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https://www.emis.de/journals/EJC/Volume_2/Abstracts/v2i1a1.tex | emis.de | CC-MAIN-2023-06 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2023-06/segments/1674764501066.53/warc/CC-MAIN-20230209014102-20230209044102-00290.warc.gz | 757,154,941 | 1,818 | %\magnification=1440
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\begin{document}
\large
Abstract for R. L. Graham and B.D. Lubachevsky,
Dense Packings of Equal Disks in an Equilateral Triangle
Previously published packings of equal disks in an equilateral
triangle have dealt with up to 21 disks.
We use a new discrete-event simulation algorithm to produce
packings for up to 34 disks.
For each $n$ in the range $22 \le n \le 34$ we present what we believe
to be the densest possible packing of $n$ equal disks in an
equilateral triangle.
For these $n$ we also list the second,
often the third and sometimes the fourth best packings
among those that we found.
In each case, the structure of the packing implies that the minimum
distance $d(n)$ between disk centers is the root of polynomial
$P_n$ with integer coefficients.
In most cases we do not explicitly compute $P_n$
but in all cases we do compute and report $d(n)$
to 15 significant decimal digits.
Disk packings in equilateral triangles differ
from those in squares or circles in that for triangles
there are an infinite number of values of $n$
for which the exact value of $d(n)$ is known,
namely, when $n$ is of the form $\Delta (k) := \frac{k(k+1)}{2}$.
It has also been conjectured that $d(n-1) = d(n)$ in this case.
Based on our computations, we present conjectured optimal packings
for seven other infinite classes of $n$, namely
\begin{eqnarray*}
n & = & \Delta (2k) +1,~\Delta (2k+1) +1,
\Delta (k+2) -2 , ~ \Delta (2k+3) -3, ~ \\
&& \Delta (3k+1)+2 ,
~ 4 \Delta (k), ~~\mbox{and}~~
2 \Delta (k+1) + 2 \Delta (k) -1 ~.
\end{eqnarray*}
We also report the best packings we found
for other values of $n$ in these forms
which are larger than 34, namely,
$n=37$, 40, 42, 43, 46, 49, 56, 57, 60, 63, 67, 71, 79, 84, 92, 93, 106, 112, 121, and 254,
and also for $n=58$, 95, 108, 175, 255, 256, 258, and 260.
We say that an infinite class of packings of $n$ disks,
$n=n(1), n(2),...n(k),...$,
is {\em tight }, if
[$1/d(n(k)+1) - 1/d(n(k))$] is
bounded away from zero as $k$ goes to infinity.
We conjecture that some of our infinite classes are tight,
others are not tight, and that there are infinitely many
tight classes.
\end{document}
|
http://web.cs.dal.ca/~jamie/teach/JBlustein/Permission/4163-permission.tex | dal.ca | CC-MAIN-2013-20 | application/x-tex | null | crawl-data/CC-MAIN-2013-20/segments/1368705955434/warc/CC-MAIN-20130516120555-00088-ip-10-60-113-184.ec2.internal.warc.gz | 300,308,281 | 1,480 | \documentclass[11pt,letterpaper]{article}
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\begin{center}
\Large
Permission to Use Copies of a Student's Assignments
\end{center}
I hereby give permission for James Blustein to use a copy of my
assignments from Computer Science course number~4163 (entitled
Human-Computer Interaction) in the term indicated below at Dalhousie
University as an example for other students in future courses.
This permission does not change the ownership of any assignment.
I acknowledge that I grant this permission entirely of my own
volition.
I have not and will not get any course credit or payment either by
granting or not granting this permission.
James Blustein acknowledges that I may rescind my permission in
writing at any time.
\[
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Term (circle one): Fall / Winter / Summer
& Year \underline{\hspace{20ex}}
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Name: & \underline{\hspace{50ex}} \\\\\\
Signature: & \underline{\hspace{50ex}}
\\ \\ \\ \\
Name of Witness: & \underline{\hspace{50ex}} \\\\\\
Signature of Witness: & \underline{\hspace{50ex}}
\end{tabular}
\right \} \mbox{Date:\,} \underline{\hspace{20ex}}
\\ \\ \\ \\
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Signature of James Blustein: & \underline{\hspace{50ex}} \\\\
Date: & \underline{\hspace{30ex}}
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|
http://www.tcl.tk/cgi-bin/tct/tip/71.tex | tcl.tk | CC-MAIN-2013-20 | application/x-latex | null | crawl-data/CC-MAIN-2013-20/segments/1368704179963/warc/CC-MAIN-20130516113619-00036-ip-10-60-113-184.ec2.internal.warc.gz | 729,234,224 | 3,011 | \documentclass[]{article}
\usepackage{amsmath,graphicx,supertabular,hyperref,tabularx,ifthen}
\title{TIP \#71: Tk Bitmap Image Improvements}
\date{October 26, 2001}
\author{Chris Nelson, Kevin Kenny, Eric Melski, Donal K. Fellows}
\urlstyle{sf}
\setlength{\parskip}{1ex}
\setlength{\parindent}{0pt}
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\newenvironment{tipabstract}{\begin{abstract}}{\end{abstract}}
\begin{document}\maketitle
\begin{center}\begin{tabularx}{\linewidth}{|r@{: }X|}\hline
\textbf{TIP \#71}&\textbf{Tk Bitmap Image Improvements}\\\hline
Author&
Chris Nelson \tipmail{chris}{pinebush.com}
\par
Kevin Kenny \tipmail{kennykb}{acm.org}
\par
Eric Melski \tipmail{ericm}{interwoven.com}
\par
Donal K. Fellows \tipmail{donal.k.fellows}{man.ac.uk}
\\
Created&Friday, \(\text{26}^{\text{th}}\) October 2001\\
Type&Project\\
Tcl Version&\(8.5\)\\
State&Withdrawn\\
Vote&Pending\\
Version&\tipversion$Revision: 1.14 $\\
Post History&\\
\hline\end{tabularx}\end{center}
\thispagestyle{empty}\pagestyle{empty}
\begin{tipabstract}
Tk has a number of pre-defined bitmaps (10 on all platforms) but it lacks a number of bitmaps useful for creating GUI elements. This TIP adds several such bitmaps (as bitmap images).
\end{tipabstract}
\tableofcontents\setcounter{page}{0}\clearpage\pagestyle{plain}
\section{New Bitmaps}
Many complex widgets like comboboxes, spinboxes, etc. require arrows pictures on buttons. While newer releases of Tk have added more widgets, there will always be some unforeseen need for new or customized widgets. One example is a menubutton which, according to the Microsoft Windows User Experience [\url{http://msdn.microsoft.com/library/default.asp?url=/library/en-us/dnwue/html/welcome.asp}], should have a downward arrow on the right side. With compound buttons, it is not hard to do:
\begingroup\small\begin{verbatim}
button .mb -text Tools -image downarrow -compound right
\end{verbatim}
\endgroup
but there is no stock down-arrow image.
I propose to add 12 bitmap images providing all four directions (up, down, left, and right) in three sizes (3x2, 5x3, and 7x4) in black. The down arrows would look something like:
\begingroup\small\begin{verbatim}
@@@@@@@ @@@@@ @@@
.@@@@@. .@@@. .@.
..@@@.. ..@..
...@...
\end{verbatim}
\endgroup
I propose the following names:
\begingroup\small\begin{verbatim}
arrow_u7x4 arrow_u5x3 arrow_u3x2
arrow_d7x4 arrow_d5x3 arrow_d3x2
arrow_l7x4 arrow_l5x3 arrow_l3x2
arrow_r7x4 arrow_r5x3 arrow_r3x2
\end{verbatim}
\endgroup
I'm mindful of the fact that adding new predefined bitmap images has the potential to collide with application-defined images or other commands but I'm unsure of the workaround for that.
\section{Reference Implementation}
SourceForge patch 475332 provided a reference implementation of a previous version of this proposal [\url{http://sf.net/tracker/?func=detail&aid=475332&group_id=12997&atid=312997}]. This version is not implemented yet.
\section{Commentary}
\textit{Donal K. Fellows \tipmail{donal.k.fellows}{man.ac.uk} writes:}
\begin{quote}
Previous versions of this TIP proposed fixing the problem using bitmaps instead of bitmap images and added an infrastructure for tracking those bitmaps. Since I think that ultimately we should be getting rid of bitmaps and instead using something based on the image infrastructure (which already has proper introspection support) those parts of this TIP have been removed. However, making the changes to effect the switch to using bitmap images instead of bitmaps for things like stippes, cursors, etc. lies outside the scope of this TIP.
\end{quote}
\textit{Donal K. Fellows \tipmail{donal.k.fellows}{man.ac.uk} writes:}
\begin{quote}
In the long period since this TIP was proposed, the world of GUIs has moved on somewhat. Although the requirement for arrows remains the same, the solutions proposed in this TIP (both originally and as it now stands) do not permit the sort of graphical snazziness that modern users tend to expect. Nor is there a sufficient range of sizes for a reasonable selection to be available for a modern display; even the largest of those arrows would look unusably tiny on my desktop! This indicates that a completely different solution is required, which in turn would be better stated as a separate TIP.
\end{quote}
\section{Copyright}
This document has been placed in the public domain.
\section{Colophon}
\textit{TIP AutoGenerator --- written by Donal K. Fellows}
\end{document}
|
http://www.isska.ch/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%20198%20ORDER%20BY%20year%20DESC%2C%20first_author%2C%20author_count%2C%20author%2C%20title&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=15&rowOffset=&wrapResults=1&citeOrder=year&citeStyle=APA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg= | isska.ch | CC-MAIN-2019-39 | application/x-latex | application/x-latex | crawl-data/CC-MAIN-2019-39/segments/1568514575513.97/warc/CC-MAIN-20190922114839-20190922140839-00203.warc.gz | 286,604,170 | 1,241 | %&LaTeX
\documentclass{article}
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\usepackage{textcomp}
\begin{document}
\section*{2018}
H{\"a}uselmann, P., \& Ganauser, R. (2018). \textit{Die Wasserf{\"a}rbung 2015 in der Schrattenfluh}.
\end{document}
|
http://ctan.math.utah.edu/ctan/tex-archive/systems/knuth/dist/tex/texbook.tex | utah.edu | CC-MAIN-2016-26 | application/x-tex | null | crawl-data/CC-MAIN-2016-26/segments/1466783399385.17/warc/CC-MAIN-20160624154959-00096-ip-10-164-35-72.ec2.internal.warc.gz | 70,205,271 | 353,442 | % This manual is copyright (C) 1984 by the American Mathematical Society.
% All rights are reserved!
% The file is distributed only for people to see its examples of TeX input,
% not for use in the preparation of books like The TeXbook.
% Permission for any other use of this file must be obtained in writing
% from the copyright holder and also from the publisher (Addison-Wesley).
\loop\iftrue
\errmessage{This manual is copyrighted and should not be TeXed}\repeat
\pausing1 \input manmac
\ifproofmode\message{Proof mode is on!}\pausing1\fi
% halftitle
\titlepage
\pageno=-1983
\null\bigskip
\line{\cmman The\hfill T\kern-10pt\lower13pt\hbox{E}\kern-5pt Xbook}
\vfill
\ifproofmode
\rightline{The fine print in the upper right-hand}
\rightline{corner of each page is a draft of intended}
\rightline{index entries; it won't appear in the real book.}
\rightline{Some index entries will be in |typewriter type|}
\rightline{and/or preceded by {\tt\char`\\} or enclosed in \<$\ldots$>, etc;}
\rightline{such typographic distinctions aren't shown here.}
\rightline{An index entry often extends for several pages;}
\rightline{the actual scope will be determined later.}
\rightline{Please note things that should be indexed but aren't.}
\fi
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\titlepage\null\vfill\eject % blank page
% title
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\null\bigskip
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^^{Knuth, Donald Ervin}
^^{Bibby, Duane Robert}
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\titlepage
\eightpoint
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\noindent\strut
This manual describes \TeX\ Version 3.0. Some
of the advanced features mentioned here are absent from earlier versions.
\medskip
\noindent
The quotation on page \sesame\ is copyright $\copyright$ 1970 by Sesame
Street, Inc., and used by permission of the Children's Television Workshop.
\medskip
\noindent
\TeX\ is a trademark of the American Mathematical Society.
\medskip
\noindent
{\manual opqrstuq} is a trademark of Addison\kern.1em--Wesley
Publishing Company.
\bigskip\medskip
\noindent
{\bf Library of Congress cataloging in publication data}
\medskip
{\tt\halign{#\hfil\cr
Knuth, Donald Ervin, 1938-\cr
\ \ \ The TeXbook.\cr
\noalign{\medskip}
\ \ \ (Computers \& Typesetting ; A)\cr
\ \ \ Includes index.\cr
\ \ \ 1.~TeX (Computer system).\ \ 2.~Computerized\cr
typesetting.\ \ 3.~Mathematics printing.\ \ I.~Title.\cr
II.\ Series:\ \ Knuth, Donald Ervin, 1938-\ \ \ \ .\cr
Computers \& typesetting ; A.\cr
Z253.4.T47K58\ \ 1986\ \ \ \ \ \ \ \ \ 686.2\char13 2544\ \ \ \ \ \ 85-30845\cr
ISBN 0-201-13447-0\cr
ISBN 0-201-13448-9 (soft)\cr}}
\vfill
\noindent
%{\sl \kern-1pt First hardcover edition, January 1986}
%{\sl \kern-1pt Incorporates the final corrections made in 1996, and a few dozen more.}
{\sl \kern-1pt Incorporates all corrections known in 2013.}
\smallskip
\noindent
Internet page {\tt http://www-cs-faculty.stanford.edu/\char`\~
knuth/abcde.html}
contains current information about this book and related books.
\smallskip
\noindent
Copyright $\copyright$ 1984, 1986 by the American Mathematical Society
\smallskip
\noindent
This book is published jointly by the American Mathematical Society
and Addison\kern.1em--Wesley Publishing Company.
All rights reserved. No part of this publication may be reproduced, stored in
a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise, without
the prior written permission of the publishers. Printed in the United
States of America. % Published simultaneously in Canada.
\medskip
\noindent
%ISBN 0-201-13448-9\par % paperback
%ISBN 0-201-13447-0\par % hardcover
ISBN-13 \enspace 978-0-201-13447-6\par\noindent
ISBN-10 \enspace\phantom{978-}0-201-13447-0\par
%33 34 35 36 37 38 39 DOC 09 08 07 06 % paperback
%18 19 20 21 22 23 24 DOC 10 09 08 07 06 % hardcover
\smallskip\noindent
Text printed in the United States
at Courier Westford in Westford, Massachusetts.\par\noindent
Nineteenth Printing, February 2012
^^{Knuth, Donald Ervin}
^^|\copyright|
\eject
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^^{Knuth, Jill Carter}
\vskip2pt
\rightline{\eightssi For your books and brochures}
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{\sc ENTLE} R{\sc EADER}: \strut This is a handbook about
\TeX, a new typesetting system intended for the creation
of beautiful books---and especially for books that contain a lot of
mathematics. By preparing a manuscript in \TeX\ format, you will be
telling a computer exactly how the manuscript is to be transformed into
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(If such claims sound too good to be true, keep in mind that they were made
by \TeX's designer, on a day when \TeX\ happened to
be working, so the statements may be biased; but read on anyway.)
This manual is intended for people who have never used \TeX\ before, as
well as for experienced \TeX\ hackers. In other words, it's supposed to
be a panacea that satisfies everybody, at the risk of satisfying nobody.
Everything you need to know about \TeX\ is explained
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If you are preparing a simple manuscript, you won't need to
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$$\vbox{\hbox{\dbend}\vskip 11pt}$$
appears at the beginning of a paragraph, it warns of a ``^{dangerous bend}''
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Brave and experienced drivers at the controls of \TeX\ will gradually enter
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details won't matter.
All that you really ought to know, before reading on, is how to get a
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manual explains what that file ought to look like so that \TeX\ will
understand it, but basic computer usage is not explained here.
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Some of the paragraphs in this manual are so esoteric that they are rated
$$\vcenter{\hbox{\dbend\kern1pt\dbend}\vskip 11pt}\;;$$
everything that was said about single dangerous-bend signs goes double
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\TeX78, and its use should rapidly fade away. Let's keep the name \TeX\
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I wish to thank the hundreds of people who have helped me to formulate
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experiences with preliminary versions of the system. My work at Stanford
has been generously supported by the ^{National Science Foundation}, the
^{Office of Naval Research}, the ^{IBM Corporation}, and the ^{System
Development Foundation}. I also wish to thank the ^{American Mathematical
Society} for its encouragement, for establishing the \TeX\ Users Group,
and for publishing the {\sl ^{TUGboat}\/} newsletter (see Appendix~J).
\medskip
\line{{\sl Stanford, California}\hfil--- D. E. K.}^^{Knuth, Don}
\line{\sl June 1983\hfil}
} % end of the special \topskip
\endchapter
`Tis pleasant, sure, to see one's name in print;
A book's a book, although there's nothing in 't.
\author ^{BYRON}, {\sl English Bards and Scotch Reviewers\/} (1809)
\bigskip
A question arose as to whether we were covering the field
that it was intended we should fill with this manual.
\author RICHARD R. ^{DONNELLEY}, {\sl Proceedings, United %
Typothet{\ae} of America\/} (1897)
\eject
% the table of contents
\titlepage
\vbox to 8pc{
\rightline{\titlefont Contents}
\vfill}
^^{Contents of this manual, table}
\def\rhead{Contents}
\tenpoint
\begingroup
\countdef\counter=255
\def\diamondleaders{\global\advance\counter by 1
\ifodd\counter \kern-10pt \fi
\leaders\hbox to 20pt{\ifodd\counter \kern13pt \else\kern3pt \fi
.\hss}}
\baselineskip 15pt plus 5pt
\def\\#1. #2. #3.{\line{\strut
\hbox to\parindent{\bf\hbox to 1em{\hss#1}\hss}%
\rm#2\diamondleaders\hfil\hbox to 2em{\hss#3}}}
\\1. The Name of the Game. 1.
\\2. Book Printing versus Ordinary Typing. 3.
\\3. Controlling \TeX. 7.
\\4. Fonts of Type. 13.
\\5. Grouping. 19.
\\6. Running \TeX. 23.
\\7. How \TeX\ Reads What You Type. 37.
\\8. The Characters You Type. 43.
\\9. \TeX's Roman Fonts. 51.
\\10. Dimensions. 57.
\\11. Boxes. 63.
\\12. Glue. 69.
\\13. Modes. 85.
\\14. How \TeX\ Breaks Paragraphs into Lines. 91.
\\15. How \TeX\ Makes Lines into Pages. 109.
\\16. Typing Math Formulas. 127.
\\17. More about Math. 139.
\\18. Fine Points of Mathematics Typing. 161.
\\19. Displayed Equations. 185.
\\20. Definitions (also called Macros). 199.
\\21. Making Boxes. 221.
\\22. Alignment. 231.
\\23. Output Routines. 251.
\eject
\vbox to 8pc{}
\\24. Summary of Vertical Mode. 267.
\\25. Summary of Horizontal Mode. 285.
\\26. Summary of Math Mode. 289.
\\27. Recovery from Errors. 295.
\null
\leftline{\indent\bf Appendices}
\\A. Answers to All the Exercises. 305.
\\B. Basic Control Sequences. 339.
\\C. Character Codes. 367.
\\D. Dirty Tricks. 373.
\\E. Example Formats. 403.
\\F. Font Tables. 427.
\\G. Generating Boxes from Formulas. 441.
\\H. Hyphenation. 449.
\\I\hskip 1pt. Index. 457.
\\J\hskip 1pt. Joining the \TeX\ Community. 483.
\null % 17 lines so far to balance the 23 on the other page
\null % 18
\null % 19
\null % 20
\null % 21
\null % 22
\null % 23
\eject
\endgroup
\beginchapter Chapter 1. The Name of\\the Game
\pageno=1 % This is page number 1, number 1,
English words like `technology' stem from a Greek root beginning with
the letters $\tau\epsilon\chi\ldots\,$; and this same Greek word means {\sl
art\/} as well as technology. Hence the name \TeX, which is an
uppercase form of $\tau\epsilon\chi$.^^{TeX (actually \TeX), meaning of}
^^|\tau|^^|\epsilon|^^|\chi|
Insiders pronounce the $\chi$ of \TeX\ as a Greek chi, not as an `x', so that
\TeX\ rhymes with the word blecchhh. It's the `ch' sound in Scottish words
like {\sl loch\/} or German words like {\sl ach\/}; it's a Spanish `j' and a
Russian `kh'. When you say it correctly to your computer, the terminal
may become slightly moist.
The purpose of this pronunciation exercise is to remind you that \TeX\ is
primarily concerned with high-quality technical manuscripts: Its emphasis is
on art and technology, as in the underlying Greek word. If you merely want
to produce a passably good document---something acceptable and basically
readable but not really beautiful---a simpler system will usually suffice.
With \TeX\ the goal is to produce the {\sl finest\/} quality; this requires
more attention to detail, but you will not find it much harder to go the
extra distance, and you'll be able to take special pride in the finished
product. ^^{beauty} % since truth is indexed, ...
On the other hand, it's important to notice another thing about \TeX's name:
The `E' is out of kilter. This ^^{logo}
displaced `E' is a reminder that \TeX\ is about typesetting, and it
distinguishes \TeX\ from other system names. In fact, ^{TEX} (pronounced
{\sl tecks\/}) is the admirable {\sl Text EXecutive\/} processor developed by
^{Honeywell Information Systems}. Since these two system names are
^^{Bemer, Robert, see TEX, ASCII}
pronounced quite differently, they should also be spelled differently. The
correct way to refer to \TeX\ in a computer file, or when using some other
medium that doesn't allow lowering of the `E', is to type `^|TeX|'. Then
there will be no confusion with similar names, and people will be
primed to pronounce everything properly.
\exercise After you have mastered the material in this book, what will
you be: a \TeX pert, or a \TeX nician?
\answer A \TeX nician (underpaid); sometimes also called a \TeX acker.
\endchapter
They do certainly give
very strange and new-fangled names to diseases.
\author ^{PLATO}, {\sl The Republic}, Book 3 (c.\ 375 B.C.) % 405c
\bigskip
Technique! The very word is like the shriek
Of outraged Art. It is the idiot name
Given to effort by those who are too weak,
Too weary, or too dull to play the game.
\author LEONARD ^{BACON}, {\sl Sophia Trenton\/} (1920) % composed at Stanford
\eject
\beginchapter Chapter 2. Book Printing\\versus\\Ordinary Typing
When you first started using a computer terminal, you probably had to adjust
to the difference between the digit `1' and the lowercase letter `l'.
When you take the next step to the level of typography that is common in
book publishing, a few more adjustments of the same kind need to be made;
your eyes and your fingers need to learn to make a few more distinctions.
In the first place, there are two kinds of ^{quotation marks} in books, but
only one kind on the typewriter. Even your computer terminal, which has
more characters than an ordinary typewriter, probably has only a
non-oriented double-quote mark (|"|), because the standard ^{ASCII} code
for computers was not invented with book publishing in mind. However, your
terminal probably does have two flavors of single-quote marks, namely |`|
and |'|; the second of these is useful also as an ^{apostrophe}.
American keyboards usually contain a left-quote character that shows up
as something like {\tt\char'22}, and an apostrophe or right-quote that
looks like {\tt\char'15} or {\tt\char'23}.
To produce double-quote marks with \TeX, you simply type two single-quote marks
of the appropriate kind. For example, to get the phrase
\begindisplay
``I understand.''
\enddisplay
(including the quotation marks) you should type
\begintt
``I understand.''
\endtt
to your computer.
A typewriter-like style of type will be used throughout this manual to indicate
\TeX\ constructions that you might type on your terminal, so that the
symbols actually typed are readily distinguishable from the output \TeX\ would
produce and from the comments in the manual itself. Here are the symbols to be
used in the examples:
\begintt
ABCDEFGHIJKLMNOPQRSTUVWXYZ
abcdefghijklmnopqrstuvwxyz
0123456789"#$%&@*+-=,.:;?!
()<>[]{}`'\||/_^~
\endtt
If your computer terminal doesn't happen to have all of these, don't
despair; \TeX\ can make~do with the ones you have. An additional symbol
\begindisplay
\]
\enddisplay
is used to stand for a {\sl ^{blank space}}, in case it is important
to emphasize that a blank space is being typed; thus, what you {\sl
really\/} type in the example above is
\begintt
``I|]understand.''
\endtt
Without such a symbol you would have
difficulty seeing the invisible parts of certain constructions. But we
won't be using `\]' very often, because spaces are usually visible enough.
Book printing differs significantly from ordinary typing with respect to
^{dashes}, ^{hyphens}, and ^{minus signs}. In good math books,
these symbols are all different; in fact there usually are at least four
different symbols:
\begindisplay
a hyphen (-);\cr
an en-dash (--);\cr
an em-dash (---);\cr
a minus sign ($-$).\cr
\enddisplay
Hyphens are used for compound words like `daughter-in-law' and `X-rated'.
^{En-dash}es are used for number ranges like `pages 13--34', and also in
contexts like `exercise 1.2.6--52'. ^{Em-dash}es are used for punctuation in
sentences---they are what we often call simply dashes. And minus signs are
used in formulas. A conscientious user of \TeX\ will be careful to distinguish
these four usages, and here is how to do it:
\begindisplay
for a hyphen, type a hyphen (|-|);\cr
for an en-dash, type two hyphens (|--|);\cr
for an em-dash, type three hyphens (|---|);\cr
for a minus sign, type a hyphen in mathematics mode (|$-$|).\cr
\enddisplay
(Mathematics mode occurs between dollar signs; it is discussed later, so you
needn't worry about it now.)
\exercise Explain how to type the following sentence to \TeX: Alice said,
``I always use an en-dash instead of a hyphen when specifying page numbers
like `480--491' in a ^{bibliography}.''
\answer |Alice said, ``I always use an en-dash instead of a hyphen when|\break
|specifying page numbers like `480--491' in a bibliography.''| \
(The wrong answer to this question ends with |'480-49l' in a bibliography."|)
\exercise What do you think happens when you type four hyphens in a row?
\answer You get em-dash and hyphen (----), which looks awful.
If you look closely at most well-printed books, you will find that certain
combinations of letters are treated as a unit. For example, this is true
of the `f' and the `i' of `find'. Such combinations are called {\sl
^{ligatures}}, and professional typesetters have traditionally been
trained to watch for letter combinations such as |ff|, |fi|, |fl|, |ffi|, and
|ffl|. \ (The reason is that words like `f{}ind' don't look very good in
most styles of type unless a ligature is substituted for the letters that
clash. It's somewhat surprising how often the traditional ligatures appear
in English; other combinations are important in other languages.)
\exercise Think of an English word that contains two ligatures.
\answer fluffier firefly fisticuffs, flagstaff fireproofing,
chiffchaff and riffraff.
The good news is that you do
{\sl not\/} have to concern yourself with ligatures: \TeX\ is perfectly
capable of handling such things by itself, using the
same mechanism that converts `|--|' into `--'. In fact, \TeX\ will also look
for combinations of adjacent letters (like `|A|' next to `|V|'\thinspace)
that ought to be moved closer together for better appearance; this is
called {\sl ^{kerning}}.
\medbreak
To summarize this chapter: When using \TeX\ for straight copy, you type
the copy as on an ordinary typewriter, except that you need to be careful
about quotation marks, the number 1, and various kinds of hyphens/dashes.
\TeX\ will automatically take care of other niceties like ligatures and
kerning.
\danger (Are you sure you should be reading this paragraph? The
``^{dangerous bend}'' sign here is meant to warn you about material that
ought to be skipped on first reading. And maybe also on second reading.
The reader-beware paragraphs sometimes refer to concepts that aren't
explained until later chapters.)
\danger If your keyboard does not contain a left-quote symbol, you can
type ^|\lq|, followed by a space if the next character is a letter, or
followed by a |\| if the next character is a space. Similarly, ^|\rq|
yields a right-quote character. Is that clear?
\begintt
\lq\lq|]I|]understand.\rq\rq\|]
\endtt
\danger In case you need to type ^{quotes within quotes}, for example a
single quote followed by a double quote, you can't simply type
\thinspace|'''|\thinspace\ because \TeX\ will interpret this as '''
(namely, double quote followed by single quote). If you have already read
Chapter~5, you might expect that the solution will be to use
grouping---namely, to type something like \thinspace|{'}''|. But it turns
out that this doesn't produce the desired result, because there is usually
less space following a single right quote than there is following a double
right quote: What you get is {'}'', which is indeed a single quote
followed by a double quote (if you look at it closely enough), but it
looks almost like three equally spaced single quotes. On the other hand,
you certainly won't want to type \thinspace|'|\]|''|, because that space
is much too large---it's just as large as the space between words---and
\TeX\ might even start a new line at such a space when making up a
paragraph! The solution is to type \thinspace|'\thinspace''|, which
produces '\thinspace'' as desired.^^|\thinspace|
\dangerexercise OK, now you know how to produce ''' and '\thinspace'';
how do you get ``\thinspace` and `{}``\thinspace?
\answer |``\thinspace`|; and either |`{}``| or |{`}``| or something similar.
Reason: There's usually less space {\sl preceding\/} a single left quote than
there is preceding a double left quote. \ (Left and right are opposites.)
\dangerexercise Why do you think the author introduced the control
sequence |\thinspace| to solve the adjacent-quotes problem, instead of
recommending the trickier construction |'$|^|\,||$''| (which also works)?
\answer Eliminating ^|\thinspace| would mean that a user need not learn
the term; but it is not advisable to minimize terminology by ``overloading''
math mode with tricky constructions. For example, a user who wishes to
take advantage of \TeX's ^|\mathsurround| feature would be thwarted by
non-mathematical uses of dollar signs. \ (Incidentally, neither |\thinspace|
nor ^|\,| are built into \TeX; both are defined in terms of more
primitive features, in Appendix~B.)
\endchapter
In modern Wit all printed Trash, is
Set off with num'rous\/ {\rm Breaks}\raise.5ex\vbox{\hrule width 2em}%
and\/ {\rm Dashes}\raise.5ex\vbox{\hrule width 1em}
% no period after the em-dash: stet!
% Sir Walter Scott ruined this quote in his edition of Swift!
\author JONATHAN ^{SWIFT}, {\sl On Poetry: A Rapsody\/} (1733) % lines 93--94
% Rapsody: stet!
\bigskip
Some compositors still object to work
in offices where type-composing machines are introduced.
\author WILLIAM STANLEY ^{JEVONS}, {\sl Political Economy\/} (1878) % sec 55
% "They are all afraid that if the work is done too easily and rapidly,
% they will not be wanted to do it."
% Jevons goes on to say that justifying and page makeup can't be done
% profitably by machines, so the employees needn't fear losing their jobs.
\eject
\beginchapter Chapter 3. Controlling\\\TeX
Your keyboard has very few keys compared to the large number of symbols that you
may want to specify. In order to make a limited keyboard sufficiently versatile,
one of the characters that you can type is reserved for special use, and
it is called the {\sl ^{escape character}}. Whenever you want to type
something that controls the format of your manuscript, or something that
doesn't use the keyboard in the ordinary way, you should type the escape
character followed by an indication of what you want to do.
Note: Some computer terminals have a key marked `|ESC|', but that is {\sl not\/}
your escape character! It is a key that sends a special message to the operating
system, so don't confuse it with what this manual calls ``escape.''
\TeX\ allows any character to be used for escapes, but the ``^{backslash}''
character `|\|' is usually adopted for this purpose, since backslashes are
reasonably convenient to type and they are rarely needed in ordinary text.
Things work out best when different \TeX\ users do things consistently,
so we shall escape via backslashes in all the examples of this manual.
Immediately after typing `|\|' (i.e., immediately after an escape
character) you type a coded command telling \TeX\ what you have in mind.
Such commands are called {\sl ^{control sequences}}. For example, you might type
^^{markup commands, see control sequences}
\begintt
\input MS
\endtt
which (as we will see later) causes \TeX\ to begin reading a file called
`|MS.tex|'; the string of characters `^|\input|' is a control sequence.
Here's another example:
\begintt
George P\'olya and Gabor Szeg\"o.
\endtt
% sic; this is the spelling used in opening pages of their famous books
% but I give the Hungarian spellings in the index
\TeX\ converts this to `George P\'olya and Gabor Szeg\"o.' There are two
^^{Polya}^^{Szego}^^{acute}^^{umlaut}
control sequences, ^|\'| and ^|\"|, here; these control sequences
have been used to place ^{accents} over some of the letters.
Control sequences come in two flavors. The first kind, like |\input|,
is called a {\sl^{control word}\/}; it
consists of an escape character followed by one or more {\sl letters}, followed
by a space or by something besides a letter. \ (\TeX\ has to know where the
control sequence ends, so you must put a space after a control word if
the next character is a letter. For example, if you type `|\inputMS|',
\TeX\ will naturally interpret this as a control word with seven
letters.) \ In case you're wondering what a ``^{letter}'' is, the answer
is that \TeX\ normally regards the 52 symbols |A...Z| and |a...z| as
letters. The digits |0...9| are {\sl not\/} considered to be
letters, so they don't appear in control sequences of the first kind.
A control sequence of the other kind, like |\'|, is called a {\sl
^{control symbol}\/}; it consists of the escape
character followed by a single {\sl nonletter}. In this case you don't
need a space to separate the control sequence from a letter that follows,
since control sequences of the second kind always have exactly one
symbol after the escape character.
\exercise What are the control sequences in `|\I'm \exercise3.1\\!|'\thinspace?
\answer |\I|, |\exercise|, and |\\|. (The last of these is of type~2, i.e.,
a control symbol, since the second backslash is not a letter; the first
backslash keeps the second one from starting its own control sequence.)
\exercise We've seen that the input |P\'olya| yields `P\'olya'. Can
you guess how the French words `math\'ematique' and `centim\`etre'
should be specified?
\answer |math\'ematique| and |centim\`etre|.^^|\'|^^|\`|
When a space comes after a control word (an all-letter control
sequence), it is ignored by
\TeX; i.e., it is not considered to be a ``real'' space belonging to the
manuscript that is being typeset. But when a space comes after a control
symbol, it's truly a space.
Now the question arises, what do you do if you actually {\sl want\/} a
space to appear after a control word? We will see later that \TeX\
treats two or more consecutive spaces as a single ^{space}, so the answer
is {\sl not\/} going to be ``type two spaces.'' The correct answer is to
type ``control space,'' ^^|\ | namely
\begintt
\|]
\endtt
(the escape character followed by a blank space); \TeX\ will treat this as
a space that is not to be ignored. Notice that |\|\] is a control
sequence of the second kind, namely a control symbol, since there is a
single nonletter (\]) following the escape character. Two consecutive
spaces are considered to be equivalent to a single space, so further
spaces immediately following |\|\] will be ignored. But if you want to
enter, say, three consecutive spaces into a manuscript you can type
`|\|\]|\|\]|\|\]'. Incidentally, typists are often taught to put two
spaces at the ends of sentences; but we will see later that \TeX\ has its
own way to produce extra space in such cases. Thus you needn't be
consistent in the number of spaces you type.
\danger Nonprinting control characters like \<return> might follow
an escape character, and these lead to distinct control sequences according
to the rules. \TeX\ is initially set up to treat |\|\<return> and
|\|\<tab> ^^|\<return>|^^|\<tab>|
the same as |\|\] (control space); these special control sequences
should probably not be redefined, because you can't see the difference
between them when you look at them in a file.
^^{carriage-return, see <return>}
It is usually unnecessary for you to use ``control space,'' since control
sequences aren't often needed at the ends of words. But here's an example
that might shed some light on the matter: This manual itself has been
typeset by \TeX, and one of the things that occurs fairly often is the
tricky ^{logo} `\TeX', which requires backspacing and lowering the E.
There's a special control word
\begintt
\TeX
\endtt
that produces the half-dozen or so instructions necessary to typeset `\TeX'.
When a phrase like `\TeX\ ignores spaces after control words.' is
desired, the manuscript renders it as follows:
\begintt
\TeX\ ignores spaces after control words.
\endtt
Notice the extra |\| following ^|\TeX|; this produces the control space
that is necessary because \TeX\ ignores spaces after control words.
Without this extra |\|, the result would have been
\begindisplay
\TeX ignores spaces after control words.
\enddisplay
On the other hand, you can't simply put |\| after |\TeX| in all contexts.
For example, consider the phrase
\begintt
the logo `\TeX'.
\endtt
In this case an extra backslash doesn't work at all; in fact,
you get a curious result if you type
\begintt
the logo `\TeX\'.
\endtt
Can you guess what happens? \ Answer: The |\'| is a control sequence denoting
an acute accent, as in our |P\'olya| example above; the effect is
therefore to put an accent over the next nonblank character,
which happens to be a period. In other words, you get an accented
period, and the result is
\begindisplay
the logo `\TeX\'.
\enddisplay
Computers are good at following instructions, but not at reading your mind.
\TeX\ understands about 900 control sequences as part of its built-in
vocabulary, and all of them are explained in this manual somewhere. But
you needn't worry about learning so many different things, because you won't
really be needing very many of them unless you are faced with unusually
complicated copy. Furthermore, the ones you do need to learn actually fall into
relatively few categories, so they can be assimilated without great difficulty.
For example, many of the control sequences are simply the names of special
characters used in math formulas; you type `^|\pi|'~to get~`$\pi$',
`^|\Pi|'~to get~`$\Pi$',
`^|\aleph|'~to get~`$\aleph$',
`^|\infty|'~to get~`$\infty$',
`^|\le|'~to get~`$\le$',
`^|\ge|'~to get~`$\ge$',
`^|\ne|'~to get~`$\ne$',
`^|\oplus|'~to get~`$\oplus$',
`^|\otimes|'~to get~`$\otimes$'.
Appendix~F contains several tables of such symbols.
\danger There's no built-in relationship between ^{uppercase} and ^{lowercase}
letters in control sequence names. For example, `|\pi|' and `|\Pi|'
and `|\PI|' and `|\pI|' are four different control words.
The 900 or so control sequences that were just mentioned actually aren't
the whole story, because it's easy to define more. For example, if you
want to substitute your own favorite names for math symbols, so that you
can remember them better, you're free to go right ahead and do it;
Chapter~20 explains how.
About 300 of \TeX's control sequences are called {\sl ^{primitive}\/}; these
are the low-level atomic operations that are not decomposable into simpler
functions. All other control sequences are defined, ultimately, in terms
of the primitive ones. For example, ^|\input| is a primitive operation,
but ^|\'| and ^|\"| are not; the latter are defined in terms of an
^|\accent| primitive.
People hardly ever use \TeX's primitive control sequences in their
manuscripts, because the primitives are $\ldots$ well $\ldots$ so
{\sl primitive}. You have to type a lot of instructions when you are
trying to make \TeX\ do low-level things; this takes time and invites
mistakes. It is generally better to make use of higher-level control
sequences that state what functions are desired, instead of typing
out the way to achieve each function each time. The higher-level control
sequences need to be defined only once in terms of primitives. For
example, |\TeX| is a control sequence that means ``typeset the \TeX\ logo'';
|\'| is a control sequence that means ``put an acute accent over the
next character''; and both of these control sequences might require different
combinations of primitives when the style of type changes. If \TeX's logo
were to change, the author would simply have to change one definition, and the
changes would appear automatically wherever they were needed. By contrast,
an enormous amount of work would be necessary to change the logo if it
were specified as a sequence of primitives each time.
At a still higher level, there are control sequences that govern the
overall format of a document. For example, in the present book the author
typed `^|\exercise|' just before stating each exercise; this |\exercise|
command was programmed to make \TeX\ do all of the following things:
\nobreak\medskip
\item\bull compute the exercise number (e.g., `3.2' for the second
exercise in Chapter~3);
\smallskip
\item\bull typeset `\thinspace{\manual\char'170\rm\kern.15em
\ninebf EXERCISE \bf3.2}' with the appropriate typefaces, on a line by
itself, and with the triangle sticking out in the left margin;
\smallskip
\item\bull leave a little extra space just before that line, or begin
a new page at that line if appropriate;
\smallskip
\item\bull prohibit beginning a new page just after that line;
\smallskip
\item\bull suppress indentation on the following line.
\medbreak\noindent
It is obviously advantageous to avoid typing all of these individual
instructions each time. And since the manual is entirely described in
terms of high-level control sequences, it could be printed in a radically
different format simply by changing a dozen or so definitions.
% and sweating over the page layout in the math and alignment chapters!
\danger How can a person distinguish a \TeX\ primitive from a control sequence
that has been defined at a higher level? There are two ways: \ (1)~The index
to this manual lists all of the control sequences that are discussed, and each
primitive is marked with an asterisk. \ (2)~You can display the meaning of a
control sequence while running \TeX\null. If you type `^|\show||\cs|'
where |\cs| is any control sequence, \TeX\ will respond with its current
meaning. For example, `|\show\input|' results in \hbox{`|> \input=\input.|'},
because |\input| is primitive. On the other hand, `|\show|^|\thinspace|' yields
\begintt
> \thinspace=macro:
->\kern .16667em .
\endtt
This means that |\thinspace| has been defined as an abbreviation for
`|\kern|~|.16667em|~'. By typing `|\show|\penalty0|\kern|' you can verify
that ^|\kern| is primitive. The results of\/ |\show| appear on your
terminal and in the ^{log file} that you get after running \TeX.
\dangerexercise Which of the control sequences |\|\] and
|\|\<return> is primitive?
\answer According to the index, |\|\] is primitive but
|\|\<return> isn't. The command `|\def\^^M{\ }|' in
Appendix~B is what actually defines |\|\<return>, since a
return is representable as |^^M|. Asking \TeX\ to |\show\^^M|
\looseness-1
produces the response `|>| |\^^M=macro:->\|\]|.|'.
In the following chapters we shall frequently discuss ``^{plain \TeX}''
format, which is a set of about 600 ^{basic control sequences} that are
defined in Appendix~B\null. These control sequences, together with the 300
or so primitives, are usually present when \TeX\ begins to process a
manuscript; that is why \TeX\ claims to know roughly 900 control sequences
when it starts. We shall see how plain \TeX\ can be used to create
documents in a flexible format that meets many people's needs, using some
typefaces that come with the \TeX\ system. However, you should keep in
mind that plain \TeX\ is only one of countless ^{formats} that can be
designed on top of \TeX's primitives; if you want some other format, it
will usually be possible to adapt \TeX\ so that it will handle whatever
you have in mind. The best way to learn is probably to start with plain
\TeX\ and to change its definitions, little by little, as you gain more
experience.
\danger Appendix E contains examples of formats that can be added to
Appendix~B for special applications; for example, there is a set of
definitions suitable for business correspondence. A complete specification
of the format used to typeset this manual also appears in Appendix~E\null.
Thus, if your goal is to learn how to design \TeX\ formats, you will
probably want to study Appendix~E while mastering Appendix~B\null. After you
have become skilled in the lore of control-sequence definition, you
will probably have developed some formats that other people will want
to use; you should then write a supplement to this manual, explaining
your style rules.
The main point of these remarks, as far as novice \TeX\ users are concerned, is
that it is indeed possible to define nonstandard \TeX\ control sequences.
When this manual says that something is part of ``plain \TeX,'' it means
that \TeX\ doesn't insist on doing things exactly that way; a person
could change the rules by changing one or more of the definitions in
Appendix~B\null. But you can safely rely on the control sequences of plain
\TeX\ until you become an experienced \TeX nical~typist.
\ddangerexercise How many different control sequences of length~2
(including the escape character) are possible? How many of length~3?
\answer There are 256 of length~2; most of these are undefined when \TeX\
begins. \ (\TeX\ allows any character to be an escape, but it does not
distinguish between control sequences that start with different escape
characters.) \
If we assume that there are 52 letters, there are exactly $52^2$
possible control sequences of length~3 (one for each pair of letters, from
|AA| to |zz|). But Chapter~7 explains how to use ^|\catcode| to change any
character into a ``^{letter}''; therefore it's possible to use any of
$256^2$ potential control sequences of length~3.
\endchapter
Syllables govern the world.
\author JOHN ^{SELDEN}, {\sl Table Talk\/} (1689) % section on Power
\bigskip
I claim not to have controlled events,
but confess plainly that events have controlled me.
\author ABRAHAM ^{LINCOLN} (1864) % letter to A. G. Hodges, April 4
\eject
\beginchapter Chapter 4. Fonts\\of Type
Occasionally you will want to change from one ^{typeface} to another, for
example if you wish to be {\bf ^{bold}} or to {\sl emphasize\/} something.
\TeX\ deals with sets of up to 256 characters called ``^{fonts}'' of type,
and control sequences are used to select a particular font. For example,
you could specify the last few words of the first sentence above
in the following way, using the plain \TeX\ format of Appendix~B:
\begintt
to be \bf bold \rm or to \sl emphasize \rm something.
\endtt
Plain \TeX\ provides the following control sequences for changing fonts:
\begindisplay
^|\rm| switches to the normal ``roman'' typeface:&Roman\cr
^|\sl| switches to a slanted roman typeface:&\sl Slanted\cr
^|\it| switches to italic style:&\it Italic\cr
^|\tt| switches to a typewriter-like face:&\tt Typewriter\cr
^|\bf| switches to an extended boldface style:&\bf Bold\cr
\enddisplay
^^{typewriter type}^^{face}
At the beginning of a run you get ^{roman type} (|\rm|) unless you specify
otherwise.
Notice that two of these faces have an ``^{oblique}'' slope for emphasis:
{\sl ^{Slanted type} is essentially the same as roman, but the letters are
slightly skewed, \it while the letters in ^{italic type} are drawn in a
different style.} \ (You can perhaps best appreciate the difference between
the roman and italic styles by contemplating {\tenu letters that are
in an unslanted italic face.}) \ Typographic conventions are presently
in a state of transition, because new technology has made it possible to
do things that used to be prohibitively expensive; people are wrestling
with the question of how much to use their new-found typographic freedom.
Slanted roman type was introduced in the 1930s, but it first became widely
used as an alternative to the conventional italic during the late 1970s.
It can be bene\-ficial in mathematical texts, since slanted
letters are distinguishable from the italic letters in math formulas.
The double use of italic type for two different purposes---for example,
when statements of theorems are italicized as well as the names of variables in
those theorems---has led to some confusion, which can now be
avoided with slanted type. People are not generally agreed about the relative
merits of slanted versus italic, but slanted type is rapidly becoming a
favorite for the titles of books and journals in bibliographies.
Special fonts are effective for emphasis, but not for sustained reading;
your eyes would tire if long portions of this manual were entirely set in
a bold or slanted or italic face. Therefore roman type accounts for the
bulk of most typeset material. But it's a nuisance to say `|\rm|' every
time you want to go back to the roman style, so \TeX\ provides an easier
way to do it, using ``^{curly brace}^^{brace}'' symbols: You can switch
fonts inside the special symbols |{| and |}|, without affecting the fonts
outside. For example, the displayed phrase at the beginning of this
chapter is usually rendered
\begintt
to be {\bf bold} or to {\sl emphasize} something.
\endtt
This is a special case of the general idea of ``^{grouping}'' that we shall
discuss in the next chapter. It's best to forget about the first way of
changing fonts, and to use grouping instead; then your \TeX\ manuscripts
will look more natural, and you'll probably
never\footnote*{Well \dots, hardly ever.} have to type `|\rm|'.
\exercise Explain how to type the bibliographic reference `Ulrich ^{Dieter},
{\sl Journal f\"ur die reine und angewandte Mathematik\/ \bf201} (1959),
37--70.' [Use grouping.]
\answer |Ulrich Dieter, {\sl Journal f\"ur die reine und angewandte|\parbreak
|Mathematik\/ \bf201} (1959), 37--70.|\par\nobreak\smallskip\noindent
It's convenient to use a single group for both |\sl| and |\bf| here. The
`|\/|' is a refinement that you might not understand until you read the
rest of Chapter~4.
We have glossed over an important aspect of quality in the preceding
discussion. Look, for example, at the {\it italicized} and {\sl slanted}
words in this sentence. Since italic and slanted styles slope to the right,
the d's stick into the spaces that separate these words from the roman
type that follows; as a result, the spaces appear to be too skimpy,
although they are correct at the base of the letters. To equalize the
effective white space, \TeX\ allows you to put the special control sequence
`^|\/|' just before switching back to unslanted letters. When you type
\begintt
{\it italicized\/} and {\sl slanted\/} words
\endtt
you get {\it italicized\/} and {\sl slanted\/} words that look better.
The `|\/|' tells \TeX\ to add an\break % makes the line tighter, to be fair
``{\sl^{italic correction}\/}'' to the
previous letter, depending on that letter; this correction is about four
times as much for an `$f$' as for a `$c$', in a typical italic font.
Sometimes the italic correction is not desirable, because other factors take
up the visual slack. The standard rule of thumb is to use |\/| just before
switching from slanted or italic to roman or bold, unless the next
character is a period or comma. For example, type
\begintt
{\it italics\/} for {\it emphasis}.
\endtt
Old manuals of style say that the ^{punctuation} after a word should be in the
{\it same\/} font as that {\it word;\/} but an italic semicolon often looks
wrong, so this convention is changing. When an italicized word occurs
just before a semicolon, the author recommends typing `|{\it word\/};|'.
\exercise {\it Explain how to typeset a\/ {\rm roman} word in the midst
of an italicized sentence.}
\answer |{\it Explain ... typeset a\/ {\rm roman} word ... sentence.}|
Note the position of the italic correction in this case.
\danger Every letter of every font has an italic correction, which you can
bring to life by typing |\/|. The correction is usually zero in unslanted
styles, but there are exceptions: To typeset a bold `{\bf f\/}' in quotes,
you should say |a| |bold| \hbox{|`{\bf f\/}'|}, lest you get a bold `{\bf f}'.
\ddangerexercise Define a control sequence |\ic| such that `|\ic c|' puts the
italic correction of character $c$ into \TeX's register |\dimen0|.
\answer |\def\ic#1{\setbox0=\hbox{#1\/}\dimen0=\wd0|\parbreak
|\setbox0=\hbox{#1}\advance\dimen0 by -\wd0}|.
\ddanger The primitive control sequence ^|\nullfont| stands for a font that
has no characters. This font is always present, in case you haven't
specified any others.
Fonts vary in size as well as in shape. For example, the font you are now
reading is called a ``10-point'' font, because certain features of its
design are 10 ^{points} apart, when measured in printers' units. \ (We
will study the point system later; for now, it should suffice to point out
that the parentheses around this sentence are exactly 10 points tall---and
the em-dash is just 10 points wide.) \ The ``^{dangerous bend}''
sections of this manual are set in 9-point type, the foot\-notes in 8-point,
^{subscripts} in 7-point or 6-point, sub-subscripts in 5-point.
Each font used in a \TeX\ manuscript is associated with a control sequence;
for example, the 10-point font in this paragraph is called ^|\tenrm|, and
the corresponding 9-point font is called ^|\ninerm|. The slanted fonts that
match |\tenrm| and |\ninerm| are called ^|\tensl| and ^|\ninesl|. These
control sequences are not built into \TeX, nor are they the actual names
of the fonts; \TeX\ users are just supposed to make up convenient names,
whenever new fonts are introduced into a manuscript. Such control
sequences are used to change typefaces.
When fonts of different sizes are used simultaneously, \TeX\ will line the
letters up according to their ``^{baseline}s.'' For example, if you type
\begintt
\tenrm smaller \ninerm and smaller
\eightrm and smaller \sevenrm and smaller
\sixrm and smaller \fiverm and smaller \tenrm
\endtt
the result is {smaller \ninerm and smaller \eightrm and smaller
\sevenrm and smaller \sixrm and smaller \fiverm and smaller}. Of course
this is something that authors and readers aren't accustomed to, because
printers couldn't do such things with traditional lead types. Perhaps
poets who wish to speak in {\fiverm a still small voice} will cause future
books to make use of frequent font variations, but nowadays it's only
an occasional font freak {\fiverm(like the author of this manual)} who
likes such experiments. One should not get too carried away by the prospect
of font switching unless there is good reason.
An alert reader might well be confused at this point because we started out
this chapter by saying that `|\rm|' is the command that switches to roman
type, but later on we said that `|\tenrm|' is the way to do it. The truth
is that both ways work. But it has become customary to set things up so that
|\rm| means ``switch to roman type in the current size'' while |\tenrm| means
``switch to roman type in the 10-point size.'' In plain \TeX\ format, nothing
but 10-point fonts are provided, so |\rm| will always get you |\tenrm|; but
in more complicated formats the meaning of\/ |\rm| will change in different
parts of the manuscript. For example, in the format used by the author to
typeset this manual, there's a control sequence `^|\tenpoint|' that causes
|\rm| to mean |\tenrm|, |\sl| to mean |\tensl|, and so on, while
`^|\ninepoint|' changes the definitions so that |\rm| means |\ninerm|,
etc. There's another control sequence used to introduce the quotations at
the end of each chapter; when the quotations are typed, |\rm| and |\sl|
temporarily stand for {\eightss 8-point unslanted sans-serif type} and
{\eightssi 8-point slanted sans-serif type}, respectively. This device of
constantly redefining the abbreviations |\rm| and |\sl|, behind the
scenes, frees the typist from the need to remember what size or style of
type is currently being used.
\exercise Why do you think the author chose the names `|\tenpoint|' and
`|\tenrm|', etc., instead of `|\10point|' and `|\10rm|'\thinspace?
\answer Control word names are made of letters, not digits.
\dangerexercise Suppose that you have typed a manuscript using slanted type for
emphasis, but your editor suddenly tells you to change all the slanted to
italic. What's an easy way to do this?
\answer Say |\def\sl{\it}| at the beginning, and delete other definitions
of\/ |\sl| that might be present in your format file (e.g., there might be
one inside a |\tenpoint| macro).
\danger Each font has an external name that identifies it with respect to
all other fonts in a particular library. For example, the font in this
sentence is called `|cmr9|', which is an abbreviation for ``^{Computer
Modern} Roman 9~point.'' ^^{cm fonts} In order to prepare \TeX\ for
using this font, the command
\begintt
\font\ninerm=cmr9
\endtt
appears in Appendix E\null. In general you say `^|\font||\cs=|\<external
font name>' to load the information about a particular font into \TeX's
memory; afterwards the control sequence |\cs| will select that font for
typesetting. Plain \TeX\ makes only sixteen fonts available initially (see
Appendix~B and Appendix~F\null), but you can use |\font| to access
anything that exists in your system's font library.
\danger It is often possible to use a font at several different sizes, by
magnifying or shrinking the character images. Each font has a so-called
^{design size}, which reflects the size it normally has by default; for
example, the design size of |cmr9| is 9~points. But on many systems there is
also a range of sizes at which you can use a particular font, by scaling its
dimensions up or down. To load a scaled font into \TeX's memory, you
simply say `|\font\cs=|\<external font name> ^|at| \<desired size>'.
For example, the command
\begintt
\font\magnifiedfiverm=cmr5 at 10pt
\endtt
brings in 5-point Computer Modern Roman at twice its normal size. \ (Caution:
Before using this `|at|' feature, you should check to make sure that your
typesetter supports the font at the size in question; \TeX\ will accept any
\<desired size> that is positive and less than 2048 points, but the final
output will not be right unless the scaled font really is available on your
printing device.)
\danger What's the difference between |cmr5| |at| |10pt| and the normal
10-point font, |cmr10|? Plenty; a well-designed font will be drawn
differently at different point sizes, and the letters will often have
different relative heights and widths, in order to enhance readability.
\begindisplay
\tenrm Ten-point type is different from%
\magnifiedfiverm\ magnif{}ied f{}ive-point type.
\enddisplay
It is usually best to scale fonts only slightly with respect to
their design size, unless the final product is going to be photographically
reduced after \TeX\ has finished with it, or unless you are trying for an
unusual effect.^^{magnification}^^{reduction}
\danger Another way to magnify a font is to specify a scale factor that is
relative to the design size. For example, the command
\begintt
\font\magnifiedfiverm=cmr5 scaled 2000
\endtt
is another way to bring in the font ^|cmr5| at double size. The scale factor
is specified as an integer that represents a magnification ratio times~1000.
Thus, a scale factor of 1200 specifies magnification by 1.2, etc.
\dangerexercise State two ways to load font |cmr10| into \TeX's memory
at half its normal size.
\answer |\font\squinttenrm=cmr10 at 5pt|\parbreak
|\font\squinttenrm=cmr10 scaled 500|
\font\onerm=cmr10 scaled\magstep1
\font\onett=cmtt10 scaled\magstep1
\font\tworm=cmr10 scaled\magstep2
\font\twott=cmtt10 scaled\magstep2
%\font\threerm=cmr10 scaled\magstep3 % such large magnifications look ugly
%\font\threett=cmtt10 scaled\magstep3 % in a book context!
\danger At many computer centers it has proved convenient to supply fonts
at magnifications that grow in geometric ratios---something like equal-tempered
tuning on a ^{piano}. The idea is to have all fonts available at their true
size as well as at magnifications 1.2 and~1.44 (which is $1.2\times1.2$);
perhaps also at magnification~1.728 ($=1.2\times1.2\times1.2$) and even
higher. Then you can magnify an entire document by 1.2 or~1.44 and still
stay within the set of available fonts. Plain \TeX\ provides the
abbreviations ^|\magstep||0| for a scale factor of 1000, |\magstep1| for a
scaled factor of 1200, |\magstep2| for 1440, and so on up to |\magstep5|.
You say, for example,
\begintt
\font\bigtenrm=cmr10 scaled\magstep2
\endtt
to load font |cmr10| at $1.2\times1.2$ times its normal size.
\begindisplay \lineskip5pt
\tenrm\llap{``}This is {\tentt cmr10} at normal size
({\tentt \char`\\magstep0}).''\cr
\onerm\llap{``}This is {\onett cmr10} scaled once by 1.2
({\onett \char`\\magstep1}).''\cr
\tworm\llap{``}This is {\twott cmr10} scaled twice by 1.2
({\twott \char`\\magstep2}).''\cr
%\threerm\llap{``}This is {\threett cmr10} scaled by
% {\threett \char`\\magstep3}.''\cr
\enddisplay
(Notice that a little magnification goes a long way.) \
There's also ^|\magstephalf|, which magnifies by $\sqrt{1.2}$,
i.e., halfway between steps 0 and~1.
\danger Chapter~10 explains how to apply magnification to an entire
document, over and above any magnification that has been specified when
fonts are loaded. For example, if you have loaded a font that is scaled
by |\magstep1| and if you also specify ^|\magnification||=\magstep2|, the
actual font used for printing will be scaled by |\magstep3|. Similarly, if
you load a font scaled by |\magstephalf| and if you also say
|\magnification=\magstephalf|, the printed results will be
scaled by |\magstep1|.
\endchapter
Type faces---like people's faces---have distinctive features
indicating aspects of character. % I don't think he was kidding
\author MARSHALL ^{LEE}, {\sl Bookmaking\/} (1965) % page 83
\bigskip
This was the Noblest Roman of them all.
\author WILLIAM ^{SHAKESPEARE}, {\sl The Tragedie %
of Julius C\ae sar\/} (1599) % Act V, Scene 5, line 68
% For Shakespeare I'm using the spelling from First Folio (1623)
% (titles not from the contents page, but the running heads in the plays)
% but act/line numbers from The Riverside Shakespeare (throughout)
\eject
\beginchapter Chapter 5. Grouping
Every once in a while it is necessary to treat part of a manuscript as a
unit, so you need to indicate somehow where that part begins and where it
ends. For this purpose \TeX\ gives special interpretation to two
``^{grouping characters},'' which (like the escape character) are
treated differently from the normal symbols that you type. We assume in
this manual that |{| and |}| are the grouping characters, since they
are the ones used in plain \TeX. ^^{curly braces, see braces}
We saw examples of grouping in the previous chapter, where it was mentioned
that font changes inside a group do not affect the fonts in force outside.
The same principle applies to almost anything else that is defined inside
a group, as we will see later; for example, if you define a control sequence
within some group, that definition will disappear when the group ends.
In this way you can conveniently instruct \TeX\ to do something unusual,
by changing its normal conventions temporarily inside of a group; since
the changes are invisible from outside the group, there is no need to worry
about messing up the rest of a manuscript by forgetting to restore the
normal conventions when the unusual construction has been finished.
Computer scientists have a name for this aspect of grouping, because it's
an important aspect of programming languages in general; they call it
``^{block structure},'' and definitions that are in force only within
a group are said to be ``^{local}'' to that group.
You might want to use grouping even when you don't care about block
structure, just to have better control over spacing. For example, let's
consider once more the control sequence ^|\TeX| that produces
the logo `\TeX' in this manual: We observed in Chapter~3 that a blank space
after this control sequence will be gobbled up unless one types
`\hbox{|\TeX\ |}', yet it is a mistake to say `|\TeX\|' when the following
character is not a blank space. Well, in {\sl all\/} cases it would be
correct to specify the simple group
\begintt
{\TeX}
\endtt
whether or not the following character is a ^{space}, because the |}| stops
\TeX\ from absorbing an optional space into |\TeX|. This might come in
handy when you're using a text editor (e.g., when replacing all occurrences
of a particular word by a control sequence). Another thing you could do is
type
\begintt
\TeX{}
\endtt
using an {\sl empty\/} group for the same purpose: The `|{}|' here is a
group of no characters, so it produces no output, but it does have the
effect of stopping \TeX\ from skipping blanks.
^^{empty group} ^^{lbrace rbrace}
\exercise Sometimes you run into a rare word like `shelfful' that looks
better as `shelf{}ful' without the `ff' ^{ligature}. How can you fool
\TeX\ into thinking that there aren't two consecutive f's in such a word?
\answer |{shelf}ful| or |shelf{}ful|, etc.; or even |shelf\/ful|, which
yields a shelf\/ful instead of a shelf{\kern0pt}ful.
In fact, the latter idea---to
insert an ^{italic correction}---is preferable because \TeX\ will ^^|\/|
reinsert the ff ligature by itself after ^{hyphenating} |shelf{}ful|. \
(Appendix~H points out that ligatures are put into a hyphenated word that
contains no ``^{explicit kerns},'' and an italic correction is an
explicit kern.) \ But the italic correction may be too much (especially in an
italic font); |shelf{|^|\kern||0pt}ful| is often best.
\dangerexercise Explain how to get three blank spaces in a row without
using `|\|\]'.^^{control space}
\answer `\]|{|\]|}|\]' or `\]|{}|\]|{}|\]', etc. Plain \TeX\ also has a
^|\space| macro, so you can type |\space\space\space|. \ (These aren't
strictly equivalent to `|\|\]|\|\]|\|\]', since they adjust the spaces by
the current ``^{space factor},'' as explained later.)
But \TeX\ also uses grouping for another, quite different, purpose, namely
to determine how much of your text is to be governed by certain control
sequences. For example, if you want to center something on a line you can type
\begintt
\centerline{This information should be centered.}
\endtt
using the control sequence ^|\centerline| defined in plain \TeX\ format.
Grouping is used in quite a few of \TeX's more intricate instructions; and
it's possible to have groups within groups within groups, as you can see
by glancing at Appendix~B\null. Complex grouping is generally unnecessary,
however, in ordinary manuscripts, so you needn't worry about it. Just
don't forget to finish each group that you've started, because a lost
`|}|' might cause trouble.
Here's an example of two groups, one ^{nested} inside the other:
\begintt
\centerline{This information should be {\it centered}.}
\endtt
As you might expect, \TeX\ will produce a centered line that also contains
italics:
$$\hbox{This information should be {\it centered}.}$$
But let's look at the example more closely: `|\centerline|' appears outside
the curly braces, while `|\it|' appears inside. Why are the two cases
different? And how can a beginner learn to remember which is which?
Answer: |\centerline| is a control sequence that applies only to the very next
thing that follows, so you want to put braces around the text that is to
be centered (unless that text consists of a single symbol or control sequence).
For example, to center the \TeX\ logo on a line, it would suffice to
type `|\centerline\TeX|', but to center the phrase `\TeX\ has groups' you
need braces: `|\centerline{\TeX\ has groups}|'. On the other hand, |\it| is
a control sequence that simply means ``change the current font''; it acts
without looking ahead, so it affects {\sl everything\/} that follows, at
least potentially. The braces surround |\it| in order to confine the font
change to a local region.
In other words, the two sets of braces in this example actually have different
functions: One serves to treat several words of the text as if they
were a single object, while the other provides local block structure.
\exercise What do you think happens if you type the following:
\begintt
\centerline{This information should be {centered}.}
\centerline So should this.
\endtt
\answer In the first case, you get the same result as if the innermost
braces had not appeared at all, because you haven't used the grouping to
change fonts or to control spacing or anything. \TeX\ doesn't mind if you
want to waste your time making groups for no particular reason.
But in the second case, the necessary braces were forgotten. You get the
letter `S' centered on a line by itself, followed by a paragraph that
begins with `o should this.' on the next line.
\exercise And how about this one?
\begintt
\centerline{This information should be \it centered.}
\endtt
\answer You get the same result as if another pair of braces were present
around `|\it centered|', except that the period is typeset from the
italic font. \ (Both periods look about the same.) \ The |\it| font
will not remain in force after the |\centerline|, but this is
something of a coincidence: \TeX\ uses the braces to determine what
text is to be centered, but then it removes the braces. The
|\centerline| operation, as defined in Appendix~B\null, puts the
resulting braceless text inside {\sl another\/} group; and that's why
|\it| disappears after |\centerline|. \ (If you don't understand this,
just don't risk leaving out braces in tricky situations, and you'll be OK.)
\smallskip
\dangerexercise Define a control sequence |\ital| so that a user could type
`|\ital{text}|' instead of `|{\it text\/}|'. Discuss the pros and cons of
|\ital| versus |\it|.
\answer |\def\ital#1{{\it#1\/}}|. \ Pro:~Users might find this easier to
learn, because it works more like |\centerline| and they don't have to
remember to make the italic correction. \ Con:~To avoid the italic correction
just before a {\it comma} or {\it period}, users should probably be taught
another control sequence; for example, with
\begintt
\def\nocorr{\kern0pt }
\endtt
a user could type `|\ital{comma} or \ital{period\nocorr},|'. The alternative
of putting a period or comma in italics, to avoid the italic correction,
doesn't look as good. A long sequence of italics would be inefficient for
\TeX, since the entire text for the argument to |\ital| must be read into
memory only to be scanned again.
\ddanger Subsequent chapters describe many primitive operations of \TeX\ for
which the locality of grouping is important. For example, if one of \TeX's
internal parameters is changed within a group, the previous contents of
that parameter will be restored when the group ends. Sometimes, however,
it's desirable to make a definition that transcends its current group. This
effect can be obtained by prefixing `^|\global|' to the definition. For
example, \TeX\ keeps the current page number in a register called~|\count0|,
and the routine that outputs a page wants to increase the
^{page number}. ^{Output routines} are always protected by enclosing them
in groups, so that they do not inadvertently mess up the rest of \TeX; but
the change to |\count0| would disappear if it were kept local to the
output group. The command
\begintt
\global\advance\count0 by 1
\endtt
^^|\advance|
solves the problem; it increases |\count0| and makes this value stick around
at the end of the output routine. In general, |\global| makes the immediately
following definition pertain to all existing groups, not just to the
innermost one.
\ddangerexercise If you think you understand local and global definitions,
here's a little test to make sure: Suppose |\c| stands for `|\count1=|',
|\g| stands for `|\global\count1=|', and |\s| stands for
`|\showthe\count1|'. What values will be shown?
\begintt
{\c1\s\g2{\s\c3\s\g4\s\c5\s}\s\c6\s}\s
\endtt
\answer |{1 {2 3 4 5} 4 6} 4|.
\ddanger Another way to obtain block structure with \TeX\ is to use the
primitives ^|\begingroup| and ^|\endgroup|. These control sequences make it
easy to begin a group within one control sequence and end it within
another. The text that \TeX\ actually executes, after control sequences
have been expanded, must have properly ^{nested groups}, i.e., groups that
don't overlap. For example,
\begintt
{ \begingroup } \endgroup
\endtt
is not legitimate.
\ddangerexercise Define control sequences |\beginthe|\<block name> and
|\endthe|\<block name> that provide a ``named'' block structure. In other
words,
\begintt
\beginthe{beguine}\beginthe{waltz}\endthe{waltz}\endthe{beguine}
\endtt
should be permissible, but not
\begintt
\beginthe{beguine}\beginthe{waltz}\endthe{beguine}\endthe{waltz}.
\endtt
\answer |\def\beginthe#1{\begingroup\def\blockname{#1}}|\parbreak
|\def\endthe#1{\def\test{#1}%|\parbreak
| \ifx\test\blockname\endgroup|\parbreak
| \else\errmessage{You should have said|\parbreak
| \string\endthe{\blockname}}\fi}|
\endchapter
I have had recourse to varieties of type,
and to braces.
\author JAMES ^{MUIRHEAD}, {\sl The Institutes of Gaius\/} (1880) % p. xii
% he actually said braces for what we call brackets
\bigskip
An encounter group is a gathering, for a few hours or a few days,
of twelve or eighteen personable, responsible, certifiably normal
and temporarily smelly people.
\author JANE ^{HOWARD}, {\sl Please Touch\/} (1970)
\eject
\beginchapter Chapter 6. Running\\\TeX
The best way to learn how to use \TeX\ is to use it. Thus, it's high time
for you to sit down at a computer terminal and interact with the \TeX\
system, trying things out to see what happens. Here are some small but
complete examples suggested for your first encounter.
^^{Running the program}
Caution: This chapter is rather a long one. Why don't you stop reading
now, and come back fresh tomorrow?
\smallskip OK, let's suppose that you're rested and excited about having a
trial run of \TeX\null. Step-by-step instructions for using it appear in this
chapter. First do this: Go to the lab where the graphic output device is,
since you will be wanting to see the output that you get---it won't really
be satisfactory to run \TeX\ from a remote location, where you can't hold
the generated documents in your own hands. Then log in; and start \TeX. \
(You may have to ask somebody how to do this on your local computer. Usually
the operating system prompts you for a command and you type `|tex|' or
`|run| |tex|' or something like that.)
When you're successful, \TeX\ will welcome you with a message such as
\begintt
This is TeX, Version 3.141 (preloaded format=plain 89.7.15)
**
\endtt
The `^|**|' is \TeX's way of asking you for an input file name.
% Incidentally, 89.7.15 was Jill's 50th birthday.
Now type `^|\relax|' (including the backslash), and ^\<return>
(or whatever is used to mean ``end-of-line'' on your terminal).
\TeX\ is all geared up for action, ready to read a long manuscript; but
you're saying that it's all right to take things easy, since this is
going to be a real simple run. In fact, |\relax| is a control sequence
that means ``do nothing.''
The machine will type another asterisk at you. This time type something
like `|Hello?|'\ and wait for another ^{asterisk}. Finally type `^|\end|',
and stand back to see what happens.
\TeX\ should respond with `^|[1]|' (meaning that it has finished page~1
of your output); then the program will halt, probably with some
indication that it has created a file called `|texput.dvi|'. \ (\TeX\
uses the name ^|texput| for its output when you haven't specified any
better name in your first line of input; and ^|dvi| stands for
``^{device independent},'' since |texput.dvi| is capable of
being printed on almost any kind of typographic output device.)
Now you're going to need some help again from your friendly local
computer hackers. They will tell you how to produce hardcopy from
|texput.dvi|. And when you see the hardcopy---Oh, glorious day!---you
will see a magnificent `Hello?'\ and the page number `1' at the bottom.
Congratulations on your first masterpiece of fine printing.
\smallbreak
The point is, you understand now how to get something through the whole cycle.
It only remains to do the same thing with a somewhat longer document.
So our next experiment will be to work from a file instead of typing
the input online.
Use your favorite text editor to create a file called ^|story.tex|
that contains the following 18 lines of text (no more, no less):
$$\halign{\hbox to\parindent{\hfil\sevenrm#\ \ }&#\hfil\cr
1&|\hrule|\cr\noalign{^^|\hrule|}
2&|\vskip 1in|\cr\noalign{^^|\vskip|^^{leading, see vskip}}
3&|\centerline{\bf A SHORT STORY}|\cr\noalign{^^|\centerline|}
4&|\vskip 6pt|\cr
5&|\centerline{\sl by A. U. Thor}|\cr\noalign{^^{Thor}}
6&|\vskip .5cm|\cr
7&|Once upon a time, in a distant|\cr
8&| galaxy called \"O\"o\c c,|\cr\noalign{^^|\"|^^|\c|}
9&|there lived a computer|\cr
10&|named R.~J. Drofnats.|\cr\noalign{^^{Drofnats}}
11&||\cr
12&|Mr.~Drofnats---or ``R. J.,'' as|\cr
13&|he preferred to be called---|\cr
14&|was happiest when he was at work|\cr
15&|typesetting beautiful documents.|\cr
16&|\vskip 1in|\cr
17&|\hrule|\cr
18&|\vfill\eject|\cr\noalign{^^|\vfill|^^|\eject|}}$$
\write16{\ifnum\pageno=\storypage
\else Redefine \string\storypage to be \the\pageno\fi}%
(Don't type the numbers at the left of these lines, of course; they are present
only for reference.) \ This example is a bit long, and more than a bit silly;
but it's no trick for a good typist like you and it will give you some
worthwhile experience, so do it. For your own good. And think about what
you're typing, as you go; the example introduces a few important features
of \TeX\ that you can learn as you're making the file.
Here is a brief explanation of what you have just typed: Lines 1 and~17
put a horizontal ^{rule} (a thin line) across the page. Lines 2 and~16
skip past one inch of space; `|\vskip|' means ``vertical skip,'' and this
extra space will separate the horizontal rules from the rest of the copy.
Lines 3 and~5 produce the title and the author name, centered, in boldface
and in slanted type. Lines 4 and~6 put extra white space between those
lines and their successors. \ (We shall discuss units of measure like
`|6pt|' and `|.5cm|' in Chapter~10.)
The main bulk of the story appears on lines 7--15, and it consists of
two ^{paragraphs}. The fact that line~11 is blank informs \TeX\ that
^^{blank line} ^^{empty line}
line~10 is the end of the first paragraph; and the `|\vskip|' on line~16
implies that the second paragraph ends on line~15, because vertical
skips don't appear in paragraphs. Incidentally, this example seems
to be quite full of \TeX\ commands; but it is atypical in that respect,
because it is so short and because it is supposed to be teaching things.
Messy constructions like |\vskip| and |\centerline| can be expected at the
very beginning of a manuscript, unless you're using a canned format, but
they don't last long; most of the time you will find yourself typing
straight text, with relatively few control sequences.
And now comes the good news, if you haven't used computer typesetting
before: You don't have to worry about where to break lines in a paragraph
(i.e., where to stop at the right margin and to begin a new line), because
\TeX\ will do that for you. Your manuscript file can contain long lines or
short lines, or both; it doesn't matter. This is especially helpful when
you make changes, since you don't have to retype anything except the words
that changed. {\sl Every time you begin a new line in your manuscript file
it is essentially the same as typing a space.} When \TeX\ has read an
entire paragraph---in this case lines 7 to~11---it will try to break up
the text so that each line of output, except the last, contains about the
same amount of copy; and it will hyphenate words if necessary to keep the
spacing consistent, but only as a last resort.
Line 8 contains the strange concoction
\begintt
\"O\"o\c c
\endtt
and you already know that |\"| stands for an ^{umlaut} accent. The
|\c| stands for a ``^{cedilla},'' so you will get `\"O\"o\c c' as the
name of that distant galaxy.
The remaining text is simply a review of the conventions that we discussed
long ago for dashes and quotation marks, except that the `|~|' signs in
lines 10 and~12 are a new wrinkle. These are called {\sl ^{ties}}, because
they tie words together; i.e., \TeX\ is supposed to treat `|~|' as a
normal space but not to break between lines there.
A good typist will use ties within names, as shown in our
example; further discussion of ties appears in Chapter~14. ^^{tilde}
Finally, line~18 tells \TeX\ to `^|\vfill|', i.e., to fill the rest of
the page with white space; and to `^|\eject|' the page, i.e., to send it
to the output file.
\smallskip Now you're ready for Experiment~2: Get \TeX\ going again.
This time when the machine says `|**|' you should answer `|story|', since
that is the name of the file where your input resides. \ (The file
could also be called by its full name `|story.tex|', but \TeX\ automatically
supplies the suffix `|.tex|' if no suffix has been specified.)
^^{file names}
You might wonder why the first prompt was `^|**|', while the subsequent
ones are `^|*|'; the reason is simply that the first thing you type to
\TeX\ is slightly different from the rest: If the first character of your
response to `|**|' is not a backslash, \TeX\ automatically inserts
`^|\input|'. Thus you can usually run \TeX\ by merely naming your input
file. \ (Previous \TeX\ systems required you to start by typing `|\input
story|' instead of `|story|', and you can still do that; but most \TeX\
users prefer to put all of their commands into a file instead of typing
them online, so \TeX\ now spares them the nuisance of starting out with
|\input| each time.) \ Recall that in Experiment~1 you typed `|\relax|';
that started with a backslash, so |\input| was not implied.
\danger There's actually another difference between `|**|' and `|*|': If the
first character after |**| is an ^{ampersand} (\thinspace`|&|'\thinspace),
\TeX\ will replace its memory with a precomputed ^{format file} before
proceeding. Thus, for example, you can type `|&plain \input story|' or
even `|&plain story|' in response to `|**|', if you are running some
version of \TeX\ that might not have the plain format preloaded.
^^{preloaded formats}
\danger Incidentally, many systems allow you to invoke \TeX\ by typing a
one-liner like `|tex story|' instead of waiting for the `|**|'; similarly,
`|tex \relax|' works for Experiment~1, and `|tex &plain story|' loads the
plain format before inputting the |story| file. You might want to try
this, to see if it works on your computer, or you might ask somebody if
there's a similar shortcut.
As \TeX\ begins to read your story file, it types `|(story.tex|', possibly
with a version number for more precise identification, depending on your
local operating system. Then it types `|[1]|', meaning that page~1 is done;
and `|)|', meaning that the file has been entirely input.
\TeX\ will now prompt you with `|*|', because the file did not contain
`^|\end|'. Enter |\end| into the computer now, and you should get a file
|story.dvi| containing a typeset version of Thor's story. As in Experiment~1,
you can proceed to convert |story.dvi| into hardcopy; go ahead and do that now.
The typeset output won't be shown here, but you can see the results by
doing the experiment personally. Please do so before reading on.
\exercise Statistics show that only 7.43 of 10 people who read this manual
actually type the |story.tex| file as recommended, but that those people
learn \TeX\ best. So why don't you join them?
\answer Laziness and/or obstinacy.
\exercise Look closely at the output of Experiment~2, and compare it to
|story.tex|\thinspace: If you followed the instructions carefully, you
will notice a typographical error. What is it, and why did it sneak in?
\answer There's an unwanted space after `called---', because (as the book
says) \TeX\ treats the end of a line as if it were a blank space. That
blank space is usually what you want, except when a line ends with a
hyphen or a dash; so you should {\sc WATCH OUT} for lines that end with
hyphens or dashes.
With Experiment 2 under your belt, you know how to make a document from a
file. The remaining experiments in this chapter are intended to help you
cope with the inevitable anomalies that you will run into later; we will
intentionally do things that will cause \TeX\ to ``squeak.''
But before going on, it's best to fix the error revealed by the previous
output (see exercise 6.2): Line~13 of the |story.tex| file should be changed to
\begintt
he preferred to be called---% error has been fixed!
\endtt
The `|%|' sign here ^^{percent} is a feature of plain \TeX\ that we haven't
discussed before: It effectively terminates a line of your input file,
without introducing the blank space that \TeX\ ordinarily inserts when
moving to the next line of input. Furthermore, \TeX\ ignores everything
that you type following a |%|, up to the end of that line in the file;
you can therefore put ^{comments} into your manuscript, knowing that the
comments are for your eyes only.
Experiment 3 will be to make \TeX\ work harder, by asking it to set the
story in narrower and narrower columns. Here's how: After starting the
program, type
\begintt
\hsize=4in \input story
\endtt
in response to the `|**|'. This means, ``Set the story in a 4-inch column.''
More precisely, ^|\hsize| is a primitive of \TeX\ that specifies the
horizontal size, i.e., the width of each normal line in the output when a
paragraph is being typeset; and ^|\input| is a primitive that causes \TeX\
to read the specified file. Thus, you are instructing the machine to
change the normal setting of\/ |\hsize| that was defined by plain \TeX, and
then to process |story.tex| under this modification.
\TeX\ should respond by typing something like `|(story.tex [1])|' as
before, followed by `|*|'. Now you should type
\begintt
\hsize=3in \input story
\endtt
and, after \TeX\ says `|(story.tex [2])|' asking for more, type three more lines
\begintt
\hsize=2.5in \input story
\hsize=2in \input story
\end
\endtt
to complete this four-page experiment.
Don't be alarmed when \TeX\ screams `|Overfull| |\hbox|' several times as
it works at the 2-inch size; that's what was supposed to go wrong during
Experiment~3. There simply is no good way to break the given
paragraphs into lines that are exactly two inches wide, without making
the spaces between words come out too large or too small. Plain \TeX\
has been set up to ensure rather strict tolerances on all of the lines it
produces:
\begindisplay
\hbox spread-1em{You don't get spaces between words narrower than this,\ and}\cr
\hbox spread+1.679895em{you don't get spaces between words wider than this.}\cr
\enddisplay
If there's no way to meet these restrictions, you get an ^{overfull box}.
And with the overfull box you also get (1)~a warning message, printed
on your terminal, and (2)~a big black bar inserted at the right of the
offending box, in your output. \ (Look at page~4 of the output from
Experiment~3; the overfull boxes should stick out like sore thumbs.
On the other hand, pages 1--3 should be perfect.)
Of course you don't want overfull boxes in your output, so \TeX\ provides
several ways to remove them; that will be the subject of our Experiment~4.
But first let's look more closely at the results of Experiment~3, since
\TeX\ reported some potentially valuable information when it was forced
to make those boxes too full; you should learn how to read this data:
\begintt
Overfull \hbox (0.98807pt too wide) in paragraph at lines 7--11
\tenrm tant galaxy called []O^^?o^^Xc, there lived||
Overfull \hbox (0.4325pt too wide) in paragraph at lines 7--11
\tenrm a com-puter named R. J. Drof-nats. ||
Overfull \hbox (5.32132pt too wide) in paragraph at lines 12--16
\tenrm he pre-ferred to be called---was hap-||
\endtt
Each overfull box is correlated with its location in your input file
(e.g., the first two were generated when processing the paragraph on
lines 7--11 of |story.tex|), and you also learn by how much the copy
sticks out (e.g., 0.98807 points).
Notice that \TeX\ also shows the contents of the overfull boxes in
abbreviated form. For example, the last one has the words `he preferred
to be called---was hap-', set in font |\tenrm| (10-point roman type);
the first one has a somewhat curious rendering of `\"O\"o\c c', because the
accents appear in strange places within that font. In general, when you
see `^|[]|' in one of these messages, it stands either for
the paragraph indentation or for some sort of complex construction;
in this particular case it stands for an umlaut that has been raised
up to cover an `O'.
\dangerexercise Can you explain the `\|' that appears after
`|lived|' in that message?
\answer It represents the heavy bar that shows up in
your output. \ (This bar wouldn't be present if\/ ^|\overfullrule| had been
set to |0pt|, nor is it present in an underfull box.)
\ddangerexercise Why is there a space before the `\|' in `|Drof-nats.
|\|'\thinspace?
\answer This is the ^|\parfillskip| space that ends the paragraph.
In plain \TeX\ the parfillskip is zero when the last line of the paragraph
is full; hence no space actually appears before the rule in the output
of Experiment~3. But all hskips show up as spaces in an overfull box
message, even if they're zero.
You don't have to take out pencil and paper in order to write down the
overfull box messages that you get before they disappear from view, since
\TeX\ always writes a ``^{transcript}'' or ``^{log file}'' that records what
happened during each session. For example, you should now have a file
called |story.log| containing the transcript of Experiment~3, as well
as a file called |texput.log| containing the transcript of Experiment~1. \
(The transcript of Experiment~2 was probably overwritten when you did
number~3.) \ Take a look at |story.log| now; you will see that the overfull
box messages are accompanied not only by the abbreviated box contents,
but also by some strange-looking data about hboxes and glue and kerns and
such things. This data gives a precise description of what's in that
overfull box; \TeX\ wizards will find such
listings important, if they are called upon to diagnose some mysterious
error, and you too may want to understand \TeX's internal code some day.
The abbreviated forms of overfull boxes show the hyphenations that
\TeX\ tried before it resorted to overfilling. The ^{hyphenation} algorithm,
which is described in Appendix~H\null, is excellent but not perfect; for
example, you can see from the messages in |story.log| that \TeX\ finds the
hyphen in `pre-ferred', and it can even hyphenate `Drof-nats'. Yet it
discovers no hyphen in `galaxy', and every once in
a~while an overfull box problem can be cured simply by giving \TeX\ a hint
about how to hyphenate some word more completely. \ (We will see later that
there are two ways to do this, either by inserting ^{discretionary hyphens}
each time as in `\hbox{|gal\-axy|}', or by saying
`\hbox{|\hyphenation{gal-axy}|}' once at the beginning of your manuscript.)
In the present example, hyphenation is not a problem, since \TeX\ found
and tried all the hyphens that could possibly have helped. The only way to
get rid of the overfull boxes is to change the tolerance, i.e., to allow
wider spaces between words. Indeed, the tolerance that plain \TeX\ uses
for wide lines is completely inappropriate for 2-inch columns; such narrow
columns simply can't be achieved without loosening the constraints, unless
you rewrite the copy to fit.
\TeX\ assigns a numerical value called ``^{badness}'' to each line that
it sets, in order to assess the quality of the spacing. The exact rules
for badness are different for different fonts, and they will be discussed
in Chapter~14; but here is the way badness works for the roman font
of plain \TeX:
\begindisplay \hbadness10000
\hbox spread-.666667em{The badness of this line is 100.}&
\quad(very tight)\cr
\hbox spread-.333333em{The badness of this line is 12.}&
\quad(somewhat tight)\cr
\hbox{The badness of this line is 0.}&
\quad(perfect)\cr
\hbox spread.5em{The badness of this line is 12.}&
\quad(somewhat loose)\cr
%\hbox spread 1em{The badness of this line is 100.}&
% \quad(loose)\cr % then "looser"
\hbox spread 1.259921em{The badness of this line is 200.}&
\quad(loose)\cr
%\hbox spread 1.713em{The badness of this line is 500.}&
% \quad(bad)\cr % then "worse"
\hbox spread 2.155em{The badness of this line is 1000.}&
\quad(bad)\cr
\hbox spread 3.684em{The badness of this line is 5000.}& % actually 4995!
\quad(awful)\cr
\enddisplay
Plain \TeX\ normally stipulates that no line's badness should exceed 200;
but in our case, the task would be impossible since
\begindisplay \hbadness 10000
`\hbox to 2in{tant galaxy called \"O\"o\c c, there}'\hskip 3em
has badness 1521;\cr
`\hbox to 2in{he preferred to be called---was}'\hskip 3em
has badness 568.\cr
\enddisplay
So we turn now to Experiment~4, in which spacing variations that are
more appropriate to narrow columns will be used.
Run \TeX\ again, and begin this time by saying
\begintt
\hsize=2in \tolerance=1600 \input story
\endtt
so that lines with badness up to 1600 will be tolerated. Hurray! There are
^^|\tolerance|
no overfull boxes this time. \ (But you do get a message about an {\sl
underfull\/} box, since \TeX\ reports all boxes whose badness exceeds
a certain threshold called ^|\hbadness|; plain \TeX\ sets |\hbadness=1000|.) \
^^{underfull box}
Now make \TeX\ work still harder by trying
\begintt
\hsize=1.5in \input story
\endtt
(thus leaving the tolerance at 1600 but making the ^{column width} still
^^{measure, see hsize}
skimpier). Alas, overfull boxes return; so try typing
\begintt
\tolerance=10000 \input story
\endtt
in order to see what happens. \TeX\ treats 10000 as if it were ``infinite''
tolerance, allowing arbitrarily wide space; thus, a tolerance of 10000 will
{\sl never\/} produce an overfull box, unless something strange occurs like
an unhyphenatable word that is wider than the column itself.
The underfull box that \TeX\ produces in the 1.5-inch case is really bad;
with such narrow limits, an occasional wide space is unavoidable. But try
\begintt
\raggedright \input story
\endtt
for a change. \ ^^|\raggedright|(This tells \TeX\ not to worry about keeping
the right margin straight, and to keep the spacing uniform within each line.) \
Finally, type
\begintt
\hsize=.75in \input story
\endtt
followed by `|\end|', to complete Experiment 4. This makes the columns
almost impossibly narrow.
\danger The output from this experiment will give you some feeling for the
problem of ^{breaking a paragraph} into approximately equal lines. When the
lines are relatively wide, \TeX\ will almost always find a good solution.
But otherwise you will have to figure out some compromise, and several
options are possible. Suppose you want to ensure that no lines have
badness exceeding~500. Then you could set |\tolerance| to some high
number, and |\hbadness=500|; \TeX\ would not produce overfull boxes, but
it would warn you about the underfull ones. Or you could set
|\tolerance=500|; then \TeX\ might produce overfull boxes. If you really
want to take corrective action, the second alternative is better, because
you can look at an overfull box to see how much sticks out; it becomes
graphically clear what remedies are possible. On the other hand, if you
don't have time to fix bad spacing---if you just want to know how bad it
is---then the first alternative is better, although it may require more
computer time.
\dangerexercise When |\raggedright| has been specified, badness reflects
the amount of space at the right margin, instead of the spacing between
words. Devise an experiment by which you can easily determine what
badness \TeX\ assigns to each line, when the |story| is set ragged-right
in 1.5-inch columns.
\answer Run \TeX\ with \hbox{|\hsize=1.5in|} \hbox{|\tolerance=10000|}
\hbox{|\raggedright|} \hbox{|\hbadness=-1|} and then |\input story|. \TeX\ will
report the badness of all lines (except the final lines of paragraphs, where
fill glue makes the badness zero).
\danger A parameter called ^|\hfuzz| allows you to ignore boxes that are only
slightly overfull. For example, if you say |\hfuzz=1pt|, a box must stick
out more than one point before it is considered erroneous. Plain \TeX\
sets |\hfuzz=0.1pt|.
\ddangerexercise Inspection of the output from Experiment~4, especially
page~3, shows that with narrow columns it would be better to allow white
space to appear before and after a dash, whenever other spaces in the
same line are being stretched. Define a ^|\dash| macro that does this.
\answer |\def\extraspace{\nobreak \hskip 0pt plus .15em\relax}|\parbreak
|\def\dash{\unskip\extraspace---\extraspace}|\par\nobreak\smallskip\noindent
(If you try this with the story at 2-inch and 1.5-inch sizes, you will
notice a substantial improvement. The |\unskip| allows people to leave a
space before typing |\dash|. \TeX\ will try to hyphenate before |\dash|,
but not before `|---|'; cf.\ Appendix~H\null. The ^|\relax| at the end of
|\extraspace| is a precaution in case the next word is `|minus|'.)
You were warned that this is a long chapter. But take heart: There's only
one more experiment to do, and then you will know enough about \TeX\ to
run it fearlessly by yourself forever after. The only thing you are still
missing is some information about how to cope with ^{error messages}---i.e.,
not just with warnings about things like overfull boxes, but with cases
where \TeX\ actually stops and asks you what to do next.
Error messages can be terrifying when you aren't prepared for them;
but they can be fun when you have the right attitude. Just remember that
you really haven't hurt the computer's feelings, and that nobody will
hold the errors against you. Then you'll find that running \TeX\ might
actually be a creative experience instead of something to dread.
The first step in Experiment 5 is to plant two intentional mistakes in the
|story.tex| file. Change line~3 to
\begintt
\centerline{\bf A SHORT \ERROR STORY}
\endtt
and change `|\vskip|' to `|\vship|' on line~2.
Now run \TeX\ again; but instead of `|story|' type `|sorry|'. The computer
should respond by saying that it can't find file |sorry.tex|, and it will
ask you to try again. Just hit \<return> this time; you'll see
that you had better give the name of a real file. So type `|story|' and
wait for \TeX\ to find one of the {\sl faux pas\/} in that file.
Ah yes, the machine will soon stop,\footnote*{Some installations of \TeX\ do
not allow interaction. In such cases all you can do is look at the error
messages in your log file, where they will appear together with the ``help''
information.} after typing something like this:
\begintt
! Undefined control sequence.
l.2 \vship
1in
?
\endtt
\write16{\ifnum\pageno=\vshippage
\else Redefine \string\vshippage to be \the\pageno\fi}%
\TeX\ begins its error messages with `|!|', and it shows what it was
reading at the time of the error by displaying two lines of context. The
top line of the pair (in this case `|\vship|'\thinspace) shows what \TeX\
has looked at so far, and where it came from (`|l.2|', i.e., line number~2);
the bottom line (in this case `|1in|'\thinspace) shows what \TeX\ has yet
to read.
The `^|?|'\ that appears after the context display means that \TeX\ wants
advice about what to do next. If you've never seen an error message before,
or if you've forgotten what sort of response is expected, you can type
`|?|'\ now (go ahead and try it!); \TeX\ will respond as follows:
\begintt
Type <return> to proceed, S to scroll future error messages,
R to run without stopping, Q to run quietly,
I to insert something, E to edit your file,
1 or ... or 9 to ignore the next 1 to 9 tokens of input,
H for help, X to quit.
\endtt
This is your menu of options. You may choose to continue in various ways:
\smallskip\item{1.}
Simply type \<return>. \TeX\ will resume its processing, after
attempting to recover from the error as best it can.
\smallbreak\item{2.} Type `|S|'. \TeX\ will proceed without
pausing for instructions if further errors arise. Subsequent error messages
will flash by on your terminal, possibly faster than you can read them, and
they will appear in your log file where you can scrutinize them at your
leisure. Thus, `|S|' is sort of like typing \<return> to every message.
\smallbreak\item{3.} Type `|R|'. This is like `|S|' but even stronger,
since it tells \TeX\ not to stop for any reason, not even if a file name
can't be found.
\smallbreak\item{4.} Type `|Q|'. This is like `|R|' but even more so,
since it tells \TeX\ not only to proceed without stopping but also to
suppress all further output to your terminal. It is a fast, but somewhat
reckless, way to proceed (intended for running \TeX\ with no operator in
attendance).
\smallbreak\item{5.} Type `|I|', followed by some text that you want to
insert. \TeX\ will read this line of text before encountering what it
would ordinarily see next. Lines inserted in this way are not assumed to
end with a blank space. ^^{inserting text online}
^^{online interaction, see interaction} ^^{interacting with TeX}
\smallbreak\item{6.} Type a small number (less than 100). \TeX\ will
delete this many characters and control sequences from whatever it is
about to read next, and it will pause again to give you another chance to
look things over. ^^{deleting tokens}
\smallbreak\item{7.} Type `|H|'. This is what you should do now and whenever
you are faced with an error message that you haven't seen for a~while. \TeX\
has two messages built in for each perceived error: a formal one and an
informal one. The formal message is printed first (e.g., `|! Undefined
control sequence.|'\thinspace); the informal one is printed if you request
more help by typing `|H|', and it also appears in your log file if you
are scrolling error messages. The informal message tries to complement the
formal one by explaining what \TeX\ thinks the trouble is, and often
by suggesting a strategy for recouping your losses.^^{help messages}
\smallbreak\item{8.} Type `|X|'. This stands for ``exit.'' It causes \TeX\
to stop working on your job, after putting the finishing touches on your
|log| file and on any pages that have already been output to your |dvi|
file. The current (incomplete) page will not be output.
\smallbreak\item{9.} Type `|E|'. This is like `|X|', but it also prepares
the computer to edit the file that \TeX\ is currently reading, at the
current position, so that you can conveniently make a change before
trying again.
\smallbreak\noindent
After you type `|H|' (or `|h|', which also works), you'll get a message
that tries to explain that the control sequence just read by \TeX\
(i.e., |\vship|) has never been assigned a meaning, and that you should
either insert the correct control sequence or you should go on as if the
offending one had not appeared.
In this case, therefore, your best bet is to type
\begintt
I\vskip
\endtt
(and \<return>), with no space after the `|I|'; this effectively replaces
|\vship| by |\vskip|. \ (Do it.)
If you had simply typed \<return> instead of
inserting anything, \TeX\ would have gone ahead and read `|1in|', which
it would have regarded as part of a paragraph to be typeset. Alternatively,
you could have typed `|3|'\thinspace; that would have deleted
`|1in|' from \TeX's input. Or you could have typed `|X|' or `|E|' in
order to correct the spelling error in your file. But it's usually
best to try to detect as many errors as you can, each time you run \TeX,
since that increases your productivity while decreasing your computer bills.
Chapter~27 explains more about the art of steering \TeX\ through
troubled text.
\dangerexercise What would have happened if you had typed `|5|' after
the |\vship| error?
\answer \TeX\ would have deleted five tokens: |1|, |i|, |n|, \],
|\centerline|. (The space was at the end of line~2, the |\centerline| at the
beginning of line~3.)
\danger You can control the level of interaction by giving commands
in your file as well as online: The \TeX\ primitives ^|\scrollmode|,
^|\nonstopmode|, and ^|\batchmode| correspond respectively to typing
`|S|', `|R|', or `|Q|' in response to an error message, and
^|\errorstopmode| puts you back into the normal level of interaction. \
(Such changes are global, whether or not they appear inside a group.) \
Furthermore, many installations have implemented a way to ^{interrupt}
\TeX\ while it is running; such an interruption causes the program to
revert to |\errorstopmode|, after which it pauses and waits for
further instructions.
What happens next in Experiment 5? \TeX\ will hiccup on the other bug that
we planted in the file. This time, however, the error message is more
elaborate, since the context appears on six lines instead of two:
\begintt
! Undefined control sequence.
<argument> \bf A SHORT \ERROR
STORY
\centerline #1->\line {\hss #1
\hss }
l.3 \centerline{\bf A SHORT \ERROR STORY}
|null
?
\endtt
You get multiline error messages like this when the error is detected
while \TeX\ is processing some higher-level commands---in this case,
while it is trying to carry out |\centerline|, which is not a primitive
operation (it is defined in plain \TeX). At first, such error
messages will appear to be complete nonsense to you, because much of what
you see is low-level \TeX\ code that you never wrote. But you can overcome
this hangup by getting a feeling for the way \TeX\ operates.
First notice that the context information always appears in pairs of lines.
As before, the top line shows what \TeX\ has just read (\thinspace
`|\bf A SHORT \ERROR|'\thinspace), then comes what it is about to read
(\thinspace`|STORY|'\thinspace). The next pair of lines shows the context
of the first two; it indicates what \TeX\ was doing just before it began to
read the others. In this case, we see that \TeX\ has just read `|#1|', which
is a special code that tells the machine to ``read the first ^{argument} that
is governed by the current control sequence''; i.e., ``now read the stuff that
^|\centerline| is supposed to center on a line.'' The definition in Appendix~B
says that |\centerline|, when applied to some text, is supposed to be carried
out by sticking that text in place of the `|#1|' in `|\line{\hss#1\hss}|'.
So \TeX\ is in the midst of this expansion of\/ |\centerline|, as well as being
in the midst of the text that is to be centered.
\looseness-1
The bottom line shows how far \TeX\ has gotten until now in the |story| file.
\ (Actually the bottom line is blank in this example; what appears to be
the bottom line is really the first of two lines of context, and it
indicates that \TeX\ has read everything including the `|}|' in line~3 of
the file.) \ Thus, the context in this error message gives us a glimpse of
how \TeX\ went about its business. First, it saw |\centerline| at the
beginning of line~3. Then it looked at the definition of\/ |\centerline| and
noticed that |\centerline| takes an ``argument,'' i.e., that |\centerline|
applies to the next character or control sequence or group that follows.
So \TeX\ read~on, and filed `|\bf A SHORT \ERROR STORY|' away as the
argument to |\centerline|. Then it began to read the expansion, as
defined in Appendix~B\null. When it reached the |#1|, it began to read
the argument it had saved. And when it reached |\ERROR|, it complained
about an undefined control sequence.
\dangerexercise Why didn't \TeX\ complain about |\ERROR| being undefined
when |\ERROR| was first encountered, i.e., before reading `|STORY}|' on line~3?
\answer A control sequence like |\centerline| might well define a control
sequence like |\ERROR| before telling \TeX\ to look at |#1|. Therefore
\TeX\ doesn't interpret control sequences when it scans an argument.
When you get a multiline error message like this, the best clues about the
source of the trouble are usually on the bottom line (since that is what
you typed) and on the top line (since that is what triggered the error
message). Somewhere in there you can usually spot the problem.
Where should you go from here? If you type `|H|' now, you'll just get
the same help message about undefined control sequences that you saw
before. If you respond by typing \<return>, \TeX\ will go
on and finish the run, producing output virtually identical to that in
Experiment~2. In other words, the conventional responses won't teach you
anything new. So type `|E|' now; this terminates the run and prepares
the way for you to fix the erroneous file. \ (On some systems, \TeX\ will
actually start up the standard text editor, and you'll be positioned at
the right place to delete `|\ERROR|'. On other systems, \TeX\ will simply
tell you to edit line~3 of file |story.tex|.) ^^{editing}
When you edit |story.tex| again, you'll notice that line~2 still contains
|\vship|; the fact that you told \TeX\ to insert |\vskip| doesn't mean
that your file has changed in any way. In general, you should correct all
errors in the input file that were spotted by \TeX\ during a run; the
log file provides a handy way to remember what those errors were.
\smallskip
Well, this has indeed been a long chapter, so let's summarize what has
been accomplished. By doing the five experiments you have learned at first
hand (1)~how to get a job printed via \TeX; (2)~how to make a file that
contains a complete \TeX\ manuscript; (3)~how to change the plain \TeX\
format to achieve columns with different widths; and (4)~how to avoid
panic when \TeX\ issues stern~warnings.
So you could now stop reading this book and go on to print a bunch of
documents. It is better, however, to continue bearing with the author
(after perhaps taking another rest), since you're just at the threshold
of being able to do a lot more. And you ought to read Chapter~7
at least, because it warns you about certain symbols that you must not
type unless you want \TeX\ to do something special. While reading the
remaining chapters it will, of course, be best for you to continue making
trial runs, using experiments of your own design.
\ddanger If you use \TeX\ format packages designed by others, your
error messages may involve many inscrutable two-line levels of macro
context. By setting ^|\errorcontextlines||=0| at the beginning of your file,
you can reduce the amount of information that is reported;
\TeX\ will show only the top and bottom pairs of context lines
together with up to |\errorcontextlines| additional two-line items. \ (If
anything has thereby been omitted, you'll also see `|...|'.) \ Chances
are good that you can spot the source of an error even when most of a
large context has been suppressed; if not, you can say
`|I\errorcontextlines=100\oops|' and try again. \ (That will usually
give you an undefined control sequence error and plenty of context.) \
Plain \TeX\ sets |\errorcontextlines=5|.
\endchapter
What we have to learn to do we learn by doing.
\author ^{ARISTOTLE}, {\sl Ethica Nicomachea\/} II (c.~325 B.C.)
\bigskip
He may run who reads.
\author ^{HABAKKUK} 2\thinspace:\thinspace2 (c.~600 B.C.) % RSV
\smallskip
He that runs may read.
\author WILLIAM ^{COWPER}, {\sl Tirocinium\/} (1785)
\eject
\beginchapter Chapter 7. How \TeX\ Reads\\What You Type
We observed in the previous chapter that an input manuscript is expressed
in terms of ``lines,'' but that these lines of input are essentially
independent of the lines of output that will appear on the finished pages.
Thus you can stop typing a line of input at any place that's convenient for
you, as you prepare or edit a file. A few other related rules have also
been mentioned:
\medskip
\item\bull A $\langle\hbox{return}\rangle$ is like a space.
\smallskip
\item\bull Two spaces in a row count as one space.
\smallskip
\item\bull A blank line denotes the end of a paragraph.
\medskip
\noindent Strictly speaking, these rules are contradictory: A blank line
is obtained by typing $\langle\hbox{return}\rangle$ twice in a row,
and this is different from typing two spaces in a row. Some day you might want
to know the {\sl real\/} rules. In this chapter and the next, we shall study
the very first stage in the transition from input to output.
\smallskip
In the first place, it's wise to have a precise idea of what your keyboard
sends to the machine. There are 256 characters that \TeX\ might encounter at
each step, in a file or in a line of text typed directly on your terminal. These
256~characters are classified into 16 categories numbered 0 to 15:
\begindisplay \def\\{\hfill}
\hfil\hidewidth\it Category\hidewidth&\it \qquad Meaning\hidewidth\cr
\noalign{\smallskip}
\\0&Escape character&(|\| in this manual)\cr
\\1&Beginning of group&(|{| in this manual)\cr
\\2&End of group&(|}| in this manual)\cr
\\3&Math shift&(|$| in this manual)\cr
\\4&Alignment tab&(|&| in this manual)\cr
\\5&End of line&(\<return> in this manual)\cr
\\6&Parameter&(|#| in this manual)\cr
\\7&Superscript&(|^| in this manual)\cr
\\8&Subscript&(|_| in this manual)\cr
\\9&Ignored character&(\<null> in this manual)\cr
10&Space&(\] in this manual)\cr
11&Letter&(|A|, \dots, |Z| and |a|, \dots, |z|)\cr
12&Other character&(none of the above or below)\cr
13&Active character&(|~| in this manual)\cr
14&Comment character&(|%| in this manual)\cr
15&Invalid character&(\<delete> in this manual)\cr
\enddisplay
^^{escape character}
^^{begin-group character}
^^{end-group character}
^^{math mode character}
^^{alignment tab}
^^{parameter}
^^{superscript}
^^{subscript}
^^{ignored character}
^^{space}
^^{letter}
^^{other character}
^^{active character}
^^{comment character}
^^{invalid character}
^^{category codes, table}
It's not necessary for you to learn these code numbers; the point is only that
\TeX\ responds to 16~different types of characters. At first this manual led
you to believe that there were just two types---the escape character and the
others---and then you were told about two more types, the grouping
symbols |{| and~|}|. In Chapter~6 you learned two more: |~| and~|%|.
Now you know that there are really~16. This is the whole truth of the
matter; no more types remain to be revealed. The category code for any
character can be changed at any time, but it is usually wise to stick to a
^^{reserved character} ^^{special character table} ^^\<null> ^^\<delete>
particular scheme.
The main thing to bear in mind is that each \TeX\ format reserves certain
characters for its own special purposes. For example, when you are using plain
\TeX\ format (Appendix~B\null), you need to know that the ten characters
\begintt
\ { } $ & # ^ _ % ~
\endtt
cannot be used in the ordinary way when you are typing;
^^{special characters}
^^{backslash}^^{left brace}^^{right brace}^^{dollar sign}^^{ampersand}
^^{hash mark}^^{hat}^^{underline}^^{percent}^^{tilde}
^^{single-character control sequences}
each of them will cause \TeX\ to do something special, as explained elsewhere
in this book. If you really need these symbols as part of your manuscript,
plain \TeX\ makes it possible for you to type
\begindisplay
|\$| for \$,\qquad |\%| for \%,\qquad |\&| for \&,\qquad
|\#| for \#,\qquad |\_| for \_\thinspace;
\enddisplay
the |\_| symbol is useful for {\it compound\_identifiers\/} in computer
^^{identifiers} ^^{computer programs}
programs. In mathematics formulas you can use |\{| and |\}| for $\{$ and
$\}$, while ^|\backslash| produces a ^{reverse slash}; for example,
\begindisplay
`|$\{a \backslash b\}$|'\quad yields\quad`$\{a\backslash b\}$'.
\enddisplay
Furthermore |\^| produces a circumflex accent (e.g., `|\^e|' yields
`\^e'\thinspace); and |\~| yields a tilde accent (e.g., `|\~n|' yields
`\~n'\thinspace).
\exercise What horrible errors appear in the following sentence?
^^{Procter} ^^{Gamble}
\begintt
Procter & Gamble's stock climbed to $2, a 10% gain.
\endtt
\answer Three forbidden characters were used. One should type
\begintt
Procter \& Gamble's ... \$2, a 10\% gain.
\endtt
(Also the facts are wrong.)
\exercise Can you imagine why the designer of plain \TeX\ decided not
to make `|\\|' the control sequence for reverse slashes?^^{backslash}
\answer Reverse slashes (backslashes) are fairly uncommon in formulas or
text, and |\\| is very easy to type; it was therefore felt best not to
reserve |\\| for such limited use. Typists can define |\\| to be whatever
they want (including |\backslash|).
\danger When \TeX\ reads a line of text from a file, or a line of text that
you entered directly on your terminal, it converts that text into a list of
``^{tokens}.'' A token is either (a)~a single character with an attached
category code, or (b)~a control sequence. For example, if the normal
conventions of plain \TeX\ are in force, the text `|{\hskip 36 pt}|' is
converted into a list of eight tokens:
\begindisplay
|{|$_1$\quad\cstok{hskip}\quad|3|$_{12}$\quad|6|$_{12}$\quad
\]$_{10}$\quad|p|$_{11}$\quad|t|$_{11}$\quad|}|$_{2}$
\enddisplay
The subscripts here are the category codes, as listed earlier: 1 for
``beginning of group,'' 12 for ``other character,'' and so on. The
\cstok{hskip} doesn't get a subscript, because it represents a control
sequence token instead of a character token. Notice that the space after
|\hskip| does not get into the token list, because it follows a
^{control word}.
\danger It is important to understand the idea of token lists, if you want
to gain a thorough understanding of \TeX, and it is convenient to learn
the concept by thinking of \TeX\ as if it were a living organism. The
individual lines of input in your files are seen only by \TeX's ``eyes''
and ``mouth''; but after that text has been gobbled up, it is sent to
\TeX's ``stomach'' in the form of a token list, and the digestive processes
that do the actual typesetting are based entirely on tokens. As far as the
stomach is concerned, the input flows in as a stream of tokens, somewhat
as if your \TeX\ manuscript had been typed all on one extremely long line.
\danger You should remember two chief things about \TeX's tokens: (1)~A
control sequence is considered to be a single object that is no longer
composed of a sequence of symbols. Therefore long control sequence names
are no harder for \TeX\ to deal with than short ones, after they have been
replaced by tokens. Furthermore, spaces are not ignored after control
sequences inside a token list; the ignore-space rule applies only in an
input file, during the time that strings of characters are being
tokenized. (2)~Once a category code has been attached to a character
token, the attachment is permanent. For example, if character `|{|' were
suddenly declared to be of category~12 instead of category~1, the
characters `|{|$_1$' already inside token lists of \TeX\ would still
remain of category 1; only newly made lists would contain `|{|$_{12}$'
tokens. In other words, individual characters receive a fixed
interpretation as soon as they have been read from a file, based on the
category they have at the time of reading. Control sequences are
different, since they can change their interpretation at any time. \TeX's
digestive processes always know exactly what a character token signifies,
because the category code appears in the token itself; but when the
digestive processes encounter a control sequence token, they must look up
the current definition of that control sequence in order to figure out
what it means.
\ddangerexercise Some of the category codes 0 to 15 will never appear as
subscripts in character tokens, because they disappear in \TeX's mouth.
For example, characters of category 0 (escapes) never get to be tokens.
Which categories can actually reach \TeX's stomach?
\answer 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13. ^{Active characters} (type 13)
are somewhat special; they behave like control sequences in most cases
(e.g., when you say `^|\let||\x=~|' or `^|\ifx||\x~|'), but they behave like
character tokens when they appear in the token list of\/ ^|\uppercase|
or ^|\lowercase|, and when unexpanded after ^|\if| or ^|\ifcat|.
\ddanger There's a program called ^|INITEX| that is used to install
\TeX, starting from scratch; |INITEX| is like \TeX\ except that it can
do even more things. It can compress ^{hyphenation} patterns into special
tables that facilitate rapid hyphenation, and it can produce ^{format}
files like `|plain.fmt|' from `|plain.tex|'. But |INITEX| needs extra
space to carry out such tasks, so it generally has less memory available
for typesetting than you would expect to find in a production version of \TeX.
\ddanger When |INITEX| begins, it knows nothing
but \TeX's primitives. All 256~characters are initially of category~12,
except that ^\<return> has category~5,
^\<space> has category~10, ^\<null> has category~9, ^\<delete> has category~15,
the 52 letters |A|$\,\ldots\,$|Z| and |a|$\,\ldots\,$|z| have category~11,
|%| and~|\| have the respective categories 14 and~0. ^^{backslash}^^{percent}
It follows that |INITEX| is initially incapable of carrying out some of
\TeX's primitives that depend on grouping; you can't use |\def| or |\hbox|
until there are characters of categories 1 and~2. The format in
Appendix~B begins with ^|\catcode| commands to provide characters of the
necessary categories; e.g.,
\begintt
\catcode`\{=1
\endtt
assigns category 1 to the |{| symbol. The |\catcode| operation is like
many other primitives of \TeX\ that we shall study later; by modifying
internal quantities like the category codes, you can adapt \TeX\ to a wide
variety of applications.
\ddangerexercise Suppose that the commands
\begintt
\catcode`\<=1 \catcode`\>=2
\endtt
appear near the beginning of a group that begins with `|{|'; these
specifications instruct \TeX\ to treat |<| and |>| as group delimiters.
According to \TeX's rules of locality, the characters |<| and |>| will
revert to their previous categories when the ^{group} ends. But should the
group end with |}| or~with~|>|\thinspace?
\answer It ends with either |>| or |}| or any character of category 2;
then the effects of all |\catcode| definitions within the group are wiped
out, except those that were ^|\global|. \TeX\ doesn't have any built-in
knowledge about how to pair up particular kinds of grouping characters.
New category codes take effect as soon as a |\catcode| assignment has been
digested. For example,
\begintt
{\catcode`\>=2 >
\endtt
is a complete group. But without the space after `|2|' it would not be
complete, since \TeX\ would have read the~`|>|' and converted it to a
token before knowing what category code was being specified; \TeX\ always
reads the token following a constant before evaluating that ^{constant}.
\ddanger Although control sequences are treated as single objects,
\TeX\ does provide a way to break them into lists of character tokens:
If you write ^|\string||\cs|,
where |\cs| is any control sequence, you get the list of characters for that
control sequence's name. For example, |\string\TeX| produces four tokens:
|\|$_{12}$, |T|$_{12}$, |e|$_{12}$, |X|$_{12}$. Each character in this token
list automatically gets category code~12 (``other''),
including the ^{backslash} that |\string| inserts to represent an escape
character. However, category~10 will be assigned to the character `\]'
(blank ^{space}) if a space character somehow sneaks into the name of a
control sequence.
\ddanger Conversely, you can go from a list of character tokens to a
control sequence by saying `^|\csname|\<tokens>^|\endcsname|'. The tokens
that appear in this construction between |\csname| and |\endcsname| may
include other control sequences, as long as those control sequences
ultimately expand into characters instead of \TeX\ primitives; the final
characters can be of any category, not necessarily letters. For example,
`|\csname TeX\endcsname|' is essentially the same as `|\TeX|'; but
`|\csname\TeX\endcsname|' is illegal, because |\TeX| expands into tokens
containing the ^|\kern| primitive. Furthermore,
`|\csname\string\TeX\endcsname|' will produce the unusual control sequence
`|\\TeX|', i.e., the token \cstok{\char`\\TeX}, which you can't ordinarily
write.
\ddangerexercise Experiment with \TeX\ to see what |\string| does when it
is followed by an ^{active character} like |~|. \ (Active characters behave
like control sequences, but they are not prefixed by an escape.) \ What
is an easy way to conduct such experiments online? What control sequence
could you put after |\string| to~obtain the single character
token~|\|$_{12}$?
\answer If you type `|\message{\string~}|' and `|\message{\string\~}|', \TeX\
responds with `|~|' and `|\~|', respectively. ^^|\message|
To get |\|$_{12}$ from |\string| you therefore need to make backslash an
active character. One way to do this is
\begintt
{\catcode`/=0 \catcode`\\=13 /message{/string\}}
\endtt
(The ``^{null control sequence}'' that you get when there are no
tokens between |\csname| and |\endcsname| is not a solution to this exercise,
because |\string| converts it to `|\csname\endcsname|'. There is, however,
another solution: If \TeX's ^|\escapechar| parameter---which will be
explained in one of the next dangerous bends---is negative or greater
than~255, then `|\string\\|' works.)
\ddangerexercise What tokens does
`|\expandafter\string\csname a\string\ b\endcsname|' produce?
(There are three spaces before the |b|. Chapter~20 explains ^|\expandafter|.)
\answer |\|$_{12}$ |a|$_{12}$ |\|$_{12}$ \]$_{10}$ |b|$_{12}$.
\ddangerexercise When |\csname| is used to define a control sequence for
the first time, that control sequence is made equivalent to |\relax|
until it is redefined. Use this fact to design a macro |\ifundefined#1|
such that, for example,
\begindisplay
|\ifundefined{TeX}|\<true text>|\else|\<false text>|\fi|
\enddisplay
expands to the \<true text> if\/ |\TeX| hasn't previously been defined,
or if\/ |\TeX| has been |\let| equal to |\relax|; it should expand
to the \<false text> otherwise. ^^|\ifundefined|
\answer |\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}|%
\hfil\break Note that a control sequence like this must be used with care;
it cannot be included in ^{conditional} text, because the |\ifx| will not
be seen when |\ifundefined| isn't expanded.
\ddanger In the examples so far, |\string| has converted control sequences
into lists of tokens that begin with |\|$_{12}$. But this backslash token isn't
really hardwired into \TeX; there's a parameter called ^|\escapechar| that
specifies what character should be used when control sequences are output
as text. The value of\/ |\escapechar| is normally \TeX's internal code for
backslash, but it can be changed if another convention is desired.
\ddanger \TeX\ has two other token-producing operations similar to the
|\string| command. If you write ^|\number|\<number>, you get the decimal
equivalent of the \<number>; and if you write ^|\romannumeral|\<number>,
you get the number expressed in lowercase ^{roman numerals}. For example,
`|\romannumeral24|' produces `|xxiv|', a list of four tokens each having
category~12. The |\number| operation is redundant when it is applied
to an explicit constant (e.g., `|\number24|' produces `|24|'); but it does
suppress leading zeros, and it can also be used with numbers that are in
\TeX's internal registers or parameters. For example, `|\number-0015|'
produces `|-15|'; and if register |\count5| holds the value 316, then
`|\number\count5|' produces `|316|'.
\ddanger The twin operations ^|\uppercase||{|\<token list>|}| and
^|\lowercase||{|\<token list>|}| go through a given token list and convert
all of the character tokens to their ``uppercase'' or ``lowercase''
equivalents. Here's how: Each of the 256 possible characters
has two associated values called the ^|\uccode| and the ^|\lccode|; these values
are changeable just as a |\catcode| is. Conversion to uppercase means
that a character is replaced by its |\uccode| value, unless the |\uccode|
value is zero (when no change is made). Conversion to lowercase is
similar, using the |\lccode|. The category codes aren't changed. When
^|INITEX| begins, all |\uccode| and |\lccode| values are zero except that
the ^{letters} |a| to~|z| and |A| to~|Z| have |\uccode| values |A| to~|Z|
and |\lccode| values |a| to~|z|.
\ddanger \TeX\ performs the |\uppercase| and |\lowercase| transformations
in its stomach, but the |\string| and |\number| and |\romannumeral|
and |\csname| operations are carried out en route to the stomach (like
macro expansion), as explained in Chapter~20.
\ddangerexercise What token list results from
`|\uppercase{a\lowercase{bC}}|'\thinspace?
\answer First |\uppercase| produces `|A\lowercase{BC}|'; then you get `|Abc|'.
\ddangerexercise \TeX\ has an internal integer parameter called ^|\year| that is
set equal to the current year number at the beginning of every job. Explain how
to use |\year|, together with |\romannumeral| and |\uppercase|, to
print a copyright notice like \year=1986
`\copyright\ \uppercase\expandafter{\romannumeral\year}'
for all jobs run in \number\year.
\answer `\thinspace|\copyright\ \uppercase\expandafter{\romannumeral\year}|%
\thinspace'. \ (This is admittedly tricky; the `^|\expandafter|' expands
the token after the `|{|', not the token after the group.)
\ddangerexercise Define a control sequence |\appendroman| with three parameters
such that |\appendroman#1#2#3| defines control sequence |#1| to
expand to a control sequence whose name is the name of control sequence
|#2| followed by the value of the positive integer |#3| expressed in roman
numerals. For example, suppose |\count20| equals 30; then
`|\appendroman\a\TeX{\count20}|' should have the same effect as
`|\def\a{\TeXxxx}|'.^^{tricky macros}
\answer (We assume that parameter |#2| is not simply an active character,
and that ^|\escapechar| is between 0 and~255.)
\begintt
\def\gobble#1{} % remove one token
\def\appendroman#1#2#3{\expandafter\def\expandafter#1\expandafter
{\csname\expandafter\gobble\string#2\romannumeral#3\endcsname}}
\endtt
\endchapter
Some bookes are to bee tasted,
others to bee swallowed,
and some few to bee chewed and disgested.
\author FRANCIS ^{BACON}, {\sl Essayes\/} (1597) % p2 of orig edition
\bigskip
`Tis the good reader that makes the good book.
\author RALPH WALDO ^{EMERSON}, {\sl Society \& Solitude\/} (1870) % Success
\eject
\beginchapter Chapter 8. The Characters\\You Type
A lot of different keyboards are used with \TeX, but few keyboards can
produce 256 different symbols. Furthermore, as we have seen, some of the
characters that you {\sl can\/} type on your ^{keyboard} are reserved for
^^{terminal keyboard}
special purposes like escaping and grouping. Yet when we studied fonts it
was pointed out that there are 256 characters per font. So how can you
refer to the characters that aren't on your keyboard, or that have been
pre-empted for formatting?
One answer is to use control sequences. For example, the plain format
of Appendix B\null, which defines |%| to be a special kind of symbol so that you
can use it for comments, defines the control sequence |\%| to mean
a ^{percent sign}.
To get access to any character whatsoever, you can type
\begindisplay
|\char|\<number>
\enddisplay
where \<number> is any number from 0 to 255 (optionally followed by a space);
you will get the corresponding character from the current font. That's how
Appendix~B handles |\%|; it defines `|\%|' to be an abbreviation for
`|\char37|', since 37 is the character code for a percent sign.
The codes that \TeX\ uses internally to represent characters are based on
``^{ASCII},'' the American Standard Code for Information Interchange.
^^{internal character codes} ^^{character codes}
Appendix~C gives full details of this code, which assigns numbers to
certain control functions as well as to ordinary letters and punctuation
marks. For example, ^\<space>${}=32$ and ^\<return>${}=13$.
There are 94~standard visible symbols, and they have been assigned code
numbers from 33 to~126, inclusive.
It turns out that `|b|' is character number 98 in ASCII. So you can
typeset the word |bubble| in a strange way by putting
\begintt
\char98 u\char98\char98 le
\endtt
into your manuscript, if the |b|-key on your keyboard is broken. \
(An optional space is ignored after constants like `|98|'.
Of course you need the |\|, |c|, |h|, |a|, and~|r| keys to type `^|\char|',
so let's hope that they are always working.)
\danger \TeX\ always uses the internal character code of Appendix~C
for the standard ASCII characters,
regardless of what external coding scheme actually appears in the files
being read. Thus, |b| is 98 inside of \TeX\ even when your computer
normally deals with ^{EBCDIC} or some other non-ASCII scheme; the \TeX\
software has been set up to convert text files to internal code, and to
convert back to the external code when writing text files.
Device-independent (^|dvi|) output files use \TeX's internal code. In
this way, \TeX\ is able to give identical results on all computers.
\danger Character code tables like those in Appendix~C often give the code
numbers in {\sl ^{octal notation}}, i.e., the radix-8 number system, in which
the digits are {\it0},~{\it1}, {\it2}, {\it3}, {\it4}, {\it5}, {\it6},
and~{\it7}.\footnote*{The author of this manual likes to use italic digits
for octal numbers, and typewriter type for hexadecimal numbers, in order
to provide a typographic clue to the underlying radix whenever possible.}
Sometimes {\sl^{hexadecimal notation}\/} is also used, in which case the
digits are |0|,~|1|, |2|, |3|, |4|, |5|, |6|, |7|, |8|, |9|, |A|, |B|, |C|,
|D|, |E|, and~|F|. For example, the octal code for `|b|' is {\it142}, and
its hexadecimal code is |62|. A ^\<number> in \TeX's language can begin
with~a~|'|, in which case it is regarded as octal, or with a |"|, when it is
regarded as hexadecimal. Thus, |\char'142| and |\char"62| are equivalent
to |\char98|. The legitimate character codes in octal notation run from
\oct0 to \oct{377}; in hexadecimal, they run from \hex0 to \hex{FF}.
^^{apostrophe}^^{doublequote}
\danger But \TeX\ actually provides another kind of \<number> that makes it
unnecessary for you to know ASCII at all! The token |`|$_{12}$ (^{left quote}),
when followed by any character token or by any control sequence token
whose name is a single character, stands for \TeX's internal code for the
character in question. For example, |\char`b| and |\char`\b| are also
equivalent to |\char98|. ^^{reverse apostrophe}
If you look in Appendix~B to see how |\%| is defined, you'll notice that
the definition is
\begintt
\def\%{\char`\%}
\endtt
instead of\/ |\char37| as claimed above.
\dangerexercise What would be wrong with |\def\%{\char`%}|?
\answer The |%| would be treated as a comment character, because its
category code is~14; thus, no |%| token or |}| token would get through
to the gullet of \TeX\ where numbers are treated. When a character is
of category 0, 5, 9, 14, or~15, the extra |\| must be used; and the
|\| doesn't hurt, so you can always use it to be safe.
\ddanger The preface to this manual points out that the author
tells little white lies from time to time. Well, if you actually
check Appendix~B you'll find that
\begintt
\chardef\%=`\%
\endtt
is the true definition of\/ |\%|. Since format designers often want to
associate a special character with a special control sequence name, \TeX\
provides the construction `^|\chardef|\<control sequence>|=|\<number>'
for numbers between 0 and 255, as an efficient alternative to
`^|\def|\<control sequence>|{\char|\<number>|}|'.
Although you can use |\char| to access any character in the current
font, you can't use it in the middle of a control sequence. For example,
if you type
\begintt
\\char98
\endtt
\TeX\ reads this as the control sequence |\\| followed by |c|, |h|, |a|,
etc., not as the control sequence |\b|.
You will hardly ever need to use |\char| when typing a manuscript, since
the characters you want will probably be available as predefined control
sequences; |\char| is primarily intended for the designers of book formats
like those in the appendices. But some day you may require a ^{special
symbol}, and you may have to hunt through a font catalog until you find
it. Once you find it, you can use it by simply selecting the appropriate
font and then specifying the character number with |\char|. For example,
the ``^{dangerous bend}'' sign used in this manual appears as character
number~127 of font ^|manfnt|, and that font is selected by the control
sequence ^|\manual|. The macros in Appendix~E therefore display dangerous
bends by saying `|{\manual\char127}|'.
We have observed that the ASCII character set includes only 94 printable
symbols; but \TeX\ works internally with 256 different character codes,
from 0 to 255, each of which is assigned to one of the sixteen categories
described in Chapter~7. If your keyboard has additional symbols, or if it
doesn't have the standard~94, the people who installed your local \TeX\ system
can tell you the correspondence between what you type and the character
number that \TeX\ receives. Some people are fortunate enough to have keys
marked `{\tentex\char'32}' and `{\tentex\char'34}' and `{\tentex\char'35}';
it is possible to install \TeX\ so that it will recognize these handy symbols
and make the typing of mathematics more pleasant. But if you do not have
such keys, you can get by with the control sequences ^|\ne|, ^|\le|,
and ^|\ge|. ^^{not-equal}^^{less-or-equal}^^{greater-or-equal}
\danger \TeX\ has a standard way to refer to the invisible characters of ASCII:
Code~0 can be typed as the sequence of three characters |^^@|, code~1 can
be typed |^^A|, and so on up to code~31, which is |^^_| (see Appendix~C\null).
If the character following |^^| has an internal code between 64 and 127, \TeX\
subtracts 64 from the code; if the code is between 0 and 63, \TeX\
adds~64. Hence code 127 can be typed |^^?|, and
the dangerous bend sign can be obtained by saying
|{\manual^^?}|. However, you must change the category code of character
127 before using it, since this character ordinarily has category~15
(^{invalid}); say, e.g., |\catcode`\^^?=12|.
^^{double hat} ^^{hat hat}
The |^^| notation is different from |\char|, because |^^| combinations are
like single characters; for example, it would not be permissible to say
|\catcode`\char127|, but |^^| symbols can even be used as letters within
control words.
\danger One of the overfull box messages in Chapter 6 illustrates the fact
that \TeX\ sometimes uses the funny |^^| convention in its output:
The umlaut character in that example appears as |^^?|, and the cedilla appears
as~|^^X|, because `\thinspace\"{}\thinspace' and `\char'30' occur in
positions \oct{177} and~\oct{30} of the ^|\tenrm| font.
\danger There's also a special convention in which |^^| is
followed by {\sl two\/} ``lowercase hexadecimal digits,'' |0|--|9| or |a|--|f|.
With this convention, all 256 characters are obtainable in a uniform
way, from |^^00| to |^^ff|. Character 127 is |^^7f|.
\danger Most of the |^^| codes are unimportant except in unusual applications.
But |^^M| is particularly noteworthy because it is code 13, the ASCII
^\<return> that \TeX\ normally places at the right end of every line of
your input file. By changing the category of~|^^M| you can obtain useful
special effects, as we shall see later. ^^{hat hat M}
\danger The control code |^^I| is also of potential interest, since it's
the ASCII ^\<tab>. Plain \TeX\ makes \<tab> act like a blank space.
\ddanger People who install \TeX\ systems for use with non-American alphabets
can make \TeX\ conform to any desired standard. For example, suppose
you have a ^{Norwegian keyboard} containing the letter {\tt\ae}, which
^^{Scandinavian letters} ^^{foreign languages}
comes in as code~241 (say). Your local format package should define
|\catcode`|{\tt\ae}|=11|; then you could have control sequences like
|\s|{\tt\ae}|rtrykk|. Your \TeX\ input files could be made readable by
American installations of \TeX\ that don't have your keyboard, by
substituting |^^f1| for character~241. \ (For example, the stated control
sequence would appear as |\s^^f1rtrykk| in the file; your American
friends should also be provided with the format that you used, with its
|\catcode`^^f1=11|.) \ Of course you should also arrange your fonts
so that \TeX's character 241 will print as {\ae}; and you should
change \TeX's hyphenation algorithm so that it will do correct
Norwegian hyphenation. The main point is that such changes are not
extremely difficult; nothing in the design of \TeX\ limits it to the
American alphabet. Fine printing is obtained by fine tuning to the
language or languages being used.
^^{keyboards, non-ASCII}
\ddanger European languages can also be accommodated effectively with
only a limited character set.
For example, let's consider Norwegian again, but suppose that you
want to use a keyboard without an {\tt\ae} character. You can arrange the
^{font metric file} so that \TeX\ will interpret |ae|, |o/|, |aa|, |AE|,
|O/|, and |AA| as ligatures that produce \ae, \o, \aa, \AE, \O, and \AA,
respectively; and you could put the characters \aa\ and \AA\ into positions
128 and~129 of the font. By setting |\catcode`/=11| you would be able to
use the ligature |o/| in control sequences like `|\ho/yre|'. \TeX's
hyphenation method is not confused by ligatures; so you could use this
scheme to operate essentially as suggested before, but with two keystrokes
occasionally replacing one. \ (Your typists would have to watch out for
the occasional times when the adjacent characters |aa|, |ae|, and |o/|
should not be treated as ligatures; also, `|\/|' would be a ^{control
word}, not a ^{control symbol}.)
\ddanger The rest of this chapter is devoted to \TeX's reading rules,
which define the conversion from text to tokens. For example, the fact
that \TeX\ ignores spaces after control words is a consequence of
the rules below, which imply among other things that spaces after control
words never become space tokens. The rules are intended to work the
way you would expect them to, so you may not wish to bother reading them;
but when you are communicating with a computer, it is nice to understand
what the machine thinks it is doing, and here's your chance.
\ddanger The input to \TeX\ is a sequence of ``^{lines}.''
Whenever \TeX\ is reading a line of text from a file, or a line of
text that you entered directly on your terminal, the computer's
reading apparatus is in one of three so-called ^{states}:
\begindisplay
\noalign{\vskip1pt}
State $N$&Beginning a new line;\cr
State $M$&Middle of a line;\cr
State $S$&Skipping blanks.\cr
\noalign{\vskip-3pt}
\enddisplay
At the beginning of every line it's in state $N$; but most of the time it's
in state $M$, and after a control word or a space it's in state $S$.
Incidentally, ``states'' are different from the ``^{modes}'' that we will
be studying later; the current {\sl state\/} refers to \TeX's eyes and
mouth as they take in characters of new text, but the current {\sl mode\/}
refers to the condition of \TeX's gastro-intestinal tract. Most of the
things that \TeX\ does when it converts characters to ^{tokens} are independent
of the current state, but there are differences when spaces or end-of-line
characters are detected (categories 10 and 5).
\ddanger \TeX\ deletes any ^\<space> characters (number 32) that occur at the
right end of an input line. Then it inserts a ^\<return> character (number~13)
at the right end of the line, except that it places nothing additional at the
end of a line that you inserted with `|I|'
during ^{error recovery}. Note that \<return> is considered to be an actual
character that is part of the line; you can obtain special effects by
changing its catcode.
\ddanger If \TeX\ sees an escape character (category 0) in any state, it
scans the entire ^{control sequence} name as follows. (a)~If there are no
more characters in the line, the name is empty (like |\csname\endcsname|).
^^{null control sequence} ^^{csname endcsname}
Otherwise (b)~if the next character is not of category~11 (letter), the
name consists of that single symbol. Otherwise (c)~the name consists of all
letters beginning with the current one and ending just before the first
nonletter, or at the end of the line. This name becomes a control sequence
token. \TeX\ goes into state~$S$ in case~(c), or in case~(b) with respect
to a character of category~10 (space); otherwise \TeX\ goes into state~$M$.
\ddanger If \TeX\ sees a superscript character (category 7) in any state,
and if that character is followed by another identical character, and if
those two equal characters are followed by a character of code
$c<128$, then they
are deleted and 64 is added~to or subtracted from the code~$c$.
\ (Thus, |^^A| is
replaced by a single character whose code is~1, etc., as explained earlier.) \
However, if the two superscript characters are immediately followed by two
of the lowercase hexadecimal digits |0123456789abcdef|, the
four-character sequence is replaced by a single character having the
specified hexadecimal code.
The replacement is carried out also if such a trio or quartet of
characters is encountered during steps (b) or~(c) of the control-sequence-name
scanning procedure described above. After the replacement is made, \TeX\
begins again as if the new character had been present all the time.
If a superscript character is not the first of such a trio or quartet, it is
handled by the following rule.
\ddanger If \TeX\ sees a character of categories 1, 2, 3, 4, 6, 8, 11, 12,
or~13,
or a character of category~7 that is not the first of a special
sequence as just
described, it converts the character to a token by attaching the category
code, and goes into state~$M$. This is the normal case; almost every
nonblank character is handled by this rule.
\ddanger If \TeX\ sees an end-of-line character (category 5), it throws
away any other information that might remain on the current line. Then if
\TeX\ is in state~$N$ (new line), the end-of-line character is converted
to the control sequence token `\cstok{par}' ^^|\par| (end of paragraph); if
\TeX\ is in state~$M$ (mid-line), the end-of-line character is converted
to a token for character~32 (`\]') of category~10 (^{space}); and if \TeX\
is in state~$S$ (skipping blanks), the end-of-line character is simply dropped.
\ddanger If \TeX\ sees a character to be ignored (category~9), it simply
bypasses that character as if it weren't there, and remains in the same state.
\ddanger If \TeX\ sees a character of category~10 (space), the action
depends on the current state. If \TeX\ is in state $N$ or $S$, the
character is simply passed by, and \TeX\ remains in the same state.
Otherwise \TeX\ is in state $M$; the character is converted to a token
of category~10 whose character code is~32, and \TeX\ enters state~$S$.
The character code in a space token is always~32.
\ddanger If \TeX\ sees a comment character (category~14), it throws away that
character and any other information that might remain on the current line.
\ddanger Finally, if \TeX\ sees an invalid character (category~15),
it bypasses that character, prints an error message, and remains in the
same state.
\ddanger If \TeX\ has nothing more to read on the current line, it goes to
the next line and enters state $N$. However, if\/ ^|\endinput| has been
specified for a file being ^|\input|, or if an |\input| file has ended,
\TeX\ returns to whatever it was reading when the |\input| command
was originally given. \ (Further details of\/ |\input| and |\endinput| are
discussed in Chapter~20.)
\ddangerexercise Test your understanding of \TeX's reading rules by answering
the following quickie questions: (a)~What is the difference between
categories 5 and~14? (b)~What is the difference between categories 3
and~4? (c)~What is the difference between categories 11 and~12? (d)~Are
spaces ignored after active characters? (e)~When a line ends with a comment
character like |%|, are spaces ignored at the beginning of the next line?
(f)~Can an ignored character appear in the midst of a control sequence name?
\answer (a)~Both characters terminate the current line; but a character of
category~5 might be converted into \]$_{10}$ or a \cstok{par} token, while
a character of category~14 never produces a token. (b)~They produce
character tokens stamped with different category numbers. For example,
|$|$_3$ is not the same token as |$|$_4$, so \TeX's digestive processes
will treat them differently. (c)~Same as~(b), plus the fact that control
sequence names treat letters differently. (d)~No. (e)~Yes; characters of
category~10 are ignored at the beginning of every line, since every line
starts in state~$N$. (f)~No.
\ddangerexercise Look again at the error message that appears on page
\vshippage. When \TeX\ reported that |\vship| was an undefined
control sequence, it printed two lines of context, showing that
it was in the midst of reading line~2 of the |story| file. At the
time of that error message, what state was \TeX\ in? What character
was it about to read next?
\answer \TeX\ had just read the control sequence |\vship|, so it
was in state~$S$, and it was just ready to read the space before `|1in|'.
Afterwards it ignored that space, since it was in state~$S$; but if
you had typed |I\obeyspaces| in response to that error message,
you would have seen the space. Incidentally, when \TeX\ prints
the ^{context of an error message}, the bottom pair of lines comes from
a text file, but the other pairs of lines are portions of token lists
that \TeX\ is reading (unless they begin with `|<*>|', when they
represent text inserted during ^{error recovery}).
\ddangerexercise Given the category codes of plain \TeX\ format,
what tokens are produced from the input line
`| $x^2$~ \TeX ^^62^^6|'\thinspace?
\answer |$|$_{3}$ |x|$_{11}$ |^|$_7$ |2|$_{12}$ |$|$_{3}$ |~|$_{13}$ \]$_{10}$
\cstok{TeX} |b|$_{11}$ |v|$_{11}$ \]$_{10}$. The final space comes from the
\<return> placed at the end of the line. Code |^^6| yields |v| only
when not followed by |0|--|9| or |a|--|f|.
The initial space is ignored, because state~$N$
governs the beginning of the line.
\ddangerexercise Consider an input file that contains exactly
three lines; the first line says `|Hi!|', while the other two lines
are completely blank. What tokens are produced when \TeX\ reads
this file, using the category codes of plain \TeX\ format?
\answer |H|$_{11}$ |i|$_{11}$ |!|$_{12}$ \]$_{10}$ \cstok{par}
\cstok{par}. The `\]' comes from the \<return> at the
end of the first line; the second and third lines each contribute
a \cstok{par}.
\ddangerexercise Assume that the category codes of plain \TeX\ are in
force, except that the characters |^^A|, |^^B|, |^^C|, |^^M| belong
respectively to categories 0, 7, 10, and 11. What tokens are produced from
the (rather ridiculous) input line `|^^B^^BM^^A^^B^^C^^M^^@\M|\]'?
(Remember that this line is followed by \<return>, which is
|^^M|; and recall that |^^@| denotes the ^\<null> character, which has
category~9 when |INITEX| begins.)
\answer The two |^^B|'s are not recognized as consecutive superscript
characters, since the first |^^B| is converted to code~2 which doesn't
equal the following character |^|. Hence
the result is seven tokens: |^^B|$_7$ |^^B|$_7$
|M|$_{11}$ \cstok{\^{}\^{}B} \]$_{10}$ |^^M|$_{11}$ \cstok{M\^{}\^{}M}.
The last of these is a control word whose name has two letters.
The \<space> after |\M| is deleted before \TeX\ inserts the \<return> token.
\ddanger The special character inserted at the end of each line needn't
be ^\<return>; \TeX\ actually inserts the current value of an integer
parameter called ^|\endlinechar|, which normally equals~13 but it can
be changed like any other parameter. If the value of\/ |\endlinechar| is
negative or greater than~255, no character is appended, and the effect is
as if every line ends with~|%| (i.e., with a comment character).
\ddanger Since it is possible to change the category codes, \TeX\ might
actually use several different categories for the same character on a single
line. For example, Appendices D and~E contain several ways to coerce \TeX\ to
process text ``^{verbatim},'' so that the author could prepare this manual
without great difficulty. \ (Try to imagine typesetting a \TeX\ manual;
backslashes and other special characters need to switch back and forth
between their normal categories and category~12!) \ Some care is needed to
get the timing right, but you can make \TeX\ behave in a variety of
different ways by judiciously changing the categories. On the other hand,
it is best not to play with the category codes very often, because you must
remember that characters never change their categories once they have become
tokens. For example, when the arguments to a macro are first scanned,
they are placed into a token list, so their categories are fixed once and
for all at that time. The author has intentionally kept the category
codes numeric instead of mnemonic, in order to discourage people from
making extensive use of\/ |\catcode| changes except in unusual
circumstances.
\ddangerexercise Appendix B defines ^|\lq| and ^|\rq| to be abbreviations
for |`| and |'| (single left and right quotes, respectively). Explain why
the definitions
\begintt
\chardef\lq=96 \chardef\rq=39
\endtt
would not be as good.
\answer Both alternatives work fine in text; in particular, they combine
as in |\lq\lq| to form ligatures. But the definition in Appendix~B works
also in connection with constants; e.g., |\char\lq\%| and
|\char\rq140| are valid. \ (Incidentally, the construction |\let\lq=`|
would not work with constants, since the quotes in a ^\<number> must
come from character tokens of category~12; after |\let\lq=`| the control
sequence token |\lq| will not expand into a character token, nor {\sl is\/}
it a character token!) ^^|\let| ^^{implicit character}
\endchapter
for life's not a paragraph
\quad
% he left a blank line here, really
And death i think is no parenthesis.
\author e.~e.~^{cummings}, {\sl since feeling is first\/} (1926)
\bigskip
This coded character set is to facilitate
the general interchange of information
among information processing systems,
communication systems, and
associated equipment.
$\ldots$ An 8-bit set was considered
but the need for more than 128 codes
in general applications was not yet evident.
\author ASA SUBCOMMITTEE X3.2, {\sl American Standard\break %
Code for Information Interchange\/^^{ASCII}} (1963)
% in {\sl Communications of the ACM\/}
\eject
\beginchapter Chapter 9. \TeX's\\Roman Fonts
When you're typing a manuscript for \TeX, you need to know what symbols
are available. The plain \TeX\ format of Appendix~B is based on the
Computer Modern fonts, which provide the characters needed to typeset a
wide variety of documents. It's time now to discuss what a person can do
with plain \TeX\ when typing straight text. We've already touched on some of
the slightly subtle things---for example, dashes and quotation marks
were considered in Chapter~2, and certain kinds of accents appeared in the
examples of Chapters 3 and~6. The purpose of this chapter is to give a
more systematic summary of the possibilities, by putting all the facts
together.
Let's begin with the rules for the normal roman font (|\rm| or |\tenrm|);
plain \TeX\ will use this font for everything unless you specify
otherwise. Most of the ordinary symbols that you need are readily available
and you can type them in the ordinary way: There's nothing special about
\begindisplay \openup1pt
the ^{letters} |A| to |Z| and |a| to |z|\cr
the ^{digits} |0| to |9|\cr
common ^{punctuation} marks |: ; ! ? ( ) [ ] ` ' - * / . , @|\cr
\enddisplay
except that \TeX\ recognizes certain combinations as ^{ligatures}:
$$\openup1pt\halign{\indent#\hfil\cr
|ff| yields ff\thinspace;$\!$\quad |fi| yields fi\thinspace;$\!$\quad
|fl| yields fl\thinspace;$\!$\quad
|ffi| yields ffi\thinspace;$\!$\quad |ffl| yields ffl\thinspace;\cr
|``| yields``\thinspace;\qquad |''| yields ''\thinspace;\qquad
|!||`| yields !`\thinspace;\qquad |?||`| yields ?`\thinspace;\cr
|--| yields --\thinspace;\qquad |---| yields ---\thinspace.\cr}$$
^^{Spanish ligatures}
You can also type |+| and |=|, to get the corresponding
symbols + and~=; but it's much better to use such characters
only in math mode, i.e., enclosed between two |$| signs, since that tells
\TeX\ to insert the proper spacing for mathematics. Math mode is
explained later; for now, it's just a good idea to remember that formulas
and text should be segregated. A non-mathematical hyphen and a non-mathematical
slash should be specified by typing `|-|' and `|/|' outside of mathematics
mode, but subtraction and division should be specified by typing `|-|' and
`|/|' between |$|~signs.
^^{Colon}
^^{Semicolon}
^^{Exclamation point}^^{Shriek, see exclamation point}
^^{Question mark}
^^{Parentheses}
^^{Brackets}
^^{Apostrophe} ^^{Reverse apostrophe}
^^{Hamza, see apostrophe} ^^{Ain, see reverse apostrophe}
^^{Hyphen} ^^{Dash}
^^{Asterisk}
^^{At sign}
^^{Virgule, see slash}
^^{Solidus, see slash}
^^{Shilling sign, see slash}
^^{Slash}
^^{Period} ^^{Full stop, see period}
^^{Comma}
^^{Plus sign}
^^{Equals sign}
The previous paragraph covers 80 of the 94 visible characters of standard
ASCII; so your keyboard probably contains at least 14 more symbols, and
you should learn to watch out for the remaining ones, since they are special.
Four of these are pre\"empted by plain \TeX; if your manuscript requires
the symbols
\begintt
$ # % &
\endtt
^^{dollar sign} ^^{sharp sign, see hash mark} ^^{number sign, see hash mark}
^^{hash mark} ^^{percent sign} ^^{ampersand}
you should remember to type them as
\begintt
\$ \# \% \&
\endtt
respectively. Plain \TeX\ also reserves the six symbols
\begintt
\ { } ^ _ ~
\endtt
^^{backslash} ^^{braces} ^^{curly braces, see braces} ^^{hat, see circumflex}
^^{circumflex} ^^{underline} ^^{tilde}
but you probably don't mind losing these, since they don't appear in
normal copy. Braces and backslashes are available via control sequences
in math mode.
\goodbreak
There are four remaining special characters in the standard ASCII set:
\begintt
" || < >
\endtt
Again, you don't really want them when you're typesetting text. \ (Double-quote
marks should be replaced either by |``| or by |''|; vertical
lines and relation signs are needed only in math mode.)
^^{double-quote mark} ^^{vertical line, see norm} ^^{norm symbol}
^^{less than sign} ^^{greater than sign}
Scholarly publications in English often refer to other languages, so
plain \TeX\ makes it possible to typeset the most commonly used ^{accents}:
$$\halign{\indent\hbox to 50pt{#\hfil}&\hbox to 35pt{#\hfil}&#\hfil\cr
\it\negthinspace Type&\it to get\cr
\noalign{\smallskip}
|\`o|&\`o&(grave accent)\cr
|\'o|&\'o&(acute accent)\cr
|\^o|&\^o&(circumflex or ``hat'')\cr
|\"o|&\"o&(umlaut or dieresis)\cr
|\~o|&\~o&(tilde or ``squiggle'')\cr
|\=o|&\=o&(macron or ``bar'')\cr
|\.o|&\.o&(dot accent)\cr
|\u o|&\u o&(breve accent)\cr
|\v o|&\v o&(h\'a\v cek or ``check'')\cr
|\H o|&\H o&(long Hungarian umlaut)\cr
|\t oo|&\t oo&(tie-after accent)\cr}$$
^^|\`| ^^{grave accent}
^^|\'| ^^{acute accent}
^^{esc hat} ^^{circumflex accent} ^^{hat accent}
^^|\"| ^^{umlaut accent} ^^{dieresis}
^^{esc tilde} ^^{tilde accent} ^^{squiggle accent}
^^|\=| ^^{macron accent} ^^{bar accent}
^^|\.| ^^{dot accent}
^^|\v| ^^{h\'a\v cek accent} ^^{check accent}
^^|\u| ^^{breve accent}
^^|\H| ^^{Hungarian umlaut}
^^|\t| ^^{tie-after accent}
^^{embellished letters, see accents}
Within the font, such accents are designed to appear at the right height
for the letter `o'; but you can use them over any letter, and \TeX\ will
raise an accent that is supposed to be taller. Notice that spaces are needed
in the last four cases, to separate the control sequences from the letters
that follow. You could, however, type `|\H{o}|' in order to avoid putting a
space in the midst of a word.
\medbreak
Plain \TeX\ also provides three accents that go underneath:
$$\halign{\indent\hbox to 50pt{#\hfil}&\hbox to 35pt{#\hfil}&#\hfil\cr
\it\negthinspace Type&\it to get\cr
\noalign{\smallskip}
|\c o|&\c o&(cedilla accent)\cr
|\d o|&\d o&(dot-under accent)\cr
|\b o|&\b o&(bar-under accent)\cr}$$
^^|\c| ^^{cedilla accent}
^^|\d| ^^{dot-under accent} ^^{emphatics, see dot-under}
^^|\b| ^^{bar-under accent}
And there are a few special letters:
$$\halign{\indent\hbox to 50pt{#\hfil}&\hbox to 35pt{#\hfil}&#\hfil\cr
\it\negthinspace Type&\it to get\cr
\noalign{\smallskip}
|\oe,\OE|&\oe,\thinspace\OE&(French ligature OE)\cr
|\ae,\AE|&\ae,\thinspace\AE&(Latin ligature and Scandinavian letter AE)\cr
|\aa,\AA|&\aa,\thinspace\AA&(Scandinavian A-with-circle)\cr
|\o,\O|&\o,\thinspace\O&(Scandinavian O-with-slash)\cr
|\l,\L|&\l,\thinspace\L&(Polish suppressed-L)\cr
|\ss|&\ss&(German ``es-zet'' or sharp S)\cr}$$
^^{Scandinavian letters} ^^{sharp S} ^^{es-zet} ^^{German} ^^{Polish}
^^{Norwegian} ^^{Danish} ^^{Swedish} ^^{suppressed-L}
^^{diphthongs, see \ae, \oe}
The |\rm| font contains also the ^{dotless letters} `\i' and `\j',
which you can obtain by typing `^|\i|' and `^|\j|'. These are needed because
`i' and `j' should lose their dots when they gain an accent. For example,
the right way to obtain `m\=\i n\u us' is to type \hbox{`|m\=\i n\u us|'}
or `|m\={\i}n\u{u}s|'.
This completes our summary of the |\rm| font. Exactly the same conventions
apply to |\bf|, |\sl|, and |\it|, so you don't have to do things differently
when you're using a different typeface. For example, |\bf\"o| yields
{\bf\"o} and |\it\&| yields {\it\&}. Isn't that nice?
\danger However, |\tt| is slightly different. You will be glad to know that
|ff|, |fi|, and so on are not treated as ligatures when you're using
^{typewriter type}; nor do you get ligatures from dashes and quote marks.
That's fine, because ordinary dashes and ordinary double-quotes are
appropriate when you're trying to imitate a typewriter. Most of the
accents are available too. But |\H|, |\.|, |\l|, and |\L| cannot be
used---the typewriter font contains other symbols in their place.
Indeed, you are suddenly allowed to type |"|, \|, |<|, and |>|;
^^{doublequote} ^^{vertical line} ^^{less than sign} ^^{greater than sign}
see Appendix~F\null. All of the letters, spaces, and other symbols in
|\tt| have the same width.
\exercise What's the non-naive way to type `na\"\i ve'\thinspace?
\answer |na\"\i ve| or |na{\"\i}ve| or |na\"{\i}ve|.
\exercise List some English words that contain accented letters.
\answer Belov\`ed prot\'eg\'e; r\^ole co\"ordinator; souffl\'es, cr\^epes,
p\^at\'es, etc.
\exercise How would you type `\AE sop's \OE uvres en fran\c cais'\thinspace?
\answer |\AE sop's \OE uvres en fran\c cais|.
\exercise Explain what to type in order to get this sentence:
{\sl Commentarii Academi\ae\ scientiarum imperialis petropolitan\ae\/}
became {\sl Akademi\t\i a Nauk SSSR, Doklady}.
\answer |{\sl Commentarii Academi\ae\ scientiarum imperialis|\hfil\break
|petropolitan\ae\/} became {\sl Akademi\t\i a Nauk SSSR, Doklady}.|
\exercise And how would you specify the names
Ernesto ^{Ces\`aro},
P\'al ^{Erd\H os},
\O ystein ^{Ore},
Stanis\l aw \'Swierczkowski, ^^{Swiercz...}
Serge\u\i\ \t Iur'ev, ^^{Iur'ev}
Mu\d hammad ibn M\^us\^a ^{al-Khw\^arizm\^\i}?
\answer |Ernesto Ces\`aro,
P\'al Erd\H os,
\O ystein Ore,
Stanis\l aw \'Swier%|\break|czkowski,
Serge\u\i\ \t Iur'ev,
Mu\d hammad ibn M\^us\^a al-Khw\^arizm\^\i.|
\dangerexercise Devise a way to typeset {\tt P\'al Erd{\bf\H{\tt o}}s}
in typewriter type.
\answer The proper umlaut is |\H|, which isn't available in |\tt|, so
it's necessary to borrow the accent from another font. For example,
\hbox{|{\tt P\'al Erd{\bf\H{\tt o}}s}|} uses a bold accent, which
is suitably dark.
The following symbols come out looking exactly the same whether you are using
|\rm|, |\sl|, |\bf|, |\it|, or |\tt|:
$$\halign{\indent#\hfil\ &\hfil#\hfil&#\hfil\cr
\it\negthinspace Type&\it to get\cr
\noalign{\smallskip}
|\dag|&\dag&(dagger or obelisk)\cr
|\ddag|&\ddag&(double dagger or diesis)\cr
|\S|&\S&(section number sign)\cr
|\P|&\P&(paragraph sign or pilcrow)\cr}$$
^^{dagger} ^^{double dagger} ^^{obelisk} ^^{obelus, see obelisk} ^^{diesis}
^^{section number sign} ^^{paragraph sign} ^^{pilcrow, see paragraph sign}
(They appear in just one style because plain \TeX\ gets them from the
math symbols font. Lots of other symbols are needed for mathematics;
we shall study them later. See Appendix~B for a few more non-math symbols.)
\exercise In plain \TeX's italic font, the `\$' sign comes out as
`{\it\$}\thinspace'.
^^{dollar sign} ^^{British pound sign} ^^{pound sterling} ^^{sterling}
This gives you a way to refer to pounds sterling, but you might want an
italic dollar sign. Can you think of a way to typeset a reference to
the book {\it Europe on {\sl\$}15.00 a day}\thinspace?
\answer |{\it Europe on {\sl\$}15.00 a day\/}|
\ddanger Appendix B shows that plain \TeX\ handles most of the accents
by using \TeX's ^|\accent| primitive. For example, |\'#1| is equivalent
to |{\accent19 #1}|, where |#1| is the argument being accented.
The general rule is that |\accent|\<number> puts
an accent over the next character; the \<number> tells where that accent
appears in the current font. The accent is assumed to be properly
positioned for a character whose height equals the ^{x-height} of the
current font; taller or shorter characters cause the accent to be raised
or lowered, taking due account of the slantedness of the fonts of accenter
and accentee. The width of the final construction is the width of the
character being accented, regardless of the width of the accent.
Mode-independent commands like font changes may appear between the accent
number and the character to be accented, but grouping operations must not
intervene. If it turns out that no suitable character is present, the
accent will appear by itself as if you had said |\char|\<number> instead
of\/ |\accent|\<number>. For example, |\'{}| produces \'{}.
\ddangerexercise Why do you think plain \TeX\ defines |\'#1| to be
`|{\accent19 #1}|' instead of simply letting |\'| be an abbreviation
for `|\accent19 |'\thinspace? \ (Why the extra
braces, and why the argument |#1|?)
\answer The extra braces keep font changes local. An argument makes the
use of\/ |\'| more consistent with the use of other accents like |\d|, which
are manufactured from other characters without using the |\accent|
primitive.
\ddanger It's important to remember that these conventions we have discussed
for accents and special letters are not built into \TeX\ itself; they belong
only to the plain \TeX\ format, which uses the Computer Modern fonts. Quite
different conventions will be appropriate when other fonts are involved;
format designers should provide rules for how to obtain accents and
special characters in their particular systems. Plain \TeX\ works well
enough when accents are infrequent, but the conventions of this chapter
are by no means recommended for large-scale applications of \TeX\ to
other languages. For example, a well-designed \TeX\ font for ^{French}
might well treat accents as ligatures, so that one could |e'crire
de cette manie`re nai"ve en franc/ais| without backslashes. (See the
remarks about Norwegian in Chapter~8.)
^^{foreign languages}
\endchapter
Let's doo't after the high Roman fashion.
\author WILLIAM ^{SHAKESPEARE}, {\sl The Tragedie of %
Anthony and Cleopatra\/} (1606) % Act IV, Scene 13, line 87
\bigskip
English is a straightforward, frank, honest, open-hearted, no-nonsense language,
which has little truck with such devilish devious devices as accents;
indeed U.S. editors and printers are often thrown into a dither
when a foreign word insinuates itself into the language.
However there is one word on which Americans seem to have closed ranks,
printing it confidently, courageously, and almost invariably
complete with accent---the cheese presented to us as M\"unster.
\smallskip
Unfortunately, ^{Munster} doesn't take an accent.
\author WAVERLEY ^{ROOT}, in the {\sl International Herald Tribune\/} (1982)
% Tuesday 18 May 82 page 8
\eject
\beginchapter Chapter 10. Dimensions
Sometimes you want to tell \TeX\ how big to make a space, or how wide to
make a line. For example, the short story of Chapter~6 used the instruction
`|\vskip .5cm|' to skip vertically by half a centimeter, and we also
said `|\hsize=4in|' to specify a horizontal size of 4~inches. It's time now
to consider the various ways such ^{dimensions} can be communicated to \TeX.
``^{Points}'' and ``^{picas}'' are the traditional units of measure for
printers and compositors in English-speaking countries, so \TeX\
understands points and picas. \TeX\ also understands inches and metric
units, as well as the continental European versions of points and picas.
Each unit of measure is given a two-letter abbreviation, as follows:
^^{units of measure, table}
$$\halign{\indent\tt#&\quad#\hfil\cr
pt&point (baselines in this manual are $12\pt$ apart)\cr
pc&pica ($\rm1\,pc=12\,pt$)\cr
in&inch ($\rm1\,in=72.27\,pt$)\cr
bp&big point ($\rm72\,bp=1\,in$)\cr
cm¢imeter ($\rm2.54\,cm=1\,in$)\cr
mm&millimeter ($\rm10\,mm=1\,cm$)\cr
dd&didot point ($\rm1157\,dd=1238\,pt$)\cr
cc&cicero ($\rm1\,cc=12\,dd$)\cr
sp&scaled point ($\rm65536\,sp=1\,pt$)\cr}$$
^^|pt|^^{point}
^^|pc|^^{pica}
^^|in|^^{inch}
^^|bp|^^{big point}
^^|cm|^^{centimeter}
^^|mm|^^{millimeter}
^^|dd|^^{didot point}^^{Didot, F. A.}
^^|cc|^^{cicero}
^^|sp|^^{scaled point}
The output of \TeX\ is firmly grounded in the metric system, using the
conversion factors shown here as exact ratios.
\exercise How many points are there in 254 centimeters?
\answer Exactly $7227\pt$.
When you want to express some physical dimension to \TeX, type it as
\begindisplay
\<optional sign>\<number>\<unit of measure>\cr
\noalign{\hbox{or}}
\<optional sign>\<digit string>|.|\<digit string>\<unit of measure>\cr
\enddisplay
where an ^\<optional sign> is either a `|+|' or a `|-|' or nothing at all,
and where a ^\<digit string> consists of zero or more consecutive
decimal digits. The `|.|'\ can also be a `|,|'\null.
For example, here are six typical dimensions:
$$\halign{\indent#\hfil&\hskip 6em#\hfil\cr
|3 in|&|29 pc|\cr
|-.013837in|&|+ 42,1 dd|\cr
|0.mm|&|123456789sp|\cr}$$
A plus sign is redundant, but some people occasionally like extra
redundancy once in a~while. Blank spaces are optional before the signs and the
numbers and the units of measure, and you can also put an optional space
after the dimension; but you should not put spaces within the digits
of a number or between the letters of the unit of measure.
\exercise Arrange those six ``typical dimensions'' into order,
from smallest to largest.
\answer $\rm-.013837\,in$, $\rm0.\,mm$, $\rm+42.1\,dd$, $\rm3\,in$,
$\rm29\,pc$, $\rm123456789\,sp$.
\ (The lines of text in this manual are 29~picas wide.)
\dangerexercise Two of the following three dimensions are legitimate
according to \TeX's rules. Which two are they? What do they mean?
Why is the other one incorrect?
\begintt
'.77pt
"Ccc
-,sp
\endtt
\answer The first is not allowed, since octal notation cannot be used with
a decimal point. The second is, however, legal, since a \<number> can be
hexadecimal according to the rule mentioned in Chapter~8; it means
$\rm12\,cc$, which is $\rm144\,dd\approx154.08124\,pt$. The third is also
accepted, since a \<digit string> can be empty; it is a complicated
way to say $\rm0\,sp$.
\smallskip
The following ``rulers'' have been typeset by \TeX\ so that you can get
some idea of how different units compare to each other. If no distortion
has been introduced during the camera work and printing processes that
have taken place after \TeX\ did its work, these rulers are highly accurate.
$$ \abovedisplayskip 15pt plus 4pt minus 4pt
\belowdisplayskip 15pt plus 4pt minus 4pt
\vbox{
\def\1{\vrule height 0pt depth 2pt}
\def\2{\vrule height 0pt depth 4pt}
\def\3{\vrule height 0pt depth 6pt}
\def\4{\vrule height 0pt depth 8pt}
\def\ruler#1#2#3{\leftline{$\vcenter{\hrule\hbox{\4#1}}\,\,\rm#2\,{#3}$}}
\def\\#1{\hbox to .125in{\hfil#1}}
\def\8{\\\1\\\2\\\1\\\3\\\1\\\2\\\1\\\4}
\ruler{\8\8\8\8}4{in}
\vskip 18pt
\def\\#1{\hbox to 10pt{\hfil#1}}
\def\8{\\\1\\\1\\\1\\\1\\\2\\\1\\\1\\\1\\\1\\\4}
\ruler{\8\8\8}{300}{pt}
\vskip 18pt
\def\\#1{\hbox to 10dd{\hfil#1}}
\def\8{\\\1\\\1\\\1\\\1\\\2\\\1\\\1\\\1\\\1\\\4}
\ruler{\8\8\8}{300}{dd}
\vskip 18pt
\def\\#1{\hbox to 5mm{\hfil#1}}
\def\8{\\\2\\\4}
\ruler{\8\8\8\8\8\8\8\8\8\8}{10}{cm}
\vskip 6pt}$$
\dangerexercise (To be worked after you know about boxes and glue and have
read Chapter~21.) \ Explain how to typeset such a $\rm10\,cm$ ^{ruler},
using \TeX.
\answer {\obeylines|\def\tick#1{\vrule height 0pt depth #1pt}|
|\def\\{\hbox to 1cm{\hfil\tick4\hfil\tick8}}|
|\vbox{\hrule\hbox{\tick8\\\\\\\\\\\\\\\\\\\\}}|
\noindent(You might also try putting ticks at every millimeter, in order %
to see how good your system is; %
some output devices can't handle 101~rules all at once.)}
\danger \TeX\ represents all dimensions internally as an integer multiple
of the tiny units called sp. Since the wavelength of visible light is
approximately $\rm100\,sp$, % in fact: violet=75sp, red=135sp!
rounding errors of a few sp make no difference to the eye.
However, \TeX\ does all of its arithmetic very carefully so that
identical results will be obtained on different computers. Different
implementations of \TeX\ will produce the same line breaks and the same
page breaks when presented with the same document, because the integer
arithmetic will be the same.
^^{machine-independence} ^^{rounding}
\danger The units have been defined here so that precise conversion to~sp
is~efficient on a wide variety of machines. In order to achieve this,
\TeX's ``pt'' has been made slightly larger than the official printer's
point, which was defined to equal exactly $\rm.013837\,in$ by the American
Typefounders Association in~1886 [cf.~National Bureau of Standards
Circular~570 (1956)]. In fact, one classical point is exactly
$.99999999\pt$, so the ``error'' is essentially one part in $10^8$.
This is more than two orders of magnitude less than the amount by which
the inch itself changed during 1959, when it shrank to $\rm2.54\,cm$ from
its former value of $\rm(1/0.3937)\,cm$; so there is no point in worrying
about the difference. The new definition $\rm72.27\,pt=1\,in$ is not only
better for calculation, it~is also easier to remember.
\danger \TeX\ will not deal with dimensions whose absolute value is
$\rm2^{30}\,sp$ or more. In other words, the ^{maximum legal dimension} is
slightly less than $16384\pt$. This is a distance of about 18.892 feet
(5.7583 meters), so it won't cramp your style.
In a language manual like this it is convenient to use ``^{angle brackets}''
in abbreviations for various constructions like \<number> and \<optional
sign> and \<digit string>. Henceforth we shall use the term ^\<dimen> to
stand for a legitimate \TeX\ dimension. For example,
\begindisplay
|\hsize=|\<dimen>
\enddisplay
will be the general way to define the column width that \TeX\ is supposed
to use. The idea is that \<dimen> can be replaced by any quantity like
`|4in|' that satisfies \TeX's grammatical rules for dimensions;
abbreviations in angle brackets make it easy to state such laws of grammar.
When a dimension is zero, you have to specify a unit of measure even
though the unit is irrelevant. Don't just say `|0|'\thinspace; say `|0pt|' or
`|0in|' or something.
\smallbreak
The 10-point size of type that you are now reading is normal in textbooks,
but you probably will often find yourself wanting a larger font. Plain \TeX\
makes it easy to do this by providing {\magnifiedfiverm ^{magnif{}ied
output}.} If you say
\begintt
\magnification=1200
\endtt
at the beginning of your manuscript, everything will be enlarged by 20\%;
i.e., it will come out at 1.2 times the normal size. Similarly,
`|\magnification=2000|' doubles everything; this actually quadruples the area of
each letter, since heights and widths are both doubled. To magnify a
document by the factor $f$, you say ^|\magnification||=|\<number>, where
the \<number> is 1000~times~$f$. This instruction must be given before the
first page of output has been completed. You cannot apply two different
magnifications to the same document.
Magnification has obvious advantages: You'll have less ^{eyestrain} when
you're ^{proofreading}; you can easily make ^{transparencies} ^^{slides}
for lectures; and you can photo-reduce magnified output, in order to minimize
the deficiencies of a ^{low-resolution printer}. Conversely, you might
even want `|\magnification=500|' in order to create a ^{pocket-size}
version of some book. ^^{squint print} But there's a slight catch:
You can't use magnification unless your printing device happens to have the
fonts that you need at the magnification you desire. In other words, you need
to find out what sizes are available before you can magnify. Most
installations of \TeX\ make it possible to print all the fonts of plain
\TeX\ if you magnify by ^|\magstep||0|,~|1|, |2|,~|3|, and perhaps~|4| or
even~|5| (see Chapter~4); but the use of large fonts can be expensive
because a lot of system memory space is often required to store the shapes.
\exercise Try printing the short story of Chapter 6 at 1.2, 1.44, and 1.728
times the normal size. What should you type to get \TeX\ to do this?
\answer For example, say `|\magnification=\magstep1 \input story \end|'
to get magnification 1200; |\magstep2| and |\magstep3| are 1440 and 1728.
Three separate runs are needed, since there can be at most one
magnification per job. The output may look funny if the fonts don't exist
at the stated magnifications.
\danger When you say |\magnification=2000|, an operation like
`|\vskip.5cm|' will actually skip $\rm1.0\,cm$ of space in the final
document. If you want to specify a dimension in terms of the final size,
\TeX\ allows you to say `^|true|' just before |pt|, |pc|, |in|, |bp|,
|cm|, |mm|, |dd|, |cc|, and |sp|. This unmagnifies the units, so that the
subsequent magnification will cancel out. For example, `|\vskip.5truecm|'
is equivalent to `|\vskip.25cm|' if you have previously said
`|\magnification=2000|'. Plain \TeX\ uses this feature in the
|\magnification| command itself: Appendix~B includes the instruction
\begintt
\hsize = 6.5 true in
\endtt
just after a new magnification has taken effect. This adjusts the line width
so that the material on each page will be $6{1\over2}$ inches wide when it
is finally printed, regardless of the magnification factor.
There will be an inch of margin at both left and right,
assuming that the paper is $8{1\over2}$ inches wide.
\danger If you use no `|true|' dimensions, \TeX's internal computations are not
affected by the presence or absence of magnification; line breaks and page
breaks will be the same, and the ^|dvi| file will change in only two places.
\TeX\ simply tells the printing routine that you want a certain magnification,
and the printing routine will do the actual enlargement when it reads the
|dvi| file.
\dangerexercise Chapter 4 mentions that fonts of different magnifications
can be used in the same job, by loading them `^|at|' different sizes.
Explain what fonts will be used when you give the commands
^^{magnified fonts} ^^|scaled|
\begintt
\magnification=\magstep1
\font\first=cmr10 scaled\magstep1
\font\second=cmr10 at 12truept
\endtt
\answer Magnification is by a factor of 1.2. Since font |\first| is |cmr10|
at $12\pt$, it will be |cmr10| at $14.4\pt$ after magnification;
font |\second| will be |cmr10| at $12\pt$. \ (\TeX\ changes
`|12truept|' into `|10pt|', and the final output magnifies it back to
$12\pt$.)
\ddanger Magnification is actually governed by \TeX's ^|\mag| primitive,
which is an integer parameter that should be positive and at~most~32768.
The value of\/ |\mag| is examined in three cases: (1)~just before the
first page is shipped to the |dvi| file; (2)~when computing a |true|
dimension; (3)~when the |dvi| file is being closed. Alternatively,
some implementations of \TeX\ produce non-|dvi| output; they examine
|\mag| in case~(2) and also when shipping out each page. Since each
document has only one magnification, the value of\/ |\mag| must not change
after it has first been examined.
\danger \TeX\ also recognizes two units of measure that are relative
rather than absolute; i.e., they depend on the current context:
\begindisplay
^|em| is the width of a ``^{quad}'' in the current font;\cr
^|ex| is the ``^{x-height}'' of the current font.\cr
\enddisplay
Each font defines its own em and ex values. In olden days, an ``em'' was
the width of an `M', but this is no longer true; ems are simply arbitrary
units that come with a font, and so are exes. The Computer Modern fonts
have the property that an em-dash is one em wide, each of the ^{digits} 0
to~9 is half an em wide, and lowercase `x' is one ex high; but these are
not hard-and-fast rules for all fonts.
The |\rm| font (^|cmr10|) of plain \TeX\ has $\rm1\,em=10\,pt$
and $\rm1\,ex\approx4.3\,pt$; the |\bf| font (^|cmbx10|) has
$\rm1\,em=11.5\,pt$ and $\rm1\,ex\approx4.44\,pt$; and the |\tt| font
(^|cmtt10|) has $\rm1\,em=10.5\,pt$ and $\rm1\,ex\approx4.3\,pt$. All of
these are ``10-point'' fonts, yet they have different em and ex values.
It~is generally best to use |em| for horizontal measurements and |ex| for
vertical measurements that depend on the current font.
\danger A \<dimen> can also refer to \TeX's internal registers or parameters.
We shall discuss registers later, and a complete definition of everything that
a ^\<dimen> can be will be given in Chapter~24. For now it will suffice to
give some hints about what is to come:
`|\hsize|' stands for the current horizontal line size,
and `|.5\hsize|' is half that amount;
`|2\wd3|' denotes twice the width of register~|\box3|;
`|-\dimen100|' is the negative of register~|\dimen100|.
\ddanger Notice that the unit names in dimensions
are not preceded by backslashes. The same is true of other so-called
^{keywords} of the \TeX\ language. Keywords can be given in uppercase letters
or in a mixture of upper and lower case; e.g., `|Pt|' is equivalent to `|pt|'.
The category codes of these letters are irrelevant; you may, for example,
be using a |p| of category~12 (other) that was generated by expanding
`|\the\hsize|' as explained in Chapter~20.
\TeX\ gives a special interpretation to keywords only when they
appear in certain very restricted contexts. For example, `|pt|' is a
keyword only when it appears after a number in a \<dimen>;
`|at|' is a keyword only when it appears after the external name of a
font in a |\font| declaration.
Here is a complete list of \TeX's keywords, in case you are wondering about
the full set: |at|, |bp|, |by|, |cc|, |cm|, |dd|, |depth|, |em|, |ex|,
|fil|, |height|, |in|, |l|, |minus|, |mm|, |mu|, |pc|, |plus|,
|pt|, |scaled|, |sp|, |spread|, |to|, |true|, |width|. ^^{reserved words}
\ (See Appendix~I for references to the contexts in which each of these is
recognized as a keyword.)
\endchapter
The methods that have hitherto been taken
to discover the measure of the Roman foot,
will, upon examination, be found so unsatisfactory, that
it is no wonder the learned are not yet agreed on that point.
$\ldots$
9 London inches are equal to 8,447 Paris inches.
\author MATTHEW ^{RAPER}, in {\sl Philosophical Transactions\/} (1760)
% ``An Enquiry into the Measure of the {\sl Roman\/} Foot,''
% {\sl Philos.\ Trans.\ \bf51} (1760), 774--823.
\bigskip
\checkequals\sesame\pageno %
Without the letter U,
units would be nits.
\author ^{SESAME STREET}{^^{Children's Television Workshop}} (1970)
\eject
\beginchapter Chapter 11. Boxes
\TeX\ makes complicated pages by starting with simple individual characters
and putting them together in larger units, and putting these together in still
larger units, and so on. Conceptually, it's a big paste-up job. The \TeX nical
terms used to describe such page construction are {\sl ^{boxes}\/} and
{\sl ^{glue}}.
Boxes in \TeX\ are two-dimensional things with a rectangular shape, having
three associated measurements called {\sl^{height}}, {\sl^{width}}, and
{\sl^{depth}}. Here is a picture of a typical box, showing its so-called
^{reference point} and ^{baseline}:
{\eightpoint
\setbox0=\hbox{$\uparrow$}
\setbox1=\hbox to \wd0{$\hss\mid\hss$} % with luck, they'll line up
\setbox2=\vbox{\copy0
\nointerlineskip \kern-.5pt \copy1
\nointerlineskip \kern-.5pt \copy1
\moveleft 1em\hbox{height}
\copy1 \nointerlineskip \kern-.5pt
\copy1 \nointerlineskip \kern-.5pt
\hbox{$\downarrow$}
\kern.2pt}
\setbox3=\vbox{\kern.2pt\copy0
\moveleft 1em\hbox{depth}
\hbox{$\downarrow$}
\kern0pt}
\setbox4=\vtop{\kern-3pt % this cancels the null text above the samplebox
\hbox{\samplebox{\ht2}{\ht3}{6em}{}%
\kern-6em
\raise3pt\hbox to 6em{\hss Baseline\hss}}
\kern3pt
\arrows{6em}{width}}
\medskip\indent
\setbox0=\hbox{$\vcenter{}$}% \ht0 is the axis height
\lower\ht0\hbox{Reference point$-$\kern-.2em$\rightarrow$\kern2pt}%
\raise\ht2\box4
\kern1.5em
\raise\ht2\vtop{\kern0pt\box2\nointerlineskip\box3}}
\medskip\noindent
From \TeX's viewpoint, a single character from a font is a box; it's one
of the simplest kinds of boxes. The font designer has decided what the
height, width, and depth of the character are, and what the symbol will
look like when it is in the box; \TeX\ uses these dimensions to paste
boxes together, and ultimately to determine the locations of the reference
points for all characters on a page. In plain \TeX's |\rm| font (|cmr10|), for
example, the letter `h' has a height of 6.9444 points, a width of 5.5555
points, and a depth of zero; the letter `g' has a height of 4.3055
points, a width of 5 points, and a depth of 1.9444 points. Only certain
special characters like parentheses have height plus depth actually equal
to 10 points, although ^|cmr10| is said to be a ``10-point'' font. You needn't
bother to learn these measurements yourself, but it's good to be aware of
the fact that \TeX\ deals with such information; then you can better
understand what the computer does to your manuscript.
The character shape need not fit inside the boundaries of its box. For example,
some characters that are used to build up larger math symbols like matrix
brackets intentionally protrude a little bit, so that they overlap
properly with the rest of the symbol. Slanted letters frequently extend a
little to the right of the box, as if the box were skewed right at the top
and left at the bottom, keeping its baseline fixed. For example, compare
the letter `g' in the |cmr10| and ^|cmsl10| fonts (|\rm| and |\sl|):
\begindisplay
\vbox to 40pt{\ifproofmode\hrule\vfill
\hsize=2.5in \baselineskip 6pt \fiverm\noindent
(A figure will be inserted here; too bad you can't see it now.
It shows two g's, as claimed.)
\vfill\hrule\fi}
\enddisplay
In both cases \TeX\ thinks that the box is 5 points wide, so both letters get
exactly the same treatment. \TeX\ doesn't have any idea where the ink will
go---only the output device knows this. But the slanted letters will be
spaced properly in spite of \TeX's lack of knowledge, because the baselines
will match up.
Actually the font designer also tells \TeX\ one other thing, the so-called
{\sl^{italic correction}\/}: A number is specified for each character,
telling roughly how far that character extends to the right of its box
boundary, plus a little to spare. For example,
the italic correction for `g' in |cmr10| is $0.1389\pt$, while in |cmsl10|
it is $0.8565\pt$. Chapter~4 points out that this correction is added to the
normal width if you type `^|\/|' just after the character. You should remember
to use |\/| when shifting from a slanted font to an unslanted one, especially
in cases like
\begintt
the so-called {\sl italic correction\/}:
\endtt
since no space intervenes here to compensate for the loss of slant.
\smallbreak
\TeX\ also deals with another simple kind of box, which might be called
a~``^{black box},'' namely, a rectangle like
`\thinspace \vrule width 4pt height 6pt depth 1.5pt \thinspace'
that is to be entirely filled with ink at printing time. You can specify any
height, width, and depth you like for such boxes---but they had better not have
too much area, or the printer might get upset. \ (Printers generally
prefer white space to black space.)
Usually these black boxes are made very skinny, so that they appear as
horizontal lines or vertical lines. Printers traditionally call such lines
``^{horizontal rules}'' and ``^{vertical rules},'' so the terms \TeX\ uses
to stand for black boxes are ^|\hrule| and ^|\vrule|. Even when the box is
square, as in `\thinspace\bull\thinspace', you must call it either an~|\hrule|
or a~|\vrule|. We shall discuss the use of ^{rule boxes} in greater
detail later. \ (See Chapter~21.)
\smallbreak
Everything on a page that has been typeset by \TeX\ is made up of simple
character boxes or rule boxes, pasted together in combination. \TeX\
pastes boxes together in two ways, either {\sl horizontally\/} or {\sl
vertically}. When \TeX\ builds a ^{horizontal list} of boxes, it lines
them up so that their reference points appear in the same horizontal row;
therefore the baselines of adjacent characters will match up as they
should. Similarly, when \TeX\ builds a ^{vertical list} of boxes, it lines
them up so that their reference points appear in the same vertical column.
% Here are some macros for making blank boxes
\def\dolist{\afterassignment\dodolist\let\next= }
\def\dodolist{\ifx\next\endlist \let\next\relax
\else \\\let\next\dolist \fi
\next}
\def\endlist{\endlist}
\def\\{\expandafter\if\space\next\ \else \setbox0=\hbox{\next}\maketypebox\fi}
\def\demobox#1{\setbox0=\hbox{\dolist#1\endlist}%
\copy0\kern-\wd0\makelightbox}
Let's take a look at what \TeX\ does behind the scenes, by comparing the
computer's methods with what you would do if you were setting metal type
by hand. In the time-tested traditional method, you choose the letters that
you~need out of a type case---the uppercase letters are in the ^{upper
case}---and you put them into a ``^{composing stick}.'' When a line is
complete, you adjust the spacing and transfer the result to the ``chase,''
where it joins the other rows of type. Eventually you lock the type up
tightly by adjusting external wedges called ``quoins.'' This isn't much
different from what \TeX\ does, except that different words are used; when
\TeX\ locks up a line, it creates what is called an ``^{hbox}''
(^{horizontal box}), because the components of the line are pieced
together horizontally. You can give an instruction like
\begintt
\hbox{A line of type.}
\endtt
in a \TeX\ manuscript; this tells the computer to take boxes for the appropriate
letters in the current font and to lock them up in an hbox. As far as \TeX\ is
concerned, the letter `A' is a box
`\thinspace\setbox0\hbox{A}\maketypebox\thinspace'
and the letter `p' is a box
`\thinspace\setbox0\hbox{p}\maketypebox\thinspace'.
So the given instruction causes \TeX\ to form the hbox
\begindisplay
\demobox{A line of type.}
\enddisplay
representing `A line of type.' The hboxes for individual lines of type are
eventually joined together by putting them into a ``^{vbox}'' (^{vertical
box}). For example, you can say
\begintt
\vbox{\hbox{Two lines}\hbox{of type.}}
\endtt
and \TeX\ will convert this into
\begindisplay%
\setbox0=\vbox{\hbox{\demobox{Two lines}}\hbox{\demobox{of type.}}}
$\vcenter{\hbox{\makelightbox\kern-\wd0\box0}}$\qquad
i.e.,\qquad$\vcenter{\vbox{\hbox{Two lines}\hbox{of type.}}}$
\enddisplay
The principal difference between \TeX's method and the old way is that metal
types are generally cast so that each character has the same height and
depth; this makes it easy to line them up by hand. \TeX's types have
variable height and depth, because the computer has no trouble lining
characters up by their baselines, and because the extra information about
height and depth helps in the positioning of accents and mathematical
symbols.
Another important difference between \TeX\ setting and hand setting is, of
course, that \TeX\ will choose line divisions automatically; you don't
have to insert ^|\hbox| and ^|\vbox| instructions unless you want to
retain complete control over where each letter goes. On the other hand,
if you do use |\hbox| and |\vbox|, you can make \TeX\ do almost everything
that Ben ^{Franklin} could do in his printer's shop. You're only giving
up the ability to make the letters come out charmingly crooked or badly
inked; for such effects you need to make a new font. \ (And of course you
lose the tactile and olfactory sensations, and the thrill of
doing everything by yourself. \TeX\ will never completely replace the
good~old~ways.)
A page of text like the one you're reading is itself a box, in \TeX's view:
It is a largish box made from a vertical list of smaller boxes representing
the lines of text. Each line of text, in turn, is a box made from a
horizontal list of boxes representing the individual characters. In more
complicated situations, involving mathematical formulas and/or complex
tables, you can have boxes within boxes within boxes $\ldots$ to any level.
But even these complicated situations arise from horizontal or vertical lists
of boxes pasted together in a simple way; all that you and \TeX\ have to
worry about is one list of boxes at a time. In fact, when you're typing
straight text, you don't have to think about boxes at all, since \TeX\ will
automatically take responsibility for assembling the character boxes into
words and the words into lines and the lines into pages. You need to be
aware of the box concept only when you want to do something out of the
ordinary, e.g., when you want to center a heading.
\danger From the standpoint of \TeX's digestive processes, a manuscript
comes in as a sequence of tokens, and the tokens are to be transformed into
a sequence of boxes. Each token of input is essentially an instruction or
a piece of an instruction; for example, the token `|A|$_{11}$' normally means,
``put a character box for the letter |A| at the end of the current hbox,
using the current font''; the token `\cstok{vskip}' normally means, ``skip
vertically in the current vbox by the \<dimen> specified in the
following tokens.''
\danger The height, width, or depth of a box might be negative, in which
case it is a ``^{shadow box}'' that is somewhat hard to draw. \TeX\ doesn't
balk at ^{negative dimensions}; it just does arithmetic as usual. For example,
the combined width of two adjacent boxes is the sum of their widths, whether
or not the widths are positive. A font designer can declare a character's
width to be negative, in which case the character acts like a ^{backspace}. \
(Languages that read from right to left could be
handled in this way, but only to a limited extent, since \TeX's line-breaking
^^{Hebrew} ^^{Arabic}
algorithm is based on the assumption that words don't have negative widths.)
\danger \TeX\ can raise or lower the individual boxes
in a horizontal list; such adjustments take care of mathematical
subscripts and superscripts, as well as the heights of accents and a few
other things. For example, here is a way to make a box that contains
the \TeX\ logo, putting it into \TeX's internal register |\box0|:
\begintt
\setbox0=\hbox{T\kern-.1667em\lower.5ex\hbox{E}\kern-.125em X}
\endtt
^^|\setbox|
Here `^|\kern||-.1667em|' means to insert blank space of $-.1667$ ems in the
current font, i.e., to back up a bit; and `^|\lower||.5ex|' means that
the box |\hbox{E}| is to be lowered by half of the current x-height, thus
offsetting that box with respect to the others. Instead of
`|\lower.5ex|' one could also say `^|\raise||-.5ex|'. Chapters 12 and~21
discuss the details of how to construct boxes for special effects;
our goal in the present chapter is merely to get a taste of the
possibilities.
\danger \TeX\ will exhibit the contents of any ^{box register}, if you
ask it to. For example, if you type `^|\showbox||0|' after setting
|\box0| to the \TeX\ logo as above, your ^{log file} will contain
the following mumbo jumbo: ^^{TeX logo}
\begintt
\hbox(6.83331+2.15277)x18.6108
.\tenrm T
.\kern -1.66702
.\hbox(6.83331+0.0)x6.80557, shifted 2.15277
..\tenrm E
.\kern -1.25
.\tenrm X
\endtt
^^{diagnostic format} ^^{internal box-and-glue representation} ^^{box displays}
The first line means that |\box0| is an hbox whose height, depth, and width
are respectively $6.83331\pt$, $2.15277\pt$, and $18.6108\pt$.
Subsequent lines beginning with `|.|'\ indicate that they are {\sl inside\/}
of a box. The first thing in this particular box is the letter~|T| in
font |\tenrm|; then comes a kern. The next item is an hbox that contains
only the letter~|E|; this box has the height, depth, and width of an |E|, and
it has been shifted downward by $2.15277\pt$ (thereby accounting for
the depth of the larger box).
\dangerexercise Why are there two dots in the `|..\tenrm E|' line here?
\answer This |E| is inside a box that's inside a box.
\danger Such displays of box contents will be discussed further in
Chapters 12 and~17.
They are used primarily for diagnostic purposes, when you are trying to figure
out exactly what \TeX\ thinks it's doing. The main reason for bringing them
up in the present chapter is simply to provide a glimpse of how \TeX\ represents
boxes in its guts. A computer program doesn't really move boxes around; it
fiddles with lists of representations of boxes.
\dangerexercise By running \TeX, figure out how it actually handles italic
corrections to characters: How are the corrections represented inside a box?
\answer The idea is to construct a box and to look inside. For example,
\begintt
\setbox0=\hbox{\sl g\/} \showbox0
\endtt
reveals that |\/| is implemented by placing a kern after the character.
Further experiment shows that this kern is inserted even when the italic
correction is zero.
\dangerexercise The ``opposite'' of \TeX's logo---namely,
T\kern+.1667em\raise.5ex\hbox{E}\kern+.125em X---is produced by
\begintt
\setbox1=\hbox{T\kern+.1667em\raise.5ex\hbox{E}\kern+.125em X}
\endtt
What would |\showbox1| show now? \ (Try to guess, without running the machine.)
\answer The height, depth, and width of the enclosing box should be just large
enough to enclose all of the contents, so the result is:
\begintt
\hbox(8.98608+0.0)x24.44484
.\tenrm T
.\kern 1.66702
.\hbox(6.83331+0.0)x6.80557, shifted -2.15277
..\tenrm E
.\kern 1.25
.\tenrm X
\endtt
(You probably predicted a height of |8.9861|; \TeX's internal calculations are
in |sp|, not |pt|/100000, so the rounding in the fifth decimal place is not
readily predictable.)
\dangerexercise Why do you think the author of \TeX\ didn't make boxes more
symmetrical between horizontal and vertical, by allowing reference points
to be inside the boundary instead of insisting that the reference point
must appear at the left edge of each box?
\answer No applications of such symmetrical boxes to English-language
printing were apparent; it seemed pointless to carry extra generality
as useless baggage that would rarely if ever be used, merely for the sake of
symmetry. In other words, the author wore a computer science cap instead
of a mathematician's mantle on the day that \TeX's boxes were born.
Time will tell whether or not this was a fundamental error!
\ddangerexercise Construct a |\demobox| macro for use in writing manuals
like this, so that an author can write `|\demobox{Tough exercise.}|'
in order to typeset `\thinspace\demobox{Tough exercise.}\thinspace'.
\answer The following solution is based on a general |\makeblankbox|
macro that prints the edges of a box using rules of given thickness
outside and inside that box; the box dimensions are those of\/ |\box0|.\par
|\def\dolist{\afterassignment\dodolist\let\next= }|\parbreak
|\def\dodolist{\ifx\next\endlist \let\next\relax|\parbreak
| \else \\\let\next\dolist \fi|\parbreak
| \next}|\par
|\def\endlist{\endlist}|\par
|\def\hidehrule#1#2{\kern-#1%|\parbreak
| \hrule height#1 depth#2 \kern-#2 }|\par
|\def\hidevrule#1#2{\kern-#1{\dimen0=#1|\parbreak
| \advance\dimen0 by#2\vrule width\dimen0}\kern-#2 }|\par
|\def\makeblankbox#1#2{\hbox{\lower\dp0\vbox{\hidehrule{#1}{#2}%|\parbreak
| \kern-#1 % overlap the rules at the corners|\parbreak
| \hbox to \wd0{\hidevrule{#1}{#2}%|\parbreak
| \raise\ht0\vbox to #1{}% set the vrule height|\parbreak
| \lower\dp0\vtop to #1{}% set the vrule depth|\parbreak
| \hfil\hidevrule{#2}{#1}}%|\parbreak
| \kern-#1\hidehrule{#2}{#1}}}}|\par
|\def\maketypebox{\makeblankbox{0pt}{1pt}}|\par
|\def\makelightbox{\makeblankbox{.2pt}{.2pt}}|\par
|\def\\{\if\space\next\ % assume that \next is unexpandable|\parbreak
| \else \setbox0=\hbox{\next}\maketypebox\fi}|\par
|\def\demobox#1{\setbox0=\hbox{\dolist#1\endlist}%|\parbreak
| \leavevmode\copy0\kern-\wd0\makelightbox}|\par
\def\frac#1/#2{\leavevmode\kern.1em
\raise.5ex\hbox{\the\scriptfont0 #1}\kern-.1em
/\kern-.15em\lower.25ex\hbox{\the\scriptfont0 #2}}
\ddangerexercise Construct a |\frac| macro such that `|\frac1/2|' yields
`\frac1/2'. \checkequals\fracexno\exno
\answer |\def\frac#1/#2{\leavevmode\kern.1em|\parbreak
|\raise.5ex\hbox{\the\scriptfont0 #1}\kern-.1em|\parbreak
|/\kern-.15em\lower.25ex\hbox{\the\scriptfont0 #2}}|
\endchapter
I have several boxes in my memory
in which I will keep them all very safe,
% he's talking about "instructions"
there shall not a one of them be lost.
\author IZAAK ^{WALTON}, {\sl The Compleat Angler\/} (1653) % beginning Chap12
% in 1654 and subsequent editions, this quote comes in Chap17
% the 1653 spelling agrees with 20th century conventions in this passage!
\bigskip
How very little does the amateur, dwelling at home at ease,
comprehend the labours and perils of the author.
\author R. L. ^{STEVENSON} and L. ^{OSBOURNE}, {\sl The Wrong Box\/} (1889)
\eject
\beginchapter Chapter 12. Glue
But there's more to the story than just boxes: There's also some magic mortar
called {\sl ^{glue}\/} that
\TeX\ uses to paste boxes together. For example, there is a little space between
the lines of text in this manual; it has been calculated so that the baselines
of consecutive lines within a paragraph are exactly 12~points apart. And
there is space between words too; such space is not an ``empty'' box, it is
part of the glue between boxes. This glue can stretch or shrink so that the
right-hand margin of each page comes out looking straight.
^^{leading, see baselineskip} ^^{skipping space, see glue}
When \TeX\ makes a large box from a horizontal or vertical list of smaller
boxes, there often is glue between the smaller boxes. Glue has three
attributes, namely its natural {\sl space}, its ability to {\sl ^{stretch}}, and
its ability to {\sl ^{shrink}}.
In order to understand how this works, consider the following example of
four boxes in a horizontal list separated by three globs of glue:
\begindisplay\eightpoint
\vbox{
\hbox{\samplebox{7mm}{8mm}{5\varunit}{width 5}%
\sampleglue{9\varunit}{space 9\cr stretch 3\cr shrink 1}%
\samplebox{3mm}{2mm}{6\varunit}{width 6}%
\sampleglue{9\varunit}{space 9\cr stretch 6\cr shrink 2}%
\samplebox{8mm}{3mm}{3\varunit}{width 3}%
\sampleglue{12\varunit}{space 12\cr stretch 0\cr shrink 0}%
\samplebox{4mm}{7mm}{8\varunit}{width 8}}
\kern6pt
\arrows{52\varunit}{width 52}}
\enddisplay
The first glue element has 9 units of space, 3 of stretch, and 1 of shrink;
the next one also has 9 units of space, but 6 units of stretch and 2 of
shrink; the last one has 12 units of space, but it is unable to stretch
or to shrink, so it will remain 12 units of space no matter what.
The total width of boxes and glue in this example, considering only the
space components of the glue, is $5+9+6+9+3+12+8=52$ units. This is called
the {\sl ^{natural width}\/} of the horizontal list; it's the preferred way to
paste the boxes together. Suppose, however, that \TeX\ is told to make the
horizontal list into a box that is 58~units wide; then the glue has to
stretch by 6~units. Well, there are $3+6+0=9$ units of stretchability present,
so \TeX\ multiplies each unit of stretchability by 6/9 in order to obtain the
extra 6~units needed. The first glob of glue becomes $9+(6/9)\times3=11$
units wide, the next becomes $9+(6/9)\times6=13$ units wide, the last remains
12 units wide, and we obtain the desired box looking like this:
\begindisplay\eightpoint
\vbox{\kern-3pt
\hbox{\samplebox{7mm}{8mm}{5\varunit}{}%
\sampleglue{11\varunit}{$9+2$}%
\samplebox{3mm}{2mm}{6\varunit}{}%
\sampleglue{13\varunit}{$9+4$}%
\samplebox{8mm}{3mm}{3\varunit}{}%
\sampleglue{12\varunit}{$12+0$}%
\samplebox{4mm}{7mm}{8\varunit}{}}
\kern6pt
\arrows{58\varunit}{width 58}}
\enddisplay
On the other hand, if \TeX\ is supposed to make a box 51 units wide from the
given list, it is necessary for the glue to shrink by a total of one unit. There
are three units of shrinkability present, so the first glob of glue would
shrink by 1/3 and the second by 2/3.
\smallbreak
The process of determining glue thickness when a box is being made from a
horizontal or vertical list is called {\sl ^{setting the glue}}. Once glue has
been set, it becomes rigid; it won't stretch or shrink any more, and the
resulting box is essentially indecomposable.
Glue will never shrink more than its stated shrinkability. For example,
the first glob of glue in our illustration will never be allowed to become
narrower than 8 units wide, and \TeX\ will never shrink the given
horizontal list to make its total width less than 49 units. But glue is
allowed to stretch arbitrarily far, whenever it has a positive stretch
component.
\exercise How wide would the glue globs be if the horizontal list in the
illustration were to be made 100 units wide?
\answer $9+16$ units, $9+32$ units, $12+0$ units. \ (But \TeX\ would
consider so much stretching to be ``infinitely bad.'')
Once you understand \TeX's concept of glue, you may well decide that it
was misnamed; real glue doesn't stretch or shrink in such ways, nor does it
contribute much space between boxes that it welds together. Another word
like ``spring'' would be much closer to the essential idea, since ^{springs}
have a natural width, and since different springs compress and expand at
different rates under tension. But whenever the author has suggested
changing \TeX's terminology, numerous people have said that they like the
word ``glue'' in spite of its inappropriateness; so the original name has
stuck.
\danger \TeX\ is somewhat reluctant to stretch glue more than the stated
stretchability; therefore you can decide how big to make each aspect of the
glue by using the following rules: \ (a)~The natural
glue space should be the amount of space that looks best. \ (b)~The glue
stretch should be the maximum amount of space that can be added to the natural
spacing before the layout begins to look bad. \ (c)~The glue shrink should
be the maximum amount of space that can be subtracted from the natural spacing
before the layout begins to look bad.
In most cases the designer of a book layout will have specified all the kinds
of glue that are to be used, so a typist will not need to decide how big any
glue attributes should be. For example, users of the plain \TeX\ format of
Appendix~B can type `|\smallskip|' when they want a little extra ^{space
between paragraphs}; a ^|\smallskip| turns out to be $3\pt$ worth of
vertical glue that can stretch or shrink by an additional~$1\pt$. Here is
a |\smallskip|: \smallskip
\noindent
Instead of sprinkling various amounts of glue throughout a manuscript,
expressing each of them explicitly in terms of points, you will find it
much better to explain your intentions more clearly by typing something
like `|\smallskip|' when you want abnormal spacing. The definition of\/
|\smallskip| can readily be changed later, in case you want such spaces to
be smaller or larger. Plain \TeX\ also provides you with `^|\medskip|',
which is worth two smallskips, and `^|\bigskip|', which is worth two medskips.
\danger A plain \TeX\ |\medskip| appears before and after each
``^{dangerous bend}'' section of this manual, so you have already seen
numerous examples of such spacing before you knew what it was called.
Vertical glue is created by writing `|\vskip|\<glue>', where ^\<glue> is
any glue specification. The usual way to specify \<glue> to \TeX\ is
\begindisplay
\<dimen> |plus|\<dimen> |minus|\<dimen>
\enddisplay
where the `|plus|\<dimen>' and `|minus|^\<dimen>' are optional and assumed
to be zero if not present; `^|plus|' introduces the amount of
stretchability, `^|minus|' introduces the amount of shrinkability.
For example, Appendix~B defines |\medskip| to be an abbreviation for
`|\vskip6pt plus2pt minus2pt|'. The normal-space component of glue must
always be given as an explicit \<dimen>, even when it is zero.
\danger Horizontal glue is created in the same way, but with ^|\hskip|
instead of\/ ^|\vskip|. For example, plain \TeX\ defines ^|\enskip| as an
abbreviation for the command `|\hskip.5em\relax|'; this skips horizontally by
one ``^{en},'' i.e., by exactly half of an em in the current font. There is no
stretching or shrinking in an |\enskip|. The control sequence ^|\relax|
after `|.5em|' prevents \TeX\ from thinking that a ^{keyword}
is present, in case the text following |\enskip| just happens to begin
with `|plus|' or `|minus|'.
One of the interesting things that can happen when glue stretches and shrinks
at different rates is that there might be glue with {\sl ^{infinite}\/}
stretchability. For example, consider again the four boxes we had at the
beginning of this chapter, with the same glue as before except that the
glue in the middle can stretch infinitely far. Now the total
stretchability is infinite; and when the line has to grow, all of the
additional space is put into the middle glue. If, for example, a box of
width 58 is desired, the middle glue expands from 9 to~15 units, and the
other spacing remains unchanged.
If such infinitely stretchable glue is placed at the left of a row of
boxes, the effect is to place them ``flush right,'' i.e., to move them
over to the ^^{right justification} ^^{centering} ^^{flush right}
rightmost boundary of the constructed box. And if you take {\sl two\/}
globs of infinitely stretchable glue, putting one at the left and one
at the right, the effect is to {\sl center\/} the list of boxes within
a larger box. This in fact is how the ^|\centerline| instruction works
in plain \TeX\null: It places infinite glue at both ends, then makes a
box whose width is the current value of\/ |\hsize|.
The short story example of Chapter 6 used infinite glue not only for
centering, but also in the ^|\vfill| instruction at the end; `|\vfill|'
essentially means ``skip vertically by zero, but with infinite stretchability.''
In other words, |\vfill| fills up the rest of the current page with
blank space.
\danger \TeX\ actually recognizes several kinds of infinity, some of which
are ``more infinite'' than others. You can say both ^|\vfil| and |\vfill|;
the second is stronger than the first. In other words, if no other
infinite stretchability is present, |\vfil| will expand to fill the remaining
space; but if both |\vfil| and |\vfill| are present simultaneously,
the |\vfill| effectively prevents |\vfil| from stretching. You can think
of it as if\/ |\vfil| has one mile of stretchability, while |\vfill| has
a trillion miles.
\danger Besides |\vfil| and |\vfill|, \TeX\ has ^|\hfil| and ^|\hfill|,
for stretching indefinitely in the horizontal direction. You can also say
^|\hss| or ^|\vss|, in order to get glue that is infinitely shrinkable as
well as infinitely stretchable. \ (The name `|\hss|' stands for ``horizontal
stretch or shrink''; `|\vss|' is its vertical counterpart.) \ Finally, the
primitives ^|\hfilneg| and ^|\vfilneg| will cancel the stretchability of
|\hfil| and |\vfil|; we shall discuss applications of these curious glues later.
\danger Here are some examples of\/ |\hfil|, using the ^|\line| macro of
plain \TeX, which creates an hbox whose width is the current |\hsize|:
^^{flush left}
\begintt
\line{This text will be flush left.\hfil}
\line{\hfil This text will be flush right.}
\line{\hfil This text will be centered.\hfil}
\line{Some text flush left\hfil and some flush right.}
\line{Alpha\hfil centered between Alpha and Omega\hfil Omega}
\line{Five\hfil words\hfil equally\hfil spaced\hfil out.}
\endtt
\dangerexercise Describe the result of \xdef\linexno{\the\exno}%
\begintt
\line{\hfil\hfil What happens now?\hfil}
\line{\hfill\hfil and now?\hfil}
\endtt
\answer `What happens now?' is placed in a line of width |\hsize|, with
twice as much space at the left as at the right; `and now?' is put flush right
on the following line.
\ddangerexercise How do the following three macros behave differently?
\begintt
\def\centerlinea#1{\line{\hfil#1\hfil}}
\def\centerlineb#1{\line{\hfill#1\hfill}}
\def\centerlinec#1{\line{\hss#1\hss}}
\endtt
\answer The first two give an ``overfull box'' if the argument doesn't fit
on a line; the third allows the argument to stick out into the margins
instead. \ (Plain \TeX's ^|\centerline| is |\centerlinec|; the stickout effect
shows up in the narrow-column experiment of Chapter~6.) \ If the argument
contains no infinite glue, |\centerlinea| and |\centerlineb| produce the same
effect; but |\centerlineb| will center an argument that contains `fil' glue.
\danger In order to specify such infinities, you are allowed to use
the special units `^|fil|', `^|fill|', and `^|filll|' in the \<dimen>
parts of a stretchability or shrinkability component. For example,
|\vfil|, |\vfill|, |\vss|, and |\vfilneg| are essentially equivalent
to the glue specifications
\begintt
\vskip 0pt plus 1fil
\vskip 0pt plus 1fill
\vskip 0pt plus 1fil minus 1fil
\vskip 0pt plus -1fil
\endtt
respectively. It's usually best to stick to the first order infinity
(fil) as much as you can, resorting to second order (fill) only when
you really need something extremely infinite. Then the ultimate order (filll) is
always available as a last resort in emergencies. \ (\TeX\ does not
provide a `^|\vfilll|' primitive, since the use of this highest infinity
is not encouraged.) \ You can use fractional multiples of infinity like
`|3.25fil|', as long as you stick to fewer than 16384 fil units. \TeX\
actually does its calculations with integer multiples of $\rm2^{-16}\,fil$
(or fill or filll); so |0.000007filll| turns out to be indistinguishable
from |0pt|, but |0.00001filll| is infinitely greater than |16383.99999fill|.
Now here's something important for all \TeX nical typists to know:
Plain \TeX\ puts extra space at the end of a ^{sentence}; furthermore,
it automatically increases the stretchability (and decreases
the shrinkability) after ^{punctuation} marks. The reason is that it's
usually better to put more space after punctuation than between two ordinary
words, when spreading a line out to reach the desired margins. Consider, for
example, the following sentences from a classic kindergarten pre-primer:
^^{Dick and Jane}
\begintt
``Oh, oh!'' cried Baby Sally. Dick and Jane laughed.
\endtt
If \TeX\ sets this at its natural width, all the spaces will be the same,
except after the quote and after `Baby Sally.':
\begindisplay
``Oh, oh!'' cried Baby Sally. Dick and Jane laughed.
\enddisplay
But if the line needs to be expanded by 5 points, 10 points, 15 points, or more,
\TeX\ will set it as
\begindisplay
\hbox spread 5pt{``Oh, oh!'' cried Baby Sally. Dick and Jane laughed.}\cr
\hbox spread 10pt{``Oh, oh!'' cried Baby Sally. Dick and Jane laughed.}\cr
\hbox spread 15pt{``Oh, oh!'' cried Baby Sally. Dick and Jane laughed.}\cr
\hbox spread 20pt{``Oh, oh!'' cried Baby Sally. Dick and Jane laughed.}\cr
\enddisplay
The glue after the comma stretches at 1.25 times the rate of the
glue between adjacent words; the glue after the period and after the |!''|
stretches at 3 times the rate. There is no glue between adjacent letters,
so individual words will always look the same. If \TeX\ had to shrink
this line to its minimum width, the result would be
\begindisplay \hfuzz 1000pt
\hbox to 0pt{``Oh, oh!'' cried Baby Sally. Dick and Jane laughed.}
\enddisplay
The glue after a ^{comma} shrinks only 80 percent as much as ordinary
inter-word glue, and after a ^{period} or ^{exclamation point} or
^{question mark} it shrinks by only one third as much.
This all makes for nice-looking output, but it unfortunately adds a bit
of a burden to your job as a typist, because \TeX's rule for determining
the end of a sentence {\sl doesn't always work}. The problem is that
a period sometimes comes in the middle of a sentence $\ldots$ like when it
is used (as here) to make an ``^{ellipsis}'' of three dots.
^^{three dots, see ellipsis} ^^{dot dot dot, see ellipsis}
Moreover, if you try to specify `$\ldots$' by typing three periods in
a row, you get `...'---the dots are too close together. One way to
handle this is to go into {\sl mathematics\/} mode, using the ^|\ldots|
control sequence defined in plain \TeX\ format. For example, if you type
\begintt
Hmmm $\ldots$ I wonder why?
\endtt
the result is `Hmmm $\ldots$ I wonder why?'. This works because
math formulas are exempt from the normal text spacing rules.
Chapter 18 has more to say about |\ldots| and related topics.
^{Abbreviations} present problems too. For example, the short story in
Chapter~6 referred to `Mr.~^{Drofnats}'; \TeX\ must be told somehow that the
period after `Mr.'\ or `Mrs.'\ or `Ms.' or `Prof.'\ or `Dr.' or `Rt.~Hon.',
etc., doesn't count as a sentence-ending ^{full stop}.
We avoided that embarrassment in Chapter~6 by typing `|Mr.~Drofnats|';
the ``^{tie}'' mark |~| ^^{tilde} tells plain \TeX\ to insert a normal
space, and to refrain from breaking between lines at that space. Another way
to get \TeX\ to put out a normal space is to type `|\|\]' (^{control
space}); e.g., `|Mr.\ Drofnats|' would be almost the same as `|Mr.~Drofnats|',
except that a line might end after the `Mr.'.
The tie mark is best for abbreviations within a name, and after several
other common abbreviations like `Fig.'\ and `cf.'\ and `vs.'\ and `resp.';
you will find that it's easy to train yourself to type `|cf.~Fig.~5|'.
In fact, it's usually wise to type |~| (instead of a space) just after a common
abbreviation that occurs in the middle of a sentence.
Manuals of style will tell you that the abbreviations `e.g.'\ and `i.e.'\
should always be followed by commas, never by spaces, so those particular
cases shouldn't need any special treatment.
The only remaining abbreviations that arise with significant frequency
occur in bibliographic references; ^{control spaces} are appropriate here.
^^{interword spacing}
If, for example, you are typing a manuscript that refers to
`Proc.\ Amer.\ Math.\ Soc.', you should say
\begintt
Proc.\ Amer.\ Math.\ Soc.
\endtt
Granted that this input looks a bit ugly, it makes the output look right.
It's one of the things we occasionally must do when dealing with a computer
that tries to be smart.
\exercise Explain how to type the following sentence: ``Mr.~\& Mrs.~^{User}
were married by Rev.~^{Drofnats}, who preached on
Matt.~19\thinspace:\thinspace3--9.''
\answer |Mr.~\& Mrs.~User
were married by Rev.~Drofnats, who preached on
Matt.~19\thinspace:\thinspace3--9.| \ (Such thin spaces are traditional
for ^{Biblical references} to chapter and verse, but you weren't really
expected to know that. Plain \TeX\ defines ^|\thinspace| to be a kern,
not glue; hence no break between lines will occur at a thinspace.)
\exercise Put the following bibliographic reference into plain \TeX\
language: Donald~E. ^{Knuth}, ``Mathematical typography,'' {\sl Bull.\
Amer.\ Math.\ Soc.\ \bf1} (1979), 337--372.
\answer |Donald~E.\ Knuth, ``Mathematical typography,'' {\sl Bull.\
Amer.\ Math.\ Soc.\ \bf1} (1979), 337--372.| \ (But the `|\|' after `|E.|'
isn't necessary, because of a rule you will learn if you venture
around the next dangerous bend.)
On the other hand, if you don't care about such refinements of spacing
you can tell plain \TeX\ to make all spaces the same, regardless of
punctuation marks, by simply typing `^|\frenchspacing|' at the beginning
of your manuscript. French spacing looks like this:
\begindisplay \frenchspacing
``Oh, oh!'' cried Baby Sally. Dick and Jane laughed.
\enddisplay
You can also shift back and forth between the two styles, either by saying
`^|\nonfrenchspacing|' to establish ^{sophisticated spacing}, or by making
your use of\/ |\frenchspacing| local to some group. For example, you might
want to use French spacing only when typing the bibliography of some document.
\danger \TeX\ doesn't consider a period or question mark or exclamation point
to be the end of a sentence if the preceding character is an uppercase
letter, since \TeX\ assumes that such uppercase letters are most likely
somebody's initials. Thus, for example, the `|\|' is unnecessary after
the~`|I.|' in `|Dr.~Livingstone~I.\ Presume|'; that particular period is
not assumed to be a full stop. ^^{Presume}
\dangerexercise What can you do to make \TeX\ recognize the ends of sentences
that do end with uppercase letters (e.g., `$\ldots$ launched by NASA.\null' or
`Did I?' or `$\ldots$ see Appendix~A.')?
\answer There are several ways; perhaps the easiest are to type
`|\hbox{NASA}.|'\ or `|NASA\null.|' \ (The ^|\null| macro is an abbreviation
for `|\hbox{}|'.)
\danger You can see the glue that \TeX\ puts between words by looking at
the contents of hboxes in the internal ^{diagnostic format} that we
discussed ^^{internal box-and-glue representation} briefly in Chapter~11.
For example, Baby Sally's exclamation begins as follows, after \TeX\ has
digested it and put it into a box, assuming |\nonfrenchspacing|:
\begintt
.\tenrm \ (ligature ``)
.\tenrm O
.\tenrm h
.\tenrm ,
.\glue 3.33333 plus 2.08331 minus 0.88889
.\tenrm o
.\tenrm h
.\tenrm !
.\tenrm " (ligature '')
.\glue 4.44444 plus 4.99997 minus 0.37036
.\tenrm c
.\tenrm r
.\tenrm i
.\tenrm e
.\tenrm d
.\glue 3.33333 plus 1.66666 minus 1.11111
.\tenrm B
.\tenrm a
.\tenrm b
.\kern-0.27779
.\tenrm y
.\glue 3.33333 plus 1.66666 minus 1.11111
.\tenrm S
.\tenrm a
.\tenrm l
.\tenrm l
.\tenrm y
.\kern-0.83334
.\tenrm .
.\glue 4.44444 plus 4.99997 minus 0.37036
\endtt
The normal ^{interword glue} in font |\tenrm| is $3.33333\pt$, plus
$1.66666\pt$ of stretchability, minus $1.11111\pt$ of shrinkability.
Notice that the interword |\glue| in this list stretches more, and shrinks
less, after the punctuation marks; and the natural space is in fact larger
at the end of each sentence. This example also shows several other things
that \TeX\ does while it processes the sample line of text: It converts
|``| and |''| into single characters, i.e., ^{ligatures}; and it inserts
small ^{kerns} in two places to improve the spacing. A ^|\kern| is similar
to glue, but it is not the same, because kerns cannot stretch or shrink;
furthermore, \TeX\ will never break a line at a kern, unless that kern is
immediately followed by glue.
\ddanger You may be wondering what \TeX's rules for interword glue really
are, exactly. For example, how did \TeX\ remember the effect of Baby Sally's
exclamation point, when quotation marks intervened before the next space?
The details are slightly tricky, but not incomprehensible. When \TeX\
is processing a horizontal list of boxes and glue, it keeps track of a positive
integer called the current ``^{space factor}.'' The space factor is normally
1000, which means that the interword glue should not be modified. If the
space factor $f$ is different from 1000, the interword glue is computed as
follows: Take the normal space glue for the current font, and add the
extra space if $f\ge2000$. \ (Each font specifies a normal space, normal
stretch, normal shrink, and extra space; for example, these quantities are
$3.33333\pt$, $1.66666\pt$, $1.11111\pt$, and $1.11111\pt$, respectively, in
^|cmr10|. We'll discuss such font parameters in greater detail later.) \
^^|\fontdimen|
Then the stretch component is multiplied by $f/1000$, while the shrink
component is multiplied by $1000/f$.
\ddanger However, \TeX\ has two parameters ^|\spaceskip| and ^|\xspaceskip|
that allow you to override the normal spacing of the current font. If
$f\ge2000$ and if\/ |\xspaceskip| is nonzero, the |\xspaceskip| glue is
used for an ^{interword space}. Otherwise if\/ |\spaceskip| is nonzero,
the |\spaceskip| glue is used, with stretch and shrink components
multiplied by $f/1000$ and $1000/f$. For example, the ^|\raggedright|
macro of plain \TeX\ uses |\spaceskip| and |\xspaceskip| to suppress all
stretching and shrinking of interword spaces.
\ddanger The space factor $f$ is 1000 at the beginning of a horizontal list,
and it is set to 1000 just after a non-character box or a math formula
has been put onto the current horizontal list. You can say
`^|\spacefactor||=|\<number>' to assign any particular value to the space
factor; but ordinarily, $f$ gets set to a number other than 1000 only when
a simple character box goes on the list. Each character has a ^{space factor
code}, and when a character whose space factor code is $g$ enters the current
list the normal procedure is simply to assign $g$ as the new space factor.
However, if $g$ is zero, $f$ is not changed; and if $f<1000<g$, the
space factor is set to~1000. \ (In other words, $f$ doesn't jump from a
value less than~1000 to a value greater than~1000 in a single step.) \
The maximum space factor is 32767 (which is much higher
than anybody would ever want to use).
\ddanger When ^|INITEX| creates a brand new \TeX, all characters have
a space factor code of~1000, except that the uppercase letters `|A|'
through~`|Z|' have code~999. \ (This slight difference is what makes
punctuation act differently after an uppercase letter; do you see why?)
\ Plain \TeX\ redefines a few of these codes using the ^|\sfcode|
primitive, which is similar to |\catcode| (see Appendix~B\null); for
example, the instructions
\begintt
\sfcode`)=0 \sfcode`.=3000
\endtt
make right parentheses ``transparent'' to the space factor, while tripling
the stretchability after periods. The |\frenchspacing| operation resets
|\sfcode`.| to 1000.
\ddanger When ligatures are formed, or when a special character is
specified via ^|\char|, the space factor code is computed from the
individual characters that generated the ligature. For example,
plain \TeX\ sets the space factor code for single-right-quote to zero, so
that the effects of punctuation will be propagated. Two adjacent
characters |''| combine to form a ligature that is in character position
\oct{042}; but the space factor code of this double-right-quote ligature
is never examined by \TeX, so plain \TeX\ does not assign any value to
|\sfcode'042|.
\ddangerexercise What are the space factors after each token
of the Dick-and-Jane example?
\answer 1000, except: 999 after |O|, |B|, |S|, |D|, and |J|; 1250 after the
comma; 3000 after the exclamation point, the right-quote marks, and the
periods. If a period had come just after the |B| (i.e., if the text had
said `|B. Sally|'), the space factor after that period would have
been~1000, not~3000.
\danger Here's the way \TeX\ goes about ^{setting the glue} when an hbox
is being wrapped up: The natural width, $x$, of the box contents is
determined by adding up the widths of the boxes and kerns inside, together
with the natural widths of all the glue inside. Furthermore the total
amount of glue stretchability and shrinkability in the box is computed;
let's say that there's a total of $y_0+y_1\,{\rm fil}+y_2\,{\rm
fill}+y_3\,{\rm filll}$ available for stretching and $z_0+z_1\,{\rm
fil}+z_2\,{\rm fill}+z_3\,{\rm filll}$ available for shrinking. Now the
natural width~$x$ is compared to the desired width~$w$. If $x=w$,
all glue gets its natural width.
Otherwise the glue will be modified, by computing a ``^{glue set
ratio}''~$r$ and a ``^{glue set order}''~$i$ in the following way: \
(a)~If $x<w$, \TeX\ attempts to stretch the contents of the box; the glue
order is the highest subscript~$i$ such that $y_i$ is nonzero, and the
glue ratio is $r=(w-x)/y_i$. (If $y_0=y_1=y_2=y_3=0$, there's no
stretchability; both $i$ and $r$ are set to zero.) \ (b)~If $x>w$, \TeX\
attempts to shrink the contents of the box in a similar way; the glue
order is the highest subscript~$i$ such that $z_i\ne0$, and the glue ratio
is normally $r=(x-w)/z_i$. However, $r$ is set to 1.0 in the case $i=0$
and $x-w>z_0$, because the maximum shrinkability must not be exceeded. \
(c)~Finally, every glob of glue in the horizontal list being boxed is
modified. Suppose the glue has natural width~$u$, stretchability~$y$, and
shrinkability~$z$, where $y$~is a $j$th order infinity and $z$~is a $k$th
order infinity. Then if $x<w$ (stretching), this glue takes the new width
$u+ry$ if $j=i$; it keeps its natural width~$u$ if $j\ne i$. If $x>w$
(shrinking), this glue takes the new width $u-rz$ if $k=i$; it keeps
its natural width~$u$ if $k\ne i$. Notice that stretching or shrinking
occurs only when the glue has the highest order of infinity that doesn't
cancel out.
\danger \TeX\ will construct an hbox that has a given width $w$ if you issue
the command `\hbox{|\hbox to |\<dimen>|{|\<contents of box>|}|}', where
$w$ is the value of the \<dimen>. For example, the ^|\line| macro discussed
earlier in this chapter is simply an abbreviation for `|\hbox to\hsize|'.
^^|to|^^|\hbox|
\TeX\ also allows you to specify the exact amount of stretching or shrinking;
the command `\hbox{|\hbox spread|\<dimen>|{|\<contents of box>|}|}'
creates a box whose width~$w$ is a given amount more ^^|spread|
than the natural width of the contents. For example, one of the boxes
displayed earlier in this chapter was generated by
\begintt
\hbox spread 5pt{``Oh, oh!'' ... laughed.}
\endtt
In the simplest case, when you just want a box to have its natural width,
you don't have to write `|\hbox spread 0pt|'; you can simply say
`|\hbox{|\<contents of box>|}|'.
\danger The ^{baseline} of a constructed hbox is the common baseline of the
boxes inside. \ (More precisely, it's the common baseline that they would
share if they weren't raised or lowered.) \ The height and depth of a
constructed hbox are determined by the maximum distances by which the
interior boxes reach above and below the baseline, respectively. The
result of\/ |\hbox| never has negative height or negative depth, but the
width can be negative.
\dangerexercise Assume that
|\box1| is $1\pt$~high, $1\pt$~deep, and $1\pt$~wide;
|\box2| is $2\pt$~high, $2\pt$~deep, and $2\pt$~wide.
A third box is formed by saying ^^|\setbox|
\begintt
\setbox3=\hbox to3pt{\hfil\lower3pt\box1\hskip-3pt plus3fil\box2}
\endtt
What are the height, depth, and width of\/ |\box3|? Describe the position
of the reference points of boxes 1 and~2 with respect to the reference
point of box~3.
\answer |\box3| is $2\pt$ high, $4\pt$ deep, $3\pt$ wide.
Starting at the reference point of\/ |\box3|, go right $.75\pt$ and down
$3\pt$ to reach the reference point of\/ |\box1|; or go right $1\pt$
to reach the reference point of\/ |\box2|.
\danger The process of setting glue for vboxes is similar to that for
hboxes; but before we study the |\vbox| operation, we need to discuss how
\TeX\ stacks boxes up vertically so that their baselines tend to be
a fixed distance apart. The boxes in a horizontal list often touch each
other, but it's usually wrong to do this in a vertical list; imagine
how awful a page would look if its lines of type were brought closer
together whenever they didn't contain tall letters, or whenever they
didn't contain any letters that descended below the baseline.
\danger \TeX's solution to this problem involves three primitives called
^|\baselineskip|, ^|\lineskip|, and ^|\lineskiplimit|. A format designer
chooses values of these three quantities by writing
\begindisplay
|\baselineskip=|\<glue>\cr
|\lineskip=|\<glue>\cr
|\lineskiplimit=|\<dimen>\cr
\enddisplay
and the interpretation is essentially this: Whenever a box is added to a
vertical list, \TeX\ inserts ``^{interline glue}'' intended to make
the distance between the baseline of the new box and the baseline of
the previous box exactly equal to the value of\/ |\baselineskip|.
But if the interline glue calculated by this rule would cause the
top edge of the new box to be closer than |\lineskiplimit| to the
bottom edge of the previous box, then |\lineskip| is used as the
interline glue. In other words, the distance between adjacent baselines
will be the |\baselineskip| setting, unless that would bring the boxes
too close together; the |\lineskip| glue will separate adjacent boxes
in the latter case.
\danger The rules for interline glue in the previous paragraph are
carried out without regard to other kinds of glue that might be present;
all vertical spacing due to explicit appearances of\/ |\vskip| and |\kern|
acts independently of the interline glue. Thus, for example, a
^|\smallskip| between two lines always makes their baselines further apart
than usual, by the amount of a |\smallskip|; it does not
affect the decision about whether |\lineskip| glue is used between
those lines.
\danger For example, let's suppose that |\baselineskip=12pt plus 2pt|,
|\lineskip=|\penalty0|3pt minus 1pt|, and |\lineskiplimit=2pt|. \ (These values
aren't particularly useful; they have simply been chosen to illustrate
the rules.) \ Suppose further that a box whose depth is $3\pt$
was most recently added to the current vertical list; we are about to
add a new box whose height is~$h$. If $h=5\pt$, the interline glue
will be $4\pt$~plus~$2\pt$, since this will make the baselines
$12\pt$~plus~$2\pt$ apart when we add $h$ and the previous depth to
the interline glue. But if $h=8\pt$, the interline
glue will be $3\pt$~minus~$1\pt$, since |\lineskip| will be
chosen in order to keep from violating the given |\lineskiplimit|
when stretching and shrinking are ignored.
\danger When you are typesetting a document that spans several pages,
it's generally best to define the |\baselineskip| so that it cannot stretch
or shrink, because this will give more uniformity to the pages. A small
variation in the distance between baselines---say only half a point---can
make a substantial difference in the appearance of the type, since it
significantly affects the proportion of white to black. On the other hand,
if you are preparing a one-page document, you might want to give the
baselineskip some stretchability, so that \TeX\ will help you fit the copy
on the page.
\dangerexercise What settings of\/ |\baselineskip|, |\lineskip|, and
|\lineskiplimit| will cause the interline glue to be a ``continuous''
function of the next box height (i.e., the interline glue will never
change a lot when the box height changes only a little)?
\answer The stretch and shrink components of\/ |\baselineskip| and
|\lineskip| should be equal, and the |\lineskiplimit| should
equal the normal |\lineskip| spacing, to guarantee continuity.
\danger A study of \TeX's ^{internal box-and-glue representation} should
help to firm up some of these ideas. Here is an excerpt from the vertical
list that \TeX\ constructed when it was typesetting this very paragraph:
\begintt
\glue 6.0 plus 2.0 minus 2.0
\glue(\parskip) 0.0 plus 1.0
\glue(\baselineskip) 1.25
\hbox(7.5+1.93748)x312.0, glue set 0.80154, shifted 36.0 []
\penalty 10000
\glue(\baselineskip) 2.81252
\hbox(6.25+1.93748)x312.0, glue set 0.5816, shifted 36.0 []
\penalty 50
\glue(\baselineskip) 2.81252
\hbox(6.25+1.75)x348.0, glue set 116.70227fil []
\penalty 10000
\glue(\abovedisplayskip) 6.0 plus 3.0 minus 1.0
\glue(\lineskip) 1.0
\hbox(149.25+0.74998)x348.0 []
\endtt
{\showboxdepth0\showboxbreadth9999\batchmode\showlists\errorstopmode}%
% The log file now contains lines like these; I copied them
% into this MS! But I deleted an `insert' for the index...
The first |\glue| in this example is the ^|\medskip| that precedes each
dangerous-bend paragraph. Then comes the ^|\parskip| glue, which is
automatically supplied before the first line of a new paragraph. Then
comes some interline glue of $1.25\pt$; it was calculated to make
a total of $11\pt$ when the height of the next box ($7.5\pt$)
and the depth of the previous box were added. \ (The previous box is not
shown---it's the bottom line of exercise~\chapno.\the\exno---but
we can deduce that its depth was $2.25\pt$.) \ The |\hbox| that
follows is the first line of this paragraph; it has been shifted right
$36\pt$ because of ^{hanging indentation}. The glue set ratio
for this hbox is 0.80154; i.e., the glue inside is stretched by 80.154\%\
of its stretchability. \ (In the case of shrinking, the ratio following
`^|glue set|' would have been preceded by `|- |'; hence we know that
stretching is involved here.) \ \TeX\ has put `|[]|' at the end of each
hbox line to indicate that there's something in the box that isn't shown.
\ (The box contents would have been displayed completely, if
^|\showboxdepth| had been set higher.) \ The ^|\penalty| indications are
used to discourage bad breaks between pages, as we will see later. The
third hbox has a glue ratio of 116.70227, which applies to
first-order-infinite stretching (i.e., fil); this results from an |\hfil|
that was implicitly inserted just before the displayed material, to fill
up the third line of the paragraph. Finally the big hbox whose height is
$149.25\pt$ causes |\lineskip| to be the interline glue. This large box
contains the individual lines of typewriter type that are displayed; they
have been packaged into a single box so that they cannot be split between
pages. Careful study of this example will teach you a lot about \TeX's
inner workings.
\danger Exception: No interline glue is inserted before or after a rule
box. You can also inhibit interline glue by saying ^|\nointerlineskip|
between boxes.
\ddanger \TeX's implementation of interline glue involves another primitive
quantity called ^|\prevdepth|, which usually contains the depth of the
most recent box on the current vertical list. However, |\prevdepth| is set
to the sentinel value $-1000\pt$ at the beginning of a vertical list, or
just after a rule box; this serves to suppress the next interline glue.
The user can change the value of\/ |\prevdepth| at any time when building
a vertical list; thus, for example, the |\nointerlineskip| macro of
Appendix~B simply expands to `|\prevdepth=-1000pt|'.
\ddanger Here are the exact rules by which \TeX\ calculates the interline
glue between boxes: Assume that a new box of height~$h$ (not a rule box)
is about to be appended to the bottom of the current vertical list, and
let $\hbox{|\prevdepth|}=p$, $\hbox{|\lineskiplimit|}=l$,
\hbox{$\hbox{|\baselineskip|}=(b$ plus~$y$ minus~$z)$}. If $p\le-1000\pt$,
no interline glue is added. Otherwise if $b-p-h\ge l$, the interline glue
`$(b-p-h)$ plus~$y$ minus~$z$' will be appended just above the new box.
Otherwise the |\lineskip| glue will be appended. Finally, |\prevdepth| is
set to the depth of the new box.
\ddangerexercise Mr.~B. L. ^{User} had an application in which he wanted
to put a number of boxes together in a vertical list, with no space
between them. He didn't want to say |\nointerlineskip| after each box;
so he decided to set |\baselineskip|, |\lineskip|, and |\lineskiplimit|
all equal to |0pt|. Did this work?
\answer Yes it did, but only because none of his boxes had a negative
height or depth. He would have been safer if he had set
|\baselineskip=-1000pt|, |\lineskip=0pt|, and
|\lineskiplimit=16383pt|. \ (Plain \TeX's ^|\offinterlineskip| macro does this.)
\danger The vertical analog of\/ |\hbox| is ^|\vbox|, and \TeX\ will obey
the commands `|\vbox to|\<dimen>' and `|\vbox spread|\<dimen>' in about
the way you would expect, by analogy with the horizontal case. However,
there's a slight complication because boxes have both height and depth
in the vertical direction, while they have only width in the
horizontal direction. The dimension in a |\vbox| command refers to
the final height of the vbox, so that, for example, `|\vbox to 50pt{...}|'
produces a box that is $50\pt$ high; this is appropriate because everything
that can stretch or shrink inside a vbox appears in the part that
contributes to the height, while the depth is unaffected by glue setting.
\danger The depth of a constructed |\vbox| is best thought of as the depth
^^{depth of box} ^^{height of box}
of the bottom box inside. Thus, a vbox is conceptually built by taking
a bunch of boxes and arranging them so that their reference
points are lined up vertically; then the reference point of the lowest
box is taken as the reference point of the whole, and the glue is set
so that the final height has some desired value.
\danger However, this description of vboxes glosses over some
technicalities that come up when you consider unusual cases. For example,
\TeX\ allows you to shift boxes in a vertical list to the right or to
the left by saying
`^|\moveright|\<dimen>\<box>' or `^|\moveleft|\<dimen>\<box>'; this is
like the ability to ^|\raise| or ^|\lower| boxes in a horizontal list, and
it implies that the reference points inside a vbox need not always lie in
a vertical line. Furthermore, it is necessary to guard against boxes that
have too much depth, lest they extend too far into the bottom margin of a
page; and later chapters will point out that vertical lists can contain
other things like penalties and marks, in addition to boxes and glue.
\ddanger Therefore, the actual rules for the depth of a constructed vbox
are somewhat \TeX nical. Here they are: Given a vertical list that
is being wrapped up via |\vbox|, the problem is to determine its
natural depth. \ (1)~If the vertical list contains no boxes, the depth
is zero. \ (2)~If there's at least one box, but if the final box is
followed by kerning or glue, possibly with intervening penalties or
other things, the depth is zero. \ (3)~If there's at least
one box, and if the final box is not followed by kerning or glue, the
depth is the depth of that box. \ (4)~However, if the depth computed
by rules (1), (2), or~(3) exceeds ^|\boxmaxdepth|, the depth will be
the current value of\/ |\boxmaxdepth|. \ (Plain \TeX\ sets |\boxmaxdepth|
to the largest possible dimension; therefore rule~(4) won't apply unless
you specify a smaller value. When rule~(4) does decrease the depth,
\TeX\ adds the excess depth to the box's natural height, essentially
moving the reference point down until the depth has been reduced to the
stated maximum.)
\danger The glue is set in a vbox just as in an hbox, by determining a
^{glue set ratio} and a ^{glue set order}, based on the difference between
the natural height~$x$ and the desired height~$w$, and based on the amounts
of stretchability and shrinkability that happen to be present.
\danger The width of a computed |\vbox| is the maximum distance by which
an enclosed box extends to the right of the reference point, taking
possible shifting into account. This width is always nonnegative.
\dangerexercise Assume that |\box1| is $1\pt$ high, $1\pt$ deep, and
$1\pt$ wide; |\box2| is $2\pt$ high, $2\pt$ deep, and $2\pt$ wide; the
baselineskip, lineskip, and lineskiplimit are all zero; and the |\boxmaxdepth|
is very large. A third box is formed by saying
\begintt
\setbox3=\vbox to3pt{\moveright3pt\box1\vskip-3pt plus3fil\box2}
\endtt
What are the height, depth, and width of\/ |\box3|? Describe the position
of the reference points of boxes 1 and~2 with respect to the reference point
of box~3.
\answer The interline glue will be zero, and the natural height is
$1+1-3+2=1\pt$ (because the depth of\/ |\box2| isn't included in the natural
height); so the glue will ultimately become |\vskip-1pt| when it's set.
Thus, |\box3| is $3\pt$ high, $2\pt$ deep, $4\pt$ wide. Its reference
point coincides with that of\/ |\box2|; to get to the reference point
of\/ |\box1| you go up $2\pt$ and right $3\pt$.
\ddangerexercise Under the assumptions of the previous exercise, but
with |\baselineskip=9pt minus3fil|, describe |\box4| after
\begintt
\setbox4=\vbox to4pt{\vss\box1\moveleft4pt\box2\vss}
\endtt
\answer The interline glue will be $6\pt$ minus $3\,{\rm fil}$; the final
depth will be zero, since |\box2| is followed by glue; the natural
height is $12\pt$; and the shrinkability is $5\,{\rm fil}$. So |\box4|
will be $4\pt$ high, $0\pt$ deep, $1\pt$ wide, and it will contain
five items: |\vskip|\penalty0\hbox{|-1.6pt|}, |\box1|, |\vskip1.2pt|,
|\moveleft4pt\box2|, |\vskip-1.6pt|. Starting at the reference point of
|\box4|, you get to the reference point of\/ |\box1| by going up $4.6\pt$,
or to the reference point of\/ |\box2| by going up $.4\pt$ and left $4\pt$.
\ (For example, you go up $4\pt$ to get to the upper left corner of
|\box4|; then down $-1.6\pt$, i.e., up $1.6\pt$, to get to the upper left
corner of\/ |\box1|; then down $1\pt$ to reach its reference point. This
problem is clearly academic, since it's rather ridiculous to include
infinite shrinkability in the baselineskip.)
\ddangerexercise Solve the previous problem but with |\boxmaxdepth=-4pt|.
\answer Now |\box4| will be $4\pt$ high, $-4\pt$ deep, $1\pt$ wide, and it
will contain |\vskip|\penalty0\hbox{|-2.4pt|}, |\box1|, |\vskip-1.2pt|,
|\moveleft4pt\box2|, |\vskip-2.4pt|. From the baseline of\/ |\box4|, go up
exactly $5.4\pt$ to reach the baseline of\/ |\box1|, or exactly $3.6\pt$
to reach the baseline of\/ |\box2|.
\danger We have observed that |\vbox| combines a bunch of boxes into
a larger box that has the same baseline as the bottom box inside.
\TeX\ has another operation called ^|\vtop|, which gives you a box like |\vbox|
but with the same baseline as the top box inside. For example,
\begintt
\hbox{Here are \vtop{\hbox{two lines}\hbox{of text.}}}
\endtt
produces
\begindisplay
Here are \vtop{\hbox{two lines}\hbox{of text.}}
\enddisplay
\ddanger You can say `|\vtop to|\<dimen>' and `|\vtop spread|\<dimen>'
just as with |\vbox|, but you should realize what such a construction
means. \TeX\ implements |\vtop| as follows: \ (1)~First a~vertical box
is formed as if\/ |\vtop| had been |\vbox|, using all of the rules for
|\vbox| as given above. \ (2)~The final height~$x$ is defined to
be zero unless the very first item inside the new vbox is a box;
in the latter case, $x$ is the height of that box. \ (3)~Let $h$
and~$d$ be the height and depth of the vbox in step~(1). \TeX\ completes
the |\vtop| by moving the reference point up or down, if necessary, so
that the box has height~$x$ and depth $h+d-x$.
\ddangerexercise Describe the empty boxes that you get from
`|\vbox to|\<dimen>|{}|' and
`|\vtop to|\<dimen>|{}|'. What are their heights, depths, and widths?
\answer |\vbox to| $x$|{}| produces height $x$;
|\vtop to| $x$|{}| produces depth $x$; the other dimensions are zero.
\ (This holds even when $x$ is negative.)
\ddangerexercise Define a macro |\nullbox#1#2#3| that produces a box
whose height, depth, and width are given by the three parameters.
The box should contain nothing that will show up in print.
\answer There are several possibilities:
\begintt
\def\nullbox#1#2#3{\vbox to#1{\vss\hrule height-#2depth#2width#3}}
\endtt
works because the rule will be of zero thickness. Less tricky is
\begintt
\def\nullbox#1#2#3{\vbox to#1{\vss\vtop to#2{\vss\hbox to#3{}}}}
\endtt
Both of these are valid with negative height and/or depth, but they do
not produce negative width. If the width might be negative, but not the
height or depth, you can use, e.g.,
|\def\nullbox#1#2#3{\hbox to#3{\hss\raise#1\null\lower#2\null}}|.
It's impossible for |\hbox| to construct a box
whose height or depth is negative; it's impossible for |\vbox| or
|\vtop| to construct a box whose width is negative.\par
However, there's actually a trivial solution to the general problem, based on
features that will be discussed later:
\begintt
\def\nullbox#1#2#3{\setbox0=\null
\ht0=#1 \dp0=#2 \wd0=#3 \box0 }
\endtt
\danger The |\vbox| operation tends to produce boxes with large height and
small depth, while |\vtop| tends to produce small height and large depth.
If you're trying to make a vertical list out of big vboxes, however, you
may not be satisfied with either |\vbox| or |\vtop|; you might well wish
that a box had two reference points simultaneously, one for the top and
one for the bottom. If such a dual-reference-point scheme were in use, one
could define interline glue based on the distance between the lower
reference point of one box and the upper reference point of its successor
in a vertical list. But alas, \TeX\ gives you only one reference point per box.
\danger There's a way out of this dilemma, using an important idea
called a ``^{strut}.'' Plain \TeX\ defines ^|\strut| to be an invisible
box of width zero that extends just enough above and below the baseline
so that you would need no interline glue at all if every line contained
a strut. \ (Baselines are $12\pt$ apart in plain \TeX; it turns out that
|\strut| is a vertical rule, $8.5\pt$ high and $3.5\pt$ deep and
$0\pt$ wide.) \ If you contrive to put a strut on the top line and
another on the bottom line, inside your large vboxes, then it's possible to
obtain the correct spacing in a larger assembly by simply letting the
boxes butt together. For example, the ^|\footnote| macro in Appendix~B
puts struts at the beginning and end of every footnote, so that the spacing
will be right when several footnotes occur together at the bottom
of some page. ^^{fitting boxes together}
\danger If you understand boxes and glue, you're ready to learn
the ^|\rlap| and ^|\llap| macros of plain \TeX; these names are abbreviations
for ``right ^{overlap}'' and ``left overlap.'' Saying `|\rlap{|\<something>|}|'
is like typesetting \<something> and then backing up as if you hadn't
typeset anything. More precisely, `|\rlap{|\<something>|}|' creates a box
of width zero, with `\<something>' appearing just at the right of that box
(but not taking up any space). The |\llap| macro is similar, but it does
the ^{backspacing} first; in other words, `|\llap{|\<something>|}|'
creates a box of width zero, with `\<something>' extending just to the
left of that box. Using typewriter type, for example, you can typeset
`{\tt\rlap/=}' by saying either `|\rlap/=|' or `|/\llap=|'. It's possible
to put text into the left margin using |\llap|, or into the right margin
using |\rlap|, because \TeX\ does not insist that the contents of a box
must be strictly confined within that box's boundaries. ^^{marginal notes}
\danger The interesting thing about |\rlap| and |\llap| is that they can be
done so simply with infinite glue. One way to define |\rlap| would be
\begintt
\def\rlap#1{{\setbox0=\hbox{#1}\copy0\kern-\wd0}}
\endtt
but there's no need to do such a lengthy computation. The actual definition
in Appendix~B is much more elegant, namely,
\begintt
\def\rlap#1{\hbox to 0pt{#1\hss}}
\endtt
and it's worth pondering why this works. Suppose, for example, that you're
doing |\rlap{g}| where the letter `g' is $5\pt$ wide. Since |\rlap| makes
an hbox of width $0\pt$, the glue represented by ^|\hss| must shrink by
$5\pt$. Well, that glue has $0\pt$ as its natural width, but it has
infinite shrinkability, so it can easily shrink to $-5\pt$; and
`|\hskip-5pt|' is exactly what |\rlap| wants in this case.
\dangerexercise Guess the definition of\/ |\llap|, without peeking at
Appendices A or~B.
\answer |\def\llap#1{\hbox to 0pt{\hss#1}}|
\dangerexercise (This is a sequel to exercise 12.\linexno,
but it's trickier.) \ Describe the result of
\begintt
\line{\hfil A puzzle.\hfilneg}
\endtt
\answer You get `A' at the extreme left and `puzzle.\null' at the extreme right,
because the space between words has the only stretchability that is finite;
the infinite stretchability cancels out. \ (In this case, \TeX's rule
about ^{infinite glue} differs from what you would get in the limit if the
value of $1\,{\rm fil}$ were finite but getting larger and larger.
The true limiting behavior would stretch the text `A~puzzle.\null' in the
same way, but it would also move that text infinitely far away beyond the right
edge of the page.)
\endchapter
There was things which he stretched,
but mainly he told the truth.
\author MARK ^{TWAIN}{^^{Clemens}}, {\sl Huckleberry Finn\/} (1884) % chap1
\bigskip
Every shape exists only because of the space around it.
% Jeder Formwert lebt nur dank seiner Umgebung.
$\ldots$ Hence there is a `right' position for every shape in every situation.
% ... Daraus geht hervor, ein bestimmter Formvert in einer bestimmten Situation
% einen bestimmten Platz verlangt, der ihm gem\"a\ss\ ist.
If we succeed in finding that position, we have done our job.
% Gelingt es uns, diese Pl\"atze zu finden,
% so nennen wir die Arbeit vollendet.
\author JAN ^{TSCHICHOLD}, {\sl Typographische Gestaltung\/} (1935) % p64
% I've borrowed from the free translation in {\sl Asymmetric Typography}
\eject
\beginchapter Chapter 13. Modes
Just as people get into different moods, \TeX\ gets into different ``modes.'' \
(Except that \TeX\ is more predictable than people.) \ There are six ^{modes}:
\medskip
\item\bull^{Vertical mode}. [Building the main vertical list, from which
the pages of output are derived.]
\smallskip\item\bull
^{Internal vertical mode}. [Building a vertical list for a vbox.]
\smallskip\item\bull
^{Horizontal mode}. [Building a horizontal list for a paragraph.]
\smallskip\item\bull
^{Restricted horizontal mode}. [Building a horizontal list for an hbox.]
\smallskip\item\bull
^{Math mode}. [Building a mathematical formula to be placed in
a horizontal list.]
\smallskip\item\bull
^{Display math mode}. [Building a mathematical formula to be placed
on a line by itself, temporarily interrupting the current paragraph.]
\medskip\noindent In simple situations, you don't need to be aware of what
mode \TeX\ is in, because the computer just does the right thing. But
when you get an error message that says `\thinspace|!|~|You| |can't| |do|
|such-and-such| |in| |restricted| |horizontal| |mode|\thinspace', a
knowledge of modes helps to explain why \TeX\ thinks you goofed.
Basically \TeX\ is in one of the vertical modes when it is preparing a list of
boxes and glue that will be placed vertically above and below one another on
the page; it's in one of the horizontal modes when it is preparing a list
of boxes and glue that will be strung out horizontally next to each other
with baselines aligned; and it's in one of the math modes when it is
reading a formula.
A play-by-play account of a typical \TeX\ job should make the mode idea clear:
At the beginning, \TeX\ is in vertical mode, ready to construct pages. If you
specify glue or a box when \TeX\ is in vertical mode, the glue or the box gets
placed on the current page below what has already been specified. For example,
the ^|\vskip| instructions in the sample run we discussed in Chapter~6
contributed vertical glue to the page; and the ^|\hrule| instructions
contributed horizontal rules at the top and bottom of the story. The
^|\centerline| commands also produced boxes that were included in the main
vertical list; but those boxes required a bit more work than the rule boxes:
\TeX\ was in vertical mode when it encountered
`|\centerline{\bf A SHORT STORY}|', and it went temporarily into restricted
horizontal mode while processing the words `|A SHORT STORY|'; then
the digestive process returned to vertical mode, after setting the
glue in the~|\centerline|~box.
Continuing with the example of Chapter 6, \TeX\ switched into horizontal
mode as soon as it read the `|O|' of `|Once upon a time|'. Horizontal mode
is the mode for making ^{paragraphs}. The entire paragraph (lines 7 to~11
of the |story| file) was input in horizontal mode; then the text was
divided into output lines of the appropriate width, those lines were
put in boxes and appended to the page (with appropriate interline glue between
them), and \TeX\ was back in vertical mode. The `|M|' on line~12 started up
horizontal mode again.
When \TeX\ is in vertical mode or internal vertical mode,
the first token of a new paragraph changes the mode to horizontal for the
duration of a paragraph. In other words, things that do not have a vertical
orientation cause the mode to switch automatically from vertical to
horizontal. This occurs when you type any character, or ^|\char| or ^|\accent|
or ^|\hskip| or |\|\] ^^{control space} or ^|\vrule| or math shift (|$|);
\TeX\ inserts the current paragraph ^{indentation} and rereads the
horizontal token as if it had occurred in horizontal mode.
\danger You can also tell \TeX\ explicitly to go into horizontal mode,
instead of relying on such implicit mode-switching, by saying `^|\indent|'
or `^|\noindent|'. For example, if line~7 of the |story| file in Chapter~6
had begun
\begintt
\indent Once upon a time, ...
\endtt
the same output would have been obtained, because `|\indent|' would have
instructed \TeX\ to begin the paragraph. And if that line had begun with
\begintt
\noindent Once upon a time, ...
\endtt
the first paragraph of the story would not have been indented. The
|\noindent| command simply tells \TeX\ to enter horizontal mode if the
current mode is vertical or internal vertical; |\indent| is similar,
but it also creates an empty box whose width is the current value of
^|\parindent|, and it puts this empty box into the current horizontal list.
Plain \TeX\ sets |\parindent=20pt|.
If you say |\indent\indent|, you get double indentation; if you say
|\noindent\noindent|, the second |\noindent| does nothing.
\dangerexercise If you say `^|\hbox||{...}|' in horizontal mode, \TeX\ will
construct the specified box and it will contribute the result to the
current paragraph. Similarly, if you say `|\hbox{...}|' in vertical mode,
\TeX\ will construct a box and contribute it to the current page.
What can you do if you want to begin a paragraph with an |\hbox|?
\answer Simply saying |\hbox{...}| won't work, since that box will just
continue the previous vertical list without switching modes. You need
to start the paragraph explicitly, and the straightforward way to
do that is to say |\indent\hbox{...}|.
