \eqref{prove1} are also true uniformly on $\mathbb{R}^d$ and in
% $L^1(\mathbb{R}^d)$
\end{thm}
\medskip
\section{Proof of Theorem \ref{chaoticss}.}
The following elementary lemma sets the stage for the proof. It will be
expressed in terms of the probability distribution
\[
g(\bw) := \frac{e^{-|\bw|^{2+\alpha}}}{\int_{\mathbb{R}^d} e^{- |\bw|^{2+\alpha}} d\bw } \ .
\]
\begin{lem} \label{basic} The following formulas hold for $F_N(\bV)$:
\begin{equation} \label{formulaeff}
F_N(\bV) = \frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})} \frac{1}{Z_N}
\int_0^\infty t^{dN-1} \prod_{j=1}^N g( \bv_j t ) \frac{dt}{t} \ ,
\end{equation}
\begin{equation} \label{formulazee}
Z_N=\frac{(2+\alpha)}{\Gamma\left(\frac{dN-1}{2+\alpha}\right)}\int_{
\mathbb{R}^{dN}}\frac{
\prod_{i=1}^Ng(\bw_i)}{|\bW|}\,d\bW
\end{equation}
and
\begin{align} \label{formulakay}
f^{(k)}_N(\bV_k) :&= \sqrt{\frac{N}{N - |\bV_k|}}
\int_{\cS^{d(N-k)-1}(\sqrt{N - |\bV_k|^2})}
F_N(\bV_k, \bV^k) d \sigma^{d(N-k)-1}(\bV^k) \\
&= \frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})Z_N } \sqrt{\frac{N}{(N -
|\bV_k|)^{dk+1}}}\times\nonumber\\
&\phantom{=\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})Z_N
}}\int_{\mathbb{R}^{d(N-k)}} \prod_{j=1}^k g\left(\frac{\bv_j
|\bW^k|
}{\sqrt{N- |\bV_k|^2}}\right) |\bW^k|^{dk-1}
\prod_{j=k+1}^N g(\bw_j) d\bW^k \ .\nonumber
\end{align}
\end{lem}
\begin{proof}
Formula \eqref{formulaeff} follows from \eqref{eff} and
\begin{equation}\label{exp}
A^{- \gamma} = \frac{1}{\Gamma(\gamma)} \int_0^\infty s^\gamma e^{-As} \frac{ds}{s} \ ,
\end{equation}
valid for all $A>0$ and $\gamma >0$ by setting
\[
A = \left(\sum_{i=1}^N
|\bv_i|^{2+\alpha}\right) \ , \ \gamma = dN-1
\]
and substituting $s =t^{2+\alpha}$. The normalization constant $\widetilde{Z}_N$
given in \eqref{normalization} is then given as
\[
\frac{2+\alpha}{ \Gamma(\frac{dN-1}{2 +\alpha})} \int_{\cS^{Nd-1}(\sqrt N)}
\int_0^\infty t^{dN-1} \prod_{j=1}^N
e^{-\left(|\bv_j| t \right)^{2+\alpha}} \frac{dt}{t} d \sigma^{Nd-1} (\bV)
\]
which, using Fubini's theorem and changing variables $\bv_j = \sqrt N \bw_j$
equals
\[
\frac{(2+\alpha)N^{\frac{Nd-1}{2}}}{\Gamma(\frac{dN-1}{2 +\alpha})}
\int_0^\infty \int_{\cS^{Nd-1}(1)} \prod_{j=1}^N e^{-( |\bw_j|t\sqrt N)
^{2+\alpha}} d \sigma^{Nd-1} (\bW) t^{dN-1} \frac{dt}{t} \ .
\]
One more variable change $r = \sqrt N t $ yields
\[
\frac{(2+\alpha)}{\Gamma(\frac{dN-1}{2 +\alpha})} \int_0^\infty
\int_{\cS^{Nd-1}(1)} \prod_{j=1}^N e^{-( |\bw_j|r ) ^{2+\alpha}} d
\sigma^{Nd-1} (\bW) r^{dN-1} \frac{dr}{r} \ .
\]
Hence
\[
Z_N = \frac{(2+\alpha)}{\Gamma(\frac{dN-1}{2 +\alpha})} \int_0^\infty
\int_{\cS^{Nd-1}(1)} \prod_{j=1}^N g ( \bw_jr) d \sigma^{Nd-1} (\bW)
r^{dN-1} \frac{dr}{r} \ .
\]
This integral is \eqref{formulazee} written in terms of polar coordinates. To
see \eqref{formulakay} we start with \eqref{kmarginal} and find
\[
f_N^{(k)}(\bV_k) =
\sqrt{\frac{N}{N -|\bV_k|}}
\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})} \frac{1}{Z_N}
\int_{\cS^{d(N-k)-1}(\sqrt{N -|\bV_k|^2})} \int_0^\infty
t^{dN-1} \prod_{j=1}^N g (\bv_j t) \frac{dt}{t} d \sigma^{d(N-k)-1}(\bV^k) \
.
\]
Once more, using Fubini's theorem and changing variables $\bv_j = \sqrt{N -
|\bV_k|^2} \bw_j, j=k+1, \dots, N$ yields
\begin{align}
\sqrt{\frac{N}{N -|\bV_k|^2}}&
\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})} \frac{1}{Z_N}
(N- |\bV_k|^2)^{\frac{d(N-k)-1}{2}} \times\nonumber\\
&\int_0^\infty t^{dN-1} \prod_{j=1}^k g(\bv_j t)
\int_{\cS^{d(N-k)-1}(1)}
\prod_{j=k+1}^N g\left(\sqrt{N - |\bV_k|^2} \bw_j t \right) d
\sigma^{d(N-k)-1}(\bW^k) \frac{dt}{t} \ . \nonumber
\end{align}
Changing variables $r = \sqrt{N - |\bV_k|^2} t$ yields the
expression
\begin{align*}
&\sqrt{\frac{N}{N - |\bV_k|^2}}
\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})} \frac{1}{Z_N}
\frac1{(N- |\bV_k|^2)^{\frac{dk}{2}}} \times\nonumber\\\
&\qquad \int_0^\infty r^{d(N-k)-1} \prod_{j=1}^k g\left(\frac{\bv_j
r}{\sqrt{N-
|\bV_k|^2}} \right) r^{dk-1}
\int_{\cS^{d(N-k)-1}(1)}
\prod_{j=k+1}^N g\left(\bw_j r \right) d \sigma^{d(N-k)-1}(\bW^k) dr
\nonumber\ .
\end{align*}
which equals
\[
\frac{2+\alpha}{\Gamma(\frac{dN-1}{2 +\alpha})Z_N} \sqrt{\frac{N}{(N -
|\bV_k|^2)^{dk+1}}}
\int_{\mathbb{R}^{d(N-k)}} \prod_{j=1}^k g\left(\frac{\bv_j |\bW^k|
}{\sqrt{N-|\bV_k|^2}} \right)|\bW^k|^{dk-1} \prod_{j=k+1}^N
g\left(\bw_j \right) d\bW^k \ .
\]
\end{proof}
\begin{lem} \label{limitnorm}
We have the following limit
\[
\lim_{N \to \infty} \int_{\mathbb{R}^{Nd}} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N
g(\bw_j) d\bW = \frac{1}{\sqrt \mu}
\]
where
\[
\mu := \int_{\mathbb{R}^d} |\bw|^2 g(\bw) d \bw \ ,
\]
so that
\[
\lim_{N \to \infty} \sqrt N \Gamma\left(\frac{dN-1}{2+\alpha}\right)Z_N
=\frac{(2+\alpha)}{\sqrt \mu}
\ .
\]
\end{lem}
\begin{proof} The proof follows with a slight modification from the law of
large numbers. We denote
\[
\mathbb{P}(A) :=\int_A \prod_{j=1}^N g(\bw_j) d\bW \ , \
\ {\rm and} \ \sigma^2 := \int_{\mathbb{R}^d} (|\bw|^2 - \mu)^2 g(\bw) d \bw \ .
\]
If
\[
A_\varepsilon =\left\{ \bW \in \mathbb{R}^{Nd} : \left| \frac{|\bW|^2}{N} -
\mu\right| > \varepsilon \right\}
\]
then it is a standard estimate that
\begin{equation} \label{tcheb}
\mathbb{P}(A_\varepsilon) \le \frac{\sigma^2}{\varepsilon^2 N} \ .
\end{equation}
From this it follows readily that
\begin{equation}
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^c_\varepsilon) \le
\int_{A^c_\varepsilon} \frac{\sqrt N}{|\bW|} \Pi_{j=1}^N g(\bw_j) d \bW
\le \frac{1}{\sqrt{\mu-\varepsilon}} \mathbb{P}(A^c_\varepsilon) \ .
\end{equation}
The set $A_\varepsilon$ can be written as $A_\varepsilon = A^< _\varepsilon \cup
A^> _\varepsilon$ where
\[
A^< _\varepsilon = \{ \bW \in \mathbb{R}^{Nd} : \frac{|\bW|^2}{N} < \mu - \varepsilon \} \ {\rm and} \
A^>_\varepsilon = \{ \bW \in \mathbb{R}^{Nd} : \frac{|\bW|^2}{N} > \mu + \varepsilon \} \ .
\]
Clearly
\[
\int_{A^>_\varepsilon} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N g(\bw_j) d \bW \le
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon) \ , \]
however, to estimate the corresponding integral in the set $A^<_\varepsilon$ is
a little bit trickier, on account of the singularity of $\frac{1}{|\bW|}$. Fix
some number $a < \sqrt{\frac{\mu - \varepsilon}{d}}$ independent of $N$ so that
for $N$ sufficiently large
\[
\int_{C^d} \frac{g(\bw)}{|\bw|^{\frac{1}{N}}} d \bw < \frac{1}{2}
\]
where $C^d =[-a,a]^d$.
Note that the cube $C^{Nd} =[-a,a]^{Nd} \subset A^<_\varepsilon$. By the
arithmetic-geometric mean inequality
\begin{equation}\label{agm}
\sqrt{\frac{N}{|\bW|^2}} \le \prod_{j=1}^N |\bw_j|^{-1/N}
\end{equation}
so that
\[
\int_{C^{Nd}} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N g(\bw_j) d \bW \le
\left(\int_{C^d} \frac{g(\bw)}{|\bw|^{\frac{1}{N}}} d \bw \right)^N \le
\frac{1}{2^N} \ .
\]
It remains to estimate the integral
\[
\int_{A^<_\varepsilon \setminus C^{Nd}} \frac{\sqrt N}{|\bW|} \Pi_{j=1}^N g(\bw_j) d \bW \ .
\]
Here we note that the ball centered at the origin of radius $a$ is a subset of
$C^{Nd}$ and hence
\[
\sqrt{\frac{N}{|\bW|^2}} \ge \frac{\sqrt N}{a}
\]
and this leads to the estimate
\[
\int_{A^<_\varepsilon\setminus C^{Nd}} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N
g(\bw_j) d \bW \le
\frac{\sqrt N}{a} \mathbb{P}(A^<_\varepsilon \setminus C^{Nd}) \ .
\]
Collecting these bounds yields
\begin{equation}\label{Ae}
\int_{A_\varepsilon} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N
g(\bw_j) d \bW \le\frac{\sqrt N}{a}
\mathbb{P}(A^<_\varepsilon \setminus C^{Nd}) +\frac{1}{2^N} +
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon)
\end{equation}
and finally
\begin{eqnarray}
\frac{1}{\sqrt{\mu+\varepsilon}} - \frac{1}{\sqrt{\mu+\varepsilon}}
\mathbb{P} (A_\varepsilon)
&\le& \int_{\mathbb{R}^{Nd}} \frac{\sqrt N}{|\bW|} \prod_{j=1}^N g(\bw_j) d
\bW
\nonumber \\ &\le& \frac{1}{\sqrt{\mu-\varepsilon}} + \frac{\sqrt N}{a}
\mathbb{P}(A^<_\varepsilon \setminus C^{Nd}) +\frac{1}{2^N} +
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon)
\end{eqnarray}
and using the estimate \eqref{tcheb} we find that
\[
\frac{1}{\sqrt{\mu+\varepsilon}}(1 - \frac{\sigma^2}{\varepsilon^2 N})
\le \int_{\mathbb{R}^{Nd}} \frac{\sqrt N}{|\bW|} \Pi_{j=1}^N g(\bw_j) d \bW \le
\frac{1}{\sqrt{\mu-\varepsilon}}(1 + \frac{\sigma^2}{\varepsilon^2 N} ) +
\frac{\sqrt N}{a} \frac{\sigma^2}{\varepsilon^2 N} +\frac{1}{2^N}
\]
By choosing $\varepsilon = N^{-1/8}$ the lemma follows.
\end{proof}
We now turn our attention to $f_N^{(k)}$. We first need the following
elementary Lemma.
\begin{lem}\label{change}
Let $\bY=(\by_1,\ldots,\by_N)$ be defined by
\[\begin{cases}
\by_i=\frac{|\bV^k|\bv_i}{\sqrt{N-|\bV_k|^2}}& 1\leq
i\leq k\\
\by_i=\bv_i& k+1\leq
i\leq N\ .
\end{cases}
\]
Then
\begin{equation}\label{Jac}
\bv_i=\by_i\sqrt{\frac{N}{|\bY|^2}}\qquad\qquad 0\leq i \leq k\ .
\end{equation}
Moreover the Jacobian determinant is given by
\[
\left|\frac{\partial (\bv_1,\ldots,\bv_k)}{\partial
(\by_1,\ldots,\by_k)}\right|=\frac{|\bY^k|^2}{N}\left(\frac{N}{|\bY|^2}
\right)^{\frac
{dk}2+1}=\frac{|\bV^k|^2}{N}\left(\frac{N^2-|\bV_k|^2}{|\bV^k|^2}\right)^{
\frac {dk}2+1}
\]
\end{lem}
\begin{proof} We have
\[
|\bY_k|^2=\frac{|\bV^k|^2|\bV_k|^2}{N-|\bV_k|^2}
\]
so that
\[
N-|\bV_k|^2=\frac{N|\bY^k|^2}{|\bY|^2}
\]
from which \eqref{Jac} follows immediately. To conclude the proof of the
Lemma it is enough to observe that
\[
\frac{\partial \bv_i}{\partial\by_j}=
\sqrt{\frac{N}{|\bY|^2}}
\left(\delta_{i,j } - \frac{y_iy_j}{|\bY|^2}\right)
\]
\end{proof}
Let now $\varphi(\bV_k)$ be a continuous function on
$\mathbb{R}^{dk}$ such that
\[
\sup_{\bV_k\in\mathbb{R}^{dk}}\varphi(\bV_k)<K\ .
\]
Using Lemma \ref{change} and \ref{formulaeff} we get
\begin{eqnarray}\label{inte}
&&\int d\bV_k\varphi(\bV_k)
f_N^{(k)}(\bV_k)=\nonumber\\
&&\qquad\frac{2+\alpha}{\Gamma(\frac{dN-1}{2
+\alpha})Z_N }\int_{\mathbb{R}^{dk}} d\bY_k\prod_{i=1}^kg(\by_i)
\int_{\mathbb{R}^{d(N-k)}}d\bY^k
\frac{\prod_{j=k+1}^N g(\by_j)}{|\bY|}%\frac{|\bY^k|}{|\bY|}
\varphi\left(\by_1\frac{\sqrt{N}}{|\bY|} , \ldots
, \by_k\frac {
\sqrt{N}}{|\bY|^2}\right):=\nonumber\\
&&\qquad\int_{\mathbb{R}^{dk}} d\by_1\cdots d\by_k\prod_{i=1}^kg(\by_i)
H_N(\by_1,\ldots,\by_k)
\end{eqnarray}
Since
\[
\frac{|\bY^k|^2}{|\bY^k|^2+|\bY_k|^2}<1
\]
we get
\[
|H_N(\by_1,\ldots,\by_k)|\leq
K\frac{\Gamma\left(\frac{d(N-k)-1}{2+\alpha}\right)Z_{N-k}}{\Gamma\left(\frac{
dN-1}{2+\alpha}\right)Z_N}\leq K'
\]
for a suitable constant $K'$. For the last inequality we have used Lemma
\ref{limitnorm}. The proof can then be completed by applying the Dominated
Convergence Theorem if we can show that
\[
\lim_{N\to\infty}H_N(\by_1,\ldots,\by_N)=\varphi\left(\frac{\by_1}{\mu},\ldots,
\frac{\by_N}{\mu}\right)
\]
pointwise in $\mathbb{R}^{dk}$. The proof of this fact follows closely that of
Lemma \ref{limitnorm}. From \eqref{Ae} it follows that
\begin{eqnarray*}
&& \int_{A_\varepsilon}d\bY^k
\prod_{j=k+1}^N
g(\by_j)\frac{\sqrt{N}}{|\bY^k|}\nonumber\sqrt{\frac{|\bY^k|^2}{|\bY^k|^2+
|\bY_k|^2}}
\left|\varphi\left(\by_1\frac{\sqrt{N}}{|\bY|} , \ldots ,
\by_k\frac{\sqrt{N}}{|\bY|}\right)-\varphi\left(\frac{\by_1}{\mu }
,\ldots,
\frac{\by_N}{\mu}\right)\right|\leq\nonumber\\
&&\qquad\qquad2K\left(\frac{\sqrt N}{a}
\mathbb{P}(A^\varepsilon \setminus C^{Nd}) +\frac{1}{2^N} +
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon)\right)
\end{eqnarray*}
while
\begin{eqnarray*}
&& \int_{\mathbb{R}^{Nd}\setminus A_\varepsilon}d\bY^k
\prod_{j=k+1}^N
g(\by_j)\frac{\sqrt{N}}{|\bY^k|}\nonumber\sqrt{\frac{|\bY^k|^2}{|\bY^k|^2+
|\bY_k|^2}}\left|\varphi\left(\by_1\frac{\sqrt{N}}{|\bY|} ,
\ldots , \by_k\frac{\sqrt{N}}{|\bY|}\right)-
\varphi\left(\frac{\by_1}{\mu }
,\ldots,
\frac{\by_N}{\mu}\right)
\right|\leq\nonumber\\
&&\qquad\qquad\frac{1}{\sqrt{\mu-\varepsilon}}\sup_{\lambda\in
I_N}|\varphi(\lambda\by_i,\ldots,\lambda\by_k)-\varphi(\by_i,\ldots,
\by_k)|
\end{eqnarray*}
where $I_N$ is the interval
\[
I_N=\left[\sqrt{\frac{N}{N(\mu+\varepsilon)+|\bY_k|^2}},
\sqrt{\frac{N}{N(\mu-\varepsilon)+|\bY_k|^2}}\right]\ .
\]
Collecting these estimates we get
\begin{eqnarray}
\biggl|H_N(\by_i,\ldots,\by_k)&-&\varphi\left(\frac{\by_1}{\mu},\ldots,
\frac{\by_N}{\mu}\right)\biggr|\leq\nonumber\\
&&\frac{2+\alpha}{\Gamma(\frac{dN-1}{2
+\alpha})\sqrt{N}Z_N }\left[ \frac{1}{\sqrt{\mu-\varepsilon}}\sup_{\lambda\in
I_N}|\varphi(\lambda\by_i,\ldots,\lambda\by_k)-\varphi\left(\frac{\by_i}{\mu},
\ldots ,
\frac{\by_k}{\mu}\right)|+\right.\nonumber\\
&&\left.2K\left(\frac{\sqrt N}{a}
\mathbb{P}(A^\varepsilon \setminus C^{Nd}) +\frac{1}{2^N} +
\frac{1}{\sqrt{\mu+\varepsilon}} \mathbb{P}(A^>_\varepsilon)\right)\right]
\end{eqnarray}
from which Theorem \ref{chaoticss} follows by choosing $\varepsilon=N^{-\frac
18}$.
\section{Extension and Remarks.}
It is easy to extend the results of the previous section in a couple of
interesting directions. We first observe that one can give a stronger
definition of chaoticity by requiring that given a sequence of functions
$F_N(\bV)$ on $\cS^{dN-1}(\sqrt{N})$, the entropy per particle of this sequence
converge to the entropy of the 1 particle marginal. More precisely if
\[
S_N=\int_{\cS^{dN-1}(\sqrt{N})}F_N(\bV)\log
F_N(\bV)d\sigma(\bV
\]
is the entropy of the $N$ particles system than
\[
\lim_{N\to \infty}\frac{S_N}{N}=\int_{\mathbb{R}^d} f(\bv)\log
f(\bv)d\bv
\]
where, as before
\[
f(\bv)=\lim_{N\to\infty} f^{(1)}_N(\bv).
\]
If this is true we say that the sequence $F_N$ is {\bf entropically chaotic},
see \cite{CCLRV}.
\begin{cor}
The sequence $F_N(\bV)$ given by \eqref{eff} is entropically chaotic and
\begin{equation}\label{SSN}
\lim_{N\to \infty} N^{-1}\int_{\cS^{dN-1}(\sqrt{N})}F_N(\bV)\log
F_N(\bV)d\sigma(\bV)=\log\left(\frac{\mu^{\frac{d}{2}}}{c}\right)-\frac{d}{
2+\alpha}=\int_{\mathbb{R}^d} f(\bv)\log f(\bv)d\bv
\end{equation}
\end{cor}
\begin{proof}
We will just report here the minor modification to the proof of Lemma
\ref{limitnorm} needed to prove the corollary. We observe that
\[
x\log x=\lim_{\delta\to 0} \frac{x^{1+\delta}-x}{\delta}
\]
Applying this to \eqref{exp} we get
\[
A^{-\gamma}\log A^{-\gamma}=\frac{1}{\Gamma(\gamma)}\int_0^\infty
s^{\gamma}\log s^{\gamma}e^{-As}ds-\gamma\psi(\gamma)A^{-\gamma}
\]
where $\psi(x)=\Gamma'(x)/\Gamma(x)$ is the digamma
function. Following the proof of Lemma \ref{basic}, we find
\begin{eqnarray}\label{SN}
\frac{S_N}{N}&=& -\frac{\log\tilde
Z_N}{N}-\frac{dN-1}{(2+\alpha)N}\psi\left(\frac{dN-1}{2+\alpha}
\right)+\nonumber\\
&&\frac{dN-1}{N}\frac{(2+\alpha)}{\Gamma\left(\frac{dN-1}{2+\alpha}\right)Z_N}
\int_ {\mathbb{R}^{dN}}\log\left(\frac{|\bW|}{\sqrt{N}}\right)\frac{
\prod_{i=1}^Ng(\bw_i)}{|\bW|}\,d\bW
\end{eqnarray}
where $\tilde Z_N=c^{N}Z_N$. Using Stirling formula we get that
\[
\lim_{N\to\infty}\left(\frac{\log\tilde
Z_N}{N}-\frac{dN-1}{(2+\alpha)N}\psi\left(\frac{dN-1}{2+\alpha}
\right)\right)=-\log c-\frac{d}{2+\alpha}
\]
Finally we need to compute the integral in the last term of \eqref{SN}.
It is very easy to adapt the proof of Lemma \ref{limitnorm} to this expression.
Indeed we have that
\[
\log\left(\frac{|\bW|}{\sqrt{N}}\right)\frac{\sqrt{N}}{|\bW|}
<\max\left(e , \frac{\log\sqrt{\mu+\epsilon}}{\sqrt{\mu+\epsilon}}\right)
\]
for $\bW\in A_\varepsilon^>$. On the other hand, the function $|(\log x)/x|$ is
increasing for $x<1$ so that from \eqref{agm} we get
\[
\left|\log\left(\frac{|\bW|}{\sqrt{N}}\right)\frac{\sqrt{N}}{|\bW|}\right|\leq
\frac{\sum_{i=1}^N\left|\log|\bw_i|\right|}{N}\prod
\frac{1}{|\bw_i|^{\frac{1}{N}}}\qquad\qquad\mathrm{for\
}\frac{|\bW|}{\sqrt{N}}<1
\]
One can now proceed like in Lemma \ref{limitnorm} with the only difference that
$a$ must be chosen such that
\[
\int_{C^d} \frac{\left|\log|\bw_i|\right|}{|\bw|^{\frac{1}{N}}} g(\bw)d \bw <
\frac{1}{2}
\]
We finally obtain that
\[
\lim_{N\to\infty}\int_
{\mathbb{R}^{dN}}\log\left(\frac{|\bW|}{\sqrt{N}}\right)\frac{
\sqrt{N}}{|\bW|}\prod_{i=1}^Ng(\bw_i)\,d\bW=\frac{\log\sqrt{\mu}}{\sqrt{\mu}}
\]
Collecting the above computation we get the first equality in \eqref{SSN}. The
second equality is immediate.
\end{proof}
A last interesting extension regards the first order correction in $\bE$. In
\cite{BCKLs}, under the assumption that the limit $|\bE|\to0$ exists, it was
shown that
\[
F_{\rm ss}(\bV,\bE)=\frac{1}{\widetilde{Z}_N}\left(F_N(\bV)+
\sum_{i=1}^N \bE\cdot\bc(\omega_i)|\bv_i|H_N(\bV)+o(|\bE|)\right)
\]
where $\bv_i=|\bv_i|\omega_i$ and
\[
H_N(\bV)=\frac{dN-1}{\left(\sum_{i=1}^N
|\bv_i|^{2+\alpha}\right)^{\frac{dN-1}{2+\alpha}+1}
}=\frac{1}{|\bv_1|^{2+\alpha}}\bv_1\cdot\nabla_{\bv_1}F_N(\bV),
\]
$\bc(\omega)$ is the unique solution of
\[
\left[(\mathrm{Id}-\cK)\bc\right](\omega)=\omega
\]
where $\cK$ is the convolution operator generated by $K$, that is
\begin{equation*}
\left(\cK\bc\right)(\omega)=\int_{\cS^{d-1}(1)}
K(\omega\cdot\omega')\bc(\omega')d\sigma^ { d-1 }
(\omega').
\end{equation*}
Since $-\bc(-\omega)$ is also a solution if $\bc(\omega)$ is, we have, by
uniqueness, that $\bc(\omega)=-\bc(-\omega)$. As a consequence
\[
\int_{\cS^{d-1}(1)}\bc(\omega')d\sigma^{d-1}(\omega')=0.
\]
Calling $h_N^{(k)}$ the marginal of $H_N$, we easily get that
\[
h_N^{(k)}(\bv_1,\ldots,\bv_k)=\frac{1}{|\bv_1|^{2+\alpha}}
\bv_1\cdot\nabla_{\bv_1}f_N^{(k)}(\bv_1,\ldots,\bv_k)
\]
It is easy to see, from \eqref{inte}, that we can take the limit for
$N\to\infty$ on both side and obtain
\[
\lim_{N\to\infty} h_N^{(k)}(\bv_1,\ldots,\bv_k)=h(\bv_1)\prod_{i=2}^k f(\bv_k)
\]
where
\[
h(\bv)=\frac{1}{|\bv_1|^{2+\alpha}}
\bv\cdot\nabla_{\bv}f(\bv)=(2+\alpha)\mu^{\frac{2+\alpha}{2}}f(\bv)
\]
Combining the above results we get that the $k$ particle marginal of $F_{\rm
ss}$ is
\begin{equation}\label{1oN}
\lim_{N\to\infty} f_{\rm
ss}^{(k)}(\bv_1,\ldots,\bv_k;\bE)=\left(1+(2+\alpha)\mu^{\frac{2+\alpha}{2}}
\sum_{i=1}^k \bE\cdot\bc(\omega_i)|\bv_i|\right)\prod_{i=1}^k f(\bv_k)
\end{equation}
This is consistent with the results on the Boltzmann equation \eqref{BE}. To
solve the steady state equation of \eqref{BE} one as to make an assumption on
the form of $\hat\bj_{\rm ss}(E)$ for small $|E$. It is natural to assume that
\begin{equation}\label{cond}
\hat\bj_{\rm ss}(E)=\tau\underline\kappa E+o(|E|)
\end{equation}
where $\underline\kappa$ is the conductivity tensor for the system with 1
particle and energy 1, that is
\[
\underline\kappa=\frac{1}{|\cS^{d-1}(1)|}\int_{\cS^{d-1}(1)}
\bc(\omega)\otimes\omega\,d\sigma^{d-1}(\omega).
\]
Under this assumption one finds that
\begin{equation}\label{1oinfty}
f_{\rm
ss}(\bv,\bE)=\left(1+(2+\alpha)\nu^{\frac{2+\alpha}{2}}
\bE\cdot\bc(\omega)|\bv|\right)\tilde f(\bv)+o(|E|)
\end{equation}
where
\[
\tilde f(\bv)=\frac{\nu^{\frac{d}{2}}}{b}e^{-(\sqrt{\nu} |\bv|)^{2+\alpha}}
\]
with $\nu$ and $b$ uniquely determined by normalization and \eqref{cond}. One
easily see that the average energy of this solution is
\[
u= \int_{\mathbb{R}^d}
|v^2|\tilde f(\bv)\,dv=\left(\frac{\nu}{\mu}\right)^{\frac{2+\alpha}{2}}
\]
so that, requiring $u=1$ we get back the large $N$ limit of the one particle
marginal of $F_{\rm ss}$.
Clearly the first order in $E$ of the $k$-fold tensor product of
\eqref{1oinfty} give us back \eqref{1oN}. We finally notice that the above
results tell us that the current per particle at small field for a large
system is $(2+\alpha)\mu^{\frac{2+\alpha}{2}}$ times the current of the one
particle system, if the energy per particle is 1.
\subsection*{Acknowledgment} We are indebted to Joel Lebowitz, Ovidiu Costin
and Eric Carlen form many enlightening discussions.
\bibliographystyle{unsrt}
\bibliography{nonequi}
\end{document}
|
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|
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| 366,892,955 | 5,587 |
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\title{Building A Movement}
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\section{Chicago—May 3}
Honoring the Haymarket Martyrs
The U.S. National Park Service has declared Chicago’s Haymarket Martyrs’ monument a National Historic Landmark and the Illinois Labor History Society (ILHS) is sponsoring a celebration, Sunday afternoon, May 3. The ceremony will take place at the former Waldheim cemetery, now called Forest Home, at 863 Desplaines Ave. in Forest Park, Ill., outside of Chicago.
The monument is a tribute to five anarchists, Albert Parsons, August Spies, Adolph Fischer, George Engle and Louis Ling, who died at the hands of the state of Illinois following a frame-up murder trial in 1887. They were convicted for a bomb thrown during a police riot although none of the accused were ever shown to have been directly connected with the device. Much like the current British Gandalf Defendants (see p. 5), the Haymarket martyrs were held responsible by their writings and utterances for an act committed by someone unknown to them.
Although it seems quite a contradiction to have anarchists honored by a government agency, official designation may be for the best.
Old Waldheim Cemetery, resting place of Emma Goldman and other radicals, had fallen on hard times recently. Security was so lax on the grounds that the Monument was vandalized and its elaborate brass trim stolen for scrap.
Leslie Orear, president of the ILHS, said a sculptor and foundry are necessary to restore the missing filigree. He added that security has improved under the site’s new owners and he was hopeful a reconstituted statue would be properly guarded.
For information and directions contact the ILHS, 28 E. Jackson, Rm. 1012, Chicago, IL 60604; (312) 663–4107; www.kentlaw.edu\Slash{}ilhs.
\section{Everywhere—May 16}
Global Street Party
People in London and Turku, Finland have proposed Saturday, May 16 as the day for a Global Street Party. These events would coincide with the 1998 G8 meeting in Birmingham, England, where world leaders from the eight largest capitalist economies will make decisions about the future of the planet and its people.
These politicians and corporate vampires, in their ceaseless drive for profit, will then fly to Geneva to celebrate the 50\textsuperscript{th} anniversary of GATT, where they will sign agreements enabling them to wrench more power and control away from local communities and siphon it into their self-appointed dictatorship.
People everywhere are rebelling against these global forces, and mid-May will see countless world-wide protests. A transnational street party has the potential to be a defining moment of resistance. Imagine the kick of taking back your street in the knowledge that all over the world similar acts of defiance are taking place.
Time is short, so please try to respond to this idea as soon as possible.
London Reclaim The Streets, PO box 9656, London N4 4NL, UK; 0171-281-4621; www.hrc.wmin.ac.uk\Slash{}campaigns\Slash{}rts.html.
\section{St. Louis—July 17–19}
Biotechnology Conference
Several environmental groups are sponsoring the first grassroots gathering on biodevastation and genetic engineering, July 17–19, at Fontbonne College in St. Louis.-Organizers say the conference will address the intertwined issues of genetic engineering, patenting of life forms and herbicide-resistant seeds, world trade in genetic material, and the monopolization of food production.
Bringing together major critics of biotechnology such as Indian physicist Vandana Shiva, author Brian Tokar, and Howard Lyman (co-defendant in the Oprah Winfrey case), the conference will host workshops and panels for environmentalists, pure food activists, and farmers.
For more information contact the Gateway Green Alliance at (314) 727–8554 or (314) 772–6463.
\section{Toronto—Aug. 17–23}
Anarchist Gathering Set For August
Toronto anarchists and radical activists are organizing Active Resistance ’98, August 17–23, designated as the ten-year anniversary of the Survival Gathering held in that city (see Summer 1988 FE for background).
Canada is home to the second-largest community of Nazi war criminals in the world, and Toronto has a disproportionate share of boneheads and fascist propagandists. It also recently witnessed a brutal wave of police killings of people of color in a city which is heavily multi-cultural.
The province of Ontario is rated the third most polluted area in North America, clearcut deforestation is rampant in the Temagami region, and the Darlington nuclear reactor was recently cited for incompetence. Campus occupations have taken place across the province by student activists frustrated with increasingly inaccessible tuition fees, and the Ontario Federation of Labor is poised for the possibility of a province-wide general strike.
The city is ripe for a creative, fun, culturally diverse, queer positive, Native solidarity, pro-feminist, youth and political prisoner-friendly, labor-aligned, anti-poverty, ecologically-minded gathering of anarchist activists taking action against oppression in Toronto, North America, and the world.
Send information requests, organizing and outreach ideas, workshop topics, fundraising suggestions, contact addresses to: Active
Resistance ’98 Toronto Planning Crew\Slash{} P.O. Box 108 Station P, Toronto, Ont., Canada M5S 2S8; 41676352763; resist62 tao.ca.
\section{Ft. Benning—Nov. 22}
Cross the Line at the SOA
What are you doing Nov. 22? The School of Americas Watch is looking for 1,000 good men and women to help permanently close the U.S. school for torture at Ft. Benning, Georgia. It is dubbed the School of Assassins by activists for training 60,000 Latin American soldiers who return home to murder, torture, rape and intimidate the poor and those working for the rights of the poor.
Last November 16, 2000 people demonstrated at Ft. Benning demanding the facility be closed. 601 people were arrested who “crossed the line” drawn by the cops and MPs at the base entrance. First-time offenders were given a letter barring them from the premises for a year. Repeat offenders, thirty in all, were given jail sentences for trespassing.
The alma mater of such U.S.-trained grotesque torturers as Salvador’s Robert “Blowtorch Bob” d’Aubisson must be shut down. Contact SOA.Watch, 1719 Irving St. NW, Washington DC 20010 and pledge to “cross the line” this November.
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Various Authors
Building A Movement
Coming Events
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\href{https://www.fifthestate.org/archive/351-summer-1998/building-a-movement}{\texttt{https://www.fifthestate.org/archive/351-summer-1998/building-a-movement}}
Fifth Estate \#351, Summer 1998
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\textbf{fifthestate.anarchistlibraries.net}
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Linear Algebra for Electrical Engineering 1
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Dr. Dennis Gulko,
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\item{} Fields: the definition of a field, complex numbers.
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author = {Афанасьев, Геннадий Фёдорович},
date = {2002-12-24},
keywords = {сигнал},
title = {Методические указания по лабораторным работам 'Устройства приема и обработки сигналов'},
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\caption{Solucion instancia dibujo10\_10.dat}\end{figure}
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\end{document}
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http://r2012.bordeaux.inria.fr/doc/gabarit_RR.tex
|
inria.fr
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CC-MAIN-2023-06
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application/x-tex
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application/x-tex
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crawl-data/CC-MAIN-2023-06/segments/1674764495012.84/warc/CC-MAIN-20230127195946-20230127225946-00579.warc.gz
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\documentclass[12pt]{article}
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% No paragraph indent or paragraph skip
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\begin{document}
\centerline{\bf Instructions pour les auteurs}
\vspace{12pt}
\centerline{ {\bf A.~Auteur}$^{\rm a}$ and {\bf B.~Auteur}$^{\rm b}$}
\vspace{12pt}
\centerline{$^{\rm a}$Departement de Mathematiques}
\centerline{A-Institut}
\centerline{A-Adresse}
\centerline{[email protected]}
\vspace{12pt}
\centerline{$^{\rm b}$Departement de Biologie}
\centerline{B-Institut}
\centerline{B-Adresse}
\centerline{[email protected]}
\vspace{24pt}
{\bf Mots clefs} : Statistique, Biologie, R\'egression.
\vspace{24pt}
Le processus de s\'election est bas\'e sur un r\'esum\'e de la pr\'esentation. Chaque r\'esum\'e sera examin\'e par un relecteur. Le processus de soumission comporte trois \'etapes :
\begin{enumerate}
\item Soumission du r\'esum\'e initial avant le 30 mars 2012
\item Notification de l'acceptation du r\'esum\'e autour du 23 avril 2012.
\item Soumission du r\'esum\'e final avant le 30 avril 2012.
\end{enumerate}
\vspace{12pt}
La longueur du r\'esum\'e ne doit pas exc\'eder 2 pages. Le r\'esum\'e doit contenir
\begin{enumerate}
\item le titre de la pr\'esentation,
\item les noms des auteurs,
\item les mots clefs,
\item Les contacts des auteurs, incluant l'adresse postale et l'email.
\end{enumerate}
Des mod\`eles Word et LaTeX pour la pr\'eparation des r\'esum\'es sont disponibles sur le site web de la conf\'erence. Une fois que le r\'esum\'e est pr\^et, merci de le convertir en format PDF. {\bf Les autres types de fichiers ne seront pas accept\'es}.
Chaque r\'esum\'e devra \^etre soumis en version \'electronique via le site web de la conf\'erence.
\vspace{12pt}
%The final abstract is to be prepared using the following format:
Le r\'esum\'e est \`a pr\'eparer en respectant les consignes suivantes :
$\bullet$ Marges : 2cm \`a gauche et \`a droite, 2cm en haut et 3cm en bas
$\bullet$ Titre en gras
$\bullet$ Nom d'auteurs en gras
$\bullet$ Pour les auteurs utilisant Word, choisir la police Times New Roman font \`a 12 pt
$\bullet$ Pour les auteurs utilisant LaTeX, choisir la classe article en 12pt
$\bullet$ 1 ligne d'espace entre le titre et le nom du premier auteur
$\bullet$ 1 ligne d'espace entre les auteurs et les affiliations
$\bullet$ 2 lignes d'espace entre les affiliations et les mots clefs
$\bullet$ 2 lignes d'espace entre les mots clefs et le corps du texte
$\bullet$ 1 ligne d'espace entre chaque paragraphe
$\bullet$ Pas d'indentation des paragraphes
$\bullet$ 1 ligne d'espace entre le corps du texte et les r\'ef\'erences [1]
$\bullet$ Ne pas inclure de num\'eros de pages
$\bullet$ Inclure les figures et les \'equations
$\bullet$ ``R\'ef\'erences'' en gras comme indiqu\'e ci-dessous
$\bullet$ R\'ef\'erences num\'erot\'ees [2], pas de ligne d'espace entre les r\'ef\'erences.
\vspace{12pt}
\parindent=0pt
{\bf R\'ef\'erences}
[1] Liquet, B., Saracco, J. (2011). A criterion for selecting the number of slices and the dimension of the model in SIR and SAVE approaches. To appear in {\it Computational Statistics}.
[2] Liquet, B., Saracco, J. (2008). Application of the bootstrap approach to the choice of dimension and the $\alpha$ parameter in the SIR$_\alpha$ method. {\it Communications in Statistics - Simulation and Computation}, {\bf 37}(6), 1198-121
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https://mirror.anarhija.net/usa.anarchistlibraries.net/mirror/w/wg/william-gillis-71-days-to-prepare-before-the-first-executive-orders.tex
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anarhija.net
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CC-MAIN-2021-10
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application/octet-stream
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application/x-tex
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crawl-data/CC-MAIN-2021-10/segments/1614178373095.44/warc/CC-MAIN-20210305152710-20210305182710-00285.warc.gz
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\title{71 Days to Prepare Before the First Executive Orders}
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While presidents almost always expand the power of their office and of the government, Donald Trump is likely to enact a degree of barefisted authoritarianism the modern United States is totally unprepared for.
Even if the corrupt and limp political elites that have so far utterly failed to stand against him unexpectedly rally a steadfast resistance in Congress Trump will still enter office with the overwhelming backing of rank and file law enforcement. A strongman unburdened by any conditioning of social norms or sufficient intelligence to understand game theoretic constraints. A man who the NYTimes reported “privately muses about all the ways he will punish his enemies after Election Day.”
Never mind the wild geopolitical catastrophes as yet undreamed, those of us in the belly of the beast have more immediate concerns to look forward to. Trump’s promised “law and order” presidency would mean a Christmas list of presents for the police. Exploding budgets and the removal of any pretense of constraining oversight from the Justice Department. Every beat cop in the nation flush with the invigorating knowledge that the President of the United States has their back, with money, legal support, public support, and ultimately the Presidential Pardon. The full extent of what is possible is dark indeed, but even moderate predictions are dire. We cannot afford to plan for the best.
In 71 days Trump will begin turning the ICE into a military operation capable of the industrial-scale ethnic cleansing he promised repeatedly. He will certainly shirk on some promises, but even if his effectiveness at getting all the millions he targets falls short, he will not miss the opportunity to demonstrate power, even if that means something as obscene as the national guard standing openly in sanctuary cities.
In 71 days Trump will approach an FBI already coursing with his fervent supporters and tell them to go ahead and do whatever it takes to get the domestic terrorists that didn’t vote for him. All the bored and overstaffed Joint Terrorism Taskforce offices the Bush administration left behind surveilling vegan potlucks will finally get to just bring the damn hippies in for questioning. The same pattern we’ve seen in countless countries when right-wing populists get into office will play out. Police raiding punk houses and roughing up anarchists for the sheer pleasure of it, finally able to assert their authority over those whose mere existence offends them. And this is the presidency when US police will be given drones with weapons.
In 71 days Trump will immediately turn his vindictive eye upon the media and every journalist he can get his fingers on. The press corp will be gutted and reporters will be threatened. The same tired procedure we’ve seen in dozens of other countries will arrive here overnight. The sort of regime where armed raids are used to conduct tax audits and incidentally bust up equipment. Trump’s number one concern with the Supreme Court justices he’ll stock it with will be — as always — their loyalty to him and their openness to allowing him to sue everyone for libel.
In 71 days Trump will inherit a vast surveillance apparatus of unparalleled scope in the world that will immediately be turned to his benefit against domestic adversaries or dissidents. He will empower those who have been stewing in outrage at their (meager) constraint. The US’ torture program will escalate. Just because he can. Just for the kick of it. American citizens will end up in Guantanamo and black sites around the world, what barriers to this have previously stood will make no sense to Trump. It will not take long, all things considered, before such American torture subjects are not just Muslim.
In 71 days Trump will start asking what can be done about that whole unruly internet thing and all those losers spreading lies. A president already aligned with Russia and with less than zero compulsion to lecture about human rights or leverage the activists within other superpowers will inexorably build a unprecedented global collaboration against Tor and internet freedom. A unified coalition that most of our existing tools were not prepared for.
We will face an America probably more reactionary and authoritarian than Italy under Berlusconi but probably less authoritarian than Germany under Hitler. The proper analogues are probably modern Hungary and Romania. Authoritarian populist “law and order” regimes with some pretense of normal modern life, riven with empowered racists and neighborhood curbstompings. A broadly mobilized reactionary populace and a shattered and demoralized opposition.
To most it will be perfectly survivable, if objectionable. To many it will be an apocalypse.
We have 71 days to prepare. To bootstrap a radical infrastructure capable of enabling survival and resistance. For ourselves and for all those likely to be targeted.
On a personal level you can:
\begin{itemize}
\item\relax
Read on security culture, opsec (also), threat modeling, best practices and frequent problems.
\item\relax
Get trained on tools of civil disobedience and street protests. From lockboxes to riot shield formations. Refamiliarize yourself with some basic activist skills
\item\relax
Learn, adopt and spread crypto tools like Signal, Torbrowser, GPG, and full-disk-encryption. Everyone can use Signal and Tor. GPG and full-disk-encryption will help you the most in the face of raids, although GPG has a steep learning curve and disk encryption can require a reinstall of your operating system. Read up on the limits and optimal ways to use Signal and Tor. Check out the EFF’s excellent guides for the basics. Read this list of other tools for a wider array of options, although beware that anything not in the EFF’s guide is problematic. Download a copy of the Tails operating system and put it on a spare USB.
\item\relax
Get a gun before dissidents and leftists with arrest records cease passing background checks. Learn about firearms if you’re ignorant. Go to the range and practice. Work out and enroll in self-defense classes focused on surviving attacks from reactionary thugs in the streets.
\item\relax
Familiarize yourself with how to use cryptocurrencies. Buy some prepaid phones with cash (don’t activate them or keep them on near your house, don’t give your actual info or call\Slash{}logon using your actual devices when activating them). Buy cash cards.
\item\relax
Buy and stockpile long-lasting emergency contraceptives and hormones that will likely be constrained. If you are trans get a letter from your physician that you’re undergoing transition, and a court order\Slash{}decree with proof of your name change, as these are likely to become requisite in some areas and also highly curtailed.
\item\relax
Get together with your closest friends and build affinity groups and networks of people capable of providing underground support.
\item\relax
Possibly start purging your social media, electronic devices and the like. Find other ways to communicate serious issues and organize.
\item\relax
Go to churches that do support work for undocumented people or provide sanctuary and start building personal ties.
\item\relax
Set up text trees or use apps like Cell 411 and Bouy to help establish communications networks to deal with raids and checkpoints, to mobilize responses to attacks and pogroms.
\item\relax
Get wifi routers, water, medical supplies, a month’s supply of dry food, and a go bag. Set up small geocaches (buried resources\Slash{}tools in waterproof containers you can reach if on the run).
\item\relax
Train anyone and everyone you can in the above skills. Get at least a dozen of your friends to use Signal. Give a training in something else to larger numbers of people at least twice before the election.
\end{itemize}
Hackers need to shift gears towards building to just keep people in contact, to keep networks up and information flowing, in the face of unified global adversaries, a hostile legal regime to NGOs, and even possibly the shutting down of BGP or ISP infrastructure. The country should be treated as a disaster zone and mesh wifi networks built as fast as we humanly can, similarly the time to swap over to heavy focus on steganography and bootstrapping devices in a hostile landscape is now. Biohackers need to get DIY contraceptives and hormones flowing asap. Activists need to build coalitions capable of acting while respecting the innate dissonance and diversity of subjective actors. The object now must be making a difference, not playing games of positioning and signalling, not focusing on personally feeling virtuous or wrapping ourselves in the comforting blankets of community. Local networks and face-to-face AFK ties will regrettably become all the more critical. Food security and basic needs infrastructure must be locked down. But most importantly we need robust networks and widely shared skills.
Many will continue chortling — flush with the ecstatic visceral delight of seeing a monster like Clinton bloodied, smug liberals distraught, and the vile establishment shaken to its foundations — mocking and unsympathetic to those foretelling cataclysm. But beware that most limited and inane sort of induction that presumes that what exists must necessarily exist. We have lived in but a tiny and incredibly unstable sliver of the full expanse of what is possible. And dystopian as our world already is there’s a hell of a lot worse possible. There is no magic force that presumes things will continue ticking much as they always have. Such shallow comfortable thinking is incapable of seeing deeper troubles, the edge cases, and the barriers to the effective region of such normalcy.
We’re looking down the barrel of a grim future — the violent reaction to the cosmopolitan complexities of global interconnection and the information era. Liberty is in retreat like few other moments in history. Around the globe numerous countries are falling to nationalism. The reassuring simplicity of authority and tribalism is a balm to the anxious and a virus that self-compounds via fear and brutality. There will be all the pressures in the world to let such crop up in the ranks of the opposition. But remember every anti-authoritarian resistance movement that has adopted or made deals with nationalism in response to nationalism has regretted it.
We live, as the curse goes, in interesting times. We have no idea how much turns on the next 71 days, how much of the future we can influence by hustling now, but we are far more likely to underestimate it than overestimate.
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The Anarchist Library (Mirror)
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Anti-Copyright
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William Gillis
71 Days to Prepare Before the First Executive Orders
November 9th, 2016
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Am 30. Januar 1933 kommt Adolf Hitler in Deutschland an die Macht. Er macht dies nicht mittels eines brutalen Staatsstreiches, bei dem er seine bewaffneten Milizen den vermeintlichen Rechtsstaat wegfegen lässt: Er wird direkt vom Präsident Hindenburg zum Kanzler ernannt. Drei Monate zuvor war der Führer des Nationalsozialismus abgeschrieben worden, nachdem seine Partei bei den Wahlen am 6.November zwei Millionen Stimmen verloren, während die Kommunistische Partei (KPD) 700.000 gewonnen hatte.
Am auf das Wahlergebnis folgenden Morgen annoncierte die Rote Fahne [das zentrale Organ der KPD] euphorisch, dass „ überall Mitglieder der Sturmabteilung (SA) aus den Reihen des Hitlerismus desertieren und sich der kommunistischen Fahne anschließen.“ Die selbe Fahne, die noch am 25 Januar 1933 während der großen Antifaschistischen Demonstration in Berlin stolz im Wind wehte, bei der 125.000 Arbeiter aufmarschierten- „eine wundervolle Jugend“,“eine Beteiligung, ein Enthusiasmus, eine Entschlossenheit, die wir noch nie gesehen hatten.““Versuchen wir die Anzahl an Kämpfern zu bewerten, die der Kolonne nützlich sein können. Fünfundneunzig Prozent, in Anbetracht ihres Alters, ihres Verhaltens beeindrucken uns als Aktivisten, die bereit sind für den bewaffneten Kampf“ wird ein Zeuge sagen, der fünf Tage später sehen wird, „wie sich die großartige Kommunistische Partei Deutschlands wie ein Stück Zucker im Wasser auflösen wird. Die erste Partei Berlins, die stärkste Sektion der kommunistischen Internationalen.“
Hitler war an der Macht und das Rote der Fahne der Arbeiter nahm die Farbe der Schmach, der Schande, der Demütigung an. Es gab keine Massenproteste, es gab keine Generalstreiks, es gab keine Zusammenstöße auf der Straße. Es gab keinen Bürgerkrieg, es gab keine Revolution. Es ist nichts Nennenswertes passiert, außer einer Reihe von Subversiven, die unter den Schlägen der braunen Pest gefallen sind. Entmutigung, Verzweiflung, Enttäuschung, Machtlosigkeit, Kappitulation, Niederlage, das ist es ,was in diesem Februar 1933 die revolutionäre Beweging durchdrang, die vom stupidesten Gehorsam und dem blinden Vertrauen in die Partei dominiert wurde. Wohin waren die abertausenden „Kameraden“ verschwunden, die Teil der verschiedenen Selbstverteidigungsmilizen waren, die allen Parteien, die sozialdemokratische mit eingeschlossen, zur Verfügung standen. Wo waren diese fünfundneunzig Prozent Aktivisten, die für den bewaffneten Kampf bereit waren?
Verschwunden, aufgelöst während einer kalten Nacht Ende Januars. An diesen furchtbaren Tagen ist es nicht das Kommunistische Manifest, ist es nicht das anarchistische Ideal, ist es nicht die Metaphysische Wahrheit die von einem dreiundzwanzigjährigen, holländischen, halbblinden Räteanhänger, Marinus Van der Lubbe, allein gegen alle verteidigt werden, sondern menschliche Gefühle wie die Würde und der Stolz. In der Nacht vom 27. zum 28. Februar schleuste er sich in den Reichstag ein und steckte ihn in einem letzten Versuch, das deutsche Proletariat zur Revolte aufzurufen, an. Ein großzügiger und vergeblicher Versuch, der nicht nur durch die Folter und Enthauptung durch seine Feinde bestraft wurde, sondern auch mit dem Unverständnis, der Verleumdung und dem Vegessen seiner eigenen Freunde belohnt wurde.
Nein! Im Land des Spartakusaufstandes von 1919, im Land, das die Wiege der Arbeiterbewegung war, protestieren und warten die Massen im Angesicht des Nazi-Schreckens, sie wählen und warten, sie demonstrieren und warten, sie schimpfen und warten, sie ertragen und warten, warten, warten\dots{} warten darauf die Meinung ihrer Anführer zu hören, dieser Funktionäre der dialektischen Wissenschaft, die am Abend des 30.Januar – als der österreichische Schmierfink gerade frisch ernannt worden war – davon überzeugt waren, dass dieser Hitler sich bald aufbrauchen würde, dass Hitler mit dem Krieg den Weg zur Revolution öffnen würde, dass Hitler es niemals wagen würde, sie zu Gesetzlosen zu erklären, dass Hitler niemals von den internatioalen Regierungen anerkannt werden würde, dass Hitler ein dunkler, brutaler Übergang sei, den die Massen durchlaufen müssten, bevor sie dann zur so sehr angestrebten roten Regierung gelangen würden.
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Die Massen warteten und hofften während die Parteichefs hinhielten und verrieten. Aber nicht das Individuum. Letzteres hat nichts zu erwarten oder zu erhoffen, es hat nur sein Gewissen, vor dem es sich zu verantworten hat und seinen Willen, den es in die Tat umzusetzen gilt. Und manchmal reicht dies, um Geschichte zu machen. Oder sie um nur dreizehn Minuten, um nur 780 Sekunden zu verfehlen.
\section{Der Handwerker}
Er hieß Georg Elser und wurde am 4. Januar 1903 in Hermaringen, einer kleinen Ortschaft im Südwesten Deutschlands geboren, bevor seine Familie dann nach Königsbronn (ebenfalls in Baden-Würtemberg) zog. Als Ältestes von vier Kindern arbeitete er schon sehr jung auf dem Bauernhof der Familie. Mit sechzehn begann er eine Ausbildug in einer Schreinerei, eine Arbeit, die er liebte und in der er ein wahrhafter Meister wurde. Dabei begriff er den qualitativen Unterschied zwischen der Arbeit eines Arbeiters, die mechanisch und sich wiederholend war und bei dem sich der Arbeiter am Fließband auslaugte und dem Beruf des Schreiners, der Gegenstände mit seinen eigenen Händen erschafft. Er arbeitete nicht nur für Geld, sondern auch um wahrhaften Kunstwerken Form zu geben. Im Lauf der Jahre voller Armut und Arbeitslosigkeit war Elser gezwungen, in der Gegend umherzuwandern und oftmals die Arbeit zu wechseln. Die Wirtschaftskrise verschonte niemanden, nicht einmal die Möbelfabrikanten, es wurde immer schwieriger. Er arbeitete auch in irgendeinder Uhrenfabrik, wobei er sich für die Mechanik der Uhren begeisterte. Schließlich kehrte er auf das dringliche Bitten seiner Familie nach Hause zurück, die kurz davor stand ihren Bauernhof zu verlieren.
Als Hitler Anfang 1933 die Macht ergriff befand Elser sich eben in Königsbronn, wo er sein Leben in mitten von tausenden von Schwierigkeiten fortsetzte. Die Arbeit wurde immer automatisierter, die menschliche Fertigkeit war nicht mehr wichtig, die Löhne brachen zusammen. Im Laufe der Jahre näherte sich Elser linken Gruppen, in denen er nie aktiv gewesen zu sein scheint. Er war kein Aktivist, er öffnete keine Bücher, er las sehr wenig die Zeitungen, er interessierte sich nicht für Politik. Es gefiel ihm ganz einfach unter Leuten wie er selbst zu sein, unter Proletariern. Er hat natürlich schon seine Mitgliedskarte der Kommunistischen Partei genommen und sich zur selben Zeit sogar dem Rotfrontkämpferbund angeschlossen, aber nur weil dies ihm ermöglichte, in der Blaskapelle dieser Organisation zu spielen. Leidenschaftlich für Musik, konnte er mehrere Instrumente spielen, unter diesen die Zither.
Georg Elser war sehr geschickt mit seinen Händen, aber besaß eine geringe „politische“ Kultur und Vorbildung. Dies war ein wahrhaftes Glück, denn sein Kopf wurde so von den marxistischen Tiraden über den historischen Materialismus und die Dialektik verschont. Man musste nicht diplomiert sein, um zu merken, was die Nazis am Treiben waren, die alltägliche Vergewaltigung jeglicher Freiheit, der auferzwungene Terror mit dem Verbannen der Parteien und Gewerkschaften, die Verschärfung der Lebensbedingungen und – ab 1938 – dem Schreckgespenst des Krieges, das immer konkreter wurde. Nicht nötig, einen durchdringenden Blick zu haben, um die Privilegien zu sehen, in denen sich die Nazifunktionäre suhlen. Und daraus alle Konsequenzen zu ziehen.
Seine Freunde erinnerten sich viel später daran, dass sich Elser nie Hitlers Reden im Radio anhörte, dass er sich weigerte den Hitlergruß zu machen und dass er einmal während einer Demonstration für Hitler sich umdrehte und sie auspfiff. Aber Georg Elser war nicht wie seine Freunde, er war nicht wie die Millionen Deutschen, die sich damit zu frieden gaben über das Naziregime zu meckern. Als einfacher und praktischer Mensch, traf er Anfang 1938 seine Entscheidung. Wie er später erklärte, kam er zum Schluss, „dass die Verhältnisse in Deutschland nur durch eine Beseitigung der augenblicklichen Führung geändert werden könnten.“ Das Individuum, Begehren und Willen, hatte seine Entscheidung getroffen: Hitler musste sterben. Der große Diktator und sein ganzer Kreis wurden damit zum Tode verurteilt; nicht durch ein Gericht des Staates, nicht durch das Urteil der Geschichte und noch weniger durch das göttliche, sondern durch einen kleinen Handwerker aus dem Schwabenland. Ein Weckruf an die Massen und ihre Organisationen.
Einsam und alleinstehend vertraute Elser niemandem seine Projekte an und suchte laut den Historikern keine Hilfe von außerhalb. Es scheint dennoch, dass ihm bei seiner Unternehmung von ein paar Individuen geholfen wurde: der Anarchist und deutsch-englische Ex-Spartakist John Olday und revolutionäre Sozialistin jüdischer Herkunft Hilda Monte, beide in Verbindung mit der Schwarzrotgruppe. Aus was diese Hilfe bestand weiß niemand wirklich. Jedenfalls musste Elser ein praktisches Problem lösen. Es musste ihm gelingen, sich dem Führer nah genug zu nähern, um ihn zu töten. Es hatten schon andere diese Idee liebgewonnen, aber alle sind auf die selbe Schwierigkeit gestoßen. Im Bewusstsein, dass er mehr gefürchtet als geliebt wurde, war Hitler von Attentaten besessen und hatte die Gewohnheit seine Programmpläne unvorhersehbar zu ändern.
Wenn seine Anwesenheit bei irgendeiner öffentlichen Versammlung angekündigt wurde, wussten nicht einmal seine nahestehendsten Kolaborateure ob er zu dem vorgesehenen Termin kommen würde. Auf diese Weise konnte kein eventuelles Durchsickern seinen Feinden helfen, die nie im Voraus wissen konnten, wo er hingehen würde.
Diese unentwegte Vorsicht hatte jedoch einen Fehler. Es gab tatsächlich einen einzigen öffentlichen, jährlichen Termin, den er um Nichts in der Welt aufgegeben, vor dem er sich nicht gedrückt hätte. Eine besondere Gedenkfeier, ein Jahrestag an den es sich zu erinnern galt, eine Rede voller Emotionen, die Feier seines ersten, fehlgeschlagenen Versuches die Macht zu ergreifen – der Putsch in München vom 8. November 1923. An jenem Tag hatte der vierunddreißigjährige Hitler an der Spitze seiner Waffenbrüder ein beeindruckendes Eintreten in den Bürgerbräukeller hingelegt, wo eine Versammlung stattfand, an der die bayerischen Autoritäten teilnahmen, wobei er einen Schuss in die Luft abgab. Er hatte ihnen angekündigt, dass ein Staatsstreich im Gange war, wobei er sie dazu einlud, sich den Nazis anzuschließen. Der Putschversuch, der allzu improvisiert war, endete am nächsten Morgen während einer Schießerei zwischen den Demonstranten, die auf dem Weg zum Kriegsministerium waren und den Ordungskräften, an deren Ende vierzehn Nazis getötet worden waren.
Ab 1933 fand sich Adolf Hitler jeden 8. November mit seinem gesamten inneren Kreis ein um an der Gedenkfeier des Putschs teilzunehmen. Umgeben von tausend frühen Nazikämpfern, mit denen er Witze und Anekdoten austauschte, versuchte der Führer dieses Jahr 1938 in seinem üblichen Redefluß die kriegerische Wut seiner Anhänger anzuheizen. Im November 1938 – 10 Monate vor dem Überfall auf Polen durch die deutschen Truppen – nahm Elser den Zug nach München und beteiligte sich unauffällig an den Festlichkeiten der Nazis. Als Hitler an jenem Abend auf die Bühne stieg, konnte er nicht wissen, dass sich vor den Türen der Brauerei sein Todfeind einfand, der bis dorthin gekommen war um sich ein Bild zu machen. Die Brauerei, die ihren Namen von Bürgerbräukeller zu Löwenbräu geändert hatte, enthielt einen enormen unterirdischen Saal, der mehr als 3000 Personen fassen konnte. Elser mischte sich unter die Menge derer, die eine Erlaubnis hatten am Ende des Abends, nach der Rede und der Abfahrt Hitlers, einzutreten, und merkte sich die Einteilung des Ortes, indem er die Sicherheitsmaßnahmen studierte, die für das Ereignis getroffen wurden. Er konstatierte unglaubliche Mängel. Ihr Verantwortlicher war Christian Weben, ein ehemaliger Türsteher vor Nachtlokalen, dem es, als glühender Nazi, nicht in den Sinn gekommen war, dass jemand Hitler bis auf den Tod hassen konnte. Die Aufmerksamkeit Elsers konzentrierte sich vorallem auf den einzigen Ort wo sich Hitler seit zu langer Zeit in Sicherheit wähnte: die Bühne. Er bemerkte eine Säule aus Stein genau dahinter, die einen großen Balkon stütze, der der Mauer entlang verlief. Unschwer zu verstehen, dass eine, im Innern der Säule platzierte, mächtige Bombe, den gesamten Balkon zusammenbrechen lassen würde, und Hitler und seine Nahestehenden unter dem Schutt begraben würde. Eine unmögliche Unternehmung für viele, aber für einen erfahrenen Handwerker nicht.
Es ist der Tag nach dem 9. und 10. November 1938, an dem sich die Nazis durch das ganze Land, aber auch teils durch Österreich und Tschechoslowakei austobten, in dem, was Kristallnacht heißen sollte. Das anti-jüdische Pogrom verstärkte Elsers Entschlossenheit noch. Er hatte ein Jahr um sein Projekt zu Ende zu bringen, und dem er sich mit Zähigkeit und Akribie widmete. Er musste sich Sprengstoff besorgen, eine Bombe mit Zeitverzögerung konstruieren, dann das Gerät im Inneren der Säule verstecken. Um das zu tun, versuchte er temporäre Arbeit in einer Waffenfabrik zu finden, dann in einer Mine und dort hatte er Erfolg. Dort, nutzte er jede Gelegenheit um sich starken Sprengstoff und Dynamit anzueignen, und auch Detonatoren habhaft zu werden. Abends, eingeschlossen in seiner Wohnung, erarbeitete er Pläne um eine raffinierte zeitverzögerte Bombe zu konstruieren.
Im April kehrte er nach München zurück um unter ruhigeren Umständen eine neue, detailliertere Zeichnung anzufertigen. Er bemerkte, dass sich auf der Etage über dem Saal Lagerräume befanden, wo er sich verstecken und aus der Nähe die Säule begutachten konnte. Sie war bedeckt mit Holz! Perfekt. Er erkundete dann die schweizerische Grenze, um einen Fluchtweg zu finden, und fand schließlich einen Abschnitt ohne Patrouillen. Sicherlich, Georg Elser wollte Hitler töten, aber er hatte auch die Absicht die an sich gerissene Freiheit zu leben und zu geniessen. Es gab keinen Opfergeist in ihm.
Am 5. August 1939 nahm Georg Elser den Zug und ließ sich ein letztes Mal in München nieder um den letzten Teil seines Projekts zu realisieren, den schwierigsten und auch den riskantesten: einen Hohlraum in den Pfeiler hinter dem Rednerpult zu graben, der groß genug ist, und dort eine tödliche Vorrichtung zu verstecken, ohne entdeckt zu werden. Er wurde zu einem Stammkunden des Löwenbräu, der bei den Nazis beliebtesten Brauerei in München. Er verbrachte dort den ganzen Tag, so dass die Kellner und Kellnerinnen aufhörten ihrem gemächlichen, teuren Kunden allzu viel Aufmerksamkeit zu schenken. Jeden Abend, blieb Elser bis zur Schließung, um dann heimlich auf die obere Etage zu schleichen, wo er sich in einem Lagerraum versteckte. Während die Räumlichkeiten leer waren, kam er heraus um am Pfeiler zu arbeiten. Beim Licht einer Taschenlampe demontierte er mit Vorsicht die Holzpanele der Säule, stellte sie zur Seite damit sie einfach wieder anzubringen waren, und begann geduldig den Stein auszuhöhlen. Inmitten der Stille, hallte der Lärm des Bildhauerbeitels, der den Stein schlug, derartig in diesem Kuppeldach, dass er gezwungen war, mit einer anstrengenden Langsamkeit zu arbeiten. Die einzelnen Schläge folgten den Intervallen mehrerer Minuten, die er versuchte mit dem Lärmen der Straße, wie dem Vorbeifahren eines Autos, in Einklang zu bringen. Jede Spur von Pulver oder Stein musste anschließend verschwinden, und das Holzpanel musste vor dem Morgengrauen perfekt an Ort und Stelle angebracht werden.
Abend für Abend widmete er sich seinem Meisterwerk.
\forcelinebreak
Er verbrachte 35 schlaflose Nächte, gebeugt von der anstrengenden Arbeit. Eines Morgens, wurde er von einem Kellner überrascht, der vor Arbeitsbeginn ankam und sofort den Wirt der Brauerei rief. Elser, der gerade dabei gewesen war zu gehen, nachdem er alles aufgeräumt hatte, entschuldigte sich, indem er sagte ein Stammkunde zu sein und dass er die Örtlichkei offen vorgefunden habe. Er bestellte einen Kaffee, trank ihn schweigend und mit kleinen Schlücken, bis er ging. Er war nicht verbrannt.
\forcelinebreak
Um seine Bombe zu preparieren, stellte er einen Zeitzünder her, indem er eine Uhr modifizierte. Der Zeitzünder konnte im Rahmen von 144 Stunden laufen, bevor er auf einen kleinen Hebel drückte, der die Vorrichtung aktivierte. Aus Skrupel fügte er zur Sicherheit einen zweiten Zeitzünder hinzu. Die Bombe war in einem eleganten Holzgehäuse untergebracht, das mit Präzision in das gegrabene Loch im Innern der Säule eingefügte war. Damit man das Tick-Tack der Uhr nicht hörte, nahm er Kork und bereitete ein Blech vor um das Innere des Holzpanels auszukleiden. Er wollte nicht, dass ein Mitglied des Personal versehentlich einen Nagel in sein Kunstwerk trieb!
Im vorangegangenen Jahr, notierte sich Elser, dass die Rede Hitler um 20.30Uhr begann, man versicherte ihm das sei eine Gewohnheit. Der Führer sprach für eineinhalb Stunden, blieb dann im Lokal um sich unter seine alten Kameraden zu mischen. Elser stellte seine Uhr, damit sie in der Hälfte der Rede auslöste, das heißt um 21.20Uhr. Der erste Versuch die Bombe zu platzieren scheiterte, und zwang ihn die Abmessungen seines Gehäuses zu reduzieren. Am Abend des 5. Novembers 1939 beendete Georg Elser sein Meisterstück. Er fügte das Gehäuse in die Säule ein, übergab das Holzpanel seinem Platz, indem er es versiegelte, und beseitigte dann jede Spur. Er verließ München bevor er zwei Tage später zurückkehrte. Am Tag vor der Ankunft des großen Diktators, näherte sich das kleine Individuum dieser Säule und legte sein Ohr an in der Hoffnung, aus der Ferne etwas zu hören. Wir können uns sein Lächeln vorstellen, während er noch ein Mal dieses wunderbare Ticken vernahm.
\section{8. November 1939}
Georg Elser las keine Zeitungen, und noch weniger in diesen fieberhaften Tagen. Andernfalls hätte er herausgefunden, dass Hitler sein üblicherweise jährliches Treffen annuliert hatte. Oder besser gesagt, er hatte seine Idee wieder geändert: er würde dorthin zurückkommen, aber viel eher als gewöhnlich. Seine Anwesenheit in Berlin war unerlässlich, deswegen würde er nur kurz nach München kommen. Seine Rede würde um 20Uhr beginnen, und nur eine kurze Stunde dauern. Aufgrund des schlechten Wetters, wurde ihm abgeraten mit dem Flugzeug zu reisen, und sich für den Zug zu entscheiden; langsamer, aber sicherer. Am Abend des 8. Novembers 1939 hörte Adolf Hitler um 21.07Uhr auf zu reden. Fünf Minuten später, die Einladungen älterer Kämpfer noch zu bleiben ablehnend, verließ er die Halle mit seiner Gefolgschaft an würdetragenden Nazis, unter ihnen der Chef der Polizei und Reichsführer SS Heinrich Himmler, der Propagandaminister Joseph Goebbels und der Chef der Sicherheitspolizei und des Sicherheitsdienstes Reinhard Heydrich. Sie waren zweifelos dabei in den Zug zu steigen, als die Explosion losging, und hörten sie selbst noch nicht einmal. Sie erfuhren von dem was passiert war erst während des kurzen Halts ihres Expresszugs nach Berlin in Nürnberg.
Um 21.20Uhr, wie vorher gesehen, hörte das Tick-Tack der Uhr Georg Elsers auf zu schlagen. Mit einem schrecklichen Rumoren brach die hinter der Bühne stehende Säule zusammen, ließ den Balkon, den sie stützte sowie das Dach einstürzen, und zerstörte das Lokal. Ein Regen holziger Trümmer, Ziegelsteine und Stahl fiel herab auf die Szene und vernebelte sie vollständig. Aber die Bühne war schon leer, und die Halle beinahe verlassen. Acht Personen starben und dreiundsechzig wurden verletzt, alles alte Nazi-Kämpfer oder Brauereiangestellte. „Das Glück des Teufels“ ein weiteres Mal an seiner Seite, mit dem Hitler prahlte, es zu besitzen. Das war dagegen nicht der Fall bei dem Individuum, welches ihn herausgefordert hatte.
\forcelinebreak
Am Morgen des 8. Novembers 1939 nahm Georg Elser den Zug nach Konstanz, an die deutsch-schweizer Grenze. In der kommenden Nacht, ging er zu Fuß in Richtung Grenze, in dem ruhigen Bereich, den er im vorherigen April entdeckt hatte. Aber mit der Invasion von Polen durch Deutschland am 1. September hatte sich die Situation grundlegend verändert. Er wurde von einer Patrouille bemerkt und festgenommen, die ihn durchsuchte. Er trug eine Mitgliedskarte der kommunistischen Partei, die Zeichnungen eines seltsamen Gerätes, das an ein Schema einer Bombe erinnerte, einen Zünder und eine Besuchskarte einer bekannten münchner Brauerei, dem Löwenbräu, mit sich.
Es ist sehr wahrscheinlich, dass Elser all diese Dinge, die ihn entschiedenerweise verdächtig machten, mit sich führte, um die helvetischen Autoritäten zu überreden ihm Asyl zu gewähren. Er ging im Gegenzug das Risiko ein, dass, sollte er in die Hände des Feindes fallen, es genau diese Objekte sein würden, die sein Ende bedeuteten.
\section{Einer}
Nach München gebracht, wurde Georg Elser von Männern der Gestapo verhört. Trotz der Schläge und der Folter, hat er nie die Version der Ereignisse geändert. Er war es, und er allein, der das Attentat vorbereitet und durchgeführt hatte. In Berlin war Hitler persönlich an der Affäre interessiert, und geriet in Zorn, als man ihm die Wort Elsers berichtete. „Wer ist der Narr, der diese Befragung führt?“, schrie er ihn an. Es war unmöglich, dass ein erbärmliches Individuum das große Reich hätte herausfordern können: die Komplexität der Aktion bewies, dass es dahinter ein breite Verschwörung geben musste, die aus\dots{} Geheimdiensten bestand, offensichtlich, und in diesem Fall den britischen. Um seine Entscheidung aufzuerlegen, schickte Hitler einen Vertrauensmann nach München, damit beauftragt die Verhöre neuzubeginnen: Heinrich Himmler.
Weder er noch all die Foltern, die er umsetzte, schafften es den Führer zu befriedigen. Elser wiederholt bis zu seinem Ende allein gehandelt zu haben, um seinen Henkern zu beweisen, dass er ganz allein es gewagt hatte Hitler anzugreifen, reproduzierte er von neuem ein Schema seiner Bombe. Himmler selbst musste schließlich offiziell die Verschwörungsthese aufgeben, und Elser, um später hingerichtet zu werden, wurde ins KZ Sachsenhausen verbracht. In Isolation, erlaubte man ihm ganz für sich an einer Werkbank zu arbeiten. Der Grund für diese anscheinend gefällige Behandlung war, dass Hitler in der Folgezeit Elser dazu bestimmte in einem Prozess der Kriegsverbrechen gegen England benutzt zu werden. Am 9. April 1945, während die amerikanischen, englischen und russischen Truppen sich immer mehr Berlin näherten, erinnerte sich Himmler an die Kühnheit des unglücklichen Schreiner-Uhrmachers, der in der Zwischenzeit nach Dachau verlegt worden war. Er gab den Befehl ihn aus der Zelle zu holen und hinzurichten. Die Neuigkeiten seines Todes gingen eine Woche später durch die deutsche Presse, und wurde einem alliierten Luftangriff zu gewiesen.
Trotz des Einsatzes der nazistischen Effizienz im Vorhinein, um die Wahrhaftigkeit der individuellen Initiative Elsers anzuzweifeln, und trotz des Geschwätzes seiner Unglückskameraden in Sachsenhausen, gemäß denen Elser, sowie Van der Lubbe, auf Befehl der Nazis selbst gehandelt haben soll, wagt es heute niemand die Aufrichtigkeit seiner Unternehmung zu leugnen. Sein Gedenken, wie das von vielen verpassten Attentaten gegen Hitler, wurde lange Zeit von Historikern verwischt, die einzig der Staatsraison gehörig waren, aber auch von bestimmten revolutionären Liebhabern der kollektiven Aktionen, welche wenig begierig darauf sind ihrer ideologischen Bewegung einen „schlechten Ruf“ zu geben.
Weil keiner unter ihnen die Beaobachtung tolerieren kann, dass die Entschlossenheit eines einzelnen Individuums, im Gegenteil zur trostosen Ohnmacht der Massen, die Geschichte hätte ändern können, indem es sie vor dem bewahrt, das als das absolut Böse definiert wurde. Um nur 13 unglückliche Minuten wurde der zweite Weltkrieg nicht abgewendet, was vielleicht Millionen Menschenleben und unaussprechliche Leiden erspart hätte. Und das, was diese Möglichkeit ergriff, das war keine erleuchtete Regierung, das war keine effektive Organisation. Das war ein kleiner Mensch, allein, oder vielleicht mit ein oder zwei Gefährten. Hier haben wir es, warum der Name Georg Elsers seit so langem vergessen ist, und hier ist auch der Grund, wieso wir ihm hier eine Huldigung geben. Nichts ist unmöglich für einen Willen, der vom Begehren getrieben wird. Und trotz der Rückschläge des Unvorhergesehenen, ist es das Tick-Tack dieses Uhrmachers, das wir noch heute vernehmen.
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Anarchistische Bibliothek
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Anticopyright
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Anonym
Dreizehn Minuten
März 2015
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Entnommen am 20.11.2019 von: \href{https://zuendlumpen.noblogs.org/post/2019/11/08/dreizehn-minuten/}{https:\Slash{}\Slash{}zuendlumpen.noblogs.org\Slash{}post\Slash{}2019\Slash{}11\Slash{}08\Slash{}dreizehn-minuten\Slash{}}
Originaltitel: \emph{13 minuti,} in \emph{Insolito sguardo}, ed. Gratis, mars 2015. Französische Übersetzung erschienen in: Avis de tempêtes. Bulletin anarchiste pour la guerre sociale; Nr. 9, September 2018. Deutsche Übersetzung aus aus dem Französischen Oktober 2019 von \emph{RumpelstilzchenEditionen.}
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\textbf{anarchistischebibliothek.org}
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Risk-sensitive filtering
involves minimisation of the expectation of an exponential in quadratic cost
criteria. As opposed to $L_2$ filtering, (termed as {\em risk-neutral filtering } in \cite{SM1}), which achieves the minimisation of a quadratic error
criteria, risk-sensitive filtering robustifies the filter against plant and
noise uncertainties by penalising all the higher-order moments of the
estimation error energy. It also allows a trade-off between optimal filtering
for the nominal model case and the average noise
situation, and robustness to worst case noise and model uncertainty by
weighting the index of the exponential by a risk-sensitive parameter.
The risk-sensitive filtering problem has been addressed for linear Gauss-Markov
signal models in \cite{Speyer}. In a
companion paper \cite{SM1} to the present
one, the problem has been solved
for a general class of discrete-time nonlinear state space signal models
via the so-called reference probability method and the linear Gauss-Markov
signal model has been treated as a special case. It has been seen that
risk-sensitive filters are closely related to $H_{\infty}$ filters \cite{jrs1}.
Also, related risk-sensitive control problems are abundant in literature
\cite{EMoore} \cite{Iain}
\cite{JBE} \cite{Whittle}.
The problem of extracting
finite-state homogeneous
Markov chains hidden in white Gaussian noise
has been studied as an off-line estimation problem using the well known
Expectation Maximisation (EM) algorithm \cite{Baum}
\cite{demp} \cite{Rabiner}.
On-line
estimation schemes for
Hidden Markov Models (HMM)
have been given in \cite{Iain1} \cite{Vmoore}.
In all these estimation schemes, the so-called
``forward variable'' \cite{Rabiner}
is the true filtered estimate which is also a conditional expectation of the
state at a certain point of time given the observations up and until that
point. The smoothed estimate of the state is obtained as a maximum-likelihood
estimate based on a fixed set of observations.
These filtering schemes are essentially related to risk-neutral
filtering for HMMs.
In this paper, we address the problem of risk-sensitive filtering and
smoothing for discrete-time Hidden Markov Models with finite-discrete states.
We derive information-state filters which are linear and finite-dimensional.
The optimising state estimate is given as the minimising argument of a
finite-dimensional sum. Also, the backward filters and unnormalised smoothed
conditional probability measures are derived.
The derivation techniques are based on
a reference probability method which has been developed in \cite{MElliot} and
used in \cite{EMoore} \cite{Iain} \cite{SM1}.
In Section 2, we describe the Hidden Markov Model, formally define
the risk-sensitive filtering problem, and then deal with the change of measure
and reformulation of the problem
in the new probability measure to achieve the
filtering and smoothing results.
In Section 3, we establish the connection
between risk-sensitive and risk-neutral filtering and Section 4 presents some
concluding remarks.
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\begin{document}
\title{プレ高数学科}\author{gbb60166}
%■■■■■■■■■■■■■ テスト領域 ■■■■■■■■■■■■■■
%\end{document}
%■■■■■■■■■■■■■ 完成品 ■■■■■■■■■■■■■■
\begin{frame}[t]
\frametitle{\ajKakko{1}\quad $3$桁の数は何通り？}
\begin{minipage}[t]{0.46\textwidth}
{\large
\ajKaku{1}%
\invisible<7->{\ajKaku{2}}%
\ajKaku{3}%
\ajKaku{4}%
\invisible<10->{\ajKaku{5}}%
\ajKaku{6}%
\invisible<4->{\ajKaku{7}}%
}の\\
$7$枚のカードを並べてできる\colorbox{CUDCream}{\textcolor{CUDBlue}{$3$桁の数}}
\vspace*{1ex}
\only<2-3>{\textcolor{CUDRed}{百の位に入れることができるカードは$7$枚}}
\only<5-6>{\textcolor{CUDRed}{十の位に入れることができるカードは$6$枚}}
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\end{minipage}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{\ajKakko{2}\quad $3$桁の偶数は何通り？}
\begin{minipage}[t]{0.46\textwidth}
{\large
\ajKaku{1}%
\alt<3-5>{\textcolor{CUDRed}{\ajKaku{2}}}{\ajKaku{2}}%
\invisible<11->{\ajKaku{3}}%
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\ajKaku{5}%
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}の\\
$7$枚のカードを並べてできる\colorbox{CUDCream}{\textcolor{CUDBlue}{$3$桁の偶数}}
\vspace*{1ex}
\only<2-3>{\textcolor{CUDRed}{一の位が偶数であればよい}}
\only<7-8>{\textcolor{CUDRed}{百の位に入れることができるカードは$6$枚}}
\only<9-10>{\textcolor{CUDRed}{十の位に入れることができるカードは$5$枚}}
\only<4-5>{\textcolor{CUDRed}{一の位に入れることができるカードは$3$枚}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\frametitle{\ajKakko{3}\quad $3$桁の奇数は何通り？}
\begin{minipage}[t]{0.46\textwidth}
{\large\vphantom{\ajKaku{9}}%■詰め物をする
\invisible<6->{\alt<3-5>{\textcolor{CUDRed}{\ajKaku{1}}}{\ajKaku{1}}}%
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\ajKaku{4}%
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$7$枚のカードを並べてできる\colorbox{CUDCream}{\textcolor{CUDBlue}{$3$桁の奇数}}
\vspace*{1ex}
\only<2-3>{\textcolor{CUDRed}{一の位が奇数であればよい}}
\only<7-8>{\textcolor{CUDRed}{百の位に入れることができるカードは$6$枚}}
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\end{minipage}
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baseline={([yshift=-24pt] current bounding box.north)}]
%\draw[help lines] (-0.3,-5) grid (10,4.2);
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\end{tikzpicture}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{\ajKakko{4}\quad $3$桁の５の倍数は何通り？}
\begin{minipage}[t]{0.46\textwidth}
{\large
\ajKaku{1}%
\invisible<11->{\ajKaku{2}}%
\ajKaku{3}%
\ajKaku{4}%
\invisible<6->{\alt<3-5>{\textcolor{CUDRed}{\ajKaku{5}}}{\ajKaku{5}}}%
\invisible<9->{\ajKaku{6}}%
\ajKaku{7}%
}の\\
$7$枚のカードを並べて
できる\colorbox{CUDCream}{\textcolor{CUDBlue}{$3$桁の５の倍数}}
\vspace*{1ex}
\only<2-3>{\textcolor{CUDRed}{一の位が\ajKaku{5}であればよい}}
\only<7-8>{\textcolor{CUDRed}{百の位に入れることができるカードは$6$枚}}
\only<9-10>{\textcolor{CUDRed}{十の位に入れることができるカードは$5$枚}}
\only<4-5>{\textcolor{CUDRed}{一の位に入れることができるカードは$1$枚}}
\end{minipage}
\hspace*{1ex}
\begin{tikzpicture}[scale=0.7, >=stealth,inner sep=0pt,
baseline={([yshift=-24pt] current bounding box.north)}]
%\draw[help lines] (-0.3,-5) grid (10,4.2);
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\Large
\foreach \x in {0,2,4}\draw(\x,0) rectangle ++(2,3);
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\draw(3,3.5) node{\small 十};
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\end{tikzpicture}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{\ajKakko{5}\quad $3$桁の$500$より大きい数は何通り？}
\begin{minipage}[t]{0.46\textwidth}
{\large
\ajKaku{1}%
\invisible<11->{\ajKaku{2}}%
\ajKaku{3}%
\ajKaku{4}%
\invisible<6->{\alt<3-5>{\textcolor{CUDRed}{\ajKaku{5}}}{\ajKaku{5}}}%
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\alt<3-5>{\textcolor{CUDRed}{\ajKaku{7}}}{\ajKaku{7}}%
}の\\
$7$枚のカードを並べてできる\\
\colorbox{CUDCream}{\textcolor{CUDBlue}{$500$より大きい数}}%手動調整
\vspace*{1ex}
\only<2-3>{\textcolor{CUDRed}{百の位が\ajKaku{5}、\kern-7pt\ajKaku{6}、
\kern-7pt\ajKaku{7}\\どれかだとよい}}
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\only<9-10>{\textcolor{CUDRed}{一の位に入れることができるカードは$5$枚}}
\only<4-5>{\textcolor{CUDRed}{百の位に入れることができるカードは$3$枚}}
\end{minipage}
\hspace*{1ex}
\begin{tikzpicture}[scale=0.7, >=stealth,inner sep=0pt,
baseline={([yshift=-24pt] current bounding box.north)}]
%\draw[help lines] (-0.3,-5) grid (10,4.2);
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
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https://cs.uwaterloo.ca/journals/JIS/VOL20/Ballot/ballot22.tex
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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}
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\begin{center}
\vskip 1cm{\LARGE\bf
On Functions Expressible as Words \\
\vskip .1in
on a Pair of Beatty Sequences
}
\vskip 1cm
\large
Christian Ballot \\
D\'epartement de Math\'ematiques et Informatique\\
Universit\'e de Caen-Normandie \\
France \\
\href{mailto:[email protected]}{\tt [email protected]} \\
\end{center}
\vskip .2 in
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\begin{abstract}
Let $a(n)=\lfloor n\a\rfloor$ and $b(n)=\lfloor n\a^2\rfloor$, where
$\a=\frac{1+\sqrt{5}}2$. Then a theorem of Carlitz et al.\ states that each
function $f$, composed of several $a$'s and $b$'s, can be expressed in
the form $c_1a+c_2b-c_3,$ where $c_1$ and $c_2$ are consecutive
Fibonacci numbers determined by the numbers of $a$'s and of $b$'s
composing $f$ and $c_3$ is a nonnegative constant. We provide
generalizations of this theorem to two infinite families of
complementary pairs of Beatty sequences. The particular case involving
`Narayana' numbers is examined in depth. The details reveal that $x_n=
\lf\a^3\lf\a^3\lf\cdots\lf\a^3\rf\cdots\rf\rf\rf$, with $n$ nested
pairs of $\lf\;\rf$, is a 7th-order linear recurrence, where $\a$ is the
dominant zero of $x^3-x^2-1$.
\end{abstract}
\section{Introduction}
\label{sec1}
If $\a$ is a positive irrational number, then $a(n)=\lfloor n\a\rfloor$ is said to be a
{\it Beatty sequence}.
A pair of Beatty sequences $a(n)=\lfloor n\a\rfloor$ and $b(n)=\lfloor n\b\rfloor$ is said to be
{\it complementary} whenever their ranges form a partition of the positive integers. A famous
theorem states that complementarity occurs if and only if $1/\a+1/\b=1$. According to Kimberling \cite{Ki2},
though this theorem was stated as a problem \cite{Bea}, it had appeared even earlier in the book \cite[p.~123]{Ray}.
The lower and upper Wythoff sequences, i.e., the sequences
$a(n)=\lfloor n\a\rfloor$ and $b(n)=\lfloor n\a^2\rfloor$, where $\a=\frac{1+\sqrt{5}}2$, are
a pair of complementary Beatty sequences. This pair has often been considered, in part because the
positions $(a(n), b(n))$ are winning positions in a variant of the game of Nim \cite{Wyt}.
Carlitz et al.~\cite{Car} and later again Kimberling \cite{Ki} studied the functions $(w(n))_{n\ge1}$, where $w=\ell_1\circ\ell_2\circ\cdots\circ\ell_s$ and
each $\ell_i$ is either the $a$ or the $b$ sequence. The following lemma was proved
\begin{lemma}\label{lem:Kim} Let $a(n)=\lf n\a\rf$ and $b(n)=\lf n\a^2\rf$, where $\a=(1+\sqrt{5})/2$. Then
\begin{align*}
a^2&=b-1,\\
ba&=a+b-1,\\
ab&=a+b,\\
b^2&=a+2b,
\end{align*} where $\ell_1\ell_2$ stands for $\ell_1\circ\ell_2$ and $\ell^2$ for $\ell\ell$.
\end{lemma}
One of the key facts is that, as $\a^2=\a+1$, we have $b(n)=a(n)+n$, for all integers $n$.
Here is essentially Theorem 13 of Carlitz et al.~\cite{Car} and the principal theorem of Kimberling \cite{Ki}, proved in both instances
by induction on $s$ using Lemma \ref{lem:Kim}.
\begin{theorem}\label{thm:Kim} Let $w=\ell_1\circ\ell_2\circ\cdots\circ\ell_s$, $(s\ge1)$, where each $\ell_i$ is
either $a$ or $b$. Assume $x$ and $y$ are, respectively, the number of $a$'s and the number of $b$'s
in $w$. Then,
$$
w(n)=F_{x+2y-2}\,a(n)+F_{x+2y-1}\,b(n)-e_w,
$$ where $e_w=F_{x+2y+1}-w(1)\ge0$ and $F_k$ denotes the $k$th Fibonacci number.
\end{theorem}
As usual, the Fibonacci sequence $(F_k)$ is defined by $F_0=0$, $F_1=1$ and $F_{k+2}=F_{k+1}+F_k$
for all integers $k$. With $f^x$ denoting the $x$-fold composite function $f\circ f\circ\cdots\circ f$,
we state a corollary from material observed by Kimberling \cite{Ki}.
\begin{cor}\label{cor:Kim} We have $a^x=F_{x-2}a+F_{x-1}b-F_{x+1}+1$ and $b^y=F_{2y-2}a+F_{2y-1}b$.
In particular, $b^y(1)=F_{2y+1}$.
\end{cor}
For each pair $a(n)=\lf n\a\rf$ and $b(n)=\lf n\b\rf$ of complementary Beatty sequences, we will always
take $\a$ to be less than $\b$. Then we necessarily have $1<\a<2<\b=\a/(\a-1)$.
This paper studies two infinite families of pairs of complementary Beatty sequences. Each of the two
families contains the Wythoff pair as its simplest case. For each family,
our main goal is to find
a sensible generalization of Theorem \ref{thm:Kim}.
Our investigation begins in Section~\ref{sec2} by looking at the pair of complementary Beatty sequences $(\a,\b)=
(\sqrt{2},2+\sqrt{2})$. That is,
$a(n)=\lfloor n\sqrt{2}\rfloor$ and $b(n)=\lfloor n(2+\sqrt{2})\rfloor$. This pair satisfies the obvious
property $b(n)=a(n)+2n$ instead of $b(n)=a(n)+n$ for the Wythoff pair. In Section~\ref{sec3}, we study the
general pair of complementary Beatty sequences where $b(n)=a(n)+rn$, $r\ge1$ any integer.
We obtain Theorem \ref{thm:R}, a most general theorem.
\medskip
The second infinite family we study stems from the observation \cite{Ba} that for all integers $q\ge2$,
the complex polynomial $x^q-x^{q-1}-1$ has a simple dominant real zero $\a$, $1<\a<2$,
and the pair $(\a,\b)$, where $\b=\a^q$, generates a complementary pair
of Beatty sequences $(a(n),b(n))$. Section~\ref{sec4} studies in detail the case $q=3$.
Then, of course, $\b$ is the cube rather than the square of $\a$,
as was the case for the Wythoff pair.
Section~\ref{sec5} is a brief section on the case $q=4$. A sixth section deals with the general case $q\ge2$, where
we reach our most general result, Theorem \ref{thm:G}.
We believe beginning with particular cases makes the transition to the general case both more
readable and more enjoyable. However, readers can skip Sections~\ref{sec2} and \ref{sec5}, if they wish.
Yet in Sections \ref{sec2}, \ref{sec4}, and \ref{sec5}, we investigate the functions $e_w$ in more detail than in the general cases.
For instance, the functions $e_w$ studied in Sections \ref{sec2} and
\ref{sec4} are nonnegative, in contrast to their general counterparts in
Theorems \ref{thm:R} and \ref{thm:G}.
\smallskip
A subsidiary investigation of the paper is the study of the sequences
$(b^y(n))_y$. This function turns out to be a second-order linear recurrence
whose characteristic polynomial is the minimal polynomial of $\b$ not just in the Wythoff case, but
in all $(a(n),a(n)+rn)$ cases for $r\ge1$, as shown in the later part of Section~\ref{sec3}.
Also, Section~\ref{sec4}, where $a(n)=\lf\a n\rf$ and $b(n)=\lf\a^3 n\rf$,
$\a^3=\a^2+1$, $\a>1$, is divided into two subsections, the second of which studies
the behavior of $b^y(1)=\lf\a^3\lf\a^3\lf\cdots\lf\a^3\rf\cdots\rf\rf\rf$, with $y$ nested pairs $\lf\;\rf$.
This sequence $(b^y(1))_y$ turns out to be a seventh-order linear recurrence. We find several
explicit formulas for it. Note that
the corresponding sequence $(a^x(1))_x$ is the constant sequence equal to $1$ as $\lf\a\rf=1$
for all complementary Beatty pairs.
\medskip
We now relate our paper to general questions and other work. We begin with the mention that Stolarsky \cite{Sto0}
compiled an extended bibliography of work linked to Beatty sequences done before 1973. A quite general question is
`what sort of behavior and structures emerge from all possible
compositions of a given set of functions?' For a single function this is the problem of analyzing
iteration (e.g., see the comments on $b^y (n)$ above). Here we examine functions of the
form
$$
g(n) = \lf n\a_i\rf
$$ (i.e., Beatty sequences), where the $\a_i$ are algebraic irrationalities.
In various
cases of interest we determine the nature of `homogeneous' compositions
$$
g_1 (g_2 (\cdots(g_k (n))\cdots).
$$
There has also been some study of `inhomogeneous' compositions such as
$$
g_1 (g_2 (n) + c_1 n + c_2 )
$$ in the Beatty context. See \cite{Por}, especially formula (1.1.4), and \cite{Fra}, in particular
Theorem 1 of \S2. Boshernitzan and Fraenkel \cite{Bos} discussed
characteristic properties of functions of the form
$$
g(n) = \lf n\a_i + \b_i\rf,
$$ but perhaps the study of arbitrarily long compositions of
such functions has not been done in any detail. Fraenkel et al.~\cite{Fra2} studied more general combinations of such functions. Cases in which the most interesting results are found frequently involve numbers $\a$
that are real algebraic integers larger than their conjugates. Even more special are
cases in which $\a$ is a Pisot number. For example, the dominant real zero of $x^q-x^{q-1}-1$
is a Pisot number for $q=2$, $3$, and $4$. In \S4, i.e., in Section~\ref{sec4}, we examine in special detail the `Narayana
case' $q=3$. Bertin et al.~\cite{Ber} published a general reference book on Pisot numbers and their relatives.
In fact, the Fibonacci ($q=2$) and Narayana cases involve a dominant zero that comes from the finite set
of `special Pisot numbers'. The significance of these numbers appears in various papers \cite{Lag, Lag2, Sch}.
Smyth \cite{Smy} provided a definitive complete determination of them.
The $\a$ of \S5 is also a special Pisot
number. In connection with \S6 we note that the dominant zero of $P(q,x)=x^q-x^{q-1}-1$
is not Pisot for $q\ge6$, and that $P(5, x) = (x^3-x-1)(x^2-x+1)$.
It has been noted that a full understanding of the Beatty sequences and Wythoff pairs
related to $\sqrt{5}$ involves recurrences of degree 4. This is the basic theme of two papers
\cite{Sto, Rid}. Here in \S4 we find that the study of Beatty sequences
corresponding to the `Narayana' cubic irrationality (one of the special Pisot numbers) inevitably
involves recurrences of degree 7. See Problems \ref{problem:3} and \ref{problem:4} of
Section~\ref{sec7}
for precise questions.
\medskip
Indeed, Section~\ref{sec7}, our final section, proposes five problems for further consideration.
\medskip
By convention, throughout the paper, the sums $\sum_{i\le j\le k}a_j$ or $\sum_{j=i}^ka_j$ are
zero whenever $k<i$. If $\a$ is an irrational real number, then the uniform distribution of
the sequence of fractional parts of the multiples of $\a$, i.e., the sequence $(\{n\a\})_{n\ge1}$, is
a well-known fact that we occasionally use.
\section{The $(\sqrt{2},2+\sqrt{2})$ case}
\label{sec2}
Thus, we now have $a(n):=\lfloor n\a\rfloor$ and $b(n):=\lfloor n\b\rfloor$, where
$\a=\sqrt{2}$ and $\b=2+\sqrt{2}$.
\begin{lemma}\label{lem:Cas1} For all integers $n\ge1$, we have
\begin{align*}
a^2(n)&=-a(n)+b(n)-d(n),\\
ba(n)&=a(n)+b(n)-d(n),\\
ab(n)&=a(n)+b(n),\\
b^2(n)&=a(n)+3b(n),
\end{align*} where
$d(n)$ is the function $\lceil\sqrt{2}\{n\sqrt{2}\}\rceil$ which is either $1$ or $2$.
\end{lemma}
\begin{proof} See the proof of the more general Lemma \ref{lem:cas-r} of the next section.
\end{proof}
\begin{theorem}\label{thm:root2} Let $w=\ell_1\circ\ell_2\circ\cdots\circ\ell_s$, $(s\ge1)$,
where each $\ell_i$ is either $a$ or $b$. If $w$ has an even number of $a$'s, say $2x$,
and $y$ $b$'s ($x\ge0$, $y\ge0$), then
\begin{equation}\label{eq:xeven}
w(n)=2^xu_y\,a(n)+2^xv_y\,b(n)-e(n),
\end{equation}
whereas, if $w$ has $2x+1$ $a$'s and $y$ $b$'s, then
\begin{equation}\label{eq:xodd}
w(n)=2^{x+1}v_{y-1}\,a(n)+2^xu_{y+1}\,b(n)-e(n),
\end{equation} where in both cases $e=e_w$ is some nonnegative integral bounded function
of $n$ that depends on $w$, and $(u_y)$ and $(v_y)$ are the recurrences with characteristic
polynomial $x^2-4x+2$ that satisfy $u_1=0$, $u_2=1$ and $v_1=1$, $v_2=3$.
\end{theorem}
\begin{proof} It is easy to verify that
\begin{equation}
\begin{aligned}
u_y+v_y & =u_{y+1}& \quad
v_y-u_y& =2v_{y-1} \\
u_y+3v_y &=v_{y+1}& \quad
u_{y+1}-2v_{y-1} &=2u_y.
\end{aligned}
\label{eq:comb}
\end{equation}
We may proceed by induction on $s$. Both (\ref{eq:xeven}) and (\ref{eq:xodd})
trivially hold for $s=1$ with $e=0$, as $v_{-1}=v_0=1/2$. Assuming the property holds for some $s\ge1$ and
$w$ is a `word' with $s$ letters, we check the property for $wa$ and $wb$.
There are four cases to treat as the form of $w$ depends on the parity of the
number of $a$'s in $w$. If $w$ has $2x$ $a$'s and $y$ $b$'s, then, by the inductive hypothesis,
$$
wa(n)=w(a(n))=2^xu_y\,a^2(n)+2^xv_y\,b(a(n))-e(a(n)),
$$ for some nonnegative bounded function $e$. Using Lemma \ref{lem:Cas1} and
gathering together the coefficients of $a(n)$ and $b(n)$, we find that
$$
wa(n)=2^x(v_y-u_y)\,a(n)+2^x(u_y+v_y)\,b(n)-e'(n),
$$ with $e'(n)=u_{y+1}d(n)+e(a(n))$. By (\ref{eq:comb}), $wa(n)=2^{x+1}v_{y-1}\,a(n)+
2^xu_{y+1}\,b(n)-e'(n)$, which is the expected result for the word $wa$. We proceed in
the same manner for $wb(n)$ and obtain
$$
wb(n)=2^x(u_y+v_y)\,a(n)+2^x(u_y+3v_y)\,b(n)-e(b(n)).
$$ We conclude using the first pair of equations of (\ref{eq:comb}).
If $w$ has $2x+1$ $a$'s and $y$ $b$'s, then the inductive hypothesis yields
$$
wa(n)=2^{x+1}v_{y-1}\,a^2(n)+2^xu_{y+1}\,ba(n)-e(a(n)).
$$ Using Lemma \ref{lem:Cas1}, we find that $wa(n)=2^x(u_{y+1}-2v_{y-1})\,a(n)+2^x(u_{y+1}+2v_{y-1})\,b(n)
-e''(n)$, where $e''(n)=2^x(2v_{y-1}+u_{y+1})d(n)+e(a(n))$. Since $v_{-1}=v_0=1/2$, we see
that $e''$ is nonnegative in all cases. We conclude using the second pair of identities of (\ref{eq:comb}),
as $u_{y+1}-2v_{y-1}=2u_y$ and $u_{y+1}+2v_{y-1}=(u_{y+1}-2v_{y-1})+4v_{y-1}=2(u_y+2v_{y-1})=2v_y$.
Finally, we obtain, in similar fashion, $wb(n)=2^x(u_{y+1}+2v_{y-1})\,a(n)+2^x(3u_{y+1}+2v_{y-1})\,b(n)-e(b(n))$.
We just saw that $u_{y+1}+2v_{y-1}=2v_y$. Moreover,
$$3u_{y+1}+2v_{y-1}=2u_{y+1}+2v_y=
u_{y+1}+(u_{y+1}+2v_y)=u_{y+1}+v_{y+1}=u_{y+2},$$
according to the two identities on the first line of (\ref{eq:comb}).
\end{proof}
\section{The general $(\a,\a+r)$ case, ($r\ge1$)}
\label{sec3}
We fix an integer $r\ge1$. If $\b=\a+r$, then we see that $b(n)=a(n)+rn$.
Solving $1/\a+1/(\a+r)=1$ leads to $\a^2+(r-2)\a-r=0$. Since $\a\in(1,2)$,
we find that $\a=1+\frac{\sqrt{r^2+4}-r}2$. Thus, $\b=1+\frac{\sqrt{r^2+4}+r}2$ and
$\b$ satisfies $\b^2-(r+2)\b+r=0$. As $r^2+4$ is never a perfect square, we note that
$\a$ and $\b$ are irrational.
\begin{lemma}\label{lem:cas-r} For all integers $n\ge1$, we have
\begin{align*}
a^2(n)&=(1-r)a(n)+b(n)-d(n),\\
ba(n)&=a(n)+b(n)-d(n),\\
ab(n)&=a(n)+b(n),\\
b^2(n)&=a(n)+(r+1)b(n),
\end{align*} where
$d(n)$ is the function $\lceil(\a+r-2)\{n\a\}\rceil$ whose range is $\{1,2,\ldots,r\}$.
\end{lemma}
\begin{proof} Using $\a^2=(2-r)\a+r$ and $\{\a n\}=\{\b n\}$, we find that
\begin{align*}
a^2(n)&=\lf\a\lf\a n\rf\rf=\lf\a^2n-\a\{\a n\}\rf=\lf\big((1-r)\a+(\a+r)\big)n-\a\{\a n\}\rf \\
&= (1-r)a(n)+b(n)+\lf(1-r)\{\a n\}+\{\a n\}-\a\{\a n\}\rf \\
&=(1-r)a(n)+b(n)-\lc(r-2+\a)\{\a n\}\rc .
\end{align*}
But the sequence $(\{\a n\})_{n\ge1}$ is dense in $(0,1)$ and $r-2+\a$ lies in $(r-1,r)$, so the range of $d$ is $\{1,2,\ldots,r\}$.
Hence, $b(a(n))=\lf(\a+r)a(n)\rf=ra(n)+a^2(n)=a(n)+b(n)-d(n)$.
Now
\begin{align*}
ab(n) &=\lf\a(\a+r)n-\a\{(\a+r)n\}\rf=\lf\a n+(\a+r)n-\a\{\a n\}\rf \\
&=a(n)+b(n) +\lf(2-\a)\{\a n\}\rf=a(n)+b(n),
\end{align*}
as both $2-\a$ and $\{\a n\}$ lie in the
interval $(0,1)$. Thus,
$$b^2(n)=\lf(\a+r)b(n)\rf=rb(n)+ab(n)=a(n)+(r+1)b(n).$$
\end{proof}
Define $(U_n^\a)_{n\ge0}$ and $(U_n^\b)_{n\ge0}$, respectively, as the fundamental
Lucas sequences associated with $(x-\a)(x-\bar\a)$ and with $(x-\b)(x-\bar\b)$,
where $\bar\a$ and $\bar\b$ are the respective algebraic conjugates of $\a$ and $\b$.
Thus,
\begin{equation}\label{eq:Luc}
U_n^\a=\frac{\a^n-\bar\a^n}{\a-\bar\a}\quad\text{ and }\quad U_n^\b=\frac{\b^n-\bar\b^n}{\b-\bar\b}.
\end{equation}
We also define two matrices
\begin{equation}\label{eq:Mat}
E_a=\left(\begin{matrix}1-r & 1\\1 & 1\end{matrix}\right)\quad\text{ and }\quad
E_b=\left(\begin{matrix}1 & 1\\1 & 1+r\end{matrix}\right).
\end{equation}
\begin{lemma}\label{lem:Mat} For all integers $x\ge0$ and $y\ge0$, we find that
$$
E_a^x=\left(\begin{matrix}U_{x+1}^\a-U_x^\a & U_x^\a \\U_x^\a & U_x^\a+rU_{x-1}^\a
\end{matrix}\right)\;\text{ and }\;
E_b^y=\left(\begin{matrix}U_y^\b-rU_{y-1}^\b & U_y^\b\\ U_y^\b & U_{y+1}^\b-U_y^\b\end{matrix}\right).
$$
\end{lemma}
\begin{proof} The characteristic polynomials of $E_a$ and $E_b$ are, respectively,
the minimal polynomials of $\a$ and $\b$. Thus, their respective eigenvalues
are $\{\a,\bar\a\}$ and $\{\b,\bar\b\}$. Thus, we may diagonalize
the two matrices $E_a$ and $E_b$ and find that
$$
E_a=P_a\left(\begin{matrix}\a & 0\\0 & \bar\a\end{matrix}\right)P_a^{-1}\quad\text{ and }\quad
E_b=P_b\left(\begin{matrix}\b & 0\\0 & \bar\b\end{matrix}\right)P_b^{-1},
$$ where we took
$$
P_a=\left(\begin{matrix}\a-1 & \bar\a-1\\1 & 1\end{matrix}\right)\quad\text{ and }\quad
P_b=\left(\begin{matrix}1 & 1\\\b-1 & \bar\b-1\end{matrix}\right),
$$ as eigenvector matrices.
We then simply calculate
$$
E_a^x=P_a\left(\begin{matrix}\a^x & 0\\0 & \bar\a^x\end{matrix}\right)P_a^{-1}\quad\text{ and }\quad
E_b^y=P_b\left(\begin{matrix}\b^y & 0\\0 & \bar\b^y\end{matrix}\right)P_b^{-1},
$$ noting that for $x=0$ or for $y=0$, the expressions in the lemma produce the identity
matrix $\left(\begin{matrix}1 & 0\\0 & 1\end{matrix}\right)$. Indeed, $U_{-1}^\a=1/r$ and
$U_{-1}^\b=-1/r$.
\end{proof}
\begin{theorem}\label{thm:R} Let $r\ge1$ be an integer, $\a=1+\frac{\sqrt{r^2+4}-r}2$ and
$\b=\a+r$. Define $a(n)=\lf\a n\rf$ and $b(n)=\lf\b n\rf$. Let
$w=\ell_1\circ\ell_2\circ\cdots\circ\ell_{x+y}$,
where $\ell_i=a$ for $x\ge0$ values and $\ell_i=b$ for the remaining $y$ values of $i$, $1\le i\le x+y$.
Then for all $n\ge1$ \begin{equation}\label{eq:R}
w(n)=s_{x,y}\,a(n)+t_{x,y}\,b(n)-d_w(n),
\end{equation}
where
\begin{align}\begin{split}\label{eq:Rcoef}
&s_{x,y}=U_x^\a U_y^\b-rU_{y-1}^\b(U_x^\a-U_{x-1}^\a)=U_x^\a U_y^\b+U_{y-1}^\b(U_{x+1}^\a-2U_x^\a),\\
&t_{x,y}=U_x^\a U_y^\b+rU_{x-1}^\a(U_y^\b-U_{y-1}^\b)=U_x^\a U_y^\b+U_{x-1}^\a(U_{y+1}^\b-2U_y^\b),
\end{split} \end{align}
$(U_x^\a)$ and $(U_y^\b)$ were defined in (\ref{eq:Luc}) and $d_w$ is an integral
and bounded function of $n$ that depends on $w$.
\end{theorem}
\begin{proof} Suppose $w(n)=s_wa(n)+t_wb(n)-d_w(n)$, where $s_w$ and $t_w$ are integers that do not depend on $n$
and $d_w$ is an integral and bounded function of $n$. Then, by Lemma \ref{lem:cas-r},
we find that
\begin{align}\begin{split}\label{eq:ab}
wa(n)&=\big((1-r)s_w+t_w\big)a(n)+(s_w+t_w)b(n)-d_{wa}(n),\\
wb(n)&=(s_w+t_w)a(n)+\big(s_w+(r+1)t_w\big)b(n)-d_{wb}(n),
\end{split}
\end{align} where $d_{wa}(n)=(s_w+t_w)d_{a^2}(n)+d_w(a(n))$ and $d_{wb}(n)=d_w(b(n))$.
Thus we see from (\ref{eq:ab}) that
$$
E_a\cdot\left(\begin{matrix}s_w\\t_w\end{matrix}\right)=\left(\begin{matrix}s_{wa}\\t_{wa}\end{matrix}\right)
\quad\text{ and }\quad E_b\cdot\left(\begin{matrix}s_w\\t_w\end{matrix}\right)=\left(\begin{matrix}s_{wb}\\t_{wb}
\end{matrix}\right),
$$ where the matrices $E_a$ and $E_b$ were defined in (\ref{eq:Mat}).
An easy induction on $x+y$ shows that for all words $w$ with $x$ letters $a$ and $y$ letters $b$ and all
$n\ge1$, $0\le\a^x\b^yn-w(n)\le\b^{x+y}$.
Indeed, if $\ell=a$ or $b$ and $\ell(n)=\lf\lambda n\rf$, then $0\le\a^x\b^y\ell(n)-w\ell(n)\le\b^{x+y}$,
by the inductive hypothesis. But $\ell(n)=\lambda n-\{\lambda n\}$, so $0\le\a^x\b^y\lambda n-
w\ell(n)\le\b^{x+y}+\a^x\b^y\{\lambda n\}\le2\b^{x+y}\le\b^{x+y+1}$, as $\b>2>\a$.
Therefore, using the triangle inequality, we find that $|w(n)-w_0(n)|\le2\b^{x+y}$ for all $w$ satisfying
the hypotheses of the theorem
and all $n\ge1$, where $w_0=a^xb^y$. Thus, it suffices to prove the theorem for the function $w=w_0$.
Since $b=0\cdot a+1\cdot b$, we can find the vector $\left(\begin{matrix}s_w\\t_w\end{matrix}\right)$
by computing the matrix product $E_a^xE_b^{y-1}\left(\begin{matrix}0\\1\end{matrix}\right)$. By
Lemma \ref{lem:Mat},
\begin{align*}
E_a^xE_b^{y-1}\left(\begin{matrix}0\\1\end{matrix}\right)
&=\left(\begin{matrix}U_{x+1}^\a-U_x^\a & U_x^\a \\U_x^\a & U_x^\a+rU_{x-1}^\a
\end{matrix}\right)\left(\begin{matrix}U_{y-1}^\b\\U_y^\b-U_{y-1}^\b\end{matrix}\right)\\
&=\left(\begin{matrix}U_x^\a U_y^\b+U_{y-1}^\b(U_{x+1}^\a-2U_x^\a)\\
U_x^\a U_y^\b+rU_{x-1}^\a(U_y^\b-U_{y-1}^\b)\end{matrix}\right).
\end{align*}
The other expressions for $s_{x,y}$ and $t_{x,y}$ in (\ref{eq:Rcoef}) are obtained using
the relations $U_{x+1}^\a=(2-r)U_x^\a+rU_{x-1}^\a$ and $(2+r)U_y^\b-rU_{y-1}^\b=U_{y+1}^\b$,
respectively. Our derivation assumed $y\ge1$. However, it is easy to check that putting $y=0$
in (\ref{eq:Rcoef}) yields $\left(\begin{matrix}s_{x,0}\\t_{x,0}\end{matrix}\right)=
\left(\begin{matrix}U_x^\a-U_{x-1}^\a\\U_{x-1}^\a\end{matrix}\right)$,
which equals $E_a^{x-1}\left(\begin{matrix}1\\0\end{matrix}\right)
=\left(\begin{matrix}s_{a^x}\\t_{a^x}\end{matrix}\right)$, by Lemma \ref{lem:Mat}.
\end{proof}
\begin{remark}
The functions $s_{x,y}$ and $t_{x,y}$ are integral linear recurrences in $x$
with characteristic polynomial the minimal polynomial of $\a$ when $y$ is fixed, and linear recurrences
in $y$ with characteristic polynomial the minimal polynomial of $\b$ when $x$ is fixed.
\end{remark}
\begin{remark}
One can easily recover $s_{x,y}$ and $t_{x,y}$ when $r=1$ or $r=2$ obtained
in Theorems \ref{thm:Kim} and \ref{thm:root2}. For instance, if $r=1$, then $U_x^\a=F_x$ and
$U_y^\b=F_{2y}$ so $s_{x,y}=F_xF_{2y}+F_{2y-2}(F_{x+1}-2F_x)=F_xF_{2y}-F_{x-2}F_{2y-2}$.
To see that $F_xF_{2y}-F_{x-2}F_{2y-2}=F_{x+2y-2}$, it is enough to observe that both
sides of the equation are linear recurrences satisfying the Fibonacci recursion once
$y$ is fixed. Thus, we are left with verifying equality, say, at $x=0$ and $x=1$.
\end{remark}
\begin{remark}
In contrast with the cases $r=1$ or $r=2$, the functions $d_w(n)$ in (\ref{eq:R})
are not necessarily always nonnegative when $r\ge3$. For instance, for $r=3$,
$$
d_{a^3}(n)=\lc(\a+1)\{\a a(n)\}\rc-\lc(\a+1)\{n\a\}\rc,
$$ which is $-1$ for $n=6$.
\end{remark}
In the Wythoff case, as we can see from Corollary \ref{cor:Kim}, the
sequences $(b^y(n))_y$ for a fixed integer $n\ge1$ are all
second-order recurrences with characteristic polynomial the minimal
polynomial of $\b=\a+1$. The next two corollaries show this phenomenon
holds for all pairs of Beatty sequences stemming from a pair
$(\a,\a+r)$, ($r\ge1$).
\begin{cor}\label{cor:byofn} For all $y\ge0$ and all $n\ge1$, we find that
$$
b^y(n)=u_ya(n)+v_yb(n),
$$ where $(u_y)$ and $(v_y)$ are both second-order linear recurrences with characteristic
polynomial $x^2-(r+2)x+r$ and initial values $u_1=0$, $u_2=1$ and $v_1=1$, $v_2=r+1$.
\end{cor}
\begin{proof} We saw in (\ref{eq:ab}) that $d_{wb}(n)=d_w(b(n))$. But for $w=b$, $d_w=0$.
Hence, we see inductively that $d_{b^y}=0$, for all $y\ge1$. Therefore,
by Theorem \ref{thm:R}, we get
$$b^y(n)=U_{y-1}^\b a(n)+(U_y^\b-U_{y-1}^\b)b(n).$$
Both $U_{y-1}^\b$ and $U_y^\b-U_{y-1}^\b$ are linear recurrences
with characteristic polynomial $x^2-(r+2)x+r$, the minimal polynomial of $\b$, and their
initial conditions are those indicated. For $y=0$, $b^y(n)=n$ and
$$U_{y-1}^\b a(n)+(U_y^\b-U_{y-1}^\b)b(n)=
U_{-1}^\b\big(a(n)-b(n)\big)=-r^{-1}(-rn)=n.$$
\end{proof}
\begin{cor}\label{cor:byof1} Given $n\ge1$, the sequence $(b^y(n))_y$ is linear recurrent with
characteristic polynomial the minimal polynomial of $\b$, namely $x^2-(r+2)x+r$. In
particular, $(b^y(1))_y=v_{y+1}$, where the sequence $(v_y)$ was defined in the statement
of Corollary \ref{cor:byofn}.
\end{cor}
\begin{proof} By Corollary \ref{cor:byofn}, once $n$ is fixed, $b^y(n)$ is a linear combination
of $u_y$ and $v_y$. Moreover, $b^y(1)=a(1)u_y+b(1)v_y=\lf\a\rf u_y+\lf\b\rf v_y=
u_y+(r+1)v_y=v_{y+1}$, since $\a\in(1,2)$ implies $\b\in(r+1,r+2)$.
\end{proof}
For future reference, we give a direct proof of Corollary \ref{cor:byof1}.
\begin{proof} Put $\nu_y=b^y(n)$. Then
\begin{align*}
\nu_{y+2}&=\lf\b^2\nu_y-\b\{\b\nu_y\}\rf= \lf(r+2)\b\nu_y-r\nu_y-\b\{\b\nu_y\}\rf \\
&= (r+2)\nu_{y+1}-r\nu_y+\lf(r+2-\b)\{\b\nu_y\}\rf =(r+2)\nu_{y+1}-r\nu_y,
\end{align*}
since, as $\b\in(r+1,r+2)$, it follows that $r+2-\b$ is in $(0,1)$.
\end{proof}
\section{The Narayana case}
\label{sec4}
Here we consider $a(n):=\lfloor \a n\rfloor$ and $b(n):=\lfloor \a^3n\rfloor$, where
$\a\doteq 1.46557$ is the dominant zero of $x^3-x^2-1$. Note that we do have $1/\a+1/\a^3=1$.
\smallskip
The Narayana sequence $(N_k)_{k\ge0}$ is the fundamental recurrence with characteristic
polynomial $x^3-x^2-1$, i.e., with initial values $0,0,1$. This sequence was used in the 14th century
to model the population growth of a herd of cows \cite{AJ}. Its OEIS number \cite{Slo} is \seqnum{A078012}.
\subsection{General results}
\begin{lemma}\label{lem:N} For all integers $n\ge1$, we have
\begin{align*}
a^2(n)&=b(n)-n-e_1(n),\\
ba(n)&=a(n)+b(n)-e_2(n),\\
ab(n)&=a(n)+b(n)-e_3(n),\\
b^2(n)&=a(n)+3b(n)-n-e_4(n),
\end{align*} where the ranges of
$e_1$ and $e_2$ are, respectively, $\{0,1,2\}$ and $\{0,1,2,3\}$, and the ranges of
$e_3$ and $e_4$ are both $\{0,1\}$.
\end{lemma}
\begin{proof}
Since $\a^3=\a^2+1$, we see that
\begin{align*}
a^2(n) &=\lfloor\a\lfloor\a n\rfloor\rfloor=
\lfloor\a^2 n-\a\{\a n\}\rfloor=
\lfloor\a^3 n-\a\{\a n\}\rfloor-n \\
& =\lfloor\lfloor\a^3 n\rfloor+\{\a^3 n\}-\a\{\a n\}\rfloor-n=
b(n)-n+\lfloor\{\a^2 n\}-\a\{\a n\}\rfloor.
\end{align*}
Clearly $-2<-\a< -\a\{\a n\}<
\{\a^2 n\}-\a\{\a n\}<\{\a^2 n\}<1$, which explains that $e_1(n)$ is either $0$, $1$ or $2$.
Similarly, $ba(n)=\lfloor\a^3\lfloor\a n\rfloor\rfloor=\lfloor\a^2\lfloor\a n\rfloor+\lfloor\a n
\rfloor\rfloor=a(n)+\lfloor\a^3n-\a^2\{\a n\}\rfloor=a(n)+b(n)+\lfloor\{\a^3n\}-\a^2\{\a n\}\rfloor$.
Note that $\a^2>2$, so $e_2(n)$ is potentially equal to $3$. Also,
\begin{align*}
ab(n)&=\lfloor\a(\a^3n-\{\a^3n\})\rfloor =\lfloor(\a^3+\a)n-\a\{\a^3n\}\rf \\
&= a(n)+b(n)+\lf\{\a^3n\}+\{\a n\}-\a\{\a^3n\}\rf=a(n)+b(n)+\lf\{\a n\}+(1-\a)\{\a^2n\}\rf.
\end{align*}
Noting
that $(1-\a)\{\a^2n\}\in(-1,0)$ explains the range of $e_3$. Finally, as
$\a^6=3\a^3+\a-1$, we see that
\begin{align*}
b^2(n)&=\lf\a^3(\a^3n-\{\a^3n\})\rf=-n+\lf3\a^3n+\a n-\a^3\{\a^3n\}\rf \\
&=a(n)+3b(n)-n+\lf(3-\a^3)\{\a^3n\}+ \{\a n\}\rf.
\end{align*}
Noting that $(3-\a^3)\{\a^3n\}\in(-1,0)$ explains the range of $e_4$.
\end{proof}
It might be worth listing the exact expressions of the functions $e_i$ found in the above proof. Namely,
\begin{align*}
e_1(n)&=\lceil\a\{\a n\}-\{\a^2n\}\rceil \\
e_2(n)&=\lceil\a^2\{\a n\}-\{\a^2n\}\rceil \\
e_3(n)&=\lceil(\a-1)\{\a^2n\}-\{\a n\}\rceil \\
e_4(n)&=\lceil(\a^3-3)\{\a^2n\}-\{\a n\}\rceil.
\end{align*}
We observe that all values in the
various ranges of the $e_i$'s are attained for some $n$.
For instance, $e_2(n)$ takes on the value
$3$, though only three times in the interval $[1,500]$.
\begin{theorem}\label{thm:N} Let $w=\ell_1\circ\ell_2\circ\cdots\circ\ell_s$, $(s\ge1)$, where each $\ell_i$ is
either $a$ or $b$. Assume $x$ and $y$ are, respectively, the number of $a$'s and the number of $b$'s
in $w$. Then,
\begin{equation}\label{eq:N}
w(n)=N_{x+3y-2}\,a(n)+N_{x+3y}\,b(n)-N_{x+3y-3}\,n-e_w(n),
\end{equation} where $e_w$ is a nonnegative bounded integral function of $n$ that depends on $w$ and satisfies
$e_w\le3pN_p$ with $p=x+3y$.
\end{theorem}
\begin{proof} The proof is by induction on $\ell\ge1$ and analogous to that of Theorem \ref{thm:H}, albeit
less demanding, and also subsumed by the proof presented in Theorem \ref{thm:G}. We only outline
the bounds on $e_w$ that Theorems \ref{thm:H}, or \ref{thm:G}, do not address.
Assume $w(n)$ satisfies (\ref{eq:N}) with $p=x+3y$.
Computing $w(a(n))$ with the expressions of $a^2(n)$ and
$b(a(n))$ from Lemma \ref{lem:N}, we find
$$w(a(n))=N_{(p+1)-2}a(n)+N_{p+1}b(n)-N_{(p+1)-3}n-e_{wa}(n),$$
where $e_{wa}(n)=N_{p-2}e_1(n)+N_pe_2(n)+e_w(a(n))$.
Since $0\le e_1\le2$ and $0\le e_2\le3$ according
to Lemma \ref{lem:N}, we deduce, with the inductive hypothesis, that
$$0\le e_{wa}(n)\le2N_{p-2}+3N_p+3pN_p\le
N_p+2N_{p+1}+3pN_p\le3N_{p+1}+3pN_{p+1}=3(p+1)N_{p+1}.$$
Similarly, $e_{wb}(n)=e_4(n)N_{p-2}+e_3(n)N_p+e_w(b(n))$.
Clearly $e_{wb}(n)\le N_{p+1}+3pN_p\le3(p+3)N_{p+3}$. Note that $N_{-1}=1$ so $e_{wa}$ and $e_{wb}$ are both
nonnegative even when $p=1$.
\end{proof}
The upper bound on the function $e_w(n)$ can be substantially reduced for some subfamilies of
sequences as the corollary below shows.
\begin{cor}\label{cor:N} We have $b^y(n)=N_{3y-2}a(n)+N_{3y}b(n)
-N_{3y-3}n-e_{(y)}(n)$ where $0\le e_{(y)}(n)\le N_{3y-2}$.
\end{cor}
\begin{proof} Theorem \ref{thm:N} gives that $b^y(n)=N_{3y-2}a(n)+N_{3y}b(n)-N_{3y-3}n-e_{(y)}(n)$ for
some nonnegative bounded function $e_{(y)}$. Thus,
$$b^{y+1}(n)=b^y(b(n))=N_{3y-2}a(b(n))+
N_{3y}b^2(n)-N_{3y-3}b(n)-e_{(y)}(b(n)),$$
which, using Lemma \ref{lem:N} and the Narayana recursion
$N_n+N_{n-2}=N_{n+1}$, yields $b^{y+1}(n)=N_{3y+1}a(n)+N_{3y+3}b(n)-N_{3y}n-N_{3y}e_4(n)-e_{(y)}(b(n))$.
On the other hand, we have $b^{y+1}(n)=N_{3y+1}a(n)+N_{3y+3}b(n)-N_{3y}n-e_{(y+1)}(n)$. Therefore,
$e_{(y+1)}(n)=N_{3y}e_4(n)+e_{(y)}(b(n))$. Thus, $e_{(y+1)}(n)-e_{(y)}(b(n))\le N_{3y}$.
Hence,
$$e_{(y)}(b(n))-e_{(y-1)}(b^2(n))\le N_{3y-3}, \ldots,
e_{(2)}(b^{y-1}(n))-e_{(1)}(b^y(n))\le N_3.$$
But $e_{(1)}=0$, so
adding those inequalities yields $e_{(y+1)}(n)\le\sum_{t=1}^yN_{3t}$. An easy induction shows
$\sum_{t=1}^yN_{3t}=N_{3y+1}$, which terminates our proof.
\end{proof}
We are curious to know whether the sequences $(b^y(n))_y$, for fixed $n$, are linear recurrences
as Corollaries \ref{cor:byofn} and \ref{cor:byof1} showed it is the case when $\b$ is of the form
$\b=\a+r$, $r$ an integer. Clearly, this will hold iff the sequences $(e_{(y)}(n))_y$ of
Corollary \ref{cor:N} are linear recurrences. The next subsection proves this is true when $n=1$.
However, before jumping to this next subsection, we fix an $n\ge1$, set $\t_y:=b^y(n)$ and
mimic
the second proof of Corollary \ref{cor:byof1} to establish that $(\t_y)$, if not
a linear recurrence, is nearly one, and with characteristic polynomial the minimal polynomial
$x^3-4x^2+3x-1$ of $A:=\a^3$. (See Lemma \ref{lem:1}.)
\begin{lemma}\label{lem:3} The sequence $(\t_y)_{y\ge0}$ satisfies the relation
$$
\t_{y+3}-4\t_{y+2}+3\t_{y+1}-\t_y=\xi_y, \text{ where } \xi_y=\lf(4-A)\{A \t_{y+1}\}-A^{-1}\{A \t_y\}\rf, (y\ge0).
$$
\end{lemma}
\begin{proof} Using $A^3-4A^2+3A-1=0$, we find that
\begin{align*}
\t_{y+3}&=\lf A\t_{y+2}\rf=\lf4\t_{y+2}+ (A-4)\lf A\t_{y+1}\rf\rf \\
&= 4\t_{y+2}+\lf(A-4)A \t_{y+1}+(4-A)\{A\t_{y+1}\}\rf.
\end{align*}
But $A^2-4A=-3+A^{-1}$
so $\t_{y+3}=4\t_{y+2}-3\t_{y+1}+\lf A^{-1}(A\t_y-\{A\t_y\})+(4-A)\{A\t_{y+1}\}\rf$,
which leads to the relation the lemma claims.
\end{proof}
We see that $\xi_y$ is either $0$ or $-1$ because $4-A\doteq 0.8521$ and $A^{-1}\doteq 0.3176$, and seems more likely
to be $0$ than $-1$.
\subsection{The sequence $(b^y(1))_y=\lf\a^3\lf\a^3\lf\cdots\lf\a^3\rf\cdots\rf\rf\rf$}
In this subsection, we write $\t_y$ for $b^y(1)$, $A$ for $\a^3$ and we define $\s_y$ as the function
\begin{equation}\label{eq:1}
\s_y:=N_{3y+3}-\sum_{k=0}^{\lf(3y-14)/12\rf}N_{3y-14-12k}.
\end{equation}
\smallskip
We intend to prove that $\s_y$ is a linear recurrence with characteristic polynomial equal
to $x^4-1$ times the minimal polynomial of $A$. This will give us a closed form for $\s_y$
from which we can see that $\s_{y+1}=\lf\a^3\s_y\rf$ for $y\ge20$. Induction will then yield the
equality of the sequences $(\s_y)$ and $(\t_y)$. The expression in (\ref{eq:1}) was found
experimentally to match the first values of $\t_y$. Using PARI, we then had checked the coincidence
of $\t_y$ and $\s_y$ for all $y$, $0\le y\le199$.
\begin{lemma}\label{lem:1} For any fixed integer $t$, the sequence $(N_{3y+t})_y$ is a third-order recurrence
with characteristic polynomial $x^3-4x^2+3x-1$.
\end{lemma}
\begin{proof} It suffices to verify that $x^3-4x^2+3x-1=(x-\a^3)
(x-\b^3)(x-\g^3)$, where $\b$ and $\g$ are, besides $\a$, the two other zeros of $x^3-x^2-1$.
Note that $\a+\b+\g=1$, $\a\b+\a\g+\b\g=0$ and $\a\b\g=1$. Thus, putting $V_n=\a^n+\b^n+\g^n$,
we find that $V_0=3$, $V_1=1$, $V_2=V_1^2-2(\a\b+\a\g+\b\g)=1$. Hence, $V_3=V_2+V_0=4$. Now
writing $W_n$ for $(\a\b)^n+(\a\g)^n+(\b\g)^n$, we see that $(W_n)$'s characteristic
polynomial is $x^3-0x^2+\a\b\g(\a+\b+\g)x-1=x^3+x-1$. Therefore, $W_3=-W_1+W_0=3$ and, consequently,
$(x-\a^3)(x-\b^3)(x-\g^3)=x^3-V_3x^2+W_3x-1=x^3-4x^2+3x-1$, as we claimed.
\end{proof}
\medskip
\begin{lemma}\label{lem:2} The sequence $(\s_y)_{y\ge0}$ satisfies the recursion
$$
\s_{y+3}-4\s_{y+2}+3\s_{y+1}-\s_y=\ep_y, \text{ where } \ep_y=\begin{cases}0,&\text{ if }y
\equiv0,1\text{ or }2\pmod 4;\\
-1,&\text{ if }y\equiv3\pmod 4.
\end{cases}
$$ In particular, $(\s_y)$ is a seventh-order recurrence with characteristic polynomial
$(x^4-1)(x^3-4x^2+3x-1)$.
\end{lemma}
\begin{proof} Define $V_y$ as $\s_{y+3}-4\s_{y+2}+3\s_{y+1}-\s_y$ for all $y$.
By (\ref{eq:1}) and Lemma \ref{lem:1}, we see that
\begin{equation}
\begin{split}
-V_y=&\sum_{k=0}^{\lf(3y-5)/12\rf}N_{3y-5-12k}-4\sum_{k=0}^{\lf(3y-8)/12\rf}N_{3y-8-12k}\\
+&3\sum_{k=0}^{\lf(3y-11)/12\rf}N_{3y-11-12k}-\sum_{k=0}^{\lf(3y-14)/12\rf}N_{3y-14-12k}.
\end{split}
\label{eq:2}
\end{equation}
By Lemma \ref{lem:1}, if all four sums in the expression of $-V_y$ above have the same
number of terms, then $V_y=0$. If $\cI$ designates the set of all intervals $J$ such that
$J\subset[m,m+1)$, for some integer $m$, then those four sums will all have $1+\lf(3y-14)/12\rf$
terms iff the interval $[(3y-14)/12,(3y-5)/12]\in \cI$. Putting $y=4\ell+r$, $0\le r\le3$,
we see
that
$$[(3y-14)/12,(3y-5)/12]\in \cI \text{ iff } [(3r-14)/12,(3r-5)/12]\in \cI,$$
which occurs iff $r=1$.
Suppose $y=4\ell$. Then $\lf(3y-5)/12\rf=\lf(3y-8)/12\rf=\lf(3y-11)/12\rf=\ell-1$ and
$\lf(3y-14)/12\rf=\ell-2$. Thus, using Lemma \ref{lem:1}, we obtain that $-V_y=N_{3y-5-12(\ell-1)}-4
N_{3y-8-12(\ell-1)}+3N_{3y-11-12(\ell-1)}=N_7-4N_4+3N_1=4-4+0=0$. Thus, $V_{4\ell}=0$ as well.
Assume now $y=4\ell+2$. Then the first sum, i.e., $\sum_{k=0}^{\lf(3y-5)/12\rf}N_{3y-5-12k}$ in
(\ref{eq:2}), contains one more term than the three others. Hence,
as $\lf(3y-5)/12\rf=\ell$, we see that
$-V_y=N_{3y-5-12\ell}=N_1=0$. Finally, suppose $y=4\ell+3$. Then $\lf(3y-5)/12\rf=\lf(3y-8)/12\rf=\ell$,
while $\lf(3y-11)/12\rf=\lf(3y-14)/12\rf=\ell-1$. Thus, $-V_y=N_{3y-5-12\ell}-4N_{3y-8-12\ell}=
N_4-4N_1=1$. Hence, $V_{4\ell+3}=-1$, which ends the proof.
\end{proof}
Since $(\s_y)$ is a seventh-order recurrence with the zeros of its characteristic polynomial all identified,
namely $\a^3$, $\b^3$, $\g^3$ and all complex fourth roots of unity, we solved a $7\times7$ linear
system and got a closed-form expression for $\s_y$. We found $\s_y=I_y+r_y$, where
\begin{align}\begin{split}
I_y&:=\frac{1}{117}\big(2N_{3y}+103N_{3y+1}+100N_{3y+2}\big),\\
r_y&:=\frac14-\frac{(-1)^y}{36}-
\frac{3i+2}{52}i^y+\frac{3i-2}{52}(-i)^y,
\end{split}\label{eq:3}\end{align} and $i=\sqrt{-1}$.
\smallskip
We may observe that $\a^3I_y-I_{y+1}$ can be made arbitrarily small, for all large enough $y$'s.
Indeed, for $t$ a fixed integer, $N_{3y+t}$ is a linear combination of $\a^{3y}$, $\b^{3y}$ and $\g^{3y}$.
But, as the absolute values of $\b$ and $\g$ are smaller than one, $\a^3N_{3y+t}-N_{3y+3+t}$ tends to $0$
as $y$ tends to infinity. The next lemma quantifies this observation.
\begin{lemma}\label{lem:E} We have $|E_y|<6\cdot10^{-5}$ for all $y\ge20$, where $E_y:=\a^3I_y-I_{y+1}$.
\end{lemma}
\begin{proof} For all $n\ge0$, we have the closed-form expression
$$
N_n=\frac{\a^n}{f'(\a)}+2\,\cR e\bigg(\frac{\b^n}{f'(\b)}\bigg),
$$ where $f'$ is the derivative of $f(x)=x^3-x^2-1$ and $\cR e(z)$ stands for the real part of a complex $z$.
Therefore, for $t=0$, $1$ or $2$,
%\begin{eqnarray*}
$$
|\a^3N_{3y+t}-N_{3y+3+t}|\le\big|2\a^3\cR e\bigg(\frac{\b^{3y+t}}{f'(\b)}\bigg)-2\cR e\bigg(\frac{\b^{3y+3+t}}{f'(\b)}\bigg)\big|\le3\a^3\frac{|\b|^{3y}}{|f'(\b)|}.
%\end{eqnarray*}
$$
Hence,
$$|\a^3I_y-I_{y+1}|\le\frac{2+103+100}{117}\cdot\frac{3\a^3}{|f'(\b)|}\cdot|\b|^{3y}<5.6\times|\b|^{3y}.$$
Since $|\b|<1$ and $5.6\times|\b|^{60}\doteq 0.0000581\cdots$, the lemma follows.
\end{proof}
We are ready to prove that the two sequences are identical.
\begin{theorem} For all $y\ge0$, $\t_y=\s_y$.
\end{theorem}
\begin{proof} It is easy to check that $r_y$ defined in (\ref{eq:3}) is of period $4$ and that
$$
r_y=\frac1{117}\times\begin{cases}17,&\text{ if }y\equiv0\pmod4;\\
46,&\text{ if }y\equiv1\pmod4;\\
35,&\text{ if }y\equiv2\pmod4;\\
19,&\text{ if }y\equiv3\pmod4.
\end{cases}
$$ (Thus, $r_y+r_{y+1}+r_{y+2}+r_{y+3}=1$ for all $y\ge0$.)
We will need the differences $Ar_y-r_{y+1}$ for all $y$'s so we compute them to three significant digits
\begin{equation}\label{eq:4}
Ar_y-r_{y+1}=117^{-1} \begin{cases}17A-46&\\
46A-35&\\
35A-19&\\
19A-17&
\end{cases}\doteq\begin{cases}0.064,&\text{ if }y\equiv0\pmod4;\\
0.938,&\text{ if }y\equiv1\pmod4;\\
0.779,&\text{ if }y\equiv2\pmod4;\\
0.366,&\text{ if }y\equiv3\pmod4.
\end{cases}
\end{equation}
Let $y\ge20$ be an integer. We suppose that $\t_k=\s_k$ for all $k$'s, $0\le k\le y$,
and proceed by induction. Thus, we need to show that $\t_{y+1}=\s_{y+1}$.
Using our inductive hypothesis, we see that $A\t_y=A\s_y=AI_y+Ar_y=I_{y+1}+E_y+Ar_y=(\s_{y+1}-r_{y+1})+E_y+Ar_y$.
That is, $A\t_y=\s_{y+1}+(Ar_y-r_{y+1}+E_y)$. By Lemma \ref{lem:E}, $|E_y|<6\cdot10^{-5}$ and, by (\ref{eq:4}),
$6\cdot10^{-5}<Ar_y-r_{y+1}<1-6\cdot10^{-5}$. Therefore, $0<Ar_y-r_{y+1}+E_y<1$ and $\lf A\t_y\rf=\s_{y+1}$.
Since, by definition, $\t_{y+1}=\lf A\t_y\rf$, the inductive step is proved. As mentioned earlier the induction
is well grounded as we checked that $\t_k=\s_k$, for all $k$, $0\le k\le 199$.
\end{proof}
\begin{remark}
The four values of $\a^3r_y-r_{y+1}$ in (\ref{eq:4}) are the limit values, rounded to three
decimals, of the fractional
parts of $\a^3\t_y$, as $y$ increases. In fact, we found that for $y=20$, $21$, $22$ and $23$, those values are
already, to three significant digits, equal to $0.064$, $0.938$, $0.779$ and $0.366$, respectively.
\end{remark}
With $\xi_y$ and $\ep_y$, respectively, defined in Lemmas \ref{lem:3} and \ref{lem:2}, we obtain the corollary
\begin{cor} For all $y\ge0$, we find that $\xi_y=\ep_y$, i.e.,
$$
\lf(4-\a^3)\{\a^3 \t_{y+1}\}-\a^{-3}\{\a^3 \t_y\}\rf=\begin{cases}0,&\text{ if }y
\equiv0,1\text{ or }2\pmod 4;\\
-1,&\text{ if }y\equiv3\pmod 4.
\end{cases}
$$
\end{cor}
And, in return, we also have an expression for the remainder function $e_{(y)}(1)$ of Corollary \ref{cor:N}.
\begin{cor} For all $y\ge0$,
$$
e_{(y)}(1)=\sum_{k=0}^{\lf(3y-14)/12\rf}N_{3y-14-12k}.
$$
\end{cor}
\begin{proof} From Corollary \ref{cor:N}, $b^y(1)=a(1)N_{3y-2}+b(1)N_{3y}-N_{3y-3}-e_{(y)}(1)$. But
$a(1)N_{3y-2}+b(1)N_{3y}-N_{3y-3}=N_{3y-2}+3N_{3y}-N_{3y-3}=N_{3y+1}+N_{3y}+N_{3y-1}=N_{3y+2}+N_{3y}=N_{3y+3}$.
Comparing with the expression of $\s_y$ in (\ref{eq:1}) yields the corollary since $\s_y=\t_y=b^y(1)$.
\end{proof}
\section{The $(\a,\a^4)$ case with $\a^4-\a^3-1=0$, $\a>1$}
\label{sec5}
Here $\a$ is the dominant zero of $x^4-x^3-1$. We find that
$\a\doteq 1.38028$. Thus, $a(n)=\lfloor n\a\rfloor$ and $b(n)=\lfloor n\a^4\rfloor$.
We denote the fundamental sequence associated with $x^4-x^3-1$ as $H=(H_k)_{k\ge0}$.
That is, $H_0=H_1=H_2=0$ and $H_3=1$ with $H_{n+4}=H_{n+3}+H_n$, for all integers $n$.
This is sequence \seqnum{A017898} in the OEIS \cite{Slo}.
\begin{lemma}\label{lem:H} For all integers $n\ge1$, we have
\begin{align*}
a^2(n)&=b(n)-n-\bigg\lfloor\frac{n}{\a}\bigg\rfloor-e_1(n),\\
ba(n)&=a(n)+b(n)-e_2(n),\\
ab(n)&=a(n)+b(n)-e_3(n),\\
b^2(n)&=a(n)+4b(n)-2n-\bigg\lfloor\frac{n}{\a}\bigg\rfloor+e_4(n),
\end{align*} where the four $e_i$'s
are bounded integral functions of $n$ with ranges $\{0,1,2,3\}$ for $e_1$ and $e_2$,
$\{0,1\}$ for $e_3$ and $\{-1,0,1\}$ for $e_4$.
\end{lemma}
In fact,
\begin{align*}
e_1(n)&=\lceil\{\frac{n}{\a}\}+\a\{\a n\}-\{\a^3 n\}\rceil \\
e_2(n)&=\lc\a^3\{\a n\}-\{\a^3 n\}\rc \\
e_3(n)&=\lceil(\a-1)\{\a^3 n\}-\{\a n\}\rceil \\
e_4(n)&=\lf(4-\a^4)\{\a^3n\}+\{\a n\}-\{\frac{n}{\a}\}\rf .
\end{align*}
The least value of $n$ for which $e_1(n)=3$ is $113$.
The least $n$ with $e_4(n)=1$ is $47$.
We omit the proof of Lemma \ref{lem:H} as it is similar in spirit to that of Lemma \ref{lem:N}.
\begin{theorem}\label{thm:H} Let $w=\ell_1\circ\ell_2\circ\cdots\circ\ell_s$, $(s\ge1)$, where each $\ell_i$ is
either $a$ or $b$. Assume $x$ and $y$ are, respectively, the number of $a$'s and the number of $b$'s
in $w$. Then $w(n)$ equals
$$
H_{x+4y-2}a(n)+H_{x+4y+1}b(n)-(H_{x+4y-3}+H_{x+4y-4})n-H_{x+4y-3}\bigg\lf\frac{n}{\a}\bigg\rf+e(n),
$$ where $e$ is a bounded integral function of $n$.
\end{theorem}
\begin{proof} We carry out an inductive proof on the number of letters $\ell$ in the word $w$.
By running the recursion defining the sequence $H$ backwards, we find that $H_{-3}=H_{-2}=0$ and
$H_{-1}=1$. Thus we easily check the result when $\ell=1$ and, using Lemma \ref{lem:H}, for $\ell=2$.
Assuming $w$ is a word with $\ell\ge2$ letters and the theorem holds for such words, we show
the theorem still holds for $wa$ and $wb$.
The inductive hypothesis gives that
\begin{eqnarray*}
wa(n)&=&H_{x+4y-2}\,a^2(n)+H_{x+4y+1}\,ba(n)\\
&-&(H_{x+4y-3}+H_{x+4y-4})a(n)-H_{x+4y-3}
\bigg\lfloor\frac{a(n)}{\a}\bigg\rfloor+e\big(a(n)\big).
\end{eqnarray*} Note that $\big\lf\frac{a(n)}{\a}\big\rf=\big\lf n-\frac{\{\a n\}}{\a}
\big\rf=n-1$. So using Lemma \ref{lem:H} and regrouping terms we obtain
\begin{eqnarray*}
wa(n)&=&(H_{x+4y+1}-H_{x+4y-3}-H_{x+4y-4})\,a(n)+(H_{x+4y-2}+H_{x+4y+1})\,b(n)\\
&-&(H_{x+4y-2}+H_{x+4y-3})n-H_{x+4y-2}\bigg\lf\frac{n}{\a}\bigg\rf
+e'(n),\end{eqnarray*} where $e'(n)=H_{x+4y-3}-H_{x+4y-2}e_1(n)+H_{x+4y+1}e_2(n)+e\big(a(n)\big)$.
But $H_{x+4y+1}-H_{x+4y-3}-H_{x+4y-4}=H_{x+4y}-H_{x+4y-4}=H_{x+4y-1}=H_{(x+1)+4y-2}$ and
$H_{x+4y-2}+H_{x+4y+1}=H_{x+4y+2}=H_{(x+1)+4y+1}$. Therefore, $wa(n)$ has the form claimed in the theorem.
\medskip
Also by the inductive hypothesis,
\begin{eqnarray*} wb(n)&=&H_{x+4y-2}\,ab(n)+H_{x+4y+1}\,b^2(n)\\
&-&(H_{x+4y-3}+H_{x+4y-4})\,b(n)-H_{x+4y-3}
\bigg\lf\frac{b(n)}{\a}\bigg\rf+e(b(n)).
\end{eqnarray*} Note that
$$
\bigg\lf\frac{b(n)}{\a}\bigg\rf=\bigg\lf\frac{(\a^5-\a)n-\{\a^4n\}}{\a}\bigg\rf=b(n)-n+\bigg\lf
\bigg(1-\frac1\a\bigg)\{\a^4n\}\bigg\rf.
$$ But $(1-1/\a)\{\a^4n\}\in(0,1)$ so $\lfloor b(n)/\a\rfloor=b(n)-n$.
Using Lemma \ref{lem:H}, the identity
$\lfloor b(n)/\a\rfloor=b(n)-n$, and regrouping like-terms, we obtain
\begin{eqnarray*}
wb(n)&=&(H_{x+4y-2}+H_{x+4y+1})a(n)\\
&+&(H_{x+4y-2}+4H_{x+4y+1}-2H_{x+4y-3}-H_{x+4y-4})b(n)\\
&-&(2H_{x+4y+1}-H_{x+4y-3})n-H_{x+4y+1}\bigg\lfloor\frac{n}{\a}\bigg\rfloor+e''(n),
\end{eqnarray*} where $e''(n)=H_{x+4y-2}e_2(n)+H_{x+4y+1}e_4(n)+e\big(b(n)\big)$ is an integral and bounded function of $n$. Using the recursion for $H$, we obtain the expected coefficients
for $a(n)$, $b(n)$, $n$ and $\big\lfloor\frac{n}{\a}\big\rfloor$. For the coefficient
of $b(n)$, put $t=x+4y+1$. Then we check that $H_{t+4}=4H_t+H_{t-3}-2H_{t-4}-H_{t-5}$.
It holds iff $H_{t+3}=3H_t+H_{t-3}-2H_{t-4}-H_{t-5}$. But
$$H_{t+3}=H_{t+2}+H_{t-1}=H_{t+1}+ H_{t-1}+H_{t-2}=H_t+H_{t-1}+H_{t-2}+H_{t-3}.$$
Thus, the identity to prove holds
iff $H_{t-1}+H_{t-2}=2H_t-2H_{t-4}-H_{t-5}$. But the latter is true as $H_t-H_{t-4}=H_{t-1}$
and $2H_{t-1}-H_{t-5}=H_{t-1}+H_{t-2}$.
\end{proof}
\section{The general case $(\a,\a^q)$, $(q\ge2)$}
\label{sec6}
Let $q\ge2$ be an integer. The polynomial $f(x):=x^q-x^{q-1}-1$ possesses a simple dominant real zero $\a>1$
\cite[Lemma 3]{Ba}.
Here, $a(n)=\lf n\a\rf$ and $b(n)=\lf n\a^q\rf$.
We denote the fundamental sequence associated with
$f(x)$ as $G=(G_k)_{k\ge0}$. That is, $G_0=G_1=\cdots=G_{q-2}=0$ and $G_{q-1}=G_q=\cdots=
G_{2q-2}=1$ as $G_{t+q}=G_{t+q-1}+G_t$.
\begin{lemma}\label{lem:geo} Let $\th$ be a zero of $x^q-x^{q-1}-1$. Then,
for all integers $n\ge m$, we find that
$$
\sum_{i=m}^n\th^i=\th^{n+q}-\th^{m+q-1}.
$$
\end{lemma}
\begin{proof} Summing the geometric series $\sum_{i=m}^n\th^i$ yields the expression
$\frac{\th^{n+1}-\th^m}{\th-1}$. But $\th^{n+1}=\th^{n+q+1}-\th^{n+q}=\th^{n+q}(\th-1)$
and $\th^m=\th^{m+q-1}(\th-1)$.
\end{proof}
\begin{lemma}\label{lem:G} For all integers $n\ge1$, we have
\begin{align*}
a^2(n)&=b(n)-n-\bigg\lf\frac{n}{\a}\bigg\rf-\cdots-\bigg\lf\frac{n}{\a^{q-3}}
\bigg\rf+O(1),\\
ba(n)&=a(n)+b(n)+O(1),\\
ab(n)&=a(n)+b(n)+O(1),\\
b^2(n)&=a(n)+q\,b(n)-(q-2)\,n-(q-3)\bigg\lf\frac{n}{\a}\bigg\rf-\cdots-\bigg\lf\frac{n}
{\a^{q-3}}\bigg\rf+O(1).
\end{align*}
\end{lemma}
\begin{proof} By Lemma \ref{lem:geo}, $\sum_{i=0}^{q-3}\a^{-i}=\sum_{i=3-q}^0\a^i=\a^q-\a^2$.
Hence,
\begin{align*}
a^2(n)&=\lf\a^2n-\a\{\a n\}\rf= \lf\a^q n-\sum_{i=0}^{q-3}\frac{n}{\a^i}-\a\{\a n\}\rf \\
&=b(n)-\sum_{i=0}^{q-3}\bigg\lf\frac{n}{\a^i}\bigg\rf+O(1).
\end{align*}
Now $ab(n)=\lf\a^q\lf\a n\rf\rf=\lf(\a^{q-1}+1)(\a n-\{\a n\})\rf=a(n)+b(n)+O(1)$.
A similar expansion
also yields our claim for $ba(n)$. The expression for $b^2(n)$ will hold if $\a^{2q}=\a+q\a^q
-\sum_{i=0}^{q-3}\a^{-i}-\sum_{i=0}^{q-2}\a^{-i}-\cdots-\a^{-0}$. That is, if $\a^{2q}=\a+q\a^q-
\sum_{j=0}^{q-3}\sum_{i=3+j-q}^0\a^i$. Now, using Lemma \ref{lem:geo} twice,
we obtain
\begin{align*}
\a+q\a^q-\sum_{j=0}^{q-3}\sum_{i=3+j-q}^0\a^i &=\a+q\a^q-\sum_{j=0}^{q-3}(\a^q-\a^{2+j}) \\
&= \a+2\a^q +(\a^{2q-1}-\a^{q+1})=-(\a^{q+1}-\a^q-\a)+(\a^q+\a^{2q-1}) \\
&=0+\a^{2q}.
\end{align*}
\end{proof}
\begin{remark}
The third bounded function $O(1)$ in the identity $ab(n)=a(n)+b(n)+O(1)$
is $\lf(1-\a)\{\a^qn\}+\{\a n\}\rf$, so it is either $0$ or $-1$.
\end{remark}
\begin{lemma}\label{lem:g} Let $n\ge m$ be integers. Then
$$
\sum_{i=m}^nG_i=G_{n+q}-G_{m+q-1}.
$$
\end{lemma}
\begin{proof} The derivative of $f(x)$ only has $0$ and $(q-1)/q$ as zeros. Since
neither $0$, nor $(q-1)/q$ is a zero of $f$, the zeros $\th_1,\ldots,\th_q$ of $f(x)$ are simple. Thus,
$G_i$ is a linear combination of the $\th_t^i$, $t=1,\ldots,q$. Hence, the lemma
is a direct consequence of Lemma \ref{lem:geo}.
\end{proof}
\begin{cor}\label{cor:g2} Let $p\ge1$, $q\ge3$ and $0\le j\le q-3$ be integers. Then
$$
\sum_{3\le i\le q-j}G_{p-i}=G_{p+q-3}-G_{p+j-1}.
$$ \end{cor}
\begin{proof} We note that $\sum_{3\le i\le q-j}G_{p-i}=\sum_{i=p+j-q}^{p-3}G_i$ and apply
Lemma \ref{lem:g}.
\end{proof}
\begin{theorem}\label{thm:G} Let $w$ be a composite function of some $a$'s and $b$'s. Putting
$p=x+qy$, where $x$ and $y$ are, respectively, the number of $a$'s and the number of $b$'s
in $w$, we find that, for all $n\ge1$,
\begin{equation}\label{eq:main}
w(n)=G_{p-2}\,a(n)+G_{p+q-3}\, b(n)-\sum_{0\le j\le q-3}c_j\bigg\lf\frac{n}{\a^j}\bigg\rf+O(1),
\end{equation} where $c_j=\sum_{3\le i\le q-j}G_{p-i}$, or alternatively $c_j=G_{p+q-3}-G_{p+j-1}$.
\end{theorem}
\begin{proof} Note that for $q=2$, the sum $\sum_{0\le j\le q-3}c_j\lf n/\a^j\rf$
is empty and equals $0$ by convention. Hence, in that case, the theorem is implied by Theorem
\ref{thm:Kim}. Thus, assume $q\ge3$. Observe that, by Corollary \ref{cor:g2},
$\sum_{3\le i\le q-j}G_{p-i}=G_{p+q-3}-G_{p+j-1}$.
We may proceed by induction on $x+y$ as was done in Theorems \ref{thm:Kim},
\ref{thm:root2}, \ref{thm:N} and \ref{thm:H}. One checks the result for $x+y=1$ directly.
For instance, if $w=a$, then taking the function $O(1)$ to be the null function and noting
that $G_{-1}=1$ and $G_{-i}=0$, $2\le i\le q-1$, we find that all coefficients $c_j$
of (\ref{eq:main}) are zero so that $G_{-1}a(n)+G_{q-2}b(n)-0+0$ is indeed $a(n)$.
\smallskip
Now suppose (\ref{eq:main}) holds for some $w$ with $x+y\ge1$ letters. Replacing $n$ by $a(n)$
in (\ref{eq:main}), using Lemma \ref{lem:G} to express $a^2(n)$ and $ba(n)$ and filling in some
constant terms into the $O(1)$ term, we find that
\begin{equation}
\begin{split}
wa(n)&=G_{p-2}\bigg(b(n)-\sum_{0\le j\le q-3}\bigg\lf\frac{n}{\a^j}\bigg\rf\bigg)+G_{p+q-3}
\big(a(n)+b(n)\big)\\
&-\sum_{0\le j\le q-3}\big(G_{p+q-3}-G_{p+j-1}\big)
\bigg\lf\frac{a(n)}{\a^j}\bigg\rf+O(1).
\end{split}
\label{eqn:wa}
\end{equation}
The coefficient of $b(n)$ is $G_{p-2}+G_{p+q-3}=G_{p+q-2}=G_{(p+1)+q-3}$, while that of
$a(n)$ is $G_{p+q-3}-(G_{p+q-3}-G_{p-1})=G_{(p+1)-2}$, as expected.
Since $\lf n/\a^{j-1}\rf-\lf a(n)/\a^j\rf$ is $0$ or $1$ for all $j\ge1$,
the remaining terms are
$$
-G_{p-2}\sum_{0\le j\le q-3}\bigg\lf\frac{n}{\a^j}\bigg\rf-\sum_{1\le j\le q-3}
\big(G_{p+q-3}-G_{p+j-1}\big)\bigg\lf\frac{n}{\a^{j-1}}\bigg\rf+O(1).
$$
But $\sum_{1\le j\le q-3}\big(G_{p+q-3}-G_{p+j-1}\big)\lf n/\a^{j-1}\rf=
\sum_{0\le j\le q-4}\big(G_{p+q-3}-G_{p+j}\big)\lf n/\a^j\rf$, so we see that
the coefficient $c_j(wa)$ of $-\lf n/\a^j\rf$ in $wa(n)$ is, for $0\le j\le q-4$,
equal to
$$(G_{p-2}+G_{p+q-3})-G_{p+j}=G_{p+q-2}-G_{p+j}=G_{(p+1)+q-3}-G_{(p+1)+j-1},$$
while $c_{q-3}(wa)=G_{p-2}=G_{(p+1)-3}$, as expected.
\smallskip
Similarly, we expand $wb(n)$, expressing $ab(n)$ and $b^2(n)$ with Lemma \ref{lem:G}, to
find that $wb(n)$ may be written as
\begin{align}
&G_{p-2}\big(a(n)+b(n)\big)+G_{p+q-3}\bigg(a(n)+q\,b(n)-\sum_{0\le i\le q-3}
(q-2-i)\bigg\lf\frac{n}{\a^i}\bigg\rf\bigg)\nonumber\\
&-\sum_{0\le j\le q-3}\big(G_{p+q-3}-G_{p+j-1}\big)
\bigg\lf\frac{b(n)}{\a^j}\bigg\rf+O(1).\label{eqn:wb}
\end{align}
The coefficient of $a(n)$, $G_{p-2}+G_{p+q-3}$, is, as expected, equal to $G_{(p+q)-2}$.
Given $j$ between $1$ and $q-3$, and noting that
$$\a^qn=(\a^{q+1}-\a)n=(\a^{q+2}-\a^2-\a)n=
\cdots=(\a^{q+j}-\a^j-\a^{j-1}-\cdots-\a)n,$$
we see that, for all $j\ge0$,
\begin{equation}\label{eq:wb}
\bigg\lf\frac{b(n)}{\a^j}\bigg\rf=b(n)-\sum_{0\le k\le j-1}\bigg\lf\frac{n}{\a^k}\bigg\rf+O(1).
\end{equation}
Therefore the (natural) coefficient of $b(n)$ in $wb(n)$ is
\begin{align}
G_{p-2}&+q\,G_{p+q-3}-\sum_{0\le j\le q-3}\big(G_{p+q-3}-G_{p+j-1}\big)\nonumber\\
&=G_{p-2}+2G_{p+q-3}+\sum_{0\le j\le q-3}G_{p-1+j}\nonumber\\
&=G_{p-2}+2G_{p+q-3}+(G_{p-1+2q-3}-G_{p+q-2})\label{eq:5}\\
&=2G_{p+q-3}+G_{p+2q-4}-(G_{p+q-2}-G_{p-2})\nonumber\\
&=G_{p+q-3}+G_{p+2q-3}-G_{p+q-3}=G_{p+2q-3}\nonumber,
\end{align} as expected, where in (\ref{eq:5}) we used Corollary \ref{cor:g2}.
\smallskip
By (\ref{eqn:wb}) and (\ref{eq:wb}), the coefficient $c_k(wb)$ is
\begin{align}
&(q-2-k)G_{p+q-3}-\sum_{k+1\le j\le q-3}(G_{p+q-3}-G_{p+j-1})\nonumber\\
\qquad&=\big(q-2-k-(q-3-k)\big)G_{p+q-3}+\sum_{j=k+1}^{q-3}G_{p-1+j}\nonumber\\
\qquad&=(G_{p+q-3}+G_{p+2q-4})-G_{p+q+k-1}\label{eq:6}\\
&=G_{(p+q)+q-3}-G_{(p+q)+k-1}\nonumber,
\end{align} which is what we intended to prove. Again, in (\ref{eq:6}), we used Corollary \ref{cor:g2}.
\end{proof}
\begin{remark}
The basis functions $a(n)$, $b(n)$, and the $\lf\frac{n}{\a^j}\rf$,
used to express $w(n)$ in Theorem \ref{thm:G}, are all integral, and so are their coefficients,
but there is no uniqueness in this property. Other choices could have been made.
For instance, in Lemma \ref{lem:G}, we could have chosen to express $a^2(n)$ as
$a(n)+\lf\frac{n}{\a^{q-2}}\rf-e(n)$, where $e(n)=\lceil(\a-1)\{\a n\}-\{n/\a^{q-2}\}\rceil\in\{0,1\}$.
Had we made this choice for $a^2(n)$, the general expression of $b^y(n)$, when $q=3$, in
Corollary \ref{cor:N}, would have taken the form $b^y(n)=(N_{3y-1}+N_{3y-6})a(n)+N_{3y-1}b(n)+N_{3y-3}
\lf n/\a\rf+O(1)$.
\end{remark}
\section{Problems for future research}
\label{sec7}
We provide some ideas for further investigation. These ideas only reflect how we see
things at the moment and are probably more a measure of our ignorance than anything else.
\medskip
\begin{problem} \label{problem:1} Find other pairs of complementary Beatty sequences, preferably infinite
families of such pairs $\big(a_s(n),b_s(n)\big)$, and discover theorems comparable to Theorems \ref{thm:R}
and \ref{thm:G} that express a word in $a_s$ and $b_s$ as nearly a linear combination of $a_s$ and $b_s$.
\end{problem}
(The referee pointed out the polynomials $x^q-x^{q-1}-x-1$ for $q\ge3$. Each such
polynomial \cite[Lemma 3]{Ba} has a simple dominant zero $\a$. In fact, for $q=3$, $\a$ is a cubic Pisot number that
can play a fundamental role in the construction of Rauzy fractals \cite{Rau}.)
\medskip
The secondary question tackled in this paper of whether the sequences $(b^y(n))_y$ are linear recurrences
leads to a simple fundamental problem.
\smallskip
\begin{problem}\label{problem:2} Let $\a>1$ be, say, a real algebraic integer of minimal polynomial $P$.
Put $f(n):=\lf\a n\rf$. Fix an integer $n\ge1$ and define, for $y$ a positive integer, $u_y$ as $f^y(n)$, where
$f^y$ is the $y$-fold composite function $f\circ\cdots\circ f$.
\smallskip
{\bf 1.} Characterize those algebraic integers $\a$ for which the sequence $(u_y)_y$ is a linear
recurrence of characteristic polynomial $P$ for all choices of $n$.
\smallskip
{\bf 2.} Characterize those algebraic integers $\a$ for which the sequence $(u_y)_y$ is a linear
recurrence for all choices of $n$. Is it necessarily true that the characteristic polynomial of
$(u_y)$ must be a multiple of $P$? Or that it must be of the form $(x^h-1)P$ for some $h\ge0$?
\smallskip
{\bf 3.} Suppose $(u_y)_y$ is a linear recurrence for $n=1$. Does it follow that it is linear recurrent
for all choices of $n$? If so, would there always be an annihilating polynomial common to all $(u_y)$ for all values
of $n\ge1$? How often would that common polynomial turn out to be the characteristic polynomial of $(u_y)$ when $n=1$? \end{problem}
\medskip
Given a pair of complementary Beatty sequences $(a,b)$, write $A$ and $B$ for the respective ranges of
the functions $a$ and $b$. For the Wythoff pair $(a,b)$, we saw that
$b^y(1)=F_{2y+1}$. Stolarsky \cite[p.~441]{Sto} observed that for all $y\ge1$, the pair $V_y=(F_{2y},F_{2y+1})$
belongs to $A\times B$ and that the vectors $V_y$ satisfy the second-order recursion $V_{y+2}=3V_{y+1}-V_y$.
Note that $V_y=(\lf F_{2y-1}\a\rf,\lf F_{2y-1}\a^2\rf)=\big(ab^{y-1}(1),b^y(1)\big)$. More generally,
the vectors $V_y=\big(ab^{y-1}(1),b^y(1)\big)$ satisfy the same second-order recurrence as $(b^y(1))_y$
for all Beatty pairs of Section~\ref{sec3}, where $\b=\a+r$, $r\ge1$. Indeed, by Corollary \ref{cor:byof1},
$(b^y(1))_y$ is a second-order recurrence, and one easily sees that $ab^{y-1}(1)=b^y(1)-rb^{y-1}(1)$.
In the Narayana case,
we saw that $(b^y(1))_y$ satisfies a 7th-order linear recurrence and it does seem experimentally that the vectors
$\big(ab^{y-1}(1),b^y(1)\big)$, all in $A\times B$, satisfy the same 7th-order recursion. These various
instances of the same phenomenon raise another problem
\smallskip
\begin{problem}\label{problem:3} Characterize pairs of complementary Beatty sequences $(a,b)$ such that the vectors
$V_y:=\big(ab^{y-1}(1),b^y(1)\big)$ satisfy a linear recurrence relation.
\end{problem}
\medskip
In the Wythoff context, Stolarsky \cite{Sto} discovered a sequence of vectors in $A\times B$ which satisfy
a fourth-order linear recurrence with characteristic polynomial $x^4-10x^3+16x^2-5x-1$ coprime to $x^2-3x+1$.
Later, Ridley \cite{Rid} found infinitely many sequences of Wythoff pairs that satisfy fourth-order
linear recurrences, one being a recurrence with characteristic polynomial $x^4-x^3-5x^2+7x-1$.
\smallskip
\begin{problem}\label{problem:4} Given a pair of complementary Beatty sequences $(a,b)$ with $\a$ algebraic,
are there general methods to generate sequences of vectors in $A\times B$ that satisfy higher-order
linear recurrences?
\end{problem}
\medskip
The referee suggested another problem, not unrelated to Problem \ref{problem:2}, which he illustrated
with an example.
\smallskip
\begin{problem} In \S4, Lemma \ref{lem:3}, we reach an identity of the form
$$
\sum_{k=0}^mc_ka_{n+k}=F(a_n,a_{n+1}),
$$ where $F$ is a floor function of a linear combination of fractional parts involving $a_n$ and $a_{n+1}$
which can only take finitely many values. As turned out the
$a_n$'s of \S4 satisfy a homogeneous linear recurrence, though of higher order. To be more
specific, the characteristic polynomial has an `additional factor' of $x^4-1$. Can any general
results of this nature be obtained?
A simple example of such recurrences is the following `almost Fibonacci' recurrence
$$
a_{n+2}=a_{n+1}+a_n +\lf k\{a_{n+1}\a\}\rf,
$$
where $k$ is a fixed positive integer, $\a= (1+\sqrt{5})/2$. Do
the $a_n$'s satisfy homogeneous linear recurrences? If so, {\bf 1.} what are the corresponding characteristic
polynomials and {\bf 2.} is the linear recurrence independent of the initial conditions?
Computer experiments suggest that (for example) for $k=11$ both
$(x^3-1)(x^2-x-1)$ and $(x^8-1)(x^2-x-1)$
are possible depending upon the initial conditions. The same sort of thing seems true for
$k = 12$ and $13$ with possibly different such recurrences for different initial conditions. Perhaps
there is always a recurrence whose characteristic polynomial has the form $x^2-x-1$
or $(x^2-x-1)(x^h-1)$ for some positive integer $h$?
\end{problem}
\section{Acknowledgments}
\label{sec8}
We are thankful to the referee in three ways. He accepted our text as it was. He offered
numerous references, with comments, to beautiful papers we were not aware of that may have connections
with our text. Those appear at the end of the introduction and enrich the present paper. Finally, he suggested that
we wrote an additional section for open investigations providing a couple of his own.
We also thank Michel Dekking who, after reading the paper online, sent
email on July 12th, 2017 to mention reference \cite{Car}, where
Theorem \ref{thm:Kim} had appeared before it reappeared in \cite{Ki}.
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\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B83; Secondary 11B37, 11B39.
\noindent \emph{Keywords: }
Beatty sequence, Wythoff pair, integer part, linear recurrence.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum {A078012}, \seqnum {A017898}.)
\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received July 26 2016;
revised versions received January 17 2017; January 28 2017.
Published in {\it Journal of Integer Sequences}, January 28 2017.
Minor revision, July 30 2017.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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\[(1-x)\mathop{P^{(\alpha+1,\beta)}_{n}\/}\nolimits\!\left(x\right)+(1+x)\mathop%
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\title{In the Open}
\author{Stanton Davis Kirkham}
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\frontmatter
The Project Gutenberg EBook of In the Open, by Stanton Davis Kirkham
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org
Title: In the Open
Author: Stanton Davis Kirkham
Release Date: April 25, 2013 [EBook \#42591]
Language: English
Character set encoding: ASCII
*** START OF THIS PROJECT GUTENBERG EBOOK IN THE OPEN ***
Produced by Greg Bergquist, Diane Monico, and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive/American Libraries.)
IN THE OPEN
INTIMATE STUDIES
AND APPRECIATIONS OF
NATURE BY
STANTON DAVIS KIRKHAM
AUTHOR OF
``WHERE DWELLS THE SOUL SERENE''
``THE MINISTRY OF BEAUTY''
``\textit{Over and above a healthy}
\textit{
curiosity, or any scientific}
\textit{
acquaintance, it is the}
\textit{
companionship of the woods}
\textit{
and fields which counts---}
\textit{
a real friendship for birds}
\textit{
and bees and flowers.}''
PAUL ELDER \& COMPANY
SAN FRANCISCO AND NEW YORK
\textit{Copyright, 1908}
\textit{
by} PAUL ELDER AND COMPANY
TO MY WIFE
MARY WILLIAMS KIRKHAM
THIS BOOK IS
AFFECTIONATELY
DEDICATED
PREFACE
\textit{There is an estate on which we pay no tax and which is not susceptible}
\textit{
of improvement. It is of indefinite extent and is to be reached by}
\textit{
taking the road to the nearest woods and fields. While this is quite as}
\textit{
valuable as any property we may possess, as a matter of fact few assert}
\textit{
their title to it.}
\textit{Nature is in herself a perpetual invitation to come into the open. The}
\textit{
woods are an unfailing resource; the mountains and the sea,}
\textit{
companionable. To count among one's friends, the birds and flowers and}
\textit{
trees is surely worth while; for to come upon a new flower is then in}
\textit{
the nature of an agreeable event, and a chance meeting with a bird may}
\textit{
lend a pleasant flavor to the day.}
CONTENTS
PREFACE v
THE POINT OF VIEW 1
SIGNS OF SPRING 11
BIRD LIFE 22
SONGS OF THE WOODS 40
WILD GARDENS 56
WEEDS 69
INSECT LORE 78
THE WAYS OF THE ANT 94
AUTUMN STUDIES 113
PASTURE STONES 127
NEIGHBORS 136
THE WINTER WOODS 153
LAUGHING WATERS 164
THE MOUNTAINS 173
THE FOREST 185
THE SEA 196
INDEX 209
\begin{quotation}
\textit{A flock of wild geese on the wing is no less than an
inspiration. When that strong-voiced, stout-hearted company
of pioneers pass overhead, our thoughts ascend and sail with
them over the roofs of the world. As band after band come
into the field of vision---minute glittering specks in the
distant blue---to cross the golden sea of the sunset and
disappear in the northern twilight, their faint melodious
honk is an Orphean strain drawing irresistibly.}\end{quotation}
AFTER THE PAINTING BY
LOUIS AGASSIZ FUERTES
[Illustration]
This text was converted to LaTeX by means of \textbf{GutenMark} software (version Jul 12 2014).
The text has been further processed by software
in the iTeX project, by Bill Cheswick.
\cleardoublepage
\tableofcontents
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\mainmatter
\pagenumbering{arabic}
\gutchapter{THE POINT OF VIEW}
Nature is in herself a perpetual invitation: the birds call, the trees beckon and the winds whisper to us. After the unfeeling pavements, the yielding springy turf of the fields has a sympathy with the feet and invites us to walk. It is good to hear again the fine long-drawn note of the meadow-lark---voice of the early year,---the first bluebird's warble, the field-sparrow's trill, the untamed melody of the kinglet---a magic flute in the wilderness---and to see the ruby crown of the beloved sprite. It is good to inhale the mint crushed underfoot and to roll between the fingers the new leaves of the sweetbrier; to see again the first anemones---the wind-children,---the mandrake's canopies, the nestling erythronium and the spring beauty, like a delicate carpet; or to seek the clintonia in its secluded haunts, and to feel the old childlike joy at sight of lady's-slippers.
It is worth while to be out-of-doors all of one day, now and then, and to really \textit{know} what is morning and what evening; to observe the progress of the day as one might attend a spectacle, though this requires leisure and a free mind. The spirit of the woods will not lend itself to a mere fair-weather devotion. You must cast in your lot with the wild and take such weather as befalls. If you do not now and then spend a day in the snow, you miss some impressions that no fair weather can give. When you have walked for a time in the spring shower, you have a new and larger sympathy with the fields. The shining leaves, glistening twigs, jeweled cobwebs and the gentle cadence of the falling rain all tell you it is no time to stay indoors.
Life in the woods sharpens the nose, the eyes, the ears. There are nose-feasts, eye-feasts, ear-feasts. What if the frost-grapes are sour---they are fair to look at. Some things are for the palate and some for the eye. The fragrance of blackberries is as delicate as the flavor, a spicy aroma, a woodsy bouquet, and to eat without seeing or smelling is to lose much. Clustered cherries, so lustrous black with their red stems, refresh the inner and the outer man. You may safely become a gourmand with respect to these wild flavors. Their virtue is of the volatile sort that will not stand bottling; it will not enter into essence or tincture. You must yourself go out and pick the cherry under a September sky and in the presence of the first glowing leaves of sumac and Virginia creeper.
Does not the bayberry revive and exhilarate the walker, as smelling-salts restore fainting women? You have but to roll the waxen berry in the fingers, or crush the leaf, to feel that indefinable thrill which belongs to the woods, to the open air---the free life. Another vigorous and stimulating odor is the fragrance of green butternuts, which contains the goodness, the sweetness, the very marrow of the woods, and calls out the natural and unaffected, as a strain of music arouses the heroic. The tartness of the barberry matches the crispness of the air and rebukes the lack of vigor in us. No true child can resist the lure of wintergreen berries, while to nibble the bark of a fresh young sassafras shoot admits us to some closer association with Nature. A whiff of balsam is an invitation to share the abandon of the woods, and awakens memories of the halcyon days, the shining hours, when nutting and berrying were the real things of life.
One who is possessed with the idea of finding a certain bird or plant is in a fair way to the discovery, and sooner or later each will come into the field of vision. How the robin discovers the worm is a mystery to be explained on the score of \textit{attention}; it is perfect concentration on a single point, with faculties trained in that direction. That the footsteps of ants were audible had not occurred to me till one day in watching the progress of the annual raid of the red ants upon the black colonies, I plainly heard the patter of their feet, as the column marched at double-quick over the floor of dry leaves. There are many sounds in Nature that only become evident when we give absolute attention, when we become all ear,---as there are things seen only when we become for the time an eye.
Sensitive and sympathetic natures rarely confuse one person with another, whereas the cold or obtuse really never see the finer distinctions in a face. They make poor observers. Any one unacquainted with birds will show by an attempted description that he has not in the least seen the bird. I have known old lumbermen who had not noticed the difference in the needles of the species of pine, nor the leaves of oaks; but they knew the difference in the quality of the wood well enough, because that appealed to their interest and held their attention.
Preparedness adds zest to the walk and enriches it, precisely as a broad culture and a fund of information enlarge the view of the traveler. Notwithstanding what may be in the woods, it takes some understanding and some interest to see it. An unprepared person will see little; an uninterested person will see nothing. To many of the villagers the wood-lot is a remote and unfamiliar wilderness, and the warblers and vireos as unknown as any tropic bird. We should at least know the kinglets by their caste-mark---whether it be red or yellow---and the oriole by the colors of his ancient line.
Given a certain preparedness even the rocks become instinct with suggestion. They are more than stone,---even historical reminders, which incite one to long and pleasing trains of thought. In the mountains I came upon a flat ledge of shale which showed ripple marks of an earlier sea than any we know, a far-off Devonian ocean which once washed this primitive beach. They had long parted company, and now the beach was up among the spruce and balsams,---such vicissitudes are there in the fortunes of all. The ancient waters had left their mark, that however high the rock might go, it should none the less speak of the mother sea. Again, the traces of glaciation on ledges and boulders appeal to the imagination with a peculiar eloquence. What a mighty cosmic plane was that which smoothed these granite ledges! It planed off New England as if it were a knot on a plank, and scattered over it the dust and chips of the workshop. These ledges serve as a fairly accurate compass, and are at least more reliable than the lichens on the trees.
Some men have an eye for trees and an inborn sympathy with these rooted giants, as if the same sap ran in their own veins. To them trees have a personality quite as animals have, and, to be sure, there are ``characters'' among trees. I knew a solitary yellow pine which towered in the landscape, the last of its race. Its vast columnar trunk seemed to loom and expand as one approached. Always there was distant music in the boughs above, a noble strain descending from the clouds. Its song was more majestic than that of any other tree, and fell upon the listening ear with the far-off cadence of the surf, but sweeter and more lyrical, as if it might proceed from some celestial harp. Though there was not a breeze stirring below, this vast tree hummed its mighty song. Apparently its branches had penetrated to another world than this, some sphere of unceasing melody.
There is a difference in the voices of trees. Some with difficulty utter any note, or answer to the storm alone; others only sigh and shiver. There are days when they gently murmur together, as if a rumor of general interest had reached them. Again the woods are silent, until one enters a grove of white pines, when on the instant a sweet low chant falls on the ear. Come upon the aspen on quiet days and it is all of a tremor, in a little ecstasy by itself, while the rest are mute. Trees change their songs with the season. In winter the whistling, rattling, roaring of hickories and oaks is a veritable witch-song, beside which the voices of midsummer days are as the cooing of doves. During a quiet snowfall, the white crystals sifting through the pines convey the idea of a gentle sociability somewhere in the branches overhead, the softly whispered and amiable gossip of pine-needles and snowflakes, old cronies who have not met in the past eight months.
The woods offer unlimited opportunity for making acquaintances, and nothing else stimulates the interest more than this. The keenest pleasure is in meeting a new bird: a rare and subtle stimulus not to be defined, to be experienced only and cherished as a memory. You stand in the midst of one of the mixed flocks of autumn---winter visitants with a sprinkling of warblers, and perhaps a blue-headed vireo and a pair of silent thrushes---and recognize old friends, with a chance of discovering a stranger. It calls out the zest for the woods like an appetite for dinner---a finer, more ethereal appetite, which is satisfied through the eye and ear. Occasionally the blue-headed vireo may be heard, though the season is far advanced, and the little Parula warbler indulges in a spiritual and melodious reverie, as if he already had visions of another spring and was communicating in a state of trance and ecstasy his prophetic thought.
One supremely mellow day the last of October, there came a pair of hermits to a secluded spot, flitting into a white oak, where they remained regarding me with round bright eyes. In due season they crossed to the pine under which I sat, whereupon one, directly over my head, began cautiously descending from branch to branch through the lower dead limbs until he was but a few feet from my face. Here he sat, regarding me in a gentle friendly way and talking to himself in an undertone---or was he talking to me? The impelling force continued to draw my little friend---it was mutual did he but know, a true case of love at sight---for at last, with an indescribable little flutter, he dropped from his perch with the evident intent of alighting upon me, but changed his course directly in my face, and with a swift motion of the wings darted into the shrubbery. Upon a near view the spell had broken, and he was again the timid solitary thrush.
It is because the wild life is so shy and elusive that the unexpected encounters have such charm. They are altogether clandestine and romantic. You may stroll time and again without the least encouragement, as though wholly ostracized from this society; and then some morning you are welcomed on every hand and admitted to the inner circle of the wood life. About the woods there is ever an enticing mystery. They invite us to enter as though they concealed some treasure we sought. A race dwells here apart, and we turn aside for that silent and refreshing company. When they speak, their speech is lyrical. There are men who have never known any friendship in Nature; others again who never outgrow the love of birds and flowers, who preserve some youthfulness and innocence which keeps them in touch with wild life. Over and above a healthy curiosity, or any scientific acquaintance, it is the \textit{companionship} of the woods and fields which counts---a real friendship for birds and bees and flowers. Let us remember the woods in the days of our youth, that we may have this unfailing resource in later years.
\gutchapter{SIGNS OF SPRING}
The approach of spring is felt, rather than reasoned about. There is that in us which rises to greet the incoming tide of the year before our eyes have apprised us of any change. Winter lies over the world much as ashes are banked on coals for the night, which nevertheless retain their heat and will be found alive and glowing in the morning. In the tropics the fire is not banked and there is no cold dawn with anticipations of the kindly blaze soon to arise, no gradual uncovering of the cheerful coals. Here in New England the dawn is rigorous and spring more welcome. The winter buds are evidence that it is not far away, and it takes but the least encouragement at any time for this latent heat and life to awake and show itself in the high blueberry twigs. Such buoyant faith has the skunk-cabbage it never entirely loses sight of spring, but exerts some spell over its muddy bed, whereby you may see that there, at least, it has already come in November.
The reddening of the twigs is in effect a prelude, and precedes the real spring as dawn precedes daylight, or twilight the night; this is the dawn of the year and these blueberry twigs its first flush. Smilax turns suddenly green as the sap circulates in its spiny stems, and the brown and sear aspect of the earth is relieved and enlivened. This early green is as refreshing to the eye as the first rhubarb to the palate.
One of the earliest signs is the little rosette of bright-colored leaves on the smaller hair-cap mosses, growing in contact with an outcropping ledge. You may see whole patches in the pastures, varying from orange to deep red, a vivid bit of color next the brown earth and looking like diminutive blossoms. Then come the fruiting spikes of the common field horsetails, poking out of some sand-bank. These signs of the awakening season appeal to the trained eye rather than to the casual glance. Such an one detects the slightest swelling of a leaf-bud, the faint reddening of a twig, the deeper green of another. The sap dripping from the freshly cut limb of a birch, or pendent from the wound in a long glittering icicle, is evidence of the quickened circulation of the earth. Among the thick mat of dry leaves you may perhaps find the delicate shoots of wood anemones, and in the swamps the tightly rolled stipes of the osmunda, like little croziers, while there is ice yet in the leaves of the pitcher-plant.
Deep lying in all men is a poetic vein which now appears on the surface. The first pussy-willows and the arrival of bluebirds arouse sentiments as common to us as the love of music: some suggestion of renewal, of awakening after the sleep of winter, which touches even the rough man and makes him kin for a day to the child. We embark each year on the sea of winter, with unquestioning faith that on its other shore spring awaits us, once more to shake the violets from her lap. When, in March, that shore looms in the distance, we feel the joy of travelers in sight of their native land. There may be rough seas, and March winds are blustery, but \textit{there} in sight, nevertheless, is that faint outline on the horizon.
No blossoming rod of Aaron could appear more miraculous than do the flowering willows. These twigs of brown and lifeless aspect suddenly burst into bloom and array themselves in exquisite silvery gray catkins, while the snow may be still on the ground. Not long after, the alders in the swamp unfold their clusters of drooping aments which have been on the tree stiff and rigid throughout the winter. Thousands of little tails are thus mysteriously hung out on the alder twigs to sway gently in the breeze, turning from a reddish hue to a sulphur-yellow as they expand and become powdered with pollen. Born into a frosty world when the feeble sun is still distant and cold, the March flowers are a link between winter and spring. But Nature has certainly relaxed her features; there is just the ghost of a smile on her icy lips.
This year I heard the bluebird's warble on the 4th of February, but did not see the bird, and heard no more till early in March, when they came in flocks. Out of the sky comes to us this liquid note, as if the heavens had opened and poured upon us their benediction. How sweet it is to the ear, what music to the heart! And when suddenly a little flock starts up from the wall or fence, how rich and welcome to the eye, long denied its modicum of color, is the blue of their backs! We have had little but artificial tastes and colors and perfumes for so long that the senses seize with avidity these first offerings---we are hungry for them.
It changes the whole aspect of things, when on some raw day the first redwing of the season appears---a vivid bit of color in the bleak swamp, a hopeful and melodious voice breaking the silence of the year. The birds are shy and elusive on their arrival and we have every year to become acquainted again. Even the robins are furtive and silent, flitting in the sheltered swamps; but the middle of March finds them calling to each other in their old jocular way. Drawn by the same subtle influence, the angleworm seems to work toward the surface about the time the robin is thinking of the lawn, till one day they meet as by appointment. If the season is late, the worm retires below where it is less frosty, and the robin takes to the sumac berries, or whatever else he can find, and defers his spring relish a little longer.
Round about there is an awakening as from an enchanted sleep; the drowsy world yawns and stretches. The highhole is in evidence, and his rattling call is calculated to awake the sleepers in that pasture at least. Soon the chipmunk is on the wall, and the woodchuck warily pokes his head from his burrow. This note of the highhole is irrepressibly exuberant and ringing with energy. If it does not prove a tonic to you, nothing else will. He is even more emphatic in his drumming. His lively tattoo goes well with his vigorous call. Time to be up and doing! \textit{Wake up! Wake up! Wake up! Wake up! Wake up!}
Presently the first flock of fox-sparrows drop down from somewhere and go to scratching among the leaves, like so many chickens. The present season a flock of perhaps fifty settled in and around a thicket on March 24th. Their bold clear notes could be heard some distance away, and drew one in that direction. Numbers of them were hopping about, and occasionally a bird would rise to a branch overhead and sing, looking like a hermit-thrush as his back was turned. The place was given over to the sparrows, and never was thicket more tuneful. There was the sound of unceasing revelry---a sylvan and melodious revelry.
At this season the impulse to expression is natural and daily becomes more evident. Even the crow begins to affect music and to show off his accomplishments. But it is \textit{Mlle}. Corbeau, and not M. Reynard, that incites him to this exhibition of vanity. You may hear him in the pine grove, apparently gargling his throat, which is meant for a gay roulade to please the ear of some dusky beauty lingering near and perhaps affecting indifference. This is only a prelude to the astonishing falsetto that sometimes follows, and which, be it hoped, may prove more acceptable to \textit{Mlle}. Corbeau than to our more critical ears. It is very evident something is going on. The large flocks of winter have given away to small and excited bands which keep up a perpetual clamor. It is no surprise, then, some day in March to detect a crow carrying twigs.
At no other time is there such concerted singing among the song-sparrows as in these first days of the arrival of any considerable flocks. From bare fields and brown hedgerows arises this simple and spontaneous expression of joy, a primitive invocation to the goddess Spring, fresh and clear and innocent as the morning itself. As they hop about among the dry weeds, one will now and then pick up a straw and hold it meditatively a moment with some premonition of the nest. Presently they will be flitting among the still leafless brambles and briers with an air of secrecy and importance. Some bright morning in March there comes to the listening ear the song of the purple finch---a wild sweet strain with the abandon of gipsy music, which thrills with its very wildness and unrestraint. Anon Phoebe arrives with dry little voice and familiar swoop after the first incautious fly.
Every season has its characteristic song. More than all others is the voice of the hyla, essentially springlike and to be associated with no other time. For several days there has been an occasional desultory chirp from the woods, when of a sudden, some clear evening, there comes out of the stillness that wonderfully sweet piping of little frogs. Fresh and ringing as child voices, it has, at a distance, a certain rhythm, a soothing cadence, which lulls the ear like the musical patter of rain-drops in summer showers. Put your ear close to one---if you can find him---and the sound is deafening, so loud and shrill it pierces to the very marrow. The small creature sits in some low shrub in the swamp, grasping a twig on either side as with tiny hands, while it inflates its air-sac from time to time and sings the love-song of its race. Heard afar, how soft and pleasing are these answering calls of the hylas which are the very voice of the evening itself.
About the time the hylas begin to sing in chorus, you may look for the appearance of the leopard-frog. He is to be heard at midday in his pond uttering a most deliberate and prolonged snore, evenly and smoothly drawn out, as if his sleep were dreamless and content. Presently there is an answering snore, full as deliberate and serene, from across the pond, followed by long intervals of silence. Very different from this somnolent song of the leopard-frog is the shrilling of garden-toads. Not every one would recognize the solemn and dusty toad of the flower-beds, that flops from under the feet in the dusk, in this brighter colored creature, floating at full length in the shallow water, his air-sac inflated before him like a parti-colored bubble. The shrilling of toads fills the air; they are under a spell, a witchery, which has set them all to chanting this single strain---high-pitched and subdued---with a sort of mild frenzy.
April brings the twittering of tree-swallows, and spreads a tinge of color like a faint red mist over the swamps. This flower of the maple is one whose virtues are seldom sung, as though the blossoms of trees counted for little. Surely the bursting of silver-gray rods into this vivid bloom is an event worthy the muse. It is not only in autumn the red maple graces the swamp. These modest blossoms of the early year---willow, alder, poplar, elm, maple---must have their place in the flower calendar, are worthy a Festival of Trees, to be associated with the song of the hyla.
Anything like an exact flower calendar is out of the question, for much depends on the locality and the season. We look for bloodroot and hepatica to follow arbutus, and yet I have on occasion found bluets several weeks in advance of these. The saxifrage is perhaps quite as early as any, though I have seen the buds of the marsh-marigold about to open on the 25th of March. Much depends on which has the more favorable spot in any locality. In a warm nook, on the 13th of April, bloodroot, hepatica, spring-beauty, early saxifrage, dicentra, wood-and rue-anemones and adder's-tongue, as well as common blue and long-spurred violets, were blooming together in profusion. The saxifrage and bloodroot might, of course, have been seen a week earlier. In the same spot several days later, columbines, miterwort and groundnut, and also sweet white violets, downy yellow and lance-leaved violets, were added to the list and were followed by bellworts and wood-betony. This was in northern New Jersey. Meanwhile I had seen only the common blue violets in the Connecticut Valley, while in eastern Massachusetts the wood-anemones were not in bloom, and the leaf of the columbine had just appeared above the soil. This particular spot was evidently a sort of natural forcing ground where the columbine was made to bloom with the bloodroot. What becomes of your flower calendar here? Looking still for signs of spring, I came full upon the fickle goddess herself.
Before we know it, the migration of warblers has begun and the keen ear detects their thin wiry notes. But this is not so much a sign as it is the fulfilling of prophecies.
\gutchapter{BIRD LIFE}
Walking through bare fields in the chill and birdless world some winter days, it is brought home to us what an \textit{essential} feature of our surroundings the birds are, what a lack there is when they are absent! A certain poverty lies over the earth; the sky is no longer complete without a swift or a martin. Birds are part of the landscape; it is they which animate it. Rarely, when it seems most destitute, a flock of snow-buntings will come swirling over the pasture, like great snowflakes driven before the blast. Again, as the wind will pick up dry snow and blow it over the field, they are off and whirling away, glittering in the pale yellow light of the winter day as they wheel and come to the ground. But their presence has redeemed and softened the austere landscape, made the earth habitable once more and the bare fields friendly and companionable.
The first snipe and plover in the spring remind us what stay-at-homes are we, what wanderers they. We must appear to them but poor mollusks, as they come and go each year on their way from Patagonia to the Arctic Circle. In how many States, in what diversity of climes they are at home! And wherever they may be they get their own living by no one's favor. This prodigious self-reliance affects one as a species of heroism, whereas it is as unconscious as the falling rain.
What familiarity with the elements and with natural features of the earth the migrating birds must acquire---with winds and clouds, with mountain chains and rivers and coast lines! They know the landmarks and guide-posts of two continents and can find their own way. The whistle of curlew, or the honk of wild geese high in the air, seems a greeting out of the clouds from these cosmopolites, to us, sitting rooted to the earth beneath. A flock of wild geese on the wing is no less than an inspiration. When that strong-voiced, stout-hearted company of pioneers pass overhead, our thoughts ascend and sail with them over the roofs of the world. As band after band come into the field of vision---minute glittering specks in the distant blue---to cross the golden sea of the sunset and disappear in the northern twilight, their faint melodious honk is an Orphean strain drawing irresistibly.
A sort of noble madness seizes the birds in the spring, so that an exodus of inconceivable extent takes place toward the North, as though the Pole were a magnet to them. There is a suggestion of epic splendor in this vast impulse, this flight of the feathered tribes of the earth. We may well ask the bobolink, What news from Brazil? and the returning plover, What of the Frozen Sea? What bird-memories do they cherish of these remote regions? It casts a halo of romance about them, that they should thus be at home in lands that may perhaps remain ever unvisited by us.
As if actuated by a sublime faith, in the midst of plenty they arise and depart, drawn ever to the remote solitudes to rear their young, like those citizens who return to their own country that their children may be born in the Fatherland. I do not know if our affinity is greater with the bob-white and the ruffed grouse, which hear no call to depart, or with these nomads of the earth. In the coldest weather, redpolls, crossbills and snow-buntings come to us as to a land of plenty. This is near enough the equator for these hardy birds---this is their genial South. It is pleasant to reflect that the falling mercury, which deprives us of the last of the summer residents, will at the same time bring us some dweller in the far North which perhaps otherwise we should not see.
The advancing season makes itself known through the songsters; they have keener perceptions and receive other intimations than come to us. Day by day, as by appointment, they reappear from Florida, from the Amazon and the Orinoco, and make themselves at home again in northern pastures. I have come to look for the tree-swallows as regularly on the 1st of April, as for the oriole on the 10th of May, as if these were calendar events of real importance. Between the middle of April and the 20th of May lie the incomparable days of the migrating warblers---days of discovery and adventure, when the torpor of indifference slips away, and, like a subtle fire in the blood, is felt that enthusiasm the years do not diminish. When, at night, the small birds pass overhead, their faint silvery ``tseeps'' come out of the silence with a weird suggestion of voices from the unseen world.
Now, the days are full of pleasing suggestions because of little birds shyly flitting with plant-down and with rootlets and dried grasses. Some are unmistakably house-hunting, and the female turns herself about in the crotch of a limb, trying if it be of the right proportions. Interest in bird life centers about this season. This is their life; the rest is a preparation or a waiting. It is only natural there should be an air of secrecy about them now. They are doing their best to conceal and elude, as indeed they must, and this necessity, being uppermost in their minds, becomes evident in their manner.
While I am watching a pair of pewees gather lichens from an old maple for their beautiful shallow nest, the barn-swallows shoot by with mud for their adobe huts. Now and then one pulls from the mud a few fine rootlets---perhaps of the white violet or gold thread growing there---and carries them off. They evidently know their trade. A chestnut warbler appears with some plant-fiber in her bill, and gives a cluck of surprise and disgust to find some one on the ground where she thought to have her secluded and private estate. She hesitates with the down still in her bill; it is evident what she must be thinking; but at length she decides to risk it, and enters the huckleberries. She has, of course, gone into the bushes a long way from the nest. One has great sympathy with the birds in their little circumventions and dissimulations, knowing their tribulations. They live among their numerous foes much as did the early settlers in this land,---that is to say, in spite of them. The weasel, the owl, and the cat---the terrible cat---are appointed to decimate the population of birds.
In the several nests of warblers, I am observing, the thrifty housewife is evidently the home-builder, whereas the male seems to take it upon himself merely to cheer and encourage her. After she has constructed a framework she settles herself in this and builds the wall around her, quite as if she were fitting a garment to herself. Her little ways while so engaged are distinctly feminine. To think that she has never been taught her trade, has perhaps never before fitted such a garment, and she is already deft and expert! The pair seem to take an almost human satisfaction in their home. Now and again they appear to talk it over together. Who can doubt they have some pleasure in this preparation, that they have bird-plans and bird-hopes?
We do not really know, a bird till we have found its nest and seen it at home. When I came upon the nest of the snowbird in the midst of a clearing in the mountains, it was like visiting the house for the first time of one I had known for years---a person of some distinction at that. It was placed high and dry on a tussock in a flaming patch of hawkweed. She had an eye for the practical, and knew better than to put her house where the cellar might be flooded. The four greenish mottled eggs were her one priceless treasure, which was to her as life itself. They were warm, and the whole aspect of the nest was sweet and inviting. It appeared to breathe some feminine element, so dainty was it, so begirt with flowers.
A humming-bird's nest that I have been watching the present season is placed on a pitch pinecone, and appears to a casual view to be the cone itself. It seems as if the bird had it in mind to simulate this or she would not have chosen such a peculiar site, for it affords no advantage from a structural point of view. If this be true it is a departure from all traditions, and shows a bird of some character and originality. In other respects it is like any humming-bird's nest---one of the most exquisite of all natural objects.
[Illustration: BIRD LIFE COPYRIGHT, 1907, BY RUDOLF EICKEMEYER]
In the course of ten days, in place of one of the eggs appeared a small and peculiarly homely object which resembled a spider as much as anything. Two days later the other egg was hatched. At this stage the bills of the young birds were very short, but day by day they lengthened and grew more needlelike. At length one bird opened its minute and shining black eyes for the first time. The other fell from the nest on the following day, before its eyes were opened, so that all it had known of life was the consciousness of hunger.
The female fed her young with much less frequency than do other birds. When so engaged she perched upon the rim of the nest and pumped the food into them after the manner of her kind. As she flew to and fro, she appeared to move always at the same speed, as if her wings were keyed to a definite rate of vibration and could not vary. Gradually the young bird emerged from its gruesome infancy, and day by day became more sylphlike. Heavy winds prevailed, but the diminutive cradle remained unharmed, though branches were everywhere blown from the trees. So was the wind tempered in that case at least, till one day the sylph left the nest, as a thistle-down might detach itself and sail away on the breeze.
Birds have their home-trees, and one whose traditions are of the pine is not drawn to build in hardwoods. The woodthrush is associated with the dogwood, as the catbird with the smilax, and the oriole with the elm. There are ancient apple orchards which have come to serve only the bees and the birds; but what temples of music in May with the hum of bees, and in June with the song of wrens! At this season you cannot do better than to set out for one of these old-time orchards, neglected of man and favored of heaven.
The virile hum of honey-and carpenter-bees descends from the flowery summits to the listener beneath, the contented music of a race dwelling overhead and nearer the skies than we. It is such an apple---Baldwin, pippin, or russet---gnarled and archaic in trunk and a bower of beauty above, which becomes the home-tree of that feathered gnome, the house-wren, a sprightly elf, living in the depths of a tree trunk and yet full to the brim of song. He may derive of the flowing sap some genial trait and takes to the apple as a swift to the chimney, or a redwing to the swamp. After the cold rains of late May have taken off the blossoms and with them the bees, the place becomes melodious with his song. It is thenceforth his estate, and he dominates it with his small personality. With him his house is his castle, and in true medieval fashion he barricades his door. Within is snug enough, but without it has a feudal and forbidding look,---a formidable barrier of twigs, erected perhaps against the house-sparrow or for fear the robber-owl may peer too closely.
In this choice of a building site the bird reveals something of itself. Contrast the wren with the phoebe, a cliff-dweller, loving the contact of the ledge itself better than any bush or tree. The song-sparrow has an eye for the wild rose and the yellow warbler for apple blossoms, but the phoebe has some austere traits which make the stern rock more congenial to her. Some birds are architects, others builders merely. The vireos are a family of artists, whereas the improvident cuckoo will not even lay a proper floor to her nest.
A look into some nests is a glance at the domestic life of a savage people, and yet we find the virtues we most esteem---patience, perseverance and fortitude. Hour in and hour out the faithful kingfisher flies from the nest to the fishing-ground, bringing each time a small fish. He is a primitive and industrious fisherman who gets an honest living by his skill and supports his family, yet he is under ban, while the dilettante whips the stream for his pleasure. The hoarse rattle of the kingfisher is an altogether barbarous chant with which he beguiles himself as with a hunting song. His is an austere temperament with no room for melody. But that he returns every year to the same nest---the ancestral hall---is evidence of some more domestic and kindly trait in his character.
This nest is an excavation in the sand, high in a bluff, and is perhaps five feet deep,---a true cave, and its inmate a cave-dweller. We have thus both cave-and cliff-dwellers among us---primitive states of man still exemplified by birds. The cave-dweller had something in common with the kingfisher, which led him to burrow in the earth for a home.
That was truly an aboriginal abode which I came upon in the spruce woods in a region of perpetual twilight. The somber spruce was relieved only by some veteran yellow birches and by ghostly patches of false miterwort on a projecting ledge. High in a birch was a small hole from which the scarlet crown and chin of a sapsucker appeared in view, as the bird thrust out his head and looked inquiringly about. A harsh imperious call brought the female, who clung to the trunk till the male came out, whereupon she dived into the hole herself, while he in turn went foraging.
Whenever the pair were absent from the nest the insatiable young were heard squealing within. It was a fearsome place that was home to these young savages, a room within a tower, lighted by a single small window far above. To think of being born and raised in the dark heart of a tree! The old birds called to each other from time to time as they hunted over the neighborhood, and their speech was as that of wild men, the very rudiments of language---rude, uncouth and evidently of few words. But, as with the speech of savages, these words were doubtless packed with meaning, whole sentences and paragraphs in themselves, of hard and practical import. Now and then the scarlet crown appeared at the entrance of the dark wigwam. Any lurking foe would be espied from there. Probably not a twig moved below but it was noticed.
While the robin and the bluebird have come to wear a half domestic look, the woodpecker is the untutored savage still. As an Indian remains an Indian, a woodpecker remains a woodpecker. When he comes to the orchard he is an interloper from the forest. He carries the stamp of the wilderness with him. Defiance is in the poise of his head; his attitude is a challenge.
The life of owls and hawks is completely \textit{savage}---a fierce, carnivorous, terrible existence which no circumstance can affect. Regarding their young with solemn ferocity, their fierce natures are not to be modified or softened in the least. A little red owl having her nest in the heart of a weeping willow, lived so secluded a life her presence was hardly suspected till she was discovered by the smaller birds dozing in a cedar. Some days later she appeared at dusk with four young owls, which she fed on large beetles. The owlets remained perched in a line on the fence while the old bird in ghostly silence departed into the night in search of food. It was wonderful to see what excess of dignity and ferocity was expressed in the personality of these little birds. As well have expected an Iroquois brave to ask for quarter. Approach them and they were on the defensive with all the tricks of appearance---staring eyes, snapping bill and uncanny wavering motion of the head. Like some phantom creature, the old bird came and went, leaping noiselessly into the darkness and reappearing as by magic.
The owlets took their beetles with avidity, swallowing them whole and gulping and gagging in the process in a manner indicative of discomfit rather than any satisfaction over the meal. Once the mother brought what in the darkness appeared to be a small mouse, and this too was swallowed by one little owl, but only after heroic and protracted efforts.
It was no great matter on the following day to gain the confidence of the young owls to a slight degree. But food was the only bond of affinity. So long as I fed them they were content to perch on my finger, fierce and solemn little ruffians, and devour bits of raw meat. Their manners remained sullen and forbidding, though they never refused to eat. Soon they lost even this slight contact with our world and disappeared into their own---the nocturnal and barbarous world of the owl.
Every year there is fresh evidence that the course of true love runs far from smoothly with the birds. A pair of yellow-throated vireos built no less than three nests one season and only succeeded in occupying the last. There were two suitors for the affection of the female, and they fought continually. The rejected lover harassed the pair while at work gathering material, and that he twice stole a march on them and actually tore down the nest appears from circumstantial evidence.
Great secrecy was observed in constructing the third nest, and the rejected one no longer harassed them. Either he had transferred his affections or been fairly vanquished. Life was strenuous and impassioned with these little birds, but see what constancy and perseverance! Fancy having two houses torn down, after completing them with your own hands, and having the courage to build still a third! There is something of the pioneer and frontiersman in this. The offspring of this pair were the children of vigorous and romantic times, and should have inherited some heroic traits.
Even if all goes well otherwise, the sanctity of the nest is liable to be profaned by the cowbird. This spring was an unusually favorable one for them. I noticed the least flycatcher and the Maryland yellow-throat mothering young cowbirds, and many vireos and warblers so engaged. It is a wary caution that leads the cowbird to choose the smaller birds for her victims.
It would be hard to say which of all the foster-mothers is the more solicitous of her charge. Now it appears to be the redeye, and again the chipping-sparrow. All alike are bent on bringing the birdling to maturity as though it were of first importance. \textit{That} cowbird shall thrive though the heavens fall. The attention seems to be in proportion to the egregious demands of the foundling. Here at least is a waif well cared for, an upstart that takes precedence over the true and lawful heirs. Another year this same adventuress will invade the nests of her adopted sisters.
The yellow warbler is perhaps the oftenest chosen. Accessible and easily found, the nest is a beautiful cup-shaped structure lodged in the fork of a fruit tree, with perchance a spray of blossoms just over it---a house of silk, a satin bower. How awkward and uncouth must the cowbird appear squatting on this fragile silken thing to lay her eggs! Doubtless she watches the yellow bird stripping the dry grass stems and gathering the pappus of last year's cattails; squats low in the grass and looks all unconcerned while she marks the tree to which the fluffy material is carried, and bides her time till the nest is ready. Strange that she should never discover in herself the home-making instinct, for even nomads have their tents. Stranger still she should never once wish to undertake the duties of motherhood.
For a time, perhaps, the young cowbird is influenced by the habit of the bird that happens to mother it, whether this be a ground-sparrow or a tree-loving flycatcher. But it grows up a cowbird with all the inheritance of that peculiar tribe, and its brief contact with a superior race leaves no impress upon it.
In spite of cowbirds and the exigencies of life the woods are full of young birds, their tails not yet grown. This is their childhood---a brief one---as the days in the nest were their infancy. They are exacting children, yet they do not clamor to be amused, but only to be fed. I have seen a young chipping-sparrow, its tail half grown, showing how recently it was from the nest, pick up a straw and carry it about. So early does the maternal instinct show itself. This straw was its doll-baby, the only plaything it could know, and this its solemn play. There is a mild and innocent expression about young birds, as there is in the faces of children, apparent to a keen vision only. They have yet to be hardened by experience and vicissitude. The countenances of the old take on an astute and alert expression. These young black-and-white creepers and chestnut warblers, now shifting for themselves for the first time, come about with gentle confidence. They creep and flit through the trees, coming nearer and nearer, until you look directly into their small innocent faces and could put your hand upon them. Then as it would seem they were about to descend like blessings upon your head, they withdraw and recede from view into the wilderness of leaves where only your thoughts may follow them.
\gutchapter{SONGS OF THE WOODS}
We are drawn ever by the voices of birds. Even such as might be called monotonous and unmelodious are none the less significant and welcome. The fine lisping notes of warblers, as they industriously hunt for their food, seem expressive of the contentment of their minds. All over the hemlock swamp I hear the voices of black-throated green warblers. Not one may appear in view, but for hours together their musical conversation continues in the treetops. From somewhere in the branches above comes the call of a nuthatch, his speech wholly dissimilar from the rest, as if he might be an inhabitant of a very different world. Almost in the ear sounds the thin wiry note of a black-and-white creeper, as he winds around the trunk of a pine and approaches with his accustomed sociability. High above the others, the trill of the pine-warbler rings clear and sweet---a more resonant instrument surely. These voices all affect us agreeably, and bring us in immediate contact with their world and with wood life. They do not touch our world, however, nor set in motion the delicate mechanism of the emotions. But let a bluebird pass overhead all unseen, warbling his celestial ``Pure! Pure! Pure!''---let that significant note fall on the ear and for reasons unknown it sinks into the soul, into the abyss of \textit{feeling}, and this as mysteriously rises in a delicious flood to the surface. Whence has the bluebird his power, that by the mere \textit{quality} of tone he can exert this spell?
Some bird voices are so positive, so emphatically cheerful, that one never hears them without feeling better for it. The chickadee in the winter woods is an instance of this. If you feel dreary, \textit{he} does not. Nothing can dampen his spirits. He hopped out of the nest a cheery little chap, and it is never otherwise with him. In all his days he has never had a regret, never transgressed any law, never been unhappy. The voice of the chewink, too, is eminently sane, a mild, buoyant utterance indicative of an even disposition. He is never more hopeful, nor less so, but always exactly the same. Perhaps the birds have not what we call \textit{feeling}, but if not, why do they express themselves? What else would prompt these songs? The clear sweet call of the bob-white is full of hope, and there is a quality of tenderness in this voice. One must believe it the outcome of the disposition and character of the bird, of some refinement of feeling; just as the raucous call of the English pheasant expresses grossness and density, and the quailing of the hawk pure savagery.
If we may speak of the temperament of birds, the thrushes must be accorded the religious temperament. They are the inspired singers; their songs are eminently \textit{sacred} music. The woodthrush appears to be actuated by other than merely commonplace and personal motives. Upon him the forest has laid its spell, and he must deliver its message. He flits about with a dignity befitting his high calling. There is no abandon in his song; he does not sing about himself---has no moods---but repeats his solemn chant. It breaks the stillness of the woods with a sort of challenge to the gay fields beyond, like the call of the muezzin from the minarets of the mosque---a summons to all twittering sparrows and chattering squirrels to be silent and listen. That such fervor, such solemnity and beauty of utterance should be unconscious and unwitting seems incredible. Stand and listen to the hermit-thrush and see if you can think idle thoughts. You must hear his message and feel the spirit of his invocation---the voice of one crying in the wilderness.
Why is the hermit moved to be thus didactic, while in the fields beyond the field-sparrow lightly trills and the merry bobolink continually bubbles over with song? Such merry jingles, such uncontrollable outbursts of melody, such a rippling, bubbling medley as comes up from the meadows, while the thrush solemnly intones within the twilight shades of the woods! Surely in view of this we may speak of the temperament and the personality of birds. If the bobolink's medley is not evidence of a light heart, then are appearances deceptive indeed. Care rests easily---if at all---upon his hopeful nature, but the burden of his song is quite as well worth heeding as is that of the thrush. One is lyric, the other didactic. The bobolink communicates his joyous and irrepressible spirits, as the thrush his serene exaltation. It is certain the wood birds are of a different temperament from the field birds. Either they are influenced to their prevailing moods by their environments, or they are attracted thereto by their own peculiarities, as men are drawn to solitude or society.
The hidden, the subtle, find voice in the veery. His is perhaps the most spiritual strain of all, himself the high priest of the mystic lore of the forest. Of the thrush family he is the consecrated member, as the robin is the worldling among them. I believe there is no other bird voice so mysterious; so impersonal is it, so spiritlike, it appears to emanate from a world of higher motive than ours. In the devotional strain of the hermit, the forest prayer is breathed on the mountains. No hymn could be less impassioned, less material, more truly \textit{spiritual} than the song of this thrush; it is nearest the speech of angels. Of all instruments the organ and the harp are alone capable of producing any such effect. On rare occasions I have heard the veery indulge in a reverie never to be forgotten. It appeared to be wholly inspired and original as though the bird were improvising like some Abt Vogler at his organ, rearing a palace of music. The motive was complex and involved, and sung so \textit{pianissimo} as to be just audible, like the love-song of the catbird, a rapt utterance which admitted one to the sacred arcana of Nature.
It is not unprecedented for a bird to depart thus from its usual song and to improvise. You may detect even the jay in this mood, though it is wholly imitative with him. The love-song of the catbird and the autumn reverie of the song-sparrow are perhaps the best instances. I am not yet wholly familiar with the songs of the robin. It appears he is still studying music, and adds a phrase or varies a theme occasionally. He is the most romantic of the thrushes; his song is more personal and less spiritual than the others. When, in early spring, the robins sing together at sundown, there is an exquisite tenderness in their notes which accords with the sweet youthfulness of the year. It is later in the season, when his mate sits upon the nest, that the robin rises to the heights of lyric beauty and pours out his soul from the top of the tallest maple in the swamp,---a brave sweet love-song, sung with dignity and without hesitation, that all his world may hear.
At dawn he is moved a little more to the rapt and religious expression of the thrushes. Something there is in the solemnity of that hour which touches the hearts of all little birds. What it is we shall perhaps never know; shall never know enough of bird life to understand what emotions they may have which so powerfully sway them and become evident in their voices. The evidence is there; the cause is to be inferred. While the birds are everywhere more or less affected by the approaching day and give voice to their feelings, there appear to be musical centers in the bird world in which the expression is more concerted than in other localities,---favored sections where this hymn to Apollo is memorable indeed and hardly to be described. It is a great chant with all its solemnity, all its impressiveness.
Beginning with the desultory calls of wood-pewees, it is taken up by song-sparrows, robins and catbirds, dominated by the devotional song of the woodthrush who appears to act as chorister. Birds seem to congregate from near and far and to inspire one another to unusual efforts. The volume and stateliness of this chant, so measured and rhythmical, carries with it vibrations of power and cannot fail to communicate its influence to the listener, be he bird or man. Here is a multitude of birds actuated by a unity of purpose, impelled by a single motive, and though every one sings his own song, the myriad voices blend in one concordant whole. To arouse suddenly from a sound sleep in the woods at dawn while this chant is in progress, is like awakening in another sphere, where sings the choir celestial. We slip from sleep into the heaven of song, and it requires another awakening to bring us to consciousness of this actual world about us.
They are the troubadours these birds, the wanderers whose souls are in their voices. What bold romantic singers are the cardinal and the rose-breasted grosbeak---the lords of song! When the cardinal comes North he appears to feel out of his element and modestly withdraws. But in the South he dominates the swamp and adjoining cotton-fields with his rollicking, melodious voice. A gay minstrel, he compels attention. These voices of the cypress swamp are clear and bright in contrast with their dismal surroundings. The bell-like note of the tufted titmouse in the treetops, and the brave, cheery song of the Carolina wren lighten those fearsome shades. The wren carries his sunshine with him. There is no minor in his song; he is never discouraged, any more than the chickadee. Day after day that voice rings true---all's well with the world. Brave voices singing in the wilderness, they lighten vaster shades than any they know of, sound their note of courage and well-being for other ears than theirs. What blessed transformation from the songless ages---from that slimy reptilian world where was no music, no song---to this unpaid minstrelsy of the woods and fields! They have served us these many years---the sweet singers, the true birds of paradise, with power to lift us from our dull, unmelodious thoughts into their harmonious world.
As I was following the course of a mountain stream through the leafless woods early in April, the silence was broken by a strange musical alarm. It was the Louisiana water-thrush, but might have been the pipes of Pan, so wild and woodland was it. The first notes were high and startlingly loud and clear, while the song descended the scale and became softer and softer till it died away. This is one of the bird voices that are untamed, that seem to belong to impersonal Nature. It is wholly savage---a piece of the wilderness, untouched by the presence of man. These voices do not strike the human and sympathetic chords, but ally one with the wilderness. Such are the cry of the loon, the melody of the ruby kinglet and the song of the winter wren. The kinglet's song has a cadence unlike any other, reminding one of water murmuring underground, and for some reason a classic suggestion, as of faun and satyr. It is more truly \textit{sylvan} than any other---sylvan in the old Greek sense, so elusive and shy it is, so mysterious.
Such voices give no evidence of self-consciousness; they are as impersonal as the winds or as the murmuring stream. But with the catbird, the thrasher and the mocking-bird, pre-eminently vocalists, there is a set and declamatory method which has the appearance of affectation. Their songs are brilliant and elaborately phrased, but they lack spontaneity, and in listening to them one wishes they had put their powers to a different use. The thrasher is particularly self-conscious and stagey, and yet he has a glorious voice. No bird has a finer \textit{quality} of tone than he shows in some of his notes---clear, mellow, vibratory as in the voices of really great tenors. It is that quality which Nature alone supplies and no cultivation nor perfection of method can give. When he speaks to his mate in an undertone his voice would melt a heart of stone. There is a time, however, when the catbird rises above any suspicion of self-consciousness and is transported, and the listener with him, in a reverie of exceeding beauty. It is a wondrous love-song, an incomparable madrigal, blending with the morning sunshine and the first green leaves of the alders, soft and low as faint murmurings of a stream, a fluid melody uttered for chosen ears.
All too soon the only bird notes are those of the redeye and the pewee. For music we have the tree-toads and cicada. The sounds of this season are rhythmic and vibratory---virile songs of the year's manhood---the mature year, lusty and vigorous. But how soon they dwindle and wane, despite this sonorous protestation, grow silent and slip into the sear and yellow, and thence into the leafless, the glittering, the sublime aspects of winter! The last of September brings with it just a reminder of the sweet and winsome sounds of spring. At this season the song-sparrow indulges in a wonderfully ecstatic reverie, a bit of wild melody charged with feeling as of some larger consciousness, some tribal memories of that musical race, now finding voice in the waning year. So continuous and varied is the theme, and withal so complex and involved as compared with his usual simple and positive lay, that one must look at him twice to make sure it is he, and not some unknown minstrel from a distant shore.
Insects are the autumn singers and take the place of birds and frogs. The crickets are as musical in their way as the thrush family, though provided with but indifferent instruments. When you consider that these crickets and locusts \textit{will} express themselves---will fill the day with song---though they are without vocal organs and must perforce do with legs and wings instead, you must respect them as musicians. It is a distinctly aboriginal music as compared with that of the birds, as tom-toms and pipes are to violins and cellos. And yet it is rhythmic withal and not wanting in sweetness. Contrast these merry crickets with the silent spider. There is no song in the annals of her race. She is unsocial and unmusical like the savage birds of prey. Yet before bees and birds had appeared on the earth there were crickets chirping. Theirs is the most ancient chant of the world---the Song of Sex.
Autumn nights are melodious with a voice, which in the distance is so like that of the hyla of early spring, though softer and more throbbing, that it is often mistaken for a kind of tree-toad. Heard near at hand it is singularly clear and almost bell-like, though ventriloquial in its elusiveness and difficult to locate, for as you approach, it ceases and is taken up by another a short distance away. Even when standing directly in front of it, it appears to come from several directions. It was only after prowling the woods with a lantern that I discovered the identity of the sweet singer, a small insect of a pale green hue, not over an inch in length, looking like a sort of locust, though classed with the crickets. The translucent wings are of a delicate ivory-white and the antennae very long.
This cricket was hanging to the edge of a grape leaf when the rays of the lantern fell upon him. He perhaps took it for moonlight, for on a sudden the wings were erected until at right angles to the body, and then, as it were automatically, and with the precision of a pendulum, they moved to and fro, partly crossing their bases and thus scraping the veins of the middle portion---and the mysterious singer of the night stood revealed.
The quality of the tone---the timbre---suggests the sound made by rubbing the rim of a glass bowl, the horny plate of the wing giving it great resonance. It appears to be pitched to A below middle C, though some may be A sharp or even B. The overtones make it difficult to determine the pitch. The chirping keeps up a good part of the night, and in the wee small hours takes on an uncertain quaver, as if the little singer had fallen asleep and were droning drowsily in its slumbers.
An insect which may be the same one---certainly an allied species---has a day-song somewhat different from this song of the night, a shrilling in place of a chirp. This is made by elevating the wings in the same manner as at night, but instead of rubbing them one across the other in regular time, they are rapidly and continuously vibrated like an electric bell. The rapidity of the vibration raises the pitch, though the quality of the tone is but little different.
There is in this day-song no suggestion of the blistering, feverish shrill of the dog-day cicada, but a far-off dreamy sound. A little before sunset it gradually gives way to that of the night. Day inevitably inspires one song and night another, as if these reacted to bring out two sets of emotions. And yet there is but one theme: the minstrels sing always of that, but serenade the fair one after one fashion by day and more serenely by the light of the stars. She, having apparently no ears, hears none the less, and perhaps detects variations in this monotonous ditty and even distinguishes the fine quality of some particular voice---some clearness of tone, some pathetic \textit{tremulo} indicative of a cricket's feelings. For is not this a song-festival of all the grasshoppers? I noticed a common short-horned grasshopper stridulating in the sunshine, which he did by taking short flights and rapidly opening and shutting his wings like an accordion. This produced a series of dry, crackling sounds as the wing was scraped against the wing-cover. After thus exhibiting his powers, a female at length came from some little distance and lit beside him, as much as to say, ``If you can sing like that I am yours forevermore.''
One feels some sympathy with these sweet singers of the fields in knowing what a little life is theirs, how short is the span. For the most part they have but a few months to sport in the sunshine. This epithalamium is at the same time a requiem. In October it rises, a universal threnody, the death-song of the insects. Over all the land, wasps and bees and butterflies fall like leaves. Death overtakes them on the wing. They lie down to sleep, like travelers lost in the snow.
\gutchapter{WILD GARDENS}
Improvement easily becomes an affectation, from which all healthy natures suffer periodic reactions that take them to the mountains and the forest, to those primeval estates loved of wild bees, of the phoebe and the wren. One feels a sympathy with those renegade plants known as garden escapes---star of Bethlehem, bouncing-bet, and the rest---which have run away from the garden for the freedom of the woods and highways. The conventionalities of spade and hoe are odious to them. They wander far from the assemblage of the elect; they will live wild and free, these Philistines, following the open road wherever it may lead, with a sort of tramp instinct. Even the staid and domestic apple will break away from the fold to seek the unregenerate society of the pastures.
The hemlock woods, the meadow and the bog are wild gardens which require no cultivating themselves, but only a certain cultivation and appreciation in us, which they repay with gentle and unfailing interest year after year. What we get from them will depend on what we take to them.
Flowers are nothing away from their haunts. We must have the field in which the clover blossomed---bees and all, the cranberry-bog, the mossy bank of the violet, the white birch on which the polyporus grew. Take, for example, the clintonia, solitary amidst fallen spruce logs on the mountain slope. Imagine it transferred to a trim garden! If you have really seen that flower of the solitudes, you have seen the mossy rock overhanging it, the spruce cones lying thick about; sniffed the balsam and heard the veery on the mountain. Or consider this mountain sheep pasture with its clumps of stunted spruce and balsam, its scattered boulders and patches of sensitive fern, its reddening sorrel and running cinquefoil; bluets lie over the ground like a light fall of snow; pasture stones are incrusted with parmelias and set in a frame of hair-cap moss and reindeer lichens, incomparable mosaics; wild strawberries nestle among dainty speedwells, half hidden under the bent grass. It is a whole, an homogeneous piece of work, like a tapestry. There is not a bog-rush nor a buttercup to be spared.
From the first fragrant spicebush to the last witch-hazel, no cultivated shrub is to be compared with them, for the virtue of the wild is not to be transplanted and is never imprisoned in flower-beds. These shrubs of the pasture have a personality derived from immemorial contact with the virgin and uncultivated soil. They have been nourished by the very juices of earth and by the bone and sinew of the mountains. If you would have the barberry, you must move the pasture itself. It is of wild gardens solely, an untamed and untamable beauty. And so it is with the dogwood, for what is this but sunshine in the May woods---rifts of light breaking here and there through the overarching green of oak and tulip trees? It were as easy to catch sunbeams as to carry this away.
The mountain is the mother of these wild gardens; a vigorous dame to bring forth so gentle a brood---as the slopes of Vesuvius produce a mellow wine which has taken only a kindly warmth from the raging heart of the volcano. All her fairest virtues have blossomed in her children; her graces would remain unsuspected but for them. Let the gods but fling down a bit of rock anywhere and presently, after a few ages, it shall dissolve into violets and anemones. Grind it to powder by the wayside and you have only made it into thistles and burdock; scatter it over the fields and it becomes daisies and sunflowers.
Imperceptibly, granite melts at its outer edge into a fringe of dicksonia and wild rose. Limestone will bring forth a richer garden than sandstone, as though, like the rock-maple, it had more sweetness in its veins than another. Some of the most delightful gardens arise from disintegrating basalt. Perchance this rock retains a little of its old volcanic heat and has more of the finer graces in its make-up than that which was coldly laid down under water. Fiery lava, tempered and mollified by Time, has become kindly and amenable. Where was only desolation, after countless days the dicentra hangs out its white flags in truce to the warring elements. The sand hillocks of the terminal moraine are the chosen land of mountain laurel, and there are untold acres where this constitutes almost the sole undergrowth. What a hanging garden, when, on a level with the eye, one continuous bloom spreads through the twilight of the woods---the single buds like miniature urns of rose quartz so delicately are they sculptured,---here a warm rosy tint and there a ghostly pallid blossom. This soil, the detritus of glacial torrents, despite its many washings, has not given up all its gold, but is rich in arbutus and in \textit{pedata} violets. It is, after all, granite, the mother-lode of the earth; granite after endless transmutations but still retaining some of its virtues.
To the first flowers belongs a charm, the most exquisite of any, something tender and appealing, as though they enshrined the fairest virtues of the year---its modesty, its purity, its sweetness---in violets, anemones and bloodroot. This charm, so elusive, has never been described, nor shall be indeed. It is like music which is a language in itself and will bear no translation. The bee must approach these with some humility and more gentleness than is shown to the sturdy blossoms of summer. They are eminently the ``gentle race'' of flowers, born in the enchanted time.
We go with hungry eyes at this season. By midsummer we have been well feasted and no longer see individual blossoms so much as masses of bloom. Bloodroot and hepatica are like the dewdrops of early morning which disappear before the sun. They can be found just once in a year; after that they will not appear the same. It is cheering to come upon such a fair company of spring beauty where but a few days since were none; to enter a stretch of woodland and find it populous with these friends of a lifetime, now returned to their old haunts. We do not commonly reflect that they have been under the snow all the while. Scattered among them, the anemones lie in drifts, like a late flurry of snow and quite as evanescent, lingering in the shadows only. These are the delicate children of April; May is their foster-mother. Contact with them is like the glimpse of a \textit{spirituelle} face. But the adder's-tongue which nestles by the brook has more fire in its veins than the rest. Its spotted leaves give it an almost feline beauty as it droops with the southern languor of the lily.
[Illustration: WILD GARDENS COPYRIGHT, 1908, BY RUDOLF EICKEMEYER]
Serenity dwells with the woodland flowers. There is about them some subtle refinement and exclusiveness. They appear fit symbols of lowliness and modesty. A strip of woodland beside the turnpike is like an ancient chapel left amid the din and hubbub of city streets. The sturdier plants, both coarse and gay, halt at the edge of the wood. Within, the light is subdued; nothing obtrudes upon the eye or ear. It is obvious that the cathedral had its origin in the forest. What a fair and devout congregation has jack-in-the-pulpit, where the Canada violet stands side by side with the medeola and the painted trillium. The medeola declines its unfertilized flower, so that its maiden life is hid from view beneath the tri-leaved canopy, and only in its mature and matronly days does it begin to ascend and take a position where the seed shall crown the plant and be in evidence. From what insect despoiler is this shy virgin so carefully hid?
It seems as if the light that penetrates these woods has undergone a change, or been deprived of some of its rays, so that the wood flowers are nourished by a finer food than the rest, as with ambrosia. It is perhaps the subdued light which inspires a certain solemn and hymn-like quality in the notes of wood birds, as in the thrushes and the altogether didactic tone of the redeye. There is here none of that self-assertiveness among the flowers that is to be observed among certain groups of plants; the competitive spirit is lacking. Solomon's-seal, bellworts and twisted-stalk, like medeola, are rather at pains to conceal themselves. There is no self-advertising among them. What could be more unassuming than goldthread and wood-sorrel? They live close to the soil of which they are the offspring---a rich, odorous soil, black with the accumulated nutriment of centuries. He must be in hot haste indeed who treads on a patch of mountain wood-sorrel, such is its mute, appealing beauty. It holds the eye and stays the foot of every saunterer in the woods.
But follow the by-roads in early summer and you shall have very different company. It is here you will find the sturdy travelers, who will go the length of any road in all weathers; and there are none more cheerful and uncomplaining. They have no fault to find; the world suits them very well. You must be prepared to greet mullein and burdock as equals. Here on the road they are as good as any; they hobnob with the rose. Wild carrot borders the dusty lanes with a fringe of lacework---a real lace from the deft hand of Nature. There is no brighter gold than the St.-John's-wort, albeit it will not pass current in the town.
The winds sow the fairest hedge by the roadside---the winds and the birds; it seems that they take kindly to these wayfarers. They are the good fairies who plant elder and blackberry and scatter the wild rose. Timothy and redtop and witch-grass are the very children of AEolus. The pollen-bearing wind mothers the grass and plantain; the seed-carrying wind distributes the thistle and willow. Birds are very willing to carry cherry-pits provided they may have the cherry for their trouble.
The breeze comes laden with thistle-down, such fragile craft embark on these untried seas with all sails set. The story of such a seed would read like a fairy tale. Has not the wind whispered daily to it as its silken sail was spread? And the seed has tugged at its moorings like any boat till these were loosed and she was off, beating in and out among the high blueberries and shadbushes of the pastures, at last sailing clear of all such reefs and ascending in air to drift out into the open. How it rises and falls on the currents, like a ship riding the long swells of the sea; again it drives free before the wind to settle down at last in some pasture. If, perchance, such a seed fall on stony ground it is no great matter. The marvelous silken sail will now fall away, for the craft has reached port, no more forever to sail these seas. On occasion one is caught in a spider's web, whereupon the spider comes out to see what luck. Evidently all is not fish that comes to her net. But the self-reliant crane's bill looks neither to bird nor beast nor again to the winds of heaven, for it does its own planting, flinging the seeds away with almost an intelligent and conscious action.
This relation between the wind and the plants of the field is an agreeable stimulus to the imagination, in a matter-of-fact day when fairies are not so common as of old. Consider how the breezes have blown the pollen of the pine and later are to help carry the seed. They thus serve the trees of the forest and the grass of the prairie. These same winds urge the fruit that it should leave the parent tree. ``Come, follow us!'' say they, and first gently draw, then roughly compel, till the apple falls. They whisper all through the summer to the leaves so green, and at length, on October days, draw them irresistibly.
Verily of wild gardens there is no end; our estates are without number. But among them all the mountain is unique, for to ascend is like going northward, and at the same time to reverse the season. One, which I climbed the middle of June, is little more than four thousand feet, and yet, whereas in the valley there were daisies and wild carrot, on the summit the wild red cherry was just in bloom. In that short distance one walked upward---or rather backward---from the middle of June to late April. Another four thousand feet would have carried one back into the depths of winter. The seasons are thus with us throughout the summer; we have only to go up in the air after them.
Warblers were nesting on the mountain slopes which would otherwise hardly have been found at that season this side of Canada, such as the black-throated blue, the magnolia and myrtle. The winter wren was fairly abundant, and on the very summit a snowbird had her nest. About half way up, the butternuts of the ravine gave way to spruce and balsam. As the ascent continued, mountain-maple and mountain-ash suggested higher latitudes. But what impressed one most was the subtle recession to the early year. The seasons having fairly begun to revolve, it was as though some power were slowly turning them back again.
Some hundred feet or more up the face of an overhanging cliff, a bower of columbines hung out into the grim ravine. They were clustered just under the brink, gems of the first water in a rude setting. The red blossoms glowed faintly against the bald cliff like rubies set in the walls of a rock temple. From under the roots of the clinging spruce a small stream slid like molten glass over the escarpment above and burst into spray, gently undulating like a fine veil, as it descended to the pool below with the dominant and strenuous song of the waterfall.
Probably honey bees do not leave their mountain meadows for this dim twilight region, though they may possibly become acquainted with these hanging gardens on their way to some bee-tree in the woods. It is left to the wandering bumblebee to fertilize most woodland flowers, and in the case of the columbine, perhaps to the humming-bird. On the same cliff were tufts of the alpine woodsia and dense patches of rock-brake---but these stand in no need of the bee.
When, at some three thousand feet, wood-anemones were blooming, summer slipped gently away and April took its place. It seemed quite natural then to find adder's-tongue and to see wake-robins and bunchberry everywhere. The last part of the ascent might have been through a swamp, so strong was the suggestion of swamp life. Spagnum grew in places along the trail, and the fern moss was in evidence on the rocks. False hellebore was abundant, and on the very top stood a poison sumac---a typical bog plant. Yet the summit was rocky and covered for the most part with stunted balsam as thickly matted together as a hedge. The mountain pokes its cold head up into the clouds, and is continually refreshed by the dews of heaven. In some unaccountable manner the swamp plants, as if guided by instinct, ascend and find their natural environment at the top.
When I descended, it was to leave spring behind with every step, not again to meet her in that year.
\gutchapter{WEEDS}
A strange analogy exists between plant life and some aspects of human life. The same stern necessity of the survival of the fittest---physical in one, and in the other mental and spiritual---seems to inhere in both. Among the weeds, competition is the dominant note, as it is in our world. In some higher circles it is sounded faintly, while untold legions of the more delicate plants---like sensitive natures---are driven to the wall, unequal to the struggle.
There are weeds whose ways suggest the arrogant monopoly, and others which recall the parasites of society. The dodder fastens upon its victim and the bindweed throttles the innocent. To withstand the severe competition of pigweed and ragweed, the garden patch requires your energy, plus its own; and the more war is waged upon these, the more does it seem to encourage the purslane, which thrives like a freebooter in this sort of warfare.
One can imagine no more irrepressible rabble than these weeds of the garden. They seem possessed almost of a conscious life, and to push and shove and scramble for place like a hard-headed, thick-skinned, piratical crew. Many of them are immigrants, the riffraff of Europe, who have found their way to our shores, some to become good citizens, and others to remain pestilent anarchists, opposed to the law and order of the kitchen-garden and rebelling against all government by the hoe. Yet how happy are the bob-whites and the tree-sparrows for the poor seeds of the ragweed when the snow lies deep. They repair to these as to an unfailing larder, which may lie between them and starvation at such times. Through some kind providence, the seeds remain into the winter to be shaken down upon the snow. The obnoxious weed of summer rises to the dignity of usefulness and becomes a food plant---grain and corn to the hungry birds.
There are weeds and there are weeds. So much depends upon the point of view; is it a weed on the lawn, or is the lawn but a background for the dandelions which star the grass? What bright day-stars are these which beam upon us from the orchards and by-roads with cheerful golden radiance! And when these shining stars have grown dim and faded from their firmament of green, there appear in their place such white wraiths of their former selves as resemble the moon seen by the light of day. They are now so many extinct suns, so many ghosts of the dandelions, soon dissolving into still less substantial state, to be spirited away on the winds.
During the summer the common dandelions gradually disappear, and at length the fall dandelions suddenly spring into prominence, poking their flower-heads up on long scapes. With commendable thrift these are closed every night, that a little pollen may not be wet by the dew. These fall flowers appear to be more numerous even than the early species. They can sustain themselves in tall grass where the latter could not, keeping their flower-heads always floating on the rising tide of green. You may see fields of red clover mixed with dandelions, while the Virginia creeper lies in scarlet splendor along stone walls, and goldenrod and asters are massed on the borders---Elysian fields surely. The play of light and color is a kind of music, and stimulates one to some inner hearing. The deaf could hear this. And were the blind to listen to the crickets' reverie, they might see these fields.
Is there anywhere a more audacious beauty than the pokeweed in autumn? It flaunts itself in your face---one of the respectable \textit{bourgeoisie} of weeds, now suddenly arrayed in this regal fashion and mocking you with its splendid beauty. A weed! Why are not roses weeds as they stand all forlorn before this voluptuous child of the people? Out of the plebeian rabble there comes here and there such a superb creature as this.
Consider the milkweeds,---a family of beauties. Something luxuriant and sensuous there is in their ample proportions. They have an excessive health, an exuberance of vitality; a full-blooded race, if you so much as break a leaf from one it bleeds like a wounded creature. From the mud, the swamp-milkweed has derived some rich hue, while the butterfly-weed in the pasture has caught the very sunshine itself and become a living flame. The great pod of the milkweed is the luxuriant fruit of this fine plant, as tropical in appearance as any mango or cocoa bean. When it is ripe, in place of a luscious flavor, it discloses a mass of finest silk, a fluffy ball. Who would guess the treasure within these grotesque pods with their long beaks, their spines and wrinkles? They are like curious old junks with a cargo of rich stuffs of the East, which children---young pirates that they are---overhaul on the high seas of the pasture and despoil of their treasure.
It is the sturdy character, if nothing more, of some weeds which constitutes their charm, for health is beautiful everywhere. Ironweed and joepye-weed are such lusty, vigorous plants, and burdock and jimson-weed. The earth \textit{shall} nourish them; they push themselves to the front; they do not live by any one's favor. How can the impoverished dust of the roadside sustain these burdocks with their incredible leaves? The richest swamp produces no such extravagant foliage. As for the ironweed, it clothes the pastures with a royal purple, so rich a hue it compels the eye, and is a kind of stimulant. One may become mildly intoxicated with such color.
In August the high-roads and by-roads are painted---stripes of gamboge and patches of delicate blue---and all because of some weeds. It would be worth while riding through the country at this season, if for no other reason than this. Vivid streaks of tansy stretch in narrow lines for rods together. Where the road skirts a pond, the eye is refreshed by the pickerel-weed, resting like aureoles above the surface of the water. In the fields beyond is the celestial blue of the chicory---so common a weed, so divine a hue; while everywhere a fringe of wild carrot trails in the dust, the lace border of that gorgeous mantle. Such laces and jewels nature provides if you are but rich enough in thought to possess them.
In the pastures mullein and thistle grow side by side, two pronounced personalities, as different as it is possible to be, yet nourished by the same soil and under the same conditions. The mullein seems to invite you to take hold of its leaves, while the thistle as plainly says, Hands off! They suggest similar types of people, one bristling and repellent, the other suave and genial. These great flannel leaves of the mullein are caressing and soft to the touch. Contact with them is agreeable, well nigh soothing. If, perchance, your feelings have been ruffled by a bellicose thistle, address yourself to the tender young leaves of the mullein and you shall feel their soothing effect.
The perfume of the Canada thistle is equal to that of most wild flowers and superior to many. It is wholly refined, with no taint of coarseness. With what vulgar effrontery a cheap perfume assails the nose. But here is a despised thistle which brings itself to notice by an influence not plebeian but patrician. You might pass this thistle day in and day out and never suspect it had any such virtue, till you had gone out of your way to cultivate a closer acquaintance. Call it a weed if you will, it has an individuality that separates it from other common plants, and by reason of which it commands attention.
Floating in nebulous masses about the blackberry thicket, the delicately conspicuous hue of the fireweed catches the eye. If you will but watch the slender pods you may now and again see one suddenly open and its four walls silently withdraw, while there emerges from the interior a phantom shape, the filmy mass of pappus-down with rows of golden seeds attached. This white cloud of silk gradually takes shape, as the mist might rise from a mountain lake, lingers a moment, and then sails away on a passing breeze---ethereal still as the mist---growing less and less, and vanishing at length, as if resolved again into the invisible.
Old gravelly roads, which meander across the pasture and seem destitute of any special beauty, are often adorned from end to end with the round-leaved spurge, of richest hue, varying from maroon to plum color. This little weed is so unpretentious, so sincerely humble and unassuming, that probably very few ever see it or are aware of its existence. It lies prone upon the earth, where, once it attracts the attention, it is seen to be a beautiful embroidery on the bare ground. Here grows the poverty-grass which on misty days is covered with dewdrops---incrusted with jewels---while more pretentious plants are not decked in any such beautiful array. The mist descends upon the poorest of them all, and makes that resplendent.
In the society of weeds there is this tendency to segregate, quite as in human society. Even the beach has its clique, a curious throng quite distinct from any of the fields, which defy the encroachment of the waves. About these coarse weeds of the beach is something peculiarly in keeping with their environment. The strange spiny fruit of the orache suggest sculpins, or some sea-shells, while the innumerable erect stems of the spreading house-leek resemble the backbone of fish. Carrying with it its air-sacs and paraphernalia of the sea, the rockweed, which is a ``weed'' of another world, grows as far up on the land as it can go, while the weeds of the beach approach the water as near as they dare. Here is the frontier, the edge of their world, and one and all would scramble over the border could they sustain life on the other side.
\gutchapter{INSECT LORE}
Apis the bee, Vespa the wasp, and Arachne the spider---these might properly figure in many a saga. Mighty are the works of the tribes of Apis, while Bombus the bumblebee befriends the pale flowers of the forest as do the winds the pine. Arachne beguiles the fly, for she is a very Medusa; the solitary wasp slays the Gorgon and lays her in the tomb she has prepared, rolling a stone over the entrance; lastly, from the body of the spider springs the race of wasps, like warriors from dragons' teeth in the days of Jason.
From the first flowering shrubs to the last goldenrod there is the hum of industry. The willows, on mild April days, resound with the roar of insect traffic. The bees push in rudely among the bunches of stamens, and the red anthers so neatly and compactly arranged are soon disheveled, the filaments bent by the myriad insect legs which scramble and kick through them. It is everywhere bustle and hurry; all are wrought to a tense degree. Life is here at a white heat---purposeful, Anglo-Saxon; yet it appears to move without friction. Occasionally a bee visits the meek-looking pistillate shrub near by, which patiently waits while the buzz and din continue uninterrupted across the path.
It is always a mystery just how the honey-bee transfers the pollen to the pollen-basket---even in view of the explanation. It appears to be scraped from one leg to the other, and gradually shifted from fore to aft by a dexterous process until lodged in the proper place, the bee remaining all the time on the wing so that the legs are moved with perfect freedom. Finally it is stowed more neatly and compactly than any pack-mule's load, and the panniers are good to see, rich and yellow as pumpkins glistening in the corn field. Doubtless the bee is careful to keep the balance and not put more in one basket than in the other. Since pollen-grains are of distinct and definite shapes in different plants, is it not possible that the insect, from its near point of view, detects these differences, and in place of so much indistinguishable dust, finds itself handling minute cubes, spheres and variously shaped blocks?
How readily bees are apprised of the blossoming of any flower. On the very instant the dwarf-sumacs open, the place hums with them. Solitary bumblebees continually scout through the woods and discover when the Indian-pipe, the shinleaf, the pipsissewa are in bloom. Only the queen bumblebee can have any memory of these flowers, as the life of the workers is but a season long. Probably they do not communicate the news, but each hunts for itself. With the honey-bees, however, this is the gossip of the hive as much as the state of the crops with farmers: ``Meadow sweet is open today!'' ``Clethra is in bloom!'' ``The first goldenrod!'' Imagine the news circulating like wildfire through the hives. Honey-bees have little time or patience to hunt up solitary and retiring flowers. They want masses of bloom, fields of blossom, having a large work to do---a city to build, a host to feed.
The bumblebee is the good angel of the woodland flowers, the visiting priest---or shall I say priestess---to all outlying parishes, calling at every ledge and gorge and dell where is any colony of blossoms or a lone settler or two. The bee discovers the pale pendent blossoms of the checkerberry under the leaves and almost prone upon the ground. In order to reach them it sometimes turns on its back upon the hemlock needles as it inserts its tongue in the flower above. In winter when you gather a checkerberry now and then in your walk you shall bestow a thought upon the buzzing priest of Flora who solemnized these nuptials. It visits every flower in the transparent groups of Indian-pipes which push their way up through the leaf mould to stand like an assembly of the pale-sheeted dead, and looks singularly rich and velvety against these stems of alabaster. Here is a botanist who knows the flora well, and takes a tithe from every blossom to which is brought a grain of pollen---the marriage fee. It is hard to believe so willing an agent is unaware of the service; that it fills an office which it does not recognize, while we, the biographers, alone perceive the relation.
Tell me, is there not something heroic in the life of the queen bumblebee? She awakens after her winter sleep, the sole survivor of her race, and bravely goes forth to collect pollen, lay her eggs and become the founder of a new race of workers. There is rude and virile romance in the life of this bee with its flavor of the forest. She is the queen-mother indeed, no mere figurehead, but strong, capable, self-reliant. Think of her retiring under the moss and leaves at the approach of winter, the last of her race; or, rather, do they all resign themselves to a sleep from which she alone is to awaken. She remains encircled by Cold---as Brunhilde was engirdled with Fire---till the sun shall cross the magic line and awaken the sleeping Amazon.
Today I split open a dead twig of sumac in which the little upholsterer-bee had laid her eggs. From the summit a well or shaft was sunk some ten inches through the central pith. This I cautiously descended by means of a jack-knife and found it partitioned into a dozen cells, in each of which lay a pupa, the pallid sleepers like mummies in their royal tombs awaiting a resurrection.
The cells were lined---upholstered---in silk and partitioned from each other by walls of chips cemented together. In some cases the pupa was being devoured by the minute larvae of a chalcid fly, and in one cell only the dried skin remained. For that pupa there was to be no resurrection into the life of the bee, but as the cell was opened, out stepped a tiny chalcid into the light of day, its dapper little person shining blue-black and its minute wings of an iridescent green.
You may see many broken twigs of sumac, elder and blackberry, perforated at the end in evidence that in the cells below are the larvae of a bee, or perhaps the pupae wrapped in their transforming slumbers. This sepulcher is sign to the chalcid fly as well. In one such that I opened were several perfect bees, beautiful little green creatures. Immediately they stepped out upon my hand and began dusting and cleaning themselves, but appeared to be troubled by the brightness, and eager to hide. When offered the open end of a tube, such as they had recently come from, they seemed glad to enter. They were not yet fitted for contact with the world of light and preferred to return to the darkness and security of their cells. A spider had concealed herself in a silken room at the mouth of one tube, perhaps seeking this privacy in which to change her skin. When their time had come to emerge, the inmates would naturally have walked into the spider's den, while the light of day appeared beyond, but for a single instant, as a faint glimmer which they were destined never to reach.
However, there is a Theseus for every monster. A spider was one day spinning her web in an outer angle of the veranda, laying the first strands, the scaffolding. Attaching one point she swung out on her line and fixed a second, aided by the breeze. Without the wind she perhaps could not have erected her scaffolding in that place. The morning sunlight caught these first threads, stretched from post to beam, and they gleamed like silver or spun glass. At length a wide space was to be bridged and she swung free at the end of a long strand. The breeze carried her to and fro, far out from under the roof, so that she remained suspended in mid-air.
But other eyes were watching her at her work. As she swung thus, self-possessed and at ease, suddenly a mud-dauber pounced upon her. The silver strand parted in the sunlight, and the spider was carried to the beam above, where the wasp apparently stung her several times. A moment after she rose in air holding the large globular spider, now paralyzed and inert, and sailed away over the treetops in the direction of her nest. The victim was to be immured in a sarcophagus of mud together with the egg of the wasp. When the egg hatched, the larva in this tomb with the body of the spider would find such gruesome state congenial enough---being of the wasps. In this case a spider the less means a wasp the more.
[Illustration: INSECT LORE COPYRIGHT, 1908, BY RUDOLF EICKEMEYER]
Late one afternoon a spider was constructing her web. She already had her first line stretched between two small shrubs. On this she crossed and recrossed several times, each trip reeling out a new strand from her spinnerets, until she had a stout cable from which the gossamer structure was to depend. From an end of this she dropped to the ground and fastened a thread, then ascended, traversed the cable and dropped lines from the other end to the twigs beneath. All were remarkably taut and firm. By crossing two she now established the center of the web---not the geometric center---and from the overhead cable spun some radii to this point and from this to the lower strand. In an incredibly short time she had lines radiating in all directions from the center like the spokes of a wheel.
She now fairly ran over these spokes paying out the strand as she laid the spiral web upon the gleaming radii. Starting at the center she traveled from left to right, passing the thread through the claw of one of the last pair of legs. By this means it was held from her as far as possible and quickly attached to each of the radii. A very short time sufficed for her to complete this spiral of perhaps a foot in diameter, and she had only to return over the ground with the final thread, on which are strung the viscid drops.
She paused as if resting, and in that moment a Social wasp descended like a fury and bore her to the ground. The wasp quickly rose holding the spider in her embrace, and returning to the bush suspended herself by one hind claw. Here she held the body of the spider with two pair of legs, and turning it about, as though it were on a spit, bit off some of the head parts with her strong jaws which worked like a pair of shears. So near was I that I could see these jaws meet and sever the thorax, which fell and glanced from a leaf a few inches below with the faintest imaginable sound. The wasp then proceeded to tear open the abdomen. The builder of gossamer bridges, who overcame space and flung her nets to the breeze, was no more. I looked again at the unfinished web and in it struggled a small fly.
In stretching the first strand the spider avails herself of the wind to some extent. When crossing from one point to another it is by no means necessary she should drop from a height equal to the distance to be crossed; for if the wind is strong enough she has but to descend a little way, and then, as it holds her out at right angles, she pays out the line and so continues moving in mid-air. As soon as she comes in contact with some object she at once attaches her thread. I have more than once observed a spider drop a short distance when there was no breeze to carry her, but by the movement of her body she imparted a slight motion to the line and thus set herself to gyrating until she finally swung across the intervening space.
The spinners of flat webs in the grass are associated with dog-days and with foggy weather, as if they spread their tents only at such times to fold them again and steal away with the appearance of the sun. As a matter of fact these spiders work in clear weather and at different hours of the day, but the web is so fine as to be next to invisible unless covered with moisture, when it at once attracts the eye, like a writing in invisible ink which becomes manifest only under the right conditions.
There are other spiders which become evident only at the approach of winter. It is something to the credit of these small spiders that, being without wings, they should still aspire to fly; whereas the ants, born with wings, are in haste to tear them off. The past year they were so in evidence on the 11th of November that I shall henceforth associate that day with the flight of the Erigone. The weather was cool, but with a suggestion of Indian summer in the air. I first noticed the spiders on top of a hill, for the bare twigs of sumacs were streaming with gossamer threads which shone like silver. From time to time little spiders descended from the upper regions and ran about over my coat. One, which was spinning threads on my sleeve, finally ran out upon my hand and, elevating its spinnerets, began paying out a line, which I could see as I held it against the sun. When this had reached a length of several feet the little spider was whisked off by the breeze and carried away.
Toward sunset a delicate network of gossamer threads covered the open pastures like a silver mesh in which the earth lay captive. These minute spiders have a way at this time of allowing the strands to be drawn from their spinnerets by the wind, until they carry sail enough to be lifted off their feet. They fly away thus on the wings of the winds, perhaps carried high above the earth by ascending currents. Lo, the hegira of the spiders!
It would appear that the Solitary wasps are more ingenious and self-reliant, and less governed by tradition, than the Social bees and wasps; for I have seen a small black one which was unable to rise on the wing with the large spider it was carrying, finally drag it up the trunk of an oak to the height of seven feet and from that vantage fly away. Such an one pulled a spider much larger than herself up on my knee and left it there, paralyzed but alive, while she made explorations, after which she returned and took it away. As I was making some notes at the time with reference to wasps, the incident made a pleasant impression, quite as though she had taken me into her confidence and had gone out of her way to reveal some facts of her life.
One day I encountered a sand-wasp which had just stung a wireworm and was dragging it over the ground. The worm, which resembled a brown twig, was three inches long and as large around as a slate-pencil, while the wasp was not over an inch and a quarter in length and very slender. Seizing the victim in her jaws and straddling it, the wasp walked along in this uncomfortable fashion, over ground strewn with pebbles and partly covered with brush. Difficulties were many, and she was kept constantly pulling, tugging and boosting to get the worm along.
At length she penetrated the brush and came out bearing the worm into an open gravelly space. Here she turned off sharply for a distance of two yards, and, after running nervously to and fro, stopped in front of a small hole. She had been over an hour dragging the worm. During that time one main direction had been followed, though never had she to my knowledge left her burden and risen above the brush and trees to get her bearings; yet she found her way unerringly, and only turned aside because of the boulders and clumps of white birch stumps. The whole distance was about forty feet in a straight line, but further as the wasp had gone.
Backing into the hole, she seized the worm and attempted to drag it in after her, but the entrance proved too small. She therefore came out and began rapidly enlarging it by seizing bits of gravel with her jaws and fore legs, rising in the air and carrying them off six or eight inches. Again she entered, and this time was able to pull the worm in after her. She remained three or four minutes in the hole, during which time she was depositing her eggs, then her head reappeared at the opening.
She now began filling in. Dropping two or more bits of gravel, she would then turn her back and rapidly scratch in dirt with her fore legs, evidently to fill up the interstices. Twice she took out a bit of gravel and carried it away, precisely as a mason might throw aside a stone that was not the right shape or size. As her head was thus inserted in the hole a black ant approached and peered into the depths. Suddenly the wasp turned and gave one look, whereupon the ant fled in haste.
When the hole was filled to the brim she tamped it down with her head. This occupied her some minutes and she appeared to take the utmost care. Gravel was then brought and piled upon the spot until it exactly resembled its surroundings. The stones carried varied in size from those as large as a buckshot to some the size of a marrowfat pea. They were lifted and carried seemingly without effort, and dropped almost before one could see what she was about. Twenty minutes were consumed in filling up the hole and restoring the surface.
On a sudden she vanished, and with her vanished the place itself where she had been at work. It was as if a trap-door had closed, and no sign was left. So carefully had she done her work and so closely imitated the surroundings, like a miser burying his gold, it was only after careful search I could again locate the spot.
Thus in the economy of Nature every insect appears to be food for some other. On the leaves of the Virginia creeper you may usually find, in early autumn, some caterpillars which have received the eggs of a small chalcid fly. These caterpillars, otherwise so large and green and awesome to the beholder, have become limp and lean and have an aged and decrepit look. They hold feebly to the vine but no longer eat anything. I brought home one of them and in a short time there emerged from its body a great number of small white grubs, fifty or more by actual count. Upon the back of their emaciated host they proceeded to spin for themselves marvelous little cocoons of white silk which they did in a very brief time. Moving their heads this way and that they spun the fine threads about themselves until they were completely enveloped. Here were a great number of little spinners, making for themselves garments of silk, and at last spinning themselves out of sight. The caterpillar now bristled with the small white cocoons which stood upon end on its back, where they were attached, and almost hid it from view.
The wary caterpillar has many foes. If it escapes the hungry warblers and vireos, there is still the army of goggle-eyed wasps and nervous ichneumons to circumvent. Yet a prodigious number survive. Were it not for their enemies they would overrun the earth. The butterflies sporting in the sunshine, and the small moths flitting about the lamp, have come through many perils, and may almost be said to have lived by their wits, so astonishing are the ruses they have devised to deceive their pursuers.
\gutchapter{THE WAYS OF THE ANT}
If you would see the ants to advantage---to your own, that is---you must turn over a pasture stone under which one of the species of small yellow ants has its nest. By thus gently removing the roof, if it is a good-sized stone, the whole colony will be in view at once. The red-ant hill presents difficulties. To dig into it or to pull it apart is quite useless, as the earth falls in and nothing is to be seen but a struggling heap of dusty and indignant ants. It rarely happens that such a hill may be built around a small boulder. If this boulder is suddenly and deftly removed, not dragged or rolled aside, but lifted clear of the hill so that the sides of the nest may not be broken in, a remarkable scene is disclosed.
I have found such an ant hill, and by removing the stone the household was placed on exhibition---but not all its secrets revealed by any means. From several large chambers, now roofless, galleries and corridors radiated in all directions. The instant the stone was lifted the ants swarmed from the galleries into these chambers, which were packed with the large cocoons. There were thousands of pupae, of a delicate brown tint, looking wonderfully clean and fresh, but with such celerity did the ants work that inside of ten minutes all were carried from view.
Among the rest were perhaps a dozen young ants, the head and thorax being white and the abdomen a pale mauve. These creatures moved feebly about, taking no interest in the proceedings, and were for the most part seized by the workers and conveyed into the galleries. Apparently they were individuals that had just emerged from their pupa-cases.
Under another large stone were two very numerous colonies living side by side, of different species. The nests were, of course, entirely separate and under opposite ends of the stone. The smaller of the two appeared to be stinging ants, for they clustered in great numbers over their small pupae, elevating their abdomens in a threatening manner like so many diminutive scorpions. The other species were large and active ants of a polished bronze hue. Their pupae were naked, which gave the nest the appearance of being filled with grains of rice.
These large ants set to work with frenzied activity and removed all of their own pupae. Then, and not until then, they swarmed over into the adjoining nest and began carrying the cocoons of the small ants back into their own nest. Now and then some small ant bolder than the rest would resist, and an individual combat ensued which ended by the large ant carrying off her small antagonist. There was, however, very little resistance of this sort, and the pillage, if such it were, continued until the remaining cocoons had all been carried over into the nest of the large ants. So few of the small ants made any resistance that it gave one the agreeable impression the larger ants were only offering assistance. But I failed to find on subsequent visits that they had returned the pupae. And although they daily brought their own pupae out of the galleries, the smaller cocoons never more came to view, and the small ants subsequently abandoned their nest. Thereafter I felt some compunction in thus disturbing a whole community for mere curiosity.
It is noticeable above all how the ants at such times take no thought for their own safety, but for that of their charge solely. Whether their interest is in any sense maternal or merely a property interest does not appear. Another feature evident in disturbing a formicary is the general harmony in which the individuals of any one colony work together. Here is no less than a catastrophe, as if the roof of one's house were suddenly to be removed and everything upset. And yet not one runs away or apparently conflicts with any other. There are no cross purposes; no two get in each other's way; but animated by a common motive, and by one only, the community proceeds with despatch to the work in hand.
Is this socialism among ants something preordained for them as the condition of their life, or is it in part an acquired tendency of the ants themselves? That they \textit{do} acquire tendencies would seem clear enough. If it should be proven that this social state is in fact the result of an evolution among them, it would be one of the most significant facts of natural history.
It serves the community admirably at any rate. But with them the individual does not count. Ants are ahead of us in one respect in that they have order without coercion. There is such harmony, such co-operation among them, they have evolved no ruling class, the queens being such only in name and more properly the mother ants. The life of the community is all, and every one looks out for it.