But suppose you want to define
a macro that expands to an hbox, where this macro is to be used in the
midst of a paragraph as well as at the beginning; then you don't want
to force users to type |\indent| before calling your macro at the
beginning of a paragraph, nor do you want to say |\indent| in the
macro itself (since that might insert unwanted indentations). One
solution to this more general problem is to say
`|\|\]^|\unskip||\hbox{...}|', since |\|\] makes the mode
horizontal while |\unskip| removes the unwanted space. Plain \TeX\
provides a ^|\leavevmode| macro, which solves this problem in what is
probably the most efficient way: |\leavevmode| is an abbreviation for
`|\unhbox\voidbox|', where |\voidbox| is a permanently empty box register.
When handling simple manuscripts, \TeX\ spends almost all of its time in
horizontal mode (making paragraphs), with brief excursions into vertical
mode (between paragraphs). A paragraph is completed when you type ^|\par|
or when your manuscript has a blank line, since a blank line is converted
to |\par| by the reading rules of Chapter~8. A paragraph also ends when
you type certain things that are incompatible with horizontal mode.
For example, the command `|\vskip 1in|' on line~16 of Chapter~6's |story|
file was enough to terminate the paragraph about `|...beautiful
documents.|'; no |\par| was necessary, since |\vskip| introduced
vertical glue that couldn't belong to the~paragraph.
If a begin-math token (|$|) appears in
horizontal mode, \TeX\ plunges into math mode and processes the formula up until
the closing `|$|', then appends the text of this formula to the current
paragraph and returns to horizontal mode. Thus, in the ``I wonder why?''\
example of Chapter~12, \TeX\ went into math mode temporarily
while processing |\ldots|, treating the dots as a formula.
However, if two consecutive begin-math tokens appear in a paragraph (|$$|),
\TeX\ interrupts the paragraph where it is, contributes the paragraph-so-far
to the enclosing vertical list, then processes a math formula in display
math mode, then contributes this formula to the enclosing list, then
returns to horizontal mode for more of the paragraph. \ (The formula to be
displayed should end with `|$$|'.) \ For example, suppose you type
\begintt
the number $$\pi \approx 3.1415926536$$ is important.
\endtt
\TeX\ goes into display math mode between the |$$|'s, and the output you
get states that the number $$\pi \approx 3.1415926536$$ is important.
^^|\pi|
\smallskip
\TeX\ ignores blank spaces and blank lines (or ^|\par| commands) when it's
in vertical or internal vertical mode, so you need not worry that
such things might change the mode or affect a printed document.
A ^{control space} (|\|\]) will, however, be regarded as the
beginning of a paragraph; the paragraph will start with a blank space
after the indentation.
\smallskip
At the end of a \TeX\ manuscript it's usually best to finish everything off
by typing `^|\bye|', which is plain \TeX's abbreviation for
`|\vfill\eject\end|'. The `|\vfill|' gets \TeX\ into vertical
mode and inserts enough space to fill up the last page; `|\eject|' outputs
that last page; and `|\end|' sends the computer into its ^{endgame} routine.
\danger \TeX\ gets into internal vertical mode when you ask it to construct
something from a vertical list of boxes (using |\vbox| or |\vtop| or |\vcenter|
or |\valign| or |\vadjust| or |\insert|). It gets into restricted
horizontal mode when you ask it to construct something from a horizontal list
of boxes (using |\hbox| or |\halign|). Box construction is discussed
in Chapters 12 and~21. We will see later that there is very little difference
between internal vertical mode and ordinary vertical mode, and very little
difference between restricted horizontal mode and ordinary horizontal mode;
but they aren't quite identical, because they have different goals.
\danger Whenever \TeX\ looks at a token of input to decide what should be
done next, the current mode has a potential influence on what that token
means. For example, ^|\kern| specifies vertical spacing in vertical mode,
but it specifies horizontal spacing in horizontal mode; a math shift
character like `|$|' causes entry to math mode from horizontal mode, but
it causes exit from math mode when it occurs in math mode; two consecutive
math shifts (|$$|) appearing in horizontal mode will initiate display
math mode, but in restricted horizontal mode they simply denote an empty
math formula. \TeX\ uses the fact that some operations are inappropriate
in certain modes to help you recover from errors that might have crept
into your manuscript. Chapters 24 to~26 explain exactly what
happens to every possible token in every possible mode.
\danger \TeX\ often interrupts its work in one mode to do some task in
another mode, after which the original mode is resumed again.
For example, you can say `|\hbox{|' in any mode; when \TeX\ digests this,
it suspends whatever else it was doing and enters restricted horizontal
mode. The matching `|}|' will eventually cause the hbox to be completed,
whereupon the postponed task will be taken up anew. In this sense \TeX\ can
be in many modes simultaneously, but only the innermost
mode influences the calculations at any time; the other modes have
been pushed out of \TeX's consciousness.
\goodbreak
\danger One way to become familiar with \TeX's modes is to consider the
following curious test file called |modes.tex|, which exercises all the
modes at once:
$$\halign{\hbox to\parindent{\hfil\sevenrm#\ \ }&#\hfil\cr
1&|\tracingcommands=1|\cr\noalign{^^|\tracingcommands|}
2&|\hbox{|\cr
3&|$|\cr
4&|\vbox{|\cr
5&|\noindent$$|\cr
6&|x\showlists|\cr
7&|$$}$}\bye|\cr}$$
The first line of ^|modes.tex| tells \TeX\ to log every command it
receives; \TeX\ will produce diagnostic data whenever
|\tracingcommands| is positive. Indeed, if you run \TeX\ on |modes.tex|
you will get a |modes.log| file that includes the following information:
\begintt
{vertical mode: \hbox}
{restricted horizontal mode: blank space }
{math shift character $}
{math mode: blank space }
{\vbox}
{internal vertical mode: blank space }
{\noindent}
{horizontal mode: math shift character $}
{display math mode: blank space }
{the letter x}
\endtt
The meaning is that \TeX\ first saw an |\hbox| token in vertical mode;
this caused it to go ahead and read the `|{|' behind the scenes.
Then \TeX\ entered restricted horizontal mode, and
saw the blank space token that resulted from the end of line~2 in the
file. Then it saw a math shift character token (still in restricted
horizontal mode), which caused a shift to math mode; another
blank space came through. Then |\vbox| inaugurated internal
vertical mode, and |\noindent| instituted horizontal mode within that; two
subsequent |$| signs led to display math mode. \ (Only the first |$| was
shown by |\tracingcommands|, because that one caused \TeX\ to look ahead
for another.)
\danger The next thing in |modes.log| after the output above is
`|{|^|\showlists||}|'. This is another handy diagnostic command that
you can use to find out things that \TeX\ ordinarily keeps to itself; it
causes \TeX\ to display the lists that are being worked on, in the current
mode and in all enclosing modes where the work has been suspended:
\begintt
### display math mode entered at line 5
\mathord
.\fam1 x
### internal vertical mode entered at line 4
prevdepth ignored
### math mode entered at line 3
### restricted horizontal mode entered at line 2
\glue 3.33333 plus 1.66666 minus 1.11111
spacefactor 1000
### vertical mode entered at line 0
prevdepth ignored
\endtt
In this case the lists represent five levels of activity, all
present at the end of line~6 of |modes.tex|.
The current mode is shown first, namely, display math mode, which began
on line~5. The current math list contains one ``^{mathord}''
object, consisting of the letter~|x| in family~1. \ (Have patience and you
will understand what that means, when you learn about \TeX's math formulas.) \
Outside of display math mode comes internal vertical mode, to which \TeX\
will return when the paragraph containing the
displayed formula is complete. The vertical list ^^|prevdepth ignored|
on that level is empty; `|prevdepth ignored|' means that |\prevdepth|
has a value $\le-1000\pt$, so that the next interline glue will be omitted
(cf.~Chapter~12). The math mode outside of this internal vertical mode has
an empty list, likewise, but the restricted horizontal mode enclosing the math
mode contains some glue. Finally, we see the main vertical mode that
encloses everything; this mode was `|entered at line 0|', i.e., before
the file |modes.tex| was input; nothing has been contributed
so far to the vertical list on this outermost level.
\dangerexercise Why is there glue in one of these lists but not in the
others?
\answer The output of\/ |\tracingcommands| shows that four blank space tokens
were digested; these originated at the ends of lines 2,~3, 4, and~5. Only
the first had any effect, since blank spaces are ignored in math formulas
and in vertical modes.
\dangerexercise After this output of\/ |\showlists|, the |modes.log| file
contains further output from |\tracingcommands|. In fact, the next two
lines of that file are
\begintt
{math shift character $}
{horizontal mode: end-group character }}
\endtt
because the `|$$|' on line 7 finishes the displayed formula, and this
resumes horizontal mode for the paragraph that was interrupted.
What do you think are the next three lines of |modes.log|\thinspace?
\answer The |end-group character| finishes the paragraph and the |\vbox|,
and |\bye| stands for `|\par\vfill...|', so the next three commands are
\begintt
{math mode: math shift character $}
{restricted horizontal mode: end-group character }}
{vertical mode: \par}
\endtt
\dangerexercise Suppose \TeX\ has generated a document without ever
leaving vertical mode. What can you say about that document?
\answer It contains only mixtures of vertical glue and horizontal rules
whose reference points appear at the left of the page; there's no text.
\ddangerexercise Some of \TeX's modes cannot immediately enclose other modes;
for example, display math mode is never directly enclosed by horizontal
mode, even though displays occur within paragraphs, because an interrupted
paragraph-so-far of horizontal mode is always completed and
removed from \TeX's memory before the processing of a displayed formula
begins. Give a complete characterization of all pairs of consecutive
modes that can occur in the output of\/ |\showlists|.
\answer Vertical mode can occur only as the outermost mode; horizontal
mode and display math mode can occur only when immediately enclosed by
vertical or internal vertical mode; ordinary math mode cannot be
immediately enclosed by vertical or internal vertical mode; all other
cases are possible.
\endchapter
Every mode of life has its conveniences.
\author SAMUEL ^{JOHNSON}, {\sl The Idler\/} (1758)
\bigskip
[Hindu musicians] have eighty-four modes,
of which thirty-six are in general use,
and each of which, it appears, has a peculiar expression,
and the power of moving some particular sentiment or affection.
\author MOUNTSTUART ^{ELPHINSTONE}, {\sl History of India\/} (1841)
% III.vii.I.297
\eject
\beginchapter Chapter 14. How \TeX\ Breaks\\Paragraphs into\\Lines
One of a typesetting system's chief duties is to take a long sequence of words
and to break it up into individual lines of the appropriate size.
For example, every paragraph of this manual has been broken into lines
that are 29~picas wide, but the author didn't have to worry about such
details when he composed the manuscript. \TeX\ chooses breakpoints
in an interesting way that considers each paragraph in its entirety; the
closing words of a paragraph can actually influence the appearance of the
first line. As a result, the spacing between words is as uniform as
possible, and the computer is able to reduce the number of times that
words must be hyphenated or formulas must be split between lines.
^^{H\&J, see hyphenation, line breaking, setting glue}
^^{justification, see setting glue, line breaking}
^^{quad left, see flush left}
^^{quad right, see flush right}
^^{quad middle, see :break}
The experiments of Chapter 6 have already illustrated the general ideas:
We discussed the notion of ``badness,'' and we ran into ``overfull'' and
``underfull'' boxes in difficult situations. We also observed that different
settings of \TeX's ^|\tolerance| parameter
will produce different effects; a higher tolerance
means that wider spaces are acceptable.
\TeX\ will find the absolutely best way to typeset any given paragraph,
according to its ideas of minimum badness. But such ``badness'' doesn't
account for everything, and if you rely entirely on
an automatic scheme you will occasionally encounter line breaks that are
not really the best on psychological grounds; this is inevitable, because
computers don't understand things the way people~do (at least not yet).
Therefore you'll sometimes want to tell the machine that certain places
are not good breakpoints. Conversely, you will sometimes want to force a
break at a particular spot. \TeX\ provides a convenient way to avoid
psychologically bad breaks, so that you will be able to obtain results of
the finest quality by simply giving a few hints to the machine.
``^{Ties}''---denoted by `|~|' in plain \TeX---are the key to
successful line breaking. ^^{auxiliary space, see tie} ^^{tilde}
Once you learn how to insert them, you will have graduated from the ranks
of ordinary \TeX nical typists to the select group of Distinguished
\TeX nicians. And it's really not difficult to train yourself to
insert occasional ties, almost without thinking, as you type a manuscript.
^^{line breaks, avoiding} ^^{breaks, avoiding bad}
When you type |~| it's the same as typing a space, except that \TeX\
won't break a line at this space. Furthermore, you shouldn't leave
any blanks next to the |~|, since they will count as additional spaces.
If you put |~| at the very end of a line in your input file, you'll get a
wider space than you want, because the \<return> that follows the |~|
produces an extra space.
We have already observed in Chapter~12 that it's generally a good idea to
type |~| after an abbreviation that does not come at the end of a sentence.
Ties also belong in several other places:
\smallskip
\item\bull In references to named parts of a document:
$$\halign{#\hfil&\hskip 80pt#\hfil\cr
|Chapter~12|&|Theorem~1.2|\cr
|Appendix~A|&|Table~\hbox{B-8}|\cr
|Figure~3|&|Lemmas 5 and~6|\cr}$$
\noindent(No |~| appears after `|Lemmas|' in the final example, since there's
no harm in having `5~and~6' at the beginning of a line. The use of\/ |\hbox|
is explained below.)
\smallbreak
\item\bull Between a person's forenames and between multiple surnames:
$$\halign{#\hfil&\hskip 40pt#\hfil\cr
|Donald~E. Knuth|&|Luis~I. Trabb~Pardo|\cr
|Bartel~Leendert van~der~Waerden|&|Charles~XII|\cr}$$
^^{Knuth} ^^{Trabb Pardo} ^^{van der Waerden} ^^{Charles XII}
\noindent
Note that it is sometimes better to hyphenate a name than to break it
between words; e.g., `Don-' and `ald~E.~Knuth' is more tolerable
than `Donald' and `E.~Knuth'. The previous rule can be regarded as a
special case of this one, since we may think of `Chapter~12' as a
compound name; another example is `|register~X|'. Sometimes a name is
so long that we dare not tie it all together, lest there be no way to
break the line: ^^{Vall\'ee Poussin}
\begintt
Charles Louis Xavier~Joseph de~la Vall\'ee~Poussin.
\endtt
\item\bull Between math symbols in apposition with nouns:
$$\halign{#\hfil\cr
|dimension~$d$ width~$w$ function~$f(x)$|\cr
|string~$s$ of length~$l$|\cr}$$
\noindent However, the last example should be compared with
\begintt
string~$s$ of length $l$~or more.
\endtt
\item\bull Between symbols in series:
$$\halign{#\hfil\cr
|1,~2, or~3|\cr
|$a$,~$b$, and~$c$.|\cr
|1,~2, \dots,~$n$.|\cr}$$
\item\bull When a symbol is a tightly bound object of a preposition:
$$\halign{#\hfil\cr
|of~$x$|\cr
|from 0 to~1|\cr
|increase $z$ by~1|\cr
|in common with~$m$.|\cr}$$
\noindent The rule does not, however, apply to compound objects:
\begintt
of $u$~and~$v$.
\endtt
\item\bull When mathematical phrases are rendered in words:
$$\halign{#\hfil&\hskip20pt#\hfil&\hskip20pt#\hfil\cr
|equals~$n$|&|less than~$\epsilon$|&|(given~$X$)|\cr
|mod~2|&|modulo~$p^e$|&|for all large~$n$|\cr}$$
\noindent Compare `|is~15|' with `|is 15~times the height|'.
\medbreak
\item\bull When cases are being enumerated within a paragraph:
^^{enumerated cases within a paragraph}
$$\halign{#\hfil\cr
|(b)~Show that $f(x)$ is (1)~continuous; (2)~bounded.|\cr}$$
\noindent It would be nice to boil all of these rules down to one or two
simple principles, and~it would be even nicer if the rules could be
automated so that keyboarding could be done without them; but subtle
semantic considerations seem to be involved. Therefore it's best to use
your own judgment with respect to ties. The computer needs your help.
A tie keeps \TeX\ from breaking at a space, but sometimes you want to
prevent the machine from breaking at a ^{hyphen} or a ^{dash}. This can be
done by using ^|\hbox|, because \TeX\ will not split up the contents of a
box; boxes are indecomposable units, once they have been constructed. We
have already illustrated this principle in the `|Table~\hbox{B-8}|'
example considered earlier. Another example occurs when you are typing
the page numbers in a ^{bibliographic reference}: It doesn't look good to
put \hbox{`22.'} on a line by itself, so you can type `|\hbox{13--22}.|'
to prohibit breaking `\hbox{13--22}.' On the other hand, \TeX\ doesn't
often choose line breaks at hyphens, so you needn't bother to insert
|\hbox| commands unless you need to correct a bad break that \TeX\ has
already made on a previous run.
\exercise Here are some phrases culled from previous chapters of this
manual. How do you think the author typed them?
\begindisplay
(cf.~Chapter~12).\cr
Chapters 12 and~21.\cr
line~16 of Chapter~6's {\tt story}\cr
lines 7 to~11\cr
lines 2,~3, 4, and~5.\cr
(2)~a big black bar\cr
All 256~characters are initially of category~12,\cr
letter~{\tt x} in family~1.\cr
the factor~$f$, where $n$~is 1000~times~$f$.\cr
\enddisplay
\answer|(cf.~Chapter~12).|\parbreak
|Chapters 12 and~21.|\parbreak
|line~16 of Chapter~6's {\tt story}|\parbreak
|lines 7 to~11|\parbreak
|lines 2,~3, 4, and~5.|\parbreak
|(2)~a big black bar|\parbreak
|All 256~characters are initially of category~12,|\parbreak
|letter~{\tt x} in family~1.|\parbreak
|the factor~$f$, where $n$~is 1000~times~$f$.|
\exercise How would you type the phrase `for all $n$ greater than
$n_0$'\thinspace?
\answer `|for all $n$~greater than~$n_0$|' avoids distracting breaks.
\exercise And how would you type `exercise 4.3.2--15'\thinspace?
\answer `|exercise \hbox{4.3.2--15}|' guarantees that there is no break
after the ^{en-dash}. But this precaution is rarely necessary, so
`|exercise 4.3.2--15|' is an acceptable answer. No |~| is needed;
`\hbox{4.3.2--15}' is so long that it causes no offense
at the beginning of a line.
\exercise Why is it better to type `|Chapter~12|' than to type
`|\hbox{Chapter 12}|'\thinspace?
\answer The space you get from |~| will stretch or shrink with the
other spaces in the same line, but the space inside an hbox has
a fixed width since that glue has already been set once and for all.
Furthermore the first alternative permits the word Chap-\break
ter to be ^{hyphenate}d.
\dangerexercise \TeX\ will sometimes break a math formula after an
equals sign. How can you stop the computer from breaking the formula
`$x=0$'\thinspace?
\answer `|\hbox{$x=0$}|' is unbreakable, and we will see later that
`|${x=0}$|' cannot be broken. Both of these solutions set the glue
surrounding the equals sign to some fixed value, but such glue normally
wants to stretch; furthermore, the |\hbox| solution might include undesirable
blank space at the beginning or end of a line, if\/ ^|\mathsurround| is
nonzero. A third solution `|$x=\nobreak0$|' avoids both defects.
\ddangerexercise Explain how you could instruct \TeX\ not to make any
breaks after explicit hyphens and dashes. \ (This is useful in
lengthy ^{bibliographies}.)
\answer |\exhyphenpenalty=10000| prohibits all such breaks, according
to the rules found later in this chapter. Similarly, |\hyphenpenalty=10000|
prevents breaks after implicit (discretionary) hyphens.
Sometimes you want to permit a line break after a `/' just as if it were
a hyphen. For this purpose plain \TeX\ allows you to say `^|\slash|';
for example, `|input\slash output|' produces `input\slash output' with
an optional break.
{\hbadness=10000
If you want to force \TeX\ to break between lines at a certain point in
^^{line breaks, forcing} ^^{breaks, forcing good}
the middle of a paragraph, just say `^|\break|'. However, that might cause the
line to be really ^^{underfull}spaced out.\break
If you want \TeX\ to fill up the right-hand part of a line
with blank space just before a forced line break,\hfil\break
without indenting the next line, say `|\hfil\break|'.\par}
\danger You may have several consecutive lines of input
for which you want the output to appear line-for-line in the same way.
One solution is to type `|\par|' at the end of each input line; but that's
somewhat of a nuisance, so plain \TeX\ provides the abbreviation
`^|\obeylines|', which causes each end-of-line in the input to be
like |\par|. After you say |\obeylines| you will get one line of
output per line of input, unless an input line ends with `|%|' or
unless it is so long that it must be broken. For example, you
probably want to use |\obeylines| if you are typesetting a ^{poem}.
Be sure to enclose |\obeylines| in a group, unless you want this
``poetry mode'' to continue to the end of your document.
\begintt
{\obeylines\smallskip
Roses are red,
\quad Violets are blue;
Rhymes can be typeset
\quad With boxes and glue.
\smallskip}
\endtt
\dangerexercise Explain the uses of\/ ^|\quad| in this poem. What would
have happened if `|\quad|' had been replaced by `^|\indent|' in both places?
\answer The second and fourth lines are indented by an additional ``quad''
of space, i.e., by one extra em in the current type style.
\ (The control sequence |\quad| does an ^|\hskip|; when \TeX\ is in
vertical mode, |\hskip| begins a new paragraph and puts glue after the
indentation.) \ If\/ |\indent| had been used instead, those lines wouldn't
have been indented any more than the first and third, because |\indent| is
implicit at the beginning of every paragraph. Double indentation on the
second and fourth lines could have been achieved by `|\indent\indent|'.
Roughly speaking, \TeX\ breaks paragraphs into lines in the following way:
Breakpoints are inserted between words or after hyphens so as to produce lines
whose badnesses do not exceed the current ^|\tolerance|. If there's no
way to insert such breakpoints, an ^{overfull box} is set. Otherwise the
breakpoints are chosen so that the paragraph is mathematically optimal, i.e.,
best possible, in
the sense that it has no more ``^{demerits}'' than you could obtain by any
other sequence of breakpoints. Demerits are based on the badnesses of
individual lines and on the existence of such things as consecutive lines
that end with hyphens, or tight lines that occur next to loose ones.
\danger But the informal description of line breaking in the previous
paragraph is an oversimplification of what really happens. The remainder
of this chapter explains the details precisely, for people who want to
apply \TeX\ in nonstandard ways. \TeX's line-breaking algorithm
has proved to be general enough to handle a surprising variety of
different applications; this, in fact, is probably the most interesting
aspect of the whole \TeX\ system. However, every paragraph from now on
until the end of the chapter is prefaced by at least one dangerous bend
sign, so you may want to learn the following material in easy stages
instead of all at once.
\ninepoint
\danger Before the lines have been broken, a paragraph inside of \TeX\
is actually a {\sl ^{horizontal list}}, i.e., a sequence of items that
\TeX\ has gathered while in horizontal mode. We have been saying
informally that a horizontal list consists of boxes and glue; the truth
is that boxes and glue aren't the whole story. Each item in a horizontal
list is one of the following types of things:\enddanger
\smallskip
\item\bull a box (a character or ligature or rule or hbox or vbox);
\item\bull a ^{discretionary break} (to be explained momentarily);
^^{break, discretionary}
\item\bull a ``^{whatsit}'' (something special to be explained later);
\item\bull vertical material (from ^|\mark| or ^|\vadjust| or ^|\insert|);
\item\bull a glob of ^{glue} (or ^|\leaders|, as we will see later);
\item\bull a ^{kern} (something like glue that doesn't stretch or shrink);
\item\bull a ^{penalty} (representing the undesirability of breaking here);
\item\bull ``^{math-on}'' (beginning a formula) or ``^{math-off}'' (ending a
formula).
\smallskip\noindent
The last four types (glue, kern, penalty, and math items)
are called {\sl ^{discardable}}, since they
may change or disappear at a line break; the first four types are
called non-discardable, since they always remain intact. Many of the
things that can appear in horizontal lists have not been touched on yet
in this manual, but it isn't necessary to understand them in order to
understand line breaking. Sooner or later you'll learn how each of the
gismos listed above can infiltrate a horizontal list; and if you want to
get a thorough understanding of \TeX's internal processes, you can always
use ^|\showlists| with various features of the language, in
order to see exactly what \TeX\ is doing.
\danger A discretionary break consists of three sequences of characters
called the {\sl pre-break}, {\sl post-break}, and {\sl no-break\/}
texts. The idea is that if a line break occurs here, the ^{pre-break text}
will appear at the end of the current line and the ^{post-break text} will
occur at the beginning of the next line; but if no break occurs, the
^{no-break text} will appear in the current line. Users can specify
^^|\discretionary|
discretionary breaks in complete generality by writing
\begindisplay
|\discretionary{|\<pre-break text>|}{|\<post-break text>|}{|\<no-break text>|}|
\enddisplay
where the three texts consist entirely of characters, boxes, and kerns.
For example, \TeX\ can hyphenate the word
`difficult' between the f's, even though this requires breaking the
`ffi' ligature into `f-' followed by an `fi' ligature, if the horizontal
list contains
\begintt
di\discretionary{f-}{fi}{ffi}cult.
\endtt
Fortunately you need not type such a mess yourself; \TeX's hyphenation algorithm
works behind the scenes, taking ^{ligatures} apart and putting them
into discretionary breaks when necessary.
\danger The most common case of a discretionary break is a simple
discretionary hyphen
\begintt
\discretionary{-}{}{}
\endtt
for which \TeX\ accepts the abbreviation `^|\-|'. The next most common case is
\begintt
\discretionary{}{}{}
\endtt
(an ``^{empty discretionary}''), which \TeX\ automatically inserts after
`|-|' and after every ligature that ends with `|-|'. In the case of plain
\TeX, empty discretionaries are therefore inserted after ^{hyphens} and
^{dashes}. \ (Each font has an associated ^|\hyphenchar|, which we can
assume for simplicity is equal to `|-|'.)
\danger When \TeX\ ^{hyphenates} words, it simply inserts discretionary
breaks into the horizontal list. For example, the words `|discretionary
hyphens|' are transformed into the equivalent of
\begintt
dis\-cre\-tionary hy\-phens
\endtt
if hyphenation becomes necessary. But \TeX\ doesn't apply its hyphenation
algorithm to any word that already contains a discretionary break;
therefore you can use explicit discretionaries to override \TeX's automatic
method, in an emergency.
\dangerexercise Before 1998, some ^{German} words changed their spelling
when split between lines. For example, `backen' became `bak-ken'
and `Bettuch' sometimes became `Bett-tuch'.
How can you instruct \TeX\ to produce such effects?
\answer |ba\ck/en| and |Be\ttt/uch|, where the macros |\ck/| and |\ttt/|
are defined by
\begintt
\def\ck/{\discretionary{k-}{k}{ck}}
\def\ttt/{tt\discretionary{-}{t}{}}
\endtt
The English word `eighteen' might deserve similar treatment.
\TeX's hyphenation algorithm will not make such spelling changes automatically.
\danger In order to save time, \TeX\ tries first to break a paragraph
into lines without inserting any discretionary hyphens. This first pass
will succeed if a sequence of breakpoints is found for which none
of the resulting lines has a badness exceeding the current value of
^|\pretolerance|. If the first pass fails, the method of Appendix~H is
used to hyphenate each word of the paragraph by inserting discretionary
breaks into the horizontal list, and a second attempt is
made using ^|\tolerance| instead of\/ |\pretolerance|. When the lines
are fairly wide, as they are in this manual, experiments show that
the first pass succeeds more than 90\% of the time, and that fewer than
2~words per paragraph need to be subjected to the hyphenation algorithm,
on the average. But when the lines are very narrow the
first pass usually fails rather quickly. Plain \TeX\ sets |\pretolerance=100|
and |\tolerance=200| as the default values. If you make |\pretolerance=10000|,
the first pass will essentially always succeed, so hyphenations will not
be tried (and the spacing may be terrible); on the other hand if you make
|\pretolerance=-1|, \TeX\ will omit the first pass and will try to
hyphenate immediately.
\danger Line breaks can occur only in certain places within a horizontal
list. Roughly speaking, they occur between words and after hyphens, but in
actuality they are permitted in the following five cases:\enddanger
\smallskip
\item{a)} at glue, provided that this glue is immediately preceded by a
non-discardable item, and that it is not part of a math formula (i.e., not
between math-on and math-off). A break ``at glue'' occurs at the left edge
of the glue space.
\smallskip
\item{b)} at a kern, provided that this kern is immediately followed by
glue, and that it is not part of a math formula.
\smallskip
\item{c)} at a math-off that is immediately followed by glue.
\smallskip
\item{d)} at a penalty (which might have been inserted automatically in a
formula).
\smallskip
\item{e)} at a discretionary break.
\smallskip\noindent
Notice that if two globs of glue occur next to each other, the second one
will never be selected as a breakpoint, since it is preceded by glue (which
is discardable).
\danger Each potential breakpoint has an associated ``penalty,'' which
represents the ``aesthetic cost'' of breaking at that place. In cases
(a), (b),~(c), the penalty is zero; in case~(d) an explicit penalty
has been specified; and in case~(e) the penalty is the current value of
^|\hyphenpenalty| if the pre-break text is nonempty, or the current value of
^|\exhyphenpenalty| if the pre-break text is empty.
Plain \TeX\ sets |\hyphenpenalty=50| and |\exhyphenpenalty=50|.
\danger For example, if you say `^|\penalty| |100|' at some point in a
paragraph, that position will be a legitimate place to break between
lines, but a penalty of 100 will be charged. If you say `\hbox{|\penalty-100|}'
you are telling \TeX\ that this is a rather good place to break, because
a negative penalty is really a ``^{bonus}''; a line that ends with a bonus
might even have ``merits'' (negative demerits).
\danger Any penalty that is 10000 or more is considered to be so large
^^{infinite penalty} that \TeX\ will never break there. At the other
extreme, any penalty that is $-10000$ or less is considered to be so small
that \TeX\ will always break there. The ^|\nobreak| macro of plain \TeX\
is simply an abbreviation for `|\penalty10000|', because this prohibits a
line break. A tie in plain \TeX\ is equivalent to `|\nobreak\|\]';
there will be no break at the glue represented by |\|\] in this
case, because glue is never a legal breakpoint when it is preceded by a
discardable item like a penalty.
\dangerexercise Guess how the ^|\break| macro is defined in plain \TeX.
\answer |\def\break{\penalty-10000 }|
\dangerexercise What happens if you say |\nobreak\break| or
|\break\nobreak|?
\answer You get a forced break as if\/ |\nobreak| were not present, because
|\break| cannot be cancelled by another penalty. In general if you
have two penalties in a row, their combined effect is the same as a single
penalty whose value is the minimum of the two original values, unless
both of those values force breaks. \ (You get two breaks from
|\break\break|; the second one creates an empty line.)
\danger When a line break actually does occur, \TeX\ removes all discardable
items that follow the break, until coming to something non-discardable,
or until coming to another chosen breakpoint. For example, a sequence of
glue and penalty items will vanish as a unit, if no boxes intervene,
unless the optimum breakpoint sequence includes one or more of the penalties.
Math-on and math-off items act essentially as kerns that contribute the spacing
specified by ^|\mathsurround|; such spacing will disappear into the line
break if a formula comes at the very end or the very beginning of a line,
because of the way the rules have been formulated above.
\ddanger The ^{badness} of a line is an integer that is approximately 100
times the cube of the ratio by which the glue inside the line must stretch
or shrink to make an hbox of the required size. For example, if the line
has a total shrinkability of 10 points, and if the glue is being
compressed by a total of 9 points, the badness is computed to be~73 (since
$100\times(9/10)^3=72.9$); similarly, a line that stretches by twice its
total stretchability has a badness of 800. But if the badness obtained by
this method turns out to be more than 10000, the value 10000 is used. \
$\bigl($See the discussion of ``^{glue set ratio}''~$r$ and ``^{glue set
order}''~$i$ in Chapter~12; if $i\ne0$, there is infinite stretchability or
shrinkability, so the badness is zero, otherwise the badness is
approximately $\min(100r^3,10000)$.$\bigr)$ \ Overfull boxes are
considered to be infinitely bad; they are avoided whenever possible.
^^{infinite badness}
\ddanger A line whose badness is 13 or more has a glue set ratio exceeding
50\%. We call such a line {\sl ^{tight}\/} if its glue had to shrink,
{\sl ^{loose}\/} if its glue had to stretch, and {\sl ^{very loose}\/}
if it had to stretch so much that the badness is 100 or more. But if the
badness is 12 or less we say that the line is {\sl ^{decent}}. Two
adjacent lines are said to be {\sl {visually incompatible}\/} if their
classifications are not adjacent, i.e., if a tight line is next to a
loose or very loose line, or if a decent line is next to a very loose one.
\ddanger \TeX\ rates each potential sequence of breakpoints by totalling
up {\sl ^{demerits}\/} that are assessed to individual lines. The goal
is to choose breakpoints that yield the fewest total demerits. Suppose that a
line has badness~$b$, and suppose that the penalty~$p$ is associated with
the breakpoint at the end of this line. As stated above, \TeX\ will not
even consider such a line if $p\ge10000$, or if $b$~exceeds the current
tolerance or pretolerance. Otherwise the demerits of such a line are
defined by the formula
\begindisplay
$\displaystyle{d=\cases{
(l+b)^2+p^2,&if $0\le p<10000$;\cr
(l+b)^2-p^2,&if $-10000<p<0$;\cr
(l+b)^2,&if $p\le-10000$.\cr}}$
\enddisplay
Here $l$ is the current value of\/ ^|\linepenalty|, a parameter that can be
increased if you want \TeX\ to try harder to keep all paragraphs to the
minimum number of lines; plain \TeX\ sets |\linepenalty=10|. For example,
a line with badness~20 ending at glue will have $(10+20)^2=900$ demerits,
if $l=10$, since there's no penalty for a break at glue. Minimizing the
total demerits of a paragraph is roughly the same as minimizing the
sum of the squares of the badnesses and penalties; this usually means
that the maximum badness of any individual line is also minimized, over
all sequences of breakpoints.
\ddangerexercise The formula for demerits has a strange discontinuity: It
seems more reasonable at first to define $d=(l+b)^2-10000^2$, in the
case $p\le-10000$. Can you account for this apparent discrepancy?
\answer Breaks are forced when $p\le-10000$, so there's no point in
subtracting a large constant whose effect on the total demerits is
known {\sl a priori}, especially when that might cause arithmetic overflow.
\ddanger Additional demerits are assessed based on pairs of adjacent lines.
If two consecutive lines are visually incompatible, in the sense explained
a minute ago, the current value of\/ ^|\adjdemerits| is added to~$d$. If two
consecutive lines end with discretionary breaks, the ^|\doublehyphendemerits|
are added. And if the second-last line of the entire paragraph ends with
a discretionary, the ^|\finalhyphendemerits| are added. Plain \TeX\ sets
up the values |\adjdemerits=10000|, |\doublehyphendemerits=10000|,
and |\finalhyphendemerits=5000|. Demerits are in units of
``badness squared,'' so the demerit-oriented parameters need to be rather
large if they are to have much effect; but tolerances and
penalties are given in the same units as badness.
\ddanger If you set ^|\tracingparagraphs||=1|, your log file will contain a
summary of \TeX's line-breaking calculations, so you can watch the tradeoffs
that occur when parameters like |\linepenalty| and |\hyphenpenalty| and
|\adjdemerits| are twiddled. The line-break data looks pretty scary at first,
but you can learn to read it with a little practice; this, in fact, is the
best way to get a solid understanding of line breaking. Here is the
trace that results from the second paragraph of the |story| file in
Chapter~6, when |\hsize=2.5in| and |\tolerance=1000|:
\begindisplay
|[]\tenrm Mr. Drofnats---or ``R. J.,'' as he pre-|\cr
|@\discretionary via @@0 b=0 p=50 d=2600|\cr
|@@1: line 1.2- t=2600 -> @@0|\cr
|ferred to be called---was hap-pi-est when |\cr
|@ via @@1 b=131 p=0 d=29881|\cr
|@@2: line 2.0 t=32481 -> @@1|\cr
|he|\cr
|@ via @@1 b=25 p=0 d=1225|\cr
|@@3: line 2.3 t=3825 -> @@1|\cr
|was at work type-set-ting beau-ti-ful doc-|\cr
|@\discretionary via @@2 b=1 p=50 d=12621|\cr
|@\discretionary via @@3 b=291 p=50 d=103101|\cr
|@@4: line 3.2- t=45102 -> @@2|\cr
|u-|\cr
|@\discretionary via @@3 b=44 p=50 d=15416|\cr
|@@5: line 3.1- t=19241 -> @@3|\cr
|ments.|\cr
|@\par via @@4 b=0 p=-10000 d=5100|\cr
|@\par via @@5 b=0 p=-10000 d=5100|\cr
|@@6: line 4.2- t=24341 -> @@5|\cr
\enddisplay
Lines that begin with `|@@|' ^^{atsign atsign} represent {\sl^{feasible
breakpoints}}, i.e., breakpoints that can be reached without any badness
exceeding the tolerance. Feasible breakpoints are numbered consecutively,
starting with |@@1|; the beginning of the paragraph is considered to be
feasible too, and it is number |@@0|. Lines that begin with `|@|' but
not `|@@|' are candidate ways to reach the feasible breakpoint that
follows; \TeX\ will select only the best candidate, when there is a choice.
Lines that do not begin with `|@|' indicate how far \TeX\ has gotten in the
paragraph. Thus, for example, we find `|@@2: line 2.0 t=32481 -> @@1|'
after `|...hap-pi-est when|' and before `|he|', so we know that feasible
breakpoint~|@@2| occurs at the space between the words |when| and |he|.
The notation `|line 2.0|' means that this feasible break comes at the end
of line~2, and that this line will be very loose. \ (The suffixes
|.0|, |.1|, |.2|, |.3| stand respectively for very loose, loose, decent,
and tight.) \ A hyphen is suffixed to the line number if that line
ends with a discretionary break, or if it is the final line of the
paragraph; for example, `|line 1.2-|' is a decent line that was hyphenated.
The notation `|t=32481|' means that the total demerits from the beginning
of the paragraph to~|@@2| are 32481, and `|-> @@1|' means that the best
way to get to |@@2| is to come from |@@1|. On the preceding line of trace
data we see the calculations for a typeset line to this point from |@@1|:
The badness is~131, the penalty is~0, hence there are 29881 demerits.
Similarly, breakpoint |@@3| presents an alternative for the second line of
the paragraph, obtained by breaking between `|he|' and `|was|'; this one
makes the second line tight, and it has only 3825 demerits when the
demerits of line~1 are added, so it appears that |@@3| will work much
better than |@@2|. However, the next feasible breakpoint (|@@4|) occurs
after `|doc-|', and the line from |@@2| to~|@@4| has only 12621 demerits,
while the line from |@@3| to~|@@4| has a whopping 103101; therefore
the best way to get from |@@0| to~|@@4| is via~|@@2|. If we regard
demerits as distances, \TeX\ is finding the ``^{shortest paths}'' from
|@@0| to each feasible breakpoint (using a variant of a well-known
algorithm for shortest paths in an acyclic graph). Finally the end of
the paragraph comes at breakpoint |@@6|, and the shortest path from
|@@0| to~|@@6| represents the best sequence of breakpoints. Following
the arrows back from~|@@6|, we deduce that~the best breaks in this
particular paragraph go through |@@5|, |@@3|, and~|@@1|.
\ddangerexercise Explain why there are 29881 demerits from |@@1| to |@@2|,
and 12621 demerits from~|@@2| to~|@@4|.
\answer $(10+131)^2+0^2+10000=29881$ and $(10+1)^2+50^2+10000=12621$.
In both cases the ^|\adjdemerits| were added because the lines were
visually incompatible (decent, then very loose, then decent); plain
\TeX's values for ^|\linepenalty| and |\adjdemerits| were used.
\ddanger If `|b=*|' ^^|*| appears in such trace data, it means that an
infeasible breakpoint had to be chosen because there was no feasible
way to keep total demerits small.
\danger We still haven't discussed the special trick that allows the
final line of a paragraph to be shorter than the others. Just before
\TeX\ begins to choose breakpoints, it does two important things: \
(1)~If the final item of the current horizontal list is glue,
^^|\unskip|
that glue is discarded. \ (The reason is that a blank space
often gets into a token list just before ^|\par| or just before |$$|,
and this blank space should not be part of the paragraph.) \ (2)~Three more
items are put at the end of the current horizontal list: |\penalty10000|
(which prohibits a line break); |\hskip\parfillskip| (which adds
``^{finishing glue}'' to the paragraph); and |\penalty-10000| (which
forces the final break). Plain \TeX\ sets ^|\parfillskip||=0pt plus1fil|,
so that the last line of each paragraph will be filled with white space
if necessary; but other settings of\/ |\parfillskip| are appropriate in
special applications. For example, the present paragraph ends flush with
the right margin, because it was typeset with |\parfillskip=0pt|;
the author didn't have to rewrite any of the text in order to make this
possible, since a long paragraph generally allows so much flexibility that
a line break can be forced at almost any point. You can have some fun
playing with paragraphs, because the algorithm for line breaking
occasionally appears to be clairvoyant. Just write paragraphs
that are long enough.\parfillskip=0pt
% the \danger macro makes this \parfillskip local!
\dangerexercise Ben ^{User} decided to say `|\hfilneg\par|' at the end of
a paragraph, intending that the negative stretchability of\/ ^|\hfilneg|
would cancel with the |\parfillskip| of plain \TeX\null. Why didn't his
bright idea work? ^^{paragraph, ending}
\answer Because \TeX\ discards a glue item that occurs just before
|\par|. Ben should have said, e.g., `|\hfilneg\ \par|'.
\dangerexercise How can you set |\parfillskip| so that the last line
of a paragraph has exactly as much white space at the right as the
first line has indentation at the left?\nobreak\hskip\parindent\hfilneg\
\answer Just say |\parfillskip|\stretch|=|\stretch|\parindent|. Of course,
\TeX\ will not be able to find appropriate line breaks unless each
paragraph is sufficiently long or sufficiently lucky; but with an
appropriate text, your output will be immaculately
symmetrical.{\parfillskip=\parindent\par}
\ddangerexercise Since \TeX\ reads an entire paragraph before it makes
any decisions about line breaks, the computer's memory capacity might
^^{capacity exceeded} be exceeded if you are typesetting the works of some
^^{Joyce, James} ^{philosopher} or modernistic novelist who writes
200-line paragraphs. Suggest a way to cope with such authors.
\answer Assuming that the author is deceased and/or set in his or her
ways, the remedy is to insert `|{\parfillskip=0pt\par\parskip=0pt\noindent}|'
in random places, after each 50 lines or so of text. \ (Every space
between words is usually a feasible breakpoint, when you get sufficiently
far from the beginning of a paragraph.)
\danger \TeX\ has two parameters called ^|\leftskip| and ^|\rightskip| that
specify glue to be inserted at the left and right of every line in a
paragraph; this glue is taken into account when badnesses and demerits are
computed. Plain \TeX\ normally keeps |\leftskip| and |\rightskip| zero,
but it has a `^|\narrower|' macro that increases both of their values by
the current ^|\parindent|. You may want to use |\narrower| when ^{quoting}
lengthy passages from a book.
\begintt
{\narrower\smallskip\noindent
This paragraph will have narrower lines than
the surrounding paragraphs do, because it
uses the ``narrower'' feature of plain \TeX.
The former margins will be restored after
this group ends.\smallskip}
\endtt
(Try it.) \ The second `^|\smallskip|' in this example ends the paragraph.
It's important to end the paragraph before ending the group, for otherwise
the effect of\/ |\narrower| will disappear before \TeX\ begins to choose
line breaks.
\dangerexercise When an entire paragraph is typeset in ^{italic} or ^{slanted}
type, it sometimes appears to be offset on the page with respect to
other paragraphs. Explain how you could use |\leftskip| and |\rightskip|
to shift all lines of a paragraph left by $1\pt$.
\answer |{\leftskip=-1pt \rightskip=1pt| \<text> |\par}|\par
\nobreak\medskip\noindent
(This applies to a full paragraph; if you want to correct only
isolated lines, you have to do it by hand.)
\dangerexercise The ^|\centerline|, ^|\leftline|, ^|\rightline|, and ^|\line|
macros of plain \TeX\ don't take |\leftskip| and |\rightskip| into
account. How could you make them do so?
\answer `|\def\line#1{\hbox to\hsize{\hskip\leftskip#1\hskip\rightskip}}|'
is the only change needed. \ (Incidentally,
^{displayed equations} don't take account of\/ |\leftskip| and |\rightskip|
either; it's more difficult to change that, because so many variations
are possible.)
\ddanger If you suspect that ^|\raggedright| setting is accomplished by
some appropriate manipulation of\/ |\rightskip|, you are correct. But some
care is necessary. For example, a person can
set |\rightskip=0pt plus1fil|, and every
line will be filled with space at the right. But this isn't a particularly
good way to make ragged-right margins, because the infinite stretchability
will assign zero badness to lines that are very short.
To do a decent job of ragged-right setting, the trick is to set
|\rightskip| so that it will stretch enough to make line breaks possible,
yet not too much, because short lines should be considered bad. Furthermore
the spaces between words should be fixed so that they do not stretch
or shrink. \ (See the definition of\/ |\raggedright| in Appendix~B\null.) \
It would also be possible to allow a little variability in the interword
glue, so that the right margin would not be quite so ragged but the
paragraphs would still have an informal appearance.
\danger \TeX\ looks at the parameters that affect
line breaking only when it is breaking lines. For example, you shouldn't
try to change the ^|\hyphenpenalty| in the middle of a paragraph,
if you want \TeX\ to penalize the hyphens in one word more than it does in
another word. The relevant values of
|\hyphenpenalty|, |\rightskip|, |\hsize|, and so on,
are the ones that are current at the end of the paragraph.
On the other hand, the width of indentation that you get
implicitly at the beginning of a paragraph or when you say `^|\indent|'
is determined by the value of\/ ^|\parindent| at the time the
indentation is contributed to the current horizontal list,
not by its value at the end of the paragraph. Similarly, penalties
that are inserted into math formulas within a paragraph are based
on the values of\/ ^|\binoppenalty| and ^|\relpenalty| that are current
at the end of each particular formula. Appendix~D contains an example
that shows how to have both ragged-right and ragged-left margins within
a single paragraph, without using |\leftskip| or |\rightskip|.
\varunit=0.989pt % getting ready to make circular insert
% \varunit=1.078pt was used with amr5: it had more letterspacing
\setbox0=\vtop{\null
\baselineskip6\varunit
\parfillskip0pt
\parshape 19
-18.25\varunit 36.50\varunit
-30.74\varunit 61.48\varunit
-38.54\varunit 77.07\varunit
-44.19\varunit 88.39\varunit
-48.47\varunit 96.93\varunit
-51.70\varunit 103.40\varunit
-54.08\varunit 108.17\varunit
-55.72\varunit 111.45\varunit
-56.68\varunit 113.37\varunit
-57.00\varunit 114.00\varunit
-56.68\varunit 113.37\varunit
-55.72\varunit 111.45\varunit
-54.08\varunit 108.17\varunit
-51.70\varunit 103.40\varunit
-48.47\varunit 96.93\varunit
-44.19\varunit 88.39\varunit
-38.54\varunit 77.07\varunit
-30.74\varunit 61.48\varunit
-18.25\varunit 36.50\varunit
\fiverm
\frenchspacing
\noindent
\hbadness 6000
\tolerance 9999
\pretolerance 0
\hyphenation{iso-peri-met-ric}
The area of a circle is a mean proportional
between any two regular and similar polygons of which one
circumscribes it and the other is isoperimetric with it.
In addition, the area of the circle is less than that of any
circumscribed polygon and greater than that of any
isoperimetric polygon. And further, of these
circumscribed polygons, the one that has the greater number of sides
has a smaller area than the one that has a lesser number;
but, on the other hand, the isoperimetric polygon that
has the greater number of sides is the larger.
\hbox to 36.50\varunit{\hss[Galileo,\thinspace1638]\hss}
}
\danger
\parshape 16
3pc 26pc
3pc 26pc
0pc 24.69pc
0pc 23.51pc
0pc 22.73pc
0pc 22.20pc
0pc 21.85pc
0pc 21.65pc
0pc 21.58pc
0pc 21.65pc
0pc 21.85pc
0pc 22.20pc
0pc 22.73pc
0pc 23.51pc
0pc 24.69pc
0pc 29pc
\vadjust{\moveright 28pc\vbox to 0pt{\vskip88pt\vskip-60\varunit
\vskip-3pt\box0\vss}}%
\strut It's possible to control the length of lines in a much more general
way, if simple changes to |\leftskip| and |\rightskip| aren't
flexible enough for your purposes. For example, a semicircular
^{hole} has been cut out of the present paragraph, in order to make
room for a circular illustration that contains some of ^{Galileo}'s
immortal words about ^{circle}s; all of the line breaks in this paragraph
and in the circular quotation were found by \TeX's line-breaking
algorithm. You can specify an essentially arbitrary paragraph
shape by saying ^|\parshape||=|\<number>, where the \<number> is
a positive integer $n$, followed by $2n$ \<dimen> specifications.
In general, `|\parshape=|$n$ $i_1$~$l_1$ $i_2$~$l_2$ $\ldots$ $i_n$~$l_n$'
specifies a paragraph whose first $n$ lines will have lengths
$l_1$, $l_2$, \dots,~$l_n$, respectively, and they will be
indented from the left margin by the respective amounts
$i_1$, $i_2$, \dots,~$i_n$. If the paragraph has fewer than
$n$ lines, the additional specifications will be ignored;
if it has more than $n$ lines, the specifications for line $n$ will
be repeated ad infinitum. You can cancel the effect of a previously
specified |\parshape| by saying `|\parshape=0|'.\parfillskip0pt
^^{illustrations, fitting copy around}
\ddangerexercise Typeset the following ^{Pascal}ian quotation in
the shape of an isosceles ^{triangle}:
``I turn, in the following treatises, to various uses of those
triangles whose generator is unity. But I leave out many more
than I include; it is extraordinary how fertile in properties
this triangle is. Everyone can try his hand.''
\answer The author's best solution is based on a variable |\dimen|
register |\x|:
\begintt
\setbox1=\hbox{I}
\setbox0=\vbox{\parshape=11 -0\x0\x -1\x2\x -2\x4\x -3\x6\x
-4\x8\x -5\x10\x -6\x12\x -7\x14\x -8\x16\x -9\x18\x -10\x20\x
\ifdim \x>2em \rightskip=-\wd1
\else \frenchspacing \rightskip=-\wd1 plus1pt minus1pt
\leftskip=0pt plus 1pt minus1pt \fi
\parfillskip=0pt \tolerance=1000 \noindent I turn, ... hand.}
\centerline{\hbox to \wd1{\box0\hss}}
\endtt
Satisfactory results are obtained with font |cmr10| when |\x| is set to
$8.9\pt$, $13.4\pt$, $18.1\pt$, $22.6\pt$, $32.6\pt$, and $47.2\pt$,
yielding triangles that are respectively 11,~9, 8, 7, 6, and~5 lines tall.
\danger You probably won't need unusual parshapes very often. But there's
a special case that occurs rather frequently, so \TeX\ provides a special
abbreviation for~it in terms of two parameters called ^|\hangindent| and
^|\hangafter|. The command `|\hangindent=|\<dimen>' specifies a so-called
^{hanging indentation}, and the command `|\hangafter=|\<number>' specifies
the duration of that indentation. Let $x$ and $n$ be the respective values
of\/ |\hangindent| and |\hangafter|, and let $h$ be the value of\/
^|\hsize|; then if $n\ge0$, hanging indentation will occur on lines $n+1$,
$n+2$, $\ldots$ of the paragraph, but if $n<0$ it will occur on lines
1,~2, \dots,~$\vert n\vert$. Hanging indentation means that lines will be
of width $h-\vert x\vert$ instead of their normal width~$h$; if $x\ge0$,
the lines will be indented at the left margin, otherwise they will be
indented at the right margin. For example, the ``dangerous bend''
paragraphs of this manual have a hanging indentation of 3~picas that lasts
for two lines; they were set with |\hangindent=3pc| and |\hangafter=-2|.
\danger Plain \TeX\ uses hanging indentation in its `^|\item|' macro, which
produces a paragraph in which every line has the same indentation as a
normal |\indent|. Furthermore, |\item| takes a parameter that is placed
into the position of the indentation on the first line. Another macro called
`^|\itemitem|' does the same thing but with double indentation.
For example, suppose you type
\begintt
\item{1.} This is the first of several cases that are being
enumerated, with hanging indentation applied to entire paragraphs.
\itemitem{a)} This is the first subcase.
\itemitem{b)} And this is the second subcase. Notice
that subcases have twice as much hanging indentation.
\item{2.} The second case is similar.
\endtt
{\let\par=\endgraf Then you get the following output:
\medskip
\item{1.} This is the first of several cases that are being
enumerated, with hanging indentation applied to entire paragraphs.
\itemitem{a)} This is the first subcase.
\itemitem{b)} And this is the second subcase. Notice
that subcases have twice as much hanging indentation.
\item{2.} The second case is similar.
\medskip}\noindent\hangindent0pt
(Indentations in plain \TeX\ are not actually as dramatic as those
displayed here; Appendix~B
says `|\parindent=20pt|', but this manual has been set with
|\parindent=36pt|.) \ It is customary to put ^|\medskip| before and after
a group of itemized paragraphs, and to say |\noindent|
before any closing remarks that apply to all of the cases.
^^{enumerated cases in separate paragraphs}
Blank lines are not needed before |\item| or |\itemitem|, since those macros
begin with |\par|.
\dangerexercise Suppose one of the enumerated cases continues for two
or more paragraphs. How can you use |\item| to get hanging indentation
on the subsequent paragraphs?
\answer |\item{}| at the beginning of each paragraph that wants hanging
indentation.
\dangerexercise Explain how to make a ``^{bullet}ed'' item that says `$\bullet$'
instead of `1.'.
\answer |\item{$\bullet$}|
\ddangerexercise The `|\item|' macro doesn't alter the right-hand margin. How
could you indent at both sides?
\answer Either change |\hsize| or |\rightskip|. The trick is to change it back
again at the end of a paragraph. Here's one way, without grouping:
\begintt
\let\endgraf=\par \edef\restorehsize{\hsize=\the\hsize}
\def\par{\endgraf \restorehsize \let\par=\endgraf}
\advance\hsize by-\parindent
\endtt
\ddangerexercise Explain how you could specify a hanging indentation
of $-2$ ems (i.e., the lines should project into the left margin),
after the first two lines of a paragraph.
\answer |\dimen0=\hsize \advance\dimen0 by 2em|\parbreak
|\parshape=3 0pt\hsize 0pt\hsize -2em\dimen0|
\danger If\/ |\parshape| and hanging indentation have both been specified,
|\parshape| takes precedence and |\hangindent| is ignored. You get the
normal paragraph shape, in which every line width is |\hsize|, when
|\parshape=0|, |\hangindent=0pt|, and |\hangafter=1|. \TeX\ automatically
restores these normal values at the end of every paragraph, and (by
local definitions) whenever it enters internal vertical mode. For example,
hanging indentation that might be present outside of a ^|\vbox| construction
won't occur inside that vbox, unless you ask for it inside.
^^{paragraph shape reset} ^^{hanging indentation reset}
\ddangerexercise Suppose you want to leave room at the right margin for
a rectangular illustration that takes up 15 lines, and you expect that
three paragraphs will go by before you have typeset enough text to get
past that illustration. Suggest a good way to do this without trial and error,
given the fact that \TeX\ resets hanging indentation.
\answer The three paragraphs can be combined into a single paragraph, if
you use `|\hfil\vadjust{\vskip\parskip}\break\indent|' instead of
`|\par|' after the first two. Then of course you say, e.g.,
|\hangindent=-50pt \hangafter=-15|. \ (The same idea can be applied in
connection with |\looseness|, if you want \TeX\ to make one of three
paragraphs looser but if you don't want to choose which one it will be.
However, long paragraphs fill \TeX's memory; please use restraint.) \
See also the next exercise.
\ddanger If ^{displayed equations} occur in a paragraph that has a nonstandard
shape, \TeX\ always assumes that the display takes up exactly three lines.
For example, a paragraph that has four lines of text, then a display, then
two more lines of text, is considered to be $4+3+2=9$ lines long; the
displayed equation will be indented and centered using the paragraph shape
information appropriate to line~6.
\ddanger \TeX\ has an internal integer variable called ^|\prevgraf| that
records the number of lines in the most recent paragraph that has been
completed or partially completed. You can use |\prevgraf| in the context of
a \<number>, and you can set |\prevgraf| to any desired nonnegative value
if you want to make \TeX\ think that it is in some particular part of the
current paragraph shape. For example, let's consider again a paragraph
that contains four lines plus a display plus two more lines. When \TeX\
starts the paragraph, it sets |\prevgraf=0|; when it starts the display,
|\prevgraf| will be~4; when it finishes the display, |\prevgraf| will
be~7; and when it ends the paragraph, |\prevgraf| will be~9. If the
display is actually one line taller than usual, you could set
|\prevgraf=8| at the beginning of the two final lines; then \TeX\ will
think that a 10-line paragraph is being made. The value of\/ |\prevgraf|
affects line breaking only when \TeX\ is dealing with nonstandard
|\parshape| or |\hangindent|.
\edef\lastex{\chapno.\the\exno}
\ddangerexercise Solve exercise \lastex\ using |\prevgraf|.
\answer Use |\hangcarryover| between paragraphs, defined as follows:
\begintt
\def\hangcarryover{\edef\next{\hangafter=\the\hangafter
\hangindent=\the\hangindent}
\par\next
\edef\next{\prevgraf=\the\prevgraf}
\indent\next}
\endtt
\ddanger You are probably convinced by now that \TeX's line-breaking algorithm
has plenty of bells and whistles, perhaps even too many.
But there's one more feature,
called ``looseness''; some day you might find yourself needing it,
when you are fine-tuning the pages of a book. If you set |\looseness=1|,
\TeX\ will try to make the current paragraph one line longer than its
optimum length, provided that there is a way to choose such breakpoints
without exceeding the tolerance you have specified for the badnesses
of individual lines. Similarly, if you set |\looseness=2|, \TeX\ will
try to make the paragraph two lines longer; and |\looseness=-1| causes an
attempt to make it shorter. The general idea is that \TeX\ first finds
breakpoints as usual; then if the optimum breakpoints produce
$n$~lines, and if the current ^|\looseness| is~$l$, \TeX\ will choose
the final breakpoints so as to make the final number of lines as close
as possible to $n+l$ without exceeding the current tolerance. Furthermore,
the final breakpoints will have fewest total demerits, considering all ways
to achieve the same number of~lines.
\ddanger For example, you can set |\looseness=1| if you want to avoid
a lonely ``^{club line}'' or ``^{widow line}'' on some page that does not
have sufficiently flexible glue, or if you want the total number of
lines in some two-column document to come out to be an even number.
It's usually best to choose a paragraph that is already pretty ``full,''
i.e., one whose last line doesn't have much white space, since such
paragraphs can generally be loosened without much harm. You might
also want to insert a ^{tie} between the last two words of that paragraph,
so that the loosened version will not end with only one ``^{widow word}'' on the
^^{orphans, see widow words}
line; this tie will cover your tracks, so that people will find it hard to
detect the fact that you have tampered with the spacing. On the other
hand, \TeX\ can take almost any sufficiently long paragraph and stretch
it a bit, without substantial harm; the present paragraph is, in fact,
one line looser than \hbox{its optimum length}.\looseness=1
\ddanger \TeX\ resets the looseness to zero at the same time as it resets
|\hangindent|, |\hangafter|, and |\parshape|.
\ddangerexercise Explain what \TeX\ will do if you set |\looseness=-1000|.
\answer It will set the current paragraph in the minimum number of lines
that can be achieved without violating the tolerance; and, given that
number of lines, it will break them optimally. \ (However, nonzero
looseness makes \TeX\ work harder, so this is not recommended if you
don't want to pay for the extra computation. You can achieve almost the
same result much more efficiently by setting ^|\linepenalty||=100|, say.)
\danger Just before switching to horizontal mode to begin scanning a
paragraph, \TeX\ inserts the glue specified by ^|\parskip| into the vertical
list that will contain the paragraph, unless that vertical list is empty so
far. For example, `|\parskip=3pt|' will cause 3~points of extra space
to be placed between paragraphs. Plain \TeX\ sets |\parskip=0pt plus1pt|;
this gives a little stretchability, but no extra space.
\danger After line breaking is complete, \TeX\ appends the lines to the
current vertical list that encloses the current paragraph, inserting
interline glue as explained in Chapter~12; this interline glue will
depend on the values of\/ ^|\baselineskip|, ^|\lineskip|, and ^|\lineskiplimit|
that are currently in force. \TeX\ will also insert penalties into
the vertical list, just before each glob of ^{interline glue}, in order to
help control page breaks that might have to be made later. For example, a
special penalty will be assessed for breaking a page between the first two
lines of a paragraph, or just before the last line, so that ``club'' or
``widow'' lines that are detached from the rest of a paragraph will not
appear all alone on a page unless the alternative is worse.
\danger Here's how interline penalties are calculated: \TeX\ has just
chosen the breakpoints for some paragraph, or for some partial paragraph
that precedes a displayed equation; and $n$~lines have been formed.
The penalty between lines $j$ and $j+1$, given a value of $j$ in the
range $1\le j<n$, is the value of\/ ^|\interlinepenalty| plus
additional charges made in special cases: The ^|\clubpenalty| is
added if $j=1$, i.e., just after the first line; then the
^|\displaywidowpenalty| or the ^|\widowpenalty| is added if $j=n-1$,
i.e., just before the last line, depending on whether or not
the current lines immediately precede a display; and finally the
^|\brokenpenalty| is added, if the $j$th line ended at a discretionary break.
(Plain \TeX\ sets |\clubpenalty=150|, |\widowpenalty=150|,
|\displaywidowpenalty=50|, and |\brokenpenalty=100|; the value of\/
|\interlinepenalty| is normally zero, but it is increased to 100 within
^{footnotes}, so that long footnotes will tend not to be broken between
pages.)
\dangerexercise Consider a five-line paragraph in which the second and fourth
lines end with hyphens. What penalties does plain \TeX\ put between the lines?
\answer 150, 100, 0, 250. \ (When the total penalty is zero, as between lines
3 and~4 in this case, no penalty is actually inserted.)
\dangerexercise What penalty goes between the lines of a two-line paragraph?
\answer |\interlinepenalty| plus |\clubpenalty| plus |\widowpenalty| (and
also plus |\brokenpenalty|, if the first line ends with a discretionary break).
\ddanger If you say ^|\vadjust||{|\<vertical list>|}| within a paragraph,
\TeX\ will insert the specified internal vertical list into the vertical
list that encloses the paragraph, immediately after whatever line
contained the position of the |\vadjust|. For example, you can say
`|\vadjust{\kern1pt}|' to increase the amount of space between lines of a
paragraph if those lines would otherwise come out too close together. \ (The
\vadjust{\vskip1pt}author
did it in the previous line, just to illustrate what happens.) \ Also,
if you want to make sure that a page break will occur immediately after a
certain line, you can say `|\vadjust{\eject}|' ^^|\eject| anywhere in that line.
\ddanger Later chapters discuss |\insert| and |\mark| commands that are
relevant to \TeX's page builder. If such commands appear within a
paragraph, they are removed from whatever horizontal lines contain them
and placed into the enclosing vertical list, together with other vertical
material from |\vadjust| commands that might be present. In the final
vertical list, each horizontal line of text is an hbox that is immediately
preceded by interline glue and immediately followed by vertical material
that has ``^{migrate}d out'' from that line (with left to right order
preserved, if there are several instances of vertical material); then
comes the interline penalty, if it is nonzero. Inserted vertical material
does not influence the ^{interline glue}.
\ddangerexercise Design a |\marginalstar| macro ^^{marginal notes}
that can be used anywhere in a paragraph. It should use |\vadjust| to
place an asterisk in the margin just to the left of the line where
|\marginalstar| occurs.
\answer The tricky part is to avoid ``opening up'' the paragraph by
adding anything to its height; yet this star is to be contributed after
a line having an unknown depth, because the depth of the line depends
on details of line breaking that aren't known until afterwards.
The following solution uses ^|\strut|, and assumes that the line containing
the marginal star does not have depth exceeding ^|\dp||\strutbox|, the
depth of a ^|\strut|.
\begintt
\def\strutdepth{\dp\strutbox}
\def\marginalstar{\strut\vadjust{\kern-\strutdepth\specialstar}}
\endtt
Here |\specialstar| is a box of height zero and depth |\strutdepth|,
and it puts an asterisk in the left margin:
\begintt
\def\specialstar{\vtop to \strutdepth{
\baselineskip\strutdepth
\vss\llap{* }\null}}
\endtt
\ddanger When \TeX\ enters ^{horizontal mode}, it will interrupt its normal
scanning to read tokens that were predefined by the command
^|\everypar||={|\<token list>|}|. For example, suppose you have said
`|\everypar={A}|'. If you type `|B|' in vertical mode, \TeX\ will shift
to horizontal mode (after contributing ^|\parskip| glue to the current
page), and a horizontal list will be initiated by inserting an empty box
of width ^|\parindent|. Then \TeX\ will read `|AB|', since it reads the
|\everypar| tokens before getting back to the `|B|' that triggered the
new paragraph. Of course, this is not a very useful illustration of
|\everypar|; but if you let your imagination run you will think of
better applications.
\ddangerexercise Use |\everypar| to define an |\insertbullets| macro: All
paragraphs in a group of the form `|{\insertbullets ...\par}|' should have a
bullet symbol `$\bullet$' as part of their indentation.
^^{bulleted lists}
\answer |\def\insertbullets{\everypar={\llap{$\bullet$\enspace}}}|\par
\nobreak\smallskip\noindent
(A similar device can be used to insert hanging indentation,
and/or to number the paragraphs automatically.)
\ddanger A paragraph of zero lines is formed if you say `|\noindent\par|'.
If\/ |\everypar| is null, such a paragraph contributes nothing except
|\parskip| glue to the current vertical list.
\ddangerexercise Guess what happens if you say `|\noindent$$...$$ \par|'.
\answer First comes |\parskip| glue (but you might not see it on the current
page if you say |\showlists|, since glue disappears at the top of each
page). Then comes the result of\/ |\everypar|, but let's assume that
|\everypar| doesn't add anything to the horizontal list, so that
you get an empty horizontal list; then there's no partial paragraph
before the display. The displayed equation follows the normal rules
(it occupies lines 1--3 of the paragraph, and uses the indentation and
length of line~2, if there's a nonstandard shape). Nothing follows the
display, since a blank space is ignored after a closing `|$$|'.\par
Incidentally, the behavior is different if you start a paragraph with
`|$$|' instead of with |\noindent$$|, ^^{display at beginning of paragraph}
since \TeX\ inserts a paragraph indentation that will appear on a line by
itself (with |\leftskip| and |\parfillskip| and |\rightskip| glue).
\ddanger Experience has shown that \TeX's line-breaking algorithm can be
harnessed to a surprising variety of tasks. Here, for example, is an application
that indicates one of the possibilities: Articles that are published in
{\sl^{Mathematical Reviews}\/\null} are generally signed with the reviewer's
name and address, and this information is typeset flush right, i.e., at
the right-hand margin. ^^{flush right}
If there is sufficient space to put such a name and address at the right of
the final line of the paragraph, the publishers can save space, and at the same
time the results look better because there are no strange gaps on the page.
\def\signed #1 (#2){{\unskip\nobreak\hfil\penalty50\hskip2em
\hbox{}\nobreak\hfil\sl#1\/ \rm(#2)
\parfillskip=0pt \finalhyphendemerits=0 \endgraf}}
\begindisplay
\vbox{\hsize 3.0in \parindent0pt
This is a case where the name and address fit in nicely with the review.
\signed A. Reviewer (Ann Arbor, Mich.)
\medskip
But sometimes an extra line must be added. \signed N. Bourbaki (Paris)}
\enddisplay
^^{Reviewer} ^^{Bourbaki}
Let's suppose that a space of at least two ems should separate the reviewer's
name from the text of the review, if they occur on the same line. We would
like to design a macro so that the examples shown above could be typed
as follows in an input file:
\begintt
... with the review. \signed A. Reviewer (Ann Arbor, Mich.)
... an extra line must be added. \signed N. Bourbaki (Paris)
\endtt
Here is one way to solve the problem:
\begintt
\def\signed #1 (#2){{\unskip\nobreak\hfil\penalty50
\hskip2em\hbox{}\nobreak\hfil\sl#1\/ \rm(#2)
\parfillskip=0pt \finalhyphendemerits=0 \par}}
\endtt
If a line break occurs at the |\penalty50|, the |\hskip2em| will disappear
and the empty |\hbox| will occur at the beginning of a line, followed by
|\hfil| glue. This yields two lines whose badness is zero; the first of these
lines is assessed a penalty of~50. But if no line break occurs at the
|\penalty50|, there will be glue of $2\em$ plus $2\,{\rm fil}$ between
the review and the name; this yields one line of badness zero. \TeX\ will
try both alternatives, to see which leads to the fewest total demerits.
The one-line solution will usually be preferred if it is feasible.
\ddangerexercise Explain what would happen if `|\hbox{}|' were left out
of the ^|\signed| macro.
\answer A break at |\penalty50| would cancel |\hskip2em\nobreak\hfil|,
so the next line would be forced to start with the reviewer's name flush left.
\ (But ^|\vadjust||{}| would actually be better than |\hbox{}|; it
uses \TeX\ more efficiently.)
\ddangerexercise Why does the |\signed| macro say
`^|\finalhyphendemerits||=0|'\thinspace?
\answer Otherwise the line-breaking algorithm might prefer two final lines to
one final line, simply in order to move a hyphen from the second-last line up
to the third-last line where it doesn't cause demerits. This in fact caused
some surprises when the |\signed| macro was being tested; |\tracingparagraphs=1|
was used to diagnose the problem.
{\hbadness=10000
\ddangerexercise In one of the paragraphs earlier in this chapter, the author
used ^|\break| to force a line break in a specific place; as a result, the
third line of that particular paragraph was really spaced out.\break
Explain why all the extra space went into the third line, instead of being
distributed impartially among the first three lines.
\answer Distributing the extra space evenly would lead to three lines of
the maximum badness (10000). It's better to have just one bad line
instead of three, since \TeX\ doesn't distinguish degrees of badness when
lines are really awful. In this particular case the ^|\tolerance| was 200,
so \TeX\ didn't try any line breaks that would stretch the first two lines;
but even if the tolerance had been raised to 10000, the optimum setting would
have had only one underfull line. If you really want to spread the
space evenly you can do so by using ^|\spaceskip| to increase the
amount of stretchability between words.
}
\ddanger If you want to avoid overfull boxes at all costs without
trying to fix them manually, you might be tempted to set
|\tolerance=10000|; this allows arbitrarily bad lines to be acceptable
in tough situations. But infinite tolerance is a bad idea, because
\TeX\ doesn't distinguish between terribly bad and preposterously
horrible lines. Indeed, a tolerance of 10000 encourages \TeX\ to
concentrate all the badness in one place, making one truly unsightly
line instead of two moderately bad ones, because a single
``write-off'' produces fewest total demerits according to the rules.
There's a much better way to get the desired effect: \TeX\ has a
parameter called ^|\emergencystretch| that is added to the assumed
stretchability of every line when badness and demerits are computed,
in cases where overfull boxes are otherwise unavoidable. If
|\emergencystretch| is positive, \TeX\ will make a third pass over a
paragraph before choosing the line breaks, when the first passes did
not find a way to satisfy the ^|\pretolerance| and ^|\tolerance|.
The effect of\/ |\emergencystretch| is to scale down the badnesses so
that large infinities are distinguishable from smaller ones. By
setting |\emergencystretch| high enough (based on |\hsize|) you can be
sure that the |\tolerance| is never exceeded; hence overfull boxes
will never occur unless the line-breaking task is truly impossible.
\ddangerexercise Devise a ^|\raggedcenter| macro (analogous to ^|\raggedright|)
that partitions the words of a paragraph into as few as possible lines
of approximately equal size and centers each individual line. Hyphenation
should be avoided if possible.
\answer |\def\raggedcenter{\leftskip=0pt plus4em \rightskip=\leftskip|%
\parbreak|\parfillskip=0pt \spaceskip=.3333em \xspaceskip=.5em|\parbreak
|\pretolerance=9999 \tolerance=9999 \parindent=0pt|\parbreak
|\hyphenpenalty=9999 \exhyphenpenalty=9999 }|
\endchapter
When the author objects to [a hyphenation]\/
he should be asked to add or cancel or substitute
a word or words that will prevent the breakage.
\smallskip
Authors who insist on even spacing always,
with sightly divisions always,
do not clearly understand the rigidity of types.
\author T. L. ^{DE VINNE}, {\sl Correct Composition\/} (1901) % p138, p206
\bigskip
In reprinting his own works, whenever [William ^{Morris}]\/
found a line that justified awkwardly, he altered the wording
solely for the sake of making it look well in print.
\smallskip
When a proof has been sent me with two or three
lines so widely spaced as to make a grey band across the page,
I have often rewritten the passage so as to fill up the lines better;
but I am sorry to say that my object has generally been so little
understood that the compositor has spoilt all the rest
of the paragraph instead of mending his former bad work.
\author GEORGE BERNARD ^{SHAW}, in {\sl The Dolphin\/} (1940) % v4.1 p80
\eject
\beginchapter Chapter 15. How \TeX\ Makes\\Lines into Pages
\tracingpages=1
\TeX\ attempts to choose desirable places to divide your document into
individual pages, and its technique for doing this usually works pretty
well. But the problem of ^{page make-up} is considerably more difficult
than the problem of line breaking that we considered in the previous chapter,
because pages often have much less flexibility than lines do. If the
vertical glue on a page has little or no ability to stretch or to shrink,
\TeX\ usually has no choice about where to start a new page; conversely, if
there is too much variability in the glue, the result will look bad because
different pages will be too irregular. Therefore if you are fussy about the
appearance of pages, you can expect to do some rewriting of the manuscript
until you achieve an appropriate balance, or you might need to fiddle
with the ^|\looseness| as described in Chapter~14; no automated system will
be able to do this as well as you.
Mathematical papers that contain a lot of displayed equations have an
advantage in this regard, because the glue that surrounds a display tends to
be quite flexible. \TeX\ also gets valuable room to maneuver when you
have occasion to use ^|\smallskip| or ^|\medskip| or ^|\bigskip| spacing
between certain paragraphs. For example, consider a page that contains
a dozen or so exercises, and suppose that there is $3\pt$ of additional
space between exercises, where this space can stretch to $4\pt$ or
shrink to $2\pt$. Then there is a chance to squeeze an extra line on the page,
or to open up the page by removing one line, in order to avoid splitting
an exercise between pages. Similarly, it is possible to use flexible
glue in special publications like membership rosters or company telephone
directories, so that individual entries need not be split between columns
or pages, yet every column appears to be the same height.
For ordinary purposes you will probably find that \TeX's automatic method
of page breaking is satisfactory. And when it occasionally gives
unpleasant results, you can force the machine to break at your favorite
place by typing `^|\eject|'. But be careful: |\eject| will cause \TeX\ to
stretch the page out, if necessary, so that the top and bottom baselines
agree with those on other pages. If you want to eject a short page,
filling it with blank space at the bottom, type `|\vfill\eject|' instead.
\danger If you say `|\eject|' in the middle of a paragraph, the paragraph
will end first, as if you typed `|\par\eject|'. But Chapter~14 mentions
that you can say `^|\vadjust||{\eject}|' in mid-paragraph, if you want to
force a page break after whatever line contains your current position
when the full paragraph is eventually broken up into lines; the rest of the
paragraph will go on the following page.
\danger To prevent a page break, you can say `^|\nobreak|' in vertical
mode, just as |\nobreak| in horizontal mode prevents breaks between lines.
For example, it is wise to say |\nobreak| between the title of a subsection
and the first line of text in that subsection. But |\nobreak| does not
cancel the effect of other commands like |\eject| that tell \TeX\ to
break; it only inhibits a break at glue that immediately follows. You
should become familiar with \TeX's rules for line breaks and page breaks
if you want to maintain fine control over everything. The remainder of
this chapter is devoted to the intimate details of page breaking.
\ninepoint
\danger \TeX\ breaks lists of lines into pages by computing badness ratings
and penalties, more or less as it does when breaking paragraphs into lines.
But pages are made up one at a time and removed from \TeX's memory; there is
no looking ahead to see how one page break will affect the next one.
In other words, \TeX\ uses a special method to find the optimum
breakpoints for the lines in an entire paragraph, but it doesn't attempt
to find the optimum breakpoints for the pages in an entire document. The
computer doesn't have enough high-speed memory capacity to remember the
contents of several pages, so \TeX\ simply chooses each page break as best
it can, by a process of ``local'' rather than ``global'' optimization.
\tracingpages=0
\danger Let's look now at the details of \TeX's page-making process.
Everything you contribute to the pages of your document is placed on
the {\sl ^{main vertical list}}, which is the sequence of items that
\TeX\ has accumulated while in vertical mode. Each item in a ^{vertical
list} is one of the following types of things:\enddanger
\smallskip
\item\bull a box (an hbox or vbox or rule);
\item\bull a ``^{whatsit}'' (something special to be explained later);
\item\bull a ^{mark} (another thing that will be explained later);
\item\bull an ^{insertion} (yet another thing that we will get to);
\item\bull a glob of ^{glue} (or ^|\leaders|, as we will see later);
\item\bull a ^{kern} (something like glue that doesn't stretch or shrink);
\item\bull a ^{penalty} (representing the undesirability of breaking here).
\smallskip\noindent
The last three types (glue, kern, and penalty items) are called ^{discardable},
for the same reason that we called them discardable in horizontal lists.
You might want to compare these specifications with the analogous rules
for the horizontal case, found in Chapter~14; it turns out that vertical
lists are just like horizontal ones except that character boxes, discretionary
breaks, ^|\vadjust| items, and math shifts cannot appear in vertical lists.
Chapter~12 exhibits a typical vertical list in \TeX's internal box-and-glue
representation.
\danger Page breaks can occur only at certain places within a vertical list.
The permissible breakpoints are exactly the same as in the horizontal case,
namely\enddanger
\smallskip
\item{a)} at glue, provided that this glue is immediately preceded by a
non-discardable item (i.e., by a box, whatsit, mark, or insertion);
\smallskip
\item{b)} at a kern, provided that this kern is immediately followed by glue;
\smallskip
\item{c)} at a penalty (which might have been inserted automatically in a
paragraph).
\smallskip\noindent
Interline glue is usually inserted automatically between the boxes of a
vertical list, as explained in Chapter~12, so there is usually a valid
breakpoint between boxes.
\danger As in horizontal lists, each potential breakpoint has an
associated penalty, which is high for undesirable breakpoints and negative
for desirable ones. The penalty is zero at glue and kern breaks, so it is
nonzero only at explicit penalty breaks. If you say `^|\penalty||-100|'
between two paragraphs, you are indicating that \TeX\ should try to break
here because the penalty is negative; a bonus of 100 points for breaking
at this place will essentially cancel up to 100 units of badness that
might be necessary to achieve such a break. A penalty of 10000 or more is
so large that it inhibits breaking; a penalty of $-10000$ or less is so
small that it forces breaking.
\danger Plain \TeX\ provides several control sequences that help to
control page breaks. For example, ^|\smallbreak|, ^|\medbreak|, and
^|\bigbreak| specify increasingly desirable places to break, having
respective penalties of $-50$, $-100$, and~$-200$; furthermore, they will
insert a ^|\smallskip|, ^|\medskip|, or ^|\bigskip| of space,
respectively, if a break is not taken. However, |\smallbreak|,
|\medbreak|, and |\bigbreak| do not increase existing glue unnecessarily;
for example, if you say |\smallbreak| just after a displayed
equation, you won't get a |\smallskip| of space in addition to the glue that
already follows a display. Therefore these commands can conveniently
be used before and after the statements of ^{theorems}, in a format for
mathematical papers. In the present manual the author has used a macro
that puts |\medbreak| before and after every dangerous-bend paragraph;
|\medbreak\medbreak| is equivalent to a single |\medbreak|, so you don't
see two medskips when one such paragraph ends and another one~begins.
\danger The ^|\goodbreak| macro is an abbreviation for `|\par\penalty-500|'.
This is a good thing to insert in your manuscript when proofreading, if
you are willing to stretch some page a little bit extra in order to improve the
following one. Later on if you make another change so that this |\goodbreak|
command does not appear near the bottom of a page, it will have no effect;
thus it is not as drastic as |\eject|.
\danger The most interesting macro that plain \TeX\ provides for page make-up
is called ^|\filbreak|. It means, roughly, ``Break the page here and fill the
bottom with blank space, unless there is room for more
copy that is itself followed by |\filbreak|.'' Thus if you put |\filbreak|
at the end of every paragraph, and if your paragraphs aren't too long,
every page break will occur between paragraphs, and \TeX\ will fit as many
paragraphs as possible on each page. The precise meaning of\/ |\filbreak|~is
\begintt
\vfil\penalty-200\vfilneg
\endtt
according to Appendix B\null; and this simple combination of \TeX's primitives
produces the desired result: If a break is taken at the |\penalty-200|,
the preceding |\vfil| will fill the bottom of the page with blank space,
and the ^|\vfilneg| will be discarded after the break; but if no break
is taken at the penalty, the |\vfil| and |\vfilneg| will cancel each other
and have no effect.
\danger Plain \TeX\ also provides a ^|\raggedbottom| command, which is a
vertical analog of\/ ^|\raggedright|: It tells \TeX\ to permit a small
amount of variability in the bottom margins on different pages, in order
to make the other spacing uniform.
\ddanger We saw in Chapter 14 that breakpoints for paragraphs are chosen
by computing ``demerits'' for each line and summing them over all lines.
The situation for pages is simpler because each page is considered separately.
\TeX\ figures the ``^{cost}'' of a page break by using the following formula:
\begindisplay
$\displaystyle{c=\cases{
p,&if $b<\infty$ and $p\le-10000$ and $q<10000$;\cr
b+p+q,&if $b<10000$ and $-10000<p<10000$ and $q<10000$;\cr
100000,&if $b=10000$ and $-10000<p<10000$ and $q<10000$;\cr
\infty,&if ($b=\infty$ or $q\ge10000$) and $p<10000$.\cr}}$
\enddisplay
Here $b$ is the ^{badness} of the page that would be formed if a break
were chosen here; $p$~is the penalty associated with the current
breakpoint; and $q$~is `^|\insertpenalties|', the sum of all penalties for
split insertions on the page, as explained below. Vertical badness is
computed by the same rules as horizontal badness; it is an integer between
0 and~10000, inclusive, except when the box is overfull, when it
is~$\infty$ (infinity).
^^{infinite badness}
\ddanger When a page is completed, it is removed from the main vertical
list and passed to an ``^{output routine},'' as we will see later; so its
boxes and glue eventually disappear from \TeX's memory. The remainder of
the main vertical list exists in two parts: First comes the ``^{current
page},'' which contains all the material that \TeX\ has considered so far
as a candidate for the next page to be broken off; then there are
``^{recent contributions},'' i.e., items that will be moved to the current
page as soon as \TeX\ finds it convenient to do so. If you say
^|\showlists|, \TeX\ will display the contents of the current page and the
recent contributions, if any, on your log file. \ (The example in
Chapter~13 doesn't show any such lists because they were both empty in
that case. Chapter~24 explains more about \TeX's timing.)
\ddanger Whenever \TeX\ is moving an item from the top of the ``recent
contributions'' to the bottom of the ``current page,'' it discards a
^{discardable item} (glue, kern, or penalty) if the current page does not
contain any boxes. This is how glue disappears at a page break. Otherwise
if a discardable item is a legitimate breakpoint, \TeX\ calculates the
cost~$c$ of breaking at this point, using the formula that we have just
discussed. If the resulting~$c$ is less than or equal to the smallest cost
seen so far on the current page, \TeX\ remembers the current
breakpoint as the best so far. And if $c=\infty$ or if $p\le-10000$, \TeX\
seizes the initiative and breaks the page at the best remembered
breakpoint. Any material on the current page following that best
breakpoint is moved back onto the list of recent contributions, where it
will be considered again; thus the ``current page'' typically gets more
than one page's worth of material before the breakpoint is chosen.
\ddanger This procedure may seem mysterious until you see it in action.
Fortunately, there is a convenient way to watch it;
you can set ^|\tracingpages||=1|, thereby instructing \TeX\ to put its
page-cost calculations into your log file. For example, here is what
appeared on the log file when the author used |\tracingpages=1| at the
beginning of the present chapter:
\begintt
%% goal height=528.0, max depth=2.2
% t=10.0 g=528.0 b=10000 p=150 c=100000#
% t=22.0 g=528.0 b=10000 p=0 c=100000#
% t=34.0 g=528.0 b=10000 p=0 c=100000#
|kern-3pt
|qquad|hbox|bgroup|rm|vdots|quad(25 similar lines are being omitted here)|egroup
% t=346.0 plus 2.0 g=528.0 b=10000 p=0 c=100000#
% t=358.0 plus 2.0 g=528.0 b=10000 p=150 c=100000#
% t=370.02223 plus 2.0 g=528.0 b=10000 p=-100 c=100000#
% t=398.0 plus 5.0 minus 2.0 g=528.0 b=10000 p=0 c=100000#
% t=409.0 plus 5.0 minus 2.0 g=528.0 b=10000 p=0 c=100000#
% t=420.0 plus 5.0 minus 2.0 g=528.0 b=10000 p=150 c=100000#
% t=431.0 plus 5.0 minus 2.0 g=528.0 b=10000 p=-100 c=100000#
% t=459.0 plus 8.0 minus 4.0 g=528.0 b=10000 p=0 c=100000#
% t=470.0 plus 8.0 minus 4.0 g=528.0 b=10000 p=0 c=100000#
% t=481.0 plus 8.0 minus 4.0 g=528.0 b=10000 p=0 c=100000#
% t=492.0 plus 8.0 minus 4.0 g=528.0 b=10000 p=0 c=100000#
% t=503.0 plus 8.0 minus 4.0 g=528.0 b=3049 p=0 c=3049#
% t=514.0 plus 8.0 minus 4.0 g=528.0 b=533 p=150 c=683#
% t=525.0 plus 8.0 minus 4.0 g=528.0 b=5 p=-100 c=-95#
% t=553.0 plus 11.0 minus 6.0 g=528.0 b=* p=0 c=*
\endtt
This trace output is admittedly not ``user-friendly'' in appearance, but
after all it comes from deep inside \TeX's bowels where things have been
reduced to numeric calculations. You can learn to read it with a little
practice, but you won't need to do so very often unless you need to
plunge into page-breaking for special applications. Here's what it
means: The first line, which starts with `|%%|', ^^{percent percent}
is written when the first box or insertion enters the current page list;
it shows the ``^{goal height}'' and the ``^{max depth}'' that will be
used for that page (namely, the current values of\/ ^|\vsize| and
^|\maxdepth|). In the present manual we have |\vsize=44pc| and
|\maxdepth=2.2pt|; dimensions in the log file are always displayed in
points. The subsequent lines, which start with a single `|%|', ^^{percent}
are written whenever a legal breakpoint is being moved from the list of
recent contributions to the current page list. Every |%|~line shows $t$,
which is the total height so far if a page break were to occur,
and $g$, which is the goal height; in this example $g$~stays fixed at
$528\pt$, but $g$ would have decreased if insertions such as footnotes had
occurred on the page. The values of~$t$ are steadily increasing from 10 to
22 to~34, etc.; baselines are $12\pt$ apart at the top of the page and
$11\pt$ apart at the bottom (where material is set in nine-point type).
We are essentially seeing one |%|~line per hbox of text being placed on
the current page. However, the |%|~lines are generated by the penalty or
glue items that follow the hboxes, not by the boxes themselves. Each
|%|~line shows also the badness~$b$, the penalty~$p$, and the cost~$c$
associated with a breakpoint; if this cost is the best so far, it is
marked with a `|#|' sign, ^^{sharp} meaning that ``this breakpoint will be
used for the current page if nothing better comes along.'' Notice that the
first 40 or so breaks all have $b=10000$, since they are so bad that \TeX\
considers them indistinguishable; in such cases $c=100000$, so \TeX\ simply
accumulates material until the page is full enough to
have $b<10000$. A penalty of 150 reflects the ^|\clubpenalty| or the
^|\widowpenalty| that was inserted as described in Chapter~14. The three
lines that say |p=-100| are the breakpoints between ``dangerous bend''
paragraphs; these came from ^|\medbreak| commands. The notation |b=|^|*| and
|c=*| on the final line means that $b$ and~$c$ are infinite; the total
height of $553\pt$ cannot be reduced to $528\pt$ by shrinking the
available glue. Therefore the page is ejected at the best previous place,
which turns out to be a pretty good break: |b=5| and |p=-100| yield a
net cost of $-95$.
\ddangerexercise Suppose the paragraph at the bottom of the example page
had been one line shorter; what page break would have been chosen?
\answer The last three page-break calculations would have been
\begintt
% t=503.0 plus 8.0 minus 4.0 g=528.0 b=3049 p=150 c=3199#
% t=514.0 plus 8.0 minus 4.0 g=528.0 b=533 p=-100 c=433#
% t=542.0 plus 11.0 minus 6.0 g=528.0 b=* p=0 c=*
\endtt
so the break would have occurred at the same place. The badness would have
been~533, but the page would still have looked tolerable. \ (On the other
hand if that paragraph had been two lines shorter instead of one,
the first two lines of the next ``dangerous bend'' paragraph
would have appeared on that page; the natural height $t=531\pt$ would have
been able to shrink to $g=528\pt$ because the three ``medskips'' on
the page would have had a total shrinkability of $6\pt$. This would certainly
have been preferable to a stretched-out page whose badness was~3049; but the
author might have seen it and written another sentence or two, so that
the paragraph would not have been broken up. After all, this manual is supposed
to be an example of good practice.)
\ddangerexercise The last two ``\kern.5pt|%| lines'' of this example show
the natural height of~$t$ jumping by $28\pt$, from 525.0 to~553.0. Explain
why there was such a big jump.
\answer The next legal break after the beginning of a dangerous bend
paragraph occurs $28\pt$ later, because there is $6\pt$ additional space for
a |\medskip|, followed by two lines of $11\pt$ each. \TeX\ does not
allow breaking between those two lines; the ^|\clubpenalty| is set briefly
to 10000 in Appendix~E\null, since the dangerous bend symbol is two lines tall.
\ddanger The ^|\maxdepth| parameter tells \TeX\ to raise the bottom box
on the page if that box has too much depth, so that the depth of the
constructed page will not exceed a specified value. \ (See the discussion of
^|\boxmaxdepth| in Chapter~12.) \ In our example |\maxdepth=2.2pt|, and
the influence of this parameter can be seen in the line that says
`|% t=370.02223|'. Ordinarily $t$~would have been 370.0 at that breakpoint;
but the hbox preceding it was unusual because it contained the letter
|j| in ^|\tt|, and a 10-point typewriter-style |j| descends $2.22223\pt$
below the baseline. Therefore \TeX\ figured badness as if the hbox were
$.02223\pt$ higher and only $2.2\pt$ deep.
\ddanger Notice that the first ``\kern.5pt|%| line'' of our example says
|t=10.0|; this is a consequence of another parameter, called ^|\topskip|.
Glue disappears at a page break, but it is desirable to produce pages
whose top and bottom baselines occur in predetermined positions, whenever
possible; therefore \TeX\ inserts special glue just before the first box
on each page. This special glue is equal to |\topskip|, except that the
natural space has been decreased by the height of the first box, or it has
been set to zero in lieu of a negative value. For example, if\/
|\topskip=20pt plus2pt|, and if the first box on the current page is
$13\pt$ tall, \TeX\ inserts `|\vskip7pt plus2pt|' just above that box.
Furthermore, if the first box is more than $20\pt$ tall, `|\vskip0pt
plus2pt|' is inserted. But this example is atypical, since the |\topskip|
glue usually has no stretchability or shrinkability; plain \TeX\ sets
|\topskip=10pt|.
\ddangerexercise Assume that |\vsize=528pt|, |\maxdepth=2.2pt|,
|\topskip=10pt|, and that no |\insert| commands are being used. \TeX\ will
make pages that are $528\pt$ high, and the following two statements will
normally be true: (a)~The baseline of the topmost box on the page will be
$10\pt$ from the top, i.e., $518\pt$ above the baseline of the page itself.
(b)~The baseline of the bottommost box on the page will coincide with the
baseline of the page itself. Explain under what circumstances (a) and~(b)
will fail.
\answer A page always contains at least one box, if there are no
insertions, since the legal breakpoints are discarded otherwise.
Statement~(a) fails if the height of the topmost box exceeds $10\pt$.
Statement~(b) fails if the depth of the bottommost box exceeds $2.2\pt$, or
if some glue or kern comes between the bottommost box and the page break
(unless that glue or kern exactly cancels the depth of the box).
\ddanger Since |\vsize|, |\maxdepth|, and |\topskip| are
parameters, you can change them at any time; what happens if you do?
Well, \TeX\ salts away the values of\/ |\vsize| and |\maxdepth| when
it prints the ``\kern.5pt|%%|~line,'' i.e., when the first box or insertion
occurs on the current page; subsequent changes to those two parameters
have no effect until the next current page is started. On the other hand,
\TeX\ looks at |\topskip| only when the first box is being contributed
to the current page. If insertions occur before the first box, the
|\topskip| glue before that box is considered to be a valid breakpoint;
this is the only case in which a completed page might not contain a box.
\ddanger You can look at the $t$ and $g$ values that are used in page
breaking by referring to the \<dimen> values `^|\pagetotal|' and
`^|\pagegoal|', respectively. You can even change them (but let's hope
that you know what you are doing). For example, the command
|\pagegoal=500pt| overrides the previously saved value of\/ |\vsize|.
Besides |\pagetotal|, which represents the accumulated natural height,
\TeX\ maintains the quantities ^|\pagestretch|, ^|\pagefilstretch|,
^|\pagefillstretch|, ^|\pagefilllstretch|, ^|\pageshrink|, and
^|\pagedepth|. When the current page contains no boxes, |\pagetotal| and
its relatives are zero and |\pagegoal| is $16383.99998\pt$ (\TeX's largest
\<dimen>); changing their values has no effect at such times. The integer
$q$ in the formula for page costs is also available for inspection and
change; it is called ^|\insertpenalties|.
\ddanger Page breaking differs from line breaking in one small respect
that deserves mention here: If you say |\eject\eject|, the second |\eject|
is ignored, because it is equivalent to |\penalty-10000| and penalties are
discarded after a page break. But if you say |\break\break| in a
paragraph, the second ^|\break| causes an empty line, because penalties
are discarded after a break in a paragraph only if they do not belong to
the final sequence of breakpoints. This technicality is unimportant in
practice, because |\break\break| isn't a good way to make an empty line;
that line will usually be an underfull hbox, since it has only the
|\leftskip| and |\rightskip| glue in it. Similarly, `|\eject\eject|'
would not be a good way to make an empty page, even if \TeX\ were to
change its rules somehow so that an |\eject| would never be ignored. The
best way to eject an ^{empty page} is to say `|\eject\line{}\vfil\eject|',
and the best way to create an ^{empty line} is
`|\break\hbox{}\hfil\break|'. Both of these avoid underfull boxes.
\danger You are probably wondering how page numbers and such things
get attached to pages. The answer is that \TeX\ allows you to do further
processing after each page break has been chosen; a special ``output
routine'' goes into action before pages actually receive their final
form. Chapter~23 explains how to construct output routines and how to
modify the output routine of plain~\TeX.
\danger Every once in a~while, \TeX\ will produce a really awful-looking
page and you will wonder what happened. For example, you might get just
one paragraph and a lot of white space, when some of the text on the
following page would easily fit into the white space. The reason for such
apparently anomalous behavior is almost always that no good page break
is possible; even the alternative that looks better to you is quite terrible
as far as \TeX\ is concerned! \TeX\ does not distinguish between two
choices that both have 10000 units of badness or more, even though some bad
breaks do look much worse than others. The solution in such cases is to
insert |\eject| or |\vfill\eject| in some acceptable spot, or to
revise the manuscript. If this problem arises frequently, however,
you probably are using a format that sets overly strict limitations
on page format; try looking at the output of\/ |\tracingpages| and
modifying some of \TeX's parameters, until you have better luck.
\danger The remainder of this chapter is about insertions: things like
footnotes and ^{illustrations}, and how they interact with page breaks.
Before we discuss the primitive operations by which \TeX\ deals with
insertions, we will take a look at the facilities that plain \TeX\ provides
at a higher level.
\danger Illustrations can be inserted in several ways using plain \TeX.
The simplest of these is called a ``^{floating topinsert}''; you say
\begindisplay
^|\topinsert|\<vertical mode material>^|\endinsert|
\enddisplay
and \TeX\ will attempt to put the vertical mode material at the top of the
current page. If there's no room for such an insertion on this page, \TeX\
will insert it at the top of the next page. The \<vertical mode material>
can contain embedded paragraphs that temporarily interrupt vertical mode
in the usual way; for example:
\begintt
\topinsert \vskip 2in
\hsize=3in \raggedright
\noindent{\bf Figure 3.} This is the caption to the
third illustration of my paper. I have left two inches
of space above the caption so that there will be room
to introduce special artwork. \endinsert
\endtt
The ^{caption} in this example will be set ^{ragged-right} in a 3-inch
column at the left of the page. Plain \TeX\ automatically adds a
``^{bigskip}'' below each topinsert; this will separate the caption from
the text. The effects of\/ |\hsize=3in| and |\raggedright| do not extend
past the |\endinsert|, since ^{grouping} is implied.
\dangerexercise Modify this example so that the caption is moved over next to
the right margin, instead of appearing at the left.
\answer |\topinsert\vskip2in\rightline{\vbox{\hsize|\stretch|...|\stretch
|artwork.}}\endinsert|
does the job. But it's slightly more efficient to avoid ^|\rightline| by
changing ^|\leftskip| as follows:
`|\leftskip=\hsize \advance\leftskip by-3in|'.
Then \TeX\ doesn't have to read the text of the caption twice.
\danger Similarly, if you say `^|\pageinsert|\stretch\<vertical mode
material>\stretch |\endinsert|', the vertical mode material will be
justified to the size of a full page (without a bigskip below it);
the result will appear on the following page.
\danger There's also `^|\midinsert|\stretch\<vertical mode material>\stretch
|\endinsert|', which tries first to insert the material in place, wherever
you happen to be, in the middle of the current page. If there is enough
room, you get the effect of^^|\bigskip|^^|\bigbreak|
\begindisplay
|\bigskip\vbox{|\<vertical mode material>|}\bigbreak|
\enddisplay
otherwise the |\midinsert| is effectively converted to a |\topinsert|.
There is a slight probability that |\midinsert| will not find the
best placement, because \TeX\ is sometimes processing text ahead of the
current page. You may want to say `^|\goodbreak|' just before |\midinsert|.
\danger You should use the commands |\topinsert|, |\pageinsert|,
|\midinsert| in vertical mode (i.e., between paragraphs), not inside of
boxes or other insertions.
\danger If you have two or more |\topinsert| or |\pageinsert| commands in
quick succession, \TeX\ may need to carry them over to several subsequent
pages; but they will retain their relative order when they are
carried over. For example, suppose you have pages that are nine inches
tall, and suppose you have already specified 4~inches of text for some
page, say page~25. Then suppose you make seven topinserts in a row, of
respective sizes $1,2,3,9,3,2,1$ inches; the 9-inch one is actually a
|\pageinsert|. What happens? Well, the first and second will appear at
the top of page 25, followed by the 4~inches of copy you have already
typed; that copy will immediately be followed by two more inches that you
type after the seven inserts. The third topinsert will appear at the top
of page~26, followed by six more inches of text; the fourth will fill
page~27; and the remaining three will appear at the top of page~28.
\dangerexercise What would happen in the example just discussed if the final
1-inch insertion were a |\midinsert| instead of a |\topinsert|?
\answer It would appear on page~25, since it does fit there. A |\midinsert|
will jump ahead of other insertions only if it is not carried over to another
page; for example, if the second 3-inch insertion were a |\midinsert|, it would
not appear on page~26, because it is converted to a |\topinsert| as soon as the
|\midinsert| macro notices that the insertion is too big for page~25.
\danger At the end of a paper, you probably want to make sure that no
insertions are lost; and at the end of a chapter, you probably want to make
sure that no insertions float into the following chapter. Plain \TeX\ will
flush out all remaining insertions, with blank space filling the bottom of
incomplete pages, if you say `|\vfill|^|\supereject|'.
\danger Besides illustrations that are inserted at the top of a page,
plain \TeX\ will also insert ^{footnotes} at the bottom of a page. The
^|\footnote| macro is provided for use within paragraphs;\footnote*{Like
this.} for example, the footnote in the present sentence was typed
in the following way:
\begintt
... paragraphs;\footnote*{Like this.} for example, ...
\endtt
There are two parameters to a |\footnote|; first comes the ^{reference mark},
which will appear both in the paragraph\footnote{**}{The author typed
`|paragraph\footnote{**}{The author ...}|' here.} and in the footnote itself,
and then comes the text of the footnote.\footnote{$^{45}$}{And
`|footnote.\footnote{$^{45}$}{And ...}|' here. The footnotes in this manual
appear in smaller type, and they are set with hanging indentation;
furthermore a smallskip occurs between footnotes on the same page. But
in plain \TeX, footnotes are typeset with the normal size of type, with
|\textindent| used for the reference mark, and without extra smallskips.
The |\textindent| macro is like |\item|, but it omits hanging
indentation.}
The latter text may be several paragraphs long, and it may contain
displayed equations and such things, but it should not involve other
^^|\textindent| ^^|\item| % these are in a split footnote, after the split!
insertions. \TeX\ will ensure that each footnote occurs at the bottom of
the same page as its reference.\footnote\dag{Printers often use the
symbols |\dag| (\dag), |\ddag| (\ddag), |\S|~(\S), and |\P|~(\P) as
reference marks; sometimes also |$\|\||$| ($\Vert$).
You can say, e.g., `|\footnote\dag{...}|'.} A long
footnote will be split, if necessary, and continued at the bottom of the
following page, as you can see in the ^^|\dag|^^|\ddag|^^|\S|^^|\P|^^|\Vert|
somewhat contrived example that appears here. Authors who are interested
in good exposition should avoid footnotes whenever possible, since
footnotes tend to be ^^{Gibbon} distracting.\footnote\ddag{Yet Gibbon's
{\sl Decline and Fall\/} would not have been the same without footnotes.}
\danger The |\footnote| macro should be used only in paragraphs or hboxes
that are contributed to \TeX's main vertical list; insertions will be lost
if they occur inside of boxes that are inside of boxes. Thus, for example,
you should not try to put a |\footnote| into a subformula of a math formula.
But it's OK to use footnotes within ^|\centerline|, e.g., ^^{Thor}
\begintt
\centerline{A paper by A. U. Thor%
\footnote*{Supported by NSF.}}
\endtt
or even on the outer level of a table entry inside an ^|\halign|.
\ddanger Topinserts work fine by themselves, and footnotes work fine by
themselves, but complications can arise when you try to mix them in
devious ways. For example, if a |\pageinsert| floats to the page that
follows a long footnote that had to be broken, both of the held-over
insertions may try to force themselves onto the same page, and an overfull
vbox may result. Furthermore, insertions cannot appear within insertions,
so you can't use |\footnote| within a |\topinsert|. If you really need a
footnote in some caption, there's a ^|\vfootnote| macro that can be used
in vertical mode. To use it, you put a reference mark like~`|*|' in the
caption, and then you say `|\vfootnote*{The footnote}|' somewhere on the
page where you guess that the caption will finally fall. In such complex
circumstances you might want to rethink whether or not you are really
using the most appropriate format for the exposition of your ideas.
\ddanger Chapter 24 explains the exact rules about ^{migration} of
vertical-mode material (like footnotes) from horizontal lists to the
enclosing vertical list. Insertions, marks, and the results of\/ ^|\vadjust|
all migrate in the same fashion.
\danger Now let's study the primitives of \TeX\ that are used to construct
macros like |\topinsert| and |\footnote|. We are about to enter behind the
scenes into a sublanguage of \TeX\ that permits users to do complex
manipulations with boxes and glue. Our discussion will be in two parts:
First we shall consider \TeX's ``^{registers},'' with which a user can
do ^{arithmetic} related to typesetting; and then we shall discuss the insertion
items that can appear in horizontal and vertical lists. Our discussion of the
first topic (registers) will be marked with single dangerous-bend signs,
since registers are of general use in advanced applications of \TeX,
whether or not they relate to insertions. But the second topic will be marked
with double dangerous-bend signs, since insertions are rather esoteric.
\danger \TeX\ has 256 registers called ^|\count||0| to |\count255|, each capable
of containing integers between $-2147483647$ and $+2147483647$, inclusive;
^^\<number>
i.e., the magnitudes should be less than $2^{31}$. \TeX\ also has 256 registers
called ^|\dimen||0| to |\dimen255|, each capable of containing a ^\<dimen>
(see Chapter~10). There are another 256 registers called ^|\skip||0| to
|\skip255|, each containing ^\<glue> (see Chapter~12); and ^|\muskip||0| to
|\muskip255|, each containing ^\<muglue> (see Chapter~18). You can assign
new values to these registers by saying
\begindisplay
|\count|\<number> |=| \<number>\cr
|\dimen|\<number> |=| \<dimen>\cr
|\skip|\<number> |=| \<glue>\cr
|\muskip|\<number> |=| \<muglue>\cr
\enddisplay
and then you can add or subtract values of the same type by saying^^|\advance|
\begindisplay
|\advance\count|\<number> |by| \<number>\cr
|\advance\dimen|\<number> |by| \<dimen>\cr
|\advance\skip|\<number> |by| \<glue>\cr
|\advance\muskip|\<number> |by| \<muglue>\cr
\enddisplay
For example, `|\dimen8=\hsize \advance\dimen8 by 1in|' sets register
|\dimen8| to an inch more than the current value of the normal line size.
\danger If infinite glue components are added, lower order infinities disappear.
For example, after the two commands
\begintt
\skip2 = 0pt plus 2fill minus 3fill
\advance\skip2 by 4pt plus 1fil minus 2filll
\endtt
the value of\/ |\skip2| will be $4\pt$ plus $2\,{\rm fill}$ minus
$2\,{\rm filll}$.
\danger Multiplication and division are possible too, but only by integers.
For example, `^|\multiply||\dimen4 by 3|' triples the value of\/ |\dimen4|, and
`^|\divide||\skip5 by 2|' cuts in half all three components of the glue that is
currently registered in |\skip5|.
You shouldn't divide by zero, nor should you multiply by numbers that
will make the results exceed the register capacities. Division of a positive
integer by a positive integer discards the remainder, and the sign of the
result changes if you change the sign of either operand. For example,
14~divided by~3 yields~4; $-14$~divided by~3 yields~$-4$;
$-14$~divided by~$-3$ yields~4. Dimension values are integer multiples
of~^{sp} (scaled points).
\danger You can use any |\count| register in the context of a ^\<number>,
any |\dimen| register in the context of a ^\<dimen>, any |\skip| register
in the context of ^\<glue>, and any |\muskip| register in the context of
^\<muglue>. For example, `|\hskip\skip1|' puts horizontal glue into a
list, using the value of\/ |\skip1|; and if\/ |\count5| is 20,
the~command `|\advance\dimen20 by\dimen\count5|' is equivalent to
`|\multiply\dimen20 by 2|'.
\danger A |\dimen| register can be used also in the context of a \<number>,
and a |\skip| register can be used as a \<dimen> or a \<number>.
\TeX\ converts \<glue> to \<dimen> by omitting the stretch and shrink
components, and it converts \<dimen> to \<number> by assuming units
of~sp (scaled points). For example, if\/ |\skip1| holds the value
$1\pt$ plus~$2\pt$, then `|\dimen1=\skip1|' sets |\dimen1| equal
to~$1\pt$; and the commands `|\count2=\dimen1|' or `|\count2=\skip1|' will
set |\count2| equal to~65536. These rules also apply to \TeX's internal
parameters; for example, `|\dimen2=\baselineskip|' will set |\dimen2| to
the natural space component of the current baselineskip glue.
\dangerexercise Test your knowledge of \TeX's registers by stating the
results of each of the following commands when they are performed in sequence:
\begintt
\count1=50 \dimen2=\count1pt \divide\count1 by 8
\skip2=-10pt plus\count1fil minus\dimen2
\multiply\skip2 by-\count1 \divide\skip2 by \dimen2 \count6=\skip2
\skip1=.5\dimen2 plus\skip2 minus\count\count1fill
\multiply\skip2 by\skip1 \advance\skip1 by-\skip2
\endtt
\answer Set |\count1| to 50,
then |\dimen2| to~$50\pt$,
then |\count1| to~6,
then |\skip2| to~$-10\pt$ plus~$6\,{\rm fil}$ minus~$50\pt$,
then |\skip2| to~$60\pt$ plus~$-36\,{\rm fil}$ minus~$-300\pt$,
then |\skip2| to~$1\,{\rm sp}$ minus~$-6\,{\rm sp}$,
then |\count6| to~1,
then |\skip1| to~$25\pt$ plus~$1\,{\rm sp}$ minus~$1\,{\rm fill}$,
then |\skip2| to~$25\pt$ minus~$-150\pt$,
then |\skip1| to~$0\pt$ plus~$1\,{\rm sp}$ minus~$1\,{\rm fill}$.
\dangerexercise What is in |\skip5| after the following three commands have
acted?
\begintt
\skip5=0pt plus 1pt
\advance\skip5 by \skip4 \advance\skip5 by -\skip4
\endtt
\answer If\/ |\skip4| has infinite stretchability, |\skip5| will be zero;
otherwise it will be $0\pt$ plus~$1\pt$.
\dangerexercise (For mathematicians.) Explain how to ^{round} |\dimen2|
to the nearest multiple of\/ |\dimen3|, assuming that |\dimen3| is positive.
\answer |\advance\dimen2 by\ifnum\dimen2<0 -\fi.5\dimen3|\parbreak
|\divide\dimen2 by\dimen3 \multiply\dimen2 by\dimen3|
\danger The registers obey \TeX's ^{group structure}. For example,
changes to |\count3| inside |{...}| will not affect the value of
|\count3| outside. Therefore \TeX\ effectively has more than 256
registers of each type. If you want the effect of a register command
to transcend its group, you must say ^|\global| when you change the value.
\dangerexercise What is in |\count1| after the following sequence of commands?
\begintt
\count1=5 {\count1=2 \global\advance\count1by\count1
\advance\count1by\count1}
\endtt
\answer |\count1| takes the values 5, then~2 (the old 5 is saved),
then~4 (which is made global), then~8 (and 4~is saved); finally the value~4 is
restored, and that is the answer. \ (For further remarks, see the discussion
of\/ |\tracingrestores| in Chapter~27.)
\danger The first ten |\count| registers, |\count0| through |\count9|,
are reserved for a special purpose:
\TeX\ displays these ten counts on your terminal whenever outputting
a page, and it transmits them to the output file as an identification
of that page. The counts are separated by decimal points on your terminal,
with trailing `|.0|' patterns suppressed. Thus, for example, if
|\count0=5| and |\count2=7| when a page is being shipped out to the
^|dvi| file, and if the other count registers are zero, \TeX\ will
type `|[5.0.7]|'. Plain \TeX\ uses |\count0| for the page number, and it
keeps |\count1| through~|\count9| equal to zero; that is why you see just
`^|[1]|' when page~1 is being output. In more complex applications the
page numbers can have further structure; ten counts are shipped out so
that there will be plenty of identification.
\danger It's usually desirable to have symbolic names for registers.
\TeX\ provides a ^|\countdef| command (similar to |\chardef|,
cf.~Chapter~8), which makes it easy to do this: You just say
\begintt
\countdef\chapno=28
\endtt
and |\chapno| is henceforth an abbreviation for |\count28|. Similar
commands ^|\dimendef|, ^|\skipdef|, and ^|\muskipdef| are available
for the other types of numeric registers. After a control sequence
has been defined by |\countdef|, it can be used in \TeX\ commands exactly
as if it were an integer ^{parameter} like |\tolerance|. Similarly,
|\dimendef| effectively creates a new dimension parameter, |\skipdef|
effectively creates a new glue parameter, and |\muskipdef| effectively
creates a new muglue parameter.
\danger Besides the numerical registers, \TeX\ also has 256 box registers
called ^|\box||0| to |\box255|. A~box register gets a value when you say
^|\setbox|\<number>|=|^\<box>; for example, `|\setbox3=\hbox{A}|' sets
|\box3| to an hbox that contains the single letter~|A|. Several other
examples of\/ |\setbox| have already appeared in Chapter~12. Chapter~10
points out that `|2\wd3|' is a \<dimen> that represents twice the width
of\/ |\box3|; similarly, ^|\ht|\<number> and ^|\dp|\<number> can be used to
refer to the height and depth of a given box register. ^^|\wd|
\danger Box registers are local to groups just as arithmetic registers are.
But there's a big difference between box registers and all the rest: When
you use a |\box|, it loses its value. For example, the construction
`|\raise2pt\box3|' in a horizontal list not only puts the contents of
|\box3| into the list after raising it by~$2\pt$, it also makes |\box3|
void. \TeX\ does this for efficiency, since it is desirable to avoid copying
the contents of potentially large boxes. If you want to use a box register
without wiping out its contents, just say `^|\copy|' instead of `|\box|';
for example, `|\raise2pt\copy3|'.
\danger Another way to use a box register is to extract the inside of
an hbox by saying `^|\unhbox|'. This annihilates the contents of the
register, like `|\box|' does, and it also removes one level of boxing. For
example, the commands
\begintt
\setbox3=\hbox{A} \setbox3=\hbox{\box3 B}
\setbox4=\hbox{A} \setbox4=\hbox{\unhbox4 B}
\endtt
put |\hbox{\hbox{A}B}| into |\box3| and |\hbox{AB}| into |\box4|.
Similarly, ^|\unvbox| unwraps a vbox. If you want to construct a large
box by accretion (e.g., a ^{table of contents}), it is best to use
|\unhbox| or |\unvbox| as in the |\setbox4| example; otherwise you use
more of \TeX's memory space, and you might even obtain boxes inside boxes
nested to such a deep level that hardware or software limits are exceeded.
\danger The operations ^|\unhcopy| and ^|\unvcopy| are related to
|\unhbox| and |\unvbox| as |\copy| is to |\box|. \ (But their
names are admittedly peculiar.)
\danger An unboxing operation ``unsets'' any glue that was set at the box's
outer level. For example, consider the sequence of commands
\begintt
\setbox5=\hbox{A \hbox{B C}} \setbox6=\hbox to 1.05\wd5{\unhcopy5}
\endtt
This makes |\box6| five percent wider than |\box5|;
the glue between |A| and |\hbox{B C}| stretches to
make the difference, but the glue inside the inner hbox does not change.
\danger A box register is either ``^{void}'' or it contains an hbox or a vbox.
There is a difference between a void register and one that contains an
empty box whose height, width, and depth are zero; for example, if\/ |\box3|
is void, you can say |\unhbox3| or |\unvbox3| or |\unhcopy3| or |\unvcopy3|,
but if\/ |\box3| is equal to |\hbox{}| you can say only |\unhbox3| or
|\unhcopy3|. If you say `|\global\setbox3=|\<box>', register |\box3| will
become ``globally void'' when it is subsequently used or unboxed.
\dangerexercise What is in register |\box5| after the following commands?
\begintt
\setbox5=\hbox{A} \setbox5=\hbox{\copy5\unhbox5\box5\unhcopy5}
\endtt
\answer |\hbox{\hbox{A}A}|. After `|\unhbox5|', |\box5| is void; |\unhcopy5|
yields nothing.
\dangerexercise And what's in |\box3| after ^^{grouping with box registers}
`|{\global\setbox3=\hbox{A}\setbox3=\hbox{}}|'?\kern-1pt
\answer |\hbox{A}|. But after `|{\global\setbox3=\hbox{A}\setbox3=\box3}|',
|\box3| will be void.
\danger If you are unsure about how \TeX\ operates on its registers, you can
experiment online by using certain `|\show|' commands. For example,
\begintt
\showthe\count1 \showthe\dimen2 \showthe\skip3
\endtt
will display the contents of\/ |\count1|, |\dimen2|, and |\skip3|; and
`^|\showbox||4|' will display the contents of\/ |\box4|. ^^|\showthe|
Box contents will appear only in the log file, unless you say
`|\tracingonline=1|'. Plain \TeX\ provides a macro `^|\tracingall|'
that turns on every possible mode of interaction, including
^|\tracingonline|. The author used these features to check the answers to
several of the exercises above.
\danger Large applications of \TeX\ make use of different sets of macros
written by different groups of people. Chaos would reign if a register
like |\count100|, say, were being used simultaneously for different purposes
in different macros. Therefore plain \TeX\ provides an ^{allocation} facility;
cooperation will replace confusion if each ^{macro writer} uses these
conventions. The idea is to say, e.g., `^|\newcount|' when you want
to dedicate a |\count| register to a special purpose. For example, the
author designed a macro called `|\exercise|' to format the exercises
in this manual, and one of the features of\/ |\exercise| is that it
computes the number of the current exercise. The format
macros in Appendix~E reserve a |\count| register for this purpose by saying
\begintt
\newcount\exno
\endtt
and then the command `|\exno=0|'
is used at the beginning of each chapter. Similarly, `|\advance\exno by1|'
is used whenever a new exercise comes along, and `|\the\exno|'
is used to typeset the current exercise number. The |\newcount| operation
assigns a unique count register to its argument |\exno|, and it defines
|\exno| with a ^|\countdef| command. All of the other format macros
are written without the knowledge of exactly which |\count| register
actually corresponds to |\exno|.
\danger Besides |\newcount|, plain \TeX\ provides ^|\newdimen|,
^|\newskip|, ^|\newmuskip|, and ^|\newbox|; there also are ^|\newtoks|,
^|\newread|, ^|\newwrite|, ^|\newfam|, and ^|\newinsert|, for
features we haven't discussed yet. Appendices~B and~E contain
several examples of the proper use of allocation. In the cases of
|\newbox|, |\newread|, etc., the allocated number is defined by |\chardef|.
For example, if the command `|\newbox\abstract|' is used to define a
box register that will contain an abstract, and if the |\newbox| operation
decides to allocate |\box45| for this purpose, then it defines the
meaning of\/ |\abstract| by saying `|\chardef\abstract=45|'. \TeX\ allows
^|\chardef|'d quantities to be used as integers, so that you can say
|\box\abstract| and |\copy\abstract|, etc. \ (There is no |\boxdef|
command.)
\newcount\notenumber
\def\clearnotenumber{\notenumber=0\relax}
\def\note{\advance\notenumber by1 \footnote{$^{\the\notenumber}$}}
\clearnotenumber
\dangerexercise Design a |\note| macro that produces footnotes numbered
sequentially. For example,\note{First note.} it should produce
the footnotes here\note{Second note.} if you type
\begintt
... example,\note{First note.} it should produce
the footnotes here\note{Second note.} if ...
\endtt
(Use |\newcount| to allocate a |\count| register for the footnotes.)
\answer |\newcount\notenumber|\parbreak
|\def\clearnotenumber{\notenumber=0\relax}|\parbreak
|\def\note{\advance\notenumber by 1|\parbreak
| \footnote{$^{\the\notenumber}$}}|
\danger Sometimes, however, you want to use a register just for temporary
storage, and you know that it won't conflict with anybody else's macros.
Registers |\count255|, |\dimen255|, |\skip255|, and |\muskip255| are
traditionally kept available for such purposes. Furthermore, plain \TeX\
reserves |\dimen0| to |\dimen9|, |\skip0| to |\skip9|, |\muskip0| to
|\muskip9|, and |\box0| to |\box9| for ``scratchwork''; these registers
are never allocated by the |\new...|\null\ operations. We have seen that
|\count0| through |\count9| are special, and |\box255| also turns out to
be special; so those registers should be avoided unless you know what you
are doing.
\ddanger Of course any register can be used for short-term purposes inside
a group (including |\count0| to |\count9| and |\box255|, and including
registers that have been allocated for other purposes), since register
changes are local to ^{groups}. However, you should be sure that \TeX\
will not output any pages before the group has ended, because output
routines might otherwise be invoked at unfortunate times. \TeX\ is
^^{output routines, when invoked}^^{page builder, when exercised}
liable to invoke an output routine whenever it tries to move something
from the list of recent contributions to the current page, because it
might discover a page break with $c=\infty$ then. Here is a list of the
times when that can happen: (a)~At the beginning or end of a paragraph,
provided that this paragraph is being contributed to the main vertical
list. (b)~At the beginning or end of a displayed equation within such a
paragraph. (c)~After completing an |\halign| in vertical mode. (d)~After
contributing a box or penalty or insertion to the main vertical list.
(e)~After an |\output| routine has ended.
\ddanger Now that we are armed with the knowledge of \TeX's flexible
registers, we can plunge into the details of insertions. There are 255
classes of insertions, |\insert0| to |\insert254|, and they are tied
to other registers of the same number. For example, |\insert100| is
connected with |\count100|, |\dimen100|, |\skip100|, and |\box100|.
Therefore plain \TeX\ provides an allocation function for insertions
as it does for registers; Appendix~B includes the command
\begintt
\newinsert\footins
\endtt
^^|\newinsert|
which defines |\footins| as the number for footnote insertions. Other
commands that deal with footnotes refer to |\count\footins|, |\dimen\footins|,
and so on. The macros for floating topinserts are similarly prefaced by
`|\newinsert\topins|', which defines |\topins| as the number of their
class. Each class of insertions is independent, but \TeX\ preserves the
order of insertions within a class. It turns out that |\footins| is class~254,
and |\topins| is class~253, but the macros do not use such numbers directly.
\def\n{\thinspace$n$}
\ddanger For our purposes let's consider a particular class of insertions
called class $n$; we will then be dealing with \TeX's primitive command
^^|\insert|
\begindisplay
|\insert|\n|{|\<vertical mode material>|}|
\enddisplay
which puts an insertion
item into a horizontal or vertical list. For this class of insertions
\begindisplay
|\box|\n\ is where the material appears when a page is output;\cr
|\count|\n\ is the magnification factor for page breaking;\cr
|\dimen|\n\ is the maximum insertion size per page;\cr
|\skip|\n\ is the extra space to allocate on a page.\cr
\enddisplay
For example, material inserted with |\insert100| will
eventually appear in |\box100|.
\ddanger Let the natural height plus depth of\/ |\insert|\n\ be $x$; then
|\count|\n\ is 1000 times the factor by which $x$~affects the page
goal. For example, plain \TeX\ sets |\count\footins=1000|, since
there is a one-to-one relationship:
A 10-point footnote effectively makes a page $10\pt$ shorter. But if we
have an application where footnotes appear in double columns, a count
value of 500 would be appropriate. One of the insertion classes in Appendix~E
makes marginal notes for proofreading purposes; in that case the
count value is zero. No actual magnification is done; |\count|\n\
is simply a number used for bookkeeping, when estimating the
costs of various page breaks.
\ddanger The first footnote on a page requires extra space, since we want
to separate the footnotes from the text, and since we want to output a
horizontal rule. Plain \TeX\ sets
`|\skip\footins=|^|\bigskipamount|'; this means that a bigskip of
extra space is assumed to be added by the output routine to any page that
contains at least one insertion of class |\footins|.
\ddanger Sometimes it is desirable to put a maximum limitation on the size
of insertions; for example, people usually don't want an entire page to
consist of footnotes. Plain \TeX\ sets |\dimen\footins=8in|; this means
that |\box\footins| is not supposed to accumulate more than 8~inches of
footnotes for any one page.
\ddanger You might want to review the page-breaking algorithm explained
at the beginning of this chapter, before reading further.
On the other hand, maybe you don't really want to read the rest of
this chapter at all, ever.
\ddanger Here now is the algorithm that \TeX\ performs when an |\insert|\n\
is moved from the ``recent contributions'' to the ``current page.'' \
(Remember that such a move does not mean that the insertion will actually take
place; the current page will be backed up later, to the breakpoint of
least cost, and only the insertions preceding that breakpoint will
actually be performed.) \ Let $g$ and $t$ be the current ^|\pagegoal| and
^|\pagetotal|; let $q$ be the ^|\insertpenalties| accumulated for the
current page; and let $d$ and $z$ be the current ^|\pagedepth| and
^|\pageshrink|. \ (The value of~$d$ is at most ^|\maxdepth|; this value
has not yet been incorporated into $t$.) \ Finally, let $x$ be the
natural height plus depth of the |\insert|\n\ that we are moving to the
current page; and let $f$ be the corresponding magnification factor,
i.e., |\count|\n\ divided by 1000.
\ninepoint
\textindent{\bf Step 1.} If there is no previous |\insert|\n\ on the
current page, decrease $g$ by $hf+w$, where $h$ is the current height plus
depth of\/ |\box|\n, and where $w$ is the natural space component of
|\skip|\n; also include the stretch and shrink components of
|\skip|\n\ in the totals for the current page (in particular, this
affects~$z$).
\medbreak
\textindent{\bf Step 2.} If a previous |\insert|\n\ on the current page
has been split, add the parameter called
^|\floatingpenalty| to~$q$, and omit Steps 3 and~4.
\medbreak
\textindent{\bf Step 3.} Test if the current insertion will fit on the page
without splitting. This means that it won't make the natural height-plus-depth
of\/ |\box|\n\ surpass |\dimen|\n, when it is added to |\box|\n\ together with
all previous |\insert|\n\ amounts on the current page; furthermore, it
means that either $xf\le0$ or $t+d+xf-z\le g$. If both tests are passed,
subtract $xf$ from~$g$ and omit Step~4.
\medbreak
\textindent{\bf Step 4.} (The current insertion will be split, at least
tentatively; but the split will not actually take place if the least-cost
page turns out to have occurred earlier than the present insertion.) \
First compute the largest amount~$v$ such that a height plus depth of $v$
will not make the total insertions into |\box|\n\ bigger than |\dimen|\n,
and such that $t+d+vf\le g$. \ (Notice that $z$ is omitted from the latter
formula, but the available shrinkability was considered in Step~3 when we
tried to avoid splitting.) \ Then find the least-cost way to split the
beginning of the vertical list of the insertion so as to obtain a box of
height~$v$. \ (Use an algorithm just like page-breaking, but without the
complexity of insertion; an additional `|\penalty-10000|' item is assumed
to be present at the end of the vertical list, to ensure that a legal
breakpoint exists.) \ Let $u$ be the natural height plus depth of that
least-cost box, and let $r$ be the penalty associated with the optimum
breakpoint. Decrease $g$ by~$uf$, and increase $q$ by~$r$. \ (If
|\tracingpages||=1|, the log file should now get a cryptic message that says
`|% split|\n\ |to| $v$|,|$u$ |p=|$r$'. For~example,^^{split insertion penalty}
\begintt
% split254 to 180.2,175.3 p=100
\endtt
^^{percent split}
means that \TeX\ has tried to split an |\insert254| to height $180.2\pt$;
the natural height-plus-depth of the best such split is $175.3\pt$,
and the penalty for breaking there is~100.)
\ddanger This algorithm is admittedly complicated, but no simpler mechanism
seems to do nearly as much. Notice that penalties of $-10000$ inside
insertions will make certain splits very attractive in Step~4, so the
user can provide hints about where to break, in difficult situations.
The algorithm provides a variety of different behaviors:
Floating insertions can be accommodated
as a special case of split insertions, by making each floating topinsert
start with a small penalty, and by having
zero as the associated |\floatingpenalty|; non-floating insertions
like footnotes are accommodated by associating larger penalties with
split insertions (see Appendix~B).
\ddanger The splitting operation mentioned in Step 4 is also available as
a primitive: `^|\vsplit|\<number> |to|\<dimen>' produces a vbox obtained
by splitting off a speci\-fied amount of material from a box register.
For example,
\begintt
\setbox200=\vsplit100 to 50pt
\endtt
sets |\box200| to a vbox whose height is $50\pt$; it goes through the
vertical list inside |\box100| (which should be a vbox) and finds the
least-cost break assuming a goal height of~$50\pt$, considering badnesses and
penalties just as in the case of page-breaking (but with $q=0$). The
algorithm uses ^|\splitmaxdepth| instead of\/ ^|\maxdepth| to govern the
maximum depth of boxes. Then it prunes the top of\/ |\box100| by removing
everything up to and including any ^{discardable} items that immediately
follow the optimum breakpoint; and it uses ^|\splittopskip| to insert new
glue before the first box inside |\box100|, just as
^|\topskip| glue appears at the top of a page. However, if the optimum
breakpoint occurs at the end of the vertical list inside |\box100|---a
`|\penalty-10000|' item is assumed to be present there---or if all items
after the optimum breakpoint are discarded, |\box100| will
be void after the |\vsplit|. And if\/ |\box100| was void before the
|\vsplit|, both |\box100| and |\box200| will be void afterwards.
\ddanger \looseness=-1
You'd better not change |\box|\n, |\count|\n, |\dimen|\n,
or |\skip|\n\ while \TeX\ is contributing insertions to the current page,
since \TeX's algorithm assumes that those quantities are static.
But you can change ^|\floatingpenalty|, |\splittopskip|, and |\splitmaxdepth|;
\TeX\ will use the values that were current just inside the closing right
brace of `|\insert|\n|{...}|' when it splits and floats insertions.
For example, Appendix~B uses |\floatingpenalty=20000| in footnote
insertions, to discourage footnotes that split before others
can start, \vadjust{\break}but |\floatingpenalty=0| in floating topinserts.
Appendix~B also uses special values of\/ |\splittopskip| and |\splitmaxdepth|,
together with ^{struts}, so that split footnotes will be typeset with
the same spacing as unsplit ones.
\ddanger The |\footnote| macro puts an |\insert| into the horizontal list
of a paragraph. After the paragraph has been broken into lines, this
insertion will move out into the vertical list just after the line that
contained it (see Chapter~14). Since there is no legal breakpoint between
that box (i.e., that line) and the insertion, \TeX\ will put the insertion
onto the page that contains the line that contains the insertion.
\ddangerexercise Study the page-breaking algorithm carefully. Is it
possible that a footnote might not appear on the same page as its reference?
\answer Yes, in severe circumstances. (1)~Previous footnotes might
have left no room for any more footnotes on the page.
(2)~If |\vadjust{\eject}| occurs on the same line as a footnote, before that
footnote, the reference will be forcibly detached. (3)~Other |\vadjust|
commands on that line could also interpose breakpoints before the insertion.
\ddanger \looseness=-1
When the best page break is finally chosen, \TeX\ removes everything after
the chosen breakpoint from the bottom of the ``current page,'' and puts it
all back at the top of the ``recent contributions.'' The
chosen breakpoint itself is placed at the very top of the recent contributions.
If it is a penalty item, the value of the penalty is recorded in
^|\outputpenalty| and the penalty in the contribution list is changed
to $10000$; otherwise |\outputpenalty| is set to 10000.
The insertions that remain on the current page are of three kinds: For
each class~$n$ there are unsplit insertions, followed possibly by
a single split insertion, followed possibly by others. If
^|\holdinginserts|$\null>0$, all insertions remain in place (so that
they might be contributed again); otherwise they are all removed from
the current page list as follows: The unsplit insertions
are appended to |\box|\n, with no ^{interline glue} between them. \
(^{Struts} should be used, as in the |\vfootnote| macro of
Appendix~B\null.) \
If a split insertion is present, it is effectively |\vsplit| to the size
that was computed previously in Step~4; the top part is treated as an
unsplit insertion, and the remainder (if any) is converted to an insertion
as if it had not been split. This remainder, followed by any other floating
insertions of the same class, is held
over in a separate place. \ (They will show up on the ``current page'' if
^|\showlists| is used while an ^|\output| routine is active; the total
number of such insertions appears in ^|\insertpenalties| during an
|\output| routine.) %\
Finally, the remaining items before the best break on the current page are put
together in a |\vbox| of height~$g$, where $g$ was the |\pagegoal| at the
time of the break, using the saved value of\/ ^|\maxdepth|; this box becomes
|\box255|. Now the user's ^|\output| ^^|\box255|
routine enters \TeX's scanner (see Chapter~23); its duty is to assemble the
final pages based on the contents of\/ |\box255| and any insertion boxes
that it knows about. The output routine will probably unbox those boxes,
so that their glue can be reset; the glue in insertion boxes usually
cooperates nicely with the glue on the rest of the page, when it is
given a chance. After the |\output| routine is finished, ^{held-over
insertion} items are placed first on the list of recent contributions, followed
by the vertical list constructed by |\output|, followed by the recent
contributions beginning with the page break. \ (Deep breath.) \ You got that?
\endchapter
Since it is impossible to foresee how [footnotes] will happen to come out
in the make-up, it is impracticable to number them from 1 up on each page.
The best way is to number them consecutively throughout an article
or by chapters in a book.
\author UNIVERSITY OF ^{CHICAGO} PRESS, {\sl Manual of Style\/} (1910) % p102
\bigskip
Don't use footnotes in your books, Don.
\author JILL ^{KNUTH} (1962)
\eject
\beginchapter Chapter 16. Typing\\Math Formulas
\TeX\ is designed to handle complex ^{mathematical expressions} in such a way
that most of them are easy to input. The basic idea is that a complicated
formula is composed of less complicated ^{formulas} put together in a
simple way; the less complicated formulas are, in turn, made up of simple
combinations of formulas that are even less complicated; and so on. Stating
this another way, if you know how to type simple formulas and how to combine
formulas into larger ones, you will be able to handle virtually any formula
at all. So let's start with simple ones and work our way up.
The simplest formula is a single letter, like `$x$', or a single number,
like `2'. In order to put these into a \TeX\ text, you type `|$x$|' and
`|$2$|', respectively. Notice that all mathematical formulas are enclosed
in special math brackets; we are using |$| as the math bracket in this
manual, in accord with the plain \TeX\ format defined in Appendix~B\null,
because mathematics is supposedly expensive.
When you type `|$x$|' the `$x$' comes out in italics, but when you
type `|$2$|' the `$2$' comes out in roman type. In general, all
characters on your keyboard have a special interpretation in math
formulas, according to the normal conventions of mathematics printing:
Letters now denote ^{italic} letters, while digits and punctuation
denote ^{roman} digits and punctuation; a hyphen ({\tt-}) now denotes a ^{minus
sign} ($-$), which is almost the same as an em-dash but not quite (see
Chapter~2). The first |$| that you type puts you into ``^{math mode}'' and the
second takes you out (see Chapter~13). So if you forget one |$| or type
one |$| too many, \TeX\ will probably become thoroughly confused and you
will probably get some sort of error message. ^^{dollarsign}
Formulas that have been typeset by a printer who is unaccustomed to
mathematics usually look quite strange to a mathematician, because a
novice printer usually gets the spacing all wrong. In order to alleviate
this problem, \TeX\ does most of its own spacing in math formulas; and it
{\sl ignores\/} any ^{spaces} that you yourself put between |$|'s. For
example, if you type `|$ x$|' and `|$ 2 $|', they will mean the same thing
as `|$x$|' and `|$2$|'. You can type `\hbox{|$(x + y)/(x - y)$|}' or
`|$(x+y) / (x-y)$|', but both will result in `$(x+y)/(x-y)$', a formula in
which there is a bit of extra space surrounding the $+$ and~$-$ signs but
none around the~/~sign. Thus, you do not have to memorize the complicated
rules of math spacing, and you are free to use blank spaces in any way you
like. Of course, spaces are still used in the normal way to mark the end
of control sequences, as explained in Chapter~3. In most circumstances
\TeX's spacing will be what a mathematician is accustomed to; but we will
see in Chapter~18 that there are control sequences by which you can
override \TeX's spacing rules if you want to.
One of the things mathematicians like to do is make their formulas look
like ^{Greek} to the uninitiated. In plain \TeX\ language you can type
`|$$\alpha, \beta, \gamma, \delta;$$|' and you will get the first four
Greek letters ^^|\alpha|^^|\beta|^^|\gamma|^^|\delta|
$$\alpha,\beta,\gamma,\delta;$$ furthermore there are uppercase Greek
letters like `$\Gamma$', which you can get by typing `|$\Gamma$|'.
^^|\Gamma| Don't feel intimidated if you aren't already familiar with Greek
letters; they will be easy to learn if you need them. The only difficulty
is that some symbols that look nearly the same must be carefully
distinguished. For example, the Greek letters ^|\nu|~($\nu$) and
^|\kappa|~($\kappa$) should not be confused with the italic letters $v$
and~$x$; the Greek ^|\phi|~($\phi$) is different from the slashed zero
called ^|\emptyset|~($\emptyset$). A~lowercase epsilon ($\epsilon$) is
quite different from the symbol used to denote membership in a set
($\in$); type `|$\epsilon$|' for $\epsilon$ and `|$\in$|' for $\in$.
^^|\epsilon| ^^|\in| Some of the lowercase Greek letters have variant
forms in plain \TeX's math italic fonts:
`|$(\phi,\theta,\epsilon,\rho)$|' yields
`$(\phi,\theta,\epsilon,\rho)$' while
`|$(\varphi,\vartheta,\varepsilon,\varrho)$|' yields
`$(\varphi,\vartheta,\varepsilon,\varrho)$'.
^^|\phi|^^|\theta|^^|\rho|^^|\varphi|^^|\vartheta|^^|\varrho|^^|\varepsilon|
Besides Greek letters, there are a lot of ^{funny symbols} like `$\approx$'
(which you get by typing `|$\approx$|') ^^|\approx|^^{special symbols for math}
and `$\mapsto$' (which you ^^|\mapsto|^^{math symbols}
get by typing `|$\mapsto$|').
A complete list of these control sequences
and the characters they correspond to appears in Appendix~F\null.
Such control sequences are allowed only in math mode, i.e., between
|$|'s, because the corresponding symbols appear in the math fonts.
\exercise What should you type to get the formula
`$\gamma+\nu\in\Gamma$'\thinspace?
\answer |$\gamma+\nu\in\Gamma$|.
\exercise Look at Appendix F to discover the control sequences for
`$\le$', `$\ge$', and~`$\ne$'. \ (These are probably the three most
commonly used math symbols that are not present on your keyboard.)
\ What does plain \TeX\ call them?
\answer ^|\le|, ^|\ge|, and ^|\ne|. \ (These are short for ``less-or-equal,''
``greater-or-equal,'' and ``not-equal.'') \ You can also use the names
^|\leq|, ^|\geq|, and ^|\neq|. \ (The fourth most common symbol is, perhaps,
`$\infty$', which stands for ``^{infinity}'' and is called `^|\infty|'.)
Now let's see how the more complex formulas get built up from simple ones.
In the first place, you can get ^{superscripts} $\rm^{(up\,high)}$ and
^{subscripts} $\rm_{(down\,low)}$
^^{indices, see subscripts}
^^{superiors, see superscripts} ^^{inferiors, see subscripts}
by using `|^|' and `|_|', as shown in the following examples:
\beginmathdemo
\it Input&\it Output\cr
\noalign{\vskip2pt}
|$x^2$|&x^2\cr
|$x_2$|&x_2\cr
|$2^x$|&2^x\cr
|$x^2y^2$|&x^2y^2\cr
|$x ^ 2y ^ 2$|&x ^ 2y ^ 2\cr
|$x_2y_2$|&x_2y_2\cr
|$_2F_3$|&_2F_3\cr
\endmathdemo
Notice that |^| and |_| apply only to the next single character.
If you want several things to be superscripted or subscripted, just enclose
them in braces:
\beginmathdemo
|$x^{2y}$|&x^{2y}\cr
|$2^{2^x}$|&2^{2^x}\cr
|$2^{2^{2^x}}$|&2^{2^{2^x}}\cr
|$y_{x_2}$|&y_{x_2}\cr
|$y_{x^2}$|&y_{x^2}\cr
\endmathdemo
The braces in these examples have been used to specify ``^{subformulas},''
i.e., simpler parts of a larger formula. \TeX\ makes a box for each
subformula, and treats that box as if it were a single symbol. Braces
also serve their usual purpose of grouping, as discussed in Chapter~5.
It is illegal to type `|x^y^z|' or `|x_y_z|'; \TeX\ will
complain of a ``double superscript'' or ``double subscript.'' You must type
`|x^{y^z}|' or `|x^{yz}|' or
`|x_{y_z}|' or `|x_{yz}|' in order to make your intention clear.
A superscript or subscript following a character applies to that character
only; but when following a subformula it applies to that whole subformula,
and it will be raised or lowered accordingly. For example,
\beginmathdemo
|$((x^2)^3)^4$|&((x^2)^3)^4\cr
|${({(x^2)}^3)}^4$|&{({(x^2)}^3)}^4\cr
\endmathdemo
In the first formula the `|^3|' and `|^4|' are superscripts on the ^{right
parentheses}, i.e., on the `|)|' characters that immediately precede them,
but in the second formula they are superscripts on the subformulas that
are enclosed in braces. The first alternative is preferable, because it is
much easier to type and it is just as easy to read.
\danger A subscript or superscript following nothing (as in the `|_2F_3|'
example on the preceding page, where the `|_2|' follows nothing) is taken
to mean a subscript or superscript of an empty subformula. Such notations
are (fortunately) rare in mathematics; but if you do encounter them it is
better to make your intention clear by showing the empty subformula
explicitly with braces. In other words, the best way to get `${}_2F_3$'
in a formula is to type `|{}_2F_3|' or `|{_2}F_3|' or `|{_2F_3}|'.
\dangerexercise What difference, if any, is there between the output of
`|$x + _2F_3$|' and the output of `|$x + {}_2F_3$|'\thinspace?
\answer In the former, the `|_2|' applies to the plus sign ($x + _2F_3$);
but in the latter, it applies to an empty subformula ($x + {}_2F_3$).
\dangerexercise Describe the differences between the outputs of `|${x^y}^z$|'
and `|$x^{y^z}$|'.
\answer The results are `${x^y}^z$' and `$x^{y^z}$'; the $z$ in the first
alternative is the same size as the $y$, but in the second it is smaller.
Furthermore, the $y$ and $z$ in the first case aren't quite at the same height.
\ (Good typists never even think of the first construction, because
mathematicians never want it.)
You can have simultaneous subscripts and superscripts, and you can specify them
in any order:
\beginmathdemo
|$x^2_3$|&x^2_3\cr
|$x_3^2$|&x_3^2\cr
|$x^{31415}_{92}+\pi$|&x^{31415}_{92}+\pi\cr
\noalign{\smallskip}
|$x_{y^a_b}^{z_c^d}$|&x_{y^a_b}^{z_c^d}\cr
\endmathdemo
Notice that simultaneous su$\rm_b^{per\kern-1pt}$scripts are positioned
over each other. However, a subscript will be ``tucked in'' slightly when it
follows certain letters; for example, `|$P_2^2$|' produces `$P_2^2$'.
If for some reason you want the left edges of both subscript and superscript
to be aligned, you can fool \TeX\ by inserting a null subformula:
`|$P{}_2^2$|' produces `$P{}_2^2$'.
The control sequence ^|\prime| stands for the symbol `$\prime$', which
is used mostly in superscripts. In fact, `$\prime$' is so big as it stands
that you would never want to use it except in a subscript or superscript,
where it occurs in a smaller size. Here are some typical examples:
\beginmathdemo
\it Input&\it Output\cr
\noalign{\vskip2pt}
|$y_1^\prime$|&y_1^\prime\cr
|$y_2^{\prime\prime}$|&y_2^{\prime\prime}\cr
|$y_3^{\prime\prime\prime}$|&y_3^{\prime\prime\prime}\cr
\endmathdemo
Since single and double primes occur rather frequently, plain \TeX\
provides a convenient abbreviation: You can simply type |'| instead
of |^\prime|, and |''| instead of |^{\prime\prime}|, and so on.
\beginmathdemo
|$f'[g(x)]g'(x)$|&f'[g(x)]g'(x)\cr
|$y_1'+y_2''$|&y_1'+y_2''\cr
|$y'_1+y''_2$|&y'_1+y''_2\cr
|$y'''_3+g'^2$|&y'''_3+g'{}^2\cr
\endmathdemo
\dangerexercise Why do you think \TeX\ treats |\prime| as a large symbol
that appears only in superscripts, instead of making it a smaller
symbol that has already been shifted up into the superscript position?
\answer The second alternative doesn't work properly when there's a
subscript at the same time as a prime. Furthermore, some mathematicians
use |\prime| also in the subscript position; they write, for example,
$F'(w,z)=\partial F(w,z)/\partial z$ and $F_\prime(w,z)=\partial F(w,z)/
\partial w$.
\dangerexercise Mathematicians sometimes use ``^{tensor notation}''
in which subscripts and superscripts are staggered, as in `$R_i{}^{jk}{}_l$'.
Explain how to achieve such an effect.
\answer |$R_i{}^{jk}{}_l$|.
Another way to get complex formulas from simple ones is to use the control
sequences ^|\sqrt|, ^|\underline|, or ^|\overline|.
^^{surds, see sqrt} ^^{vinculum, see overline}
Like |^| and |_|, these operations apply to the character or subformula
that follows them:
\beginmathdemo
|$\sqrt2$|&\sqrt2\cr
|$\sqrt{x+2}$|&\sqrt{x+2}\cr
|$\underline4$|&\underline4\cr
|$\overline{x+y}$|&\overline{x+y}\cr
|$\overline x+\overline y$|&\overline x+\overline y\cr
|$x^{\underline n}$|&x^{\underline n}\cr
|$x^{\overline{m+n}}$|&x^{\overline{m+n}}\cr
|$\sqrt{x^3+\sqrt\alpha}$|&\sqrt{x^3+\sqrt\alpha}\cr
\endmathdemo
You can also get cube roots `$\root3\of{\phantom{h}}$' and similar things
by using ^|\root|:
\beginmathdemo
|$\root 3 \of 2$|&\root 3 \of 2\cr
|$\root n \of {x^n+y^n}$|&\root n \of {x^n+y^n}\cr
|$\root n+1 \of a$|&\root n+1 \of a\cr
\endmathdemo
\danger The |\sqrt| and |\underline| and |\overline| operations are able to
place lines above or below subformulas of any size or shape; the bar lines
change their size and position, so that they are long enough to cover the
subformula, and high enough or low enough not to bump into it. For example,
consider `|\overline|~|l|' ($\,\overline l\,$) versus `|\overline|~|m|'
($\,\overline m\,$): The first has a shorter bar line, and this line has
been raised higher than the bar in the second. Similarly, the bar in
`|\underline|~|y|' ($\,\underline y\,$) is lower than the bar in
`|\underline|~|x|' ($\,\underline x\,$); and square root signs appear in a
variety of positions based on the height and depth of what is being
|\sqrt|'d: $\sqrt a + \sqrt d + \sqrt y$. \TeX\ knows the height, depth, and
width of every letter and every subformula, because it considers them to be
boxes, as explained in Chapter~11. If you have a formula in which there is
only one |\sqrt|, or only one |\overline| or |\underline|, the normal
positioning rules work fine; but sometimes you want to have uniformity
between different members of a complex formula. For example, you might want to
typeset `$\sqrt{\mathstrut a}+\sqrt{\mathstrut d}+\sqrt{\mathstrut y}$',
putting all square roots in the same vertical position. There's an easy way
to do this, using the control sequence ^|\mathstrut| as follows:
\begintt
$\sqrt{\mathstrut a}+\sqrt{\mathstrut d}+\sqrt{\mathstrut y}$.
\endtt
A |\mathstrut| is an invisible box whose width is zero; its height and depth
are the height and depth of a parenthesis `('. Therefore subformulas
that contain |\mathstrut| will always have the same height and depth,
unless they involve more complicated constructions like subscripts and
superscripts. Chapter~18 discusses more powerful operations called ^|\smash|
and ^|\phantom| by which you can obtain complete control over the positioning
of roots and similar signs.
\exercise Test your understanding of what you have read so far in this chapter
by explaining what should be typed to get the following formulas. \ (Be
sure to check your answer with Appendix~A to confirm that you're right.)
$$\hbox to\hsize{\indent
$\displaystyle
10^{10}\hfil 2^{n+1}\hfil (n+1)^2\hfil \sqrt{1-x^2}\hfil
\overline{w+\overline z}\hfil p_1^{e_1}\hfil a_{b_{c_{d_e}}}\hfil
\root3\of{h''_n(\alpha x)}\hfil$}$$
\answer |10^{10}|; \stretch|2^{n+1}|; \stretch|(n+1)^2|; \stretch|\sqrt{1-x^2}|;
\stretch|\overline{w+\overline z}|; \stretch|p_1^{e_1}|; \stretch
|a_{b_{c_{d_e}}}|; \stretch|\root3\of{h''_n(\alpha x)}|.
\ (Of course, you should enclose these formulas in dollar signs so that
\TeX\ will process them in math mode. Superscripts and subscripts can be
given in either order; for example, |h''_n| and |h_n''| both work the
same. You should not leave out any of the braces shown here; for example,
`|$10^10$|' would yield `$10^10$'. But it doesn't hurt to insert
additional braces around letters or numbers, as in `|({n}+{1})^{2}|'. The
indicated blank spaces are necessary unless you use extra braces;
otherwise \TeX\ will complain about undefined control sequences
|\overlinez| and |\alphax|.)
\exercise What mistake did B. C. ^{Dull} discover after he typed the
following?
\begintt
If$ x = y$, then $x$ is equal to $y.$
\endtt
\answer He got `If$ x = y\ldots$' because he forgot to leave a space
after `|If|'; ^{spaces} disappear between dollar signs. He should
also have ended the sentence with `|$y$.|'; punctuation that belongs
to a sentence should not be included in a formula, as we will see
in Chapter~18. \ (But you aren't expected to know that yet.)
\exercise Explain how to type the following sentence:
\begindisplay
Deleting an element from an $n$-tuple leaves an $(n-1)$-tuple.
\enddisplay
\answer |Deleting an element from an $n$-tuple leaves an $(n-1)$-tuple.|
\exercise List all the italic letters that descend below the baseline.
\ (These are the letters for which |\underline| will lower its bar line.)
\answer $Q,f,g,j,p,q,y$. \ (The analogous ^{Greek} letters are
^^{italic letters with descenders} ^^{descenders}
$\beta,\gamma,\zeta,\eta,\mu,\xi,\rho,\phi,\varphi,\chi,\psi$.)
We have discussed the fact that the characters you type have special meanings
in math mode, but the examples so far are incomplete; they don't reveal all
the power that is at your fingertips just after you press the `|$|' key.
It's time now to go back to basics: Let us make a systematic survey of
what each character does, when it is used in a formula.
The 52 ^{letters} (|A| to |Z| and |a| to |z|) denote italic symbols
($A$~to~$Z$ and $a$~to~$z$), which a mathematician would call ``^{variables}.''
\TeX\ just calls them ``^{ordinary symbols},'' because they make up the
bulk of math formulas. There are two variants of lowercase L in plain \TeX,
namely `$l$' (which you get by simply typing `|l|') and `$\ell$'
(which you get by typing `^|\ell|'). Although mathematicians commonly
write something that looks like `$\ell$' in their manuscripts, they
do so only to distinguish it from the numeral~`1'. This
distinguishability problem is not present in printed mathematics, since an
italic `$l$' is quite different from a~`1'; therefore it is traditional to
use `$l$' unless `$\ell$' has been specifically requested.
Plain \TeX\ also treats the 18 characters
\begintt
0 1 2 3 4 5 6 7 8 9 ! ? . || / ` @ "
\endtt
^^{digits}^^{numerals}
as ordinary symbols; i.e., it doesn't insert any extra space when these
symbols occur next to each other or next to letters. Unlike the letters,
these 18 characters remain in roman type when they appear in formulas.
There's nothing special for you to remember about them, except that the
^{vertical line} `\|' has special uses that we shall discuss later.
Furthermore, you should be careful to distinguish between `oh' and `zero':
The italic letter~$O$ is almost never used in formulas unless it appears
just before a left parenthesis, as in `$O(n)$'; ^^{big-$O$ notation}
and the numeral~$0$ is almost never used just before a left parenthesis
unless it is preceded by another digit, as in `$10(n-1)$'. Watch for
left parentheses and you'll be $0K$. \ (Lowercase o's also tend to
appear only before left parentheses; type `|x_0|' instead of `|x_o|',
since the formula `$x_0$' is generally more correct than `$x_o$'.)
The three characters |+|, |-|, and |*|
are called ``^{binary operations},'' because they operate on two parts of
a formula. For example, |+|~is a ^{plus sign}, which is used for the sum
of two numbers; |-|~is a ^{minus sign}. The ^{asterisk}~(|*|) is rarer
^^{star, see asterisk}
in mathematics, but it also behaves as a binary operation. Here are
some examples of how \TeX\ typesets binary operations when they appear
next to ordinary symbols:
\beginmathdemo
\it Input&\it Output\cr
\noalign{\vskip2pt}
|$x+y-z$|&x+y-z\cr
|$x+y*z$|&x+y*z\cr
|$x*y/z$|&x*y/z\cr
\endmathdemo
Notice that |-| and |*| produce quite different math symbols from what you
get in normal text: The ^{hyphen}~(-) becomes a minus sign~($-$), and
the raised asterisk~(*) drops down to a lower level~($*$).
\danger \TeX\ does not treat |/| as a binary operation, even though a
^{slash} stands for division (which qualifies as a binary operation on
mathematical grounds). The reason is that printers traditionally put extra
space around the symbols $+$, $-$, and~$*$, but not around~$/$. If \TeX\ were to
typeset |/| as a binary operation, the formula `|$1/2$|' would come out
`$1\mathbin/2$', which is wrong; so \TeX\ considers |/| to be an ordinary
symbol.
\danger Appendix F lists many more binary operations, for which you type
control sequences instead of single characters. Here are some examples:
\beginmathdemo
|$x\times y\cdot z$|&x\times y\cdot z\cr
|$x\circ y\bullet z$|&x\circ y\bullet z\cr
|$x\cup y\cap z$|&x\cup y\cap z\cr
|$x\sqcup y\sqcap z$|&x\sqcup y\sqcap z\cr
|$x\vee y\wedge z$|&x\vee y\wedge z\cr
|$x\pm y\mp z$|&x\pm y\mp z\cr
\endmathdemo
It is important to distinguish $\times$ (^|\times|) from $X$ (|X|)
and from $x$ (|x|); to distinguish $\cup$ (^|\cup|) from $U$ (|U|)
and from $u$ (|u|); to distinguish $\vee$ (^|\vee|) from $V$ (|V|)
and from $v$ (|v|); to distinguish $\circ$ (^|\circ|) from $O$ (|O|)
and from $o$ (|o|). ^^|\cdot|^^|\bullet|^^|\cap|^^|\sqcup|^^|\sqcap|^^|\wedge|
^^{cross, see dagger, times}
^^|\pm|^^|\mp| The symbols `$\lor$' and `$\land$' can also be called
^|\lor| and ^|\land|, since they frequently stand for binary operations
that are called ``^{logical~or}'' and ``^{logical~and}.''
\danger Incidentally, binary operations are treated as ordinary symbols
if they don't occur between two quantities that they can operate on.
For example, no extra space is inserted next to the $+$, $-$, and~$*$
in cases like the following:
\beginmathdemo
|$x=+1$|&x=+1\cr
|$3.142-$|&3.142-\cr
|$(D*)$|&(D*)\cr
\endmathdemo
Consider also the following examples, which show that binary
operations can be used as ordinary symbols in superscripts and subscripts:
\beginmathdemo
|$K_n^+,K_n^-$|&K_n^+,K_n^-\cr
|$z^*_{ij}$|&z^*_{ij}\cr
|$g^\circ \mapsto g^\bullet$|&g^\circ \mapsto g^\bullet\cr
|$f^*(x) \cap f_*(y)$|&f^*(x) \cap f_*(y)\cr
\endmathdemo
\dangerexercise How would you obtain the formulas `$z^{*2}$'
and `$h_*'(z)$'\thinspace?
\answer |$z^{*2}$| and |$h_*'(z)$|.
Plain \TeX\ treats the four characters |=|, |<|, |>|, and |:|\ as
``^{relations}'' because they express a relationship between two
quantities. For example, `${x<y}$' means that $x$~is less than~$y$.
Such relationships have a rather different meaning from binary
operations like $+$, and the symbols are typeset somewhat differently:
\beginmathdemo
|$x=y>z$|&x=y>z\cr
|$x:=y$|&x:=y\cr
|$x\le y\ne z$|&x\le y\ne z\cr
|$x\sim y\simeq z$|&x\sim y\simeq z\cr
|$x\equiv y\not\equiv z$|&x\equiv y\not\equiv z\cr
|$x\subset y\subseteq z$|&x\subset y\subseteq z\cr
\endmathdemo
^^|\le|^^|\ne|^^|\simeq|^^{colon}^^{equals}^^{lessthan}^^{greaterthan}
^^{colonequals}^^|\equiv|^^|\not|^^|\subset|^^|\subseteq|^^|\sim|
^^{hooks, see subset, supset} ^^{wiggle, see sim}
(The last several examples show some of the many other
relational symbols that plain \TeX\ makes available via control sequences;
see Appendix~F.)
The two characters `|,|' (^{comma}) and `|;|' (^{semicolon}) are treated
as ^{punctuation marks in formulas}; this means that \TeX\ puts a little
extra space after them, but not before them.
\beginmathdemo
|$f(x,y;z)$|&f(x,y;z)\cr
\endmathdemo
It isn't customary to put extra space after a `|.|'\ (^{period}) in
math formulas, so \TeX\ treats a period as an ordinary symbol.
If you want the `|:|'\ character to be treated as a punctuation mark
instead of as a relation, just call it ^|\colon|:
\beginmathdemo
|$f:A\to B$|&f:A\to B\cr
|$f\colon A\to B$|&f\colon A\to B\cr
\endmathdemo
^^|\to|
If you want to use a comma as an ordinary symbol (e.g., when it
appears in a large number), just put it in braces; \TeX\ treats
anything in braces as an ordinary symbol. For instance,
\beginmathdemo
|$12,345x$|&12,345x\qquad\rm(wrong)\cr
|$12{,}345x$|&12{,}345x\qquad\,\rm(right)\cr
\endmathdemo
\dangerexercise What's an easy way to get a raised dot in a decimal
constant (e.g., `$3{\cdot}1416$')?
\answer |$3{\cdot}1416$|. \ (One of the earlier examples in this
chapter showed that ^|\cdot| is a binary operation; putting it in braces
makes it act like an ordinary symbol.)\par
If you have lots of constants like this, for example in a table, there's a way
to make ordinary periods act like |\cdot| symbols: Just define
^|\mathcode||`.| to be |"0201|, assuming that the fonts of plain \TeX\ are
being used. However, this could be dangerous, since ordinary
periods are used frequently in displayed equations; the |\mathcode| change
should be confined to places where every period is to be a |\cdot|.
So far we have considered letters, other ordinary symbols, binary operations,
relations, and punctuation marks; hence we have covered almost every key on
the typewriter. There are just a few more: The characters `|(|' ^^{lparen}
and `|[|' ^^{lbracket} are called ``^{openings},'' while `|)|' ^^{rparen} and
^^{fences, see opening, closing, delimiters}
`|]|' ^^{rbracket} are called ``^{closings}''; these act pretty much like
ordinary symbols, but they help \TeX\ to decide when a binary operation is
not really being used in a binary way. Then there is the character~|'|,
which we know is used as an abbreviation for |\prime| superscripts.
Finally, we know that plain \TeX\ reserves the other ten characters:
\begintt
\ $ % # & ~ { } _ ^
\endtt
These are not usable for symbols in math mode unless their ^|\catcode|
values are changed (see Chapter~7). Although |{| and |}| specify
grouping, the control sequences `|\{|' and `|\}|' ^^|\lbrace|^^|\rbrace| can
be used to get `$\{$' as an opening and `$\}$' as a closing.
\ddanger All of these math mode interpretations are easily changeable, since
each character has a ^|\mathcode|, as explained in Chapter~17; none of
the conventions are permanently built into \TeX\null. However, most of them are
so standard that it is usually unwise to make many changes, except perhaps
in the interpretations of |`|, |"|, and |@|. ^^{at sign} ^^{leftquote}
^^{doublequote}
The special characters |^| and |_| that designate superscripts
^^{circumflex}^^{underbar}
and subscripts should not be used except in formulas. Similarly,
the names of math symbols like |\alpha| and |\approx|, and the
control sequences for math operations like |\overline|, must not
invade ordinary text. \TeX\ uses these facts to detect ^{missing dollar
signs} in your input, before such mistakes cause too much trouble. For
example, suppose you were to type
\begintt
The smallest $n such that $2^n>1000$ is~10.
\endtt
\TeX\ doesn't know that you forgot a `|$|' after the first `|n|', because
it doesn't understand English; so it finds a ``formula'' between the
first two |$| signs:
\begindisplay
The smallest $n such that $
\enddisplay
after which it thinks that `|2|' is part of the text. But then the |^|
reveals an inconsistency; \TeX\ will automatically insert a~|$| before
the~|^|, and you will get an error message. In this way the computer has
gotten back into synch, and the rest of the document can be typeset as if
nothing had happened.
\danger Conversely, a blank line or ^|\par| is not permitted in math mode.
This gives \TeX\ another way to recover from a missing~|$|; such
errors will be confined to the paragraph in which they occur.
\danger If for some reason you cannot use |^| and |_| for superscripts
and subscripts, because you have an unusual keyboard or because you need
|^| for French accents or something, plain \TeX\ lets you type ^|\sp|
and ^|\sb| instead. For example, `|$x\sp2$|' is another way to get `$x\sp2$'.
On the other hand, some people are lucky enough to have keyboards that
contain additional symbols besides those of standard ASCII.
^^{character set}
When such symbols are available, \TeX\ can be set up to make math
typing a bit more pleasant. For example, at the author's installation there
are keys labeled \up\ and~\dn\ that produce visible symbols
(these make superscripts and subscripts look much nicer
on the screen); there are keys for the relations {\tentex\char'34},
{\tentex\char'35}, and {\tentex\char'32} (these save time); and there are
about two dozen more keys that occasionally come in handy. \ (See Appendix~C.)
^^{uparrow}^^{downarrow}^^{leq}^^{geq}^^{neq}
\danger Mathematicians are fond of using ^{accents} over letters, because
this is often an effective way to indicate relationships between
mathematical objects, and because it greatly extends the number of available
symbols without increasing the number of necessary fonts.
Chapter~9 discusses the use of accents in ordinary text, but mathematical
accents are somewhat different, because spacing is not the same; \TeX\ uses
special conventions for accents in formulas, so that the two sorts of
accents will not be confused with each other. The following math accents
are provided by plain~\TeX:
\beginmathdemo
|$\hat a$|&\hat a\cr
|$\check a$|&\check a\cr
|$\tilde a$|&\tilde a\cr
|$\acute a$|&\acute a\cr
|$\grave a$|&\grave a\cr
|$\dot a$|&\dot a\cr
|$\ddot a$|&\ddot a\cr
|$\breve a$|&\breve a\cr
|$\bar a$|&\bar a\cr
|$\vec a$|&\vec a\cr
\endmathdemo
^^|\hat|^^|\check|^^|\tilde|^^|\acute|^^|\grave|
^^|\dot|^^|\ddot|^^|\breve|^^|\bar|^^|\vec|
The first nine of these are called |\^|, |\v|, |\~|, |\'|, |\`|, |\.|,
|\"|, |\u|, and |\=|, respectively, when they appear in text; |\vec| is an
accent that appears only in formulas. \TeX\ will complain if you try to
use |\^| or |\v|, etc., in formulas, or if you try to use |\hat| or
|\check|, etc., in ordinary text.
\danger It's usually a good idea to define special control sequences for
accented letters that you need frequently. For example, you can put
^^|\def|
\begintt
\def\Ahat{{\hat A}}
\def\chat{{\hat c}}
\def\scheck{{\check s}}
\def\xtilde{{\tilde x}}
\def\zbar{{\bar z}}
\endtt
at the beginning of a manuscript that uses the symbols $\hat A$, $\hat c$,
$\check s$, $\tilde x$, and $\bar z$ more than, say, five times. This
saves you a lot of keystrokes, and it makes the manuscript easier to
read. Chapter~20 explains how to define control sequences.
\danger When the letters $i$ and $j$ are accented in math formulas, ^{dotless}
symbols $\imath$ and $\jmath$ should be used under the accents. These
symbols are called ^|\imath| and ^|\jmath| in plain \TeX. Thus, for example,
a paper that uses `$\hat\imath$' and `$\hat\jmath$' ought to begin with
the following definitions:
\begintt
\def\ihat{{\hat\imath}}
\def\jhat{{\hat\jmath}}
\endtt
\def\Ahat{{\hat A}}
\danger You can put ^{accents on top of accents}, making symbols like
$\skew6\hat\Ahat$ that might cause a mathematician to
squeal with ecstasy. However, it takes a bit of finesse to get the
upper accent into a position that looks right, because the designer of a
font for mathematics usually tells \TeX\ to position math accents in
special ways for special letters. Plain \TeX\ provides a control sequence
called ^|\skew| that makes it fairly easy to shift superaccents into their
proper place. For example, `|\skew6\hat\Ahat|' was used to produce the
symbol above. The number `|6|' in this example was chosen by trial and
error; `|5|'~seems to put the upper accent a bit too far left, while
`|7|'~makes it a bit too far right, at least in the author's opinion.
The idea is to fiddle with the amount of skew until you find what
pleases you best.
\danger It's possible, in fact, to put math accents on any subformula, not
just on single characters or accented characters. But there's usually not
much point in doing so, because \TeX\ just centers the accent over the
whole subformula. For example, `|$\hat{I+M}$|' yields `$\hat{I+M}$'. In
particular, a |\bar| accent always stays the same size; it's not like
^|\overline|, which grows with the formula under it. Some people prefer
the longer line from |\overline| even when it applies to only a single
letter; for example, `|$\bar z+\overline z$|' produces `$\bar z+\overline
z$', and you can take your pick when you define |\zbar|. However, plain
\TeX\ does provide two accents that grow; they are called ^|\widehat| and
^|\widetilde|:
\beginmathdemo
|$\widehat x,\widetilde x$|&\tenmath\widehat x,\widetilde x\cr
|$\widehat{xy},\widetilde{xy}$|&\tenmath\widehat{xy},\widetilde{xy}\cr
|$\widehat{xyz},\widetilde{xyz}$|&\tenmath\widehat{xyz},\widetilde{xyz}\cr
\endmathdemo
The third example here shows the maximum size available.
\def\ghat{{\hat g}}
\exercise This has been another long chapter; but cheer up, you have learned
a lot! Prove it by explaining what to type in order to get the formulas
$e^{-x^2}$, $D\sim p^\alpha M+l$, and $\ghat\in(H^{\pi_1^{-1}})'$. \ (In
the last example, assume that a control sequence |\ghat| has already been
defined, so that |\ghat| produces the accented letter $\ghat$.)
\answer |$e^{-x^2}$|, |$D\sim p^\alpha M+l$|, and |$\ghat\in(H^{\pi_1^{-1}})'$|.
\ (If you are reading the dangerous bend sections, you know that the
recommended way to define |\ghat| is `|\def\ghat{{\hat g}}|'.)
\endchapter
Producing ^{Greek} letters is as easy as $\pi$.
You just type |... as easy as $\pi$.|
\author LESLIE ^{LAMPORT}, {\sl The ^{L\kern-.2em\raise.6ex\hbox{a}%
\kern-.1em\TeX} Document Preparation System\/} (1983)
% Note: the final manual has a slightly different wording on p43.
% It's now called "LaTeX: A Document Preparation System" (1986)
% But I decided to cite the original, partly because I have
% no smallcaps sans-serif `A' to match the new LaTeX logo!
\bigskip
\TeX\ has no regard for the glories of the Greek tongue---\/
as far as it is concerned, Greek letters are just additional weird symbols,
and they are allowed\/ {\rm only} in math mode.
In a pinch you can get the output $\tau\epsilon\chi$ by typing %
|$\tau\epsilon\chi$|,
but if you're actually setting Greek text, you will be using
a different version of \TeX, designed for a keyboard with Greek letters on it,
and you shouldn't even be reading this manual,
which is undoubtedly all English to you.
\author MICHAEL ^{SPIVAK}, {\sl The Joy of \TeX\/} (1982)
\eject
\beginchapter Chapter 17. More about Math
Another thing mathematicians like to do is make fractions---and they
like to build symbols up on top of each other in a variety of different ways:
\begindisplay
$\displaystyle
{1\over2}\qquad{\rm and}\qquad{n+1\over3}\qquad{\rm and}\qquad
{n+1\choose3}\qquad{\rm and}\qquad\sum_{n=1}^3 Z_n^2\,.$
\enddisplay
You can get these four formulas as displayed equations by typing
`|$$1\over2$$|' and
`|$$n+1\over3$$|' and
`|$$n+1\choose3$$|' and
`|$$\sum_{n=1}^3 Z_n^2$$|';
we shall study the simple rules for such constructions in this chapter.
^^|\sum|^^|\choose|
First let's look at ^{fractions}, which use the `^|\over|' notation. The
control sequence |\over| applies to everything in the formula unless you
use braces to enclose it in a specific subformula; in the latter
^^{stacked fractions, see over}
case, |\over| applies to everything in that subformula.
\begindisplaymathdemo
\it Input&\it Output\cr
\noalign{\vskip-3pt}
|$$x+y^2\over k+1$$|&x+y^2\over k+1\cr
\noalign{\vskip2pt}
|$${x+y^2\over k}+1$$|&{x+y^2\over k}+1\cr
\noalign{\vskip-1pt}
|$$x+{y^2\over k}+1$$|&x+{y^2\over k}+1\cr
\noalign{\vskip-1pt}
|$$x+{y^2\over k+1}$$|&x+{y^2\over k+1}\cr
\noalign{\vskip-3pt}
|$$x+y^{2\over k+1}$$|&x+y^{2\over k+1}\cr
\endmathdemo
You aren't allowed to use |\over| twice in the same subformula; instead of
typing something like
`|a \over b \over 2|', you must specify what goes over what:
\begindisplaymathdemo
\noalign{\vskip3pt}
|$${a\over b}\over 2$$|&{a\over b}\over 2\cr
|$$a\over{b\over 2}$$|&a\over{b\over 2}\cr
\endmathdemo
Unfortunately, both of these alternatives look pretty awful. Mathematicians
tend to ``overuse'' |\over| when they first begin to typeset their own work
on a system like \TeX. A good typist or copy editor will convert fractions
to a ``^{slashed form},'' whenever a built-up construction would be too
small or too crowded. For example, the last two cases should be treated
as follows:
\begindisplaymathdemo
\noalign{\vskip3pt}
|$$a/b \over 2$$|&a/b \over 2\cr
|$$a \over b/2$$|&a \over b/2\cr
\endmathdemo
Conversion to slashed form takes a little bit of mathematical knowhow, since
^{parentheses} sometimes need to be inserted in order to preserve the meaning
of the formula. Besides substituting `|/|' for~`|\over|', the two parts
of the fraction should be put in parentheses unless they are single
symbols; for example, $a\over b$~becomes simply~$a/b$, but
$a+1\over b$ becomes $(a+1)/b$, and $a+1\over b+1$ becomes
${(a+1)/(b+1)}$. Furthermore, the entire fraction should generally
be enclosed in parentheses if it appears next to something else;
for example, ${a\over b}x$ becomes $(a/b)x$. If you are a typist without
mathematical training, it's best to ask the author of the manuscript
for help, in doubtful cases; you might also tactfully suggest that
unsightly fractions be avoided altogether in future manuscripts.
\exercise What's a better way to render the formula $x+y^{2\over k+1}$?
\answer $x+y^{2/(k+1)}$\quad(|$x+y^{2/(k+1)}$|).
\exercise Convert `${a+1\over b+1}x$' to slashed form.
\answer $((a+1)/(b+1))x$\quad(|$((a+1)/(b+1))x$|).
\exercise What surprise did B. L. ^{User} get when he typed `|$$x = (y^2\over
k+1)$$|'\thinspace?
\answer He got the displayed formula$$x=(y^2\over k+1)$$ because he forgot
that an unconfined |\over| applies to everything. \ (He should probably
have typed `|$$x=\left(y^2\over k+1\right)$$|', using ideas that will be
presented later in this chapter; this not only makes the parentheses
larger, it keeps the `$x=$' out of the fraction, because |\left| and
|\right| introduce subformulas.)
\def\cents{\hbox{\rm\rlap/c}}
\exercise How can you make `$7{1\over2}\cents$'? \ (Assume that
the control sequence |\cents| yields~`$\cents$'.)^^{money}^^{cents}
\answer `|$7{1\over2}\cents$|' or `|7$1\over2$\cents|'. \ (Incidentally,
the definition used here was |\def\cents{\hbox{\rm\rlap/c}}|.)
^^|\rlap|^^|\cents|
The examples above show that letters and other symbols sometimes get
smaller when they appear in fractions, just as they get smaller when they
are used as exponents. It's about time that we studied \TeX's method for
choosing the sizes of things. \TeX\ actually has eight different
^{styles} in which it can treat formulas, namely
$$\halign{\indent#\hfil\quad&#\hfil\cr
display style&(for formulas displayed on lines by themselves)\cr
text style&(for formulas embedded in the text)\cr
script style&(for formulas used as superscripts or subscripts)\cr
scriptscript style&(for second-order superscripts or subscripts)\cr}$$
^^{display style}^^{text style}^^{script style}^^{scriptscript style}
and four other ``^{cramped}'' styles that are almost the same except that
exponents aren't raised quite so much. For brevity we shall refer to the
eight styles as
\begindisplay
$\displaystyle D,\ D',\ T,\ T',\ S,\ S',\ \SS,\ \SS',$
\enddisplay
where $D$ is display style, $D'$ is cramped display style, $T$~is text style,
etc. \TeX\ also uses three different ^{sizes of type for mathematics};
they are called ^{text size}, ^{script size}, and ^{scriptscript size}.
The normal way to typeset a formula with \TeX\ is to enclose it in dollar
signs |$|$\,\ldots\,$|$|; this yields the formula in text style
(style~$T$). Or you can enclose it in double dollar signs |$$|$\,\ldots\,$|$$|;
this displays the formula in display style (style~$D$). The subformulas of
a formula might, of course, be in different styles. Once you know
the style, you can determine the size of type that \TeX\ will use:
$$\everycr{\noalign{\penalty10000}}
\halign{\indent#\hfil\qquad&#\hfil&\quad#\llap(like this)\hfil\cr
If a letter is in style&then it will be set in\cr
\noalign{\vskip 2pt}
$D,D',T,T'$&text size&\cr
$S,S'$&script size&\sevenrm\cr
$\SS,\SS'$&scriptscript size&\fiverm\cr}$$
There is no ``$\it SSS$'' style or ``scriptscriptscript'' size; such tiny
symbols would be even less readable than the scriptscript ones. Therefore
\TeX\ stays with scriptscript size as the minimum:
$$\halign{\indent\hbox to 1.3in{#\hfil}&\hbox to 1.2in{#\hfil}&#\hfil\cr
In a formula&the superscript&and the subscript\cr
of style&style is&style is\cr
\noalign{\vskip 2pt}
$D,T$&$S$&$S'$\cr
$D',T'$&$S'$&$S'$\cr
$S,\SS$&$\SS$&$\SS'$\cr
$S',\SS'$&$\SS'$&$\SS'$\cr}$$
For example, if |x^{a_b}| is to be typeset in style $D$, then |a_b| will
be set in style~$S$, and {\tt b}~in style~$\SS'$; the result is
`$\displaystyle x^{a_b}$'.
So far we haven't seen any difference between styles $D$ and $T$. Actually
there is a slight difference in the positioning of exponents, although
script size is used in each case: You get
$\displaystyle x^2$~in $D$~style and $x^2$~in $T$~style and \vbox to 0pt{
\vss\hbox{$\displaystyle{\atop x^2}$}\kern0pt}~in $D'$ or $T'$~style---do
you see the difference? But there is a big distinction between $D$ style and
$T$ style when it comes to fractions:
$$\halign{\indent\hbox to 1.3in{#\hfil}&\hbox to 1.2in{#\hfil}&#\hfil\cr
In a formula&the style of the&and the style of the\cr
$\alpha$|\over|$\,\beta$ of style&numerator $\alpha$ is&denominator
$\beta$ is\cr
\noalign{\vskip 2pt}
$D$&$T$&$T'$\cr
$D'$&$T'$&$T'$\cr
$T$&$S$&$S'$\cr
$T'$&$S'$&$S'$\cr
$S,\SS$&$\SS$&$\SS'$\cr
$S',\SS'$&$\SS'$&$\SS'$\cr}$$
^^{numerator}^^{denominator}
Thus if you type `|$1\over2$|' (in a text) you get $1\over2$, namely style
$S$ over style~$S'$; but if you type
`|$$1\over2$$|' you get $$1\over2$$ (a displayed formula), which is style
$T$ over style $T'$.
\danger While we're at it, we might as well finish the style rules:
^|\underline| does not change the style. ^{Math accents}, and the operations
^|\sqrt| and ^|\overline|, change uncramped styles to their cramped
counterparts; for example, $D$ changes to $D'$, but $D'$ stays as it was.
\dangerexercise State the style and size of each part of the formula
$\displaystyle \sqrt{p_2^{e'}}$, assuming that the formula itself is in
style~$D$.
\answer Style $D'$ is used for the subformula $p_2^{e'}$, hence style~$S'$
is used for the superscript~$e'$ and the subscript~2, and style~$\SS'$
is used for the supersuperscript prime. The square root sign and the $p$
appear in text size; the 2 and the~$e$ appear in script size; and the
$\prime$ is in scriptscript size.
Suppose you don't like the style that \TeX\ selects by its automatic style
rules. Then you can specify the style you want by typing ^|\displaystyle|
or ^|\textstyle| or ^|\scriptstyle| or ^|\scriptscriptstyle|; the style
that you select will apply until the end of the formula or subformula, or
until you select another style. For example,
`|$$n+\scriptstyle n+\scriptscriptstyle n.$$|' produces the display
$$n+\scriptstyle n+\scriptscriptstyle n.$$
This is a rather silly example, but it does show
that the plus signs get smaller too, as the style changes. \TeX\ puts no
space around + signs in script styles.
Here's a more useful example of style changes: Sometimes you need to
typeset a ``^{continued fraction}'' made up of many other fractions,
all of which are supposed to be in display style:
$$a_0+{1\over\displaystyle a_1+
{\strut 1\over\displaystyle a_2+
{\strut 1\over\displaystyle a_3+
{\strut 1\over a_4}}}}$$
In order to get this effect, the idea is to type
\begintt
$$a_0+{1\over\displaystyle a_1+
{\strut 1\over\displaystyle a_2+
{\strut 1\over\displaystyle a_3+
{\strut 1\over a_4}}}}$$
\endtt
(The control sequence ^|\strut| has been used to make the denominators
taller; this is a refinement that will be discussed in
Chapter~18. Our concern now is with the style commands.) \
Without the appearances of\/ |\strut| and |\displaystyle| in this formula,
the result would be completely different:
$$a_0+{1\over a_1+{1\over
a_2+{1\over a_3+{1\over a_4}}}}$$
\danger These examples show that the numerator and denominator of a fraction
are generally centered with respect to each other. If you prefer to have
the numerator or denominator appear ^{flush left}, put `^|\hfill|' after
it; or if you prefer ^{flush right}, put `|\hfill|' at the left. For
example, if the first three appearances of `|1\over|' in the previous
example are replaced by `|1\hfill\over|', you get the display
$$a_0+{1\hfill\over\displaystyle a_1+
{\strut1\hfill\over\displaystyle a_2+
{\strut1\hfill\over\displaystyle a_3+
{\strut1\over a_4}}}}$$
(a format for continued fractions that many authors prefer). This works
because |\hfill| stretches at a faster rate than the glue that is
actually used internally by \TeX\ when it centers the numerators
and denominators.
\TeX\ has another operation `^|\atop|', which is like |\over| except that
it leaves out the fraction line:
\begindisplaymathdemo
|$$x\atop y+2$$|&x\atop y+2\cr
\endmathdemo
The plain \TeX\ format in Appendix B also defines `^|\choose|', which is
like |\atop| but it encloses the result in parentheses:
\begindisplaymathdemo
|$$n\choose k$$|&n\choose k\cr
\endmathdemo
It is called |\choose| because it's
a common notation for the so-called ^{binomial coefficient}
that tells how many ways there are to choose $k$~things out of $n$~things.
You can't mix |\over| and |\atop| and |\choose| with each other.
For example, `|$$n \choose k \over 2$$|' is illegal; you must use
grouping, to get either `|$${n\choose k}\over2$$|' or
`|$$n\choose{k\over2}$$|', i.e.,
\begindisplay
$\displaystyle{{n\choose k}\over2}\qquad{\rm or}\qquad {n\choose{k\over2}}.$
\enddisplay
The latter formula, incidentally, would look better as
`|$$n\choose k/2$$|' or `|$$n\choose{1\over2}k$$|', yielding
\begindisplay
$\displaystyle{n\choose k/2}\qquad{\rm or}\qquad{n\choose{1\over2}k}.$
\enddisplay
\medskip
\exercise As alternatives to $\displaystyle{{n\choose k}\over2}$,
discuss how you could obtain the two displays
\begindisplay\abovedisplayskip=0pt\belowdisplayskip=0pt
$\displaystyle
{1\over2}{n\choose k}
\qquad{\rm and}\qquad
{\displaystyle{n\choose k}\over2}.$
\enddisplay
\answer |$${1\over2}{n\choose k}$$|;
|$$\displaystyle{n\choose k}\over2$$|.
All of these braces are necessary.
\bigbreak
\exercise Explain how to specify the displayed formula
$${p \choose 2}x^2 y^{p-2} - {1 \over 1-x}{1 \over 1-x^2}.$$
\answer |$${p \choose 2} x^2 y^{p-2} - {1 \over 1-x}{1 \over 1-x^2}.$$|
\danger \TeX\ has a generalized version of\/ |\over| and |\atop| in which you
specify the exact thickness of the line rule by typing
`^|\above|\<dimen>'. For example,
\begintt
$$\displaystyle{a\over b}\above1pt\displaystyle{c\over d}$$
\endtt
will produce a ^{compound fraction} with a heavier ($1\pt$ thick) rule as
its main bar:
$${\displaystyle{a\over b}\above 1pt\displaystyle{c\over d}}.$$
This sort of thing occurs primarily in textbooks on elementary mathematics.
\goodbreak
Mathematicians often use the sign $\sum$ to stand for ``^{summation}''
and the sign $\int$ to stand for ``^{integration}.'' If you're a typist but not
a mathematician, all you need to remember is that ^|\sum| stands for
$\sum$ and ^|\int| for $\int$; these abbreviations appear in Appendix~F
together with all the other symbols, in case you forget. Symbols like
$\sum$ and $\int$ (and a few others like $\bigcup$ and $\prod$ and $\oint$
and~$\bigotimes$, all listed in Appendix~F) are called {\sl ^{large operators}},
^^{collective signs, see large operators} ^^{sigma signs, see sum}
and you type them just as you type ordinary symbols or letters. The
difference is that \TeX\ will choose a {\sl larger\/} large operator in
display style than it will in text style. For example,
$$\halign{\indent#\hfil\qquad yields\qquad&$#\hfil$\qquad&#\hfil\cr
|$\sum x_n$|&\sum x_n&($T$ style)\cr
\noalign{\vskip3pt}
|$$\sum x_n$$|&\displaystyle\sum x_n&($D$ style).\cr}$$
A displayed |\sum| usually occurs with ``^{limits},'' i.e., with
subformulas that are to appear above and below it. You type limits just
as if they were superscripts and subscripts; for example, if you want
$$\sum_{n=1}^m$$
you type either `|$$\sum_{n=1}^m$$|' or `|$$\sum^m_{n=1}$$|'. According
to the normal conventions of mathematical typesetting, \TeX\ will change
this to `$\sum_{n=1}^m$' (i.e., without limits) if it occurs in text
style rather than in display style.
Integrations are slightly different from summations, in that the superscripts
and subscripts are not set as limits even in display style:
$$\halign{\indent\hbox to2.3in{#\hfil}\hbox to.6in{yields\hfil}&
$#\hfil$\qquad&#\hfil\cr
|$\int_{-\infty}^{+\infty}$|&\int_{-\infty}^{+\infty}&($T$ style)\cr
\noalign{\vskip3pt}
|$$\int_{-\infty}^{+\infty}$$|&\displaystyle\int_{-\infty}^{+\infty}&
($D$ style).\cr}$$
\danger Some printers prefer to set limits above and below $\int$ signs;
this takes more space on the page, but it
gives a better appearance if the subformulas are complex, because it
keeps them out of the way of the rest of the formula. Similarly, limits
are occasionally desirable in text style or script style; but some
printers prefer not to set limits on displayed $\sum$ signs. You can change
\TeX's convention by simply typing `^|\limits|' or `^|\nolimits|' immediately
after the large operator.
For example,
$$\halign{\indent\hbox to2.3in{#\hfil}\hbox to.6in{yields\hfil}&
$\displaystyle{#}$\hfil\cr
|$$\int\limits_0^{\pi\over2}$$|&\int\limits_0^{\pi\over2}\cr
\noalign{\vskip 4pt}
|$$\sum\nolimits_{n=1}^m$$|&\sum\nolimits_{n=1}^m\cr}$$
\ddanger If you say `|\nolimits\limits|' (presumably because some macro
like |\int| specifies |\nolimits|, but you do want them), the last word
takes precedence. There's also a command `^|\displaylimits|' that can be
used to restore \TeX's normal conventions; i.e., the limits will be
displayed only in styles $D$ and $D'$.
\danger Sometimes you need to put two or more rows of limits under a large
operator; you can do this with `^|\atop|'. For example, if you want
the displayed formula
$$\sum_{\scriptstyle0\le i\le m\atop\scriptstyle0<j<n}P(i,j)$$
the correct way to type it is
\begintt
$$\sum_{\scriptstyle0\le i\le m\atop\scriptstyle0<j<n}P(i,j)$$
\endtt
(perhaps with a few more spaces to make it look nicer in the manuscript
file). The instruction `^|\scriptstyle|' was necessary here,
twice---otherwise the lines `$0\le i\le m$' and `$0<j<n$' would have been in
scriptscript size, which is too small. This is another instance of a rare
case where \TeX's automatic style rules need to be overruled.
\exercise How would you type the displayed formula $\displaystyle
\sum_{i=1}^p\sum_{j=1}^q\sum_{k=1}^ra_{ij}b_{jk}c_{ki}$\enspace?
\answer |$$\sum_{i=1}^p\sum_{j=1}^q\sum_{k=1}^ra_{ij}b_{jk}c_{ki}$$|.
\dangerexercise And how would you handle $\displaystyle
\sum_{{\scriptstyle1\le i\le p\atop\scriptstyle1\le j\le q}
\atop\scriptstyle1\le k\le r}a_{ij}b_{jk}c_{ki}$\enspace?
\answer |$$\sum_{{\scriptstyle 1\le i\le p \atop \scriptstyle 1\le j\le q}
\atop \scriptstyle 1\le k\le r} a_{ij} b_{jk} c_{ki}$$|.
Since mathematical formulas can get horribly large, \TeX\ has to have some
way to make ever-larger symbols. For example, if you type
\begintt
$$\sqrt{1+\sqrt{1+\sqrt{1+
\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+x}}}}}}}$$
\endtt
the result shows a variety of available ^{square-root signs}:
\begindisplay
$\displaystyle\sqrt{1+\sqrt{1+\sqrt{1+
\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+x}}}}}}}$
\enddisplay
The three largest signs here are all essentially the same, except for a
vertical segment `\vbox{\hbox{\tenex\char'165}\vss}' that gets repeated as
often as necessary to reach the desired size; but the smaller signs are
distinct characters found in \TeX's math fonts.
A similar thing happens with parentheses and other so-called
``^{delimiter}'' symbols. For example, here are some of the different sizes of
^^{fences, see delimiters}
^{parentheses} and ^{braces} that plain \TeX\ might use in formulas:
\begindisplay
$\displaystyle
\left(\vbox to 27pt{}\left(\vbox to 24pt{}\left(\vbox to 21pt{}
\Biggl(\biggl(\Bigl(\bigl(({\scriptstyle({\scriptscriptstyle(\hskip3pt
)})})\bigr)\Bigr)\biggr)\Biggr)\right)\right)\right)
\left\{\vbox to 27pt{}\left\{\vbox to 24pt{}\left\{\vbox to 21pt{}
\Biggl\{\biggl\{\Bigl\{\bigl\{\{{\scriptstyle\{{\scriptscriptstyle\{\hskip3pt
\}}\}}\}\bigr\}\Bigr\}\biggr\}\Biggr\}\right\}\right\}\right\}$
\enddisplay
The three largest pairs in each case are made with repeatable extensions,
so they can become as large as necessary.
^^{pieces of symbols}
Delimiters are important to mathematicians, because they provide good
visual clues to the underlying structure of complex expressions; they delimit
the boundaries of individual subformulas. Here is a list of the 22~basic
delimiters provided by plain \TeX:
\begindisplay
\it Input&\it Delimiter\cr
\noalign{\vskip2pt}
|(|&left parenthesis: $($\cr
|)|&right parenthesis: $)$\cr
|[| or ^|\lbrack|&left bracket: $[$\cr
|]| or ^|\rbrack|&right bracket: $]$\cr
|\{| or ^|\lbrace|&left curly brace: $\{$\cr
|\}| or ^|\rbrace|&right curly brace: $\}$\cr
^|\lfloor|&left floor bracket: $\lfloor$\cr
^|\rfloor|&right floor bracket: $\rfloor$\cr
^|\lceil|&left ceiling bracket: $\lceil$\cr
^|\rceil|&right ceiling bracket: $\rceil$\cr
^|\langle|&left angle bracket: $\langle$\cr
^|\rangle|&right angle bracket: $\rangle$\cr
|/|&slash: $/$\cr
^|\backslash|&reverse slash: $\backslash$\cr
\| or ^|\vert|&vertical bar: $\vert$\cr
|\|\| or ^|\Vert|&double vertical bar: $\Vert$\cr
^|\uparrow|&upward arrow: $\uparrow$\cr
^|\Uparrow|&double upward arrow: $\Uparrow$\cr
^|\downarrow|&downward arrow: $\downarrow$\cr
^|\Downarrow|&double downward arrow: $\Downarrow$\cr
^|\updownarrow|&up-and-down arrow: $\updownarrow$\cr
^|\Updownarrow|&double up-and-down arrow: $\Updownarrow$\cr
\enddisplay
^^{bent bars, see langle, rangle} ^^{curly braces, see lbrace, rbrace}
^^{leftbracket}^^{rightbracket}^^{leftbrace}^^{rightbrace}^^{/}
In some cases, there are two ways to get the same delimiter; for example,
you can specify a left bracket by typing either `|[|' or `|\lbrack|'. The
latter alternative has been provided because the symbol `|[|' is not
readily available on all computer keyboards. Remember, however,
that you should never try to specify a left brace or right brace simply by
typing `|{|' or `|}|'; the |{| and |}| symbols are reserved for grouping.
The right way is to type `|\{|' or `|\}|' or `|\lbrace|' or `|\rbrace|'.
In order to get a slightly larger version of any of these symbols, just
precede them by `^|\bigl|' (for opening delimiters) or `^|\bigr|' (for
closing ones). This makes it easier to read formulas that contain
delimiters inside delimiters:
\beginlongmathdemo
\it Input&\it Output\cr
\noalign{\vskip2pt}
|$\bigl(x-s(x)\bigr)\bigl(y-s(y)\bigr)$|&
\bigl(x-s(x)\bigr)\bigl(y-s(y)\bigr)\cr
|$\bigl[x-s[x]\bigr]\bigl[y-s[y]\bigr]$|&
\bigl[x-s[x]\bigr]\bigl[y-s[y]\bigr]\cr
|$\bigl|\|| |\||x|\||-|\||y|\|| \bigr|\||$|&
\bigl\vert\vert x\vert-\vert y\vert\bigr\vert\cr
|$\bigl\lfloor\sqrt A\bigr\rfloor$|&
\bigl\lfloor\sqrt A\bigr\rfloor\cr
\endmathdemo
The |\big| delimiters are just enough bigger than ordinary ones so that
the difference can be perceived, yet small enough to be used in the text
of a paragraph. Here are all~22 of them, in the ordinary size and in
the |\big| size:
\begindisplay
$(\,)\,[\,]\,\{\,\}\,\lfloor\,\rfloor\,\lceil\,\rceil\,\langle\,\rangle
\,/\,\backslash\,\vert\,\Vert\,\uparrow\,\Uparrow\,\downarrow\,\Downarrow
\,\updownarrow\,\Updownarrow$\cr
\noalign{\smallskip}
$\bigl(\,\bigr)\,\bigl[\,\bigr]\,\bigl\{\,\bigr\}\,\bigl\lfloor
\,\bigr\rfloor\,\bigl\lceil\,\bigr\rceil\,\bigl\langle\,\bigr\rangle
\,\big/\,\big\backslash\,\big\vert\,\big\Vert\,\bigm\uparrow\,\bigm\Uparrow
\,\bigm\downarrow\,\bigm\Downarrow\,\bigm\updownarrow\,\bigm\Updownarrow$\cr
\enddisplay
You can also type ^|\Bigl| and ^|\Bigr| to get larger symbols suitable for
displays:
\begindisplay
$\Bigl(\,\Bigr)\,\Bigl[\,\Bigr]\,\Bigl\{\,\Bigr\}\,\Bigl\lfloor
\,\Bigr\rfloor\,\Bigl\lceil\,\Bigr\rceil\,\Bigl\langle\,\Bigr\rangle
\,\Big/\,\Big\backslash\,\Big\vert\,\Big\Vert\,\Bigm\uparrow\,\Bigm\Uparrow
\,\Bigm\downarrow\,\Bigm\Downarrow\,\Bigm\updownarrow\,\Bigm\Updownarrow$
\enddisplay
These are 50\% taller than their |\big| counterparts. Displayed formulas
most often use delimiters that are even taller (twice the size of\/ |\big|);
such delimiters are constructed by ^|\biggl| and ^|\biggr|, and they
look like this:
\begindisplay
$\biggl(\,\biggr)\,\biggl[\,\biggr]\,\biggl\{\,\biggr\}\,\biggl\lfloor
\,\biggr\rfloor\,\biggl\lceil\,\biggr\rceil\,\biggl\langle\,\biggr\rangle
\,\bigg/\,\bigg\backslash\,\bigg\vert\,\bigg\Vert\,\biggm\uparrow
\,\biggm\Uparrow\,\biggm\downarrow\,\biggm\Downarrow\,\biggm\updownarrow
\,\biggm\Updownarrow$
\enddisplay
Finally, there are ^|\Biggl| and ^|\Biggr| versions, 2.5 times as tall
as the |\bigl| and |\bigr| delimiters:
\begindisplay
$\Biggl(\,\Biggr)\,\Biggl[\,\Biggr]\,\Biggl\{\,\Biggr\}\,\Biggl\lfloor
\,\Biggr\rfloor\,\Biggl\lceil\,\Biggr\rceil\,\Biggl\langle\,\Biggr\rangle
\,\Bigg/\,\Bigg\backslash\,\Bigg\vert\,\Bigg\Vert\,\Biggm\uparrow
\,\Biggm\Uparrow\,\Biggm\downarrow\,\Biggm\Downarrow\,\Biggm\updownarrow
\,\Biggm\Updownarrow$
\enddisplay
\medskip
\exercise Guess how to type the formula $\displaystyle
\biggl({\partial^2\over\partial x^2}+{\partial^2\over\partial y^2}
\biggr)\bigl\vert\varphi(x+iy)\bigr\vert^2=0$, in display style,
using |\bigg| delimiters for the large parentheses. \ (The symbols $\partial$
and $\varphi$ that appear here are called ^|\partial| and ^|\varphi|.)
\answer |$\displaystyle\biggl({\partial^2\over\partial x^2}+|\hfil\break
|{\partial^2\over\partial y^2}\biggr)\bigl|\||\varphi(x+iy)\bigr|\||^2=0$|.
\dangerexercise In practice, |\big| and |\bigg| delimiters are used much
more often than |\Big| and |\Bigg| ones. Why do you think this is true?
\answer Formulas that are more than one line tall are usually two lines tall,
not 1$1\over2$ or 2$1\over2$ lines tall.
\danger A |\bigl| or |\Bigl| or |\biggl| or |\Biggl| delimiter is an
^{opening}, like a left parenthesis;
a |\bigr| or |\Bigr| or |\biggr| or |\Biggr| delimiter is a
^{closing}, like a right parenthesis. Plain \TeX\ also provides
^|\bigm| and ^|\Bigm| and ^|\biggm| and ^|\Biggm| delimiters, for use
in the middle of formulas; such a delimiter plays the r\^ole of a ^{relation},
like an equals sign, so \TeX\ puts a bit of space on either side of it.
\beginlongmathdemo
|$\bigl(x\in A(n)\bigm|\||x\in B(n)\bigr)$|&
\tenmath\bigl(x\in A(n)\bigm\vert x\in B(n)\bigr)\cr
\noalign{\vskip2pt}
|$\bigcup_n X_n\bigm\|\||\bigcap_n Y_n$|&
\tenmath\bigcup_n X_n\bigm\Vert\bigcap_n Y_n\cr
\endmathdemo
^^|\bigcup|^^|\bigcap|^^|\verticalline|^^|\in|
You can also say just ^|\big| or ^|\Big| or ^|\bigg| or ^|\Bigg|; this produces
a delimiter that acts as an ordinary variable. It is used primarily with
slashes and backslashes, as in the following example.
\beginlongmathdemo
\noalign{\vskip-2pt}
|$${a+1\over b}\bigg/{c+1\over d}$$|&
\tenmath\displaystyle{a+1\over b}\bigg/{c+1\over d}\cr
\endmathdemo
\dangerexercise What's the professional way to type
$\tenmath\bigl(x+f(x)\bigr)\big/\bigl(x-f(x)\bigr)$? \ (Look closely.)
\answer |$\bigl(x+f(x)\bigr) \big/ \bigl(x-f(x)\bigr)$|. \ Notice especially
the `|\big/|'; an ordinary ^{slash} would look too small between the
|\big| parentheses.
\TeX\ has a built-in mechanism that figures out how tall a pair of delimiters
needs to be, in order to enclose a given subformula; so you can use this
method, instead of deciding whether a delimiter should be |\big| or
|\bigg| or whatever. All you do is say
\begindisplay
^|\left|\<delim$_1$>\<subformula>^|\right|\<delim$_2$>
\enddisplay
and \TeX\ will typeset the subformula, putting the specified delimiters at
the left and the right. The size of the delimiters will be just big enough
to cover the subformula. For example, in the display
\beginlongdisplaymathdemo
|$$1+\left(1\over1-x^2\right)^3$$|&1+\left(1\over1-x^2\right)^3\cr
\endmathdemo
\TeX\ has chosen |\biggl(| and |\biggr)|, because smaller delimiters
would be too small for this particular fraction. A simple formula like
`|$\left(x\right)$|' yields just `$\left(x\right)$'; thus, |\left| and
|\right| sometimes choose delimiters that are smaller than |\bigl| and |\bigr|.
Whenever you use |\left| and |\right| they must pair up with each other,
just as braces do in groups. You can't have |\left| in one formula
and |\right| in another, nor are you allowed to type things like
`|\left(...{...\right)...}|' or
`|\left(...\begingroup...\right)...\endgroup|'.
This restriction makes sense, because \TeX\ needs to typeset the
subformula that appears between |\left| and |\right| before it can decide
how big to make the delimiters. But it is worth explicit mention here,
because you do {\sl not\/} have to match ^{parentheses} and ^{brackets}, etc.,
^^{crotchets, see brackets}
when you are not using |\left| and |\right|: \TeX\ will not complain if
you input a formula like `|$[0,1)$|' or even `|$)($|' or just `|$)$|'.\
(And it's a good thing \TeX\ doesn't, for such unbalanced formulas occur
surprisingly often in mathematics papers.) \ Even when you do use |\left|
and |\right|, \TeX\ doesn't look closely at the particular delimiters that
you happen to choose; thus, you can type strange things like `|\left)|'
and/or `|\right(|' if you know what you're doing. Or even if you don't.
The |\over| operation in the example displayed above does not involve the
`|1+|' at the beginning of the formula; this happens because |\left| and
|\right| have the function of ^{grouping}, in addition to their function
of delimiter-making. Any definitions that you happen to make between
|\left| and |\right| will be local, as if braces had appeared around the
enclosed subformula.
\exercise Use |\left| and |\right| to typeset the following display
(with ^|\phi| for $\phi$):
$$\pi(n)=\sum_{k=2}^n\left\lfloor\phi(k)\over k-1\right\rfloor.$$
\answer |$$\pi(n)=\sum_{k=2}^n\left\lfloor\phi(k)\over k-1\right\rfloor.$$|
At this point you are probably wondering why you should bother learning about
|\bigl| and |\bigr| and their relatives, when |\left| and |\right| are there
to calculate sizes for you automatically. Well, it's true that |\left|
and |\right| are quite handy, but there are at least three situations in which
you will want to use your own wisdom when selecting the proper delimiter size:
\ (1)~Sometimes |\left| and |\right| choose a smaller delimiter than you want.
For example, we used |\bigl| and |\bigr| to produce $\bigl\vert\vert x\vert-
\vert y\vert\bigr\vert$ in one of the previous illustrations; |\left| and
|\right| don't make things any bigger than necessary, so
`|$\left|\||\left|\||x\right|\||-\left|\||y\right|\||\right|\||$|'
yields only
`$\left\vert \left\vert x\right\vert -\left\vert y\right\vert \right\vert$'.
\ (2)~Sometimes |\left| and |\right| choose a larger delimiter than you want.
This happens most frequently when they enclose a large operator in a display;
for example, compare the following two formulas:
\beginlongdisplaymathdemo
\noalign{\vskip 6pt}
|$$\left( \sum_{k=1}^n A_k \right)$$|&\left( \sum_{k=1}^n A_k \right)\cr
\noalign{\vskip 3pt}
|$$\biggl( \sum_{k=1}^n A_k \biggr)$$|&\biggl( \sum_{k=1}^n A_k \biggr)\cr
\endmathdemo
The rules of\/ |\left| and |\right| cause them to enclose the ^|\sum| together
with its ^{limits}, but in special cases like this it looks better to let
the limits hang out a~bit; |\bigg| delimiters are better here.
\ (3)~Sometimes you need to break a huge displayed
formula into two or more separate lines, and you want to make sure that
its opening and closing delimiters have the same size; but you can't use
|\left| on the first line and |\right| on the last, since |\left| and
|\right| must occur in pairs. The solution is to use |\Biggl| (say) on
the first line and |\Biggr| on the last.
\danger Of course, one of the advantages of\/ |\left| and |\right| is that
they can make arbitrarily large delimiters---much bigger than |\biggggg|!
The slashes and angle brackets do have a maximum size, however; if you
ask for really big versions of those symbols you will get the largest
ones available.
\exercise Prove that you have mastered delimiters: Coerce \TeX\ into
producing the formula
$$\pi(n)=\sum_{m=2}^n\left\lfloor\biggl(\sum_{k=1}^{m-1}\bigl\lfloor
(m/k)\big/\lceil m/k\rceil\bigr\rfloor\biggr)^{-1}\right\rfloor.$$
\answer |$$\pi(n)=\sum_{m=2}^n\left\lfloor\biggl(\sum_{k=1}^{m-1}\bigl|
\hfil\break
|\lfloor(m/k)\big/\lceil m/k\rceil\bigr\rfloor\biggr)^{-1}\right\rfloor.$$|
\danger If you type `|.|'\ after |\left| or |\right|, instead of
specifying one of the basic delimiters, you get a so-called ^{null
delimiter} (which is blank). Why on earth would anybody want that, you may
ask. Well, you sometimes need to produce formulas that contain only one
large delimiter. For example, the display
$$\vert x\vert=\cases{x,&if $x\ge0$\cr
-x,&if $x<0$\cr}$$
has a `$\{$' but no `$\}$'. It can be produced by a construction of the form
\begindisplay
|$$|\||x|\||=\left\{ ... \right.$$|
\enddisplay
Chapter 18 explains how to fill in the `\hbox{|...|}' to finish this
construction; let's just notice for now that the `|\right.|'\ makes it
possible to have an invisible right delimiter to go with the visible
left brace.
\ddanger A null delimiter isn't completely void; it is an empty box
whose width is a \TeX\ parameter called ^|\nulldelimiterspace|.
We will see later that null delimiters are inserted next to fractions.
Plain \TeX\ sets |\nulldelimiterspace=1.2pt|.
You can type `|<|' or `|>|' as convenient abbreviations for ^|\langle| and
^|\rangle|, when \TeX\ is looking for a delimiter. For example,
`|\bigl<|' is equivalent to `|\bigl\langle|', and `|\right>|' is
equivalent to `|\right\rangle|'. Of course `|<|' and `|>|' ordinarily
produce the ^{less-than} and ^{greater-than} relations `${<}\,{>}$', which
are quite different from ^{angle brackets} `$\langle\,\rangle$'.
\danger Plain \TeX\ also makes available a few more delimiters, which were
not listed in the basic set of~22 because they are sort of special.
The control sequences ^|\arrowvert|, ^|\Arrowvert|, and ^|\bracevert| produce
delimiters made from the repeatable parts of the vertical arrows, double
vertical arrows, and large braces, respectively, without the arrowheads
or the curly parts of the braces. They produce results similar to
^|\vert| or ^|\Vert|, but they are surrounded by more white space and
they have a different weight. You can also use ^|\lgroup| and ^|\rgroup|,
which are constructed from braces without the middle parts; and
^|\lmoustache| and ^|\rmoustache|, ^^{moustaches}
which give you the top and bottom halves of large braces. For example,
here are the |\Big| and |\bigg| versions of\/ |\vert|, |\Vert|, and
these seven special delimiters:
$$\halign{\indent$#\hfil$\cr
\ldots\Big\vert\ldots\Big\Vert
\ldots\Big\arrowvert\ldots\Big\Arrowvert\ldots\Big\bracevert
\ldots\Big\lgroup\ldots\Big\rgroup\ldots\Big\lmoustache\ldots\Big\rmoustache
\ldots\,;\cr
\noalign{\smallskip}
\ldots\bigg\vert\ldots\bigg\Vert
\ldots\bigg\arrowvert\ldots\bigg\Arrowvert\ldots\bigg\bracevert
\ldots\bigg\lgroup\ldots\bigg\rgroup\ldots\bigg\lmoustache\ldots\bigg\rmoustache
\ldots\,.\cr}$$
Notice that |\lgroup| and |\rgroup| are rather like bold parentheses, with
sharper bends at the corners; this makes them attractive for certain large
displays. But you cannot use them exactly like parentheses, because
they are available only in large sizes (|\Big|~or~more).
\ddanger Question: What happens if a ^{subscript} or ^{superscript}
follows a large delimiter? Answer:~That's a good question. After a |\left|
delimiter, it is the first subscript or superscript of the enclosed
subformula, so it is effectively preceded by |{}|. After a |\right|
delimiter, it is a subscript or superscript of the entire |\left...\right|
subformula. And after a |\bigl| or |\bigr| or |\bigm| or |\big| delimiter,
it applies only to that particular delimiter. Thus, `|\bigl(_2|' works
quite differently from `|\left(_2|'.
\danger If you look closely at the examples of math typesetting in this
chapter, you will notice that large parentheses and brackets are
symmetric with respect to an invisible horizontal line that runs a little
bit above the ^{baseline}; when a delimiter gets larger, its height and
depth both grow by the same amount. This horizontal line is called the
{\sl^{axis}\/} of the formula; for example, a formula in the text of the
present paragraph would have an axis at this level: $\hskip 2em\over$. The
bar line in every fraction is centered on the axis, regardless of the size
of the numerator or denominator.
\danger Sometimes it is necessary to create a special box that should be
centered vertically with respect to the axis. \ (For example, the
`$\vert x\vert=\bigl\{\,\ldots$' example above was done with such a box.) \
\TeX\ provides a simple way to do this: You just say
\begindisplay
|\vcenter{|\<vertical mode material>|}|
\enddisplay
and the vertical mode material will be packed into a box just as if
^|\vcenter| had been ^|\vbox|. Then the box will be raised or lowered until
its top edge is as far above the axis as the bottom edge is below.
\ddanger The concept of ``axis'' is meaningful for \TeX\ only in math
formulas, not in ordinary text; therefore \TeX\ allows you to use
|\vcenter| only in math mode. If you really need to center something
vertically in horizontal mode, the solution is to say `|$\vcenter{...}$|'.
\ (Incidentally, the constructions `|\vcenter| |to|\<dimen>'
and `|\vcenter| |spread|\<dimen>' are legal too, in math mode;
vertical glue is always set by the rules for |\vbox| in
Chapter~12. But |\vcenter| by itself is usually sufficient.)
\danger Any box can be put into a formula by simply saying ^|\hbox| or
|\vbox| or ^|\vtop| or ^|\box| or ^|\copy| in the normal way, even when
you are in math mode. Furthermore you can use ^|\raise| or ^|\lower|, as
if you were in horizontal mode, and you can insert vertical rules with
^|\vrule|. Such constructions, like |\vcenter|, produce boxes that can be
used like ordinary symbols in math formulas.
\ddanger Sometimes you need to make up your own symbols, when you run across
something unusual that doesn't occur in the fonts. If the new symbol
occurs only in one place, you can use |\hbox| or |\vcenter| or something
to insert exactly what you want; but if you are defining a macro for
general use, you may want to use different constructions in different
styles. \TeX\ has a special feature called ^|\mathchoice| that comes
to the rescue in such situations: You write
\begindisplay
|\mathchoice{|\<math>|}{|\<math>|}{|\<math>|}{|\<math>|}|
\enddisplay
where each \<math> specifies a subformula. \TeX\ will choose the first
subformula in style $D$ or~$D'$, the second in style $T$ or~$T'$, the
third in style $S$ or~$S'$, the fourth in style $\SS$ or $\SS'$.
\ (\TeX\ actually typesets all four subformulas, before it chooses the
final one, because the actual style is not always known at the time a
|\mathchoice| is encountered; for example, when you type `|\over|' you often
change the style of everything that has occurred earlier in the formula.
Therefore |\mathchoice| is somewhat expensive in terms of time and space,
and you should use it only when you're willing to pay the price.)
\ddangerexercise Guess what output is produced by the following commands:
\begintt
\def\puzzle{{\mathchoice{D}{T}{S}{SS}}}
$$\puzzle{\puzzle\over\puzzle^{\puzzle^\puzzle}}$$
\endtt
\answer A displayed formula equivalent to |$${D}{{T}\over{T}^{{S}^{SS}}}$$|.
\ddangerexercise Devise a `^|\square|' macro that produces a
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox{\vrule width.#2pt height#1pt \kern#1pt \vrule}
\hrule height.#2pt}}}}%
`$\,\sqr34\,$' for use in math formulas. The box should be symmetrical
with respect to the axis, and its inside dimensions should be $3\pt$ in
display and text styles, $2.1\pt$ in script styles, and $1.5\pt$ in
scriptscript styles. The rules should be $0.4\pt$ thick in
display and text styles, $0.3\pt$ thick otherwise.
\answer |\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt|\parbreak
| \hbox{\vrule width.#2pt height#1pt \kern#1pt|\parbreak
| \vrule width.#2pt}|\parbreak
| \hrule height.#2pt}}}}|\parbreak
|\def\square{\mathchoice\sqr34\sqr34\sqr{2.1}3\sqr{1.5}3}|
\ddanger Plain \TeX\ has a macro called ^|\mathpalette| that is useful
for |\mathchoice| constructions; `|\mathpalette\a{xyz}|' expands to
the four-pronged array of choices
`|\mathchoice|\stretch|{\a|\stretch|\displaystyle|\stretch|{xyz}}|\stretch
|...|\stretch|{\a|\stretch|\scriptscriptstyle|\stretch|{xyz}}|\stretch'.
Thus the first argument to |\mathpalette| is a control sequence whose
first argument is a style selection. Appendix~B contains several examples
that show how |\mathpalette| can be applied. \ (See in particular the
definitions of\/ |\phantom|, |\root|, and |\smash|; the ^{congruence sign}
^|\cong| ($\cong$) is also constructed from $=$ and $\sim$ using
|\mathpalette|.)
^^{constructing new math symbols}
^^{math symbols, construction of}
\ddanger At the beginning of this chapter we discussed the commands
|\over|, |\atop|, |\choose|, and |\above|. These are special cases of
\TeX's ``^{generalized fraction}'' feature, which includes also the
three primitives
\begindisplay
|\overwithdelims|\<delim$_1$>\<delim$_2$>\cr
|\atopwithdelims|\<delim$_1$>\<delim$_2$>\cr
|\abovewithdelims|\<delim$_1$>\<delim$_2$>\<dimen>\cr
\enddisplay
The third of these is the most general, as it encompasses all of the other
generalized fractions: ^|\overwithdelims| uses a ^{fraction} bar whose
thickness is the default for the current size, and ^|\atopwithdelims| uses
an invisible fraction bar whose thickness is zero, while
^|\abovewithdelims| uses a bar whose thickness is specified explicitly.
\TeX\ places the immediately preceding subformula (the ^{numerator}) over
the immediately following subformula (the ^{denominator}), separated by a
bar line of the desired thickness; then it puts \<delim$_1$> at the left
and \<delim$_2$> at the right. For example, `^|\choose|' is equivalent to
`|\atopwithdelims()|'. If you define |\legendre| to be
`|\overwithdelims()|', you can typeset the ^{Legendre symbol}
\def\legendre{\overwithdelims()}%
`$a\legendre b$' by saying `|{a\legendre b}|'. The size of the surrounding
delimiters depends only on the style, not on the size of the fractions;
larger delimiters are used in styles $D$ and~$D'$ (see Appendix~G\null). The
simple commands ^|\over|, ^|\atop|, and ^|\above| are equivalent to the
corresponding `|withdelims|' commands when the delimiters are null; for
example, `|\over|' is an abbreviation for `|\overwithdelims..|'.
\def\euler{\atopwithdelims<>}
\ddangerexercise Define a control sequence |\euler| so that the
^{Eulerian number} $n\euler k$ will be produced when you type `|{n\euler k}|'
in a formula.
\answer|\def\euler{\atopwithdelims<>}|.
\ddanger Appendix G explains exactly how \TeX\ computes the desired size
of delimiters for |\left| and~|\right|. The general idea is that delimiters
are vertically centered with respect to the ^{axis}; hence, if we want
to cover a subformula between |\left| and |\right| that extends $y_1$~units
above the axis and $y_2$~units below, we need to make a delimiter whose
height plus depth is at least $y$~units, where $y=2\max(y_1,y_2)$.
It is usually best not to cover the formula completely, however,
but just to come close; so \TeX\ allows you to specify
two parameters, the ^|\delimiterfactor|~$f$ (an~integer) and the
^|\delimitershortfall|~$\delta$ (a~dimension). The minimum delimiter size
is taken to be at least $y\cdot f/1000$, and at least $y-\delta$. Appendix~B
sets $f=901$ and $\delta=5\pt$. Thus, if $y=30\pt$, the plain \TeX\ format
causes the delimiter to be more than $27\pt$ tall; if $y=100\pt$, the
corresponding delimiter will be at least $95\pt$ tall.
\danger So far we have been discussing the rules for typing math formulas,
but we haven't said much about how \TeX\ actually goes about converting
its input into lists of boxes and glue. Almost all of the control
sequences that have been mentioned in Chapters 16 and~17 are ``high level''
features of the plain \TeX\ format; they are not built into \TeX\ itself.
Appendix~B defines those control sequences in terms of more primitive
commands that \TeX\ actually deals with. For example, `|\choose|' is
an abbreviation for `|\atopwithdelims()|'; Appendix~B not only introduces
|\choose|, it also tells \TeX\ where to find the delimiters |(| and~|)|
in various sizes. The plain \TeX\ format defines all of the special
characters like |\alpha| and~|\mapsto|, all of the special accents like
|\tilde| and~|\widehat|, all of the large operators like |\sum| and~|\int|,
and all of the delimiters like |\lfloor| and~|\vert|. Any of these things
can be redefined, in order to adapt \TeX\ to other mathematical styles
and/or to other fonts.
\danger The remainder of this chapter discusses the low-level commands
that \TeX\ actually obeys behind the scenes. Every paragraph on the next
few pages is marked with double dangerous bends, so you should skip to
Chapter~18 unless you are a glutton for \TeX nicalities.
\ninepoint
\ddanger All characters that are typeset in math mode belong to one of
sixteen {\sl^{families} of fonts}, numbered internally from 0 to~15. Each
of these families consists of three fonts: one for text size, one for
script size, and one for scriptscript size. The commands ^|\textfont|,
^|\scriptfont|, and ^|\scriptscriptfont| are used to specify the members
of each family. For example, ^{family~0} in the plain \TeX\ format is
used for roman letters, and Appendix~B contains the instructions
\begintt
\textfont0=\tenrm
\scriptfont0=\sevenrm
\scriptscriptfont0=\fiverm
\endtt
to set up this family: The 10-point roman font (^|\tenrm|) is used for
normal symbols, 7-point roman (^|\sevenrm|) is used for subscripts, and
5-point roman (^|\fiverm|) is used for sub-subscripts. Since there are up to
256~characters per font, and 3~fonts per family, and 16~families, \TeX\ can
access up to 12,288 characters in any one formula (4096 in~each of
the three sizes). Imagine that.
\ddanger A definition like |\textfont|\<family number>|=|\<font identifier>
is local to the group that contains it, so you can easily change family
membership from one set of conventions to another and back again. Furthermore
you can put any font into any family; for example, the command
\begintt
\scriptscriptfont0=\scriptfont0
\endtt
makes sub-subscripts in family~0 the same size as the subscripts currently
are. \TeX\ doesn't check to see if the families are sensibly organized; it
just follows instructions. \ (However, fonts cannot be used in families
2 and~3 unless they contain a certain number of special parameters, as we
shall see later.) \ Incidentally, \TeX\ uses ^|\nullfont|, which contains
no characters, for each family member that has not been defined.
\ddanger During the time that a math formula is being read,
\TeX\ remembers each symbol as being ``character position so-and-so in
family number such-and-such,'' but it does not take note of what fonts
are actually in the families until reaching the end of the formula.
Thus, if you have loaded a font called |\Helvetica| that contains Swiss-style
numerals, and if you say something like
\begintt
$\textfont0=\tenrm 9 \textfont0=\Helvetica 9$
\endtt
you will get two 9's in font |\Helvetica|, assuming that \TeX\ has been
set up to take 9's from family~0. The reason is that |\textfont0|
is~|\Helvetica| at the end of the formula, and that's when it counts. On
the other hand, if you say
\begintt
$\textfont0=\tenrm 9 \hbox{$9\textfont0=\Helvetica$}$
\endtt
the first 9 will be from |\tenrm| and the second from |\Helvetica|, because
the formula in the hbox will be typeset before it is incorporated into
the surrounding formula.
\ddangerexercise If you say `|${\textfont0=\Helvetica 9}$|', what
font will be used for the~9?
\answer The |\textfont0| that was current at the beginning of the formula
will be used, because this redefinition is local to the braces. \
(It would be a different story if `^|\global||\textfont|' had appeared instead;
that would have changed the meaning of\/ |\textfont0| at all levels.)
\ddanger Every ^{math character} is given an identifying code
number between 0 and~4095, obtained by adding 256~times the family number
to the position number. This is easily expressed in ^{hexadecimal
notation}, using one hexadecimal digit for the family and two for the
character; for example, \hex{24A} stands for character~\hex{4A} in
family~2. Each character is also assigned to one of eight classes,
^^{classes of math characters, table} ^^{math codes} ^^{table of ...}
numbered 0 to~7, as follows:
$$\halign{\indent#\hfil&\quad#\hfil&\quad#\hfil&
\hskip4em#\hfil&\quad#\hfil&\quad#\hfil\cr
\it \kern-2pt Class&\it Meaning&\kern-2pt\it Example&
\it \kern-2pt Class&\it Meaning&\kern-2pt\it Example\cr
\noalign{\vskip2pt}
0&Ordinary&|/|&
4&Opening&|(|\cr
1&Large operator&|\sum|&
5&Closing&|)|\cr
2&Binary operation&|+|&
6&Punctuation&|,|\cr
3&Relation&|=|&
7&Variable family&|x|\cr
}$$
^^{large operator}^^{binary operation}^^{relation}^^{opening}^^{closing}
^^{punctuation}^^{variable family}
Classes 0 to 6 tell what ``part of speech'' the character belongs to, in
math-printing language; class~7 is a special case discussed below. The class
number is multiplied by 4096 and added to the character number, and this
is the same as making it the leading digit of a four-digit hexadecimal
number. For example, Appendix~B defines |\sum| to be the math character
\hex{1350}, meaning that it is a large operator (class~1) found in position
\hex{50} of family~3.
\ddangerexercise The ^|\oplus| and ^|\bullet| symbols ($\oplus$ and $\bullet$)
are binary operations that appear in positions 8 and~15 (decimal)
of family~2, when the fonts of plain~\TeX\ are being used. Guess
what their math character codes are. \ (This is too easy.)
\answer \hex{2208} and \hex{220F}.
\ddanger Class 7 is a special case that allows math symbols to change families.
It behaves exactly like class~0, except that the specified family is
replaced by the current value of an integer parameter called ^|\fam|,
provided that |\fam| is a legal family number (i.e., if it lies between
0 and~15). \TeX\ automatically sets |\fam=-1| whenever math mode is entered;
therefore class~7 and class~0 are equivalent unless |\fam| has been
given a new value. Plain \TeX\ changes |\fam| to~0 when the user
types `^|\rm|'; this makes it convenient to get roman letters in formulas,
as we will see in Chapter~18, since letters belong to class~7. \ (The
control sequence |\rm| is an abbreviation for `|\fam=0 \tenrm|'; thus,
|\rm| causes
|\fam| to become zero, and it makes |\tenrm| the ``^{current font}.''
In horizontal mode, the |\fam| value is irrelevant and the current font
governs the typesetting of letters; but in math mode, the current font is
irrelevant and the |\fam| value governs the letters. The current font
affects math mode only if\/ |\|\] is used ^^{control space} or if
dimensions are given in ^|ex| or ^|em| units;
it also has an effect if an |\hbox| appears inside a formula, since
the contents of an hbox are typeset in horizontal mode.)
\ddanger The interpretation of characters in math mode is defined by a
table of~256 ``mathcode'' values; these table entries can be changed
by the ^|\mathcode| command, just as the category codes are changed
by ^|\catcode| (see Chapter~7). Each mathcode specifies class, family, and
character position, as described above. For example, Appendix~B contains
the commands
\begintt
\mathcode`<="313C
\mathcode`*="2203
\endtt
which cause \TeX\ to treat the character `|<|' in math mode as a relation
^^{less than} (class~3) found in position \hex{3C} of family~1, and to treat an
^{asterisk} `|*|' as a binary operation found in position~3 of family~2.
The initial value of\/ |\mathcode`b| is \hex{7162}; thus, |b|~is character
\hex{62} in ^{family~1} (italics), and its family will vary with |\fam|.
\ (|INITEX| starts out with |\mathcode|$\,x=x$ for all characters~$x$
that are neither
letters nor digits. The ten digits have |\mathcode|$\,x=x+\hbox{\hex{7000}}$;
the 52 letters have |\mathcode|$\,x=x+\hbox{\hex{7100}}$.) \
\TeX\ looks at the mathcode only when it is typesetting a character whose
catcode is 11~(letter) or 12~(other), or when it encounters a character that
is given explicitly as ^|\char|\<number>.
\ddanger A |\mathcode| can also have the special value \hex{8000}, which
causes the character to behave as if it has catcode~13 (active). Appendix~B
uses this feature to make |'| ^^{apostrophe} expand to |^{|^|\prime||}| in a
slightly tricky way. The mathcode of |'| does not ^^{active math character}
interfere with the use of |'| in ^{octal} constants.
\ddanger The mathcode table allows you to refer indirectly to any character in
any family, with the touch of a single key. You can also specify a math
character code directly, by typing ^|\mathchar|, which is analogous to
^|\char|. For example, the command `|\mathchar"1ABC|' specifies a
character of class~1, family~10 (\hex A), and position \hex{BC}. A~hundred
or so definitions like
\begintt
\def\sum{\mathchar"1350 }
\endtt
would therefore suffice to define the special symbols of plain \TeX\null. But
there is a better way: \TeX\ has a primitive command ^|\mathchardef|,
which relates to |\mathchar| just as ^|\chardef| does to |\char|.
Appendix~B has a hundred or so definitions like
\begintt
\mathchardef\sum="1350
\endtt
to define the special symbols. A |\mathchar| must be between 0 and 32767
(\hex{7FFF}).
\ddanger A character of class~1, i.e., a ^{large operator} like |\sum|, will
be vertically centered with respect to the axis when it is typeset. Thus,
the large operators can be used with different sizes of type. This vertical
adjustment is not made for symbols of the other classes.
\ddanger \TeX\ associates classes with subformulas as well as with individual
characters. Thus, for example, you can treat a complex construction as if
it were a binary operation or a relation, etc., if you want to. The
commands ^|\mathord|, ^|\mathop|, ^|\mathbin|, ^|\mathrel|, ^|\mathopen|,
^|\mathclose|, and ^|\mathpunct| are used for this purpose; each of them
is followed either by a single character or by a subformula in braces.
For example, |\mathopen\mathchar"1234| is equivalent to |\mathchar"4234|,
because |\mathopen| forces class~4 (opening). In the formula
`|$G\mathbin:H$|', the ^{colon} is~treated as a binary operation.
And Appendix~B constructs large opening symbols by
defining ^|\bigl||#1| to be an abbreviation for
\begintt
\mathopen{\hbox{$\left#1 ...\right.$}}
\endtt
There's also an eighth classification, ^|\mathinner|, which is not
normally used for individual symbols; fractions and ^|\left||...|^|\right|
constructions are treated as ``inner'' subformulas, which means that
they will be surrounded by additional space in certain circumstances.
All other subformulas are generally treated as ordinary symbols,
whether they are formed by |\overline| or |\hbox| or |\vcenter| or
by simply being enclosed in braces. Thus, |\mathord| isn't really
a necessary part of the \TeX\ language; instead of typing
`|$1\mathord,234$|' you can get the same effect from `|$1{,}234$|'.
\ddangerexercise Commands like |\mathchardef\alpha="010B| are used in
Appendix~B to define the lowercase ^{Greek} letters. Suppose that you want
to extend plain \TeX\ by putting ^{boldface math italic} letters
in family~9, analogous to the normal math italic letters in family~1.
\ (Such fonts aren't available in stripped down versions of \TeX, but
let's assume that they exist.) \ Assume that the control sequence
|\bmit| has been defined as an abbreviation for `|\fam=9|'; hence
`|{\bmit b}|' will give a boldface math italic~|b|. What change to the
definition of\/ |\alpha| will make |{\bmit\alpha}| produce a boldface~alpha?
\checkequals\bmiexno\exno
\answer |\mathchardef\alpha="710B|. Incidentally, |{\rm\alpha}| will
then give a spurious result, because character position \hex{0B} of
roman fonts does not contain an alpha; you should warn
your users about what characters they are allowed to type under the
influence of special conventions like ^|\rm|.
\ddanger ^{Delimiters} are specified in a similar but more complicated
way. Each character has not only a~|\catcode| and a~|\mathcode| but also
a~^|\delcode|, which is either negative (for characters that should not
act as delimiters) or less than \hex{1000000}. In other words,
nonnegative delcodes consist of six hexadecimal digits. The first three
digits specify a ``small'' variant of the delimiter, and the last three
specify a ``large'' variant. For example, the command
\begintt
\delcode`x="123456
\endtt
means that if the letter |x| is used as a delimiter, its small variant
is found in position \hex{23} of family~1, and its large variant is found
in position \hex{56} of family~4. If the small or large variant is
given as |000|, however (position~0 of ^{family~0}), that variant is ignored.
\TeX\ looks at the delcode when a character follows ^|\left| or ^|\right|,
or when a character follows one of the ^|withdelims| commands; a
negative delcode leads to an error message, but otherwise \TeX\ finds
a suitable delimiter by first trying the small variant and then
the large. \ (Appendix~G discusses this process in more detail.) \
For example, Appendix~B contains the commands
\begintt
\delcode`(="028300 \delcode`.=0
\endtt
which specify that the small variant of a left parenthesis is found in
position \hex{28} of family~0, and that the large variant is in position~0
of family~3; also, a period has no variants, hence `|\left.|'\ will produce
a ^{null delimiter}. There actually are several different left parenthesis
symbols in family~3; the smallest is in position~0, and the others are
linked together by information that comes with the font. All delcodes
are~$-1$ until they are changed by a |\delcode| command.
\ddangerexercise Appendix~B defines |\delcode`<| so that there is a
shorthand notation for ^{angle brackets}. Why do you think Appendix~B
doesn't go further and define |\delcode`{|?
\answer If\/ |\delcode`{| were set to some nonnegative delimiter code, you
would get no error message when you wrote something like `|\left{|'.
This would be bad because strange effects would happen when certain
subformulas were given as arguments to macros, or when they appeared
in alignments. But it has an even worse defect, because a user who
gets away with `|\left{|' is likely to try also `|\bigl{|', which
fails miserably.
\ddanger A delimiter can also be given directly, as `^|\delimiter|\<number>'.
In this case the number can be as high as \hex{7FFFFFF}, i.e., seven
hexadecimal digits; the leading digit specifies a class, from 0 to~7,
as in a |\mathchar|. For example, Appendix~B contains the definition
\begintt
\def\langle{\delimiter"426830A }
\endtt
and this means that ^|\langle| is an opening (class 4) whose small
variant is \hex{268} and whose large variant is \hex{30A}. When |\delimiter|
appears after |\left| or |\right|, the class digit is ignored; but
when |\delimiter| occurs in other contexts, i.e., when \TeX\ isn't
looking for a delimiter, the three rightmost digits are dropped and
the remaining four digits act as a |\mathchar|. For example, the expression
`|$\langle x$|' is treated as if it were `|$\mathchar"4268 x$|'.
\ddangerexercise What goes wrong if you type
`|\bigl\delimiter"426830A|'\thinspace?
\answer Since |\bigl| is defined as a macro with one parameter,
it gets just `|\delimiter|' as the argument. You have to write
`|\bigl{\delimiter"426830A}|' to make this work. On the other hand,
|\left| will balk if the following character is a left brace. Therefore
it's best to have control sequence names for all delimiters.
\ddanger Granted that these numeric conventions for |\mathchar| and
|\delimiter| are not beautiful, they sure do pack a lot of information into
a small space. That's why \TeX\ uses them for low-level definitions inside
formats. Two other low-level primitives also deserve to be mentioned:
^|\radical| and ^|\mathaccent|. Plain \TeX\ makes ^{square root signs}
and math accents available by giving the commands
\begintt
\def\sqrt{\radical"270370 }
\def\widehat{\mathaccent"362 }
\endtt
and several more like them. The idea is that |\radical| is followed by
a delimiter code and |\mathaccent| is followed by a math character code,
so that \TeX\ knows the family and character positions for the symbols
used in radical and accent constructions. Appendix~G gives precise
information about the positioning of these characters. By changing the
definitions, \TeX\ could easily be extended so that it would typeset a
variety of different radical signs and a variety of different accent
signs, if such symbols were available in the fonts.
^^{surd signs, see radical}
\ddanger Plain \TeX\ uses ^{family~1} for math italic letters, ^{family~2} for
ordinary math symbols, and ^{family~3} for large symbols. \TeX\ insists that
^^{math fonts}
the fonts in families 2 and~3 have special ^|\fontdimen| parameters,
which govern mathematical spacing according to the rules in Appendix~G\null;
the ^|cmsy| and ^|cmex| ^{symbol fonts} have these parameters, so
their assignment to families 2 and~3 is almost mandatory. \ (There is, however,
a way to modify the parameters of any font, using the ^|\fontdimen| command.) \
^|INITEX| initializes the mathcodes of all ^{letters} |A| to~|Z| and |a| to~|z|
so that they are symbols of class~7 and family~1; that's why
it is natural to use family~1 for math italics. Similarly, the digits |0|
to~|9| are class~7 and family~0. None of the other families
is treated in any special way by \TeX. Thus, for example, plain \TeX\ puts
^{text italic} in family~4, slanted roman in family~5, bold roman in family~6,
and typewriter type in family~7, but any of these numbers could be
switched around. There is a macro ^|\newfam|, analogous to |\newbox|,
that will assign symbolic names to families that aren't already used.
\ddanger When \TeX\ is in horizontal mode, it is making a horizontal list;
in vertical mode, it is making a vertical list. Therefore it should come
as no great surprise that \TeX\ is making a ^{math list} when it is in
^{math mode}. The contents of horizontal lists were explained in Chapter~14,
and the contents of vertical lists were explained in Chapter~15; it's time
now to describe what math lists are made of. Each item in a math list
is one of the following types of things:\enddanger
\smallskip
\item\bull an ^{atom} (to be explained momentarily);
\item\bull horizontal material (a rule or discretionary or penalty or
``whatsit'');
\item\bull vertical material (from |\mark| or |\insert| or |\vadjust|);
\item\bull a glob of ^{glue} (from |\hskip| or |\mskip| or |\nonscript|);
\item\bull a ^{kern} (from |\kern| or |\mkern|);
\item\bull a ^{style change} (from |\displaystyle|, |\textstyle|, etc.);
\item\bull a ^{generalized fraction} (from |\above|, |\over|, etc.);
\item\bull a ^{boundary} (from |\left| or |\right|);
\item\bull a four-way ^{choice} (from ^|\mathchoice|).
\ddanger The most important items are called {\sl atoms}, and they have
three parts: a {\sl^{nucleus}}, a {\sl^{superscript}}, and a {\sl^{subscript}}.
For example, if you type
\begintt
(x_i+y)^{\overline{n+1}}
\endtt
in math mode, you get a math list consisting of five atoms:
$($, $x_i$, $+$, $y$, and~$)^{\overline{n+1}}$. The nuclei of these atoms
are $($, $x$, $+$, $y$, and~$)$; the subscripts are empty except for the
second atom, which has subscript~$i$; the superscripts are empty except for the
last atom, whose superscript is~$\overline{n+1}$. This superscript is
itself a math list consisting of one atom, whose nucleus is~$n+1$; and that
nucleus is a math list consisting of three atoms.
\ddanger There are thirteen kinds of atoms, each of which might act
differently in a formula; for example, `$($' is an Open atom because
^^{atomic types, table}
it comes from an opening. Here is a complete list of the different kinds:
$$\halign{\indent#\hfil&\enskip#\hfil\cr
Ord&is an ordinary atom like `$x$'\thinspace;\cr
Op&is a large operator atom like `$\sum$'\thinspace;\cr
Bin&is a binary operation atom like `$+$'\thinspace;\cr
Rel&is a relation atom like `$=$'\thinspace;\cr
Open&is an opening atom like `$($'\thinspace;\cr
Close&is a closing atom like `$)$'\thinspace;\cr
Punct&is a punctuation atom like `$,$'\thinspace;\cr
Inner&is an inner atom like `$1\over2$'\thinspace;\cr
Over&is an overline atom like `$\overline x$'\thinspace;\cr
Under&is an underline atom like `$\underline x$'\thinspace;\cr
Acc&is an accented atom like `$\hat x$'\thinspace;\cr
Rad&is a radical atom like `$\sqrt2$'\thinspace;\cr
Vcent&is a vbox to be centered, produced by |\vcenter|.\cr
}$$
\ddanger An atom's nucleus, superscript, and subscript are called its
{\sl ^{fields}}, and there are four possibilities for each of these fields.
A field can be\enddanger
\smallskip
\item\bull empty;
\item\bull a math symbol (specified by family and position number);
\item\bull a box; or
\item\bull a math list.
\smallskip\noindent
For example, the Close atom $)^{\overline{n+1}}$ considered above has an
empty subscript field; its nucleus is the symbol `$)$', which is
character~\hex{28} of family~0 if the conventions of plain \TeX\ are
in force; and its superscript field is the math list $\overline{n+1}$.
The latter math list consists of an Over atom whose nucleus
is the math list $n+1$; and that math list, in turn, consists of
three atoms of types Ord, Bin, Ord.
\ddanger You can see \TeX's view of a math list by typing ^|\showlists|
in math mode. ^^{internal list format}
For example, after `|$(x_i+y)^{\overline{n+1}}\showlists|' your log
file gets the following curious data:
\begindisplay
|\mathopen|\cr
|.\fam0 (|\cr
|\mathord|\cr
|.\fam1 x|\cr
|_\fam1 i|\cr
\noalign{\penalty-500}
|\mathbin|\cr
|.\fam0 +|\cr
\noalign{\penalty-500}
|\mathord|\cr
|.\fam1 y|\cr
\noalign{\penalty-500}
|\mathclose|\cr
|.\fam0 )|\cr
|^\overline|\cr
|^.\mathord|\cr
|^..\fam1 n|\cr
|^.\mathbin|\cr
|^..\fam0 +|\cr
|^.\mathord|\cr
|^..\fam0 1|\cr
\enddisplay
In our previous experiences with |\showlists| we observed that there can
be boxes within boxes, and that each line in the log file is
prefixed by dots to indicate its position in the hierarchy. Math lists
have a slightly more complex structure; therefore a dot is used to denote
the nucleus of an atom, a~`|^|' is used for the superscript field, and
a~`|_|' is used for the subscript field. Empty fields are not shown. Thus,
for example, the Ord atom~$x_i$ is represented here by three lines
`|\mathord|', `|.\fam1 x|', and `|_\fam1 i|'.
\ddanger Certain kinds of atoms carry additional information besides their
nucleus, subscript, and superscript fields: An Op atom will be marked
`^|\limits|' or `^|\nolimits|' if the normal ^|\displaylimits|
convention has been overridden; a Rad atom contains
a delimiter field to specify what radical sign is to be used; and an Acc atom
contains the family and character codes of the accent symbol.
\ddanger When you say ^|\hbox||{...}| in math mode, an Ord atom is placed
on the current math list, with the hbox as its nucleus. Similarly,
^|\vcenter||{...}| produces a Vcent atom whose nucleus is a box. But in
most cases the nucleus of an atom will be either a symbol or a math list.
You can experiment with |\showlists| to discover how other things like
fractions and mathchoices are represented internally.
\ddanger Chapter~26 contains complete details of how math lists are
constructed. As soon as math mode ends (i.e., when the closing `|$|'
occurs), \TeX\ dismantles the current math list and converts it into a
horizontal list. The rules for this conversion are spelled out in
Appendix~G\null. You can see ``before and after'' representations of such math
typesetting by ending a formula with `|\showlists$\showlists|'; the first
|\showlists| will display the math list, and the second will show the
(possibly complex) horizontal list that is manufactured from it.
\endchapter
The learning time is short. A few minutes gives the general flavor, and
typing a page or two of a paper generally uncovers most of the misconceptions.
\author ^{KERNIGHAN} and ^{CHERRY}, {\sl A System for %
Typesetting Mathematics\/} (1975)
% in {\sl Communications of the ACM\/} p152
\bigskip
Within a few hours (a few days at most)
a typist with no math or typesetting experience
can be taught to input even the most complex equations.
\author PETER J. ^{BOEHM}, {\sl Software and Hardware Considerations %
for a\break Technical Typesetting System\/} (1976)
% in {\sl IEEE Transactions on Professional Communication\/} PC-19, pp15--19
\eject
\beginchapter Chapter 18. Fine Points of\\Mathematics\\Typing
We have discussed most of the facilities needed to construct math
formulas, but there are several more things a good mathematical typist
will want to watch for. After you have typed a dozen or so formulas using
the basic ideas of Chapters 16 and~17, you will find that it's easy to
visualize the final appearance of a mathematical expression as you type
it. And once you have gotten to that level, there's only a little bit
more to learn before you are producing formulas as beautiful as any the
world has ever seen; tastefully applied touches of \TeX nique will add a
professional polish that works wonders for the appearance and readability
of the books and papers that you type. This chapter talks about such
tricks, and it also fills in a few gaps by mentioning some aspects of math
that didn't fit comfortably into~Chapters~16~and~17.
\subsection Punctuation. When a formula is followed by a ^{period}, ^{comma},
^{semicolon}, ^{colon}, ^{question mark}, ^{exclamation point}, etc., put the
^{punctuation} {\sl after\/} the |$|, when the formula is in the text; but
put the punctuation {\sl before\/} the |$$| when the formula is displayed.
For example,
\begintt
If $x<0$, we have shown that $$y=f(x).$$
\endtt
\TeX's spacing rules within paragraphs work best when the
^{punctuation marks} are not considered to be part of the formulas.
Similarly, don't ever type anything like
\begintt
for $x = a, b$, or $c$.
\endtt
It should be
\begintt
for $x = a$, $b$, or $c$.
\endtt
(Better yet, use a ^{tie}: `|or~$c$|'.) \ The reason is that \TeX\ will
typeset expression `|$x|~|=|~|a,|~|b$|' as a single formula, so it will
put a ``^{thin space}'' between the comma and the $b$. This space will
not be the same as the space that \TeX\ puts after the
comma {\sl after\/} the $b$, since spaces between words are always bigger than
thin spaces. Such unequal spacing looks bad, but when you type things right
the spacing will look good.
Another reason for not typing `|$x| |=| |a,| |b$|' is that it inhibits the
possibilities for breaking lines in a paragraph: \TeX\ will never break at
the space between the comma and the |b| because breaks after commas in
formulas are usually wrong. For example, in the equation
`|$x|~|=|~|f(a,|~|b)$|' we certainly don't want to put `$x=f(a,$' on one
line and `$b)$' on the next.
Thus, when typing formulas in the text of a paragraph, keep the math properly
segregated: Don't take operators like $-$ and $=$ outside of the |$|'s,
and keep commas inside the formula if they are truly part of the formula.
But if a comma or period or other punctuation mark belongs linguistically
to the sentence rather than to the formula, leave it outside the |$|'s.
\exercise Type this: $R(n,t)=O(t^{n/2})$, as $t\to0^+$.
\answer |$R(n,t)=O(t^{n/2})$, as $t\to0^+$.| \
(N.B.: `|O(|', not `|0('|.)
\danger Some mathematical styles insert a bit of extra space around
formulas to separate them from the text. For example, when copy is
being produced on an ordinary typewriter that doesn't have italic
letters, the best technical typists have traditionally put an extra
blank space before and after each formula, because this provides a
useful visual distinction. You might find it helpful to think of
each |$| as a symbol that has the potential of adding a little space
to the printed output; then the rule about excluding sentence
punctuation from formulas may be easier to remember.
\ddanger \TeX\ does, in fact, insert additional ^{space} before and after each
formula; the amount of such space is called ^|\mathsurround|, which is
a \<dimen>-valued parameter. For example, if you set |\mathsurround=1pt|,
each formula will effectively be 2~points wider ($1\pt$ at each side):
$$\baselineskip13pt\halign{\indent\mathsurround=#pt
For $x=a$, $b$, or $c$.\hfil&\quad(#)\hfil\cr
1&|\mathsurround=1pt|\cr 0&|\mathsurround=0pt|\cr}$$
This extra space will disappear into the left or right margin if the formula
occurs at the beginning or end of a line. The value of\/ |\mathsurround| that
is in force when \TeX\ reads the closing |$| of a formula is used at both
left and right of that formula. Plain \TeX\ takes |\mathsurround=0pt|, so
you won't see any extra space unless you are using some other format, or
unless you change |\mathsurround| yourself.
\subsection Non-italic letters in formulas. The names of algebraic variables
are usually italic or Greek letters, but common mathematical functions
like `log' are always set in ^{roman type}. The best way to deal with such
constructions is to make use of the following 32~control sequences (all of
which are defined in plain \TeX\ format, see Appendix~B):
\begintt
\arccos \cos \csc \exp \ker \limsup \min \sinh
\arcsin \cosh \deg \gcd \lg \ln \Pr \sup
\arctan \cot \det \hom \lim \log \sec \tan
\arg \coth \dim \inf \liminf \max \sin \tanh
\endtt
^^|\arccos| ^^|\cos| ^^|\csc| ^^|\exp| ^^|\ker| ^^|\limsup| ^^|\min|
^^|\sinh| ^^|\arcsin| ^^|\cosh| ^^|\deg| ^^|\gcd| ^^|\lg| ^^|\ln| ^^|\Pr|
^^|\sup| ^^|\arctan| ^^|\cot| ^^|\det| ^^|\hom| ^^|\lim| ^^|\log| ^^|\sec|
^^|\tan| ^^|\arg| ^^|\coth| ^^|\dim| ^^|\inf| ^^|\liminf| ^^|\max|
^^|\sin| ^^|\tanh|
These control sequences lead to roman type with appropriate
spacing:\def\sep{&\hskip-1em}
\beginlongmathdemo
\it Input\sep\it Output\cr
\noalign{\vskip2pt}
|$\sin2\theta=2\sin\theta\cos\theta$|\sep\sin2\theta=2\sin\theta\cos\theta\cr
|$O(n\log n\log\log n)$|\sep O(n\log n\log\log n)\cr
|$\Pr(X>x)=\exp(-x/\mu)$|\sep\Pr(X>x)=\exp(-x/\mu)\cr
|$$\max_{1\le n\le m}\log_2P_n$$|\sep
\displaystyle{\max_{1\le n\le m}\log_2P_n}\cr
\noalign{\vskip2pt}
|$$\lim_{x\to0}{\sin x\over x}=1$$|\sep\displaystyle{\lim_{x\to0}
{\sin x\over x}=1}\cr
\endmathdemo
^^|\mu|
The last two formulas, which are displays, show that some of the special
control sequences are treated by \TeX\ as ``large operators'' with limits
just like $\sum$: The subscript on |\max| is not treated like the subscript
on |\log|. Subscripts and superscripts will become limits
when they are attached to |\det|, |\gcd|, |\inf|, |\lim|, |\liminf|,
|\limsup|, |\max|, |\min|, |\Pr|, and |\sup|, in display style.
\exercise Express the following display in plain \TeX\ language, using
`^|\nu|' for `$\nu$':
$$p_1(n)=\lim_{m\to\infty}\sum_{\nu=0}^\infty
\bigl(1-\cos^{2m}(\nu!^n\pi/n)\bigr).$$
\answer |$$p_1(n)=\lim_{m\to\infty}\sum_{\nu=0}^\infty|\parbreak
| \bigl(1-\cos^{2m}(\nu!^n\pi/n)\bigr).$$|\par
\smallskip\noindent $\bigl[$Mathematicians may enjoy
interpreting this formula; cf.~G.~H. ^{Hardy},
{\sl Messenger of Mathematics\/ \bf35} (1906), 145--146.$\bigr]$
\danger If you need roman type for some mathematical function or operator
that isn't included in plain \TeX's list of~32, it is easy to define a
new control sequence by mimicking the definitions in Appendix~B\null. Or, if
you need roman type just for a ``one shot'' use, it is even easier to
get what you want by switching to ^|\rm| type, as follows:
\beginlongmathdemo
|$\sqrt{{\rm Var}(X)}$|&\sqrt{{\rm Var}(X)}\cr
|$x_{\rm max}-x_{\rm min}$|&x_{\rm max}-x_{\rm min}\cr
|${\rm LL}(k)\Rightarrow{\rm LR}(k)$|&{\rm LL}(k)\Rightarrow{\rm LR}(k)\cr
|$\exp(x+{\rm constant})$|&\exp(x+{\rm constant})\cr
|$x^3+{\rm lower\ order\ terms}$|&x^3+{\rm lower\ order\ terms}\cr
\endmathdemo
Notice the uses of `|\|\]' ^^{control space} in the last case;
without them, the result would have been `$x^3+{\rm lower order terms}$',
because ordinary blank ^{spaces} are ignored in math mode.