On warm afternoons early in September you may look for the swarming of the queens, when myriads of ants sail into the air in their desultory marriage flight. In apparently endless succession they pass, every now and then one alighting, whereupon begins the curious part of the performance, for they run rapidly about, throwing themselves upon their backs to squirm from side to side after the manner of a dog scratching. They then get upon all sixes and continue running to and fro. After these contortions the wings wear a most disheveled appearance, and, as the process continues, become more and more crumpled, until at length one or more are missing.
Sometimes in sheer desperation an ant will lie on her back and revolve rapidly in this position. In some cases the wings seem to resist all attempts to remove them and the ants redouble their efforts. Their frenzy appears to know no bounds; they fairly stand on their heads and repeatedly fall over miniature precipices and into Lilliputian crevices in their blind determination to tear off the wings. Again they seem to use their legs as though trying to twist off a wing. It is the most fanatical performance to be witnessed among insects.
Such dogged persistence must sooner or later attain its end, and presently the ant is seen running about wingless or perhaps with only a torn stub left. The behavior is no longer frantic as before, but she now moves about as if enjoying great relief. During one such flight great numbers came down into a gravelly path through a huckleberry patch. They apparently avoided the bushes on either hand, and chose to alight in the path, for it was alive with ants twisting and turning and wriggling upon their backs in the gravel. Others, having gotten rid of their wings, were attempting to go head foremost into the ground, possibly with a view of laying their eggs, or merely because the soil was their natural element.
Around the formicary itself the workers were grouped \textit{en masse}, endeavoring either to restrain the new brood of queens in the old colony or to coerce them into leaving. They appeared to drive them as a squad of police might force back a crowd. But it is manifestly difficult to interpret their motives with any assurance, and it is more likely they were provoking them to flight. At such times they ascend the branches of a bush and collect in excited little groups on the buds and flowers around the females, as if determined they should go. No doubt it is an exciting day with them, a sort of Labor Day demonstration. In this case it is the womenfolk who are thus bent on asserting their rights and doing as they will. But why, having once ascended into the larger world and the liberty of winged creatures, must they insist on tearing off this means of freedom to become crawling, laborious insects? They appear to hear two calls, one from above and the other of the earth, earthy, and to obey the latter. But it is with them the race and the future---always the future.
To an ant a tree is a forest in itself. Ascending its mammoth trunk to the upper regions, she follows the great highways of the branches, out into the unknown and trackless wilderness of leaves in pursuit of her game---the aphid. She knows well in what wild and solitary uplands to look for this mountain-goat.
The under side of maple leaves affords good pasturage to numerous green aphids which there browse contentedly in the pleasant shade and under the watchful eyes of the small brown ants that herd them. The aphids are all sizes and ages, though as to age the difference is probably but a few days. With a glass, the process of ``milking'' may be observed, the ants merely stroking the aphids with their antennae. Two small tubes, like sap quills, protrude from the back of the aphid, and from time to time minute glistening drops are seen to exude from these tubes and are removed by the ants in attendance. Surely, to the ant here is the land of milk and honey. They move constantly to and fro among the aphids, now and then stopping to stroke one. Apparently they detect by some signs which are ready to yield the sweet fluid. Their presence appears to be agreeable to the aphids and is never in the least resented. After long watching with the glass, I have never seen anything akin to insubordination. Pluck the leaf ever so gently and hold it in a proper position, the difference is at once apparent to the aphids, for there begins an exodus, and large and small troop up the stem of the leaf and so on to whatsoever it may be attached; nor does it cease until they have deserted to the last one.
But the life of ants is by no means given over to these bucolic pursuits. While the meadow-ants seem to be in the pastoral stage, the red species and the large black ones are hunters and warriors. The most sanguinary conflict I have witnessed was a battle of the ants. Two armies of the same black species met on the floor of a neighbor's barn. The battle lasted throughout several days, and both sides fought with indescribable ferocity. Where they came from was a mystery, as no such colonies of ants had ever been seen thereabouts.
They appeared to be of the species \textit{Formica pennsylvanica} which nests in trees, but these do not occur in very large colonies, whereas the contending hosts upon the barn floor were as the Tartar hordes. The floor was strewn with struggling pairs and with the dead and injured, and always fresh forces were arriving.
The persistence with which they fought is only to be compared to that of bulldogs, while they showed the ferocity of weasels. Once let an ant get another by the thorax and she would continue crunching and sawing until she had severed the head, notwithstanding in the meantime one or several of her own legs had been cut off by her antagonist. This was the usual outcome of the various individual combats.
From time to time I placed pairs of combatants on the slide of a dissecting lens, and through the glass observed them as in an arena. It was a miniature combat of gladiators, but with no appeal for mercy on the part of the vanquished. Much evidently depended on the best hold, as in wrestling, for there was no dislodging an ant once she had secured it. Under the lens the comparatively great strength and the skill and relentless ferocity of these miniature warriors became more evident and was astonishing to witness.
A bird's-eye view of the battle-field revealed no plan of action nor any directing genius. It was every one for himself---or rather herself---but there was absolute unity of purpose. Occasionally some could be seen running about with the heads of the vanquished suspended on their antennae, whereon the jaws had closed in the death-struggle, not again to be relaxed. These ants appeared to seek no relief from such a monstrous encumbrance, nor seemingly was any offered by their comrades. Others were crawling on an uneven number of legs in search of new foes. The cause of such a conflict among ants of the same species remains a mystery---one of the many mysteries.
Every year the red ants raid the common blacks for the purpose of making slaves---a most highhanded proceeding. This season I came upon the invading host marching up the road about ten in the morning of July 28th. The invasion had but lately begun, as the ants were carrying no pupae; it was the skirmish line. As the column advanced, frequent and rapid communication took place between individuals and stragglers who were coming back. Later, when the raid was well under way, there was little of this. The nest of the red ants was by the side of a path in the woods which led out to the wagon road, while the negroes were domiciled some distance up this lane. Now the column of red ants followed the path and the road the entire way, in place of going directly through the bushes, though it doubled the distance, which thus amounted to some fifty yards.
Red ants were soon pouring out of the various openings in the nest of the blacks, carrying both pupae and larvae, and rarely one passed with a bunch of small white eggs. Several black queens came out of the nest, and as they emerged were set upon by red ants, which tried to hold them by their wings. They managed, however, to throw off their assailants, and ran under my feet, where they were followed by a score of black workers, all of whom crowded under the soles of my shoes as I stood on the loose gravel. At noon I timed the ants and found that, on the average, forty pupae and larvae were carried past a given point every minute. Two unbroken columns now extended the entire distance between the nests, one advancing and the other returning.
Occasionally one passed carrying a portion of a black ant, a head and thorax, or an abdomen. Again, one would appear with a live black, which, when liberated by me, frantically made her escape. Very young negroes when carried off were never injured. On one occasion several red ants were struggling with a black, and among them was a black who fought against her own friend. This is the only case in which I saw a black ant help the enemy in this way---a traitor, evidently, but presumably one whose pupa had been captured the year before and reared in slavery. Whereas the red ants always came to each other's assistance, the blacks rarely did so.
By five o'clock the raid was practically over for the day. It ceased as suddenly as it had begun. Early in the struggle a slender, straggling column had diverged from the main line, about half way between the nests. I now found the entire body of ants moving in this new direction. The one raid over, they had undertaken another upon a colony of blacks some twenty-five yards distant, and were transporting the pupae and larvae at about the same rate as before. To reach this nest, the column must cross the wagon road, and here a number were crushed from time to time by passing vehicles. But the marching army passed by with the stolen pupae and paid no heed to their wounded comrades. This second foray ceased before nightfall.
The following morning by ten o'clock the raid had been renewed and a great stream of ants were bearing away pupae as before. Whenever the column moved over dry leaves its progress was distinctly audible, a rustling sound suggesting the curiously dry \textit{crik crik} of a serpent. The footfall of the ants was as incessant as the patter of rain; a barefooted insect host, a rabble of \textit{sans culottes}, and the sound of their marching feet reached my listening ears, as it were in the clouds above them.
On the fourth day the slavers began kidnapping the blacks themselves and carrying them unharmed to the nest. Quite often I found them carrying individuals of their own species. These may have been deserters or they may have been ants from some other community, who, learning of the raid, thought to be present at the final sack and perhaps share in the spoils. A still more puzzling thing was the fact that some few red ants bore negroes in the wrong direction,---that is, from the red back to the black colony. I have noticed on former occasions that the raid may become thus complicated toward its close as if the ants, drunk with victory, were beside themselves.
On the 7th of August the raid was directed against a new negro colony some distance further down the road. It was carried on with something like the usual vigor until the 25th of the month, when it apparently ceased. The first nests of blacks, in which some few ants remained, were no longer molested, though the besieging army passed them on its way to the field of operation. Thus the series of raids of this one colony of red ants continued for nearly a month.
I found no less than three other raids in progress at this time, among widely separated communities, so that the marauding spirit was contagious among them and spread like the war fever. The red warriors were everywhere in arms and bent on pillage. One hill, being free from grass, offered a clear view of what was going on at the doorway at least. Here the black workers---the slaves of a former raid---were carrying out bits of gravel, while the train of red ants entered, bearing the stolen pupae from the pillaged nest. The red ants were at this time bringing some large queen pupae which they had great difficulty in getting over the ground. As they approached the entrance, the black workers deposited their bits of gravel and ran to their assistance. Several blacks which remained near the entrance seemed to act thus as porters, while others about the top of the hill were engaged as laborers.
Stopping work at about five o'clock, the train of red ants melted away before one's eyes. They dropped their task very much as a gang of men do when the whistle blows. Their day at that sort of labor was therefore only about seven or eight hours, as if some of the principles of Labor Union were in vogue among these brigands. They would kidnap only so many hours a day. The slaves, however, kept at work until dusk. Perhaps the red ants continued inside the nest, disposing of the pupae captured during the day, but they brought in none after five o'clock.
Three days had elapsed from the close of this raid when, for some reason, the entire colony of red ants deserted the hill, carrying the newly captured slaves and their pupae with them. They took up their abode under a cement walk, an unusual place for red ants, and a week of incessant labor was consumed in carrying the black ants and pupae to the new site. This was, then, a \textit{bona fide} exodus of an entire community.
Under the cement walk to which the colony of red ants had migrated with their slaves were numerous nests of small brown ants. These swarmed one sultry afternoon, and as they came pouring out of the cracks in the walk and clustered on the surface, the fierce red ants fell upon them with fury, slaying hundreds and leaving most of the bodies on the walk, though many were carried away. This I took to be a veritable hunting expedition. Like some other ``sportsmen,'' they appeared to kill more than they wanted, and the little heaps of winged dead were left to be scattered by a gust of wind.
On the following day a new chapter opened in the history of this remarkable colony, for I found them attacking a large negro colony some distance away. Contrary to custom, the blacks defended their nests with spirit, and at first seemed to hold their own. Not divining what was to follow, I was surprised to find the red ants carrying away no pupae. But the next day it was made plain enough, for the red ants appeared in a compact column bearing pupae and slaves, which but a week before they had deposited under the walk, and which they were now moving for the third time. Was this a second exodus or had the move to the walk been merely an expedient until they should find a more suitable place? Without further ado they invaded the nest, and four distinct colonies (the red ants held slaves of a previous year), one red and three black, with all larvae and pupae and some eggs, were thus housed together. One may imagine the feelings of the unfortunate community on finding not only an invading army of freebooters, but that some thousands of their own cousins, children and all, were come bag and baggage to live with them.
Now the marching column passed close by the nests of the little brown ants which had been their hunting-ground of the few past days. They were too engrossed in carrying pupae to follow the chase, but I found three of their slaves posted by some small holes in the cement through which the brown ants left their nests. These negroes remained near the opening, and, as the brown ants appeared, would reach over the edge and pull one forth which was soon crushed and tossed aside. During the several hours that I watched them the three slaves remained so engaged. From time to time they would run about among the wounded, and picking up one here or there, apparently give it a nip.
This final move occupied some eight days, and nothing further transpired in the history of this colony,---that is, above ground. The war fever subsided as suddenly as it had arisen, and the erstwhile warriors were perhaps become peaceful educators of the slaves now being born into captivity with only some vague instinct of freedom, some race memory handed down from the halcyon days before the advent of the red Tartar.
If the sluggard is to go to the ant, then let it not be to the red ant, nor again to the slave, but to some Syrian species known to Solomon, which stored up provender for the winter, or to the little brown ant which herds the aphid. Huber relates that he found the slave-making ant of Europe (\textit{P. rufescens}) unable to feed itself, so that, if isolated, it would miserably starve in the midst of plenty. Not to such an ant, then, should the sluggard go, but to that wise yellow species which, declares Lubbock, actually brought in and cared for the eggs of an aphid through the winter, and carried out the young aphids in the spring to their proper food plant. Certainly should we ever attain to the dignity of wings, there will be no occasion to emulate the ant, which, being born into that freedom, tears them from its body, the rest of its days to crawl upon the earth.
\gutchapter{AUTUMN STUDIES}
Early in August we are surprised each year by the glowing leaves on the tupelo, a little patch of scarlet gleaming in the swamp, while the high blueberry is still in fruit and the silver-rod is making its appearance. By the time the wood-lilies have faded in the huckleberry pasture, the red bunchberries add their bit of color to the carpet on the edge of the swamp. The large berries of the clintonia turn that rare shade of blue which they retain but a short time, growing darker as they ripen. This delicate bloom appears later on the berries of the smilax, the frost-grapes, the savin and the viburnums; but in the clintonia there is an admixture of some tint lacking in these, which gives a finer blue, as though there were reflected here some remoter depths of the heavens, a bit of ethereal and celestial color imprisoned for a moment. Mountain-holly is now in its prime, its berries of a deep cherry, perhaps one of the richest reds to be found in nature, as those of the clintonia present one of the rarest blues, equaled only by gentians and bluebirds. Both berries, of course, wear their true colors only in their prime and lose them on becoming overripe. In the swamps the little yellow and brown cyperus is in flower and the leaves of the small, pale St.-John's-wort have reddened to a brilliant hue, while young bullfrogs and pickerel-frogs sun themselves on the lily-pads and dream away the mellow hours.
While the dog-days are disappointing in respect to bird life, there are compensations. The charm of this season lies in the mushrooms. Though these last through October, they are more in evidence in August, and take on prominence then because of a diminishing flora and the withdrawal from view of a large number of birds. It is a second spring---hot, moist and fungus---a blooming of the mushroom world. Old stumps and dead branches blossom gaily, and bring forth a tropic flora. Decay is seen to be the matrix of beauty. The logs of corduroy roads through the swamp are incrusted with a shelf fungus (\textit{P. versicolor}) of marvelous hues. These, spread like open fans, are fastened to the wood by the pileus itself, as by the handle. Some are banded in seal-brown and amber, the surface having the lustrous, changeful effects of a cat's eye. Others are striped in violet and deep green; still others in green and mauve, and some in ochre and tawny hues, while over all there is a play of light as on watered silk.
It requires somewhat of the heroic spirit to discover whether a mushroom is edible or not. But we may feast our eyes on the amanita, and all other mushrooms, with no fear of consequences. The mycologist seems to overlook the finer and esthetic value of mushrooms. They are beautiful to look upon---surely this is one important qualification. What more attractive these misty days than the deadly amanita---the ``destroying angel''? How it gleams in the woods! How it lures with its terrible beauty! But they who are tempted to taste must be wholly given over to the pleasures of the table. It was not made for the stomach, but to be digested and assimilated by mental processes alone and the perception of beauty thereby nourished and sustained.
How clean and wholesome is the pasture mushroom---\textit{the} mushroom---with its white flesh, pink gills, and cap from which the skin peels as readily as from a fig. The same field is often sprinkled over with puffballs looking as fresh as new-laid eggs, as they poke out of the close-cropped turf. Some species are thus eminently wholesome and inviting, while others have a loathsome fungoid personality and affect one like the sight of reptiles. They express the fact that they are of the \textit{lower orders}---the slimy world. Mushrooms are indeed almost as varied in outline and color as flowers. Red species of russula vie with the rose, with ripe cherries, or the cheeks of Bartlett pears, while the green russula is of richer, more velvety hue than any unripe fruit. The grotesque forms of boleti have a kind of fascination. One comes to distinguish minute differences and to cherish these odd and sometimes graceful shapes, as a connoisseur might his bronzes or antique vases.
Many of the mosses are fruiting at this season, but they, for the most part, belong to that mysterious and unfathomable world of the compound microscope. Yet here are some, be it said with joy, that so proclaim themselves as to be known of all men. Such we can take home to us as friends of our leisure and landmarks in our excursions. These at least we have reclaimed from science. In the shadowy sea of Latin names these few green isles appear---peat-moss, broom-moss, hair-cap and fern-moss. Like miniature smilax are the mniums, marvelous little trailing beauties, while of all vegetable elves the silvery bryum has the greatest witchery, with young drooping pea-green capsules like so many fairy pipes. A miniature jungle is the fern-moss, a forest of tree ferns at our very doors---Ceylon and Java in our wood lot. It is only a difference of dimension. A patch of this is as rich and luxuriant as any jungle of bamboos on the lower slope of the Himalaya, and a spider might as easily lose himself in one as a man in the other.
[Illustration: AUTUMN STUDIES COPYRIGHT, 1908, BY RUDOLF EICKEMEYER]
With what a fine garment of green does Nature clothe the trunks of swamp-maples and some black birches. It is a true woodland costume befitting their sylvan life; a snug garment tightly wrapped about the trunk as though to protect the vital parts of the body while the extremities are bared to the winds. Woven in woodland looms of mosses and lichens, it forever replenishes itself, the holes mended and the bare spots renewed as by deft and invisible weavers.
Where do the birds go in August? Never an oriole's note nor a bluebird's warble. All the more we appreciate the faithful redeye and the wood-pewee. The importunate twittering of young birds with their speckled breasts and half-grown tails is in evidence; they at least do not hesitate to make themselves known. But in September are bright days when there come waves of birds. The returning warblers rove in little bands, and companies of young field-and chipping-sparrows flit in and out among the bayberries and alight in the path.
In their dull, autumn colors the warblers have an unfamiliar look. They come disguised in winter cloaks which, if you do not know their little mannerisms, may be effective enough. With provoking celerity they flit in and out the thick foliage, and you dance attendance; now this way and now that, stumbling over pasture stones or plunging into the midst of blackberry and rose thickets, to be detained at last by the persuasive catbrier. Again you go forth to find the game has stolen away and not a warbler is to be seen. Such are the exigencies of bird study in September; yet in a few days other flocks may arrive. Every faintest clue is valuable to the ornithologist who honestly refrains from the gun. Were it not for the peculiar jerking of the tail, one would hardly recognize the yellowpoll in his dull suit. The fly-catchers frequently declare their identity through mannerisms. Were it not for difference of manner and voice, the phoebe and the pewee might easily be confused; so also the redeye and the warbling vireo. I have known the redeye for years, but can never make out his \textit{red} eye, unless it be a glass one.
Now comes the winter wren, peeping and prying round about a mossy tussock like a little mouse, but far more self-contained. His wee tail is elevated and his whole demeanor pert. What a picture he makes, prying about in the hair-caps, his head little higher than the capsules,---a ruddy, rich-hued, speckled little fellow. If only he would give us a measure of that fabled song, that Orphean strain of the far North and of the mountain tops, which is denied to dwellers on these lower levels! There are songs to be heard only on Parnassus.
These are the days of journeying seeds. In spring it was blowing pollen; in early autumn, mushroom spores; and now winged seeds flying before the wind. Those of the hop-hornbeam are done up in little papery bags which, though incapable of an extended flight, manage to sail out and away from the parent tree. Even the small seeds of birch and alder, compact as they are, have wings provided,---for no ambitious flight, to be sure, but a gentle excursion only, such as the broad-winged maple seed may take when its hour arrives. Acorns will fall directly below the tree, perhaps roll some little distance on uneven ground and lie in rich confusion---a symbol of plenty. For any further transportation they must depend upon the wings of the jay and the feet of the squirrel. In this respect the sweet acorns of the white oak have the better chance, while at the same time they run the greater risk of being eaten. Jays constantly carry acorns, and may frequently drop them. Gray squirrels bury them, and recover a surprising number later when the snow is on the ground. They know wherein the white are superior and are as well informed about acorns as are we about apples or the varieties of squash. The white oak acorn is to them Hubbard squash or Baldwin apple.
When Nature planned that the nut trees should bear as they do, she doubtless considered the squirrel and the boy that was to be. She had no idea of deriving a thousand seedlings from a hickory, but perhaps one only, and allowing for those that should come to naught, the boys and the squirrels might have the rest---to say nothing of weevils, which get ahead of both when it comes to chestnuts, being on hand to lay their eggs in the flower. When the boy arrives, it is to find them already in possession---surely nine-tenths of the law in this case. The chestnut-bur was seemingly designed as a means of protection rather than of transportation,---unless it be that in remote times the tertiary monkey got them in his coat, or perhaps slyly pelted the mastodon with these monster burs, and they were thus conveyed, as now a dog will carry beggar-ticks. As a protection it does not serve against its most insidious foe, the larva of the weevil, which works not from without but from within. Nature has treated the butternut better by surrounding it with a husk, as food for the grubs, which are content to go no deeper. One is a case of armed resistance, the other of diplomacy, and diplomacy wins.
How evidently all Nature is flowing. It is as though we stood on the banks of a river and saw pass---today arbutus, tomorrow, columbines, and later, goldenrod. The last is hardly gone before the advance guard of skunk-cabbage appears again. Autumn nourishes a vigorous brood---whole acres of wild sunflowers, acres again of joepye-weed, and salt marshes aglow with the great rose-mallow. Presently there will be only asters and goldenrod---everywhere purple and gold; royal robes worn not for long, to give way to the sober dress of early winter---a monk's garb.
Early in September the common brakes turn, imparting a faint glow to the woods. Dicksonia has a brighter hue, and patches surrounding a pasture boulder fairly seem to emit light. But this is as nothing to the splendor of cinnamon-ferns in the open bogs, now dry, and the spagnum withered and sear. It is as if the smouldering earth-fires leapt at the touch of autumn and glowed in these stately fronds. In the woods is always a predominance of yellow at this season; so lately somber and damp, heavy with the mustiness and humidity of the dog-days, they are now full of imprisoned sunshine. As by a touch of enchantment, the falling of the lower leaves on all shrubbery and in brier thickets has suddenly given us distances, larger perspective and new vistas, where before we were hedged in between dense green walls. Aspen, shadbush, blackberry, birch and hickory all incline to yellow, mottled and speckled more or less with brown. Ochre, umber, sienna, gamboge are on Nature's palette; soon she will replace these with crimson and scarlet. Already there is a touch of vermilion in the brilliant poison-ivy; and she has spilled drops of scarlet everywhere on the outskirts of the woods, along a wall, over a fence, up in a pine, in the very midst of a radiant gleaming hickory---wherever the Virginia creeper grows.
Nature works deftly, at first with delicate brush touching a shadbush, a clump of osmunda, or again only a leaf, a spot of color, a patch here and a streak there; but the day of transfiguration approaches. Early October sees the stag-horn sumacs fairly scintillate with color. At last the whole color-box is upset and runs red down a hillside huckleberry patch, meeting a yellow streak in a ravine and spreading out over the swamps, a sea of scarlet and gold. Every year Nature starts out in this modest fashion and ends in an upset and riot of color. We should know her ways by this time, but though her plan is the same she varies the details infinitely and there are always surprises. These same earth-fires which blazed in the osmunda now glow deep red in the dwarf sumacs---a dull, fierce flame, as if for the nonce Pluto's fires shone through the thin shell of earth. The poison-ivy is in its glory, and no tupelo, no sugar-maple, can rival its scarlet and vermilion. Earth indeed wears a jewel now. But there is nowhere a warmer, mellower tint than the shadbush has caught and held,---not brilliant nor showy, not a shining mark in the woods, but a cheery sight that warms the cockles of your heart. Little clumps of the maple-leaved viburnum are now of a delicate smoky pink, while the ash turns an indescribable hue---a greenish maroon or purplish green if such there be.
Already the hickory leaves are falling, detaching themselves one by one and floating leisurely to earth. It will now be our gentle pleasure to walk through crisp and rustling leaves. Barberries are ripe, and old-fashioned folk gather them for jelly or preserve them in molasses, wherein they are as so many shoe-pegs drowned in sweetness. The solitary sandpiper comes again to preside briefly over the ponds---a lone, wild spirit. Little flocks of coots scud low over the water, and in the dark, spongy humus of the hemlock swamp, red squirrels are digging caches and concealing the small cones, a dozen or more in a place. Such are the signs of the times.
Yet another sign---the last effort of the dying year---is the witch-hazel, which sheds its leaves and stands arrayed in yellow blossoms. A brave suggestion is this flower of the late autumn, blossoming when all else is in the sear and yellow, that it may bear seed in another year. When all others have given up and are retreating, this one comes forth as much as to say it is never too late. There is a very witchery in the crinkled yellow flower born of the old year in a frosty world; a borean child brought hither on the wings of the North wind; a sturdy blossom that will not show itself till it hears the music of rustling leaves.
Late in autumn the white pines shed their needles and lay down a new carpet. No turning of the old here, but every year another---fresh, wholesome, fragrant; a plain, well-wearing groundwork that never offends the eye and on which is traced from time to time a rare and original design. It is now a scarlet tupelo or a maple leaf dropped here and there, and again a creeping mitchella with a red berry or two, or a clump of ground-pine and a drift of beech and scarlet oak leaves. On occasion appears a solitary gleaming amanita. Over the rich seal-brown of ancient hemlock stumps is a tracery of the gray-green cladonia with its scarlet fruiting cups. What are Tabriz, Daghestan, Bokhara and the rest to this? These odorous pine-needles are the magic carpet which gently conveys one into the sylvan world of faun and nymph. Now it is a sunbath we want rather than a cold dip,---to bask in the warmth like any cottontail. To lie in some sheltered spot while the frost is taking off the last leaves, and become saturated with sunlight, is a mellowing process, and ripens one,---as tomatoes are ripened on the window-sill or grapes on the trellis.
As the vivid hues of the red maple fade in the swamp and are replaced by the soft silvery gray and purplish sheen of the bark, the oaks on the hillside become ruddy. The coloring is rich and subdued, rather than brilliant and glowing as at first---mahogany and maroon set off by the purple mists of Indian summer. And now at last branches are bare and leaves rustle underfoot.
\gutchapter{PASTURE STONES}
In New England pastures, the boulders are as much in harmony with their environment as any tree or shrub. They have the appearance of having grown here, quite as naturally as the bayberry and the sweet fern, and are kindred of the savin, and the low-spreading juniper which circles round them and hugs the stone like the lichen itself. The migrant boulders from the North are congenial to these hardy northern plants which reflect the somber character of the rock.
A field that has been entirely cleared of its pasture stones and left to stand thus, somehow looks barren and deserted. You feel you would like to restore a boulder here and there and invite the juniper and the bayberry to return. There is character in these ancient pasture stones, and they cannot be removed without depriving the landscape of that which they imparted; it is no longer virile and forceful, but tame and meek as though shorn of its strength.
If you would build your house on truly historic ground, lay it on foundation of pasture stones, and incorporate, as it were, Time itself into the structure. This is to let the very elements work for you. On many a farm the boulders are as good a crop as any; when they are gathered into the walls to give room for one more lucrative, this value at least of the farm is still represented. The fields have produced but one crop of boulders, and only the ages could mature this. If the pastures must lose this ancient beauty, let the house gain by it. Build it into your chimney. Take it to your hearth that it may not be lost. Let the boulder tell its story by the light of the hickory logs.
There is a rustic notion that boulders somehow \textit{grow}, in some inexplicable manner enlarging like puff balls and drawing sustenance from the earth---and what could be more puzzling to the uninitiated than the presence of these pasture stones? His was an ingenious mind who conjured up that remote ice age from this fragmentary evidence and derived a history from these scattered letters and elliptical sentences. It was like tracing the stars to their origin.
It takes a bold imagination, indeed, to see these familiar fields and woods overlaid with a mile's thickness of ice; to recognize here in this present landscape a very Greenland, redeemed and made hospitable. There was need of a solid foundation of fact, patiently garnered, before such an arch of fancy could be sprung. What chaos and desolation once reigned here, only these boulders can tell. Here was a frozen waste as barren as the face of the moon. But beneath lay the soil that was to nurture the violet and the hepatica. There was a fine satisfaction in riding a miracle like this to earth, to corner it and see it resolve itself into the working of natural laws.
Nature appears as intent on breaking up the old rocks as in forming new ones. The ledge is, after all, but a mass of masonry in which huge blocks are set without mortar and as closely and evenly as jewels. What a lathe was that ancient glacier in which to turn and smooth these rough gems; or rather a great file which rasped their edges and corners. In rectangular blocks that have weathered, the decay is deeper at the corners, so that a cubical block tends to become a sphere as it diminishes. Frost is the stone-cutter, who scatters his chips over the world; Rain, the giant who is bent on turning these into soil. Consider what power lay in this tongue of ice which licked up the crumbs of the earth; carried Canada into New England and New England into New York, depositing its burden as gently as the petal falls from a rose.
Boulders are to be considered veterans of glacial times, which carry still the scars of that strenuous day. What tales they have to tell of that mammoth conflict, that prehistoric incursion of the Arctic hosts, but only to very good listeners are they unfolded. You must needs have a sympathetic ear to become their confidant. The unconscious rock assumes dignity in view of its past, as though here were an imprisoned earth-spirit, proceeding thus through the strenuous life to some ultimate freedom. Sermons in stones indeed! A terminal moraine is the most ancient battle-ground of the world. Here are the very heroes themselves, stretched upon the field in imperturbable granite, as certain others were fixed in the heavens as constellations. To walk among them is to see in fancy the advent of the wall of ice, mile-high, which buried the primitive jungle forever. Here the great glacier began its retreat, and over the spot there broods a silence, as over historic ground once the theater of great actions. After untold centuries, the wild rose and the hay-scented fern cluster round the boulder, and dandelions star the grass.
I please myself with imagining the venerable pasture stones to have been observant of events and to have retained the memory of it all, as the Colosseum might have memories of Rome, or the Sphinx of Egypt and the desert. Such have seen races live out their lives and disappear. That every dog has his day might well be a maxim among these ancient ones of the earth who saw a tropic jungle resolve itself into an Arctic solitude and as slowly give way to a temperate zone. I salute the pasture stone as having witnessed the advent of man upon the earth. It is difficult to associate the tertiary animals with anything but the museum, or to realize that those preposterous Paleozoic reptiles were ever other than fossils. But here is a weather-beaten observer that was actually contemporary with that life, to us so intangible and shadowy; that knew the ancestor of the horse, and ages before the separation from the mother ledge, it may be, was wont to see the sky darkened by flying reptiles.
They were fashioned roughly, these boulders, cast in a rude mould, as if they had emerged from chaos itself before form had become defined. The sea would have all the pebbles on its shore of a size and shape. It takes a block from the cliff and turns it in its lathe that it may become a polished sphere, as in that larger and cosmic lathe the planets are turned. On the beach are innumerable stones that look as much alike as so many eggs. But no two pasture stones are the same. They were turned in no such precise lathe as the sea's, but by a rough-handed force, which here planed a surface and there gouged a depression. Pasture stones are thus almost as individual in appearance as men. Here is one squat like a toad, one humpbacked as a dromedary, another flat as a cake---a mere slab of granite. They are wrinkled and deformed, as so many gnomes, and covered with excrescences---razor-backed or round-shouldered, lopsided or with protruding paunch, while the great solitary boulders rise from the pasture, massive domes and pinnacles of granite.
But none are polished, none are symmetrical; nowhere is there an ellipsoid, such as the sea loves to turn, but rough outlines always. Frequently one surface is rounded; the work of making a sphere was begun but progressed only thus far. Again, two surfaces may be approximately parallel and the remainder rough and angular. Commonly it is an affair of many angles, all unequal, and of a multitude of curves of different radii. It is cast in a mould it would be difficult to classify. With the multiform aspects of crystals, they are still not so varied as these pasture stones. For crystals, for leaves, for snowflakes, there are definite patterns. But the boulder is a thing by itself, subject to other laws and formed under a different order of architecture---or under no order---but the will of the glacier, which has left here and there the marks of its icy fingers.
There is a suggestion of friendliness in the way the lichens clothe these stones, as though Nature aimed to cover the scars she could not heal, or to hang them with such rich medallions as the parmelia in token of that ancient service. Here are colors such as only Time can mix,---shades which are the work of centuries, unspeakably softened and mellowed, like ivory and meerschaum and bronze. In its day the Acropolis may have been glaring and crude in tone; the raw marble, fresh from the quarry, needed these centuries to subdue and mellow it. It has acquired a tender beauty unknown to that classic day which saw it in its splendor. Some such service has been rendered to the pasture stone and the ledge. When the Archaean granite was poured out from the depths it must have worn a new and crude look, albeit so fresh and clean. Then it was but so much raw feldspar and quartz and mica. But it has long been wooed by the air and the water, by moss and lichen; the years have lent it beauty, softened its curves, rounded its angles and brought it the richness of age.
Boulders are sometimes clothed with a larger growth. I have in mind one, from whose apex springs a maple at least half a century old. It lies at the head of a swamp, and in autumn this tree is always one of the first to turn. In August when the tupelos show signs of change, the maple is already glowing with color. The tree springs from the very summit of the rock while its main root reaches through a split some fifteen feet to the earth. Looking across the swamp, it appears to crown the boulder with a noble dignity---a landmark in the country round---as if reflecting those elementary forces which conspired to bring about this unusual condition,---the glacier which brought the boulder, the winds which carried the maple seed, the frost which split the rock.
After their many vicissitudes, the boulders have settled down upon the bosom of the pasture and come to be a fixture in the landscape. This present age is to them the serene and mellow autumn of their troubled life. Their day is a thousand years. But they are melting into soil---as icicles dissolve in the sun---in that measureless and yet imperceptible thaw which melts granite. The pasture land is perhaps the dust of a still more primitive race whose life has been transmuted into the dandelion and the thistle.
\gutchapter{NEIGHBORS}
All wild animals are wary and suspicious, even when they do not prey upon one another. What friend has the rabbit, the chipmunk or the weasel? They lead friendless lives and die tragic deaths. Why should not a rabbit gossip with a woodchuck, for instance? One would think their common danger might draw them together, and that they might perhaps learn a little woodcraft one of the other. But caste is nowhere stronger than in the woods. They do not sit at meat together unless, indeed, one is himself the repast.
Like a subtle atmosphere the spirit of the wild pervades the forest. Whoever enters comes under its spell. In the woods the dog tends to revert to the wolf, and savage instincts come to light. On the street he may pay no heed to people, will move in and out among them, himself a bit of civilization; but let him leave the village and go into the woods, and he is suspicious and on his guard.
We have so fostered this attitude of fear and distrust that our wild neighbors are at best but casual acquaintances, if not complete strangers to us. We are like sharpshooters ambushed around the outposts of an encampment. A stray inmate pokes his head out of the trenches and essays to go to the spring for water. Perhaps we let him drink and make a note of that, then---whiz! we let fly at him. We discover what he has had for dinner and a few other trifling matters---and we get his skin. His ways remain strange to us and his language no more familiar than Choctaw. Sometimes we catch him and put him in a cage. But what can be learned of a poor, sullen prisoner fretting away his life with terrible thoughts of distant sunlight and running streams and friendly woods?
The acquaintance of a wild animal is not to be made with a gun. Practically nothing is learned in this way; it is difficult enough to know them without this barrier. But never to have loved the wild things is to have lost much---to have lived less. Any dolt can shoot an animal and have a bag of bones for his pains, but to win over such a creature in the smallest degree implies a victory, and is evidence of the redeeming power of the heart. There is a rare pleasure in encountering deer when you have no designs upon them. Such furtive meetings are in themselves adequate. They have the fascination of lovely faces seen for a fleeting moment in a crowd, instantly to be lost sight of. How little we really know about the lives of animals. We can surmise a few things and imagine a great many, but we \textit{know} next to nothing. Perhaps there is not so very much to know. Their emotions are not complex but simple; their lives run in narrow grooves. That they suffer, much as we suffer, is certain, and the main thing is to be kind. It is impossible to come upon a wild animal and watch it unobserved without deriving a subtle impression foreign to our usual life. There is something in the free, savage existence which is a shock to the thought-burdened, educated mind, and breaks for a moment its prison of glass.
A glen to which I often go is, like most others in the sequestered woods, really populous, while being to all appearances quite deserted. Its inhabitants are closely associated with the brook; they drink at it and all their lives hear its song. This glen is \textit{their} world, and yet they possess it and live in it in virtue of persistent self-effacement.
There are mice and shrews, chipmunks, red and gray squirrels, a woodchuck or two, a skunk, a little gray rabbit, a weasel and a mink. Far from being alone, you are watched by numerous unblinking eyes. From the grass, the rocks, the trees, motionless and in silence these creatures are observing you.
The squirrels have overcome somewhat their hereditary fear, doubtless because we are more kindly disposed to them. As I take my lunch from my pocket, thinking to eat it alone, a chipmunk approaches and sniffs at the package as I put it down. The aroma of bread and butter tickles his nostrils, suggesting some unaccustomed variety of fare, and presently he loses all fear and begins tearing the paper. After a little coaxing he takes a piece of bread from my hand, licking the butter off first with his small pink tongue. He has no sooner eaten it than another chipmunk appears and sniffs the whiskers of the first one. He, too, is overcome by the seductive aroma, and apparently receives some assurances, for he cautiously approaches and takes a morsel of bread. The package is returned to my pocket, and both chipmunks climb in without hesitation, tear off the paper and help themselves. Meanwhile a third arrives, having somehow learned of the good cheer, and it is not long before all three are scrambling over me.
One cold February day, when no gray squirrels were to be seen, and the snow lay deep in the glen, a solitary red squirrel appeared and looked long in my direction. Then by as direct a course as the ground would permit, he came toward me, over the intervening boulders, until he reached the one on which I sat, whereupon he immediately ate the bits of apple I gave him. He had been with me some little time when I chanced to look over my shoulder, and there at my elbow was the mink. The squirrel saw him at once and made off toward the trees. The mink appeared to take no notice of him, but his presence had evidently disturbed the harmony of the occasion.
The red squirrel stands in no awe of man, but he is as untamable as anything in the woods, none the less. Sit quietly under the hemlocks and the chances are that before long he will be scolding at you from somewhere in the tree tops. Presently he will come down the trunk, head foremost, moving mechanically with little jerks, as though pulled by a string, his hind legs stretched straight out above him. Down almost to the ground he comes, holding himself well out from the tree and eyeing you inquisitively. Suddenly he turns and scurries up the tree, chippering volubly meanwhile, to rush out on a limb and continue the denunciation, adding emphasis with his tail with which he seems to gesticulate.
There is no merrier sight in the woods than a pair of gray squirrels in a frisky mood; it is unmistakable fun. The gray is averse to the coniferous woods and the red prefers them; thus each has its territory. Apparently the red is more self-contained and readily amuses himself. He is of a more caustic mood; his fun is not so childlike and guileless. Nor is he himself, for there is a dark streak in his make-up, a certain taint in his disposition and always a satirical note in his laughter among the tree tops.
Eight inches or more of snow, and a hard crust, and it becomes poor pickings for the wild things. Here and there are holes where the gray squirrel has been prospecting. Near by, in most cases, lies the cup of an acorn and strips of shell, showing the squirrel went directly to the right place. It is to be observed how many of these excavations are under pines, sometimes several under a single tree. As late as the 1st of April I have noticed a gray squirrel busy under a pignut, burying the nuts which had lain on the ground through the winter. He would first rapidly shuck them, then dig a small hole, force them well into the earth with a vigorous push with his jaws, and as rapidly cover them again. In this way he would bury a dozen in as many minutes, and then make off through the woods.
Between the squirrels and the mink family the difference is as much a matter of disposition as of structure. The mink is the evil genius of the place. His character has written itself in his physiognomy, glitters in his eye and shows itself in the serpentine motion of his head. His silence speaks. But his presence is agreeable in a way, for it is a touch of that savage nature we do not otherwise get without going back into the wilderness. A squirrel reveals his candor in his inquisitiveness and in his noisy ways; curiosity gets the better of his fears. These psychologic differences are as marked with animals as with men.
I once surprised the weasel in this glen, with a young robin in her mouth which she had just taken from the nest and was carrying home for her family. She dropped the bird when I threw a stone, whereupon I stood by the dead robin and waited, anticipating her return, for I knew the weasel's boldness of old. Almost immediately the sinister-looking creature poked her head from the bushes and, without hesitation, approached and seized the bird where it lay between my feet. Another stone caused her to drop it again before she had gone far. This time I moved the robin some little distance away and stood beside it as before. Soon the weasel reappeared, and going to the spot where she had last dropped it, became visibly excited on finding it gone. She then began rapidly following the scent, like a hound, and at length by a circuitous course, approached, and again took the bird from under my feet.
Almost every fine day in autumn the woodchuck is to be met. He emerges from the bushes with deliberation and ambles out into the open where there is a little clover to tempt him, his tawny legs showing in strong contrast with his grayish back and scraggly black tail. His enjoyment is evident; the sun feels good to him. He is a chilly body, and, like the snakes, cannot get any too much warmth. Now he sits upon his haunches and takes a deliberate survey, then pokes some greens into his mouth with his forepaws. If his sharp ears bring him no suspicious sound, he drops upon all fours and goes to browsing again.
No one has explained why the woodchuck holes up so early in the autumn and comes out at such an unseasonable time in the spring. He goes in while there is still plenty to eat, and reappears when there is scarcely anything to be had. Possibly the habit was acquired in some remote past when the winter may have come earlier in the year, and the woodchucks, being a conservative race and loath to change their ways, have never adapted themselves, but go to bed now as it were in the middle of the afternoon and get up before daybreak, impelled to this early rising by hunger. Soon we shall be walking over his head, but it will not disturb his nap. He will have rolled himself up in a ball for a four or five months' snooze in company with all the little frogs and snakes---a sleepy crowd. The chipmunk is likewise a chilly body, but he is not going to fast---not he---so he lays in a good store of chestnuts and makes all snug for the cold weather.
While the moral of the ant and the grasshopper will doubtless always hold good, there is little incentive for the grasshopper to become thrifty as few would live to enjoy the results. But the woodchuck might well profit by the example of the chipmunk, who loves his comfort and a well-stocked larder in which to snooze away the winter months, a round of dinners and after-dinner naps. Besides his hordes of beech and chestnuts, he is credited with gathering the seeds of the buttercup as well as buckwheat and grass seed. I have seen him on the tips of witch-hazel twigs biting off the nutlets of the preceding year. He has some variety at his table then. The buttercups must be in the nature of a delicacy---his sweetcakes perhaps.
As the weather grows colder the vegetation seems to droop hourly, the bare earth becoming visible, except where the dry leaves have roofed themselves over the huckleberry bushes or in the thick tangle of briers. The rabbit must feel himself rather too much in evidence as the ground is thus exposed, and perforce relies more on his protective coloration to escape notice. An adept at dissimulation, he turns into a stump and remains so indefinitely. Yet looking at him recently, as he sat motionless on some dry leaves among the bare stems of the blackcap raspberries, I was struck with how poor a refuge his colors really do afford when once your eye is upon him. At the first glance, and before he had come into the mental vision as a rabbit, he appeared as a small grayish stump covered with buff-tinted shelf fungi. But the moment I looked sharply at him, he was a rabbit in every detail. His colors did not greatly harmonize with the oak leaves on which he sat, yet he allowed me to approach and walk around him. It is all a matter of the attention; by remaining quiet the animal does not arrest the eye readily, but once this is directed upon him the disguise is seen to be very thin.
Save for his nose, which wobbled slightly, he was motionless as a stone. After some time his ear moved gently, much as a leaf is turned over by the wind, but his eye never winked and its expression was one of extreme alertness. On too near an approach he made off in haste. Noting his direction, I followed to see if I could again locate him. For some time no rabbit was visible, when I chanced again upon a little gray stump covered with buff-tinted fungi, which appeared this time on the pine-needles and just within the charmed precincts of the briers.
I produced an apple as a peace offering and in token of my good-will and desire to be of service to the tribe of gray rabbits. He remained like a stone while the bits of apple descended about him and lay at a tempting distance. At last there was a more vigorous wobbling of the nose, the long ears moved---as a leaf turns---and with two little hops he approached and accepted the token, and we were brought together in amity in the silent woods. A humble offering, indeed, but it served for the moment to bring me in touch with the wild and to strike a common chord. The seemingly impassable barrier of caste, which lies between man and the wild things, was crossed, and we broke bread together.
After a light fall of snow it is instructive to read what the rabbit has written in his diary. Such scattered notes as he leaves are wholly personal and do not seem to imply interest in anything but himself. You may see where he has hopped through his runways and stopped now and then when the necessity appealed to him of removing certain briers to keep the passageway clear. Sometimes it is a stem of the catbrier; again a rose or blackberry. In every case it is cut obliquely and as sharply and neatly as with a knife. Frequently stems are severed thickly set with thorns and prickers, and the wonder is how he closed his teeth upon them without getting an unpleasant mouthful. Hundreds of cuts reveal never a slip or break, but each is sharply defined as if done by one stroke of a razor. His track shows places where he sat upon his haunches, and where he stood up to reach the buds of a stunted wild apple; again he followed the shore of the pond and nibbled the small willows and clethra. Occasionally he appears to have cut a large brier merely for practice in using his teeth.
Rabbit and fox are outlaws and without rights. They are hunted to death; hence they live by their wits if they live at all. It has become second nature to them to proceed indirectly, to break the scent and double on their tracks whenever occasion offers. The fox knows few foes besides men and dogs, but the rabbit must circumvent owls, weasels, minks and foxes as well. Hence I bow to the rabbit as to a superior intelligence: one deeply versed in the ancient lore of woodcraft and possessing knowledge as yet unrevealed to us. Does he carry some charm whereby the earth opens and receives him in need, some tarn hut in which he becomes invisible, or does the fabled St.-John's-wort exercise for his race a special protection? What shall fill the place of the wild things when they are swept from the earth? Why not tolerate an occasional fox if only to hear him yap, and to have the assurance that there is still this much untamed?
In such a timid world, where fear of man is so large a factor, one is struck by the least evidence of self-assurance. In view of this I entertain a covert admiration for the skunk. Fear rests lightly on his shoulders. Meet him in the woods, teetering along, and he is the less concerned of the two. His imperturbability is his leading characteristic. In this he is the very opposite of the coon. But he knows how terrible is the weapon he carries, how vulnerable the nose of man. The nose is the point of attack; he would slay you through your olfactories. It is seldom any one says a good word for the skunk. He must needs be a villain and a chicken thief who smells thus to heaven. Yet in fact there are bolder thieves in town than he, with more sinister designs on the hen-roost. It is impolite to mention him, as though his name were as unsavory as his odor. Men deal more kindly with his memory, for he is permitted to undergo a commercial transfiguration, to rise triumphant from the vat, henceforth to be taken to our bosoms as Alaska sable.
The skunk receives no credit for the countless beetles he grubs from the earth. No more does the mole who suffers for the sins of the meadow-mouse. They are victims of prejudice. When I see a mole emerge from the earth, I feel I am looking upon an inhabitant of another sphere---the underworld; one as strange to me as I am to him. What use has he for the sun? He cares not for celestial light, but for subterranean fires only.
In the pond above the glen is a colony of muskrats. It antedates the memory of the oldest inhabitants, and the muskrats were in all probability the first settlers themselves. The huts, which lie scattered through the sedge and cattails, are some of them flat while others are high and dome-shaped. Their number does not seem to vary much from year to year, whereas muskrats are said to be very prolific. What, then, becomes of all the young? I have never known of any one trapping or killing them in this pond. It may be the old mink in the glen, and many another, make this their hunting-ground and thus keep down the number.
These queer neighbors pique our curiosity. What manner of life do they lead indoors? They take some rude pleasure and have dull animal thoughts perhaps. As you stamp upon the ice and slap your hands to keep from freezing, the muskrat sits serenely below enjoying the comforts of the pond, and quite unaware the mercury has dropped to zero. He has built him a house and stocked his cellar, and what cares he. As snug as a mouse in a cheese, he has taken the precaution to make his home of his favorite dish. Let the world freeze, then, if it will, he nibbles the walls of his room till it thaws again. Consider the interior of that dwelling, what a murky house is there, its front door under water and never a window.
Muskrats repair and enlarge their huts in the fall, and perhaps subsequently gnaw out as much from the inside as they add to the exterior. The walls are made of grass and sedge roots, together with spatter-docks and bur-reeds. During the summer you might not suspect the presence of one, hidden as they are in the cattails and rank growth of sedge. As the vegetation dies down in autumn, the huts loom proportionately, so that they come prominently into view by November; and then, on some fine cold morning, in place of the reedy pond, appears a sheet of ice with isolated domes rising here and there. From these, the muskrat and his family travel to their feeding-grounds. They have chosen their estate at the bottom of the pond---rich lands for which none contend with them.
In fact our wild neighbors all live in a dim world of shadows, in which they lurk like phantoms. They have retreated into the night, and for days together you may not meet one. But the new fallen snow reveals their presence.
\gutchapter{THE WINTER WOODS}
The first snow-storm of the season never becomes an old story. It retains its charm indefinitely, to all original minds at least, and to such as have cherished any degree of simplicity. Here is a mimic invasion of an elemental beauty which conquers us by reason of its very gentleness. We are soothed and beguiled into submission. Tempestuous winds call forth our resistance; we front them with set teeth. But who can resist the silent snow descending as if to lay the world under a soft enchantment? The woods are renewed and reclothed in virgin purity. It is as if old scores were wiped out and the world were again a spotless thing.
What can be more companionable than the falling snow? Its touch is so caressing, its advent so silent in the open, its voice so pleasing as it sifts through the pine-needles. The first solitary flakes approach with the gentle effect of preparing one for the miracle to ensue. A calm settles over all, as though these were indeed the messengers of peace.
Recently there fell such a clinging and abundant snow as comes perhaps only once in a season, and some years not at all. The woods were literally buried and saplings everywhere bent to the ground beneath its weight. It enveloped the pines until they became miniature Alps in the landscape, while among the oaks were gleaming corridors and marble halls. The open, barren aspect common to winter was gone, and the dense walls had shut in again as in summer, but now crystalline and dazzling.
This is perhaps Nature's greatest transformation. In a single night have been erected such palaces as were never seen in Persia. What a bold, free hand wrought here! In the thousand domes and arches is a massive architecture, relieved by the utmost delicacy, as though Nature said, ``Behold, I show you a miracle.'' A miracle indeed! Here have wrought the genii of the air while mortals slept, and all that was to be heard was the rustling of their wings. At such times the woods grow suddenly strange and unfamiliar. They so lend themselves to the enchantment we are lost in our own wood-lot. Familiar paths are obliterated by pendulous boughs drooping to the earth, while in the pasture tree-sparrows hop upon the snow among the protruding tops of the tallest ragweeds.
Realize if you can in your walk, over how many sleepers you step all unknown; how many woodchucks in their burrows, and frogs in the mud under the ice; how many torpid snakes and dozing chipmunks. Here is an enchanted household---underground. They are at peace and their timid hearts know no fear. The dreaming toad has no terror of writhing blacksnakes, and the snoozing woodchuck has forgotten the dog. Presently they will awake to hunger and fear again. Woodchucks will be up long before breakfast, to go shivering in the cold dawn of the year waiting for the table to be spread. Snakes do not come out till the sun is well up, to lie basking in the noonday heat, catching the first unwary grasshoppers.
Every fresh snowfall makes some revelation of its own, recording crepuscular journeys and prowlings in the night. The broad track of the skunk meanders in and out among the bushes. That he had no definite direction, took never a straight course, nor apparently did he hurry, is in itself evidence of his phlegmatic temperament and leisurely habit of mind. Footprints of the ruffed grouse show that he has on his snow-shoes, inasmuch as they are feathered, broad and lobed rather than angular. The squirrel leaves evidence of his impetuous ways, moving always impulsively, and the snow makes plain record of the fact. Tracks of deer seem to bespeak their innocence, as that of the fox might be said to have a sinister purport, doubtless because the hoof prints have a gentle suggestion and imply the herbivorous diet.
In the winter walk the eye finds relatively so little to hold it, that it rivets itself upon minute details, dissecting that which might pass unnoticed at other seasons. Form and outline come into prominence while color is in abeyance. We must now perforce judge the trees by this standard. Who shall describe the winter beauty of the beech as it stands stripped and naked to the winds like an athlete, every muscle and sinew in evidence, every outline expressive of reserve power and self-assurance---a clean-limbed, stout-hearted tree, dauntless before all gales? Its trunk is a superb torso, and with its roots it reaches down to the heart of the earth, draws sustenance therefrom and derives heat from that deep-lying warmth below all frost lines. No parasite this, no surface weed, but the sturdy child of Earth herself, suckled by a Spartan mother. Look upon an ancient beech, bared thus to the storm, and the chest involuntarily expands, as though we too should take firmer hold somewhere and stand more erect. The shellbark is as shaggy, raw-boned and loose-jointed as the beech is trim and closely knit. Its bare branches are not clean-cut against the sky but swollen and distorted like knotted hands of toil---horny, crooked fingers upraised to the heavens. What rude strength is their portion who stand thus alone and derive from the earth as befits the stalwart---buffeting, solitary and unyielding, the winter gales.
[Illustration: WINTER COPYRIGHT, 1908, BY RUDOLF EICKEMEYER]
As the trees are leafless, the bark is now more in evidence. Moosewood looks slender and striped as a ribbon-snake, and limbs of the hop-hornbeam have the appearance of sinews. Where a black and a white oak stand near together, the difference in color is as evident as between a negro and a white man. The white birch is to the winter woods what the dogwood is in spring, the maple in autumn. How is it the ancients did not metamorphose the fairest of all nymphs into this tree, so distinctly feminine is its beauty? Portions of bark outlast the wood, and are to be found standing erect and empty. The tree has departed, bequeathing its fair skin in token of a vanished loveliness. Now and then the yellow birch is seen in all its beauty, the golden inner bark shining through a silver filigree. To look at this tree is like looking at a picture or reading a poem: one feels somehow refreshed. Nor is the black birch without charm; its bark has a dusky beauty, and again shows fine wood colors and metallic tints similar to the black cherry. This fine luster the birch has in an eminent degree while most trees show it only on their small branches, if at all.
Club-mosses appear to be a lesser growth of pines, a pygmy folk dwelling at the feet of the elder race. Here are miniature trunks and branches bearing miniature cones, perfect little conifers no higher than a chickadee. Ground-pine and trailing Christmas green thrive together on the bank, the latter with stems a yard long, which, while they grow at one end, die at the other. These little plants are crisp and green and refresh the eye on winter days, as does the Christmas fern, which affords a pleasant encounter at a time when one meets few acquaintances. It has, moreover, a certain charm of its own which doubtless lies in the crispness of the fronds and clear-cut outlines of the pinnae. The marginal shield-fern is another acquaintance to be looked for on the winter walk, and everywhere the hardy polypody, which is as much a child of winter as the little spiny cladonia that clusters about its roots and clings to the same granite ledge.
Let there come a warm rain, the high blueberries redden their twigs and the lichens renew their tints---quite as though Nature had softened her heart. These lichens suddenly become conspicuous with a sort of gentle prominence, and mildly compel attention; on the oaks the yellow cetraria, on the white pines, olive, slate-colored and blue-green parmelias. Had faun and satyr thus carved upon the forest trees the name of some fair Rosalind among the nymphs, they could not have wrought in more fitting and altogether sylvan characters.
A common necessity and hardship hold the birds together in closer bonds so that they are impelled to consort in little roving bands---chickadees, creepers, kinglets and nuthatches, with often a single downy woodpecker accompanying them. If one chance to drop a morsel he will descend to the ground in search of it. He will not waste a spider's egg, so severe has been the lesson in economy. In zero weather the jay forgets to be saucy, and if there is a glaze on the snow, his native impertinence seems to ooze from him, and he becomes meek enough. Taking a weazened acorn from the tree, he holds the nut with one claw, and with vigorous taps of his bill tears it open. After extracting the frozen kernel, he drops the shell with a trace of his customary impertinence, as though feeling in somewhat better spirits for even this poor repast. A bone nailed to a tree is inducement for him to stay near the house, but not when he can get acorns readily.
The board may fairly creak with its weight of partridgeberries, beechnuts and acorns, many of the latter crushed and available, and then in a night this plentiful feast is put out of sight under a six-inch layer of snow, to which the next day adds a glaze as if to seal irrevocably the doom of all bob-whites. A fast has been declared in effect, as peremptorily as by any medieval pope, to be broken only with an occasional leaf bud or the poor seeds of the ragweed. But the good sun is a trusty friend, and snow is only so much water. Presently berries and acorns again come into view.
There is no more touching note in nature than the bob-white's at this season, as wandering together in the snow in search of their scanty fare they utter from time to time those low but distinct calls in which they seemingly express their solicitude. June itself has no sweeter song than this note of the winter woods, albeit it is such a plaintive one: mother-notes these, and child-voices of the hunted, full of a wild pathos,---tender voices which to us have been but the inarticulate cries of the dumb. The birds feed frequently on the crushed acorns lying in the path, and the jay at times participates to the extent of taking an acorn from the feast and eating it in the branches above, where he is a good sentinel, though prone to imitate the quailing of the red-shouldered hawk when the feast is at its height, to the general discomfiture and alarm of the diners below.
Birds become less suspicious as the mercury falls, and they are hard pressed for food. The snow around the ragweeds is thickly covered with the tracks of bob-whites, like those of chickens, broad and firm, but with hardly any hind toe mark at all, as though they walked about on tiptoe. Very different from these are the long, triangular tracks of the jays, showing where they have hopped upon the snow. It is thus fairly tramped down and strewn with leaves and chaff where the bob-whites have fed, leaving these husks in token of their frugal meal. Such seed must be very small provender for these birds---much like a diet of crumbs for a hungry man. Goldfinches, juncos and tree-sparrows seek the same meager repast. The musical flocks of redpolls fare better in the alders around the pond. These are not to be seen every day, any more than the pine-siskins---perhaps not at all during several years. But occasionally an enormous flock will arrive and settle in the alders with all the chattering and commotion of a social and hungry company. As the seeds are shaken down upon the ice, the birds soon leave the bushes, and are under the table, so to speak.
Crossbills have the easier time, feeding as they do on the seeds of the pine, for these are always available. No sound seems better to accord with the spirit of a still cold winter day than this faint crackling of opening cones, forced asunder by the shearing motion of the peculiar bills of these birds. Surely here is an adaptation to definite ends. Nature produces a cone that cannot readily be opened, and, as if relenting, produces a bird to open it. The wings of the seeds come zigzagging to the ground as the feast continues overhead---all that is destined to be planted.
The lumbermen come into the woods with the crossbills, and everywhere is heard the winter music of the ax. It is good music enough, but it has a sinister purport, and the swish and boom of falling trees is a sad refrain. Ancient pines are laid low, singing to the last their brave and beautiful song, which seems to come, not directly from overhead, but remotely from the empyrean, as though it issued from the distant Court of the Winds. Of the pantheon of trees the village elm is the last to hold our homage; we have dethroned our idols. As the sound of the ax breaks the stillness, I find myself instinctively turning in the opposite direction, to escape that which is soon to follow---the swan-song of the forest primeval.
\gutchapter{LAUGHING WATERS}
There are days when the sea is austere and unapproachable, when its mood is too lofty and severe. But the pond, fringed with alders and button-bushes, smiles in the sunshine and is friendly and inviting. It is more on the level of our every-day thought. Not always are we consoled by the vast and sublime, and we crave even more the companionable and social aspects of Nature. Grim though the surroundings of granite ledge and somber pines, the nestling pond is winsome, notwithstanding. Never forbidding, never altogether distant in its mood, even though frozen, it is a cheerful and alluring personality to which we are drawn from afar.
About a pond as about a mountain there is a kind of magnetism. A new field of discovery, there is ever the hope that from a new scene we shall gain a fresh impression. Every pond holds out this possibility and invites exploration of its shores, as if \textit{there} were the promised land. But over and above this is that element of personality, a charm purely feminine, and eluding any attempt to hold it.
Peculiarly sensitive to light and air, a pond is susceptible of little moods that do not come to the sea. It is the eye of the landscape. Dawn, high noon and dusk are each reflected there. Its afternoon mood is not like that of the morning any more than is our own. The more passive it is, the more perfectly it reflects the heavens. At all time it draws to itself light from the sky, and when the surrounding woods are swallowed in the advancing darkness, still gleams with a faint opalescence. These pale glimmers illumine the bogs, where a pool has caught and retained the daylight, or rather the spectral light of dawn. One appears to look through this serene and reflecting surface into the heart of some other wood, darkly mysterious and impenetrable, which vanishes when the wind blows, as if the curtain were drawn.
Gently as snowflakes, the leaves detach themselves and settle on the ponds, to sail away like diminutive barks upon those friendly seas. Numberless sails of scarlet and gold softly scud before the breeze, threading the inlets between the button-bushes and crowding the miniature bays; oriental craft these, of rich aspect; caciques and royal barges upon some Golden Horn. Here and there, one more venturesome steers boldly out into the open, carried by favoring winds, and makes some foreign port among the lily-pads. You may become enamored of a winsome pond on October days, a mystical beauty veiled in autumn haze, only to find her mood changed for the reserve and uncommunicativeness of winter.
When the pond freezes over we experience something of that feeling which comes with the first snow, a delightful sense of novelty, briefly entertained each season. The water has suddenly lost its mobility and become passive and expressionless, as one in a hypnotic state. A great calm has settled upon the earth; the winter sleep is in the air and the ponds have succumbed with the woodchuck. Only the chickadees, scolding and gossiping in the pitch-pines, seem to be awake and unaffected by the change. A cold bluish light pervades the leafless woods, reflected from the snow and appearing to emanate from the ground rather than the sky. The earth is wrapped in silence, yet it is not austere nor repellent. One \textit{feels} this stillness, which appeals to some sixth sense, and is more acceptable at times than any music,---is itself the most heavenly music.
Far across the valley the steam of a passing locomotive rises slowly, and then, like the opening of a flower, unfolds in snow-white voluptuous petals and remains as if carved in the still air. A shaft of light reaches the eye from a distant pool of molten silver at the base of purple hills. All around are little sparkling lights of icicles, flashing their pure rays in the sun. It is the magic water, the protean thing so full of light, laughter and music. Once it was laughter; now in the silence it is light.
All at once the pond is alive with skaters, its solitary aspect transformed by this merry invasion. Boys cutting figure eights suggest whirligigs. Myriad black figures, clear cut in the pale light, move in and out with undulating rhythm, as on a surface of polished steel. The pond, now more companionable than ever, becomes a playground, and we never so much as reflect upon the strangeness of it. Something there is in this unbending on the part of Nature which puts us in a good humor, for certainly people are never more good-natured than on the ice. Their habitual stiffness melts away as readily as ice melts in the sun. They experience a thaw and become democratic.
To skate over meadows and into inaccessible bogs gives one a taste for exploration. It is a new freedom and perhaps the next thing to flying. Seen through the clear ``black'' ice, familiar objects have an added interest; the pebbles on the bottom, the spagnum, the lily-pads, all give the impression of being severed from our world, though so plainly in view. The skater glides in and out amongst cassandra and andromeda, clethra and black alders---wintry jungles, enlivened only by red winterberries---where in summer is the haunt of the rose pogonia and the white-fringed orchis. Who would imagine now that the swamp was capable of producing anything so exquisite, that it held beneath the ice the seeds of such beauty?
The most friendly voice in Nature is the song of the brook. Not the wind in the pines, not the voice of the sea, can compare with this for true sociability. These are always somewhat remote, somewhat mystical in our ears, but the song of the brook is cheerfulness itself. Its \textit{bonhomie} is irresistible. It gradually prevails over any whim and wins us to a sociable and contented mood. Though the world may seem discordant enough, there is always this wholesome note.
No two brooks are alike. As the result of the character of the country through which they flow, they impress one as having strongly defined personalities. A creek flowing sluggishly through the alluvial districts of the South is insipid compared to a mountain stream in New England. Your mountain brook is a strong, salient personality which dominates the landscape. It sweeps in bold curves about the base of cliffs, and contracts into a mere mill race cut in the distorted schist and gneiss. Its suggestion is wholly of savage strength, a rude, forceful thing of the wilderness; its song a masterful strain, a triumphant chant of power. Again, there are merry little streams tinkling in the sunlight.
In cutting down its channel, the brook may reach a stratum seemingly richer than any above, so that in April its banks become a garden. While scarcely a flower is to be seen on the hillsides, the fertile floor of the ravine is carpeted with spurred violets, groundnut and spring beauties.
One such as this falls into a glen over a little precipice, spreading itself out like a fine veil which ceaselessly undulates in the breeze, and now and again floats away in mist ere it can reach the pool below. Under the overhanging rock, Alpine woodsia and cliff-brake thickly cluster, while on narrow shelves are hanging gardens of dicentra, and in the crannies, little patches of mountain saxifrage.
Below is a golden sheen where the spicebush is in flower, and a shimmer of pale green about the early willows. From the glen comes the song of the ruby kinglet, bubbling up and dying away. Incomparably wild, it seems to express the abandon of a spirit ever free. All the while the companionable brook gurgles and tinkles its reposeful melody, and the white veil of the waterfall undulates softly in its dark cavern. The air is full of that indescribable suggestion of spring, which is like hashish, and casts a glamor over the world. Gradually one is imbued with a sylvan consciousness and attains to a rapt and intimate point of view.
It is curious, as one follows down the ravine, to hear the different voices. The brook seems as if inhabited by a number of spirits throughout its length, some whispering, some laughing, others singing. Not only are the voices pitched in various keys, but the quality of tone differs essentially. Some are loud and portentous; others, melodious, liquid gurgles. In one place the voice implies an intimate and confidential mood, so gentle, so exquisite, that the full import of the musical conversation is felt only in midstream,---whispers and murmurs which have almost a ventriloquial effect.
Countless bubbles glide down the current and vanish one by one. Sunbeams dance over the rapids and out upon the pool, and then, as the sun goes under a cloud, the stream as quickly takes on a somber mood. Presently comes the melodious patter of rain-drops on the ground, an even, sustained note, very different from any voice of the brook as it dimples and answers the rain, one soft voice replying to the other. Already little pools form in hollows of the rock and reflect light, so that the face of Nature is perceptibly brighter.
Considering this aspect of the streams, it is easy to see how the primitive mind came to personify them, since the brooks have motion, voice and expression, ripple and laugh in the sunshine and are responsive to the wind and the sky. They are still divinities to the fisherman with whom he comes into an ever closer affiliation, as gentle and poetic as he may be qualified to enjoy. The murmuring waters, the whispering trees, the silver and cupreous gleams of trout are the facts with which he becomes enamored, while he loses affinity with the world, which slips into the background.
\gutchapter{THE MOUNTAINS}
He knew the mountains, who said, ``I will lift up mine eyes unto the hills from whence cometh my help''; knew them in some intimate, spiritual way, for his words imply a noble association and companionship. Wordsworth understood them in this way, but not as the mountaineer knows them. They are ethereal dream-mountains the poet sees, rather than actual rock and soil.
On the horizon the mountains wrap themselves in mysterious light and color and seem invested with certain qualities which they lose near at hand,---as a cloud, so beautiful an object floating in the heavens, is but a fog bank once we are enveloped in it. Distance does actually lend enchantment. The range beyond has always some attraction this one lacks. In truth, mountains are illusory objects, and, to the most matter-of-fact point of view, are something more than rock. That marvelous purple of the distant hills, assumed as an imperial robe, slips away as we approach, and we find them dressed in plain brown homespun. Never do we as much as touch the hem of that royal mantle.
A symbol of the unchangeable, they are none the less marvelously sensitive to the play of light, and thus appear to vary with the conditions of the atmosphere. There are days when they seem to approach, and times again when they recede and become distant and nebulous. This magic-play of light proceeded from their birth, and goes on forever, the unceasing illusion, the beautiful witchery.
From the violet shadows of their bases they rise through a stratum of ethereal blue to emerge glistening white. Now they are savage and defiant in their somber shadows, ramparts and battlements; again, opalescent, lying like cumulous clouds on the horizon. What a vast bulk is yonder spur, massy and ponderous in this light, but tomorrow it may appear immaterial as thistle-down and to hang suspended in the ambient air. In the morning the crags and cliffs stand out naked and dazzling on the great rock mass of the peak; yet before night every detail may be obliterated and the mountain appear a lowering mass, dull and grim.
It is with mountains somewhat as it is with people---there must be perspective if they are to appear all serene and beautiful. In the distant chain the details are lost, and we receive a single distinct impression of serenity, as though they stood there a type of the fixed and eternal. But in among them there are everywhere signs of convulsion, everywhere evidence of change and decay. It is in the distance, then, that the poet loves them best, as a beautiful vision, which lures and beckons him. It is to these he lifts his eyes, from these he receives his inspiration, for they are ethereal and opalescent and play upon his fancy, provoking him to subtle thoughts of the Ideal, rose-colored as themselves.
They who do not live where they can see the mountains miss somewhat in their lives, as do they who never hear the sea. It would seem as though one or the other were essential to a normal human environment, providing that changeful beauty which forever stimulates the imagination. We necessarily lift up our eyes to the mountains with some corresponding elevation of thought. Again, from those desolate heaps of granite we receive the suggestion of something immutable and permanent---delusion though it may be. Whatever convulsions they may have known were birth throes and growing pains. Venerable beyond human conception, their life is measured, not in years, nor yet in centuries, but in epochs and eons of time; and out of this inconceivable antiquity, with its tumultuous youth, has come repose at last---a serene old age.
One readily understands in the mountains how the old myths of the gods and giants arose. Why should not the gods have dwelt on Olympus---and here in the Rockies as well? What place more fitting? A setting, stern and heroic, and not altogether hospitable to the puny race of man. There are places of such sublimity and desolation, you feel you have looked in upon Olympus when the gods were away, and that any moment they may return with their thunderbolts. Wandering alone in these regions is like an excursion into legendary lore---and one would better wander alone, for in our deepest moments the mountains are company enough.
One companion you may have---should have, in the mountains---a horse, a kindly and sociable animal, who knows your foibles as you know his, and is willing to humor them. He must be a trail-horse, sure-footed and not finicky about fording mountain streams. If you do not come into some renewed sense of freedom, if the solitude does not speak to you, if you do not become better acquainted with yourself, it is because you really have not surrendered to the genius of the hills but have come preoccupied with other and lesser things. Thoreau did not so greatly exaggerate when he said one must make his will and settle his affairs before he was ready to walk.
One does not tire of sauntering through the mountains. They seem always to invite. Mystery lurks in the ravines. There is no sound but the distant tinkle of a cow-bell, which is pleasant music. Over the ranges and the velvet folds of the \textit{mesa} the lights and shadows play like a passing smile.
Though the ideal eludes on a nearer view, we nevertheless derive some larger sense of freedom from personal contact with the range. The foot must know the trail; and this association yields that which no road can ever give---a good understanding with the mountain itself. As far as the eye can see, neither fence, nor house, nor road; only the somber forest, the naked ledge. While this tramping over trails hardens the muscles, it toughens also the sinews of the mind. One has mountain thoughts as well as mountain air. The single drop of aboriginal blood tingles in the veins, while the tendency is strong to revert to the wild and to a more rude and savage life. There is experienced some furtive desire, as of a wild animal, to scurry away into these grim ravines, or to leap from crag to crag with the bighorn,---presumably a sort of mountain madness, which is dispelled on the descent to the village.
Who can hear the wild song of the ouzel and not feel an answering thrill? Perched upon a rock in the midst of the rapids, he is the incarnation of all that is untamed, a wild spirit of the mountain stream, as free as a rain-drop or a sunbeam. How solitary he is, a lone little bird, flitting from rock to rock through the desolate gorge, like some spirit in a Stygian world. Yet he sings continually as he takes his solitary way along the stream, and bursts of melody, so eery and sylvan as to fire the imagination, come to the ear, sounding above the roar of the torrent. Like Orpheus, he seeks in the nether world of that wild gorge for his Eurydice, now dashing through the rapids, now peering into some pool, as if to discover her fond image in its depths, and calling ever to lure her thence from that dark retreat up into the world of light and love. This bird, more than all others, embodies the wild. In him the spirit of the mountain finds a voice.
Here we make the acquaintance of the rocks as no where else. One discovers their individuality and comes to feel that even they may be companionable. They have much to say if only one can hear it; but like the aged, their conversation is all of the past. The foibles of their youth are still to be traced in faulting and non-conformity. How tumultuous was that youth; how serene their old age! Stratified or volcanic, each tells its own story. The sandstone cliffs speak of the sea, which preceded them, and of which they are the sediment merely. Upon that shore no human eye ever looked, and yet it is registered here, as the ruins of Mitla record a race unknown to history. The cliff is a chapter in a biography written before the advent of man. Long after the sea had disappeared, some convulsions upheaved the strata and threw them on end. Here and there in the canyons glimpses are to be had of the granite or porphyry which underlies the sandstone---the very corner-stone of the hills. It is as though one had come upon the most ancient papyrus of the world or unearthed the first Babylonian inscription.
It seems incredible the stream should have sawed its way through so many feet of rock and produced the canyon. Day and night it eats its way inward, like a saw cutting to the heart of a forest tree. But see what the rain will do---so gentle a thing as the falling rain. Together they have hewn the cliffs, which are like vast rock tombs with their Egyptian massiveness. A filmy cloud floats down the gorge, trailing along the edge of the precipices, an intangible and shadowy form, spiritlike and ethereal, receiving the rays of the setting sun and becoming golden and then rose-colored, and dissolving away at last into the invisible. This fugitive, shadowy thing, this bit of mist, is the mountain sculptor.
The rocks were the prototype of the temple, as was the forest of the Gothic cathedral, the date-palm of the Byzantine dome. But there worships here only the canyon-wren. He is the high priest who lifts up his voice in these rock temples---a sweet utterance delivered with the usual abandon of the wrens.
Above the cliffs, on the precipitous slopes, is the impress of still another agent. The ledge, smoothed as by a plane, and the scattered boulders amidst the dead timber and small aspens, give it an appearance of extreme desolation. Here, where now the Indian paint-brush glows in summer, the glacier crept snail-like down the mountain, from its cradle in some \textit{cirque} above the forest. Timber-line is the frontier, the boundary between the verdant world and the land of snow and ice.
It was the glaciers which in the days of their strength chiseled the lake basins every one, and began the great canyons on which the streams have been at work ever since. At the same time they laid out the \textit{moraines}, like so many parks, where the pines and the spruce have planted themselves. They did the rough work and prepared the great rock masses for the finer work of the rain and frost and wind---as the stone-cutter precedes the sculptor.
These lakes in many cases became the glacial meadows of today, which are like jewels set in the vast matrix of rock. Out of elemental changes, terrible in their immensity, came some of the most charming of all wild gardens,---as a rainbow follows a thunder-storm. These serene and altogether beautiful aspects of Nature were the outcome of tumult and passion---earthquakes, avalanches, lava-flows, glaciers, and now these idyllic meadows, beloved of bees and blossoms.
There is a certain canyon hereabout which is closed abruptly at one end by a precipice, over which descends a considerable stream. This fall is a thing of beauty, and so holds the eye that few think of scaling the cliffs to see what may be beyond. But, as it happens, there lies above, and sundered from the world beneath, one of the most delightful little valleys in the Rockies---a long, narrow defile, flanked by perpendicular cliffs of pink and red and buff sandstone.
All day the black-headed grosbeak sings in the aspens, dropping from one reverie into another. You may hear the voice of the green-tailed towhee, and the canyon-wren singing from his rock temple. The stream winds along the floor of the little valley, which is some eight thousand feet above the sea, now through quaking aspens and now under spruce, and its voice is as the murmuring of pines. This is the haunt of the shooting-star and the Alpine mertensia, delicate and exquisite blossoms, wooed by fugitive sunbeams and by the floating mist; which dwell in a subdued and tempered light amidst the Alpine silence, as in some floral cloister. Such are the rare and beautiful places of earth, which the mountain barriers defend and the clouds veil, as if they cherished here the last vestige of the fading youth and innocence of the old world.
There are days when the clouds shut down upon the little valley, veiling it from mortal eyes. The cliffs and buttes seem to float in air; the trail becomes a path to the clouds. You have only to go up on some ridge, and the pinnacles, looming in the fog, appear to be forlorn rocks in mid-ocean. It is the isolation again of the sea and of the desert.
At such times one receives impressions from the mountains which bring to mind the ocean, as if these retained memories---as they still bear traces---of the waters which gave them birth. This relation, once so intimate, is now sundered and only to be inferred. Where is the ancient sea which mothered the Rockies? The desert is its vast bed, now unoccupied. It vanished forever, leaving its impress upon the mountains. And now this sea-child is in its dotage, and it too dwindles and wanes century by century. But the fog still recalls the mother-sea, and out of the forgotten past conjures up little waves to dance upon a primordial beach.
\gutchapter{THE FOREST}
One who is accustomed only to our eastern woods can have little idea of the true forest as it occurs in the Sierra Nevada, which is a world of itself, as distinct from any idea of the ``woods'' as the snow peaks, the colossal granite domes and the great canyons of the Sierra are different from the mild topography of the Berkshires.
Here is a forest primeval such as was never known east of the Cascade, not, at least, since that remote period when the sequoia flourished in Greenland. Man wanders, a mere pygmy, in a Brobdingnagian world of vast columnar trunks. This is the true home of the great conifers, the sequoia, silver fir, sugar-pine and Douglas spruce,---the magnificent of the earth. There is no wilderness of saplings as in the woods, and the general openness of the forest is remarkable, so that one has far-reaching vistas through splendid arches and is able to appreciate the size and character of individual trees.
Distinct from all others, the sequoias are a race apart. The big tree and the redwood of the Coast Range are the only surviving members of that ancient family, the giants of the foreworld. Their immense trunks might be the fluted columns of some noble order of architecture, surviving its builders like the marble temples of Greece,---columns three hundred feet high and thirty feet through at the base. Such a vast nave, such majestic aisles, such sublime spires, only the forest cathedrals know. Symmetrical silver firs, giant cedars and spruce grow side by side with sugar-pines of vast and irregular outline, whose huge branches, like outstretched arms, hold aloft the splendid cones---such is the ancient wood.
It is doubtful if these giant conifers are really as companionable as our eastern beeches and maples and oaks. The company is almost too grandiose; their dignity is overpowering. One could never, for instance, form such a pleasant acquaintance with a great sugar-pine as with a slender white birch. Fatherly white oaks and village elms seem to ally themselves with man as protecting deities of the wood. But this great race of trees has little affinity with our world. Be that as it may, there is perhaps no loftier association in Nature than contact with the forest. It is the force of a tremendous personality---calm, inspiring, majestic. Like the sea, it is not to be grasped in its entirety, and the mind responds to it, as some giant sugar-pine to the wind. These sequoias, which may easily be from two to four thousand years old, have seen men come and go as so many squirrels, or as bubbles on the stream; they have outlived empires, and may again.
As the forest inspires in sensitive minds the religious sentiment, so does it impose upon all alike, silence. Self-effacement is the law. Wild animals merge into their environment and have acquired protective coloration through force of necessity. The Indian has come to imitate them; it has become second nature to him to move stealthily, to stand and sit immovable for long at a time, to speak little. To the woodsman, silence is more congenial than speech; his wood life has made him alert; he has the habit of listening, and talk interferes.
Another influence is for sanity. It cannot fail to communicate a little of its imperturbable calm, that stable equilibrium of the granite ledge and the great tree trunks. There being none of the external and artificial excitations which constantly play on the mind in cities, a tremendous force of complex suggestion is removed, and the thought naturally works more simply and directly. The multiplicity of desires lies dormant. Everything conspires for simplicity, as in the city all things are in conspiracy against it.
A certain resourcefulness is the portion of the woodsman, a little of the independence and dexterity of the Indian, but more than this, an intellectual and spiritual resourcefulness. It devolves upon him in the solitude to become acquainted with himself---to be his own friend. A sturdy content grows out of this association with the forest. He does not require to be amused. It does not necessarily promote an unsocial state, but it does make him independent of much society. Thus the forest has its finer or spiritual influence.
Even greater is the suggestion of primitive vigor. The display of vast rude strength induces a robust state of mind quite as readily as the open-air life gives appetite and sleep. With the savage this influence is direct and may almost be classed as instinct. With the refined and cultivated mind it must first pierce the outer shell, the veneer, and filter into the subconscious depths, as the sunlight penetrates the forest twilight and brings to life dormant seeds lying there. A new class of ideas comes to life. The seeds of thought planted long ago in the nomadic period of evolution---in the hunter stage---germinate under the forest influence and send forth shoots. It is memory---the race-memory---coming blindly to the surface, and amounts to a reversion, not so great, however, but it may be wholesome. We speak of men being \textit{animal} when they are sensual or dissipated, unmindful that animals are neither, but eminently sane, rendering a complete and unconscious obedience to the laws of Nature. Some men make the mistake of trying to take the city to the wilderness, and, as a result, get neither one nor the other. The forest has its luxuries, and they consist, in a measure, of freedom from those things considered luxuries in the city.
Here in the Sierras we live in a wickiup, a sort of a roofless wigwam. The camp overlooks the forest in which the canyons and ranges are as folds and wrinkles. Neighbors are few, for animals conceal themselves, while song-birds are not properly of the forest, but seek the clearing and the settlement. An Oregon snowbird has her nest near by and comes hopping about on her marketing expeditions. A pair of lazuli-finches also live on the edge of the clearing, and the male is, perhaps, the most beautiful bird in the forest. His demure little mate is seldom seen, as she is preoccupied with her domestic cares, but he constantly flits about in the chaparral, where he gleams in the sunlight like a jewel.
One other neighbor we have, an Audubon hermit-thrush, which might be a voice merely---like Echo haunting the mountain---and no bird at all. He appears to sing in the twilight only, and his song, like that of all thrushes, is spiritual and unworldly. A single white lily, tall and branching, stands near the camp, and day after day opens its ghostly racemes in the dusk to white moths which come flitting out of the forest like winged Psyches; and with the opening of the spirit-like flower comes the vesper song of the thrush.
Night in the forest is a spell, an enchantment. It descends suddenly and envelopes us in darkness, tangible and real. The wickiup stands at the edge of a little clearing, and, as we roll ourselves in our blankets, we seem to float in inky blackness, while the pines are like beetling cliffs against the starlit heavens. Darkness and light confront each other; it is as if we hovered between them and had made our camp for the night on the borderland. But with the dawn, that luminous world has vanished and we are again under the familiar pines.
One is impressed most by the wonderful stillness of the night. Not only is the world blotted out in the enveloping darkness, but it is voiceless, and there prevails absolute silence. Rarely this is broken by the yapping of coyotes, or a dry twig snaps sharply under the foot of some animal.
Not until the wind rises does the forest recover its voice. During the day there is always music; it is as constant as noise in the city. Impalpable currents descend from the empyrean to caress only the tops of the tallest pines, coming no nearer to earth than this, and while all is silent below there arises a distant chant in the tree tops, which have been touched by an invisible hand and made to respond to moods of the sky. Full and resonant, yet with that muffled quality of tone which makes it appear always to come from a distance, the rhythmic force of this chant sways one like the vibrations of an orchestra. Starting at some center, as if at a signal, these tremulous waves of sound recede farther and farther into the forest and die away in a sigh.
Here the tendency grows on one to wander in the early morning and again in late afternoon, to become crepuscular, like the animals, and to stay in camp in the middle of the day. Deer do not stir abroad in the heat, nor do fish bite, nor birds sing. This love of dawn and twilight is partly inspired by fear of man, but it is none the less natural. At daybreak the deer go down the canyons to the salt-licks, as surreptitiously as nymphs going to bathe. It is their witching hour, as midnight is the owls'.
To arise at dawn should be an occasion; to make it usual would mean the sacrifice of the more subtle impressions, the mind is so readily blunted by the habitual.
Like a black mantle the great forest lies over the earth as I roll myself in my blankets beside the fire. That little flaring light appears to be the only one in this dark wilderness, reclaiming a minute portion of space and making it habitable. Wherever one may be in the forest, it is only necessary to gather a few dry sticks and strike a match. The signal summons the genii, servant of the woodsman. More properly one should use a flint, or rub two sticks together. He allies himself with man against the hosts of darkness and defies the wilderness; a merry fellow, his laugh may be heard in the crackling flames. All through the night he entertains with his merry gossip and with pictures he shows in the fire. At times he reveals his own glowing face in the embers, but quickly assumes the head of a bear or a lynx, or melts away in the flames, to reappear presently in another spot.
When I awake, the morning-star hangs low in the heavens like a great lamp, its light an infinitely pure and serene radiance with no suggestion of heat or combustion, made to appeal to some higher vision. A heap of cold gray ashes is all that is left of the fire, in the center a single glowing spot, which may have been the eye of the genii of the night. The black mantle has been lifted, and the earth is illumined by a faint glow, as if solely by the reflected rays of that planet. Unspeakably soft is this light, the forerunner of the dawn, in which the forest is bathed and from which one derives a peculiar satisfaction.
Imperceptibly, almost, it fades, and is replaced by one of a different quality---the light of day---which creeps over the world until at length one is aware that that other, which was neither of the night nor of the day, has gone. Long pale lines of fog and fleecy banks of clouds now evolve upon the horizon. The earth remains suffused in this cold light, which fascinates and still repels, making the ranges look distant and severe, and giving to the whole face of Nature an unsympathetic look. It is the beauty of marble, a Gorgon beauty, which chills the heart. In that scene is no note of human passion. Those pale clouds, cold and gray as the ashes of the fire, seem to lure to some beyond, as if they would draw one from the world of life and warmth to some region of cold and death.
Presently comes a faint blush in the sky and over the hills, a new warmth of light, as if blood now ran in those marble veins. It is the \textit{foreglow}, which is to the sunrise what the afterglow is to the sunset. Color is again born into the world, and the earth is once more alive and sympathetic. As the sun rises, dawn, the exquisite dawn, the most ethereal thing that mortal eyes shall ever behold, flees away into the uttermost parts of space. The mystical, alluring quality slowly dies, and it is once more the matter-of-fact light of day.
With the appearance of the sun these subtle impressions vanish, like a dream vague and unreal. Nature reasserts herself in the robust sense of existence; now the smell of frying bacon, the comforting effect of the morning coffee in a tin cup, are the real and important things. Physical life is enough in itself---so concentrated, vigorous, aggressive it is. The mere breathing, seeing, tasting are more in themselves than is possible under other conditions. How good the resiny odor of the forest! How exhilarating the scene in its pure savagery! How stimulating the morning air! How the stream lures as I get down the trout-rod, and climbing out on a sugar-pine log cast a brown hackle on the swirling glassy flood!
\gutchapter{THE SEA}
The sea ever baffles description. It is a living thing, pulsating with energy, and, possessed of a subtle consciousness, elusive and full of moods---changeable as woman and as incomprehensible. Now it is tender and appealing; again distant and cold. Perhaps it is because of its essentially feminine traits that it so beguiles. Certainly it fascinates as nothing else fascinates in Nature.
There is what may be called a \textit{sense} of the sea, which is indefinable. No lesser body of water, no other aspect of Nature affords this. It is in the air, like a touch of autumn, and we know it as much through feeling as through seeing. The coast is saturated for some distance inland with this presence of the sea, much as the beach is soaked with salt water. It is music and poetry to the soul and as elusive as they, wrapping us in dreams and yielding fugitive glimpses of that which we may never grasp, but which skirts, like a beautiful phantom, the mind's horizon. Like music, it is an opiate, and unlocks for us new states of mind in which we wander, as in halls of alabaster and mother-of-pearl, but where, alas, we may not linger. We can as readily sound the ocean as fathom the feelings it inspires. It is too deep for thought. As often as the sea speaks to us of the birth of Venus and of Joy, so also does it remind of Prometheus bound and the thrall of Nature.
Who can recall those impressions of the sea which were his as a child---a relish, a vividness, perhaps never experienced in after life? What wonderful thing was the pure white sand; what fascinating objects the sea-shells---and the boom of the surf, what thrilling music! No longer is it that simple strain, but inwrought with hopes and fears and memories. The children on the beach play in an ocean of their own; we cannot put foot on their shore, try as we will. Sometimes, as the salty fragrance is wafted over the sands, one is on the point of regaining that lost consciousness, and then it eludes and is gone. Never again shall we find that alluring and altogether wonderful sea upon which we happened in childhood. Yet who knows but in some auspicious moment we may come upon one still more entrancing.
With an east wind the sea is always musical. It breaks forth in its solemn chant, as though the wind were an influence that awakened memories of the immeasurable past, and inspired this primitive song. From a distance it comes like a rhythmical murmur upon the horizon, and it is strange how this sound will fall upon unheeding ears, and then with what suddenness one becomes aware of it. At times it loses its rhythmical character and becomes a sort of recitative. One imagines the venerable sea to be muttering of its epic past---to be relating that wonderful saga.
Yesterday the sea was glass. It lay tranquil as if never again could its surface be ruffled. So indefinite was the sky-line it was difficult to tell which was sky and which water,---a dream-ocean, a charming vision, which was to dissolve like a mirage of the desert.
This morning how it was changed! Up from the shore came a muffled and ominous growl. As one approached, this ceased, and there was instead the spitting and hissing of little waves---a sound of irritation and suppressed anger. The sea was leaden, aggressive, formidable. It was as if some troubled spirit had entered there---it was possessed of a devil. This unrest is savage and terrible like that of a caged tiger. The eye turns with relief to the imperturbable rock, which seems to confine and restrain the angry waters. The granite rests in unalterable calm, sphinxlike, on the edge of the watery desert. It stands for the constant and enduring, as it forever confronts the inconstant and changeful sea. They are two opposing forces: the sea coy, arch, coquettish, now bewitching and full of her beautiful wiles, now disdainful and imperious, again mad, tempestuous, hurling herself in her wild passion; the granite grim, massive, unconquerable.
Late in the afternoon the wind is blowing from the north, the sky has cleared and the sea is sapphire, dotted with whitecaps; yesterday, opal, this morning leaden, and later, sapphire. It is no longer formidable, rather is it cold and distant. The \textit{face} of the waters is a peculiarly pertinent figure of speech, for the sea is as a face reflecting all moods. In the glare of noonday, ocean and landscape seem to discharge themselves of feeling,---that is to say, they are barren to the eye and unproductive of feeling in us. But in the atmosphere of sunset and twilight they are again expressive. The quality of light may be compared to the \textit{timbre} of sound. Sometimes---as at noon---it is like the blare of brass, and, again, it has the softness of wood-winds, the tenderness of violins and cellos.
The receding day carries with it the disquieting influences, and night exorcises the demons of unrest. They scurry away with the sunset clouds on the horizon like fleeing witches. As if in obedience to some silent command, the sea becomes passive. He must be distraught indeed who can look at it now without coming under the spell of the hour---the serene hour. It is as if the passion and strife of life had been succeeded by the beautiful calm of death. To gaze on the mute and motionless ocean at ebb-tide is to be inevitably inspired to reflection, so potent is the suggestion of repose. Apparently the forces of Nature have conspired together for peace.
Death? Nay, rather transfiguration, for now the sea is illumined by a golden radiance. Stretches of burnished copper and molten gold merge one into the other; areas again of liquid silver, and beyond, the vast ethereal blue. Out of the coves shadows come creeping and stealing over the water, silently advancing to overwhelm the rose and copper and gold, while these recede and slip out to sea, growing fainter and fainter until they are absorbed in the all-pervading dusk. In the succeeding darkness one beholds, not the sea, but a vast bottomless pit, Dantesque and terrible.
Above all else it is the immense vigor of the sea which appeals to us. We are made to feel the play of cosmic forces. The long stretch of rocky coast is rude and Titanic; the expanse of ocean suggests that chaos from which the earth has gradually been redeemed. The waters piling themselves up are as elemental and chaotic as nebulae or the seething envelopes of the sun. It is incredible they should be hitched to the gentle moon, and should follow that pale phantom like a leashed panther, now purring, now growling, but obedient always. The mountains impress one with their age, the sea with its agelessness. Here at least is something which appears superior to Time. It is no more youth than it is age---the formless, without beginning and without end, but always that superabundant vigor, power, freedom.
Denuded woodland and disfigured landscape bring to mind that iron Necessity which it is not pleasant to see advertised. But the sea is unimproved. It is the universal solvent, and dissolves the trivial, the commonplace, the mean, and gives an heroic cast to whatever it touches. One needs, however, to observe it from the shore and to have that vantage which is derived from being on land. In mid-ocean it is too entirely dominant---there is nothing to afford contrast. It is like the moon---so fair at a distance, such desolation upon its surface. One can be alone on the mountains and find them friendly, but who would choose to be alone in mid-ocean? There is a sense of isolation, a disassociation, as if one had, in fact, severed connection with earthly affairs altogether; hour after hour and day after day the same inscrutable desert of water, which begins everywhere and ends nowhere.
Yet how inviting it appears when the glittering sunbeams dance on a gently rippling surface. It seems an expression of irrepressible gaiety as if all the joyousness in Nature had come to the surface here. The twinkling dance of the innocent waves---who can recall the tragedies now?
[Illustration: THE SEA COPYRIGHT, 1908, BY RUDOLF EICKEMEYER]
The gulls appear to enjoy some favoritism, as though they were kin to the sea---its very own. To them it is altogether friendly; they find it always congenial. Whether the breeze blows north or south, it is all the same. In the last gale it was next to impossible to keep one's feet in the full force of the wind, but the gulls sustained themselves with ease. Over the gray-green sea the clouds appeared to rest like a cowl. The thunder of the waves drowned all else and shut one off from the world; consciousness was swallowed up in the din and tumult. In vast mountainous billows the swirling waters rushed for the shore and dissolved in spray. I stood in the lee of the rocks, bracing myself against the gale---a reed shaken by the wind---and saw flocks of coots riding at ease in the maelstrom beyond. Always facing the wind, they sank into the troughs and rose again, were lost to view as the crests broke over them, and reappeared in the old position. Ships would have dragged their anchors where these coots rode at ease, anchored by heaven knows what power.
Where the surf broke with its terrible thunder, countless crabs, urchins, starfish and whelk reposed in the rockweed and Irish moss. Were they aware of the storm? Did the anemones shut their doors or open them wider in view of a feast?
The marvelous pools in which they live have no resemblance to the surface of the sea, but suggest the bottom of the deep---limpid, dark and still. Each is a world by itself, inhabited by a strange order of beings: dull nomads, which drift with the waves, or cling, they know not how, to something, they know not what. If there is any event in their life it is the rise of the tide. In all likelihood they do not know our day and night, are not impressed by these phenomena; but the flood is their day, the ebb their night. Small whelk stud the rich background of sea-mosses like precious stones, some gamboge, some orange, others white as marble or banded with black. There are colonies of sertularia tinted a delicate mauve, solitary sea-urchins of heliotrope, and starfish, some luminous pink, others deep rose-madder. These hues are characteristic of sea life, as of lichens and mushrooms and the lower orders in general; not crude colors, red and blue, but delicate gradations. Now and again a single jellyfish, stranded by the receding tide, a spectral diaphanous creature, hovers ghostlike in the liquid atmosphere of his strange world. It is all of an antediluvian and prehistoric character, associated with the beginning of things---with an age of fishes rather than an age of man. The deathless sea takes no note of the flight of time; it still brings forth only brood upon brood of slimy, goggle-eyed things.
What a harvest, this of the sea! After a storm all craft put out. The lobstermen in their dories take in the lobster-pots and replenish the bait, while the dory rises and sinks on the long swells. Fleets of mackerel boats and schooners bound for the Banks after cod and haddock creep along the horizon-line. On the beach men rake up the Irish moss, flung ashore in the storm, and spread it on old sails to bleach in the sun. Others haul kelp for the fields, while women gather driftwood. So great a resource is the ocean; so many gleaners there are.
The sea is humanized and redeemed somewhat by the presence of these workers. It is agreeable to reflect that while it nourishes them, they in turn do not mar it. Man communicates the character of his mind and aims to the landscape; enriches it by his labor on the farm, and disfigures it again in a thousand ways, till it is as barren and sordid as his own thoughts. But upon the deep he makes no impression. It is virgin ever. It overpowers him by its stern music---lifts him for a time a little above the sordid and commonplace. The sailor ashore is not the same man he is out there. He must needs have courage, for he must meet the sea. Portuguese, Swedes, Finns---poor stuff for poems in their sailor boarding-houses ashore. But hear how they face the winter gales. Learn the actual experience which makes up that life. The sea invests the poorest, meanest man with heroic qualities. That is his stage; there he looms large. Oil-skins and sou'wester are but his make-up.
I take home a piece of driftwood, for no ordinary fire, but to kindle the imagination, for it is saturated with memories and carries with it the enchantment of the sea. To light this is to set in motion a sort of magic-play. True \textit{driftwood} has been seasoned by the waters and mellowed by the years. Not any piece of a lobster-pot, or pleasure yacht, or, for that matter, of any modern craft at all is driftwood. It must have come from the timber of a vessel built in the olden time when copper bolts were used, so that the wood is impregnated with copper salts. That is merely the chemistry of it. The wood is saturated with sunshine and moonlight as well, with the storms and calms of the sea---its passions, its subtle moods; more than this, it absorbed of the human life whose destiny was involved with the vessel---the tragedy, the woe. It had two lives---a forest life and a sea life. By force of tragedy alone it became driftwood. Winter and summer the sea sang its brave songs over the boat and chanted her requiem at last as she lay on the ledge. This fragment drifted ashore out of the wreck of a vessel, out of the wreck of great hopes, out of the passion of the sea.
Driftwood, then, is to be lighted in a spirit of reverence. No ordinary blaze, rather is it an altar fire to Poseidon, to whom were immolated the victims; to Aphrodite born of the waves. Rather is it the funeral pyre of a sea-bird, now to rise again from its ashes. It is not to warm the hands, this magic sea-fire, which has borrowed the emerald and sapphire and azure of the waters and reflects still the phosphorescent gleam which lay in the wake of the vessel, but to kindle some feeling and to nurture vague dreams. To set match to this pyre is to invoke the spirit of the deep, to hear the crooning of some distant surf, the hissing of the fretful spray; to conjure up again the wondrous opaline sea.
Somewhere on this phantom ocean rides a phantom bark with all sails set, which reflect, now a rose-pink, now the faintest imaginable golden sheen, and disappear in the dusk. Perchance there flits over the mind a haunting recollection of that lost sea of childhood---that sea of virgin impressions---to vanish also into the dusk of oblivion.
\gutchapter{INDEX}
Abdomen, 86, 95, 105.
Acorns, 120, 142, 160, 161.
Acquired tendency, 97.
Adder's-tongue, 20, 61, 67.
AEolus, 64.
Afterglow, 194.
Age, 201.
Agelessness, 201.
Age of fishes, 204.
Air-sac, 18, 19, 77.
Alaska sable, 150.
Alders, 14, 20, 50, 120, 162, 164, 168.
Alpine woodsia, 67, 170.
Alps, 154.
Altar fire, 207.
Amanita, 115, 126.
Amazon, 25.
Aments, 14.
Andromeda, 168.
Anemones, 1, 13, 20, 21, 59, 61, 67, 203.
Animals, 136-138, 142, 146, 187, 189, 191, 192.
Antennae, 52, 101, 103.
Anthers, 78.
Ant hill, 94, 108.
Ants, 4, 88, 91, 94-107, 109, 111, 112, 145.
Aphid, 100, 101, 112.
Aphrodite, 207.
Apis, 78.
Apple, wild, 148.
April, 142, 169.
Arachne, 78.
Arbutus, 20, 60, 121.
Arctic, 131.
Arctic Circle, 23.
Armies, 102, 107, 111.
Ash, 66, 124.
Aspen, 7, 123.
Aspens, 181, 182.
Asters, 71, 122.
August, 113, 114, 117, 134.
Autumn, 20, 45, 51, 72, 92, 118, 119, 122, 125, 134, 135, 144, 152, 157, 166, 196.
Audubon hermit-thrush, 190.
Avalanches, 182.
Balsam, 3, 6, 57, 66, 68.
Bamboos, 117.
Banks, the, 205.
Barberry, 3, 58, 124.
Bare branches, 126.
Bark, 157, 158.
Barn-swallows, 26.
Basalt, 59.
Battle, 102.
Battlefield, 103.
Bayberry, 3, 118, 127.
Beach, 76, 132, 183, 196, 197, 205.
Bear, 193.
Beech, 126, 156, 157, 186.
Beechnuts, 145, 160.
Bees, 30, 31, 51, 55-57, 60, 67, 78-80, 82, 83, 89, 182.
Beetles, 34, 35, 150.
Bee-tree, 67.
Beggar-ticks, 121.
Bellworts, 21, 62.
Bent grass, 57.
Berkshires, 185.
Berries, 113, 114, 126, 161.
Bighorn, 178.
Big tree, 186.
Billows, 203.
Bindweed, 69.
Birch, 12, 32, 33, 57, 90, 117, 120, 123, 157, 158, 186.
Black alders, 168.
Black-and-white creepers, 39, 40.
Black ant, 91, 96, 102, 104-110.
Blackberries, 2, 63, 75, 118, 123, 148.
Black birch, 117, 158.
Blackcap raspberries, 146.
Black cherry, 158.
Black colonies, 4.
Black-headed grosbeak, 182.
Black oak, 157.
Blacksnakes, 155.
Black-throated blue warbler, 66.
Black-throated green warblers, 40.
Bloodroot, 20, 21, 60.
Blueberry, 11, 12, 64, 113, 159.
Bluebird, 1, 13, 14, 34, 41, 114, 117.
Blue-headed vireo, 8.
Bluets, 20, 57.
Blue violets, 20.
Boat, 207.
Bobolink, 24, 43.
Bob-white, 24, 42, 70, 160-162.
Bog, 56, 57, 122, 165, 168.
Bog-rush, 57.
Bokhara, 126.
Boleti, 116.
Bombus, 78.
Boulders, 6, 57, 91, 94, 122, 127-135, 140, 181.
Bouncing-bet, 56.
Brakes, 122.
Brazil, 24.
Briers, 147, 148.
Brook, 61, 138, 168-171.
Broom-moss, 117.
Brown ants, 109, 111, 112.
Brown hackle, 195.
Brush, 90.
Bryum, 117.
Buckwheat, 145.
Bulldogs, 102.
Bullfrogs, 114.
Bumblebee, 67, 78, 80.
Bunchberry, 67.
Bunchberries, 113.
Burdock, 59, 63, 73.
Bur-reeds, 152.
Burrow, 16, 155.
Burs, 121.
Buttercup, 57.
Buttercup seed, 145.
Butterflies, 55, 93.
Butterfly-weed, 72.
Butternut, 3, 66, 121.
Buttes, 183.
Button-bushes, 164, 165.
Byroads, 71, 73.
Caches, 125.
Calms, 207.
Camp, 189-192.
Canada, 66, 130.
Canada thistle, 75.
Canada violet, 62.
Canyons, 180-182, 185, 189, 192.
Canyon-wren, 180, 182.
Capsules, 119.
Captivity, 112.
Cardinal, 47.
Carolina wren, 47.
Carpenter-bees, 30.
Carrot, 63, 66, 74.
Cascade, 185.
Cassandra, 168.
Cat, 27.
Catbird, 30, 44-46, 49.
Catbrier, 118, 148.
Caterpillars, 92, 93.
Catkins, 13.
Cattails, 38, 150, 152.
Cedar, 34.
Cedars, 186.
Cells, 82, 83.
Cetraria, 159.
Ceylon, 117.
Chalcid fly, 82, 83, 92.
Chant, 46, 47, 169, 191, 198.
Chaos, 201.
Chaparral, 190.
Checkerberry, 80.
Cherry, 2, 3, 64, 66, 113, 158.
Chestnut-bur, 121.
Chestnuts, 121, 145.
Chestnut warbler, 26, 39.
Chewink, 41.
Chickadee, 41, 47, 158, 159, 166.
Chickory, 74.
Chipmunk, 15, 136, 139, 140, 144, 145, 155.
Chipping-sparrow, 37, 38, 118.
Chirping, 53.
Christmas fern, 158.
Christmas green, 158.
Cicada, 50, 53.
Cinnamon fern, 122.
Cinquefoil, 57.
Cirque, 181.
City, 189.
Cladonia, 126, 159.
Claw, 86.
Clearing, 190.
Clethra, 80, 148, 168.
Cliff, 66, 67, 132, 169, 174, 179-183.
Cliff-brake, 170.
Clintonia, 1, 57, 113, 114.
Cloud, 173, 174, 180, 183, 194, 200.
Clover, 57, 71.
Club-mosses, 158.
Coast Range, 186.
Cocoa-bean, 72.
Cocoons, 93, 95, 96.
Cod, 205.
Colony, 94, 95, 97, 99, 102, 107-111.
Columbines, 21, 66, 67, 121.
Column, 104-106, 110, 111.
Combat, 103.
Companionship, 10, 173.
Competition, 69.
Cones, 125, 158, 162, 163, 186.
Conifers, 158, 185, 186.
Coniferous, 141.
Connecticut Valley, 21.
Constellations, 130.
Coon, 149.
Coots, 124, 203.
Copper salts, 206.
Co-operation, 98.
Corridors, 95.
Cotton-fields, 47.
Cottontail, 126.
Court of the Winds, 163.
Coves, 200.
Cow-bell, 177.
Cowbird, 36-38.
Coyotes, 191.
Crabs, 203.
Cranberry-bog, 57.
Crane's-bill, 65.
Creek, 169.
Creepers, 39, 159.
Crest, 203.
Crickets, 51, 52, 54, 72.
Crop, 128.
Crossbills, 24, 162, 163.
Crow, 16, 17.
Crystals, 133.
Cuckoo, 31.
Curlew, 23.
Cyperus, 114.
Daghestan, 126.
Daisies, 59, 66.
Dandelions, 70, 71, 131, 135.
Dawn, 45, 47, 191-194.
Daybreak, 192.
Death-song, 55.
Deep, 204, 205, 207.
Deer, 138, 156, 192.
Dell, 80.
Desert, 131, 183, 198, 199.
Deserters, 107.
Devonian, 5.
Dexterity, 188.
Dicentra, 20, 59, 170.
Dicksonia, 59, 122.
Dodder, 69.
Dog, 136.
Dog-days, 87, 114, 122.
Dogwood, 30, 58, 157.
Dory, 205.
Douglas spruce, 185.
Doves, 7.
Downy woodpecker, 159.
Downy yellow violets, 21.
Driftwood, 205-207.
Dromedary, 132.
Drumming, 16.
Dwarf-sumac, 80, 124.
Earthquake, 182.
Earth-spirit, 130.
East wind, 198.
Ebb-tide, 200, 204.
Echo, 190.
Edible, 115.
Eggs, 81, 82, 84, 85, 91, 93, 99, 105, 110, 112, 121, 132, 160.
Egypt, 131.
Elder, 63.
Elm, 20, 30, 163, 186.
Elysian fields, 71.
English pheasant, 42.
Epithalamium, 54.
Erigone, 88.
Erythronium, 1, 20.
Europe, 70.
Eurydice, 179.
Evolution, 97.
Exodus, 101, 109, 110.
\textit{Face} of the waters, 199.
Fall, 151.
Fall dandelions, 71.
False hellebore, 68.
Farm, 128.
Faulting, 179.
Faun, 49, 126, 159.
February, 140.
Feeding-grounds, 152.
Feldspar, 134.
Fern, 57, 131, 158.
Fern moss, 68, 117.
Fields, 22, 42, 43, 48, 57, 59, 71, 72, 74, 76, 79, 115, 127-129, 205.
Field-sparrow, 1, 43, 118.
Filaments, 78.
Finch, 18, 190.
Fir, 185, 186.
Fire, 192-194.
Fireweed, 75.
Fish, 77, 192.
Fisherman, 171.
Flight, 88, 98-100, 120.
Flood, 204.
Florida, 25.
Fly, 78, 86.
Flycatcher, 37, 119.
Fog, 173, 183, 184, 194.
Forest, 34, 42, 44, 56, 62, 65, 78, 82, 100, 117, 136, 163, 178, 181, 185, 187-193, 195.
Forest cathedrals, 186.
Food plant, 112.
Foreglow, 194.
\textit{Formica pennsylvanica}, 102.
Formicary, 97, 99.
Fox, 148, 149, 156.
Fox-sparrows, 16.
Freedom, 112, 177, 201.
Frogs, 18, 19, 51, 114, 144, 155.
Fossils, 131.
Fronds, 159.
Frost, 129, 135, 181.
Frost-grapes, 2, 113.
Fruit, 65, 72, 76.
Fungus, 114, 147.
Funeral pyre, 207.
Gale, 203.
Galleries, 95, 96.
Garden, 56, 58, 59, 65, 67, 69, 70, 169, 170.
Garden escapes, 56.
Gentians, 114.
Glacial times, 130.
Glacier, 129, 131, 133, 135, 181, 182.
Gleaners, 205.
Glen, 138-140, 143, 150, 151, 169, 170.
Gneiss, 169.
Goldenrod, 71, 78, 80, 122.
Goldfinch, 162.
Gold thread, 26, 63.
Gorge, 80, 178, 180.
Granite, 6, 59, 60, 130, 132, 134, 135, 159, 175, 180, 187, 199.
Granite domes, 185.
Grape, 52, 126.
Grass, 57, 64, 65, 70, 71, 76, 87, 108, 131, 139.
Grasshoppers, 54, 145, 155.
Grass roots, 152.
Grass seed, 145.
Gray squirrels, 120, 139, 140-142.
Greece, 186.
Greenland, 129, 185.
Green-tailed towhee, 182.
Grosbeak, 47, 182.
Groundnut, 21, 169.
Ground-pine, 158.
Grouse, 24, 156.
Grove, 17.
Grubs, 92, 121.
Gulls, 202, 203.
Hackle, 195.
Haddock, 205.
Hair-cap moss, 12, 57, 117, 119.
Hardwoods, 30.
Hawk, 34, 42, 161.
Hawkweed, 28.
Hay-scented fern, 131.
Hegira, 89.
Hemlocks, 56, 81, 126, 140.
Hepatica, 20, 60, 129.
Hermit-thrush, 8, 16, 43, 44, 190.
Hickory, 7, 121, 123, 124, 128.
Highhole, 15, 16.
Highroads, 73.
Highways, 56, 100.
Hillocks, 59.
Hills, 94, 109, 167, 173, 177, 180, 194.
Himalaya, 117.
Hive, 80.
Home-trees, 30.
Honey bees, 67, 79, 80.
Hop-hornbeam, 119, 157.
Horse, 131, 177.
Horsetails, 12.
Host, 93.
Houseleek, 77.
House-sparrow, 31.
House-wren, 30.
Hound, 143.
Huber, 112.
Huckleberries, 26, 99, 123, 145.
Humming-bird, 28, 29, 67.
Humus, 125.
Hunter stage, 189.
Hunters, 102.
Hunting expedition, 110.
Hunting-ground, 111.
Huts, muskrat, 150-152.
Hyla, 18-20, 51.
Ice, 129, 130, 151, 152, 155, 162, 167, 168, 181.
Ichneumons, 93.
Icicles, 135, 167.
Immigrants, 70.
Independence, 188.
Indian, 187, 188.
Indian paint-brush, 181.
Indian-pipe, 80, 81.
Indian summer, 88, 126.
Instinct, 188.
Invasion, 104.
Irish moss, 203, 205.
Iron weed, 73.
Jack-in-the-pulpit, 62.
Java, 117.
Jaws, 86, 91, 103, 142.
Jay, 45, 120, 160, 161.
Jellyfish, 204.
Jimson-weed, 73.
Joepye-weed, 73, 122.
Juncos, 162.
June, 161.
Jungle, 117, 130, 131, 168.
Juniper, 127.
Kelp, 205.
Kidnapping, 107.
Kingfisher, 32.
Kinglet, 1, 5, 48, 49, 159, 170.
Laborers, 108.
Lady's-slipper, 1.
Lake, 75, 181.
Lake basins, 181.
Lance-leaved violets, 21.
Larvae, 82, 83, 105, 106, 110, 121.
Laurel, 59.
Lava, 59.
Lava-flow, 182.
Lazuli-finch, 190.
Leaf bud, 160.
Least flycatcher, 37.
Ledge, 5, 6, 31, 33, 80, 129, 132, 134, 159, 164, 178, 181, 187, 207.
Leopard-frog, 19.
Lichens, 6, 26, 57, 117, 127, 133, 134, 159, 204.
Lily, 61, 190.
Lily-pads, 114, 166, 168.
Limestone, 59.
Lobstermen, 205.
Lobster-pots, 205, 206.
Locusts, 51, 52.
Long-spurred violets, 20.
Loon, 48.
Love-song, 18, 44, 45, 50.
Lubbock, 112.
Lumbermen, 163.
Luster, 158.
Lynx, 193.
Mackerel boats, 205.
Magnolia warbler, 66.
Mallow, 122.
Mandrake, 1.
Mango, 72.
Maple, 19, 20, 26, 45, 59, 66, 101, 117, 120, 124-126, 134, 135, 157, 186.
Maple-leaved viburnum, 124.
Marble, 134.
Marriage flight, 98.
Marshes, 122.
Marsh-marigold, 20.
Martin, 22.
Maryland yellow-throat, 37.
Massachusetts, 21.
Mastodon, 121.
Meadow-ants, 102.
Meadow-lark, 1.
Meadow-mouse, 150.
Meadows, 43, 56, 67, 168, 182.
Meadow sweet, 80.
Medeola, 62.
Mertensia, 183.
\textit{Mesa}, 177.
Mica, 134.
Mice, 139.
Milking, 101.
Milkweeds, 72.
Mink, 139, 140, 142, 149, 151.
Minstrelsy, 48.
Mint, 1.
Mirage, 198.
Mist, 75, 76, 170, 180, 183.
Mitchella, 126.
Miterworts, 21, 32.
Mitla, 179.
Mixed flocks, 8.
Mniums, 117.
Mocking-bird, 49.
Mole, 150.
Mollusks, 23.
Monkey, 121.
Moon, 129, 201, 202.
Moosewood, 157.
Moraine, 59, 130, 181.
Morning-star, 193.
Mosses, 12, 57, 68, 82, 116, 117, 134.
Mother ants, 98.
Moths, 93, 190.
Mountain-ash, 66.
Mountain-goat, 100.
Mountain-holly, 113.
Mountain-maple, 66.
Mountains, 56-58, 65, 66, 68, 119, 164, 173-177, 179, 181, 183, 184, 190, 201, 202.
Mountain stream, 169, 177, 178.
Mountain thoughts, 178.
Mouse, 35, 119.
Mud-dauber, 84.
Mullein, 63, 74.
Mushrooms, 114-116, 204.
Muskrats, 150-152.
Myrtle warbler, 66.
Natural history, 97.
Nebulae, 201.
Negroes, 104, 105, 107, 111.
Nest, 26-29, 31-34, 36-38, 41, 45, 66, 84, 95, 96, 102, 104-111, 190.
New England, 6, 11, 127, 130, 169.
New Jersey, 21.
New York, 130.
Night, 190, 200.
Non-conformity, 179.
North, 127.
November, 152.
Nuthatch, 40, 159.
Nuts, 142.
Nymph, 126, 157, 159, 192.
Oaks, 5, 7, 8, 58, 89, 120, 126, 154, 157, 159, 186.
Ocean, 183, 197, 199-201, 205, 207.
October, 114, 123, 166.
Oil-skins, 206.
Olympus, 176.
Opal, 199.
Open, the, 153, 166.
Orache, 77.
Orchards, 30, 34, 71.
Orchis, 168.
Oregon snowbird, 190.
Orinoco, 25.
Oriole, 5, 25, 30, 117.
Ornithologist, 118.
Orphean strain, 119.
Orpheus, 178.
Osmunda, 13, 123, 124.
Outlaws, 148.
Ouzel, 178.
Owl, 27, 31, 34, 35, 148, 192.
Paint-brush, 181.
Painted trillium, 62.
Paleozoic reptiles, 131.
Panther, 201.
Pappus, 38, 75.
Papyrus, 180.
Parasite, 157.
Parmelia, 57, 133, 159.
Parnassus, 119.
Partridgeberries, 160.
Parula warbler, 8.
Pastoral stage, 102.
Pasture, 15, 22, 25, 56-58, 64, 72-74, 76, 88, 113, 127, 128, 135, 155.
Pasture mushroom, 115.
Pasture stones, 57, 94, 118, 127, 128, 131, 132, 134.
Patagonia, 23.
Peak, 174, 185.
Peat-moss, 117.
Pebbles, 132.
Pedata violets, 60.
Persia, 154.
Personality, 43.
Petal, 130, 167.
Pewees, 26, 46, 50, 118, 119.
Pheasant, 42.
Phoebe, 18, 31, 56, 119.
Pickerel-frogs, 114.
Pickerel-weed, 74.
Pignut, 142.
Pigweed, 69.
Pileus, 114.
Pine, 5, 7, 9, 17, 28, 30, 40, 65, 78, 123, 125, 142, 154, 158, 162-164, 166, 168, 181, 183, 185-187, 191.
Pine-needles, 7, 126, 147, 153.
Pine-siskins, 162.
Pine-warbler, 40.
Pinnae, 159.
Pipes of Pan, 48.
Piping, 18.
Pipsissewa, 80.
Pistillate, 79.
Pitcher-plant, 13.
Pitch-pine, 28, 166.
Planets, 132.
Plantain, 64.
Plant-fiber, 26.
Planting, 65.
Plant life, 69.
Plover, 22, 24.
Pod, 72, 73, 75.
Poison-ivy, 123, 124.
Poison sumac, 68.
Pokeweed, 72.
Pollen, 14, 65, 71, 79, 81, 119.
Pollen-basket, 79.
Pollen-bearing, 64.
Pollen-grains, 79.
Polyporus, 57.
Polypody, 159.
Pond, 19, 74, 124, 148, 150-152, 162, 164-167.
Pool, 165, 167, 170, 171, 179, 203.
Poplar, 20.
Porphyry, 180.
Porters, 108.
Poseidon, 207.
Poverty-grass, 76.
Prairie, 65.
Precipice, 169, 180, 182.
Protective coloration, 146, 187.
Puffballs, 116, 128.
Pumpkins, 79.
Pupa, 82, 83, 95, 96, 104-106, 108-111.
Pupa-cases, 95.
Purple finch, 18.
Purslane, 69.
Pussy-willows, 13.
Queen bumblebee, 80, 81.
Queen mother, 82.
Queens, 98, 99, 105, 108.
Quartz, 134.
Rabbit, 136, 139, 145-149.
Racemes, 190.
Ragweed, 69, 70, 155, 161.
Raid, 104, 106-109.
Rainbow, 182.
Rain, 130.
Range, 173, 177, 189, 194.
Rapids, 171, 178, 179.
Raspberries, 146.
Ravine, 66, 123, 169, 170, 177, 178.
Red ants, 4, 102, 104-110, 112.
Red cherry, 66.
Red clover, 71.
Redeye, 37, 50, 62, 118, 119.
Red maple, 20, 126.
Red owl, 34.
Redpolls, 24, 162.
Red-shouldered hawk, 161.
Red squirrel, 125, 139-141.
Redtop, 64.
Redwing, 15, 31.
Redwood, 186.
Reindeer lichens, 57.
Reptiles, 132.
Requiem, 54, 207.
Resourcefulness, 188.
Reverie, 72, 182.
Ribbon-snake, 157.
Ripple marks, 5.
Roads, 76.
Robin, 4, 15, 34, 44-46, 143.
Rock-brake, 67.
Rockies, 176, 182, 183.
Rock-maple, 59.
Rocks, 179, 180, 183, 199, 203.
Rockweed, 77, 203.
Rome, 131.
Rose, 31, 59, 63, 64, 72, 118, 130, 131, 148.
Rose-breasted grosbeak, 47.
Rose-mallow, 122.
Rose pogonia, 168.
Round-leaved spurge, 76.
Ruby kinglet, 48, 170.
Rue-anemone, 20.
Ruffed grouse, 24, 156.
Runways, 148.
Rush, 57.
Russula, 116.
Saga, 198.
Sailor, 206.
Salt-licks, 192.
Sand, 197.
Sandpiper, 124.
Sandstone, 59, 179, 180, 182.
Sand-wasp, 89.
Sanity, 187.
Saplings, 154, 185.
Sapphire, 199.
Sapsucker, 33.
Sassafras, 3.
Satyr, 49, 159.
Savage, 188.
Savin, 113, 127.
Saxifrage, 20, 170.
Scapes, 71.
Scarlet oak, 126.
Schist, 169.
Schooners, 205.
Scorpions, 95.
Sculpins, 77.
Sea, 77, 132, 133, 164, 165, 168, 175, 179, 182, 183, 187, 196-208.
Sea-bird, 207.
Sea-fire, 207.
Sea life, 204.
Sea-mosses, 204.
Sea-shells, 77, 197.
Sea-urchins, 203, 204.
Sedge, 150, 152.
Sedge roots, 152.
Sediment, 179.
Seed, 64, 65, 70, 75, 119, 120, 125, 135, 145, 161-163.
Seed-carrying, 64.
Seedlings, 121.
Self-effacement, 187.
\textit{Sense} of the sea, 196.
Sensitive fern, 57.
September, 118, 122.
Sequoia, 185-187.
Serpent, 106.
Sertularia, 204.
Shadbush, 64, 123, 124.
Shale, 5.
Shelf fungus, 114, 146.
Shellbark, 157.
Shield-fern, 159.
Shinleaf, 80.
Shooting-star, 183.
Shore, 198, 202, 203.
Shrews, 139.
Shrilling, 19, 53.
Sierra Nevada, 185, 189.
Silence, 187, 191.
Silver fir, 185, 186.
Silver-rod, 113.
Simplicity, 188.
Sky, 165, 171, 191, 194, 198, 199.
Skunk, 149, 155.
Skunk-cabbage, 11, 122.
Slave-making ant, 112.
Slavery, 106.
Slaves, 104, 108-112.
Smilax, 12, 30, 113, 117.
Snakes, 143, 155.
Snipe, 22.
Snow, 70, 140, 141, 147, 152-156, 160, 161, 166, 181.
Snowbird, 27, 66, 190.
Snow-buntings, 22, 24.
Snowflakes, 8, 22, 133, 153, 165.
Snow-storm, 153.
Socialism, 96.
Social wasp, 86, 89.
Solitary wasps, 89.
Solitude, 177, 188.
Solomon's-seal, 62.
Song, 18-20, 31, 32, 41-43, 45-51, 53, 55, 67, 119, 138, 161, 163, 168-170, 178, 190, 198.
Song-birds, 189.
Song-sparrows, 17, 31, 45, 50.
Sorrel, 57.
Sou'wester, 206.
Spagnum, 68, 122, 168.
Sparrows, 16, 17, 31, 37, 38, 42, 43, 45, 46, 50, 70, 118, 155, 162.
Spatter-docks, 152.
Speedwells, 57.
Spicebush, 58, 170.
Spider, 29, 51, 65, 78, 83-89, 117.
Spider's egg, 160.
Spider-web, 64, 84.
Spinnerets, 85, 88, 89.
Spinning, 84, 88, 93.
Spores, 119.
Spray, 203, 207.
Spring, 11-14, 17, 21, 24, 45, 50, 51, 68, 112, 114, 119, 144, 157, 170.
Spring beauty, 1, 20, 61, 169.
Spruce, 6, 32, 57, 66, 67, 181, 182, 185, 186.
Spur, 174.
Spurge, 76.
Spurred violets, 169.
Squirrels, 42, 120, 121, 125, 139-142, 156, 187.
Stag-horn sumac, 123.
Stamens, 78.
Starfish, 203, 204.
Star of Bethlehem, 56.
Stars, 128.
Stinging ants, 95.
St.-John's-wort, 63, 114, 149.
Storm, 205, 206.
Strand, 85, 89.
Strata, 179.
Strawberries, 57.
Stream, 48-50, 67, 169, 170, 177, 178, 180-182, 187, 195.
Stridulating, 54.
Stumps, 114.
Sugar-maple, 124.
Sugar-pine, 185-187, 195.
Sumac, 3, 15, 68, 80, 82, 83, 88, 123, 124.
Summer, 60, 66, 67, 70, 71, 126, 152, 154, 168, 181, 207.
Sunflowers, 59, 122.
Sunrise, 194.
Sunset, 194, 199.
Surf, 7, 203, 207.
Swallows, 19, 25, 26.
Swamp milkweed, 72.
Swamps, 13, 15, 18-20, 31, 40, 45, 47, 68, 73, 113, 114, 123, 125, 126, 134, 168.
Swan-song, 163.
Swarming, 98.
Sweetbrier, 1.
Sweet fern, 127.
Swift, 22, 30.
Sylvan, 49.
Tabriz, 126.
Tansy, 74.
Temperament, 43.
Temperate zone, 131.
Tendency, 97.
Tertiary animals, 131.
Thaw, 135, 151, 168.
Thicket, 16, 75, 118, 122.
Thistle-down, 30, 64.
Thistles, 59, 64, 74, 75, 135.
Thorax, 86, 95, 102, 105.
Thoreau, 177.
Thrasher, 49.
Thread, 84, 87, 88.
Thrushes, 8, 9, 16, 30, 42-46, 48, 51, 62, 190.
Thunder-storm, 182.
Tide, 204.
Tiger, 199.
Timber, 181.
\textit{Timbre}, 200.
Timothy, 64.
Toads, 19, 50, 132, 155.
Towhee, 182.
Track, 148, 156, 161, 162.
Trail, 177, 178, 183.
Trail-horse, 177.
Transmutations, 60.
Tree ferns, 117.
Tree-sparrows, 70, 155, 162.
Tree-swallows, 19, 25.
Tree-toads, 50, 52.
Trillium, 62.
Trough, 203.
Trout, 172.
Trout-rod, 195.
Trunk, 141, 158, 185, 186, 188.
Tufted titmouse, 47.
Tulip trees, 58.
Tupelo, 113, 124, 125, 134.
Twilight, 192, 199.
Twisted-stalk, 62.
Twittering, 19.
Undergrowth, 59.
Upholsterer-bee, 82.
Urchins, 203, 204.
Veery, 44.
Vespa, 78.
Vessel, 206, 207.
Vesuvius, 58.
Viburnum, 113, 124.
Violets, 13, 20, 21, 26, 57, 59, 60, 62, 129, 169.
Vireos, 5, 8, 31, 36, 37, 93.
Virginia creeper, 3, 71, 92, 123.
Viscid drops, 86.
Volcano, 58.
Wake-robins, 67.
Warblers, 5, 8, 21, 25-27, 31, 37, 39, 40, 66, 93, 118.
Warriors, 102, 103, 108, 111.
Wasps, 55, 78, 84-86, 89-91, 93.
Waterfall, 67, 170.
Water-thrush, 48.
Waves, 76, 184, 198, 202-204, 207.
Weasels, 27, 102, 136, 143, 149.
Web, 64, 84, 85, 87.
Weeds, 69, 70, 72-74, 76, 77.
Weevil, 121.
Whelk, 203, 204.
White birch, 57, 90, 157, 186.
Whitecaps, 199.
White-fringed orchis, 168.
White oak, 8, 120, 157, 186.
White pines, 7, 125.
White violets, 21, 26.
Wickiup, 189, 190.
Wigwam, 189.
Wild animals, 136.
Wild apple, 148.
Wild gardens, 56, 58, 65, 182.
Wild geese, 23.
Wilderness, 34, 43, 48, 100, 142, 169, 189, 193.
Wild, the, 137, 147, 179.
Willows, 13, 20, 34, 64, 78, 148, 170.
Winds, 49, 63-65, 71, 78, 84, 87, 89, 110, 117, 119, 125, 135, 153, 156, 165, 166, 168, 171, 181, 191, 198, 199, 203.
Wings, 88, 89, 98, 99, 105, 112, 120, 125, 163.
Winter, 11, 13, 14, 17, 22, 50, 66, 70, 82, 88, 112, 118, 122, 142, 144, 145, 154, 162, 166, 207.
Winterberries, 168.
Winter buds, 11.
Winter gales, 157, 206.
Wintergreen, 3.
Winter music, 163.
Winter visitants, 8.
Winter walk, 156, 159.
Winter woods, 157, 161.
Winter wren, 49, 66, 119.
Wireworm, 89.
Witch-hazel, 58, 125, 145.
Witch-grass, 64.
Wolf, 136.
Wood-anemones, 13, 20, 21, 67.
Wood-betony, 21.
Woodchuck, 15, 136, 143-145, 155, 166.
Woodcraft, 136, 149.
Woodland, 201.
Woodland birds, 62.
Woodland flowers, 61, 62, 67, 80.
Wood life, 187.
Wood-lilies, 113.
Woodpecker, 34, 160.
Wood-pewee, 118.
Woods, 32, 38, 41-43, 47, 48, 52, 56, 61, 62, 67, 80, 104, 115, 122-124, 129, 136-138, 140-142, 147, 149, 153, 154, 157, 163, 165, 166, 185, 186.
Woodsia, 67, 170.
Woodsman, 187, 188, 193.
Wood-sorrel, 63.
Woodthrush, 30, 42, 46.
Wordsworth, 173.
Workers, 80, 81, 95, 99, 105, 108.
Worm, 4, 15, 89-91.
Wounded, 111.
Wreck, 207.
Wrens, 30, 31, 47, 49, 56, 66, 119, 181.
Yapping, 191.
Yellow ants, 94, 112.
Yellow birch, 32, 158.
Yellow pine, 6.
Yellowpoll, 119.
Yellow-throated vireo, 36.
Yellow violets, 21.
Yellow warbler, 31, 37.
\gutchapter{Transcriber's Notes}
Page 27: Changed ``mail'' to ``male''.
(Orig: the mail seems to take it upon himself)
\gutchapter{End of the Project Gutenberg EBook of In the Open, by Stanton Davis Kirkham}
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\documentclass{article}
\usepackage{ols,epsfig}
\usepackage{url}
\begin{document}
%don't want date printed
\date{}
\title{\Large \bf SE Linux Tutorial for Linux Kongress}
\author{
Russell Coker \url=<[email protected]>=,\\
{\normalsize \url=http://www.coker.com.au/=} \\
} % end author
\maketitle
% You have to do this to suppress page numbers. Don't ask.
\thispagestyle{empty}
\section{Introduction}
The aim of this tutorial is an advanced hands-on training in administering
NSA Security Enhanced Linux~\cite{ols2001:loscocco-smalley}. The Debian
distribution is used because it's support for SE Linux is better than any
other distribution, and because I did most of the Debian development work
regarding SE Linux and know it well. Most of the material covered here should
apply to other distributions when they have support for SE Linux.
To install a package on a Debian system the command \emph{dpkg -i package.deb}
is used.
At the start of this tutorial you will have a workstation with a root password
of "1234" and a set of Debian packages in the \emph{/root/pkg} directory that
can be installed. Also in the \emph{/usr/src} directory there will be a
complete archive of kernel source.
\section{Kernel Build}
The first stage of installing SE Linux is to build a kernel with support for
it. This involves firstly applying the \emph{LSM} kernel patch from
\url=http://lsm.immunix.org/= which includes the SE Linux patch. For Debian
this patch is in the \emph{kernel-patch-2.4-lsm} and
\emph{kernel-patch-2.5-lsm} packages for the 2.4.x and 2.5.x kernels
respectively.
Install the package \emph{kernel-patch-2.4-lsm}.
In the /usr/src directory you will find the file \emph{linux-2.4.19.tar.bz2}.
Extract it with the command \emph{tar xvjf linux-2.4.19.tar.bz2}. Now
change to the \emph{linux-2.4.19} directory that has been created and apply the
kernel patch with the command \emph{/usr/src/kernel-patches/all/apply/lsm}.
After the patch is applied run the command \emph{make menuconfig} to
configure the kernel. There is a new section \emph{Security options} which
contains the options for SE Linux and other security options. For SE Linux
turn on the options for \emph{Capabilities Support} (which SE Linux requires),
\emph{NSA SELinux Support}, and \emph{NSA SELinux Development Module}.
If a kernel is compiled with Development support then it boots into
\emph{permissive mode} by default, which means that it writes log messages
instead of preventing operations. This is essential when you first setup
SE Linux, otherwise a mistake in the policy could render the machine
unbootable.
We will not compile the kernel now due to lack of time.
\section{First Stage of Installation (kernel and policy)}
The first stage of installing SE Linux is to install the \emph{login} program
from the package \emph{login\_20000902-12.se1\_i386.deb} which is needed to
assign the correct security context to the user when they login.
After this is installed then install the kernel image from
\emph{kernel-image-2.4.19lsm\_1\_i386.deb}. But DO NOT REBOOT!
It is important not to reboot until all the other files are ready for SE
operation to avoid the problems of a partially working SE setup.
The next packages to install are the \emph{selinux} package that has the core
SE Linux administrative programs, and the \emph{selinux-policy-default} package
which has sample security policy files. These are in
\emph{selinux\_2002070313-9\_i386.deb} and
\emph{selinux-policy-default\_2002070313-9\_all.deb} respectively, and must be
installed in this order.
The \emph{selinux-policy-default} package will ask you a number of questions
about policy files to remove. Your workstation has a fairly minimal
configuration at the moment, so you can remove any policy files that don't
interest you (apart from \emph{named.te} and \emph{irc.te} which we need
for later exercises), later in the tutorial you may install other packages
so you may want to leave the policy files in place for such packages). Also
please note that the \emph{sendmail.te} policy conflicts with that for other
mail servers so it must be removed.
If you accidentally remove a \emph{.te} file that you need, then later you can
copy it to \emph{/etc/selinux/domains/program/} from
\emph{/usr/share/selinux/policy/default/domains/program/} and run
"\emph{make -C /etc/selinux load}".
After you have finished deciding which policy files to remove the package
will then compile and install the policy. Then it will label every file on
the file system with a security \emph{type}. The core of SE Linux is
\emph{Domain Type} access control (known as \emph{DT}). DT relies on every
object (file, directory, network port, etc) having a security type associated
with it, and every process being in a security \emph{domain}. So the labelling
of all the files on the file system is a very important part of the
installation!
The final package to install in this phase is dpkg (the Debian packge manager).
The package contains a modified version of dpkg that will label files with
the correct SE Linux \emph{type} after installing them.
Now reboot the machine to use the new kernel.
\section{Second Stage of Installation}
In the previous section you installed the bare minimum SE Linux functionality,
the SE kernel support and a policy for the kernel to load. The policy package
relabelled all files on the file system with correct types for you. You also
installed a login program to allow you to login with the correct context, and a
modified version of \emph{dpkg} to allow you to install programs in the
correct context.
Now login to the machine, after entering your root password you will see the
following: \\
\begin{verbatim}
lyta login: root
Password:
Last login: Fri Aug 16 03:38:02 2002 on vc/2
Linux server 2.4.19lsm #1 Tue Aug 6 14:53:07 CEST 2002 i686 unknown unknown GNU/Linux
Debian GNU/Linux comes with ABSOLUTELY NO WARRANTY, to the extent
permitted by applicable law.
Your default security context is root:user_r:user_t
Do you want to enter a new security context?[n]
\end{verbatim}
Now this means that it will log you in with the \emph{context} of
\emph{root:user\_r:user\_t} by default.
The security context is comprised of three parts, the first part is the
\emph{identity} which is the Unix username that you used to login. The next
part is the \emph{role} which is one of the roles that are assigned to the
identity (an identity can have one role or multiple roles), role names
customarily end in \emph{\_r}. The final part of the security context is the
\emph{domain}. The domain name ends in \emph{\_t}, it is the determining factor
for all SE Linux security decisions. Each role is limited to a certain set
of domains.
Therefore your \emph{identity} (your Unix username) limits your choice of
\emph{role}, which determines which \emph{domain} you can use, and therefore
what access you get to the system.
Now the security context of \emph{root:user\_r:user\_t} is not what you want,
you want to be able to continue doing administration work and configuring all
aspects of the system, the \emph{user\_t} domain is very limited. This means
that you need to type "y" at the prompt to enter a new security context,
giving the following result:
\begin{verbatim}
Your default security context is root:user_r:user_t
Do you want to enter a new security context?[n] y
Enter role (return for default) [user_r]
\end{verbatim}
At this time you must type "sysadm\_r" to select the system administration
domain giving the following:
\begin{verbatim}
Do you want to enter a new security context?[n] y
Enter role (return for default) [user_r] sysadm_r
Enter type (return for default) [sysadm_t]
\end{verbatim}
Now you have to press ENTER to select the default domain of \emph{sysadm\t}
and you get a root shell with full administrative privileges.
However there is a problem! At the time we labelled the files on the file
system the machine was running without SE Linux, so any files created after
the file system was labelled but before we booted a SE kernel will not have
a type associated with them (that means any files created during the
shut-down process). To fix this we have to run "make -C /etc/selinux relabel".
Now you want put SE Linux in enforcing mode. The program \emph{avc\_toggle}
is used to switch between \emph{permissive} mode (where messages are logged
and nothing else happens), and \emph{enforcing} mode (where actions prohibited
by policy are denied. Run \emph{avc\_toggle} now to put the machine in
enforcing mode, and notice that it informs you of the new state by printing
\emph{enforcing} on the command line. Also a message such as the following
will be logged by the kernel and appear on your screen:
avc: granted { avc\_toggle } for pid=16153 exe=/usr/bin/avc\_toggle scontext=root:sysadm\_r:sysadm\_t tcontext=system\_u:system\_r:kernel\_t tclass=system
Now go to another virtual console and login as \emph{root} with the default
security context of \emph{root:user\_r:user\_t}.
Notice that in the session where you are logged in with the \emph{sysadm\_t}
domain you can run "\emph{ls -l ~root}", \emph{dmesg}, and "\emph{ps aux}" and
get the results that you are used to. Now compare the results of the same
commands in the session that is logged in with the \emph{user\_t} domain.
When logged in as a non-administrative domain the \emph{ps} command will only
show you processes in the same domain and in certain other domains that you
have access to. You can not see daemon processes, so if the system is running
a daemon that has a security hole a hostile user will be less able to discover
it's existence or know how to attack it. Also the root home directory is
configured such that non-administrative users can not write to it (otherwise
they could replace the \emph{.bashrc} with a script that does something
nasty). Finally \emph{dmesg} is denied from most users so that they can't
see details of system hardware, and when their actions are blocked by the
security policy they can't see why.
Now you have verified that SE Linux is working, try running \emph{avc\_toggle}
again and notice that the machine has re-entered \emph{permissive} mode, this
means that the login with the \emph{user\_t} domain can see all processes with
\emph{ps}, run \emph{dmesg}, etc. After verifying this run \emph{avc\_toggle}
once more to leave the machine in \emph{enforcing} mode.
At any time if you need to know the mode the machine is in you can run the
program \emph{avc\_enforcing}, and it will tell you. Try it now.
The next time you boot the machine you want it to run in enforcing mode, to
do this run the command "\emph{ln -s /sbin/avc\_toggle /etc/rc.boot}" to make
\emph{avc\_toggle} be run at boot time.
For the purposes of rest of the tutorial make sure you leave the machine in
enforcing mode.
\section{Installing Utility Programs}
Now you have a machine that has every process running in a security
\emph{domain} and every file and directory has a \emph{type} associated with
it. But you are unable to determine what the domains and types are!
To do this you need to install some new packages, however the regular
\emph{dpkg} command will not work when SE Linux is in enforcing mode. There is
a wrapper called \emph{se\_dpkg} which sets the correct security context that
you can use instead. These are \emph{shellutils} for a modified version of
\emph{id} and adds the \emph{runas} command, \emph{procps} for a modified
\emph{ps}, and \emph{fileutils} adds the \emph{chcon} command and has modified
versions of \emph{cp}, \emph{mv}, \emph{install}, \emph{ls}, \emph{mkdir},
\emph{mkfifo}, and \emph{mknod}.
Install the packages \emph{shellutils\_2.0.12-2.se1\_i386.deb},
\emph{procps\_2.0.7-10.se1\_i386.deb}, and
\emph{fileutils\_4.1.10-2.se1\_i386.deb} now by using the \emph{se\_dpkg}
command. This command will prompt you for the password matching the
\emph{identity} that you are using (in this case the root password).
Run the command \emph{id} from both of your root login sessions, notice how
it reports the different security context used for the processes in each
session.
Run the command "\emph{ls --context /}" and observe the different types being
used for the files.
Run the command "\emph{ps ax --context}" from the \emph{sysadm\_t} login and
observe the different security contexts being used for the daemons.
Run the command "\emph{ls --context `tty`}" from both sessions and observe
that the tty devices have a different type for each login. This prevents
unauthorised access to a tty. To test this use the \emph{wall} command, note
that when you run \emph{wall} as \emph{sysadm\_t} all users see it, but when
you run it as \emph{user\_t} only other users in \emph{user\_t} see it.
Now login as \emph{user\_r} on another console and run \emph{irc} to connect
to the IRC server. Go to the other console and run "\emph{ps ax --context}"
and note how the IRC program runs in the \emph{user\_irc\_t} domain, now run
"\emph{ls -atr --context}" to see how the file that the IRC client creates
has a different context. This is so that the IRC client has no access to
files or directories under your home directory, it can create and modify it's
own configuration files and nothing else. Then if someone cracks the IRC
client it can't kill any programs you run, or read or modify any files you
own (apart from it's own configuration). After writing this policy I spent
some time in IRC as \emph{root} and had some amusing discussions with IRC
users who seemed to think it was a security risk and could not be convinced
otherwise.
Now join the \#se channel, we will use that for distributing policy samples,
and for some discussion during the tutorial. Also if you wish to discuss in
German with your friends please join the \#de channel as well (unfortunately I
can't read German so I won't be on that channel).
\section{Installing Modified System Programs}
Now that you have installed the utility programs you next have to install
modified versions of some system programs. Two important programs are
\emph{cron} and \emph{logrotate}.
We need a modified version of \emph{cron} to run the user cron jobs in the
correct context, and it has to check the \emph{type} of the crontab file to
ensure that it hasn't been created by someone who is not authorised for that
cron domain.
We need a modified \emph{logrotate} so that when it creates a new log file it
gives it the same \emph{type} as the old file that has been renamed.
Install the packages \emph{cron\_3.0pl1-72.se3\_i386.deb} and
\emph{logrotate\_3.5.9-10.se1\_i386.deb} now.
\section{Creating User Accounts}
On a SE Linux machine you may want to have users with a UID=0 (root users) who
are not permitted to use the \emph{sysadm\_r} for administrative access. To
keep them out they have to be prevented from accessing \emph{/etc/shadow},
otherwise they could run crack on the passwords.
This of course requires a mechanism for changing the password. There are
wrappers for the programs \emph{passwd}, \emph{chsh}, \emph{chfn}, \emph{vipw},
and \emph{useradd}, these are \emph{spasswd}, \emph{schsh}, \emph{schfn},
\emph{svipw}, and \emph{suseradd} respectively. The main wrapper programs
\emph{spasswd}, \emph{schsh}, and \emph{schfn} do not allow specifying any
parameters (so you can't change the password, shell, or finger details for
anyone else's account). The \emph{svipw}, and \emph{suseradd} wrappers allow
specifying parameters but can only be run by the administrator in domain
\emph{sysadm\_t}. If you are logged in as root with administrative
privileges then you can change someone else's password with \emph{sadminpaswd}.
Now change the root password with \emph{spasswd}, create a new user for your
own use with \emph{suseradd}, and edit the user's details with \emph{svipw}.
After adding the new user you need to assign them to a role (or roles). To
do this firstly edit the file \emph{/etc/selinux/users}. If you have an
account named \emph{john} that you want to allow for the \emph{user\_r} role
then you add "\emph{user john roles user\_r;}" to the file. If you want to
grant access to the role \emph{sysadm\_r} as well then add
"\emph{user john roles \{ user\_r sysadm\_r \};}". After saving the changes
apply them with the command "\emph{make -C /etc/selinux load}". Note that
the default role is \emph{user\_r}, if a user is not permitted any other role
then there is no need to specifically add them to the policy. However adding
the user to the policy allows syslog messages about denied actions to include
the user name (which makes it easier to determine which user is doing things
you don't like), so it's handy to have.
After adding an entry for a user in the \emph{users} file you must set them
a default security context for their login sessions, this is determined by
the file \emph{/etc/security/default\_context}. In the case of an account
\emph{john} you would add \emph{john:user\_r:user\_t} to the
\emph{/etc/security/default\_context} file.
The next thing to do is to relabel their home directory to the correct type
through the following commands:
\begin{verbatim}
find /home/john | xargs chcon -h system\_u:object\_r:user\_home\_t
chcon -h system\_u:object\_r:user\_home\_dir\_t /home/john
\end{verbatim}
Now install \emph{ssh} from \emph{ssh\_3.4p1-2.se1\_i386.deb}, create a new
account named \emph{test} that is only authorised for the \emph{user\_r}
domain and tell the people on the computer next to yours the password and
the IP address of your machine so that they can login.
Now login to each other's machines and have a look around. Try various
methods of tracing what the other person is doing on your machine as
\emph{sysadm\_t} while they (as \emph{user\_t}) can't see what you are
deoing. Please play nice when logging into each other's machines (just in
case they made a mistake).
\section{Starting and Stopping Daemons}
When a daemon is run it has to be run in the correct context to ensure that it
gets access to the resources it needs, and that it is denied access to
everything it shouldn't be accessing.
The daemon start scripts are run in the context
\emph{system\_u:system\_r:initrc\_t}, and there are policy rules to cause a
transition from the \emph{initrc\_t} domain to the domain the daemon runs in.
Inspect the file \emph{/etc/selinux/domains/program/named.te} and notice that
it has \emph{daemon\_domain(named)} on line 14. The \emph{daemon\_domain}
macro is defined on line 932 of \emph{/etc/selinux/macros/global\_macros.te}
which calls the macro \emph{daemon\_base\_domain} on line 901 of the same file.
The relevant line is line 907 which calls the \emph{domain\_auto\_trans} macro
to setup an automatic domain transition. When the \emph{initrc\_t} domain
executes a file of type \emph{named\_exec\_t} it will be run in the new domain
\emph{named\_t}.
For more information on SE Linux policy see Stephen D.\ Smalley's
paper~\cite{smalley}.
When logged in under the \emph{sysadm\_t} domain you can stop a daemon in the
usual fashion. To start a daemon you need to use the \emph{run\_init} program
to start it in the correct context, \emph{run\_init} will run a specified
program in the \emph{system\_u:system\_r:initrc\_t} context. Stop the
\emph{named} daemon now with the command "\emph{/etc/init.d/bind stop}", and
restart it with the command "\emph{run\_init /etc/init.d/bind start}".
Now use the command "\emph{ps ax --context | grep named}" to verify that it is
running and in the correct domain.
\section{Adding a New User Domain}
To add a new user domain \emph{second\_t} and role \emph{second\_r}
edit the file \emph{/etc/selinux/domains/user.te} add the following:
\begin{verbatim}
full_user_role(second)
allow system_r second_r;
allow sysadm_r second_r;
\end{verbatim}
The first line creates the domain \emph{second\_t}, and the types
\emph{second\_home\_dir\_t} and \emph{second\_home\_t} (for the home directory
and for files under the home directory respectively),
\emph{second\_tmp\_t} for files created under /tmp,
\emph{second\_tmpfs\_t} for shared memory created in the context of tmpfs, and
\emph{second\_tty\_device\_t} and \emph{second\_devpts\_t} for user labelling
of tty devices and pseudo-tty devices respectively. It also creates all the
basic policy rules for using these types.
The reason for having the types \emph{second\_home\_dir\_t} and
\emph{second\_home\_t} is so that the IRC client can be given access to ~user
which is of type \emph{second\_home\_dir\_t} but not to ~user/Mail which is of
type \emph{second\_home\_t}.
SE Linux does not internally support any type of object orientation,
inheritance of domains/types, etc. Also there is currently no policy language
that supports such features (one could be written, but no-one has done so yet).
So to get the features we need for easily creating new domains etc we use
\emph{M4} macros.
Now create a new user for use in this domain with the \emph{suseradd} program,
add them to the \emph{/etc/selinux/users} file with an entry that gives them
access to the role \emph{second\_r} (and no other roles), then run
"\emph{make -C /etc/selinux load}" to apply the new policy. Then run the
following commands to apply the correct labels to the home directory:
\begin{verbatim}
find /home/user | xargs chcon -h system_u:object_r:second_home_t
chcon -h system_u:object_r:second_home_dir_t /home/user
\end{verbatim}
The next thing to do is set the default domain for the new role, to do this
edit the file \emph{/etc/security/default\_type} and add the line
"\emph{second\_r:second\_t}". This is for the login prompt for the domain.
Now try logging in as the new user!
\section{Setting up a Chroot Environment}
Now we want to setup a chroot environment running the old Debian version
named \emph{Potato} from the \emph{user\_t} domain.
The administrator of the chroot environment can allow other people to
login as root with administrative privileges (they can add accounts, change
passwords etc), but they can't escape from the chroot. Also some of the files
in the chroot environment can be set as read-only, so that the administrative
user inside the chroot can't change then. This is both a protection against
having the chroot environment "\emph{cracked}", and a way of limiting what
the root user inside the chroot can do.
Firstly add the line "\emph{chroot(user, potato)}" to
\emph{/etc/selinux/domains/user.te}. Then create the file \\
\emph{/etc/selinux/domains/misc/custom.te} with the line
"\emph{file\_type\_auto\_trans(user\_t, src\_t, potato\_ro\_t)}". Now load the
policy.
Now login as \emph{root} in the \emph{user\_r} role and extract the archive
\emph{/usr/src/potato.tar.bz2} with the command
"\emph{cd /usr/src ; tar xvjf potato.tar.bz2}". This archive includes the
script \emph{mountit} as follows:
\begin{verbatim}
#!/bin/bash -e
if [ "$1" == "mount" ]; then
cd $2
mount -n none -t proc proc
mount -n /dev dev --bind
fi
if [ "$1" == "umount" ]; then
cd $2
umount -n `pwd`/proc/
umount -n `pwd`/dev
fi
\end{verbatim}
Also the script \emph{setperms} as follows:
\begin{verbatim}
#!/bin/bash -e
# the first parameter is the full path to the root directory
# the second parameter is the name of the chroot (the second parameter to
# the chroot() macro
echo cd $1
IDENTITY=`id -c | cut -f1 -d:`
BASECON=$IDENTITY:object_r:$2
find home root tmp var | xargs -l300 chcon ${BASECON}_rw_t
find etc/init.d | xargs -l300 chcon ${BASECON}_dropdown_t etc/init.d
chcon ${BASECON}_ro_t var var/lib
find /var/lib/dpkg | xargs -l300 chcon ${BASECON}_ro_t
chcon ${BASECON}_super_entry_t entry
\end{verbatim}
The files in the chroot are currently labelled as \emph{potato\_ro\_t} due to
the "\emph{file\_type\_auto\_trans(user\_t, src\_t, potato\_ro\_t)}" which
causes any files or directories created under a \emph{src\_t} directory by
the \emph{user\_t} domain to be labelled as \emph{potato\_ro\_t}.
So the before we try running anything we have to use the \emph{setperms}
script to label some of the files and directories as allowing writes, the
command is "\emph{/usr/src/potato/setperms /usr/src/potato potato}".
Now to get the chroot going you first have to mount \emph{/proc} and
\emph{/mount} under the chroot with the \emph{mountit} script, the command
"\emph{/usr/src/potato/mountit mount /usr/src/potato}" will do this.
Before you try logging in to the chroot environment you have to change the
root password, the command "\emph{cd /usr/src/potato ; chroot . passwd}" will
do this.
Now we have the files setup for a chroot environment and a password in place,
all we have to do is allow logins to it by starting a syslogd daemon and a ssh
daemon. I have configured the ssh daemon to run on port 222 (so it doesn't
conflict with the main sshd), so all you have to do is run
"\emph{cd /usr/src/potato ; chroot . /etc/init.d/sysklogd start}" to start the
system log, and
"\emph{cd /usr/src/potato ; chroot . /etc/init.d/ssh start}" to start sshd.
\begin{thebibliography}{99}
\bibitem{ols2001:loscocco-smalley}\emph{Meeting Critical Security Objectives with Security-Enhanced Linux} \\
Peter A.\ Loscocco, NSA, [email protected] \\
Stephen D.\ Smalley, NAI Labs, [email protected] \\
{\tt \small \url=http://www.nsa.gov/selinux/ottawa01-abs.html/=}
\bibitem{smalley}\emph{Configuring the SELinux Policy} \\
Stephen D.\ Smalley, NAI Labs, [email protected] \\
{\tt \small \url=http://www.nsa.gov/selinux/policy2-abs.html/=}
\end{thebibliography}
\end{document}
|
http://floridaclimateinstitute.org/refbase/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%202177%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=20&rowOffset=&wrapResults=1&citeOrder=creation-date&citeStyle=APA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg=
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%&LaTeX
\documentclass{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
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\begin{thebibliography}{1}
\bibitem{Nowak+Greenfield2018} Nowak, D. J., \& Greenfield, E. J. (2018). Declining urban and community tree cover in the United States. \textit{Urban Forestry \& Urban Greening}, \textit{32}, 32--55.
\end{thebibliography}
\end{document}
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http://ctan.sharelatex.com/tex-archive/macros/latex/contrib/flipbook/flipbook.sty
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sharelatex.com
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text/x-tex
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https://projects.mako.cc/source/beamer-mako/blob_plain/35f4f87927ef257ef8610ebfb685877aab69d9fb:/template/beamerouterthemedecolines.sty
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mako.cc
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https://ctan.math.washington.edu/tex-archive/info/examples/Einfuehrung/06-03-107.ltx
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washington.edu
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%%
%% Ein Beispiel der DANTE-Edition
%%
%% 1. Auflage
%%
%% Beispiel 06-03-107 auf Seite 291.
%%
%% Copyright (C) 2011 Herbert Voss
%%
%% 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.
%%
%%
%% ====
% Show page(s) 1
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\documentclass[]{exaarticle}
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\StartShownPreambleCommands
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\begin{tabular}{SS[table-align-exponent=false]}\toprule
\emph{Spalte} & \emph{Spalte} \\\midrule
1.2e3 & 1.2e3\\
1.234e56 & 1.234e56\\\bottomrule
\end{tabular}
\end{document}
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https://tec.ucc.asn.au/UCC%20Stuff/Documents/Emails/2020Announce2.tex
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ucc.asn.au
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% Created 2020-03-22 Sun 17:11
% Intended LaTeX compiler: pdflatex
\documentclass{scrartcl}
\usepackage[T1]{fontenc}
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\usepackage{hyperref}
\usepackage{minted}
\author{tec}
\date{\today}
\title{\textls*[70]{\textsc{ucc}}-Announce email 2}
\hypersetup{
pdfauthor={tec},
pdftitle={\textls*[70]{\textsc{ucc}}-Announce email 2},
pdfkeywords={},
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pdfcreator={Emacs 26.3 (Org mode 9.4)},
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\urlstyle{same}
\begin{document}
\maketitle
\tableofcontents
Dear \textls*[70]{\textsc{ucc}}ans,
Over the past week, there have been several important developments. This email is a long one, but also an important one for the club.
\section{The new clubroom offer}
\label{sec:orgf281a79}
On Tuesday (2020-03-17) the committee met to discuss the new clubroom offer in Guild commercial space. We decided that acceptance of the offer would only be suitable if the following conditions could be met:
\begin{itemize}
\item \textls*[70]{\textsc{ucc}} will be guaranteed a space until the next relocations
\item The room satisfies basic power, networking, and security requirements
\item This room is of sufficient size that we can retain our equipment
\item Amendment to the current Tenancy Allocation Outcome, which stated that no space would be offered
\end{itemize}
Unfortunately, despite repeated attempts to establish further communications with the Societies Council following the offer, we failed to receive a response. Without the opportunity to discuss these matters, and with the appeals deadline closing, we felt it was not possible to accept this offer; particularly as no details had been formally presented.
Hence, on Friday, the committee submitted a letter to appeal the original Tenancy decision which stated that \textls*[70]{\textsc{ucc}} would not be offered a clubroom for the 2020-2022 allocation period.
\section{The Appeal}
\label{sec:org6ba7516}
We have just submitted the \emph{first substantive appeal}. For your interest I have attached the following documents:
\begin{itemize}
\item Our original application
\item Tenancy's response
\item Our appeal
\end{itemize}
Should this appeal fail, rest assured that there are other layers we can move to.
Given the recent closures of University and Guild spaces such as Cameron Hall, any decisions regarding the future of \textls*[70]{\textsc{ucc}}'s tenancy may not be known for some time. This means there will be time for you to help.
It would be much appreciated if you could write a statement in support of \textls*[70]{\textsc{ucc}}. Should you do so, consider mentioning aspects such as
\begin{itemize}
\item The impact the club has had on you personally
\item The value you think the club provides to students in general
\item The necessity of a room for the club
\end{itemize}
Consider sharing such a letter to the following groups:
\begin{itemize}
\item The Tenancy Committee
\item The Guild Executive
\item cc: [email protected]
\end{itemize}
This is a pain, so \href{mailto:[email protected],[email protected],[email protected],[email protected],[email protected],[email protected],[email protected][email protected]}{click here for a pre-addressed email}.
If you are a \textls*[70]{\textsc{ucc}} alumni then consider also sending the email to Guild Development and Alumni Relations at [email protected].
For \textls*[70]{\textsc{ucc}} alumni, you may also consider writing to,
\begin{itemize}
\item Alumni Relations: [email protected]
\item Guild Development: [email protected]
\item Convocation: [email protected]
\end{itemize}
\section{Closure of Cameron Hall}
\label{sec:org96e7cb3}
In response to the \textls*[70]{\textsc{covid}}-19 situation, the Guild has announced that it is taking the following steps to protect the health and safety of students:
\begin{itemize}
\item Cancelling all events
\item Closing all Guild spaces (\textbf{i.e. Cameron Hall})
\item Postponing tenancy allocations
\end{itemize}
This decision is set to be re-evaluated in \textbf{\textbf{six months}}, which is around the time of Guild Elections. Until that time, it this means that:
\begin{itemize}
\item You will be unable to access Cameron Hall
\item The tenancy allocation will be put on hold
\end{itemize}
Moving forward, we will be ramping up our online events, and thanks to [\textls*[70]{\textsc{trs}}] we have \href{https://meetings.ucc.asn.au}{meetings.ucc.asn.au} available for member use.
To quote the landing page, this is an ``open-source web conferencing server. You can create your own rooms to host sessions, or join others using a short and convenient link.''
\section{New LinkedIn group}
\label{sec:org4927207}
While separate to all of the drama above, \textls*[70]{\textsc{ucc}} now has a LinkedIn group, in addition to a re-vamped company page.
\begin{itemize}
\item \href{https://www.linkedin.com/company/universitycomputerclub/}{\textls*[70]{\textsc{ucc}} LinkedIn page}
\item \href{https://www.linkedin.com/groups/13844386/}{\textls*[70]{\textsc{ucc}} LinkedIn Group}
\end{itemize}
This should provide another avenue for \textls*[70]{\textsc{ucc}}ans to keep in touch long after leaving the club :)
This is a tumultuous time for \textls*[70]{\textsc{ucc}}. We will continue to endeavour to keep you abreast of the situation, though with the six month tenancy postponement, the time pressure has considerably lessened.
Stay well,
Timothy.
\end{document}
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http://theanarchistlibrary.org/library/johann-most-anarchist-communism.tex
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theanarchistlibrary.org
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\title{Anarchist Communism}
\date{1889}
\author{Johann Most}
\subtitle{}
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{\usekomafont{title}{\huge Anarchist Communism\par}}%
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\par
Anarchism is a world view, a philosophy of society; indeed the philosophy of society, for whoever considers the world and human life in their profoundest senses and their complete development, and then decides on the societal form of greatest desirability, cannot but decide for anarchism. Every other form is a half-measure and a patchwork.
Is anarchism desirable? Well, who does not seek freedom? What man, unless willing to declare himself in bondage, would care to call any control agreeable? Think about it!
Is anarchism possible? The failure of attempts to attain freedom does not mean the cause is lost. The facts that the struggle for freedom is clearer and stronger than ever before, that today there are different preconditions to achieving the goal, and that we therefore stand nearer anarchy than had been hoped — prove a development of the desire to wash from the face of the earth what is authoritarian.
Anarchists are socialists because they want the improvement of society, and they are communists because they are convinced that such a transformation of society can only result from the establishment of a commonwealth of property.
The aims of anarchists and true communists are identical. Why, then, are anarchists not satisfied to call themselves socialists or communists? Because they do not want to be confused with people who misappropriate these words, as many people do nowadays, and because they believe communism would be an incomplete, less-than-desirable system if not infused with the spirit of anarchism.
Communists and anarchists also agree on tactics. He who negates present society, and seeks social conditions based on the sharing of property, is a revolutionary whether he calls himself an anarchist or a communist. But anarchists are not bloodhounds who speak with levity of revolution by murder and arson. They make revolutionary propaganda because they know the privileged class can never be overturned peacefully.
The anarchists, on behalf of the proletariat, therefore consider it necessary to show the proletariat that it will have to win a gigantic battle before it realizes its goals. The anarchists prepare for social revolution and use every means — speech, writing, or deed, whichever is more to the point — to accelerate revolutionary development.
Can anyone, who honestly supports the proletariat, blame them for that? The fact that, as a consequence, capitalists, police, press, clergy, and other hypocrites and philistines hate us with all their hearts, all their minds, all their souls, and all their strength all the time — we can readily understand.
But it seems unnatural that at every step we meet fanatical hostility inside the labor movement, accompanied by bullheaded stupidity. The greatest stumbling block to anarchism among the non-anarchist socialists, which causes much of the discord, is the “free contract.” Yet one need not put oneself into a different world — neither Mars nor in Utopia — to see how the free contract would work. Take, for example, the International Postal Union. The national postal organizations join of their own free will and can withdraw in the same way. These contracting parties agree to what they will provide one another, in order to achieve service of the highest practicality and greatest efficiency. International law lacks precedent for compelling a violator be taken to court.
Nevertheless, “free contract” works — because, since every breach of promise carries with it damage to the breacher, it behooves every contracting party not to violate the contract. If irregularities arise, conferences agree on adjustments. This institution, a model for free association, is not an isolated example. People who have little else in common form groups, trusts, and pools — organizations musical, gymnastic, commercial, protective, educational, and political; and associations for the advancement of arts and science — in all countries, despite contradictory natures of the parties, and despite the fact that the parties cannot be forced to fulfill the agreements. Everything done in these agreements is done because of advantage to each member.
Absurd the claim that these organizations could not work without control by a higher power! Indeed, whenever and wherever government has interfered, it has disturbed and obstructed the organizations. Moreover, where this kind of intervention is happening, the organizations agitate with supreme energy for its abolition.
In a society of the free and equal there can be nothing but the free contract; cooperation by force violates freedom and equality. The gist of the matter is whether, in a society of the future, the various organizations (created and operating according to free contracts) are to be centralized or of a federal nature. We are for federalism as necessary and right, because experience has taught us that centralization must end in monstrous total-power accumulation in a few hands; centralization causes abuse of power, dominating by a few, and loss of freedom by many. In addition, we see nothing useful or necessary in centralization. If we hope and even assume that the social question will be answered through communism, and not in this or that country but in the world, any thought of centralization must be a monstrosity. Think of a bakers’ central commission, meeting in Washington, prescribing the bakers of Peking and Melbourne the size and amount of the rolls they are to bake.
Since the people of the future will not be old-fashioned fools, they will not fall into such nonsense. They will regulate their affairs as practice and experience teach. The shortsighted object. Freedom is now enjoyed in economic affairs, they say, and since government does not interfere, freedom has caused abuses. We accept this argument of our enemies and with it teach them something better. That is, economic freedom abused by private property has created the social question. Private property, guarded by the state, increasingly exploits the poor; and the poor less and less use what they produce. If the government did not wholeheartedly maintain this swindle, the masses would not suffer it.
Yes, the state is the organized power of property. Therefore the unpropertied must destroy the state, eliminate private property, and establish ownership in common.
Communism, contrary to the liberal-bourgeois tradition, needs no state to achieve its freedom and equality. Communism finds the force of the state disturbing and restrictive.
Now we come to the main objection to communism, that in it the individual gives himself up to the whole and leads no existence of his own — a thought fit to frighten away the original characters and throw a scare even into common philistines with no individuality to lose. We need do no more than repeat: only under communism does the individual become himself and lead his own life. Conversely, does anarchism isolate people and dissolve society? No. Our discussions show: the individual develops fullest in the system of ownership-in-common. Anarchism also does not prohibit the cooperation of some, many, or all — whichever is desirable — for the achievement of common goals.
Above all, what socialist, without flushing with shame, maintains he is not a revolutionary? We say: none!.
And the revolutionary favors constant propagation of principles. While we have entertained the contention that a deed may make more propaganda than hundreds of speeches, thousands of articles, and tens of thousands of pamphlets, we have held that an arbitrary act of violence will not necessarily have such an effect.
In short, propaganda-by-the-deed has not become our hobbyhorse, which we ride to the neglect of other propaganda. If on the one side we do not harbor the illusion that the entire proletariat must be enlightened before it can be called into battle, so on the other we do not doubt that as much enlightenment as possible must be produced with oral and printed agitation.
Fortunately, no country was ever more suited for anarchist agitation than present-day America. Here nobody wants to experiment further with the people’s state. It has been more than a century; it has experienced the profoundest fiasco [the civil war]; and future state-makers had better learn the lesson. Whoever looks at America will see: the ship is powered by stupidity, corruption, or prejudice. Long has the government disgusted noble and intelligent natures; they avoid voting; and they are, even if they don’t know it, anarchists.
The sharp-minded observer, the upright character, and the independent thinker see in the people’s state a crude superstition and are ready to listen to the anarchists. Finally, whatever else may be said, this much is for sure: the welfare of humanity, which the future can and will bring, lies in communism. It excludes in logical ways all authority and servitude, and therefore equals anarchy. The way to the goal is the social revolution. By energetic, relentless, international action, it will destroy class rule and establish a free society based on cooperative organization of production. Long Live the Social Revolution!
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The Anarchist Library
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\begin{center}
Johann Most
Anarchist Communism
1889
\bigskip
Retrieved on April 25, 2009 from \href{http://dwardmac.pitzer.edu/Anarchist\_Archives/bright/most/anarcom.html}{dwardmac.pitzer.edu}
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\textbf{theanarchistlibrary.org}
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% a plain TeX file for a 14 page document
\magnification=\magstep1
\baselineskip=15pt
\normalbaselineskip=15pt
\input ../inputs/paperB
\def\diff{\mathbin{\mkern-1.5mu\setminus\mkern-1.5mu}}% for \setminus
\def\pu{{\cal P}(u)}
\def\pv{{\cal P}(v)}
\def\puv{{\cal P}(u,v)}
\def\th{\theta}
\def\mtg{{\rm mult}(\th, G)}
\def\mt#1{{\rm mult}(\th, #1)}
\def\match{\mathop{\hbox{$\mu$}}\nolimits}
\def\mgx{\match(G,x)}
\def\mx#1{\match(#1,x)}
\authors={C. D. Godsil}
\runninghead={Algebraic Matching Theory}
\font\smcp=cmcsc8
\headline={\ifnum\pageno>1%
{\smcp the electronic journal of combinatorics 2 (1995),
\#R8\hfill\folio} \fi}
\pagefoot={\hss}
\rewritexrffilefalse
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\null\vskip0.5cm
\centerline{\bf ALGEBRAIC MATCHING THEORY}
\vskip1.0cm
\centerline
{C. D. Godsil
\footnote{$^1$}{Support from grant OGP0009439 of the National
Sciences and Engineering Council of Canada is gratefully
acknowledged.}}
\vskip0.5cm
\centerline{Department of Combinatorics and Optimization}
\centerline{University of Waterloo}
\centerline{Waterloo, Ontario}
\centerline{Canada N2L 3G1}
\centerline{\tt [email protected]}
\bigskip
\centerline{Submitted: July 6, 1994;\quad Accepted: April 11, 1995.}
\vskip1.0cm
\bigskip\noindent{\narrower
{\bf Abstract:} The number of vertices missed by a maximum matching in
a graph $G$ is the multiplicity of zero as a root of the matchings
polynomial $\mgx$ of $G$, and hence many results in matching theory
can be expressed in terms of this multiplicity. Thus, if $\mtg$ denotes
the multiplicity of $\th$ as a zero of $\mgx$, then Gallai's lemma
is equivalent to the assertion that if $\mt{G\diff u}<\mtg$ for
each vertex $u$ of $G$, then $\mtg=1$.
This paper extends a number of results in matching theory to results
concerning $\mtg$, where $\th$ is not necessarily zero. If $P$ is a
path in $G$ then $G\diff P$ denotes the graph got by deleting the
vertices of $P$ from $G$. We prove that $\mt{G\diff P}\ge\mtg-1$, and
we say $P$ is {\sl $\th$-essential}\/ when equality holds. We show
that if, all paths in $G$ are $\th$-essential, then $\mtg=1$. We define
$G$ to be {\sl $\th$-critical}\/ if all vertices in $G$ are
$\th$-essential and $\mtg=1$. We prove that if $\mtg=k$ then there is
an induced subgraph $H$ with exactly $k$ $\th$-critical components,
and the vertices in $G\diff H$ are covered by $k$ disjoint paths.
\medskip\noindent
AMS Classification Numbers: 05C70, 05E99\par}
\section{intro}% 1
Introduction
A {\sl $k$-matching} in a graph $G$ is a matching with exactly $k$
edges and the number of $k$-matchings in $G$ is denoted by by
$p(G,k)$. If $n=|V(G)|$ we define the matchings polynomial $\mgx$ by
$$
\mgx:=\sum_{k\ge0}(-1)^kp(G,k)x^{n-2k}.
$$
(Here $p(G,0)=1$.) By way of example, the matchings polynomial of the
path on four vertices is $x^4-3x^2+1$. The matchings polynomial is
related to the characteristic polynomial $\phi(G,x)$ of $G$, which is
defined to be the characteristic polynomial of the adjacency matrix of
$G$. In particular $\phi(G,x)=\mgx$ if and only if $G$ is a forest
[\r{cg-ig}:~Corollary~4.2]. Also the matchings polynomial of any
connected graph is a factor of the characteristic polynomial of some
tree. (For this, see \p{path-tree} below.)
Let $\mtg$ denote the multiplicity of $\th$ as a zero of $\mgx$. If
$\th=0$ then $\mtg$ is the number of vertices in $G$ missed by a
maximum matching. Consequently many classical results in the theory
of matchings provide information related to ${\rm mult}(0,G)$. We
refer in particular to Gallai's lemma and the Edmonds-Gallai structure
theorem, which we now discuss briefly.
A vertex $u$ of $G$ is {\sl $\th$-essential}\/ if $\mt{G\diff u}<\mtg$.
So a vertex is $0$-essential if and only if it is missed by some
maximum matching of $G$. Gallai's lemma is the assertion that if $G$
is connected, $\th=0$ and every vertex is $\th$-essential then
$\mtg=1$. (A more traditional expression of this result is given in
[\r{lv-pl}:~\S3.1].) A vertex is {\sl $\th$-special}\/ if it is not
$\th$-essential but has a neighbour which is $\th$-essential. The
Edmonds-Gallai structure in large part reduces to the assertion that
if $\th=0$ and $v$ is a $\th$-special vertex in $G$ then a vertex $u$
is $\th$-essential in $G$ and if and only if it is $\th$-essential in
$G\diff v$. (For more information, see [\r{lv-pl}:~\S3.2].) One aim
of the present paper is to investigate the extent to which these
results are true when $\th\ne0$.
There is a second source of motivation for our work. Heilman and Lieb
proved that if $G$ has a Hamilton path then all zeros of $\mgx$ are
simple. (This is an easy consequence of \p{path-del} below.) Since
all known vertex-transitive graphs have Hamilton paths we are lead to
ask whether there is a vertex-transitive graph $G$ such that $\mgx$
has a multiple zero. As we will see, it is easy to show that if $\th$
is a zero of $\mgx$ and $G$ is vertex-transitive then every vertex of
$G$ is $\th$-essential. Hence, if we could prove Gallai's lemma for
general zeros of the matchings polynomial, we would have a negative
answer to this question.
\section{idents}% 2
Identities
The first result provides the basic properties of the matchings
polynomial $\mgx$. We write $u\sim v$ to denote that the vertex $u$
is adjacent to the vertex $v$. For the details see, for example,
[\r{cgbk}:~Theorem~1.1].
\result{recurs}% 2.1
Theorem. The matchings polynomial satisfies the following identities:
\item{(a)} $\match(G\cup H,x)=\mgx\match(H,x)$,
\item{(b)} $\mgx=\match(G\diff e,x)-\match(G\diff uv,x)$
if $e=\{u,v\}$ is an edge of $G$,
\item{(c)} $\mgx =x\match(G\diff u,x)
-\sum_{i\sim u}\match(G\diff ui,x)$, if $u\in V(G)$,
\item{(d)} ${d\over dx}\mgx
=\sum_{i\in V(G)}\match(G\diff i,x)$.\qed
Let $G$ be a graph with a vertex $u$. By $\pu$ we denote the set of
paths in $G$ which start at $u$. The {\sl path tree}\/ $T(G,u)$ of
$G$ relative to $u$ has $\pu$ as its vertex set, and two paths are
adjacent if one is a maximal proper subpath of the other. Note that
each path in $\pu$ determines a path starting with $u$ in $T(G,u)$ and
with same length. We will usually denote them by the same symbol.
The following result is taken from [\r{cgbk}:~Theorem~6.1.1].
\result{path-tree}% 2.2
Theorem. Let $u$ be a vertex in the graph $G$ and let $T=T(G,u)$ be
the path tree of $G$ with respect to $u$. Then
$$
{\match(G\diff u,x)\over\mgx}={\match(T\diff u,x)\over\match(T,x)}
$$
and, if $G$ is connected, then $\mgx$ divides $\match(T,x)$.\qed
Because the matchings polynomial of a tree is equal to the
characteristic polynomial of its adjacency matrix, its zeros are real;
consequently \p{path-tree} implies that the zeros of the matchings
polynomial of $G$ are real, and also that they are interlaced by the
zeros of $\match(G\diff u,x)$, for any vertex $u$. (By interlace, we
mean that, between any two zeros of $\mgx$, there is a zero of
$\match(G\diff u,x)$. This implies in particular that the
multiplicity of a zero $\th$ in $\mgx$ and $\match(G\diff u,x)$ can
differ by at most one.) For a more extensive discussion of these matters,
see [\r{cgbk}:~\S6.1].
We will need a strengthening of the first claim in \p{path-tree}.
\result{pt-cor}% 2.3
Corollary. Let $u$ be a vertex in the graph $G$ and let $T=T(G,u)$ be
the path tree of $G$ with respect to $u$. If $P\in\pu$ then
$$
{\match(G\diff P,x)\over\mgx}={\match(T\diff P,x)\over\match(T,x)}.
$$
\proof
We proceed by induction on the number of vertices in $P$. If $P$ has
only one vertex, we appeal to the theorem. Suppose then that $P$ has
at least two vertices in it, and that $v$ is the end vertex of $P$
other than $u$. Let $Q$ be the path $P\diff v$ and let $H$ denote
$G\diff Q$. Then
$$
{\match(G\diff P,x)\over\mgx}
={\match(G\diff P,x)\over\match(G\diff Q,x)}
{\match(G\diff Q,x)\over\mgx}
={\match(T(H,v)\diff v,x)\over\match(T(H,v),x)}
{\match(T\diff Q,x)\over\match(T,x)},
$$
where the second equality follows by induction. Now $T(H,v)$ is one
component of $T(G,u)\diff Q$, and if we delete the vertex $v$ from
this component from $T(G,u)\diff Q$, the graph that results is
$T(G,u)\diff P$. Consequently
$$
{\match(T(H,v)\diff v,x)\over\match(T(H,v),x)}
={\match(T\diff P,x)\over\match(T\diff Q,x)}.
$$
The results follows immediately from this.\qed
Let $\puv$ denote the set of paths in $G$ which start at $u$ and
finish at $v$. The following result will be one of our main tools.
It is a special case of [\r{h-l}:~Theorem~6.3].
\result{HL}% 2.4
Lemma (Heilmann and Lieb). Let $u$ and $v$ be vertices in the graph
$G$. Then
$$
\match(G\diff u,x)\match(G\diff v,x)
-\mgx\match(G\diff uv,x)
=\sum_{P\in\puv}\match(G\diff P,x)^2.\qed
$$
This lemma has a number of important consequences. In
[\r{cg-rgp}:~Section~4] it is used to show that $\mtg$ is a lower
bound on the number of paths needed to cover the vertices of $G$, and
that the number of distinct zeros of $\mgx$ is an upper bound on the
length of a longest path. For our immediate purposes, the following
will be the most useful.
\result{path-del}% 2.5
Corollary. If $P$ is a path in the graph $G$ then $\match(G\diff
P,x)/\mgx$ has only simple poles. In other words, for any zero $\th$
of $\mgx$ we have $$
\mt{G\diff P}\ge\mtg-1.
$$
\proof
Suppose $k=\mtg$. Then, by interlacing, $\mt{G\diff u}\ge k-1$ for
any vertex $u$ of $G$ and $\mt{G\diff uv}\ge k-2$. Hence the
multiplicity of $\th$ as a zero of
$$
\match(G\diff u,x)\match(G\diff v,x)
-\mgx\match(G\diff uv,x)
$$
is at least $2k-2$. It follows from \p{HL} that $\mt{G\diff P}\ge k-1$
for any path $P$ in $\puv$.\qed
\section{ess-vps}% 3
Essential Vertices and Paths
Let $\th$ be a zero of $\mgx$. A path $P$ of $G$ is {\sl
$\th$-essential}\/ if $\mt{G\diff P}<\mtg$. (We will often be
concerned with the case where $P$ is a single vertex.) A vertex is
{\sl $\th$-special} if it is not $\th$-essential and is adjacent to an
$\th$-essential vertex. A graph is {\sl $\th$-primitive}\/ if and
only if every vertex is $\th$-essential and it is
{\sl $\th$-critical}\/ if it is $\th$-primitive and $\mtg=1$. (When $\th$
is determined by the context we will often drop the prefix `$\th$-'
from these expressions.) If $\th=0$ then a $\th$-critical graph is
the same thing as a factor-critical graph.
The next result implies that a vertex-transitive graph is $\th$-primitive
for any zero $\th$ of its matchings polynomial.
\result{essvx}% 3.1
Lemma. Any graph has at least one essential vertex.
\proof
Let $\th$ be a zero of $\mgx$ with multiplicity $k$. Then $\th$ has
multiplicity $k-1$ as a zero of $\match'(G,x)$. Since
$$
\match'(G,x)=\sum_{u\in V(G)}\match(G\diff u,x)
$$
we see that if $\mt{G\diff u}\ge k$ for all vertices $u$ of $G$ then
$\th$ must have multiplicity at least $k$ as a zero of $\match'(G,x)$.\qed
\result{essnbr}% 3.2
Lemma. If $\th\ne0$ then any $\th$-essential vertex $u$ has a neighbour
$v$ such that the path $uv$ is essential.
\proof
Assume $\th\ne0$ and let $u$ be a $\th$-essential vertex. Since
$$
\mgx=x\mx{G\diff u}-\sum_{i\sim u}\mx{G\diff ui}
$$
we see that if $\mt{G\diff ui}\ge\mtg$ for all neighbours $i$ of $u$
then $\mt{G\diff u}\ge\mtg$.\qed
Note that the vertex $v$ is not essential in $G\diff u$. However it
follows from the next lemma that the vertex $v$ in the above lemma
must be essential in $G$; accordingly if $\th\ne0$ then any essential
vertex must have an essential neighbour.
\result{no-essp}% 3.3
Lemma. If $v$ is not an essential vertex of $G$ then no path with $v$
as an end-vertex is essential.
\proof
Assume $k=\mtg$. If $v$ is not essential then $\mt{G\diff v}\ge k$
and so, for any vertex $u$ not equal to $v$, the multiplicity of $\th$
as a zero of
$$
\match(G\diff u,x)\match(G\diff v,x)-\mgx\match(G\diff uv,x)
$$
is at least $2k-1$. By \p{HL} we deduce that it is at least $2k$ and
that $\mt{G\diff P}\ge k$ for all paths $P$ in $\pv$.\qed
We now need some more notation. Suppose that $G$ is a graph and $\th$
is a zero of $\match(G,x)$ with positive multiplicity $k$. A vertex
$u$ of $G$ is {\sl $\th$-positive}\/ if $\mt{G\diff u}=k+1$ and {\sl
$\th$-neutral}\/ if $\mt{G\diff u}=k$. (The `negative' vertices will
still be referred to as essential.) Note that, by interlacing,
$\mt{G\diff u}$ cannot be greater than $k+1$.
\result{posvs}% 3.4
Lemma. Let $G$ be a graph and $u$ a vertex in $G$ which is not essential.
Then $u$ is positive in $G$ if and only if some neighbour of it is
essential in $G\diff u$.
\proof
From \p{recurs}(c) we have
$$
\mgx =x\match(G\diff u,x)
-\sum_{i\sim u}\match(G\diff ui,x).\idno{vdel}
$$
If $\mt{G\diff u}=k+1$ and $\mt{G\diff ui}\ge k+1$ for all neighbours
$i$ of $u$ then it follows that $\mtg\ge k+1$ and $u$ is not positive.
On the other hand, suppose $u$ is not essential in $G$ and $v$ is a
neighbour of $u$ which is essential in $G\diff u$. From the previous
lemma we see that the path $uv$ is not essential and thus $\mt{G\diff
uv}\ge\mtg$. As $v$ is essential in $G\diff u$ it follows that
$\mt{G\diff u}>\mtg$.\qed
We say that $S$ is an {\sl extremal}\/ subtree of the tree $T$ if $S$
is a component of $T\diff v$ for some vertex $v$ of $G$.
\result{extree}% 3.5
Lemma. Let $S$ be an extremal subtree of $T$ that is
inclusion-minimal subject to the condition that $\mt{S}\ne0$, and let
$v$ be the vertex of $T$ such that $S$ is a component of $T\diff v$.
Then $v$ is $\th$-positive in $T$.
\proof
Let $u$ be the vertex of $S$ adjacent to $v$ and let $e$ be the edge
$\{u,v\}$. Then $T\diff e$ has exactly two components, one of which
is $S$. Denote the other by $R$.
By hypothesis $\mt{S'}=0$ for any component $S'$ of $S\diff u$,
therefore $\mt{S\diff u}=0$ by \p{recurs}(a) and so $u$ is essential
in $S$. Since $S$ is a component of $T\diff v$ it follows that $u$ is
essential in $T\diff v$. If we can show that $v$ is not essential
then $v$ must be positive in $T$, by the previous lemma.
Suppose $\mt{T}=m$. By interlacing $\mt{T\diff u}\ge m-1$ and, as
$$
\mt{T\diff u}=\mt{R}+\mt{S\diff u}=\mt{R},
$$
we find that $\mt{R}\ge m-1$. By parts (a) and (b) of \p{recurs} we have
$$
\mx T=\mx{R}\mx{S}-\mx{R\diff v}\mx{S\diff u}
$$
and so, since the multiplicity of $\th$ as a zero of $\mx{R}\mx{S}$ is
at least $m$, we deduce that the multiplicity of $\th$ as a zero of
$\mx{R\diff v}\mx{S\diff u}$ is at least $m$. Since $\mt{S\diff u}=0$,
it follows that $\mt{R\diff v}\ge m$. On the other hand
$$
\mt{T\diff v}=\mt{R\diff v}+\mt{S}=\mt{R\diff v}+1,
$$
therefore $\mt{T\diff v}\ge m+1$ and $v$ is positive in $T$.\qed
\result{Neu}% 3.6
Corollary (Neumaier). Let $T$ be a tree and let $\th$ be a zero of $\mx{T}$.
The following assertions are equivalent:
\item{(a)} $\mt{S}=0$ for all extremal subtrees of $T$,
\item{(b)} $T$ is $\th$-critical,
\item{(c)} $T$ is $\th$-primitive.
\proof
Since $T\diff v$ is a disjoint union of extremal subtrees for any
vertex $v$ in $T$, we see that if (a) holds then $\mt{T\diff v}=0$ for
any vertex $v$. Hence $T$ is $\th$-critical and therefore it is also
$\th$-primitive. If $T$ is $\th$-primitive then no vertex in $T$ is
$\th$-positive, whence \p{extree} implies that (a) holds.\qed
\p{Neu} combines Theorem~3.1 and Corollary~3.3 from [\r{neu}]. Note that
the equivalence of (b) and (c) when $\th=0$ is Gallai's lemma for trees.
\result{simpz}% 3.7
Lemma. Let $G$ be a connected graph. If $u\in V(G)$ and all paths in
$G$ starting at $u$ are essential then $G$ is critical.
\proof
If all paths in $\pu$ are essential then \p{no-essp} implies that
all vertices in $G$ are essential. Hence $G$ is primitive, and
it only remains for us to show that $\mtg=1$.
Let $T=T(G,u)$ be the path tree of $G$ relative to $u$. From
\p{path-tree} we see that a path $P$ from $\pu$ is essential in $G$ if
and only if it is essential in $T$. So our hypothesis implies that
all paths in $T$ which start at $u$ are essential, whence \p{no-essp}
yields that all vertices in $T$ are essential. Hence $T$ is
$\th$-primitive and therefore, by \p{Neu}, $\th$ is a simple zero of
$\match(T,x)$. Using \p{path-tree} again we deduce that $\mtg=1$.\qed
\result{ess-path}% 3.8
Lemma. If $u$ and $v$ are essential vertices in $G$ and $v$ is not
essential in $G\diff u$ then there is a $\th$-essential path in $\puv$.
\proof
Assume $\mtg=k$. Our hypotheses imply that $\mt{G\diff uv}\ge k-1$.
If no path in $\puv$ is essential then, by \p{HL}, the multiplicity
of $\th$ as a zero of
$$
\mx{G\diff u}\mx{G\diff v}-\mgx\mx{G\diff uv}
$$
is at least $2k$. Since $\th$ has multiplicity $2k-1$ as a zero of
$\mgx\mx{G\diff uv}$ it must also have multiplicity at least $2k-1$ as
a zero of $\mx{G\diff u}\mx{G\diff v}$. Hence $u$ and $v$ cannot both
be essential.\qed
If $u$ and $v$ are essential in $G$ then $v$ is essential in $G\diff
u$ if and only if $u$ is essential in $G\diff v$. Thus the hypothesis
of \p{ess-path} is symmetric in $u$ and $v$, despite appearances.
\result{pths-vxs}% 3.9
Corollary. Let $G$ be a tree, let $\th$ be a zero of $\mgx$ and let
$u$ be a vertex in $G$. Then all paths in $\pu$ are essential if and
only if all vertices in $G$ are essential.
\proof
It follows from \p{no-essp} that if all paths in $\pu$ are essential
then all vertices in $G$ are essential. Suppose conversely that all
vertices in $G$ are essential. By \p{Neu} it follows that $\mtg=1$.
Hence the hypotheses of \p{ess-path} are satisfied by any two vertices
in $G$, and so any two vertices are joined by an essential path.
Since $G$ is a tree the path joining any two vertices is unique and
therefore all paths in $\pu$ are essential.\qed
\section{struct}% 4
Structure Theorems
We now apply the machinery we have developed in the previous section.
\result{deCaen}% 4.1
Lemma (De Caen [\r{ddc}]). Let $u$ and $v$ be adjacent vertices in a
bipartite graph. If $u$ is $0$-essential then $v$ is $0$-special.
\proof
Suppose that $u$ and $v$ are $0$-essential neighbours in the bipartite
graph $G$. As $uv$ is a path, using \p{path-del} we find that
$$
{\rm mult}(0,G\diff uv)\ge{\rm mult}(0,G)-1={\rm mult}(0,G\diff u),
$$
and therefore $v$ is not essential in $G\diff u$. It follows from
\p{ess-path} that there is a $0$-essential path $P$ in $G$ joining
$u$ to $v$.
We now show that $P$ must have even length. From this it will follow
that $P$ together with the edge $uv$ forms an odd cycle, which is
impossible. From the definition of the matchings polynomial we see
that ${\rm mult}(0, H)$ and $|V(H)|$ have the same parity for any
graph $H$. As
$$
{\rm mult}(0,G\diff P)={\rm mult}(0,G)-1
$$
we deduce that $|V(G)|$ and $|V(G\diff P)|$ have different parity and
therefore $P$ has even length.\qed
In the above proof we showed that a $0$-essential path in a graph must
have even length. Consequently no edge, viewed as a path of length
one, can ever be $0$-essential. It follows that $K_1$ is the only
connected graph such that all paths are $0$-essential. In general any
graph which is minimal subject to its matchings polynomial having a
particular zero $\th$ will have the property that all its paths are
$\th$-essential.
\p{deCaen} is not hard to prove without reference to the matchings
polynomial. Note that it implies that in any bipartite graph there is
a vertex which is covered by every maximal matching, and consequently
that a bipartite graph with at least one edge cannot be $0$-primitive.
As noted by de Caen [\r{ddc}], this leads to a very simple inductive
proof of K\"onig's lemma.
Our next result is a partial analog to the Edmonds-Gallai structure
theorem. See, e.g., [\r{lv-pl}:~Chapter~3.2].
%\vbox{
\result{edgall}% 4.2
Theorem. Let $\th$ be a zero of $\mgx$ with non-zero multiplicity $k$ and
let $a$ be a positive vertex in $G$. Then:
\item{(a)} if $u$ is essential in $G$ then it is essential in $G\diff a$;
\item{(b)} if $u$ is positive in $G$ then it is essential or positive in
$G\diff a$;
\item{(c)} if $u$ is neutral in $G$ then it is essential or neutral in
$G\diff a$.
%\par}
\proof
If $\mt{G\diff u}=k-1$ and $\mt{G\diff a}=k+1$, it follows by
interlacing that $\mt{G\diff au}=k$. Hence $u$ is essential in
$G\diff a$. Now suppose that $u$ is positive in $G$. If $\mt{G\diff
au}\ge k+1$ then $\th$ has multiplicity at least $2k+1$ as a zero of
$p(x)$ where
$$
p(x) := \mx{G\diff u}\mx{G\diff a}-\mgx\mx{G\diff au}.\idno{edg}
$$
By \p{HL}, the multiplicity of $\th$ as a zero of $p(x)$
must be even. It follows that this multiplicity must be at least
$2k+2$ and hence that $\th$ has multiplicity at least $2k+2$ as a zero
of $\mgx\mx{G\diff au}$. Therefore $\mt{G\diff au}\ge k+2$ and so, by
interlacing, $\mt{G\diff au}=k+2$ and $u$ is positive in $G\diff a$.
If $\mt{G\diff ua}=k+2$ and $u$ is neutral in $G$, then the
multiplicity of $\th$ as a zero of $p(x)$ is at least $2k+1$ and
therefore at least $2k+2$, but this implies that $\th$ is a zero of
$\mx{G\diff u}\mx{G\diff a}$ with multiplicity at least $2k+2$. Thus
we conclude that $u$ is neutral or essential in $G\diff a$.\qed
We note that \p{edgall}(a) holds even if $a$ is only neutral. If $a$
is neutral and $u$ is essential in $G$ but not in $G\diff a$ then
$\th$ has multiplicity at least $2k-1$ as a zero of \preveq/ and so
must have multiplicity at least $2k$ as a zero of $\mgx\mx{G\diff
au}$. Hence its multiplicity as a zero of
$\match(G\diff u,x)\match(G\diff a,x)$ is at least $2k$,
which is impossible.
\medbreak
The following consequence of \p{edgall} and the previous remark was proved
for trees by Neumaier. (See [\r{neu}:~Theorem~3.4(iii)].)
\result{neutree}% 4.3
Corollary. Any special vertex is positive.
\proof
Suppose that $a$ is special in $G$, and that $u$ is a neighbour of $a$
which is essential in $G$. By part (a) of the theorem and the remark
above, $u$ is essential in $G\diff a$ and therefore, by \p{posvs},
$a$ is positive in $G$.\qed
\p{simpz} implies that if $G$ is not $\th$-critical then it contains a
path, $P$ say, that is not essential. If we delete $P$ from $G$ then
the multiplicity of $\th$ as a zero of $\mgx$ cannot decrease. Hence
we may successively delete `inessential' paths from $G$, to obtain a
graph $H$ such that $\mt{H}\ge\mtg$ and all paths in $H$ are
essential. If $k=\mt{H}$ then, by \p{simpz} again, $H$ contains
exactly $k$ critical components. The following result is a sharpening
of this observation, since it implies that if $\mtg=k$ we may produce
a graph with $k$ critical components by deleting $k$ vertex disjoint
paths from $G$,
\result{critcomp}% 4.4
Lemma. Let $G$ be a graph, let $\th$ be a zero of $\mgx$ and let $u$
be a $\th$-essential vertex of $G$. Suppose that there is a path in
$\pu$ which is not $\th$-essential. Then there is a path $P$ in $G$
starting at $u$ such that $\mt{G\diff P}=\mtg$ and some component $C$
of $G\diff P$ is critical. All vertices of $C$ are essential in $G$.
\proof
Suppose that there are paths in $\pu$ which are not essential, choose
one of minimum length and call it $P$. Let $v$ be the end-vertex of
$P$ other than $u$ and let $P'$ be the path $P\diff v$. Then $P'$ is
essential, hence
$$
\mt{G\diff P'}=\mtg-1
$$
and, as $P$ is not essential,
$$
\mt{G\diff P}\ge \mtg.
$$
But we get $G\diff P$ from $G\diff P'$ by deleting the single vertex
$v$, therefore $\mt{G\diff P}=\mtg$ and $v$ is positive in $G\diff
P'$. Consequently, by \p{posvs}, there is an essential vertex $u_1$
adjacent to $v$ in $(G\diff P')\diff v=G\diff P$.
We now prove by induction on the number of vertices that, if the
conditions of the lemma hold, then there is a path $P$ and a component
$C$ of $G\diff P$ as claimed and, further, there is a vertex $w$ in $C$
adjacent to the end-vertex of $P$ distinct from $u$ such that all paths
in $C$ that start at $w$ are essential in $C$.
Let $H$ denote $G\diff P$. If all paths in $H$ starting at $u_1$ are
essential then, by \p{simpz}, the component $C$ of $H$
that contains $u_1$ is critical. If $Q$ is a path in $C$ starting at
$u_1$ then $\mt{C\diff Q}<\mt{C}$; this implies that the path formed
by the concatenation of $P$ and $Q$ is essential in $G$ and hence, by
\p{no-essp}, that all vertices in $C$ are essential in $G$.
Thus we may suppose that there is a path in $H$ starting at $u_1$ that
is not essential. Because $H$ has fewer vertices than $G$, we may
assume inductively that there is a path $Q$ in $H$ starting at
$u_1$ such that $\mt{H}=\mt{H\diff Q}$ and a critical component $C$ of
$H\diff Q$ that contains a neighbour $w$ of the end-vertex of $Q$ distinct
from $u_1$. Further all the paths in $C$ that start at $w$ are essential.
Let $PQ$ denote the path formed by concatenating $P$ and $Q$. Then
all claims of the lemma hold for $G$, $PQ$, $u$ and $C$.\qed
The two results which follow provide a strengthening of the
observation that the zeros of the matchings polynomial of a graph with
a Hamilton path are simple.
\result{gcd}% 4.5
Lemma. Suppose that $u$ and $v$ are adjacent vertices in $G$ such
that $\mx{G\diff u}$ and $\mx{G\diff uv}$ have no common zero. Then
$\mgx$ and $\mx{G\diff u}$ have no common zero, and therefore both
polynomials have have only simple zeros.
\proof
Assume by way of contradiction that $\th$ is a common zero of $\mgx$
and $\mx{G\diff u}$. If $\mtg>1$ then by \p{path-del} we see that
$\th$ is a zero of $\mx{G\diff u}$ and $\mx{G\diff uv}$. If
$\mt{G\diff u}>1$ then $\mt{G\diff uv}>0$, by interlacing. Hence
$$
\mtg=\mt{G\diff u}=1
$$
and so $u$ is a neutral vertex in $G$. It
follows from \p{posvs} that no neighbour of $u$ can be essential in
$G\diff u$ and consequently $\mt{G\diff uv}>0$.\qed
A simple induction argument on the length of $P$ yields the following.
\result{simples}% 4.7
Corollary. Let $H$ be an induced subgraph of $G$ and suppose that there
is a vertex $u$ in $H$ and a path $P$ in $G$ such that
$$
V(H)\cap V(P)=u,\qquad V(H)\cup V(P)=V(G).
$$
If $\match(H,x)$ and $\match(H\diff u,x)$ have no common zero then all
zeros of $G$ are simple.\qed
Note that the path $P$ in this corollary does not have to be an induced path.
One consequence of it is that if a graph has a Hamilton path then the zeros
of its matchings polynomial are all simple. However this result shows that
there will be many other graphs with all zeros simple.
\section{evs}% 5
Eigenvectors
Let $G$ be a graph with adjacency matrix $A=A(G)$. We view an eigenvector
$f$ of $A$ with eigenvalue $\th$ as a function on $V(G)$ such that
$$
\th f(u)=\sum_{i\sim u} f(i).
$$
We denote the characteristic polynomial of $G$ by $\phi(G,x)$. (It is
defined to be $\det(xI-A(G))$.) We recall that for forests the
characteristic and matchings polynomials are equal. Our first result
follows from the proof of Theorem~5.2 in [\r{cg-mw}].
\result{maxf}% 5.1
Lemma. Let $\th$ be an eigenvalue of the graph $G$ and let $u$ be a vertex in $G$. Then the maximum value of $f(u)^2$ as $f$ ranges over the
eigenvectors of $G$ with eigenvalue $\th$ and norm one is equal to
$\phi(G\diff u,\th)/\phi'(G,\th)$.\qed
\result{neu-ess}% 5.2
Corollary (Neumaier [\r{neu}: Theorem~3.4]). Let $T$ be a tree and
let $\th$ be a zero of its matchings polynomial. Then a vertex $u$ is
essential if and only if there is an eigenvector $f$ of $T$ such that
$f(u)\ne0$.\qed
\result{tree-essv}% 5.3
Theorem. Let $T$ be a tree, let $\th$ be a zero of $\mx{T}$ and let
$a$ be a vertex of $T$ which is not essential. Then a vertex is
essential in $T\diff a$ if and only if it is essential in $T$.
Further, if $a$ is positive then it has an essential neighbour.
\proof
Let $W$ be the eigenspace of $T$ belonging to $\th$ and let $W_a$ be the
corresponding eigenspace of $T\diff a$. Then $W_a$ is the direct sum of
the eigenspaces of the component of $T\diff a$ belonging to $\th$ and $W$
is the subspace formed by the vectors $f$ such that
$$
\sum_{i\sim a} f(i)=0.
$$
If $a$ is neutral then $W=W_a$ and so $T$ and $T\diff a$ have the same
essential vertices. If $a$ is positive then $W$ is a proper subspace of
$W_a$, whence it follows that there are vectors in $W_a$ which are not
zero on all neighbours of $a$. For each vector in $W_a$ there is an
eigenvector in $W$ with the same support on $T\diff a$. Hence $a$ has an
essential neighbour and any vertex which is essential in $T\diff a$ is
also essential in $T$.\qed
\p{tree-essv} is a strengthening of a result of Neumaier
[\r{neu}:~Corollary~3.5]. Suppose that $T$ is a tree with exactly $s$
special vertices and $\mt{T}=k$. Then \p{tree-essv} together with \p{edgall}
implies that we may successively delete the special vertices, obtaining a
forest $F$ with no special vertices and $\mt{F}=k+s$. Hence any component
of $F$ is either $\th$-critical or does not have $\th$ as a zero of its
matchings polynomial. Therefore $F$ has exactly $k+s$ $\th$-critical
components, and these components form an induced subgraph of $T$.
\section{qqq}%
Questions
Many problems remain. Here are some.
\medbreak
\item{(1)} Must a positive vertex be special when $\th\ne0$?
(If $\th=0$ then all vertices which are not essential are positive.)
\item{(2)} What can be said of the graphs where every pair of vertices are
joined by at least one essential path? (Or of the graphs with a vertex $u$
such that all vertices can be joined to $u$ by an essential path?)
\item{(3)} Must a $\th$-primitive graph be $\th$-critical?
\medbreak\noindent
It might be interesting to investigate the case $\th=1$ in depth.
\nonumsection
References
\ref{agt}
N. Biggs,
{\sl Algebraic Graph Theory}.
(Cambridge U.\ P., Cambridge) 1974.
\ref{ddc}
D. de Caen,
On a theorem of K\"onig on bipartite graphs,
{\sl J.\ Comb.\ Inf.\ System Sci.\ \bf13} (1988) 127.
\ref{cg-mw}
C. D. Godsil,
Matchings and walks in graphs,
{\sl J.\ Graph Theory, \bf 5}, (1981) 285--297.
\ref{cg-ig}
C. D. Godsil and I. Gutman,
On the theory of the matching polynomial,
{\sl J.\ Graph Theory, \bf 5} (1981), 137--144.
\ref{cg-rgp}
C. D. Godsil,
Real graph polynomials,
in {\sl Progress in Graph Theory},
edited by J.~A. Bondy and U.~S.~R. Murty,
(Academic Press, Toronto) 1984, pp.\ 281--293.
\ref{cgbk}
C. D. Godsil,
{\sl Algebraic Combinatorics}.
(Chapman and Hall, New York) 1993.
\ref{h-l}
O. J. Heilmann and E. H. Lieb,
Theory of monomer-dimer systems,
{\sl Commun.\ Math.\ Physics, \bf 25} (1972), 190--232.
\ref{lv-pl}
L. Lov\'asz and M. D. Plummer,
{\sl Matching Theory}.
Annals Discrete Math.\ 29, (North-Holland, Amsterdam) 1986.
\ref{neu}
A. Neumaier,
The second largest eigenvalue of a tree,
{\sl Linear Algebra Appl. \bf48} (1982) 9--25.
\bye
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%\setbindingcorrection{3mm}
%%%%%%% Questo test è usato appunto per collaudare diversi stili,
%%%%%%% non per comporre una vera tesi.
%%%%%%% Non usarlo mai, solo perché qui è usato!
\ifclassica%
{\begin{dedica}
A mio padre
\textdagger\ A mio nonno Pino
\end{dedica}\fi
%%%%%%% Fine esperimento
\sommario
La pressione barometrica di Giove viene misurata
mediante un metodo originale messo a punto dai candidati, che si basa
sul rilevamento telescopico della pressione.
% \paginavuota % funziona anche senza specificare l'opzione classica
\ringraziamenti
I candidati ringraziano vivamente il Granduca di Toscana per i mezzi
messi loro a disposizione, ed il signor Von Braun, assistente del
prof.~Albert Einstein, per le informazioni riservate che egli ha
gentilmente fornito loro, e per le utili discussioni che hanno permesso
ai candidati di evitare di riscoprire l'acqua calda.
\tablespagetrue\figurespagetrue % normalmente questa riga non serve ed e' commentata
\indici
%%%%%%%% Altro esperimento con l'opzione classica
%%%%%%%% Non usare mai anche se qui lo si è fatto!
%%%%%%%% Oltretutto funziona solo se si è specificata la lingua greca fra le opzioni.
%%%%%%%% Commentare fra \ifclassica fino a \fi compresi.
\ifclassica
\begin{citazioni}
\textit{testo testo testo\\testo testo testo}
[\textsc{G.\ Leopardi}, Operette Morali]\vspace{1em}
\textgreek{>all'a p'anta <o k'eraunos d'' >oiak'izei}
[\textsc{Eraclito}, fr.\ D-K 134]
\end{citazioni}
\fi
%%%%%%%% fine esperimento
\mainmatter
\part{Prima Parte}
\chapter{Introduzione generale}
\section{Principi generali}
Il problema della determinazione della pressione barometrica dell'atmosfera di
Giove non ha ricevuto finora una soluzione soddisfacente, per l'elementare
motivo che il pianeta suddetto si trova ad una distanza tale che i mezzi attuali
non consentono di eseguire una misura diretta.
Conoscendo però con grande precisione le orbite dei satelliti principali di
Giove, e segnatamente le orbite dei satelliti medicei, è possibile eseguire
delle misure indirette, che fanno ricorso alla nota formula \cite{gal}:
\[
\Phi = K\frac{\Xi^2 +\Psi\ped{max}}{1+\gei\Omega}
\]
dove le varie grandezze hanno i seguenti significati:
\begin{enumerate}
\item
$\Phi$ angolo di rivoluzione del satellite in radianti se $K=1$, in gradi se
$K=180/\pi$;
\item
$\Xi$ eccentricità dell'orbita del satellite; questa è una grandezza priva
di dimensioni;
\item
$\Psi\ped{max}$ rapporto fra il semiasse maggiore ed il semiasse minore
dell'orbita del satellite, nelle condizioni di massima eccentricità;
poiché le dimensioni di ciascun semiasse sono $[l]=\unit{km}$, la grandezza
$\Psi\ped{max}$ {è} adimensionata;
\item
$\Omega$ velocità istantanea di rotazione; si ricorda che è $[\Omega]=%
\unit{rad}\unit{s}^{-1}$;
\item bisogna ancora ricordarsi che $10^{-6}\unit{m}$ equivalgono a 1\unit{\micro m}.
\end{enumerate}
%
Le grandezze in gioco sono evidenziate nella figura \ref{fig:orbita}.
\begin{figure}[ht]\centering
\setlength{\unitlength}{0.01\textwidth}
\begin{picture}(40,30)(30,0)
\put(50,15){\scalebox{1}[0.7]{\circle{25}}}
\put(47,15){\circle*{1}}
\put(30,0){\line(0,1){30}}
\put(30,30){\line(1,0){40}}
\put(70,30){\line(0,-1){30}}
\put(70,0){\line(-1,0){40}}
\end{picture}
\caption{Orbita del generico satellite; si noti l'eccentricità dell'orbita rispetto al pianeta.}\label{fig:orbita}
\end{figure}
Per misurare le grandezze che compaiono in questa formula è necessario
ricorrere ad un pirometro con una resistenza di 120\unit{M\ohm}, altrimenti gli
errori di misura sono troppo grandi, ed i risultati completamente falsati.
\section{I satelliti medicei}
I satelliti medicei, come noto, sono quattro ed hanno dei periodi di rivoluzione
attorno al pianeta Giove che vanno dai sette giorni alle tre settimane.
Essi furono per la prima volta osservati da uno dei candidati mentre
sperimentava l'efficacia del tubo occhiale che aveva appena inventato
rielaborando una idea sentita di seconda mano da un viaggiatore appena arrivato
dai Paesi Bassi.
%\blankpagestyle{headings}
%\lipsum[1-2]
\chapter{Il barometro}
\section{Generalità}
\begin{interlinea}{0.87} Il barometro, come dice il nome, serve per
misurare la pesantezza; più precisamente la pesantezza dell'aria
riferita all'unità di superficie.
\end{interlinea}
\begin{interlinea}{2} Studiando il fenomeno fisico si può concludere
che in un dato punto grava il peso della colonna d'aria che lo
sovrasta, e che tale colonna è tanto più grave quanto maggiore
è la superficie della sua base; il rapporto fra il peso e la base
della colonna si chiama pressione e si misura in once toscane al cubito
quadrato, \cite{tor1}; nel Ducato di Savoia la misura in once al piede
quadrato è quasi uguale, perché colà usano un piede molto
grande, che è simile al nostro cubito.
\end{interlinea}
\subsection{Forma del barometro}
Il barometro consta di un tubo di vetro chiuso ad una estremità e
ripieno di mercurio, capovolto su di un vaso anch'esso ripieno di
mercurio; mediante un'asta graduata si può misurare la distanza fra
il menisco del mercurio dentro il tubo e la superficie del mercurio
dentro il vaso; tale distanza è normalmente di 10 pollici toscani,
\cite{tor1,tor2}, ma la misura può variare se si usano dei pollici
diversi; è noto infatti che gl'huomini sogliono avere mani di
diverse grandezze, talché anche li pollici non sono egualmente
lunghi.
\section{Del mercurio}
Il mercurio è un a sostanza che si presenta come un liquido, ma ha il colore
del metallo. Esso è pesantissimo, tanto che un bicchiere, che se fosse pieno
d'acqua, sarebbe assai leggiero, quando invece fosse ripieno di mercurio,
sarebbe tanto pesante che con entrambe le mani esso necessiterebbe di essere
levato in suso.
Esso mercurio non trovasi in natura nello stato nel quale è d'uopo che sia
per la costruzione dei barometri, almeno non trovasi così abbondante come
sarebbe necessario.
\setcounter{footnote}{25}
Il Monte Amiata, che è locato nel territorio del Ducato%
\footnote{Naturalmente stiamo parlando del Granducato di Toscana.%
\ifclassica\NoteWhiteLine\fi
} del nostro Eccellentissimo et Illustrissimo Signore Granduca di Toscana\footnote{Cosimo IV de' Medici.}, è uno dei
luoghi della terra dove può rinvenirsi in gran copia un sale rosso, che
nomasi \emph{cinabro}, dal quale con artifizi alchemici, si estrae il mercurio
nella forma e nella consistenza che occorre per la costruzione del barometro
terrestre%
\ifclassica
\nota{Nota senza numero\dots
\dots e che va a capo.
}\fi.
La densità del mercurio è molto alta e varia con la temperatura come
può desumersi dalla tabella \ref{t:1}.
Il mercurio gode della sorprendente qualità et proprietà, cioè che esso
diventa tanto solido da potersene fare una testa di martello et infiggere
chiodi aguzzi nel legname.
\begin{table}[htp] % crea un floating body col nome Tabella nella
% didascalia
\centering % comando necessario per centrare la tabella
\begin{tabular}% % inizio vero e proprio della tabella
{rrrrrr} % parametri di incolonnamento
\hline\hline % filetti orizzontali sopra la tabella
% intestazione della tabella
\multicolumn{3}{c}{\rule{0pt}{2.5ex}Temperatura} % \rule serve per lasciare
& \multicolumn{3}{c}{Densità} \\ % un po' di spazio sopra le parole
&\unit{\gradi C} & & & $\unit{t/m^3}$ & \\
\hline% % Filetto orizzontale per separare l'intestazione
\hspace*{1.3em}& 0 & & & 13,8 & \\ % I numeri sono incolonnati %
& 10 & & & 13,6 & \\ % a destra; le colonne vuote
& 50 & & & 13,5 & \\ % servono per centrare le colonne
&100 & & & 13,3 & \\ % numeriche sotto le intestazioni
\hline \hline % Filetti di fine tabella
\end{tabular}
\caption[Densità del mercurio]{Densità del mercurio. Si può fare molto meglio usando il pacchetto \textsf{booktabs}.} \label{t:1} % didascalia con label
\end{table}
%\selectlanguage{italian}
\begin{osservazione}\normalfont
Questa proprietà si manifesta quando esso è estremamente freddo, come
quando lo si immerge nella salamoia di sale e ghiaccio che usano li maestri
siciliani per confetionare li sorbetti, dei quali sono insuperabili artisti.
\end{osservazione}
Per nostra fortuna, questo grande freddo, che necessita per la confetione de
li sorbetti, molto raramente, se non mai, viene a formarsi nelle terre del
Granduca Eccellentissimo, sicché non vi ha tema che il barometro di mercurio
possa essere ruinato dal grande gelo e non indichi la pressione giusta, come
invece deve sempre fare uno strumento di misura, quale è quello che è
descritto costì.\cite{duane1964}
\chapter{Il listato del pacchetto \texttt{topcoman.sty}}
\listing{topcoman.sty}
\begin{thebibliography}{9}
\bibitem{gal} G.~Galilei, {\em Nuovi studii sugli astri medicei}, Manuzio,
Venetia, 1612.
\bibitem{tor1} E.~Torricelli, in ``La pressione barometrica'', {\em Strumenti
Moderni}, Il Porcellino, Firenze, 1606.
\bibitem{tor2} E.~Torricelli e A.~Vasari, in ``Delle misure'', {\em Atti Nuovo
Cimento}, vol.~III, n.~2 (feb. 1607), p.~27--31.
\bibitem{duane1964} Duane J.T., \emph{Learning Curve Approach To Reliability
Monitoring}, IEEE Transactions on Aerospace, Vol. 2, pp. 563-566, 1964
\end{thebibliography}
\end{document}
% altri riferimenti da usare come esempi.
\bibitem{chiesa2008} Chiesa S., \emph{Affidabilità, sicurezza e manutenzione
nel progetto dei sistemi}, CLUT, gennaio 2008
\bibitem{chiesa2}Chiesa S., Fioriti M., Fusaro R., \emph{On Board System
Technological Level Improvement Effect on UAV MALE}
\bibitem{bigliano2010} Bigliano M., \emph{Sicurezza nell'installazione di un velivolo
senza pilota MALE; applicazione di metodologia di Zonal Safety
Analysis al velivolo del Progetto SAvE}, Politecnico di Torino,
maggio 2010
\bibitem{astrid2012} Chiesa S., Di Meo G.A., Fioriti M., Medici G., Viola N.,
\emph{ASTRID - Aircraft on board Systems sizing and TRade-off
analysis in Initial Design}, Research Bulletin, Warsaw University
of Technology, Institute of Aeronautics and Applied Mechanics,
p. 1-28, 17-19, ottobre 2012
|
https://texample.net/media/tikz/examples/TEX/eratosthenes-sieve.tex
|
texample.net
|
CC-MAIN-2021-17
|
text/x-tex
|
text/x-matlab
|
crawl-data/CC-MAIN-2021-17/segments/1618038077818.23/warc/CC-MAIN-20210414125133-20210414155133-00513.warc.gz
| 661,814,372 | 7,192 |
%%% Sieve of Eratosthenes
%%% ---------------------
%%%
%%% Step 1: Create a list of consecutive integers 2...n
%%% Step 2: Let p=2 be the first prime number
%%% Step 3: Mark multiples of p as non-prime.
%%% Step 4: First number > p not marked as non-prime is prime.
%%% Step 5: Repeat from Step 3 with this new prime.
%%%
%%% ---------------------------------------------------------------
%%% Set-up: Choose desired output:
%%% 1. Animation: GIF, or PDF
%%% 2. Paper version: Step by step) or Final state.
%%%
%%% Author: Peter Grill
%\def\ShowStepByStep{}% Comment out if only want final result
%%% Choose if want animated version. This overrides \ShowStepByStep
\def\AnimateSieve{}% Comment out if don't want animated version
%%% Chose if want animated gif image instead. Overrides \AnimateSieve
%\def\AnimatedGif{}%
%
%%% The \AnimatedGif option produces a PDF with each page containing
%%% a single frame. To convert this to a GIF, use the following, where
%%% convert is part of ImageMagik
%%%
%%% pdfcrop SieveOfEratosthenes.pdf
%%% convert -verbose -delay 100 -loop 0 -density 400
%%% SieveOfEratosthenes-crop.pdf
%%% SieveOfEratosthenes.gif
%%% ---------------------------------------------------------------
%%% Customize: Choose size: NumberOfColumns x NumberOfRows
%%% Other options may need tweaking based on size settings
%%%
%%% Note that if the product of \NumOfColumns x \NumOfRows
%%% is greater than 100, the \FramesToHoldAtEnd should be
%%% larger than 25. No check below is made of this, but will
%%% result in some of larger primes not being highlighted at
%%% the end of the cycle (if this is not large enough).
\def\NumOfColumns{10}% See note above if product of
\def\NumOfRows{10}% NumOfColumns and NumOfRows > 100.
%% \FramesToHoldAtEnd should be larger than the number of primes
%% so that they can get highlighted at the end of the process
\def\FrameRate{1}%
\def\FramesToHoldAtStart{3}%
\def\FramesToHoldAtEnd{25}% 25 is enough for 10x10
\def\Scale{0.6}% May need tweaking..
\def\MinipageScale{1.0}%
\def\MinipageScaleForStepByStep{0.49}%
%% Without this scale adjustment for the animated GIF,
%% the image is quite large.
\def\ScaleForAnimatedGif{0.6}%
\def\PrimeColor{yellow}% Shade for primes found previously
\def\NewPrimeColor{cyan}% Shade for prime just found
\def\NewPrimeText{blue}% Color for primes in list
\def\NonPrimeColor{red}% Shade for non-primes
%% List of Primes is typeset into a \node of this width.
\def\TextWidth{2.0cm}%
%%% ---------------------------------------------------------------
\ifdefined\AnimatedGif
\documentclass[border=2pt,multi=true]{standalone}
\else\ifdefined\AnimateSieve
\documentclass{article}
\else
\documentclass{article}% for paper version
\fi\fi
%%%<
\usepackage{verbatim}
%%%>
\begin{comment}
:Title: Sieve of Eratosthenes
:Tags: Animations;Mathematics
:Author: Peter Grill
:Slug: eratosthenes-sieve
The sieve of Eratosthenes is an ancient algorithm for finding all prime
numbers up to any given limit. It iteratively marks the multiples of each
prime as composite (i.e. not prime). It starts with the multiples of 2.
A good way for showing the algorithm is using animations. Here the
animate package is used together with TikZ. The darker the shade of red
the higher number of times that number got marked as non-prime (i.e.,
has a higher number of prime factors). At each step, the newly found prime
is highlighted in blue, and the list of current primes is shown.
The PDF animation can be produced with the following settings:
%\def\ShowStepByStep{}% Ignored in this case
\def\AnimateSieve{}%
%\def\AnimatedGif{}% Must be commented
For a 10x10, which requires a long compiling time, use
\def\NumOfColumns{10}%
\def\NumOfRows{10}%
\def\FramesToHoldAtEnd{25}% 25 is enough for 10x10
Note that this requires a PDF viewer capable of showing animations
(such as Acrobat).
A gif animation can also be produced, for example with the following
settings:
%\def\ShowStepByStep{}% Ignored in this case
%\def\AnimateSieve{}% Ignored in this case
\def\AnimatedGif{}%
\def\NumOfColumns{5}%
\def\NumOfRows{5}%
\def\FramesToHoldAtEnd{10}%
and postprocessed
pdfcrop erathostenes-sieve.pdf
convert -verbose -delay 100 -loop 0 -density 150 erathostenes-sieve-crop.pdf
erathostenes-sieve.gif
Furthermore, there's a paper version:
\def\ShowStepByStep{}%
%\def\AnimateSieve{}% This MUST be commented
%\def\AnimatedGif{}% This MUST be commented
\def\NumOfColumns{7}%
\def\NumOfRows{7}%
\def\FramesToHoldAtEnd{10}%
Non-square sizes can be produced as well by adjusting the value of
\NumOfColumns and \NumOfRows.
This has been originally posted on
http://tex.stackexchange.com/q/44673/213
as answer to a question by the user azetina.
\end{comment}
%%% Can use the following to show the final state for large
%%% (tested up to 53x52)
%\usepackage[paperwidth=35in,paperheight=35in]{geometry}
\usepackage{geometry}
\usepackage{microtype}% Allow comma into margin in list of primes
\usepackage{xstring}% String comparison
\usepackage{tikz}% Drawing
\usetikzlibrary{calc}% Coordinate calculations
\usetikzlibrary{backgrounds}% Apply shading on background layer
\ifdefined\AnimatedGif
\usepackage{animate}% no controls, or looping needed
\def\AnimateSieve{}% Simplifies code
% if this is set for \AnimatedGif as well.
\def\Scale{\ScaleForAnimatedGif}% Otherwise GIF is too large
% Simplifies code below if we just redefine these two from the
% animate package so that they do don't much.
\renewenvironment{animateinline}[1]{\begingroup}{\endgroup}%
\renewcommand{\newframe}[1][]{\newpage}%
\else% Note: This \else is skipped for \AnimatedGif
\ifdefined\AnimateSieve%
\usepackage[loop,controls]{animate}% looped animation
\let\ShowStepByStep\relax% Ensure that \ShowStepByStep is undefined
\else% Print version
\usepackage{animate}% provides whiledo (could include ifthen)
% Simplifies code below if we just redefine these two from the
% animate package so that they do don't much.
\renewenvironment{animateinline}[1]{\begingroup}{\endgroup}%
\renewcommand{\newframe}[1][]{\newpage}%
\ifdefined\ShowStepByStep
\def\MinipageScale{\MinipageScaleForStepByStep}%
\fi
\fi
\fi
%%% ---------------------------------------------------------------
%%% Should not need to adjust anything below this line
%%%
\pgfmathtruncatemacro{\MaxNumber}{\NumOfRows*\NumOfColumns}%
\pgfmathtruncatemacro{\MaxValue}{sqrt(\MaxNumber)}%
% Choose opacity so that we can have the max number of shades
\pgfmathsetmacro{\Opacity}{1.0/min(20,\MaxValue-1)}%
%% The Sieve algorithm requires that once a number is marked
%% as non-prime (i.e., was a multiple of some other number)
%% we don't need to check multiples of that number as they
%% have already been marked as non-prime.
%%
%% Usually one would use an array and set a flag. But since
%% variables with numbers are difficult with TeX, we can
%% define a node named with the number that is non-prime.
%% Then just check that the node exists to see if it was
%% marked as non-prime.
\makeatletter
% Mark number as either "Prime" or "NonPrime".
\newcommand*{\MarkNumber}[2][NonPrime]{\node (#1#2) {}}% #1=prefix, #2=num
% http://tex.stackexchange.com/questions/37709/
\newcommand{\IfNumberAlreadyMarked}[4][NonPrime]{% #1=prefix, #2=num
\pgfutil@ifundefined{pgf@sh@ns@#1#2}{#4}{#3}%
}
% http://tex.stackexchange.com/questions/20655/
\newcommand*\@nameundef[1]{%
\global\expandafter\let\csname #1\endcsname\@undefined%
}
%% Since we repeat the process from the beginning for the animated
%% version, use this to clear the nodes so that the numbers are
%% not marked as multiples of a number from the previous run.
\newcommand{\ClearAllNumberedNodeNames}{%
\foreach \i in {1,...,\MaxValue}{%
\@nameundef{pgf@sh@ns@NonPrime\i}%
\@nameundef{pgf@sh@ns@Prime\i}%
}%
}
\makeatother
%% The Sieve algorithm skips multiples of numbers already marked as
%% non-prime. So, to number the individual steps, need to use
%% a counter.
%% i.e., Step 4 is processing multiples of 5 (since we skip 4).
\newcounter{StepNumber}%
%%% ---------------------------------------------------------------
%%%
%%% Titles and Labels
%%%
\newcommand\ListOfPrimes{}
\newcommand\AddToListOfPrimes[2][fill=\PrimeColor]{%
\IfStrEq{\ListOfPrimes}{}{%
\def\Separator{}% First member of list of primes
}{%
\def\Separator{, }% Subsequent member of list of primes
}%
%
\FillCellForGivenNumber[#1]{#2};%
\global\edef\ListOfPrimes{\ListOfPrimes\Separator#2}%
\MarkNumber[Prime]{#2};%
}
\newcommand*{\ClearListOfPrimes}{%
\ClearAllNumberedNodeNames;%
\renewcommand{\ListOfPrimes}{}%
}
\newcommand*{\Title}{%
{\noindent\Large%
\textbf{Sieve of Eratosthenes}~%
($\NumOfColumns \times \NumOfRows$)%
}%
}
\newcommand*{\SubTitleInitial}{%
\noindent\textbf{Step \theStepNumber}: Numbers from 2 \ldots\MaxNumber%
}%
\newcommand*{\SubTitle}[1]{% For animation
\noindent\textbf{Step \theStepNumber}:~%
Eliminating multiples of \textcolor{\NewPrimeText}{\textbf{#1}}%
}
\newcommand*{\SubTitlePastTense}[1]{% For step by step
\noindent\textbf{Step \theStepNumber}:~%
Eliminated multiples of \textcolor{\NewPrimeText}{\textbf{#1}}%
}
\newcommand*{\SubTitleFinal}{%
\IfEq{\the\value{StepNumber}}{0}{%
% We are only showing the final result, so no steps to label.
% This is when we are not animating (nor showing step by step)
}{%
\noindent\textbf{Step \theStepNumber}: Remaining are prime.%
}%
}
\newcommand*{\AddTitleNode}{%
\ifdefined\AnimateSieve% Otherwise don't need title each time
\node [above, yshift=1.0ex]
at ($(0,0)!0.5!(\NumOfColumns,0)$) {\Title}
\fi%
}
\newcommand*{\AddSubTitleNode}[1]{%
\IfStrEq{#1}{\empty}{%
% This is the final hold frame where we are showing the primes
\node [right] at (-1,0) {\SubTitleFinal}
}{%
\ifdefined\AnimateSieve%
\node at ($(0,0)!0.5!(\NumOfColumns,0)$) {\SubTitle{#1}}
\else%
\node [right] at (-1,0) {\SubTitlePastTense{#1}}
\fi%
}%
}
\newcommand*{\AddInitialSubTitleNode}[1]{%
\ifdefined\AnimateSieve%
\node at ($(0,0)!0.5!(\NumOfColumns,0)$) {\SubTitleInitial}
\else%
\node [right] at (-1,0) {\SubTitleInitial}
\fi%
}
\newcommand*{\Phantom}[1]{}%
\newcommand*{\ShowListOfPrimesNode}{%
\IfStrEq{\ListOfPrimes}{}{%
%% Empty list of primes, so don't want to show anything.
%% Just add phantom spacing
\renewcommand*{\Phantom}[1]{\phantom{##1}}%
}{%
\renewcommand*{\Phantom}[1]{##1}%
}%
\node [below right, xshift=0.5em, yshift=-0.5ex, align=left,
text width=\TextWidth] at (\NumOfColumns,-1)
{\Phantom{\textbf{Primes:}}};
\node [below right, xshift=0.2em, yshift=-3.5ex, align=left,
text width=\TextWidth] at (\NumOfColumns,-1)
{\Phantom{\textbf{\textcolor{\NewPrimeText}%
{\raggedleft\ListOfPrimes}}}};
}
%%% ---------------------------------------------------------------
%%%
%%% Step 1: Create a list of integers 2...n
%%%
\newcommand*{\DrawGridWithNumbers}{%
\begin{scope}[draw=gray, thick]% Add numbers to each node
\draw (0,-1) -- ($(0,-\NumOfRows-1)$);
\foreach \col in {1,...,\NumOfColumns} {%
\draw (\col,-1) -- ($(\col,-\NumOfRows-1)$);
\draw (0,-1) -- (\NumOfColumns,-1);
\foreach \row in {1,...,\NumOfRows}{%
\pgfmathtruncatemacro{\value}{\col+\NumOfColumns*(\row-1)}
\IfEq{\value}{1}{
%% Suppress number 1 from being printed since first
%% step of Sieve of Eratosthenes algorithm is to
%% create a list of integers 2...n
}{
\node at ($(\col,-\row)-(0.5,0.5)$) {\textbf{\value}};
}
\draw (0,-\row-1) -- (\NumOfColumns,-\row-1);
}
}
\end{scope}
%% Since we just drew the grid we should ensure that none
%% of the numbered nodes exist (i.e., that no numbers
%% are marked as non-prime. And reset list of primes.
\ClearListOfPrimes;
\ClearAllNumberedNodeNames;
\ShowListOfPrimesNode;
}
\newcommand*{\FillCellForGivenNumber}[2][]{%
%% #1 = fill options
%% #2 = number
%%
\pgfmathtruncatemacro{\Column}{mod(#2,\NumOfColumns)}%
\IfEq{\Column}{0}{\pgfmathtruncatemacro{\Column}{\NumOfColumns}}{}%
\pgfmathtruncatemacro{\Row}{(#2-1)/\NumOfColumns+1}%
\begin{scope}[on background layer]
\fill [#1]
(\Column-1,-\Row) --
($(\Column-1,-\Row)+(1,0)$) --
($(\Column-1,-\Row)+(1,-1)$) --
(\Column-1,-\Row-1) --
cycle;
\end{scope}
}
\newcommand*{\ColorMultiplesOf}[2][0]{%
%% If only 1 arg is given (i.e., #1=0), then
%% #2 = the multiple for which the coloring is applied
%%
%% If two args are given (i.e., #1 != 0) then
%% #1 = Value of \MaxMultiple (used for animated version)
%% In the two arg case we run the entire sequence
%% from the beginning up until the multiple #1*#2
%% is reached.
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\documentclass[12pt,reqno]{article}
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\newcommand{\N}{\mathbb{N}}
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\begin{document}
\begin{center}
\includegraphics[width=4in]{logo129.eps}
\end{center}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
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\begin{center}
\vskip 1cm{\LARGE\bf
Redundance in the Signed $m$-Bonacci \\
\vskip .1in
Numeration System
}
\vskip 1cm
\large
Zuzana Mas\'akov\'a and \'Elise Vandomme\\
Department of Mathematics FNSPE\\
Czech Technical University in Prague\\
Trojanova 13 \\
120 00 Praha 2\\
Czech Republic\\
\href{mailto:[email protected]}{\tt [email protected]} \\
\end{center}
\vskip .2 in
\begin{abstract}
We study redundance in the signed $m$-bonacci numeration system. We
determine the number of representations of 0 in this system and describe
the automaton accepting the language of such representations. We reveal
a surprising connection between Fibonacci and Tribonacci numbers.
\end{abstract}
\section{Introduction}
Integers are usually represented as sums of powers of a base $q\in\N$ for
$q\geq 2$. In particular, every $x\in\N$ can be written as $x=\sum_{j=1}^{n}a_jq^{j-1}$.
If $a_j\in\{0,1,\dots,q-1\}$ and $a_n\neq 0$, then such a representation of $x$ is unique. One can replace the sequence $(q^{j-1})_{j\geq 1}$ of powers of the base by any linear recurrence sequence $(G_j)_{j\geq 1}$. For details about such representations for a general linear recurrence sequence $(G_j)$ and conditions on uniqueness, see~\cite{PeTi89}
or~\cite[Chapter 7]{lothaire2}.
Perhaps the first example of a possibility of representing integers in these so-called linear numeration systems is the well-known Zeckendorf numeration system~\cite{Lekkerkerker}, where for $(G_j)$ we take the sequence of Fibonacci numbers $G_j=F_j$, where $F_1=1$, $F_2=2$, $F_{j+2}=F_{j+1}+F_j$ for $j\geq 1$.
The representation of an integer $x$ in the form $x=\sum_{j=1}^{n}a_jF^{j}$ is unique if $a_j\in\{0,1\}$ where $a_j\cdot a_{j-1}=0$, i.e., the sequence of coefficients does not contain consecutive 1's. This representation can be obtained by the greedy algorithm. Relaxing the condition on consecutive 1's, the system becomes redundant even with the alphabet of digits $\{0,1\}$. The function $R(n)$ expressing the number of representations of $n\in\N$ has been studied by several authors. Berstel~\cite{Berstel01} provided a formula for calculating $R(n)$ based on the greedy representation of $n$. A different way of computation was given by Edson and Zamboni~\cite{EdZa04}. Koc\'abov\'a et al.~\cite{KoMaPe05} and Stockmeyer~\cite{Stock} studied the function $R(n)$ in more detail.
A similar study was performed for different linear recurrence sequences~\cite{GrKiTi}. The $m$-bonacci sequence as a natural generalization of the Fibonacci case $m=2$ was considered by Koc\'abov\'a et al.~\cite{KoMaPe07} and Edson and Zamboni~\cite{EdZa06}. Second-order linear recurrences were studied by Edson~\cite{Edson}. In all these cases, the alphabet of digits was chosen to be the minimal one allowing representation of every integer. In the case of the $m$-bonacci system, this means the alphabet $\{0,1\}$.
One can extend the study even more, to considering redundance when the alphabet contains more than the necessary number of digits.
Considering such representations may have practical consequences, e.g.,\ for the possibility to define algorithms for parallel addition or algorithms for fast multiplication using non-adjacent form or representations of minimal weight. Parallel addition for a numeration system with integer base $q\geq 3$ was given by Avizienis~\cite{Avizienis}. For a signed binary system (with the alphabet $\{-1,0,1\}$), the algorithm was described by
Chow and Robertson~\cite{ChowRob}. An algorithm for parallel addition in the Fibonacci system with alphabet $\{-1,0,1\}$ was given by Frougny et al.~\cite{FrPeSv}. Minimal weight expansions were studied, for example, by Heuberger~\cite{Heuberger} and Grabner~\cite{GrHe06}.
Similar questions are examined by Frougny and Steiner~\cite{FroSt} in systems with real algebraic base $\beta$ which are directly connected to linear numeration systems~\cite{Frougny92fr}.
In this paper we study the linear numeration system based on the $m$-bonacci sequence over the alphabet of digits $\{-1,0,1\}$.
We show that the number $r(k)$ of representations of $0$ in the $m$-bonacci system over $\{-1,0,1\}$ of length at most $k$ is also a linear recurrence sequence and we determine the corresponding recurrence of order $m+1$. The proof of this result uses the $m$-bonacci numeration system with the alphabet $\{0,1,2\}$.
We also present the automaton accepting precisely the representations of $0$ over $\{-1,0,1\}$.
\section{Representations}
\begin{definition}
Let $m\in\N$ with $m\ge 2$.
Let $(T_k)_{k\ge 1}$ denote the $m$-bonacci sequence defined by
$$
T_k= 2^{k-1} \text{ for } k\in\{1,\ldots,m\}\ \text{ and }\ T_{k}= \sum_{j=1}^{m} T_{k-j}\text{ for }k> m.
$$
\end{definition}
\begin{definition}\label{d:rep}
Let $A\subset{\mathbb N}$ be an alphabet.
A positive integer $n$ is \emph{represented} by $(a_\ell a_{\ell-1}\cdots a_1)$ if $n=\sum_{j=1}^\ell a_j T_j$ with $a_j\in A$ and $a_\ell\ne 0$.
In that case, $\ell$ is the \emph{length} of the representation.
The number $n$ represented by $(a_\ell a_{\ell-1}\cdots a_1)$ is called the \emph{value} of $(a_\ell a_{\ell-1}\cdots a_1)$ and is denoted by $\val(a_\ell a_{\ell-1}\cdots a_1)$.
Let $p_A(n)$ denote the number of representations of $n$ over the alphabet $A$ and by $p_A(n,i)$ the number of representations of $n$ of length exactly $i$ over the alphabet $A$. If the alphabet is clear from the context, we simply write $p(n)$ and $p(n,i)$.
\end{definition}
Our aim is to enumerate the representations of 0 in the $m$-bonacci system with the alphabet $B=\{-1,0,1\}$. Conveniently, we transfer the question to determine
the number of representations in the system with the alphabet $A=\{0,1,2\}$ of the number $N_k$, where $(N_k)_{k\ge 1}$ is the running sum of the sequence $(T_k)_{k\ge 1}$, i.e.,
$$
N_{k}= \sum_{j=1}^{k} T_j\ \text{ for } k\ge 1.
$$
We set $N_0:=0$. In fact, adding 1 to every digit in a representation of 0 over $B=\{-1,0,1\}$ of length $k$, we obtain a representation of $N_k$ over the alphabet $A=\{0,1,2\}$. The exact relation of the two systems with alphabets $A$ and $B$ is explained in Section~\ref{sec:signed}.
First, let us present some computer calculations of $p(N_k)$ for the $m$-bonacci system with $m=2,3,4$, and $5$.
We count the number of representations of $N_k$ in the $m$-bonacci numeration system using the alphabet $A=\{0,1,2\}$.
\begin{table}[H]
\begin{tabular}{rr}
& $m=2$\\ \hline
$N_i$ & $p(N_i)$ \\ \hline
$1$ & $1$ \\
$3$ & $2$ \\
$6$ & $4$ \\
$11$ & $7$ \\
$19$ & $13$ \\
$32$ & $24$ \\
$53$ & $44$ \\
$87$ & $81$ \\
$142$ & $149$ \\
$231$ & $274$ \\
$375$ & $504$ \\
$608$ & $927$ \\
$985$ & $1705$ \\
$1595$ & $3136$ \\
$2582$ & $5768$ \\
$4179$ & $10609$ \\
$6763$ & $19513$ \\
$10944$ & $35890$ \\
$17709$ & $66012$ \\
$28655$ & $121415$ \\
\end{tabular}
$\quad$
\begin{tabular}{rr}
& $m=3$\\ \hline
$N_i$ & $p(N_i)$ \\ \hline
$1$ & $1$ \\
$3$ & $1$ \\
$7$ & $2$ \\
$14$ & $4$ \\
$27$ & $6$ \\
$51$ & $9$ \\
$95$ & $15$ \\
$176$ & $25$ \\
$325$ & $40$ \\
$599$ & $64$ \\
$1103$ & $104$ \\
$2030$ & $169$ \\
$3735$ & $273$ \\
$6871$ & $441$ \\
$12639$ & $714$ \\
$23248$ & $1156$ \\
$42761$ & $1870$ \\
$78651$ & $3025$ \\
$144663$ & $4895$ \\
$266078$ & $7921$ \\
\end{tabular}
$\quad$
\begin{tabular}{rr}
& $m=4$\\ \hline
$N_i$ & $p(N_i)$ \\ \hline
$1$ & $1$ \\
$3$ & $1$ \\
$7$ & $1$ \\
$15$ & $2$ \\
$30$ & $4$ \\
$59$ & $6$ \\
$115$ & $8$ \\
$223$ & $11$ \\
$431$ & $17$ \\
$832$ & $27$ \\
$1605$ & $41$ \\
$3095$ & $60$ \\
$5967$ & $88$ \\
$11503$ & $132$ \\
$22174$ & $200$ \\
$42743$ & $301$ \\
$82391$ & $449$ \\
$158815$ & $669$ \\
$306127$ & $1001$ \\
$590080$ & $1502$ \\
\end{tabular}
$\quad$
\begin{tabular}{rr}
& $m=5$\\ \hline
$N_i$ & $p(N_i)$ \\ \hline
$1$ & $1$ \\
$3$ & $1$ \\
$7$ & $1$ \\
$15$ & $1$ \\
$31$ & $2$ \\
$62$ & $4$ \\
$123$ & $6$ \\
$243$ & $8$ \\
$479$ & $10$ \\
$943$ & $13$ \\
$1855$ & $19$ \\
$3648$ & $29$ \\
$7173$ & $43$ \\
$14103$ & $61$ \\
$27727$ & $84$ \\
$54511$ & $116$ \\
$107167$ & $164$ \\
$210686$ & $236$ \\
$414199$ & $340$ \\
$814295$ & $485$ \\
\end{tabular}
\end{table}
\begin{remark}
Note the surprising fact in the column of results for the case $m=2$: The number of representations of $N_k=\sum_{j=1}^kF_j=F_{k+2}-2$ is equal to the $k$-th Tribonacci number $T_k$.
This observation was first made by Vla\v sic~\cite{Vlasic}. This statement is included in Proposition~\ref{prop:recurrence} as the case $m=2$.
\end{remark}
\section{Representations in the $m$-bonacci system over $\{0,1,2\}$}\label{sec:012}
\begin{proposition}\label{prop:recurrence}
Let $A= \{0,1,2\}$, $k,m\in\N$ with $m\ge 2$ and $k\ge 1$. The number $p(N_k)$ of representations of $N_k$ in the $m$-bonacci numeration system over the alphabet $A$ satisfies the recurrence relation
\begin{equation}\label{eq:recurrence}
p_A(N_k)= p_A(N_{k-1})+p_A(N_{k-m})+p_A(N_{k-m-1}) \text{ for } k> m+1
\end{equation}
with $p_A(N_k)= 1$ for $k\in\{0,\ldots, m-1\}$, $p_A(N_m)=2$ and $p_A(N_{m+1})=4$.
\end{proposition}
The proof of this result relies on the following lemma.
\begin{lemma}
Let $k\in \N$. The following inequalities hold:
\begin{equation}\label{eq:N_less_T}
N_k < T_k + T_{k+1}\le T_{k+2} \text{ for }k\ge 1,
\end{equation}
\begin{equation}\label{eq:2N_less_N}
2N_{k-2} < N_k \text{ for } k \ge 3,
\end{equation}
\begin{equation}\label{eq:N_less_2T}
N_k < 2T_{k+1} \text{ for }k\ge 1.
\end{equation}
\end{lemma}
\begin{proof}
For the first inequality, we proceed by induction. It is easy to check that $N_1< T_1+T_2 \le T_3$. Let $k\ge 2$ and assume the results hold for $k-1$. We have by definition and induction hypothesis that
$N_{k}= T_k+N_{k-1} < T_k+ T_{k+1} \le T_{k+2}$.
All the other inequalities follow from the first one.
For the second inequality with $k\ge 3$, we have
$$
N_k = T_k + T_{k-1} + N_{k-2} > N_{k-2} + T_{k-1}+ N_{k-2}> 2N_{k-2}.
$$
For the last inequality, we just note that $T_k<T_{k+1}$ for all $k\ge 1$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:recurrence}]
First note that \eqref{eq:N_less_T} and~\eqref{eq:2N_less_N} imply that every representation of $N_k$ has length in $\{k-1,k,k+1\}$. So
\begin{equation}\label{eq:lengths}
p(N_k)=p(N_k,k+1)+p(N_k,k)+p(N_k,k-1) \text{ for } k\geq 1.
\end{equation}
Obviously, by definition of $N_k$, $(1^k)$ is one of the representations of $N_k$.
We first take care of the initial conditions. For $k\in\{1,\ldots, m-1\}$, the number $N_k=2^k-1$ has a unique representation that is $(1^k)$. For $k=m$, we have $N_m=2^m-1 = T_{m+1}$. Therefore, $N_m$ can be represented only by $(1^m)$ or $(10^m)$. Finally, for $k=m+1$, we can use the $m$-bonacci recurrence to write
$$
N_{m+1} =\sum_{j=1}^{m+1}T_j = T_{m+2}+ T_1 =2 T_{m+1} = 2\sum_{j=1}^{m}T_j .
$$
Thus the representations of $N_{k+1}$ are four, namely $(1^{m+1})$, $(10^{m}1)$, $(20^m)$, and $(2^m)$. It is easy to check that these are the only ones.
We can summarize it in Table~\ref{t}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{ccccc}
$k$ & $p(N_k)$ & $p(N_k,k+1)$ & $p(N_k,k)$ & $p(N_k,k-1)$ \\ \midrule
1 & 1 & 0 & 1 & 0 \\\midrule
\vdots & \vdots & \vdots & \vdots & \vdots \\\midrule
$m-1$ & 1 & 0 & 1 & 0 \\\midrule
$m$ & 2 & 1 & 1 & 0 \\\midrule
$m+1$ & 4 & 1 & 2 & 1 \\\midrule
\end{tabular}
\end{center}
\caption{Initial conditions for the sequences $p(N_k)$, $p(N_k,k+1)$, $p(N_k,k)$, and $p(N_k,k-1)$.}\label{t}
\end{table}
For $k\geq m+2$, we will distinguish three cases according to the possible lengths of the representations.
\noindent \textbf{Case 1:} consider the representations $(a_{k+1}\cdots a_1)$ of $N_k$ of length $k+1$ and assume $k\ge m$. In that case, $a_{k+1}\ne 2$ because otherwise $N_k=\val(2a_k\cdots a_1)\ge 2T_{k+1}$, which is in contradiction with~\eqref{eq:N_less_2T}.
So $a_{k+1}=1$ and $\val(a_k\cdots a_1)=N_k-T_{k+1}=N_k-\sum_{j=0}^{m-1}T_{k-j}=N_{k-m}$. This means that $p(N_k,k+1)$ is equal to the number of representations of $N_{k-m}$ of length at most $k$. Since $k-m+1<k$, this number is equal to $p(N_{k-m})$.
Thus
\begin{equation}\label{eq:rec_k+1}
p(N_k,k+1)=p(N_{k-m}) \text{ for }k\ge m.
\end{equation}
\noindent\textbf{Case 2:} consider the representations $(a_k\cdots a_1)$ of $N_k$ of length $k$
and assume $k\ge m+1$.
If $a_k=2$, then
$\val(a_{k-1}\cdots a_1)=N_k-2T_k = N_{k-m-1}$. The number of such representations is equal to the number of representations of $N_{k-m-1}$, and since $k-m<k-1$, this is equal to $p(N_{k-m-1})$.
If $a_k=1$, then
$\val(a_{k-1}\cdots a_1)=N_k-T_k = N_{k-1}.$ So the number of such representations is equal to $p(N_{k-1},k-1)+p(N_{k-1},k-2)$.
In conclusion, we get
\begin{align*}
p(N_k,k) & = p(N_{k-m-1})+p(N_{k-1},k-1)+p(N_{k-1},k-2)\\
& = p(N_{k-m-1})+p(N_{k-1})-p(N_{k-1},k).
\end{align*}
Since $k\geq m+1$, we can use~\eqref{eq:rec_k+1} to obtain $p(N_{k-1},k) = p(N_{k-m-1})$ and thus
\begin{equation}\label{eq:rec_k}
p(N_k,k) = p(N_{k-1})\text{ for }k\ge m+1.
\end{equation}
\noindent \textbf{Case 3:} consider the representations $(a_{k-1}\cdots a_1)$ of $N_k$ of length $k-1$
and assume $k\ge m+2$.
We have $a_{k-1}=a_{k-2}=\cdots =a_{k-m+1}= 2$.
Indeed, let $\ell \in\{1,\ldots,m-1\}$. If the prefix of length $\ell-1$ of the representation is a concatenation of $2$'s (i.e., $a_{k-1}=\cdots = a_{k+1-\ell}= 2$), then $a_{k-\ell}=2$.
Otherwise, $a_{k-\ell}$ is equal to $1$ or $0$ and
\begin{align*}
2 N_{k-\ell-1}&\ge \val(a_{k-\ell-1}\cdots a_1) = N_k - 2 \sum_{j=1}^{\ell-1} T_{k-j} - a_{k-\ell}T_{k-\ell} \ge\\
& \ge N_k - 2 \sum_{j=1}^{\ell-1} T_{k-j} - T_{k-\ell} = N_{k-\ell-1} + T_k - \sum_{j=1}^{\ell-1} T_{k-j} = \\
& = N_{k-\ell-1} + \sum_{j=k-m}^{k-\ell} T_{j} > 2N_{k-\ell-1} %\tag{by Eq.~\eqref{eq:N_less_T}}
\end{align*}
which is a contradiction. Note that for the last inequality we have used \eqref{eq:N_less_T} which is justified since $k-\ell-1 \geq k-m \geq 2$.
So the representation of $N_k$ is of the form $(a_{k-1}\cdots a_{1})= (2^{m-1}a_{k-m}\cdots a_{1})$ and
$$
\val(a_{k-m}\cdots a_{1})= N_k-2\sum_{j=k-m+1}^{k-1}T_j = N_{k-m}+T_{k} - \sum_{j=1}^{m-1}T_{k-j} = N_{k-m} + T_{k-m}.
$$
If $a_{k-m}=2$, then $\val(a_{k-m-1}\cdots a_1)=N_{k-m-1}$. The number of representations of $N_k$ of this form is $p(N_{k-m-1},k-m-1)+p(N_{k-m-1},k-m-2)$.
If $a_{k-m}=1$, then $\val(a_{k-m-1}\cdots a_1)=N_{k-m}$, and the number of corresponding representations is $p(N_{k-m},k-m-1)$.
Finally, $a_{k-m}=0$ is not possible because it would imply
$$
2 N_{k-m-1}\ge \val(a_{k-m-1}\cdots a_1)=N_{k-m}+T_{k-m}= N_{k-m-1}+2T_{k-m}
$$
which is in contradiction with \eqref{eq:N_less_2T}. Here we use that $k-m-1\geq 1$ from the assumption.
Putting everything together, we obtain
\begin{equation}\label{eq:aux_k-1}
\begin{aligned}
p(N_k,k\!-\!1)= &\ p(N_{k-m-1},k\!-\!m\!-\!1)+p(N_{k-m-1},k\!-\!m\!-\!2)+p(N_{k-m},k\!-\!m\!-\!1)\\
=&\ p(N_{k-m-1}) - p(N_{k-m-1},k\!-\!m)+ p(N_{k-m},k\!-\!m\!-\!1).
\end{aligned}
\end{equation}
Now suppose that $k\in\{m+2,\ldots, 2m\}$. Then $1\leq k-m-1\leq m-1$ and from the initial conditions in Table~\ref{t} we have
$p(N_{k-m-1},k-m)=0$ and $p(N_{k-m},k-m-1)=0$. It follows from~\eqref{eq:aux_k-1} that $p(N_k,k-1)=p(N_{k-m-1})$.
With \eqref{eq:rec_k+1} and \eqref{eq:rec_k}, it implies that
\begin{equation}\label{eq:rec_basis}
p(N_k)=p(N_{k-1})+p(N_{k-m})+p(N_{k-m-1})\text{ for } k\in\{m+2,\ldots, 2m\}.
\end{equation}
Now assume that $k> 2m$. Using \eqref{eq:rec_k+1} and \eqref{eq:rec_k} we deduce from~\eqref{eq:aux_k-1} that
\begin{align*}
p(N_k,k-1)= &\ p(N_{k-m-1}) - p(N_{k-m-1},k-m)+\\
&\ + p(N_{k-m})- p(N_{k-m},k-m+1)-p(N_{k-m},k-m)=\\
= &\ p(N_{k-m-1}) - p(N_{k-2m-1})
+ p(N_{k-m})- p(N_{k-2m}) - p(N_{k-m-1})\\
= &\ p(N_{k-m})- p(N_{k-2m}) - p(N_{k-2m-1}).
\end{align*}
Therefore, for $k>2m$, we have
$$
p(N_k)=p(N_{k-1})+p(N_{k-m})+ p(N_{k-m})- p(N_{k-2m}) - p(N_{k-2m-1}).
$$
To conclude the argument, an easy induction on $k$ shows the claim
$$
p(N_k)=p(N_{k-1})+p(N_{k-m})+p(N_{k-m-1})\text{ for } k> m+1,
$$
since the base cases are treated in \eqref{eq:rec_basis}.
\end{proof}
\section{Representations of zero in the signed $m$-bonacci system}\label{sec:signed}
We are interested in the representations of $0$ over the balanced alphabet $B=\{-1,0,1\}$. We let ${\mathcal Z}$ denote the set of such representations. For a fixed $k\geq 1$, we further let ${\mathcal Z}_k$ denote the set of representations of 0 of length at most $k$. Let us describe the number of such representations, i.e.,
$r(k)=\#{\mathcal Z}_k$.
In order to determine $r(k)$ for $k\geq 2m+1$, let us present the correspondence between the $m$-bonacci numeration systems with the alphabet of digits $A=\{0,1,2\}$ and $B=\{-1,0,1\}$ and then derive the recurrence for the sequence $(r(k))_{k\geq 1}$.
\begin{proposition}
Let $A=\{0,1,2\}$ and $B=\{-1,0,1\}$, $k,m\in\N$ with $m\ge 2$ and $k\ge 1$.
The number $r(k)$ of representations of $0$ of length at most $k$ in the $m$-bonacci numeration system over the alphabet $B$ is given by
$$
r(k)=\sum_{j=0}^k p_B(0,j) = p_A(N_{k})- p_A(N_{k-m}) = p_A(N_{k-1})+ p_A(N_{k-m-1})\text{ for }k\ge m+1
$$
and $r(k)=p_A(N_k)=1$ for $k\in\{0,\ldots, m\}$.
\end{proposition}
\begin{proof}
Any representation $(b_k\cdots b_1)$ of $0$ over $B=\{-1,0,1\}$ (with possible leading zeros) is in one-to-one correspondence with
a representation $(a_k\cdots a_1)$ of $N_{k}$ of length at most $k$ over the alphabet $A=\{0,1,2\}$, setting $a_j=b_j+1$ for $j=1,\dots,k$.
By~\eqref{eq:lengths}, we have $r(k)= p_A(N_k,k)+p_A(N_k,k-1)=p_A(N_k)-p_A(N_k,k+1)$.
Using~\eqref{eq:rec_k+1}, we have $r(k)=p_A(N_k)-p_A(N_{k-m})$. The last equality in the statement of the proposition follows from the recurrence on $p_A(N_k)$.
\end{proof}
\begin{corollary}\label{coro:r}
The number $r(k)$ of representations of $0$ of length at most $k$ in the $m$-bonacci numeration system over the alphabet $B=\{-1,0,1\}$ satisfies
\begin{equation}\label{eq:rek-rk}
r(k)=r(k-1)+r(k-m)+r(k-m-1) \text{ for }k\ge m+1
\end{equation}
and $r(k)=1$ for $k\in\{0,\ldots, m\}$.
\end{corollary}
\begin{remark}\label{rem:r}
For further use, let us determine a set of initial values of the sequence $(r(k))_{k\geq 0}$. Obviously, the only representation of zero of length $k\leq m$ is $(0^k)$, which is in accordance with the initial conditions of the recurrence~\eqref{eq:rek-rk}.
If $k=m+1$, we have, besides the representation $(0^{m+1})$ also $(1\bar{1}^m)$ and $(\bar{1}1^m)$, where we use $\bar{1}$ for $-1$ for lucidity of notation.
We have $r(m+1)=3=r(m)+r(1)+r(0)$. For $m+2\leq k\leq 2m$ we have from~\eqref{eq:rek-rk} that
$r(k)=r(k-1)+2$, i.e.,\ $r(m+j)=2j+1$ for $j=1,\dots,m$.
\end{remark}
\begin{figure}
\centering
\begin{tikzpicture}[auto, initial text=, >=latex]
\node[state, accepting, initial,initial where=below] (v0) at (0, 0) {$i$};
\node[state] (v1) at (2, 0) {$v_1$};
\node[state] (v2) at (-2,0) {$u_1$};
\node[state] (v3) at (4, 0) {$v_2$};
\node[state] (v5) at (-4, 0) {$u_2$};
\path[->] (v0) edge[loop above] node {$0$} ();
\path[->] (v0) edge node[above] {$1$} (v1);
\path[->] (v0) edge node[above] {$-1$} (v2);
\path[->] (v1) edge node[above] {$-1$} (v3);
\path[->] (v2) edge node[above] {$1$} (v5);
\path[->] (v3) edge[bend right=30] node[above] {$-1$} (v0);
\path[->] (v3) edge[bend left=30] node[below] {$0$} (v1);
\path[->] (v5) edge[bend left=30] node[above] {$1$} (v0);
\path[->] (v5) edge[bend right=30] node[below] {$0$} (v2);
\end{tikzpicture}
\caption{Automaton accepting representations of $0$ over the alphabet $\{-1,0,1\}$ in the Fibonacci numeration system.}
\label{fig:automata-Fibonacci}
\end{figure}
An automaton accepting the representations of zero in the signed Fibonacci numeration system ($m=2$, digits $-1,0,1$) was described by Frougny~\cite[Chap.~7]{lothaire2}; see Figure~\ref{fig:automata-Fibonacci}.
The number of representations of $0$ of length at most $k$ is exactly the number of representations with $k$ symbols where we allow leading zeros.
Hence it corresponds to the number of length-$k$ accepting paths in the automaton.
Here, we present an automaton recognizing representations of zero in the signed $m$-bonacci numeration system for $m\geq 2$.
\begin{proposition}\label{prop:automat}
Let $\mathcal A=(Q,B,\delta,I,T)$ be a deterministic finite automaton with the set of states $Q=\{i,u_1,\dots,u_m,v_1,\dots,v_m\}$,
alphabet $B=\{-1,0,1\}$, the set of initial and terminal states $I=T=\{i\}$. Define the transition function $\delta: Q\times\{-1,0,1\}\to Q$,
$$
\begin{aligned}
(i,0) &\mapsto i,\quad (i,-1)\mapsto u_1,\quad (i,1)\mapsto v_1\\
(u_j,1)&\mapsto u_{j+1},\quad (u_m,0)\mapsto u_1,\quad (u_m,1) \mapsto i\\
(v_j,-1)&\mapsto v_{j+1},\quad (v_m,0)\mapsto v_1,\quad (v_m,-1) \mapsto i,
\end{aligned}
$$
see Figure~\ref{fig:automata-mbonacci}. Then $\mathcal A$ recognizes precisely the representations of 0
in the $m$-bonacci numeration system over the alphabet $B$.