\danger You can also use ^|\hbox| instead of\/ |\rm| to get roman letters
into formulas. For example, four of the last five formulas can be
generated by
\beginlongmathdemo
|$\sqrt{\hbox{Var}(X)}$|&\sqrt{\hbox{Var}(X)}\cr
|$\hbox{LL}(k)\Rightarrow\hbox{LR}(k)$|&\hbox{LL}(k)\Rightarrow\hbox{LR}(k)\cr
|$\exp(x+\hbox{constant})$|&\exp(x+\hbox{constant})\cr
|$x^3+\hbox{lower order terms}$|&x^3+\hbox{lower order terms}\cr
\endmathdemo
In this case `|\|\]' isn't necessary, because the material in an
|\hbox| is processed in horizontal mode, when spaces are significant.
But such uses of\/ |\hbox| have two disadvantages: \ (1)~The contents of the
box will be typeset in the same size, whether or not the box occurs as a
subscript; for example, `|$x_{\hbox{max}}$|' yields `$x_{\hbox{max}}$'.
\ (2)~The font that's used inside |\hbox| will be the ``^{current font},''
so it might not be roman. For example, if you are typesetting the statement
of some theorem that is in slanted type, and if that theorem refers
to `|$\sqrt{\hbox{Var}(X)}$|', you will get the unintended result
`{\sl$\sqrt{\hbox{Var}(X)}$}'. In order to make sure that an |\hbox| uses
roman type, you need to specify |\rm|, e.g., `|$\sqrt{\hbox{\rm Var}(X)}$|';
and then the |\hbox| serves no purpose. We will see later, however, that
|\hbox| can be very useful in displayed formulas.
\ddangerexercise When the displayed formula
`|$$\lim_{n\to\infty}x_n {\rm\ exists} \iff|\break
|\limsup_{n\to\infty}x_n = \liminf_{n\to\infty}x_n.$$|' is typeset with
the standard macros of plain \TeX, you get
$$\lim_{n\to\infty}x_n{\rm\ exists}\iff
\limsup_{n\to\infty}x_n=\liminf_{n\to\infty}x_n.$$
But some people prefer a different notation: Explain how you could change
the definitions of\/ ^|\limsup| and ^|\liminf| so that the display would be
$$
\def\limsup{\mathop{\overline{\rm lim}}}
\def\liminf{\mathop{\underline{\rm lim}}}
\lim_{n\to\infty}x_n{\rm\ exists}\iff
\limsup_{n\to\infty}x_n=\liminf_{n\to\infty}x_n.$$
\answer |\def\limsup{\mathop{\overline{\rm lim}}}|\parbreak
|\def\liminf{\mathop{\underline{\rm lim}}}|\par
\smallskip\noindent
[Notice that the limits `$n\to\infty$' appear at different levels, in both
of the displays, because `sup' and the underbar descend below the baseline.
It is possible to unify the limit positions by using ^{phantoms}, as explained
later in this chapter. For example,
\begintt
\def\limsup{\mathop{\vphantom{\underline{}}\overline{\rm lim}}}
\endtt
would give lower limits in the same position as |\liminf|.]
\danger The word `mod' is also generally set in roman type, when it occurs
in formulas; but this word needs more care, because it is used in two
different ways that require two different treatments.
Plain \TeX\ provides two different control sequences,
^|\bmod| and ^|\pmod|, for the two cases: |\bmod| is to be used when
`mod' is a ^{binary operation} (i.e., when it occurs between two quantities,
like a plus sign usually does), and |\pmod| is to be used when
`mod' occurs parenthetically at the end of a formula. For example,
\beginmathdemo
|$\gcd(m,n)=\gcd(n,m\bmod n)$|&\gcd(m,n)=\gcd(n,m\bmod n)\cr
|$x\equiv y+1\pmod{m^2}$|&x\equiv y+1\pmod{m^2}\cr
\endmathdemo
The `|b|' in `|\bmod|' stands for ``binary''; the `|p|' in `|\pmod|' stands
for ``parenthesized.'' Notice that |\pmod| inserts its own parentheses;
the quantity that appears after `mod' in the parentheses should be
enclosed in braces, if it isn't a single symbol.
\dangerexercise What did poor B. L. ^{User} get when he typed
`|$x\equiv0 (\pmod y^n)$|'\thinspace?
\answer $x\equiv0(\pmod y^n)$. He should have typed
`|$x\equiv0\pmod{y^n}$|'.
\dangerexercise Explain how to produce \lower12pt\null\
$\smash{\displaystyle{n\choose k}\equiv{\lfloor n/p\rfloor\choose
\lfloor k/p\rfloor}{n\bmod p\choose k\bmod p}\pmod p.}$
\answer |$${n\choose k}\equiv{\lfloor n/p\rfloor\choose|\parbreak
| \lfloor k/p\rfloor}{n\bmod p\choose k\bmod p}\pmod p.$$|
\danger The same mechanism that works for roman type in formulas can be used
to get other styles of type as well. For example, ^|\bf| yields ^{boldface}:
\beginmathdemo
|$\bf a+b=\Phi_m$|&\bf a+b=\Phi_m\cr
\endmathdemo
Notice that whole formula didn't become emboldened in this example; the
`$+$' and `$=$' stayed the same. Plain \TeX\ sets things up so
that commands like |\rm| and |\bf| will affect only the uppercase letters
|A|~to~|Z|, the lowercase letters |a|~to~|z|, the digits |0|~to~|9|,
the uppercase Greek letters |\Gamma| to~|\Omega|, and math ^{accents}
like ^|\hat| and ^|\tilde|. Incidentally, no braces were used in this
example, because |$|'s have the effect of grouping; |\bf| changes the
current font, but the change is local, so it does not affect the font that
was current outside the formula.
\ddanger The bold fonts available in plain \TeX\ are ``bold roman,'' rather
than ``bold italic,'' because the latter are rarely needed. However, \TeX\
could readily be set up to make use of bold math italics, if desired
(see Exercise 17.\bmiexno). A more extensive set of math fonts would also
include ^{script}, ^{Fraktur}, and ``^{blackboard bold}'' styles; plain
\TeX\ doesn't have these, but other formats like \AmSTeX\ do. ^^{AMS-TeX}
^^{German black letters}
\danger Besides |\rm| and |\bf|, you can say ^|\cal| in formulas to get
uppercase letters in a ``^{calligraphic}'' style. For example, `|$\cal
A$|' produces `$\cal A$' and `|$\cal Z$|' produces `$\cal Z$'. But beware:
This works only with the letters |A| to |Z|; you'll get weird results if
you apply |\cal| to lowercase or Greek letters.
\danger There's also ^|\mit|, which stands for ``^{math italic}.'' This
affects ^{uppercase Greek}, so that you get
$\mit(\Gamma,\Delta,\Theta,\Lambda,\Xi,\Pi,\Sigma,\Upsilon,\Phi,\Psi,\Omega)$
instead of $(\Gamma,\ldots,\Omega)$. When~|\mit| is in effect, the
ordinary letters |A| to |Z| and |a| to |z| are not changed; they are set
in italics as usual, because they ordinarily come from the math italic
font. Conversely, uppercase Greek letters and math accents are unaffected
by |\rm|, because they ordinarily come from the roman font. Math accents
should not be used when the |\mit| family has been selected, because the
math italic font contains no accents.
\dangerexercise Type the formula $\bf\bar x^{\rm T}Mx={\rm0}\iff x=0$,
using as few keystrokes as possible. ^^{boldface numbers in math}
\ (The first `0' is roman, the second is bold. The superscript `T' is roman.)
\answer |$\bf\bar x^{\rm T}Mx={\rm0}\iff x=0$|. \ (If you typed a space between
|\rm| and~|0|, you wasted a keystroke; but don't feel guilty about it.)
\dangerexercise Figure out how to typeset
`$S\subseteq\mit\Sigma\iff S\in\cal S$'.
\answer |$S\subseteq{\mit\Sigma}\iff S\in{\cal S}$|. In this case the
braces are redundant and could be eliminated; but you shouldn't try to do
{\sl everything\/} with fewest keystrokes, or you'll outsmart yourself
some day.
\danger Plain \TeX\ also allows you to type ^|\it|, ^|\sl|, or ^|\tt|, if
you want text italic, slanted, or typewriter letters to occur in a math
formula. However, these fonts are available only in text size, so you
should not try to use them in subscripts.
\danger If you're paying attention, you probably wonder why both
|\mit| and |\it| are provided; the answer is that |\mit| is ``math italic''
(which is normally best for formulas), and |\it| is ``^{text italic}'' (which
is normally best for running text).
\beginmathdemo
|$This\ is\ math\ italic.$|&This\ is\ math\ italic.\cr
|{\it This is text italic.}|&\hbox{\it This is text italic.}\cr
\endmathdemo
The math italic letters are a little wider, and the spacing is different;
this works better in most formulas, but it fails spectacularly when
you try to type certain italic words like `$different$' using math mode
(`|$different$|'). A wide `$f$' is usually desirable in formulas, but it
is undesirable in text. Therefore wise typists
use |\it| in a math formula that is supposed
to contain an actual italic word. Such cases almost never occur in
classical mathematics, but they are common when ^{computer programs}
are being typeset, since programmers often use multi-letter ``^{identifiers}'':
\beginmathdemo
|$\it last:=first$|&\it last:=first\cr
|$\it x\_coord(point\_2)$|&\it x\_coord(point\_2)\cr
\endmathdemo
The first of these examples shows that \TeX\ recognizes the ^{ligature}
`{\it fi\/}' when text italic occurs in a math formula;
the other example illustrates the use of short ^{underlines} to break
up identifier names. ^^{control-underline}
When the author typeset this manual, he used `|$\it SS$|' to refer to
style~$\SS$, since `|$SS$|' makes the $S$'s too far apart: $SS$.
\dangerexercise What plain \TeX\ commands will produce the following display?
$$\tenmath
{\it available}+\sum_{i=1}^n\max\bigl({\it full}(i),{\it reserved}(i)\bigr)
={\it capacity}.$$
\answer |$${\it available}+\sum_{i=1}^n\max\bigl({\it full}(i),|\parbreak
|{\it reserved}(i)\bigr)={\it capacity}.$$|
\smallskip\noindent [If\/ |\it| had been used throughout
the formula, the subscript~$i$ and superscript~$n$ would have caused error
messages saying `^|\scriptfont| |4| |is| |undefined|',
since plain \TeX\ makes |\it| available only in text size.]
\ddangerexercise How would you go about typesetting the following computer
program, using the macros of plain \TeX?
$$\vbox{\let\par=\endgraf
\obeylines\sfcode`;=3000
{\bf for $j:=2$ step $1$ until $n$ do}
\quad {\bf begin} ${\it accum}:=A[j]$; $k:=j-1$; $A[0]:=\it accum$;
\quad {\bf while $A[k]>\it accum$ do}
\qquad {\bf begin} $A[k+1]:=A[k]$; $k:=k-1$;
\qquad {\bf end};
\quad $A[k+1]:=\it accum$;
\quad {\bf end}.
}$$
\answer |{\obeylines \sfcode`;=3000|^^|\sfcode|\parbreak
|{\bf for $j:=2$ step $1$ until $n$ do}|\parbreak
|\quad {\bf begin} ${\it accum}:=A[j]$; $k:=j-1$; $A[0]:=\it accum$;|\parbreak
|\quad {\bf while $A[k]>\it accum$ do}|\parbreak
|\qquad {\bf begin} $A[k+1]:=A[k]$; $k:=k-1$;|\parbreak
|\qquad {\bf end};|\parbreak
|\quad $A[k+1]:=\it accum$;|\parbreak
|\quad {\bf end}.\par}|\par
\smallskip\noindent
[This is something like the ``poetry'' example in Chapter~14, but much
more difficult. Some manuals of style say that ^{punctuation} should inherit
the font of the preceding character, so that three kinds of semicolons
should be typeset; e.g., these experts recommend `$k:=j-1$; \
$A[0]:={}${\it accum;} \ {\bf end;}'. The author heartily disagrees.]
\subsection Spacing between formulas. ^{Displays} often contain more than one
formula; for example, an equation is frequently accompanied by a ^{side
condition}:
$$F_n=F_{n-1}+F_{n-2},\qquad n\ge2.$$
In such cases you need to tell \TeX\ how much space to put after the comma,
because \TeX's normal spacing conventions would bunch things together;
without special precautions you would get
$$F_n=F_{n-1}+F_{n-2}, n\ge2.$$
The traditional hot-metal technology for printing has led to some ingrained
standards for situations like this, based on what printers call a ``^{quad}''
of space. Since these standards seem to work well in practice, \TeX\ makes
it easy for you to continue the tradition: When you type `^|\quad|' in plain
\TeX\ format, you get a printer's quad of space in the horizontal direction.
Similarly, `^|\qquad|' gives you a double quad (twice as much); this
is the normal spacing for situations like
the $F_n$ example above. Thus, the recommended procedure is to type
\begintt
$$ F_n = F_{n-1} + F_{n-2}, \qquad n \ge 2. $$
\endtt
It is perhaps worth reiterating that \TeX\ ignores all the spaces in math
mode (except, of course, the space after `|\qquad|', which is needed
to distinguish between `|\qquad|~|n|' and `|\qquadn|'); so the same result
would be obtained if you were to leave out all but one space:
\begintt
$$F_n=F_{n-1}+F_{n-2},\qquad n\ge2.$$
\endtt
Whenever you want spacing that differs from the normal conventions, you must
specify it explicitly by using control sequences such as |\quad| and |\qquad|.
\danger A quad used to be a square piece of blank type, $1\em$ wide and $1\em$
tall---approximately the size of a capital M, as explained in Chapter~10. This
tradition has not been fully retained: The control sequence |\quad| in plain
\TeX\ is simply an abbreviation for `|\hskip|~|1|^|em||\relax|', so \TeX's
quad has width but no height.
\danger You can use |\quad| in text as well as in formulas; for example,
Chapter~14 illustrates how |\quad| applies to poetry. When |\quad| appears
in a formula it stands for one~em in the current text font, independent of the
current math size or style or family. Thus, for example, |\quad| is just
as wide in a subscript as it is on the main line of a formula.
Sometimes a careless author will put two formulas next to each other in
the text of a paragraph. For example, you might find a sentence like this:
\begindisplay
The ^{Fibonacci} numbers satisfy $F_n=F_{n-1}+F_{n-2}$, \ $n\ge2$.
\enddisplay
Everybody who teaches proper ^{mathematical} ^{style} is agreed that formulas
ought to be separated by words, not just by commas; the author of that
sentence should at least have said `for $n\ge2$', not simply `$n\ge2$'.
But alas, such lapses are commonplace, and many prominent mathematicians
are hopelessly addicted to clusters of formulas. If we are not allowed to change
their writing style, we can at least insert extra space where they
neglected to insert an appropriate word. An additional interword space
generally works well in such cases; for example, the sentence above was
typeset thus:
\begintt
... $F_n=F_{n-1}+F_{n-2}$, \ $n\ge2$.}$$
\endtt
The `|\|\]' ^^{control space} here gives a visual separation that
partly compensates for the bad style.
\exercise Put the following paragraph into \TeX\ form, treating punctuation
and spacing carefully; also insert ^{ties} to prevent bad line breaks.
\begindisplay\baselineskip13pt
\vbox{\raggedright\hsize=310pt\parindent=0pt
Let $H$~be a Hilbert space, \
$C$~a closed bounded convex subset of~$H$, \
$T$~a nonexpansive self map of~$C$.
Suppose that as $n\to\infty$, \ $a_{n,k}\to0$ for each~$k$,
and $\gamma_n=\sum_{k=0}^\infty(a_{n,k+1}-a_{n,k})^+\to0$.
Then for each $x$~in~$C$, \
$A_nx=\sum_{k=0}^\infty a_{n,k}T^kx$ converges weakly
to a fixed point of~$T$.
} % taken from Bull. AMS 82 (1976), p 959; chosen by AMS in '78 for demo
\enddisplay
\answer |Let $H$~be a Hilbert space, \
$C$~a closed bounded convex subset of~$H$, \
$T$~a nonexpansive self map of~$C$.
Suppose that as $n\to\infty$, \ $a_{n,k}\to0$ for each~$k$,
and $\gamma_n=\sum_{k=0}^\infty(a_{n,k+1}-|\allowbreak|a_{n,k})^+\to0$.
Then for each $x$~in~$C$, \
$A_nx=\sum_{k=0}^\infty a_{n,k}T^kx$ converges weakly
to a fixed point of~$T$.|\par
[If any mathematicians are reading this, they might either appreciate
or resent the following attempt to edit the given paragraph
into a more acceptable style: ``%
Let $C$~be a closed, bounded, convex subset of a Hilbert space~$H$,
and let $T$~be a nonexpansive self map of~$C$.
Suppose that as $n\to\infty$, we have $a_{n,k}\to0$ for each~$k$,
and $\gamma_n=\sum_{k=0}^\infty(a_{n,k+1}-a_{n,k})^+\to0$.
Then for each $x$~in~$C$, the infinite sum
$A_nx=\sum_{k=0}^\infty a_{n,k}T^kx$ converges weakly
to a fixed point of~$T$.'']
\subsection Spacing within formulas. Chapter 16 says that \TeX\ does
automatic ^{spacing} of math formulas so that they look right, and this is
almost true. But occasionally you must give \TeX\ some help. The number of
possible math formulas is vast, and \TeX's spacing rules are rather
simple, so it is natural that exceptions should arise. Of course, it is
desirable to have fine units of spacing for this purpose, instead of the
big chunks that arise from |\|\], |\quad| and |\qquad|.
The basic elements of space that \TeX\
puts into formulas are called {\sl ^{thin spaces}}, {\sl ^{medium
spaces}}, and {\sl ^{thick spaces}}. In order to get a feeling for these
units, let's take a look at the $F_n$ example again: Thick spaces occur
just before and after the = sign, and also before and after the $\ge$\thinspace;
medium spaces occur just before and after the $+$ sign. Thin spaces are
slightly smaller, but noticeable; it's a thin space that makes the
difference between `loglog' and `$\log\log$'. The normal space between
words of a paragraph is approximately equal to two thin spaces.
\TeX\ inserts thin spaces, medium spaces, and thick spaces into formulas
automatically, but you can add your own spacing whenever you want to,
by using the control sequences ^^|\,|^^|\!|^^|\;|^^|\>|
$$\halign{\indent#\hfil&\quad#\hfil\cr
|\,|&thin space \ (normally 1/6 of a quad);\cr
|\>|&medium space \ (normally 2/9 of a quad);\cr
|\;|&thick space \ (normally 5/18 of a quad);\cr
|\!|&negative thin space \ (normally $-1/6$ of a quad).\cr}$$
In most cases you can rely on \TeX's spacing while you are typing a manuscript,
and you'll want to insert or delete space with these four control sequences
only in rare circumstances after you see what comes out.
\ddanger We observed a minute ago that |\quad| spacing does not
change with the style of formula, nor does it depend on the math font
families that are being used. But thin spaces, medium spaces, and thick
spaces do get bigger and smaller as the size of type gets bigger and
smaller; this is because they are defined in terms of ^\<muglue>, a~special
brand of glue intended for math spacing. You specify \<muglue> just
as if it were ordinary glue, except that the units are given in terms of
`^|mu|' (math units) instead of~|pt| or~|cm| or something else. For
example, Appendix~B contains the definitions
\begintt
\thinmuskip = 3mu
\medmuskip = 4mu plus 2mu minus 4mu
\thickmuskip = 5mu plus 5mu
\endtt
^^|\thinmuskip|^^|\medmuskip|^^|\thickmuskip|
and this defines the thin, medium, and thick spaces that \TeX\ inserts
into formulas. According to these specifications, thin spaces in plain
\TeX\ do not stretch or shrink; medium spaces can stretch a little, and
they can shrink to zero; thick spaces can stretch a lot, but they never shrink.
\ddanger There are 18 mu to an em, where the em is taken from family~2
(the math symbols family). In other words, ^|\textfont|~|2| defines the em
value for |mu| in display and text styles; ^|\scriptfont|~|2| defines the
em for script size material; and ^|\scriptscriptfont|~|2| defines it for
scriptscript size.
\ddanger You can insert math glue into any formula just by giving
the command `^|\mskip|\<muglue>'. For example, `|\mskip 9mu plus 2mu|'
inserts one half em of space, in the current size, together with some
stretchability. Appendix~B defines `|\,|' to be an abbreviation for
`|\mskip\thinmuskip|'. Similarly, you can use the command `^|\mkern|'
when there is no stretching or shrinking; `|\mkern18mu|' gives one em of
horizontal space in the current size. \TeX\ insists that |\mskip| and
|\mkern| be used only with |mu|; conversely, ^|\hskip| and ^|\kern| (which
are also allowed in formulas) must never give units in |mu|.
Formulas involving ^{calculus} look best when an extra thin space appears
before $dx$ ^^{dx} or~$dy$ or~$d\,$whatever; but \TeX\ doesn't do this
automatically. Therefore a well-trained typist will remember to insert
`|\,|' in examples like the following:
\beginmathdemo
\it Input&\it Output\cr
\noalign{\vskip2pt}
|$\int_0^\infty f(x)\,dx$|&\int_0^\infty f(x)\,dx\cr
|$y\,dx-x\,dy$|&y\,dx-x\,dy\cr
|$dx\,dy=r\,dr\,d\theta$|&dx\,dy=r\,dr\,d\theta\cr
|$x\,dy/dx$|&x\,dy/dx\cr \endmathdemo Notice that no `|\,|' was desirable
after the `|/|' in the last example. Similarly, there's no need for
`|\,|' in cases like
\begindisplaymathdemo
|$$\int_1^x{dt\over t}$$|&\int_1^x{dt\over t}\cr
\endmathdemo
since the $dt$ appears all by itself in the numerator of a fraction; this
detaches it visually from the rest of the formula.
\exercise Explain how to handle the display
$$\int_0^\infty{t-ib\over t^2+b^2}e^{iat}\,dt=e^{ab}E_1(ab),\qquad a,b>0.$$
\answer |$$\int_0^\infty{t-ib\over t^2+b^2}e^{iat}\,dt=|\parbreak
| e^{ab}E_1(ab),\qquad a,b>0.$$|
\danger When physical ^{units} appear in a formula, they should be set in roman
type and separated from the preceding material by a thin space:
\beginmathdemo
|$55\rm\,mi/hr$|&55\rm\,mi/hr\cr
|$g=9.8\rm\,m/sec^2$|&g=9.8\rm\,m/sec^2\cr
|$\rm1\,ml=1.000028\,cc$|&\rm1\,ml=1.000028\,cc\cr
\endmathdemo
\dangerexercise Typeset the following display, assuming that `^|\hbar|'
generates `$\hbar$':
$$\hbar=1.0545\times10^{-27}\rm\,erg\,sec.$$
\answer |$$\hbar=1.0545\times10^{-27}\rm\,erg\,sec.$$|
\danger Thin spaces should also be inserted after ^{exclamation points}
(which stand for the ``^{factorial}'' operation in a formula), if the next
character is a letter or a number or an opening delimiter:
\beginmathdemo
|$(2n)!/\bigl(n!\,(n+1)!\bigr)$|&(2n)!/\bigl(n!\,(n+1)!\bigr)\cr
\noalign{\vskip6pt}
|$${52!\over13!\,13!\,26!}$$|&\displaystyle{52!\over13!\,13!\,26!}\cr
\endmathdemo
Besides these cases, you will occasionally encounter formulas in which
the symbols are bunched up too tightly, or where too much white space
appears, because of certain unlucky combinations of shapes. It's usually
impossible to anticipate optical glitches like this until you see the first
proofs of what you have typed; then you get to use your judgment about how
to add finishing touches that provide extra beauty, clarity, and finesse.
A tastefully applied `|\,|' or `|\!|'\ will open things up or close things
together so that the reader won't be distracted from the mathematical
significance of the formula. ^{Square root} signs and ^{multiple integrals} are
often candidates for such fine tuning. Here are some examples of situations
to look out for:
\beginmathdemo
|$\sqrt2\,x$|&\sqrt2\,x\cr
|$\sqrt{\,\log x}$|&\sqrt{\,\log x}\cr
|$O\bigl(1/\sqrt n\,\bigr)$|&O\bigl(1/\sqrt n\,\bigr)\cr
|$[\,0,1)$|&[\,0,1)\cr
|$\log n\,(\log\log n)^2$|&\log n\,(\log\log n)^2\cr
|$x^2\!/2$|&x^2\!/2\cr
|$n/\!\log n$|&n/\!\log n\cr
|$\Gamma_{\!2}+\Delta^{\!2}$|&\Gamma_{\!2}+\Delta^{\!2}\cr
|$R_i{}^j{}_{\!kl}$|&R_i{}^j{}_{\!kl}\cr
|$\int_0^x\!\int_0^y dF(u,v)$|&\int_0^x\!\int_0^y dF(u,v)\cr
\noalign{\vskip6pt}
|$$\int\!\!\!\int_D dx\,dy$$|&\displaystyle{\int\!\!\!\int_D dx\,dy}\cr
\endmathdemo
^^|\Gamma|^^|\Delta|^^|\intint|
In each of these formulas the omission of\/ |\,| or |\!|\ would lead to
somewhat less satisfactory results.
\ddanger Most of these examples where thin-space corrections are desirable
arise because of chance coincidences. For example, the superscript in
|$x^2/2$| leaves a hole before the slash ($x^2/2$); a negative thin
space helps to fill that hole. The positive thin space in |$\sqrt{\,\log x}$|
compensates for the fact that `$\log x$' begins with a tall, unslanted
letter; and so on. But two of the examples involve corrections that
were necessary because \TeX\ doesn't really know a great deal about
mathematics: \ (1)~In the formula
|$\log n(\log\log n)^2$|, \TeX\ inserts no thin space before the left
parenthesis, because there are similar formulas like |$\log n(x)$| where
no such space is desired. \ (2)~In the formula |$n/\log n$|, \TeX\
automatically inserts an unwanted thin~space before |\log|, since the slash is
treated as an ordinary symbol, and since a~thin space is usually desirable
between an ordinary symbol and an operator like |\log|.
\ddanger In fact, \TeX's rules for spacing in formulas are fairly simple.
A formula is converted to a math list as described at the end of Chapter~17,
and the math list consists chiefly of ``^{atoms}'' of eight basic types:
^{Ord}~(^{ordinary}), ^{Op}~(^{large operator}), ^{Bin}~(^{binary operation}),
^{Rel}~(^{relation}), ^{Open}~(^{opening}), ^{Close}~(^{closing}),
^{Punct}~(punctuation), ^^{punctuation} and ^{Inner}~(a delimited
subformula). Other kinds of atoms, which arise from commands like
^|\overline| or ^|\mathaccent| or ^|\vcenter|, etc., are all treated as
type~Ord; ^{fractions} are treated as type~Inner. The following table is
used to determine the spacing between pairs of adjacent atoms:
$$\baselineskip0pt\lineskip0pt
\halign to\hsize
{\strut\hbox to\parindent{\it#\hfil}& % for the legend "Left atom"
#\hfil\quad& % for the row labels
#\hfil\tabskip 0pt plus 10pt& % for the rule at the left
\hbox to 25pt{\tt\hss#\hss}& % for column 1
\hbox to 25pt{\tt\hss#\hss}& % for column 2
\hbox to 25pt{\tt\hss#\hss}& % for column 3
\hbox to 25pt{\tt\hss#\hss}& % for column 4
\hbox to 25pt{\tt\hss#\hss}& % for column 5
\hbox to 25pt{\tt\hss#\hss}& % for column 6
\hbox to 25pt{\tt\hss#\hss}& % for column 7
\hbox to 25pt{\tt\hss#\hss}& % for column 8
#\hfil\tabskip0pt\cr % for the rule at the right
\noalign{\vskip-6pt} % it just happens that there's extra white space
&&&&\multispan7\hss\it Right atom\hss\cr
\noalign{\vskip3pt}
&&&\rm Ord&\rm Op&\rm Bin&\rm Rel&\rm Open&\rm Close&\rm Punct&\rm Inner\cr
\noalign{\vskip2pt}
\omit&&\multispan{10}\leaders\hrule\hfil\cr
\omit\vbox to 2pt{}&&\vrule&&&&&&&&&\vrule\cr
&Ord&\vrule&0&1&(2)&(3)&0&0&0&(1)&\vrule\cr
&Op&\vrule&1&1&*&(3)&0&0&0&(1)&\vrule\cr
&Bin&\vrule&(2)&(2)&*&*&(2)&*&*&(2)&\vrule\cr
Left&Rel&\vrule&(3)&(3)&*&0&(3)&0&0&(3)&\vrule\cr
atom&Open&\vrule&0&0&*&0&0&0&0&0&\vrule\cr
&Close&\vrule&0&1&(2)&(3)&0&0&0&(1)&\vrule\cr
&Punct&\vrule&(1)&(1)&*&(1)&(1)&(1)&(1)&(1)&\vrule\cr
&Inner&\vrule&(1)&1&(2)&(3)&(1)&0&(1)&(1)&\vrule\cr
\omit\vbox to 2pt{}&&\vrule&&&&&&&&&\vrule\cr
\omit&&\multispan{10}\leaders\hrule\hfil\cr}$$
^^{spacing table} ^^{math spacing table}
Here 0, 1, 2, and 3 stand for no space, thin space, medium space, and
thick space, respectively; the table entry is parenthesized if the space
is to be inserted only in display and text styles, not in script and
scriptscript styles. For example, many of the entries in the Rel row
and the Rel column are `{\tt(3)}'; this means that thick spaces are normally
inserted before and after relational symbols like `=', but not in
subscripts. Some of the entries in the table are `{\tt*}'; such cases
never arise, because Bin atoms must be preceded and followed by atoms
compatible with the nature of binary operations. Appendix~G contains
precise details about how math lists are converted to horizontal lists;
this conversion is done whenever \TeX\ is about to leave math mode, and the
inter-atomic spacing is inserted at that time.
\ddanger For example, the displayed formula specification
\begintt
$$x+y=\max\{x,y\}+\min\{x,y\}$$
\endtt
will be transformed into the sequence of atoms
\def\\#1{\vbox to 33pt{\vbox to 22pt{\vfill\hrule
\hbox{\vrule\hskip-.4pt$#1$\hskip-.4pt\vrule}}\hrule\vfill}}%
\begindisplay
\vbox{\vskip-11pt\hbox{$
\\x\;\;\\+\;\;\\y\;\;\\=\;\;\\\max\;\;\\\{\;\;
\\x\;\;\\,\;\;\\y\;\;\\\}\;\;\\+\;\;\\\min\;\;\\\{
\;\;\\x\;\;\\,\;\;\\y\;\;\\\}$}\vskip-11pt}
\enddisplay
of respective types Ord, Bin, Ord, Rel, Op, Open, Ord, Punct, Ord, Close,
Bin, Op, Open, Ord, Punct, Ord, and Close.
Inserting spaces according to the table gives
$$\def\0{\thinspace}
\def\1{\thinspace{\tt\bslash,}\thinspace}
\def\2{\thinspace{\tt\bslash>}\thinspace}
\def\3{\thinspace{\tt\bslash;}\thinspace}
\halign{\indent\hfil#\cr
Ord\2Bin\2Ord\3Rel\3Op\0Open\0Ord\0Punct\1Ord\0Close\2\qquad\cr
Bin\2Op\0Open\0Ord\0Punct\1Ord\0Close\cr}$$
and the resulting formula is
$$\vbox{\vskip-11pt\hbox{$
\\x\>\\+\>\\y\;\\=\;\\\max\\\{
\\x\\,\,\\y\\\}\>\\+\>\\\min\\\{
\\x\\,\,\\y\\\}$}\vskip-11pt}$$
i.e.,$$x+y=\max\{x,y\}+\min\{x,y\}\rlap{\quad.}$$
This example doesn't involve subscripts or superscripts; but subscripts and
superscripts merely get attached to atoms without changing the atomic type.
\ddangerexercise Use the table to determine what spacing \TeX\
will insert between the atoms of the formula `|$f(x,y)<x^2+y^2$|'.
\answer There are ten atoms (the first is $f$ and last is $y^2$); their types,
and the interatomic spacing, are respectively
\begindisplay \def\0{\thinspace}%
\def\1{\thinspace{\tt\bslash,}\thinspace}%
\def\2{\thinspace{\tt\bslash>}\thinspace}%
\def\3{\thinspace{\tt\bslash;}\thinspace}
Ord\0Open\0Ord\0Punct\1Ord\0Close\3Rel\3Ord\2Bin\2Ord.
\enddisplay
\ddanger The plain \TeX\ macros ^|\bigl|, ^|\bigr|, ^|\bigm|, and ^|\big|
all produce identical delimiters; the only difference between them is that
they may lead to different spacing, because they make the delimiter into
different types of atoms: |\bigl| produces an Open atom, |\bigr| a~Close,
|\bigm| a~Rel, and |\big| an~Ord. On the other hand, when a subformula
appears between ^|\left| and ^|\right|, it is typeset by itself and placed
into an Inner atom. Therefore it is possible that a subformula enclosed by
|\left| and |\right| will be surrounded by more space than there would be
if that subformula were enclosed by |\bigl| and~|\bigr|. For example, Ord
followed by Inner (from |\left|) gets a thin space, but Ord followed by
Open (from |\bigl|) does not. The rules in Chapter~17 imply that the
construction `^|\mathinner||{\bigl({|\<subformula>|}\bigr)}|' within any
formula produces a result exactly equivalent to
`|\left(|\<subformula>|\right)|', when the \<subformula> doesn't end
with Punct, except that the ^{delimiters} are forced to
be of the |\big| size regardless of the height and depth of the subformula.
\danger \TeX's spacing rules sometimes fail when `\|' and `|\|\|'
appear in a formula, because $\vert$ and $\Vert$ are treated as ordinary symbols
^^{verticalline}^^|\verticalline|
instead of as delimiters. For example, consider the formulas
\beginlongmathdemo
|$|\||-x|\||=|\||+x|\||$|&\vert-x\vert=\vert+x\vert\cr
|$\left|\||-x\right|\||=\left|\||+x\right|\||$|&
\left\vert-x\right\vert=\left\vert+x\right\vert\cr
|$\lfloor-x\rfloor=-\lceil+x\rceil$|&\lfloor-x\rfloor=-\lceil+x\rceil\cr
\endmathdemo
In the first case the spacing is wrong because \TeX\ thinks that the plus
sign is computing the sum of `$\vert$' and `$x$'. The use of\/ |\left| and
|\right| in the second example puts \TeX\ on the right track. The third
example shows that no such corrections are needed with other delimiters,
because \TeX\ knows whether they are openings or closings.
\ddangerexercise Some perverse mathematicians use ^{brackets} backwards,
to denote ``^{open intervals}.''
Explain how to type the following bizarre formula: % from MR53 #3451
$\left]-\infty,T\right[\times\left]-\infty,T\right[$.
\answer |$\left]-\infty,T\right[\times\left]-\infty,T\right[$|. \ (Or one could
say ^|\mathopen| and ^|\mathclose| instead of\/ |\left| and |\right|;
then \TeX\ would not choose the size of the delimiters, nor would it consider
the subformulas to be of type Inner.) \
% that formula was quoted from MR review of paper by Mario Marino
% in Ricerche Mat. 24 (1975), no.~1, 152--171
Open intervals are more clearly expressed in print
by using parentheses instead of reversed brackets; for example,
compare `$(-\infty,T)\times(-\infty,T)$' to the given formula.
\ddangerexercise Study Appendix G and determine what spacing will be used
in the formula `|$x++1$|'. Which of the plus signs will be regarded
as a ^{binary operation}?
\answer The first |+| will become a Bin atom, the second an Ord; hence
the result is $x$, medium space, $+$, medium space, $+$, no space, 1.
\subsection Ellipses\/ {\rm(``three dots'')}. ^^{ellipses}
Mathematical copy looks much nicer if you are careful about how groups of
^{three dots} are typed in formulas and text. Although it looks fine to
type `|...|'\ on a typewriter that has fixed spacing, the result looks too
crowded when you're using a printer's fonts: `|$x...y$|' results in
`$x...y$', and such close spacing is undesirable except in subscripts or
superscripts.
An ellipsis can be indicated by two different kinds of dots, one higher
than the other; the best mathematical traditions distinguish between
these two possibilities. It is generally correct to produce formulas like
\begindisplay
$\displaystyle x_1+\cdots+x_n\qquad {\rm and}\qquad (x_1,\ldots,x_n),$
\enddisplay
but wrong to produce formulas like
\begindisplay
$\displaystyle x_1+\ldots+x_n\qquad {\rm and}\qquad (x_1,\cdots,x_n).$
\enddisplay
The plain \TeX\ format of Appendix B allows you to solve the ``three dots''
problem very simply, and everyone will be envious of the beautiful formulas
that you produce. The idea is simply to type ^|\ldots| when you want three
low dots~($\,\ldots\,$), and ^|\cdots| when you want three vertically centered
dots~($\,\cdots\,$).
In general, it is best to use |\cdots| between $+$ and $-$ and~$\times$
signs, and also between $=$~signs or $\le$~signs or $\subset$ signs or other
similar relations. Low dots are used between ^{commas}, and when things
are juxtaposed with no signs between them at all. For example:
\beginmathdemo
|$x_1+\cdots+x_n$|&x_1+\cdots+x_n\cr
|$x_1=\cdots=x_n=0$|&x_1=\cdots=x_n=0\cr
|$A_1\times\cdots\times A_n$|&A_1\times\cdots\times A_n\cr
|$f(x_1,\ldots,x_n)$|&f(x_1,\ldots,x_n)\cr
|$x_1x_2\ldots x_n$|&x_1x_2\ldots x_n\cr
|$(1-x)(1-x^2)\ldots(1-x^n)$|&(1-x)(1-x^2)\ldots(1-x^n)\cr
|$n(n-1)\ldots(1)$|&n(n-1)\ldots(1)\cr
\endmathdemo
\exercise Type the formulas `$x_1+x_1x_2+\cdots+x_1x_2\ldots x_n$' and
`$(x_1,\ldots,x_n)\cdot(y_1,\ldots,y_n)=x_1y_1+\cdots+x_ny_n$'.
\ [{\sl Hint:\/} A single raised dot is called `^|\cdot|'.]
\answer |$x_1+x_1x_2+\cdots+x_1x_2\ldots x_n$| \ and\hfil\break
|$(x_1,\ldots,x_n)\cdot(y_1,\ldots,y_n)=x_1y_1+\cdots+x_ny_n$|.
But there's an important special case in which |\ldots| and |\cdots|
don't give the correct spacing, namely when they appear at the very end
of a formula, or when they appear just before a closing delimiter like~`|)|'.
In such situations an extra ^{thin space} is needed. For example, consider
sentences like this:
\begindisplay
Prove that $(1-x)^{-1}=1+x+x^2+\cdots\,$.\cr
Clearly $a_i<b_i$ for $i=1$,~2, $\ldots\,$,~$n$.\cr
The coefficients $c_0$,~$c_1$, \dots,~$c_n$ are positive.\cr
\enddisplay
To get the first sentence, the author typed
\begintt
Prove that $(1-x)^{-1}=1+x+x^2+\cdots\,$.
\endtt
Without the `^|\,|' the period would have come too close to the |\cdots|.
Similarly, the second sentence was typed thus:
\begintt
Clearly $a_i<b_i$ for $i=1$,~2, $\ldots\,$,~$n$.
\endtt
Notice the use of ^{ties}, which prevent bad line breaks as explained in
Chapter~14. Such ellipses are extremely common in some forms of mathematical
writing, so plain \TeX\ allows you to say just `^|\dots|' as an abbreviation
for `|$\ldots\,$|' in the text of a paragraph. The third sentence
can therefore be typed
\begintt
The coefficients $c_0$,~$c_1$, \dots,~$c_n$ are positive.
\endtt
\exercise B. C. ^{Dull} tried to take a shortcut by typing the
second example this way:
\begintt
Clearly $a_i<b_i$ for~$i=1, 2, \ldots, n$.
\endtt
What's so bad about that?
\answer The commas belong to the sentence, not to the formula; his
decision to put them into math mode meant that \TeX\ didn't put large
enough spaces after them. Also, his formula `$i=1, 2, \ldots, n$' allows
no breaks between lines, except after the $=$, so he's risking
overfull box problems. But suppose the sentence had been more terse:
\begindisplay
Clearly $a_i<b_i$ \ ($i=1,2,\ldots,n$).
\enddisplay
Then his idea would be basically correct:
\begintt
Clearly $a_i<b_i$ \ ($i=1,2,\ldots,n$).
\endtt
\exercise How do you think the author typed the ^{footnote} in Chapter 4
of this book?
\answer $\ldots$ |never\footnote*{Well \dots, hardly ever.} have| $\ldots$
\subsection Line breaking. When you have formulas in a paragraph, \TeX\
may have to break them between lines. This is a necessary evil, something
like the hyphenation of words; we want to avoid it unless the alternative
is worse. ^^{line breaking in math} ^^{breaking formulas between lines}
A formula will be broken only after a relation symbol like $=$
or~$<$ or $\to$, or after a binary operation symbol like $+$ or $-$ or
$\times$, where the relation or binary operation is on the ``outer level'' of
the formula (i.e., not enclosed in |{...}| and not part of an `|\over|'
construction). For example, if you type
\begintt
$f(x,y) = x^2-y^2 = (x+y)(x-y)$
\endtt
in mid-paragraph, there's a chance that \TeX\ will break after either of the
|=|~signs (it prefers this) or after the~|-| or~|+| or~|-| (in an emergency).
But there won't be a break after the comma in any case---commas
after which breaks are desirable shouldn't appear between |$|'s.
If you don't want to permit breaking in this example except after the
|=|~signs, you could type
\begintt
$f(x,y) = {x^2-y^2} = {(x+y)(x-y)}$
\endtt
because these additional braces ``freeze'' the ^{subformulas}, putting them
into unbreakable boxes in which the glue has been set to its natural width.
But it isn't necessary to bother worrying about such things unless \TeX\
actually does break a formula badly, since the chances of this are
pretty slim.
\danger A ``^{discretionary multiplication sign}'' is allowed in formulas:
If you type `|$(x+y)\*(x-y)$|', \TeX\ will treat the ^|\*| something like
the way it treats \hbox{|\-|}; namely, a line break will be allowed at
that place, with the hyphenation penalty. However, instead of inserting a
hyphen, \TeX\ will insert a $\times$ sign in text size.
\danger If you do want to permit a break at some point in the outer level
of a formula, you can say ^|\allowbreak|. For example, if the formula
\begintt
$(x_1,\ldots,x_m,\allowbreak y_1,\ldots,y_n)$
\endtt
appears in the text of a paragraph, \TeX\ will allow it to be broken into the
two pieces `$(x_1,\ldots,x_m,$' and `$y_1,\ldots,y_n)$'.
\ddanger The penalty for breaking after a Rel atom is called ^|\relpenalty|,
and the penalty for breaking after a Bin atom is called ^|\binoppenalty|.
Plain \TeX\ sets |\relpenalty=500| and |\binoppenalty=700|. You can change
the penalty for breaking in any particular case by typing `^|\penalty|\<number>'
immediately after the atom in question; then the number you have specified
will be used instead of the ordinary penalty. For example, you can prohibit
breaking in the formula `$x=0$' by typing `|$x=\nobreak0$|', since
^|\nobreak| is an abbreviation for `|\penalty10000 |'.
\ddangerexercise Is there any difference between the results of
`|$x=\nobreak0$|' and `|${x=0}$|'?
\answer Neither formula will be broken between lines, but the thick spaces
in the second formula will be set to their natural width while the thick
spaces in the first formula will retain their stretchability.
\ddangerexercise How could you prohibit all breaks in formulas, by making only
a few changes to the macros of plain \TeX?
\answer Set ^|\relpenalty||=10000| and ^|\binoppenalty||=10000|.
And you also need to change the definitions of\/ ^|\bmod| and ^|\pmod|,
which insert their own penalties.
\subsection Braces. A variety of different notations have sprung up involving
the symbols `$\{$' and `$\}$'; plain \TeX\ includes several control
sequences that help you cope with formulas involving such things.
^^{braces} ^^{leftbrace} ^^{rightbrace}
In simple situations, braces are used to indicate a ^{set} of objects;
for example, `$\{a,b,c\}$' stands for the set of three objects $a$, $b$,
and~$c$. There's nothing special about typesetting such formulas, except
that you must remember to use |\{| and |\}| for the braces:
^^|\leftbrace| ^^|\rightbrace|
\beginmathdemo
|$\{a,b,c\}$|&\{a,b,c\}\cr
|$\{1,2,\ldots,n\}$|&\{1,2,\ldots,n\}\cr
|$\{\rm red,white,blue\}$|&\{\rm red,white,blue\}\cr
\endmathdemo
A slightly more complex case arises when a set is indicated by giving a
generic element followed by a specific condition; for example, `$\{\,x\mid
x>5\,\}$' stands for the set of all objects $x$ that are greater than~5.
In such situations the control sequence ^|\mid| should be used for the
^{vertical bar}, and thin spaces should be inserted inside the braces:
\beginmathdemo
|$\{\,x\mid x>5\,\}$|&\{\,x\mid x>5\,\}\cr
|$\{\,x:x>5\,\}$|&\{\,x:x>5\,\}\cr
\endmathdemo
(Some authors prefer to use a ^{colon} instead of `$\mid$', as in the second
example here.) \ When the delimiters get larger, as in
\begindisplay
$\displaystyle\bigl\{\,\bigl(x,f(x)\bigr)\bigm\vert x\in D\,\bigr\}$
\enddisplay
they should be called ^|\bigl|, ^|\bigm|, and~^|\bigr|; for example,
the formula just given would be typed
\begintt
\bigl\{\,\bigl(x,f(x)\bigr)\bigm|char`||x\in D\,\bigr\}
\endtt
and formulas that involve still larger delimiters would use ^|\Big| or
^|\bigg| or~even ^|\Bigg|, as explained in Chapter~17.
\exercise How would you typeset the formula
$\bigl\{\,x^3\bigm\vert h(x)\in\{-1,0,+1\}\,\bigr\}$?
\answer |$\bigl\{\,x^3\bigm|\||h(x)\in\{-1,0,+1\}\,\bigr\}$|.
\dangerexercise Sometimes the condition that defines a set is given as
a fairly long English description, not as a formula; for example, consider
`$\{\,p\mid p$~and $p+2$ are prime$\,\}$'. An hbox would do the job:
\begintt
$\{\,p\mid\hbox{$p$ and $p+2$ are prime}\,\}$
\endtt
but a long formula like this is troublesome in a paragraph, since an hbox
cannot be broken between lines, and since the glue inside the |\hbox| does
not vary with the interword glue in the line that contains it. Explain how
the given formula could be typeset with line breaks allowed. [{\sl Hint:\/}
Go back and forth between math ^{mode} and horizontal mode.]
\answer |$\{\,p\mid p$~and $p+2$ are prime$\,\}$|, assuming that
^|\mathsurround| is zero. The more difficult alternative
`|$\{\,p\mid p\ {\rm and}\ p+2\rm\ are\ prime\,\}$|' is not a solution,
because line breaks do not occur at |\|\] ^^|\space|
(or at glue of any kind) within math formulas. Of course it may be best to
display a formula like this, instead of breaking it between lines.
Displayed formulas often involve another sort of brace, to indicate a choice
between various alternatives, as in the construction
\begindisplay
$\displaystyle\vert x\vert=\cases{x,&if $x\ge0$;\cr -x,&otherwise.\cr}$
\enddisplay
^^{selection, see cases} ^^{alternatives, see cases} ^^{choices, see cases}
You can typeset it with the control sequence ^|\cases|:
\begintt
$$|char`||x||=\cases{x,&if $x\ge0$;\cr
-x,&otherwise.\cr}$$
\endtt
Look closely at this example and notice that it uses the character |&|,
^^{ampersand} which we said in Chapter~7 was reserved for special purposes.
Here for the first time in this manual we have an example of why |&|~is
so special: Each of the cases has two parts, and the~|&| separates those
parts. To the left of the~|&| is a math formula that is implicitly
enclosed in |$...$|; to the right of the~|&| is ordinary text, which is
{\sl not\/} implicitly enclosed in |$...$|. For example, the `|-x,|' in
the second line will be typeset in math mode, but the `|otherwise|' will
be typeset in horizontal mode. Blank spaces after the~|&| are ignored.
There can be any number of cases, but there usually are at least two.
Each case should be followed by ^|\cr|. Notice that the |\cases| construction
typesets its own `$\{$'; there is no corresponding `$\}$'.
\exercise Typeset the display \lower12pt\null\
$\smash{\displaystyle
f(x)=\cases{1/3&if $0\le x\le1$;\cr 2/3&if $3\le x\le4$;\cr 0&elsewhere.\cr}
}$
\answer |$$f(x)=\cases{1/3&if $0\le x\le1$;\cr 2/3&if $3\le x\le4$;\cr|\hfil
\break|0&elsewhere.\cr}$$|
\danger You can insert `^|\noalign||{|$\langle$vertical mode
material$\rangle$|}|' just after any \kern-1pt|\cr| within |\cases|, as
explained in Chapter~22, because |\cases| is an application of the general
alignment constructions considered in that chapter. For example, the
command `|\noalign{\vskip2pt}|' can be used to put a little extra space
between two of the cases.
\danger ^{Horizontal braces} will be set over or under parts of a displayed
formula if you use the control sequences ^|\overbrace| or ^|\underbrace|.
Such constructions are considered to be large operators like |\sum|, so you
can put limits above them or below them by specifying superscripts or
subscripts, as in the following examples:
\beginlongdisplaymathdemo
\noalign{\vskip9pt}
|$$\overbrace{x+\cdots+x}^{k\rm\;times}$$|&
\overbrace{x+\cdots+x}^{k\rm\;times}\cr
\noalign{\vskip-6pt}
|$$\underbrace{x+y+z}_{>\,0}.$$|&
\underbrace{x+y+z}_{>\,0}.\cr
\endmathdemo
\subsection Matrices. Now comes the fun part. Mathematicians in many different
disciplines like to construct rectangular arrays of formulas that have been
arranged in rows and columns; such an ^{array} is called a {\sl^{matrix}}.
Plain \TeX\ provides a ^|\matrix| control sequence that makes it convenient
to deal with the most common types of matrices.
For example, suppose that you want to specify the display
$$A=\left(\matrix{x-\lambda&1&0\cr
0&x-\lambda&1\cr
0&0&x-\lambda\cr}\right).$$
All you do is type
\begintt
$$A=\left(\matrix{x-\lambda&1&0\cr
0&x-\lambda&1\cr
0&0&x-\lambda\cr}\right).$$
\endtt
^^|\lambda|
This is very much like the |\cases| construction we looked at earlier;
each row of the matrix is followed by~|\cr|, and `|&|'~signs are used
between the individual entries of each row. Notice, however, that you are
supposed to put your own |\left| and |\right| delimiters around the matrix;
this makes |\matrix| different from |\cases|, which inserts a big `$\{$'
automatically. The reason is that |\cases| always involves a left brace,
but different delimiters are used in different matrix constructions. On
the other hand, parentheses are used more often than other delimiters, so
you can write ^|\pmatrix| if you want plain \TeX\ to fill in the
parentheses for you; the example above then reduces to
\begintt
$$A=\pmatrix{x-\lambda&...&x-\lambda\cr}.$$
\endtt
\dangerexercise Typeset the display \ \lower12pt\null
$\tenpoint\smash{\displaystyle
\left\lgroup\matrix{a&b&c\cr d&e&f\cr}\right\rgroup
\left\lgroup\matrix{u&x\cr v&y\cr w&z\cr}\right\rgroup
}$, \
using ^|\lgroup| and ^|\rgroup|.
\answer |$$\left\lgroup\matrix{a&b&c\cr d&e&f\cr}\right\rgroup|\hfil\break
|\left\lgroup\matrix{u&x\cr v&y\cr w&z\cr}\right\rgroup$$|.
\danger The individual entries of a matrix are normally centered in columns.
Each column is made as wide as necessary to accommodate the entries it
contains, and there's a ^{quad} of space between columns.
If you want to put something ^{flush right} in its column, precede it
by ^|\hfill|; if you want to put something ^{flush left} in its column,
follow it by~|\hfill|.
\danger Each entry of a matrix is treated separately from the others,
and it is typeset as a math formula in text style. Thus, for example,
if you say |\rm| in one entry, it does not affect the others.
Don't try to say `|{\rm x&y}|'.
Matrices often appear in the form of generic patterns that use ^{ellipses}
(i.e., dots) to indicate rows or columns that are left out. You can typeset
such matrices by putting the ellipses into rows and/or columns of their own.
Plain \TeX\ provides ^|\vdots| (vertical dots) and ^|\ddots| (diagonal dots)
as companions to ^|\ldots| for constructions like this. For example, the
^{generic matrix}
$$A=\pmatrix{a_{11}&a_{12}&\ldots&a_{1n}\cr
a_{21}&a_{22}&\ldots&a_{2n}\cr
\vdots&\vdots&\ddots&\vdots\cr
a_{m1}&a_{m2}&\ldots&a_{mn}\cr}$$
is easily specified:
\begintt
$$A=\pmatrix{a_{11}&a_{12}&\ldots&a_{1n}\cr
a_{21}&a_{22}&\ldots&a_{2n}\cr
\vdots&\vdots&\ddots&\vdots\cr
a_{m1}&a_{m2}&\ldots&a_{mn}\cr}$$
\endtt
\medskip
\exercise How can you get \TeX\ to produce the ^{column vector} ^^{vector}
\lower18pt\null\ $\smash{\displaystyle
\pmatrix{y_1\cr \vdots\cr y_k\cr}
}$\quad?
\answer |\pmatrix{y_1\cr \vdots\cr y_k\cr}|.
\danger Sometimes a matrix is bordered at the top and left by formulas
that give labels to the rows and columns. Plain \TeX\ provides a special
macro called ^|\bordermatrix| for this situation. For example, the display
$$\tenmath
M=\bordermatrix{&C&I&C'\cr C&1&0&0\cr I&b&1-b&0\cr C'&0&a&1-a\cr}$$
is obtained when you type
\begintt
$$M=\bordermatrix{&C&I&C'\cr
C&1&0&0\cr I&b&1-b&0\cr C'&0&a&1-a\cr}$$
\endtt
The first row gives the upper labels, which appear above the big left
and right parentheses; the first column gives the left labels, which are
typeset flush left, just before the matrix itself. The first column in
the first row is normally blank. Notice that |\bordermatrix| inserts
its own parentheses, like |\pmatrix| does.
\danger It's usually inadvisable to put matrices into the text of a paragraph,
because they are so big that they are better displayed. But occasionally
you may want to specify a small matrix like $1\,1\choose0\,1$, which you can
^^|\choose| ^^{matrix, small}
typeset for example as `|$1\,1\choose0\,1$|'. Similarly, the small matrix
$\bigl({a\atop l}{b\atop m}{c\atop n}\bigr)$ can be typeset as
\begintt
$\bigl({a\atop l}{b\atop m}{c\atop n}\bigr)$
\endtt
^^|\atop| The |\matrix| macro does not produce small arrays of this sort.
\subsection Vertical spacing. If you want to tidy up an unusual formula,
you know already how to move things farther apart or closer together, by
using positive or negative thin spaces. But such spaces affect only the
horizontal dimension; what if you want something to be moved higher
or lower? That's an advanced topic.
\danger Appendix B provides a few macros that can be used to fool \TeX\
into thinking that certain formulas are larger or smaller than they really
are; such tricks can be used to move other parts of the formula up or down
or left or right. For example, we have already discussed the use of
^|\mathstrut| in Chapter~16 and ^|\strut| in Chapter~17; these invisible
boxes caused \TeX\ to put square root signs and the denominators of
continued fractions into different positions than usual.
\danger If you say `^|\phantom||{|\<subformula>|}|' in any formula, plain
\TeX\ will do its spacing as if you had said simply
`|{|\<subformula>|}|', but the subformula itself will be invisible. Thus,
for example, `|\phantom{0}2|' takes up just as much space as `|02|' in the
current style, but only the~|2| will actually appear on the page. If you
want to leave blank space for a ^{new symbol} that has exactly the same
size as $\sum$, but if you are forced to put that symbol in by hand for
some reason, `|\mathop{\phantom\sum}|' will leave exactly the right amount
of blank space. \ (The `^|\mathop|' here makes this phantom behave like
|\sum|, i.e., as a large operator.)
\danger Even more useful than |\phantom| is ^|\vphantom|, which makes
an invisible box whose height and depth are the same as those of
the corresponding |\phantom|, but
the width is zero. Thus, |\vphantom| makes a vertical ^{strut} that can
increase a formula's effective height or depth. Plain \TeX\ defines
|\mathstrut| to be an abbreviation for `|\vphantom(|'. There's also
^|\hphantom|, which has the width of a |\phantom|, but its height
and depth are zero.
\danger Plain \TeX\ also provides `^|\smash||{|\<subformula>|}|', a macro
that yields the same result as `|{|\<subformula>|}|' but makes the height and
depth zero. By using both |\smash| and |\vphantom| you can typeset any
subformula and give it any desired nonnegative height and depth. For example,
\begintt
\mathop{\smash\limsup\vphantom\liminf}
\endtt
produces a large operator that says `$\limsup$', but its height and depth
are those of\/ ^|\liminf| (i.e., the depth is zero). ^^|\limsup|
\def\undertext#1{$\underline{\hbox{#1}}$}
\ddangerexercise If you want to underline some text, you could use a macro like
\begintt
\def\undertext#1{$\underline{\hbox{#1}}$}
\endtt
to do the job. \undertext{But} \undertext{this} \undertext{doesn't}
\undertext{always} \undertext{work} \undertext{right}. Discuss better
alternatives. ^^{underlined text}
\answer |\def|\stretch|\undertext|\stretch|#1{$\underline|\stretch
|{\smash|\stretch|{\hbox|\stretch|{#1}}}$}| will underline the
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}%
words and cross \undertext{through} the descenders; or you could insert
|\vphantom{y}| before the |\hbox|, thereby lowering all of the underlines
to a position below all descenders. Neither of these gives exactly what is
wanted. \ (See also ^|\underbar| in Appendix~B\null.) \ Underlining is actually
not very common in fine typography, since font changes usually work just
as well or better, when you want to emphasize something. If you really want
underlined text, it's best to have a special font in which all
the letters are underlined.
\ddanger You can also use ^|\raise| and ^|\lower| to adjust the vertical
positions of boxes in formulas. For example, the formula
`|$2^{\raise1pt\hbox{$\scriptstyle n$}}$|' will have its superscript~$n$
one point higher than usual ($2^{\raise1pt\hbox{$\scriptstyle n$}}$ instead of
$2^n$). Note that it was necessary to say ^|\scriptstyle| in this example,
since the contents of an ^|\hbox| will normally be in text style even when
that hbox appears in a superscript, and since |\raise| can be used only in
connection with a box. This method of positioning is not used extremely
often, but it is sometimes helpful if the ^|\root| macro doesn't put its
argument in a suitable place. For example,
\begindisplay
|\root\raise|\<dimen>|\hbox{$\scriptscriptstyle|\<argument>|$}\of...|
\enddisplay
will move the argument up by a given amount.
\ddanger Instead of changing the sizes of subformulas, or using |\raise|,
you can also control vertical spacing by changing the parameters
that \TeX\ uses when it is converting math lists to horizontal lists.
These parameters are described in Appendix~G\null; you need to be careful when
changing them, because such changes are ^{global} (i.e., not local to groups).
Here is an example of how such a change might be made: Suppose that you
are designing a format for ^{chemical typesetting}, and that you expect to be
setting a lot of formulas like `$\rm Fe_2^{+2}Cr_2O_4$'. You may not like the
fact that the subscript in~$\rm Fe_2^{+2}$ is lower than the subscript
in~$\rm Cr_2$; and you don't want to force users to type monstrosities like
\begintt
$\rm Fe_2^{+2}Cr_2^{\vphantom{+2}}O_4^{\vphantom{+2}}$
\endtt
just to get the formula
$\rm Fe_2^{+2}Cr_2^{\vphantom{+2}}O_4^{\vphantom{+2}}$
with all subscripts at the same level. Well, all you need to do is set
`^|\fontdimen||16\tensy=2.7pt|' and `|\fontdimen17\tensy=2.7pt|', assuming
that ^|\tensy| is your main symbol font (|\textfont2|); this lowers all
normal ^{subscripts} to a position $2.7\pt$ below the baseline, which is
enough to make room for a possible superscript that contains a plus sign.
Similarly, you can adjust the positioning of ^{superscripts} by changing
|\fontdimen14\tensy|. There are parameters for the position of the ^{axis line},
the positions of ^{numerator} and ^{denominator} in a generalized ^{fraction},
the spacing above and below ^{limits}, the default ^{rule thickness}, and so
on. Appendix~G gives precise details.
\subsection Special features for math hackers. \TeX\ has a few more primitive
operations for math mode that haven't been mentioned yet. They are
occasionally useful if you are designing special formats.
\ddanger If a glue or kern specification is immediately preceded by
`^|\nonscript|', \TeX\ will not use that glue or kern in
script or scriptscript styles. Thus, for example, the sequence
`|\nonscript\;|' produces exactly the amount of space specified by
`{\tt(3)}' in the spacing table for mathematics that appeared earlier
in this chapter.
\ddanger Whenever \TeX\ has scanned a |$| and is about to read a math formula
that appears in text, it will first read another list of tokens that has
been predefined by the command ^|\everymath||={|\<token list>|}|. \ (This is
analogous to |\everypar|, which was described in Chapter~14.) \ Similarly,
you can say ^|\everydisplay||={|\<token list>|}| to predefine a list of tokens
for \TeX\ to read just after it has scanned an opening |$$|, i.e., just
before reading a formula that is to be displayed. With |\everymath| and
|\everydisplay|, you can set up special conventions that you wish to apply
to all formulas.
\subsection Summary. We have discussed more different kinds of formulas in
this chapter than you will usually find in any one book of mathematics.
If you have faithfully done the exercises so far, you can face almost
any formula with confidence.
\newcount\chalcount \chalcount=0
\outer\def\challenge{\d@nger\chall}
\outer\def\cchallenge{\dd@nger\chall}
\def\chall{\global\advance\chalcount by1
\dexercise \hbox{Challenge number \the\chalcount:\enskip}\ignorespaces}
\danger But here are a few more exercises, to help you review what you
have learned. Each of the following ``challenge formulas'' illustrates one
or more of the principles already discussed in this chapter. The author
confesses that he is trying to trip you~up on several of these.
Nevertheless, if you try each one before looking at the answer, and if
you're alert for traps, you should find that these formulas provide a good
way to consolidate and complete your knowledge.
\challenge Explain how to type the phrase `$n^{\rm th}$ root', where
`$n^{\rm th}$' is treated as a mathematical formula with a superscript in
roman type.
\answer |$n^{\rm th}$ root|. \ (Incidentally, it is also acceptable
to type `|$n$th|', getting `$n$th', in such situations; the fact that
the $n$ is in italics distinguishes it from the suffix. Typed manuscripts
generally render this with a hyphen, but `$n$-th' is frowned on nowadays
when an italic~$n$ is available.) ^^{nth}
\challenge $\qquad\tenmath{\bf S^{\rm-1}TS=dg}(\omega_1,\ldots,\omega_n)
=\bf\Lambda$.
\answer |${\bf S^{\rm-1}TS=dg}(\omega_1,|\stretch|\ldots,|\stretch|\omega_n)
=\bf\Lambda$|.
\ $\bigl($Did you notice the difference between ^|\omega| ($\omega$)
and~|w| ($w$)?$\bigr)$
\challenge $\qquad\tenmath\Pr(\,m=n\mid m+n=3\,)$.
\answer |$\Pr(\,m=n\mid m+n=3\,)$|. \ (Analogous to a set.) ^^|\Pr|
\challenge $\qquad\tenmath\sin18^\circ={1\over4}(\sqrt5-1)$.^^{degrees}
\answer |$\sin18^\circ={1\over4}(\sqrt5-1)$|. ^^|\circ|
\challenge $\qquad\tenmath k=1.38065\times10^{-16}\rm\,erg\,K^{-1}$.
\answer |$k=1.38065\times10^{-16}\rm\,erg\,K^{-1}$|.
\challenge $\qquad\tenmath \bar\Phi\subset NL_1^*/N=\bar L_1^*
\subseteq\cdots\subseteq NL_n^*/N=\bar L_n^*$.
\answer |$\bar\Phi\subset NL_1^*/N=\bar L_1^*|\parbreak
| \subseteq\cdots\subseteq NL_n^*/N=\bar L_n^*$|.
\challenge $\qquad\tenmath I(\lambda)=\int\!\!\int_Dg(x,y)e^{i\lambda h(x,y)}
\,dx\,dy$. % cf. Math. Comp. 37 (1981), 509
\answer |$I(\lambda)=\int\!\!\int_Dg(x,y)e^{i\lambda h(x,y)}\,dx\,dy$|.\hfil
\break
(Although three |\!|'s work out best between consecutive integral signs in
displays, the text style seems to want only two.) ^^{double integral}
^^{integral, multiple}
\challenge $\qquad\tenmath
\int_0^1\!\cdots\int_0^1f(x_1,\ldots,x_n)\,dx_1\ldots\,dx_n$.
\answer |$\int_0^1\!\cdots\int_0^1f(x_1,\ldots,x_n)\,dx_1\ldots\,dx_n$|.
\challenge Here's a display.
$$\tenmath x_{2m}\equiv\cases{Q(X_m^2-P_2W_m^2)-2S^2&($m$ odd)\cr
\noalign{\vskip2pt}
P_2^2(X_m^2-P_2W_m^2)-2S^2&($m$ even)\cr}\pmod N.$$
\answer |$$x_{2m}\equiv\cases{Q(X_m^2-P_2W_m^2)-2S^2&($m$ odd)\cr|\parbreak
| \noalign{\vskip2pt} % spread the lines apart a little|\parbreak
| P_2^2(X_m^2-P_2W_m^2)-2S^2&($m$ even)\cr}\pmod N.$$|
\challenge And another. % ACP Eq. 1.2.9--33
$$\tenmath (1+x_1z+x_1^2z^2+\cdots\,)\ldots(1+x_nz+x_n^2z^2+\cdots\,)
={1\over(1-x_1z)\ldots(1-x_nz)}.$$
\answer |$$(1+x_1z+x_1^2z^2+\cdots\,)\ldots(1+x_nz+x_n^2z^2+\cdots\,)|\parbreak
| ={1\over(1-x_1z)\ldots(1-x_nz)}.$$| \ (Notice the uses of\/ |\,|.)
\challenge And another. % Eq. 1.2.9--9
$$\tenmath \prod_{j\ge0}\biggl(\sum_{k\ge0}a_{jk}z^k\biggr)
=\sum_{n\ge0}z^n\,\Biggl(\sum_
{\scriptstyle k_0,k_1,\ldots\ge0\atop
\scriptstyle k_0+k_1+\cdots=n}
a_{0k_0}a_{1k_1}\ldots\,\Biggr).$$
\answer |$$\prod_{j\ge0}\biggl(\sum_{k\ge0}a_{jk}z^k\biggr)|\parbreak
| =\sum_{n\ge0}z^n\,\Biggl(\sum_|\parbreak
| {\scriptstyle k_0,k_1,\ldots\ge0\atop|\parbreak
| \scriptstyle k_0+k_1+\cdots=n}|\parbreak
| a_{0k_0}a_{1k_1}\ldots\,\Biggr).$$|\par
\nobreak\smallskip\noindent Some people would prefer to have the latter
parentheses larger; but |\left| and |\right| come out a bit too large in this
case. It's not difficult to define ^|\bigggl| and ^|\bigggr| macros, analogous
to the definitions of\/ |\biggl| and |\biggr| in Appendix~B.
\challenge And, % cf ACP vol1 p64
$$\tenmath {(n_1+n_2+\cdots+n_m)!\over n_1!\,n_2!\ldots n_m!}
={n_1+n_2\choose n_2}{n_1+n_2+n_3\choose n_3}
\ldots{n_1+n_2+\cdots+n_m\choose n_m}.$$
\answer |$${(n_1+n_2+\cdots+n_m)!\over n_1!\,n_2!\ldots n_m!}|\parbreak
| ={n_1+n_2\choose n_2}{n_1+n_2+n_3\choose n_3}|\parbreak
| \ldots{n_1+n_2+\cdots+n_m\choose n_m}.$$|
\challenge Yet another display. % found in Chaundy et al
$$\tenmath \def\\#1#2{(1-q^{#1_#2+n})} % to save typing
\Pi_R{a_1,a_2,\ldots,a_M\atopwithdelims[]b_1,b_2,\ldots,b_N}
=\prod_{n=0}^R{\\a1\\a2\ldots\\aM\over\\b1\\b2\ldots\\bN}.$$
\answer |$$\def\\#1#2{(1-q^{#1_#2+n})} % to save typing|\parbreak
|\Pi_R{a_1,a_2,\ldots,a_M\atopwithdelims[]b_1,b_2,\ldots,b_N}|\parbreak
| =\prod_{n=0}^R{\\a1\\a2\ldots\\aM\over\\b1\\b2\ldots\\bN}.$$|
^^|\atopwithdelims|
\challenge And another.
$$\tenmath \sum_{p\rm\;prime}f(p)=\int_{t>1}f(t)\,d\pi(t).$$
\answer |$$\sum_{p\rm\;prime}f(p)=\int_{t>1}f(t)\,d\pi(t).$$|
\challenge Still another.
$$\tenmath \{\underbrace{\overbrace{\mathstrut a,\ldots,a}
^{k\;a\mathchar`'\rm s},
\overbrace{\mathstrut b,\ldots,b}
^{l\;b\mathchar`'\rm s}}_{k+l\rm\;elements}\}.$$
\answer |$$\{\underbrace{\overbrace{\mathstrut a,\ldots,a}|\parbreak
| ^{k\;a\mathchar`'\rm s},|\parbreak
| \overbrace{\mathstrut b,\ldots,b}|\parbreak
| ^{l\;b\mathchar`'\rm s}}_{k+l\rm\;elements}\}.$$|\par
\smallskip\noindent Notice how ^{apostrophes} (instead of primes) were obtained.
\challenge Put a ^|\smallskip| between the rows of matrices in the
compound matrix ^^{compound matrix}
$$\tenmath \pmatrix{\pmatrix{a&b\cr c&d\cr}&
\pmatrix{e&f\cr g&h\cr}\cr
\noalign{\smallskip}
0&\pmatrix{i&j\cr k&l\cr}\cr}.$$
\answer |$$\pmatrix{\pmatrix{a&b\cr c&d\cr}&|\parbreak
| \pmatrix{e&f\cr g&h\cr}\cr|\parbreak
| \noalign{\smallskip}|\parbreak
| 0&\pmatrix{i&j\cr k&l\cr}\cr}.$$|
\challenge Make the columns ^{flush left} here. % cf Polya/Szego VII.43.2
$$\tenmath \det\left\vert\,\matrix{
c_0&c_1\hfill&c_2\hfill&\ldots&c_n\hfill\cr
c_1&c_2\hfill&c_3\hfill&\ldots&c_{n+1}\hfill\cr
c_2&c_3\hfill&c_4\hfill&\ldots&c_{n+2}\hfill\cr
\,\vdots\hfill&\,\vdots\hfill&
\,\vdots\hfill&&\,\vdots\hfill\cr
c_n&c_{n+1}\hfill&c_{n+2}\hfill&\ldots&c_{2n}\hfill\cr
}\right\vert>0.$$
\answer |$$\det\left|\||\,\matrix{|\parbreak
| c_0&c_1\hfill&c_2\hfill&\ldots&c_n\hfill\cr|\parbreak
| c_1&c_2\hfill&c_3\hfill&\ldots&c_{n+1}\hfill\cr|\parbreak
| c_2&c_3\hfill&c_4\hfill&\ldots&c_{n+2}\hfill\cr|\parbreak
| \,\vdots\hfill&\,\vdots\hfill&|\parbreak
| \,\vdots\hfill&&\,\vdots\hfill\cr|\parbreak
| c_n&c_{n+1}\hfill&c_{n+2}\hfill&\ldots&c_{2n}\hfill\cr|\parbreak
| }\right|\||>0.$$|
\cchallenge The main problem here is to prime the $\sum$.
^^|\sum prime| ^^{=def}
$$\tenmath \mathop{{\sum}'}_{x\in A}f(x)\mathrel{\mathop=^{\rm def}}
\sum_{\scriptstyle x\in A\atop\scriptstyle x\ne0}f(x).$$
\answer |$$\mathop{{\sum}'}_{x\in A}f(x)\mathrel{\mathop=^{\rm def}}|\parbreak
| \sum_{\scriptstyle x\in A\atop\scriptstyle x\ne0}f(x).$$|\par
\smallskip\noindent
This works because |{\sum}| is type Ord (so its superscript is not set
above), but ^|\mathop||{{\sum}'}| is type Op (so its subscript is set below).
The limits are centered on $\sum'$, however, not on $\sum$. If you don't
like that, the remedy is more difficult; one solution is to use
|\sumprime_{x\in A}| where ^|\sumprime| is defined as follows:
\par\nobreak\medskip
|\def\sumprime_#1{\setbox0=\hbox{$\scriptstyle{#1}$}|\parbreak
| \setbox2=\hbox{$\displaystyle{\sum}$}|\parbreak
| \setbox4=\hbox{${}'\mathsurround=0pt$}|\parbreak
| \dimen0=.5\wd0 \advance\dimen0 by-.5\wd2|\parbreak
| \ifdim\dimen0>0pt|\parbreak
| \ifdim\dimen0>\wd4 \kern\wd4 \else\kern\dimen0\fi\fi|\parbreak
| \mathop{{\sum}'}_{\kern-\wd4 #1}}|
\cchallenge You may be ready now for this display.
$$\tenmath 2\uparrow\uparrow k\mathrel{\mathop=^{\rm def}}
2^{2^{2^{\cdot^{\cdot^{\cdot^2}}}}}
\vbox{\hbox{$\Big\}\scriptstyle k$}\kern0pt}.$$
\answer |$$2\uparrow\uparrow k\mathrel{\mathop=^{\rm def}}|\parbreak
| 2^{2^{2^{\cdot^{\cdot^{\cdot^2}}}}}|\parbreak
| \vbox{\hbox{$\Big\}\scriptstyle k$}\kern0pt}.$$|\par
\cchallenge And finally, when you have polished off all the other examples,
here's the ultimate test. Explain how to obtain the ^{commutative diagram}
% from Invent. Math. 70 (1982), 34 % with "typo" corrected 99.12.03
$$\tenmath\def\normalbaselines{\baselineskip20pt\lineskip1pt\lineskiplimit0pt }
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow
\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{\noalign{\vskip6pt}&&&&&&0\cr
&&&&&&\mapdown{}\cr
0&\mapright{}&{\cal O}_C&\mapright\iota&
\cal E&\mapright\rho&\cal L&\mapright{}&0\cr
&&\Big\Vert&&\mapdown\phi&&\mapdown\psi\cr
0&\mapright{}&{\cal O}_C&\mapright\pi&
\pi_*{\cal O}_D&\mapright\delta&
R^1f_*{\cal O}_V(-D)&\mapright{}&0\cr
&&&&&&\mapdown{\theta_i\otimes\gamma^{-1}}\cr
&&&&&&\hidewidth R^1f_*\bigl({\cal O}
_V(-iM)\bigr)\otimes\gamma^{-1}\hidewidth\cr
&&&&&&\mapdown{}\cr
&&&&&&0\cr\noalign{\vskip6pt}}$$
using ^|\matrix|. \ (Many of the entries are blank.)
\answer If you have to do a lot of commutative diagrams, you will want to
define some macros like those in the first few lines of this solution.
The ^|\matrix| macro resets the baselines to ^|\normalbaselines|, because
other commands like |\openup| might have changed them, so
we redefine |\normalbaselines| in this solution. Some of the things
shown here haven't been explained yet, but Chapter~22 will reveal all.
\smallskip
|$$\def\normalbaselines{\baselineskip20pt|\parbreak
| \lineskip3pt \lineskiplimit3pt }|\parbreak
|\def\mapright#1{\smash{|\parbreak
| \mathop{\longrightarrow}\limits^{#1}}}|\parbreak
|\def\mapdown#1{\Big\downarrow|\parbreak
| \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}|\parbreak
|\matrix{&&&&&&0\cr|\parbreak
| &&&&&&\mapdown{}\cr|\parbreak
| 0&\mapright{}&{\cal O}_C&\mapright\iota&|\parbreak
| \cal E&\mapright\rho&\cal L&\mapright{}&0\cr|\parbreak
| &&\Big\Vert&&\mapdown\phi&&\mapdown\psi\cr|\parbreak
| 0&\mapright{}&{\cal O}_C&\mapright\pi&|\parbreak
| \pi_*{\cal O}_D&\mapright\delta&|\parbreak
| R^1f_*{\cal O}_V(-D)&\mapright{}&0\cr|\parbreak
| &&&&&&\mapdown{\theta_i\otimes\gamma^{-1}}\cr|\parbreak
| &&&&&&\hidewidth R^1f_*\bigl({\cal O}|\parbreak
| _V(-iM)\bigr)\otimes\gamma^{-1}|^|\hidewidth||\cr|\parbreak
| &&&&&&\mapdown{}\cr|\parbreak
| &&&&&&0\cr}$$|
\subsection Words of advice. The number of different notations is
enormous and still growing, so you will probably continue to find
new challenges as you continue to type mathematical papers. It's a
good idea to keep a personal notebook in which you record all of
the non-obvious formulas that you have handled successfully,
showing both the final output and what you typed to get it.
Then you'll be able to refer back to those solutions when you
discover that you need to do something similar, a few months later.
If you're a mathematician who types your own papers, you have now learned
^^{author, typesetting by}
how to get enormously complex formulas into print, and you can do so
without going through an intermediary who may somehow distort their
meaning. But please, don't get too carried away by your newfound talent;
the fact that you are able to typeset your formulas with \TeX\ doesn't
necessarily mean that you have found the best notation for communicating
with the readers of your work. Some notations will be unfortunate even
when they are beautifully formatted.
\endchapter
Mathematicians are like Frenchmen:\/
% Die Mathematiker sind eine Art Franzosen:
whenever you say something to them, they translate it into their own language,
% redet man zu ihnen, so u"bersetzen sie es in ihre Sprache,
and at once it is something entirely different.
% und dann ist es alsobald ganz etwas Anders.
\author ^{GOETHE}, {\sl Maxims and Reflexions\/} (1829)
% see Schriften der Goethe-Gesellschaft, vol 21, pp 266 and 389
\bigskip
The best notation is no notation;
whenever it is possible to avoid the use of a complicated alphabetic apparatus,
avoid it.
A good attitude to the preparation of written mathematical exposition
is to pretend that it is spoken.
Pretend that you are explaining the subject to a friend
on a long walk in the woods, with no paper available;
fall back on symbolism only when it is really necessary.
\author PAUL ^{HALMOS}, {\sl How to Write Mathematics\/} (1970)
% in {\sl L'Enseignement Math\'ematique\/}
% vol 16, 123--152; section 15; reprinted in AMS pub "How to Write Math"
\eject
\beginchapter Chapter 19. Displayed Equations
By now you know how to type mathematical formulas so that \TeX\ will handle
them with supreme elegance; your knowledge of math typing is nearly complete.
But there is one more part to the story, and the purpose of this chapter
is to present the happy ending. We have discussed how to deal with individual
formulas; but ^{displays} often involve a whole bunch of different formulas,
or different pieces of a huge formula, and it's a bit of a problem to
lay them out so that they line up properly with each other. Fortunately,
large displays generally fall into a few simple patterns.
\subsection One-line displays. Before plunging into the general question
of display layout, let's recapitulate what we have already covered. If
you type `|$$|\<formula>|$$|', \TeX\ will display the formula
^^{dollardollar} in flamboyant display style, centering it on a line by
itself. We have also noted in Chapter~18 that it's possible to display two
short formulas at once, by typing
`|$$|\<formula$_1$>^|\qquad|\<formula$_2$>|$$|'; this reduces the
two-formula problem to a one-formula problem. You get the two formulas
separated by two quads of space, the whole being centered on a line.
Displayed equations often involve ordinary text. Chapter~18 explains
how to get roman type into formulas without leaving math mode, but the
best way to get text into a display is to put it into an ^|\hbox|.
There needn't even be any math at all; to typeset
$$\hbox{Displayed Text}$$
you can simply say `|$$\hbox{Displayed Text}$$|'. But here's a more interesting
example:
$$X_n=X_k \qquad\hbox{if and only if}\qquad
Y_n=Y_k \quad\hbox{and}\quad Z_n=Z_k.$$
Formulas and text were combined in this case by typing
\begintt
$$X_n=X_k \qquad\hbox{if and only if}\qquad
Y_n=Y_k \quad\hbox{and}\quad Z_n=Z_k.$$
\endtt
Notice that |\qquad| appears around `if and only if', but a single ^|\quad|
surrounds `and'; this helps to indicate that the $Y$ and~$Z$ parts of the
display are related more closely to each other than to the $X$~part.
Consider now the display
$$Y_n=X_n\bmod p \quad\hbox{and}\quad Z_n=X_n\bmod q
\qquad\hbox{for all }n\ge0.$$
Can you figure out how to type this? One solution is
\begintt
$$Y_n=X_n\bmod p \quad\hbox{and}\quad Z_n=X_n\bmod q
\qquad\hbox{for all }n\ge0.$$
\endtt
Notice that a space has been left after `|all|' in the hbox here, since spaces
disappear when they are out in formula-land. But there's a simpler and more
logical way to proceed, once you get used to \TeX's idea of modes: You can type
\begintt
... \qquad\hbox{for all $n\ge0$.}$$
\endtt
Wow---that's math mode inside of horizontal mode inside of display
math mode. But in this way your manuscript mirrors what you are trying to
accomplish, while the previous solution (with the space after `|all|')
looks somewhat forced.
\exercise Typeset the following four displays (one at a time):
$$\openup1\jot\displaylines{
\sum_{n=0}^\infty a_nz^n\qquad\hbox{converges if}\qquad
\vert z\vert<\Bigl(\limsup_{n\to\infty}
\root n\!\of{\vert a_n\vert}\,\Bigr)^{-1}.\cr
{f(x+\Delta x)-f(x)\over\Delta x}\to f'(x)\qquad\hbox{as $\Delta x\to0$.}\cr
\noalign{\vskip2pt}
\Vert u_i\Vert=1,\qquad u_i\cdot u_j=0\quad\hbox{if $i\ne j$.}\cr
\it\hbox{The confluent image of}\quad\left\{
\matrix{\hbox{an arc}\hfill\cr\hbox{a circle}\hfill\cr
\hbox{a fan}\hfill\cr}
\right\}\quad\hbox{is}\quad\left\{
\matrix{\hbox{an arc}\hfill\cr\hbox{an arc or a circle}\hfill\cr
\hbox{a fan or an arc}\hfill\cr}\right\}.\cr
\noalign{\vskip-8pt}}$$
^^|\Delta| ^^|\Vert|
% the last example comes from Proc AMS 55 (1976), 410, with typos corrected
\answer |$$\sum_{n=0}^\infty a_nz^n\qquad\hbox{converges if}\qquad|\parbreak
| |\||z|\||<\Bigl(\limsup_{n\to\infty}\root n\!\of{|\||a_n|\|^^|\root|
|}\,\Bigr)^{-1}.$$|\kern-.33pt\par
\smallskip
|$${f(x+\Delta x)-f(x)\over\Delta x}\to f'(x)|\parbreak
| \qquad\hbox{as $\Delta x\to0$.}$$|\par
\smallskip
|$$\|\||u_i\|\||=1,\qquad u_i\cdot u_j=0\quad\hbox{if $i\ne j$.}$$|\par
\smallskip
|$$\it\hbox{The confluent image of}\quad\left\{|\parbreak
| \matrix{\hbox{an arc}\hfill\cr\hbox{a circle}\hfill\cr|\parbreak
| \hbox{a fan}\hfill\cr}|\parbreak
| \right\}\quad\hbox{is}\quad\left\{|\parbreak
| \matrix{\hbox{an arc}\hfill\cr|\parbreak
| \hbox{an arc or a circle}\hfill\cr|\parbreak
| \hbox{a fan or an arc}\hfill\cr}\right\}.$$|\par
\smallskip\noindent
The first example includes |\!| and |\,| to give slightly refined spacing;
but the point of the problem was to illustrate the hbox, not to fuss over
such extra details.
The last example can be done much more simply using the ideas of
Chapter~22, if you don't mind descending to the level of \TeX\ primitives;
for example, the first matrix could be replaced by ^^|\halign|
\begintt
\,\vcenter{\halign{#\hfil\cr an arc\cr a circle\cr a fan\cr}}\,
\endtt
and the second is similar.
\dangerexercise Sometimes display style is too grandiose, when the formula
being displayed is
$$y={1\over2}x$$
or something equally simple. One day B. L. ^{User} tried to remedy this by
typing it as `|$$y={\scriptstyle1\over\scriptstyle2}x$$|', but the
resulting formula
$$y={\scriptstyle1\over\scriptstyle2}x$$
wasn't at all what he had in mind. What's the right way to get ^^{one half}
^^{1/2--unslashed form} simply `$y={1\over2}x$' when you don't want big
^{fractions in displays}?
\answer |$$\textstyle y={1\over2}x$$|. \ (Switching to text style is
especially common in multiline formulas. For example, you will probably
find occasions to use ^|\textstyle| on both sides of the |&|'s within
an ^|\eqalign|.)
\dangerexercise What difference, if any, is there between the result of
typing `|$$|\<formula>|$$|' and the result of typing
`|$$\hbox{$|\<formula>|$}$$|'\thinspace?
\answer The latter formula will be in text style, not display style.
And even if you do type `|$$\hbox{$\displaystyle{|\<formula>|}$}$$|', the
results are not quite the same, as we will see later: \TeX\ will compress
the glue in `|$$|\<formula>|$$|' if the formula is too wide to fit on
a line at its natural width, but the glue inside |\hbox{...}| is frozen
at its natural width. The |\hbox| version also invokes |\everymath|.
\dangerexercise You may have noticed that most of the displays in this
manual are not centered; displayed material is usually aligned at the
left with the paragraph indentation, as part of the book design, because
this is an unusual book. Explain how you could typeset a formula like
$$\leftline{\indent$\displaystyle
1-{1\over2}+{1\over3}-{1\over4}+\cdots=\ln2$}$$
^^{displays, non-centered}
that is off-center in this way.
\answer One solution is to put the formula in an hbox that occupies a full line:
\begintt
$$\leftline{\indent$\displaystyle
1-{1\over2}+{1\over3}-{1\over4}+\cdots=\ln2$}$$
\endtt
But this takes a bit of typing. If you make the definitions
\begintt
\def\leftdisplay#1$${\leftline{\indent$\displaystyle{#1}$}$$}
\everydisplay{\leftdisplay}
\endtt
you can type `|$$|\<formula>|$$|' as usual, and the formatting will be
inserted automatically. \ (This doesn't work with equation numbers;
Appendix~D illustrates how to handle them as well.)
If you've had previous experience typing mathematical papers, you probably
have been thinking, ``What about ^{equation numbers}? When is this book
going to talk about them?'' Ah yes, now is the time to discuss those sneaky
little labels that appear off to the side of displays. If you type
\begindisplay
|$$|\<formula>|\eqno|\<formula>|$$|
\enddisplay
\TeX\ will display the first formula and it will also put an equation number
(the second formula) at the right-hand margin. For example,
\begintt
$$x^2-y^2 = (x+y)(x-y).\eqno(15)$$
\endtt
^^|\eqno|
will produce this:
$$x^2-y^2 = (x+y)(x-y).\eqno(15)$$
You can also get equation numbers at the left-hand margin, with ^|\leqno|.
For example,
\begintt
$$x^2-y^2 = (x+y)(x-y).\leqno(16)$$
\endtt
will produce this:
$$x^2-y^2 = (x+y)(x-y).\leqno(16)$$
Notice that you always give the equation number second, even when it is going
to appear at the left. Everything from the |\eqno| or |\leqno| command to
the |$$| that ends the display is the equation number. Thus, you're not
allowed to have two equation numbers in the same display; but there's a
way to get around that restriction, as we'll see later.
\danger Nowadays people are using right-hand equation numbers more and more,
because a display most often comes at the end of a sentence or clause, and
the right-hand convention keeps the number from intruding into the clause.
Furthermore, it's often possible to save space when a displayed equation
follows a short text line, since less space is needed above the display;
such savings are not possible with |\leqno|, because there's no room for
overlap. For example, there is less space above display~(15) than there is
above~(16) in our illustrations of\/ |\eqno| and |\leqno|, although the
formulas and text are otherwise identical.
\danger If you look closely at (15) and (16) above, you can see that the
displayed formulas have been centered without regard to the presence of
the equation numbers. But when a formula is large, \TeX\ makes sure that
it does not interfere with its number; the equation number may even be
placed on a line by itself.
\exercise How would you produce the following display?
$$\prod_{k\ge0}{1\over(1-q^kz)}=
\sum_{n\ge0}z^n\bigg/\!\!\prod_{1\le k\le n}(1-q^k).\eqno(16')$$
\answer |$$\prod_{k\ge0}{1\over(1-q^kz)}=|\parbreak
| \sum_{n\ge0}z^n\bigg/\!\!\prod_{1\le k\le n}(1-q^k).\eqno(16')$$|
\dangerexercise Equation numbers are math formulas, typeset in text style.
So how can you get an equation number like `\hbox{(3--1)}'
(with an ^{en-dash})?
\answer |\eqno\hbox{(3--1)}|.
\ddangerexercise B. L. ^{User} tried typing `|\eqno(*)|' and `|\eqno(**)|',
and he was pleased to discover that this produced the equation numbers
`$(*)$' and `$(**)$'. \ [He had been a bit worried that they would come out
`(*)' and `(**)' instead.] \ But then a few months later he tried
`|\eqno(***)|' and got a surprise. What was it?
\answer When you type an ^{asterisk} in math mode, plain \TeX\ considers
|*| to be a binary operation. In the cases `|(*)|' and `|(**)|', the
binary operations are converted to type~Ord, because they don't appear in
a binary context; but the middle asterisk in `|(***)|' remains of type~Bin.
So the result was `$(***)$'. To avoid the extra medium spaces, you can
type `|\eqno(*{*}*)|'; or you can change ^|\mathcode||`*|, if you never use
|*| as a binary operation.
\ddanger Somewhere in this manual there ought to be a description of exactly
how \TeX\ displays formulas; i.e., how it centers them, how it places the
equation numbers, how it inserts extra space above and below, and so on.
Well, now is the time for those rules to be stated. They are somewhat
complex, because they interact with things like |\parshape|, and because
they involve several parameters that haven't been discussed yet. The purpose
of the rules is to explain exactly what sorts of boxes, glue, and
penalties are placed onto the current ^{vertical list} when a display occurs.
\ddanger If a display occurs after, say, four lines of a paragraph, \TeX's
internal register called ^|\prevgraf| will be equal to~4 when the display
starts. The display will be assumed to take three lines, so |\prevgraf|
will become~7 when the paragraph is resumed at the end of the display
(unless you have changed |\prevgraf| in the meantime). \TeX\ assigns
special values to three \<dimen> parameters immediately after the opening
|$$| is sensed: ^|\displaywidth| and ^|\displayindent| are set to the
line width~$z$ and the shift amount~$s$ for line number |\prevgraf|${}+2$,
based on the current paragraph shape or hanging indentation. \ (Usually
|\displaywidth| is the same as ^|\hsize|, and |\displayindent| is zero,
but the paragraph shape can vary as described in Chapter~14.) \ Furthermore,
^|\predisplaysize| is set to the effective width~$p$ of the line preceding
the display, as follows: If there was no previous line (e.g., if the |$$|
was preceded by ^|\noindent| or by the closing |$$| of another display),
$p$~is set to $-16383.99999\pt$ (i.e., to the smallest legal dimension,
$-$^|\maxdimen|). Otherwise \TeX\ looks inside the hbox that was formed
by the previous line, and sets $p$ to the position of the right edge of
the rightmost box inside that hbox, plus the indentation by which the
enclosing hbox has been moved right, plus two ems in the current font.
However, if this value of~$p$ depends on the fact that glue in that hbox
was stretching or shrinking---for example, if the ^|\parfillskip| glue is
finite, so that the material preceding it has not been set at its natural
width---then $p$~is set to |\maxdimen|. \ (This doesn't happen
often, but it keeps \TeX\ machine independent, since $p$~never depends on
quantities that may be rounded differently on different computers.) \
Notice that |\displaywidth| and |\displayindent| are not
affected by |\leftskip| and |\rightskip|, but
|\predisplaysize| is. The values of\/ |\displaywidth|,
|\displayindent|, and |\predisplaysize|
will be used by \TeX\ after the displayed formula has been
read, as explained below; your program can examine them and/or change
them, if you want the typesetting to be done differently.
\ddanger After a display has been read, \TeX\ converts it from a math list
to a horizontal list~$h$ in display style, as explained in
Appendix~G\null. An equation number, if present, is processed in text
style and put into an hbox~$a$ with its natural width. Now the fussy
processing begins: Let $z$, $s$, and~$p$ be the current values of\/\
|\displaywidth|, |\displayindent|, and
|\predisplaysize|. Let $q$ and~$e$ be zero if there is no equation
number; otherwise let~$e$ be the width of the equation number, and let~$q$
be equal to $e$~plus one quad in the symbols font (i.e., in
^|\textfont||2|). Let $w_0$ be the natural width of the displayed
formula~$h$. If $w_0+q\le z$, list~$h$~is packaged in an hbox~$b$ having
its natural width~$w_0$. But if $w_0+q>z$ (i.e., if the display is too
wide to fit at its natural width), \TeX\ performs the following ``^{squeeze
routine}'': If $e\ne0$ and if there is enough shrinkability in the
displayed formula~$h$ to reduce its width to $z-q$, then list $h$ is
packaged in an hbox~$b$ of width~$z-q$. Otherwise $e$ is set to zero, and
list~$h$ is packaged in a (possibly overfull) hbox~$b$ of width $\min(w_0,z)$.
\ddanger (Continuation.) \ \TeX\ tries now to center the display without
regard to the equation number. But if such centering would make it too close
to that number (where ``too close'' means that the space between them is less
than the width~$e$), the equation is either centered in the remaining space
or placed as far from the equation number as possible. The latter alternative
is chosen only if the first item on list~$h$ is glue, since \TeX\ assumes that
such glue was placed there in order to control the spacing precisely.
But let's state the rules more formally: Let~$w$ be the width of box~$b$.
\TeX\ computes a displacement~$d$, to be used later when positioning box~$b$,
by first setting $d={1\over2}(z-w)$. If $e>0$ and if $d<2e$, then $d$~is
reset to ${1\over2}(z-w-e)$ or to zero, where zero is chosen if list~$h$
begins with a glue item.
\ddanger (Continuation.) \ \TeX\ is now ready to put things onto the current
vertical list, just after the material previously constructed for the
paragraph-so-far. First comes a ^{penalty} item, whose cost is an integer
parameter called ^|\predisplaypenalty|. Then comes glue. If $d+s\le p$,
or if there was a left equation number (|\leqno|), \TeX\ sets $g_a$ and
$g_b$ to glue items specified by the parameters ^|\abovedisplayskip|
and ^|\belowdisplayskip|, respectively; otherwise $g_a$ and $g_b$ become
glue items corresponding to ^|\abovedisplayshortskip| and
^|\belowdisplayshortskip|. \ [Translation: If the predisplaysize is short
enough so that it doesn't overlap the displayed formula, the glue above and
below the display will be ``short'' by comparison with the glue that is
used when there is an overlap.] \ If $e=0$ and if there is an |\leqno|,
the equation number is appended as an hbox by itself, shifted right~$s$ and
preceded by interline glue as usual; an infinite penalty is also appended,
to prevent a page break between this number and the display. Otherwise
a glue item~$g_a$ is placed on the vertical list.
\ddanger (Continuation.) \ Now comes the displayed equation itself. If
$e\ne0$, the equation number box~$a$ is combined with the formula box~$b$ as
follows: Let~$k$ be a kern of width $z-w-e-d$. In the |\eqno| case, box~$b$
is replaced by an hbox containing $(b,k,a)$; in the |\leqno| case, box~$b$
is replaced by an hbox containing $(a,k,b)$, and $d$~is set to zero. In all
cases, box~$b$ is then appended to the vertical list, shifted right by~$s+d$.
\ddanger (Continuation.) \ The final task is to append the glue or the
equation number that follows the display. If there was an |\eqno| and if
$e=0$, an infinite penalty is placed on the vertical list, followed by the
equation number box~$a$ shifted right by $s+z$ minus its width, followed
by a penalty item whose cost is the value of\/ ^|\postdisplaypenalty|.
Otherwise a penalty item for the |\postdisplaypenalty| is appended
first, followed by a glue item for~$g_b$ as specified above. \TeX\ now
adds~3 to |\prevgraf| and returns to horizontal mode, ready to resume the
paragraph.
\ddanger One consequence of these rules is that you can force an equation
number to appear on a line by itself by making its width zero, i.e.,
by saying either `|\eqno|^|\llap||{$|\<formula>|$}|' or
`|\leqno|^|\rlap||{$|\<formula>|$}|'. This makes $e=0$, and
the condition $e=0$ controls \TeX's positioning logic, as explained
in the rules just given.
\ddanger Plain \TeX\ sets |\predisplaypenalty=10000|, because fine
printers traditionally shun displayed formulas at the very top of a page.
You can change |\predisplaypenalty| and
|\postdisplaypenalty| if you want to encourage or discourage
page breaks just before or just after a display. For example,
`\hbox{|$$\postdisplaypenalty=|}\allowbreak\hbox{|-10000|\<formula>|$$|}'
will force a page break, putting the formula at the bottom line. It is better
to force a ^{page break} this way than to say ^|\eject| right after |$$...$$|;
such an eject (which follows the |\belowdisplayskip| glue below the
display) causes the page to be short, because it leaves unwanted glue at
the bottom.
\ddangerexercise Read the rules carefully and deduce the final position
of `$x=y$' in the formula
\begintt
$$\quad x=y \hskip10000pt minus 1fil \eqno(5)$$
\endtt
assuming that there is no hanging indentation. Also consider |\leqno|
instead of\/ |\eqno|.
\answer Assuming that |\hsize| is less than $10000\pt$, the natural width of
this equation will be too large to fit on a line; also, |\quad| specifies
glue at the left. Therefore `$x=y$' will appear exactly $1\em$ from
the left, and `(5)' will appear flush right. \ (The widths will satisfy
^^{displays, non-centered} $w=z-q$, $d=0$, $k=q-e=18\rm\,mu$.) \
In the case of\/ |\leqno|, `(5)' will appear flush left, followed by
one quad of space in |\textfont2|, followed by one quad of space in the
current text font, followed by `$x=y$'.
\ddanger \TeX\ also allows ``^{alignment displays},'' which are not
processed in math mode because they contain no formulas at the outer
level. An alignment display is created by commands of the general form
\begindisplay
|$$|\<assignments>|\halign{|\<alignment>|}|\<assignments>|$$|
\enddisplay
where the \<assignments> are optional things like parameter changes that do not
produce any math lists. In such displays, the |\halign| is processed exactly
as if it had appeared in vertical mode, and it will construct a vertical
list~$v$ as usual, except that each row of the alignment will be shifted
right by the ^|\displayindent|. After the alignment and the closing
assignments have been processed, \TeX\ will put a ^|\predisplaypenalty|
item and some ^|\abovedisplayskip| glue on the main vertical list, followed
by~$v$, followed by a ^|\postdisplaypenalty| item and ^|\belowdisplayskip|
glue. Thus, alignment displays are essentially like ordinary alignments,
except that they can interrupt paragraphs; furthermore, they are embedded in
glue and penalties just like other displays. The ^|\displaywidth| and
^|\predisplaysize| do not affect the result, although you could use
those parameters in your ^|\halign|. An entire alignment display is considered
to be only three lines long, as far as ^|\prevgraf| is concerned.
\subsection Multi-line displays. OK, the use of displayed formulas is
very nice. But when you try typing a lot of manuscripts you will run into
some displays that don't fit the simple pattern of a one-line formula with
or without an equation number. Plain \TeX\ provides special control
sequences that will cover most of the remaining cases.
Multi-line displays usually consist of several equations that should be
lined up by their `$=$'~signs, as in
$$\eqalign{X_1+\cdots+X_p&=m,\cr
Y_1+\cdots+Y_q&=n.\cr}$$
The recommended procedure for such a display is to use ^|\eqalign|,
which works with special markers |&| ^^{ampersand} and ^|\cr| that we
have already encountered in connection with |\cases| and |\matrix|
in Chapter~18. Here's how to type this particular one:
\begintt
$$\eqalign{X_1+\cdots+X_p&=m,\cr
Y_1+\cdots+Y_q&=n.\cr}$$
\endtt
There can be any number of equations in an |\eqalign|; the general pattern is
$$\halign{\indent#&#\hfil\cr
|\eqalign{|&\<left-hand side$_1$>|&|\<right-hand side$_1$>|\cr|\cr
&\<left-hand side$_2$>|&|\<right-hand side$_2$>|\cr|\cr
\noalign{\vskip-2pt}
&\qquad\vdots\cr
&\<left-hand side$_n$>|&|\<right-hand side$_n$>|\cr}|\cr}$$
where each \<right-hand side> starts with the symbol on which you want
alignment to occur. For example, every right-hand side often begins
with an $=$~sign. The equations will be typeset in display style.
\exercise In practice, the left-hand sides of aligned formulas are often
blank, and the alignment is often done with respect to other symbols
as well as~$=$. For example, the following display is typical; see if you
can guess how the author typed it:
$$\eqalign{T(n)\le T(2^{\lceil\lg n\rceil})
&\le c(3^{\lceil\lg n\rceil}-2^{\lceil\lg n\rceil})\cr
&<3c\cdot3^{\lg n}\cr
&=3c\,n^{\lg3}.\cr}$$ % from v2 p279
\answer (Note in particular that the final `|.|'\ comes {\sl before\/} the
final `|\cr|'.)
\begintt
$$\eqalign{T(n)\le T(2^{\lceil\lg n\rceil})
&\le c(3^{\lceil\lg n\rceil}-2^{\lceil\lg n\rceil})\cr
&<3c\cdot3^{\lg n}\cr
&=3c\,n^{\lg3}.\cr}$$
\endtt
The result of\/ |\eqalign| is a vertically centered box. This makes it easy to
get a formula like
$$\left\{
\eqalign{\alpha&=f(z)\cr \beta&=f(z^2)\cr \gamma&=f(z^3)\cr}
\right\}\qquad\left\{
\eqalign{x&=\alpha^2-\beta\cr y&=2\gamma\cr}\right\}.$$ % meaningless
You simply use |\eqalign| twice in the same line:
\begintt
$$\left\{
\eqalign{\alpha&=f(z)\cr \beta&=f(z^2)\cr \gamma&=f(z^3)\cr}
\right\}\qquad\left\{
\eqalign{x&=\alpha^2-\beta\cr y&=2\gamma\cr}\right\}.$$
\endtt
\exercise Try your hand at the numbered two-line display % Polya/Szego V.29
$$\eqalign{P(x)&=a_0+a_1x+a_2x^2+\cdots+a_nx^n,\cr
P(-x)&=a_0-a_1x+a_2x^2-\cdots+(-1)^na_nx^n.\cr}\eqno(30)$$
[{\sl Hint:\/} Use the fact that |\eqalign| produces a vertically centered
box; the equation number `(30)' is supposed to appear halfway between
the two lines.]
\answer |$$\eqalign{P(x)&=a_0+a_1x+a_2x^2+\cdots+a_nx^n,\cr|\parbreak
| P(-x)&=a_0-a_1x+a_2x^2-\cdots+(-1)^na_nx^n.\cr}\eqno(30)$$|\par
\exercise What happens if you forget the |&| in one equation of an |\eqalign|?
\answer Both sides of that equation are considered to be on the left, so
you get results that look like this:
$$\openup-\jot
\left\{\eqalign{\alpha&=f(z)\cr \beta&=f(z^2)\cr \gamma=f(z^3)\cr}
\right\}.$$
\danger Multi-line formulas sometimes fit together in odd ways, and you'll
find that every once in a~while you will want to move certain lines farther
apart or closer together. If you type `^|\noalign||{|^|\vskip|\<glue>|}|'
after any |\cr|, \TeX\ will insert the given amount of extra glue just
after that particular line. For example,
\begintt
\noalign{\vskip3pt}
\endtt
will put $3\pt$ of additional space between lines. You can also change the
amount of space before the first line, in the same way.
The next level of complexity occurs when you have several aligned
equations with several equation numbers. Or perhaps some of the
lines are numbered and others are not:
$$\eqalignno{(x+y)(x-y)&=x^2-xy+yx-y^2\cr
&=x^2-y^2;&(4)\cr
(x+y)^2&=x^2+2xy+y^2.&(5)\cr}$$
For this situation plain \TeX\ provides ^|\eqalignno|; you use it like
|\eqalign|, but on each line that you want an equation number you add
`|&|\<equation number>' just before the |\cr|. The example above was
generated by
\begintt
$$\eqalignno{(x+y)(x-y)&=x^2-xy+yx-y^2\cr
&=x^2-y^2;&(4)\cr
(x+y)^2&=x^2+2xy+y^2.&(5)\cr}$$
\endtt
Notice that the second |&| is omitted unless there's an equation number.
And there's also ^|\leqalignno|, which puts equation numbers at the left.
In this case it is appropriate to move the `(4)' to the beginning
of its equation:
$$\leqalignno{(x+y)(x-y)&=x^2-xy+yx-y^2&(4)\cr
&=x^2-y^2;\cr
(x+y)^2&=x^2+2xy+y^2.&(5)\cr}$$
Although the equation numbers appear at the left, you are still supposed to
input them at the right, just as you do with |\leqno|; in other words,
you should type
`|$$\leqalignno{(x+y)(x-y)&...&(4)\cr...}$$|' to get the previous display.
Caution: |\eqalignno| and |\leqalignno| both center the set of equations
without regard to the widths of the equation numbers. If the equations or
their numbers get too wide, they might overlap, yet no error message will
be given.
\exercise Typeset the following display: ^^|\gcd|
$$\leqalignno{\gcd(u,v)&=\gcd(v,u);&(9)\cr
\gcd(u,v)&=\gcd(-u,v).&(10)\cr}$$ % v2 p316
\answer |$$\leqalignno{\gcd(u,v)&=\gcd(v,u);&(9)\cr|\parbreak
| \gcd(u,v)&=\gcd(-u,v).&(10)\cr}$$|
\exercise And here's another one to try, just to keep in practice: ^^|\int|
$$\vbox{
\eqalignno{\biggl(\int_{-\infty}^\infty e^{-x^2}\,dx\biggr)^2
&=\int_{-\infty}^\infty\int_{-\infty}^\infty
e^{-(x^2+y^2)}\,dx\,dy\cr
&=\int_0^{2\pi}\int_0^\infty e^{-r^2}r\,dr\,d\theta\cr
&=\int_0^{2\pi}\biggl(-{e^{-r^2}\over2}
\bigg\vert_{r=0}^{r=\infty}\,\biggr)\,d\theta\cr
&=\pi.&(11)\cr}
}$$ % cf Joy of TeX
\answer %
|$$\eqalignno{\biggl(\int_{-\infty}^\infty e^{-x^2}\,dx\biggr)^2|\parbreak
| &=\int_{-\infty}^\infty\int_{-\infty}^\infty|\parbreak
| e^{-(x^2+y^2)}\,dx\,dy\cr|\parbreak
| &=\int_0^{2\pi}\int_0^\infty e^{-r^2}r\,dr\,d\theta\cr|\parbreak
| &=\int_0^{2\pi}\biggl(-{e^{-r^2}\over2}|\parbreak
| \bigg|\||_{r=0}^{r=\infty}\,\biggr)\,d\theta\cr|\parbreak
| &=\pi.&(11)\cr}$$| ^^|\bigg|
\danger Although |\eqalign| and |\eqalignno| look nearly the same, there's
really a fundamental distinction between them: |\eqalign| makes a single,
vertically centered box, which is no wider than it needs to be; but
|\eqalignno| generates a set of lines that have the full display width
(reaching all the way to both margins). Thus, for example, you can use
|\eqalign| several times in a display, but |\eqalignno| can appear only
once. If you try to use ^|\eqno| in conjunction with |\eqalign|,
you get a decent result, but if you try to use |\eqno| in connection
with |\eqalignno| you'll get some sort of weird error message(s).
\ddanger The definitions in Appendix~B reveal why |\eqalign| and |\eqalignno|
behave differently: |\eqalign| is an
abbreviation for ^|\vcenter||{|^|\halign||{...}}|, while
|\eqalignno| is an abbreviation for |\halign to\displaywidth{...}|;
thus the |\eqalignno| macro generates an ``^{alignment display}.''
\ddanger This difference between |\eqalign| and |\eqalignno| has two
interesting consequences. \ (1)~It's impossible to break an |\eqalign|
between pages, but an |\eqalignno| can be broken. In fact, you can
{\sl force\/} a ^{page break} after a particular line if you insert
`^|\noalign||{|^|\break||}|'
after the |\cr| for that line. You can prohibit {\sl all\/} breaks
in an |\eqalignno| if you set ^|\interdisplaylinepenalty||=10000|; or you
can enclose the whole works in a ^|\vbox|:
\begintt
$$\vbox{\eqalignno{...}}$$
\endtt
(2) You can also insert a line of text between two equations, without
losing the alignment. For example, consider the two displays
$$\eqalignno{x&=y+z\cr
\noalign{\hbox{and}}
x^2&=y^2+z^2.\cr}$$
These were actually generated as a single display by typing
\begintt
$$\eqalignno{x&=y+z\cr
\noalign{\hbox{and}}
x^2&=y^2+z^2.\cr}$$
\endtt
Therefore the fact that their $=$ signs line up is not just a lucky
coincidence. Sometimes you will want to adjust the spacing above or below
such a line of inserted text, by putting a |\vskip| or two inside of the
|\noalign{...}|. Incidentally, this example also shows that it is
possible to use |\eqalignno| without giving any equation numbers.
\ddangerexercise What happens if\/ |\eqalign| is substituted for
|\eqalignno| in this last example?
\answer You get the displayed box
$$\eqalign{x&=y+z\cr
\noalign{\hbox{and}}
x^2&=y^2+z^2.\cr}$$
Reason: The `and' occurs at the left of the |\eqalign| box, not at the
left of the whole display, and the |\eqalign| box is centered as usual.
\ddangerexercise Our friend Ben ^{User} got into trouble again when he tried to
move an equation number up higher than its usual position, by typing this:
^^|\raise|
\begintt
$$\eqalignno{...&\raise6pt\hbox{(5)}\cr}$$
\endtt
What was his oversight, and what could he have done instead?
\answer By raising the equation number, he increased the line height,
so \TeX\ put extra space between that line and the previous line
when it calculated the inter-line glue. If he had said
`^|\smash||{\raise...}|', he wouldn't have had that problem.
\danger For other types of displays, plain \TeX\ provides ^|\displaylines|,
which lets you display any number of formulas in any way you want,
without any alignment. The general form is
$$\halign{\indent\hfil#&#\hfil\cr
|$$\displaylines{|&\<displayed formula$_1$>|\cr|\cr
&\<displayed formula$_2$>|\cr|\cr
\noalign{\vskip-2pt}
&\qquad\vdots\cr
&\<displayed formula$_n$>|\cr}$$|\cr}$$
Each formula will be centered, because |\displaylines| puts ^|\hfil| at
the left and the right of each line; you can override this centering to
get things flush left or flush right by inserting ^|\hfill|, which takes
precedence over |\hfil|.
\dangerexercise Use |\displaylines| to typeset the three-line display
$$\displaylines{\hfill x\equiv x;\hfill\llap{(1)}\cr
\hfill\hbox{if}\quad x\equiv y\quad\hbox{then}\quad
y\equiv x;\hfill\llap{(2)}\cr
\hfill\hbox{if}\quad x\equiv y\quad\hbox{and}\quad
y\equiv z\quad\hbox{then}\quad
x\equiv z.\hfill\llap{(3)}\cr}$$
\answer |$$\displaylines{\hfill x\equiv x;\hfill\llap{(1)}\cr|\parbreak
| \hfill\hbox{if}\quad x\equiv y\quad\hbox{then}\quad|\parbreak
| y\equiv x;\hfill\llap{(2)}\cr|\parbreak
| \hfill\hbox{if}\quad x\equiv y\quad\hbox{and}\quad|\parbreak
| y\equiv z\quad\hbox{then}\quad|\parbreak
| x\equiv z.\hfill\llap{(3)}\cr}$$|\par\medskip\noindent
There's also a trickier solution, which begins with
\begintt
$$\displaylines{x\equiv x;\hfil\llap{(1)}\hfilneg\cr
\endtt
\danger If you look closely at the multi-line displays in this chapter,
you'll see that the baselines are farther apart than they are in normal
text; mathematics publishers generally do this in order to make the
displays easier to read. In accordance with this tradition, |\eqalign| and
its relatives automatically increase the ^|\baselineskip|. If~you are
making a multi-line display with \TeX's primitive ^|\halign| command,
instead of using one of the plain \TeX\ macros, you might want to make
this same baseline adjustment, and you can do it easily by saying
`|$$\openup1\jot \halign{...}$$|'. The ^|\openup| macro increases
^|\lineskip| and ^|\lineskiplimit| as well as |\baselineskip|. If
you say `|\openup2\jot|', the lines are spread apart 2 extra units, where
plain \TeX\ opens things up in units of $3\pt$. Since |$$...$$| acts as a
^{group}, the effect of\/ |\openup| will disappear when the display is
finished. Any \<dimen> can follow |\openup|, but it's customary to express the
amount symbolically in terms of a ^|\jot| instead of using absolute units;
^^{generic coding}
then your manuscript can be used with a variety of different formats.
\ddanger Plain \TeX's ^|\displaylines|, ^|\eqalignno|, and ^|\leqalignno|
macros begin with `|\openup1\jot|'. If you don't want the lines to be
opened up, you can cancel this by saying, e.g.,
`|$$\openup-1\jot \eqalignno{...}$$|', because |\openup| has a cumulative
effect.
\ddanger Suppose that you have decided to make a homegrown display having
the general form `|$$\openup1\jot \halign{...}$$|'; and for convenience,
let's suppose that the normal conventions of plain \TeX\ are in force, so
that |\jot=3pt| and |\baselineskip=12pt|. Then the |\openup| macro
changes the baselineskip distance to $15\pt$. It follows that the baseline
of the text line that immediately precedes the display will be $15\pt$
above the topmost baseline of the display, plus the ^|\abovedisplayskip|.
But when the paragraph resumes, its next baseline will be only $12\pt$
below the bottom baseline of the display, plus the ^|\belowdisplayskip|,
because the |\baselineskip| parameter will have reverted to its normal
value. The |\eqalignno| and |\displaylines| macros say
`|\noalign{\vskip|$-d$|}|' before their first lines, where $d$ is
the net amount of opening-up, in order to compensate for this difference.
\subsection Long formulas. Our discussion of mathematics typing is almost
complete; we need to deal with just one more problem: What should be
done when a formula is so long that it doesn't fit on a single line?
For example, suppose that you encounter the equation
$$\hfuzz=20pt % overfull box tolerated here
\sigma(2^{34}-1,2^{35},1)=
-3+(2^{34}-1)/2^{35}+2^{35}\!/(2^{34}-1)+7/2^{35}(2^{34}-1)
-\sigma(2^{35},2^{34}-1,1).$$ % from v2, 1st ed, p76
You'll have to break it up somehow; \TeX\ has done its best to squeeze
everything together by shrinking the spaces next to the $+$ and~$-$ signs
to zero, but still the line has come out overfull.
Let's try to break that equation just before the `$+7$'. One common way to
do this is to type
\begintt
$$\eqalign{\sigma(2^{34}-1,2^{35},1)
&=-3+(2^{34}-1)/2^{35}+2^{35}\!/(2^{34}-1)\cr
&\qquad+7/2^{35}(2^{34}-1)-\sigma(2^{35},2^{34}-1,1).\cr}$$
\endtt
which yields
$$\eqalign{\sigma(2^{34}-1,2^{35},1)
&=-3+(2^{34}-1)/2^{35}+2^{35}\!/(2^{34}-1)\cr
&\qquad+7/2^{35}(2^{34}-1)-\sigma(2^{35},2^{34}-1,1).\cr}$$
The idea is to treat a long one-line formula as a two-line formula,
using |\qquad| on the second line so that the second part of the formula
appears well to the right of the $=$~sign on the first line.
\exercise Explain how to deal with the following display. % v2 p107
$$\eqalignno{x_nu_1+\cdots+x_{n+t-1}u_t
&=x_nu_1+(ax_n+c)u_2+\cdots\cr
&\qquad+\bigl(a^{t-1}x_n+c(a^{t-2}+\cdots+1)\bigr)u_t\cr
&=(u_1+au_2+\cdots+a^{t-1}u_t)x_n+h(u_1,\ldots,u_t).
\quad&(47)\cr}$$
\answer |$$\eqalignno{x_nu_1+\cdots+x_{n+t-1}u_t|\parbreak
| &=x_nu_1+(ax_n+c)u_2+\cdots\cr|\parbreak
| &\qquad+\bigl(a^{t-1}x_n+c(a^{t-2}+\cdots+1)\bigr)u_t\cr|\parbreak
| &=(u_1+au_2+\cdots+a^{t-1}u_t)x_n+h(u_1,\ldots,u_t).|\parbreak
| \quad&(47)\cr}$$|\par\noindent
You weren't expected to insert the `|\quad|' on the last line; such
refinements usually can't be anticipated until you see the first proofs.
But without that |\quad| the `(47)' would occur half a quad closer to the
formula.
\danger It's quite an art to decide how to ^{break long displayed formulas}
into several lines; \TeX\ never attempts to break them, because no set of
rules is really adequate. The author of a mathematical manuscript is
generally the best judge of what to do, since break positions depend on
subtle factors of mathematical exposition. For example, it is often
desirable to emphasize some of the symmetry or other structure that
underlies a formula, and such things require a solid understanding of
exactly what is going on in that formula.
\begingroup\ninepoint
\danger Nevertheless, it is possible to state a few rules of thumb about
how to deal with long formulas in displays, since there are some
principles that the best mathematical typesetters tend to follow:\enddanger
\smallskip
\textindent{a)}Although formulas within a paragraph always break {\sl after\/}
binary operations and relations, displayed formulas always break {\sl before\/}
binary operations and relations. Thus, we didn't end the first line of
our $\sigma(\,\ldots\,)$ example with `|(2^{34}-1)+|'; we ended it with
`|(2^{34}-1)|' and began the second line with `|+|'.
\smallskip
\textindent{b)}When an equation is broken before a binary operation, the second
line should start at least two quads to the right of where the innermost
subformula containing that binary operation begins on the first line.
For example, if you wish to break
\begindisplay
|$$\sum_{0<k<n}\left(|\<formula$_1$>|+|\<formula$_2$>|\right)$$|
\enddisplay
at the plus sign between \<formula$_1$> and \<formula$_2$>, it is almost
mandatory to have the plus sign on the second line appear somewhat to the
right of the large left parenthesis that corresponds to `|\left(|'.
\endgroup
\danger In the example just considered, special care is needed to break the
formula into two lines, because ^|\left| and ^|\right| delimiters cannot be
used in isolation; you can't have only |\left| in one line of a formula
and only |\right| in the second. Furthermore, you'll want the two delimiters
to be of the same size, even though they occur in different lines. The best
solution is usually to choose the delimiter size yourself; for example,
you could type
\begindisplay
|$$\eqalign{\sum_{0<k<n}\biggl(&|\<formula$_1$>|\cr|\cr
| &\qquad{}+|\<formula$_2$>|\biggr)\cr}$$|\cr
\enddisplay
if\/ ^|\bigg| delimiters are best. Notice that the |&|~markers don't occur
at $=$~signs in this example, they just mark a point of alignment.
\danger There's another way to break long formulas, sometimes called the
{\sl^{two-line}\/} form. The idea is to put the first part of the formula
almost ^{flush left}, and to put the second part almost ^{flush right},
where ``almost flush'' means ``one quad away.'' Thus, the two-line form of
the long $\sigma(\,\ldots\,)$ equation considered earlier is
$$\displaylines{\quad\sigma(2^{34}-1,2^{35},1)
=-3+(2^{34}-1)/2^{35}+2^{35}\!/(2^{34}-1)\hfill\cr
\hfill{}+7/2^{35}(2^{34}-1)-\sigma(2^{35},2^{34}-1,1).\quad\cr}$$
It isn't difficult to get this two-line effect with ^|\displaylines|:
\begintt
$$\displaylines{\quad\sigma(2^{34}-1,2^{35},1)
=-3+(2^{34}-1)/2^{35}+2^{35}\!/(2^{34}-1)\hfill\cr
\hfill{}+7/2^{35}(2^{34}-1)-\sigma(2^{35},2^{34}-1,1).\quad\cr}$$
\endtt
An extra `|{}|' was typed on the second line here so that \TeX\ would know
that the `|+|' is a binary operation. The two-line form is especially
recommended for equations that have a long left-hand side; in that case the
break generally comes just before the~$=$~sign.
\dangerexercise Typeset the following display:
$$\displaylines{\quad\sum_{1\le j\le n}{1\over
(x_j-x_1)\ldots(x_j-x_{j-1})(x-x_j)(x_j-x_{j+1})
\ldots(x_j-x_n)}\hfill\cr
\hfill={1\over(x-x_1)\ldots(x-x_n)}.\quad(27)\cr}$$ % v2 p80
\answer |$$\displaylines{\quad\sum_{1\le j\le n}{1\over|\parbreak
| (x_j-x_1)\ldots(x_j-x_{j-1})(x-x_j)(x_j-x_{j+1})|\parbreak
| \ldots(x_j-x_n)}\hfill\cr|\parbreak
| \hfill={1\over(x-x_1)\ldots(x-x_n)}.\quad(27)\cr}$$|
\ddangerexercise If it is necessary to typeset a huge fraction like
^^{fraction, huge}
$$\def\\#1;{(#1;q^2)_\infty}
q^{{1\over2}n(n+1)}\\ea;\\eq/a;\\caq/e;\\cq^2\!/ae;
\over(e;q)_\infty(cq/e;q)_\infty$$
in a single narrow column, you might have to break up the numerator and
resort to
\begindisplay
$\displaystyle{\def\\#1;{(#1;q^2)_\infty}
\displaystyle{q^{{1\over2}n(n+1)}\\ea;\\eq/a;\qquad\atop
\hfill\\caq/e;\\cq^2\!/ae;}
\over(e;q)_\infty(cq/e;q)_\infty}$
\enddisplay
How would you specify the latter fraction to \TeX?
% cf SIAM J Math Anal 7 (1976) p333; even longer ones appear on p334
\answer |$$\def\\#1;{(#1;q^2)_\infty} % to save typing|\parbreak
|\displaystyle{q^{{1\over2}n(n+1)}\\ea;\\eq/a;\qquad\atop|\parbreak
| \hfill\\caq/e;\\cq^2\!/ae;}|\parbreak
|\over(e;q)_\infty(cq/e;q)_\infty$$|
\endchapter
When a formula is too long for the page-width
and has to be broken into successive lines
(and we are now, of course, speaking of displayed formulae),
it should be broken, if possible, at the end of a natural `phrase';
if, for example, it is a much-bracketed formula,
it should be broken at the end of one of the major brackets
and not at an inner symbol.
This natural phrasing (as in music or speech)
makes for intelligibility between writer and reader
and should not be left to the compositor.
An author, when he finds himself writing a longish formula,
should indicate a convenient point of fracture in case of need.
\author ^{CHAUNDY}, ^{BARRETT}, and ^{BATEY}, %
{\sl The Printing of Mathematics\/} (1954) % p38
\bigskip
Some authors use display with discretion,
some run even extremely long, complicated equations into the text,
while others tend to display every equation in the paper.
The tendency to overdisplay is probably more predominant
than the tendency to underdisplay;
for this reason it is possible for the copy editor to shorten
(and even improve) papers by running displayed material into text. $\ldots$
On the other hand, there are occasions when the copy editor needs
to suggest the display of complicated expressions that have been run into text,
particularly when it would involve a bad break at the end of a text line.
\author ELLEN ^{SWANSON}, {\sl Mathematics into Type\/} (1971) % p41
\eject
\beginchapter Chapter 20. Definitions\\(also called Macros)
You can often save time typing math formulas by letting ^{control sequences}
stand for constructions that occur frequently in a particular manuscript.
For example, if some document uses the vector `$(x_1,\ldots,x_n)$' a lot,
you can type
\begintt
\def\xvec{(x_1,\ldots,x_n)}
\endtt
and |\xvec| will henceforth be an abbreviation for `|(x_1,\ldots,x_n)|'.
Complex displays \rlap{like}
$$\def\xvec{(x_1,\ldots,x_n)}
\sum_{\xvec\ne(0,\ldots,0)} \bigl(f\xvec+g\xvec\bigr)$$
can then be typed simply as
\begintt
$$\sum_{\xvec\ne(0,\ldots,0)} \bigl(f\xvec+g\xvec\bigr)$$
\endtt
instead of in a tedious long form. By ^{defining a control sequence}
like |\xvec|, you not only cut down on the number of keystrokes that you need
to make, you also reduce your chances of introducing typographical errors
and inconsistencies. ^^{abbreviations, see macros}
Of course, you usually won't be making a definition just to speed up the
typing of one isolated formula; that doesn't gain anything, because time goes by
when you're deciding whether or not to make a definition, and when you're
typing the definition itself. The real payoff comes when some cluster of
symbols is used dozens of times throughout a manuscript. A wise typist will
look through a document before typing anything, thereby getting a feeling
for what sorts of problems will arise and what sorts of definitions will
be helpful. For example, Chapter~16 recommends that the control sequence
|\Ahat| be defined at the beginning of any manuscript that makes frequent
use of the symbol~$\hat A$.
Abbreviations like |\xvec| turn out to be useful in many applications of
computers, and they have come to be known as {\sl^{macros}\/} because they
are so powerful; one little macro can represent an enormous amount of
material, so it has a sort of macroscopic effect. System programs like
\TeX\ that are designed to deal with macro definitions are said to {\sl
expand\/} the user's macros; for example, |\xvec| expands into
|(x_1,\ldots,x_n)|, and ^|\ldots| in turn is a macro that expands into
|\mathinner{\ldotp\ldotp\ldotp}|. Thus, |\xvec| is actually an
abbreviation for `|(x_1,\mathinner{\ldotp\ldotp\ldotp},x_n)|'. \ (The
expansion stops here, because ^|\mathinner| is a primitive control
sequence of \TeX, and because |\ldotp| has been defined with
^|\mathchardef|; thus |\mathinner| and |\ldotp| are not macros.)
\TeX\ users generally build up their own personal ^{library of macros} for
things that they want to do in different documents. For example, it is common
to have a file called |macros.tex| that contains definitions of your favorite
special control sequences, perhaps together with commands that load your
favorite special fonts, etc. If you begin a document with the command
\begintt
\input macros
\endtt
then \TeX\ will read all those definitions, ^^|\input|
saving you all the trouble of retyping them. Of course, \TeX's memory is
limited, and it takes time to read a file, so you shouldn't put thousands
of definitions into |macros.tex|. A large collection of macro definitions
(e.g., the set of definitions in Appendix~B) is called a {\sl^{format}\/}
(e.g., ``plain \TeX\ format''); \TeX\ has a special way to input
a format at high speed, assuming that the format doesn't change very often.
The |\xvec| and |\Ahat| examples apply to math formulas, but you can make
good use of macro definitions even when you aren't doing any math at all.
For example, if you are using \TeX\ for ^{business correspondence},
you can have a |\yours| macro that stands for `Sincerely yours, A.~U.
^{Thor}'. If you often write ^{form letters} you can have macros that
generate entire sentences or paragraphs or groups of paragraphs.
The ^{Internal Revenue Service} could, for example, make use of the
following two macros:
\begintt
\def\badcheck{A penalty has been added because your
check to us was not honored by your bank.\par}
\def\cheater{A penalty of 50\% of the underpaid tax
has been added for fraud.\par}
\endtt
Simple macro definitions, like these, start with `|\def|'; then comes the
control sequence name, e.g., `|\badcheck|'; and then comes the replacement
text enclosed in `|{|' and~`|}|'. The ^{braces} do not represent
^{grouping} in this case; they simply show the extent of the replacement
text in the definition. You could, of course, define a macro that
includes actual braces in its replacement text, as long as those braces
match each other properly. For example, `|\def\xbold{{\bf x}}|'
makes |\xbold| an abbreviation for `|{\bf x}|'.
\exercise
Write a |\punishment| macro that prints 100 lines containing the message
`I must not talk in class.' \ [{\sl Hint:\/} First write a macro |\mustnt|
that prints the message once; then write a macro |\five| that prints it
five times.] \checkequals\punishexno\exno
\answer |\def\mustnt{I must not talk in class.\par}|\parbreak
|\def\five{\mustnt\mustnt\mustnt\mustnt\mustnt}|\parbreak
|\def\twenty{\five\five\five\five}|\parbreak
|\def\punishment{\twenty\twenty\twenty\twenty\twenty}|\par
\smallskip\noindent Solutions to more complicated problems of this type are
discussed later.
\dangerexercise What is the expansion of\/ |\puzzle|, given the following
definitions?
\begintt
\def\a{\b}
\def\b{A\def\a{B\def\a{C\def\a{\b}}}}
\def\puzzle{\a\a\a\a\a}
\endtt
\answer |ABCAB|. \ (The first |\a| expands into |A\def\a{B...}|; this redefines
|\a|, so the second |\a| expands into |B...|, etc.) \ At least, that's what
happens if\/ |\puzzle| is encountered when \TeX\ is building a list. But if
|\puzzle| is expanded in an ^|\edef| or ^|\message| or something like that,
we will see later that the interior |\def| commands are not performed
while the expansion is taking place, and the control sequences following
|\def| are expanded; so the result is an infinite string
\begintt
A\def A\def A\def A\def A\def A\def A\def A\def A...
\endtt
which causes \TeX\ to abort because the program's input stack is finite.
This example points out that a control sequence (e.g., |\b|) need not be
defined when it appears in the replacement text of a definition. The example
also shows that \TeX\ doesn't expand a macro until it needs to.
\danger As soon as you get the hang of simple macros like those illustrated
above, you will probably begin to think, ``Boy, wouldn't it be nice if I could
have a macro in which some of the text in the expansion is changeable? I'd like
to be able to stick different things into the middle of that text.'' Well,
\TeX\ has good news for you: Control sequences can be defined in terms of
{\sl^{parameters}}, and you can supply {\sl^{arguments}\/} that will be
substituted for the parameters.
\danger For example, let's consider |\xvec| again. Suppose that you not only
refer to `$(x_1,\ldots,x_n)$', but you also make frequent use of
`$(y_1,\ldots,y_n)$' and other similar things. Then you might want to type
\begintt
\def\row#1{(#1_1,\ldots,#1_n)}
\endtt
\def\row#1{(#1_1,\ldots,#1_n)}%
after which |\row x| will produce `$\row x$' and |\row y| will produce
`$\row y$'. The symbol |#1| ^^{sharpsign} stands for the first parameter to
the macro, and when you say `|\row|~|x|' the |x| is a so-called argument that
will be inserted in place of the |#1|'s in the replacement text. In this
case the argument consists of a single letter, |x|. You can also say
|\row\alpha|, in which case the argument will be the control sequence
^|\alpha|, and the result will be `$\row\alpha$'. If you want the argument
to contain more than one symbol or control sequence, you can simply enclose
it in ^{braces}; for example, |\row{x'}| yields $\row{x'}$. The argument
in this case is |x'| (without the braces). Incidentally, if you say
|\row{{x'}}|, you get $\row{{x'}}$; the reason is that only one pair of
braces is stripped off when the argument is collected, and
$({x'}_1,\ldots,{x'}_n)$ is what you get from
|({x'}_1,\ldots,{x'}_n)| in math mode, according to the rules
of Chapter~16. ^^{apostrophe}
\dangerexercise Continuing this example, what is the result of
|$\row{\bf x}$|?
\answer \def\row#1{(#1_1,\ldots,#1_n)}$\row{\bf x}$. Note that the
subscripts are bold here, because the expansion |(\bf x_1,\ldots,\bf x_n)|
doesn't ``turn off'' ^|\bf|. To prevent this, one should write
|\row{{\bf x}}|; or (better), |\row\xbold|, in conjunction with
|\def\xbold{{\bf x}}|.
\danger The notation `|#1|' suggests that there might be an opportunity to have
more than one parameter, and indeed there is. You can write, for example,
\begintt
\def\row#1#2{(#1_1,\ldots,#1_#2)}
\endtt
\def\row#1#2{(#1_1,\ldots,#1_#2)}%
after which `|\row xn|' would be the proper protocol for `$\row xn$'. There can
be as many as nine parameters, |#1| to~|#9|, and when you use them you must
number them~in order. For example, you can't use |#5| in a definition unless
the previous parameter in~that definition was called |#4|. \ (This restriction
applies only to the initial statement of parameters, before the replacement
text starts; the stated parameters can be used any number of times, in any
order, in the replacement text itself.)
\danger A control sequence has only one definition at a
time, so the second definition of\/ |\row| would supersede the first one if
both had appeared in the same document. Whenever \TeX\ encounters a macro
that it wants to expand, it uses the most recent definition. However,
definitions are ^{local} to the group that contains them; old definitions
will be restored in the usual way when a ^{group} ends.
\danger Caution: When you define a macro with simple parameters, as in
these examples, you must be careful not to put blank spaces before the
`|{|' that begins the replacement text. For example, `|\def\row #1 #2 {...}|'
will not give the same result as `|\def\row#1#2{...}|', because the spaces
after |#1| and~|#2| tell \TeX\ to look for arguments that are followed by
spaces. \ (Arguments can be ``delimited'' in a fairly general way, as
explained below.) \ But the space after |\row| is optional, as usual,
because \TeX\ always disregards spaces after control words. After
you have said `|\def\row#1#2{...}|', you are allowed to put spaces
between the arguments (e.g., `|\row x n|'), because \TeX\ doesn't
use single spaces as undelimited arguments.
\danger The following exercise is particularly recommended for people who
want to learn to write \TeX\ macros. Even if you have gotten into the
dangerous habit of skimming other exercises, you should try your hand
at this one.
\dangerexercise Extending exercise 20.\punishexno, write a ``generalized
punishment'' macro that has two parameters, so that |\punishment{run}{the
halls}| will produce 100 paragraphs that say `I~must not run in the halls.'
\answer The catch is that the parameters have to percolate down to the
|\mustnt| macro, if you extend the previous answer:
\begintt
\def\mustnt#1#2{I must not #1 in #2.\par}
\def\five#1#2{\mustnt{#1}{#2}...\mustnt{#1}{#2}}
\def\twenty#1#2{\five{#1}{#2}...\five{#1}{#2}}
\def\punishment#1#2{\twenty{#1}{#2}...\twenty{#1}{#2}}
\endtt
When you pass parameters from one macro to another in this way, you need to
enclose them in braces as shown. But actually this particular solution
punishes \TeX\ much more than it needs to, because it takes a lot of
time to copy the parameters and read them again and again. There's a
much more efficient way to do the job, by defining control sequences:
\begintt
\def\mustnt{I must not \doit\ in \thatplace.\par}
\def\punishment#1#2{\def\doit{#1}\def\thatplace{#2}%
\twenty\twenty\twenty\twenty\twenty}
\endtt
and by defining |\five| and |\twenty| without parameters as before.
You can also delve more deeply into \TeX nicalities, constructing solutions
that are more efficient yet; \TeX\ works even faster when macros
communicate with each other via ^{boxes}.
^^{efficient macros} ^^{communication between macros}
For example,
\begintt
\def\mustnt{\copy0 }
\def\punishment#1#2{\setbox0=
\vbox{\strut I must not #1 in #2.\strut}%
\twenty\twenty\twenty\twenty\twenty}
\endtt
sets 100 identical paragraphs at high speed, because \TeX\ has to
process the paragraph and break it into lines only once. It's much faster
to ^{copy a box} than to build it up from scratch. \ (The ^{struts} in
this example keep the interbaseline distances correct between boxed
paragraphs, as explained in Chapter~12. Two struts are used, for if the
message takes more than one line there will be a strut at both top
and bottom. If it were known that each sentence will occupy only a single
line, no struts would be needed, because interline glue is added as
usual when a box created by |\copy| is appended to the current vertical list.)
\ninepoint % the rest of this chapter is all dangerous
\ddanger \TeX\ also allows you to define macros whose parameters are delimited
in quite a general way; you needn't always enclose arguments in braces.
For example,
\begintt
\def\cs #1. #2\par{...}
\endtt
defines a control sequence |\cs| with two parameters, and its two arguments
will be determined as follows: |#1| will consist of all tokens between
|\cs| and the next subsequent appearance of `|.|\]' (period and space);
|#2| will consist of all tokens between that `|.|\]' and the next
|\par| token. \ (The ^|\par| might be given explicitly, or it might be
generated by a blank line as explained in Chapter~8.) \ For example, when
\TeX\ expands
\begintt
\cs You owe \$5.00. Pay it.\par
\endtt
the first argument is `|You owe \$5.00|' and the second is `|Pay it.|'.
The period in `|\$5.00|' doesn't stop |#1|, in this example,
because \TeX\ keeps going until finding a period that is followed immediately
by a space.
\ddanger Furthermore, an argument will not stop when its delimiter is enclosed
in braces, because that would produce unbalanced braces. For example, in
\begintt
\def\cs #1.#2\par{...}
\endtt
the first argument is now delimited by a single period, so |#1| would be
`|You owe \$5|' and the |#2| would be `|00. Pay it.|' if\/ |\cs| were
invoked as above. But
\begintt
\cs You owe {\$5.00}. Pay it.\par
\endtt
satisfactorily hides the first period, making it part of argument |#1|,
which becomes \hbox{`|You owe {\$5.00}|'}.
\ddanger If you are designing a format for mathematical papers, you will
probably want to include a macro for the statement of ^{theorems},
definitions, lemmas, corollaries, and such things. For example, you might
want to typeset a statement like\enddanger
\proclaim Theorem 1. \TeX\ has a powerful macro capability.\par
\noindent from the input
\begintt
\proclaim Theorem 1. \TeX\ has a powerful macro capability.\par
\endtt
In fact, plain \TeX\ includes a ^|\proclaim| macro that does just that;
its definition is
\begintt
\def\proclaim #1. #2\par{\medbreak
\noindent{\bf#1.\enspace}{\sl#2}\par\medbreak}
\endtt
^^{enunciations, see proclaim} ^^{enspace}
so the arguments are delimited exactly as in our first |\cs| example. The
replacement text here uses |\medbreak| to separate the proclaimed
paragraph from what precedes and follows; the title of the proclamation is
set in bold face type, while the text itself is set slanted. \ (The
actual definition of\/ |\proclaim| in Appendix~B is not quite the same as
this; the final |\medbreak| has been modified so that a break between
pages will be discouraged immediately following the statement of a
theorem. Hence a short theorem will tend to appear at the top of a page
rather than at the bottom.)
\ddanger By making changes to the |\proclaim| macro, you can change the
format of all the proclamations in your paper, without changing the text
of the paper itself. For example, you could produce something like\enddanger
\medbreak
\font\tencsc=cmcsc10
\noindent {\tencsc Theorem 1:}\enspace
{\it\TeX\ has a powerful macro capability.}
\goodbreak\medbreak\noindent
by making simple alterations to the replacement text of\/ |\proclaim|,
assuming that you have a ``^{caps and small caps}'' font. \TeX\ is
intended to support ^{higher-level languages for composition} in which all of
the control sequences that a user actually types are macros rather than
\TeX\ primitives. The ideal is to be able to describe important classes of
documents in terms of their components, without mentioning actual fonts or
point sizes or details of spacing; a single ^{style-independent document}
^^{format-independent document}^^{generic coding}
can then be set in many different styles.
\ddanger Now that we have seen a number of examples, let's look at the
precise rules that govern \TeX\ macros. Definitions have the general form
\begindisplay
|\def|\<control sequence>\<parameter text>|{|\<replacement text>|}|
\enddisplay
where the \<parameter text> contains no ^{braces}, and where all occurrences
of |{| and |}| in the \<replacement text> are properly nested. Furthermore
the |#| symbol has a special significance: In the \<parameter text>, the
first appearance of |#| must be followed by~|1|, the next by~|2|, and
so on; up to nine |#|'s are allowed. In the \<replacement text> each~|#|
must be followed by a digit that appeared after~|#| in the \<parameter text>,
or else the~|#| should be followed by another~|#|. The latter case stands
for a single~|#| token when the macro is expanded; the former case stands
for insertion of the corresponding argument.
\ddanger For example, let's consider a ``random'' definition that doesn't
do anything useful except that it does exhibit \TeX's rules. The definition
\begintt
\def\cs AB#1#2C$#3\$ {#3{ab#1}#1 c##\x #2}
\endtt
says that the control sequence |\cs| is to have a parameter text consisting of
nine tokens
\begindisplay
|A|$_{11}$, \ |B|$_{11}$, \ |#1|, \ |#2|, \ |C|$_{11}$, \
|$|$_3$, \ |#3|, \ \cstok{\char`$}, \ \]$_{10}$
\enddisplay
(assuming the ^{category codes} of plain \TeX), and a replacement text
of twelve tokens
\begindisplay
|#3|, \ |{|$_1$, \ |a|$_{11}$, \ |b|$_{11}$, \ |#1|, \
|}|$_2$, \ |#1|, \ \]$_{10}$, \ |c|$_{11}$, \ |#|$_6$, \ \cstok{x}, \ |#2|.
\enddisplay
Henceforth when \TeX\ reads the control sequence |\cs| it will expect that
the next two tokens will be |A|$_{11}$ and |B|$_{11}$ (otherwise you will
get the error message `|Use| |of| |\cs| |doesn't| |match| |its|
|definition|'); then comes argument~|#1|, followed by argument~|#2|,
then~|C|$_{11}$, then~|$|$_3$, then argument~|#3|, then |\$|, and finally
a space token. It is customary to use the word ``argument'' to mean the
string of tokens that gets substituted for a parameter; parameters appear
in a definition, and arguments appear when that definition is used. \ (For
the purposes of these rules, we are extending Chapter~7's definition of
^{token}: In addition to control sequences and (character code, category
code) pairs, \TeX\ also recognizes ``^{parameter tokens},'' denoted here
by |#1|~to~|#9|. Parameter tokens can appear only in token lists for macros.)
\ddanger How does \TeX\ determine where an argument stops, you ask.
Answer: There are two cases. A {\sl^{delimited parameter}\/} is followed
in the \<parameter text> by one or more non-parameter tokens, before
reaching the end of the parameter text or the next parameter token;
in this case the corresponding argument is the shortest (possibly empty)
sequence of tokens with properly nested |{...}| groups that is followed in
the input by this particular list of non-parameter tokens. \ (Category
codes and character codes must both match, and control sequence names
must be the same.) \ An {\sl^{undelimited parameter}\/} is followed immediately
in the \<parameter text> by a parameter token, or it occurs at the very end
of the parameter text; in this case the corresponding argument is the
next nonblank token, unless that token is `|{|', when the argument will
be the entire |{...}| group that follows. In both cases, if the argument
found in this way has the form `|{|\<nested tokens>|}|', where \<nested
tokens> stands for any sequence of tokens that is properly nested with
respect to braces, the outermost braces enclosing the argument are removed
and the \<nested tokens> will remain. For example, let's continue with
|\cs| as defined above and suppose that the subsequent text contains
\begintt
\cs AB {\Look}C${And\$ }{look}\$ 5.
\endtt
Argument |#1| will be the token \cstok{Look}, since |#1| is an undelimited
parameter (it is followed immediately by~|#2| in the definition);
in this case \TeX\ ignores the blank space after |B|, and strips the
braces off~of |{\Look}|. Argument~|#2| will be empty, since |C$| follows
immediately. And argument~|#3| will be the thirteen tokens corresponding
to the text |{And\$|\]|}{look}|, because |#3| is to be followed by `|\$|\]',
and because the first occurrence of `|\$|\]' is within braces. Even though
argument~|#3| begins with a left brace and ends with a right brace, the
braces are not removed, since that would leave the unnested tokens `|And\$
}{look|'. The net effect then, after substituting arguments for
parameters in the replacement text, will be that \TeX\ will next read the
token list
\begintt
{And\$ }{look}{ab\Look}\Look|]c#\x5.
\endtt
The space \] here will be part of the resulting token list, even though
it follows the control word |\Look|, because ^{spaces} are removed
after ^{control word} tokens only when \TeX\ first converts input lines to
token lists as described in Chapter~8.
\ddangerexercise The example definition of\/ |\cs| includes a |##| in
its replacement text, but the way |##| is actually used in that example
is rather pointless. Give an example of a definition where |##| serves
a useful purpose. ^^{sharp sharp}
\answer The |##| feature is indispensable when the replacement text of
a definition contains other definitions. For example, consider
\begintt
\def\a#1{\def\b##1{##1#1}}
\endtt
after which `|\a!|'\ will expand to `|\def\b#1{#1!}|'. We will see later
that |##| is also important for alignments; see, for example, the definition
of\/ |\matrix| in Appendix~B.
\ddanger A special extension is allowed to these rules: If the very
last character of the \<parameter text> is~|#|, so that this~|#| is
immediately followed by~|{|, \TeX\ will behave as if the~|{| had been
inserted at the right end of both the parameter text and the replacement text.
For example, if you say `|\def\a#1#{\hbox to #1}|', the
subsequent text `|\a3pt{x}|' will expand to `|\hbox to 3pt{x}|', because
the argument of\/ |\a| is delimited by a left brace.
^^{dimensions as arguments}
\ddanger Tokens that precede the first parameter token in the \<parameter
text> of a definition are required to follow the control sequence; in
effect, they become part of the control sequence name. For example, the author
might have said
\begintt
\def\TeX/{...}
\endtt
instead of defining ^|\TeX| without the slash. Then it would be necessary
to type |\TeX/| each time the \TeX\ logo is desired, but the new definition
would have the advantage that spaces are {\sl not\/} ignored after
`|\TeX/|'. You can use this idea to define macros that are intended to
be used in sentences, so that users don't have to worry about the possible
disappearance of ^{spaces}.
\ddangerexercise Define a control sequence |\a| such that |\a{...}| expands
to |\b{...}|, and such that \TeX\ gives an error message if\/ |\a| is not
immediately followed by a left brace.
\answer |\def\a#{\b}|.
\ddanger Complicated macros have a habit of behaving differently from what
you expect, when you first define them, even though \TeX's rules are
not especially complicated. If you have trouble understanding why some
|\def| doesn't work the way you think it should, help is available:
You can set |\tracingmacros=1|, whereupon \TeX\ will write something in
your log file whenever it expands a macro, and whenever it has read a
macro argument. For example, if\/ ^|\tracingmacros| is positive when
\TeX\ processes the |\cs| example above, it will put the following
four lines into the log: ^^{debugging macros}
\begintt
\cs AB#1#2C$#3\$ ->#3{ab#1}#1 c##\x #2
#1<-\Look
#2<-
#3<-{And\$ }{look}
\endtt
\ddanger In all of the rules stated above, `|{|' and `|}|' and `|#|' stand
for any characters whose ^{category codes} are respectively 1, 2, and 6 in
the token list when \TeX\ reads the macro definition; there's nothing
sacred about the particular symbols that plain \TeX\ uses to denote
grouping and parameters. You can even make use of several different
characters with these category codes, all at the same time.
\ddangerexercise Suppose that `|[|', `|]|', and `|!|'\ have the respective
catcodes 1,~2, and~6, as do `|{|',~`|}|', and~`|#|'. See if you can guess
what the following definition means:
\begintt
\def\!!1#2![{!#]#!!2}
\endtt
What token list will result when `|\! x{[y]][z}|' is expanded?
\answer Let's go slowly on this one, so that the answer will give enough
background to answer all similar questions. The \<parameter text> of the
definition consists of the three tokens |#1|, |#2|, |[|$_1$; the
\<replacement text> consists of the six tokens |{|$_1$, |#|$_6$, |]|$_2$,
|!|$_6$, |#2|, |[|$_1$. \ (When two tokens of category~6 occur in the
replacement text, the character code of the second one survives; the
character code of a category-6 character is otherwise irrelevant. Thus,
`|\def\!#1!2#[{##]!!#2]|' would produce an essentially identical
definition.) \ When expanding the given token list, argument~|#1| is
|x|$_{11}$, since it is undelimited. Argument~|#2| is delimited by~|[|$_1$,
which is different from~|{|$_1$, so it is set provisionally to |{[y]]|;
but the outer ``braces'' are stripped off, so |#2|~reduces to the
three tokens |[|$_1$, |y|$_{11}$,~|]|$_2$. The result of the expansion
is therefore
\begindisplay
|{|$_1$ |#|$_6$ |]|$_2$ |!|$_6$ |[|$_1$ |y|$_{11}$ |]|$_2$
|[|$_1$ |z|$_{11}$ |}|$_2$.
\enddisplay
Incidentally, if you display this with ^|\tracingmacros||=1|, \TeX\ says
\begintt
\!!1#2[->{##]!!#2[
#1<-x
#2<-[y]
\endtt
Category codes are not shown, but a character of category~6 always
appears twice in succession. A parameter token in the replacement text
uses the character code of the final parameter in the parameter text.
^^{token lists, as displayed by TeX}
\ddanger In practice, we all make mistakes. And one of the most common
typographic errors is to forget a~`|}|', or to insert an extra~`|{|',
somewhere in an argument to a macro. If \TeX\ were to follow the rules
blindly in such a case, it would have to keep absorbing more and
more tokens in hopes of finding the end of the argument. But a mistyped
argument is unending, like so many arguments in real life (sigh); so
\TeX\ would have to go on until the end of the file, or (more likely)
until tokens completely fill the computer's memory. In either case, a
single typographical error would have ruined the run, and the user would
be forced to start over. Therefore \TeX\ has another rule, intended
to confine such errors to the paragraph in which they occur: {\sl The
token `\thinspace^|\par|' is not allowed to occur as part of an argument},
unless you explicitly tell \TeX\ that |\par| is OK. Whenever \TeX\ is
about to include |\par| as part of an argument, it will abort the current
macro expansion and report that a ``^{runaway} argument'' has been found.
\ddanger If you actually want a control sequence to allow arguments with
|\par| tokens, you can define it to be a ``long'' macro by saying
`^|\long|' just before `|\def|'. For example, the |\bold| macro
defined by
\begintt
\long\def\bold#1{{\bf#1}}
\endtt
is capable of setting several paragraphs in boldface type. \ (However,
such a macro is not an especially good way to typeset bold text.
It would be better to say, e.g.,
\begintt
\def\beginbold{\begingroup\bf}
\def\endbold{\endgroup}
\endtt
because this doesn't fill \TeX's memory with a long argument.)
\ddanger The |\par|-forbidding mechanism doesn't catch all conceivable
missing-brace errors, however; you might forget the |}| at the end of a
|\def|, and the same problem would arise. In this case it's harder to
confine the error, because |\par| is a useful thing in replacement
texts; we wouldn't want to forbid |\par| there, so \TeX\ has another
mechanism: When a macro definition is preceded by `^|\outer|', the
corresponding control sequence will not be allowed to appear in any place
where tokens are being absorbed at high speed. ^^{forbidden control sequence}
An |\outer| macro cannot appear in an argument (not even when |\par| is
allowed), nor can it appear in the parameter text or the replacement text
of a definition, nor in the ^{preamble} to an alignment, nor in ^{conditional
text} that is being skipped over. If an |\outer| macro does show up in
such places, \TeX\ stops what it is doing and reports either a
``runaway'' situation or an ``^{incomplete}'' conditional. The ^{end of an
input file} or alignment template ^^{endtemplate}
is also considered to be |\outer| in this sense; for example, a
file shouldn't end in the middle of a definition. If you are designing a
format for others to use, you can help them detect errors before too much
harm is done, by using |\outer| with all control sequences that should
appear only at ``quiet times'' within a document. For example, Appendix~B
defines ^|\proclaim| to be |\outer|, since a user shouldn't be stating a
theorem as part of a definition or argument or preamble.
\ddanger We have now seen that |\def| can be preceded by |\long|
or~|\outer|, and it can also be preceded by ^|\global| if the definition
is supposed to transcend its group. These three prefixes can be applied
to |\def| in any order, and they can even appear more than once.
\TeX\ also has a ^|\gdef| primitive that is equivalent to |\global\def|.
Thus, for example,
\begintt
\long\outer\global\long\def
\endtt
means the same thing as `|\outer\long\gdef|'.
\ddanger So far in this manual we have encountered several ways to
assign a meaning to a control sequence. For example,
$$\halign{\indent#\hfil\quad&#\hfil\cr
|\font\cs=|\<external font name>&makes |\cs| a font identifier;\cr
|\chardef\cs=|\<number>&makes |\cs| a character code;\cr
|\countdef\cs=|\<number>&makes |\cs| a |\count| register;\cr
|\def\cs...{...}|&makes |\cs| a macro.\cr
\noalign{\medskip \hbox{It's time now to reveal another important command
of this type:} \medskip}
|\let\cs=|\<token>&gives |\cs| the token's current meaning.\cr}$$
^^|\let|
If the \<token> is another control sequence, |\cs| will acquire the
same significance as that control sequence. For example, if you say
`|\let\a=\def|', you could then say `|\a\b...{...}|' to define a macro~|\b|,
because |\a| would behave like \TeX's primitive |\def| command.
If you say
\begintt
\let\a=\b \let\b=\c \let\c=\a
\endtt
you have interchanged the former meanings of\/ |\b| and |\c|. And if you say
\begintt
\outer\def\a#1.{#1:}
\let\b=\a
\endtt
the effect is exactly the same as `|\outer\def\b#1.{#1:} \let\a=\b|'.
\ddanger If the \<token> in a |\let| is a single character---i.e.,
if it is a (character code, category code) pair---then
the control sequence will behave to a certain extent like that
character; but there are some differences. For example, after
`|\let\zero=0|' you can't use |\zero| in a numerical constant, because
\TeX\ requires the tokens in a numerical constant to be digits, after
macro expansion; |\zero| is not a macro, so it doesn't expand. However,
such uses of\/ |\let| have their value, as we will see later.
\ddangerexercise Is there a significant difference between `|\let\a=\b|'
and `|\def\a{\b}|'?
\answer Yes indeed. In the first case, |\a| receives the meaning of\/~|\b|
that is current at the time of the |\let|. In the second case, |\a|~becomes
a~macro that will expand into the token~|\b| whenever |\a|~is used,
so it has the meaning of\/~|\b| that is current at the time of use.
You need |\let|, if you want to interchange the meanings of\/ |\a| and~|\b|.
\ddangerexercise Experiment with \TeX\ to discover the answers to the
following questions: (a)~If the control sequence ^|\par| has been redefined
(e.g., `|\def\par{\endgroup\par}|'\thinspace), is |\par| still forbidden
to appear in an argument? \ (b)~If you say |\let\xpar=\par|, is |\xpar|
also forbidden in an argument?
\answer (a) Yes. \ (b) No; any other control sequence can appear
(except those declared as |\outer| macros).
\ddanger \TeX\ also allows the construction
`^|\futurelet||\cs|\<token$_1$>\<token$_2$>', which has the effect of
`|\let\cs = |\<token$_2$>\<token$_1$>\<token$_2$>'. The idea is that you can
say, for example, `|\futurelet\a\b|' at the end of the replacement text of
a macro; \TeX~will set |\a| to the token that follows the macro, after
which |\b| will be expanded. The control sequence~|\b| can continue the
processing, and it can examine |\a| to see what's coming up next.
^^{looking ahead}
\danger The next thing a person wants, after getting used to macros with
parameters, is the ability to write macros that change their behavior
depending on current conditions. \TeX\ provides a variety of primitive
commands for this purpose. The general form of such ``^{conditional text}'' is
\begindisplay
|\if|\<condition>\<true text>|\else|\<false text>|\fi|
\enddisplay
where the \<true text> is skipped unless the \<condition> is true, and
the \<false text> is skipped unless the \<condition> is false.
If the \<false text> is empty, you can omit the~^|\else|. The
`|\if|\<condition>' part of this construction begins with a control
sequence whose first two letters are `|if|'; for example,
\begintt
\ifodd\count0 \rightpage \else\leftpage \fi
\endtt
^^|\ifodd|
specifies a condition that is true when \TeX's integer register ^|\count||0|
is odd. Since \TeX\ generally keeps the current ^{page number} in
|\count0|, the macro |\rightpage| will be expanded in this example if
the page number is odd, while |\leftpage| will be expanded if the
page number is even. Conditional commands always end with a final `^|\fi|'.
\danger Conditionals are primarily intended for experienced \TeX\ users,
who want to define high-level macros; therefore
the remaining paragraphs in this chapter are headed by ``double
dangerous bends.'' Do not feel guilty about skipping right to Chapter~21;
in other words, imagine that the manual says `|\ifexperienced|' right
here, and that there is a matching `|\fi|' at the end of the present chapter.
\ddanger Before we discuss \TeX's repertoire of\/ |\if...|\ commands, let's
look at another example, so that the general ideas will be clear. Suppose
that the |\count| register |\balance| holds an amount that somebody
has paid in excess of his or her income tax; this amount is given in
pennies, and it might be positive, negative, or zero. Our immediate goal
will be to write a \TeX\ macro that generates a suitable statement for the
^{Internal Revenue Service} to include as part of a letter to that person,
based on the amount of the balance. The statement will be quite different
for positive balances than for negative ones, so we can exploit \TeX's ability
to act conditionally:
\begintt
\def\statement{\ifnum\balance=0 \fullypaid
\else\ifnum\balance>0 \overpaid
\else\underpaid
\fi
\fi}
\endtt
Here ^|\ifnum| is a conditional command that compares two numbers; the
|\statement| macro reduces to |\fullypaid| if the balance is zero, and so on.
\ddanger It is vastly important to notice the spaces after the |0|'s in
this construction. If the example had said
\begintt
...=0\fullypaid...
\endtt
then \TeX\ would have begun to expand `|\fullypaid|' before it knew the
value of the constant |0|, because |\fullypaid| might start with a~|1| or
something that would change the number. \ (After all, `|01|' is a perfectly
acceptable \<number>, in \TeX's eyes.) \ In this particular case the
program would still have worked, because we will see in a moment that
|\fullypaid| begins with the letter~|Y|; thus, the only problem caused by
the missing space would be that \TeX\ would go slower, since it would have
to skip over the whole expansion of\/ |\fullypaid| instead of just skipping
|\fullypaid| as a single, unexpanded token. But in other situations a
missing space like this might cause \TeX\ to expand macros when you don't
want any expansion, and such anomalies can cause subtle and confusing errors.
For best results, {\sl always put a blank space after a numeric constant\/};
this blank space tells \TeX\ that the constant is complete, and
such a space will never ``get through'' to the output. In fact, when you
don't have a blank ^{space after a constant}, \TeX\ actually has to do
more work, because each constant continues until a non-digit has been
read; if this non-digit is not a space, \TeX\ takes the token you did have
and backs it up, ready to be read again. \ (On the other hand, the author
often omits the space when a constant is immediately followed by some
other character, because extra spaces do look funny in the file;
aesthetics are more important than~efficiency.)
\ddangerexercise Continuing the IRS example, assume that |\fullypaid|
and |\underpaid| are defined as follows:
\begintt
\def\fullypaid{Your taxes are fully paid---thank you.}
\def\underpaid{{\count0=-\balance
\ifnum\count0<100
You owe \dollaramount, but you need not pay it, because
our policy is to disregard amounts less than \$1.00.
\else Please remit \dollaramount\ within ten days,
or additional interest charges will be due.\fi}}
\endtt
Write a macro |\overpaid| to go with these, assuming that |\dollaramount|
is a macro that generates the contents of\/ |\count0| in dollars and cents.
Your macro should say that a check will be mailed under separate cover,
unless the amount is less than \$1.00, in which case the person must
specifically request a check.
\answer |\def\overpaid{{\count0=\balance|\parbreak
| You have overpaid your tax by \dollaramount.|\parbreak
| \ifnum\count0<100 It is our policy to refund|\parbreak
| such a small amount only if you ask for it.|\parbreak
| \else A check for this amount is being mailed|\parbreak
| under separate cover.\fi}}|
\ddangerexercise Write a |\dollaramount| macro, to complete the Internal
Revenue |\statement|.
\answer The tricky part is to get the zero in an amount like `|$2.01|'.
\begintt
\def\dollaramount{\count2=\count0 \divide\count2 by100
\$\number\count2.%
\multiply\count2 by-100 \advance\count2 by\count0
\ifnum \count2<10 0\fi
\number\count2 }
\endtt
\ddanger Now let's make a complete survey of \TeX's conditional commands.
Some of them involve features that have not yet been introduced in this
manual.\enddanger
\nobreak\medskip
\item\bull^|\ifnum|\<number$_1$>\<relation>\<number$_2$>\quad
(compare two integers)
\nobreak\smallskip\noindent
The ^\<relation> must be either `|<|$_{12}$' or `|=|$_{12}$' or `|>|$_{12}$'.
The two integer numbers are compared to each other in the usual way, and
the result is true or false accordingly.
\medbreak
\item\bull^|\ifdim|\<dimen$_1$>\<relation>\<dimen$_2$>\quad
(compare two dimensions)
\nobreak\smallskip\noindent
This is like |\ifnum|, but it compares two \<dimen> values. For example,
to test whether the value of\/ |\hsize| exceeds $100\pt$, you can say
`|\ifdim\hsize>100pt|'.
\medbreak
\item\bull^|\ifodd|\<number>\quad(test for odd integer)
\nobreak\smallskip\noindent
The condition is true if the \<number> is odd, false if it is even.
\medbreak
\item\bull^|\ifvmode|\quad(test for vertical mode)
\nobreak\smallskip\noindent
True if \TeX\ is in vertical mode or internal vertical mode (see Chapter~13).
\medbreak
\item\bull^|\ifhmode|\quad(test for horizontal mode)
\nobreak\smallskip\noindent
True if \TeX\ is in horizontal mode or restricted horizontal mode
(see Chapter~13).
\medbreak
\item\bull^|\ifmmode|\quad(test for math mode)
\nobreak\smallskip\noindent
True if \TeX\ is in math mode or display math mode (see Chapter~13).
\medbreak
\item\bull^|\ifinner|\quad(test for an internal mode)
\nobreak\smallskip\noindent
True if \TeX\ is in internal vertical mode, or restricted
horizontal mode, or (nondisplay) math mode (see Chapter~13).
\medbreak
\item\bull^|\if|\<token$_1$>\<token$_2$>\quad(test if character codes agree)
\nobreak\smallskip\noindent
\TeX\ will expand macros following |\if| until two unexpandable tokens are
found. If either token is a control sequence, \TeX\ considers it to have
character code~256 and category code~16, unless the current equivalent of
that control sequence has been |\let| equal to a non-active character
token. In this way, each token specifies a (character~code,
\hbox{category}~code) pair. The condition is true if the character codes
are equal, independent of the category codes. For example, after
|\def\a{*}| and |\let\b=*| and |\def\c{/}|, the tests `|\if*\a|' and
`|\if\a\b|' will be true, but `|\if\a\c|' will be false. Also
`|\if\a\par|' will be false, but `|\if\par\let|' will be true. % Beresford=true
\medbreak
\item\bull^|\ifcat|\<token$_1$>\<token$_2$>\quad(test if category codes agree)
\nobreak\smallskip\noindent
This is just like |\if|, but it tests the ^{category codes}, not the character
codes. ^{Active characters} have category~13, but you have to say
`^|\noexpand|\<active character>' in order to suppress expansion when you
are looking at such characters with |\if| or |\ifcat|. For example, after
\begintt
\catcode`[=13 \catcode`]=13 \def[{*}
\endtt
the tests `|\ifcat\noexpand[\noexpand]|' and `|\ifcat[*|' will be true,
but the test `|\ifcat\noexpand[*|' will be false.
\medbreak
\item\bull^|\ifx|\<token$_1$>\<token$_2$>\quad(test if tokens agree)
\nobreak\smallskip\noindent
In this case, \TeX\ does {\sl not\/} expand control sequences when it
looks at the two tokens. The condition is true if (a)~the two tokens are
not macros, and they both represent the same (character code, category
code) pair or the same \TeX\ primitive or the same ^|\font| or ^|\chardef|
or ^|\countdef|, etc.; or if (b)~the two tokens are macros, and they both
have the same status with respect to ^|\long| and ^|\outer|, and they both
have the same parameters and ``top level'' expansion. For example, after
`|\def\a{\c}| |\def\b{\d}| |\def\c{\e}| |\def\d{\e}| |\def\e{A}|', an
|\ifx| test will find |\c| and |\d| equal, but not |\a| and~|\b|, nor |\d|
and~|\e|, nor any other combinations of\/ |\a|, |\b|, |\c|, |\d|, |\e|.
\medbreak
\item\bull^|\ifvoid|\<number>, ^|\ifhbox|\<number>, ^|\ifvbox|\<number>\quad
(test a box register)
\nobreak\smallskip\noindent
The \<number> should be between 0 and 255. The condition is true if that
|\box| is void or contains an hbox or a vbox, respectively (see Chapter~15).
\medbreak
\item\bull^|\ifeof|\<number>\quad(test for end of file)
\nobreak\smallskip\noindent
The \<number> should be between 0 and 15. The condition is true unless the
corresponding input stream is open and not fully read. \ (See the command
^|\openin| below.)
\medbreak
\item\bull^|\iftrue|, ^|\iffalse|\quad(always true or always false)
\nobreak\smallskip\noindent
These conditions have a predetermined outcome. But they turn out to be
useful in spite of this, as explained below.
\medbreak
Finally, there's one more conditional construction, which is somewhat different
from the rest because it is capable of making a many-way branch:
\begindisplay
\llap{\bull\enspace}^|\ifcase|\<number>\<text for case 0>|\or|\<text
for case 1>|\or|$\;\cdots$\cr
| \or|\<text for case $n$>|\else|\<text for all other cases>|\fi|\cr
\enddisplay
Here there are $n+1$ cases separated by $n$ ^|\or|'s, where $n$ can be any
nonnegative number. The \<number> selects the text that will be used. Once
again the ^|\else| part is optional, if you don't want to specify any text
for cases when the \<number> is negative or greater than~$n$.
\ddangerexercise Design a |\category| macro that prints a character's current
category code symbolically, given a one-character control sequence for
that character. For example, if the category codes of plain \TeX\ are in
force, `|\category\\|' should expand to `|escape|', and `|\category\a|'
should expand to `|letter|'.
\answer |\def\category#1{\ifcase\catcode`#1|\parbreak
| escape\or begingroup\or endgroup\or math\or|\parbreak
| align\or endline\or parameter\or superscript\or|\parbreak
| subscript\or ignored\or space\or letter\or|\parbreak
| otherchar\or active\or comment\or invalid\fi}|\par
\ddangerexercise Test yourself on the following questions to see if you
understand certain borderline situations: After the definitions
`|\def\a{}| |\def\b{**}| |\def\c{True}|',
which of the following are true?
(a)~`|\if\a\b|';
(b)~`|\ifcat\a\b|';
(c)~`|\ifx\a\b|';
(d)~`|\if\c|';
(e)~`|\ifcat\c|';
(f)~`|\ifx\ifx\ifx|'.
(g)~`|\if\ifx\a\b\c\else\if\a\b\c\fi\fi|'.
\answer (a,b)~True. (c,d)~False. (e,f)~True. In case~(e), the \<true text>
starts with `|ue|'. (g)~The |\ifx| is false and the inner |\if| is true;
so the outer |\if| becomes `|\if True...|', which is false. \
(Interestingly, \TeX\ knows that the outer |\if| is false even before it
has looked at the |\fi|'s that close the |\ifx| and the inner |\if|.)
\ddanger Notice that all of the control sequences for conditionals
begin with |\if...|, and they all have a matching~|\fi|. This convention---that
|\if...|\ pairs up with |\fi|---makes it easier to see the nesting
of conditionals within your program. The ^{nesting} of\/ |\if...\fi| is
independent of the nesting of |{...}|; thus, you can begin or end a ^{group}
in the middle of a conditional, and you can begin or end a conditional
in the middle of a group. Extensive experience with macros has shown that
such independence is important in applications; but it can also lead to
confusion if you aren't careful.
\ddanger It's sometimes desirable to pass information from one macro to
another, and there are several ways to do this: ^^{communication between
macros} by passing it as an argument, by putting it into a register, or by
defining a control sequence that contains the information. For example,
the macros |\hphantom|, |\vphantom|, and ^|\phantom| in Appendix~B are
quite similar, so the author ^^{Knuth} wanted to do most of the work in
another macro |\phant| that would be common to all three. Somehow |\phant|
was to be told what kind of phantom was desired. The first approach was to
define control sequences |\hph| and |\vph| something like this:
\begintt
\def\hphantom{\ph YN} \def\vphantom{\ph NY} \def\phantom{\ph YY}
\def\ph#1#2{\def\hph{#1}\def\vph{#2}\phant}
\endtt
after which |\phant| could test `|\if Y\hph|' and `|\if Y\vph|'. This
worked, but there were various ways to make it more efficient; for example,
`|\def\hph{#1}|' could be replaced by `|\let\hph=#1|', avoiding macro expansion.
An even better idea then suggested itself:
\begintt
\def\yes{\if00} \def\no{\if01}
\def\hphantom{\ph\yes\no}...\def\phantom{\ph\yes\yes}
\def\ph#1#2{\let\ifhph=#1\let\ifvph=#2\phant}
\endtt
after which |\phant| could test `|\ifhph|' and `|\ifvph|'. \ (This
construction was tried before |\iftrue| and |\iffalse| were part of the
\TeX\ language.) \ The idea worked fine, so the author started to use
|\yes| and |\no| in a variety of other situations. But then one day a
complex conditional failed, because it contained an |\ifhph|-like test
inside another conditional:
\begintt
\if... \ifhph...\fi ... \else ... \fi
\endtt
Do you see the problem that developed? When the \<true text> of the
outermost conditional was executed, everything worked fine, because
|\ifhph| was either |\yes| or |\no| and it expanded into either |\if00| or
|\if01|. But when the \<true text> was skipped, the |\ifhph| was not
expanded, so the first |\fi| was mistakenly paired with the first |\if|;
everything soon went haywire. That's when ^|\iftrue| and ^|\iffalse|
were put into the language, in place of\/ |\yes| and |\no|; now
|\ifhph| is either |\iftrue| or |\iffalse|, so \TeX\ will match it
properly with a closing~|\fi|, whether or not it is being skipped over.
\ddanger To facilitate |\if...|~constructions, plain \TeX\ has a
^|\newif| macro, such that after you say `|\newif\ifabc|' three
control sequences will be defined: |\ifabc| (for testing the switch),
|\abctrue| (for making the switch true), and |\abcfalse| (for making
it false). The |\phantom| problem is now solved in Appendix~B by writing
\begintt
\newif\ifhph \newif\ifvph
\def\hphantom{\hphtrue\vphfalse\phant}
\endtt
and with similar definitions of\/ |\vphantom| and |\phantom|. There is no
longer any need for a |\ph| macro; again |\phant| tests |\ifhph| and
|\ifvph|. Appendix~E contains other examples of conditionals created
by~|\newif|. New conditionals are initially false.
\ddanger Caution: Don't say anything like `|\let\ifabc=\iftrue|' in
conditional text. If \TeX\ skips over this command, it will think that
both |\ifabc| and |\iftrue| require a matching |\fi|, since the |\let| is
not being executed! Keep such commands buried inside macros, so that \TeX\
will see the `|\if...|'\ only when it is not skipping over the text that it
is reading.
\ddanger \TeX\ has 256 ``^{token list registers}'' called ^|\toks||0|
through |\toks255|, so that token lists can easily be shuffled around
without passing them through \TeX's reading apparatus.
There's also a ^|\toksdef| instruction so that, e.g.,
\begintt
\toksdef\catch=22
\endtt
makes |\catch| equivalent to |\toks22|. Plain \TeX\ provides a ^|\newtoks|
macro that allocates a new token list register; it is analogous to
|\newcount|. Token list registers behave like the ^{token list parameters}
|\everypar|, |\everyhbox|, |\output|, |\errhelp|, etc. To assign
a new value to a token list parameter or register, you say either
\begindisplay
\<token variable>|={|\<replacement text>|}|\cr
\llap{or }\<token variable>|=|\<token variable>\cr
\enddisplay
where ^\<token variable> means either a token list parameter or a
control sequence defined by |\toksdef| or |\newtoks|, or an
explicit register designation `|\toks|\<number>'.
\ddanger Everyone who makes extensive use of a powerful macro facility
encounters situations when the macros do surprising things. We have
already mentioned the possibility of setting |\tracingmacros=1|, in order
to see when \TeX\ expands macros and what arguments it finds. There's also
another helpful way to watch what \TeX\ is doing: If you set
^|\tracingcommands||=1|, \TeX\ will show every command that it executes,
as we saw in Chapter~13. Furthermore, if you set
|\tracingcommands=2|, \TeX\ will show all conditional commands and
their outcomes, as well as the unconditional commands that are actually
performed or expanded. This diagnostic information goes into your log
file. You can also see it on your terminal, if you say
^|\tracingonline||=1|. \ (Incidentally, if you make |\tracingcommands|
greater than~2, you get the same information as when it equals~2.) \
Similarly, ^|\tracingmacros||=2| will trace |\output|, |\everypar|, etc.
\ddanger One way to understand the occasional strangeness of macro
operation is to use the tracing features just described, so that you can
watch what \TeX\ does in slow motion. Another way is to learn the rules for
how macros are expanded; we shall now discuss those rules.
\ddanger \TeX's mastication process converts your input to a long token
list, as explained in Chapter~8; and its digestive processes work
strictly on this token list. When \TeX\ encounters a control sequence in
the token list, it looks up the current meaning, and in certain cases
it will expand that token into a sequence of other tokens before
continuing to read. The expansion process applies to macros and to
certain other special primitives like |\number| and |\if| that we shall
consider momentarily. Sometimes, however, the ^{expansion} is not carried out;
for example, when \TeX\ is taking care of a |\def|, the \<control sequence>,
the \<parameter text>, and the \<replacement text> of that |\def| are
not subject to expansion. Similarly, the two tokens after |\ifx| are
never expanded. A complete list of occasions when tokens are not expanded
appears later in this chapter; you can use it for reference in an emergency.
\ddanger Now let's consider the control sequences that are expanded whenever
expansion has not been inhibited. Such control sequences
fall into several classes:\enddanger
\nobreak\medskip
\textindent\bull Macros. When a macro is expanded, \TeX\ first determines its
arguments (if any), as explained earlier in this chapter. Each argument
is a token list; the tokens are not expanded when they are being
accepted as arguments. Then \TeX\ replaces the macro and its arguments
by the replacement text.
\smallbreak
\textindent\bull Conditionals. When an |\if...| is expanded, \TeX\ reads
ahead as far as necessary to determine whether the condition is true or
false; and if false, it skips ahead (keeping track of\/ |\if...\fi| nesting)
until finding the |\else|, |\or|, or~|\fi| that ends the skipped text.
Similarly, when |\else|, |\or|, or~|\fi| is expanded, \TeX\ reads to the
end of any text that ought to be skipped. The ``expansion'' of a conditional
is empty. \ (Conditionals always reduce the number of tokens that are seen by
later stages of the digestive process, while macros usually increase the
number of tokens.)
\smallbreak
\textindent\bull ^|\number|\<number>. When \TeX\ expands |\number|, it reads
the \<number> that follows (expanding tokens as it goes); the final
expansion consists of the ^{decimal representation} of that number,
preceded by `|-|' if negative.
\smallbreak
\textindent\bull ^|\romannumeral|\<number>. This is like |\number|, but
the expansion consists of lowercase roman numerals. For example,
`|\romannumeral 1984|' produces `|mcmlxxxiv|'. The expansion is
empty if the number is zero or negative.
\smallbreak
\textindent\bull ^|\string|\<token>. \TeX\ first reads the \<token> without
expansion. If a control sequence token appears, its |\string| expansion
consists of the control sequence name (including ^|\escapechar| as an escape
character, if the control sequence isn't simply an active character).
Otherwise the \<token> is a character token, and its character code
is retained as the expanded result.
\smallbreak
\textindent\bull ^|\jobname|. The expansion is the name that \TeX\ has chosen
for this job. For example, if \TeX\ is putting its output on files
|paper.dvi| and |paper.log|, ^^|dvi| then |\jobname| expands to `|paper|'.
\smallbreak
\textindent\bull ^|\fontname|\<font>. The expansion is the external file name
corresponding to the given font; e.g., `|\fontname\tenrm|' might expand
to `|cmr10|' (five tokens). If the font is not being used at its design size,
the ``^{at size}'' also appears in the expansion. A ^\<font> is either an
identifier defined by ^|\font|; or |\textfont|\<number>,
|\scriptfont|\<number>, or |\scriptscriptfont|\<number>; or |\font|, which
denotes the current font.
\smallbreak
\textindent\bull ^|\meaning|\<token>. \TeX\ expands this to the sequence of
characters that would be displayed on your terminal by the commands
`|\let\test=|\<token> |\show\test|'. For example, `|\meaning A|' usually
expands to `|the letter A|'; `|\meaning\A|' after `|\def\A#1B{\C}|'
expands to `|macro:#1B->\C |'.
\smallbreak
\textindent\bull ^|\csname||...|^|\endcsname|. When \TeX\ expands |\csname|
it reads to the matching |\endcsname|, expanding tokens as it goes;
only character tokens should remain after this expansion has taken place.
Then the ``expansion'' of the entire |\csname...\endcsname| text
will be a single control sequence token, defined to be like |\relax| if
its meaning is currently undefined.
\smallbreak
\textindent\bull ^|\expandafter|\<token>. \TeX\ first reads the token that
comes immediately after |\expandafter|, without expanding it;
let's call this token~$t$. Then \TeX\ reads the token that comes after~$t$
(and possibly more tokens, if that token has an argument), replacing it by
its expansion. Finally \TeX\ puts~$t$ back in front of that expansion.
\smallbreak
\textindent\bull ^|\noexpand|\<token>. The expansion is the token itself;
but that token is interpreted as if its meaning were `|\relax|' if it is a
control sequence that would ordinarily be expanded by \TeX's expansion rules.
\smallbreak
\textindent\bull ^|\topmark|, ^|\firstmark|, ^|\botmark|, ^|\splitfirstmark|,
and ^|\splitbotmark|. \kern-1.7pt % This saves an overfull box (March 27, 1983)
The expansion is the token list in the corresponding ``^{mark}'' register
(see Chapter~23).
\smallbreak
\textindent\bull ^|\input|\<file name>. The expansion is null; but \TeX\
prepares to read from the specified file before looking at any more
tokens from its current source.
\smallbreak
\textindent\bull ^|\endinput|. The expansion is null. The next time \TeX\ gets
to the end of an |\input| line, it will stop reading from the file containing
that line.
\smallbreak
\textindent\bull ^|\the|\<internal quantity>. The expansion is a list of
tokens representing the current value of one of \TeX's variables, as
explained below. For example, `|\the\skip5|' might expand into
`|5.0pt plus 2.0fil|' (17~tokens).
\ddanger The powerful |\the| operation has many subcases, so we shall discuss
them one at a time. A variety of internal numeric quantities
can be brought up front:\enddanger
\nobreak\medskip
\textindent\bull |\the|\<parameter>, where \<parameter> is the name of
one of \TeX's ^{integer parameters} (e.g., |\the\widowpenalty|),
^{dimension parameters} (e.g., |\the\parindent|), ^{glue parameters}
(e.g., |\the\leftskip|), or ^{muglue parameters} (e.g., |\the\thinmuskip|).
\smallbreak
\textindent\bull |\the|\<register>, where \<register> is the name of
one of \TeX's integer ^{registers} (e.g., |\the\count|\stretch|0|),
dimension registers (e.g., |\the\dimen169|), glue registers (e.g.,
|\the\skip255|), or muglue registers (e.g., |\the\muskip\count2|).
\smallbreak
\textindent\bull |\the|\<codename>\<8-bit number>, where \<codename>
stands for either ^|\catcode|, ^|\mathcode|, ^|\lccode|, ^|\uccode|,
^|\sfcode|, or ^|\delcode|. For example, |\the\mathcode`/| produces the
current (integer) math code value for a slash.
\smallbreak
\textindent\bull |\the|\<special register>, where \<special register> is
one of the integer quantities ^|\prevgraf|, ^|\deadcycles|, ^|\insertpenalties|,
^|\inputlineno|, ^|\badness|,
or ^|\parshape| (denoting only the number of lines of\/ |\parshape|); or
one of the dimensions ^|\pagetotal|, ^|\pagegoal|, ^|\pagestretch|,
^|\pagefilstretch|, ^|\pagefillstretch|, ^|\pagefilllstretch|, ^|\pageshrink|,
^|\pagedepth|. In horizontal modes you can also refer to a special integer,
|\the\spacefactor|; in vertical modes there's a special dimension,
|\the\prevdepth|.
\smallbreak
\textindent\bull |\the|^|\fontdimen|\<parameter number>\<font>.
This produces a dimension; for example, parameter~6 of a font is its
``^{em}'' value, so `|\the\fontdimen6\tenrm|' yields `|10.0pt|' (six tokens).
\smallbreak
\textindent\bull |\the|^|\hyphenchar|\<font>, |\the|^|\skewchar|\<font>.
These produce the corresponding integer values defined for the specified font.
\smallbreak
\textindent\bull |\the|^|\lastpenalty|, |\the|^|\lastkern|, |\the|^|\lastskip|.
These yield the amount of penalty, kerning, glue, or
muglue in the final item on the current list, provided that the item is a
penalty, kern, or glue, respectively; otherwise they yield `|0|' or `|0.0pt|'.
\smallbreak
\textindent\bull |\the|\<defined character>, where \<defined character> is
a control sequence that has been given an integer value with ^|\chardef| or
^|\mathchardef|; the result is that integer value, in decimal notation.
\ddanger In all of the cases listed so far, |\the| produces a result that
is a sequence of ^{ASCII} character tokens. Category code~12 (``other'') is
assigned to each token, except that character code~32 gets category~10
(``space''). The same rule is used to assign ^{category codes} to the
tokens produced by ^|\number|, ^|\romannumeral|, ^|\string|, ^|\meaning|,
^|\jobname|, and ^|\fontname|.
\ddanger There also are cases in which |\the| produces non-character tokens,
either a font identifier like |\tenrm|, or an arbitrary token list:
\textindent\bull |\the|\<font> produces a font identifier that selects the
specified font. For example, `|\the\font|' is a control sequence corresponding
to the current font.
\goodbreak\smallskip
\textindent\bull |\the|\<token variable> produces a copy of the token list that
is the current value of the variable. For example, you can expand
`|\the\everypar|' and `|\the\toks5|'.
\ddanger \TeX's primitive command `^|\showthe|' will display on your
terminal exactly what `|\the|' would produce in an expanded definition;
the expansion is preceded by `|> |' and followed by a period. For example,
`|\showthe\parindent|' will display
\begintt
> 20.0pt.
\endtt
if the plain \TeX\ paragraph indentation is being used.
\ddanger Here now is the promised list of all cases when expandable tokens
are not expanded. Some of the situations involve primitives that
haven't been discussed yet, but we'll get to them
eventually. Expansion is suppressed at the following times:\enddanger
\nobreak\medskip
\item\bull When tokens are being deleted during ^{error recovery} (see
Chapter~6).
\smallskip
\item\bull When tokens are being skipped because conditional text is being
ignored.
\smallskip
\item\bull When \TeX\ is reading the arguments of a macro.
\smallskip
\item\bull When \TeX\ is reading a control sequence to be defined by
^|\let|, ^|\futurelet|, ^|\def|, ^|\gdef|, ^|\edef|, ^|\xdef|, ^|\chardef|,
^|\mathchardef|, ^|\countdef|, ^|\dimendef|, ^|\skipdef|, ^|\muskipdef|,
^|\toksdef|, ^|\read|, and ^|\font|.
\smallskip
\item\bull When \TeX\ is reading argument tokens for ^|\expandafter|,
^|\noexpand|, ^|\string|, ^|\meaning|, ^|\let|, ^|\futurelet|, ^|\ifx|,
^|\show|, ^|\afterassignment|, ^|\aftergroup|.
\smallskip
\item\bull When \TeX\ is absorbing the parameter text of a |\def|, |\gdef|,
|\edef|, or |\xdef|.
\smallskip
\item\bull When \TeX\ is absorbing the replacement text of a |\def| or
|\gdef| or ^|\read|; or the text of a ^{token variable} like ^|\everypar|
or ^|\toks||0|; or the token list for ^|\uppercase| or ^|\lowercase| or
^|\write|. \ (The token list for |\write| will be expanded later, when it
is actually output to a file.)
\smallskip
\item\bull When \TeX\ is reading the preamble of an alignment, except after
a token for the primitive command |\span| or when reading the \<glue>
after ^|\tabskip|.
\smallskip
\item\bull Just after a |$|$_3$ token that begins math mode, to see if
another~|$|$_3$ follows.
\smallskip
\item\bull Just after a |`|$_{12}$ token that begins an ^{alphabetic constant}.
\ddanger Sometimes you will find yourself wanting to define new macros
whose replacement text has been expanded, based on current conditions,
instead of simply copying the replacement text verbatim. \TeX\ provides
the ^|\edef| (expanded definition) command for this purpose, and also
^|\xdef| (which is equivalent to |\global\edef|). The general format
is the same as for |\def| and |\gdef|, but \TeX\ blindly expands the tokens
of the replacement text according to the expansion rules above. For
example, consider
\begintt
\def\double#1{#1#1}
\edef\a{\double{xy}} \edef\a{\double\a}
\endtt
Here the first |\edef| is equivalent to `|\def\a{xyxy}|' and the second is
equivalent to `|\def\a{xyxyxyxy}|'. All of the other kinds of expansion
will take place too, including conditionals; for example,
\begintt
\edef\b#1#2{\ifmmode#1\else#2\fi}
\endtt
gives a result equivalent to `|\def\b#1#2{#1}|' if \TeX\ is in math
mode at the time of the |\edef|, otherwise the result is equivalent to
`|\def\b#1#2{#2}|'.
\ddanger Expanded definitions that are made with |\edef| or |\xdef| continue
to expand tokens until only unexpandable tokens remain, except that
token lists produced by `^|\the|' are not expanded further. Furthermore
a token following `^|\noexpand|' will not be expanded, since its ability
to expand has been nullified. These two operations can be used to control
^^{expansion, avoiding} what gets expanded and what doesn't.
\ddanger Suppose, for example, that you
want to define |\a| to be equal to~|\b| (expanded) followed by~|\c|
(not expanded) followed by |\d| (expanded), assuming that |\b| and
|\d| are simple macros without parameters. There are two easy ways to do it:
\begintt
\edef\a{\b\noexpand\c\d}
\toks0={\c} \edef\a{\b\the\toks0 \d}
\endtt
And it's even possible to achieve the same effect without using either
|\noexpand| or |\the|; a reader who wants to learn more about \TeX's
expansion mechanism is encouraged to try the next three exercises.
\ddangerexercise Figure out a way to define |\a| as in the previous paragraph,
without using \TeX's primitives `|\noexpand|' and `|\the|'.
\answer One idea is to say
\begintt
\let\save=\c \let\c=0 \edef\a{\b\c\d} \let\c=\save
\endtt
because control sequences equivalent to characters are not expandable.
However, this doesn't expand occurrences of~|\c|
that might be present in the expansions of\/ |\b| and~|\d|. Another way,
which is free of this defect, is
\begintt
\edef\next#1#2{\def#1{\b#2\d}} \next\a\c
\endtt
(and it's worth a close look!).
\ddangerexercise Continuing the example of expansion avoidance, suppose that
you want to expand |\b| completely until only unexpandable tokens are left,
but you don't want to expand |\c| at all, and you want to expand |\d|
only one level. For example, after |\def\b{\c\c}| and |\def\c{*}| and
|\def\d{\b\c}| the goal would be to get the effect of\/ |\def\a{**\c\b\c}|.
How can such a partial expansion be achieved, using ^|\the|?
\answer |\toks0={\c} \toks2=\expandafter{\d}|\parbreak
|\edef\a{\b\the\toks0 \the\toks2 }|
\smallskip\noindent
(Notice that ^|\expandafter| expands the token after the left brace here.)
\ddangerexercise Solve the previous exercise without |\the| or |\noexpand|.
\ (This is difficult.)
\answer The following shouldn't be taken too seriously, but it does work:
^^|\span|
\begintt
{\setbox0=\vbox{\halign{#{\c\span\d}\cr
\let\next=0\edef\next#1{\gdef\next{\b#1}}\next\cr}}}
\let\a=\next
\endtt
\ddanger \TeX's primitive commands ^|\mark||{...}|, ^|\message||{...}|,
^|\errmessage||{...}|, ^|\special||{...}|, and ^|\write|\<number>|{...}|
all expand the token lists in braces almost exactly as |\edef| and |\xdef| do.
However, a macro parameter
character like~|#| should not be duplicated in such commands; you need
to say |##| within an |\edef|, but only |#| within a |\mark|. The
|\write| command is somewhat special, because its token list is first
read without expansion; expansion occurs later, when the tokens
are actually being written to a file.
\ddangerexercise Compare the following two definitions:
\begintt
\def\a{\iftrue{\else}\fi}
\edef\b{\iftrue{\else}\fi}
\endtt
Which of them yields an ^{unmatched left brace}? (This is tricky.)
\answer Neither one, although |\a| will behave like an unmatched left
brace when it is expanded. The definition of\/ |\b| is {\sl not complete},
because it expands to `|\def\b{{}|'; \TeX\ will continue to read ahead,
looking for another right brace, possibly discovering a runaway
definition! It's impossible to define a macro that has unmatched braces.
But you {\sl can\/} say |\let\a={|; Appendix~D discusses several
other ^{brace tricks}.
\def\rhead{Chapter \chapno: Definitions (aka Macros)% my little joke
\gdef\rhead{Chapter \chapno: Definitions (also called Macros)}}
\ddanger \TeX\ has the ability to read individual lines of text from up
to~16 files at once, in addition to the files that are being |\input|.
To initiate reading such an auxiliary file, you should say
\begindisplay
^|\openin|\<number>|=|\<file name>
\enddisplay
where the \<number> is between 0 and 15. \ (Plain \TeX\ allocates
input stream numbers 0~through~15 with the ^|\newread| command, which
is analogous to |\newbox|.) \ In most installations
of \TeX, the extension `^|.tex|' will be appended to the file name,
as with ^|\input|, if no extension is given explicitly. If the file
cannot be found, \TeX\ will give no error message; it will simply consider
that the input stream is not open, and you can test this condition
with ^|\ifeof|. When you're done with a file, you can say
\begindisplay
^|\closein|\<number>
\enddisplay
and the file associated with that input stream number will be closed,
i.e., returned to its initial condition, if such a file was open.
To get input from an open file, you say
\begindisplay
^|\read|\<number>^|to|\<control sequence>
\enddisplay
and the control sequence is defined to be a parameterless macro whose
replacement text is the contents of the next line read from the
designated file. This line is converted to a token list, using the
procedure of Chapter~8, based on the current category codes.
Additional lines are read, if necessary, until an equal number of
left and right braces has been found. An empty line is implicitly appended
to the end of a file that is being |\read|. ^^{empty line at end of file}
If the \<number> is not between 0 and~15, or if no such file is open, or if
the file has ended, input will be from the terminal; \TeX\ will prompt
the user unless the \<number> is negative. The macro definition will be
local unless you say |\global\read|.
\ddanger For example, it's easy to have ^{dialogs with the user}, by
using |\read| together with the ^|\message| command (which
writes an expanded token list on the terminal and in the log file):
\begintt
\message{Please type your name:}
\read16 to\myname
\message{Hello, \myname!}
\endtt
The |\read| command in this case will print `|\myname=|' and it will wait
for a response; the response will be echoed on the log file.
The `|\myname=|' would have been omitted if `|\read16|' had been
`|\read-1|'.
\ddangerexercise The |\myname| example just given doesn't work quite right,
because the \<return> at the end of the line gets translated into a
space. Figure out how to fix that glitch.
\answer One way is to redefine |\catcode`\^^M=9| (ignored) just before
the |\read|, so that the \<return> will be ignored. Another solution is
to redefine ^|\endlinechar||=-1|, so that no character is put at the
end of the line. Or you could try to be tricky by stripping off the
space with macro expansion as follows:
\begintt
\def\stripspace#1 \next{#1}
\edef\myname{\expandafter\stripspace\myname\next}
\endtt
The latter solution doesn't work if the user types `|%|' at the end of
his~or her name, or if the name contains control sequences.
\ddangerexercise Continuing the previous example, define a macro
|\MYNAME| that contains the letters of\/ |\myname| all in ^{uppercase
letters}. For example, if\/ |\myname| expands to |Arthur|, |\MYNAME|
should expand to |ARTHUR|. Assume that |\myname| contains only
letters and spaces in its expansion.
\answer Here are two solutions:
\begintt
\def\next#1\endname{\uppercase{\def\MYNAME{#1}}}
\expandafter\next\myname\endname
|smallskip\edef\next{\def\noexpand\MYNAME{\myname}}
\uppercase\expandafter{\next}
\endtt
\ddanger Appendices B, D, and E contain numerous examples of how to make
macros do useful things. Let's close this chapter by presenting a few
examples that show how \TeX\ can actually be used as a primitive ^{programming}
language, if you want to achieve special effects, and if you don't care very
much about computer costs.
\ddanger Plain \TeX\ contains a |\loop...\repeat| construction, which works
like this: You say `^|\loop|~$\alpha$~|\if...|~$\beta$~|\repeat|', where
$\alpha$ and~$\beta$ are any sequences of commands, and where |\if...|\
is any conditional test (without a matching |\fi|). \TeX\ will first
do~$\alpha$; then if the condition is true, \TeX\ will do~$\beta$ and
repeat the whole process again starting with~$\alpha$. If the
condition ever turns out to be false, the loop will stop. For example,
here is a program that carries out a little dialog in which \TeX\ waits for
the user to type `|Yes|' or `|No|': ^^{repeating commands, see :loop}
\begintt
\def\yes{Yes } \def\no{No } \newif\ifgarbage
\loop\message{Are you happy? }
\read-1 to\answer
\ifx\answer\yes\garbagefalse % the answer is Yes
\else\ifx\answer\no\garbagefalse % the answer is No
\else\garbagetrue\fi\fi % the answer is garbage
\ifgarbage\message{(Please type Yes or No.)}
\repeat
\endtt
\ddangerexercise Use the |\loop...\repeat| mechanism to construct a
general |\punishment| macro that repeats any given paragraph any given number of
times. For example,
\begintt
\punishment{I must not talk in class.}{100}
\endtt
should produce the results desired in exercise 20.\punishexno.
\answer (Here's a solution that also numbers the lines, so that the number of
repetitions is easily verifiable.
The only tricky part about this answer is the use of\/ ^|\endgraf|,
which is a substitute for |\par| because |\loop| is not a ^|\long| macro.)
\begintt
\newcount\n
\def\punishment#1#2{\n=0
\loop\ifnum\n<#2 \advance\n by1
\item{\number\n.}#1\endgraf\repeat}
\endtt
\let\plaind=\d
\newif\ifprime \newif\ifunknown
\newcount\n \newcount\p \newcount\d \newcount\a
\def\primes#1{2,~3% assume that #1 is at least 3
\n=#1 \advance\n by-2 % n more to go
\p=5 % odd primes starting with p
\loop\ifnum\n>0 \printifprime\advance\p by2 \repeat}
\def\printp{, % we will invoke \printp if p is prime
\ifnum\n=1 and~\fi % this precedes the last value
\number\p \advance\n by -1 }
\def\printifprime{\testprimality \ifprime\printp\fi}
\def\testprimality{{\d=3 \global\primetrue
\loop\trialdivision \ifunknown\advance\d by2 \repeat}}
\def\trialdivision{\a=\p \divide\a by\d
\ifnum\a>\d \unknowntrue\else\unknownfalse\fi
\multiply\a by\d
\ifnum\a=\p \global\primefalse\unknownfalse\fi}
\ddanger The first thirty prime numbers are \primes{30}. You may not
find this fact very startling; but you may be surprised to learn that
the previous sentence was typeset by saying
\begintt
The first thirty prime numbers are \primes{30}.
\endtt
\TeX\ did all of the calculation by expanding the |\primes| macro, so the author
is pretty sure that the list of ^{prime numbers} given above is quite free
of typographic errors. Here is the set of macros that did it:
\begindisplay
^|\newif||\ifprime \newif\ifunknown % boolean variables|\cr
^|\newcount||\n \newcount\p \newcount\d \newcount\a % integer variables|\cr
|\def\primes#1{2,~3% assume that #1 is at least 3|\cr
| \n=#1 \advance\n by-2 % n more to go|\cr
| \p=5 % odd primes starting with p|\cr
| \loop\ifnum\n>0 \printifprime\advance\p by2 \repeat}|\cr
|\def\printp{, % we will invoke \printp if p is prime|\cr
| \ifnum\n=1 and~\fi % `and' precedes the last value|\cr
| \number\p \advance\n by -1 }|\cr
|\def\printifprime{\testprimality \ifprime\printp\fi}|\cr
|\def\testprimality{{\d=3 \global\primetrue|\cr
| \loop\trialdivision \ifunknown\advance\d by2 \repeat}}|\cr
\noalign{\penalty-500}
|\def\trialdivision{\a=\p \divide\a by\d|\cr
| \ifnum\a>\d \unknowntrue\else\unknownfalse\fi|\cr
| \multiply\a by\d|\cr
| \ifnum\a=\p \global\primefalse\unknownfalse\fi}|\cr
\enddisplay
^^|\multiply|^^|\divide|^^|\advance|^^|\newcount|
The computation is fairly straightforward, except that it involves
a loop inside a loop; therefore |\testprimality| introduces an extra
set of braces, to keep the inner loop control from interfering with
the outer loop. The braces make it necessary to say `^|\global|' when
|\ifprime| is being set true or false. \TeX\ spent more time constructing
that sentence than it usually spends on an entire page; the
|\trialdivision| macro was expanded 132 times. % cpu time was 4 sec
\let\d=\plaind
\ddanger The |\loop| macro that does all these wonderful things is
actually quite simple. It puts the code that's supposed to be repeated
into a control sequence called |\body|, and then another control
sequence iterates until the condition is false:
\begintt
\def\loop#1\repeat{\def\body{#1}\iterate}
\def\iterate{\body\let\next=\iterate\else\let\next=\relax\fi\next}
\endtt
The expansion of\/ |\iterate| ends with the expansion of\/ |\next|; therefore
\TeX\ is able to remove |\iterate| from its memory before
invoking |\next|, and the memory does not fill up during a long loop.
Computer scientists call this ``^{tail recursion}.''
\ddanger The |\hex| macro below, which converts count register |\n| to
^{hexadecimal notation}, illustrates a {\sl recursive\/} control structure
in which many copies of\/ |\hex| can be active simultaneously.
^{Recursion} works better than simple |\loop| ^{iteration} in this
application because the hexadecimal digits are discovered from right to
left, while they must be output from left to right.
\ (The number in |\n| should be $\ge0$.)
\begintt
\def\hex{{\count0=\n \divide\n by16
\ifnum\n>0 \hex\fi \count2=\n \multiply\count2 by-16
\advance\count0 by\count2 \hexdigit}}
\def\hexdigit{\ifnum\count0<10 \number\count0
\else\advance\count0 by-10 \advance\count0 by`A \char\count0 \fi}
\endtt
\ddanger Our final example is a macro that computes the number of nonblank
tokens in its argument; for example, `|\length{argument}|' expands to
`|8|'. This illustrates yet another aspect of macro technique.
\begintt
\def\length#1{{\count0=0 \getlength#1\end \number\count0}}
\def\getlength#1{\ifx#1\end \let\next=\relax
\else\advance\count0 by1 \let\next=\getlength\fi \next}
\endtt
\endchapter
By this time [37 A.D.] the influence of ^{Macro} had become supreme.
\author ^{TACITUS}, {\sl Annals\/} (c.\thinspace120 A.D.) % book VI, ch 45
\bigskip
% Oh, you want a definition.
I hate definitions.
\author BENJAMIN ^{DISRAELI}, {\sl Vivian Grey\/} (1826) % Book II, Chapter 6
\eject
\beginchapter Chapter 21. Making Boxes
In Chapters 11 and 12 we discussed the principles of boxes and glue, and by now
we have seen many applications of those concepts. You can get by in most cases
with the boxes that \TeX\ manufactures automatically with its paragraph
builder, its page builder, and its math formula processor; but if you want to
do nonstandard things, you have the option of making boxes by yourself.
For example, Chapter~14 points out that you can keep something from being
hyphenated or split between lines if you enclose it in an ^|\hbox|;
Chapter~19 points out that |\hbox| allows you to get ordinary text into
a displayed equation. ^^|\vbox|
\danger The purpose of the present chapter is to nail down whatever details
about boxes haven't been covered yet. Fortunately, there isn't much more
to discuss; we have already mentioned most of the rules, so this chapter is
fairly short. In fact, the previous chapters have dealt with almost
everything except the rules about rules.
\danger To make a ^{rule box} (i.e., a solid ^{black rectangle}), you
type `^|\hrule|' in vertical mode or `^|\vrule|' in horizontal mode,
followed by any or all of the specifications `^|width|\<dimen>',
`^|height|\<dimen>', `^|depth|\<dimen>', in any order. For example, if
\begintt
\vrule height4pt width3pt depth2pt
\endtt
appears in the middle of a paragraph, \TeX\ will typeset the black box
`\thinspace\vrule height4pt width3pt depth2pt\thinspace'. If you
specify a dimension twice, the second specification overrules the first.
If you leave a dimension unspecified, you get the following by default:
$$\halign{\indent#\hfil&\qquad\hfil#\hfil&\qquad\hfil#\hfil\cr
&|\hrule|&|\vrule|\cr
width&\tt*&$0.4\pt$\cr
height&$0.4\pt$&\tt*\cr
depth&$0.0\pt$&\tt*\cr}$$
Here `{\tt*}' means that the actual dimension depends on the context;
the rule will extend to the boundary of the smallest box or alignment
that encloses it.
\hrule
\danger For example, the author typed `|\hrule|' just before typing this
paragraph, and you can see what happened: A horizontal rule, $0.4\pt$
thick, was extended across the page, because the vertical box that
encloses it turned out to be just that wide. \ (In fact, the
vertical box that encloses it is the page itself.) \ Another example
appears immediately after this paragraph, where you can see the result
of
\begintt
\hrule width5cm height1pt \vskip1pt \hrule width6cm
\endtt
\TeX\ does not put ^{interline glue} between rule boxes and their neighbors
in a vertical list, so these two rules are exactly $1\pt$ apart.
\hrule width5cm height1pt \vskip1pt \hrule width6cm
\dangerexercise B. L. ^{User} didn't want one of his horizontal rules to
touch the left margin, so he put it in a box and moved it right, like this:
\begintt
\moveright 1in \vbox{\hrule width3in}
\endtt
^^|\moveright|
But he found that this produced more space above and below the rule
than when he had simply said `|\hrule width 4in|' with no |\vbox|.
Why did \TeX\ insert more space, and what should he have done to
avoid it?
\answer The interline skip is added for vboxes, but not for rules; he
forgot to say ^|\nointerlineskip|, before and after the |\moveright|
construction.
\danger If you specify all three dimensions of a rule, there's no essential
difference between |\hrule| and |\vrule|, since both will produce exactly
the same black box. But you must call it an |\hrule| if you want to put
it in a vertical list, and you must call it a |\vrule| if you want to put
it in a horizontal list, regardless of whether it actually looks like
a horizontal rule or a vertical rule or neither. If you say |\vrule| in
vertical mode, \TeX\ starts a new paragraph; if you say |\hrule| in
horizontal mode, \TeX\ stops the current paragraph and returns to
vertical mode.
\danger The dimensions of a rule can be negative; for example, here's
a rule whose height is $3\pt$ and whose depth is $-2\pt$:
`\thinspace\vrule height3pt depth-2pt width1in\thinspace'.
However, a rule is invisible unless its height plus depth is positive
and its width is positive. A rule whose width is negative cannot be
seen, but it acts like a ^{backspace} when it appears in a horizontal list.
\dangerexercise Explain how the author probably obtained the rule
`\thinspace\vrule height3pt depth-2pt width1in\thinspace' in the
previous paragraph. [{\sl Hint:\/} It's one inch long.]
\answer |\vrule height3pt depth-2pt width1in|. Notice that it was necessary
to call it a |\vrule| since it appeared in horizontal mode.
\ddanger Now let's summarize all of the ways there are to specify boxes
explicitly to \TeX. \ (1)~A character by itself makes a character box, in
horizontal mode; this character is taken from the current font. \
(2)~The commands |\hrule| and |\vrule| make rule boxes, as just explained.
\ (3)~Otherwise you can make hboxes and vboxes, which fall under the
generic term ^\<box>. A \<box> has one of the following seven forms:
$$\halign{\indent#\hfil&\quad(see Chapter #)\hfil\cr
^|\hbox|\<box specification>|{|\<horizontal material>|}|&12\cr
^|\vbox|\<box specification>|{|\<vertical material>|}|&12\cr
^|\vtop|\<box specification>|{|\<vertical material>|}|&12\cr
^|\box|\<register number>&15\cr
^|\copy|\<register number>&15\cr
^|\vsplit|\<register number>|to|\<dimen>&15\cr
^|\lastbox|&21\cr}$$
Here a ^\<box specification> is either `^|to|\<dimen>' or
`^|spread|\<dimen>' or empty; this governs the setting of glue in
the horizontal or vertical lists inside the box, as explained in
Chapter~12. A \<register number> is between 0 and 255; after you say
|\box|, that register becomes void, but after |\copy| the register
is unchanged, as explained in Chapter~15. The |\vsplit| operation
is also explained in Chapter~15. In math modes an additional
type of box is available:
^|\vcenter|\<box specification>|{|\<vertical material>|}| (see Chapter~17).
\ddanger The bottom line of the table above refers to ^|\lastbox|,
a primitive operation that hasn't been mentioned before. If the last item
on the current horizontal list or vertical list is an hbox or vbox, it
is removed from the list and it becomes the |\lastbox|; otherwise
|\lastbox| is void. This operation is allowed in ^{internal vertical
mode}, horizontal mode, and restricted horizontal mode, but you cannot use
it to take a box from the current page in vertical mode. In math modes,
|\lastbox| is always void. At the beginning of a paragraph,
`|{\setbox0=\lastbox}|' removes the ^{indentation box}.
\ddanger The operation ^|\unskip| is something like |\lastbox|, except that
it applies to glue instead of to boxes. If the last thing on the current list
is a glue item (or ^{leaders}, as explained below), it is removed. You
can't remove glue from the current page by using |\unskip| in vertical
mode, but you can say `|\vskip-|^|\lastskip|', which has almost
the same effect.
\ddanger Chapters 24 to 26 present summaries of all \TeX's operations
in all modes, and when those summaries mention a `\<box>' they mean one
of the seven possibilities just listed. For example, you can say
`|\setbox|\<register number>|=|\<box>' in any mode, and you can say
`|\moveright|\<dimen>\<box>' in vertical modes. But you can't say
`|\setbox|\<register number>|=C|' or `|\moveright|\<dimen>|\hrule|';
if you try either of these, \TeX\ will complain that a \<box> was
supposed to be present. Characters and rules are so special, they
aren't regarded as \<box>es.
\ddangerexercise Define a control sequence |\boxit| so that
`|\boxit{|\<box>|}|' yields the given box surrounded by 3~points of space and
by ruled lines on all four sides.
$$\def\boxit#1{\vbox{\hrule\hbox{\vrule\kern3pt
\vbox{\kern3pt#1\kern3pt}\kern3pt\vrule}\hrule}}
\setbox4=\vbox{\hsize 23pc \noindent \strut For example, the sentence you
are now reading was typeset as part of the displayed formula
|$$\boxit{\boxit{\box4}}$$|, where box~4 was created by typing
`|\setbox4=\vbox{\hsize 23pc \noindent \strut For example,
the sentence you are now reading ... \strut}|'.\strut}
\boxit{\boxit{\box4}}$$
\answer |\def\boxit#1{\vbox{\hrule\hbox{\vrule\kern3pt|\parbreak
| \vbox{\kern3pt#1\kern3pt}\kern3pt\vrule}\hrule}}|\par
\smallskip\noindent
(The resulting box does not have the baseline of the original one;
you have to work a little bit harder to get that.)
\danger Let's look also at what can go inside a box. An hbox contains
a horizontal list; a vbox contains a vertical list. Both kinds of lists
are made up primarily of things like boxes, glue, kerns, and penalties,
as we have seen in Chapters 14 and~15. But you can also include some
special things that we haven't discussed yet, namely ``leaders''
and ``whatsits.'' Our goal in the rest of this chapter will be to
study how to make use of such exotic items.
\danger The dots you see before your eyes here
\xleaders\hbox to 6pt{$\hss\cdot\hss$}\hfil\
are called {\sl^{leaders}\/}\break
because they lead your eyes across the page; such things are often
used in indexes or tables of contents. The general idea is to repeat a
box as many times as necessary to fill up some given space. \TeX\ treats
leaders as a special case of glue; no, wait, it's the other way
around: \TeX\ treats glue as a special case of leaders. Ordinary glue
fills space with nothing, while leaders fill space with any
desired thing. In horizontal mode you can say
\begindisplay
^|\leaders|\<box or rule>|\hskip|\<glue>
\enddisplay
and the effect will be the same as if you had said just `|\hskip|\<glue>',
except that the space will be occupied by copies of the specified
\<box or rule>. The glue stretches or shrinks in the usual way. For example,
\begintt
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
\line{Alpha\leaderfill Omega}
\line{The Beginning\leaderfill The Ending}
\endtt
will produce the following two lines:
$$\vbox{
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
\line{Alpha\leaderfill Omega}
\line{The Beginning\leaderfill The Ending}
}$$
Here `|\hbox to 1em{\hss.\hss}|' specifies a box one em wide, with a
period in its center; the control sequence |\leaderfill| then causes this
box to be replicated when filling space in the |\line| box. \ (Plain \TeX's
^|\line| macro makes an hbox whose width is the |\hsize|.)
\danger Notice that the ^{dots} in the two example lines appear exactly
above each other. This is not a coincidence; it's a consequence of the
fact that the |\leaders| operation acts something like a window that lets
you see part of an infinite row of boxes. If the words `Alpha' and `Omega'
are replaced by longer words, the number of dots might be different
but the ones that you see will be in the same places as before. The infinitely
replicated boxes are lined up so that they touch each other, and so that,
if you could see them all, one of them would have the same reference point as
the smallest enclosing box. Thus, |\leaders| will put a box flush with the
left edge of an enclosing box, if you start the leaders there; but you
won't get a box flush right unless the width of the enclosing box is
exactly divisible by the width of the repeated box. If~the repeated box
has width~$w$, and if the space to be filled is at least~$2w$, then you
will always see at least one copy of the box; but if the space is less
than~$2w$ the box may not appear, because boxes in the infinite
row are typeset only when their entire width falls into the available space.
\ddanger When leaders are isolated from each other, you might not
want them to be aligned as just described, so \TeX\ also provides
for ^{nonaligned leaders}. In this case a box of width~$w$ will be
copied $q$~times when the space to be filled is at least~$qw$ and
less than $(q+1)w$; furthermore, the results will be centered in the
available space. There are two kinds of nonaligned leaders in \TeX,
namely ^|\cleaders| (centered leaders) and ^|\xleaders| (expanded
leaders). Centered leaders pack the boxes tightly next to each other
leaving equal amounts of blank space at the left and right; expanded
leaders distribute the extra space equally between the $q+1$ positions
adjacent to the $q$~boxes. For example, let's suppose that a $10\pt$-wide
box is being used in leaders that are supposed to fill $56\pt$ of space.
Five copies of the box will be used; |\cleaders| will produce $3\pt$ of
space, then the five boxes, then another $3\pt$ of space. But
|\xleaders| will produce $1\pt$ of space, then the first box, then
another $1\pt$ of space, then the second box, \dots, then the fifth box,
and $1\pt$ of space.
\ddangerexercise Suppose that a $10\pt$-wide box is to fill $38\pt$ of
space starting $91\pt$ from the left of its enclosing box. How many
copies of the box will be produced by |\leaders|, |\cleaders|, and
|\xleaders|? Where will the boxes be positioned, relative to the
left edge of the enclosing box, in each of the three cases?
\answer |\leaders|: two boxes starting at $100\pt$, $110\pt$.\par
|\cleaders|: three boxes starting at $95\pt$, $105\pt$, $115\pt$.\par
|\xleaders|: three boxes starting at $93\pt$, $105\pt$, $117\pt$.
\ddangerexercise Assuming that the `|.|'\ in the |\leaderfill| macro on the
previous page is only $0.2\em$ wide, there is $0.4\em$ of blank space
at both sides of the one-em box. Therefore the |\leaders| construction
will leave between $0.4\em$ and $1.4\em$ of blank space between the
periods and the text at either end. Redefine |\leaderfill| so that
the amount of blank space at either end will be between $0.1\em$ and
$1.1\em$, but the leaders on adjacent lines will still be aligned
with each other.
\answer |\def\leaderfill{\kern-0.3em\leaders\hbox to 1em{\hss.\hss}%|\parbreak
| \hskip0.6em plus1fill \kern-0.3em }|
\danger Instead of giving a \<box> in the leaders construction, you can
give a \<rule>, which means either |\hrule| or |\vrule|, followed
by optional ^|height|, ^|width|, and ^|depth| specifications as
usual. The rule will then be made as wide as the corresponding~\<glue>.
This is a case where |\hrule| makes sense in horizontal mode, because
it gives a ^{horizontal rule in text}. For example, if the
|\leaderfill| macro in our earlier illustration is changed to
\begintt
\def\leaderfill{ \leaders\hrule\hfill\ }
\endtt
then the results look like this:
$$\vbox{
\def\leaderfill{ \leaders\hrule\hfill\ }
\line{Alpha\leaderfill Omega}
\line{The Beginning\leaderfill The Ending}
}$$
When a rule is used instead of a box, it fills the space completely, so
there's no difference between |\leaders|, |\cleaders|, and |\xleaders|.
\ddangerexercise What does |\leaders\vrule\hfill| produce?
\answer Since no |height| or |depth| specification follows the |\vrule|,
the height and depth are `|*|'; i.e., the rule extends to the smallest
enclosing box. This usually makes a heavy black band, which is too
horrible to demonstrate here. However, it does work if you know that the
enclosing box is sufficiently small; and |\leaders\vrule\vfill| works fine in
vertical mode.
\ddanger Leaders work in vertical mode as well as in horizontal mode. In this
case vertical glue (e.g., |\vskip|\<glue> or |\vfill|) is used instead of
horizontal glue, and |\leaders| produces boxes that are aligned so that the
top of each repeated box has the same vertical position as the top
of the smallest enclosing box, plus a multiple of the height-plus-depth
of the repeated box. No interlineskip glue is placed between boxes in
vertical leaders; the boxes are just stacked right on top of each other.
\ddanger If you specify horizontal leaders with a box whose width isn't
positive, or if you specify vertical leaders with a box whose
height-plus-depth isn't positive, \TeX\ silently ignores the leaders
and produces ordinary glue instead.
\ddangerexercise Explain how you can end a paragraph with a rule that
is at least $10\pt$ long and extends all the way to the right
margin, like this:
\null\nobreak\leaders\hrule\hskip10pt plus1filll\ \par
\answer For example, say
\begintt
\null\nobreak\leaders\hrule\hskip10pt plus1filll\ \par
\endtt
The `|\|\]' provides extra glue that is wiped out by the implied |\unskip|
at the end of every paragraph (see Chapter~14), and the `|\null\nobreak|'
makes sure that the leaders do not disappear at a line break; `^|filll|'
overtakes the ^|\parfillskip| glue.
\ddanger Horizontal leaders differ slightly from horizontal glue, because
they have height and depth when \TeX\ calculates the size of the enclosing
box (even though the number of replications might be zero). Similarly,
vertical leaders have width.
\ddangerexercise Demonstrate how to produce the following `\TeX ture'
$$\hbox to 2.5in{\cleaders\vbox to .5in{\cleaders\hbox{\TeX}\vfil}\hfil}$$
by using vertical leaders inside of horizontal leaders. \ (The \TeX\ logo
has been put into a rectangular box, and copies of this box have been packed
together tightly.) ^^{TeX logo}
\answer |$$\hbox to 2.5in{\cleaders|\parbreak
| \vbox to .5in{\cleaders\hbox{\TeX}\vfil}\hfil}$$|
\ddangerexercise Use vertical leaders to solve exercise 20.\punishexno.
\answer We assume that a strut is $12\pt$ tall, and that 50 lines
fit on a page:
\begintt
\setbox0=\hbox{\strut I must not talk in class.}
\null\cleaders\copy0\vskip600pt\vfill\eject % 50 times on page 1;
\null\cleaders\box0\vskip600pt\bye % 50 more on page 2.
\endtt
The ^|\null| keeps glue (and leaders) from disappearing at the top of
the page.
\danger The ^|\overbrace| and ^|\underbrace| macros of plain \TeX\ are
constructed by combining characters with rules. Font ^|cmex10| contains
four symbols $\bracelu\;\braceld\;\bracerd\;\braceru$, each of which has
depth zero and height equal to the thickness of a rule that joins them
properly. Therefore it's easy to define ^|\upbracefill| and ^|\downbracefill|
macros so that you can obtain, e.g., ^^{braces}
\begindisplay
\vbox{\kern4pt
\hbox to 100pt{\downbracefill}\hbox to 50pt{\upbracefill}
}
\enddisplay
by saying `|\hbox to 100pt{\downbracefill}\hbox to 50pt{\upbracefill}|'
in vertical mode. Details of those macro definitions appear in Appendix~B.
\ddanger The definition of\/ ^|\overrightarrow| in Appendix B is more complex
than that of\/ |\overbrace|, because it involves a box instead of a rule. The
fonts of plain \TeX\ are designed so that symbols like $\leftarrow$ and
$\to$ can be extended with ^{minus signs}; similarly, $\Leftarrow$ and
^^{arrows}
$\Rightarrow$ can be extended with ^{equals signs}. However, you can't
simply put the characters next to each other, because that leaves
gaps (`${\leftarrow}{-}{-}$' and `${\Leftarrow}{=}{=}$'); it is necessary
to backspace a little between characters. An additional complication
arises because the extension line in a long arrow might need to be some
non-integer number of minus signs long. To solve this problem, the
^|\rightarrowfill| macro in Appendix~B uses |\cleaders| with a repeatable
box consisting of the middle 10 units of a minus sign, where one unit
is ${1\over18}\em$. The leaders are preceded and followed by $-$ and~$\to$;
there's enough backspacing to compensate for up to 5~units of extra space,
fore and aft, that |\cleaders| might leave blank. In this way a macro is
obtained such that
\begintt
\hbox to 100pt{\rightarrowfill}
\endtt
yields `\hbox to 100pt{\rightarrowfill}'.
\ddanger Now we know all about leaders. What about ^{whatsits}? Well,
whatsits have been provided as a general mechanism by which important
special printing applications can be handled as extensions to \TeX\null.
It's possible for system wizards to modify the \TeX\ program, without
changing too much of the code, so that new features can be accommodated
at high speed instead of encoding them in macros. The author hopes that
such extensions will not be made very often, because he doesn't want
incompatible pseudo-\TeX\ systems to proliferate; yet he realizes that
certain special books deserve a special treatment. Whatsits make it
possible to incorporate new things into boxes without bending the existing
conventions too much. But they make applications less portable from
one machine to another.
\ddanger Two kinds of whatsits are defined as part of all \TeX\
implementations. They aren't really ^{extensions to \TeX}, but
they are coded as if they were, so that they provide a model of how
other extensions could be made. The first of these is connected with
output to text files, and it involves the \TeX\ primitive commands
^|\openout|, ^|\closeout|, ^|\write|, and ^|\immediate|. The second is
connected with special instructions that can be transmitted to
printing devices, via \TeX's ^|\special| command.
\ddanger The ability to write text files that can later be input by other
programs (including \TeX) makes it possible to take care of tables of
contents, indexes, and many other things. You can say
`|\openout|\<number>|=|\<file name>' and `|\closeout|\<number>' by analogy
with the |\openin| and |\closein| commands of Chapter~20; the
\<number> must be between 0 and~15. The filename is usually extended with
`^|.tex|' if it has no extension. There is a |\write| command that writes
one line to a file, analogous to the |\read| command that reads one line;
you say
\begindisplay
|\write|\<number>|{|\<token list>|}|
\enddisplay
and the material goes out to the file that corresponds to the given stream
number. If the \<number> is negative or greater than~15, or if the
specified stream has no file open for output, the output goes to the
user's ^{log file}, and to the ^{terminal} unless the number is negative.
Plain \TeX\ has a ^|\newwrite| command that allocates output stream numbers
from 0 to~15. Output streams are completely independent of input streams.
\ddanger However, the output actually takes place in a delayed fashion;
the |\openout|, |\closeout|, and |\write| commands that you give are not
performed when \TeX\ sees them. Instead, \TeX\ puts these commands into
whatsit items, and places them into the current horizontal
or vertical or math list that is being built. No actual output will occur
until this whatsit is eventually shipped out to the |dvi| file, as part
of a larger box. The reason for this delay is that |\write| is often
used to make an index or table of contents, and the exact page on which
a particular item will appear is generally unknown when the |\write|
instruction occurs in mid-paragraph. \TeX\ is usually working ahead,
reading an entire paragraph before breaking it into lines, and accumulating
more than enough lines to fill a page before deciding what goes on
the page, as explained in Chapters 14 and~15. Therefore a deferred writing
mechanism is the only safe way to ensure the validity of page number
references.
\ddanger The \<token list> of a |\write| command is first stored in a
whatsit without performing any macro expansion; the macro expansion takes
place later, when \TeX\ is in the middle of a ^|\shipout| operation. For
example, suppose that some paragraph in your document contains the text
\begintt
... For \write\inx{example: \the\count0}example, suppose ...
\endtt
Then the horizontal list for the paragraph will have a whatsit just before
the word `|example|', and just after the interword space following~`|For|'.
This whatsit item contains the unexpanded token list
`|example: \the\count0|'. It sits dormant while the paragraph is being
broken into lines and put on the current page. Let's suppose that
this word `|example|' (or some hyphenated initial part of it, like
`|ex-|') is shipped out on page~256. Then \TeX\ will write the line
\begintt
example: 256
\endtt
on output stream |\inx|, because the `|\the\count0|' will be expanded
at that time. Of course, |\write| commands are usually generated by
macros; they are rarely typed explicitly in mid-paragraph.
\ddanger \TeX\ defers |\openout| and |\closeout| commands by putting them
into whatsits too; thus, the relative order of output commands will be
preserved, unless boxes are shipped out in some other order due to
insertions or such things.
\ddanger Sometimes you don't want \TeX\ to defer a |\write| or
|\openout| or |\closeout|. You could say, e.g., `|\shipout\hbox{\write...}|',
but that would put an unwanted empty page in your |dvi| file. So \TeX\ has
another feature that gets around this problem: If you type `^|\immediate|'
just before |\write| or |\openout| or |\closeout|, the operation will
be performed immediately, and no whatsit will be made. For example,
\begintt
\immediate\write16{Goodbye}
\endtt
prints `|Goodbye|' on your terminal. Without the |\immediate|, you wouldn't
see the `|Goodbye|' until the current list was output. \ (In fact, you
might never see it; or you may see it more than once, if the
current list goes into a box that was copied.) \ An `|\immediate\write16|'
differs from ^|\message| in that |\write| prints the text on a line by itself;
the results of several |\message| commands might appear on the same line,
separated by spaces.
\ddanger The \<token list> of a |\write| ought to be rather short, since
it makes one line of output. Some implementations of \TeX\ are unable to
write long lines; if you want to write a lot of stuff, just give
several |\write| commands. Alternatively, you can set \TeX's ^|\newlinechar|
parameter to the ASCII code number of some character that you want to
stand for ``begin a new line''; then \TeX\ will begin a new line whenever it
would ordinarily output that character to a file.
For example, one way to output two lines to the terminal
in a single |\write| command is to say
\begintt
\newlinechar=`\^^J
\immediate\write16{Two^^Jlines.}
\endtt
Each |\write| command produces output in the form that \TeX\ always uses
to display ^{token lists symbolically}: Characters represent
themselves (except that you get duplicated characters like |##| for macro
parameter characters^^{hash}); unexpandable control sequence tokens
produce their names, preceded by the ^|\escapechar| and followed by a
space (unless the name is an active character or a control sequence formed
from a single nonletter).
\ddanger \TeX\ ignores |\write|, |\openout|, and |\closeout| whatsits that
appear within boxes governed by ^{leaders}. If you are upset about this,
you shouldn't be.
\ddanger Since the \<token list> of a deferred |\write| is expanded at a
fairly random time (when |\shipout| occurs), you should be careful about
what control sequences it is allowed to contain. The techniques of Chapter~20
for controlling macro expansion often come in handy with respect to |\write|.
\ddangerexercise Suppose that you want to |\write| a token list that
involves a macro |\chapno|, containing the current chapter number, as well as
`|\the\count0|' which refers to the current page. You want |\chapno| to
be expanded immediately, because it might change before the token list
is written; but you want |\the\count0| to be expanded at the time of
|\shipout|. How can you manage this?
\answer |{\let|\stretch|\the=0\edef|\stretch|\next|\stretch
|{\write|\stretch|\cont|\stretch|{|\<token list>|}}\next|\stretch|}| will
expand everything but |\the| when the |\write| command is given.
\ddanger Now let's wrap up our study of boxes by considering one more feature.
The command `^|\special||{|\<token list>|}|' can be given in any
mode. Like |\write|, it puts its token list into a whatsit; and like
|\message|, it expands the token list immediately. This token list will be
output to the |dvi| file with the other typesetting commands that
\TeX\ produces. Therefore it is implicitly associated with a particular
position on the page, namely the reference point that would have been present
if a box of height, depth, and width zero had appeared in place of the
whatsit. The \<token list> in a |\special| command should consist of a
keyword followed if necessary by a space and appropriate arguments. For example,
\begintt
\special{halftone pic1}
\endtt
might mean that a ^{picture} on file |pic1| should be inserted ^^{halftones}
on the current page, with its reference point at the current position.
\TeX\ doesn't look at the token list to see if it makes any sense; the
list is simply copied to the output. However, you should be careful not to
make the list too long, or you might overflow \TeX's string memory.
The |\special| command enables you to make use of special equipment that
might be available to you, e.g., for printing books in glorious
\TeX ni^{color}.
\ddanger Software programs that convert |dvi| files to printed or
displayed output should be able to fail gracefully when they don't
recognize your special keywords. Thus, |\special| operations should
never do anything that changes the current position. Whenever you use
|\special|, you are taking a chance that your output file will not be
printable on all output devices, because all |\special| functions are
extensions to \TeX. However, the author anticipates that certain standards
for common graphic operations will emerge in the \TeX\ user community,
after careful experiments have been made by different groups of people;
then there will be a chance for some uniformity in the use of
|\special| extensions.
\ddanger \TeX\ will report the badness of glue setting in a box if
you ask for the numeric quantity
^|\badness| after making a box. For example, you might say
\begintt
\setbox0=\line{\trialtexta}
\ifnum\badness>250 \setbox0=\line{\trialtextb}\fi
\endtt
The badness is between 0 and 10000
unless the box is overfull, when |\badness=1000000|.
\endchapter
If age or weaknes doe prohibyte bloudletting,
you must use boxing.
\author PHILIP ^{BARROUGH}, {\sl The Methode of Phisicke\/} (1583) % page 6
\bigskip
The only thing that never looks right is a rule.
There is not in existence a page with a rule on it
that cannot be instantly and obviously improved
by taking the rule out.
% "Even dashes, cherished as they are by authors who cannot punctuate,
% spoil a page. They are generally merely ignorant substitutes for colons."
\author GEORGE BERNARD ^{SHAW}, in {\sl The Dolphin\/} (1940) % v4.1 p81
\eject
\beginchapter Chapter 22. Alignment
Printers charge extra when you ask them to typeset ^{tables}, and they do so
for good reason: Each table tends to have its own peculiarities, so it's
necessary to give some thought to each one, and to fiddle with alternative
approaches until finding something that looks good and communicates well.
However, you needn't be too frightened of doing tables with \TeX, since plain
\TeX\ has a ``tab'' feature that handles simple situations pretty much like
you would do them on a typewriter. Furthermore, \TeX\ has a powerful
alignment mechanism that makes it possible to cope with extremely complex
tabular arrangements. Simple cases of these ^{alignment} operations will
suffice for the vast majority of applications.
Let's consider ^{tabbing} first. If you say `^|\settabs| $n$ ^|\columns|',
plain \TeX\ makes it easy to produce lines that are divided into $n$~equal-size
^{columns}. Each line is specified by typing
\begindisplay
|\+|\<text$_1$>|&|\<text$_2$>|&|$\,\cdots\,$|\cr|
\enddisplay
where \<text$_1$> will start flush with the left margin,
\<text$_2$> will start at the left of the second column, and so on. Notice
that `^|\+|' starts the line. The final column is followed by `^|\cr|',
which old-timers will recognize as an abbreviation for the ``^{carriage
return}'' operation on typewriters that had carriages. For example,
consider the following specification:
\begintt
\settabs 4 \columns
\+&&Text that starts in the third column\cr
\+&Text that starts in the second column\cr
\+\it Text that starts in the first column, and&&&
the fourth, and&beyond!\cr
\endtt
After `|\settabs|\stretch|4\columns|' each |\+| line is divided into quarters,
so the result~is
\medskip
\settabs 4 \columns
\+&&Text that starts in the third column\cr
\+&Text that starts in the second column\cr
\+\it Text that starts in the first column, and&&&
the fourth, and&beyond!\cr
\def\tick{\kern-0.2pt % that's half the rule width
\vbox to 0pt{\kern-36pt\leaders\hbox{\vrule height1pt\vbox to4pt{}}\vfil}}
\vskip-\baselineskip
\+\tick&\tick&\tick&\tick&\tick\cr
\medskip
This example merits careful study because it illustrates several things.
(1)~The `|&|' ^^{ampersand} is like the {\sc TAB} key on many typewriters;
it tells \TeX\ to advance to the next tab position, where there's a tab at
the right edge of each column. In this example, \TeX\ has set up four tabs,
indicated by the dashed lines; a dashed line is also shown at the left
margin, although there isn't really a tab there. (2)~But `|&|' isn't
exactly like a mechanical typewriter {\sc TAB}, because it first backs up
to the beginning of the current column before advancing to the next. In
this way you can always tell what column you're tabbing to, by counting
the number of |&|'s; that's handy, because variable-width type otherwise
makes it difficult to know whether you've passed a tab position or not.
Thus, on the last line of our example, three |&|'s were typed in order to
get to column~4, even though the text had already extended into column~2
and perhaps into column~3. (3)~You can say `|\cr|' before you have
specified a complete set of columns, if the remaining columns are blank.
(4)~The |&|'s are different from tabs in another way, too: \TeX\ ignores
^{spaces} after~`|&|', hence you can conveniently finish a column by typing
`|&|'~at the end of a line in your input file, without worrying that an
extra blank space will be introduced there. \ (The second-last line of the
example ends with~`|&|', and there is an implicit blank space following that
symbol; if \TeX\ hadn't ignored that space, the words `the fourth'
wouldn't have started exactly at the beginning of the fourth column.) \
Incidentally, plain \TeX\ also ignores spaces after `|\+|', so that the
first column is treated like the others. (5) The `^|\it|' in the last line
of the example causes only the first column to be italicized, even though
no ^{braces} were used to confine the range of italics, because \TeX\
implicitly inserts braces around each individual entry of an alignment.
\danger Once you have issued a |\settabs| command, the tabs remain set until you
reset them, even though you go ahead and type ordinary paragraphs as usual.
But if you enclose |\settabs| in |{...}|, the tabs defined inside a group
don't affect the tabs outside; `^|\global||\settabs|' is not permitted.
\danger Tabbed lines usually are used between paragraphs, in the same
places where you would type ^|\line| or ^|\centerline| to get lines with
a special format. But it's also useful to put |\+|~lines inside a |\vbox|;
this makes it convenient to specify ^{displays} that contain aligned
material. For example, if you type
\begintt
$$\vbox{\settabs 3 \columns
\+This is&a strange&example\cr
\+of displayed&three-column&format.\cr}$$
\endtt
you get the following display:
$$\vbox{\settabs 3 \columns
\+This is&a strange&example\cr
\+of displayed&three-column&format.\cr}$$
In this case the first column doesn't appear flush left, because \TeX\
centers a box that is being displayed. Columns that end with |\cr| in
a |\+|~line are put into a box with their natural width; so the first
and second columns here are one-third of the |\hsize|, but the third column
is only as wide as the word `example'. We have used |$$| ^^{dollardollar}
in this construction even though no mathematics is involved, because |$$|
does other useful things; for example, it centers the box, and it inserts
space above and below.
People don't always want tabs to be equally spaced, so there's another
way to set them, by typing `|\+|\<sample line>|\cr|' immediately after
`|\settabs|'. In this case tabs are placed at the positions
of the |&|'s in the ^{sample line}, and the sample line itself does not appear
in the output. For example,
\begintt
\settabs\+\indent&Horizontal lists\quad&\cr % sample line
\+&Horizontal lists&Chapter 14\cr
\+&Vertical lists&Chapter 15\cr
\+&Math lists&Chapter 17\cr
\endtt
causes \TeX\ to typeset the following three lines of material:
\nobreak\medskip
\settabs\+\indent&Horizontal lists\quad&\cr
\+&Horizontal lists&Chapter 14\cr
\+&Vertical lists&Chapter 15\cr
\+&Math lists&Chapter 17\cr
\medbreak\noindent
The |\settabs| command in this example makes column~1 as wide as a paragraph
^^{indention, see indentation}
indentation; and column~2 is as wide as `Horizontal lists' plus one quad of
space. ^^|\quad| Only two tabs are set in this case, because only two |&|'s
appear in the sample line. \ (A sample line might as well end with~|&|,
because the text following the last tab isn't used for anything.)
The first line of a table can't always be used as a sample line, because it
won't necessarily give the correct tab positions. In a large table you have
to look ahead and figure out the biggest entry in each column; the sample
line is then constructed by typing the widest first column, the widest
second column, etc., omitting the last column. Be sure to include some
extra space between columns in the sample line, so that the columns
won't touch each other.
\def\frac#1/#2{\leavevmode\kern.1em
\raise.5ex\hbox{\the\scriptfont0 #1}\kern-.1em
/\kern-.15em\lower.25ex\hbox{\the\scriptfont0 #2}}
\exercise Explain how to typeset the following table [from Beck,
Bertholle, and Child, {\sl Mastering the Art of French Cooking\/}
(New York: Knopf, 1961)]: % p283
^^{Beck, Simone} ^^{Bertholle, Louisette} ^^{Child, Julia}
\nobreak\medskip
\settabs\+\indent&10\frac1/2 lbs.\qquad&\it Servings\qquad&\cr
\+&\negthinspace\it Weight&\it Servings&
{\it Approximate Cooking Time\/}*\cr
\smallskip
\+&8 lbs.&6&1 hour and 50 to 55 minutes\cr
\+&9 lbs.&7 to 8&About 2 hours\cr
\+&9\frac1/2 lbs.&8 to 9&2 hours and 10 to 15 minutes\cr
\+&10\frac1/2 lbs.&9 to 10&2 hours and 15 to 20 minutes\cr
\smallskip
\+&* For a stuffed goose,
add 20 to 40 minutes to the times given.\cr
\answer Notice the uses of `|\smallskip|' here to separate the table heading
and footing from the table itself; such refinements are often worthwhile.
\begintt
\settabs\+\indent&10\frac1/2 lbs.\qquad&\it Servings\qquad&\cr
\+&\negthinspace\it Weight&\it Servings&
{\it Approximate Cooking Time\/}*\cr
\smallskip
\+&8 lbs.&6&1 hour and 50 to 55 minutes\cr
\+&9 lbs.&7 to 8&About 2 hours\cr
\+&9\frac1/2 lbs.&8 to 9&2 hours and 10 to 15 minutes\cr
\+&10\frac1/2 lbs.&9 to 10&2 hours and 15 to 20 minutes\cr
\smallskip
\+&* For a stuffed goose,
add 20 to 40 minutes to the times given.\cr
\endtt
\def\frac#1/#2{\leavevmode\kern.1em
\raise.5ex\hbox{\the\scriptfont0 #1}\kern-.1em
/\kern-.15em\lower.25ex\hbox{\the\scriptfont0 #2}}%
The title line specifies `|\it|' three times, because each entry between
tabs is treated as a group by \TeX; you would get error messages galore
if you tried to say something like \hbox{`|\+&{\it Weight&Servings&...}\cr|'}.
The `^|\negthinspace|' in the title line is a small backspace that
compensates for the slant in an italic {\it W\/}; the author inserted
this somewhat unusual correction after seeing how the table looked
without it, on the first proofs. \ (You weren't supposed to think of this,
but it has to be mentioned.) \ See exercise 11.\fracexno\ for the `|\frac|'
macro; it's better to say `\frac1/2' than `$1\over2$', in a cookbook.\par
Another way to treat this table would be to display it in a vbox, instead
of including a first column whose sole purpose is to specify indentation.
\ninepoint % it's all dangerous from here to end of chapter
\danger If you want to put something ^{flush right} in its column, just type
`^|\hfill|' before it; and be sure to type `|&|' after it, so that
\TeX\ will be sure to move the information all the way until it touches
the next tab. Similarly, if you want to ^{center} something in its
column, type `|\hfill|' before it and `|\hfill&|' after it. For example,
\begintt
\settabs 2 \columns
\+\hfill This material is set flush right&
\hfill This material is centered\hfill&\cr
\+\hfill in the first half of the line.&
\hfill in the second half of the line.\hfill&\cr
\endtt
produces the following little table:\enddanger
\nobreak\medskip
\settabs 2 \columns
\+\hfill This material is set flush right&
\hfill This material is centered\hfill&\cr
\+\hfill in the first half of the line.&
\hfill in the second half of the line.\hfill&\cr
\danger The |\+| macro in Appendix~B works
by putting the \<text> for each column that's followed by~|&|
into an hbox as follows:
\begindisplay
|\hbox to |\<column width>|{|\<text>|\hss}|
\enddisplay
The ^|\hss| means that the text is normally flush left, and that it can
extend to the right of its box. Since |\hfill| is ``more infinite'' than
|\hss| in its ability to stretch, it has the effect of right-justifying or
centering as stated above. Note that |\hfill| doesn't shrink, but |\hss|
does; if the text doesn't fit in its column, it will stick out at the right.
You could avoid this by adding |\hskip| |0pt| |minus-1fil|; then
an oversize text would produce an overfull box.
You could also center some text by putting `|\hss|' before it and just
`|&|' after it; in that case the text would be allowed to extend to the
left and right of its column.
The last column of a |\+|~line (i.e., the column entry that is
followed by |\cr|) is treated differently: The
\<text> is simply put into an hbox with its natural~width.\looseness=-1
\danger ^{Computer programs} present difficulties of a different kind, since
some people like to adopt a style in which the tab positions change from
line to line. For example, consider the following program fragment:
$$\vbox{\+\bf if $n<r$ \cleartabs&\bf then $n:=n+1$\cr
\+&\bf else &{\bf begin} ${\it print\_totals}$; $n:=0$;\cr
\+&&{\bf end};\cr
\+\bf while $p>0$ do\cr
\+\quad\cleartabs&{\bf begin} $q:={\it link}(p)$;
${\it free\_node}(p)$; $p:=q$;\cr
\+&{\bf end};\cr}$$
Special tabs have been set up so that `{\bf then}' and `{\bf else}' appear
one above the other, and so do `{\bf begin}' and `{\bf end}'. It's possible
to achieve this by setting up a new sample line whenever a new tab position
is needed; but that's a tedious job, so plain \TeX\ makes it a little simpler.
Whenever you type |&| to the right of all existing tabs, the effect is to
set a new tab there, in such a way that the column just completed will have
its natural width. Furthermore, there's an operation `^|\cleartabs|' that
resets all tab positions to the right of the current column. Therefore the
computer program above can be \TeX ified as follows:
\begindisplay
|$$\vbox{\+\bf if $n<r$ \cleartabs&\bf then $n:=n+1$\cr|\cr
| \+&\bf else &{\bf begin} ${\it print\_totals}$; $n:=0$;\cr|\cr
| \+&&{\bf end};\cr|\cr
| |\<The remaining part is left as an exercise>|}$$|\cr
\enddisplay
\dangerexercise Complete the example computer program by specifying three more
|\+|~lines.
\answer In such programs it seems best to type |\cleartabs| just before |&|,
whenever it is desirable to reset the old tabs. Multiletter identifiers look
best when set in ^{text italics} with ^|\it|, as explained in Chapter~18.
Thus, the following is recommended:
\begintt
\+\bf while $p>0$ do\cr
\+\quad\cleartabs&{\bf begin} $q:={\it link}(p)$;
${\it free\_node}(p)$; $p:=q$;\cr
\+&{\bf end};\cr
\endtt
\danger Although |\+| lines can be used in vertical boxes, you must never
use |\+| inside of another |\+| line. The |\+| macro is intended for
simple applications only.
\ddanger The |\+| and |\settabs| macros of Appendix B keep track of tabs by
maintaining register |\box|^|\tabs| as a box full of empty boxes whose
widths are the column widths in reverse order. Thus you can examine the
tabs that are currently set, by saying `^|\showbox||\tabs|'; this puts
the column widths into your log file, from right to left. For example,
after `|\settabs\+\hskip100pt&\hskip200pt&\cr\showbox\tabs|', \TeX\
will show the lines
\begintt
\hbox(0.0+0.0)x300.0
.\hbox(0.0+0.0)x200.0
.\hbox(0.0+0.0)x100.0
\endtt
\ddangerexercise Study the |\+| macro in Appendix B and figure out how to
change it so that tabs work as they do on a mechanical typewriter
(i.e., so that `|&|' always moves to the next tab that lies strictly
to the right of the current position). Assume that the user
doesn't backspace past previous tab positions; for example, if the input is
\hbox{`|\+&&\hskip-2em&x\cr|'}, do not bother to put `x' in the first or
second column, just put it at the beginning of the third column. \
(This exercise is a bit difficult.)
\answer Here we retain the idea that |&| inserts a new tab, when there
are no tabs to the right of the current position. Only one of the macros
that are used to process |\+|~lines needs to be changed; but
(unfortunately) it's the most complex one:
\begintt
\def\t@bb@x{\if@cr\egroup % now \box0 holds the column
\else\hss\egroup \dimen@=0\p@
\dimen@ii=\wd0 \advance\dimen@ii by1sp
\loop\ifdim \dimen@<\dimen@ii
\global\setbox\tabsyet=\hbox{\unhbox\tabsyet
\global\setbox1=\lastbox}%
\ifvoid1 \advance\dimen@ii by-\dimen@
\advance\dimen@ii by-1sp \global\setbox1
=\hbox to\dimen@ii{}\dimen@ii=-1pt\fi
\advance\dimen@ by\wd1 \global\setbox\tabsdone
=\hbox{\box1\unhbox\tabsdone}\repeat
\setbox0=\hbox to\dimen@{\unhbox0}\fi
\box0}
\endtt
\danger \TeX\ has another important way to make tables, using an operation
called ^|\halign| (``horizontal alignment''). In this case the table format
is based on the notion of a {\sl^{template}}, not on tabbing; the idea
is to specify a separate environment for the text in each column.
Individual entries are inserted into their templates, and presto, the
table is complete.
\danger For example, let's go back to the Horizontal/Vertical/Math list
example that appeared earlier in this chapter; we can specify it with
|\halign| instead of with tabs. The new specification is
\begintt
\halign{\indent#\hfil&\quad#\hfil\cr
Horizontal lists&Chapter 14\cr
Vertical lists&Chapter 15\cr
Math lists&Chapter 17\cr}
\endtt
and it produces exactly the same result as the old one. This example
deserves careful study, because |\halign| is really quite simple once
you get the hang of it. The first line contains the {\sl ^{preamble}\/} to
the alignment, which is something like the sample line used to set tabs
for~|\+|. In this case the preamble contains two templates, namely
`|\indent#\hfil|' for the first column and `|\quad#\hfil|' for the
second. Each template contains exactly one appearance of `|#|', ^^{sharp}
and it means ``stick the text of each column entry in this place.''
Thus, the first column of the line that follows the preamble becomes
\begintt
\indent Horizontal lists\hfil
\endtt
when `|Horizontal lists|' is stuffed into its template; and the second
column, similarly, becomes `|\quad Chapter 14\hfil|'. The question is,
why |\hfil|? Ah, now we get to the interesting point of the whole thing:
\TeX\ reads an entire |\halign{...}| specification into its memory
before typesetting anything, and it keeps track of the maximum width
of each column, assuming that each column is set without stretching or
shrinking the glue. Then it goes back and puts every entry into a box,
setting the glue so that each box has the maximum column width. That's
where the |\hfil| comes in; it stretches to fill up the extra space in
narrower entries.
\dangerexercise What table would have resulted if the template for the
first column in this example had been `|\indent\hfil#|' instead of
`|\indent#\hfil|'?
\answer \par\nobreak\vskip-\baselineskip
\halign{\indent\hfil#&\quad#\hfil\cr
Horizontal lists&Chapter 14\cr\noalign{\nobreak}
Vertical lists&Chapter 15\cr\noalign{\nobreak}
Math lists&Chapter 17\qquad (i.e., the first column would be
right-justified)\cr}
\danger Before reading further, please make sure that you understand the
idea of templates in the example just presented. There are several
important differences between |\halign| and~|\+|: (1)~|\halign| calculates
^^{halign compared to tabbing}
the maximum column widths automatically; you don't have to guess what the
longest entries will be, as you do when you set tabs with a sample line.
(2)~Each |\halign| does its own calculation of column widths; you have to
do something special if you want two different |\halign| operations to
produce identical alignments. By contrast, the |\+| operation remembers tab
positions until they are specifically reset; any number of paragraphs and
even |\halign| operations can intervene between |\+|'s, without affecting
the tabs. (3)~Because |\halign| reads an entire table in order to
determine the maximum column widths, it is unsuitable for huge tables
that fill several pages of a book. By contrast, the~|\+|~operation deals
with one line at a time, so it places no special demands on \TeX's memory.
\ (However, if you have a huge table, you should probably define your own
special-purpose macro for each line instead of relying on the general
|\+|~operation.) (4)~|\halign| takes less computer time than |\+|~does,
because |\halign| is a built-in command of \TeX, while |\+|~is a macro
that has been coded in terms of\/ |\halign| and various other primitive
operations. (5)~Templates are much more versatile than tabs, and they can
save you a lot of typing. For example, the Horizontal/Vertical/Math list
table could be specified more briefly by noticing that there's common
information in the columns:
\begintt
\halign{\indent# lists\hfil&\quad Chapter #\cr
Horizontal&14\cr Vertical&15\cr Math&17\cr}
\endtt
You could even save two more keystrokes by noting that the chapter numbers
all start with `|1|'\thinspace! \ (Caution: It takes more time to think of
optimizations like this than to type things in a straightforward way;
do it only if you're bored and need something amusing to keep up
your interest.)\ (6)~On the other hand, templates are no substitute for
tabs when the tab positions are continually varying, as in the
computer program example.