\end{proposition}
\begin{figure}
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[auto, initial text=, >=latex]
\node[state, accepting, initial,initial where=below] (v0) at (0, 0) {$i$};
\node[state] (v1) at (2, 0) {$v_1$};
\node[state] (v3) at (4, 0) {$v_2$};
\node[state] (m) at (5.5, 0) {$v_{m-1}$};
\node[state] (v4) at (7.5, 0) {$v_m$};
\node[state] (v2) at (-2,0) {$u_1$};
\node[state] (v5) at (-4, 0) {$u_2$};
\node[state] (n) at (-5.5, 0) {$u_{m-1}$};
\node[state] (v6) at (-7.5, 0) {$u_m$};
\path[->] (v0) edge[loop above] node {$0$} ();
\path[->] (v0) edge node[above] {$1$} (v1);
\path[->] (v0) edge node[above] {$-1$} (v2);
\path[->] (v1) edge node[above] {$-1$} (v3);
\path[->] (m) edge node[above] {$-1$} (v4);
\path[->] (v2) edge node[above] {$1$} (v5);
\path[->] (n) edge node[above] {$1$} (v6);
\path[->] (v4) edge[bend right=30] node[above] {$-1$} (v0);
\path[->] (v4) edge[bend left=30] node[below] {$0$} (v1);
\path[->] (v6) edge[bend left=30] node[above] {$1$} (v0);
\path[->] (v6) edge[bend right=30] node[below] {$0$} (v2);
\path[dashed] (v3) edge (m);
\path[dashed] (v5) edge (n);
\end{tikzpicture}
}
\caption{Automata accepting representations of $0$ over the alphabet $\{-1,0,1\}$ in the $m$-bonacci numeration system with $m\ge 2$.}
\label{fig:automata-mbonacci}
\end{figure}
In order to justify that the accepting paths correspond precisely to the representations of 0, we study the adjacency matrix $\mathbb A$ of the automaton ${\mathcal A}$. In an adjacency matrix of the automaton, the element ${\mathbb A}_{j,k}$ is defined as 1 if there is an edge from the state $j$ to the state $k$, and 0 otherwise. Fixing the ordering of states of ${\mathcal A}$ from $1$ to $2m+1$ as $\{i,u_1,\dots,u_m,v_1,\dots,v_m\}$, we have
$$
{\mathbb A}=
\begin{pmatrix}
1 & v^T & v^T \\
u & C & O \\
u & O & C
\end{pmatrix},
$$
where $v^T=(1 0 \dots 0)\in \Z^m$, $u^T=(0 \dots 0 1)\in\Z^m$, $O\in\Z^{m\times m}$ is the zero matrix and
$$
C = \begin{pmatrix}
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & & \vdots \\
\vdots & & \ddots & & 0 \\
0 & & & 0 & 1 \\
1 & 0 & \cdots & 0 & 0
\end{pmatrix}\in\Z^{m\times m}
$$
is the companion matrix of the polynomial $x^m-1$.
\begin{lemma}\label{l:charpolA}
The characteristic polynomial of ${\mathbb A}$ is equal to
$$
f(x)=(x^m-1)(x^{m+1}-x^m-x-1)=x^{2m+1}-x^{2m}-2x^{m+1}+x+1.
$$
\end{lemma}
\begin{proof}
For the computation of the characteristic polynomial $f(x)=-\det({\mathbb A}-xI)$, we use the following statement about determinants of block matrices~\cite{Silvester}.
{\it
Let $E,F,G,H$ be complex matrices of dimensions $p\times p$, $p\times q$, $q\times p$, $q\times q$, with $H$ invertible.
Then for the determinant of
$\begin{pmatrix}
E & F \\
G & H
\end{pmatrix}$
we have
\begin{equation}\label{eq:odeterminantech}
\det\begin{pmatrix}
E & F \\
G & H
\end{pmatrix} =
\det(E-FH^{-1}G)\det(H)\,.
\end{equation}
}
\noindent
We use the statement~\eqref{eq:odeterminantech} in two steps. First set
$$
E=\begin{pmatrix}
1-x & v^T \\
u & H
\end{pmatrix},\
F=\begin{pmatrix}
v^T \\
O
\end{pmatrix},\
G=\begin{pmatrix}
u & O
\end{pmatrix},
$$
$$
H=C-xI=\begin{pmatrix}
-x & 1 & 0 & \cdots & 0 \\
0 & -x & 1 & & \vdots \\
\vdots & & \ddots & & 0 \\
0 & & & -x & 1 \\
1 & 0 & \cdots & 0 & -x
\end{pmatrix}.
$$
Note that $\det(H)=(-1)^m(x^m-1)$. Also note that
the product $FH^{-1}G=(h_{j,k})\in \Z^{(m+1)\times(m+1)}$ is a matrix that is null everywhere except
for $c_{1,1}=h_{1,m}$ with $(h_{j,k})=H^{-1}=\det(H)^{-1}{\rm adj}(H)$ where ${\rm adj}(H)$ is the adjugate matrix.
Since the adjugate matrix is computed as the transpose of the cofactor matrix, we have
$$
c_{1,1}=h_{1,m}=\frac{1}{(-1)^m(x^m-1)} (-1)^{m+1}\det\begin{pmatrix}
1 & 0 & \dots & 0 \\
-x & 1 & \ddots & \vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \dots & -x & 1
\end{pmatrix} = \frac{-1}{x^m-1}\,.
$$
At this stage, we have
$$
\det({\mathbb A}-xI) = \det(E-FH^{-1}G)\det(H) =
\det(H)\det\begin{pmatrix}
1-x - c_{1,1} & v^T \\
u & H
\end{pmatrix}.
$$
We use~\eqref{eq:odeterminantech} again, with redefined blocks $E,F,G$, $E=(1-x-c_{1,1})$, namely $F= v^T$, $G= u$.
Realizing that $FH^{-1}G=(c_{1,1})$, we derive
$$
\begin{aligned}
-f(x) &=\det({\mathbb A}-xI) =
\det(H)^2 (1-x-2c_{1,1}) =\\
&= (x^m-1)^2\big(1-x-\frac{-2}{x^m-1}\big)=-(x^m-1)(x^{m+1}-x^m-x-1),
\end{aligned}
$$
which proves the lemma.
\end{proof}
Let us now demonstrate that the automaton presented in Figure~\ref{fig:automata-mbonacci} recognizes precisely the representations of 0 in the $m$-bonacci system over the alphabet $B=\{-1,0,1\}$.
\begin{proof}[Proof of Proposition~\ref{prop:automat}]
The language accepted by the automaton ${\mathcal A}$ depicted in Figure~\ref{fig:automata-mbonacci} can be described by the regular expression
$$
0^*\big(1(\bar{1}^{m-1}0)^*\bar{1}^m+\bar{1}(1^{m-1}0)^*1^m\big)^*0^*
$$
It can be directly checked with the use of the $m$-bonacci recurrence that every word of the above form corresponds to
a representation of $0$ in the sense of Definition~\ref{d:rep}. For every fixed $k$ we thus have
${\mathcal L}_k(\mathcal A) \subseteq {\mathcal Z}_k$.
In order to check the opposite inclusion, it suffices to show that
$$
\#{\mathcal L}_k(\mathcal A) = \#{\mathcal Z}_k = r(k).
$$
Consider the adjacency matrix ${\mathbb A}$ of the automaton ${\mathcal A}$.
An accepting path in the automaton is a path both starting and ending in the initial state $i$. The number of such paths of length $k$, say $s(k)$, is equal to $({\mathbb A}^k)_{11}$. We thus have $s(k):=\#{\mathcal L}_k(\mathcal A)=({\mathbb A}^k)_{11}$. We will show that $s(k)=r(k)$ by verifying that both sequences satisfy the same linear recurrence of order $2m+1$ with the same initial conditions $s(j)=r(j)$ for $j=0,\dots,2m$.
The characteristic polynomial $f$ of the matrix ${\mathbb A}$ is computed in Lemma~\ref{l:charpolA}. By the Cayley-Hamilton theorem, we have
$$
f(\mathbb A)={\mathbb A}^{2m+1}-{\mathbb A}^{2m}-2{\mathbb A}^{m+1}+{\mathbb A}+I = O,
$$
where $I,O\in\Z^{(2m+1)\times(2m+1)}$ are the identity and zero matrix, respectively. Multiplying the above by ${\mathbb A}^k$ for $k\geq 0$,
and considering the sequence $s(k)=({\mathbb A}^k)_{11}$, we have
$$
s(k+2m+1)-s(k+2m)-2s(k+m+1)+s(k+1)+s(k)=0 \quad\text{ for } k\geq 0.
$$
Let us verify that the sequence $r(k)$ satisfies the same recurrence. From Corollary~\ref{coro:r} we have for $k\geq 0$ that
$$
\begin{aligned}
r(k+m+1)-r(k+m)-r(k+1)-r(k)&=0,\\
r(k+2m+1)-r(k+2m)-r(k+m+1)-r(k+m)&=0.
\end{aligned}
$$
Subtracting the two, we obtain
$$
r(k+2m+1)-r(k+2m)-2r(k+m+1)+r(k+1)+r(k)=0 \quad\text{ for } k\geq 0,
$$
as desired.
The initial values $r(0),\dots,r(2m)$ are determined in Remark~\ref{rem:r} as
$r(j)=1$ for $0\leq j\leq m$ and $r(j+m)=2j+1$ for $0\leq j\leq m$.
It remains to determine $s(0),\dots, s(2m)$. For that, we check directly the automaton ${\mathcal A}$. We see that $s(k)=r(k)$ for $0\leq k\leq 2m$ which implies that $s(k)=r(k)$ for every $k\geq 0$.
\end{proof}
\section{Characteristic polynomial}\label{sec:polynom}
When studying growth of the sequences $p_A(N_k)$, $r(k)$ counting representations
in the $m$-bonacci system, one is interested in the zeros of the characteristic polynomial of the corresponding recurrence relation. From Proposition~\ref{prop:recurrence} and Corollary~\ref{coro:r} we see that both $p_A(N_k)$ and $r(k)$ satisfy the same recurrence with characteristic polynomial
$f(x)=x^{m+1}-x^m-x-1$.
Quadrinomials whose coefficients take values only in $\{\pm1\}$ have been considered by several authors, e.g., Ljunggren~\cite{Ljunggren}, Finch and Jones~\cite{FiJo}, or Mills~\cite{Mills}. It is not difficult to derive the following result.
\begin{proposition}
Let $m\geq 2$. Then $\chi(x)=x^{m+1}-x^m-x-1$ is irreducible over $\Q$ if and only if $m\not\equiv 3\pmod 4$. If $m=4k+3$ for $k\geq 1$, then we have
$\chi(x)=(x^2+1)\psi(x)$ where for $k=1$ we have $\psi(x)=(x^3-x^2+1)(x^3-x-1)$, and for $k\geq 2$, $\psi\in\Z[x]$ is irreducible over $\Q$.
\end{proposition}
\begin{proof}
It follows from Theorem~2 in the paper by Finch and Jones~\cite{FiJo} that the quadrinomial $\chi(x)=x^{m+1}-x^m-x-1$ is irreducible over $\Q$ if and only if $m\not\equiv 3\pmod 4$.
The factorization of the polynomial $\chi$ in the reducible case can be deduced from the results of Ljunggren~\cite{Ljunggren} and Mills~\cite{Mills}.
\end{proof}
\begin{figure}
\centering
\includegraphics[width=0.32\textwidth]{Polynom4.png}\
\includegraphics[width=0.32\textwidth]{Polynom8.png}\
\includegraphics[width=0.32\textwidth]{Polynom33.png}
\caption{Zeros of the polynomial $\chi(x)=x^{m+1}-x^m-x-1$ for $m=3$, $m=7$ and $m=32$. The zeros $\lambda$ with $\Re\lambda>0$ are of modulus greater than $1$, the zeros with negative real part lie inside the unit disk.}\label{f:roots}
\end{figure}
\section{Acknowledgments}
This work was supported by the project CZ.02.1.01/0.0/0.0/16\_019/0000778
from European Regional Development Fund.
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\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2020 {\it Mathematics Subject Classification}:
Primary 11A67; Secondary 11B39.
\noindent \emph{Keywords: } linear numeration system, $m$-bonacci number.
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received May 6 2021;
revised version received June 2 2021; June 4 2021.
Published in {\it Journal of Integer Sequences},
June 4 2021.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in
\end{document}
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% The following is a LaTeX source file for the paper (to appear in J.
% Stat. Phys.) `Shock Fluctuations in the Two-Dimensional Asymmetric
% Simple Exclusion Process' by Alexander/Cheng/Janowsky/Lebowitz
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\author{Francis J. Alexander\thanks{Supported in part by a Rutgers University
Excellence Fellowship. Current address: Center for Nonlinear Studies,
MS-B258, Los Alamos National Lab, Los Alamos, NM 87545},
Zheming Cheng\thanks{Current address: Program Development Corp.,
300~Hamilton Ave., Suite~409, White Plains, NY 10601},\\
Steven A. Janowsky\thanks{Supported in part by National
Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS
90-07206}\hspace{.25em} and
Joel L. Lebowitz\\[5pt]
Departments of Mathematics and Physics\\
Rutgers University\\New Brunswick, NJ 08903}
\title{Shock Fluctuations in the Two-Dimensional Asymmetric Simple
Exclusion Process\thanks{Supported in part by National Science
Foundation grant DMR 89-18903.}}
\date{February 26, 1992}
\begin{document}\maketitle
\vspace{-.75cm}
\renewcommand{\baselinestretch}{1.1}\large\normalsize
\begin{abstract}
We study via computer simulations (using various serial and parallel updating
techniques) the time evolution of shocks, particularly the shock width
$\sigma(t)$, in several versions of the two-dimensional asymmetric simple
exclusion process (ASEP). The basic dynamics of this process consists of
particles jumping independently to empty neighboring lattice sites with rates
$p_{\rm up} = p_{\rm down} = p_\perp$, $p_{\rm left} < p_{\rm right}$. If the
system is initially divided into two regions with densities $\rho_{\rm left} <
\rho_{\rm right}$, the boundary between the two regions corresponds to a shock
front. Macroscopically the shock remains sharp and moves with a constant
velocity $v_{\rm shock} = (p_{\rm right} - p_{\rm left})(1 - \rho_{\rm left} -
\rho_{\rm right})$. We find that microscopic fluctuations cause $\sigma$ to
grow as $t^\beta$, $\beta\approx 1/4$. This is consistent with theoretical
expectations.
We also study the nonequilibrium stationary states of the ASEP on a periodic
lattice, where we break translation invariance by reducing the jump rates
across the bonds between two neighboring columns of the system by a factor
$r$. We find that for fixed overall density $\rho_{\rm avg}$ and reduction
factor $r$ sufficiently small (depending on $\rho_{\rm avg}$ and the jump
rates) the system segregates into two regions with densities $\rho_1$ and
$\rho_2=1-\rho_1$, where these densities do not depend on the overall density
$\rho_{\rm avg}$. The boundary between the two regions is again
macroscopically sharp. We examine the shock width and the variance in the
shock position in the stationary state, paying particular attention to the
scaling of these quantities with system size. This scaling behavior shows
many of the same features as the time-dependent scaling discussed above,
providing an alternate determination of the result $\beta\approx 1/4$.\\[3pt]
\noindent
KEY WORDS: Stochastic particle systems; shock waves; surface growth
\end{abstract}
\clearpage
\section{Introduction}
Macroscopic equations that describe fluid flow result from an averaging over
the rapidly fluctuating microscopic motion of a large number of
molecules~\cite{LPS}. This procedure yields deterministic (typically
nonlinear) partial differential equations for the conserved quantities
(momentum, energy, mass density)---which vary slowly on the microscopic scale.
The familiar hydrodynamic equations of motion (Navier-Stokes and Euler) which
describe the dynamics of fluids are of this form.
These equations describe quite well what happens to fluids on a large scale
when the flow is smooth. Problems arise when gradients in the hydrodynamic
variables become very large and the assumptions made in the derivation of the
hydrodynamic equations break down---for example, where discontinuities
(shocks) in macroscopic variables (such as the density) appear. We would like
to gain a better understanding of these situations both at the macroscopic and
microscopic levels.
\subsection{The Burgers Equation}
One of the simplest examples of a nonlinear macroscopic equation with a single
conserved quantity is the Burgers equation,
\begin{equation}\label{Burger1}
\frac{\partial {\bf u}}{\partial t} = -{\bf u}\cdot\! \nabla{\bf u} +
\nu\Delta {\bf u},
\end{equation}
originally proposed to study turbulence with ${\bf u}$ representing a
velocity field~\cite{Burgers,Tatsumi}; in our considerations, ${\bf u}$
will represent a scalar density field. What makes the Burgers equation
interesting is that initially smooth density profiles can evolve after a
finite time into traveling wave fronts. The transitions between low and
high density regions in these fronts occur in very narrow spatial
regions---regions with width proportional to $\sqrt{\nu}$. In the limit
$\nu\rightarrow 0$ the profile becomes discontinuous and we say that
shocks form. If $\nu$ is finite but small ({\em i.e.}\ microscopic) we
may still refer to the narrow transition region as a shock. These
shocks as given by the Burgers equation move with a deterministic
velocity. We are interested in what happens to shocks when viewed on
the microscopic level.
The non-viscous Burgers equation, with $\nu=0$, can be derived
rigorously from a number of computationally efficient particle
models~\cite{LPS,Spohn}. The field ${\bf u}$ represents the space and time
rescaled particle configurations, {\em i.e.}\ the hydrodynamic limit
with Euler scaling in which time and space are scaled by a fixed ratio.
(Certain special cases with other scalings can result in limits with
$\nu> 0$.) Here we use the asymmetric simple exclusion process. On a
lattice, particles hop to unoccupied neighboring sites with a drift
(asymmetry) along one of the lattice directions. The hard-core exclusion
(only one particle per site) is the source of the nonlinearity and
provides for some interesting phenomena. Even on the microscopic scale
shocks form. At this level, instead of traveling with some definite
deterministic velocity, there is a a fluctuating velocity, due to the
initial conditions and the dynamics, that must be superposed upon the
macroscopic one. Therefore, the location of the shock will deviate from
the location given by the Burgers equation by some fluctuating
quantity~\cite{DKPS}.
\subsection{The ASEP}
The ASEP is a continuous time stochastic process in which particles
occupy sites of the lattice ${\Bbb Z}^d$ and move according to simple
rules. Configurations in this process are denoted by $\eta \in
\{ 0,1 \}^{{\Bbb Z}^d}$, where the individual site occupation variable $\eta
(\vec{\bf r}) = 1$ if $\vec{\bf r}$ is occupied, and $0$ if unoccupied.
An exclusion rule prevents more than one particle from simultaneously
inhabiting the same site. Independently and randomly, each particle
waits for an exponentially distributed time with mean 1 and attempts to
jump to a neighboring site. If the target site is unoccupied, then the
jump succeeds; if not, then it fails. An asymmetry enhances jump
attempts in one direction and induces a net particle current. In the
one-dimensional model, a particle attempts to jump to the right with
rate $p_{\rm right}$ and to the left with rate $p_{\rm left}$, $p_{\rm
right}>p_{\rm left}$, $p_{\rm right}+p_{\rm left} = 1$. In higher
dimensions the asymmetry along the $x$ axis persists, but the jump rates
along both directions of the perpendicular axes are equal (symmetric)
and given by $p_\perp$ such that $p_{\rm left}+p_{\rm right}+2(d-1)
p_\perp =1$, recalling that $d$ is the dimension. Shocks will now
correspond to $(d-1)$-dimensional fronts which will fluctuate in space
and time.
\subsection{Shock Growth $=$ Surface Growth; KPZ approach}
We may also interpret the shock evolution in the $d$-dimensional ASEP as a
model of $(d-1)$-dimensional surface growth where holes are driven to the left
and stick to the ``hole substrate,'' with the surface of this substrate
traveling to the right. This interpretation is particularly useful given the
recent interest in surface deposition models and the kinetic roughening of
surfaces~\cite{KS}. The model we describe here is complicated for analysis as
well as for simulation since we are interested in the statistics of the shock
(= surface) and its dependence on a priori {\em unknown} properties of the
ASEP. In other surface problems one usually assumes that the particles (or
holes) hitting the surface are uncorrelated or have {\em known} correlations.
One can then construct a stochastic partial differential equation which models
the surface dynamics.
Typically the approach to such problems in surface roughening models is to
introduce a Langevin type equation which governs the local interface position
$h(\vec{\bf r},t)$, the usual choice being the Kardar-Parisi-Zhang (KPZ)
equation~\cite{KPZ}:
\begin{equation}\label{KPZ1}
\frac{\partial h}{\partial t} = v_0 + \nu\Delta h + \frac{\lambda}{2}
(\nabla h)^2 + \zeta(\vec{\bf r},t).
\end{equation}
The random noise term $\zeta(\vec{\bf r},t)$ is such that
\begin{equation}
\langle \zeta (\vec{\bf r},t) \rangle = 0,
\end{equation}
but otherwise is chosen according to the specific nature of the model being
studied~\cite{Medina}.
The KPZ equation provides a phenomenologically based description of a growing
surface. Each term represents a different aspect of the growth process: the
constant $v_0$ is the growth rate for a completely flat interface. The
Laplacian term accounts for surface restructuring as particles diffuse on the
surface and move to fill gaps, while the nonlinear gradient term describes an
inclination dependent growth rate. The noise term represents growth due to
fluctuations at the microscopic level; without this term the KPZ equation
(\ref{KPZ1}) can be related to the Burgers equation (\ref{Burger1}) via the
transformation ${\bf u} = -\nabla h$. Higher order terms are not included in
the KPZ equation because they are irrelevant in the renormalization group
sense; without loss of generality we change variables $h\rightarrow h-v_0t$
and take $v_0=0$.
\subsubsection{Linearized KPZ}
We believe that the one-dimensional KPZ equation, with the nonlinear
term absent ($\lambda=0$), describes the interface behavior of the
two-dimensional ASEP. (This reduces to a noisy, linear diffusion
equation.) We argue this on several grounds. In the first place, the
nonlinear term arises from the dependence of surface growth on the local
orientation. In the ASEP, however, the shock velocity is independent of
surface orientation and so averaging over orientations results in
cancellation of the nonlinear term~\cite{Spohn}.
Secondly, in the case of weak asymmetry, where $p_\perp$ is fixed, $p_{\rm
right} - p_{\rm left} = \epsilon$, space is rescaled by $\epsilon$ and time is
rescaled by $\epsilon^2$, it is known rigorously that the equation describing
the interface fluctuations in the limit $\epsilon\rightarrow0$ is indeed just
the linear KPZ equation~\cite{Ravi}
\begin{equation}\label{eq:ravishan}
\frac{\partial h(r,t)}{\partial t} = \nu\Delta h(r,t) + \zeta(r,t),
\end{equation}
with $\zeta(r,t)$ Gaussian white noise: $\langle \zeta (r,t) \rangle = 0$ and
\begin{equation}
\langle \zeta(r,t) \zeta (r',t') \rangle = K\delta(r-r')\delta(t-t'),
\end{equation}
where $K$ depends on the asymptotic densities on either side of the
interface. One can solve (\ref{eq:ravishan}) exactly by using Fourier
transforms; the result for the shock width starting with an initially
flat interface in a strip of width $W$ is~\cite{Family}
\begin{equation}\label{scalingform}
\sigma(t) \sim \left\{
\begin{array}{l@{\quad}l}
t^\beta, &t\ll W^{\alpha/\beta},\\[3pt]
W^\alpha, &t\gg W^{\alpha/\beta};
\end{array}\right.
\end{equation}
where
\begin{equation}
\sigma^2(t) \equiv \frac1W\int dr\,\left(h(r,t)-\bar{h}(t)\right)^2,
\qquad
\bar{h}(t) \equiv \frac1W\int dr\, h(r,t),
\end{equation}
$W$ is the width of the system (length of the interface) and the scaling
exponents are
\begin{equation}\label{alphabeta}
\alpha = 1/2, \qquad \beta = 1/4.
\end{equation}
We shall see later that these exponents are indeed consistent with the results
of our simulations---although the approach to scaling behavior can take a
very long time.
\subsubsection{Logarithmic Correction}
In fact there is a defect in the above analysis; the two-dimensional ASEP is
known to exhibit superdiffusive behavior~\cite{SvB} and thus one should not
expect a linear diffusion equation to accurately model it. Analysis similar
to that of \cite{vB} indicates that the correct behavior is~\cite{vB:private}
\begin{equation}
\sigma(t) \sim t^{1/4} (\log t)^{1/3}\quad{\rm for}\quad
t^{1/4} (\log t)^{1/3} \ll W^{1/2},
\end{equation}
while for $t^{1/4} (\log t)^{1/3} \gg W^{1/2}$ the result remains $W^{1/2}$.
This correction is sufficiently small that it will be unobservable in any
numerical simulation likely to be done before the next century, and
(\ref{eq:ravishan}) remains a reasonable approximation.
\section{Models}
We studied two basic classes of systems undergoing ASEP dynamics: The
time-dependent behavior of an effectively infinite system, and the stationary
states of a finite periodic model.
\subsection{Two-Dimensional ASEP---Time-Dependent Model}
The specific system we considered for our time dependent studies consisted of
a $W\times L$ lattice with periodic boundary conditions in the vertical
(perpendicular to the field) direction. Parallel to the field no effort was
made to apply suitable boundary conditions as simulations were always ended
prior to the arrival of information about the boundaries. In most cases we
chose the jumps to be totally asymmetric in the parallel direction, {\em
i.e.}\ we took $p_{\rm left} = 0$.
\subsubsection*{Simulation of a Continuous Time Process}
Since each lattice site has an independent exponentially distributed
waiting time for attempting a jump, the probability of two sites
attempting to jump at the same time is zero. Thus we can simulate the
ASEP dynamics in discrete time by choosing at each time step one site
where we attempt to schedule a jump. All sites in $[1,W]\times [1,L-1]$
are chosen with equal probability. If the chosen site is occupied, we
pick a direction $\in \{\rm up,\ down,\ right,\ left\}$ with
probabilities $p_\perp$, $p_\perp$, $p_{\rm right}$ and $p_{\rm left}$,
respectively. If the nearest neighbor site in the chosen direction is
unoccupied, the particle jumps to that site; if it is occupied, the
particle does not move for the given time step.
\subsection{Stationary Model}
Here we consider ASEP dynamics on an $W\times L$ torus. We break the
translation invariance of this periodic system by inserting a blockage into
the system between columns $L$ and 1, which reduces the probability of a
particle traveling between those two columns---a set of ``slow bonds'' which
act as a traffic jam for the particles. In the language of driven diffusive
systems, the introduction of ``slow bonds'' is similar to altering the driving
field at this one column~\cite{AL}. This blockage is analogous to a
restriction in a pipe through which fluid flows; the corresponding model also
provides an example of the dramatic global effects caused by a local
perturbation in conservative systems which do not satisfy detailed
balance~\cite{GLMS}. It also provides an alternate method for observing the
same behavior as in the time-dependent model. The one dimensional version of
this system was examined in~\cite{JL}.
More specifically, we reduce the jump rates between columns $L$ and 1 by a
factor $r$, $0\le r\le1$. For $r=1$ the model is translation invariant. For
$r=0$ the model is fully blocked; the stationary state has density one behind
the blockage and density zero in front of it; there is no current flowing
through the system. For $0<r<1$ the model has nontrivial behavior, with the
stationary state satisfying the requirement that the current through any
column of bonds must be independent of the column location.
In this model with a blockage we are not particularly interested in the time
evolution of the shock but instead study the properties of the shock in the
stationary state, and how those properties depend on system size.
\subsubsection{Simulation of ASEP dynamics with blockage}
The simulation of the ASEP dynamics in the periodic system with a blockage is
basically the same as that in the ordinary model. All sites in $[1,W]\times
[1,L]$ are chosen with equal probability. If the chosen site is occupied, we
pick a direction $\in \{\rm up,\ down,\ right,\ left \}$ with probabilities
$p_\perp$, $p_\perp$, $p_{\rm right}$ and $p_{\rm left}$, respectively. We
now must distinguish between the blockage and non-blockage columns. If the
chosen site is in column $1$ or $L$ and the chosen direction is left or right,
respectively, then the particle is attempting to jump the blockage and the
attempt is completed (assuming the destination site is vacant) with
probability $r$. Other jumps take place in the same fashion as above but are
completed (assuming the destination site is vacant) with probability $1$.
Of course we must allow an adequate time for this system to evolve so that we
can be confident that the properties we observe are indicative of the
stationary state. The necessary time is determined simply by observing the
time evolution of shock width, position, etc., and waiting until they reach
asymptotic values. For all system sizes except for the very largest a
substantial ``safety'' factor was also included.
\subsubsection{Parallel and semi-parallel models}
In order to examine very large system sizes, we devised various modifications
of the ASEP that would permit effective utilization of vector and massively
parallel supercomputers. Unfortunately attempting to perform updates on all
sites at once is impossible as this leads to two or more particles vying for
the same site. Sublattice updating is a traditional technique; however, it
may introduce spurious correlations into the model. We did have limited
success with the following variations on sublattice updating:
\begin{itemize}
\item
We were able to consider 32 different realizations of the same size system at
once by treating each bit of a 32 bit word as an individual system. Although
the choices for which sites to update were the same for all the systems, the
choice of directions in each system were independent. We were thus able to
improve the statistics over that which would have otherwise been available;
the 32 (correlated) systems appeared to provide data equivalent to that which
would have been given by approximately 10 truly independent systems. Most of
our data sampling (for both time-dependent and stationary models) made use of
this procedure.
\item
We considered a partial sublattice updating where 32 sites in each column
(spaced $W/32$ apart) were updated simultaneously. {\em Qualitatively} the
model remained the same, but significant correlations were introduced at
vertical distances of $W/32$, $W/16$, etc. None of our quantitative data
makes use of this procedure.
\item
In an attempt to reduce the correlations introduced by a fixed sublattice
updating, we decided to examine the possibility of using a sublattice which at
each step had a random origin and spacing. First fix the system width $W$ as
$W = a w_1 w_2 w_3 w_4$, where all the factors $\{w_i\}$ are relatively prime.
Thus we have four possible sublattice spacings available. Updating proceeds
by first choosing a sublattice spacing (stride) $w_i$ with probability
$w_i\big/\sum w_i$, then choosing a site, and finally updating that site and
all (vertical) translates of that site by multiples of $w_i$. For $W =
9\times10\times11\times13 = 12870$ ($a=1$) this allowed us to update between
990 and 1430 sites simultaneously using vectorized instructions on a Cray YMP.
This large speedup allowed us to examine systems that were much larger than we
could have otherwise considered.
\end{itemize}
\section{Time-Dependent Results}
\subsection{Shock identity}
When studying systems with shocks, we can either wait for shocks to
develop from smooth initial data, or else we can impose them on the
initial data by hand. We opted for the latter approach to facilitate
computer simulation. In either case we must be able to identify the
location of the shock. We begin by giving a particular operational
definition of what is meant by a shock at the microscopic level in one
dimension.
Consider the integer lattice ${\Bbb Z}$. On each site $x \geq L/2$ we
independently place a particle with probability $\rho_0$. All sites $x
< L/2$ are empty. Asymmetry in the jump rates will drive particles to
the right. At time $t=0$ we label the leftmost particle in the system
the ``first'' particle. As the system evolves, there is no reordering
of particles since jumps are to nearest neighbor sites only (no
crossing). Therefore, this ``first particle'' label is permanently
attached to the same particle. If we view the process from the first
particle, then to the left the density is always zero while to the
right, the density of particles approaches $\rho_0$ at an exponentially
(in space) fast rate, where the decay length is $1/\log(p_{\rm
right}/p_{\rm left})$~\cite{Ferrari}. Therefore, we have an abrupt
change in particle density (over a few sites)---a shock. Accordingly,
we define the shock to be located at the site containing the first
particle~\cite{Wick}, and we know that even on the microscopic scale the
shock remains sharp. Note that this picture breaks down when $p_{\rm
right} = p_{\rm left}$; in this case the behavior is purely diffusive
and the average distance between the first and second particles grows
like $t^{1/2}$.
In two dimensions this picture needs modification. The asymmetry drives
particles to the right while allowing for diffusion in the vertical direction.
What results is a right-moving, rough interface across which there is an
abrupt change in average particle density. We also refer to this fluctuating
interface as a shock. Now, however, the first particle in each row is no
longer a permanent label as a result of perpendicular diffusion. But the
basic principle is still valid, and the most natural and computationally
efficient way to characterize the shock (when the density is zero to the left)
is to look at the distribution of first particles in different rows (different
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\caption{Example of a particle configuration and the resulting shock
position $h(y,t)$.}\label{shockdef}
\end{center}\end{figure}
$y$ values)---see Figure~\ref{shockdef}. The instantaneous width is then
defined by fluctuations in these locations $h(y,t)$ about their average:
\begin{equation}
\sigma (t) \equiv \left(\frac1W \sum_{y=1}^W (h(y,t)-{\bar h}(t))^2
\right)^{1/2},
\end{equation}
where ${\bar h}(t)$ is the average location of the first particles,
\begin{equation}
{\bar h}(t) = \frac1W \sum_{y=1}^W h(y,t).
\end{equation}
\subsection{Results}
With the exception of some limited runs to check the gross behavior with
varied parameters, all of our time-dependent simulations were carried
out with an initial particle density of $\rho_0 = .5$ and with jump
rates $p_{\rm right}=.75$, $p_{\rm left}=0$ and $p_\perp =.125$. The
field drove particles to the right and holes to the left. Since
particles eventually moved away from the left wall (and were not
replenished) and holes moved away from the right wall (and were not
replenished), we were left with particle substrate on the right of the
system moving to the left, and the hole substrate on the left, moving to
the right. Particle-hole symmetry (at $\rho_0 = .5$) implies they
should have the same behavior. With this approach we were able to study
the statistics of two nearly independent shocks at once.
In Figure~\ref{width}
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\label{width}
\end{center}\end{figure}
we compare the width of the shock interface for system widths $W=256$ and
$W=1024$; in each case the data is averaged over 75 independent runs. The
available run time was not long enough for the shock width to saturate,
although the effect of the finite system width is apparent as we see the
$W=256$ and $W=1024$ curves separate for $t>400$. We expect that for $W$
large enough there will an ``asymptotic'' time regime for the growth of
$\sigma(t)$ before it saturates, with behavior given by equations
(\ref{scalingform}--\ref{alphabeta}).
The results in Figure~\ref{width} indicate that the growth of fluctuations has
not reached the asymptotic (in time) regime. If we think of the exponent
$\beta$ characterizing this growth as a time-dependent quantity, then $\beta
\to .17$ for the longest times that we were able to observe (in the $W=1024$
system), and is apparently still increasing. It may be that the long time
behavior will still be consistent with the predictions of the linear KPZ
equation given in (\ref{scalingform}--\ref{alphabeta}), but that the approach
to this asymptotic behavior is extremely slow.
There appear to be two reasons for this slow convergence. First, the shock
width at $t=0$ does not strictly vanish. There are fluctuations inherent to
the initializing process of setting down particles randomly with probability
$\rho_0$. This accounts for a ``zero point fluctuation'' of a few lattice
spacings. We can eliminate this effect by starting with an artificially
perfectly flat interface. The true asymptotic behavior will not be affected
by this.
This, however, is not the main problem---the main difficulty is that there are
natural, short wavelength fluctuations which result from the dynamics and form
at an early time. Letting $\langle\cdot\rangle$ represent an average over the
random dynamics, {\em i.e.}\ a sampling average, we see that there is a
natural separation distance $\langle |(h(y,t) - h(y+1,t))| \rangle$ between
the shock locations in neighboring rows, which builds up rapidly and then
saturates. If the perpendicular jump rate is small, then this length can be
quite large. In this case, $p_\perp \ll p_{\rm left}+ p_{\rm right}$,
neighboring rows are essentially noninteracting for long periods of time. The
first particles in each row then execute a random walk relative to each other
and can become quite distant. As their distance increases, it becomes more
probable to have a transition from one row to another which will reduce the
distance between the first particles in the two rows. These perpendicular
transitions prevent the first particles in neighboring rows from moving
arbitrarily far from each other (confinement), and thus $\langle |h(y,t) -
h(y+1,t)| \rangle$ approaches an asymptotic value. For the jump rates we
used, we found that this natural extension was on the order of a few lattice
spacings, just enough to mask the overall shock broadening which was typically
less than 8 lattice spacings.
This latter problem could not be eliminated with a simple adjustment of the
initial conditions or jump rates. Therefore, we looked at other manifestations
of shock broadening which in the asymptotic time regime should be equivalent
to the shock width defined above. Defining
\begin{equation}
G(m,t) = \langle h(y,t) h(y+m,t) \rangle - \langle h(y,t)\rangle^2,
\end{equation}
we see that $G(0,t) = \sigma^2(t)$. For $m=1$ or $2$, $G(m,t)$ represents the nearest
(row) neighbor, $m=1$, and next nearest neighbor, $m=2$, truncated first
particle correlation functions. These functions have none of the $t=0$
fluctuations mentioned above and will be affected to a much lesser extent by
the short-wavelength fluctuations. In fact, $G(1)$ differs from $G(0)$ by
exactly the nearest-neighbor fluctuations:
\begin{equation}
G(0,t) - G(1,t) = \frac12 \left\langle (h(y,t) - h(y+1,t))^2 \right\rangle.
\end{equation}
In Figure~\ref{g1k} we show these functions and include the width ($m=0$) for
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\caption{Time dependence of $G(m,t)$ for $W=1024$.
$m=0 (\Diamond)$, $m=1(+)$, $m=2(\Box )$. Line has slope 1/4.}\label{g1k}
\end{center}\end{figure}
comparison. Note that for $m= 1$ and $m=2$ that the ``effective growth
exponent,'' characterized by the slope of the tangent to this curve, is
actually greater than $.25$ and is decreasing.
For the width $m=0$ it is increasing.
As well as studying $G(m,t)$ for fixed $m$ and varying $t$, we considered the
complementary case of fixed $t$ and varying $m$. The results appear in
Figure~\ref{cortim1k}.
\begin{figure}\begin{center}
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\caption{Correlations along the shock at times 36 ($\diamond $),
133(+), and 662($\Box $).}\label{cortim1k}
\end{center}\end{figure}
The self-correlation $(m=0)$ is the square of the shock width.
As $G(m,t)$ decays exponentially in $m$ (for $m$ not too large), we can
define a correlation length along the shock, $\xi_{\parallel}(t)$, determined
by
\begin{equation}
G(m,t) \sim \exp [-m\xi_{\parallel}(t)]
\end{equation}
for small $m$. In Figure~\ref{corlen1k}
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\caption{Growth of correlation length. $W=1024$. Line has slope
1/2.}\label{corlen1k}
\end{center}\end{figure}
we show time time evolution of the correlation length. The asymptotic behavior
corresponds to a diffusive growth: $\xi_{\parallel} \sim t^{1/2}$; this is in
agreement with the evolution determined by equation~(\ref{eq:ravishan}). Note
that the initial behavior is much more rapid as the short wavelength
fluctuations develop.
\subsubsection{Linearized surface equation}
The short wavelength fluctuations in the ASEP are of the same order of
magnitude as the shock width and therefore mask its growth. We checked to see
if this behavior persisted in a discretized version of the linear (1d) KPZ
equation, where we know the asymptotics exactly. Consider the discretization
of (\ref{eq:ravishan}):
\begin{equation} \label{eq:linapp}
h(y,t+1)-h(y,t) = D [ h(y-1,t) -2 h(y,t) + h(y+1,t) ] + \gamma \zeta (y,t).
\end{equation}
The noise term $\zeta$ is Gaussian with covariance
\begin{equation}
\langle \zeta(y,t) \zeta(y',t')\rangle = \delta(y-y')\delta(t-t').
\label{eq:linnoise}
\end{equation}
We solved for time dependence of the shock width and $m=1$ correlations by
simulating the process governed by (\ref{eq:linapp}) and (\ref{eq:linnoise}).
The size of the system (corresponding to the length of the interface which is
the dimension of the ASEP perpendicular to the field, {\em i.e.}\ $W$) was
10000 sites, and we averaged over 50 independent samples. The diffusion
constant is $D=0.5$, and the noise amplitude is $\gamma =0.5$. These
parameters were chosen to compare with the results of the ASEP simulations.
The initial conditions on $h(y,t)$ were equivalent to what would result from
the initialization process outlined above with a density $\rho_0 = 0.5$.
What we found was qualitatively very similar to the simulations of the ASEP,
and is presented in Figure~\ref{phenom}.
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\caption{$G(m,t)$ for a discretized linear KPZ equation,
$m=0 (\diamond )$ and $m=1(+)$. Solid line has slope 1/4.}\label{phenom}
\end{center}\end{figure}
Namely, we observed an initial regime of fast growth in the nearest neighbor
correlation function $G(1,t)$ and an eventual tapering off of the effective
time-dependent exponent. Meanwhile, for $G(0,t)$ (shock width squared) we
found that the effective growth rate {\em increased} with time consistent with
an exponent $\beta = 1/4$.
\subsubsection{Compatibility with Existing Theory}
We have presented computer simulations of the two dimensional asymmetric
simple exclusion model. The growth of shock width in the ASEP is consistent
with the current theory, but the natural width inherent to the dynamics make
it difficult strengthen this claim. Since similar behavior is also observed
in simulations of the discretized interface equation, we are, however,
supportive of the linear theory.
\section{Stationary Results}
When we study the stationary state of our periodic model with a blockage, we
have an advantage over the time-dependent case in that we need not worry
about initial conditions---only the total particle number is relevant.
Thus we need not worry about the best way to produce a shock, {\em e.g.}\
whether we should use a checkerboard pattern or simple product measure.
However, this lessens the freedom we have in determining the type of shock
that results. Since a shock in the stationary state must have no net drift,
the densities on either side must be symmetric with respect to density 1/2
(the shock velocity, which must be zero in the stationary state, is given by
\(v_{\rm shock} = 1 - \rho_{\rm left} - \rho_{\rm right}\)). Thus the simple
technique of identifying the shock by the first particle in each row, valid
for $\rho_{\rm left} = 0$, will not be effective.
\subsection{Shock identity}
The difficulty of identifying the shock also appeared in the
one-dimensional case~\cite{JL}, where a so-called second class
particle~\cite{BCFG} was used to track the shock. The second class
particle is an extra particle added to the system, which is treated as a
hole in exchanges with particles and as a particle in exchanges with
holes; this does not change the dynamics of the original particles.
When the second class particle is in a high density region (of ordinary
particles), it is forced to the left by the particles jumping (to the
right) and landing on it; when it is in a low density region it moves to
the right.
In two dimensions one can also add second class particles to the system, but
as they can diffuse from row to row we no longer have a single second class
particle associated with the shock position in a given row. One possibility is
to add a large number of second class particles to the system and determine
their distribution, unfortunately a computationally intensive procedure.
The second class particle is actually a much more powerful tool than we need
for determining the shock position, so we discard it in favor of something
simpler. The motion of the second class particle is basically that of a
biased random walk with a drift towards the shock position; instead of second
class particles we introduce shadow particles whose dynamics is {\em exactly}
that of a biased random walk with a drift towards the shock position. These
shadow particles do not affect the motion of the ordinary particles but move
in a ``potential'' determined by the ASEP configuration. Specifically, after
each sweep of Monte-Carlo updates, each shadow particle (of which there is one
per row) moves according to the following rule:
\begin{equation}
h(y,t+1) = \left\{
\begin{array}{ll}
h(y,t)&\mbox{with probability $1/4$},\\[3pt]
h(y,t) - s(h(y,t),y; t) &\mbox{with probability $3/4$},
\end{array}\right.
\end{equation}
where $s(x,y; t) = 1$ if there is a particle at site $(x,y)$ at time $t$, and
$s(x,y; t) = -1$ if there is no particle there. Thus the shadow particles
move to the left in regions of high density and to the right in regions of low
density, driving them towards the shock where these regions meet. The
probabilities 1/4 and 3/4 were chosen simply for convenience; other similar
rules for the shadow particle evolution were tried but did not yield
significantly different behavior.
Note that we continue to label the location of the surface by $h(y,t)$, even
though our definition of this location has changed from the time-dependent
case, where we made use of the first particle position, since both definitions
represent the same physical idea.
Ideally, one would like to allow the ``shadow'' random walk to evolve for a
long time for each given ASEP configuration; under such conditions it is clear
that the shadow particles will accurately identify the shock position,
provided a shock does in fact exist. The ratio of one shadow update per ASEP
Monte Carlo step is a compromise between the need to accurately identify the
shock postion and the desire to reduce the computational load of tracking it.
\subsubsection*{Quantities studied}
We were interested in studying both the shock profile as well as its
fluctuations. To this end we computed the following quantities from the
sampled shock positions $h(y,t)$, where $\langle\cdot\rangle$ represents a
sampling ({\em i.e.}\ time) average:
\begin{itemize}
\item the average shock position
\(\displaystyle \langle\bar{h}\rangle = \left\langle \frac1W \sum_{y=1}^W
h(y,t) \right\rangle
= \frac1W \sum_{y=1}^W \langle h(y,t) \rangle
\),
\item the variance in the shock position
\(\displaystyle \langle\delta^2\rangle = \left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle
\),
\item the average shock width
\(\displaystyle \langle\sigma\rangle = \left\langle\left(\frac1W \sum_{y=1}^W
(h(y,t)-{\bar h}(t))^2 \right)^{1/2}\right\rangle
\),
\item the rms shock width
\(\displaystyle \langle\sigma^2\rangle^{1/2} = \left\langle\left(\frac1W
\sum_{y=1}^W (h(y,t)-{\bar h}(t))^2 \right)\right\rangle^{1/2}
\),
\item the truncated height-height correlation function
\(\displaystyle \langle G(m)\rangle = \left\langle h(0,t)h(m,t) - \bar{h}^2(t)
\right\rangle =
\left\langle \frac1W \sum_{y=1}^W \left(h(y,t) h(y+m,t) - \bar{h}^2(t)\right)
\right\rangle\).
\end{itemize}
\subsection{Results}
We attempted to determine the behavior of the shock width as we varied the
size of our system. We found that if either the system size $L$ or the system
width $W$ was taken very large, the shock width approached an asymptotic value.
Thus we were able to reduce our two parameter system to a single parameter, by
considering the regimes $L\gg W$ and $L\ll W$.
For $L\gg W$, the finite width of the system prevents the shock from growing
indefinitely. This behavior should correspond to growth saturation in the
time-dependent case. We present our data for a system with $\rho_{\rm avg} =
0.5$, $p_{\rm right} = 0.75$, $p_\perp = .125$ and $r=0.25$ in
Figure~\ref{width-vs-width}.
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\caption{Shock width {\em vs.}\ system width $W$ for $L\gg W$. Solid line
has slope 0.5.}
\label{width-vs-width}
\end{figure}
The error bars (three symbols are plotted for each measurement: the actual
value and that value shifted up or down by the error bound) represent
statistical error based on the approximate number of independent samples
selected from the steady state in each system. To fully saturate the shock
width required that we consider a system length of $3200$ for a system width
of $180$; typically we had $L\gtrsim 16W$.
We only plot the data for the rms shock width; the average shock width behaved
similarly. Along with $\langle\sigma^2\rangle^{1/2}$ we plot the nearest
neighbor correlation $\langle G(1)\rangle^{1/2}$ which should have the same
asymptotic behavior as the shock width. The difference between the two is
due to the short-wavelength fluctuations, just as in the time-dependent case.
It is clear that we are just beginning to access the asymptotic behavior; our
results are consistent with $\alpha=0.5$ but cannot be considered conclusive.
For $L\ll W$, the finite width of the system is irrelevant to the width of the
shock. In this case the system length $L$ determines the shock width. We
present our data for a system with $\rho_{\rm avg} = 0.5$, $p_{\rm right} =
0.75$, $p_\perp = .125$ and $r=0.125$ in Figure~\ref{width-vs-length}.
\begin{figure}
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\end{center}
\caption{Shock width {\em vs.}\ system length $L$ for $L\ll W$. Solid line
has slope 0.25.}
\label{width-vs-length}
\end{figure}
To reach the asymptotic shock width for the longest system, $L=360$, required
a system width of $720$; typically we had $W\gtrsim 2L$.
If one associates length with time, we should expect to see the same exponent
$\beta$ as we saw in the time-dependent case, {\em i.e.}
\begin{equation}
\langle\sigma^2\rangle^{1/2} \sim L^\beta \sim t^\beta \sim \sigma(t).
\end{equation}
Indeed, our results are consistent with $\beta=0.25$.
In an attempt to examine systems larger than $720\times 360$, we used the
random-sublattice updating technique described earlier. With a system width
of $12870$ we were able to examine lengths up to $800$ and be quite sure that
the finite width of the system was not affecting the shock width.
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\caption{Shock width {\em vs.}\ system length $L$ for the semi-parallel update
model. Solid line has slope 0.25.}
\label{width-vs-length-cray}
\end{figure}
We present the data from our semi-parallel model in
Figure~\ref{width-vs-length-cray}. The parameters are the same as for
Figure~\ref{width-vs-length}, and the data is in fact very similar.
We should be somewhat careful in interpreting the results from our
semi-parallel model, because there are differences from the serial model. In
Figure~\ref{hhcor} we plot the height-height correlation function $G$ for the
ASEP with length $L=64$ and width $W=256$, along with the semi-parallel model
with the same length.
\begin{figure}
\begin{center}
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\caption{Height-height correlation function.}
\label{hhcor}
\end{figure}
Although both systems have an initial exponential decay, the serial model has a
significantly larger ``dip'' of negatively correlated surface heights than
that which we see in the semi-parallel model.
The basic structure of the shock seems to be fairly impervious to the details
of the model. The scaling behavior of its intrinsic width, for example, does
not depend upon the overall density $\rho_{\rm avg}$. On the other hand, the
location $\bar{h}$ and the fluctuations of that location {\em do} depend upon
$\rho_{\rm avg}$, specifically, whether or not $\rho_{\rm avg}=0.5$. This is
the same behavior observed in the one-dimensional model, where the shock
fluctuations were reduced when there was particle-hole symmetry~\cite{JL}.
For a system where $L\gg W$, we would expect the system to be effectively
one-dimensional, and we recover the one-dimensional results~\cite{JL}. For
$\rho_{\rm avg} \ne 0.5$, we expect $\left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle^{1/2}$ to scale like $L^{1/2}$. In
Figure~\ref{fluc5625} we plot
\begin{figure}
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\caption{Rescaled shock fluctuation: $W^{1/2}\left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle^{1/2}$ {\em vs.}\ system length $L$.
Average density different from 1/2. The dotted line has slope 1/2.}
\label{fluc5625}
\end{figure}
the shock fluctuation, rescaled by the square root of the system width,
against the system length. The system parameters are $\rho_{\rm avg} =
0.5625$, $p_{\rm right}=0.75$, $p_\perp=.125$ and $r=0.25$. It is
clear from this data that
\begin{equation}
\langle\delta^2\rangle = \left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle \sim \frac{L}{W}.
\end{equation}
The collapse of the data shows that this scaling is valid even for quite
narrow systems ($W=4$). Thus the $\rho_{\rm avg} \ne 0.5$, two-dimensional
system behaves as a collection of $W$ independent one-dimensional systems, at
least as far as the overall fluctuations of the shock position is concerned.
The situation is different when $\rho_{\rm avg} = 0.5$. The only difference
between the data in Figure~\ref{fluc50} and Figure~\ref{fluc5625} is that in
Figure~\ref{fluc50} we have $\rho_{\rm avg} = 0.5$.
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\caption{Rescaled shock fluctuation: $W^{1/2}\left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle^{1/2}$ {\em vs.}\ system length $L$.
Average density is 1/2. Dotted lines have slope 1/4 and 1/3.}
\label{fluc50}
\end{figure}
Here the rescaled data for different system widths superimpose on each other
only for $L$ small (compared with $W$). Examining the data in
Figure~\ref{fluc50} carefully, we see that the fluctuations for each width
cross over from $L^{1/4}$ behavior to $L^{1/3}$ behavior, with the crossover
point increasing with $W$. The fact that the data superimposes for small $L$
indicates that in this regime fluctuations scale as $L^{1/4}/W^{1/2}$, while
the $L^{1/3}$ behavior scales with a power of $W$ different from $1/2$. To
get a better handle on this phenomenon, we plot the rescaled fluctuations
against the system width
in Figure~\ref{flucvw-res}. For $L\gg W$ the slope approaches $-1/6$, which
indicates that the fluctuations scale as $L^{1/3}W^{-1/2 -1/6} =
L^{1/3}/W^{2/3}$. While we do not have a complete understanding of this
exponent, its makes sense from general scaling arguments as the crossover from
$L^{1/4}$ to $L^{1/3}$ occurs when
\begin{equation}
L^{1/4}/W^{1/2} \sim L^{1/3}/W^{2/3},
\end{equation}
or when $L\sim W^2$---{\em i.e.}\ we observe one-dimensional behavior when
particles have a chance to diffuse across the entire width of the system.
\begin{figure}
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\caption{Rescaled shock fluctuation: $W^{1/2}\left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle^{1/2}$ {\em vs.}\ system width $W$.
Average density is 1/2.}
\label{flucvw-res}
\end{figure}
We can also think of the system at $L\gg W$ as a pseudo-one-dimensional system
where the noise is reduced from that in the one-dimensional ASEP because of
the average over the width. If we insert a noise intensity parameter $I$ into
the (one-dimensional) KPZ equation (\ref{KPZ1}), {\em i.e.}\ the fluctuating
Burgers equation~\cite{Medina,vB}, yielding
\begin{equation}
\frac{\partial h}{\partial t} = \nu\frac{\partial^2 h}{\partial x^2} +
\frac{\lambda}{2} \left(\frac{\partial h}{\partial x}\right)^2
+ I\zeta(x,t),
\end{equation}
then effectively $I\propto W^{-1/2}$. A scaling analysis of Amar and
Family~\cite{AF} indicates that fluctuations in $h$ are proportional to
$I^{4/3}$, so that we obtain fluctuations proportional to $W^{-2/3}$. This
factor multiplies $L^{1/3}$ which is the behavior of the fluctuations in the
one-dimensional system~\cite{JL,GP,Dit,CLS,vB}. This is confirmed by
simulations we have performed on the one-dimensional ASEP~\cite{inprogress}.
For hole density $\rho_h$ (=$1-\rho$) we find that the fluctuations in the
shock position scale as $\rho_h^{2/3} t^{1/3}$, which is consistent with
\cite{AF} since $I\sim \rho_h^2$.
This crossover point also makes sense from the point of view of understanding
the $L^{1/4}/W^{1/2}$ behavior of the fluctuations in the shock position, and
connects this with the $L^{1/4}$ behavior of the shock width. For $L\ll W^2$
treat each row of the system as an almost independent one-dimensional ASEP.
By ``almost'' we mean we allow coupling only through the total density in each
row; otherwise the rows are treated independently.
If a particular row has a density of $\rho_{\rm row}$, its (local) shock
position will have variance proportional to $|\rho_{\rm row} -
1/2|L$~\cite{JL}. The diffusive coupling between rows will produce
fluctuations in $\rho_{\rm row} - 1/2$ that are $O(L^{-1/2})$. Thus the
typical deviation of the local shock from the center of the system is
$\left(O(L^{-1/2})L\right)^{1/2} = O(L^{1/4})$. This is the correct
contribution to the shock width. The overall shock position is the average of
the local shock position; we are treating each of the $W$ local positions as
an independent random variable, so the standard deviation of the average is
$W^{-1/2}O(L^{1/4})$. Of course this neglects the $L^{1/3}$ term which must
eventually be larger.
Our results for the stationary model are summarized in Table~\ref{table}.
\begin{table}
\caption{Scaling behavior for the variance in shock position
$\langle\delta^2\rangle$ and shock width $\langle\sigma^2\rangle^{1/2}$.}
\label{table}
\[
\renewcommand{\arraystretch}{1.4}
\begin{array}{l||c|c|c|c|}
\cline{2-5}
&\multicolumn{2}{c|}{\rho_{\rm avg} = 1/2}
&\multicolumn{2}{c|}{\rho_{\rm avg} \ne 1/2}\\
\cline{2-5}
& L\gg W & L \ll W & L\gg W & L \ll W\\
\hline
\langle\delta^2\rangle & \left(L/W^2\right)^{2/3} &
\left(L/W^2\right)^{1/2}&\multicolumn{2}{c|}{L/W}\\
\hline
\langle\sigma^2\rangle^{1/2} & W^{1/2} & L^{1/4} & W^{1/2} & L^{1/4}\\
\hline
\end{array}
\]
\end{table}
\section{Discussion}
\subsection{Relationship Between Stationary and Time-Dependent Models}
Studying the time-dependent behavior of the ASEP is significantly
different from studying the stationary states of the ASEP with a
blockage. While there is no rigorous argument that the same exponents
should describe both the time-dependent and size-dependent scaling of
the shock width, it is not suprising to expect that they are the same,
considering that the underlying physics is identical in both models.
Namely, the transit time for traversal of the distance from block to
shock is of order $L$, so that if the time dependence is $t^\beta$,
the length dependence should be $L^\beta$. The experimental
correspondence is unmistakable---Figure~\ref{g1k}, where we plot the
correlations $G$ {\em vs.}\ time $t$, and
Figure~\ref{width-vs-length}, where we plot the correlations {\em
vs.}\ system length $L$, are virtually copies of each other.
Of course, our only interpretations of this phenomenon are heuristic.
Although it is clear that fluctuations due to the transit time will
produce $L^\beta$ behavior, we can not exclude the existence of
stronger noise sources which would overwhelm this behavior---which in
fact we do observe when particle-hole symmetry is broken.
\subsection{Difficulties}
The precise determination of scaling exponents, particularly small ones, is
often not a straightforward task. This is especially true if one is limited in
analyzing the range over which the scaling holds. In our case the constraints
on simulation run-time, due to lattice size and computer memory
considerations, proved crucial.
There were several options available to us which would have alleviated these
problems, but all involved tampering with the ASEP dynamics, usually by
destroying (or introducing spurious) correlations---small perturbations in
models such as the ASEP that have a conservation law but do not satisfy
detailed balance can have dramatic global effects~\cite{GLMS}. It was
precisely the ASEP, and not some approximate variant, that we wished to study.
As a result, we were forced to simulate a truly two-dimensional problem and
not just some restricted (effectively one dimensional) domain containing the
shock surface.
\subsection{Parallel {\em vs.}\ Serial Models}
The future of large-scale scientific computing will rely on massively parallel
computers and thus systems with parallel dynamics. As opposed to certain
disciplines where the utilization of parallelism has been difficult, there are
no inherent problems with the use of parallelism in physics---certainly ``real
world'' dynamics are parallel. However, although they may not accurately
represent the real universe, serial models are generally more amenable to
analytical study~\cite{Spohn}.
We attempted to utilize parallel models that differed only very slightly from
the original serial models. Even so, we observed behavior that was
significantly different in certain respects. Thus the parallel dynamics must
be checked carefully for consistency with the serial dynamics.
\subsection{Other Models}
\subsubsection{Non-infinite temperature}
The ASEP dynamics can be viewed as the infinite temperature limit of a driven
diffusive system where the jump rates of the particles depend on the local
environment. Models with temperature are clearly more realistic models than
those without it, and the immediate question is whether or not the behavior we
have observed is limited to the infinite temperature case.
In one dimension we have found no significant difference in the behavior
between infinite and finite temperature models. The addition of temperature
makes the determination of the dependence of the current on the density more
difficult, but once this has been done, the shock fluctuations scale as
predicted by the fluctuating Burgers equation (with the appropriate $J$ {\em
vs.}\ $\rho$ behavior)~\cite{inprogress}.
In two dimensions many driven diffusive systems exhibit a phase
transition~\cite{KLS,GMD};
the low temperature behavior is qualitatively different from the ASEP.
At higher temperatures, the behavior is at least qualitatively similar to the
ASEP, in that shocks form and we can have segregation perpendicular to the
field as in our stationary model~\cite{AL}, but we have no detailed data on
the behavior of the interface.
\subsubsection{Non-lattice models ({\em e.g.}\ molecular dynamics)}
In addition to studying more realistic lattice models, it would be interesting
to examine the behavior of models in continuous space. Specifically, we would
like to study via nonequilibrium molecular dynamics simulations~\cite{EM} the
statistics of shocks that form when a compressible fluid is forced to flow
through a pinched tube.
\subsection*{Acknowledgments}
We thank Henk van Beijeren and Herbert Spohn for their contributions to this
work. This work was conducted using the computational resources of the
Pittsburgh Supercomputing Center.
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\end{document}
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\definecolor{shadecolor}{gray}{0.95}
\setlength{\parindent}{0pt}
\pagestyle{fancy}
\setlength{\headheight}{15pt}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
% begin the document
\begin{document}
% actually insert the title.
\maketitle
%%%% Cite any collaboration here %%%
Cited collaboration: N/A
\begin{enumerate}
% 1
\item Say in Silicon Valley, 35\% of engineers program in Java and 28\% of the engineers who program in Java also program in C++. Furthermore, 40\% of engineers program in C++.
\begin{enumerate}[label=\alph*.]
\item What is the probability that a randomly selected engineer programs in Java and C++?
\item What is the conditional probability that a randomly selected engineer programs in Java given that they program in C++?
\end{enumerate}
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 2
\item A website wants to detect if a visitor is a robot or a human. They give the visitor five CAPTCHA tests that are hard for robots but easy for humans. If the visitor fails one of the tests, they are flagged as a robot. The probability that a human succeeds at a single test is 0.95, while a robot only succeeds with probability 0.3. Assume all tests are independent. The percentage of visitors on this website that are robots is 5\%; all other visitors are human.
\begin{enumerate}[label=\alph*.]
\item If a visitor is actually a robot, what is the probability they get flagged (the probability they fail at least one test)?
\item If a visitor is human, what is the probability they get flagged?
\item Suppose a visitor gets flagged. Using your answers from part (a) and (b), what is the probability that the visitor is a robot?
\item If a visitor is human, what is the probability that they pass exactly three of the five tests?
\item Building off of your answer from part (d), what is the probability that a visitor with unknown identity passes exactly three of the five tests?
\end{enumerate}
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 3
\item Say all computers either run operating system W or X. A computer running operating system W is twice as likely to get infected with a virus as a computer running operating system X. If 70\% of all computers are running operating system W, what percentage of computers infected with a virus are running operating system W?
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 4
\item The Superbowl institutes a new way to determine which team receives the kickoff first. The referee chooses with equal probability one of three coins. Although the coins look identical, they have probability of heads 0.1, 0.5 and 0.9, respectively. Then the referee tosses the chosen coin 3 times. If more than half the tosses come up heads, one team will kick off; otherwise, the other team will kick off. If the tosses resulted in the sequence H, T, H, what is the probability that the fair coin was actually used?
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 5
\item After a long night of programming, you have built a powerful, but slightly buggy, email spam filter. When you don't encounter the bug, the filter works very well, always marking a spam email as SPAM and always marking a non-spam email as GOOD. Unfortunately, your code contains a bug that is encountered 10\% of the time when the filter is run on an email. When the bug is encountered, the filter always marks the email as GOOD. As a result, emails that are actually spam will be erroneously marked as GOOD when the bug is encountered. Let $p$ denote the probability that an email is actually non-spam, and let $q$ denote the conditional probability that an email is non-spam given that it is marked as GOOD by the filter.
\begin{enumerate}[label=\alph*.]
\item Determine $q$ in terms of $p$.
\item Using your answer from part (a), explain mathematically whether $q$ or $p$ is greater. Also, provide an intuitive justification for your answer.
\end{enumerate}
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 6
\item Two cards are randomly chosen without replacement from an ordinary deck of 52 cards. Let $E$ be the event that both cards are Aces. Let $F$ be the event that the Ace of Spades is one of the chosen cards, and let $G$ be the event that at least one Ace is chosen.
\begin{enumerate}[label=\alph*.]
\item Compute $P(E \mid F)$.
\item Are $E$ and $F$ independent? Justify your answer using your response to part (a).
\item Compute $P(E \mid G)$.
\end{enumerate}
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 7
\item Your colleagues in a comp-bio lab have sequenced DNA from a large population in order to understand how a gene ($G$) influences two particular traits ($T_1$ and $T_2$). They find that $P(G) = 0.6$, $P(T_1\|G) = 0.7$, and $P(T_2\|G) = 0.9$. They also observe that if a subject does not have the gene $G$, they express neither $T_1$ nor $T_2$. The probability of a patient having both $T_1$ and $T_2$ given that they have the gene $G$ is 0.63.
\begin{enumerate}[label=\alph*.]
\item Are $T_1$ and $T_2$ conditionally independent given $G$?
\item Are $T_1$ and $T_2$ conditionally independent given $G^C$?
\item What is $P(T_1)$?
\item What is $P(T_2)$?
\item Are $T_1$ and $T_2$ independent?
\end{enumerate}
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 8
\item The color of a person's eyes is determined by a pair of eye-color genes, as follows:
\begin{itemize}
\item if both of the eye-color genes are blue-eyed genes, then the person will have blue eyes
\item if one or more of the genes is a brown-eyed gene, then the person will have brown eyes
\end{itemize}
A newborn child independently receives one eye-color gene from each of its parents, and the gene it receives from a parent is equally likely to be either of the two eye-color genes of that parent. Suppose William and both of his parents have brown eyes, but William's sister (Claire) has blue eyes. (We assume that blue and brown are the only eye-color genes.)
\begin{enumerate}[label=\alph*.]
\item What is the probability that William possesses a blue-eyed gene?
\item Suppose that William's wife has blue eyes. What is the probability that their first child will have blue eyes?
\end{enumerate}
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 9
\item Consider the following algorithm for betting in roulette (\url{https://en.wikipedia.org/wiki/Roulette}). At each round (``spin''), you bet \$1 on a color (``red'' or ``black''). If that color comes up on the wheel, you keep your bet AND win \$1; otherwise, you lose your bet.
\begin{enumerate}[label=\roman*.]
\item Bet \$1 on ``red''
\item If ``red'' comes up on the wheel (with probability 18/38), then you win \$1 (and keep your original \$1 bet) and you \textbf{immediately} quit (i.e., you do not do step (iii) below).
\item If ``red'' did not come up on the wheel (with probability 20/38), then you lose your initial \$1 bet. But, then you bet \$1 on ``red'' on \textit{each} of the next \textbf{two} spins of the wheel. After those two spins, you quit (no matter what the outcome of the next two spins).
\end{enumerate}
Let $X$ denote your ``winnings'' when you quit, i.e., the total amount of money won minus any amounts lost while playing. This value may be negative.
\begin{enumerate}
\item Determine $P(X > 0)$.
\item Determine $E[X]$. (Rhetorical question: Would you play this game?)
\end{enumerate}
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 10
\item See Problem Set \#2 for a full write-up of this problem. You will submits parts (a) and (b) as code in Gradescope, and write up answers to parts (c) and (d) (and optionally part (e)) in this document.
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 11
\item \textbf{[Written, Extra Credit]} Suppose we want to write an algorithm \texttt{fairRandom} for randomly generating a 0 or a 1 with equal probability (= 0.5). Unfortunately, all we have available to us is a function:
~~~~\texttt{def unknownRandom() -> int}
that randomly generates bits, where on each call a 1 is returned with some unknown probability $p$ that need not be equal to 0.5 (and a 0 is returned with probability $1 - p$).
Consider the following algorithm for \texttt{fairRandom}:
\begin{minted}[frame=single]{python}
def fairRandom():
r1, r2 = 0, 0 # set r1 = 0 and r2 = 0
while True:
r1 = unknownRandom()
r2 = unknownRandom()
if (r1 != r2): break
return r2
\end{minted}
\begin{enumerate}[label=\alph*.]
\item Show mathematically that \texttt{fairRandom} does indeed return a 0 or a 1 with equal probability.
\vfill
\begin{center}
\textcolor{gray}{\textit{Continued on next page\dots}}
\end{center}
\newpage
\item Say we want to simplify the function, so we write the \texttt{simpleRandom} function below. Would the \texttt{simpleRandom} function also generate 0's and 1's with equal probability? Explain why or why not. Determine P(\texttt{simpleRandom} returns 1) in terms of $p$.
\end{enumerate}
\begin{minted}[frame=single]{python}
def simpleRandom():
r1, r2 = 0, 0 # set r1 = 0 and r2 = 0
r1 = unknownRandom()
while True:
r2 = unknownRandom()
if (r1 != r2): break
return r2
\end{minted}
\begin{shaded}
\begin{answer}
\end{answer}
\end{shaded}
\newpage
% 12
\item \textbf{[Coding, Extra Credit]} Please submit your article as code through Gradescope. No need to write anything here.
\end{enumerate}
\end{document}
|
http://dlmf.nist.gov/28.10.E1.tex
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nist.gov
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crawl-data/CC-MAIN-2014-10/segments/1394011038777/warc/CC-MAIN-20140305091718-00063-ip-10-183-142-35.ec2.internal.warc.gz
| 52,769,727 | 680 |
\[\frac{2}{\pi}\int_{{0}}^{{\ifrac{\pi}{2}}}\mathop{\cos\/}\nolimits\!\left(2h%
\mathop{\cos\/}\nolimits z\mathop{\cos\/}\nolimits t\right)\mathop{\mathrm{ce}%
_{{2n}}\/}\nolimits\!\left(t,h^{2}\right)dt=\frac{A_{0}^{{2n}}(h^{2})}{\mathop%
{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}\mathop{%
\mathrm{ce}_{{2n}}\/}\nolimits\!\left(z,h^{2}\right),\]
|
http://dlmf.nist.gov/1.17.E16.tex
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nist.gov
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crawl-data/CC-MAIN-2017-22/segments/1495463607848.9/warc/CC-MAIN-20170524152539-20170524172539-00621.warc.gz
| 97,417,731 | 663 |
\[\mathop{\delta\/}\nolimits\!\left(x-a\right)=\int_{-\infty}^{\infty}\mathop{%
\mathrm{Ai}\/}\nolimits\!\left(t-x\right)\mathop{\mathrm{Ai}\/}\nolimits\!%
\left(t-a\right)\mathrm{d}t.\]
|
http://www.mathematik.tu-dortmund.de/lsiii/cms/bibtex/92917238.tex
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tu-dortmund.de
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crawl-data/CC-MAIN-2019-13/segments/1552912201885.28/warc/CC-MAIN-20190319032352-20190319054352-00420.warc.gz
| 330,492,440 | 809 |
@INPROCEEDINGS{DamanikOuazziHronTurek2009,
author = {Damanik, H. and Ouazzi, A. and Hron, J. and Turek, S.},
editor = {Ambrosio},
title = {A monolithic {FEM} approach for temperature and shear dependent viscosity in viscoelastic flow},
booktitle = {7th EUROMECH Solid Mechanics Conference},
publisher = {ACM Press},
year = {2009},
}
|
https://wiki.starship-factory.ch/Vereinskram/Formulare/schluesselvertrag.tex
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starship-factory.ch
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CC-MAIN-2022-05
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crawl-data/CC-MAIN-2022-05/segments/1642320305341.76/warc/CC-MAIN-20220128013529-20220128043529-00570.warc.gz
| 662,076,178 | 1,241 |
\documentclass[paper=A4,foldmarks=H,border=5pt]{scrlttr2}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{fancyvrb}
\usepackage[german]{babel}
\begin{document}
\begin{wrapfigure}{R}{0.2\textwidth}
\vspace{-30pt}
\includegraphics[width=.2\textwidth]{../Corporate_Identity/Briefvorlage/logo1farbig.pdf}
\end{wrapfigure}
\hspace{1cm}
\textbf{\huge Schlüsselantrag }\\[1.5em]
\begin{tabular}{l@{}p{2in}p{3in}}
Mitglied: &\dotfill&\\[2em]
Unterstützer:$\;\;$&(Name)&(Unterschrift)\\[1.5em]
&\dotfill&\dotfill\\[1.5em]
&\dotfill&\dotfill\\[1.5em]
&\dotfill&\dotfill
\end{tabular}
\vspace{4em}
Interesse: \dotfill\\[1em]
\dotfill
\vspace{2em}
\input{tmp.tex}
\vspace{3em}
\begin{tabular}{l@{}p{2.5in}}
Ort/Datum &\dotfill\\[2em]
Mitglied &\dotfill\\[2em]
Unterschrift&\dotfill
\end{tabular}
\vspace{1em}
\hspace{-5.5em} Cut here \dotfill
\vspace{1.5em}
\textbf{\huge Schlüsselquittung }
\vspace{0.5em}
Die Starship Factory hat das Schlüsseldepot von 50 CHF erhalten. Es wurde ein Schlüssel an das unten stehende Mitglied
ausgehändigt. Das Schlüsseldepot wird bei Rückgabe des intakten Schlüssels zurückgezahlt.
\vspace{1.5em}
\begin{tabular}{l@{}p{2.5in}}
Mitglied &\dotfill\\[1em]
Ort/Datum &\dotfill\\[2em]
Unterschrift&\dotfill
\end{tabular}
\end{document}
|
https://www.emis.de/journals/EJC/Volume_15/Abstracts/v15i1r31.abs.tex
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emis.de
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| 782,337,288 | 1,034 |
\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Xueliang Li and Yan Liu}
%
%
\medskip
\noindent
%
%
{\bf Nonexistence of Triples of Nonisomorphic Connected Graphs with Isomorphic Connected $P_3$-graphs}
%
%
\vskip 5mm
\noindent
%
%
%
%
In the paper ``Broersma and Hoede, {\it Path graphs}, J. Graph Theory
{\bf 13} (1989) 427-444", the authors asked a problem whether there is
a triple of mutually nonisomorphic connected graphs which have an
isomorphic connected $P_3$-graph. In this paper, we show that there is
no such triple, and thus completely solve this problem.
\bye
|
http://build.openvpn.net/doxygen/latex/structsettings__t.tex
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openvpn.net
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| 46,912,472 | 1,668 |
\hypertarget{structsettings__t}{}\subsection{settings\+\_\+t Struct Reference}
\label{structsettings__t}\index{settings\+\_\+t@{settings\+\_\+t}}
{\ttfamily \#include $<$service.\+h$>$}
\subsubsection*{Data Fields}
\begin{DoxyCompactItemize}
\item
T\+C\+H\+AR \hyperlink{structsettings__t_a42376d1376edba6dac152e505b5177db}{exe\+\_\+path} \mbox{[}M\+A\+X\+\_\+\+P\+A\+TH\mbox{]}
\item
T\+C\+H\+AR \hyperlink{structsettings__t_a7d3f6a2f9c73a9c41fff9bdff69c5d01}{config\+\_\+dir} \mbox{[}M\+A\+X\+\_\+\+P\+A\+TH\mbox{]}
\item
T\+C\+H\+AR \hyperlink{structsettings__t_ab73e9de01c93a6ca31ce2eb79a2cc8ad}{ext\+\_\+string} \mbox{[}16\mbox{]}
\item
T\+C\+H\+AR \hyperlink{structsettings__t_a088b83e62804f0717fcb0d67352b5c71}{log\+\_\+dir} \mbox{[}M\+A\+X\+\_\+\+P\+A\+TH\mbox{]}
\item
T\+C\+H\+AR \hyperlink{structsettings__t_a73cbc7e839af82bb046b2e0119873512}{ovpn\+\_\+admin\+\_\+group} \mbox{[}\hyperlink{service_8h_ac7c0207aa5a0e10d378be03b68041350}{M\+A\+X\+\_\+\+N\+A\+ME}\mbox{]}
\item
D\+W\+O\+RD \hyperlink{structsettings__t_ab0aa05b71ae5c8d3ed0b123dd1b830f3}{priority}
\item
B\+O\+OL \hyperlink{structsettings__t_a1ec5d7f57adfd0df6150c9aae42fb484}{append}
\end{DoxyCompactItemize}
\subsubsection{Detailed Description}
Definition at line 64 of file service.\+h.
\subsubsection{Field Documentation}
\mbox{\Hypertarget{structsettings__t_a1ec5d7f57adfd0df6150c9aae42fb484}\label{structsettings__t_a1ec5d7f57adfd0df6150c9aae42fb484}}
\index{settings\+\_\+t@{settings\+\_\+t}!append@{append}}
\index{append@{append}!settings\+\_\+t@{settings\+\_\+t}}
\paragraph{\texorpdfstring{append}{append}}
{\footnotesize\ttfamily B\+O\+OL settings\+\_\+t\+::append}
Definition at line 71 of file service.\+h.
Referenced by Get\+Openvpn\+Settings(), and Service\+Start\+Automatic().
\mbox{\Hypertarget{structsettings__t_a7d3f6a2f9c73a9c41fff9bdff69c5d01}\label{structsettings__t_a7d3f6a2f9c73a9c41fff9bdff69c5d01}}
\index{settings\+\_\+t@{settings\+\_\+t}!config\+\_\+dir@{config\+\_\+dir}}
\index{config\+\_\+dir@{config\+\_\+dir}!settings\+\_\+t@{settings\+\_\+t}}
\paragraph{\texorpdfstring{config\+\_\+dir}{config\_dir}}
{\footnotesize\ttfamily T\+C\+H\+AR settings\+\_\+t\+::config\+\_\+dir\mbox{[}M\+A\+X\+\_\+\+P\+A\+TH\mbox{]}}
Definition at line 66 of file service.\+h.
Referenced by Check\+Config\+Path(), Get\+Openvpn\+Settings(), and Service\+Start\+Automatic().
\mbox{\Hypertarget{structsettings__t_a42376d1376edba6dac152e505b5177db}\label{structsettings__t_a42376d1376edba6dac152e505b5177db}}
\index{settings\+\_\+t@{settings\+\_\+t}!exe\+\_\+path@{exe\+\_\+path}}
\index{exe\+\_\+path@{exe\+\_\+path}!settings\+\_\+t@{settings\+\_\+t}}
\paragraph{\texorpdfstring{exe\+\_\+path}{exe\_path}}
{\footnotesize\ttfamily T\+C\+H\+AR settings\+\_\+t\+::exe\+\_\+path\mbox{[}M\+A\+X\+\_\+\+P\+A\+TH\mbox{]}}
Definition at line 65 of file service.\+h.
Referenced by Get\+Openvpn\+Settings(), Handle\+Block\+D\+N\+S\+Message(), Run\+Openvpn(), and Service\+Start\+Automatic().
\mbox{\Hypertarget{structsettings__t_ab73e9de01c93a6ca31ce2eb79a2cc8ad}\label{structsettings__t_ab73e9de01c93a6ca31ce2eb79a2cc8ad}}
\index{settings\+\_\+t@{settings\+\_\+t}!ext\+\_\+string@{ext\+\_\+string}}
\index{ext\+\_\+string@{ext\+\_\+string}!settings\+\_\+t@{settings\+\_\+t}}
\paragraph{\texorpdfstring{ext\+\_\+string}{ext\_string}}
{\footnotesize\ttfamily T\+C\+H\+AR settings\+\_\+t\+::ext\+\_\+string\mbox{[}16\mbox{]}}
Definition at line 67 of file service.\+h.
Referenced by Get\+Openvpn\+Settings(), and Service\+Start\+Automatic().
\mbox{\Hypertarget{structsettings__t_a088b83e62804f0717fcb0d67352b5c71}\label{structsettings__t_a088b83e62804f0717fcb0d67352b5c71}}
\index{settings\+\_\+t@{settings\+\_\+t}!log\+\_\+dir@{log\+\_\+dir}}
\index{log\+\_\+dir@{log\+\_\+dir}!settings\+\_\+t@{settings\+\_\+t}}
\paragraph{\texorpdfstring{log\+\_\+dir}{log\_dir}}
{\footnotesize\ttfamily T\+C\+H\+AR settings\+\_\+t\+::log\+\_\+dir\mbox{[}M\+A\+X\+\_\+\+P\+A\+TH\mbox{]}}
Definition at line 68 of file service.\+h.
Referenced by Get\+Openvpn\+Settings(), and Service\+Start\+Automatic().
\mbox{\Hypertarget{structsettings__t_a73cbc7e839af82bb046b2e0119873512}\label{structsettings__t_a73cbc7e839af82bb046b2e0119873512}}
\index{settings\+\_\+t@{settings\+\_\+t}!ovpn\+\_\+admin\+\_\+group@{ovpn\+\_\+admin\+\_\+group}}
\index{ovpn\+\_\+admin\+\_\+group@{ovpn\+\_\+admin\+\_\+group}!settings\+\_\+t@{settings\+\_\+t}}
\paragraph{\texorpdfstring{ovpn\+\_\+admin\+\_\+group}{ovpn\_admin\_group}}
{\footnotesize\ttfamily T\+C\+H\+AR settings\+\_\+t\+::ovpn\+\_\+admin\+\_\+group\mbox{[}\hyperlink{service_8h_ac7c0207aa5a0e10d378be03b68041350}{M\+A\+X\+\_\+\+N\+A\+ME}\mbox{]}}
Definition at line 69 of file service.\+h.
Referenced by Get\+Openvpn\+Settings(), Run\+Openvpn(), and Validate\+Options().
\mbox{\Hypertarget{structsettings__t_ab0aa05b71ae5c8d3ed0b123dd1b830f3}\label{structsettings__t_ab0aa05b71ae5c8d3ed0b123dd1b830f3}}
\index{settings\+\_\+t@{settings\+\_\+t}!priority@{priority}}
\index{priority@{priority}!settings\+\_\+t@{settings\+\_\+t}}
\paragraph{\texorpdfstring{priority}{priority}}
{\footnotesize\ttfamily D\+W\+O\+RD settings\+\_\+t\+::priority}
Definition at line 70 of file service.\+h.
Referenced by Get\+Openvpn\+Settings(), Run\+Openvpn(), and Service\+Start\+Automatic().
The documentation for this struct was generated from the following file\+:\begin{DoxyCompactItemize}
\item
/root/openvpn/src/openvpnserv/\hyperlink{service_8h}{service.\+h}\end{DoxyCompactItemize}
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https://melusine.eu.org/syracuse/B/BaseCollege/Sixieme/diveucl/div/exo55.tex?enregistrement=ok
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eu.org
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| 464,117,670 | 1,070 |
%@Auteur: François Meria\par
Compléter le tableau suivant en mettant une croix (\og $\times$
\fg) dans la case correspondante.
\vskip 0.5cm
\begin{center}
\newcolumntype{Z}{>{\centering\arraybackslash}X}
\begin{tabularx}{160mm}{|Z||Z|Z|Z|Z|Z|}
\hline
& \multicolumn{5}{c|}{\texttt{divisible par}} \\
\cline{2-6}
& & & & & \\
\texttt{Nombre} & 2 & 3 & 5 & 9 & 10 \\
& & & & & \\\hline
\hline
& & & & & \\
15009 & & & & & \\ \hline
& & & & & \\
36270 & & & & & \\ \hline
& & & & & \\
39045 & & & & & \\ \hline
& & & & & \\
888840 & & & & & \\ \hline
& & & & & \\
362002 & & & & & \\ \hline
& & & & & \\
111222 & & & & & \\ \hline
\end{tabularx}
\end{center}
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https://ctan.math.washington.edu/tex-archive/info/examples/tlc2/8-8-3.ltx
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washington.edu
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| 247,771,748 | 1,149 |
%%
%% The LaTeX Companion, 2ed (second printing August 2004)
%%
%% Example 8-8-3 on page 509.
%%
%% 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}
\setlength\textwidth{111.0pt}
%%
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$A_B \neq \mathscr{A}_\mathscr{B} \neq \mathcal{A}_\mathcal{B}$
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https://tug.org/PSTricks/Examples/Logos/ponto.tex
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tug.org
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CC-MAIN-2023-14
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application/x-tex
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crawl-data/CC-MAIN-2023-14/segments/1679296948673.1/warc/CC-MAIN-20230327154814-20230327184814-00383.warc.gz
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https://faculty.math.illinois.edu/~murphyrf/teaching/M220-S2011/e3.tex
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illinois.edu
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CC-MAIN-2022-49
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\documentclass[11 pt]{article}
\pagestyle{empty}
\usepackage{graphicx}
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\begin{document}
\begin{center}
\textbf{MATH 220 \hfill Test 3 \hfill Spring 2011} \\
\end{center}
\vspace{.25in}
\noindent
\textbf{Name}\ \rule{3.5in}{.4pt}
\noindent
\vspace{0.05in}
\begin{center}
(circle your TA discussion section)
\end{center}
\vspace{0.03in}
\begin{tabular}{lllll}
$\triangleright$ & \textbf{AD1}, TR 1:00-1:50, Sarah Son & \hspace{0.2in} & $\triangleright$ & \textbf{AD2}, TR 1:00-1:50, Daniel Hockensmith\\
$\triangleright$ & \textbf{AD4}, TR 1:00-1:50, Sogol Jahanbekam & \hspace{0.2in} & $\triangleright$ & \textbf{AD5}, TR 2:00-2:50, Daniel Hockensmith\\
$\triangleright$ & \textbf{AD7}, TR 3:00-3:50, Ners\'{e}s Aramyan & \hspace{0.2in} & $\triangleright$ & \textbf{AD8}, MW 11:00-12:50, Austin Rochford\\
$\triangleright$ & \textbf{AD9}, MW 9:00-10:50, Ben Reiniger & \hspace{0.2in} & \\
\end{tabular}
\vspace{0.1in}
\hrulefill
\noindent
\begin{itemize}
\item Sit in your assigned seat (shown below).
\item Do not open this test booklet until I say \textit{START}.
\item Turn off all electronic devices and put away all items except a pen/pencil and an eraser.
\item You must show sufficient work to justify each answer.
\item While the test is in progress, we will not answer questions concerning the test material.
\item Quit working and close this test booklet when I say \textit{STOP}.
\item Quickly turn in your test to me or a TA and show your Student ID.
\end{itemize}
\hrulefill
\vspace{0.08in}
\begin{center}
\begin{scriptsize}
\begin{tabular}{ccccccccccccccccccccccccccc}
263 & 264 & 265 & 266 & 267 & 268 & 269 & 270 & $\bullet$ & 271 & 272 & 273 & & & & & 278 & 279 & $\bullet$ & 280 & 281 & 282 & 283 & 284 & 285 & 286 & 287 \\[6pt]
& 240 & 241 & 242 & 243 & 244 & 245 & 246 & $\bullet$ & 247 & 248 & 249 & 250 & 251 & 252 & 253 & 254 & 255 & $\bullet$ & 256 & 257 & 258 & 259 & 260 & 261 & 262 & \\[6pt]
& 217 & 218 & 219 & 220 & 221 & 222 & 223 & $\bullet$ & 224 & 225 & 226 & 227 & 228 & 229 & 230 & 231 & 232 & $\bullet$ & 233 & 234 & 235 & 236 & 237 & 238 & 239 & \\[6pt]
& 194 & 195 & 196 & 197 & 198 & 199 & 200 & $\bullet$ & 201 & 202 & 203 & 204 & 205 & 206 & 207 & 208 & 209 & $\bullet$ & 210 & 211 & 212 & 213 & 214 & 215 & 216 & \\[6pt]
& 171 & 172 & 173 & 174 & 175 & 176 & 177 & $\bullet$ & 178 & 179 & 180 & 181 & 182 & 183 & 184 & 185 & 186 & $\bullet$ & 187 & 188 & 189 & 190 & 191 & 192 & 193 & \\[6pt]
& 148 & 149 & 150 & 151 & 152 & 153 & 154 & $\bullet$ & 155 & 156 & 157 & 158 & 159 & 160 & 161 & 162 & 163 & $\bullet$ & 164 & 165 & 166 & 167 & 168 & 169 & 170 & \\[6pt]
& $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & 139 & 140 & 141 & 56 & 143 & 144 & 13 & 146 & 147 & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & \\[6pt]
& 116 & 117 & 118 & 119 & 120 & 121 & 122 & $\bullet$ & 123 & 124 & 125 & 126 & 127 & 132 & 145 & 130 & 131 & $\bullet$ & 16 & 133 & 134 & 135 & 136 & 137 & 138 & \\[6pt]
& 93 & 94 & 95 & 96 & 97 & 98 & 99 & $\bullet$ & 100 & 101 & 102 & 103 & 128 & 105 & 106 & 107 & 108 & $\bullet$ & 109 & 110 & 111 & 112 & 113 & 114 & 115 & \\[6pt]
& 70 & 71 & 72 & 73 & 74 & 75 & 76 & $\bullet$ & 77 & 78 & 79 & 80 & 81 & 82 & 83 & 84 & 85 & $\bullet$ & 86 & 87 & 88 & 89 & 90 & 91 & 92 & \\[6pt]
& 47 & 48 & 49 & 50 & 51 & 52 & 53 & $\bullet$ & 54 & 55 & 104 & 57 & 58 & 59 & 60 & 61 & 62 & $\bullet$ & 63 & 64 & 65 & 66 & 67 & 68 & 69 & \\[6pt]
& 24 & 25 & 26 & 27 & 28 & 29 & 30 & $\bullet$ & 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & $\bullet$ & 40 & 41 & 42 & 43 & 44 & 45 & 46 & \\[6pt]
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & $\bullet$ & & & & & & & & & & $\bullet$ & 17 & 18 & 19 & 20 & 21 & 22 & 23 & \\[6pt]
\end{tabular}
\end{scriptsize}
\vspace{0.15in}
\fbox{FRONT OF ROOM -- 314 Altgeld Hall}
\end{center}
\newpage
\noindent
\begin{enumerate}
\item (8 points) Determine an appropriate linear approximation of the function $\D{f(x)=\sqrt{x}}$ and use it to approximate $\D{\sqrt{26.3}}$. Write your answer in decimal form.
\vfill
\item (6 points) Precisely state \textit{Rolle's Theorem}.
\vfill
\newpage
\item (8 points) Evaluate the following limit. Be sure to use proper notation throughout your evaluation of this limit. Simplify your answer.\\[0.2in]
$\D{\lim_{n \to \infty} \sum_{k=1}^{n} \left( \frac{17}{4n} - \frac{5k}{2n^2} \right)}$
\vfill
\item (12 points) Suppose $f$ is an even function, $g$ is an odd function, and $f$ and $g$ are each integrable on the interval $[-3,3]$. Given that $\D{\int_{0}^{3} f(x) \, dx = 5 \,}$ and $\D{\, \int_{0}^{3} g(x) \, dx = 4}$, evaluate the following definite integrals.
\begin{enumerate}
\item $\D{\int_{3}^{0} g(x) \, dx}$
\vspace{0.2in}
\item $\D{\int_{3}^{3} f(x) \, dx}$
\vspace{0.2in}
\item $\D{\int_{-3}^{3} \left(2f(x) + 4g(x)\right) \, dx}$
\vspace{0.5in}
\item $\D{\int_{-3}^{3} \left(4 + \left(g(x)\right)^5\right) \, dx}$
\vspace*{0.2in}
\end{enumerate}
\newpage
\item (9 points each) Evaluate the following definite integrals. Simplify each answer.
\begin{enumerate}
\item $\D{\int_{\pi/3}^{\pi/2} \left( 12+6\sin{x} \right) \, dx}$
\vfill
\item $\D{\int_{0}^{2} \left(6x^2 + 3e^{-x}\right) \, dx}$
\vfill
\end{enumerate}
\newpage
\item (8 points each) Evaluate the following indefinite integrals.
\begin{enumerate}
\item $\D{\int \frac{6x^3+4x^2+5x}{x^2} \, dx}$
\vfill
\item $\D{\int \frac{1}{x\sqrt{\ln{x}}} \, dx}$
\vfill
\newpage
\item $\D{\int \tan^5{x} \sec^4{x} \, dx}$
\vfill
\end{enumerate}
\item (6 points) Evaluate the following indefinite integral.\\[0.2in]
$\D{\int x^2 \sqrt{x + 1} \, dx}$
\vfill
\newpage
\item (6 points each) The intersection points on the graphs of $\D{f(x) = x^2 + 2}$ and $\D{g(x) = 3x + 6}$ occur at $x=-1$ and at $x=4$. Let \textbf{R} be the finite region bounded by the graphs of $f(x)$ and $g(x)$. Set up, but do not evaluate, definite integrals which represent the given quantities. Use proper notation.
\begin{enumerate}
\item The area of \textbf{R}.
\vfill
\item The volume of the solid obtained when \textbf{R} is revolved around the vertical line $x=10$.
\vfill
\item The volume of the solid obtained when \textbf{R} is revolved around the $x$-axis.
\vfill
\end{enumerate}
\end{enumerate}
\newpage
\textbf{Students -- do not write on this page!}
\vspace{0.5in}
\begin{enumerate}
\item[1.] (8 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[2.] (6 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[3.] (8 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[4.] (12 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[5a.] (9 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[5b.] (9 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[6a.] (8 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[6b.] (8 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[6c.] (8 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[7.] (6 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[8a.] (6 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[8b.] (6 points) \hrulefill \hspace{4in} \vspace{0.15in}
\item[8c.] (6 points) \hrulefill \hspace{4in} \vspace{0.15in}
\end{enumerate}
\vspace{0.3in}
\textbf{TOTAL (100 points) \hrulefill \hspace{4in}} \vspace{0.15in}
\end{document}
|
http://dlmf.nist.gov/9.6.E9.tex
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nist.gov
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CC-MAIN-2017-17
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application/x-tex
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| 98,161,754 | 728 |
\[\mathop{\mathrm{Bi}\/}\nolimits'\!\left(-z\right)=(z/\sqrt{3})\left(\mathop{J_%
{-2/3}\/}\nolimits\!\left(\zeta\right)+\mathop{J_{2/3}\/}\nolimits\!\left(%
\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}\mathop{{H^{(1)}_%
{2/3}}\/}\nolimits\!\left(\zeta\right)+e^{-\pi i/3}\mathop{{H^{(2)}_{2/3}}\/}%
\nolimits\!\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/3%
}\mathop{{H^{(1)}_{-2/3}}\/}\nolimits\!\left(\zeta\right)+e^{\pi i/3}\mathop{{%
H^{(2)}_{-2/3}}\/}\nolimits\!\left(\zeta\right)\right).\]
|
https://pharmascope.org/ijrps/article/download/1912/3125
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pharmascope.org
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\begin{document}
\def\RunningAuthor{Dharmajit Pattanayak et al}
\firstPage{899}
\articleType{Original Article}
\receivedDate{05.07.2019}
\acceptedDate{18.10.2019}
\revisedDate{09.10.2019}
\journalVolume{2020, 11}
\journalIssue{1}
\journalDoi{ijrps.v11i1.1912}
\copyrightYear{2020}
\def\authorCount{4}
\def\affCount{2}
\def\journalTitle{International Journal of Research in Pharmaceutical Sciences}
\title{Formulation and Evaluation of Floating and Mucoadhesive tablets containing Glipizide.}
\author{Jagdish~K~Arun\textsuperscript{1},
Dharmajit~Pattanayak\textsuperscript{*}\textsuperscript{1},
Shrivastava~B\textsuperscript{1},
Ramesh~Adepu\textsuperscript{2}~\\[5pt]{\textsuperscript{1}School of Pharmaceutical Sciences, Jaipur National University, Jaipur, Rajasthan, India }~\\{\textsuperscript{2}Department of Pharmaceutics, Vikas College of Pharmaceutical Sciences, Suryapet, TS, India}}
\begin{abstract}
In the present study, Glipizide, a drug mainly preferred for type-II diabetes, is formulated in the form of floating mucoadhesive tablets to improve its bioavailability. Hydroxy Propyl Methyl Cellulose K200M, Sodium Carboxy Methyl Cellulose, Carbopol 974P, Karaya gum, Chitosan, and Xanthan gum were used as mucoadhesive polymers in designing of the floating mucoadhesive tablets. Different proportions of glipizide and polymer were used to prepare tablets. Pre-compression evaluation studies evaluated the powder blend of Glipizide mucoadhesive tablets (Pre-compression blend). It concluded that the blend had good flow property and better compressibility by interpreting the data obtained from the test. Hence the floating mucoadhesive tablets were prepared by direct compression technique. The results of floating lag time, and buoyancy studies suggested that formulations had a satisfactory floating ability. The release profile of the active pharmaceutical ingredient (glipizide) from the prepared dosage form indicated a controlled and enhanced drug release for a period of 12hrs. \textit{An in-vivo} study done for selected formulation. By the interpretation of data obtained from all the evaluation studies (Pre-compression test, floating property, drug release profile, \& \textit{in-vivo} study) concluded that formulation GF8 containing drug: Carbopol 974P (1:2) was optimized. The drug release kinetics of the formulation GF8 followed the Higuchi model with a regression value of 0.993.
\end{abstract}\def\keywordstitle{Keywords}
\begin{keywords}Glipizide,\newline Mucoadhesive,\newline Floating,\newline Mucoadhesive polymers
\end{keywords}
\twocolumn[ \maketitle {\printKwdAbsBox}]
\makeatletter\textsuperscript{*}Corresponding Author\par Name:\ Dharmajit~Pattanayak~\\ Phone:\ 9533727372~\\ Email:\ [email protected]
\par\vspace*{-11pt}\hrulefill\par{\fontsize{12}{14}\selectfont ISSN: 0975-7538}\par%
\textsc{DOI:}\ \href{https://doi.org/10.26452/\@journalDoi}{\textcolor{blue}{\underline{\smash{https://doi.org/10.26452/\@journalDoi}}}}\par%
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\section{Introduction}
The most acceptable route of drug administration is the oral route, which gives more patient compliance with systemic drug release. The majority of drugs (\ensuremath{\geq } 90\%) are administered through an oral dosage form. In conventional dosage form, the formulation resides in the stomach for a limited period. Due to these properties, the active pharmaceutical ingredients having an absorption window in stomach hampered the bioavailability. Oral controlled dosage form represents the widest advisable dosage form to design the controlled and prolonged drug action to achieve various benefits and minimizing the cons of conventional therapy. Conventional dosage form gives different patterns of drug release rate, which is majorly affected by gastric residence time (GRT) / gastric emptying time (GET) for formulations. \unskip~\citep{632472:14811446,632472:14811439,632472:14811422}.\ensuremath{^{}}
Gastric emptying time for tablets (Formulation) can be modified by altering the different parameters of tablets like density, size, and gas generating agents, which help the dosage form to retain in the stomach for a prolonged interval of time \unskip~\citep{632472:14811426}. The gastro retentive dosage form is the most advisable technique to enhance the residence time of dosage form in the stomach \unskip~\citep{632472:14811430} .Gastro retentive drug delivery system (GRDDS) can be elaborated as a modified technique of dosage form which can remain in the upper region of gastrointestinal track(GIT) for a long duration of time by altering the gastric emptying time as well as release the drug for a systemic and controlled manner, and then metabolized \unskip~\citep{632472:14811450}.In the present scenario, the different approaches for the gastro retentive drug delivery system are established to enhance the upper GI residence time. The primary objective of the gastro retentive drug delivery system is to overcome the problems related to other oral drug delivery systems, which will be more beneficial towards the patients . \unskip~\citep{632472:14811441,632472:14811431,632472:14811436}.\ensuremath{^{}}
There are different parameters listed out which effects on the GRT of a dosage form among theme one parameter is 'fluid level.' The fluid level in the stomach is not constant always. This creates a problem for GRDDS to float for a desired duration of time and to give drug release in a controlled and systemic way in the stomach \unskip~\citep{632472:14811432}.\ensuremath{^{}} GRDDS is effective for a drug which is having high absorption properties in the stomach or for the drug, which unstable in alkaline pH due to poor solubility and a narrow of absorption window \unskip~\citep{632472:14811442}. The GRDDS have low bulk density than fluids of upper GIT due to which it floats for a prolong period without hampered by gastric emptying time. Due to prolong the floating time in the fluid of upper GIT, the desired amount of drug can release from the dosage form slowly \unskip~\citep{632472:14811449}. So to overcome the problems of low gastric retention time, a new delivery system is designed, which is a combination of the Floating \& Mucoadhesive technique. In the present work Glipizide, an antidiabetic drug is formulated with a different type of controlled release and mucoadhesive polymers in different concentrations to optimize a formulation, which will help to overcome the above-said problems.
\section{Materials and Methods}
Glipizide was obtained from Triveni Chemicals through the supplier. The polymers like HPMC K200M \unskip~\citep{632472:14811443}, sodium carboxymethylcellulose (NaCMC), Carbopol 974P, Karaya gum, Chitosan, Xanthan gum and other remaining excipients like sodium bicarbonate, magnesium stearate, talc, lactose too obtained from S. D Fine Chemicals. The remaining additives utilized in this work belong to the laboratory scale.
\textbf{Precompression evaluation: }
\textbf{Solubility Studies}
The solubility of Glipizide ,\unskip~\citep{632472:14811427,632472:14811433},was studied in 0.1N HCL (pH 1.2) solution by phase equilibrium method. In a 20 ml vial, 10 mL above resolution, an excessive amount of drug was taken. The above vials were sealed by closures (rubber caps) and mixed properly by rotary shaker for overnight at the reasonable climatic condition. After that, the drug solution was passed into 0.2 \ensuremath{\mu}m Whatmann's paper. Followed by it was scanned through UV spectrophotometer 227nm. The calibration curve for Glipizide was done using the above acidic solution, and the solubility of Glipizide was estimated from the slope of the calibration curve. The same method was carried out for 3 times to find out the mean of the result.
\textbf{Drug-excipient compatibility studies }
\textbf{Fourier transform infrared spectroscopic studies: }
FT-IR spectrophotometer was performed to check the compatibility between the drug-excipient by the non-thermal analysis. The spectrum for the sample was scanned in the frequency of 450-4000 cm\ensuremath{^{-1}}.
\textbf{Pre-compression Evaluation}
Preformulation study is a group of studies which deals with the physicochemical parameter of the drug, also helps in designing of dosage form, and also provides an outline for the selection of pharmaceutical additives or excipients.
\textbf{Compressibility index}
It reflects the assessment for inter particulate interactions of powder. The compressibility index (percentage compressibility) of the API was calculated by using the following formula.
\begin{eqnarray*}\style{font-size:10px}{I\;=(\frac{DT\;-\;Db\;}{DT})100} \end{eqnarray*}
Where, I = Compressibility index
Dt= Tapped density of sample.
D\ensuremath{_{b}}= Bulk density of the sample.
\textbf{Hausner's ratio}
It reflects the flow properties of the powder sample and is calculated by the following formula
\begin{eqnarray*}\style{font-size:10px}{I\;=(\frac{DT\;-\;Db\;}{DT})100} \end{eqnarray*}
Where, H =Hausner's ratio
D\ensuremath{_{t}}= Tapped density of the sample.
D\ensuremath{_{b}}= Bulk density of the sample.
\textbf{Angle of repose}
It also reflects the flow property of the powder sample and determined from the height and radius of the pile obtained by the powder sample. It is expressed as\textbf{\space }
\begin{eqnarray*}\style{font-size:10px}{\theta\;=\tan-1\left(\frac hr\right)} \end{eqnarray*}
Where\textbf{, }\ensuremath{\theta }= angle of repose
h = height in cm
r = radius in cm.
\textbf{Preparation of Floating mucoadhesive tablets}
The direct compression method is opted to formulate Floating mucoadhesive tablets containing Glipizide; Different batches were developed by changing the ratio of HPMC K200M, Na CMC, Carbopol 974P, Karaya gum, Chitosan, and Xanthan gum. Sodium bicarbonate is helped to float the tablets. Talc and magnesium stearate and lactose are used as a lubricant, glidant, and diluent, respectively. The drug, polymers, sodium bicarbonate, and lactose were correctly mixed for 15 min until they formed a homogeneous mixture. Followed by talc and Magnesium Stearate are added as lubricating agents. The above powder mixture was combined homogeneously by using a polyethylene bag. Finally, the tablets were prepared by a 6 mm diameter to die in a 9-station rotary punching machine (Lab Press, India). Table~\ref{tw-a5c56f171cdf} shows the different formulation approaches\unskip~\citep{632472:14811438}.
\textbf{\space }
\begin{table*}[!htbp]
\caption{\boldmath {\textbf{The Composition of Floating Mucoadhesive Tablets Of Glipizide}} }
\label{tw-a5c56f171cdf}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.12350000000000001\linewidth-2\tabcolsep}p{\dimexpr.05430000000000001\linewidth-2\tabcolsep}p{\dimexpr.05369999999999997\linewidth-2\tabcolsep}p{\dimexpr.05289999999999996\linewidth-2\tabcolsep}p{\dimexpr.05549999999999999\linewidth-2\tabcolsep}p{\dimexpr.05340000000000003\linewidth-2\tabcolsep}p{\dimexpr.04849999999999998\linewidth-2\tabcolsep}p{\dimexpr.04520000000000004\linewidth-2\tabcolsep}p{\dimexpr.05109999999999998\linewidth-2\tabcolsep}p{\dimexpr.0531\linewidth-2\tabcolsep}p{\dimexpr.0484\linewidth-2\tabcolsep}p{\dimexpr.04279999999999999\linewidth-2\tabcolsep}p{\dimexpr.0448\linewidth-2\tabcolsep}p{\dimexpr.0489\linewidth-2\tabcolsep}p{\dimexpr.046800000000000015\linewidth-2\tabcolsep}p{\dimexpr.04409999999999999\linewidth-2\tabcolsep}p{\dimexpr.0446\linewidth-2\tabcolsep}p{\dimexpr.045600000000000005\linewidth-2\tabcolsep}p{\dimexpr.04279999999999999\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Ingredients & GF \mbox{}\protect\newline 1 & GF \mbox{}\protect\newline 2 & GF \mbox{}\protect\newline 3 & GF \mbox{}\protect\newline 4 & GF \mbox{}\protect\newline 5 & GF \mbox{}\protect\newline 6 & GF \mbox{}\protect\newline 7 & GF \mbox{}\protect\newline 8 & GF \mbox{}\protect\newline 9 & GF \mbox{}\protect\newline 10 & GF \mbox{}\protect\newline 11 & GF \mbox{}\protect\newline 12 & GF \mbox{}\protect\newline 13 & GF \mbox{}\protect\newline 14 & GF \mbox{}\protect\newline 15 & GF \mbox{}\protect\newline 16 & GF \mbox{}\protect\newline 17 & GF \mbox{}\protect\newline 18\\
\tblmidrule
Glipizide &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5 &
5\\
HPMC \mbox{}\protect\newline K200 M &
5 &
10 &
15 &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
-\\
Na CMC &
- &
- &
- &
5 &
10 &
15 &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
-\\
Carbopol \mbox{}\protect\newline 974P &
- &
- &
- &
- &
- &
- &
5 &
10 &
15 &
- &
- &
- &
- &
- &
- &
- &
- &
-\\
Karaya gum &
- &
- &
- &
- &
- &
- &
- &
- &
- &
5 &
10 &
15 &
- &
- &
- &
- &
- &
-\\
Chitosan &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
5 &
10 &
15 &
- &
- &
-\\
Xanthan gum &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
- &
5 &
10 &
15\\
NaHCO3 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10 &
10\\
Magnesium stearate &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4 &
4\\
Talc &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3 &
3\\
Lactose &
73 &
68 &
63 &
73 &
68 &
63 &
73 &
68 &
63 &
73 &
68 &
63 &
73 &
68 &
63 &
73 &
68 &
63\\
Total Weight &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100 &
100\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\textbf{\textbf{Post- compression Evaluation}}
\textbf{Physicochemical characterization for prepared formulation}
The formulated Glipizide Floating mucoadhesive tablets were studied for the \textbf{\textbf{\space }}following mentioned test \textbf{, }
\begin{enumerate}
\item \relax Weight variation
\item \relax Thickness
\item \relax Hardness
\item \relax Friability
\item \relax Drug Content
\end{enumerate}
\textbf{Weight variation}
This test is carried out by a random selection of 20 tablets and followed by weighed accurately. The mean weight of 20 tablets calculated and followed by compared with the weight of the tablet individually. \unskip~\citep{632472:14811444,632472:14811424,632472:14811423} .Not more than two tablets should not deviate from the average weight by \ensuremath{\pm} 10 \%. The percent deviation was calculated as follows:
\% Deviation =
\begin{eqnarray*}\style{font-size:10px}{(\frac{Individual\;weight\;-\;Mean\;weight\;}{Mean\;weight})\times100} \end{eqnarray*}
\textbf{Tablet Thickness}
From the production level, physical shape (thickness and diameter) for the tablets was monitored properly. Various parameters have an impact on thickness like compression force, the configuration of the die, and the rotation of machine per minute (RPM) of the compression machine. Hence these criterions are essential for acceptance of formulation, tablet uniformity, and packaging. The Digital Vernier caliper\ensuremath{^{}} was utilized to examine the physical shape of the tablets \unskip~\citep{632472:14811448}. The tablets (10 tablets for each formulation) were selected randomly, and the mean was calculated. The standard deviation for thickness was calculated.
\textbf{Tablet Hardness}
The hardness of a dosage form explains as a force necessitates / adequate to break it in two parallel plates. Tablets need durable resistance power for additional mechanical shocks. Every batch of pills was taken (6 tablets), and hardness was estimated by Monsanto hardness tester, and the average was calculated \unskip~\citep{632472:14811445}. It is expressed in Kg/cm\ensuremath{^{2}}\textit{.}
\textbf{Friability}
The hardness of a tablet is not sufficient to express the resistance due to loosening their crown positions during compression. Accordingly, to cross-check the strength of the tablet, another measure for the tablet was proposed i.e., friability (Roche friability). A set of pills that were selected for the test is to get the mechanical force (shock and abrasion). Roche friability, which rotates at 25 rpm speed for 4 minutes. \unskip~\citep{632472:14811429}.
The initial weight of tablets was noted down before the test. After the test, the pills were then de-dusted and reweighed. They are finally expressed in the percentage of friability.
Friability (\%) =
$\style{font-size:8px}{\begin{array}{l}\begin{array}{l}\frac{Initial\;weight\;of\;10\;tablets\;{\textendash}\;final\;weight\;of\;10\;tablets\;(W)}{The\;initial\;weight\;of\;10\;tablets\;(Wo)}\end{array}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times100\end{array}} $
Where, W\ensuremath{_{o}} is the initial weight of the tablets (Preweighed)
W is the final weight of the tablets (Reweighed)\textbf{\space }
\textbf{ Assay}
Randomized selection of tablets (6 Tablet for each formulation) was carried out to find out the percentage of active pharmaceutical ingredient available in each formulation. Equivalent to the weight of each formulation, the powder was taken and dissolved in 100 ml of 0.1N HCl by stirring for 10 min. The membrane filter (0.45 \ensuremath{\mu }) was used for straining the above solution and diluted suitably followed by measuring absorbance by using a UV-Visible spectrophotometer at 237nm using pH 6.8 phosphate buffers \unskip~\citep{632472:14811447}.
\textit{\textbf{In vitro}} \textbf{ Buoyancy studies}
It mainly reflects two parameters i.e., floating lag time and total floating time. These two values are finding out by putting formulation in 0.1N HCL. The Floating Lag Time (FLT) is denoted as the time taken by the formulation to reach the top of the medium, and the Total Floating Time (TFT) was determined by noting down the duration of floating time \unskip~\citep{632472:14811442}.
\textit{\textbf{In vitro }} \textbf{release studies}
The release study was carried out by the USP type II dissolution test apparatus. In Floating mucoadhesive tablets were release the medicaments from a single surface, so to maintain an \textit{in vitro} gastric condition, the formulation was partially covered by an impermeable membrane and followed by it was adhered to a glass slide (2x2 cm) cyanoacrylate adhesive. Then the slide was placed in the 900 ml of pH 1.2 HCl buffer, and paddle speed was 50 rpm at a temperature of 37 \ensuremath{\pm} 0.5 \ensuremath{^\circ}C. A fixed amount of test solution withdrawn at fixed time intervals up to 12 h and scanned by spectra at 237nm after appropriate dilution \unskip~\citep{632472:14811451,632472:14811421}.
\textit{\textbf{\space }} \textit{\textbf{In vitro }} \textbf{bioadhesion strength}
Ultra Test Tensile strength tester was used to measure the adhesion strength of tablets. 25 kg load cell was set. A membrane was attached to an adaptor, and the Floating mucoadhesive formulation (sample tablet) was attached to adaptor having a similar size using a by adhesive. 100 \ensuremath{\mu}l of 1\% w/v mucin solution was applied on the membrane, and immediately the formulation was allowed to come in contact with the mucosa. After a certain period, the upper adapter was withdrawn at 0.5 mm/sec until the tablet was completely separate from the membrane. The area under the force-distance curve was helped to determine the work of adhesion.
Force of adhesion =
\begin{eqnarray*}\style{font-size:10px}{{\;(\frac{Bioadhesion\;strength\;}{\;\;\;\;\;\;\;\;\;\;\;\;\;1000\;\;\;\;\;\;\;\;\;\;\;\;})\times}9\cdot8} \end{eqnarray*}
Bond strength =
\begin{eqnarray*}\style{font-size:10px}{\frac{Force\;of\;adhesion}{Surface\;area}} \end{eqnarray*}
\textbf{Moisture absorption}
Before the study Floating, mucoadhesive tablets (6 from each formulation) were dehumidified by vacuum oven, if any, followed by immediately partially covered with a water-insoluble backing membrane. The above tablets were exposed to the surface of the agar media (5\% m/v) for 1h. \unskip~\citep{632472:14811425}. After the completion of the specified time, the weight of formulation was noted moisture absorption was calculated:
\% Moisture Absorption =
\textbf{\space }
\begin{eqnarray*}\style{font-size:10px}{(\frac{Final\;weight\;-Initial\;weight\;}{Initial\;weight})\times100} \end{eqnarray*}
\textbf{Kinetic analysis of dissolution data}
To analyze the \textit{in vitro} release data various dynamic models were used like Zero-order model (Cumulative \% drug released versus time), First-order model (Log cumulative percent drug remaining versus time), Higuchi's model {\textendash}Cumulative percent drug released versus square root of time, Korsmeyer equation / Peppa's model {\textendash} Log cumulative \% drug released versus log time \unskip~\citep{632472:14811428,632472:14811440,632472:14811437,632472:14811434}.
\textit{\textbf{In vivo}} \textbf{ studies - Pharmacokinetic studies}
To determine the peak plasma concentration, pharmacokinetic studies were carried out. The \textit{In vivo } studies were carried out on male Wistar rats weighing range from 250-300 gm. They were housed in polypropylene cages and had free access to food and water. The formulation for the test was formulated according to the doses of anti-diabetic drugs, which were calculated as per the bodyweight of animals. The proposed proof on the animal was approved by the Institutional Animal Ethical Committee (IEAC), which is recognized by the Committee for the Purpose of Control and Supervision of Experiments on Animal (CPCSEA). The optimized Floating mucoadhesive matrix tablets were administered orally. Blood samples were collected for over 24h according to a predetermined sample collection schedule. Various pharmacokinetic parameters like C max, T max, AUC were determined \unskip~\citep{632472:14811434,632472:14811435} \textbf{.}
\section{Results and Discussion}
The solubility studies indicated that the drug has less solubility in water as compared to methanol and 0.1N HCl.
\textbf{Drug {\textendash}Polymer Compatibility Studies by FTIR}
Drug polymer compatibility studies were performed by FTIR (Fourier transform infrared spectroscopy). From the FTIR data, it was evident that the drug and excipients doses not have any interactions\textbf{. } Hence they were compatible, as mentioned in Figure~\ref{f-aeb029410a9d}\&Figure~\ref{f-fb35a8c85695}.
\bgroup
\fixFloatSize{images/9cbf9821-82eb-48b5-92b8-eedb15945d7c-upicture1.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/9cbf9821-82eb-48b5-92b8-eedb15945d7c-upicture1.png}{\includegraphics{images/9cbf9821-82eb-48b5-92b8-eedb15945d7c-upicture1.png}}{}
\makeatother
\caption{\boldmath {FTIR of Glipizide pure drug,Pre-compressionEvaluation:}}
\label{f-aeb029410a9d}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/cb82baf7-4688-4308-b964-81718cc6b543-upicture2.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/cb82baf7-4688-4308-b964-81718cc6b543-upicture2.png}{\includegraphics{images/cb82baf7-4688-4308-b964-81718cc6b543-upicture2.png}}{}
\makeatother
\caption{\boldmath {FTIR Spectra of Mixture Drug+Polymer Physical}}
\label{f-fb35a8c85695}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/80eb9f83-c982-4cbe-9d19-98998e5c3ecf-upicture3.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/80eb9f83-c982-4cbe-9d19-98998e5c3ecf-upicture3.png}{\includegraphics{images/80eb9f83-c982-4cbe-9d19-98998e5c3ecf-upicture3.png}}{}
\makeatother
\caption{\boldmath {Angle of repose for the obtained formulation}}
\label{f-6542df2a90af}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/eca537b2-1e5f-4993-9760-480338d81fd0-upicture4.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/eca537b2-1e5f-4993-9760-480338d81fd0-upicture4.png}{\includegraphics{images/eca537b2-1e5f-4993-9760-480338d81fd0-upicture4.png}}{}
\makeatother
\caption{\boldmath {Carr's Index for the obtained formulation}}
\label{f-adf728857cd1}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/03851034-c7df-49bc-b04a-cbc4d9d5db91-upicture5.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/03851034-c7df-49bc-b04a-cbc4d9d5db91-upicture5.png}{\includegraphics{images/03851034-c7df-49bc-b04a-cbc4d9d5db91-upicture5.png}}{}
\makeatother
\caption{\boldmath {Hausner's Ratio for the obtained formulation}}
\label{f-f4af69fe29d0}
\end{figure}
\egroup
The angle of repose for all formulations was in the range of 22.29 to 29.36, as mentioned in Figure~\ref{f-6542df2a90af}. This suggested that the powder blend has excellent to moderate flow property. The above results showed that the pre-compressed mixture has a good Compressibility index and Hausner's ratio, as mentioned in Figure~\ref{f-adf728857cd1} \&Figure~\ref{f-f4af69fe29d0}. This is an indication that the mixture had good compression properties
\textbf{Post-compression Evaluation}
\begin{table*}[!htbp]
\caption{\boldmath {Evaluation of floating mucoadhesive tablets of Glipizide} }
\label{tw-92c9ceade7de}
\centering
\begin{threeparttable}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.1373\linewidth-2\tabcolsep}p{\dimexpr.12\linewidth-2\tabcolsep}p{\dimexpr.127\linewidth-2\tabcolsep}p{\dimexpr.13110000000000003\linewidth-2\tabcolsep}p{\dimexpr.1296\linewidth-2\tabcolsep}p{\dimexpr.13489999999999998\linewidth-2\tabcolsep}p{\dimexpr.13370000000000003\linewidth-2\tabcolsep}p{\dimexpr.08640000000000004\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Formulation \mbox{}\protect\newline Code & Thickness \mbox{}\protect\newline (mm) & Average Weight \mbox{}\protect\newline (mg) & Hardness \mbox{}\protect\newline (Kg/cm2) & Friability \mbox{}\protect\newline (\%) & Content uniformity \mbox{}\protect\newline (\%) & Floating \mbox{}\protect\newline Lag time \mbox{}\protect\newline (Sec) \mbox{}\protect\newline & Total Floating \mbox{}\protect\newline time(hr)\\
\tblmidrule
GF1 &
4.21\ensuremath{\pm}0.22 &
97.32\ensuremath{\pm}0.11 &
6.1\ensuremath{\pm}0.09 &
0.55\ensuremath{\pm}0.07 &
98.36\ensuremath{\pm}0.33 &
102\ensuremath{\pm}0.09 &
{\textgreater}12\\
GF2 &
4.39\ensuremath{\pm}0.29 &
96.10\ensuremath{\pm}0.09 &
6.5\ensuremath{\pm}0.05 &
0.35\ensuremath{\pm}0.09 &
94.15\ensuremath{\pm}0.40 &
95\ensuremath{\pm}0.04 &
{\textgreater}12\\
GF3 &
4.20\ensuremath{\pm}0.18 &
99.39\ensuremath{\pm}0.15 &
6.8\ensuremath{\pm}0.09 &
0.48\ensuremath{\pm}0.05 &
95.29\ensuremath{\pm}0.29 &
62\ensuremath{\pm}0.05 &
{\textgreater}12\\
GF4 &
4.96\ensuremath{\pm}0.27 &
95.48\ensuremath{\pm}0.14 &
6.3\ensuremath{\pm}0.04 &
0.15\ensuremath{\pm}0.08 &
99.15\ensuremath{\pm}0.31 &
50\ensuremath{\pm}0.07 &
{\textgreater}12\\
GF5 &
4.33\ensuremath{\pm}0.10 &
99.83\ensuremath{\pm}0.08 &
6.9\ensuremath{\pm}0.08 &
0.19\ensuremath{\pm}0.04 &
97.39\ensuremath{\pm}0.28 &
45\ensuremath{\pm}0.10 &
{\textgreater}12\\
GF6 &
4.75\ensuremath{\pm}0.19 &
100.3\ensuremath{\pm}0.16 &
6.0\ensuremath{\pm}0.03 &
0.49\ensuremath{\pm}0.06 &
99.25\ensuremath{\pm}0.36 &
36\ensuremath{\pm}0.08 &
{\textgreater}12\\
GF7 &
4.19\ensuremath{\pm}0.17 &
99.25\ensuremath{\pm}0.13 &
6.4\ensuremath{\pm}0.06 &
0.52\ensuremath{\pm}0.07 &
98.25\ensuremath{\pm}0.37 &
30\ensuremath{\pm}0.06 &
{\textgreater}12\\
GF8 &
4.62\ensuremath{\pm}0.21 &
97.64\ensuremath{\pm}0.10 &
6.7\ensuremath{\pm}0.04 &
0.39\ensuremath{\pm}0.06 &
100.0\ensuremath{\pm}0.31 &
35\ensuremath{\pm}0.11 &
{\textgreater}12\\
GF9 &
4.81\ensuremath{\pm}0.29 &
99.47\ensuremath{\pm}0.27 &
6.1\ensuremath{\pm}0.08 &
0.31\ensuremath{\pm}0.05 &
95.12\ensuremath{\pm}0.27 &
120\ensuremath{\pm}0.08 &
{\textgreater}12\\
GF10 &
4.67\ensuremath{\pm}0.17 &
95.36\ensuremath{\pm}0.21 &
6.8\ensuremath{\pm}0.08 &
0.30\ensuremath{\pm}0.04 &
98.64\ensuremath{\pm}0.35 &
110\ensuremath{\pm}0.06 &
{\textgreater}12\\
GF11 &
4.15\ensuremath{\pm}0.27 &
98.61\ensuremath{\pm}0.09 &
6.5\ensuremath{\pm}0.10 &
0.28\ensuremath{\pm}0.06 &
99.20\ensuremath{\pm}0.39 &
125\ensuremath{\pm}0.12 &
{\textgreater}12\\
GF12 &
4.21\ensuremath{\pm}0.09 &
95.92\ensuremath{\pm}0.16 &
6.1\ensuremath{\pm}0.08 &
0.21\ensuremath{\pm}0.08 &
95.10\ensuremath{\pm}0.34 &
110\ensuremath{\pm}0.09 &
{\textgreater}12\\
GF13 &
4.79\ensuremath{\pm}0.19 &
97.18\ensuremath{\pm}0.15 &
6.9\ensuremath{\pm}0.06 &
0.15\ensuremath{\pm}0.07 &
99.67\ensuremath{\pm}0.32 &
80\ensuremath{\pm}0.08 &
{\textgreater}12\\
GF14 &
4.69\ensuremath{\pm}0.16 &
99.86\ensuremath{\pm}0.11 &
6.4\ensuremath{\pm}0.09 &
0.28\ensuremath{\pm}0.04 &
98.33\ensuremath{\pm}0.27 &
60\ensuremath{\pm}0.10 &
{\textgreater}12\\
GF15 &
4.91\ensuremath{\pm}0.28 &
100.0\ensuremath{\pm}0.08 &
6.3\ensuremath{\pm}0.13 &
0.19\ensuremath{\pm}0.03 &
99.49\ensuremath{\pm}0.30 &
75\ensuremath{\pm}0.13 &
{\textgreater}12\\
GF16 &
4.86\ensuremath{\pm}0.24 &
97.85\ensuremath{\pm}0.15 &
6.0\ensuremath{\pm}0.07 &
0.40\ensuremath{\pm}0.05 &
97.11\ensuremath{\pm}0.38 &
56\ensuremath{\pm}0.06 &
{\textgreater}12\\
GF17 &
4.44\ensuremath{\pm}0.21 &
99.90\ensuremath{\pm}0.19 &
6.5\ensuremath{\pm}0.05 &
0.32\ensuremath{\pm}0.09 &
100.0\ensuremath{\pm}0.25 &
35\ensuremath{\pm}0.08 &
{\textgreater}12\\
GF18 &
4.52\ensuremath{\pm}0.23 &
96.98\ensuremath{\pm}0.23 &
6.7\ensuremath{\pm}0.09 &
0.10\ensuremath{\pm}0.08 &
97.84\ensuremath{\pm}0.29 &
25\ensuremath{\pm}0.05 &
{\textgreater}12\\
\tblbottomrule
\end{tabulary}\par
\begin{tablenotes}\footnotesize
\item{Each value represents the mean\ensuremath{\pm}SD (n=3)}
\end{tablenotes}
\end{threeparttable}
\end{table*}
The thickness of the prepared tablets was, as mentioned in Table~\ref{tw-92c9ceade7de} poses in the range between 4.15mm to 4.96mm. The weight variation was in the limit as specified in I.P. The maximum, and minimum hardness of tablets was 6.9Kg/cm\ensuremath{^{2}}, and 6Kg/cm\ensuremath{^{2, }} respectively, mentioned in Table~\ref{tw-92c9ceade7de}. This is an optimum hardness for floating mucoadhesive tablet. The friability study depicted that all formulations tend to withstand handling and packing. The maximum floating lag time was 125Sec, as mentioned in Table~\ref{tw-92c9ceade7de}.
\textit{\textbf{In vitro }} \textbf{release studies}
\bgroup
\fixFloatSize{images/cbd1cecd-07ba-4c79-9422-770d533f4624-upicture6.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/cbd1cecd-07ba-4c79-9422-770d533f4624-upicture6.png}{\includegraphics{images/cbd1cecd-07ba-4c79-9422-770d533f4624-upicture6.png}}{}
\makeatother
\caption{\boldmath {In vitro Dissolution study of RT 1 to RT 9}}
\label{f-2315606043b8}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/a8f64d07-fe31-42e5-94f1-2da5e51f2a58-upicture7.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/a8f64d07-fe31-42e5-94f1-2da5e51f2a58-upicture7.png}{\includegraphics{images/a8f64d07-fe31-42e5-94f1-2da5e51f2a58-upicture7.png}}{}
\makeatother
\caption{\boldmath {In vitro Dissolution study of RT 10 to RT 18}}
\label{f-fc22fc409956}
\end{figure}
\egroup
The minimum drug release was observed for formulation GF6, which contains drug: Na CMC in the ratio of 1:3. This may be the reason to provide a high amount of polymer. The maximum drug release is found in formulation GF8, which contains drugs: carbopol 974P in ratio 1:2, as mentioned in Figure~\ref{f-2315606043b8} \&Figure~\ref{f-fc22fc409956}. This shows that the drug: carbopol 974P in ratio 1:2 is optimum to achieve a mucoadhesive and free tablet.
In the same way, the moisture absorption and adhesion strength properties presented inTable~\ref{tw-2f733934cf0c}for the GF8 formulation because the evaluation results of GF8 are better than others, and it also represents the good adhesion strength due to the optimum concentration of carbopol 974P.
\begin{table*}[!htbp]
\caption{\boldmath {\textbf{Moisture absorption, adhesion strength values of selected formulations. } \textbf{\space }} }
\label{tw-2f733934cf0c}
\centering
\begin{threeparttable}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2275\linewidth-2\tabcolsep}p{\dimexpr.20150000000000002\linewidth-2\tabcolsep}p{\dimexpr.29849999999999994\linewidth-2\tabcolsep}p{\dimexpr.27250000000000004\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}{\multirow{2}{*}{Formulation Code}} & {\multirow{2}{*}{Moisture absorption}} & \multicolumn{2}{p{\dimexpr(.57099999999999994\linewidth-2\tabcolsep)}}{\cAlignHack Bioadhesion strength}\\
\rowcolor{kwdboxcolor} & & Peak detachment force (N) & Work of adhesion \mbox{}\protect\newline (mJ)\\
\tblmidrule
GF8 &
46\ensuremath{\pm}0.25 &
3.6\ensuremath{\pm}0.22 &
12.42\ensuremath{\pm}6.16\\
\tblbottomrule
\end{tabulary}\par
\begin{tablenotes}\footnotesize
\item{ Each value represents the mean\ensuremath{\pm}SD (n=3)}
\end{tablenotes}
\end{threeparttable}
\end{table*}
\textbf{\space }
\bgroup
\fixFloatSize{images/5c70809f-3208-48fd-8901-13d07cd31424-upicture8.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/5c70809f-3208-48fd-8901-13d07cd31424-upicture8.png}{\includegraphics{images/5c70809f-3208-48fd-8901-13d07cd31424-upicture8.png}}{}
\makeatother
\caption{\boldmath {Zero-order plot of optimized formulation}}
\label{f-753e54da5df9}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/3cccf9bb-4497-4c5b-9bc6-61d1d18cd035-upicture9.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/3cccf9bb-4497-4c5b-9bc6-61d1d18cd035-upicture9.png}{\includegraphics{images/3cccf9bb-4497-4c5b-9bc6-61d1d18cd035-upicture9.png}}{}
\makeatother
\caption{\boldmath {Higuchi plot of optimized formulation}}
\label{f-dba2c472a208}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/dc60ce0c-2a70-4da1-b509-2bc189c7b4d0-upicture10.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/dc60ce0c-2a70-4da1-b509-2bc189c7b4d0-upicture10.png}{\includegraphics{images/dc60ce0c-2a70-4da1-b509-2bc189c7b4d0-upicture10.png}}{}
\makeatother
\caption{\boldmath {Koresmeyer-Peppasplot of optimized formulation}}
\label{f-1b34bd6eceec}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/c6320f98-a176-449b-ab7c-ef7aba41be35-upicture11.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/c6320f98-a176-449b-ab7c-ef7aba41be35-upicture11.png}{\includegraphics{images/c6320f98-a176-449b-ab7c-ef7aba41be35-upicture11.png}}{}
\makeatother
\caption{\boldmath {First order plot of optimized formulation}}
\label{f-b4d26ad879eb}
\end{figure}
\egroup
\begin{table*}[!htbp]
\caption{\boldmath {\textbf{ Pharmacokinetic parameters of optimized formulation}} }
\label{tw-33fc373fa907}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.221\linewidth-2\tabcolsep}p{\dimexpr.375\linewidth-2\tabcolsep}p{\dimexpr.404\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}S.No & Parameter & Glipizide \\
\tblmidrule
1 &
Cmax &
629.0 ng/mL (\ensuremath{\pm}94.2) \\
2 &
T max(hr) &
3.92hours(\ensuremath{\pm}0.89) \\
3 &
AUC &
3430ng\ensuremath{\cdot}h/Ml(\ensuremath{\pm} 882) \\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\textbf{Release kinetics}
Data of \textit{in vitro} release studies of formulations, which was showing better drug release, was applied for different release kinetics such as zero, first-order kinetics, Higuchi, and Korsmeyer Peppas of Glipizide release from mucoadhesive tablets given in Figures~\ref{f-753e54da5df9}, \ref{f-dba2c472a208} and~\ref{f-1b34bd6eceec} \&Figure~\ref{f-b4d26ad879eb}.
Based on all studies, GF8 formulation was found to be better when compared with all other formulations. This formulation was following the Higuchi mechanism with a regression value of 0.993.
\textit{\textbf{In vivo}} \textbf{ Studies - Pharmacokinetic Studies}
The pharmacokinetics parameters are mentioned inTable~\ref{tw-33fc373fa907}, Mean time to reach peak drug concentration (T\ensuremath{_{max}}) and maximum drug concentration (Cmax) were 3.92 hours and 629.0 mg/mL, respectively. The values for C\ensuremath{_{max}}, T\ensuremath{_{max}}, AUC were represents the sustain release pattern, as mentioned in
Table~\ref{tw-33fc373fa907}.
\section{Conclusion}
Glipizide, was formulated as Floating mucoadhesive tablets to improve its bioavailability. HPMC K200 M, Na CMC, Carbopol 974P, Karaya gum, Chitosan, Xanthan gum were selected as polymers. The pre-compression blend of Glipizide Floating mucoadhesive tablets was characterized with respect to all the pre-compression parameters. It found that all the results reflected that the blend was having a good flow of nature and better compression properties. Peak detachment force (N) and work of adhesion were also represented good adhesion activity.
Glipizide GF8 formulation was considered as an optimized formulation because of proper drug release (99.11 \%) in 12 hours, Moisture absorption(46\ensuremath{\pm}0.25), Peak detachment force (N) (3.6\ensuremath{\pm}0.22N), Work of adhesion (12.42\ensuremath{\pm}6.16mJ). GF8 formulation follows the Higuchi mechanism with a regression value of 0.993. The in-vivo pharmacokinetic studies showed that the drug reaches the maximum concentration in 3.92 hr. The C\ensuremath{_{max}} and AUC data predicts that the drug has excellent oral bioavailability. Further studies can be carried out using different drugs to correlate the data.
\section{Acknowledgement}
The authors would like to convey sincere thanks to Vikas College of Pharmaceutical Sciences and the Bengal School of Technology for their esteemed support towards this work.
\bibliographystyle{pharmascope_apa-custom}
\bibliography{\jobname}
\end{document}
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\title{The religion of green anarchy}
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This text was originally sent to “Black and Green Review”. They never responded.
Disclaimer—in North America, which is my context, land defense struggles are often from indigenous perspectives, and they are struggles based on reclaiming or defending land from the state. I can only speak from a settler perspective, and my critique is specifically of land defense through a futurist lens and the deification of nature as it is practiced by settlers in North America.
\section{The Religion of Green Anarchy: a critique, a question, and a proposal}
Many land defense struggles in North America focus on the purity of the wild when coming from a settler green anarchist perspective. Based on the propaganda and analysis that comes from this perspective and out of these struggles, we seek to defend these spaces from industrial civilization (and through this, colonial expansion) because we are defending the last ‘wild’ areas, from which we can subsist. This belief in ‘wild’ and ‘untouched’ spaces is not only unfounded, but falls into the creation of a morality of the wild, which takes on a religious tone. This religious tone can be broken down into: a) ‘good’ wilderness vs ‘bad’ wilderness, and b) preservation of a utopia or ‘heaven’ for future generations. Oftentimes, settlers in North America lack a coherent culture – there is no North American culture outside of capitalism. This religious tone can be understood as a response to this cultureless void, as we try to create a context for ourselves—an anchoring for our identities.
When we approach land defense struggles from this moralist and future-oriented perspective, we limit the potential of these struggles. The primary drive of engaging in land defense struggles for future generations can prefigure the struggles themselves. This leads to an acceptance of concessions and defeats, as we are able to convince ourselves that a failed land defense is contributing to a culture of resistance, with which the future generations can engage. What would these struggles look like were we to see them as book-ended by our life and death, breaking from the limitations of morality, culture, or the future generations? What trajectory would a land defense take if individual sensory experience were the guiding principal?
\section{CRITIQUE}
\textbf{1) Wild and untouched spaces don’t exist, and agriculture isn’t the original sin.}
The definition of a pure space is often tied up in settler misconceptions of the ‘pure’ native and hunter-gatherer societies – as those untouched by colonization or agriculture. This ignores that many nations, pre-contact, managed wild spaces. Examples of this are maintaining burnsites for berry-picking, creating clam gardens, and complex territorial management and distribution. This isn’t meant to be a generalization about all nations—just examples of how some nations, pre-contact, interacted with the wilderness in ways that are similar to agriculture.
Green anarchist analysis and critiques cite agriculture as the beginning of the end of hunter-gatherer lifestyles. I don’t contest that agriculture necessitated more sedentary lifestyles, acted as a colonial force, and created delayed-return lifestyles and economies that eventually resulted in increased domestication. However, it was not the driving force that enabled and created these storylines or this history. Desire for ‘power-over’ through enforcing hierarchy is a more likely culprit, and comes from individuals who facilitate a certain society. This isn’t to say that hierarchy and desire for power over are inherent in nature—not at all—but to caution against the strong correlation between nature and perfection, agriculture and humanness\Slash{}domestication—these are false comparisons. Nature is imperfect, and it’s incorrect to fetishize the natural world as being pre-domination. Destroying agriculture or returning to a hunter-gatherer lifestyle will not destroy society and capitalism. This is evidenced by the hunter-gatherer societies who reproduce systems of morality and norms similar to those in North American capitalist culture. Perhaps the ways in which these moralities or norms were reinforced is different from our current society (for instance, shaming as opposed to prison), but it’s important to note that these norms existed, and were socially-imposed.
Pure wilderness is a civilized concept, and shouldn’t be used to determine which territories warrant defense, or as a wildness to return to.
\textbf{2) Nature is a whore.}
\emph{\textbf{a) Purity of the wild as morality}}.\forcelinebreak Within green anarchist analysis, ‘pure’ wildness (we can replace ‘pure’ with ‘undomesticated’ or ‘wild’) is deserving of preservation and defense, whereas ‘impure’ (domesticated) wildness is not. This is evidenced by the struggles which receive the most attention from the settler community—largely, anti-resource extraction land defense in undomesticated areas, with clean drinking water, a focus on preserving intact salmon runs, etc. This moralization of nature, and distinction between good and bad nature, allow for good nature to accumulate value (as it is defended) and bad nature to depreciate in value (as it is undefended), leading to the commodification of nature.
We also see this language of morality and purity in green anarchist publications, as they create standards for living a ‘good’ vs ‘bad’ life. A morally ‘good’ life is a hunter-gatherer lifestyle, while a morally ‘bad’ life is one that resides in or relies on the city and industry, or practices agriculture. The fact that projects such as the Feralculture Project, which rely on capitalism and colonialism, are lauded and uncriticized, demonstrates that the only barometer for green anarchist morality is pure vs impure wilderness.
This obsession with a ‘pure’ wilderness is very similar to judeo-christian obsession with the pure and untainted body—virginity. As the human hand corrupts nature through management and development, the heathen hand corrupts the woman through sexuality and desire.
\emph{\textbf{b) “This was here before you, and will be here after you.”}}\forcelinebreak “The future is primitive, whether we see it or not.” –Black and Green Review \#2
This moralization of nature—good vs bad nature, some worthy of defense and other that is tainted by human contact—is always presented in the context of future generations. Again, when we look at the propaganda put out by land defense struggles, it is frequently through the lens of preservation for the future—not the present, individual sensory experience.
This is dangerous not only in that it nurtures an unreal hope that there is a better and happier future awaiting us should we lead morally-correct lives, but because it places our struggle in this future as opposed to the present. This becomes the same hope as religion– one that allows us to withstand the banal and depressing day-to-day through the guarantee that at some point on the future, all will be better. Green anarchist moralists live either in the past—by idealizing hunter-gatherer societies—or in the future—by hoping for a ‘primitive future’. Similar to many religions, where one lives life not for now or even in now, but for a life after death. The present becomes time killed reading and learning about the lives of saints, the life of god, the mythics and stories of the bible.
This futurist mentality is particularly dangerous for people with uteruses. Since the highest value of life is set for the future generations, our bodies have unfortunately and oftentimes unintentionally been transformed into tools for the green anarchist project. Any hope or value that anarchists place upon the future generations originates from a reliance on the principle that every individual, if exposed to the correct conditions, experiences, and ideas, will identify with anarchist principles. As any person who spends time with children will know, they are their own people. Even if they grow up in a co-operative and caring environment, surrounded by a strong critique of society and power, they may still turn out to be individuals who profit from power and hierarchy. They may turn out to be our enemies, not our allies. This critique of the valorization of the child is brought forth in Baeden, a journal of queer nihilism, in the following statement:
“All political positions, he argues, represent themselves as doing what is best for the children. Politicians, whatever their parties or leanings, universally frame their debates around the question of what policies are best for the children, who keeps the Child safest, or what type of world we want to be building for our children. The centrality of the Child in the field of the political is not limited to electoral politics or political parties. Nationalist groups organize themselves around a necessity to preserve a future for their children, while anarchist and communist revolutionaries concern themselves with revolutionary organizing meant to create a better world for future generations. Politicians concern themselves with different children depending on their varying from ideologies, but the Child stays constant as a universal Möbius strip, inverting itself and flipping so as to be the unquestioned and untouchable universal value of all politics. Politics, however supposedly radical, is simply the universal movement of submission to the ideal of the future—to preserve, maintain and upgrade the structures of society and to proliferate them through time all for the sake of the children.”
The majority of land defense struggles are strongly defined by this concept of ‘for the future generations’, and the idea that we struggle against industrial civilization for the future, not for now. Though I am uncertain of the origin point of this reasoning, it is my experience that it is frequently referenced in native land-defense struggles. It is not logical to take the same perspective of native land-defense and super-impose it onto our lives and our struggles as settlers. The term ‘ally’, though corrupted by settler-guilt and identity politic olympics, originally meant two groups of different origins fighting for a common outcome. With this definition in mind, it does not make sense for settlers to appropriate various indigenous understandings of historical rooting and ‘fighting for the future generations’. A native friend once explained to me that colonization had so thoroughly eroded her current community that it was impossible for her to conceive of fighting for anything other than the future generations, because she believed that healing would require more than one generation. This is fucking intense, but to claim this as a settler reasoning for struggle would require a lot more reflection and intention than it is ever attributed.
One explanation for why settlers hold on to this concept is that it provides a generally-understood answer for the question of why we engage in land defense struggles, and has become widespread as a reasoning. As a result, this perspective precedes the struggles themselves and influences how they play out. If we invest ourselves in the future and can see our struggles, regardless of their outcome, as contributing to a culture and history of conflict, we are more likely to concede defeat or compromise before we reach our goals.
What if we fought as though our lives depended on it? Not the lives of our children, or our friend’s children, but our lives, right now, in the present? This would make for a very different type of struggle. Potentially short-lived, but that’s the way it is with uncompromising struggles.This is neither a critique nor a conclusion, but a question.
\section{PROPOSAL}
Everything on this earth has been touched, in one way or another, by humans and society. According to green anarchist morality everything is impure. This doesn’t mean that a polluted river, or an abandoned city lot is undeserving of protection or defense. In contrast, if you look at land defense through a non-humanist perspective, these impure areas are worthwhile to defend in that they in no way can be beneficial to humanity or society—the mercury-laden soil can’t produce medicine or food, but is still valuable to the multitude of species that exist within it.
A proposal for how to value the impurity of the wild would be to destroy any attempt to create culture in the cultureless void. This cultural void is a gift and a step closer to life free of imposed morality, cultural stigma and codes. And what a gift, an identity formed only by the individual (and their subjective experiences)! We should embrace our lack of culture, this void, instead of trying to fill it with god and religion by another name. We can do so by trying to destroy any attempt to create this culture through participation in the human strike.
The human strike “\dots{}defines a type of strike that involves the whole life and not only its professional side, that acknowledges exploitation in all the domains and not only at work. Human strike can be a revolt within a revolt, an unarticulated refusal, an excess of work or the total refusal of any labour, depending on the situation. There is no orthodoxy for it. If strikes are made in order to improve specific aspects of the workers’ conditions, they are always a means to an end.”
The creation of culture in green anarchy\Slash{}land defense struggles is a reaction to a cultureless void. The human strike in terms of land defense can be seen as refusing to acknowledge ‘futurism’ by refusing to participate in creation of a culture through reproduction and faith in the future generations. It can take the form of refusing to participate in the morality of the wild by refusing to act for the future generations, and acting only for ourselves and our individual sensory experiences. “But human strike is a pure means, a way to create an immediate present here where there is nothing but waiting, projecting, expecting, hoping\dots{} To produce the present is not to produce the future.”
Participating in the human strike is to not allow our bodies to become tools for the struggle for the future, through either reproduction2 or by dedicating them to tasks that facilitate a society that maintains itself through coercing its participant into relying on a hope3 for the future. Another aspect would be a destruction of capitalism, society, and culture, while at the same time recognizing and disassembling the trap of the culture of green anarchy\Slash{}morality of the wild.
Concretely, an example of how to engage with this definition of the human strike would be to occupy land, and practice the skills for subsistence and life independent of society, without acknowledging the state. Independence from society can mean learning to sustain oneself outside of it. In cities, this often takes the form of stealing and scamming. These are and beautiful anti-social ways of surviving—they destroy the relationship to and power of money, rendering it ridiculous, as well as the individual actualizing their desires in conflict with society. Hunting, fishing, harvesting wild foods, etc. achieve the same goals as scamming and stealing, as long as they are acted without consent of the state (poaching, for example). These forms of subsistence make a mockery of money, while also allowing the individual to thrive independent of and in contradiction to the state and society.
By learning these skills, we doubly participate in the human strike by dedicating our time to something that is completely irrelevant and useless to capitalism and society. Building a log cabin, snaring rabbits, or harvesting maple water–these things we do for our enjoyment and our enjoyment only. This is a passive form of resistance, in that it is just diverting our energy from production for society towards our own end goals, our own desires, our own joys. This could fall into the trap of drop-out culture, but differs in that it understands the necessity of defending the areas of our enjoyment against incursion, and attack on resource extraction projects that threaten our ability to continue to live unmediated by the state. Through pairing dedication of time to joyful projects irrelevant to capitalism with refusal to seek consent from the state and a strong investment in land defense, this can become part of a coherent and conflictual life.
At this specific time in North America, power is accumulated in the resource extraction projects throughout the North. To participate in the human strike would mean attacking where power is accumulated and where the state’s intervention is weakest. Rural and more northern areas, where these resource extraction projects are located, have a less developed infrastructure for surveillance and repression than cities.
A very regional and specific example of how to engage with the human strike would be to occupy land that has been slated for development, use the land to learn and practice subsistence skills that you enjoy, and then fiercely defend it from the state, without concessions or compromise. Practically, if you wanted to participate in anti-colonial governance structures, this could take the form of seeking permission\Slash{}complicity from hereditary governance structures, and occupying land for subsistence purposes. The goal of this occupation would primarily be conflict, not preservation. These spaces may be short-lived, but this would transform land defense from a pseudo-religious\Slash{}future-oriented project into the daily action of our desires.
These spaces would also not be isolated from urban struggles, but a complement to them. Though power is accumulated in these resource extraction projects up north, there are still ties to the urban environments that provide the workers, house development offices, and plan the projects themselves. These occupied spaces could also become refuge for those who are avoiding repression. There are no cameras or randomized ID checks in the forest or the mountains. Search parties have little success trying to find people who don’t want to be found.
This seems like a pretty extravagant proposal. To occupy land, learn the subsistence skills that give us joy, and then militantly defend it. All with the understanding that we do this for ourselves, for our own individual sensory experience, with no reliance on the future generations or with the safety blanket of ‘we’re contributing to a culture of resistance’. There are already several land defense camps which demonstrate aspects of this proposal, primarily from a First Nations land reclamation perspective. Unist’ot’en, Madii Lii, Lax U’u’la, and the Standing Rock Sioux anti-pipeline camps are all examples of successful and inspiring struggles. The above critiques and proposal are not geared towards these struggles, but towards settler intervention and green anarchist analysis that comes out about these struggles. Part of the destruction of society and capitalism is acting from a place of decolonization\Slash{}anti-colonialism (colonialism facilitates capitalism, capitalism facilitates colonialism). Any land occupation that occurs in North America, if it is to be successful in not reproducing the power structures of capitalism and society, must include an anti-colonial\Slash{}decolonization analysis. This would mean creating links with the pre-existing land defense projects, finding affinity with the individuals whose territory is under attack, and figuring out where our struggles overlap.
This is also not a proscription for how to participate in the human strike and land defense, but a proposal, such that those who feel affinity with the ideas presented can choose to participate.
For specific examples of shaming as a method of ensuring social norm compliance, you can read the “Use of Humour in Hunter-Gatherer Governance” section of Peter Gray’s “Play as a Foundation of Hunter-Gatherer Social Existence”.
Having children doesn’t exclude you from participating in the human strike, as I am defining it. If you want to have children, and it gives you immediate joy, that is centering your body on your own experience. It is the investment in the children and hope that they will somehow contribute to a struggle in the future that contributes to a creation of culture.
\begin{quote}
“Despite the madness of war, we lived for a world that would be different. Do you really think that, without the hope that such a world is possible, that the rights of man would be restored again, we could stand the concentration camp even for one day? It is that very hope that makes people go without a murmur to the gas chambers, keeps them from risking revolt, paralyses them into numb inactivity. It is hope that breaks down family ties, makes mothers renounce their children, or wives sell their bodies for bread, or husbands to kill. It is hope that compels man to hold on to one more day of life, because that day may be the day of liberation\dots{}Never before in the history of mankind has hope been stronger than man, but never also has it done so much harm as it has in this war, in this concentration camp. We were never taught how to give up on hope, and this is why today we perish in gas chambers.”—Tadeusz Borowski, Auschwitz, our home (a letter)
\end{quote}
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Library.Anarhija.Net
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The religion of green anarchy
May 10, 2017
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\href{https://actforfree.nostate.net/?p=27129}{actforfree.nostate.net}
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\textbf{lib.anarhija.net}
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\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{eurosym}
\usepackage[english]{babel}
\begin{document}
\title{Mon titre}
\author{Votre nom ici}
\maketitle
\section{Introduction}
Ceci est l'intro
\section{Autre section}
Une formule: $f(x) = ax+b$ \bigskip
Une autre formule:
$$g(x) = x^2+b$$
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\section{Extensions of the Base Logic}
In this section we discuss some additional constructions that we define within and on top of the base logic.
These are not ``extensions'' in the sense that they change the proof power of the logic, they just form useful derived principles.
\subsection{Derived Rules about Base Connectives}
We collect here some important and frequently used derived proof rules.
\begin{mathparpagebreakable}
\inferhref{L{\"o}b}{Loeb}
{}
{(\later\prop\Ra\prop) \proves \prop}
\infer{}
{\prop \Ra \propB \proves \prop \wand \propB}
\infer{}
{\prop * \Exists\var.\propB \provesIff \Exists\var. \prop * \propB}
\infer{}
{\prop * \All\var.\propB \proves \All\var. \prop * \propB}
\infer{}
{\always(\prop*\propB) \provesIff \always\prop * \always\propB}
\infer{}
{\always(\prop \Ra \propB) \proves \always\prop \Ra \always\propB}
\infer{}
{\always(\prop \wand \propB) \proves \always\prop \wand \always\propB}
\infer{}
{\always(\prop \wand \propB) \provesIff \always(\prop \Ra \propB)}
\infer{}
{\later(\prop \Ra \propB) \proves \later\prop \Ra \later\propB}
\infer{}
{\later(\prop \wand \propB) \proves \later\prop \wand \later\propB}
\infer{}
{\TRUE \proves \plainly\TRUE}
\end{mathparpagebreakable}
Noteworthy here is the fact that Löb induction can be derived from $\later$-introduction and the fact that we can take fixed-points of functions where the recursive occurrences are below $\later$.%
\footnote{See \url{https://en.wikipedia.org/wiki/L\%C3\%B6b\%27s_theorem}.}
Furthermore, $\TRUE \proves \plainly\TRUE$ can be derived via $\plainly$ commuting with universal quantification ranging over the empty type $0$.
To derive that existential quantifiers commute with separating conjunction requires an intermediate step using a magic wand: From $P * \exists x, Q \vdash \Exists x. P * Q$ we can deduce $\Exists x. Q \vdash P \wand \Exists x. P * Q$ and then proceed via $\exists$-elimination.
\subsection{Persistent Propositions}
We call a proposition $\prop$ \emph{persistent} if $\prop \proves \always\prop$.
These are propositions that ``do not own anything'', so we can (and will) treat them like ``normal'' intuitionistic propositions.
Of course, $\always\prop$ is persistent for any $\prop$.
Furthermore, by the proof rules given in \Sref{sec:proof-rules}, $\TRUE$, $\FALSE$, $t = t'$ as well as $\ownGhost\gname{\mcore\melt}$ and $\mval(\melt)$ are persistent.
Persistence is preserved by conjunction, disjunction, separating conjunction as well as universal and existential quantification and $\later$.
\subsection{Timeless Propositions and Except-0}
One of the troubles of working in a step-indexed logic is the ``later'' modality $\later$.
It turns out that we can somewhat mitigate this trouble by working below the following \emph{except-0} modality:
\[ \diamond \prop \eqdef \later\FALSE \lor \prop \]
Except-0 satisfies the usual laws of a ``monadic'' modality (similar to, \eg the update modalities):
\begin{mathpar}
\inferH{ex0-mono}
{\prop \proves \propB}
{\diamond\prop \proves \diamond\propB}
\axiomH{ex0-intro}
{\prop \proves \diamond\prop}
\axiomH{ex0-idem}
{\diamond\diamond\prop \proves \diamond\prop}
\begin{array}[c]{rMcMl}
\diamond{(\prop * \propB)} &\provesIff& \diamond\prop * \diamond\propB \\
\diamond{(\prop \land \propB)} &\provesIff& \diamond\prop \land \diamond\propB \\
\diamond{(\prop \lor \propB)} &\provesIff& \diamond\prop \lor \diamond\propB
\end{array}
\begin{array}[c]{rMcMl}
\diamond{\All x. \prop} &\provesIff& \All x. \diamond{\prop} \\
\diamond{\Exists x. \prop} &\provesIff& \Exists x. \diamond{\prop} \\
\diamond\always{\prop} &\provesIff& \always\diamond{\prop} \\
\diamond\later\prop &\proves& \later{\prop}
\end{array}
\end{mathpar}
In particular, from \ruleref{ex0-mono} and \ruleref{ex0-idem} we can derive a ``bind''-like elimination rule:
\begin{mathpar}
\inferH{ex0-elim}
{\prop \proves \diamond\propB}
{\diamond\prop \proves \diamond\propB}
\end{mathpar}
This modality is useful because there is a class of propositions which we call \emph{timeless} propositions, for which we have
\[ \timeless{\prop} \eqdef \later\prop \proves \diamond\prop \]
In other words, when working below the except-0 modality, we can \emph{strip
away} the later from timeless propositions (using \ruleref{ex0-elim}):
\begin{mathpar}
\inferH{ex0-timeless-strip}{\timeless{\prop} \and \prop \proves \diamond\propB}
{\later\prop \proves \diamond\propB}
\end{mathpar}
In fact, it turns out that we can strip away later from timeless propositions even when working under the later modality:
\begin{mathpar}
\inferH{later-timeless-strip}{\timeless{\prop} \and \prop \proves \later \propB}
{\later\prop \proves \later\propB}
\end{mathpar}
This follows from $\later \prop \proves \later\FALSE \lor \prop$, and then by straightforward disjunction elimination.
The following rules identify the class of timeless propositions:
\begin{mathparpagebreakable}
\infer
{\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop \land \propB}}
\infer
{\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop \lor \propB}}
\infer
{\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop * \propB}}
\infer
{\vctx \proves \timeless{\prop}}
{\vctx \proves \timeless{\always\prop}}
\infer
{\vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop \Ra \propB}}
\infer
{\vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop \wand \propB}}
\infer
{\vctx,\var:\type \proves \timeless{\prop}}
{\vctx \proves \timeless{\All\var:\type.\prop}}
\infer
{\vctx,\var:\type \proves \timeless{\prop}}
{\vctx \proves \timeless{\Exists\var:\type.\prop}}
\axiom{\timeless{\TRUE}}
\axiom{\timeless{\FALSE}}
\infer
{\text{$\term$ or $\term'$ is a discrete OFE element}}
{\timeless{\term =_\type \term'}}
\infer
{\text{$\melt$ is a discrete OFE element}}
{\timeless{\ownM\melt}}
\infer
{\text{$\melt$ is an element of a discrete camera}}
{\timeless{\mval(\melt)}}
\end{mathparpagebreakable}
\subsection{Dynamic Composeable Higher-Order Resources}
\label{sec:composeable-resources}
The base logic described in \Sref{sec:base-logic} works over an arbitrary camera $\monoid$ defining the structure of the resources.
It turns out that we can generalize this further and permit picking cameras ``$\iFunc(\Prop)$'' that depend on the structure of propositions themselves.
Of course, $\Prop$ is just the syntactic type of propositions; for this to make sense we have to look at the semantics.
Furthermore, there is a composability problem with the given logic: if we have one proof performed with camera $\monoid_1$, and another proof carried out with a \emph{different} camera $\monoid_2$, then the two proofs are actually carried out in two \emph{entirely separate logics} and hence cannot be combined.
Finally, in many cases just having a single ``instance'' of a camera available for reasoning is not enough.
For example, when reasoning about a dynamically allocated data structure, every time a new instance of that data structure is created, we will want a fresh resource governing the state of this particular instance.
While it would be possible to handle this problem whenever it comes up, it turns out to be useful to provide a general solution.
The purpose of this section is to describe how we solve these issues.
\paragraph{Picking the resources.}
The key ingredient that we will employ on top of the base logic is to give some more fixed structure to the resources.
To instantiate the logic with dynamic higher-order ghost state, the user picks a family of locally contractive bifunctors $(\iFunc_i : \COFEs^\op \times \COFEs \to \CMRAs)_{i \in \mathcal{I}}$.
(This is in contrast to the base logic, where the user picks a single, fixed camera that has a unit.)
From this, we construct the bifunctor defining the overall resources as follows:
\begin{align*}
\GName \eqdef{}& \nat \\
\textdom{ResF}(\ofe^\op, \ofe) \eqdef{}& \prod_{i \in \mathcal I} \GName \fpfn \iFunc_i(\ofe^\op, \ofe)
\end{align*}
We will motivate both the use of a product and the finite partial function below.
$\textdom{ResF}(\ofe^\op, \ofe)$ is a camera by lifting the individual cameras pointwise, and it has a unit (using the empty finite partial function).
Furthermore, since the $\iFunc_i$ are locally contractive, so is $\textdom{ResF}$.
Now we can write down the recursive domain equation:
\[ \iPreProp \cong \UPred(\textdom{ResF}(\iPreProp, \iPreProp)) \]
Here, $\iPreProp$ is a COFE defined as the fixed-point of a locally contractive bifunctor, which exists and is unique up to isomorphism by \thmref{thm:america_rutten}, so we obtain some object $\iPreProp$ such that:
\begin{align*}
\Res &\eqdef \textdom{ResF}(\iPreProp, \iPreProp) \\
\iProp &\eqdef \UPred(\Res) \\
\wIso &: \iProp \nfn \iPreProp \\
\wIso^{-1} &: \iPreProp \nfn \iProp \\
\wIso(\wIso^{-1}(x)) &\eqdef x \\
\wIso^{-1}(\wIso(x)) &\eqdef x
\end{align*}
Now we can instantiate the base logic described in \Sref{sec:base-logic} with $\Res$ as the chosen camera:
\[ \Sem{\Prop} \eqdef \UPred(\Res) \]
We obtain that $\Sem{\Prop} = \iProp$.
Effectively, we just defined a way to instantiate the base logic with $\Res$ as the camera of resources, while providing a way for $\Res$ to depend on $\iPreProp$, which is isomorphic to $\Sem\Prop$.
We thus obtain all the rules of \Sref{sec:base-logic}, and furthermore, we can use the maps $\wIso$ and $\wIso^{-1}$ \emph{in the logic} to convert between logical propositions $\Sem\Prop$ and the domain $\iPreProp$ which is used in the construction of $\Res$ -- so from elements of $\iPreProp$, we can construct elements of $\Sem{\textlog M}$, which are the elements that can be owned in our logic.
\paragraph{Proof composability.}
To make our proofs composeable, we \emph{generalize} our proofs over the family of functors.
This is possible because we made $\Res$ a \emph{product} of all the cameras picked by the user, and because we can actually work with that product ``pointwise''.
So instead of picking a \emph{concrete} family, proofs will assume to be given an \emph{arbitrary} family of functors, plus a proof that this family \emph{contains the functors they need}.
Composing two proofs is then merely a matter of conjoining the assumptions they make about the functors.
Since the logic is entirely parametric in the choice of functors, there is no trouble reasoning without full knowledge of the family of functors.
Only when the top-level proof is completed we will ``close'' the proof by picking a concrete family that contains exactly those functors the proof needs.
\paragraph{Dynamic resources.}
Finally, the use of finite partial functions lets us have as many instances of any camera as we could wish for:
Because there can only ever be finitely many instances already allocated, it is always possible to create a fresh instance with any desired (valid) starting state.
This is best demonstrated by giving some proof rules.
So let us first define the notion of ghost ownership that we use in this logic.
Assuming that the family of functors contains the functor $\Sigma_i$ at index $i$, and furthermore assuming that $\monoid_i = \Sigma_i(\iPreProp, \iPreProp)$, given some $\melt \in \monoid_i$ we define:
\[ \ownGhost\gname{\melt:\monoid_i} \eqdef \ownM{(\ldots, \emptyset, i:\mapsingleton \gname \melt, \emptyset, \ldots)} \]
This is ownership of the pair (element of the product over all the functors) that has the empty finite partial function in all components \emph{except for} the component corresponding to index $i$, where we own the element $\melt$ at index $\gname$ in the finite partial function.
We can show the following properties for this form of ownership:
\begin{mathparpagebreakable}
\inferH{res-alloc}{\text{$G$ infinite} \and \melt \in \mval_{M_i}}
{ \TRUE \proves \upd \Exists\gname\in G. \ownGhost\gname{\melt : M_i}
}
\and
\inferH{res-update}
{\melt \mupd_{M_i} B}
{\ownGhost\gname{\melt : M_i} \proves \upd \Exists \meltB\in B. \ownGhost\gname{\meltB : M_i}}
\inferH{res-empty}
{\text{$\munit$ is a unit of $M_i$}}
{\TRUE \proves \upd \ownGhost\gname\munit}
\axiomH{res-op}
{\ownGhost\gname{\melt : M_i} * \ownGhost\gname{\meltB : M_i} \provesIff \ownGhost\gname{\melt\mtimes\meltB : M_i}}
\axiomH{res-valid}
{\ownGhost\gname{\melt : M_i} \Ra \mval_{M_i}(\melt)}
\inferH{res-timeless}
{\text{$\melt$ is a discrete OFE element}}
{\timeless{\ownGhost\gname{\melt : M_i}}}
\end{mathparpagebreakable}
Below, we will always work within (an instance of) the logic as described here.
Whenever a camera is used in a proof, we implicitly assume it to be available in the global family of functors.
We will typically leave the $M_i$ implicit when asserting ghost ownership, as the type of $\melt$ will be clear from the context.
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\documentclass[a4paper, oneside, 10pt]{article}
\usepackage[english]{babel}
\usepackage[unicode]{hyperref}
\usepackage[utf8x]{inputenc}
\usepackage{listings}
\usepackage{graphicx}
\graphicspath{{media/}}
\date{\today}
\title{}
\author{}
\begin{document}
\section{\texorpdfstring{Bruno Sano}{Bruno Sano}}
\label{sec:bruno_sano}
\includegraphics[keepaspectratio=true,width=0.8\textwidth]{bie5782/01_curso_atual/alunos/trabalho_final/bsano00/foto_criada_em_2014-02-06_a_s_10.50}
Mestrando em ecologia, Instituto de Biociências, USP.
O Título do meu projeto é: “Taxa de herbivoria em diferentes modelos de restauração ambiental na Planície de Caraguatatuba”, orientado pela Professora Adriana Martini.
\href{media/bie5782/01_curso_atual/alunos/trabalho_final/bsano00/9.2.r}{9.2.r}
\href{media/bie5782/01_curso_atual/alunos/trabalho_final/bsano00/7b.r}{7b.r}
\href{media/bie5782/01_curso_atual/alunos/trabalho_final/bsano00/7.2.r}{7.2.r}
\href{media/bie5782/01_curso_atual/alunos/trabalho_final/bsano00/exer5.txt}{exer5.txt}
\href{media/bie5782/01_curso_atual/alunos/trabalho_final/bsano00/exer4.txt}{exer4.txt}
\href{media/bie5782/01_curso_atual/alunos/trabalho_final/bsano00/exe1.txt}{exe1.txt}
\subsubsection{\texorpdfstring{Trabalho Final}{Trabalho Final}}
\label{sec:trabalho_final}
\subparagraph{\texorpdfstring{Proposta A}{Proposta A}}
\label{sec:proposta_a}
Pensei em uma função que calculasse quantas mudas são necessárias para fazer um plantio em uma área de X ha ou metros quadrados.
O input poderia ser a metragem do terreno, o sistema de plantio (com nucleação, isoladas, ou misto), tamanho do espaçamento (se a pessoa quiser algum), tamanho das bordas.
O output seria o número de árvores.
\subparagraph{\texorpdfstring{Proposta B}{Proposta B}}
\label{sec:proposta_b}
Uma outra função seria para calcular quanto de água uma casa ou estabelecimento pode juntar se eles utilizarem calhas para capturar água da chuva.
O input poderia ser a precipitação média do local, cidade e a área do telhado.
O output seria quantidade de água economizada, economia em reais, tempo para recuperar o tempo investido, gráficos.
Bruno , achei a proposta A um tanto simples, pois, pelo que entendi, a sua função iria realizar essencialmente algumas operações de divisão. Acredito que você possa desenvolver a proposta B, que implicará em operações um pouco mais complexas e terá um output mais interessante.
----\emph{\href{mailto:[email protected]}{ Cristiane}}
\subsubsection{\texorpdfstring{Função coleta.agua.chuva}{Funao coletaaguachuva}}
\label{sec:funao_coletaaguachuva}
\lstset{frame=single}
\begin{lstlisting}
coleta.agua.chuva=function(precip, ate, tt, dtr, per, ua, gma, pa) #argumentos da funcao ##
{
vol=round(precip*ate*0.8) ##volume de agua coletado=precipitacao*area do telhado*aproveitamento de coleta 80% ##
#Investimento dependendo do tipo de telhado#
if(tt=="UA") ##se o tipo do telhado for de 1 agua##
{
I=(per*23 + ##perimetro do telhado x valor do metro da calha##
(per/3)*4.85 + ##a cada 3m de calha precisa de um suporte x valor do suporte##
10.6+14+130+170+350+ ##valor emenda+pe+separador folha+separador fluxo+cisterna 1000L##
dtr*10 + ##distancia do telhado ate o resertorio x valor do cano##
1000) ##mao de obra##
}
if(tt=="DA") ##se o tipo do telhado for de 2 aguas##
{
I=(per*23 + ##perimetro do telhado x valor do metro da calha##
(per/3)*4.85 + ##a cada 3m de calha precisa de um suporte x valor do suporte##
10.6+14+20+130+170+350+(2*19.2)+ ##valor emenda+pe+esquadro+separador folha+separador fluxo+cisterna 1000L+2cabeceiras##
dtr*10 + ##distancia do telhado ate o resertorio x valor do cano##
1000) ##mao de obra##
}
if(tt=="QA") ##se o tipo do telhado for de 4 aguas##
{
I=(per*23 + ##perimetro do telhado x valor do metro da calha##
(per/3)*4.85 + ##a cada 3m de calha precisa de um suporte x valor do suporte##
10.6+14+(4*20)+130+170+350+ ##valor emenda+pe+esquadro+separador folha+separador fluxo+cisterna 1000L##
dtr*10 + ##distancia do telhado ate o resertorio x valor do cano##
1000) ##mao de obra##
}
##uso da agua##
if(ua=="E") ##se o uso da agua for para uso externo##
{
If=round(I) ##Investimento final é igual ao investimento inicial##
econ=round((gma*0.15*pa)*12*2) ##economia é igual ao gasto mensal de agua x a porcentagem do consumo da casa que pode ser substituido pela agua coleta x preco da agua x 12 meses x economia com o esgoto##
tempr= round(If/econ) ##tempo de retorno do investimento e igual ao investimento dividido pela economia em anos##
}
if(ua=="ED") ##se o uso da agua for para uso externo e descarga do banheiro##
{
If=round((I+300+200+(dtr*10+1000))) ##o investimento é igual ao investimento inicial+preco de outra caixa+preco da bomba+preco canos+custo com reforma##
econ=round((gma*0.40*pa)*12*2) ##economia é igual ao gasto mensal de agua x a porcentagem do consumo da casa que pode ser substituido pela agua coleta x preco da agua x 12 meses x economia com o esgoto##
tempr= round(If/econ) ##tempo de retorno do investimento e igual ao investimento dividido pela economia em anos##
}
resultado=data.frame("vol.colet(L)/ano"=vol,"Invest(R$)"=If,"Economia(R$)/ano"=econ,"tempo.retor(anos)"=tempr,row.names = NULL)
return(resultado)
}
\end{lstlisting}
\subsubsection{\texorpdfstring{Help coleta.agua.chuva}{Help coletaaguachuva}}
\label{sec:help_coletaaguachuva}
\lstset{frame=single}
\begin{lstlisting}
coleta.agua.chuva package: nenhum R Documentation
Cáculo do investimento e tempo de retorno financeiro para captação de água da chuva.
Description:
coleta.agua.chuva é uma função que calcula o volume de água da chuva que pode ser captada pelo telhado do imóvel, o investimento necessário para implementar o projeto, a economia gerada e o tempo de retorno do investimento financeiro, a partir de valores de mercado do ano de 2014.
Usage:
coleta.agua.chuva(precip, ate, tt, dtr, per, ua, gma, pa)
Arguments:
precip precipitação media mensal da região (mm).
ate área do telhado (m2).
tt tipo de telhado do imovel. "UA"=uma água, "DA"=duas águas,"QA"=quatro águas.
dtr distancia entre o telhado e o reservatorio (m).
per perímetro do telhado em que a água pode ser captada(m).
ua tipo de utilização da água coleta. "E"=uso externo, "ED"=uso externo e descarga.
gma gasto mensal de água (mil L).
pa preco da água na região (R$)
Details:
Para calcular o volume de água que pode ser captado pelo telhado basta multiplicar a precipitação media do local pela área do telhado. Porem temos que considerar que não é possível utilizar 100% desse volume, cerca de 20% é descartada.
Tipo de telhado:
uma água-telhado que é possível captar água em apenas um dos lados
duas águas-telhado que é possível captar água em dois lados
quatro águas-telhado que é possível captar água em quatro lados
O tipo de telhado influencia o material utilizado para implementar o projeto, consequentemente, no calculo do investimento.
Uso da água:
"E"-Uso da água nas partes externas do imóvel
"ED"-Uso da água nas partes externas e nas descargas dos banheiros
O tipo de uso da água influencia a mão-de-obra, reforma da casa, e material utilizado no projeto, consequentemente, no calculo do investimento.
O calculo do preço da mão-de-obra e materiais utilizados tiveram como base a cidade de São Paulo.
O tempo de retorno não leva em consideração a inflação ao longo dos anos.
Values:
coleta.agua.chuva retorna um data frame com o volume(L) de água que pode ser coletado por mês/ano, o investimento(R$) para implementar o projeto, a economia(R$) por ano e o tempo para recuperar o investimento(anos).
Author:
Bruno Sano
References:
ANAUGER - Disponível em: <http://www.anauger.com.br> Acesso em: 15/05/2014
AQUASTOCK - Água da Chuva. Sistema de Reaproveitamento da Água da Chuva. Disponível em: <http://www.engeplasonline.com.br> Acesso em: 15/05/2014.
Companhia de Saneamento Básico do Estado de São Paulo - SABESP . Disponível em:<www.sabesp.com.br> Acesso 15/05/2014.
Instituto Nacional de Meteorologia - INMET. Disponível em <http://www.inmet.gov.br> Acesso em: 15/05/2014. .
Examples:
coleta.agua.chuva(125, 60, "QA", 5, 18, "E", 20,4)
coleta.agua.chuva(115,80,"DA",5,30,"ED",20,3)
coleta.agua.chuva(125, 60, "UA", 5, 18, "E", 20,4)
coleta.agua.chuva(115,80,"DA",5,30,"ED",20,3)
\end{lstlisting}
\end{document}
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https://www.apmep.fr/IMG/tex/Corrige-brevet-Polynesie-sept-2012.tex
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apmep.fr
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CC-MAIN-2021-39
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| 676,291,416 | 7,134 |
\documentclass[10pt]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{fourier}
\usepackage[scaled=0.875]{helvet}
\renewcommand{\ttdefault}{lmtt}
\usepackage{amsmath,amssymb,makeidx}
\usepackage[normalem]{ulem}
\usepackage{fancybox}
\usepackage{tabularx}
\usepackage{ulem}
\usepackage{pifont}
\usepackage{dcolumn}
\usepackage{textcomp}
\usepackage{lscape}
\usepackage{multirow}
\newcommand{\euro}{\eurologo{}}
\usepackage{graphics}
\usepackage{pstricks,pst-plot,pstricks-add}
\setlength\paperheight{297mm}
\setlength\paperwidth{210mm}
\setlength{\textheight}{23,5cm}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}
\renewcommand{\theenumi}{\textbf{\arabic{enumi}}}
\renewcommand{\labelenumi}{\textbf{\theenumi.}}
\renewcommand{\theenumii}{\textbf{\alph{enumii}}}
\renewcommand{\labelenumii}{\textbf{\theenumii.}}
\newcommand{\vect}[1]{\mathchoice%
{\overrightarrow{\displaystyle\mathstrut#1\,\,}}%
{\overrightarrow{\textstyle\mathstrut#1\,\,}}%
{\overrightarrow{\scriptstyle\mathstrut#1\,\,}}%
{\overrightarrow{\scriptscriptstyle\mathstrut#1\,\,}}}
\def\Oij{$\left(\text{O},~\vect{\imath},~\vect{\jmath}\right)$}
\def\Oijk{$\left(\text{O},~\vect{\imath},~\vect{\jmath},~\vect{k}\right)$}
\def\Ouv{$\left(\text{O},~\vect{u},~\vect{v}\right)$}
\setlength{\voffset}{-1,5cm}
\usepackage{fancyhdr}
\usepackage[dvips]{hyperref}
\hypersetup{%
pdfauthor = {APMEP},
pdfsubject = {Corrigé du brevet des collèges},
pdftitle = {Polynésie septembre 2012},
allbordercolors = white}
\usepackage[frenchb]{babel}
\usepackage[np]{numprint}
\begin{document}
\setlength\parindent{0mm}
\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small Corrigé du brevet des collèges}
\lfoot{\small{Polynésie}}
\rfoot{\small{septembre 2012}}
\renewcommand \footrulewidth{.2pt}
\pagestyle{fancy}
\thispagestyle{empty}
\begin{center}
{\Large \textbf{\decofourleft~Corrigé du brevet des collèges Polynésie ~\decofourright\\
septembre 2012}}
\vspace{0,5cm}
\textbf{Durée : 2 heures} \end{center}
\vspace{0,5cm}
\textbf{\large Activités numériques \hfill 12 points}
\medskip
\textbf{Exercice 1 :}
\medskip
%On donne le programme de calcul suivant :
%
%\medskip
%
%\begin{center}
%\begin{tabular}{|@{~$\bullet~~$}l l|}\hline
%&Choisir un nombre.\\
%&Lui ajouter 1.\\
%&Calculer le carré de cette somme.\\
%&Enlever 16 au résultat obtenu.\\ \hline
%\end{tabular}
%\end{center}
\begin{enumerate}
\item
\begin{enumerate}
\item %Vérifier que, lorsque le nombre de départ est 4, on obtient comme résultat $9$.
$4 \to 4 + 1 = 5 \to 5^2 = 25 \to 25 - 16 = 9$.
\item %Lorsque le nombre de départ est $(- 3)$. quel résultat obtient-on ?
$- 3 \to -3 + 1 = - 2 \to (- 2)^2 = 4 \to 4 - 16 = - 12$.
\item %Le nombre de départ étant, exprimer le résultat final en fonction de $x$,
%On appelle $P$ cette expression.
$x \to x + 1 \to (x + 1 )^2 \to (x + 1)^2 - 16 = P$
\item %Vérifier que $P = x^2 + 2x - 15$.
$P = (x + 1)^2 - 16 = x^2 + 2x + 1 - 16 = x^2 + 2x - 15$.
\end{enumerate}
\item
\begin{enumerate}
\item %Vérifier que $(x - 3)(x + 5) = P$.
on a $P = (x + 1)^2 - 16 = (x + 1)^2 - 4^2 = (x + 1 + 4)(x + 1 - 4) = (x + 5)(x - 3)$.
\item %Quels nombres peut-on choisir au départ pour que le résultat final soit $0$ ?
Avec la forme factorisée on voit que $P = 0$ si $x + 5 = 0$ ou si $x - 3 = 0$, soit pour $x = - 5$ ou $x = 3$.
%Justifier votre réponse.
\end{enumerate}
\end{enumerate}
\medskip
\textbf{Exercice 2 :}
\medskip
%Cet exercice est un questionnaire à choix multiples.
%
%Pour chaque question quatre réponses sont proposées mais \textbf{une seule est exacte}.
%
%Pour chacune des questions, écrire sur votre copie le numéro de la question et la lettre A, B, C ou D correspondant à la réponse choisie. Aucune justification n'est demandée.
%
%\medskip
%
%\begin{tabularx}{\linewidth}{|>{\small}m{4cm}|*{4}{>{\centering \arraybackslash}X|}}\hline
%Question &Réponse A &Réponse B &Réponse C &Réponse D\\ \hline
%\textbf{1.} L'écriture sous forme scientifique de $10^2 \times 21 \times 10^{- 7}$ est :&
%$21 \times 10^{-3}$& $2,1 \times 10^{9}$& $2,1 \times 10^{- 4}$& \small $0,21 \times 10^{- 3}$\\ \hline
%\textbf{2.} Le premier quartile Q$_{1}$ de la série de valeurs : 58 ; 55 ; 61 ; 70 ;
%61 ; 65 ; 58 ; 55 ; 72 est :& 61&58&55&2\\ \hline
%\textbf{3.} $\sqrt{500}$ est égale à :&
%$10\sqrt{5}$&$100\sqrt{5}$&$22,36$& $50$\\ \hline
%\textbf{4.} Les solutions de l'inéquation
%$- 2x + 5 \geqslant 7$ sont les nombres $x$ tels que : &$x \geqslant 1$&$x \leqslant 1$&$x \geqslant - 1$&$x \leqslant - 1$\\ \hline
%\end{tabularx}
\begin{enumerate}
\item $10^2 \times 21 \times 10^{- 7} = 21 \times 10^{- 5} = 2,1 \times 10^{- 4}$. Réponse C.
\item Réponse B.
\item $\sqrt{500} = \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10\sqrt{5}$. Réponse A.
\item $- 2x + 5 \geqslant 7$ d’où $5 - 7 \geqslant 2x$ ou $- 2 \geqslant 2x$ et $- 1 \geqslant x$ ou encore $x \leqslant - 1$. Réponse D.
\end{enumerate}
\medskip
\textbf{Exercice 3 :}
\medskip
%Une usine de Moorea fabrique du jus de fruits.
%
%Soit $C$ une fonction qui, à une quantité de jus fabriquée en litre(s) associe le coût de fabrication en F.
%
%On a représenté ci-dessous la fonction $C$ pour une quantité de jus comprise entre $0$ et 130 litres.
%
%\medskip
%
%\begin{center}
%\psset{xunit=0.085cm,yunit=0.0085cm}
%\begin{pspicture}(-5,-50)(135,800)
%\psaxes[linewidth=1.5pt,Dx=10,Dy=100]{->}(0,0)(-5,-50)(135,800)
%%\psplot[plotpoints=8000,linewidth=1.25pt,linecolor=blue]{0}{130}{x 3 exp 3 div x dup mul 67.5 mul sub x 2525 mul add 0.019017 neg mul 600 add}
%\pscurve[linewidth=1.25pt,linecolor=blue](0,600)(10,660)(20,678)(30,680)(40,662)(50,628)(60,575)(70,525)(80,470)(90,430)(100,400)(110,395)(120,410)(130,455)
%\end{pspicture}
%\end{center}
%
%À l'aide du graphique ci-dessus, répondre aux questions suivantes :
%
%\medskip
\begin{enumerate}
\item
\begin{enumerate}
\item %Donner le coût de fabrication de 100 litres de jus.
La droite d’équation $x = 100$ coupe la courbe en un point d’ordonnée 400.
\item %Pour quelle(s) quantité(s) de jus, le coût de fabrication est-il supérieur à 550~F ?
Le coût de fabrication est supérieur à 550~F pour $0 \leqslant x \leqslant 65$.
\end{enumerate}
\item
\begin{enumerate}
\item %Donner l'image de 85 par la fonction $C$.
On lit à peu près $C(85) = 450$.
\item On lit $C(75) = 500$.
\item %Donner le(s) antécédent(s) de $600$ par la fonction $C$.
Les antécédents de 600 semblent être 0 et 50.
\end{enumerate}
\end{enumerate}
\vspace{0,5cm}
\textbf{\large Activités géométriques \hfill 12 points}
\medskip
\textbf{Exercice 1 :}
\medskip
Un sculpteur fabrique un \og umete \fg{} en bois (récipient) ayant la forme d'une demi-sphère de rayon 15~cm (\emph{l'épaisseur du umete est supposée négligeable}).
\medskip
\begin{enumerate}
\item %Vérifier que la valeur exacte du volume du umete est égale à \np{2250}$\pi$~ cm$^3$.
Le volume d’une boule étant $\dfrac{4}{3}\pi R^3$, celle de l’umete est $\dfrac{2}{3}\pi \times 15^3 = 2 \times 15 \times 15 \times 5\pi = \np{2250}\pi$~cm$^3$.
\item %Dans cette question, toute trace de recherche, même incomplète, sera prise en compte dans l'évaluation.
%Pourra-t-on verser dans ce umete 7 litres de lait de coco sans déborder ? Justifier.
$\np{2250}\pi \approx \np{7068,58}$~cm$^3$ soit plus de 7,06~dm$^3$ ou 7,06~L. L’umete ne débordera pas.
%\begin{tabular}{l l l}
%\textbf{Rappels}&$\bullet~~$&Le volume d'une boule de rayon $r$ est : $V = \dfrac{4}{3}\pi r^3$.\\
%&$\bullet~~$&1 litre = 1 dm$^3$\\
%\end{tabular}
\end{enumerate}
\medskip
\textbf{Exercice 2 :}
\medskip
%\textbf{Dans tout cet exercice, les figures ne sont pas à l'échelle.}
%
%Moana décide de participer à un triathlon. Il prend connaissance des parcours des. trois épreuves : natation, cyclisme et course à pied.
%
%\medskip
\begin{enumerate}
\item 1\up{re} épreuve : la natation
%Le départ se fait sur la plage au point D, Les triathlètes doivent contourner une bouée située au point B, puis rejoindre la plage au point A.
%
%On donne AB = 800~m et AD = \np{2341}~m et (AB) $\perp$ (BD).
%
%\medskip
%
%\begin{center}
%\psset{unit=1cm}
%\begin{pspicture}(8,4)
%\def\vague{\pscurve(0,0)(0.5,0.2)(1,-0.2)(1.5,0.2)}
%\psline[linestyle=dashed](1,1)(5,3.5)(6.5,1)%DBA
%\psline(1,1)(6.5,1)
%\uput[d](1,1){D} \uput[ur](5,3.5){B} \uput[d](6.5,1){A}
%\rput(3.75,0.5){Plage}
%\rput(7,1.8){\vague}
%\rput(1,2){\vague} \rput(1.8,3.5){\vague} \rput(1.8,1.3){\vague} \rput(3.7,2){\vague}\rput(6.5,3){\vague} \rput(7,4){\vague}
%\end{pspicture}
%\end{center}
\begin{enumerate}
\item %Calculer la longueur du parcours \og natation \fg{} représenté par DB + BA. Donner la réponse arrondie au mètre.
Le triangle ABD est rectangle en B ; le théorème de Pythagore permet d’écrire :
$\text{AD}^2 = \text{AB}^2 + \text{BD}^2$ d’où $\text{BD}^2 = \text{AD}^2 - \text{AB}^2 = \np{2341}^2 - 800^2 = \np{4840281}$.
Donc BD $ = \sqrt{4840281} \approx \np{2200,06}$ soit \np{2200} au mètre près.
Le parcours natation fait donc AB + BD = 800 + \np{2200} = \np{3000}~m, soit 3~km.
\item %Calculer $\sin \widehat{\text{ADB}}$ ; en déduire la mesure de l'angle $\widehat{\text{ADB}}$ arrondie au degré.
Dans le triangle ABD rectangle en B, on a $\sin \widehat{\text{ADB}} = \dfrac{\text{AB}}{\text{AD}} = \dfrac{800}{\np{2341}} \approx 0,342$.
La calculatrice donne $\widehat{\text{ADB}} \approx 19,98$ soit 20\degres au degré près.
\end{enumerate}
\item 2\up{e} épreuve :
%\parbox{0.55\linewidth}{Le circuit \og vélo \fg{} est un arc de cercle de centre O.\\
%Le départ a lieu au point M et l'arrivée au point N.\\
%Un spectateur situé en O voit le premier tronçon $\widearc{\text{MC}}$
%parcouru par le cycliste sous un angle de $81,5~\degres$).\\
% On souhaite déterminer la mesure de l'angle $\widehat{\text{MNC}}$.}\hfill
%\parbox{0.4\linewidth}{\psset{unit=1cm}
%\begin{pspicture}(-2,-2)(2,2)
%\SpecialCoor
%\psarc(0,0){2}{-180}{55}
%\psline[linestyle=dashed](0;0)(2;180)(2;55)(2;-95)(0;0)
%\SpecialCoor
%\uput[u](2;180){M}\uput[ur](2;55){N}\uput[dl](2;-95){C}\uput[ur](0;0){O}
%\psarc(2;55){0.3}{-155}{-105}\psarc(0;0){0.3}{-180}{-101}
%\rput(0.7;-134){81,5~\degres}
%\end{pspicture}
%}
%
\begin{enumerate}
\item
\textbf{Dans cet exercice}, pour déterminer la mesure de l'angle $\widehat{\text{MNC}}$, laquelle des quatre propriétés suivantes faut-il utiliser ? Choisir et \textbf{recopier la propriété sur votre copie}.
%
%\medskip
%
%\setlength\parindent{8mm}
%\begin{itemize}
%\item[$\bullet~~$] Si deux angles inscrits dans un cercle interceptent le même arc, alors ils ont la même mesure,
%\item[$\bullet~~$] Si un triangle est inscrit dans un cercle et a pour côté un diamètre de ce cercle, alors c'est un triangle rectangle.
%\item[$\bullet~~$] Dans un cercle, si un angle inscrit et un angle au centre interceptent le même arc, alors la mesure de l'angle inscrit est la moitié de la mesure de l'angle au centre,
%\item[$\bullet~~$] Dans un triangle isocèle, les angles à la base ont la même mesure.
%\end{itemize}
%\setlength\parindent{0mm}
Dans un cercle, si un angle inscrit et un angle au centre interceptent le même arc, alors la mesure de l'angle inscrit est la moitié de la mesure de l'angle au centre.
\item %Donner alors la mesure de l'angle $\widehat{\text{MNC}}$.
D’après la propriété précédente, on a $\widehat{\text{MNC}} = \dfrac{\widehat{\text{MOC}}}{2} = 40,75$\degres.
\end{enumerate}
\item 3\up{e} épreuve : la course à pied
%\begin{center}
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%\uput[u](4,1){10~km}\uput[d](4,0){10~km}
%\end{pspicture}
%\end{center}
%
%Le circuit \og course à pied \fg{)} est un aller-retour de 20~km (10 km à l'aller et 10 km au retour).
%
%Pour le trajet aller, qui s'effectue dans le sens du vent, Moana estime que sa vitesse moyenne sera de 16 km/h.
%
%Pour le trajet retour,à cause du vent de face et de la fatigue Moana pense courir à la vitesse moyenne de 10 km/h.
%
%Peut-on affirmer que sa vitesse moyenne sera de 13 km/h sur l'ensemble du circuit \og course à pied \fg ? Justifier votre réponse.
%\medskip
%
%\emph{L'évaluation de cette question tiendra compte des observations et étapes de recherche, même incomplètes ; les faire apparaître sur votre copie.}
Le temps $t_a$ de l’aller est égal à $\dfrac{10}{16} = \dfrac{5}{8} = 0,625$~(h).
Le temps $t_r$ du retour est égal à $t_r = \dfrac{10}{10} = 1$~(h).
Il pense donc faire les 20 km en 1,625~h, soit à une vitesse moyenne de $\dfrac{20}{1,625} \approx 12,3$ km/h donc moins que les 13 km/h espérés.
\end{enumerate}
\vspace{0,5cm}
\textbf{\large Problème \hfill 12 points}
\medskip
\begin{center}\textbf{Partie A} \end{center}
%Un bijoutier achète un lot de 220 perles de Tahiti.
%
%Un contrôleur qualité s'intéresse à leurs formes (ronde ou baroque) et à leurs couleurs (grise ou verte).
%
%\medskip
%
%\setlength\parindent{8mm}
%\begin{itemize}
%\item[$\bullet~~$] 35\,\% des perles sont de couleur verte, et parmi celles~ci $13$ sont de forme ronde.
%\item[$\bullet~~$] Il y a 176 perles de forme baroque,
%\end{itemize}
%\setlength\parindent{0mm}
%
%\medskip
%
%Il note les résultats dans la feuille de calcul ci-dessous
%
%\medskip
\begin{center}
\begin{tabularx}{0.7\linewidth}{|c|*{4}{>{\centering \arraybackslash}X|}}\hline
&A &B &C &D\\ \hline
1 & &Rondes &Baroques&Total\\ \hline
2 &Grises &31 &112 &143\\ \hline
3 &Vertes &13 &64 &77\\ \hline
4 &Total &44 &176 & 220\\ \hline
\end{tabularx}
\end{center}
\begin{enumerate}
\item %Pour obtenir le nombre de perles vertes à partir des informations données dans l'énoncé, quelle formule doit-il saisir en D3 ? Parmi les quatre formules proposées, recopier sur votre copie la bonne formule :
%\medskip
%\begin{tabularx}{\linewidth}{*{4}{>{\centering \arraybackslash}X}}
%\fbox{=D4*1,35}&\fbox{220*35 / 100}&\fbox{=D4 * 0,35}&\fbox{=B3 + C3}\\
%\end{tabularx}
\fbox{=D4 * 0,35}
\item %Compléter le tableau ci-dessus.
Voir au dessus.
\item %On choisit au hasard une perle de ce lot.
\begin{enumerate}
\item %Quelle est la probabilité pour que cette perle soit de forme baroque ?
La probabilité est égale à $\dfrac{176}{220} = \dfrac{16 \times 11}{20 \times 11} = \dfrac{16}{20} = \dfrac{4}{5} = 0,8 = 80\,\%$.
\item %Quelle est la probabilité de tirer une perle baroque verte ?
La probabilité est égale à $\dfrac{77}{220} = \dfrac{7 \times 11}{20 \times 11} = \dfrac{7}{20} = 0,35 = 35\,\%$.
\end{enumerate}
\end{enumerate}
\begin{center}\textbf{Partie B} \end{center}
%Ce bijoutier se fournit chez un perliculteur de l'archipel des Gambier.
%
%L'acheminement vers Tahiti des lots de perles, s'effectue selon deux tarifs :
%
%\medskip
%
%\setlength\parindent{8mm}
%\begin{itemize}
%\item[$\bullet~~$] Tarif \og Ho' \fg{} : \np{2300}~F par lot.
%\item[$\bullet~~$] Tarif \og Piti \fg{} : \np{7000}~F fixe et $900$~F par lot
%\end{itemize}
%\setlength\parindent{0mm}
%
%\medskip
\begin{enumerate}
\item %Calculer, pour chaque tarif, le montant de l'acheminement de 4 lots.
Tarif Ho’ : $4 \times \np{2300} = \np{9200}$~F ;
Tarif Piti : $\np{7000} + 4 \times 900 = \np{10600}$~F.
\item %On note $x$ le nombre de lots de perles expédié(s).
\begin{enumerate}
\item %Exprimer, en fonction de $x$, le montant de l'acheminement avec le tarif Ho'.
Le tarif est égal à $\np{2300}x$
\item %Exprimer, en fonction de $x$, le montant de l'acheminement avec le tarif Piti.
Le tarif est égal à $\np{7000} + 900x$
\end{enumerate}
\newpage
\begin{center}\textbf{Cette page est à rendre avec la copie}
\end{center}
\item
\begin{enumerate}
\item %Soit $f$ et $g$ les deux fonctions définies par :
%\[f : x \longmapsto \np{2300}x \quad \text{et}\quad g \longmapsto 900x + \np{7000}.\]
%
%\textbf{Dans le repère ci-dessous} (à rendre avec la copie), construire les représentations graphiques des fonctions $f$ et $g$.
\item %Par lecture graphique, déterminer à partir de combien de lots expédiés, le tarif Piti est plus avantageux pour le bijoutier que le tarif Ho'e.
On constate que pour $x >5$ le tarif Piti devient le moins onéreux.
\medskip
%\emph{Vous ferez apparaître, sur le dessin, les tracés nécessaires pour justifier votre réponse.}
\end{enumerate}
\end{enumerate}
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Return-Path: <[email protected]>
Delivery-Date:
Received: from ouse.cl.cam.ac.uk (user pa10007 (rfc931)) by swan.cl.cam.ac.uk
with SMTP (PP-6.5) to cl; Tue, 21 Mar 1995 14:27:17 +0000
To: [email protected]
From: [email protected]
Subject: Booklist
Date: Tue, 21 Mar 1995 14:27:04 +0000
Sender: [email protected]
Message-ID: <"swan.cl.cam.:253320:950321142738"@cl.cam.ac.uk>
Sebastian
Please may I have the source of the booklist for updating and
issue to enquiries.
Also 0 201 52372 8 Thinking in Postscript by Gleen reid is out of
print and Addison-Wesley say unlikely to be reprinted.
Please can you add this to my report for Baskerville and delete
from the booklist.
Thanks
Peter
PS Don't forget I still need unzip for discs 8 and 9, I have
an outstanding request for a copy.
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Note: This is a working paper which will be expanded/updated frequently.
The directory
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has a pdf copy of this article and the complete Rmd file.
\section{Problem}\label{problem}
We study differentiability of the multidimensional scaling loss function
rStress (De Leeuw (2016)), defined as
\begin{equation}
\sigma_r(x):=\sum_{i=1}^n w_i(\delta_i-(x'A_ix)^r)^2
\end{equation}
for some \(r>0\). Here the \(w_i\) are positive weights and the
\(\delta_i\) are positive dissimilarities. The matrices \(A_i\) are
positive semi-definite, and the quantities \(x'A_ix\) are squared
distances.
Clearly if \(x'A_ix>0\) for all \(i\) the loss function is
differentiable. De Leeuw (1984) proves directional differentiability for
\(r=\frac12\) and he shows that at a local minimum we generally have
\(x'A_ix>0\). We investigate if and how this results generalizes to
\(\sigma_r\).
\section{Directional Derivatives}\label{directional-derivatives}
Define the directional derivative \[
d\sigma_r(x,y):=\lim_{\epsilon\downarrow 0}\frac{\sigma_r(x+\epsilon y)-\sigma_r(x)}{\epsilon}.
\] For our computations we need
\begin{align*}
I_+(x)&:=\{i\mid x'A_ix>0\},\\
I_0(x)&:=\{i\mid x'A_ix=0\}.
\end{align*}
Then
\begin{multline*}
\frac{\sigma_r(x+\epsilon y)-\sigma_r(x)}{\epsilon}=-4r\sum_{i\in I_+}w_i(\delta_i-(x'A_ix)^r)(x'A_ix)^{r-1}y'A_ix\\
-2\epsilon^{2r-1}\sum_{i\in I_0}w_i\delta_i(y'A_iy)^r+\epsilon^{4r-1}\sum_{i\in I_0}w_i(y'A_iy)^{2r}
+\frac{o(\epsilon)}{\epsilon},
\end{multline*}
and thus \[
d\sigma_r(x,y)=
\begin{cases}
-4r\sum_{i=1}^nw_i(\delta_i-(x'A_ix)^r)(x'A_ix)^{r-1}y'A_ix&\text { if }r>\frac12,\\
-4r\sum_{i\in I_+}w_i(\delta_i-(x'A_ix)^r)(x'A_ix)^{r-1}y'A_ix-2\sum_{i\in I_0}w_i\delta_i(y'A_iy)^r&\text { if }r=\frac12,\\
+\infty&\text{ if }r<\frac12.
\end{cases}
\]
\section{Results}\label{results}
From our computations we derive the following results.
\textbf{Theorem 1:} If \(r>\frac12\) then \(\sigma_r\) is differentiable
at \(x\). If \(\sigma_r\) has a local minimum at \(x\) then \[
\sum_{i=1}^nw_i\delta_i(x'A_ix)^{r-1}A_ix=\sum_{i=1}^nw_i(x'A_ix)^{2r-1}A_ix.
\]
\textbf{Theorem 2:} If \(r=\frac12\) then \(\sigma_r\) is directionally
differentiable at \(x\) in every direction \(y\). If \(\sigma_r\) has a
local minimum at \(x\) then \[
\sum_{i\in I_+(x)}w_i\delta_i(x'A_ix)^{r-1}A_ix=\sum_{i\in I_+(x)}w_i(x'A_ix)^{2r-1}A_ix.
\] and \(I_0(x)=\emptyset\).
\textbf{Theorem 3:} If \(r<\frac12\) then \(\sigma_r\) is directionally
differentiable only in those directions \(y\) with \(y'A_iy=0\) for all
\(i\in I_0(x)\).
Thus for \(r=\frac12\) we have non-zero distances and differentiability
at local minima, for \(r>\frac12\) it is quite possible that local
minima with zero distances exist, and for \(r>\frac12\) rStress is not
even directionally differentiable at points with zero distances.
\section{Local Maximum}\label{local-maximum}
We can also generalize a result of De Leeuw (1993) to rStress.
\textbf{Theorem 4:} \(\sigma_r\) has a local maximum at \(x\) if and
only if \(x=0\).
\textbf{Proof:} If \(x=0\) then
\[\sigma_r(x+\epsilon y)-\sigma_r(x)=-2\epsilon^{2r}\left\{\sum_{i=1}^nw_i\delta_i(y'Ay)^r-\frac12\epsilon^{2r}\sum_{i=1}^nw_i(y'A_iy)^{2r}\right\}.\]
It follows that if \[
\frac12\epsilon^{2r}\leq\frac{\sum_{i=1}^nw_i\delta_i(y'Ay)^r}{\sum_{i=1}^nw_i(y'A_iy)^{2r}}
\] we have \(\sigma(x+\epsilon y)-\sigma(x)\leq 0\). So, although
\(\sigma_r\) may not even directionally differentiable at \(x=0\), it
does decrease in all directions and is thus a local minimum.
Converse, suppose \(\sigma_r\) has a local maximum at \(x\not= 0\). Then
\[
\sigma_r(\epsilon x)=\sum_{i=1}^nw_i\delta_i^2-2\theta\sum_{i=1}^nw_i\delta_i(x'Ax)^r+\theta^2\sum_{i=1}^nw_i(x'A_ix)^{2r},
\] with \(\theta:=\epsilon^{2r}\). Thus \(\sigma_r\) is a convex
quadratic in \(\theta\) and it cannot have a local maximum on the ray
through \(x\). \textbf{QED} \$\$
\section{NEWS}\label{news}
001 01/14/16 -- First upload
002 01/15/16 -- Added local maximum result
003 02/08/16 -- Corrected some typos
\section*{References}\label{references}
\addcontentsline{toc}{section}{References}
\hypertarget{refs}{}
\hypertarget{ref-deleeuwux5fAux5f84f}{}
De Leeuw, J. 1984. ``Differentiability of Kruskal's Stress at a Local
Minimum.'' \emph{Psychometrika} 49: 111--13.
\url{http://www.stat.ucla.edu/~deleeuw/janspubs/1984/articles/deleeuw_A_84f.pdf}.
\hypertarget{ref-deleeuwux5fRux5f93c}{}
---------. 1993. ``Fitting Distances by Least Squares.'' Preprint Series
130. Los Angeles, CA: UCLA Department of Statistics.
\url{http://www.stat.ucla.edu/~deleeuw/janspubs/1993/reports/deleeuw_R_93c.pdf}.
\hypertarget{ref-deleeuwux5fEux5f16a}{}
---------. 2016. ``Minimizing rStress Using Majorization.''
\url{http://rpubs.com/deleeuw/142619}.
\end{document}
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%&LaTeX
\documentclass{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{Arsentieva2017} Ars{\'e}ntieva, N. (2017). La filosof{\'\i}a del eros, alcahueter{\'\i}a y magia en {\textquotedblleft}La Celestina{\textquotedblright} y {\textquotedblleft}El caballero de Olmedo{\textquotedblright}. En F. Toro Ceballos (Ed.), \textit{Due{\~n}as, cortesanas y alcahuetas: {\textquotedblleft}Libro de buen amor{\textquotedblright}, {\textquotedblleft}La Celestina{\textquotedblright} y {\textquotedblleft}La lozana andaluza{\textquotedblright}} (pp. 25--34). Alcal{\'a} la Real: Ayuntamiento de Alcal{\'a} la Real.
\end{thebibliography}
\end{document}
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\immediate\write18{tex spath3.dtx}
\documentclass{ltxdoc}
\usepackage[T1]{fontenc}
\usepackage{trace}
\usepackage{lmodern}
\usepackage{morefloats}
\usepackage{tikz}
\usetikzlibrary{knots}
\usepackage[numbered]{hypdoc}
\definecolor{lstbgcolor}{rgb}{0.9,0.9,0.9}
\usepackage{listings}
\lstloadlanguages{[LaTeX]TeX}
\lstset{breakatwhitespace=true,breaklines=true,language=TeX}
\usepackage{fancyvrb}
\newenvironment{example}
{\VerbatimEnvironment
\begin{VerbatimOut}{example.out}}
{\end{VerbatimOut}
\begin{center}
\setlength{\parindent}{0pt}
\fbox{\begin{minipage}{.9\linewidth}
\lstset{breakatwhitespace=true,breaklines=true,language=TeX,basicstyle=\small}
\lstinputlisting[]{example.out}
\end{minipage}}
\fbox{\begin{minipage}{.9\linewidth}
\centering
\input{example.out}
\end{minipage}}
\end{center}
}
\providecommand*{\url}{\texttt}
\GetFileInfo{spath3.sty}
\title{The \textsf{knots} Package: Documentation}
\author{Andrew Stacey \\ \url{[email protected]}}
\date{\fileversion~from \filedate}
\begin{document}
\maketitle
\section{Introduction}
The \Verb+knots+ package is a TikZ library for drawing knot (and similar) diagrams.
It provides a few useful styles and node shapes but its main contribution is the \Verb+knot+ environment.
The \Verb+knot+ environment allows you to draw some strands of a knot (or braid or tangle or whatever -- we shall use the imprecise term ``knot'' to refer to any similar diagram) and then to modify the crossings via a simple interface.
The main part of this package was developed in response to a question on the \href{http://tex.stackexchange.com}{TeX-SX} site by Jamie Vicary.
The original question was \href{http://tex.stackexchange.com/q/32125/86}{Braid diagrams in TikZ}.
Jamie's comment (quoted below) was the inspiration for the mechanism of the \Verb+knot+ environment.
\begin{quotation}
[It] would be really cool if it was possible to draw the curves, let TikZ calculate all the intersection points automatically, and then tell it to redraw the intersections according to an under/over specification... do you think this is in the realm of plausibility?
\end{quotation}
\section{Examples}
Let us begin with an example.
To use the library, simply load the \Verb+tikz+ package and add \Verb+knots+ to the list of TikZ libraries that you load.
For example, the following in your preamble would work:
\begin{verbatim}
\usepackage{tikz}
\usetikzlibrary{knots}
\end{verbatim}
Let's draw a simple tangle (actually a braid).
\begin{example}
\begin{tikzpicture}
\draw[red,ultra thick] (0,0) .. controls +(1,0) and +(-1,0) .. (2,1) .. controls +(1,0) and +(-1,0) .. (4,0);
\draw[blue,ultra thick] (0,1) .. controls +(1,0) and +(-1,0) .. (2,0) .. controls +(1,0) and +(-1,0) .. (4,1);
\end{tikzpicture}
\end{example}
Now a common way to draw crossings for knots is to draw a gap in the under strand through which the over strand passes.
One way to achieve this in TikZ is to draw the over strand twice, the first time with a thicker line width and the colour of the background.
We'll draw it twice, once with background a different colour to illustrate this.
\begin{example}
\begin{tikzpicture}
\draw[red,ultra thick] (0,0) .. controls +(1,0) and +(-1,0) .. (2,1) .. controls +(1,0) and +(-1,0) .. (4,0);
\draw[pink,double=blue,ultra thick,double distance=1.6pt] (0,1) .. controls +(1,0) and +(-1,0) .. (2,0) .. controls +(1,0) and +(-1,0) .. (4,1);
\draw[xshift=5cm,red,ultra thick] (0,0) .. controls +(1,0) and +(-1,0) .. (2,1) .. controls +(1,0) and +(-1,0) .. (4,0);
\draw[xshift=5cm,white,double=blue,ultra thick,double distance=1.6pt] (0,1) .. controls +(1,0) and +(-1,0) .. (2,0) .. controls +(1,0) and +(-1,0) .. (4,1);
\end{tikzpicture}
\end{example}
Now the problem with this method is that there is no way to draw the red and blue paths so that the blue is the over strand at the first crossing and the red at the second.
Either the blue path is always on top (as shown) or the red.
One way to resolve this is to split the paths and draw one of them in segments:
\begin{example}
\begin{tikzpicture}
\draw[red,ultra thick] (0,0) .. controls +(1,0) and +(-1,0) .. (2,1);
\draw[white,double=blue,ultra thick,double distance=1.6pt] (0,1) .. controls +(1,0) and +(-1,0) .. (2,0) .. controls +(1,0) and +(-1,0) .. (4,1);
\draw[white,double=red,double distance=1.6pt,ultra thick] (2,1) .. controls +(1,0) and +(-1,0) .. (4,0);
\end{tikzpicture}
\end{example}
Another method (employed by the \Verb+braids+ package) is to break the under path either side of the crossing and not draw it there.
This means that the order of drawing doesn't matter.
Both of these methods have their drawbacks (particularly for general knots as opposed to the more structured braids) in that they require a detailed knowledge of the pieces of the paths and the positions of the crossings.
As pointed out by Jamie Vicary in the above-{}quoted comment, TikZ should be able to compute these itself.
That's what this package does.
Let's do the above example using this package.
The main changes to the drawing are that we use the command \Verb+\strand+ rather than \Verb+\draw+ and we enclose it in the \Verb+knot+ environment.
When initially drawing the strands it is useful to provide the option \Verb+draft mode=strands+.
That's because the detailed computation can take a little time and so it is best only to do it when necessary.
So on first run through we get the following.
\begin{example}
\begin{tikzpicture}
\begin{knot}[
draft mode=strands
]
\strand[red,thick] (0,0) .. controls +(1,0) and +(-1,0) .. (2,1) .. controls +(1,0) and +(-1,0) .. (4,0);
\strand[blue,thick] (0,1) .. controls +(1,0) and +(-1,0) .. (2,0) .. controls +(1,0) and +(-1,0) .. (4,1);
\end{knot}
\end{tikzpicture}
\end{example}
Once we're happy with the positioning of the strands, we change the option \Verb+draft mode=strands+ to \Verb+draft mode=crossings+.
\begin{example}
\begin{tikzpicture}
\path (2,1.5) (2,-.5);
\begin{knot}[
draft mode=crossings,
clip width=5,
]
\strand[red,ultra thick] (0,0) .. controls +(1,0) and +(-1,0) .. (2,1) .. controls +(1,0) and +(-1,0) .. (4,0);
\strand[blue,ultra thick] (0,1) .. controls +(1,0) and +(-1,0) .. (2,0) .. controls +(1,0) and +(-1,0) .. (4,1);
\end{knot}
\end{tikzpicture}
\end{example}
The \Verb+\path (2,1.5) (2,-.5);+ is to extend the bounding box of the picture a little upwards.
The extra pieces are not used when computing the bounding box of the picture so that it doesn't change position on the page.
The extra information is that the strands and the crossings have been numbered.
The crossings have also been rendered (the \Verb+clip width+ option enlarges the crossing gap to make it more obvious).
Unfortunately, for both crossings the red path is on top.
We need to flip one of these crossings (the first).
To do this, we either use the command \Verb+flipcrossings+ or the \Verb+flip crossing+ key.
\begin{example}
\begin{tikzpicture}
\path (2,1.5) (2,-.5);
\begin{knot}[
draft mode=crossings,
clip width=5,
flip crossing=1,
]
\strand[red,ultra thick] (0,0) .. controls +(1,0) and +(-1,0) .. (2,1) .. controls +(1,0) and +(-1,0) .. (4,0);
\strand[blue,ultra thick] (0,1) .. controls +(1,0) and +(-1,0) .. (2,0) .. controls +(1,0) and +(-1,0) .. (4,1);
\end{knot}
\end{tikzpicture}
\end{example}
Once we're happy with it, we remove the \Verb+draft mode+ option to render it in its final form.
\begin{example}
\begin{tikzpicture}
\begin{knot}[
clip width=5,
flip crossing=1,
]
\strand[red,ultra thick] (0,0) .. controls +(1,0) and +(-1,0) .. (2,1) .. controls +(1,0) and +(-1,0) .. (4,0);
\strand[blue,ultra thick] (0,1) .. controls +(1,0) and +(-1,0) .. (2,0) .. controls +(1,0) and +(-1,0) .. (4,1);
\end{knot}
\end{tikzpicture}
\end{example}
Here's a more complicated example.
\begin{example}
\begin{tikzpicture}
\node (A) at (0,4) [draw,minimum width=30pt,minimum height=10pt,thick] {};
\begin{knot}[
clip width=5,
clip radius=8pt,
]
\strand [thick,only when rendering/.style={dashed}] (0,0)
to [out=up, in=down] (-1,1)
to [out=up, in=down] (0,2)
to [out=up, in=down] (-1.2,4)
to [out=up, in=down, looseness=0.7] (0,5.5)
to [out=up, in=down] (-2,7);
\strand [thick] (-1,0)
to [out=up, in=down] (1,2)
to [out=up, in=down] (A.south);
\strand [thick,blue] (1,0)
to [out=up, in=down] (-1,2)
to [out=up, in=down] (1.5,4)
to [out=up, in=right] (0,5.5)to [out=left, in=up] (-2,4)
to [out=down, in=up] (-2,0);
\strand [thick] (A.150)
to [out=up, in=down] (0.7,5.5)
to [out=up, in=down] (0,7);
\strand [thick] (A.30)
to [out=up, in=down] (-1,6)
to [out=up, in=down] (2,7);
\flipcrossings{6,2,9,5,11}
\end{knot}
\end{tikzpicture}
\end{example}
One feature about this example is the \Verb+only when rendering+ key.
The gaps are made by drawing the strand again with extra thickness in the background colour.
If the \Verb+dashed+ option were always in play for that strand, the gap would be dashed which would spoil the effect.
So the \Verb+only when rendering+ key gathers those options (such as a dash pattern) which should only be applied to the rendered strand and not to the redraw that creates the gap.
Here's another example.
\begin{example}
\newcommand{\motif}[1]{
to ++(180+#1:0.50) arc (270+#1:150+#1:0.15)
to ++( 60+#1:0.50) arc (-30+#1:150+#1:0.15)
to ++(240+#1:0.25) arc (150+#1:330+#1:0.25)
to ++( 60+#1:0.55) arc (150+#1: 30+#1:0.20)
}
\newcommand{\celticknot}{\motif{0}\motif{120}\motif{240}}
\begin{tikzpicture}
\begin{knot}[
line width=2pt,
line join=round,
clip width=2,
scale=5,
consider self intersections,
ignore endpoint intersections=false,
background color=white,
only when rendering/.style={
draw=red,
double=white,
double distance=6pt,
line cap=round,
}
]
\strand (0,0) \celticknot;
\flipcrossings{1,3,6,8,10}
\end{knot}
\end{tikzpicture}
\end{example}
In this case the strand is a single path.
In the standard case crossings are only considered between separate strands (since the algorithm used by TikZ means that a strand intersects itself infinitely often).
The key \Verb+consider self intersections+ gets round this by ``exploding'' the strand into segments and considering each as a separate path with regard to finding the intersections.
A path consists of a series of lines and B\'ezier cubics.
The ``explosion'' of a path uses this decomposition.
Unfortunately, even that is not always enough as it is possible for a B\'ezier cubic to self-intersect.
The \Verb+consider self intersections+ also splits these B\'ezier curves in two to ensure that this doesn't happen\footnote{Computing when this is strictly necessary is difficult so it splits more than it needs to to ensure that enough are done.}.
To disable this, use the \Verb+consider self intersections=no splits+ option.
This is the recommended option.
\begin{example}
\begin{tikzpicture}
\begin{knot}[
consider self intersections,
draft mode=crossings,
]
\strand (0,0) .. controls +(3,1) and +(-3,1) .. (1,0);
\end{knot}
\begin{knot}[
xshift=3cm,
consider self intersections=no splits,
draft mode=crossings,
]
\strand (0,0) .. controls +(3,1) and +(-3,1) .. (1,0);
\end{knot}
\end{tikzpicture}
\end{example}
Finally, given that TikZ has to do some heavy computation to find the intersections, it is worth considering using the \Verb+external+ library of TikZ to avoid having to do this on every run.
\section{Usage}
\subsection{The \texttt{knot} Environment}
\DescribeMacro{knot}
\DescribeMacro{\strand}
This package provides a \Verb+knot+ environment for including in a \Verb+tikzpicture+ to render a knot.
This takes an optional argument which is passed to \Verb+\tikzset+ and can be used to configure the knot.
Within that environment, specific strands are defined using the \Verb+\strand+ command (in place of a \Verb+\path+ or \Verb+\draw+).
Further keys can be specified on the strands.
\DescribeMacro{flip crossing}
\DescribeMacro{\flipcrossings}
Specifying the crossings to be flipped can be done either using the key \Verb+flip crossing+ or the macro \Verb+\flipcrossings+.
The latter can take a comma separated list of crossings to flip.
The former takes a single crossing but can be extended to a comma separated list using the \Verb+.list+ handler as in \Verb+flip crossing/.list={1,2,3}+ (this is what \Verb+\flipcrossings+ does internally).
\DescribeMacro{\redraw}
There is also a macro \Verb+\redraw+ which redraws a strand in the neighbourhood of a point.
This is effectively what happens for the crossings and can be used to fix something that wasn't done correctly by the main algorithm.
It takes two arguments, the strand number and the point at which to render the strand, as in \Verb+\redraw{2}{(1,1)}+.
\subsection{Keys}
The various keys are as follows.
The majority of the keys are in the \Verb+/tikz/knot diagram+ family, but it does its best to pass unknown keys down to \Verb+/tikz/+.
The keys processed by the \Verb+knot+ environment are automatically in this family but the keys processed by the \Verb+\strand+ command are not.
If a standard key (in the option to the \Verb+knot+ environment) doesn't work, try prefixing it with \Verb+/tikz/+ or \Verb+/pgf/+.
\begin{itemize}
\item \DescribeMacro{every knot diagram}
The style \Verb+every knot diagram+ is executed at the start of the \Verb+knot+ environment.
Note that it is inside the \Verb+knot diagram+ family so if setting it outside (say, in the preamble) use \Verb+\tikzset{knot diagram/every knot diagram}+.
\item \DescribeMacro{name}
The crossings of a knot are given coordinates of the form \Verb+<name> <number>+.
The default name is \Verb+knot+.
The \Verb+name+ key renames it.
\item \DescribeMacro{every strand}
The contents of \Verb+every strand+ are applied to every strand.
By default this contains the \Verb+draw+ key so if resetting it you should probably ensure that it still has the \Verb+draw+ key.
Note that it is inside the \Verb+knot diagram+ family so if setting it outside (say, in the preamble) use \Verb+\tikzset{knot diagram/every knot diagram}+.
\item \DescribeMacro{only when rendering}
The key \Verb+only when rendering={<style>}+ is applied to the strand only when it is actually drawn and not when it is used to cut out part of the underlying path.
Note that there are actually two versions of this key: one in the \Verb+knot diagram+ family and one in the \Verb+/tikz+ family.
This is so that it works equally well in the argument to the \Verb+knot+ environment and the \Verb+\strand+ command.
\item \DescribeMacro{ignore endpoint intersections}
When a strand is split into pieces then the intersection library finds ``false positives'' in that neighbouring pieces intersect at their endpoints.
The code ignores such intersections between neighbouring pieces.
The \Verb+ignore endpoint intersections+ (which is true by default) means that all intersections that are near an endpoint are ignored whether or not they are with the next or previous piece of that strand.
The celtic knot example of the previous section sets this to false to ensure that it gets all intersections.
\item \DescribeMacro{consider self intersections}
The key \Verb+consider self intersections=<option>+ handles the splitting facility so that strands can self intersect.
The options are \Verb+true+, \Verb+false+, and \Verb+no splits+.
It is \Verb+false+ at the start, and the default option is \Verb+true+.
The difference between \Verb+true+ and \Verb+no splits+ is as to whether segments are further split to avoid \emph{all} self intersections.
\item \DescribeMacro{clip radius}
The \Verb+clip radius=<length>+ is the radius of the clip region around each crossing.
\item \DescribeMacro{end tolerance}
The \Verb+end tolerance=<length>+ is the distance at which an intersection is considered as being near an endpoint (for simplicity, it uses the \(\ell^1\)--metric).
\item \DescribeMacro{clip width}
The \Verb+clip width=<factor>+ is the multiplier for the thickness of the ``wipeout'' path relative to the line width of the actual path.
\item \DescribeMacro{flip crossing}
The \Verb+flip crossing=<number>+ key flips the \Verb+<number>+ crossing.
\item \DescribeMacro{background colour} \DescribeMacro{background color}
The keys \Verb+background colour=<colour>+ and \Verb+background color=<color>+ set the background colour for the crossings.
\item \DescribeMacro{intersection <number>}
The style in \Verb+intersection <number>+ is applied just before that crossing is drawn so can be used to modify the crossing style just for that crossing.
\item \DescribeMacro{draft mode}
The key \Verb+draft mode=<option>+ sets the different styles for aiding with rendering the knot.
The options are \Verb+strands+, \Verb+crossings+, or \Verb+off+.
The \Verb+strands+ option just renders the strands with no crossings.
The \Verb+crossings+ option renders the crossings and labels the strands and crossings.
The \Verb+off+ option renders the crossings without the labels.
\item \DescribeMacro{draft/crossing label}
The style \Verb+draft/crossing label+ is applied to each of the crossing labels.
The labels are actually \Verb+pin+s attached to coordinates at the crossings.
This style is applied to the \Verb+pin+ itself.
The default is:
\begin{lstlisting}
overlay,
fill=white,
fill opacity=.5,
text opacity=1,
text=blue,
pin edge={blue,<-}
\end{lstlisting}
\item \DescribeMacro{draft/strand label}
The style \Verb+draft/strand label+ is applied to each of the strand labels.
The default is:
\begin{lstlisting}
overlay,
circle,
draw=purple,
fill=white,
fill opacity=.5,
text opacity=1,
text=purple,
inner sep=0pt
\end{lstlisting}
\end{itemize}
\subsection{Other Styles}
The other things defined by this package are for drawing knot diagrams when the user knows in advance either the locations of the crossings or can arrange that the paths are drawn in the correct order.
In these circumstances the \Verb+knot+ environment is overkill.
\DescribeMacro{knot}
\DescribeMacro{knot gap}
This \Verb+knot=<colour>+ style sets up a doubled path with inner colour the given colour (or the current draw colour if not given) and outer colour the knot background colour.
The width of the inner line is the current line width and the full width is controlled by the \Verb+knot gap=<factor>+ key which is initially set to \(3\) (thus giving a line's width either side).
\begin{example}
\begin{tikzpicture}[knot gap=7]
\draw[thin,knot=red] (-1,-1) -- (1,1);
\draw[thin,knot=red] (-1,1) -- (1,-1);
\begin{scope}[xshift=2.5cm]
\draw[knot=red] (-1,-1) -- (1,1);
\draw[knot=red] (-1,1) -- (1,-1);
\end{scope}
\begin{scope}[xshift=5cm]
\draw[thick,knot=red] (-1,-1) -- (1,1);
\draw[thick,knot=red] (-1,1) -- (1,-1);
\end{scope}
\end{tikzpicture}
\end{example}
\DescribeMacro{knot crossing}
\DescribeMacro{knot over cross}
\DescribeMacro{knot under cross}
\DescribeMacro{knot horiz}
\DescribeMacro{knot vert}
This package also defines some node shapes to help draw knot and link diagrams.
The idea with these is to place a node of the appropriate type at each crossing and then link them accordingly.
The node shapes are \Verb+knot crossing+, \Verb+knot over cross+, \Verb+knot under cross+, \Verb+knot vert+, \Verb+knot horiz+.
The two styles \Verb+knot over cross+ and \Verb+knot under cross+ are crossings, \Verb+knot vert+ and \Verb+knot horiz+ are for when resolving the crossings in a diagram.
By judicious use of the \Verb+\foreach+ command, a family of resolved link diagrams can be produced.
\begin{example}
\begin{tikzpicture}[every node/.style={draw,red}]
\node[knot over cross] at (1,0) {};
\node[knot under cross] at (2,0) {};
\node[knot vert] at (3,0) {};
\node[knot horiz] at (4,0) {};
\end{tikzpicture}
\end{example}
The node \Verb+knot crossing+ is not meant to be drawn, it is an empty shape.
Its value is in that it defines more anchors than the usual rectangle shape.
For each of the 8 main compass directions, it defines anchors at 2, 4, 8, 16, and 32 times further out.
This can be useful for designing curves that enter and exit the crossing gracefully at particular directions.
When using this node shape, the crossing itself is easiest to draw by using the \Verb+center+ anchor for the strands that form the over cross.
\begin{example}
\begin{tikzpicture}[every path/.style={red,thick}, every node/.style={transform shape, knot crossing, inner sep=1.5pt}]
\node[rotate=45] (tl) at (-1,1) {};
\node[rotate=-45] (tr) at (1,1) {};
\node (m) at (0,-1) {};
\node (b) at (0,-2) {};
\draw (b) .. controls (b.4 north west) and (m.4 south west) .. (m.center);
\draw (b.center) .. controls (b.4 north east) and (m.4 south east) .. (m);
\draw (m) .. controls (m.8 north west) and (tl.3 south west) .. (tl.center);
\draw (m.center) .. controls (m.8 north east) and (tr.3 south east) .. (tr);
\draw (tl.center) .. controls (tl.16 north east) and (tr.16 north west) .. (tr);
\draw (b) .. controls (b.16 south east) and (tr.16 north east) .. (tr.center);
\draw (b.center) .. controls (b.16 south west) and (tl.16 north west) .. (tl);
\draw (tl) -- (tr.center);
\end{tikzpicture}
\end{example}
\section{Other Relevant Packages by the Same Author}
Another tool for drawing knot diagrams is provided by the \Verb+hobby+ package which draws smooth curves through a prescribed set of points.
This can be combined with the facilities of this package but also has some features of its own that make it suitable for drawing knot diagrams.
For braids themselves, there is the \Verb+braids+ package which allows input specification in the form of a word in the braid group.
\end{document}
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\label{doc_BIGPICTURE_md_md_doc_BIGPICTURE}%
\Hypertarget{doc_BIGPICTURE_md_md_doc_BIGPICTURE}%
Elektra solves a non-\/trivial issue\+: how to abstract configuration in a way that software can be integrated and reconfiguration can be automated. Elektra solves this problem in a holistic way. Read \mbox{\hyperlink{doc_WHY_md}{why Elektra}} for an explanation of why such a solution is necessary.\hypertarget{doc_BIGPICTURE_md_autotoc_md1542}{}\doxysection{Virtual File System Analogy}\label{doc_BIGPICTURE_md_autotoc_md1542}
If you know virtual file systems, you already know a very similar solution to a very similar problem (otherwise first read about what a virtual file system is \href{https://en.wikipedia.org/wiki/Virtual_file_system}{\texttt{ here}}).
Before file systems (or for devices without operating system) software simply wrote at discs (think of {\ttfamily dd of=/dev/sda}) to persistently store data. This obviously does not work with multiple applications. To allow multiple applications to access data in {\ttfamily /dev/sda}, a file system structures them in a way that every application knows where its bits are. In analogy configuration files are application-\/specific initialization that cannot be shared with other applications. So as first steps we need to have a \char`\"{}file system\char`\"{} that describes the content of a configuration file in a uniform way. In Elektra \mbox{\hyperlink{src_plugins_README_md}{plugins}} represent file systems\+: they know how the data in configuration files should be interpreted.
For file systems, the API is {\ttfamily open}, {\ttfamily read}, {\ttfamily write}, and then {\ttfamily close}. For configuration key-\/value access is more suitable because values are so small that a single read/write always suffices. Thus Elektra \href{https://doc.libelektra.org/api/latest/html}{\texttt{ has a key-\/value API}} with {\ttfamily kdb\+Open}, {\ttfamily kdb\+Get}, {\ttfamily kdb\+Set} and {\ttfamily kdb\+Close}. Not every application is written in C, thus many {\ttfamily bindings} were written to access file systems. For example, in C++ you have {\ttfamily fstream}, and in Java {\ttfamily File\+Reader}. Also Elektra provides different bindings\+: In C++ you have a class {\ttfamily KDB} and can use {\ttfamily kdb.\+get} or {\ttfamily kdb.\+set}. Furthermore, every language has native support for the language\textquotesingle{}s iterators which make Elektra easier to use.
Furthermore, command-\/line tools like {\ttfamily cat} and {\ttfamily ls} provide an additional interface to the content of file systems for users and administrators. In Elektra also \mbox{\hyperlink{doc_help_kdb_md}{command-\/line tools}} for the analog purpose exist\+: With {\ttfamily kdb cp}, {\ttfamily kdb mv}, and {\ttfamily kdb ls} some command-\/line tools operating on file systems are mimicked.
There is not one file system satisfying every need, e.\+g., {\ttfamily proc}, {\ttfamily tmpfs}, {\ttfamily nfs} and {\ttfamily ext4} have quite different use cases and are needed at the same time in parallel. In analogy, different configuration file format (parsers) have different advantages and disadvantages. For example, {\ttfamily /etc/passwd} or {\ttfamily /etc/hosts} are structured, while many other configuration files are semi-\/structured.
To have multiple file systems present at the same time a virtual file system is able to {\ttfamily mount (2)} concrete file systems and thus give applications a way to uniformly access them. Similarly, Elektra also implements a \mbox{\hyperlink{doc_help_elektra-mounting_md}{mount}} functionality in its core. In Elektra a \href{/home/jenkins/workspace/libelektra-release/doc/CONTRACT.ini}{\texttt{ contract}} specifies the obligations between plugins.
Sometimes, it is even useful to have multiple file systems at the same mount point, so called {\ttfamily stacked} file systems. They allow you to combine features of different file systems. For example, \href{https://wiki.archlinux.org/index.php/ECryptfs}{\texttt{ e\+Cryptfs}} allows you to encrypt directories or files. In Elektra, stacking plugins is a core feature and heavily used to avoid feature-\/bloated plugins. For example, the \mbox{\hyperlink{md_src_plugins_crypto_README_src_plugins_crypto_README_md}{crypto plugin}} allows you to encrypt individual keys or the \mbox{\hyperlink{md_src_plugins_iconv_README_src_plugins_iconv_README_md}{iconv plugin}} to change the character encoding.
In file systems metadata describes information about files, e.\+g. when they were last accessed and who they are owned by ({\ttfamily ls -\/l}). In the same way Elektra has metadata that describe individual key-\/value pairs. In Elektra metadata is \href{/home/jenkins/workspace/libelektra-release/doc/METADATA.ini}{\texttt{ defined globally}} but implemented in many \mbox{\hyperlink{src_plugins_README_md}{plugins}}.
Implementations of file systems is not an easy task. The idea of FUSE (Filesystem in Userspace) is to make file system development easier by having the conveniences of userspace together with a helper library {\ttfamily libfuse}. In particular this allowed developers to use any programming language and easier abstractions. Elektra also tries hard to make plugin development simple. For example, special \mbox{\hyperlink{src_plugins_README_md}{interpreter plugins}} enable developers to also write plugins in different languages. Furthermore, \mbox{\hyperlink{src_libs_README_md}{other libraries}} also assist in creating plugins.
Of course not every feature of virtual file systems or Elektra has an analogy in the other system. If they would solve the same problem, one of them would be useless. The main differences are\+:
\begin{DoxyItemize}
\item API (get/set vs. read/write)
\item commit semantics\+: one {\ttfamily kdb\+Set} can change many configuration files atomically. This is important if you want, e.\+g., a new host in {\ttfamily /etc/hosts} and use this host in some other configuration files.
\item \mbox{\hyperlink{doc_help_elektra-namespaces_md}{namespaces}} there are many places where the same configuration is stored. All of these configuration files have the same semantics and they override each other (think of command-\/line arguments, {\ttfamily /etc}, {\ttfamily \$\+HOME/.config}, ...)
\item Elektra interacts closely with the program execution environment such as command-\/line parsing. The namespace {\ttfamily proc} is specifically reserved for this purpose.
\item in Elektra it is possible to create holes (files without directories above them) which are needed because of these override semantics\+: we want to be able to override a single value without duplicating the whole skeleton.
\item validation\+: in Elektra you can describe how valid configuration should look like and reject invalid configuration.
\item and much more...
\end{DoxyItemize}\hypertarget{doc_BIGPICTURE_md_autotoc_md1543}{}\doxysection{Further Readings}\label{doc_BIGPICTURE_md_autotoc_md1543}
\begin{DoxyItemize}
\item \mbox{\hyperlink{doc_COMPILE_md}{Compile}} and \mbox{\hyperlink{doc_INSTALL_md}{Install}} Elektra
\item Then continue reading the \mbox{\hyperlink{md_doc_tutorials_README_doc_tutorials_README_md}{tutorials}}
\item Read about bindings
\item Read about tools
\item Look into \mbox{\hyperlink{doc_help_elektra-glossary_md}{the glossary}}.
\item Start reading about \mbox{\hyperlink{doc_help_kdb_md}{command-\/line tools}}
\end{DoxyItemize}
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%% Copyright (C) 2004 Frank Mittelbach, Michel Goossens,
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%% It may be distributed and/or modified under the conditions
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\documentclass[11pt,a4paper]{article}
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\usepackage[usenames,dvipsnames]{xcolor}
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\lhead{\footnotesize\bf Introduction to Network Analysis ({\color{magenta}INA})}
\rhead{\footnotesize\bf Homework {\color{magenta}\#3}}
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\rhead{\footnotesize\bf Spring 2016/17}
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{\huge\bf Homework {\color{magenta}\#3}}
\end{center}
\vspace*{0.025in}
% \section*{Homework details}
\paragraph{} % This homework is only partial and will be updated in the coming days.
This homework is complete and will not be changed. The homework does not require a lot of writing, but may require a lot of thinking. It does not require a lot of processing power, but may require efficient programming. It accounts for $12.5\%$ of the course grade. All questions and comments regarding the homework should be directed to \href{https://piazza.com}{Piazza}.
\section*{Submission details}
\paragraph{} This homework is due on {\bf\color{magenta} May 2nd} at 3:00pm, while late days expire on {\bf\color{magenta} May 5th} at 3:00pm. The homework must be submitted as a hard-copy in the submission box in front of R 2.49 and also as an electronic version to \href{https://ucilnica.fri.uni-lj.si/course/view.php?id=183}{eUcilnica}. It can be prepared in either English or Slovene and either written by hand or typed on a computer. The hard-copy should include (1)~this cover sheet with filled out time of the submission and signed honor code, (2)~short answers to the questions, which can also demand proofs, tables, plots, diagrams and other, and (3)~a printout of all the code required to complete the exercises. The electronic submission should include only (1)~answers to the questions in a single file and (2)~all the code in a format of the specific programming language. Note that hard-copies will be graded, while electronic submissions will be used for plagiarism detection. The homework is considered submitted only when both versions have been submitted. Failing to include this honor code in the submission will result in {\bf\color{LimeGreen} 10\% deduction}. Failing to submit all the developed code to \href{https://ucilnica.fri.uni-lj.si/course/view.php?id=183}{eUcilnica} will result in {\bf\color{LimeGreen} 50\%~deduction}.
\section*{Honor code}
\paragraph{} The students are strongly encouraged to discuss the homework with other classmates and form study groups. Yet, each student must then solve the homework by herself or himself without the help of others and should be able to redo the homework at a later time. In other words, the students are encouraged to collaborate, but should not copy from one another. Referring to any solutions obtained from classmates, course books, previous years, found online or other, is considered an honor code violation. Also, stating any part of the solutions in class or on \href{https://piazza.com}{Piazza} is considered an honor code violation. Finally, failing to name the correct study group members, or filling out the wrong date or time of the submission, is also considered an honor code violation. Honor code violation will not be tolerated. Any student violating the honor code will be reported to {\bf\color{LimeGreen} faculty disciplinary committee} and vice dean for education.
\vspace*{0.033in}
\paragraph{Name \& SID:} \rule{4.5in}{0.5pt}
\paragraph{Study group:} \rule{4.5in}{0.5pt}
\paragraph{Date \& time:} \rule{2.5in}{0.5pt}
\paragraph{} I acknowledge and accept the honor code.
\paragraph{Signature:} \rule{2.5in}{0.5pt}
\pagebreak
\section{Graph Laplacian \totals{0.75}}
\paragraph{} Let $n$ be the number of nodes in an undirected network and let $m$ be the number of links. Graph Laplacian $L$ is $n\times n$ matrix defined as $L=D-A$, where $A$ is the network adjacency matrix and $D$ the diagonal matrix with node degrees $\{k_i\}$ along its diagonal. Link incidence matrix $B$ is $m\times n$ matrix defined as $B_{ij}=1$ if $j$ is the first endpoint of $i$-th link, $B_{ij}=-1$ if $j$ is the second endpoint of $i$-th link, and $B_{ij}=0$ otherwise. \hint{Arbitrarily designate one endpoint to be the first one and the other to be the second one.} First show that $L=B^TB$. Using this equality further show that all eigenvalues of $L$ are non-negative and that vector of all ones is an eigenvector of $L$. These results prove useful in spectral community detection~\cite{Fie73,New10}.
\subsection*{What to submit?}
\paragraph{} Show that the equality holds \points{0.25}. Give proof of the non-negativity of the eigenvalues of $L$ \points{0.25} and show that vector of all ones is an eigenvector of $L$ \points{0.25}.
\section{Ring graph modularity \total}
\paragraph{} Imagine a graph with $n$ nodes positioned on a ring thus each node is linked to its two neighbors (see~\figref{ring}). Let the graph be partitioned into $c$ consecutive clusters with $n_c=n/c$ nodes each. Compute modularity $Q$~\cite{GN02} of such partition and express it in terms of $n_c$ and~$n$. \hint{See lecture handouts for the definition of modularity.} Find the size of clusters $n_c$ that optimizes modularity $Q$ and express it in terms of $n$.
\begin{figure}[h] \centering
\includegraphics[width=0.275\textwidth]{ring}
\caption{{\bf Ring graph with $n=36$, $n_c=4$ and $Q=0.64$}}
\label{fig:ring}
\end{figure}
\subsection*{What to submit?}
\paragraph{} Derive the expression for modularity $Q$ of a ring graph \points{0.5} and the optimal size of clusters $n_c$ according to modularity $Q$ \points{0.5}.
\section{Who's the winner? \totals{4.5}}
\paragraph{} Community detection is one of the most popular research areas of network science~\cite{New12}. Indeed, literary hundreds of community detection algorithms have been proposed in the literature in the last two decades~\cite{For10,FH16}. These include hierarchical clustering, spectral methods (e.g.\ \href{http://www.cs.utexas.edu/users/dml/Software/graclus.html}{Graclus}), modularity optimization (e.g.\ \href{https://sites.google.com/site/findcommunities/}{Louvain}), map equation algorithms (e.g.\ \href{http://mapequation.org/code.html}{Infomap}), statistical methods (e.g.\ \href{http://www.oslom.org}{OSLOM}), link clustering (e.g.\ \href{https://github.com/bagrow/linkcomm}{Links}), label propagation (e.g.\ \href{http://www.cs.bris.ac.uk/~steve/networks/copra/}{COPRA}), random walks (e.g.\ \href{https://www-complexnetworks.lip6.fr/~latapy/PP/walktrap.html}{Walktrap}), clique percolation (e.g.\ \href{http://www.lce.hut.fi/~mtkivela/kclique.html}{SCP}) and many others (e.g.\ \href{http://www.michelecoscia.com/?page_id=42}{DEMON}). Your task is to compare the accuracy, robustness and uncertainty of selected three algorithms. These should include at least \href{https://sites.google.com/site/findcommunities/}{Louvain} and \href{http://mapequation.org/code.html}{Infomap}, and also an algorithm of your own choice which should not be from the same class of methods as the mentioned two. \hint{If you are unable to compile any of the required algorithm implementations on your machine, and there is no equivalent implementation within your programming library, you should write to \href{https://piazza.com}{Piazza} and ask for an appropriate alternative. Code required to solve this exercise will likely consist of several ad hoc scripts that will have to be run sequentially.}
\begin{figure}[t] \centering
\subfloat[{\bf Girvan-Newman} benchmark graphs]{\includegraphics[width=0.425\textwidth]{GN}\label{fig:GN}}
\subfloat[{\bf Lancichinetti} benchmark graphs]{\includegraphics[width=0.475\textwidth]{LFR}\label{fig:LFR}}
\caption{{\bf Synthetic benchmark graphs with planted partition for $\mu=0.1$}}
\label{fig:benchmarks}
\end{figure}
\begin{description}
\item[(i)] Implement a variant of Girvan-Newman synthetic benchmark graphs with planted partition~\cite{GN02}. The graphs consist of three groups of $24$ nodes each, while the expected degree of each node is $20$ (see~\figref{GN}). The group structure is controlled by a mixing parameter $\mu$. For $\mu=0$, all links are placed within the groups, while for $\mu=1$, all links are placed between the groups. Apply all three community detection algorithms to $25$ benchmark graph realizations with $\mu$ equal to $0$, $0.1$, $0.2$, $0.3$, $0.4$ and $0.5$. For each algorithm and each value of $\mu$, compute normalized mutual information between the planted partitions and the detected community structures, and average the results. \hint{See lecture handouts for the definition of normalized mutual information.} Plot community detection accuracy of all three algorithms on a single plot with $\mu$ on the horizontal axis and the average normalized mutual information on the vertical axis. Which algorithm comes out on top? Briefly discuss the results by comparing the performance of different algorithms.
\item[(ii)] Consider a more realistic \href{http://lovro.lpt.fri.uni-lj.si/ina/LFR.zip}{Lancichinetti synthetic benchmark graphs} with planted partition \cite{LFR08}. \hint{See lecture handouts for the description of benchmark graphs.} The graphs consist of $1000$ nodes (see~\figref{LFR}), while the group structure is again controlled by a mixing parameter $\mu$. Apply all three community detection algorithms to $25$ benchmark graph realizations with $\mu$ equal to $0$, $0.2$, $0.4$, $0.6$ and $0.8$. Plot community detection accuracy of all three algorithms on a single plot with $\mu$ on the horizontal axis and the average normalized mutual information on the vertical axis. Which algorithm comes out on top now? Why? Briefly discuss the results by comparing the performance of different algorithms.
\item[(iii)] Consider an Erd\H{o}s-R\'{e}nyi random graph~\cite{ER59} that lacks community structure. Community detection algorithms should be robust enough to detect this and output each connected component of the graph as a single community. Apply all three community detection algorithms to $25$ random graph realizations with $1000$ nodes and the average degree equal to $8$, $16$, $24$, $32$ and $40$. Plot community detection robustness of all three algorithms on a single plot with the average degree on the horizontal axis and the average normalized variation of information on the vertical axis. \hint{See lecture handouts for the definition of normalized variation of information.} Which algorithms appear robust to random structure? Why? Briefly discuss the results by comparing the robustness of different algorithms.
\item[(iv)] Consider \href{http://lovro.lpt.fri.uni-lj.si/ina/dolphins}{Lusseau bottlenose dolphins network}~\cite{LSBHSD03} with a known sociological division into two groups. Apply each community detection algorithm $25$ times and analyze community detection uncertainty. More precisely, compute pair-wise normalized variation of information between the detected community structures and average the results. State the average normalized variation of information for all three algorithms and briefly discuss the results. Which algorithms appear most deterministic?
\item[(v)] Given all the knowledge gained above, which algorithm would you choose for your course project if needed? State the weaknesses of each algorithm and finally select the winner.
\end{description}
\subsection*{What to submit?}
\begin{description}
\item[(i)] Give a printout of the benchmark graph implementation \points{0.25}. Plot community detection accuracy of all three algorithms \points{3\times 0.25}. Give an answer to the question and briefly comment on the results \points{0.25}.
\item[(ii)] Plot community detection accuracy of all three algorithms \points{3\times 0.25}. Give answers to both questions and briefly comment on the results \points{0.25}.
\item[(iii)] Plot community detection robustness of all three algorithms \points{3\times 0.25}. Give answers to both questions and briefly comment on the results \points{0.25}.
\item[(iv)] State community detection uncertainty of all three algorithms \points{3\times 0.25}. Give an answer to the question and briefly comment on the results \points{0.25}.
\item[(v)] State the weaknesses of each algorithm and give a brief answer to the question \points{0.25}.
\end{description}
\section{Peers, ties and the Internet \totals{3.25}}
\paragraph{} Link prediction is probably the most common application of network analysis techniques. For given nodes $i$ and $j$, link prediction methods compute an index $s_{ij}$ that should be high for $i$ and $j$ that are likely to connect, and low for all other pairs of $i$ and $j$. You will be investigating three link prediction methods that are based on different structural properties of real networks.
\begin{enumerate}
\item Scale-free degree distribution of real networks is believed to be the consequence of preferential attachment~\cite{BA99} that states that nodes are more likely to connect to high degree nodes. The preferential attachment index~\cite{LK07} is thus defined as $s_{ij}=k_ik_j$, where $k_i$ is the degree of node $i$.
\item Small-world networks are characterized by an abundance of triangles~\cite{WS98}, which can be explained by triadic closure. Hence, nodes are more likely to connect if they share many common neighbors. The Adamic-Adar index~\cite{AA03} takes into account also that it is more likely to share a high degree neighbor. It is defined as $s_{ij}=\sum_{x\in \Gamma_i\cap\Gamma_j}\frac{1}{\log k_x}$, where $\Gamma_i$ is the set of neighbors of node $i$.
\item Many real networks consist of communities of densely linked nodes with only few links between the communities~\cite{GN02}. Links are thus more likely to appear within communities than between. Let $\{C\}$ be the communities revealed by \href{https://sites.google.com/site/findcommunities/}{Louvain modularity optimization}~\cite{BGLL08} and let $c_i$ be the community label of node $i$. Furthermore, let $n_c$ and $m_c$ be the number of nodes and links within the community $C$. Then, the community index is defined as $s_{ij}=m_c/{n_c \choose 2}$ for $c_i=c_j$, whereas $s_{ij}=0$ for $c_i\neq c_j$. \hint{If you are unable to compile the required algorithm implementation on your machine, and there is no equivalent implementation within your programming library, you should write to \href{https://piazza.com}{Piazza} and ask for an appropriate alternative.}
\end{enumerate}
\begin{figure}[t] \centering
\includegraphics[width=0.66\textwidth]{circles}
\caption{{\bf Communities in Facebook social circles revealed with Louvain method}}
\label{fig:circles}
\end{figure}
\begin{description}
\item[(x)] Assume that you apply a link prediction method to all unlinked pairs of nodes in a real network and later evaluate between which pairs of nodes the links actually occurred. Considering the density of real networks, what would be the expected classification accuracy of a method that simply predicts that no links will occur?
\item[(y)] Implement the following framework for evaluating link prediction methods. For a given network and link prediction index $s$, randomly sample $\frac{m}{10}$ pairs of nodes that are not yet linked and store them into $L_N$. These will serve as negative examples for the prediction. Next, randomly remove $\frac{m}{10}$ links from the network and store them into $L_P$. These will serve as positive examples for the prediction. Finally, compute the link prediction index $s$ for all pairs of nodes in $L_N\cup L_P$. Link prediction is usually evaluated using area under the ROC curve (AUC), which can be defined as the probability that a randomly chosen pair of nodes in $L_P$ has higher value of $s$ than a randomly chosen pair of nodes in $L_N$. Note that random guessing gives $\frac{1}{2}$. To compute AUC, randomly sample $\frac{m}{10}$ pairs of nodes from $L_P$ and $\frac{m}{10}$ pairs from $L_N$ with repetitions, and compare their indices $s$. Let $m'$ be the number of times when the value of $s$ for the pair of nodes from $L_P$ is larger than the value for the pair of nodes form $L_N$, and let $m''$ be the number of times when both values are equal. Then, AUC $=\frac{m'+m''/2}{m/10}$.
\item[(z)] Compute the average AUC over several runs for all three link prediction methods above applied to an Erd\H{o}s-R\'{e}nyi random graph~\cite{ER59} with $25000$ nodes and the average degree $10$, and to three real networks. These are \href{http://lovro.lpt.fri.uni-lj.si/ina/gnutella}{Gnutella peer-to-peer file sharing network}, a small sample of \href{http://lovro.lpt.fri.uni-lj.si/ina/circles}{Facebook social circles network} (see~\figref{circles}) and \href{http://lovro.lpt.fri.uni-lj.si/ina/nec}{nec overlay map} of the Internet. \hint{Although some networks are directed, treat them as undirected.} Which method comes out on top for each individual network? Why? Briefly discuss the results and compare the performance of methods on random graphs and real networks.
\end{description}
\subsection*{What to submit?}
\begin{description}
\item[(x)] Give a brief answer to the question \points{0.25}.
\item[(y)] Give a printout of the framework implementation \points{0.75}.
\item[(z)] For each link prediction method, state the average AUC obtained for random graphs and real networks \points{3\times 0.5}. Answer both questions for each individual network and briefly comment on the results \points{3\times 0.25}.
\end{description}
\section{Get at least 70\% right! \totals{1.5}}
\paragraph{} You are given a \href{http://lovro.lpt.fri.uni-lj.si/ina/aps_2008_2013}{citation network} of scientific papers published by the American Physical Society between the years $2008$ and $2013$. The papers were published in ten journals, which represent the metadata information you would like to infer from the structure of citation network. More precisely, you would like to predict the correct journal of all papers published in the year $2013$ based on their citation patterns and the journal information of all papers published between the years $2008$ and $2012$. Predicting the paper's journal to be the most frequent journal in the neighborhood of the corresponding node gives $\approx 65\%$ classification accuracy, while your task is to propose a strategy that gives at least $\approx 70\%$ classification accuracy. The strategy can use any network analysis technique as long as it scales better than $\mathcal{O}(n^2)$.
\subsection*{What to submit?}
\paragraph{} Describe your strategy and briefly explain its rationale \points{0.25}. State the average classification accuracy obtained over several runs and compare results with the baseline \points{0.75}. Print out any code you might have used or describe how you solved the exercise \points{0.5}.
\bibliographystyle{alpha}
\bibliography{biblio}
\end{document}
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\iteman{ZMATH 2015d.00672}
\itemau{Boester, Timothy; Lehrer, Richard}
\itemti{Visualizing algebraic reasoning.}
\itemso{Kaput, James J. (ed.) et al., Algebra in the early grades. London: Routledge (ISBN 978-0-8058-5472-5/hbk; 978-0-8058-5473-2/pbk). Studies in Mathematical Thinking and Learning Series, 211-234 (2008).}
\itemab
From the text: According to {\it A. D. Aleksandrov}, {\it A. N. Kolmogorov}, and {\it M. A. Lavrent'ev} [Mathematics, its content, methods, and meaning. Cambridge, MA: MIT Press (1969)], ``Arithmetic and geometry are the two roots from which has grown the whole of mathematics" (p. 24). Algebra is generally understood as having derived from the arithmetical root. In Chapter 9, the authors highlight algebra's indebtedness to the geometric root of mathematics, noting that ``spatial structure serves as a potentially important springboard to algebraic reasoning, but also that algebraic reasoning supports coming to `see' lines and other geometric elements in new lights." Their argument is not historical but rather psychological: ``Visualization bootstraps algebraic reasoning and algebraic generalization promotes `seeing' new spatial structure".
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\itemcc{H10 G40 C30 E40}
\itemut{algebraic reasoning; visualization; geometry; generalization; spatial structure; research; design studies; grade 6; lower secondary; meta-representational competence; representational competencies; graphical representations; coordinates; modes of representation; complicated sorting; comparing the slopes of lines; ratio; patterns in tables; similarity; early education in algebra}
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\title{What to Keep and How to Analyze It: Data Curation and Data Analysis with
Multiple Phases}
\author{Alyssa Goodman}
\affil{Affiliation not available}
\author{Vinay Kashyap}
\affil{Affiliation not available}
\author{Ashish Mahabal}
\affil{Affiliation not available}
\author{Xiao-Li Meng}
\affil{Affiliation not available}
\author{Aleksandra Slavkovic}
\affil{Affiliation not available}
\author{Aneta Siemiginowska}
\affil{Affiliation not available}
\author{Alberto Pepe}
\affil{Affiliation not available}
\author{Christine L. Borgman}
\affil{Affiliation not available}
\author{Paul Groth}
\affil{Affiliation not available}
\author{Yolanda Gil}
\affil{Affiliation not available}
\author{David W. Hogg}
\affil{Affiliation not available}
\author{Kyle Cranmer}
\affil{Affiliation not available}
\author{Rosanne Di Stefano}
\affil{Affiliation not available}
\date{\today}
\maketitle
\subsection{Overview}\label{overview}
This open document is being used to describe and record the events at
the
\href{http://www.radcliffe.harvard.edu/exploratory-seminars/what-to-keep-and-how-to-analyze-it}{Radcliffe
Exploratory Seminar on Data Curation and Analysis}, to be held at the
Radcliffe Institute for Advanced Study, May 9-10 2013.
\href{https://drive.google.com/folderview?id=0BxIRxiTe1u6BdWh1eDhZemt4dlU\&usp=sharing}{This
Google Drive Directory} should be used to deposit all files contributed
by participants before and during the meeting. (Click ``Open in Drive''
on your browser to make a new folder, e.g.~with your name as its name.)
\href{https://docs.google.com/document/d/1sHuFqYKVJ-y0TUrbai3x9wJXA0nrQviifKseIH1t6ME/edit?usp=sharing}{This
Google Doc} is used for collaborative real-time note-taking.
\textbf{ABSTRACT:} Rapid advances in technology have allowed us to
collect vast amounts of data in myriad fields and forms, but our ability
to manage and analyze these data has not kept pace. As a result, the
amount of data collected far exceeds what can be analyzed and, often,
what can be archived. These issues only become more pressing as data
collection accelerates. Astronomers and astrophysicists, for example,
collect terabytes of data per night; the phrase ``drowning in a data
tsunami'' is increasingly used to describe this situation. The issues of
what to keep and what to distribute are surprisingly complex, even when
we put aside technological issues such as long-term storage and
retrieval. A central challenge is the fundamental conflict between
reducing the size of data and preserving information for future
scientific inquires and statistical analyses. Complicating matters
further, the parties/teams involved in the entire data collection,
curation, and analysis process often have only limited communication
with each other owing to the sequential nature of this process. This
seminar brings together a core group of leading experts and emerging
scholars in information and natural sciences to discuss, debate, and
design principles and strategies to address this grand challenge, which
increasingly affects almost every aspect of science and society.
\textbf{GOAL:} By gathering experts from information and natural
sciences, we aim to start building a set of principles and methods that
will allow us to understand such problems and to provide better
preprocessing, analyses, and data preservation, especially in the
context of the natural sciences. The ultimate goals of this research
include providing methods for assessing the validity of such
collaborative analyses, guidance on statistically-principled
preprocessing, and a rich new theory of statistical learning and
inference with multiple parties. We believe that this collaboration will
simultaneously sow the seeds for innovative mathematical theory and shed
light on directly usable guidelines for the construction and curation of
scientific databases.
\subsection{Draft Schedule of Events, May 9-10,
2013}\label{draft-schedule-of-events-may-9-10-2013}
Location: Room 112, Radcliffe Gymnasium, Radcliffe Yard, 18 Mason
Street, Cambridge, MA
(\href{https://maps.google.com/maps?client=safari\&oe=UTF-8\&q=42.376227,-71.122744\&ie=UTF-8\&hq=\&hnear=0x89e37769beff1187:0x14c949af7a2810e0,42.376227,-71.122744\&gl=us\&ei=v2eKUb_hFK7C0AHb24CYDg\&ved=0CDMQ8gEwAA}{Red
pin on this map} marks the front door of the Radcliffe Gymnasium--zoom
in!)
\subsubsection{Day 1 (Thursday, May 9)}\label{day-1-thursday-may-9}
8:30 AM - 9:00 AM Continental Breakfast
9:00 AM Introductory remarks and welcome address
\textbf{SESSION I} 9:15 AM -- 12:30 PM \emph{Quantitative and
qualitative perspectives on multiphase science -- Beginning a dialogue}
9:15-11:45 Introductions: each of 16 participants will answer the
following questions (5 min/person, including short discussions \& coffee
break, total of 2.5 hours.)
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
What about your background gives you an interest in data curation?
\item
What do you think is the most important opportunity good data curation
offers? (Please just one!)
\item
What do you think is the biggest danger facing scientific research
today if we don't improve data curation? ((Please just one!)
\end{enumerate}
Coffee Break at appropriate stopping point during the above, at roughly
at 10:30.
11:45-12:30 Introduction to solutions proposed in the literature (Part
I) Presented by: Meng, Borgman, Crosas, Pepe et al. (TBD)
12:30 PM -- 1:30 PM Lunch
1:30-2:00 Introduction to solutions proposed in the literature (Part II)
Presented by: Meng, Borgman, Crosas, Pepe et al. (TBD)
\textbf{SESSION II} 2:00 PM -- 5:00 PM Specific challenges in data
curation, provenance, and multiphase analysis
2:00 PM--4:00 PM
Roughly 40 minutes for each of the topics below (as amended at the
Workshop). \emph{Suggested} discussion leaders indicated, but changes
can and will(!) be made to respond to participant suggestions. Each
workshop attendee will each ``sign up'' (at lunchtime) for 3 discussions
total, to be held within groups of roughly 5 or 6 people each. Multiple
rooms will be available, and a schedule of which discussions will take
place in which room will be made on-the-fly, and posted here. There will
be three ``blocks'' of 40 minutes, with two or three topics to choose
from within each block.
\begin{itemize}
\tightlist
\item
group collaboration challenges (Cranmer/Hogg)
\item
provenance, what's realistic? (Hedstrom/Pepe)
\item
storage, ideas on what to keep, sociological \& algorithmic approaches
(Groth/Blocker)
\item
can statistics help? (Slavkovic/Siemiginowska)
\item
the divide between theory and practice: what we should do, versus what
we do do (Goodman/Borgman)
\item
what has \& has not worked in Astronomy? (DiStefano/Kashyap/Mahabal)
\item
working with \& educating the community of data producers (Gil/Crosas)
\end{itemize}
4:00-4:20 Coffee Break
4:30-5:30 Group discussion of smaller group's discussions, used to
refine plans for Day 2.
6:30 PM Group Dinner at NuBar, Cambridge (in the Sheraton Commander)
\subsubsection{Day 2 (Friday, May 10)}\label{day-2-friday-may-10}
8:30 AM - 9:00 AM Continental Breakfast
\textbf{SESSION III} 9:00 AM -- 12:30 PM Where can we connect?
Addressing foundational issues from interdisciplinary perspectives
12:30 PM -- 1:30 PM Lunch
\textbf{SESSION IV} 1:30 PM -- 5:00 PM What can we do together?
Identifying opportunities for collaboration
6:30 PM Group Dinner (social event, location TBD)
\subsection{Participants}\label{participants}
\textbf{Alexander Blocker}, Statistics
--\href{http://www.awblocker.com}{Bio}-- Email [email protected]
\textbf{Christine L. Borgman}, Information Science
--\href{http://polaris.gseis.ucla.edu/cborgman/Chriss_Site/Bio.html}{Bio}--
Email: [email protected]
\textbf{Kyle Cranmer}, Particle Physics
--\href{http://physics.as.nyu.edu/object/kylecranmer.html}{Bio}-- Email
[email protected]
\textbf{Merce Crosas}, Data Science
--\href{http://projects.iq.harvard.edu/seamlessastronomy/people/merce-crosas-0}{Bio}--
Email [email protected]
\textbf{Rosanne DiStefano}, Astrophysics
--\href{http://astronomy.fas.harvard.edu/people/rosanne-di-stefano}{Bio}--
Email [email protected]
\textbf{Yolanda Gil}, Information Science
--\href{http://www.isi.edu/~gil/cv/short-bio.html}{Bio}-- Email
[email protected]; [email protected]
\textbf{Alyssa Goodman}, Astrophysics, Visualization
--\href{https://www.cfa.harvard.edu/~agoodman/newweb/about.html}{Bio}--
Email [email protected]
\textbf{Paul Groth}, Computer Science
--\href{http://www.few.vu.nl/~pgroth/Site/Welcome.html}{Bio}-- Email
[email protected]
\textbf{David Hogg}, Astrophysics, Data Science
--\href{http://cosmo.nyu.edu/hogg/bio.html}{Bio}-- Email
[email protected]
\textbf{Vinay Kashyap}, Astrophysics, Statistics
--\href{http://hea-www.cfa.harvard.edu/~kashyap/}{Bio}-- Email
[email protected]
\textbf{Margaret Hedstrom} Information Science
--\href{http://en.wikipedia.org/wiki/Margaret_Hedstrom}{Bio}-- Email
[email protected]
\textbf{Ashish Mahabal}, Astrophysics
--\href{http://www.astro.caltech.edu/~aam/}{Bio}-- Email
[email protected]
\textbf{Xiao-Li Meng},
Statistics--\href{http://www.stat.harvard.edu/faculty_page.php?page=meng.html}{Bio}--
Email [email protected]
\textbf{Alberto Pepe} --\href{http://albertopepe.com/resume}{Bio}--
Email [email protected]
\textbf{Aneta Siemiginowska}, Astrophysics, Statistics
--\href{http://hea-www.harvard.edu/~aneta/HomePage.html}{Bio}-- Email
[email protected]
\textbf{Aleksandra B. Slavkovic}, Statistics
--\href{http://sites.stat.psu.edu/~sesa/}{Bio}-- Email [email protected]
\href{}{} Click
\href{mailto:[email protected],\%[email protected],\%[email protected],\%[email protected],\%[email protected],\%[email protected],\%[email protected],\%[email protected],\%[email protected],[email protected],[email protected],[email protected],[email protected],[email protected],[email protected],\%[email protected],\%[email protected],\%[email protected],\%[email protected]}{here}
to email all Workshop Participants at once.