\danger Let's do a more interesting table, to get more
experience with |\halign|. Here is another example based on the
^{Beck}/^{Bertholle}/^{Child} book cited earlier:
$$\vbox{\openup2pt
\halign{\hfil\bf#&\quad\hfil\it#\hfil&\quad\hfil#\hfil&
\quad\hfil#\hfil&\quad#\hfil\cr
\sl American&\sl French&\sl Age&\sl Weight&\sl Cooking\cr
\noalign{\vskip-2pt}
\sl Chicken&\sl Connection&\sl(months)&\sl(lbs.)&\sl Methods\cr
\noalign{\smallskip}
Squab&Poussin&2&\frac3/4 to 1&Broil, Grill, Roast\cr
Broiler&Poulet Nouveau&2 to 3&1\frac1/2 to 2\frac1/2&Broil, Grill, Roast\cr
Fryer&Poulet Reine&3 to 5&2 to 3&Fry, Saut\'e, Roast\cr
Roaster&Poularde&5\frac1/2 to 9&Over 3&Roast, Poach, Fricassee\cr
Fowl&Poule de l'Ann\'ee&10 to 12&Over 3&Stew, Fricassee\cr
Rooster&Coq&Over 12&Over 3&Soup stock, Forcemeat\cr}}$$
Note that, except for the title lines, the first column is set right-justified
in boldface type; the middle columns are centered; the second column
is centered and in italics; the final column is left-justified. We would
like to be able to type the rows of the table as simply as possible; hence,
for example, it would be nice to be able to specify the bottom row by
typing only
\begintt
Rooster&Coq&Over 12&Over 3&Soup stock, Forcemeat\cr
\endtt
without worrying about type styles, centering, and so on. This not only
cuts down on keystrokes, it also reduces the chances for making typographical
errors. Therefore the template for the first column should be
`|\hfil\bf#|'; for the second column it should be `|\hfil\it#\hfil|' to
get the text centered and italicized; and so on. We also need to allow
for space between the columns, say one quad. {\it Voil\`a! La typographie
est sur la table:\/}\looseness=-1
\begindisplay
|\halign{\hfil\bf#&\quad\hfil\it#\hfil&\quad\hfil#\hfil&|\cr
| \quad\hfil#\hfil&\quad#\hfil\cr|\cr
\ \<the title lines>\cr
| Squab&Poussin&2&\frac3/4 to 1&Broil, Grill, Roast\cr|\cr
| ... Forcemeat\cr}|\cr
\enddisplay
As with the |\+| operation, spaces are ignored after |&|, in the preamble
as well as in the individual rows of the table. Thus, it is convenient
to end a long row with `|&|' when the~row takes up more than one line
in your input file.
\dangerexercise How was the `{\bf Fowl}' line typed? \ (This is too easy.)
\answer |Fowl&Poule de l'Ann\'ee&10 to 12&Over 3&Stew, Fricassee\cr|
\danger The only remaining problem in this example is to specify the title
lines, which have a different format from the others. In this case the style
is different only because the typeface is slanted, so there's no special
difficulty; we just type
\begintt
\sl American&\sl French&\sl Age&\sl Weight&\sl Cooking\cr
\sl Chicken&\sl Connection&\sl(months)&\sl(lbs.)&\sl Methods\cr
\endtt
It is necessary to say `|\sl|' each time, because each individual entry
of a table is implicitly enclosed in braces.
\danger The author used `^|\openup||2pt|' to increase the distance between
baselines in the ^{poultry} table; a discriminating reader will notice
that there's also a bit of extra space between the title line and the
other lines. This extra space was inserted by typing
`^|\noalign||{\smallskip}|' just after the title line. In general, you can say
\begindisplay
|\noalign{|\<vertical mode material>|}|
\enddisplay
just after any |\cr| in an |\halign|; \TeX\ will simply copy the vertical
mode material, without subjecting it to alignment, and it will appear
in place when the |\halign| is finished. You can use |\noalign| to
insert extra space, as here, or to insert penalties that affect page
breaking, or even to insert lines of text (see Chapter~19). Definitions
inside the braces of\/ |\noalign{...}| are local to that group.
\danger The |\halign| command also makes it possible for you to adjust
the spacing between columns so that a table will fill a specified area.
You don't have to decide that the ^{inter-column space} is a quad; you can
let \TeX\ make the decisions, based on how wide the columns come out,
because \TeX\ puts ``^{tabskip glue}'' between columns. This tabskip glue
is usually zero, but you can set it to any value you like by saying
`^|\tabskip||=|\<glue>'. For example,
let's do the poultry table again, but with the beginning of the
specification changed as follows:
\begintt
\tabskip=1em plus2em minus.5em
\halign to\hsize{\hfil\bf#&\hfil\it#\hfil&\hfil#\hfil&
\hfil#\hfil&#\hfil\cr
\endtt
The main body of the table is unchanged, but the |\quad| spaces have been
removed from the preamble, and a nonzero |\tabskip| has been specified
instead. Furthermore `|\halign|' has been changed to `|\halign
to\hsize|'; this means that each line of the table will be put into a
box whose width is the current value of\/ ^|\hsize|, i.e., the horizontal
line width usually used in paragraphs. The resulting table looks like this:
$$\vbox{\openup2pt
\tabskip=1em plus2em minus.5em
\halign to\hsize{\hfil\bf#&\hfil\it#\hfil&\hfil#\hfil&
\hfil#\hfil&#\hfil\cr
\sl American&\sl French&\sl Age&\sl Weight&\sl Cooking\cr
\noalign{\vskip-2pt}
\sl Chicken&\sl Connection&\sl(months)&\sl(lbs.)&\sl Methods\cr
\noalign{\smallskip}
Squab&Poussin&2&\frac3/4 to 1&Broil, Grill, Roast\cr
Broiler&Poulet Nouveau&2 to 3&1\frac1/2 to 2\frac1/2&Broil, Grill, Roast\cr
Fryer&Poulet Reine&3 to 5&2 to 3&Fry, Saut\'e, Roast\cr
Roaster&Poularde&5\frac1/2 to 9&Over 3&Roast, Poach, Fricassee\cr
Fowl&Poule de l'Ann\'ee&10 to 12&Over 3&Stew, Fricassee\cr
Rooster&Coq&Over 12&Over 3&Soup stock, Forcemeat\cr}}$$
\danger In general, \TeX\ puts tabskip glue before the first column, after
the last column, and between the columns of an alignment. You can specify
the final aligned size by saying `|\halign to|\<dimen>' or
`|\halign spread|\<dimen>', ^^|to| ^^|spread|
just as you can say `|\hbox to|\<dimen>' and `|\hbox spread|\<dimen>'.
This specification governs the setting of the tabskip glue; but it does
not affect the setting of the glue within column entries. \ (Those
entries have already been packaged into boxes having the maximum
natural width for their columns, as described earlier.)
\ddanger Therefore `|\halign| |to| |\hsize|' will do nothing if the
tabskip glue has no stretchability or shrinkability, except that it will
cause \TeX\ to report an ^{underfull} or ^{overfull} box. An overfull box
occurs if the tabskip glue can't shrink to meet the
given specification; in this case you get a warning on the terminal
and in your log file, but there is no ``^{overfull rule}'' to mark the
oversize table on the printed output. The warning message shows a
``^{prototype row}'' (see Chapter~27).
\danger The poultry example just given used the same tabskip glue
everywhere, but you can vary it by resetting ^|\tabskip| within the
preamble. The tabskip glue that is in force when \TeX\ reads the
`|{|' following |\halign| will be used before the first column;
the tabskip glue that is in force when \TeX\ reads the `|&|' after
the first template will be used between the first and second
columns; and so on. The tabskip glue that is in force when \TeX\
reads the |\cr| after the last template will be used after the
last column. For example, in
\begintt
\tabskip=3pt
\halign{\hfil#\tabskip=4pt& #\hfil&
\hbox to 10em{\hss\tabskip=5pt # \hss}\cr ...}
\endtt
the preamble specifies aligned lines that will consist of the following
seven parts:
\begindisplay
tabskip glue $3\pt$;\cr
first column, with template `|\hfil#|';\cr
tabskip glue $4\pt$;\cr
second column, with template `|#\hfil|';\cr
tabskip glue $4\pt$;\cr
third column, with template `|\hbox to 10em{\hss# \hss}|';\cr
tabskip glue $5\pt$.\cr
\enddisplay
\ddanger \TeX\ copies the templates without interpreting them except to
remove any |\tabskip| glue specifications. More precisely, the tokens of the
preamble are passed directly to the templates without macro expansion;
\TeX\ looks only for `|\cr|' commands, `|&|', `|#|', `|\span|', and
`|\tabskip|'. The \<glue> following `|\tabskip|' is scanned in~the usual
way (with macro expansion), and the corresponding tokens are not included~in
the current template. Notice that, in the example above, the space
after `|5pt|' also disappeared. The fact that |\tabskip=5pt| occurred
inside an extra level of braces did not make the definition local, since
\TeX\ didn't ``see'' those braces; similarly, if\/ |\tabskip| had been
preceded by `|\global|', \TeX\ wouldn't have made a global definition, it
would just have put `|\global|' into the template. All assignments to
|\tabskip| within the preamble are local to the |\halign| (unless
^|\globaldefs| is positive), so the value of\/ |\tabskip| will be $3\pt$ again
when this particular |\halign| is completed.
\ddanger When `^|\span|' appears in a preamble, it causes the next token
to be expanded, i.e., ``ex-span-ded,'' before \TeX\ reads on.
\def\\{{\it c\/}}
\dangerexercise Design a preamble for the following table:
$$\halign to\hsize{\sl#\hfil\tabskip=.5em plus.5em&
#\hfil\tabskip=0pt plus.5em&
\hfil#\tabskip=1em plus2em&
\sl#\hfil\tabskip=.5em plus.5em&
#\hfil\tabskip=0pt plus.5em&
\hfil#\tabskip=0pt\cr
England&P. Philips&1560--1628&
Netherlands&J. P. Sweelinck&1562--1621\cr
&J. Bull&\\1563--1628&
&P. Cornet&\\1570--1633\cr
Germany&H. L. Hassler&1562--1612&
Italy&G. Frescobaldi&1583--1643\cr
&M. Pr\ae torius&1571--1621&
Spain&F. Correa de Arauxo&\\1576--1654\cr
France&J. Titelouze&1563--1633&
Portugal&M. R. Coelho&\\1555--\\1635\cr}$$
The tabskip glue should be zero at the left and right of each line; it should be
$1\em$ plus $2\em$ in the center; and it should be $.5\em$
plus $.5\em$ before the names, $0\em$ plus $.5\em$ before
the dates. Assume that the lines of the table will be specified by, e.g.,
\begintt
France&J. Titelouze&1563--1633&
Portugal&M. R. Coelho&\\1555--\\1635\cr
\endtt
where `|\\|' has been predefined by `|\def\\{{\it c\/}}|'.
^^{organists}
^^{Cornet, Peeter} ^^{Philips, Peter} ^^{Sweelinck, Jan Pieterszoon}
^^{Bull, John} ^^{Titelouze, Jehan} ^^{Hassler, Hans Leo}
^^{Pr\ae torius [Schultheiss], Michael} ^^{Frescobaldi, Girolamo}
^^{Coelho, Manuel Rodrigues} % so listed in Lisbon & Rio, contrary to Groves!
^^{Correa de Arauxo, Francisco}
% The idea for this table came from The Organ and its Music, by Peeters and
% Vente (Antwerp, 1971); but their data was so flaky, I'm not citing them...
\answer |$$\halign to\hsize{\sl#\hfil\tabskip=.5em plus.5em&|\parbreak
| #\hfil\tabskip=0pt plus.5em&|\parbreak
| \hfil#\tabskip=1em plus2em&|\parbreak
| \sl#\hfil\tabskip=.5em plus.5em&|\parbreak
| #\hfil\tabskip=0pt plus.5em&|\parbreak
| \hfil#\tabskip=0pt\cr ...}$$|
\medskip
\ddangerexercise Design a preamble so that the table ^^{Welsh conjugation}
$$\def\welshverb#1={{\bf#1} = }
\halign to\hsize{\welshverb#\hfil\tabskip=1em plus1em&
\welshverb#\hfil&\welshverb#\hfil\tabskip=0pt\cr
rydw i=I am&ydw i=am I&roeddwn i=I was\cr
rwyt ti=thou art&wyt ti=art thou&roeddet ti=thou wast\cr
mae e=he is&ydy e=is he&roedd e=he was\cr
mae hi=she is&ydy hi=is she&roedd hi=she was\cr
rydyn ni=we are&ydyn ni=are we&roedden ni=we were\cr
rydych chi=you are&ydych chi=are you&roeddech chi=you were\cr
maen nhw=they are&ydyn nhw=are they&roedden nhw=they were\cr}$$
can be specified by typing lines like
\begintt
mae hi=she is&ydy hi=is she&roedd hi=she was\cr
\endtt
\answer The trick is to define a new macro for the preamble:
\begintt
$$\def\welshverb#1={{\bf#1} = }
\halign to\hsize{\welshverb#\hfil\tabskip=1em plus1em&
\welshverb#\hfil&\welshverb#\hfil\tabskip=0pt\cr ...}$$
\endtt
\setbox0=\vbox{\lineskip0pt
\tabskip=0pt plus1fil\halign to\hsize{\tabskip=0pt\strut
\hfil#: &\vtop{\parindent=0pt\hsize=16em\hangindent.5em\strut#\strut}\cr
\omit\hfil\sevenrm B.C.&\cr
397&War between Syracuse and Carthage\cr
396&Aristippus of Cyrene and An\-tis\-the\-nes of Athens (philosophers)\cr
395&Athens rebuilds the Long Walls\cr
394&Battles of Coronea and Cnidus\cr
\\393&Plato's
{\sl Apology\/};
Xenophon's
{\sl Memorabilia\/};
Aristophanes'
{\sl Ecclesiazus\ae\/}\cr
391--87&Dionysius subjugates south Italy\cr
391&Isocrates opens his school\cr
390&Evagoras Hellenizes Cyprus\cr
387&``King's Peace''; Plato visits Ar\-chy\-tas of Taras (mathematician)
and Dionysius I\cr
386&Plato founds the Academy\cr
383&Spartans occupy Cadmeia at Thebes\cr
380&Isocrates' {\sl Panegyricus\/}\cr}}
\medskip
\ddangerexercise \hsize=13pc
The line breaks in the second column of the table at the right were chosen
by \TeX\ so that the second column was exactly 16~ems wide. Furthermore,
the author specified one of the rows of the table by typing
$$\halign{#\hfil\cr
|\\393&Plato's {\sl Apology\/};|\cr
| Xenophon's|\cr
| {\sl Memorabilia\/};|\cr
| Aristophanes'|\cr
| {\sl Ecclesiazus\ae\/}\cr|\cr}$$
Can you guess what preamble was used in the alignment? \ [The data comes
from Will ^{Durant}'s {\sl The Life of Greece\/} (Simon \& Schuster, 1939).]
^^{Aristippus of Cyrene} ^^{Antisthenes of Athens}
^^{Plato} ^^{Xenophon} ^^{Aristophanes} ^^{Dionysius I of Syracuse}
^^{Isocrates} ^^{Evagoras of Salamis} ^^{Archytas of Taras}
\strut\vadjust{\vbox to 0pt{\vss\box0\kern0pt}}% insert the aligned table
\answer |\hfil#: &\vtop{\parindent=0pt\hsize=16em|\parbreak
| \hangindent.5em\strut#\strut}\cr|\par\nobreak\medskip\noindent
With such narrow measure and such long words, the ^|\tolerance| should probably
also have been increased to, say, 1000 inside the ^|\vtop|; luckily it turned
^^|\strut| out that a higher tolerance wasn't needed.\par
{\sl Note:\/} The stated preamble solves the problem and demonstrates
that \TeX's line-breaking capability can be used within tables. But this
particular table is not really a good example of the use of\/ |\halign|,
because \TeX\ could typeset it directly, using ^|\everypar| in an
appropriate manner to set up the hanging indentation, and using |\par|
instead of\/ |\cr|. For example, one could say
\begintt
\hsize20em \parindent0pt \clubpenalty10000 \widowpenalty10000
\def\history#1&{\hangindent4.5em
\hbox to4em{\hss#1: }\ignorespaces}
\everypar={\history} \def\\{\leavevmode{\it c\/}}
\endtt
which spares \TeX\ all the work of\/ |\halign| but yields essentially the
same result. ^^|\leavevmode|
\danger Sometimes a template will apply perfectly to all but one or two of
the entries in a column. For example, in the exercise just given, the
colons in the first column of the alignment were supplied by the
template `|\hfil#:|\]'; but the very first entry in that column,
`{\sevenrm B.C.}', did not have a colon. \TeX\ allows you to escape from
the stated template in the following way: If the very first token
of an alignment entry is `^|\omit|' (after macro expansion), then
the template of the preamble is omitted; the trivial template `|#|'
is used instead. For example, `{\sevenrm B.C.}' was put into the table above
by typing `|\omit\hfil\sevenrm B.C.|' immediately after the preamble.
You can use |\omit| in any column, but it must come first; otherwise \TeX\
will insert the template that was defined in the preamble.
\ddanger If you think about what \TeX\ has to do when it's processing
|\halign|, you'll realize that the timing of certain actions is critical.
Macros are not expanded when the preamble is being read, except as
described earlier; but once the |\cr| at the end of the preamble has been
sensed, \TeX\ must look ahead to see if the next token is |\noalign| or
|\omit|, and macros are expanded until the next non-space token is found.
If the token doesn't turn out to be |\noalign| or |\omit|, it is put
back to be read again, and \TeX\ begins to read the template (still
expanding macros). The template has two parts, called the $u$ and~$v$ parts,
where $u$~precedes the~`|#|' and $v$~follows~it. When \TeX\ has finished
the $u$~part, its reading mechanism goes back to the token that was
neither |\noalign| nor |\omit|, and continues to read the entry until
getting to the |&| or~|\cr| that ends the entry; then the $v$~part of
the template is read. A special internal operation called ^|\endtemplate|
is always placed at the end of the $v$~part; this causes \TeX\ to put
the entry into an ``^{unset box}'' whose glue will be set later when
the final column width is known. Then \TeX\ is ready for another entry;
it looks ahead for |\omit| (and also for |\noalign|, after~|\cr|) and
the process continues in the same way.
\ddanger One consequence of the process just described is that it may be
dangerous to begin an entry of an alignment with |\if...|, ^^{conditionals}
or with any macro that will expand into a replacement text whose first token
is |\if...|; the reason is that the condition will be evaluated before the
template has been read. \ (\TeX\ is still looking to see whether an |\omit|
will occur, when the |\if| is being expanded.) \ For example, if\/ ^|\strut|
has been defined to be an abbreviation for
\begindisplay
^|\ifmmode|\<text for math modes>|\else|\<text for nonmath modes>|\fi|
\enddisplay
and if\/ |\strut| appears as the first token in some alignment entry,
then \TeX\ will expand it into the \<text for nonmath modes> even though
the template might be `|$#$|', because \TeX\ will not yet be in math
mode when it is looking for a possible |\omit|. Chaos will
probably ensue. Therefore the replacement text for |\strut| in
Appendix~B is actually
\begintt
\relax\ifmmode...
\endtt
and `|\relax|' has also been put into all other macros that might suffer
from such timing problems. Sometimes you do want \TeX\ to expand a
conditional before a template is inserted, but careful macro designers
watch out for cases where this could cause trouble.
\newdimen\digitwidth \setbox0=\hbox{\sixrm0} \digitwidth=\wd0
\danger When you're typesetting ^{numerical tables}, it's common practice
to line up the ^{decimal points} in a column. For example, if two numbers
like `0.2010' and `297.1' both appear in the same column, you're supposed
to produce `$\catcode`?=13 \def?{\kern\digitwidth}
??0.2010 \atop 297.1???$'. This result isn't especially pleasing to the
eye, but that's what people do, so you might have to conform to the practice.
One way to handle this is to treat the column as two columns, somewhat as
|\eqalign| treats one formula as two formulas; the `.'\ can be placed at
the beginning of the second half-column. But the author usually prefers to
use another, less sophisticated method, which takes advantage of the fact
that the digits 0,~1, \dots,~9 have the same width in most fonts: You can
choose a character that's not used elsewhere in the table, say `|?|',
and change it to an ^{active character} that produces a blank space exactly
equal to the width of a digit. Then it's usually no chore to put such nulls
into the table entries so that each column can be regarded as either
centered or right-justified or left-justified. For example, `|??0.2010|'
and `|297.1???|'\ have the same width, so their decimal points will line
up easily. Here is one way to set up `|?|'\ for this purpose:
\begintt
\newdimen\digitwidth
\setbox0=\hbox{\rm0}
\digitwidth=\wd0
\catcode`?=\active
\def?{\kern\digitwidth}
\endtt
The last two definitions should be local to some ^{group}, e.g., inside a
|\vbox|, so that `|?|'\ will resume its normal behavior when the table is
finished. ^^|\active|
\danger Let's look now at some applications to mathematics. Suppose first
that you want to typeset the small table
$$\vbox{\halign{$\hfil#$ =&&\ \hfil#\hfil\cr
n\phantom)&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&\dots\cr
{\cal G}(n)&1&2&4&3&6&7&8&16&18&25&32&11&64&31&128&10&256&5&512&28&
1024&\dots\cr}}$$ % The Grundy function for SYM [cf. Winning Ways p441]
as a ^{display}ed equation. A brute force approach using |\eqalign| or
|\atop| is cumbersome because ${\cal G}(n)$ and $n$ don't always have the
same number of digits. It would be much nicer to type
\begindisplay
|$$\vbox{\halign{|\<preamble>|\cr|\cr
| n\phantom)&0&1&2&3& ... &20&\dots\cr|\cr
| {\cal G}(n)&1&2&4&3& ... &1024&\dots\cr}}$$|\cr
\enddisplay
for some \<preamble>. On the other hand, the \<preamble> is sure to be
long, since this table has 23 columns; so it looks as though |\settabs|
and |\+| will be easier. \TeX\ has a handy feature that helps a lot
in cases like this: Preambles often have a periodic structure,
^^{periodic preambles} ^^{cyclic preambles} ^^{ampersand ampersand}
and if you put an extra `|&|' ^^{ampersand} just before one of the templates,
\TeX\ will consider the preamble to be an infinite sequence that begins
again at the marked template when the |\cr| is reached. For example,
\begindisplay
$t_1\,$|&|$\,t_2\,$|&|$\,t_3\,$|&&|$\,t_4\,$|&|$\,t_5\,$|\cr|
\ is treated like \
$t_1\,$|&|$\,t_2\,$|&|$\,t_3\,$|&|$\,t_4\,$|&|$\,t_5\,$%
|&|$\,t_4\,$|&|$\,t_5\,$|&|$\,t_4\,$|&|$\,\,\cdots\,$\cr
\noalign{\hbox{and}}
|&|$t_1\,$|&|$\,t_2\,$|&|$\,t_3\,$|&|$\,t_4\,$|&|$\,t_5\,$|\cr|
\ is treated like \
$t_1\,$|&|$\,t_2\,$|&|$\,t_3\,$|&|$\,t_4\,$|&|$\,t_5\,$%
|&|$\,t_1\,$|&|$\,t_2\,$|&|$\,t_3\,$|&|$\,\,\cdots\,.$\cr
\enddisplay
The tabskip glue following each template is copied with that template.
The preamble will grow as long as needed, based on the number of columns
actually used by the subsequent alignment entries. Therefore all it takes is
\begintt
$\hfil#$ =&&\ \hfil#\hfil\cr
\endtt
to make a suitable \<preamble> for the ${\cal G}(n)$ problem.
\ddanger Now suppose that the task is to typeset three pairs of displayed
formulas, with all of the =~signs lined up: % cf. ACP Section 3.3.4
$$\vcenter{\openup1\jot \halign{
$\hfil#$&&${}#\hfil$&\qquad$\hfil#$\cr
V_i&=v_i-q_iv_j,&X_i&=x_i-q_ix_j,&
U_i&=u_i,\qquad\hbox{for $i\ne j$};\cr
V_j&=v_j,&X_j&=x_j,&
U_j&=u_j+\sum_{i\ne j}q_iu_i.\cr}}\eqno(23)$$
It's not easy to do this with three ^|\eqalign|'s, because the $\sum$ with
a subscript `$i\ne j$' makes the right-hand pair of formulas bigger than the
others; the baselines won't agree unless ``^{phantoms}'' are put into the
other two |\eqalign|'s (see Chapter~19). Instead of using |\eqalign|,
which is defined in Appendix~B to be a macro that uses |\halign|, let's
try to use |\halign| directly. The natural way to approach this display is
^^|\jot| to type
\begindisplay
|$$\vcenter{\openup1\jot \halign{|\<preamble>|\cr|\cr
| |\<first line>|\cr |\<second line>|\cr}}\eqno(23)$$|\cr
\enddisplay
because the ^|\vcenter| puts the lines into a box that is properly centered
with respect to the equation number `(23)'; the ^|\openup| macro puts a
bit of extra space between the lines, as mentioned in Chapter~19.
\ddanger OK, now let's figure out how to type the \<first line> and \<second
line>. The usual convention is to put `|&|' before the symbols that we want
to line up, so the obvious solution is to type
\begintt
V_i&=v_i-q_iv_j,&X_i&=x_i-q_ix_j,&
U_i&=u_i,\qquad\hbox{for $i\ne j$};\cr
V_j&=v_j,&X_j&=x_j,&
U_j&=u_j+\sum_{i\ne j}q_iu_i.\cr
\endtt
Thus the alignment has six columns. We could take common elements into
the preamble (e.g., `|V_|' and `|=v_|'), but that would be too error-prone
and too tricky.
\ddanger The remaining problem is to construct a preamble to
support those lines. To the left of the =~signs we want the column to
be filled at the left; to the right of the =~signs we want it to be filled
at the right. There's a slight complication because we are breaking a math
formula into two separate pieces, yet we want the result to have the same
spacing as if it were one formula. Since we're putting the `|&|' just before
a relation, the solution is to insert `|{}|' ^^{lbrace rbrace}
at the beginning of the right-hand formula; \TeX\ will put the proper
space before the equals sign in `|${}=...$|', but it puts no space before
the equals sign in `|$=...$|'. Therefore the desired \<preamble> is
\begintt
$\hfil#$&${}#\hfil$&
\qquad$\hfil#$&${}#\hfil$&
\qquad$\hfil#$&${}#\hfil$
\endtt
The third and fourth columns are like the first and second, except for
the |\qquad| that separates the equations; the fifth and sixth columns
are like the third and fourth. Once again we can use the handy `|&&|'
shortcut ^^{ampersand ampersand} to reduce the preamble to
\begintt
$\hfil#$&&${}#\hfil$&\qquad$\hfil#$
\endtt
With a little practice you'll find that it becomes easy to compose
preambles as you are typing a manuscript that needs them. However, most
manuscripts don't need them, so it may be a~while before you acquire even
a little practice in this regard.
\ddangerexercise Explain how to produce the following display:
$$\openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
$\hfil#$&$\hfil{}#{}$&
$\hfil#$&$\hfil{}#{}$&
$\hfil#$&$\hfil{}#{}$&
$\hfil#$&${}#\hfil$\tabskip=0pt plus1fil&
\llap{#}\tabskip=0pt\cr
10w&+&3x&+&3y&+&18z&=1,&(9)\cr
6w&-&17x&&&-&5z&=2.&(10)\cr}$$ % cf. ACP Eqs. 4.5.2-17,18
\answer The equation is divided into separate parts for terms and
plus/minus signs, and tabskip glue is used to center it:
\begintt
$$\openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
$\hfil#$&$\hfil{}#{}$&
$\hfil#$&$\hfil{}#{}$&
$\hfil#$&$\hfil{}#{}$&
$\hfil#$&${}#\hfil$\tabskip=0pt plus1fil&
\llap{#}\tabskip=0pt\cr
10w&+&3x&+&3y&+&18z&=1,&(9)\cr
6w&-&17x&&&-&5z&=2.&(10)\cr}$$
\endtt
\ddanger The next level of complexity occurs when some entries of a table
span two or more columns. \TeX\ provides two ways to handle this. First
^^{spanned columns in tables}
there's ^|\hidewidth|, which plain \TeX\ defines to be equivalent to
\begintt
\hskip-1000pt plus 1fill
\endtt
In other words, |\hidewidth| has an extremely negative ``natural width,''
but it will stretch without limit. If you put |\hidewidth| at the right of
some entry in an alignment, the effect is to ignore the width of this
entry and to let it stick out to the right of its box. \ (Think about it;
this entry won't be the widest one, when |\halign| figures the column
width.) \ Similarly, if you put |\hidewidth| at the left of an entry, it will
stick out to the left; and you can put |\hidewidth| at both left and right,
as we'll see later.
\ddanger The second way to handle table entries that span columns is to use
the ^|\span| primitive, which can be used instead of `|&|' in any
line of the table. \ (We've already seen that |\span| means ``expand'' in
preambles; but outside of preambles its use is
completely different.) \ When `|\span|' appears in place of `|&|',
the material before and after the |\span| is processed in the ordinary
way, but afterward it is placed into a single box instead of two boxes.
The width of this combination box is the sum of the individual column
widths plus the width of the tabskip glue between them; therefore the
spanning box will line up with non-spanning boxes in other rows.
\ddanger For example, suppose that there are three columns, with the
respective templates $u_1\,$|#|$\,v_1\,$|&| $u_2\,$|#|$\,v_2\,$|&|
$u_3\,$|#|$\,v_3$; suppose that the column widths are $w_1$, $w_2$,~$w_3$;
suppose that $g_0$,~$g_1$, $g_2$,~$g_3$ are the tabskip glue widths after
the glue has been set; and suppose that the line
\begindisplay
$a_1$|\span|$\,\,a_2$|\span|$\,\,a_3$|\cr|
\enddisplay
has appeared in the alignment. Then the material for
`$u_1a_1v_1u_2a_2v_2u_3a_3v_3$' (i.e., the result `$u_1a_1v_1$' of
column~1 followed by the results of columns 2 and~3) will be placed into
an hbox of width $w_1+g_1+ w_2+g_2+w_3$. That hbox will be preceded by
glue of width~$g_0$ and it will be followed by glue of width~$g_3$, in the
larger hbox that contains the entire aligned line.
\ddanger You can use ^|\omit| in conjunction with |\span|. For example,
if we continue with the notation of the previous paragraph, the line
\begindisplay
|\omit|$\,a_1\,$|\span|$\,a_2\,$|\span\omit|$\,a_3\,$|\cr|
\enddisplay
would put the material for `$a_1u_2a_2v_2a_3$' into the hbox just considered.
\ddanger It's fairly common to span several columns and to omit all their
templates, so plain \TeX\ provides a ^|\multispan| macro that spans
a given number of columns. For example, `|\multispan3|' expands into
`|\omit\span\omit\span\omit|'. If the number of spanned columns is
greater than~9, you must put it in braces, e.g., `|\multispan{13}|'.
\ddanger The preceding paragraphs are rather abstract, so let's look at
an example that shows what |\span| actually does. Suppose you type
\begintt
$$\tabskip=3em
\vbox{\halign{&\hrulefill#\hrulefill\cr
first&second&third\cr
first-and-second\span\omit&\cr
&second-and-third\span\omit\cr
first-second-third\span\omit\span\omit\cr}}$$
\endtt
The preamble specifies arbitrarily many templates equal to
`|\hrulefill#\hrulefill|'; the ^|\hrulefill| macro is like |\hfill|
except that the blank space is filled with a horizontal rule. Therefore
you can see the filling in the resulting alignment, which shows the
spanned columns:
$$\tabskip=3em
\vbox{\halign{&\hrulefill#\hrulefill\cr
first&second&third\cr
first-and-second\span\omit&\cr
&second-and-third\span\omit\cr
first-second-third\span\omit\span\omit\cr}}$$
The rules stop where the tabskip glue separates columns. You don't see
rules in the first line, since the entries in that line were the widest
in their columns. However, if the tabskip glue had been $1\em$ instead
of $3\em$, the table would have looked like this:
$$\tabskip=1em
\vbox{\halign{&\hrulefill#\hrulefill\cr
first&second&third\cr
first-and-second\span\omit&\cr
&second-and-third\span\omit\cr
first-second-third\span\omit\span\omit\cr}}$$
\ddangerexercise Consider the following table, which is called
^{Walter's worksheet}: ^^{IRS}
^^{Green, Walter} % from instructions to form 1040 (1982), p13
$$\halign{\indent
\hfil# &#\hfil&\quad#&\ \hfil#&\ \hfil#\cr
1&Adjusted gross income\dotfill\span\omit\span&\$4,000\cr
2&Zero bracket amount for&\cr
&a single individual\dotfill\span\omit&\$2,300\cr
3&Earned income\dotfill\span\omit&\underbar{ 1,500}\cr
4&Subtract line 3 from line 2\dotfill\span\omit\span&\underbar{ 800}\cr
5&Add lines 1 and 4. Enter here\span\omit\span\cr
&and on Form 1040, line 35\dotfill\span\omit\span&\$4,800\cr}
$$
Define a preamble so that the following specification will produce
Walter's worksheet.
$$\halign{\indent#\hfil\cr
|\halign{|\<preamble>|\cr|\cr
| 1&Adjusted gross income\dotfill\span\omit\span&\$4,000\cr|\cr
| 2&Zero bracket amount for&\cr|\cr
| &a single individual\dotfill\span\omit&\$2,300\cr|\cr
| 3&Earned income\dotfill\span\omit&\underbar{ 1,500}\cr|\cr
| 4&Subtract line 3 from line 2\dotfill|\cr
| \span\omit\span&\underbar{ 800}\cr|\cr
| 5&Add lines 1 and 4. Enter here\span\omit\span\cr|\cr
| &and on Form 1040, line 35\dotfill\span\omit\span&\$4,800\cr}|\cr
}$$
(The macro ^|\dotfill| is like |\hrulefill| but it fills with dots;
the macro ^|\underbar| puts its argument into an hbox and underlines it.)
\answer |\hfil# &#\hfil&\quad#&\ \hfil#&\ \hfil#\cr|
\ddanger Notice the ``early'' appearance of\/ ^|\cr| in line~2 of the
previous exercise. You needn't have the same number of columns in every
line of an alignment; `|\cr|' means that there are no more columns
in the current line.
\medskip
\ddangerexercise Explain how to typeset the ^{generic matrix}
$\smash{\pmatrix{a_{11}&a_{12}&\ldots&a_{1n}\cr
a_{21}&a_{22}&\ldots&a_{2n}\cr
\multispan4\dotfill\cr
a_{m1}&a_{m2}&\ldots&a_{mn}\cr}.}$
\answer |\pmatrix{a_{11}&a_{12}&\ldots&a_{1n}\cr|\parbreak
| a_{21}&a_{22}&\ldots&a_{2n}\cr|\parbreak
| \multispan4\dotfill\cr|\parbreak
| a_{m1}&a_{m2}&\ldots&a_{mn}\cr}|
\bigskip\medskip
\ddanger The presence of spanned columns adds a complication to \TeX's
rules for calculating column widths; instead of simply choosing the
maximum natural width of the column entries, it's also necessary to
make sure that the sum of certain widths is big enough to accommodate
spanned entries. So here is what \TeX\ actually does: First, if
any pair of adjacent columns is always spanned as a unit (i.e., if
there's a |\span| between them whenever either one is used), these
two columns are effectively merged into one and the tabskip glue between
them is set to zero. This reduces the problem to the case that
every tab position actually occurs at a boundary. Let there be $n$
columns remaining after such reductions, and for $1\le i\le j\le n$ let
$w_{ij}$ be the maximum natural width of all entries that span columns $i$
through~$j$, inclusive; if there are no such spanned entries, let
$w_{ij}=-\infty$. \ (The merging of dependent columns guarantees that, for
each~$j$, there exists $i\le j$ such that $w_{ij}>-\infty$.) \ Let $t_k$
be the natural width of the tabskip glue between columns $k$ and~$k+1$,
for $1\le k<n$. Now the final width $w_j$ of column~$j$ is determined by
the formula
\begindisplay
$\displaystyle w_j=\max_{1\le i\le j}\textstyle\bigl(w_{ij}
-\sum_{i\le k<j}(w_k+t_k)\bigr)$
\enddisplay
for $j=1$, 2, \dots, $n$ (in this order). It follows that
$w_{ij}\le w_i+t_i+\cdots+t_{j-1}+w_j$, for all $i\le j$,
as desired. After the widths~$w_j$ are determined, the tabskip amounts
may have to stretch or shrink; if they shrink, $w_{ij}$ might turn out to
be more than the final width of a box that spans columns $i$ through~$j$,
hence the glue in such a box might shrink.
\ddanger These formulas usually work fine, but sometimes they produce
undesirable effects. For example, suppose that $n=3$, $w_{11}=w_{22}=w_{33}
=10$, $w_{12}=w_{23}=-\infty$, and $w_{13}=100$; in other words, the columns
by themselves are quite narrow, but there's a big wide entry that's
supposed to span all three columns. In this case \TeX's formula makes
$w_1=w_2=10$ but $w_3=80-t_1-t_2$, so all the excess width is allocated
to the third column. If that's not what you want, the remedy is~to use
^|\hidewidth|, or to increase the natural width of the tabskip glue
between columns.
\ddanger The next level of complexity that occurs in tables is the
appearance of horizontal and vertical ruled lines. People who know
how to make ^{ruled tables} are generally known as \TeX\ Masters.
^^{TeX Masters} Are you ready?
\ddanger If you approach vertical rules in the wrong manner, they can be
difficult; but there {\sl is\/} a decent way to get them into tables
without shedding too many tears. The first step is to say
`^|\offinterlineskip|', which means that there will be no blank space
between lines; \TeX\ cannot be allowed to insert ^{interline glue} in its
normal clever way, because each line is supposed to contain a ^|\vrule|
that abuts another ^|\vrule| in the neighboring lines above and/or below.
We will put a strut into every line, by including one in the preamble;
then each line will have the proper height and depth, and there will be no
need for interline glue. \TeX\ puts every column entry of an alignment into an
hbox whose height and depth are set equal to the height and depth of the
entire line; therefore |\vrule| commands will extend to the top and bottom
of the lines even when their height and/or depth are not specified.
\ddanger A ``column'' should be allocated to every vertical rule, and such
a column can be assigned the template `|\vrule#|'. Then you obtain a
vertical rule by simply leaving the column entries blank, in the normal
lines of the alignment; or you can say `|\omit|' if you want to omit the
rule in some line; or you can say `|height 10pt|' if you want a
nonstandard height; and so on.
\ddanger Here is a small table that illustrates the points just made.
\ [The data appeared in an article by A. H. ^{Westing}, {\sl BioScience\/
\bf31} (1981), 523--524.]
\def\BC{\hbox to2em{ \sc B.C.\hss}}%
\def\AD{\hbox to2em{ \sc A.D.\hss}}%
$$\hbox to\hsize{\vbox{\halign{\indent#\hfil\cr
|\vbox{\offinterlineskip|\cr
|\hrule|\cr
|\halign{&\vrule#&|\cr
| \strut\quad\hfil#\quad\cr|\cr
|height2pt&\omit&&\omit&\cr|\cr
|&Year\hfil&&World Population&\cr|\cr
|height2pt&\omit&&\omit&\cr|\cr
|\noalign{\hrule}|\cr
|height2pt&\omit&&\omit&\cr|\cr
|&8000\BC&&5,000,000&\cr|\cr
|&50\AD&&200,000,000&\cr|\cr
|&1650\AD&&500,000,000&\cr|\cr
|&1850\AD&&1,000,000,000&\cr|\cr
|&1945\AD&&2,300,000,000&\cr|\cr
|&1980\AD&&4,400,000,000&\cr|\cr
|height2pt&\omit&&\omit&\cr}|\cr
|\hrule}|\cr
}}\hfill\vbox{\offinterlineskip
\halign{&\vrule#&
\strut\quad\hfil#\quad\cr
\multispan5\hrulefill\cr
height2pt&\omit&&\omit&\cr
&Year\hfil&&World Population&\cr
height2pt&\omit&&\omit&\cr
\multispan5\hrulefill\cr
height2pt&\omit&&\omit&\cr
&8000\BC&&5,000,000&\cr
&50\AD&&200,000,000&\cr
&1650\AD&&500,000,000&\cr
&1850\AD&&1,000,000,000&\cr
&1945\AD&&2,300,000,000&\cr
&1980\AD&&4,400,000,000&\cr
height2pt&\omit&&\omit&\cr
\multispan5\hrulefill\cr}}}$$
In this example the first, third, and fifth columns are reserved for vertical
rules. Horizontal rules are obtained by saying `^|\hrule|' outside the
|\halign| or `^|\noalign||{\hrule}|' inside it, because the |\halign| appears
in a vbox whose width is the full table width. The horizontal
rules could also have been specified by saying `^|\multispan||5\hrulefill|'
inside the |\halign|, since that would produce a rule that spans all
five columns.
\ddanger The only other nonobvious thing about this table is the
inclusion of several lines that say
`|height2pt&\omit&&\omit&\cr|'; can you see what they do? The |\omit|
instructions mean that there's no numerical information, and they
also suppress the ^|\strut| from the line; the `|height2pt|' makes the
first |\vrule| $2\pt$ high, and the other two rules will follow suit.
Thus, the effect is to extend the vertical rules by two points, where
they touch the horizontal rules. This is a little touch that improves
the appearance of boxed tables; look for it as a mark of quality.
\ddangerexercise Explain why the lines of this table say `|&\cr|' instead of
just `|\cr|'.
\answer `|\cr|' would have omitted the final column, which is a vertical rule.
\ddanger Another way to get vertical rules into tables is to typeset without
them, then back up (using negative glue) and insert them.
\ddanger Here is another table; this one has become a classic, ever since
Michael ^{Lesk} published it as one of the first examples in his report
on a program to format tables [Bell Laboratories Computing Science
Technical Report {\bf 49} (1976)]. It illustrates several typical
problems that arise in connection with boxed information. In order to
demonstrate \TeX's ability to adapt a table to different circumstances,
tabskip glue is used here to adjust the column widths; the table appears
twice, once generated by `|\halign|~|to125pt|' and once by
`|\halign|~|to200pt|', with nothing else changed. ^^{AT\&T}
$$\hbox to\hsize{%
\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to125pt{\strut#&\vrule#\tabskip=1em plus2em&
\hfil#&\vrule#&\hfil#\hfil&\vrule#&
\hfil#&\vrule#\tabskip=0pt\cr\tablerule
&&\multispan5\hfil AT\&T Common Stock\hfil&\cr\tablerule
&&\omit\hidewidth Year\hidewidth&&
\omit\hidewidth Price\hidewidth&&
\omit\hidewidth Dividend\hidewidth&\cr\tablerule
&&1971&&41--54&&\$2.60&\cr\tablerule
&& 2&&41--54&&2.70&\cr\tablerule
&& 3&&46--55&&2.87&\cr\tablerule
&& 4&&40--53&&3.24&\cr\tablerule
&& 5&&45--52&&3.40&\cr\tablerule
&& 6&&51--59&&.95\rlap*&\cr\tablerule
\noalign{\smallskip}
&\multispan7* (first quarter only)\hfil\cr
}}\hfil
\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to200pt{\strut#&\vrule#\tabskip=1em plus2em&
\hfil#&\vrule#&\hfil#\hfil&\vrule#&
\hfil#&\vrule#\tabskip=0pt\cr\tablerule
&&\multispan5\hfil AT\&T Common Stock\hfil&\cr\tablerule
&&\omit\hidewidth Year\hidewidth&&
\omit\hidewidth Price\hidewidth&&
\omit\hidewidth Dividend\hidewidth&\cr\tablerule
&&1971&&41--54&&\$2.60&\cr\tablerule
&& 2&&41--54&&2.70&\cr\tablerule
&& 3&&46--55&&2.87&\cr\tablerule
&& 4&&40--53&&3.24&\cr\tablerule
&& 5&&45--52&&3.40&\cr\tablerule
&& 6&&51--59&&.95\rlap*&\cr\tablerule
\noalign{\smallskip}
&\multispan7* (first quarter only)\hfil\cr}}}$$
The following specification did the job:
\begindisplay
|\vbox{\tabskip=0pt \offinterlineskip|\cr
|\def\tablerule{\noalign{\hrule}}|\cr
|\halign to|\<dimen>|{\strut#& \vrule#\tabskip=1em plus2em&|\cr
| \hfil#& \vrule#& \hfil#\hfil& \vrule#&|\cr
| \hfil#& \vrule#\tabskip=0pt\cr\tablerule|\cr
|&&\multispan5\hfil AT\&T Common Stock\hfil&\cr\tablerule|\cr
|&&\omit\hidewidth Year\hidewidth&&|\cr
| \omit\hidewidth Price\hidewidth&&|\cr
| \omit\hidewidth Dividend\hidewidth&\cr\tablerule|\cr
|&&1971&&41--54&&\$2.60&\cr\tablerule|\cr
|&& 2&&41--54&&2.70&\cr\tablerule|\cr
|&& 3&&46--55&&2.87&\cr\tablerule|\cr
|&& 4&&40--53&&3.24&\cr\tablerule|\cr
|&& 5&&45--52&&3.40&\cr\tablerule|\cr
|&& 6&&51--59&&.95\rlap*&\cr\tablerule \noalign{\smallskip}|\cr
|&\multispan7* (first quarter only)\hfil\cr}}|\cr
\enddisplay
Points of interest are: (1)~The first column contains a strut; otherwise
it would have been necessary to put a strut on the lines that say
`AT\&T' and `(first quarter only)', since those lines omit the templates
of all other columns that might have a built-in strut. (2)~`^|\hidewidth|'
is used in the title line so that the width of columns will be affected
only by the width of the numeric data. (3)~`^|\rlap|' is used so that
the asterisk doesn't affect the alignment of the numbers. (4)~If the
tabskip specification had been `|0em plus3em|' instead of `|1em plus2em|',
the alignment wouldn't have come out right, because `AT\&T Common Stock'
would have been wider than the natural width of everything it spanned; the
excess width would all have gone into the `Dividend' column.
\ddangerexercise Explain how to add $2\pt$ more space above and below
`AT\&T Common Stock'.
\answer One way is to include two lines just before and after the title
line, saying `|\omit&height2pt&\multispan5&\cr|'. Another way is to
put |\bigstrut| into some column of the title line, for some appropriate
invisible box |\bigstrut| of width zero. Either way makes the table
look better.
\ddangerexercise Typeset the following chart, making it exactly 36em wide:
^^{family tree}
^^{Bohning [Knuth], Louise Marie}
^^{Ehlert [Bohning], Pauline Anna Marie}
^^{B\"ohning, Martin John Henry}
^^{Wischmeyer [Ehlert], Clara Louise}
^^{Ehlert, Ernst Fred}
^^{Blase [B\"ohning], Maria Dorothea}
^^{B\"ohning, Jobst Heinrich}
$$\vbox{\tabskip=0pt \offinterlineskip
\halign to 36em{\tabskip=0pt plus1em#&
#\hfil&#&#\hfil&#&#\hfil&#\tabskip=0pt\cr
&&&&&\strut J. H. B\"ohning, 1838&\cr
&&&&\multispan3\hrulefill\cr
&&&\strut M. J. H. B\"ohning, 1882&\vrule\cr
&&\multispan3\hrulefill\cr
&&\vrule&&\vrule&\strut M. D. Blase, 1840&\cr
&&\vrule&&\multispan3\hrulefill\cr
&\strut L. M. Bohning, 1912&\vrule\cr
\multispan3\hrulefill\cr
&&\vrule&&&\strut E. F. Ehlert, 1845&\cr
&&\vrule&&\multispan3\hrulefill\cr
&&\vrule&\strut P. A. M. Ehlert, 1884&\vrule\cr
&&\multispan3\hrulefill\cr
&&&&\vrule&\strut C. L. Wischmeyer, 1850&\cr
&&&&\multispan3\hrulefill\cr
}}$$
\answer The trick is to have ``empty'' columns at the extreme left and right;
then the |\hrulefill|'s are able to span the tabskip glue.
\begintt
$$\vbox{\tabskip=0pt \offinterlineskip
\halign to 36em{\tabskip=0pt plus1em#&
#\hfil&#&#\hfil&#&#\hfil&#\tabskip=0pt\cr
&&&&&\strut J. H. B\"ohning, 1838&\cr
&&&&\multispan3\hrulefill\cr
&&&\strut M. J. H. B\"ohning, 1882&\vrule\cr
&&\multispan3\hrulefill\cr
&&\vrule&&\vrule&\strut M. D. Blase, 1840&\cr
&&\vrule&&\multispan3\hrulefill\cr
&\strut L. M. Bohning, 1912&\vrule\cr
\multispan3\hrulefill\cr
&&\vrule&&&\strut E. F. Ehlert, 1845&\cr
&&\vrule&&\multispan3\hrulefill\cr
&&\vrule&\strut P. A. M. Ehlert, 1884&\vrule\cr
&&\multispan3\hrulefill\cr
&&&&\vrule&\strut C. L. Wischmeyer, 1850&\cr
&&&&\multispan3\hrulefill\cr}}$$
\endtt
\ddanger If you're having trouble ^{debugging} an alignment, it sometimes
helps to put `^|\ddt|' at the beginning and end of the templates in your
preamble. This is an undefined control sequence that causes \TeX\ to stop,
displaying the rest of the template. When \TeX\ stops, you can use
|\showlists| and other commands to see what the machine thinks it's doing.
If \TeX\ doesn't stop, you know that it never reached that part of the
template.
\ddanger It's possible to have alignments within alignments. Therefore when
\TeX\ sees a `|&|' or `|\span|' or `|\cr|', it needs some way to decide which
alignment is involved. The rule is that an entry ends when `|&|' or `|\span|'
or `|\cr|' occurs at the same level of braces that was current when the
entry began; i.e., there must be an equal number of left and right ^{braces}
in every entry. For example, in the line
\begintt
\matrix{1&1\cr 0&1\cr}&\matrix{0&1\cr 0&0\cr}\cr
\endtt
\TeX\ will not resume the template for the first column when it is
scanning the argument to |\matrix|, because the |&|'s and |\cr|'s in
that argument are enclosed in braces. Similarly, |&|'s and |\cr|'s in
the preamble do not denote the end of a template unless the resulting
template would have an equal number of left and right braces.
\ddanger You have to be careful with the use of |&| and ^|\span| and ^|\cr|,
^^{ampersand}
because these tokens are intercepted by \TeX's scanner even when it is
not expanding macros. For example, if you say `|\let\x=\span|' in the
midst of an alignment entry, \TeX\ will think that the `|\span|' ends
the entry, so |\x| will become equal to the first token following the
`|#|' in the template. You can hide this |\span| by putting it in
braces; e.g., `|{\global\let\x=\span}|'. \ (And Appendix~D explains how to
avoid |\global| here.)
\ddanger Sometimes people forget the |\cr| on the last line of an
alignment. This can cause mysterious effects, because \TeX\ is not
clairvoyant. For example, consider the following apparently simple case:
\begintt
\halign{\centerline{#}\cr
A centered line.\cr
And another?}
\endtt
(Notice the missing |\cr|.) \ A curious thing happens here when \TeX\
processes the erroneous line, so please pay attention. The template
begins with `|\centerline{|', so \TeX\ starts to scan the argument to
|\centerline|. Since there's no `|\cr|' after the question mark, the `|}|'
after the question mark is treated as the end of the argument to
|\centerline|, {\sl not\/} as the end of the |\halign|. \TeX\ isn't going
to be able to finish the alignment unless the subsequent text has
the form `|...{...\cr|'. Indeed, an entry like `|a}b{c|' is legitimate
with respect to the template `|\centerline{#}|', since it yields
`|\centerline{a}b{c}|'; \TeX\ is correct when it gives no error message in
this case. But the computer's idea of the current situation is
different from the user's, so a puzzling error message will probably occur
a few lines later.
\ddanger To help avoid such situations, there's a primitive command
^|\crcr| that acts exactly like |\cr| except that it does nothing
when it immediately follows a |\cr| or a |\noalign{...}|. Thus, when
you write a macro like |\matrix|, you can safely insert |\crcr|
at the end of the user's argument; this will cover up an error if the
user forgot the final |\cr|, and it will cause no harm if the final
|\cr| was present.
\ddanger Are you tired of typing |\cr|? ^^{cr, avoiding}
You can get plain \TeX\ to insert an automatic |\cr| at the end of
each input line in the following way: ^^|\begingroup|
\begindisplay
|\begingroup \let\par=\cr \obeylines %|\cr
|\halign{|\<preamble>\cr
\ \ \ \<first line of alignment>\cr
\ \ \ \ \dots\cr
\ \ \ \<last line of alignment>\cr
| }\endgroup|\cr
\enddisplay
This works because ^|\obeylines| makes the ASCII ^\<return> into
an active character that uses the current meaning of\/ ^|\par|, and
plain \TeX\ puts \<return> at the end of an input line (see Chapter~8).
If you don't want a~|\cr| at the end of a certain line,
just type `|%|' and the corresponding |\cr|
will be ``commented out.'' ^^{percent} \ (This special mode doesn't
work with ^|\+| lines, since |\+| is a macro whose argument is delimited
by the token `|\cr|', not simply by a token that has the same meaning
as~|\cr|. ^^{delimited arguments} But you can redefine |\+| to overcome
this hurdle, if you want to. For example, define a macro |\alternateplus|
that is just like |\+| except that its argument is delimited by the active
character |^^M|; then include the command `|\let\+=\alternateplus|' as
part of\/ |\obeylines|.)
\danger The control sequence ^|\valign| is analogous to |\halign|, but
rows and columns change r\^oles. In this case |\cr| marks the bottom of
a column, and the aligned columns are vboxes that are put together in
horizontal mode. The individual entries of each column are vboxed with
depth zero (i.e., as if\/ ^|\boxmaxdepth| were zero, as explained in
Chapter~12); the entry heights for each row of a |\valign| are maximized
in the same fashion as the entry widths for each column of an~|\halign|
are maximized. The ^|\noalign| operation can now be used to insert
horizontal mode material between columns; the ^|\span| operation now
spans rows. ^^{spanned rows in tables} People usually work with \TeX\
at least a year before they find their first application for |\valign|;
and then it's usually a one-row `|\valign{\vfil#\vfil\cr...}|'.
But the general mechanism is there if you need it.
\endchapter
If sixteen pennies are arranged in the form of a square
there will be the same number of pennies in every row, every column,
and each of the two long diagonals.
Can you do the same with twenty pennies?
\author HENRY ERNEST ^{DUDENEY}, {\sl The Best Coin Problems\/} (1909)
% Strand Magazine, July 1909, page 83; answer in August 1909, page 240
\immediate\write\ans{}
\immediate\write\ans{\string\ansno\chapno.$\infty$:}
\copytoblankline (Solution to Dudeney's problem.) \
Let |\one| and |\two| be macros that produce a vertical list
denoting one and two pennies, respectively. The problem can be
solved with ^|\valign| as follows:
\begintt
\valign{\vfil#&\vfil#&\vfil#&\vfil#\cr
\two&\one&\one&\one\cr
\one&\one&\two&\one\cr
\one&\one&\one&\two\cr
\one&\two&\one&\one\cr}
\endtt
Since |\valign| transposes rows and columns, the result is\quad
\setbox0=\hbox{\vbox{
\def\pennytop{\hbox to 24pt{\manual\char'130\hfil}}%
\def\pennyedge{\hbox{\manual\char'133}}%
\def\one{\pennytop\pennyedge}%
\def\two{\one\pennyedge}%
\baselineskip0pt\lineskip0pt\tabskip=14pt
\hbox{\valign{\vfil#&\vfil#&\vfil#&\vfil#\cr
\two&\one&\one&\one\cr
\one&\one&\two&\one\cr
\one&\one&\one&\two\cr
\one&\two&\one&\one\cr}}\kern-11pt}}%
\ht0=0pt \dp0=11pt \box0.
\bigskip
It was she who controlled the whole of the Fifth Column.
\author AGATHA ^{CHRISTIE}, {\sl N or M?\/} (1941) % chapter 5, part 1
\eject
\beginchapter Chapter 23. Output Routines
We investigated \TeX's page-building technique in Chapter 15, where we
discussed the basic two-stage strategy that is used: \TeX\ gathers
material until it has accumulated more than will fit on a page; then it
spews out one page of data, based on what it thinks is the best breakpoint
between pages; then it returns to gather material for the next page in
the same way. Page numbers, headings, and similar things are attached
after each page has been ejected, by a special sequence of \TeX\ commands
called the current {\sl^{output routine}}.
Plain \TeX\ has an output routine that takes care of ordinary jobs. It
handles the simple things that most manuscripts require, and it also
copes with more complicated things like the insertions made with
^|\footnote| and ^|\topinsert|, as described in the dangerous bends of
Chapter~15. We shall begin the present chapter by discussing how to make
simple changes to the behavior of plain \TeX's output routine; then we
shall turn to the details of how to define output routines that do more
complex tasks.
If you run \TeX\ without modifying the ^{plain \TeX\ format}, you get
^^{page format, modifying}
pages that are numbered at the bottom; and each page will be approximately
8$1\over2$~inches wide and 11~inches tall, including 1-inch margins at
all four sides. This format is suitable for preprints of
technical papers, but you might well want to change it, especially if
you are not using \TeX\ to make a preprint of a technical paper.
For example, we saw in the experiments of Chapter 6 that the width of the
material on a page can be changed by giving a different value to the
horizontal line size, ^|\hsize|. Plain \TeX\ format says `|\hsize=6.5in|',
in order to obtain 8.5-inch pages with 1-inch margins; you can change
|\hsize| to whatever you want. Similarly, you can control the vertical
size of a page by changing ^|\vsize|. Plain \TeX\ sets |\vsize=8.9in|
(not |9in|, since |\vsize| doesn't include the space for page numbers at the
bottom of each page); if you say `|\vsize=4in|' you will get shorter pages,
with only 4 inches of copy per sheet. It's best not to monkey with
|\hsize| and |\vsize| except at the very beginning of a job, or after you
have ejected all pages from \TeX's memory.
If you want your output to be positioned differently when it is ultimately
printed, you can offset it by giving nonzero values to ^|\hoffset| and
^|\voffset|. For example,
\begintt
\hoffset=.5in \voffset=1.5in
\endtt
will move the output half an inch to the right of its normal position, and
1.5 inches down. You should be careful not to offset the output so much
that it falls off the edge of the physical medium on which it is being
printed, unless you know that such out-of-bounds activity won't cause trouble.
\TeX\ is often used to typeset announcements, ^{brochures}, or other
documents for which ^{page numbers} are inappropriate. If you say
\begintt
\nopagenumbers
\endtt
at the beginning of your manuscript, plain \TeX\ will refrain from inserting
numbers at the bottom of each page.
\danger In fact, ^|\nopagenumbers| is a special case of a much more general
mechanism by which you can control headings and footings. The plain \TeX\
output routine puts out a special line of text called the {\sl^{headline}\/}
at the top of each page, and another special line of text called the
{\sl^{footline}\/} at the bottom. The headline is normally blank, and
the footline is normally a centered page number, but you can specify any
headline and footline that you want by redefining the control sequences
^|\headline| and ^|\footline|. For example, ^^|\hrulefill|
\begintt
\headline={\hrulefill}
\endtt
will put a horizontal rule at the top of every page. The basic idea is
that plain \TeX\ puts `|\line{\the\headline}|' at the top and
`|\line{\the\footline}|' at the bottom, with blank lines separating these
extra lines from the other material. \ (Recall that ^|\line| is an
abbreviation for `|\hbox to\hsize|'; hence the headline and footline are
put into boxes as wide as the normal lines on the page itself.) \ The
normal value of\/ |\headline| is `|\hfil|', so that no heading is visible.
The |\nopagenumbers| macro described earlier is simply an abbreviation for
`|\footline={\hfil}|'.
\danger The normal value of\/ |\footline| is `|\hss\tenrm\folio\hss|';
this centers the page number on a line, using font ^|\tenrm|, because
^|\folio| is a control sequence that produces the number of the current
page in text form.
\danger The page number appears in \TeX's internal register |\count0|,
as explained in Chapter~15, and plain \TeX\ makes ^|\pageno| an
abbreviation for ^|\count0|. Thus you can say `|\pageno=100|' if you want
the next page of your output to be number~100. The |\folio| macro
converts negative page numbers to ^{roman numerals}; if your manuscript
begins with `|\pageno=-1|', the pages will be numbered i, ii, iii, iv,
v,~etc. In fact, Appendix~B defines |\folio| to be an abbreviation for
^^|\romannumeral|^^|\number|
\begintt
\ifnum\pageno<0 \romannumeral-\pageno \else\number\pageno \fi
\endtt
\danger It is important to include the name of each font explicitly
whenever you are defining a headline or footline, because an output routine
in \TeX\ can come into action at somewhat unpredictable times. For example,
suppose that |\footline| had been set to `|\hss\folio\hss|', without
specifying |\tenrm|; then the page number would be typeset in whatever
font happens to be current when \TeX\ decides to output a page.
Mysterious effects can occur in such cases, because \TeX\ is typically
in the midst of page~101 when it is outputting page~100.
\dangerexercise Explain how to put ^{en-dashes} around the page numbers in a
plain \TeX\ job. For example, `\hbox{ -- 4 -- }' should appear at the bottom
of page~4.
\answer |\footline={\hss\tenrm-- \folio\ --\hss}|
\danger Here is an example of a headline in which the page numbers
appear at the top. Furthermore, odd-numbered and even-numbered pages are
treated differently:
\begintt
\nopagenumbers % suppress footlines
\headline={\ifodd\pageno\rightheadline \else\leftheadline\fi}
\def\rightheadline{\tenrm\hfil RIGHT RUNNING HEAD\hfil\folio}
\def\leftheadline{\tenrm\folio\hfil LEFT RUNNING HEAD\hfil}
\voffset=2\baselineskip
\endtt
English-language books traditionally have ^{odd-numbered pages} on the
right and ^{even-numbered pages} on the left. Text that appears as a
headline on several pages is often called a ``^{running head}.'' When
you use headlines, it is generally wise to set ^|\voffset| to the
equivalent of two lines of text, as shown in this example, so that there
will still be a margin of one inch at the top of your output pages.
\dangerexercise Suppose that you're using \TeX\ to typeset your ^{r\'esum\'e},
which is several pages long. Explain how to define |\headline| so that
the first page is headed by `{\bf R\'ESUM\'E}', centered in boldface type,
while each subsequent page has a headline like this: ^^{Thor}
$$\line{R\'esum\'e of A. U. Thor \dotfill\ Page 2}$$
\answer |\headline={\ifnum\pageno=1 \hss\tenbf R\'ESUM\'E\hss|\parbreak
| \else\tenrm R\'esum\'e of A. U. Thor \dotfill\ Page \folio\fi}|
\smallskip\noindent (You should also say |\nopagenumbers| and
|\voffset=2\baselineskip|.)
\danger If you don't change the |\vsize|, all of the headlines and footlines
will occur in the same place regardless of the contents of the page
between them. Thus, for example, if you are using ^|\raggedbottom| as
explained in Chapter~15, so that pages do not always contain the same
amount of text, the raggedness will occur above the footline; the
footline won't move up. If you do change ^|\vsize|, the footline position
will change correspondingly, while the headline will stay put.
\ddanger The rest of this chapter is intended for people who want an
output format that is substantially different from what plain \TeX\ provides.
Double dangerous bends are used in all of the subsequent paragraphs, because
you should be familiar with the rest of \TeX\ before you plunge into these
final mysteries of the language. Chapter~22 taught you how to be a \TeX\
Master, i.e., a person who can produce complicated tables using |\halign|
and |\valign|; the following material will take you all the way to the
rank of ^{Grandmaster}, i.e., a person who can design output routines.
When you are ready for this rank, you will be pleased to discover
that---like alignments---output routines are not really so mysterious as
they may seem at first.
\ninepoint % it's all \ddangerous from here on
\ddanger Let's begin by recapping some of the rules at the end of Chapter~15.
\TeX\ periodically chooses to output a page of information, by breaking
its main vertical list at what it thinks is the best place, and at such times
it enters internal vertical mode and begins to read the commands in the
current |\output| routine. When the output routine begins, ^|\box255|
contains the page that \TeX\ has completed; the output routine is supposed to
do something with this vbox. When the output routine ends, the list of
items that it has constructed in internal vertical mode is placed just
before the material that follows the page break. In this way \TeX's
page-break decisions can effectively be changed: Some or all of the material
on the broken-off page can be removed and carried forward to the next page.
\ddanger The current ^|\output| routine is defined as a token list
parameter, just like ^|\everypar| or ^|\errhelp|, except that \TeX\
automatically inserts a begin-group symbol~`|{|' at the beginning
and an end-group symbol~`|}|' at the end. These ^{grouping characters}
help to keep the output routine from interfering with what
\TeX\ was doing when the page break was chosen; for example, an output
routine often changes the ^|\baselineskip| when it puts a headline
or footline on a page, and the extra ^{braces} keep this change local.
If no |\output| routine has been specified, or if the user has said
`|\output={}|', \TeX\ supplies its own routine, which is essentially
equivalent to `|\output={\shipout\box255}|'; this outputs the page without
any headline or footline, and without changing the page number.
^^{default output routine}^^|\shipout|
\ddanger \TeX's primitive command |\shipout|\<box> is what actually
causes output. It sends the contents of the box to the |dvi| file,
which is \TeX's main output file; after \TeX\ has finished,
the ^|dvi| file will contain a compact device-independent encoding of
instructions that specify exactly what should be printed. When a
box is shipped out, \TeX\ displays the values of\/ |\count0| through
|\count9| on your terminal, ^^|\count0| as explained in Chapter~15; these
ten counters are also recorded in the |dvi| file, where they can be used
to identify the page. All of the ^|\openout|, ^|\closeout|, and ^|\write|
commands that appear inside of the \<box> are performed in their natural
order as that box is being shipped out. Since a |\write| command
expands macros, as explained in Chapter~21, \TeX's scanning mechanism
might detect syntax errors while a |\shipout| is in progress. If
^|\tracingoutput| is nonzero at the time of a |\shipout|, the contents
of the \<box> being shipped are written into your log file in symbolic
form. You can say |\shipout| anywhere, not only in an output routine.
\ddanger The delayed aspect of\/ |\write| imposes a noteworthy restriction:
It is necessary to be sure that all macros that might appear within the
text of a |\write| are properly defined when a |\shipout| command is
given. For example, the plain \TeX\ format in Appendix~B temporarily makes
spaces active and says `|\global\let|\]|=|^|\space|'; the reason is that
^|\obeyspaces| might be in force during a |\write| command, so a definition
for \] as an active character should exist during the next |\shipout|,
even though \TeX\ might no longer be making ^{spaces active} at that time.
\ddanger Chapter 15 points out that \TeX\ gives special values to
certain internal registers and parameters, in addition to |\box255|,
just before the output routine begins. Insertions are put into their
own vboxes, and ^|\insertpenalties| is set equal to the total number of
heldover insertions; furthermore the ^|\outputpenalty| parameter is
set to the value of the penalty at the current breakpoint. An output
routine can be made to do special things when these quantities have
special values. For example, the output routine of plain \TeX\ recognizes
a ^|\supereject| (which ejects all held-over insertions) by the fact that
|\supereject| causes |\outputpenalty| to be $-20000$, and by using
|\insertpenalties| to decide if any insertions are being held over.
\ddanger The default output routine, `|\shipout\box255|', illustrates one
extreme in which nothing is put into the vertical list that is carried over
to the next page. The other extreme is
\begintt
\output={\unvbox255
\ifnum\outputpenalty<10000 \penalty\outputpenalty\fi}
\endtt
which ships nothing out and puts {\sl everything\/} back onto the main
vertical list. \ (The command `^|\unvbox||255|' takes the completed page
out of its box, and the command `|\penalty\outputpenalty|' reinserts the
penalty at the chosen breakpoint.) \ This makes a seamless join between
the completed page and the subsequent material, because \TeX\ has still
not discarded glue and penalties at the breakpoint when it invokes
an |\output| routine; hence \TeX\ will go back and reconsider the page
break. If the |\vsize| hasn't changed, and if all insertions have been
held in place, the same page break will be found; but it will be found
much faster than before, because the vertical list has already been
constructed---the paragraphing doesn't need to be done again. Of course,
an output routine like this makes \TeX\ spin its wheels endlessly, so
it~is of no use except as an example of an extreme case.
\ddanger To prevent such looping, your output routine should always
make progress of some sort whenever it comes into play. If you make a mistake,
\TeX\ may be able to help you diagnose the error, because a special
loop-detection mechanism has been built in:
There is an internal integer variable called ^|\deadcycles|,
which is cleared to zero after every |\shipout| and increased by~1
just before every |\output|. Thus, |\deadcycles| keeps track of how
many times an output routine has been initiated since the most recent
|\shipout|, unless you change the value of\/ |\deadcycles| yourself.
There's also an integer parameter called |\maxdeadcycles|, which plain
\TeX\ sets to~25. If\/ |\deadcycles| is greater than or equal to
|\maxdeadcycles| when your output routine is about to be started (i.e.,
when |\deadcycles| is about to be increased), \TeX\ issues an error
message and performs the ^{default output routine} instead of yours.
\ddanger When your output routine is finished, |\box255| should be void.
In other words, you must do something with the information in that
box; it should either be shipped out or put into some other place.
Similarly, ^|\box255| should be void when \TeX\ is getting ready to
fill it with a new page of material, just before starting an output
routine. If\/ |\box255| is nonvoid at either of those times, \TeX\ will
complain that you are misusing this special register, and the
register contents will be destroyed.
\ddanger But let's not talk forever about borderline cases and special
parameters; let's look at some real examples. The output routine of
plain \TeX, found in Appendix~B\null, is set up by saying
`|\output={\plainoutput}|', where ^|\plainoutput| is an abbreviation for
\begintt
\shipout\vbox{\makeheadline
\pagebody
\makefootline}
\advancepageno
\ifnum\outputpenalty>-20000 \else\dosupereject\fi
\endtt
Let us consider this ``program'' one line at a time:\enddanger
\medskip
\textindent{1)} The ^|\makeheadline| macro constructs a vbox of height
and depth zero in such a way that the headline is properly positioned
above the rest of the page. Its actual code is ^^|\headline|^^|\nointerlineskip|
\begintt
\vbox to 0pt{\vskip-22.5pt
\line{\vbox to8.5pt{}\the\headline}\vss}
\nointerlineskip
\endtt
The magic constant $-22.5\pt$ is equal to $\bigl(\hbox{topskip}-\hbox{height of
strut}-2(\hbox{baselineskip})\bigr)$, i.e., $10\pt-8.5\pt-24\pt$;
^^{strut}^^|\vss| ^^|\line| this places the reference point of the headline
exactly $24\pt$ above the reference point of the top line on the page,
unless the headline or the top line are excessively large.
\medbreak
\textindent{2)} The ^|\pagebody| macro is an abbreviation for
\begintt
\vbox to\vsize{\boxmaxdepth=\maxdepth \pagecontents}
\endtt
The value of\/ ^|\boxmaxdepth| is set to ^|\maxdepth| so that the vbox
will be constructed under the assumptions that \TeX's page builder has
used to set up |\box255|.
\medbreak
\textindent{3)} The ^|\pagecontents| macro produces a vertical list for
everything that belongs on the main body of the page, namely the contents
of\/ |\box255| together with illustrations (inserted at the top)
and footnotes (inserted at the bottom): ^^|\topins| ^^|\footnote|
\begintt
\ifvoid\topins \else\unvbox\topins\fi
\dimen0=\dp255 \unvbox255
\ifvoid\footins\else % footnote info is present
\vskip\skip\footins
\footnoterule
\unvbox\footins\fi
\ifraggedbottom \kern-\dimen0 \vfil \fi
\endtt
Here ^|\topins| and ^|\footins| are the insertion class numbers for
the two kinds of insertions used in plain \TeX; if more classes of
^{insertions} are added, |\pagecontents| should be changed accordingly.
Notice that the boxes are unboxed so that the glue coming from
insertions can help out the glue on the main page. The ^|\footnoterule|
macro in Appendix~B places a dividing line between the page and its
footnotes; it makes a net contribution of $0\pt$ to the height of
the vertical list. ^{Ragged-bottom setting} is achieved by inserting ^^|\vfil|
^{infinite glue}, which overpowers the stretchability of\/ ^|\topskip|.
\medbreak
\textindent{4)} The ^|\makefootline| macro puts ^|\footline| into
its proper position:
\begintt
\baselineskip=24pt
\line{\the\footline}
\endtt
\medbreak
\textindent{5)} The ^|\advancepageno| macro normally advances ^|\pageno|
by~$+1$; but if\/ |\pageno| is negative (for roman numerals), the advance is
by~$-1$. The new value of\/ |\pageno| will be appropriate for the next time
the output routine is called into action. ^^|\global|^^|\advance|
\begintt
\ifnum\pageno<0 \global\advance\pageno by-1
\else \global\advance\pageno by 1 \fi
\endtt
\medbreak
\textindent{6)} Finally, the ^|\dosupereject| macro is designed to clear
out any insertions that have been held over, whether they are illustrations
or footnotes or both: ^^|\insertpenalties| ^^|\supereject|
\begintt
\ifnum\insertpenalties>0
\line{} \kern-\topskip \nobreak
\vfill\supereject\fi
\endtt
The mysterious negative ^|\kern| here cancels out the natural space of the
^|\topskip| glue that goes above the empty |\line|; that empty line box
prevents the ^|\vfill| from disappearing into a page break. The vertical
list that results from |\dosupereject| is placed on \TeX's list of things
to put out next, just after the straggling insertions have been
reconsidered as explained in Chapter~15. Hence another super-eject will
occur, and the process will continue until no insertions remain.
\ddangerexercise Explain how to change the output routine of plain \TeX\
so that it will produce twice as many pages. The material that would
ordinarily go on pages 1,~2, 3,~etc., should go onto pages 1,~3, 5,~\dots;
and the even-numbered pages should be entirely blank except for the
headline and footline. \ (Imagine that photographs will be mounted
on those blank pages later.)
\answer |\output={\plainoutput\blankpageoutput}|\parbreak
|\def\blankpageoutput{\shipout\vbox{\makeheadline|\parbreak
| \vbox to\vsize{}\makefootline}\advancepageno}|
\ddanger Suppose now that ^{double-column} format is desired.
More precisely, let's attempt to modify plain \TeX\ so that it sets type
in columns whose width is ^|\hsize||=3.2in|. Each actual page of
output should contain two such columns separated by $0.1\rm\,in$ of
space; thus the text area of each page will still be 6.5~inches wide.
The headlines and footlines should span both columns, but the columns
themselves should contain independent insertions as if they were the
facing pages of a book. In other words, each column should contain
its own footnotes and its own illustrations; we do not have to change
the ^|\pagebody| macro. ^^{two-column format} ^^{multicolumn format}
\ddanger In order to solve this problem, let us first introduce a
new dimension register called ^|\fullhsize| that represents the
width of an entire page.
\begintt
\newdimen\fullhsize
\fullhsize=6.5in \hsize=3.2in
\def\fullline{\hbox to\fullhsize}
\endtt
The ^|\makeheadline| and ^|\makefootline| macros should be modified
so that they use `^|\fullline|' instead of `^|\line|'.
\ddanger The new output routine will make use of a control sequence
|\lr| that is set to either `|L|' or `|R|', according as the
next column belongs at the left or at the right of the next page.
When a left column has been completed, the output routine simply
saves it in a box register; when a right column has been completed,
the routine outputs both columns and increases the page number.
^^|\advancepageno|
\begintt
\let\lr=L \newbox\leftcolumn
\output={\if L\lr
\global\setbox\leftcolumn=\columnbox \global\let\lr=R
\else \doubleformat \global\let\lr=L\fi
\ifnum\outputpenalty>-20000 \else\dosupereject\fi}
\def\doubleformat{\shipout\vbox{\makeheadline
\fullline{\box\leftcolumn\hfil\columnbox}
\makefootline}
\advancepageno}
\def\columnbox{\leftline{\pagebody}}
\endtt
The |\columnbox| macro uses |\leftline| in order to ensure that it produces a
box whose width is |\hsize|. The width of\/ |\box255| is usually, but not
always, equal to |\hsize| at the beginning of an output routine; any other
width would louse up the format.
\ddanger When double-column setting ends, there's a 50-50 chance that the
final column has fallen at the left, so it will not yet have been output.
The code
\begintt
\supereject
\if R\lr \null\vfill\eject\fi
\endtt
supplies an empty right-hand column in this case, ensuring that all
of the accumulated material will be printed. It's possible to do
fancier column balancing on the last page, but the details are tricky
if footnotes and other insertions need to be accommodated as well.
Appendix~E includes the macros that were used to balance the columns at the
end of the index in Appendix~I\null, and to start two-column format in mid-page.
\ddangerexercise How should the example above be modified if you want
^{three-column output}?
\answer Set |\hsize=2.1in|, allocate `|\newbox\midcolumn|', and use the
following code:
\begintt
\output={\if L\lr
\global\setbox\leftcolumn=\columnbox \global\let\lr=M
\else\if M\lr
\global\setbox\midcolumn=\columnbox \global\let\lr=R
\else \tripleformat \global\let\lr=L\fi\fi
\ifnum\outputpenalty>-20000 \else\dosupereject\fi}
\def\tripleformat{\shipout\vbox{\makeheadline
\fullline{\box\leftcolumn\hfil\box\midcolumn\hfil\columnbox}
\makefootline}
\advancepageno}
\endtt
At the end, |\supereject| and say `|\if L\lr \else\null\vfill\eject\fi|'
twice.
\ddanger Since \TeX's output routine lags behind its page-construction
activity, you can get erroneous results if you change the |\headline| or the
|\footline| in an uncontrolled way. For example, suppose that you are
typesetting a book, and that the format you are using allows chapters
to start in the middle of a page; then it would be a mistake to change
the ^{running headline} at the moment you begin a new chapter, since the
next actual page of output might not yet include anything from the new
chapter. Consider also the task of typesetting a dictionary or a
membership roster; a well-designed reference book displays the current range
of entries at the top of each page or pair of pages, so that it is
easy for readers to thumb through the book when they are searching for
isolated words or names. But \TeX's asynchronous output mechanism
makes it difficult, if not impossible, to determine just what range
of entries is actually present on a page.
\ddanger Therefore \TeX\ provides a way to put ``^{marks}'' into a list;
these marks inform the output routine about the range of information on
each page. The general idea is that you can say ^^|\mark|
\begindisplay
|\mark{|\<mark text>|}|
\enddisplay
in the midst of the information you are typesetting, where the
\<mark text> is a token list that is expanded as in the commands
|\edef|, |\message|, etc. \TeX\ puts an internal representation of
the mark text into the list it is building; then later on, when a
completed page is packed into |\box255|, \TeX\ allows the output
routine to refer to the first and last mark texts on that page.
\ddanger The best way to think of this is probably to imagine that
\TeX\ generates an arbitrarily long vertical list of boxes, glue,
and other items such as penalties and marks. Somehow that long vertical
list gets divided up into pages, and the pages are made available to
the output routine, one at a time. Whenever a page is put in |\box255|,
\TeX\ sets up the value of three quantities that act essentially
like macros:\enddanger
\smallskip
\item\bull ^|\botmark| is the mark text most recently encountered on the
page that was just boxed;
\item\bull ^|\firstmark| is the mark text that was first encountered on
the page that was just boxed;
\item\bull ^|\topmark| has the value that |\botmark| had just before the
current page was boxed.
\smallbreak\noindent
Before the first page, all three of these are null, i.e., they expand
to nothing. When there is no mark on a page, all three are equal to
the previous |\botmark|.
\ddanger For example, suppose that your manuscript includes exactly four
marks, and that the pages are broken in such a way that |\mark{|$\alpha$|}|
happens to fall on page~2, |\mark{|$\beta$|}| and |\mark{|$\gamma$|}|
on page~4, and |\mark{|$\delta$|}| on page~5. Then
$$\halign{\indent\hfil#\hfil&&\qquad\hfil#\hfil\cr
On page&|\topmark| is&|\firstmark| is&|\botmark| is\cr
\noalign{\vskip2pt}
1&null&null&null\cr
2&null&$\alpha$&$\alpha$\cr
3&$\alpha$&$\alpha$&$\alpha$\cr
4&$\alpha$&$\beta$&$\gamma$\cr
5&$\gamma$&$\delta$&$\delta$\cr
6&$\delta$&$\delta$&$\delta$\cr}$$
\ddanger When you use a |\mark| command in vertical mode, \TeX\ puts a
mark into the main vertical list. When you use a |\mark| command in
horizontal mode, \TeX\ treats it as vertical mode material like
^|\vadjust| and ^|\insert|; i.e., after the paragraph has been broken
into lines, each mark will go into the main vertical list just after the
box for the line where that mark originally appeared. If
you use |\mark| in restricted horizontal mode, the mark may migrate out to
the enclosing vertical list in the same way that |\insert| and |\vadjust|
items do (see Chapter~24); but a mark that is locked too deeply inside
a box will not ^{migrate}, so it will never appear as a |\firstmark|
or |\botmark|. Similarly, a |\mark| that occurs in internal vertical mode
goes into a vbox, and it is not accessible in the main vertical list.
\ddanger Chapter 15 discusses the ^|\vsplit| command, which allows you to
break up vertical lists by yourself. This operation sometimes provides
a useful alternative to \TeX's ordinary page-building mechanism.
For example, if you simply want to typeset some material in two columns
of equal height, you can put that material into a vbox, then |\vsplit|
the box into two pieces; no output routine is needed at all. The
|\vsplit| operation sets up the values of two macro-like quantities
that were not mentioned in Chapter~15: ^|\splitfirstmark| and
^|\splitbotmark| expand to the mark texts of the first and last
marks that appear in the vertical list that was split off by the most
recent |\vsplit| command. Both quantities are null if there were no
such marks. The values of\/ |\topmark|, |\firstmark|, |\botmark|,
|\splitfirstmark|, and |\splitbotmark| are global; i.e., they are
not affected by \TeX's ^{grouping} mechanism.
\ddanger Most dictionaries use the equivalent of\/ |\firstmark| and |\botmark|
to give ^{guide words} at the top of each pair of facing pages. For example,
if the definition of the word `type' starts on page~1387 and continues
onto page~1388, the guide word on page~1387 (a right-hand page)
will be `type'; but the guide word at the top of page~1388 (a left-hand page)
will be the next word in the dictionary (e.g., `typecast') even though
the top of page~1388 is about `type'.
\ddanger The dictionary scheme works fine for dictionaries, since a reader
should start reading each dictionary entry at its beginning. But a different
scheme is appropriate for a technical book like the author's {\sl^{Art
of Computer Programming}}, ^^{Knuth} where Section~1.2.8 (for example)
starts in the middle of page~78, but the top of page~78 contains
exercises 19--24 of Section~1.2.7. The headline at the top of page~78
refers to `1.2.7', because that will help somebody who is searching
for exercise 1.2.7--22. Notice that the dictionary convention would put
`1.2.8' at the top of page~78, but that would be appropriate only if
Section~1.2.8 had begun exactly at the top of that page.
\ddanger Continuing this example from {\sl The Art of Computer Programming},
let's suppose that the \TeX\ manuscript for Section~1.2.8 begins with
a macro call like
\begintt
\beginsection 1.2.8. Fibonacci Numbers.
\endtt
How should |\beginsection| be defined? Here is one attempt:
\begintt
\def\beginsection #1. #2.
{\sectionbreak
\leftline{\sectionfont #1. #2}
\mark{#1}
\nobreak\smallskip\noindent}
\endtt
The |\sectionbreak| macro should encourage \TeX\ either to break the page at the
current position, or to leave a goodly amount of blank space; e.g.,
|\sectionbreak| might be an abbreviation for `|\penalty-200| |\vskip18pt|
|plus4pt| |minus6pt|'. The |\beginsection| macro ends with commands that
suppress indentation of the first paragraph in the section. But the thing
that concerns us with respect to output routines is the |\mark| command
that follows |\leftline|. In the example we have been considering, the
beginning of Section~1.2.8 would insert `|\mark{1.2.8}|' into the main
vertical list just after the box containing the title of that section.
\ddanger Is such a |\mark| adequate? Unfortunately, no, not even if we
assume for simplicity that at most one section begins on each page.
The page that contains the beginning of Section~1.2.8 will then have
|\topmark=1.2.7| and |\firstmark=1.2.8|, regardless of whether or not the
section starts at the very top of the page. What we want in this application
is a cross between |\topmark| and |\firstmark|: something that will reflect
the mark text that represents the state of affairs just after the first
line of the page. And \TeX\ doesn't provide that.
\ddanger The solution is to emit the |\mark| just before the |\sectionbreak|,
instead of just after the |\leftline|. Then |\topmark| will always reflect
the truth about the section that is current at the top line. \ (Think about it.)
\ddanger However, the format for {\sl The Art of Computer Programming\/}
is more complex than this. On left-hand pages, the section number in the
headline is supposed to reflect the situation at the top of the page,
as we have just discussed, but on right-hand pages it is supposed to refer
to the bottom of the page. Our solution to the previous problem made
|\topmark| correct for the top, but it can make |\botmark| incorrect
at the bottom. In order to satisfy both requirements, it is necessary
to pack more information into the marks. Here's one way to solve the problem:
\begintt
\def\beginsection #1. #2.
{\mark{\currentsection \noexpand\else #1}
\sectionbreak
\leftline{\sectionfont #1. #2}
\mark{#1\noexpand\else #1} \def\currentsection{#1}
\nobreak\smallskip\noindent}
\def\currentsection{} % the current section number
\endtt
The idea is to introduce two marks, one just before the section break
and one just after the section has begun. Furthermore each mark has two
parts; the mark just before the potential break between Sections 1.2.7
and~1.2.8 is `|1.2.7\else 1.2.8|', while the one just after that potential
break is `|1.2.8\else 1.2.8|'. It follows that the section number corresponding
to the bottom of a page is the left component of\/ |\botmark|; the section
number corresponding to the top of a page is the right component of\/
|\topmark|. The |\rightheadline| macro can make use of
`^|\iftrue||\botmark\fi|' to read the left component, and the
|\leftheadline| macro can say `^|\expandafter|^|\iffalse||\topmark\fi|' to
read the right component.
\ddangerexercise B. C. ^{Dull} used a construction very much like the one above,
but he put the second |\mark| just before the |\leftline| instead of
just after it. What went wrong?
\answer He forgot that ^{interline glue} is inserted automatically before
the |\leftline|; this permits a legal breakpoint between the |\mark| and
the |\leftline| box, according to the rules of page breaking in Chapter~15.
One cure would be to say ^|\nobreak| just after the |\mark|; but it's
usually best to put marks and ^{insertions} just {\sl after\/} boxes.
\ddangerexercise The marks in the previous construction have the form
`$\alpha$|\else|$\,\beta$', where $\alpha$ and $\beta$ are two independent
pieces of information. The `|\else|' makes it possible to select
either $\alpha$ or $\beta$ by means of\/ ^|\iftrue| and ^|\iffalse|.
Generalize this idea: Suppose that you have an application in which
marks are supposed to carry five independent pieces of information,
and that each mark has the form `$\alpha_0$|\or|$\,\alpha_1$|\or|%
$\,\alpha_2$|\or|$\,\alpha_3$|\or|$\,\alpha_4$'. Explain how to select
any one of the five $\alpha$'s from such a mark.
\answer Say, for example, |\ifcase2\expandafter\relax\botmark\fi| to
read part $\alpha_2$ of\/ |\botmark|. Another solution puts the five
components into five parameters of a macro, analogous to the method
used by |\inxcheck| later in this chapter; but the |\ifcase| approach
is usually more efficient, because it lets \TeX\ pass over the unselected
components at high speed.
\ddanger Let's conclude our discussion of output routines by considering an
application to indexes, such as the index to this manual that appears
^^{index marks} ^^{index example}
in Appendix~I\null. The most complicated entries in such an index will
look something like this:
\begindisplay
Main entry, 4, 6, 8--10, 12, 14--16,\cr
\qquad 18--22, 24--28, 30.\cr
\quad first subsidiary entry, 1--3, 6, 10--11,\cr
\qquad 15, 21, 24, 28.\cr
\quad second subsidiary entry, 1, 3, 6--7,\cr
\qquad 10, 15, 21, 25, 28, 31.\cr
\enddisplay
Main entries and subsidiary entries are typeset ^{ragged-right},
with two ems of hanging indentation after the first line; subsidiary entries
are indented one em on the first line. Our goal will be to typeset
such material from input that looks like this:
\begintt
\beginindex
...
Main entry, 4, 6, 8--10, 12, 14--16, 18--22, 24--28, 30.
\sub first subsidiary entry, 1--3, 6, 10--11, 15, 21, 24, 28.
\sub second subsidiary entry, 1, 3, 6--7, 10, 15, 21, 25, %
28, 31.
...
\endindex
\endtt
where `|...|'\ stands for other entries. Each line of input normally specifies
one main entry or one subsidiary entry; if an entry is so long that it doesn't
fit on a single input line, `\]|%|' is typed at the end of the line so
that it merges with the following one.
\ddanger The interesting thing about this index problem is that it is
desirable to set up a system of marks so that the output routine can insert
special lines of text when an entry has been broken between columns or
pages. For example, if a page break occurs between any of the six lines
of typeset output shown above, the output routine should emit the special
line
\begindisplay
Main entry ({\it continued}\thinspace):
\enddisplay
and if a page break occurs within a subsidiary entry, an additional special
line
\begindisplay
\quad subsidiary entry ({\it continued}\thinspace):
\enddisplay
should also appear. The solution below produces marks so that |\botmark|
will be null if a break occurs between main entries; it will be
`|Main entry|' if a break occurs after lines 1, 2, or~4 of the six
example output lines; it will be `|Main entry\sub first subsidiary entry|'
if a break occurs after line~3 (within the first subsidiary entry); and it will
be `|Main entry\sub second subsidiary entry|' if a break occurs after line~5.
\ddanger The reader may wish to try solving this problem before looking at
the solution, because it will then be easier to appreciate the subtler
issues that are involved. \ (Go ahead: Try to define a macro |\beginindex|
that does the ragged-right setting and produces the specified marks. Turn
back to the previous page to study the problem carefully, before peeking
at the answer.)
^^|\everypar| ^^|\futurelet| ^^|\exhyphenpenalty| ^^|\raggedright|
^^|\hangindent|
\begintt
\def\beginindex{\begingroup
\parindent=1em \maxdepth=\maxdimen
\def\par{\endgraf \futurelet\next\inxentry}
\obeylines \everypar={\hangindent 2\parindent}
\exhyphenpenalty=10000 \raggedright}
\def\inxentry{\ifx\next\sub \let\next=\subentry
\else\ifx\next\endindex \let\next=\vfill
\else\let\next=\mainentry \fi\fi \next}
\def\endindex{\mark{}\break\endgroup}
\let\sub=\indent \newtoks\maintoks \newtoks\subtoks
\def\mainentry#1,{\mark{}\noindent
\maintoks={#1}\mark{\the\maintoks}#1,}
\def\subentry\sub#1,{\mark{\the\maintoks}\indent
\subtoks={#1}\mark{\the\maintoks\sub\the\subtoks}#1,}
\endtt
Even if you have read this solution, you probably want an explanation of
what it does, because it uses ``\TeX tics'' that have not appeared before
in this manual.\enddanger
\smallskip
\textindent{1)} The |\beginindex| macro uses ^|\begingroup| to keep other
changes local; thus, it won't be necessary to restore ^|\parindent| and
^|\maxdepth|, etc., to their former values when the index is finished.
The |\maxdepth| parameter is set to ^|\maxdimen|, which is essentially
infinite, so that |\box255| will have the true depth of the last box that
it contains; we will use this fact below. \ (It is safe to disable
|\maxdepth| in this way, since the entries in an index can be assumed
to have reasonably small depth.) \ Notice that ^|\obeylines| is used, so
that ^|\par| will effectively be inserted at the end of every line of
input. The meaning of\/ |\par| is changed so that it does more than usual:
First it does ^|\endgraf|, which is \TeX's ordinary |\par| operation;
then it sets |\next| to the first token of the next line, after which the
macro |\inxentry| will be expanded.
\smallskip
\textindent{2)} When |\inxentry| comes into play it looks at |\next|
to decide what to do. There are three cases: If\/ |\next| is `|\sub|',
the line will be treated as a subsidiary entry; if\/ |\next| is `|\endindex|',
the next commands executed will be `|\vfill|\allowbreak
|\mark{}|\allowbreak|\break|\allowbreak|\endgroup|';
otherwise the line will be treated as a main entry.
\smallskip
\textindent{3)} The text of a main entry is put into parameter |#1| of
|\mainentry|; this parameter is delimited by a comma. The first thing that
|\mainentry| does is `|\mark{}|', which clears the mark in case of a
break between entries. Then comes `^|\noindent|', which causes \TeX\ to
go into horizontal mode and to emit ^|\parskip| glue. \ (The |\parskip|
glue will be a legal breakpoint between lines; it will later be followed
by interline glue, when the first line of the main entry has been typeset
by \TeX's paragraphing routine.) \ Then another |\mark| is put into
the paragraph itself; this one contains the text of the main entry,
and a ^|\toks| register called ^|\maintoks| is used to ^{inhibit
expansion} of the mark text. When the paragraph is completed and broken
into lines, this particular mark will immediately follow the box for the
paragraph's first line, so it will be the |\botmark| if a page break
occurs anywhere within the paragraph.
\smallskip
\textindent{4)} A similar construction is used for |\subentry|, but the
mark is more complicated. The |\maintoks| register will still contain the
main entry. The text for the subsidiary entry is added using another
token list register, |\subtoks|. Since |\sub| has been defined to equal
^|\indent|, it will not be expanded in this |\mark|.
\ddanger The macros just defined will typeset entries that contain the
necessary marks; now we must construct an output routine that uses
these marks in the desired way, to insert new lines that say `({\it
continued}\thinspace)' as mentioned above. Again, the reader is advised to
try solving this problem before looking at the following solution.
\begintt
\output={\dimen0=\dp255 \normaloutput
\expandafter\inxcheck\botmark\sub\end}
\def\inxcheck#1\sub#2\end{\def\next{#1}%
\ifx\next\empty % do nothing if \botmark is null
\else\noindent #1\continued % `Main entry (continued):'
\def\next{#2}%
\ifx\next\empty % nothing more if \botmark has no \sub
\else\let\sub=\continued \indent #2\fi
\advance\dimen0 by-\prevdepth \kern\dimen0 \fi}
\def\continued{ ({\it continued}\thinspace):\endgraf}
\endtt
This coding is a bit more subtle than usual. It assumes that |\normaloutput|
takes care of shipping out |\box255| (possibly putting it into multicolumn
format) and advancing the page number; then comes new stuff, which is
performed by |\inxcheck|. The |\inxcheck| macro is invoked in an interesting
way that allows |\botmark| to be separated into its components. If\/ |\botmark|
^^{macro arguments}
is null, argument~|#1| to |\inxcheck| will be null; hence |\next| will be
found equivalent to ^|\empty|. \ (Plain \TeX\ says `|\def\empty{}|' in
order to accommodate situations like this.) \ If\/ |\botmark| doesn't contain
the token~|\sub|, argument~|#1| will be the contents of\/ |\botmark| while
|#2|~will be null. Otherwise, if\/ |\botmark| has the form $\alpha$|\sub|$\,
\beta$, argument~|#1| will be $\alpha$ and |#2|~will be `$\beta$|\sub|'.
\ddanger If\/ |\botmark| isn't null, the |\inxcheck| macro produces one or
more lines of text that will be contributed to \TeX's main vertical list
at the position of the page break. And here's where the most subtle point
arises: There will be ^{interline glue} at the page break, computed on
the basis of the depth of the box that preceded the break. That depth
is known to the output routine, since it's the depth of\/ |\box255|. \ (The
value of\/ ^|\maxdepth| was made infinite for precisely this reason.) \
Therefore the |\inxcheck| macro can insert a ^|\kern| to compensate for
the difference in depth between the old box and the one that will be
inserted before the interline glue that has already been computed. Without
this |\kern|, the spacing would be wrong. The reader should study this
example carefully, to understand the reasoning behind the |\kern| command,
before designing an output routine that inserts new boxes between
random lines of output.
\ddangerexercise Modify this construction so that continuation lines are
inserted only in the left columns of even-numbered pages, assuming
two-column format.
\answer |\output={\dimen0=\dp255 \normaloutput|\parbreak
| \ifodd\pageno\else\if L\lr|\parbreak
| \expandafter\inxcheck\botmark\sub\end\fi\fi}|
\smallskip\noindent In this case the |\normaloutput| macro should be the
two-column output routine that was described earlier in this chapter, beginning
with `|\if L\lr|' and ending with `|\let\lr=L\fi|'. \ (There is no need
to test for |\supereject|.)
\ddangerexercise True or false: The |\inxcheck| macro in this
example contributes at most two lines of output to the main vertical list.
\answer False. If the text of the main and/or subsidiary entry is lengthy,
a continuation line may actually become two or more lines. \ (Incidentally,
hanging indentation will then occur, because the |\everypar| command---which
was set up outside the |\output| routine---is effective inside.) \ The
|\vsize| must be large enough to accommodate all continuation lines plus
at least one more line of index material, or else infinite looping will occur.
\ddanger When \TeX\ sees an ^|\end| command, it terminates the job only
if the main vertical list has been entirely output and if\/ ^|\deadcycles||=0|.
Otherwise it inserts the equivalent of
\begintt
\line{} \vfill \penalty-'10000000000
\endtt
into the main vertical list, and prepares to read the `|\end|' token again.
This has the effect of invoking the output routine repeatedly until
everything has been shipped out. In particular, the last column of two-column
format will not be lost.
\ddanger It is possible to devise output routines that always leave a residue
on the main vertical list, yet they never allow |\deadcycles| to
increase. In such a case \TeX\ will never come to an end! An output
routine can recognize that it is being invoked by \TeX's endgame, because
of the highly negative |\outputpenalty| caused by the special
^|\penalty-'10000000000|. At such times the output routine should modify
its behavior, if necessary, so that a happy ending will ensue.
\endchapter
I think you will like them,
when you shall see them on a beautiful quarto page,
where a neat rivulet of text
shall meander through a meadow of margin.
'Fore Gad they will be the most elegant things of their kind!
\author RICHARD BRINSLEY ^{SHERIDAN}, %
{\sl The School for Scandal\/} (1777) % Act I Sc 1
\bigskip
The influence of technical changes upon outputs
through variation in the general investment level {\cmman\char'14}
is so small that actually it could have been neglected.
\author WASSILY W. ^{LEONTIEF}\kern-1pt, %
{\sl The Structure of American Economy, 1919--1929\/} (1941)
\eject
\beginchapter Chapter 24. Summary of\\Vertical\\Mode
^^{vertical mode}
The whole \TeX\ language has been presented in the previous chapters;
we have finally reached the end of our journey into previously
uncharted territory. Hurray! Victory! Now it is time to take a more
systematic look at what we have encountered: to consider the facts in an
orderly manner, rather than to mix them up with informal examples and
applications as we have been doing. A child learns to speak a language
before learning formal rules of grammar, but the rules of grammar come
in handy later on when the child reaches adulthood. The purpose of
this chapter---and of the two chapters that follow---is to present a
precise and concise summary of the language that \TeX\ understands, so
that mature users will be able to communicate as effectively as possible
with the machine.
We will be concerned in these chapters solely with \TeX's {\sl^{primitive}\/}
operations, rather than with the higher-level features of plain \TeX\ format
that most people deal with. Therefore novice users should put off reading
Chapters 24--26 until they feel a need to know what goes on inside the
computer. Appendix~B contains a summary of plain \TeX, together with a
ready-reference guide to the things that most people want to know about
\TeX\ usage. The best way to get an overview of \TeX\ from a high level
is to turn to the opening pages of Appendix~B.
\medskip\ninepoint
Our purpose here, however, is to survey the low-level parts of \TeX\ on
which higher-level superstructures have been built, in order to provide
a detailed reference for people who do need to know the details.
The remainder of this chapter is set in small type, like that of the
present paragraph, since it is analogous to material that is marked
``doubly dangerous'' in other chapters. Instead of using dangerous bend
signs repeatedly, let us simply agree that Chapters 24--26 are dangerous
by definition.
\medbreak
\TeX\ actually has a few features that didn't seem to be worth mentioning
in previous chapters, so they will be introduced here as part of our
complete survey. If there is any disagreement between something that was
said previously and something that will be said below, the facts in the
present chapter and its successors should be regarded as better
approximations to the ^{truth}.
\medbreak
We shall study \TeX's digestive processes, i.e., what \TeX\ does with
the lists of tokens that arrive in its ``stomach.'' Chapter~7 has
described the process by which input files are converted to lists of
tokens in \TeX's ``mouth,'' and Chapter~20 explained how expandable tokens
^^{anatomy of TeX}
are converted to unexpandable ones in \TeX's ``gullet'' by a process
similar to regurgitation. When unexpandable tokens finally reach \TeX's
gastro-intestinal tract, the real activity of typesetting begins, and
that is what we are going to survey in these summary chapters.
Each token that arrives in \TeX's tummy is considered to be a {\sl^{command}\/}
that the computer will obey. For example, the letter `{\tt L}' is a command
to typeset an `L' in the current font; `|\par|' tells \TeX\ to
finish a paragraph. \TeX\ is always in one of six modes, as
described in Chapter~13, and a command sometimes means different things
in different modes. The present chapter is about vertical mode (and
internal vertical mode, which is almost the same): We shall
discuss \TeX's response to every primitive command, when that command occurs
in vertical mode. Chapters 25 and~26 characterize horizontal mode and math mode
in a similar way, but those chapters are shorter than this one because
many commands have the same behavior in all modes; the rules for such
commands will not be repeated thrice, they will appear only once.
Some commands have ^{arguments}. In other words, one or more of the tokens
that follow a command might be used to modify that command's behavior,
and those tokens are not considered to be commands themselves. For example,
when \TeX\ processes the sequence of tokens that corresponds to
`|\dimen2=2.5pt|', it considers only the first token `|\dimen|' to be a
command; the next tokens are swept up as part of the operation, because \TeX\
needs to know what |\dimen| register is to be set equal to what \<dimen> value.
We shall define \TeX's parts of speech by using a modified form of the
grammatical notation that was introduced about 1960 by John ^{Backus} and
Peter ^{Naur} for the definition of computer languages. Quantities in
^{angle brackets} will either be explained in words or they will be defined
by {\sl^{syntax rules}\/} that show exactly how they are formed from other
quantities. For example,
\beginsyntax
<unit of measure>\is<optional spaces><internal unit>
\alt<optional {\tt true}><physical unit>
\endsyntax
defines a \<unit of measure> to be either an occurrence of
\<optional spaces> followed by an \<internal unit>, or \<optional {\tt true}>
followed by \<physical unit>. The symbol `\is' in a syntax rule
means ``is defined to be,'' and `\alt' means ``or.''
Sometimes a syntax rule is ^{recursive}, in the sense that the right-hand
side of the definition involves the quantity being defined. For example,
the rule
\beginsyntax
<optional spaces>\is<empty>\alt<space token><optional spaces>
\endsyntax
defines the grammatical quantity called \<optional spaces> to be either
\<empty>, or a \<space token> followed by \<optional spaces>. The quantity
^\<empty> stands for ``nothing,'' i.e., for no tokens at all; hence the
syntax rule just given is a formalized way of saying that \<optional
spaces> stands for a sequence of zero or more spaces.
The alternatives on the right-hand side of a syntax rule need not consist
entirely of quantities in angle brackets. Explicit tokens can be used
as well. For example, the rule
\beginsyntax
<plus or minus>\is|+|$_{12}$\alt|-|$_{12}$
\endsyntax
says that \<plus or minus> stands for a ^{character token} that is either
a plus sign or a~minus sign, with category code~12.
We shall use a special convention for ^{keywords}, since the actual syntax
of a keyword is somewhat technical. Letters in typewriter type like
`\[pt]' will stand for
\begindisplay
\<optional spaces>\<p or P>\<t or T>,
\enddisplay
where \<p or P> denotes any non-active character token for either |p| or~|P|
(independent of the category code), and where \<t or T> is similar.
When a control sequence like `|\dimen|' is used in the syntax rules below,
it stands for any token whose current meaning is the same as the meaning
that |\dimen| had when \TeX\ started up. Other tokens can be given this
same meaning, using |\let| or |\futurelet|, and the meaning of the
control sequence |\dimen| itself may be redefined by the user, but
the syntax rules take no note of this; they just use `|\dimen|'
as a way of referring to a particular primitive command of \TeX. \ (This
notation is to be distinguished from `\cstok{dimen}', which stands
for the control sequence token whose actual name is |dimen|; see
Chapter~7.) ^^{boxed words}
Control sequences sometimes masquerade as characters, if their meaning has
been assigned by |\let| or |\futurelet|. For example, Appendix~B says
\begintt
\let\bgroup={ \let\egroup=}
\endtt
and these commands make ^|\bgroup| and ^|\egroup| act somewhat like left
and right ^{curly} ^{braces}. Such control sequences are called ``^{implicit
characters}''; they are interpreted in the same way as characters, when
\TeX\ acts on them as commands, but not always when they appear in
arguments to commands. For example, the command `|\let\plus=+|' does not
make |\plus| an acceptable substitute for the character token `|+|$_{12}$'
in the syntax rule for \<plus or minus> given above, nor does the command
`|\let\p=p|' make |\p| acceptable as part of the keyword \[pt]. When
\TeX's syntax allows both explicit and implicit characters, the rules
below will be careful to say so, explicitly.
The quantity ^\<space token>, which was used in the syntax of \<optional
spaces> above, stands for an explicit or implicit space. In other words,
it denotes either a character token of category~10, or a control sequence
or active character whose current meaning has been made equal to such a
token by |\let| or |\futurelet|.
It will be convenient to use the symbols `|{|', `|}|', and `|$|' to stand
for any explicit or implicit character tokens of the respective categories
1, 2, and~3, whether or not the actual character codes are braces or dollar
signs. Thus, for example, plain~\TeX's |\bgroup| is an example of a `|{|',
and so are the tokens `|{|$_1$' and `|(|$_1$'; but `|{|$_{12}$' is not.
The last few paragraphs can be summarized by saying that the alternatives
on the right-hand sides of \TeX's formal syntax rules are made from one
or more of the following things: (1)~syntactic quantities like
\<optional spaces>; (2)~explicit character tokens like |+|$_{12}$;
(3)~keywords like \[pt]; (4)~control sequence names like |\dimen|;
or (5)~the special symbols |{|, |}|, |$|.
\medbreak
Let us begin our study of \TeX's syntax by discussing the precise meanings
of quantities like \<number>, \<dimen>, and \<glue> that occur frequently
as arguments to commands. The most important of these is \<number>,
which specifies an integer value. Here's exactly what a \<number> is:
\beginsyntax
<number>\is<optional signs><unsigned number>
<optional signs>\is<optional spaces>
\alt<optional signs><plus or minus><optional spaces>
<unsigned number>\is<normal integer>\alt<coerced integer>
<normal integer>\is<internal integer>
\alt<integer constant><one optional space>
\alt|'|$_{12}$<octal constant><one optional space>
\alt|"|$_{12}$<hexadecimal constant><one optional space>
\alt|`|$_{12}$<character token><one optional space>
<integer constant>\is<digit>\alt<digit><integer constant>
<octal constant>\is<octal digit>\alt<octal digit><octal constant>
<hexadecimal constant>\is<hex digit>\alt<hex digit><hexadecimal constant>
<octal digit>\is|0|$_{12}$\alt|1|$_{12}$\alt|2|$_{12}$\alt|3|$_{12}$\alt%
|4|$_{12}$\alt|5|$_{12}$\alt|6|$_{12}$\alt|7|$_{12}$
<digit>\is<octal digit>\alt|8|$_{12}$\alt|9|$_{12}$
<hex digit>\is<digit>\alt|A|$_{11}$\alt|B|$_{11}$\alt|C|$_{11}$\alt%
|D|$_{11}$\alt|E|$_{11}$\alt|F|$_{11}$
\alt|A|$_{12}$\alt|B|$_{12}$\alt|C|$_{12}$\alt%
|D|$_{12}$\alt|E|$_{12}$\alt|F|$_{12}$
<one optional space>\is<space token>\alt<empty>
<coerced integer>\is<internal dimen>\alt<internal glue>
\endsyntax
The value of a \<number> is the value of the corresponding \<unsigned number>,
times~$-1$ for every minus sign in the \<optional signs>.
An ^{alphabetic constant} denotes the character code in a
^\<character token>; \TeX\ does not expand this token, which should either
be a (character~code, category~code) pair, or an active character, or
a control sequence whose name consists of a single character.
\ (See Chapter~20 for a complete list of all situations in which \TeX\ does
not expand tokens.) \ An \<integer constant> must not be immediately followed by
a \<digit>; in other words, if several digits appear consecutively, they
are all considered to be part of the same \<integer constant>. A similar
remark applies to the quantities \<octal constant> and \<hexadecimal constant>.
The quantity ^\<one optional space> is \<empty> only if it has to be;
i.e., \TeX\ looks for \<one optional space> by reading a token and backing
up if a \<space token> wasn't there.
\ddangerexercise Can you think of a reason why you might want `|A|$_{12}$'
to be a \<hex digit> even though the letter {\tt A} has category~11? \
(Don't worry if your answer is ``no.''\thinspace)
\answer If\/ |\cs| has been defined by ^|\chardef| or ^|\mathchardef|, \TeX\
uses ^{hexadecimal notation} when it expands ^|\meaning||\cs|, and it
assigns category~12 to each digit of the expansion. You might have an
application in which you want the last part of the expansion to be treated
as a \<number>. \ (This is admittedly an obscure reason.)
The definition of \<number> is now complete except for the three quantities
called \<internal integer>, \<internal dimen>, and \<internal glue>, which
will be explained later; they represent things like parameters and registers.
For example, |\count1| and |\tolerance| and |\hyphenchar\tenrm| are
internal integers; |\dimen10| and |\hsize| and |\fontdimen6\tenrm| are
internal dimensions; |\skip100| and |\baselineskip| and |\lastskip| are
internal glue values. An internal dimension can be ``coerced'' to be an
integer by assuming units of scaled points. For example, if\/ |\hsize=100pt|
^^{coerce <dimen> to <number>}
^^{coerce <glue> to <dimen>}
and if\/ |\hsize| is used in the context of a \<number>, it denotes the
integer value 6553600. Similarly, an internal glue value can be coerced to
be an integer by first coercing it to be a dimension (omitting the
stretchability and shrinkability), then coercing that dimension.
\smallskip
Let's turn now to the syntax for \<dimen>, and for \<mudimen> its cousin:
\beginsyntax
<dimen>\is<optional signs><unsigned dimen>
<unsigned dimen>\is<normal dimen>\alt<coerced dimen>
<coerced dimen>\is<internal glue>
<normal dimen>\is<internal dimen>\alt<factor><unit of measure>
<factor>\is<normal integer>\alt<decimal constant>
<decimal constant>\is|.|$_{12}$\alt|,|$_{12}$
\alt<digit><decimal constant>
\alt<decimal constant><digit>
<unit of measure>\is<optional spaces><internal unit>
\alt<optional {\tt true}><physical unit><one optional space>
<internal unit>\is[em]<one optional space>\alt[ex]<one optional space>
\alt<internal integer>\alt<internal dimen>\alt<internal glue>
<optional {\tt true}>\is[true]\alt<empty>
<physical unit>\is[pt]\alt[pc]\alt[in]\alt[bp]\alt[cm]\alt[mm]\alt[dd]\alt%
[cc]\alt[sp]\vadjust{\vskip 3pt minus 2pt}
<mudimen>\is<optional signs><unsigned mudimen>
<unsigned mudimen>\is<normal mudimen>\alt<coerced mudimen>
<coerced mudimen>\is<internal muglue>
<normal mudimen>\is<factor><mu unit>
<mu unit>\is<optional spaces><internal muglue>\alt[mu]<one optional space>
\endsyntax
When `|true|' is present, the factor is multiplied by~1000 and divided by
the ^|\mag| parameter. Physical units are defined in Chapter~10; |mu| is
explained in Chapter~18.
\goodbreak
Encouraged by our success in mastering the precise syntax of the quantities
\<number>, \<dimen>, and \<mudimen>, let's tackle \<glue> and \<muglue>:
\beginsyntax
<glue>\is<optional signs><internal glue>
\alt<dimen><stretch><shrink>
<stretch>\is[plus]<dimen>\alt[plus]<fil dimen>\alt<optional spaces>
<shrink>\is[minus]<dimen>\alt[minus]<fil dimen>\alt<optional spaces>
<fil dimen>\is<optional signs><factor><fil unit><optional spaces>
<fil unit>\is[fil]\alt<fil unit>[l]
<muglue>\is<optional signs><internal muglue>
\alt<mudimen><mustretch><mushrink>
<mustretch>\is[plus]<mudimen>\alt[plus]<fil dimen>\alt<optional spaces>
<mushrink>\is[minus]<mudimen>\alt[minus]<fil dimen>\alt<optional spaces>
\endsyntax
\TeX\ makes a large number of internal quantities accessible so that a
format designer can influence \TeX's behavior. Here is a list of all
these quantities, except for the parameters (which will be listed later).
\beginsyntax
<internal integer>\is<integer parameter>\alt<special integer>\alt^|\lastpenalty|
\alt<countdef token>\alt^|\count|<8-bit number>\alt<codename><8-bit number>
\alt<chardef token>\alt<mathchardef token>\alt^|\parshape|\alt^|\inputlineno|
\alt^|\hyphenchar|<font>\alt^|\skewchar|<font>\alt^|\badness|
<special integer>\is^|\spacefactor|\alt^|\prevgraf|
\alt^|\deadcycles|\alt^|\insertpenalties|
<codename>\is^|\catcode|\alt^|\mathcode|
\alt^|\lccode|\alt^|\uccode|\alt^|\sfcode|\alt^|\delcode|
<font>\is<fontdef token>\alt^|\font|\alt<family member>
<family member>\is<font range><4-bit number>
<font range>\is^|\textfont|\alt^|\scriptfont|\alt^|\scriptscriptfont|
<internal dimen>\is<dimen parameter>\alt<special dimen>\alt^|\lastkern|
\alt<dimendef token>\alt^|\dimen|<8-bit number>
\alt<box dimension><8-bit number>\alt^|\fontdimen|<number><font>
<special dimen>\is^|\prevdepth|\alt^|\pagegoal|\alt^|\pagetotal|
\alt^|\pagestretch|\alt^|\pagefilstretch|\alt^|\pagefillstretch|
\alt^|\pagefilllstretch|\alt^|\pageshrink|\alt^|\pagedepth|
<box dimension>\is^|\ht|\alt^|\wd|\alt^|\dp|
<internal glue>\is<glue parameter>\alt^|\lastskip|
\alt<skipdef token>\alt^|\skip|<8-bit number>
<internal muglue>\is<muglue parameter>\alt^|\lastskip|
\alt<muskipdef token>\alt^|\muskip|<8-bit number>
\endsyntax
A ^\<countdef token> is a control sequence token in which the control
sequence's current meaning has been defined by ^|\countdef|; the other
quantities ^\<dimendef token>, etc.,
^^\<skipdef token>^^\<muskipdef token>^^\<chardef token>^^\<mathchardef token>
^^\<toksdef token> are defined similarly. A \<fontdef
token> refers to a definition by ^|\font|, or it can be the predefined font
identifier called ^|\nullfont|. When a \<countdef token> is used as an
internal integer, it denotes the value of the corresponding ^|\count|
register, and similar statements hold for \<dimendef token>, \<skipdef
token>, \<muskipdef token>. When a \<chardef token> or \<mathchardef
token> is used as an internal integer, it denotes the value in the
^|\chardef| or ^|\mathchardef| itself. An ^\<8-bit number> is a \<number>
whose value is between 0~and $2^8-1=255$; a ^\<4-bit number> is similar.
^^\<15-bit number> ^^\<27-bit number>
\TeX\ allows |\spacefactor| to be an internal integer only in horizontal
modes; |\prevdepth| can be an internal dimension only in vertical modes;
|\lastskip| can be \<internal muglue> only in math mode when the current
math list ends with a muglue item; and |\lastskip| cannot be \<internal
glue> in such a case. When |\parshape| is used as an internal integer, it
denotes only the number of controlled lines, not their sizes or
indentations. The seven special dimensions |\pagetotal|, |\pagestretch|,
and so on are all zero when the current page contains no boxes, and
|\pagegoal| is |\maxdimen| at such times (see Chapter~15).
\smallskip
From the syntax rules just given, it's possible to deduce exactly what
happens to ^{spaces} when they are in the vicinity of numerical quantities:
\TeX\ allows a \<number> or \<dimen> to be preceded by arbitrarily many
spaces, and to be followed by at most one space; however, there is no
optional space after a \<number> or \<dimen> that ends with an unexpandable
control sequence. For example, if \TeX\ sees `|\space\space24\space\space|' when
it is looking for a \<number>, it gobbles up the first three spaces, but
the fourth one survives; similarly, one space remains when `|24pt\space\space|'
and `|\dimen24\space\space|' and `|\pagegoal\space|' are treated as
\<dimen> values.
\ddangerexercise Is `|24\space\space pt|' a legal \<dimen>?
\answer Yes; any number of spaces can precede any keyword.
\ddangerexercise Is there any difference between `|+\baselineskip|',
`|- -\baselineskip|', and `|1\baselineskip|', when \TeX\ reads
them as \<glue>?
\answer The first two have the same meaning; but the third coerces
|\baselineskip| to a \<dimen> by suppressing the stretchability
and shrinkability that might be present.
\ddangerexercise What \<glue> results from |"DD DDPLUS2,5 \spacefactor\space|,
assuming the conventions of plain \TeX, when |\spacefactor| equals 1000?
\answer The natural width is $221\rm\,dd$ (which \TeX\ rounds to
$15497423\rm\,sp$ and displays as |236.47191pt|).
The stretchability is $2500\rm\,sp$, since an
internal integer is coerced to a dimension when it appears as an
^^{coerce <number> to <dimen>}
\<internal unit>. The shrinkability is zero. Notice that the final |\space|
is swallowed up as part of the optional ^{spaces} of the \<shrink> part
in the syntax for \<glue>. \ (If |PLUS| had been |MINUS|, the final |\space|
would {\sl not\/} have been part of this \<glue>!)
Let's turn now to \TeX's ^{parameters}, which the previous chapters have
introduced one at a time; it will be convenient to assemble them
all together. An ^\<integer parameter> is one of the following tokens:
\begindisplay\belowdisplayskip=3pt plus 6pt \abovedisplayskip=3pt plus 1pt%
\openup.15pt
^|\pretolerance|\quad(badness tolerance before hyphenation)\cr
^|\tolerance|\quad(badness tolerance after hyphenation)\cr
^|\hbadness|\quad(badness above which bad hboxes will be shown)\cr
^|\vbadness|\quad(badness above which bad vboxes will be shown)\cr
^|\linepenalty|\quad(amount added to badness of every line in a paragraph)\cr
^|\hyphenpenalty|\quad(penalty for line break after discretionary hyphen)\cr
^|\exhyphenpenalty|\quad(penalty for line break after explicit hyphen)\cr
^|\binoppenalty|\quad(penalty for line break after binary operation)\cr
^|\relpenalty|\quad(penalty for line break after math relation)\cr
^|\clubpenalty|\quad(penalty for creating a club line at bottom of page)\cr
^|\widowpenalty|\quad(penalty for creating a widow line at top of page)\cr
^|\displaywidowpenalty|\quad(ditto, before a display)\cr
^|\brokenpenalty|\quad(penalty for page break after a hyphenated line)\cr
^|\predisplaypenalty|\quad(penalty for page break just before a display)\cr
^|\postdisplaypenalty|\quad(penalty for page break just after a display)\cr
^|\interlinepenalty|\quad(additional penalty for page break between lines)\cr
^|\floatingpenalty|\quad(penalty for insertions that are split)\cr
\noalign{\goodbreak}
^|\outputpenalty|\quad(penalty at the current page break)\cr
^|\doublehyphendemerits|\quad(demerits for consecutive broken lines)\cr
^|\finalhyphendemerits|\quad(demerits for a penultimate broken line)\cr
^|\adjdemerits|\quad(demerits for adjacent incompatible lines)\cr
^|\looseness|\quad(change to the number of lines in a paragraph)\cr
^|\pausing|\quad(positive if pausing after each line is read from a file)\cr
^|\holdinginserts|\quad(positive if insertions remain dormant in output box)\cr
^|\tracingonline|\quad(positive if showing diagnostic info on the terminal)\cr
^|\tracingmacros|\quad(positive if showing macros as they are expanded)\cr
^|\tracingstats|\quad(positive if showing statistics about memory usage)\cr
^|\tracingparagraphs|\quad(positive if showing line-break calculations)\cr
^|\tracingpages|\quad(positive if showing page-break calculations)\cr
^|\tracingoutput|\quad(positive if showing boxes that are shipped out)\cr
^|\tracinglostchars|\quad(positive if showing characters not in the font)\cr
^|\tracingcommands|\quad(positive if showing commands
before they are executed)\cr
^|\tracingrestores|\quad(positive if showing deassignments when groups end)\cr
^|\language|\quad(the current set of hyphenation rules)\cr
^|\uchyph|\quad(positive if hyphenating words beginning with capital letters)\cr
^|\lefthyphenmin|\quad(smallest fragment at beginning of hyphenated word)\cr
^|\righthyphenmin|\quad(smallest fragment at end of hyphenated word)\cr
^|\globaldefs|\quad(nonzero if overriding |\global| specifications)\cr
^|\defaulthyphenchar|\quad(^|\hyphenchar| value when a font is loaded)\cr
^|\defaultskewchar|\quad(^|\skewchar| value when a font is loaded)\cr
^|\escapechar|\quad(escape character in the output of
control sequence tokens)\cr
^|\endlinechar|\quad(character placed at the right end of an input line)\cr
^|\newlinechar|\quad(character that starts a new output line)\cr
^|\maxdeadcycles|\quad(upper bound on |\deadcycles|)\cr
^|\hangafter|\quad(hanging indentation changes after this many lines)\cr
^|\fam|\quad(the current family number)\cr
^|\mag|\quad(magnification ratio, times 1000)\cr
^|\delimiterfactor|\quad(ratio for variable delimiters, times 1000)\cr
^|\time|\quad(current time of day in minutes since midnight)\cr
^|\day|\quad(current day of the month)\cr
^|\month|\quad(current month of the year)\cr
^|\year|\quad(current year of our Lord)\cr
^|\showboxbreadth|\quad(maximum items per level when boxes are shown)\cr
^|\showboxdepth|\quad(maximum level when boxes are shown)\cr
^|\errorcontextlines|\quad(maximum extra context shown when errors occur)\cr
\enddisplay
The first few of these parameters have values in units of ``badness'' and
``penalties'' that affect line breaking and page breaking. Then come
demerit-oriented parameters; demerits are essentially given in units of
``badness squared,'' so those parameters tend to have larger values.
By contrast, the next few parameters (|\looseness|, |\pausing|, etc.)\
generally have quite small values (either $-1$ or 0 or 1 or~2).
Miscellaneous parameters complete the set. \TeX\ computes the date and time
when it begins a job, if the operating system provides such information; but
afterwards the clock does not keep ticking: The user can change |\time|
just like any ordinary parameter. Chapter~10 points out that |\mag| must
not be changed after \TeX\ is committed to a particular magnification.
\goodbreak\noindent
A ^\<dimen parameter> is one of the following:
\begindisplay\openup.15pt
^|\hfuzz|\quad(maximum overrun before overfull hbox messages occur)\cr
^|\vfuzz|\quad(maximum overrun before overfull vbox messages occur)\cr
^|\overfullrule|\quad(width of rules appended to overfull boxes)\cr
^|\emergencystretch|\quad(reduces badnesses on final pass of line-breaking)\cr
^|\hsize|\quad(line width in horizontal mode)\cr
^|\vsize|\quad(page height in vertical mode)\cr
^|\maxdepth|\quad(maximum depth of boxes on main pages)\cr
^|\splitmaxdepth|\quad(maximum depth of boxes on split pages)\cr
^|\boxmaxdepth|\quad(maximum depth of boxes on explicit pages)\cr
^|\lineskiplimit|\quad(threshold where |\baselineskip| changes
to |\lineskip|)\cr
^|\delimitershortfall|\quad(maximum space not covered by a delimiter)\cr
^|\nulldelimiterspace|\quad(width of a null delimiter)\cr
^|\scriptspace|\quad(extra space after subscript or superscript)\cr
^|\mathsurround|\quad(kerning before and after math in text)\cr
^|\predisplaysize|\quad(length of text preceding a display)\cr
^|\displaywidth|\quad(length of line for displayed equation)\cr
^|\displayindent|\quad(indentation of line for displayed equation)\cr
^|\parindent|\quad(width of\/ |\indent|)\cr
^|\hangindent|\quad(amount of hanging indentation)\cr
^|\hoffset|\quad(horizontal offset in |\shipout|)\cr
^|\voffset|\quad(vertical offset in |\shipout|)\cr
\enddisplay
And the possibilities for ^\<glue parameter> are:
\begindisplay\openup.15pt
^|\baselineskip|\quad(desired glue between baselines)\cr
^|\lineskip|\quad(interline glue if\/ |\baselineskip| isn't feasible)\cr
^|\parskip|\quad(extra glue just above paragraphs)\cr
^|\abovedisplayskip|\quad(extra glue just above displays)\cr
^|\abovedisplayshortskip|\quad(ditto, following short lines)\cr
^|\belowdisplayskip|\quad(extra glue just below displays)\cr
^|\belowdisplayshortskip|\quad(ditto, following short lines)\cr
^|\leftskip|\quad(glue at left of justified lines)\cr
^|\rightskip|\quad(glue at right of justified lines)\cr
^|\topskip|\quad(glue at top of main pages)\cr
^|\splittopskip|\quad(glue at top of split pages)\cr
^|\tabskip|\quad(glue between aligned entries)\cr
^|\spaceskip|\quad(glue between words, if nonzero)\cr
^|\xspaceskip|\quad(glue between sentences, if nonzero)\cr
^|\parfillskip|\quad(additional |\rightskip| at end of paragraphs)\cr
\enddisplay
Finally, there are three permissible ^\<muglue parameter> tokens:
\begindisplay\openup.15pt
^|\thinmuskip|\quad(thin space in math formulas)\cr
^|\medmuskip|\quad(medium space in math formulas)\cr
^|\thickmuskip|\quad(thick space in math formulas)\cr
\enddisplay
All of these quantities are explained in more detail somewhere else in
this book, and you can use Appendix~I to find out where.
\TeX\ also has parameters that are token lists. Such parameters do not
enter into the definitions of \<number> and such things, but we might as
well list them now so that our tabulation of parameters is complete.
A ^\<token parameter> is any of:
\begindisplay\openup.15pt
^|\output|\quad(the user's output routine)\cr
^|\everypar|\quad(tokens to insert when a paragraph begins)\cr
^|\everymath|\quad(tokens to insert when math in text begins)\cr
^|\everydisplay|\quad(tokens to insert when display math begins)\cr
^|\everyhbox|\quad(tokens to insert when an hbox begins)\cr
^|\everyvbox|\quad(tokens to insert when a vbox begins)\cr
^|\everyjob|\quad(tokens to insert when the job begins)\cr
^|\everycr|\quad(tokens
to insert after every ^|\cr| or nonredundant ^|\crcr|)\cr
^|\errhelp|\quad(tokens that supplement an |\errmessage|)\cr
\enddisplay
That makes a total of 103 parameters of all five kinds.
\ddangerexercise Explain how |\everyjob| can be non-null when a job begins.
\answer If it was non-null when a ^|\dump| operation occurred. Here's
^^|\jobname|
a nontrivial example, which sets up ^|\batchmode| and puts ^|\end| at the
end of the input file:
\begintt
\everyjob={\batchmode\input\jobname\end}
\endtt
It's time now to return to our original goal, namely to study the commands
that are obeyed by \TeX's digestive organs. Many commands are
carried out in the same way regardless of the current mode. The most
important commands of this type are called {\sl^{assignments}}, since
they assign new values to the meaning of control sequences or to \TeX's
internal quantities. For example, `|\def\a{a}|' and `|\parshape=1 5pt 100pt|'
and `|\advance\count20 by-1|' and `|\font\ff = cmff at 20pt|' are all
assignments, and they all have the same effect in all modes.
Assignment commands often include an~^|=|~sign, ^^{equals sign}
but in all cases this sign is optional; you can leave it out if you
don't mind the fact that the resulting \TeX\ code might not look quite
like an assignment.
\beginsyntax
<assignment>\is<non-macro assignment>\alt<macro assignment>
<non-macro assignment>\is<simple assignment>
\alt^|\global|<non-macro assignment>
<macro assignment>\is<definition>\alt<prefix><macro assignment>
<prefix>\is|\global|\alt^|\long|\alt^|\outer|
<equals>\is<optional spaces>\alt<optional spaces>\thinspace|=|$_{12}$
\endsyntax
This syntax shows that every assignment can be prefixed by |\global|, but
only macro-definition assignments are allowed to be prefixed by |\long|
or |\outer|. Incidentally, if the |\globaldefs| parameter is positive at the
time of the assignment, a prefix of\/ |\global| is automatically implied;
but if\/ ^|\globaldefs| is negative at the time of the assignment, a prefix
of\/ |\global| is ignored. If\/ |\globaldefs| is zero (which it usually~is),
the appearance or nonappearance of\/ |\global| determines whether or not
a global assignment is made.
\beginsyntax
<definition>\is<def><control sequence><definition text>
<def>\is^|\def|\alt^|\gdef|\alt^|\edef|\alt^|\xdef|
<definition text>\is<parameter text><left brace><balanced text><right brace>
\endsyntax
Here ^\<control sequence> denotes a token that is either a control sequence
or an active character; ^\<left brace> and ^\<right brace> are explicit
character tokens whose category codes are respectively of types 1 and~2.
The ^\<parameter text> contains no \<left brace> or \<right brace> tokens,
and it obeys the rules of Chapter~20. All occurrences of \<left brace>
and \<right brace> tokens within the ^\<balanced text> must be properly nested
like \hbox{parentheses}. A |\gdef| command is equivalent to |\global\def|,
and |\xdef| is equivalent to |\global\edef|. \TeX\ reads the
\<control sequence> and \<parameter text> tokens and the opening \<left brace>
without expanding them; it expands the \<balanced text>\allowbreak\<right
brace> tokens only in the case of\/ |\edef| and |\xdef|.
Several commands that we will study below have a syntax somewhat like
that of a definition, but the \<parameter text> is replaced by an
arbitrary sequence of spaces and `|\relax|' commands, and the
\<left brace> token can be implicit:
\beginsyntax
<filler>\is<optional spaces>\alt<filler>^|\relax|<optional spaces>
<general text>\is<filler>|{|<balanced text><right brace>
\endsyntax
The main purpose of a \<general text> is to specify the \<balanced text>
inside.
Many different kinds of assignments are possible, but they fall into
comparatively few patterns, as indicated by the following syntax rules:
\beginsyntax
<simple assignment>\is<variable assignment>\alt<arithmetic>
\alt<code assignment>\alt<let assignment>\alt<shorthand definition>
\alt<fontdef token>\alt<family assignment>\alt<shape assignment>
\alt^|\read|<number>[to]<optional spaces><control sequence>
\alt^|\setbox|<8-bit number><equals><filler><box>
\alt^|\font|<control sequence><equals><file name><at clause>
\alt<global assignment>
<variable assignment>\is<integer variable><equals><number>
\alt<dimen variable><equals><dimen>
\alt<glue variable><equals><glue>
\alt<muglue variable><equals><muglue>
\alt<token variable><equals><general text>
\alt<token variable><equals><filler><token variable>
<arithmetic>\is^|\advance|<integer variable><optional {\tt by}><number>
\alt|\advance|<dimen variable><optional {\tt by}><dimen>
\alt|\advance|<glue variable><optional {\tt by}><glue>
\alt|\advance|<muglue variable><optional {\tt by}><muglue>
\alt^|\multiply|<numeric variable><optional {\tt by}><number>
\alt^|\divide|<numeric variable><optional {\tt by}><number>
<optional {\tt by}>\is[by]\alt\<optional spaces>
<integer variable>\is<integer parameter>\alt<countdef token>
\alt^|\count|<8-bit number>
<dimen variable>\is<dimen parameter>\alt<dimendef token>
\alt^|\dimen|<8-bit number>
<glue variable>\is<glue parameter>\alt<skipdef token>
\alt^|\skip|<8-bit number>
<muglue variable>\is<muglue parameter>\alt<muskipdef token>
\alt^|\muskip|<8-bit number>
<token variable>\is<token parameter>\alt<toksdef token>
\alt^|\toks|<8-bit number>
<numeric variable>\is<integer variable>\alt<dimen variable>
\alt<glue variable>\alt<muglue variable>% I want to force a page break here!
\endgraf\penalty-500\syntaxrule% this defeats the \beginsyntax trickery
<at clause>\is[at]<dimen>\alt[scaled]<number>\alt<optional spaces>
<code assignment>\is<codename><8-bit number><equals><number>
<let assignment>\is^|\futurelet|<control sequence><token><token>
\alt^|\let|<control sequence><equals><one optional space><token>
<shorthand definition>\is^|\chardef|<control sequence><equals><8-bit number>
\alt^|\mathchardef|<control sequence><equals><15-bit number>
\alt<registerdef><control sequence><equals><8-bit number>
<registerdef>\is^|\countdef|\alt^|\dimendef|\alt^|\skipdef|\alt%
^|\muskipdef|\alt^|\toksdef|
<family assignment>\is<family member><equals><font>
<shape assignment>\is^|\parshape|<equals><number><shape dimensions>
\endsyntax
The \<number> at the end of a \<code assignment> must not be negative, except in
the case that a |\delcode| is being assigned. Furthermore, that \<number> should
be at most 15~for |\catcode|, 32768~for |\mathcode|, 255~for |\lccode| or
|\uccode|, 32767~for |\sfcode|, and $2^{24}-1$~for |\delcode|. In a
\<shape assignment> for which the \<number> is $n$, the ^\<shape dimensions>
are \<empty> if $n\le0$, otherwise they consist of $2n$ consecutive
occurrences of \<dimen>. \TeX\ does not expand tokens when it scans the
arguments of\/ |\let| and |\futurelet|.
\ddangerexercise We discussed the distinction between explicit and ^{implicit
character tokens} earlier in this chapter. Explain how you can make the
control sequence |\cs| into an implicit space, using (a)~|\futurelet|,
(b)~|\let|.
\answer (a) |\def\\#1\\{}\futurelet\cs\\|\]|\\|. (b) |\def\\{\let\cs= }\\|\].
\ (There are many other solutions.)
All of the assignments mentioned so far will obey \TeX's grouping structure;
i.e., the changed quantities will be restored to their former values
when the current group ends, unless the change was global. The remaining
assignments are different, since they affect \TeX's global font tables
or hyphenation tables, or they affect certain control variables of such
^^{global parameters}
an intimate nature that grouping would be inappropriate. In all of the
following cases, the presence or absence of\/ |\global| as a prefix has no
effect.
\beginsyntax
<global assignment>\is<font assignment>
\alt<hyphenation assignment>
\alt<box size assignment>
\alt<interaction mode assignment>
\alt<intimate assignment>
<font assignment>\is^|\fontdimen|<number><font><equals><dimen>
\alt^|\hyphenchar|<font><equals><number>
\alt^|\skewchar|<font><equals><number>
<hyphenation assignment>\is^|\hyphenation|<general text>
\alt^|\patterns|<general text>
<box size assignment>\is<box dimension><8-bit number><equals><dimen>
<interaction mode assignment>\is^|\errorstopmode|\alt^|\scrollmode|
\alt^|\nonstopmode|\alt^|\batchmode|
<intimate assignment>\is<special integer><equals><number>
\alt<special dimen><equals><dimen>
\endsyntax
When a |\fontdimen| value is assigned, the \<number> must be positive and
not greater than the number of parameters in the font's metric information
file, unless that font information has just been loaded into \TeX's
memory; in the latter case, you are allowed to increase the number of
parameters (see Appendix~F\null). The \<special integer> and \<special
dimen> quantities were listed above when we discussed internal integers
and dimensions. When |\prevgraf| is set to a \<number>, the number must
not be negative.
The syntax for ^\<file name> is not standard in \TeX, because different
operating systems have different conventions. You should ask your local
system wizards for details on just how they have decided to implement file
names. However, the following principles should hold universally:
A~\<file name> should consist of \<optional spaces> followed by explicit
character tokens (after expansion). A sequence of six or fewer ordinary
letters and/or digits followed by a space should be a file name that works
in essentially the same way on all installations of\/ \TeX\null. Uppercase
letters are not considered equivalent to their lowercase counterparts in
file names; for example, if you refer to fonts |cmr10| and |CMR10|, \TeX\
will not notice any similarity between them, although it might input the
same font metric file for both fonts.
\TeX\ takes precautions so that constructions like `|\chardef\cs=10\cs|' and
`|\font\cs=name\cs|' won't expand the second |\cs| until
the assignments are done.
Our discussion of assignments is complete except that the |\setbox|
assignment involves a quantity called \<box> that has not yet been
defined. Here is its syntax:
\beginsyntax
<box>\is^|\box|<8-bit number>\alt^|\copy|<8-bit number>
\alt^|\lastbox|\alt^|\vsplit|<8-bit number>[to]<dimen>
\alt^|\hbox|<box specification>|{|<horizontal mode material>|}|
\alt^|\vbox|<box specification>|{|<vertical mode material>|}|
\alt^|\vtop|<box specification>|{|<vertical mode material>|}|
<box specification>\is[to]<dimen><filler>
\alt[spread]<dimen><filler>\alt<filler>
\endsyntax
The |\lastbox| operation is not permitted in math modes, nor is it allowed
in vertical mode when the main vertical list has been entirely contributed
to the current page. But it is allowed in horizontal modes and in
internal vertical mode; in such modes it refers to (and removes) the
last item of the current list, provided that the last item is an hbox or~vbox.
The three last alternatives for a \<box> present us with a new situation:
The ^\<horizontal mode material> in an |\hbox| and the
^\<vertical mode material> in a |\vbox| can't simply be swallowed up
in one command like an \<8-bit number> or a \<dimen>; thousands of
commands may have to be executed before that box is constructed and
before the |\setbox| command can be completed.
Here's what really happens: A command like
\begindisplay
|\setbox|\<number>|=\hbox to|\<dimen>|{|\<horizontal mode material>|}|
\enddisplay
causes \TeX\ to evaluate the \<number> and the \<dimen>, and to put those
values on a ``stack'' for safe keeping. Then \TeX\ reads the `|{|' (which
stands for an explicit or implicit begin-group character, as explained
earlier), and this initiates a new level of grouping. At this point
\TeX\ enters restricted horizontal mode and proceeds to execute commands
in that mode. An arbitrarily complex box can now be constructed;
the fact that this box is eventually destined for a |\setbox| command
has no effect on \TeX's behavior while the box is being built. Eventually,
when the matching `|}|' appears, \TeX\ restores values that were
changed by assignments in the group just ended; then it packages the hbox
(using the size that was saved on the stack), and completes the
|\setbox| command, returning to the mode it was in at the time of
the |\setbox|.
\smallbreak
Let us now consider other commands that, like assignments, are
obeyed in basically the same way regardless of \TeX's current mode.
\def\\{\smallbreak\textindent{$\bull$}}
\\^|\relax|.\enskip
This is an easy one: \TeX\ does nothing.
\\|}|.\enskip
This one is harder, because it depends on the current group. \TeX\ should
now be working on a group that began with |{|; and it knows why it
started that group. So it does the appropriate finishing actions, undoes
the effects of non-global assignments, and leaves the group.
At this point \TeX\ might leave its current mode and return to a mode that
was previously in effect.
\\^|\begingroup|.\enskip
When \TeX\ sees this command, it enters a group that must be terminated
by |\endgroup|, not by |}|. The mode doesn't change.
\\^|\endgroup|.\enskip
\TeX\ should currently be processing a group that began with |\begingroup|.
Quantities that were changed by non-global assignments in that group
are restored to their former values. \TeX\ leaves the group,
but stays in the same mode.
\\^|\show|\stretch\<token>,
\stretch\stretch\stretch^|\showbox|\stretch\<8-bit number>,
\stretch\stretch\stretch^|\showlists|,
\stretch\stretch\stretch^|\showthe|$\langle$internal quantity$\rangle$.\enskip
These commands are intended to help you figure out what \TeX\ thinks it
is doing. The tokens following |\showthe| should be anything that can
follow |\the|, as explained in Chapter~20.
\ddangerexercise Review the rules for what can follow |\the| in
Chapter~20, and construct a formal syntax that defines ^\<internal
quantity> in a way that fits with the other syntax rules we have been
discussing.
\answer \<internal quantity>\is\<internal integer>\alt
\<internal dimen>\parbreak\qquad\alt\<internal glue>\alt\<internal muglue>\alt
\<internal nonnumeric>\parbreak \<internal nonnumeric>\is\<token
variable>\alt \<font>
\\^|\shipout|\<box>.\enskip
After the \<box> is formed---possibly by constructing it explicitly and
changing modes during the construction, as explained for |\hbox| earlier---its
contents are sent to the ^|dvi|~file (see Chapter~23).
\\^|\ignorespaces|\stretch$\langle$optional spaces$\rangle$.
\TeX\ reads (and expands) tokens, doing nothing until reaching one that is
not a \<space token>.
\\^|\afterassignment|\<token>.\enskip
The \<token> is saved in a special place; it will be inserted
back into the input just after the next assignment command has been
performed. An assignment need not follow immediately; if another
|\afterassignment| is performed before the next assignment, the second one
overrides the first. If the next assignment is a ^|\setbox|, and if the
assigned \<box> is |\hbox| or |\vbox| or |\vtop|, the \<token> will be
inserted just after the |{| in the box construction, not after the |}|;
it will also come just before any tokens inserted by ^|\everyhbox| or
^|\everyvbox|.
\\^|\aftergroup|\<token>.\enskip
The \<token> is saved on \TeX's stack; it will be inserted back into the
input just after the current group has been completed and its local
assignments have been undone. If several |\aftergroup| commands occur
in the same group, the corresponding commands will be scanned in the
same order; for example, `|{\aftergroup\a\aftergroup\b}|' yields `|\a\b|'.
\\^|\uppercase|\<general text>, ^|\lowercase|\<general text>.\enskip
The \<balanced text> in the general text is converted to uppercase form
or to lowercase form using the |\uccode| or |\lccode| table,
as explained in Chapter~7; no expansion is done. Then \TeX\ will
read that \<balanced text> again.
\\^|\message|\<general text>, ^|\errmessage|\<general text>.\enskip
The balanced text (with expansion) is written on the user's terminal,
using the format of error messages in the case of\/ |\errmessage|.
In the latter case the ^|\errhelp| tokens will be shown if they are nonempty
and if the user asks for help.
\\^|\openin|\stretch$\langle$4-bit number$\rangle$\stretch\<equals>\stretch
\<filename>, \ ^|\closein|\stretch$\langle$4-bit number$\rangle$.
These commands open or close the specified input stream, for use in
|\read| assignments as explained in Chapter~20.
\\|\immediate\openout|\<4-bit number>\<equals>\<filename>\kern-.4pt,
|\immediate\closeout|\allowbreak\<4-bit number>.\enskip ^^|\immediate|
The specified output stream is opened or closed, for use in |\write|
commands, as explained in Chapter~21.
\\|\immediate\write|\<number>\<general text>.\enskip
The balanced text is written on the file that corresponds to the
specified stream number, provided that such a file is open. Otherwise it
is written on the user's terminal and on the log file. \ (See Chapter~21;
the terminal is omitted if the \<number> is negative.)
\medbreak
That completes the list of mode-independent commands, i.e., the commands
that do not directly affect the lists that \TeX\ is building. When \TeX\
is in vertical mode or internal vertical mode, it is constructing a
vertical list; when \TeX\ is in horizontal mode or restricted horizontal
mode, it is constructing a horizontal list; when \TeX\ is in math mode
or display math mode, it is constructing---guess what---a math list. In
each of these cases we can speak of the ``current list''; and there are
some commands that operate in essentially the same way, regardless of the
mode, except that they deal with different sorts of lists:
\\^|\openout|\<4-bit number>\<equals>\<filename>,
^|\closeout|\<4-bit number>,
^|\write|\allowbreak\<number>\<general text>.\enskip
These commands are recorded into a ``whatsit'' item, which is appended to
the current list. The command will be performed later, during any
|\shipout| that applies to this list, unless the list is part of
a box inside ^{leaders}.
\\^|\special|\<general text>.\enskip
The balanced text is expanded and put into a ``whatsit'' item, which is
appended to the current list. The text will eventually appear in the
^|dvi|~file as an instruction to subsequent software (see Chapter~21).
\\^|\penalty|\<number>.\enskip
A penalty item carrying the specified number is appended to the current list.
In vertical mode, \TeX\ also exercises the page builder (see~below).
\\^|\kern|\<dimen>, ^|\mkern|\<mudimen>.\enskip
A kern item carrying the specified dimension is appended to the current list.
In vertical modes this denotes a vertical space; otherwise it denotes a
horizontal space. An |\mkern| is allowed only in math modes.
\\^|\unpenalty|, ^|\unkern|, ^|\unskip|.\enskip
If the last item on the current list is respectively of type penalty,
kern, or glue (possibly including ^{leaders}), that item is removed from
the list. However, like |\lastbox|, these commands are not permitted in
vertical mode if the main vertical list-so-far has been entirely contributed
to the current page, since \TeX\ never removes items from the current page.
\\^|\mark|\<general text>.\enskip
The balanced text is expanded and put into a mark item, which is appended to
the current list. The text may eventually become the replacement text
for ^|\topmark|, ^|\firstmark|, ^|\botmark|, ^|\splitfirstmark|, and/or
^|\splitbotmark|, if this mark item ever gets into a vertical list. \ (Mark
items can appear in horizontal lists and math lists, but they have no
effect until they ``migrate'' out of their list. The ^{migration process}
is discussed below and in Chapter~25.)
\\^|\insert|\<8-bit number>\<filler>|{|\<vertical mode material>|}|;
the \<8-bit number> must not be~255. \ The `|{|' causes \TeX\ to enter
internal vertical mode and a new level of grouping. When the matching~`|}|'
is sensed, the vertical list is put into an insertion item that is
appended to the current list using the values of\/ ^|\splittopskip|,
^|\splitmaxdepth|, and ^|\floatingpenalty| that were current in the group
just ended. \ (See Chapter~15.) \ This insertion item leads ultimately to
a page insertion only if it appears in \TeX's main vertical list, so it
will have to ``^{migrate}'' there if it starts out in a horizontal list or
a math list. \TeX\ also exercises the page builder (see below), after an
|\insert| has been appended in vertical~mode.
\\^|\vadjust|\<filler>|{|\<vertical mode material>|}|.\enskip
This is similar to |\insert|; the constructed vertical list goes into an
adjustment item that is appended to the current list. However,
|\vadjust| is not allowed in vertical modes. When an adjustment item
migrates from a horizontal list to a vertical list, the vertical list
inside the adjustment item is ``unwrapped'' and put directly into
the enclosing list.
\medbreak
\centerline{$*\qquad*\qquad*$}
\medskip\noindent
Almost everything we have discussed so far in this chapter could equally
well have appeared in a chapter entitled ``Summary of Horizontal Mode''
or a chapter entitled ``Summary of Math Mode,'' because \TeX\ treats
all of the commands considered so far in essentially the same way
regardless of the current mode. Chapters 25 and~26 are going to be
a lot shorter than the present one, since it will be unnecessary
to repeat all of the mode-independent rules.
But now we come to commands that are mode-dependent;
we shall conclude this chapter by discussing what \TeX\ does with the
remaining commands, when in vertical mode or internal vertical mode.
One of the things characteristic of vertical mode is the page-building
operation described in Chapter~15. \TeX\ periodically takes material
that has been put on the main vertical list and moves it from the
``contribution list'' to the ``current page.'' At such times the output
routine might be invoked. We shall say that \TeX\ {\sl exercises
the ^{page builder}\/} whenever it tries to empty the current
contribution list. The concept of contribution list exists only in
the outermost vertical mode, so nothing happens when \TeX\ exercises
the page builder in internal vertical mode.
Another thing characteristic of vertical modes is the ^{interline glue}
that is inserted before boxes, based on the values of\/ |\prevdepth|
and ^|\baselineskip| and ^|\lineskip| and ^|\lineskiplimit| as explained
in Chapter~12. If a command changes ^|\prevdepth|, that fact is
specifically mentioned below. The |\prevdepth| is initially set to
$-1000\pt$, a special value that inhibits interline glue, whenever \TeX\
begins to form a vertical list, except in the case of\/ |\halign| and
|\noalign| when the interline glue conventions of the outer list continue
inside the inner one.
\\^|\vskip|\<glue>, ^|\vfil|, ^|\vfill|, ^|\vss|, ^|\vfilneg|.\enskip
A glue item is appended to the current vertical list.
\\\<leaders>\<box or rule>\<vertical skip>.\enskip
Here ^\<vertical skip> refers to one of the five glue-appending commands
just mentioned. The formal syntax for \<leaders> and for \<box or rule> is
\beginsyntax
<leaders>\is|\leaders|\alt|\cleaders|\alt|\xleaders|
<box or rule>\is<box>\alt<vertical rule>\alt<horizontal rule>
<vertical rule>\is|\vrule|<rule specification>
<horizontal rule>\is|\hrule|<rule specification>
<rule specification>\is<optional spaces>\alt<rule dimension><rule specification>
<rule dimension>\is[width]<dimen>\alt[height]<dimen>\alt[depth]<dimen>
\endsyntax
A glue item that produces ^{leaders} is appended to the current list.
\\^\<space token>.\enskip
Spaces have no effect in vertical modes.
\\^\<box>.\enskip
The box is constructed, and if the result is void nothing happens.
Otherwise the current vertical list receives (1)~interline glue, followed by
(2)~the new box, followed by (3)~vertical material that ^{migrates} out of the
new box (if the \<box> was an ^|\hbox| command). Then ^|\prevdepth| is
set to the new box's depth, and \TeX\ exercises the page builder.
\\^|\moveleft|\<dimen>\<box>, ^|\moveright|\<dimen>\<box>.\enskip
This acts exactly like an unadorned \<box> command, except that the new box
being appended to the vertical list is also shifted left or right by the
specified amount.
\\^|\unvbox|\<8-bit number>, ^|\unvcopy|\<8-bit number>.\enskip
If the specified box register is void, nothing happens. Otherwise that
register must contain a vbox. The vertical list inside that box is
appended to the current vertical list, without changing it in any way.
The value of\/ |\prevdepth| is not affected. The box register becomes void
after |\unvbox|, but it remains unchanged by |\unvcopy|.
\\\<horizontal rule>.\enskip
The specified ^{rule} is appended to the current list. Then |\prevdepth|
is set to $-1000\pt$; this will prohibit interline glue when the next box
is appended to the list.
\\^|\halign|\<box specification>|{|\<alignment material>|}|.\enskip
The ^\<alignment material> consists of a preamble followed by zero or more
lines to be aligned; see Chapter~22. \TeX\ enters a new level of grouping,
represented by the `|{|' and `|}|', within which changes to ^|\tabskip|
will be confined. The alignment material can also contain optional
occurrences of `^|\noalign|\<filler>|{|\<vertical mode material>|}|'
between lines; this adds another level of grouping. \TeX\ operates in
internal vertical mode while it works on the material in |\noalign| groups
and when it appends lines of the alignment; the resulting internal
vertical list will be appended to the enclosing vertical list after the
alignment is completed, and the page builder will be exercised. The value
of\/ |\prevdepth| at the time of the |\halign| is used at the beginning of
the internal vertical list, and the final value of\/ |\prevdepth| is carried
to the enclosing vertical list when the alignment is completed, so that the
interline glue is calculated properly at the beginning and end of the
alignment. \TeX\ also enters an additional level of grouping when it
works on each individual entry of the alignment, during which time it acts
in restricted horizontal mode; the individual entries will be hboxed as
part of the final alignment, and their vertical material will ^{migrate}
to the enclosing vertical list. The commands |\noalign|, ^|\omit|,
^|\span|, ^|\cr|, ^|\crcr|, and |&| (where |&| denotes an explicit or
implicit character of category~4) are intercepted by the alignment
process, en route to \TeX's stomach, so they will not appear as commands in
the stomach unless \TeX\ has lost track of what alignment they belong to.
\\^|\indent|.\enskip
The ^|\parskip| glue is appended to the current list, unless \TeX\ is
in internal vertical mode and the current list is empty.
Then \TeX\ enters unrestricted horizontal mode, starting the horizontal
list with an empty hbox whose width is ^|\parindent|. The ^|\everypar|
tokens are inserted into \TeX's input. The page builder is~exercised.
When the paragraph is eventually completed, horizontal mode will come to
an end as described in Chapter~25.
\\^|\noindent|.\enskip
This is exactly like |\indent|, except that \TeX\ starts out in horizontal
mode with an empty list instead of with an indentation.
\\^|\par|.\enskip
The primitive |\par| command has no effect when \TeX\ is in vertical mode,
except that the page builder is exercised in case something is present
on the contribution list, and the paragraph shape parameters are cleared.
\\|{|.\enskip
A character token of category 1, or a control sequence like~|\bgroup|
that has been |\let| equal to such a character token, causes \TeX\ to
start a new level of ^{grouping}. When such a group ends---with `|}|'---\TeX\
will undo the effects of non-global assignments without leaving whatever
mode it is in at that time.
\\Some commands are incompatible with vertical mode because they are
intrinsically horizontal. When the following commands appear in
vertical modes they cause \TeX\ to begin a ^{new paragraph}:
\beginsyntax
<horizontal command>\is<letter>\alt<otherchar>\alt^|\char|\alt<chardef token>
\alt^|\noboundary|\alt^|\unhbox|\alt^|\unhcopy|\alt^|\valign|\alt^|\vrule|
\alt^|\hskip|\alt^|\hfil|\alt^|\hfill|\alt^|\hss|\alt^|\hfilneg|
\alt^|\accent|\alt^|\discretionary|\alt^|\-|\alt^|\|\]\alt|$|
\endsyntax
Here \<letter> and \<otherchar> stand for explicit or implicit character
tokens of categories 11 and~12. If any of these tokens occurs as a
command in vertical mode or internal vertical mode, \TeX\ automatically
performs an |\indent| command as explained above. This leads into
horizontal mode with the |\everypar| tokens in the input, after
which \TeX\ will see the \<horizontal command> again.
\\^|\end|.\enskip
This command is not allowed in internal vertical mode. In regular vertical
mode it terminates \TeX\ if the main vertical list is empty and
^|\deadcycles=0|. Otherwise \TeX\ backs up the |\end| command so that it can
be read again; then it exercises the page builder, after appending a
box/glue/penalty combination that will force the output routine to act.
\ (See the end of Chapter~23.)
\\^|\dump|.\enskip
(Allowed only in ^|INITEX|, not in production versions of \TeX.) \
This command is treated exactly like |\end|, but it must not appear
inside a group. It outputs a format file that can be loaded into \TeX's
memory at comparatively high speed to restore the current status.
\\None of the above: If any other primitive command of \TeX\ occurs
in vertical mode, an error message will be given, and \TeX\ will try
to recover in a reasonable way. For example, if a superscript or
subscript symbol appears, or if any other inherently mathematical
command is given, \TeX\ will try to insert a `|$|' (which will start a
paragraph and enter math mode). On the other hand if a totally
misplaced token like ^|\endcsname| or |\omit| or |\eqno| or |#| appears
in vertical mode, \TeX\ will simply ignore~it, after reporting
the error. You might enjoy trying to type some really stupid input,
just to see what happens. \ (Say `|\tracingall|' first, as explained
in Chapter~27, in order to get maximum information.)
\endchapter
The first and most striking feature is the Verticality of composition,
as opposed to the Horizontality of all anterior structural modes.
\author COCKBURN ^{MUIR}, {\sl Pagan or Christian?\/} (1860) % p61
\bigskip
Sometimes when I have finished a book I give a summary of the whole of it.
\author ROBERT WILLIAM ^{DALE}, {\sl Nine Lectures on Preaching} (1878)
% viii.231
\eject
\beginchapter Chapter 25. Summary of\\Horizontal Mode
^^{horizontal mode}
Continuing the survey that was begun in Chapter 24, let us investigate
exactly what \TeX's digestive processes can do, when \TeX\ is building
lists in horizontal mode or in restricted horizontal mode.
\ninepoint
\def\\{\smallbreak\textindent{$\bull$}}
\medbreak
\centerline{$*\qquad*\qquad*$}
\medskip\noindent
Three asterisks, just like those that appear here, can be found near the
end of Chapter~24.
Everything preceding the three asterisks in that chapter applies to
horizontal mode as well as to vertical mode, so we need not repeat
all those rules. In particular, Chapter~24 explains assignment commands,
and it tells how kerns, penalties, marks, insertions, adjustments,
and ``whatsits'' are put into horizontal lists. Our present goal
is to consider the commands that have an intrinsically horizontal
flavor, in the sense that they behave differently in horizontal
mode than they do in vertical or math modes.
One of the things characteristic of horizontal mode is the ``^{space
factor},'' which modifies the width of spaces as described in Chapter~12.
If a command changes the value of\/ ^|\spacefactor|, that fact is
specifically noted here. The space factor is initially set to~1000, when
\TeX\ begins to form a horizontal list, except in the case of\/ |\valign| and
|\noalign| when the space factor of the outer list continues inside the
inner one.
\\^|\hskip|\<glue>, ^|\hfil|, ^|\hfill|, ^|\hss|, ^|\hfilneg|.\enskip
A glue item is appended to the current horizontal list.
\\\<leaders>\<box or rule>\<horizontal skip>.\enskip
Here ^\<horizontal skip> refers to one of the five glue-appending commands
just mentioned; the formal syntax for \<leaders> and for \<box or rule> is
given in Chapter~24. A glue item that produces ^{leaders} is appended.
\\^\<space token>.\enskip
Spaces append glue to the current list; the exact amount of glue depends on
|\spacefactor|, the current font, and the |\spaceskip| and |\xspaceskip|
parameters, as described in Chapter~12.
\\|\|\].\enskip ^^{control space}
A control-space command appends glue to the current list, using the same amount
that a \<space token> inserts when the space factor is 1000.
\\^\<box>.\enskip
The box is constructed, and if the result is void nothing happens.
Otherwise the new box is appended to the current list, and
the space factor is~set~to~1000.
\\^|\raise|\<dimen>\<box>, ^|\lower|\<dimen>\<box>.\enskip
This acts just like an unadorned \<box> command, except that the new box
being appended to the horizontal list is also shifted up or down by the
specified amount.
\\^|\unhbox|\<8-bit number>, ^|\unhcopy|\<8-bit number>.\enskip
If the specified box register is void, nothing happens. Otherwise that
register must contain an hbox. The horizontal list inside that box is
appended to the current horizontal list, without changing it in any way.
The value of\/ |\spacefactor| is not affected. The box register becomes void
after |\unhbox|, but it remains unchanged by |\unhcopy|.
\\\<vertical rule>.\enskip
The specified ^{rule} is appended to the current list, and the |\spacefactor|
is set to 1000.
\\^|\valign|\<box specification>|{|\<alignment material>|}|.\enskip
The ^\<alignment material> consists of a preamble followed by zero or more
columns to be aligned; see Chapter~22. \TeX\ enters a new level of grouping,
represented by the `|{|' and `|}|', within which changes to ^|\tabskip|
will be confined. The alignment material can also contain optional
occurrences of `|\noalign|\<filler>|{|\<horizontal mode material>|}|'
between columns; this adds another level of grouping. \TeX\ operates in
restricted horizontal mode while it works on the material in ^|\noalign|
\vadjust{\eject}% squeeze another line onto page 285, we need it on page 286!
groups and when it appends columns of the \hbox{alignment}; the resulting
internal horizontal list will be appended to the enclosing horizontal list
after the alignment is completed. The value of\/ |\spacefactor| at the time
of the |\valign| is used at the beginning of the internal horizontal list,
and the final value of\/ |\spacefactor| is carried to the enclosing
horizontal list when the alignment is completed. The space factor is set
to 1000 after each column; hence it affects the results only in |\noalign|
groups. \TeX\ also enters an additional level of grouping when it works
on each individual entry of the alignment, during which time it acts in
internal vertical mode; the individual entries will be vboxed as part of
the final alignment.
%No room for the following redundant remarks:
% The commands |\noalign|, |\omit|, |\span|, |\cr|,
%|\crcr|, and |&| (where |&| denotes an explicit or implicit character of
%category~4) are intercepted by the alignment process, en route to \TeX's
%stomach, so they will not appear as commands in the stomach unless \TeX\
%has lost track of what alignment they belong to.
\\^|\indent|.\enskip
An empty box of width ^|\parindent| is appended to the current list, and
the space factor is set to 1000.
\\^|\noindent|.\enskip
This command has no effect in horizontal modes.
\\^|\par|.\enskip
The primitive |\par| command, also called ^|\endgraf| in plain \TeX,
does nothing in restricted horizontal mode. But it terminates horizontal
mode: The current list is finished off by doing ^|\unskip| ^|\penalty10000|
^^|\parfillskip|
|\hskip\parfillskip|, then it is broken into lines as explained in Chapter~14,
and \TeX\ returns to the enclosing vertical or internal vertical mode.
The lines of the paragraph are appended to the enclosing vertical list,
interspersed with interline glue and interline penalties, and with
the ^{migration} of vertical material that was in the horizontal list.
Then \TeX\ exercises the page builder.
\\|{|.\enskip
A character token of category 1, or a control sequence like~|\bgroup|
that has been |\let| equal to such a character token, causes \TeX\ to
start a new level of ^{grouping}. When such a group ends---with `|}|'---\TeX\
will undo the effects of non-global assignments without leaving whatever
mode it is in at that time.
\\Some commands are incompatible with horizontal mode because they are
intrinsically vertical. When the following commands appear in unrestricted
horizontal mode, they cause \TeX\ to conclude the current paragraph:
^^{paragraph end, implied}
\beginsyntax
<vertical command>\is^|\unvbox|\alt^|\unvcopy|\alt^|\halign|\alt^|\hrule|
\alt^|\vskip|\alt^|\vfil|\alt^|\vfill|\alt^|\vss|\alt^|\vfilneg|%
\alt^|\end|\alt^|\dump|
\endsyntax
The appearance of a \<vertical command> in restricted horizontal mode is
forbidden, but in regular horizontal mode it causes \TeX\ to insert the
token \cstok{par} into the input; after reading and expanding this \cstok{par}
token, \TeX\ will see the \<vertical command> token again. \ (The current
meaning of the control sequence ^|\par| will be used; \cstok{par} might no
longer stand for \TeX's |\par| primitive.)
\\\<letter>, \<otherchar>, \kern-1pt^|\char|\<8-bit number>, \<chardef token>,
\kern-1pt^|\noboundary|.\enskip
The most common commands of all are the character commands that tell
\TeX\ to append a character to the current horizontal
list, using the current font.
If two or more commands of this type occur in succession, \TeX\ processes
them all as a unit, converting to ligatures and/or
inserting kerns as directed by the font information. \ (Ligatures and
kerns may be influenced by invisible ``boundary'' characters at the left
and right, unless |\noboundary| appears.) \ Each character
command adjusts ^|\spacefactor|, using
the ^|\sfcode| table as described in Chapter~12.
In unrestricted horizontal mode, a
`|\discretionary{}{}{}|' item is appended after a character whose code is
the ^|\hyphenchar| of its font, or after a ligature formed from a sequence
that ends with such a character. ^^|\discretionary|
\\^|\accent|\<8-bit number>\<optional assignments>.
Here ^\<optional assignments> stands for zero or more \<assignment>
commands other than ^|\setbox|.
If the assignments are not followed by a \<character>, where
\<character> stands for any of the commands just discussed in the previous
paragraph, \TeX\ treats |\accent| as if it were |\char|, except that
the space factor is set to 1000. Otherwise the character that follows
the assignment is accented by the character that corresponds to the
\<8-bit number>. \ (The purpose of the intervening assignments is to
allow the accenter and accentee to be in different fonts.) \ If the
accent must be moved up or down, it is put into an hbox that is
raised or lowered. Then the accent is effectively superposed on the
character by means of kerns, in such a way that the width of the accent
does not influence the width of the resulting horizontal list.
Finally, \TeX\ sets |\spacefactor=1000|.
\\^|\/|.\enskip
If the last item on the current list is a character or ligature, an
explicit kern for its ^{italic correction} is appended.
\\^|\discretionary|\<general text>\<general text>\<general text>.\enskip
The three general texts are processed in restricted horizontal mode. They
should contain only fixed-width things; hence they aren't really very
general in this case. More precisely, the horizontal list formed by each
discretionary general text must consist only of characters, ligatures,
kerns, boxes, and rules; there should be no glue or penalty items, etc.
This command appends a discretionary item to the current list; see
Chapter~14 for the meaning of a discretionary item. The space factor is
not changed.
\\^|\-|.\enskip
This ``discretionary hyphen'' command is defined in Appendix H.
\\^|\setlanguage|\<number>.\enskip
See the conclusion of Appendix H.
\\|$|.\enskip
A ``^{math shift}'' character causes \TeX\ to enter math mode or display math
mode in the following way:
\TeX\ looks at the following token without expanding it. If that token
is a~|$| and if \TeX\ is currently in unrestricted horizontal mode,
then \TeX\ breaks the current paragraph into lines as explained above
(unless the current list is empty), returns to the enclosing vertical
mode or internal vertical mode, calculates values like |\prevgraf|
and |\displaywidth| and |\predisplaysize|, enters a new level of grouping,
inserts the |\everydisplay| tokens into the input, exercises the page
builder, processes `\<math mode material>|$$|' in display math mode, puts
the display into the enclosing vertical list as explained in Chapter~19
(letting vertical material ^{migrate}), exercises the page builder again,
increases |\prevgraf| by~3, and resumes horizontal mode again, with an
empty list and with the space factor equal to~1000. \ (You got that?) \
Otherwise \TeX\ puts the looked-at token back into the
input, enters a new level of grouping, inserts the |\everymath| tokens,
and processes `\<math mode material>|$|'; the math mode material
is converted to a horizontal list and appended to the current list,
surrounded by ``math-on'' and ``math-off'' items, and the space factor
is set to~1000. One consequence of these rules is that `|$$|' in
restricted horizontal mode simply yields an empty math formula.
\\None of the above: If any other primitive command of \TeX\ occurs
in horizontal mode, an error message will be given, and \TeX\ will try
to recover in a reasonable way. For example, if a superscript or
subscript symbol appears, or if any other inherently mathematical
command is given, \TeX\ will try to insert a `|$|' just before the
offending token; this will enter math mode.
%No room for the following redundant remarks:
%On the other hand if a totally
%misplaced token like |\endcsname| or |\omit| or |\eqno| or |#| appears
%in horizontal mode, \TeX\ will simply ignore it, after reporting
%the error. You might enjoy trying to type some really stupid input,
%just to see what happens. \ (Say `|\tracingall|' first, as explained
%in Chapter~27, in order to get maximum information.)
\endchapter
\strut{\rm Otherwise.} %
You may reduce all\/ \kern.5pt{\rm Verticals} into\/ {\rm Horizontals}.
\author JOSEPH ^{MOXON}, {\sl A Tutor to Astronomie and Geographie\/} (1659)
\bigskip
\strut\tt!~You can't use `\bslash moveleft' in horizontal mode.
\author \TeX\ (1982)
\eject
\beginchapter Chapter 26. Summary of\\Math Mode
^^{math mode}
To conclude the survey that was begun in Chapter 24, let us investigate
exactly what \TeX's digestive processes can do when \TeX\ is building
lists in math mode or in display math mode.
\ninepoint
\def\\{\smallbreak\textindent{$\bull$}}
\medbreak
\centerline{$*\qquad*\qquad*$}
\medskip\noindent
Three asterisks, just like those that appear here, can be found near the
end of Chapter~24.
Everything preceding the three asterisks in that chapter applies to
math mode as well as to vertical mode, so we need not repeat
all those rules. In particular, Chapter~24 explains assignment commands,
and it tells how kerns, penalties, marks, insertions, adjustments,
and ``whatsits'' are put into math lists. Our present goal
is to consider the commands that have an intrinsically mathematical
flavor, in the sense that they behave differently in math
mode than they do in vertical or horizontal modes.
Math lists are somewhat different from \TeX's other lists because they
contain three-pronged ``^{atoms}'' (see Chapter~17). Atoms come in thirteen
flavors: Ord, Op, Bin, Rel, Open, Close, Punct, Inner, Over, Under, Acc,
Rad, and Vcent. Each atom contains three ``^{fields}'' called its ^{nucleus},
^{superscript}, and ^{subscript}; and each field is either empty or is filled
with a math symbol, a box, or a subsidiary math list. Math symbols, in turn,
have two components: a family number and a position number.
It's convenient to introduce a few more rules of syntax, in order to specify
what goes into a math list:
\beginsyntax
<character>\is<letter>\alt<otherchar>\alt^|\char|<8-bit number>\alt%
<chardef token>
<math character>\is^|\mathchar|<15-bit number>\alt<mathchardef token>
\alt|\delimiter|<27-bit number>
<math symbol>\is<character>\alt<math character>
<math field>\is<filler><math symbol>\alt<filler>|{|<math mode material>|}|
<delim>\is<filler>^|\delimiter|<27-bit number>
\alt<filler><letter>\alt<filler><otherchar>
\endsyntax
We have already seen the concept of \<character> in Chapter~25.
Indeed, characters are \TeX's staple food: The vast majority of all
commands that reach \TeX's digestive processes in horizontal mode are instances
of the \<character> command, which specifies a number between 0 and~255 that
causes \TeX\ to typeset the corresponding character in the current font.
When \TeX\ is in math mode or display math mode, a \<character> command
takes on added significance: It specifies a number between 0 and~$32767=
2^{15}-1$. This is done by replacing the character number by its
^|\mathcode| value. If the
|\mathcode| value turns out to be $32768=\null$\hex{8000}, however,
the \<character>
is replaced by an ^{active character} token having the original character
code (0 to~255); \TeX\ forgets the original \<character> and expands this
active character according to the rules of Chapter~20.
A \<math character> defines a 15-bit number either by specifying it
directly with ^|\mathchar| or in a previous ^|\mathchardef|, or by
specifying a 27-bit |\delimiter| value; in the latter case, the least
significant 12~bits are discarded.
It follows that every \<math symbol>, as defined by the syntax above,
specifies a 15-bit number, i.e., a number between 0 and~32767.
Such a number can be represented in the form $4096c+256f+a$, where
$0\le c<8$, $0\le f<16$, and $0\le a<256$. If $c=7$, \TeX\ changes $c$~to~0;
and in this case if the current
value of\/ ^|\fam| is between 0 and~15, \TeX\ also replaces $f$~by~|\fam|.
This procedure yields, in all cases, a class number~$c$ between 0~and~6,
a family number~$f$ between 0~and~15, and a position number~$a$ between
0 and~255. \ (\TeX\ initializes the value of\/ |\fam| by implicitly
putting the assignment `|\fam=-1|' at the very beginning of\/ |\everymath|
and |\everydisplay|. Thus, the substitution of\/ |\fam| for~$f$ will occur
only if the user has explicitly changed~|\fam| within the formula.)
A \<math field> is used to specify the nucleus, superscript, or subscript
of an atom. When a \<math field> is a \<math symbol>, the $f$ and~$a$
numbers of that symbol go into the atomic field. Otherwise the \<math field>
begins with a~`|{|', which causes \TeX\ to enter a new level
of grouping and to begin a new math list; the ensuing \<math mode material>
is terminated by a~`|}|', at which point the group ends and the resulting
math list goes into the atomic field. If the math list turns out to be
simply a single Ord atom without subscripts or superscripts,
or an Acc whose nucleus is an Ord, the
enclosing braces are effectively removed.
A \<delim> is used to define both a ``small character'' $a$ in family~$f$
and a ``large character''~$b$ in family~$g$, where $0\le a,b\le255$
and $0\le f,g\le15$; these character codes are used to construct variable-size
^{delimiters}, as explained in Appendix~G\null. If the \<delim> is given
explicitly in terms of a 27-bit number, the desired codes are obtained
by interpreting that number as
$c\cdot2^{24}+f\cdot2^{20}+a\cdot2^{12}+g\cdot2^8+b$, ignoring the value
of~$c$. Otherwise the delimiter is specified as a \<letter> or
\<otherchar> token, and the 24-bit ^|\delcode| value of that character is
interpreted as $f\cdot2^{20}+a\cdot2^{12}+g\cdot2^8+b$.
\smallskip
Now let's study the individual commands as \TeX\ obeys them in math mode,
considering first the ones that have analogs in vertical and/or horizontal
mode:
\\^|\hskip|\<glue>, ^|\hfil|, ^|\hfill|, ^|\hss|, ^|\hfilneg|,
^|\mskip|\<muglue>.\enskip
A glue item is appended to the current math list.
\\\<leaders>\<box or rule>\<horizontal skip>.\enskip
Here ^\<horizontal skip> refers to one of the first five glue-appending
commands just mentioned; the formal syntax for \<leaders> and
for \<box or rule> is given in Chapter~24. A glue item that produces
^{leaders} is appended to the current list.
\\^|\nonscript|. A special glue item of width zero is appended; it will
have the effect of cancelling the following item on the list, if that item
is glue and if the |\nonscript| is eventually typeset in ``script style''
or in ``scriptscript style.''
\\^|\noboundary|. This command is redundant and therefore has no
effect; boundary ligatures are automatically disabled in math modes.
\\^\<space token>.\enskip
Spaces have no effect in math modes.
\\|\|\].\enskip ^^{control space}
A control-space command appends glue to the current list, using the same amount
that a \<space token> inserts in horizontal mode when the space factor is 1000.
\\^\<box>.\enskip
The box is constructed, and if the result is void nothing happens.
Otherwise a new Ord atom is appended to the current math list, and the
box becomes its nucleus.
\\^|\raise|\<dimen>\<box>, ^|\lower|\<dimen>\<box>.\enskip
This acts just like an unadorned \<box> command, except that the new box
being put into the nucleus is also shifted up or down by the specified amount.
\\^|\vcenter|\stretch\<box specification>\stretch
|{|\<vertical mode material>|}|.\enskip
A vbox is formed as if `|\vcenter|' had been `|\vbox|'. Then a new
^{Vcent} atom is appended to the current math list, and the box becomes
its nucleus.
\\\<vertical rule>.\enskip
A ^{rule} is appended to the current list (not as an atom).
\\^|\halign|\<box specification>|{|\<alignment material>|}|.\enskip
This command is allowed only in display math mode, and only when the current
math list is empty. The alignment is carried out exactly as if it were done in
the enclosing vertical mode (see Chapter~24), except that the lines are shifted
right by the ^|\displayindent|. The closing `|}|' may be followed by
optional \<assignment> commands other than ^|\setbox|,
after which `|$$|'~must conclude
the display. \TeX\ will insert the ^|\abovedisplayskip| and
^|\belowdisplayskip| glue before and after the result of the alignment.
\\^|\indent|.\enskip
An empty box of width ^|\parindent| is appended to the current list, as the
nucleus of a new Ord atom.
\\^|\noindent|.\enskip
This command has no effect in math modes.
\\|{|\<math mode material>|}|.\enskip
A character token of category 1, or a control sequence like~|\bgroup|
that has been |\let| equal to such a character token, causes \TeX\ to
start a new level of ^{grouping} and also to begin work on a new
math list. When such a group ends---with `|}|'---\TeX\
uses the resulting math list as the nucleus of a new Ord atom that
is appended to the current list. If the resulting math list is a
single Acc atom, however (i.e., an accented quantity),
that atom itself is appended.
\\\<math symbol>.\enskip
(This is the most common command in math mode; see the syntax near the
beginning of this chapter.) \ A math symbol determines three values,
$c$,~$f$, and~$a$, as explained earlier. \TeX\ appends an atom to the current
list, where the atom is of type Ord, Op, Bin, Rel, Open, Close, or Punct,
according as the value of~$c$ is 0,~1, 2, 3, 4, 5, or~6. The nucleus of
this atom is the math symbol defined by $f$ and~$a$.
\\^\<math atom>\<math field>.\enskip
A \<math atom> command is any of the following:
\begindisplay
|\mathord|\alt|\mathop|\alt|\mathbin|\alt|\mathrel|\alt|\mathopen|\cr
\qquad\alt|\mathclose|\alt|\mathpunct|\alt|\mathinner|%
\alt|\underline|\alt|\overline|\cr
\enddisplay
\TeX\ processes the \<math field>, then appends a new atom of the
specified type to the current list; the nucleus of this atom contains
the specified field.
\\^|\mathaccent|\<15-bit number>\<math field>.\enskip
\TeX\ converts the \<15-bit number> into $c$, $f$, and~$a$ as it does
with any |\mathchar|. Then it processes the \<math field> and appends a
new Acc atom to the current list. The nucleus of this atom contains
the specified field; the accent character in this atom contains $(a,f)$.
\\^|\radical|\<27-bit number>\<math field>.\enskip
\TeX\ converts the \<27-bit number> into $a$, $f$, $b$, and~$g$ as it does
with any |\delimiter|. Then it processes the \<math field> and appends a
new Rad atom to the current list. The nucleus of this atom contains
the specified field; the delimiter field in this atom contains
$(a,f)$ and $(b,g)$.
\\^\<superscript>\<math field>.\enskip
A \<superscript> command is an explicit or implicit character token of
category~7. If the current list does not end with an atom, a new Ord atom
with all fields empty is appended; thus the current list will end with an
atom, in~all cases. The superscript field of this atom should be empty; it
is made nonempty by changing it to the result of the specified \<math field>.
\\^\<subscript>\<math field>.\enskip
A \<subscript> command is an explicit or implicit character token of
category~8. It acts just like a \<superscript> command, except, of course,
that it affects the subscript field instead of the superscript field.
\\^|\displaylimits|, ^|\limits|, ^|\nolimits|.\enskip These commands are
allowed only if the current list ends with an Op atom. They modify a
special field in that Op atom, specifying what conventions should be used
with respect to limits. The normal value of that field is |\displaylimits|.
\\^|\/|.\enskip
A kern of width zero is appended to the current list. \ (This will have the
effect of adding the italic correction to the previous character, if the
italic correction wouldn't normally have been added.)
\\^|\discretionary|\<general text>\<general text>\<general text>.
This command is treated just as in horizontal mode (see Chapter~25), but the
third \<general text> must produce an empty list.
\\^|\-|.\enskip
This command is usually equivalent to `|\discretionary{-}{}{}|';
the `|-|' is therefore interpreted as a ^{hyphen}, not as a minus sign.
\ (See Appendix~H.)
\def\s{\hskip0pt plus1pt}
\\^|\mathchoice|\s
$\langle$filler$\rangle$\s|{|\s$\langle$math mode material$\rangle$\s|}|\s
$\langle$filler$\rangle$\s|{|\s$\langle$math mode material$\rangle$\s|}|\break
$\langle$filler$\rangle$|{|$\langle$math mode material$\rangle$|}|
$\langle$filler$\rangle$|{|$\langle$math mode material$\rangle$|}|.
Four math lists, which are defined as in the
second alternative of a \<math field>, are
recorded in a ``choice item'' that is appended to the current list.
\\^|\displaystyle|, ^|\textstyle|, ^|\scriptstyle|,
^|\scriptscriptstyle|.\enskip A style-change item that corresponds to the
specified style is appended to the current list.
\\^|\left|\<delim>\<math mode material>^|\right|\<delim>.\enskip
\TeX\ begins a new group, and processes the \<math mode material>
by starting out with a new math list that begins with a left boundary
item containing the first delimiter. This group must be terminated by
`|\right|', at which time the internal math list is completed with a
right boundary item containing the second delimiter. Then \TeX\
appends an Inner atom to the current list; the nucleus of this atom
contains the internal math list.
\\\<generalized fraction command>.\enskip
This command takes one of six forms:
\begindisplay
^|\over|\alt^|\atop|\alt^|\above|\<dimen>\cr
\qquad\alt^|\overwithdelims|\<delim>\<delim>\cr
\qquad\alt^|\atopwithdelims|\<delim>\<delim>\cr
\qquad\alt^|\abovewithdelims|\<delim>\<delim>\<dimen>\cr
\enddisplay
(See Chapter 17.) \ When \TeX\ sees a \<generalized fraction command> it
takes the entire current list and puts it into the numerator field of a
generalized fraction item. The denominator field of this new item is
temporarily empty; the left and right delimiter fields are set equal
to the specified delimiter codes. \TeX\ saves this generalized fraction
item in a special place associated with the current level of math mode
processing. \ (There should be no other generalized fraction item in
that special place, because constructions like `|a\over b\over c|' are
illegal.) \ Then \TeX\ makes the current list empty and continues to
process commands in math mode. Later on, when the current level of math
mode is completed (either by coming to a~`|$|' or a~`|}|' or a~|\right|,
depending on the nature of the current group), the current list will be
moved into the denominator field of the generalized fraction item that
was saved; then that item, all by itself, will take the place of the
entire list. However, in the special case that the current list began
with |\left| and will end with |\right|, the boundary items will be
extracted from the numerator and denominator of the generalized fraction,
and the final list will consist of three items: left boundary,
generalized fraction, right boundary. \ (If you want to watch the process
by which math lists are built, you might find it helpful to type `^|\showlists|'
while \TeX\ is processing the denominator of a generalized fraction.)
\\^\<eqno>\<math mode material>|$|.\enskip
Here \<eqno> stands for either ^|\eqno| or ^|\leqno|; these commands
are allowed only in display math mode. Upon reading \<eqno>, \TeX\
enters a new level of grouping, inserts the ^|\everymath| tokens,
and enters non-display math mode to put the \<math mode material>
into a math list. When that math list is completed, \TeX\ converts it
to a horizontal list and puts the result into a box that will be used
as the equation number of the current display. The closing |$|~token
will be put back into the input, where it will terminate the display.
\\|$|.\enskip
If \TeX\ is in display math mode, it reads one more token, which must
also be~|$|. In either case, the math-shift command terminates the current
level of math mode processing and ends the current group, which should
have begun with either~|$| or~\<eqno>. Once the math list is finished, it
is converted into a horizontal list as explained in Appendix~G\null. \TeX\
scans \<one optional space> after completing a displayed formula; this is
usually the implicit space at the end of a line in the input file.
\\^|\unhbox|\<8-bit number>, ^|\unhcopy|\<8-bit number>.\enskip
The specified box register must be void. Nothing happens.
\\None of the above: If any other primitive command of \TeX\ occurs in
math mode, an error message will be given, and \TeX\ will try to recover
in a reasonable way. For example, if a |\par| command appears, or if any
other inherently non-mathematical command is given, \TeX\ will try to
insert a `|$|' just before the offending token; this will lead out of math
mode. On the other hand if a totally misplaced token like |\endcsname| or
|\omit| or |#| appears in math mode, \TeX\ will simply ignore it, after
reporting the error. You might enjoy trying to type some really stupid
input, just to see what happens. \ (Say `|\tracingall|' first, as
explained in Chapter~27, in order to get maximum information.)
\ddangerexercise ^{Powers of ten}: The whole \TeX\ language has now been
summarized completely. To~demonstrate how much you know, name all of the ways
you can think of in which the numbers 10, ^^{Derek, Bo} 100, 1000, 10000,
and 100000 have special significance to \TeX.
\answer Radix 10 notation is used for numeric constants and for the output
of numeric data. The first 10 |\count| registers are displayed at each
|\shipout|, and their values are recorded on the |dvi| file at such times.
% Also, TeX was first implemented on a DEC-10.
% The \catcode for <space> is 10.
% My birthday is January 10.
% Can this all be just a coincidence?
A box whose glue has stretched or shrunk to its stated stretchability
or shrinkability has badness 100; this badness value separates ``loose''
boxes from ``very loose'' or ``underfull'' ones. \TeX\ will scroll up
to 100 errors in a single paragraph before giving up (see Chapter~27).
The normal values of\/ |\spacefactor| and |\mag| are 1000. A~|\prevdepth|
value of $-1000\pt$ suppresses interline glue. The badness rating of a
box is at most 10000, except that the |\badness| of
an overfull box is 1000000. |INITEX| initializes |\tolerance| to
10000, thereby making all line breaks feasible. Penalties of 10000 or more
prohibit breaks; penalties of $-10000$ or less make breaks mandatory. The
cost of a page break is 100000, if the badness is 10000 and if the
associated penalties are less than 10000 in magnitude (see Chapter~15).
\ddangerexercise ^{Powers of two}: Name all of the ways
you can think of in which the numbers 8, 16, 32, 64, 128, 256, \dots\
have special significance to \TeX.
\answer \TeX\ allows constants to be expressed in radix 8 (octal)
or radix~16 (hexadecimal) notation, and it uses hexadecimal notation to
display |\char| and |\mathchar| codes. There are 16 families for math
fonts, 16 input streams for |\read|, 16 output streams for |\write|.
A |\catcode| value must be less than~16. The notation |^^?|, |^^@|,
|^^A| specifies characters whose codes differ by~64
from the codes of |?|, |@|, |A|; this convention applies only to
characters with ASCII codes less than~128. There are 256 possible characters,
hence 256 entries in each of the |\catcode|, |\mathcode|,
|\lccode|, |\uccode|, |\sfcode|, and |\delcode| tables. All
|\lccode|, |\uccode|, and |\char| values
must be less than~256. A font has at most 256 characters. There are
256~|\box| registers, 256~|\count| registers, 256~|\dimen| registers,
256~|\skip| registers, 256~|\muskip| registers, 256~|\toks| registers,
256~hyphenation tables.
The ``at size'' of a font must be less than~$2048\pt$, i.e.,~$2^{11}\pt$.
Math delimiters are encoded by multiplying the math~code of the ``small
character'' by~$2^{12}$. The magnitude of
a~\<dimen> value must be less than~$16384\pt$, i.e.,~$2^{14}\pt$;
similarly, the \<factor> in a~\<fil dimen> must be less than~$2^{14}$.
A~|\mathchar| or |\spacefactor| or |\sfcode| value must be less than~$2^{15}$;
a~|\mathcode| or |\mag| value must be less than or equal to~$2^{15}$,
and $2^{15}$ denotes an ``active'' math character. There
are $2^{16}\rm\,sp$ per~pt. A~|\delcode| value
must be less than~$2^{24}$; a~|\delimiter|, less than $2^{27}$.
The |\end| command sometimes contributes
a penalty of $-2^{30}$ to the current page. A~\<dimen> must be less than
$2^{30}\rm\,sp$ in absolute value; a~\<number> must be
less than $2^{31}$ in absolute value.
\endchapter
Mathematics is known in the trade as {\rm difficult,} or {\rm penalty, copy}
because it is slower, more difficult, and more expensive to set in type
than any other kind of copy normally occurring in books and journals.
\author UNIVERSITY OF ^{CHICAGO} PRESS, {\sl A Manual of Style\/} %
(1969) % 12th edition, page 295
\bigskip
The tale of Math is a complex one,
and it resists both a simple plot summary
and a concise statement of its meaning.
\author PATRICK K. ^{FORD}, {\sl The Mabinogi\/} (1977) % p89
% from his introduction to "Math Son of Mathonwy"
\eject
\beginchapter Chapter 27. Recovery from\\Errors
OK, everything you need to know about \TeX\ has been explained---unless you
happen to be fallible. If you don't plan to make any errors, don't bother to
read this chapter. Otherwise you might find it helpful to make use of some
of the ways that \TeX\ tries to pinpoint bugs in your manuscript.
In the trial runs you did when reading Chapter 6, you learned the general
form of ^{error messages}, and you also learned the various ways in which
you can respond to \TeX's complaints. With practice, you will be able to
correct most errors ``online,'' as soon as \TeX\ has detected them, by
inserting and deleting a few things. The right way to go about this is to
be in a mellow mood when you approach \TeX, and to regard the error
messages that you get as amusing puzzles---``Why did the machine do
that?''---rather than as personal insults.
\TeX\ knows how to issue more than a hundred different sorts of error messages,
and you probably never will encounter all of them, because some types of
mistakes are very hard to make. We discussed the ``undefined control sequence''
error in Chapter~6; let's take a look at a few of the others now.
If you misspell the name of some unit of measure---for example, if you
type `|\hsize=4im|' instead of `|\hsize=4in|'---you'll get an error
message that looks something like this:
\begintt
! Illegal unit of measure (pt inserted).
<to be read again>
i
<to be read again>
m
<*> \hsize=4im
\input story
?
\endtt
^^|Illegal unit|
\TeX\ needs to see a legal unit before it can proceed; so in this case it has
implicitly inserted `|pt|' at the current place in the input, and it has
set |\hsize=4pt|.
What's the best way to ^{recover} from such an error? Well, you should
always type `|H|' or `|h|' to see the help message, if you aren't sure
what the error message means. Then you can look at the lines of context
and see that \TeX\ will read `|i|' and then `|m|' and then `~|\input|
|story|~' if you simply hit \<return> and carry on. Unfortunately, this
easy solution isn't very good, because the `|i|' and~`|m|' will be typeset
as part of the text of a new paragraph. A much more graceful recovery is
possible in this case, by first typing~`|2|'. This tells \TeX\ to discard
the next two tokens that it reads; and after \TeX\ has done so, it will
stop again in order to give you a chance to look over the new situation.
Here is what you will see:
\begintt
<recently read> m
|indent
<*> \hsize=4im
\input story
?
\endtt
Good; the `|i|' and `|m|' are read and gone. But if you hit \<return> now,
\TeX\ will `|\input story|' and try to typeset the |story.tex| file with
|\hsize=4pt|; that won't be an especially exciting experiment, because it
will simply produce dozens of overfull boxes, one for every syllable
of the story. Once again there's a better way: You can insert the command
that you had originally intended, by typing
\begintt
I\hsize=4in
\endtt
now. This instructs \TeX\ to change |\hsize| to the correct value, after
which it will |\input story| and you'll be on your way.
\exercise Ben ^{User} typed `|8|', not `|2|', in response to the
error message just considered; his idea was to delete `|i|', `|m|',
`|\input|', and the five letters of `|story|'. But \TeX's response was
\begintt
<*> \hsize=4im \input stor
y
\endtt
Explain what happened.
\answer He forgot to count the space; \TeX\ deleted `|i|', `|m|',
`\]', `|\input|', and four letters. (But all is not lost; he can
type `|1|' or `|2|', then \<return>, and after being prompted by `|*|' he
can enter a new line of input.)
\TeX\ usually tries to recover from errors either by ignoring a command
that it doesn't understand, or by inserting something that will keep it
happy. For example, we saw in Chapter~6 that \TeX\ ignores an
undefined control sequence; and we just observed that \TeX\ inserts
`|pt|' when it needs a physical unit of measure.
Here's another example where \TeX\ puts something in:
\begintt
! Missing $ inserted.
<inserted text>
$
<to be read again>
^
l.11 the fact that 32768=2^
{15} wasn't interesting
? H
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.
\endtt
^^{help message} ^^|Missing|
(The user has forgotten to enclose a formula in |$| signs, and \TeX\
has tried to recover by inserting one.) \ In this case the \<inserted text>
is explicitly shown, and it has not yet been read; by contrast, our
previous example illustrated a case where \TeX\ had already internalized
the `|pt|' that it had inserted. Thus the user has a chance here to
remove the inserted~`|$|' before \TeX\ really sees it.
What should be done? The error in this example occurred before \TeX\ noticed
anything wrong; the characters `32768=2' have already been typeset in
horizontal mode. There's no way to go back and cancel the past, so
the lack of proper spacing around the `=' cannot be fixed. Our goal of
error recovery in this case is therefore not to produce perfect
output; we want rather to proceed in some way so that \TeX\ will pass by
the present error and detect subsequent ones. If we were simply to hit
\<return> now, our aim would not be achieved, because \TeX\ would
typeset the ensuing text as a math formula: `$^{15} wasn't interesting
\ldots$'\thinspace; another error would be detected when the paragraph is
found to end before any closing~`|$|' has appeared. On the other hand,
there's a more elaborate way to recover, namely to type~`|6|' and then
`|I$^{15}$|'; this deletes `|$^{15}|' and inserts a
correct partial formula. But that's more complicated than necessary.
The best solution in this case is to type just~`|2|' and then go on;
\TeX\ will typeset the incorrect equation `32768=215', but the important
thing is that you will be able to check out the rest of the document as if
this error hadn't occurred.
\dangerexercise Here's a case in which a backslash was inadvertently omitted:
\begintt
! Missing control sequence inserted.
<inserted text>
\inaccessible
<to be read again>
m
l.10 \def m
acro{replacement}
\endtt
% The help message issued by TeX refers to "exercise 27.2 in The TeXbook"!
\TeX\ needs to see a control sequence after `|\def|', so it has inserted one
that will allow the processing to continue. \ (This control sequence is
shown as `^|\inaccessible|', but it has no relation to any control
sequence that you can actually specify in an error-free manuscript.) \
If you simply hit \<return> at this point, \TeX\ will define the
inaccessible control sequence, but that won't do you much good;
later references to |\macro| will be undefined. Explain how to
recover from this error so that the effect will be the same as if
line~10 of the input file had said `|\def\macro{replacement}|'.
\answer First delete the unwanted tokens, then insert what you want:
Type `|6|' and then `|I\macro|'. \ (Incidentally, there's a sneaky
way to get at the |\inaccessible| control sequence by typing
\begintt
I\garbage{}\let\accessible=
\endtt
in response to an error message like this. The author ^^{Knuth} designed
\TeX\ in such a way that you can't destroy anything by playing such
nasty tricks.)
\dangerexercise When you use the `|I|' option to respond to an error message,
the rules of Chapter~8 imply that \TeX\ removes all spaces from the
right-hand end of the line. Explain how you can use the `|I|'~option
to insert a ^{space}, in spite of this fact.
\answer `|I|\]|%|' does the trick, if |%| is a ^{comment character}.
Some of the toughest errors to deal with are those in which you make a
mistake on line~20 (say), but \TeX\ cannot tell that anything is amiss
until it reaches line~25 or so. For example, if you forget a `|}|' that
completes the argument to some macro, \TeX\ won't notice any problem
until reaching the end of the next paragraph. In such cases you probably
have lost the whole paragraph; but \TeX\ will usually be able to get
straightened out in time to do the subsequent paragraphs as if
nothing had happened. A ``^{runaway argument}'' will be displayed,
and by looking at the beginning of that text you should be able to
figure out where the missing `|}|' belongs.
It's wise to remember that the first error in your document may well spawn
spurious ``errors'' later on, because anomalous commands can inflict
serious injury on \TeX's ability to cope with the subsequent material.
But most of the time you will find that a single run through the
machine will locate all of the places in which your input conflicts
with \TeX's rules.
When your error is due to misunderstanding rather than mistyping,
the situation is even more serious: \TeX's error messages will probably
not be very helpful, even if you ask \TeX\ for help.
If you have unknowingly redefined an important control sequence---for
example, if you have said `|\def\box{...}|'---all sorts of strange
disasters might occur. Computers
aren't clairvoyant, and \TeX\ can only explain what looks wrong from
its own viewpoint; such an explanation is bound to be mysterious
unless you can understand the machine's attitude. The solution
to this problem is, of course, to seek human counsel and advice;
or, as a last resort, to read the instructions in Chapters 2, 3, \dots, 26.
\newcount\serialnumber
\def\firstnumber{\serialnumber=0 }
\def\nextnumber{\advance \serialnumber by 1
\number\serialnumber)\nobreak\hskip.2em }
\dangerexercise J. H. ^{Quick} (a student) once defined the following
set of macros:
\begintt
\newcount\serialnumber
\def\firstnumber{\serialnumber=0 }
\def\nextnumber{\advance \serialnumber by 1
\number\serialnumber)\nobreak\hskip.2em }
\endtt
Thus he could type, for example,
\begintt
\firstnumber
\nextnumber xx, \nextnumber yy, and \nextnumber zz
\endtt
and \TeX\ would typeset
\firstnumber
`\nextnumber xx, \nextnumber yy, and \nextnumber zz'.
Well, this worked fine, and he showed the macros to his buddies.
But several months later he received a frantic phone call; one of his
friends had just encountered a really ^{weird error} message:
\begintt
! Missing number, treated as zero.
<to be read again>
c
l.107 \nextnumber minusc
ule chances of error
?
\endtt
Explain what happened, and advise Quick what to do.
\answer The `^|minus|' of `|minuscule|' was treated as part of
the |\hskip| command in |\nextnumber|. Quick should put `|\relax|'
at the end of his macro. \ (The ^{keywords} ^|l|, ^|plus|, |minus|,
^|width|, ^|depth|, or ^|height| might just happen to occur in text
when \TeX\ is reading a glue specification or a rule specification;
designers of general-purpose macros should guard against this.
If you get a `^|Missing number|' error and you can't guess
why \TeX\ is looking for a number, plant the instruction
`|\tracingcommands=1|' shortly before the error point; your log file
will show what command \TeX\ is working~on.)
Sooner or later---hopefully sooner---you'll get \TeX\ to process your
whole file without stopping once to complain. But maybe the output
still won't be right; the mere fact that \TeX\ didn't stop doesn't mean
that you can avoid proofreading. At this stage it's usually easy to
see how to fix typographic errors by correcting the input. Errors of
layout can be overcome by using methods we have discussed before:
Overfull boxes can be cured as described in Chapter~6; bad breaks can
be avoided by using ties or |\hbox| commands as discussed in Chapter~14;
math formulas can be improved by applying the principles of
Chapters 16--19.
But your output may contain seemingly inexplicable errors. For example,
if you have specified a font at some magnification that is not supported
by your printing software, \TeX\ will not know that there is any
problem, but the program that converts your |dvi| file to hardcopy
might not tell you that it has substituted an ``approximate'' font
for the real one; the resultant spacing may look quite horrible.
If you can't find out what went wrong, try the old trick of simplifying
your program: Remove all the things that do work, until you obtain
the shortest possible input file that fails in the same way as the
original. The shorter the file, the easier it will be for you or somebody
else to pinpoint the problem.
Perhaps you'll wonder why \TeX\ didn't put a blank space in some
position where you think you typed a space. Remember that \TeX\ ignores
^{spaces} that follow control words, when it reads your file.
\ (\TeX\ also ignores a space after a \<number> or a \<unit of
measure> that appears as an argument to a primitive command; but if
you are using properly designed macros, such rules will not concern
you, because you will probably not be using primitive commands~directly.)
\ddanger On the other hand, if you are designing macros, the task of
troubleshooting can be a lot more complicated. For example, you may
discover that \TeX\ has emitted three blank spaces when it processed some
long sequence of complicated code, consisting of several dozen commands.
How can you find out where those spaces crept in? The answer is to set
`^|\tracingcommands||=1|', as mentioned in Chapter~13. This tells \TeX\ to
put an entry in your log file whenever it begins to execute a primitive
command; you'll be able to see when the command is `|blank| |space|'.
\danger Most implementations of \TeX\ allow you to ^{interrupt} the program
in some way. This makes it possible to diagnose the causes of ^{infinite
loops}. \TeX\ switches to ^|\errorstopmode| when interrupted; hence
you have a chance to insert commands into the input: You can abort the
run, or you can ^|\show| or change the current contents of control sequences,
registers, etc. You can also get a feeling for where \TeX\ is spending most
of its time, if you happen to be using an inefficient macro, since random
interrupts will tend to occur in whatever place \TeX\ visits most often.
\danger Sometimes an error is so bad that \TeX\ is forced to quit
prematurely. For example, if you are running in ^|\batchmode| or
^|\nonstopmode|, \TeX\ makes an ``^{emergency stop}'' if it needs
input from the terminal; this happens when a necessary file cannot
be opened, or when no ^|\end| command was found in the input
document. Here are some of the messages you might get just before
\TeX\ gives up the ghost: \enddanger
{\ninepoint
\def\fatal#1. {\medbreak{\tt#1.}\par\nobreak\smallskip\noindent\ignorespaces}
\fatal
Fatal format file error; I'm stymied.
^^|Fatal format file error|
This means that the preloaded format you have specified cannot be used,
because it was prepared for a different version of \TeX.
\fatal
That makes 100 errors; please try again.
\TeX\ has scrolled past 100 errors since the last paragraph ended, so
it's probably in an~endless ^{loop}. ^^{infinite loop}
\fatal
Interwoven alignment preambles are not allowed.
^^|Interwoven alignment preambles|
If you have been so devious as to get this message, you will understand
it, and you will deserve no sympathy.
\fatal
I can't go on meeting you like this.
^^|I can't go on|
A previous error has gotten \TeX\ out of whack. Fix it and try again.
\fatal
This can't happen.
^^|This can't happen|
Something is wrong with the \TeX\ you are using. Complain fiercely.
\goodbreak}
\danger There's also a dreadful message that \TeX\ issues only with
great reluctance. But it can happen:
\begintt
TeX capacity exceeded, sorry.
\endtt
^^|TeX capacity exceeded|
This, alas, means that you have tried to stretch \TeX\ too far. The
message will tell you what part of \TeX's memory has become overloaded;
one of the following fourteen things will be mentioned:
\begindisplay
|number of strings|\qquad(names of control sequences and files)\cr
|pool size|\qquad(the characters in such names)\cr
|main memory size|\qquad(boxes, glue, breakpoints, token lists,
characters, etc.)\cr
|hash size|\qquad(control sequence names)\cr
|font memory|\qquad(font metric data)\cr
|exception dictionary|\qquad(hyphenation exceptions)\cr
|input stack size|\qquad(simultaneous input sources)\cr
|semantic nest size|\qquad(unfinished lists being constructed)\cr
|parameter stack size|\qquad(macro parameters)\cr
|buffer size|\qquad(characters in lines being read from files)\cr
|save size|\qquad(values to restore at group ends)\cr
|text input levels|\qquad(|\input| files and error insertions)\cr
|grouping levels|\qquad(unfinished groups)\cr
|pattern memory|\qquad(hyphenation pattern data)\cr
\enddisplay
The current amount of memory available will also be shown.
\danger If you have a job that doesn't overflow \TeX's capacity, yet
you want to see just how closely you have approached the limits,
just set ^|\tracingstats| to a positive value before the end of your
job. The log file will then conclude with a report on your actual
usage of the first eleven things named above (i.e., the number of strings,
\dots, the save size), in that order. ^^{stack positions}
Furthermore, if you set |\tracingstats| equal to 2~or~more, \TeX\
will show its current memory usage whenever it
does a ^|\shipout| command. Such statistics are broken into two
parts; `|490&5950|' means, for example, that 490 words are being used
for ``large'' things like boxes, glue, and
breakpoints, while 5950 words are being used for ``small'' things like
tokens and characters.
\danger What can be done if \TeX's capacity is exceeded? All of the
above-listed components of the capacity can be increased, provided
that your computer is large enough; in fact, the space necessary to
increase one component can usually be obtained by decreasing some
other component, without increasing the total size of \TeX\null.
If you have an especially important application, you may be able
to convince your local system people to provide you with a special
\TeX\ whose capacities have been hand-tailored to your needs.
But before taking such a drastic step, be sure that you are using
\TeX\ properly. If you have specified a gigantic paragraph or
a gigantic alignment that spans more than one page, you should
change your approach, because \TeX\ has to read all the way to the
end before it can complete the line-breaking or the alignment
calculations; this consumes huge amounts of memory space. If you have
built up an enormous macro library, you should remember that \TeX\
has to remember all of the replacement texts that you define; therefore
if memory space is in short supply, you should load only the macros
that you need. \ (See Appendices B and~D\null, for ideas on how to make
macros more compact.)
\danger Some erroneous \TeX\ programs will overflow any finite
memory capacity. For example, after `|\def\recurse{(\recurse)}|', the
^^{recursion} use of\/ |\recurse| will immediately bomb out:
\begintt
! TeX capacity exceeded, sorry [input stack size=80].
\recurse ->(\recurse
)
\recurse ->(\recurse
)
...
\endtt
The same sort of error will obviously occur no matter how much you increase
\TeX's input stack size.
\ddanger The special case of ``^|save size|'' capacity exceeded is one
of the most troublesome errors to correct, especially if you run into
the error only on long jobs. \TeX\ generally uses up two words of save
size whenever it performs a non-global assignment to some quantity
whose previous value was not assigned at the same level of ^{grouping}.
When macros are written properly, there will rarely be a need for
more than 100 or~so things on the ``^{save stack}''; but it's possible to
make save stack usage grow without limit if you make both local and
^{global assignments} to the same variable. You can figure out what \TeX\
puts on the save stack by setting ^|\tracingrestores||=1|; then your
log file will record information about whatever is removed from the
stack at the end of a group. For example, let |\a| stand for the
command `|\advance\day|~|by|~|1|'; let |\g| stand for
`|\global\advance\day|~|by|~|1|'; and consider the following commands:
\begintt
\day=1 {\a\g\a\g\a}
\endtt
The first |\a| sets |\day=2| and remembers the old value |\day=1| by putting
it on the save stack. The first |\g| sets |\day=3|, globally; nothing needs
to go on the save stack at the time of a global assignment. The next |\a|
sets |\day=4| and remembers the old value |\day=3| on the save stack.
Then |\g| sets |\day=5|; then |\a| sets |\day=6| and remembers |\day=5|.
^^{right brace}
Finally the `|}|' causes \TeX\ to go back through the save stack; if
|\tracingrestores=1| at this point, the log file will get the following
data:
\begintt
{restoring \day=5}
{retaining \day=5}
{retaining \day=5}
\endtt
Explanation: The |\day| parameter is first restored to its global value~5.
Since this value is global, it will be retained, so the other saved values
(|\day=3| and |\day=1|) are essentially ignored. Moral: If you find \TeX\
retaining a lot of values, you have a set of macros that could cause
the save stack to overflow in large enough jobs. To prevent this, it's
usually wise to be consistent in your assignments to each variable that
you use; the assignments should either be global always or local always.
\ddanger \TeX\ provides several other kinds of tracing in addition to
|\tracingstats| and |\tracingrestores|: We have already discussed
|\tracingcommands| in Chapters 13 and~20, |\tracingparagraphs| in
Chapter~14, |\tracingpages| in Chapter~15, and |\tracingmacros| in
Chapter~20. There is also ^|\tracinglostchars|, which (if positive)
causes \TeX\ to record each time a character has been dropped because
it does not appear in the current font; and ^|\tracingoutput|, which
(if positive) causes \TeX\ to display in symbolic form the contents
of every box that is being shipped out to the ^|dvi|~file. ^^|\shipout|
The latter allows you to see if things have been typeset properly, if
you're trying to decide whether some anomaly was caused by \TeX\ or
by some other software that acts on \TeX's output.
\danger When \TeX\ displays a box as part of diagnostic output, the amount
of data is controlled by two parameters called ^|\showboxbreadth| and
^|\showboxdepth|. The first of these, which plain \TeX\ sets equal to~5,
tells the maximum number of items shown per level; the second, which plain
\TeX\ sets to~3, tells the deepest level. For example, a small box
whose full contents are ^^{internal box format} ^^{symbolic box format}
\begintt
\hbox(4.30554+1.94444)x21.0, glue set 0.5
.\hbox(4.30554+1.94444)x5.0
..\tenrm g
.\glue 5.0 plus 2.0
.\tenrm || (ligature ---)
\endtt
will be abbreviated as follows when |\showboxbreadth=1| and |\showboxdepth=1|:
^^{ligature} ^^{em-dash}
\begintt
\hbox(4.30554+1.94444)x21.0, glue set 0.5
.\hbox(4.30554+1.94444)x5.0 []
.etc.
\endtt
And if you set |\showboxdepth=0|, you get only the top level:
\begintt
\hbox(4.30554+1.94444)x21.0, glue set 0.5 []
\endtt
(Notice how `^|[]|' and `^|etc.|'~indicate that the data has been
truncated.)
\danger A nonempty hbox is considered ``^{overfull}'' if its ^{glue}
cannot shrink to achieve the specified size, provided that ^|\hbadness| is
less than~100 or that the excess width (after shrinking by the maximum
amount) is more than ^|\hfuzz|. It is ``^{tight}'' if its glue shrinks and
the ^{badness} exceeds |\hbadness|; it is ``^{loose}'' if its glue
stretches and the badness exceeds |\hbadness| but is not greater than~100;
it is ``^{underfull}'' if its glue stretches and the badness is greater
than |\hbadness| and greater than~100. Similar remarks apply to nonempty
vboxes. \TeX\ prints a warning message and displays the offending box,
whenever such anomalies are discovered. Empty boxes are never considered
to be anomalous.
\ddanger When an ^{alignment} is ``overfull'' or ``tight'' or ``loose'' or
``underfull,'' you don't get a warning message for every aligned line;
you get only one message, and \TeX\ displays a {\sl^{prototype row}\/}
(or, with ^|\valign|, a {\sl prototype column\/}).
For example, suppose you say `|\tabskip=0pt plus10pt
\halign to200pt{&#\hfil\cr...\cr}|', ^^|\halign| and suppose that the aligned
material turns out to make two columns of widths $50\pt$ and $60\pt$,
respectively. Then you get the following message:
\begintt
Underfull \hbox (badness 2698) in alignment at lines 11--18
[] []
\hbox(0.0+0.0)x200.0, glue set 3.0
.\glue(\tabskip) 0.0 plus 10.0
.\unsetbox(0.0+0.0)x50.0
.\glue(\tabskip) 0.0 plus 10.0
.\unsetbox(0.0+0.0)x60.0
.\glue(\tabskip) 0.0 plus 10.0
\endtt
The ``unset boxes'' in a prototype row show the individual column widths.
In this case the ^{tabskip glue} has to stretch 3.0 times its stretchability,
in order to reach the $200\pt$ goal, so the box is underfull. \ (According
to the formula in Chapter~14, the badness of this situation is~2700; \TeX\
actually uses a similar but more efficient formula, so it computes a
badness of~2698.) \ Every line of the alignment will be underfull, but
only the prototype row will be displayed in a warning message.
``^{Overfull rules}'' are never appended to the lines of overfull alignments.
\ddanger The |\tracing...|\ commands put all of their output into your log
file, unless the ^|\tracingonline| parameter is positive; in the latter
case, all diagnostic information goes to the terminal as well as to the
log file. Plain \TeX\ has a ^|\tracingall| macro that turns on the
maximum amount of tracing of all kinds. It not only sets~up
|\tracingcommands|, |\tracingrestores|, |\tracingparagraphs|, and so on,
it also sets |\tracingonline=1|, and it sets ^|\showboxbreadth| and
^|\showboxdepth| to extremely high values, so that the entire contents of
all boxes will be displayed.
\ddanger Some production versions of \TeX\ have been streamlined for
speed. These implementations don't look at the values of the parameters
|\tracingparagraphs|, |\tracingpages|, |\tracingstats|, and
|\tracingrestores|, because \TeX\ runs faster when it doesn't have
to maintain statistics or keep tabs on whether tracing is required.
If you want all of \TeX's diagnostic tools, you should be sure to
use the right version.
\ddanger If you set ^|\pausing||=1|, \TeX\ will give you a chance to edit
each line of input as it is read from the file. In this way you can
make temporary patches (e.g., you can insert |\show...|\ commands)
while you're troubleshooting, without changing the actual contents
of the file, and you can keep \TeX\ running at human speed.
Final hint: When working on a long manuscript, it's best to prepare
only a few pages at a time. Set up a ``^{galley}'' file and a ``^{book}''
file, and enter your text on the galley file. \ (Put control
information that sets up your basic format at the beginning of this
file; an example of |galley.tex| appears in Appendix~E\null.) \
After the galleys come out looking right, you can append them to the
book file; then you can run the book file through \TeX\ occasionally,
in order to see how the pages really fit together. For example,
when the author prepared this manual, he did one chapter at a time,
and the longer chapters were split into subchapters.
\ddangerexercise Final exercise: Find all of the ^{lies} in this
manual, and all of the ^{jokes}.
\answer If this exercise isn't just a joke, the title of this
appendix is a lie.
\line{Final exhortation: G{\sc O} {\sc FORTH} now and create
{\sl masterpieces of the publishing art!\/}}
\endchapter
Who can understand his errors?
\author ^^{Biblical}{\sl Psalm 19\thinspace:\thinspace12\/} (c.~1000 B.C.)
\bigskip
It is one thing, to shew a Man that he is in an Error,
and another, to put him in possession of Truth.
\author JOHN ^{LOCKE}, {\sl An Essay Concerning Humane Understanding\/} (1690)
% bk 4 ch 7 sec 11
\eject
\beginchapter Appendix A. Answers to\\All the\\Exercises
The preface to this manual points out the wisdom of trying to figure out
each exercise before you look up the answer here. But these answers are intended
to be read, since they occasionally provide additional information that
you are best equipped to understand when you have just worked on a problem.
\immediate\closeout\ans % this makes the answers file ready
\ninepoint
\input answers
\endchapter
If you can't solve a problem,
you can always look up the answer.
But please, try first to solve it by yourself;
then you'll learn more and you'll learn faster.
\author DONALD E. ^{KNUTH}, {\sl The \TeX book\/} (1984)
\bigskip
How answer you for your selues?
\author WILLIAM ^{SHAKESPEARE}, {\sl Much Adoe About Nothing\/} (1598)
% Act IV, Scene 2, line 25
\eject
\beginchapter Appendix B. Basic\\Control\\Sequences
Let's begin this appendix with a chart that summarizes plain \TeX's
^^{summary of plain TeX} ^^|\+| ^^{tabbing} ^^{accents}
conventions.
\def\sep{\medskip\hrule width\hsize\medskip}
\medskip\smallskip
\hrule height .61803pt
\kern 1pt
\hrule
\medskip
\line{\strut Characters that are reserved for special purposes:\hfil
|\|\hfil|{|\hfil|}|\hfil|$|\hfil|&|\hfil|#|\hfil|%|\hfil|^|\hfil|_|\hfil|~|}
\sep
\halign to\hsize{\strut\hfil#\hfil\tabskip\z@ plus10pt&
\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil\tabskip\z@skip\cr
|\rm roman,|&|{\sl slanted},|&|{\bf boldface},|&|{\it italic\/} type|\cr
roman,&{\sl slanted},&{\bf boldface},&{\it italic\/} type\cr}
\sep
\halign to\hsize{\strut\hfil#\hfil\tabskip\z@ plus10pt&
\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil\tabskip\z@skip\cr
|``|&|''|&|--|&|---|&|?||`|&|!||`|&|\$|&|\#|&|\&|&|\%|&
|\ae|&|\AE|&|\oe|&|\OE|&|\aa|&|\AA|&|\ss|&|\o|&|\O|\cr
``&''&--&---&?`&!`&\$&\#&\&&\%&\ae&\AE&\oe&\OE&\aa&\AA&\ss&\o&\O\cr}
\sep
\halign to\hsize{\strut\hfil#\hfil\tabskip\z@ plus10pt&
\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil\tabskip\z@skip\cr
|\`a|&|\'e|&|\^o|&|\"u|&|\=y|&|\~n|&|\.p|&|\u\i|&
|\v s|&|\H\j|&|\t\i u|&|\b k|&|\c c|&|\d h|\cr
\`a&\'e&\^o&\"u&\=y&\~n&\.p&\u\i&\v s&\H\j&\t\i u&\b k&\c c&\d h\cr}
\sep
\halign to\hsize{\strut\hfil#\hfil\tabskip\z@ plus10pt&
\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil\tabskip\z@skip\cr
|\l|&|\L|&|\dag|&|\ddag|&|\S|&|\P|&|{\it\$|&
|\&}|&|\copyright|&|\TeX|&|\dots|\cr
\l&\L&\dag&\ddag&\S&\P&\phantom{\tt////}\it\$&
\it\&\phantom{\tt/}&\copyright&\TeX&\dots\cr}
\sep
\line{\strut Line break controls:\hfil
|\break|\hfil|\nobreak|\hfil|\allowbreak|\hfil|\hbox{unbreakable}|}
\line{\strut|dis\-cre\-tion\-ary hy\-phens|\hfil|virgule\slash breakpoint|}
\sep
\settabs 2\columns
\+\strut Breakable horizontal spaces:& Unbreakable horizontal spaces:\cr
\+|\|\] \ normal interword space& |~| \ normal interword space\cr
\+|\enskip| \ this\enskip much& |\enspace | \ this\enspace much\cr
\+|\quad | \ this\quad much& |\thinspace | \ this\thinspace much\cr
\+|\qquad | \ this\qquad much& |\negthinspace| \ this\negthinspace much\cr
\+\strut|\hskip| \<arbitrary dimen>& |\kern| \<arbitrary dimen>\cr
\sep
\smallskip
\+\strut Vertical spaces:\hfill
|\smallskip| $\vcenter{\hrule width2em\smallskip\hrule}$\hfill&
|\medskip| $\vcenter{\hrule width3em\medskip\hrule}$\hfill
|\bigskip| $\vcenter{\hrule width4em\bigskip\hrule}$&\cr
\smallskip
\sep
\line{\strut Page break controls:\hfil|\eject|\hfil|\supereject|\hfil
|\nobreak|\hfil|\goodbreak|\hfil|\filbreak|}
\+\strut Vertical spaces and good breakpoints:&
|\smallbreak|\hfill|\medbreak|\hfill|\bigbreak|&\cr
\sep
\settabs 4 \columns
\hbox{\strut|\settabs 4 \columns|}
\hbox{|\+Here's an example&of\hfill some &tabbing:&\hrulefill&\cr|}
\+Here's an example&of\hfill some &tabbing:&\hrulefill&\cr
\+\hfill|\hrulefill |&\hrulefill&\hfill|\dotfill |&\dotfill&\cr
\+\hfill|\leftarrowfill |&\leftarrowfill&
\hfill|\rightarrowfill |&\rightarrowfill&\cr
\+\hfill|\upbracefill |&\upbracefill&
\hfill|\downbracefill |&\downbracefill&\cr
\smallskip
\line{\strut More general alignments use |\halign|, |\valign|, |\omit|,
|\span|, and |\multispan|.}
\sep
\line{\strut Examples of the principal conventions for text layout
appear on the next page.}
\sep\unskip
\kern 1pt
\hrule height .61803pt
\eject
\begingroup\parindent\z@ \obeylines \baselineskip12pt plus 1pt
|% This test file generates the output shown on the opposite page.|
|% It's a bit complex because it tries to illustrate lots of stuff.|
|% TeX ignores commentary (like this) that follows a `%' sign.|
| |
|% First the standard output style is changed slightly:|
^|\hsize||=29pc % The lines in this book are 29 picas wide.|
^|\vsize||=42pc % The page body is 42 picas (not counting footlines).|
^|\footline||={\tenrm Footline\quad\dotfill\quad Page \folio}|
^|\pageno||=1009 % This is the starting page number (don't ask why).|
|% See Chapter 23 for the way to make other page format changes via|
|% \hoffset, \voffset, \nopagenumbers, \headline, or \raggedbottom.|
| |
^|\topglue|| 1in % This makes an inch of blank space (1in=2.54cm).|
^|\centerline||{\bf A Bold, Centered Title}|
^|\smallskip|| % This puts a little extra space after the title line.|
^|\rightline||{\it avec un sous-titre \`a la fran\c caise}|
|% Now we use \beginsection to introduce part 1 of the document.|
^|\beginsection|| 1. Plain \TeX nology % The next line must be blank!|
| |^^{blank line}
|The first paragraph of a new section is not indented.|
^|\TeX||\ recognizes the end of a paragraph when it comes to a blank|
|line in your manuscript file. % or to a `\par': See below.|^^|\par|
| |^^|\footnote|^^|\tt|^^|\char|
|Subsequent paragraphs {\it are\/} indented.\footnote*{The amount|
| of indentation can be changed by changing a parameter called|
|{\tt\char`\\parindent}. Turn the page for a summary of \TeX's most|
|important parameters.} (See?) The computer breaks a paragraph's|
|text into lines in an interesting way---see reference~[1]---and h%|
| yphenates words automatically when necessary.|^^{percent}
| |
^|\midinsert|| % This begins inserted material, e.g., a figure.|
|\narrower\narrower % This brings the margins in (see Chapter 14).|
^|\noindent|| |^|\llap||{``}If there hadn't been room for this material on|
|the present page, it would have been inserted on the next one.''|
|\endinsert % This ends the insertion and the effect of \narrower.|
| |
^|\proclaim|| Theorem T. The typesetting of $math$ is discussed in|
|Chapters 16--19, and math symbols are summarized in Appendix~F.|
| |
|\beginsection 2. Bibliography\par % `\par' acts like a blank line.|
^|\frenchspacing|| % (Chapter 12 recommends this for bibliographies.)|
^|\item||{[1]} D.~E. Knuth and M.~F. Plass, ``Breaking paragraphs|^^{Plass}
|into lines,'' {\sl Softw. pract. exp. \bf11} (1981), 1119--1184.|^^{Knuth}
^|\bye|| % This is the way the file ends, not with a \bang but a \bye.|
\eject\endgroup
{\parindent 20pt
\topglue 1in % This makes an inch of blank space (1in=2.54cm).
\centerline{\bf A Bold, Centered Title}
\smallskip % This puts a little extra space between lines here.
\rightline{\it avec un sous-titre \`a la fran\c caise}
% Now we use \beginsection to introduce part 1 of the document.
\beginsection 1. Plain \TeX nology % The next line must be blank!
The first paragraph of a new section is not indented.
\TeX\ recognizes the end of a paragraph when it comes to a blank
line in your manuscript file. % or to a `\par': See below.
Subsequent paragraphs {\it are\/} indented.*{\spacefactor=3000}
(See?) The computer breaks a paragraph's
text into lines in an interesting way---see reference~[1]---and h%
yphenates words automatically when necessary.
\midinsert % This begins inserted material, e.g., a figure.
\narrower\narrower % This brings the margins in (see Chapter 14).
\noindent \llap{``}If there hadn't been room for this material on
the present page, it would have been inserted on the next one.''
\endinsert % This ends the insertion and the effect of \narrower.
\proclaim Theorem T. The typesetting of $math$ is discussed in
Chapters 16--19, and math symbols are summarized in Appendix~F.
\beginsection 2. Bibliography\par% `\par' is just like blank line.
\frenchspacing % (Chapter 12 recommends this for bibliographies.)
\item{[1]} D.~E. Knuth and M.~F. Plass, ``Breaking paragraphs
into lines,'' {\sl Softw. pract. exp. \bf11} (1981), 1119--1184.
\vfill
\hrule width 2in \kern 2.6pt
\textindent{*}\strut
The amount of indentation can be changed by changing a parameter called
{\tt\char`\\parindent}. Turn the page for a summary of \TeX's most
important parameters.\par
\baselineskip 24pt
\line{Footline\quad\dotfill\quad Page 1009}
\eject}
The preceding example illustrates most of the basic things
that you can do directly with plain \TeX, but it does not provide an
exhaustive list. Thus, it uses |\centerline| and |\rightline|,
but not |\leftline| or |\line|; it uses |\midinsert|, but not
|\topinsert| or |\pageinsert|; it uses |\smallskip|, but not
|\medskip| or |\bigskip|; it uses |\llap| but not |\rlap|, |\item|
but not |\itemitem|, |\topglue| but not |\hglue|. It does not
illustrate |\raggedright| setting of paragraphs; it does not use
^|\obeylines| or ^|\obeyspaces| to shut off \TeX's automatic
formatting. ^^{as is, see obeylines, obeyspaces}
All such control sequences are explained later in this appendix, and
further information can be found by looking them up in the index.
The main purpose of the example is to serve as a reminder of the
repertoire of possibilities.
Most of the control sequences used in the example are defined by
macros of plain \TeX\ format, but three of them are ^{primitive}, i.e.,
built~in: `|\par|' (end of paragraph), `|\noindent|' (beginning of
non-indented paragraph), and `|\/|' (italic correction). The example
also assigns values to two of \TeX's primitive parameters, namely
|\hsize| and |\vsize|. \TeX\ has scores of parameters, all of which
are listed in Chapter~24, but only a few of them are of special
concern to the majority of \TeX\ users. Here are examples of how
you might want to give new values to the most important ^{parameters}
other than |\hsize| and |\vsize|:
\par\begingroup\nobreak\medskip\parindent\z@ \obeylines
^|\tolerance||=500|\quad(\TeX\ will tolerate lines whose badness is %
rated 500 or less.)
^|\looseness||=1|\quad(The next paragraph will be one line longer than %
usual.)
^|\parindent||=4mm|\quad(Paragraphs will be indented by four millimeters.)
^|\hoffset||=1.5in|\quad(All output will be shifted right by one and a %
half inches.)
^|\voffset||=24pt|\quad(All output will be shifted down by 24 points.)
^|\baselineskip||=11pt plus.1pt|\quad(Baselines will be $11\pt$ apart, %
or a bit more.)
^|\parskip||=3pt plus1pt minus.5pt|\quad(Extra space will %
precede each paragraph.)
\endgroup\medbreak\noindent
Plain \TeX\ uses |\parindent| also to control the amount of
indentation provided by ^|\item|, ^|\itemitem|, and ^|\narrower|.
\danger The remainder of this appendix is devoted to the details of the
plain \TeX\ format, which is a set of macros that come with normal
implementations of \TeX. These macros serve three basic purposes:
\ (1)~They make \TeX\ usable, because \TeX's primitive capabilities
operate at a very low level. A~``virgin'' \TeX\ system that has no
macros is like a newborn baby that has an immense amount to learn about
the real world; but it is capable of learning fast. \ (2)~The plain
\TeX\ macros provide a basis for more elaborate and powerful formats
tailored to individual tastes and applications. You can do a lot with
plain \TeX, but pretty soon you'll want to do even more. \ (3)~The macros
also serve to illustrate how additional formats can be designed.
\ninepoint
Somewhere in your computer system you should be able to find a file
called ^|plain.tex| that contains exactly what has been preloaded into
the running \TeX\ system that you use. Our purpose in the rest of this
appendix will be to discuss the contents of |plain.tex|. However, we
will not include a verbatim description, because some parts of that file
are too boring, and because the actual macros have been ``^{optimized}''
^^{efficiency}
with respect to memory space and running time. Unoptimized versions of the
macros are easier for humans to understand, so we shall deal with those;
|plain.tex| contains equivalent constructions that work better on a machine.
So here's the plan for the rest of Appendix~B\null: We shall go through
the contents of |plain.tex|, interspersing an edited transcription of
that file with comments about noteworthy details. When we come to
macros whose usage has not yet been explained---for example, somehow
|\vglue| and |\beginsection| never made it into Chapters 1 through~27---we
shall consider them from a user's viewpoint; but most of the time we shall
be addressing the issues from the standpoint of a macro designer.
\subsection The code tables. A format's first duty is to establish
^|\catcode| values. This is necessary because, for example, a |\def|
command can't be used until there are characters like |{| and~|}| of
categories 1 and~2. The ^|INITEX| program (which reads |plain.tex|
so that \TeX\ can be initialized) begins without knowing any grouping
characters; hence |plain.tex| starts out as follows:
\beginlines
|% This is the plain TeX format that's described in The TeXbook.|
|% N.B.: A version number is defined at the very end of this file;|
|% please change that number whenever the file is modified!|
|% And don't modify the file under any circumstances.|
\smallbreak
|\catcode`\{=1 % left brace is begin-group character|
|\catcode`\}=2 % right brace is end-group character|
|\catcode`\$=3 % dollar sign is math shift|
|\catcode`\&=4 % ampersand is alignment tab|
|\catcode`\#=6 % hash mark is macro parameter character|
|\catcode`\^=7 \catcode`\^^K=7 % circumflex and uparrow for superscripts|
|\catcode`\_=8 \catcode`\^^A=8 % underline and downarrow for subscripts|
|\catcode`\^^I=10 % ASCII tab is treated as a blank space|
|\chardef\active=13 \catcode`\~=\active % tilde is active|
|\catcode`\^^L=\active \outer\def^^L{\par} % ASCII form-feed is \outer\par|
\smallskip
|\message{Preloading the plain format: codes,}|
\endlines
These instructions set up the nonstandard characters
|^^K| and |^^A| for ^{superscripts} and ^{subscripts}, in addition to~|^|
and~|_|, so that people with extended character sets can use
{\tentex\char'13} and {\tentex\char1} as recommended in Appendix~C\null.
^^{uparrow char} ^^{downarrow char}
Furthermore |^^I| (^{ASCII} ^\<tab>) is given category~10 (space);
and |^^L| (ASCII
\<formfeed>) becomes an active character that will detect runaways on
files that have been divided into ``^{file pages}'' by ^\<formfeed>
characters. The control sequence ^|\active| is defined to yield the
constant~13; this is the one category code that seems to deserve a
symbolic name, in view of its frequent use in constructing
special-purpose macros.
When |INITEX| begins, category 12 (other) has been assigned to all~256
possible characters, except that the 52~letters |A...Z| and |a...z| are
category~11 (letter), and a few other assignments equivalent to the
following have been made:
\begintt
\catcode `\\ =0 \catcode`\ =10 \catcode `\% =14
\catcode`\^^@=9 \catcode`\^^M=5 \catcode`\^^?=15
\endtt
Thus `|\|' is already an escape character, ^^{backslash}
`\]' is a space, ^^\<space>
and `|%|' is available for comments on the first line of the file;
ASCII ^\<null> is ignored, ASCII ^\<return> is an end-of-line character,
and ASCII ^\<delete> is invalid.
The ^|\message| command shown above prints a progress report on the terminal
when |plain.tex| is being input by |INITEX|. Later on comes
`|\message{registers,}|' and several other messages, but we won't mention
them specifically. The terminal will eventually display something like
this when initialization is complete:
\begintt
** plain
(plain.tex Preloading the plain format: codes, registers,
parameters, fonts, more fonts, macros, math definitions,
output routines, hyphenation (hyphen.tex))
* \dump
Beginning to dump on file plain.fmt
\endtt
followed by a variety of statistics about what fonts were loaded, etc.
If you want to make a new format |super.tex| that adds more features
to |plain.tex|, it's best not to make a new file containing all the
plain stuff, or even to |\input plain|; just type `|&plain super|' in
response to |INITEX|'s ^|**| prompt, ^^{ampersand} ^^|fmt|
to input |plain.fmt| at high speed.
After the opening |\message|, |plain.tex| goes on to define a control
sequence ^|\dospecials| that lists all the characters whose catcodes should
probably be changed to~12 (other) when copying things verbatim:
\beginlines
|\def\dospecials{\do\ \do\\\do\{\do\}\do\$\do\&%|
| \do\#\do\^\do\^^K\do\_\do\^^A\do\%\do\~}|
\endlines
(Appendix E illustrates how to use |\dospecials|.) \ The ASCII codes for
\<null>, \<tab>, \<linefeed>, \<formfeed>, \<return>, and \<delete>
have not been included in the list.
At this point |plain.tex| completes its initialization of category codes
by setting |\catcode`\@=11|, thereby making the character `|@|' behave
temporarily like a letter. The command |\catcode`\@=12| will appear later,
hence ^{at-sign characters} will act just like ordinary punctuation
marks when \TeX\ is running. The idea is to make it easy for plain \TeX\
to have ^{private control sequences} that cannot be redefined by ordinary
users; all such control sequences will have at least one `|@|' in
their names.
The next job is to set up the ^|\mathcode| table:
\beginlines
|\mathcode`\^^@="2201 \mathcode`\^^A="3223 \mathcode`\^^B="010B|
|\mathcode`\^^C="010C \mathcode`\^^D="225E \mathcode`\^^E="023A|
|\mathcode`\^^F="3232 \mathcode`\^^G="0119 \mathcode`\^^H="0115|
|\mathcode`\^^I="010D \mathcode`\^^J="010E \mathcode`\^^K="3222|
|\mathcode`\^^L="2206 \mathcode`\^^M="2208 \mathcode`\^^N="0231|
|\mathcode`\^^O="0140 \mathcode`\^^P="321A \mathcode`\^^Q="321B|
|\mathcode`\^^R="225C \mathcode`\^^S="225B \mathcode`\^^T="0238|
|\mathcode`\^^U="0239 \mathcode`\^^V="220A \mathcode`\^^W="3224|
|\mathcode`\^^X="3220 \mathcode`\^^Y="3221 \mathcode`\^^Z="8000|
|\mathcode`\^^[="2205 \mathcode`\^^\="3214 \mathcode`\^^]="3215|
|\mathcode`\^^^="3211 \mathcode`\^^_="225F \mathcode`\^^?="1273|
|\mathcode`\ ="8000 \mathcode`\!="5021 \mathcode`\'="8000|
|\mathcode`\(="4028 \mathcode`\)="5029 \mathcode`\*="2203|
|\mathcode`\+="202B \mathcode`\,="613B \mathcode`\-="2200|
|\mathcode`\.="013A \mathcode`\/="013D \mathcode`\:="303A|
|\mathcode`\;="603B \mathcode`\<="313C \mathcode`\=="303D|
|\mathcode`\>="313E \mathcode`\?="503F \mathcode`\[="405B|
|\mathcode`\\="026E \mathcode`\]="505D \mathcode`\_="8000|
|\mathcode`\{="4266 \mathcode`\|\||="026A \mathcode`\}="5267|
\endlines
A mathcode is relevant only when the corresponding category code is
11 or~12; therefore many of these codes will rarely be looked at. For
example, the math code for |^^M| specifies the character |\oplus|,
but it's hard to imagine a user who would want |^^M| (ASCII ^\<return>)
to produce an~$\oplus$ sign in the middle of a math formula, since plain
\TeX\ appends |^^M| to the end of every line of input. The math codes
have been set up here, however, to be entirely consistent with the
extended character set presented in Appendix~C and the Computer Modern
fonts described in Appendix~F\null. |INITEX| has done the rest of the
work, as far as mathcodes are concerned: It has set |\mathcode|$\,x=
x+\hex{7000}$ for each of the ten digits $x={}$|`0| to |`9|;
|\mathcode|$\,x=x+\hex{7100}$ for each of the 52~letters; and
|\mathcode|$\,x=x$ for all other values of~$x$.
There's no need to change the ^|\uccode| and ^|\lccode| tables. |INITEX|
has made |\uccode`X=`X|, |\uccode`x=`X|, |\lccode`X=`x|, |\lccode`x=`x|,
and it has made similar assignments for all other letters. The codes are
zero for all nonletters. These tables are used by \TeX's ^|\uppercase|
and |\lowercase| operations, and the hyphenation algorithm also looks
at |\lccode| (see Appendix~H\null). Changes should be made only in format
packages that set \TeX\ up for languages with more than~26 letters
(see Chapter~8).
Next comes the ^|\sfcode| table, which |INITEX| has initialized entirely
to 1000, except that |\sfcode`X=999| for each of the 26 uppercase letters.
Some characters are made ``transparent'' by setting
\beginlines
|\sfcode`\)=0 \sfcode`\'=0 \sfcode`\]=0 % won't change the space factor|
\endlines
and the |\nonfrenchspacing| macro will be used later to change the sfcodes
of special punctuation marks. \ (Chapter~12 explains what an |\sfcode| does.)
The last code table is called ^|\delcode|, and again it's necessary to
change only a few values. |INITEX| has made all delimiter codes equal to
$-1$, which means that no characters are recognized as delimiters in
formulas. But there's an exception: The value |\delcode`\.=0| has been
prespecified, so that `|.|'\ ^^{period} stands for a ``^{null delimiter}.''
\ (See Chapter~17.) \ Plain format sets up the following nine values,
based on the delimiters available in Computer Modern:
\beginlines
|\delcode`\(="028300 \delcode`\/="02F30E \delcode`\)="029301|
|\delcode`\[="05B302 \delcode`\|\||="26A30C \delcode`\]="05D303|
|\delcode`\<="26830A \delcode`\\="26E30F \delcode`\>="26930B|
\endlines
It's important to note that |\delcode`\{| and |\delcode`\}| have been left
equal to~$-1$. If those codes were set to certain values, a user would
^^{left brace} ^^{right brace} be able to type, e.g.,
`|\big{|' to get a big left brace; but it would be a big mistake.
The reason is that ^{braces} are used for grouping, when supplying
arguments to macros; all sorts of strange things can happen if you try
to use them both as math delimiters and group delimiters.
At this point the |plain.tex| file contains several definitions
\beginlines
|\chardef\@ne=1 \chardef\tw@=2 \chardef\thr@@=3 \chardef\sixt@@n=16|
|\chardef\@cclv=255 \mathchardef\@cclvi=256|
|\mathchardef\@m=1000 \mathchardef\@M=10000 \mathchardef\@MM=20000|
\endlines
which allow ``private'' control sequences ^|\@ne|, |\tw@|, etc.,
to be used as abbreviations for commonly used constants 1,~2,~\dots; this
convention makes \TeX\ run a little faster, ^^{optimization} ^^{efficiency}
and it means that the macros will consume slightly less memory space.
The usage of these abbreviations will not, however, be shown below
unless necessary; we shall pretend that `|1|\]' appears instead of\/ |\@ne|,
`|10000|\]' instead of\/ |\@M|, and so on, since that makes the programs
more readable. \ (Notice that the long form of\/ |\@ne| is `|1|\]' including
a space, because \TeX\ looks for and removes a space following a constant.)
\subsection Allocation of registers. The second major part of the
|plain.tex| file provides a foundation on which systems of independently
developed macros can coexist peacefully without interfering in their
usage of ^{registers}. The idea is that macro writers should abide by the
^^{macro conventions}
following ground rules: (1)~Registers numbered 0~to~9 are always free for
temporary ``^{scratch}'' use, but their values are always assumed to be
clobbered whenever any other macro might get into control. \ (This applies
to registers like |\dimen0|, |\toks0|, |\skip1|, |\box3|, etc.; but \TeX\
has already reserved |\count0| through |\count9| for page number
identification.) \ (2)~The registers |\count255|, |\dimen255|, and
|\skip255| are freely available in the same way.
\ (3)~All assignments to the scratch registers whose numbers are
1,~3, 5, 7, and~9 should be ^|\global|; all assignments to the
other scratch registers (0,~2, 4, 6, 8,~255) should be non-|\global|.
\ (This prevents the phenomenon of ``^{save stack buildup}'' discussed
in Chapter~27.)
\ (4)~Furthermore, it's possible to
use any register in a group, if you ensure that \TeX's grouping
mechanism will restore the register when you're done with the group, and
if you are certain that other macros will not make global assignments
to that register when you need it. \ (5)~But when a register is used
by several macros, or over long spans of time, it should be allocated
by |\newcount|, |\newdimen|, |\newbox|, etc. \ (6)~Similar remarks
apply to ^{input/output streams} used by ^|\read| and ^|\write|, to math
^{families} used by ^|\fam|, to sets of hyphenation rules used by
^|\language|, and to insertions (which require
^|\box|, ^|\count|, ^|\dimen|, and ^|\skip| registers all having the
same number).
Some handy abbreviations are introduced at this point so that the macros
below will have easy access to scratch registers:
\beginlines
|\countdef\count@=255 \toksdef\toks@=0 \skipdef\skip@=0|
|\dimendef\dimen@=0 \dimendef\dimen@i=1 \dimendef\dimen@ii=2|
\endlines
Here now are the macros that provide allocation for quantities of more
permanent value. These macros use registers
|\count10| through |\count20| to hold the numbers that were allocated
most recently; for example, if\/ |\newdimen| has just reserved |\dimen15|,
the value of\/ |\count11| will be~15. However, the rest of the world
is not supposed to ``know'' that |\count11| has anything to do with
|\dimen| registers. There's a special counter called ^|\allocationnumber|
that will be equal to the most recently allocated number, after every
|\newcount|, |\newdimen|, \dots, |\newinsert| operation; macro packages
are supposed to refer to |\allocationnumber| if they want to find out what
number was allocated. The inside story of how allocation is actually performed
should be irrelevant when the allocation macros are used at a higher level;
you mustn't assume that |plain.tex| really does allocation in any
particular way.
\beginlines
|\count10=22 % this counter allocates \count registers 23, 24, 25, ...|
|\count11=9 % this counter allocates \dimen registers 10, 11, 12, ...|
|\count12=9 % this counter allocates \skip registers 10, 11, 12, ...|
|\count13=9 % this counter allocates \muskip registers 10, 11, 12, ...|
|\count14=9 % this counter allocates \box registers 10, 11, 12, ...|
|\count15=9 % this counter allocates \toks registers 10, 11, 12, ...|
|\count16=-1 % this counter allocates input streams 0, 1, 2, ...|
|\count17=-1 % this counter allocates output streams 0, 1, 2, ...|
|\count18=3 % this counter allocates math families 4, 5, 6, ...|
|\count19=0 % this counter allocates language codes 1, 2, 3, ...|
|\count20=255 % this counter allocates insertions 254, 253, 252, ...|
|\countdef\insc@unt=20 % nickname for the insertion counter|
|\countdef\allocationnumber=21 % the most recent allocation|
|\countdef|^|\m@ne||=22 \m@ne=-1 % a handy constant|
|\def|^|\wlog||{\immediate\write-1} % this will write on log file (only)|
\smallbreak
|\outer\def|^|\newcount||{\alloc@0\count\countdef\insc@unt}|
|\outer\def|^|\newdimen||{\alloc@1\dimen\dimendef\insc@unt}|
|\outer\def|^|\newskip||{\alloc@2\skip\skipdef\insc@unt}|
|\outer\def|^|\newmuskip||{\alloc@3\muskip\muskipdef\@cclvi}|
|\outer\def|^|\newbox||{\alloc@4\box\chardef\insc@unt}|
|\let\newtoks=\relax % this allows plain.tex to be read in twice|
|\outer\def\newhelp#1#2{\newtoks#1#1=\expandafter{\csname#2\endcsname}}|
|\outer\def|^|\newtoks||{\alloc@5\toks\toksdef\@cclvi}|
|\outer\def|^|\newread||{\alloc@6\read\chardef\sixt@@n}|
|\outer\def|^|\newwrite||{\alloc@7\write\chardef\sixt@@n}|
|\outer\def|^|\newfam||{\alloc@8\fam\chardef\sixt@@n}|
|\outer\def|^|\newlanguage||{\alloc@9\language\chardef\@cclvi}|
\smallbreak
|\def\alloc@#1#2#3#4#5{\global\advance\count1#1 by 1|\parbreak%
| \ch@ck#1#4#2% make sure there's still room|\parbreak%
| \allocationnumber=\count1#1 \global#3#5=\allocationnumber|\parbreak%
| \wlog{\string#5=\string#2\the\allocationnumber}}|
\smallbreak
|\outer\def|^|\newinsert||#1{\global\advance\insc@unt by-1|\parbreak%
| \ch@ck0\insc@unt\count \ch@ck1\insc@unt\dimen|\parbreak%
| \ch@ck2\insc@unt\skip \ch@ck4\insc@unt\box|\parbreak%
| \allocationnumber=\insc@unt|\parbreak%
| \global\chardef#1=\allocationnumber|\parbreak%
| \wlog{\string#1=\string\insert\the\allocationnumber}}|
\smallbreak
|\def\ch@ck#1#2#3{\ifnum\count1#1<#2%|\parbreak%
| \else\errmessage{No room for a new #3}\fi}|^^|No room|
\endlines
The `|\alloc@|' macro does most of the work of allocation. It puts
a message like `|\maxdimen=\dimen10|' into the log file after
|\newdimen| has allocated a place for the |\dimen| register that
will be called |\maxdimen|; such information might be useful when
difficult macros are being ^{debugged}.
A ^|\newhelp| macro has been provided to aid in creating home-made help
texts: You can say, e.g., |\newhelp\helpout{This is a help message.}|, and
then give the command `^|\errhelp||=\helpout|' just before issuing an
^|\errmessage|. This method of creating help texts makes efficient
use of \TeX's ^{memory}, because it puts the text into a control sequence
name where it doesn't take up space that is needed for tokens.
The |plain| file now goes ahead and allocates registers for important constants:
\beginlines
|\newdimen\maxdimen \maxdimen=16383.99999pt|
|\newskip\hideskip \hideskip=-1000pt plus1fill|
|\newskip\centering \centering=0pt plus 1000pt minus 1000pt|
|\newdimen\p@ \p@=1pt % this saves macro space and time|
|\newdimen\z@ \z@=0pt| % likewise
|\newskip\z@skip \z@skip=0pt plus0pt minus0pt|
|\newbox\voidb@x % permanently void box register|
\endlines
The control sequence ^|\maxdimen| stands for the largest permissible
\<dimen>. Alignment macros that appear below will make use of special glue
values called ^|\hideskip| and ^|\centering|. {\sl N.B.: These three constants
must not be changed under any circumstances\/}; you should either ignore
them completely or just use them and enjoy them.
In fact, the next four constant registers (^|\p@|, ^|\z@|, ^|\z@skip|,
and ^|\voidb@x|) have been given private names so that they are untouchable.
The control sequence |\p@| is used several dozen times as an abbreviation
for `|pt |', and |\z@| is used quite often to stand for either `|0pt |' or
`|0 |'; the use of such abbreviations saves almost 10\% of the space
needed to store the tokens in plain \TeX's macros. But we shall stick to
^^{optimization}
the unabbreviated forms below, so that the macros are more readable.
A different sort of allocation comes next:
^^|\string|^^|\escapechar|^^|\uppercase|^^|\iftrue|^^|\iffalse|
\beginlines
|\outer\def\newif#1{\count@=\escapechar \escapechar=-1|\parbreak%
| |^|\expandafter||\expandafter\expandafter|\parbreak%
| \def\@if#1{true}{\let#1=\iftrue}%|\parbreak%
| \expandafter\expandafter\expandafter|\parbreak%
| \def\@if#1{false}{\let#1=\iffalse}%|\parbreak%
| \@if#1{false}\escapechar=\count@} % the condition starts out false|\parbreak%
|\def\@if#1#2{|^|\csname||\expandafter\if@\string#1#2\endcsname}|\parbreak%
|{\uccode`1=`i \uccode`2=`f \uppercase{\gdef\if@12{}}} % `if' is required|
\endlines
For example, the command |\newif\ifalpha| creates a trio of control
sequences called |\alphatrue|, |\alphafalse|, and |\ifalpha|
(see Chapter~20).
\subsection Parameters. |INITEX| sets almost all of the numeric registers
and parameters equal to zero; it makes all of the token registers and parameters
empty; and it makes all of the box registers void. But there are a few
^^{parameters, default values} ^^{default values of parameters}
exceptions: ^|\mag| is set initially to~1000, ^|\tolerance| to~10000,
^|\maxdeadcycles| to~25, ^|\hangafter| to~1, ^|\escapechar| to~|`\\|, and
^|\endlinechar| to~|`\^^M|. Plain \TeX\ assigns new parameter values as
follows:
\beginlines
^|\pretolerance||=100 |^|\tolerance||=200 |^|\hbadness||=1000 |%
^|\vbadness||=1000|
^|\linepenalty||=10 |^|\hyphenpenalty||=50 |^|\exhyphenpenalty||=50|
^|\binoppenalty||=700 |^|\relpenalty||=500|
^|\clubpenalty||=150 |^|\widowpenalty||=150 |^|\displaywidowpenalty||=50|
^|\brokenpenalty||=100 |^|\predisplaypenalty||=10000|
^|\doublehyphendemerits||=10000 |^|\finalhyphendemerits||=5000 |%
^|\adjdemerits||=10000|
^|\tracinglostchars||=1 |^|\uchyph||=1 |^|\delimiterfactor||=901|
^|\defaulthyphenchar||=`\- |^|\defaultskewchar||=-1 |%
^|\newlinechar||=-1|
^|\showboxbreadth||=5 |^|\showboxdepth||=3 |^|\errorcontextlines||=5|
\smallbreak
^|\hfuzz||=0.1pt |^|\vfuzz||=0.1pt |^|\overfullrule||=5pt|
^|\hsize||=6.5in |^|\vsize||=8.9in |^|\parindent||=20pt|
^|\maxdepth||=4pt |^|\splitmaxdepth||=\maxdimen |^|\boxmaxdepth||=\maxdimen|
^|\delimitershortfall||=5pt |^|\nulldelimiterspace||=1.2pt |%
^|\scriptspace||=0.5pt|
\smallbreak
^|\parskip||=0pt plus 1pt|
^|\abovedisplayskip||=12pt plus 3pt minus 9pt|
^|\abovedisplayshortskip||=0pt plus 3pt|
^|\belowdisplayskip||=12pt plus 3pt minus 9pt|
^|\belowdisplayshortskip||=7pt plus 3pt minus 4pt|
^|\topskip||=10pt |^|\splittopskip||=10pt|
^|\parfillskip||=0pt plus 1fil|
\smallbreak
^|\thinmuskip||=3mu|
^|\medmuskip||=4mu plus 2mu minus 4mu|
^|\thickmuskip||=5mu plus 5mu|
\endlines
(Some parameters are set by \TeX\ itself as it runs, so it is inappropriate
to initialize them: ^|\time|, ^|\day|, ^|\month|, and ^|\year| are established
at the beginning of a job; ^|\outputpenalty| is given a value when an
|\output| routine is invoked; ^|\predisplaysize|, ^|\displaywidth|,
and ^|\displayindent| get values just before a display is processed;
and the values ^|\looseness||=0|, ^|\hangindent||=0pt|, ^|\hangafter||=1|,
^|\parshape||=0| are assigned at the end of a paragraph and when \TeX\ enters
internal vertical mode.)
The parameters ^|\baselineskip|, ^|\lineskip|, and ^|\lineskiplimit| have not
been initialized here, but a macro called ^|\normalbaselines| is defined below;
this macro sets |\baselineskip=\normalbaselineskip|,
|\lineskip=\normallineskip|, and |\lineskiplimit=\normallineskiplimit|.
An indirect approach like this has been used so that several different
type sizes may be handled, as illustrated in Appendix~E\null. Plain \TeX\
deals exclusively with $10\pt$ type, but it supports extension to other styles.
Some ``^{pseudo parameters}'' come next. These quantities behave just like
internal parameters of \TeX, and users are free to change them in the same
way, but they are part of the plain \TeX\ format
rather than primitives of the language. ^^{parameters, pseudo}
\beginlines
|\newskip|^|\smallskipamount|| % the amount of a \smallskip|
| \smallskipamount=3pt plus1pt minus1pt|
|\newskip|^|\medskipamount|| % the amount of a \medskip|
| \medskipamount=6pt plus2pt minus2pt|
|\newskip|^|\bigskipamount|| % the amount of a \bigskip|
| \bigskipamount=12pt plus4pt minus4pt|
|\newskip|^|\normalbaselineskip|| % normal value of \baselineskip|
| \normalbaselineskip=12pt|
|\newskip|^|\normallineskip|| % normal value of \lineskip|
| \normallineskip=1pt|
|\newdimen|^|\normallineskiplimit|| % normal value of \lineskiplimit|
| \normallineskiplimit=0pt|
|\newdimen|^|\jot|| % unit of measure for opening up displays|
| \jot=3pt|
|\newcount|^|\interdisplaylinepenalty|| % interline penalty in \displaylines|
| \interdisplaylinepenalty=100|
|\newcount|^|\interfootnotelinepenalty|| % interline penalty in footnotes|
| \interfootnotelinepenalty=100|
\endlines
\subsection Font information. Now |plain.tex| brings in the data that \TeX\
needs to know about how to typeset lots of characters in lots of different
fonts. First the ^|\magstep| macros are defined, to support font scaling:
\beginlines
|\def|^|\magstephalf||{1095 }|
|\def\magstep#1{\ifcase#1 1000\or|
| 1200\or 1440\or 1728\or 2074\or 2488\fi\relax}|
\weakendlines
(Incidentally, |\magstep| doesn't use ^|\multiply| to compute values,
since it is supposed to expand to a ^\<number> en route to \TeX's
``stomach''; ^^{anatomy} |\multiply| wouldn't work, because
it is an assignment command, performed only in the stomach.)
One of the main things that distinguishes one format from another
is the fact that each format gives \TeX\ the necessary knowledge about
a certain family of typefaces. In this case the ^{Computer Modern} types
described in Appendix~F are taken as a basis, although there is a provision
for incorporating other styles.
\beginlines
|\font|^|\tenrm||=cmr10 \font\preloaded=cmr9 \font\preloaded=cmr8|
\nobreak
|\font\sevenrm=cmr7 \font\preloaded=cmr6 \font\fiverm=cmr5|
\smallskip
|\font\teni=cmmi10 \font\preloaded=cmmi9 \font\preloaded=cmmi8|
\nobreak
|\font\seveni=cmmi7 \font\preloaded=cmmi6 \font\fivei=cmmi5|
\smallbreak
|\font\tensy=cmsy10 \font\preloaded=cmsy9 \font\preloaded=cmsy8|
\nobreak
|\font\sevensy=cmsy7 \font\preloaded=cmsy6 \font\fivesy=cmsy5|
\nobreak\smallskip
|\font\tenex=cmex10|
\smallbreak
|\font\tenbf=cmbx10 \font\preloaded=cmbx9 \font\preloaded=cmbx8|
\nobreak
|\font\sevenbf=cmbx7 \font\preloaded=cmbx6 \font\fivebf=cmbx5|
\smallskip
|\font\tensl=cmsl10 \font\preloaded=cmsl9 \font\preloaded=cmsl8|
\nobreak
|\font\tentt=cmtt10 \font\preloaded=cmtt9 \font\preloaded=cmtt8|
\nobreak
|\font\tenit=cmti10 \font\preloaded=cmti9 \font\preloaded=cmti8|
\nobreak
|\font\preloaded=cmss10 \font\preloaded=cmssq8|
\nobreak
|\font\preloaded=cmssi10 \font\preloaded=cmssqi8|
\smallskip
|\font\preloaded=cmr7 scaled \magstep4 % for titles|
\nobreak
|\font\preloaded=cmtt10 scaled \magstep2|
\nobreak
|\font\preloaded=cmssbx10 scaled \magstep2|
\smallskip
|% Additional \preloaded fonts can be specified here.|
|% (And those that were \preloaded above can be eliminated.)|
|\let\preloaded=\undefined % preloaded fonts must be declared anew later.|
\endlines
Notice that most of the fonts have been called ^|\preloaded|; but the
control sequence |\preloaded| is made undefined at the very end, so those
fonts cannot be used directly. There are two reasons for this strange
approach: First, it is desirable to keep the total number of fonts
of plain \TeX\ relatively small, because plain \TeX\ is a sort of
standard format; it shouldn't cost much for someone to acquire all the
fonts of plain \TeX\ in addition to those he really wants. Second, it
is desirable on many computer systems to preload the information for
most of the fonts that people will actually be using, since this saves
a lot of machine time. The |\preloaded| font
information goes into \TeX's memory, where it will come alive instantly
if the user defines the corresponding |\font| again. For example, the
book format in Appendix~E says `|\font\ninerm=cmr9|'; after that
assignment has been obeyed, the control sequence |\ninerm| will
identify the |cmr9| font, whose information does not have to be loaded again.
The exact number and nature of fonts that are preloaded is unimportant;
the only important thing needed for standardization between machines
is that sixteen basic fonts (|cmr10|, |cmr7|, \dots,~|cmti10|) should
actually be loaded. The |plain.tex| files used on different machines
can be expected to differ widely with respect to preloaded fonts,
since the choice of how many fonts to preload and the selection of the
most important fonts depends on local conditions.
For example, at the author's university it is useful to preload a
font that contains the Stanford seal, but that particular font is not
very popular at Berkeley.
Most of these fonts have the default values of\/ ^|\hyphenchar| and
^|\skewchar|, namely |`-| and |-1|; but the math italic and math symbol fonts
have special |\skewchar| values, which are defined next:
\beginlines
|\skewchar\teni='177 \skewchar\seveni='177 \skewchar\fivei='177|
|\skewchar\tensy='60 \skewchar\sevensy='60 \skewchar\fivesy='60|
\endlines
Once the fonts are loaded, they are also grouped into families for
use in math setting, and shorthand names like ^|\rm| and~^|\it| are defined:
\beginlines
|\textfont0=\tenrm \scriptfont0=\sevenrm |^|\scriptscriptfont||0=\fiverm|
\nobreak
|\def\rm{\fam0 \tenrm}|
^|\textfont||1=\teni |^|\scriptfont||1=\seveni \scriptscriptfont1=\fivei|
\nobreak
|\def|^|\mit||{\fam1 } \def|^|\oldstyle||{\fam1 \teni}|
|\textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy|
\nobreak
|\def|^|\cal||{\fam2 }|
|\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex|
\smallbreak
|\newfam\itfam \def\it{\fam\itfam\tenit} \textfont\itfam=\tenit|
|\newfam\slfam \def|^|\sl||{\fam\slfam\tensl} \textfont\slfam=\tensl|
|\newfam\bffam \def|^|\bf||{\fam\bffam\tenbf} \textfont\bffam=\tenbf|
\nobreak
| \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf|
^|\newfam||\ttfam \def|^|\tt||{\fam\ttfam\tentt} \textfont\ttfam=\tentt|
\endlines
\subsection Macros for text. The fifth part of |plain.tex| introduces
macros that do basic formatting unrelated to mathematics. First come
some macros that were promised above:
\beginlines
|\def|^|\frenchspacing||{\sfcode`\.=1000 \sfcode`\?=1000 \sfcode`\!=1000|
\nobreak
| \sfcode`\:=1000 \sfcode`\;=1000 |^|\sfcode||`\,=1000 }|
|\def|^|\nonfrenchspacing||{\sfcode`\.=3000 \sfcode`\?=3000 \sfcode`\!=3000|
\nobreak
| \sfcode`\:=2000 \sfcode`\;=1500 \sfcode`\,=1250 }|
|\def|^|\normalbaselines||{\lineskip=\normallineskip|
\nobreak
| \baselineskip=\normalbaselineskip \lineskiplimit=\normallineskiplimit}|
\endlines
The next macros are simple but vital. ^^{control tab}^^{control return}
^^{control space} First |\|\<tab> and |\|\<return> are defined so that they
expand to |\|\<space>; this helps to prevent confusion, since all three
cases look identical when displayed on most computer terminals.
Then the macros ^|\lq|, ^|\rq|, ^|\lbrack|, and ^|\rbrack| are defined,
for people who have difficulty typing quotation marks and/or brackets. The
control sequences ^|\endgraf|
and ^|\endline| are made equivalent to \TeX's primitive ^|\par| and~^|\cr|
operations, since it is often useful to redefine the meanings of\/ |\par|
and~|\cr| themselves. Then come the definitions of\/ ^|\space| (a blank space),
^|\empty| (a~list of no tokens), and ^|\null| (an~empty hbox). Finally,
^|\bgroup| and ^|\egroup| are made to provide ``implicit'' grouping characters
that turn out to be especially useful in macro definitions. \ (See Chapters
24--26 and Appendix~D for information about ^{implicit characters}.)
\beginlines
|\def\^^I{\ } \def\^^M{\ }|
|\def\lq{`} \def\rq{'} \def\lbrack{[} \def\rbrack{]}|
|\let\endgraf=\par \let\endline=\cr|
|\def\space{ } \def\empty{} \def\null{\hbox{}}|
|\let\bgroup={ \let\egroup=}|
\endlines
Something a bit tricky comes up now in the definitions of\/ ^|\obeyspaces|
and ^|\obeylines|, since \TeX\ is only ``half obedient'' while these
definitions are half finished:
\beginlines
|\def\obeyspaces{\catcode`\ =\active}|
|{\obeyspaces\global\let =\space}|
|{\catcode`\^^M=\active % these lines must end with `%'|
\nobreak
| \gdef\obeylines{\catcode`\^^M=\active \let^^M=\par}%|
\nobreak
| \global\let^^M=\par} % this is in case ^^M appears in a \write|
\endlines
The |\obeylines| macro says `|\let^^M=\par|' instead of `|\def^^M{\par}|'
because the |\let| technique allows constructions such as
`|\let\par=\cr| |\obeylines| ^|\halign||{...}|' in which ^|\cr|'s need
not be given within the alignment.
The ^|\loop||...|^|\repeat| macro provides for iterative operations
as illustrated at the end of Chapter~20. In this macro and several others,
the control sequence `^|\next|' is given a temporary value that is not
going to be needed later; thus, |\next| acts like a ``^{scratch control
sequence}.''
\beginlines
|\def\loop#1\repeat{\def\body{#1}\iterate}|
|\def\iterate{\body \let\next=\iterate \else\let\next=\relax\fi \next}|
|\let\repeat=\fi % this makes \loop...\if...\repeat skippable|
\endlines
Spacing is the next concern. The macros ^|\enskip|, ^|\quad|, and ^|\qquad|
provide spaces that are legitimate breakpoints within a paragraph;
^|\enspace|, ^|\thinspace|, and ^|\negthinspace| produce space that cannot
cause a break (although the space will disappear if it occurs just
next to certain kinds of breaks). All six of these spaces are relative
to the current font. You can get horizontal space that
never disappears by saying `^|\hglue|\<glue>'; this space is able to
stretch or shrink. Similarly, there's a vertical analog,
`^|\vglue|\<glue>'.
The ^|\nointerlineskip| macro suppresses interline glue that would
ordinarily be inserted before the next box in vertical mode; this is a
``one shot'' macro, but ^|\offinterlineskip| is more drastic---it sets
things up so that future ^{interline glue} will be present, but zero.
There also are macros for potentially breakable vertical spaces:
^|\smallskip|, ^|\medskip|, and ^|\bigskip|.^^|\topglue|
\beginlines
|\def\enskip{\hskip.5em\relax} \def\enspace{\kern.5em }|
|\def\quad{\hskip1em\relax} \def\qquad{\hskip2em\relax}|
|\def\thinspace{\kern .16667em } \def\negthinspace{\kern-.16667em }|
\smallbreak
|\def\hglue{|^|\afterassignment||\hgl@\skip@=}|
|\def\hgl@{\leavevmode \count@=\spacefactor \vrule width0pt|
| \nobreak\hskip\skip@ \spacefactor=\count@}|
|\def\vglue{\afterassignment\vgl@\skip@=}|
|\def\vgl@{\par \dimen@=\prevdepth \hrule height0pt|
| \nobreak\vskip\skip@ \prevdepth=\dimen@}|
|\def\topglue{\nointerlineskip \vglue-\topskip \vglue} % for top of page|
\smallbreak
|\def\nointerlineskip{\prevdepth=-1000pt }|
|\def\offinterlineskip{\baselineskip=-1000pt|
| \lineskip=0pt \lineskiplimit=\maxdimen}|
\smallbreak
|\def\smallskip{\vskip\smallskipamount}|
|\def\medskip{\vskip\medskipamount}|
|\def\bigskip{\vskip\bigskipamount}|
\endlines
Speaking of breakpoints, the following macros introduce ^{penalty} markers
that make breaking less, or more, desirable. The ^|\break|, ^|\nobreak|,
and ^|\allowbreak| macros are intended for use in any mode; the |~|~(tie)
and ^|\slash| (hyphen-like~`/') macros are intended for horizontal mode.
The others are intended only for vertical mode, i.e., between paragraphs,
so they begin with |\par|.
\beginlines
|\def\break{\penalty-10000 } \def\nobreak{\penalty10000 }|
|\def\allowbreak{\penalty0 }|
|\def~{\penalty10000\ }|
|\def\slash{/\penalty\exhyphenpenalty}|
\smallbreak
|\def|^|\filbreak||{\par\vfil\penalty-200\vfilneg}|
|\def|^|\goodbreak||{\par\penalty-500 }|
|\def|^|\eject||{\par\penalty-10000 }|
|\def|^|\supereject||{\par\penalty-20000 }|
\smallbreak
|\def|^|\removelastskip||{\ifdim\lastskip=0pt \else\vskip-\lastskip\fi}|
|\def|^|\smallbreak||{\par \ifdim\lastskip<\smallskipamount|\parbreak%
| \removelastskip \penalty-50 \smallskip \fi}|
|\def|^|\medbreak||{\par \ifdim\lastskip<\medskipamount|\parbreak%
| \removelastskip \penalty-100 \medskip \fi}|
|\def|^|\bigbreak||{\par \ifdim\lastskip<\bigskipamount|\parbreak%
| \removelastskip \penalty-200 \bigskip \fi}|
\endlines
Boxes are next: ^|\line|, ^|\leftline|, ^|\rightline|, and ^|\centerline|
produce boxes of the full line width, while ^|\llap| and ^|\rlap| make
boxes whose effective width is zero. The ^|\underbar| macro puts its
argument into an hbox with a straight line at a fixed distance under it.
\beginlines
|\def\line{\hbox to\hsize}|
|\def\leftline#1{\line{#1\hss}} \def\rightline#1{\line{\hss#1}}|
|\def\centerline#1{\line{\hss#1\hss}}|
\smallskip
|\def\llap#1{\hbox to 0pt{\hss#1}} \def\rlap#1{\hbox to 0pt{#1\hss}}|
\smallskip
|\def\m@th{\mathsurround=0pt }|
|\def\underbar#1{$\setbox0=\hbox{#1} \dp0=0pt|\parbreak%
| \m@th \underline{\box0}$}|
\endlines
(Notice that |\underbar| uses math mode to do its job, although the
operation is essentially non-mathematical in nature. A few of the other
macros below use math mode in similar ways; thus, \TeX's mathematical
abilities prove to be useful even when no mathematical typesetting
is actually being done. A special control sequence ^|\m@th| is
used to ``turn off'' ^|\mathsurround| when such constructions are
being performed.)
\smallbreak
A ^|\strut| is implemented here as a rule of width zero, since this
takes minimum space and time in applications where numerous struts are
present.
\beginlines
|\newbox\strutbox|
|\setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width0pt}|
|\def\strut{\relax\ifmmode\copy\strutbox\else\unhcopy\strutbox\fi}|
\weakendlines
The `^|\relax|' in this macro and in others below is necessary in case
|\strut| appears first in an alignment entry, because \TeX\ is in a
somewhat unpredictable mode at such times (see Chapter~22).
The ^|\ialign| macro provides for ^{alignments} when it is necessary to
be sure that ^|\tabskip| is initially zero. The ^|\hidewidth| macro can be
used essentially as |\hfill| in alignment entries that are permitted to
``stick out'' of their column. There's also ^|\multispan|, which permits
alignment entries to span one or more columns.
\beginlines
|\def\ialign{\everycr={}\tabskip=0pt \halign} % initialized \halign|
|\def\hidewidth{\hskip\hideskip}|
\smallbreak
|\newcount\mscount|
|\def\multispan#1{\omit\mscount=#1\relax\loop\ifnum\mscount>1 \sp@n\repeat}|%
\kern-10pt\null
|\def\sp@n{\span\omit \advance\mscount by -1 }|
\endlines
Now we get to the ``^{tabbing}'' macros, which are more complicated than
anything else in plain \TeX. They keep track of the tab positions by
maintaining boxes full of empty boxes having the specified widths.
\ (The best way to understand these macros is probably to watch
them in action on simple examples, using |\tracingall|.)
\beginlines
|\newif\ifus@ \newif\if@cr|
|\newbox\tabs \newbox\tabsyet \newbox\tabsdone|
\smallbreak
|\def|^|\cleartabs||{\global\setbox\tabsyet=\null \setbox\tabs=\null}|
|\def|^|\settabs||{\setbox\tabs=\null \futurelet\next\sett@b}|
|\let\+=\relax % in case this file is being read in twice|
|\def\sett@b{\ifx\next\+\def\nxt{\afterassignment\s@tt@b\let\nxt}%|\parbreak%
| \else\let\nxt=\s@tcols\fi|\parbreak%
| \let\next=\relax \nxt} % turn off \outerness|
|\def\s@tt@b{\let\nxt=\relax \us@false\m@ketabbox}|
|\outer\def|^|\+||{\tabalign} \def\tabalign{\us@true \m@ketabbox}|
|\def\s@tcols#1\columns{\count@=#1 \dimen@=\hsize|\parbreak%
| \loop \ifnum\count@>0 \@nother \repeat}|
|\def\@nother{\dimen@ii=\dimen@ \divide\dimen@ii by\count@|\parbreak%
| \setbox\tabs=\hbox{\hbox to\dimen@ii{}\unhbox\tabs}%|\parbreak%
| \advance\dimen@ by-\dimen@ii \advance\count@ by -1 }|
\smallbreak
|\def\m@ketabbox{\begingroup|\parbreak%
| \global\setbox\tabsyet=\copy\tabs \global\setbox\tabsdone=\null|\parbreak%
| \def\cr{\@crtrue\crcr\egroup\egroup|\parbreak%
| \ifus@ \unvbox0 |^|\lastbox||\fi \endgroup|\parbreak%
| \setbox\tabs=\hbox{\unhbox\tabsyet|^|\unhbox||\tabsdone}}%|\parbreak%
| \setbox0=\vbox\bgroup\@crfalse \ialign\bgroup&\t@bbox##\t@bb@x\crcr}|
\smallbreak
|\def\t@bbox{\setbox0=\hbox\bgroup}|
|\def\t@bb@x{\if@cr\egroup % now \box0 holds the column|\parbreak%
| \else\hss\egroup \global\setbox\tabsyet=\hbox{\unhbox\tabsyet|\parbreak%
| \global\setbox1=\lastbox}% now \box1 holds its size|\parbreak%
| \ifvoid1 \global\setbox1=\hbox to\wd0{}%|\parbreak%
| \else\setbox0=\hbox to\wd1{\unhbox0}\fi|\parbreak%
| \global\setbox\tabsdone=\hbox{\box1\unhbox\tabsdone}\fi|\parbreak%
| \box0}|
\endlines
The macro |\+| has been declared `^|\outer|' here, so that \TeX\ will
be better able to detect runaway arguments and definitions (see Chapter~20).
A non-|\outer| version, called ^|\tabalign|, has also been provided
in case it is necessary to use |\+| in some ``inner'' place.
You can use |\tabalign| just like |\+|, except after ^|\settabs|.
\textindent{$\bullet$} Paragraph shapes of a limited but important kind
are provided by ^|\item|, ^|\itemitem|, and ^|\narrower|. There are also
two macros that haven't been mentioned before: \ (1)~^|\hang| causes
hanging indentation by the normal amount of\/ ^|\parindent|, after the
first line; thus, the entire paragraph will be indented by the same
amount (unless it began with ^|\noindent|). \ (2)~^|\textindent||{stuff}|
is like ^|\indent|, but it puts the `|stuff|' into the indentation, flush
right except for an en~space; it also removes spaces that might follow
the right brace in `|{stuff}|'. For example, the present paragraph
was typeset by the commands `|\textindent{$\bullet$} Paragraph shapes
...|'; the opening `P' occurs at the normal position for a paragraph's
first letter. ^^|\bullet| ^^|\ignorespaces|
\beginlines
|\def\hang{\hangindent\parindent}|
|\def\item{\par\hang\textindent}|
|\def\itemitem{\par\indent \hangindent2\parindent \textindent}|
|\def\textindent#1{\indent\llap{#1\enspace}\ignorespaces}|
|\def\narrower{\advance\leftskip by\parindent|\parbreak%
| \advance\rightskip by\parindent}|
\endlines
The ^|\beginsection| macro is intended to mark the beginning of a
new major subdivision in a document; to use it, you say
`|\beginsection|\<section title>' followed by a blank line (or~|\par|).
The macro first emits glue and penalties, designed to start a new page
if the present page is nearly full; then it makes a ^|\bigskip| and
puts the section title flush left on a line by itself, in boldface type.
The section title is also displayed on the terminal.
After a ^|\smallskip|, with page break prohibited, a ^|\noindent|
command is given; this suppresses indentation in the next paragraph,
i.e., in the first paragraph of the new section.
\ (However, the next ``paragraph'' will be empty if vertical mode
material immediately follows the |\beginsection| command.) ^^|\message|
\beginlines
|\outer\def\beginsection#1\par{\vskip0pt plus.3\vsize\penalty-250|\parbreak%
| \vskip0pt plus-.3\vsize\bigskip\vskip\parskip|\parbreak%
| \message{#1}\leftline{\bf#1}\nobreak\smallskip\noindent}|
\endlines
Special statements in a mathematical paper are often called ^{theorems},
lemmas, definitions, axioms, postulates, remarks, corollaries, algorithms,
facts, conjectures, or some such things, and they generally are
given special typographic treatment. The ^|\proclaim| macro, which
was illustrated earlier in this appendix and also in Chapter~20,
puts the title of the proclamation in boldface, then sets the rest
of the paragraph in slanted type. The paragraph is followed by
something similar to ^|\medbreak|, except that the amount of penalty
is different so that page breaks are discouraged:
\beginlines
|\outer\def\proclaim #1. #2\par{\medbreak|\parbreak%
| \noindent{\bf#1.\enspace}{\sl#2\par}%|\parbreak%
| \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi}|
\endlines
^{Ragged-right setting} is initiated by restricting the spaces between
words to have a fixed width, and by putting variable space at the right
of each line. You should not call ^|\raggedright| until your text font
has already been specified; it is assumed that the ragged-right material
will not be in a variety of different sizes. \ (If this assumption is
not valid, a different approach should be used: ^|\fontdimen| parameters
3 and~4 of the fonts you will be using should be set to zero, by saying,
e.g., `|\fontdimen3\tenrm=0pt|'. These parameters specify the stretchability
and shrinkability of ^{interword spaces}.) \ ^^|\spaceskip| ^^|xspaceskip|
A special macro ^|\ttraggedright| should be used for ragged-right setting
in ^{typewriter type}, since the spaces between words are generally
bigger in that style. \ (Spaces are already unstretchable and
unshrinkable in font~|cmtt|.)
\beginlines
|\def\raggedright{\rightskip=0pt plus2em|\parbreak%
| \spaceskip=.3333em \xspaceskip=.5em\relax}|
|\def\ttraggedright{\tt\rightskip=0pt plus2em\relax}|
\endlines
Now we come to special symbols and accents, which depend primarily on
the characters available in the Computer Modern fonts. Different
constructions will be necessary if other styles of type are used.
When a symbol is built up by forming a box, the ^|\leavevmode| macro
is called first; this starts a new paragraph, if \TeX\ is in vertical
mode, but does nothing if \TeX\ is in horizontal mode or math mode.
\beginlines
|\chardef\%=`\% \chardef\&=`\& \chardef\#=`\# \chardef\$=`\$|%
^^{control percent}^^{control ampersand}^^{control hash}^^{control dollar}
|\chardef|^|\ae||="1A \chardef|^|\oe||="1B \chardef|^|\o||="1C| %
|\chardef|^|\ss||="19|
|\chardef|^|\AE||="1D \chardef|^|\OE||="1E \chardef|^|\O||="1F|
|\chardef|^|\i||="10 \chardef|^|\j||="11 % dotless letters|
|\def|^|\aa||{\accent'27a} \def|^|\l||{\char'40l}|
\smallbreak
|\def\leavevmode{\unhbox\voidb@x} % begins a paragraph, if necessary|
|\def\_{\leavevmode \kern.06em \vbox{\hrule width0.3em}}|^^{control underline}
|\def|^|\L||{\leavevmode\setbox0=\hbox{L}\hbox to\wd0{\hss\char'40L}}|
|\def|^|\AA||{\leavevmode\setbox0=\hbox{!}\dimen@=\ht0 |%
|\advance\dimen@ by-1ex|\parbreak%
| \rlap{\raise.67\dimen@\hbox{\char'27}}A}|
\smallbreak
|\def|^|\mathhexbox||#1#2#3{\leavevmode \hbox{$\m@th \mathchar"#1#2#3$}}|
|\def|^|\dag||{\mathhexbox279} \def|^|\ddag||{\mathhexbox27A}|
|\def|^|\S||{\mathhexbox278} \def|^|\P||{\mathhexbox27B} |%
|\def|^|\Orb||{\mathhexbox20D}|
\smallbreak
|\def|^|\oalign||#1{\leavevmode\vtop{\baselineskip0pt \lineskip.25ex|\parbreak%
| \ialign{##\crcr#1\crcr}}} \def\o@lign{\lineskiplimit=0pt \oalign}|
|\def|^|\ooalign||{\lineskiplimit=-\maxdimen \oalign} % chars over each other|
|{\catcode`p=12 \catcode`t=12 \gdef\\#1pt{#1}} \let|^|\getf@ctor||=\\|
|\def\sh@ft#1{\dimen@=#1 \kern\expandafter\getf@ctor\the\fontdimen1\font|\parbreak%
| \dimen@} % kern by #1 times the current slant|
|\def|^|\d||#1{{\o@lign{\relax#1\crcr\hidewidth\sh@ft{-1ex}.\hidewidth}}}|
|\def|^|\b||#1{{\o@lign{\relax#1\crcr\hidewidth\sh@ft{-3ex}%|\parbreak%
| \vbox to.2ex{\hbox{\char'26}\vss}\hidewidth}}}|
|\def|^|\c||#1{{\setbox0=\hbox{#1}\ifdim\ht0=1ex \accent'30 #1%|\parbreak%
| \else\ooalign{\unhbox0\crcr\hidewidth\char'30\hidewidth}\fi}}|
|\def|^|\copyright||{{\ooalign{\hfil\raise.07ex\hbox{c}\hfil\crcr\Orb}}}|
\smallbreak
|\def|^|\dots||{\relax\ifmmode\ldots\else$\m@th \ldots\,$\fi}|
|\def|^|\TeX||{T\kern-.1667em \lower.5ex\hbox{E}\kern-.125em X}|
\smallbreak
|\def|^|\`||#1{{\accent"12 #1}} \def|^|\'||#1{{\accent"13 #1}}|
|\def|^|\v||#1{{\accent"14 #1}} \def|^|\u||#1{{\accent"15 #1}}|
^^{esc hat}|\def|^|\=||#1{{\accent"16 #1}} \def\^#1{{\accent"5E #1}}|
|\def|^|\.||#1{{\accent"5F #1}} \def|^|\H||#1{{\accent"7D #1}}|
^^{esc tilde}|\def\~||#1{{\accent"7E #1}} \def|^|\"||#1{{\accent"7F #1}}|
|\def|^|\t||#1{{\edef\next{\the\font}\the\textfont1\accent"7F\next#1}}|
\endlines
At this point three alternative control-symbol ^{accents} are defined,
suitable for keyboards with extended character sets (cf.~Appendix~C):
\beginlines
|\let\^^_=\v \let\^^S=\u \let\^^D=\^|
\endlines
Various ways to fill space with ^{leaders} are provided next.
\beginlines
|\def|^|\hrulefill||{\leaders\hrule\hfill}|
|\def|^|\dotfill||{\cleaders\hbox{$\m@th \mkern1.5mu . \mkern1.5mu$}\hfill}|
|\def|^|\rightarrowfill||{$\m@th \smash- \mkern-7mu|\parbreak%
| \cleaders\hbox{$\mkern-2mu \smash- \mkern-2mu$}\hfill|\parbreak%
| \mkern-7mu \mathord\rightarrow$}|
|\def|^|\leftarrowfill|%
|{$\m@th \mathord\leftarrow \mkern-7mu|\parbreak%
| \cleaders\hbox{$\mkern-2mu \smash- \mkern-2mu$}\hfill|\parbreak%
| \mkern-7mu \smash-$}|
\smallbreak
|\mathchardef\braceld="37A \mathchardef\bracerd="37B|
|\mathchardef\bracelu="37C \mathchardef\braceru="37D|
|\def|^|\upbracefill||{$\m@th|\parbreak%
| \setbox0=\hbox{$\braceld$}%|\parbreak%
| \bracelu\leaders\vrule height\ht0 depth0pt\hfill\bracerd|\parbreak%
| \braceld\leaders\vrule height\ht0 depth0pt\hfill\braceru$}|
|\def|^|\downbracefill||{$\m@th|\parbreak%
| \setbox0=\hbox{$\braceld$}%|\parbreak%
| \braceld\leaders\vrule height\ht0 depth0pt\hfill\braceru|\parbreak%
| \bracelu\leaders\vrule height\ht0 depth0pt\hfill\bracerd$}|
\endlines
\smallbreak
Finally, the fifth section of |plain.tex| closes by defining ^|\bye|:
\beginlines
^|\outer||\def\bye{\par\vfill\supereject\end} % the recommended way to stop|
\endlines
\subsection Macros for math. The sixth section of |plain.tex| is the
longest; but it will suffice to give only excerpts here, because most
of it is simply a tedious listing of special symbols together with their
font locations, and the same information appears in Appendix~F.
Some rudimentary things come first: The control sequences ^|\sp| and ^|\sb|
are provided for people who can't easily type |^| and |_|; there are
four control symbols that provide spacing corrections; a ``^{discretionary
times sign}'' ^|\*| is defined; and then there's an interesting set of
macros that convert |f'''| into |f^{\prime\prime\prime}|:
\beginlines
|\let\sp=^ \let\sb=_ {\catcode`\_=\active \global\let_=\_}|\parbreak%
|\def|^|\,||{\mskip\thinmuskip} \def|^|\!||{\mskip-\thinmuskip}|\parbreak%
|\def|^|\>||{\mskip\medmuskip} \def|^|\;||{\mskip\thickmuskip}|
|\def\*{\discretionary{\thinspace\the\textfont2\char2}{}{}}|
|{\catcode`\^^Z=\active \gdef^^Z{\not=}} % ^^Z is like \ne in math|
\smallbreak
|{\catcode`\'=\active \gdef'{^\bgroup\prim@s}}|\parbreak%
|\def\prim@s{\prime\futurelet\next\pr@m@s}|\parbreak%
|\def\pr@m@s{\ifx'\next\let\nxt\pr@@@s|%
| \else\ifx^\next\let\nxt\pr@@@t|\parbreak%
| \else\let\nxt\egroup\fi\fi \nxt}|\parbreak%
|\def\pr@@@s#1{\prim@s} \def\pr@@@t#1#2{#2\egroup}|
\endlines
The next job is to define ^{Greek letters} and other symbols of type ^{Ord}.
Uppercase Greek letters are assigned hexadecimal codes of the form \hex{7xxx},
so that they will change families when ^|\fam| changes.
Three dots `$\,\cdots\,$' are used here and below to indicate that
additional symbols, having similar definitions, are listed in Appendix~F.
\beginlines
|\mathchardef|^|\alpha||="010B|\qquad$\cdots$\qquad|\mathchardef\omega="0121|
|\mathchardef|^|\Gamma||="7000|\qquad$\cdots$\qquad|\mathchardef\Omega="700A|
|\mathchardef|^|\aleph||="0240|\qquad$\cdots$\qquad%
|\mathchardef\spadesuit="027F|
|\def|^|\hbar||{{\mathchar'26\mkern-9muh}}|
|\def|^|\surd||{{\mathchar"1270}}|
|\def|^|\angle||{{\vbox{\ialign{$\m@th\scriptstyle##$\crcr|\parbreak%
| \not\mathrel{\mkern14mu}\crcr \noalign{\nointerlineskip}|\parbreak%
| \mkern2.5mu\leaders\hrule height.34pt\hfill\mkern2.5mu\crcr}}}}|
\endlines
Large operators are assigned hexadecimal codes of the form \hex{1xxx}:
\beginlines
|\mathchardef\smallint="1273|
|\mathchardef|^|\sum||="1350|\qquad$\cdots$\qquad|\mathchardef\biguplus="1355|
|\mathchardef\intop="1352 \def|^|\int||{\intop\nolimits}|
|\mathchardef\ointop="1348 \def\oint{\ointop\nolimits}|
\endlines
Integral signs get special treatment so that their limits won't be set
above and below.
\smallskip
Binary operations are next; nothing exciting here.
\beginlines
|\mathchardef\pm="2206|\qquad$\cdots$\qquad|\mathchardef\amalg="2271|
\endlines
Relations are also fairly straightforward, except for the ones that are
constructed from other characters. The ^|\mapstochar| is a character
`$\mapstochar\mskip5mu$' of width zero that is quite useless by itself,
but it combines with right arrows to make ^|\mapsto| `$\mapsto$' and
^|\longmapsto| `$\longmapsto$'. Similarly, ^|\not| is a relation character of
width zero that puts a slash over the character that follows. When two
relations are adjacent in a math formula, \TeX\ puts no space between them.
\beginlines
|\mathchardef\leq="3214|\qquad$\cdots$\qquad|\mathchardef\perp="323F|
|\def|^|\joinrel||{\mathrel{\mkern-3mu}}|
|\def|^|\relbar||{\mathrel{\smash-}} \def|^|\Relbar||{\mathrel=}|
|\def|^|\longrightarrow||{\relbar\joinrel\rightarrow}|\parbreak%
|\def|^|\Longrightarrow||{\Relbar\joinrel\Rightarrow}|
|\def|^|\longleftarrow||{\leftarrow\joinrel\relbar}|\parbreak%
|\def|^|\Longleftarrow||{\Leftarrow\joinrel\Relbar}|
|\def|^|\longleftrightarrow||{\leftarrow\joinrel\rightarrow}|\parbreak%
|\def|^|\Longleftrightarrow||{\Leftarrow\joinrel\Rightarrow}|
|\mathchardef\mapstochar="3237 \def\mapsto{\mapstochar\rightarrow}|\parbreak%
|\def\longmapsto{\mapstochar\longrightarrow}|
|\mathchardef\lhook="312C \def|^|\hookrightarrow||{\lhook\joinrel\rightarrow}|
|\mathchardef\rhook="312D \def|^|\hookleftarrow||{\leftarrow\joinrel\rhook}|
\smallbreak
|\def|^|\neq||{\not=} \def|^|\models||{\mathrel|\||\joinrel=}|
|\def|^|\bowtie||{\mathrel\triangleright\joinrel\mathrel\triangleleft}|
\weakendlines
After defining characters ^|\ldotp| and ^|\cdotp| that act as math
punctuation, it is easy to define ^|\ldots| and ^|\cdots| macros that
\vadjust{\penalty-50}% fairly good breakpoint (on July 27, 1983)
give the proper spacing in most circumstances. Vertical and diagonal
dots (^|\vdots| and ^|\ddots|) are also provided here:
\beginlines
|\mathchardef\ldotp="613A\mathchardef\cdotp="6201\mathchardef\colon="603A|
|\def\ldots{\mathinner{\ldotp\ldotp\ldotp}}|
|\def\cdots{\mathinner{\cdotp\cdotp\cdotp}}|
|\def\vdots{\vbox{\baselineskip=4pt \lineskiplimit=0pt|
| \kern6pt \hbox{.}\hbox{.}\hbox{.}}}|
|\def\ddots{\mathinner{\mkern1mu\raise7pt\vbox{\kern7pt\hbox{.}}\mkern2mu|
| \raise4pt\hbox{.}\mkern2mu\raise1pt\hbox{.}\mkern1mu}}|
\endlines
Most of the math accents are handled entirely by the ^|\mathaccent|
primitive, but a few of the variable-width ones are constructed the
hard way:
\beginlines
|\def\acute{\mathaccent"7013 }|\qquad$\cdots$\qquad%
|\def\ddot{\mathaccent"707F }|
|\def|^|\widetilde||{\mathaccent"0365 } \def|^|\widehat||{\mathaccent"0362 }|
|\def|^|\overrightarrow||#1{\vbox{\m@th\ialign{##\crcr|\parbreak%
| \rightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}|\parbreak%
| $\hfil\displaystyle{#1}\hfil$\crcr}}}|
|\def|^|\overleftarrow||#1{\vbox{\m@th\ialign{##\crcr|\parbreak%
| \leftarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}|\parbreak%
| $\hfil\displaystyle{#1}\hfil$\crcr}}}|
|\def|^|\overbrace|%
|#1{\mathop{\vbox{\m@th\ialign{##\crcr\noalign{\kern3pt}|\parbreak%
| \downbracefill\crcr\noalign{\kern3pt\nointerlineskip}|\parbreak%
| $\hfil\displaystyle{#1}\hfil$\crcr}}}|^|\limits||}|
|\def|^|\underbrace||#1{\mathop{\vtop{\m@th\ialign{##\crcr|\parbreak%
| $\hfil\displaystyle{#1}\hfil$\crcr|%
|\noalign{\kern3pt\nointerlineskip}|\parbreak%
| \upbracefill\crcr\noalign{\kern3pt}}}}\limits}|
|\def|^|\skew||#1#2#3{{\muskip0=#1mu \mkern.5\muskip0|\parbreak%
| #2{\mkern-.5\muskip0{#3}\mkern.5\muskip0}\mkern-.5\muskip0}{}}|
\endlines
Now we come to 24 delimiters that can change their size: ^^|\delimiter|
\beginlines
|\def\langle{\delimiter"426830A } \def\rangle{\delimiter"526930B }|
|\def\lbrace{\delimiter"4266308 } \def\rbrace{\delimiter"5267309 }|
|\def\lceil{\delimiter"4264306 } \def\rceil{\delimiter"5265307 }|
|\def\lfloor{\delimiter"4262304 } \def\rfloor{\delimiter"5263305 }|
|\def\lgroup{\delimiter"462833A } \def\rgroup{\delimiter"562933B }|
|\def\lmoustache{\delimiter"437A340 } \def\rmoustache{\delimiter"537B341 }|%
\kern-2pt
|\def\uparrow{\delimiter"3222378 } \def\Uparrow{\delimiter"322A37E }|
|\def\downarrow{\delimiter"3223379 } \def\Downarrow{\delimiter"322B37F }|
|\def\updownarrow{\delimiter"326C33F } \def\arrowvert{\delimiter"026A33C }|
|\def\Updownarrow{\delimiter"326D377 } \def\Arrowvert{\delimiter"026B33D } |
|\def\vert{\delimiter"026A30C } \def\Vert{\delimiter"026B30D } |
|\def\backslash{\delimiter"026E30F } \def\bracevert{\delimiter"077C33E }|
\endlines
The `^|\big||...|^|\Bigg|' macros produce specific sizes: ^^|\Big|^^|\bigg|
\beginlines
|\def\bigl{\mathopen\big} \def\bigm{\mathrel\big} \def\bigr{\mathclose\big}|
|\def\Bigl{\mathopen\Big} \def\Bigm{\mathrel\Big} \def\Bigr{\mathclose\Big}|
|\def\biggl{\mathopen\bigg} \def\Biggl{\mathopen\Bigg}|
|\def\biggr{\mathclose\bigg} \def\Biggr{\mathclose\Bigg}|
|\def\biggm{\mathrel\bigg} \def\Biggm{\mathrel\Bigg}|
|\def\big#1{{\hbox{$\left#1\vbox to 8.5pt{}\right.\n@space$}}}|
|\def\Big#1{{\hbox{$\left#1\vbox to 11.5pt{}\right.\n@space$}}}|
|\def\bigg#1{{\hbox{$\left#1\vbox to 14.5pt{}\right.\n@space$}}}|
|\def\Bigg#1{{\hbox{$\left#1\vbox to 17.5pt{}\right.\n@space$}}}|
|\def\n@space{\nulldelimiterspace=0pt \m@th}|
\endlines
There are a few other simple abbreviations related to delimiters:
\beginlines
|\def|^|\choose||{\atopwithdelims()}|
|\def|^|\brack||{\atopwithdelims[]}|
|\def|^|\brace||{\atopwithdelims\{\}}|
|\def|^|\sqrt||{\radical"270370 }|
\endlines
And now we come to something more interesting. The ^|\mathpalette| operation
constructs a formula in all four styles; it is applied here in the
implementation of\/ ^|\phantom|, ^|\smash|, ^|\root|, and other
operations. (Actually |\phantom| and |\smash| are not perfect: They
assume that the current style is uncramped.)
\beginlines
|\def\mathpalette#1#2{\mathchoice{#1\displaystyle{#2}}|\parbreak%
| {#1\textstyle{#2}}{#1\scriptstyle{#2}}{#1\scriptscriptstyle{#2}}}|
\smallbreak
|\newbox\rootbox|
|\def\root#1\of{\setbox\rootbox=|\parbreak%
| \hbox{$\m@th \scriptscriptstyle{#1}$}\mathpalette\r@@t}|
|\def\r@@t#1#2{\setbox0=\hbox{$\m@th #1\sqrt{#2}$}|\parbreak%
| \dimen@=\ht0 \advance\dimen@ by-\dp0|\parbreak%
| \mkern5mu \raise.6\dimen@\copy\rootbox \mkern-10mu \box0}|
\smallbreak
|\newif\ifv@ \newif\ifh@|
|\def|^|\vphantom||{\v@true\h@false\ph@nt}|
|\def|^|\hphantom||{\v@false\h@true\ph@nt}|
|\def\phantom{\v@true\h@true\ph@nt}|
|\def\ph@nt{\ifmmode\def\next{\mathpalette\mathph@nt}%|\parbreak%
| \else\let\next=\makeph@nt\fi \next}|
|\def\makeph@nt#1{\setbox0=\hbox{#1}\finph@nt}|
|\def\mathph@nt#1#2{\setbox0=\hbox{$\m@th#1{#2}$}\finph@nt}|
|\def\finph@nt{\setbox2=\null \ifv@ \ht2=\ht0 \dp2=\dp0 \fi|\parbreak%
| \ifh@ \wd2=\wd0 \fi \box2 }|
|\def\mathstrut{\vphantom(}|
\smallbreak
|\def\smash{\relax % \relax, in case this comes first in \halign|\parbreak%
| \ifmmode\def\next{\mathpalette\mathsm@sh}\else\let\next\makesm@sh|\parbreak%
| \fi \next}|
|\def\makesm@sh#1{\setbox0=\hbox{#1}\finsm@sh}|
|\def\mathsm@sh#1#2{\setbox0=\hbox{$\m@th#1{#2}$}\finsm@sh}|
|\def\finsm@sh{\ht0=0pt \dp0=0pt \box0 }|
\smallbreak
|\def|^|\cong||{\mathrel{\mathpalette\@vereq\sim}} % \sim over =|\parbreak%
|\def\@vereq#1#2{\lower.5pt\vbox{\lineskiplimit\maxdimen|%
| \lineskip-.5pt|\parbreak%
| \ialign{$\m@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}}|
|\def|^|\notin||{\mathrel{\mathpalette\c@ncel\in}}|\parbreak%
|\def\c@ncel#1#2{\m@th\ooalign{$\hfil#1\mkern1mu/\hfil$\crcr$#1#2$}}|\parbreak%
|\def|^|\rightleftharpoons||{\mathrel{\mathpalette\rlh@{}}}|\parbreak%
|\def\rlh@#1{\vcenter{\m@th\hbox{\ooalign{\raise2pt|\parbreak%
| \hbox{$#1\rightharpoonup$}\crcr $#1\leftharpoondown$}}}}|
|\def|^|\buildrel||#1\over#2{\mathrel{\mathop{\kern0pt #2}\limits^{#1}}}|
|\def|^|\doteq||{\buildrel\textstyle.\over=}|
\endlines
These definitions illustrate how other built-up
symbol combinations could be defined to work in all four styles.
Alternate names are defined now:
\beginlines
|\let|^|\ne||=\neq \let|^|\le||=\leq \let|^|\ge||=\geq|
^^{esc vert}^^{esc lbrace}^^{esc rbrace}%
|\let\{=\lbrace \let\|\||=\Vert \let\}=\rbrace|
|\let|^|\to||=\rightarrow \let|^|\gets||=\leftarrow \let|^|\owns||=\ni|
|\let|^|\land||=\wedge \let|^|\lor||=\vee \let|^|\lnot||=\neg|
|\def|^|\iff||{\;\Longleftrightarrow\;}|
\endlines
The 32 common functions whose names generally appear in roman letters are listed
in Chapter~18. Only a few of the definitions need to be shown here:
^^|\arccos| ^^|\cos| ^^|\csc| ^^|\exp| ^^|\ker| ^^|\limsup| ^^|\min|
^^|\sinh| ^^|\arcsin| ^^|\cosh| ^^|\deg| ^^|\gcd| ^^|\lg| ^^|\ln| ^^|\Pr|
^^|\sup| ^^|\arctan| ^^|\cot| ^^|\det| ^^|\hom| ^^|\lim| ^^|\log| ^^|\sec|
^^|\tan| ^^|\arg| ^^|\coth| ^^|\dim| ^^|\inf| ^^|\liminf| ^^|\max|
^^|\sin| ^^|\tanh|
\beginlines
|\def\arccos{|^|\mathop||{\rm arccos}|^|\nolimits||}|
\hskip100pt$\cdots$\qquad|\def\tanh{|^|\mathop||{\rm tanh}|^|\nolimits||}|
|\def\det{\mathop{\rm det}}|\qquad$\cdots$\qquad|\def\sup{\mathop{\rm sup}}|
|\def\liminf{\mathop{\rm lim\,inf}} \def\limsup{\mathop{\rm lim\,sup}}|
\smallskip
|\def|^|\bmod||{\nonscript\mskip-\medmuskip \mkern5mu|\parbreak%
| |^|\mathbin||{\rm mod} \penalty900 \mkern5mu \nonscript\mskip-\medmuskip}|
|\def|^|\pmod||#1{\allowbreak \mkern18mu ({\rm mod}\,\,#1)}|
\endlines
The definition of\/ ^|\matrix| goes to some pains to ensure that two
$n$-rowed matrices will have the same height and the same depth, unless
at least one of their rows is unusually big. The definition of
^|\bordermatrix| is even more complicated, but it seems to work
reasonably well; it uses a constant ^|\p@renwd| that represents the
width of a big extensible left parenthesis.
\beginlines
|\def\matrix#1{\null\,|^|\vcenter||{\normalbaselines\m@th|\parbreak%
| \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr|\parbreak%
| \mathstrut\crcr\noalign{\kern-\baselineskip}|\parbreak%
| #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}|
\smallbreak
|\newdimen\p@renwd \setbox0=\hbox{\tenex B} \p@renwd=\wd0|
|\def\bordermatrix#1{\begingroup \m@th|\parbreak%
| \setbox0=\vbox{\def\cr|%
|{\crcr\noalign{\kern2pt\global\let\cr=\endline}}|\parbreak%
| \ialign{$##$\hfil\kern2pt\kern\p@renwd|%
|&\thinspace\hfil$##$\hfil|\parbreak%
| &&\quad\hfil$##$\hfil\crcr|\parbreak%
| \omit\strut\hfil\crcr\noalign{\kern-\baselineskip}|\parbreak%
| #1\crcr\omit\strut\cr}}|\parbreak%
| \setbox2=\vbox{\unvcopy0 \global\setbox1=\lastbox}|\parbreak%
| \setbox2=\hbox{\unhbox1 \unskip \global\setbox1=\lastbox}|\parbreak%
| \setbox2=\hbox{$\kern\wd1\kern-\p@renwd \left( \kern-\wd1|\parbreak%
| \global\setbox1=\vbox{\box1\kern2pt}|\parbreak%
| \vcenter{\kern-\ht1 \unvbox0 \kern-\baselineskip} \,\right)$}|\parbreak%
| \null\;\vbox{\kern\ht1\box2}\endgroup}|
\endlines
The next macros are much simpler:
\beginlines
|\def|^|\cases||#1{\left\{\,\vcenter{\normalbaselines\m@th|\parbreak%
| \ialign{$##\hfil$&\quad##\hfil\crcr#1\crcr}}\right.}|
|\def|^|\pmatrix||#1{\left( \matrix{#1} \right)}|
\endlines
Finally there are macros for displayed equations:
\beginlines
|\def|^|\openup||{\afterassignment\@penup\dimen@=}|\parbreak%
|\def\@penup{\advance\lineskip\dimen@|\parbreak%
| \advance\baselineskip\dimen@ \advance\lineskiplimit\dimen@}|
|\def|^|\eqalign||#1{\null\,\vcenter{\openup1\jot \m@th|\parbreak%
| \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil|\parbreak%
| \crcr#1\crcr}}\,}|
\smallbreak
|\newif\ifdt@p|
|\def\displ@y{\global\dt@ptrue \openup1\jot \m@th|\parbreak%
| |^|\everycr||{\noalign{\ifdt@p \global\dt@pfalse|%
| \ifdim\prevdepth>-1000pt|\parbreak%
| \vskip-\lineskiplimit \vskip\normallineskiplimit \fi|\parbreak%
| \else \penalty|^|\interdisplaylinepenalty|| \fi}}}|
|\def\@lign{\tabskip=0pt\everycr={}} % restore inside \displ@y|
|\def|^|\displaylines||#1{\displ@y \tabskip=0pt|\parbreak%
| \halign{\hbox to\displaywidth{|%
|$\hfil\@lign\displaystyle##\hfil$}\crcr|\parbreak%
| #1\crcr}}|
\smallbreak
|\def|^|\eqalignno||#1{\displ@y \tabskip=|^|\centering|\parbreak%
| \halign to\displaywidth{\hfil$\@lign\displaystyle{##}$\tabskip=0pt|\parbreak%
| &$\@lign\displaystyle{{}##}$\hfil\tabskip=\centering|\parbreak%
| &\llap{$\@lign##$}\tabskip=0pt\crcr|\parbreak%
| #1\crcr}}|
|\def|^|\leqalignno||#1{\displ@y \tabskip=\centering|\parbreak%
| \halign to\displaywidth{\hfil$\@lign\displaystyle{##}$\tabskip=0pt|\parbreak%
| &$\@lign\displaystyle{{}##}$\hfil\tabskip=\centering|\parbreak%
| &\kern-\displaywidth\rlap{$\@lign##$}\tabskip=\displaywidth\crcr|\parbreak%
| #1\crcr}}|
\endlines
The value of\/ |\lineskiplimit| is assumed to be |\normallineskiplimit|
plus the accumulated amount of ``opening up.'' Thus, the |\vskip|
instructions in |\displ@y| will compensate for the fact that the
first baseline of an alignment is separated by an opened-up baselineskip
from the last line preceding the display.
\subsection Macros for output. The |plain.tex| file also contains the output
routine described in Chapters 15 and~23. First there are simple
facilities related to page numbers, headings, and footings:
\beginlines
|\countdef|^|\pageno||=0 \pageno=1 % first page is number 1|
|\newtoks|^|\headline|| \headline={\hfil} % headline is normally blank|
|\newtoks|^|\footline|| \footline={\hss\tenrm\folio\hss}|\parbreak%
| % footline is normally a centered page number in font \tenrm|
|\def|^|\folio||{\ifnum\pageno<0 \romannumeral-\pageno \else\number\pageno \fi}|
|\def|^|\nopagenumbers||{\footline={\hfil}} % blank out the footline|
|\def|^|\advancepageno||{\ifnum\pageno<0 \global\advance\pageno by -1|\parbreak%
| \else\global\advance\pageno by 1 \fi} % increase |\||pageno|\|
\smallbreak
|\newif\ifr@ggedbottom|
|\def|^|\raggedbottom||{\topskip10pt plus60pt \r@ggedbottomtrue}|
|\def|^|\normalbottom||{\topskip10pt \r@ggedbottomfalse} % undoes \raggedbottom|
\endlines
The ^|\footnote| macro has a few subtle features that can best be appreciated
by someone who reads Chapter~15 very carefully. It also uses some
^|\bgroup| and ^|\futurelet| and ^|\aftergroup| trickery, so that the footnote
text does not need to be a ^{parameter} to |\vfootnote|:
\beginlines
^|\newinsert||\footins|
|\def\footnote#1{\let\@sf=\empty|%
| % parameter #2 (the text) is read later|\parbreak%
| \ifhmode\edef\@sf{\spacefactor=\the\spacefactor}\/\fi|\parbreak%
| #1\@sf\vfootnote{#1}}|
|\def|^|\vfootnote||#1{\insert\footins\bgroup|\parbreak%
| \interlinepenalty=\interfootnotelinepenalty|\parbreak%
| \splittopskip=\ht\strutbox % top baseline for broken footnotes|\parbreak%
| \splitmaxdepth=\dp\strutbox \floatingpenalty=20000|\parbreak%
| \leftskip=0pt \rightskip=0pt \spaceskip=0pt \xspaceskip=0pt|\parbreak%
| \textindent{#1}\footstrut\futurelet\next\fo@t}|
|\def\fo@t{\ifcat\bgroup\noexpand\next \let\next\f@@t|\parbreak%
| \else\let\next\f@t\fi \next}|
|\def\f@@t{\bgroup\aftergroup\@foot\let\next}|
|\def\f@t#1{#1\@foot}|
|\def\@foot{\strut\egroup}|
|\def\footstrut{\vbox to\splittopskip{}}|
|\skip\footins=\bigskipamount % space added when footnote is present|
|\count\footins=1000 % footnote magnification factor (1 to 1)|
|\dimen\footins=8in % maximum footnotes per page|
\endlines
^{Floating insertions} are handled by doing an ^|\insert| whose
vertical list consists of a penalty item followed by a single box:
\beginlines
|\newinsert\topins \newif\ifp@ge \newif\if@mid|
|\def|^|\topinsert||{\@midfalse\p@gefalse\@ins}|
|\def|^|\midinsert||{\@midtrue\@ins}|
|\def|^|\pageinsert||{\@midfalse\p@getrue\@ins}|
|\skip\topins=0pt % no space added when a topinsert is present|
|\count\topins=1000 % magnification factor (1 to 1)|
|\dimen\topins=\maxdimen % no limit per page|
|\def\@ins{\par\begingroup\setbox0=\vbox\bgroup} % start a \vbox|
|\def\endinsert{\egroup % finish the \vbox|\parbreak%
| \if@mid \dimen@=\ht0 \advance\dimen@ by\dp\z@ |%
|\advance\dimen@ by12\p@|\parbreak%
| \advance\dimen@ by\pagetotal \advance\dimen@ by-\pageshrink|\parbreak%
| \ifdim\dimen@>\pagegoal \@midfalse\p@gefalse\fi\fi|\parbreak%
| \if@mid \bigskip \box0 \bigbreak|\parbreak%
| \else\insert\topins{\penalty100 % floating insertion|\parbreak%
| |^|\splittopskip||=0pt |^|\splitmaxdepth||=\maxdimen |%
^|\floatingpenalty||=0|\parbreak%
| \ifp@ge \dimen@=\dp0|\parbreak%
| \vbox to\vsize{\unvbox0 \kern-\dimen@} % depth is zero|\parbreak%
| \else \box0 \nobreak\bigskip\fi}\fi\endgroup}|
\endlines
Most of the ^|\output| routine appears in Chapter 23; it is given here
in full:
\beginlines
|\output={\plainoutput}|
|\def|^|\plainoutput|%
|{\shipout\vbox{\makeheadline\pagebody\makefootline}%|\parbreak%
| \advancepageno|\parbreak%
| \ifnum\outputpenalty>-20000 \else\dosupereject\fi}|
|\def|^|\pagebody||{\vbox to\vsize{\boxmaxdepth=\maxdepth \pagecontents}}|
|\def|^|\makeheadline||{\vbox to 0pt{\vskip-22.5pt|\parbreak%
| \line{\vbox to8.5pt{}\the\headline}\vss}\nointerlineskip}|
|\def|^|\makefootline||{\baselineskip=24pt \lineskiplimit=0pt|\parbreak%
| \line{\the\footline}}|
|\def|^|\dosupereject|%
|{\ifnum\insertpenalties>0 % something is being held over|\parbreak%
| \line{}\kern-\topskip\nobreak\vfill\supereject\fi}|
\smallbreak
|\def|^|\pagecontents||{\ifvoid\topins\else\unvbox\topins\fi|\parbreak%
| \dimen@=\dp255 \unvbox255|\parbreak%
| \ifvoid\footins\else % footnote info is present|\parbreak%
| \vskip\skip\footins \footnoterule \unvbox\footins\fi|\parbreak%
| \ifr@ggedbottom \kern-\dimen@ \vfil \fi}|
|\def|^|\footnoterule||{\kern-3pt|\parbreak%
| \hrule width 2truein \kern 2.6pt} % the \hrule is .4pt high|
\endlines
\subsection Hyphenation and everything else. The last part of |plain.tex|
reads the hyphenation patterns and exceptions found on file ^|hyphen.tex|
(see Appendix~H\null); then it defines a few miscellaneous macros, sets
up ^|\rm| type, and that's all!
\beginlines
^|\lefthyphenmin||=2 |^|\righthyphenmin||=3 % disallow x- or -xx breaks|
|\input hyphen % the hyphenation patterns and exceptions|
\nobreak\smallskip
|\def|^|\magnification||{|^|\afterassignment||\m@g\count@}|
|\def\m@g{\mag=\count@|\parbreak%
| \hsize6.5truein\vsize8.9truein\dimen\footins8truein}|
\smallbreak
|\def|^|\loggingall|%
|{\tracingcommands=2 \tracingstats=2|\parbreak%
| \tracingpages=1 \tracingoutput=1 \tracinglostchars=1 |\parbreak%
| \tracingmacros=2 \tracingparagraphs=1 \tracingrestores=1 |\parbreak%
| \showboxbreadth=\maxdimen \showboxdepth=\maxdimen}|
|\def|^|\tracingall||{\tracingonline=1 \loggingall}|
\smallbreak
|\def|^|\showhyphens|%
|#1{\setbox0=\vbox{\parfillskip0pt \hsize=\maxdimen \tenrm|\parbreak%
| \pretolerance=-1 \tolerance=-1 \hbadness=0 \showboxdepth=0 \ #1}}|
\smallbreak
|\normalbaselines\rm % select roman font|
|\nonfrenchspacing % punctuation affects the spacing|
|\catcode`@=12 % at signs are no longer letters|
\nobreak\smallskip
|\def|^|\fmtname||{plain}|
|\def\fmtversion{3.141592653} % identifies the current format|
\endlines
The format name and version number are recorded in control sequences,
in order to help the people who might have to explain why something
doesn't work. Macro files like |plain.tex| should not be changed in any way,
except with respect to preloaded fonts, unless the changes are authorized
by the author of the macros.
\endchapter
The purpose of a programming system is to make a computer easy to use.
To do this, it furnishes languages and various facilities
that are in fact programs invoked and controlled by language features.
But these facilities are bought at a price:
the external description of a programming system is ten to twenty times
as large as the external description of the computer system itself.
The user finds it far easier to specify any particular function,
but there are far more to choose from,
and far more options and formats to remember.
\author FREDERICK P. ^{BROOKS}, JR., {\sl The Mythical Man Month\/} (1975)
% p 43
\bigskip
When someone says, ``I want a programming language
in which I need only say what I wish done,''
give him a lollipop.
\author ALAN ^{PERLIS}, {\sl Epigrams on Programming\/} (1982)
% SIGPLAN Notices 17,9 (September 82), 7--13.
% There are many more, like "Editing is a rewording activity."
\eject
\beginchapter Appendix C. Character\\Codes
\ninepoint
Different computers tend to have different ways of representing the
characters in files of text, but \TeX\ gives the same results on
all machines, because it converts everything to a standard internal
code when it reads a file. \TeX\ also converts back from its internal
representation to the appropriate external code, when it writes
a file of text; therefore most users need not be aware of the fact
that the ^{codes} have actually switched back and forth inside the machine.
The purpose of this appendix is to define \TeX's internal code,
which has the same characteristics on all implementations of \TeX.
The existence of such a code is important, because it
makes \TeX\ constructions ``portable.'' For example, \TeX\ allows
^{alphabetic constants} like |`b| to be used as numbers; the fact
that |`b| always denotes the integer~98 means that we can
write machine-independent macros that decide, for instance, whether
a given character is a digit between |0| and |9|.
Furthermore the internal code of \TeX\ also survives in its ^|dvi|
output files, which can be printed by software that knows nothing
about where the |dvi| data originated; essentially the same
output will be obtained from all implementations of \TeX, regardless
of the host computer, because the |dvi| data is expressed in a
machine-independent code.
\smallskip
\TeX's internal code is based on the American Standard Code for
Information Interchange, known popularly as ``^{ASCII}.'' There are
128 codes, numbered 0~to~127; we conventionally express the numbers
in octal notation, from \oct{000} to \oct{177}, or in
hexadecimal notation, from \hex{00} to \hex{7F}. Thus, the value of
|`b| is normally called \oct{142} or \hex{62}, not 98. In the
ASCII scheme, codes \oct{000} through \oct{040} and
code \oct{177} are~assigned to special functions; for example,
code \oct{007} is called |BEL|, and it means ``Ring the bell.''
The other 94 codes are assigned to visible symbols. Here is a
chart that shows ASCII codes in such a way that octal and hexadecimal
equivalents can eas |
https://genkuroki.github.io/documents/20160501StirlingFormula/20160501StirlingFormula-0.31.tex | github.io | CC-MAIN-2019-39 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2019-39/segments/1568514576345.90/warc/CC-MAIN-20190923084859-20190923110859-00549.warc.gz | 479,970,479 | 74,916 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
((中略))
%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.
%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.
%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.
%5月5日Ver.0.5: 誤りの訂正と様々な追加(全17頁).
%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).
%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).
%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).
%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).
%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).
%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).
%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}).
%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.
%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.
%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.
%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.
%5月24日Ver.0.17(61頁): 細かな誤りの訂正.
%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).
%6月3日Ver.0.19(76頁): 細かい訂正と追加.
%Laplace-Stieltjes変換に関する漸近挙動に緩変動函数付きの
%Tauber型定理に関するかなり詳しい解説を追加した
%(\secref{sec:Laplace-Stieltjes}, \secref{sec:Tauber-Stieltjes}).
%他にも色々追加した.
%6月4日Ver.0.20(81頁): Stirling-Binetの公式に関する\secref{sec:Binet1}を追加した.
%その節の結果のまとめがこの時点では書きかけの\secref{sec:Binet2}の最初にある.
%6月23日Ver.0.20.1: このノートの内容は変化無し.
%この場所でこのノートの続編が公開されていることを\secref{sec:intro}の直前で
%アナウンスしておいた.
%6月29日Ver.0.21(86頁): 正規分布関連の確率分布に関する
%\theoremref{theorem:chi-square}, \theoremref{theorem:t},
%\theoremref{theorem:normal-t}, \theoremref{theorem:normal-t2},
%\theoremref{theorem:F}を追加した.
6月30日Ver.0.22: 細かな訂正と追記.
\secref{sec:Beta1st}を大幅に書き直した.
7月1日Ver.0.23(89頁): \secref{sec:Betas}を追加した.
7月4日Ver.0.24: \secref{sec:y=xt}の凡ミスを訂正した.
7月4日Ver.0.25(91頁): 不偏分散の直交変換による取り扱いに関する\secref{sec:Y_k}を追加した.
7月30日Ver.0.26(94頁): 多項分布とPearsonのカイ2乗統計量と多次元正規分布に関する
\secref{sec:Pearson}を追加した.
8月27日Ver.0.27(94頁): 細かい修正と追加.
9月11日Ver.0.28(96頁): \secref{sec:Gamma-Gauss-Stirling}の誤りを修正した.
9月12日Ver.0.29(96頁): この更新記録を大幅に削った.
更新の歴史については公開した古い版を参照して欲しい.
9月12日Ver.0.29a: 微修正.
10月4日Ver.0.30(97頁): \secref{sec:Poisson-Gamma}を書き直した.
2017年1月22日Ver.0.31(98ページ)：「Taylorの定理{\bf に}証明の仕方」となっていたのを直した
(\secref{sec:Taylor}).
「関数」を「函数」に統一した.
Riemann-Lebesgueの定理の説明を詳しくした(\secref{sec:Riemann-Lebesgue}).
}
\\[\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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%\bigskip\bigskip
\paragraph{古い版}
このノートの古い版が次の場所で公開されている:
\begin{center}
{\small
\href
{http://www.math.tohoku.ac.jp/~kuroki/LaTeX/20160501StirlingFormula/}
{\tt http://www.math.tohoku.ac.jp/{\textasciitilde}kuroki/LaTeX/20160501StirlingFormula/}
}
\end{center}
最初のVer.0.1は3ページしかなかった.
%\bigskip
\paragraph{続編}
このノートの続編が次の場所で公開されている:
\begin{center}
{\small
\href
{http://www.math.tohoku.ac.jp/~kuroki/LaTeX/20160616KullbackLeibler.pdf}
{\tt http://www.math.tohoku.ac.jp/{\textasciitilde}kuroki/LaTeX/20160616KullbackLeibler.pdf}
}
\end{center}
この続編ではKullback-Leibler情報量(相対エントロピーの $-1$ 倍)と
Sanovの定理を扱っており, Sanovの定理から,
Boltzmann因子($e^{-\beta E_i}$),
Gibbs分布(カノニカル分布, $e^{-\beta E_i}q_i/Z$)
が経験分布として自然に現われることを示している.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{section}{-1} % 最初の節番号を0にする
\section{はじめに}
\label{sec:intro}
{\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{ガンマ分布の特性函数と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
\[
f_{n/2,2}(y) =
\begin{cases}
\dfrac{e^{-y/2}y^{n/2-1}}{\Gamma(n/2)2^{n/2}} & \qquad (y>0), \\
0 & \qquad (y\leqq 0).
\end{cases}
\]
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
\[
\lim_{n\to\infty}
\sum_{k=0}^n
g\left(\frac{k-np}{\sqrt{npq}}\right)
\binom{n}{k}p^k q^{n-p}
=
\int_{-\infty}^\infty g(y) \frac{e^{-y^2/2}}{\sqrt{2\pi}}\,dy.
\]
$g(y)$ 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
\[
\lim_{n\to\infty}
P\left(a\leqq \frac{X_n-np}{\sqrt{npq}}\leqq b\right)
=
\int_a^b \frac{e^{-y^2/2}}{\sqrt{2\pi}}\,dy.
\]
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}
\label{sec:Gamma-Gauss-Stirling}
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: $k=0,1,2,\ldots$ のとき
\begin{align*}
\int_{-\infty}^\infty e^{-y^2/2}y^{2k}\,dy
&=2\int_0^\infty e^{-y^2/2} (y^2)^k \,dy
=2\int_0^\infty e^{-x} (2x)^k \sqrt{2}\,\frac{x^{-1/2}}{2}\,dx
\\ &
=2^k\sqrt{2}\int_0^\infty e^{-x} x^{k-1/2}\,dx
=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,
\(
\int_{-\infty}^\infty e^{-y^2/2}\,dy
=\int_{-\infty}^\infty e^{-y^2/2}y^2\,dy
=\sqrt{2\pi}
\),
\[
\qquad
\int_{-\infty}^\infty e^{-y^2/2}y^4\,dy = 3\sqrt{2\pi}, \qquad
\int_{-\infty}^\infty e^{-y^2/2}y^6\,dy = 15\sqrt{2\pi}.
\]
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 $n+\sqrt{n}\,y=n(1+y/\sqrt{n})$ 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}\,y} \left( 1+\frac{y}{\sqrt{n}} \right)^n \,dy.
\end{align*}
被積分函数の対数を $\phi_n(y)$ と書くと:
\begin{align*}
\phi_n(y)
&=n\log\left(1+\frac{y}{\sqrt{n}}\right)-\sqrt{n}\,y
=-\frac{y^2}{2} + \frac{y^3}{3\sqrt{n}}-\frac{y^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 $0$ に収束する量である.
ゆえに
\begin{align*}
e^{-\sqrt{n}\,y} \left( 1+\frac{y}{\sqrt{n}} \right)^n
&=e^{-y^2/2}
\exp\left( \frac{y^3}{3\sqrt{n}}-\frac{y^4}{4n}+ o\left(\frac{1}{n}\right) \right)
\\ &
=e^{-y^2/2}
\left(
1
+\frac{y^3}{3\sqrt{n}}
-\frac{y^4}{4n}
+\frac{1}{2}\left( \frac{y^3}{3\sqrt{n}} \right)^2
+o\left(\frac{1}{n}\right)
\right)
\\ &
=e^{-y^2/2}
\left(
1
+\frac{y^3}{3\sqrt{n}}
-\frac{y^4}{4n}
+\frac{y^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 $y$ について奇函数になることに注意せよ.
奇函数と $e^{-y^2/2}$ の積の $-\infty<y<\infty$ における
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_{-\sqrt{n}}^\infty
e^{-\sqrt{n}\,y} \left( 1+\frac{y}{\sqrt{n}} \right)^n\,dx
& \sim
\int_{-\infty}^{\infty}
e^{-y^2/2}
\left(
1
-\frac{y^4}{4n}
+\frac{y^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}
\label{sec:Riemann-Lebesgue}
$f(x)$ 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,
\[
|\widehat{f}(x+h)-\widehat{f}(x)|
\leqq\int_\R|e^{ihx}-1||f(x)|\,dx
\longrightarrow
\int_\R|e^{i0x}-1||f(x)|\,dx = 0
\qquad(h\to 0).
\]
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で $L^1$ 近似されることから,
階段函数の場合に制限されたRiemann-Lebesgueの定理の証明に帰着する%
\footnote{$\R$ 上の可積分函数列 $f_n$ が $\R$ 上の可積分函数 $f$ に $L^1$ 収束する
とは $n\to\infty$ のとき $\int_\R |f_n(x)-f(x)|\,dx\to 0$ が成立することである.}.
その理由は以下の通り.
正の実数 $\eps>0$ と $\R$ 上の可積分函数 $f,g$ について,
$\int_\R|f(x)-g(x)|\,dx\leqq\eps$ と $\lim_{|p|\to\infty}\int_\R e^{-ipx}g(x)\,dx=0$
が成立していれば,
\begin{align*}
\left|\int_\R e^{-ipx}f(x)\,dx\right|
&\leqq
\left|\int_\R e^{-ipx}g(x)\,dx\right|
+ \left|\int_\R e^{-ipx}(f(x)-g(x))\,dx\right|
\\ &
\leqq
\left|\int_\R e^{-ipx}g(x)\,dx\right| + \int_\R|f(x)-g(x)|\,dx
\leqq
\left|\int_\R e^{-ipx}g(x)\,dx\right| + \eps
\end{align*}
なので
\begin{align*}
\limsup_{|p|\to\infty}\left|\int_\R e^{-ipx}f(x)\,dx\right| \leqq \eps
\end{align*}
となる.
そして, $\R$ 上の任意の可積分函数 $f$ に対して階段函数列 $f_n$ で $f$ に $L^1$ 収束
するものが存在するので, 一般の可積分函数に関するRiemann-Lebesgueの定理の証明は
階段函数に関するRiemann-Lebesgueの定理の証明に帰着する.
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.
階段函数は $1_I$ 型の函数の一次結合である.
ゆえに, 階段函数に関するRiemann-Lebesgueの定理の証明は $1_I$ 型の函数に
関するRiemann-Lebesgueの定理の証明に帰着する.
その場合の証明は次のようにほぼ自明である:
\[
\widehat{1_I}(p)
= \int_a^b e^{-ipx}\,dx
= \frac{e^{-ipb}-e^{-ipa}}{-ip}
\]
なので, $\widehat{1_I}(p)\to 0$ ($|p|\to\infty$).
一般の可積分函数に関するRiemann-Lebesgueの定理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: Gauss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を使わずにすむ積分変数の変換による計算}
\label{sec:y=xt}
以下のように計算すると積分の順序交換と1変数の置換積分のみを使って
Gauss積分を計算できる.
$y=x\tan\theta$ によって $y$ から $\theta$ に積分変数を変換すると,
\[
dy = \frac{d\theta}{\cos^2\theta}, \qquad
x^2+y^2=x^2(1+\tan^2\theta) = \frac{x^2}{\cos^2\theta}
\]
より,
\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} \exp\left(-\frac{x^2}{\cos^2\theta}\right)\,\frac{x}{\cos^2\theta}\,d\theta
\right)\,dx
\\ &
=4
\int_0^{\pi/2}
\left(
\int_0^\infty \exp\left(-\frac{x^2}{\cos^2\theta}\right)\,\frac{x}{\cos^2\theta}\,dx
\right)\,d\theta
=4
\int_0^{\pi/2}
\left[
\frac{\exp\left(-\frac{x^2}{\cos^2\theta}\right)}{-2}
\right]_{x=0}^{x=\infty}
\,d\theta
\\ &
=4
\int_0^{\pi/2}\frac{1}{2}\,d\theta
=4\frac{1}{2}\frac{\pi}{2}
=\pi.
\end{align*}
3つ目の等号で積分の順序交換を行なった.
$y=xt$ によって $y$ から $t$ に積分変数を変換すること
によって本質的に同じ計算を実行することもできる:
\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^\infty e^{-(1+t^2)x^2}\,x \,dt
\right)\,dx
\\ &
=4
\int_0^\infty
\left(
\int_0^\infty e^{-(1+t^2)x^2}\,x \,dx
\right)\,dt
=4
\int_0^\infty\left[\frac{e^{-(1+t^2)x^2}}{-2(1+t^2)}\right]_{x=0}^{x=\infty}\,dt
\\ &
=2
\int_0^\infty\frac{dt}{1+t^2}
=2[\arctan t]_0^\infty
=2\frac{\pi}{2}
=\pi.
\end{align*}
3つ目の等号で積分の順序交換を行ない, 6つ目の等号で $\arctan t$ の導函数が $1/(1+t^2)$
であることを使った%
\footnote{$t=\tan\theta$ のとき $dt/d\theta = 1+\tan^2\theta=1+t^2$ なので,
$\theta=\arctan t$ の導函数は $d\theta/dt = 1/(1+t^2)$ になる.
そのことから, $\arctan t = \int_0^t dt/(1+t^2)$ となることもわかる.}.
\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.
\subsection{類似の積分}
$a\geqq 0$ に対する次の公式を証明しよう:
\begin{align*}
\int_{-\infty}^\infty
\exp\left(-\frac{1}{2}\left(x^2+\frac{a^2}{x^2}\right)\right)\,dx
= \sqrt{2\pi}\,e^{-a}.
\tag{$*$}
\end{align*}
積分範囲を $x>0$ に制限し,
$y=x-a/x$ で $x>0$ を $-\infty<y<\infty$ に置換積分すれば計算できる.
任意の $y\in\R$ ついて $x-a/x=y$ は $x>0$ で
\begin{align*}
x=\frac{1}{2}(y+\sqrt{y^2+4a})
\end{align*}
と一意的に解けて
\begin{align*}
dx = \frac{1}{2}\left(1 + \frac{y}{\sqrt{y^2+4a}}\right)\,dy.
\end{align*}
が成立している. ゆえに
\begin{align*}
\int_{-\infty}^\infty \exp\left(-\frac{1}{2}\left(x-\frac{a}{x}\right)^2\right)\,dx
&
=2\int_0^\infty \exp\left(-\frac{1}{2}y^2\right)
\frac{1}{2}\left(1 + \frac{y}{\sqrt{y^2+4a}}\right)\,dy
\\ &
=\int_{-\infty}^\infty e^{-y^2/2}\,dy
=\sqrt{2\pi}.
\end{align*}
2番目の等号で $y/\sqrt{y^2+4a}$ が $y$ の奇函数であることを使った.
一方
\begin{align*}
-\frac{1}{2}\left(x-\frac{a}{x}\right)^2
=-\frac{1}{2}\left(x^2+\frac{a^2}{x^2}\right)+a
\end{align*}
なので
\begin{align*}
\int_{-\infty}^\infty \exp\left(-\frac{1}{2}\left(x-\frac{a}{x}\right)^2\right)\,dx
=e^a\int_{-\infty}^\infty \exp\left(-\frac{1}{2}\left(x^2+\frac{a^2}{x^2}\right)\right)\,dx.
\end{align*}
以上を合わせると($*$)が得られる.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
\[
\frac{1}{\Gamma(s)}
=e^{\gamma s}s\prod_{n=1}^\infty\left[ \left(1+\frac{s}{n}\right)e^{-s/n} \right]
\]
を{\bf Weierstrass の公式}と呼ぶ.
右辺の無限積は任意の $s\in\C$ に対して収束する.
以上によって収束が示された極限の逆数を $F(s)$ と書くことにする:
\[
F(s)=\lim_{n\to\infty}\frac{n!n^s}{s(s+1)\cdots(s+n)}.
\]
このとき
\[
F(s+1)
=\lim_{n\to\infty}
\frac{ns}{s+1+n}\frac{n!n^s}{s(s+1)\cdots(s+n)}
=sF(s), \quad
F(1)=\frac{n!\,n}{(n+1)!}=1.
\]
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直接に得る方法
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{Stirling-Binet の公式(1)}
\label{sec:Binet1}
以下の解説はほぼ
E.~T.~Whittaker and G.~N.~Watson, A course of modern analysis (1927)
からの引き写しである.
この本は様々な古典的な公式が大量に載っている非常に楽しい本である.
ガンマ函数の対数微分をディガンマ函数(digamma函数)と呼び, $\psi(s)$ と表わす:
\[
\psi(s)=\frac{d}{ds}\log\Gamma(s)=\frac{\Gamma'(s)}{\Gamma(s)}.
\]
さらにディガンマ函数の導函数 $\psi'(s)$ をトリガンマ函数(trigamma函数)と呼ぶ.
ガンマ函数の無限乗積展開(Weierstrassの公式)より,
\[
\log\Gamma(s)
=
-\gamma s - \log s
-\sum_{n=1}^\infty\left[\log\left(1+\frac{s}{n}\right)-\frac{s}{n}\right].
\]
ここで $\gamma$ はEuler定数である.
両辺を項別微分することによって次を得る:
\[
\psi(s)=\frac{d}{ds}\log\Gamma(s)
=
-\gamma - \frac{1}{s}
-\sum_{n=1}^\infty\left[\frac{1}{n+s}-\frac{1}{n} \right].
\]
さらにもう一度項別微分すると
\[
\psi'(s)
=\frac{1}{s^2}+\sum_{n=1}^\infty\frac{1}{(n+s)^2}
=\sum_{n=0}^\infty\frac{1}{(n+s)^2}.
\]
このようにして, ガンマ函数の無限乗積展開から,
ディガンマ函数とトリガンマ函数の部分分数展開が得られる.
以下の目標はディガンマ函数に関する公式を経由して,
$\log\Gamma(s)$ に関するStirling-Binetの公式を示すことである.
長くて地道な計算になるが, 内容的には
ディガンマ函数の部分分数展開からディガンマ函数の積分表示式を得て,
それを積分して $\log\Gamma(z)$ の表示を得るだけの単純な計算である.
まず, Euler定数の積分表示式
\[
\gamma
%= \int_0^1 \frac{1-e^{-u}}{u}\,du - \int_1^\infty \frac{e^{-u}}{u}\,du
= \int_0^\infty\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\,dt
\]
を示そう.
Euler定数の定義に $1/k=\int_0^1 x^{k-1}\,dx$
と $\log n=\int_1^n du/u$ を代入すると,
\begin{align*}
\gamma
&
=\lim_{n\to\infty}\left[\sum_{k=1}^n \frac{1}{k} - \log n\right]
=\lim_{n\to\infty}\left[\sum_{k=1}^n \int_0^1 x^{k-1}\,dx - \int_1^n \frac{du}{u}\right]
\\ &
=\lim_{n\to\infty}\left[\int_0^1\frac{1-x^n}{1-x}\,dx - \int_1^n \frac{du}{u}\right]
=\lim_{n\to\infty}\left[\int_0^1\frac{1-(1-y)^n}{y}\,dy - \int_1^n \frac{du}{u}\right]
\\ &
=\lim_{n\to\infty}\left[\int_0^n\frac{1-(1-u/n)^n}{u}\,du - \int_1^n \frac{du}{u}\right]
\\ &
=\lim_{n\to\infty}
\left[\int_0^1\frac{1-(1-u/n)^n}{u}\,du - \int_1^n \frac{(1-u/n)^n}{u}\,du\right]
\\ &
= \int_0^1 \frac{1-e^{-u}}{u}\,du - \int_1^\infty \frac{e^{-u}}{u}\,du
\\ &
= \lim_{\delta\searrow 0}
\left[ \int_\delta^1 \frac{du}{u} - \int_\delta^\infty \frac{e^{-u}}{u}\,du \right]
%\\ &
= \lim_{\delta\searrow 0}
\left[
\int_\Delta^1 \frac{du}{u}
-\int_\Delta^\delta \frac{du}{u}
-\int_\delta^\infty \frac{e^{-u}}{u}\,du
\right]
\\ &
= \lim_{\delta\searrow 0}
\left[ \int_\Delta^1 \frac{du}{u} - \int_\delta^\infty \frac{e^{-u}}{u}\,du \right]
= \lim_{\delta\searrow 0}
\left[
\int_\delta^\infty \frac{e^{-t}}{1-e^{-t}}\,dt
- \int_\delta^\infty \frac{e^{-t}}{t}\,dt
\right]
\\ &
= \lim_{\delta\searrow 0}
\int_\delta^\infty
\left(
\frac{e^{-t}}{1-e^{-t}}
-\frac{e^{-t}}{t}
\right) \,dt
\\ &
= \int_0^\infty
\left(
\frac{e^{-t}}{1-e^{-t}}
-\frac{e^{-t}}{t}
\right)\,dt.
\end{align*}
1つ目の等号はEuler定数の定義であり, 2つ目の等号で積分の代入を行ない,
3つ目の等号で等比数列の和 $1+x+\cdots+x^{n-1}=(1-x^n)/(1-x)$ を使った.
さらに4つ目の等号で積分変数の変換 $y=1-x$ を行ない,
5つ目の等号でさらに $y=t/n$ とおき,
6つ目の等号で $0$ から $1$ までの積分と $1$ から $\infty$ までの積分を分けた.
6つ目の等号の右辺は $n\to\infty$ の極限が取れる形式になっていることに注意せよ.
9つ目の等号で $0<\Delta=1-e^{-\delta}<\delta$ とおいた.
10個目の等号で $\delta\searrow 0$ のとき
$\int_\Delta^\delta du/u=\log(\delta/(1-e^{-\delta}))\to 0$ となることを使った.
11個目の等号で1つ目の積分で $u=1-e^{-t}$ とおき, 2つ目の積分で $u=t$ とおいた.
次に, Euler定数の積分表示式に似ているディガンマ函数の積分表示式
\[
\psi(s)=\frac{d}{ds}\log\Gamma(s)
=\int_0^\infty\left(\frac{e^{-t}}{t}-\frac{e^{-st}}{1-e^{-t}}\right)\,dt.
\]
を示そう(Gaussによるディガンマ函数の無限積分表示).
ディガンマ函数の部分分数展開で,
上で証明したEuler定数の積分表示とよく使われる公式
\[
\frac{1}{c^s} = \frac{1}{\Gamma(s)}\int_0^\infty e^{-ct}t^{s-1}\,dt
\qquad (\Re c>0)
\tag{$\$$}
\]
の $s=1$ の場合を $c=s,s+n,n$ に適用した結果を使うと,
\begin{align*}
\psi(s)
&
=\frac{d}{ds}\log\Gamma(s)
\\ &
=
-\int_0^\infty\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\,dt
-\int_0^\infty e^{-st}\,dt
+\sum_{n=1}^\infty \int_0^\infty (e^{-nt}-e^{-(n+s)t})\,dt
\\ &
=
\int_0^\infty\left(\frac{e^{-t}}{t}-\frac{e^{-t}}{1-e^{-t}}\right)\,dt
+\lim_{n\to\infty}
\int_0^\infty \frac{e^{-t}-e^{-nt}-e^{-st}+e^{-(s+n)t}}{1-e^{-t}}\,dt
\\ &
=
\int_0^\infty\left(\frac{e^{-t}}{t}-\frac{e^{-st}}{1-e^{-t}}\right)\,dt
-\lim_{n\to\infty}
\int_0^\infty \frac{1-e^{-st}}{1-e^{-t}} e^{-nt} \,dt
\\ &
=
\int_0^\infty\left(\frac{e^{-t}}{t}-\frac{e^{-st}}{1-e^{-t}}\right)\,dt
\end{align*}
3つ目の等号で
\begin{align*}
&
e^{-t}+e^{-2t}+\cdots+e^{-(n-1)t}=\frac{e^{-t}-e^{-nt}}{1-e^{-t}},
\\ &
e^{-st}+e^{-(s+1)t}+e^{-(s+2)t}+\cdots+e^{-(s+n-1)t}=\frac{e^{-st}-e^{(s+n)t}}{1-e^{-t}}
\end{align*}
を使った. これでディガンマ函数の積分表示式が証明された.
以下の目標はディガンマ函数に関する公式を積分して $\log\Gamma(z)$ に
関する公式を得ることである.
ここで公式
\[
\int_0^\infty \frac{e^{-t}-e^{-zt}}{t}\,dt = \log z
\tag{$*$}
\]
を示しておこう. 左辺を $f(z)$ と書くと, $f(1)=0$ でかつ,
$f'(z)=\int_0^\infty e^{-zt}\,dt=1/z$ なので $f(z)=\log z$ となる.
上で証明したディガンマ函数の積分表示式で $s=z+1$ とおいて,
被積分函数の2つ目の項の分子分母に $e^t$ をかけると:
\[
\psi(z+1)=\frac{d}{dz}\log\Gamma(z+1)
=\int_0^\infty\left( \frac{e^{-t}}{t}-\frac{e^{-zt}}{e^t-1}\right)\,dt.
\]
これにすぐ上の段落で証明した公式を適用すると,
\begin{align*}
\psi(z+1)
=\log z + \int_0^\infty\left(\frac{e^{-zt}}{t}-\frac{e^{-zt}}{e^t-1}\right)\,dt
=\log z - \int_0^\infty\left(-\frac{1}{t}+\frac{1}{e^t-1}\right)e^{-zt}\,dt.
\end{align*}
一時的に $f(t)=1/(e^t-1)$ とおくと $f(-t)=-f(t)-1$ より $1/2+f(t)$ は奇函数
になる. さらに $1/2-1/t+f(t)$ は $t=0$ で正則になり,
\[
\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}
=\frac{t}{12}-\frac{t^3}{720}+\frac{t^5}{30240}+O(t^7)
\tag{$*$}
\]
となることもわかる. 左辺の函数の $1/t$ 倍は $0<t<\infty$ で有界である.
上の $\psi(z+1)$ の表示の積分の括弧の中がこれになるようにするために,
\[
\int_0^\infty \frac{1}{2}e^{-zt}\,dt = \frac{1}{2z}
\]
を使うと
\[
\psi(z+1)=\frac{d\log\Gamma(z+1)}{dz}
=\log z+\frac{1}{2z}
-\int_0^\infty\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}\right)e^{-zt}\,dt.
\]
$\log\Gamma(2)=\log 1=0$ なので, この式を $1$ から $z$ まで積分すると,
\begin{align*}
\log\Gamma(z+1)
&=z\log z-z+1 + \frac{1}{2}\log z
+\int_0^\infty\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}\right)\frac{e^{-zt}-e^{-t}}{t}\,dt.
\end{align*}
$\log\Gamma(z+1)=\log z+\log\Gamma(z)$ より,
\[
\log\Gamma(z)
=z\log z-z+1 - \frac{1}{2}\log z
+\int_0^\infty\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}\right)\frac{e^{-zt}-e^{-t}}{t}\,dt.
\]
右辺の積分を評価するために
\[
I(z)
=\int_0^\infty\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}\right)e^{-zt}\frac{dt}{t}
\]
とおく. $\Gamma(1/2)=\sqrt{\pi}$ より,
\[
\log\sqrt{\pi}=\frac{1}{2}+I\left(\frac{1}{2}\right)-I(1).
\]
一方, $I(1)$ の定義式で $t$ を $t/2$ で置き換えると,
\[
I(1)
=\int_0^\infty\left(\frac{1}{2}-\frac{2}{t}+\frac{1}{e^{t/2}-1}\right)e^{-t/2}\frac{dt}{t}
\]
なので
\begin{align*}
I\left(\frac{1}{2}\right)-I(1)
=\int_0^\infty\left(\frac{1}{t}-\frac{e^{t/2}}{e^t-1}\right)e^{-t/2}\frac{dt}{t}
=\int_0^\infty\left(\frac{e^{-t/2}}{t}-\frac{1}{e^t-1}\right)\frac{dt}{t}
\end{align*}
であるから, 今度は $I(1)$ の定義式の方を使うと,
\begin{align*}
I\left(\frac{1}{2}\right)
&
=\int_0^\infty\left(
\frac{e^{-t/2}}{t}-\frac{1}{e^t-1}
+\frac{e^{-t}}{2}-\frac{e^{-t}}{t}+\frac{e^{-t}}{e^t-1}
\right)\frac{dt}{t}
\\ &
=\int_0^\infty\left(
\frac{e^{-t/2}-e^{-t}}{t}-\frac{e^{-t}}{2}
\right)\frac{dt}{t}
=\int_0^\infty\left(
\frac{e^{-t/2}-e^{-t}}{t^2}-\frac{e^{-t}}{2t}
\right)\,dt.
\end{align*}
ここで
\begin{align*}
&
-\frac{d}{dt}\frac{e^{-t/2}-e^{-t}}{t}
=\frac{e^{-t/2}-e^{-t}}{t^2}
+\frac{e^{-t/2}/2-e^{-t}}{t},
\\ &
-\frac{e^{-t/2}/2-e^{-t}}{t}-\frac{e^{-t}}{2t}
=-\frac{1}{2}\frac{e^{-t/2}-e^{-t}}{t}
\end{align*}
に注意すれば $I(1/2)$ の計算は($*$)に帰着できることがわかる:
\begin{align*}
I\left(\frac{1}{2}\right)
&
=-\left[\frac{e^{-t/2}-e^{-t}}{t}\right]_{t=0}^{t=\infty}
+\frac{1}{2}\int_0^\infty\frac{e^{-t}-e^{-t/2}}{t}\,dt
=\frac{1}{2}+\frac{1}{2}\log\frac{1}{2}
=\frac{1}{2}-\log\sqrt{2}.
\end{align*}
したがって,
\[
-I(1)
= -\frac{1}{2}-I\left(\frac{1}{2}\right)+\log\sqrt{\pi}.
= -1 + \log\sqrt{2\pi}.
\]
以上をまとめると,
\[
\log\Gamma(z)
=z\log z-z+1-\frac{1}{2}\log z + I(z)-I(1)
=z\log z-z+\log\sqrt{\frac{2\pi}{z}}+I(z).
\]
$\log\Gamma(z+1)=\log z+\log\Gamma(z)$ より,
\[
\log\Gamma(z+1)=z\log z - z + \log\sqrt{2\pi z}+I(z).
\]
$I(z)$ の定義式を再掲すると
\[
I(z)=
\int_0^\infty\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}\right)e^{-zt}\frac{dt}{t}.
\]
この積分の被積分函数の括弧の内側の $1/t$ 倍は $0<t<\infty$ で有界である.
ゆえにある正の実数 $M$ が存在して $\Re z>0$ において $I(z)$ は上から次のように評価される:
\[
|I(z)|\leqq M\int_0^\infty e^{-(\Re z)t}\,dt = \frac{M}{\Re z}.
\]
これは $z>0$ に対して,
\[
\log\Gamma(z+1)=z\log z-z+\log\sqrt{2\pi z}+O\left(\frac{1}{z}\right)
\qquad (z\to\infty).
\]
これでStirlingの公式
\[
\Gamma(z+1) = z^z e^{-z}\sqrt{2\pi z} \,(1+O(1/z)) \qquad (z\to\infty)
\]
が再証明された. より精密には($\#$)と($\$$)より,
\begin{align*}
I(z)
&=\int_0^\infty
e^{-zt}
\left(\frac{1}{12}-\frac{t^2}{720}+\frac{t^4}{30240}+O(t^6)\right)\,dt
\\ &
=\frac{\Gamma(1)}{12z}-\frac{\Gamma(3)}{720z^3}+\frac{\Gamma(5)}{30240z^5}
+O\left(\frac{1}{z^7}\right)
\\ &
=\frac{1}{12z}-\frac{1}{360z^3}+\frac{1}{1260z^5}+O\left(\frac{1}{z^7}\right)
\qquad (z\to\infty).
\end{align*}
すなわち
\[
\Gamma(z+1)=z^z e^{-z}\sqrt{2\pi z}\,
\exp\left(\frac{1}{12z}-\frac{1}{360z^3}+\frac{1}{1260z^5}+O\left(\frac{1}{z^7}\right)\right)
\qquad (z\to\infty).
\]
特に
\[
\Gamma(z+1)=z^z e^{-z}\sqrt{2\pi z}\,
\left(1+\frac{1}{12z}+O\left(\frac{1}{z^2}\right)\right)
\qquad (z\to\infty)
\]
と第1補正項 $1/(12n)$ も得られた.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stirling-Binet の公式(2) 書きかけ}
\label{sec:Binet2}
前節の結果は以下のようにまとめられる:
\begin{align*}
&
\log\Gamma(z+1)=z\log z-z+\log\sqrt{2\pi z}+I(z),
\\ &
I(z)=\int_0^\infty\left(\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}\right)e^{-zt}\frac{dt}{t},
\\ &
\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}
=\frac{t}{12}-\frac{t^3}{720}+\frac{t^5}{30240}+O(t^7).
\end{align*}
この公式は E.~T.~Whittaker and G.~N.~Watson, A course of modern analysis (1927)
の12・31節で ``Binet's first expression for $\log\Gamma(z)$ in terms of an infinite integral''
と呼ばれている. 以下では12・32節に書いてある ``Binet's second expression'' を紹介しよう.
すなわち,
\[
I(z)=2\int_0^\infty \frac{\arctan(t/z)}{e^{2\pi t}-1}\,dt
\]
という公式の証明を目指そう.
\bigskip{\Large\bf 書きかけ}\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{特性函数}
ガンマ分布の特性函数は次の形になる:
\begin{align*}
\varphi_{\tau,\alpha}(t)
=\frac{1}{\tau^\alpha\Gamma(\alpha)}\int_0^\infty e^{itx}e^{-x/\tau}x^{\alpha-1}\,dx
=(1-i\tau t)^{-\alpha}.
\end{align*}
証明するためには $\varphi_{\tau,\alpha}'(t)=i\alpha\tau(1-i\tau t)^{-1}\varphi_{\tau,\alpha}(t)$
を示せば十分である. そのことは以下のようにして示される:
\begin{align*}
\tau^\alpha\Gamma(\alpha)\varphi_{\tau,\alpha}'(t)
&
=\int_0^\infty i e^{itx} e^{-x/\tau} x^\alpha\,dx
%\\ &
= \frac{i}{it-\tau^{-1}}
\int_0^\infty \frac{\d}{\d x}(e^{itx} e^{-x/\tau}) x^\alpha\,dx
\\ &
=\frac{-i\tau}{1-i\tau t}
\int_0^\infty \frac{\d}{\d x}(e^{itx} e^{-x/\tau}) x^\alpha\,dx
%\\ &
=\frac{i\tau}{1-i\tau t}
\int_0^\infty e^{itx} e^{-x/\tau} \frac{\d}{\d x}x^\alpha\,dx
\\ &
=\frac{i\alpha\tau}{1-i\tau t}
\int_0^\infty e^{itx} e^{-x/\tau} x^{\alpha-1}\,dx
=\frac{i\alpha\tau}{1-i\tau t}
\tau^\alpha\Gamma(\alpha)\varphi_{\tau,\alpha}(t).
\end{align*}
1つめの等号で積分記号化での微分を行い,
4つめの等号で部分積分を使った.
特性函数の形から次の再生性がただちに導かれる.
\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.
\paragraph{カイ2乗分布}
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:
\[
f_{2,n/2}(x)\,dx
= \frac{e^{-x/2} x^{n/2-1}}{2^{n/2}\Gamma(n/2)}\,dx
= \frac{e^{-x/2}(x/2)^{n/2}}{\Gamma(n/2)}\frac{dx}{x}.
\]
カイ2乗分布は自由度 $n$ について再生性を持つ.
\begin{theorem}[標準正規分布からカイ2乗分布が得られること]
\label{theorem:chi-square}
確率変数 $X_1,X_2,\ldots$ は独立同分布の確率変数列であり,
各々は標準正規分布にしたがうと仮定する.
このとき $Y=X_1^2+\cdots+X_n^2$ は自由度 $n$ のカイ2乗分布にしたがう.
\end{theorem}
\begin{proof}
次を示せば十分である:
\[
E[f(Y)] = \text{const.}\, \int_0^\infty f(y)e^{-y/2} y^{n/2-1}\,dy.
\]
これを示そう.
\begin{align*}
E[f(Y)]
&
=E[f(X_1^2+\cdots+X_n^2)]
\\ &
=\frac{1}{(2\pi)^{n/2}} \int\!\!\cdots\!\!\int_{\R^n}
f(x_1^2+\cdots+x_n^2)e^{-(x_1^2+\cdots+x_n^2)/2}\,dx_1\cdots dx_n
\\ &
=\frac{A_{n-1}}{(2\pi)^{n/2}} \int_0^\infty f(r^2)e^{-r^2/2} r^{n-1}\,dr
\\ &
=\frac{A_{n-1}}{2(2\pi)^{n/2}} \int_0^\infty f(y)e^{-y/2} y^{(n-1)/2} y^{-1/2}\,dy
\\ &
=\frac{A_{n-1}}{2(2\pi)^{n/2}} \int_0^\infty f(y)e^{-y/2} y^{n/2-1} \,dy.
\end{align*}
3つめの等号で $r=\sqrt{x_1^2+\cdots+x_n^2}$ とおき,
$\R^n$ における微小体積要素が $n-1$ 次元単位球面上の微小面積と $r^{n-1}\,dr$ の積
になることを使い, $n-1$ 次元単位球面の面積を $A_{n-1}$ と書いた.
4つめの等号で $r=y^{1/2}$, $dr=(1/2)y^{-1/2}\,dy$ とおいた.
\qed
\end{proof}
\begin{remark}
以上の計算によって, $n-1$ 次元単位球面の面積 $A_{n-1}$ に関して,
\[
A_{n-1} = \frac{2\pi^{n/2}}{\Gamma(n/2)}
\]
が成立することも示されたことになる.
これは\secref{sec:MB2}での計算結果と一致している.
\qed
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{多項分布とPearsonのカイ2乗統計量と多次元正規分布}
\label{sec:Pearson}
$K=(K_1,\ldots,K_r)$ は多項分布に従う離散型ベクトル値確率変数であるとする.
すなわち, $p_i>0$, $\sum_{i=1}^r p_i=1$ であるとし,
実数 $k_1,\ldots,k_r$ に対して,
$K=(k_1,\ldots,k_r)$ となる確率は, $k_i$ がすべて非負の整数で $\sum_{i=1}^r k_i=n$
のとき
\[
P(K=(k_1,\ldots,k_r)) = \frac{n!}{k_1!\cdots k_r!} p_1^{k_1}\cdots p_r^{k_r}
\]
であり, それ以外のとき $0$ であるとする.
\begin{example}[サイコロ]
$1$ から $6$ までの目が同じ確率で出るサイコロを $n$ 回ふったときに
$i$ の目が出た回数を $K_i$ と表わすと, $K=(K_1,\ldots,K_6)$ は
$r=6$, $p_i=1/6$ の多項分布にしたがう.
一般の多項分布も同様に理解できる%
\footnote{確率変数の話はサイコロをふる話だと思っていると理解し易いと思う.
確率変数はプログラミングにおける「乱数」のことだと思ってもよい.
様々な分布を持つ確率変数を考えることは様々な「乱数」を考えることと
同じだと思ってよい.}.
\qed
\end{example}
確率の総和が $1$ になることは多項定理
\[
\sum_{k_1+\cdots+k_r=m} \frac{m!}{k_1!\cdots k_r!} x_1^{k_1}\cdots x_r^{k_r}
= (x_1+\cdots+x_r)^m
\]
を使えば確認できる. 多項定理は二項定理と同様の考え方で証明される.
もしくは二項定理を用いた $m$ に関する帰納法で証明される.
$K_i$ の平均は $\mu_i=np_i$ になる:
\[
\mu_i
=E[K_i]
=\sum_{k_1+\cdots+k_r=n} \frac{n!}{k_1!\cdots k_r!} p_1^{k_1}\cdots p_r^{k_r} k_i
=np_i(p_1+\cdots+p_r)^{n-1}
=np_i.
\]
3つ目の等号で多項定理を使った.
$K_i$ の分散は $\sigma_i^2=np_i(1-p_i)$ になる:
\begin{align*}
&
E[K_i(K_i-1)]
=\sum_{k_1+\cdots+k_r=n} \frac{n!}{k_1!\cdots k_r!} p_1^{k_1}\cdots p_r^{k_r} k_i(k_i-1)
\\ & \hphantom{E[K_i(K_i-1)]}
=n(n-1)p_i^2(p_1+\cdots+p_r)^{n-2}
=n(n-1)p_i^2,
\\
&
\sigma_i^2
= E[K_i^2]-\mu_i^2 = E[K_i(K_i-1)]+\mu_i-\mu_i^2
\\ & \hphantom{\sigma_i^2}
=n(n-1)p_i^2 + np_i - n^2p_i^2
=np_i(1-p_i).
\end{align*}
2つ目の等号で多項定理を使った.
$i\ne j$ のとき $K_i$ と $K_j$ の共分散は $\sigma_{ij}=\sigma_{ji}=-np_ip_j$ になる:
\begin{align*}
\sigma_{ij}
&
=E[K_i K_j]-\mu_i\mu_j
=\sum_{k_1+\cdots+k_r=n} \frac{n!}{k_1!\cdots k_r!} p_1^{k_1}\cdots p_r^{k_r} k_i k_j
-\mu_i\mu_j
\\ &
=n(n-1)p_ip_j-n^2p_ip_j
=-np_ip_j.
\end{align*}
3つ目の等号で多項定理を使った.
したがってベクトル値確率変数 $X=(X_1,\ldots,X_r)$ を
\[
X_i = \frac{K_i-np_i}{\sqrt{np_i}}
\]
と定めると, $X_i$ の平均は $0$ になり, 分散は
\[
p_{ii} = \frac{np_i(1-p_i)}{np_i} = 1-p_i = 1-\sqrt{p_i}\,\sqrt{p_i}
\]
になり, $i\ne j$ のとき $X_i$ と $X_j$ の共分散は
\[
p_{ij}=p_{ji}=\frac{-np_ip_j}{n\sqrt{p_i}\,\sqrt{p_j}}=-\sqrt{p_i}\,\sqrt{p_j}
\]
になる. すなわち $X=(X_1,\ldots,X_r)$ の分散共分散行列 $P=[p_{ij}]$ は
\[
P = E + aa^T, \qquad
a =
\begin{bmatrix}
\sqrt{p_1} \\ \vdots \\ \sqrt{p_r}
\end{bmatrix}
\]
の形になる. ここで $E$ は単位行列であり, $a^T$ は列ベクトル $a$ の転置である.
$\sum_{i=1}^r p_i=1$ より, $a$ は単位ベクトルになる.
列ベクトル $v\in\R^r$ に対して,
\[
Pv = v - \bra a,v\ket a
\]
は $a$ の直交補空間への $v$ の直交射影になる($r=3$ の場合の図を描いてみよ).
ここでEuclid内積を $\bra\ ,\ \ket$ と書いた.
$P$ が単位ベクトル $a$ の直交補空間への直交射影を表現する行列であることから,
$P^2=P$ となり, $P$ のランクが $r-1$ になることがわかる%
\footnote{この結果はPearsonのカイ2乗統計量が $n\to\infty$ で
カイ2乗分布にしたがう確率変数に(弱)収束することを示すためのキーになる.}.
\begin{definition}[Pearsonのカイ2乗統計量]
多項分布にしたがう確率変数 $K=(K_1,\ldots,K_r)$ から定まる
次の確率変数を{\bf Pearsonのカイ2乗統計量}と呼ぶ:
\[
Y=\sum_{i=1}^r X_i^2 = \sum_{i=1}^r \frac{(K_i-np_i)^2}{np_i}
\]
これは{\bf カイ2乗分布}にしたがう確率変数ではない. しかし次の定理が成立している.
\qed
\end{definition}
\begin{theorem}
Pearsonのカイ2乗統計量は $n\to\infty$ で自由度 $r-1$ のカイ2乗分布に
したがう確率変数に(弱)収束する%
\footnote{この結果はよく使われているPearsonのカイ2乗検定の基礎になっている.
このノートにこの節を追加しようと思った動機は,
入門的な統計学の教科書には
「$n$ が大きなとき, どうしてPearsonのカイ2乗統計量をカイ2乗分布で近似してよいのか」
に関する説明がないように見えたからである.}.
\end{theorem}
\begin{proof}
多次元版の中心極限定理%
\footnote{多次元版中心極限定理も1次元版中心極限定理と同様の方法で証明される.
すなわち特性函数の $n\to\infty$ が正規分布の特性函数に収束することを示せばよい.}%
より, $X=(X_1,\ldots,X_r)$ は平均 $0$, 分散共分散行列が $P$ の
多次元正規分布に(弱)収束する.
したがって, $X=(X_1,\ldots,X_r)$ が
平均 $0$, 分散共分散行列 $P$ を持つ多次元正規分布にしたがうとき,
\[
Y = \sum_{i=1}^r X_i^2
\]
が自由度 $r-1$ のカイ2乗分布にしたがうことを示せばよい.
そのことを示すためには次の一般的な補題を示せば十分である.
\qed
\end{proof}
\begin{lemma}
ベクトル値確率変数 $X=(X_1,\ldots,X_r)$ が
平均 $0$, 分散共分散行列 $P$ を持つ多次元正規分布にしたがうとき,
$P^2=P$ かつ $P$ のランクが $s$ ならば,
$Y=\sum_{i=1}^r X_i^2$ は自由度 $s$ のカイ2乗分布にしたがう.
\end{lemma}
\begin{proof}
一般に分散共分散行列 $P$ は実対称行列になる.
$P^2=P$ ならば $P$ の固有値は $0$ と $1$ になり,
固有値 $1$ の重複度と $P$ のランクは一致する.
ゆえにある直交行列 $U$ が存在して,
\[
U^T PU=U^{-1}PU=\diag(\underbrace{1,\ldots,1}_{s},0,\ldots,0).
\]
$P$, $U$ の $(i,j)$ 成分をそれぞれ $p_{ij}$, $u_{ij}$ と書き,
\[
Z_i = \sum_{j=1}^r u_{ji} X_j
\]
とおく. このとき,
$X=(X_1,\ldots,X_r)$ から $Z=(Z_1,\ldots,Z_r)$ への変換は直交変換なので
\[
Y = \sum_{i=1}^r X_i^2 = \sum_{i=1}^r Z_i^2
\]
が成立し, 直交行列 $U$ の取り方より,
\[
E[Z_i Z_l]
=
\sum_{j,k=1}^r u_{ji}E[X_j X_k]u_{kl}
=
\sum_{j,k=1}^r u_{ji}p_{jk}u_{kl}
=
\begin{cases}
1 & (1\leqq i=l\leqq s), \\
0 & (\text{その他の場合}).
\end{cases}
\tag{\&}
\]
確率変数を成分に持つ行列まで期待値汎函数 $E[\ ]$ を拡張すると以上の計算を以下の
ように書くことができる:
\begin{align*}
&
\begin{bmatrix}
Z_1 & \cdots & Z_r
\end{bmatrix}
=
\begin{bmatrix}
X_1 & \cdots & X_r
\end{bmatrix}
U,
\\ &
E\left[
\begin{bmatrix}
Z_1 \\ \vdots \\ Z_r
\end{bmatrix}
\begin{bmatrix}
Z_1 & \cdots & Z_r
\end{bmatrix}
\right]
=
U^T
E\left[
\begin{bmatrix}
X_1 \\ \vdots \\ X_r
\end{bmatrix}
\begin{bmatrix}
X_1 & \cdots & X_r
\end{bmatrix}
\right]
U
\\ & \qquad
=U^T P U
=\diag(\underbrace{1,\ldots,1}_{s},0,\ldots,0).
\end{align*}
公式($\&$)より, $Z_1,\ldots,Z_s$ は独立同分布で各々が標準正規分布にしたがい,
$Z_{s+1},\ldots,Z_r$ は $0$ に台を持つデルタ分布にしたがうこと
(確率 $1$ で $Z_{s+1}=\cdots=Z_r=0$ となること)がわかる.
ゆえに\theoremref{theorem:chi-square}より
\[
\sum_{i=1}^r Z_i^2 = Z_1^2 +\cdots+Z_s^2 \qquad\text{(almost sure)}
\]
は自由度 $s$ のカイ2乗分布にしたがう.
これで示すべきことが示された.
\qed
\end{proof}
\begin{remark}[多次元正規分布]
非負の固有値を持つ $r$ 次の実対称行列 $A$ に対して,
$\R^r$ 値の確率変数 $X=(X_1,\ldots,X_r)$ が
平均 $0$, 分散共分散行列 $A$ の多次元正規分布にしたがうとは,
その特性函数が次の形になることであると定義できる:
\[
E\left[e^{i\bra t,X\ket}\right] = \exp\left(-\frac{1}{2}\bra t, A t\ket\right)
\qquad (t\in\R^r).
\tag{$*$}
\]
ここで $\bra\ ,\ \ket$ は $\R^r$ の標準Euclid内積である.
このスタイルであれば分散共分散行列 $A$ が可逆でなくても多次元正規分布が定義される.
最も極端な場合として $A=0$ のとき $X$ は $(0,\ldots,0)$ に台を持つデルタ分布に
したがう.
$\sigma_1>0,\ldots,\sigma_s>0$,
$A=\diag(\sigma_1^2,\ldots,\sigma_s^2,0,\ldots,0)$ のとき,
$X_1,\ldots,X_r$ は独立であり,
$i=1,\ldots,s$ に対する $X_i$ は平均 $0$, 分散 $\sigma_i^2$ の正規分布にしたがい,
$i=s+1,\ldots,r$ に対する $X_i$ は $0$ に台を持つデルタ分布にしたがう.
一般の場合は直交変換によってそのような場合に帰着する.
特に任意の非負実対称行列 $A$ を分散共分散行列に持つ
多次元正規分布が存在することがわかる.
$A$ が可逆ならば, $\R^r$ 上の有界連続函数 $f(x)$ について,
\[
E[f(X)]
=
\frac{1}{\sqrt{\det(2\pi A)}}
\int_{\R^r}f(x) \exp\left(-\frac{1}{2}\bra x, A^{-1} x\ket\right)
\,dx
\]
となる. ここで $dx$ は $\R^r$ 上のLebesgue測度である.
このとき($*$)が成立することは $A$ を直交行列で対角化することによって示される.
\qed
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 $n$ の $t$ 分布の平均と分散をそれぞれ $\mu_n$, $\sigma_n^2$ と書くと、
\[
\mu_n = 0 \quad (n>1), \qquad
\sigma_n^2 = \frac{n}{n-2} \quad (n>2).
\]
になる. 自由度無限大の極限で $t$ 分布は標準正規分布に収束する.
自由度 $1$ の $t$ 分布は{\bf Cauchy分布}とも呼ばれており,
平均も有限の分散も持たない確率分布の典型例になっている.
自由度 $2$ の $t$ 分布は平均 $0$ を持つが, 分散は無限大になる.
\begin{theorem}[標準正規分布とカイ2乗分布から $t$ 分布が得られること]
\label{theorem:t}
$Z$, $Y$ は独立な確率変数であり,
$Z$ は標準正規分布にしたがい,
$Y$ は自由度 $n$ のカイ2乗分布にしたがうと仮定する.
このとき
\[
T = \frac{Z}{\sqrt{Y/n}}
\]
は自由度 $n$ の $t$ 分布にしたがう.
\end{theorem}
\begin{proof}
次を示せば十分である:
\[
E[f(T)]
=\text{const.}\,
\int_{-\infty}^\infty f(t)\left(1+\frac{t^2}{n}\right)^{-(n+1)/2}\,dt.
\]
これを示そう. $a_n=1/(\sqrt{2\pi}\, 2^{n/2}\Gamma(n/2))$ とおくと,
\begin{align*}
E[f(T)]
&
=E\left[f\left(\frac{Z}{\sqrt{Y/n}}\right)\right]
=a_n\int_0^\infty\left(
\int_{-\infty}^\infty f\left(\frac{z}{\sqrt{y/n}}\right) e^{-z^2/2}e^{-y/2}y^{n/2-1}\,dz
\right)\,dy
\\ &
=a_n\int_0^\infty\left(
\int_{-\infty}^\infty f\left(\frac{z}{\sqrt{y/n}}\right) e^{-(y+z^2)/2}y^{n/2-1}\,dz
\right)\,dy
\\ &
=\frac{a_n}{\sqrt{n}}\int_0^\infty\left(
\int_{-\infty}^\infty f(t) e^{-(1+t^2/n)y/2}y^{(n+1)/2-1}\,dt
\right)\,dy
\\ &
=\frac{a_n}{\sqrt{n}}\int_{-\infty}^\infty f(t)\left(
\int_0^\infty e^{-(1+t^2/n)y/2}y^{(n+1)/2-1}\,dy
\right)\,dt
\\ &
=\frac{2^{(n+1)/2}\Gamma((n+1)/2)a_n}{\sqrt{n}}
\int_{-\infty}^\infty f(t)\left(1+\frac{t^2}{n}\right)^{-(n+1)/2}\,dt.
\end{align*}
ここで, 2つめの等号は標準正規分布とカイ2乗分布の定義から得られる.
4つめの等号では $z=t\sqrt{y/n}$ とおいた($z^2=yt^2/n$, $dz=y^{1/2}dt/\sqrt{n}$).
6つめの等号で次の公式を使った:
\[
\int_0^\infty e^{-\alpha y} y^{s-1}\,dy
=\int_0^\infty e^{-x} \left(\frac{x}{\alpha}\right)^{s-1} \frac{dx}{\alpha}
=\alpha^{-s}\Gamma(s)
\qquad (\alpha,s>0).
\]
証明のためには必要な計算であるが, さらに,
\[
\frac{2^{(n+1)/2}\Gamma((n+1)/2)a_n}{\sqrt{n}}
=\frac{2^{(n+1)/2}\Gamma((n+1)/2)}{\sqrt{n}\,\sqrt{2\pi}\, 2^{n/2}\Gamma(n/2)}
=\frac{\Gamma((n+1)/2)}{\sqrt{n\pi}\,\Gamma(n/2)}
=c_n.
\]
これで確認するべきことがすべて確認された.
\qed
\end{proof}
\begin{theorem}[正規分布から $t$ 分布が得られること]
\label{theorem:normal-t}
$X_1,X_2,\ldots$ が独立同分布な確率変数列であり,
各々が平均 $\mu$, 分散 $\sigma^2$ の正規分布にしたがうとき,
\[
M_n=\frac{1}{n}\sum_{k=1}^n X_k, \qquad
U_n^2= \frac{1}{n-1}\sum_{k=1}^n(X_k-M_n)^2, \qquad
T_n = \frac{M_n-\mu}{U_n/\sqrt{n}}
\tag{$*$}
\]
とおくと, $M_n$ と $U_n$ は独立になり,
$M_n$ は平均 $\mu$, 分散 $\sigma^2/n$ の正規分布にしたがい,
$(n-1)U_n^2/\sigma^2$ は自由度 $n-1$ のカイ2乗分布にしたがい,
$T_n$ は自由度 $n-1$ の $t$ 分布にしたがう.
($U_n\geqq 0$ と仮定した.)
\qed
\end{theorem}
\begin{remark}[$T_n$ の出処]
上の定理の設定のもとで, $E[M_n]=\mu$, $E[U_n^2]=\sigma^2$ となる.
$\mu$, $\sigma^2$ は
それぞれ母集団平均(population mean), 母集団分散(population variant)と呼ばれ,
$M_n$, $U_n^2$ は
それぞれ標本平均(sample mean), 不偏分散(unbiased variant)と呼ばれている.
正規分布の再生性より, $M_n$ は平均 $\mu$, 分散 $\sigma^2/n$ の正規分布
にしたがう. ゆえに
\[
T_n = \frac{M_n-\mu}{U_n/\sqrt{n}}
\]
に類似の確率変数
\[
Z_n = \frac{M_n-\mu}{\sigma/\sqrt{n}}
\]
は標準正規分布にしたがう. 上で述べたことは,
$Z_n$ の分母の母集団標準偏差 $\sigma$ を
経験的に得られた不偏標準偏差 $U_n$ で置き換えると,
標準正規分布ではなく, 自由度 $n-1$ の $t$ 分布にしたがうようになるということである.
母集団分散 $\sigma^2$ がわかっている場合には
標準正規分布になる $Z_n$ を使えるが,
そうでない場合には $Z_n$ を使えない.
そこで母集団の分散 $\sigma^2$ の代わりに
経験的に得られた不偏分散 $U_n^2$ を代用すると,
確率分布は正規分布よりも裾野が太い $t$ 分布になってしまうのである.
\qed
\end{remark}
\begin{remark}[自由度の大きな $t$ 分布が標準正規分布で近似されること]
$n\to\infty$ で
\[
\left(1+\frac{t^2}{n}\right)^{-(n+1)/2}
=
\left(1+\frac{t^2}{n}\right)^{-1/2}
\left(1-\frac{t^2}{n}+O\left(\frac{1}{n^2}\right)\right)^{n/2}
\longrightarrow
e^{-t^2/2}
\]
となるので, 自由度無限大の極限で $t$ 分布の確率密度函数は
標準正規分布の確率密度函数に収束する.
自由度が大きな $t$ 分布は標準正規分布で近似される.
\qed
\end{remark}
\begin{proof}[\theoremref{theorem:normal-t}の証明]
必要ならば $X_k$ を $X_k-\mu$ で置き換えることによって,
$\mu=0$ であると仮定できる.
以下では $\mu=0$ の場合のみを扱う.
まず, $X_1,X_2,\ldots$ は独立同分布な確率変数列であり,
各々が平均 $0$ と有限の分散 $\sigma^2$ を持つと仮定し,
正規分布であると仮定せずにどの程度のことが言えるかを調べよう.
($*$)のように $M_n$, $U_n$ を定める.
($U_n\geqq 0$ としておく.)
$Y_n=\sqrt{n}\,M_n=(X_1+\cdots+X_n)/\sqrt{n}$ とおく.
正規直交座標系 $(X_1,\ldots,X_n)$ を
$Y_n$ を含む別の正規直交座標系 $(Y_1,\ldots,Y_n)$ に座標変換できる%
\footnote{$Y_n$ に対応する単位ベクトル $(1,1,\ldots,1)/\sqrt{n}$ を
含む正規直交系を作り, その正規直交系に対応する座標系を取ればよい.
$(Y_1,\ldots,Y_{n-1})$ は $Y_n$ に対応する単位ベクトルの直交補空間
上の正規直交座標系になる.
$Y_k$ の具体的な取り方の例については\secref{sec:Y_k}を参照せよ.}.
このとき多項式の計算として
\[
\sum_{k=1}^n X_k^2 = \sum_{k=1}^n Y_k^2
\]
が成立している%
\footnote{直交変換でノルムの2乗が保たれる.}%
ので,
\begin{align*}
\sum_{k=1}^n(X_k-M_n)^2
&
=\sum_{k=1}^n (X_k^2 - 2M_n X_k + M_n^2)
=\sum_{k=1}^n X_k^2 -2M_n\sum_{k=1}^n X_k + n M_n^2
\\ &
=\sum_{k=1}^n X_k^2 -nM_n^2
=\sum_{k=1}^n Y_k^2 - Y_n^2
=\sum_{k=1}^{n-1} Y_k^2.
\end{align*}
さらに, 座標変換の仕方より $E[Y_k^2]=\sigma^2$ となるので,
\[
E\left[\sum_{k=1}^n(X_k-M_n)^2\right]
=E\left[\sum_{k=1}^{n-1} Y_k^2\right]
=(n-1)\sigma^2
\]
となることもわかる%
\footnote{$Y_k$ を使わない直接的計算でこの結果を確認することも容易である.}.
これより $E[U_n^2]=\sigma^2$ となることもわかる.
以上の結果は $X_k$ が正規分布でなくても成立している.
以下では, $X_k$ たちが平均 $0$, 分散 $\sigma^2$ の正規分布に
したがうと仮定しよう.
このとき正規分布の確率密度函数の形より, $Y_k$ たちも独立同分布になり,
各々が平均 $0$, 分散 $\sigma^2$ の正規分布にしたがう.
ゆえに
\[
U_n^2 = \frac{1}{n-1}\sum_{k=1}^n (X_k-M_n)^2 = \frac{1}{n-1}\sum_{k=1}^{n-1} Y_k^2, \qquad
M_n = \frac{1}{n}\sum_{k=1}^n X_k = \frac{1}{\sqrt{n}} Y_n
\]
は独立になり, \theoremref{theorem:chi-square}より,
\[
\frac{n-1}{\sigma^2}U_n^2 = \frac{1}{\sigma^2}\sum_{k=1}^{n-1} Y_k^2
\]
は自由度 $n-1$ のカイ2乗分布になる.
したがって\theoremref{theorem:t}より
\[
\frac{\dfrac{M_n-\mu}{\sigma/\sqrt{n}}}{\sqrt{\dfrac{(n-1)U_n^2/\sigma^2}{n-1}}}
=\frac{M_n-\mu}{U_n/\sqrt{n}}
\]
が自由度 $n-1$ の $t$ 分布にしたがうことがわかる.
\qed
\end{proof}
\begin{theorem}[正規分布から $t$ 分布が得られること(\theoremref{theorem:normal-t})の一般化]
\label{theorem:normal-t2}
確率変数たち $X_{s,k}$ ($s=1,\ldots,r$, $k=1,\ldots,n_s$) は独立であり,
各 $s$ ごとに $X_{s,k}$ たちは
平均 $\mu_s$, 分散 $\sigma_s^2$ の正規分布にしたがっていると仮定する.
このとき
\begin{alignat*}{2}
&
M_s = \frac{1}{n_s}\sum_{k=1}^{n_s} X_{s,k},
& \qquad &
U_s^2 = \frac{1}{n_s-1}\sum_{k=1}^{n_s}(X_{s,k}-M_s)^2,
\\ &
Z=
\frac
{\dsum_{s=1}^r(M_s-\mu_s)}
{\sqrt{\dsum_{s=1}^r\frac{\sigma_s^2}{n_s}}},
& \qquad &
Y=\sum_{s=1}^r \frac{n_s-1}{\sigma_s^2}U_s^2
=\sum_{s=1}^r\frac{1}{\sigma_s^2}\sum_{k=1}^{n_s}(X_{s,k}-M_s)^2,
\\ &
n=\sum_{s=1}^r n_s,
& \qquad &
T = \frac{Z}{\sqrt{Y/(n-r)}}
\end{alignat*}
とおくと, $T$ は自由度 $n-r$ の $t$ 分布にしたがう%
\footnote{もしも $\sigma_1^2=\cdots=\sigma_r^2=\sigma^2$ ならば
$T$ の定義中の分子分母の $\sigma$ がキャンセルし,
$Z$ は母集団分散 $\sigma^2$ の情報抜きに計算できる量になることに注意せよ.}.
\end{theorem}
\begin{proof}
\theoremref{theorem:normal-t}の証明より,
$M_s$ と $U_s^2$ は独立になり,
$M_s$ は平均 $\mu_s$, 分散 $\sigma_s^2/n_s$ の正規分布にしたがい,
$(n_s-1)U_s^2/\sigma_s^2$ は自由度 $n_s-1$ のカイ2乗分布にしたがうことがわかる.
特に $Z$ と $Y$ は独立になる.
正規分布の再生性より,
$\sum_{s=1}^{n_s}(M_s-\mu_s)$ は平均 $0$,
分散 $\sum_{s=1}^r(\sigma_s^2/n_s)$ の正規分布にしたがうので,
$Z$ は標準正規分布にしたがう.
カイ2乗分布の再生性より, $Y$ は自由度 $n-r$ のカイ2乗分布にしたがう.
ゆえに, \theoremref{theorem:t}より,
$T$ は自由度 $n-r$ の $t$ 分布にしたがう.
\qed
\end{proof}
上の定理を $s=2$, $\mu_1=\mu$, $\mu_2=-\mu$, $\sigma_1=\sigma_2$ の場合に
適用することによって次の系がただちに得られる.
\begin{cor}
確率変数 $X_{s,k}$ ($s=1,2$, $k=1,\ldots,n_s$)たちは独立同分布であり,
平均 $\mu$, 分散 $\sigma$ の正規分布にしたがっているとする.
このとき
\begin{align*}
&
M_s=\frac{1}{n}\sum_{k=1}^{n_s}X_{s,k},
\qquad
Y'=\sum_{s=1}^2\sum_{k=1}^{n_s}(X_{s,k}-M_s)^2,
\\ &
T=\frac{M_1-M_2}{\sqrt{\dfrac{1}{n_1}+\dfrac{1}{n_2}}\,\sqrt{\dfrac{Y'}{n_1+n_2-2}}}
\end{align*}
とおくと, $T$ は自由度 $n_1+n_2-2$ の $t$ 分布にしたがう%
\footnote{この結果は分散が等しい正規分布にしたがう2つのサンプルの
平均が等しいと言えるどうかの検定に使われる.}.
\qed
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{不偏分散の直交変換による取り扱いについて}
\label{sec:Y_k}
$X_1,X_2,\ldots$ は独立同分布な確率変数列であり,
それぞれの平均は $0$ で分散は $\sigma^2$ であると仮定する%
\footnote{平均が $\mu\ne 0$ の場合には $X_k$ を $X_k-\mu$ で置き換えて考えれば同様である.}.
このときサイズ $n$ の標本 $X_1,\ldots,X_n$ の標本平均 $\Xbar_n$ と不偏分散 $U_n^2$ が
\[
\Xbar_n = \frac{1}{n} \sum_{k=1}^n X_k, \qquad
U_n^2 = \frac{1}{n-1}\sum_{k=1}^n (X_k - \Xbar_n)^2
\]
と定義される. さらに次が成立していることに注意せよ:
\begin{align*}
\sum_{k=1}^n (X_k - \Xbar_n)^2 = \sum_{k=1}^n X_k^2 - n\Xbar_n^2.
\end{align*}
確率変数 $Y_1,\ldots,Y_n$ を次のように定義する:
\begin{align*}
&
Y_n = \frac{1}{\sqrt{n}}\sum_{k=1}^n X_k = \sqrt{n}\,\Xbar_n,
\\ &
Y_k = \frac{1}{\sqrt{k(k+1)}}\left(\sum_{j=1}^k X_j - k X_{k+1}\right)
\quad (k=1,2,\ldots,n-1).
\end{align*}
$n\times n$ 行列 $A=[a_{ij}]$ を
\[
A=
\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 & 1 \\
-1 & 1 & 1 & \cdots & 1 & 1 \\
0 &-2 & 1 & \cdots & 1 & 1 \\
0 & 0 &-3 & \ddots & \vdots & \vdots \\
\vdots & \vdots & \ddots & \ddots & 1 & 1 \\
0 & 0 & \cdots & 0 & -(n-1) & 1 \\
\end{bmatrix}
\begin{bmatrix}
1/\sqrt{2} \\
& 1/\sqrt{6} \\
& & 1/\sqrt{12} \\
& & & \ddots \\
& & & & 1/\sqrt{n(n+1)} \\
& & & & & \sqrt{n} \\
\end{bmatrix}
\]
と定めると, $Y_j$ たちの定義は
\[
Y_j = \sum_{i=1}^n a_{ij} X_i
\]
と書ける. さらに行列 $A$ は直交行列であることもわかる%
\footnote{${}^t AA$ が単位行列になることを直接確認できる.
まず行列 $A$ の定義式の左側の行列の列ベクトルたちが互いに直交する
ことを確認せよ.}.
ゆえに $\sum_{i=1}^n a_{ki}a_{li}=\delta_{kl}$ が成立するので
\begin{align*}
\sum_{i=1}^n Y_i^2
= \sum_{i,k,l} a_{ki}a_{li} X_k X_l
= \sum_{k,l}\delta_{kl} X_k X_l
= \sum_{k=1}^n X_k^2.
\end{align*}
さらに, $E[X_i X_j]=\sigma^2\delta_{ij}$ も使うと,
\begin{align*}
E[Y_k Y_l]
=\sum_{j,j} a_{ki}a_{lj}E[X_i X_j]
=\sigma^2 \sum_{i,j} a_{ki}a_{li}\delta_{ij}
=\sigma^2 \sum_{i=1}^n a_{ki}a_{li}
=\sigma^2 \delta_{kl}.
\end{align*}
以上の公式を使うと,
\begin{align*}
\sum_{k=1}^n X_k^n - n\Xbar_n^2
= \sum_{k=1}^n Y_k - Y_n^2
= \sum_{k=1}^{n-1} Y_k.
\end{align*}
ゆえに
\begin{align*}
U_n^2
=\frac{1}{n-1}\left(\sum_{k=1}^n X_k^2 - n\Xbar^2\right)
=\frac{1}{n-1}\sum_{k=1}^{n-1}Y_k^2.
\end{align*}
不偏分散が $n-1$ 個の和の $n-1$ の分の1の形で書けた.
さらに
\begin{align*}
E[U_n^2] = \frac{1}{n-1}\sum_{k=1}^{n-1}E[Y_k^2]
=\frac{1}{n-1}(n-1)\sigma^2=\sigma^2.
\end{align*}
もしも $X_k$ たちが独立で平均 $0$, 分散 $\sigma^2$ の正規分布にしたがっているならば,
$Y_k$ たちも独立で平均 $0$, 分散 $\sigma^2$ の正規分布にしたがうことも
容易に確かめられる. この場合には $\Xbar_n=Y_n/\sqrt{n}$ は平均 $0$, 分散 $\sigma^2/n$ の
正規分布にしたがい, $(n-1)U_n^2/\sigma^2=\sum_{k=1}^{n-1}Y_k^2/\sigma^2$ は
自由度 $n-1$ のカイ二乗分布にしたがい, それらは独立になる.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{第一種および第二種ベータ分布と $F$ 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.
第一種ベータ分布の確率密度函数で $x$ に $x/(1+x)$ ($x>0$)を代入すると
\begin{align*}
f_{\alpha,\beta}\left(\frac{x}{1+x}\right)\frac{dx}{(1+x)^2}
=
\frac{1}{B(\alpha,\beta)}\,
\frac{x^{\alpha-1}}{(1+x)^{\alpha+\beta}}\,dx
\qquad (x>0)
\end{align*}
となる. これは第二種ベータ分布の確率密度函数である.
整理の途中で $1-x/(1+x)=1/(1+x)$ を使った.
さらに, $m,n>0$ とし,
第二種ベータ分布の確率密度函数の $x$ に $mx/n$ ($x>0$) を代入すると,
上と同様にして,
\begin{align*}
f_{\alpha,\beta}\left( \frac{mx/n}{1+mx/n} \right) \frac{(m/n)\,dx}{(1+mx/n)^2}
=
\frac{1}{B(\alpha,\beta)}\,
\frac{(mx/n)^\alpha}{(1+mx/n)^{\alpha+\beta}}
\,\frac{dx}{x}
\qquad (x>0)
\end{align*}
となる. これは, $\alpha=m/2$, $\beta=n/2$ のとき, 次の形になる:
\[
g_{m,n}(x)\,dx
=
\frac{1}{B(m/2,n/2)}\,
\frac{(mx/n)^{m/2}}{(1+mx/n)^{(m+n)/2}}
\,\frac{dx}{x}
\qquad (x>0).
\]
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.
パラメーター $m,n$ の $F$ 分布の
平均と分散をそれぞれ $\mu_{m,n}$, $\sigma_{m,n}^2$ と書くと,
\[
\mu_{m,n}=\frac{n}{n-2} \quad (n>2), \qquad
\sigma_{m,n}^2 = \frac{2n^2(m+n-2)}{m(n-2)^2(n-4)} \quad (n>4)
\]
になる.
$X$ がパラメーター $m,n$ の $F$ 分布にしたがうならば,
$(mX/n)/(1+mX/n)$ はパラメーター $m/2,n/2$ の第一種ベータ分布にしたがい,
$mX/n$ はパラメーター $m/2,n/2$ の第二種ベータ分布にしたがう.
\begin{theorem}[カイ2上分布から $F$ 分布が得られること]
\label{theorem:F}
独立な確率変数 $Y$, $Z$ が
それぞれ自由度 $m$, $n$ のカイ2乗分布にしたがうとき,
\[
X = \frac{Y/m}{Z/n}
\]
はパラメーター $m,n$ の $F$ 分布にしたがう.
したがって, $Y_k$, $Z_l$ が
独立同分布な確率変数であり,
各々が標準正規分布にしたがうとき,
\[
X=
\frac
{\left(\sum_{k=1}^m Y_k^2\right)/m}
{\left(\sum_{l=1}^n Z_l^2\right)/n}
\]
はパラメーター $m,n$ の $F$ 分布にしたがう.
\end{theorem}
\begin{proof}
後半の主張は前半の主張と\theoremref{theorem:chi-square}から得られる.
前半の主張を示すためには
\[
E[f(X)]
=\text{const.}\,
\int_0^\infty
f(x)
\frac{(mx/n)^{m/2}}{(1+mx/n)^{(m+n)/2}}
\,\frac{dx}{x}
\]
を示せばよい. $a=[2^{(m+n)/2}\Gamma(m/2)\Gamma(n/2)]^{-1}$ とおくと,
\begin{align*}
E[f(X)]
&
=E\left[f\left(\frac{Y/m}{Z/n}\right)\right]
=a\int_0^\infty\left(
\int_0^\infty f\left(\frac{y/m}{z/n}\right) e^{-(y+z)/2} y^{m/2-1} z^{n/2-1}\,dy
\right)\,dz
\\ &
=a\int_0^\infty\left(
\int_0^\infty f(x) e^{-(1+mx/n)z/2}
\left(\frac{m}{n}xz\right)^{m/2-1} z^{n/2-1} \frac{m}{n}z\,dx
\right)\,dz
\\ &
=a\int_0^\infty f(x)
\left(\left(\frac{mx}{n}\right)^{m/2}
\int_0^\infty e^{-(1+mx/n)z/2} z^{(m+n)/2-1}\,dz
\right)\frac{dx}{x}
\\ &
=2^{(m+n)/2}\Gamma\left(\frac{m+n}{2}\right)a
\int_0^\infty f(x)
\left(\frac{mx}{n}\right)^{m/2} \left(1+\frac{mx}{n}\right)^{-(m+n)/2}
\frac{dx}{x}
\end{align*}
3つめの等号で $y/m=(z/n)x$ とおいた($y=(mx/n)z$, $dy=(m/n)z\,dx$).
5つめの等号で次の一般的な公式を使った:
\[
\int_0^\infty e^{-\alpha z} z^{s-1}\,dy
=\int_0^\infty e^{-t} \left(\frac{t}{\alpha}\right)^{s-1} \frac{dt}{\alpha}
=\alpha^{-s}\Gamma(s)
\qquad (\alpha,s>0).
\]
これで示すべきことは示された. さらに
\[
2^{(m+n)/2}\Gamma\left(\frac{m+n}{2}\right)a
=\frac{2^{(m+n)/2}\Gamma((m+n)/2)}{2^{(m+n)/2}\Gamma(m/2)\Gamma(n/2)}
=\frac{1}{B(m/2,n/2)}.
\]
これで確認するべきことがすべて確認された.
\qed
\end{proof}
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.
このことは以下のように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{ガンマ分布と第一種と第二種のベータ分布の関係}
\label{sec:Betas}
ガンマ函数とベータ函数の関係式
\[
\Gamma(p)\Gamma(q) = \Gamma(p+q)B(p,q)
\]
を二通りの経路で証明してみよう. ここで
\[
\Gamma(s)=\int_0^\infty e^{-x}x^{s-1}\,dx, \qquad
B(p,q)
=\int_0^1 t^{p-1}(1-t)^{q-1}\,dt
=\int_0^\infty \frac{u^{\alpha-1}\,du}{(1+u)^{\alpha+\beta}}.
\]
ベータ函数 $B(p,q)$ の
前者の表示の被積分函数は本質的に第一種ベータ分布の確率密度函数であり,
後者の表示の被積分函数は本質的に第二種ベータ分布の確率密度函数である.
\paragraph{第一種ベータ分布と関係する表示を先に出す方法}
$x=zt$, $y=z(1-t)$ と変数変換すると
\begin{align*}
&
\int_0^\infty\int_0^\infty f(x,y)e^{-(x+y)}x^{p-1}y^{q-1}\,dx\,dy
\\ & \qquad\qquad
=\int_0^1 dt \int_0^\infty dz\, f(zt,z(1-t))e^{-z}z^{p+q-1}t^{p-1}(1-t)^{q-1}.
\end{align*}
ゆえに, もしも $f(x,y)=f(x/y)$ が成立しているならば
\begin{align*}
&
\int_0^\infty\int_0^\infty f\left(\frac{x}{y}\right)e^{-(x+y)}x^{p-1}y^{q-1}\,dx\,dy
\\ & \qquad\qquad
=\int_0^1 dt \int_0^\infty dz\, f\left(\frac{t}{1-t}\right)e^{-z}z^{p+q-1}t^{p-1}(1-t)^{q-1}
\\ & \qquad\qquad\qquad\qquad
=\Gamma(p+q)\int_0^1 f\left(\frac{t}{1-t}\right)t^{p-1}(1-t)^{q-1}\,dt.
\end{align*}
この表示は定数倍を除いて第一種ベータ分布における $f(t/(1-t))$ の期待値に一致する.
特に $f(x/y)=1$ のとき $\Gamma(p)\Gamma(q)=\Gamma(p+q)B(p,q)$ となることがわかる.
さらに $t/(1-t)=u$ すなわち $t=u/(1+u)$ とおくと
\begin{align*}
&
\int_0^\infty\int_0^\infty f\left(\frac{x}{y}\right)e^{-(x+y)}x^{p-1}y^{q-1}\,dx\,dy
\\ & \qquad\qquad
=\Gamma(p+q)\int_0^\infty f(u)
\left(\frac{u}{1+u}\right)^{p-1}\left(\frac{1}{1+u}\right)^{q-1}\frac{du}{(1+u)^2}
\\ & \qquad\qquad\qquad\qquad
=\Gamma(p+q)\int_0^\infty f(u)\frac{u^{p-1}\,du}{(1+u)^{p+q}}
\end{align*}
この表示は定数倍を除いて第二種ベータ分布における $f(u)$ の期待値に一致する.
$F$ 分布は第二種ベータ分布のスケール変換なので,
これはスケール変換の違いを除いて,
$F$ 分布における期待値に一致しているとみなすこともできる.
\paragraph{第二種ベータ分布と関係する表示を先に出す方法}
$x=uy$ によって $x$ から $u$ に積分変数を変換すると,
\begin{align*}
&
\int_0^\infty\int_0^\infty f(x,y)e^{-(x+y)}x^{p-1}y^{q-1}\,dx\,dy
\\ & \qquad\qquad
=\int_0^\infty du \int_0^\infty dy\, f(uy,y) e^{-(1+u)y}y^{p+q-1}u^{\alpha-1}.
\end{align*}
さらに $(1+u)y=z$ すなわち $y=z/(1+u)$ によって $y$ から $z$ に積分変数を変換すると,
\begin{align*}
&
\int_0^\infty\int_0^\infty f(x,y)e^{-(x+y)}x^{p-1}y^{q-1}\,dx\,dy
\\ & \qquad\qquad
=\int_0^\infty du \int_0^\infty dz\,
f\left(\frac{uz}{1+u},\frac{z}{1+u}\right) e^{-z}z^{p+q-1} \frac{u^{p-1}\,du}{(1+u)^{p+q}}.
\end{align*}
ここで $f(x,y)=f(x/y)$ と仮定すると,
\begin{align*}
%&
\int_0^\infty\int_0^\infty f\left(\frac{x}{y}\right)e^{-(x+y)}x^{p-1}y^{q-1}\,dx\,dy
%\\ & \qquad\qquad
=\Gamma(p+q)\int_0^\infty f(u)\frac{u^{p-1}\,du}{(1+u)^{p+q}}.
\end{align*}
上で述べた理由によって, これはスケール変換の違いを除いて,
$F$ 分布における $f$ の期待値に一致しているとみなすこともできる.
以上の計算によって, ガンマ分布にしたがう独立な確率変数の比は
第二種ベータ分布にしたがうことがわかる.
そして正規分布にしたがう独立な確率変数の和はガンマ分布にしたがう.
これらのシンプルな事実を合わせると,
正規分布とカイ2乗分布, $t$ 分布, $F$ 分布の関係が自然に得られる.
以上で使った積分変数たちのあいだには以下の関係がある:
\[
x:y = t:(1-t) = u:1
\]
$x,y$ はガンマ分布の積分変数, $t$ は第一種ベータ分布の積分変数,
$u$ は第二種ベータ分布の確率変数である%
\footnote{筆者は以上の事実に気付いてから,
ベータ函数の二種類の表示を平等に扱いたくなった(2016年7月1日).}.
さらにガンマ函数の積分表示の積分変数の $x$ をそれぞれ $x^2$ に置き換えた式
\[
\Gamma(s)=2\int_{0^\infty} e^{-x^2}x^{2s-1}\,dx
\]
から出発して, $x=r\cos\theta$, $y=r\sin\theta$ もしくは $y=x\tan\theta$ と
積分変数を変換する経路で計算することによって三角函数を用いた
積分表示との関係も明瞭になる.
\paragraph{極座標変換の場合}
$x=r\cos\theta$, $y=r\sin\theta$ によって積分変数を極座標変換すると,
\begin{align*}
&
4\int_0^\infty\int_0^\infty g(x,y)e^{-(x^2+y^2)}x^{2p-1}y^{2q-1}\,dx\,dy
\\ & \qquad\qquad
=
4\int_0^{\pi/2}d\theta\int_0^\infty dr\,
g(r\cos\theta,r\sin\theta)\,e^{-r^2}r^{2(p+q)-1} (\cos\theta)^{2p-1}(\sin\theta)^{2q-1}
\end{align*}
ゆえに $g(x,y)=g(y/x)$ のとき
\begin{align*}
&
4\int_0^\infty\int_0^\infty g\left(\frac{y}{x}\right)e^{-(x^2+y^2)}x^{2p-1}y^{2q-1}\,dx\,dy
\\ & \qquad\qquad
=
\Gamma(p+q)\cdot
2\int_0^{\pi/2} g(\tan\theta)\,(\cos\theta)^{2p-1}(\sin\theta)^{2q-1}\,d\theta.
\end{align*}
特に $g(y/x)=1$ のときより
\[
B(p,q) = 2\int_0^{\pi/2} (\cos\theta)^{2p-1}(\sin\theta)^{2q-1}\,d\theta
\]
となることがわかる.
\paragraph{正接を使う座標変換の場合}
$y=x\tan\theta$ によって $y$ から $\theta$ に積分変数を変換すると,
\begin{align*}
&
4\int_0^\infty\int_0^\infty g(x,y)e^{-(x^2+y^2)}x^{2p-1}y^{2q-1}\,dx\,dy
\\ & \qquad
=
4\int_0^{\pi/2}d\theta\int_0^\infty dx\,
g(x,x\tan\theta)\,e^{-(1+\tan^2\theta)x^2}x^{2(p+q)-1}
(\tan\theta)^{2q-1}(1+\tan^2\theta).
\end{align*}
さらに $x=r/\sqrt{1+\tan^2\theta}$ で $x$ から $r$ に積分変数を変換すると,
\begin{align*}
&
4\int_0^\infty\int_0^\infty g(x,y)e^{-(x^2+y^2)}x^{2p-1}y^{2q-1}\,dx\,dy
\\ &
=
4\int_0^{\pi/2}d\theta\int_0^\infty dr\,
g\left(\frac{r}{\sqrt{1+\tan^2\theta}},\frac{r\tan\theta}{\sqrt{1+\tan^2\theta}}\right)\,
e^{-r^2}r^{2(p+q)-1}
\frac{(\tan\theta)^{2q-1}}{(1+\tan^2\theta)^{p+q-1}}
\end{align*}
ゆえに $g(x,y)=g(y/x)$ のとき
\begin{align*}
&
4\int_0^\infty\int_0^\infty g\left(\frac{y}{x}\right)e^{-(x^2+y^2)}x^{2p-1}y^{2q-1}\,dx\,dy
\\ & \qquad\qquad
=
\Gamma(p+q)\cdot
2\int_0^{\pi/2} g(\tan\theta)\,\frac{(\tan\theta)^{2q-1}}{(1+\tan^2\theta)^{p+q-1}}\,d\theta.
\end{align*}
以上の結果は上で求めた極座標変換の場合の公式と同じものである. なぜならば
\[
\frac{(\tan\theta)^{2q-1}}{(1+\tan^2\theta)^{p+q-1}}
=
(\cos\theta)^{2p-1}(\sin\theta)^{2q-1}.
\]
この公式は左辺に $\tan\theta=\sin\theta/\cos\theta$ を代入しても示せるし,
右辺に
\[
\cos^2\theta=\frac{1}{1+\tan^2\theta}, \qquad
\sin^2\theta=\frac{\tan^2\theta}{1+\tan^2\theta}
\]
を代入しても示せる. 後者の公式は上の方で説明した $t=u/(1+u)$ に対応している.
積分変数たちのあいだには以下の関係がある:
\[
x:y = \cos\theta:\sin\theta = 1:\tan\theta.
\]
この関係は, $x,y$ の立場を交換すると%
\footnote{上の方でも $f(x,y)=f(y/x)$ と仮定しておくべきだったかもしれない.},
ちょうど上で説明した $x:y=t:(1-t)=u:1$ に対応している
($t=\sin^2\theta$, $1-t=\cos^2\theta$, $u=\tan^2\theta$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 函数と佐藤 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}
二項分布からPoisson分布を導こう.
$0<\mu<n\in\Z$ であるとし,
$N_n$ は独立試行回数 $n$, 確率 $p=\mu/n$ の二項分布に従う確率変数であるとする:
\begin{align*}
P(N_n=k)
&= \left(\frac{\mu}{n}\right)^k \left(1-\frac{\mu}{n}\right)^{n-k} \binom{n}{k}
\\ &
= \left(1-\frac{\mu}{n}\right)^n \frac{\mu^k}{k!}
\left(1-\frac{\mu}{n}\right)^{-k} \left(1-\frac{1}{n}\right)\cdots\left(1-\frac{k-1}{n}\right)
\end{align*}
このとき, $n\to\infty$ で
\begin{align*}
P(N_n=k)
&
= \left(1-\frac{\mu}{n}\right)^n \frac{\mu^k}{k!}
\left(1-\frac{\mu}{n}\right)^{-k} \left(1-\frac{1}{n}\right)\cdots\left(1-\frac{k-1}{n}\right)
\longrightarrow
e^{-\mu}\frac{\mu^k}{k!}.
\end{align*}
離散的確率変数 $N$ が $k=0,1,2,\ldots$ について
\begin{align*}
P(N=k) = e^{-\mu}\frac{\mu^k}{k!}
\end{align*}
を満たすとき, $N$ はパラメーター $\mu$ のPoisson分布に従うと言う.
二項分布に従う確率変数 $N_n$ は $n\to\infty$ でPoisson分布に弱収束する.
これは次のように解釈される.
単位時間あたり平均して $\mu$ 回の事象が発生するようにしたい.
そのためには $N$ 分の単位時間あたり独立に確率 $\mu/N$ で事象が発生する
ようにすればよい. このとき単位時間あたりに発生する事象の回数は
確率変数 $N_n$ で表わされる.
$\mu$ と比較して $N$ が十分に大きなとき,
$N_n$ の従う確率分布はPoisson分布でよく近似される.
Poisson分布は単位時間当たりに事象が起こる回数を意味する確率分布として
よく使われる.
パラメーター $\mu$ のPoisson分布の平均と分散はともに $\mu$ である:
\begin{align*}
&
(\text{$N$ の平均})
=E[N]
= \sum_{k=0}^\infty k e^{-\mu}\frac{\mu^k}{k!}
= \mu e^{-\mu} \sum_{k=1}^\infty \frac{\mu^{k-1}}{(k-1)!}
= \mu,
\\ &
E[N(N-1)]
= \mu^2 e^{-\mu} \sum_{k=2}^\infty \frac{\mu^{k-2}}{(k-2)!}
= \mu^2,
\\ &
E[N^2] = E[N(N-1)]+E[N]=\mu^2+\mu,
\\ &
(\text{$N$ の分散})=E[(N-\mu)^2] = E[N^2]-\mu^2 = \mu.
\end{align*}
Poisson分布の特性函数は次のようになる:
\begin{align*}
E[e^{itN}]
&= e^{-\mu} \sum_{k=0}^\infty e^{itk} \frac{\mu^k}{k!}
%\\ &
= \exp(\mu(e^{it}-1))
= \left(\exp(e^{it}-1)\right)^\mu.
\end{align*}
これより, Poisson分布はパラメーター $\mu$ について再生性を持つことがわかる.
ゆえに, 中心極限定理より, $\mu$ を大きくすると,
\begin{align*}
X = \frac{N-\mu}{\sqrt{\mu}}
\end{align*}
は標準正規分布に近似的に従い,
\begin{align*}
Y = \frac{(N-\mu)^2}{\mu}
\end{align*}
は自由度 $1$ のカイ二乗分布に近似的に従う%
\footnote{%
カイ二乗検定の文脈では,
$N$ を観測値(observed)と解釈し,
$\mu$ を期待値(expectated)と解釈し,
それらをそれぞれ $O$, $E$ と表わし, $Y=(O-E)^2/E$
%\begin{align*}
%\frac{(O-E)^2}{E}
%\end{align*}
のように書くことがある.}.
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 $\tau=1$ のガンマ分布と呼ぶのであった:
\[
f_{k+1,1}(\mu)\,d\mu
=\frac{e^{-\mu}\mu^k}{k!}\,d\mu
\qquad (\mu>0).
\]
平均は $\mu=k+1$, 分散は $k+1$ になり,
最頻値は $\mu=k$ になる.
このことから, 単位時間の観測で事象が $k$ 回起こったならば,
単位時間あたりに事象が起こる回数の平均値 $\mu$ の推定値が
shape $\alpha=k+1$, scale $\tau=1$ のガンマ分布にしたがっていると
みなすことは場合によっては十分に合理的でありそうなことがわかる%
\footnote{共役事前分布の話.}.
単位時間あたりの事象生成回数の平均値が $\mu$ のとき,
時間幅 $t>0$ の場合にPoisson分布を拡張するとき,
時間幅 $T$ のあいだに観測される事象の回数 $N_t$ は
パラメーター $\mu t$ のPoisson分布に従うと考えられる:
\begin{align*}
P(N_t=k) = e^{-\mu t}\frac{(\mu t)^k}{k!}
\qquad (k=0,1,2,3,\ldots).
\end{align*}
形状 $\alpha=k+1>0$, スケール $\tau=1/t$ のガンマ分布の確率密度函数は
次のようになる:
\begin{align*}
f_{k+1,1/t}(\mu)\,d\mu
=e^{-\mu t} \frac{(\mu t)^k}{k!}\,t\,d\mu.
\end{align*}
この分布の平均は $\mu=(k+1)/t$ になり,
分散は $(k+1)/t^2$ になり,
最頻値は $\mu=k/t$ になる.
ゆえに $k\approx \mu_0 t$ ($\mu_0>0$ は一定)のとき,
$t\to\infty$ で平均と最頻値はともに $\mu_0$ に収束し,
分散は $0$ に収束する.
%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%
%\footnote{共役事前分布の話.}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
$g(x)$ が有界連続函数のとき,
$g(X)$ 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{不定積分のTauber型定理}
\begin{theorem}
$f(t)$ は $t>0$ で定義された正値函数でかつ単調減少または単調増加%
\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.
「単調増加」についても不等式の無きを逆にするだけでまったく同様である.}%
していると仮定し, $\alpha,a>0$ 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}
まず, $f$ が単調減少である場合を扱う.
$f$ が単調減少函数であることより, 任意の $c>1$ に対して,
\[
\frac{\int_0^{cx} f(t)\,dt-\int_0^x f(t)\,dt}{cx-x}
\leqq
f(x)
\leqq
\frac{\int_0^x f(t)\,dt-\int_0^{c^{-1}x} f(t)\,dt}{x-c^{-1}x}.
\tag{1}
\]
これの全体を $ax^{\alpha-1}$ で割ると,
\[
\frac{\dfrac{\int_0^{cx} f(t)\,dt}{ax^\alpha}-\dfrac{\int_0^x f(t)\,dt}{ax^\alpha}}{c-1}
\leqq
\frac{f(x)}{ax^{\alpha-1}}
\leqq
\frac{\dfrac{\int_0^x f(t)\,dt}{ax^{\alpha}}-\dfrac{\int_0^{c^{-1}x} f(t)\,dt}{ax^\alpha}}{1-c^{-1}}.
\tag{2}
\]
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.},
\[
\frac{c^\alpha-1}{c-1}
\leqq
\liminf_{x\to\infty}\frac{f(x)}{ax^{\alpha-1}}
\leqq
\limsup_{x\to\infty}\frac{f(x)}{ax^{\alpha-1}}
\leqq
\frac{1-c^{-\alpha}}{1-c^{-1}}.
\tag{3}
\]
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 $f$ が単調減少の場合に示すべきことが示された.
次に $f$ が単調増加の場合を扱おう.
$f$ が単調増加の場合には(1),(2)で不等号の向きを逆にした結果が得られる.
ゆえに, (3)で $\liminf$ と $\limsup$ を交換して,
不等号の向きを逆にした結果が得られる.
そのことに注意すれば,
$f$ が単調増加の場合に示すべき結果が同様に得られることがわかる.
\qed
\end{proof}
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}
$a_1,a_2,a_3,\ldots$ は正値数列で単調減少または単調増加しているとし,
$a,\alpha>0$ 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変換のTauber型定理}
\label{sec:Tauber-Laplace}
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 $t>0$ で定義された正値函数で単調減少または単調増加しているとし,
$\alpha,a>0$ 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}
$a_0,a_1,a_2,\ldots$ は正値数列で単調減少または単調増加しているとし,
$\alpha,a>0$ 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%
\footnote{\secref{sec:Laplace-Stieltjes}, \secref{sec:Tauber-Stieltjes}で
漸近挙動に緩変動函数が含まれているより一般的の場合に関する解説を書いておいた.
そちらの解説の方が準備を十分にしているので証明も簡明になっている.}.
\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%
\footnote{\secref{sec:Tauber-Laplace}や\secref{sec:Tauber-Stieltjes}で
解説したタイプのTauber型定理はWallisの公式の一般化だとみなせる.}.
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
\[
F(0.99) \approx 0.494098, \qquad
F(0.999)\approx 0.500124.
\]
$F(x)$ 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Laplace-Stieltjes変換}
\label{sec:Laplace-Stieltjes}
以下は\secref{sec:Tauber-Stieltjes}のための準備である.
$F(x)$ は $\R$ 上の右連続%
\footnote{$F(x)$ が右連続(もしくは右側から連続)である
とは $\lim_{\eps\searrow 0}F(x+\eps)=F(x)$ が成立していることである.}%
かつ単調非減少%
\footnote{$F(x)$ が単調非減少とは $x\leqq x'$ ならば $F(x)\leqq F'(x)$ が
成立していることである.}%
であると仮定する.
さらに, もしも $\lim_{x\to-\infty}F(x)=0$ と
$\lim_{x\to\infty}F(x)=1$ が成立しているならば,
$F(x)$ は{\bf 累積確率分布函数}もしくは単に{\bf 分布函数}と呼ぶことがある.
そのとき $F(x)$ の値は「$x$ 以下の値になる確率」だと解釈される.
以下では $F(x)$ が累積確率分布函数になっているとは仮定しない.
$F(x)$ が単調非減少であることから, 任意の $x\in\R$ において,
左からの極限 $F(x-0)=\lim_{\eps\searrow0}F(x-\eps)$ が存在することがわかる
(上に有界な実数の集合は上限を持つから).
さらに $F(x)$ は右連続なので $F(x)=F(x+0)=\lim_{\eps\searrow}F(x+\eps)$ が
成立している. $F(x)-F(x-0)>0$ となることと
函数 $F$ が点 $x$ で不連続なことは同値である.
すなわち $F(x)$ は不連続点 $x$ で $F(x)-F(x-0)>0$ の分だけ上にジャンプしている.
このことから $F(x)$ の不連続点は高々可算個であることがわかる.
$F(x)$ に関するLebesgue-Stieltjes積分は
\[
\mu_F((a,b]) = F(b)-F(a) \qquad (-\infty\leqq a<b)
\]
を満たすBorel測度 $\mu_F$ (一意的に存在する)に関する積分として定義される.
すなわち, Borel集合 $A$ について
\[
\int_A f(x)\,dF(x)
=
\int_A f(x)\,\mu_F(dx)
\]
と書き, これをLebesgue-Stieltjes積分と呼ぶ.
測度 $\mu_F(dx)$ の代わりに $dF(x)$ と書くことが多い.
$F(x)$ が $x=a$ で連続なことと $\mu_F(\{a\})=0$ は同値になり,
\[
F(b)-F(a) = \int_{(a,b]}\,dF(x) = \int_{[a,b]}\,dF(x)
\qquad (a<b)
\]
が成立している. 2つ目の等号は $a=b$ でかつ $x=a$ で $F(x)$ が不連続な
とき成立しないことに注意せよ.
もしも $\R$ 上のBorel測度 $\mu$ が任意の $x\in\R$ について $\mu((-\infty,x])<\infty$
を満たしているならば $F(x)=\mu((-\infty,x])$ とおくと $F(x)$ は右連続な単調非減少函数
で $\lim_{x\to-\infty}F(x)=0$ を満たしており, $\mu=\mu_F$ が成立している.
区間 $[a,b]$ ($a<b$)におけるRiemann-Stieltjes積分は
$[a,b]$ の分割
\[
a=x_0< x_1<x_2<\cdots<x_n=b, \qquad c_i\in[x_{i-1},x_i]
\]
を取り,
\[
\int_a^b f(x)dF(x)
=\lim \sum_{i=1}^n f(c_i)(F(x_i)-F(x_{i-1}))
\]
で定義される. ここで右辺の極限は分割を細かくする極限である.
$f(x)$ が連続函数ならば
\[
\int_a^b f(x)\,dF(x) = \int_{[a,b]} f(x)\,\mu_F(dx)
\]
が成立している.
以下ではさらに $F(0)=0$ と仮定する.
$\mu$ が $(0,\infty)$ 上のBorel測度のとき
\[
M(s) = \int_0^\infty e^{-sx}\,\mu(dx)
\]
を測度 $\mu$ の Laplace変換と呼ぶ. より一般に
\[
\int_0^\infty e^{-sx}f(x)\,\mu(dx)
\]
を $f(x)\,\mu(dx)$ のLaplace変換と呼ぶ.
\begin{remark}
\label{remark:ReduceToFiniteMeas}
$\mu$ が有限測度($\mu((0,\infty))<\infty$)のとき
\[
M(s)=\int_0^\infty e^{-sx}\,\mu(dx)<\infty
\qquad (s\geqq 0).
\]
$\mu$ が有限測度でない場合であっても, $M(\sigma)<\infty$ ならば
$e^{-\sigma x}\mu(dx)$ が有限測度を定めるので,
無限測度のLaplace変換に関する問題の多くが
有限測度のLaplace変換の場合に帰着することがわかる.
さらに $e^{-\sigma x}\mu(dx)/M(\sigma)$ が確率測度を
定めることから, 多くの問題が確率測度の場合に帰着することもわかる.
\qed
\end{remark}
\begin{prop}
$\mu$, $\nu$ は $(0,\infty)$ 上のBorel測度のとき,
十分大きな $s$ について
\[
\int_0^\infty e^{-sx}\,\mu(dx)
=
\int_0^\infty e^{-sx}\,\nu(dx)
<\infty
\]
が成立しているならば $\mu=\nu$ となる.
すなわち, 十分大きな $s$ について $\mu,\nu$ のLaplace変換が有限な値に
収束してかつ等しいならば, 2つの測度 $\mu,\nu$ は一致する.
\end{prop}
\begin{proof}
\remarkref{remark:ReduceToFiniteMeas}のアイデアを使おう.
$A=\int_0^\infty e^{-\sigma x}\,\mu(dx)=\int_0^\infty e^{-\sigma x}\,\nu(dx)<\infty$
のとき, $\mu(dx)$, $\nu(dx)$ の
それぞれを $e^{-\sigma x}\,\mu(dx)/A$, $e^{-\sigma x}\,\nu(dx)/A$ で
置き換えることによって, $\mu$ と $\nu$ は確率測度であると仮定できる.
このとき, 任意の $s\geqq 0$ に対してそれらのLaplace変換は
有限な値に収束する.
同相写像 $f:(0,\infty)\to(0,1)$, $x\mapsto y$ を
$f(x)=y=e^{-x}$ と定め,
$(0,1)$ 上の確率測度 $\mu',\nu'$ を
\[
\mu'(A) = \mu(f^{-1}(A)), \qquad
\nu'(A) = \nu(f^{-1}(A))
\]
と定めると%
\footnote{たとえば $\mu(dx)=\rho(x)\,dx$ のとき $\mu'(dy)=\rho(-\log y)y^{-1}\,dy$.},
\[
\int_0^\infty e^{-sx}\,\mu(dx)
=
\int_0^\infty e^{-sx}\,\nu(dx)
\]
は次のように書き直される(置換積分):
\[
\int_0^1 y^s \,\mu'(dy)
=
\int_0^1 y^s \,\nu'(dy)
\qquad (s\geqq 0).
\]
特に $\mu'$ と $\nu'$ のモーメントがすべて等しい.
ゆえに $\mu'=\nu'$ すなわち $\mu=\nu$ であることがわかる.
\qed
\end{proof}
\begin{remark}[モーメント問題について]
有限区間上の確率測度はそのモーメント全体によって一意的に決定される%
\footnote{Stone-Weierstrassの多項式近似定理を使って証明できる.
連続函数の多項式による一様近似が可能なのは
有限閉区間上に限ることに注意せよ.
\lemmaref{lemma:MomentConv-DistConv}の証明と同様の方法を使って証明できる.}.
しかし無限区間上の確率測度の場合にはそうではない.
たとえば対数正規分布はそのモーメントたちだけから一意に決定されない.
対数標準正規分布の確率密度函数は次で与えられる:
\[
f(x)\,dx = \frac{e^{-(\log x)^2/2}}{\sqrt{2\pi}}\frac{dx}{x} \quad (x>0),
\qquad f(x)=0 \quad (x\leqq 0).
\]
$s(y)$ は周期 $1$ を持つ実数値奇函数でその絶対値は常に $1$ 以下であるとする.
たとえば $s(y)=a\sin(2\pi y)$ ($|a|\leqq 1$) であるとする. そして
\[
g(x)\,dx=f(x)(1+s(\log x))\,dx
\]
とおく. $s(x)$ の絶対値は常に $1$ 以下なので $g(x)\geqq 0$ となる.
この $g(x)\,dx$ が $\R$ 上の確率測度を定め,
対数標準正規分布 $f(x)\,dx$ と同じモーメントたちを持つことを示したい.
そのためには $k=0,1,2,\ldots$ に対して
\[
\int_0^\infty x^k f(x)s(\log x)\,dx
= \frac{1}{\sqrt{2\pi}}\int_0^\infty
x^k e^{-(\log x)^2/2} s(\log x)\,\frac{dx}{x}
= 0
\]
を示せば十分である. 積分変数を $x=e^{y+k}$ と置換すると,
$s(y)$ が周期 $1$ を持つことより,
\begin{align*}
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty
e^{ky+k^2}e^{-(y+k)^2/2}s(y+k)\,dy
=
\frac{e^{k^2/2}}{\sqrt{2\pi}}
\int_{-\infty}^\infty e^{-y^2/2}s(y)\,dy
\end{align*}
$s(y)$ は奇函数だったのでこの積分は $0$ になる.
これで $g(x)\,dx$ は確率測度を定めそのモーメントたち
は対数標準正規分布 $f(x)\,dx$ のモーメントたちに等しいことがわかった.
以上の例は,
Willium Feller,
An Introduction to Probability Theory and Its Applications Vol.~2,
First Edition (1970)のVII.3のp.227の例の引き写しである.
そこには「この興味深い例は C.~C.~Heyde による」と書いてある.
さらにその下には以下のように書いてある:
\begin{enumerate}
\item[(1)] $\R$ 上の確率分布の $k$ 次のモーメントを $\mu_k$ と書くとき,
\[
\sum_{n=1}^\infty \frac{1}{\sqrt[2n]{\mu_{2n}}} = \infty
\]
ならば, モーメントたちからもとの確率分布が一意に決まる(Carlemanの定理).
\item[(2)] それより弱い結果: 偶数次のモーメントたちから得られるべき級数
\[
\sum_{n=0}^\infty \mu_{2n}\frac{z^{2n}}{(2n)!}
\]
の収束半径が $0$ より大きいならば(すなわちある $z>0$ に対して収束するならば),
モーメントたちからもとの確率分布が一意的に決定される(第6節XV.4).
\end{enumerate}
要するに高次のモーメントたちの増大度が十分小さければ
モーメントたちからもとの確率分布は一意的に決定される.
対数標準正規分布の $k$ 次のモーメント $\mu_k$
は $x=e^{y+k}$ と変数変換することによって,
\begin{align*}
\mu_k
&=
\int_0^\infty x^k \frac{e^{-(\log x)^2/2}}{\sqrt{2\pi}}\frac{dx}{x}
=
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{ky+k^2}e^{-(y+k)^2/2}\,dt
\\ &
=
\frac{e^{k^2/2}}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-y^2/2}\,dt
=
e^{k^2/2}
\end{align*}
と計算され, 次数 $k$ の2次函数の指数函数の速さで急速に増大することがわかる.
(その増大度は $k!\sim \exp(k\log k - k + (1/2)\log k + \log\sqrt{2\pi k})$ より真に大きい.)
確率変数 $X$ の $k$ 次のモーメントを $\mu_k$ の定めるべき級数
\[
\sum_{k=0}^\infty \mu_k \frac{z^k}{k!}
\tag{$\#$}
\]
の収束半径が正であることと上の(2)のべき級数の収束半径が正であることは同値である.
なぜならば $E[|X|^{2n-1}]\leqq 1+E[X^{2n}]=1+\mu_{2n}$ となるからである:
\begin{align*}
E[|X|^{2n-1}]
&
= E[1_{|X|<1}(X)|X|^{2n-1}] + E[1_{|X|\geqq 1}(X)|X|^{2n-1}]
\\ &
\leqq 1 + E[1_{|X|\geqq 1}(X)|X|^{2n-1}]
\leqq 1 + E[|X|^{2n}]
=1+\mu_{2n}.
\end{align*}
ゆえに, べき級数($\#$)の収束半径が正ならば, ある $r>0$ が存在して,
\[
E[e^{zX}] = \sum_{k=0}^\infty \mu_k \frac{z^k}{k!}
\qquad
(|z|<r)
\]
が成立する. このことから $X$ の特性函数 $E[e^{itX}]$ ($t\in \R$) が
モーメントたちから一意的に決まることがわかる.
確率分布はその特性函数から一意的に決まるので,
べき級数($\#$)の収束半径が $0$ でないならば,
モーメントたちによって確率分布が一意的に決まることがわかった.
\qed
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Laplace-Stieltjes変換のTauber型定理}
\label{sec:Tauber-Stieltjes}
\begin{lemma}
\label{lemma:MomentConv-DistConv}
$\mu_n$, $\mu$ は $(0,1)$ 上の有限Borel測度であるとし,
\[
\lim_{n\to\infty}
\int_0^1 y^k\,\mu_n(dy)
=
\int_0^1 y^k\,\mu(dy)
\qquad
(k=0,1,2,\ldots)
\]
が成立していると仮定する.
このとき $\mu(\{y\})=0$ となるすべての点 $y\in(0,1)$ において
\[
\lim_{n\to\infty} \mu_n((0,y])=\mu((0,y])
\]
となる. すなわち, 右連続単調非減少函数たち
を $F_n(y)=\mu_n((0,y])$, $F(y)=\mu((0,y])$ と定めると,
$F$ が連続になるすべての点 $y\in(0,1)$ において
\[
\lim_{n\to\infty} F_n(y)=F(y)
\]
となる.
\end{lemma}
\begin{proof}
$\mu_n$, $\mu$ が有限測度であることと
仮定の $k=0$ の場合より, ある定数 $C>0$ が存在して
\[
\mu_n((0,1))\leqq C \quad (n=1,2,3,\ldots),
\qquad
\mu((0,1))\leqq C
\]
となる. 仮定より, すべての多項式函数 $p(y)$ について
\[
\lim_{n\to\infty}
\int_0^1 p(y)\,\mu_n(dy)
=
\int_0^1 p(y)\,\mu(dy)
\qquad
(k=0,1,2,\ldots)
\tag{1}
\]
が成立している.
任意に $\eps>0$ を取って固定する.
$a\in(0,1)$ は $\mu(\{a\})=0$ を満たしているとし,
$0<\delta<\min\{a,1-a\}$ と仮定する.
$\mu(\{a\})=0$ より $\mu((0,a))=\mu((0,a])=F(a)$ となる.
$\R$ 上の連続函数 $g_\delta(y)$, $h_\delta(y)$ を次のように定める:
\[
g_\delta(y)=
\begin{cases}
1 & (y\leqq a-\delta), \\
(a-y)/\delta & (a-\delta\leqq y\leqq a), \\
0 & (a\leqq y),
\end{cases}
\quad
h_\delta(y)=
\begin{cases}
1 & (y\leqq a), \\
1-(y-a)/\delta & (a\leqq y\leqq a+\delta), \\
0 & (a+\delta\leqq y).
\end{cases}
\]
さらに次のように定める:
\[
g(y)=
\begin{cases}
1 & (y<a), \\
0 & (a\leqq y),
\end{cases}
\qquad
h_\delta(y)=
\begin{cases}
1 & (y\leqq a), \\
0 & (a< y).
\end{cases}
\]
このとき, 以下が成立している:
\begin{align*}
&
0\leqq g_\delta(y)\leqq g(y)\leqq h(y)\leqq h_\delta(y)\leqq 1,
%\\ &
\qquad
\lim_{\delta\searrow 0}g_\delta(y)=g(y),
\qquad
\lim_{\delta\searrow 0}h_\delta(y)=h(y).
\end{align*}
ゆえにLebesgueの収束定理と $\mu((0,a))=\mu((0,a])=F(a)$ より
\[
\lim_{\delta\searrow 0}\int_0^1 g_\delta(y)\,\mu(dy)
=F(a),
\qquad
\lim_{\delta\searrow 0}\int_0^1 h_\delta(y)\,\mu(dy)
=F(a).
\]
ゆえに十分 $\delta>0$ を小さく取って,
\[
\left|\int_0^1 g_\delta(y)\,\mu(dy)-F(a) \right|\leqq\eps,
\qquad
\left|\int_0^1 h_\delta(y)\,\mu(dy)-F(a) \right|\leqq\eps
\tag{2}
\]
となるようにできる.
Stone-Weierstrassの多項式近似定理によって
次を満たす多項式函数 $P(y)$ と $Q(y)$ が存在することがわかる%
\footnote{Stone-Weierstrass の多項式近似定理より,
$|P(y)-(g_\delta(y)-\eps/2)|\leqq\eps/2$ ($y\in(0,1)$) を
満たす多項式函数 $P(y)$ が存在する. $Q(y)$ についても同様.}:
\begin{align*}
&
P(y)\leqq g_\delta(y)\leqq g(y)\leqq h(y)\leqq h_\delta(y)\leqq Q(y), \qquad
\\ &
|P(y)-g_\delta(y)|\leqq\eps, \qquad
|Q(y)-h_\delta(y)|\leqq\eps \qquad (y\in(0,1)).
\end{align*}
このとき
\begin{align*}
&
\int_0^1 P(y)\,\mu_n(dy)
\leqq
\mu_n((0,a])=F_n(a)
\leqq
\int_0^1 Q(y)\,\mu_n(dy),
\tag{3}
\\ &
\left|\int_0^1 P(y)\,\mu(dy)-\int_0^1 g_\delta(y)\,\mu(dy)\right|\leqq C\eps,
\tag{4}
\\ &
\left|\int_0^1 Q(y)\,\mu(dy)-\int_0^1 h_\delta(y)\,\mu(dy)\right|\leqq C\eps.
\tag{5}
\end{align*}
ゆえに(3)で(1)を用いると
\[
\int_0^1 P(y)\,\mu(dy)
\leqq
\liminf_{n\to\infty}F_n(a)
\leqq
\limsup_{n\to\infty}F_n(a)
\leqq
\int_0^1 Q(y)\,\mu(dy).
\]
ところが, (2),(4),(5)より $\eps>0$ を小さくすると $\int_0^1 P(y)\,\mu(dy)$ と
$\int_0^1 Q(y)\,\mu(dy)$ はいくらでも $F(a)$ に近付く. ゆえに
\[
\lim_{n\to\infty} F_n(a)=F(a)
\]
となることがわかる. これで示すべきことが示された.
\qed
\end{proof}
\begin{example}
上の補題において $F$ が連続な点 $y$ で $F_n(y)$ が $F(y)$ に収束することを
示したが, $F(y)$ が不連続な点では $F_n(y)$ が $F(y)$ に収束するとは
限らない. たとえば $a\in(0,1)$, $0<\eps<1-a$ に対して,
連続函数たち $F_\eps(y)$ と右連続函数 $F(y)$ を
\[
F_\eps(y)=
\begin{cases}
0 & (0<y\leqq a) \\
(y-a)/\eps & (a\leqq y\leqq a+\eps) \\
1 & (a+\eps\leqq y<1), \\
\end{cases}
\quad
F(y)=
\begin{cases}
0 & (0<y\leqq a) \\
1 & (a\leqq y<1) \\
\end{cases}
\]
と定めると,
\begin{align*}
&
\int_0^1 y^k dF_\eps(y)
=\frac{1}{\eps}\int_a^{a+\eps} y^k\,dy
=\frac{1}{k+1}\frac{(a+\eps)^{k+1}-a^{k+1}}{\eps},
%\\ &
\qquad
\int_0^1 y^k dF(y) = a^k
\end{align*}
となるので,
\[
\lim_{\eps\searrow 0}\int_0^1 y^k dF_\eps(y)
=\int_0^1 y^k dF(y)
\]
となり, $a$ 以外の $y\in(0,1)$ について $\lim_{\eps\searrow 0}F_\eps(y)=F(y)$
となる. しかし $F_\eps(a)=0$, $F(a)=1$ なので $y=a$ ではそうならない.
\qed
\end{example}
\begin{lemma}
\label{lemma:Laplace-Cumulative}
$F_n(x)$, $F(x)$ は右連続単調非減少函数で $x=0$ で $0$ になるものであるとする.
このとき, 十分大きな $\lambda$ に対して,
$\int_0^\infty e^{-\lambda x}\,dF_n(x)$ ($n=1,2,3,\ldots$)
と $\int_0^\infty e^{-\lambda x}\,dF(x)$ が有限の値に収束し,
\[
\lim_{n\to\infty}
\int_0^\infty e^{-\lambda x} \,dF_n(x)
=
\int_0^\infty e^{-\lambda x} \,dF(x)
\]
が成立しているならば, 函数 $F$ が連続なすべての点 $x>0$ において
\[
\lim_{n\to\infty}F_n(x)=F(x)
\]
となる.
\end{lemma}
\begin{proof}
$\mu_n$, $\mu$ は
$\mu_n((a,b])=F_n(b)-F_n(a)$, $\mu((a,b])=F(b)-F(a)$ ($a<b$) を
満たすBorel測度であるとする.
仮定より, ある $\sigma>0$ について
$A_n=\int_0^\infty e^{-\sigma x}\,\mu_n(dx)$ たち
と $A=\int_0^\infty e^{-\sigma x}\,\mu(dx)$ は有限の値になる.
測度 $\widetilde\mu_n$, $\widetilde\mu$ を
$\widetilde\mu_n(dx)=e^{-\sigma x}\mu_n(dx)$,
$\widetilde\mu(dx)=e^{-\sigma x}\mu(dx)$
と定めると, $\widetilde\mu_n((0,\infty))=A_n$,
$\widetilde\mu((0,\infty))=A$ となる.
$\mu_n$, $\mu$ の代わりに $\widetilde\mu_n$, $\widetilde\mu$
を考えることに $\mu_n$, $\mu$ は有限測度であると仮定してよい.
そのとき, 変数変換 $y=e^{-x}$ によって,
この補題における $(0,\infty)$ 上の問題を
$(0,1)$ 上の問題に関する\lemmaref{lemma:MomentConv-DistConv}に帰着できる.
\qed
\end{proof}
\begin{definition}
函数 $L(x)$ が $x\to\infty$ に
おける{\bf 緩変動函数}(slowly varying function)であるとは,
任意の $c>0$ に対して
\[
L(cx)\sim L(x) \quad (x\to\infty),
\qquad\text{すなわち}\quad
\lim_{x\to\infty}\frac{L(cx)}{L(x)}=1
\]
が成立していることである. たとえば $(\log x)^\beta$ は緩変動函数である.
\qed
\end{definition}
\begin{theorem}
$F(x)$ は $x\geqq 0$ における右連続な単調増加(非減少)函数であり,
$F(0)=0$ を満たしているものであり,
$\alpha>0$ であるとし, $L(x)$ は $x\to\infty$ における緩変動函数であるとする.
このとき
\[
M(\lambda) := \int_0^\infty e^{-\lambda x}\,dF(x)
\sim
\lambda^{-\alpha}L(\lambda^{-1})
\qquad (\lambda\searrow 0)
\]
ならば
\[
F(x)
\sim \frac{M(x^{-1})}{\Gamma(\alpha+1)}
\sim \frac{x^\alpha L(x)}{\Gamma(\alpha+1)}
\qquad (x\to\infty)
\]
が成立する.
\end{theorem}
\begin{proof}
連続函数函数 $G(x)$ を
\[
G(x) = \frac{x^\alpha}{\Gamma(\alpha+1)} \quad (x\geqq 0), \qquad
G(x) = 0 \quad (x<0)
\]
と定めると, $c>0$ に対して
\[
\int_0^\infty e^{-cx}\,dG(x)
= \frac{\alpha}{\Gamma(\alpha+1)}
\int_0^\infty e^{-cx} x^{\alpha-1}\,dx
= \frac{c^{-\alpha}}{\Gamma(\alpha)}\int_0^\infty e^{-t}t^{\alpha-1}\,dt
= c^{-\alpha}
\]
が成立する. 2つ目の等号で $x=t/c$ とおいた.
$c>0$ のとき, $M(\lambda)$ の漸近挙動に関する仮定より,
\[
\frac{M(c\lambda)}{M(\lambda)}
\sim
\frac{c^{-\alpha}\lambda^{-\alpha}L(c^{-1}\lambda^{-1})}{\lambda^{-\alpha}L(\lambda^{-1})}
\sim
c^{-\alpha}
=\int_0^\infty e^{-cx}dG(x)
\qquad (\lambda\searrow 0).
\]
さらに $M(\lambda)$ の定義より,
\[
\frac{M(c\lambda)}{M(\lambda)}
=\frac{1}{M(\lambda)} \int_0^\infty e^{-c\lambda x}dF(x)
=\int_0^\infty e^{-cx}d\left(\frac{F(x/\lambda)}{M(\lambda)}\right).
\]
ゆえに
\[
\lim_{\lambda\searrow 0}
\int_0^\infty e^{-cx}d\left(\frac{F(x/\lambda)}{M(\lambda)}\right)
=\int_0^\infty e^{-cx}dG(x)
\qquad (c>0).
\]
したがって, \lemmaref{lemma:Laplace-Cumulative}より,
\[
\lim_{\lambda\searrow 0}\frac{F(x/\lambda)}{M(\lambda)}
=G(x)=\frac{x^\alpha}{\Gamma(\alpha+1)}
\qquad (x>0)
\]
となる. すなわち
\[
F(x/\lambda)\sim \frac{M(\lambda)x^\alpha}{\Gamma(\alpha+1)}
\qquad (\lambda\searrow 0).
\]
$x=1$ とおき, $\lambda$ を $x^{-1}$ で置き換えることによって,
\[
F(x)
\sim \frac{M(x^{-1})}{\Gamma(\alpha+1)}
\sim \frac{x^\alpha L(x)}{\Gamma(\alpha+1)}
\qquad (x\to\infty)
\]
が得られる. \qed
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{付録: Taylorの定理の証明の仕方}
\label{sec:Taylor}
「(函数)=(Taylor展開の途中まで)+(剰余項)」の形式の公式を{\bf Taylorの定理}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
\[
f(x)=f(a)+f'(a)\int_a^x\!\!\!dx_1+f''(a)\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2
+\int_a^x\!\!\!dx_1\int_a^{x_1}\!\!\!dx_2\int_a^{x_2}f'''(x_3)\,dx_3.
\]
さらに同じ操作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(次々に一つ上の式を使うことによって)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!},
\\ &
\qquad\qquad\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\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 $k=1$ のとき, 積分の値は
線分 $\{\, x_1 \mid a\leqq x_1\leqq x\,\}$ の長さになる.
\item $k=2$ のとき, 逐次積分の値は
頂点 $(a,a),(x,a),(x,x)$ を持つ直角二等辺三角形
\[
\{\,(x_1,x_2)\mid a\leqq x_2\leqq x_1\leqq x\,\}
\]
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 $k=3$ のとき, 逐次積分の値は
頂点 $(a,a,a),(x,a,a),(x,x,a),(x,x,x)$ を持つ四面体
\[
\{\,(x_1,x_2,x_3)\mid a\leqq x_3\leqq x_2\leqq x_1\leqq x\,\}
\]
の体積, すなわち立方体の体積 $(x-a)^3$ 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 $x_3\leqq x_2\leqq x_1$ の以外の
順番に並んでいるすべての $a\leqq x_1,x_2,x_3\leqq x$ について
積分しなければいけない.
$x_1,x_2,x_3$ の個数は $3$ 個なのでそれらの並べ方の総数は $3!$ 通りある.
\item 一般の $k$ の場合も以上と同様である.
逐次積分の値は頂点
\[
(a,a,a,\ldots,a),(x,a,a,\ldots,a),(x,x,a,\ldots,a),\ldots,(x,x,x,\ldots,x)
\]
を持つ $k$ 次元単体%
\footnote{点, 線分, 三角形, 四面体の $k$ 次元版を $k$ 次元単体(simplex)と呼ぶ.}
\[
\{\,(x_1,x_2,\ldots,x_k)\mid a\leqq x_k\leqq\ldots\leqq x_2\leqq x_1\leqq x\,\}
\]
の体積になる.
$x_1,x_2,\ldots,x_k$ の並べ方の総数は $k!$ 通りなので
逐次積分の値は $k$ 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!}
すなわち, $1$ の $k$ 回の逐次積分の結果は $k$ 次元立方体の体積の $k!$ 分の1に
なるので分母に $k!$ が現われる.
以上のまとめ:
\begin{align*}
&
f(x)=\sum_{k=0}^{n-1}f^{(k)}(a)\frac{(x-a)^k}{k!} + R_n,
\\ &
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.
\end{align*}
これを積分剰余項型の{\bf Taylorの定理}と呼ぶことにする.
\begin{remark*}
次のように考えてもよい.
$n$ 階の導函数 $f^{(n)}(x)$ を $n$ 回逐次積分すれば $f(x)$ が得られるはずである.
しかし, 積分定数を考慮すれば $x$ について $n-1$ 次以下の項が生じることになる.
その結果, 以下のような公式が得られる:
\[
f(x) = a_0+a_1(x-a)+a_2(x-a)^2+\cdots+a_{n-1}(x-a)^{n-1}+R_n.
\]
ここで $R_n$ は上のように定義された $f^{(n)}(x)$ を $n$ 回逐次積分したものである.
この式の両辺を $k=0,1,\ldots,n-1$ 回微分して $x=a$ とおけば $R_n$ から来る項
は $0$ になるので, $f^{(k)}(a)=k!a_k$ が得られる.
すなわち $a_k=f^{(k)}(a)/k!$ である.
以上の計算の仕方は,
形式的に $f(x)=\sum_{k=0}^\infty a_k(x-a)^k$ とおいて
両辺を繰り返し微分して $x=a$ とおくことによって $a_k$ を決定する方法と同じだが,
剰余項 $R_n$ の正体が明瞭にわかっているので
Taylor級数の収束性に関わる論理的なギャップが生じない.
このように剰余項付きのTaylorの定理は
「$n$ 階の導函数 $f^{(n)}(x)$ を $n$ 回逐次積分すればもとの $f(x)$ が得られるはずだ」
という非常にもっともな考え方から素直に得られるのである.
\qed
\end{remark*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{剰余項の絶対値の上からの評価とTaylor展開の具体例}
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%
\footnote{$x=0$ におけるTaylor展開をMaclaurin展開と呼ぶことがある.}.
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絶対値の上からの評価(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 $n!$ と同じ程度の場合にはTaylor展開は $|x-a|<1$ で $f(x)$ 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.
\[
(1+x)^\alpha = \sum_{k=0}^\infty\binom{\alpha}{k}x^k \quad (|x|<1), \qquad
\log(1+x) = \sum_{k=1}^\infty(-1)^{k-1}\frac{x^k}{k} \quad (|x|<1).
\]
次のTaylor展開もよく使われる:
\[
-\log(1-x) = \sum_{k=1}^\infty \frac{x^k}{k} \quad (|x|<1).
\]
これを拡張した公式
\[
\Li_r(x)=\sum_{k=1}^\infty\frac{x^k}{k^r} \qquad (|x|<1,\ r=1,2,3,\ldots)
\]
で定義される函数 $\Li_r(x)$ は
$r$ 次の多重対数函数(polylogarithm, 通称ポリログ)と呼ばれている.
特に $\Li_2(x)$ は dilogarithm (通称ダイログ)と,
$\Li_3(x)$ の場合には trilogarithm (通称トリログ)と呼ばれている.
このとき
\[
\frac{d\Li_r(x)}{dx}=\frac{\Li_{r-1}(x)}{x} \quad (r\geqq 2), \qquad
\Li_1(x)=-\log(1-x)=\int_0^x \frac{dx_1}{1-x_1}
\]
なので, 多重対数函数は
\[
\operatorname{Li}_r(x)
=\int_0^x\frac{dx_r}{x_r}\cdots\int_0^{x_3}\frac{dx_2}{x_2}
\int_0^{x_2}\frac{dx_1}{1-x_1}
=\mathop{\int\cdots\int}\limits_{0<x_1<\cdots<x_r<x}
\frac{dx_1}{1-x_1}\frac{dx_2}{x_2}\cdots\frac{dx_r}{x_r}
\]
と逐次積分表示される.
$2$ つ目の等号で $0<x<1$ を仮定した.
以上を合わせるとRiemannのゼータ函数の
$2$ 以上の整数 $r$ における特殊値の積分表示が得られる:
\[
\zeta(r)
=\sum_{n=1}^\infty\frac{1}{n^r}
=\Li_r(1)
=\mathop{\int\cdots\int}\limits_{0<x_1<\cdots<x_r<1}
\frac{dx_1}{1-x_1}\frac{dx_2}{x_2}\cdots\frac{dx_r}{x_r}
\qquad (r=2,3,4,\ldots).
\]
かなり話題を脱線させてしまったので, Taylorの定理の話に戻ろう.
大抵の場合, 剰余項の $R_n$ 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 $1$ の逐次積分が $(n-1)$ 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 $(F(s),G(s))$ の軌跡の曲線を描いて,
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 $t_1$ の存在を
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%
\footnote{Rollの定理は「閉区間上の連続函数が最大値と最小値を持つ」という結果から従う.
もしくはRollの定理を経由せずに直接的に次のように考えてもよい.
$s$ は $a$ と $t$ のあいだを動くものとし,
点 $(F(a),G(a))$ と点 $(F(t),G(t))$ を結ぶ直線 $\ell$ から
軌跡上の点 $(F(s),G(s))$ への距離が最大になる時刻 $s=t_1$ が存在する.
このとき速度ベクトル $(F'(t_1),G'(t_1))$ が直線 $\ell$ に平行になる
ことを示せる.
}%
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{(t-a)^2}{2}-f'''(a)\frac{(t-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)}
\tag{$*$}
\\
=&\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.
これで $4$ 回微分可能な函数に関する微分剰余項型のTaylorの定理が証明された.
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%
\footnote{もしも $f$ が $C^4$ 級ならば $\lim_{t\to a}f^{(4)}(t_4)=f^{(4)}(a)$
となるので, うしろから2つ目の等号の右辺を使う必要はない.
しかし, 我々は $f^{(4)}$ の連続性を仮定せずに,
$f$ は4回微分可能だと仮定していただけだったのでそうしなければいけなくなった.}.
$t_4$ 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``small order''型Taylorの定理も証明できた.
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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http://www.rasmusen.org/g406/chapters/08-monopoly.tex | rasmusen.org | CC-MAIN-2022-40 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-40/segments/1664030334332.96/warc/CC-MAIN-20220925004536-20220925034536-00511.warc.gz | 97,116,964 | 32,031 | %\documentclass[12pt]{book}
\input{../preamble-regulation.tex}
%------------------------------------------------------------
\begin{document}
\parindent 16pt
%--------------------------------------
\setcounter{chaptercounter}{7}
\refstepcounter{chaptercounter} \label{chap07}
\setcounter{page}{1}
\pagestyle{fancy} \lhead[\ref{chap07}--\thepage]{} \chead{Monopoly} \rhead[]
{\ref{chap07}--\thepage} \cfoot{}
%--------------------------------------
\vspace*{\fill}
\begin{center}
\begin{Huge}
{\bf \ref{chap07}: Monopoly} \label{s05.0} \\
\end{Huge}
\vspace*{\fill}
\begin{quotation}
\begin{Large}
{\it ``He that withholdeth corn, the people shall curse him: but blessing shall be upon the head of
him that selleth it." }
\end{Large}
\end{quotation}
\vspace*{\fill}
\includegraphics[width=\linewidth]{fig06-bond-US-Steel.jpg}
\end{center}
\vspace*{\fill}
{\small \hfill {\it October 12, 2019}}\\
{\small \hfill \url{http://rasmusen.org/g406/reg-rasmusen.htm}, [email protected]}
%------------------------------------------------------
\newpage
\noindent
{\bf \ref{chap07}.1. Choosing Price and Output to Maximize Profits}\label{s05.1}
Monopoly is a classic source of market failure. Being the only firm in the industry means that the firm has market power, defined as the ability to increase the price without losing all its customers. All firms have this to some extent, but if market power is significant, the firm may decide to restrict output in order to drive up the price. In diagrams, we depict market power as a downward-sloping demand curve facing the individual firm, as opposed to a downward-sloping demand curve for the market as a whole.
Since the demand curve facing it slopes downward, a firm with market power can increase its producer surplus by reducing output and raising the price. This power is not value-creating in itself. It just shifts surplus from consumer to producer. Moreover, since the producer purposely foregoes some sales that would generate surplus, to push up the price, consumers lose more than producers gain. Still another problem is that producers may devote effort to acquiring market power, creating additional waste.\footnote{An important exception is effort in the form of innovation. If the inventor of a new product is given a monopoly on it, that actually raises surplus if he wouldn't invent the product otherwise, so it's worth providing him with profit as an incentive. If a firm in a competitive market puts effort into discovering a new market as yet unserved, so it has a monopoly, that too is innovation that deserves reward. }
A firm without market power simply takes the market price as given and to maximize profit compares the cost of output to that price. It increases output $Q$ until the marginal cost equals the price: $MC(Q) =P$. A firm with market power needs to think about {\bf marginal revenue}\index{marginal revenue}: the change in revenue from an increase in quantity. Marginal revenue equals the market price if the firm has no market power: sell one more unit, and revenue increases by the price. Marginal revenue is less than the price for a firm with market power: produce one more unit, and the price drops a little.
%\vspace*{24pt}
\noindent
%\begin{minipage}[c]{\linewidth}
\begin{figure}[htb]
\begin{center}
{\sc Figure \ref{chap07}.\ref{fig05-1.pdf} \\
Marginal Revenue }
% \vspace*{-12pt}
\refstepcounter{figurecounter} \label{fig05-1.pdf}
\includegraphics[width=.8\linewidth]{fig05-1-monopoly.jpg}
\end{center}
\end{figure}
%\vspace*{24pt}
The firm's tradeoff is between (a) selling more quantity and (b) keeping the price high, as Figure \ref{chap07}.\ref{fig05-1.pdf} shows. If $Q=4$ and $P=8$ and the firm increases its output to $Q= 5$, revenue won't rise by 8, because the price will fall. The price falls from 8 to 7, so revenue only rises from 32 to 35, an increase of 3. The old 4 units are now sold at 7 instead of 8, so they are collecting 28 for the firm instead of 32, a loss of 4. The new unit is collecting the new price of 7, so it is adding 7. There is a net gain of 3 in revenue. That gain is the marginal revenue: the change in revenue when quantity rises.
The firm's marginal cost is 4 at $Q=4$, however, so increasing output to 5 reduces producer surplus.
To maximize surplus, the seller should set $Q$ so that marginal revenue equals marginal cost. Since the
marginal revenue curve lies below the demand curve, this means that the marginal revenue curve will hit marginal cost before the demand curve, so the MR=MC rule produces less output than the P=MC rule.
The simplest case is when the seller has zero marginal cost. Then its task is to set $MR=0$, which
means to maximize revenue because it picks quantity so that the slope of the revenue function (which is marginal revenue) is zero. An example is a stadium selling tickets to a football game. If the
stadium is not filled up, the marginal cost of admitting an extra person is approximately zero. To
maximize profit, however, some seats should be left unfilled. Increasing ticket sales would require
a drop in price that would actually reduce revenue. The stadium example illustrates why market power
leads to market failure: it would cost nothing to let more people enjoy the football game, but the
seller keeps them out anyway.
We often use linear (straight line diagonal) demand curves in diagrams. When demand is linear, marginal
revenue is linear too, with twice as steep a slope (as in Figure \ref{chap07}.1).\footnote{Calculus
shows this. If demand is $P= a- b P$, then revenue is $R= PQ =(a- b Q)Q = aQ- bQ^2$,
and marginal revenue is $\frac{dR}{dQ} = a - 2bQ$, just like demand but with a 2 so it descends
twice as fast.} In drawing, remember that if the demand curve hits the quantity axis at $Q=10$,
then the marginal revenue curve hits at $Q=5$. Also, though the demand curve becomes flat at $P=0$, the marginal revenue curve keeps going down below the $P=0$ axis. That is because marginal revenue can be negative, which just means that increasing output reduces revenue because the quantity rises by less than the price falls.
Another way to think of pricing with market power is using elasticities. The{\bf price elasticity of
demand} (which is easier to remember if you think of it as ``price sensitivity of demand'') is
\begin{equation}
Elasticity = \frac{\text{ \% change in quantity demanded}}{ \text{\% change in price}}
\end{equation}
Iff the quantity demanded falls by 20\% after the price rises by 10\%, the elasticity of demand
is -2. Economists often just say that the elasticity is ``2'',
because the quantity demanded falls with price so we can assume the elasticity is negative. A firm with
market power is trading off price against quantity, so the elasticity is relevant to its decision.
If the elasticity is -1, marginal revenue is zero: if the firm increases output by 1\%, the price
falls by 1\% and revenue is unchanged. We say that demand is {\bf inelastic} \index{inelastic}if
the price elasticity of demand is less than 1, in which case the quantity demanded falls less than
the price rises. A monopoly never wants to choose a price and quantity where demand is inelastic,
because reducing sales at that point by 1\% would drive up the price more than 1\% and thus raise
revenue--- and better yet, total production cost would because of the 1\% reduction. Demand is {\bf perfectly inelastic}
\index{perfectly inelastic} if quantity demanded does not change at all when the price increases. If demand is perfectly inelastic, the demand curve is vertical and the elasticity is 0.
A monopoly would want to raise its price if demand
were perfectly inelastic. Demand is {\bf perfectly elastic} if the quantity changes infinitely (to
zero) as the price rises. Then, the demand curve facing the firm has an elasticity of $-$infinity
and is perfectly flat, just as under perfect competition.
A general rule relating the monopoly markup of price over marginal cost to the elasticity of demand
is the {\bf Lerner Rule}\index{Lerner Rule}.\footnote{The original paper is Abba P.
Lerner, \index{Lerner} ``The Concept of Monopoly and the Measurement of Monopoly Power,''
{\it Review
of Economic Studies}, 1: 157--175 (1934). A good verbal discussion of its modern use is Kenneth G.
Elzinga,\index{Elzinga} \& David E. Mills,\index{Mills}
\href{http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1884993}{``The Lerner Index of
Monopoly Power: Origins and Uses,'' }{\it American Economic Review: Papers and Proceedings},
101(3): 558--564 (2011). } It says that at the profit-maximizing price:
\footnote{The rule $MR=MC$ can be written as $\frac{dP}{dQ} Q + P =MC$, so $-\frac{dP}{dQ}Q = P -MC$
and $-\frac{dP}{dQ}\frac{Q}{P} = \frac{P -MC}{P}$, but $Elasticity = \frac{dQ}{dP}\frac{P}{Q}$ and
so we get the Lerner Rule. }
\begin{equation} \label{lerner1}
\frac{ P - MC}{P} = -\left( \frac{ 1}{ Elasticity} \right)
\end{equation}
Remember: at the optimal price, demand must have gotten elastic, more negative than $-1$.
If the elasticity of demand is $-2$, the Lerner Rule tells us that the price is such that $\frac{ P - MC}{P} = -\left( \frac{ 1}{ Elasticity} \right) = - \frac{1}{-2}$, so $P-MC = \frac{P}{2}$ and $P = 2MC$. If the elasticity of demand is $-1.5$, the price
is three times marginal cost, by the same working out of the equation.
The Lerner Rule can be used in two ways. The firm can use it to help choose its price. The
analyst can use it to estimate the elasticity of demand facing the firm. The expression $\frac{P-MC}
{P} $ on the left-hand-side of equation \eqref{lerner1} is known as the {\bf Lerner Index} of
market power. Its biggest possible value is 1 (if MC=0, so $\frac{P-MC}{P}=1 $) and its smallest
possible value is 0 (if $P=MC$). An analyst who saw that a firm's price was 20 and its marginal
cost was 15 could use equation \eqref{lerner1} to conclude that the Lerner Index was $\frac{P-MC}
{P}= \frac{20-15}{20} = .25$ and the elasticity of demand was $-4$. Thus,
to find out the elasticity of demand facing the firm, the measure of market power we are after, we
just need to know the price and the marginal cost.
\bigskip
\noindent
{\bf \ref{chap07}.2: Oligopoly and the Cournot Model}\label{s05.2}
Price fixing is a form of {\bf collusion}\index{collusion}--- firms agreeing on prices, outputs, or
sales territories--- and is {\it per se} illegal. Even if firms do not collude, however, prices and
profits can still be above marginal cost if entry into the industry is difficult and the
number of firms is small. If the cola industry has just two firms, then even if they never discuss
prices with each other, they know that it is to their mutual benefit to keep prices high and that a
price cut of one of them will be followed by a price cut of the other. Without communication between
them, though, it is hard to agree on a price in the first place and even harder to monitor cheating
than under a cartel. Thus, we expect prices in a two-firm industry--- a {\bf duopoly}
\index{duopoly}--- or a several-firm industry--- an {\bf oligopoly}\index{oligopoly}--- to be lower
than in a monopoly or cartel, but higher than in a perfectly competitive industry. Note, however, that this
depends on the {\bf incumbent firms}\index{incumbent firms}--- the ones already in the industry---
being safe from {\bf entrants}\index{entrants}--- new firms who start selling in the industry
attracted by the high profits. If it is easy to start up a new business in the industry ({\bf free
entry})\index{free entry} then even a monopoly could not get away with earning high profits very long.
The {\bf Cournot model}\index{Cournot model} is the basic model economists use to explain why
prices fall with the number of firms when there is not free entry. In the model, each firm chooses
its output given its expectations about the other firms' outputs. The market price is
determined by how much consumers will pay for the total output of all the firms.
Let's look at a specific example of a Cournot model.
Let there be $n$ firms, each firm producing its output $q_i$ $ i = 1, \ldots, n$ with no
fixed cost and with a constant marginal cost of 20. Let market demand be
\begin{equation} \label{eqx}
Q = 140-P,
\end{equation}
where $Q$ is the total output: $Q = q_1+q_2+ \ldots q_n$. Firm $i$'s profit function is then
\begin{equation} \label{eqx}
\pi_i = Pq_i - 20q_i .
\end{equation}
\bigskip
%-------------------------------------------------------
\newpage
\noindent
{\bf One Firm: Monopoly}
First, suppose $n=1$, so Firm 1 is a monopoly. Then, $P = 140-q_1$, so Firm 1's profit, shown
in Figure \ref{chap07}.\ref{fig06-6-profit.jpg}, is
\begin{equation} \label{eqx}
\pi_1 = (140-q_1)q_1 - 20q_1 = (140q_1 -q_1^2) - 20q_1
\end{equation}
\begin{center}
\vspace*{24pt}
\begin{minipage}[c]{.8\linewidth}
\begin{center}
{\sc Figure \ref{chap07}.\ref{fig06-6-profit.jpg} \\
Monopoly Profits with One Seller } \\
\refstepcounter{figurecounter} \label{fig06-6-profit.jpg}
\includegraphics[width=\linewidth]{fig06-6-profit.jpg}
\end{center}
\end{minipage}
%\vspace*{24pt}
\end{center}
To maximize the firm's profit, use a little calculus. Take profit's derivative with respect to
$q_1$ and set the derivative equal to zero, the equation called the {\bf first-order condition}
\index{first-order condition} for maximization. That yields
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = (140 -2q_1 ) - 20 =0
\end{equation}
which solving out for $q_1$ yields $q_1 = 60$. Then, using the demand function, $P=80$, so $\pi_1 =
(80)(60) - 20(60) = 3,600.$
This is the same result as we would get by setting marginal revenue equal to marginal cost,
because the last equation can be rewritten as
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = (Marginal \; revenue) - Marginal \; cost =0
\end{equation}
Recall that with linear demand,the marginal revenue curve slopes down twice as fast as the demand
curve, so with $P=140-Q$, we get $MR = 140-2Q$. So calculus is a way to generate that rule.
\bigskip
%-------------------------------------------------------
\noindent
{\bf Two Firms: Duopoly--- The Asymmetric-Costs Case}
So much for monopoly. Let's do the same thing for duopoly. We will start with an asymmetric case, in which Firm 1 has a cost advantage over Firm 2: Firm 1's marginal cost is $c_1=20$ and Firm 2's is $c_2=40$.
As before, $P = 140-(q_1+q_2)$, so Firm
1's profit is:
\begin{equation} \label{eqx}
\pi_1 (q_1)= (140-q_1-q_2)q_1 - 20q_1 = (140q_1 -q_1^2 - q_2q_1) - 20q_1.
\end{equation}
Take profit's derivative with respect to $q_1$ and set it to zero:
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = (140 -2q_1 -q_2) - 20 =0.
\end{equation}
This gives us the {\bf reaction function } \index{reaction function} or {\bf reaction curve}
\index{reaction curve} for $q_1$ as a function
of $q_2$. The more output Firm 1 expects from Firm 2, the smaller will Firm 1 choose its own output
to be:
\begin{equation} \label{eqx}
q_1 (q_2) = 60 -\frac{q_2}{2}.
\end{equation}
At the extremes, if Firm 1 thinks Firm 2 will set $q_2=0$, then Firm 1 chooses (``reacts with'')
its monopoly output of 60; but if Firm 1 thinks Firm 2 will set $q_2=120$ and drive the price down
to marginal cost, then Firm 1 chooses to produce zero. Firm 1's reaction curve goes
from 60 on the y-axis to 120 on the x-axis.
\bigskip
Now let's do the same for Firm 2. Firm
2's profit is:
\begin{equation} \label{eqx}
\pi_2(q_2) = (140-q_1-q_2)q_2 - 40q_2 = (140q_2 -q_2^2 - q_2q_1) - 40q_2.
\end{equation}
Take profit's derivative with respect to $q_2$ and set it to zero:
\begin{equation} \label{eqx}
\frac{d \pi_2}{d q_2} = (140 - 2q_2 - q_1 ) - 40 =0.
\end{equation}
This gives us the {\bf reaction function } for $q_2$ as a function
of $q_1$. The more output Firm 2 expects from Firm 1, the smaller will Firm 2 choose its own output
to be:
\begin{equation} \label{eqx}
q_2 (q_1) = 50 -\frac{q_1}{2}.
\end{equation}
At the extremes, if Firm 2 thinks Firm 1 will set $q_1=0$, then Firm 2 chooses (``reacts using'')
its monopoly output of 50 ($not$ Firm 1's monopoly output, which is 60---why not?). If Firm 2 thinks Firm 1 will set $q_1=100$ and drive the price down
to Firm 2's marginal cost of 40, then Firm 2 chooses to produce zero. Firm 2's reaction curve goes
from 100 on the y-axis to 50 on the x-axis.
What is the value of $q_2$ that Firm 2 will actually pick?
The two reaction equations give us two equations for two unknowns, $q_1$ and $q_2$. We can substitute to get:
\begin{equation} \label{eqx}
q_2 (q_1) = 50 - \frac{60-\frac{q_2}{2} } {2} = 50 - 30+ \frac{q_2}{4} = 20+\frac{q_2}{4}.
\end{equation}
Solving this last equation yields $ q_2^*= \frac{80}{3} = 26 \frac{2}{3}$. We can substitute $q_2^*$ into $q_1(q_2) $ to get $q_1^*= 60 - \frac{\frac{80}{3} }{2} =\frac{140}{3}= 46 \frac{2}{3}$.
Using the demand function, $P = 140- \frac{140}{3} - \frac{80}{3} =67 \frac{1}{3} $.
%\vspace*{24pt}
\begin{center}
\begin{minipage}[c]{.8\linewidth}
\begin{center}
{\sc Figure \ref{chap07}.\ref{fig05-2a-asym-cournot.jpg} \\
Asymmetric Cournot Duopoly Reaction Curves } \\
\medskip
\refstepcounter{figurecounter} \label{fig05-2a-asym-cournot.jpg}
\includegraphics[width=\linewidth]{fig05-2a-asym-cournot.jpg}
\end{center}
\end{minipage}
\end{center}
\vspace*{24pt}
\bigskip
%-------------------------------------------------------
\noindent
{\bf Two Firms: Duopoly--- The Symmetric-Costs Case}
Let's do the same thing for a duopoly in which both firms have the same cost curves. This is simpler than the asymmetric-cost Cournot duopoly, but its ease is deceptive, because it doesn't show so clearly how each firm is acting independently to maximize its own profits. As before, $P = 140-(q_1+q_2)$, so Firm
1's profit is
\begin{equation} \label{eqx}
\pi_1 = (140-q_1-q_2)q_1 - 20q_1 = (140q_1 -q_1^2 - q_2q_1) - 20q_1
\end{equation}
Again take profit's derivative with respect to $q_1$ and set it to zero.
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = (140 -2q_1 -q_2) - 20 =0
\end{equation}
This gives us a reaction function (as shown in Figure \ref{chap07}.\ref{fig06-2.jpg}) for $q_1$ as a function
of $q_2$. The more output Firm 1 expects from Firm 2, the smaller will Firm 1 choose its own output
to be.
\begin{equation} \label{eqx}
q_1 (q_2) = 60 -\frac{q_2}{2}.
\end{equation}
At the extremes, if Firm 1 thinks Firm 2 will set $q_2=0$, then Firm 1 chooses (``reacts with'')
the monopoly output of 60; but if Firm 1 thinks Firm 2 will set $q_2=120$ and drive the price down
to marginal cost, then Firm 1 chooses to produce zero. That is why Firm 1's reaction curve goes
from 60 on the y-axis to 120 on the x-axis.
%\vspace*{24pt}
\begin{center}
\begin{minipage}[c]{.8\linewidth}
\begin{center}
{\sc Figure \ref{chap07}.\ref{fig06-2.jpg} \\
Cournot Duopoly Reaction Curves } \\
\medskip
\refstepcounter{figurecounter} \label{fig06-2.jpg}
\includegraphics[width=\linewidth]{fig06-2.jpg}
\end{center}
\end{minipage}
\end{center}
\vspace*{24pt}
What is the value of $q_2$ that Firm 2 will actually pick? If we were to set up the same problem for Firm 2's choice of $q_2$ to maximize
profit and solve the two first order conditions together, it turns out that we would find that $q_1=
q_2$. Figure \ref{chap07}.\ref{fig06-2.jpg} shows Firm 2's reaction curve, and how it crosses Firm
1's reaction curve at $q_1=q_2=40$. Rather than finding the equation for Firm 2's reaction
curve, though, we can set $q_2=q_1$ in Firm 1's reaction curve, so
\begin{equation} \label{eqx}
q_1 = 60 -\frac{q_1}{2},
\end{equation}
which solves out to $q_1 = 40$. As a result, $q_2=q_1=40$ and $Q = q_1+q_2 = 80.$ Using the demand
function, $P = 60$. Profits are $pi_1 = \pi_2 = (60)(40) - 20 (40) = 1,600$ each. Duopoly industry
profit is 3,200, which is below the monopoly industry profit of 3,600, but above the competitive
profit of 0 that would result if $P =MC =20$.
Figure \ref{chap07}.\ref{fig06-2.jpg} shows another line going from 30 on the y-axis to 30 on the
x-axis, labelled ``Cartel Output''. This is the combinations of $q_1$ and $q_2$ that add up to 60,
the monopoly output. Any combination on that line maximizes industry profit. The natural cartel
outcome if collusion were legal would be $q_1=q_2=30$, but if Firm 1 is better at bargaining and
threatens to break up a cartel, it could get $q_1=32, q_2=28$. The price would be $P=80$, so Firm
2's profit would be (80-20) (28) = 1,680, better than Firm 2 could get from Cournot duopoly if it
refused Firm 1's offer. That kind of threatening and bargaining, though, is one reason cartels
often do not form or form but break up as the result of bickering.
\bigskip
%-------------------------------------------------------
\noindent
{\bf Three or More Firms: Oligopoly to Competition}
The beauty of the Cournot model is that it can be extended to any number of firms. Now let there
be $n$ firms, so
$P = 140-(q_1+q_2 + \ldots + q_n)$, and Firm 1's profit is
\begin{equation} \label{eqx}
\pi_1 = \left(140- (q_1+q_2+\ldots q_n) \right)q_1 - 20q_1 = \left(140q_1 -q_1^2 - q_1 (
q_2+\ldots q_n)\right) - 20q_1
\end{equation}
Again take profit's derivative with respect to $q_1$ and set it to zero.
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = \left(140 -2q_1 - ( q_2+\ldots q_n)\right) - 20 =0
\end{equation}
Firm 1's reaction curve is
\begin{equation} \label{eqx}
q_1 = 60 - \frac{( q_2+\ldots q_n)}{2}.
\end{equation}
If there are 3 firms the reaction curves
are actually reaction planes in 3-dimensional space. The equations are $q_1 =
60 - \frac{q_2 + q_3}{2}$, $q_2 = 60 - \frac{q_1 + q_3}{2}$, and $q_3 = 60 -
\frac{q_2 + q_3}{2}$.
Since the firms all are identical and solve the same kind of profit maximization problem, it turns
out that $q_1 = q_2 = \ldots = q_n$, so $( q_2+\ldots q_n) = (n-1) q_1$ and we can write
\begin{equation} \label{eqx}
q_1 = 60 -\frac{(n-1) q_1}{2} = \frac{120}{n+1}
\end{equation}
Table \ref{chap07}.\ref{tab05-04} shows what happens as the number of firms increases from 1 to 99.
\vspace*{24pt}
\noindent
\begin{minipage}[c]{\linewidth}
\begin{center}
\refstepcounter{tablecounter} \label{tab05-04}
{\sc Table \ref{chap07}.\ref{tab05-04} \\
Cournot Industry Output Rises with the Number of Firms}\\
\bigskip
\begin{tabular}{l|cr|cr r}
\hline
\hline
& & &&& \\
Number of & Output &Profit & Total & Price & Industry \\
firms ($n$) &per firm & per firm & output &&profit \\
& & &&& \\
\hline
& & &&& \\
\: 1 & 60& 3,600 &60&80 &3,600 \\
\: 2 & 40 & 1,600 & 80& 60& 3,200 \\
\: 3 & 30 & 900 &90 & 50 &2,700 \\
\: 4 & 24 & 576 & 96 &44 &2,304 \\
\: 5 &20 & 200 & 100 &40 &2,000 \\
\: 9 & 12 & 144 &108 & 32& 1,296 \\
99 &$\approx 1$ & $\approx 1$ & $\approx 119$ & $\approx 21$ & $\approx 143$ \\
& & &&& \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\vspace*{12pt}
Table \ref{chap07}.\ref{tab05-04} shows that if the number of firms becomes large, the Cournot
model predicts that the price will approach marginal cost and profits will approach zero. It is
always true that as the number of firms increases, output per firm falls, industry output rises, and
the price falls.
\bigskip
%-------------------------------------------------------
\noindent
{\bf \ref{chap07}.3: Price Fixing} \label{s05.5}
Price fixing is the most frequent
antitrust violation, and perhaps the most important. The two most common kinds of cases involve
industries for homogeneous products that have few sellers and bidders in auctions, whether auctions
to sell objects where the high bid wins or auctions for procurement contracts where the low bid
wins. (A {\bf homogeneous}\index{homogeneous} product such as salt is the same whichever firm
produces it; a {\bf heterogeneous }\index{heterogeneous} of {\bf differentiated}
\index{differentiated}product such as perfume differs across firms.) Probably most cases go
undetected because they are on a local scale and too small for federal authorities to intervene
(though there do exist state antitrust laws as well). Even when there is little fear of punishment,
however, cartels have a problem: making sure members obey the cartel rules.\footnote{By the way, the
now-common practice of referring to drug gangs as drug cartels misuses the word completely. A cartel
is a group of sellers that agree to all sell at the same price. If a group of gangs agree to all
sell cocaine at a certain price, that is a cartel. If they compete ferociously using price cuts and
automatic weapons, that is the opposite of a cartel.}
\vspace*{16pt}
\noindent
\begin{minipage}[c]{\linewidth}
\begin{center}
\begin{footnotesize}
\refstepcounter{tablecounter} \label{table5-2}
{\sc Table \ref{chap07}.\ref{table5-2}\\
The Prisoner's Dilemma}\\
\bigskip
\begin{tabular}{lll ccc}
\hline
\hline
\multicolumn{6}{l}{\it } \\
& & &\multicolumn{3}{c}{\bf Column}\\
& & & {\it Not Confess} & & {\it Confess} \\
\multicolumn{6}{l}{\it } \\
& & {\it Not Confess} & -1,-1 & & -10, 0 \\ & {\bf Row} && & & \\
& & {\it Confess} & 0,-10 & & {\bf -8,-8} \\
\multicolumn{6}{l}{\it } \\
\multicolumn{6}{l}{\it Payoffs to: (Row,Column) } \\
\multicolumn{6}{l}{\it } \\
\hline
\hline
\end{tabular}
\end{footnotesize}
\end{center}
\end{minipage}
\vspace*{16pt}
The cartel enforcement problem is a good example of the Prisoner's Dilemma\index{Prisoner's
Dilemma}. In the original Prisoner's Dilemma, illustrated in Table \ref{chap07}.\ref{table5-2}, two
criminals, Mr. Row and Mr. Column, have been captured and accused of carrying out a crime together.
They are kept in separate cells and each offered a choice: Confess or Not Confess. If neither
confess, there is still enough evidence to sentence each to 1 year in prison. If both confess, they
will each get 8 years in prison. But if one confesses and the other does not, the confessor gets off
free and the nonconfessor is sentenced to 10 years. Clearly the best thing for the prisoners jointly
is to Not Confess. For each one acting individually, however, Confess is a dominant strategy. Think
of Prisoner Row's choice. If Prisoner Column confesses,Row gets 8 years in prison by confessing
too, and 10 years if he chooses Not Confess, so he should confess. If Prisoner Column doesn't
confess, Row gets 0 years in prison by confessing, and 1 years if he chooses Not Confess, so he
should confess. Either way, Prisoner Row does better by confessing. And so they both confess and
get a total of 16 years in prison.
\vspace*{24pt}
\noindent
\begin{minipage}[c]{\linewidth}
\begin{center}
\refstepcounter{tablecounter} \label{table5-3}
{\sc Table \ref{chap07}.\ref{table5-3}\\
The Price-Fixer's Dilemma}\\
\bigskip
\begin{tabular}{lllccc}
\hline
\hline
\multicolumn{6}{l}{\it } \\
& & &\multicolumn{3}{c}{\bf Column}\\
& & & {\it High Price} & & {\it Low Price} \\
\multicolumn{6}{l}{\it } \\
& & {\it High Price} & 400, 400 & & 200, 500 \\
& {\bf Row} && & & \\
& & {\it Low Price} & 500, 200 & & {\bf 300, 300} \\
\multicolumn{6}{l}{\it } \\
\multicolumn{6}{l}{\it Payoffs to: (Row, Column) }\\
\multicolumn{6}{l}{\it } \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\vspace*{24pt}
Now consider price fixers Row and Column in Table \ref{chap07}.\ref{table5-3}. They have a constant
marginal cost of \$4 per unit for a differentiated product. If they each charge the cartel price of
\$8 that they have agreed upon each will sell 100 units (for a profit of \$800 - \$400 each), but
if they both sell at the low, discounted price of \$5 each will sell 300 units (for a profit of
\$1,500 - \$1,200 each). If one sells at the low price and one sells at the high price, then the
low-priced seller sells 500 and the high-priced seller sells 50. The product is differentiated,
which is why even a high-priced seller still gets some sales.
If both price fixers charge high prices, they each will have profits of 400. If both choose low
prices, they will each get 300. But if one prices high and the other prices low, the low-price firm
has profits of 500 and the high-price firm has profits of 200. Clearly the best thing for the firms
jointly is to price high. For each one acting individually, however, Low Price is a dominant
strategy. Think of Row's choice. If Column prices high, Row's profits are 400 if he too prices
high, but 500 if he prices low. If Column prices low, Row's profits are 200 if he prices high,
but 300 if he prices low. Either way, Row does better by pricing low. And so both firms price
low, and industry profits are 600 instead of the 800 they could be if both stuck to the cartel
price.
%\begin{figure}[htb]
%\end{figure}
If Row and Column could make a legal agreement to keep prices high, their task would be much
easier, but the courts would not enforce such a contract even before the Sherman
Act.\footnote{Courts in the common law countries of England the United States would not enforce
cartel contracts, as being ``against public policy'' like contracts to commit crimes. Germany, on
the other hand, did enforce cartel contracts, which were common. See Steven B. Webb \href{http://www.jstor.org/stable/2120181 }{``Tariffs, Cartels, Technology, and Growth in the German Steel Industry, 1879 to 1914,''} {\it Journal of Economic History}
40: 309-330 (1980). }
Because of the antitrust laws, they cannot even hire an accounting firm to audit each other and
find out if someone has been cheating on the cartel. To even detect cheating, price fixers have to
rely on gossip from customers--- who would like to stir up competition--- and fluctuations in their
own demand. If Row's sales go down, however, he cannot tell whether it is because Column has been
making secret price cuts to steal customers or because of random fluctuations in market demand. For
this reason, cartels tend to break down eventually.
I
suggested earlier that industries with homogeneous products would have the most successful cartels. The reason will now be clearer: if the
product sold by each firm is identical, it is easier to police a cartel and detect when one member
is trying to undercut the others. Prices are most easily compared when the product is homogeneous:
Row's salt should sell a pound of salt at the same price as Column, but if Honda and Toyota tried to
fix prices, should a Honda Accord sell for the same price as a Toyota Camry, particularly if
different sellers include different options such as leather seating? In addition, if products are
heterogeneous, a seller can gain an advantage by increasing the quality of his product or giving the
consumer a better version at the same price.
If there are few sellers, it is easier to come to an agreement. When there are many firms, one
firm can stay out of the cartel and still profit from the high prices generated by the cartel's
reduced output, so there is a free rider problem. If the good is sold by auction, then the cartel
members can see who won and perhaps even see the price, depending on the auction rules. This aids
in detecting who cheated on the cartel agreement. If Row wins the auction for a government
contract this week, and Column was supposed to win, Row has some explaining to do. Thus, the
structure of the market affects whether successful cartels form.
%-------------------------------------------------------
\bigskip
\noindent
{\bf \ref{chap07}.4: Regulating Monopoly: The History of Antitrust Law} \label{s06.1}
What to do about market power has long been controversial. Perhaps the earliest discussion is in Aristotle's {\it Politics},\href{http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0058:book=1:section= 1259a&highlight=thales}{1.1259a}, where he writes about Thales, the first philosopher, and a ``man of Sicily".
\begin{bigquote}
Thales, so the story goes, because of his poverty was taunted with the uselessness of philosophy; but from his knowledge of astronomy he had observed while it was still winter that there was going to be a large crop of olives, so he raised a small sum of money and paid round deposits for the whole of the olive-presses in Miletus and Chios, which he hired at a low rent as nobody was running him up; and when the season arrived, there was a sudden demand for a number of presses at the same time, and by letting them out on what terms he liked he realized a large sum of money, so proving that it is easy for philosophers to be rich if they choose, but this is not what they care about. Thales then is reported to have thus displayed his wisdom, but as a matter of fact this device of taking an opportunity to secure a monopoly is a universal principle of business; hence even some states have recourse to this plan as a method of raising revenue when short of funds: they introduce a monopoly of marketable goods.
There was a man in Sicily who used a sum of money deposited with him to buy up all the iron from the iron mines, and afterwards when the dealers came from the trading- centers he was the only seller, though he did not greatly raise the price, but all the same he made a profit of a hundred talents on his capital of fifty. When Dionysius came to know of it he ordered the man to take his money with him but clear out of Syracuse on the spot,6 since he was inventing means of profit detrimental to the tyrant's own affairs. Yet really this device is the same as the discovery of Thales, for both men alike contrived to secure themselves a monopoly.
\end{bigquote}
Modern regulation of monopolies started in
1603, when Queen Elizabeth I of England granted her groom a legal monopoly on selling playing cards.
A court voided the grant, saying that monopolies were counter to the common law. ({\it Darcy v.
Allein}, 77 Eng. Rep. 1260 (K.B. 1603)) Another source of market power was {\bf price-fixing
agreements}\index{price-fixing agreements} or {\bf cartels}:\index{cartels} agreements between
sellers to jointly keep their prices high. Forming a cartel was not a criminal offense, but the
courts would not enforce an agreement ``in restraint of trade'' unless it could be shown that the
agreement benefitted the public. An example of such a beneficial agreement would be if someone sold
a law firm under the condition that he not start a new law firm in the same city and steal back his
old clients.
\begin{wrapfigure}{L}{2.4in}
\noindent
\begin{boxedminipage}[c]{\linewidth}
\renewcommand{\baselinestretch}{.9}
\begin{footnotesize}
\refstepcounter{sidebarcounter} \label{sidebar6-4}
\noindent
{\sc Box \ref{chap07}.\ref{sidebar6-4}\\
An Indiana Cement Conspiracy }\\
\hspace*{16pt}
Gus B.
(Butch) Nuckols of Builder's Concrete and Supply Co. agreed with another company, IMI, to fix
cement prices from 2000 till 2004. Since cement is so heavy, competition tends to be local,
high profits will not attract new entrants for some time. The FBI found that meetings were held
at the Nuckols horse barn in
Fishers, Indiana to discuss price, discounts, and conditions of sale.
\hspace*{16pt}IMI made \$225 million from the conspiracy. In a
plea bargain, Nuckols turned on his co-conspirators in exchange for a lighter sentence: a
\$50,000 fine personally, a 14-month prison sentence, and a fine of \$4 million for his company.
\hspace*{16pt}Four executives of IMI pled guilty too. They received fines of one to two
hundred thousand dollars, and went to jail for 5 months and the IMI itself was fined \$29
million.
\end{footnotesize}
\end{boxedminipage}
\end{wrapfigure}
Excluding cartels from the benefits of contract rights does have an effect, because members of a
cartel are always tempted to cheat by reducing prices to increase the quantity they sell, at the
expense of other members of the cartel.
In 1890, the Sherman Act\index{Sherman Act} was passed in the United States, and it remains one
of the two most important antitrust laws. Section 1 of the Act, in its 2010 form (the fine limit
has been increased from the 1890 version), says
\begin{bigquote}
{\bf Every contract}, combination in the form of trust or otherwise, or conspiracy, {\bf in
restraint of trade or commerce among the several States, or with foreign nations, is declared to be
illegal.} Every person who shall make any contract or engage in any combination or conspiracy hereby
declared to be illegal shall be deemed guilty of a felony, and, on conviction thereof, shall be
punished by fine not exceeding \$100,000,000 if a corporation, or, if any other person, \$1,000,000,
or by imprisonment not exceeding 10 years, or by both said punishments, in the discretion of the
court. (U.S.C. Title 15, Chapter 1, \S 1)
\end{bigquote}
Section 1 is aimed directly at price-fixing agreements. At least two parties must be involved,
and they must make some definite agreement to restraint trade by such actions as directly
restricting output or raising prices. It says:
\begin{bigquote}
{\bf Every person who shall monopolize, or attempt to monopolize,} or combine or conspire with any
other person or persons, to monopolize {\bf any part of the trade or commerce among the several
States,} or with foreign nations, {\bf shall be deemed guilty of a felony}, and, on conviction
thereof, shall be punished by fine not exceeding \$100,000,000 if a corporation, or, if any other
person, \$1,000,000, or by imprisonment not exceeding 10 years, or by both said punishments, in the
discretion of the court. (U.S.C. Title 15, Chapter 1, \S 2, as of 2010)
\end{bigquote}
Section 2 is harder to interpret than Section 1. It is directed not just against agreements, but
also at single parties who try to monopolize a market. What ``monopolize'' means is left unclear,
and so has had to be interpreted by executive branch policy and by the courts. Early interpretation
ruled that if firms merged to form a monopoly, that was not ``monopolizing'' and so was legal.
Courts treat Section 1 and Section 2 violations differently. Section 1 violations are {\bf per se}
\index{per se} illegal, meaning that they are illegal even if they do not cause any harm. If two gas
stations agree on a minimum price, they violate Section 1 even if they can show that there are many
other competing gas stations and they did not hurt any consumers. Section 2 violations, on the
other hand, are subject to {\bf the rule of reason}\index{the rule of reason}: if the defendant can
show that his actions were not really intended to monopolize or did not have a bad effect, he can
escape punishment.
Since ``monopolizing'' is a vague term, the
Clayton Act\index{Clayton Act} was passed in 1914 to try to pin down monopolizing practices more
clearly. One section, which incorporates amendments in the 1930's called the {\bf Robinson-Patman
Act}\index{Robinson-Patman Act}\footnote{For more on this, see Donald S. Clark, ``The Robinson-
Patman Act:
General Principles, Commission Proceedings,
and Selected Issues,''\url{http://www.ftc.gov/speeches/other/patman.shtm}, June 7, 1995.} attacks
{\bf price discrimination}: \index{price discrimination}
\begin{bigquote}
{\bf It shall be unlawful for any person} engaged in commerce, in the course of such commerce,
either directly or indirectly, {\bf to discriminate in price between different purchasers of
commodities of like grade and quality}, where either or any of the purchases involved in such
discrimination are in commerce, where such commodities are sold for use, consumption, or resale
within the United States or any Territory thereof or the District of Columbia or any insular
possession or other place under the jurisdiction of the United States, and {\bf where the effect of
such discrimination may be substantially to lessen competition or tend to create a monopoly} in any
line of commerce, {\bf or to injure, destroy, or prevent competition with any person who either
grants or knowingly receives the benefit of such discrimination, or with customers of either of
them} \ldots (U.S.C. Title 15, Chapter 1, \S 13)
\end{bigquote}
Section 13 does not apply to every case where a seller charges different customers different
prices, only to cases where the effect is to substantially lessen competition. {\bf Predatory
pricing} \index{Predatory pricing} is the best example of this: a monopoly charges below-cost prices
in some markets to drive out competing firms and high prices in other markets where competitors
do not operate.
Another section of the Clayton Act tackles special contractual provisions that could help a firm
monopolize a market. One of these is {\bf exclusive dealing contracts} \index{exclusive dealing
contracts}or {\bf total-requirements contracts}\index{total-requirements contracts} under which the
buyer must buy the good exclusively from the one seller, even if his competitor offers a lower
price:
\begin{bigquote}
It shall be unlawful for any person engaged in commerce, in the course of such commerce, to lease or
make a sale or contract for sale of goods, ... on the condition, agreement, or understanding that
the lessee or purchaser thereof shall not use or deal in the goods, wares, merchandise, machinery,
supplies, or other commodities of a competitor or competitors of the lessor or seller, where the
effect \ldots may be to substantially lessen competition or tend to create a monopoly in any line of
commerce. (U.S.C. Title 15, Chapter 1, \S 14)
\end{bigquote}
The most important part of the Clayton Act deals with mergers.
\begin{bigquote}
No person engaged in commerce or in any activity affecting commerce shall acquire, directly or
indirectly, the whole or any part of the stock or other share capital and no person subject to the
jurisdiction of the Federal Trade Commission shall acquire the whole or any part of the assets of
another person engaged also in commerce or in any activity affecting commerce, where in any line of
commerce or in any activity affecting commerce in any section of the country, the effect of such
acquisition may be substantially to lessen competition, or to tend to create a monopoly. (15 U.S.C.
\S 18)
\end{bigquote}
As with other Clayton Act provisions, this is subject to the rule of reason. Mergers are legal
unless they substantially lessen competition. Merger regulation is the most active area of
antitrust enforcement in the United States today. Businesses over a certain size which wish to merge
are required to notify the Justice Department and the Federal Trade Commission, who scrutinize the
merger for anticompetitive effects. Either agency may then challenge the legality of the merger in
court. Ordinarily, if the agencies decide the merger would substantially reduce competition, the
companies either drop the merger or change its terms in consultation with the agencies. If some
cities are only served by two chain store chains, for example, a merger of those two chains might
be required to sell some of their stores in the ``captive'' cities to a third chain to preserve
competition. Occasionally firms that wish to merge think that the FTC and Justice are so
unreasonable that they say they will merge anyway. In that case, the agency goes to a federal
court and tries to persuade a judge to issue an injunction blocking the merger on the grounds
that it would substantially lessen competition.
A major purpose of the Clayton Act was actually to reduce competition in one area of the economy:
labor markets. The Sherman Act and the common law had been used to punish labor unions, which are
cartels in which workers agree to sell their labor jointly at a price above the market level. These
are heavily regulated by labor laws, but are exempt from antitrust laws and much of labor law is
devoted to encouraging workers to organize, to increase the price at which they sell their labor,
and to be able to use exclusive-dealing contracts to shut out workers not in the union. The Clayton
Act says:
\begin{bigquote}
The labor of a human being is not a commodity or article of commerce. {\bf Nothing contained in the
antitrust laws shall be construed to forbid the existence and operation of labor, agricultural, or
horticultural organizations, instituted for the purposes of mutual help, and not having capital
stock or conducted for profit,} or to forbid or restrain individual members of such organizations
from lawfully carrying out the legitimate objects thereof; {\bf nor shall such organizations, or the
members thereof, be held or construed to be illegal combinations or conspiracies in restraint of
trade,} under the antitrust laws. (15 U.S.C. \S 17)
\end{bigquote}
As one would expect, since labor cartels are legal, they are common. They are heavily regulated,
however, by the National Labor Relations Board, which uses a multitude of rules to determine how a
labor union may be organized, what happens if competing unions both wish to one company's workers,
whether the union can require the firm to fire competing workers who refuse to the join the union,
and so forth.
An example that shows the market power of a labor cartel can exercise is
Local 1 of the International Alliance of Theatrical Stage Employees in New York City. Carnegie Hall
has five stagehands, who hang the lights, move furniture around, and so forth. The highest-paid is
the props manager, who earns
\$423,000 cash per year plus \$107,000 in benefits and deferred compensation. The lowest-paid, an
electrician, earns \$403,000. Admittedly, the workweek can last up to 80 hours, during busy times
during the heavy season, but the contrast to the incomes of aspiring musicians is amazing. Of
course, Carnegie Hall pays especially well.
Average stagehand compensation at Avery Fisher Hall and the Alice Tully Hall in Lincoln Center
is only \$290,000 per year. But a successful monopoly can do well even in a market with demand as
seemingly fragile as classical music backstage
labor.\footnote{\href{http://www.northjersey.com/news/opinions/68164552.html?page=all} {``For
Backstage Labor, Rich Rewards,''} James Ahearn, {\it Northjersey.com} (November 1, 2009).}
One of the most important things to note about antitrust law is that it does not make monopoly
illegal. Rather, it makes {\bf monopolizing}\index{monopolizing} illegal. If a company grows to
dominate its industry because it has low costs or good products, that is perfectly legal. Moreover,
the firm can restrict its output by following the MR(Q) = MC(Q) rule and that is legal too. There is
still market failure, but the potential for government failure in breaking up large, successful
firms is too great to make a law against mere size a good idea. Letting the government control the
firm's prices and practices is also too fraught with risk. Both sources of government failure are
acute: the government would not know how to run the business, and the government might have the
objective of favoring politically powerful groups. Thus, the law focuses on practices which add to a
firm's market power for reasons that do not help consumers, practices such as practically all price-
fixing agreements and some mergers. The law goes well beyond price-fixing and mergers to suspicious
practices such as predatory pricing, tie-in requirements, and exclusive dealing, but those
practices, while still monitored and sometimes punished, are not the focus of antitrust activity.
\bigskip
%-------------------------------------------------------
\noindent
{\bf Antitrust Law in Europe}
Article 101 (formerly article 81) of the Treaty on the Functioning of the European Union
prohibits cartels and other ``concerted practices,'' in a way comparable to the Sherman Act's
Section 1.\footnote{Treaty on the Functioning of the European Union (consolidated text), Official
Journal of the European Union September 5, 2008,\url{http://eur-
lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:C:2008:115:0047:0199:EN:PDF.}}
Article 102 is like the Sherman Act's Section 2. It says, ``Any abuse by one or more undertakings of
a dominant position within the common market... shall be prohibited as incompatible with the common
market insofar as it may affect trade between Member States.''\footnote{For more, see
Baumgardner\index{Baumgardner}, Larry (2005) ``Antitrust Law in the European Union
The law is changing---but to what effect?'' {\it Graziadio Business Report}, 11, 3. \url{
http://gbr.pepperdine.edu/2010/08/antitrust-law-in-the-european-union/}.} The laws are enforced
by the European Commission and by the antitrust authorities of each of the 20-plus nations of the
European Union.
EU antitrust policy is administered by the {\bf Competition Commissioner}\index{Competition
Commissioner}, one of 27 commissioners of the {\bf European Commission}\index{European Commission}.
He is appointed by the {\bf Council of the European Union}\index{Council of the European Union}
(informally, the {\bf Council of Ministers})\index{Council of Ministers} which is composed of one
representative of each EU country.\footnote{EU governance is confusing. There also exists a Council
of Europe and a European Council, which are different from the Council of the European Union.} In
addition, member countries continue to have their own antitrust agencies, e.g. the United Kingdom's
\url{http://www.competition-commission.org.uk/}{Competition Commission}.
The European Union's main legislative branch is the {\bf European Parliament}\index{European
Parliament}, which is elected directly by voters according to a country's population. {\bf EU
directives}\index{EU directives} are rules which member countries are supposed to implement by
passing their own national laws and regulations in accord with the rule. {\bf EU regulations}
\index{EU regulations} are rules which take effect immediately and bind every member country (note
that ``regulation'' has a particular legal meaning here, contrasting with directives). The
European Parliament does not make antitrust law, the rules for which are issued by the European
Commission and the Council of Ministers.\footnote{European Parliament, ``General Competition Policy
and Concerted
Practices,''\url{http://www.europarl.europa.eu/parliament/expert/displayFtu.do?language=en&id=
74&ftuId=FTU_3.3.1.html} (September 2006). }
Regulations say that the Competition Commissioner can block any merger which ``significantly impede
effective competition,'' a standard similar to the Clayton Act's and equally open to widely
varying interpretations.\footnote{Council Regulation 139/2004 on The Control of Concentrations
Between Undertakings (the EC Merger Regulation), art. 1, 2004 O.J. (L 24) 1.} The Commission has
issued Merger Guidelines based on how many firms are in a market to let businesses know which kinds
of mergers it is likely to challenge.\footnote{Guidelines on the Assessment of Horizontal Mergers
under the Council Regulation on the Control of Concentrations between Undertakings, 2004 O.J. (C 31)
5.}
In 2001 the EU blocked a merger of General Electric and Honeywell, even though American authorities
had already approved it. GE's CEO said, ``The European regulators' demands exceeded anything I or
our European advisers imagined and differed sharply from antitrust counterparts in the U.S. and
Canada.''\footnote{Andrew Ross Sorkin, \href{http://www.nytimes.com/2004/05/30/business/dealbook-a-trustbuster-s-song-is-ending-but-a-coda-is-possible.html}{``A Trustbuster's Song Is Ending, but a Coda Is Possible,''}
{\it New York Times}, May 30, 2004, C3.} The personal views of whoever is Competition
Commissioner are extremely important. Another notable case involved Microsoft, which in 2004 was
fined \$600 million for anti-competitive tactics, a much more severe penalty than in its settlement
with U.S. antitrust authorities.
\bigskip
%-------------------------------------------------------
\noindent
{\bf Enforcement}
The first way the antitrust laws are enforced is the same way as most federal laws are enforced:
by the {\bf Department of Justice}.\index{Department of Justice} One division of the Justice
Department is the {\bf Antitrust Division}, \index{Antitrust Division}which has a staff of
economists and lawyers ready to find and prosecute violations of the Sherman and Clayton Acts. The
Justice Department can bring both criminal and civil actions.
\begin{wrapfigure}{L}{.5\linewidth}
\noindent
\begin{boxedminipage}[c]{\linewidth}
\renewcommand{\baselinestretch}{.9}
\begin{footnotesize}
\refstepcounter{sidebarcounter} \label{sidebar05-2}
\noindent
{\sc Box \ref{chap07}.\ref{sidebar05-2} \\
A Rogue's Gallery of Price Fixers, and Regulators}\\
\includegraphics[height=.18\linewidth]{fig05BjornSjaastadOdfjell.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05brooks_2.jpg}\hfill
\includegraphics[height=.18\linewidth]{fig05davidge_2.jpeg}\hfill
\includegraphics[height=.18\linewidth]{fig05debney.jpg}\hfill
\includegraphics[height=.18\linewidth]{fig05kim.jpg}\\
\includegraphics[height=.18\linewidth]{fig05koehler.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05whitacre.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05tennant.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05taubman.jpg} \hfill
%\includegraphics[height=.18\linewidth]{fig05sommer.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05rpratt.jpg}\\
\includegraphics[height=.18\linewidth]{fig05baye-michael.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05carlto.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05farrell.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05katz_michael.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05Salinger-Michael.jpg}
\end{footnotesize}
\end{boxedminipage}
\end{wrapfigure}
Violations of the Sherman Act can be prosecuted as {\bf criminal violations}\index{criminal
violations}. This means that the accused must be found guilty ``beyond a reasonable doubt,'' and
may be punished with prison time of up to 20 years (since Sarbanes-Oxley raised the limit in 2002)
as well as fines. The {\bf reasonable doubt standard}\index{reasonable doubt standard} is a high
hurdle. The government may prefer to bring the case as a {\bf civil action}\index{civil action}.\footnote{Examples of complaints by the Justice Department and the FTC are:
\href{http://www.jdsupra.com/legalnews/complaint-in-us-v-apple-and-hachette-e-26600/}{ ``Complaint: United States v. Apple Inc., Hachette Book Group, et al.''
United States District Court for the Southern District of New York'' (April 12, 2011) } and
\href{http://www.ftc.gov/os/caselist/0610197/complaint.pdf}{``TC Group-Riverstone-Carlyle-Kinder Morgan Complaint''} (2007).}
There, the firm or person must be found to have done something illegal {\bf by the preponderance
of the evidence}\index{by the preponderance of the evidence}; that is, more likely than not. A
civil action can result in fines, and also in orders from the court called {\bf injunctions}
\index{injunctions}that can call for the firm to have special government oversight, to cease some
business practice, or even to split up into smaller firms. Or, the government and the firm may
settle a suit by a {\bf consent decree}\index{consent decree}, an agreement enforced by the court
under which the firm agrees to behave in a certain way.
The other agency that enforces antitrust law is the {\bf Federal Trade Commission}\index{Federal
Trade Commission} or FTC, which was created in 1914 at the same time as the Clayton Act was passed.
The FTC only brings civil cases, not criminal cases. If it brings a complaint, the complaint is
judged by an FTC hearing overseen by an {\bf administrative law judge} \index{administrative law
judge}from within the FTC who recommends a course of action to the Commission's five members at the
end of the hearing. The Commission then votes on a decision similar to a court injunction.
Besides the two agencies, private parties can enforce the antitrust laws through civil actions. The
private party, typically a customer or a competitor of the company charged with wrongdoing, must
show that it has been harmed, and by how much. Ordinarily if someone is found liable in a civil suit
he must pay the party who brought the suit (the plaintiff) the damage the court decides he caused.
(Punitive damages are possible, but unusual.) The antitrust laws, however, say that private
parties can collect {\bf treble damages}:\index{treble damages} that is, if the plaintiff can prove
he lost \$1 million dollars due to the defendant's monopolizing practices, he can collect \$3
million from the defendant.\footnote{On the other hand, treble damages aren't always high. A jury
agreed that the National Football League (NFL) did ``willfully acquire or maintain monopoly
power,'' but estimated the resulting damage to the rival United States Football League at only a
nominal \$1 (it would have been in financial trouble anyway). When trebled, that came to a measly
\$3 award. \href{http://sportsillustrated.cnn.com/vault/article/magazine/MAG1065099/index.htm}
{``The Award Was Only Token:
The USFL Proved Part of Its Case against the NFL only to See the Jury Throw the Winners for a
Loss,''}, Craig Neff, {\it Sports Illustrated} (August 11, 1986).} Defendants are rarely sentenced
to prison time in antitrust cases and criminal and civil fines are low relative to the potential
gains from violations, so the treble damages from private suits are perhaps the biggest punishment
violators face. Often, private parties can wait until the government has proved a case and then
pile on with private suits, much of the work having been done for them.
Who can sue for damages is an interesting legal question. The U.S. Supreme Court ruled in {\it
Hanover Shoe, Inc. v. United Shoe Machinery Corp.} (392 U.S. 481 [1968]) that the direct purchaser
of the overpriced product can sue for the price difference even if it resold the product and passed
the cost on to its own customers. Thus, if a cartel of plastic companies sold plastic to bottle
companies which then raised their prices to soft drink companies, the bottle companies could sue for
the entire plastic overcharge. What about the indirect purchasers, the soft drink companies? The
U.S. Supreme Court ruled in {\it Illinois Brick Co. v. Illinois} (431 U.S. 720 [1977]) that only
the direct purchasers could sue, since otherwise there would be duplicative suits and it would be
hard to track down all the indirect purchasers. Many states, however, have passed ``Illinois Brick
repealer'' statutes that allow the indirect purchasers to sue in state court.
\footnote{See Dennis W. Carlton, , \index{Carlton}\href{http://www.jstor.org/stable/30033739}
{``Does Antitrust Need To Be Modernized?''} {\it The Journal of Economic Perspectives}, 21(3):
155--176 (Summer 2007); Edward D. Cavanagh,\index{Cavanagh}\href{http://heinonline.org/HOL/Page?handle=hein.journals/lyclr17&div=8&g_sent=1&collection=journals}{ ``Illinois Brick: A Look Back
and a Look Ahead,''} {\it Loyola Consumer Law Review}, 17: 1--51 (2004).}
Why have this threefold enforcement? We must go back to government failure and the principal-
agent problem. Each of these three enforcers--- the Justice Department, the FTC, and private
parties--- has different incentives. The Justice Department is headed by the Attorney-
General\index{Attorney-General}, who is appointed by the President and who can be fired by him.
Thus, the Justice Department is subject to political influence. This is not an entirely bad thing,
because the President has, after all, been elected to execute the laws in a certain way, so if his
degree of enforcement is more severe or more lax, that is the result of voter choice. There is
potential, however, for favoritism in prosecutions.
Thus, when Congress established the Federal Trade Commission in 1914, they set it up as an {\bf
independent agency}\index{independent agency}. What this means is that although its five members are
appointed by the President, they cannot be fired by him, and their seven-year terms are staggered so
that when the Presidency changes parties, the President inherits the old Commission members
appointed by his predecessor. He must wait for their terms to end before he can appoint his own
people. Moreover, no more than three of the five members may be of the same political party.
If the Justice Department decides not to investigate a tying arrangement or challenge a merger,
the FTC can do so instead, and vice versa. Moreover, if both of them are too reluctant to enforce
the laws, private parties can bring treble-damage suits. Thus, the overlapping authority can
overcome the problem of one agency, an agent for the voters as principal, is slack in its effort.
In practice there do not seem to have been important policy differences between the FTC and the
Justice Department. They cooperate well, and they split investigation of proposed mergers so each
can specialize in certain industries rather than duplicating effort.\footnote{For more on the history and politics and anti-trust regulation, see
\href{http://www.nber.org/chapters/c12565.pdf}{``Antitrust and Regulation,''}
Dennis W. Carlton \& Randal C. Picker, a chapter in {\it Economic Regulation and Its Reform: What
Have We Learned?} National Bureau of Economic Research (2011).}
\bigskip
\noindent
{\bf \ref{chap07}.5: Mergers} \label{s06.2}
Certain business action might be either
surplus-increasing or surplus-reducing, and so are subject to the ``rule of reason'' in antitrust
law.