\subsection{Participant Introductory
Comments}\label{participant-introductory-comments}
\textbf{Hogg}--\href{http://astrometry.net}{Astrometry.net}--just did it
because they could, now they see it's important. Q. What's the
difference between provenance and metadata? Very interested in being
able to reconstruct the reasons people took data. How did they do what
they did? In Astronomy--you never get the photons again--one chance
only. Strong relationship between data curation and software curation.
Knowledge about data is embedded within the software.
\textbf{Gil} more information about the data, the more we can write
software and intelligent systems that can assist scientists; provenance
standards--including metadata from people with different expertise;
opportunity--discovery informatics--helping scientists with intelligent
assistance; how to preserve connections between data \& models? (e.g.
\href{http://creativemachines.cornell.edu/eureqa\%20from\%20Cornell}{``Eureqa''
system}); will post link to workshop on discovery informatics; many
problems are social rather than technological
\begin{itemize}
\tightlist
\item
Crosas mentions system similar to Eureqa in social science
\item
DOE-funded software innovation project at Michigan mentioned by
Hedstrom--looking for ``models'' of data--do they need to be
``scientific'' or ``physical'' or can they just be ``statistical''
(with no \emph{a priori} knowledge of phenomenology).
\item
Cranmer: thinking about the ``multi-phase'' nature of modeling--many
steps are taken along the say, some may have more ``statistical''
nature and some more ``scientific''--the distinction between these two
is clearer at the higher level, where scientific models are needed to
connect disparate data sets/results
\item
(Hedstrom) Is social science (and maybe life science?) different--in
that there isn't necessarily an underlying theory--maybe there it's
more statistical/empirical from the start
\end{itemize}
\textbf{Groth} -- offers definition of provenance information-- rough
quote (fix later): ``Provenance is information about entities,
activities, and people involved in producing a piece of data or thing,
which can be used to form assessments about its quality, reliability or
trustworthiness.'' (one
\href{https://www.w3.org/2002/09/wbs/46974/prov-vocabulary-usage-survey/results}{source})
We need ``Data Connoisseurs'' obsessed with data \& provenance. Biggest
opportunity: (re-)using other people's data, assuming it has enough
provenance to use! Danger: more bad results, due to poor use of others'
data--negative impact on science overall
\begin{itemize}
\tightlist
\item
question from Cranmer--will the run-of-the-mill physicists/scientists
ever use provenance systems properly? (psychology/sociology
questions!)
\item
(Hedstrom) what do connoisseurs need, in comparison with ``regular''
people
\item
Hogg-difference between ``goods'' and ``data''--data is easier to
spoof, harder to control
\end{itemize}
\textbf{Pepe} -- mention of CDS \emph{manual} curation, enriching
documents with hand-done links to astronomical sources; discussion of
\href{http://arxiv.org/abs/1111.3983}{ADS All-Sky Survey}, which uses
manual curation to make all-sky heat maps of the sky, showing where it's
been studied, when, and why; then discussion of automated
image-extraction and solving for the images positions/scale
(astroreferencing), and how that led back to manual curation, using
Zooniverse to create the ``oldAstronomy'' platform where citizens will
enter the necessary metadata to make images useful. Point about
\href{http://authorea.com}{authorea.com} is that it will allow for
automated provenance in the future.
\begin{itemize}
\tightlist
\item
question about reliability of classification from Kayshap--answer is 3
people do each image
\item
Hedstrom mention of \href{http://ebird.org}{ebird} as another good
citizen science platform, in addition to
\href{http://zooniverse.org}{Zooniverse}
\item
Hogg mention of value of people (connoisseurs) talking about data
sets, Pepe mentions ``TALK'' page at Zooniverse, that looks great \&
performs this function
\end{itemize}
\textbf{Kayshap} -- work as a calibration scientist for Chandra is
relevant to this discussion; perspective from someone who thinks about
the ``measurement to data'' part of this situation; regime of ``small
data'' where every photon matters needs special consideration, because
each bit is precious, unlike the case with ``big data'' sets
\textbf{Mahabal} -- transient science (in big surveys)- must make
decisions in \textasciitilde{}real-time about what is interesting;
archival information is critical -- to put the new data into context of
old data, immediately, in order to make these decisions; idea of a
``portfolio'' for each object
\textbf{Siemiginowska} -- high-energy astrophysics has an amazing
archive, \href{http://heasarc.gsfc.nasa.gov}{``HEASARC''}, 50 years
worth of data, to use this it is critical to understand the metadata and
to have the original software used to reduce the data (but danger is
that the old s/w is not usable, so how does one reproduce the analysis?)
Danger: what about people who use a hybrid system of computers and
``paper'' to do their research--even young postdocs--and then their
research cannot be reproduced? Shouldn't we \textbf{educate} people to
work in a different way to make their research reproducible. 2nd issue:
people who write their own s/w vs.~people who write/use software meant
for a group--the latter is typically better-documented \& what is value
in that?
\textbf{Borgman} -- big data is hard, but little data is harder, since
there's so much less consistency and regularity in the small data case.
Basic premises: invest in metadata on ingest OR ``google model'' chuck
it all in \& try to make sense of it ``later.'' (\ldots{}then there's
``digital archaeology''..) Opportunity: Grab data early in the life
cycle. That's also the DANGER--if you do this early, and the data are
``dirty,'' they won't get cleaned up early. AG silent comment: that's
like putting GPS on digital cameras, built-in!
\textbf{Crosas} -- her IQSS team builds tools to solve all these
problems! trying to add-on functionality to alleviate extra burdens on
researchers ``at the end''. Opportunity: ``Sustainable science.''
Automating the process as much as possible will help this happen.
\textbf{Hedstrom} -- started as an archivist, including archiving paper,
and ``electronic records'' Interested now in ``general'' vs.
``(discipline-) specific'' tools for archiving. IGERT program trains
many varieties of students. Works with different communities of
scientists (life sciences where data deposit is required with
publications to materials scientists who have no idea, essentially, what
data reuse means). Works on \href{http://sead-data.net/resources}{SEAD},
Sustainable Environment Actionable Data--massive data integration
problem. Most important opportunity: re-create an environment where
researchers can do research--less time on ``data wrangling''. Biggest
danger: false conclusions from messy data.
\textbf{Cranmer} --described re-analysis of archived high-energy physics
data (and software) that could have led to Nobel Prize (if particle hd
been found in the data!). Quick discussion of ``Collaborative
Statistical Modeling'' (see
\href{http://en.wikipedia.org/wiki/Kyle_Cranmer}{link}), showed
(network) graph of multi-phase analysis of a huge amount of data by a
very large amount of people, leading to better statistical limits on
\href{http://en.wikipedia.org/wiki/Higgs_boson}{Higgs Boson}. Also,
service (prototype) where theorists can go get data to test their
models.
\begin{itemize}
\tightlist
\item
Groth mentions
\href{http://www.imi.europa.eu/content/european-lead-factory}{European
LEAD factory} from pharmaceutical companies that led people run models
over data \& get results on how predictive they are, without direct
access to all the data.
\end{itemize}
\textbf{Di Stefano} --opportunities in data sets that can find important
phenomena like gravitational lensing (Pan-STARRS, LSST), only a tiny
fraction of what's possible have been realized--lack of curation seems
to be the problem
\textbf{Slavkovic} --more after lunch
\subsection{Contributed links}\label{contributed-links}
\begin{itemize}
\tightlist
\item
5-minute data/code sharing survey from the Harvard-Smithsonian Center
for Astrophysics, April 2013
http://projects.iq.harvard.edu/seamlessastronomy/book/three-highlighted-graphs-spring-2013-cfa-data-code-sharing-survey
\item
Christine Borgman's
\href{http://polaris.gseis.ucla.edu/cborgman/Chriss_Site/Courses.html}{Course
slides}, which Yolanda Gil praises as fabulous compendia of relevant
literature
\end{itemize}
\subsection{Appendix 1: Original Workshop
Justification}\label{appendix-1-original-workshop-justification}
With the dramatic increases in the size, diversity, and complexity of
data available for scientific discoveries, medical advances, education
reforms and evidence-based policy making, the entire enterprise of
scientific quantitative inquiry has been presented with unprecedented
challenges and opportunities. In particular, the vast majority of
current quantitative inquires are not made by a single individual or
even a single team. The final scientific inference and, more generally,
quantitative learning is a result of a multi-party effort, with
teams/parties entering the process sequentially over several phases
(e.g.~data collection, processing, curation, and analysis). Due to
practical constraints such as resource limitations and confidentiality,
each team involved in a given phase may not have full knowledge of the
assumptions made by, and resources available to, those coming before or
after it. This fact compels all of us involved in the production and
preservation of scientific data to rethink the traditional paradigms of
statistical analysis and data preservation. These have been built around
two ideas: (1) the academic paper as the primary repository of
scientific knowledge and information, and (2) the analysis of data
beginning (and ending) with a single team, who has essentially full
knowledge of the data's origins and all assumptions made in its genesis.
Shifts in the scientific landscape call for revision of both of these
ideas. Projects in astronomy, biology, ecology, and social sciences (to
name a small sampling) are increasingly focused on building databases
for future analyses as a primary objective. These projects must decide
what levels of preprocessing to apply to their data and what additional
information to provide to their users. Clearly, providing all of the
original data allows the most flexibility in subsequent analyses. In
practice, the journey from raw data to a complete analysis is typically
too intricate and problematic for the majority of users, who instead
choose to use preprocessed output. Unfortunately, decisions made at this
stage can be quite treacherous from a statistical perspective because of
the potential for serious information loss and/or information
distortion.
Scientific data released to end-users almost always undergo editing,
imputation, and other forms of preprocessing before they are analyzed.
When such steps are taken, the data analysis becomes a collaborative
endeavor by all parties involved in data collection, preprocessing, and
analysis. Such settings are rife with subtleties and pitfalls. Teams
subsequently handling those data do not and often cannot have a perfect
understanding of the entire phenomenon at hand; the final results will
inevitably contain some combination of their judgments, and some
preprocessing can irreversibly destroy information from the raw data. By
gathering experts from information and natural sciences, we aim to start
building a set of principles and methods that will allow us to
understand such problems and to provide better preprocessing, analyses,
and data preservation, especially in the context of the natural
sciences. The ultimate goals of this research include providing methods
for assessing the validity of such collaborative analyses, guidance on
statistically-principled preprocessing, and a rich new theory of
statistical learning and inference with multiple parties. We believe
that this collaboration will simultaneously sow the seeds for innovative
mathematical theory and shed light on directly usable guidelines for the
construction and curation of scientific databases.
Defects incurred by earlier parties may cause more damage than those in
subsequent analyses, just as problems in the data collection stages are
usually harder to address than problems in the analysis stage. This is
especially true when some of those steps are ``irreversible''. An
example of great current interest in astronomy and astrophysics concerns
the use of data from Chandra X-ray Observatory. As described in the
Chandra documentation (http://cxc.harvard.edu/ciao/dictionary/sdp.html),
the ``Chandra data'' come with different level of processing, from Level
0 ``raw data'', which are not recommended for analysis, to Level 3
``higher lever information'' available to public, where the Level 2 data
processing is considered to be irreversible, which was defined as ``By
`irreversible' we mean that information that has been lost cannot be
regained from the L2 products alone.'' Evidently judgments have been
made regards what to retain and what to discard, and as such assessing
their impact on the subsequent analyses is of great importance for the
so called V\&V (Verification and Validation) process. Indeed, the
question of ``what to keep'' has been a much debated and discussed topic
in the rapidly growing literature on data curation, yet currently there
is few collaboration between fields with overlapping interests in this
area. For example, statisticians have been largely absent such
discussion and debates.
Such collaborations would appear quite natural given the complementary
strengths of the participants. Literature in the field of data curation
has largely focused on describing how scientists use data, their
motivations for data sharing, and the organizational and cultural issues
involved in implementing better data curation practices. Simultaneously,
computer scientists are developing technical solutions to enable
tracking of data provenance and easier access to scientific resources,
to name only a few directions. Statisticians are interested in
developing principled statistical methods for these situations. These
lines of research are distinct, but they provide necessary complements
for each other and could benefit immensely from greater communication
and collaboration.
As a specific example of the fundamental restructuring needed to address
the aforementioned grand challenge, consider the current paradigm for
conducting and evaluating statistical inferences. Statisticians are
trained to regard their mathematical models as approximations to a true
underlying reality. Consequently, these models are typically not
designed to capture the journey from data collection to data analysis.
This is very problematic because such journeys necessarily involve
judgments and data preprocessing from other teams. If the assumptions
made and procedures used in this preprocessing phase are incompatible
with those used in the final analysis (so-called ``uncongeniality'' in
the literature of statistical analysis), then the current statistical
framework is ineffective, or, at worst, entirely inapplicable. In
particular, standard notions such as estimation consistency and
unbiasedness become misguided mathematical idealizations. They are
misguided because they do not take into account the fact that even if
every team in this sequence has reached the perfect answer given their
available information and resources, the lack of mutual knowledge can
still make the final output significantly inferior to that possible
using all the information available to every team. Yet it is clear that
we still can and should have a theoretical foundation for comparing
different methods in such environments. In mathematical terms, we need
to reformulate our criteria by taking into account additional practical
constraints and then seek the most effective methods, instead of
comparing methods using a criterion that none can ever satisfy. A
general statistical framework for this purpose is now being built. This
development can greatly benefit from the input and perspectives of the
data curation community, which has a much better understanding of the
practical constraints and goals involved in these collaborative research
settings.
Conversely, approaches to data curation would benefit greatly from the
involvement of statisticians. Scientists and librarians alike often rely
on general principles of future utility to base decisions on what to
select and on what to keep, rather than on analyses of the actual trends
in data or on demonstrated utility. As a concrete example, at the Center
for Embedded Networked Sensing, a five-university NSF Science and
Technology Center based at UCLA, the involvement of a statistician (Mark
Hansen, a suggested Seminar attendee) midway through the Center's
lifespan radically changed the course of data collection and data
curation. Scientists changed their data collection, storage, and
retrieval methods, and involved their information science partners in
developing better data curation and management methods.
In a nutshell, Radcliffe Exploratory Seminar provides an ideal forum for
intense interdisciplinary exchanges on emerging challenges that truly
require collaborations from multiple disciplines in order to make
meaningful headways. As far as we are aware, if funded, this would be
the first workshop that brings leading computer scientists, information
scientists, natural scientists, and statisticians under one roof to
address some of most intellectually stimulating and practically
challenging problems of the information age.
\subsection{Ten Simple Rules for Good Data
Citizenship}\label{ten-simple-rules-for-good-data-citizenship}
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%% Last modified: Fri Dec 29 11:59:22 EST 2000 mdukhon (address &logo location)
%% Wed Dec 23 13:43:14 EST 1998 milun & amit (colorlogo)
%% Wed Dec 23 15:52:24 EST 1998 milun (more font playing)
%% Tue May 18 15:03:09 EDT 1999 milun (made public)
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\input texinfo @c -*-texinfo-*-
@c %**start of header
@setfilename xboard.info
@settitle XBoard
@c %**end of header
@include version.texi
@ifinfo
@format
INFO-DIR-SECTION Games
START-INFO-DIR-ENTRY
* xboard: (xboard). An X Window System graphical chessboard.
END-INFO-DIR-ENTRY
@end format
@end ifinfo
@titlepage
@title XBoard
@page
@vskip 0pt plus 1filll
@include copyright.texi
@end titlepage
@ifset man
.TH xboard 6 "$Date: " "GNU"
.SH NAME
.PP
xboard @- X graphical user interface for chess
.SH SYNOPSIS
.PP
.B xboard [options]
.br
.B xboard -ics -icshost hostname [options]
.br
.B xboard -ncp [options]
.br
.B |pxboard
.br
.B cmail [options]
@end ifset
@node Top
@top Introduction
@cindex introduction
@ifset man
.SH DESCRIPTION
@end ifset
XBoard is a graphical chessboard that can serve as a
user interface to chess engines (such as GNU Chess), the
Internet Chess Servers,
electronic mail correspondence chess, or your own collection of saved games.
This manual documents version @value{VERSION} of XBoard.
@menu
* Major modes:: The main things XBoard can do.
* Menus:: Menus, buttons, and keys.
* Options:: Command options supported by XBoard.
* Chess Servers:: Using XBoard with an Internet Chess Server (ICS).
* Firewalls:: Connecting to a chess server through a firewall.
* Environment:: Environment variables.
* Limitations:: Known limitations and/or bugs.
* Problems:: How and where to report any problems you run into.
* Contributors:: People who have helped developing XBoard.
* CMail:: Using XBoard for electronic correspondence chess.
* Other programs:: Other programs you can use with XBoard.
@ifnottex
* Copyright:: Copyright notice for this manual.
@end ifnottex
* Copying:: The GNU General Public License.
* Index:: Index of concepts and symbol names.
@end menu
@node Major modes
@chapter Major modes
@cindex Major modes
XBoard always runs in one of four major modes. You select the
major mode from the command line when you start up XBoard.
@table @asis
@item xboard [options]
As an interface to GNU Chess or another chess engine running on your
machine, XBoard lets you play a game against the machine,
set up arbitrary positions, force variations, watch a game between two
chess engines, interactively analyze your stored games or set up and
analyze arbitrary positions. (Note: Not all chess engines support
analysis.)
@item xboard -ics -icshost hostname [options]
As Internet Chess Server (ICS) interface, XBoard
lets you play against other ICS users, observe games
they are playing, or review games that have recently finished. Most
of the ICS "wild" chess variants are supported, including bughouse.
@item xboard -ncp [options]
XBoard can also be used simply
as an electronic chessboard to play through games. It will read and
write game files and allow you to play through variations
manually. You can use it to browse games off the net or review games
you have saved. These features are also available in the other modes.
@item |pxboard
If you want to pipe games into XBoard, use the supplied shell
script @file{pxboard}. For example, from the news reader @file{xrn},
find a message with one or more games in it, click the Save button,
and type @samp{|pxboard} as the file name.
@item cmail [options]
As an interface to electronic mail correspondence chess, XBoard
works with the cmail program. See @ref{CMail} below for
instructions.
@end table
@node Menus
@chapter Menus, buttons, and keys
@cindex Menus
To move a piece, you can drag it with the left mouse button, or you
can click the left mouse button once on the piece, then once more on
the destination square. To drop a new piece on a square (when
applicable), press the middle or the right mouse button over the
square and select from the pop-up menu. In cases where you can drop
either a white or black piece, use the middle button (or shift+right)
for white and the right button (or shift+middle) for black. When you
are playing a bughouse game on an Internet Chess Server, a list of the
off-board pieces that each player has available is shown in the window
title after the player's name; in addition, the piece menus show the
number of pieces available of each type. From version 4.3.14 on, it is
also possible in crazyhouse, bughouse or shogi to dag and drop pieces
to the board from the holdings squares displayed next to the board.
All other XBoard commands are available from the menu bar. The most
frequently used commands also have shortcut keys or on-screen buttons.
When XBoard is iconized, its graphical icon is a white knight if
it is White's turn to move, a black knight if it is Black's turn. See
Iconize in @ref{Keys} below if you have problems getting this
feature to work.
@menu
* File Menu:: Accessing external games and positions.
* Mode Menu:: Selecting XBoard's mode.
* Action Menu:: Talking to the chess engine or ICS opponents.
* Step Menu:: Controlling the game.
* Options Menu:: User preferences.
* Help Menu:: Getting help.
* Keys:: Other shortcut keys.
@end menu
@node File Menu
@section File Menu
@cindex File Menu
@cindex Menu, File
@table @asis
@item New Game
@cindex New Game, Menu Item
Resets XBoard and the chess engine to the beginning of a new chess
game. The @kbd{r} key is a keyboard equivalent. In Internet Chess
Server mode, clears the current state of XBoard, then
resynchronizes with the ICS by sending a refresh command. If you want to
stop playing, observing, or examining an ICS game, use an
appropriate command from the Action menu, not @samp{New Game}.
@xref{Action Menu}.
@item New Shuffle Game
@cindex New Shuffle Game, Menu Item
Similar to @samp{New Game}, but allows you to specify a particular initial position
(according to a standardized numbering system)
in chess variants which use randomized opening positions (e.g. Chess960).
The selected opening position will persistently be chosen on any following
New Game command until you use this menu to select another. Selecting
position number -1 will produce a newly randomized position on any new game.
Using this menu item in variants that normally do not shuffle their opening position
does cause these variants to become shuffle variants until you use the
@samp{New Shuffle Game} menu to explicitly switch the randomization off,
or select a new variant.
@item New Variant
@cindex New variant, Menu Item
Allows you to select a new chess variant in non-ICS mode.
(In ICS play, the ICS is responsible for deciding which variant will be played,
and XBoard adapts automatically.) If you play with an engine, the engine must
be able to play the selected variant, or the command will be ignored.
XBoard supports all major variants, such as xiangqi, shogi, chess, chess960,
Capablanca Chess, shatranj, crazyhouse, bughouse.
(But not every board size has built-in bitmaps for un-orthodox pieces!)
@item Load Game
@cindex Load Game, Menu Item
Plays a game from a record file. The @kbd{g} key is a keyboard equivalent.
A pop-up dialog prompts you for the file name. If the file contains more
than one game, a second pop-up dialog
displays a list of games (with information drawn from their PGN tags, if
any), and you can select the one you want. Alternatively, you can load the
Nth game in the file directly, by typing the number @kbd{N} after the
file name, separated by a space.
The game file parser will accept PGN (portable game notation),
or in fact almost any file that contains moves in algebraic
notation.
Notation of the form @samp{P@@f7}
is accepted for piece-drops in bughouse games;
this is a nonstandard extension to PGN.
If the file includes a PGN position (FEN tag), or an old-style
XBoard position diagram bracketed by @samp{[--} and @samp{--]}
before the first move, the game starts from that position. Text
enclosed in parentheses, square brackets, or curly braces is assumed to
be commentary and is displayed in a pop-up window. Any other
text in the file is ignored. PGN variations (enclosed in
parentheses) are treated as comments; XBoard is not able to walk
variation trees.
The nonstandard PGN tag [Variant "varname"] functions similarly to
the -variant command-line option (see below), allowing games in certain chess
variants to be loaded. There is also a heuristic to
recognize chess variants from the Event tag, by looking for the strings
that the Internet Chess Servers put there when saving variant ("wild") games.
@item Load Next Game
@cindex Load Next Game, Menu Item
Loads the next game from the last game record file you loaded.
The shifted @kbd{N} key is a keyboard equivalent.
@item Load Previous Game
@cindex Load Previous Game, Menu Item
Loads the previous game from the last game record file you
loaded. The shifted @kbd{P} key is a keyboard equivalent.
Not available if the last game was loaded from a pipe.
@item Reload Same Game
@cindex Reload Same Game, Menu Item
Reloads the last game you loaded.
Not available if the last game was loaded from a pipe.
@item Save Game
@cindex Save Game, Menu Item
Appends a record of the current game to a file.
A pop-up dialog
prompts you for the file name. If the game did not begin with
the standard starting position, the game file includes the
starting position used. Games are saved in the PGN (portable
game notation) format, unless the oldSaveStyle option is true,
in which case they are saved in an older format that is specific
to XBoard. Both formats are human-readable, and both can be
read back by the @samp{Load Game} command.
Notation of the form @samp{P@@f7}
is accepted for piece-drops in bughouse games;
this is a nonstandard extension to PGN.
@item Copy Game
@cindex Copy Game, Menu Item
Copies a record of the current game to an internal clipboard in PGN
format and sets the X selection to the game text. The game can be
pasted to another application (such as a text editor or another copy
of XBoard) using that application's paste command. In many X
applications, such as xterm and emacs, the middle mouse button can be
used for pasting; in XBoard, you must use the Paste Game command.
@item Paste Game
@cindex Paste Game, Menu Item
Interprets the current X selection as a game record and loads it, as
with Load Game.
@item Load Position
@cindex Load Position, Menu Item
Sets up a position from a position file. A pop-up dialog prompts
you for the file name. If the file contains more than one saved
position, and you want to load the Nth one, type the number N
after the file name, separated by a space. Position files must
be in FEN (Forsythe-Edwards notation), or in the format that the
Save Position command writes when oldSaveStyle is turned on.
@item Load Next Position
@cindex Load Next Position, Menu Item
Loads the next position from the last position file you loaded.
@item Load Previous Position
@cindex Load Previous Position, Menu Item
Loads the previous position from the last position file you
loaded. Not available if the last position was loaded from a pipe.
@item Reload Same Position
@cindex Reload Same Position, Menu Item
Reloads the last position you loaded.
Not available if the last position was loaded from a pipe.
@item Save Position
@cindex Save Game, Menu Item
Appends a diagram of the current position to a file.
A pop-up dialog
prompts you for the file name. Positions are saved in
FEN (Forsythe-Edwards notation) format unless the @code{oldSaveStyle}
option is true, in which case they are saved in an older,
human-readable format that is specific to XBoard. Both formats
can be read back by the @samp{Load Position} command.
@item Copy Position
@cindex Copy Position, Menu Item
Copies the current position to an internal clipboard in FEN format and
sets the X selection to the position text. The position can be pasted
to another application (such as a text editor or another copy of
XBoard) using that application's paste command. In many X
applications, such as xterm and emacs, the middle mouse button can be
used for pasting; in XBoard, you must use the Paste Position command.
@item Paste Position
@cindex Paste Position, Menu Item
Interprets the current X selection as a FEN position and loads it, as
with Load Position.
@item Mail Move
@itemx Reload CMail Message
@cindex Mail Move, Menu Item
@cindex Reload CMail Message, Menu Item
See @ref{CMail}.
@item Exit
@cindex Exit, Menu Item
Exits from XBoard. The shifted @kbd{Q} key is a keyboard equivalent.
@end table
@node Mode Menu
@section Mode Menu
@cindex Menu, Mode
@cindex Mode Menu
@table @asis
@item Machine White
@cindex Machine White, Menu Item
Tells the chess engine to play White.
@item Machine Black
@cindex Machine Black, Menu Item
Tells the chess engine to play Black.
@item Two Machines
@cindex Two Machines, Menu Item
Plays a game between two chess engines.
@item Analysis Mode
@cindex Analysis Mode, Menu Item
XBoard tells the chess engine to start analyzing the current game/position
and shows you the analysis as you move pieces around.
Note: Some chess engines do not support Analysis mode.
To set up a position to analyze, you do the following:
1. Select Edit Position from the Mode Menu
2. Set up the position. Use the middle and right buttons to
bring up the white and black piece menus.
3. When you are finished, click on either the Black or White
clock to tell XBoard which side moves first.
4. Select Analysis Mode from the Mode Menu to start the analysis.
The analysis function can also be used when observing games on an ICS
with an engine loaded (zippy mode); the engine then will analyse
the positions as they occur in the observed game.
@item Analyze File
@cindex Analyze File, Menu Item
This option lets you load a game from a file (PGN, XBoard format, etc.)
and analyze it. When you select this menu item, a pop-up window appears
and asks for a file name to load.
If the file contains multiple games, another pop up appears that lets
you select which game you wish to analyze.
After a game is loaded, use the XBoard arrow buttons to step
forwards/backwards through the game and watch the analysis.
Note: Some chess engines do not support Analysis mode.
@item ICS Client
@cindex ICS Client, Menu Item
This is the normal mode when XBoard
is connected to a chess server. If you have moved into
Edit Game or Edit Position mode, you can select this option to get out.
To use xboard in ICS mode, run it in the foreground with the -ics
option, and use the terminal you started it from to type commands and
receive text responses from the chess server. See
@ref{Chess Servers} below for more information.
XBoard activates some special position/game editing features when you
use the @kbd{examine} or @kbd{bsetup} commands on ICS and you have
@samp{ICS Client} selected on the Mode menu. First, you can issue the
ICS position-editing commands with the mouse. Move pieces by dragging
with mouse button 1. To drop a new piece on a square, press mouse
button 2 or 3 over the square. This brings up a menu of white pieces
(button 2) or black pieces (button 3). Additional menu choices let
you empty the square or clear the board. Click on the White or Black
clock to set the side to play. You cannot set the side to play or
drag pieces to arbitrary squares while examining on ICC, but you can
do so in @kbd{bsetup} mode on FICS. In addition, the menu commands
@samp{Forward}, @samp{Backward}, @samp{Pause}, and @samp{Stop Examining}
have special functions in this mode; see below.
@item Edit Game
@cindex Edit Game, Menu Item
Allows you to make moves for both Black and White, and to change
moves after backing up with the @samp{Backward} command. The clocks do
not run.
In chess engine mode, the chess engine continues to check moves for legality
but does not participate in the game. You can bring the chess engine
into the game by selecting @samp{Machine White}, @samp{Machine Black},
or @samp{Two Machines}.
In ICS mode, the moves are not sent to the ICS: @samp{Edit Game} takes
XBoard out of ICS Client mode and lets you edit games locally.
If you want to edit games on ICS in a way that other ICS users
can see, use the ICS @kbd{examine} command or start an ICS match
against yourself.
@item Edit Position
@cindex Edit Position, Menu Item
Lets you set up an arbitrary board position.
Use mouse button 1 to drag pieces to new squares, or to delete a piece
by dragging it off the board or dragging an empty square on top of it.
To drop a new piece on a square, press mouse button 2 or 3 over the
square. This brings up a menu of white pieces (button 2) or
black pieces (button 3). Additional menu choices let you empty the
square or clear the board. You can set the side to play next by
clicking on the word White or Black at the top of the screen.
Selecting @samp{Edit Position} causes XBoard to discard
all remembered moves in the current game.
In ICS mode, changes made to the position by @samp{Edit Position} are
not sent to the ICS: @samp{Edit Position} takes XBoard out of
@samp{ICS Client} mode and lets you edit positions locally. If you want to
edit positions on ICS in a way that other ICS users can see, use
the ICS @kbd{examine} command, or start an ICS match against yourself.
(See also the ICS Client topic above.)
@item Training
@cindex Training, Menu Item
Training mode lets you interactively guess the moves of a game for one
of the players. You guess the next move of the game by playing the
move on the board. If the move played matches the next move of the
game, the move is accepted and the opponent's response is auto-played.
If the move played is incorrect, an error message is displayed. You
can select this mode only while loading a game (that is, after
selecting @samp{Load Game} from the File menu). While XBoard is in
@samp{Training} mode, the navigation buttons are disabled.
@item Show Game List
@cindex Show Game List, Menu Item
Shows or hides the list of games generated by the last @samp{Load Game}
command.
@item Show Move History
@cindex Show Move History, Menu Item
Shows or hides a list of moves of the current game.
This list allows you to move the display to any earlier position in the game
by clicking on the corresponding move.
@item Show Engine Output
@cindex Show Engine Output, Menu Item
Shows or hides a window in which the thinking output of any loaded engines
is displayed.
@item Edit Tags
@cindex Edit Tags, Menu Item
Lets you edit the PGN (portable game notation)
tags for the current game. After editing, the tags must still conform to
the PGN tag syntax:
@example
<tag-section> ::= <tag-pair> <tag-section>
<empty>
<tag-pair> ::= [ <tag-name> <tag-value> ]
<tag-name> ::= <identifier>
<tag-value> ::= <string>
@end example
@noindent
See the PGN Standard for full details. Here is an example:
@example
[Event "Portoroz Interzonal"]
[Site "Portoroz, Yugoslavia"]
[Date "1958.08.16"]
[Round "8"]
[White "Robert J. Fischer"]
[Black "Bent Larsen"]
[Result "1-0"]
@end example
@noindent
Any characters that do not match this syntax are silently ignored. Note that
the PGN standard requires all games to have at least the seven tags shown
above. Any that you omit will be filled in by XBoard
with @samp{?} (unknown value), or @samp{-} (inapplicable value).
@item Edit Comment
@cindex Edit Comment, Menu Item
Adds or modifies a comment on the current position. Comments are
saved by @samp{Save Game} and are displayed by @samp{Load Game},
@samp{Forward}, and @samp{Backward}.
@item ICS Input Box
@cindex ICS Input Box, Menu Item
If this option is set in ICS mode,
XBoard
creates an extra window that you can use for typing in ICS commands.
The input box is especially useful if you want to type in something long or do
some editing on your input, because output from ICS doesn't get mixed
in with your typing as it would in the main terminal window.
@item Pause
@cindex Pause, Menu Item
Pauses updates to the board, and if you are playing against a chess engine,
also pauses your clock. To continue, select @samp{Pause} again, and the
display will automatically update to the latest position.
The @samp{P} button and keyboard @kbd{p} key are equivalents.
If you select Pause when you are playing against a chess engine and
it is not your move, the chess engine's clock
will continue to run and it will eventually make a move, at which point
both clocks will stop. Since board updates are paused, however,
you will not see the move until you exit from Pause mode (or select Forward).
This behavior is meant to simulate adjournment with a sealed move.
If you select Pause while you are observing or examining a game on a
chess server, you can step backward and forward in the current history
of the examined game without affecting the other observers and
examiners, and without having your display jump forward to the latest
position each time a move is made. Select Pause again to reconnect
yourself to the current state of the game on ICS.
If you select @samp{Pause} while you are loading a game, the game stops
loading. You can load more moves manually by selecting @samp{Forward}, or
resume automatic loading by selecting @samp{Pause} again.
@end table
@node Action Menu
@section Action Menu
@cindex Menu, Action
@cindex Action, Menu
@table @asis
@item Accept
@cindex Accept, Menu Item
Accepts a pending match offer. If there is more than one offer
pending, you will have to type in a more specific command
instead of using this menu choice.
@item Decline
@cindex Decline, Menu Item
Declines a pending offer (match, draw, adjourn, etc.). If there
is more than one offer pending, you will have to type in a more
specific command instead of using this menu choice.
@item Call Flag
@cindex Call Flag, Menu Item
Calls your opponent's flag, claiming a win on time, or claiming
a draw if you are both out of time. You can also call your
opponent's flag by clicking on his clock or by pressing the
keyboard @kbd{t} key.
@item Draw
@cindex Draw, Menu Item
Offers a draw to your opponent, accepts a pending draw offer
from your opponent, or claims a draw by repetition or the 50-move
rule, as appropriate. The @kbd{d} key is a keyboard equivalent.
@item Adjourn
@cindex Adjourn, Menu Item
Asks your opponent to agree to adjourning the current game, or
agrees to a pending adjournment offer from your opponent.
@item Abort
@cindex Abort, Menu Item
Asks your opponent to agree to aborting the current game, or
agrees to a pending abort offer from your opponent. An aborted
game ends immediately without affecting either player's rating.
@item Resign
@cindex Resign, Menu Item
Resigns the game to your opponent. The shifted @kbd{R} key is a
keyboard equivalent.
@item Stop Observing
@cindex Stop Observing, Menu Item
Ends your participation in observing a game, by issuing the ICS
observe command with no arguments. ICS mode only.
@item Stop Examining
@cindex Stop Examining, Menu Item
Ends your participation in examining a game, by issuing the ICS
unexamine command. ICS mode only.
@item Adjudicate to White
@itemx Adjudicate to Black
@itemx Adjudicate Draw
@cindex Adjudicate to White, Menu Item
@cindex Adjudicate to Black, Menu Item
@cindex Adjudicate Draw, Menu Item
Terminate an ongoing game in Two-Machines mode (including match mode),
with as result a win for white, for black, or a draw, respectively.
The PGN file of the game will accompany the result string
by the comment "user adjudication".
@end table
@node Step Menu
@section Step Menu
@cindex Step Menu
@cindex Menu, Step
@table @asis
@item Backward
@cindex Backward, Menu Item
@cindex <, Button
Steps backward through a series of remembered moves.
The @samp{[<]} button and the @kbd{b} key are equivalents,
as is turning the mouse wheel towards you.
In addition, pressing the Control key steps back one move, and releasing
it steps forward again.
In most modes, @samp{Backward} only lets you look back at old positions;
it does not retract moves. This is the case if you are playing against
a chess engine, playing or observing a game on an ICS, or loading a game.
If you select @samp{Backward} in any of these situations, you will not
be allowed to make a different move. Use @samp{Retract Move} or
@samp{Edit Game} if you want to change past moves.
If you are examining an ICS game, the behavior of @samp{Backward}
depends on whether XBoard is in Pause mode. If Pause mode is
off, @samp{Backward} issues the ICS backward command, which backs up
everyone's view of the game and allows you to make a different
move. If Pause mode is on, @samp{Backward} only backs up your local
view.
@item Forward
@cindex Forward, Menu Item
@cindex >, Button
Steps forward through a series of remembered moves (undoing the
effect of @samp{Backward}) or forward through a game file. The
@samp{[>]} button and the @kbd{f} key are equivalents,
as is turning the mouse wheel away from you.
If you are examining an ICS game, the behavior of Forward
depends on whether XBoard is in Pause mode. If Pause mode is
off, @samp{Forward} issues the ICS forward command, which moves
everyone's view of the game forward along the current line. If
Pause mode is on, @samp{Forward} only moves your local view forward,
and it will not go past the position that the game was in when
you paused.
@item Back to Start
@cindex Back to Start, Menu Item
@cindex <<, Button
Jumps backward to the first remembered position in the game.
The @samp{[<<]} button and the shifted @kbd{B} key are equivalents.
In most modes, Back to Start only lets you look back at old
positions; it does not retract moves. This is the case if you
are playing against a local chess engine, playing or observing a game on
a chess server, or loading a game. If you select @samp{Back to Start} in any
of these situations, you will not be allowed to make different
moves. Use @samp{Retract Move} or @samp{Edit Game} if you want to change past
moves; or use Reset to start a new game.
If you are examining an ICS game, the behavior of @samp{Back to
Start} depends on whether XBoard is in Pause mode. If Pause mode
is off, @samp{Back to Start} issues the ICS @samp{backward 999999}
command, which backs up everyone's view of the game to the start and
allows you to make different moves. If Pause mode is on, @samp{Back
to Start} only backs up your local view.
@item Forward to End
@cindex Forward to End, Menu Item
@cindex >>, Button
Jumps forward to the last remembered position in the game. The
@samp{[>>]} button and the shifted @kbd{F} key are equivalents.
If you are examining an ICS game, the behavior of @samp{Forward to
End} depends on whether XBoard is in Pause mode. If Pause mode
is off, @samp{Forward to End} issues the ICS @samp{forward 999999}
command, which moves everyone's view of the game forward to the end of
the current line. If Pause mode is on, @samp{Forward to End} only moves
your local view forward, and it will not go past the position
that the game was in when you paused.
@item Revert
@cindex Revert, Menu Item
If you are examining an ICS game and Pause mode is off, issues
the ICS command @samp{revert}.
@item Truncate Game
@cindex Truncate Game, Menu Item
Discards all remembered moves of the game beyond the current
position. Puts XBoard into @samp{Edit Game} mode if it was not there
already.
@item Move Now
@cindex Move Now, Menu Item
Forces the chess engine to move immediately. Chess engine mode only.
@item Retract Move
@cindex Retract Move, Menu Item
Retracts your last move. In chess engine mode, you can do this only
after the chess engine has replied to your move; if the chess engine is still
thinking, use @samp{Move Now} first. In ICS mode, @samp{Retract Move}
issues the command @samp{takeback 1} or @samp{takeback 2}
depending on whether it is your opponent's move or yours.
@end table
@node Options Menu
@section Options Menu
@cindex Menu, Options
@cindex Options Menu
@table @asis
@item Flip View
@cindex Flip View, Menu Item
Inverts your view of the chess board for the duration of the
current game. Starting a new game returns the board to normal.
The @kbd{v} key is a keyboard equivalent.
@item Adjudications
@cindex Adjudications, Menu Item
Pops up a sub-menu where you can enable or disable various adjudications
that XBoard can perform in engine-engine games.
You can instruct XBoard to detect and terminate the game on checkmate
or stalemate, even if the engines would not do so, to verify engine
result claims (forfeiting engines that make false claims), rather than
naively following the engine, to declare draw on positions
which can never be won for lack of mating material, (e.g. KBK),
or which are impossible to win unless the opponent seeks its own demise
(e.g. KBKN).
For these adjudications to work, @samp{Test Legality} should be switched on.
It is also possible to instruct XBoard to enforce a 50-move or 3-fold-repeat
rule and automatically declare draw (after a user-adjustable number of moves
or repeats) even if the engines are prepared to go on.
It is also possible to have XBoard declare draw on games that seem to drag on
forever, or adjudicate a loss if both engines agree (for 3 consecutive moves) that one
of them is behind more than a user-adjustable score threshold.
For the latter adjudication to work, XBoard should be able to properly understand
the engine's scores. To facilitate the latter, you can inform xboard here if
the engines report scores from the viewpoint of white, or from that of their own color.
@item Engine Settings
Pops up a sub-menu where you can set some engine parameters common to most engines,
such as hash-table size, tablebase cache size, maximum number of processors
that SMP engines can use, and where to find the Polyglot adapter needed
to run UCI engines under XBoard. The feature that allows setting of these parameters on
engines is new since XBoard 4.3.15, so not many WinBoard engines respond
to it yet, but UCI engines should.
It is also possible to specify a GUI opening book here, i.e. an opening
book that XBoard consults for any position a playing engine gets in.
It then forces the engine to play the book move, rather than to think up its own,
if that position is found in the book.
The book can switched on and off independently for either engine.
@item Time Control
@cindex Time Control, Menu Item
Pops up a sub-menu where you can set the time-control parameters interactively.
Allows you to select classical or incremental time controls,
set the moves per session, session duration, and time increment.
Also allows specification of time-odds factors for one or both engines.
If an engine is given a time-odds factor N, all time quota it gets,
be it at the beginning of a session or through the time increment or
fixed time per move, will be divided by N.
@item Always Queen
@cindex Always Queen, Menu Item
If this option is off, XBoard brings up a dialog
box whenever you move a pawn to the last rank, asking what piece
you want to promote it to. If the option is true, your pawns are
always promoted to queens. Your opponent can still under-promote.
@item Animate Dragging
@cindex Animate Dragging, Menu Item
If Animate Dragging is on, while you are dragging a piece with the
mouse, an image of the piece follows the mouse cursor.
If Animate Dragging is off, there is no visual feedback while you are
dragging a piece, but if Animate Moving is on, the move will be
animated when it is complete.
@item Animate Moving
@cindex Animate Moving, Menu Item
If Animate Moving is on, all piece moves are animated. An image of the
piece is shown moving from the old square to the new square when the
move is completed (unless the move was already animated by Animate Dragging).
If Animate Moving is off, a moved piece instantly disappears from its
old square and reappears on its new square when the move is complete.
@item Auto Comment
@cindex Auto Comment, Menu Item
If this option is on, any remarks made on ICS while you are observing or
playing a game are recorded as a comment on the current move. This includes
remarks made with the ICS commands @kbd{say}, @kbd{tell}, @kbd{whisper},
and @kbd{kibitz}.
Limitation: remarks that you type yourself are not recognized;
XBoard scans only the output from ICS, not the input you type to it.
@item Auto Flag
@cindex Auto Flag, Menu Item
If this option is on and one player runs out of time
before the other,
XBoard
will automatically call his flag, claiming a win on time.
In ICS mode, Auto Flag will only call your opponent's flag, not yours,
and the ICS may award you a draw instead of a win if you have
insufficient mating material. In local chess engine mode,
XBoard
may call either player's flag and will not take material into account.
@item Auto Flip View
@cindex Auto Flip View, Menu Item
If the Auto Flip View option is on when you start a game, the board
will be automatically oriented so that your pawns move from the bottom
of the window towards the top.
@item Auto Observe
@cindex Auto Observe, Menu Item
If this option is on and you add a player to your @code{gnotify}
list on ICS, XBoard will automatically observe all of that
player's games, unless you are doing something else (such as
observing or playing a game of your own) when one starts.
The games are displayed
from the point of view of the player on your gnotify list; that is, his
pawns move from the bottom of the window towards the top.
Exceptions: If both players in a game are on your gnotify list, if
your ICS
@code{highlight}
variable is set to 0, or if the ICS you are using does not
properly support observing from Black's point of view,
you will see the game from White's point of view.
@item Auto Raise Board
@cindex Auto Raise Board, Menu Item
If this option is on, whenever a new game begins, the chessboard window
is deiconized (if necessary) and raised to the top of the stack of windows.
@item Auto Save
@cindex Auto Save, Menu Item
If this option is true, at the end of every game XBoard prompts
you for a file name and appends a record of the game to the file
you specify.
Disabled if the @code{saveGameFile} command-line
option is set, as in that case all games are saved to the specified file.
@xref{Load and Save options}.
@item Blindfold
@cindex Blindfold, Menu Item
If this option is on, XBoard displays the board as usual but does
not display pieces or move highlights. You can still move in the
usual way (with the mouse or by typing moves in ICS mode), even though
the pieces are invisible.
@item Flash Moves
@cindex Flash Moves, Menu Item
If this option is on, whenever a move is completed, the moved piece flashes.
The number of times to flash is set by the flashCount command-line
option; it defaults to 3 if Flash Moves is first turned on from the menu.
If you are playing a game on an ICS, the board is always
oriented at the start of the game so that your pawns move from
the bottom of the window towards the top. Otherwise, the starting
orientation is determined by the @code{flipView} command line option;
if it is false (the default), White's pawns move from bottom to top
at the start of each game; if it is true, Black's pawns move from
bottom to top. @xref{User interface options}.
@item Get Move List
@cindex Get Move List, Menu Item
If this option is on, whenever XBoard
receives the first board of a new ICS game (or a different game from
the one it is currently displaying), it
retrieves the list of past moves from the ICS.
You can then review the moves with the @samp{Forward} and @samp{Backward}
commands
or save them with @samp{Save Game}. You might want to
turn off this option if you are observing several blitz games at once,
to keep from wasting time and network bandwidth fetching the move lists over
and over.
When you turn this option on from the menu, XBoard
immediately fetches the move list of the current game (if any).
@item Highlight Last Move
@cindex Highlight Last Move, Menu Item
If Highlight Last Move is on, after a move is made, the starting and
ending squares remain highlighted. In addition, after you use Backward
or Back to Start, the starting and ending squares of the last move to
be unmade are highlighted.
@item Move Sound
@cindex Move Sound, Menu Item
If this option is on, XBoard alerts you by playing a sound
after each of your opponent's moves (or after every
move if you are observing a game on the Internet Chess Server).
The sound is not played after moves you make or moves read from a
saved game file. By default, the
sound is the terminal bell, but on some systems you can change it
to a sound file using the soundMove option; see below.
If you turn on this option when using XBoard with the Internet
Chess Server, you will probably want to give the
@kbd{set bell 0}
command to the ICS, since otherwise the ICS will ring the terminal bell
after every move (not just yours). (The @file{.icsrc} file
is a good place for this; see @ref{ICS options}.)
@item ICS Alarm
@cindex ICS Alarm, Menu Item
When this option is on, an alarm sound is played when your clock
counts down to the icsAlarmTime (by default, 5 seconds) in an ICS
game. For games with time controls that include an increment, the
alarm will sound each time the clock counts down to the icsAlarmTime.
By default, the alarm sound is the terminal bell, but on some systems
you can change it to a sound file using the soundIcsAlarm option; see
below.
@item Old Save Style
@cindex Old Save Style, Menu Item
If this option is off, XBoard saves games in PGN
(portable game notation) and positions in FEN (Forsythe-Edwards
notation). If the option is on, a save style that is compatible
with older versions of XBoard is used instead.
The old position style is more human-readable
than FEN; the old game style has no particular advantages.
@item Periodic Updates
@cindex Periodic Updates, Menu Item
If this option is off (or if
you are using a chess engine that does not support periodic updates),
the analysis window
will only be updated when the analysis changes. If this option is
on, the Analysis Window will be updated every two seconds.
@item Ponder Next Move
@cindex Ponder Next Move, Menu Item
If this option is off, the chess engine will think only when it is on
move. If the option is on, the engine will also think while waiting
for you to make your move.
@item Popup Exit Message
@cindex Popup Exit Message, Menu Item
If this option is on, when XBoard wants to display a message just
before exiting, it brings up a modal dialog box and waits for you to
click OK before exiting. If the option is off, XBoard prints the
message to standard error (the terminal) and exits immediately.
@item Popup Move Errors
@cindex Popup Move Errors, Menu Item
If this option is off, when you make an error in moving (such as
attempting an illegal move or moving the wrong color piece), the
error message is displayed in the message area. If the option is
on, move errors are displayed in small pop-up windows like other errors.
You can dismiss an error pop-up either by clicking its OK button or by
clicking anywhere on the board, including down-clicking to start a move.
@item Premove
@cindex Premove, Menu Item
If this option is on while playing a game on an ICS, you can register
your next planned move before it is your turn. Move the piece with
the mouse in the ordinary way, and the starting and ending squares
will be highlighted with a special color (red by default). When it is
your turn, if your registered move is legal, XBoard will send it to
ICS immediately; if not, it will be ignored and you can make a
different move. If you change your mind about your premove, either
make a different move, or double-click on any piece to cancel the move
entirely.
@item Quiet Play
@cindex Quiet Play, Menu Item
If this option is on, XBoard will automatically issue an ICS
@kbd{set shout 0}
command whenever you start a game and a
@kbd{set shout 1}
command whenever you finish one. Thus, you will not be distracted
by shouts from other ICS users while playing.
@item Show Coords
@cindex Show Coords, Menu Item
If this option is on, XBoard displays algebraic coordinates
along the board's left and bottom edges.
@item Hide Thinking
@cindex Hide Thinking, Menu Item
If this option is off, the chess engine's notion of the score and best
line of play from the current position is displayed as it is
thinking. The score indicates how many pawns ahead (or if negative,
behind) the chess engine thinks it is. In matches between two
machines, the score is prefixed by @samp{W} or @samp{B} to indicate
whether it is showing White's thinking or Black's, and only the thinking
of the engine that is on move is shown.
@item Test Legality
@cindex Test Legality, Menu Item
If this option is on, XBoard tests whether the moves you try to make
with the mouse are legal and refuses to let you make an illegal move.
Moves loaded from a file with @samp{Load Game} are also checked. If
the option is off, all moves are accepted, but if a local chess engine
or the ICS is active, they will still reject illegal moves. Turning
off this option is useful if you are playing a chess variant with
rules that XBoard does not understand. (Bughouse, suicide, and wild
variants where the king may castle after starting on the d file are
generally supported with Test Legality on.)
@end table
@node Help Menu
@section Help Menu
@cindex Menu, Help
@cindex Help Menu
@table @asis
@item Info XBoard
@cindex Info XBoard, Menu Item
Displays the XBoard documentation in info format. For this feature to
work, you must have the GNU info program installed on your system, and
the file @file{xboard.info} must either be present in the current
working directory, or have been installed by the @samp{make install}
command when you built XBoard.
@item Man XBoard
@cindex Man XBoard, Menu Item
Displays the XBoard documentation in man page format. For this
feature to work, the file @file{xboard.6} must have been installed by
the @samp{make install} command when you built XBoard, and the
directory it was placed in must be on the search path for your
system's @samp{man} command.
@item Hint
@cindex Hint, Menu Item
Displays a move hint from the chess engine.
@item Book
@cindex Book, Menu Item
Displays a list of possible moves from the chess engine's opening
book. The exact format depends on what chess engine you are using.
With GNU Chess 4, the first column gives moves, the second column
gives one possible response for each move, and the third column shows
the number of lines in the book that include the move from the first
column. If you select this option and nothing happens, the chess
engine is out of its book or does not support this feature.
@item About XBoard
@cindex About XBoard, Menu Item
Shows the current XBoard version number.
@end table
@node Keys
@section Other Shortcut Keys
@cindex Keys
@cindex Shortcut keys
@table @asis
@item Iconize
Pressing the @kbd{i} or @kbd{c} key iconizes XBoard. The graphical
icon displays a white knight if it is White's move, or a black knight
if it is Black's move. If your X window manager displays only text
icons, not graphical ones, check its documentation; there is probably
a way to enable graphical icons. If you get black and white reversed,
we would like to hear about it; see @ref{Problems} below for
instructions on how to report this problem.
@end table
You can add or remove shortcut keys using the X resources
@code{form.translations}. Here is an example of what would go in your
@file{.Xresources} file:
@example
XBoard*form.translations: \
Shift<Key>?: AboutGameProc() \n\
<Key>y: AcceptProc() \n\
<Key>n: DeclineProc() \n\
<Key>i: NothingProc()
@end example
@noindent
Binding a key to @code{NothingProc} makes it do nothing, thus removing
it as a shortcut key. The XBoard commands that can be bound to keys
are:
@example
AbortProc, AboutGameProc, AboutProc, AcceptProc, AdjournProc,
AlwaysQueenProc, AnalysisModeProc, AnalyzeFileProc,
AnimateDraggingProc, AnimateMovingProc, AutobsProc, AutoflagProc,
AutoflipProc, AutoraiseProc, AutosaveProc, BackwardProc,
BlindfoldProc, BookProc, CallFlagProc, CopyGameProc, CopyPositionProc,
DebugProc, DeclineProc, DrawProc, EditCommentProc, EditGameProc,
EditPositionProc, EditTagsProc, EnterKeyProc, FlashMovesProc,
FlipViewProc, ForwardProc, GetMoveListProc, HighlightLastMoveProc,
HintProc, Iconify, IcsAlarmProc, IcsClientProc, IcsInputBoxProc,
InfoProc, LoadGameProc, LoadNextGameProc, LoadNextPositionProc,
LoadPositionProc, LoadPrevGameProc, LoadPrevPositionProc,
LoadSelectedProc, MachineBlackProc, MachineWhiteProc, MailMoveProc,
ManProc, MoveNowProc, MoveSoundProc, NothingProc, OldSaveStyleProc,
PasteGameProc, PastePositionProc, PauseProc, PeriodicUpdatesProc,
PonderNextMoveProc, PopupExitMessageProc, PopupMoveErrorsProc,
PremoveProc, QuietPlayProc, QuitProc, ReloadCmailMsgProc,
ReloadGameProc, ReloadPositionProc, RematchProc, ResetProc,
ResignProc, RetractMoveProc, RevertProc, SaveGameProc,
SavePositionProc, ShowCoordsProc, ShowGameListProc, ShowThinkingProc,
StopExaminingProc, StopObservingProc, TestLegalityProc, ToEndProc,
ToStartProc, TrainingProc, TruncateGameProc, and TwoMachinesProc.
@end example
@node Options
@chapter Options
@cindex Options
@cindex Options
This section documents the command-line options to XBoard. You can
set these options in two ways: by typing them on the shell command
line you use to start XBoard, or by setting them as X resources
(typically in your @file{.Xresources} file). Many of the options
cannot be changed while XBoard is running; others set the initial
state of items that can be changed with the @ref{Options} menu.
Most of the options have both a long name and a short name. To turn a
boolean option on or off from the command line, either give its long
name followed by the value true or false
(@samp{-longOptionName true}), or give just the short name to turn the
option on (@samp{-opt}), or the short name preceded by @samp{x} to
turn the option off (@samp{-xopt}). For options that take strings or
numbers as values, you can use the long or short option names
interchangeably.
Each option corresponds to an X resource with the same name, so
if you like, you can set options in your @file{.Xresources} file
or in a file named @file{XBoard} in your home directory.
For options that have two names, the longer one is the name of
the corresponding X resource; the short name is not recognized.
To turn a boolean option on or off as an
X resource, give its long name followed by the value
true or false (@samp{XBoard*longOptionName: true}).
@menu
* Chess engine options:: Controlling the chess engine.
* UCI + WB Engine Settings:: Setting some very common engine parameters
* ICS options:: Connecting to and using ICS.
* Load and Save options:: Input/output options.
* User interface options:: Look and feel options.
* Adjudication Options:: Control adjudication of engine-engine games.
* Other options:: Miscellaneous.
@end menu
@node Chess engine options
@section Chess Engine Options
@cindex options, Chess engine
@cindex Chess engine options
@table @asis
@item -tc or -timeControl minutes[:seconds]
@cindex tc, option
@cindex timeControl, option
Each player begins with his clock set to the @code{timeControl} period.
Default: 5 minutes.
The additional options @code{movesPerSession} and @code{timeIncrement}
are mutually exclusive.
@item -mps or -movesPerSession moves
@cindex mps, option
@cindex movesPerSession, option
When both players have made @code{movesPerSession} moves, a
new @code{timeControl} period is added to both clocks. Default: 40 moves.
@item -inc or -timeIncrement seconds
@cindex inc, option
@cindex timeIncrement, option
If this option is specified, @code{movesPerSession} is ignored.
Instead, after each player's move, @code{timeIncrement} seconds are
added to his clock.
Use @samp{-inc 0} if you want to require the entire
game to be played in one @code{timeControl} period, with no increment.
Default: -1, which specifies @code{movesPerSession} mode.
@item -clock/-xclock or -clockMode true/false
@cindex clock, option
@cindex clockMode, option
Determines whether or not to display the chess clocks. If clockMode is
false, the clocks are not shown, but the side that is to play next
is still highlighted. Also, unless @code{searchTime}
is set, the chess engine still keeps track of the clock time and uses it to
determine how fast to make its moves.
@item -st or -searchTime minutes[:seconds]
@cindex st, option
@cindex searchTime, option
Tells the chess engine to spend at most the given amount of time
searching for each of its moves. Without this option, the chess engine
chooses its search time based on the number of moves and amount
of time remaining until the next time control.
Setting this option also sets clockMode to false.
@item -depth or -searchDepth number
@cindex sd, option
@cindex searchDepth, option
Tells the chess engine to look ahead at most the given number of moves
when searching for a move to make. Without this option, the chess
engine chooses its search depth based on the number of moves and
amount of time remaining until the next time control. With the option,
the engine will cut off its search early if it reaches the specified depth.
@item -firstNPS number
@itemx -secondNPS number
@cindex firstNPS, option
@cindex secondNPS, option
Tells the chess engine to use an internal time standard based on its node count,
rather then wall-clock time, to make its timing decisions.
The time in virtual seconds should be obtained by dividing the node count
through the given number, like the number was a rate in nodes per second.
Xboard will manage the clocks in accordance with this, relying on the number
of nodes reported by the engine in its thinking output. If the given number equals zero,
it can obviously not be used to convert nodes to seconds, and the time reported
by the engine is used to decrement the XBoard clock in stead. The engine is supposed to
report in CPU time it uses, rather than wall-clock time, in this mode. This option
can provide fairer conditions for engine-engine matches on heavily loaded machines,
or with very fast games (where the wall clock is too inaccurate).
@code{showThinking} must be on for this option to work. Default: -1 (off).
Not many engines might support this yet!
@item -firstTimeOdds factor
@itemx -secondTimeOdds factor
@cindex firstTimeOdds, option
@cindex secondTimeOdds, option
Reduces the time given to the mentioned engine by the given factor.
If pondering is off, the effect is indistinguishable from what would happen
if the engine was running on an n-times slower machine. Default: 1.
@item -timeOddsMode mode
@cindex timeOddsMode, option
This option determines how the case is handled where both engines have a time-odds handicap.
If mode=1, the engine that gets the most time will always get the nominal time,
as specified by the time-control options, and its opponent's time is renormalized accordingly.
If mode=0, both play with reduced time. Default: 0.
@item -hideThinkingFromHuman true/false
Controls the Hide Thinking option. @xref{Options Menu}. Default: true.
(Replaces the Show-Thinking option of older xboard versions.)
@item -thinking/-xthinking or -showThinking true/false
@cindex thinking, option
@cindex showThinking, option
Forces the engine to send thinking output to xboard.
Used to be the only way to control if thinking output was displayed
in older xboard versions,
but as the thinking output in xboard 4.3 is also used for several other
purposes (adjudication, storing in PGN file) the display of it is now controlled
by the new option Hide Thinking. @xref{Options Menu}. Default: false.
(But if xboard needs the thinking output for some purpose,
it makes the engine send it despite the setting of this option.)
@item -ponder/-xponder or -ponderNextMove true/false
@cindex ponder, option
@cindex ponderNextMove, option
Sets the Ponder Next Move menu option. @xref{Options Menu}. Default: true.
@item -smpCores number
Specifies the maximum number of CPUs an SMP engine is allowed to use.
Only works for engines that support the WinBoard-protocol cores feature.
@item -mg or -matchGames n
@cindex mg, option
@cindex matchGames, option
Automatically runs an n-game match between two chess engines,
with alternating colors.
If the @code{loadGameFile} or @code{loadPositionFile} option is set,
XBoard
starts each game with the given opening moves or the given position;
otherwise, the games start with the standard initial chess position.
If the @code{saveGameFile} option is set, a move record for the
match is appended to the specified file. If the @code{savePositionFile}
option is set, the final position reached in each game of the match is appended
to the specified file. When the match is over, XBoard
displays the match score and exits. Default: 0 (do not run a match).
@item -mm/-xmm or -matchMode true/false
@cindex mm, option
@cindex matchMode, option
Setting @code{matchMode} to true is equivalent to setting
@code{matchGames} to 1.
@item -sameColorGames n
@cindex sameColorGames, option
Automatically runs an n-game match between two chess engines,
without alternating colors.
Otherwise the same applies as for the @samp{-matchGames} option,
over which it takes precedence if both are specified. (See there.)
Default: 0 (do not run a match).
@item -fcp or -firstChessProgram program
@cindex fcp, option
@cindex firstChessProgram, option
Name of first chess engine.
Default: @file{Fairy-Max}.
@item -scp or -secondChessProgram program
@cindex scp, option
@cindex secondChessProgram, option
Name of second chess engine, if needed.
A second chess engine is started only in Two Machines (match) mode.
Default: @file{Fairy-Max}.
@item -fb/-xfb or -firstPlaysBlack true/false
@cindex fb, option
@cindex firstPlaysBlack, option
In games between two chess engines, firstChessProgram normally plays
white. If this option is true, firstChessProgram plays black. In a
multi-game match, this option affects the colors only for the first
game; they still alternate in subsequent games.
@item -fh or -firstHost host
@itemx -sh or -secondHost host
@cindex fh, option
@cindex firstHost, option
@cindex sh, option
@cindex secondHost, option
Hosts on which the chess engines are to run. The default for
each is @file{localhost}. If you specify another host, XBoard
uses @file{rsh} to run the chess engine there. (You can substitute a
different remote shell program for rsh using the @code{remoteShell}
option described below.)
@item -fd or -firstDirectory dir
@itemx -sd or -secondDirectory dir
@cindex fd, option
@cindex firstDirectory, option
@cindex sd, option
@cindex secondDirectory, option
Working directories in which the chess engines are to be run.
The default is "", which means to run the chess engine
in the same working directory as XBoard
itself. (See the CHESSDIR environment variable.)
This option is effective only when the chess engine is being run
on the local host; it does not work if the engine is run remotely
using the -fh or -sh option.
@item -initString string
@itemx -secondInitString string
@cindex initString, option
@cindex secondInitString, option
The string that is sent to initialize each chess engine for a new game.
Default:
@example
new
random
@end example
@noindent
Setting this option from the command line is tricky, because you must
type in real newline characters, including one at the very end.
In most shells you can do this by
entering a @samp{\} character followed by a newline. It is easier to set
the option from your @file{.Xresources} file; in that case you can
include the character sequence @samp{\n} in the string, and it will
be converted to a newline.
If you change this option, don't remove the @samp{new}
command; it is required by all chess engines to
start a new game.
You can remove the @samp{random} command if you like; including it
causes GNU Chess 4 to randomize its move selection slightly so that it
doesn't play the same moves in every game. Even without
@samp{random}, GNU Chess 4 randomizes its choice of moves from its
opening book. Many other chess engines ignore this command entirely
and always (or never) randomize.
You can also try adding other commands to the initString; see the
documentation of the chess engine you are using for details.
@item -firstComputerString string
@itemx -secondComputerString string
@cindex firstComputerString, option
@cindex secondComputerString, option
The string that is sent to the chess engine if its opponent is another
computer chess engine. The default is @samp{computer\n}. Probably the
only useful alternative is the empty string (@samp{}), which keeps the
engine from knowing that it is playing another computer.
@item -reuse/-xreuse or -reuseFirst true/false
@itemx -reuse2/-xreuse2 or -reuseSecond true/false
@cindex reuse, option
@cindex reuseFirst, option
@cindex reuse2, option
@cindex reuseSecond, option
If the option is false,
XBoard kills off the chess engine after every game and starts
it again for the next game.
If the option is true (the default),
XBoard starts the chess engine only once
and uses it repeatedly to play multiple games.
Some old chess engines may not work properly when
reuse is turned on, but otherwise games will start faster if it is left on.
@item -firstProtocolVersion version-number
@itemx -secondProtocolVersion version-number
@cindex firstProtocolVersion, option
@cindex secondProtocolVersion, option
This option specifies which version of the chess engine communication
protocol to use. By default, version-number is 2. In version 1, the
"protover" command is not sent to the engine; since version 1 is a
subset of version 2, nothing else changes. Other values for
version-number are not supported.
@item -firstScoreAbs true/false
@itemx -secondScoreAbs true/false
@cindex firstScoreAbs, option
@cindex secondScoreAbs, option
If this option is set, the score reported by the engine is taken to be
that in favor of white, even when the engine plays black.
Important when XBoard uses the score for adjudications, or in PGN reporting.
@item -niceEngines priority
@cindex niceEngines, option
This option allows you to lower the priority of the engine processes,
so that the generally insatiable hunger for CPU time of chess engines does not interfere so much
with smooth operation of WinBoard (or the rest of your system).
Negative values could increase the engine priority, which is not recommended.
@item -firstOptions string
@itemx -secondOptions string
@cindex firstOptions, option
@cindex secondOptions, option
The given string is a comma-separated list of (option name=option value) pairs,
like the following example: "style=Karpov,blunder rate=0".
If the options announced by the engine at startup through the feature commands of WinBoard protocol
matches one of the option names (i.e. "style" or "blunder rate"),
it would be set to the given value (i.e. "Karpov" or 0)
through a corresponding option command to the engine.
This provided that the type of the value (text or numeric) matches as well.
@item -firstNeedsNoncompliantFEN string
@itemx -secondNeedsNoncompliantFEN string
@cindex firstNeedsNoncompliantFEN, option
@cindex secondNeedsNoncompliantFEN, option
The castling rights and e.p. fields of the FEN sent to the mentioned engine
with the setboard command will be replaced by the given string. This can for
instance be used to run engines that do not understand Chess960 FENs in
variant fischerandom, to make them at least understand the opening position,
through setting the string to "KQkq -". (Note you also have to give the e.p. field!)
Other possible applications are to provide work-arounds for engines that want to see
castling and e.p. fields in variants that do not have castling or e.p.
(shatranj, courier, xiangqi, shogi) so that WinBoard would normally omit them
(string = "- -"), or to add variant-specific fields that are not yet supported by WinBoard
(e.g. to indicate the number of checks in 3check).
@end table
@node UCI + WB Engine Settings
@section UCI + WB Engine Settings
@cindex Engine Settings
@cindex Settings, Engine
@table @asis
@item -fUCI or -firstIsUCI true/false
@itemx -sUCI or -secondIsUCI true/false
@cindex fUCI, option
@cindex sUCI, option
@cindex firstIsUCI, option
@cindex secondIsUCI, option
Indicates if the mentioned engine executable file is an UCI engine,
and should be run with the aid of the Polyglot adapter rather than directly.
Xboard will then pass the other UCI options and engine name to Polyglot
through a .ini temporary file created for the purpose.
@item -PolyglotDir filename
@cindex PolyglotDir, option
Gives the name of the directory in which the Polyglot adapter for UCI engines expects its files.
Default: "/usr/local/share/polyglot".
@item -usePolyglotBook true/false
@cindex usePolyglotBook, option
Specifies if the Polyglot book should be used.
@item -PolyglotBook filename
@cindex PolyglotBook, option
Gives the filename of the opening book that Polyglot should use.
From XBoard 4.3.15 on, native WinBoard engines will also use the opening book specified here,
provided the @code{usePolyglotBook} option is set to true,
and the option @code{firstHasOwnBookUCI} or @code{secondHasOwnBookUCI} applying to the engine
is set to false.
The engine will be kept in force mode as long as the current position is in book,
and XBoard will select the book moves for it. Default "".
@item -fNoOwnBookUCI or -firstXBook or -firstHasOwnBookUCI true/false
@itemx -sNoOwnBookUCI or -secondXBook or -secondHasOwnBookUCI true/false
@cindex fNoOwnBookUCI, option
@cindex sNoOwnBookUCI, option
@cindex firstHasOwnBookUCI, option
@cindex secondHasOwnBookUCI, option
@cindex firstXBook, option
@cindex secondXBook, option
Indicates if the mentioned engine has its own opening book it should play from,
rather than using the external book through XBoard. Default: false.
@item -defaultHashSize n
@cindex defaultHashSize, option
Sets the size of the hash table to n MegaBytes. Together with the EGTB cache size
this number is also used to calculate the memory setting of WinBoard engines,
for those that support the memory feature of WinBoard protocol. Default: 64.
@item -defaultCacheSizeEGTB n
@cindex defaultCacheSizeEGTB, option
Sets the size of the EGTB cache to n MegaBytes. Together with the hash-table size
this number is also used to calculate the memory setting of WinBoard engines,
for those that support the memory feature of WinBoard protocol. Default: 4.
@item -defaultPathEGTB filename
@cindex defaultPathEGTB, option
Gives the name of the directory where the end-game tablebases are installed, for UCI engines.
Default: "/usr/local/share/egtb".
@item -egtFormats string
@cindex egtFormats, option
Specifies which end-game tables are installed on the computer, and where.
The argument is a comma-separated list of format specifications,
each specification consisting of a format name, a colon, and a directory path name,
e.g. "nalimov:/usr/local/share/egtb".
If the name part matches that of a format that the engine requests through a feature command,
xboard will relay the path name for this format to the engine through an egtpath command.
One egtpath command for each matching format will be sent.
Popular formats are "nalimov" DTM tablebases and "scorpio" bitbases.
Default: "".
@end table
@node ICS options
@section ICS options
@cindex ICS options
@cindex Options, ICS
@table @asis
@item -ics/-xics or -internetChessServerMode true/false
@cindex ics, option
@cindex internetChessServerMode, option
Connect with an Internet Chess Server to play chess against its
other users, observe games they are playing, or review games
that have recently finished. Default: false.
@item -icshost or -internetChessServerHost host
@cindex icshost, option
@cindex internetChessServerHost, option
The Internet host name or address of the chess server to connect
to when in ICS mode. Default: @code{chessclub.com}.
Another popular chess server to try is @code{freechess.org}.
If your site doesn't have a working Internet name server, try
specifying the host address in numeric form.
You may also need
to specify the numeric address when using the icshelper option
with timestamp or timeseal (see below).
@item -icsport or -internetChessServerPort port-number
@cindex icsport, option
@cindex internetChessServerPort, option
The port number to use when connecting to a chess server in ICS
mode. Default: 5000.
@item -icshelper or -internetChessServerHelper prog-name
@cindex icshelper, option
@cindex internetChessServerHelper, option
An external helper program used to communicate with the chess server.
You would set it to "timestamp" for ICC (chessclub.com) or
"timeseal" for FICS (freechess.org), after
obtaining the correct version of timestamp or timeseal for your
computer. See "help timestamp" on ICC and "help timeseal" on FICS.
This option is shorthand for @code{-useTelnet -telnetProgram program}.
@item -telnet/-xtelnet or -useTelnet true/false
@cindex telnet, option
@cindex useTelnet, option
This option is poorly named; it should be called useHelper.
If set to true, it instructs XBoard to run an external
program to communicate with the Internet Chess Server.
The program to use is given by the telnetProgram option.
If the option is
false (the default), XBoard opens a TCP socket and uses its own
internal implementation of the telnet protocol to communicate with the
ICS. @xref{Firewalls}.
@item -telnetProgram prog-name
@cindex telnetProgram, option
This option is poorly named; it should be called helperProgram.
It gives the name of the telnet program to be used with
the @code{gateway} and @code{useTelnet} options. The default is
@file{telnet}. The telnet program is invoked with the value of
@code{internetChessServerHost} as its first argument and the value
of @code{internetChessServerPort} as its second argument.
@xref{Firewalls}.
@item -gateway host-name
@cindex gateway, option
If this option is set to a host name, XBoard communicates with the
Internet Chess Server by using @file{rsh} to run
the @code{telnetProgram} on the given host,
instead of using its own internal implementation
of the telnet protocol. You can substitute a different remote shell
program for @file{rsh} using the @code{remoteShell} option described below.
@xref{Firewalls}.
@item -internetChessServerCommPort or -icscomm dev-name
@cindex internetChessServerCommPort, option
@cindex icscomm, option
If this option is set, XBoard communicates with the ICS through
the given character I/O device instead of opening a TCP connection.
Use this option if your system does not have any kind of
Internet connection itself (not even a SLIP or PPP connection),
but you do have dial-up access (or a hardwired terminal line) to
an Internet service provider from which you can telnet to the ICS.
The support for this option in XBoard is minimal. You need to
set all communication parameters and tty modes before you enter
XBoard.
Use a script something like this:
@example
stty raw -echo 9600 > /dev/tty00
xboard -ics -icscomm /dev/tty00
@end example
Here replace @samp{/dev/tty00} with the name of the device that your
modem is connected to. You might have to add several more
options to these stty commands. See the man pages for @file{stty}
and @code{tty} if you run into problems. Also, on many systems stty
works on its standard input instead of standard output, so you
have to use @samp{<} instead of @samp{>}.
If you are using linux, try starting with the script below.
Change it as necessary for your installation.
@example
#!/bin/sh -f
# configure modem and fire up XBoard
# configure modem
(
stty 2400 ; stty raw ; stty hupcl ; stty -clocal
stty ignbrk ; stty ignpar ; stty ixon ; stty ixoff
stty -iexten ; stty -echo
) < /dev/modem
xboard -ics -icscomm /dev/modem
@end example
@noindent
After you start XBoard in this way, type whatever commands are
necessary to dial out to your Internet provider and log in.
Then telnet to ICS, using a command like
@kbd{telnet chessclub.com 5000}.
Important: See the paragraph below about extra echoes,
in @ref{Limitations}.
@item -icslogon or -internetChessServerLogonScript file-name
@cindex icslogon, option
@cindex internetChessServerLogonScript, option
@cindex .icsrc
Whenever XBoard connects to the Internet Chess Server,
if it finds a file with the name given in this option, it feeds the
file's contents to the ICS as commands. The default file name
is @file{.icsrc}.
Usually the first two lines of the file should be
your ICS user name and password.
The file can be either in $CHESSDIR, in XBoard's working
directory if CHESSDIR is not set, or in your home directory.
@item -msLoginDelay delay
@cindex msLoginDelay, option
If you experience trouble logging on to an ICS when using the
@code{-icslogon} option, inserting some delay between characters
of the logon script may help. This option adds @code{delay}
milliseconds of delay between characters. Good values to try
are 100 and 250.
@item -icsinput/-xicsinput or -internetChessServerInputBox true/false
@cindex icsinput, option
@cindex internetChessServerInputBox, option
Sets the ICS Input Box menu option. @xref{Mode Menu}. Default: false.
@item -autocomm/-xautocomm or -autoComment true/false
@cindex autocomm, option
@cindex autoComment, option
Sets the Auto Comment menu option. @xref{Options Menu}. Default: false.
@item -autoflag/-xautoflag or -autoCallFlag true/false
@cindex autoflag, option
@cindex autoCallFlag, option
Sets the Auto Flag menu option. @xref{Options Menu}. Default: false.
@item -autobs/-xautobs or -autoObserve true/false
@cindex autobs, option
@cindex autoObserve, option
Sets the Auto Observe menu option. @xref{Options Menu}. Default: false.
@item -autoKibitz
@cindex autoKibitz, option
Enables kibitzing of the engines last thinking output (depth, score, time, speed, PV)
before it moved
to the ICS, in zippy mode. The option @code{showThinking} must be switched on for
this option to work.
Also diverts similar kibitz information of an opponent engine that is playing you
through the ICS to the engine-output window, as if the engine was playing locally.
@item -moves/-xmoves or -getMoveList true/false
@cindex moves, option
@cindex getMoveList, option
Sets the Get Move List menu option. @xref{Options Menu}. Default: true.
@item -alarm/-xalarm or -icsAlarm true/false
@cindex alarm, option
@cindex icsAlarm, option
Sets the ICS Alarm menu option. @xref{Options Menu}. Default: true.
@item -icsAlarmTime ms
@cindex icsAlarmTime, option
Sets the time in milliseconds for the ICS Alarm menu option.
@xref{Options Menu}. Default: 5000.
@item lowTimeWarning true/false
@cindex lowTimeWarning, option
Controls a color change of the board as a warning your time is running out.
@xref{Options Menu}. Default: false.
@item -pre/-xpre \fRor\fB -premove true/false
@cindex pre, option
@cindex premove, option
Sets the Premove menu option. @xref{Options Menu}. Default: true.
@item -quiet/-xquiet or -quietPlay true/false
@cindex quiet, option
@cindex quietPlay, option
Sets the Quiet Play menu option. @xref{Options Menu}. Default: false.
@item -colorizeMessages or -colorize
@cindex Colors
@cindex colorize, option
Setting colorizeMessages
to true tells XBoard to colorize the messages received from
the ICS. Colorization works only if your xterm
supports ISO 6429 escape sequences for changing text colors.
@item -colorShout foreground,background,bold
@itemx -colorSShout foreground,background,bold
@itemx -colorChannel1 foreground,background,bold
@itemx -colorChannel foreground,background,bold
@itemx -colorKibitz foreground,background,bold
@itemx -colorTell foreground,background,bold
@itemx -colorChallege foreground,background,bold
@itemx -colorRequest foreground,background,bold
@itemx -colorSeek foreground,background,bold
@itemx -colorNormal foreground,background,bold
@cindex Colors
@cindex colorShout, option
@cindex colorSShout, option
@cindex colorChannel1, option
@cindex colorChannel, option
@cindex colorKibitz, option
@cindex colorTell, option
@cindex colorChallenge, option
@cindex colorRequest, option
@cindex colorSeek, option
@cindex colorNormal, option
These options set the colors used when colorizing ICS messages.
All ICS messages are grouped into one of these categories:
shout, sshout, channel 1, other channel, kibitz, tell, challenge,
request (including abort, adjourn, draw, pause, and takeback), or
normal (all other messages).
Each foreground or background argument can be one of the following:
black, red, green, yellow, blue, magenta, cyan, white, or default.
Here ``default'' means the default foreground or background color of
your xterm. Bold can be 1 or 0. If background is omitted, ``default''
is assumed; if bold is omitted, 0 is assumed.
Here is an example of how to set the colors in your @file{.Xresources} file.
The colors shown here are the default values; you will get
them if you turn @code{-colorize} on without specifying your own colors.
@example
xboard*colorizeMessages: true
xboard*colorShout: green
xboard*colorSShout: green, black, 1
xboard*colorChannel1: cyan
xboard*colorChannel: cyan, black, 1
xboard*colorKibitz: magenta, black, 1
xboard*colorTell: yellow, black, 1
xboard*colorChallenge: red, black, 1
xboard*colorRequest: red
xboard*colorSeek: blue
xboard*colorNormal: default
@end example
@item -soundProgram progname
@cindex soundProgram, option
@cindex Sounds
If this option is set to a sound-playing program that is installed and
working on your system, XBoard can play sound files when certain
events occur, listed below. The default program name is "play". If
any of the sound options is set to "$", the event rings the terminal
bell by sending a ^G character to standard output, instead of playing
a sound file. If an option is set to the empty string "", no sound is
played for that event.
@item -soundShout filename
@itemx -soundSShout filename
@itemx -soundChannel filename
@itemx -soundKibitz filename
@itemx -soundTell filename
@itemx -soundChallenge filename
@itemx -soundRequest filename
@itemx -soundSeek filename
@cindex soundShout, option
@cindex soundSShout, option
@cindex soundChannel, option
@cindex soundKibitz, option
@cindex soundTell, option
@cindex soundChallenge, option
@cindex soundRequest, option
@cindex soundSeek, option
These sounds are triggered in the same way as the colorization events
described above. They all default to "", no sound. They are played
only if the colorizeMessages is on.
@item -soundMove filename
@cindex soundMove, option
This sound is used by the Move Sound menu option. Default: "$".
@item -soundIcsAlarm filename
@cindex soundIcsAlarm, option
This sound is used by the ICS Alarm menu option. Default: "$".
@item -soundIcsWin filename
@cindex soundIcsWin, option
This sound is played when you win an ICS game. Default: "" (no sound).
@item -soundIcsLoss filename
@cindex soundIcsLoss, option
This sound is played when you lose an ICS game. Default: "" (no sound).
@item -soundIcsDraw filename
@cindex soundIcsDraw, option
This sound is played when you draw an ICS game. Default: "" (no sound).
@item -soundIcsUnfinished filename
@cindex soundIcsUnfinished, option
This sound is played when an ICS game that you are participating in is
aborted, adjourned, or otherwise ends inconclusively. Default: "" (no
sound).
Here is an example of how to set the sounds in your @file{.Xresources} file:
@example
xboard*soundShout: shout.wav
xboard*soundSShout: sshout.wav
xboard*soundChannel1: channel1.wav
xboard*soundChannel: channel.wav
xboard*soundKibitz: kibitz.wav
xboard*soundTell: tell.wav
xboard*soundChallenge: challenge.wav
xboard*soundRequest: request.wav
xboard*soundSeek: seek.wav
xboard*soundMove: move.wav
xboard*soundIcsWin: win.wav
xboard*soundIcsLoss: lose.wav
xboard*soundIcsDraw: draw.wav
xboard*soundIcsUnfinished: unfinished.wav
xboard*soundIcsAlarm: alarm.wav
@end example
@end table
@node Load and Save options
@section Load and Save options
@cindex Options, Load and Save
@cindex Load and Save options
@table @asis
@item -lgf or -loadGameFile file
@itemx -lgi or -loadGameIndex index
@cindex lgf, option
@cindex loadGameFile, option
@cindex lgi, option
@cindex loadGameIndex, option
If the @code{loadGameFile} option is set, XBoard loads the specified
game file at startup. The file name @file{-} specifies the standard
input. If there is more than one game in the file, XBoard
pops up a menu of the available games, with entries based on their PGN
(Portable Game Notation) tags.
If the @code{loadGameIndex} option is set to @samp{N}, the menu is suppressed
and the N th game found in the file is loaded immediately.
The menu is also suppressed if @code{matchMode} is enabled or if the game file
is a pipe; in these cases the first game in the file is loaded immediately.
Use the @file{pxboard} shell script provided with XBoard if you
want to pipe in files containing multiple games and still see the menu.
If the loadGameIndex specifies an index -1, this triggers auto-increment
of the index in @code{matchMode}, which means that after every game the
index is incremented by one, causing each game of the match to be played
from the next game in the file. Similarly, specifying an index value of -2
causes the index to be incremented every two games, so that each game
in the file is used twice (with reversed colors).
The @code{rewindIndex} option causes the index to be reset to the
first game of the file when it has reached a specified value.
@item -rewindIndex n
Causes a position file or game file to be rewound to its beginning after n
positions or games in auto-increment @code{matchMode}.
See @code{loadPositionIndex} and @code{loadGameIndex}.
default: 0 (no rewind).
@item -td or -timeDelay seconds
@cindex td, option
@cindex timeDelay, option
Time delay between moves during @samp{Load Game}. Fractional seconds
are allowed; try @samp{-td 0.4}. A time delay value of -1 tells
XBoard not to step through game files automatically. Default: 1
second.
@item -sgf or -saveGameFile file
@cindex sgf, option
@cindex saveGameFile, option
If this option is set, XBoard appends a record of every game
played to the specified file. The file name @file{-} specifies the
standard output.
@item -autosave/-xautosave or -autoSaveGames true/false
@cindex autosave, option
@cindex autoSaveGames, option
Sets the Auto Save menu option. @xref{Options Menu}. Default: false.
Ignored if @code{saveGameFile} is set.
@item -lpf or -loadPositionFile file
@itemx -lpi or -loadPositionIndex index
@cindex lpf, option
@cindex loadPositionFile, option
@cindex lpi, option
@cindex loadPositionIndex, option
If the @code{loadPositionFile} option is set, XBoard loads the
specified position file at startup. The file name @file{-} specifies the
standard input. If the @code{loadPositionIndex} option is set to N,
the Nth position found in the file is loaded; otherwise the
first position is loaded.
If the loadPositionIndex specifies an index -1, this triggers auto-increment
of the index in @code{matchMode}, which means that after every game the
index is incremented by one, causing each game of the match to be played
from the next position in the file. Similarly, specifying an index value of -2
causes the index to be incremented every two games, so that each position
in the file is used twice (with the engines playing opposite colors).
The @code{rewindIndex} option causes the index to be reset to the
first position of the file when it has reached a specified value.
@item -spf or -savePositionFile file
@cindex spf, option
@cindex savePositionFile, option
If this option is set, XBoard appends the final position reached
in every game played to the specified file. The file name @file{-}
specifies the standard output.
@item -pgnExtendedInfo true/false
@cindex pgnExtendedInfo, option`
If this option is set, WinBoard saves depth, score and time used for each
move that the engine found as a comment in the PGN file.
Default: false.
@item -pgnEventHeader string
@cindex pgnEventHeader, option`
Sets the name used in the PGN event tag to string.
Default: "Computer Chess Game".
@item -saveOutOfBookInfo true/false
@cindex saveOutOfBookInfo, option`
Include the information on how the engine(s) game out of its opening book in a special 'annotator' tag with the PGN file.
@item -oldsave/-xoldsave or -oldSaveStyle true/false
@cindex oldsave, option
@cindex oldSaveStyle, option
Sets the Old Save Style menu option. @xref{Options Menu}. Default: false.
@item -gameListTags string
@cindex gameListTags, option
The character string lists the PGN tags that should be printed in the
Game List, and their order. The meaning of the codes is e=event,
s=site, d=date, o=round, p=players, r=result, w=white Elo, b=black Elo,
t=time control, v=variant, a=out-of-book info.
Default: "eprd"
@end table
@node User interface options
@section User interface options
@cindex User interface options
@cindex Options, User interface
@table @asis
@item -display
@itemx -geometry
@itemx -iconic
@cindex display, option
@cindex geometry, option
@cindex iconic, option
These and most other standard Xt options are accepted.
@item -noGUI
@cindex noGUI, option
Suppresses all GUI functions of XBoard
(to speed up automated ultra-fast engine-engine games, which you donít want to watch).
There will be no board or clock updates, no printing of moves,
and no update of the icon on the task bar in this mode.
@item -movesound/-xmovesound or -ringBellAfterMoves true/false
@cindex movesound, option
@cindex bell, option
@cindex ringBellAfterMoves, option
Sets the Move Sound menu option. @xref{Options Menu}. Default: false.
For compatibility with old XBoard versions, -bell/-xbell are also
accepted as abbreviations for this option.
@item -exit/-xexit or -popupExitMessage true/false
@cindex exit, option
@cindex popupExitMessage, option
Sets the Popup Exit Message menu option. @xref{Options Menu}. Default: true.
@item -popup/-xpopup or -popupMoveErrors true/false
@cindex popup, option
@cindex popupMoveErrors, option
Sets the Popup Move Errors menu option. @xref{Options Menu}. Default: false.
@item -queen/-xqueen or -alwaysPromoteToQueen true/false
@cindex queen, option
@cindex alwaysPromoteToQueen, option
Sets the Always Queen menu option. @xref{Options Menu}. Default: false.
@item -legal/-xlegal or -testLegality true/false
@cindex legal, option
@cindex testLegality, option
Sets the Test Legality menu option. @xref{Options Menu}. Default: true.
@item -size or -boardSize (sizeName | n1,n2,n3,n4,n5,n6,n7)
@cindex size, option
@cindex boardSize, option
@cindex board size
Determines how large the board will be, by selecting the pixel size
of the pieces and setting a few related parameters.
The sizeName can be one of: Titanic, giving 129x129 pixel pieces,
Colossal 116x116, Giant 108x108, Huge 95x95, Big 87x87, Large 80x80, Bulky 72x72,
Medium 64x64, Moderate 58x58, Average 54x54, Middling 49x49, Mediocre
45x45, Small 40x40, Slim 37x37, Petite 33x33, Dinky 29x29, Teeny 25x25,
or Tiny 21x21.
Pieces of all these sizes are built into XBoard.
Other sizes can
be used if you have them; see the pixmapDirectory and bitmapDirectory
options.
The default depends on the size of your screen; it is approximately the
largest size that will fit without clipping.
You can select other sizes or vary other layout parameters by providing
a list of comma-separated values (with no spaces) as the argument.
You do not need to provide all the values; for any you omit from the
end of the list, defaults are taken from the nearest built-in size.
The value @code{n1} gives the piece size, @code{n2} the width of the
black border
between squares, @code{n3} the desired size for the
clockFont, @code{n4} the desired size for the coordFont,
@code{n5} the desired size for the default font,
@code{n6} the smallLayout flag (0 or 1),
and @code{n7} the tinyLayout flag (0 or 1).
All dimensions are in pixels.
If the border between squares is eliminated (0 width), the various
highlight options will not work, as there is nowhere to draw the highlight.
If smallLayout is 1 and @code{titleInWindow} is true,
the window layout is rearranged to make more room for the title.
If tinyLayout is 1, the labels on the menu bar are abbreviated
to one character each and the buttons in the button bar are made narrower.
@item -coords/-xcoords or -showCoords true/false
@cindex coords, option
@cindex showCoords, option
Sets the Show Coords menu option. @xref{Options Menu}. Default: false.
The @code{coordFont} option specifies what font to use.
@item -autoraise/-xautoraise or -autoRaiseBoard true/false
@cindex autoraise, option
@cindex autoRaiseBoard, option
Sets the Auto Raise Board menu option. @xref{Options Menu}. Default: true.
@item -autoflip/-xautoflip or -autoFlipView true/false
@cindex autoflip, option
@cindex autoFlipView, option
Sets the Auto Flip View menu option. @xref{Options Menu}. Default: true.
@item -flip/-xflip or -flipView true/false
@cindex flip, option
@cindex flipView, option
If Auto Flip View is not set, or if you are observing but not participating
in a game, then the positioning of the board at the start of each game
depends on the flipView option. If flipView is false (the default),
the board is positioned so that the white pawns move from the bottom to the
top; if true, the black pawns move from the bottom to the top.
In any case, the Flip menu option (see @ref{Options Menu})
can be used to flip the board after
the game starts.
@item -title/-xtitle or -titleInWindow true/false
@cindex title, option
@cindex titleInWindow, option
If this option is true, XBoard displays player names (for ICS
games) and game file names (for @samp{Load Game}) inside its main
window. If the option is false (the default), this information is
displayed only in the window banner. You probably won't want to
set this option unless the information is not showing up in the
banner, as happens with a few X window managers.
@item -buttons/-xbuttons or -showButtonBar True/False
@cindex buttons, option
@cindex showButtonBar, option
If this option is False, xboard omits the [<<] [<] [P] [>] [>>] button
bar from the window, allowing the message line to be wider. You can
still get the functions of these buttons using the menus or their keyboard
shortcuts. Default: true.
@item -mono/-xmono or -monoMode true/false
@cindex mono, option
@cindex monoMode, option
Determines whether XBoard displays its pieces and squares with
two colors (true) or four (false). You shouldn't have to
specify @code{monoMode}; XBoard will determine if it is necessary.
@item -flashCount count
@itemx -flashRate rate
@itemx -flash/-xflash
@cindex flashCount, option
@cindex flashRate, option
@cindex flash, option
@cindex xflash, option
These options enable flashing of pieces when they
land on their destination square.
@code{flashCount}
tells XBoard how many times to flash a piece after it
lands on its destination square.
@code{flashRate}
controls the rate of flashing (flashes/sec).
Abbreviations:
@code{flash}
sets flashCount to 3.
@code{xflash}
sets flashCount to 0.
Defaults: flashCount=0 (no flashing), flashRate=5.
@item -highlight/-xhighlight or -highlightLastMove true/false
@cindex highlight, option
@cindex highlightLastMove, option
Sets the Highlight Last Move menu option. @xref{Options Menu}. Default: false.
@item -blind/-xblind or -blindfold true/false
@cindex blind, option
@cindex blindfold, option
Sets the Blindfold menu option. @xref{Options Menu}. Default: false.
@item -clockFont font
@cindex clockFont, option
@cindex Font, clock
The font used for the clocks. If the option value is a pattern
that does not specify the font size, XBoard tries to choose an
appropriate font for the board size being used.
Default: -*-helvetica-bold-r-normal--*-*-*-*-*-*-*-*.
@item -coordFont font
@cindex coordFont, option
@cindex Font, coordinates
The font used for rank and file coordinate labels if @code{showCoords}
is true. If the option value is a pattern that does not specify
the font size, XBoard tries to choose an appropriate font for
the board size being used.
Default: -*-helvetica-bold-r-normal--*-*-*-*-*-*-*-*.
@item -font font
@cindex font, option
@cindex Font
The font used for popup dialogs, menus, comments, etc.
If the option value is a pattern that does not specify
the font size, XBoard tries to choose an appropriate font for
the board size being used.
Default: -*-helvetica-medium-r-normal--*-*-*-*-*-*-*-*.
@item -fontSizeTolerance tol
@cindex fontSizeTolerance, option
In the font selection algorithm, a nonscalable font will be preferred
over a scalable font if the nonscalable font's size differs
by @code{tol} pixels
or less from the desired size. A value of -1 will force
a scalable font to always be used if available; a value of 0 will
use a nonscalable font only if it is exactly the right size;
a large value (say 1000) will force a nonscalable font to always be
used if available. Default: 4.
@item -bm or -bitmapDirectory dir
@itemx -pixmap or -pixmapDirectory dir
@cindex bm, option
@cindex bitmapDirectory, option
@cindex pixmap, option
@cindex pixmapDirectory, option
These options control what piece images xboard uses. The XBoard
distribution includes one set of pixmap pieces in xpm format, in the
directory @file{pixmaps}, and one set of bitmap pieces in xbm format,
in the directory @file{bitmaps}. Pixmap
pieces give a better appearance on the screen: the white pieces have
dark borders, and the black pieces have opaque internal details. With
bitmaps, neither piece color has a border, and the internal details
are transparent; you see the square color or other background color
through them.
If XBoard is configured and compiled on a system that includes libXpm,
the X pixmap library, the xpm pixmap pieces are compiled in as the
default. A different xpm piece set can be selected at runtime with
the @code{pixmapDirectory} option, or a bitmap piece set can be selected
with the @code{bitmapDirectory} option.
If XBoard is configured and compiled on a system that does not include
libXpm (or the @code{--disable-xpm} option is given to the configure
program), the bitmap pieces are compiled in as the default. It is not
possible to use xpm pieces in this case, but pixmap pieces in another
format called "xim" can be used by giving the @code{pixmapDirectory} option.
Or again, a different bitmap piece set can be selected with the
@code{bitmapDirectory} option.
Files in the @code{bitmapDirectory} must be named as follows:
The first character of a piece bitmap name gives the piece it
represents (@samp{p}, @samp{n}, @samp{b}, @samp{r}, @samp{q}, or @samp{k}),
the next characters give the size in pixels, the
following character indicates whether the piece is
solid or outline (@samp{s} or @samp{o}),
and the extension is @samp{.bm}.
For example, a solid 80x80 knight would be named @file{n80s.bm}.
The outline bitmaps are used only in monochrome mode.
If bitmap pieces are compiled in and the bitmapDirectory is missing
some files, the compiled in pieces are used instead.
If the bitmapDirectory option is given,
it is also possible to replace xboard's icons and menu checkmark,
by supplying files named @file{icon_white.bm}, @file{icon_black.bm}, and
@file{checkmark.bm}.
For more information about pixmap pieces and how to get additional
sets, see @ref{zic2xpm} below.
@item -whitePieceColor color
@itemx -blackPieceColor color
@itemx -lightSquareColor color
@itemx -darkSquareColor color
@itemx -highlightSquareColor color
@itemx -lowTimeWarningColor color
@cindex Colors
@cindex whitePieceColor, option
@cindex blackPieceColor, option
@cindex lightSquareColor, option
@cindex darkSquareColor, option
@cindex highlightSquareColor, option
@cindex lowTimeWarningColor, option
Colors to use for the pieces, squares, and square highlights.
Defaults:
@example
-whitePieceColor #FFFFCC
-blackPieceColor #202020
-lightSquareColor #C8C365
-darkSquareColor #77A26D
-highlightSquareColor #FFFF00
-premoveHighlightColor #FF0000
-lowTimeWarningColor #FF0000
@end example
On a grayscale monitor you might prefer:
@example
-whitePieceColor gray100
-blackPieceColor gray0
-lightSquareColor gray80
-darkSquareColor gray60
-highlightSquareColor gray100
-premoveHighlightColor gray70
-lowTimeWarningColor gray70
@end example
@item -drag/-xdrag or -animateDragging true/false
@cindex drag, option
@cindex animateDragging, option
Sets the Animate Dragging menu option. @xref{Options Menu}. Default: true.
@item -animate/-xanimate or -animateMoving true/false
@cindex animate, option
@cindex animateMoving, option
Sets the Animate Moving menu option. @xref{Options Menu}. Default: true.
@item -animateSpeed n
@cindex -animateSpeed, option
Number of milliseconds delay between each animation frame when Animate
Moves is on.
@end table
@node Adjudication Options
@section Adjudication Options
@cindex Options, adjudication
@table @asis
@item -adjudicateLossThreshold n
@cindex adjudicateLossThreshold, option
If the given value is non-zero, XBoard adjudicates the game as a loss
if both engines agree for a duration of 6 consecutive ply that the score
is below the given score threshold for that engine. Make sure the score
is interpreted properly by XBoard,
using @code{-firstScoreAbs} and @code{-secondScoreAbs} if needed.
Default: 0 (no adjudication)
@item -adjudicateDrawMoves n
@cindex adjudicateDrawMoves, option
If the given value is non-zero, XBoard adjudicates the game as a draw
if after the given number of moves it was not yet decided. Default: 0 (no adjudication)
@item -checkMates true/false
@cindex checkMates, option
If this option is set, XBoard detects all checkmates and stalemates,
and ends the game as soon as they occur.
Legality-testing must be switched on for this option to work.
Default: true
@item -testClaims true/false
@cindex testClaims, option
If this option is set, XBoard verifies all result claims made by engines,
and those who send false claims will forfeit the game because of it.
Legality-testing must be switched on for this option to work. Default: true
@item -materialDraws true/false
@cindex materialDraws, option
If this option is set, XBoard adjudicates games as draws when there is
no sufficient material left to inflict a checkmate.
This applies to KBKB with like bishops (any number, actually), and to KBK, KNK and KK.
Legality-testing must be switched on for this option to work. Default: true
@item -trivialDraws true/false
@cindex trivialDraws, option
If this option is set, XBoard adjudicates games as draws that cannot be
usually won without opponent cooperation. This applies to KBKB with unlike bishops,
and to KBKN, KNKN, KNNK, KRKR and KQKQ. The draw is called after 6 ply into these end-games,
to allow quick mates that can occur in some exceptional positions to be found by the engines.
KQKQ does not really belong in this category, and might be taken out in the future.
(When bitbase-based adjudications are implemented.)
Legality-testing must be on for this option to work. Default: false
@item -ruleMoves n
@cindex ruleMoves, option
If the given value is non-zero, XBoard adjudicates the game as a draw after the given
number of consecutive reversible moves. Engine draw claims are always accepted after 50 moves,
irrespective of the given value of n.
@item -repeatsToDraw n
If the given value is non-zero, xboard adjudicates the game as a draw if a position
is repeated the given number of times. Engines draw claims are always accepted after 3 repeats,
(on the 3rd occurrence, actually), irrespective of the value of n.
Beware that positions that have different castling or en-passant rights do not count
as repeats, XBoard is fully e.p. and castling aware!
@end table
@node Other options
@section Other options
@cindex Options, miscellaneous
@table @asis
@item -ncp/-xncp or -noChessProgram true/false
@cindex ncp, option
@cindex noChessProgram, option
If this option is true, XBoard acts as a passive chessboard; it
does not start a chess engine at all. Turning on this option
also turns off clockMode. Default: false.
@item -mode or -initialMode modename
@cindex mode, option
@cindex initalMode, option
If this option is given, XBoard selects the given modename
from the Mode menu after starting and (if applicable) processing the
loadGameFile or loadPositionFile option. Default: "" (no selection).
Other supported values are
MachineWhite, MachineBlack, TwoMachines, Analysis,
AnalyzeFile, EditGame, EditPosition, and Training.
@item -variant varname
@cindex variant, option
Activates preliminary, partial support for playing chess variants
against a local engine or editing variant games. This flag is not
needed in ICS mode. Recognized variant names are:
@example
normal Normal chess
wildcastle Shuffle chess, king can castle from d file
nocastle Shuffle chess, no castling allowed
fischerandom Fischer Random shuffle chess
bughouse Bughouse, ICC/FICS rules
crazyhouse Crazyhouse, ICC/FICS rules
losers Lose all pieces or get mated (ICC wild 17)
suicide Lose all pieces including king (FICS)
giveaway Try to have no legal moves (ICC wild 26)
twokings Weird ICC wild 9
kriegspiel Opponent's pieces are invisible
atomic Capturing piece explodes (ICC wild 27)
3check Win by giving check 3 times (ICC wild 25)
shatranj An ancient precursor of chess (ICC wild 28)
xiangqi Chinese Chess (on a 9x10 board)
shogi Japanese Chess (on a 9x9 board & piece drops)
capablanca Capablanca Chess (10x8 board, with Archbishop
and Chancellor pieces)
gothic similar, with a better initial position
caparandom An FRC-like version of Capablanca Chess (10x8)
janus A game with two Archbishops (10x8 board)
courier Medieval intermediate between shatranj and
modern Chess (on 12x8 board)
falcon Patented 10x8 variant with two Falcon pieces
berolina Pawns capture straight ahead, and move diagonal
cylinder Pieces wrap around the board edge
knightmate King moves as Knight, and vice versa
super Superchess (shuffle variant with 4 exo-pieces)
fairy A catchall variant in which all piece types
known to XBoard can participate (8x8)
unknown Catchall for other unknown variants
@end example
NOT ALL BOARDSIZES PROVIDE A COMPLETE SET OF BUILT-IN BITMAPS FOR ALL
UN-ORTHODOX PIECES, though. Only in @code{boardSize} middling and bulky
all 22 piece types are provided, while -boardSize petite has most
of them. Archbishop, Chancellor and Amazon are supported in every
size from petite to bulky. Kings or Amazons are substituted for
missing bitmaps. You can still play variants needing un-orthodox
pieces in other board sizes providing your own bitmaps through the
@code{bitmapDirectory} or @code{pixmapDirectory} options.
In the shuffle variants, XBoard now does shuffle the pieces, although
you can still do it by hand using Edit Position. Some variants are
supported only in ICS mode, including bughouse, and
kriegspiel. The winning/drawing conditions in crazyhouse (off-board
interposition on mate) are not fully understood, but losers, suicide,
giveaway, atomic, and 3check should be OK.
Berolina and cylinder chess can only be played with legality testing off.
In crazyhouse, XBoard now does keep
track of off-board pieces. In shatranj it does implement the baring
rule when mate detection is switched on.
@item -boardHeight N
@cindex boardHeight, option
Allows you to set a non-standard number of board ranks in any variant.
If the height is given as -1, the default height for the variant is used.
Default: -1
@item -boardWidth N
@cindex boardWidth, option
Allows you to set a non-standard number of board files in any variant.
If the width is given as -1, the default width for the variant is used.
With a non-standard width, the initial position will always be an empty board,
as the usual opening array will not fit.
Default: -1
@item -holdingsSize N
@cindex holdingsSize, option
Allows you to set a non-standard size for the holdings in any variant.
If the size is given as -1, the default holdings size for the variant is used.
The first N piece types will go into the holdings on capture, and you will be
able to drop them on the board in stead of making a normal move. If size equals 0,
there will be no holdings.
Default: -1
@item -defaultFrcPosition N
@cindex defaultFrcPosition, option
Specifies the number of the opening position in shuffle games like Chess960.
A value of -1 means the position is randomly generated by XBoard
at the beginning of every game.
Default: -1
@item -pieceToSquareTable string
@cindex pieceToSquareTable, option
The characters that are used to represent the piece types XBoard knows in FEN
diagrams and SAN moves. The string argument has to have an even length
(or it will be ignored), as white and black pieces have to be given separately
(in that order). The last letter for each color will be the King.
The letters before that will be PNBRQ and then a whole host of fairy pieces
in an order that has not fully crystallized yet (currently FEACWMOHIJGDVSLU,
F=Ferz, Elephant, A=Archbishop, C=Chancellor, W=Wazir, M=Commoner, O=Cannon,
H=Nightrider). You should list at least all pieces that occur in the variant
you are playing. If you have less than 44 characters in the string, the pieces
not mentioned will get assigned a period, and you will not be able to distinguish
them in FENs. You can also explicitly assign pieces a period, in which case they
will not be counted in deciding which captured pieces can go into the holdings.
A tilde '~' as a piece name does mean this piece is used to represent a promoted
Pawn in crazyhouse-like games, i.e. on capture it turns back onto a Pawn.
A '+' similarly indicates the piece is a shogi-style promoted piece, that should
revert to its non-promoted version on capture (rather than to a Pawn).
Note that promoted pieces are represented by pieces 11 further in the list.
You should not have to use this option often: each variant has its own default
setting for the piece representation in FEN, which should be sufficient in normal use.
Default: ""
@item -debug/-xdebug or -debugMode true/false
@cindex debug, option
@cindex debugMode, option
Turns on debugging printout.
@item -debugFile filename or -nameOfDebugFile filename
@cindex debugFile, option
@cindex nameOfDebugFile, option
Sets the name of the file to which WinBoard saves debug information
(including all communication to and from the engines).
@item -engineDebugOutput number
@cindex engineDebugOutput, option
Specifies how WinBoard should handle unsolicited output from the engine,
with respect to saving it in the debug file.
The output is further (hopefully) ignored.
If number=0, WinBoard refrains from writing such spurious output to the debug file.
If number=1, all engine output is written faithfully to the debug file.
If number=2, any protocol-violating line is prefixed with a '#' character,
as the engine itself should have done if it wanted to submit info for inclusion in the debug file.
This option is provided for the benefit of applications that use the debug file
as a source of information, such as the broadcaster of live games TLCV / TLCS.
Such applications can be protected from spurious engine output that might otherwise confuse them.
@item -rsh or -remoteShell shell-name
@cindex rsh, option
@cindex remoteShell, option
Name of the command used to run programs remotely. The default
is @file{rsh} or @file{remsh}, determined when XBoard is
configured and compiled.
@item -ruser or -remoteUser user-name
@cindex ruser, option
@cindex remoteUser, option
User name on the remote system when running programs with the
@code{remoteShell}. The default is your local user name.
@item -userName username
@cindex userName, option
Name under which the Human player will be listed in the PGN file.
Default is the login name on your local computer.
@item -delayBeforeQuit number
@itemx -delayAfterQuit number
@cindex delayBeforeQuit, option
@cindex delayAfterQuit, option
These options specify how long WinBoard has to wait before sending a termination signal to rogue engine processes, that do not want to react to the 'quit' command. The second one determines the pause after killing the engine, to make sure it dies.
@end table
@node Chess Servers
@chapter Chess Servers
@cindex ICS
@cindex ICS, addresses
@cindex Internet Chess Server
An @dfn{Internet Chess Server}, or @dfn{ICS}, is a place on the
Internet where people can get together to play chess, watch other
people's games, or just chat. You can use either @code{telnet} or a
client program like XBoard to connect to the server. There are
thousands of registered users on the different ICS hosts, and it is
not unusual to meet 200 on both chessclub.com and freechess.org.
Most people can just type @kbd{xboard -ics} to start XBoard as an ICS
client. Invoking XBoard in this way connects you to the Internet
Chess Club (ICC), a commercial ICS. You can log in there as a guest
even if you do not have a paid account. To connect to the largest
Free ICS (FICS), use the command @kbd{xboard -ics -icshost freechess.org}
instead, or substitute a different host name to connect to your
favorite ICS.
For a full description of command-line options that control
the connection to ICS and change the default values of ICS options, see
@ref{ICS options}.
While you are running XBoard as an ICS client,
you use the terminal window that you started XBoard from
as a place to type in commands and read information that is
not available on the chessboard.
The first time you need to use the terminal is to enter your login name
and password, if you are a registered player. (You don't need to do
this manually; the @code{icsLogon} option can do it for you.
@pxref{ICS options}.) If you are not registered,
enter @kbd{g} as your name, and the server will pick a
unique guest name for you.
Some useful ICS commands
include
@table @kbd
@item help <topic>
@cindex help, ICS command
to get help on the given <topic>. To get a list of possible topics type
@dfn{help} without topic. Try the help command before you ask other
people on the server for help.
For example @kbd{help register} tells you how to become a registered
ICS player.
@item who <flags>
@cindex who, ICS command
to see a list of people who are logged on. Administrators
(people you should talk to if you have a problem) are marked
with the character @samp{*}, an asterisk. The <flags> allow you to
display only selected players: For example, @kbd{who of} shows a
list of players who are interested in playing but do not have
an opponent.
@item games
@cindex games, ICS command
to see what games are being played
@item match <player> [<mins>] [<inc>]
to challenge another player to a game. Both opponents get <mins> minutes
for the game, and <inc> seconds will be added after each move.
If another player challenges you, the server asks if you want to
accept the challenge; use the @kbd{accept} or @kbd{decline} commands
to answer.
@item accept
@itemx decline
@cindex accept, ICS command
@cindex decline, ICS command
to accept or decline another player's offer.
The offer may be to start a new game, or to agree to a
@kbd{draw}, @kbd{adjourn} or @kbd{abort} the current game. @xref{Action Menu}.
If you have more than one pending offer (for example, if more than one player
is challenging you, or if your opponent offers both a draw and to adjourn the
game), you have to supply additional information, by typing something
like @kbd{accept <player>}, @kbd{accept draw}, or @kbd{draw}.
@item draw
@itemx adjourn
@itemx abort
@cindex draw, ICS command
@cindex adjourn, ICS command
@cindex abort, ICS command
asks your opponent to terminate a game by mutual agreement. Adjourned
games can be continued later.
Your opponent can either @kbd{decline} your offer or accept it (by typing the
same command or typing @kbd{accept}). In some cases these commands work
immediately, without asking your opponent to agree. For example, you can
abort the game unilaterally if your opponent is out of time, and you can claim
a draw by repetition or the 50-move rule if available simply by typing
@kbd{draw}.
@item finger <player>
@cindex finger, ICS command
to get information about the given <player>. (Default: yourself.)
@item vars
@cindex vars, ICS command
to get a list of personal settings
@item set <var> <value>
@cindex set, ICS command
to modify these settings
@item observe <player>
@cindex observe, ICS command
to observe an ongoing game of the given <player>.
@item examine
@itemx oldmoves
@cindex examine, ICS command
@cindex oldmoves, ICS command
to review a recently completed game
@end table
Some special XBoard features are activated when you are
in examine mode on ICS. See the descriptions of the menu commands
@samp{Forward}, @samp{Backward}, @samp{Pause}, @samp{ICS Client},
and @samp{Stop Examining} on the @ref{Step Menu}, @ref{Mode Menu}, and
@ref{Options Menu}.
@node Firewalls
@chapter Firewalls
By default, XBoard communicates with an Internet Chess Server
by opening a TCP socket directly from the machine it is running on
to the ICS. If there is a firewall between your machine and the ICS,
this won't work. Here are some recipes for getting around common
kinds of firewalls using special options to XBoard.
Important: See the paragraph in the below about extra echoes, in
@ref{Limitations}.
Suppose that you can't telnet directly to ICS, but you can telnet
to a firewall host, log in, and then telnet from there to ICS.
Let's say the firewall is called @samp{firewall.example.com}. Set
command-line options as follows:
@example
xboard -ics -icshost firewall.example.com -icsport 23
@end example
@noindent
Or in your @file{.Xresources} file:
@example
XBoard*internetChessServerHost: firewall.example.com
XBoard*internetChessServerPort: 23
@end example
@noindent
Then when you run XBoard in ICS mode, you will be prompted
to log in to the firewall host. This works because port 23 is the
standard telnet login service. Do so, then telnet to ICS, using a
command like @samp{telnet chessclub.com 5000}, or whatever command
the firewall provides for telnetting to port 5000.
If your firewall lets you telnet (or rlogin) to remote hosts but
doesn't let you telnet to port 5000, you may be able to connect to the
chess server on port 23 instead, which is the port the telnet program
uses by default. Some chess servers support this (including
chessclub.com and freechess.org), while some do not.
If your chess server does not allow connections on port 23 and your
firewall does not allow you to connect to other ports, you may be able
to connect by hopping through another host outside the firewall that
you have an account on. For instance, suppose you have a shell
account at @samp{foo.edu}. Follow the recipe above, but instead of
typing @samp{telnet chessclub.com 5000} to the firewall, type
@samp{telnet foo.edu} (or @samp{rlogin foo.edu}), log in there, and
then type @samp{telnet chessclub.com 5000}.
Suppose that you can't telnet directly to ICS, but you can use rsh
to run programs on a firewall host, and that host can telnet to ICS.
Let's say the firewall is called @samp{rsh.example.com}. Set
command-line options as follows:
@example
xboard -ics -gateway rsh.example.com -icshost chessclub.com
@end example
@noindent
Or in your @file{.Xresources} file:
@example
XBoard*gateway: rsh.example.com
XBoard*internetChessServerHost: chessclub.com
@end example
Then when you run XBoard in ICS mode, it will connect to
the ICS by using @file{rsh} to run the command
@samp{telnet chessclub.com 5000} on host @samp{rsh.example.com}.
Suppose that you can telnet anywhere you want, but you have to
run a special program called @file{ptelnet} to do so.
First, we'll consider the easy case, in which
@samp{ptelnet chessclub.com 5000} gets you to the chess server.
In this case set command line options as follows:
@example
xboard -ics -telnet -telnetProgram ptelnet
@end example
@noindent
Or in your @file{.Xresources} file:
@example
XBoard*useTelnet: true
XBoard*telnetProgram: ptelnet
@end example
@noindent
Then when you run XBoard in ICS mode, it will issue the
command @samp{ptelnet chessclub.com 5000} to connect to the ICS.
Next, suppose that @samp{ptelnet chessclub.com 5000} doesn't work;
that is, your @file{ptelnet} program doesn't let you connect to
alternative ports. As noted above, your chess server may allow you to
connect on port 23 instead. In that case, just add the option
@samp{-icsport ""} to the above command, or add
@samp{XBoard*internetChessServerPort:} to your @file{.Xresources} file.
But if your chess server doesn't let you connect on port 23, you will have
to find some other host outside the firewall and hop through it. For
instance, suppose you have a shell account at @samp{foo.edu}. Set
command line options as follows:
@example
xboard -ics -telnet -telnetProgram ptelnet -icshost foo.edu -icsport ""
@end example
@noindent
Or in your @file{.Xresources} file:
@example
XBoard*useTelnet: true
XBoard*telnetProgram: ptelnet
XBoard*internetChessServerHost: foo.edu
XBoard*internetChessServerPort:
@end example
@noindent
Then when you run XBoard in ICS mode, it will issue the
command @samp{ptelnet foo.edu} to connect to your account at
@samp{foo.edu}. Log in there, then type @samp{telnet chessclub.com 5000}.
ICC timestamp and FICS timeseal do not work through some
firewalls. You can use them only if your firewall gives a clean TCP
connection with a full 8-bit wide path. If your firewall allows you
to get out only by running a special telnet program, you can't use
timestamp or timeseal across it. But if you have access to a
computer just outside your firewall, and you have much lower netlag
when talking to that computer than to the ICS, it might be worthwhile
running timestamp there. Follow the instructions above for hopping
through a host outside the firewall (foo.edu in the example),
but run timestamp or timeseal on that host instead of telnet.
Suppose that you have a SOCKS firewall that will give you a clean
8-bit wide TCP connection to the chess server, but only after you
authenticate yourself via the SOCKS protocol. In that case, you could
make a socksified version of XBoard and run that. If you are using
timestamp or timeseal, you will to socksify it, not XBoard; this may
be difficult seeing that ICC and FICS do not provide source code for
these programs. Socksification is beyond the scope of this document,
but see the SOCKS Web site at http://www.socks.permeo.com/.
If you are missing SOCKS, try http://www.funbureau.com/.
@node Environment
@chapter Environment variables
@cindex Environment variables
@cindex CHESSDIR
Game and position files are found in a directory named by the
@code{CHESSDIR} environment variable. If this variable is not set, the
current working directory is used. If @code{CHESSDIR} is set,
XBoard actually changes its working directory to
@code{$CHESSDIR}, so any files written by the chess engine
will be placed there too.
@node Limitations
@chapter Limitations and known bugs
@cindex Limitations
@cindex Bugs
There is no way for two people running copies of XBoard to play
each other without going through an Internet Chess Server.
Under some circumstances, your ICS password may be echoed when you log on.
If you are connecting to the ICS by running telnet on an Internet
provider or firewall host, you may find that each line you type is
echoed back an extra time after you hit @key{Enter}. If your Internet
provider is a Unix system, you can probably turn its echo off by
typing @kbd{stty -echo} after you log in, and/or typing
@key{^E}@key{Enter} (Ctrl+E followed by the Enter key) to the telnet
program after you have logged into ICS. It is a good idea to do this
if you can, because the extra echo can occasionally confuse XBoard's
parsing routines.
The game parser recognizes only algebraic notation.
Many of the following points used to be limitations in XBoard 4.2.7 and earlier,
but are now fixed:
The internal move legality tester in XBoard 4.3.xx does look at the game history,
and is fully aware of castling or en-passant-capture rights. It permits castling with
the king on the d file because this is possible in some "wild 1" games on ICS.
The piece-drop menu does not check piece drops in bughouse to see if you actually hold
the piece you are trying to drop. But this way of dropping pieces should be considered
an obsolete feature, now that pieces can be dropped by dragging them from the holdings
to the board. Anyway, if you would attempt an illegal move when using a chess engine or the ICS,
WinBoard will accept the error message that comes back, undo the move, and let you try another.
FEN positions saved by XBoard do include correct information about whether castling or
en passant are legal, and also handle the 50-move counter.
The mate detector does not understand that non-contact mate is not really mate in bughouse.
The only problem this causes while playing is minor: a "#" (mate indicator) character will
show up after a non-contact mating move in the move list. XBoard will not assume the game
is over at that point, not even when the option Detect Mates is on.
Edit Game mode always uses the rules of the selected variant,
which can be a variant that uses piece drops.
You can load and edit games that contain piece drops.
The (obsolete) piece menus are not active,
but you can perform piece drops by dragging pieces from the holdings.
Edit Position mode does not allow you to edit the crazyhouse holdings properly.
You cannot drag pieces to the holding, and using the popup menu to put pieces
there does not adapt the holding counts and leads to an inconsistent state.
Set up crazyhouse positions by loading / pasting a bFEN, from there you can set the holdings.
Fischer Random castling is fully understood.
You can enter castlings by dragging the King on top of your Rook.
You can probably also play Fischer Random successfully on ICS by typing
castling moves into the ICS Interaction window.
The menus may not work if your keyboard is in Caps Lock or Num Lock mode.
This seems to be a problem with the Athena menu widget,
not an XBoard bug.
Also see the ToDo file included with the distribution for many other
possible bugs, limitations, and ideas for improvement that have been
suggested.
@node Problems
@chapter Reporting problems
@cindex Bugs
@cindex Bug reports
@cindex Reporting bugs
@cindex Problems
@cindex Reporting problems
Report bugs and problems with XBoard to @code{<bug-xboard@@gnu.org>}.
Please use the @file{script} program to start a typescript, run
XBoard with the @samp{-debug} option, and include the typescript
output in your message.
Also tell us what kind of machine and what operating system version
you are using. The command @samp{uname -a} will often tell you this.
Here is a sample of approximately what you should type:
@example
script
uname -a
./configure
make
./xboard -debug
exit
mail bug-xboard@@gnu.org
Subject: Your short description of the problem
Your detailed description of the problem
~r typescript
.
@end example
The WinBoard / XBoard 4.3 line is being developed by H.G. Muller
independently of the GNU Savannah xboard project.
Bug reports on this version, and suggestions for improvements and additions,
are best posted in the WinBoard forum,
WinBoard-development section (http://www.open-aurec.com/wbforum).
If you improve XBoard, please send a message about your changes,
and we will get in touch with you about merging them in
to the main line of development.
Also see our Web site at http://savannah.gnu.org/projects/xboard/.
@node Contributors
@chapter Authors and contributors
@cindex Authors
@cindex Contributors
Tim Mann has been responsible for XBoard versions 1.3 and beyond, and
for WinBoard, a port of XBoard to Microsoft Win32 (Windows NT and
Windows 95). H.G.Muller is responsible for version 4.3.
Mark Williams contributed the initial (WinBoard-only) implementation
of many new features added to both XBoard and WinBoard in version
4.1.0, including copy/paste, premove, icsAlarm, autoFlipView, training
mode, auto raise, and blindfold. Ben Nye contributed X copy/paste
code for XBoard.
Hugh Fisher added animated piece movement to XBoard, and Henrik Gram
(henrikg@@funcom.com) added it to WinBoard. Frank McIngvale added
click/click moving, the Analysis modes, piece flashing, ZIICS import,
and ICS text colorization to XBoard. Jochen Wiedmann ported XBoard to
the Amiga, creating AmyBoard, and converted the documentation to
texinfo. Elmar Bartel contributed the new piece bitmaps introduced in
version 3.2. John Chanak contributed the initial implementation of
ICS mode. The color scheme and the old 80x80 piece bitmaps were taken
from Wayne Christopher's @code{XChess} program.
Chris Sears and Dan Sears wrote the original XBoard. They were
responsible for versions 1.0 through 1.2.
Evan Welsh wrote @code{CMail}. Patrick Surry helped in designing,
testing, and documenting CMail.
Alessandro Scotti added many elements to the user interface of WinBoard,
including the board textures and font-based rendering, the evaluation-graph,
move-history and engine-output window.
He was also responsible for adding the UCI support.
H.G. Muller made WinBoard castling- and e.p.-aware,
added variant support with adjustable board sizes,
the crazyhouse holdings, and the fairy pieces.
In addition he added most of the adjudication options,
made WinBoard more robust in dealing with buggy and crashing engines,
and extended time control with a time-odds and node-count-based modes.
Most of the options that initially were WinBoard only have now been back-ported to XBoard.
Michel van den Bergh provided the code for reading Polyglot opening books.
Arun Persaud worked with H.G. Muller to combine all the features of the never-released WinBoard 4.2.8
of the Savannah project (mainly by Daniel Mehrmann),
and the never-released 4.3.16 into a unified WinBoard 4.4,
which is now available both from the Savannah web site and the WinBoard forum.
@node CMail
@chapter CMail
@cindex cmail
The @file{cmail} program can help you play chess by email with opponents of
your choice using XBoard as an interface.
You will usually run @file{cmail} without giving any options.
@menu
* CMail options:: Invoking CMail.
* CMail game:: Starting a CMail game.
* CMail answer:: Answering a move.
* CMail multi:: Multiple games in one message.
* CMail completion:: Completing a game.
* CMail trouble:: Known CMail problems.
@end menu
@node CMail options
@section CMail options
@table @asis
@item -h
Displays @file{cmail} usage information.
@item -c
Shows the conditions of the GNU General Public License.
@xref{Copying}.
@item -w
Shows the warranty notice of the GNU General Public License.
@xref{Copying}.
@item -v
@itemx -xv
Provides or inhibits verbose output from @file{cmail} and XBoard,
useful for debugging. The
@code{-xv}
form also inhibits the cmail introduction message.
@item -mail
@itemx -xmail
Invokes or inhibits the sending of a mail message containing the move.
@item -xboard
@itemx -xxboard
Invokes or inhibits the running of XBoard on the game file.
@item -reuse
@itemx -xreuse
Invokes or inhibits the reuse of an existing XBoard to display the
current game.
@item -remail
Resends the last mail message for that game. This inhibits running
XBoard.
@item -game <name>
The name of the game to be processed.
@item -wgames <number>
@itemx -bgames <number>
@itemx -games <number>
Number of games to start as White, as Black or in total. Default is 1 as
white and none as black. If only one color is specified then none of the
other color is assumed. If no color is specified then equal numbers of
White and Black games are started, with the extra game being as White if an
odd number of total games is specified.
@item -me <short name>
@itemx -opp <short name>
A one-word alias for yourself or your opponent.
@item -wname <full name>
@itemx -bname <full name>
@itemx -name <full name>
@itemx -oppname <full name>
The full name of White, Black, yourself or your opponent.
@item -wna <net address>
@itemx -bna <net address>
@itemx -na <net address>
@itemx -oppna <net address>
The email address of White, Black, yourself or your opponent.
@item -dir <directory>
The directory in which @file{cmail} keeps its files. This defaults to the
environment variable @code{$CMAIL_DIR} or failing that, @code{$CHESSDIR},
@file{$HOME/Chess} or @file{~/Chess}. It will be created if it does not exist.
@item -arcdir <directory>
The directory in which @file{cmail} archives completed games. Defaults to
the environment variable @code{$CMAIL_ARCDIR} or, in its absence, the same
directory as cmail keeps its working files (above).
@item -mailprog <mail program>
The program used by cmail to send email messages. This defaults to the
environment variable @code{$CMAIL_MAILPROG} or failing that
@file{/usr/ucb/Mail}, @file{/usr/ucb/mail} or @file{Mail}. You will need
to set this variable if none of the above paths fit your system.
@item -logFile <file>
A file in which to dump verbose debugging messages that are invoked with
the @samp{-v}
option.
@item -event <event>
The PGN Event tag (default @samp{Email correspondence game}).
@item -site <site>
The PGN Site tag (default @samp{NET}).
@item -round <round>
The PGN Round tag (default @samp{-}, not applicable).
@item -mode <mode>
The PGN Mode tag (default @samp{EM}, Electronic Mail).
@item Other options
Any option flags not listed above are passed through to XBoard.
Invoking XBoard through CMail changes the default values of two XBoard
options: The default value for @samp{-noChessProgram} is changed to
true; that is, by default no chess engine is started. The default
value for @samp{-timeDelay} is changed to 0; that is, by default
XBoard immediately goes to the end of the game as played so far,
rather than stepping through the moves one by one. You can still set
these options to whatever values you prefer by supplying them on
CMail's command line. @xref{Options}.
@end table
@node CMail game
@section Starting a CMail Game
Type @file{cmail} from a shell to start a game as white. After an opening
message, you will be prompted for a game name, which is optional---if you
simply press @key{Enter}, the game name will take the form
@samp{you-VS-opponent}. You will next be prompted for the short name
of your opponent. If you haven't played this person before, you will also
be prompted for his/her email address. @file{cmail} will then invoke
XBoard in the background. Make your first move and select
@samp{Mail Move} from the @samp{File} menu. @xref{File Menu}. If all is well,
@file{cmail} will mail a copy of the move to your opponent. If you select
@samp{Exit} without having selected @samp{Mail Move} then no move will be
made.
@node CMail answer
@section Answering a Move
When you receive a message from an opponent containing a move in one of
your games, simply pipe the message through @file{cmail}. In some mailers
this is as simple as typing @kbd{| cmail} when viewing the message, while in
others you may have to save the message to a file and do @kbd{cmail < file}
at the command line. In either case @file{cmail} will display the game using
XBoard. If you didn't exit XBoard when you made your first move
then @file{cmail} will do its best to use the existing XBoard instead
of starting a new one. As before, simply make a move and select
@samp{Mail Move} from the @samp{File} menu. @xref{File Menu}. @file{cmail}
will try to use the
XBoard that was most recently used to display the current game. This
means that many games can be in progress simultaneously, each with its own
active XBoard.
If you want to look at the history or explore a variation, go ahead, but
you must return to the current position before XBoard will allow you
to mail a move. If you edit the game's history you must select
@samp{Reload Same Game} from the @samp{File} menu to get back to the original
position, then make the move you want and select @samp{Mail Move}.
As before, if you decide you aren't ready to make a move just yet you can
either select @samp{Exit} without sending a move or just leave
XBoard running until you are ready.
@node CMail multi
@section Multi-Game Messages
It is possible to have a @file{cmail} message carry more than one game.
This feature was implemented to handle IECG (International Email Chess
Group) matches, where a match consists of one game as white and one as black,
with moves transmitted simultaneously. In case there are more general uses,
@file{cmail} itself places no limit on the number of black/white games
contained in a message; however, XBoard does.