Section 1 of the Sherman Act makes illegal ``every contract, combination in the
form of trust or otherwise, or conspiracy, in restraint of trade,'' while Section 2 makes it illegal
to
``monopolize, or attempt to monopolize, or combine or conspire with any other person or persons, to
monopolize.''
Not only are there lots
of monopolizing practices, but for each one we could talk about several surplus-increasing
motivations and
several surplus-reducing motivations and how common each of these is. This makes
monopolizing a fun and important area for academic researchers, but it runs into diminishing
returns. Prosecutors are not eager to bring complicated cases, and judges do not look forward to
having to decide them, since the cases turn on difficult arguments being explained by expert
witnesses using mathematics and game theory. This is not laziness, but rationality; prosecutors and
courts have limited resources and if they can deal with five price-fixing cases in the time
required to analyze one predatory pricing case, they should do it--- particularly since they might
end up punishing someone for predatory pricing when he actually was engaged in surplus-increasing
severe price competition. In practice, by far the most important monopolizing practice regulated
is merger, since unlimited merger would clearly hurt competition severely and the least justifiable
mergers are easy to detect. The illegality of other practices is important, though, because they
too are sometimes punished and the mere threat of government action curtails the most abusive
behavior.
The most important suspicious practice is the merger of two or more firms. After it became
clear that the courts would use the Sherman Act to punish cartels firms turned to merger, and {\bf
the Great Merger Wave} \index{the Great Merger Wave} took place around 1900, mergers that involved a
substantial fraction of U.S. manufacturing.\footnote{See George Bittlingmayer,\index{Bittlingmayer} \href{http://heinonline.org/HOL/Page?handle=hein.journals/jlecono28&div=9&g_sent=1&collection=journals}{
``Did Antitrust Policy Cause the Great Merger Wave?'' } {\it The Journal of Law and Economics},
28: 77--118 (April 1985).} The hope was that the courts would interpret the Sherman Act to
mean that although agreements between firms to raise prices were illegal, it would still be legal to
merge the firms--- in which case, as has been explained, it is not illegal for a firm to use its
market power to raise its own prices unilaterally. In response, the courts began using the Sherman
Act against mergers in 1904 in the {\it Northern Securities}\index{Northern Securities} case and in
1914 the Clayton Act clarified that mergers could be illegal monopolization, saying, you will
recall, that:
\begin{bigquote}
\hspace*{16pt} No person \ldots shall acquire, directly or indirectly, the whole or any part of the
stock or other share capital and no person\ldots shall acquire the whole or any part of the assets
of another person \ldots where\ldots in any activity affecting commerce in any section of the
country, the effect of such acquisition may be substantially to lessen competition, or to tend to
create a monopoly.(15 U.S.C. \S 18)
\end{bigquote}
Mergers {\it can be}, not {\it are}, illegal monopolization. It is hard enough to think of surplus-
maximizing reasons for competitors to agree with each other to raise prices that the law makes such
agreements per se illegal. It is easy, though, to think of surplus-maximizing reasons for two
competitors to agree to merge. The new merged firm might have lower costs, either because of
economies of scale or because one of the firms has managers who are better at managing. On the other
hand, with fewer firms in the industry, competition falls, so the new firm might have cost curves
just the same or worse but be better able to raise prices, a bad motivation.
\bigskip
\noindent
{\bf Concentration Levels}
Since the number of firms matters to the amount of competition, it is useful to quantify the {\bf concentration}
\index{concentration} of a market: how few firms sell in it.
A simple measure is the {\bf four-firm concentration ratio}\index{four-firm concentration ratio}:
the percentage of market sales by the top four firms. Thus, if the firms in an industry have market
shares of 30\%, 25\%, 20\%, 15\%, 5\%, 4\%, and 1\%, the four-firm concentration ratio is
90\%. That is rather crude, since it doesn't care about how evenly the top four firms split the
market. The same level would be reached by an industry in which one firm had an 87\% market shares
and 13 other firms had shares of 1\% each.
\begin{footnotesize}
\vspace*{24pt}
\noindent
\begin{minipage}[c]{.8\linewidth}
\refstepcounter{tablecounter} \label{tab05-05}
\begin{center}
{\sc Table \ref{chap07}.\ref{tab05-05} \\
Concentration Ratios }\\
\bigskip
\begin{tabular}{lr | r| rrrr | r }
\hline
\hline
& & & & & & & \\
Industry & NAICS & Estab's & \multicolumn{4}{|c|}{No. of Firms } & Herf. \\
& Code & & 4 & 8 &20 & 50 & Index\\
& & & & & & & \\
\hline
& & & & & & & \\
Food Manufacturing& 311 & 21,355 & 14 &22&37& 50 &102\\
Sugar and Confectionery & 3113 & 1,631 &37 &49&66& 83 & 667 \\
Product Manufacturing & & & & & & & \\
Sugar Manufacturing& 31131 & 37 &59 &78&96& 100 &904\\
Beet Sugar Manufacturing & 311313 & 12 & 81 &98&100& 100 &x\\
& & & & & & & \\
\hline
& & & & & & & \\
Dog and Cat Food & 31111& 199 & 71 & 83&92& 98& 2,325 \\
Manufacturing & & & & & & & \\
Retail Bakeries &311812 & 6,101& 3 &6 &9&18 & 7 \\
Accommodation and & 72 & 634,361 & 5 & 10&17 & 23 &x \\
Food Services & & & & & & & \\
Hotels (except casino hotels) & 72111 & 48,108 &22 &28 & 35& 42 &x\\
and motels & & & & & & & \\
National Commercial & 5221101 & 46,809 & 55 &69 &83 & 90&x \\
Banks & & & & & & & \\
& & & & & & & \\
\hline
\hline
\end{tabular}\\
\end{center}
{\it Notes:} \url{http://factfinder.census.gov/}. One company can have many establishments, e.g. Hilton hotels. For some industries
the Herfindahl Index is not published.
\end{minipage}
\end{footnotesize}
\vspace*{24pt}
A more sophisticated approach is the {\bf Herfindahl Index}\index{Herfindahl
Index}:\footnote{The government and others call this the ``{\bf Herfindahl-Hirschman Index}
\index{Herfindahl-Hirschman Index}'' or ``HHI'' in documents, but that's too long a name, and in
common practice government people and economists just say ``Herfindahl Index''--- Too bad for
Professor Hirschman, a more important scholar than Herfindahl, but better for everyone else.}
The Herfindahl is calculated by summing the squares of the individual firms' market shares, and
thus gives proportionately greater weight to the larger market shares. Table
\ref{chap07}.\ref{tab05-09} shows how that works out. Table \ref{chap07}.\ref{tab05-05} also has
examples. When using the Herfindahl, the agencies consider both the post-merger level of the
Herfindahl and the increase in the Herfindahl resulting from the merger. The increase in the
Herfindahl is equal to twice the product of the market shares of the merging firms. \footnote{The
Herfindahl Index does have a theoretical justification. If the firms are Cournot competitors with
different marginal costs, they will have different market shares. The Herfindahl Index equals the
weighted average of their market shares multiplied by the industry elasticity of demand (and
multiplied by -10,000). }
\begin{center}
\begin{minipage}[c]{ \linewidth}
\begin{center}
\begin{footnotesize}
\refstepcounter{tablecounter} \label{tab05-09}
{\sc Table \ref{chap07}.\ref{tab05-09} \\
Herfindahl Examples} \\
\bigskip
\begin{tabular}{llr}
\hline
\hline
& & \\
Industry & Computation & Herfindahl Index\\
& & \\
\hline
& & \\
Entirely monopolized &(100*100) & 10,000 \\
& & \\
One firm has 70\% of sales and the other has 30\% & (70*70+30*30) & 5,800 \\
& & \\
Two firms split the market evenly &(50*50 + 50*50) &5,000\\
& & \\
One firm has 50\% and 50 others have 1\% each & (50*50+ 50*1*1) &2,550 \\
& & \\
Five firms split the market evenly & (5*(20*20)) &2,000\\
& & \\
Ten firms split the market evenly &(10*(10*10)) & 1,000 \\
& & \\
100 firms split the market evenly & (100*(1*1)) & 100 \\
& & \\
\hline
\hline
\end{tabular}
\end{footnotesize}
\end{center}
\end{minipage}
\end{center}
Thus, antitrust law allows mergers, but is suspicious of them. Federal law requires any two large
firms that wish to merge to submit information to the FTC and the Justice Department on the likely
effects on price. If the FTC or Justice object to the merger, they can try to block it.
Since the statute is not clear on what it means for a merger to substantially reduce competition,
the agencies have to decide what mergers to block and the courts have to decide whether the agencies
are interpreting the law correctly. Companies would like to know which mergers are going to be
allowed, since setting up a merger only to have it blocked by the FTC results in a lot of wasted
effort. Thus, the FTC and Justice have issued regulations declaring what kinds of mergers are
likely to be blocked. A regulation called ``Horizontal Merger Guidelines'' was issued in 2010 for
{\bf horizontal mergers}\index{horizontal mergers}, which are mergers between firms selling
competing products.\footnote{U.S. Department of Justice and the Federal Trade Commission,
\url{http://www.justice.gov/atr/public/guidelines/hmg-2010.html} {``Horizontal Merger
Guidelines,''} (August 9, 2010).} An example would be a merger between Ford Motor Company and
Toyota, which both sell cars. Another memo, ``Non-Horizontal Merger Guidelines,'' not revised
since 1984, covers vertical and conglomerate mergers. \footnote{U.S. Department of Justice,
\url{http://www.justice.gov/atr/public/guidelines/2614.htm}{``Non-Horizontal Merger Guidelines,''}
(June 14, 1984). } {\bf Vertical mergers} \index{Vertical mergers} merge two companies when one
company (the {\bf upstream firm}\index{upstream firm}) sells to another company (the {\bf downstream
firm}\index{downstream firm} which in turn sells to consumers. An example would be a merger of
U.S. Steel and with the Ford Motor Company. {\bf Conglomerate mergers} \index{Conglomerate
mergers} merge companies that sell unrelated products, such as a merger of the Ford Motor Company
with Pepsi. Horizontal mergers generally reduce competition, and the main question is whether
they substantially reduce it. Conglomerate mergers do not reduce competition, since the two firms
are in different markets and market concentration is unchanged by the merger. Vertical mergers are
the most complicated to analyze, since they can reduce competition but in more subtle
ways.\footnote{ One's first thought is that once the firms merge vertically, the upstream firm will
stop selling to competitors of the downstream firm, so downstream competition will be reduced. The
flaw in that reasoning is that if the upstream firm found it profitable to sell to all the
downstream firms before the merger, it will still be profitable afterwards, so trying to stifle
competition there would actually hurt the merged firm's profits. There are counter-arguments, but
they would take too long to explain.}
The {\it 2010 Horizontal Guidelines}\index{ 2010 Horizontal Guidelines} try to inform businesses
what mergers will be allowed without tying the hands of the antitrust agencies. The {\it
Guidelines} talk about objective criteria for market concentration, for example, but they are also
careful to say that if the Justice Department finds a memo saying ``Of course, the main reason why
we're merging is so we can increase prices and gouge consumers,'' the merger can be declared
illegal. The {\it Guidelines} are written for companies trying to comply with the law in good faith.
An unprincipled firm cannot say, ``But the guidelines didn't say I couldn't do X!'' and be excused.
The {\it Guidelines} establish that what the agencies really care about is whether the merger will
raise prices, which is what economists focus on.
Earlier antitrust policy focussed on a more legalistic definition of the market, trying to use a
more objective and predictable measure of how concentrated the market was. An example of such an
approach (not a real example--- just a simple one for discussion) would be to say that two firms
can merge if and only if they wouldn't have over 30\% of the market after the merger. This
rule would make for neither good economics nor good law.
The economic problem is that such a simple rule ignores whether the merger really affects market
power or not. In one market, two firms with market shares of 16\% would be forbidden to merge even
though neither could raise price above cost because entry by new competitors is easy. In another
market, two firms with market shares of 14\% would be allowed to merger even though there
exists only one other firm in the market (with the remaining 72\% of sales) so a triopoly would be
converted to a duopoly.
The legal problem is that the rule is not actually so simple. It depends completely on how one
defines ``market''. Lawyers can spend endless hours arguing about that. In the Cellophane Case,
was Dupont selling in the market for ``cellophane'' (of which it had a 75\% market share) or
``flexible packaging material'' (of which cellophane was only 20\%). {\it U.S. v. E. I. du Pont},
351 U.S. 377 (1956). In the ReaLemon Case, was Borden selling in the market for lemon juice
in plastic lemons or did the market include natural lemons too? (are ReaLemons like real lemons?)
{\it Borden. v. Federal Trade Commission},
{674 F.2d 498} (6th
Cir. 1982). In the 2000's, were Whole Foods and Wild Oats competing with ``premium, natural, and
organic supermarkets'' or with
``grocery stores and supermarkets''? {\it FTC v. Whole Foods Markets, Inc.},
{
533 F.3d 869} (D.C. Cir. 2008).
The {\it Guidelines} use a different approach to decide what group of products makes up ``a
market'' when trying to figure out how many firms currently are in the market of the proposed
merger. The
{\bf Hypothetical Monopolist Test}\index{Hypothetical Monopolist Test} asks whether if a single
firm was the only seller of a group of products, it ``likely would impose at least a small but
significant and non-transitory increase in price ({\bf ``SSNIP''})\index{SSNIP} on at least one
product in the market, including at least one product sold by one of the merging firms.'' A ten-
percent increase counts as significant. So, perhaps would be a 4\% increase. The Guidelines are
vague on that point.
The {\it Guidelines} make use of the Herfindahl Index as a way to screen mergers. They say:
\begin{bigquote}
\hspace*{16pt}The Agencies generally classify markets into three types:
\begin{itemize}
\item Unconcentrated Markets: HHI below 1,500
\item Moderately Concentrated Markets: HHI between 1,500 and 2,500
\item Highly Concentrated Markets: HHI above 2,500
\end{itemize}
\hspace*{16pt}The Agencies employ the following general standards for the relevant markets they have
defined:
\begin{itemize}
\item Small Change in Concentration: Mergers involving an increase in the HHI of less than 100
points are unlikely to have adverse competitive effects and ordinarily require no further analysis.
\item Unconcentrated Markets: Mergers resulting in unconcentrated markets are unlikely to have
adverse competitive effects and ordinarily require no further analysis.
\item Moderately Concentrated Markets: Mergers resulting in moderately concentrated markets that
involve an increase in the HHI of more than 100 points potentially raise significant competitive
concerns and often warrant scrutiny.
\item Highly Concentrated Markets: Mergers resulting in highly concentrated markets that involve
an increase in the HHI of between 100 points and 200 points potentially raise significant
competitive concerns and often warrant scrutiny. Mergers resulting in highly concentrated markets
that involve an increase in the HHI of more than 200 points will be presumed to be likely to enhance
market power. The presumption may be rebutted by persuasive evidence showing that the merger is
unlikely to enhance market power.
\end{itemize}
\end{bigquote}
%\end{minipage}
Consider an example.
One firm has 50\% and 50 others have 1\% each.
The Herfindahl is 2,550 (50*50+ 50*1*1) so the industry is {\it concentrated}.
Can 12 of the small firms merge?
They would contribute 144 to the Herfindahl instead of 12,
an increase of 132.
Such a merger {\it often warrants scrutiny} because it is in a {\it concentrated industry} even tho
it is small.
Concentration measures are not really what we're worried about, but they are numbers, and
relatively objective numbers, and that tends to focus the attention of policymakers, lawyers,
judges, and people generally. The subjective element gets shifted, but it doesn't disappear. In fact
it is crucial. Where it goes is into the definition of ``market'' that you use to measure the market
shares that go into the Herfindahl Index. If Microsoft's market is ``software'', its market
share is moderate. If the market is ``operating systems'' its market share is huge. If the
market is ``perfectly Windows-compatible operating systems'', it is an absolute monopoly.
Despite all of the details on measuring concentration,
the {\it Guidelines} say not to rely too much on the Herfindahl Index:
\begin{bigquote}
The purpose of these thresholds is not to provide a rigid screen to separate competitively benign
mergers from anticompetitive ones, although high levels of concentration do raise concerns. Rather,
they provide one way to identify some mergers unlikely to raise competitive concerns and some others
for which it is particularly important to examine whether other competitive factors confirm,
reinforce, or counteract the potentially harmful effects of increased concentration.
\end{bigquote}
%-------------------------------------------------------
\bigskip
\noindent
{\bf The Staples-Office Depot Merger}
When Staples and Office Depot proposed to merge in 1997, they presented evidence to the FTC and
the Justice Department that their costs would fall enough to increase total surplus. The two firms
were by far the biggest office supply firms, though, so it was clear their market power would
increase. The companies argued that the market should include anybody selling office supplies,
including big-box companies like Wal-Mart and K-Mart, but this case was not decided by looking at
concentration ratios.\footnote{See Harrington et al,; and Federal Trade Commission,
\url{http://www.ftc.gov/opa/1997/04/stapdep.shtm}{``FTC Rejects Proposed Settlement in
Staples/Office Depot Merger:
Says Deal Would Still Violate Antitrust Laws and Lead to Higher Prices for Office Supplies,''}
(April 4, 1997); John Broder, \href{http://www.nytimes.com/1997/07/01/business/office-depot-and-staples-merger-halted.html}{ ``Office Depot and Staples Merger Halted,''} {\it The New York Times}
(July 1, 1997). }
The companies said their joint costs would fall. The Justice Department let the FTC decide what to
do, and the FTC was skeptical about how big the cost savings would be. The FTC said that average
cost would fall by 1.4\% and the merger would raise prices by 7.3\%. Staples
said that if there weren't any cost savings at all, prices would rise by 2.4\%, but there
would be cost savings, so the ultimate effect would be a 2.2\% fall in prices.
What about total surplus, though? Obviously, if Staples were right it would rise because both
consumer and producer surplus would rise. But what if the FTC was right?
Staples and the FTC went to court to decide whether the merger was legal. The judge ruled that it
was not. He ruled that total surplus was not relevant, only consumer surplus, and that the merger
would raise prices and so would violate the Clayton Act. Indeed, Professor Carlton reports that only
New Zealand and Canada have antitrust laws which hold total surplus to be the goal, not consumer
surplus.\footnote{See p.\ 157 of Carlton,\index{Carlton} Dennis W.,
\href{http://www.jstor.org/stable/30033739}{``Does Antitrust Need to Be Modernized?''} {\it The
Journal of Economic Perspectives}, 21: 155--176 (Summer 2007). }
Thus, companies that want to merge have to show not only that their costs would fall but that
costs would fall enough that even consumers would benefit. Most mergers can pass this test, but not
all, as the Staples case shows.
%-------------------------------------------------------
\bigskip
%-------------------------------------------------------
\noindent
{\bf \ref{chap07}.6: Conclusion} \label{s06.5}
The theme in this chapter has been that there are business practices such as merging with another
firm that sometimes have good, surplus-increasing motivations and sometimes have bad,
surplus-reducing (but producer-surplus-{\it increasing}) motivations. The enforcement agencies
and the judges have to decide which is the real motive in a given situation. Knowing some economics
is clearly helpful to figuring that out. A major purpose of antitrust law is to deter egregiously
bad behavior, but to the extent that the law is successful, we would not actually observe that bad
behavior, and in fact we do not. Cartels can be kept secret and still work, but mergers and
predatory pricing cannot, so the cases most commonly brought involve secret price cutting.
Antitrust law addresses the market failure caused by market power. It does not address high
monopoly prices directly. Having a monopoly is legal, and so is charging high prices using market
power. Because of government failure, it would be imprudent for the antitrust authorities to try to
regulate prices or to break up companies simply because they are large and successful. Rather,
antitrust law addresses the creation of market power by means which do not create surplus, only
redistribute it. In this chapter, we have looked at the behavior of firms with market power, whether
in isolation as a monopoly or in strategic interactions with each other as a duopoly or oligopoly.
We've also looked at how antitrust laws are enforced and at its first priority: preventing price
fixing. In the next chapter we will look at other monopoly-creating behaviors such as mergers and
predatory pricing.
%-----------------------------
%\vfill
%\cleardoublepage
\begin{center}
{\sc Review Questions }
\end{center}
\begin{enumerate}
\item How does a firm with market power choose price and output to maximize
profits?
\item How does the Prisoner's Dilemma describe the incentive problem of cartels?
\item What is the Cournot Model and how does one solve for its equilibrium price?
\item
Under what conditions are mergers illegal in the United States?
\item How would you decide whether a merger would increase market power too
much?
\item How has the law treated cartels in the 19th century and in the present?
\item How do the Sherman Act and the Clayton Act differ?
\item How is antitrust law enforced in Europe and in the United States?
\end{enumerate}
%\newpage
% \vspace{1in}
%\cleardoublepage
\begin{center}
{\sc Readings }
\end{center}
\begin{enumerate}
\item
\href{https://www.law360.com/articles/875833/print?section=competition} {``\$54B Deal Will Spawn Daunting Anthem-Cigna, Expert Says,''} {\it Law360}.
\item
\href{http://www.wsj.com/articles/eu-to-fine-truck-makers-over-price-fixing-and-other-collusion-1468861361} { `` EU To Fine Truck Makers over Price-Fixing and Other Collusion:
Total Fines of Around \$3.3 Billion after a Probe by Antitrust Authorities,'' } {\it The Wall Street Journal}.
\item
\href{http://www.bbc.com/news/business-35028380} { ``Canada's Maple Syrup 'Rebels' ,''} {\it The BBC}.
\item
\href{http://www.economist.com/news/finance-and-economics/21568364-how-antitrust-economists-are-getting-better-spotting-cartels-scam-busters}{``The Scam Busters:
How Antitrust Economists Are Getting Better at Spotting Cartels,''} {\it The Economist}.
\item
\href{https://www.washingtonpost.com/opinions/george-will-a-strike-against-rent-seeking/2014/12/31/ba5a1686-9109-11e4-ba53-a477d66580ed_story.html?utm_term=.dd6d67cc1820}{``A Strike against Rent-Seeking," } George Will, {\it The Washington Post}.
\end{enumerate}
\end{document}
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\begin{document}
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\title{\textbf{COVID -19: Emerging out of the Pandemic Situation }}
\author{Sakshi~~Kumeriya\textsuperscript{1},
Bharat~Rathi\textsuperscript{*}\textsuperscript{2},
Renu~Rathi\textsuperscript{3},
Mujahid~Khan\textsuperscript{1}~\\[5pt]{\textsuperscript{1}Mahatma Gandhi Ayurved College, Hospital \& Research Centre, Datta Meghe Institute of Medical Sciences (Deemed to be University), Sawangi(M), Wardha {\textendash} 442107, Maharashtra, India}~\\{\textsuperscript{2}Department of Rasashastra \& Bhaishajya Kalpana, Mahatma Gandhi Ayurveda College Hospital \& Research Centre, Datta Meghe Institute of Medical Sciences (Deemed to be University), Sawangi(M), Wardha {\textendash} 442107, Maharashtra, India}~\\{\textsuperscript{3}Department of Kaumarbhritya, Mahatma Gandhi Ayurveda College Hospital \& Research Centre, Datta Meghe Institute of Medical Sciences (Deemed to be University), Sawangi(M), Wardha {\textendash} 442107, Maharashtra, India}}
\begin{abstract}
The novel SARS-CoV-2 Coronavirus that emerged in the city of Wuhan, China, last year and spread all over the world countries and caused the worst ever pandemic situation. This deadly virus held the entire world to be in high alert. Considering its seriousness, the World Health Organization declared a public health emergency of international concern. Govt. of India and its all allied organizations are working together and trying hard to control it and avoid the situation of community transmission. Nationwide lockdown helped to control the transmission but after unlock it transmitted speedily in the community, which is the alarming sign for everyone. In this review, the attempt is made to focus on published research articles on coronavirus disease, mode of transmission and disinfection measures, the current state of research, treatment protocol so those common people can understand its gravity and follow the measures stringently.
\end{abstract}\def\keywordstitle{Keywords}
\begin{keywords}Coronavirus outbreak,\newline COVID-19,\newline virus transmission,\newline treatment protocol
\end{keywords}
\twocolumn[ \maketitle {\printKwdAbsBox}]
\makeatletter\textsuperscript{*}Corresponding Author\par Name:\ Bharat~Rathi~\\ Phone:\ +91- 9011058301~\\ Email:\ [email protected]
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\section{Introduction}
Coronaviruses are a group of viruses responsible for developing a range of illnesses from the common cold to more severe ailments such as Severe Acute Respiratory Syndrome (SARS) and the Middle East Respiratory Syndrome (MERS) which leads to severe intestinal and respiratory illnesses. In the beginning, animals were the major source for the growth and spread of coronaviruses, but later 2003 onwards, this fact has been changed and viruses infected humans and spread from human to human which turned into a global pandemic within a small period of time \unskip~\citep{943902:20858358}.
Center for Disease and Control Prevention has identified the most common human coronaviruses, HKU1, OC43, NL63 and 229E \unskip~\citep{943902:20858357}. In the same series coronavirus disease (COVID-19), first identified in Wuhan city of China is caused by the recently discovered coronavirus serotype-2 which is zoonotic in nature but now spreading human to human through airborne droplets all over the world causing thousands of lives and global concern \unskip~\citep{943902:20858354}. According to the World Health Organization (WHO) situation information, the worldwide distribution of COVID-19 is intensifying each and every day. According to the data updated on 8\ensuremath{^{th}} Aug 2020, the confirmed cases of COVID 19 have reached whooping 2 Crore with 7 lakhs deaths throughout the world. WHO classified the threat of COVID19 transmission as very high at a global level \unskip~\citep{943902:20858343}. The incubation period of the disease is expected to be between 2 to 15 days, and its transmission from asymptomatic cases has been reported \unskip~\citep{943902:20858350}. As the screening and detection methods have grown up in most of the countries, more COVID 19 cases are predicted to be detected in the days ahead and death rate may swell up from these infected individuals.
In such a pandemic situation, it's become compulsory for each and every individual to know what is right and what is wrong and how to come out of this pandemic situation. Need to adopt strict disinfection measures to avoid further transmission of the disease and save lives. Hence in this article attempt is made to know and aware about the disinfection measures to be adopted by the common people, mode of transmission, the current state of research, treatment protocol so those common people can understand its gravity and follow the measures stringently.
\section{Materials and Methods}
In the present review relevant references and published scientific research papers related to COVID-19 were searched on Pubmed, Scopus, Web of science, Medknow, Ayushdhara and Google scholar by using relevant keywords such as Novel Coronavirus, COVID-19, corona viral outbreak etc. All the references were reviewed, compiled, analyzed and discussed thoroughly for the in-depth understanding of the concept of what is right and what wrong measures for COVID-19 pandemic.
\textbf{Observation \& Results}
The outcome from this review reveals that the COVID-19 transmission is rapidly occurring in human throughout the globe resulting in severe casualties. Scientist all over the globe is involved in finding its mutation risks source of virus spread which is yet to be identified. Detail source of the virus, epidemiology and transmission enforced a big challenge, which emphasizes the need for advance studies in the near future.
\textbf{Clinical and Epidemiological Characteristics features}
Huang \textit{et al}. from China tried to comprehend the disease characteristics and outcomes. For this, they collected data of 41 confirmed patients of COVID-19 and analyzed it. They were surprised to know that many of the earlier identified cases were associated with the animal market and Huanan seafood situated in Wuhan, which gives a confirm clue of connection of the new virus with an animal. They also noticed the resemblance between the SARS and COVID-19 as both diseases leads to fetal pneumonia. But still, the pathophysiology of the new disease remains unknown. The initial study revealed that most of the infected patients were males above 60 yrs of age and quite often visited the seafood market or in contact with a person suffering from respiratory problems. \unskip~\citep{943902:20858345}
\textbf{Case Identifications and mode of Virus Transmission }
Various measures have been suggested for the clinical assessment of the signs and symptoms of COVID-19. Naso-pharyngeal and Oro-pharyngeal swabs, laboratory investigations and radiological tests are few suggested measures to confirm the diagnosis and other allied complications. Phan observed the possible transmission of the virus from human to human within the family members who were having the travel history from red zone area to green zone area and stayed with other family members or came in contact with other individuals \unskip~\citep{943902:20858344}.
\textbf{Coronavirus outbreak as an international emergency}
Till 23\ensuremath{^{rd}} of January, the occurrence and spread of COVID-19 cases was limited to China only and not well-thought-out as Public Health Emergency. Till that date, it was considered that this disease leads to symptoms of severe respiratory illness and nearly five hundred cases were declared confirm in China and neighbouring countries like Thailand, South Korea and Japan. Considering its further spread in other provinces, China Government had taken strict measures such as isolation of suspected cases, ban on travelling, public awareness about diagnosis and treatment. But despite such measures, infected cases were continued to increase day after day and turned into an epidemic \unskip~\citep{943902:20858341}. New Cases deaths were also reported among travellers in China and other countries. Considering the seriousness of the disease, on January 30\ensuremath{^{th}} 2020, World Health Organization (WHO) declared coronavirus disease as a global public health disaster of international concern, which pointed out the worldwide spread of the disease and required synchronized universal support to control the outbreak. \unskip~\citep{943902:20858359,943902:20858351}
\textbf{Measures to be adopted }
The Government heads in Wuhan, as well as other countries, have made much effort to control the virus which included imposing a nationwide ban on wildlife trade in markets, closing the animal and seafood markets and performing sanitation and disinfection of public places. Major countries like Japan, Germany, France, UK, and USA announced the suspension of the closure of airports, ban on public transportation like bus and railways to prevent further disease transmission. In India, the Government imposed order 144, which banned the gathering of 5 or more persons in public places. The Centers for Disease Control and Prevention suggested preventive measures in order to control the infection in public places and healthcare supporters. The preventive measures included to wash the hands with soap and water for at least 30 seconds or with alcohol-based hand sanitizers frequently, to wear a face mask whenever we go out of the home, encouraged work from home instead of going to the office, to avoid touching the mouth, nose and eyes with unwashed hands, to avoid touching doors, handles, to avoid the gathering in marriages, conferences, sports events or funerals. Medicated fumigation is helpful as an air purifier, germ killer-bacteriostatic, prevents vector-borne diseases. \unskip~\citep{943902:20858346,943902:20858347}. World Health Organizations and various Government organizations engaged in health promotion campaigns as a part of educating the people to minimize the COVID-19 infection transmission to others.
\textbf{COVID-19 scenario in India}
As of today, across the globe, the total tally of corona virus-positive cases has crossed whopping 2 Crores and death toll to 7.20 lakhs. Daily new cases are occurring more than 3 lakhs and daily deaths are crossed 7000. The worst affecting country is the USA where the death tally crossed to 1.63 lakhs due to coronavirus disease. In India, the total number of coronavirus cases jumped to 21 lakhs with 42000 deaths. Maharashtra proved to be worst affected state with 5 Lakhs COVID-19 cases with more than 18000 deaths followed by Tamilnadu, Andhrapradesh, Karnataka and Delhi among the most affected states. To control the spread of coronavirus, Govt. of India has declared complete lockdown across India for 4 times with essential services exempted. Later Prime minister let the decision to state governments to take the decision of lockdown according to the COVID-19 situation in the respective states. This helped the Govt. to control the situation and limited the number of COVID-19 cases.
\textbf{Treatment protocol and recovery rate }
Till today, neither any specific medicine for coronavirus infection is currently available, nor any vaccine is invented to prevent COVID-19 and the researches on vaccine development programs are going on war foot level \unskip~\citep{943902:20858342}. After the individual gets infected, the treatment and resurgence depend upon the severity of the sign and symptoms and related complications. It is very much encouraging that most of the infected cases are cured. Recovery rate in India is far better as compared to global recovery rate.
All the developing countries, including China and India, are engaged in clinical trials in order to find a satisfactory solution for COVID-19. Some compounds, including chloroquine/hydroxychloroquine, remdesivir, ritonavir, lopinavir, interferon-\ensuremath{\beta }, azithromycin, ribavirin have come out as promising alternatives to treat the COVID-19. These drugs block the virus from entering host cells, check viral replication, and ease exacerbation of the host's immune response.
\unskip~\citet{943902:20858352} patients treated with this drug have shown a significant drop in fever, progress in CT images of lung and taken a shorter time to cure as compared to parallel groups. \unskip~\citep{943902:20858348,943902:20858356}. Another anti-viral drug under trial is favilavir, which demonstrated efficacy in treating 70-patients of COVID-19 with minor side effects. For the first time, National Medical Products Administration of China has given the approval to this drug to treat COVID-19 patients. Research Labs of many pharmaceutical companies are engaged in COVID-19 vaccine development \unskip~\citep{943902:20858355}. Dept of AYUSH, Govt of India has undertaken no. of trials on traditional medicines considering their role in immunity boosting. Several measures and medicines are mentioned in Ayurveda to combat with the COVID-19 like a pandemic. Use of kitchen herbs, spices like turmeric, black pepper and herbal tea with ginger may protect from coronavirus effectively \unskip~\citep{943902:20858360,943902:20858349}
\section{Discussion}
Considering the Human-to-human coronavirus transmission, it is expected that more COVID-19 cases will be reported in the near future, which can cause serious trouble to worldwide public health organization and financial losses \unskip~\citep{943902:20858353}. Government organizations all over the world are engaged to implement strict preventive measures, and investigating the source of the disease to know more about the virus's characteristics, mode of transmission and illness severity. \unskip~\citep{943902:20858355}.
It is the need of the hour to have a better understanding of the currents updates of a new virus. It is up to the countries to provide reliable data with an open mind, as well as conducting clinical trials on the reported cases. So that an effective medicine can be invented without delay to save the lives, at the same time, countries should keep attention to work on improving the preventive measures implemented to decrease the number of COVID-19 and transmissions. Till the invention of new medicine, the vaccine is only last hope of every people for getting out from COVID{\textendash}19 pandemic. But as per experts, it has said that if any vaccine is invented now, then it nearly took 1-2 years to hit it to the market.
\section{Conclusion}
In such a pandemic situation, it's a duty of each and every individual to know what is right and what is wrong and follow the disinfection measures meticulously to avoid further transmission of the disease. Although the scientists are involved in searching the effective medicine to overcome the virus and some evidence indicated the positive results, it will take time to validate and standardize these agents to prove their therapeutic value in human beings.
\section*{Acknowledgement}Author would like to thank DMIMSU for motivating and providing all necessary help for writing this article.
\textbf{Funding Support}
DMIMS (Deemed to be University), Sawangi (Meghe), Wardha.
\textbf{Conflict of Interest}
The authors declare that they have no conflict of interest for this study.
\bibliographystyle{pharmascope_apa-custom}
\bibliography{\jobname}
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https://umbertomichieli.github.io/download/slides/slides_bachelor_template.tex | github.io | CC-MAIN-2023-14 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296949331.26/warc/CC-MAIN-20230330132508-20230330162508-00349.warc.gz | 660,210,892 | 2,152 | % Compile it with PDFLatTeX
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\title[Correlation analysis of EEG and EMG]{\textsc{Analisi della correlazione tra segnali EEG ed EMG}}
\author[Umberto Michieli]{Laureando: Umberto Michieli\\Relatore: Leonardo Badia\\Correlatrice: Giulia Cisotto}
\date[18/07/2016]{18/07/2016\\Anno accademico 2015/2016}
\institute[]{Corso di Laurea in Ingegneria dell'Informazione\\ Dipartimento di Ingegneria dell'Informazione}
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\includegraphics[scale=0.2]{images/dei_logo.pdf} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \includegraphics[scale=0.07]{images/logo_unipd.pdf}\end{flushleft}
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\begin{document}
\transduration{1}
\frame{\titlepage} %------------------------------ TITLE PAGE
\section*{Sommario}
\begin{frame} %------------------------------ OUTLINE
\transwipe[direction=0] % book-like transition
\frametitle{Sommario}
\tableofcontents
\end{frame}
\section{Obiettivo}
\begin{frame}
\transwipe[direction=0]
%\frametitle{}
EEG=Elettroencefalogramma\\
EMG=Elettromiogramma\\
APB=muscolo abduttore breve del pollice
\begin{block}{}
Gli eventi patologici \textit{burst} nell'EMG sono provocati da uno stimolo cerebrale o locale? \\ \textcolor{blu}{$\Longrightarrow$} Analisi di correlazione e coerenza tra EEG ed EMG
\end{block}
EMG APB sano, attività di fondo \quad \quad EMG APB con eventi burst
\end{frame}
\section{Conclusioni e sviluppi futuri}
\begin{frame}
\transwipe[direction=0]
\frametitle{}
\begin{block}{Conclusioni}
\begin{itemize}
\item massimo di correlazione con EEG in anticipo in media di 10 ms su EMG, confermando la fisiologia umana
\item coerenza caso sano: picchi di coerenza a 20 Hz e a frequenze inferiori
\item coerenza caso patologico: picchi di coerenza significativa a 20 e attorno a 35-40 Hz
\end{itemize}
\end{block}
\begin{block}{Sviluppi futuri}
\begin{itemize}
\item allargare lo studio per rendere le stime più robuste
\item verificare se gli eventi \textit{burst} sono volontari oppure no
\end{itemize}
\end{block}
\end{frame}
\end{document} |
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\otsake{Esimerkki 1}
Eksponenttifunktio matemaatikoiden tapaan, ts. Neperin luku kursivoituna:
Syöte \verb+\ee{2x}+, tulos $\ee{2x}$.
\otsake{Esimerkki 2}
Muissa maissa kuin Suomessa ei käytetä määrätyn integraalin
sijoitusmerkkinä isoa vinoviivaa eikä tätä ole {\LaTeX}issakaan
valmiina. Sen voi itse määritellä makroksi, jolloin käyttö on
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Syöte \verb+\sij{0}{\infty}{\frac{\sin x}{\ee{x}}}+, tulos
\[
\sij{0}{\infty}{\frac{\sin x}{\ee{x}}}
\]
\otsake{Esimerkki 3}
Kahden harjoitustehtävän tulostus käyttäen ympäristön \verb+teht+
määrittelyä, jossa näytetään tehtävien koodinumero hakasuluissa:
\begin{teht}{int53}
Laske a)~osittaisintegroinnilla ja b)~sopivaa trigonometrian kaavaa
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<p>
<strong>This is an old revision of the document!</strong>
</p>
<hr />
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\lhead{\Large \bf Keywords: Nanocopter}
\rhead{\Large \bf List of Publications}
\renewcommand{\headrulewidth}{1pt}
\begin{document}
\tabcolsep 0.5cm
\begin{center}
\end{center}
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\Large {\bf Conference and Workshop Papers}\\[-8mm]
\tabcolsep 0.4cm
\begin{center}
\begin{enumerate}[{[C}1{]}]
\normalsize
\normalsize
\item \parbox[t]{150mm}{
\rm
O. Dunkley, J. Engel, J. Sturm and D. Cremers,\\ {\bf Visual-Inertial Navigation for a Camera-Equipped 25g Nano-Quadrotor,}\\ {\em{IROS2014 Aerial Open Source Robotics Workshop}},\ 2014.}
\end{enumerate}
\end{center}
\Large {\bf MastersThesis}\\[-8mm]
\tabcolsep 0.4cm
\begin{center}
\begin{enumerate}[{[M}1{]}]
\normalsize
\normalsize
\item \parbox[t]{150mm}{
\rm
Oliver Montague Welton Dunkley,\\ {\bf Visual Inertial Control of a Nano-Quadrotor,}\\ Technical University Munich,\ Germany,\ |
http://ljk.imag.fr/Seminars/soutenances_en.tex | imag.fr | CC-MAIN-2018-47 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-47/segments/1542039748901.87/warc/CC-MAIN-20181121133036-20181121155036-00317.warc.gz | 201,436,978 | 14,794 | \subsection{LJK Academic Defenses}
\vspace{0.5cm}{\Large 2018}
\begin{itemize}
\item Jeudi 20 D\'ecembre Mr Alexandre DERVILLE
D\'eveloppement d'algorithmes de m\'etrologie d\'edi\'es \`a la caract\'erisation de nano-objets \`a partir d'informations h\'et\'erog\`enes
\item Lundi 19 Novembre Mr Luciano GERVASONI
Contributions to the formalization and implementation of spatial urban indices using open data: application to urban sprawl studies
\item Jeudi 8 Novembre Mr Alexandre BLERON
Rendu stylis\'e de sc\`enes 3D anim\'ees temps-r\'eel
\item Lundi 22 Octobre Mr Jeremy WAMBECKE
Visualisation de donn\'ees temporelles personnelles
\item Mardi 16 Octobre Mr Jocelyn MEYRON
Transport optimal semi-discret et applications en optique anidolique
\item Mardi 25 Septembre Mr Alexandre VIEIRA
Commande optimale des syst\`emes de compl\'ementarit\'e lin\'eaire
\item Mercredi 19 Septembre Mme Stefanie WUHRER
Deformation Models for Human Shape Analysis.
\item Lundi 17 Septembre Mr Duong Hung PHAM
Contributions \`a l'Analyse des Signaux Multicomposantes: Synchrosqueezing et M\'ethodes Associ\'ees
\item Mardi 3 Juillet Mr Jean Baptiste ORFILA
Evaluation de la confiance dans les architectures de s\'ecurit\'e
\item Lundi 25 Juin Mr Guillaume LOUBET
Efficient models for representing sub-pixel appearances
\item Mardi 22 Mai Mr St\'ephane LATHUILIÈRE
Mod\`eles profonds de regression et applications \`a la vision par ordinateur pour l'interaction homme-robot (Deep Regression Models and Computer Vision Applications for Multi-person Human-Robot Interaction)
\item Lundi 26 Mars Mr REMI PELLEREJ
Étude et d\'eveloppement d'algorithmes d'assimilation de donn\'ees variationnelle adapt\'es aux mod\`eles coupl\'es oc\'ean-atmosph\`ere
\item Vendredi 16 Mars Mr Laurent DOYEN (Universit\'e Grenoble Alpes)
Imperfect Maintenance: Stochastic Modelling, Statistical Inference, Software Development and Real Case Study
\item Jeudi 15 F\'evrier Mr Charles PELLETIER
Étude math\'ematique du couplage oc\'ean-atmosph\`ere incluant les \'echelles turbulentes
\item Jeudi 1 F\'evrier Mr Alexandre HOFFMAN
Docking Flexible Proteins using Polynomial Expansions
\item Jeudi 11 Janvier Mr Dmitrii OSTROVSKI
Adaptive Signal Recovery by Convex Optimization
\end{itemize}
\vspace{0.5cm}{\Large 2017}
\begin{itemize}
\item Vendredi 22 D\'ecembre Mr Chun-Hsiang TSOU
Identification d'une cible par l'\'electro-localisation
\item Mercredi 20 D\'ecembre Mr Qi XUE
Mathematical and numerical study of the inverse problem of electro-seismicity in porous media
\item Mardi 19 D\'ecembre Mr Abdelfatah GTET
Asymptotic Analysis of Plasmonic Resonances of some Metallic Structures
\item Mardi 12 D\'ecembre
\item Mardi 12 D\'ecembre Mr Kevin POLISANO
Mod\'elisation de textures anisotropes par la transform\'ee en ondelettes monog\`enes, et super-r\'esolution de lignes 2-D
\item Vendredi 8 D\'ecembre Mr Tibor STANKO
Reconstruction de surfaces lisses maill\'ees \`a partir de capteurs inertiels
\item Mercredi 6 D\'ecembre Mr Adnane BOUKHAYMA
Surface Motion Capture Animation
\item Jeudi 28 Septembre Mr J\'er\^ome LELONG (GINP)
Quelques contributions aux m\'ethodes num\'eriques probabilistes et \`a la mod\'elisation stochastique
\item Mardi 26 Septembre Mr JOSHI BIKASH
Scalable Algorithms for Large-scale Machine Learning Problems: Application to Multi-class Classification and Asynchronous Distributed Optimization.
\item Vendredi 15 Septembre Mr Ludovic METIVIER (Charg\'e de recherche, CNRS)
Development of numerical methods for seismic imaging using the full waveform
\item Vendredi 15 Septembre Mr Achmad CHOIRUDDIN
S\'election de variables pour des processus ponctuels spatiaux
\item Mardi 4 Juillet Mr Adrien MAZOYER
Mod\`eles de mutation : \'etude param\'etrique et estimation param\'etrique
\item Lundi 3 Juillet Mme Margaux VAUTHRIN
Etude de quelques mod\`eles en imagerie photoacoustique
\item Vendredi 30 Juin Mme Dinh Huong PHAM
Bases mixtes ondelettes-gaussiennes pour le calcul de structures \'electroniques
\item Vendredi 23 Juin Mr Federico PIERUCCI
Nonsmooth Optimization for Statistical Learning with Structured Matrix Regularization
\item Vendredi 7 Avril Mr Benoit ARBELOT
Transferts d'apparence en espace image bas\'es sur des propri\'et\'es texturelles
\item Lundi 3 Avril Mr Thomas CAPELLE
Recherche sur des m´ethodes d'optimisation pour la mise en place de mod`eles int ´egr´es de transport et usage des sols
\item Vendredi 31 Mars Mr Benjamin ALLAIN
: Suivi volum\'etrique de formes 3D non rigides
\item Lundi 6 F\'evrier Mr Mattis PAULIN
De l'apprentissage de repr\'esentations visuelles robustes aux invariances pour la classification et la recherche d'images
\item Mercredi 1 F\'evrier Mme Patricia TENCALIEC
Developments in statistics applied to hydrometeorology: imputation of streamflow data and semiparametric precipitation modeling
\item Vendredi 27 Janvier Mr Li WANG
Algorithms and Criteria for Volumetric Centroidal Voronoi Tessellations
\item Jeudi 26 Janvier Mr J\'er\^ome MALICK (Charg\'e de Recherche, CNRS)
Variational-analysis look at combinatorial optimization and other selected topics in optimization
\end{itemize}
\vspace{0.5cm}{\Large 2016}
\begin{itemize}
\item Jeudi 22 D\'ecembre Mr Okba HAMITOU
Pr\'econditionnement du solveur it\'eratif CARP-CG pour la r\'esolution du probl\`eme de propagation des ondes dans les milieux visco-\'elastiques
\item Lundi 19 D\'ecembre Mme Aina FRAU-PASCUAL
Mod\`eles statistiques pour l'analyse des modalit\'es d'imagerie par r\'esonance magn\'etique ASL et BOLD pour \'etudier le fonctionnement et les maladies c\'er\'ebrales.
\item Jeudi 15 D\'ecembre Mr Gilles DAVIET
Mod\`eles et algorithmes pour la simulation du contact frottant dans les mat\'eriaux complexes : application aux milieux fibreux et granulaires
\item Mercredi 14 D\'ecembre
Appearance Modelling for 4D Multi-View Representations
\item Lundi 12 D\'ecembre Mme Shreyas SAXENA
Learning representations for visual recognition
\item Vendredi 9 D\'ecembre Mr Romain HUG
Analyse math\'ematique et convergence d'un algorithme pour le transport optimal dynamique Mathematical analysis and convergence of an algorithm for optimal transport problem
\item Jeudi 8 D\'ecembre Mr Nelson FEYEUX
Transport optimal pour l'assimilation de donn\'ees images
\item Vendredi 2 D\'ecembre Mr Pierre ETORE
Quelques contributions \`a l'\'etude et \`a la simulation des diffusions asym\'etriques
\item Mardi 29 Novembre Mr Jose Eduardo MORALES
Ondes localis\'ees dans des syst\`emes m\'ecaniques discrets excitables .
\item Vendredi 25 Novembre Mme Zofia TRSTANOVA
L'analyse math\'ematique et algorithmique de la dynamique de Langevin modifi\'e
\item Jeudi 10 Novembre Mme Armelle BAUER (Universit\'e Grenoble Alpes)
Mod\'elisation anatomique utilisateur sp\'ecifique et animation temps-r\'eel. Application \`a l'apprentissage de l'anatomie
\item Lundi 24 Octobre Mme Camille SCHRECK
D\'eformation interactive de papier virtuel
\item Lundi 17 Octobre Mr Laurent GILQUIN
Echantillonnages Monte Carlo et quasi-Monte Carlo pour l'estimation des indices de Sobol'. Application \`a un mod\`ele de transport-urbanisme.
\item Lundi 3 Octobre Mr Pierre Luc MANTEAUX
Simulation et contr\^ole de ph\'enom\`enes physiques
\item Mardi 27 Septembre Mr Mehdi Pierre DAOU
D\'eveloppement d'une m\'ethodologie de couplage multimod\`ele avec changements de dimension. Validation sur un cas-test r\'ealiste.
\item Vendredi 23 Septembre Mr Philippe WEINZAEPFEL (Universit\'e Grenoble Alpes)
Motion in action: optical flow estimation and action localization in videos
\item Mardi 5 Juillet Mme Thi To Nhu DANG
Estimation des indices de stabilit\'e et d'autosimilarit\'e par variations de puissances n\'egatives
\item Mardi 28 Juin Mr Jean Yves COURTONNE
"Evaluation environnementale de territoires \`a travers l'analyse de fili\`eres - La comptabilit\'e biophysique pour l'aide \`a la d\'ecision d\'elib\'erative"
\item Vendredi 24 Juin Mme Charlotte DION (Universit\'e Grenoble Alpes)
Estimation non-param\'etrique de la densit\'e de variables al\'eatoires cach\'ees
\item Jeudi 16 Juin Mr L\'eo ALLEMAND-GIORGIS
Visualisation de champs scalaires guid\'ee par la topologie
\item Vendredi 8 Avril Mme Morgane HENRY
Transport optimal et ondelettes : nouveaux algorithmes et applications \`a l'image
\item Lundi 14 Mars Mr Mahamar DICKO (UGA)
Numerical methods for the resolution of surface PDE. Application to embryogenegis
\end{itemize}
\vspace{0.5cm}{\Large 2015}
\begin{itemize}
\item Jeudi 17 D\'ecembre Mr Laurent DEBREU (Charg\'e de Recherche, INRIA)
Mod\'elisation num\'erique de l'oc\'ean
\item Mercredi 16 D\'ecembre Mr hugo LOI (INRIA)
Synth\`ese programmable de textures vectorielles e't application \`a la cartographie
\item Lundi 14 D\'ecembre Mr Faouzi TRIKI (Universit\'e Joseph Fourier)
Nano-optics Modeling and inverse problems
\item Mercredi 9 D\'ecembre Mr Burak EKICI
Certification de programmes avec des effets calculatoires
\item Mardi 24 Novembre Mme Marianne CLAUSEL (Universit\'e Grenoble 1)
Quelques contributions \`a l'\'etude de la mod\'elisation de l'anisotropie et \`a l'estimation des propri\'et\'es de m\'emoire d'une s\'erie temporelle
\item Lundi 23 Novembre Mr Matthias RABATEL (UJF)
Mod\'elisation dynamique d'un assemblage de floes rigides
\item Lundi 23 Novembre Mme Adela BARBULESCU
G\'en\'eration de la Prosodie Audio-Visuelle pour les Acteurs Virtuels Expressifs
\item Vendredi 20 Novembre Mr Franck HETROY-WHEELER (Grenoble INP)
Segmentation and skeleton methods for digital shape understanding
\item Jeudi 15 Octobre Mr Simon NANTY (CEA Cadarache)
Quantification des incertitudes et analyse de sensibilit\'e pour codes de calcul \`a entr\'ees fonctionnelles et d\'ependantes
\item Jeudi 8 Octobre Mr Federico ZERTUCHE
Utilisation de simulateurs multi-fid\'elit\'e pour les \'etudes d'incertitudes dans les codes de caclul
\item Lundi 28 Septembre Mr Pierre-Olivier LAMARE (Universit\'e de Grenoble)
Contr\^ole de Syst\`emes Hyperboliques par Analyse Lyapunov
\item Jeudi 24 Septembre Mr Pierre-Jean MEYER
Invariance and symbolic control of cooperative systems for temperature regulation in intelligent buildings
\item Mercredi 16 Septembre Mme Ester MARIUCCI (Universit\'e Joseph Fourier)
Quelques r\'esultats d'\'equivalence asymptotique pour des exp\'eriences statistiques dans un cadre non param\'etrique
\item Mercredi 22 Juillet Mr Danila POTAPOV (INRIA GRENOBLE)
Supervised Learning Approaches for Automatic Structuring of Videos
\item Lundi 20 Juillet Mr Dan ONEATA (INRIA)
Mod\`eles robustes et efficaces pour la reconnaissance d'actions et leur localisation
\item Jeudi 16 Juillet Mr Vincent ACARY (INRIA)
"Analysis, simulation and control of nonsmooth dynamical systems"
\item Mardi 7 Juillet Mme Chlo\'e MIMEAU (Universit\'e de Grenoble)
Conception et mise en œuvre d'une m\'ethode vortex hybride-fronti\`eres immerg\'ees en milieux solides-fluides-poreux : Application au contr\^ole passif d'\'ecoulement
\item Vendredi 3 Juillet Mme Konstantina CHARMPI (Universit\'e de Grenoble)
M\'ethodes statistiques pour la fouille de donn\'ees dans les bases de donn\'ees de g\'enomique
\item Jeudi 2 Juillet Mr Martin GUAY (Universit\'e de Grenoble)
Free-form sketching of poses and movements for expressive character animation.
\item Lundi 29 Juin Mr Boris THIBERT (Ma\^itre de Conf\'erence, Universit\'e Joseph Fourier)
R\'esolution effective de probl\`emes g\'eom\'etriques non lin\'eaires
\item Vendredi 26 Juin Mr Romain CASATI (INRIA)
Quelques contributions \`a la mod\'elisation num\'erique de structures \'elanc\'ees pour l'informatique graphique
\item Mardi 17 Mars Mr Abdelaziz DJELOUAH (Universit\'e de Grenoble)
Segmentation multi-vues d'objet
\item Jeudi 12 Mars Mr Julien ANDRE (Universit\'e de Grenoble)
Mod\'elisation g\'eom\'etrique de surfaces pour des applications photom\'etriques
\item Jeudi 12 Mars Mr R\'emi BROUET (Universit\'e de Grenoble)
Interactions gestuelles multi-point et g\'eom\'etrie d\'eformable pour l'\'edition 3D sur \'ecran tactile
\item Mardi 3 F\'evrier Mr No\'e BERNABEU (Universit\'e de Grenoble)
Math\'ematiques, Sciences et technologies de l'information, Informatique
\end{itemize}
\vspace{0.5cm}{\Large 2014}
\begin{itemize}
\item Jeudi 18 D\'ecembre Mr Vineet GANDHI (Universit\'e de Grenoble)
Automatic Rush Generation with Application to Theatre Performances
\item Jeudi 18 D\'ecembre Mr Louis CUEL (Universit\'e de Grenoble)
discrete geometric inference
\item Mercredi 10 D\'ecembre Mr Arnaud EMILIEN (Universit\'e de Grenoble)
Cr\'eation int\'eractive de mondes virtuels: Combiner g\'en\'eration proc\'edurale et contr\^ole utilisateur intuitif.
\item Lundi 8 D\'ecembre Mr Abdoulaye SAMAKE (Universit\'e de Grenoble)
Large scale nonconforming domain decomposition methods
\item Jeudi 4 D\'ecembre Mr Jean-Matthieu ETANCELIN (Universit\'e de Grenoble)
Couplage de mod\`eles, algorithmes multi-\'echelles et calcul hybride
\item Vendredi 28 Novembre Mme Nadia MORSLI (Universit\'e de Grenoble)
Inf\'erence non param\'etrique pour les mod\`eles gibbsiens de processus ponctuels spatiaux
\item Mercredi 26 Novembre Mr St\'ephane VEYS (Universit\'e de Grenoble)
Un framework de calcul pour la m\'ethode des bases r\'eduites : applications \`a des probl\`emes non-lin\'eaires multiphysiques
\item Lundi 24 Novembre Mr Ali-Hamadi DICKO (INRIA)
Construction of musculoskeletal systems for anatomical simulation
\item Lundi 17 Novembre Mr Gildas MAZO (Universit\'e de Grenoble)
Construction et Estimation de copules en grande dimension
\item Lundi 17 Novembre Mr Quentin MERIGOT (CNRS)
De l'inf\'erence g\'eom\'etrique au transport optimal num\'erique
\item Mardi 4 Novembre Mme Sofia ZAOURAR (Universit\'e de Grenoble)
Optimisation convexe non-diff\'erentiable et m\'ethodes de d\'ecomposition en recherche op\'erationnelle
\item Mardi 21 Octobre Mr J\'er\'emie DEMANGE (Universit\'e de Grenoble)
Sch\'emas num\'eriques d'advection et de propagation d'ondes de gravit\'e pour les mod\`eles de circulation oc\'eanique
\item Lundi 20 Octobre Mr Simon COURTEMANCHE (INRIA)
Analyse et Simulation des Mouvements Optimaux en Escalade
\item Jeudi 16 Octobre Mme Meryam KRIT (Universit\'e de Grenoble)
Goodness-of-fit tests in reliability: Weibull distribution and imperfect maintenance models
\item Jeudi 2 Octobre Mr Pierre JOLIVET (Universit\'e de Grenoble)
M\'ethodes de d\'ecomposition de domaine. Application au calcul haute performance
\item Vendredi 26 Septembre Mr Eric HEITZ (Universit\'e de Grenoble)
Apparence multi-\'echelles pour le rendu r\'ealiste et efficace des surfaces complexes
\item Jeudi 25 Septembre Mr Kol\'e KEITA (Universit\'e de Grenoble)
Mod\'elisation math\'ematique et analyse num\'erique des mod\`eles de type Bloch pour les bo\^ites quantiques
\item Lundi 22 Septembre Mr Lukas JAKABCIN (Universit\'e de Grenoble)
Mod\'elisation, analyse et simulation num\'erique de solides combinant plasticit\'e, rupture et dissipation visqueuse.
\item Mardi 22 Juillet Mr Ramazan Gokberk CINBIS (Universit\'e de Grenoble)
Classification d'images et localisation d'objets par des m\'ethodes de type noyau de Fisher Anglais: "Fisher kernel based models for image classification and object localization"
\item Vendredi 11 Juillet Mr Vincent CHABOT (Universit\'e de Grenoble)
Étude de repr\'esentations parcimonieuses des statistiques d'erreur d'observation pour diff\'erentes m\'etriques. Application \`a l'assimilation de donn\'ees images.
\item Mardi 8 Juillet Mr Flavien BOUSSUGE (Universit\'e de Grenoble)
Idealization of CAD assemblies for FE structural analyses
\item Mardi 24 Juin Mr Cyril SOLER (Charg\'e de Recherche, INRIA)
Analyses et Mod\`eles pour la Synth\`ese d'Images
\item Vendredi 16 Mai Mr Lionel REVERET (Charg\'e de Recherche, INRIA)
“Measurements and Models for Motion Capture”
\item Mardi 4 F\'evrier Mr Pierre-Antoine BOUTTIER (Universit\'e de Grenoble)
Assimilation variationnelle de donn\'ees altim\'etriques dans le mod\`ele oc\'eanique NEMO : Exploration d el'affet des non-lin\'earit\'es dans une configuration simplifi\'ees \`a haute r\'esolution.
\item Lundi 6 Janvier Mr Zeynep AKATA (Universit\'e de Grenoble)
Analyse du contenu des images \`a grande \'echelle et une nouvelle approche de l'apprentissage Z\'ero-Shot
\end{itemize}
\vspace{0.5cm}{\Large 2013}
\begin{itemize}
\item Mardi 10 D\'ecembre Mlle Christine BAKHOUS (Universit\'e de Grenoble)
Mod\`eles d'encodage parcimonieux de l'activit\'e c\'er\'ebrale mesur\'ee par IRM fonctionnelle
\item Vendredi 6 D\'ecembre Mr C\'edric ZANNI (Universit\'e de Grenoble)
Mod\'elisation implicite par squelette et Applications
\item Jeudi 28 Novembre Mlle Ma\"elle NODET (UJF)
Probl\`emes inverses pour l’environnement : outils, m\'ethodes et applications
\item Mardi 26 Novembre Mr Antoine DELEFORGE (Universit\'e de Grenoble)
"Projection d’espaces acoustiques: une approche par apprentissage automatis\'e de la s\'eparation et de la localisation de sources sonores"
\item Mardi 19 Novembre Mr Antoine GIRARD (UJF)
Approches computationnelles pour l'analyse et le contr\^ole des syst\`emes hybrides
\item Vendredi 15 Novembre Mr Bertrand BONAN (Universit\'e de Grenoble)
Assimilation de donn\'ees pour l'initialisation et l'estimation de param\`etres d'un mod\`ele d'\'evolution de calotte polaire
\item Jeudi 7 Novembre Mr Alexandre DEROUET-JOURDAN (Universit\'e de Grenoble)
Inversion statique de fibres : de la g\'eom\'etrie de courbes 3d \`a l'\'equilibre d'une assembl\'ee de tiges m\'ecaniques en contact frottant
\item Lundi 4 Novembre Mr Thomas OBERLIN (Universit\'e de Grenoble)
Analyse de Signaux Multicomposantes : Contributions \`a la D\'ecomposition Modale Empirique, aux Repr\'esentations temps-fr\'equence et au Synchrosqueezing
\item Lundi 28 Octobre Mlle Manel TAYACHI (Universit\'e de Grenoble)
Couplage de mod\`eles de dimensions h\'et\'erog\`enes et application en hydrodynamique
\item Mardi 22 Octobre Mlle Madison GIACOFCI (Universit\'e de Grenoble)
Classification non supervis\'ee et s\'election de variables dans les mod\`eles mixtes fonctionnels. Applications a la biologie mol\'eculaire
\item Mercredi 9 Octobre Mr Mohamad BELOUNI (Universit\'e de Grenoble)
Plans d'exp\'erience optimaux en regression appliqu\'ee \`a la pharmacocin\'etique
\item Lundi 7 Octobre Mr Jonathan EL-METHNI (Universit\'e de Grenoble)
Contributions \`a l'estimation de quantiles extr\^emes. Applications \`a des donn\'ees environnementales
\item Lundi 23 Septembre Mr Mathieu HUARD (Universit\'e de Grenoble)
Mod\'elisation G\'eom\'etrique et Reconstruction de Formes Instrument\'ees par des Capteurs d'Orientation
\item Lundi 23 Septembre Mlle Ga\"elle CHASTAING (Universit\'e de Grenoble)
Analyse de sensibilit\'e et variables d'entr\'ee d\'ependantes
\item Lundi 8 Juillet Mr Vincent CHABANNES (Universit\'e de Grenoble)
Vers la simulation des \'ecoulements sanguins
\item Mercredi 3 Juillet Mr Adrien BERNHARDT (Universit\'e de Grenoble)
Mod\`eles pour la cr\'eation 3D interactive intuitive
\item Vendredi 14 Juin Mr Jordi SANCHEZ-RIERA (Universit\'e de Grenoble)
Capacit\'es audiovisuelles en robot humano\"ide NAO
\item Lundi 27 Mai Mr Jean-Marc GRATIEN (Universit\'e de Grenoble)
Programmation g\'en\'erative appliqu\'ee au prototypage d'Applications performantes sur des architectures massivement parall\`eles pour l'approximation volumes finis de syst\`emes physiques complexes
\item Lundi 15 Avril Mr Mohamed Amin BENSASSI (Universit\'e de Grenoble)
Analyse et Contr\^ole de Syst\`emes Dynamiques Polynomiaux
\end{itemize}
\vspace{0.5cm}{\Large 2012}
\begin{itemize}
\item Mercredi 12 D\'ecembre Mr David CHEREL (Universit\'e de Grenoble)
D\'ecomposition de domaine pour des syst\`emes issus des \'equations de Navier-Stokes
\item Lundi 3 D\'ecembre Mlle Estelle DUVEAU (Universit\'e de Grenoble)
Mesure de surface 3D pour la caract\'eris aplication ) la caract\'erisation du vieillissement chez la souris
\item Mercredi 28 Novembre Mr Samuel MARTIN (Universit\'e de Grenoble)
Coordination et robustesse des syst\`emes dynamiques multi-agents
\item Lundi 26 Novembre Mr Roland DENIS (Universit\'e de Grenoble)
Mod\'elisation et simulation de l'effet Leidenfrost dans les micro-gouttes
\item Mardi 20 Novembre Mr Visesh CHARI
Estimation de la forme d'objets sp\'eculaires \`a partir d'un syst\`eme multi-vues.
\item Lundi 19 Novembre Mr Madhi MOHAMMADBAGHER (Universit\'e de Grenoble)
Apparence Mat\'erielle : repr\'esentation et rendu photo-r\'ealiste
\item Vendredi 16 Novembre Mr Jean-Yves TISSOT (LJK / MOISE)
Soutenance de th\`ese : Sur la d\'ecomposition ANOVA et l'estimation des indices de Sobol'. Application \`a un mod\`ele d'\'ecosyst\`eme marin
\item Jeudi 15 Novembre Mr Alexandre JANON (Universit\'e de Grenoble)
Analyse de sensibilit\'e, r\'eduction de dimension. Application \`a l'oc\'eanographie
\item Mardi 30 Octobre Mr Laurent BELCOUR (Universit\'e de Grenoble)
A frequency Analysis of Light Transport, from Theory to Implementation
\item Mardi 30 Octobre Mr Avinash SHARMA (INRIA)
Repr\'esentation et enregistrement de formes visuelles 3D \`a l'aide du Laplacien de graphe et du noyau de la chaleur
\item Vendredi 26 Octobre Mr Thomas MENSINK (Universit\'e de Grenoble)
Learning Image Classification and Retrieval Models
\item Jeudi 25 Octobre Mr Adrien GAIDON (Universit\'e de Grenoble)
Structured Models for Action Recognition in Real-world Videos - Mod\`eles Structur\'es pour la Reconnaissance d'Actions dans des Vid\'eos R\'ealistes
\item Jeudi 25 Octobre Mr Guillaume BOUSQUET (Universit\'e de Grenoble)
D\'eformation et d\'ecoupe interactive de solides \`a g\'eom\'etrie complexe.
\item Vendredi 19 Octobre Mr Ma\"el BOSSON (Universit\'e de Grenoble)
Adaptative algorithms for computational chemistry and interactive modeling
\item Jeudi 18 Octobre Mr Pierre-Yves GIRES
Interaction hydrodynamique entre deux v\'esicules dans un cisaillement simple.
\item Lundi 30 Juillet Mr Antoine LETOUZEY (INRIA)
Mod\'elisation 4D \`a partir de plusieurs cam\'eras
\item Lundi 16 Juillet Mr C\'edric CAGNIART
Acquisition de surfaces d\'eformables \`a partir d'un syst\`eme multicamera calibr\'e
\item Jeudi 21 Juin Mr Brice BOYER (Universit\'e de Grenoble)
Multiplication matricielle efficace et conception logicielle pour la biblioth\`eque de calcul exact LinBox
\item Mercredi 30 Mai Mlle Svetlana ARTEMOVA (Universit\'e de Grenoble)
Algorithmes adaptatifs pour la simulation mol\'eculaire (Adaptative algorithms for molecular simulation)
\item Jeudi 24 Mai Mr Nassim JIBAI (Universit\'e de Grenoble)
Lissage Multi-\'echelle sur GPU des Images et Volumes avec Pr\'eservation des D\'etails
\item Mardi 22 Mai Mr Alexandre CONINX (Universit\'e de Grenoble)
Visualisation interactive de grands volumes de donn\'ees incertaines : pour une approche perceptive
\item Mercredi 4 Avril Mr Emmanuel PRADOS (INRIA)
Recherches en reconstruction 3D photom\'etrique
\end{itemize}
\vspace{0.5cm}{\Large 2011}
\begin{itemize}
\item Jeudi 15 D\'ecembre Mr Medhi BENALLEGUE (Universit\'e de Montpellier)
Contr\^oleur Miroir : Approche Calculatoire Bio-Inspir\'ee pour l'Imitation de mouvements humains de Marche par un Robot Humano\"ide
\item Mardi 13 D\'ecembre Mlle Lamiae AZIZI (Universit\'e de Grenoble)
Champs al\'eatoires de Markov cach\'es pour la cartographie du risque en \'epid\'emiologie
\item Mercredi 7 D\'ecembre Mr Sibt Ul HUSSAIN (Grenoble INP)
Apprentissage machine pour la d\'etection des objets
\item Jeudi 1 D\'ecembre EXPOSÉS D'ÉQUIPE : ÉQUIPE FIGAL (LJK)
Pr\'esentation du package R EBSpat pour la simulation/estimation de mod\`eles de Gibbs de type plus proches voisins. ET: Zeros complexes de la constante de normalisation pour le mod\`ele de Potts pour une famille de graphes self-duaux.
\item Vendredi 25 Novembre Mr Romain ARCILA (Universit\'e de Lyon)
S\'equence de maillages : classification et m\'ethodes de segmentation
\item Vendredi 18 Novembre
\item Lundi 14 Novembre Mr Charles DE ROUSIERS (Universit\'e de Grenoble)
Rendu Temps-R\'eel de Mod\`eles Complexes
\item Mardi 27 Septembre Mr R\'egis PERRIER (Universit\'e de Grenoble)
Estimation de l'attitude d'un satellite \`a l'aide de cam\'eras pushbroom et de capteurs stellaires
\item Mardi 27 Septembre Mr Hedi HARZALLAH (UdG)
Contribution \`a la d\'etection et \`a la reconnaissance d'objets dans les images.
\item Mardi 20 Septembre Mr Damien ROHMER
Soutenance de th\`ese: G\'eometrie active pour l'animation et la mod\'elisation
\item Mardi 12 Juillet Mr Adrien MAGNI (UdG)
M\'ethodes particulaires avec remaillage : analyse num\'erique, nouveaux sch\'emas et applications pour la simulation d'\'equations de transport.
\item Mardi 12 Juillet Mr Cyril CRASSIN (UdG)
GigaVoxels : un pipeline de rendu bas\'e Voxel pour l'exploration efficace de sc\`enes larges et d\'etaill\'ees.
\item Jeudi 7 Juillet Pierre BENARD (UdG)
Stylisation temporellement coh\'erente d’animations 3D bas\'ee sur des textures.
\item Mercredi 20 Avril Mr Samir TOUZANI (Univ. de Grenoble)
M\'ethodes de surface de r\'eponse bas\'ees sur la d\'ecomposition de la variance fonctionnelle et application \`a l'analyse de sensibilit\'e.
\item Mercredi 6 Avril Mr Aymen LAADHARI (UJF)
Mod\'elisation num\'erique de la dynamique des globules rouges par la m\'ethode des fonctions de niveau.
\item Jeudi 31 Mars Mlle Emilie NEVEU (UJF)
Application des m\'ethodes multigrilles \`a l'assimilation variationnelle de donn\'ees en g\'eophysique.
\item Lundi 21 F\'evrier Mr Benjamin PETIT (UdG)
T\'el\'epr\'esence, immersion et interactions pour la reconstruction 3D temps-r\'eel.
\item Jeudi 17 F\'evrier Mr Sylvain MEIGNEN (Grenoble INP)
Diff\'erentes approches non lin\'eaires multi-\'echelles pour l'analyse des signaux et des image.
\item Jeudi 17 F\'evrier Mr Pierre-Edouard LANDES (UdG)
Extraction d'information pour l'\'edition et la synth\`ese par l'exemple en rendu expressif.
\end{itemize}
\vspace{0.5cm}{\Large 2010}
\begin{itemize}
\item Jeudi 9 D\'ecembre Mme M\'elanie CORNILLAC (Univ. de Grenoble)
Morphing multir\'esolution de courbes.
\item Mardi 7 D\'ecembre <a href="mailto:Florence.Forbes@inrialpes.fr"> Mme Florence FORBES</a> (CR, INRIA Rh\^one-Alpes)
Mod\`eles et inf\'erence pour des syst\`emes stochastiques structur\'es.
\item Vendredi 3 D\'ecembre Mr Antoine GERBAUD (UJF)
Mod\'elisation de r\'eseaux d'interactions par des graphes al\'eatoires.
\item Jeudi 2 D\'ecembre Mr Kiran VARANASI (UJF)
Mod\'elisation spatio-temporelle des sc\`enes dynamiques 3D \`a partir des donn\'ees visuelles.
\item Lundi 29 Novembre <a href="mailto:OPalombi@chu-grenoble.fr"> Mr Olivier PALOMBI</a> (MCF, UJF)
Vers une nouvelle repr\'esentation du Savoir Anatomique.
\item Mardi 23 Novembre Mr Jean-Fran\c cois COEURJOLLY (UPMF)
Sur quelques r\'esultats d'inf\'erence pour les processus fractionnaires et les processus ponctuels de Gibbs.
\item Vendredi 19 Novembre Mlle Adeline PIHUIT (Univ. de Grenoble)
Croquis interactifs pour l'enseignement de l'anatomie.
\item Mercredi 17 Novembre Mr Laurent GARDES (UPMF)
Contributions \`a la th\'eorie des valeurs extr\^emes et \`a la r\'eduction de dimension pour la r\'egression.
\item Lundi 8 Novembre Mr C\'edric DOUCET (Univ. de Grenoble)
Hi\'erarchies d'\'el\'ements finis mixtes pour les maillages hybrides maintenus conformes par des pyramides quadangulaires et solution des syst\`emes d'\'equations lin\'eaires creux pour l'approximation quasi-stationnaire en \'electromagn\'etisme.
\item Vendredi 5 Novembre Mme Hedlena BEZERRA (Univ. de Grenoble)
Le rendu expressif assist\'e par l'ordinateur.
\item Lundi 25 Octobre Mr Innocent SOUOPGUI (Univ. de Grenoble)
Assimilation d'images pour les fluides g\'eophysiques.
\item Mardi 19 Octobre Mr Jamil DRARENI (Univ. de Montr\'eal)
Exploitation de contraintes photom\'etriques et g\'eom\'etriques en vision. Application au suivi, au calibrage et \`a la reconstruction.
\item Lundi 18 Octobre Mr Vassil KHALIDOV (Univ. de Grenoble)
Mod\`eles de m\'elanges conjugu\'es pour la mod\'elisation de la perception visuelle et auditive.
\item Lundi 18 Octobre Mr Joerg LIEBELT (Univ. de Grenoble)
D\'etection de classes d'objets et estimation de leur poses \`a partir de mod\`eles 3D synth\'etiques.
\item Mercredi 13 Octobre Mr Alexandre LEKINA (Univ. de Grenoble)
Estimation non-param\'etrique des quantiles extr\^emes conditionnels.
\item Lundi 27 Septembre Mr Matthieu GUILLAUMIN (Universit\'e de Grenoble)
Donn\'ees multimodales pour l'analyse d'image.
\item Mardi 14 Septembre Mlle Ramya NARASIMHA (Universit\'e de Grenoble)
daM\'ethodes d'estimation de la profondeur par mise en correspondance st\'er\'eoscopique \`a l'aide de champs al\'eatoires coupl\'es.
\item Lundi 13 Septembre Mr Souleymane KADRI HAROUNA (Universit\'e de Grenoble)
Ondelettes pour la prise en compte de conditions aux limites en turbulence incompressible.
\item Samedi 31 Juillet Mr Alexander KLASER (Grenoble INP)
Learning Human Actions in Videos.
\item Mardi 20 Juillet <a href="mailto:Jean-Guillaume.Dumas@imag.fr"> Mr Jean-Guillaume DUMAS</a> (MCF)
Contributions au calcul exact intensif.
\item Mercredi 30 Juin Mr Everton HERMANN (UJF)
Simulations Physiques Interactives sur des Architectures Multi-Core et Multi-GPU
\item Mardi 22 Juin Mr Ibrahim CHEDDADI (UJF)
Mod\'elisation num\'erique d'\'ecoulements de mousse.
\item Vendredi 11 Juin Mr Zaher EL CHAMI (UJF)
Rehaussement de la parole par des techniques de s\'eparation de sources bi-capteurs.
\item Lundi 17 Mai Mr Thierry STEIN (UJF)
Rendu narratif en synth\`ese d'images
\item Mardi 27 Avril Mme Anna URBANSKA-MARSZALEK (UJF)
Algorithmes hybrides et adaptatifs pour l'alg\`ebre lin\'eaire exacte.
\end{itemize}
\vspace{0.5cm}{\Large 2009}
\begin{itemize}
\item Jeudi 17 D\'ecembre Mr Mohammed MIRI (Grenoble INP)
D\'eveloppement stochastique et formules ferm\'ees de prix pour les options europ\'eennes.
\item Mercredi 2 D\'ecembre Mr R\'emi RONFARD (INRIA Rh\^one-Alpes)
Analyse automatique de film - Des s\'equences d'images aux s\'equences d'actions.
\item Lundi 30 Novembre <a href="mailto:Luc.Biard@imag.fr"> Mr Luc BIARD</a> (MC, UJF)
Mod\'elisation g\'eom\'etrique et reconstruction de surfaces.
\item Vendredi 27 Novembre Mr Azmi MAKHLOUF (Grenoble INP)
R\'egularit\'e fractionnaire et analyse stochastique de discr\'etisations; Algorithme adaptatif de simulation en risque de cr\'edit.
\item Jeudi 26 Novembre Mr Florent CADOUX (UJF)
M\'ethodes d'optimisation pour la dynamique non-r\'eguli\`ere.
\item Jeudi 12 Novembre Mr Lionel BABOUD (UJF)
Repr\'esentations alternatives du d\'etail visuel pour le rendu en temps-r\'eel.
\item Jeudi 15 Octobre Mr Adrien BOUSSEAU (UJF)
Manipulations d'image expressives pour une vari\'et\'e de repr\'esentations visuelles.
\item Lundi 25 Mai Mr Thomas MILCENT (UJF)
Une approche eul\'erienne du couplage fluide-structure, analyse math\'ematique et applications en biom\'ecanique.
\item Jeudi 30 Avril Anatoli IOUDITSKI (LJK)
Sur les conditions v\'erifiables d'estimation parcimonieuse par la minimisation l1
\item Jeudi 30 Avril Mr Fr\'ed\'eric HUGUET (UJF)
Mod\'elisation et calcul du flot de sc\`ene st\'er\'eoscopique par une m\'ethode variationnelle.
\item Mardi 21 Avril Mr Elie BRETIN (Grenoble INP)
Mouvements par courbure moyenne et m\'ethode de champ de phase.
\item Vendredi 13 F\'evrier Mr Christian BOUCHENY (UJF)
Visualisation scientifique interactive de grands volumes de donn\'ees : pour une approche perceptive.
\end{itemize}
\vspace{0.5cm}{\Large 2008}
\begin{itemize}
\item Mardi 9 D\'ecembre Mlle Claire TAUVEL (UJF)
Optimisation stochastique \`a grande \'echelle.
\item Lundi 8 D\'ecembre <a href="mailto:Sophie.Lambert@imag.fr"> Mme Sophie LAMBERT-LACROIX</a> (MCF, UJF)
Processus et champs al\'eatoires. Donn\'ees de grandes dimensions et sciences du vivant.
\item Vendredi 28 Novembre Mlle Diane LARLUS-LARRONDO (Grenoble INP)
Cr\'eation et utilisation de vocabulaires visuels pour la cat\'egorisation d'images et la segmentation de classes d'objets.
\item Mercredi 26 Novembre <a href="mailto:Emmanuel.Maitre@imag.fr"> Mr Emmanuel MAITRE</a> (MCF, UJF)
Equations de transport, Level Set et m\'ecanique eul\'erienne. Application au couplage fluide-structure.
\item Mardi 25 Novembre <a href="mailto:fran\c cois.faure@imag.fr"> Mr Fran\c cois FAURE</a> (MCF, UJF)
Simulation physique interactive pour la synth\`ese d'images.
\item Lundi 24 Novembre Mr David VANDERHAEGHE (UJF)
Distributions coh\'erentes de primitives pour le rendu expressif de sc\`enes anim\'ees et le rendu endemi-tons.
\item Lundi 24 Novembre Mme Jo\"elle THOLLOT (Grenoble INP)
Un ensemble d'outils et techniques pour la communication visuelle.
\item Lundi 17 Novembre Mr Florian LEMARIE (UJF)
Algorithmes de Schwarz et couplage oc\'ean-atmosph\`ere.
\item Vendredi 24 Octobre Mr Eli LAUCOIN (UJF)
D\'eveloppement du parall\'elisme des m\'ethodes num\'eriques adaptatives pour un code industriel de simulation en m\'ecanique des fuides.
\item Jeudi 23 Octobre Mr Morgan BRASSEL (UJF)
Instabilit\'es de forme en croissance cristalline.
\item Lundi 20 Octobre Mr Daniel WEINLAND (INPG)
Action Representation and Recognition
\item Jeudi 16 Octobre Mlle Claire BOST (UJF)
M\'ethodes Level-Set et p\'enalisation pour le calcul d'interactions fluide-structure.
\item Mercredi 1 Octobre Mr Yann DIJOUX (INP Grenoble)
Mod\`eles d'\^ages virtuel et de risques concurrents pour la maintenance imparfaite.
\item Jeudi 25 Septembre Mr Mathieu COQUERELLE (INP Grenoble)
Calcul d'interaction fluide-structure par m\'ethode de vortex et application en synth\`ese d'images.
\item Mardi 16 Septembre Mr Elmar EISEMANN (UJF)
Repr\'esentations optimis\'ees pour l'acc\'el\'eration des requ\^etes d'affichage et collisions.
\item Vendredi 27 Juin Mr David ROGER (UJF)
R\'eflexions sp\'eculaires en temps interactif dans les sc\`enes dynamiques.
\item Jeudi 26 Juin Mr Robin GIRARD (UJF)
R\'eduction de dimension en statistique et applicatoin en imagerie hyper-spectrale.
\item Mardi 24 Juin Mr Matthieu NESME (UJF)
Milieu m\'ecanique d\'eformable multir\'esolution, pour la simulation interactive.
\item Mercredi 7 Mai Mr Christophe DAMERVAL (UJF)
Ondelettes pour la d\'etection de caract\'eristiques en traitement d'images. Applications \`a la d\'etection de r\'egions d'int\'er\^et.
\item Lundi 11 F\'evrier Mr Pau GARGALLO I PIRACES (INP Grenoble)
Contributions \`a l'approche bay\'esienne pour la st\'er\'eovision multi-vues.
\item Jeudi 17 Janvier Mme Sonia HEDLI-GRICHE (UPMF)
Estimation de l'op\'erateur de r\'egression pour des donn\'ees fonctionnelles et des erreurs corr\'el\'ees.
\end{itemize}
\vspace{0.5cm}{\Large 2007}
\begin{itemize}
\item Mercredi 19 D\'ecembre Mr Farid BENINEL (ENSAI, Bruz)
Contribution \`a l'analyse discriminante g\'en\'eralis\'ee, aux distributions exactes et au positionnement multidimentionnel (MDS).
\item Vendredi 7 D\'ecembre Mme Ir\`ene GANNAZ (UJF)
Estimation par ondelettes dans les mod\`eles partiellement lin\'eaires.
\item Vendredi 30 Novembre Mme Carine LUCAS (UJF)
Effets de petites \'echelles, du tenseur des contraintes, des conditions au fond et \`a la surface sur les \'equations de Saint-Venant.
\item Jeudi 22 Novembre <a href="mailto:Jerome.Monnier.@imag.fr"> Mr J\'er\^ome MONNIER</a> (MCF, INP Grenoble)
Mod\`eles num\'eriques directs et inverses d'\'ecoulements de fluides.
\item Jeudi 8 Novembre Mr Ehouarn SIMON (UJF)
Assimilation variationnelle de donn\'ees pour des mod\`eles embo\^it\'es.
\item Mardi 30 Octobre Mr Matthieu VIGNES (UJF)
Mod\`eles Markoviens graphiques pour la fusion de donn\'ees individuelles et d'interactions : applications \`a la classification de g\`enes.
\item Jeudi 25 Octobre Mlle Claire SCHEID (UJF)
Analyse th\'eorique et num\'erique au voisinage du point triple en \'electromouillage.
\item Mercredi 24 Octobre Mr William CASTAINGS (UJF)
Analyse de sensibilit\'e et estimation de param\`etres pour la mod\'elisation hydrologique : potentiel et limitations des m\'ethodes variationnelles.
\item Jeudi 11 Octobre Mme Anne BILGOT (UJF)
M\'ethodes locales d'identification de surfaces de discontinuit\'e \`a partir de projections tronqu\'ees pour l'imagerie interventionnelle.
\item Mercredi 10 Octobre Mlle Juliette BLANCHET (UJF)
Mod\`eles markoviens et extensions pour la classification de donn\'ees complexes.
\item Mercredi 3 Octobre Mr Marc HONNORAT (INP Grenoble)
Assimilation de donn\'ees Lagrangiennes pour la simulation num\'erique en hydraulique fluviale.
\item Jeudi 5 Juillet Mme Nathalie SPRYNSKI (UJF)
Reconstruction de courbes et de surfaces \`a partir de donn\'ees tangentielles.
\item Mercredi 27 Juin Mr S\'ebastien KOLB (INP Grenoble)
Th\'eorie des bifurcations appliqu\'ee \`a l'analyse de la dynamique du vol des h\'elicopt\`eres.
\item Lundi 12 F\'evrier Mr Isidore Paul AKAM BITA (UJF)
Sur l'application de l'analyse en composantes ind\'ependantes \`a la compression des images multi composantes.
\end{itemize}
\vspace{0.5cm}{\Large 2006}
\begin{itemize}
\item Mercredi 13 D\'ecembre Lisl WEYNANS (UJF)
M\'ethode particulaire multi-niveaux pour la dynamique des gaz, application au calcul d'\'ecoulements multifluides.
\item Vendredi 20 Octobre Alexandre BROUSTE (UJF)
Etude d'un processus bifractal et application statistique en g?ologie.
\item Jeudi 28 Septembre Charles BOUVEYRON (UJF)
Mod\'elisation et classification des donn\'ees de grande dimension : application \`a l'analyse d'images.
\item Mercredi 27 Septembre Mr Cl\'ement PERNET
Alg\`ebre lin\'eaire exacte efficace : le calcul du polyn\^ome caract\'eristique.
\item Mardi 18 Juillet Aude RONDEPIERRE
Algorithmes hybrides pour le contr\^ole optimal des syst\`emes non lin\'eaires.
\item Lundi 27 Mars Mr Erwan DERIAZ
Ondelettes pour la simulation des \'ecoulements fluides incompressibles en turbulence.
\end{itemize}
\vspace{0.5cm}{\Large 2005}
\begin{itemize}
\item Mercredi 7 D\'ecembre Basile SAUVAGE (INPG)
D\'eformation de courbes et surfaces multir\'esolution sous contraintes.
\item Mercredi 30 Novembre Laurent TOURNIER (INPG.)
Etude et mod\'elisation math\'ematique de r\'eseaux de r\'egulation g\'en\'etiques et m\'etaboliques.
\item Lundi 14 Novembre Claire CHAUVIN (INPG)
Les ondelettes comme fonctions de base dans le calcul de structures \'electroniques.
\item Mercredi 26 Octobre Hakim BOUMARAF (UJF)
S\'eparation aveugle de m\'elanges convolutifs de sources.
\item Mardi 20 Septembre Julie PEYRE (UJF)
Analyse statistique des donn\'ees issues des biopuces \`a ADN.
\item Mercredi 20 Juillet Etienne FARCOT
Etude d'une classe d'\'equations diff\'erentielles affines par morceaux mod\'elisant des r\'eseaux de r\'egulation biologique.
\item Jeudi 30 Juin Vincent GUIGUES (UJF)
Inf\'erence statistique pour l'optimisation stochastique. Application en finance et en gestion de production.
\item Lundi 25 Avril Lin WU (UJF)
M\'ethodes variationnelles pour des mod\`eles structure-fonction de plantes : identification de param\`etre, contr\^ole et assimilation de donn\'ees.
\item Lundi 25 Avril Claire LAUVERNET (UJF)
Assimilation variationnelle des observations de t\'el\'ed\'etection dans les mod\`eles de fonctionnement de la v\'eg\'etation : utilisation du mod\`ele adjoint et prise en compte des contraintes spatiales.
\item Mardi 4 Janvier Karim DAHIA (UJF)
Nouvelles m\'ethodes en filtrage particulaire. Application au recalage de navigation inertielle.
\end{itemize}
\vspace{0.5cm}{\Large 2004}
\begin{itemize}
\item Mardi 21 D\'ecembre C\'eline ROBERT (UJF)
D\'eveloppement et comparaison de m\'ethodes d'assimilation de donn\'ees de rang r\'eduit dans un mod\`ele de circulation oc\'eanique : application \`a d'Oc\'ean Pacifique Tropical.
\item Lundi 13 D\'ecembre Alex YVART (INPG)
Mod\'elisation hi\'erarchique de surfaces \`a partir de maillages poly\'edriques et applications.
\item Lundi 22 Novembre Laurent DOYEN (INPG)
Mod\'elisation et \'evaluation de l'efficacit\'e de la maintenance des syst\`emes r\'eparales.
\item Jeudi 30 Septembre Mr Antoine GIRARD
Analyse algorithmique des syst\`emes hybrides.
\item Mardi 28 Septembre Olivier LE CADET (INPG)
M\'ethodes d'ondelettes pour la segmentation d'images. Applications \`a l'imagerie m\'edicale et au tatouage d'images.
\item Lundi 27 Septembre Jocelyn ETIENNE (INPG)
Simulation num\'erique d'\'ecoulements gravitaires \`a fortes diff\'erences de densit\'e. Application aux avalanches.
\end{itemize}
\vspace{0.5cm}{\Large 2003}
\begin{itemize}
\item Vendredi 12 D\'ecembre Fran\c cois GANNAZ (U.J.F.)
Approximation de convexes par des polytopes et d\'ecomposition approch\'ee de normes
\item Vendredi 12 D\'ecembre Cyril MAZAURIC (UJF)
Assimilation de donn\'ees pour les mod\`eles d'hydraulique fluviale. Estimation de param\`etres, analyse de sensibilit\'e et d\'ecomposition de domaine
\item Mardi 9 D\'ecembre Ayman MOURAD (INPG)
Description topologique de l'architecture fibreuse et mod\'elisation m\'ecanique du myocarde
\item Mardi 2 D\'ecembre Sophie ACHARD (UJF)
Mesures de d\'ependance pour la s\'eparation aveugle de sources. Application aux m\'elanges post non lin\'eaires
\item Lundi 17 Novembre Guillaume ALLEGRE (UJF)
Repr\'esentation g\'eom\'etrique des arrangements de droites du plan
\item Vendredi 3 Octobre Vo\"ichita MAXIM (UJF)
Restauration de signaux bruit\'es observ\'es sur des plans d'exp\'erience al\'eatoires
\item Mardi 2 Septembre J\'erome BIGOT (UJF)
Recalage de signaux et analyse de variance fonctionnelle par ondelettes. Applications au domaine biom\'edical
\end{itemize}
\vspace{0.5cm}{\Large 2002}
\begin{itemize}
\item Vendredi 6 D\'ecembre
Simulation des circuits electroniques mixtes RF/Analogiques/Num\'eriques exit\'es par des signaux \`a modulation complexe
\item Lundi 2 D\'ecembre Fr\'ed\'eric BÉRINGER (INPG)
Contributions \`a la r\'esolution d'\'equations diff\'erentielles non lin\'eaires scalaires par la m\'ethode du polygone de Newton
\item Mardi 8 Octobre Blaise FAUGERAS (UJF)
Assimilation variationnelle de donn\'ees dans un mod\`ele coupl\'e oc\'ean-biog\'eochimie
\item Lundi 30 Septembre Maryline BRUEL
Etude de m\'ethodes math\'ematiques pour la r\'esolution et l'analyse de syst\`emes d'\'equations mod\'elisant le comportement dynamique des circuits thermo-hydrauliques
\item Mardi 10 Septembre Mihaela MIRICA-RUSE (UJF)
Contribution \`a l'\'etude des syst\`emes
\item Vendredi 7 Juin Olivier GAUDOIN (MC-INPG)
Pr\'esentation de travaux en vue de l' habilitation \`a diriger des recherches
\item Vendredi 8 Mars D\'elia JIROVEANU (LMC-EDP et UNIVERSITE DE L'OUEST DE TIMISOARA (ROUMANIE))
Analyse math\'ematique et num\'erique de certains mod\`eles de viscosit\'e turbulente
\end{itemize}
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http://yasuda.homeip.net/insomnia/archives/oldstandard-ex.tex | homeip.net | CC-MAIN-2019-51 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2019-51/segments/1575541319511.97/warc/CC-MAIN-20191216093448-20191216121448-00274.warc.gz | 257,792,522 | 1,807 | % -*- coding: utf-8; -*-
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Вопросъ о разрывѣ Татьяны съ Онѣгинымъ, несмотря на всю
ясность. Пушкинскаго стиха, выросъ у насъ въ своего рода
\glqq гамлетовскую проблему\grqq,
къ которой постоянно возвращается критическая мысль.
Онъ, по выраженію Достоевскаго, имѣетъ въ нашей литературѣ
\glqq своего рода исторію весьма характерную\grqq.
Иначе говоря, вокругъ него сталкивались и боролись
различныя враждующія теченія соціально-политической мысли.
И совсѣмъ не случайно выразителями двухъ наиболѣе
противоположныхъ взглядовъ на этотъ предметъ являются
представители враждующихъ міросозерцаній
"---соціалистъ-радикалъ Бѣлинскій и
націоналистъ-славянофилъ Достоевскій.
\vspace{1em}%
\small%
\hfill%
\parbox[t]{.8\textwidth}{%
\textit{А. И. Ванновский}~~Зерколо судьбы. (Сон Татьяны.)\\
в кн.: \textit{С. А. Небольсин}~~Пушкин и европейская традиция.
Историко-теоретические работы. М. 1999.}%
\end{document}
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https://authorea.com/users/312879/articles/443421/download_latex | authorea.com | CC-MAIN-2020-40 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-40/segments/1600400187390.18/warc/CC-MAIN-20200918092913-20200918122913-00341.warc.gz | 389,871,686 | 2,849 | \documentclass[10pt]{article}
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\title{The Characteristics of High Altitude Pulmonary Edema in Naqu at the
Altitude of 4,500 m}
\author[1]{Jiahuan Xu}%
\author[2]{Lianggang Lv}%
\author[3]{Bo He}%
\author[3]{Guoqiang Wang}%
\author[3]{zhuoma Bianba}%
\author[1]{Delei Kong}%
\affil[1]{China Medical University First Hospital}%
\affil[2]{People's Hospital of Naqu}%
\affil[3]{People’s Hospital of Naqu}%
\vspace{-1em}
\date{\today}
\begingroup
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\selectlanguage{english}
\begin{abstract}
Purpose: High altitude pulmonary edema (HAPE) has rapid onset and
development, and may be life-threatening if not treated in time. It is
important to correctly identify and recognize the characteristics of
HAPE and to provide timely treatment. We aimed to summarize the
characteristics of patients with HAPE by analyzing their general
information, clinical symptoms, examination and laboratory results.
Methods: 429 HAPE patients were enrolled in our study, and 200 of them
were divided into three groups according to the period from their
arrival in Naqu to the onset of the disease. We collected and analyzed
the demographic information, results of laboratory tests, imaging and
electrocardiography of all the participants at admission. The patients'
results of laboratory tests and imaging at admission were compared with
those at discharge. The results of blood routine were compared among
different groups. Results: Most of the HAPE patients were male
(90.21\%). The average white blood cell and neutrophil counts, alanine
aminotransferase and aspartate aminotransferase levels, uric acid level,
lactic dehydrogenase and creatine kinase levels were increased among
HAPE patients at admission. The counts of white blood cell, neutrophil
and lymphocyte, and the concentration of hemoglobin in HAPE patients at
admission were higher than those at discharge (p\textless{}0.05).The
counts of white blood cell and neutrophil were higher in the patients
who developed the disease within 1 day than in those who developed the
disease in more than 7 days (p\textless{}0.05). Conclusion: The
proportion of males in HAPE patients was higher. Hepatocyte and
myocardium in HAPE patients were more likely to be damaged. The white
blood cell and neutrophil counts were significantly higher than normal
range, and their increment was reduced as the incubation period
extended.%
\end{abstract}%
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\includegraphics[width=0.70\columnwidth]{figures/Figure-1/Figure-1}
\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
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\title{Abusing Resilience}
\date{February 2020}
\author{APS}
\subtitle{The Filipino in the face of Disaster}
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On the afternoon of January 12 the Taal Volcano began spewing ash and smoke from its ancient caldera. Within hours a massive evacuation effort was launched to get people out of harm’s way. A comrade was among the people fleeing the scene. Government offices and schools were understandably closed due to the disaster, but BPO centers around the areas most affected by the ensuing ashfall had the gall to call their workers back to work.
We’ve seen this story before:
A calamity or some other misfortune affects a large area of the nation and we get reports of people calling in to work being praised for “their dedication to their jobs” despite the obvious risks. The true story is most likely that they literally couldn’t afford to be gone that shift. They might not get administrative sanctions or attendance memos for being absent, thought that still happens, but they still won’t be paid for that workday. No work, no pay, right? But, this isn’t to say that the supervisors and managers frantically calling their employees to work are bad people. This is bigger than any one person.
When you have someone who lives completely on what they make per hour worked, they have little choice but to show up for work. This is the greatest triumph of modern capitalism over the human spirit. I remember someone calling money “survival notes” because it literally does mean whether or not you survive in this society. Because we live inside it! It’s become a very efficient way for the rich business owner and investor to value profits over human lives.
With slavery, you own the person, end of discussion. In feudalism, you own the land, you get part of the produce of that land. But with capitalism? Oh, boy, you not only own the place where they work, you also own the places where they spend their hard-earned survival-notes at!
That’s how you get people to show up at work soaking in rainwater after braving the elements for two hours to get to a job that pays less than a hundred pesos an hour. That’s how you get people to stay to watch over what little property they have in the face of a raging volcano. That’s how you get people to value profit over human lives, most especially if that life is their own.
So no, it isn’t surprising that there’d be people who’d come to work on the apocalypse. Capitalism has made our world so absurd that it would actually make sense.
So here’s to the working-class heroes who instead of going to work went out to help in whatever way they could, even if it’s something as natural as getting your family to safety.
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The Anarchist Library
\smallskip
Anti-Copyright
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APS
Abusing Resilience
The Filipino in the face of Disaster
February 2020
\bigskip
\href{http://libcom.org/blog/abusing-resilience-filipino-face-disaster-01032020}{libcom.org}
\bigskip
\textbf{theanarchistlibrary.org}
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http://nbarth.net/notes/src/phil/constraints-structure.tex | nbarth.net | CC-MAIN-2019-18 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2019-18/segments/1555578529472.24/warc/CC-MAIN-20190420080927-20190420102927-00320.warc.gz | 126,425,291 | 2,145 | \chapter{Constrained systems are more structured}
This is especially a discussion of how Backus's FP language is more tractible,
by being less powerful and lacking self-reference.
There is a duality between objects and their structure;
the more general or powerful an object, the less you can say about it,
while the more constrained an object, the more you can say about it:
the more structure it has.
One might quip ``slavery is freedom''.
Saunders Mac~Lane writes (in a comment in ``Categories for the Working Mathematician''):
``...good general theory does not search for the maximum generality, but for the right generality.''; I would say: ``sometimes, say more about less''.
\section{Computer examples}
A good example is programming languages versus markup languages.
At one end, XML is very simple (it is a markup language);
basically, it's a syntax for trees.
As such, you can analyze them and work on them very easily:
\begin{itemize}
\item it is very easy to write an XML parser
\item it is very easy to check if two XML documents are equal
\end{itemize}
They can be completely understood by reading them and parsing them:
you do not need to do any further computation.
At the other end, any Turing/von Neumann-type language is very complex.
\begin{itemize}
\item it is relatively hard to write a C compiler
\item it is impossible to algorithmically determine properties of a program:
you cannot determine if two programs are equal (produce the same output) or halt
\end{itemize}
You cannot understand a program without running it (at least mentally).
This is particularly pronounced with reflective languages, like self-modifying code:
you can't analyze the behavior of such a program by simply reading its code
-- not only its behavior but its \emph{structure} can only be determined by running it!
As the example of the halting problem demonstrates, there are limits on what you can determine about programs via a program.
In between, constrained languages (like regular expressions or SQL)
are powerful enough to be quite useful, but constrained enough to do
meaningful changes and analysis:
one can transform SQL queries, for instance, or rewrite regular expressions.
\subsection{Backus's Function-level programming}
John Backus's function-level programming (and his FP language) is a fascinating example:
traditional (von Neumann style) programming languages are ``too powerful'':
they are a very general class of programs and thus you can't do much with them:
you can't analyze them, you can't optimize them well, you can't parallelize them and so forth.
Instead, in Backus's language FP, you start with atoms, and then build functions from them.
The rules for combining atoms are specified in a fixed set of higher order functions
(``functional forms'');
you can program new functional forms in a \emph{separate} meta-language, Formal FP.
This constraint means that, for a fixed set of functional forms, the possible programs
are a module over the algebra of functional forms.
This is much more constrained than a general programming language,
and hence easier to study and more engineerable,
precisely because it lacks self-reference
(it has a hierarchy of meta-programming, instead of meta-meta = meta).
Backus's Turing award lecture, ``Can Programming Be Liberated from the von Neumann Style?'',
is interesting itself; I've outlined my main interest in it above though.
|
https://theanarchistlibrary.org/library/william-gillis-beyond-the-hellish-choice-of-process-documents-or-social-capitalism.tex | theanarchistlibrary.org | CC-MAIN-2022-49 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-49/segments/1669446710698.62/warc/CC-MAIN-20221129132340-20221129162340-00382.warc.gz | 630,510,531 | 18,221 | \documentclass[DIV=12,%
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\title{Beyond the Hellish Choice of Process Documents or Social Capitalism}
\date{June 10th, 2015}
\author{William Gillis}
\subtitle{}
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One of the best things about \emph{The Utopia of Rules: On Technology, Stupidity, and the Secret Joys of Bureaucracy} is that David Graeber finally tackles issues directly relevant to anarchists. While his prior work has had value, it’s also largely been about rather obvious topics and punctuated with a need to apologize for or defend anarchism. Graeber has rarely written \emph{to} us. His usual intended audience is much broader, much more liberal, and this has led to a kind of ever-present defensiveness and basics-covering that bogs everything down and taxes one’s patience. You can only read about the liberatory power of direct democracy so many times before your eyes roll away permanently. And yet, suddenly, seemingly out of nowhere comes a book that grapples with John Zerzan, Foucault, structural violence, the Tyranny of Structurelessness, and even name-checks “no future”.
It’s kinda like Graeber has finally caught up with Anarchism circa a few decades ago \dots{}or maybe just visited the West Coast.
On a less snarky note, I get the impression that Graeber is being badly pressed to write by the eldritch forces he made deals with. And now, having churned out the mandatory and completely unremarkable “\emph{I Founded Occupy But Not Really}” book of anecdotes and arguments to reach shitty liberals, he’s taken to rambling about anything he cares about in a sloppy but slightly more sincere and original fashion. As fond as I’ve grown of \emph{struggling for academic acceptance while nursing a poor-kid grudge Graeber}, this is more in the direction of \emph{geeky hanging in the back of the infoshop with other anarchists Graeber}. There’s still a few liberals in the conversation, and he hasn’t stopped trying to persuade them, but he’s speaking to us as well.
Any anarchist who’s ever attended a meeting knows the gut-clenching and heart-sinking horror attendant to the dark invocation of “process” or “the process document.” Here be monsters indeed. Otherworldly catastrophes of the mind, songs of torment the singers cannot quit, rips in the fabric of sanity that eat even the strongest among us. A few hours later our bodies are regurgitated, what brains remain turned to a blinking traumatized gruel. Perhaps, if we are “lucky”, we return with a couple simple and utterly meaningless responsibilities scribbled into our planners. Cheap gems snatched from the bowels of the process monster.
The Utopia of Rules is not a magical dagger capable of piercing the exoskeleton of this beast and freeing us forever more, but it does have productive things to say about the nature of the monster and its power, and Graeber uses it as a starting point to examine our world as a whole and recast anarchist critiques of the existing power structures and psychologies.
Why are we pulled to add more rules, more process? And why does so little get accomplished the more we add? These are simple, seemingly deafeningly obvious dynamics we’re all familiar with, but they’re too infrequently interrogated explicitly or probed for deeper dynamics.
Along the way in The Utopia of Rules Graeber conjures a number of reframings that are particularly succulent, possibly even useful. My favorite of which is that the best way to define the police is as armed bureaucrats. Having never really been about stopping trespasses between citizens so much as maintaining the power system, cops are in so many ways the violent force by which the state asserts its need to categorize and make complex or chaotic contexts simple.
\begin{quote}
The police truncheon is precisely the point where the state’s bureaucratic imperative for imposing simple administrative schemes and its monopoly on violence come together\emph{.}
\end{quote}
This is of course not fundamentally new territory, libertarian theorists have a long and rich discourse on the computational limits faced by states and corporations, and the irrationality and violence that result. And, closer to Graeber’s audience, James C. Scott has written extensively on the state’s need for a certain type of directed legibility. But Graeber’s subject here is far more sweeping.
Graeber takes bureaucracy to denote a very wide array of formal and impersonal systematizing, so sweeping a definition that there can be no pat answers. Impersonal systematizing is after all not inherently a source of stupidity and irrational logjams, it can seriously augment accessibility and transparency. And we \emph{enjoy} making certain things less personal. Sometimes you want to focus your attention on complexities in other places. If the picture of bureaucratic activist meetings forever is repulsive, certainly so too is a life made of nothing but friend and roommate drama. Sometimes you just want to go off and write songs or equations or travel on your own and not have to deal deeply with people to do so. Impersonalizing interactions with other people is often a necessity. Think of what a hellish world it would be if we had to listen to everyone’s life story before completing some trivial interaction with them. Or — worse — being confined to the small town hell of communes and land projects (ie anarchist suburbia) where there are no strangers to ever meet and every last interaction is baked with piles of implicit social tensions.
While any systematizing necessarily involves simplifications from full case-by-case particulars those simplifications can be useful, they \emph{can} free creative energy from having to detail out or navigate informal and personal systems and end up allowing greater creative complexity in other places.
The crux is that while collective process documents, food stamp applications, and corporate paperwork may have originally been intended to provide greater transparency and accessibility, such bureaucracies tend to promptly move to make themselves indispensable by monopolizing access to the information they were intended to make available. This doesn’t always occur in directly hypocritical ways, but through a roundabout creep due to the game theoretic pressures on all parties. The centralization that used to force transparency is too great a target to pass up and so is inexorably captured by power. Those capable of dominating the new \emph{formal} power structure may be different from those who ruled the informal power structure prior, but the power relations remain. Rules made and frozen in place to stop arbitrary personal authority end up leading to a negative sum game in which both sides can only compete to enact more constraints to stop whoever currently benefits more by the current formalization, until everything organic is choked out or the rules are bypassed or they’re just violently overridden by whoever has significant power from sources external to the bureaucracy.
Outright violence is even more distortionary, it severs and strips away our capacity to recognize or integrate vast amounts of context and complexities, often in the interest of making things easier for the wielders of such violence. Violence creates ridiculous simplifications and has limited capacity to process complex realities. Our present world of poisonous bureaucracy thus emerges in two ways: to assist the violent in forcibly simplifying the realities around them, but \emph{also} to seek to rectify its own idiocy by providing a way for violent systems to accept and process complex realities in a manner slightly less stupid than raw violence.
What’s most frustrating about The Utopia of Rules is that it’s a book about complexity theory that stubbornly refuses to come out and admit that’s what’s being covered. Granted this refusal means today’s culture of math-hating leftists will actually read the book, but one can’t help but be frustrated at the refusal to shed all the self-indulgent and tenuous handwaving about “play” and “fantasy” and just get to some of the obvious roots of the whole affair.
Of course the populism is for a reason: Graeber’s attempting some rather audacious restructuring of the political landscape here and while I doubt he will be successful — too many doors have closed on his face since he refused to play ball with the Marxists after \emph{Debt\dots{}} — it’s an interesting attempt to be sure. Graeber flirts with setting Bureaucracy, and all it signifies as a suppressor of imagination and possibility, as a major if not \emph{the} nemesis of the left. The question here is whether or not that’s remotely true beyond anarchist circles. I remain unconvinced there’s anything of value in the left outside anarchism, or much meaningful overlap between the two. And I don’t just mean in terms of stuffy unimaginative bureaucratic hell that “leftism” connotes for most of the world these days. Even taking into account anti-authoritarian strands like council communism and autonomism, the primary lens of the left has always been political, and the anarchist lens ethical. They think exclusively in terms of sweeping macrostructures and we think in terms of the underlying psychological and interpersonal dynamics of which those macrostructures are epiphenomena. As such it’s incredibly hard to convey critiques of things like formal process or organizationalism to a leftist, inclined as they are to sweep over such “particulars” with their eyes only on the big institutional bugaboos.
\begin{quote}
\textbf{Communist:} “We’re all on the same team, we both want a classless society, so why won’t you obey us — I mean ally with us.”
\textbf{Anarchist:} “Uh we want so much more than a merely classless society, we want a world where people don’t control and limit each other.”
\textbf{Communist:} “Hold the fuck up.”
\end{quote}
I can’t help but be suspicious of any left that might be repolarized in the spirit of ’68 as an opposition to bureaucracy. It’s not enough to reject structure and organizationalism, if we are to call ourselves anarchists as anything other than a joke we must tackle informal power dynamics too. And this will necessarily involve navigating the tensions at play in attempts at openness that so often lead to bureaucratic feedback loops. A left that doesn’t strike at the root, at power relations themselves, will only ever be able to approximate the advances of anarchism, and thus allying with them only acts to chain us down. What we’ve seen play out time and time again is the fossils of the left being forced kicking and screaming to adopt new modules of analysis on particular issues, but never delving beyond such themselves. They may eventually arrive at similar conclusions as anarchists, but only through the pressures of history, rarely if ever in advance of them. And when it comes to as deep a root as interpersonal power dynamics themselves we’ve seen that the old patriarchs who litter the movement have little interest in anything that leaves them no dynamics of power to hold onto.
One of the chief charges Graeber levels at bureaucracy is its violence to imagination and general suppression or imprisonment of creativity, inquiry, science, invention, etc. Yet this kind of suppression is hardly unique to formal bureaucracy, it’s on display in virtually every case of interpersonal abuse. As anarcha-feminists have long argued there’s an asymmetry to acts of imagination in social hierarchies. Those on the top do less than those on the bottom, and those on the bottom end up having their imaginative energies channeled and directed by those on the top. Not necessarily in the marxist sense of expropriated labor, but in the sense that the imaginative lives of those on the bottom revolve around those on the top. The caged person spends their time thinking about the cage and the mental states of their jailer. All their modeling, all their creativity is channeled by the conditions of their enslavement. Or made incapable of interacting with the rest of the universe beyond the cage. What I remember most about being homeless as a child was sitting in waiting rooms terrified and bored while filling out paperwork or watching my mother anxiously fill out paperwork. Constantly trying to think of all the possible things that could go wrong, all the possible reasons we could be fucked over, and what next steps we might have to take. Desperately pleading with the social workers to make sure there wasn’t some other form we didn’t know about. The faceless bureaucracy creates a system that can only work because those under its thumb do all the contextualization and intellectual heavy-lifting. All the tentative sorting of complicated realities into categories and check-boxes. But this asymmetry is not unique to formalized systems or macroscopic systems, it exists in interpersonal relations too and a leftism that internalizes opposition to bureaucracy as yet another module will still fall short of the anarchist critique.
Still the macroscopic is certainly important. One of the most potent questions Graeber poses in The Utopia of Rules is the very good question: Why has there been ANY innovation in our bureaucracy-strangled world? This is the sort of question I wish people would ask more often. Why on earth has there even been what scant innovation there has been despite our regime of intellectual property? (Which we know quite well dramatically suppresses innovation.) Why has innovation even happened despite the state capturing almost all basic research in a crippled academia and working hard to limit any scientific education that wasn’t just tradeschool bullshit? Why and how has anything persisted or filtered through?
Part of Graeber’s answer is that the innovation crest around the 50s was the result of tax structures whereby corporations looking to decrease their taxable income faced the choice of either reinvesting in either their workforce or research and chose research. This is an interesting nuance to the typical story that states only poured funds into research because some idealists had managed to get the Soviet Union to invest in basic research, if ultimately for propaganda purposes, driving a negative-sum competition between governments that often hurt their long-term interest in controlling their populaces.
Today, of course, the natural allegiances of power have reasserted themselves and basic research has grown an ever-smaller portion of budgets as those in power press endlessly to eradicate it completely.
Graeber however uses some rather demagoguish rhetoric to paint the suppression of science as more desperate than it really is. As a physicist I found this disgustingly underhanded and problematic in its misrepresentation. Graeber holds up quantum mechanics and general relativity as the last great advancements of physics, but the reality is incredible advancements have continued, they just haven’t been popularized or focused on the same way by the public. In part since quantum mechanics and general relativity are where physics’ insights started to diverge from the common intuitions of every day people. When your intuitions from biologically inherited heuristics or everyday experience are deeply misguided it takes a lot of work to update them and it becomes impossible to accurately Explain It Like I’m Five. The universe is under no obligation to organize itself according to our intuitions, and the reorganization of our minds necessary to understand it can involve some complex work. This is why, after all, so much nonsense flies around the public discourse about modern physics, from quantum mysticism to that phrase universally abhorred by physicists, “the god particle.” Yet major advances have continued. Emmy Noether’s work on symmetry was just as titanic as quantum mechanics and relativity but no one outside physics wanted to hear about some abstract mathematics from a girl. Field theory saw absolutely significant and inspired work in the creation of chromodynamics in the 70s, string theory has advanced so unexpectedly and awe-inspiringly in certain respects that mathematicians are still shellshocked, holographic and AdS\Slash{}CFT have been so successful it’s terrifying, and black hole research continues to spit out astonishing and challenging insights. Don’t even get me started on quantum error correction and entanglement entropy. If anything the tiny almost-entirely-choked-out-of-existence physics community has suffered from a singularity of \emph{too much} advancement. We’re so overwhelmed and there are so few of us allowed to exist that parsing these developments back into a language the intentionally-badly-educated public can understand is daunting. There’s simply no way to do so as sound bytes or even quick lectures without spawning even more terrible misperceptions than currently float around.
In general Graeber flounders when attempting to examine the intersection of science and complexity. His askance, “\emph{why no cure for the common cold and cancer?”} is more inane and embarrassing than provocative. It shouldn’t be hip to just blithely handwave away science’s insights into issues of relative complexity. Not all problems are on the same level. Many of the early predictions or dreams about future developments date back to before we had things like television, when there was a lot less knowledge about their complexity and so it was reasonable to group things like “a cure for all diseases” in with “a machine that plays all music.” The low hanging fruit was promptly harvested but the things that were revealed as many many many orders of magnitude harder have remained in our minds as in some rough sense equivalent to the invention of television. Why can’t you just throw more researchers at it? Well what if what you’re asking couldn’t be solved on a mathematical complexity level by more researchers than there are atoms in the sun? This kind of thinking treats science as a magic box and scientists as either chumps or capricious magicians. It’s the sort of smug ignorance that declares silly shit like, “The NSA has lots more money to pour into research than the anarchist squatter hackers writing crypto, so they can surely break any encryption.”
Enough of that please.
Still, obviously there are an intense amount of institutional chains holding science down. Graeber focuses at length on the role of forced competition under bureaucracy within the sciences and academia and this is certainly impedes science and is all kinds of fucked up for those put through the wringer, but I’d argue that the real underlying dynamic that competition is but one symptom of is \emph{immediatism} — a pressure by the power structures that have captured research for immediate results or measurements that is deeply hostile to theory and imagination. Measuring scientific research from an administrative perspective is as silly and impossible a task as claiming that science is a simple procedure rather than an underlying orientation or desire that gets instantiated in complex ways. Scientists are hardly unaware of the horrors that have accrued from attempts to quickly “measure” how much science is being done. The superficiality of such is the same beast as the superficiality at play in popscience journalism, the pressures of the state, of capital, and of bureaucracy’s need for instantaneous simple visibility without work on the part of the observer is a matter of fetishized immediacy.
The macrosystems of our society are obviously deeply opposed to the disruptive effects of science, hence why they insist so strongly on continuing to trot out long abandoned scientific models or postulates as truths. It never stops being amusing that the symbol the Dawkinsite Atheists use as self representation is an image of the atom as orbited by discrete electrons that was invalidated a century ago, that has no real value even as an approximation, and that no one, not even chemical engineers, use. But this is tied to cultural and personal pressures within our society for immediate results, immediate “understanding”, etc that are antithetical to science. This is deeper than just a bureaucratic need to simplify for the stupid authorities. It’s an entitled demand for the suppression of any complexity we as individuals don’t feel like engaging with.
Graeber turns around at one point and nuances in an optimistic way that, “inconvenient discoveries cannot be suppressed,” but like \emph{please tell that to the primitivsts!} Or to their allies in power who have successfully suppressed many inventions and discoveries via intellectual property and are currently strangling almost every field of science that isn’t reducible to a highly-manageable and docile pet that only engineers nonthreatening consumer goods. There are reasons to be slightly optimistic about tendencies for discoveries to creep out to the periphery, but we must be realistic about the challenges we face and the bald-facedness of power’s hostility to disruption from science and technology. They’re already openly stumping about outlawing encryption and general purpose computing. And citizen science is being increasingly outlawed.
The last step in the ideology of conservatism has finally been revealed these last two decades as the destruction of technology and science. Making sure none of this imagination stuff can ever destabilize or disrupt the power relations between humans ever again.
And here Graeber brings up the welcome reminder of how reactionary Marx was. Too often we forget that Marx was ideologically opposed to theory and theorizing. And juxtaposed them against a call for immediacy in all arenas (after you’ve first taken the time to read his big books presumably). Marx’s hostility to imaginative blueprints and insistence on immediate means and inclinations mark deep parallels with the most problematic currents in today’s insurrectos and nihlists. Such violent hostility to theorizing rather is an understandable defensive move. If Marx \emph{had} been down with modeling and theorizing about root dynamics he’d have been capable of understanding Bakunin’s critique. Or perhaps it’s better to say that if Marx’s \emph{followers} had stopped for a moment and looked ahead seriously or in depth theoretically they would have seen the jaws of the monster that ended up swallowing them. Marxism’s point of departure from anarchism is then, in Graeber’s account, its pillars of anti-theory, anti-imagination, and anti-utopian reactionism.
Claiming you’ve already found the correct theory (or perfect clarity regarding your own desires) is a dead-end to thought, permanently removing any need to compare with other models. Of course there might be a most optimal theory — to the point where all others are essentially trash — but marxism takes shortcuts in its quickness to generate an explanation and then avoid all investigation into alternative models. Such an “anti-theory” approach is ultimately about stripping away meta context.
One of the things I found most interesting in Graeber’s book is his account of the way the word “imagination” has been plucked from all embedded context and wafted away to mean some kind of disconnected flight of fancy. In today’s paradigm there’s this free floating imagination thing and then there’s immediatist proceduralism. Despite the fact that “imagination” originally meant something more like creativity and the search for possibilities. Since in our world there are to be No Other Possibilities, “imagination” can only be left meaning self-delusion. In the same way that “without rulers” can only be left meaning “fractured rulership where everyone attempts to rule everyone else”. This is what they want to do to us. To kill all science or hope and shove everything into either highly manageable engineering or into tractionless art and escapism.
I want to take a moment and turn this around on one of Graeber’s most favored canards, however. In the early 00s with the explosion of activism accompanying the counter-globalization movement we started hearing repeated refrains that “anarchism is a process.” Bullshit. Such a horrifying defanging of anarchy could only come from shellshocked cultists to “the process document.” Stockholm syndrome is not an argument. It’s ludicrous to suggest anarchism is some kind of simple process or formula we just have to cleave to. If we all follow “the process” that’ll be anarchy? What utter bureaucratization of our hopes and desires. Anarchy is an ideal, a value — not a strict method \emph{OR} an end point. No, anarchism is not about not some terminal utopia, but the notion that our only choices are a fetishized idealic endpoint or an immediatist procedurism is an absurd dichotomy. A \emph{direction} on a manifold is different from an \emph{endpoint} or a \emph{local gradient}. Anarchy is not a thing you do or a process, nor a place you arrive at or an elaborate blueprint, but \emph{a value, a direction}, that can motivate different tactics, strategies, and processes in different contexts. And that never terminates at some arbitrary endpoint. There is no “good enough” that anarchists would ever settle for.
The same is of course true for science. As the string theorist Sean Carroll has loudly pointed out the institutional pressures are to turn science into an immediatist mechanical method that can be continually checked up on and controlled, and this has suppressed science’s path-exploration. It can be important to let one’s rigorous modeling grow out and not just instantaneously prune it demanding immediately verifiable experimental results. This is a topic I’ve been writing in depth on for years now and my “Science As Radicalism” will be out pretty soon.
In fairness, Graeber is certainly quite aware of some of the complications inherent in such a sweeping conceptual bundle as “bureaucracy” and his whole book is about trying to pick them apart. His primary instinct though is to turn to the tropes of anthropology to do so.
One of his main lenses is that of play and games, which he wants to distinguish as two separate concepts. The former more free-form and the latter a temporarily agreed upon structure or process. Calvinball versus chess.
Bureaucracies often emerge in part, Graber argues, from a fear of play. \emph{And sometimes this is legitimate}. Top-down play is horrifying, literally terrorism. The sociopath playing with a tortured mouse. When play happens in the context of strong and directed power relations it’s a form of abuse, and the impression it leaves is one of capriciousness, randomness, cruelty, etc. Games on the other hand create sandboxes, they confine what we need to think of to a limited subset of variables (or they bias the things we already pay tons of attention to in new directions), allowing us to stretch our brains around entirely new contexts and creatively build new strategies or ways of thinking. As such bureaucracies are often an attempt to limit the arbitrariness wielded by those with power.
This lens on the interrelation of the creativity in play with the violence attendant to sociopathy is highly overdue. I admire Graeber’s very practical take on the right being enviably clearheaded about this. Power and order (or ossified structures of low variability) are grounded in violence whereas creativity is fun but destabilizing. The conservative fear of random violence from destabilization leads them to try to confine both creativity and violence, and often treat the two inseparably. In a bureaucratic world the only ones capable of true creativity are the violent. Hence the shrinking of all art and imagination in conservative culture to things centered on or arising through violence. The conservative position is that violence and creativity are hugely beguiling and attractive, and the only tolerable way to let people engage in them is in order to suppress other violence\Slash{}creativity. In the reactionary mind imagination unleashed can \emph{only} lead to violence and destruction. Conservatives wish to use violence to minimize this. Fascists to embrace it. But both swallow the lack of alternatives wholesale. And both appreciate the nature of the present system far better than liberals.
Graeber seeks to navigate these tensions by gravitating towards a definition of freedom as both play and transparent rules. But I think this is insufficiently nuanced. Freedom is not mere negative freedom, freedom to wallow in disconnected delusions and fantasies locally. The only coherent notion of freedom is positive freedom, the freedom to act. And freedom requires means to transmit one’s intentions into actions, to impress one’s desires upon the world. Turning your head to soup may give a ton of local degrees of freedom, but it makes it impossible for that freedom to extend outward in effects. The difference between rules-as-constraining and rules-as-enabling must be judged in terms of their efficiency at maximizing the capacity for agents to act. Recognizing the “rules” of gravity makes it easier to walk. And negotiating some kind of process for mediating in conflicts makes it easier to avoid acting at loggerheads with one another.
I would and have argued that power is about cutting off degrees of freedom, ossifying, isolating, and generally reducing informational content. It’s nice how well this approach is independently generated by Graeber who notes that in many ways solitary confinement is one of the ultimate expressions of this tendency, cutting off a node from communication, or stopping it from questioning and exploring the space of possible models and dreams.
The natural next step in this is to talk about the internet.
Yet Graeber, while a fan, is dismissive about the internet not measuring up in comparison to promises of jetpacks. In one passage he astoundingly dismisses the internet as merely a “super fast and globally accessible combination of library, post office and mail order catalog.”
It’s always a bit stunning when someone in the anarchist milieu reveals their sheer level of antiquated disconnect. It’s like when John Zerzan says, “I can’t understand what anyone would ever find appealing about the internet, email is quite boring” There’s too much being glossed over in a willful shallowness. Folks will get to the point where people are directly connecting their brain tissue and these grandpas will describe it as, “a really fast form of texting.”
Graeber’s dismissal at the developments of the internet era is a sharp hypocrisy. One second he’s going on about the importance of imagination and the next second he’s getting mad about our increasing complexity of thought and the advent of the meta. Simulation, information tech, etc. are but phantasms to Graeber. I mean it’s jaw-dropping to hear an anthropologist write off exponential feedback in cultural complexity as no big deal. Or ignore the fecund possibilities for resistance that such a singularity of social complexity opens up. Of all the concessions to make in the service of rhetoric this really seems extreme.
Graeber presents the internet as a series of filling out forms, which is a cute lens for maybe a couple seconds until you really think about it. What the internet is really about is communicating better, in expanding a vast array of possible avenues by which to communicate, about establishing new languages and protocols. Any language requires condensing or translating down the concepts in our heads into what limited impressions of these can survive in words. So yes, there are whole new means, whole new paradigms available to communicate and within them we will need to express desires, and these must in some sense fit within protocols or languages in order to be parsed by anyone, and so this sometimes involves filling our name and password in boxes. And yes, we are focusing a chunk of our expanding desires on economic particulars (the arrangement of things) and thus we need economic languages.
Within the programming world there have been pressures to cut down on bureaucracy and form-filling by automating. But what radicals in tech have quickly discovered is that this removes agency. The push within the cryptography or cypherpunk community fighting the NSA has been to make crypto tools more accessible by removing distracting options, to get things down to “one big button” for security or trust. But this is insane. Human trust is an incredibly complex array of dynamics impossible to automatedly parallel without input from the user; no tool will meet every threat model out of box. To get something that works for each person and doesn’t fuck over certain demographics we need users to make choices. And this requires checkboxes and the like. How these are presented is of course no small issue and there’s lots of room for transparency as well as better communication and education. Because, yes, people will need to integrate some comprehension of the tools they’re using to use them best. But this is like comparing learning how to chip stone tools with filling out foodstamps paperwork. With proper tools the user has full agency and comes to an appreciation of its dynamics sufficient for them; with bureaucracy someone else makes those decisions and you are forced to make haphazard guesses about how to fit yourself into those boxes. The internet, the programming and cypherpunk worlds, have generated many examples drifting in either direction.
So why then does Graeber make this mistake with the internet? What is chaining him into adopting this sloppy rhetoric? Well he wants to argue that modern technological advancement is a farce, and this argument is deeply tied to old counter-globalization narratives. But — to don my market anarchist hat briefly — the fact of the matter is that the limited degree of free trade that \emph{does} exist under neoliberal globalization \emph{is} improving a lot of those factories established in the global south. Eventually they run out of people willing to work at the same or lower standards and improvement is forced to continue. I am certainly not remotely excusing the horrors of neoliberalism, or the intolerable slowness by which such tiny progress occurs, or arguing that it occurs everywhere. But there just isn’t some infinite pool of non-american workers in dispossessed destitution with no other options and they haven’t yet locked down a sufficiently large outright slave population. Although countries like Malaysia and Bangladesh are effectively slave colonies there is resistance on a lot of fronts that is limiting the capacity for such on a global scale. Of course the capitalists will and are trying to find new ways to harvest slaves but there are counter-pressures. There’s a reason a lot of exploitation has to be dressed up in the trappings of market freedom and personal agency. It’s not cuz they’re omniscient conspirators creating an ever more unassailable slavery, it’s because they’re under pressures that are only increasing. It’s an empirical fact that wealth is slipping out into the families of sweatshop workers and we’ve seen that wealth turn around and force improvements to conditions. When sweatshop owners are beheaded by their workers I grin — admitting that advancements in factory technologies have happened rather than just constantly churning through an endless supply of slave workers does not mean lessening our hostility to neoliberal slavery. We certainly don’t need to turn to preposterous arguments and rhetoric claiming that no meaningful or positive technological developments are happening.
While Graeber has the intellectual honesty to have said nice things about markets in the past and repeats some of these in The Utopia of Rules, I’ll just direct you to my coverage of this in my review of \emph{Debt} rather than rehash the same points here. What I will say is that Graeber’s acknowledgement that impersonal relations can be convenient and called for should lead toward a better appreciation for the appeal of freed markets.
Indeed many left market anarchists have in the last decade taken to critiquing anarcho-communists on the grounds that markets provide \emph{less} room for informal power dynamics, that they can make things clearer and more accessible and traversible to those autistics and the like who are systemically marginalized in the typical social capitalism of communal societies. This is new and productive work that I would love to see more discourse on. What we choose to invest cognitive or symbolic complexity in and what we simplify by default in order to explore complexity in other areas is a deep and important discussion for anarchists and there will obviously be no universal answers.
Graeber is at pains to point out that the interpretive depth of a subject does not relate to its significance. The most interesting dynamics are not always the most important, and frequently the really basic realities get overlooked by social theorists hungry for things they can spend forever nuancing or complicating.
The problem is that by appealing to a very big and abstract bundle of associations like “bureaucracy” and using the messy language of anthropologists and social theorists Graeber is partially engaging in precisely this sin. He’s aware of it, and it enables him to flesh out an entire book tracing the complexities (and thus fend off the lovecraftian horrors he’s made deals with for another year), but it creeps in as a kind of sloppiness time and time again.
It’s not enough to merely map out the idea space traversing around endlessly, but to probe it and restructure models of it to get down to the roots. Graeber seems to desperately want to do that, which is highly encouraging, and he gets part of the way in audaciously whittling away towards the underlying dynamics but he frustratingly stops short before tackling the issues of informational complexity in any rigorous way.
Still you should read this book. It’s pretty good.
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The Anarchist Library
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Anti-Copyright
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\begin{center}
William Gillis
Beyond the Hellish Choice of Process Documents or Social Capitalism
June 10th, 2015
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https:\Slash{}\Slash{}c4ss.org\Slash{}content\Slash{}38077
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\textbf{theanarchistlibrary.org}
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https://personal.math.ubc.ca/~gerg/teaching/613-Winter2011/template.tex | ubc.ca | CC-MAIN-2022-33 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-33/segments/1659882571745.28/warc/CC-MAIN-20220812170436-20220812200436-00311.warc.gz | 430,940,455 | 6,588 | \documentclass[12pt,reqno]{amsart}
\usepackage{fullpage}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{times}
\usepackage{graphicx}
\vfuzz=2pt
% some "funny lines" referred to later:
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
{ \theoremstyle{remark}\newtheorem*{remark}{Remark} }
\newcommand{\class}{MATH 613E: Topics in Analytic Number Theory}
\newcommand{\population}[2]{There are #2 students giving lectures, and also #2 students writing up notes from lectures. Everybody in the class is very #1 about this.}
\newcommand{\tex}{{\tt .tex}}
\begin{document}
\title[Short version of title]{Title of the article}
% or if you want, simply \title{Title of the article}
\author{Your Name}
\email{[email protected]}
\maketitle
\begin{abstract}
Here's a \LaTeX\ template.
\end{abstract}
\section{Introduction}
\label{first section}
The content of this manuscript is not interesting. The purpose of this manuscript is to provide a template for using \LaTeX\ to complete your written assignment for \class. It can also help as a general \LaTeX primer. For example, look
at
this sentence in the file { \tt template.tex} to see that
\LaTeX
\ mostly ignores
white
space. (If you're keen, look closely at the \tex\ file to figure out why the ``\LaTeX'' got stuck to the word "primer" in the 3rd sentence of this article, but was fine in the sentence before that. Now look at the sentence before this one to see how \LaTeX\ processes quotation marks.) In general, comparing the output to the \tex\ file will teach you (if you don't already know) how do to all sorts of things.
A blank line in the \tex\ file makes a new paragraph. Don't try to do line breaks or page breaks yourself; \LaTeX\ does that automatically.
You can {\em emphasize parts of the text}. If you really want to control how text looks, you can {\it put things in italics} or {\bf in boldface} or in a {\tt fixed-width font}.
You can make comments in the \tex\ file that don't appear in the output, if it helps you.
% Here is one such comment.
Between this sentence and the previous one % and also right in the middle of this one!
is one such comment.
In Section~\ref{math}, we'll talk about typesetting mathematics; in Section~\ref{autonumber}, which starts on page~\pageref{autonumber}, we'll talk about how to automatically number equations, theorems, and sections (some automatically generated numbers appear in this very sentence). Section~\ref{macros} discusses macros, which are definitions of your own commands. Finally, in Section~\ref{how to write} we'll suggest a format for your writing assignment for \class.
% Minor point: each ~ makes a space where a line break is not allowed.
Feel free to experiment with changing your own copy of this file to see what happens; an original copy will remain on the course web page should you need it again.
\section{Typesetting math}
\label{math}
\subsection{Some basics}
Math can be typeset two different ways:
\begin{enumerate}
\item It can be typeset {\em inline}, such as this: $c^2 = a^2+b^2-2ab\cos c$.
\item It can be {\em displayed}, such as this:
\[
c^2 = a^2+b^2-2ab\cos c.
\]
\end{enumerate}
\begin{lem}
Some things look different when typset inline or displayed: $\int_0^1 \frac{2x}{x^2+1} \, dx = \sum_{k=1}^\infty (-1)^{k-1}/k$, but:
\label{display lemma}
\[
\int_0^1 \frac{2x}{x^2+1} \, dx = \sum_{k=1}^\infty (-1)^{k-1}/k.
\]
\end{lem}
There are ways to override this: $\displaystyle \int_0^1 \frac{2x}{x^2+1} \, dx = \sum_{k=1}^\infty (-1)^{k-1}/k$ or $\int_0^1 {\displaystyle \frac{2x}{x^2+1} } \, dx = \sum_{k=1}^\infty (-1)^{k-1}/k$ or $\int_0^1 \frac{2x}{x^2+1} \, dx = \sum\limits_{k=1}^\infty (-1)^{k-1}/k$ or
\[
\textstyle \int_0^1 \frac{2x}{x^2+1} \, dx = \sum_{k=1}^\infty (-1)^{k-1}/k
\]
or
\[
\int\limits_0^1 \tfrac{2x}{x^2+1} \, dx = \sum\nolimits_{k=1}^\infty (-1)^{k-1}/k.
\]
Notice that \LaTeX\ is good at making line breaks in prose but can sometimes get overwhelmed by line breaks in inline math.
Greek letters like $\alpha,\beta,\gamma,\dots$ are available, as well as those capitals $\Gamma,\Delta,\dots$ that don't just look like Roman letters. Lots of other symbols are defined as well---Google around to find them.
\newcommand{\li}{\mathop{\rm li}} % \mathop makes LaTeX treat the "li" like a math operator, like sin and exp. There's a similar thing, \mathrel, for relations like < and \sim.
\begin{remark}
We analytic number theorists use certain notation like $\log (x+\sin x) = \log x + O(1)$ and $\log x\ll_\epsilon x^\epsilon$ and $(x-1)^3 \gg x^3$ and $\pi(x) \sim \li(x)$ and $\pi(x) - \li(x) = o\big( x(\log x)^{-A} \big)$ a lot.
\end{remark}
In this course there are a lot of sums with multiple conditions of summation, so it's useful to know how to make operators like
\begin{equation}
\sum_{\substack{ n\le x \\ n\equiv 3\mod 4}} \quad\text{and}\qquad \prod_{\substack{ p>p_0 \\ \text{$p$ prime} \\ p\equiv 3\pmod 4}}.
\label{bad mods}
\end{equation}
\begin{prop}
\label{subs and supers proposition}
Subscripts, superscripts, and other doodads don't need brackets or spaces when they're a single symbol:
\begin{equation*}
\int_{a}^{b} \frac{8}{9} t_{3}^{\pi} \,dt = \int_a^b \frac89 t_3^\pi \,dt
\end{equation*}
But they do when they're more than one symbol:
\[
\int_{3/4}^{2b} \frac{2x}{x^2+1} t_{odd} \,dt \ne \int_3/4^2b \frac 2x x^2+1 t_odd \,dt
\]
\end{prop}
You can make big parentheses and other delimiters by hand,
\[
\bigg( \Big[ \big\{ ( 4 ) \big\} \Big] \bigg) + \Big( ( \bigg( \big( 7 \bigg) \big) \Big) ),
\]
or you can let \LaTeX\ do it for you:
\[
\left( \sum_{n\le x} \left[ \left\{ \ell \right\} \left\{ 3^k \right\} x^{2^k} \right] \right) + \left( \prod_p \left[ 1 - 1/p^2 \right\} \right.
\]
\subsection{Displays with multiple lines}
If you have a very long {\it expression}, you can put it on multiple lines:
\begin{multline*}
\frac1{(1-x)^2} = \sum_{k=0}^\infty (k+1)x^k = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5+7x^6+8x^7 \\ +9x^8 +10x^9+11x^{10} + 12x^{11} + 13x^{12} + 14x^{13} + \cdots
\end{multline*}
If you have a series of {\it equations or inequalities}, you can put them on multiple lines and align them:
\begin{align*}
\frac1{(1-x)^2} &= \sum_{k=0}^\infty (k+1)x^k \\
&= 1 + \sum_{k=1}^\infty (k+1)x^k \\
&= 1+2x+\sum_{k=2}^\infty (k+1)x^k = \cdots
\end{align*}
You can split an expression by hand if it's necessary:
\begin{align*}
\frac1{(1-x)^2} &= \sum_{k=0}^\infty (k+1)x^k \\
&= 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5+7x^6+8x^7 \\
&\qquad{}+ 9x^8 +10x^9+11x^{10} + 12x^{11} + 13x^{12} + 14x^{13} + \cdots \\
&= \bigg( \sum_{k=0}^\infty x^k \bigg)^2.
\end{align*}
\section{Automatic numbering}
\label{autonumber}
In my opinion, the most indispensable part of \LaTeX\ is its ability to automatically number equations, theorems (and lemmas etc.), sections, page numbers, and bibliographic references---so that if you need to move material around while editing, the numerical references will automatically update themselves. Compare the \tex\ file to the output to see how this is done. Some examples already appeared back on page~\pageref{first section} in Section~\ref{first section}; here are some others.
\begin{thm}
Every even integer is followed by an odd integer.
\label{even then odd}
\end{thm}
\begin{proof}
If $n$ is an even integer, then set
\begin{equation}
m = n+1;
\label{consecutives}
\end{equation}
then $m$ is odd.
\end{proof}
\begin{thm}
Every odd integer is followed by an even integer.
\label{odd then even}
\end{thm}
\begin{proof}
If $n$ is an odd integer, and $m$ is defined as in equation~\eqref{consecutives},
then $m$ is even.
\end{proof}
\begin{remark}
In Theorems~\ref{even then odd} and~\ref{odd then even}, the word ``followed'' can be replaced by ``preceded'', although the proofs would need to be redone. See \cite {ExampleBook} for lots of facts not particularly related to this. You might also look at \cite[page 217] {ExamplePaper}, although I have no idea what's there.
\end{remark}
\begin{cor}
There are almost exactly as many even integers as odd integers between $1$ and $x$.
\end{cor}
Some of the ``funny lines'' at the beginning of the \tex\ file control how theorems and their ilk are numbered: the system used in this document is for theorem numbers to start over in every section (2.1, 2.2, 2.3, then 3.1, etc.). Also, theorems, corollaries, propositions, and lemmas all use the same counter, so that after Lemma~\ref {display lemma} comes Proposition~\ref{subs and supers proposition}, not Proposition~\ref {display lemma}. (This makes it easier for readers to find things.) {\em Equation} numbering, on the other hand, goes sequentially throughout the document (not resetting every section) on its own counter. All of these choices can be changed if you want, but I recommend the given settings.
\section{Macros}
\label{macros}
You can define your own macros to make repetitive phrases easier to type. For example, if you are reading this, you're probably taking \class. \population{ecstatic}{7} Those macros were defined at the beginning of the \tex\ file. Another macro is re-defined right after this sentence in the \tex\ file.
\renewcommand{\mod}[1]{{\ifmmode\text{\rm\ (mod~$#1$)}\else\discretionary{}{}{\hbox{ }}\rm(mod~$#1$)\fi}}
If you're like me and don't like the way the {\tt mod} and {\tt pmod} commands work, as in equation~\eqref{bad mods}, you can use the definition just before this sentence in the \tex\ file (which is complicated so that it will work correcly both inline and displayed). Now the {\tt\textbackslash mod} command displays like this:
\begin{equation}
\sum_{\substack{ n\le x \\ n\equiv 3\mod 4}}
\end{equation}
You can put macro definitions and re-definitions anywhere in the \tex\ file, and they will be active from that point on. You can even put them inside sub-environments to make them temporarily in force if you really want to:
\[
\alpha+\beta=\beta+\alpha
\]
\begin{lem}
\renewcommand{\alpha}{ALPHA}
$\alpha+\beta=\beta+
\renewcommand{\alpha}{{\rm cx}}
\alpha$.
\end{lem}
$
{
\renewcommand{\alpha}{1st\ Greek\ letter}
\alpha
}
+\beta=\beta+
\alpha
$
\[
\alpha+\beta=\beta+\alpha
\]
That being said, for most purposes macros can simply be put at the top of the \tex\ file.
\begin{remark}
By this time, if you've been playing around with the \tex\ file, you might have gotten some errors while compiling (for example, you forget a closing bracket or something). The only helpful thing I can really say about that is that there are only a few types of errors that come up frequently, and you'll learn to figure out how to fix them. Occasionally you'll have to decipher why a correctly compiling file doesn't do what you think it should do; hopefully the above examples will help a bit with that. All answers are known, by someone!
\end{remark}
\section{Suggested structure of your article}
\label{how to write}
% itemize
It's probably still a bit unclear exactly what you're supposed to produce for the written part of your course assessment: we've been using a lot of words---notes, expository article, etc.---that don't all match with one another. Here's my attempt at clarifying these expectations:
\begin{quote}
You should produce an article, on the topic assigned to you, that could serve as a (short) chapter in a good textbook on the subject of the course. The article should contain everything that you would say in class, if you had all the time you wanted to lecture on the topic.
\end{quote}
Some observations on this goal:
\begin{itemize}
\item The article should be written in complete sentences and paragraphs, with mathematics inserted where appropriate. It shouldn't be a series of equations with no explanation. See your favorite textbook for an example.
\item The article shouldn't be a historical record of what the lecturer actually said. You'll have more time to write the article than the lecturer had to give the lectures, and so you'll be able to give a more complete and polished account.
\item On the other hand, the article doesn't need to be written like a paper to be published in a research journal. Even if you had infinite time to lecture on the topic, that doesn't necessarily mean you have to include every single detail. Many intermediate results might be sufficiently covered in the prerequisite analytic number theory course, or in a topic selected by another lecturer. In fact, it might simply be that a particular technical lemma is so complicated that it would detract from the communication of the main topic. Don't use this as an excuse to leave out pertinent details, but it is an option where appropriate---just communicate clearly to the reader that this is what you've done.
\item Include specific references to textbooks or published papers (supplemented, if appropriate, by excellent internet resources).
\end{itemize}
Here's a suggestion on how to organize your article (other outlines are also possible).
\begin{itemize}
\item Introduce the topic
\begin{itemize}
\item Something the reader already knows that's related to the new topic
\item What new question do we want to answer?
\item Give context and history
\item What is conjectured?
\end{itemize}
\item State the main result(s) (perhaps there has been a series of related results---some of them can be stated rigorously but without proving each and every one)
\item Describe the overall strategy of the proof
\item Break out some technical parts of the proof into preliminary lemmas and propositions (try to describe their role in the overall proof as you go)
\item Give the proof(s) of the main result(s)
\item Mention possible improvements and directions for future research
\item Bibliography
\end{itemize}
\subsection*{The bottom line}
If you want to run your ideas by me, get a reality check from me, or just ask for advice for a starting point, you are always welcome to ask me in class or in my office, or to email me.
\begin{thebibliography}{99} % don't worry about the 99
\bibitem{ExamplePaper}
E.~Edelman, ``The probability that a random real Gaussian matrix has $k$ real eigenvalues, related distributions, and the circular law'', {\em J.~Multivariate~Anal.} {\bf 60} (1997), no.~2, 202--232.
\bibitem{ExampleBook}
H.~L.~Montgomery and R.~C.~Vaughan, {\em Multiplicative Number Theory I: Classical Theory}, Cambridge University Press (2007).
\end{thebibliography}
\end{document}
|
https://ctan.math.washington.edu/tex-archive/info/examples/PSTricks_en/15-04-25.ltx | washington.edu | CC-MAIN-2022-27 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-27/segments/1656103619185.32/warc/CC-MAIN-20220628233925-20220629023925-00760.warc.gz | 242,279,927 | 1,090 | %%
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http://porocila.imfm.si/2008/mat/clani/bokal.tex | imfm.si | CC-MAIN-2023-14 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296943750.71/warc/CC-MAIN-20230322051607-20230322081607-00138.warc.gz | 39,088,090 | 3,218 | \clan {Drago Bokal}
%--------------------------------------------------------
% A. objavljene znanstvene monografije
%--------------------------------------------------------
%\begin{skupina}{A}
%\disertacija
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\magisterij
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\monografija
% {AVTORJI}
% {NASLOV}
% {ZALOZBA}
% {KRAJ} {DRZAVA} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% B. raziskovalni clanki sprejeti v objavo v znanstvenih
% revijah in v zbornikih konferenc
%--------------------------------------------------------
%\begin{skupina}{B}
%\sprejetoRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA}
%\sprejetoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% C. raziskovalni clanki objavljeni v znanstvenih revijah
% in v zbornikih konferenc
%--------------------------------------------------------
%\begin{skupina}{C}
%\objavljenoRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\objavljenoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
%\end{skupina}
\begin{skupina}{C}
\objavljenoZbornik
% 1.08: {\bf 3}. BOKAL, Drago, FIJAV\v{Z}, Ga\v{s}per, HAREJ, Bor,
%TARANENKO, Andrej, \v{Z}AGAR, Klemen. A modular hybrid approach to
%employee timetabling. V: 7th International Conference on the Practice
%and Theory of Automated Timetabling - PATAT 2008, Universit\'{e} de
%Montr\'{e}al, August 18 - 22, 2008. {\it Complete program}. Montr\'{e}al:
%Universit\'{e} de Montr\'{e}al, 2008, 12 str. http://w1.cirrelt.ca/
%\~{}patat2008/PATAT\_7\_PROCEEDINGS/Papers/Fijavz-HA3b. pdf.
%$[$COBISS.SI-ID 14940249$]$\\
%POZOR: Bibliografija2008.tex > 2008\mat\clani\bokal.tex 700/80:
%Ne najdem podatka {DRZAVA}
%POZOR: Bibliografija2008.tex > 2008\mat\clani\bokal.tex 700/81:
%Ne najdem podatka {ZBORNIK}
{\crta, G.~Fijav\v{z}, B.~Harej, A.~Taranenko, K.~\v{Z}agar}
{A modular hybrid approach to employee timetabling}
{7$^{\rm th}$ International Conference on the Practice and Theory of Automated Timetabling -- PATAT 2008}
{Montr\'{e}al} {Kanada} {avgust} {2008}
{Complete program} {12 str}
\objavljenoRevija
% 1.01: {\bf 1}. BOKAL, Drago, FIJAV\v{Z}, Ga\v{s}per, WOOD, David Richard.
%The minor crossing number of graphs with an excluded minor. {\it Electron.
%j. comb. (On line)}. $[$Online ed.$]$, 2008, vol. 15, no. 1, r4 (13 str.).
%http://www.combinatorics.org/Volume\_15/PDF/v15i1r4.pdf.
%$[$COBISS.SI-ID 14499417$]$\\
{\crta, G.~Fijav\v{z}, D.~R.~Wood}
{The minor crossing number of graphs with an excluded minor}
{Electron.\ J. Combin.\ (On line)} {15} {2008} {1} {13 str.}
\objavljenoRevija
% 1.01: %list {\bf 2}. REDDING, David W., HARTMANN, Klass, MIMOTO, Aki,
%BOKAL, Drago, DEVOS, Matt, MOOERS, Arne {\O}. Evolutionarily distinctive
%species often capture more phylogenetic diversity than expected. {\it J.
%theor. biol.}, 2008, vol. 251, iss. 4, str. 606-615.
%http://dx.doi.org/10.1016/j.jtbi.2007.12.006. $[$COBISS.SI-ID 14945625$]$\\
{D.~W.~Redding, K.~Hartmann, A.~Mimoto, \crta, M.~DeVos, A.~\O., Mooers}
{Evolutionarily distinctive species often capture more phylogenetic diversity than expected}
{J. Theoret.\ Biol.} {251} {2008} {4} {606--615}
\end{skupina}
%--------------------------------------------------------
% D. urednistvo v znanstvenih revijah in zbornikih
% znanstvenih konferenc
%--------------------------------------------------------
%\begin{skupina}{D}
%\urednikRevija
% {OPIS}
% {REVIJA}
%\urednikZbornik
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% E. organizacija mednarodnih in domacih znanstvenih
% srecanj
%--------------------------------------------------------
%\begin{skupina}{E}
%\organizacija
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% F. vabljena predavanja na tujih ustanovah in
% mednarodnih konferencah
%--------------------------------------------------------
%\begin{skupina}{F}
%\predavanjeUstanova
% {NASLOV}
% {OPIS}
% {USTANOVA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\predavanjeKonferenca
% {NASLOV}
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% G. aktivne udelezbe na mednarodnih in domacih
% konferencah
%--------------------------------------------------------
%\begin{skupina}{G}
%\konferenca
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
\begin{skupina}{G}
\konferenca
% 1.08: {\bf 3}. BOKAL, Drago, FIJAV\v{Z}, Ga\v{s}per, HAREJ, Bor, TARANENKO, Andrej, \v{Z}AGAR, Klemen. A modular hybrid approach to employee timetabling. V: 7th International Conference on the Practice and Theory of Automated Timetabling - PATAT 2008, Universit\'{e} de Montr\'{e}al, August 18 - 22, 2008. {\it Complete program}. Montr\'{e}al: Universit\'{e} de Montr\'{e}al, 2008, 12 str. http://w1.cirrelt.ca/\~{}patat2008/PATAT\_7\_PROCEEDINGS/Papers/Fijavz-HA3b. pdf. $[$COBISS.SI-ID 14940249$]$\\
%POZOR: Bibliografija2008.tex > 2008\mat\clani\bokal.tex 700/178: Ne najdem podatka {DRZAVA}
{A modular hybrid approach to employee timetabling}
{7$^{\rm th}$ International Conference on the Practice and Theory of Automated Timetabling -- PATAT 2008}
{Montr\'{e}al} {Kanada} {avgust} {2008}
\konferenca
% 1.12: {\bf 4}. BOKAL, Drago, FIJAV\v{Z}, Ga\v{s}per, WOOD, David Richard. Planar decompositions and the minor crossing number of graphs. V: SIAM conference on discrete mathematics, June 16-19, 2008, University of Vermont, Burlington, Vermont. {\it Final program and abstracts}. Philadelphia: SIAM, 2008, str. 45. http://www.siam.org/meetings/dm08/abstractsDM08.pdf. $[$COBISS.SI-ID 14887769$]$\\
%POZOR: Bibliografija2008.tex > 2008\mat\clani\bokal.tex 714/186: Ne najdem podatka {DRZAVA}
%POZOR: Bibliografija2008.tex > 2008\mat\clani\bokal.tex 714/187: Ne najdem podatka {MESEC}
{Planar decompositions and the minor crossing number of graphs}
{SIAM conference on discrete mathematics}
{Burlington} {ZDA} {junij} {2008}
\konferenca
% 1.12: {\bf 4}. BOKAL, Drago, FIJAV\v{Z}, Ga\v{s}per, WOOD, David Richard. Planar decompositions and the minor crossing number of graphs. V: SIAM conference on discrete mathematics, June 16-19, 2008, University of Vermont, Burlington, Vermont. {\it Final program and abstracts}. Philadelphia: SIAM, 2008, str. 45. http://www.siam.org/meetings/dm08/abstractsDM08.pdf. $[$COBISS.SI-ID 14887769$]$\\
%POZOR: Bibliografija2008.tex > 2008\mat\clani\bokal.tex 714/186: Ne najdem podatka {DRZAVA}
%POZOR: Bibliografija2008.tex > 2008\mat\clani\bokal.tex 714/187: Ne najdem podatka {MESEC}
{On some results relating cuts and crossing numbers}
{Aspectos Combinatorios y Computacionales de Optimizaci\'{o}n, Topolog\'{i}a y Algebra}
{Oaxaca} {Mehika} {december} {2008}
\end{skupina}
%--------------------------------------------------------
% H. strokovni clanki
%--------------------------------------------------------
%\begin{skupina}{H}
%\clanekRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\clanekZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% I. razno
%--------------------------------------------------------
%\begin{skupina}{I}
%\razno
% {OPIS}
%\end{skupina}
%\begin{skupina}{I}
%POZOR: Bibliografija2008.tex > 2008\mat\clani\bokal.tex 740/228: Stevilo neopredeljenih zadetkov: 2
%\razno
% 1.25: {\bf 5}. JURI\v{S}I\'{C}, Aleksandar, BATAGELJ, Vladimir, LUK\v{S}I\v{C}, Primo\v{z}, ORBANI\'{C}, Alen, \v{Z}EROVNIK, Janez, BOKAL, Drago, TRONTELJ, Zvonko. Aplikativni dose\v{z}ki. V: KLAV\v{Z}AR, Sandi (ur.), MR\v{C}UN, Janez (ur.). {\it Predstavitev In\v{s}tituta za matematiko, fiziko in mehaniko}. Ljubljana: IMFM, 2008, str. 49-59. $[$COBISS.SI-ID 14691417$]$\\
%\razno
% 2.13: {\bf 6}. OMAHEN, Gregor, VALEN\v{C}I\v{C}, Leon, BREGAR, Zvonko, PERME, Jo\v{z}e, MATVOZ, Dejan, ZLATAREV, Georgi, KERNJAK, Maja, BOKAL, Drago, JEREBIC, Janja{\it . Vrednotenje in optimiranje nalo\v{z}b v dolgoro\v{c}nih na\v{c}rtih razvoja prenosnega omre\v{z}ja : \v{s}tudija \v{s}t. 1922}. Ljubljana: Elektroin\v{s}titut Milan Vidmar, 2008. 89 f. $[$COBISS.SI-ID 30225925$]$\\
%\end{skupina}
%--------------------------------------------------------
% tuji gosti
%--------------------------------------------------------
%\begin{seznam}
%\gost {IME} {TRAJANJE} {USTANOVA} {KRAJ} {DRZAVA} {MESEC} {LETO} {POVABILO}
%\end{seznam}
%--------------------------------------------------------
% gostovanja
%--------------------------------------------------------
%\begin{seznam}
%\gostovanje {Drago Bokal} {7 dni} {University of South Carolina} {Columbia, SC} {ZDA} {junij} {2008}
%\gostovanje {Drago Bokal} {7 dni} {University of Waterloo} {Waterloo, ON} {Kanada} {junij} {2008}
%\gostovanje {Drago Bokal} {14 dni} {Centro de Investigaci\'{o}n en Matemam\'{a}ticas, A.C.} {Guanajuato} {Mehika} {julij} {2008}
%\end{seznam}
|
http://www.ma.utexas.edu/users/allcock/research/symmsurf1.tex | utexas.edu | CC-MAIN-2017-30 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2017-30/segments/1500550977093.96/warc/CC-MAIN-20170728183650-20170728203650-00092.warc.gz | 477,590,693 | 3,878 | % Hyperbolic surfaces with prescribed symmetry
% Daniel Allcock
% April 2005
\documentclass[12pt]{amsart}
\usepackage{amssymb}
\overfullrule=10pt
\newcommand{\e}{\varepsilon}
\newcommand{\aut}{\mathop{\rm Aut}\nolimits}
\newcommand{\isom}{\mathop{\rm Isom}\nolimits}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\newcommand\Z{\mathbb{Z}} % integers
\newcommand\sset{\subseteq}
% theorem-like environments
\theoremstyle{plain}
\newtheorem*{theorem}{Theorem}
\newtheorem*{lemma}{Lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\title[Prescribed symmetry groups]{Hyperbolic surfaces with prescribed
infinite symmetry groups}
%
\author{Daniel Allcock}
\address{Department of Mathematics\\University of Texas at Austin\\Austin, TX 78712}
\email{[email protected]}
\urladdr{http://www.math.utexas.edu/\textasciitilde allcock}
%
\date{April 17, 2005}
\thanks{Partly supported by NSF grant DMS-024512.}
%
\begin{abstract}
For any countable group $G$ whatsoever, there is a complete
hyperbolic surface whose isometry group is $G$. The argument is
elementary.
\end{abstract}
\subjclass[2000]{Primary 51M09, Secondary 20F99}
\maketitle
Greenberg proved in 1972 \cite{greenberg} that for any finite group
$G$ there is a closed hyperbolic 2-manifold having isometry group
isomorphic to~$G$. Kojima proved the corresponding result for
hyperbolic 3-manifolds \cite{kojima}. Following work of Long and Reid
\cite{long-reid}, Belolipetsky and
Lubotzky extended this to hyperbolic manifolds of arbitrary dimension
\cite{BL}. Our purpose here is to give a generalization in a
different direction, which also provides an elementary proof of
Greenberg's original theorem.
\begin{theorem}
\label{thm}
For any countable group $G$, there is a complete hyperbolic
surface $X$ having isometry group isomorphic to $G$. When $G$ is
finite, $X$ may be taken to be closed.
\end{theorem}
We build $X$ by gluing together pairs of pants. A pair of pants is a
surface homeomorphic to the 2-sphere minus three open disks whose
closures are disjoint, equipped with a hyperbolic metric under which
the boundary curves are geodesic. (All boundary curves in this note
are assumed geodesic.) Up to isometry, there is a
unique pair of pants whose boundary lengths are any three given
positive numbers. The standard way to build pairs of pants is by
doubling right-angled hexagons; see \cite{thurston}. This
construction makes the following lemma obvious.
\begin{lemma}
\label{lem}
There exists $\e>0$ such that if $P$ is a pair of pants with all its
boundary curves of length${}<\e$, then
(1) the boundary curves are
the only closed geodesics on $P$ of length${}<\e$, and (2) any
geodesic interval in $P$ of length${}<\e$ with its endpoints in $\partial
P$ lies entirely in $\partial P$.
\qed
\end{lemma}
We call a closed geodesic on a hyperbolic surface short if it has
length${}<\e$, and very short if it has length${}<\e/2$. The point of
the lemma is that on a hyperbolic surface obtained by gluing together
pairs of pants with all short boundary curves, the short curves are
exactly the curves along which the pairs of pants were glued. (1)
deals with short closed geodesics that lie in a single pair of pants
and (2) rules out the existence of any others.
\begin{proof}[Proof of the theorem:]
Let $I$ be a set indexing a sequence of group elements $a_i$ that
generate $G$; by
enlarging $I$ we may suppose $|I|\geq2$. The Cayley graph $C$ of $G$
has automorphism group equal to $G$, and we will use $C$ as a sort of
framework on which to build $X$. Explicitly, $C$ has vertex set
$G$, and for each $g\in G$ and each $i\in I$ there is a directed edge
from $g$ to $ga_i$ labeled `$i$'. An automorphism of $C$ is an
automorphism of the underlying graph which preserves the directions
and labels of edges. $G$ acts on $C$ by left-multiplication, and it
is easy to see that $G$ is all of $\aut C$. (An automorphism $g$ of
$C$ permutes the vertices in a manner commuting with
$G$'s right-multiplication action on itself; since this action is
transitive, $g$ is completely determined by what it does to any one
vertex.)
We choose positive numbers $\a_i$, $\b_i<\e/2$ for $i\in I$, all
distinct. We build for each $i$ a compact hyperbolic surface $E_i$
with two boundary curves, of lengths $\a_i$ and $\b_i$. We may
suppose that these are the only very short curves on $E_i$, say by
building $E_i$ from two pairs of pants, with boundary curves of
lengths $\a_i$, $\e/2$ and $\e/2$ (resp. $\b_i$, $\e/2$ and $\e/2$),
glued together along their boundary curves of length $\e/2$. We call
$E_i$ the $i$th `edge surface'. Next we build a complete hyperbolic
surface $V$ with $2\cdot|I|$ boundary curves, of lengths $\a_i$ and
$\b_i$, say by gluing a sequence of pairs of pants together along
boundary curves which are short but not very short. The shortness of
the boundary curves of the pairs of pants assures us that the only very short
geodesics on $V$ are its boundary curves. We may suppose that each
of the pairs of pants has boundary components of three distinct
lengths; this lets us arrange for $V$ to have trivial isometry group,
as follows. $\isom(V)$ must permute the short geodesics of $V$, hence
permute the pairs of pants, hence carry to itself the unique one of
these pairs of pants with a boundary curve of length $\a_0$, $0$ being some fixed
element of $I$. A pair of pants with boundary curves of three
distinct lengths has isometry group $\Z/2$, and by rotating the gluing
of an adjacent pair of pants, we can make sure that the nontrivial
isometry does not extend to an isometry of $V$. We call $V$
the `vertex surface'.
Now we build $X$. We take one copy of the vertex surface for each
vertex of $C$ and attach one copy of the edge surface $E_i$ for each
edge of $C$ labeled $i$. We attach the length $\a_i$ (resp. $\b_i$)
boundary curve of each copy of $E_i$ to the length $\a_i$
(resp. $\b_i$) boundary curve of the copy of $V$ from which the edge
originates (resp. to which it points). This may be done in a manner
compatible with the action of $G$ on $C$, so that $G\sset\isom X$.
To see that $G$ is all of $\isom X$, suppose $g:X\to X$ is an
isometry. Then it permutes the very short geodesics of $X$ and hence
the components of their complement, which are the (interiors of the)
edge and vertex surfaces we used to build $X$. It must carry
components with two boundary curves to other such components, so it
carries edge (resp. vertex) surfaces to edge (resp. vertex) surfaces.
Since the $\a_i$ and $\b_i$ are all distinct, it carries edges
labeled $i$ to other such edges, and preserves their directions.
Therefore $g$ acts on $C$; by composing $g$ with an element of $G$ we
may suppose that $g$ fixes a vertex of $C$. Since we chose $V$ to
have no isometries, $g$ must be the identity.
It is clear that $X$ is complete; it may be taken to be
compact when $G$ is finite by taking $I$ finite and $V$ compact.
\end{proof}
\begin{thebibliography}{99}
\bibitem{BL}
M. Belolipetsky and A. Lubotzky, Finite Groups and Hyperbolic
Manifolds, to appear in {\it Invent. Math.}
\bibitem{greenberg}
L. Greenberg, Maximal groups and signatures,
{\it Discontinuous groups and Riemann surfaces (Proc. Conf.,
Univ. Maryland, College Park, Md., 1973)}, Ann. of Math. Studies,
No. 79, Princeton University Press, 1974, pp. 207--226
\bibitem{kojima}
S. Kojima, Isometry transformations of hyperbolic 3-manifolds, {\it
Topology and its Applications} {\bf 29} (1988) 297--307
\bibitem{long-reid}
D. D. Long and A. W. Reid, On asymmetric hyperbolic manifolds, to appear
{\it Math. Proc. Camb. Phil. Soc.}
\bibitem{thurston}
W. Thurston, {\it Three-dimensional geometry and topology.},
ed. Silvio Levy. Princeton Mathematical Series, 35.
Princeton University Press, 1997
\end{thebibliography}
\end{document}
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http://ctan.unixbrain.com/info/examples/Math/6-13-1.ltx | unixbrain.com | CC-MAIN-2013-20 | text/x-tex | null | crawl-data/CC-MAIN-2013-20/segments/1368696382396/warc/CC-MAIN-20130516092622-00060-ip-10-60-113-184.ec2.internal.warc.gz | 63,991,621 | 996 | %%
%% Der Mathematiksatz mit LaTeX, 1. Auflage 2009
%%
%% Example 6-13-1 on page 121.
%%
%% Copyright (C) 2009 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.
%%
\documentclass[]{ttctexa}
\pagestyle{empty}
\setcounter{page}{6}
\setlength\textwidth{170.59192pt}
\AtBeginDocument{\setlength\parindent{0pt}}
\StartShownPreambleCommands
\usepackage{amsmath}
\newcommand*\Modulo{\operatorname{Modulo}}
\StopShownPreambleCommands
\begin{document}
\begin{align*}
y &= 135\Modulo17=16\\
y &= 135\mathrm{Modulo}17=16
\end{align*}
\end{document}
|
https://git.tetalab.org/tTh/FloatImg/raw/commit/34ab825670cccac66250a6483b61557220de52c3/doc/the_floatimg_hack.tex | tetalab.org | CC-MAIN-2022-27 | text/plain | application/x-tex | crawl-data/CC-MAIN-2022-27/segments/1656103646990.40/warc/CC-MAIN-20220630001553-20220630031553-00588.warc.gz | 339,035,227 | 19,724 | \documentclass[a4paper,10pt]{article}
% \listfiles % pour le debug
\usepackage[french]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{makeidx}
\usepackage{listings}
% \lstset{frame=single} % dessin d'un cadre autour du listing
\lstset{basicstyle=\ttfamily\small}
\lstset{aboveskip=0.222em,belowskip=0.222em}
\usepackage{babel} % ?
\usepackage{graphicx} % for gnuplot ylabel rotate
\usepackage{pifont} % caractères rigolos
\usepackage{enumitem}
\setitemize[1]{label={\ding{87}}}
\frenchbsetup{CompactItemize=false}
% \usepackage{color}
% \usepackage{url}
\usepackage{xspace}
\usepackage[verbose]{layout}
\setlength \parskip {0.35em}
\makeatletter
% explication de ce truc ?
\def\verbatim@font{\normalfont\ttfamily\small}
\makeatother
\setlength{\hoffset}{0em}
\setlength{\textheight}{640pt}
\setlength{\textwidth}{422pt}
\setlength{\marginparwidth}{10pt}
\makeindex
% ------ a few new commands
\newcommand{\interparagraphe { \vspace{60pt} } }
% -------------------------------------------------------------------
\title{Floating images processing}
\author{tTh}
\begin{document}
\maketitle
\section{Image flottante ?}
Mais de quoi parle-t-on exactement ?
% XXX XXX XXX\vspace{1em}
Traditionnellement, les valeurs des pixels dans les images
informatiques sont mémorisées sur 8 bits, un octet\index{octet},
soit 256 valeurs différentes.
Ceci dit, on trouve parfois des images codées sur 16 bits par
composante, mais c'est loin d'être le cas général.
J'ai donc souhaité aller plus loin, et coder chaque canal de
chaque pixel en virgule flottante sur 32bits, le type
\texttt{float}\index{float} du langage C. Ce qui correspond
à la norme IEEE 754-1985.
% XXX XXX XXX\vspace{1em}
Attention, tout le code que nous allons voir ensemble est en
perpétuelle évolution\footnote{voir page \pageref{TODO}},
et sa fiablité (surtout sur certains aspects mathématiques)
reste à démontrer\index{valgrind}.
Mais le service après-vente est assez réactif. Du moins
pour ceux qui suivent le canal \texttt{\#tetalab} sur le réseau
IRC de Freenode.
\textbf{Attention !} ce document commence par une bonne rafale
de technique parfois hardue\footnote{gni?}.
Vous avez parfaitement le droit de sauter directement à
la page \pageref{outils} pour quelque chose de plus concret.
% -------------------------------------------------------------------
\setlength \parskip {0em}
\tableofcontents
\pagebreak
\setlength \parskip {0.40em}
% \layout \pagebreak
% -------------------------------------------------------------------
\section{Théorie}\index{théorie}
Pour le moment, seule la quête de l'empirisme absolu a été
visée. Les justifications mathématiques attendront le retour
du schmod777. Ceci dit, rien ne nous empêche d'aller consulter
Wikipedia~:
\begin{quotation}
An IEEE 754 32-bit base-2 floating-point variable has
a maximum value of
$(2 - 2^{-23}) \times 2^{127} \approx 3.4028235 \times 10^{38}$.
All integers with 7 or fewer decimal digits, and any $2^{n}$ for
a whole number $-149 \leq n \leq 127$, can be converted exactly into
an IEEE 754 single-precision floating-point value.
In the IEEE 754-2008 standard, the 32-bit base-2 format is
officially referred to as binary32; it was called single in
IEEE 754-1985.
\end{quotation}
Ce qui nous conduit à estimer qu'il est possible de cumuler environ
quelques milliers d'images standard à 256 niveaux, sans trop avoir
à se soucier des éventuelles pertes de précision. Mais ça demande
à être confirmé par des esprits supérieurs.
\subsection{Dynamique}\index{dynamique}
Dynamique, précision et \textsl{macheps} ?
\subsection{Pixel négatif ?}
Il est très difficle d'imaginer une lumière négative. Sauf peut-être
si nous songeons à des coefficients d'absorption, ou un canal
\textsl{alpha} qui inverserait les valeurs ?
% -------------------------------------------------------------------
\section{Premier exemple}\index{exemple}\label{exemple}
\textsc{FloatImg} a débuté sous la forme de quelques fonctions
basiques en C, gérant la structure des données d'image en mémoire
et sur disque. Ça a été imaginé de façon presque empirique,
mais nous sommes tous là pour améliorer les choses, dans
la mesure de nos moyens.
Nous allons donc directement rentrer au cœur du problème,
en écrivant quelques lignes de code.
Pour commencer par quelque chose de simple,
nous allons créer une image RGB\index{RGB} complètement noire,
puis l'enregistrer dans un fichier \texttt{.fimg}\index{.fimg},
un format complètement inconnu, puisque je viens de l'inventer
à l'instant même.
Enfin, non, il y a déja longtemps, avant la pandémie.
Tout d'abord, nous devons déclarer et garnir quelques variables
pour gérer la machinerie interne.
\begin{lstlisting}
int width = 640, height = 480;
char *fname = "exemple.fimg";
FloatImg fimg;
\end{lstlisting}
Ensuite, nous enchainerons trois étapes : la création de l'image
en mémoire centrale, l'initialisation des valeurs de chaque pixel à 0.0,
et pour conclure, l'enregistrement dans un fichier\footnote{Au format
ésotérique, mais très véloce.} binaire.
\begin{lstlisting}
foo = fimg_create(&fimg, width, height, FIMG_TYPE_RGB);
if (foo) {
fprintf(stderr, "create floatimg -> %d\n", foo);
exit(1);
}
fimg_clear(&fimg);
foo = fimg_dump_to_file(&fimg, fname, 0);
if (foo) {
fprintf(stderr, "dump fimg -> %d\n", foo);
exit(1);
}
\end{lstlisting}
Une fois ce code enrobé dans un \texttt{main()}, compilé et exécuté,
nous pouvons entrevoir, grâce au logiciel
\texttt{fimgstats} (voir page \pageref{fimgstats}),
le résultat sous forme de chiffres divers, et/ou inutiles~:
\begin{verbatim}
$ ./fimgstats quux.img
----------- numbers from 'quux.img' :
640 480 3 0x7f3718c4f010 0x7f3718d7b010 0x7f3718ea7010
surface 307200
mean values:
R 0.000000
G 0.000000
B 0.000000
A 0.000000
max value 0.000000
\end{verbatim}
Nous avons donc sous la main une mécanique qui ne demande qu'à
faire des trucs futiles et des images qui clignotent.
La suite vers la page \pageref{codaz}.
Vous trouverez dans le répertoire \texttt{tools/}\index{tools/}
d'autres exemples de mise en œuvre des fonctions disponibles
sous formes d'outils en ligne de commande,
lesquels sont décrits en page \pageref{outils}.
% -------------------------------------------------------------------
\section{Installation}
Sauf indications contraires, ces instructions se réfèrent à
une distribution Debian\index{Debian} récente, mais ça marche
quasiment pareil avec Fedora\index{Fedora}.
\textit{Attention, ça devient un peu gore\dots}
\subsection{Prérequis}
Vous devez, en dehors des outils classiques (gcc, Awk, make\dots),
avoir quelques bibliothèques installées\footnote{Les \texttt{-dev}
en plus pour Debian et dérivées}~:
\textsf{
libv4l2, libpnglite, libtiff,
libnetpbm\footnote{package libnetpbm10-dev},
libz\footnote{package zlib1g-dev}, libcurses,
libcfitsio-dev
} % end of textsf
éventuellement avec le \textsf{-dev} correspondant,
et probablement d'autres choses.
Il est même quasiment certain que Bash soit indispensable, tout
comme \textsc{gnu}/make\index{make}.
Une connaissance de base de l'utilisation du shell\index{shell}
et de l'écriture de Makefile's sera un plus.
Il faut aussi savoir où trouver le code.
\subsection{Compilation}
Un script \texttt{build.sh} permet de construire approximativement
le bouzin. Il est loin d'être parfait\footnote{Il doit être possible
de faire un Makefile récursif, mais\dots}.
Dans chacun des répertoires à traiter, ce script devrait trouver
un Makefile et un fichier \texttt{t.c} qui est le source de la cible
par défaut du make.
Pour le moment, la procédure d'installation est un peu rude,
pour ne pas dire clairement sommaire.
Si le résultat de l'étape compilation vous semble correct,
vous pouvez copier les deux fichiers \texttt{floatimg.h} et
\texttt{libfloatimg.a} dans un emplacement approprié, par exemple
\texttt{/usr/local/include} et \texttt{/usr/local/lib}.
Le script \texttt{install.sh}, à la racine du projet, est censé
faciliter un peu la chose. Il prend également en compte la copie
des divers binaires du dossier \texttt{tools/} (cf page \pageref{outils})
dans le répertoire prévu à cet effet~: \texttt{/usr/local/bin}.
Il reste enfin quelques exemples d'utilisation des outils de la
ligne de commande depuis un shell
dans le répertoire \texttt{scripts/}.
Ils sont décrits plus en détail page
\pageref{scripts}, et c'est à vous de les adapter à votre \textsl{usecase}.
Faites-en ce que vous voulez.
% =================================================================
\section{Utilisation coté codeur}\label{codaz}
Classiquement, il y a un fichier \texttt{.h} à inclure dans chacun
de vos codes source,
\texttt{floatimg.h}, généralement logé dans \texttt{/usr/local/include}
contenant un certain nombre de définition de structures, de macros,
de constantes\footnote{À l'ancienne, via le pré-processeur}
et les prototypes des fonctions utilisables par vos logiciels.
Au niveau du code source, ces fonctions sont approximativement
classées en deux catégories : \texttt{lib/} et \texttt{funcs/}.
La première contient les choses qui sont relativement figées,
et la seconde celles qui risquent de bouger. Cette classification
est en fait indécement arbitraire.
\subsection{Structures, macros\dots}
Les pixels flottants d'une image résidant en mémoire centrale
sont décrits par un ensemble
de données (certains appelent ça des \textsl{metadatas}) regroupées
dans une jolie structure que nous allons examiner dès maintenant.
\begin{lstlisting}
/* in memory descriptor */
typedef struct {
int magic;
int width;
int height;
int type;
float fval;
int count;
float *R, *G, *B, *A;
int reserved;
} FloatImg;
\end{lstlisting}\index{FloatImg}
Le premier champ, \texttt{magic}, servira un de ces jours à
robustifier l'ensemble du machin.
Les deux suivants sont \textsl{obvious}.
Le troisième est le type d'image : pour le moment, il y en a % trois
un certain nombre
qui sont définis\footnote{et plus ou moins bien gérés\dots} :
gris, rgb et rgba/rgbz\index{rgba}\index{rgbz}.
Les constantes adéquates sont dans \texttt{floatimg.h}
\begin{lstlisting}
#define FIMG_TYPE_GRAY 1
#define FIMG_TYPE_RGB 3
#define FIMG_TYPE_RGBA 4
#define FIMG_TYPE_RGBZ 99
\end{lstlisting}
Un peu plus loin, nous avons les pointeurs vers les
différents \textsl{pixmaps} de l'image. En principe l'organisation
interne de ces zones est improbable, puisque elle dérive
d'idées approximatives. C'est cette utilisation constructive de larache
qui fait que seuls les champs documentés de cette structure ne sont
pas explosifs.
Mais revenons aux choses sérieuses\dots
Les deux champs suivants (fval et count) sont à la disposition du
\textsl{yuser}
qui peut jouer avec à loisir pour faire, par exemple, ce genre de
chose : imaginons un périphérique de capture qui nous fournisse des
images en gris sur 4 bits (et linéaire).
Et que nous voulions cumuler\index{cumul}
quelques images...
Le champ \textsl{count} sera mis à 0 et
le champ \textsl{fval} sera initialisé à 15.0
(qui est la valeur maximale que peut renvoyer ce capteur).
Ensuite, dans la boucle capture/cumul, \textsl{count} sera
incrémenté à chaque passe, et nous aurons donc, en finale,
toutes les informations nécessaires pour exploiter au mieux la dynamique
de notre image dans les étapes ultérieures, puisque la valeur
maximale théorique est égale à $fval * count$.
La fonction \texttt{fimg\_printhead(FloatImg *h)} affiche
sommairement le contenu de ce descripteur,
et \texttt{fimg\_describe(FloatImg *head, char *txt)} propose
un affichage plus détaillé. Ça peut parfois aider.
Une bonne partie des fonctions que nous allons voir est indéterministe.
Ce qui veut dire, en langage de tous les soirs, que ça risque de ne
pas être la même chose dans l'avenir. Mais après tout, ce n'est
encore qu'un concept en devenir, n'est-ce pas ?
% ----------------------------------
\subsection{Les fondations}\index{lib/}
La première chose que nous devons absolument voir est la gestion
dynamique de la mémoire qui sera occupée par tous ces pixels flottants,
ce qui est un sujet parfois délicat\footnote{GC or not GC ?}.
Elle est donc faite, à la base, par ces deux fonctions~:
\begin{lstlisting}
int fimg_create(FloatImg *fimg, int w, int h, int type);
int fimg_destroy(FloatImg *fimg);
\end{lstlisting}
L'appelant doit lui-même gérer le descripteur d'image (une structure
C décrite plus haut) en le considérant comme un type semi-opaque dont
la forme \emph{peut} varier.
Certains membres de cette structure sont
documentés dans ce document, et les autres sont dangereux à
toucher. Les types d'images actuellement gérés sont les trois grands
classiques : gray, rgb et rgba. et expliquées quelques lignes plus haut.
Comme vous allez le voir plus loin, il y a plein de fonctions qui
prennent en argument deux images: une source et une destination.
Et dans la plupart des cas, ces deux images doivent être compatibles,
c'est à dire même type et mêmes dimensions.
\begin{lstlisting}
/* return 0 if compatible */
int fimg_images_not_compatible(FloatImg *a, FloatImg *b);
\end{lstlisting}
C'est bien beau d'être enfin résident en mémoire centrale, mais
pouvoir aussi exister à long terme en étant stocké dans la matrice
est tout aussi pertinent. Il y a deux opérations qui supportent le
reste des transits ram/ps.
Le format des fichiers est décrit page \pageref{formatfimg}.
\begin{lstlisting}
int fimg_dump_to_file(FloatImg *fimg, char *fname, int notused);
int fimg_load_from_dump(char *fname, FloatImg *where);
\end{lstlisting}
Recharger une image depuis un fichier nécessite que celle-ci et
l'image de destination en mémoire
ait précisément les mêmes caractéristiques
(taille, type...), donc l'image en ram doit être
pré-allouée. On peut connaitre ces valeurs en appelant
\texttt{int fimg\_fileinfos(char *fname, int datas[3])}.
Si tout s'est bien passé (valeur retournée égale à 0),
on va trouver la largeur dans \texttt{datas[0]},
la hauteur dans \texttt{datas[1]} et le type dans
\texttt{datas[2]}\footnote{La fonction
\texttt{fimg\_type\_is\_valid(int type)} peut vous aider\dots}.
Je sais aussi que certains d'entre vous aiment la facilité, aussi
je vais vous révéler l'existence d'un nouveau truc bien plus
simple, une fonction qui enchaine ces deux actions
(allocation, puis lecture), et s'utilise
comme ça :
\begin{lstlisting}
FloatImg head;
memset(&head, 0, sizeof(FloatImg));
foo = fimg_create_from_dump("lena.fimg", &head);
\end{lstlisting}
Si la valeur retournée est différente de 0, c'est que quelque
chose s'est probablement mal passé.
Certains messages sont parfois explicites.
% _________
\subsection{Dessiner}
Bon, vous avez une image latente, et vous souhaitez dessiner dessus
(ou dedans ?) avec vos encres flottantes ?
Il y a des fonctions pour ça, par exemple~:
\begin{lstlisting}
int fimg_plot_rgb(FloatImg *head, int x, int y, float r, float g, float b);
\end{lstlisting}
Les paramètres sont explicites, mais leur validité doit être
sévèrement controlée par l'appelant. Il y a une fonction
soeur, \texttt{fimg\_add\_rgb}\index{fimg\_add\_rgb},
qui ajoute du rgb à un pixel, laquelle fonction a d'ailleurs été
à la base de la seconde génération de la
photographie\index{photographie} en cumul\index{cumul}.
Inversement, la fonction
\texttt{fimg\_get\_rgb(FloatImg *head, int x, int y, float *rgb)}
permet de lire les valeurs des trois canaux d'un pixel donné.
Là aussi, il n'y a aucun contrôle sur la validité des valeurs
$x$ et $y$ de la demande.
Quand au canal \textsl{alpha}\index{alpha}, il est pour le moment
superbement ignoré. Ceci dit, on vient de me faire remarquer qu'il
peut être utilisable aussi pour faire du
\textsl{z-buffer}\index{z-buffer}\index{rgbz}\dots
% ----------------------------------
\subsection{Contraste}\index{contraste}\label{contraste}
Certaines opérations d'ajustement du contraste d'une image
semblent cohérentes avec la notion d'image flottante.
Certaines d'entre elles, les plus simples, sont disponibles.
Les autres sont à imaginer.
\begin{figure}[h]
\input{cos01.tex} % XXX XXX XXX
\caption{Correcteur cos01}
\end{figure}
Ils prennent chacun trois paramètres, d'abord les images
source et destination (\texttt{* FloatImg}),
ensuite le troisième
qui est un nombre en double précision donnant la valeur
maximale \textsl{supposée} de l'image source,
valeur qui peut être déterminée de plusieurs manières.
\begin{lstlisting}
/* source in lib/contrast.c */
int fimg_square_root(FloatImg *s, FloatImg *d, double maxval);
int fimg_power_2(FloatImg *s, FloatImg *d, double maxval);
int fimg_cos_01(FloatImg *s, FloatImg *d, double maxval);
int fimg_cos_010(FloatImg *s, FloatImg *d, double maxval);
\end{lstlisting}
Si vous souhaitez rajouter votre propre méthode de modification
de contraste, il y a quelques explication en page \pageref{exemplefunc}.
Mais rien ne vous oblige à le faire. Sauf vos envies.
\begin{figure}[h]
\input{cos010.tex} % XXX XXX XXX
\caption{Correcteur cos010}
\end{figure}
Rappelons qu'il est possible pour un logiciel applicatif
comme \texttt{grabvidseq} (cf page \pageref{grabvidseq})
de renseigner deux champs du descripteur d'image avec des
données pertinentes.
Ces deux champs sont \textit{fval} et \textit{count}.
Dans ce cas particulier, le premier indique la valeur
maximale du capteur, et le second sert à compter le
nombre de capture\footnote{Et c'est bien géré aussi
dans l'upscaling.} effectuées.
La fonction
\texttt{fimg\_normalize(FloatImg *fi, double maxima, int notused);}
tente de gérer ce cas d'utilisation. Son ajout au captureur d'images
floues sera probablement le bienvenue. Je me suis bien rendu compte
à l'usage\footnote{Une histoire de \textit{fonderie}, un logiciel
censé faire des films flous à partir d'images floues} en situation
festive qu'il manquait des données dans la chaine de traitement.
L'autre façon de procéder est d'explorer notre image à la
recherche de la valeur maximale.
La fonction \texttt{float fimg\_get\_maxvalue(\&fimg)} est
prévue pour ça de façon sommaire.
C'est actuellement la méthode utilisée par l'outil qui
sert à faire les modifications de contraste (page \pageref{fimgfx}).
On pourra aussi envisager d'utiliser
\texttt{fimg\_get\_minmax\_rgb(FloatImg *head, float mmvals[6])},
qui permet un contrôle bien plus fin des dérives.
La prochaine étape consistera à trouver une façon de faire
une égalisation\index{égalisation} par histogramme\index{histogramme}
qui respecte, dans toute sa futilité, le concept\index{concept}
de pixel flottant. \textsl{Et c'est pas gagné...}
% ----------------------------------
\subsection{Géométrie}\index{géométrie}\label{geometrie}
Très prochainement, le retour du blitter\index{blitter}.
Et pour attendre, un truc improbable, voire même
inutile, en fait l'inverse de l'upscaling.
\begin{lstlisting}
int fimg_halfsize_0(FloatImg *src, FloatImg *dst, int notused);
int fimg_halfsize_1(FloatImg *src, FloatImg *dst, int notused);
\end{lstlisting}
Attention lors de l'appel, le descripteur \texttt{dst} ne doit pas
contenir d'image, et doit être effacé avec un bon
\texttt{memset(\&result, 0, sizeof(FloatImg));} bien senti.
La première propose un résultat très moyen : il n'y a pas d'interpolation,
alors que la seconde semble bien mieux.
\begin{lstlisting}
int fimg_extract_0(FloatImg *src, FloatImg *dst, int x, int y);
\end{lstlisting}
Contrairement à la fonction précédente, celle-ci demande absolument une
image de destination initialisée aux dimensions (largeur et hauteur)
désirées.
\begin{lstlisting}
int fimg_rotate_90(FloatImg *src, FloatImg *dst, int notused);
\end{lstlisting}
Rotation\index{rotation} de 90 degrés dans le sens
horlogique\footnote{ou trigonométrique,le code et la doc
ne semblent pas d'accord.} d'une image RGB.
L'image de destination peut être soit vierge, soit pré-allouée
aux bonnes dimensions (échange W et H).
% ----------------------------------
\subsection{Format du fichier \textsc{fimg}}\index{format}\label{formatfimg}
D'un design très empirique, c'est certainement à revoir pour l'avenir.
Tout d'abord pour normaliser l'endianess et le packing\dots
\begin{lstlisting}
typedef struct {
char magic[8];
int w, h, t;
} FimgFileHead;
\end{lstlisting}
\dots Mais aussi pour faciliter l'ajout de métadonnées, telles que
la valeur maximale, la date de création, une longueur d'onde,
et bien plus encore.
Le champ \texttt{magic[8]} doit contenir une valeur magique.
Le champ \texttt{t} (le type de l'image) doit avoir les trois octets
de poids fort à 0.
% ----------------------------------
\subsection{Exportation \& Importation}\index{export}\label{export}
Notre format de fichier étant totalement inconnu, il nous
faut bien exporter nos images en quelque chose de plus
connu. Bien entendu, c'est toujours affaire de compromis
entre précision de valeurs et taille des fichiers.
Et dans le sens inverse, il serait bien de savoir importer
le monde extérieur dans nos sombres caves à pixel.
Il faut quand même reconnaitre que c'est un peu la jungle dans les
formats de fichiers d'image, ce qui explique le retard
dans ce domaine\dots
\subsubsection{Vers PNM}\index{PNM}
Nous avons ici 16 bits par composante, soit 65536 valeurs différentes,
ce qui est bien au-delà de ce que peuvent percevoir vos yeux.
Hélas, c'est au prix
d'une taille énorme sur les fichiers. D'un autre coté,
l'utilisation du codage \textsc{ascii}\index{ascii}
(alors qu'on pourrait mettre du binaire, plus compact) y est
pour quelque chose.
\begin{lstlisting}
int fimg_save_as_pnm(FloatImg *head, char *fname, int flags);
\end{lstlisting}
Le bit \texttt{0} du paramètre \texttt{flags} mis à \texttt{1} demande
à la fonction de faire la mise à l'échelle avec le couple
\textsl{fvalue/count} décrit plus haut dans cette doc.
Et si il est à zéro, c'est la fonction de recherche de valeur
maximale (cf page \pageref{contraste}) qui est utilisée.
Le bit \texttt{1} permettra bientôt\index{vaporware} de demander
l'enregistrement de métadonnées\index{metadata} pertinentes, telle
que l'epochtime de l'enregistrement.
Les autres bits ne sont pas utilisés et doivent être à zéro.
\subsubsection{Vers PNG}\index{PNG}
Actuellement, on peut enregistrer uniquement en mode RGB,
8 bits par composante,
mais on a quand même une bonne compression, ça compense.
J'utilise \textsl{libpnglite} avec qui j'ai un peu de mal à suivre.
Mais je me soigne. Le mode 16 bits va bientôt arriver.
On peut aussi songer à l'export de metadatas.
\begin{lstlisting}
int fimg_save_as_png(FloatImg *src, char *outname, int flags);
\end{lstlisting}
Tous les flags doivent être à zéro. Sinon, ça foire parfois.
Et en fait (mars 2021) je ne suis pas très content de
\texttt{pnglite}, donc un de ces jours\footnote{Rendez-nous notre
Mixou !}, je prendrais cinq jours pour régler le souci.
\subsubsection{Vers TIFF}\index{TIFF}
Le format canonique de la PAO\index{PAO} du siècle dernier. Il permet
de gérer une foultitude de formats numériques. C'est aussi un format
classique proposé par les gros scanners corporates.
\begin{lstlisting}
int fimg_write_as_tiff(FloatImg *src, char *outname, int flags);
\end{lstlisting}
Tous les flags doivent être à zéro. Pour le moment.
\subsubsection{Vers FITS}\index{FITS}
Ce format est essentiellement utilisé pour stocker des images
d'astronomie, donc il peut aussi servir pour des images floues.
Cette partie est basée sur la bibliothèque \texttt{cfitsio} de
la NASA.
\begin{lstlisting}
int fimg_save_R_as_fits(FloatImg *src, char *outname, int flags);
int fimg_save_G_as_fits(FloatImg *src, char *outname, int flags);
int fimg_save_B_as_fits(FloatImg *src, char *outname, int flags);
\end{lstlisting}
Bizarrement, l'image est stockée \textsl{upside-down} et je ne
sais pas encore comment régler ce petit détail.
Tous les flags doivent être à zéro.
\subsection{Utilitaires}
Commençons par quelques petits trucs pour nous faciliter la vie
dans des domaines annexes,
tels que l'interprétation d'arguments dans la ligne de commande ou un
fichier de configuration.
\begin{lstlisting}
int parse_WxH(char *str, int *pw, int *ph)
int parse_double(char *str, double *dptr)
\end{lstlisting}
La fonction \texttt{int format\_from\_extension(char *fname)} examine un
nom de fichier tel que \texttt{lena.xxx}, et retourne, si la partie
\texttt{xxx} est connue, un éventuel nombre positif, dont les valeurs sont
déclarées dans floatimg.h
le valeureux.
Les extensions actuellement connues sont : fimg, png, pnm, fits et tiff.
To be continued\index{XXX}\dots
\subsection{Effets}\index{sfx}
Quelques routines qui servent futilement à \textsl{brotcher} les images
en tripotant les flux visuels chromatiques.
\begin{lstlisting}
int fimg_killcolors_a(FloatImg *fimg, float fval);
int fimg_killcolors_b(FloatImg *fimg, float fval);
int fimg_colors_mixer_a(FloatImg *fimg, float fval);
\end{lstlisting}
\subsection{Glitches}\index{glitch}
Un \textsl{glitch} peut-il être classé dans la catégorie des effets
spéciaux ou non ? \textsc{Hmha}, non. un fx est paramétrable
et surtout répétitif. Un glitch est quasiment souvbent un phénomène
aléatoire\index{drand48} et tout aussi paramétrable.
J'ai commencé à étudier ces objets étranges quand j'ai commencé
sur l'interpolator\index{interpolator} à l'automne 2020.
% ----------------------------------
\subsection{Filtrages}\index{filtrage}
Pour commencer, il faut que je réfléchisse au traitement
des bordures des images.
Ensuite que je débuggue\index{bug} ces deux fonctions~:
\begin{lstlisting}
int fimg_lissage_2x2(FloatImg *img);
int fimg_killborders(FloatImg *img);
\end{lstlisting}
Bon, oké, ça marche ? Passons à l'tape suivante.
La convolution avec une matrice 3x3, c'est possible.
Et pas trop compliqué à faire.
Bon, il reste le souci avec les bordures, souci qui ne peut
être que temporaire, mais ésotérique à fixer.
Passons maintenant aux choses sérieuses, et définissons la
description d'un filtre 3x3.
\begin{lstlisting}
typedef struct {
float matrix[9];
float mult;
float offset;
} FimgFilter3x3;
\end{lstlisting}
L'usage des champs \texttt{mult} et \texttt{offset} n'est pas
clairement défini. Le prototype de la fonction de filtrage
non plus, mais assez simpe quand même. Source et destination
ne peuvent désigner la même image, et le champ \texttt{matrix}
du filtre doit contenir des valeurs cohérentes.
\begin{lstlisting}
int fimg_filter_3x3(FloatImg *src, FloatImg *dst, FimgFilter3x3 *filtr)
\end{lstlisting}
Comme dans la plupart des cas, la gestion des valeurs négatives
de pixel est laissé au hasard. Quoique, il doit bien exister
quelques solutions de contournement : clamping ou shift ?
\textsl{To be continued\index{XXX}\dots}
% ----------------------------------
\subsection{Exemple de fonction}\index{exemple}\label{exemplefunc}
Nous allons maintenant écrire une fonction intégrable dans
le répertoire \texttt{funcs/}.
Elle aura donc accès aux \textsl{internals}%
\footnote{que je peux décider de changer n'importe quand}
de \textsc{FloatImg},
une chose qui est en principe interdit aux programmes
\textsl{enduser}. Soyez prudents.
Cette fonction va faire quelque chose
à partir d'une image source et d'une valeur, et écrire le
résultat dans une image de destination.
Pour simplifier les choses, nous n'allons traiter que les
images de type \textsc{FIMG\_TYPE\_RGB}, de loin le plus
répandu par les temps qui courent.
\begin{lstlisting}
int fimg_example(FloatImg *s, FloatImg *d, float value)
{
int size, index;
if ( s->type!=FIMG_TYPE_RGB || d->type!=FIMG_TYPE_RGB) {
perror("fimg_example");
return -1;
}
size = s->width * s->height;
for (idx=0; idx<size; idx++) {
d->R[idx] = s->R[idx] - value;
d->G[idx] = s->G[idx] - value;
d->B[idx] = s->B[idx] - value;
}
return 0;
}
\end{lstlisting}
Je vous laisse imaginer les dégats que peut faire cette
fonction en utilisation réelle. Mieux, je vous propose
d'essayer par vous-même.
En particulier tout le reste du code qui suppose qu'un pixel
ne peut \textbf{pas} être négatif va peut-être exploser de rire.
Vous pouvez aussi remarquer qu'il n'y a pas de controle
de cohérence sur les dimensions des deux images, malgré l'existence
de fonctions prévues à cet effet..
% -------------------------------------------------------------------
\section{Les outils}\label{outils}
\textsf{3615mavie} : sur des projets comme celui-ci, qui travaillent
in-fine sur des objets que l'on peut considérer comme « physiques »,
il est important de passer à une utilisation
normale\footnote{Il y a une vie en dehors de git.} et construire
des trucs qui mettent en action le code primitif.
Ces machins ont en commun quelques options bien pratiques~:
\texttt{-h} pour avoir un résumé des options disponibles,
\texttt{-v} qui augmente la puissance de bavardage, et
\texttt{-K nn.nn} pour un paramètre flottant.
Dans un avenir incertain, il existera des pages de man\index{man}.
% ---------------------
\subsection{mkfimg}\index{mkfimg}\label{mkfimg}
Propose la création d'un fichier contenant une image de « teinte »
constante (ou pas).
Cette notion de teinte est assez inconsistante pour le moment,
mais ça n'est pas si grave que ça.
\begin{verbatim}
tth@debian:~/Devel/FloatImg/tools$ ./mkfimg -h
Usage: mkfimg [options] quux.fimg width height
-k N.N give a float parameter
-t type howto make the pic
black, drand48...
-v increase verbosity
\end{verbatim}
La plupart des types d'image générée prennent un paramètre flottant qui
devra être donné avec l'option \texttt{-k F.F} avec une valeur par défaut
à $1.0$.
\begin{description} \index{XXX}
\item [black/gray/grey:] efface avec 0.0 (black) ou avec la valeur
\texttt{-k} (gray).
\item [drand48:] beaucoup de bruit dans chacun des canaux.
\item [hdeg/vdeg:] dégradé du noir au blanc (relatif à \texttt{-k}).
\end{description}
% ---------------------
\subsection{png2fimg}\index{png2fimg}\label{png2fimg}
Grosse panne\index{bug} à réparer.
\begin{verbatim}
tth@debian:~/TMP/floatimg$ png2fimg A.png foo.fimg
error in 'fimg_create_from_png' : read png -> -1 File error
png2fimg : err -1, abort.
\end{verbatim}
Il faut peut-être envisager le passage à \texttt{libpng}\index{libpng}.
\subsection{fimgstats}\index{fimgstats}\label{fimgstats}
Affichage de quelques valeurs calculées à partir du contenu d'un fichier
\texttt{.fimg}\index{.fimg}.
\begin{verbatim}
usage : fimgstats [options] file.fimg
-c make a machinable csv
-v increase verbosity
\end{verbatim}
À vrai dire, je ne sais pas encore quelles métriques seront utiles
en première approche, alors commençont par le plus simple,
les valeurs moyennes de chaque composante.
Puis nous rajouterons\footnote{Les patchs sont les bienvenus}
le calcul de la variance\index{variance}. Les compétences
de \texttt{schmod777} sont attendues au dd2\index{dd2}.
% ---------------------
\subsection{fimgfx}\index{fimgfx}\label{fimgfx}
Ce programme, \textit{en cours de création\index{XXX}}, applique
un effet spécial à une image.
À l'heure actuelle\footnote{janvier 2019, vers 13:37}, nous avons
déja quelques ajustements basiques de contraste, qui ne tiennent
pas vraiment compte du contenu de l'image.
\begin{verbatim}
tth@daubian:~/Devel/FloatImg/tools$ ./fimgfx -v -h
--- fimg special effects ---
cos01 cos010 pow2 sqrt gray0 cmixa xper desat
\end{verbatim}
Certaines de ces opérations ont besoin d'un paramètre flottant.
Celui-ci peut être fixé avec l'option \texttt{-k}.
Une liste détaillée des opérations possibles
sera lisible avec le sélecteur \texttt{-L}.
\begin{description}
\item [Ajustements de contraste:] cos01 cos010 pow2 sqrt
\item [Distorsions chromatiques:] gray0
\item [Déformations géométriques:] r90
\end{description}
Et pour les aventureux, la commande \texttt{xper} (abréviation
de \textsl{expérimental}) permet de tester la plus récente tentative
de friture du moment.
% ---------------------
\subsection{fimgops}\index{fimgops}\label{fimgops}
Quelques opérations diverses entre deux images, qui doivent être
de la même taille, et uniquement du type \textsl{RGB}. Certaines
de ces opérations peuvent avoir un effet étrange sur vos images,
par exemple si un pixel se retrouve avec une valeur négative.
\begin{verbatim}
usage:
fimgops [options] A.fimg B.fimg operator D.fimg
operators:
add 1
sub 2
mix 3
mul 4
mini 5
maxi 6
options:
-g convert output to gray
-k N.N set float value
-v increase verbosity
-X explosive action
\end{verbatim}
Pour des operateurs paramétrable (comme \texttt{mix}), le paramêtre
flottant doit être fourni en utilisant l'option \texttt{-k}.
La véracité mathématique n'est pas garantie. Et n'oubliez pas que
les valeurs négatives peuvent être la cause de \textsl{glitches}
de qualitay.
% -------------------------
\subsection{fimg2png, fimg2pnm, fimg2tiff, fimg2fits}
\index{fimg2png}\label{fimg2png}
\index{fimg2pnm}\label{fimg2pnm}
\index{fimg2tiff}\label{fimg2tiff}
\index{fimg2fits}\label{fimg2fits}
Quelques petits proggies pour exporter notre format\index{.fimg} secret
vers des choses plus directement utilisables. À condition que le
code soit écrit et documenté.
D'un autre coté, écrire un greffon d'import/export pour
Gimp\index{Gimp} ou ImageMagick\index{ImageMagick} ou Krita\index{krita}
ne devrait pas être trop difficile. Des volontaires ?
\textsl{D'ailleurs, pourquoi $N$ logiciels indépendants alors q'un
seul devrait être nécessaire ?}
\subsection{fimg2gray}\index{fimg2gray}\label{fimg2gray}
Nous avons vu dans ce document que chaque image flottante pouvait
avoir plusieurs plans de réalité. Il ne faut en négliger aucun.
Il faut quand même deviner que pour passer de l'espace RGB\index{RGB}
à une abstraction linéaire mono-dimensionnelle, il existe une foultitude
de méthodes, toutes plus légitimes que les autres.
\index{procrastination}
Et face à l'incertitude du choix, j'ai reporté l'écriture de ce
logiciel aux calendes grecques, voire même plus tard.
\subsection{cumulfimgs}\index{cumulfimgs}\label{cumulfimgs}
Cet outil accumule\index{cumul} une quantité d'images flottantes
(même taille et même type) afin d'obtenir
un flou de meilleure qualité. Aucune mise à l'échelle n'etant
effctuée, les pixels de sortie peuvent atteindre des valeurs
considérables\footnote{Prévoir une gestion des \textsf{overflows} ?}
\begin{verbatim}
tth@delirium:~/Devel/FloatImg/tools$ ./cumulfimgs -h
usage :
cumulfimgs a.fimg b.fimg c-fimg ...
cumulator options :
-v increase verbosity
-o name of output file
-g convert to gray level
\end{verbatim}
Le nom par défaut du fichier résultant est \texttt{out.fimg}.
L'exportation "multiformat" est pour bientôt.
% -------------------------------------------------------------------
\section{TODO}\index{TODO}\label{TODO}\
\index{XXX}
Il reste plein de choses à faire pour que ce soit vraiment utilisable,
surtout dans un contexte artistique à grande porosité.
C'est par ces frottements de techniques ayant du sens que les
choses seront acquises.
\begin{itemize}
\item Import/export au format \textsc{tiff}\index{TIFF}.
\item Remplacer le « fait-maison » par \textsc{libnetpnm}\index{pnm}.
\textsl{[en cours]}.
\item Compléter les traitements mathémathiques (eg le gamma\index{gamma}).
\item Formaliser les codes d'erreur. \textbf{Urgent}.
\item Faire une passe complète de Valgrind\index{valgrind}.
\item Intégrer la fonderie et l'interpolator.
\item Vérifier le gestion des images mono-canal.
\end{itemize}
% -------------------------------------------------------------------
\section{Exemples pour yusers}\index{exemple}
Nous allons \textsl{essayer d'improviser} un exemple presque réel,
avec un peu de rache\index{rache} dedans, et beaucoup de simplification.
Ce qui est autorisé dans les exemples, mais dans la vrai vie, il ne faut
jamais négliger le traitement des éventuelles erreurs.
Nous savons générer une image contenant des pixels aux valeurs
probablement aléatoires, avec la commande \texttt{mkfimg},
qui utilise le \texttt{drand48}\index{drand48} de \textsc{posix}\index{POSIX}.
Maintenant, posons-nous une question de statisticien : ue se passe-t-il si
nous faisons la somme de plusieurs centaines de ces images ?
\begin{lstlisting}
#!/bin/bash
ACCU="quux.fimg"
TMPF="tmp.fimg"
DIMS="320 240"
mkfimg $ACCU $DIMS
for i in {0..1000}
do
mkfimg -t drand48 ${TMPF} ${DIMS}
fname=$( printf "xx%04d.pnm" $i )
fimgops $ACCU $TMPF add $ACCU
fimg2pnm -v -g $ACCU $fname
done
convert -delay 10 xx*.pnm foo.gif
\end{lstlisting}
Voilà, si les choses se passent mal, vous allez découvrir
que votre \texttt{drand48} n'est pas si "drand" que ça.
Et ce n'est pas à moi d'en tirer les conclusions...
\subsection{Scripts}\index{scripts}\label{scripts}
Le script bash\index{bash} \texttt{scripts/shoot.sh} est un front-end
encore un peu rudimentaire
vers le programme de capture d'image décrit page \pageref{grabvidseq}.
Il utilise deux fichiers dans le répertoire de travail~:
\textit{reglages} et \textit{compteur}. Le premier est, en fait,
un bout de shell affectant quelques variables, ou plutôt, les surchargeant.
Voici un exemple de réglage~:
\begin{lstlisting}
OPTIONS="${OPTIONS} -v -c pow2 "
SHOW="yes"
NBRE=1000
PERIOD=0
OFORMAT="p_%04d.png"
\end{lstlisting}
La première ligne demande, en plus des options par défaut, plus de
bavardage, et un changement de contraste. La seconde demande
l'affichage de la photo. Les deux suivantes demandent la
capture de 1000 images à la cadence méga-blast.
La dernière est moins simple~: \texttt{man sprintf}\index{printf}
pour comprendre.
Quand au second fichier, il contient un compteur (stocké en ascii) qui
est incrémenté après chaque capture réussie. Et ce compteur est
utilisable par la variable \texttt{OFORMAT} que nous avons
vue quelques lignes plus haut.
\lstinputlisting[language=sh]{../scripts/shoot.sh}
\subsection{Fonderie}\index{fonderie}\label{fonderie}
Ce projet externe\footnote{... pour le moment, j'ai des soucis sur
l'architecture du \textbf{pipdeprod} à adopter\dots} est destiné à la confection
de films flous\index{film} à partir de photos floues.
Le script \texttt{scripts/echomix.sh} est une première expérimentation
en bash, utilisant deux outils en \textsc{cli},
le premier pouvant salement brotcher une image, et le second capable de
mélanger harmonieusement deux images, la balance est équilibrée.
Il s'agit donc d'un petit programme écrit en Bash\index{bash}, un langage
dont la connaissance est, pour moi, indispendable à qui veut faire des
images kitchies\index{kitchy}. Mais ne vous inquiétez pas, c'est en
fait assez simple à comprendre. Et comprendre, c'est apprendre.
Voici donc le script, décomposé et expliqué :
\begin{verbatim}
#!/bin/bash
SRCDIR="Fist"
DSTDIR="Pong"
FTMP="/dev/shm/tmp.fimg"
FDST="/dev/shm/foo.fimg"
# count the nomber of picz in the source directory
NBRE=$(ls -1 ${SRCDIR}/*.fimg | wc -l)
# compute the echo picz offset
OFFS=$(( NBRE / 4 ))
\end{verbatim}
Dans ce préliminaire logiciel, nous avons nommés le répertoire
\textsc{srcdir} contenant les captures d'image au format fimg, le répertoire
\textsc{dstdir} dans lequel seront rangées les images calculées,
et l'emplacement de deux fichiers de travail.
Les quelques lignes suivantes, qui semble bien magiques, ne sont en fait
que de la magie Unix\index{Unix}. Elles nous permettent d'avoir
\textsc{nbre}, le nombre d'images à traiter, et \textsc{offs}, un décalage
dépendant du nombre d'image. Muni de toutes ces informations, nous
pouvons rentrer dans le lard du sujet, la boucle qui travaille.
\begin{verbatim}
# MAIN LOOP
for idx in $(seq 0 $NBRE)
do
# build the two input filenames ...
#
imgA=$(printf "$SRCDIR/%04d.fimg" $idx)
vb=$(( $(( idx + OFFS )) % NBRE))
imgB=$(printf "$SRCDIR/%04d.fimg" $vb)
# ... and the output filename
#
dst=$(printf "%s/%05d.png" ${DSTDIR} $idx)
\end{verbatim}
Dans cette première partie de la boucle nous avons construit plusieurs
noms de fichier à partir du rang de la boucle en cours d'exécution,
des deux valeurs \textsc{nbre} et \textsc{offs} calculées en préambule.
\begin{verbatim}
# trying to autocompute the mixing coefficient
#
compute=" s(${idx} / 16) "
K=$(echo $compute | bc -l)
printf " %25s => %8.3f\n" "$compute" $K
\end{verbatim}
Cette seconde partie sert à calculer avec la commande
\texttt{bc}\index{bc}%
\footnote{\texttt{bc}, c'est vraiment un truc à découvrir.}
un coefficient variable en fonction du temps :
$sin(idx/16)$ afin d'avoir une oscillation du coefficient entre
-1.0 et 1.0, deux valeurs probablement glitchantes.
\begin{verbatim}
# do the hard floating computation
#
fimgfx -v cos010 ${imgB} ${FTMP}
fimgops -k ${K} ${FTMP} ${imgA} mix ${FDST}
\end{verbatim}
Étape suivante, étape cruciale : le brassage d'une multitude de
pixels flottants.
Tout d'abord, nous faisons subir à l'image-echo
(\texttt{imgB}, définie au début du script) un distorsion
chromatique de type \textsl{cos010}, le résultat étant écrit
dans un fichier temporaire. Ensuite, nous mixons l'image
primaire et son echo en utilisant le rapport de mixage
calculé quelques lignes plus haut.
\begin{verbatim}
# write the output as PNG for video encoding
#
fimg2png ${FDST} ${dst}
done
\end{verbatim}
Et en fin de boucle, nous convertissons le résultat de nos
savants calculs au format PNG, et écrivons le fichier dans le répertoire
de destination fixé au début.
C'est le moment de passer la main à ffmpeg\index{ffmpeg}.
C'est juste une POC\index{POC}, et une implémentation bien plus
complète écrite en \textbf{C}\index{C} est déja en chantier,
avec une complexité prévue à un niveau assez réjouissant.
% -------------------------------------------------------------------
\section{Video for Linux}\index{v4l2}
Donc, maintenant, nous savons un peu tripoter ces images flottantes.
Et nous devons nous poser une question fondamentale\footnote{primitive ?}
sur la provenance de ces données prétendant être des images.
En fait, notre désir secret (enfin, surtout le mien)
est la découverte des choses cachées du
monde qui nous entoure. Nous voulons des images du \textbf{réel} et
pour cela, l'outil le plus commun, le plus répandu,
est la webcam\index{webcam}. L'universelle webcam. Et l'incontournable
v4l2.
\subsection{grabvidseq}\index{grabvidseq}\label{grabvidseq}
Un logiciel en évolution (trop ?) lente, qui permet déja la capture
d'images en
\textsl{longue pose} selon la méthode du cumul\index{cumul}, et
devrait bientôt retrouver sa capacité à enregistrer des
séquences d'images.
\begin{verbatim}
tth@debian:~/Devel/FloatImg/v4l2$ ./grabvidseq -h
options :
-c quux contrast adjustement
-d /dev/? select video device
-g convert to gray
-n NNN how many frames ?
-O ./ set Output dir
-o bla set output filename
-p NN.N delay between frames
-r 90 mode portrait
-s WxH size of capture
-u try upscaling...
-v increase verbosity
-X arg Xperiment option
\end{verbatim}
La plupart de ces options ont un usage quasi-évident.
L'option \texttt{-s} doit correspondre à une des
résolutions possibles de votre capteur. Le type du
fichier en sortie (option \texttt{-o}) est déterminé par
l'extension du nom.
Actuellement
seulement \texttt{.fimg}, \texttt{.pnm}, \texttt{.fits},
\texttt{.tiff} et \texttt{.png}
sont reconnus.
La conversion en gris (option \texttt{-g}) mérite un
peu plus de travail, et une paramétrisation plus facile.
L'ajustement de contraste (option\texttt{-c}) est
vaguement expliqué page \pageref{contraste}.
L'option \texttt{-X} me permet d'intégrer des \textit{fritures}
expérimentales dans le binaire, et ne doit donc pas être
utilisée dans des scripts si on a des visions à long (ou même)
terme.
\subsubsection{Upscaling}\index{upscaling}\label{upscaling}
La fonction que j'ai appelée \textsl{upscaling} est un petit
hack qui permet de doubler artificiellement la résolution
de l'image, en profitant du fait que l'on est capable
de prendre $N$ images en rafale.
Pour être rigoureux dans la prise de vue, ce $N$ doit
être un multiple de 4, surtout si le nombre de capture est faible.
N'hésitez pas à faire des essais, le résultat est parfois
aléatoire, surtout avec une caméra qui bouge.
\textbf{Là, il manque un schéma\dots}
\subsection{video-infos}\index{video-infos}\label{video-infos}
Que contient, que peut faire mon périphérique \textsl{àlc} ?
Quelles sont ses possibilités de réglage ?
\begin{verbatim}
tth@debian:~/Devel/FloatImg$ v4l2/video-infos -h
Options :
-d select the video device
-K nnn set the K parameter
-l list video devices
-T bla add a title
-v increase verbosity
\end{verbatim}
Je me sens obligé d'avouer qu'il reste quelques points mystérieux dans
l'\textsc{api} de \textsc{v4l2}, et donc, que ce que raconte
ce logiciel doit être pris avec des pincettes. En particulier
la liste des résolutions disponibles.
\subsection{nc-camcontrol}
Ajustement \textsl{Brightness Contrast Saturation Hue\dots}
% -------------------------------------------------------------------
\section{À l'extérieur}
\subsection{ImageMagick}\index{ImageMagick}
Pour afficher notre format .fimg exotique avec \texttt{display}, vous
devez mettre ce bout de XML\index{XML} dans le fichier
\texttt{\$HOME/.magick/delegates.xml}~:
\begin{lstlisting}
<?xml version="1.0" encoding="UTF-8"?>
<delegatemap>
<delegate decode="fimg" command="fimg2png '%i' '%o'"/>
</delegatemap>
\end{lstlisting}
C'est juste un hack rapide, qui ne fonctionne pas très bien avec
d'autres commande de IM, comme identify, qui a tendance à
raconter un peu n'importe quoi, puisqu'elle se base sur le
résultat de la conversion\dots
Je compte donc sur le bouquin de \textsl{Brunus} pour avancer\dots
\subsection{Gimp}\index{Gimp}
Mmmmm... Ça semble un peu plus compliqué.
La documentation à ce sujet me semble ésotérique.
D'un autre coté, il
faut faire ça en \textbf{C}, ce qui ne peut pas être négatif.
\subsection{ffmpeg}\index{ffmpeg}
Un petit aide-mémoire pour encoder vos superbes
timelapses\index{timelapse} :
\begin{lstlisting}
ffmpeg -nostdin \
-y -r 30 -f image2 -i stereo/S%04d.png \
-c:v libx264 \
-pix_fmt yuv420p \
-tune film \
stereo.mp4
\end{lstlisting}
\subsection{Et encore ?}\index{krita}\index{geeqie}
Il y a d'autres logiciels pour lesquels écrire une fonction d'importation
serait bien~: \textsl{Geeqie}, un visualiseur d'image fort pratique, ou
\textsl{Krita} qui semble avoir les faveurs de
dessinateurs de talent\footnote{Oui, David, c'est à toi que je pense.}.
% -------------------------------------------------------------------
% ===================================================================
\section{Le flou temporel}
Et si nous jouions sur l'axe du temps ?
Nous avons plus ou moins la magie du cumul sur la prise de vue
d'\textbf{image} en enchainant plusieurs capture d'image.
Maintenant, voyons ce que l'on peut faire à partir de plusieurs images.
On peut d'abord penser faire une moyenne (ou la somme, en fait) de toutes
ces images. Mais ce n'est qu'une façon déguisée de faire du cumul.
C'est à ce moment que nous changeons l'axe de vue du défi.
\subsection{textsl{moving average}}
Moyenne mobile.
\subsection{Interpolator}\index{interpolator}
Juste des calculs pas si simple que ça.
\subsection{Déviance}
Là, nous tombons dans la troiD de haut niveau, avec plein de maths
à l'intérieur.
% ===================================================================
\section{Et pour la suite ?}
En fait, je fait de la photo par la méthode du « cumul »\index{cumul}
depuis plusieurs années. Une webcam\index{webcam},
un Linux\index{Linux}, et ça \textsl{juste marche}.
Sauf que c'est quand même un peu galère à déplacer, il faut
avoir un shell pour déclencher, c'est pas facile à utiliser
en mode portnawak\dots
L'idée est donc de construire un appareil autonome, basé sur un Raspi et
une webcam \textsc{usb}\index{USB}, pilotable par \textsc{lirc}\index{LIRC},
alimenté par une (grosse) batterie et permettant d'aller
faire des images au bord d'un lac ou dans la campagne de l'Ariège.
% -------------------------------------------------------------------
\printindex
\end{document}
|
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author={N. Nicodemo and G. Naithani and K. Drossos and T. Virtanen and Roberto Saletti},
booktitle={28th European Signal Processing Conference ({EUSIPCO})},
title={Memory Requirement Reduction of Deep Neural Networks Using Low-bit Quantization of Parameters},
year={2020},
pages={},
doi={},
ISSN={},
month={Aug.},} |
http://porocila.imfm.si/2006/mat/clani/dolzan.tex | imfm.si | CC-MAIN-2023-14 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296943695.23/warc/CC-MAIN-20230321095704-20230321125704-00249.warc.gz | 48,473,402 | 996 | \clan {David Dolžan}
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%--------------------------------------------------------
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{}
{The n-insertive subgroups of units}
{Bull.\ Austral.\ Math.\ Soc.}
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% 1.01: {\bf 1}. DOL\v{Z}AN, David. Multiplicative sets of idempotents in a finite ring. {\it J. algebra}, 2006, vol. 304, no. 1, str. 271-277. http://dx.doi.org/10.1016/j.jalgebra.2006.03.022. $[$COBISS.SI-ID 14131289$]$\\
{}
{Multiplicative sets of idempotents in a finite ring}
{J. Algebra} {304} {2006} {1} {271--277}
\end{skupina}
|
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\title{On the US-Israeli Invasion of Lebanon}
\date{August 19, 2006}
\author{Noam Chomsky}
\subtitle{}
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Though there are many interacting factors, the immediate issue that lies behind the latest US-Israeli invasion of Lebanon remains, I believe, what it was in the four preceding invasions: the Israel-Palestine conflict. In the most important case, the devastating US-backed 1982 Israeli invasion was openly described in Israel as a war for the West Bank, undertaken to put an end to annoying PLO calls for a diplomatic settlement (with the secondary goal of imposing a client regime in Lebanon). There are numerous other illustrations. Despite the many differences in circumstances, the July 2006 invasion falls generally into the same pattern.
Among mainstream American critics of Bush administration policies, the favored version is that “We had always approached [conflict between Israel and its neighbors] in a balanced way, assuming that we could be the catalyst for an agreement,” but Bush II regrettably abandoned that neutral stance, causing great problems for the United States (Middle East specialist and former diplomat Edward Walker, a leading moderate). The actual record is quite different: For over 30 years, Washington has unilaterally barred a peaceful political settlement, with only slight and brief deviations.
The consistent rejectionism can be traced back to the February 1971 Egyptian offer of a full peace treaty with Israel, in the terms of official US policy, offering nothing for the Palestinians. Israel understood that this peace offer would put an end to any security threat, but the government decided to reject security in favor of expansion, then mostly into northeastern Sinai. Washington supported Israel’s stand, adhering to Kissinger’s principle of “stalemate”: force, not diplomacy. It was only 8 years later, after a terrible war and great suffering, that Washington agreed to Egypt’s demand for withdrawal from its territory.
Meanwhile the Palestinian issue had entered the international agenda, and a broad international consensus had crystallized in favor of a two-state settlement on the pre-June 1967 border, perhaps with minor and mutual adjustments. In December 1975, the UN Security Council agreed to consider a resolution proposed by the Arab “confrontation states” with these provisions, also incorporating the basic wording of UN 242. The US vetoed the resolution. Israel’s reaction was to bomb Lebanon, killing over 50 people in Nabatiye, calling the attack “preventive” – presumably to “prevent” the UN session, which Israel boycotted.
The only significant exception to consistent US-Israeli rejectionism was in January 2001, when Israeli and Palestinian negotiators came close to agreement in Taba. But the negotiations were called off by Israeli Prime Minister Barak four days early, ending that promising effort. Unofficial but high-level negotiations continued, leading to the Geneva Accord of December 2002, with similar proposals. It was welcomed by most of the world, but rejected by Israel and dismissed by Washington (and, reflexively, the US media and intellectual classes).
Meanwhile US-backed Israeli settlement and infrastructure programs have been “creating facts on the ground” in order to undermine potential realization of Palestinian national rights. Throughout the Oslo years, these programs continued steadily, with a sharp peak in 2000: Clinton’s final year, and Barak’s. The current euphemism for these programs is “disengagement” from Gaza and “convergence” in the West Bank – in Western rhetoric, Ehud Olmert’s courageous program of withdrawal from the occupied territories. The reality, as usual, is quite different.
The Gaza “disengagement” was openly announced as a West Bank expansion plan. Having turned Gaza into a disaster area, sane Israeli hawks realized that there was no point leaving a few thousand settlers taking the best land and scarce resources, protected by a large part of the IDF. It made more sense to send them to the West Bank and Golan Heights, where new settlement programs were announced, while turning Gaza into “the world’s largest prison,” as Israeli human rights groups accurately call it. West Bank “Convergence” formalizes these programs of annexation, cantonization, and imprisonment. With decisive US support, Israel is annexing valuable lands and the most important resources of the West Bank (primarily water), while carrying out settlement and infrastructure projects that divide the shrinking Palestinian territories into unviable cantons, virtually separated from one another and from whatever pitiful corner of Jerusalem will be left to Palestinians. All are to be imprisoned as Israel takes over the Jordan Valley, and of course any other access to the outside world.
All of these programs are recognized to be illegal, in violation of numerous Security Council resolutions and the unanimous decision of the World Court any part of the “separation wall” that is built to “defend” the settlements is “ipso facto” illegal (U.S. Justice Buergenthal, in a separate declaration). Hence about 80–85\% of the wall is illegal, as is the entire “convergence” program. But for a self-designated outlaw state and its clients, such facts are minor irrelevancies.
Currently, the US and Israel demand that Hamas accept the 2002 Arab League Beirut proposal for full normalization of relations with Israel after withdrawal in accord with the international consensus. The proposal has long been accepted by the PLO, and it has also been formally accepted by the “supreme leader” of Iran, Ayatollah Khamenei. Sayyed Hassan Nasrallah has made it clear that Hezbollah would not disrupt such an agreement if it is accepted by Palestinians. Hamas has repeatedly indicated its willingness to negotiate in these terms.
The facts are doctrinally unacceptable, hence mostly suppressed. What we see, instead, is the stern warning to Hamas by the editors of the New York Times that their formal agreement to the Beirut peace plan is “an admission ticket to the real world, a necessary rite of passage in the progression from a lawless opposition to a lawful government.” Like others, the NYT editors fail to mention that the US and Israel forcefully reject this proposal, and are alone in doing so among relevant actors. Furthermore, they reject it not merely in rhetoric, but far more importantly, in deeds. We see at once who constitutes the “lawless opposition” and who speaks for them. But that conclusion cannot be expressed, even entertained, in respectable circles.
The only meaningful support for Palestinians facing national destruction is from Hezbollah. For this reason alone it follows that Hezbollah must be severely weakened or destroyed, just as the PLO had to be evicted from Lebanon in 1982. But Hezbollah is too deeply embedded within Lebanese society to be eradicated, so Lebanon too must be largely destroyed. An expected benefit for the US and Israel was to enhance the credibility of threats against Iran by eliminating a Lebanese-based deterrent to a possible attack. But none of this turned out as planned. Much as in Iraq, and elsewhere, Bush administration planners have created catastrophes, even for the interests they represent. That is the primary reason for the unprecedented criticism of the administration among the foreign policy elite, even before the invasion of Iraq.
In the background lie more far-reaching and lasting concerns: to ensure what is called “stability” in the reigning ideology. “Stability,” in simple words, means obedience. “Stability” is undermined by states that do not strictly follow orders, secular nationalists, Islamists who are not under control (in contrast, the Saudi monarchy, the oldest and most valuable US ally, is fine), etc. Such “destabilizing” forces are particularly dangerous when their programs are attractive to others, in which case they are called “viruses” that must be destroyed. “Stability” is enhanced by loyal client states. Since 1967, it has been assumed that Israel can play this role, along with other “peripheral” states. Israel has become virtually an off-shore US military base and high-tech center, the natural consequence of its rejection of security in favor of expansion in 1971, and repeatedly since. These policies are subject to little internal debate, whoever holds state power. The policies extend world-wide, and in the Middle East, their significance is enhanced by one of the leading principles of foreign policy since World War II (and for Britain before that): to ensure control over Middle East energy resources, recognized for 60 years to be “a stupendous source of strategic power” and “one of the greatest material prizes in world history.”
The standard Western version is that the July 2006 invasion was justified by legitimate outrage over capture of two Israeli soldiers at the border. The posture is cynical fraud. The US and Israel, and the West generally, have little objection to capture of soldiers, or even to the far more severe crime of kidnapping civilians (or of course to killing civilians). That had been Israeli practice in Lebanon for many years, and no one ever suggested that Israel should therefore be invaded and largely destroyed. Western cynicism was revealed with even more dramatic clarity as the current upsurge of violence erupted after Palestinian militants captured an Israeli soldier, Gilad Shalit, on June 25. That too elicited huge outrage, and support for Israel’s sharp escalation of its murderous assault on Gaza. The scale is reflected in casualties: in June, 36 Palestinian civilians were killed in Gaza; in July, the numbers more than quadrupled to over 170, dozens of them children. The posture of outrage was, again, cynical fraud, as demonstrated dramatically, and conclusively, by the reaction to Israel’s kidnapping of two Gaza civilians, the Muamar brothers, one day before, on June 24. They disappeared into Israel’s prison system, joining the hundreds of others imprisoned without charge — hence kidnapped, as are many of those sentenced on dubious charges. There was some brief and dismissive mention of the kidnapping of the Muamar brothers, but no reaction, because such crimes are considered legitimate when carried out by “our side.” The idea that this crime would justify a murderous assault on Israel would have been regarded as a reversion to Nazism.
The distinction is clear, and familiar throughout history: to paraphrase Thucydides, the powerful are entitled to do as they wish, while the weak suffer as they must.
We should not overlook the progress that has been made in undermining the imperial mentality that is so deeply rooted in Western moral and intellectual culture as to be beyond awareness. Nor should we forget the scale of what remains to be achieved, tasks that must be undertaken in solidarity and cooperation by people in North and South who hope to see a more decent and civilized world.
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The Anarchist Library
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Anti-Copyright
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Noam Chomsky
On the US-Israeli Invasion of Lebanon
August 19, 2006
\bigskip
Retrieved on 1\textsuperscript{st} October 2021 from \href{https://chomsky.info/20060819/}{chomsky.info}
Published in \emph{Al-Adab.}
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\textbf{theanarchistlibrary.org}
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\begin{document}
\begin{thebibliography}{1}
\bibitem{Galindo2004} Galindo, C. (2004). Amor y dinero en {\textquotedblleft}La Celestina{\textquotedblright}. En S. Fern{\'a}ndez, \& C. E. Armijo (Eds.), \textit{A quinientos a{\~n}os de {\textquotedblleft}La Celestina{\textquotedblright} (1499-1999)} (pp. 81--90). M{\'e}xico: UNAM.
\end{thebibliography}
\end{document}
|
https://doc.libelektra.org/api/pr/latex/autotoc_md83.tex | libelektra.org | CC-MAIN-2020-40 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-40/segments/1600400206133.46/warc/CC-MAIN-20200922125920-20200922155920-00772.warc.gz | 354,671,010 | 1,366 |
\begin{DoxyItemize}
\item infos = Information about the camel plugin is in keys below
\item infos/author = René Schwaiger \href{mailto:[email protected]}{\tt sanssecours@me.\+com}
\item infos/licence = B\+SD
\item infos/needs =
\item infos/provides = storage/camel
\item infos/recommends =
\item infos/placements = getstorage setstorage
\item infos/status = maintained shelltest unittest nodep preview unfinished concept discouraged
\item infos/metadata =
\item infos/description = A very basic plugin that reads and writes a very small subset of Y\+A\+ML
\end{DoxyItemize}\hypertarget{autotoc_md83_src_plugins_camel_README_md}{}\section{Camel}\label{autotoc_md83_src_plugins_camel_README_md}
\hypertarget{autotoc_md83_autotoc_md84}{}\subsection{Introduction}\label{autotoc_md83_autotoc_md84}
This plugin reads configuration data specified in a {\bfseries very limited} subset of the data serialization language \href{http://www.yaml.org}{\tt Y\+A\+ML}.\hypertarget{autotoc_md83_autotoc_md85}{}\subsection{Examples}\label{autotoc_md83_autotoc_md85}
\hypertarget{autotoc_md83_autotoc_md86}{}\subsubsection{Basic Usage}\label{autotoc_md83_autotoc_md86}
``{\ttfamily @section autotoc\+\_\+md87 Mount camel plugin to cascading namespace}/tests/camel` sudo kdb mount config.\+yaml /tests/camel camel
kdb set /tests/camel/key value kdb get /tests/camel/key \#$>$ value
kdb set /tests/camel/kittens \char`\"{}warm \& fuzzy\char`\"{} kdb get /tests/camel/kittens \#$>$ warm \& fuzzy
kdb set /tests/camel/empty \char`\"{}\char`\"{}
kdb export /tests/camel camel \#$>$ \{ \#$>$ \char`\"{}empty\char`\"{} \+: \char`\"{}\char`\"{} \#$>$ , \char`\"{}key\char`\"{} \+: \char`\"{}value\char`\"{} \#$>$ , \char`\"{}kittens\char`\"{} \+: \char`\"{}warm \& fuzzy\char`\"{} \#$>$ \}
kdb rm -\/r /tests/camel sudo kdb umount /tests/camel ```\hypertarget{autotoc_md83_autotoc_md88}{}\subsection{Limitations}\label{autotoc_md83_autotoc_md88}
Currently this plugin {\bfseries should not be used by anyone}. |
https://wiki.horde.org/Project/Sesha?actionID=export&format=tex | horde.org | CC-MAIN-2020-34 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-34/segments/1596439735885.72/warc/CC-MAIN-20200804220455-20200805010455-00469.warc.gz | 528,364,653 | 2,111 | \documentclass{article}
\usepackage{ulem}
\pagestyle{headings}
\begin{document}
\tableofcontents
SeshaBugsPeopleDescriptionSesha for Horde 3 +++Sesha for Horde 5 +++UsageFuture DevelopmentResources
\part{Sesha}
Sesha is a simple online inventory application based on the Horde framework.
Sesha has been used in several locations to assist in network management by providing storage for an inventory of hardware and software assets. However it is sufficiently generic to be applicable to any set of inventory items.
\section{Bugs}
List any tickets on http://bugs.horde.org/\footnote{http://bugs.horde.org/} that cover this issue or are relevant to it.
\section{People}
\begin{itemize}
\item{Bo Daley}
\item{Andrew Coleman}
\item{Johannes Huebner (huebner - at - b1-systems.de) is working on share-based, personalized inventory lists (assign inventory to people)}
\end{itemize}
\section{Description}
\subsection{Sesha for Horde 3 +++}
Sesha is currently available in Horde git. https://github.com/horde/sesha\footnote{https://github.com/horde/sesha}
\subsection{Sesha for Horde 5 +++}
Sesha has been ported to Horde 5 and is scheduled to be released together with the next minor update.\newline
At various points in the past Sesha has been able to hook into Whups so that tickets can be raised for inventory items. This feature currently requires uncommenting an API line in the registry.local.php file.
'sesha' => array(\newline
'name' => \_("Inventory"),\newline
\textsl{ Uncomment this line if you want Sesha to provide queue and version\newline
} names instead of Whups:\newline
// 'provides' => array('tickets/listQueues',\newline
'tickets/getQueueDetails', 'tickets/listVersions',\newline
'tickets/getVersionDetails'),\newline
'menu\_parent' => 'devel',\newline
),
This makes sesha categories show up in whups as ticket queues and stock items show up as queue versions.\newline
Warning, this hides your existing whups queues and versions.
A better configurable version is proposed in ticket \#10927 and ticket \#109278
\subsection{Usage}
To start you should add a set of Properties to describe your inventory items. Click the 'Manage Properties' tab under 'Admin'.
You can then add a set of Categories to place your items in (available by clicking the 'Manage Categories' tab). Each Category can be attached to one or more Properties.
For example, the 'Laptops' Category could be attached to a set of Properties such as: Manufacturer, Model, IP Address, etc.
\subsection{Future Development}
Plans and suggestions for further development
* Sesha items and categories as kronolith resources / resource groups\newline
* Better control which sesha items show up as whups bug queues\newline
* tagging support\newline
* Sesha could also be extended to provide storage for other applications requiring inventory or stock (eg. shopping carts).
\section{Resources}
TBD\ldots
\noindent\rule{\textwidth}{1pt}
Back to the Project List\footnote{http://example.com/index.php?page=Project}
\end{document}
|
https://fifthestate.anarchistlibraries.net/library/380-spring-2009-periodicals-received.tex | anarchistlibraries.net | CC-MAIN-2020-34 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-34/segments/1596439737019.4/warc/CC-MAIN-20200806180859-20200806210859-00441.warc.gz | 278,859,386 | 2,946 | \documentclass[DIV=12,%
BCOR=0mm,%
headinclude=false,%
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\finalhyphendemerits=10000
% given that we said footinclude=false, this should be safe
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\title{Periodicals Received}
\date{}
\author{Fifth Estate Collective}
\subtitle{}
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{\usekomafont{title}{\huge Periodicals Received\par}}%
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\textbf{Anarchist Studies} vol. 16 \#1 (2008)
c\Slash{}o Lawrence \& Wishart,
99a Wallis Rd,
London E9 5LN, UK
www.lwbooks.co.uk\Slash{}journals\Slash{}anarchiststudies\Slash{}contents.html
£20.00 subscription
\textbf{Anarchy: A Journal of Desire Armed} \#66 (Winter\Slash{}Fall 2008)
PO Box 3488,
Berkeley, CA 94703
anarchymag.org
\$20 \Slash{} 5 issue subscription
\textbf{Anchorage Anarchy} \#12 (\#13 December 2008)
BAD Press,
PO Box 230332,
Anchorage AK 99523
[email protected]
\$1 single copy
\textbf{Anarcho-Syndicalist Review} \#51 (Winter 2009)
PO Box 42531,
Philadelphia PA 19101
www.syndicalist.org
\$15 subscription
\textbf{Communicating Vessels} \#20 (Fall\Slash{}Winter 2008–2009)
3527 NE 15\textsuperscript{th} Ave,
Portland OR 97212
free sample copy; \$16 subscription
\textbf{Earth First! Journal} vol. 29 \#3 (March-April 2009)
PO Box 3023,
Tucson, AZ 85702
www.earthfirstjournal.org
\$25 subscription
\textbf{The Nuclear Resister} \#152 (Jan 24, 2009)
PO Box 43383,
Tucson, AZ 85733
www.nuclearresister.org
\$15 subscription
\textbf{Profane Existence} \#57 (Summer 2008)
PO Box 18051,
Minneapolis MN 55418
www.profaneexistence.org
\$30 subscription
\textbf{Student Insurgent} \#19.5
EMU Suite One,
University of Oregon,
Eugene, OR 97403
\$15 subscription
% begin final page
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\strut
\vfill
\begin{center}
Fifth Estate Collective
Periodicals Received
\bigskip
\href{https://www.fifthestate.org/archive/380-spring-2009/periodicals-received}{\texttt{https://www.fifthestate.org/archive/380-spring-2009/periodicals-received}}
Fifth Estate \#380, Spring 2009
\bigskip
\textbf{fifthestate.anarchistlibraries.net}
\end{center}
% end final page with colophon
\end{document}
|
https://textfiles.vistech.net/sex/asiaa_2.sty | vistech.net | CC-MAIN-2022-40 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-40/segments/1664030337668.62/warc/CC-MAIN-20221005203530-20221005233530-00269.warc.gz | 594,367,677 | 10,112 | asiaa_2.sty
Amy, the Thai Tease
by Gerry Kanne
Part 2
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
"What the hell is this?" Bob thought, pulling the card away from the
staples. She HAD known what was going on! She'd seen him sweating and
flushed; probably she'd heard a moan escape when he was cumming. The
sweet fox had known all along and never said anything! If only she had
made some sign, said something or even looked him square in the eye, then
he could have.... The simple thought of the juicy oriental coed
approving of his sexual act aroused him. He felt like getting in the car
and going now. But why give him this? Was this her way of telling him
to go jerk off with someone else? No, he remembered the twinkle in her
eye as she smiled when had left his office in the morning. This must be
her way of inviting him to ... to what? What was she planning? How did
she know to bring the card with her today? Why did she even have this
free pass? Was she a strip joint fan?
Highly unlikely. As he turned the card over few times to study it for
clues, he noticed that there was no day or time on the card, so she
wasn't trying to set up a date. He went back to her homework and quickly
read the second page, then went back and scanned the paper again. It was
simply the assigned topic, well done, and as much as he wished there was
something more, he found no "I want to fuck you, teacher," message. He
brought the paper up to his face and inhaled deeply. No scent but that
of wood pulp.
During the rest of the evening, Bob held the card and daydreamed about
Amy, twice turning back to stare her paper and the delicate curved script
on it. No other papers got corrected, and while he was able to get to
sleep, it was between sticky sheets that were moistened twice that night.
Getting ready for class the next morning, Bob decided that it would be
best to ignore the card sitting in his wallet and not visit the
nightclub. He was afraid of being seen and recognized in a place like
that. Even though it wouldn't get him fired, after all what he did on
his own time was his business, he didn't want to be known as the sex-mad
professor. Rumors had a way of spreading quickly through the faculty and
students, and he would lose the respect of many students who looked up to
him. Besides, he didn't really want coeds flashing and propositioning
him to 'improve' their grades, in spite of his fantasies. It was also
possible that Amy was trying to trap him in some way and then blackmail
him. It would be best just not to go, and hope Amy wouldn't say anything
to anyone. He would have to be extra careful in her presence to do
nothing suggestive and make sure he kept his eyes in head, no matter how
she dressed.
But while Bob was busy making virtuous decisions, the primal part of his
unconscious mind, a part that connected to the world only through
hormones, was already set on what he would do. Early in the evening,
card in his wallet, Bob left his house and got in the car to drive across
town. There was an empty feeling in his stomach and his palms were
sweating nervously. "Why are you doing this? What if SHE's there? How
will you be able to face her in class?" Suddenly, almost without
realizing how he got there, a bright sign announced "Girls! All Live!
All Nude!".
After parking his car in the lot outside, he looked around, got out and
quickly walked to the door. Just inside the entrance, he was stopped by
a man sitting at a table. "Ten dollar admission, two drink minimum,
sir." Bob took out the card and gave it to the man. He took it and
turned it over to see the note on the back. "You're a friend of Amy's,
huh? When you're that lucky, I don't see why you should get in free
too," as he stamped Bob's hand. "You still have to pay for the drinks,
though. Enjoy the show."
Bob entered the club, walking by a guy who seemed seven feet of solid
muscle, and found himself a seat in a dark back corner. The lights were
low, so no one was likely to recognize him, assuming anyone was looking
anywhere besides the stage. A slim, young blonde with small tits was at
the height of her act, lying on her side on floor of the stage doing leg
lifts. Her pussy was fleeced with hair noticeably darker than the hair
on her head. She was good looking, but not the dish Bob had come for.
The doorman had known Amy, so Bob expected to see her on stage. The idea
of seeing her up there had him excited, but he wasn't sure he wanted her
to know he was watching.
As the music started up again, another girl appeared on the stage backlit
by a spotlight. She stood with her legs apart, light shining through,
her hair a misty frame around her head. On the beat, she turned sideways
presenting a profile in silhouette. She leaned forward a bit, thrusting
her chest forward and arching her back to present the classic lines of a
pin-up. Lights came up revealing Amy in a skimpy costume. She wore
black and strings of pearls. Her lacy black bustier pushed her white
breasts up, making a shelf for the long strings of pearls that rolled
over them and flowed down between her breasts. Below was a black garter
belt with matching stockings and 5-inch heels to show off her calves, the
seams of the stockings leading the eyes up to the firm curves of her ass.
A tiny black lace g-string was partly hidden in the front by a loop of
pearls that circled her waist. From this loop hung 5 strands of pearls
reaching to mid-thigh, forming a sort of bead-curtain loincloth that
flashed the scrap of translucent lace covering her crotch with every
gyration of her hips.
As Amy danced, she slowly peeled off her top and teased her nipples by
rolling the pearls back and forth over them. As she showed off the
stiffening tips of her breasts to the crowd, she locked eyes briefly with
each person in the audience. When her eyes reached Bob, she stopped and
ran her hands up from her inner thighs to the ends of her nipples, now
fully engorged on the up-turned tips of her quivering breasts. Even
sitting near the back of the dimly lit room, Bob knew Amy had seen and
recognized him. Blood rushed to his face, and if the light had been
better, it would have been plain to anyone that he was blushing as
furiously as any virgin. However, blood also rushed to his cock which
pulsed in time to Amy's bumps and grinds. When the black g-string
dropped to the stage floor, Amy bent over, legs spread, her shaved cunt
aimed right at her teacher. She looked over her shoulder at Bob with a
sultry gaze, her hair covering half her face and ran her hands up the
back of her legs, following the stocking seam from her knees up to the
garter strap and then spreading her tight cheeks wide. The smile and
wink of an eye that followed almost knocked Bob off his chair. He
reached for his soft drink to wet his suddenly dry throat and found his
hand shaking so much he had to set the glass down on the table and lean
forward to drink.
Bob's desire for the petite Asian had him literally shaking. He wanted
very much to get her alone, as she had been the day before in his office.
Here he couldn't do anything, not even secretly jerking off. He was sure
that would be an excellent way to get tossed out of the joint. Besides,
he wanted to do much more than simply look, especially after the way she
was teasing him from the stage.
Amy began to move around the audience off stage, picking up tips while
doing a little bump and grind for each man. In the dim light of Bob's
back corner, Amy strutted up and began to undulate her hips, teasing a
tip out of his wallet. Bob grinned wide enough to cross the Mississippi,
but wasn't sure what to say to his student, so he stared at her jiggling
tits. The round mounds stood out from her chest with no sag at all. Her
full, up-turned nipples trapped his eyes. The aureoles stood out forming
their own walnut-sized mounds topped by stiff nipples. The flesh
quivering to Amy's movements made Bob's mouth literally water. He looked
up and found Amy's dark eyes locked on his.
As he swallowed, Amy spoke up. "Aren't you even going to tip me? You
look happy to see me," she said and bit softly on her lower lip while her
arms squeezed her breasts together, pushing her nipples out even farther,
all the while staring at him.
"Oh, sure," Bob said, and pulled a ten spot out of his wallet. "I really
appreciated the free pass, and your show was great." He took a deep
breath and continued "You're the most beautiful--the most gorgeous woman
I've ever seen in person," he said raspy with nervousness and breaking
contact with her eyes only to be trapped by the white smooth valley
between her breasts.
"Down here," Amy said, and Bob's eyes were momentarily again struck by
the intense stare of dark eyes from the East before her finger, wagging
in front of her face began to trace a line downwards pushing over her
full lips and down the soft throat to between the soft cushions of her
breasts. Her forearm passed over her tit, mashing it down, and when it
popped free, its inflamed nipple snapping out, Bob found that his cock,
hard as iron, had suddenly pulsed and become stiffer. He could have used
it for batting practice with a hard ball with no problem at all.
"Down.." Amy said, her finger indenting her soft stomach and poking
briefly into her navel before traveling across the smooth flat skin
between her hips.
"...here," she finished, her fingers now toying with the pearls that were
her only covering before moving on. Bob now was hypnotized by the magic
place behind the swinging strands of pearls. The shimmery pearls rolled
over a mohawk strip of trimmed pussy hair, pushing the strands down only
to have them pop up before the next string of pearls repeated the
process. Below, her cunt was shaved bare and smooth, the inner lips just
visible between the pouting outer fold. Unconsciously leaning down
toward the area of his dreams, Bob detected a faint, musky odor. The
clacking of the of the pearls, audible above the music this close, mixed
with the scent and sight of his student's pussy carried him away from the
noisy club, and his entire world became a few square inches of nubile
flesh. The pearls pattered against the bare cunt lips, perhaps creating
the arousal that Bob smelled. One strand hit squarely between the lips
and stuck for an instant before dropping away glistening with juice.
This woman was wet and ready.
"Down here, silly," Amy repeated for the third time, and Bob saw her
stretching out a garter two-thirds up on her right thigh. He came to
himself and reached slowly towards the thigh with great deliberation,
afraid his hand might stray to the moist lips inches away. The backs of
his fingers brushed over the smooth skin of Amy's inner thigh as they
pulled the garter out to receive ten dollar bill. The skin was slightly
cool to the touch, but the back of his hand felt waves of heat coming
from the pussy above as if it were a space heater. The pearl strings ran
across his hand as he pulled it back, leaving a small trail of dampness
that cooled in the open air.
"If you like what you see, and I can tell that you do," she said, her
eyes shining with the reflected glow of neon, "you can have a private
dance for only $50 in our 1-on-1 booth."
"Sounds great," Bob's cock replied as he reached back to pull out his
wallet a second time. If she had said $5,000 for getting in a pit with
10 starving, rabid Dobermans, his response would have been the same.
Sexual desire had shorted out the rational center of his brain and the
need of his hard dick fogged out all of his earlier shyness. To be alone
with his fantasy girl in a situation where her role was to tease and
arouse him directly touched a deep unconscious need that broke through
the restraints imposed by his role as teacher to this sultry student.
His cock pulsed, wanting attention as he handed over the money. All he
could think of was his body pressed against Amy's soft, warm skin while
his dick was encased in the hot, moist grip of her cunt.
He heard her instructions to go to a door at the side of the room, but
his mind was screaming for joy! His imagination was in overdrive,
producing images of his hand reaching out and pressing, kneading and
pulling on those puckered nipples as he bends down to kiss her full lips.
How far could he go? His imagination made fantasies become realities,
and Bob trembled.
In a few minutes, Bob found himself in small room. He sat in the simple
chair against the wall and stared at through the clear glass partition 2
feet in front of him that separated his small cubby hole from a larger
dance area. This was not what he had expected at all. Sure, it was
private, but he had certainly thought that "private" also implied
something more intimate. Talk about safe sex! This glass barrier could
protect a herd of pachyderms. The whole set up seemed designed so that a
man, perhaps for a suitable tip, could masturbate and the dancer would
not have to worry about the guy getting too carried away. Bob felt very
uncomfortable in the situation, thinking that Amy would type him as some
strange pervert bent on preying on students. Adding a sharp edge to the
situation was his recent memory of the girl's smoldering dark eyes and
her full red lips in a smile as he ogled her naked breasts and wet pussy.
If he didn't know Amy, then he could have gone with the situation and let
his desire take control. However, he wanted to keep his lusts secret
from the co-ed, although how that was possible, now that he'd accepted
this invitation, was a bit of a problem. He wondered how many men had
come to this room and heard her high, girlish voice tinkling out
obscenities, urging them to orgasm.
Now Amy came out. Her tits jiggled and bounced with each step; the
bright yellow body suit molded itself to her skin without restraining the
fluid movements of her supple muscles and without completely obscuring
the dark puffy buds of her nipples. Matching yellow stockings wrapped
around the curves of her legs. "Hi there," she said, popping a cassette
into a tape deck, " Make yourself comfortable. I hope you enjoy
yourself." Her voice came over a speaker on Bob's side, but he could see
no microphone.
Bob was already enjoying the olive skin above the stocking top that
curved up and out when it reached her low-slung ass. The high cut bottom
of the body suit set off her soft, fleshy bottom, her olive skin making a
nice contrast to the bright yellow of the narrow band that disappeared
between her cheeks. Even if he couldn't touch her, Bob was enjoying the
view.
Then music started and the light in Amy's side of the booth went out,
replaced by a black light that set the co-ed's yellow garb glowing. Bob
could dimly make out her face and the bare skin of her thighs, but her
glowing legs and the outline of her tits jiggling to her movements were
the focus of his attention. His cock pulsed and swelled when her thick
nipples atop puffy areolae were clearly outlined in profile, the sheer
yellow fabric molding perfectly to their shape. At regular intervals a
strobe light flashed, revealing the nubile student in poses that seared
his retinas. He saw her kneading her crotch or grabbing her breasts
through the thin, semi-transparent material. One flash of the light
revealed her tongue out, wetting her upper lip, bright red with lipstick.
He also noticed that her eyes were closed, the lids lightly colored with
yellow eyeshadow matching her costume.
Amy's closed eyes gave him the courage to unzip his trousers and pry out,
a little painfully, his stiff prick. As the stiff organ snapped out, he
grabbed it and began to seriously stroke it, the head an angry purple.
He was intensely aware of his student a few feet away from his naked
dick. As he pumped and pulled the obscenely swollen organ, Amy began to
spray herself with a shower head on a hose. Bob didn't know where it had
come from, but the water sparkled in the black light and encased the sexy
co-ed's body in a silvery halo. It also, he quickly noticed, turned the
bodysuit completely transparent. His pants were below his knees now as
his hand pumped his cock more insistently as Amy bent over, her ass
inches from the glass, the bare lips of her pussy showing clearly through
the soaked fabric.
The erotic, wet-dream world created by the black light, music, water and
sex goddess body of the young Thai lady became more charged when Amy
changed the spray of water to pulsing jets. She stroked the hose in time
to the bursts of compact water that she aimed at her face and open mouth.
Bob's unconscious immediately turned the water splattering and drenching
the student's face into his cum. He pressed against the glass, moving
his head back only when his breath began to fog his view of her full red
lips and the stray lock of hair stuck to her cheek. The swollen head of
his penis flattened against the cool glass as his body pressed forward,
trying to reach the biological magnet attracting it, the Asian dreamgirl
working as hard as she could to get his dick hard and his balls pumping
cum. She was no longer his student or even a real person, for Bob's mind
had found both the source and goal of the lust that had driven the lives
of creatures alive before the hominid family was ever considered by the
forces of evolution.
Amy turned to face him in the flickering strobe light and as one pulled
at a puckered nipple the other sensuously gathered the water from one
cheek and popped into her mouth. Her almond eyes were still closed when
her finger began to fuck her shiny red lips which were pursed around the
substitute cock. At the same time, her other hand stroked down her body
and slid beneath the thin transparent strip of cloth which stuck wetly to
her pussy. First one and then two fingers slipped between the naked cunt
lips deep into her wet hole. Amy began to gyrate her hips and masturbate
in time to the deep low sound of the bass line in the music, driving her
fingers hard and deep into her creaming pussy. Her beautiful face,
sparkling with drops of water that could be lust-created sweat, was drawn
in passion while she licked and bit the finger in her mouth.
Bob steadily and firmly stroked his cock, his knees bent and his hips
thrust forward so that the engorged organ bobbed in the air in front of
his student. He moved his other hand between his legs, the palm pressing
the soft wrinkled skin of his ball sack and massaging the two grape-sized
orbs inside as his fingers rubbed the stiff hidden base of his dick back
between his legs. His lust was amplified by the uninhibited eroticism
displayed by the girl finger fucking herself a mere foot or two away.
The mushroom head flared wide and became smooth and shiny. The rod in
his hand became steel-rigid and his muscles tensed in pleasure, but
inside he felt a relaxing as a passage opened up for his building current
of cum. His cock, now huge and swollen, waved openly in angry power in
front of its focus, the Asian coed bitch-in-heat. His pounding fist now
ran on automatic, blurring up and down the stiff pulsing dick.
A strong flow of cum was being pressured up his penis when the music
suddenly stopped and Amy dropped to her knees in front of him, face up,
tongue wrapped around her finger, eyes open and staring into his. If the
eyes are mirrors of the soul, here animal passion had clearly claimed
both, and in the meeting of these two spirits, Bob felt a snarling,
frenzied lust which could never be bettered but only matched by the most
savage of animal matings. Thick white gouts of passion slammed and
spattered the glass in front of his Oriental student's face over and over
again while Bob's lips curled back and his breath came strong vocal
pants. Lava flows of cum ran down the glass only to be fed from the
source once more. Amy stuck out her wet tongue and wagged it at the
pumping prick, then bit her full lower lip, her eyes never leaving Bob's.
"Cunt. Slut," Bob's voice growled out as he pulled out the last of the
cum to drip on floor. Amy smiled behind the sperm-smeared glass.
Where did THAT come from, Bob wondered, his rational mind now taking
control. He quickly took in the sleazy scene. His pants were down
around his ankles; both hand were in his crotch cupping his balls and
holding his penis, cum oozing out the tip. The glass looked like a water
balloon filled with cum had hit it, and Amy was on the other side, the
coed with the shaved cunt and a bodysuit that covered everything but hid
nothing. His temples, armpits and back were wet, and the smell of semen
and sweat filled his nostrils.
How could he have done this in front of her? His conscience, formed many
years ago by parents and pulpit, came down hard. He turned to the side
away from Amy and groped for his trousers. His cock had not yet had time
to wilt, so he had to painfully bend the still massive organ and
forcefully stuff it in his briefs. He did not notice Amy's avid stare.
"Wow! I didn't even take off my clothes yet. I told you it was a good
show," Amy remarked smugly. "Hey, wait!" she cried when she saw her
professor moving toward the door before he had even put his clothes back
together. "Can't you stay around for a while? I have a special show to
do later, but after you can drive me home. I don't want to take a cab."
Bob heard her muffled voice through the glass, but he couldn't bring
himself to turn around and look at the girl. But wait, she said
something about taking her home?
"I could do that, if you want," he said hesitantly over his shoulder.
"Great," Amy replied and gave the glass a kick to get him to turn around.
"See you later," she said, her fingers pulling her stiff nipples through
the yellow fabric. |
http://wiki.axiom-developer.org/axiom-website/CATS/richhyper400-499.input.pamphlet | axiom-developer.org | CC-MAIN-2017-26 | unk | application/x-tex | crawl-data/CC-MAIN-2017-26/segments/1498128320057.96/warc/CC-MAIN-20170623114917-20170623134917-00678.warc.gz | 445,145,956 | 93,154 | \documentclass{article}
\usepackage{axiom}
\setlength{\textwidth}{400pt}
\begin{document}
\title{\$SPAD/src/input richhyper400-499.input}
\author{Albert Rich and Timothy Daly}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\begin{chunk}{*}
)set break resume
)sys rm -f richhyper400-499.output
)spool richhyper400-499.output
)set message auto off
)clear all
--S 1 of 526
t0400:= 1/(a+b*sech(x))
--R
--R
--R 1
--R (1) -------------
--R b sech(x) + a
--R Type: Expression(Integer)
--E 1
--S 2 of 526
r0400:= x/a-2*b*atan((a-b)*tanh(1/2*x)/(a^2-b^2)^(1/2))/a/(a^2-b^2)^(1/2)
--R
--R
--R x
--R (b - a)tanh(-) +---------+
--R 2 | 2 2
--R 2b atan(--------------) + x\|- b + a
--R +---------+
--R | 2 2
--R \|- b + a
--R (2) ---------------------------------------
--R +---------+
--R | 2 2
--R a\|- b + a
--R Type: Expression(Integer)
--E 2
--S 3 of 526
a0400:= integrate(t0400,x)
--R
--R
--R (3)
--R [
--R b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b - a
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R +
--R 2 3 2 3 3 2
--R (2a b - 2a )sinh(x) + (2a b - 2a )cosh(x) + 2b - 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x)
--R +
--R 2b cosh(x) + a
--R +
--R +-------+
--R | 2 2
--R x\|b - a
--R /
--R +-------+
--R | 2 2
--R a\|b - a
--R ,
--R +---------+
--R | 2 2 +---------+
--R (a sinh(x) + a cosh(x) + b)\|- b + a | 2 2
--R 2b atan(---------------------------------------) + x\|- b + a
--R 2 2
--R b - a
--R ----------------------------------------------------------------]
--R +---------+
--R | 2 2
--R a\|- b + a
--R Type: Union(List(Expression(Integer)),...)
--E 3
--S 4 of 526
m0400a:= a0400.1-r0400
--R
--R
--R (4)
--R +---------+
--R | 2 2
--R b\|- b + a
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b - a
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R +
--R 2 3 2 3 3 2
--R (2a b - 2a )sinh(x) + (2a b - 2a )cosh(x) + 2b - 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R a
--R +
--R x
--R +-------+ (b - a)tanh(-)
--R | 2 2 2
--R - 2b\|b - a atan(--------------)
--R +---------+
--R | 2 2
--R \|- b + a
--R /
--R +---------+ +-------+
--R | 2 2 | 2 2
--R a\|- b + a \|b - a
--R Type: Expression(Integer)
--E 4
--S 5 of 526
d0400a:= D(m0400a,x)
--R
--R
--R (5)
--R 2 2
--R (b sinh(x) + (2b cosh(x) + 2b)sinh(x) + b cosh(x) + 2b cosh(x) + b)
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 2 2
--R - b sinh(x) + (- 2b cosh(x) + 2b)sinh(x) - b cosh(x) + 2b cosh(x) - b
--R /
--R 2 2 2 2
--R (a b - a )sinh(x) + ((2a b - 2a )cosh(x) + 2b - 2a b)sinh(x)
--R +
--R 2 2 2 2
--R (a b - a )cosh(x) + (2b - 2a b)cosh(x) + a b - a
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 2 2 2 2
--R (- a b - a )sinh(x) + ((- 2a b - 2a )cosh(x) - 2b - 2a b)sinh(x)
--R +
--R 2 2 2 2
--R (- a b - a )cosh(x) + (- 2b - 2a b)cosh(x) - a b - a
--R Type: Expression(Integer)
--E 5
--S 6 of 526
m0400b:= a0400.2-r0400
--R
--R
--R (6)
--R +---------+ x
--R | 2 2 (b - a)tanh(-)
--R (a sinh(x) + a cosh(x) + b)\|- b + a 2
--R 2b atan(---------------------------------------) - 2b atan(--------------)
--R 2 2 +---------+
--R b - a | 2 2
--R \|- b + a
--R --------------------------------------------------------------------------
--R +---------+
--R | 2 2
--R a\|- b + a
--R Type: Expression(Integer)
--E 6
--S 7 of 526
d0400b:= D(m0400b,x)
--R
--R
--R (7)
--R 2 2
--R (b sinh(x) + (2b cosh(x) + 2b)sinh(x) + b cosh(x) + 2b cosh(x) + b)
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 2 2
--R - b sinh(x) + (- 2b cosh(x) + 2b)sinh(x) - b cosh(x) + 2b cosh(x) - b
--R /
--R 2 2 2 2
--R (a b - a )sinh(x) + ((2a b - 2a )cosh(x) + 2b - 2a b)sinh(x)
--R +
--R 2 2 2 2
--R (a b - a )cosh(x) + (2b - 2a b)cosh(x) + a b - a
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 2 2 2 2
--R (- a b - a )sinh(x) + ((- 2a b - 2a )cosh(x) - 2b - 2a b)sinh(x)
--R +
--R 2 2 2 2
--R (- a b - a )cosh(x) + (- 2b - 2a b)cosh(x) - a b - a
--R Type: Expression(Integer)
--E 7
--S 8 of 526
t0401:= 1/(a+b*sech(x))^2
--R
--R
--R 1
--R (8) ------------------------------
--R 2 2 2
--R b sech(x) + 2a b sech(x) + a
--R Type: Expression(Integer)
--E 8
--S 9 of 526
r0401:= x/a^2-2*b^3*atan((a-b)*tanh(1/2*x)/(a^2-b^2)^(1/2))/a^2/_
(a^2-b^2)^(3/2)-4*b*atan((a-b)*tanh(1/2*x)/(a^2-b^2)^(1/2))/_
a^2/(a^2-b^2)^(1/2)+b^2*sinh(x)/a/(a^2-b^2)/(b+a*cosh(x))
--R
--R
--R (9)
--R x
--R (b - a)tanh(-)
--R 3 3 4 2 2 2
--R ((2a b - 4a b)cosh(x) + 2b - 4a b )atan(--------------)
--R +---------+
--R | 2 2
--R \|- b + a
--R +
--R +---------+
--R 2 2 3 3 2 | 2 2
--R (- a b sinh(x) + (a b - a )x cosh(x) + (b - a b)x)\|- b + a
--R /
--R +---------+
--R 3 2 5 2 3 4 | 2 2
--R ((a b - a )cosh(x) + a b - a b)\|- b + a
--R Type: Expression(Integer)
--E 9
--S 10 of 526
a0401:= integrate(t0401,x)
--R
--R
--R (10)
--R [
--R 3 3 2
--R (a b - 2a b)sinh(x)
--R +
--R 3 3 4 2 2
--R ((2a b - 4a b)cosh(x) + 2b - 4a b )sinh(x)
--R +
--R 3 3 2 4 2 2 3 3
--R (a b - 2a b)cosh(x) + (2b - 4a b )cosh(x) + a b - 2a b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b - a
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R +
--R 2 3 2 3 3 2
--R (2a b - 2a )sinh(x) + (2a b - 2a )cosh(x) + 2b - 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x)
--R +
--R 2b cosh(x) + a
--R +
--R 2 3 2
--R (a b - a )x sinh(x)
--R +
--R 2 3 3 2 3
--R ((2a b - 2a )x cosh(x) + (2b - 2a b)x + 2b )sinh(x)
--R +
--R 2 3 2 3 2 3 2 3
--R (a b - a )x cosh(x) + ((2b - 2a b)x + 2b )cosh(x) + (a b - a )x
--R +
--R 2
--R 2a b
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R /
--R 3 2 5 2 3 2 5 2 3 4
--R (a b - a )sinh(x) + ((2a b - 2a )cosh(x) + 2a b - 2a b)sinh(x)
--R +
--R 3 2 5 2 2 3 4 3 2 5
--R (a b - a )cosh(x) + (2a b - 2a b)cosh(x) + a b - a
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R ,
--R
--R 3 3 2
--R (2a b - 4a b)sinh(x)
--R +
--R 3 3 4 2 2
--R ((4a b - 8a b)cosh(x) + 4b - 8a b )sinh(x)
--R +
--R 3 3 2 4 2 2 3 3
--R (2a b - 4a b)cosh(x) + (4b - 8a b )cosh(x) + 2a b - 4a b
--R *
--R +---------+
--R | 2 2
--R (a sinh(x) + a cosh(x) + b)\|- b + a
--R atan(---------------------------------------)
--R 2 2
--R b - a
--R +
--R 2 3 2
--R (a b - a )x sinh(x)
--R +
--R 2 3 3 2 3
--R ((2a b - 2a )x cosh(x) + (2b - 2a b)x + 2b )sinh(x)
--R +
--R 2 3 2 3 2 3 2 3
--R (a b - a )x cosh(x) + ((2b - 2a b)x + 2b )cosh(x) + (a b - a )x
--R +
--R 2
--R 2a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R /
--R 3 2 5 2 3 2 5 2 3 4
--R (a b - a )sinh(x) + ((2a b - 2a )cosh(x) + 2a b - 2a b)sinh(x)
--R +
--R 3 2 5 2 2 3 4 3 2 5
--R (a b - a )cosh(x) + (2a b - 2a b)cosh(x) + a b - a
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R ]
--R Type: Union(List(Expression(Integer)),...)
--E 10
--S 11 of 526
m0401a:= a0401.1-r0401
--R
--R
--R (11)
--R 2 3 4 4 3 2 2
--R ((a b - 2a b)cosh(x) + a b - 2a b )sinh(x)
--R +
--R 2 3 4 2 4 3 2 5 2 3
--R ((2a b - 4a b)cosh(x) + (4a b - 8a b )cosh(x) + 2b - 4a b )
--R *
--R sinh(x)
--R +
--R 2 3 4 3 4 3 2 2
--R (a b - 2a b)cosh(x) + (3a b - 6a b )cosh(x)
--R +
--R 5 2 3 4 4 3 2
--R (2b - 3a b - 2a b)cosh(x) + a b - 2a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b - a
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R +
--R 2 3 2 3 3 2
--R (2a b - 2a )sinh(x) + (2a b - 2a )cosh(x) + 2b - 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R a
--R +
--R 2 3 4 4 3 2 2
--R ((- 2a b + 4a b)cosh(x) - 2a b + 4a b )sinh(x)
--R +
--R 2 3 4 2 4 3 2 5
--R (- 4a b + 8a b)cosh(x) + (- 8a b + 16a b )cosh(x) - 4b
--R +
--R 2 3
--R 8a b
--R *
--R sinh(x)
--R +
--R 2 3 4 3 4 3 2 2
--R (- 2a b + 4a b)cosh(x) + (- 6a b + 12a b )cosh(x)
--R +
--R 5 2 3 4 4 3 2
--R (- 4b + 6a b + 4a b)cosh(x) - 2a b + 4a b
--R *
--R x
--R +-------+ (b - a)tanh(-)
--R | 2 2 2
--R \|b - a atan(--------------)
--R +---------+
--R | 2 2
--R \|- b + a
--R +
--R 2 2 3 2 2 3 2
--R a b sinh(x) + (2a b cosh(x) + 2a b )sinh(x)
--R +
--R 2 2 2 3 4 2 2 3 2
--R (a b cosh(x) + 4a b cosh(x) + 2b + a b )sinh(x) + 2a b cosh(x)
--R +
--R 4 2 2 3
--R (2b + 2a b )cosh(x) + 2a b
--R *
--R +---------+ +-------+
--R | 2 2 | 2 2
--R \|- b + a \|b - a
--R /
--R 4 2 6 3 3 5 2
--R ((a b - a )cosh(x) + a b - a b)sinh(x)
--R +
--R 4 2 6 2 3 3 5 2 4 4 2
--R ((2a b - 2a )cosh(x) + (4a b - 4a b)cosh(x) + 2a b - 2a b )sinh(x)
--R +
--R 4 2 6 3 3 3 5 2
--R (a b - a )cosh(x) + (3a b - 3a b)cosh(x)
--R +
--R 2 4 4 2 6 3 3 5
--R (2a b - a b - a )cosh(x) + a b - a b
--R *
--R +---------+ +-------+
--R | 2 2 | 2 2
--R \|- b + a \|b - a
--R Type: Expression(Integer)
--E 11
--S 12 of 526
d0401a:= D(m0401a,x)
--R
--R
--R (12)
--R 3 3 4 2 6
--R (- a b + a b )sinh(x)
--R +
--R 3 3 4 2 2 4 3 3 5
--R ((- 4a b + 4a b )cosh(x) - 4a b + 4a b )sinh(x)
--R +
--R 3 3 4 2 5 2
--R (- 4a b + 5a b - 2a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 2 4 3 3 4 2
--R (- 9a b + 11a b - 4a b )cosh(x) - 3a b + 4a b - 4a b + 2a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 3 5 3 2 4 3 3 4 2 5 2
--R (4a b - 8a b)cosh(x) + (4a b + 8a b - 20a b - 4a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 2 4
--R (12a b - 20a b - 4a b )cosh(x) + 4a b - 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 3 3 4 2 5 4
--R (11a b - 5a b - 12a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 3
--R (26a b - 2a b - 36a b - 12a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 2
--R (18a b + 16a b - 38a b - 28a b - 4a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 6 5
--R (4b + 16a b - 18a b - 18a b - 8a b )cosh(x) + 4b - 2a b
--R +
--R 2 4 3 3 4 2
--R - 4a b - 5a b + a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 3 4 2 5 5
--R (8a b - 4a b - 8a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 4
--R (24a b - 4a b - 28a b - 12a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 3
--R (24a b + 12a b - 32a b - 36a b - 8a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 5 2
--R (8b + 20a b - 12a b - 32a b - 20a b - 4a b)cosh(x)
--R +
--R 6 2 4 3 3 4 2 5 2 4 3 3
--R (8b - 4a b - 16a b - 8a b )cosh(x) + 4a b - 4a b - 4a b
--R *
--R sinh(x)
--R +
--R 3 3 4 2 5 6 2 4 3 3 4 2 5 5
--R (2a b - a b - 2a b)cosh(x) + (7a b - a b - 8a b - 4a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 4
--R (9a b + 4a b - 10a b - 14a b - 4a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 5 3
--R (4b + 8a b - 2a b - 14a b - 12a b - 4a b)cosh(x)
--R +
--R 6 5 3 3 4 2 5 2
--R (4b + 2a b - 10a b - 9a b - 2a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 3 3
--R (4a b - a b - 5a b - 4a b )cosh(x) + a b - 2a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 3 3 4 2 6 3 3 4 2 2 4 3 3 5
--R (a b + a b )sinh(x) + ((4a b + 4a b )cosh(x) + 4a b + 4a b )sinh(x)
--R +
--R 3 3 4 2 5 2 2 4 3 3 4 2
--R (4a b + 5a b + 2a b)cosh(x) + (9a b + 11a b + 4a b )cosh(x)
--R +
--R 5 2 4 3 3 4 2
--R 3a b + 4a b + 4a b + 2a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 3 5 3 2 4 3 3 4 2 5 2
--R (- 4a b + 8a b)cosh(x) + (- 4a b + 8a b + 20a b - 4a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 2 4
--R (12a b + 20a b - 4a b )cosh(x) + 4a b + 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 3 3 4 2 5 4
--R (- 11a b - 5a b + 12a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 3
--R (- 26a b - 2a b + 36a b - 12a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 2
--R (- 18a b + 16a b + 38a b - 28a b + 4a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 6 5
--R (- 4b + 16a b + 18a b - 18a b + 8a b )cosh(x) + 4b + 2a b
--R +
--R 2 4 3 3 4 2
--R - 4a b + 5a b + a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 3 4 2 5 5
--R (- 8a b - 4a b + 8a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 4
--R (- 24a b - 4a b + 28a b - 12a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 3
--R (- 24a b + 12a b + 32a b - 36a b + 8a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 5 2
--R (- 8b + 20a b + 12a b - 32a b + 20a b - 4a b)cosh(x)
--R +
--R 6 2 4 3 3 4 2 5 2 4 3 3
--R (8b - 4a b + 16a b - 8a b )cosh(x) + 4a b + 4a b - 4a b
--R *
--R sinh(x)
--R +
--R 3 3 4 2 5 6 2 4 3 3 4 2 5 5
--R (- 2a b - a b + 2a b)cosh(x) + (- 7a b - a b + 8a b - 4a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 4
--R (- 9a b + 4a b + 10a b - 14a b + 4a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 5 3
--R (- 4b + 8a b + 2a b - 14a b + 12a b - 4a b)cosh(x)
--R +
--R 6 5 3 3 4 2 5 2
--R (4b - 2a b + 10a b - 9a b + 2a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 3 3
--R (4a b + a b - 5a b + 4a b )cosh(x) - a b + 2a b
--R /
--R 5 3 6 2 7 8 2
--R (a b - a b - a b + a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 3 5 4 4 5 3 6 2
--R (2a b - 2a b - 2a b + 2a b)cosh(x) + a b - a b - a b + a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 3 6 2 7 8 3
--R (4a b - 4a b - 4a b + 4a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 2
--R (12a b - 12a b - 12a b + 12a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 2 6 3 5
--R (12a b - 12a b - 12a b + 12a b )cosh(x) + 4a b - 4a b
--R +
--R 4 4 5 3
--R - 4a b + 4a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 3 6 2 7 8 4
--R (6a b - 6a b - 6a b + 6a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 3
--R (24a b - 24a b - 24a b + 24a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 2
--R (34a b - 34a b - 32a b + 32a b - 2a b + 2a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 7
--R (20a b - 20a b - 16a b + 16a b - 4a b + 4a b)cosh(x) + 4a b
--R +
--R 2 6 3 5 4 4 5 3 6 2
--R - 4a b - 2a b + 2a b - 2a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 6 2 7 8 5
--R (4a b - 4a b - 4a b + 4a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 4
--R (20a b - 20a b - 20a b + 20a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 3
--R (36a b - 36a b - 32a b + 32a b - 4a b + 4a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 2
--R (28a b - 28a b - 16a b + 16a b - 12a b + 12a b)cosh(x)
--R +
--R 7 2 6 3 5 4 4 5 3 6 2 2 6
--R (8a b - 8a b + 4a b - 4a b - 12a b + 12a b )cosh(x) + 4a b
--R +
--R 3 5 4 4 5 3
--R - 4a b - 4a b + 4a b
--R *
--R sinh(x)
--R +
--R 5 3 6 2 7 8 6
--R (a b - a b - a b + a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 5
--R (6a b - 6a b - 6a b + 6a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 4
--R (13a b - 13a b - 11a b + 11a b - 2a b + 2a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 3
--R (12a b - 12a b - 4a b + 4a b - 8a b + 8a b)cosh(x)
--R +
--R 7 2 6 3 5 4 4 5 3 6 2 7 8 2
--R (4a b - 4a b + 6a b - 6a b - 9a b + 9a b - a b + a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 3 5 4 4
--R (4a b - 4a b - 2a b + 2a b - 2a b + 2a b)cosh(x) + a b - a b
--R +
--R 5 3 6 2
--R - a b + a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 5 3 6 2 7 8 2
--R (- a b - a b + a b + a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 3 5 4 4 5 3 6 2
--R (- 2a b - 2a b + 2a b + 2a b)cosh(x) - a b - a b + a b + a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 3 6 2 7 8 3
--R (- 4a b - 4a b + 4a b + 4a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 2
--R (- 12a b - 12a b + 12a b + 12a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 2 6 3 5 4 4
--R (- 12a b - 12a b + 12a b + 12a b )cosh(x) - 4a b - 4a b + 4a b
--R +
--R 5 3
--R 4a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 3 6 2 7 8 4
--R (- 6a b - 6a b + 6a b + 6a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 3
--R (- 24a b - 24a b + 24a b + 24a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 2
--R (- 34a b - 34a b + 32a b + 32a b + 2a b + 2a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 7
--R (- 20a b - 20a b + 16a b + 16a b + 4a b + 4a b)cosh(x) - 4a b
--R +
--R 2 6 3 5 4 4 5 3 6 2
--R - 4a b + 2a b + 2a b + 2a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 6 2 7 8 5
--R (- 4a b - 4a b + 4a b + 4a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 4
--R (- 20a b - 20a b + 20a b + 20a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 3
--R (- 36a b - 36a b + 32a b + 32a b + 4a b + 4a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 2
--R (- 28a b - 28a b + 16a b + 16a b + 12a b + 12a b)cosh(x)
--R +
--R 7 2 6 3 5 4 4 5 3 6 2 2 6
--R (- 8a b - 8a b - 4a b - 4a b + 12a b + 12a b )cosh(x) - 4a b
--R +
--R 3 5 4 4 5 3
--R - 4a b + 4a b + 4a b
--R *
--R sinh(x)
--R +
--R 5 3 6 2 7 8 6
--R (- a b - a b + a b + a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 5
--R (- 6a b - 6a b + 6a b + 6a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 4
--R (- 13a b - 13a b + 11a b + 11a b + 2a b + 2a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 3
--R (- 12a b - 12a b + 4a b + 4a b + 8a b + 8a b)cosh(x)
--R +
--R 7 2 6 3 5 4 4 5 3 6 2 7 8 2
--R (- 4a b - 4a b - 6a b - 6a b + 9a b + 9a b + a b + a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 3 5 4 4
--R (- 4a b - 4a b + 2a b + 2a b + 2a b + 2a b)cosh(x) - a b - a b
--R +
--R 5 3 6 2
--R a b + a b
--R Type: Expression(Integer)
--E 12
--S 13 of 526
m0401b:= a0401.2-r0401
--R
--R
--R (13)
--R 2 3 4 4 3 2 2
--R ((2a b - 4a b)cosh(x) + 2a b - 4a b )sinh(x)
--R +
--R 2 3 4 2 4 3 2 5 2 3
--R ((4a b - 8a b)cosh(x) + (8a b - 16a b )cosh(x) + 4b - 8a b )
--R *
--R sinh(x)
--R +
--R 2 3 4 3 4 3 2 2
--R (2a b - 4a b)cosh(x) + (6a b - 12a b )cosh(x)
--R +
--R 5 2 3 4 4 3 2
--R (4b - 6a b - 4a b)cosh(x) + 2a b - 4a b
--R *
--R +---------+
--R | 2 2
--R (a sinh(x) + a cosh(x) + b)\|- b + a
--R atan(---------------------------------------)
--R 2 2
--R b - a
--R +
--R 2 3 4 4 3 2 2
--R ((- 2a b + 4a b)cosh(x) - 2a b + 4a b )sinh(x)
--R +
--R 2 3 4 2 4 3 2 5
--R (- 4a b + 8a b)cosh(x) + (- 8a b + 16a b )cosh(x) - 4b
--R +
--R 2 3
--R 8a b
--R *
--R sinh(x)
--R +
--R 2 3 4 3 4 3 2 2
--R (- 2a b + 4a b)cosh(x) + (- 6a b + 12a b )cosh(x)
--R +
--R 5 2 3 4 4 3 2
--R (- 4b + 6a b + 4a b)cosh(x) - 2a b + 4a b
--R *
--R x
--R (b - a)tanh(-)
--R 2
--R atan(--------------)
--R +---------+
--R | 2 2
--R \|- b + a
--R +
--R 2 2 3 2 2 3 2
--R a b sinh(x) + (2a b cosh(x) + 2a b )sinh(x)
--R +
--R 2 2 2 3 4 2 2 3 2
--R (a b cosh(x) + 4a b cosh(x) + 2b + a b )sinh(x) + 2a b cosh(x)
--R +
--R 4 2 2 3
--R (2b + 2a b )cosh(x) + 2a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R /
--R 4 2 6 3 3 5 2
--R ((a b - a )cosh(x) + a b - a b)sinh(x)
--R +
--R 4 2 6 2 3 3 5 2 4 4 2
--R ((2a b - 2a )cosh(x) + (4a b - 4a b)cosh(x) + 2a b - 2a b )sinh(x)
--R +
--R 4 2 6 3 3 3 5 2
--R (a b - a )cosh(x) + (3a b - 3a b)cosh(x)
--R +
--R 2 4 4 2 6 3 3 5
--R (2a b - a b - a )cosh(x) + a b - a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R Type: Expression(Integer)
--E 13
--S 14 of 526
d0401b:= D(m0401b,x)
--R
--R
--R (14)
--R 3 3 4 2 6
--R (- a b + a b )sinh(x)
--R +
--R 3 3 4 2 2 4 3 3 5
--R ((- 4a b + 4a b )cosh(x) - 4a b + 4a b )sinh(x)
--R +
--R 3 3 4 2 5 2
--R (- 4a b + 5a b - 2a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 2 4 3 3 4 2
--R (- 9a b + 11a b - 4a b )cosh(x) - 3a b + 4a b - 4a b + 2a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 3 5 3 2 4 3 3 4 2 5 2
--R (4a b - 8a b)cosh(x) + (4a b + 8a b - 20a b - 4a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 2 4
--R (12a b - 20a b - 4a b )cosh(x) + 4a b - 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 3 3 4 2 5 4
--R (11a b - 5a b - 12a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 3
--R (26a b - 2a b - 36a b - 12a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 2
--R (18a b + 16a b - 38a b - 28a b - 4a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 6 5
--R (4b + 16a b - 18a b - 18a b - 8a b )cosh(x) + 4b - 2a b
--R +
--R 2 4 3 3 4 2
--R - 4a b - 5a b + a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 3 4 2 5 5
--R (8a b - 4a b - 8a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 4
--R (24a b - 4a b - 28a b - 12a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 3
--R (24a b + 12a b - 32a b - 36a b - 8a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 5 2
--R (8b + 20a b - 12a b - 32a b - 20a b - 4a b)cosh(x)
--R +
--R 6 2 4 3 3 4 2 5 2 4 3 3
--R (8b - 4a b - 16a b - 8a b )cosh(x) + 4a b - 4a b - 4a b
--R *
--R sinh(x)
--R +
--R 3 3 4 2 5 6 2 4 3 3 4 2 5 5
--R (2a b - a b - 2a b)cosh(x) + (7a b - a b - 8a b - 4a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 4
--R (9a b + 4a b - 10a b - 14a b - 4a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 5 3
--R (4b + 8a b - 2a b - 14a b - 12a b - 4a b)cosh(x)
--R +
--R 6 5 3 3 4 2 5 2
--R (4b + 2a b - 10a b - 9a b - 2a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 3 3
--R (4a b - a b - 5a b - 4a b )cosh(x) + a b - 2a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 3 3 4 2 6 3 3 4 2 2 4 3 3 5
--R (a b + a b )sinh(x) + ((4a b + 4a b )cosh(x) + 4a b + 4a b )sinh(x)
--R +
--R 3 3 4 2 5 2 2 4 3 3 4 2
--R (4a b + 5a b + 2a b)cosh(x) + (9a b + 11a b + 4a b )cosh(x)
--R +
--R 5 2 4 3 3 4 2
--R 3a b + 4a b + 4a b + 2a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 3 5 3 2 4 3 3 4 2 5 2
--R (- 4a b + 8a b)cosh(x) + (- 4a b + 8a b + 20a b - 4a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 2 4
--R (12a b + 20a b - 4a b )cosh(x) + 4a b + 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 3 3 4 2 5 4
--R (- 11a b - 5a b + 12a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 3
--R (- 26a b - 2a b + 36a b - 12a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 2
--R (- 18a b + 16a b + 38a b - 28a b + 4a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 6 5
--R (- 4b + 16a b + 18a b - 18a b + 8a b )cosh(x) + 4b + 2a b
--R +
--R 2 4 3 3 4 2
--R - 4a b + 5a b + a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 3 4 2 5 5
--R (- 8a b - 4a b + 8a b)cosh(x)
--R +
--R 2 4 3 3 4 2 5 4
--R (- 24a b - 4a b + 28a b - 12a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 3
--R (- 24a b + 12a b + 32a b - 36a b + 8a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 5 2
--R (- 8b + 20a b + 12a b - 32a b + 20a b - 4a b)cosh(x)
--R +
--R 6 2 4 3 3 4 2 5 2 4 3 3
--R (8b - 4a b + 16a b - 8a b )cosh(x) + 4a b + 4a b - 4a b
--R *
--R sinh(x)
--R +
--R 3 3 4 2 5 6 2 4 3 3 4 2 5 5
--R (- 2a b - a b + 2a b)cosh(x) + (- 7a b - a b + 8a b - 4a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 4
--R (- 9a b + 4a b + 10a b - 14a b + 4a b)cosh(x)
--R +
--R 6 5 2 4 3 3 4 2 5 3
--R (- 4b + 8a b + 2a b - 14a b + 12a b - 4a b)cosh(x)
--R +
--R 6 5 3 3 4 2 5 2
--R (4b - 2a b + 10a b - 9a b + 2a b)cosh(x)
--R +
--R 5 2 4 3 3 4 2 5 3 3
--R (4a b + a b - 5a b + 4a b )cosh(x) - a b + 2a b
--R /
--R 5 3 6 2 7 8 2
--R (a b - a b - a b + a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 3 5 4 4 5 3 6 2
--R (2a b - 2a b - 2a b + 2a b)cosh(x) + a b - a b - a b + a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 3 6 2 7 8 3
--R (4a b - 4a b - 4a b + 4a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 2
--R (12a b - 12a b - 12a b + 12a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 2 6 3 5
--R (12a b - 12a b - 12a b + 12a b )cosh(x) + 4a b - 4a b
--R +
--R 4 4 5 3
--R - 4a b + 4a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 3 6 2 7 8 4
--R (6a b - 6a b - 6a b + 6a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 3
--R (24a b - 24a b - 24a b + 24a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 2
--R (34a b - 34a b - 32a b + 32a b - 2a b + 2a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 7
--R (20a b - 20a b - 16a b + 16a b - 4a b + 4a b)cosh(x) + 4a b
--R +
--R 2 6 3 5 4 4 5 3 6 2
--R - 4a b - 2a b + 2a b - 2a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 6 2 7 8 5
--R (4a b - 4a b - 4a b + 4a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 4
--R (20a b - 20a b - 20a b + 20a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 3
--R (36a b - 36a b - 32a b + 32a b - 4a b + 4a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 2
--R (28a b - 28a b - 16a b + 16a b - 12a b + 12a b)cosh(x)
--R +
--R 7 2 6 3 5 4 4 5 3 6 2 2 6
--R (8a b - 8a b + 4a b - 4a b - 12a b + 12a b )cosh(x) + 4a b
--R +
--R 3 5 4 4 5 3
--R - 4a b - 4a b + 4a b
--R *
--R sinh(x)
--R +
--R 5 3 6 2 7 8 6
--R (a b - a b - a b + a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 5
--R (6a b - 6a b - 6a b + 6a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 4
--R (13a b - 13a b - 11a b + 11a b - 2a b + 2a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 3
--R (12a b - 12a b - 4a b + 4a b - 8a b + 8a b)cosh(x)
--R +
--R 7 2 6 3 5 4 4 5 3 6 2 7 8 2
--R (4a b - 4a b + 6a b - 6a b - 9a b + 9a b - a b + a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 3 5 4 4
--R (4a b - 4a b - 2a b + 2a b - 2a b + 2a b)cosh(x) + a b - a b
--R +
--R 5 3 6 2
--R - a b + a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 5 3 6 2 7 8 2
--R (- a b - a b + a b + a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 3 5 4 4 5 3 6 2
--R (- 2a b - 2a b + 2a b + 2a b)cosh(x) - a b - a b + a b + a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 3 6 2 7 8 3
--R (- 4a b - 4a b + 4a b + 4a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 2
--R (- 12a b - 12a b + 12a b + 12a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 2 6 3 5 4 4
--R (- 12a b - 12a b + 12a b + 12a b )cosh(x) - 4a b - 4a b + 4a b
--R +
--R 5 3
--R 4a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 3 6 2 7 8 4
--R (- 6a b - 6a b + 6a b + 6a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 3
--R (- 24a b - 24a b + 24a b + 24a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 2
--R (- 34a b - 34a b + 32a b + 32a b + 2a b + 2a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 7
--R (- 20a b - 20a b + 16a b + 16a b + 4a b + 4a b)cosh(x) - 4a b
--R +
--R 2 6 3 5 4 4 5 3 6 2
--R - 4a b + 2a b + 2a b + 2a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 6 2 7 8 5
--R (- 4a b - 4a b + 4a b + 4a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 4
--R (- 20a b - 20a b + 20a b + 20a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 3
--R (- 36a b - 36a b + 32a b + 32a b + 4a b + 4a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 2
--R (- 28a b - 28a b + 16a b + 16a b + 12a b + 12a b)cosh(x)
--R +
--R 7 2 6 3 5 4 4 5 3 6 2 2 6
--R (- 8a b - 8a b - 4a b - 4a b + 12a b + 12a b )cosh(x) - 4a b
--R +
--R 3 5 4 4 5 3
--R - 4a b + 4a b + 4a b
--R *
--R sinh(x)
--R +
--R 5 3 6 2 7 8 6
--R (- a b - a b + a b + a )cosh(x)
--R +
--R 4 4 5 3 6 2 7 5
--R (- 6a b - 6a b + 6a b + 6a b)cosh(x)
--R +
--R 3 5 4 4 5 3 6 2 7 8 4
--R (- 13a b - 13a b + 11a b + 11a b + 2a b + 2a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 3
--R (- 12a b - 12a b + 4a b + 4a b + 8a b + 8a b)cosh(x)
--R +
--R 7 2 6 3 5 4 4 5 3 6 2 7 8 2
--R (- 4a b - 4a b - 6a b - 6a b + 9a b + 9a b + a b + a )cosh(x)
--R +
--R 2 6 3 5 4 4 5 3 6 2 7 3 5 4 4
--R (- 4a b - 4a b + 2a b + 2a b + 2a b + 2a b)cosh(x) - a b - a b
--R +
--R 5 3 6 2
--R a b + a b
--R Type: Expression(Integer)
--E 14
--S 15 of 526
t0402:= 1/(a+b*sech(x))^3
--R
--R
--R 1
--R (15) ----------------------------------------------
--R 3 3 2 2 2 3
--R b sech(x) + 3a b sech(x) + 3a b sech(x) + a
--R Type: Expression(Integer)
--E 15
--S 16 of 526
r0402:= x/a^3-6*b^3*atan((a-b)*tanh(1/2*x)/(a^2-b^2)^(1/2))/_
(a^2-b^2)^(3/2)/a^3-6*b*atan((a-b)*tanh(1/2*x)/(a^2-b^2)^(1/2))/_
a^3/(a^2-b^2)^(1/2)-b^3*(a^2+2*b^2)*atan((a-b)*tanh(1/2*x)/_
(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)/a^3-1/2*b^3*_
sinh(x)/a^2/(a^2-b^2)/(b+a*cosh(x))^2+3/2*b^4*_
sinh(x)/a^2/(a^2-b^2)^2/(b+a*cosh(x))+3*_
b^2*sinh(x)/a^2/(a^2-b^2)/(b+a*cosh(x))
--R
--R
--R (16)
--R 2 5 4 3 6 2 6 3 4 5 2
--R (4a b - 10a b + 12a b)cosh(x) + (8a b - 20a b + 24a b )cosh(x)
--R +
--R 7 2 5 4 3
--R 4b - 10a b + 12a b
--R *
--R x
--R (b - a)tanh(-)
--R 2
--R atan(--------------)
--R +---------+
--R | 2 2
--R \|- b + a
--R +
--R 2 4 4 2 5 3 3
--R ((- 3a b + 6a b )cosh(x) - 2a b + 5a b )sinh(x)
--R +
--R 2 4 4 2 6 2 5 3 3 5
--R (2a b - 4a b + 2a )x cosh(x) + (4a b - 8a b + 4a b)x cosh(x)
--R +
--R 6 2 4 4 2
--R (2b - 4a b + 2a b )x
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R /
--R 5 4 7 2 9 2 4 5 6 3 8 3 6
--R (2a b - 4a b + 2a )cosh(x) + (4a b - 8a b + 4a b)cosh(x) + 2a b
--R +
--R 5 4 7 2
--R - 4a b + 2a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R Type: Expression(Integer)
--E 16
--S 17 of 526
a0402:= integrate(t0402,x)
--R
--R
--R (17)
--R [
--R 2 5 4 3 6 4
--R (2a b - 5a b + 6a b)sinh(x)
--R +
--R 2 5 4 3 6 6 3 4 5 2 3
--R ((8a b - 20a b + 24a b)cosh(x) + 8a b - 20a b + 24a b )sinh(x)
--R +
--R 2 5 4 3 6 2
--R (12a b - 30a b + 36a b)cosh(x)
--R +
--R 6 3 4 5 2 7 2 5 4 3 6
--R (24a b - 60a b + 72a b )cosh(x) + 8b - 16a b + 14a b + 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 4 3 6 3
--R (8a b - 20a b + 24a b)cosh(x)
--R +
--R 6 3 4 5 2 2
--R (24a b - 60a b + 72a b )cosh(x)
--R +
--R 7 2 5 4 3 6 6 3 4 5 2
--R (16b - 32a b + 28a b + 24a b)cosh(x) + 8a b - 20a b + 24a b
--R *
--R sinh(x)
--R +
--R 2 5 4 3 6 4 6 3 4 5 2 3
--R (2a b - 5a b + 6a b)cosh(x) + (8a b - 20a b + 24a b )cosh(x)
--R +
--R 7 2 5 4 3 6 2
--R (8b - 16a b + 14a b + 12a b)cosh(x)
--R +
--R 6 3 4 5 2 2 5 4 3 6
--R (8a b - 20a b + 24a b )cosh(x) + 2a b - 5a b + 6a b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b - a
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R +
--R 2 3 2 3 3 2
--R (2a b - 2a )sinh(x) + (2a b - 2a )cosh(x) + 2b - 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x)
--R +
--R 2b cosh(x) + a
--R +
--R 2 4 4 2 6 4
--R (2a b - 4a b + 2a )x sinh(x)
--R +
--R 2 4 4 2 6 5 3 3 5
--R (8a b - 16a b + 8a )x cosh(x) + (8a b - 16a b + 8a b)x
--R +
--R 5 3 3
--R 8a b - 14a b
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 4 2 6 2
--R (12a b - 24a b + 12a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R ((24a b - 48a b + 24a b)x + 24a b - 42a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2
--R (8b - 12a b + 4a )x + 12b - 18a b - 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 4 4 2 6 3
--R (8a b - 16a b + 8a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3 2
--R ((24a b - 48a b + 24a b)x + 24a b - 42a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2
--R ((16b - 24a b + 8a )x + 24b - 36a b - 24a b )cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R (8a b - 16a b + 8a b)x + 16a b - 34a b
--R *
--R sinh(x)
--R +
--R 2 4 4 2 6 4
--R (2a b - 4a b + 2a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3 3
--R ((8a b - 16a b + 8a b)x + 8a b - 14a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2 2
--R ((8b - 12a b + 4a )x + 12b - 18a b - 12a b )cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R ((8a b - 16a b + 8a b)x + 16a b - 34a b )cosh(x)
--R +
--R 2 4 4 2 6 2 4 4 2
--R (2a b - 4a b + 2a )x + 6a b - 12a b
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R /
--R 5 4 7 2 9 4
--R (2a b - 4a b + 2a )sinh(x)
--R +
--R 5 4 7 2 9 4 5 6 3 8 3
--R ((8a b - 16a b + 8a )cosh(x) + 8a b - 16a b + 8a b)sinh(x)
--R +
--R 5 4 7 2 9 2
--R (12a b - 24a b + 12a )cosh(x)
--R +
--R 4 5 6 3 8 3 6 5 4 9
--R (24a b - 48a b + 24a b)cosh(x) + 8a b - 12a b + 4a
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 7 2 9 3
--R (8a b - 16a b + 8a )cosh(x)
--R +
--R 4 5 6 3 8 2
--R (24a b - 48a b + 24a b)cosh(x)
--R +
--R 3 6 5 4 9 4 5 6 3 8
--R (16a b - 24a b + 8a )cosh(x) + 8a b - 16a b + 8a b
--R *
--R sinh(x)
--R +
--R 5 4 7 2 9 4 4 5 6 3 8 3
--R (2a b - 4a b + 2a )cosh(x) + (8a b - 16a b + 8a b)cosh(x)
--R +
--R 3 6 5 4 9 2 4 5 6 3 8
--R (8a b - 12a b + 4a )cosh(x) + (8a b - 16a b + 8a b)cosh(x)
--R +
--R 5 4 7 2 9
--R 2a b - 4a b + 2a
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R ,
--R
--R 2 5 4 3 6 4
--R (2a b - 5a b + 6a b)sinh(x)
--R +
--R 2 5 4 3 6 6 3 4 5 2 3
--R ((8a b - 20a b + 24a b)cosh(x) + 8a b - 20a b + 24a b )sinh(x)
--R +
--R 2 5 4 3 6 2
--R (12a b - 30a b + 36a b)cosh(x)
--R +
--R 6 3 4 5 2 7 2 5 4 3 6
--R (24a b - 60a b + 72a b )cosh(x) + 8b - 16a b + 14a b + 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 4 3 6 3
--R (8a b - 20a b + 24a b)cosh(x)
--R +
--R 6 3 4 5 2 2
--R (24a b - 60a b + 72a b )cosh(x)
--R +
--R 7 2 5 4 3 6 6 3 4 5 2
--R (16b - 32a b + 28a b + 24a b)cosh(x) + 8a b - 20a b + 24a b
--R *
--R sinh(x)
--R +
--R 2 5 4 3 6 4 6 3 4 5 2 3
--R (2a b - 5a b + 6a b)cosh(x) + (8a b - 20a b + 24a b )cosh(x)
--R +
--R 7 2 5 4 3 6 2
--R (8b - 16a b + 14a b + 12a b)cosh(x)
--R +
--R 6 3 4 5 2 2 5 4 3 6
--R (8a b - 20a b + 24a b )cosh(x) + 2a b - 5a b + 6a b
--R *
--R +---------+
--R | 2 2
--R (a sinh(x) + a cosh(x) + b)\|- b + a
--R atan(---------------------------------------)
--R 2 2
--R b - a
--R +
--R 2 4 4 2 6 4
--R (a b - 2a b + a )x sinh(x)
--R +
--R 2 4 4 2 6 5 3 3 5
--R (4a b - 8a b + 4a )x cosh(x) + (4a b - 8a b + 4a b)x
--R +
--R 5 3 3
--R 4a b - 7a b
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 4 2 6 2
--R (6a b - 12a b + 6a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R ((12a b - 24a b + 12a b)x + 12a b - 21a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2
--R (4b - 6a b + 2a )x + 6b - 9a b - 6a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 4 4 2 6 3
--R (4a b - 8a b + 4a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3 2
--R ((12a b - 24a b + 12a b)x + 12a b - 21a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2
--R ((8b - 12a b + 4a )x + 12b - 18a b - 12a b )cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R (4a b - 8a b + 4a b)x + 8a b - 17a b
--R *
--R sinh(x)
--R +
--R 2 4 4 2 6 4
--R (a b - 2a b + a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3 3
--R ((4a b - 8a b + 4a b)x + 4a b - 7a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2 2
--R ((4b - 6a b + 2a )x + 6b - 9a b - 6a b )cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R ((4a b - 8a b + 4a b)x + 8a b - 17a b )cosh(x)
--R +
--R 2 4 4 2 6 2 4 4 2
--R (a b - 2a b + a )x + 3a b - 6a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R /
--R 5 4 7 2 9 4
--R (a b - 2a b + a )sinh(x)
--R +
--R 5 4 7 2 9 4 5 6 3 8 3
--R ((4a b - 8a b + 4a )cosh(x) + 4a b - 8a b + 4a b)sinh(x)
--R +
--R 5 4 7 2 9 2 4 5 6 3 8
--R (6a b - 12a b + 6a )cosh(x) + (12a b - 24a b + 12a b)cosh(x)
--R +
--R 3 6 5 4 9
--R 4a b - 6a b + 2a
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 7 2 9 3 4 5 6 3 8 2
--R (4a b - 8a b + 4a )cosh(x) + (12a b - 24a b + 12a b)cosh(x)
--R +
--R 3 6 5 4 9 4 5 6 3 8
--R (8a b - 12a b + 4a )cosh(x) + 4a b - 8a b + 4a b
--R *
--R sinh(x)
--R +
--R 5 4 7 2 9 4 4 5 6 3 8 3
--R (a b - 2a b + a )cosh(x) + (4a b - 8a b + 4a b)cosh(x)
--R +
--R 3 6 5 4 9 2 4 5 6 3 8 5 4
--R (4a b - 6a b + 2a )cosh(x) + (4a b - 8a b + 4a b)cosh(x) + a b
--R +
--R 7 2 9
--R - 2a b + a
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R ]
--R Type: Union(List(Expression(Integer)),...)
--E 17
--S 18 of 526
m0402a:= a0402.1-r0402
--R
--R
--R (18)
--R 4 5 6 3 8 2 3 6 5 4 7 2
--R (2a b - 5a b + 6a b)cosh(x) + (4a b - 10a b + 12a b )cosh(x)
--R +
--R 2 7 4 5 6 3
--R 2a b - 5a b + 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 5 6 3 8 3
--R (8a b - 20a b + 24a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2
--R (24a b - 60a b + 72a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 3 6 5 4
--R (24a b - 60a b + 72a b )cosh(x) + 8a b - 20a b + 24a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 5 6 3 8 4
--R (12a b - 30a b + 36a b)cosh(x)
--R +
--R 3 6 5 4 7 2 3
--R (48a b - 120a b + 144a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 2
--R (68a b - 166a b + 194a b + 12a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 9 2 7
--R (40a b - 92a b + 100a b + 24a b )cosh(x) + 8b - 16a b
--R +
--R 4 5 6 3
--R 14a b + 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 5 6 3 8 5
--R (8a b - 20a b + 24a b)cosh(x)
--R +
--R 3 6 5 4 7 2 4
--R (40a b - 100a b + 120a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 3
--R (72a b - 172a b + 196a b + 24a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 2
--R (56a b - 116a b + 108a b + 72a b )cosh(x)
--R +
--R 9 2 7 4 5 6 3 8 3 6 5 4
--R (16b - 16a b - 12a b + 72a b )cosh(x) + 8a b - 20a b + 24a b
--R *
--R sinh(x)
--R +
--R 4 5 6 3 8 6 3 6 5 4 7 2 5
--R (2a b - 5a b + 6a b)cosh(x) + (12a b - 30a b + 36a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 4
--R (26a b - 61a b + 68a b + 12a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 3
--R (24a b - 44a b + 32a b + 48a b )cosh(x)
--R +
--R 9 4 5 6 3 8 2
--R (8b - 24a b + 55a b + 6a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 2 7 4 5 6 3
--R (8a b - 16a b + 14a b + 12a b )cosh(x) + 2a b - 5a b + 6a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b - a
--R *
--R +-------+
--R | 2 2
--R \|b - a
--R +
--R 2 3 2 3 3 2
--R (2a b - 2a )sinh(x) + (2a b - 2a )cosh(x) + 2b - 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R a
--R +
--R 4 5 6 3 8 2
--R (- 4a b + 10a b - 12a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2 7 4 5 6 3
--R (- 8a b + 20a b - 24a b )cosh(x) - 4a b + 10a b - 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 5 6 3 8 3
--R (- 16a b + 40a b - 48a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2
--R (- 48a b + 120a b - 144a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 3 6 5 4
--R (- 48a b + 120a b - 144a b )cosh(x) - 16a b + 40a b - 48a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 5 6 3 8 4
--R (- 24a b + 60a b - 72a b)cosh(x)
--R +
--R 3 6 5 4 7 2 3
--R (- 96a b + 240a b - 288a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 2
--R (- 136a b + 332a b - 388a b - 24a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 9 2 7
--R (- 80a b + 184a b - 200a b - 48a b )cosh(x) - 16b + 32a b
--R +
--R 4 5 6 3
--R - 28a b - 24a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 5 6 3 8 5
--R (- 16a b + 40a b - 48a b)cosh(x)
--R +
--R 3 6 5 4 7 2 4
--R (- 80a b + 200a b - 240a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 3
--R (- 144a b + 344a b - 392a b - 48a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 2
--R (- 112a b + 232a b - 216a b - 144a b )cosh(x)
--R +
--R 9 2 7 4 5 6 3 8 3 6
--R (- 32b + 32a b + 24a b - 144a b )cosh(x) - 16a b + 40a b
--R +
--R 5 4
--R - 48a b
--R *
--R sinh(x)
--R +
--R 4 5 6 3 8 6
--R (- 4a b + 10a b - 12a b)cosh(x)
--R +
--R 3 6 5 4 7 2 5
--R (- 24a b + 60a b - 72a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 4
--R (- 52a b + 122a b - 136a b - 24a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 3
--R (- 48a b + 88a b - 64a b - 96a b )cosh(x)
--R +
--R 9 4 5 6 3 8 2
--R (- 16b + 48a b - 110a b - 12a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 2 7 4 5 6 3
--R (- 16a b + 32a b - 28a b - 24a b )cosh(x) - 4a b + 10a b - 12a b
--R *
--R x
--R +-------+ (b - a)tanh(-)
--R | 2 2 2
--R \|b - a atan(--------------)
--R +---------+
--R | 2 2
--R \|- b + a
--R +
--R 4 4 6 2 3 5 5 3 5
--R ((3a b - 6a b )cosh(x) + 2a b - 5a b )sinh(x)
--R +
--R 4 4 6 2 2 3 5 5 3 2 6
--R (12a b - 24a b )cosh(x) + (20a b - 44a b )cosh(x) + 8a b
--R +
--R 4 4
--R - 20a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 4 6 2 3 3 5 5 3 2
--R (18a b - 36a b )cosh(x) + (56a b - 116a b )cosh(x)
--R +
--R 2 6 4 4 6 2 7 3 5 5 3
--R (52a b - 106a b - 12a b )cosh(x) + 16a b - 30a b - 10a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 4 6 2 4 3 5 5 3 3
--R (12a b - 24a b )cosh(x) + (68a b - 134a b )cosh(x)
--R +
--R 2 6 4 4 6 2 2
--R (108a b - 198a b - 36a b )cosh(x)
--R +
--R 7 3 5 5 3 8 2 6 4 4
--R (64a b - 98a b - 68a b )cosh(x) + 12b - 10a b - 32a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 4 6 2 5 3 5 5 3 4
--R (3a b - 6a b )cosh(x) + (38a b - 71a b )cosh(x)
--R +
--R 2 6 4 4 6 2 3
--R (92a b - 158a b - 36a b )cosh(x)
--R +
--R 7 3 5 5 3 2
--R (80a b - 102a b - 116a b )cosh(x)
--R +
--R 8 2 6 4 4 6 2 7 3 5 5 3
--R (24b + 4a b - 109a b - 6a b )cosh(x) + 16a b - 32a b - 5a b
--R *
--R sinh(x)
--R +
--R 3 5 5 3 5 2 6 4 4 6 2 4
--R (8a b - 14a b )cosh(x) + (28a b - 46a b - 12a b )cosh(x)
--R +
--R 7 3 5 5 3 3
--R (32a b - 34a b - 58a b )cosh(x)
--R +
--R 8 2 6 4 4 6 2 2
--R (12b + 14a b - 74a b - 12a b )cosh(x)
--R +
--R 7 3 5 5 3 2 6 4 4
--R (16a b - 22a b - 24a b )cosh(x) + 6a b - 12a b
--R *
--R +---------+ +-------+
--R | 2 2 | 2 2
--R \|- b + a \|b - a
--R /
--R 7 4 9 2 11 2 6 5 8 3 10
--R (2a b - 4a b + 2a )cosh(x) + (4a b - 8a b + 4a b)cosh(x)
--R +
--R 5 6 7 4 9 2
--R 2a b - 4a b + 2a b
--R *
--R 4
--R sinh(x)
--R +
--R 7 4 9 2 11 3
--R (8a b - 16a b + 8a )cosh(x)
--R +
--R 6 5 8 3 10 2
--R (24a b - 48a b + 24a b)cosh(x)
--R +
--R 5 6 7 4 9 2 4 7 6 5 8 3
--R (24a b - 48a b + 24a b )cosh(x) + 8a b - 16a b + 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 7 4 9 2 11 4
--R (12a b - 24a b + 12a )cosh(x)
--R +
--R 6 5 8 3 10 3
--R (48a b - 96a b + 48a b)cosh(x)
--R +
--R 5 6 7 4 9 2 11 2
--R (68a b - 132a b + 60a b + 4a )cosh(x)
--R +
--R 4 7 6 5 8 3 10 3 8 5 6 9 2
--R (40a b - 72a b + 24a b + 8a b)cosh(x) + 8a b - 12a b + 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 7 4 9 2 11 5
--R (8a b - 16a b + 8a )cosh(x)
--R +
--R 6 5 8 3 10 4
--R (40a b - 80a b + 40a b)cosh(x)
--R +
--R 5 6 7 4 9 2 11 3
--R (72a b - 136a b + 56a b + 8a )cosh(x)
--R +
--R 4 7 6 5 8 3 10 2
--R (56a b - 88a b + 8a b + 24a b)cosh(x)
--R +
--R 3 8 5 6 7 4 9 2 4 7 6 5 8 3
--R (16a b - 8a b - 32a b + 24a b )cosh(x) + 8a b - 16a b + 8a b
--R *
--R sinh(x)
--R +
--R 7 4 9 2 11 6 6 5 8 3 10 5
--R (2a b - 4a b + 2a )cosh(x) + (12a b - 24a b + 12a b)cosh(x)
--R +
--R 5 6 7 4 9 2 11 4
--R (26a b - 48a b + 18a b + 4a )cosh(x)
--R +
--R 4 7 6 5 8 3 10 3
--R (24a b - 32a b - 8a b + 16a b)cosh(x)
--R +
--R 3 8 5 6 7 4 9 2 11 2
--R (8a b + 4a b - 30a b + 16a b + 2a )cosh(x)
--R +
--R 4 7 6 5 10 5 6 7 4 9 2
--R (8a b - 12a b + 4a b)cosh(x) + 2a b - 4a b + 2a b
--R *
--R +---------+ +-------+
--R | 2 2 | 2 2
--R \|- b + a \|b - a
--R Type: Expression(Integer)
--E 18
--S 19 of 526
d0402a:= D(m0402a,x)
--R
--R
--R (19)
--R 5 5 6 4 7 3 8 2 4 6 5 5 6 4
--R (- 3a b + 3a b + 6a b - 6a b )cosh(x) - a b + a b + 4a b
--R +
--R 7 3
--R - 4a b
--R *
--R 8
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 2
--R (- 18a b + 18a b + 36a b - 36a b )cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 3 7 4 6
--R (- 24a b + 24a b + 60a b - 60a b )cosh(x) - 6a b + 6a b
--R +
--R 5 5 6 4
--R 24a b - 24a b
--R *
--R 7
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 3
--R (- 40a b + 42a b + 79a b - 84a b + 6a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 2
--R (- 94a b + 100a b + 214a b - 229a b + 18a b )cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R (- 58a b + 64a b + 163a b - 178a b + 36a b - 18a b )cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R - 10a b + 12a b + 40a b - 45a b + 18a b - 12a b
--R *
--R 6
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 4
--R (- 30a b + 42a b + 54a b - 84a b + 36a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9
--R (- 116a b + 164a b + 242a b - 362a b + 132a b + 12a b)
--R *
--R 3
--R cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R (- 126a b + 198a b + 288a b - 468a b + 252a b - 36a b )
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R (- 48a b + 96a b + 114a b - 234a b + 228a b - 84a b )cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R - 8a b + 20a b + 14a b - 44a b + 72a b - 36a b
--R *
--R 5
--R sinh(x)
--R +
--R 5 5 7 3 9 5
--R (30a b - 75a b + 90a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 4
--R (60a b + 90a b - 180a b - 195a b + 390a b + 60a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R 30a b + 264a b - 171a b - 558a b + 753a b + 114a b
--R +
--R 9
--R 18a b
--R *
--R 3
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 6a b + 276a b - 144a b - 543a b + 804a b - 3a b
--R +
--R 8 2
--R 54a b
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 132a b - 78a b - 234a b + 411a b - 60a b + 72a b
--R +
--R 8 2
--R - 18a b
--R *
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b - 18a b - 36a b + 78a b - 21a b + 30a b - 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 6
--R (82a b - 42a b - 184a b + 84a b + 120a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 5
--R (304a b - 64a b - 736a b + 136a b + 600a b + 120a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R 430a b + 146a b - 1144a b - 272a b + 1212a b + 456a b
--R +
--R 9
--R 72a b
--R *
--R 4
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 304a b + 400a b - 924a b - 740a b + 1316a b + 556a b
--R +
--R 8 2 9
--R 264a b + 24a b
--R *
--R 3
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R 112a b + 344a b - 412a b - 584a b + 774a b + 234a b
--R +
--R 7 3 8 2
--R 396a b + 36a b
--R *
--R 2
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 16b + 128a b - 88a b - 176a b + 204a b - 12a b
--R +
--R 6 4 7 3
--R 276a b + 12a b
--R *
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 16b - 8a b - 8a b + 14a b - 26a b + 72a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 7
--R (72a b - 42a b - 159a b + 84a b + 90a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9
--R (326a b - 116a b - 770a b + 245a b + 510a b + 120a b)
--R *
--R 6
--R cosh(x)
--R +
--R 3 7 5 5 6 4 7 3 8 2 9
--R (594a b - 1467a b + 18a b + 1098a b + 594a b + 108a b)
--R *
--R 5
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 546a b + 324a b - 1368a b - 627a b + 1134a b + 1026a b
--R +
--R 8 2 9
--R 468a b + 72a b
--R *
--R 4
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R 256a b + 440a b - 628a b - 764a b + 570a b + 654a b
--R +
--R 7 3 8 2 9
--R 789a b + 240a b + 18a b
--R *
--R 3
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 48b + 240a b - 108a b - 288a b + 90a b - 18a b
--R +
--R 6 4 7 3 8 2
--R 654a b + 273a b + 54a b
--R *
--R 2
--R cosh(x)
--R +
--R 10 2 8 3 7 4 6 5 5 6 4
--R 48b + 24a b - 18a b - 180a b + 249a b + 138a b
--R +
--R 7 3 8 2
--R 60a b - 6a b
--R *
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b - 6a b - 48a b + 32a b + 25a b + 22a b - 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 8
--R (30a b - 18a b - 66a b + 36a b + 36a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 7
--R (156a b - 60a b - 366a b + 126a b + 228a b + 60a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R 342a b - 30a b - 816a b + 60a b + 528a b + 348a b
--R +
--R 9
--R 72a b
--R *
--R 6
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 384a b + 144a b - 870a b - 306a b + 504a b + 720a b
--R +
--R 8 2 9
--R 360a b + 72a b
--R *
--R 5
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R 216a b + 276a b - 402a b - 492a b + 102a b + 546a b
--R +
--R 7 3 8 2 9
--R 678a b + 300a b + 36a b
--R *
--R 4
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 48b + 192a b - 24a b - 192a b - 120a b - 72a b
--R +
--R 6 4 7 3 8 2 9
--R 594a b + 438a b + 132a b + 12a b
--R *
--R 3
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 48b + 24a b + 72a b - 54a b - 270a b + 228a b
--R +
--R 6 4 7 3 8 2
--R 240a b + 180a b + 36a b
--R *
--R 2
--R cosh(x)
--R +
--R 9 3 7 4 6 5 5 6 4 7 3
--R (48a b - 72a b + 18a b + 6a b + 108a b + 36a b )cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4
--R 12a b - 6a b - 24a b + 24a b + 12a b
--R *
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 9
--R (5a b - 3a b - 11a b + 6a b + 6a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 8
--R (29a b - 11a b - 68a b + 23a b + 42a b + 12a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2 9
--R (74a b - 8a b - 173a b + 14a b + 105a b + 78a b + 18a b)
--R *
--R 7
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R 98a b + 28a b - 208a b - 65a b + 92a b + 181a b + 102a b
--R +
--R 9
--R 24a b
--R *
--R 6
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 64a b + 68a b - 94a b - 122a b - 33a b + 150a b + 213a b
--R +
--R 8 2 9
--R 114a b + 18a b
--R *
--R 5
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R 16b + 56a b + 14a b - 44a b - 84a b - 45a b + 186a b
--R +
--R 7 3 8 2 9
--R 189a b + 78a b + 12a b
--R *
--R 4
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R 16b + 16a b + 40a b - 22a b - 116a b + 53a b + 102a b
--R +
--R 7 3 8 2 9
--R 115a b + 42a b + 6a b
--R *
--R 3
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b + 6a b - 24a b - 4a b - 23a b + 64a b + 47a b
--R +
--R 8 2
--R 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 2 8
--R (12a b + 2a b - 26a b + 4a b + 17a b + 18a b )cosh(x) + 2a b
--R +
--R 4 6 6 4
--R - 5a b + 6a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 5 5 6 4 7 3 8 2 4 6 5 5 6 4 7 3
--R ((3a b + 3a b - 6a b - 6a b )cosh(x) + a b + a b - 4a b - 4a b )
--R *
--R 8
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 2
--R (18a b + 18a b - 36a b - 36a b )cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 3 7 4 6 5 5
--R (24a b + 24a b - 60a b - 60a b )cosh(x) + 6a b + 6a b - 24a b
--R +
--R 6 4
--R - 24a b
--R *
--R 7
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 3
--R (40a b + 42a b - 79a b - 84a b - 6a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 2
--R (94a b + 100a b - 214a b - 229a b - 18a b )cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R (58a b + 64a b - 163a b - 178a b - 36a b - 18a b )cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 10a b + 12a b - 40a b - 45a b - 18a b - 12a b
--R *
--R 6
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 4
--R (30a b + 42a b - 54a b - 84a b - 36a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 3
--R (116a b + 164a b - 242a b - 362a b - 132a b + 12a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2 2
--R (126a b + 198a b - 288a b - 468a b - 252a b - 36a b )cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R (48a b + 96a b - 114a b - 234a b - 228a b - 84a b )cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R 8a b + 20a b - 14a b - 44a b - 72a b - 36a b
--R *
--R 5
--R sinh(x)
--R +
--R 5 5 7 3 9 5
--R (- 30a b + 75a b - 90a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 4
--R (- 60a b + 90a b + 180a b - 195a b - 390a b + 60a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R - 30a b + 264a b + 171a b - 558a b - 753a b + 114a b
--R +
--R 9
--R - 18a b
--R *
--R 3
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R (- 6a b + 276a b + 144a b - 543a b - 804a b - 3a b - 54a b )
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R (132a b + 78a b - 234a b - 411a b - 60a b - 72a b - 18a b )
--R *
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b + 18a b - 36a b - 78a b - 21a b - 30a b - 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 6
--R (- 82a b - 42a b + 184a b + 84a b - 120a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 5
--R (- 304a b - 64a b + 736a b + 136a b - 600a b + 120a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R - 430a b + 146a b + 1144a b - 272a b - 1212a b + 456a b
--R +
--R 9
--R - 72a b
--R *
--R 4
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R - 304a b + 400a b + 924a b - 740a b - 1316a b + 556a b
--R +
--R 8 2 9
--R - 264a b + 24a b
--R *
--R 3
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R - 112a b + 344a b + 412a b - 584a b - 774a b + 234a b
--R +
--R 7 3 8 2
--R - 396a b + 36a b
--R *
--R 2
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R - 16b + 128a b + 88a b - 176a b - 204a b - 12a b - 276a b
--R +
--R 7 3
--R 12a b
--R *
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 16b + 8a b - 8a b - 14a b - 26a b - 72a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 7
--R (- 72a b - 42a b + 159a b + 84a b - 90a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 6
--R (- 326a b - 116a b + 770a b + 245a b - 510a b + 120a b)cosh(x)
--R +
--R 3 7 5 5 6 4 7 3 8 2 9 5
--R (- 594a b + 1467a b + 18a b - 1098a b + 594a b - 108a b)cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R - 546a b + 324a b + 1368a b - 627a b - 1134a b + 1026a b
--R +
--R 8 2 9
--R - 468a b + 72a b
--R *
--R 4
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R - 256a b + 440a b + 628a b - 764a b - 570a b + 654a b
--R +
--R 7 3 8 2 9
--R - 789a b + 240a b - 18a b
--R *
--R 3
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R - 48b + 240a b + 108a b - 288a b - 90a b - 18a b - 654a b
--R +
--R 7 3 8 2
--R 273a b - 54a b
--R *
--R 2
--R cosh(x)
--R +
--R 10 2 8 3 7 4 6 5 5 6 4 7 3
--R 48b + 24a b + 18a b - 180a b - 249a b + 138a b - 60a b
--R +
--R 8 2
--R - 6a b
--R *
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b + 6a b - 48a b - 32a b + 25a b - 22a b - 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 8
--R (- 30a b - 18a b + 66a b + 36a b - 36a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 7
--R (- 156a b - 60a b + 366a b + 126a b - 228a b + 60a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2 9
--R (- 342a b - 30a b + 816a b + 60a b - 528a b + 348a b - 72a b)
--R *
--R 6
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R - 384a b + 144a b + 870a b - 306a b - 504a b + 720a b
--R +
--R 8 2 9
--R - 360a b + 72a b
--R *
--R 5
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R - 216a b + 276a b + 402a b - 492a b - 102a b + 546a b
--R +
--R 7 3 8 2 9
--R - 678a b + 300a b - 36a b
--R *
--R 4
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R - 48b + 192a b + 24a b - 192a b + 120a b - 72a b - 594a b
--R +
--R 7 3 8 2 9
--R 438a b - 132a b + 12a b
--R *
--R 3
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R 48b - 24a b + 72a b + 54a b - 270a b - 228a b + 240a b
--R +
--R 7 3 8 2
--R - 180a b + 36a b
--R *
--R 2
--R cosh(x)
--R +
--R 9 3 7 4 6 5 5 6 4 7 3 2 8
--R (48a b - 72a b - 18a b + 6a b - 108a b + 36a b )cosh(x) + 12a b
--R +
--R 3 7 4 6 5 5 6 4
--R 6a b - 24a b - 24a b + 12a b
--R *
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 9
--R (- 5a b - 3a b + 11a b + 6a b - 6a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 8
--R (- 29a b - 11a b + 68a b + 23a b - 42a b + 12a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2 9 7
--R (- 74a b - 8a b + 173a b + 14a b - 105a b + 78a b - 18a b)cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R - 98a b + 28a b + 208a b - 65a b - 92a b + 181a b - 102a b
--R +
--R 9
--R 24a b
--R *
--R 6
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R - 64a b + 68a b + 94a b - 122a b + 33a b + 150a b - 213a b
--R +
--R 8 2 9
--R 114a b - 18a b
--R *
--R 5
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R - 16b + 56a b - 14a b - 44a b + 84a b - 45a b - 186a b
--R +
--R 7 3 8 2 9
--R 189a b - 78a b + 12a b
--R *
--R 4
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R 16b - 16a b + 40a b + 22a b - 116a b - 53a b + 102a b
--R +
--R 7 3 8 2 9
--R - 115a b + 42a b - 6a b
--R *
--R 3
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R (24a b - 6a b - 24a b + 4a b - 23a b - 64a b + 47a b - 18a b )
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 2 8
--R (12a b - 2a b - 26a b - 4a b + 17a b - 18a b )cosh(x) - 2a b
--R +
--R 4 6 6 4
--R 5a b - 6a b
--R /
--R 8 5 9 4 10 3 11 2 12 13 3
--R (2a b - 2a b - 4a b + 4a b + 2a b - 2a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 2
--R (6a b - 6a b - 12a b + 12a b + 6a b - 6a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R (6a b - 6a b - 12a b + 12a b + 6a b - 6a b )cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 2a b - 2a b - 4a b + 4a b + 2a b - 2a b
--R *
--R 6
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 4
--R (12a b - 12a b - 24a b + 24a b + 12a b - 12a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 3
--R (48a b - 48a b - 96a b + 96a b + 48a b - 48a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 2
--R (72a b - 72a b - 144a b + 144a b + 72a b - 72a b )cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R (48a b - 48a b - 96a b + 96a b + 48a b - 48a b )cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 12a b - 12a b - 24a b + 24a b + 12a b - 12a b
--R *
--R 5
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 5
--R (30a b - 30a b - 60a b + 60a b + 30a b - 30a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (150a b - 150a b - 300a b + 300a b + 150a b - 150a b)
--R *
--R 4
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R 294a b - 294a b - 582a b + 582a b + 282a b - 282a b
--R +
--R 12 13
--R 6a b - 6a
--R *
--R 3
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 282a b - 282a b - 546a b + 546a b + 246a b - 246a b
--R +
--R 11 2 12
--R 18a b - 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 132a b - 132a b - 246a b + 246a b + 96a b - 96a b
--R +
--R 10 3 11 2
--R 18a b - 18a b
--R *
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 24a b - 24a b - 42a b + 42a b + 12a b - 12a b + 6a b
--R +
--R 10 3
--R - 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 6
--R (40a b - 40a b - 80a b + 80a b + 40a b - 40a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (240a b - 240a b - 480a b + 480a b + 240a b - 240a b)
--R *
--R 5
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R 576a b - 576a b - 1128a b + 1128a b + 528a b - 528a b
--R +
--R 12 13
--R 24a b - 24a
--R *
--R 4
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 704a b - 704a b - 1312a b + 1312a b + 512a b - 512a b
--R +
--R 11 2 12
--R 96a b - 96a b
--R *
--R 3
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 456a b - 456a b - 768a b + 768a b + 168a b - 168a b
--R +
--R 10 3 11 2
--R 144a b - 144a b
--R *
--R 2
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R 144a b - 144a b - 192a b + 192a b - 48a b + 48a b
--R +
--R 9 4 10 3
--R 96a b - 96a b
--R *
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6 8 5
--R 16a b - 16a b - 8a b + 8a b - 32a b + 32a b + 24a b
--R +
--R 9 4
--R - 24a b
--R *
--R 3
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 7
--R (30a b - 30a b - 60a b + 60a b + 30a b - 30a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (210a b - 210a b - 420a b + 420a b + 210a b - 210a b)
--R *
--R 6
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R 594a b - 594a b - 1152a b + 1152a b + 522a b - 522a b
--R +
--R 12 13
--R 36a b - 36a
--R *
--R 5
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 870a b - 870a b - 1560a b + 1560a b + 510a b - 510a b
--R +
--R 11 2 12
--R 180a b - 180a b
--R *
--R 4
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 696a b - 696a b - 1044a b + 1044a b + 6a b - 6a b
--R +
--R 10 3 11 2 12 13
--R 336a b - 336a b + 6a b - 6a
--R *
--R 3
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R 288a b - 288a b - 252a b + 252a b - 342a b + 342a b
--R +
--R 9 4 10 3 11 2 12
--R 288a b - 288a b + 18a b - 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6
--R 48a b - 48a b + 48a b - 48a b - 222a b + 222a b
--R +
--R 8 5 9 4 10 3 11 2
--R 108a b - 108a b + 18a b - 18a b
--R *
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 24a b - 24a b - 42a b + 42a b + 12a b - 12a b + 6a b
--R +
--R 10 3
--R - 6a b
--R *
--R 2
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 8
--R (12a b - 12a b - 24a b + 24a b + 12a b - 12a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 7
--R (96a b - 96a b - 192a b + 192a b + 96a b - 96a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R 312a b - 312a b - 600a b + 600a b + 264a b - 264a b
--R +
--R 12 13
--R 24a b - 24a
--R *
--R 6
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 528a b - 528a b - 912a b + 912a b + 240a b - 240a b
--R +
--R 11 2 12
--R 144a b - 144a b
--R *
--R 5
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 492a b - 492a b - 648a b + 648a b - 168a b + 168a b
--R +
--R 10 3 11 2 12 13
--R 312a b - 312a b + 12a b - 12a
--R *
--R 4
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R 240a b - 240a b - 96a b + 96a b - 480a b + 480a b
--R +
--R 9 4 10 3 11 2 12
--R 288a b - 288a b + 48a b - 48a b
--R *
--R 3
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6
--R 48a b - 48a b + 120a b - 120a b - 312a b + 312a b
--R +
--R 8 5 9 4 10 3 11 2
--R 72a b - 72a b + 72a b - 72a b
--R *
--R 2
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 48a b - 48a b - 48a b + 48a b - 48a b + 48a b + 48a b
--R +
--R 10 3
--R - 48a b
--R *
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 12a b - 12a b - 24a b + 24a b + 12a b - 12a b
--R *
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 9
--R (2a b - 2a b - 4a b + 4a b + 2a b - 2a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 8
--R (18a b - 18a b - 36a b + 36a b + 18a b - 18a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 12
--R 66a b - 66a b - 126a b + 126a b + 54a b - 54a b + 6a b
--R +
--R 13
--R - 6a
--R *
--R 7
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 126a b - 126a b - 210a b + 210a b + 42a b - 42a b
--R +
--R 11 2 12
--R 42a b - 42a b
--R *
--R 6
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 132a b - 132a b - 150a b + 150a b - 90a b + 90a b
--R +
--R 10 3 11 2 12 13
--R 102a b - 102a b + 6a b - 6a
--R *
--R 5
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 72a b - 72a b + 6a b - 6a b - 198a b + 198a b + 90a b
--R +
--R 10 3 11 2 12
--R - 90a b + 30a b - 30a b
--R *
--R 4
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6 8 5
--R 16a b - 16a b + 64a b - 64a b - 122a b + 122a b - 10a b
--R +
--R 9 4 10 3 11 2 12 13
--R 10a b + 50a b - 50a b + 2a b - 2a
--R *
--R 3
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 24a b - 24a b - 6a b + 6a b - 54a b + 54a b + 30a b
--R +
--R 10 3 11 2 12
--R - 30a b + 6a b - 6a b
--R *
--R 2
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 10 3 11 2 5 8
--R (12a b - 12a b - 18a b + 18a b + 6a b - 6a b )cosh(x) + 2a b
--R +
--R 6 7 7 6 8 5 9 4 10 3
--R - 2a b - 4a b + 4a b + 2a b - 2a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 8 5 9 4 10 3 11 2 12 13 3
--R (- 2a b - 2a b + 4a b + 4a b - 2a b - 2a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 2
--R (- 6a b - 6a b + 12a b + 12a b - 6a b - 6a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 5 8
--R (- 6a b - 6a b + 12a b + 12a b - 6a b - 6a b )cosh(x) - 2a b
--R +
--R 6 7 7 6 8 5 9 4 10 3
--R - 2a b + 4a b + 4a b - 2a b - 2a b
--R *
--R 6
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 4
--R (- 12a b - 12a b + 24a b + 24a b - 12a b - 12a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 3
--R (- 48a b - 48a b + 96a b + 96a b - 48a b - 48a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 2
--R (- 72a b - 72a b + 144a b + 144a b - 72a b - 72a b )cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R (- 48a b - 48a b + 96a b + 96a b - 48a b - 48a b )cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 12a b - 12a b + 24a b + 24a b - 12a b - 12a b
--R *
--R 5
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 5
--R (- 30a b - 30a b + 60a b + 60a b - 30a b - 30a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (- 150a b - 150a b + 300a b + 300a b - 150a b - 150a b)
--R *
--R 4
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R - 294a b - 294a b + 582a b + 582a b - 282a b - 282a b
--R +
--R 12 13
--R - 6a b - 6a
--R *
--R 3
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R - 282a b - 282a b + 546a b + 546a b - 246a b - 246a b
--R +
--R 11 2 12
--R - 18a b - 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 132a b - 132a b + 246a b + 246a b - 96a b - 96a b
--R +
--R 10 3 11 2
--R - 18a b - 18a b
--R *
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4 10 3
--R - 24a b - 24a b + 42a b + 42a b - 12a b - 12a b - 6a b - 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 6
--R (- 40a b - 40a b + 80a b + 80a b - 40a b - 40a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (- 240a b - 240a b + 480a b + 480a b - 240a b - 240a b)
--R *
--R 5
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R - 576a b - 576a b + 1128a b + 1128a b - 528a b - 528a b
--R +
--R 12 13
--R - 24a b - 24a
--R *
--R 4
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R - 704a b - 704a b + 1312a b + 1312a b - 512a b - 512a b
--R +
--R 11 2 12
--R - 96a b - 96a b
--R *
--R 3
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 456a b - 456a b + 768a b + 768a b - 168a b - 168a b
--R +
--R 10 3 11 2
--R - 144a b - 144a b
--R *
--R 2
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R - 144a b - 144a b + 192a b + 192a b + 48a b + 48a b
--R +
--R 9 4 10 3
--R - 96a b - 96a b
--R *
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R - 16a b - 16a b + 8a b + 8a b + 32a b + 32a b - 24a b - 24a b
--R *
--R 3
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 7
--R (- 30a b - 30a b + 60a b + 60a b - 30a b - 30a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (- 210a b - 210a b + 420a b + 420a b - 210a b - 210a b)
--R *
--R 6
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R - 594a b - 594a b + 1152a b + 1152a b - 522a b - 522a b
--R +
--R 12 13
--R - 36a b - 36a
--R *
--R 5
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R - 870a b - 870a b + 1560a b + 1560a b - 510a b - 510a b
--R +
--R 11 2 12
--R - 180a b - 180a b
--R *
--R 4
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 696a b - 696a b + 1044a b + 1044a b - 6a b - 6a b
--R +
--R 10 3 11 2 12 13
--R - 336a b - 336a b - 6a b - 6a
--R *
--R 3
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R - 288a b - 288a b + 252a b + 252a b + 342a b + 342a b
--R +
--R 9 4 10 3 11 2 12
--R - 288a b - 288a b - 18a b - 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6
--R - 48a b - 48a b - 48a b - 48a b + 222a b + 222a b
--R +
--R 8 5 9 4 10 3 11 2
--R - 108a b - 108a b - 18a b - 18a b
--R *
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4 10 3
--R - 24a b - 24a b + 42a b + 42a b - 12a b - 12a b - 6a b - 6a b
--R *
--R 2
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 8
--R (- 12a b - 12a b + 24a b + 24a b - 12a b - 12a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 7
--R (- 96a b - 96a b + 192a b + 192a b - 96a b - 96a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R - 312a b - 312a b + 600a b + 600a b - 264a b - 264a b
--R +
--R 12 13
--R - 24a b - 24a
--R *
--R 6
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R - 528a b - 528a b + 912a b + 912a b - 240a b - 240a b
--R +
--R 11 2 12
--R - 144a b - 144a b
--R *
--R 5
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 492a b - 492a b + 648a b + 648a b + 168a b + 168a b
--R +
--R 10 3 11 2 12 13
--R - 312a b - 312a b - 12a b - 12a
--R *
--R 4
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R - 240a b - 240a b + 96a b + 96a b + 480a b + 480a b
--R +
--R 9 4 10 3 11 2 12
--R - 288a b - 288a b - 48a b - 48a b
--R *
--R 3
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6
--R - 48a b - 48a b - 120a b - 120a b + 312a b + 312a b
--R +
--R 8 5 9 4 10 3 11 2
--R - 72a b - 72a b - 72a b - 72a b
--R *
--R 2
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R - 48a b - 48a b + 48a b + 48a b + 48a b + 48a b - 48a b
--R +
--R 10 3
--R - 48a b
--R *
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 12a b - 12a b + 24a b + 24a b - 12a b - 12a b
--R *
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 9
--R (- 2a b - 2a b + 4a b + 4a b - 2a b - 2a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 8
--R (- 18a b - 18a b + 36a b + 36a b - 18a b - 18a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 12
--R - 66a b - 66a b + 126a b + 126a b - 54a b - 54a b - 6a b
--R +
--R 13
--R - 6a
--R *
--R 7
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3 11 2
--R - 126a b - 126a b + 210a b + 210a b - 42a b - 42a b - 42a b
--R +
--R 12
--R - 42a b
--R *
--R 6
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4 10 3
--R - 132a b - 132a b + 150a b + 150a b + 90a b + 90a b - 102a b
--R +
--R 11 2 12 13
--R - 102a b - 6a b - 6a
--R *
--R 5
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R - 72a b - 72a b - 6a b - 6a b + 198a b + 198a b - 90a b
--R +
--R 10 3 11 2 12
--R - 90a b - 30a b - 30a b
--R *
--R 4
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6 8 5
--R - 16a b - 16a b - 64a b - 64a b + 122a b + 122a b + 10a b
--R +
--R 9 4 10 3 11 2 12 13
--R 10a b - 50a b - 50a b - 2a b - 2a
--R *
--R 3
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R - 24a b - 24a b + 6a b + 6a b + 54a b + 54a b - 30a b
--R +
--R 10 3 11 2 12
--R - 30a b - 6a b - 6a b
--R *
--R 2
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 10 3 11 2 5 8
--R (- 12a b - 12a b + 18a b + 18a b - 6a b - 6a b )cosh(x) - 2a b
--R +
--R 6 7 7 6 8 5 9 4 10 3
--R - 2a b + 4a b + 4a b - 2a b - 2a b
--R Type: Expression(Integer)
--E 19
--S 20 of 526
m0402b:= a0402.2-r0402
--R
--R
--R (20)
--R 4 5 6 3 8 2
--R (4a b - 10a b + 12a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2 7 4 5 6 3
--R (8a b - 20a b + 24a b )cosh(x) + 4a b - 10a b + 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 5 6 3 8 3
--R (16a b - 40a b + 48a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2
--R (48a b - 120a b + 144a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 3 6 5 4
--R (48a b - 120a b + 144a b )cosh(x) + 16a b - 40a b + 48a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 5 6 3 8 4
--R (24a b - 60a b + 72a b)cosh(x)
--R +
--R 3 6 5 4 7 2 3
--R (96a b - 240a b + 288a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 2
--R (136a b - 332a b + 388a b + 24a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 9 2 7
--R (80a b - 184a b + 200a b + 48a b )cosh(x) + 16b - 32a b
--R +
--R 4 5 6 3
--R 28a b + 24a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 5 6 3 8 5
--R (16a b - 40a b + 48a b)cosh(x)
--R +
--R 3 6 5 4 7 2 4
--R (80a b - 200a b + 240a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 3
--R (144a b - 344a b + 392a b + 48a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 2
--R (112a b - 232a b + 216a b + 144a b )cosh(x)
--R +
--R 9 2 7 4 5 6 3 8 3 6
--R (32b - 32a b - 24a b + 144a b )cosh(x) + 16a b - 40a b
--R +
--R 5 4
--R 48a b
--R *
--R sinh(x)
--R +
--R 4 5 6 3 8 6 3 6 5 4 7 2 5
--R (4a b - 10a b + 12a b)cosh(x) + (24a b - 60a b + 72a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 4
--R (52a b - 122a b + 136a b + 24a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 3
--R (48a b - 88a b + 64a b + 96a b )cosh(x)
--R +
--R 9 4 5 6 3 8 2
--R (16b - 48a b + 110a b + 12a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 2 7 4 5 6 3
--R (16a b - 32a b + 28a b + 24a b )cosh(x) + 4a b - 10a b + 12a b
--R *
--R +---------+
--R | 2 2
--R (a sinh(x) + a cosh(x) + b)\|- b + a
--R atan(---------------------------------------)
--R 2 2
--R b - a
--R +
--R 4 5 6 3 8 2
--R (- 4a b + 10a b - 12a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2 7 4 5 6 3
--R (- 8a b + 20a b - 24a b )cosh(x) - 4a b + 10a b - 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 5 6 3 8 3
--R (- 16a b + 40a b - 48a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2
--R (- 48a b + 120a b - 144a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 3 6 5 4
--R (- 48a b + 120a b - 144a b )cosh(x) - 16a b + 40a b - 48a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 5 6 3 8 4
--R (- 24a b + 60a b - 72a b)cosh(x)
--R +
--R 3 6 5 4 7 2 3
--R (- 96a b + 240a b - 288a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 2
--R (- 136a b + 332a b - 388a b - 24a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 9 2 7
--R (- 80a b + 184a b - 200a b - 48a b )cosh(x) - 16b + 32a b
--R +
--R 4 5 6 3
--R - 28a b - 24a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 5 6 3 8 5
--R (- 16a b + 40a b - 48a b)cosh(x)
--R +
--R 3 6 5 4 7 2 4
--R (- 80a b + 200a b - 240a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 3
--R (- 144a b + 344a b - 392a b - 48a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 2
--R (- 112a b + 232a b - 216a b - 144a b )cosh(x)
--R +
--R 9 2 7 4 5 6 3 8 3 6
--R (- 32b + 32a b + 24a b - 144a b )cosh(x) - 16a b + 40a b
--R +
--R 5 4
--R - 48a b
--R *
--R sinh(x)
--R +
--R 4 5 6 3 8 6
--R (- 4a b + 10a b - 12a b)cosh(x)
--R +
--R 3 6 5 4 7 2 5
--R (- 24a b + 60a b - 72a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 4
--R (- 52a b + 122a b - 136a b - 24a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 3
--R (- 48a b + 88a b - 64a b - 96a b )cosh(x)
--R +
--R 9 4 5 6 3 8 2
--R (- 16b + 48a b - 110a b - 12a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 2 7 4 5 6 3
--R (- 16a b + 32a b - 28a b - 24a b )cosh(x) - 4a b + 10a b - 12a b
--R *
--R x
--R (b - a)tanh(-)
--R 2
--R atan(--------------)
--R +---------+
--R | 2 2
--R \|- b + a
--R +
--R 4 4 6 2 3 5 5 3 5
--R ((3a b - 6a b )cosh(x) + 2a b - 5a b )sinh(x)
--R +
--R 4 4 6 2 2 3 5 5 3 2 6
--R (12a b - 24a b )cosh(x) + (20a b - 44a b )cosh(x) + 8a b
--R +
--R 4 4
--R - 20a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 4 6 2 3 3 5 5 3 2
--R (18a b - 36a b )cosh(x) + (56a b - 116a b )cosh(x)
--R +
--R 2 6 4 4 6 2 7 3 5 5 3
--R (52a b - 106a b - 12a b )cosh(x) + 16a b - 30a b - 10a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 4 6 2 4 3 5 5 3 3
--R (12a b - 24a b )cosh(x) + (68a b - 134a b )cosh(x)
--R +
--R 2 6 4 4 6 2 2
--R (108a b - 198a b - 36a b )cosh(x)
--R +
--R 7 3 5 5 3 8 2 6 4 4
--R (64a b - 98a b - 68a b )cosh(x) + 12b - 10a b - 32a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 4 6 2 5 3 5 5 3 4
--R (3a b - 6a b )cosh(x) + (38a b - 71a b )cosh(x)
--R +
--R 2 6 4 4 6 2 3
--R (92a b - 158a b - 36a b )cosh(x)
--R +
--R 7 3 5 5 3 2
--R (80a b - 102a b - 116a b )cosh(x)
--R +
--R 8 2 6 4 4 6 2 7 3 5 5 3
--R (24b + 4a b - 109a b - 6a b )cosh(x) + 16a b - 32a b - 5a b
--R *
--R sinh(x)
--R +
--R 3 5 5 3 5 2 6 4 4 6 2 4
--R (8a b - 14a b )cosh(x) + (28a b - 46a b - 12a b )cosh(x)
--R +
--R 7 3 5 5 3 3
--R (32a b - 34a b - 58a b )cosh(x)
--R +
--R 8 2 6 4 4 6 2 2
--R (12b + 14a b - 74a b - 12a b )cosh(x)
--R +
--R 7 3 5 5 3 2 6 4 4
--R (16a b - 22a b - 24a b )cosh(x) + 6a b - 12a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R /
--R 7 4 9 2 11 2 6 5 8 3 10
--R (2a b - 4a b + 2a )cosh(x) + (4a b - 8a b + 4a b)cosh(x)
--R +
--R 5 6 7 4 9 2
--R 2a b - 4a b + 2a b
--R *
--R 4
--R sinh(x)
--R +
--R 7 4 9 2 11 3
--R (8a b - 16a b + 8a )cosh(x)
--R +
--R 6 5 8 3 10 2
--R (24a b - 48a b + 24a b)cosh(x)
--R +
--R 5 6 7 4 9 2 4 7 6 5 8 3
--R (24a b - 48a b + 24a b )cosh(x) + 8a b - 16a b + 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 7 4 9 2 11 4
--R (12a b - 24a b + 12a )cosh(x)
--R +
--R 6 5 8 3 10 3
--R (48a b - 96a b + 48a b)cosh(x)
--R +
--R 5 6 7 4 9 2 11 2
--R (68a b - 132a b + 60a b + 4a )cosh(x)
--R +
--R 4 7 6 5 8 3 10 3 8 5 6 9 2
--R (40a b - 72a b + 24a b + 8a b)cosh(x) + 8a b - 12a b + 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 7 4 9 2 11 5
--R (8a b - 16a b + 8a )cosh(x)
--R +
--R 6 5 8 3 10 4
--R (40a b - 80a b + 40a b)cosh(x)
--R +
--R 5 6 7 4 9 2 11 3
--R (72a b - 136a b + 56a b + 8a )cosh(x)
--R +
--R 4 7 6 5 8 3 10 2
--R (56a b - 88a b + 8a b + 24a b)cosh(x)
--R +
--R 3 8 5 6 7 4 9 2 4 7 6 5 8 3
--R (16a b - 8a b - 32a b + 24a b )cosh(x) + 8a b - 16a b + 8a b
--R *
--R sinh(x)
--R +
--R 7 4 9 2 11 6 6 5 8 3 10 5
--R (2a b - 4a b + 2a )cosh(x) + (12a b - 24a b + 12a b)cosh(x)
--R +
--R 5 6 7 4 9 2 11 4
--R (26a b - 48a b + 18a b + 4a )cosh(x)
--R +
--R 4 7 6 5 8 3 10 3
--R (24a b - 32a b - 8a b + 16a b)cosh(x)
--R +
--R 3 8 5 6 7 4 9 2 11 2
--R (8a b + 4a b - 30a b + 16a b + 2a )cosh(x)
--R +
--R 4 7 6 5 10 5 6 7 4 9 2
--R (8a b - 12a b + 4a b)cosh(x) + 2a b - 4a b + 2a b
--R *
--R +---------+
--R | 2 2
--R \|- b + a
--R Type: Expression(Integer)
--E 20
--S 21 of 526
d0402b:= D(m0402b,x)
--R
--R
--R (21)
--R 5 5 6 4 7 3 8 2 4 6 5 5 6 4
--R (- 3a b + 3a b + 6a b - 6a b )cosh(x) - a b + a b + 4a b
--R +
--R 7 3
--R - 4a b
--R *
--R 8
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 2
--R (- 18a b + 18a b + 36a b - 36a b )cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 3 7 4 6
--R (- 24a b + 24a b + 60a b - 60a b )cosh(x) - 6a b + 6a b
--R +
--R 5 5 6 4
--R 24a b - 24a b
--R *
--R 7
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 3
--R (- 40a b + 42a b + 79a b - 84a b + 6a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 2
--R (- 94a b + 100a b + 214a b - 229a b + 18a b )cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R (- 58a b + 64a b + 163a b - 178a b + 36a b - 18a b )cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R - 10a b + 12a b + 40a b - 45a b + 18a b - 12a b
--R *
--R 6
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 4
--R (- 30a b + 42a b + 54a b - 84a b + 36a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9
--R (- 116a b + 164a b + 242a b - 362a b + 132a b + 12a b)
--R *
--R 3
--R cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R (- 126a b + 198a b + 288a b - 468a b + 252a b - 36a b )
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R (- 48a b + 96a b + 114a b - 234a b + 228a b - 84a b )cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R - 8a b + 20a b + 14a b - 44a b + 72a b - 36a b
--R *
--R 5
--R sinh(x)
--R +
--R 5 5 7 3 9 5
--R (30a b - 75a b + 90a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 4
--R (60a b + 90a b - 180a b - 195a b + 390a b + 60a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R 30a b + 264a b - 171a b - 558a b + 753a b + 114a b
--R +
--R 9
--R 18a b
--R *
--R 3
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 6a b + 276a b - 144a b - 543a b + 804a b - 3a b
--R +
--R 8 2
--R 54a b
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 132a b - 78a b - 234a b + 411a b - 60a b + 72a b
--R +
--R 8 2
--R - 18a b
--R *
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b - 18a b - 36a b + 78a b - 21a b + 30a b - 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 6
--R (82a b - 42a b - 184a b + 84a b + 120a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 5
--R (304a b - 64a b - 736a b + 136a b + 600a b + 120a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R 430a b + 146a b - 1144a b - 272a b + 1212a b + 456a b
--R +
--R 9
--R 72a b
--R *
--R 4
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 304a b + 400a b - 924a b - 740a b + 1316a b + 556a b
--R +
--R 8 2 9
--R 264a b + 24a b
--R *
--R 3
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R 112a b + 344a b - 412a b - 584a b + 774a b + 234a b
--R +
--R 7 3 8 2
--R 396a b + 36a b
--R *
--R 2
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 16b + 128a b - 88a b - 176a b + 204a b - 12a b
--R +
--R 6 4 7 3
--R 276a b + 12a b
--R *
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 16b - 8a b - 8a b + 14a b - 26a b + 72a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 7
--R (72a b - 42a b - 159a b + 84a b + 90a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9
--R (326a b - 116a b - 770a b + 245a b + 510a b + 120a b)
--R *
--R 6
--R cosh(x)
--R +
--R 3 7 5 5 6 4 7 3 8 2 9
--R (594a b - 1467a b + 18a b + 1098a b + 594a b + 108a b)
--R *
--R 5
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 546a b + 324a b - 1368a b - 627a b + 1134a b + 1026a b
--R +
--R 8 2 9
--R 468a b + 72a b
--R *
--R 4
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R 256a b + 440a b - 628a b - 764a b + 570a b + 654a b
--R +
--R 7 3 8 2 9
--R 789a b + 240a b + 18a b
--R *
--R 3
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 48b + 240a b - 108a b - 288a b + 90a b - 18a b
--R +
--R 6 4 7 3 8 2
--R 654a b + 273a b + 54a b
--R *
--R 2
--R cosh(x)
--R +
--R 10 2 8 3 7 4 6 5 5 6 4
--R 48b + 24a b - 18a b - 180a b + 249a b + 138a b
--R +
--R 7 3 8 2
--R 60a b - 6a b
--R *
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b - 6a b - 48a b + 32a b + 25a b + 22a b - 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 8
--R (30a b - 18a b - 66a b + 36a b + 36a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 7
--R (156a b - 60a b - 366a b + 126a b + 228a b + 60a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R 342a b - 30a b - 816a b + 60a b + 528a b + 348a b
--R +
--R 9
--R 72a b
--R *
--R 6
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 384a b + 144a b - 870a b - 306a b + 504a b + 720a b
--R +
--R 8 2 9
--R 360a b + 72a b
--R *
--R 5
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R 216a b + 276a b - 402a b - 492a b + 102a b + 546a b
--R +
--R 7 3 8 2 9
--R 678a b + 300a b + 36a b
--R *
--R 4
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 48b + 192a b - 24a b - 192a b - 120a b - 72a b
--R +
--R 6 4 7 3 8 2 9
--R 594a b + 438a b + 132a b + 12a b
--R *
--R 3
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 48b + 24a b + 72a b - 54a b - 270a b + 228a b
--R +
--R 6 4 7 3 8 2
--R 240a b + 180a b + 36a b
--R *
--R 2
--R cosh(x)
--R +
--R 9 3 7 4 6 5 5 6 4 7 3
--R (48a b - 72a b + 18a b + 6a b + 108a b + 36a b )cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4
--R 12a b - 6a b - 24a b + 24a b + 12a b
--R *
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 9
--R (5a b - 3a b - 11a b + 6a b + 6a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 8
--R (29a b - 11a b - 68a b + 23a b + 42a b + 12a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2 9
--R (74a b - 8a b - 173a b + 14a b + 105a b + 78a b + 18a b)
--R *
--R 7
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R 98a b + 28a b - 208a b - 65a b + 92a b + 181a b + 102a b
--R +
--R 9
--R 24a b
--R *
--R 6
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 64a b + 68a b - 94a b - 122a b - 33a b + 150a b + 213a b
--R +
--R 8 2 9
--R 114a b + 18a b
--R *
--R 5
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R 16b + 56a b + 14a b - 44a b - 84a b - 45a b + 186a b
--R +
--R 7 3 8 2 9
--R 189a b + 78a b + 12a b
--R *
--R 4
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R 16b + 16a b + 40a b - 22a b - 116a b + 53a b + 102a b
--R +
--R 7 3 8 2 9
--R 115a b + 42a b + 6a b
--R *
--R 3
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b + 6a b - 24a b - 4a b - 23a b + 64a b + 47a b
--R +
--R 8 2
--R 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 2 8
--R (12a b + 2a b - 26a b + 4a b + 17a b + 18a b )cosh(x) + 2a b
--R +
--R 4 6 6 4
--R - 5a b + 6a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 5 5 6 4 7 3 8 2 4 6 5 5 6 4 7 3
--R ((3a b + 3a b - 6a b - 6a b )cosh(x) + a b + a b - 4a b - 4a b )
--R *
--R 8
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 2
--R (18a b + 18a b - 36a b - 36a b )cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 3 7 4 6 5 5
--R (24a b + 24a b - 60a b - 60a b )cosh(x) + 6a b + 6a b - 24a b
--R +
--R 6 4
--R - 24a b
--R *
--R 7
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 3
--R (40a b + 42a b - 79a b - 84a b - 6a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 2
--R (94a b + 100a b - 214a b - 229a b - 18a b )cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R (58a b + 64a b - 163a b - 178a b - 36a b - 18a b )cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R 10a b + 12a b - 40a b - 45a b - 18a b - 12a b
--R *
--R 6
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 4
--R (30a b + 42a b - 54a b - 84a b - 36a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 3
--R (116a b + 164a b - 242a b - 362a b - 132a b + 12a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2 2
--R (126a b + 198a b - 288a b - 468a b - 252a b - 36a b )cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R (48a b + 96a b - 114a b - 234a b - 228a b - 84a b )cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R 8a b + 20a b - 14a b - 44a b - 72a b - 36a b
--R *
--R 5
--R sinh(x)
--R +
--R 5 5 7 3 9 5
--R (- 30a b + 75a b - 90a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 4
--R (- 60a b + 90a b + 180a b - 195a b - 390a b + 60a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R - 30a b + 264a b + 171a b - 558a b - 753a b + 114a b
--R +
--R 9
--R - 18a b
--R *
--R 3
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R (- 6a b + 276a b + 144a b - 543a b - 804a b - 3a b - 54a b )
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R (132a b + 78a b - 234a b - 411a b - 60a b - 72a b - 18a b )
--R *
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b + 18a b - 36a b - 78a b - 21a b - 30a b - 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 6
--R (- 82a b - 42a b + 184a b + 84a b - 120a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 5
--R (- 304a b - 64a b + 736a b + 136a b - 600a b + 120a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2
--R - 430a b + 146a b + 1144a b - 272a b - 1212a b + 456a b
--R +
--R 9
--R - 72a b
--R *
--R 4
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R - 304a b + 400a b + 924a b - 740a b - 1316a b + 556a b
--R +
--R 8 2 9
--R - 264a b + 24a b
--R *
--R 3
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R - 112a b + 344a b + 412a b - 584a b - 774a b + 234a b
--R +
--R 7 3 8 2
--R - 396a b + 36a b
--R *
--R 2
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R - 16b + 128a b + 88a b - 176a b - 204a b - 12a b - 276a b
--R +
--R 7 3
--R 12a b
--R *
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5
--R 16b + 8a b - 8a b - 14a b - 26a b - 72a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 7
--R (- 72a b - 42a b + 159a b + 84a b - 90a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 6
--R (- 326a b - 116a b + 770a b + 245a b - 510a b + 120a b)cosh(x)
--R +
--R 3 7 5 5 6 4 7 3 8 2 9 5
--R (- 594a b + 1467a b + 18a b - 1098a b + 594a b - 108a b)cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R - 546a b + 324a b + 1368a b - 627a b - 1134a b + 1026a b
--R +
--R 8 2 9
--R - 468a b + 72a b
--R *
--R 4
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R - 256a b + 440a b + 628a b - 764a b - 570a b + 654a b
--R +
--R 7 3 8 2 9
--R - 789a b + 240a b - 18a b
--R *
--R 3
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R - 48b + 240a b + 108a b - 288a b - 90a b - 18a b - 654a b
--R +
--R 7 3 8 2
--R 273a b - 54a b
--R *
--R 2
--R cosh(x)
--R +
--R 10 2 8 3 7 4 6 5 5 6 4 7 3
--R 48b + 24a b + 18a b - 180a b - 249a b + 138a b - 60a b
--R +
--R 8 2
--R - 6a b
--R *
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R 24a b + 6a b - 48a b - 32a b + 25a b - 22a b - 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 8
--R (- 30a b - 18a b + 66a b + 36a b - 36a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 7
--R (- 156a b - 60a b + 366a b + 126a b - 228a b + 60a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2 9
--R (- 342a b - 30a b + 816a b + 60a b - 528a b + 348a b - 72a b)
--R *
--R 6
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3
--R - 384a b + 144a b + 870a b - 306a b - 504a b + 720a b
--R +
--R 8 2 9
--R - 360a b + 72a b
--R *
--R 5
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4
--R - 216a b + 276a b + 402a b - 492a b - 102a b + 546a b
--R +
--R 7 3 8 2 9
--R - 678a b + 300a b - 36a b
--R *
--R 4
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R - 48b + 192a b + 24a b - 192a b + 120a b - 72a b - 594a b
--R +
--R 7 3 8 2 9
--R 438a b - 132a b + 12a b
--R *
--R 3
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R 48b - 24a b + 72a b + 54a b - 270a b - 228a b + 240a b
--R +
--R 7 3 8 2
--R - 180a b + 36a b
--R *
--R 2
--R cosh(x)
--R +
--R 9 3 7 4 6 5 5 6 4 7 3 2 8
--R (48a b - 72a b - 18a b + 6a b - 108a b + 36a b )cosh(x) + 12a b
--R +
--R 3 7 4 6 5 5 6 4
--R 6a b - 24a b - 24a b + 12a b
--R *
--R sinh(x)
--R +
--R 5 5 6 4 7 3 8 2 9 9
--R (- 5a b - 3a b + 11a b + 6a b - 6a b)cosh(x)
--R +
--R 4 6 5 5 6 4 7 3 8 2 9 8
--R (- 29a b - 11a b + 68a b + 23a b - 42a b + 12a b)cosh(x)
--R +
--R 3 7 4 6 5 5 6 4 7 3 8 2 9 7
--R (- 74a b - 8a b + 173a b + 14a b - 105a b + 78a b - 18a b)cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R - 98a b + 28a b + 208a b - 65a b - 92a b + 181a b - 102a b
--R +
--R 9
--R 24a b
--R *
--R 6
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3
--R - 64a b + 68a b + 94a b - 122a b + 33a b + 150a b - 213a b
--R +
--R 8 2 9
--R 114a b - 18a b
--R *
--R 5
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R - 16b + 56a b - 14a b - 44a b + 84a b - 45a b - 186a b
--R +
--R 7 3 8 2 9
--R 189a b - 78a b + 12a b
--R *
--R 4
--R cosh(x)
--R +
--R 10 9 2 8 3 7 4 6 5 5 6 4
--R 16b - 16a b + 40a b + 22a b - 116a b - 53a b + 102a b
--R +
--R 7 3 8 2 9
--R - 115a b + 42a b - 6a b
--R *
--R 3
--R cosh(x)
--R +
--R 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2
--R (24a b - 6a b - 24a b + 4a b - 23a b - 64a b + 47a b - 18a b )
--R *
--R 2
--R cosh(x)
--R +
--R 2 8 3 7 4 6 5 5 6 4 7 3 2 8
--R (12a b - 2a b - 26a b - 4a b + 17a b - 18a b )cosh(x) - 2a b
--R +
--R 4 6 6 4
--R 5a b - 6a b
--R /
--R 8 5 9 4 10 3 11 2 12 13 3
--R (2a b - 2a b - 4a b + 4a b + 2a b - 2a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 2
--R (6a b - 6a b - 12a b + 12a b + 6a b - 6a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R (6a b - 6a b - 12a b + 12a b + 6a b - 6a b )cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 2a b - 2a b - 4a b + 4a b + 2a b - 2a b
--R *
--R 6
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 4
--R (12a b - 12a b - 24a b + 24a b + 12a b - 12a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 3
--R (48a b - 48a b - 96a b + 96a b + 48a b - 48a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 2
--R (72a b - 72a b - 144a b + 144a b + 72a b - 72a b )cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R (48a b - 48a b - 96a b + 96a b + 48a b - 48a b )cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 12a b - 12a b - 24a b + 24a b + 12a b - 12a b
--R *
--R 5
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 5
--R (30a b - 30a b - 60a b + 60a b + 30a b - 30a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (150a b - 150a b - 300a b + 300a b + 150a b - 150a b)
--R *
--R 4
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R 294a b - 294a b - 582a b + 582a b + 282a b - 282a b
--R +
--R 12 13
--R 6a b - 6a
--R *
--R 3
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 282a b - 282a b - 546a b + 546a b + 246a b - 246a b
--R +
--R 11 2 12
--R 18a b - 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 132a b - 132a b - 246a b + 246a b + 96a b - 96a b
--R +
--R 10 3 11 2
--R 18a b - 18a b
--R *
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 24a b - 24a b - 42a b + 42a b + 12a b - 12a b + 6a b
--R +
--R 10 3
--R - 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 6
--R (40a b - 40a b - 80a b + 80a b + 40a b - 40a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (240a b - 240a b - 480a b + 480a b + 240a b - 240a b)
--R *
--R 5
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R 576a b - 576a b - 1128a b + 1128a b + 528a b - 528a b
--R +
--R 12 13
--R 24a b - 24a
--R *
--R 4
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 704a b - 704a b - 1312a b + 1312a b + 512a b - 512a b
--R +
--R 11 2 12
--R 96a b - 96a b
--R *
--R 3
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 456a b - 456a b - 768a b + 768a b + 168a b - 168a b
--R +
--R 10 3 11 2
--R 144a b - 144a b
--R *
--R 2
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R 144a b - 144a b - 192a b + 192a b - 48a b + 48a b
--R +
--R 9 4 10 3
--R 96a b - 96a b
--R *
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6 8 5
--R 16a b - 16a b - 8a b + 8a b - 32a b + 32a b + 24a b
--R +
--R 9 4
--R - 24a b
--R *
--R 3
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 7
--R (30a b - 30a b - 60a b + 60a b + 30a b - 30a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (210a b - 210a b - 420a b + 420a b + 210a b - 210a b)
--R *
--R 6
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R 594a b - 594a b - 1152a b + 1152a b + 522a b - 522a b
--R +
--R 12 13
--R 36a b - 36a
--R *
--R 5
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 870a b - 870a b - 1560a b + 1560a b + 510a b - 510a b
--R +
--R 11 2 12
--R 180a b - 180a b
--R *
--R 4
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 696a b - 696a b - 1044a b + 1044a b + 6a b - 6a b
--R +
--R 10 3 11 2 12 13
--R 336a b - 336a b + 6a b - 6a
--R *
--R 3
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R 288a b - 288a b - 252a b + 252a b - 342a b + 342a b
--R +
--R 9 4 10 3 11 2 12
--R 288a b - 288a b + 18a b - 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6
--R 48a b - 48a b + 48a b - 48a b - 222a b + 222a b
--R +
--R 8 5 9 4 10 3 11 2
--R 108a b - 108a b + 18a b - 18a b
--R *
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 24a b - 24a b - 42a b + 42a b + 12a b - 12a b + 6a b
--R +
--R 10 3
--R - 6a b
--R *
--R 2
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 8
--R (12a b - 12a b - 24a b + 24a b + 12a b - 12a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 7
--R (96a b - 96a b - 192a b + 192a b + 96a b - 96a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R 312a b - 312a b - 600a b + 600a b + 264a b - 264a b
--R +
--R 12 13
--R 24a b - 24a
--R *
--R 6
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 528a b - 528a b - 912a b + 912a b + 240a b - 240a b
--R +
--R 11 2 12
--R 144a b - 144a b
--R *
--R 5
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 492a b - 492a b - 648a b + 648a b - 168a b + 168a b
--R +
--R 10 3 11 2 12 13
--R 312a b - 312a b + 12a b - 12a
--R *
--R 4
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R 240a b - 240a b - 96a b + 96a b - 480a b + 480a b
--R +
--R 9 4 10 3 11 2 12
--R 288a b - 288a b + 48a b - 48a b
--R *
--R 3
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6
--R 48a b - 48a b + 120a b - 120a b - 312a b + 312a b
--R +
--R 8 5 9 4 10 3 11 2
--R 72a b - 72a b + 72a b - 72a b
--R *
--R 2
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 48a b - 48a b - 48a b + 48a b - 48a b + 48a b + 48a b
--R +
--R 10 3
--R - 48a b
--R *
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 12a b - 12a b - 24a b + 24a b + 12a b - 12a b
--R *
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 9
--R (2a b - 2a b - 4a b + 4a b + 2a b - 2a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 8
--R (18a b - 18a b - 36a b + 36a b + 18a b - 18a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 12
--R 66a b - 66a b - 126a b + 126a b + 54a b - 54a b + 6a b
--R +
--R 13
--R - 6a
--R *
--R 7
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R 126a b - 126a b - 210a b + 210a b + 42a b - 42a b
--R +
--R 11 2 12
--R 42a b - 42a b
--R *
--R 6
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R 132a b - 132a b - 150a b + 150a b - 90a b + 90a b
--R +
--R 10 3 11 2 12 13
--R 102a b - 102a b + 6a b - 6a
--R *
--R 5
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 72a b - 72a b + 6a b - 6a b - 198a b + 198a b + 90a b
--R +
--R 10 3 11 2 12
--R - 90a b + 30a b - 30a b
--R *
--R 4
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6 8 5
--R 16a b - 16a b + 64a b - 64a b - 122a b + 122a b - 10a b
--R +
--R 9 4 10 3 11 2 12 13
--R 10a b + 50a b - 50a b + 2a b - 2a
--R *
--R 3
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R 24a b - 24a b - 6a b + 6a b - 54a b + 54a b + 30a b
--R +
--R 10 3 11 2 12
--R - 30a b + 6a b - 6a b
--R *
--R 2
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 10 3 11 2 5 8
--R (12a b - 12a b - 18a b + 18a b + 6a b - 6a b )cosh(x) + 2a b
--R +
--R 6 7 7 6 8 5 9 4 10 3
--R - 2a b - 4a b + 4a b + 2a b - 2a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 8 5 9 4 10 3 11 2 12 13 3
--R (- 2a b - 2a b + 4a b + 4a b - 2a b - 2a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 2
--R (- 6a b - 6a b + 12a b + 12a b - 6a b - 6a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 5 8
--R (- 6a b - 6a b + 12a b + 12a b - 6a b - 6a b )cosh(x) - 2a b
--R +
--R 6 7 7 6 8 5 9 4 10 3
--R - 2a b + 4a b + 4a b - 2a b - 2a b
--R *
--R 6
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 4
--R (- 12a b - 12a b + 24a b + 24a b - 12a b - 12a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 3
--R (- 48a b - 48a b + 96a b + 96a b - 48a b - 48a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 2
--R (- 72a b - 72a b + 144a b + 144a b - 72a b - 72a b )cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R (- 48a b - 48a b + 96a b + 96a b - 48a b - 48a b )cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 12a b - 12a b + 24a b + 24a b - 12a b - 12a b
--R *
--R 5
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 5
--R (- 30a b - 30a b + 60a b + 60a b - 30a b - 30a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (- 150a b - 150a b + 300a b + 300a b - 150a b - 150a b)
--R *
--R 4
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R - 294a b - 294a b + 582a b + 582a b - 282a b - 282a b
--R +
--R 12 13
--R - 6a b - 6a
--R *
--R 3
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R - 282a b - 282a b + 546a b + 546a b - 246a b - 246a b
--R +
--R 11 2 12
--R - 18a b - 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 132a b - 132a b + 246a b + 246a b - 96a b - 96a b
--R +
--R 10 3 11 2
--R - 18a b - 18a b
--R *
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4 10 3
--R - 24a b - 24a b + 42a b + 42a b - 12a b - 12a b - 6a b - 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 6
--R (- 40a b - 40a b + 80a b + 80a b - 40a b - 40a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (- 240a b - 240a b + 480a b + 480a b - 240a b - 240a b)
--R *
--R 5
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R - 576a b - 576a b + 1128a b + 1128a b - 528a b - 528a b
--R +
--R 12 13
--R - 24a b - 24a
--R *
--R 4
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R - 704a b - 704a b + 1312a b + 1312a b - 512a b - 512a b
--R +
--R 11 2 12
--R - 96a b - 96a b
--R *
--R 3
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 456a b - 456a b + 768a b + 768a b - 168a b - 168a b
--R +
--R 10 3 11 2
--R - 144a b - 144a b
--R *
--R 2
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R - 144a b - 144a b + 192a b + 192a b + 48a b + 48a b
--R +
--R 9 4 10 3
--R - 96a b - 96a b
--R *
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R - 16a b - 16a b + 8a b + 8a b + 32a b + 32a b - 24a b - 24a b
--R *
--R 3
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 7
--R (- 30a b - 30a b + 60a b + 60a b - 30a b - 30a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12
--R (- 210a b - 210a b + 420a b + 420a b - 210a b - 210a b)
--R *
--R 6
--R cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R - 594a b - 594a b + 1152a b + 1152a b - 522a b - 522a b
--R +
--R 12 13
--R - 36a b - 36a
--R *
--R 5
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R - 870a b - 870a b + 1560a b + 1560a b - 510a b - 510a b
--R +
--R 11 2 12
--R - 180a b - 180a b
--R *
--R 4
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 696a b - 696a b + 1044a b + 1044a b - 6a b - 6a b
--R +
--R 10 3 11 2 12 13
--R - 336a b - 336a b - 6a b - 6a
--R *
--R 3
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R - 288a b - 288a b + 252a b + 252a b + 342a b + 342a b
--R +
--R 9 4 10 3 11 2 12
--R - 288a b - 288a b - 18a b - 18a b
--R *
--R 2
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6
--R - 48a b - 48a b - 48a b - 48a b + 222a b + 222a b
--R +
--R 8 5 9 4 10 3 11 2
--R - 108a b - 108a b - 18a b - 18a b
--R *
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4 10 3
--R - 24a b - 24a b + 42a b + 42a b - 12a b - 12a b - 6a b - 6a b
--R *
--R 2
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 8
--R (- 12a b - 12a b + 24a b + 24a b - 12a b - 12a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 7
--R (- 96a b - 96a b + 192a b + 192a b - 96a b - 96a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2
--R - 312a b - 312a b + 600a b + 600a b - 264a b - 264a b
--R +
--R 12 13
--R - 24a b - 24a
--R *
--R 6
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3
--R - 528a b - 528a b + 912a b + 912a b - 240a b - 240a b
--R +
--R 11 2 12
--R - 144a b - 144a b
--R *
--R 5
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 492a b - 492a b + 648a b + 648a b + 168a b + 168a b
--R +
--R 10 3 11 2 12 13
--R - 312a b - 312a b - 12a b - 12a
--R *
--R 4
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5
--R - 240a b - 240a b + 96a b + 96a b + 480a b + 480a b
--R +
--R 9 4 10 3 11 2 12
--R - 288a b - 288a b - 48a b - 48a b
--R *
--R 3
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6
--R - 48a b - 48a b - 120a b - 120a b + 312a b + 312a b
--R +
--R 8 5 9 4 10 3 11 2
--R - 72a b - 72a b - 72a b - 72a b
--R *
--R 2
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R - 48a b - 48a b + 48a b + 48a b + 48a b + 48a b - 48a b
--R +
--R 10 3
--R - 48a b
--R *
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4
--R - 12a b - 12a b + 24a b + 24a b - 12a b - 12a b
--R *
--R sinh(x)
--R +
--R 8 5 9 4 10 3 11 2 12 13 9
--R (- 2a b - 2a b + 4a b + 4a b - 2a b - 2a )cosh(x)
--R +
--R 7 6 8 5 9 4 10 3 11 2 12 8
--R (- 18a b - 18a b + 36a b + 36a b - 18a b - 18a b)cosh(x)
--R +
--R 6 7 7 6 8 5 9 4 10 3 11 2 12
--R - 66a b - 66a b + 126a b + 126a b - 54a b - 54a b - 6a b
--R +
--R 13
--R - 6a
--R *
--R 7
--R cosh(x)
--R +
--R 5 8 6 7 7 6 8 5 9 4 10 3 11 2
--R - 126a b - 126a b + 210a b + 210a b - 42a b - 42a b - 42a b
--R +
--R 12
--R - 42a b
--R *
--R 6
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 8 5 9 4 10 3
--R - 132a b - 132a b + 150a b + 150a b + 90a b + 90a b - 102a b
--R +
--R 11 2 12 13
--R - 102a b - 6a b - 6a
--R *
--R 5
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R - 72a b - 72a b - 6a b - 6a b + 198a b + 198a b - 90a b
--R +
--R 10 3 11 2 12
--R - 90a b - 30a b - 30a b
--R *
--R 4
--R cosh(x)
--R +
--R 2 11 3 10 4 9 5 8 6 7 7 6 8 5
--R - 16a b - 16a b - 64a b - 64a b + 122a b + 122a b + 10a b
--R +
--R 9 4 10 3 11 2 12 13
--R 10a b - 50a b - 50a b - 2a b - 2a
--R *
--R 3
--R cosh(x)
--R +
--R 3 10 4 9 5 8 6 7 7 6 8 5 9 4
--R - 24a b - 24a b + 6a b + 6a b + 54a b + 54a b - 30a b
--R +
--R 10 3 11 2 12
--R - 30a b - 6a b - 6a b
--R *
--R 2
--R cosh(x)
--R +
--R 4 9 5 8 6 7 7 6 10 3 11 2 5 8
--R (- 12a b - 12a b + 18a b + 18a b - 6a b - 6a b )cosh(x) - 2a b
--R +
--R 6 7 7 6 8 5 9 4 10 3
--R - 2a b + 4a b + 4a b - 2a b - 2a b
--R Type: Expression(Integer)
--E 21
--S 22 of 526
t0403:= 1/(a+b*sech(x)^2)
--R
--R
--R 1
--R (22) --------------
--R 2
--R b sech(x) + a
--R Type: Expression(Integer)
--E 22
--S 23 of 526
r0403:= x/a-b^(1/2)*atanh(b^(1/2)*tanh(x)/(a+b)^(1/2))/a/(a+b)^(1/2)
--R
--R
--R +-+
--R +-+ tanh(x)\|b +-----+
--R - \|b atanh(-----------) + x\|b + a
--R +-----+
--R \|b + a
--R (23) ------------------------------------
--R +-----+
--R a\|b + a
--R Type: Expression(Integer)
--E 23
--S 24 of 526
a0403:= integrate(t0403,x)
--R
--R
--R (24)
--R [
--R +-----+
--R | b
--R |-----
--R \|b + a
--R *
--R log
--R 2 2 2
--R (4a b + 4a )sinh(x) + (8a b + 8a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (4a b + 4a )cosh(x) + 8b + 12a b + 4a
--R *
--R +-----+
--R | b
--R |-----
--R \|b + a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 4a b + 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (8a b + 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2 2
--R (4a b + 2a )cosh(x) + 8b + 8a b + a
--R /
--R 4 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2
--R (6a cosh(x) + 4b + 2a)sinh(x)
--R +
--R 3 4
--R (4a cosh(x) + (8b + 4a)cosh(x))sinh(x) + a cosh(x)
--R +
--R 2
--R (4b + 2a)cosh(x) + a
--R +
--R 2x
--R /
--R 2a
--R ,
--R +-------+
--R | b
--R +-------+ (2b + 2a) |- -----
--R | b \| b + a
--R - |- ----- atan(----------------------------------------------------) + x
--R \| b + a 2 2
--R a sinh(x) + 2a cosh(x)sinh(x) + a cosh(x) + 2b + a
--R --------------------------------------------------------------------------]
--R a
--R Type: Union(List(Expression(Integer)),...)
--E 24
--S 25 of 526
m0403a:= a0403.1-r0403
--R
--R
--R (25)
--R +-----+
--R | b +-----+
--R |----- \|b + a
--R \|b + a
--R *
--R log
--R 2 2 2
--R (4a b + 4a )sinh(x) + (8a b + 8a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (4a b + 4a )cosh(x) + 8b + 12a b + 4a
--R *
--R +-----+
--R | b
--R |-----
--R \|b + a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 4a b + 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (8a b + 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2 2
--R (4a b + 2a )cosh(x) + 8b + 8a b + a
--R /
--R 4 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2
--R (6a cosh(x) + 4b + 2a)sinh(x)
--R +
--R 3 4
--R (4a cosh(x) + (8b + 4a)cosh(x))sinh(x) + a cosh(x)
--R +
--R 2
--R (4b + 2a)cosh(x) + a
--R +
--R +-+
--R +-+ tanh(x)\|b
--R 2\|b atanh(-----------)
--R +-----+
--R \|b + a
--R /
--R +-----+
--R 2a\|b + a
--R Type: Expression(Integer)
--E 25
--S 26 of 526
d0403a:= D(m0403a,x)
--R
--R
--R (26)
--R 4 3 2 2
--R b sinh(x) + 4b cosh(x)sinh(x) + (6b cosh(x) + 2b)sinh(x)
--R +
--R 3 4 2
--R (4b cosh(x) + 4b cosh(x))sinh(x) + b cosh(x) + 2b cosh(x) + b
--R *
--R 2
--R tanh(x)
--R +
--R 4 3 2 2
--R - b sinh(x) - 4b cosh(x)sinh(x) + (- 6b cosh(x) + 2b)sinh(x)
--R +
--R 3 4 2
--R (- 4b cosh(x) + 4b cosh(x))sinh(x) - b cosh(x) + 2b cosh(x) - b
--R /
--R 4 3
--R a b sinh(x) + 4a b cosh(x)sinh(x)
--R +
--R 2 2 2
--R (6a b cosh(x) + 4b + 2a b)sinh(x)
--R +
--R 3 2 4
--R (4a b cosh(x) + (8b + 4a b)cosh(x))sinh(x) + a b cosh(x)
--R +
--R 2 2
--R (4b + 2a b)cosh(x) + a b
--R *
--R 2
--R tanh(x)
--R +
--R 2 4 2 3
--R (- a b - a )sinh(x) + (- 4a b - 4a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((- 6a b - 6a )cosh(x) - 4b - 6a b - 2a )sinh(x)
--R +
--R 2 3 2 2
--R ((- 4a b - 4a )cosh(x) + (- 8b - 12a b - 4a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R (- a b - a )cosh(x) + (- 4b - 6a b - 2a )cosh(x) - a b - a
--R Type: Expression(Integer)
--E 26
--S 27 of 526
m0403b:= a0403.2-r0403
--R
--R
--R (27)
--R +-+
--R +-+ tanh(x)\|b
--R \|b atanh(-----------)
--R +-----+
--R \|b + a
--R +
--R -
--R +-------+
--R | b +-----+
--R |- ----- \|b + a
--R \| b + a
--R *
--R +-------+
--R | b
--R (2b + 2a) |- -----
--R \| b + a
--R atan(----------------------------------------------------)
--R 2 2
--R a sinh(x) + 2a cosh(x)sinh(x) + a cosh(x) + 2b + a
--R /
--R +-----+
--R a\|b + a
--R Type: Expression(Integer)
--E 27
--S 28 of 526
d0403b:= D(m0403b,x)
--R
--R
--R (28)
--R 4 3 2 2
--R b sinh(x) + 4b cosh(x)sinh(x) + (6b cosh(x) + 2b)sinh(x)
--R +
--R 3 4 2
--R (4b cosh(x) + 4b cosh(x))sinh(x) + b cosh(x) + 2b cosh(x) + b
--R *
--R 2
--R tanh(x)
--R +
--R 4 3 2 2
--R - b sinh(x) - 4b cosh(x)sinh(x) + (- 6b cosh(x) + 2b)sinh(x)
--R +
--R 3 4 2
--R (- 4b cosh(x) + 4b cosh(x))sinh(x) - b cosh(x) + 2b cosh(x) - b
--R /
--R 4 3
--R a b sinh(x) + 4a b cosh(x)sinh(x)
--R +
--R 2 2 2
--R (6a b cosh(x) + 4b + 2a b)sinh(x)
--R +
--R 3 2 4
--R (4a b cosh(x) + (8b + 4a b)cosh(x))sinh(x) + a b cosh(x)
--R +
--R 2 2
--R (4b + 2a b)cosh(x) + a b
--R *
--R 2
--R tanh(x)
--R +
--R 2 4 2 3
--R (- a b - a )sinh(x) + (- 4a b - 4a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((- 6a b - 6a )cosh(x) - 4b - 6a b - 2a )sinh(x)
--R +
--R 2 3 2 2
--R ((- 4a b - 4a )cosh(x) + (- 8b - 12a b - 4a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R (- a b - a )cosh(x) + (- 4b - 6a b - 2a )cosh(x) - a b - a
--R Type: Expression(Integer)
--E 28
--S 29 of 526
t0404:= 1/(a+b*sech(x)^2)^2
--R
--R
--R 1
--R (29) -------------------------------
--R 2 4 2 2
--R b sech(x) + 2a b sech(x) + a
--R Type: Expression(Integer)
--E 29
--S 30 of 526
r0404:= x/a^2-2*b^(1/2)*atanh(b^(1/2)*tanh(x)/(a+b)^(1/2))/a^2/(a+b)^(1/2)+_
1/2*b^(1/2)*(a+2*b)*atanh(b^(1/2)*tanh(x)/(a+b)^(1/2))/a^2/_
(a+b)^(3/2)-b*tanh(x)/a^2/(a+b*sech(x)^2)+1/2*b*(a+2*b)*_
tanh(x)/a^2/(a+b)/(a+b*sech(x)^2)
--R
--R
--R (30)
--R +-+
--R 2 2 2 +-+ tanh(x)\|b
--R ((- 2b - 3a b)sech(x) - 2a b - 3a )\|b atanh(-----------)
--R +-----+
--R \|b + a
--R +
--R 2 2 2 +-----+
--R (- a b tanh(x) + (2b + 2a b)x sech(x) + (2a b + 2a )x)\|b + a
--R /
--R 2 2 3 2 3 4 +-----+
--R ((2a b + 2a b)sech(x) + 2a b + 2a )\|b + a
--R Type: Expression(Integer)
--E 30
--S 31 of 526
a0404:= integrate(t0404,x)
--R
--R
--R (31)
--R [
--R 2 4 2 3
--R (2a b + 3a )sinh(x) + (8a b + 12a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((12a b + 18a )cosh(x) + 8b + 16a b + 6a )sinh(x)
--R +
--R 2 3 2 2
--R ((8a b + 12a )cosh(x) + (16b + 32a b + 12a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R (2a b + 3a )cosh(x) + (8b + 16a b + 6a )cosh(x) + 2a b + 3a
--R *
--R +-----+
--R | b
--R |-----
--R \|b + a
--R *
--R log
--R 2 2 2
--R (4a b + 4a )sinh(x) + (8a b + 8a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (4a b + 4a )cosh(x) + 8b + 12a b + 4a
--R *
--R +-----+
--R | b
--R |-----
--R \|b + a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 4a b + 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (8a b + 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2 2
--R (4a b + 2a )cosh(x) + 8b + 8a b + a
--R /
--R 4 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2
--R (6a cosh(x) + 4b + 2a)sinh(x)
--R +
--R 3 4
--R (4a cosh(x) + (8b + 4a)cosh(x))sinh(x) + a cosh(x)
--R +
--R 2
--R (4b + 2a)cosh(x) + a
--R +
--R 2 4 2 3
--R (4a b + 4a )x sinh(x) + (16a b + 16a )x cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 2
--R ((24a b + 24a )x cosh(x) + (16b + 24a b + 8a )x + 8b + 4a b)sinh(x)
--R +
--R 2 3
--R (16a b + 16a )x cosh(x)
--R +
--R 2 2 2
--R ((32b + 48a b + 16a )x + 16b + 8a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 2 4 2 2 2 2
--R (4a b + 4a )x cosh(x) + ((16b + 24a b + 8a )x + 8b + 4a b)cosh(x)
--R +
--R 2
--R (4a b + 4a )x + 4a b
--R /
--R 3 4 4 3 4 3
--R (4a b + 4a )sinh(x) + (16a b + 16a )cosh(x)sinh(x)
--R +
--R 3 4 2 2 2 3 4 2
--R ((24a b + 24a )cosh(x) + 16a b + 24a b + 8a )sinh(x)
--R +
--R 3 4 3 2 2 3 4
--R ((16a b + 16a )cosh(x) + (32a b + 48a b + 16a )cosh(x))sinh(x)
--R +
--R 3 4 4 2 2 3 4 2 3 4
--R (4a b + 4a )cosh(x) + (16a b + 24a b + 8a )cosh(x) + 4a b + 4a
--R ,
--R
--R 2 4 2 3
--R (- 2a b - 3a )sinh(x) + (- 8a b - 12a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((- 12a b - 18a )cosh(x) - 8b - 16a b - 6a )sinh(x)
--R +
--R 2 3 2 2
--R ((- 8a b - 12a )cosh(x) + (- 16b - 32a b - 12a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R (- 2a b - 3a )cosh(x) + (- 8b - 16a b - 6a )cosh(x) - 2a b - 3a
--R *
--R +-------+
--R | b
--R +-------+ (2b + 2a) |- -----
--R | b \| b + a
--R |- ----- atan(----------------------------------------------------)
--R \| b + a 2 2
--R a sinh(x) + 2a cosh(x)sinh(x) + a cosh(x) + 2b + a
--R +
--R 2 4 2 3
--R (2a b + 2a )x sinh(x) + (8a b + 8a )x cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 2
--R ((12a b + 12a )x cosh(x) + (8b + 12a b + 4a )x + 4b + 2a b)sinh(x)
--R +
--R 2 3
--R (8a b + 8a )x cosh(x)
--R +
--R 2 2 2
--R ((16b + 24a b + 8a )x + 8b + 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 2 4 2 2 2 2
--R (2a b + 2a )x cosh(x) + ((8b + 12a b + 4a )x + 4b + 2a b)cosh(x)
--R +
--R 2
--R (2a b + 2a )x + 2a b
--R /
--R 3 4 4 3 4 3
--R (2a b + 2a )sinh(x) + (8a b + 8a )cosh(x)sinh(x)
--R +
--R 3 4 2 2 2 3 4 2
--R ((12a b + 12a )cosh(x) + 8a b + 12a b + 4a )sinh(x)
--R +
--R 3 4 3 2 2 3 4
--R ((8a b + 8a )cosh(x) + (16a b + 24a b + 8a )cosh(x))sinh(x)
--R +
--R 3 4 4 2 2 3 4 2 3 4
--R (2a b + 2a )cosh(x) + (8a b + 12a b + 4a )cosh(x) + 2a b + 2a
--R ]
--R Type: Union(List(Expression(Integer)),...)
--E 31
--S 32 of 526
m0404a:= a0404.1-r0404
--R
--R
--R (32)
--R 2 2 2 2 3 4
--R ((2a b + 3a b)sech(x) + 2a b + 3a )sinh(x)
--R +
--R 2 2 2 2 3 3
--R ((8a b + 12a b)cosh(x)sech(x) + (8a b + 12a )cosh(x))sinh(x)
--R +
--R 2 2 2 3 2 2 2
--R ((12a b + 18a b)cosh(x) + 8b + 16a b + 6a b)sech(x)
--R +
--R 2 3 2 2 2 3
--R (12a b + 18a )cosh(x) + 8a b + 16a b + 6a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 3 2 2
--R ((8a b + 12a b)cosh(x) + (16b + 32a b + 12a b)cosh(x))
--R *
--R 2
--R sech(x)
--R +
--R 2 3 3 2 2 3
--R (8a b + 12a )cosh(x) + (16a b + 32a b + 12a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 4 3 2 2 2 2
--R (2a b + 3a b)cosh(x) + (8b + 16a b + 6a b)cosh(x) + 2a b
--R +
--R 2
--R 3a b
--R *
--R 2
--R sech(x)
--R +
--R 2 3 4 2 2 3 2 2 3
--R (2a b + 3a )cosh(x) + (8a b + 16a b + 6a )cosh(x) + 2a b + 3a
--R *
--R +-----+
--R | b +-----+
--R |----- \|b + a
--R \|b + a
--R *
--R log
--R 2 2 2
--R (4a b + 4a )sinh(x) + (8a b + 8a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (4a b + 4a )cosh(x) + 8b + 12a b + 4a
--R *
--R +-----+
--R | b
--R |-----
--R \|b + a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 4a b + 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (8a b + 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2 2
--R (4a b + 2a )cosh(x) + 8b + 8a b + a
--R /
--R 4 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2
--R (6a cosh(x) + 4b + 2a)sinh(x)
--R +
--R 3 4
--R (4a cosh(x) + (8b + 4a)cosh(x))sinh(x) + a cosh(x)
--R +
--R 2
--R (4b + 2a)cosh(x) + a
--R +
--R 2 2 2 2 3 4
--R ((4a b + 6a b)sech(x) + 4a b + 6a )sinh(x)
--R +
--R 2 2 2 2 3 3
--R ((16a b + 24a b)cosh(x)sech(x) + (16a b + 24a )cosh(x))sinh(x)
--R +
--R 2 2 2 3 2 2 2
--R ((24a b + 36a b)cosh(x) + 16b + 32a b + 12a b)sech(x)
--R +
--R 2 3 2 2 2 3
--R (24a b + 36a )cosh(x) + 16a b + 32a b + 12a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 3 2 2
--R ((16a b + 24a b)cosh(x) + (32b + 64a b + 24a b)cosh(x))
--R *
--R 2
--R sech(x)
--R +
--R 2 3 3 2 2 3
--R (16a b + 24a )cosh(x) + (32a b + 64a b + 24a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 4 3 2 2 2 2
--R (4a b + 6a b)cosh(x) + (16b + 32a b + 12a b)cosh(x) + 4a b
--R +
--R 2
--R 6a b
--R *
--R 2
--R sech(x)
--R +
--R 2 3 4 2 2 3 2 2 3
--R (4a b + 6a )cosh(x) + (16a b + 32a b + 12a )cosh(x) + 4a b + 6a
--R *
--R +-+
--R +-+ tanh(x)\|b
--R \|b atanh(-----------)
--R +-----+
--R \|b + a
--R +
--R 2 4 2 3
--R 2a b sinh(x) + 8a b cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R (12a b cosh(x) + 8a b + 4a b)sinh(x)
--R +
--R 2 3 2 2 2 4
--R (8a b cosh(x) + (16a b + 8a b)cosh(x))sinh(x) + 2a b cosh(x)
--R +
--R 2 2 2 2
--R (8a b + 4a b)cosh(x) + 2a b
--R *
--R tanh(x)
--R +
--R 3 2 2 2 2 2
--R ((8b + 4a b )sech(x) + 8a b + 4a b)sinh(x)
--R +
--R 3 2 2 2 2
--R ((16b + 8a b )cosh(x)sech(x) + (16a b + 8a b)cosh(x))sinh(x)
--R +
--R 3 2 2 2 2 2 2 2 2
--R ((8b + 4a b )cosh(x) + 4a b )sech(x) + (8a b + 4a b)cosh(x) + 4a b
--R *
--R +-----+
--R \|b + a
--R /
--R 3 2 4 2 4 5 4
--R ((4a b + 4a b)sech(x) + 4a b + 4a )sinh(x)
--R +
--R 3 2 4 2 4 5 3
--R ((16a b + 16a b)cosh(x)sech(x) + (16a b + 16a )cosh(x))sinh(x)
--R +
--R 3 2 4 2 2 3 3 2 4 2
--R ((24a b + 24a b)cosh(x) + 16a b + 24a b + 8a b)sech(x)
--R +
--R 4 5 2 3 2 4 5
--R (24a b + 24a )cosh(x) + 16a b + 24a b + 8a
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 4 3 2 3 3 2 4
--R ((16a b + 16a b)cosh(x) + (32a b + 48a b + 16a b)cosh(x))
--R *
--R 2
--R sech(x)
--R +
--R 4 5 3 3 2 4 5
--R (16a b + 16a )cosh(x) + (32a b + 48a b + 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 4 4 2 3 3 2 4 2 3 2
--R (4a b + 4a b)cosh(x) + (16a b + 24a b + 8a b)cosh(x) + 4a b
--R +
--R 4
--R 4a b
--R *
--R 2
--R sech(x)
--R +
--R 4 5 4 3 2 4 5 2 4 5
--R (4a b + 4a )cosh(x) + (16a b + 24a b + 8a )cosh(x) + 4a b + 4a
--R *
--R +-----+
--R \|b + a
--R Type: Expression(Integer)
--E 32
--S 33 of 526
d0404a:= D(m0404a,x)
--R
--R
--R (33)
--R 2 3 2 3 2 8
--R (a b sech(x) - a b )sinh(x)
--R +
--R 2 3 2 3 2 7
--R (8a b cosh(x)sech(x) - 8a b cosh(x))sinh(x)
--R +
--R 2 3 2 4 2 3 2 3 2 2 2 3
--R (28a b cosh(x) + 8a b + 4a b )sech(x) - 28a b cosh(x) - 8a b
--R +
--R 3 2
--R - 4a b
--R *
--R 6
--R sinh(x)
--R +
--R 2 3 3 4 2 3 2
--R (56a b cosh(x) + (48a b + 24a b )cosh(x))sech(x)
--R +
--R 3 2 3 2 3 3 2
--R - 56a b cosh(x) + (- 48a b - 24a b )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 3 4 4 2 3 2 5 4
--R 70a b cosh(x) + (120a b + 60a b )cosh(x) + 16b + 16a b
--R +
--R 2 3
--R 6a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 2 3 3 2 2 4 2 3
--R - 70a b cosh(x) + (- 120a b - 60a b )cosh(x) - 16a b - 16a b
--R +
--R 3 2
--R - 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 5 4 2 3 3
--R 56a b cosh(x) + (160a b + 80a b )cosh(x)
--R +
--R 5 4 2 3
--R (64b + 64a b + 24a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 5 2 3 3 2 3
--R - 56a b cosh(x) + (- 160a b - 80a b )cosh(x)
--R +
--R 4 2 3 3 2
--R (- 64a b - 64a b - 24a b )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 3 6 4 2 3 4
--R 28a b cosh(x) + (120a b + 60a b )cosh(x)
--R +
--R 5 4 2 3 2 4 2 3
--R (96b + 96a b + 36a b )cosh(x) + 8a b + 4a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 6 2 3 3 2 4
--R - 28a b cosh(x) + (- 120a b - 60a b )cosh(x)
--R +
--R 4 2 3 3 2 2 2 3 3 2
--R (- 96a b - 96a b - 36a b )cosh(x) - 8a b - 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 7 4 2 3 5
--R 8a b cosh(x) + (48a b + 24a b )cosh(x)
--R +
--R 5 4 2 3 3 4 2 3
--R (64b + 64a b + 24a b )cosh(x) + (16a b + 8a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 7 2 3 3 2 5
--R - 8a b cosh(x) + (- 48a b - 24a b )cosh(x)
--R +
--R 4 2 3 3 2 3 2 3 3 2
--R (- 64a b - 64a b - 24a b )cosh(x) + (- 16a b - 8a b )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 8 4 2 3 6
--R a b cosh(x) + (8a b + 4a b )cosh(x)
--R +
--R 5 4 2 3 4 4 2 3 2 2 3
--R (16b + 16a b + 6a b )cosh(x) + (8a b + 4a b )cosh(x) + a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 8 2 3 3 2 6
--R - a b cosh(x) + (- 8a b - 4a b )cosh(x)
--R +
--R 4 2 3 3 2 4 2 3 3 2 2 3 2
--R (- 16a b - 16a b - 6a b )cosh(x) + (- 8a b - 4a b )cosh(x) - a b
--R *
--R 4
--R tanh(x)
--R +
--R 4 2 3 4 2 3 3 2 2 3 2 4
--R ((2a b + 3a b )sech(x) + (4a b + 5a b )sech(x) + 4a b + 4a b)
--R *
--R 8
--R sinh(x)
--R +
--R 4 2 3 4
--R (16a b + 24a b )cosh(x)sech(x)
--R +
--R 2 3 3 2 2 3 2 4
--R (32a b + 40a b )cosh(x)sech(x) + (32a b + 32a b)cosh(x)
--R *
--R 7
--R sinh(x)
--R +
--R 4 2 3 2 4 2 3 4
--R ((56a b + 84a b )cosh(x) + 16a b + 12a b )sech(x)
--R +
--R 2 3 3 2 2 2 3 3 2 2
--R ((112a b + 140a b )cosh(x) + 24a b + 20a b )sech(x)
--R +
--R 3 2 4 2 2 3 3 2 4
--R (112a b + 112a b)cosh(x) + 16a b + 32a b + 16a b
--R *
--R 6
--R sinh(x)
--R +
--R 4 2 3 3 4 2 3 4
--R ((112a b + 168a b )cosh(x) + (96a b + 72a b )cosh(x))sech(x)
--R +
--R 2 3 3 2 3 2 3 3 2
--R ((224a b + 280a b )cosh(x) + (144a b + 120a b )cosh(x))
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 3 2 3 3 2 4
--R (224a b + 224a b)cosh(x) + (96a b + 192a b + 96a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 4 2 3 4 4 2 3 2
--R (140a b + 210a b )cosh(x) + (240a b + 180a b )cosh(x)
--R +
--R 5 4 2 3
--R 16b + 28a b + 18a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 4 2 3 3 2 2
--R (280a b + 350a b )cosh(x) + (360a b + 300a b )cosh(x)
--R +
--R 4 2 3 3 2
--R 16a b + 40a b + 30a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 4 2 3 3 2 4 2
--R (280a b + 280a b)cosh(x) + (240a b + 480a b + 240a b)cosh(x)
--R +
--R 4 2 3 3 2 4
--R 32a b + 64a b + 56a b + 24a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 2 3 5 4 2 3 3
--R (112a b + 168a b )cosh(x) + (320a b + 240a b )cosh(x)
--R +
--R 5 4 2 3
--R (64b + 112a b + 72a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 5 2 3 3 2 3
--R (224a b + 280a b )cosh(x) + (480a b + 400a b )cosh(x)
--R +
--R 4 2 3 3 2
--R (64a b + 160a b + 120a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 5 2 3 3 2 4 3
--R (224a b + 224a b)cosh(x) + (320a b + 640a b + 320a b)cosh(x)
--R +
--R 4 2 3 3 2 4
--R (128a b + 256a b + 224a b + 96a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 3 6 4 2 3 4
--R (56a b + 84a b )cosh(x) + (240a b + 180a b )cosh(x)
--R +
--R 5 4 2 3 2 4 2 3
--R (96b + 168a b + 108a b )cosh(x) + 16a b + 12a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 6 2 3 3 2 4
--R (112a b + 140a b )cosh(x) + (360a b + 300a b )cosh(x)
--R +
--R 4 2 3 3 2 2 2 3 3 2
--R (96a b + 240a b + 180a b )cosh(x) + 24a b + 20a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 6 2 3 3 2 4 4
--R (112a b + 112a b)cosh(x) + (240a b + 480a b + 240a b)cosh(x)
--R +
--R 4 2 3 3 2 4 2 2 3 3 2
--R (192a b + 384a b + 336a b + 144a b)cosh(x) + 16a b + 32a b
--R +
--R 4
--R 16a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 2 3 7 4 2 3 5
--R (16a b + 24a b )cosh(x) + (96a b + 72a b )cosh(x)
--R +
--R 5 4 2 3 3 4 2 3
--R (64b + 112a b + 72a b )cosh(x) + (32a b + 24a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 7 2 3 3 2 5
--R (32a b + 40a b )cosh(x) + (144a b + 120a b )cosh(x)
--R +
--R 4 2 3 3 2 3 2 3 3 2
--R (64a b + 160a b + 120a b )cosh(x) + (48a b + 40a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 7 2 3 3 2 4 5
--R (32a b + 32a b)cosh(x) + (96a b + 192a b + 96a b)cosh(x)
--R +
--R 4 2 3 3 2 4 3
--R (128a b + 256a b + 224a b + 96a b)cosh(x)
--R +
--R 2 3 3 2 4
--R (32a b + 64a b + 32a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 4 2 3 8 4 2 3 6
--R (2a b + 3a b )cosh(x) + (16a b + 12a b )cosh(x)
--R +
--R 5 4 2 3 4 4 2 3 2
--R (16b + 28a b + 18a b )cosh(x) + (16a b + 12a b )cosh(x)
--R +
--R 4 2 3
--R 2a b + 3a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 8 2 3 3 2 6
--R (4a b + 5a b )cosh(x) + (24a b + 20a b )cosh(x)
--R +
--R 4 2 3 3 2 4 2 3 3 2 2
--R (16a b + 40a b + 30a b )cosh(x) + (24a b + 20a b )cosh(x)
--R +
--R 2 3 3 2
--R 4a b + 5a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 8 2 3 3 2 4 6
--R (4a b + 4a b)cosh(x) + (16a b + 32a b + 16a b)cosh(x)
--R +
--R 4 2 3 3 2 4 4
--R (32a b + 64a b + 56a b + 24a b)cosh(x)
--R +
--R 2 3 3 2 4 2 3 2 4
--R (16a b + 32a b + 16a b)cosh(x) + 4a b + 4a b
--R *
--R 2
--R tanh(x)
--R +
--R 4 2 3 4 2 3 3 2 2 3 2 4
--R ((- 2a b - 3a b )sech(x) + (- 5a b - 7a b )sech(x) - 3a b - 4a b)
--R *
--R 8
--R sinh(x)
--R +
--R 4 2 3 4
--R (- 16a b - 24a b )cosh(x)sech(x)
--R +
--R 2 3 3 2 2 3 2 4
--R (- 40a b - 56a b )cosh(x)sech(x) + (- 24a b - 32a b)cosh(x)
--R *
--R 7
--R sinh(x)
--R +
--R 4 2 3 2 2 3 4
--R ((- 56a b - 84a b )cosh(x) + 4a b )sech(x)
--R +
--R 2 3 3 2 2 4 2 3 3 2 2
--R ((- 140a b - 196a b )cosh(x) - 8a b - 12a b + 4a b )sech(x)
--R +
--R 3 2 4 2 2 3 3 2
--R (- 84a b - 112a b)cosh(x) - 8a b - 12a b
--R *
--R 6
--R sinh(x)
--R +
--R 4 2 3 3 2 3 4
--R ((- 112a b - 168a b )cosh(x) + 24a b cosh(x))sech(x)
--R +
--R 2 3 3 2 3
--R (- 280a b - 392a b )cosh(x)
--R +
--R 4 2 3 3 2
--R (- 48a b - 72a b + 24a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 3 2 3 3 2
--R (- 168a b - 224a b)cosh(x) + (- 48a b - 72a b )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 4 2 3 4 2 3 2 5 4
--R (- 140a b - 210a b )cosh(x) + 60a b cosh(x) + 16b + 36a b
--R +
--R 2 3
--R 14a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 4
--R (- 350a b - 490a b )cosh(x)
--R +
--R 4 2 3 3 2 2 5 2 3 3 2
--R (- 120a b - 180a b + 60a b )cosh(x) - 16b + 50a b + 22a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 4 2 3 3 2 2 4
--R (- 210a b - 280a b)cosh(x) + (- 120a b - 180a b )cosh(x) - 16a b
--R +
--R 2 3 3 2 4
--R - 16a b + 14a b + 8a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 2 3 5 2 3 3
--R (- 112a b - 168a b )cosh(x) + 80a b cosh(x)
--R +
--R 5 4 2 3
--R (64b + 144a b + 56a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 5
--R (- 280a b - 392a b )cosh(x)
--R +
--R 4 2 3 3 2 3
--R (- 160a b - 240a b + 80a b )cosh(x)
--R +
--R 5 2 3 3 2
--R (- 64b + 200a b + 88a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 5 2 3 3 2 3
--R (- 168a b - 224a b)cosh(x) + (- 160a b - 240a b )cosh(x)
--R +
--R 4 2 3 3 2 4
--R (- 64a b - 64a b + 56a b + 32a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 3 6 2 3 4
--R (- 56a b - 84a b )cosh(x) + 60a b cosh(x)
--R +
--R 5 4 2 3 2 2 3
--R (96b + 216a b + 84a b )cosh(x) + 4a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 6
--R (- 140a b - 196a b )cosh(x)
--R +
--R 4 2 3 3 2 4
--R (- 120a b - 180a b + 60a b )cosh(x)
--R +
--R 5 2 3 3 2 2 4 2 3 3 2
--R (- 96b + 300a b + 132a b )cosh(x) - 8a b - 12a b + 4a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 6 2 3 3 2 4
--R (- 84a b - 112a b)cosh(x) + (- 120a b - 180a b )cosh(x)
--R +
--R 4 2 3 3 2 4 2 2 3 3 2
--R (- 96a b - 96a b + 84a b + 48a b)cosh(x) - 8a b - 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 2 3 7 2 3 5
--R (- 16a b - 24a b )cosh(x) + 24a b cosh(x)
--R +
--R 5 4 2 3 3 2 3
--R (64b + 144a b + 56a b )cosh(x) + 8a b cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 7
--R (- 40a b - 56a b )cosh(x)
--R +
--R 4 2 3 3 2 5
--R (- 48a b - 72a b + 24a b )cosh(x)
--R +
--R 5 2 3 3 2 3
--R (- 64b + 200a b + 88a b )cosh(x)
--R +
--R 4 2 3 3 2
--R (- 16a b - 24a b + 8a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 7 2 3 3 2 5
--R (- 24a b - 32a b)cosh(x) + (- 48a b - 72a b )cosh(x)
--R +
--R 4 2 3 3 2 4 3
--R (- 64a b - 64a b + 56a b + 32a b)cosh(x)
--R +
--R 2 3 3 2
--R (- 16a b - 24a b )cosh(x)
--R *
--R sinh(x)
--R +
--R 4 2 3 8 2 3 6
--R (- 2a b - 3a b )cosh(x) + 4a b cosh(x)
--R +
--R 5 4 2 3 4 2 3 2 4 2 3
--R (16b + 36a b + 14a b )cosh(x) + 4a b cosh(x) - 2a b - 3a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 8 4 2 3 3 2 6
--R (- 5a b - 7a b )cosh(x) + (- 8a b - 12a b + 4a b )cosh(x)
--R +
--R 5 2 3 3 2 4
--R (- 16b + 50a b + 22a b )cosh(x)
--R +
--R 4 2 3 3 2 2 2 3 3 2
--R (- 8a b - 12a b + 4a b )cosh(x) - 5a b - 7a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 8 2 3 3 2 6
--R (- 3a b - 4a b)cosh(x) + (- 8a b - 12a b )cosh(x)
--R +
--R 4 2 3 3 2 4 4 2 3 3 2 2
--R (- 16a b - 16a b + 14a b + 8a b)cosh(x) + (- 8a b - 12a b )cosh(x)
--R +
--R 3 2 4
--R - 3a b - 4a b
--R /
--R 3 4 4 3 4 4 3 5 2 2 5 2 6
--R ((2a b + 2a b )sech(x) + (4a b + 4a b )sech(x) + 2a b + 2a b)
--R *
--R 8
--R sinh(x)
--R +
--R 3 4 4 3 4
--R (16a b + 16a b )cosh(x)sech(x)
--R +
--R 4 3 5 2 2 5 2 6
--R (32a b + 32a b )cosh(x)sech(x) + (16a b + 16a b)cosh(x)
--R *
--R 7
--R sinh(x)
--R +
--R 3 4 4 3 2 2 5 3 4 4 3 4
--R ((56a b + 56a b )cosh(x) + 16a b + 24a b + 8a b )sech(x)
--R +
--R 4 3 5 2 2 3 4 4 3 5 2 2
--R ((112a b + 112a b )cosh(x) + 32a b + 48a b + 16a b )sech(x)
--R +
--R 5 2 6 2 4 3 5 2 6
--R (56a b + 56a b)cosh(x) + 16a b + 24a b + 8a b
--R *
--R 6
--R sinh(x)
--R +
--R 3 4 4 3 3
--R (112a b + 112a b )cosh(x)
--R +
--R 2 5 3 4 4 3
--R (96a b + 144a b + 48a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 3
--R (224a b + 224a b )cosh(x)
--R +
--R 3 4 4 3 5 2
--R (192a b + 288a b + 96a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 3 4 3 5 2 6
--R (112a b + 112a b)cosh(x) + (96a b + 144a b + 48a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 4 4 3 4
--R (140a b + 140a b )cosh(x)
--R +
--R 2 5 3 4 4 3 2 6 2 5
--R (240a b + 360a b + 120a b )cosh(x) + 32a b + 64a b
--R +
--R 3 4 4 3
--R 44a b + 12a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 4
--R (280a b + 280a b )cosh(x)
--R +
--R 3 4 4 3 5 2 2 2 5 3 4
--R (480a b + 720a b + 240a b )cosh(x) + 64a b + 128a b
--R +
--R 4 3 5 2
--R 88a b + 24a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 4 4 3 5 2 6 2
--R (140a b + 140a b)cosh(x) + (240a b + 360a b + 120a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R 32a b + 64a b + 44a b + 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 4 4 3 5
--R (112a b + 112a b )cosh(x)
--R +
--R 2 5 3 4 4 3 3
--R (320a b + 480a b + 160a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3
--R (128a b + 256a b + 176a b + 48a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 5
--R (224a b + 224a b )cosh(x)
--R +
--R 3 4 4 3 5 2 3
--R (640a b + 960a b + 320a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2
--R (256a b + 512a b + 352a b + 96a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 5 4 3 5 2 6 3
--R (112a b + 112a b)cosh(x) + (320a b + 480a b + 160a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R (128a b + 256a b + 176a b + 48a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 4 4 3 6
--R (56a b + 56a b )cosh(x)
--R +
--R 2 5 3 4 4 3 4
--R (240a b + 360a b + 120a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 2 2 5
--R (192a b + 384a b + 264a b + 72a b )cosh(x) + 16a b
--R +
--R 3 4 4 3
--R 24a b + 8a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6
--R (112a b + 112a b )cosh(x)
--R +
--R 3 4 4 3 5 2 4
--R (480a b + 720a b + 240a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 2 3 4
--R (384a b + 768a b + 528a b + 144a b )cosh(x) + 32a b
--R +
--R 4 3 5 2
--R 48a b + 16a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 6 4 3 5 2 6 4
--R (56a b + 56a b)cosh(x) + (240a b + 360a b + 120a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 2 4 3 5 2
--R (192a b + 384a b + 264a b + 72a b)cosh(x) + 16a b + 24a b
--R +
--R 6
--R 8a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 4 4 3 7
--R (16a b + 16a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5
--R (96a b + 144a b + 48a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 3
--R (128a b + 256a b + 176a b + 48a b )cosh(x)
--R +
--R 2 5 3 4 4 3
--R (32a b + 48a b + 16a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 7
--R (32a b + 32a b )cosh(x)
--R +
--R 3 4 4 3 5 2 5
--R (192a b + 288a b + 96a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 3
--R (256a b + 512a b + 352a b + 96a b )cosh(x)
--R +
--R 3 4 4 3 5 2
--R (64a b + 96a b + 32a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 4 3 5 2 6 5
--R (16a b + 16a b)cosh(x) + (96a b + 144a b + 48a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 3
--R (128a b + 256a b + 176a b + 48a b)cosh(x)
--R +
--R 4 3 5 2 6
--R (32a b + 48a b + 16a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 4 4 3 8 2 5 3 4 4 3 6
--R (2a b + 2a b )cosh(x) + (16a b + 24a b + 8a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 4
--R (32a b + 64a b + 44a b + 12a b )cosh(x)
--R +
--R 2 5 3 4 4 3 2 3 4 4 3
--R (16a b + 24a b + 8a b )cosh(x) + 2a b + 2a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 8 3 4 4 3 5 2 6
--R (4a b + 4a b )cosh(x) + (32a b + 48a b + 16a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 4
--R (64a b + 128a b + 88a b + 24a b )cosh(x)
--R +
--R 3 4 4 3 5 2 2 4 3 5 2
--R (32a b + 48a b + 16a b )cosh(x) + 4a b + 4a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 8 4 3 5 2 6 6
--R (2a b + 2a b)cosh(x) + (16a b + 24a b + 8a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 4
--R (32a b + 64a b + 44a b + 12a b)cosh(x)
--R +
--R 4 3 5 2 6 2 5 2 6
--R (16a b + 24a b + 8a b)cosh(x) + 2a b + 2a b
--R *
--R 2
--R tanh(x)
--R +
--R 3 4 4 3 5 2 4 4 3 5 2 6 2
--R (- 2a b - 4a b - 2a b )sech(x) + (- 4a b - 8a b - 4a b)sech(x)
--R +
--R 5 2 6 7
--R - 2a b - 4a b - 2a
--R *
--R 8
--R sinh(x)
--R +
--R 3 4 4 3 5 2 4
--R (- 16a b - 32a b - 16a b )cosh(x)sech(x)
--R +
--R 4 3 5 2 6 2
--R (- 32a b - 64a b - 32a b)cosh(x)sech(x)
--R +
--R 5 2 6 7
--R (- 16a b - 32a b - 16a )cosh(x)
--R *
--R 7
--R sinh(x)
--R +
--R 3 4 4 3 5 2 2 2 5 3 4 4 3
--R (- 56a b - 112a b - 56a b )cosh(x) - 16a b - 40a b - 32a b
--R +
--R 5 2
--R - 8a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 2 3 4 4 3 5 2
--R (- 112a b - 224a b - 112a b)cosh(x) - 32a b - 80a b - 64a b
--R +
--R 6
--R - 16a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 2 4 3 5 2 6 7
--R (- 56a b - 112a b - 56a )cosh(x) - 16a b - 40a b - 32a b - 8a
--R *
--R 6
--R sinh(x)
--R +
--R 3 4 4 3 5 2 3
--R (- 112a b - 224a b - 112a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2
--R (- 96a b - 240a b - 192a b - 48a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 3
--R (- 224a b - 448a b - 224a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R (- 192a b - 480a b - 384a b - 96a b)cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 3
--R (- 112a b - 224a b - 112a )cosh(x)
--R +
--R 4 3 5 2 6 7
--R (- 96a b - 240a b - 192a b - 48a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 4 4 3 5 2 4
--R (- 140a b - 280a b - 140a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 2 6
--R (- 240a b - 600a b - 480a b - 120a b )cosh(x) - 32a b
--R +
--R 2 5 3 4 4 3 5 2
--R - 96a b - 108a b - 56a b - 12a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 4
--R (- 280a b - 560a b - 280a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 2 2 5
--R (- 480a b - 1200a b - 960a b - 240a b)cosh(x) - 64a b
--R +
--R 3 4 4 3 5 2 6
--R - 192a b - 216a b - 112a b - 24a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 4
--R (- 140a b - 280a b - 140a )cosh(x)
--R +
--R 4 3 5 2 6 7 2 3 4 4 3
--R (- 240a b - 600a b - 480a b - 120a )cosh(x) - 32a b - 96a b
--R +
--R 5 2 6 7
--R - 108a b - 56a b - 12a
--R *
--R 4
--R sinh(x)
--R +
--R 3 4 4 3 5 2 5
--R (- 112a b - 224a b - 112a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 3
--R (- 320a b - 800a b - 640a b - 160a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 5 2
--R (- 128a b - 384a b - 432a b - 224a b - 48a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 5
--R (- 224a b - 448a b - 224a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 3
--R (- 640a b - 1600a b - 1280a b - 320a b)cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6
--R (- 256a b - 768a b - 864a b - 448a b - 96a b)cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 5
--R (- 112a b - 224a b - 112a )cosh(x)
--R +
--R 4 3 5 2 6 7 3
--R (- 320a b - 800a b - 640a b - 160a )cosh(x)
--R +
--R 3 4 4 3 5 2 6 7
--R (- 128a b - 384a b - 432a b - 224a b - 48a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 4 4 3 5 2 6
--R (- 56a b - 112a b - 56a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 4
--R (- 240a b - 600a b - 480a b - 120a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 5 2 2
--R (- 192a b - 576a b - 648a b - 336a b - 72a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2
--R - 16a b - 40a b - 32a b - 8a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 6
--R (- 112a b - 224a b - 112a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 4
--R (- 480a b - 1200a b - 960a b - 240a b)cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6 2
--R (- 384a b - 1152a b - 1296a b - 672a b - 144a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R - 32a b - 80a b - 64a b - 16a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 6
--R (- 56a b - 112a b - 56a )cosh(x)
--R +
--R 4 3 5 2 6 7 4
--R (- 240a b - 600a b - 480a b - 120a )cosh(x)
--R +
--R 3 4 4 3 5 2 6 7 2 4 3
--R (- 192a b - 576a b - 648a b - 336a b - 72a )cosh(x) - 16a b
--R +
--R 5 2 6 7
--R - 40a b - 32a b - 8a
--R *
--R 2
--R sinh(x)
--R +
--R 3 4 4 3 5 2 7
--R (- 16a b - 32a b - 16a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 5
--R (- 96a b - 240a b - 192a b - 48a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 5 2 3
--R (- 128a b - 384a b - 432a b - 224a b - 48a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2
--R (- 32a b - 80a b - 64a b - 16a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 7
--R (- 32a b - 64a b - 32a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 5
--R (- 192a b - 480a b - 384a b - 96a b)cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6 3
--R (- 256a b - 768a b - 864a b - 448a b - 96a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R (- 64a b - 160a b - 128a b - 32a b)cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 7
--R (- 16a b - 32a b - 16a )cosh(x)
--R +
--R 4 3 5 2 6 7 5
--R (- 96a b - 240a b - 192a b - 48a )cosh(x)
--R +
--R 3 4 4 3 5 2 6 7 3
--R (- 128a b - 384a b - 432a b - 224a b - 48a )cosh(x)
--R +
--R 4 3 5 2 6 7
--R (- 32a b - 80a b - 64a b - 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 4 4 3 5 2 8
--R (- 2a b - 4a b - 2a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6
--R (- 16a b - 40a b - 32a b - 8a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 5 2 4
--R (- 32a b - 96a b - 108a b - 56a b - 12a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 2 3 4 4 3 5 2
--R (- 16a b - 40a b - 32a b - 8a b )cosh(x) - 2a b - 4a b - 2a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 8
--R (- 4a b - 8a b - 4a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 6
--R (- 32a b - 80a b - 64a b - 16a b)cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6 4
--R (- 64a b - 192a b - 216a b - 112a b - 24a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 2 4 3 5 2 6
--R (- 32a b - 80a b - 64a b - 16a b)cosh(x) - 4a b - 8a b - 4a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 8
--R (- 2a b - 4a b - 2a )cosh(x)
--R +
--R 4 3 5 2 6 7 6
--R (- 16a b - 40a b - 32a b - 8a )cosh(x)
--R +
--R 3 4 4 3 5 2 6 7 4
--R (- 32a b - 96a b - 108a b - 56a b - 12a )cosh(x)
--R +
--R 4 3 5 2 6 7 2 5 2 6 7
--R (- 16a b - 40a b - 32a b - 8a )cosh(x) - 2a b - 4a b - 2a
--R Type: Expression(Integer)
--E 33
--S 34 of 526
m0404b:= a0404.2-r0404
--R
--R
--R (34)
--R 2 2 2 2 3 4
--R ((2a b + 3a b)sech(x) + 2a b + 3a )sinh(x)
--R +
--R 2 2 2 2 3 3
--R ((8a b + 12a b)cosh(x)sech(x) + (8a b + 12a )cosh(x))sinh(x)
--R +
--R 2 2 2 3 2 2 2
--R ((12a b + 18a b)cosh(x) + 8b + 16a b + 6a b)sech(x)
--R +
--R 2 3 2 2 2 3
--R (12a b + 18a )cosh(x) + 8a b + 16a b + 6a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 3 2 2
--R ((8a b + 12a b)cosh(x) + (16b + 32a b + 12a b)cosh(x))
--R *
--R 2
--R sech(x)
--R +
--R 2 3 3 2 2 3
--R (8a b + 12a )cosh(x) + (16a b + 32a b + 12a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 4 3 2 2 2 2
--R (2a b + 3a b)cosh(x) + (8b + 16a b + 6a b)cosh(x) + 2a b
--R +
--R 2
--R 3a b
--R *
--R 2
--R sech(x)
--R +
--R 2 3 4 2 2 3 2 2 3
--R (2a b + 3a )cosh(x) + (8a b + 16a b + 6a )cosh(x) + 2a b + 3a
--R *
--R +-+
--R +-+ tanh(x)\|b
--R \|b atanh(-----------)
--R +-----+
--R \|b + a
--R +
--R 2 2 2 2 3 4
--R ((- 2a b - 3a b)sech(x) - 2a b - 3a )sinh(x)
--R +
--R 2 2 2 2 3 3
--R ((- 8a b - 12a b)cosh(x)sech(x) + (- 8a b - 12a )cosh(x))sinh(x)
--R +
--R 2 2 2 3 2 2 2
--R ((- 12a b - 18a b)cosh(x) - 8b - 16a b - 6a b)sech(x)
--R +
--R 2 3 2 2 2 3
--R (- 12a b - 18a )cosh(x) - 8a b - 16a b - 6a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 3 2 2
--R ((- 8a b - 12a b)cosh(x) + (- 16b - 32a b - 12a b)cosh(x))
--R *
--R 2
--R sech(x)
--R +
--R 2 3 3 2 2 3
--R (- 8a b - 12a )cosh(x) + (- 16a b - 32a b - 12a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 4 3 2 2 2
--R (- 2a b - 3a b)cosh(x) + (- 8b - 16a b - 6a b)cosh(x)
--R +
--R 2 2
--R - 2a b - 3a b
--R *
--R 2
--R sech(x)
--R +
--R 2 3 4 2 2 3 2 2 3
--R (- 2a b - 3a )cosh(x) + (- 8a b - 16a b - 6a )cosh(x) - 2a b - 3a
--R *
--R +-------+
--R | b +-----+
--R |- ----- \|b + a
--R \| b + a
--R *
--R +-------+
--R | b
--R (2b + 2a) |- -----
--R \| b + a
--R atan(----------------------------------------------------)
--R 2 2
--R a sinh(x) + 2a cosh(x)sinh(x) + a cosh(x) + 2b + a
--R +
--R 2 4 2 3
--R a b sinh(x) + 4a b cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R (6a b cosh(x) + 4a b + 2a b)sinh(x)
--R +
--R 2 3 2 2 2 4
--R (4a b cosh(x) + (8a b + 4a b)cosh(x))sinh(x) + a b cosh(x)
--R +
--R 2 2 2 2
--R (4a b + 2a b)cosh(x) + a b
--R *
--R tanh(x)
--R +
--R 3 2 2 2 2 2
--R ((4b + 2a b )sech(x) + 4a b + 2a b)sinh(x)
--R +
--R 3 2 2 2 2
--R ((8b + 4a b )cosh(x)sech(x) + (8a b + 4a b)cosh(x))sinh(x)
--R +
--R 3 2 2 2 2 2 2 2 2
--R ((4b + 2a b )cosh(x) + 2a b )sech(x) + (4a b + 2a b)cosh(x) + 2a b
--R *
--R +-----+
--R \|b + a
--R /
--R 3 2 4 2 4 5 4
--R ((2a b + 2a b)sech(x) + 2a b + 2a )sinh(x)
--R +
--R 3 2 4 2 4 5 3
--R ((8a b + 8a b)cosh(x)sech(x) + (8a b + 8a )cosh(x))sinh(x)
--R +
--R 3 2 4 2 2 3 3 2 4 2
--R ((12a b + 12a b)cosh(x) + 8a b + 12a b + 4a b)sech(x)
--R +
--R 4 5 2 3 2 4 5
--R (12a b + 12a )cosh(x) + 8a b + 12a b + 4a
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 4 3 2 3 3 2 4 2
--R ((8a b + 8a b)cosh(x) + (16a b + 24a b + 8a b)cosh(x))sech(x)
--R +
--R 4 5 3 3 2 4 5
--R (8a b + 8a )cosh(x) + (16a b + 24a b + 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 4 4 2 3 3 2 4 2 3 2
--R (2a b + 2a b)cosh(x) + (8a b + 12a b + 4a b)cosh(x) + 2a b
--R +
--R 4
--R 2a b
--R *
--R 2
--R sech(x)
--R +
--R 4 5 4 3 2 4 5 2 4 5
--R (2a b + 2a )cosh(x) + (8a b + 12a b + 4a )cosh(x) + 2a b + 2a
--R *
--R +-----+
--R \|b + a
--R Type: Expression(Integer)
--E 34
--S 35 of 526
d0404b:= D(m0404b,x)
--R
--R
--R (35)
--R 2 3 2 3 2 8
--R (a b sech(x) - a b )sinh(x)
--R +
--R 2 3 2 3 2 7
--R (8a b cosh(x)sech(x) - 8a b cosh(x))sinh(x)
--R +
--R 2 3 2 4 2 3 2 3 2 2 2 3
--R (28a b cosh(x) + 8a b + 4a b )sech(x) - 28a b cosh(x) - 8a b
--R +
--R 3 2
--R - 4a b
--R *
--R 6
--R sinh(x)
--R +
--R 2 3 3 4 2 3 2
--R (56a b cosh(x) + (48a b + 24a b )cosh(x))sech(x)
--R +
--R 3 2 3 2 3 3 2
--R - 56a b cosh(x) + (- 48a b - 24a b )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 3 4 4 2 3 2 5 4
--R 70a b cosh(x) + (120a b + 60a b )cosh(x) + 16b + 16a b
--R +
--R 2 3
--R 6a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 2 3 3 2 2 4 2 3
--R - 70a b cosh(x) + (- 120a b - 60a b )cosh(x) - 16a b - 16a b
--R +
--R 3 2
--R - 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 5 4 2 3 3
--R 56a b cosh(x) + (160a b + 80a b )cosh(x)
--R +
--R 5 4 2 3
--R (64b + 64a b + 24a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 5 2 3 3 2 3
--R - 56a b cosh(x) + (- 160a b - 80a b )cosh(x)
--R +
--R 4 2 3 3 2
--R (- 64a b - 64a b - 24a b )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 3 6 4 2 3 4
--R 28a b cosh(x) + (120a b + 60a b )cosh(x)
--R +
--R 5 4 2 3 2 4 2 3
--R (96b + 96a b + 36a b )cosh(x) + 8a b + 4a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 6 2 3 3 2 4
--R - 28a b cosh(x) + (- 120a b - 60a b )cosh(x)
--R +
--R 4 2 3 3 2 2 2 3 3 2
--R (- 96a b - 96a b - 36a b )cosh(x) - 8a b - 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 7 4 2 3 5
--R 8a b cosh(x) + (48a b + 24a b )cosh(x)
--R +
--R 5 4 2 3 3 4 2 3
--R (64b + 64a b + 24a b )cosh(x) + (16a b + 8a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 7 2 3 3 2 5
--R - 8a b cosh(x) + (- 48a b - 24a b )cosh(x)
--R +
--R 4 2 3 3 2 3 2 3 3 2
--R (- 64a b - 64a b - 24a b )cosh(x) + (- 16a b - 8a b )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 8 4 2 3 6
--R a b cosh(x) + (8a b + 4a b )cosh(x)
--R +
--R 5 4 2 3 4 4 2 3 2 2 3
--R (16b + 16a b + 6a b )cosh(x) + (8a b + 4a b )cosh(x) + a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 8 2 3 3 2 6
--R - a b cosh(x) + (- 8a b - 4a b )cosh(x)
--R +
--R 4 2 3 3 2 4 2 3 3 2 2 3 2
--R (- 16a b - 16a b - 6a b )cosh(x) + (- 8a b - 4a b )cosh(x) - a b
--R *
--R 4
--R tanh(x)
--R +
--R 4 2 3 4 2 3 3 2 2 3 2 4
--R ((2a b + 3a b )sech(x) + (4a b + 5a b )sech(x) + 4a b + 4a b)
--R *
--R 8
--R sinh(x)
--R +
--R 4 2 3 4
--R (16a b + 24a b )cosh(x)sech(x)
--R +
--R 2 3 3 2 2 3 2 4
--R (32a b + 40a b )cosh(x)sech(x) + (32a b + 32a b)cosh(x)
--R *
--R 7
--R sinh(x)
--R +
--R 4 2 3 2 4 2 3 4
--R ((56a b + 84a b )cosh(x) + 16a b + 12a b )sech(x)
--R +
--R 2 3 3 2 2 2 3 3 2 2
--R ((112a b + 140a b )cosh(x) + 24a b + 20a b )sech(x)
--R +
--R 3 2 4 2 2 3 3 2 4
--R (112a b + 112a b)cosh(x) + 16a b + 32a b + 16a b
--R *
--R 6
--R sinh(x)
--R +
--R 4 2 3 3 4 2 3 4
--R ((112a b + 168a b )cosh(x) + (96a b + 72a b )cosh(x))sech(x)
--R +
--R 2 3 3 2 3 2 3 3 2
--R ((224a b + 280a b )cosh(x) + (144a b + 120a b )cosh(x))
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 3 2 3 3 2 4
--R (224a b + 224a b)cosh(x) + (96a b + 192a b + 96a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 4 2 3 4 4 2 3 2
--R (140a b + 210a b )cosh(x) + (240a b + 180a b )cosh(x)
--R +
--R 5 4 2 3
--R 16b + 28a b + 18a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 4 2 3 3 2 2
--R (280a b + 350a b )cosh(x) + (360a b + 300a b )cosh(x)
--R +
--R 4 2 3 3 2
--R 16a b + 40a b + 30a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 4 2 3 3 2 4 2
--R (280a b + 280a b)cosh(x) + (240a b + 480a b + 240a b)cosh(x)
--R +
--R 4 2 3 3 2 4
--R 32a b + 64a b + 56a b + 24a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 2 3 5 4 2 3 3
--R (112a b + 168a b )cosh(x) + (320a b + 240a b )cosh(x)
--R +
--R 5 4 2 3
--R (64b + 112a b + 72a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 5 2 3 3 2 3
--R (224a b + 280a b )cosh(x) + (480a b + 400a b )cosh(x)
--R +
--R 4 2 3 3 2
--R (64a b + 160a b + 120a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 5 2 3 3 2 4 3
--R (224a b + 224a b)cosh(x) + (320a b + 640a b + 320a b)cosh(x)
--R +
--R 4 2 3 3 2 4
--R (128a b + 256a b + 224a b + 96a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 3 6 4 2 3 4
--R (56a b + 84a b )cosh(x) + (240a b + 180a b )cosh(x)
--R +
--R 5 4 2 3 2 4 2 3
--R (96b + 168a b + 108a b )cosh(x) + 16a b + 12a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 6 2 3 3 2 4
--R (112a b + 140a b )cosh(x) + (360a b + 300a b )cosh(x)
--R +
--R 4 2 3 3 2 2 2 3 3 2
--R (96a b + 240a b + 180a b )cosh(x) + 24a b + 20a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 6 2 3 3 2 4 4
--R (112a b + 112a b)cosh(x) + (240a b + 480a b + 240a b)cosh(x)
--R +
--R 4 2 3 3 2 4 2 2 3 3 2
--R (192a b + 384a b + 336a b + 144a b)cosh(x) + 16a b + 32a b
--R +
--R 4
--R 16a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 2 3 7 4 2 3 5
--R (16a b + 24a b )cosh(x) + (96a b + 72a b )cosh(x)
--R +
--R 5 4 2 3 3 4 2 3
--R (64b + 112a b + 72a b )cosh(x) + (32a b + 24a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 7 2 3 3 2 5
--R (32a b + 40a b )cosh(x) + (144a b + 120a b )cosh(x)
--R +
--R 4 2 3 3 2 3 2 3 3 2
--R (64a b + 160a b + 120a b )cosh(x) + (48a b + 40a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 7 2 3 3 2 4 5
--R (32a b + 32a b)cosh(x) + (96a b + 192a b + 96a b)cosh(x)
--R +
--R 4 2 3 3 2 4 3
--R (128a b + 256a b + 224a b + 96a b)cosh(x)
--R +
--R 2 3 3 2 4
--R (32a b + 64a b + 32a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 4 2 3 8 4 2 3 6
--R (2a b + 3a b )cosh(x) + (16a b + 12a b )cosh(x)
--R +
--R 5 4 2 3 4 4 2 3 2
--R (16b + 28a b + 18a b )cosh(x) + (16a b + 12a b )cosh(x)
--R +
--R 4 2 3
--R 2a b + 3a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 8 2 3 3 2 6
--R (4a b + 5a b )cosh(x) + (24a b + 20a b )cosh(x)
--R +
--R 4 2 3 3 2 4 2 3 3 2 2
--R (16a b + 40a b + 30a b )cosh(x) + (24a b + 20a b )cosh(x)
--R +
--R 2 3 3 2
--R 4a b + 5a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 8 2 3 3 2 4 6
--R (4a b + 4a b)cosh(x) + (16a b + 32a b + 16a b)cosh(x)
--R +
--R 4 2 3 3 2 4 4
--R (32a b + 64a b + 56a b + 24a b)cosh(x)
--R +
--R 2 3 3 2 4 2 3 2 4
--R (16a b + 32a b + 16a b)cosh(x) + 4a b + 4a b
--R *
--R 2
--R tanh(x)
--R +
--R 4 2 3 4 2 3 3 2 2 3 2 4
--R ((- 2a b - 3a b )sech(x) + (- 5a b - 7a b )sech(x) - 3a b - 4a b)
--R *
--R 8
--R sinh(x)
--R +
--R 4 2 3 4
--R (- 16a b - 24a b )cosh(x)sech(x)
--R +
--R 2 3 3 2 2 3 2 4
--R (- 40a b - 56a b )cosh(x)sech(x) + (- 24a b - 32a b)cosh(x)
--R *
--R 7
--R sinh(x)
--R +
--R 4 2 3 2 2 3 4
--R ((- 56a b - 84a b )cosh(x) + 4a b )sech(x)
--R +
--R 2 3 3 2 2 4 2 3 3 2 2
--R ((- 140a b - 196a b )cosh(x) - 8a b - 12a b + 4a b )sech(x)
--R +
--R 3 2 4 2 2 3 3 2
--R (- 84a b - 112a b)cosh(x) - 8a b - 12a b
--R *
--R 6
--R sinh(x)
--R +
--R 4 2 3 3 2 3 4
--R ((- 112a b - 168a b )cosh(x) + 24a b cosh(x))sech(x)
--R +
--R 2 3 3 2 3
--R (- 280a b - 392a b )cosh(x)
--R +
--R 4 2 3 3 2
--R (- 48a b - 72a b + 24a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 3 2 3 3 2
--R (- 168a b - 224a b)cosh(x) + (- 48a b - 72a b )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 4 2 3 4 2 3 2 5 4
--R (- 140a b - 210a b )cosh(x) + 60a b cosh(x) + 16b + 36a b
--R +
--R 2 3
--R 14a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 4
--R (- 350a b - 490a b )cosh(x)
--R +
--R 4 2 3 3 2 2 5 2 3 3 2
--R (- 120a b - 180a b + 60a b )cosh(x) - 16b + 50a b + 22a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 4 2 3 3 2 2 4
--R (- 210a b - 280a b)cosh(x) + (- 120a b - 180a b )cosh(x) - 16a b
--R +
--R 2 3 3 2 4
--R - 16a b + 14a b + 8a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 2 3 5 2 3 3
--R (- 112a b - 168a b )cosh(x) + 80a b cosh(x)
--R +
--R 5 4 2 3
--R (64b + 144a b + 56a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 5
--R (- 280a b - 392a b )cosh(x)
--R +
--R 4 2 3 3 2 3
--R (- 160a b - 240a b + 80a b )cosh(x)
--R +
--R 5 2 3 3 2
--R (- 64b + 200a b + 88a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 5 2 3 3 2 3
--R (- 168a b - 224a b)cosh(x) + (- 160a b - 240a b )cosh(x)
--R +
--R 4 2 3 3 2 4
--R (- 64a b - 64a b + 56a b + 32a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 3 6 2 3 4
--R (- 56a b - 84a b )cosh(x) + 60a b cosh(x)
--R +
--R 5 4 2 3 2 2 3
--R (96b + 216a b + 84a b )cosh(x) + 4a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 6
--R (- 140a b - 196a b )cosh(x)
--R +
--R 4 2 3 3 2 4
--R (- 120a b - 180a b + 60a b )cosh(x)
--R +
--R 5 2 3 3 2 2 4 2 3 3 2
--R (- 96b + 300a b + 132a b )cosh(x) - 8a b - 12a b + 4a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 6 2 3 3 2 4
--R (- 84a b - 112a b)cosh(x) + (- 120a b - 180a b )cosh(x)
--R +
--R 4 2 3 3 2 4 2 2 3 3 2
--R (- 96a b - 96a b + 84a b + 48a b)cosh(x) - 8a b - 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 2 3 7 2 3 5
--R (- 16a b - 24a b )cosh(x) + 24a b cosh(x)
--R +
--R 5 4 2 3 3 2 3
--R (64b + 144a b + 56a b )cosh(x) + 8a b cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 7
--R (- 40a b - 56a b )cosh(x)
--R +
--R 4 2 3 3 2 5
--R (- 48a b - 72a b + 24a b )cosh(x)
--R +
--R 5 2 3 3 2 3
--R (- 64b + 200a b + 88a b )cosh(x)
--R +
--R 4 2 3 3 2
--R (- 16a b - 24a b + 8a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 7 2 3 3 2 5
--R (- 24a b - 32a b)cosh(x) + (- 48a b - 72a b )cosh(x)
--R +
--R 4 2 3 3 2 4 3
--R (- 64a b - 64a b + 56a b + 32a b)cosh(x)
--R +
--R 2 3 3 2
--R (- 16a b - 24a b )cosh(x)
--R *
--R sinh(x)
--R +
--R 4 2 3 8 2 3 6
--R (- 2a b - 3a b )cosh(x) + 4a b cosh(x)
--R +
--R 5 4 2 3 4 2 3 2 4 2 3
--R (16b + 36a b + 14a b )cosh(x) + 4a b cosh(x) - 2a b - 3a b
--R *
--R 4
--R sech(x)
--R +
--R 2 3 3 2 8 4 2 3 3 2 6
--R (- 5a b - 7a b )cosh(x) + (- 8a b - 12a b + 4a b )cosh(x)
--R +
--R 5 2 3 3 2 4
--R (- 16b + 50a b + 22a b )cosh(x)
--R +
--R 4 2 3 3 2 2 2 3 3 2
--R (- 8a b - 12a b + 4a b )cosh(x) - 5a b - 7a b
--R *
--R 2
--R sech(x)
--R +
--R 3 2 4 8 2 3 3 2 6
--R (- 3a b - 4a b)cosh(x) + (- 8a b - 12a b )cosh(x)
--R +
--R 4 2 3 3 2 4 4 2 3 3 2 2
--R (- 16a b - 16a b + 14a b + 8a b)cosh(x) + (- 8a b - 12a b )cosh(x)
--R +
--R 3 2 4
--R - 3a b - 4a b
--R /
--R 3 4 4 3 4 4 3 5 2 2 5 2 6
--R ((2a b + 2a b )sech(x) + (4a b + 4a b )sech(x) + 2a b + 2a b)
--R *
--R 8
--R sinh(x)
--R +
--R 3 4 4 3 4
--R (16a b + 16a b )cosh(x)sech(x)
--R +
--R 4 3 5 2 2 5 2 6
--R (32a b + 32a b )cosh(x)sech(x) + (16a b + 16a b)cosh(x)
--R *
--R 7
--R sinh(x)
--R +
--R 3 4 4 3 2 2 5 3 4 4 3 4
--R ((56a b + 56a b )cosh(x) + 16a b + 24a b + 8a b )sech(x)
--R +
--R 4 3 5 2 2 3 4 4 3 5 2 2
--R ((112a b + 112a b )cosh(x) + 32a b + 48a b + 16a b )sech(x)
--R +
--R 5 2 6 2 4 3 5 2 6
--R (56a b + 56a b)cosh(x) + 16a b + 24a b + 8a b
--R *
--R 6
--R sinh(x)
--R +
--R 3 4 4 3 3
--R (112a b + 112a b )cosh(x)
--R +
--R 2 5 3 4 4 3
--R (96a b + 144a b + 48a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 3
--R (224a b + 224a b )cosh(x)
--R +
--R 3 4 4 3 5 2
--R (192a b + 288a b + 96a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 3 4 3 5 2 6
--R (112a b + 112a b)cosh(x) + (96a b + 144a b + 48a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 4 4 3 4
--R (140a b + 140a b )cosh(x)
--R +
--R 2 5 3 4 4 3 2 6 2 5
--R (240a b + 360a b + 120a b )cosh(x) + 32a b + 64a b
--R +
--R 3 4 4 3
--R 44a b + 12a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 4
--R (280a b + 280a b )cosh(x)
--R +
--R 3 4 4 3 5 2 2 2 5 3 4
--R (480a b + 720a b + 240a b )cosh(x) + 64a b + 128a b
--R +
--R 4 3 5 2
--R 88a b + 24a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 4 4 3 5 2 6 2
--R (140a b + 140a b)cosh(x) + (240a b + 360a b + 120a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R 32a b + 64a b + 44a b + 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 4 4 3 5
--R (112a b + 112a b )cosh(x)
--R +
--R 2 5 3 4 4 3 3
--R (320a b + 480a b + 160a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3
--R (128a b + 256a b + 176a b + 48a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 5
--R (224a b + 224a b )cosh(x)
--R +
--R 3 4 4 3 5 2 3
--R (640a b + 960a b + 320a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2
--R (256a b + 512a b + 352a b + 96a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 5 4 3 5 2 6 3
--R (112a b + 112a b)cosh(x) + (320a b + 480a b + 160a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R (128a b + 256a b + 176a b + 48a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 4 4 3 6
--R (56a b + 56a b )cosh(x)
--R +
--R 2 5 3 4 4 3 4
--R (240a b + 360a b + 120a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 2 2 5
--R (192a b + 384a b + 264a b + 72a b )cosh(x) + 16a b
--R +
--R 3 4 4 3
--R 24a b + 8a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6
--R (112a b + 112a b )cosh(x)
--R +
--R 3 4 4 3 5 2 4
--R (480a b + 720a b + 240a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 2 3 4
--R (384a b + 768a b + 528a b + 144a b )cosh(x) + 32a b
--R +
--R 4 3 5 2
--R 48a b + 16a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 6 4 3 5 2 6 4
--R (56a b + 56a b)cosh(x) + (240a b + 360a b + 120a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 2 4 3 5 2
--R (192a b + 384a b + 264a b + 72a b)cosh(x) + 16a b + 24a b
--R +
--R 6
--R 8a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 4 4 3 7
--R (16a b + 16a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5
--R (96a b + 144a b + 48a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 3
--R (128a b + 256a b + 176a b + 48a b )cosh(x)
--R +
--R 2 5 3 4 4 3
--R (32a b + 48a b + 16a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 7
--R (32a b + 32a b )cosh(x)
--R +
--R 3 4 4 3 5 2 5
--R (192a b + 288a b + 96a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 3
--R (256a b + 512a b + 352a b + 96a b )cosh(x)
--R +
--R 3 4 4 3 5 2
--R (64a b + 96a b + 32a b )cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 4 3 5 2 6 5
--R (16a b + 16a b)cosh(x) + (96a b + 144a b + 48a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 3
--R (128a b + 256a b + 176a b + 48a b)cosh(x)
--R +
--R 4 3 5 2 6
--R (32a b + 48a b + 16a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 4 4 3 8 2 5 3 4 4 3 6
--R (2a b + 2a b )cosh(x) + (16a b + 24a b + 8a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 4
--R (32a b + 64a b + 44a b + 12a b )cosh(x)
--R +
--R 2 5 3 4 4 3 2 3 4 4 3
--R (16a b + 24a b + 8a b )cosh(x) + 2a b + 2a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 8 3 4 4 3 5 2 6
--R (4a b + 4a b )cosh(x) + (32a b + 48a b + 16a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 4
--R (64a b + 128a b + 88a b + 24a b )cosh(x)
--R +
--R 3 4 4 3 5 2 2 4 3 5 2
--R (32a b + 48a b + 16a b )cosh(x) + 4a b + 4a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 8 4 3 5 2 6 6
--R (2a b + 2a b)cosh(x) + (16a b + 24a b + 8a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 4
--R (32a b + 64a b + 44a b + 12a b)cosh(x)
--R +
--R 4 3 5 2 6 2 5 2 6
--R (16a b + 24a b + 8a b)cosh(x) + 2a b + 2a b
--R *
--R 2
--R tanh(x)
--R +
--R 3 4 4 3 5 2 4 4 3 5 2 6 2
--R (- 2a b - 4a b - 2a b )sech(x) + (- 4a b - 8a b - 4a b)sech(x)
--R +
--R 5 2 6 7
--R - 2a b - 4a b - 2a
--R *
--R 8
--R sinh(x)
--R +
--R 3 4 4 3 5 2 4
--R (- 16a b - 32a b - 16a b )cosh(x)sech(x)
--R +
--R 4 3 5 2 6 2
--R (- 32a b - 64a b - 32a b)cosh(x)sech(x)
--R +
--R 5 2 6 7
--R (- 16a b - 32a b - 16a )cosh(x)
--R *
--R 7
--R sinh(x)
--R +
--R 3 4 4 3 5 2 2 2 5 3 4 4 3
--R (- 56a b - 112a b - 56a b )cosh(x) - 16a b - 40a b - 32a b
--R +
--R 5 2
--R - 8a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 2 3 4 4 3 5 2
--R (- 112a b - 224a b - 112a b)cosh(x) - 32a b - 80a b - 64a b
--R +
--R 6
--R - 16a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 2 4 3 5 2 6 7
--R (- 56a b - 112a b - 56a )cosh(x) - 16a b - 40a b - 32a b - 8a
--R *
--R 6
--R sinh(x)
--R +
--R 3 4 4 3 5 2 3
--R (- 112a b - 224a b - 112a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2
--R (- 96a b - 240a b - 192a b - 48a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 3
--R (- 224a b - 448a b - 224a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R (- 192a b - 480a b - 384a b - 96a b)cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 3
--R (- 112a b - 224a b - 112a )cosh(x)
--R +
--R 4 3 5 2 6 7
--R (- 96a b - 240a b - 192a b - 48a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 4 4 3 5 2 4
--R (- 140a b - 280a b - 140a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 2 6
--R (- 240a b - 600a b - 480a b - 120a b )cosh(x) - 32a b
--R +
--R 2 5 3 4 4 3 5 2
--R - 96a b - 108a b - 56a b - 12a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 4
--R (- 280a b - 560a b - 280a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 2 2 5
--R (- 480a b - 1200a b - 960a b - 240a b)cosh(x) - 64a b
--R +
--R 3 4 4 3 5 2 6
--R - 192a b - 216a b - 112a b - 24a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 4
--R (- 140a b - 280a b - 140a )cosh(x)
--R +
--R 4 3 5 2 6 7 2 3 4 4 3
--R (- 240a b - 600a b - 480a b - 120a )cosh(x) - 32a b - 96a b
--R +
--R 5 2 6 7
--R - 108a b - 56a b - 12a
--R *
--R 4
--R sinh(x)
--R +
--R 3 4 4 3 5 2 5
--R (- 112a b - 224a b - 112a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 3
--R (- 320a b - 800a b - 640a b - 160a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 5 2
--R (- 128a b - 384a b - 432a b - 224a b - 48a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 5
--R (- 224a b - 448a b - 224a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 3
--R (- 640a b - 1600a b - 1280a b - 320a b)cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6
--R (- 256a b - 768a b - 864a b - 448a b - 96a b)cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 5
--R (- 112a b - 224a b - 112a )cosh(x)
--R +
--R 4 3 5 2 6 7 3
--R (- 320a b - 800a b - 640a b - 160a )cosh(x)
--R +
--R 3 4 4 3 5 2 6 7
--R (- 128a b - 384a b - 432a b - 224a b - 48a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 4 4 3 5 2 6
--R (- 56a b - 112a b - 56a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 4
--R (- 240a b - 600a b - 480a b - 120a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 5 2 2
--R (- 192a b - 576a b - 648a b - 336a b - 72a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2
--R - 16a b - 40a b - 32a b - 8a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 6
--R (- 112a b - 224a b - 112a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 4
--R (- 480a b - 1200a b - 960a b - 240a b)cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6 2
--R (- 384a b - 1152a b - 1296a b - 672a b - 144a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R - 32a b - 80a b - 64a b - 16a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 6
--R (- 56a b - 112a b - 56a )cosh(x)
--R +
--R 4 3 5 2 6 7 4
--R (- 240a b - 600a b - 480a b - 120a )cosh(x)
--R +
--R 3 4 4 3 5 2 6 7 2 4 3
--R (- 192a b - 576a b - 648a b - 336a b - 72a )cosh(x) - 16a b
--R +
--R 5 2 6 7
--R - 40a b - 32a b - 8a
--R *
--R 2
--R sinh(x)
--R +
--R 3 4 4 3 5 2 7
--R (- 16a b - 32a b - 16a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 5
--R (- 96a b - 240a b - 192a b - 48a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 5 2 3
--R (- 128a b - 384a b - 432a b - 224a b - 48a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2
--R (- 32a b - 80a b - 64a b - 16a b )cosh(x)
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 7
--R (- 32a b - 64a b - 32a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 5
--R (- 192a b - 480a b - 384a b - 96a b)cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6 3
--R (- 256a b - 768a b - 864a b - 448a b - 96a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6
--R (- 64a b - 160a b - 128a b - 32a b)cosh(x)
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 7
--R (- 16a b - 32a b - 16a )cosh(x)
--R +
--R 4 3 5 2 6 7 5
--R (- 96a b - 240a b - 192a b - 48a )cosh(x)
--R +
--R 3 4 4 3 5 2 6 7 3
--R (- 128a b - 384a b - 432a b - 224a b - 48a )cosh(x)
--R +
--R 4 3 5 2 6 7
--R (- 32a b - 80a b - 64a b - 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 4 4 3 5 2 8
--R (- 2a b - 4a b - 2a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6
--R (- 16a b - 40a b - 32a b - 8a b )cosh(x)
--R +
--R 6 2 5 3 4 4 3 5 2 4
--R (- 32a b - 96a b - 108a b - 56a b - 12a b )cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 2 3 4 4 3 5 2
--R (- 16a b - 40a b - 32a b - 8a b )cosh(x) - 2a b - 4a b - 2a b
--R *
--R 4
--R sech(x)
--R +
--R 4 3 5 2 6 8
--R (- 4a b - 8a b - 4a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 6
--R (- 32a b - 80a b - 64a b - 16a b)cosh(x)
--R +
--R 2 5 3 4 4 3 5 2 6 4
--R (- 64a b - 192a b - 216a b - 112a b - 24a b)cosh(x)
--R +
--R 3 4 4 3 5 2 6 2 4 3 5 2 6
--R (- 32a b - 80a b - 64a b - 16a b)cosh(x) - 4a b - 8a b - 4a b
--R *
--R 2
--R sech(x)
--R +
--R 5 2 6 7 8
--R (- 2a b - 4a b - 2a )cosh(x)
--R +
--R 4 3 5 2 6 7 6
--R (- 16a b - 40a b - 32a b - 8a )cosh(x)
--R +
--R 3 4 4 3 5 2 6 7 4
--R (- 32a b - 96a b - 108a b - 56a b - 12a )cosh(x)
--R +
--R 4 3 5 2 6 7 2 5 2 6 7
--R (- 16a b - 40a b - 32a b - 8a )cosh(x) - 2a b - 4a b - 2a
--R Type: Expression(Integer)
--E 35
--S 36 of 526
t0405:= (1+sech(x)^2)^(1/2)
--R
--R
--R +------------+
--R | 2
--R (36) \|sech(x) + 1
--R Type: Expression(Integer)
--E 36
--S 37 of 526
r0405:= asin(1/2*2^(1/2)*tanh(x))+atanh(tanh(x)/(2-tanh(x)^2)^(1/2))
--R
--R
--R +-+
--R tanh(x) \|2 tanh(x)
--R (37) atanh(-----------------) + asin(-----------)
--R +--------------+ 2
--R | 2
--R \|- tanh(x) + 2
--R Type: Expression(Integer)
--E 37
--S 38 of 526
a0405:= integrate(t0405,x)
--R
--R
--R (38)
--R log
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6 2
--R |------------------------------------- - sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2cosh(x)sinh(x) - cosh(x) - 1
--R +
--R -
--R log
--R 2 2
--R (- sinh(x) - 2cosh(x)sinh(x) - cosh(x) - 1)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) + 4)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) + 8cosh(x))sinh(x) + cosh(x) + 4cosh(x) - 1
--R +
--R 4
--R *
--R atan
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6 2
--R |------------------------------------- - sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2cosh(x)sinh(x) - cosh(x) - 1
--R /
--R 2
--R /
--R 2
--R Type: Union(Expression(Integer),...)
--E 38
--S 39 of 526
m0405:= a0405-r0405
--R
--R
--R (39)
--R log
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6 2
--R |------------------------------------- - sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2cosh(x)sinh(x) - cosh(x) - 1
--R +
--R -
--R log
--R 2 2
--R (- sinh(x) - 2cosh(x)sinh(x) - cosh(x) - 1)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) + 4)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) + 8cosh(x))sinh(x) + cosh(x) + 4cosh(x) - 1
--R +
--R tanh(x)
--R - 2atanh(-----------------)
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R +
--R 4
--R *
--R atan
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6 2
--R |------------------------------------- - sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2cosh(x)sinh(x) - cosh(x) - 1
--R /
--R 2
--R +
--R +-+
--R \|2 tanh(x)
--R - 2asin(-----------)
--R 2
--R /
--R 2
--R Type: Expression(Integer)
--E 39
--S 40 of 526
d0405:= D(m0405,x)
--R
--R
--R (40)
--R 14 13 2 12
--R - sinh(x) - 6cosh(x)sinh(x) + (- 11cosh(x) - 12)sinh(x)
--R +
--R 3 11
--R (4cosh(x) - 48cosh(x))sinh(x)
--R +
--R 4 2 10
--R (39cosh(x) - 24cosh(x) - 66)sinh(x)
--R +
--R 5 3 9
--R (38cosh(x) + 144cosh(x) - 156cosh(x))sinh(x)
--R +
--R 6 4 2 8
--R (- 27cosh(x) + 204cosh(x) + 102cosh(x) - 140)sinh(x)
--R +
--R 7 5 3 7
--R (- 72cosh(x) - 96cosh(x) + 432cosh(x) - 288cosh(x))sinh(x)
--R +
--R 8 6 4 2
--R (- 27cosh(x) - 336cosh(x) - 36cosh(x) - 16cosh(x) - 301)
--R *
--R 6
--R sinh(x)
--R +
--R 9 7 5 3
--R 38cosh(x) - 96cosh(x) - 552cosh(x) + 288cosh(x)
--R +
--R - 430cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 10 8 6 4
--R 39cosh(x) + 204cosh(x) - 36cosh(x) + 312cosh(x)
--R +
--R 2
--R 285cosh(x) - 328
--R *
--R 4
--R sinh(x)
--R +
--R 11 9 7 5
--R 4cosh(x) + 144cosh(x) + 432cosh(x) + 288cosh(x)
--R +
--R 3
--R 828cosh(x) - 48cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 12 10 8 6
--R - 11cosh(x) - 24cosh(x) + 102cosh(x) - 16cosh(x)
--R +
--R 4 2
--R 285cosh(x) + 560cosh(x) - 48
--R *
--R 2
--R sinh(x)
--R +
--R 13 11 9 7
--R - 6cosh(x) - 48cosh(x) - 156cosh(x) - 288cosh(x)
--R +
--R 5 3
--R - 430cosh(x) - 48cosh(x) - 48cosh(x)
--R *
--R sinh(x)
--R +
--R 14 12 10 8
--R - cosh(x) - 12cosh(x) - 66cosh(x) - 140cosh(x)
--R +
--R 6 4 2
--R - 301cosh(x) - 328cosh(x) - 48cosh(x)
--R *
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R +
--R 14 13 2 12
--R sinh(x) + 6cosh(x)sinh(x) + (11cosh(x) + 7)sinh(x)
--R +
--R 3 11
--R (- 4cosh(x) + 28cosh(x))sinh(x)
--R +
--R 4 2 10
--R (- 39cosh(x) + 14cosh(x) + 30)sinh(x)
--R +
--R 5 3 9
--R (- 38cosh(x) - 84cosh(x) + 84cosh(x))sinh(x)
--R +
--R 6 4 2 8
--R (27cosh(x) - 119cosh(x) + 6cosh(x) + 114)sinh(x)
--R +
--R 7 5 3 7
--R (72cosh(x) + 56cosh(x) - 144cosh(x) + 240cosh(x))sinh(x)
--R +
--R 8 6 4 2
--R (27cosh(x) + 196cosh(x) - 36cosh(x) + 24cosh(x) + 205)
--R *
--R 6
--R sinh(x)
--R +
--R 9 7 5 3
--R - 38cosh(x) + 56cosh(x) + 120cosh(x) - 240cosh(x)
--R +
--R 270cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 10 8 6 4
--R - 39cosh(x) - 119cosh(x) - 36cosh(x) - 276cosh(x)
--R +
--R 2
--R - 189cosh(x) + 163
--R *
--R 4
--R sinh(x)
--R +
--R 11 9 7 5
--R - 4cosh(x) - 84cosh(x) - 144cosh(x) - 240cosh(x)
--R +
--R 3
--R - 508cosh(x) + 52cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 12 10 8 6
--R 11cosh(x) + 14cosh(x) + 6cosh(x) + 24cosh(x)
--R +
--R 4 2
--R - 189cosh(x) - 222cosh(x) + 84
--R *
--R 2
--R sinh(x)
--R +
--R 13 11 9 7
--R 6cosh(x) + 28cosh(x) + 84cosh(x) + 240cosh(x)
--R +
--R 5 3
--R 270cosh(x) + 52cosh(x) + 24cosh(x)
--R *
--R sinh(x)
--R +
--R 14 12 10 8 6
--R cosh(x) + 7cosh(x) + 30cosh(x) + 114cosh(x) + 205cosh(x)
--R +
--R 4 2
--R 163cosh(x) + 84cosh(x) + 36
--R *
--R 2
--R tanh(x)
--R +
--R 14 13 2 12
--R - 2sinh(x) - 12cosh(x)sinh(x) + (- 22cosh(x) - 14)sinh(x)
--R +
--R 3 11
--R (8cosh(x) - 56cosh(x))sinh(x)
--R +
--R 4 2 10
--R (78cosh(x) - 28cosh(x) - 60)sinh(x)
--R +
--R 5 3 9
--R (76cosh(x) + 168cosh(x) - 168cosh(x))sinh(x)
--R +
--R 6 4 2 8
--R (- 54cosh(x) + 238cosh(x) - 12cosh(x) - 228)sinh(x)
--R +
--R 7 5 3 7
--R (- 144cosh(x) - 112cosh(x) + 288cosh(x) - 480cosh(x))sinh(x)
--R +
--R 8 6 4 2 6
--R (- 54cosh(x) - 392cosh(x) + 72cosh(x) - 48cosh(x) - 410)sinh(x)
--R +
--R 9 7 5 3
--R (76cosh(x) - 112cosh(x) - 240cosh(x) + 480cosh(x) - 540cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 10 8 6 4
--R 78cosh(x) + 238cosh(x) + 72cosh(x) + 552cosh(x)
--R +
--R 2
--R 378cosh(x) - 326
--R *
--R 4
--R sinh(x)
--R +
--R 11 9 7 5
--R 8cosh(x) + 168cosh(x) + 288cosh(x) + 480cosh(x)
--R +
--R 3
--R 1016cosh(x) - 104cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 12 10 8 6
--R - 22cosh(x) - 28cosh(x) - 12cosh(x) - 48cosh(x)
--R +
--R 4 2
--R 378cosh(x) + 444cosh(x) - 168
--R *
--R 2
--R sinh(x)
--R +
--R 13 11 9 7
--R - 12cosh(x) - 56cosh(x) - 168cosh(x) - 480cosh(x)
--R +
--R 5 3
--R - 540cosh(x) - 104cosh(x) - 48cosh(x)
--R *
--R sinh(x)
--R +
--R 14 12 10 8 6
--R - 2cosh(x) - 14cosh(x) - 60cosh(x) - 228cosh(x) - 410cosh(x)
--R +
--R 4 2
--R - 326cosh(x) - 168cosh(x) - 72
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 12 11 2 10
--R 8sinh(x) + 48cosh(x)sinh(x) + (112cosh(x) + 90)sinh(x)
--R +
--R 3 9
--R (112cosh(x) + 420cosh(x))sinh(x)
--R +
--R 4 2 8
--R (- 8cosh(x) + 690cosh(x) + 412)sinh(x)
--R +
--R 5 3 7
--R (- 160cosh(x) + 240cosh(x) + 1328cosh(x))sinh(x)
--R +
--R 6 4 2 6
--R (- 224cosh(x) - 780cosh(x) + 1072cosh(x) + 804)sinh(x)
--R +
--R 7 5 3 5
--R (- 160cosh(x) - 1320cosh(x) - 1072cosh(x) + 1704cosh(x))sinh(x)
--R +
--R 8 6 4 2
--R (- 8cosh(x) - 780cosh(x) - 2456cosh(x) + 156cosh(x) + 616)
--R *
--R 4
--R sinh(x)
--R +
--R 9 7 5 3
--R 112cosh(x) + 240cosh(x) - 1072cosh(x) - 1488cosh(x)
--R +
--R 1120cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 10 8 6 4
--R 112cosh(x) + 690cosh(x) + 1072cosh(x) + 156cosh(x)
--R +
--R 2
--R 1008cosh(x) + 450
--R *
--R 2
--R sinh(x)
--R +
--R 11 9 7 5
--R 48cosh(x) + 420cosh(x) + 1328cosh(x) + 1704cosh(x)
--R +
--R 3
--R 1120cosh(x) + 756cosh(x)
--R *
--R sinh(x)
--R +
--R 12 10 8 6 4
--R 8cosh(x) + 90cosh(x) + 412cosh(x) + 804cosh(x) + 616cosh(x)
--R +
--R 2
--R 450cosh(x) + 180
--R *
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R +
--R 12 11 2 10
--R - 8sinh(x) - 48cosh(x)sinh(x) + (- 112cosh(x) - 68)sinh(x)
--R +
--R 3 9
--R (- 112cosh(x) - 320cosh(x))sinh(x)
--R +
--R 4 2 8
--R (8cosh(x) - 532cosh(x) - 224)sinh(x)
--R +
--R 5 3 7
--R (160cosh(x) - 192cosh(x) - 744cosh(x))sinh(x)
--R +
--R 6 4 2 6
--R (224cosh(x) + 600cosh(x) - 656cosh(x) - 432)sinh(x)
--R +
--R 7 5 3 5
--R (160cosh(x) + 1024cosh(x) + 488cosh(x) - 920cosh(x))sinh(x)
--R +
--R 8 6 4 2 4
--R (8cosh(x) + 600cosh(x) + 1248cosh(x) - 496cosh(x) - 592)sinh(x)
--R +
--R 9 7 5 3
--R - 112cosh(x) - 192cosh(x) + 488cosh(x) - 16cosh(x)
--R +
--R - 1064cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 10 8 6 4
--R - 112cosh(x) - 532cosh(x) - 656cosh(x) - 496cosh(x)
--R +
--R 2
--R - 944cosh(x) - 396
--R *
--R 2
--R sinh(x)
--R +
--R 11 9 7 5
--R - 48cosh(x) - 320cosh(x) - 744cosh(x) - 920cosh(x)
--R +
--R 3
--R - 1064cosh(x) - 744cosh(x)
--R *
--R sinh(x)
--R +
--R 12 10 8 6 4
--R - 8cosh(x) - 68cosh(x) - 224cosh(x) - 432cosh(x) - 592cosh(x)
--R +
--R 2
--R - 396cosh(x) - 72
--R *
--R 2
--R tanh(x)
--R +
--R 12 11 2 10
--R 16sinh(x) + 96cosh(x)sinh(x) + (224cosh(x) + 136)sinh(x)
--R +
--R 3 9
--R (224cosh(x) + 640cosh(x))sinh(x)
--R +
--R 4 2 8
--R (- 16cosh(x) + 1064cosh(x) + 448)sinh(x)
--R +
--R 5 3 7
--R (- 320cosh(x) + 384cosh(x) + 1488cosh(x))sinh(x)
--R +
--R 6 4 2 6
--R (- 448cosh(x) - 1200cosh(x) + 1312cosh(x) + 864)sinh(x)
--R +
--R 7 5 3 5
--R (- 320cosh(x) - 2048cosh(x) - 976cosh(x) + 1840cosh(x))sinh(x)
--R +
--R 8 6 4 2 4
--R (- 16cosh(x) - 1200cosh(x) - 2496cosh(x) + 992cosh(x) + 1184)sinh(x)
--R +
--R 9 7 5 3
--R (224cosh(x) + 384cosh(x) - 976cosh(x) + 32cosh(x) + 2128cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 10 8 6 4
--R 224cosh(x) + 1064cosh(x) + 1312cosh(x) + 992cosh(x)
--R +
--R 2
--R 1888cosh(x) + 792
--R *
--R 2
--R sinh(x)
--R +
--R 11 9 7 5
--R 96cosh(x) + 640cosh(x) + 1488cosh(x) + 1840cosh(x)
--R +
--R 3
--R 2128cosh(x) + 1488cosh(x)
--R *
--R sinh(x)
--R +
--R 12 10 8 6 4
--R 16cosh(x) + 136cosh(x) + 448cosh(x) + 864cosh(x) + 1184cosh(x)
--R +
--R 2
--R 792cosh(x) + 144
--R /
--R 14 13 2 12
--R sinh(x) + 6cosh(x)sinh(x) + (11cosh(x) + 7)sinh(x)
--R +
--R 3 11
--R (- 4cosh(x) + 28cosh(x))sinh(x)
--R +
--R 4 2 10
--R (- 39cosh(x) + 14cosh(x) + 30)sinh(x)
--R +
--R 5 3 9
--R (- 38cosh(x) - 84cosh(x) + 84cosh(x))sinh(x)
--R +
--R 6 4 2 8
--R (27cosh(x) - 119cosh(x) + 6cosh(x) + 114)sinh(x)
--R +
--R 7 5 3 7
--R (72cosh(x) + 56cosh(x) - 144cosh(x) + 240cosh(x))sinh(x)
--R +
--R 8 6 4 2 6
--R (27cosh(x) + 196cosh(x) - 36cosh(x) + 24cosh(x) + 205)sinh(x)
--R +
--R 9 7 5 3
--R - 38cosh(x) + 56cosh(x) + 120cosh(x) - 240cosh(x)
--R +
--R 270cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 10 8 6 4
--R - 39cosh(x) - 119cosh(x) - 36cosh(x) - 276cosh(x)
--R +
--R 2
--R - 189cosh(x) + 163
--R *
--R 4
--R sinh(x)
--R +
--R 11 9 7 5
--R - 4cosh(x) - 84cosh(x) - 144cosh(x) - 240cosh(x)
--R +
--R 3
--R - 508cosh(x) + 52cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 12 10 8 6 4
--R 11cosh(x) + 14cosh(x) + 6cosh(x) + 24cosh(x) - 189cosh(x)
--R +
--R 2
--R - 222cosh(x) + 84
--R *
--R 2
--R sinh(x)
--R +
--R 13 11 9 7 5
--R 6cosh(x) + 28cosh(x) + 84cosh(x) + 240cosh(x) + 270cosh(x)
--R +
--R 3
--R 52cosh(x) + 24cosh(x)
--R *
--R sinh(x)
--R +
--R 14 12 10 8 6
--R cosh(x) + 7cosh(x) + 30cosh(x) + 114cosh(x) + 205cosh(x)
--R +
--R 4 2
--R 163cosh(x) + 84cosh(x) + 36
--R *
--R +-------------------------------------+
--R +--------------+ | 2 2
--R | 2 | 2sinh(x) + 2cosh(x) + 6
--R \|- tanh(x) + 2 |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 12 11 2 10
--R - 8sinh(x) - 48cosh(x)sinh(x) + (- 112cosh(x) - 68)sinh(x)
--R +
--R 3 9
--R (- 112cosh(x) - 320cosh(x))sinh(x)
--R +
--R 4 2 8
--R (8cosh(x) - 532cosh(x) - 224)sinh(x)
--R +
--R 5 3 7
--R (160cosh(x) - 192cosh(x) - 744cosh(x))sinh(x)
--R +
--R 6 4 2 6
--R (224cosh(x) + 600cosh(x) - 656cosh(x) - 432)sinh(x)
--R +
--R 7 5 3 5
--R (160cosh(x) + 1024cosh(x) + 488cosh(x) - 920cosh(x))sinh(x)
--R +
--R 8 6 4 2 4
--R (8cosh(x) + 600cosh(x) + 1248cosh(x) - 496cosh(x) - 592)sinh(x)
--R +
--R 9 7 5 3
--R - 112cosh(x) - 192cosh(x) + 488cosh(x) - 16cosh(x)
--R +
--R - 1064cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 10 8 6 4
--R - 112cosh(x) - 532cosh(x) - 656cosh(x) - 496cosh(x)
--R +
--R 2
--R - 944cosh(x) - 396
--R *
--R 2
--R sinh(x)
--R +
--R 11 9 7 5
--R - 48cosh(x) - 320cosh(x) - 744cosh(x) - 920cosh(x)
--R +
--R 3
--R - 1064cosh(x) - 744cosh(x)
--R *
--R sinh(x)
--R +
--R 12 10 8 6 4
--R - 8cosh(x) - 68cosh(x) - 224cosh(x) - 432cosh(x) - 592cosh(x)
--R +
--R 2
--R - 396cosh(x) - 72
--R *
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R Type: Expression(Integer)
--E 40
--S 41 of 526
t0406:= (1-sech(x)^2)^(1/2)
--R
--R
--R +--------------+
--R | 2
--R (41) \|- sech(x) + 1
--R Type: Expression(Integer)
--E 41
--S 42 of 526
r0406:= coth(x)*log(cosh(x))*(tanh(x)^2)^(1/2)
--R
--R
--R +--------+
--R | 2
--R (42) coth(x)log(cosh(x))\|tanh(x)
--R Type: Expression(Integer)
--E 42
--S 43 of 526
a0406:= integrate(t0406,x)
--R
--R
--R 2cosh(x)
--R (43) log(- -----------------) - x
--R sinh(x) - cosh(x)
--R Type: Union(Expression(Integer),...)
--E 43
--S 44 of 526
m0406:= a0406-r0406
--R
--R
--R +--------+
--R | 2 2cosh(x)
--R (44) - coth(x)log(cosh(x))\|tanh(x) + log(- -----------------) - x
--R sinh(x) - cosh(x)
--R Type: Expression(Integer)
--E 44
--S 45 of 526
d0406:= D(m0406,x)
--R
--R
--R (45)
--R +--------+
--R | 2
--R sinh(x)\|tanh(x)
--R +
--R 3 2 2
--R cosh(x)coth(x)tanh(x) + (cosh(x)coth(x) - cosh(x))tanh(x)
--R +
--R - cosh(x)coth(x)tanh(x)
--R *
--R log(cosh(x))
--R +
--R 2
--R - coth(x)sinh(x)tanh(x)
--R /
--R +--------+
--R | 2
--R cosh(x)\|tanh(x)
--R Type: Expression(Integer)
--E 45
--S 46 of 526
t0407:= (-1+sech(x)^2)^(1/2)
--R
--R
--R +------------+
--R | 2
--R (46) \|sech(x) - 1
--R Type: Expression(Integer)
--E 46
--S 47 of 526
r0407:= coth(x)*log(cosh(x))*(-tanh(x)^2)^(1/2)
--R
--R
--R +----------+
--R | 2
--R (47) coth(x)log(cosh(x))\|- tanh(x)
--R Type: Expression(Integer)
--E 47
--S 48 of 526
a0407:= integrate(t0407,x)
--R
--R
--R (48) 0
--R Type: Union(Expression(Integer),...)
--E 48
--S 49 of 526
m0407:= a0407-r0407
--R
--R
--R +----------+
--R | 2
--R (49) - coth(x)log(cosh(x))\|- tanh(x)
--R Type: Expression(Integer)
--E 49
--S 50 of 526
d0407:= D(m0407,x)
--R
--R
--R (50)
--R 3 2 2
--R - cosh(x)coth(x)tanh(x) + (- cosh(x)coth(x) + cosh(x))tanh(x)
--R +
--R cosh(x)coth(x)tanh(x)
--R *
--R log(cosh(x))
--R +
--R 2
--R coth(x)sinh(x)tanh(x)
--R /
--R +----------+
--R | 2
--R cosh(x)\|- tanh(x)
--R Type: Expression(Integer)
--E 50
--S 51 of 526
t0408:= (-1-sech(x)^2)^(1/2)
--R
--R
--R +--------------+
--R | 2
--R (51) \|- sech(x) - 1
--R Type: Expression(Integer)
--E 51
--S 52 of 526
r0408:= -atan(tanh(x)/(-2+tanh(x)^2)^(1/2))-_
atanh(coth(x)*(-2+tanh(x)^2)^(1/2))
--R
--R
--R +------------+
--R | 2 tanh(x)
--R (52) - atanh(coth(x)\|tanh(x) - 2 ) - atan(---------------)
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R Type: Expression(Integer)
--E 52
--S 53 of 526
a0408:= integrate(t0408,x)
--R
--R
--R (53)
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 + (- \|- 1 + 2)(%e ) - \|- 1
--R - 2log(---------------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R +---+ \|- (%e ) - 6(%e ) - 1 - \|- 1 (%e ) - \|- 1
--R - \|- 1 log(-------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 + (- \|- 1 - 2)(%e ) - \|- 1
--R 2log(---------------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +---+
--R \|- 1
--R *
--R log
--R +----------------------+
--R +---+ x 2 +---+ | x 4 x 2 x 4
--R (- 2\|- 1 (%e ) - 2\|- 1 )\|- (%e ) - 6(%e ) - 1 + 2(%e )
--R +
--R x 2
--R - 8(%e ) - 2
--R /
--R x 4
--R (%e )
--R /
--R 2
--R Type: Union(Expression(Integer),...)
--E 53
--S 54 of 526
m0408:= a0408-r0408
--R
--R
--R (54)
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 + (- \|- 1 + 2)(%e ) - \|- 1
--R - 2log(---------------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R +---+ \|- (%e ) - 6(%e ) - 1 - \|- 1 (%e ) - \|- 1
--R - \|- 1 log(-------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 + (- \|- 1 - 2)(%e ) - \|- 1
--R 2log(---------------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +---+
--R \|- 1
--R *
--R log
--R +----------------------+
--R +---+ x 2 +---+ | x 4 x 2 x 4
--R (- 2\|- 1 (%e ) - 2\|- 1 )\|- (%e ) - 6(%e ) - 1 + 2(%e )
--R +
--R x 2
--R - 8(%e ) - 2
--R /
--R x 4
--R (%e )
--R +
--R +------------+
--R | 2 tanh(x)
--R 2atanh(coth(x)\|tanh(x) - 2 ) + 2atan(---------------)
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R /
--R 2
--R Type: Expression(Integer)
--E 54
--S 55 of 526
d0408:= D(m0408,x)
--R
--R
--R (55)
--R +---+ 2 x 6 +---+ 2 x 4
--R - 9\|- 1 coth(x) (%e ) - 21\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 2 +---+ 2
--R - 9\|- 1 coth(x) (%e ) - \|- 1 coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R +---+ 2 +---+ x 6
--R (18\|- 1 coth(x) + 9\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 4
--R (42\|- 1 coth(x) + 21\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 2 +---+ 2 +---+
--R (18\|- 1 coth(x) + 9\|- 1 )(%e ) + 2\|- 1 coth(x) + \|- 1
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R 2 x 8 2 x 6 2 x 4
--R - 7coth(x) (%e ) - 48coth(x) (%e ) - 44coth(x) (%e )
--R +
--R 2 x 2 2
--R - 12coth(x) (%e ) - coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R 2 x 8 2 x 6
--R (14coth(x) + 7)(%e ) + (96coth(x) + 48)(%e )
--R +
--R 2 x 4 2 x 2 2
--R (88coth(x) + 44)(%e ) + (24coth(x) + 12)(%e ) + 2coth(x) + 1
--R *
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R +
--R x 6 x 4 x 2 3
--R (7coth(x)(%e ) + 13coth(x)(%e ) + 7coth(x)(%e ) + coth(x))tanh(x)
--R +
--R 2 x 6 2 x 4
--R (14coth(x) - 7)(%e ) + (26coth(x) - 13)(%e )
--R +
--R 2 x 2 2
--R (14coth(x) - 7)(%e ) + 2coth(x) - 1
--R *
--R 2
--R tanh(x)
--R +
--R x 6 x 4 x 2
--R (- 7coth(x)(%e ) - 13coth(x)(%e ) - 7coth(x)(%e ) - coth(x))
--R *
--R tanh(x)
--R +
--R 2 x 6 2 x 4
--R (- 28coth(x) + 7)(%e ) + (- 52coth(x) + 13)(%e )
--R +
--R 2 x 2 2
--R (- 28coth(x) + 7)(%e ) - 4coth(x) + 1
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6
--R - 9\|- 1 coth(x)(%e ) - 30\|- 1 coth(x)(%e )
--R +
--R +---+ x 4 +---+ x 2 +---+
--R - 30\|- 1 coth(x)(%e ) - 10\|- 1 coth(x)(%e ) - \|- 1 coth(x)
--R *
--R 3
--R tanh(x)
--R +
--R +---+ 2 +---+ x 8
--R (- 18\|- 1 coth(x) + 9\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 6
--R (- 60\|- 1 coth(x) + 30\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 4
--R (- 60\|- 1 coth(x) + 30\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 2 +---+ 2 +---+
--R (- 20\|- 1 coth(x) + 10\|- 1 )(%e ) - 2\|- 1 coth(x) + \|- 1
--R *
--R 2
--R tanh(x)
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4
--R 9\|- 1 coth(x)(%e ) + 30\|- 1 coth(x)(%e ) + 30\|- 1 coth(x)(%e )
--R +
--R +---+ x 2 +---+
--R 10\|- 1 coth(x)(%e ) + \|- 1 coth(x)
--R *
--R tanh(x)
--R +
--R +---+ 2 +---+ x 8 +---+ 2 +---+ x 6
--R (36\|- 1 coth(x) - 9\|- 1 )(%e ) + (120\|- 1 coth(x) - 30\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 4
--R (120\|- 1 coth(x) - 30\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 2 +---+ 2 +---+
--R (40\|- 1 coth(x) - 10\|- 1 )(%e ) + 4\|- 1 coth(x) - \|- 1
--R /
--R 2 x 6 2 x 4 2 x 2 2
--R (7coth(x) (%e ) + 13coth(x) (%e ) + 7coth(x) (%e ) + coth(x) )
--R *
--R 2
--R tanh(x)
--R +
--R 2 x 6 2 x 4
--R (- 14coth(x) - 7)(%e ) + (- 26coth(x) - 13)(%e )
--R +
--R 2 x 2 2
--R (- 14coth(x) - 7)(%e ) - 2coth(x) - 1
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R +---+ 2 x 8 +---+ 2 x 6
--R - 9\|- 1 coth(x) (%e ) - 30\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 4 +---+ 2 x 2 +---+ 2
--R - 30\|- 1 coth(x) (%e ) - 10\|- 1 coth(x) (%e ) - \|- 1 coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R +---+ 2 +---+ x 8
--R (18\|- 1 coth(x) + 9\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 6
--R (60\|- 1 coth(x) + 30\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 4
--R (60\|- 1 coth(x) + 30\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 2 +---+ 2 +---+
--R (20\|- 1 coth(x) + 10\|- 1 )(%e ) + 2\|- 1 coth(x) + \|- 1
--R *
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R Type: Expression(Integer)
--E 55
--S 56 of 526
t0409:= (a+b*sech(x)^2)^(1/2)
--R
--R
--R +--------------+
--R | 2
--R (56) \|b sech(x) + a
--R Type: Expression(Integer)
--E 56
--S 57 of 526
r0409:= -b^(1/2)*atan(coth(x)*(a+b*sech(x)^2)^(1/2)/b^(1/2))+_
a^(1/2)*atanh(a^(1/2)*tanh(x)/(a+b*sech(x)^2)^(1/2))
--R
--R
--R +--------------+
--R +-+ | 2
--R +-+ tanh(x)\|a +-+ coth(x)\|b sech(x) + a
--R (57) \|a atanh(-----------------) - \|b atan(------------------------)
--R +--------------+ +-+
--R | 2 \|b
--R \|b sech(x) + a
--R Type: Expression(Integer)
--E 57
--S 58 of 526
a0409:= integrate(t0409,x)
--R
--R
--R (58)
--R [
--R -
--R a
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2
--R sinh(x) + 2cosh(x)sinh(x) + cosh(x)
--R +
--R +---+ +-+
--R 2\|- b \|a
--R *
--R log
--R 4 3
--R (2b + 2a)sinh(x) + (8b + 8a)cosh(x)sinh(x)
--R +
--R 2 2
--R ((12b + 12a)cosh(x) + 2a)sinh(x)
--R +
--R 3
--R ((8b + 8a)cosh(x) + 4a cosh(x))sinh(x)
--R +
--R 4 2
--R (2b + 2a)cosh(x) + 2a cosh(x)
--R *
--R +---+ +-+
--R \|- b \|a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 2a b + 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (4a b + 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2
--R (2a b + 2a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 6 2 2 5
--R (2b - a )sinh(x) + (12b - 6a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((30b - 15a )cosh(x) - 2b - 4a b - 3a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((40b - 20a )cosh(x) + (- 8b - 16a b - 12a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (30b - 15a )cosh(x)
--R +
--R 2 2 2 2
--R (- 12b - 24a b - 18a )cosh(x) - 4a b - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5 2 2 3
--R (12b - 6a )cosh(x) + (- 8b - 16a b - 12a )cosh(x)
--R +
--R 2
--R (- 8a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (2b - a )cosh(x) + (- 2b - 4a b - 3a )cosh(x)
--R +
--R 2 2 2
--R (- 4a b - 3a )cosh(x) - a
--R *
--R +-+
--R \|a
--R +
--R 2 6 2 5
--R (- 2a b - 2a )sinh(x) + (- 12a b - 12a )cosh(x)sinh(x)
--R +
--R 2 2 2 4
--R ((- 30a b - 30a )cosh(x) - 6a b - 4a )sinh(x)
--R +
--R 2 3 2
--R ((- 40a b - 40a )cosh(x) + (- 24a b - 16a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 2 2
--R (- 30a b - 30a )cosh(x) + (- 36a b - 24a )cosh(x)
--R +
--R 2
--R - 2a
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3
--R (- 12a b - 12a )cosh(x) + (- 24a b - 16a )cosh(x)
--R +
--R 2
--R - 4a cosh(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4
--R (- 2a b - 2a )cosh(x) + (- 6a b - 4a )cosh(x)
--R +
--R 2 2
--R - 2a cosh(x)
--R *
--R +---+
--R \|- b
--R /
--R 2 4 2 3
--R (2a b + a )sinh(x) + (8a b + 4a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R ((12a b + 6a )cosh(x) + 2a b + 2a )sinh(x)
--R +
--R 2 3 2
--R ((8a b + 4a )cosh(x) + (4a b + 4a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2
--R (2a b + a )cosh(x) + (2a b + 2a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 6
--R (- 2b - 2a b - a )sinh(x)
--R +
--R 2 2 5
--R (- 12b - 12a b - 6a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((- 30b - 30a b - 15a )cosh(x) - 2b - 6a b - 3a )
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3
--R (- 40b - 40a b - 20a )cosh(x)
--R +
--R 2 2
--R (- 8b - 24a b - 12a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (- 30b - 30a b - 15a )cosh(x)
--R +
--R 2 2 2 2
--R (- 12b - 36a b - 18a )cosh(x) - 4a b - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5
--R (- 12b - 12a b - 6a )cosh(x)
--R +
--R 2 2 3 2
--R (- 8b - 24a b - 12a )cosh(x) + (- 8a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (- 2b - 2a b - a )cosh(x) + (- 2b - 6a b - 3a )cosh(x)
--R +
--R 2 2 2
--R (- 4a b - 3a )cosh(x) - a
--R *
--R +-+
--R \|a
--R +
--R a
--R *
--R log
--R 2 2
--R (- b sinh(x) - 2b cosh(x)sinh(x) - b cosh(x) - a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - b sinh(x) - 4b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 6b cosh(x) + 3b + a)sinh(x)
--R +
--R 3 4
--R (- 4b cosh(x) + (6b + 2a)cosh(x))sinh(x) - b cosh(x)
--R +
--R 2
--R (3b + a)cosh(x) + a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x)
--R +
--R 3 4
--R 4cosh(x) sinh(x) + cosh(x)
--R /
--R +-+
--R 2\|a
--R ,
--R
--R -
--R a
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2
--R sinh(x) + 2cosh(x)sinh(x) + cosh(x)
--R +
--R a
--R *
--R log
--R 2 2
--R (- b sinh(x) - 2b cosh(x)sinh(x) - b cosh(x) - a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - b sinh(x) - 4b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 6b cosh(x) + 3b + a)sinh(x)
--R +
--R 3 4
--R (- 4b cosh(x) + (6b + 2a)cosh(x))sinh(x) - b cosh(x)
--R +
--R 2
--R (3b + a)cosh(x) + a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x)
--R +
--R 3 4
--R 4cosh(x) sinh(x) + cosh(x)
--R +
--R -
--R +-+ +-+
--R 4\|a \|b
--R *
--R atan
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2 +-+ +-+
--R (2sinh(x) + 4cosh(x)sinh(x) + 2cosh(x) )\|a \|b
--R /
--R +-+
--R 2\|a
--R ]
--R Type: Union(List(Expression(Integer)),...)
--E 58
--S 59 of 526
m0409a:= a0409.1-r0409
--R
--R
--R (59)
--R -
--R a
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2
--R sinh(x) + 2cosh(x)sinh(x) + cosh(x)
--R +
--R +---+ +-+
--R 2\|- b \|a
--R *
--R log
--R 4 3
--R (2b + 2a)sinh(x) + (8b + 8a)cosh(x)sinh(x)
--R +
--R 2 2
--R ((12b + 12a)cosh(x) + 2a)sinh(x)
--R +
--R 3
--R ((8b + 8a)cosh(x) + 4a cosh(x))sinh(x)
--R +
--R 4 2
--R (2b + 2a)cosh(x) + 2a cosh(x)
--R *
--R +---+ +-+
--R \|- b \|a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 2a b + 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (4a b + 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2
--R (2a b + 2a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 6 2 2 5
--R (2b - a )sinh(x) + (12b - 6a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((30b - 15a )cosh(x) - 2b - 4a b - 3a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((40b - 20a )cosh(x) + (- 8b - 16a b - 12a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4 2 2 2
--R (30b - 15a )cosh(x) + (- 12b - 24a b - 18a )cosh(x)
--R +
--R 2
--R - 4a b - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5 2 2 3
--R (12b - 6a )cosh(x) + (- 8b - 16a b - 12a )cosh(x)
--R +
--R 2
--R (- 8a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (2b - a )cosh(x) + (- 2b - 4a b - 3a )cosh(x)
--R +
--R 2 2 2
--R (- 4a b - 3a )cosh(x) - a
--R *
--R +-+
--R \|a
--R +
--R 2 6 2 5
--R (- 2a b - 2a )sinh(x) + (- 12a b - 12a )cosh(x)sinh(x)
--R +
--R 2 2 2 4
--R ((- 30a b - 30a )cosh(x) - 6a b - 4a )sinh(x)
--R +
--R 2 3 2 3
--R ((- 40a b - 40a )cosh(x) + (- 24a b - 16a )cosh(x))sinh(x)
--R +
--R 2 4 2 2
--R (- 30a b - 30a )cosh(x) + (- 36a b - 24a )cosh(x)
--R +
--R 2
--R - 2a
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3
--R (- 12a b - 12a )cosh(x) + (- 24a b - 16a )cosh(x)
--R +
--R 2
--R - 4a cosh(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4 2 2
--R (- 2a b - 2a )cosh(x) + (- 6a b - 4a )cosh(x) - 2a cosh(x)
--R *
--R +---+
--R \|- b
--R /
--R 2 4 2 3
--R (2a b + a )sinh(x) + (8a b + 4a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R ((12a b + 6a )cosh(x) + 2a b + 2a )sinh(x)
--R +
--R 2 3 2
--R ((8a b + 4a )cosh(x) + (4a b + 4a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2
--R (2a b + a )cosh(x) + (2a b + 2a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 6
--R (- 2b - 2a b - a )sinh(x)
--R +
--R 2 2 5
--R (- 12b - 12a b - 6a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((- 30b - 30a b - 15a )cosh(x) - 2b - 6a b - 3a )sinh(x)
--R +
--R 2 2 3
--R (- 40b - 40a b - 20a )cosh(x)
--R +
--R 2 2
--R (- 8b - 24a b - 12a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (- 30b - 30a b - 15a )cosh(x)
--R +
--R 2 2 2 2
--R (- 12b - 36a b - 18a )cosh(x) - 4a b - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5
--R (- 12b - 12a b - 6a )cosh(x)
--R +
--R 2 2 3 2
--R (- 8b - 24a b - 12a )cosh(x) + (- 8a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (- 2b - 2a b - a )cosh(x) + (- 2b - 6a b - 3a )cosh(x)
--R +
--R 2 2 2
--R (- 4a b - 3a )cosh(x) - a
--R *
--R +-+
--R \|a
--R +
--R a
--R *
--R log
--R 2 2
--R (- b sinh(x) - 2b cosh(x)sinh(x) - b cosh(x) - a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - b sinh(x) - 4b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 6b cosh(x) + 3b + a)sinh(x)
--R +
--R 3 4
--R (- 4b cosh(x) + (6b + 2a)cosh(x))sinh(x) - b cosh(x)
--R +
--R 2
--R (3b + a)cosh(x) + a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x)
--R +
--R 3 4
--R 4cosh(x) sinh(x) + cosh(x)
--R +
--R +--------------+
--R +-+ | 2
--R tanh(x)\|a +-+ +-+ coth(x)\|b sech(x) + a
--R - 2a atanh(-----------------) + 2\|a \|b atan(------------------------)
--R +--------------+ +-+
--R | 2 \|b
--R \|b sech(x) + a
--R /
--R +-+
--R 2\|a
--R Type: Expression(Integer)
--E 59
--S 60 of 526
--d0409a:= D(m0409a,x)
--E 60
--S 61 of 526
m0409b:= a0409.2-r0409
--R
--R
--R (60)
--R -
--R a
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2
--R sinh(x) + 2cosh(x)sinh(x) + cosh(x)
--R +
--R a
--R *
--R log
--R 2 2
--R (- b sinh(x) - 2b cosh(x)sinh(x) - b cosh(x) - a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - b sinh(x) - 4b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 6b cosh(x) + 3b + a)sinh(x)
--R +
--R 3 4
--R (- 4b cosh(x) + (6b + 2a)cosh(x))sinh(x) - b cosh(x)
--R +
--R 2
--R (3b + a)cosh(x) + a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x)
--R +
--R 3 4
--R 4cosh(x) sinh(x) + cosh(x)
--R +
--R +--------------+
--R +-+ | 2
--R tanh(x)\|a +-+ +-+ coth(x)\|b sech(x) + a
--R - 2a atanh(-----------------) + 2\|a \|b atan(------------------------)
--R +--------------+ +-+
--R | 2 \|b
--R \|b sech(x) + a
--R +
--R -
--R +-+ +-+
--R 4\|a \|b
--R *
--R atan
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2 +-+ +-+
--R (2sinh(x) + 4cosh(x)sinh(x) + 2cosh(x) )\|a \|b
--R /
--R +-+
--R 2\|a
--R Type: Expression(Integer)
--E 61
--S 62 of 526
--d0409b:= D(m0409b,x)
--E 62
--S 63 of 526
t0410:= 1/(1+sech(x)^2)^(1/2)
--R
--R
--R 1
--R (61) ---------------
--R +------------+
--R | 2
--R \|sech(x) + 1
--R Type: Expression(Integer)
--E 63
--S 64 of 526
r0410:= atanh(tanh(x)/(2-tanh(x)^2)^(1/2))
--R
--R
--R tanh(x)
--R (62) atanh(-----------------)
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R Type: Expression(Integer)
--E 64
--S 65 of 526
a0410:= integrate(t0410,x)
--R
--R
--R (63)
--R log
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6 2
--R |------------------------------------- - sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2cosh(x)sinh(x) - cosh(x) - 1
--R +
--R -
--R log
--R 2 2
--R (- sinh(x) - 2cosh(x)sinh(x) - cosh(x) - 1)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) + 4)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) + 8cosh(x))sinh(x) + cosh(x) + 4cosh(x) - 1
--R /
--R 2
--R Type: Union(Expression(Integer),...)
--E 65
--S 66 of 526
m0410:= a0410-r0410
--R
--R
--R (64)
--R log
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6 2
--R |------------------------------------- - sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2cosh(x)sinh(x) - cosh(x) - 1
--R +
--R -
--R log
--R 2 2
--R (- sinh(x) - 2cosh(x)sinh(x) - cosh(x) - 1)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) + 4)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) + 8cosh(x))sinh(x) + cosh(x) + 4cosh(x) - 1
--R +
--R tanh(x)
--R - 2atanh(-----------------)
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R /
--R 2
--R Type: Expression(Integer)
--E 66
--S 67 of 526
d0410:= D(m0410,x)
--R
--R
--R (65)
--R 8 7 6
--R - 2sinh(x) - 4cosh(x)sinh(x) - 9sinh(x)
--R +
--R 3 5
--R (4cosh(x) - 10cosh(x))sinh(x)
--R +
--R 4 2 4
--R (4cosh(x) + 9cosh(x) - 10)sinh(x)
--R +
--R 5 3 3
--R (4cosh(x) + 20cosh(x) )sinh(x)
--R +
--R 4 2 2
--R (9cosh(x) + 20cosh(x) - 3)sinh(x)
--R +
--R 7 5 8
--R (- 4cosh(x) - 10cosh(x) + 6cosh(x))sinh(x) - 2cosh(x)
--R +
--R 6 4 2
--R - 9cosh(x) - 10cosh(x) - 3cosh(x)
--R *
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R +
--R 8 7 6
--R - 2sinh(x) - 4cosh(x)sinh(x) - 12sinh(x)
--R +
--R 3 5
--R (4cosh(x) - 12cosh(x))sinh(x)
--R +
--R 4 2 4
--R (4cosh(x) + 12cosh(x) - 18)sinh(x)
--R +
--R 5 3 3 4 2 2
--R (4cosh(x) + 24cosh(x) )sinh(x) + (12cosh(x) + 36cosh(x) )sinh(x)
--R +
--R 7 5 8 6 4
--R (- 4cosh(x) - 12cosh(x) )sinh(x) - 2cosh(x) - 12cosh(x) - 18cosh(x)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 10 9 2 8
--R sinh(x) + 4cosh(x)sinh(x) + (5cosh(x) + 5)sinh(x)
--R +
--R 7 4 2 6
--R 16cosh(x)sinh(x) + (- 6cosh(x) + 12cosh(x) + 13)sinh(x)
--R +
--R 5 3 5
--R (- 8cosh(x) - 16cosh(x) + 16cosh(x))sinh(x)
--R +
--R 6 4 2 4
--R (- 6cosh(x) - 34cosh(x) - 5cosh(x) + 27)sinh(x)
--R +
--R 5 3 3
--R (- 16cosh(x) - 16cosh(x) + 24cosh(x))sinh(x)
--R +
--R 8 6 4 2 2
--R (5cosh(x) + 12cosh(x) - 5cosh(x) - 6cosh(x) + 18)sinh(x)
--R +
--R 9 7 5 3
--R (4cosh(x) + 16cosh(x) + 16cosh(x) + 24cosh(x) + 36cosh(x))sinh(x)
--R +
--R 10 8 6 4 2
--R cosh(x) + 5cosh(x) + 13cosh(x) + 27cosh(x) + 18cosh(x)
--R *
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R +
--R 10 9 2 8
--R sinh(x) + 4cosh(x)sinh(x) + (5cosh(x) + 8)sinh(x)
--R +
--R 7 4 2 6
--R 22cosh(x)sinh(x) + (- 6cosh(x) + 12cosh(x) + 20)sinh(x)
--R +
--R 5 3 5
--R (- 8cosh(x) - 22cosh(x) + 34cosh(x))sinh(x)
--R +
--R 6 4 2 4
--R (- 6cosh(x) - 40cosh(x) - 12cosh(x) + 22)sinh(x)
--R +
--R 5 3 3
--R (- 22cosh(x) - 52cosh(x) + 26cosh(x))sinh(x)
--R +
--R 8 6 4 2 2
--R (5cosh(x) + 12cosh(x) - 12cosh(x) + 8cosh(x) + 27)sinh(x)
--R +
--R 9 7 5 3
--R (4cosh(x) + 22cosh(x) + 34cosh(x) + 26cosh(x) + 42cosh(x))sinh(x)
--R +
--R 10 8 6 4 2
--R cosh(x) + 8cosh(x) + 20cosh(x) + 22cosh(x) + 27cosh(x) + 18
--R /
--R 8 7 6
--R 2sinh(x) + 4cosh(x)sinh(x) + 12sinh(x)
--R +
--R 3 5
--R (- 4cosh(x) + 12cosh(x))sinh(x)
--R +
--R 4 2 4
--R (- 4cosh(x) - 12cosh(x) + 18)sinh(x)
--R +
--R 5 3 3
--R (- 4cosh(x) - 24cosh(x) )sinh(x)
--R +
--R 4 2 2 7 5
--R (- 12cosh(x) - 36cosh(x) )sinh(x) + (4cosh(x) + 12cosh(x) )sinh(x)
--R +
--R 8 6 4
--R 2cosh(x) + 12cosh(x) + 18cosh(x)
--R *
--R +-------------------------------------+
--R +--------------+ | 2 2
--R | 2 | 2sinh(x) + 2cosh(x) + 6
--R \|- tanh(x) + 2 |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 10 9 2 8
--R - sinh(x) - 4cosh(x)sinh(x) + (- 5cosh(x) - 8)sinh(x)
--R +
--R 7 4 2 6
--R - 22cosh(x)sinh(x) + (6cosh(x) - 12cosh(x) - 20)sinh(x)
--R +
--R 5 3 5
--R (8cosh(x) + 22cosh(x) - 34cosh(x))sinh(x)
--R +
--R 6 4 2 4
--R (6cosh(x) + 40cosh(x) + 12cosh(x) - 22)sinh(x)
--R +
--R 5 3 3
--R (22cosh(x) + 52cosh(x) - 26cosh(x))sinh(x)
--R +
--R 8 6 4 2 2
--R (- 5cosh(x) - 12cosh(x) + 12cosh(x) - 8cosh(x) - 27)sinh(x)
--R +
--R 9 7 5 3
--R (- 4cosh(x) - 22cosh(x) - 34cosh(x) - 26cosh(x) - 42cosh(x))
--R *
--R sinh(x)
--R +
--R 10 8 6 4 2
--R - cosh(x) - 8cosh(x) - 20cosh(x) - 22cosh(x) - 27cosh(x) - 18
--R *
--R +--------------+
--R | 2
--R \|- tanh(x) + 2
--R Type: Expression(Integer)
--E 67
--S 68 of 526
t0411:= 1/(1-sech(x)^2)^(1/2)
--R
--R
--R 1
--R (66) -----------------
--R +--------------+
--R | 2
--R \|- sech(x) + 1
--R Type: Expression(Integer)
--E 68
--S 69 of 526
r0411:= log(sinh(x))*tanh(x)/(tanh(x)^2)^(1/2)
--R
--R
--R tanh(x)log(sinh(x))
--R (67) -------------------
--R +--------+
--R | 2
--R \|tanh(x)
--R Type: Expression(Integer)
--E 69
--S 70 of 526
a0411:= integrate(t0411,x)
--R
--R
--R 2sinh(x)
--R (68) log(- -----------------) - x
--R sinh(x) - cosh(x)
--R Type: Union(Expression(Integer),...)