@node CMail completion
@section Completing a Game
Because XBoard can detect checkmate and stalemate, @file{cmail}
handles game termination sensibly. As well as resignation, the
@samp{Action} menu allows draws to be offered and accepted for
@file{cmail} games.
For multi-game messages, only unfinished and just-finished games will be
included in email messages. When all the games are finished, they are
archived in the user's archive directory, and similarly in the opponent's
when he or she pipes the final message through @file{cmail}. The archive
file name includes the date the game was started.
@node CMail trouble
@section Known CMail Problems
It's possible that a strange conjunction of conditions may occasionally
mean that @file{cmail} has trouble reactivating an existing
XBoard. If this should happen, simply trying it again should work.
If not, remove the file that stores the XBoard's PID
(@file{game.pid}) or use the @samp{-xreuse} option to force
@file{cmail} to start a new XBoard.
Versions of @file{cmail} after 2.16 no longer understand the old file format
that XBoard used to use and so cannot be used to correspond with
anyone using an older version.
Versions of @file{cmail} older than 2.11 do not handle multi-game messages,
so multi-game correspondence is not possible with opponents using an older
version.
@node Other programs
@chapter Other programs you can use with XBoard
@cindex Other programs
Here are some other programs you can use with XBoard
@menu
* GNU Chess:: The GNU Chess engine.
* Fairy-Max:: The Fairy-Max chess engine.
* HoiChess:: The HoiChess chess engine.
* Crafty:: The Crafty chess engine.
* zic2xpm:: The program used to import chess sets from ZIICS.
@end menu
@node GNU Chess
@section GNU Chess
The GNU Chess engine is available from:
ftp://ftp.gnu.org/gnu/gnuchess/
You can use XBoard to play a game against GNU Chess, or to
interface GNU Chess to an ICS.
@node Fairy-Max
@section Fairy-Max
Fairy-Max is a derivative from the once World's smallest Chess program micro-Max,
which measures only about 100 lines of source code.
The main difference with micro-Max is that Fairy-Max loads its move-generator
tables from a file, so that the rules for piece movement can be easily configured
to implement unorthodox pieces.
Fairy-Max can therefore play a large number of variants, normal Chess being one of those.
In addition it plays Knightmate, Capablanca and Gothic Chess, Shatranj, Courier Chess,
Cylinder chess, Berolina Chess, while the user can easily define new variants.
It can be obtained from:
http://home.hccnet.nl/h.g.muller/dwnldpage.html
@node HoiChess
@section HoiChess
HoiChess is a not-so-very-strong Chess engine, which comes with a derivative HoiXiangqi,
able to play Chinese Chess. It can be obtained from the standard Linux repositories
through:
sudo apt-get install hoichess
@node Crafty
@section Crafty
Crafty is a chess engine written by Bob Hyatt.
You can use XBoard to play a game against Crafty, hook Crafty up
to an ICS, or use Crafty to interactively analyze games and positions
for you.
Crafty is a strong, rapidly evolving chess program. This rapid
pace of development is good, because it means Crafty is always
getting better. This can sometimes cause problems with
backwards compatibility, but usually the latest version of Crafty
will work well with the latest version of XBoard.
Crafty can be obtained from its author's FTP site:
ftp://ftp.cis.uab.edu/hyatt/.
To use Crafty with XBoard, give the -fcp and -fd options as follows, where
<crafty's directory> is the directory in which you installed Crafty
and placed its book and other support files.
@node zic2xpm
@section zic2xpm
The ``zic2xpm'' program is used to import chess sets from the ZIICS(*)
program into XBoard. ``zic2xpm'' is part of the XBoard distribution.
ZIICS is available from:
ftp://ftp.freechess.org/pub/chess/DOS/ziics131.exe
To import ZIICS pieces, do this:
@table @asis
@item 1. Unzip ziics131.exe into a directory:
@example
unzip -L ziics131.exe -d ~/ziics
@end example
@item 2. Use zic2xpm to convert a set of pieces to XBoard format.
For example, let's say you want to use the
FRITZ4 set. These files are named ``fritz4.*'' in the ZIICS distribution.
@example
mkdir ~/fritz4
cd ~/fritz4
zic2xpm ~/ziics/fritz4.*
@end example
@item 3. Give XBoard the ``-pixmap'' option when starting up, e.g.:
@example
xboard -pixmap ~/fritz4
@end example
Alternatively, you can add this line to your @file{.Xresources} file:
@example
xboard*pixmapDirectory: ~/fritz4
@end example
@end table
(*) ZIICS is a separate copyrighted work of Andy McFarland.
The ``ZIICS pieces'' are copyrighted works of their respective
creators. Files produced by ``zic2xpm'' are for PERSONAL USE ONLY
and may NOT be redistributed without explicit permission from
the original creator(s) of the pieces.
@ifnottex
@node Copyright
@unnumbered Copyright
@include copyright.texi
@end ifnottex
@node Copying
@unnumbered GNU GENERAL PUBLIC LICENSE
@include gpl.texinfo
@c noman
@node Index
@unnumbered Index
@printindex cp
@contents
@c end noman
@bye
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\documentclass[border=5pt,multi=true]{standalone}
\usepackage{standalone}
\usepackage{ifthen}
\usepackage{microtype}
\usepackage{lineno}
\usepackage{xspace}
\usepackage{caption}
\usepackage{graphicx}
\usepackage{color}
\usepackage{colortbl}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
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\usepackage{hypcap}
\usepackage{cite}
\usepackage{mciteplus}
\usepackage{longtable}
\usepackage{mciteplus}
\usepackage{float}
\begin{document}
\begin{tabular}{lc}
\hline
Parameter & Value \\
\hline
$N_{\mathrm{sig}}$ & \,2357\,$\pm$\,67 \\
$N_{\mathrm{pk}}$ & \,1047\,$\pm$\,84 \\
$N_{\mathrm{npk}}$ & \,2007\,$\pm$\,116\\
$N_{\mathrm{ D\xspace\xspace^0\xspace\!\rightarrow\xspace K\xspace\xspace^-\xspace\pi\xspace\xspace^+\xspace\pi\xspace\xspace^+\xspace\pi\xspace\xspace^-\xspace\xspace}}$ & \,83\,575\,$\pm$\,334\\
$N_{\mathrm{npk}}^{K\pi\pi\pi}$ & \,38\,346\,$\pm$\,257\\
$\sigma$ & 7.17\,$\pm$\,0.03\, $ {\mathrm{\,Me\kern -0.1em V\!/}c^2}$ \xspace \\
$C_{N_{\mathrm{pk}}, N_{\mathrm{npk}}}$ & -78\% \\
$C_{N_{\mathrm{sig}}, N_{\mathrm{pk}}}$ & 27\% \\
$C_{N_{\mathrm{sig}}, N_{\mathrm{npk}}}$ & -48\% \\
$\sigma_{\cal B( D\xspace\xspace^0\xspace\!\rightarrow\xspace K\xspace\xspace^-\xspace\pi\xspace\xspace^+\xspace\mu\xspace^+\xspace\mu\xspace^-\xspace\xspace)}$ & 2.9\% \\
$\sigma_{\cal B( D\xspace\xspace^0\xspace\!\rightarrow\xspace K\xspace\xspace^-\xspace\pi\xspace\xspace^+\xspace\mu\xspace^+\xspace\mu\xspace^-\xspace\xspace)}$, if $N_{\mathrm{pk}}$ fixed & 2.8\% \\
$\sigma_{\cal B( D\xspace\xspace^0\xspace\!\rightarrow\xspace K\xspace\xspace^-\xspace\pi\xspace\xspace^+\xspace\mu\xspace^+\xspace\mu\xspace^-\xspace\xspace)}$, if $N_{\mathrm{pk}}$ and $N_{\mathrm{npk}}$ fixed & 2.4\% \\
\hline
\end{tabular}
\end{document}
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\documentclass{ltxdockit}[2010/09/26]
\usepackage[utf8]{inputenc}
\usepackage[american]{babel}
\usepackage[strict,autostyle=once]{csquotes}
\usepackage{tabularx}
\usepackage{booktabs}
\usepackage{graphicx}
\usepackage{shortvrb}
\usepackage{needspace}
\usepackage{pifont}
\MakeAutoQuote{«}{»}
\MakeAutoQuote*{<}{>}
\MakeShortVerb{\|}
\titlepage{%
title={The \sty{csquotes} Package},
subtitle={Context Sensitive Quotation Facilities},
url={http://www.ctan.org/pkg/csquotes/},
author={Philipp Lehman, Joseph Wright},
email={[email protected]},
revision={v5.1e},
date={2015/04/15}}
\hypersetup{%
pdftitle={The csquotes Package},
pdfsubject={Context Sensitive Quotation Facilities},
pdfauthor={Philipp Lehman, Joseph Wright},
pdfkeywords={tex, e-tex, latex, quote, quotation,
smart quotes, babel, polyglossia, multilingual typesetting}}
\AtBeginToc{\setcounter{tocdepth}{1}}
\AtEndToc{\setcounter{tocdepth}{5}}
% tables
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\newcolumntype{Q}{>{\raggedright}p{70pt}}
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% samples
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\newcommand{\@pseudofootnote}[1]{%
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\newenvironment{quotesample}[2][]
{\setquotestyle[#1]{#2}%
\setcounter{@pseudofootnote}{0}%
\let\footnote\@pseudofootnote
\trivlist
\leftskip\parindent
\small}
{\endtrivlist}
\makeatother
% hyphenation
\hyphenation{%
star-red
New-speak
thres-hold
re-pre-sent-ing
hand-ling
}
\begin{document}
\printtitlepage
\tableofcontents
\listoftables
\section{Introduction}
\label{int}
\subsection[About]{About \sty{csquotes}}
This package provides advanced facilities for inline and display quotations. It is designed for a wide range of tasks ranging from the most simple applications to the more complex demands of formal quotations. The facilities include commands, environments, and user"=definable <smart quotes> which dynamically adjust to their context. Quotation marks are switched automatically if quotations are nested and can adjust to the current language. There are additional features designed to cope with the more specific demands of academic writing. All quote styles as well as the optional active quotes are freely configurable.
\subsection{License}
Copyright © 2003--2011 Philipp Lehman, 2015 Joseph Wright. Permission is granted to copy, distribute and\slash or modify this software under the terms of the \lppl, version 1.3c or any later version.\fnurl{http://www.latex-project.org/lppl/}
\subsection{Contributions}
The multilingual support of this package depends on user contributions. If you want to contribute a quote style, please refer to \secref{cfg:sty} for an overview of the components a quote style is composed of and to \tabref{tab:out} for a list of commands which should be used in the definition of quote styles.
\section{Package Options}
\label{opt}
All package options are given in \keyval syntax.
\subsection{Main Options}
\label{opt:opt}
\begin{optionlist}
\boolitem[false]{strict}
This option turns all package warnings into error messages. This is useful to ensure that no problem will go unnoticed when finalizing a document. The short form \opt{strict} is equivalent to \kvopt{strict}{true}.
\optitem[american]{style}{\prm{style}}
This option selects a fixed quotation style. The style is used throughout the document unless it is changed manually, see \secref{bas:set} for details. This option implicitly sets \kvopt{autostyle}{false}. Please refer to \tabref{tab:sty,tab:als} for a list of available quote styles and aliases. See \secref{cfg:sty,cfg:als,use:spl} for instructions on adding or modifying quote styles.
\optitem[tryonce]{autostyle}{\opt{true}, \opt{false}, \opt{try}, \opt{once}, \opt{tryonce}}
This option controls multilingual support. It requires either the \sty{babel} package or the \sty{polyglossia} package.\footnote{Note that \sty{polyglossia} support is currently in a preliminary state because \sty{polyglossia} is lacking a proper interface for other packages. In practice, this means that \sty{csquotes} can detect the language (\eg \texttt{english}) but not the language variant (\eg \texttt{british}).} \kvopt{autostyle}{true} continuously adapts the quote style to the current document language; \opt{once} will only adapt the style once so that it matches the main language of the document. \kvopt{autostyle}{try} and \opt{tryonce} are similar to \opt{true} and \opt{once} if multilingual support is available but will not issue any warnings if not (\ie if neither \sty{babel} nor \sty{polyglossia} have been loaded). The short form \opt{autostyle} is equivalent to \kvopt{autostyle}{true}. See also \secref{bas:set}.
\begin{table}
\tablesetup
\begin{tabularx}{\columnwidth}{@{}P@{}X@{}}
\toprule
\multicolumn{1}{@{}H}{Option key} & \multicolumn{1}{@{}H}{Possible values} \\
\cmidrule(r){1-1}\cmidrule{2-2}
croatian & quotes, guillemets, guillemets\*\\
danish & quotes, guillemets, topquotes \\
english & american, british\\
french & quotes, quotes\*, guillemets, guillemets\*\\
german & quotes, guillemets, swiss \\
italian & guillemets, quotes \\
norwegian & guillemets, quotes \\
portuguese & portuguese, brazilian \\
spanish & spanish, mexican \\
swedish & quotes, guillemets, guillemets\*\\
\bottomrule
\end{tabularx}
\caption[Language Options]{Language Options Defined by Default}
\label{tab:lng}
\end{table}
\varitem{language}{\prm{variant}}
Use the language options listed in \tabref{tab:lng} to choose a style variant if there is more than one for a certain language. The first variant in the list is the default for the respective style. In the following example, the \opt{autostyle} option causes \sty{csquotes} to continuously adapt the quote style to the current language using the default style for that language. The defaults for German and Norwegian are changed:
\begin{ltxcode}
\usepackage[english,ngerman]{babel}
\usepackage[autostyle,german=guillemets,norwegian=quotes]{csquotes}
\end{ltxcode}
%
Note that \sty{babel}'s language name is \opt{ngerman} here whereas \sty{csquotes} uses \opt{german}. When selecting a quote style automatically, this package will essentially normalize the language names first, using a list of aliases which map languages to quote styles. See \tabref{tab:als} for details. \Tabref{tab:lng} indicates the language options defined by default. Additional options may be defined in the configuration file. See \secref{cfg:opt,use:spl} for details. Also see \secref{hnt:pre} for some additional notes concerning the predefined styles.
\intitem[2]{maxlevel}
This option controls the maximum nesting level of quotations. By default, this package supports two levels of quotations, outer and inner ones, and issues an error if quotations are nested more deeply. Alternatively, it can reuse the outer and inner quotes on further quotation levels. This alternative behavior is enabled implicitly when increasing the value of the \opt{maxlevel} option. The minimum is~\texttt{2}.
\boolitem[true]{autopunct}
This option controls whether the quotation commands scan ahead for trailing punctuation marks. See \secref{bas:txt:reg,bas:blk:reg,int:txt:reg,int:blk:reg} for details. Also see \cmd{DeclareAutoPunct} in \secref{cfg:dap} and \secref{cfg:aux} on how to configure and use the punctuation look-ahead feature. The short form \opt{autopunct} is equivalent to \kvopt{autopunct}{true}.
\intitem[3]{threshold}
This option defines the number of lines or words the block quotation facilities use as a threshold when determining whether a quotation should be typeset in inline or in display mode. It corresponds to the \cmd{SetBlockThreshold} command. See \secref{bas:blk:reg, cfg:blk} for further details.
\optitem[lines]{thresholdtype}{\opt{lines}, \opt{words}}
This option selects the block threshold type. With \kvopt{thresholdtype}{lines}, the block quotation facilities will determine the number of lines required to typeset the quotation; with \kvopt{thresholdtype}{words}, they count the number of words in the quotation.\footnote{The word counting code is based on an idea by Donald Arseneau.} The default threshold setup is 3~lines. If you prefer something like 50~words, set \kvopt{threshold}{50} and \kvopt{thresholdtype}{words}. See \secref{bas:blk:reg} for further details.
\boolitem[true]{parthreshold}
This option fine-tunes the block threshold detection. If enabled, any explicit paragraph or line break in the quotation will trigger the threshold, \ie the quotation will be typeset in display mode regardless of its length. If disabled, explicit paragraph or line breaks are applied normally if \kvopt{thresholdtype}{lines}, and treated as a regular word boundary if \kvopt{thresholdtype}{words}. The short form \opt{parthreshold} is equivalent to \kvopt{parthreshold}{true}.
\boolitem[true]{splitcomp}
This option is applicable with \kvopt{thresholdtype}{words} only. It fine-tunes the handling of compounds with explicit hyphens. If enabled, compounds are split up at all hyphens and the components are counted as separate words. If disabled, compounds are treated as a single word. The short form \opt{splitcomp} is equivalent to \kvopt{splitcomp}{true}.
\boolitem[false]{csdisplay}
By default, the block quotation facilities will automatically force inline quotations in footnotes, parboxes, minipages, and floats. Setting this option to \texttt{true} will permit context-sensitive switching to display quotations in these contexts, as in the text body. The short form \opt{csdisplay} is equivalent to \kvopt{csdisplay}{true}. This option may also be set locally with \cmd{csdisplaytrue} and \cmd{csdisplayfalse}, respectively.
\boolitem[false]{debug}
This option controls the debugging feature of the block quotation facilities. If enabled, all block quotation commands will output diagnostic messages to the transcript file. These messages indicate the calculated line\slash word count, explicit paragaph or line breaks detected in the quotation text, and the final inline\slash display determination. The short form \opt{debug} is equivalent to \kvopt{debug}{true}.
\end{optionlist}
\subsection{Compatibility Options}
\label{opt:leg}
\begin{optionlist}
\optitem{version}{\prm{version}, \opt{4.4}, \opt{3.6}, \opt{3.0}}
This option is an attempt at making \sty{csquotes} backwards compatible with earlier versions, at least to some extent. Currently, it supports backwards compatibility with version 4.4, which covers 3.7--4.4, version 3.6, which covers 3.1--3.6, and version 3.0. Older versions are not supported. It is possible to specify any arbitrary \prm{version}, even if it is not explicitly listed above. In this case, the package will keep increasing the \prm{version} number until it either finds a suitable compatibility mode or hits the current version number. For example, when specifying \kvopt{version}{4.0}, \sty{csquotes} will increase the version number until it hits 4.4, and activate the v4.4 compatibility mode because this is the highest version still compatible with the requested 4.0 release. This implies that \sty{csquotes} may be invoked in a, to some extent, <future proof> way by making the version which is current at the time of writing sticky. If future versions break backwards compatibility, they will automatically activate the best compatibility mode. If not, the \opt{version} option will simply have no effect.
\optitem{babel}{\opt{true}, \opt{false}, \opt{try}, \opt{once}, \opt{tryonce}}
An older name of the \opt{autostyle} option. This option is depreciated.
\end{optionlist}
\section{Basic Interface}
\label{bas}
This package supports two ways to tag quotations: built-in commands and active quotes defined in the document preamble or the configuration file. This section will introduce the basic commands, active quotes are discussed in \secref{act}. When working with automated citations, you might also want to learn about the integrated quotation facilities presented in \secref{cit}.
\subsection{Quoting Regular Text}
\label{bas:reg}
The most basic command will simply enclose its argument in quotation marks:
\begin{ltxsyntax}
\cmditem{enquote}{text}
\cmditem*{enquote*}{text}
Like all quotation facilities, this command is context sensitive. Depending on the nesting level, it will toggle between outer and inner quotation marks with plain and nested quotations. The starred version of this command skips directly to the inner level. If multilingual support is enabled, the style of all quotation marks will be adapted to the current language.
\end{ltxsyntax}
\subsection{Quoting Text in a Foreign Language}
\label{bas:bbl}
To facilitate typesetting quotations in a foreign language, \sty{csquotes} provides two commands which combine \cmd{enquote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\begin{ltxsyntax}
\cmditem{foreignquote}{lang}{text}
\cmditem*{foreignquote*}{lang}{text}
This command combines \cmd{enquote} with \cmd{foreignlanguage}. It switches hyphenation patterns and enables the extra definitions provided by \sty{babel}\slash\sty{polyglossia} for \prm{lang}, which must be a language name known to the respective package. The quotation marks will match the language of the quoted piece of text.
\cmditem{hyphenquote}{lang}{text}
\cmditem*{hyphenquote*}{lang}{text}
This command combines \cmd{enquote} with the \env{hyphenrules} environment, that is, it merely switches hyphenation patterns. The quotation marks will match the language of the text surrounding the quotation.
\end{ltxsyntax}
\subsection{Formal Quoting of Regular Text}
\label{bas:txt:reg}
Formal quotations are typically accompanied by a citation indicating the source of the quoted text. The following command is an extended version of \cmd{enquote} which encloses the quoted piece of text in quotation marks and adds a citation after the quotation:
\begin{ltxsyntax}
\cmditem{textquote}[cite][punct]{text}<tpunct>
\cmditem*{textquote*}[cite][punct]{text}<tpunct>
The \prm{text} may be any arbitrary piece of text to be enclosed in quotation marks. The optional arguments \prm{cite} and \prm{punct} specify the citation and any terminal punctuation of \prm{text}. \prm{tpunct} denotes trailing punctuation after the command. If the \opt{autopunct} option is enabled, the quotation commands will scan ahead for punctuation marks immediately following their last argument and can move them around if required. See \secref{cfg:aux} on how to change the way these arguments are handled and \secref{use:hok} for reasons why you may want to specify the punctuation as a separate argument. The starred version of this command skips directly to the inner quotation level. Here are some usage examples:
\begin{ltxcode}
\textquote{...}
\textquote[][.]{...}
\textquote[Doe 1990, 67]{...}
\textquote[{\cite[67]{doe90}}]{...}
\end{ltxcode}
%
Note the use of the optional arguments in the examples above. As seen in the second example, \prm{cite} is required whenever \prm{punct} is used, even if it is empty. Also keep in mind that an optional argument containing square brackets must be wrapped in an additional pair of curly braces as shown in the last example. When working with automated citations, you might also want to learn about the integrated quotation facilities presented in \secref{cit}.
\end{ltxsyntax}
\subsection{Formal Quoting of Text in a Foreign Language}
\label{bas:txt:bbl}
There are two additional commands which combine \cmd{textquote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\begin{ltxsyntax}
\cmditem{foreigntextquote}{lang}[cite][punct]{text}<tpunct>
\cmditem*{foreigntextquote*}{lang}[cite][punct]{text}<tpunct>
This command combines \cmd{textquote} with \cmd{foreignlanguage}. Apart from the language, the arguments are handled as with \cmd{textquote}.
\cmditem{hyphentextquote}{lang}[cite][punct]{text}<tpunct>
\cmditem*{hyphentextquote*}{lang}[cite][punct]{text}<tpunct>
This command combines \cmd{textquote} with the \env{hyphenrules} environment. Apart from the language, the arguments are handled as with \cmd{textquote}.
\end{ltxsyntax}
\subsection{Block Quoting of Regular Text}
\label{bas:blk:reg}
Formal requirements in academic writing frequently demand that quotations be embedded in the text if they are short but set off as a distinct and typically indented paragraph, a so-called block quotation, if they are longer than a certain number of lines or words. In the latter case no quotation marks are inserted. The following command deals with this requirement automatically:
\begin{ltxsyntax}
\cmditem{blockquote}[cite][punct]{text}<tpunct>
This command determines the length of the \prm{text}. If the length exceeds a certain threshold, the \prm{text} will be typeset in display mode, \ie as a block quotation. If not, \cmd{blockquote} will behave like \cmd{textquote}. Depending on the \opt{thresholdtype} option, the threshold may be based on the number of lines required to typeset the \prm{text} or on the number of words in the \prm{text}. If the \opt{parthreshold} option has been enabled, any explicit paragraph or line break in the \prm{text} will trigger the threshold, \ie it will be typeset in display mode regardless of its length. The default threshold setup is three lines with \opt{parthreshold} enabled. The default environment used for display quotations is the \env{quote} environment. See \secref{opt:opt, cfg:blk} on how to change these parameters. Note that \sty{csquotes} will force inline quotations in footnotes, parboxes, minipages, and floats by default. Use the \opt{csdisplay} option from \secref{opt:opt} to change this behavior. The optional arguments \prm{cite} and \prm{punct} specify the citation and any terminal punctuation of the \prm{text}. \prm{tpunct} denotes trailing punctuation after the command. If the \opt{autopunct} option is enabled, the quotation commands will scan ahead for punctuation marks immediately following their last argument and can move them around if required. See \secref{cfg:aux} on how to change the way these arguments are handled and \secref{use:hok} for reasons why you may want to specify the punctuation as a separate argument.
\end{ltxsyntax}
\subsection{Block Quoting of Text in a Foreign Language}
\label{bas:blk:bbl}
The following commands combine \cmd{blockquote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\begin{ltxsyntax}
\cmditem{foreignblockquote}{lang}[cite][punct]{text}<tpunct>
This command behaves like \cmd{foreignquote} if the quotation is short. If it exceeds the threshold, it will be wrapped in an \env{otherlanguage*} environment which is in turn wrapped in a block quotation environment. The arguments are handled as with \cmd{blockquote}.
\cmditem{hyphenblockquote}{lang}[cite][punct]{text}<tpunct>
This command behaves like \cmd{hyphenquote} if the quotation is short. If it exceeds the threshold, it will be wrapped in a \env{hyphenrules} environment which is in turn wrapped in a block quotation environment. The arguments are handled as with \cmd{blockquote}.
\cmditem{hybridblockquote}{lang}[cite][punct]{text}<tpunct>
This command behaves like \cmd{hyphenquote} if the quotation is short. If it exceeds the threshold, the command behaves like \cmd{foreignblockquote}. In other words, it combines features of \cmd{foreignblockquote} and \cmd{hyphenblockquote}. The arguments are handled as with \cmd{blockquote}.
\end{ltxsyntax}
\subsection{Selecting Quote Styles}
\label{bas:set}
Quote styles may be selected manually at any point in the document body by way of the following command:
\begin{ltxsyntax}
\cmditem{setquotestyle}[variant]{style}
\cmditem*{setquotestyle}{alias}
\cmditem*{setquotestyle}*
The regular form of this command selects a quote style and disables multilingual support. Its mandatory argument may be a quote style or an alias. If it is a quote style, the optional argument indicates the style variant. The starred version, which takes no arguments, enables multilingual support. Please refer to \tabref{tab:sty,tab:als} for a list of available styles, style variants, and language aliases.
\end{ltxsyntax}
\section{Active Quotes}
\label{act}
This package also supports active characters corresponding to the commands presented in \secref{bas}. Roughly speaking, an active character is a single character which functions as a command. Like the corresponding control sequences, active quotes are fully"=fledged markup elements which verify the nesting level and issue an error if quotations are nested in an invalid way. If multilingual support is enabled, the style of all quotation marks will be adapted to the current language. The commands presented in the following allocate such active quotes. They may be used in the configuration file, the preamble, or the document body. Note that all characters are automatically checked for validity as they are allocated. This package will reject characters which are unsuitable as active quotes. See \secref{hnt:val} for details on the characters which may be used as active quotes.
\subsection{Quoting Regular Text}
\label{act:reg}
\cmd{MakeOuterQuote} and \cmd{MakeInnerQuote} define active quotes which print outer and inner quotation marks. Both take one mandatory argument, the character serving as both opening and closing mark:
\begin{ltxsyntax}
\cmditem{MakeOuterQuote}{character}
\cmditem{MakeInnerQuote}{character}
\cmd{MakeAutoQuote} defines active quotes which toggle between outer and inner quotations automatically. The two mandatory arguments serve as opening and closing mark and must be distinct:
\cmditem{MakeAutoQuote}{character 1}{character 2}
\cmditem*{MakeAutoQuote*}{character 1}{character 2}
All active quotes defined with \cmd{MakeAutoQuote} work like \cmd{enquote}. Those defined with \cmd{MakeOuterQuote} and \cmd{MakeInnerQuote} cover only a part of this functionality. The former correspond to the outer level of \cmd{enquote} whereas the latter correspond to the starred version. \cmd{MakeAutoQuote*} is similar to \cmd{MakeInnerQuote}, i.e. it corresponds to \cmd{enquote*}.
\end{ltxsyntax}
\subsection{Quoting Text in a Foreign Language}
\label{act:bbl}
These commands combine \cmd{MakeAutoQuote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\begin{ltxsyntax}
\cmditem{MakeForeignQuote}{lang}{character 1}{character 2}
\cmditem*{MakeForeignQuote*}{lang}{character 1}{character 2}
The active quotes defined with the above commands are similar in concept and function to \cmd{foreignquote} and \cmd{foreignquote*}, respectively.
\cmditem{MakeHyphenQuote}{lang}{character 1}{character 2}
\cmditem*{MakeHyphenQuote*}{lang}{character 1}{character 2}
The active quotes defined with the above commands are similar in concept and function to \cmd{hyphenquote} and \cmd{hyphenquote*}, respectively.
\end{ltxsyntax}
\subsection{Block Quoting of Regular Text}
\label{act:blk:reg}
\cmd{MakeBlockQuote} defines active quotes which will set quotations inline or as a separate paragraph, depending on their length. This command takes three mandatory arguments which must be distinct:
\begin{ltxsyntax}
\cmditem{MakeBlockQuote}{character 1}{delimiter}{character 2}
The arguments are checked for validity, see \secref{hnt:val} for details. All active quotes defined with \cmd{MakeBlockQuote} behave essentially the same as \cmd{blockquote}, but the handling of the citation is slightly different. \prm{character 1} will serve as the opening mark in the source file, \prm{character 2} as the closing one. The character indicated by the middle argument \prm{delimiter} will serve as a delimiter separating the quoted text from the citation which is given last as the active quotes are used:
\begin{ltxcode}
\MakeBlockQuote{<}{|}{>}
...
<text|citation>
\end{ltxcode}
%
If the delimiter is omitted, the entire text between the opening and the closing mark will be treated as quotation text.
\end{ltxsyntax}
\subsection{Block Quoting of Text in a Foreign Language}
\label{act:blk:bbl}
These commands combine \cmd{MakeBlockQuote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\begin{ltxsyntax}
\cmditem{MakeForeignBlockQuote}{lang}{character 1}{delimiter}{character 2}
The active quotes defined with this command are similar in concept and function to \cmd{foreignblockquote}. The behavior of the delimiter character is similar to \cmd{MakeBlockQuote}.
\cmditem{MakeHyphenBlockQuote}{lang}{character 1}{delimiter}{character 2}
The active quotes defined with this command are similar in concept and function to \cmd{hyphenblockquote}. The behavior of the delimiter character is similar to \cmd{MakeBlockQuote}.
\cmditem{MakeHybridBlockQuote}{lang}{character 1}{delimiter}{character 2}
The active quotes defined with this command are similar in concept and function to \cmd{hybridblockquote}. The behavior of the delimiter character is similar to \cmd{MakeBlockQuote}.
\end{ltxsyntax}
\subsection{Controlling Active Quotes}
\label{act:ctl}
The commands introduced above merely allocate active quotes, but these characters are not immediately made active. All allocated quotes are automatically enabled at the beginning of the document body. If any active quotes are allocated in the document body, they need to be enabled with \cmd{EnableQuotes}. The following commands control the state of the active quotes within a local scope.
\begin{ltxsyntax}
\csitem{EnableQuotes}
Enables all active quotes by redefining the allocated characters and making them active. It also restores them when disabled, set to verbatim, or overwritten.
\csitem{DisableQuotes}
Disables all active quotes by restoring the original category codes and definitions of all allocated characters.
\csitem{VerbatimQuotes}
Switches to verbatim active quotes. All active quotes will be printed verbatim until their default behavior is restored with \cmd{EnableQuotes}.
\csitem{DeleteQuotes}
Disables and deallocates all active quotes, i.e. performs a complete reset of all allocated characters so that they may be newly defined.
\end{ltxsyntax}
\section{Integrated Interface}
\label{cit}
The commands presented in this section are extended versions of some of those discussed in \secref{bas}. They differ from their counterparts in that they integrate automated citations into their syntax. Instead of adding \cmd{cite} manually, you pass the citation arguments to the respective quotation command. See \secref{cfg:blk} on how to use a command other than \cmd{cite} to handle the citations.
\subsection{Formal Quoting of Regular Text}
\label{int:txt:reg}
The basic integrated command is an extended version of \cmd{textquote}:
\begin{ltxsyntax}
\cmditem{textcquote}[prenote][postnote]{key}[punct]{text}<tpunct>
\cmditem*{textcquote*}[prenote][postnote]{key}[punct]{text}<tpunct>
The \prm{text} may be any arbitrary piece of text to be enclosed in quotation marks. The optional arguments \prm{cite} and \prm{punct} specify the citation and any terminal punctuation of \prm{text}. \prm{tpunct} denotes trailing punctuation after the command. If the \opt{autopunct} option is enabled, the quotation commands will scan ahead for punctuation marks immediately following their last argument and can move them around if required. See \secref{cfg:aux} on how to change the way these arguments are handled and \secref{use:hok} for reasons why you may want to specify the punctuation as a separate argument. The starred version of this command skips directly to the inner quotation level. The remaining arguments are handed over to \cmd{cite}. Note that \cmd{cite} normally supports one optional argument only. \prm{prenote} is only available in conjunction with the \sty{natbib}, \sty{jurabib}, and \sty{biblatex} packages. How these arguments are handled depends on the citation command. With \sty{natbib} and \sty{biblatex}, \prm{prenote} is in fact a notice such as <see>. With jurabib, this argument has a different function by default. The argument \prm{postnote}, which is always available, indicates the citation postnote. This is usually a page number. \prm{key} is the citation key. See \secref{cfg:blk,cfg:aux} on how to customize the citation.
\end{ltxsyntax}
\subsection{Formal Quoting of Text in a Foreign Language}
\label{int:txt:bbl}
The following commands combine \cmd{textcquote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\begin{ltxsyntax}
\cmditem{foreigntextcquote}{lang}[prenote][postnote]{key}[punct]{text}<tpunct>
\cmditem*{foreigntextcquote*}{lang}[prenote][postnote]{key}[punct]{text}<tpunct>
This command combines \cmd{textcquote} with \cmd{foreignlanguage}. The handling of the arguments is similar to \cmd{textcquote}.
\cmditem{hyphentextcquote}{lang}[prenote][postnote]{key}[punct]{text}<tpunct>
\cmditem*{hyphentextcquote*}{lang}[prenote][postnote]{key}[punct]{text}<tpunct>
This command combines \cmd{textcquote} with the \env{hyphenrules} environment. The handling of the arguments is similar to \cmd{textcquote}.
\end{ltxsyntax}
\subsection{Block Quoting of Regular Text}
\label{int:blk:reg}
Block quotations may be combined with automated citations by using the extended version of \cmd{blockquote}:
\begin{ltxsyntax}
\cmditem{blockcquote}[prenote][postnote]{key}[punct]{text}<tpunct>
The difference between \cmd{blockcquote} and \cmd{blockquote} is that there are three citation arguments instead of one. The handling of these citation arguments is similar to \cmd{textcquote}; see \secref{int:txt:reg} for details. Also see \secref{cfg:blk,cfg:aux,use:hok} on how to customize block quotations.
\end{ltxsyntax}
\subsection{Block Quoting of Text in a Foreign Language}
\label{int:blk:bbl}
The following commands combine \cmd{blockcquote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\begin{ltxsyntax}
\cmditem{foreignblockcquote}{lang}[prenote][postnote]{key}[punct]{text}<tpunct>
This command combines \cmd{blockcquote} with \cmd{foreignlanguage}. Long quotations will be wrapped in an \env{otherlanguage*} environment. The handling of the citation arguments is similar to \cmd{textcquote}.
\cmditem{hyphenblockcquote}{lang}[prenote][postnote]{key}[punct]{text}<tpunct>
This command combines \cmd{blockcquote} with the \env{hyphenrules} environment. The handling of the citation arguments is similar to \cmd{textcquote}.
\cmditem{hybridblockcquote}{lang}[prenote][postnote]{key}[punct]{text}<tpunct>
This command behaves like \cmd{hyphenblockcquote} if the quotation is short, and like \cmd{foreignblockquote} if it is long. The handling of the citation arguments is similar to \cmd{textcquote}.
\end{ltxsyntax}
\section{Display Environments}
\label{env}
The environments introduced in this section will typeset quotations as a separate paragraph which looks exactly like a long quotation set by means of the block quotation facilities. Use them for quotations which are to be presented as a separate paragraph regardless of their length. Note that these environments are not replacements for the standard \env{quote} environment in the strict sense. They function as an additional layer on top of the latter, just like the block quotation facilities. The advantage of using these environments instead of resorting to the standard \env{quote} environment is that they are configurable, support citations, and will always be in sync with the block quotation facilities with respect to the configuration options discussed in \secref{cfg:blk,cfg:aux}.
\subsection{Basic Display Environments}
\label{env:bas}
The arguments of all display environments are generally appended to the \cmd{begin} section of the environment:
\begin{ltxsyntax}
\envitem{displayquote}[cite][punct]
The optional arguments \prm{cite} and \prm{punct} specify the citation and any terminal punctuation of the quotation. See \secref{cfg:blk,cfg:aux} on how to customize this environment. Also see \secref{cfg:aux} on how to change the way the optional arguments are handled and \secref{use:hok} for reasons why you may want to specify the punctuation as a separate argument. Trailing white space at the end of the environment is removed automatically. There are two additional environments which combine \env{displayquote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\envitem{foreigndisplayquote}{lang}[cite][punct]
This environment combines \env{displayquote} with \env{otherlanguage*}. Apart from the language, the arguments are handled as with \env{displayquote}.
\envitem{hyphendisplayquote}{lang}[cite][punct]
This environment combines \env{displayquote} with \env{hyphenrules}. Apart from the language, the arguments are handled as with \env{displayquote}.
\end{ltxsyntax}
\subsection{Integrated Display Environments}
\label{env:cit}
The following environment is an extended version of \env{displayquote}:
\begin{ltxsyntax}
\envitem{displaycquote}[prenote][postnote]{key}[punct]
The difference between \env{displaycquote} and its more basic counterpart is that there are three citation arguments instead of one. The placement of the citation is similar to \env{displayquote}. The handling of the citation arguments is similar to \cmd{textcquote}, see \secref{int:txt:reg} for details. See \secref{cfg:blk,cfg:aux} on how to customize this environment. Also see \secref{cfg:aux} on how to change the way the optional arguments are handled and \secref{use:hok} for reasons why you may want to specify the punctuation as a separate argument. There are two environments which combine \env{displaycquote} with the language switches of the \sty{babel} or the \sty{polyglossia} package:
\envitem{foreigndisplaycquote}{lang}[prenote][postnote]{key}[punct]
This environment combines \env{displaycquote} with \env{otherlanguage*}. Apart from the language, the arguments are handled as with \env{displaycquote}.
\envitem{hyphendisplaycquote}{lang}[prenote][postnote]{key}[punct]
This environment combines \env{displaycquote} with \env{hyphenrules}. Apart from the language, the arguments are handled as with \env{displaycquote}.
\end{ltxsyntax}
\section{Auxiliary Commands}
\label{aux}
When quoting text in a formal way, any changes applied to the quoted material, such as omissions, insertions, or alterations, are typically marked as such by using the ellipsis mark and square brackets or parentheses. Use the following commands to indicate such changes in formal quotations:
\begin{ltxsyntax}
\cmditem{textelp}{}
\cmditem*{textelp}{text}
\cmditem*{textelp}*{text}
When used with an empty \prm{text} argument, this command prints an ellipsis symbol to indicate the omission of material from the quoted material. When used with a non-empty argument, the ellipsis symbol is followed by the \prm{text} enclosed in square brackets to indicate that the \prm{text} has been added after the omitted material. The starred version reverts the order, \ie it prints the \prm{text} followed by an ellipsis symbol to indicate that the \prm{text} has been added before the omitted material. In sum, there are three ways to use this command:
\begin{ltxcode}[escapechar={\%},escapebegin={\rmfamily}]
\textelp{} %= \textelp{} %
\textelp{text} %= \textelp{text} %
\textelp*{text} %= \textelp*{text} %
\end{ltxcode}
%
The insertion of text or individual letters may be indicated with the following command:
\cmditem{textins}{text}
\cmditem*{textins}*{text}
By default, \cmd{textins} will enclose the \prm{text} added to the quoted material in square brackets. The starred version is intended for minor changes, such as the capitalization of a word, which are required to adapt the quoted material to the new context in which it is quoted.
\begin{ltxcode}[escapechar={\%},escapebegin={\rmfamily}]
\textins{text} %= \textins{text} %
\textins*{T}ext %= \textins*{T}ext %
\end{ltxcode}
%
See \secref{cfg:elp} on how to configure the appearance of ellipses and insertions.
\end{ltxsyntax}
\section{Configuration}
\label{cfg}
If available, this package will load the configuration file \path{csquotes.cfg}. You may use this file to define new quote styles and aliases or redefine existing ones.
\begin{table}
\tablesetup
\begin{tabularx}{\columnwidth}{@{}P@{}X@{}}
\toprule
\multicolumn{1}{@{}H}{Quote style} & \multicolumn{1}{@{}H}{Style variants} \\
\cmidrule(r){1-1}\cmidrule{2-2}
croatian & quotes, guillemets, guillemets\*\\
danish & quotes, guillemets \\
dutch & -- \\
english & american, british\\
finnish & -- \\
french & quotes, quotes\*, guillemets, guillemets\*\\
german & quotes, guillemets, swiss \\
greek & -- \\
italian & guillemets, quotes \\
norwegian & guillemets, quotes \\
portuguese & portuguese, brazilian \\
russian & -- \\
spanish & spanish, mexican \\
swedish & quotes, guillemets, guillemets\*\\
\bottomrule
\end{tabularx}
\caption[Styles and Variants]{Quote Styles and Style Variants Defined by Default}
\label{tab:sty}
\end{table}
\subsection{Defining Quote Styles}
\label{cfg:sty}
Use the following command to define quote styles and style variants:
\begin{ltxsyntax}
\cmditem{DeclareQuoteStyle}[variant]{style}[outer init][inner init]\\
{opening outer mark}[middle outer mark]{closing outer mark}[kern]\\
{opening inner mark}[middle inner mark]{closing inner mark}
This command may be used in the configuration file or in the document preamble. The term <outer> refers to the first quotation level, <inner> means quotations within another quotation. A <middle mark> is a quotation mark inserted at the beginning of every paragraph within a quotation spanning multiple paragraphs. In most cases, the arguments defining the quotation marks will simply contain one of the commands listed in \tabref{tab:out}. If both an outer and an inner quotation begin or end simultaneously, the kerning specified by the value \prm{kern} will be inserted between the adjoining quotation marks. While this value can be given in any unit known to \tex, it is advisable to use the relative, font"=dependent unit <em> instead of absolute units such as points, inches, or millimeters. Note that \prm{kern} is used as a fallback value only. If the font provides kerning data for the respective pair of quotation marks the font's kerning takes precedence. \prm{outer init} and \prm{inner init} are all"=purpose hooks initializing the quote style. Selecting a quote style will make these hooks available to all quotation commands without expanding them. The execution of \prm{outer init} will take place immediately before the opening outer quote is inserted, but inside the group formed by the quotation. \prm{inner init} is executed before the opening inner quote is inserted. It is advisable to avoid any global assignments in this context to prevent interference with other styles. Whenever \prm{inner init} is used \prm{outer init} has to be given as well, even if the argument is empty. Refer to \tabref{tab:sty} for a list of all predefined quote styles and their variants. These are the backend styles only, see also \tabref{tab:als} for a list of language aliases. See \secref{use:spl} for some examples as well as an illustration of how quote styles, aliases, and package options interact.
\end{ltxsyntax}
\subsection{Defining Quote Aliases}
\label{cfg:als}
The following command defines quote aliases:
\begin{ltxsyntax}
\cmditem{DeclareQuoteAlias}[variant]{style}{alias}
\cmditem*{DeclareQuoteAlias}{first-level alias}{second-level alias}
This command may be used in the configuration file or in the document preamble. The alias may point to a backend style or to another alias. Most language aliases refer to a backend style, but some point to an intermediate alias instead. If the alias is defined for the sake of the \sty{babel} or the \sty{polyglossia} package, its name must be identical to the language name used by \sty{babel}\slash\sty{polyglossia}, \ie the expansion of \cmd{languagename}. See \secref{use:spl} for an illustration of how quote styles, aliases, and package options interact. A list of all aliases defined by default is given in \tabref{tab:als}.
\end{ltxsyntax}
\begin{table}
\tablesetup
\begin{tabularx}{\columnwidth}{@{}Q@{}X@{}Q@{}X@{}}
\toprule
\multicolumn{1}{@{}H}{Alias} & \multicolumn{1}{@{}H}{Backend style or alias} &
\multicolumn{1}{@{}H}{Alias} & \multicolumn{1}{@{}H}{Backend style or alias} \\
\cmidrule(r){1-1}\cmidrule(r){2-2}\cmidrule(r){3-3}\cmidrule{4-4}
american & english/american &
naustrian & austrian \\
australian & english/british &
newzealand & english/british \\
austrian & german/quotes &
ngerman & german \\
brazil & brazilian &
norsk & norwegian \\
brazilian & portuguese/brazilian &
norwegian & norwegian/guillemets \\
british & english/british &
nynorsk & norwegian \\
canadian & english/american &
portuges & portuguese \\
croatian & croatian/quotes &
portuguese & portuguese/portuguese \\
danish & danish/quotes &
spanish & spanish/spanish \\
english & english/american &
swedish & swedish/quotes \\
french & french/quotes &
swiss & german/swiss \\
german & german/quotes &
UKenglish & british \\
italian & italian/guillemets &
USenglish & american \\
mexican & spanish/mexican &
\\
\bottomrule
\end{tabularx}
\caption[Language Aliases]{Language Aliases Defined by Default}
\label{tab:als}
\end{table}
\subsection{Defining Package Options}
\label{cfg:opt}
The following command creates a new package option based on a key\slash value syntax. It takes one mandatory argument, the quote style name:
\begin{ltxsyntax}
\cmditem{DeclareQuoteOption}{style}
When using the new option, the name of the quote style will serve as the key. The value may be any style variant defined for the respective style. The package option will select a variant by defining an alias pointing to the desired backend style. This command is available in the configuration file only. See \secref{use:spl} for an illustration of how quote styles, aliases, and package options interact.
\end{ltxsyntax}
\subsection{Executing Package Options}
\label{cfg:exe}
Apart from passing options to this package as it is loaded, you may also execute options using the following command:
\begin{ltxsyntax}
\cmditem{ExecuteQuoteOptions}{key=value,\,\dots}
This command permits presetting package options in the configuration file. It may also be used in the document preamble.
\end{ltxsyntax}
\subsection{Defining Quotes for \pdf Strings}
\label{cfg:pdf}
The following command specifies the quotation marks for \pdf strings:
\begin{ltxsyntax}
\cmditem{DeclarePlainStyle}{opening outer mark}{closing outer mark}\\
{opening inner mark}{closing inner mark}
This command may be used in the configuration file or in the document preamble. By default, outer quotations get straight double quotes and inner quotations straight single quotes. See \secref{hnt:pdf} for additional hints concerning \pdf strings.
\end{ltxsyntax}
\subsection{Configuring Quotations and Citations}
\label{cfg:blk}
The following commands change the default values used by various quotation facilities of this package. The commands affected by these parameters are indicated in \tabref{tab:blk}.
\begin{table}
\tablesetup
\let\+\tickmarkyes
\let\_\tickmarkno
\begin{tabularx}{\columnwidth}{@{}X@{}*{21}{@{}c}@{}}
\toprule
\multicolumn{1}{@{}H}{Parameter} & \multicolumn{21}{@{}H}{Command or environment} \\
\cmidrule{2-22}
& \vcmd{enquote}
& \vcmd{foreignquote}
& \vcmd{hyphenquote}
& \vcmd{textquote}
& \vcmd{foreigntextquote}
& \vcmd{hyphentextquote}
& \vcmd{textcquote}
& \vcmd{foreigntextcquote}
& \vcmd{hyphentextcquote}
& \vcmd{blockquote}
& \vcmd{foreignblockquote}
& \vcmd{hyphenblockquote}
& \vcmd{blockcquote}
& \vcmd{foreignblockcquote}
& \vcmd{hyphenblockcquote}
& \venv{displayquote}
& \venv{foreigndisplayquote}
& \venv{hyphendisplayquote}
& \venv{displaycquote}
& \venv{foreigndisplaycquote}
& \venv{hyphendisplaycquote} \\
\cmidrule(r){1-1}\cmidrule{2-22}
Threshold &\_&\_&\_&\_&\_&\_&\_&\_&\_&\+&\+&\+&\+&\+&\+&\_&\_&\_&\_&\_&\_\\
Environment &\_&\_&\_&\_&\_&\_&\_&\_&\_&\+&\+&\+&\+&\+&\+&\+&\+&\+&\+&\+&\+\\
Cite command &\_&\_&\_&\_&\_&\_&\+&\+&\+&\_&\_&\_&\+&\+&\+&\_&\_&\_&\+&\+&\+\\
\bottomrule
\end{tabularx}
\caption[Configurable Parameters]{Scope of Configurable Parameters}
\label{tab:blk}
\end{table}
\begin{ltxsyntax}
\cmditem{SetBlockThreshold}{integer}
\cmditem{SetBlockEnvironment}{environment}
\cmditem{SetCiteCommand}{command}
\cmd{SetBlockThreshold} defines the number of lines or words the block quotation facilities use as a threshold when determining whether a quotation should be typeset in inline or in display mode. The default is three. \cmd{SetBlockEnvironment} specifies the environment used for block and display quotations. It takes the name of an existing environment as its argument. The default is the \env{quote} environment provided by most document classes. The argument to \cmd{SetCiteCommand} specifies a replacement for \cmd{cite} which will be used by all integrated quotation facilities to handle citations. It must be a single command which takes one or two optional arguments followed by a mandatory one, the citation key. The default is \cmd{cite}. The citation commands of the \sty{natbib}, \sty{jurabib}, and \sty{biblatex} packages, which take two optional arguments, are supported.
\end{ltxsyntax}
\subsection{Hooks for Quotations and Citations}
\label{cfg:aux}
The appearance of quotes may be configured at a low level by redefining the hooks introduced below. This section will give an overview of their syntax. See \secref{use:hok} for practical examples. The quotation facilities which are responsive to these hooks are indicated in \tabref{tab:aux}. Also see \secref{cfg:tst} for tests which may be useful when redefining the hooks.
\begin{table}
\tablesetup
\let\+\tickmarkyes
\let\_\tickmarkno
\begin{tabularx}{\columnwidth}{@{}X@{}*{21}{@{}c}@{}}
\toprule
\multicolumn{1}{@{}H}{Hook} & \multicolumn{21}{@{}H}{Command or environment} \\
\cmidrule{2-22}
& \vcmd{enquote}
& \vcmd{foreignquote}
& \vcmd{hyphenquote}
& \vcmd{textquote}
& \vcmd{foreigntextquote}
& \vcmd{hyphentextquote}
& \vcmd{textcquote}
& \vcmd{foreigntextcquote}
& \vcmd{hyphentextcquote}
& \vcmd{blockquote}
& \vcmd{foreignblockquote}
& \vcmd{hyphenblockquote}
& \vcmd{blockcquote}
& \vcmd{foreignblockcquote}
& \vcmd{hyphenblockcquote}
& \venv{displayquote}
& \venv{foreigndisplayquote}
& \venv{hyphendisplayquote}
& \venv{displaycquote}
& \venv{foreigndisplaycquote}
& \venv{hyphendisplaycquote} \\
\cmidrule(r){1-1}\cmidrule{2-22}
\cmd{mktextquote} &\_&\_&\_&\+&\+&\+&\+&\+&\+&\+&\+&\+&\+&\+&\+&\_&\_&\_&\_&\_&\_\\
\cmd{mkblockquote} &\_&\_&\_&\_&\_&\_&\_&\_&\_&\+&\+&\+&\+&\+&\+&\_&\_&\_&\_&\_&\_\\
\cmd{mkbegdispquote} &\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\+&\+&\+&\+&\+&\+\\
\cmd{mkenddispquote} &\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\_&\+&\+&\+&\+&\+&\+\\
\cmd{mkcitation} &\_&\_&\_&\+&\+&\+&\_&\_&\_&\+&\+&\+&\_&\_&\_&\+&\+&\+&\_&\_&\_\\
\cmd{mkccitation} &\_&\_&\_&\_&\_&\_&\+&\+&\+&\_&\_&\_&\+&\+&\+&\_&\_&\_&\+&\+&\+\\
\bottomrule
\end{tabularx}
\caption[Auxiliary Hooks]{Availability of Auxiliary Hooks}
\label{tab:aux}
\end{table}
\begin{ltxsyntax}
\cmditem{mkcitation}{cite}
All facilities which take a \prm{cite} argument will pass it to the \cmd{mkcitation} hook, which may be redefined to format the citation. \cmd{mkcitation} will only be executed if there is a citation. The default behavior is to separate the citation from the preceding text by an interword space and enclose it in parentheses. This is equivalent to the following definition:
\begin{ltxcode}[showspaces=true]
\newcommand*{<<\mkcitation>>}[1]{ (#1)}
\end{ltxcode}
\cmditem{mkccitation}{cite code}
The integrated quotation facilities use \cmd{mkccitation} instead of \cmd{mkcitation}. The default behavior of this command is to separate the citation from the preceding text by an interword space. This is equivalent to the following definition:
\begin{ltxcode}[showspaces=true]
\newcommand*{<<\mkccitation>>}[1]{ #1}
\end{ltxcode}
\cmditem{mktextquote}{open}{text}{close}{punct}{tpunct}{cite}
The \cmd{mktextquote} hook controls the layout of all text quotations. This hook is used by \cmd{textquote} and related commands from \secref{bas:txt:reg,bas:txt:bbl,int:txt:reg,int:txt:bbl}. \cmd{blockquote} and related commands from \secref{bas:blk:reg,bas:blk:bbl,int:blk:reg,int:blk:bbl} use this hook for short quotations. It takes six arguments which may be arranged according to the desired output:
\begin{argumentlist}{00}
\item[\#1] The opening quotation mark.
\item[\#2] The \prm{text} argument of the command.
\item[\#3] The closing quotation mark.
\item[\#4] The optional \prm{punct} argument of the command. If there is no \prm{punct} argument, this parameter is empty.
\item[\#5] Trailing \prm{tpunct} punctuation immediately after the command. If there is no such punctuation or if the \opt{autopunct} feature is disabled, this parameter is empty.
\item[\#6] The optional \prm{cite} argument of the command, wrapped in \cmd{mkcitation}. If there is no \prm{cite} argument, this parameter is empty. With integrated quotation commands, this parameter is the citation code, wrapped in \cmd{mkccitation}.
\end{argumentlist}
%
By default, \cmd{mktextquote} encloses the \prm{punct} argument in the quotation marks along with the \prm{text} and inserts the \prm{cite} argument or the citation code before any trailing \prm{tpunct} punctuation. This is equivalent to the following definition:
\begin{ltxcode}
\newcommand{<<\mktextquote>>}[6]{#1#2#4#3#6#5}
\end{ltxcode}
%
The way in which \cmd{mktextquote} hooks into the formatting process is best seen when looking at an example. The commands
\begin{ltxcode}
\textquote[<<cite>>]{<<short quote>>}
\textcquote[<<55>>]{<<key1>>}[<<.>>]{<<short quote>>}
\blockcquote[<<87>>]{<<key2>>}{<<short quote>>}<<.>>
\end{ltxcode}
%
would execute \cmd{mktextquote} with the following arguments:
\begin{ltxcode}
\mktextquote{open}{<<short quote>>}{close}{}{}{\mkcitation{<<cite>>}}
\mktextquote{open}{<<short quote>>}{close}{<<.>>}{}{\mkccitation{\cite[<<55>>]{<<key1>>}}}
\mktextquote{open}{<<short quote>>}{close}{}{<<.>>}{\mkccitation{\cite[<<87>>]{<<key2>>}}}
\end{ltxcode}
%
where \cmd{cite} is the command selected with \cmd{SetCiteCommand} and \texttt{open}\slash\texttt{close} are internal macros which print the opening and closing quotation marks. Note that these internal macros are fully"=fledged markup elements with grouping and nesting control. They must be placed in the correct order, otherwise \sty{csquotes} will report errors about unbalanced groups or invalidly nested quotations. Since the \prm{text} should obviously be enclosed in the quotation marks, the parameter order |#1#2#3| is effectively fixed. The parameters |#4|, |#5|, |#6| may be placed freely.
\cmditem{mkblockquote}{text}{punct}{tpunct}{cite}
The \cmd{mkblockquote} hook controls the layout of all block quotations. This hook is used by \cmd{blockquote} and related commands from \secref{bas:blk:reg,bas:blk:bbl,int:blk:reg,int:blk:bbl} for long quotations. It takes four arguments which may be arranged according to the desired output:
\begin{argumentlist}{00}
\item[\#1] The \prm{text} argument of the command.
\item[\#2] The optional \prm{punct} argument of the command. If there is no \prm{punct} argument, this parameter is empty.
\item[\#3] Trailing \prm{tpunct} punctuation immediately after the command. If there is no such punctuation or if the \opt{autopunct} feature is disabled, this parameter is empty.
\item[\#4] The optional \prm{cite} argument of the command, wrapped in \cmd{mkcitation}. If there is no \prm{cite} argument, this parameter is empty. With integrated quotation commands, this parameter is the citation code, wrapped in \cmd{mkccitation}.
\end{argumentlist}
%
By default, \cmd{mkblockquote} inserts the \prm{cite} argument or the citation code immediately after the \prm{text} and adds any trailing \prm{tpunct} punctuation at the very end. This is equivalent to the following definition:
\begin{ltxcode}
\newcommand{<<\mkblockquote>>}[4]{#1#2#4#3}
\end{ltxcode}
\cmditem{mkbegdispquote}{punct}{cite}
\cmditem{mkenddispquote}{punct}{cite}
The \cmd{mkbegdispquote} and \cmd{mkenddispquote} hooks are used by \env{displayquote} and related environments from \secref{env:bas,env:cit}. These hooks take two arguments:
\begin{argumentlist}{00}
\item[\#1] The \prm{punct} argument passed to the \cmd{begin} line of the environment. If there is no \prm{punct} argument, this parameter is empty.
\item[\#2] The \prm{cite} argument passed to the environment, wrapped in \cmd{mkcitation}. If there is no \prm{cite} argument, this parameter is empty. With integrated quotation environments, this parameter is the citation code, wrapped in \cmd{mkccitation}.
\end{argumentlist}
%
By default, \cmd{mkenddispquote} adds the \prm{punct} argument as well as the \prm{cite} argument or the citation code at the very end of the quotation. \cmd{mkbegdispquote} does not insert anything be default. This is equivalent to the following definition:
\begin{ltxcode}
\newcommand{<<\mkbegdispquote>>}[2]{}
\newcommand{<<\mkenddispquote>>}[2]{#1#2}
\end{ltxcode}
%
See \secref{use:hok} for practical examples.
\end{ltxsyntax}
\subsection{Additional Tests in Quotation Hooks}
\label{cfg:tst}
The commands in this section increase the flexibility of the hooks discussed in \secref{cfg:aux}. For example, it may be desirable to adjust the format of a citation depending on the way the corresponding quotation is typeset. It may also be useful to known if the quotation ends with a punctuation mark.
\begin{ltxsyntax}
\cmditem{ifpunctmark}{character}{true}{false}
Expands to \prm{true} if preceeded by the punctuation mark \prm{character}, and to \prm{false} otherwise. The \prm{character} may be a period, a comma, a semicolon, a colon, an exclamation mark, or a question mark. Note that this test is only available in the definition of the hooks from \secref{cfg:aux}.
\cmditem{ifpunct}{true}{false}
Expands to \prm{true} if preceeded by any punctuation mark, and to \prm{false} otherwise. Note that this test is only available in the definition of the hooks from \secref{cfg:aux}.
\cmditem{ifterm}{true}{false}
Expands to \prm{true} if preceeded by a terminal punctuation mark (period, exclamation mark, or question mark), and to \prm{false} otherwise. Note that this test is only available in the definition of the hooks from \secref{cfg:aux}.
\cmditem{iftextpunctmark}{text}{character}{true}{false}
Executes \prm{true} if the \prm{text} ends with the punctuation mark \prm{character}, and to \prm{false} otherwise. The \prm{character} may be a period, a comma, a semicolon, a colon, an exclamation mark, or a question mark. This command is robust.
\cmditem{iftextpunct}{text}{true}{false}
Executes \prm{true} if the \prm{text} ends with any punctuation mark, and to \prm{false} otherwise. This command is robust.
\cmditem{iftextterm}{text}{true}{false}
Executes \prm{true} if the \prm{text} ends with a terminal punctuation mark (period, exclamation mark, or question mark), and to \prm{false} otherwise. This command is robust.
\cmditem{ifblockquote}{true}{false}
Expands to \prm{true} in all block and display quotations, and to \prm{false} otherwise.
\cmditem{ifblank}{string}{true}{false}
This generic command, which is provided by the \sty{etoolbox} package, expands to \prm{true} if the \prm{string} is blank (empty or spaces), and to \prm{false} otherwise. This is useful to test for an empty argument in the definition of the \cmd{mk...quote} commands. Note that this test is redundant in the definition of the citation hooks because they are only executed if there is a citation.
\csitem{unspace}
Removes preceding whitespace, \ie removes all skips and penalties from the end of the current horizontal list.
\end{ltxsyntax}
\subsection{Configuring Punctuation Look-Ahead}
\label{cfg:dap}
\begin{ltxsyntax}
\cmditem{DeclareAutoPunct}{characters}
This command defines the punctuation marks to be considered by the quotation commands as they scan ahead for punctuation. Note that \prm{characters} is an undelimited list of characters. Valid \prm{characters} are period, comma, semicolon, colon, exclamation and question mark. The default setting is:
\begin{lstlisting}[style=latex]{}
\DeclareAutoPunct{.,;:!?}
\end{lstlisting}
%
This definition is restored automatically whenever the \opt{autopunct} package option is set to \texttt{true}. Executing |\DeclareAutoPunctuation{}| is equivalent to setting \kvopt{autopunct}{false}, \ie it disables this feature.
\end{ltxsyntax}
\subsection{Configuring Ellipses}
\label{cfg:elp}
The appearance of ellipses and insertions formatted with the auxiliary commands from \secref{aux} is controlled by five hooks. When \cmd{textelp} is used with an empty argument (ellipsis only), it will execute \cmd{mktextelp}. When used with a non-empty \prm{text} argument (ellipsis and insertion), the \prm{text} will be passed as an argument to \cmd{mktextelpins}. The starred form will pass the \prm{text} to \cmd{mktextinselp} instead. These are the default definitions:
\begin{ltxcode}[showspaces=true]
\newcommand{<<\mktextelp>>}{[\textellipsis\unkern]}
\newcommand{<<\mktextelpins>>}[1]{[\textellipsis\unkern] [#1]}
\newcommand{<<\mktextinselp>>}[1]{[#1] [\textellipsis\unkern]}
\end{ltxcode}
%
The \cmd{textins} command passes its \prm{text} argument to \cmd{mktextins} for further processing. The starred variant of \cmd{textins} uses \cmd{mktextmod} instead. These are the default definitions:
\begin{ltxcode}[showspaces=true]
\newcommand{<<\mktextins>>}[1]{[#1]}
\newcommand{<<\mktextmod>>}[1]{[#1]}
\end{ltxcode}
%
You may redefine the above hooks to change the format of the printed output. For example, if you prefer replacements to be indicated by «[\textellipsis text]» rather than «[\textellipsis\unkern] [text]», redefine \cmd{mktextelpins} accordingly:
\begin{ltxcode}
\newcommand{<<\mktextelpins>>}[1]{[\textellipsis #1]}
\end{ltxcode}
%
The \cmd{unkern} in the default definitions is required because \cmd{textellipsis} adds asymmetric kerning by default. The kerning after the final dot is similar to the spacing between the dots, which is fine if \cmd{textellipsis} is followed by any text, but undesirable if it is enclosed in brackets.
\section{Usage Notes}
\label{use}
\subsection{Adding a New Quote Style}
\label{use:spl}
This section will give some comprehensive examples of how to define new quote styles. The examples presented here will only make use of the most basic components a quote style can be composed of. The main point is to illustrate the interaction of quote styles, variants, aliases, and package options. To get started, consider a simple house style which may be selected by way of the package option \opt{style} or the command \cmd{setquotestyle}:
\begin{ltxcode}
\DeclareQuoteStyle{house}
{\textquotedblleft}{\textquotedblright}
{\textquoteleft}{\textquoteright}
\end{ltxcode}
%
Now suppose that we wanted to add a quote style for an imaginary language called Newspeak and that there were two quote styles commonly used in Newspeak, an official one and an unofficial one. In this case, we need two backend styles implemented as variants of the \opt{newspeak} style, \opt{newspeak/official} and \opt{newspeak/unofficial}:
\needspace{3\baselineskip}
\begin{ltxcode}
\DeclareQuoteStyle[official]{newspeak}
{\textquotedblleft}{\textquotedblright}
{\textquoteleft}{\textquoteright}
\DeclareQuoteStyle[unofficial]{newspeak}
{\textquotedblright}{\textquotedblleft}
{\textquoteright}{\textquoteleft}
\end{ltxcode}
%
The official variant should be the default for this style. There is no need to copy the definition of the \opt{official} variant to accomplish that. We simply define an alias labeled \opt{newspeak} which points to the desired variant:
\begin{ltxcode}
\DeclareQuoteAlias[official]{newspeak}{newspeak}
\end{ltxcode}
%
The reason why we are using variants and aliases instead of two independent styles will become clear in a moment. Suppose that the \sty{babel} package offered support for Newspeak, but this language was known to \sty{babel} as \opt{otherspeak}:
\begin{ltxcode}
\DeclareQuoteAlias{newspeak}{otherspeak}
\end{ltxcode}
%
This is an example of a second"=level alias pointing to a first"=level alias. If the current language is \opt{otherspeak}, the above aliases will be expanded as follows:
\begin{lstlisting}[style=plain]{}
otherspeak = newspeak = newspeak/official
\end{lstlisting}
%
We also define a new package option to choose a style variant:
\begin{ltxcode}
\DeclareQuoteOption{newspeak}
\end{ltxcode}
%
This will add a new package option with a key called \opt{newspeak}. The value of this option may be any variant of the \opt{newspeak} style defined in the configuration file. In this example, there are two possible values: \opt{official} and \opt{unofficial}. To select the default or the alternative style for the entire document we use:
\begin{ltxcode}
\usepackage[style=newspeak]{csquotes}
\usepackage[style=newspeak,newspeak=unofficial]{csquotes}
\end{ltxcode}
%
To select the default or the alternative style with multilingual support we use:
\begin{ltxcode}
\usepackage[babel]{csquotes}
\usepackage[babel,newspeak=unofficial]{csquotes}
\end{ltxcode}
%
The base style must be implemented as an alias in this case since the \opt{newspeak} option will select a variant by redefining and thus overwriting the \opt{newspeak} alias. Since the \opt{otherspeak} alias points to \opt{newspeak} and not directly to any backend style, using the \opt{newspeak} option will also have the desired effect if multilingual support is enabled. Note that there are some style names which have a special meaning. See \secref{hnt:pre} for details.
\subsection{Using Quotation and Citation Hooks}
\label{use:hok}
Style guides for writers usually make detailed provisions concerning the formatting of quotations and citations, including rules dealing with punctuation placement. This section will discuss some typical usage scenarios, using hooks and other facilities introduced in \secref{cfg:aux,cfg:tst,cfg:dap}. In the examples below, we assume the following input:
\begin{ltxcode}
\textquote[citation][.]{This is a complete sentence}
\textquote[citation][]{This is an incomplete sentence}.
\end{ltxcode}
%
We start off with semantically strict punctuation placement, \ie terminal punctuation is enclosed in the quotation marks if it in fact is part of the quotation, and printed after the closing mark if it is not. Our first sample cases will use German quotation marks, but the formatting convention, as far as punctuation placement is concerned, is very common. We assume citations in footnotes in the first example, hence the desired output is as follows:
\begin{quotesample}{german}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2#4#3#5#6}
\item \textquote[citation][.]{This is a complete sentence}
\item \textquote[citation][]{This is an incomplete sentence}.
\end{quotesample}
%
This is accomplished by the following definitions:
\begin{ltxcode}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2#4#3#5#6}
\end{ltxcode}
%
In some cases, slightly different placement of the punctuation in the second line is requested:
\begin{quotesample}{german}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2#4#3#6#5}
\item \textquote[citation][.]{This is a complete sentence}
\item \textquote[citation][]{This is an incomplete sentence}.
\end{quotesample}
%
This is accomplished by the following definitions:
\begin{ltxcode}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2#4#3#6#5}
\end{ltxcode}
%
Let's switch to American quotation marks in the next examples. When using parenthetical citations, the terminal punctuation mark is typically placed at the very end of the entire sentence, after the closing quotation mark and the citation, even if it is part of the original quotation:
\begin{quotesample}{american}
\renewcommand{\mkcitation}[1]{ (#1)}
\renewcommand{\mktextquote}[6]{#1#2#3#6#4#5}
\item \textquote[citation][.]{This is a complete sentence}
\item \textquote[citation][]{This is an incomplete sentence}.
\end{quotesample}
%
This is accomplished by the following definitions:
\begin{ltxcode}
\renewcommand{\mkcitation}[1]{ (#1)}
\renewcommand{\mktextquote}[6]{#1#2#3#6#4#5}
\end{ltxcode}
%
The American quotation style is special in that it requires that a period or a comma immediately after a closing quotation mark be moved inside the quotes, even if it is not part of the original quotation. Given the above input (and assuming citations in footnotes in the next example), we need the same output in both cases:
\begin{quotesample}{american}
\DeclareAutoPunct{.,}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2#4#5#3#6}
\item \textquote[citation][.]{This is a complete sentence}
\item \textquote[citation][]{This is an incomplete sentence}.
\end{quotesample}
%
This is accomplished by the following definitions:
\begin{ltxcode}
\DeclareAutoPunct{.,}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2#4#5#3#6}
\end{ltxcode}
%
The style usually seen in French books uses semantically strict punctuation placement. The unusual aspect of this style is the footnote mark, which is placed inside the quotes, before the terminal punctuation mark. Given the above input, we need the following output:
\begin{quotesample}{french}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2#6#4#3#5}
\item \textquote[citation][.]{This is a complete sentence}
\item \textquote[citation][]{This is an incomplete sentence}.
\end{quotesample}
%
This is accomplished by the following definitions:
\begin{ltxcode}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2#6#4#3#5}
\end{ltxcode}
%
Note that the spacing of the quotation marks, as is common in French typography, is handled by the quote style, as defined with \cmd{DeclareQuoteStyle}. There is no need to deal with these details when redefining \cmd{mktextquote} and similar hooks. \cmd{mktextquote} will get the quotation marks plus spacing as parameters |#1| and |#3|.
In addition to language and style specific adaptions, the formatting hooks discussed in this section may be used to further automate the quoting process. For example, they can be configured to automatically insert an ellipsis mark when quoting a sentence truncated at the end. In the following examples, we assume these lines of input:
\begin{ltxcode}
\textquote[citation]{This is an incomplete sentence}.
\textquote[citation][.]{This is a complete sentence}
\textquote[citation]{This is a complete sentence.}
\end{ltxcode}
%
Let's assume American-style quotes combined with citations in footnotes. Our previous definition for that was:
\begin{ltxcode}
\renewcommand{\mktextquote}[6]{#1#2#4#5#3#6}
\end{ltxcode}
%
We need to insert an ellipsis mark if the \prm{punct} argument of the citation command is empty, or if it was omitted. Parameter |#4| will be blank in both cases:
\begin{ltxcode}
\renewcommand{\mktextquote}[6]{%
#1#2<<\ifblank{#4}{ \textelp{}}{#4}>>#5#3#6}
\end{ltxcode}
%
Given the above input, this definition will yield the following output:
\begin{quotesample}{american}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2\ifblank{#4}{ \textelp{}}{#4}#5#3#6}
\item \textquote[citation]{This is an incomplete sentence}.
\item \textquote[citation][.]{This is a complete sentence}
\item \textquote[citation]{This is a complete sentence.}
\end{quotesample}
%
The first two cases are handled correctly but the third one needs more tuning: there is a spurious
ellipsis because the final period is not passed to \cmd{textquote} as a separate \prm{punct} argument, but included in the quotation. To fix that, we add an additional \cmd{ifpunct} test to check if the quoted text ends with a punctuation mark, and omit the ellipsis if this is the case:
\begin{ltxcode}
\renewcommand{\mktextquote}[6]{%
#1#2<<\ifblank{#4}{\ifpunct{}{ \textelp{}}}{#4}>>#5#3#6}
\end{ltxcode}
%
The \cmd{ifpunct} test will check the last character in |#2| and omit the ellipsis when detecting a punctuation mark. This yields the desired output:
\begin{quotesample}{american}
\renewcommand{\mkcitation}[1]{\footnote{#1}}
\renewcommand{\mktextquote}[6]{#1#2\ifpunct{}{\ifblank{#4}{ \textelp{}}}{#4}#5#3#6}
\item \textquote[citation]{This is an incomplete sentence}.
\item \textquote[citation][.]{This is a complete sentence}
\item \textquote[citation]{This is a complete sentence.}
\end{quotesample}
%
When using automated citations, it is convenient to employ the integrated quotations commands from \secref{cit}. For example, instead of this:
\begin{ltxcode}
\textquote[{\cite[55]{key}}][.]{This is a complete sentence}
\textquote[{\cite[55]{key}}][]{This is an incomplete sentence}.
\end{ltxcode}
%
you might use \cmd{textcquote} instead of \cmd{textquote}:
\begin{ltxcode}
\textcquote[55]{key}[.]{This is a complete sentence}
\textcquote[55]{key}{This is an incomplete sentence}.
\end{ltxcode}
%
The definition of the punctuation hooks is the same in both cases, but citations are set up in a slightly different way in the second case, using \cmd{mkccitation} instead of \cmd{mkcitation}. The first line in the example below is intended for parenthetical citations, the second one for citations in footnotes:
\begin{ltxcode}
\renewcommand{\mkccitation}[1]{ (#1)}
\renewcommand{\mkccitation}[1]{\footnote{#1}}
\end{ltxcode}
%
Advanced citation packages such as \sty{natbib} and \sty{biblatex} provide dedicated commands for various types of citations. In this case, it is advisable to hand control of citations to the respective citation command, \ie \cmd{mkccitation} is used to add spacing where required but will leave the rest to the citation command. In the case of \sty{biblatex}, this boils down to:
\begin{ltxcode}
\SetCiteCommand{\parencite}
\renewcommand{\mkccitation}[1]{ #1}
\end{ltxcode}
%
for parenthetical citations and
\begin{ltxcode}
\SetCiteCommand{\footcite}
\renewcommand{\mkccitation}[1]{#1}
\end{ltxcode}
%
for citations in footnotes.
\section{Hints and Caveats}
\label{hnt}
\subsection{Input Encodings}
\label{hnt:inc}
The active quotes provided by this package may depend on or benefit from the \sty{inputenc}\slash \sty{inputenx} packages under certain circumstances. As long as the active quotes are in the range 0--127, there is no benefit in loading \sty{inputenc}. If you are using an 8-bit input encoding such as \opt{latin1}, \sty{inputenc} is required for the quotes to function properly in a \emph{verbatim} context. It should therefore be loaded before any active quotes are allocated (not necessarily before \sty{csquotes} is loaded). The macro-level \utf support of this package builds on the \path{utf8} module of the inputenc package. When using this encoding, make sure that inputenc is loaded with the \opt{utf8} option. Do not use the \opt{utf8x} option as this would implicitly load the \sty{ucs} package which is not supported by \sty{csquotes}. \utf encoding will be detected automatically. All commands discussed in \secref{act} work as usual with this encoding. \xetex and \luatex in native \utf mode will also work as expected and do not require any additional packages. See also \secref{hnt:mod}.
\subsection{Output Encodings}
\label{hnt:onc}
The \acr{OT1} font encoding, which is the default output encoding of \latex, merely includes the quotation marks used in English. You will need an encoding like \acr{T1}, \acr{LY1}, or, with \xetex or \luatex, \acr{EU1}\slash \acr{EU2}, in order to get guillemets or baseline quotation marks. This package deliberately refrains from providing any workarounds for the \acr{OT1} legacy encoding. If you need \acr{T1} or some other extended encoding for some of the quotation marks, you will most likely need it anyway to get proper hyphenation for the respective language. See \tabref{tab:out} for a list of common quotation marks included in \acr{T1}, \acr{LY1}, and \acr{EU1}\slash \acr{EU2}.
\begin{table}
\tablesetup
\begin{tabularx}{\columnwidth}{@{}X@{}p{60pt}X@{}p{60pt}@{}}
\toprule
\multicolumn{2}{@{}H}{Double quotation marks} & \multicolumn{2}{H}{Single quotation marks} \\
\cmidrule(r){1-2}\cmidrule(l){3-4}
\multicolumn{1}{@{}H}{Command} & \multicolumn{1}{@{}H}{Example} &
\multicolumn{1}{H}{Command} & \multicolumn{1}{@{}H}{Example} \\
\cmidrule(r){1-1}\cmidrule(r){2-2}\cmidrule(lr){3-3}\cmidrule{4-4}
\cmd{textquotedblleft} & \mksample[60]{\textquotedblleft} &
\cmd{textquoteleft} & \mksample[60]{\textquoteleft} \\
\cmd{textquotedblright} & \mksample[60]{\textquotedblright} &
\cmd{textquoteright} & \mksample[60]{\textquoteright} \\
\cmd{quotedblbase} & \mksample[60]{\quotedblbase} &
\cmd{quotesinglbase} & \mksample[60]{\quotesinglbase} \\
\cmd{guillemotleft} & \mksample[60]{\guillemotleft} &
\cmd{guilsinglleft} & \mksample[60]{\guilsinglleft} \\
\cmd{guillemotright} & \mksample[60]{\guillemotright} &
\cmd{guilsinglright} & \mksample[60]{\guilsinglright} \\
\bottomrule
\end{tabularx}
\caption[Quotation Marks]{Quotation Marks Provided by Encodings \acr{T1}, \acr{LY1}, \acr{EU1}\slash \acr{EU2}}
\label{tab:out}
\end{table}
\subsection{Valid Active Quotes}
\label{hnt:val}
In general, an active quote may be any single character with category code 12 or 13, or any multibyte \utf sequence representing a single character. However, there are a few exceptions: numbers, punctuation marks, the apostrophe, the hyphen, and all characters which are part of the \latex syntax are rejected. In sum, the following characters will be considered as reserved by this package: \texttt{A--Z a--z 0--9 . , ; : ! ? \textquotesingle\ - \# \$ \% \& \textasciicircum\ \textunderscore\ \textasciigrave\, \textasciitilde\ \textbackslash\ @ \textasteriskcentered\ \{ \} [ ]}
\subsection{Invalid Nesting and Unbalanced Active Quotes}
\label{hnt:grp}
Every quotation forms a group which includes both the quoted piece of text and the quotation marks. This package tracks the nesting level of all quotations and thus allows for basic validation. If quotations are nested in an invalid way, it will issue an error message. Keep in mind that the active quotes are more than a convenient way to enter quotation marks. They are fully"=fledged markup elements which imply grouping as well, hence they must always be balanced and must not interfere with other group boundaries. This package ensures that an error is triggered if quotes are unbalanced or nested in an invalid way. However, note that packages generally can not catch low-level errors caused by grouping mistakes, nor do they have any control over the wording of generic error messages.
\subsection{Active Quotes in Special Contexts}
\label{hnt:mod}
All quotation commands are designed for use in text mode and will issue an error message in math mode. Note that all active quotes retain their original function in math mode. It is perfectly possible to use a character like the greater-than symbol as an active quote without interfering with math mode. In a verbatim context, the active quotes will normally be disabled. If a character is in the range 128--255, its original function is restored so that the \sty{inputenc}\slash \sty{inputenx} package may handle it in verbatim environments. This feature is available with the standard verbatim environments and with those provided by (or defined via) the packages \sty{verbatim}, \sty{fancyvrb}, \sty{moreverb}, and \sty{alltt}. This also applies to the \cmd{verb} command and the \sty{shortvrb} package. The \sty{listings} package provides dedicated support for extended input encodings. When using this package, activate its <extended characters> option and specify the input encoding. Note that \sty{listings} does not support macro-level \utf decoding. \xetex and \luatex in native \utf mode will work as expected and do not require the \sty{inputenc} package.
Some care is still required when choosing active quotes. Note that you normally cannot use active characters in the argument to commands expecting a string of characters, such as \cmd{input}, \cmd{label}, or \cmd{cite}. There are two packages which try to remedy this situation: the \sty{babel} package and the \sty{underscore} package (when loaded with the \opt{strings} option). Both packages redefine several standard commands affected by this general problem. If any one of these packages is loaded, \sty{csquotes} will take advantage of all improvements automatically. Unfortunately, both packages patch a different set of commands and neither one covers all possibly vulnerable commands.
\subsection{\pdf Strings and \sty{hyperref} Support}
\label{hnt:pdf}
This package interfaces with the \sty{hyperref} package as \pdf strings such as bookmarks are generated. See \secref{cfg:pdf} on how to configure the quotation marks used in \pdf strings. Support for \pdf strings is only available with the basic facilities presented in \secref{bas:reg,bas:bbl} as well as \secref{act:reg,act:bbl}. Be advised that the way \sty{hyperref} builds \pdf strings imposes severe limitations on the capabilities of all commands. Most notably, the nesting level of quotations cannot be tracked in this context. Nested quotations will generally get outer marks, but you may use starred commands or active inner quotes to request inner marks explicitly. If quotation marks are to be included in the document properties of a \pdf file, you must use \cmd{hypersetup} to specify the strings. The replacement mechanism will not function within the optional argument to \cmd{usepackage}. See the \sty{hyperref} manual for information further details.
\subsection{Footnotes Inside Quotations}
\label{hnt:ftn}
This package will automatically reset the nesting level within any footnote included in a quotation. If the \sty{babel} or the \sty{polyglossia} package has been loaded, it will also reset the language. The language of the footnote text including the hyphenation patterns will match the language of the text surrounding the quotation. This applies to parboxes, minipages, and floats as well.
\subsection{Using \sty{csquotes} with \sty{babel}'s Shorthands}
\label{hnt:bbl}
The commands discussed in \secref{aut:amk} may be combined with the shorthands of the \sty{babel} package such that \sty{babel} provides the user interface and \sty{csquotes} the backend.\footnote{This also applies to the \sty{polyglossia} package if the \opt{babelshorthands} option of \sty{polyglossia} has been enabled.} For example, the German module of the \sty{babel} package defines, amongst other things, the shorthands |"`| and |"'|. Such shorthands are input aids, \ie physical markup elements with a fixed definition. Typing |"`| is a short way of saying |\quotedblbase| but it is not different in concept. These shorthands can be transformed into <smart quotes> which behave like \cmd{enquote}. Here is a simple ad~hoc solution suitable for documents with only one language:
\begin{ltxexample}[upquote]
\documentclass{...}
\usepackage[german]{babel}
\usepackage[babel=once]{csquotes}
\defineshorthand{"`}{\openautoquote}
\defineshorthand{"'}{\closeautoquote}
\end{ltxexample}
%
It is possible to move such definitions to \path{csquotes.cfg}. In this case, the code is slightly more complex because it needs to be more general:
\begin{ltxcode}[upquote]
\AtEndPreamble{%
\@ifpackageloaded{babel}
{\iflanguage{german}
{\declare@shorthand{german}{"`}{\openautoquote}%
\declare@shorthand{german}{"'}{\closeautoquote}}
{}}
{}}
\end{ltxcode}
%
This code redefines the shorthands only if the \sty{babel} package has been loaded and the main language is \texttt{german}. Note that \sty{babel}'s shorthands are language"=specific. The way they are configured and handled is technically and conceptually different from the active quotes discussed in \secref{act}. Active quotes are defined globally and automatically adapt to the current language. With \sty{babel}, each language has its own set of shorthands. Also note that \sty{babel} uses \cmd{AtBeginDocument} to initialize the main document language, including the corresponding shorthands. We use \cmd{AtEndPreamble} to defer the code until the end of the preamble. This way, we can be sure that \sty{babel} has been loaded but that the main document language has not been initialized yet. See the \sty{babel} manual for further details. The \cmd{AtEndPreamble} command is provided by the \sty{etoolbox} package.
\subsection{Miscellaneous Notes about the Predefined Styles}
\label{hnt:pre}
There are three styles which serve a special purpose: \opt{default}, \opt{fallback}, and \opt{debug}. The \opt{default} style is a dynamic alias pointing to the default quote style used if the multilingual interface is not enabled. The package option \opt{style} and the command \cmd{setquotestyle} will redefine this alias. The \opt{fallback} style is a backend style used as a fallback whenever the multilingual interface is enabled but there is no matching quote style for the current language. It will print bold question marks by default. The \opt{debug} style will not print quotation marks but the current quote level as a bold number. This style may be selected using the \opt{style} option or the \cmd{setquotestyle} command. It is intended for debugging only.
All variants of the \opt{french} style use spaced out guillemets as outer marks. The style variant \opt{quotes} uses double quotes as inner marks. The starred variant \opt{quotes*} is similar to its regular counterpart except that it will also space out the inner marks. The \opt{guillemets} variant employs spaced out guillemets on all levels. It will also insert guillemets at the beginning of every paragraph inside a quotation spanning multiple paragraphs. In addition to that, two adjoining marks at the end of a quotation are replaced by a single one; if two nested quotations end simultaneously, the second closing mark is omitted automatically. The starred variant \opt{guillemets*} is similar to its regular counterpart, differing only in the middle mark inserted at the beginning of every paragraph. The regular variant uses a left"=pointing guillemet whereas the starred one uses a right"=pointing one.
\section{Author Interface}
\label{aut}
The following sections discuss the programmer interface to the \sty{csquotes} package as well as some details of the implementation. They are intended for class and package authors who want to interface with this package.
\subsection{Controlling Active Quotes}
\label{aut:ctl}
The author commands in this section behave essentially like the corresponding user commands discussed in \secref{act:ctl}. The only difference is that they work quietly behind the scenes without writing any notices to the transcript file. The scope of these commands is local so that all changes may be confined to a group. Note that the active quotes are enabled at the beginning of the document body. Under no circumstances will this package make any characters active in the document preamble. You will only need the following commands when dealing with active quotes at the beginning of or in the document body.
\begin{ltxsyntax}
\csitem{@enablequotes}
This command enables all characters allocated as active quotes. It also restores their definitions if they were disabled or accidentally overwritten. With single-byte encodings, this command (re)defines all allocated characters and makes them active. With \utf encoding, it redefines the internal macro used by the \sty{inputenc} package to typeset the respective \utf sequence (\cmd{u8:}\prm{character}). \utf characters in the range 0--127 are handled as with single-byte encodings. When using a \tex engine with native \utf support, such as \xetex, all characters are handled as with single-byte encodings.
\csitem{@disablequotes}
This command restores the \emph{status quo ante} of all active quotes. With single-byte encodings, there are two possible cases. (1) If a character had already been active when it was allocated as active quote, its former definition is restored. (2) If a character had not been active when it was allocated, its former category code is restored. With \utf encoding, this command restores the former definition of the internal macro used by the \sty{inputenc} package to typeset the respective \utf sequence. \utf characters in the range 0--127 are handled as with single-byte encodings. When using a \tex engine with native \utf support, such as \xetex, all characters are handled as with single-byte encodings.
\csitem{@verbatimquotes}
For verbatim environments and similar applications, use this command rather than \cmd{@disablequotes}. It redefines the active quotes in a way that is better suited for verbatim typesetting. With single-byte encodings, it will do one of the following things. (1) If a character is in the range 0--127, it is redefined such that it expands to itself with category code 12. (2) If a character is in the range 128--255, there are two possibilities. (a) If it had already been active when it was allocated, its former definition is restored. (b) If it had not been active before, it is redefined such that it expands to itself with its former category code.
Characters in the range 0--127 are added to the \cmd{dospecials} list. Characters in the range 128--255 remain active, permitting the \sty{inputenc} package to typeset them verbatim (due to case 2a, which implies that you must load \sty{inputenc} before allocating active quotes). Case 2b is usually undesirable in verbatim environments. If \sty{inputenc} is loaded, however, this should not happen. With \utf encoding, this command restores the former definition of the internal macro used by the \sty{inputenc} package to typeset the respective \utf sequence. \utf characters in the range 0--127 are handled as with single-byte encodings. When using a \tex engine with native \utf support, such as \xetex, all characters are handled as with single-byte encodings.
Due to case 1, \cmd{@verbatimquotes} itself is independent of any \cmd{dospecials} processing. You may typeset all active quotes verbatim by using this command exclusively. The advantage of this approach is that it does not require any category code changes, hence this command may also be used to modify an argument after it has been read. Also note that the standard \latex verbatim environments as well as all environments provided by or defined via the packages \sty{verbatim}, \sty{fancyvrb}, \sty{moreverb}, and \sty{alltt} are catered for automatically. This also applies to the \cmd{verb} command and the \sty{shortvrb} package.
\csitem{@deletequotes}
This command implicitly executes \cmd{@disablequotes} and deallocates all active quotes, which results in a complete reset of all active quotes so that they may be newly defined. This command should be used with care because the reset is not visible to the user. Using \cmd{DeleteQuotes} may be preferable.
\end{ltxsyntax}
\subsection{Active Quotes in a Strings-Only Context}
\label{aut:str}
A possible problem with active characters are strings-only contexts, i.\,e. cases in which an active character is used in the formation of a control sequence name. A typical example is the \cmd{label} command which expects a string of characters. Any active character may break \cmd{label} when used in its argument. There are two packages which try to remedy this situation, albeit in different ways: \sty{babel} and \sty{underscore}.
The \sty{babel} package defines the switch \cmd{if@safe@actives} and patches several standard commands such that the switch is set to \opt{true} while they process their arguments. The approach taken by the \sty{underscore} package is slightly different. If \sty{underscore} is loaded with the \opt{strings} option, it patches several commands such that \cmd{protect} is equivalent to \cmd{string} while the arguments are processed. If any one of these packages is loaded, \sty{csquotes} will take advantage of that automatically. Unfortunately, both packages patch a different set of commands and neither one covers all possibly vulnerable commands. If \sty{babel} is loaded, for example, you may use active quotes in the argument of \cmd{label}, but not in the argument of \cmd{input}. If you load \sty{underscore} with its \opt{strings} option, active quotes may also be used in the argument of \cmd{input}.
When writing a package which may have to process user"=supplied arguments in a strings-only context, there are two ways to deal with active quotes. Taking the approach of the \sty{babel} package, you may do the following:
\begin{ltxcode}
\let\if@safe@actives\iftrue
\end{ltxcode}
%
This is best done in a group. If grouping is not feasible, you must ensure that the switch is properly restored. In contrast to using \cmd{@safe@activestrue}, this approach works even if \sty{babel} is not loaded. However, note that you must take three states into account when restoring the switch in this case: true, false, and undefined. Taking the approach of the \sty{underscore} package, you may also do the following:
\begin{ltxcode}
\let\@@protect\protect \let\protect\string
\end{ltxcode}
%
This could either be done in a group or without any grouping, but followed by \cmd{restore@protect}. The first approach works with the active characters of the \sty{babel} and the \sty{underscore} packages. The second one works with the \sty{underscore} and the \sty{at} packages. Unfortunately, the active characters of the \sty{inputenc} package support neither of the above"=mentioned techniques. As far as \sty{csquotes} is concerned, it does not matter which approach you take. In both cases all active quotes expand to themselves with category code 12. With macro-level \utf support, \utf encoded active quotes expand to a string of characters with category code 12. This string will be valid \utf. In a verbatim \cmd{write} operation, you should employ one of the techniques discussed in this section rather than \cmd{@verbatimquotes}, which is geared to verbatim typesetting.
\subsection{Block Quotations}
\label{aut:blk}
The block quotation facilities need to typeset all quotations twice. The first pass is required to measure the length of the quotation. The actual typesetting takes place on the second pass, in a format depending on the result of the first one. In order to prevent any side"=effects of the first (trial) pass, the \sty{csquotes} package (1) performs the first pass inside a group, (2) employs checkpointing to freeze all LaTeX counters, and (3) sets \cmd{if@filesw} to \texttt{false}. However, it can not prevent side"=effects caused by commands that (1) make any global assignments which are not overwritten on the second pass (for example, by way of \cmd{g@addto@macro}), (2) increment counters globally in a way that circumvents \latex's counter commands, or (3) do not check \cmd{if@filesw} every time they are about to write to an auxiliary file. If you observe any malfunctions related to the trial pass (for example, if counters are incremented twice or if an item appears twice in a list), use \cmd{BlockquoteDisable} to redefine or disable the affected command temporarily.
\begin{ltxsyntax}
\cmditem{BlockquoteDisable}{code}
The \prm{code} may be arbitrary \latex code which redefines vulnerable commands locally such that they work differently during the trial pass. The \prm{code} itself should obviously not include any global assignments. This solution should be considered as a last ressort but may be the quickest way to fix a vulnerable package. Note that there is no need to escape parameter characters by doubling them in the \prm{code} argument. Simply use this command like \cmd{AtBeginDocument} and similar hooks.
\end{ltxsyntax}
\subsection{Registering Quotation Marks}
\label{aut:sfc}
In order to track punctuation marks inside quotations, this package requires that all quotation marks be transparent to the space factor, \ie that they have a space factor code of zero. This setting is specific to the output encoding. Settings for the encodings \acr{OT1}, \acr{OT2}, \acr{OT4}, \acr{T1}, \acr{LY1}, \acr{LGR}, \acr{T2A}, \acr{T2B}, \acr{T2C}, \acr{LCY}, \acr{X2}, and \acr{EU1}\slash \acr{EU2} are provided by default. Other encodings may be set up in the configuration file using the following command:
\begin{ltxsyntax}
\cmditem{DeclareQuoteGlyph}{encoding}{position}
\end{ltxsyntax}
The \prm{encoding} is the name of the output encoding, for example \texttt{OT1}. This string corresponds to the identifiers used when loading the \sty{fontenc} package. The \prm{position} argument is an integer indicating the position of a glyph in this encoding. You need to reqister all quotation marks in the output encoding, using one declaration for each glyph. As an example, these are the settings for the \acr{OT1} encoding:
\begin{ltxcode}
\DeclareQuoteGlyph{OT1}{34}% = \textquotedblright
\DeclareQuoteGlyph{OT1}{39}% = \textquoteright
\DeclareQuoteGlyph{OT1}{92}% = \textquotedblleft
\DeclareQuoteGlyph{OT1}{96}% = \textquoteleft
\end{ltxcode}
%
The \prm{position} argument may use any notation accepted by \tex in integer assignments, \eg \texttt{171} in decimal or \verb+"00AB+ in hexadecimal notation. See the settings in \path{csquotes.def} for further examples. The advantage of registering quotes with the above command (rather than adjusting the space factor codes globally) is that the declarations are only used locally inside quotations and will not affect any other part of the document.
\subsection{Automatic Quotation Marks}
\label{aut:amk}
The commands in this section provide access to the automatic quotation facilities at a slightly lower level than the user commands in \secref{bas:reg,act:reg}. In contrast to the commands discussed in \secref{aut:imk}, the facilities in this section are fully"=fledged markup elements which verify the nesting level and issue an error if quotations are nested in an invalid way. They form groups and must always be balanced, see \secref{hnt:grp} for details. In other words, the facilities in this section are semantic markup elements, the ones in \secref{aut:imk} are physical markup elements.
\begin{ltxsyntax}
\csitem{openautoquote} Opens a nestable quotation.
\csitem{closeautoquote} Closes a nestable quotation.
\end{ltxsyntax}
In terms of their function, the above commands correspond to the regular versions of \cmd{enquote} and \cmd{MakeAutoQuote}. The following commands correspond to the starred variants \cmd{enquote*} and \cmd{MakeAutoQuote*}:
\begin{ltxsyntax}
\csitem{openinnerquote} Opens an inner quotation.
\csitem{closeinnerquote} Closes an inner quotation.
\end{ltxsyntax}
The above commands may be used to implement an alternative user interface. For example, you can combine them with the shorthands of the \sty{babel} package such that \sty{babel} provides the user interface and \sty{csquotes} the backend. See \secref{hnt:bbl} and the \sty{babel} manual for details.
\subsection{Internal Quotation Marks}
\label{aut:imk}
The commands in this section print the quotation marks of the current style, as defined with \cmd{DeclareQuoteStyle}, without any grouping or nesting control. The quotation marks reflect all changes to the quotation style. If the multilingual interface is enabled, they are also synced with the current language.
\begin{ltxsyntax}
\csitem{textooquote} Prints the opening outer quotation mark of the currently active quote style.
\csitem{textcoquote} Prints the closing outer quotation mark.
\csitem{textmoquote} Prints the middle outer quotation mark.
\csitem{textoiquote} Prints the opening inner quotation mark.
\csitem{textciquote} Prints the closing inner quotation mark.
\csitem{textmiquote} Prints the middle inner quotation mark.
\end{ltxsyntax}
Note that the initialization hooks for the respective quotation style are not executed automatically. They may be accessed separately:
\begin{ltxsyntax}
\csitem{initoquote} Executes the outer initialization hook.
\csitem{initiquote} Executes the inner initialization hook.
\end{ltxsyntax}
The scope of these hooks must always be confined to a group.
\section{Revision History}
This revision history is a list of changes relevant to users of this package. Changes of a more technical nature which do not affect the user interface or the behavior of the package are not included in the list. If an entry in the revision history states that a feature has been \emph{extended}, this indicates a syntactically backwards compatible modification, such as the addition of an optional argument to an existing command. Entries stating that a feature has been \emph{modified}, \emph{renamed}, or \emph{removed} demand attention. They indicate a modification which may require changes to existing documents in some, hopefully rare, cases. The \opt{version} option from \secref{opt:opt} may be helpful in this case. The numbers on the right indicate the relevant section of this manual.
\begin{changelog}
\begin{release}{5.1e}{2015-04-15}
\item New maintainer: Joseph Wright
\item Update \opt{danish} quote styles
\end{release}
\begin{release}{5.1d}{2011-10-22}
\item Slightly modified quote style \opt{italian/guillemets}
\item Made variant \opt{guillemets} the default for \opt{italian}
\item Fixed \xetex kerning issue
\end{release}
\begin{release}{5.1c}{2011-03-25}
\item Fixed spurious language reset
\end{release}
\begin{release}{5.1b}{2011-01-20}
\item Fixed conflict with \sty{polyglossia} and French
\end{release}
\begin{release}{5.1a}{2011-01-11}
\item Fix for paragraph environments inside \cmd{blockquote}
\end{release}
\begin{release}{5.1}{2010-11-19}
\item Added package option \opt{debug}\see{opt:opt}
\item Added package option \opt{threshold}\see{opt:opt}
\item Added package option \opt{thresholdtype}\see{opt:opt}
\item Added package option \opt{parthreshold}\see{opt:opt}
\item Added package option \opt{splitcomp}\see{opt:opt}
\item Improved block quotation facilities\see{bas:blk:reg}
\item Improved integrated block quotation facilities\see{int:blk:reg}
\end{release}
\begin{release}{5.0c}{2010-09-21}
\item Added package option \opt{csdisplay}\see{opt:opt}
\item Added \cmd{csdisplaytrue} and \cmd{csdisplayfalse}\see{opt:opt}
\item Fixed conflict with \sty{polyglossia}
\end{release}
\begin{release}{5.0b}{2010-08-06}
\item Fixed issue with \sty{babel}'s active punctuation marks
\end{release}
\begin{release}{5.0a}{2010-06-09}
\item Fixed bug related to middle quote marks
\item Minor internal update for biblatex
\end{release}
\begin{release}{5.0}{2010-06-02}
\item Renamed package option \opt{babel} to \opt{autostyle}\see{opt:opt}
\item Added compatibility option \opt{babel}\see{opt:leg}
\item Added preliminary \sty{polyglossia} interface\see{opt:opt}
\item Added package option \opt{autopunct}\see{opt:opt}
\item Added package option \opt{version}\see{opt:leg}
\item Added \cmd{hybridblockquote}\see{bas:blk:bbl}
\item Added \cmd{hybridblockcquote}\see{int:blk:bbl}
\item Added \cmd{MakeHybridBlockQuote}\see{act:blk:bbl}
\item Added \cmd{DeclareAutoPunct}\see{cfg:dap}
\item Extended \cmd{textquote}\see{bas:txt:reg}
\item Extended \cmd{foreigntextquote}\see{bas:txt:bbl}
\item Extended \cmd{hyphentextquote}\see{bas:txt:bbl}
\item Extended \cmd{blockquote}\see{bas:blk:reg}
\item Extended \cmd{foreignblockquote}\see{bas:blk:bbl}
\item Extended \cmd{hyphenblockquote}\see{bas:blk:bbl}
\item Extended \cmd{textcquote}\see{int:txt:reg}
\item Extended \cmd{foreigntextcquote}\see{int:txt:bbl}
\item Extended \cmd{hyphentextcquote}\see{int:txt:bbl}
\item Extended \cmd{blockcquote}\see{int:blk:reg}
\item Extended \cmd{foreignblockcquote}\see{int:blk:bbl}
\item Extended \cmd{hyphenblockcquote}\see{int:blk:bbl}
\item Added \cmd{mktextquote}\see{cfg:aux}
\item Added \cmd{mkblockquote}\see{cfg:aux}
\item Added \cmd{mkbegdispquote}\see{cfg:aux}
\item Added \cmd{mkenddispquote}\see{cfg:aux}
\item Removed \cmd{mkpretextpunct}\see{cfg:aux}
\item Removed \cmd{mkmidtextpunct}\see{cfg:aux}
\item Removed \cmd{mkfintextpunct}\see{cfg:aux}
\item Removed \cmd{mkpreblockpunct}\see{cfg:aux}
\item Removed \cmd{mkmidblockpunct}\see{cfg:aux}
\item Removed \cmd{mkfinblockpunct}\see{cfg:aux}
\item Removed \cmd{mkpredisppunct}\see{cfg:aux}
\item Removed \cmd{mkmiddisppunct}\see{cfg:aux}
\item Removed \cmd{mkfindisppunct}\see{cfg:aux}
\item Removed \cmd{quotetext}\see{cfg:aux}
\item Removed \cmd{quoteblock}\see{cfg:aux}
\item Renamed \cmd{ifquotepunct} to \cmd{ifpunct}\see{cfg:tst}
\item Renamed \cmd{ifquoteterm} to \cmd{ifterm}\see{cfg:tst}
\item Added \cmd{ifpunctmark}\see{cfg:tst}
\item Removed \cmd{ifquotecolon}\see{cfg:tst}
\item Removed \cmd{ifquotecomma}\see{cfg:tst}
\item Removed \cmd{ifquoteexclam}\see{cfg:tst}
\item Removed \cmd{ifquoteperiod}\see{cfg:tst}
\item Removed \cmd{ifquotequestion}\see{cfg:tst}
\item Removed \cmd{ifquotesemicolon}\see{cfg:tst}
\item Renamed \cmd{ifstringblank} to \cmd{ifblank}\see{cfg:tst}
\item Added \cmd{unspace}\see{cfg:tst}
\item Added \cmd{textelp}\see{aux}
\item Added \cmd{textins}\see{aux}
\item Added \cmd{mktextelp}\see{cfg:elp}
\item Added \cmd{mktextelpins}\see{cfg:elp}
\item Added \cmd{mktextinselp}\see{cfg:elp}
\item Added \cmd{mktextins}\see{cfg:elp}
\item Added \cmd{mktextmod}\see{cfg:elp}
\item Expanded documentation\see{use:hok}
\item Removed \cmd{cquote} legacy alias (use \cmd{textcquote} instead)\see{int:txt:reg}
\item Removed \cmd{foreigncquote} legacy alias (use \cmd{foreigntextcquote})\see{int:txt:bbl}
\item Removed \cmd{hyphencquote} legacy alias (use \cmd{hyphentextcquote})\see{int:txt:bbl}
\item Removed \cmd{RestoreQuotes} legacy alias (use \cmd{EnableQuotes})\see{act:ctl}
\item Removed \cmd{@restorequotes} legacy alias (use \cmd{@enablequotes})\see{aut:ctl}
\item Added support for \acr{EU2} encoding\see{aut:sfc}
\item Added quote style for Croatian
\end{release}
\begin{release}{4.4d}{2010-02-06}
\item Added quote style for Dutch
\item Added quote style for Finnish
\item Added quote style for Greek
\item Added support for \acr{LGR} encoding\see{aut:sfc}
\end{release}
\begin{release}{4.4c}{2009-09-23}
\item Fixed incompatibility with \sty{inputenx} package
\end{release}
\begin{release}{4.4b}{2009-07-24}
\item Fixed bug in glyph declarations for \acr{EU1} encoding\see{aut:sfc}
\end{release}
\begin{release}{4.4a}{2009-07-04}
\item Added support for \acr{EU1} encoding\see{aut:sfc}
\item Added quote style for Brazilian
\item Added preliminary quote style for Portuguese
\end{release}
\begin{release}{4.4}{2009-05-30}
\item Added package option \opt{maxlevel}\see{opt:opt}
\item Added special quote style \opt{debug}\see{hnt:pre}
\item Added \cmd{DeclareQuoteGlyph}\see{aut:sfc}
\item Added support for \acr{OT2} and \acr{OT4} encodings\see{aut:sfc}
\item Added support for \acr{T2A}, \acr{T2B}, \acr{T2C}, \acr{X2} encodings\see{aut:sfc}
\item Added support for \acr{LCY} encoding\see{aut:sfc}
\item Added quote style for Russian
\end{release}
\begin{release}{4.3}{2008-11-23}
\item Made package option \kvopt{babel}{tryonce} the default setting\see{opt:opt}
\item Internal updates for \path{biblatex} package
\end{release}
\begin{release}{4.2}{2008-06-24}
\item Upgrade to \sty{etoolbox} 1.6
\end{release}
\begin{release}{4.1}{2008-04-11}
\item Fixed timing issue with active quotes introduced in 4.0
\end{release}
\begin{release}{4.0}{2008-03-02}
\item \etex now mandatory requirement
\item New dependency on \sty{etoolbox} package
\item Added package option \opt{spanish}\see{opt:opt}
\item Added variant \opt{mexican} to style \opt{spanish}\see{opt:opt}
\item Removed variant \opt{oldstyle} from \opt{english} style\see{opt:opt}
\item Removed variant \opt{oldstyle} from \opt{french} style\see{opt:opt}
\item Removed variant \opt{imprimerie} from \opt{french} style\see{opt:opt}
\item Expanded documentation\see{hnt:bbl}
\item Added \cmd{openautoquote} and \cmd{closeautoquote}\see{aut:amk}
\item Added \cmd{openinnerquote} and \cmd{closeinnerquote}\see{aut:amk}
\item Moved predefined styles, variants, options to \path{csquotes.def}
\item Added more hints and examples to \path{csquotes.cfg}
\item Added extended \pdf bookmarks to this manual
\end{release}
\begin{release}{3.8}{2008-01-05}
\item Added variant \opt{guillemets*} to style \opt{swedish}\see{opt:opt}
\item Added language alias \opt{australian}\see{cfg:als}
\item Added language alias \opt{newzealand}\see{cfg:als}
\item Internal improvements
\end{release}
\begin{release}{3.7}{2007-03-25}
\item Added package option \kvopt{babel}{try}\see{opt:opt}
\item Added package option \kvopt{babel}{tryonce}\see{opt:opt}
\item Added \cmd{MakeAutoQuote*}\see{act:reg}
\item Added \cmd{MakeForeignQuote*}\see{act:bbl}
\item Added \cmd{MakeHyphenQuote*}\see{act:bbl}
\item Added \cmd{mkpretextpunct}\see{cfg:aux}
\item Added \cmd{mkpreblockpunct}\see{cfg:aux}
\item Added \cmd{mkpredisppunct}\see{cfg:aux}
\item Removed compatibility code for \cmd{blockcite} legacy command
\item Internal updates for \path{biblatex} package
\end{release}
\begin{release}{3.6}{2006-11-09}
\item Added \cmd{BlockquoteDisable}\see{aut:blk}
\item Fix for \sty{amsmath} package (active quotes in \env{split} and other environments)
\item Fix for \sty{endnotes} package (endnotes in block quotations)
\item Revised Spanish quote style
\end{release}
\begin{release}{3.5}{2006-08-24}
\item Exchanged definitions of French \opt{quotes} and \opt{quotes*} variants\see{hnt:pre}
\item Internal updates for \path{inputenc} 1.1\,b (2006-05-05)
\end{release}
\begin{release}{3.4}{2006-04-02}
\item Stricter validation of user"=defined active characters\see{hnt:val}
\item Author interface now documented in this manual\see{aut}
\item Added documentation of \cmd{@enablequotes}\see{aut:ctl}
\item Added documentation of \cmd{@disablequotes}\see{aut:ctl}
\item Added documentation of \cmd{@verbatimquotes}\see{aut:ctl}
\item Added documentation of \cmd{@deletequotes}\see{aut:ctl}
\item Added documentation concerning string handling \see{aut:str}
\item Added documentation of interface to internal marks\see{aut:imk}
\end{release}
\begin{release}{3.3}{2006-02-27}
\item Added support for \utf encoded active quotes\see{hnt:inc}
\item Modified active quotes, category codes 7, 8 no longer valid\see{hnt:val}
\item Modified delimiters, category codes 3, 4, 7, 8 no longer valid\see{hnt:val}
\item Active quotes may now be defined in the document body\see{act:ctl}
\item Renamed \cmd{RestoreQuotes} to \cmd{EnableQuotes}\see{act:ctl}
\item Added \cmd{DeleteQuotes}\see{act:ctl}
\item Added \cmd{VerbatimQuotes}\see{act:ctl}
\item Added \cmd{ExecuteQuoteOptions}\see{cfg:exe}
\item Added package option \kvopt{babel}{once}\see{opt:opt}
\item Added new style variant for French\see{hnt:pre}
\item Improved nesting control of active block quotes
\item Made active block quotes robust
\end{release}
\begin{release}{3.2}{2005-12-05}
\item Added quote style for Spanish
\item Fixed bug in \sty{hyperref} interface
\end{release}
\begin{release}{3.1}{2005-08-29}
\item Added \cmd{textquote}\see{bas:txt:reg}
\item Added \cmd{foreigntextquote}\see{bas:txt:bbl}
\item Added \cmd{hyphentextquote}\see{bas:txt:bbl}
\item Renamed \cmd{cquote} to \cmd{textcquote}\see{int:txt:reg}
\item Renamed \cmd{foreigncquote} to \cmd{foreigntextcquote}\see{int:txt:bbl}
\item Renamed \cmd{hyphencquote} to \cmd{hyphentextcquote}\see{int:txt:bbl}
\item Extended \cmd{textcquote}\see{int:txt:reg}
\item Extended \cmd{foreigntextcquote}\see{int:txt:bbl}
\item Extended \cmd{hyphentextcquote}\see{int:txt:bbl}
\item Modified environment \env{displayquote}\see{env:bas}
\item Modified environment \env{foreigndisplayquote}\see{env:bas}
\item Modified environment \env{hyphendisplayquote}\see{env:bas}
\item Extended environment \env{displaycquote}\see{env:cit}
\item Extended environment \env{foreigndisplaycquote}\see{env:cit}
\item Extended environment \env{hyphendisplaycquote}\see{env:cit}
\item Added \cmd{mkmidtextpunct}\see{cfg:aux}
\item Added \cmd{mkfintextpunct}\see{cfg:aux}
\item Added \cmd{mkmiddisppunct}\see{cfg:aux}
\item Added \cmd{mkfindisppunct}\see{cfg:aux}
\item Added auxiliary environment \env{quotetext}\see{cfg:aux}
\item Added detection of paragraphs to all block quotation facilities\see{bas:blk:reg}
\item \cmd{ifquote...} now usable in \cmd{mkcitation} and \cmd{mkccitation}\see{cfg:tst}
\item Terminal punctuation now evaluated by all quotation facilities
\item Prevent undesirable |?`| and |!`| ligatures in \acr{T1} encoding
\item Always adjust space factor codes of backend quotes
\end{release}
\begin{release}{3.0}{2005-07-14}
\item Extended \cmd{blockquote}\see{bas:blk:reg}
\item Extended \cmd{foreignblockquote}\see{bas:blk:bbl}
\item Extended \cmd{hyphenblockquote}\see{bas:blk:bbl}
\item Extended \cmd{setquotestyle}\see{bas:set}
\item Added \cmd{cquote}\see{int:txt:reg}
\item Added \cmd{foreigncquote}\see{int:txt:bbl}
\item Added \cmd{hyphencquote}\see{int:txt:bbl}
\item Added \cmd{blockcquote}\see{int:blk:reg}
\item Added \cmd{foreignblockcquote}\see{int:blk:bbl}
\item Added \cmd{hyphenblockcquote}\see{int:blk:bbl}
\item Added environment \env{displayquote}\see{env:bas}
\item Added environment \env{foreigndisplayquote}\see{env:bas}
\item Added environment \env{hyphendisplayquote}\see{env:bas}
\item Added environment \env{displaycquote}\see{env:cit}
\item Added environment \env{foreigndisplaycquote}\see{env:cit}
\item Added environment \env{hyphendisplaycquote}\see{env:cit}
\item Modified \cmd{DeclarePlainStyle}\see{cfg:pdf}
\item Added \cmd{SetCiteCommand}\see{cfg:blk}
\item Renamed \cmd{blockcite} to \cmd{mkcitation}\see{cfg:aux}
\item Added \cmd{mkccitation}\see{cfg:aux}
\item Added \cmd{mkmidblockpunct}\see{cfg:aux}
\item Added \cmd{mkfinblockpunct}\see{cfg:aux}
\item Added \cmd{ifquotepunct}\see{cfg:tst}
\item Added \cmd{ifquoteterm}\see{cfg:tst}
\item Added \cmd{ifquoteperiod}\see{cfg:tst}
\item Added \cmd{ifquotecomma}\see{cfg:tst}
\item Added \cmd{ifquotesemicolon}\see{cfg:tst}
\item Added \cmd{ifquotecolon}\see{cfg:tst}
\item Added \cmd{ifquoteexclam}\see{cfg:tst}
\item Added \cmd{ifquotequestion}\see{cfg:tst}
\item Added \cmd{ifstringblank}\see{cfg:tst}
\item Added evaluation of terminal punctuation within block quotations
\item With \cmd{nonfrenchspacing}, adjust space factor codes of backend quotes
\item Improved nesting control when running under \etex
\end{release}
%\begin{release}{2.8}{2005-05-11}
%\item Added \cmd{DisableQuotes}\see{act:ctl}
%\item Fixed bug causing kerning restoration to fail in some rare cases
%\end{release}
%
%\begin{release}{2.7}{2005-04-13}
%\item Use the font's kerning pairs for adjoining quotes, if available\see{cfg:sty}
%\item Renamed \cmd{setblockthreshold} to \cmd{SetBlockThreshold}\see{cfg:blk}
%\item Renamed \cmd{setblockenvironment} to \cmd{SetBlockEnvironment}\see{cfg:blk}
%\item Provided more useful default definition of \cmd{blockcite}\see{cfg:aux}
%\item Improved handling of adjoining quotes with respect to line breaking
%\item When restoring active quotes, restore catcodes of delimiters as well
%\item Improved workaround for \cmd{uppercase} and some \sty{babel} languages
%\item Issue error message on quote mismatch regardless of \opt{strict} option
%\item Issue \sty{hyperref} warning with block quotation commands in \pdf strings
%\item Fixed bug in \cmd{DeclareQuoteStyle} and \cmd{DeclareQuoteAlias}
%\end{release}
%
%\begin{release}{2.6}{2005-02-24}
%\item Always reset quote style, even for inner quotations
%\item Fixed bug preventing hyphenation in certain places
%\end{release}
%
%\begin{release}{2.5}{2004-12-04}
%\item Added \cmd{MakeBlockQuote}\see{act:blk:reg}
%\item Added \cmd{MakeForeignBlockQuote}\see{act:blk:bbl}
%\item Added \cmd{MakeHyphenBlockQuote}\see{act:blk:bbl}
%\item Added \cmd{ifblockquote}\see{cfg:tst}
%\item Modified \cmd{blockquote}\see{bas:blk:reg}
%\item Modified \cmd{foreignblockquote}\see{bas:blk:bbl}
%\item Modified \cmd{hyphenblockquote}\see{bas:blk:bbl}
%\item Changed default threshold for block quotations\see{cfg:blk}
%\item Improved math mode compatibility\see{hnt:mod}
%\item Improved verbatim compatibility\see{hnt:mod}
%\item Improved backend and active character handling
%\item Improved validation of user"=defined active characters
%\item Fixed bug suppressing kerning after block quotations
%\item Issue error message with nested block quotations
%\end{release}
%
%\begin{release}{2.4}{2004-11-01}
%\item Prevent use of \cmd{RestoreQuotes} in preamble\see{act:ctl}
%\item Fixed bug causing premature expansion of backend quote macros
%\item Fixed bug suppressing kerning before closing quotes
%\end{release}
%
%\begin{release}{2.3}{2004-09-18}
%\item Reduced default kerning between adjoining curved quotes
%\item Fixed bug with \cmd{DeclareQuoteStyle} in preamble
%\end{release}
%
%\begin{release}{2.2}{2004-07-13}
%\item Extended \cmd{DeclareQuoteStyle}\see{cfg:sty}
%\item Added initialization hook for inner quotations\see{cfg:sty}
%\item Added support for middle inner quotes\see{cfg:sty}
%\item Rearranged French quote styles, removing two variants\see{hnt:pre}
%\item Added new style variant for French\see{hnt:pre}
%\item Fixed bug causing stacking of reset hook for footnotes
%\item Fixed bug preventing hyphenation in certain places
%\item Fixed kerning issue specific to \textsc{ec} fonts
%\end{release}
%
%\begin{release}{2.1}{2004-06-15}
%\item Added auxiliary environment \env{quoteblock}\see{cfg:aux}
%\item Added support for language reset in footnotes\see{hnt:ftn}
%\item Disable active characters in \cmd{verb} and \env{verbatim}\see{hnt:mod}
%\item Disable active characters in \cmd{index} and \cmd{glossary}\see{hnt:mod}
%\item Added package option and style variants for Norwegian
%\item Removed some uncertain quote styles and aliases
%\item Rearranged quote styles and aliases
%\end{release}
%
%\begin{release}{2.0}{2004-06-04}
%\item Added \cmd{blockquote}\see{bas:blk:reg}
%\item Added \cmd{foreignblockquote}\see{bas:blk:bbl}
%\item Added \cmd{hyphenblockquote}\see{bas:blk:bbl}
%\item Added \cmd{setblockthreshold}\see{cfg:blk}
%\item Added \cmd{setblockenvironment}\see{cfg:blk}
%\item Added auxiliary command \cmd{blockcite}\see{cfg:aux}
%\item Extended \cmd{DeclareQuoteStyle}\see{cfg:sty}
%\item Added initialization hook for outer quotations\see{cfg:sty}
%\item Added support for middle outer quotes\see{cfg:sty}
%\item Added support for kerning between adjoining quotes\see{cfg:sty}
%\item Disable active characters in math and display math mode\see{hnt:mod}
%\item Revised and improved error recovery\see{opt:opt}
%\item Added package option \opt{strict}\see{opt:opt}
%\item Added package option and new style variants for French
%\item Added package option and new style variant for Italian
%\item Added new style variant for English
%\end{release}
%
%\begin{release}{1.7}{2004-05-14}
%\item Added \cmd{setquotestyle}\see{bas:set}
%\item Modified \cmd{DeclarePlainStyle}\see{cfg:pdf}
%\item Improved quote handling in \pdf strings\see{hnt:pdf}
%\item Amended default French quote style\see{hnt:pre}
%\end{release}
%
%\begin{release}{1.5}{2004-02-27}
%\item Reset quote nesting level in footnotes within quotations\see{hnt:ftn}
%\end{release}
%
%\begin{release}{1.4}{2003-12-13}
%\item Added \cmd{MakeForeignQuote}\see{act:bbl}
%\item Added \cmd{MakeHyphenQuote}\see{act:bbl}
%\item Added \cmd{RestoreQuotes}\see{act:ctl}
%\item Improved \sty{hyperref} interface\see{hnt:pdf}
%\end{release}
%
%\begin{release}{1.0}{2003-09-14}
%\item Initial public release
%\end{release}
\end{changelog}
\end{document}
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%%
%% The LaTeX Graphics Companion, 2ed (first printing May 2007)
%%
%% Example 6-7-38 on page 445.
%%
%% Copyright (C) 2007 Michel Goossens, Frank Mittelbach, Denis Roegel, Sebastian Rahtz, Herbert Vo\ss
%%
%% 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}
\setlength\textwidth{348.0pt}
\StartShownPreambleCommands
\usepackage{pst-dbicons}
\seticonparams{entity}{shadow=true,fillcolor=black!30,fillstyle=solid}
\seticonparams{attribute}{fillcolor=black!10,fillstyle=solid}
\seticonparams{relationship}{shadow=true,fillcolor=black!20,fillstyle=solid}
\StopShownPreambleCommands
\begin{document}
\setlength\attrdist{2.5em}
\entity{Person}\hspace{8cm}\entity{Company}
\attributeof{Person}{30}[key]{Name} \attributeof{Person}{90}[mv]{Nickname}
\attributeof{Person}[4em]{150}{phone}[phone\_no]
\attributeof{Person}[2em]{270}[mv]{wt}[weight\_at]
\attributeof{wt}{240}{date} \attributeof{wt}{300}{weight}
\relationshipbetween{Person}{Company}{worksat}[works\_at]
\end{document}
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pdfauthor = {APMEP},
pdfsubject = {Baccalauréat STG},
pdftitle = {ACA--ACC La Réunion juin 2005},
allbordercolors = white}
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\begin{document}
\setlength\parindent{0mm}
\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small Baccalauréat STT ACC-ACA juin 2005}
\lfoot{\small{La Réunion}}
\rfoot{\small{juin 2005}}
\pagestyle{fancy}
\thispagestyle{empty}
\begin{center}{\Large \textbf{\decofourleft~Baccalauréat STT ACA - ACC La Réunion~\decofourright\\juin 2005}}
\vspace{0,25cm}
\end{center}
\textbf{\textsc{Exercice 1} \hfill 9 points }
\medskip
Dans une entreprise de nettoyage industriel créée voici quatre ans, on a relevé l'ancienneté des $120$ techniciens et techniciennes de surface y travaillant :
\begin{center}
\begin{tabularx}{0.6\linewidth}{|*{2}{>{\centering \arraybackslash}X|}}\hline
Ancienneté en mois & Nombre \\ \hline
[0 ; 6] &37\\ \hline
[6 ; 12] &23\\ \hline
[12 ; 24] &19\\ \hline
[24 ; 36] &6\\ \hline
[36 ; 48] &35\\ \hline
\end{tabularx}
\end{center}
\begin{enumerate}
\item En considérant les intervalles d'ancienneté comme étant les classes d'une série statistique à une variable et le nombre d'employés comme étant leurs effectifs correspondants.
\begin{enumerate}
\item Calculer les centres de ces classes.
\item Calculer l'ancienneté moyenne en mois de ces $120$ employés.
\end{enumerate}
\emph{Dans toute la suite de cet exercice, les probabilités seront données à} $10$^{-2}$ \emph{près}.
\item Un (ou une) employé(e) est choisi(e) au hasard.
Quelle est la probabilité qu'il ou qu'elle ait une ancienneté inférieure à un an.
\item Chacune de ces $120$ personnes a un contrat de travail.
Dans cette entreprise on rencontre 3 types de contrat de travail différents :
Le Contrat à Durée Déterminée (C. D. D)
Le Contrat à Durée Indéterminée à mi-temps (C. D. I./M. T.)
Le Contrat à Durée Indéterminee à temps complet (C. D. I./T. C.)
Les C. D. D. représentent $30$\:\% de l'effectif total et 75\,\% des C. D. D. sont à des femmes.
Les C. D. I/T. C. représentent la moitié de l'effectif total et $60$\:\% des C. D. I./T. C. sont à des hommes
Un seul homme a un C. D. I./M. T.
\begin{enumerate}
\item Recopier et compléter le tableau suivant
\begin{center}
\begin{tabularx}{\linewidth}{|c|*{4}{>{\centering \arraybackslash}X|}}\cline{2-5}
\multicolumn{1}{c|}{} &C.D.D. & C.D.I./M.T. & C.D.I./T.C. & Total\\\hline
Hommes & & & & \\\hline
Femmes & & & & \\\hline
Total & & & &120 \\\hline
\end{tabularx}
\end{center}
\item On choisit au hasard une personne parmi ces $120$ employés.
On considère les évènements suivants :
A : \og La personne a un C. D. D. \fg{} ;
B : \og La personne est une femme \fg{} ;
C : \og La personne a un C. D. D. ou est une femme \fg{}.
Calculer la probabilité de chacun des évènements A, B et C.
\end{enumerate}
\item
\begin{enumerate}
\item Parmi les hommes, quelle est la probabilité d'en choisir un ayant un C. D. I./T. C. ?
\item Parmi les femmes, quelle est la probabilité d'en choisir une ayant un C. D. I./T. C. ?
\item Que remarque-t-on ?
\end{enumerate}
\end{enumerate}
\vspace{0,5cm}
\textbf{\textsc{Exercice 2} \hfill 11 points}
\medskip
\textbf{Partie A}
\medskip
Pour participer à la finale du jeu \og Super Game \fg{}, organisée par un magasin de jeu vidéo, deux enfants, Ulysse et Victor, s'entraînent chaque jour, pendant les vacances. Pour être sélectionné, un joueur doit obtenir un minimum de \np{2000} points avant la date de la finale et contacter l'organisateur qui l'inscrit alors sur la liste des participants au concours.
Le premier jour de son \og entraînement \fg{}, Ulysse, féru de jeu vidéo, obtient un très bon score de \np{1500}~points.
Victor, qui est plus jeune, marque \np{1000}~points. Au fur et à mesure des jours, Ulysse remarque que, quotidiennement, son score progresse de $3$\:\% alors que celui de Victor augmente de $70$~points.
On note $u_{0}$ et $v_{0}$ les scores obtenus respectivement par Ulysse et Victor le premier jour de leur entraînement, soit le 30 juin (on a donc $u_{0} = \np{1500}$ et $v_{0} = \np{1000}$).
De même, $u_{n}$ et $v_{n}$ correspondront aux scores obtenus le $n$ juillet.
\medskip
\begin{enumerate}
\item Calculer $u_{1},~u_{2}$ et $u_{3}$ en arrondissant le score au point supérieur.
\item Calculer $v_{1},~v_{2}$ et $v_{3}$.
\item Quelle est la nature de chacune des suites $\left(u_{n}\right)$ et $\left(v_{n}\right)$ ? Justifier.
\item La finale a lieu le 14 juillet. Qui sera sélectionné pour y participer ? Justifier la réponse par un calcul.
\item À l'aide de la calculatrice, pour chacun des enfants, déterminer la date à laquelle il aura atteint le score fatidique des \np{2000} points.
\end{enumerate}
\vspace{0,25cm}
\textbf{Partie B}
\medskip
On considère la fonction $f$ définie sur [0~ ;~15] par
\[f(x) = x^3 - 10x^2 + 25x + \np{1200}.\]
\begin{enumerate}
\item
\begin{enumerate}
\item On note $f'$ la dérivée de la fonction $f$ sur [0~;~15]. Déterminer $f'(x)$.
\item Montrer que $f'(x) = (3x - 5)(x -5)$.
\item Étudier le signe de $f'$ et dresser le tableau de variations de la fonction $f$ sur [0~ ;~15].
\end{enumerate}
\item La courbe $\mathcal{C}$ jointe en annexe est la représentation graphique de la fonction $f$précédente.
À l'aide de cette courbe, résoudre graphiquement l'équation $f(x) = \np{2000}$.
On fera apparaître sur l'annexe, que l'on joindra à la copie, le tracé utilisé.
\item Un troisième joueur, Fabrice, s'est également entraîné à partir du 30 juin pour la finale du jeu \og Super Game \fg. La fonction précédente correspond aux points obtenus par Fabrice, où $x$ représente le nombre de jours écoulés depuis le 30 juin.
\begin{enumerate}
\item Que représente $f(0)$ ?
\item Utiliser le résultat de la \textbf{question 2.} pour déterminer si ce joueur sera sélectionné pour la finale du 14 juillet. À quelle date ?
Qui, entre Ulysse, Victor et Fabrice sera sélectionné le premier ?
\end{enumerate}
\end{enumerate}
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\bibitem{Graf_etal2016} Graf, R., Baumann, M., Horn, R., \& Brethauer, L. (2016). Energy Autarky and Civic Activism: LCA results as decision support (Presentation). Kyoto.
\end{thebibliography}
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