--E 70
--S 71 of 526
m0411:= a0411-r0411
--R
--R
--R +--------+
--R 2sinh(x) | 2
--R (log(- -----------------) - x)\|tanh(x) - tanh(x)log(sinh(x))
--R sinh(x) - cosh(x)
--R (69) ---------------------------------------------------------------
--R +--------+
--R | 2
--R \|tanh(x)
--R Type: Expression(Integer)
--E 71
--S 72 of 526
d0411:= D(m0411,x)
--R
--R
--R +--------+
--R | 2
--R cosh(x)\|tanh(x) - cosh(x)tanh(x)
--R (70) -----------------------------------
--R +--------+
--R | 2
--R sinh(x)\|tanh(x)
--R Type: Expression(Integer)
--E 72
--S 73 of 526
t0412:= 1/(-1+sech(x)^2)^(1/2)
--R
--R
--R 1
--R (71) ---------------
--R +------------+
--R | 2
--R \|sech(x) - 1
--R Type: Expression(Integer)
--E 73
--S 74 of 526
r0412:= log(sinh(x))*tanh(x)/(-tanh(x)^2)^(1/2)
--R
--R
--R tanh(x)log(sinh(x))
--R (72) -------------------
--R +----------+
--R | 2
--R \|- tanh(x)
--R Type: Expression(Integer)
--E 74
--S 75 of 526
a0412:= integrate(t0412,x)
--R
--R
--R (73) 0
--R Type: Union(Expression(Integer),...)
--E 75
--S 76 of 526
m0412:= a0412-r0412
--R
--R
--R tanh(x)log(sinh(x))
--R (74) - -------------------
--R +----------+
--R | 2
--R \|- tanh(x)
--R Type: Expression(Integer)
--E 76
--S 77 of 526
d0412:= D(m0412,x)
--R
--R
--R cosh(x)tanh(x)
--R (75) - --------------------
--R +----------+
--R | 2
--R sinh(x)\|- tanh(x)
--R Type: Expression(Integer)
--E 77
--S 78 of 526
t0413:= 1/(-1-sech(x)^2)^(1/2)
--R
--R
--R 1
--R (76) -----------------
--R +--------------+
--R | 2
--R \|- sech(x) - 1
--R Type: Expression(Integer)
--E 78
--S 79 of 526
r0413:= atan(tanh(x)/(-2+tanh(x)^2)^(1/2))
--R
--R
--R tanh(x)
--R (77) atan(---------------)
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R Type: Expression(Integer)
--E 79
--S 80 of 526
a0413:= integrate(t0413,x)
--R
--R
--R (78)
--R +----------------------+
--R +---+ | x 4 x 2 x 2
--R +---+ \|- 1 \|- (%e ) - 6(%e ) - 1 + (%e ) + 1
--R \|- 1 log(--------------------------------------------)
--R +---+ x 2
--R \|- 1 (%e )
--R +
--R -
--R +---+
--R \|- 1
--R *
--R log
--R +----------------------+
--R x 2 | x 4 x 2 +---+ x 4
--R (2(%e ) + 2)\|- (%e ) - 6(%e ) - 1 + 2\|- 1 (%e )
--R +
--R +---+ x 2 +---+
--R - 8\|- 1 (%e ) - 2\|- 1
--R /
--R +---+ x 4
--R \|- 1 (%e )
--R /
--R 2
--R Type: Union(Expression(Integer),...)
--E 80
--S 81 of 526
m0413:= a0413-r0413
--R
--R
--R (79)
--R +----------------------+
--R +---+ | x 4 x 2 x 2
--R +---+ \|- 1 \|- (%e ) - 6(%e ) - 1 + (%e ) + 1
--R \|- 1 log(--------------------------------------------)
--R +---+ x 2
--R \|- 1 (%e )
--R +
--R -
--R +---+
--R \|- 1
--R *
--R log
--R +----------------------+
--R x 2 | x 4 x 2 +---+ x 4
--R (2(%e ) + 2)\|- (%e ) - 6(%e ) - 1 + 2\|- 1 (%e )
--R +
--R +---+ x 2 +---+
--R - 8\|- 1 (%e ) - 2\|- 1
--R /
--R +---+ x 4
--R \|- 1 (%e )
--R +
--R tanh(x)
--R - 2atan(---------------)
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R /
--R 2
--R Type: Expression(Integer)
--E 81
--S 82 of 526
d0413:= D(m0413,x)
--R
--R
--R (80)
--R +---+ x 6 +---+ x 4 +---+ x 2 +---+
--R (5\|- 1 (%e ) + 11\|- 1 (%e ) + 7\|- 1 (%e ) + \|- 1 )
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R x 8 x 6 x 4 x 2
--R 3(%e ) + 22(%e ) + 28(%e ) + 10(%e ) + 1
--R *
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R +
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (- 3(%e ) - 19(%e ) - 9(%e ) - 1)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2 +---+
--R 5\|- 1 (%e ) + 36\|- 1 (%e ) + 42\|- 1 (%e ) + 12\|- 1 (%e ) + \|- 1
--R /
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (3(%e ) + 19(%e ) + 9(%e ) + 1)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R - 5\|- 1 (%e ) - 36\|- 1 (%e ) - 42\|- 1 (%e ) - 12\|- 1 (%e )
--R +
--R +---+
--R - \|- 1
--R *
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R Type: Expression(Integer)
--E 82
--S 83 of 526
t0414:= 1/(a+b*sech(x)^2)^(1/2)
--R
--R
--R 1
--R (81) -----------------
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R Type: Expression(Integer)
--E 83
--S 84 of 526
r0414:= atanh(a^(1/2)*tanh(x)/(a+b*sech(x)^2)^(1/2))/a^(1/2)
--R
--R
--R +-+
--R tanh(x)\|a
--R atanh(-----------------)
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R (82) ------------------------
--R +-+
--R \|a
--R Type: Expression(Integer)
--E 84
--S 85 of 526
a0414:= integrate(t0414,x)
--R
--R
--R (83)
--R -
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2
--R sinh(x) + 2cosh(x)sinh(x) + cosh(x)
--R +
--R log
--R 2 2
--R (- b sinh(x) - 2b cosh(x)sinh(x) - b cosh(x) - a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - b sinh(x) - 4b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 6b cosh(x) + 3b + a)sinh(x)
--R +
--R 3 4
--R (- 4b cosh(x) + (6b + 2a)cosh(x))sinh(x) - b cosh(x)
--R +
--R 2
--R (3b + a)cosh(x) + a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2 3
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x) + 4cosh(x) sinh(x)
--R +
--R 4
--R cosh(x)
--R /
--R +-+
--R 2\|a
--R Type: Union(Expression(Integer),...)
--E 85
--S 86 of 526
m0414:= a0414-r0414
--R
--R
--R (84)
--R -
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2
--R sinh(x) + 2cosh(x)sinh(x) + cosh(x)
--R +
--R log
--R 2 2
--R (- b sinh(x) - 2b cosh(x)sinh(x) - b cosh(x) - a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - b sinh(x) - 4b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 6b cosh(x) + 3b + a)sinh(x)
--R +
--R 3 4
--R (- 4b cosh(x) + (6b + 2a)cosh(x))sinh(x) - b cosh(x)
--R +
--R 2
--R (3b + a)cosh(x) + a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2 3
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x) + 4cosh(x) sinh(x)
--R +
--R 4
--R cosh(x)
--R +
--R +-+
--R tanh(x)\|a
--R - 2atanh(-----------------)
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R /
--R +-+
--R 2\|a
--R Type: Expression(Integer)
--E 86
--S 87 of 526
d0414:= D(m0414,x)
--R
--R
--R (85)
--R 8 2 2 2 6
--R - a b sinh(x) + (4a b cosh(x) - 2b - 2a b - 2a )sinh(x)
--R +
--R 2 5
--R (- 4b - 4a b)cosh(x)sinh(x)
--R +
--R 4 2 2 2 2
--R - 6a b cosh(x) + (2b + 2a b + 2a )cosh(x) - 2b - 5a b
--R +
--R 2
--R - 4a
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 2 3
--R ((8b + 8a b)cosh(x) + 4a cosh(x))sinh(x)
--R +
--R 6 2 2 4
--R 4a b cosh(x) + (2b + 2a b + 2a )cosh(x)
--R +
--R 2 2 2
--R (4b + 10a b)cosh(x) - 4a b - 2a
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3 2
--R ((- 4b - 4a b)cosh(x) + 4a cosh(x) + (8a b + 4a )cosh(x))
--R *
--R sinh(x)
--R +
--R 8 2 2 6
--R - a b cosh(x) + (- 2b - 2a b - 2a )cosh(x)
--R +
--R 2 2 4 2 2
--R (- 2b - 5a b - 4a )cosh(x) + (- 4a b - 2a )cosh(x)
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 3 8 2 3 7
--R (4a b + a )sinh(x) + (8a b + 2a )cosh(x)sinh(x)
--R +
--R 2 2 3 6
--R (8a b + 6a b + 3a )sinh(x)
--R +
--R 2 3 3 2 2 3 5
--R ((- 8a b - 2a )cosh(x) + (16a b + 12a b + 2a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 2 3 2 2
--R (- 8a b - 2a )cosh(x) + (- 8a b - 6a b - 3a )cosh(x) + 4a b
--R +
--R 3
--R 5a
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 5 2 2 3 3
--R (- 8a b - 2a )cosh(x) + (- 32a b - 24a b - 4a )cosh(x)
--R +
--R 3
--R - 2a cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 4 2 3 2 2
--R (- 8a b - 6a b - 3a )cosh(x) + (- 8a b + 2a )cosh(x) + 10a b
--R +
--R 3
--R 5a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 7 2 2 3 5
--R (8a b + 2a )cosh(x) + (16a b + 12a b + 2a )cosh(x)
--R +
--R 3 3 2 3
--R - 2a cosh(x) + (- 4a b - 2a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 8 2 2 3 6
--R (4a b + a )cosh(x) + (8a b + 6a b + 3a )cosh(x)
--R +
--R 2 3 4 2 3 2 2 2 3
--R (4a b + 5a )cosh(x) + (10a b + 5a )cosh(x) + 8a b + 8a b + 2a
--R *
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R +
--R 2 3 6
--R (- 4a b - 2a )sinh(x)
--R +
--R 2 3 2 2 2 3 4
--R ((4a b + 2a )cosh(x) - 8a b - 8a b - 4a )sinh(x)
--R +
--R 3 3
--R 4a cosh(x)sinh(x)
--R +
--R 2 3 4 2 2 2 2 3
--R ((4a b + 2a )cosh(x) + (16a b + 16a b)cosh(x) - 4a b - 2a )
--R *
--R 2
--R sinh(x)
--R +
--R 3 3 2 3 2 3 6
--R (4a cosh(x) + (8a b + 4a )cosh(x))sinh(x) + (- 4a b - 2a )cosh(x)
--R +
--R 2 2 3 4 2 3 2
--R (- 8a b - 8a b - 4a )cosh(x) + (- 4a b - 2a )cosh(x)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 10 2 9
--R - a b sinh(x) - 4a b cosh(x)sinh(x)
--R +
--R 2 2 2 2 3 8
--R (- 5a b cosh(x) - 2a b + a b + a )sinh(x)
--R +
--R 2 3 7
--R (- 8a b + 2a )cosh(x)sinh(x)
--R +
--R 2 4 2 2 2 2 2 3
--R (6a b cosh(x) + (- 8a b - 4a b)cosh(x) + 4a b + 9a b + 3a )
--R *
--R 6
--R sinh(x)
--R +
--R 2 5 2 3 3
--R 8a b cosh(x) + (8a b - 2a )cosh(x)
--R +
--R 2 2 3
--R (8a b + 12a b + 2a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 6 2 2 3 4
--R 6a b cosh(x) + (20a b + 6a b - 2a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (- 4a b - a b - 3a )cosh(x) + 14a b + 11a b + 5a
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 5 2 2 3 3
--R (8a b - 2a )cosh(x) + (- 16a b - 8a b - 4a )cosh(x)
--R +
--R 2 2 3
--R (16a b + 8a b - 2a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 8 2 2 6
--R - 5a b cosh(x) + (- 8a b - 4a b)cosh(x)
--R +
--R 2 2 3 4 2 2 3 2
--R (- 4a b - a b - 3a )cosh(x) + (4a b - 6a b + 2a )cosh(x)
--R +
--R 3 2 2 3
--R 8b + 8a b + 12a b + 5a
--R *
--R 2
--R sinh(x)
--R +
--R 2 9 2 3 7
--R - 4a b cosh(x) + (- 8a b + 2a )cosh(x)
--R +
--R 2 2 3 5 2 2 3 3
--R (8a b + 12a b + 2a )cosh(x) + (16a b + 8a b - 2a )cosh(x)
--R +
--R 3 2 3
--R (16b + 16a b - 2a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 10 2 2 3 8
--R - a b cosh(x) + (- 2a b + a b + a )cosh(x)
--R +
--R 2 2 3 6 2 2 3 4
--R (4a b + 9a b + 3a )cosh(x) + (14a b + 11a b + 5a )cosh(x)
--R +
--R 3 2 2 3 2 2 2 3
--R (8b + 8a b + 12a b + 5a )cosh(x) + 8a b + 8a b + 2a
--R *
--R +-+
--R \|a
--R /
--R 2 3 6
--R (4a b + 2a )sinh(x)
--R +
--R 2 3 2 2 2 3 4
--R ((- 4a b - 2a )cosh(x) + 8a b + 8a b + 4a )sinh(x)
--R +
--R 3 3
--R - 4a cosh(x)sinh(x)
--R +
--R 2 3 4 2 2 2 2
--R (- 4a b - 2a )cosh(x) + (- 16a b - 16a b)cosh(x) + 4a b
--R +
--R 3
--R 2a
--R *
--R 2
--R sinh(x)
--R +
--R 3 3 2 3
--R (- 4a cosh(x) + (- 8a b - 4a )cosh(x))sinh(x)
--R +
--R 2 3 6 2 2 3 4
--R (4a b + 2a )cosh(x) + (8a b + 8a b + 4a )cosh(x)
--R +
--R 2 3 2
--R (4a b + 2a )cosh(x)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 10 2 9
--R a b sinh(x) + 4a b cosh(x)sinh(x)
--R +
--R 2 2 2 2 3 8
--R (5a b cosh(x) + 2a b - a b - a )sinh(x)
--R +
--R 2 3 7
--R (8a b - 2a )cosh(x)sinh(x)
--R +
--R 2 4 2 2 2 2 2 3
--R (- 6a b cosh(x) + (8a b + 4a b)cosh(x) - 4a b - 9a b - 3a )
--R *
--R 6
--R sinh(x)
--R +
--R 2 5 2 3 3
--R - 8a b cosh(x) + (- 8a b + 2a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 12a b - 2a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 6 2 2 3 4
--R - 6a b cosh(x) + (- 20a b - 6a b + 2a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (4a b + a b + 3a )cosh(x) - 14a b - 11a b - 5a
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 5 2 2 3 3
--R (- 8a b + 2a )cosh(x) + (16a b + 8a b + 4a )cosh(x)
--R +
--R 2 2 3
--R (- 16a b - 8a b + 2a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 8 2 2 6
--R 5a b cosh(x) + (8a b + 4a b)cosh(x)
--R +
--R 2 2 3 4 2 2 3 2
--R (4a b + a b + 3a )cosh(x) + (- 4a b + 6a b - 2a )cosh(x)
--R +
--R 3 2 2 3
--R - 8b - 8a b - 12a b - 5a
--R *
--R 2
--R sinh(x)
--R +
--R 2 9 2 3 7
--R 4a b cosh(x) + (8a b - 2a )cosh(x)
--R +
--R 2 2 3 5
--R (- 8a b - 12a b - 2a )cosh(x)
--R +
--R 2 2 3 3 3 2 3
--R (- 16a b - 8a b + 2a )cosh(x) + (- 16b - 16a b + 2a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 10 2 2 3 8
--R a b cosh(x) + (2a b - a b - a )cosh(x)
--R +
--R 2 2 3 6 2 2 3 4
--R (- 4a b - 9a b - 3a )cosh(x) + (- 14a b - 11a b - 5a )cosh(x)
--R +
--R 3 2 2 3 2 2 2 3
--R (- 8b - 8a b - 12a b - 5a )cosh(x) - 8a b - 8a b - 2a
--R *
--R +-+
--R \|a
--R *
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R Type: Expression(Integer)
--E 87
--S 88 of 526
t0415:= (1+sech(x)^2)^(3/2)
--R
--R
--R +------------+
--R 2 | 2
--R (86) (sech(x) + 1)\|sech(x) + 1
--R Type: Expression(Integer)
--E 88
--S 89 of 526
r0415:= 2*asin(1/2*2^(1/2)*tanh(x))+atanh(tanh(x)/(2-tanh(x)^2)^(1/2))+_
1/2*tanh(x)*(2-tanh(x)^2)^(1/2)
--R
--R
--R (87)
--R +--------------+ +-+
--R tanh(x) | 2 \|2 tanh(x)
--R 2atanh(-----------------) + tanh(x)\|- tanh(x) + 2 + 4asin(-----------)
--R +--------------+ 2
--R | 2
--R \|- tanh(x) + 2
--R -------------------------------------------------------------------------
--R 2
--R Type: Expression(Integer)
--E 89
--S 90 of 526
a0415:= integrate(t0415,x)
--R
--R
--R (88)
--R 6 5 2 4
--R sinh(x) + 6cosh(x)sinh(x) + (15cosh(x) + 5)sinh(x)
--R +
--R 3 3
--R (20cosh(x) + 20cosh(x))sinh(x)
--R +
--R 4 2 2
--R (15cosh(x) + 30cosh(x) + 7)sinh(x)
--R +
--R 5 3 6
--R (6cosh(x) + 20cosh(x) + 14cosh(x))sinh(x) + cosh(x)
--R +
--R 4 2
--R 5cosh(x) + 7cosh(x) + 3
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 8 7 2 6
--R - sinh(x) - 8cosh(x)sinh(x) + (- 28cosh(x) - 8)sinh(x)
--R +
--R 3 5
--R (- 56cosh(x) - 48cosh(x))sinh(x)
--R +
--R 4 2 4
--R (- 70cosh(x) - 120cosh(x) - 18)sinh(x)
--R +
--R 5 3 3
--R (- 56cosh(x) - 160cosh(x) - 72cosh(x))sinh(x)
--R +
--R 6 4 2 2
--R (- 28cosh(x) - 120cosh(x) - 108cosh(x) - 16)sinh(x)
--R +
--R 7 5 3 8
--R (- 8cosh(x) - 48cosh(x) - 72cosh(x) - 32cosh(x))sinh(x) - cosh(x)
--R +
--R 6 4 2
--R - 8cosh(x) - 18cosh(x) - 16cosh(x) - 5
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6 2
--R |------------------------------------- - sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2cosh(x)sinh(x) - cosh(x) - 1
--R +
--R 6 5 2 4
--R - sinh(x) - 6cosh(x)sinh(x) + (- 15cosh(x) - 5)sinh(x)
--R +
--R 3 3
--R (- 20cosh(x) - 20cosh(x))sinh(x)
--R +
--R 4 2 2
--R (- 15cosh(x) - 30cosh(x) - 7)sinh(x)
--R +
--R 5 3 6
--R (- 6cosh(x) - 20cosh(x) - 14cosh(x))sinh(x) - cosh(x)
--R +
--R 4 2
--R - 5cosh(x) - 7cosh(x) - 3
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 8 7 2 6
--R sinh(x) + 8cosh(x)sinh(x) + (28cosh(x) + 8)sinh(x)
--R +
--R 3 5
--R (56cosh(x) + 48cosh(x))sinh(x)
--R +
--R 4 2 4
--R (70cosh(x) + 120cosh(x) + 18)sinh(x)
--R +
--R 5 3 3
--R (56cosh(x) + 160cosh(x) + 72cosh(x))sinh(x)
--R +
--R 6 4 2 2
--R (28cosh(x) + 120cosh(x) + 108cosh(x) + 16)sinh(x)
--R +
--R 7 5 3 8
--R (8cosh(x) + 48cosh(x) + 72cosh(x) + 32cosh(x))sinh(x) + cosh(x)
--R +
--R 6 4 2
--R 8cosh(x) + 18cosh(x) + 16cosh(x) + 5
--R *
--R log
--R 2 2
--R (- sinh(x) - 2cosh(x)sinh(x) - cosh(x) - 1)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) + 4)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) + 8cosh(x))sinh(x) + cosh(x) + 4cosh(x) - 1
--R +
--R 6 5 2 4
--R 8sinh(x) + 48cosh(x)sinh(x) + (120cosh(x) + 40)sinh(x)
--R +
--R 3 3
--R (160cosh(x) + 160cosh(x))sinh(x)
--R +
--R 4 2 2
--R (120cosh(x) + 240cosh(x) + 56)sinh(x)
--R +
--R 5 3 6
--R (48cosh(x) + 160cosh(x) + 112cosh(x))sinh(x) + 8cosh(x)
--R +
--R 4 2
--R 40cosh(x) + 56cosh(x) + 24
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 8 7 2 6
--R - 8sinh(x) - 64cosh(x)sinh(x) + (- 224cosh(x) - 64)sinh(x)
--R +
--R 3 5
--R (- 448cosh(x) - 384cosh(x))sinh(x)
--R +
--R 4 2 4
--R (- 560cosh(x) - 960cosh(x) - 144)sinh(x)
--R +
--R 5 3 3
--R (- 448cosh(x) - 1280cosh(x) - 576cosh(x))sinh(x)
--R +
--R 6 4 2 2
--R (- 224cosh(x) - 960cosh(x) - 864cosh(x) - 128)sinh(x)
--R +
--R 7 5 3
--R (- 64cosh(x) - 384cosh(x) - 576cosh(x) - 256cosh(x))sinh(x)
--R +
--R 8 6 4 2
--R - 8cosh(x) - 64cosh(x) - 144cosh(x) - 128cosh(x) - 40
--R *
--R atan
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6 2
--R |------------------------------------- - sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2cosh(x)sinh(x) - cosh(x) - 1
--R /
--R 2
--R +
--R 2 2
--R (8sinh(x) + 16cosh(x)sinh(x) + 8cosh(x) + 8)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3 2 2
--R - 8sinh(x) - 32cosh(x)sinh(x) + (- 48cosh(x) - 32)sinh(x)
--R +
--R 3 4 2
--R (- 32cosh(x) - 64cosh(x))sinh(x) - 8cosh(x) - 32cosh(x) - 8
--R /
--R 6 5 2 4
--R 2sinh(x) + 12cosh(x)sinh(x) + (30cosh(x) + 10)sinh(x)
--R +
--R 3 3
--R (40cosh(x) + 40cosh(x))sinh(x)
--R +
--R 4 2 2
--R (30cosh(x) + 60cosh(x) + 14)sinh(x)
--R +
--R 5 3 6 4
--R (12cosh(x) + 40cosh(x) + 28cosh(x))sinh(x) + 2cosh(x) + 10cosh(x)
--R +
--R 2
--R 14cosh(x) + 6
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2sinh(x) + 2cosh(x) + 6
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 8 7 2 6
--R - 2sinh(x) - 16cosh(x)sinh(x) + (- 56cosh(x) - 16)sinh(x)
--R +
--R 3 5
--R (- 112cosh(x) - 96cosh(x))sinh(x)
--R +
--R 4 2 4
--R (- 140cosh(x) - 240cosh(x) - 36)sinh(x)
--R +
--R 5 3 3
--R (- 112cosh(x) - 320cosh(x) - 144cosh(x))sinh(x)
--R +
--R 6 4 2 2
--R (- 56cosh(x) - 240cosh(x) - 216cosh(x) - 32)sinh(x)
--R +
--R 7 5 3 8
--R (- 16cosh(x) - 96cosh(x) - 144cosh(x) - 64cosh(x))sinh(x) - 2cosh(x)
--R +
--R 6 4 2
--R - 16cosh(x) - 36cosh(x) - 32cosh(x) - 10
--R Type: Union(Expression(Integer),...)
--E 90
--S 91 of 526
--m0415:= a0415-r0415
--E 91
--S 92 of 526
--d0415:= D(m0415,x)
--E 92
--S 93 of 526
t0416:= (1-sech(x)^2)^(3/2)
--R
--R
--R +--------------+
--R 2 | 2
--R (89) (- sech(x) + 1)\|- sech(x) + 1
--R Type: Expression(Integer)
--E 93
--S 94 of 526
r0416:= 1/2*coth(x)*(tanh(x)^2)^(1/2)*(2*log(cosh(x))-tanh(x)^2)
--R
--R
--R +--------+
--R 2 | 2
--R (2coth(x)log(cosh(x)) - coth(x)tanh(x) )\|tanh(x)
--R (90) ---------------------------------------------------
--R 2
--R Type: Expression(Integer)
--E 94
--S 95 of 526
a0416:= integrate(t0416,x)
--R
--R
--R (91)
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) + 2)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) + 4cosh(x))sinh(x) + cosh(x) + 2cosh(x) + 1
--R *
--R 2cosh(x)
--R log(- -----------------)
--R sinh(x) - cosh(x)
--R +
--R 4 3 2 2
--R - x sinh(x) - 4x cosh(x)sinh(x) + (- 6x cosh(x) - 2x + 2)sinh(x)
--R +
--R 3 4
--R (- 4x cosh(x) + (- 4x + 4)cosh(x))sinh(x) - x cosh(x)
--R +
--R 2
--R (- 2x + 2)cosh(x) - x
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) + 2)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) + 4cosh(x))sinh(x) + cosh(x) + 2cosh(x) + 1
--R Type: Union(Expression(Integer),...)
--E 95
--S 96 of 526
m0416:= a0416-r0416
--R
--R
--R (92)
--R 4 3
--R - 2coth(x)sinh(x) - 8cosh(x)coth(x)sinh(x)
--R +
--R 2 2
--R (- 12cosh(x) - 4)coth(x)sinh(x)
--R +
--R 3
--R (- 8cosh(x) - 8cosh(x))coth(x)sinh(x)
--R +
--R 4 2
--R (- 2cosh(x) - 4cosh(x) - 2)coth(x)
--R *
--R log(cosh(x))
--R +
--R 4 3
--R coth(x)sinh(x) + 4cosh(x)coth(x)sinh(x)
--R +
--R 2 2
--R (6cosh(x) + 2)coth(x)sinh(x)
--R +
--R 3
--R (4cosh(x) + 4cosh(x))coth(x)sinh(x)
--R +
--R 4 2
--R (cosh(x) + 2cosh(x) + 1)coth(x)
--R *
--R 2
--R tanh(x)
--R *
--R +--------+
--R | 2
--R \|tanh(x)
--R +
--R 4 3 2 2
--R 2sinh(x) + 8cosh(x)sinh(x) + (12cosh(x) + 4)sinh(x)
--R +
--R 3 4 2
--R (8cosh(x) + 8cosh(x))sinh(x) + 2cosh(x) + 4cosh(x) + 2
--R *
--R 2cosh(x)
--R log(- -----------------)
--R sinh(x) - cosh(x)
--R +
--R 4 3 2 2
--R - 2x sinh(x) - 8x cosh(x)sinh(x) + (- 12x cosh(x) - 4x + 4)sinh(x)
--R +
--R 3 4
--R (- 8x cosh(x) + (- 8x + 8)cosh(x))sinh(x) - 2x cosh(x)
--R +
--R 2
--R (- 4x + 4)cosh(x) - 2x
--R /
--R 4 3 2 2
--R 2sinh(x) + 8cosh(x)sinh(x) + (12cosh(x) + 4)sinh(x)
--R +
--R 3 4 2
--R (8cosh(x) + 8cosh(x))sinh(x) + 2cosh(x) + 4cosh(x) + 2
--R Type: Expression(Integer)
--E 96
--S 97 of 526
d0416:= D(m0416,x)
--R
--R
--R (93)
--R 7 6 2 5
--R 2sinh(x) + 12cosh(x)sinh(x) + (30cosh(x) + 6)sinh(x)
--R +
--R 3 4
--R (40cosh(x) + 16cosh(x))sinh(x)
--R +
--R 4 2 3
--R (30cosh(x) + 4cosh(x) + 6)sinh(x)
--R +
--R 5 3 2
--R (12cosh(x) - 24cosh(x) + 20cosh(x))sinh(x)
--R +
--R 6 4 2 5
--R (2cosh(x) - 26cosh(x) + 22cosh(x) + 2)sinh(x) - 8cosh(x)
--R +
--R 3
--R 8cosh(x)
--R *
--R +--------+
--R | 2
--R \|tanh(x)
--R +
--R 6 2 5
--R 2cosh(x)coth(x)sinh(x) + 12cosh(x) coth(x)sinh(x)
--R +
--R 3 4
--R (30cosh(x) + 6cosh(x))coth(x)sinh(x)
--R +
--R 4 2 3
--R (40cosh(x) + 24cosh(x) )coth(x)sinh(x)
--R +
--R 5 3 2
--R (30cosh(x) + 36cosh(x) + 6cosh(x))coth(x)sinh(x)
--R +
--R 6 4 2
--R (12cosh(x) + 24cosh(x) + 12cosh(x) )coth(x)sinh(x)
--R +
--R 7 5 3
--R (2cosh(x) + 6cosh(x) + 6cosh(x) + 2cosh(x))coth(x)
--R *
--R 3
--R tanh(x)
--R +
--R 2 6
--R (2cosh(x)coth(x) - 2cosh(x))sinh(x)
--R +
--R 2 2 2 5
--R (12cosh(x) coth(x) - 12cosh(x) )sinh(x)
--R +
--R 3 2 3 4
--R ((30cosh(x) + 6cosh(x))coth(x) - 30cosh(x) - 6cosh(x))sinh(x)
--R +
--R 4 2 2 4 2
--R ((40cosh(x) + 24cosh(x) )coth(x) - 40cosh(x) - 24cosh(x) )
--R *
--R 3
--R sinh(x)
--R +
--R 5 3 2 5
--R (30cosh(x) + 36cosh(x) + 6cosh(x))coth(x) - 30cosh(x)
--R +
--R 3
--R - 36cosh(x) - 6cosh(x)
--R *
--R 2
--R sinh(x)
--R +
--R 6 4 2 2 6
--R (12cosh(x) + 24cosh(x) + 12cosh(x) )coth(x) - 12cosh(x)
--R +
--R 4 2
--R - 24cosh(x) - 12cosh(x)
--R *
--R sinh(x)
--R +
--R 7 5 3 2
--R (2cosh(x) + 6cosh(x) + 6cosh(x) + 2cosh(x))coth(x)
--R +
--R 7 5 3
--R - 2cosh(x) - 6cosh(x) - 6cosh(x) - 2cosh(x)
--R *
--R 2
--R tanh(x)
--R +
--R 6 2 5
--R - 2cosh(x)coth(x)sinh(x) - 12cosh(x) coth(x)sinh(x)
--R +
--R 3 4
--R (- 30cosh(x) - 6cosh(x))coth(x)sinh(x)
--R +
--R 4 2 3
--R (- 40cosh(x) - 24cosh(x) )coth(x)sinh(x)
--R +
--R 5 3 2
--R (- 30cosh(x) - 36cosh(x) - 6cosh(x))coth(x)sinh(x)
--R +
--R 6 4 2
--R (- 12cosh(x) - 24cosh(x) - 12cosh(x) )coth(x)sinh(x)
--R +
--R 7 5 3
--R (- 2cosh(x) - 6cosh(x) - 6cosh(x) - 2cosh(x))coth(x)
--R *
--R tanh(x)
--R *
--R log(cosh(x))
--R +
--R 6 2 5
--R - 3cosh(x)coth(x)sinh(x) - 18cosh(x) coth(x)sinh(x)
--R +
--R 3 4
--R (- 45cosh(x) - 9cosh(x))coth(x)sinh(x)
--R +
--R 4 2 3
--R (- 60cosh(x) - 36cosh(x) )coth(x)sinh(x)
--R +
--R 5 3 2
--R (- 45cosh(x) - 54cosh(x) - 9cosh(x))coth(x)sinh(x)
--R +
--R 6 4 2
--R (- 18cosh(x) - 36cosh(x) - 18cosh(x) )coth(x)sinh(x)
--R +
--R 7 5 3
--R (- 3cosh(x) - 9cosh(x) - 9cosh(x) - 3cosh(x))coth(x)
--R *
--R 5
--R tanh(x)
--R +
--R 2 6
--R (- cosh(x)coth(x) + cosh(x))sinh(x)
--R +
--R 2 2 2 5
--R (- 6cosh(x) coth(x) + 6cosh(x) )sinh(x)
--R +
--R 3 2 3 4
--R ((- 15cosh(x) - 3cosh(x))coth(x) + 15cosh(x) + 3cosh(x))sinh(x)
--R +
--R 4 2 2 4 2
--R ((- 20cosh(x) - 12cosh(x) )coth(x) + 20cosh(x) + 12cosh(x) )
--R *
--R 3
--R sinh(x)
--R +
--R 5 3 2 5
--R (- 15cosh(x) - 18cosh(x) - 3cosh(x))coth(x) + 15cosh(x)
--R +
--R 3
--R 18cosh(x) + 3cosh(x)
--R *
--R 2
--R sinh(x)
--R +
--R 6 4 2 2 6
--R (- 6cosh(x) - 12cosh(x) - 6cosh(x) )coth(x) + 6cosh(x)
--R +
--R 4 2
--R 12cosh(x) + 6cosh(x)
--R *
--R sinh(x)
--R +
--R 7 5 3 2 7
--R (- cosh(x) - 3cosh(x) - 3cosh(x) - cosh(x))coth(x) + cosh(x)
--R +
--R 5 3
--R 3cosh(x) + 3cosh(x) + cosh(x)
--R *
--R 4
--R tanh(x)
--R +
--R 6 2 5
--R 3cosh(x)coth(x)sinh(x) + 18cosh(x) coth(x)sinh(x)
--R +
--R 3 4
--R (45cosh(x) + 9cosh(x))coth(x)sinh(x)
--R +
--R 4 2 3
--R (60cosh(x) + 36cosh(x) )coth(x)sinh(x)
--R +
--R 5 3 2
--R (45cosh(x) + 54cosh(x) + 9cosh(x))coth(x)sinh(x)
--R +
--R 6 4 2
--R (18cosh(x) + 36cosh(x) + 18cosh(x) )coth(x)sinh(x)
--R +
--R 7 5 3
--R (3cosh(x) + 9cosh(x) + 9cosh(x) + 3cosh(x))coth(x)
--R *
--R 3
--R tanh(x)
--R +
--R 7 6
--R - 2coth(x)sinh(x) - 12cosh(x)coth(x)sinh(x)
--R +
--R 2 5
--R (- 30cosh(x) - 6)coth(x)sinh(x)
--R +
--R 3 4
--R (- 40cosh(x) - 24cosh(x))coth(x)sinh(x)
--R +
--R 4 2 3
--R (- 30cosh(x) - 36cosh(x) - 6)coth(x)sinh(x)
--R +
--R 5 3 2
--R (- 12cosh(x) - 24cosh(x) - 12cosh(x))coth(x)sinh(x)
--R +
--R 6 4 2
--R (- 2cosh(x) - 6cosh(x) - 6cosh(x) - 2)coth(x)sinh(x)
--R *
--R 2
--R tanh(x)
--R /
--R 6 2 5 3 4
--R 2cosh(x)sinh(x) + 12cosh(x) sinh(x) + (30cosh(x) + 6cosh(x))sinh(x)
--R +
--R 4 2 3
--R (40cosh(x) + 24cosh(x) )sinh(x)
--R +
--R 5 3 2
--R (30cosh(x) + 36cosh(x) + 6cosh(x))sinh(x)
--R +
--R 6 4 2 7 5
--R (12cosh(x) + 24cosh(x) + 12cosh(x) )sinh(x) + 2cosh(x) + 6cosh(x)
--R +
--R 3
--R 6cosh(x) + 2cosh(x)
--R *
--R +--------+
--R | 2
--R \|tanh(x)
--R Type: Expression(Integer)
--E 97
--S 98 of 526
t0417:= (-1+sech(x)^2)^(3/2)
--R
--R
--R +------------+
--R 2 | 2
--R (94) (sech(x) - 1)\|sech(x) - 1
--R Type: Expression(Integer)
--E 98
--S 99 of 526
r0417:= -1/2*coth(x)*(-tanh(x)^2)^(1/2)*(2*log(cosh(x))-tanh(x)^2)
--R
--R
--R +----------+
--R 2 | 2
--R (- 2coth(x)log(cosh(x)) + coth(x)tanh(x) )\|- tanh(x)
--R (95) -------------------------------------------------------
--R 2
--R Type: Expression(Integer)
--E 99
--S 100 of 526
a0417:= integrate(t0417,x)
--R
--R
--R (96) 0
--R Type: Union(Expression(Integer),...)
--E 100
--S 101 of 526
m0417:= a0417-r0417
--R
--R
--R +----------+
--R 2 | 2
--R (2coth(x)log(cosh(x)) - coth(x)tanh(x) )\|- tanh(x)
--R (97) -----------------------------------------------------
--R 2
--R Type: Expression(Integer)
--E 101
--S 102 of 526
d0417:= D(m0417,x)
--R
--R
--R (98)
--R 3 2 2
--R 2cosh(x)coth(x)tanh(x) + (2cosh(x)coth(x) - 2cosh(x))tanh(x)
--R +
--R - 2cosh(x)coth(x)tanh(x)
--R *
--R log(cosh(x))
--R +
--R 5 2 4
--R - 3cosh(x)coth(x)tanh(x) + (- cosh(x)coth(x) + cosh(x))tanh(x)
--R +
--R 3 2
--R 3cosh(x)coth(x)tanh(x) - 2coth(x)sinh(x)tanh(x)
--R /
--R +----------+
--R | 2
--R 2cosh(x)\|- tanh(x)
--R Type: Expression(Integer)
--E 102
--S 103 of 526
t0418:= (-1-sech(x)^2)^(3/2)
--R
--R
--R +--------------+
--R 2 | 2
--R (99) (- sech(x) - 1)\|- sech(x) - 1
--R Type: Expression(Integer)
--E 103
--S 104 of 526
r0418:= atan(tanh(x)/(-2+tanh(x)^2)^(1/2))+_
2*atanh(coth(x)*(-2+tanh(x)^2)^(1/2))-_
1/2*tanh(x)*(-2+tanh(x)^2)^(1/2)
--R
--R
--R (100)
--R +------------+
--R | 2 tanh(x)
--R 4atanh(coth(x)\|tanh(x) - 2 ) + 2atan(---------------)
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R +
--R +------------+
--R | 2
--R - tanh(x)\|tanh(x) - 2
--R /
--R 2
--R Type: Expression(Integer)
--E 104
--S 105 of 526
a0418:= integrate(t0418,x)
--R
--R
--R (101)
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (12(%e ) + 28(%e ) + 20(%e ) + 4)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R - 20\|- 1 (%e ) - 64\|- 1 (%e ) - 72\|- 1 (%e ) - 32\|- 1 (%e )
--R +
--R +---+
--R - 4\|- 1
--R *
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 + (- \|- 1 + 2)(%e ) - \|- 1
--R log(---------------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +---+ x 6 +---+ x 4 +---+ x 2 +---+
--R (3\|- 1 (%e ) + 7\|- 1 (%e ) + 5\|- 1 (%e ) + \|- 1 )
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R x 8 x 6 x 4 x 2
--R 5(%e ) + 16(%e ) + 18(%e ) + 8(%e ) + 1
--R *
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 - \|- 1 (%e ) - \|- 1
--R log(-------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (- 12(%e ) - 28(%e ) - 20(%e ) - 4)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R 20\|- 1 (%e ) + 64\|- 1 (%e ) + 72\|- 1 (%e ) + 32\|- 1 (%e )
--R +
--R +---+
--R 4\|- 1
--R *
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 + (- \|- 1 - 2)(%e ) - \|- 1
--R log(---------------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +---+ x 6 +---+ x 4 +---+ x 2 +---+
--R (- 3\|- 1 (%e ) - 7\|- 1 (%e ) - 5\|- 1 (%e ) - \|- 1 )
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R x 8 x 6 x 4 x 2
--R - 5(%e ) - 16(%e ) - 18(%e ) - 8(%e ) - 1
--R *
--R log
--R +----------------------+
--R +---+ x 2 +---+ | x 4 x 2 x 4
--R (- 2\|- 1 (%e ) - 2\|- 1 )\|- (%e ) - 6(%e ) - 1 + 2(%e )
--R +
--R x 2
--R - 8(%e ) - 2
--R /
--R x 4
--R (%e )
--R +
--R +----------------------+
--R +---+ x 6 +---+ x 4 | x 4 x 2 x 8
--R (8\|- 1 (%e ) + 8\|- 1 (%e ) )\|- (%e ) - 6(%e ) - 1 + 8(%e )
--R +
--R x 6 x 4
--R 32(%e ) + 8(%e )
--R /
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (6(%e ) + 14(%e ) + 10(%e ) + 2)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R - 10\|- 1 (%e ) - 32\|- 1 (%e ) - 36\|- 1 (%e ) - 16\|- 1 (%e )
--R +
--R +---+
--R - 2\|- 1
--R Type: Union(Expression(Integer),...)
--E 105
--S 106 of 526
m0418:= a0418-r0418
--R
--R
--R (102)
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (12(%e ) + 28(%e ) + 20(%e ) + 4)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R - 20\|- 1 (%e ) - 64\|- 1 (%e ) - 72\|- 1 (%e ) - 32\|- 1 (%e )
--R +
--R +---+
--R - 4\|- 1
--R *
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 + (- \|- 1 + 2)(%e ) - \|- 1
--R log(---------------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +---+ x 6 +---+ x 4 +---+ x 2 +---+
--R (3\|- 1 (%e ) + 7\|- 1 (%e ) + 5\|- 1 (%e ) + \|- 1 )
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R x 8 x 6 x 4 x 2
--R 5(%e ) + 16(%e ) + 18(%e ) + 8(%e ) + 1
--R *
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 - \|- 1 (%e ) - \|- 1
--R log(-------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (- 12(%e ) - 28(%e ) - 20(%e ) - 4)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R 20\|- 1 (%e ) + 64\|- 1 (%e ) + 72\|- 1 (%e ) + 32\|- 1 (%e )
--R +
--R +---+
--R 4\|- 1
--R *
--R +----------------------+
--R | x 4 x 2 +---+ x 2 +---+
--R \|- (%e ) - 6(%e ) - 1 + (- \|- 1 - 2)(%e ) - \|- 1
--R log(---------------------------------------------------------)
--R x 2
--R (%e )
--R +
--R +---+ x 6 +---+ x 4 +---+ x 2 +---+
--R (- 3\|- 1 (%e ) - 7\|- 1 (%e ) - 5\|- 1 (%e ) - \|- 1 )
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R x 8 x 6 x 4 x 2
--R - 5(%e ) - 16(%e ) - 18(%e ) - 8(%e ) - 1
--R *
--R log
--R +----------------------+
--R +---+ x 2 +---+ | x 4 x 2 x 4
--R (- 2\|- 1 (%e ) - 2\|- 1 )\|- (%e ) - 6(%e ) - 1 + 2(%e )
--R +
--R x 2
--R - 8(%e ) - 2
--R /
--R x 4
--R (%e )
--R +
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (- 12(%e ) - 28(%e ) - 20(%e ) - 4)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R 20\|- 1 (%e ) + 64\|- 1 (%e ) + 72\|- 1 (%e ) + 32\|- 1 (%e )
--R +
--R +---+
--R 4\|- 1
--R *
--R +------------+
--R | 2
--R atanh(coth(x)\|tanh(x) - 2 )
--R +
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (- 6(%e ) - 14(%e ) - 10(%e ) - 2)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R 10\|- 1 (%e ) + 32\|- 1 (%e ) + 36\|- 1 (%e ) + 16\|- 1 (%e )
--R +
--R +---+
--R 2\|- 1
--R *
--R tanh(x)
--R atan(---------------)
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R +
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (3(%e ) + 7(%e ) + 5(%e ) + 1)tanh(x)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R - 5\|- 1 (%e ) - 16\|- 1 (%e ) - 18\|- 1 (%e ) - 8\|- 1 (%e )
--R +
--R +---+
--R - \|- 1
--R *
--R tanh(x)
--R *
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R +
--R +----------------------+
--R +---+ x 6 +---+ x 4 | x 4 x 2 x 8
--R (8\|- 1 (%e ) + 8\|- 1 (%e ) )\|- (%e ) - 6(%e ) - 1 + 8(%e )
--R +
--R x 6 x 4
--R 32(%e ) + 8(%e )
--R /
--R +----------------------+
--R x 6 x 4 x 2 | x 4 x 2
--R (6(%e ) + 14(%e ) + 10(%e ) + 2)\|- (%e ) - 6(%e ) - 1
--R +
--R +---+ x 8 +---+ x 6 +---+ x 4 +---+ x 2
--R - 10\|- 1 (%e ) - 32\|- 1 (%e ) - 36\|- 1 (%e ) - 16\|- 1 (%e )
--R +
--R +---+
--R - 2\|- 1
--R Type: Expression(Integer)
--E 106
--S 107 of 526
d0418:= D(m0418,x)
--R
--R
--R (103)
--R +---+ 2 x 18 +---+ 2 x 16
--R 129\|- 1 coth(x) (%e ) + 1663\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 14 +---+ 2 x 12
--R 7290\|- 1 coth(x) (%e ) + 13482\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 10 +---+ 2 x 8
--R 12348\|- 1 coth(x) (%e ) + 6216\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 6 +---+ 2 x 4
--R 1806\|- 1 coth(x) (%e ) + 302\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 2 +---+ 2
--R 27\|- 1 coth(x) (%e ) + \|- 1 coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R +---+ 2 +---+ x 18
--R (- 258\|- 1 coth(x) - 129\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 16
--R (- 3326\|- 1 coth(x) - 1663\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 14
--R (- 14580\|- 1 coth(x) - 7290\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 12
--R (- 26964\|- 1 coth(x) - 13482\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 10
--R (- 24696\|- 1 coth(x) - 12348\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 8
--R (- 12432\|- 1 coth(x) - 6216\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 6
--R (- 3612\|- 1 coth(x) - 1806\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 4
--R (- 604\|- 1 coth(x) - 302\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 2 +---+ 2 +---+
--R (- 54\|- 1 coth(x) - 27\|- 1 )(%e ) - 2\|- 1 coth(x) - \|- 1
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R 2 x 20 2 x 18 2 x 16
--R 127coth(x) (%e ) + 2046coth(x) (%e ) + 11789coth(x) (%e )
--R +
--R 2 x 14 2 x 12
--R 30168coth(x) (%e ) + 38118coth(x) (%e )
--R +
--R 2 x 10 2 x 8 2 x 6
--R 26388coth(x) (%e ) + 10706coth(x) (%e ) + 2616coth(x) (%e )
--R +
--R 2 x 4 2 x 2 2
--R 379coth(x) (%e ) + 30coth(x) (%e ) + coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R 2 x 20 2 x 18
--R (- 254coth(x) - 127)(%e ) + (- 4092coth(x) - 2046)(%e )
--R +
--R 2 x 16 2 x 14
--R (- 23578coth(x) - 11789)(%e ) + (- 60336coth(x) - 30168)(%e )
--R +
--R 2 x 12 2 x 10
--R (- 76236coth(x) - 38118)(%e ) + (- 52776coth(x) - 26388)(%e )
--R +
--R 2 x 8 2 x 6
--R (- 21412coth(x) - 10706)(%e ) + (- 5232coth(x) - 2616)(%e )
--R +
--R 2 x 4 2 x 2 2
--R (- 758coth(x) - 379)(%e ) + (- 60coth(x) - 30)(%e ) - 2coth(x) - 1
--R *
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R +
--R 2 x 18 2 x 16 2 x 14
--R - 127coth(x) (%e ) - 903coth(x) (%e ) - 2646coth(x) (%e )
--R +
--R 2 x 12 2 x 10 2 x 8
--R - 4210coth(x) (%e ) - 4004coth(x) (%e ) - 2352coth(x) (%e )
--R +
--R 2 x 6 2 x 4 2 x 2
--R - 850coth(x) (%e ) - 182coth(x) (%e ) - 21coth(x) (%e )
--R +
--R 2
--R - coth(x)
--R *
--R 6
--R tanh(x)
--R +
--R 2 x 18 2 x 16
--R (508coth(x) + 127)(%e ) + (3612coth(x) + 903)(%e )
--R +
--R 2 x 14 2 x 12
--R (10584coth(x) + 2646)(%e ) + (16840coth(x) + 4210)(%e )
--R +
--R 2 x 10 2 x 8
--R (16016coth(x) + 4004)(%e ) + (9408coth(x) + 2352)(%e )
--R +
--R 2 x 6 2 x 4
--R (3400coth(x) + 850)(%e ) + (728coth(x) + 182)(%e )
--R +
--R 2 x 2 2
--R (84coth(x) + 21)(%e ) + 4coth(x) + 1
--R *
--R 4
--R tanh(x)
--R +
--R x 18 x 16 x 14
--R - 254coth(x)(%e ) - 1806coth(x)(%e ) - 5292coth(x)(%e )
--R +
--R x 12 x 10 x 8
--R - 8420coth(x)(%e ) - 8008coth(x)(%e ) - 4704coth(x)(%e )
--R +
--R x 6 x 4 x 2
--R - 1700coth(x)(%e ) - 364coth(x)(%e ) - 42coth(x)(%e ) - 2coth(x)
--R *
--R 3
--R tanh(x)
--R +
--R 2 x 18 2 x 16
--R - 1016coth(x) (%e ) - 7224coth(x) (%e )
--R +
--R 2 x 14 2 x 12
--R - 21168coth(x) (%e ) - 33680coth(x) (%e )
--R +
--R 2 x 10 2 x 8 2 x 6
--R - 32032coth(x) (%e ) - 18816coth(x) (%e ) - 6800coth(x) (%e )
--R +
--R 2 x 4 2 x 2 2
--R - 1456coth(x) (%e ) - 168coth(x) (%e ) - 8coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R x 18 x 16 x 14
--R 254coth(x)(%e ) + 1806coth(x)(%e ) + 5292coth(x)(%e )
--R +
--R x 12 x 10 x 8
--R 8420coth(x)(%e ) + 8008coth(x)(%e ) + 4704coth(x)(%e )
--R +
--R x 6 x 4 x 2
--R 1700coth(x)(%e ) + 364coth(x)(%e ) + 42coth(x)(%e ) + 2coth(x)
--R *
--R tanh(x)
--R +
--R 2 x 18 2 x 16
--R (1016coth(x) - 254)(%e ) + (7224coth(x) - 1806)(%e )
--R +
--R 2 x 14 2 x 12
--R (21168coth(x) - 5292)(%e ) + (33680coth(x) - 8420)(%e )
--R +
--R 2 x 10 2 x 8
--R (32032coth(x) - 8008)(%e ) + (18816coth(x) - 4704)(%e )
--R +
--R 2 x 6 2 x 4
--R (6800coth(x) - 1700)(%e ) + (1456coth(x) - 364)(%e )
--R +
--R 2 x 2 2
--R (168coth(x) - 42)(%e ) + 8coth(x) - 2
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R +---+ 2 x 20 +---+ 2 x 18
--R 129\|- 1 coth(x) (%e ) + 1276\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 16 +---+ 2 x 14
--R 4881\|- 1 coth(x) (%e ) + 9908\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 12 +---+ 2 x 10
--R 11998\|- 1 coth(x) (%e ) + 9100\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 8 +---+ 2 x 6
--R 4382\|- 1 coth(x) (%e ) + 1324\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 4 +---+ 2 x 2 +---+ 2
--R 241\|- 1 coth(x) (%e ) + 24\|- 1 coth(x) (%e ) + \|- 1 coth(x)
--R *
--R 6
--R tanh(x)
--R +
--R +---+ 2 +---+ x 20
--R (- 516\|- 1 coth(x) - 129\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 18
--R (- 5104\|- 1 coth(x) - 1276\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 16
--R (- 19524\|- 1 coth(x) - 4881\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 14
--R (- 39632\|- 1 coth(x) - 9908\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 12
--R (- 47992\|- 1 coth(x) - 11998\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 10
--R (- 36400\|- 1 coth(x) - 9100\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 8
--R (- 17528\|- 1 coth(x) - 4382\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 6
--R (- 5296\|- 1 coth(x) - 1324\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 4
--R (- 964\|- 1 coth(x) - 241\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 2 +---+ 2 +---+
--R (- 96\|- 1 coth(x) - 24\|- 1 )(%e ) - 4\|- 1 coth(x) - \|- 1
--R *
--R 4
--R tanh(x)
--R +
--R +---+ x 20 +---+ x 18
--R 258\|- 1 coth(x)(%e ) + 2552\|- 1 coth(x)(%e )
--R +
--R +---+ x 16 +---+ x 14
--R 9762\|- 1 coth(x)(%e ) + 19816\|- 1 coth(x)(%e )
--R +
--R +---+ x 12 +---+ x 10
--R 23996\|- 1 coth(x)(%e ) + 18200\|- 1 coth(x)(%e )
--R +
--R +---+ x 8 +---+ x 6
--R 8764\|- 1 coth(x)(%e ) + 2648\|- 1 coth(x)(%e )
--R +
--R +---+ x 4 +---+ x 2 +---+
--R 482\|- 1 coth(x)(%e ) + 48\|- 1 coth(x)(%e ) + 2\|- 1 coth(x)
--R *
--R 3
--R tanh(x)
--R +
--R +---+ 2 x 20 +---+ 2 x 18
--R 1032\|- 1 coth(x) (%e ) + 10208\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 16 +---+ 2 x 14
--R 39048\|- 1 coth(x) (%e ) + 79264\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 12 +---+ 2 x 10
--R 95984\|- 1 coth(x) (%e ) + 72800\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 8 +---+ 2 x 6
--R 35056\|- 1 coth(x) (%e ) + 10592\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 4 +---+ 2 x 2 +---+ 2
--R 1928\|- 1 coth(x) (%e ) + 192\|- 1 coth(x) (%e ) + 8\|- 1 coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R +---+ x 20 +---+ x 18
--R - 258\|- 1 coth(x)(%e ) - 2552\|- 1 coth(x)(%e )
--R +
--R +---+ x 16 +---+ x 14
--R - 9762\|- 1 coth(x)(%e ) - 19816\|- 1 coth(x)(%e )
--R +
--R +---+ x 12 +---+ x 10
--R - 23996\|- 1 coth(x)(%e ) - 18200\|- 1 coth(x)(%e )
--R +
--R +---+ x 8 +---+ x 6
--R - 8764\|- 1 coth(x)(%e ) - 2648\|- 1 coth(x)(%e )
--R +
--R +---+ x 4 +---+ x 2 +---+
--R - 482\|- 1 coth(x)(%e ) - 48\|- 1 coth(x)(%e ) - 2\|- 1 coth(x)
--R *
--R tanh(x)
--R +
--R +---+ 2 +---+ x 20
--R (- 1032\|- 1 coth(x) + 258\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 18
--R (- 10208\|- 1 coth(x) + 2552\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 16
--R (- 39048\|- 1 coth(x) + 9762\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 14
--R (- 79264\|- 1 coth(x) + 19816\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 12
--R (- 95984\|- 1 coth(x) + 23996\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 10
--R (- 72800\|- 1 coth(x) + 18200\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 8
--R (- 35056\|- 1 coth(x) + 8764\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 6
--R (- 10592\|- 1 coth(x) + 2648\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 4
--R (- 1928\|- 1 coth(x) + 482\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 2 +---+ 2 +---+
--R (- 192\|- 1 coth(x) + 48\|- 1 )(%e ) - 8\|- 1 coth(x) + 2\|- 1
--R /
--R 2 x 18 2 x 16 2 x 14
--R 127coth(x) (%e ) + 903coth(x) (%e ) + 2646coth(x) (%e )
--R +
--R 2 x 12 2 x 10 2 x 8
--R 4210coth(x) (%e ) + 4004coth(x) (%e ) + 2352coth(x) (%e )
--R +
--R 2 x 6 2 x 4 2 x 2
--R 850coth(x) (%e ) + 182coth(x) (%e ) + 21coth(x) (%e )
--R +
--R 2
--R coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R 2 x 18 2 x 16
--R (- 254coth(x) - 127)(%e ) + (- 1806coth(x) - 903)(%e )
--R +
--R 2 x 14 2 x 12
--R (- 5292coth(x) - 2646)(%e ) + (- 8420coth(x) - 4210)(%e )
--R +
--R 2 x 10 2 x 8
--R (- 8008coth(x) - 4004)(%e ) + (- 4704coth(x) - 2352)(%e )
--R +
--R 2 x 6 2 x 4
--R (- 1700coth(x) - 850)(%e ) + (- 364coth(x) - 182)(%e )
--R +
--R 2 x 2 2
--R (- 42coth(x) - 21)(%e ) - 2coth(x) - 1
--R *
--R +----------------------+
--R | x 4 x 2
--R \|- (%e ) - 6(%e ) - 1
--R +
--R +---+ 2 x 20 +---+ 2 x 18
--R - 129\|- 1 coth(x) (%e ) - 1276\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 16 +---+ 2 x 14
--R - 4881\|- 1 coth(x) (%e ) - 9908\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 12 +---+ 2 x 10
--R - 11998\|- 1 coth(x) (%e ) - 9100\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 8 +---+ 2 x 6
--R - 4382\|- 1 coth(x) (%e ) - 1324\|- 1 coth(x) (%e )
--R +
--R +---+ 2 x 4 +---+ 2 x 2 +---+ 2
--R - 241\|- 1 coth(x) (%e ) - 24\|- 1 coth(x) (%e ) - \|- 1 coth(x)
--R *
--R 2
--R tanh(x)
--R +
--R +---+ 2 +---+ x 20
--R (258\|- 1 coth(x) + 129\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 18
--R (2552\|- 1 coth(x) + 1276\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 16
--R (9762\|- 1 coth(x) + 4881\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 14
--R (19816\|- 1 coth(x) + 9908\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 12
--R (23996\|- 1 coth(x) + 11998\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 10
--R (18200\|- 1 coth(x) + 9100\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 8
--R (8764\|- 1 coth(x) + 4382\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 6
--R (2648\|- 1 coth(x) + 1324\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 4
--R (482\|- 1 coth(x) + 241\|- 1 )(%e )
--R +
--R +---+ 2 +---+ x 2 +---+ 2 +---+
--R (48\|- 1 coth(x) + 24\|- 1 )(%e ) + 2\|- 1 coth(x) + \|- 1
--R *
--R +------------+
--R | 2
--R \|tanh(x) - 2
--R Type: Expression(Integer)
--E 107
--S 108 of 526
t0419:= (a+b*sech(x)^2)^(3/2)
--R
--R
--R +--------------+
--R 2 | 2
--R (104) (b sech(x) + a)\|b sech(x) + a
--R Type: Expression(Integer)
--E 108
--S 109 of 526
r0419:= -3/2*a*b^(1/2)*atan(coth(x)*(a+b*sech(x)^2)^(1/2)/b^(1/2))-_
1/2*b^(3/2)*atan(coth(x)*(a+b*sech(x)^2)^(1/2)/b^(1/2))+_
a^(3/2)*atanh(a^(1/2)*tanh(x)/(a+b*sech(x)^2)^(1/2))+_
1/2*b*(a+b*sech(x)^2)^(1/2)*tanh(x)
--R
--R
--R (105)
--R +-+
--R +-+ tanh(x)\|a
--R 2a\|a atanh(-----------------)
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R +
--R +--------------+
--R | 2 +--------------+
--R +-+ coth(x)\|b sech(x) + a | 2
--R (- b - 3a)\|b atan(------------------------) + b tanh(x)\|b sech(x) + a
--R +-+
--R \|b
--R /
--R 2
--R Type: Expression(Integer)
--E 109
--S 110 of 526
a0419:= integrate(t0419,x)
--R
--R
--R (106)
--R [
--R 2 3 6 2 3 5
--R (- 2a b - a )sinh(x) + (- 12a b - 6a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((- 30a b - 15a )cosh(x) - 4a b - 3a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((- 40a b - 20a )cosh(x) + (- 16a b - 12a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2
--R (- 30a b - 15a )cosh(x) + (- 24a b - 18a )cosh(x) - 2a b
--R +
--R 3
--R - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (- 12a b - 6a )cosh(x) + (- 16a b - 12a )cosh(x)
--R +
--R 2 3
--R (- 4a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4
--R (- 2a b - a )cosh(x) + (- 4a b - 3a )cosh(x)
--R +
--R 2 3 2 3
--R (- 2a b - 3a )cosh(x) - a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (2a b + 2a b + a )sinh(x)
--R +
--R 2 2 3 7
--R (16a b + 16a b + 8a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3 6
--R ((56a b + 56a b + 28a )cosh(x) + 4a b + 8a b + 4a )sinh(x)
--R +
--R 2 2 3 3
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3
--R (24a b + 48a b + 24a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (140a b + 140a b + 70a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (60a b + 120a b + 60a )cosh(x) + 2a b + 10a b + 6a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3 3
--R (80a b + 160a b + 80a )cosh(x)
--R +
--R 2 2 3
--R (8a b + 40a b + 24a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (56a b + 56a b + 28a )cosh(x)
--R +
--R 2 2 3 4
--R (60a b + 120a b + 60a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (12a b + 60a b + 36a )cosh(x) + 4a b + 4a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (16a b + 16a b + 8a )cosh(x)
--R +
--R 2 2 3 5
--R (24a b + 48a b + 24a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (8a b + 40a b + 24a )cosh(x) + (8a b + 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (2a b + 2a b + a )cosh(x) + (4a b + 8a b + 4a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (2a b + 10a b + 6a )cosh(x) + (4a b + 4a )cosh(x) + a
--R *
--R +-+
--R \|a
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2 +-+
--R (sinh(x) + 2cosh(x)sinh(x) + cosh(x) )\|a
--R +
--R 2 2 6
--R (2b + 7a b + 3a )sinh(x)
--R +
--R 2 2 5
--R (12b + 42a b + 18a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((30b + 105a b + 45a )cosh(x) + 4b + 15a b + 9a )sinh(x)
--R +
--R 2 2 3
--R (40b + 140a b + 60a )cosh(x)
--R +
--R 2 2
--R (16b + 60a b + 36a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (30b + 105a b + 45a )cosh(x)
--R +
--R 2 2 2 2 2
--R (24b + 90a b + 54a )cosh(x) + 2b + 9a b + 9a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5
--R (12b + 42a b + 18a )cosh(x)
--R +
--R 2 2 3 2 2
--R (16b + 60a b + 36a )cosh(x) + (4b + 18a b + 18a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (2b + 7a b + 3a )cosh(x) + (4b + 15a b + 9a )cosh(x)
--R +
--R 2 2 2 2
--R (2b + 9a b + 9a )cosh(x) + a b + 3a
--R *
--R +-------------------------------------+
--R | 2 2
--R +---+ +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|- b \|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 3 8
--R (- 2b - 8a b - 7a b - 3a )sinh(x)
--R +
--R 3 2 2 3 7
--R (- 16b - 64a b - 56a b - 24a )cosh(x)sinh(x)
--R +
--R 3 2 2 3 2 3 2
--R (- 56b - 224a b - 196a b - 84a )cosh(x) - 4b - 20a b
--R +
--R 2 3
--R - 28a b - 12a
--R *
--R 6
--R sinh(x)
--R +
--R 3 2 2 3 3
--R (- 112b - 448a b - 392a b - 168a )cosh(x)
--R +
--R 3 2 2 3
--R (- 24b - 120a b - 168a b - 72a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 3 4
--R (- 140b - 560a b - 490a b - 210a )cosh(x)
--R +
--R 3 2 2 3 2 3 2
--R (- 60b - 300a b - 420a b - 180a )cosh(x) - 2b - 16a b
--R +
--R 2 3
--R - 36a b - 18a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 3 5
--R (- 112b - 448a b - 392a b - 168a )cosh(x)
--R +
--R 3 2 2 3 3
--R (- 80b - 400a b - 560a b - 240a )cosh(x)
--R +
--R 3 2 2 3
--R (- 8b - 64a b - 144a b - 72a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 3 6
--R (- 56b - 224a b - 196a b - 84a )cosh(x)
--R +
--R 3 2 2 3 4
--R (- 60b - 300a b - 420a b - 180a )cosh(x)
--R +
--R 3 2 2 3 2 2 2
--R (- 12b - 96a b - 216a b - 108a )cosh(x) - 4a b - 16a b
--R +
--R 3
--R - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 3 7
--R (- 16b - 64a b - 56a b - 24a )cosh(x)
--R +
--R 3 2 2 3 5
--R (- 24b - 120a b - 168a b - 72a )cosh(x)
--R +
--R 3 2 2 3 3
--R (- 8b - 64a b - 144a b - 72a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 32a b - 24a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 3 8
--R (- 2b - 8a b - 7a b - 3a )cosh(x)
--R +
--R 3 2 2 3 6
--R (- 4b - 20a b - 28a b - 12a )cosh(x)
--R +
--R 3 2 2 3 4
--R (- 2b - 16a b - 36a b - 18a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (- 4a b - 16a b - 12a )cosh(x) - a b - 3a
--R *
--R +---+
--R \|- b
--R *
--R log
--R 4 3
--R (2b + 2a)sinh(x) + (8b + 8a)cosh(x)sinh(x)
--R +
--R 2 2
--R ((12b + 12a)cosh(x) + 2a)sinh(x)
--R +
--R 3
--R ((8b + 8a)cosh(x) + 4a cosh(x))sinh(x)
--R +
--R 4 2
--R (2b + 2a)cosh(x) + 2a cosh(x)
--R *
--R +---+ +-+
--R \|- b \|a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 2a b + 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (4a b + 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2
--R (2a b + 2a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 6 2 2 5
--R (2b - a )sinh(x) + (12b - 6a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((30b - 15a )cosh(x) - 2b - 4a b - 3a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((40b - 20a )cosh(x) + (- 8b - 16a b - 12a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (30b - 15a )cosh(x)
--R +
--R 2 2 2 2
--R (- 12b - 24a b - 18a )cosh(x) - 4a b - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5 2 2 3
--R (12b - 6a )cosh(x) + (- 8b - 16a b - 12a )cosh(x)
--R +
--R 2
--R (- 8a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (2b - a )cosh(x) + (- 2b - 4a b - 3a )cosh(x)
--R +
--R 2 2 2
--R (- 4a b - 3a )cosh(x) - a
--R *
--R +-+
--R \|a
--R +
--R 2 6 2 5
--R (- 2a b - 2a )sinh(x) + (- 12a b - 12a )cosh(x)sinh(x)
--R +
--R 2 2 2 4
--R ((- 30a b - 30a )cosh(x) - 6a b - 4a )sinh(x)
--R +
--R 2 3 2
--R ((- 40a b - 40a )cosh(x) + (- 24a b - 16a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 2 2
--R (- 30a b - 30a )cosh(x) + (- 36a b - 24a )cosh(x)
--R +
--R 2
--R - 2a
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3
--R (- 12a b - 12a )cosh(x) + (- 24a b - 16a )cosh(x)
--R +
--R 2
--R - 4a cosh(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4
--R (- 2a b - 2a )cosh(x) + (- 6a b - 4a )cosh(x)
--R +
--R 2 2
--R - 2a cosh(x)
--R *
--R +---+
--R \|- b
--R /
--R 2 4 2 3
--R (2a b + a )sinh(x) + (8a b + 4a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R ((12a b + 6a )cosh(x) + 2a b + 2a )sinh(x)
--R +
--R 2 3 2
--R ((8a b + 4a )cosh(x) + (4a b + 4a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2
--R (2a b + a )cosh(x) + (2a b + 2a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 6
--R (- 2b - 2a b - a )sinh(x)
--R +
--R 2 2 5
--R (- 12b - 12a b - 6a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((- 30b - 30a b - 15a )cosh(x) - 2b - 6a b - 3a )
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3
--R (- 40b - 40a b - 20a )cosh(x)
--R +
--R 2 2
--R (- 8b - 24a b - 12a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (- 30b - 30a b - 15a )cosh(x)
--R +
--R 2 2 2 2
--R (- 12b - 36a b - 18a )cosh(x) - 4a b - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5
--R (- 12b - 12a b - 6a )cosh(x)
--R +
--R 2 2 3 2
--R (- 8b - 24a b - 12a )cosh(x) + (- 8a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (- 2b - 2a b - a )cosh(x) + (- 2b - 6a b - 3a )cosh(x)
--R +
--R 2 2 2
--R (- 4a b - 3a )cosh(x) - a
--R *
--R +-+
--R \|a
--R +
--R 2 3 6 2 3 5
--R (2a b + a )sinh(x) + (12a b + 6a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((30a b + 15a )cosh(x) + 4a b + 3a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((40a b + 20a )cosh(x) + (16a b + 12a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2
--R (30a b + 15a )cosh(x) + (24a b + 18a )cosh(x) + 2a b
--R +
--R 3
--R 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (12a b + 6a )cosh(x) + (16a b + 12a )cosh(x)
--R +
--R 2 3
--R (4a b + 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4
--R (2a b + a )cosh(x) + (4a b + 3a )cosh(x)
--R +
--R 2 3 2 3
--R (2a b + 3a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (- 2a b - 2a b - a )sinh(x)
--R +
--R 2 2 3 7
--R (- 16a b - 16a b - 8a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3
--R ((- 56a b - 56a b - 28a )cosh(x) - 4a b - 8a b - 4a )
--R *
--R 6
--R sinh(x)
--R +
--R 2 2 3 3
--R (- 112a b - 112a b - 56a )cosh(x)
--R +
--R 2 2 3
--R (- 24a b - 48a b - 24a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (- 140a b - 140a b - 70a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (- 60a b - 120a b - 60a )cosh(x) - 2a b - 10a b - 6a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (- 112a b - 112a b - 56a )cosh(x)
--R +
--R 2 2 3 3
--R (- 80a b - 160a b - 80a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 40a b - 24a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (- 56a b - 56a b - 28a )cosh(x)
--R +
--R 2 2 3 4
--R (- 60a b - 120a b - 60a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (- 12a b - 60a b - 36a )cosh(x) - 4a b - 4a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (- 16a b - 16a b - 8a )cosh(x)
--R +
--R 2 2 3 5
--R (- 24a b - 48a b - 24a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (- 8a b - 40a b - 24a )cosh(x) + (- 8a b - 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (- 2a b - 2a b - a )cosh(x) + (- 4a b - 8a b - 4a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (- 2a b - 10a b - 6a )cosh(x) + (- 4a b - 4a )cosh(x) - a
--R *
--R +-+
--R \|a
--R *
--R log
--R 2 2
--R (- 2b sinh(x) - 4b cosh(x)sinh(x) - 2b cosh(x) - 2a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - 2b sinh(x) - 8b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 12b cosh(x) + 6b + 2a)sinh(x)
--R +
--R 3 4
--R (- 8b cosh(x) + (12b + 4a)cosh(x))sinh(x) - 2b cosh(x)
--R +
--R 2
--R (6b + 2a)cosh(x) + 2a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x)
--R +
--R 3 4
--R 4cosh(x) sinh(x) + cosh(x)
--R *
--R +-+
--R \|a
--R +
--R 3 2 2 6
--R (- 2b - 4a b - 2a b)sinh(x)
--R +
--R 3 2 2 5
--R (- 12b - 24a b - 12a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 4
--R ((- 30b - 60a b - 30a b)cosh(x) + 2b - 6a b - 4a b)sinh(x)
--R +
--R 3 2 2 3
--R (- 40b - 80a b - 40a b)cosh(x)
--R +
--R 3 2 2
--R (8b - 24a b - 16a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 4
--R (- 30b - 60a b - 30a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (12b - 36a b - 24a b)cosh(x) + 2a b - 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 5
--R (- 12b - 24a b - 12a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (8b - 24a b - 16a b)cosh(x) + (4a b - 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 6 3 2 2 4
--R (- 2b - 4a b - 2a b)cosh(x) + (2b - 6a b - 4a b)cosh(x)
--R +
--R 2 2 2
--R (2a b - 2a b)cosh(x)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 8
--R (2b + 4a b + 2a b)sinh(x)
--R +
--R 3 2 2 7
--R (16b + 32a b + 16a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 6
--R ((56b + 112a b + 56a b)cosh(x) + 12b + 14a b + 6a b)sinh(x)
--R +
--R 3 2 2 3
--R (112b + 224a b + 112a b)cosh(x)
--R +
--R 3 2 2
--R (72b + 84a b + 36a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 4
--R (140b + 280a b + 140a b)cosh(x)
--R +
--R 3 2 2 2 3 2 2
--R (180b + 210a b + 90a b)cosh(x) - 6b + 8a b + 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 5
--R (112b + 224a b + 112a b)cosh(x)
--R +
--R 3 2 2 3
--R (240b + 280a b + 120a b)cosh(x)
--R +
--R 3 2 2
--R (- 24b + 32a b + 24a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 6
--R (56b + 112a b + 56a b)cosh(x)
--R +
--R 3 2 2 4
--R (180b + 210a b + 90a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (- 36b + 48a b + 36a b)cosh(x) - 2a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 7
--R (16b + 32a b + 16a b)cosh(x)
--R +
--R 3 2 2 5
--R (72b + 84a b + 36a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (- 24b + 32a b + 24a b)cosh(x) + (- 4a b + 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 8 3 2 2 6
--R (2b + 4a b + 2a b)cosh(x) + (12b + 14a b + 6a b)cosh(x)
--R +
--R 3 2 2 4 2 2 2
--R (- 6b + 8a b + 6a b)cosh(x) + (- 2a b + 2a b)cosh(x)
--R *
--R +-+
--R \|a
--R /
--R 6 5
--R (4b + 2a)sinh(x) + (24b + 12a)cosh(x)sinh(x)
--R +
--R 2 4
--R ((60b + 30a)cosh(x) + 8b + 6a)sinh(x)
--R +
--R 3 3
--R ((80b + 40a)cosh(x) + (32b + 24a)cosh(x))sinh(x)
--R +
--R 4 2 2
--R ((60b + 30a)cosh(x) + (48b + 36a)cosh(x) + 4b + 6a)sinh(x)
--R +
--R 5 3
--R ((24b + 12a)cosh(x) + (32b + 24a)cosh(x) + (8b + 12a)cosh(x))
--R *
--R sinh(x)
--R +
--R 6 4 2
--R (4b + 2a)cosh(x) + (8b + 6a)cosh(x) + (4b + 6a)cosh(x) + 2a
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 8 2 2 7
--R (- 4b - 4a b - 2a )sinh(x) + (- 32b - 32a b - 16a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 6
--R ((- 112b - 112a b - 56a )cosh(x) - 8b - 16a b - 8a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((- 224b - 224a b - 112a )cosh(x) + (- 48b - 96a b - 48a )cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 4
--R (- 280b - 280a b - 140a )cosh(x)
--R +
--R 2 2 2 2 2
--R (- 120b - 240a b - 120a )cosh(x) - 4b - 20a b - 12a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 5
--R (- 224b - 224a b - 112a )cosh(x)
--R +
--R 2 2 3 2 2
--R (- 160b - 320a b - 160a )cosh(x) + (- 16b - 80a b - 48a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 6
--R (- 112b - 112a b - 56a )cosh(x)
--R +
--R 2 2 4
--R (- 120b - 240a b - 120a )cosh(x)
--R +
--R 2 2 2 2
--R (- 24b - 120a b - 72a )cosh(x) - 8a b - 8a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 7 2 2 5
--R (- 32b - 32a b - 16a )cosh(x) + (- 48b - 96a b - 48a )cosh(x)
--R +
--R 2 2 3 2
--R (- 16b - 80a b - 48a )cosh(x) + (- 16a b - 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 8 2 2 6
--R (- 4b - 4a b - 2a )cosh(x) + (- 8b - 16a b - 8a )cosh(x)
--R +
--R 2 2 4 2 2 2
--R (- 4b - 20a b - 12a )cosh(x) + (- 8a b - 8a )cosh(x) - 2a
--R ,
--R
--R 2 3 6 2 3 5
--R (- 2a b - a )sinh(x) + (- 12a b - 6a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((- 30a b - 15a )cosh(x) - 4a b - 3a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((- 40a b - 20a )cosh(x) + (- 16a b - 12a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2
--R (- 30a b - 15a )cosh(x) + (- 24a b - 18a )cosh(x) - 2a b
--R +
--R 3
--R - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (- 12a b - 6a )cosh(x) + (- 16a b - 12a )cosh(x)
--R +
--R 2 3
--R (- 4a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4
--R (- 2a b - a )cosh(x) + (- 4a b - 3a )cosh(x)
--R +
--R 2 3 2 3
--R (- 2a b - 3a )cosh(x) - a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (2a b + 2a b + a )sinh(x)
--R +
--R 2 2 3 7
--R (16a b + 16a b + 8a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3 6
--R ((56a b + 56a b + 28a )cosh(x) + 4a b + 8a b + 4a )sinh(x)
--R +
--R 2 2 3 3
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3
--R (24a b + 48a b + 24a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (140a b + 140a b + 70a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (60a b + 120a b + 60a )cosh(x) + 2a b + 10a b + 6a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3 3
--R (80a b + 160a b + 80a )cosh(x)
--R +
--R 2 2 3
--R (8a b + 40a b + 24a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (56a b + 56a b + 28a )cosh(x)
--R +
--R 2 2 3 4
--R (60a b + 120a b + 60a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (12a b + 60a b + 36a )cosh(x) + 4a b + 4a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (16a b + 16a b + 8a )cosh(x)
--R +
--R 2 2 3 5
--R (24a b + 48a b + 24a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (8a b + 40a b + 24a )cosh(x) + (8a b + 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (2a b + 2a b + a )cosh(x) + (4a b + 8a b + 4a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (2a b + 10a b + 6a )cosh(x) + (4a b + 4a )cosh(x) + a
--R *
--R +-+
--R \|a
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2 +-+
--R (sinh(x) + 2cosh(x)sinh(x) + cosh(x) )\|a
--R +
--R 2 3 6 2 3 5
--R (2a b + a )sinh(x) + (12a b + 6a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((30a b + 15a )cosh(x) + 4a b + 3a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((40a b + 20a )cosh(x) + (16a b + 12a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2
--R (30a b + 15a )cosh(x) + (24a b + 18a )cosh(x) + 2a b
--R +
--R 3
--R 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (12a b + 6a )cosh(x) + (16a b + 12a )cosh(x)
--R +
--R 2 3
--R (4a b + 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4
--R (2a b + a )cosh(x) + (4a b + 3a )cosh(x)
--R +
--R 2 3 2 3
--R (2a b + 3a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (- 2a b - 2a b - a )sinh(x)
--R +
--R 2 2 3 7
--R (- 16a b - 16a b - 8a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3
--R ((- 56a b - 56a b - 28a )cosh(x) - 4a b - 8a b - 4a )
--R *
--R 6
--R sinh(x)
--R +
--R 2 2 3 3
--R (- 112a b - 112a b - 56a )cosh(x)
--R +
--R 2 2 3
--R (- 24a b - 48a b - 24a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (- 140a b - 140a b - 70a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (- 60a b - 120a b - 60a )cosh(x) - 2a b - 10a b - 6a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (- 112a b - 112a b - 56a )cosh(x)
--R +
--R 2 2 3 3
--R (- 80a b - 160a b - 80a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 40a b - 24a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (- 56a b - 56a b - 28a )cosh(x)
--R +
--R 2 2 3 4
--R (- 60a b - 120a b - 60a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (- 12a b - 60a b - 36a )cosh(x) - 4a b - 4a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (- 16a b - 16a b - 8a )cosh(x)
--R +
--R 2 2 3 5
--R (- 24a b - 48a b - 24a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (- 8a b - 40a b - 24a )cosh(x) + (- 8a b - 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (- 2a b - 2a b - a )cosh(x) + (- 4a b - 8a b - 4a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (- 2a b - 10a b - 6a )cosh(x) + (- 4a b - 4a )cosh(x) - a
--R *
--R +-+
--R \|a
--R *
--R log
--R 2 2
--R (- 2b sinh(x) - 4b cosh(x)sinh(x) - 2b cosh(x) - 2a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - 2b sinh(x) - 8b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 12b cosh(x) + 6b + 2a)sinh(x)
--R +
--R 3 4
--R (- 8b cosh(x) + (12b + 4a)cosh(x))sinh(x) - 2b cosh(x)
--R +
--R 2
--R (6b + 2a)cosh(x) + 2a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x)
--R +
--R 3 4
--R 4cosh(x) sinh(x) + cosh(x)
--R *
--R +-+
--R \|a
--R +
--R 2 2 6
--R (- 4b - 14a b - 6a )sinh(x)
--R +
--R 2 2 5
--R (- 24b - 84a b - 36a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((- 60b - 210a b - 90a )cosh(x) - 8b - 30a b - 18a )sinh(x)
--R +
--R 2 2 3
--R (- 80b - 280a b - 120a )cosh(x)
--R +
--R 2 2
--R (- 32b - 120a b - 72a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (- 60b - 210a b - 90a )cosh(x)
--R +
--R 2 2 2 2 2
--R (- 48b - 180a b - 108a )cosh(x) - 4b - 18a b - 18a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5
--R (- 24b - 84a b - 36a )cosh(x)
--R +
--R 2 2 3
--R (- 32b - 120a b - 72a )cosh(x)
--R +
--R 2 2
--R (- 8b - 36a b - 36a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (- 4b - 14a b - 6a )cosh(x) + (- 8b - 30a b - 18a )cosh(x)
--R +
--R 2 2 2 2
--R (- 4b - 18a b - 18a )cosh(x) - 2a b - 6a
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a \|b |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 3 8
--R (4b + 16a b + 14a b + 6a )sinh(x)
--R +
--R 3 2 2 3 7
--R (32b + 128a b + 112a b + 48a )cosh(x)sinh(x)
--R +
--R 3 2 2 3 2 3 2
--R (112b + 448a b + 392a b + 168a )cosh(x) + 8b + 40a b
--R +
--R 2 3
--R 56a b + 24a
--R *
--R 6
--R sinh(x)
--R +
--R 3 2 2 3 3
--R (224b + 896a b + 784a b + 336a )cosh(x)
--R +
--R 3 2 2 3
--R (48b + 240a b + 336a b + 144a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 3 4
--R (280b + 1120a b + 980a b + 420a )cosh(x)
--R +
--R 3 2 2 3 2 3 2
--R (120b + 600a b + 840a b + 360a )cosh(x) + 4b + 32a b
--R +
--R 2 3
--R 72a b + 36a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 3 5
--R (224b + 896a b + 784a b + 336a )cosh(x)
--R +
--R 3 2 2 3 3
--R (160b + 800a b + 1120a b + 480a )cosh(x)
--R +
--R 3 2 2 3
--R (16b + 128a b + 288a b + 144a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 3 6
--R (112b + 448a b + 392a b + 168a )cosh(x)
--R +
--R 3 2 2 3 4
--R (120b + 600a b + 840a b + 360a )cosh(x)
--R +
--R 3 2 2 3 2 2 2
--R (24b + 192a b + 432a b + 216a )cosh(x) + 8a b + 32a b
--R +
--R 3
--R 24a
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 3 7
--R (32b + 128a b + 112a b + 48a )cosh(x)
--R +
--R 3 2 2 3 5
--R (48b + 240a b + 336a b + 144a )cosh(x)
--R +
--R 3 2 2 3 3
--R (16b + 128a b + 288a b + 144a )cosh(x)
--R +
--R 2 2 3
--R (16a b + 64a b + 48a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 3 8
--R (4b + 16a b + 14a b + 6a )cosh(x)
--R +
--R 3 2 2 3 6
--R (8b + 40a b + 56a b + 24a )cosh(x)
--R +
--R 3 2 2 3 4
--R (4b + 32a b + 72a b + 36a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (8a b + 32a b + 24a )cosh(x) + 2a b + 6a
--R *
--R +-+
--R \|b
--R *
--R atan
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2 +-+ +-+
--R (2sinh(x) + 4cosh(x)sinh(x) + 2cosh(x) )\|a \|b
--R +
--R 3 2 2 6
--R (- 2b - 4a b - 2a b)sinh(x)
--R +
--R 3 2 2 5
--R (- 12b - 24a b - 12a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 4
--R ((- 30b - 60a b - 30a b)cosh(x) + 2b - 6a b - 4a b)sinh(x)
--R +
--R 3 2 2 3
--R (- 40b - 80a b - 40a b)cosh(x)
--R +
--R 3 2 2
--R (8b - 24a b - 16a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 4
--R (- 30b - 60a b - 30a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (12b - 36a b - 24a b)cosh(x) + 2a b - 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 5
--R (- 12b - 24a b - 12a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (8b - 24a b - 16a b)cosh(x) + (4a b - 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 6 3 2 2 4
--R (- 2b - 4a b - 2a b)cosh(x) + (2b - 6a b - 4a b)cosh(x)
--R +
--R 2 2 2
--R (2a b - 2a b)cosh(x)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 8
--R (2b + 4a b + 2a b)sinh(x)
--R +
--R 3 2 2 7
--R (16b + 32a b + 16a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 6
--R ((56b + 112a b + 56a b)cosh(x) + 12b + 14a b + 6a b)sinh(x)
--R +
--R 3 2 2 3
--R (112b + 224a b + 112a b)cosh(x)
--R +
--R 3 2 2
--R (72b + 84a b + 36a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 4
--R (140b + 280a b + 140a b)cosh(x)
--R +
--R 3 2 2 2 3 2 2
--R (180b + 210a b + 90a b)cosh(x) - 6b + 8a b + 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 5
--R (112b + 224a b + 112a b)cosh(x)
--R +
--R 3 2 2 3
--R (240b + 280a b + 120a b)cosh(x)
--R +
--R 3 2 2
--R (- 24b + 32a b + 24a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 6
--R (56b + 112a b + 56a b)cosh(x)
--R +
--R 3 2 2 4
--R (180b + 210a b + 90a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (- 36b + 48a b + 36a b)cosh(x) - 2a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 7
--R (16b + 32a b + 16a b)cosh(x)
--R +
--R 3 2 2 5
--R (72b + 84a b + 36a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (- 24b + 32a b + 24a b)cosh(x) + (- 4a b + 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 8 3 2 2 6
--R (2b + 4a b + 2a b)cosh(x) + (12b + 14a b + 6a b)cosh(x)
--R +
--R 3 2 2 4 2 2 2
--R (- 6b + 8a b + 6a b)cosh(x) + (- 2a b + 2a b)cosh(x)
--R *
--R +-+
--R \|a
--R /
--R 6 5
--R (4b + 2a)sinh(x) + (24b + 12a)cosh(x)sinh(x)
--R +
--R 2 4
--R ((60b + 30a)cosh(x) + 8b + 6a)sinh(x)
--R +
--R 3 3
--R ((80b + 40a)cosh(x) + (32b + 24a)cosh(x))sinh(x)
--R +
--R 4 2 2
--R ((60b + 30a)cosh(x) + (48b + 36a)cosh(x) + 4b + 6a)sinh(x)
--R +
--R 5 3
--R ((24b + 12a)cosh(x) + (32b + 24a)cosh(x) + (8b + 12a)cosh(x))
--R *
--R sinh(x)
--R +
--R 6 4 2
--R (4b + 2a)cosh(x) + (8b + 6a)cosh(x) + (4b + 6a)cosh(x) + 2a
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 8 2 2 7
--R (- 4b - 4a b - 2a )sinh(x) + (- 32b - 32a b - 16a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 6
--R ((- 112b - 112a b - 56a )cosh(x) - 8b - 16a b - 8a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((- 224b - 224a b - 112a )cosh(x) + (- 48b - 96a b - 48a )cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 4
--R (- 280b - 280a b - 140a )cosh(x)
--R +
--R 2 2 2 2 2
--R (- 120b - 240a b - 120a )cosh(x) - 4b - 20a b - 12a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 5
--R (- 224b - 224a b - 112a )cosh(x)
--R +
--R 2 2 3 2 2
--R (- 160b - 320a b - 160a )cosh(x) + (- 16b - 80a b - 48a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 6
--R (- 112b - 112a b - 56a )cosh(x)
--R +
--R 2 2 4
--R (- 120b - 240a b - 120a )cosh(x)
--R +
--R 2 2 2 2
--R (- 24b - 120a b - 72a )cosh(x) - 8a b - 8a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 7 2 2 5
--R (- 32b - 32a b - 16a )cosh(x) + (- 48b - 96a b - 48a )cosh(x)
--R +
--R 2 2 3 2
--R (- 16b - 80a b - 48a )cosh(x) + (- 16a b - 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 8 2 2 6
--R (- 4b - 4a b - 2a )cosh(x) + (- 8b - 16a b - 8a )cosh(x)
--R +
--R 2 2 4 2 2 2
--R (- 4b - 20a b - 12a )cosh(x) + (- 8a b - 8a )cosh(x) - 2a
--R ]
--R Type: Union(List(Expression(Integer)),...)
--E 110
--S 111 of 526
m0419a:= a0419.1-r0419
--R
--R
--R (107)
--R 2 3 6 2 3 5
--R (- 2a b - a )sinh(x) + (- 12a b - 6a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((- 30a b - 15a )cosh(x) - 4a b - 3a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((- 40a b - 20a )cosh(x) + (- 16a b - 12a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2
--R (- 30a b - 15a )cosh(x) + (- 24a b - 18a )cosh(x) - 2a b
--R +
--R 3
--R - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (- 12a b - 6a )cosh(x) + (- 16a b - 12a )cosh(x)
--R +
--R 2 3
--R (- 4a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4
--R (- 2a b - a )cosh(x) + (- 4a b - 3a )cosh(x)
--R +
--R 2 3 2 3
--R (- 2a b - 3a )cosh(x) - a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (2a b + 2a b + a )sinh(x)
--R +
--R 2 2 3 7
--R (16a b + 16a b + 8a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3 6
--R ((56a b + 56a b + 28a )cosh(x) + 4a b + 8a b + 4a )sinh(x)
--R +
--R 2 2 3 3
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3
--R (24a b + 48a b + 24a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (140a b + 140a b + 70a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (60a b + 120a b + 60a )cosh(x) + 2a b + 10a b + 6a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3 3
--R (80a b + 160a b + 80a )cosh(x)
--R +
--R 2 2 3
--R (8a b + 40a b + 24a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (56a b + 56a b + 28a )cosh(x)
--R +
--R 2 2 3 4
--R (60a b + 120a b + 60a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (12a b + 60a b + 36a )cosh(x) + 4a b + 4a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (16a b + 16a b + 8a )cosh(x)
--R +
--R 2 2 3 5
--R (24a b + 48a b + 24a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (8a b + 40a b + 24a )cosh(x) + (8a b + 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (2a b + 2a b + a )cosh(x) + (4a b + 8a b + 4a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (2a b + 10a b + 6a )cosh(x) + (4a b + 4a )cosh(x) + a
--R *
--R +-+
--R \|a
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2 +-+
--R (sinh(x) + 2cosh(x)sinh(x) + cosh(x) )\|a
--R +
--R 2 2 6 2 2 5
--R (2b + 7a b + 3a )sinh(x) + (12b + 42a b + 18a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((30b + 105a b + 45a )cosh(x) + 4b + 15a b + 9a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((40b + 140a b + 60a )cosh(x) + (16b + 60a b + 36a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (30b + 105a b + 45a )cosh(x)
--R +
--R 2 2 2 2 2
--R (24b + 90a b + 54a )cosh(x) + 2b + 9a b + 9a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5 2 2 3
--R (12b + 42a b + 18a )cosh(x) + (16b + 60a b + 36a )cosh(x)
--R +
--R 2 2
--R (4b + 18a b + 18a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (2b + 7a b + 3a )cosh(x) + (4b + 15a b + 9a )cosh(x)
--R +
--R 2 2 2 2
--R (2b + 9a b + 9a )cosh(x) + a b + 3a
--R *
--R +-------------------------------------+
--R | 2 2
--R +---+ +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|- b \|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 3 8
--R (- 2b - 8a b - 7a b - 3a )sinh(x)
--R +
--R 3 2 2 3 7
--R (- 16b - 64a b - 56a b - 24a )cosh(x)sinh(x)
--R +
--R 3 2 2 3 2 3 2
--R (- 56b - 224a b - 196a b - 84a )cosh(x) - 4b - 20a b
--R +
--R 2 3
--R - 28a b - 12a
--R *
--R 6
--R sinh(x)
--R +
--R 3 2 2 3 3
--R (- 112b - 448a b - 392a b - 168a )cosh(x)
--R +
--R 3 2 2 3
--R (- 24b - 120a b - 168a b - 72a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 3 4
--R (- 140b - 560a b - 490a b - 210a )cosh(x)
--R +
--R 3 2 2 3 2 3 2
--R (- 60b - 300a b - 420a b - 180a )cosh(x) - 2b - 16a b
--R +
--R 2 3
--R - 36a b - 18a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 3 5
--R (- 112b - 448a b - 392a b - 168a )cosh(x)
--R +
--R 3 2 2 3 3
--R (- 80b - 400a b - 560a b - 240a )cosh(x)
--R +
--R 3 2 2 3
--R (- 8b - 64a b - 144a b - 72a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 3 6
--R (- 56b - 224a b - 196a b - 84a )cosh(x)
--R +
--R 3 2 2 3 4
--R (- 60b - 300a b - 420a b - 180a )cosh(x)
--R +
--R 3 2 2 3 2 2 2
--R (- 12b - 96a b - 216a b - 108a )cosh(x) - 4a b - 16a b
--R +
--R 3
--R - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 3 7
--R (- 16b - 64a b - 56a b - 24a )cosh(x)
--R +
--R 3 2 2 3 5
--R (- 24b - 120a b - 168a b - 72a )cosh(x)
--R +
--R 3 2 2 3 3
--R (- 8b - 64a b - 144a b - 72a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 32a b - 24a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 3 8
--R (- 2b - 8a b - 7a b - 3a )cosh(x)
--R +
--R 3 2 2 3 6
--R (- 4b - 20a b - 28a b - 12a )cosh(x)
--R +
--R 3 2 2 3 4
--R (- 2b - 16a b - 36a b - 18a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (- 4a b - 16a b - 12a )cosh(x) - a b - 3a
--R *
--R +---+
--R \|- b
--R *
--R log
--R 4 3
--R (2b + 2a)sinh(x) + (8b + 8a)cosh(x)sinh(x)
--R +
--R 2 2
--R ((12b + 12a)cosh(x) + 2a)sinh(x)
--R +
--R 3
--R ((8b + 8a)cosh(x) + 4a cosh(x))sinh(x)
--R +
--R 4 2
--R (2b + 2a)cosh(x) + 2a cosh(x)
--R *
--R +---+ +-+
--R \|- b \|a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 2a b + 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (4a b + 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2
--R (2a b + 2a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 6 2 2 5
--R (2b - a )sinh(x) + (12b - 6a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((30b - 15a )cosh(x) - 2b - 4a b - 3a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((40b - 20a )cosh(x) + (- 8b - 16a b - 12a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4 2 2 2
--R (30b - 15a )cosh(x) + (- 12b - 24a b - 18a )cosh(x)
--R +
--R 2
--R - 4a b - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5 2 2 3
--R (12b - 6a )cosh(x) + (- 8b - 16a b - 12a )cosh(x)
--R +
--R 2
--R (- 8a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (2b - a )cosh(x) + (- 2b - 4a b - 3a )cosh(x)
--R +
--R 2 2 2
--R (- 4a b - 3a )cosh(x) - a
--R *
--R +-+
--R \|a
--R +
--R 2 6 2 5
--R (- 2a b - 2a )sinh(x) + (- 12a b - 12a )cosh(x)sinh(x)
--R +
--R 2 2 2 4
--R ((- 30a b - 30a )cosh(x) - 6a b - 4a )sinh(x)
--R +
--R 2 3 2 3
--R ((- 40a b - 40a )cosh(x) + (- 24a b - 16a )cosh(x))sinh(x)
--R +
--R 2 4 2 2
--R (- 30a b - 30a )cosh(x) + (- 36a b - 24a )cosh(x)
--R +
--R 2
--R - 2a
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3
--R (- 12a b - 12a )cosh(x) + (- 24a b - 16a )cosh(x)
--R +
--R 2
--R - 4a cosh(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4 2 2
--R (- 2a b - 2a )cosh(x) + (- 6a b - 4a )cosh(x) - 2a cosh(x)
--R *
--R +---+
--R \|- b
--R /
--R 2 4 2 3
--R (2a b + a )sinh(x) + (8a b + 4a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R ((12a b + 6a )cosh(x) + 2a b + 2a )sinh(x)
--R +
--R 2 3 2
--R ((8a b + 4a )cosh(x) + (4a b + 4a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2
--R (2a b + a )cosh(x) + (2a b + 2a )cosh(x) + a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 6
--R (- 2b - 2a b - a )sinh(x)
--R +
--R 2 2 5
--R (- 12b - 12a b - 6a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((- 30b - 30a b - 15a )cosh(x) - 2b - 6a b - 3a )sinh(x)
--R +
--R 2 2 3
--R (- 40b - 40a b - 20a )cosh(x)
--R +
--R 2 2
--R (- 8b - 24a b - 12a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (- 30b - 30a b - 15a )cosh(x)
--R +
--R 2 2 2 2
--R (- 12b - 36a b - 18a )cosh(x) - 4a b - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5
--R (- 12b - 12a b - 6a )cosh(x)
--R +
--R 2 2 3 2
--R (- 8b - 24a b - 12a )cosh(x) + (- 8a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (- 2b - 2a b - a )cosh(x) + (- 2b - 6a b - 3a )cosh(x)
--R +
--R 2 2 2
--R (- 4a b - 3a )cosh(x) - a
--R *
--R +-+
--R \|a
--R +
--R 2 3 6 2 3 5
--R (2a b + a )sinh(x) + (12a b + 6a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((30a b + 15a )cosh(x) + 4a b + 3a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((40a b + 20a )cosh(x) + (16a b + 12a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2 3
--R ((30a b + 15a )cosh(x) + (24a b + 18a )cosh(x) + 2a b + 3a )
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (12a b + 6a )cosh(x) + (16a b + 12a )cosh(x)
--R +
--R 2 3
--R (4a b + 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4 2 3 2
--R (2a b + a )cosh(x) + (4a b + 3a )cosh(x) + (2a b + 3a )cosh(x)
--R +
--R 3
--R a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (- 2a b - 2a b - a )sinh(x)
--R +
--R 2 2 3 7
--R (- 16a b - 16a b - 8a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3 6
--R ((- 56a b - 56a b - 28a )cosh(x) - 4a b - 8a b - 4a )sinh(x)
--R +
--R 2 2 3 3
--R (- 112a b - 112a b - 56a )cosh(x)
--R +
--R 2 2 3
--R (- 24a b - 48a b - 24a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (- 140a b - 140a b - 70a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (- 60a b - 120a b - 60a )cosh(x) - 2a b - 10a b - 6a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (- 112a b - 112a b - 56a )cosh(x)
--R +
--R 2 2 3 3
--R (- 80a b - 160a b - 80a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 40a b - 24a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (- 56a b - 56a b - 28a )cosh(x)
--R +
--R 2 2 3 4
--R (- 60a b - 120a b - 60a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (- 12a b - 60a b - 36a )cosh(x) - 4a b - 4a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (- 16a b - 16a b - 8a )cosh(x)
--R +
--R 2 2 3 5
--R (- 24a b - 48a b - 24a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (- 8a b - 40a b - 24a )cosh(x) + (- 8a b - 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (- 2a b - 2a b - a )cosh(x) + (- 4a b - 8a b - 4a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (- 2a b - 10a b - 6a )cosh(x) + (- 4a b - 4a )cosh(x) - a
--R *
--R +-+
--R \|a
--R *
--R log
--R 2 2
--R (- 2b sinh(x) - 4b cosh(x)sinh(x) - 2b cosh(x) - 2a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - 2b sinh(x) - 8b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 12b cosh(x) + 6b + 2a)sinh(x)
--R +
--R 3 4
--R (- 8b cosh(x) + (12b + 4a)cosh(x))sinh(x) - 2b cosh(x)
--R +
--R 2
--R (6b + 2a)cosh(x) + 2a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x)
--R +
--R 3 4
--R 4cosh(x) sinh(x) + cosh(x)
--R *
--R +-+
--R \|a
--R +
--R 2 3 6 2 3 5
--R (- 4a b - 2a )sinh(x) + (- 24a b - 12a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((- 60a b - 30a )cosh(x) - 8a b - 6a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((- 80a b - 40a )cosh(x) + (- 32a b - 24a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2
--R (- 60a b - 30a )cosh(x) + (- 48a b - 36a )cosh(x) - 4a b
--R +
--R 3
--R - 6a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (- 24a b - 12a )cosh(x) + (- 32a b - 24a )cosh(x)
--R +
--R 2 3
--R (- 8a b - 12a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4
--R (- 4a b - 2a )cosh(x) + (- 8a b - 6a )cosh(x)
--R +
--R 2 3 2 3
--R (- 4a b - 6a )cosh(x) - 2a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (4a b + 4a b + 2a )sinh(x)
--R +
--R 2 2 3 7
--R (32a b + 32a b + 16a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3 6
--R ((112a b + 112a b + 56a )cosh(x) + 8a b + 16a b + 8a )sinh(x)
--R +
--R 2 2 3 3
--R (224a b + 224a b + 112a )cosh(x)
--R +
--R 2 2 3
--R (48a b + 96a b + 48a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (280a b + 280a b + 140a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (120a b + 240a b + 120a )cosh(x) + 4a b + 20a b + 12a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (224a b + 224a b + 112a )cosh(x)
--R +
--R 2 2 3 3
--R (160a b + 320a b + 160a )cosh(x)
--R +
--R 2 2 3
--R (16a b + 80a b + 48a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3 4
--R (120a b + 240a b + 120a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (24a b + 120a b + 72a )cosh(x) + 8a b + 8a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (32a b + 32a b + 16a )cosh(x)
--R +
--R 2 2 3 5
--R (48a b + 96a b + 48a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (16a b + 80a b + 48a )cosh(x) + (16a b + 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (4a b + 4a b + 2a )cosh(x) + (8a b + 16a b + 8a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (4a b + 20a b + 12a )cosh(x) + (8a b + 8a )cosh(x) + 2a
--R *
--R +-+
--R \|a
--R *
--R +-+
--R tanh(x)\|a
--R atanh(-----------------)
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R +
--R 2 2 6 2 2 5
--R (2b + 7a b + 3a )sinh(x) + (12b + 42a b + 18a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((30b + 105a b + 45a )cosh(x) + 4b + 15a b + 9a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((40b + 140a b + 60a )cosh(x) + (16b + 60a b + 36a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (30b + 105a b + 45a )cosh(x)
--R +
--R 2 2 2 2 2
--R (24b + 90a b + 54a )cosh(x) + 2b + 9a b + 9a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5 2 2 3
--R (12b + 42a b + 18a )cosh(x) + (16b + 60a b + 36a )cosh(x)
--R +
--R 2 2
--R (4b + 18a b + 18a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (2b + 7a b + 3a )cosh(x) + (4b + 15a b + 9a )cosh(x)
--R +
--R 2 2 2 2
--R (2b + 9a b + 9a )cosh(x) + a b + 3a
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a \|b |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 3 8
--R (- 2b - 8a b - 7a b - 3a )sinh(x)
--R +
--R 3 2 2 3 7
--R (- 16b - 64a b - 56a b - 24a )cosh(x)sinh(x)
--R +
--R 3 2 2 3 2 3 2
--R (- 56b - 224a b - 196a b - 84a )cosh(x) - 4b - 20a b
--R +
--R 2 3
--R - 28a b - 12a
--R *
--R 6
--R sinh(x)
--R +
--R 3 2 2 3 3
--R (- 112b - 448a b - 392a b - 168a )cosh(x)
--R +
--R 3 2 2 3
--R (- 24b - 120a b - 168a b - 72a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 3 4
--R (- 140b - 560a b - 490a b - 210a )cosh(x)
--R +
--R 3 2 2 3 2 3 2
--R (- 60b - 300a b - 420a b - 180a )cosh(x) - 2b - 16a b
--R +
--R 2 3
--R - 36a b - 18a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 3 5
--R (- 112b - 448a b - 392a b - 168a )cosh(x)
--R +
--R 3 2 2 3 3
--R (- 80b - 400a b - 560a b - 240a )cosh(x)
--R +
--R 3 2 2 3
--R (- 8b - 64a b - 144a b - 72a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 3 6
--R (- 56b - 224a b - 196a b - 84a )cosh(x)
--R +
--R 3 2 2 3 4
--R (- 60b - 300a b - 420a b - 180a )cosh(x)
--R +
--R 3 2 2 3 2 2 2
--R (- 12b - 96a b - 216a b - 108a )cosh(x) - 4a b - 16a b
--R +
--R 3
--R - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 3 7
--R (- 16b - 64a b - 56a b - 24a )cosh(x)
--R +
--R 3 2 2 3 5
--R (- 24b - 120a b - 168a b - 72a )cosh(x)
--R +
--R 3 2 2 3 3
--R (- 8b - 64a b - 144a b - 72a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 32a b - 24a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 3 8
--R (- 2b - 8a b - 7a b - 3a )cosh(x)
--R +
--R 3 2 2 3 6
--R (- 4b - 20a b - 28a b - 12a )cosh(x)
--R +
--R 3 2 2 3 4
--R (- 2b - 16a b - 36a b - 18a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (- 4a b - 16a b - 12a )cosh(x) - a b - 3a
--R *
--R +-+
--R \|b
--R *
--R +--------------+
--R | 2
--R coth(x)\|b sech(x) + a
--R atan(------------------------)
--R +-+
--R \|b
--R +
--R 2 6 2 5
--R (- 2b - a b)sinh(x) + (- 12b - 6a b)cosh(x)sinh(x)
--R +
--R 2 2 2 4
--R ((- 30b - 15a b)cosh(x) - 4b - 3a b)sinh(x)
--R +
--R 2 3 2 3
--R ((- 40b - 20a b)cosh(x) + (- 16b - 12a b)cosh(x))sinh(x)
--R +
--R 2 4 2 2 2
--R (- 30b - 15a b)cosh(x) + (- 24b - 18a b)cosh(x) - 2b
--R +
--R - 3a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3
--R (- 12b - 6a b)cosh(x) + (- 16b - 12a b)cosh(x)
--R +
--R 2
--R (- 4b - 6a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4
--R (- 2b - a b)cosh(x) + (- 4b - 3a b)cosh(x)
--R +
--R 2 2
--R (- 2b - 3a b)cosh(x) - a b
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R tanh(x)\|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 8
--R (2b + 2a b + a b)sinh(x)
--R +
--R 3 2 2 7
--R (16b + 16a b + 8a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 6
--R ((56b + 56a b + 28a b)cosh(x) + 4b + 8a b + 4a b)sinh(x)
--R +
--R 3 2 2 3
--R (112b + 112a b + 56a b)cosh(x)
--R +
--R 3 2 2
--R (24b + 48a b + 24a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 4
--R (140b + 140a b + 70a b)cosh(x)
--R +
--R 3 2 2 2 3 2 2
--R (60b + 120a b + 60a b)cosh(x) + 2b + 10a b + 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 5
--R (112b + 112a b + 56a b)cosh(x)
--R +
--R 3 2 2 3
--R (80b + 160a b + 80a b)cosh(x)
--R +
--R 3 2 2
--R (8b + 40a b + 24a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 6
--R (56b + 56a b + 28a b)cosh(x)
--R +
--R 3 2 2 4
--R (60b + 120a b + 60a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (12b + 60a b + 36a b)cosh(x) + 4a b + 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 7
--R (16b + 16a b + 8a b)cosh(x)
--R +
--R 3 2 2 5
--R (24b + 48a b + 24a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (8b + 40a b + 24a b)cosh(x) + (8a b + 8a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 8 3 2 2 6
--R (2b + 2a b + a b)cosh(x) + (4b + 8a b + 4a b)cosh(x)
--R +
--R 3 2 2 4 2 2 2 2
--R (2b + 10a b + 6a b)cosh(x) + (4a b + 4a b)cosh(x) + a b
--R *
--R tanh(x)
--R *
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R +
--R 3 2 2 6
--R (- 2b - 4a b - 2a b)sinh(x)
--R +
--R 3 2 2 5
--R (- 12b - 24a b - 12a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 4
--R ((- 30b - 60a b - 30a b)cosh(x) + 2b - 6a b - 4a b)sinh(x)
--R +
--R 3 2 2 3 3 2 2
--R ((- 40b - 80a b - 40a b)cosh(x) + (8b - 24a b - 16a b)cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 4
--R (- 30b - 60a b - 30a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (12b - 36a b - 24a b)cosh(x) + 2a b - 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 5
--R (- 12b - 24a b - 12a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (8b - 24a b - 16a b)cosh(x) + (4a b - 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 6 3 2 2 4
--R (- 2b - 4a b - 2a b)cosh(x) + (2b - 6a b - 4a b)cosh(x)
--R +
--R 2 2 2
--R (2a b - 2a b)cosh(x)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 8 3 2 2 7
--R (2b + 4a b + 2a b)sinh(x) + (16b + 32a b + 16a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 6
--R ((56b + 112a b + 56a b)cosh(x) + 12b + 14a b + 6a b)sinh(x)
--R +
--R 3 2 2 3
--R (112b + 224a b + 112a b)cosh(x)
--R +
--R 3 2 2
--R (72b + 84a b + 36a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 4
--R (140b + 280a b + 140a b)cosh(x)
--R +
--R 3 2 2 2 3 2 2
--R (180b + 210a b + 90a b)cosh(x) - 6b + 8a b + 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 5
--R (112b + 224a b + 112a b)cosh(x)
--R +
--R 3 2 2 3
--R (240b + 280a b + 120a b)cosh(x)
--R +
--R 3 2 2
--R (- 24b + 32a b + 24a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 6
--R (56b + 112a b + 56a b)cosh(x)
--R +
--R 3 2 2 4
--R (180b + 210a b + 90a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (- 36b + 48a b + 36a b)cosh(x) - 2a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 7 3 2 2 5
--R (16b + 32a b + 16a b)cosh(x) + (72b + 84a b + 36a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (- 24b + 32a b + 24a b)cosh(x) + (- 4a b + 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 8 3 2 2 6
--R (2b + 4a b + 2a b)cosh(x) + (12b + 14a b + 6a b)cosh(x)
--R +
--R 3 2 2 4 2 2 2
--R (- 6b + 8a b + 6a b)cosh(x) + (- 2a b + 2a b)cosh(x)
--R *
--R +-+
--R \|a
--R /
--R 6 5
--R (4b + 2a)sinh(x) + (24b + 12a)cosh(x)sinh(x)
--R +
--R 2 4
--R ((60b + 30a)cosh(x) + 8b + 6a)sinh(x)
--R +
--R 3 3
--R ((80b + 40a)cosh(x) + (32b + 24a)cosh(x))sinh(x)
--R +
--R 4 2 2
--R ((60b + 30a)cosh(x) + (48b + 36a)cosh(x) + 4b + 6a)sinh(x)
--R +
--R 5 3
--R ((24b + 12a)cosh(x) + (32b + 24a)cosh(x) + (8b + 12a)cosh(x))
--R *
--R sinh(x)
--R +
--R 6 4 2
--R (4b + 2a)cosh(x) + (8b + 6a)cosh(x) + (4b + 6a)cosh(x) + 2a
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 8 2 2 7
--R (- 4b - 4a b - 2a )sinh(x) + (- 32b - 32a b - 16a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 6
--R ((- 112b - 112a b - 56a )cosh(x) - 8b - 16a b - 8a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((- 224b - 224a b - 112a )cosh(x) + (- 48b - 96a b - 48a )cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 4
--R (- 280b - 280a b - 140a )cosh(x)
--R +
--R 2 2 2 2 2
--R (- 120b - 240a b - 120a )cosh(x) - 4b - 20a b - 12a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 5
--R (- 224b - 224a b - 112a )cosh(x)
--R +
--R 2 2 3 2 2
--R (- 160b - 320a b - 160a )cosh(x) + (- 16b - 80a b - 48a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 6
--R (- 112b - 112a b - 56a )cosh(x)
--R +
--R 2 2 4 2 2 2
--R (- 120b - 240a b - 120a )cosh(x) + (- 24b - 120a b - 72a )cosh(x)
--R +
--R 2
--R - 8a b - 8a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 7 2 2 5
--R (- 32b - 32a b - 16a )cosh(x) + (- 48b - 96a b - 48a )cosh(x)
--R +
--R 2 2 3 2
--R (- 16b - 80a b - 48a )cosh(x) + (- 16a b - 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 8 2 2 6
--R (- 4b - 4a b - 2a )cosh(x) + (- 8b - 16a b - 8a )cosh(x)
--R +
--R 2 2 4 2 2 2
--R (- 4b - 20a b - 12a )cosh(x) + (- 8a b - 8a )cosh(x) - 2a
--R Type: Expression(Integer)
--E 111
--S 112 of 526
--d0419a:= D(m0419a,x)
--E 112
--S 113 of 526
m0419b:= a0419.2-r0419
--R
--R
--R (108)
--R 2 3 6 2 3 5
--R (- 2a b - a )sinh(x) + (- 12a b - 6a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((- 30a b - 15a )cosh(x) - 4a b - 3a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((- 40a b - 20a )cosh(x) + (- 16a b - 12a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2
--R (- 30a b - 15a )cosh(x) + (- 24a b - 18a )cosh(x) - 2a b
--R +
--R 3
--R - 3a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (- 12a b - 6a )cosh(x) + (- 16a b - 12a )cosh(x)
--R +
--R 2 3
--R (- 4a b - 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4
--R (- 2a b - a )cosh(x) + (- 4a b - 3a )cosh(x)
--R +
--R 2 3 2 3
--R (- 2a b - 3a )cosh(x) - a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (2a b + 2a b + a )sinh(x)
--R +
--R 2 2 3 7
--R (16a b + 16a b + 8a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3 6
--R ((56a b + 56a b + 28a )cosh(x) + 4a b + 8a b + 4a )sinh(x)
--R +
--R 2 2 3 3
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3
--R (24a b + 48a b + 24a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (140a b + 140a b + 70a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (60a b + 120a b + 60a )cosh(x) + 2a b + 10a b + 6a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3 3
--R (80a b + 160a b + 80a )cosh(x)
--R +
--R 2 2 3
--R (8a b + 40a b + 24a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (56a b + 56a b + 28a )cosh(x)
--R +
--R 2 2 3 4
--R (60a b + 120a b + 60a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (12a b + 60a b + 36a )cosh(x) + 4a b + 4a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (16a b + 16a b + 8a )cosh(x)
--R +
--R 2 2 3 5
--R (24a b + 48a b + 24a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (8a b + 40a b + 24a )cosh(x) + (8a b + 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (2a b + 2a b + a )cosh(x) + (4a b + 8a b + 4a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (2a b + 10a b + 6a )cosh(x) + (4a b + 4a )cosh(x) + a
--R *
--R +-+
--R \|a
--R *
--R log
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2 +-+
--R (sinh(x) + 2cosh(x)sinh(x) + cosh(x) )\|a
--R +
--R 2 3 6 2 3 5
--R (2a b + a )sinh(x) + (12a b + 6a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((30a b + 15a )cosh(x) + 4a b + 3a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((40a b + 20a )cosh(x) + (16a b + 12a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2 3
--R ((30a b + 15a )cosh(x) + (24a b + 18a )cosh(x) + 2a b + 3a )
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (12a b + 6a )cosh(x) + (16a b + 12a )cosh(x)
--R +
--R 2 3
--R (4a b + 6a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4 2 3 2
--R (2a b + a )cosh(x) + (4a b + 3a )cosh(x) + (2a b + 3a )cosh(x)
--R +
--R 3
--R a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (- 2a b - 2a b - a )sinh(x)
--R +
--R 2 2 3 7
--R (- 16a b - 16a b - 8a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3 6
--R ((- 56a b - 56a b - 28a )cosh(x) - 4a b - 8a b - 4a )sinh(x)
--R +
--R 2 2 3 3
--R (- 112a b - 112a b - 56a )cosh(x)
--R +
--R 2 2 3
--R (- 24a b - 48a b - 24a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (- 140a b - 140a b - 70a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (- 60a b - 120a b - 60a )cosh(x) - 2a b - 10a b - 6a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (- 112a b - 112a b - 56a )cosh(x)
--R +
--R 2 2 3 3
--R (- 80a b - 160a b - 80a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 40a b - 24a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (- 56a b - 56a b - 28a )cosh(x)
--R +
--R 2 2 3 4
--R (- 60a b - 120a b - 60a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (- 12a b - 60a b - 36a )cosh(x) - 4a b - 4a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (- 16a b - 16a b - 8a )cosh(x)
--R +
--R 2 2 3 5
--R (- 24a b - 48a b - 24a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (- 8a b - 40a b - 24a )cosh(x) + (- 8a b - 8a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (- 2a b - 2a b - a )cosh(x) + (- 4a b - 8a b - 4a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (- 2a b - 10a b - 6a )cosh(x) + (- 4a b - 4a )cosh(x) - a
--R *
--R +-+
--R \|a
--R *
--R log
--R 2 2
--R (- 2b sinh(x) - 4b cosh(x)sinh(x) - 2b cosh(x) - 2a)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 4 3
--R - 2b sinh(x) - 8b cosh(x)sinh(x)
--R +
--R 2 2
--R (- 12b cosh(x) + 6b + 2a)sinh(x)
--R +
--R 3 4
--R (- 8b cosh(x) + (12b + 4a)cosh(x))sinh(x) - 2b cosh(x)
--R +
--R 2
--R (6b + 2a)cosh(x) + 2a
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + 6cosh(x) sinh(x)
--R +
--R 3 4
--R 4cosh(x) sinh(x) + cosh(x)
--R *
--R +-+
--R \|a
--R +
--R 2 3 6 2 3 5
--R (- 4a b - 2a )sinh(x) + (- 24a b - 12a )cosh(x)sinh(x)
--R +
--R 2 3 2 2 3 4
--R ((- 60a b - 30a )cosh(x) - 8a b - 6a )sinh(x)
--R +
--R 2 3 3 2 3 3
--R ((- 80a b - 40a )cosh(x) + (- 32a b - 24a )cosh(x))sinh(x)
--R +
--R 2 3 4 2 3 2 2
--R (- 60a b - 30a )cosh(x) + (- 48a b - 36a )cosh(x) - 4a b
--R +
--R 3
--R - 6a
--R *
--R 2
--R sinh(x)
--R +
--R 2 3 5 2 3 3
--R (- 24a b - 12a )cosh(x) + (- 32a b - 24a )cosh(x)
--R +
--R 2 3
--R (- 8a b - 12a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 3 6 2 3 4
--R (- 4a b - 2a )cosh(x) + (- 8a b - 6a )cosh(x)
--R +
--R 2 3 2 3
--R (- 4a b - 6a )cosh(x) - 2a
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 3 8
--R (4a b + 4a b + 2a )sinh(x)
--R +
--R 2 2 3 7
--R (32a b + 32a b + 16a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 2 2 3 6
--R ((112a b + 112a b + 56a )cosh(x) + 8a b + 16a b + 8a )sinh(x)
--R +
--R 2 2 3 3
--R (224a b + 224a b + 112a )cosh(x)
--R +
--R 2 2 3
--R (48a b + 96a b + 48a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 3 4
--R (280a b + 280a b + 140a )cosh(x)
--R +
--R 2 2 3 2 2 2 3
--R (120a b + 240a b + 120a )cosh(x) + 4a b + 20a b + 12a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 3 5
--R (224a b + 224a b + 112a )cosh(x)
--R +
--R 2 2 3 3
--R (160a b + 320a b + 160a )cosh(x)
--R +
--R 2 2 3
--R (16a b + 80a b + 48a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 3 6
--R (112a b + 112a b + 56a )cosh(x)
--R +
--R 2 2 3 4
--R (120a b + 240a b + 120a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (24a b + 120a b + 72a )cosh(x) + 8a b + 8a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 7
--R (32a b + 32a b + 16a )cosh(x)
--R +
--R 2 2 3 5
--R (48a b + 96a b + 48a )cosh(x)
--R +
--R 2 2 3 3 2 3
--R (16a b + 80a b + 48a )cosh(x) + (16a b + 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 8 2 2 3 6
--R (4a b + 4a b + 2a )cosh(x) + (8a b + 16a b + 8a )cosh(x)
--R +
--R 2 2 3 4 2 3 2 3
--R (4a b + 20a b + 12a )cosh(x) + (8a b + 8a )cosh(x) + 2a
--R *
--R +-+
--R \|a
--R *
--R +-+
--R tanh(x)\|a
--R atanh(-----------------)
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R +
--R 2 2 6 2 2 5
--R (2b + 7a b + 3a )sinh(x) + (12b + 42a b + 18a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((30b + 105a b + 45a )cosh(x) + 4b + 15a b + 9a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((40b + 140a b + 60a )cosh(x) + (16b + 60a b + 36a )cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (30b + 105a b + 45a )cosh(x)
--R +
--R 2 2 2 2 2
--R (24b + 90a b + 54a )cosh(x) + 2b + 9a b + 9a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5 2 2 3
--R (12b + 42a b + 18a )cosh(x) + (16b + 60a b + 36a )cosh(x)
--R +
--R 2 2
--R (4b + 18a b + 18a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (2b + 7a b + 3a )cosh(x) + (4b + 15a b + 9a )cosh(x)
--R +
--R 2 2 2 2
--R (2b + 9a b + 9a )cosh(x) + a b + 3a
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a \|b |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 3 8
--R (- 2b - 8a b - 7a b - 3a )sinh(x)
--R +
--R 3 2 2 3 7
--R (- 16b - 64a b - 56a b - 24a )cosh(x)sinh(x)
--R +
--R 3 2 2 3 2 3 2
--R (- 56b - 224a b - 196a b - 84a )cosh(x) - 4b - 20a b
--R +
--R 2 3
--R - 28a b - 12a
--R *
--R 6
--R sinh(x)
--R +
--R 3 2 2 3 3
--R (- 112b - 448a b - 392a b - 168a )cosh(x)
--R +
--R 3 2 2 3
--R (- 24b - 120a b - 168a b - 72a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 3 4
--R (- 140b - 560a b - 490a b - 210a )cosh(x)
--R +
--R 3 2 2 3 2 3 2
--R (- 60b - 300a b - 420a b - 180a )cosh(x) - 2b - 16a b
--R +
--R 2 3
--R - 36a b - 18a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 3 5
--R (- 112b - 448a b - 392a b - 168a )cosh(x)
--R +
--R 3 2 2 3 3
--R (- 80b - 400a b - 560a b - 240a )cosh(x)
--R +
--R 3 2 2 3
--R (- 8b - 64a b - 144a b - 72a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 3 6
--R (- 56b - 224a b - 196a b - 84a )cosh(x)
--R +
--R 3 2 2 3 4
--R (- 60b - 300a b - 420a b - 180a )cosh(x)
--R +
--R 3 2 2 3 2 2 2
--R (- 12b - 96a b - 216a b - 108a )cosh(x) - 4a b - 16a b
--R +
--R 3
--R - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 3 7
--R (- 16b - 64a b - 56a b - 24a )cosh(x)
--R +
--R 3 2 2 3 5
--R (- 24b - 120a b - 168a b - 72a )cosh(x)
--R +
--R 3 2 2 3 3
--R (- 8b - 64a b - 144a b - 72a )cosh(x)
--R +
--R 2 2 3
--R (- 8a b - 32a b - 24a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 3 8
--R (- 2b - 8a b - 7a b - 3a )cosh(x)
--R +
--R 3 2 2 3 6
--R (- 4b - 20a b - 28a b - 12a )cosh(x)
--R +
--R 3 2 2 3 4
--R (- 2b - 16a b - 36a b - 18a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (- 4a b - 16a b - 12a )cosh(x) - a b - 3a
--R *
--R +-+
--R \|b
--R *
--R +--------------+
--R | 2
--R coth(x)\|b sech(x) + a
--R atan(------------------------)
--R +-+
--R \|b
--R +
--R 2 2 6
--R (- 4b - 14a b - 6a )sinh(x)
--R +
--R 2 2 5
--R (- 24b - 84a b - 36a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 4
--R ((- 60b - 210a b - 90a )cosh(x) - 8b - 30a b - 18a )sinh(x)
--R +
--R 2 2 3
--R (- 80b - 280a b - 120a )cosh(x)
--R +
--R 2 2
--R (- 32b - 120a b - 72a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 4
--R (- 60b - 210a b - 90a )cosh(x)
--R +
--R 2 2 2 2 2
--R (- 48b - 180a b - 108a )cosh(x) - 4b - 18a b - 18a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 5
--R (- 24b - 84a b - 36a )cosh(x)
--R +
--R 2 2 3
--R (- 32b - 120a b - 72a )cosh(x)
--R +
--R 2 2
--R (- 8b - 36a b - 36a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 6 2 2 4
--R (- 4b - 14a b - 6a )cosh(x) + (- 8b - 30a b - 18a )cosh(x)
--R +
--R 2 2 2 2
--R (- 4b - 18a b - 18a )cosh(x) - 2a b - 6a
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a \|b |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 3 8
--R (4b + 16a b + 14a b + 6a )sinh(x)
--R +
--R 3 2 2 3 7
--R (32b + 128a b + 112a b + 48a )cosh(x)sinh(x)
--R +
--R 3 2 2 3 2 3 2
--R (112b + 448a b + 392a b + 168a )cosh(x) + 8b + 40a b
--R +
--R 2 3
--R 56a b + 24a
--R *
--R 6
--R sinh(x)
--R +
--R 3 2 2 3 3
--R (224b + 896a b + 784a b + 336a )cosh(x)
--R +
--R 3 2 2 3
--R (48b + 240a b + 336a b + 144a )cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 3 4
--R (280b + 1120a b + 980a b + 420a )cosh(x)
--R +
--R 3 2 2 3 2 3 2
--R (120b + 600a b + 840a b + 360a )cosh(x) + 4b + 32a b
--R +
--R 2 3
--R 72a b + 36a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 3 5
--R (224b + 896a b + 784a b + 336a )cosh(x)
--R +
--R 3 2 2 3 3
--R (160b + 800a b + 1120a b + 480a )cosh(x)
--R +
--R 3 2 2 3
--R (16b + 128a b + 288a b + 144a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 3 6
--R (112b + 448a b + 392a b + 168a )cosh(x)
--R +
--R 3 2 2 3 4
--R (120b + 600a b + 840a b + 360a )cosh(x)
--R +
--R 3 2 2 3 2 2 2
--R (24b + 192a b + 432a b + 216a )cosh(x) + 8a b + 32a b
--R +
--R 3
--R 24a
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 3 7
--R (32b + 128a b + 112a b + 48a )cosh(x)
--R +
--R 3 2 2 3 5
--R (48b + 240a b + 336a b + 144a )cosh(x)
--R +
--R 3 2 2 3 3
--R (16b + 128a b + 288a b + 144a )cosh(x)
--R +
--R 2 2 3
--R (16a b + 64a b + 48a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 3 8
--R (4b + 16a b + 14a b + 6a )cosh(x)
--R +
--R 3 2 2 3 6
--R (8b + 40a b + 56a b + 24a )cosh(x)
--R +
--R 3 2 2 3 4
--R (4b + 32a b + 72a b + 36a )cosh(x)
--R +
--R 2 2 3 2 2 3
--R (8a b + 32a b + 24a )cosh(x) + 2a b + 6a
--R *
--R +-+
--R \|b
--R *
--R atan
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a 2
--R \|a |------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2
--R - 2a cosh(x)sinh(x) - a cosh(x) - a
--R /
--R 2 2 +-+ +-+
--R (2sinh(x) + 4cosh(x)sinh(x) + 2cosh(x) )\|a \|b
--R +
--R 2 6 2 5
--R (- 2b - a b)sinh(x) + (- 12b - 6a b)cosh(x)sinh(x)
--R +
--R 2 2 2 4
--R ((- 30b - 15a b)cosh(x) - 4b - 3a b)sinh(x)
--R +
--R 2 3 2 3
--R ((- 40b - 20a b)cosh(x) + (- 16b - 12a b)cosh(x))sinh(x)
--R +
--R 2 4 2 2 2
--R (- 30b - 15a b)cosh(x) + (- 24b - 18a b)cosh(x) - 2b
--R +
--R - 3a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3
--R (- 12b - 6a b)cosh(x) + (- 16b - 12a b)cosh(x)
--R +
--R 2
--R (- 4b - 6a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4
--R (- 2b - a b)cosh(x) + (- 4b - 3a b)cosh(x)
--R +
--R 2 2
--R (- 2b - 3a b)cosh(x) - a b
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R tanh(x)\|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 8
--R (2b + 2a b + a b)sinh(x)
--R +
--R 3 2 2 7
--R (16b + 16a b + 8a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 6
--R ((56b + 56a b + 28a b)cosh(x) + 4b + 8a b + 4a b)sinh(x)
--R +
--R 3 2 2 3
--R (112b + 112a b + 56a b)cosh(x)
--R +
--R 3 2 2
--R (24b + 48a b + 24a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 4
--R (140b + 140a b + 70a b)cosh(x)
--R +
--R 3 2 2 2 3 2 2
--R (60b + 120a b + 60a b)cosh(x) + 2b + 10a b + 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 5
--R (112b + 112a b + 56a b)cosh(x)
--R +
--R 3 2 2 3
--R (80b + 160a b + 80a b)cosh(x)
--R +
--R 3 2 2
--R (8b + 40a b + 24a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 6
--R (56b + 56a b + 28a b)cosh(x)
--R +
--R 3 2 2 4
--R (60b + 120a b + 60a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (12b + 60a b + 36a b)cosh(x) + 4a b + 4a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 7
--R (16b + 16a b + 8a b)cosh(x)
--R +
--R 3 2 2 5
--R (24b + 48a b + 24a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (8b + 40a b + 24a b)cosh(x) + (8a b + 8a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 8 3 2 2 6
--R (2b + 2a b + a b)cosh(x) + (4b + 8a b + 4a b)cosh(x)
--R +
--R 3 2 2 4 2 2 2 2
--R (2b + 10a b + 6a b)cosh(x) + (4a b + 4a b)cosh(x) + a b
--R *
--R tanh(x)
--R *
--R +--------------+
--R | 2
--R \|b sech(x) + a
--R +
--R 3 2 2 6
--R (- 2b - 4a b - 2a b)sinh(x)
--R +
--R 3 2 2 5
--R (- 12b - 24a b - 12a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 4
--R ((- 30b - 60a b - 30a b)cosh(x) + 2b - 6a b - 4a b)sinh(x)
--R +
--R 3 2 2 3 3 2 2
--R ((- 40b - 80a b - 40a b)cosh(x) + (8b - 24a b - 16a b)cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 4
--R (- 30b - 60a b - 30a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (12b - 36a b - 24a b)cosh(x) + 2a b - 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 5
--R (- 12b - 24a b - 12a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (8b - 24a b - 16a b)cosh(x) + (4a b - 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 6 3 2 2 4
--R (- 2b - 4a b - 2a b)cosh(x) + (2b - 6a b - 4a b)cosh(x)
--R +
--R 2 2 2
--R (2a b - 2a b)cosh(x)
--R *
--R +-------------------------------------+
--R | 2 2
--R | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 3 2 2 8 3 2 2 7
--R (2b + 4a b + 2a b)sinh(x) + (16b + 32a b + 16a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 3 2 2 6
--R ((56b + 112a b + 56a b)cosh(x) + 12b + 14a b + 6a b)sinh(x)
--R +
--R 3 2 2 3
--R (112b + 224a b + 112a b)cosh(x)
--R +
--R 3 2 2
--R (72b + 84a b + 36a b)cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R 3 2 2 4
--R (140b + 280a b + 140a b)cosh(x)
--R +
--R 3 2 2 2 3 2 2
--R (180b + 210a b + 90a b)cosh(x) - 6b + 8a b + 6a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2 5
--R (112b + 224a b + 112a b)cosh(x)
--R +
--R 3 2 2 3
--R (240b + 280a b + 120a b)cosh(x)
--R +
--R 3 2 2
--R (- 24b + 32a b + 24a b)cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 3 2 2 6
--R (56b + 112a b + 56a b)cosh(x)
--R +
--R 3 2 2 4
--R (180b + 210a b + 90a b)cosh(x)
--R +
--R 3 2 2 2 2 2
--R (- 36b + 48a b + 36a b)cosh(x) - 2a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 2 7 3 2 2 5
--R (16b + 32a b + 16a b)cosh(x) + (72b + 84a b + 36a b)cosh(x)
--R +
--R 3 2 2 3 2 2
--R (- 24b + 32a b + 24a b)cosh(x) + (- 4a b + 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 2 8 3 2 2 6
--R (2b + 4a b + 2a b)cosh(x) + (12b + 14a b + 6a b)cosh(x)
--R +
--R 3 2 2 4 2 2 2
--R (- 6b + 8a b + 6a b)cosh(x) + (- 2a b + 2a b)cosh(x)
--R *
--R +-+
--R \|a
--R /
--R 6 5
--R (4b + 2a)sinh(x) + (24b + 12a)cosh(x)sinh(x)
--R +
--R 2 4
--R ((60b + 30a)cosh(x) + 8b + 6a)sinh(x)
--R +
--R 3 3
--R ((80b + 40a)cosh(x) + (32b + 24a)cosh(x))sinh(x)
--R +
--R 4 2 2
--R ((60b + 30a)cosh(x) + (48b + 36a)cosh(x) + 4b + 6a)sinh(x)
--R +
--R 5 3
--R ((24b + 12a)cosh(x) + (32b + 24a)cosh(x) + (8b + 12a)cosh(x))
--R *
--R sinh(x)
--R +
--R 6 4 2
--R (4b + 2a)cosh(x) + (8b + 6a)cosh(x) + (4b + 6a)cosh(x) + 2a
--R *
--R +-------------------------------------+
--R | 2 2
--R +-+ | 2a sinh(x) + 2a cosh(x) + 4b + 2a
--R \|a |-------------------------------------
--R | 2 2
--R \|sinh(x) - 2cosh(x)sinh(x) + cosh(x)
--R +
--R 2 2 8 2 2 7
--R (- 4b - 4a b - 2a )sinh(x) + (- 32b - 32a b - 16a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 6
--R ((- 112b - 112a b - 56a )cosh(x) - 8b - 16a b - 8a )sinh(x)
--R +
--R 2 2 3 2 2
--R ((- 224b - 224a b - 112a )cosh(x) + (- 48b - 96a b - 48a )cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 4
--R (- 280b - 280a b - 140a )cosh(x)
--R +
--R 2 2 2 2 2
--R (- 120b - 240a b - 120a )cosh(x) - 4b - 20a b - 12a
--R *
--R 4
--R sinh(x)
--R +
--R 2 2 5
--R (- 224b - 224a b - 112a )cosh(x)
--R +
--R 2 2 3 2 2
--R (- 160b - 320a b - 160a )cosh(x) + (- 16b - 80a b - 48a )cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 2 6
--R (- 112b - 112a b - 56a )cosh(x)
--R +
--R 2 2 4 2 2 2
--R (- 120b - 240a b - 120a )cosh(x) + (- 24b - 120a b - 72a )cosh(x)
--R +
--R 2
--R - 8a b - 8a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 7 2 2 5
--R (- 32b - 32a b - 16a )cosh(x) + (- 48b - 96a b - 48a )cosh(x)
--R +
--R 2 2 3 2
--R (- 16b - 80a b - 48a )cosh(x) + (- 16a b - 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 8 2 2 6
--R (- 4b - 4a b - 2a )cosh(x) + (- 8b - 16a b - 8a )cosh(x)
--R +
--R 2 2 4 2 2 2
--R (- 4b - 20a b - 12a )cosh(x) + (- 8a b - 8a )cosh(x) - 2a
--R Type: Expression(Integer)
--E 113
--S 114 of 526
--d0419b:= D(m0419b,x)
--E 114
--S 115 of 526
t0420:= x/sech(x)^(3/2)-1/3*x*sech(x)^(1/2)
--R
--R
--R 2
--R - x sech(x) + 3x
--R (109) ------------------
--R +-------+
--R 3sech(x)\|sech(x)
--R Type: Expression(Integer)
--E 115
--S 116 of 526
r0420:= -2/9*(2*cosh(x)-3*x*sinh(x))/(1/cosh(x))^(1/2)
--R
--R
--R 6x sinh(x) - 4cosh(x)
--R (110) ---------------------
--R +-------+
--R | 1
--R 9 |-------
--R \|cosh(x)
--R Type: Expression(Integer)
--E 116
--S 117 of 526
a0420:= integrate(t0420,x)
--R
--R
--R >> Error detected within library code:
--R integrate: implementation incomplete (constant residues)
--R
--R Continuing to read the file...
--R
--E 117
--S 118 of 526
m0420:= a0420-r0420
--R
--R
--R +-------+
--R | 1
--R 9a0420 |------- - 6x sinh(x) + 4cosh(x)
--R \|cosh(x)
--R (111) ----------------------------------------
--R +-------+
--R | 1
--R 9 |-------
--R \|cosh(x)
--R Type: Expression(Integer)
--E 118
--S 119 of 526
d0420:= D(m0420,x)
--R
--R
--R 2 2
--R - x sinh(x) - 2x cosh(x)
--R (112) --------------------------
--R +-------+
--R | 1
--R 3cosh(x) |-------
--R \|cosh(x)
--R Type: Expression(Integer)
--E 119
--S 120 of 526
t0421:= x/sech(x)^(5/2)-3/5*x/sech(x)^(1/2)
--R
--R
--R 2
--R - 3x sech(x) + 5x
--R (113) -------------------
--R 2 +-------+
--R 5sech(x) \|sech(x)
--R Type: Expression(Integer)
--E 120
--S 121 of 526
r0421:= -4/25/sech(x)^(5/2)+2/5*x*sinh(x)/sech(x)^(3/2)
--R
--R
--R 10x sech(x)sinh(x) - 4
--R (114) ----------------------
--R 2 +-------+
--R 25sech(x) \|sech(x)
--R Type: Expression(Integer)
--E 121
--S 122 of 526
--a0421:= integrate(t0421,x)
--E 122
--S 123 of 526
--m0421:= a0421-r0421
--E 123
--S 124 of 526
--d0421:= D(m0421,x)
--E 124
--S 125 of 526
t0422:= x/sech(x)^(7/2)-5/21*x*sech(x)^(1/2)
--R
--R
--R 4
--R - 5x sech(x) + 21x
--R (115) --------------------
--R 3 +-------+
--R 21sech(x) \|sech(x)
--R Type: Expression(Integer)
--E 125
--S 126 of 526
r0422:= -2/441*(18*cosh(x)^3+70*cosh(x)-63*x*sinh(x)*cosh(x)^2-_
105*x*sinh(x))/(1/cosh(x))^(1/2)
--R
--R
--R 2 3
--R (126x cosh(x) + 210x)sinh(x) - 36cosh(x) - 140cosh(x)
--R (116) -------------------------------------------------------
--R +-------+
--R | 1
--R 441 |-------
--R \|cosh(x)
--R Type: Expression(Integer)
--E 126
--S 127 of 526
a0422:= integrate(t0422,x)
--R
--R
--R >> Error detected within library code:
--R integrate: implementation incomplete (constant residues)
--R
--R Continuing to read the file...
--R
--E 127
--S 128 of 526
m0422:= a0422-r0422
--R
--R
--R (117)
--R +-------+
--R | 1 2 3
--R 441a0422 |------- + (- 126x cosh(x) - 210x)sinh(x) + 36cosh(x)
--R \|cosh(x)
--R +
--R 140cosh(x)
--R /
--R +-------+
--R | 1
--R 441 |-------
--R \|cosh(x)
--R Type: Expression(Integer)
--E 128
--S 129 of 526
d0422:= D(m0422,x)
--R
--R
--R 2 2 4 2
--R (- 15x cosh(x) - 5x)sinh(x) - 6x cosh(x) - 10x cosh(x)
--R (118) ----------------------------------------------------------
--R +-------+
--R | 1
--R 21cosh(x) |-------
--R \|cosh(x)
--R Type: Expression(Integer)
--E 129
--S 130 of 526
t0423:= x^2/sech(x)^(3/2)-1/3*x^2*sech(x)^(1/2)
--R
--R
--R 2 2 2
--R - x sech(x) + 3x
--R (119) ------------------
--R +-------+
--R 3sech(x)\|sech(x)
--R Type: Expression(Integer)
--E 130
--S 131 of 526
r0423:= -8/9*x/sech(x)^(3/2)-16/27*%i*cosh(x)^(1/2)*_
EllipticF(1/2*%i*x,2)*sech(x)^(1/2)+16/27*sinh(x)/sech(x)^(1/2)+_
2/3*x^2*sinh(x)/sech(x)^(1/2)
--R
--R There are no library operations named EllipticF
--R Use HyperDoc Browse or issue
--R )what op EllipticF
--R to learn if there is any operation containing " EllipticF " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticF with argument type(s)
--R Polynomial(Complex(Fraction(Integer)))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 131
--S 132 of 526
--a0423:= integrate(t0423,x)
--E 132
--S 133 of 526
--m0423:= a0423-r0423
--E 133
--S 134 of 526
--d0423:= D(m0423,x)
--E 134
--S 135 of 526
t0424:= csch(a+b*x)
--R
--R
--R (120) csch(b x + a)
--R Type: Expression(Integer)
--E 135
--S 136 of 526
r0424:= -acoth(cosh(a+b*x))/b
--R
--R
--R acoth(cosh(b x + a))
--R (121) - --------------------
--R b
--R Type: Expression(Integer)
--E 136
--S 137 of 526
a0424:= integrate(t0424,x)
--R
--R
--R (122)
--R - log(sinh(b x + a) + cosh(b x + a) + 1)
--R +
--R log(sinh(b x + a) + cosh(b x + a) - 1)
--R /
--R b
--R Type: Union(Expression(Integer),...)
--E 137
--S 138 of 526
m0424:= a0424-r0424
--R
--R
--R (123)
--R - log(sinh(b x + a) + cosh(b x + a) + 1)
--R +
--R log(sinh(b x + a) + cosh(b x + a) - 1) + acoth(cosh(b x + a))
--R /
--R b
--R Type: Expression(Integer)
--E 138
--S 139 of 526
d0424:= D(m0424,x)
--R
--R
--R (124)
--R 3 2
--R - sinh(b x + a) - 2cosh(b x + a)sinh(b x + a)
--R +
--R 2 3
--R (cosh(b x + a) - 1)sinh(b x + a) + 2cosh(b x + a) - 2cosh(b x + a)
--R /
--R 2 2
--R (cosh(b x + a) - 1)sinh(b x + a)
--R +
--R 3 4
--R (2cosh(b x + a) - 2cosh(b x + a))sinh(b x + a) + cosh(b x + a)
--R +
--R 2
--R - 2cosh(b x + a) + 1
--R Type: Expression(Integer)
--E 139
--S 140 of 526
t0425:= csch(a+b*x)^(1/2)
--R
--R
--R +-------------+
--R (125) \|csch(b x + a)
--R Type: Expression(Integer)
--E 140
--S 141 of 526
r0425:= -2*%i*csch(a+b*x)^(1/2)*EllipticF(-1/4*%pi+1/2*%i*(a+b*x),2)*_
(%i*sinh(a+b*x))^(1/2)/b
--R
--R There are no library operations named EllipticF
--R Use HyperDoc Browse or issue
--R )what op EllipticF
--R to learn if there is any operation containing " EllipticF " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticF with argument type(s)
--R Expression(Complex(Integer))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 141
--S 142 of 526
a0425:= integrate(t0425,x)
--R
--R
--R x
--R ++ +--------------+
--I (126) | \|csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 142
--S 143 of 526
--m0425:= a0425-r0425
--E 143
--S 144 of 526
--d0425:= D(m0425,x)
--E 144
--S 145 of 526
t0426:= csch(a+b*x)^(3/2)
--R
--R
--R +-------------+
--R (127) csch(b x + a)\|csch(b x + a)
--R Type: Expression(Integer)
--E 145
--S 146 of 526
r0426:= -2*cosh(a+b*x)*csch(a+b*x)^(1/2)/b-2*%i*_
EllipticE(-1/4*%pi+1/2*%i*(a+b*x),2)/b/csch(a+b*x)^(1/2)/_
(%i*sinh(a+b*x))^(1/2)
--R
--R There are no library operations named EllipticE
--R Use HyperDoc Browse or issue
--R )what op EllipticE
--R to learn if there is any operation containing " EllipticE " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticE with argument type(s)
--R Expression(Complex(Integer))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 146
--S 147 of 526
a0426:= integrate(t0426,x)
--R
--R
--R x
--R ++ +--------------+
--I (128) | csch(%R b + a)\|csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 147
--S 148 of 526
--m0426:= a0426-r0426
--E 148
--S 149 of 526
--d0426:= D(m0426,x)
--E 149
--S 150 of 526
t0427:= csch(a+b*x)^(5/2)
--R
--R
--R 2 +-------------+
--R (129) csch(b x + a) \|csch(b x + a)
--R Type: Expression(Integer)
--E 150
--S 151 of 526
r0427:= -2/3*cosh(a+b*x)*csch(a+b*x)^(3/2)/b+2/3*%i*csch(a+b*x)^(1/2)*_
EllipticF(-1/4*%pi+1/2*%i*(a+b*x),2)*(%i*sinh(a+b*x))^(1/2)/b
--R
--R There are no library operations named EllipticF
--R Use HyperDoc Browse or issue
--R )what op EllipticF
--R to learn if there is any operation containing " EllipticF " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticF with argument type(s)
--R Expression(Complex(Integer))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 151
--S 152 of 526
a0427:= integrate(t0427,x)
--R
--R
--R x
--R ++ 2 +--------------+
--I (130) | csch(%R b + a) \|csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 152
--S 153 of 526
--m0427:= a0427-r0427
--E 153
--S 154 of 526
--d0427:= D(m0427,x)
--E 154
--S 155 of 526
t0428:= 1/csch(a+b*x)^(1/2)
--R
--R
--R 1
--R (131) ----------------
--R +-------------+
--R \|csch(b x + a)
--R Type: Expression(Integer)
--E 155
--S 156 of 526
r0428:= -2*%i*EllipticE(-1/4*%pi+1/2*%i*_
(a+b*x),2)/b/csch(a+b*x)^(1/2)/(%i*sinh(a+b*x))^(1/2)
--R
--R There are no library operations named EllipticE
--R Use HyperDoc Browse or issue
--R )what op EllipticE
--R to learn if there is any operation containing " EllipticE " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticE with argument type(s)
--R Expression(Complex(Integer))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 156
--S 157 of 526
a0428:= integrate(t0428,x)
--R
--R
--R x
--R ++ 1
--I (132) | ----------------- d%R
--R ++ +--------------+
--I \|csch(%R b + a)
--R Type: Union(Expression(Integer),...)
--E 157
--S 158 of 526
--m0428:= a0428-r0428
--E 158
--S 159 of 526
--d0428:= D(m0428,x)
--E 159
--S 160 of 526
t0429:= 1/csch(a+b*x)^(3/2)
--R
--R
--R 1
--R (133) -----------------------------
--R +-------------+
--R csch(b x + a)\|csch(b x + a)
--R Type: Expression(Integer)
--E 160
--S 161 of 526
r0429:= 2/3*cosh(a+b*x)/b/csch(a+b*x)^(1/2)+2/3*%i*csch(a+b*x)^(1/2)*_
EllipticF(-1/4*%pi+1/2*%i*(a+b*x),2)*(%i*sinh(a+b*x))^(1/2)/b
--R
--R There are no library operations named EllipticF
--R Use HyperDoc Browse or issue
--R )what op EllipticF
--R to learn if there is any operation containing " EllipticF " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticF with argument type(s)
--R Expression(Complex(Integer))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 161
--S 162 of 526
a0429:= integrate(t0429,x)
--R
--R
--R x
--R ++ 1
--I (134) | ------------------------------- d%R
--R ++ +--------------+
--I csch(%R b + a)\|csch(%R b + a)
--R Type: Union(Expression(Integer),...)
--E 162
--S 163 of 526
--m0429:= a0429-r0429
--E 163
--S 164 of 526
--d0429:= D(m0429,x)
--E 164
--S 165 of 526
t0430:= 1/csch(a+b*x)^(5/2)
--R
--R
--R 1
--R (135) ------------------------------
--R 2 +-------------+
--R csch(b x + a) \|csch(b x + a)
--R Type: Expression(Integer)
--E 165
--S 166 of 526
r0430:= 2/5*cosh(a+b*x)/b/csch(a+b*x)^(3/2)+6/5*%i*_
EllipticE(-1/4*%pi+1/2*%i*(a+b*x),2)/b/csch(a+b*x)^(1/2)/_
(%i*sinh(a+b*x))^(1/2)
--R
--R There are no library operations named EllipticE
--R Use HyperDoc Browse or issue
--R )what op EllipticE
--R to learn if there is any operation containing " EllipticE " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticE with argument type(s)
--R Expression(Complex(Integer))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 166
--S 167 of 526
a0430:= integrate(t0430,x)
--R
--R
--R x
--R ++ 1
--I (136) | -------------------------------- d%R
--R ++ 2 +--------------+
--I csch(%R b + a) \|csch(%R b + a)
--R Type: Union(Expression(Integer),...)
--E 167
--S 168 of 526
--m0430:= a0430-r0430
--E 168
--S 169 of 526
--d0430:= D(m0430,x)
--E 169
--S 170 of 526
t0431:= x*csch(a+b*x)
--R
--R
--R (137) x csch(b x + a)
--R Type: Expression(Integer)
--E 170
--S 171 of 526
r0431:= -(2*x*atanh(exp(a+b*x))*b+polylog(2,-exp(a+b*x))-_
polylog(2,exp(a+b*x)))/b^2
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 171
--S 172 of 526
a0431:= integrate(t0431,x)
--R
--R
--R x
--R ++
--I (138) | %R csch(%R b + a)d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 172
--S 173 of 526
--m0431:= a0431-r0431
--E 173
--S 174 of 526
--d0431:= D(m0431,x)
--E 174
--S 175 of 526
t0432:= x^2*csch(a+b*x)
--R
--R
--R 2
--R (139) x csch(b x + a)
--R Type: Expression(Integer)
--E 175
--S 176 of 526
r0432:= -2*(x^2*atanh(exp(a+b*x))*b^2+polylog(2,-exp(a+b*x))*x*b-_
polylog(2,exp(a+b*x))*x*b-polylog(3,-exp(a+b*x))+_
polylog(3,exp(a+b*x)))/b^3
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 176
--S 177 of 526
a0432:= integrate(t0432,x)
--R
--R
--R x
--R ++ 2
--I (140) | %R csch(%R b + a)d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 177
--S 178 of 526
--m0432:= a0432-r0432
--E 178
--S 179 of 526
--d0432:= D(m0432,x)
--E 179
--S 180 of 526
t0433:= x^3*csch(a+b*x)
--R
--R
--R 3
--R (141) x csch(b x + a)
--R Type: Expression(Integer)
--E 180
--S 181 of 526
r0433:= -(2*x^3*atanh(exp(a+b*x))*b^3+3*polylog(2,-exp(a+b*x))*x^2*b^2-_
3*polylog(2,exp(a+b*x))*x^2*b^2-6*polylog(3,-exp(a+b*x))*x*b+_
6*polylog(3,exp(a+b*x))*x*b+6*polylog(4,-exp(a+b*x))-_
6*polylog(4,exp(a+b*x)))/b^4
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 181
--S 182 of 526
a0433:= integrate(t0433,x)
--R
--R
--R x
--R ++ 3
--I (142) | %R csch(%R b + a)d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 182
--S 183 of 526
--m0433:= a0433-r0433
--E 183
--S 184 of 526
--d0433:= D(m0433,x)
--E 184
--S 185 of 526
t0434:= x*csch(a+b*x)^2
--R
--R
--R 2
--R (143) x csch(b x + a)
--R Type: Expression(Integer)
--E 185
--S 186 of 526
r0434:= -x*coth(a+b*x)/b+log(sinh(a+b*x))/b^2
--R
--R
--R log(sinh(b x + a)) - b x coth(b x + a)
--R (144) --------------------------------------
--R 2
--R b
--R Type: Expression(Integer)
--E 186
--S 187 of 526
a0434:= integrate(t0434,x)
--R
--R
--R (145)
--R 2 2
--R (sinh(b x + a) + 2cosh(b x + a)sinh(b x + a) + cosh(b x + a) - 1)
--R *
--R 2sinh(b x + a)
--R log(- -----------------------------)
--R sinh(b x + a) - cosh(b x + a)
--R +
--R 2
--R - 2b x sinh(b x + a) - 4b x cosh(b x + a)sinh(b x + a)
--R +
--R 2
--R - 2b x cosh(b x + a)
--R /
--R 2 2 2 2 2 2
--R b sinh(b x + a) + 2b cosh(b x + a)sinh(b x + a) + b cosh(b x + a) - b
--R Type: Union(Expression(Integer),...)
--E 187
--S 188 of 526
m0434:= a0434-r0434
--R
--R
--R (146)
--R 2 2
--R (- sinh(b x + a) - 2cosh(b x + a)sinh(b x + a) - cosh(b x + a) + 1)
--R *
--R log(sinh(b x + a))
--R +
--R 2 2
--R (sinh(b x + a) + 2cosh(b x + a)sinh(b x + a) + cosh(b x + a) - 1)
--R *
--R 2sinh(b x + a)
--R log(- -----------------------------)
--R sinh(b x + a) - cosh(b x + a)
--R +
--R 2
--R (b x coth(b x + a) - 2b x)sinh(b x + a)
--R +
--R (2b x cosh(b x + a)coth(b x + a) - 4b x cosh(b x + a))sinh(b x + a)
--R +
--R 2 2
--R (b x cosh(b x + a) - b x)coth(b x + a) - 2b x cosh(b x + a)
--R /
--R 2 2 2 2 2 2
--R b sinh(b x + a) + 2b cosh(b x + a)sinh(b x + a) + b cosh(b x + a) - b
--R Type: Expression(Integer)
--E 188
--S 189 of 526
d0434:= D(m0434,x)
--R
--R
--R (147)
--R 2 4
--R (- b x coth(b x + a) + coth(b x + a) + b x - 1)sinh(b x + a)
--R +
--R 2
--R - 4b x cosh(b x + a)coth(b x + a) + 4cosh(b x + a)coth(b x + a)
--R +
--R (4b x - 4)cosh(b x + a)
--R *
--R 3
--R sinh(b x + a)
--R +
--R 2 2
--R (- 6b x cosh(b x + a) + 2b x)coth(b x + a)
--R +
--R 2 2
--R (6cosh(b x + a) - 2)coth(b x + a) + (6b x - 6)cosh(b x + a) + 2b x
--R *
--R 2
--R sinh(b x + a)
--R +
--R 3 2
--R (- 4b x cosh(b x + a) + 4b x cosh(b x + a))coth(b x + a)
--R +
--R 3
--R (4cosh(b x + a) - 4cosh(b x + a))coth(b x + a)
--R +
--R 3
--R (4b x - 4)cosh(b x + a) + 4b x cosh(b x + a)
--R *
--R sinh(b x + a)
--R +
--R 4 2 2
--R (- b x cosh(b x + a) + 2b x cosh(b x + a) - b x)coth(b x + a)
--R +
--R 4 2
--R (cosh(b x + a) - 2cosh(b x + a) + 1)coth(b x + a)
--R +
--R 4 2
--R (b x - 1)cosh(b x + a) + 2b x cosh(b x + a) + b x + 1
--R /
--R 4 3
--R b sinh(b x + a) + 4b cosh(b x + a)sinh(b x + a)
--R +
--R 2 2
--R (6b cosh(b x + a) - 2b)sinh(b x + a)
--R +
--R 3 4
--R (4b cosh(b x + a) - 4b cosh(b x + a))sinh(b x + a) + b cosh(b x + a)
--R +
--R 2
--R - 2b cosh(b x + a) + b
--R Type: Expression(Integer)
--E 189
--S 190 of 526
t0435:= x^2*csch(a+b*x)^2
--R
--R
--R 2 2
--R (148) x csch(b x + a)
--R Type: Expression(Integer)
--E 190
--S 191 of 526
r0435:= -(b^2*x^2+x^2*coth(a+b*x)*b^2-2*x*log(1-exp(2*a+2*b*x))*b-_
polylog(2,exp(2*a+2*b*x)))/b^3
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 191
--S 192 of 526
a0435:= integrate(t0435,x)
--R
--R
--R x
--R ++ 2 2
--I (149) | %R csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 192
--S 193 of 526
--m0435:= a0435-r0435
--E 193
--S 194 of 526
--d0435:= D(m0435,x)
--E 194
--S 195 of 526
t0436:= x^3*csch(a+b*x)^2
--R
--R
--R 3 2
--R (150) x csch(b x + a)
--R Type: Expression(Integer)
--E 195
--S 196 of 526
r0436:= -1/2*(2*b^3*x^3+2*x^3*coth(a+b*x)*b^3-6*x^2*log(1-exp(2*a+2*b*x))*_
b^2-6*x*polylog(2,exp(2*a+2*b*x))*b+3*polylog(3,exp(2*a+2*b*x)))/b^4
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 196
--S 197 of 526
a0436:= integrate(t0436,x)
--R
--R
--R x
--R ++ 3 2
--I (151) | %R csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 197
--S 198 of 526
--m0436:= a0436-r0436
--E 198
--S 199 of 526
--d0436:= D(m0436,x)
--E 199
--S 200 of 526
t0437:= x*csch(a+b*x)^3
--R
--R
--R 3
--R (152) x csch(b x + a)
--R Type: Expression(Integer)
--E 200
--S 201 of 526
r0437:= x*atanh(exp(1)^(a+b*x))/b-1/2*csch(a+b*x)/b^2-_
1/2*x*coth(a+b*x)*csch(a+b*x)/b+1/2*polylog(2,-exp(1)^(a+b*x))/b^2-_
1/2*polylog(2,exp(1)^(a+b*x))/b^2
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 201
--S 202 of 526
a0437:= integrate(t0437,x)
--R
--R
--R x
--R ++ 3
--I (153) | %R csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 202
--S 203 of 526
--m0437:= a0437-r0437
--E 203
--S 204 of 526
--d0437:= D(m0437,x)
--E 204
--S 205 of 526
t0438:= x^2*csch(a+b*x)^3
--R
--R
--R 2 3
--R (154) x csch(b x + a)
--R Type: Expression(Integer)
--E 205
--S 206 of 526
r0438:= -acoth(cosh(a+b*x))/b^3+x^2*atanh(exp(1)^(a+b*x))/b-_
x*csch(a+b*x)/b^2-1/2*x^2*coth(a+b*x)*csch(a+b*x)/b+_
x*polylog(2,-exp(1)^(a+b*x))/b^2-x*polylog(2,exp(1)^(a+b*x))/b^2-_
polylog(3,-exp(1)^(a+b*x))/b^3+polylog(3,exp(1)^(a+b*x))/b^3
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 206
--S 207 of 526
a0438:= integrate(t0438,x)
--R
--R
--R x
--R ++ 2 3
--I (155) | %R csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 207
--S 208 of 526
--m0438:= a0438-r0438
--E 208
--S 209 of 526
--d0438:= D(m0438,x)
--E 209
--S 210 of 526
t0439:= (c+d*x)*csch(a+b*x)
--R
--R
--R (156) (d x + c)csch(b x + a)
--R Type: Expression(Integer)
--E 210
--S 211 of 526
r0439:= -(2*atanh(exp(a+b*x))*c*b+2*atanh(exp(a+b*x))*d*x*b+_
d*polylog(2,-exp(a+b*x))-d*polylog(2,exp(a+b*x)))/b^2
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 211
--S 212 of 526
a0439:= integrate(t0439,x)
--R
--R
--R x
--R ++
--I (157) | (%R d + c)csch(%R b + a)d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 212
--S 213 of 526
--m0439:= a0439-r0439
--E 213
--S 214 of 526
--d0439:= D(m0439,x)
--E 214
--S 215 of 526
t0440:= (c+d*x)^2*csch(a+b*x)
--R
--R
--R 2 2 2
--R (158) (d x + 2c d x + c )csch(b x + a)
--R Type: Expression(Integer)
--E 215
--S 216 of 526
r0440:= -2*(c+d*x)^2*atanh(exp(1)^(a+b*x))/b-2*d*(c+d*x)*_
polylog(2,-exp(1)^(a+b*x))/b^2+2*d*(c+d*x)*_
polylog(2,exp(1)^(a+b*x))/b^2+2*d^2*_
polylog(3,-exp(1)^(a+b*x))/b^3-2*d^2*_
polylog(3,exp(1)^(a+b*x))/b^3
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 216
--S 217 of 526
a0440:= integrate(t0440,x)
--R
--R
--R x
--R ++ 2 2 2
--I (159) | (%R d + 2%R c d + c )csch(%R b + a)d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 217
--S 218 of 526
--m0440:= a0440-r0440
--E 218
--S 219 of 526
--d0440:= D(m0440,x)
--E 219
--S 220 of 526
t0441:= (c+d*x)^3*csch(a+b*x)
--R
--R
--R 3 3 2 2 2 3
--R (160) (d x + 3c d x + 3c d x + c )csch(b x + a)
--R Type: Expression(Integer)
--E 220
--S 221 of 526
r0441:= -2*(c+d*x)^3*atanh(exp(1)^(a+b*x))/b-3*d*(c+d*x)^2*_
polylog(2,-exp(1)^(a+b*x))/b^2+3*d*(c+d*x)^2*_
polylog(2,exp(1)^(a+b*x))/b^2+6*d^2*(c+d*x)*_
polylog(3,-exp(1)^(a+b*x))/b^3-6*d^2*(c+d*x)*_
polylog(3,exp(1)^(a+b*x))/b^3-6*d^3*_
polylog(4,-exp(1)^(a+b*x))/b^4+6*d^3*_
polylog(4,exp(1)^(a+b*x))/b^4
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 221
--S 222 of 526
a0441:= integrate(t0441,x)
--R
--R
--R x
--R ++ 3 3 2 2 2 3
--I (161) | (%R d + 3%R c d + 3%R c d + c )csch(%R b + a)d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 222
--S 223 of 526
--m0441:= a0441-r0441
--E 223
--S 224 of 526
--d0441:= D(m0441,x)
--E 224
--S 225 of 526
t0442:= (c+d*x)*csch(a+b*x)^2
--R
--R
--R 2
--R (162) (d x + c)csch(b x + a)
--R Type: Expression(Integer)
--E 225
--S 226 of 526
r0442:= -(c+d*x)*coth(a+b*x)/b+d*log(sinh(a+b*x))/b^2
--R
--R
--R d log(sinh(b x + a)) + (- b d x - b c)coth(b x + a)
--R (163) ---------------------------------------------------
--R 2
--R b
--R Type: Expression(Integer)
--E 226
--S 227 of 526
a0442:= integrate(t0442,x)
--R
--R
--R (164)
--R 2 2
--R d sinh(b x + a) + 2d cosh(b x + a)sinh(b x + a) + d cosh(b x + a)
--R +
--R - d
--R *
--R 2sinh(b x + a)
--R log(- -----------------------------)
--R sinh(b x + a) - cosh(b x + a)
--R +
--R 2
--R - 2b d x sinh(b x + a) - 4b d x cosh(b x + a)sinh(b x + a)
--R +
--R 2
--R - 2b d x cosh(b x + a) - 2b c
--R /
--R 2 2 2 2 2 2
--R b sinh(b x + a) + 2b cosh(b x + a)sinh(b x + a) + b cosh(b x + a) - b
--R Type: Union(Expression(Integer),...)
--E 227
--S 228 of 526
m0442:= a0442-r0442
--R
--R
--R (165)
--R 2 2
--R - d sinh(b x + a) - 2d cosh(b x + a)sinh(b x + a) - d cosh(b x + a)
--R +
--R d
--R *
--R log(sinh(b x + a))
--R +
--R 2 2
--R d sinh(b x + a) + 2d cosh(b x + a)sinh(b x + a) + d cosh(b x + a)
--R +
--R - d
--R *
--R 2sinh(b x + a)
--R log(- -----------------------------)
--R sinh(b x + a) - cosh(b x + a)
--R +
--R 2
--R ((b d x + b c)coth(b x + a) - 2b d x)sinh(b x + a)
--R +
--R ((2b d x + 2b c)cosh(b x + a)coth(b x + a) - 4b d x cosh(b x + a))
--R *
--R sinh(b x + a)
--R +
--R 2
--R ((b d x + b c)cosh(b x + a) - b d x - b c)coth(b x + a)
--R +
--R 2
--R - 2b d x cosh(b x + a) - 2b c
--R /
--R 2 2 2 2 2 2
--R b sinh(b x + a) + 2b cosh(b x + a)sinh(b x + a) + b cosh(b x + a) - b
--R Type: Expression(Integer)
--E 228
--S 229 of 526
d0442:= D(m0442,x)
--R
--R
--R (166)
--R 2
--R ((- b d x - b c)coth(b x + a) + d coth(b x + a) + b d x - d + b c)
--R *
--R 4
--R sinh(b x + a)
--R +
--R 2
--R (- 4b d x - 4b c)cosh(b x + a)coth(b x + a)
--R +
--R 4d cosh(b x + a)coth(b x + a) + (4b d x - 4d + 4b c)cosh(b x + a)
--R *
--R 3
--R sinh(b x + a)
--R +
--R 2 2
--R ((- 6b d x - 6b c)cosh(b x + a) + 2b d x + 2b c)coth(b x + a)
--R +
--R 2
--R (6d cosh(b x + a) - 2d)coth(b x + a)
--R +
--R 2
--R (6b d x - 6d + 6b c)cosh(b x + a) + 2b d x + 2b c
--R *
--R 2
--R sinh(b x + a)
--R +
--R 3
--R ((- 4b d x - 4b c)cosh(b x + a) + (4b d x + 4b c)cosh(b x + a))
--R *
--R 2
--R coth(b x + a)
--R +
--R 3
--R (4d cosh(b x + a) - 4d cosh(b x + a))coth(b x + a)
--R +
--R 3
--R (4b d x - 4d + 4b c)cosh(b x + a) + (4b d x + 4b c)cosh(b x + a)
--R *
--R sinh(b x + a)
--R +
--R 4 2
--R (- b d x - b c)cosh(b x + a) + (2b d x + 2b c)cosh(b x + a) - b d x
--R +
--R - b c
--R *
--R 2
--R coth(b x + a)
--R +
--R 4 2
--R (d cosh(b x + a) - 2d cosh(b x + a) + d)coth(b x + a)
--R +
--R 4 2
--R (b d x - d + b c)cosh(b x + a) + (2b d x + 2b c)cosh(b x + a) + b d x
--R +
--R d + b c
--R /
--R 4 3
--R b sinh(b x + a) + 4b cosh(b x + a)sinh(b x + a)
--R +
--R 2 2
--R (6b cosh(b x + a) - 2b)sinh(b x + a)
--R +
--R 3 4
--R (4b cosh(b x + a) - 4b cosh(b x + a))sinh(b x + a) + b cosh(b x + a)
--R +
--R 2
--R - 2b cosh(b x + a) + b
--R Type: Expression(Integer)
--E 229
--S 230 of 526
t0443:= (c+d*x)^2*csch(a+b*x)^2
--R
--R
--R 2 2 2 2
--R (167) (d x + 2c d x + c )csch(b x + a)
--R Type: Expression(Integer)
--E 230
--S 231 of 526
r0443:= -(c+d*x)^2/b-(c+d*x)^2*coth(a+b*x)/b+2*d*(c+d*x)*_
log(1-exp(1)^(2*a+2*b*x))/b^2+d^2*polylog(2,exp(1)^(2*a+2*b*x))/b^3
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 231
--S 232 of 526
a0443:= integrate(t0443,x)
--R
--R
--R x
--R ++ 2 2 2 2
--I (168) | (%R d + 2%R c d + c )csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 232
--S 233 of 526
--m0443:= a0443-r0443
--E 233
--S 234 of 526
--d0443:= D(m0443,x)
--E 234
--S 235 of 526
t0444:= (c+d*x)^3*csch(a+b*x)^2
--R
--R
--R 3 3 2 2 2 3 2
--R (169) (d x + 3c d x + 3c d x + c )csch(b x + a)
--R Type: Expression(Integer)
--E 235
--S 236 of 526
r0444:= -(c+d*x)^3/b-(c+d*x)^3*coth(a+b*x)/b+3*d*(c+d*x)^2*_
log(1-exp(1)^(2*a+2*b*x))/b^2+3*d^2*(c+d*x)*_
polylog(2,exp(1)^(2*a+2*b*x))/b^3-3/2*d^3*_
polylog(3,exp(1)^(2*a+2*b*x))/b^4
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 236
--S 237 of 526
a0444:= integrate(t0444,x)
--R
--R
--R x
--R ++ 3 3 2 2 2 3 2
--I (170) | (%R d + 3%R c d + 3%R c d + c )csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 237
--S 238 of 526
--m0444:= a0444-r0444
--E 238
--S 239 of 526
--d0444:= D(m0444,x)
--E 239
--S 240 of 526
t0445:= (c+d*x)*csch(a+b*x)^3
--R
--R
--R 3
--R (171) (d x + c)csch(b x + a)
--R Type: Expression(Integer)
--E 240
--S 241 of 526
r0445:= (c+d*x)*atanh(exp(1)^(a+b*x))/b-1/2*d*csch(a+b*x)/b^2-_
1/2*(c+d*x)*coth(a+b*x)*csch(a+b*x)/b+1/2*d*_
polylog(2,-exp(1)^(a+b*x))/b^2-1/2*d*_
polylog(2,exp(1)^(a+b*x))/b^2
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 241
--S 242 of 526
a0445:= integrate(t0445,x)
--R
--R
--R x
--R ++ 3
--I (172) | (%R d + c)csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 242
--S 243 of 526
--m0445:= a0445-r0445
--E 243
--S 244 of 526
--d0445:= D(m0445,x)
--E 244
--S 245 of 526
t0446:= (c+d*x)^2*csch(a+b*x)^3
--R
--R
--R 2 2 2 3
--R (173) (d x + 2c d x + c )csch(b x + a)
--R Type: Expression(Integer)
--E 245
--S 246 of 526
r0446:= -d^2*acoth(cosh(a+b*x))/b^3+(c+d*x)^2*atanh(exp(1)^(a+b*x))/b-_
d*(c+d*x)*csch(a+b*x)/b^2-1/2*(c+d*x)^2*coth(a+b*x)*csch(a+b*x)/b+_
d*(c+d*x)*polylog(2,-exp(1)^(a+b*x))/b^2-_
d*(c+d*x)*polylog(2,exp(1)^(a+b*x))/b^2-_
d^2*polylog(3,-exp(1)^(a+b*x))/b^3+_
d^2*polylog(3,exp(1)^(a+b*x))/b^3
--R
--R There are no library operations named polylog
--R Use HyperDoc Browse or issue
--R )what op polylog
--R to learn if there is any operation containing " polylog " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named
--R polylog with argument type(s)
--R PositiveInteger
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 246
--S 247 of 526
a0446:= integrate(t0446,x)
--R
--R
--R x
--R ++ 2 2 2 3
--I (174) | (%R d + 2%R c d + c )csch(%R b + a) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 247
--S 248 of 526
--m0446:= a0446-r0446
--E 248
--S 249 of 526
--d0446:= D(m0446,x)
--E 249
--S 250 of 526
t0447:= csch(x)^(1/2)
--R
--R
--R +-------+
--R (175) \|csch(x)
--R Type: Expression(Integer)
--E 250
--S 251 of 526
r0447:= -2*%i*csch(x)^(1/2)*EllipticF(-1/4*%pi+1/2*%i*x,2)*(%i*sinh(x))^(1/2)
--R
--R There are no library operations named EllipticF
--R Use HyperDoc Browse or issue
--R )what op EllipticF
--R to learn if there is any operation containing " EllipticF " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticF with argument type(s)
--R Expression(Complex(Integer))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 251
--S 252 of 526
a0447:= integrate(t0447,x)
--R
--R
--R x
--R ++ +--------+
--I (176) | \|csch(%R) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 252
--S 253 of 526
--m0447:= a0447-r0447
--E 253
--S 254 of 526
--d0447:= D(m0447,x)
--E 254
--S 255 of 526
t0448:= (-csch(x)^2)^(1/2)
--R
--R
--R +----------+
--R | 2
--R (177) \|- csch(x)
--R Type: Expression(Integer)
--E 255
--S 256 of 526
r0448:= acsc(tanh(x))
--R
--R
--R (178) acsc(tanh(x))
--R Type: Expression(Integer)
--E 256
--S 257 of 526
a0448:= integrate(t0448,x)
--R
--R
--R +---+ x +---+ x
--R (179) - \|- 1 log(%e + 1) + \|- 1 log(%e - 1)
--R Type: Union(Expression(Integer),...)
--E 257
--S 258 of 526
m0448:= a0448-r0448
--R
--R
--R +---+ x +---+ x
--R (180) - \|- 1 log(%e + 1) + \|- 1 log(%e - 1) - acsc(tanh(x))
--R Type: Expression(Integer)
--E 258
--S 259 of 526
d0448:= D(m0448,x)
--R
--R
--R (181)
--R +------------+
--R +---+ x | 2 x 2 2 x 2
--R 2\|- 1 %e tanh(x)\|tanh(x) - 1 + (- (%e ) + 1)tanh(x) + (%e ) - 1
--R ----------------------------------------------------------------------
--R +------------+
--R x 2 | 2
--R ((%e ) - 1)tanh(x)\|tanh(x) - 1
--R Type: Expression(Integer)
--E 259
--S 260 of 526
t0449:= (csch(x)^2)^(1/2)
--R
--R
--R +--------+
--R | 2
--R (182) \|csch(x)
--R Type: Expression(Integer)
--E 260
--S 261 of 526
r0449:= -acoth(cosh(x))*csgn(1/sinh(x))
--R
--R There are no library operations named csgn
--R Use HyperDoc Browse or issue
--R )what op csgn
--R to learn if there is any operation containing " csgn " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named csgn
--R with argument type(s)
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 261
--S 262 of 526
a0449:= integrate(t0449,x)
--R
--R
--R (183) - log(sinh(x) + cosh(x) + 1) + log(sinh(x) + cosh(x) - 1)
--R Type: Union(Expression(Integer),...)
--E 262
--S 263 of 526
--m0449:= a0449-r0449
--E 263
--S 264 of 526
--d0449:= D(m0449,x)
--E 264
--S 265 of 526
t0450:= (csch(x)^3)^(1/2)
--R
--R
--R +--------+
--R | 3
--R (184) \|csch(x)
--R Type: Expression(Integer)
--E 265
--S 266 of 526
r0450:= -2*(csch(x)^3)^(1/2)*(coth(x)+_
%i*EllipticE(-1/4*%pi+1/2*%i*x,2)/(%i*sinh(x))^(1/2))*sinh(x)^2
--R
--R There are no library operations named EllipticE
--R Use HyperDoc Browse or issue
--R )what op EllipticE
--R to learn if there is any operation containing " EllipticE " in
--R its name.
--R
--R Cannot find a definition or applicable library operation named
--R EllipticE with argument type(s)
--R Expression(Complex(Integer))
--R PositiveInteger
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 266
--S 267 of 526
a0450:= integrate(t0450,x)
--R
--R
--R x +---------+
--R ++ | 3
--I (185) | \|csch(%R) d%R
--R ++
--R Type: Union(Expression(Integer),...)
--E 267
--S 268 of 526
--m0450:= a0450-r0450
--E 268
--S 269 of 526
--d0450:= D(m0450,x)
--E 269
--S 270 of 526
t0451:= (csch(x)^4)^(1/2)
--R
--R
--R +--------+
--R | 4
--R (186) \|csch(x)
--R Type: Expression(Integer)
--E 270
--S 271 of 526
r0451:= -cosh(x)*csgn(1/sinh(x)^2)/sinh(x)
--R
--R There are no library operations named csgn
--R Use HyperDoc Browse or issue
--R )what op csgn
--R to learn if there is any operation containing " csgn " in its
--R name.
--R
--R Cannot find a definition or applicable library operation named csgn
--R with argument type(s)
--R Expression(Integer)
--R
--R Perhaps you should use "@" to indicate the required return type,
--R or "$" to specify which version of the function you need.
--E 271
--S 272 of 526
a0451:= integrate(t0451,x)
--R
--R
--R 2
--R (187) - -----------------------------------------
--R 2 2
--R sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R Type: Union(Expression(Integer),...)
--E 272
--S 273 of 526
--m0451:= a0451-r0451
--E 273
--S 274 of 526
--d0451:= D(m0451,x)
--E 274
--S 275 of 526
t0452:= 1/(a+%i*a*csch(a+b*x))
--R
--R
--R %i
--R (188) - ----------------------
--R a csch(b x + a) - %i a
--R Type: Expression(Complex(Integer))
--E 275
--S 276 of 526
r0452:= x/a+%i*cosh(a+b*x)/a/b/(1-%i*sinh(a+b*x))
--R
--R
--R b x sinh(b x + a) - cosh(b x + a) + %i b x
--R (189) ------------------------------------------
--R a b sinh(b x + a) + %i a b
--R Type: Expression(Complex(Integer))
--E 276
--S 277 of 526
a0452:= integrate(t0452,x)
--R
--R
--R b x sinh(b x + a) + b x cosh(b x + a) + %i b x + 2%i
--R (190) ----------------------------------------------------
--R a b sinh(b x + a) + a b cosh(b x + a) + %i a b
--R Type: Union(Expression(Complex(Integer)),...)
--E 277
--S 278 of 526
m0452:= a0452-r0452
--R
--R
--R (191)
--R 2
--R (cosh(b x + a) + 2%i)sinh(b x + a) + cosh(b x + a) + %i cosh(b x + a) - 2
--R /
--R 2
--R a b sinh(b x + a) + (a b cosh(b x + a) + 2%i a b)sinh(b x + a)
--R +
--R %i a b cosh(b x + a) - a b
--R Type: Expression(Complex(Integer))
--E 278
--S 279 of 526
d0452:= D(m0452,x)
--R
--R
--R (192)
--R 4 3
--R sinh(b x + a) + (2cosh(b x + a) + %i)sinh(b x + a)
--R +
--R 2
--R (2%i cosh(b x + a) + 1)sinh(b x + a)
--R +
--R 3 2
--R (- 2cosh(b x + a) - %i cosh(b x + a) + 2cosh(b x + a) + %i)
--R *
--R sinh(b x + a)
--R +
--R 4 3 2
--R - cosh(b x + a) - 2%i cosh(b x + a) + cosh(b x + a) + 2%i cosh(b x + a)
--R /
--R 4 3
--R a sinh(b x + a) + (2a cosh(b x + a) + 4%i a)sinh(b x + a)
--R +
--R 2 2
--R (a cosh(b x + a) + 6%i a cosh(b x + a) - 6a)sinh(b x + a)
--R +
--R 2
--R (2%i a cosh(b x + a) - 6a cosh(b x + a) - 4%i a)sinh(b x + a)
--R +
--R 2
--R - a cosh(b x + a) - 2%i a cosh(b x + a) + a
--R Type: Expression(Complex(Integer))
--E 279
--S 280 of 526
t0453:= 1/(a-%i*a*csch(a+b*x))
--R
--R
--R %i
--R (193) ----------------------
--R a csch(b x + a) + %i a
--R Type: Expression(Complex(Integer))
--E 280
--S 281 of 526
r0453:= x/a-%i*cosh(a+b*x)/a/b/(1+%i*sinh(a+b*x))
--R
--R
--R b x sinh(b x + a) - cosh(b x + a) - %i b x
--R (194) ------------------------------------------
--R a b sinh(b x + a) - %i a b
--R Type: Expression(Complex(Integer))
--E 281
--S 282 of 526
a0453:= integrate(t0453,x)
--R
--R
--R b x sinh(b x + a) + b x cosh(b x + a) - %i b x - 2%i
--R (195) ----------------------------------------------------
--R a b sinh(b x + a) + a b cosh(b x + a) - %i a b
--R Type: Union(Expression(Complex(Integer)),...)
--E 282
--S 283 of 526
m0453:= a0453-r0453
--R
--R
--R (196)
--R 2
--R (cosh(b x + a) - 2%i)sinh(b x + a) + cosh(b x + a) - %i cosh(b x + a) - 2
--R /
--R 2
--R a b sinh(b x + a) + (a b cosh(b x + a) - 2%i a b)sinh(b x + a)
--R +
--R - %i a b cosh(b x + a) - a b
--R Type: Expression(Complex(Integer))
--E 283
--S 284 of 526
d0453:= D(m0453,x)
--R
--R
--R (197)
--R 4 3
--R sinh(b x + a) + (2cosh(b x + a) - %i)sinh(b x + a)
--R +
--R 2
--R (- 2%i cosh(b x + a) + 1)sinh(b x + a)
--R +
--R 3 2
--R (- 2cosh(b x + a) + %i cosh(b x + a) + 2cosh(b x + a) - %i)
--R *
--R sinh(b x + a)
--R +
--R 4 3 2
--R - cosh(b x + a) + 2%i cosh(b x + a) + cosh(b x + a) - 2%i cosh(b x + a)
--R /
--R 4 3
--R a sinh(b x + a) + (2a cosh(b x + a) - 4%i a)sinh(b x + a)
--R +
--R 2 2
--R (a cosh(b x + a) - 6%i a cosh(b x + a) - 6a)sinh(b x + a)
--R +
--R 2
--R (- 2%i a cosh(b x + a) - 6a cosh(b x + a) + 4%i a)sinh(b x + a)
--R +
--R 2
--R - a cosh(b x + a) + 2%i a cosh(b x + a) + a
--R Type: Expression(Complex(Integer))
--E 284
--S 285 of 526
t0454:= (3+3*%i*csch(x))^(1/2)
--R
--R
--R +---------------+
--R (198) \|3%i csch(x) + 3
--R Type: Expression(Complex(Integer))
--E 285
--S 286 of 526
r0454:= 2*3^(1/2)*atan((-1+%i*csch(x))^(1/2))*_
coth(x)/(-1+%i*csch(x))^(1/2)/(1+%i*csch(x))^(1/2)
--R
--R
--R +-+ +--------------+
--R 2\|3 coth(x)atan(\|%i csch(x) - 1 )
--R (199) -----------------------------------
--R +--------------+ +--------------+
--R \|%i csch(x) - 1 \|%i csch(x) + 1
--R Type: Expression(Complex(Integer))
--E 286
--S 287 of 526
a0454:= integrate(t0454,x)
--R
--R
--R (200)
--R -
--R +-+
--R \|3
--R *
--R log
--R +-----------------------------------------+
--R | 3 +-+
--R |----------------------------------------- - \|3 sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ +-+
--R - \|3 cosh(x) - %i\|3
--R /
--R sinh(x) + cosh(x)
--R +
--R +-+
--R \|3
--R *
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3 2
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x) - 2%i cosh(x)
--R +
--R - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 3 +-+ +-+ 2
--R - %i\|3 sinh(x) + (- 3%i\|3 cosh(x) - \|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+
--R (- 3%i\|3 cosh(x) - 2\|3 cosh(x) + %i\|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - %i\|3 cosh(x) - \|3 cosh(x) + %i\|3 cosh(x) + 2\|3
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R Type: Union(Expression(Complex(Integer)),...)
--E 287
--S 288 of 526
m0454:= a0454-r0454
--R
--R
--R (201)
--R -
--R +-+ +--------------+ +--------------+
--R \|3 \|%i csch(x) - 1 \|%i csch(x) + 1
--R *
--R log
--R +-----------------------------------------+
--R | 3 +-+
--R |----------------------------------------- - \|3 sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ +-+
--R - \|3 cosh(x) - %i\|3
--R /
--R sinh(x) + cosh(x)
--R +
--R +-+ +--------------+ +--------------+
--R \|3 \|%i csch(x) - 1 \|%i csch(x) + 1
--R *
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x)
--R +
--R 2
--R - 2%i cosh(x) - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 3 +-+ +-+ 2
--R - %i\|3 sinh(x) + (- 3%i\|3 cosh(x) - \|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+
--R (- 3%i\|3 cosh(x) - 2\|3 cosh(x) + %i\|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - %i\|3 cosh(x) - \|3 cosh(x) + %i\|3 cosh(x) + 2\|3
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+ +--------------+
--R - 2\|3 coth(x)atan(\|%i csch(x) - 1 )
--R /
--R +--------------+ +--------------+
--R \|%i csch(x) - 1 \|%i csch(x) + 1
--R Type: Expression(Complex(Integer))
--E 288
--S 289 of 526
d0454:= D(m0454,x)
--R
--R
--R (202)
--R 2 2 6
--R (- 6csch(x) + 6coth(x) - 6)sinh(x)
--R +
--R 2 2 5
--R (- 36cosh(x)csch(x) + 36cosh(x)coth(x) - 36cosh(x))sinh(x)
--R +
--R 2 2 2 2
--R (- 90cosh(x) - 6)csch(x) + (90cosh(x) + 6)coth(x)
--R +
--R 2
--R - 90cosh(x) - 6
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 120cosh(x) - 24cosh(x))csch(x)
--R +
--R 3 2 3
--R (120cosh(x) + 24cosh(x))coth(x) - 120cosh(x) - 24cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 90cosh(x) - 36cosh(x) + 36)csch(x)
--R +
--R 4 2 2 4
--R (90cosh(x) + 36cosh(x) - 36)coth(x) - 90cosh(x)
--R +
--R 2
--R - 36cosh(x) + 36
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 36cosh(x) - 24cosh(x) + 72cosh(x))csch(x)
--R +
--R 5 3 2 5
--R (36cosh(x) + 24cosh(x) - 72cosh(x))coth(x) - 36cosh(x)
--R +
--R 3
--R - 24cosh(x) + 72cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 6cosh(x) - 6cosh(x) + 36cosh(x) - 24)csch(x)
--R +
--R 6 4 2 2 6
--R (6cosh(x) + 6cosh(x) - 36cosh(x) + 24)coth(x) - 6cosh(x)
--R +
--R 4 2
--R - 6cosh(x) + 36cosh(x) - 24
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 2 +-+ 6
--R (6%i\|3 csch(x) - 6%i\|3 coth(x) + 6%i\|3 )sinh(x)
--R +
--R +-+ 2 +-+ 2
--R 36%i\|3 cosh(x)csch(x) - 36%i\|3 cosh(x)coth(x)
--R +
--R +-+
--R 36%i\|3 cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R +-+ 2 +-+ 2
--R (90%i\|3 cosh(x) - 6%i\|3 )csch(x)
--R +
--R +-+ 2 +-+ 2 +-+ 2 +-+
--R (- 90%i\|3 cosh(x) + 6%i\|3 )coth(x) + 90%i\|3 cosh(x) - 6%i\|3
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2
--R (120%i\|3 cosh(x) - 24%i\|3 cosh(x))csch(x)
--R +
--R +-+ 3 +-+ 2
--R (- 120%i\|3 cosh(x) + 24%i\|3 cosh(x))coth(x)
--R +
--R +-+ 3 +-+
--R 120%i\|3 cosh(x) - 24%i\|3 cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (90%i\|3 cosh(x) - 36%i\|3 cosh(x) - 24%i\|3 )csch(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (- 90%i\|3 cosh(x) + 36%i\|3 cosh(x) + 24%i\|3 )coth(x)
--R +
--R +-+ 4 +-+ 2 +-+
--R 90%i\|3 cosh(x) - 36%i\|3 cosh(x) - 24%i\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (36%i\|3 cosh(x) - 24%i\|3 cosh(x) - 48%i\|3 cosh(x))csch(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (- 36%i\|3 cosh(x) + 24%i\|3 cosh(x) + 48%i\|3 cosh(x))coth(x)
--R +
--R +-+ 5 +-+ 3 +-+
--R 36%i\|3 cosh(x) - 24%i\|3 cosh(x) - 48%i\|3 cosh(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (6%i\|3 cosh(x) - 6%i\|3 cosh(x) - 24%i\|3 cosh(x) + 24%i\|3 )
--R *
--R 2
--R csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (- 6%i\|3 cosh(x) + 6%i\|3 cosh(x) + 24%i\|3 cosh(x) - 24%i\|3 )
--R *
--R 2
--R coth(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R 6%i\|3 cosh(x) - 6%i\|3 cosh(x) - 24%i\|3 cosh(x) + 24%i\|3
--R *
--R +--------------+
--R atan(\|%i csch(x) - 1 )
--R +
--R 2 6
--R (- 6csch(x) - 6)sinh(x)
--R +
--R 2 5
--R ((- 36cosh(x) + 18%i)csch(x) - 36cosh(x) + 18%i)sinh(x)
--R +
--R 2 2 2
--R (- 90cosh(x) + 90%i cosh(x) - 3)csch(x) - 90cosh(x)
--R +
--R 90%i cosh(x) - 3
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2
--R (- 120cosh(x) + 180%i cosh(x) - 12cosh(x) - 30%i)csch(x)
--R +
--R 3 2
--R - 120cosh(x) + 180%i cosh(x) - 12cosh(x) - 30%i
--R *
--R 3
--R sinh(x)
--R +
--R 4 3 2
--R - 90cosh(x) + 180%i cosh(x) - 18cosh(x) - 90%i cosh(x)
--R +
--R 24
--R *
--R 2
--R csch(x)
--R +
--R 4 3 2
--R - 90cosh(x) + 180%i cosh(x) - 18cosh(x) - 90%i cosh(x) + 24
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 3 2
--R - 36cosh(x) + 90%i cosh(x) - 12cosh(x) - 90%i cosh(x)
--R +
--R 48cosh(x) + 12%i
--R *
--R 2
--R csch(x)
--R +
--R 5 4 3 2
--R - 36cosh(x) + 90%i cosh(x) - 12cosh(x) - 90%i cosh(x)
--R +
--R 48cosh(x) + 12%i
--R *
--R sinh(x)
--R +
--R 6 5 4 3
--R - 6cosh(x) + 18%i cosh(x) - 3cosh(x) - 30%i cosh(x)
--R +
--R 2
--R 24cosh(x) + 12%i cosh(x) - 12
--R *
--R 2
--R csch(x)
--R +
--R 6 5 4 3
--R - 6cosh(x) + 18%i cosh(x) - 3cosh(x) - 30%i cosh(x)
--R +
--R 2
--R 24cosh(x) + 12%i cosh(x) - 12
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 6
--R (3%i\|3 csch(x) + 3%i\|3 )sinh(x)
--R +
--R +-+ +-+ 2 +-+ +-+ 5
--R ((18%i\|3 cosh(x) + 6\|3 )csch(x) + 18%i\|3 cosh(x) + 6\|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+ 2
--R (45%i\|3 cosh(x) + 30\|3 cosh(x) - 3%i\|3 )csch(x)
--R +
--R +-+ 2 +-+ +-+
--R 45%i\|3 cosh(x) + 30\|3 cosh(x) - 3%i\|3
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R (60%i\|3 cosh(x) + 60\|3 cosh(x) - 12%i\|3 cosh(x) - 24\|3 )
--R *
--R 2
--R csch(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R 60%i\|3 cosh(x) + 60\|3 cosh(x) - 12%i\|3 cosh(x) - 24\|3
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 3 +-+ 2
--R 45%i\|3 cosh(x) + 60\|3 cosh(x) - 18%i\|3 cosh(x)
--R +
--R +-+ +-+
--R - 72\|3 cosh(x) - 18%i\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 4 +-+ 3 +-+ 2
--R 45%i\|3 cosh(x) + 60\|3 cosh(x) - 18%i\|3 cosh(x)
--R +
--R +-+ +-+
--R - 72\|3 cosh(x) - 18%i\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 4 +-+ 3
--R 18%i\|3 cosh(x) + 30\|3 cosh(x) - 12%i\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 72\|3 cosh(x) - 36%i\|3 cosh(x) + 12\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 5 +-+ 4 +-+ 3
--R 18%i\|3 cosh(x) + 30\|3 cosh(x) - 12%i\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 72\|3 cosh(x) - 36%i\|3 cosh(x) + 12\|3
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 5 +-+ 4
--R 3%i\|3 cosh(x) + 6\|3 cosh(x) - 3%i\|3 cosh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - 24\|3 cosh(x) - 18%i\|3 cosh(x) + 12\|3 cosh(x) + 12%i\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 6 +-+ 5 +-+ 4 +-+ 3
--R 3%i\|3 cosh(x) + 6\|3 cosh(x) - 3%i\|3 cosh(x) - 24\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 18%i\|3 cosh(x) + 12\|3 cosh(x) + 12%i\|3
--R *
--R +--------------+ +--------------+
--R \|%i csch(x) - 1 \|%i csch(x) + 1
--R +
--R 2 2 6
--R (- 3%i coth(x) csch(x) - 3coth(x) )sinh(x)
--R +
--R 2 2 5
--R (- 18%i cosh(x)coth(x) csch(x) - 18cosh(x)coth(x) )sinh(x)
--R +
--R 2 2
--R (- 45%i cosh(x) - 3%i)coth(x) csch(x)
--R +
--R 2 2
--R (- 45cosh(x) - 3)coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 60%i cosh(x) - 12%i cosh(x))coth(x) csch(x)
--R +
--R 3 2
--R (- 60cosh(x) - 12cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 45%i cosh(x) - 18%i cosh(x) + 18%i)coth(x) csch(x)
--R +
--R 4 2 2
--R (- 45cosh(x) - 18cosh(x) + 18)coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 18%i cosh(x) - 12%i cosh(x) + 36%i cosh(x))coth(x)
--R *
--R csch(x)
--R +
--R 5 3 2
--R (- 18cosh(x) - 12cosh(x) + 36cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 3%i cosh(x) - 3%i cosh(x) + 18%i cosh(x) - 12%i)coth(x)
--R *
--R csch(x)
--R +
--R 6 4 2 2
--R (- 3cosh(x) - 3cosh(x) + 18cosh(x) - 12)coth(x)
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 2 6
--R (- 3\|3 coth(x) csch(x) + 3%i\|3 coth(x) )sinh(x)
--R +
--R +-+ 2 +-+ 2 5
--R (- 18\|3 cosh(x)coth(x) csch(x) + 18%i\|3 cosh(x)coth(x) )sinh(x)
--R +
--R +-+ 2 +-+ 2
--R (- 45\|3 cosh(x) + 3\|3 )coth(x) csch(x)
--R +
--R +-+ 2 +-+ 2
--R (45%i\|3 cosh(x) - 3%i\|3 )coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2
--R (- 60\|3 cosh(x) + 12\|3 cosh(x))coth(x) csch(x)
--R +
--R +-+ 3 +-+ 2
--R (60%i\|3 cosh(x) - 12%i\|3 cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (- 45\|3 cosh(x) + 18\|3 cosh(x) + 12\|3 )coth(x) csch(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (45%i\|3 cosh(x) - 18%i\|3 cosh(x) - 12%i\|3 )coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (- 18\|3 cosh(x) + 12\|3 cosh(x) + 24\|3 cosh(x))coth(x)
--R *
--R csch(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (18%i\|3 cosh(x) - 12%i\|3 cosh(x) - 24%i\|3 cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+ 2
--R (- 3\|3 cosh(x) + 3\|3 cosh(x) + 12\|3 cosh(x) - 12\|3 )coth(x)
--R *
--R csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (3%i\|3 cosh(x) - 3%i\|3 cosh(x) - 12%i\|3 cosh(x) + 12%i\|3 )
--R *
--R 2
--R coth(x)
--R *
--R +--------------+
--R \|%i csch(x) - 1
--R /
--R +-+ 2 +-+ 6
--R (\|3 csch(x) + \|3 )sinh(x)
--R +
--R +-+ 2 +-+ 5
--R (6\|3 cosh(x)csch(x) + 6\|3 cosh(x))sinh(x)
--R +
--R +-+ 2 +-+ 2 +-+ 2 +-+ 4
--R ((15\|3 cosh(x) + \|3 )csch(x) + 15\|3 cosh(x) + \|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ 3
--R (20\|3 cosh(x) + 4\|3 cosh(x))csch(x) + 20\|3 cosh(x)
--R +
--R +-+
--R 4\|3 cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (15\|3 cosh(x) + 6\|3 cosh(x) - 6\|3 )csch(x)
--R +
--R +-+ 4 +-+ 2 +-+
--R 15\|3 cosh(x) + 6\|3 cosh(x) - 6\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (6\|3 cosh(x) + 4\|3 cosh(x) - 12\|3 cosh(x))csch(x)
--R +
--R +-+ 5 +-+ 3 +-+
--R 6\|3 cosh(x) + 4\|3 cosh(x) - 12\|3 cosh(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+ 2
--R (\|3 cosh(x) + \|3 cosh(x) - 6\|3 cosh(x) + 4\|3 )csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R \|3 cosh(x) + \|3 cosh(x) - 6\|3 cosh(x) + 4\|3
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 6
--R (- 3%i csch(x) - 3%i)sinh(x)
--R +
--R 2 5
--R (- 18%i cosh(x)csch(x) - 18%i cosh(x))sinh(x)
--R +
--R 2 2 2 4
--R ((- 45%i cosh(x) + 3%i)csch(x) - 45%i cosh(x) + 3%i)sinh(x)
--R +
--R 3 2 3
--R (- 60%i cosh(x) + 12%i cosh(x))csch(x) - 60%i cosh(x)
--R +
--R 12%i cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2 4
--R (- 45%i cosh(x) + 18%i cosh(x) + 12%i)csch(x) - 45%i cosh(x)
--R +
--R 2
--R 18%i cosh(x) + 12%i
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 18%i cosh(x) + 12%i cosh(x) + 24%i cosh(x))csch(x)
--R +
--R 5 3
--R - 18%i cosh(x) + 12%i cosh(x) + 24%i cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 3%i cosh(x) + 3%i cosh(x) + 12%i cosh(x) - 12%i)csch(x)
--R +
--R 6 4 2
--R - 3%i cosh(x) + 3%i cosh(x) + 12%i cosh(x) - 12%i
--R *
--R +--------------+ +--------------+
--R \|%i csch(x) - 1 \|%i csch(x) + 1
--R Type: Expression(Complex(Integer))
--E 289
--S 290 of 526
t0455:= (3-3*%i*csch(x))^(1/2)
--R
--R
--R +-----------------+
--R (203) \|- 3%i csch(x) + 3
--R Type: Expression(Complex(Integer))
--E 290
--S 291 of 526
r0455:= 2*3^(1/2)*atan((-1-%i*csch(x))^(1/2))*_
coth(x)/(-1-%i*csch(x))^(1/2)/(1-%i*csch(x))^(1/2)
--R
--R
--R +-+ +----------------+
--R 2\|3 coth(x)atan(\|- %i csch(x) - 1 )
--R (204) --------------------------------------
--R +----------------+ +----------------+
--R \|- %i csch(x) - 1 \|- %i csch(x) + 1
--R Type: Expression(Complex(Integer))
--E 291
--S 292 of 526
a0455:= integrate(t0455,x)
--R
--R
--R (205)
--R +-+
--R \|3
--R *
--R log
--R +-----------------------------------------+
--R | 3 +-+
--R |----------------------------------------- + \|3 sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ +-+
--R \|3 cosh(x) - %i\|3
--R /
--R sinh(x) + cosh(x)
--R +
--R -
--R +-+
--R \|3
--R *
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 3 +-+ +-+ 2
--R %i\|3 sinh(x) + (3%i\|3 cosh(x) - \|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+
--R (3%i\|3 cosh(x) - 2\|3 cosh(x) - %i\|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R %i\|3 cosh(x) - \|3 cosh(x) - %i\|3 cosh(x) + 2\|3
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R Type: Union(Expression(Complex(Integer)),...)
--E 292
--S 293 of 526
m0455:= a0455-r0455
--R
--R
--R (206)
--R +-+ +----------------+ +----------------+
--R \|3 \|- %i csch(x) - 1 \|- %i csch(x) + 1
--R *
--R log
--R +-----------------------------------------+
--R | 3 +-+
--R |----------------------------------------- + \|3 sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ +-+
--R \|3 cosh(x) - %i\|3
--R /
--R sinh(x) + cosh(x)
--R +
--R -
--R +-+ +----------------+ +----------------+
--R \|3 \|- %i csch(x) - 1 \|- %i csch(x) + 1
--R *
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 3 +-+ +-+ 2
--R %i\|3 sinh(x) + (3%i\|3 cosh(x) - \|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+
--R (3%i\|3 cosh(x) - 2\|3 cosh(x) - %i\|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R %i\|3 cosh(x) - \|3 cosh(x) - %i\|3 cosh(x) + 2\|3
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+ +----------------+
--R - 2\|3 coth(x)atan(\|- %i csch(x) - 1 )
--R /
--R +----------------+ +----------------+
--R \|- %i csch(x) - 1 \|- %i csch(x) + 1
--R Type: Expression(Complex(Integer))
--E 293
--S 294 of 526
d0455:= D(m0455,x)
--R
--R
--R (207)
--R 2 2 6
--R (- 6csch(x) + 6coth(x) - 6)sinh(x)
--R +
--R 2 2 5
--R (- 36cosh(x)csch(x) + 36cosh(x)coth(x) - 36cosh(x))sinh(x)
--R +
--R 2 2 2 2
--R (- 90cosh(x) - 6)csch(x) + (90cosh(x) + 6)coth(x)
--R +
--R 2
--R - 90cosh(x) - 6
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 120cosh(x) - 24cosh(x))csch(x)
--R +
--R 3 2 3
--R (120cosh(x) + 24cosh(x))coth(x) - 120cosh(x) - 24cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 90cosh(x) - 36cosh(x) + 36)csch(x)
--R +
--R 4 2 2 4
--R (90cosh(x) + 36cosh(x) - 36)coth(x) - 90cosh(x)
--R +
--R 2
--R - 36cosh(x) + 36
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 36cosh(x) - 24cosh(x) + 72cosh(x))csch(x)
--R +
--R 5 3 2 5
--R (36cosh(x) + 24cosh(x) - 72cosh(x))coth(x) - 36cosh(x)
--R +
--R 3
--R - 24cosh(x) + 72cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 6cosh(x) - 6cosh(x) + 36cosh(x) - 24)csch(x)
--R +
--R 6 4 2 2 6
--R (6cosh(x) + 6cosh(x) - 36cosh(x) + 24)coth(x) - 6cosh(x)
--R +
--R 4 2
--R - 6cosh(x) + 36cosh(x) - 24
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 2 +-+ 6
--R (6%i\|3 csch(x) - 6%i\|3 coth(x) + 6%i\|3 )sinh(x)
--R +
--R +-+ 2 +-+ 2
--R 36%i\|3 cosh(x)csch(x) - 36%i\|3 cosh(x)coth(x)
--R +
--R +-+
--R 36%i\|3 cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R +-+ 2 +-+ 2
--R (90%i\|3 cosh(x) - 6%i\|3 )csch(x)
--R +
--R +-+ 2 +-+ 2 +-+ 2 +-+
--R (- 90%i\|3 cosh(x) + 6%i\|3 )coth(x) + 90%i\|3 cosh(x) - 6%i\|3
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2
--R (120%i\|3 cosh(x) - 24%i\|3 cosh(x))csch(x)
--R +
--R +-+ 3 +-+ 2
--R (- 120%i\|3 cosh(x) + 24%i\|3 cosh(x))coth(x)
--R +
--R +-+ 3 +-+
--R 120%i\|3 cosh(x) - 24%i\|3 cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (90%i\|3 cosh(x) - 36%i\|3 cosh(x) - 24%i\|3 )csch(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (- 90%i\|3 cosh(x) + 36%i\|3 cosh(x) + 24%i\|3 )coth(x)
--R +
--R +-+ 4 +-+ 2 +-+
--R 90%i\|3 cosh(x) - 36%i\|3 cosh(x) - 24%i\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (36%i\|3 cosh(x) - 24%i\|3 cosh(x) - 48%i\|3 cosh(x))csch(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (- 36%i\|3 cosh(x) + 24%i\|3 cosh(x) + 48%i\|3 cosh(x))coth(x)
--R +
--R +-+ 5 +-+ 3 +-+
--R 36%i\|3 cosh(x) - 24%i\|3 cosh(x) - 48%i\|3 cosh(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (6%i\|3 cosh(x) - 6%i\|3 cosh(x) - 24%i\|3 cosh(x) + 24%i\|3 )
--R *
--R 2
--R csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (- 6%i\|3 cosh(x) + 6%i\|3 cosh(x) + 24%i\|3 cosh(x) - 24%i\|3 )
--R *
--R 2
--R coth(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R 6%i\|3 cosh(x) - 6%i\|3 cosh(x) - 24%i\|3 cosh(x) + 24%i\|3
--R *
--R +----------------+
--R atan(\|- %i csch(x) - 1 )
--R +
--R 2 6
--R (6csch(x) + 6)sinh(x)
--R +
--R 2 5
--R ((36cosh(x) + 18%i)csch(x) + 36cosh(x) + 18%i)sinh(x)
--R +
--R 2 2 2
--R (90cosh(x) + 90%i cosh(x) + 3)csch(x) + 90cosh(x)
--R +
--R 90%i cosh(x) + 3
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2
--R (120cosh(x) + 180%i cosh(x) + 12cosh(x) - 30%i)csch(x)
--R +
--R 3 2
--R 120cosh(x) + 180%i cosh(x) + 12cosh(x) - 30%i
--R *
--R 3
--R sinh(x)
--R +
--R 4 3 2
--R 90cosh(x) + 180%i cosh(x) + 18cosh(x) - 90%i cosh(x)
--R +
--R - 24
--R *
--R 2
--R csch(x)
--R +
--R 4 3 2
--R 90cosh(x) + 180%i cosh(x) + 18cosh(x) - 90%i cosh(x) - 24
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 3 2
--R 36cosh(x) + 90%i cosh(x) + 12cosh(x) - 90%i cosh(x)
--R +
--R - 48cosh(x) + 12%i
--R *
--R 2
--R csch(x)
--R +
--R 5 4 3 2
--R 36cosh(x) + 90%i cosh(x) + 12cosh(x) - 90%i cosh(x)
--R +
--R - 48cosh(x) + 12%i
--R *
--R sinh(x)
--R +
--R 6 5 4 3
--R 6cosh(x) + 18%i cosh(x) + 3cosh(x) - 30%i cosh(x)
--R +
--R 2
--R - 24cosh(x) + 12%i cosh(x) + 12
--R *
--R 2
--R csch(x)
--R +
--R 6 5 4 3
--R 6cosh(x) + 18%i cosh(x) + 3cosh(x) - 30%i cosh(x)
--R +
--R 2
--R - 24cosh(x) + 12%i cosh(x) + 12
--R *
--R +----------------+ +----------------+
--R \|- %i csch(x) - 1 \|- %i csch(x) + 1
--R +
--R 2 2 6
--R (3%i coth(x) csch(x) - 3coth(x) )sinh(x)
--R +
--R 2 2 5
--R (18%i cosh(x)coth(x) csch(x) - 18cosh(x)coth(x) )sinh(x)
--R +
--R 2 2
--R (45%i cosh(x) + 3%i)coth(x) csch(x)
--R +
--R 2 2
--R (- 45cosh(x) - 3)coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (60%i cosh(x) + 12%i cosh(x))coth(x) csch(x)
--R +
--R 3 2
--R (- 60cosh(x) - 12cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (45%i cosh(x) + 18%i cosh(x) - 18%i)coth(x) csch(x)
--R +
--R 4 2 2
--R (- 45cosh(x) - 18cosh(x) + 18)coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (18%i cosh(x) + 12%i cosh(x) - 36%i cosh(x))coth(x) csch(x)
--R +
--R 5 3 2
--R (- 18cosh(x) - 12cosh(x) + 36cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (3%i cosh(x) + 3%i cosh(x) - 18%i cosh(x) + 12%i)coth(x)
--R *
--R csch(x)
--R +
--R 6 4 2 2
--R (- 3cosh(x) - 3cosh(x) + 18cosh(x) - 12)coth(x)
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 6
--R (- 3%i\|3 csch(x) - 3%i\|3 )sinh(x)
--R +
--R +-+ +-+ 2 +-+ +-+
--R ((- 18%i\|3 cosh(x) + 6\|3 )csch(x) - 18%i\|3 cosh(x) + 6\|3 )
--R *
--R 5
--R sinh(x)
--R +
--R +-+ 2 +-+ +-+ 2
--R (- 45%i\|3 cosh(x) + 30\|3 cosh(x) + 3%i\|3 )csch(x)
--R +
--R +-+ 2 +-+ +-+
--R - 45%i\|3 cosh(x) + 30\|3 cosh(x) + 3%i\|3
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+
--R - 60%i\|3 cosh(x) + 60\|3 cosh(x) + 12%i\|3 cosh(x)
--R +
--R +-+
--R - 24\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - 60%i\|3 cosh(x) + 60\|3 cosh(x) + 12%i\|3 cosh(x) - 24\|3
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 3 +-+ 2
--R - 45%i\|3 cosh(x) + 60\|3 cosh(x) + 18%i\|3 cosh(x)
--R +
--R +-+ +-+
--R - 72\|3 cosh(x) + 18%i\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 4 +-+ 3 +-+ 2
--R - 45%i\|3 cosh(x) + 60\|3 cosh(x) + 18%i\|3 cosh(x)
--R +
--R +-+ +-+
--R - 72\|3 cosh(x) + 18%i\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 4 +-+ 3
--R - 18%i\|3 cosh(x) + 30\|3 cosh(x) + 12%i\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 72\|3 cosh(x) + 36%i\|3 cosh(x) + 12\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 5 +-+ 4 +-+ 3
--R - 18%i\|3 cosh(x) + 30\|3 cosh(x) + 12%i\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 72\|3 cosh(x) + 36%i\|3 cosh(x) + 12\|3
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 5 +-+ 4
--R - 3%i\|3 cosh(x) + 6\|3 cosh(x) + 3%i\|3 cosh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - 24\|3 cosh(x) + 18%i\|3 cosh(x) + 12\|3 cosh(x) - 12%i\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 6 +-+ 5 +-+ 4 +-+ 3
--R - 3%i\|3 cosh(x) + 6\|3 cosh(x) + 3%i\|3 cosh(x) - 24\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R 18%i\|3 cosh(x) + 12\|3 cosh(x) - 12%i\|3
--R *
--R +----------------+ +----------------+
--R \|- %i csch(x) - 1 \|- %i csch(x) + 1
--R +
--R +-+ 2 +-+ 2 6
--R (3\|3 coth(x) csch(x) + 3%i\|3 coth(x) )sinh(x)
--R +
--R +-+ 2 +-+ 2 5
--R (18\|3 cosh(x)coth(x) csch(x) + 18%i\|3 cosh(x)coth(x) )sinh(x)
--R +
--R +-+ 2 +-+ 2
--R (45\|3 cosh(x) - 3\|3 )coth(x) csch(x)
--R +
--R +-+ 2 +-+ 2
--R (45%i\|3 cosh(x) - 3%i\|3 )coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2
--R (60\|3 cosh(x) - 12\|3 cosh(x))coth(x) csch(x)
--R +
--R +-+ 3 +-+ 2
--R (60%i\|3 cosh(x) - 12%i\|3 cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (45\|3 cosh(x) - 18\|3 cosh(x) - 12\|3 )coth(x) csch(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (45%i\|3 cosh(x) - 18%i\|3 cosh(x) - 12%i\|3 )coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (18\|3 cosh(x) - 12\|3 cosh(x) - 24\|3 cosh(x))coth(x) csch(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (18%i\|3 cosh(x) - 12%i\|3 cosh(x) - 24%i\|3 cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+ 2
--R (3\|3 cosh(x) - 3\|3 cosh(x) - 12\|3 cosh(x) + 12\|3 )coth(x)
--R *
--R csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (3%i\|3 cosh(x) - 3%i\|3 cosh(x) - 12%i\|3 cosh(x) + 12%i\|3 )
--R *
--R 2
--R coth(x)
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R /
--R +-+ 2 +-+ 6
--R (\|3 csch(x) + \|3 )sinh(x)
--R +
--R +-+ 2 +-+ 5
--R (6\|3 cosh(x)csch(x) + 6\|3 cosh(x))sinh(x)
--R +
--R +-+ 2 +-+ 2 +-+ 2 +-+ 4
--R ((15\|3 cosh(x) + \|3 )csch(x) + 15\|3 cosh(x) + \|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ 3
--R (20\|3 cosh(x) + 4\|3 cosh(x))csch(x) + 20\|3 cosh(x)
--R +
--R +-+
--R 4\|3 cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2 +-+ 4
--R (15\|3 cosh(x) + 6\|3 cosh(x) - 6\|3 )csch(x) + 15\|3 cosh(x)
--R +
--R +-+ 2 +-+
--R 6\|3 cosh(x) - 6\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (6\|3 cosh(x) + 4\|3 cosh(x) - 12\|3 cosh(x))csch(x)
--R +
--R +-+ 5 +-+ 3 +-+
--R 6\|3 cosh(x) + 4\|3 cosh(x) - 12\|3 cosh(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+ 2
--R (\|3 cosh(x) + \|3 cosh(x) - 6\|3 cosh(x) + 4\|3 )csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R \|3 cosh(x) + \|3 cosh(x) - 6\|3 cosh(x) + 4\|3
--R *
--R +----------------+ +----------------+
--R \|- %i csch(x) - 1 \|- %i csch(x) + 1
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 6
--R (- 3%i csch(x) - 3%i)sinh(x)
--R +
--R 2 5
--R (- 18%i cosh(x)csch(x) - 18%i cosh(x))sinh(x)
--R +
--R 2 2 2 4
--R ((- 45%i cosh(x) + 3%i)csch(x) - 45%i cosh(x) + 3%i)sinh(x)
--R +
--R 3 2 3
--R (- 60%i cosh(x) + 12%i cosh(x))csch(x) - 60%i cosh(x)
--R +
--R 12%i cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2 4
--R (- 45%i cosh(x) + 18%i cosh(x) + 12%i)csch(x) - 45%i cosh(x)
--R +
--R 2
--R 18%i cosh(x) + 12%i
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 18%i cosh(x) + 12%i cosh(x) + 24%i cosh(x))csch(x)
--R +
--R 5 3
--R - 18%i cosh(x) + 12%i cosh(x) + 24%i cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 3%i cosh(x) + 3%i cosh(x) + 12%i cosh(x) - 12%i)csch(x)
--R +
--R 6 4 2
--R - 3%i cosh(x) + 3%i cosh(x) + 12%i cosh(x) - 12%i
--R *
--R +----------------+ +----------------+
--R \|- %i csch(x) - 1 \|- %i csch(x) + 1
--R Type: Expression(Complex(Integer))
--E 294
--S 295 of 526
t0456:= (-3+3*%i*csch(x))^(1/2)
--R
--R
--R +---------------+
--R (208) \|3%i csch(x) - 3
--R Type: Expression(Complex(Integer))
--E 295
--S 296 of 526
r0456:= -2*3^(1/2)*atan((-1-%i*csch(x))^(1/2))*_
coth(x)/(-1-%i*csch(x))^(1/2)/(-1+%i*csch(x))^(1/2)
--R
--R
--R +-+ +----------------+
--R 2\|3 coth(x)atan(\|- %i csch(x) - 1 )
--R (209) - -------------------------------------
--R +----------------+ +--------------+
--R \|- %i csch(x) - 1 \|%i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 296
--S 297 of 526
a0456:= integrate(t0456,x)
--R
--R
--R (210)
--R +-+
--R %i\|3
--R *
--R log
--R +-----------------------------------------+
--R | 3 +-+
--R |----------------------------------------- + \|3 sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ +-+
--R \|3 cosh(x) - %i\|3
--R /
--R sinh(x) + cosh(x)
--R +
--R -
--R +-+
--R %i\|3
--R *
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 3 +-+ +-+ 2
--R %i\|3 sinh(x) + (3%i\|3 cosh(x) - \|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+
--R (3%i\|3 cosh(x) - 2\|3 cosh(x) - %i\|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R %i\|3 cosh(x) - \|3 cosh(x) - %i\|3 cosh(x) + 2\|3
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R Type: Union(Expression(Complex(Integer)),...)
--E 297
--S 298 of 526
m0456:= a0456-r0456
--R
--R
--R (211)
--R +-+ +----------------+ +--------------+
--R %i\|3 \|- %i csch(x) - 1 \|%i csch(x) - 1
--R *
--R log
--R +-----------------------------------------+
--R | 3 +-+
--R |----------------------------------------- + \|3 sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ +-+
--R \|3 cosh(x) - %i\|3
--R /
--R sinh(x) + cosh(x)
--R +
--R -
--R +-+ +----------------+ +--------------+
--R %i\|3 \|- %i csch(x) - 1 \|%i csch(x) - 1
--R *
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 3 +-+ +-+ 2
--R %i\|3 sinh(x) + (3%i\|3 cosh(x) - \|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+
--R (3%i\|3 cosh(x) - 2\|3 cosh(x) - %i\|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R %i\|3 cosh(x) - \|3 cosh(x) - %i\|3 cosh(x) + 2\|3
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+ +----------------+
--R 2\|3 coth(x)atan(\|- %i csch(x) - 1 )
--R /
--R +----------------+ +--------------+
--R \|- %i csch(x) - 1 \|%i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 298
--S 299 of 526
d0456:= D(m0456,x)
--R
--R
--R (212)
--R 2 2 6
--R (6csch(x) - 6coth(x) + 6)sinh(x)
--R +
--R 2 2 5
--R (36cosh(x)csch(x) - 36cosh(x)coth(x) + 36cosh(x))sinh(x)
--R +
--R 2 2 2 2
--R (90cosh(x) + 6)csch(x) + (- 90cosh(x) - 6)coth(x)
--R +
--R 2
--R 90cosh(x) + 6
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (120cosh(x) + 24cosh(x))csch(x)
--R +
--R 3 2 3
--R (- 120cosh(x) - 24cosh(x))coth(x) + 120cosh(x) + 24cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (90cosh(x) + 36cosh(x) - 36)csch(x)
--R +
--R 4 2 2 4
--R (- 90cosh(x) - 36cosh(x) + 36)coth(x) + 90cosh(x)
--R +
--R 2
--R 36cosh(x) - 36
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (36cosh(x) + 24cosh(x) - 72cosh(x))csch(x)
--R +
--R 5 3 2 5
--R (- 36cosh(x) - 24cosh(x) + 72cosh(x))coth(x) + 36cosh(x)
--R +
--R 3
--R 24cosh(x) - 72cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (6cosh(x) + 6cosh(x) - 36cosh(x) + 24)csch(x)
--R +
--R 6 4 2 2 6
--R (- 6cosh(x) - 6cosh(x) + 36cosh(x) - 24)coth(x) + 6cosh(x)
--R +
--R 4 2
--R 6cosh(x) - 36cosh(x) + 24
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 2 +-+ 6
--R (- 6%i\|3 csch(x) + 6%i\|3 coth(x) - 6%i\|3 )sinh(x)
--R +
--R +-+ 2 +-+ 2
--R - 36%i\|3 cosh(x)csch(x) + 36%i\|3 cosh(x)coth(x)
--R +
--R +-+
--R - 36%i\|3 cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R +-+ 2 +-+ 2
--R (- 90%i\|3 cosh(x) + 6%i\|3 )csch(x)
--R +
--R +-+ 2 +-+ 2 +-+ 2 +-+
--R (90%i\|3 cosh(x) - 6%i\|3 )coth(x) - 90%i\|3 cosh(x) + 6%i\|3
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2
--R (- 120%i\|3 cosh(x) + 24%i\|3 cosh(x))csch(x)
--R +
--R +-+ 3 +-+ 2 +-+ 3
--R (120%i\|3 cosh(x) - 24%i\|3 cosh(x))coth(x) - 120%i\|3 cosh(x)
--R +
--R +-+
--R 24%i\|3 cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (- 90%i\|3 cosh(x) + 36%i\|3 cosh(x) + 24%i\|3 )csch(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (90%i\|3 cosh(x) - 36%i\|3 cosh(x) - 24%i\|3 )coth(x)
--R +
--R +-+ 4 +-+ 2 +-+
--R - 90%i\|3 cosh(x) + 36%i\|3 cosh(x) + 24%i\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (- 36%i\|3 cosh(x) + 24%i\|3 cosh(x) + 48%i\|3 cosh(x))csch(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (36%i\|3 cosh(x) - 24%i\|3 cosh(x) - 48%i\|3 cosh(x))coth(x)
--R +
--R +-+ 5 +-+ 3 +-+
--R - 36%i\|3 cosh(x) + 24%i\|3 cosh(x) + 48%i\|3 cosh(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (- 6%i\|3 cosh(x) + 6%i\|3 cosh(x) + 24%i\|3 cosh(x) - 24%i\|3 )
--R *
--R 2
--R csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (6%i\|3 cosh(x) - 6%i\|3 cosh(x) - 24%i\|3 cosh(x) + 24%i\|3 )
--R *
--R 2
--R coth(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R - 6%i\|3 cosh(x) + 6%i\|3 cosh(x) + 24%i\|3 cosh(x) - 24%i\|3
--R *
--R +----------------+
--R atan(\|- %i csch(x) - 1 )
--R +
--R 2 6
--R (6%i csch(x) + 6%i)sinh(x)
--R +
--R 2 5
--R ((36%i cosh(x) - 18)csch(x) + 36%i cosh(x) - 18)sinh(x)
--R +
--R 2 2 2
--R (90%i cosh(x) - 90cosh(x) + 3%i)csch(x) + 90%i cosh(x)
--R +
--R - 90cosh(x) + 3%i
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2
--R (120%i cosh(x) - 180cosh(x) + 12%i cosh(x) + 30)csch(x)
--R +
--R 3 2
--R 120%i cosh(x) - 180cosh(x) + 12%i cosh(x) + 30
--R *
--R 3
--R sinh(x)
--R +
--R 4 3 2
--R 90%i cosh(x) - 180cosh(x) + 18%i cosh(x) + 90cosh(x)
--R +
--R - 24%i
--R *
--R 2
--R csch(x)
--R +
--R 4 3 2
--R 90%i cosh(x) - 180cosh(x) + 18%i cosh(x) + 90cosh(x) - 24%i
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 3 2
--R 36%i cosh(x) - 90cosh(x) + 12%i cosh(x) + 90cosh(x)
--R +
--R - 48%i cosh(x) - 12
--R *
--R 2
--R csch(x)
--R +
--R 5 4 3 2
--R 36%i cosh(x) - 90cosh(x) + 12%i cosh(x) + 90cosh(x)
--R +
--R - 48%i cosh(x) - 12
--R *
--R sinh(x)
--R +
--R 6 5 4 3
--R 6%i cosh(x) - 18cosh(x) + 3%i cosh(x) + 30cosh(x)
--R +
--R 2
--R - 24%i cosh(x) - 12cosh(x) + 12%i
--R *
--R 2
--R csch(x)
--R +
--R 6 5 4 3
--R 6%i cosh(x) - 18cosh(x) + 3%i cosh(x) + 30cosh(x)
--R +
--R 2
--R - 24%i cosh(x) - 12cosh(x) + 12%i
--R *
--R +-----------------------------------------+
--R +----------------+ | 3
--R \|- %i csch(x) - 1 |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 6
--R (3\|3 csch(x) + 3\|3 )sinh(x)
--R +
--R +-+ +-+ 2 +-+ +-+
--R ((18\|3 cosh(x) + 6%i\|3 )csch(x) + 18\|3 cosh(x) + 6%i\|3 )
--R *
--R 5
--R sinh(x)
--R +
--R +-+ 2 +-+ +-+ 2
--R (45\|3 cosh(x) + 30%i\|3 cosh(x) - 3\|3 )csch(x)
--R +
--R +-+ 2 +-+ +-+
--R 45\|3 cosh(x) + 30%i\|3 cosh(x) - 3\|3
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+
--R 60\|3 cosh(x) + 60%i\|3 cosh(x) - 12\|3 cosh(x)
--R +
--R +-+
--R - 24%i\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R 60\|3 cosh(x) + 60%i\|3 cosh(x) - 12\|3 cosh(x) - 24%i\|3
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 3 +-+ 2
--R 45\|3 cosh(x) + 60%i\|3 cosh(x) - 18\|3 cosh(x)
--R +
--R +-+ +-+
--R - 72%i\|3 cosh(x) - 18\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 4 +-+ 3 +-+ 2
--R 45\|3 cosh(x) + 60%i\|3 cosh(x) - 18\|3 cosh(x)
--R +
--R +-+ +-+
--R - 72%i\|3 cosh(x) - 18\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 4 +-+ 3
--R 18\|3 cosh(x) + 30%i\|3 cosh(x) - 12\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 72%i\|3 cosh(x) - 36\|3 cosh(x) + 12%i\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 5 +-+ 4 +-+ 3
--R 18\|3 cosh(x) + 30%i\|3 cosh(x) - 12\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 72%i\|3 cosh(x) - 36\|3 cosh(x) + 12%i\|3
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 5 +-+ 4
--R 3\|3 cosh(x) + 6%i\|3 cosh(x) - 3\|3 cosh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - 24%i\|3 cosh(x) - 18\|3 cosh(x) + 12%i\|3 cosh(x) + 12\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 6 +-+ 5 +-+ 4
--R 3\|3 cosh(x) + 6%i\|3 cosh(x) - 3\|3 cosh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - 24%i\|3 cosh(x) - 18\|3 cosh(x) + 12%i\|3 cosh(x) + 12\|3
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R *
--R +--------------+
--R \|%i csch(x) - 1
--R +
--R 2 2 6
--R (- 3%i coth(x) csch(x) + 3coth(x) )sinh(x)
--R +
--R 2 2 5
--R (- 18%i cosh(x)coth(x) csch(x) + 18cosh(x)coth(x) )sinh(x)
--R +
--R 2 2 2 2
--R ((- 45%i cosh(x) - 3%i)coth(x) csch(x) + (45cosh(x) + 3)coth(x) )
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 60%i cosh(x) - 12%i cosh(x))coth(x) csch(x)
--R +
--R 3 2
--R (60cosh(x) + 12cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 45%i cosh(x) - 18%i cosh(x) + 18%i)coth(x) csch(x)
--R +
--R 4 2 2
--R (45cosh(x) + 18cosh(x) - 18)coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 18%i cosh(x) - 12%i cosh(x) + 36%i cosh(x))coth(x) csch(x)
--R +
--R 5 3 2
--R (18cosh(x) + 12cosh(x) - 36cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 3%i cosh(x) - 3%i cosh(x) + 18%i cosh(x) - 12%i)coth(x) csch(x)
--R +
--R 6 4 2 2
--R (3cosh(x) + 3cosh(x) - 18cosh(x) + 12)coth(x)
--R *
--R +-----------------------------------------+
--R +----------------+ | 3
--R \|- %i csch(x) - 1 |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 2 6
--R (- 3\|3 coth(x) csch(x) - 3%i\|3 coth(x) )sinh(x)
--R +
--R +-+ 2 +-+ 2 5
--R (- 18\|3 cosh(x)coth(x) csch(x) - 18%i\|3 cosh(x)coth(x) )sinh(x)
--R +
--R +-+ 2 +-+ 2
--R (- 45\|3 cosh(x) + 3\|3 )coth(x) csch(x)
--R +
--R +-+ 2 +-+ 2
--R (- 45%i\|3 cosh(x) + 3%i\|3 )coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2
--R (- 60\|3 cosh(x) + 12\|3 cosh(x))coth(x) csch(x)
--R +
--R +-+ 3 +-+ 2
--R (- 60%i\|3 cosh(x) + 12%i\|3 cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (- 45\|3 cosh(x) + 18\|3 cosh(x) + 12\|3 )coth(x) csch(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (- 45%i\|3 cosh(x) + 18%i\|3 cosh(x) + 12%i\|3 )coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (- 18\|3 cosh(x) + 12\|3 cosh(x) + 24\|3 cosh(x))coth(x)
--R *
--R csch(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (- 18%i\|3 cosh(x) + 12%i\|3 cosh(x) + 24%i\|3 cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+ 2
--R (- 3\|3 cosh(x) + 3\|3 cosh(x) + 12\|3 cosh(x) - 12\|3 )coth(x)
--R *
--R csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (- 3%i\|3 cosh(x) + 3%i\|3 cosh(x) + 12%i\|3 cosh(x) - 12%i\|3 )
--R *
--R 2
--R coth(x)
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R /
--R +-+ 2 +-+ 6
--R (\|3 csch(x) + \|3 )sinh(x)
--R +
--R +-+ 2 +-+ 5
--R (6\|3 cosh(x)csch(x) + 6\|3 cosh(x))sinh(x)
--R +
--R +-+ 2 +-+ 2 +-+ 2 +-+ 4
--R ((15\|3 cosh(x) + \|3 )csch(x) + 15\|3 cosh(x) + \|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ 3
--R (20\|3 cosh(x) + 4\|3 cosh(x))csch(x) + 20\|3 cosh(x)
--R +
--R +-+
--R 4\|3 cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (15\|3 cosh(x) + 6\|3 cosh(x) - 6\|3 )csch(x)
--R +
--R +-+ 4 +-+ 2 +-+
--R 15\|3 cosh(x) + 6\|3 cosh(x) - 6\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (6\|3 cosh(x) + 4\|3 cosh(x) - 12\|3 cosh(x))csch(x)
--R +
--R +-+ 5 +-+ 3 +-+
--R 6\|3 cosh(x) + 4\|3 cosh(x) - 12\|3 cosh(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+ 2
--R (\|3 cosh(x) + \|3 cosh(x) - 6\|3 cosh(x) + 4\|3 )csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R \|3 cosh(x) + \|3 cosh(x) - 6\|3 cosh(x) + 4\|3
--R *
--R +-----------------------------------------+
--R +----------------+ | 3
--R \|- %i csch(x) - 1 |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 6
--R (- 3%i csch(x) - 3%i)sinh(x)
--R +
--R 2 5
--R (- 18%i cosh(x)csch(x) - 18%i cosh(x))sinh(x)
--R +
--R 2 2 2 4
--R ((- 45%i cosh(x) + 3%i)csch(x) - 45%i cosh(x) + 3%i)sinh(x)
--R +
--R 3 2 3
--R (- 60%i cosh(x) + 12%i cosh(x))csch(x) - 60%i cosh(x)
--R +
--R 12%i cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 45%i cosh(x) + 18%i cosh(x) + 12%i)csch(x)
--R +
--R 4 2
--R - 45%i cosh(x) + 18%i cosh(x) + 12%i
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 18%i cosh(x) + 12%i cosh(x) + 24%i cosh(x))csch(x)
--R +
--R 5 3
--R - 18%i cosh(x) + 12%i cosh(x) + 24%i cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 3%i cosh(x) + 3%i cosh(x) + 12%i cosh(x) - 12%i)csch(x)
--R +
--R 6 4 2
--R - 3%i cosh(x) + 3%i cosh(x) + 12%i cosh(x) - 12%i
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R *
--R +--------------+
--R \|%i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 299
--S 300 of 526
t0457:= (-3-3*%i*csch(x))^(1/2)
--R
--R
--R +-----------------+
--R (213) \|- 3%i csch(x) - 3
--R Type: Expression(Complex(Integer))
--E 300
--S 301 of 526
r0457:= -2*3^(1/2)*atan((-1+%i*csch(x))^(1/2))*_
coth(x)/(-1-%i*csch(x))^(1/2)/(-1+%i*csch(x))^(1/2)
--R
--R
--R +-+ +--------------+
--R 2\|3 coth(x)atan(\|%i csch(x) - 1 )
--R (214) - ------------------------------------
--R +----------------+ +--------------+
--R \|- %i csch(x) - 1 \|%i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 301
--S 302 of 526
a0457:= integrate(t0457,x)
--R
--R
--R (215)
--R -
--R +-+
--R %i\|3
--R *
--R log
--R +-----------------------------------------+
--R | 3 +-+
--R |----------------------------------------- - \|3 sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ +-+
--R - \|3 cosh(x) - %i\|3
--R /
--R sinh(x) + cosh(x)
--R +
--R +-+
--R %i\|3
--R *
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3 2
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x) - 2%i cosh(x)
--R +
--R - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 3 +-+ +-+ 2
--R - %i\|3 sinh(x) + (- 3%i\|3 cosh(x) - \|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+
--R (- 3%i\|3 cosh(x) - 2\|3 cosh(x) + %i\|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - %i\|3 cosh(x) - \|3 cosh(x) + %i\|3 cosh(x) + 2\|3
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R Type: Union(Expression(Complex(Integer)),...)
--E 302
--S 303 of 526
m0457:= a0457-r0457
--R
--R
--R (216)
--R -
--R +-+ +----------------+ +--------------+
--R %i\|3 \|- %i csch(x) - 1 \|%i csch(x) - 1
--R *
--R log
--R +-----------------------------------------+
--R | 3 +-+
--R |----------------------------------------- - \|3 sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ +-+
--R - \|3 cosh(x) - %i\|3
--R /
--R sinh(x) + cosh(x)
--R +
--R +-+ +----------------+ +--------------+
--R %i\|3 \|- %i csch(x) - 1 \|%i csch(x) - 1
--R *
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x)
--R +
--R 2
--R - 2%i cosh(x) - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 3 +-+ +-+ 2
--R - %i\|3 sinh(x) + (- 3%i\|3 cosh(x) - \|3 )sinh(x)
--R +
--R +-+ 2 +-+ +-+
--R (- 3%i\|3 cosh(x) - 2\|3 cosh(x) + %i\|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - %i\|3 cosh(x) - \|3 cosh(x) + %i\|3 cosh(x) + 2\|3
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+ +--------------+
--R 2\|3 coth(x)atan(\|%i csch(x) - 1 )
--R /
--R +----------------+ +--------------+
--R \|- %i csch(x) - 1 \|%i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 303
--S 304 of 526
d0457:= D(m0457,x)
--R
--R
--R (217)
--R 2 2 6
--R (6csch(x) - 6coth(x) + 6)sinh(x)
--R +
--R 2 2 5
--R (36cosh(x)csch(x) - 36cosh(x)coth(x) + 36cosh(x))sinh(x)
--R +
--R 2 2 2 2
--R (90cosh(x) + 6)csch(x) + (- 90cosh(x) - 6)coth(x)
--R +
--R 2
--R 90cosh(x) + 6
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (120cosh(x) + 24cosh(x))csch(x)
--R +
--R 3 2 3
--R (- 120cosh(x) - 24cosh(x))coth(x) + 120cosh(x) + 24cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (90cosh(x) + 36cosh(x) - 36)csch(x)
--R +
--R 4 2 2 4
--R (- 90cosh(x) - 36cosh(x) + 36)coth(x) + 90cosh(x)
--R +
--R 2
--R 36cosh(x) - 36
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (36cosh(x) + 24cosh(x) - 72cosh(x))csch(x)
--R +
--R 5 3 2 5
--R (- 36cosh(x) - 24cosh(x) + 72cosh(x))coth(x) + 36cosh(x)
--R +
--R 3
--R 24cosh(x) - 72cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (6cosh(x) + 6cosh(x) - 36cosh(x) + 24)csch(x)
--R +
--R 6 4 2 2 6
--R (- 6cosh(x) - 6cosh(x) + 36cosh(x) - 24)coth(x) + 6cosh(x)
--R +
--R 4 2
--R 6cosh(x) - 36cosh(x) + 24
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 2 +-+ 6
--R (- 6%i\|3 csch(x) + 6%i\|3 coth(x) - 6%i\|3 )sinh(x)
--R +
--R +-+ 2 +-+ 2
--R - 36%i\|3 cosh(x)csch(x) + 36%i\|3 cosh(x)coth(x)
--R +
--R +-+
--R - 36%i\|3 cosh(x)
--R *
--R 5
--R sinh(x)
--R +
--R +-+ 2 +-+ 2
--R (- 90%i\|3 cosh(x) + 6%i\|3 )csch(x)
--R +
--R +-+ 2 +-+ 2 +-+ 2 +-+
--R (90%i\|3 cosh(x) - 6%i\|3 )coth(x) - 90%i\|3 cosh(x) + 6%i\|3
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2
--R (- 120%i\|3 cosh(x) + 24%i\|3 cosh(x))csch(x)
--R +
--R +-+ 3 +-+ 2 +-+ 3
--R (120%i\|3 cosh(x) - 24%i\|3 cosh(x))coth(x) - 120%i\|3 cosh(x)
--R +
--R +-+
--R 24%i\|3 cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (- 90%i\|3 cosh(x) + 36%i\|3 cosh(x) + 24%i\|3 )csch(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (90%i\|3 cosh(x) - 36%i\|3 cosh(x) - 24%i\|3 )coth(x)
--R +
--R +-+ 4 +-+ 2 +-+
--R - 90%i\|3 cosh(x) + 36%i\|3 cosh(x) + 24%i\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (- 36%i\|3 cosh(x) + 24%i\|3 cosh(x) + 48%i\|3 cosh(x))csch(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (36%i\|3 cosh(x) - 24%i\|3 cosh(x) - 48%i\|3 cosh(x))coth(x)
--R +
--R +-+ 5 +-+ 3 +-+
--R - 36%i\|3 cosh(x) + 24%i\|3 cosh(x) + 48%i\|3 cosh(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (- 6%i\|3 cosh(x) + 6%i\|3 cosh(x) + 24%i\|3 cosh(x) - 24%i\|3 )
--R *
--R 2
--R csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (6%i\|3 cosh(x) - 6%i\|3 cosh(x) - 24%i\|3 cosh(x) + 24%i\|3 )
--R *
--R 2
--R coth(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R - 6%i\|3 cosh(x) + 6%i\|3 cosh(x) + 24%i\|3 cosh(x) - 24%i\|3
--R *
--R +--------------+
--R atan(\|%i csch(x) - 1 )
--R +
--R 2 6
--R (- 6%i csch(x) - 6%i)sinh(x)
--R +
--R 2 5
--R ((- 36%i cosh(x) - 18)csch(x) - 36%i cosh(x) - 18)sinh(x)
--R +
--R 2 2
--R (- 90%i cosh(x) - 90cosh(x) - 3%i)csch(x)
--R +
--R 2
--R - 90%i cosh(x) - 90cosh(x) - 3%i
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 120%i cosh(x) - 180cosh(x) - 12%i cosh(x) + 30)
--R *
--R 2
--R csch(x)
--R +
--R 3 2
--R - 120%i cosh(x) - 180cosh(x) - 12%i cosh(x) + 30
--R *
--R 3
--R sinh(x)
--R +
--R 4 3 2
--R - 90%i cosh(x) - 180cosh(x) - 18%i cosh(x)
--R +
--R 90cosh(x) + 24%i
--R *
--R 2
--R csch(x)
--R +
--R 4 3 2
--R - 90%i cosh(x) - 180cosh(x) - 18%i cosh(x) + 90cosh(x)
--R +
--R 24%i
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 3
--R - 36%i cosh(x) - 90cosh(x) - 12%i cosh(x)
--R +
--R 2
--R 90cosh(x) + 48%i cosh(x) - 12
--R *
--R 2
--R csch(x)
--R +
--R 5 4 3 2
--R - 36%i cosh(x) - 90cosh(x) - 12%i cosh(x) + 90cosh(x)
--R +
--R 48%i cosh(x) - 12
--R *
--R sinh(x)
--R +
--R 6 5 4 3
--R - 6%i cosh(x) - 18cosh(x) - 3%i cosh(x) + 30cosh(x)
--R +
--R 2
--R 24%i cosh(x) - 12cosh(x) - 12%i
--R *
--R 2
--R csch(x)
--R +
--R 6 5 4 3
--R - 6%i cosh(x) - 18cosh(x) - 3%i cosh(x) + 30cosh(x)
--R +
--R 2
--R 24%i cosh(x) - 12cosh(x) - 12%i
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R +
--R 2 2 6
--R (3%i coth(x) csch(x) + 3coth(x) )sinh(x)
--R +
--R 2 2 5
--R (18%i cosh(x)coth(x) csch(x) + 18cosh(x)coth(x) )sinh(x)
--R +
--R 2 2
--R (45%i cosh(x) + 3%i)coth(x) csch(x)
--R +
--R 2 2
--R (45cosh(x) + 3)coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (60%i cosh(x) + 12%i cosh(x))coth(x) csch(x)
--R +
--R 3 2
--R (60cosh(x) + 12cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (45%i cosh(x) + 18%i cosh(x) - 18%i)coth(x) csch(x)
--R +
--R 4 2 2
--R (45cosh(x) + 18cosh(x) - 18)coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (18%i cosh(x) + 12%i cosh(x) - 36%i cosh(x))coth(x) csch(x)
--R +
--R 5 3 2
--R (18cosh(x) + 12cosh(x) - 36cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (3%i cosh(x) + 3%i cosh(x) - 18%i cosh(x) + 12%i)coth(x)
--R *
--R csch(x)
--R +
--R 6 4 2 2
--R (3cosh(x) + 3cosh(x) - 18cosh(x) + 12)coth(x)
--R *
--R +-----------------------------------------+
--R | 3
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+ 2 +-+ 6
--R (- 3\|3 csch(x) - 3\|3 )sinh(x)
--R +
--R +-+ +-+ 2 +-+ +-+
--R ((- 18\|3 cosh(x) + 6%i\|3 )csch(x) - 18\|3 cosh(x) + 6%i\|3 )
--R *
--R 5
--R sinh(x)
--R +
--R +-+ 2 +-+ +-+ 2
--R (- 45\|3 cosh(x) + 30%i\|3 cosh(x) + 3\|3 )csch(x)
--R +
--R +-+ 2 +-+ +-+
--R - 45\|3 cosh(x) + 30%i\|3 cosh(x) + 3\|3
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+
--R - 60\|3 cosh(x) + 60%i\|3 cosh(x) + 12\|3 cosh(x)
--R +
--R +-+
--R - 24%i\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - 60\|3 cosh(x) + 60%i\|3 cosh(x) + 12\|3 cosh(x) - 24%i\|3
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 3 +-+ 2
--R - 45\|3 cosh(x) + 60%i\|3 cosh(x) + 18\|3 cosh(x)
--R +
--R +-+ +-+
--R - 72%i\|3 cosh(x) + 18\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 4 +-+ 3 +-+ 2
--R - 45\|3 cosh(x) + 60%i\|3 cosh(x) + 18\|3 cosh(x)
--R +
--R +-+ +-+
--R - 72%i\|3 cosh(x) + 18\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 4 +-+ 3
--R - 18\|3 cosh(x) + 30%i\|3 cosh(x) + 12\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 72%i\|3 cosh(x) + 36\|3 cosh(x) + 12%i\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 5 +-+ 4 +-+ 3
--R - 18\|3 cosh(x) + 30%i\|3 cosh(x) + 12\|3 cosh(x)
--R +
--R +-+ 2 +-+ +-+
--R - 72%i\|3 cosh(x) + 36\|3 cosh(x) + 12%i\|3
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 5 +-+ 4
--R - 3\|3 cosh(x) + 6%i\|3 cosh(x) + 3\|3 cosh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - 24%i\|3 cosh(x) + 18\|3 cosh(x) + 12%i\|3 cosh(x) - 12\|3
--R *
--R 2
--R csch(x)
--R +
--R +-+ 6 +-+ 5 +-+ 4
--R - 3\|3 cosh(x) + 6%i\|3 cosh(x) + 3\|3 cosh(x)
--R +
--R +-+ 3 +-+ 2 +-+ +-+
--R - 24%i\|3 cosh(x) + 18\|3 cosh(x) + 12%i\|3 cosh(x) - 12\|3
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R +
--R +-+ 2 +-+ 2 6
--R (3\|3 coth(x) csch(x) - 3%i\|3 coth(x) )sinh(x)
--R +
--R +-+ 2 +-+ 2 5
--R (18\|3 cosh(x)coth(x) csch(x) - 18%i\|3 cosh(x)coth(x) )sinh(x)
--R +
--R +-+ 2 +-+ 2
--R (45\|3 cosh(x) - 3\|3 )coth(x) csch(x)
--R +
--R +-+ 2 +-+ 2
--R (- 45%i\|3 cosh(x) + 3%i\|3 )coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R +-+ 3 +-+ 2
--R (60\|3 cosh(x) - 12\|3 cosh(x))coth(x) csch(x)
--R +
--R +-+ 3 +-+ 2
--R (- 60%i\|3 cosh(x) + 12%i\|3 cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (45\|3 cosh(x) - 18\|3 cosh(x) - 12\|3 )coth(x) csch(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (- 45%i\|3 cosh(x) + 18%i\|3 cosh(x) + 12%i\|3 )coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (18\|3 cosh(x) - 12\|3 cosh(x) - 24\|3 cosh(x))coth(x) csch(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (- 18%i\|3 cosh(x) + 12%i\|3 cosh(x) + 24%i\|3 cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+ 2
--R (3\|3 cosh(x) - 3\|3 cosh(x) - 12\|3 cosh(x) + 12\|3 )coth(x)
--R *
--R csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R (- 3%i\|3 cosh(x) + 3%i\|3 cosh(x) + 12%i\|3 cosh(x) - 12%i\|3 )
--R *
--R 2
--R coth(x)
--R *
--R +--------------+
--R \|%i csch(x) - 1
--R /
--R +-+ 2 +-+ 6
--R (\|3 csch(x) + \|3 )sinh(x)
--R +
--R +-+ 2 +-+ 5
--R (6\|3 cosh(x)csch(x) + 6\|3 cosh(x))sinh(x)
--R +
--R +-+ 2 +-+ 2 +-+ 2 +-+ 4
--R ((15\|3 cosh(x) + \|3 )csch(x) + 15\|3 cosh(x) + \|3 )sinh(x)
--R +
--R +-+ 3 +-+ 2 +-+ 3
--R (20\|3 cosh(x) + 4\|3 cosh(x))csch(x) + 20\|3 cosh(x)
--R +
--R +-+
--R 4\|3 cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R +-+ 4 +-+ 2 +-+ 2
--R (15\|3 cosh(x) + 6\|3 cosh(x) - 6\|3 )csch(x)
--R +
--R +-+ 4 +-+ 2 +-+
--R 15\|3 cosh(x) + 6\|3 cosh(x) - 6\|3
--R *
--R 2
--R sinh(x)
--R +
--R +-+ 5 +-+ 3 +-+ 2
--R (6\|3 cosh(x) + 4\|3 cosh(x) - 12\|3 cosh(x))csch(x)
--R +
--R +-+ 5 +-+ 3 +-+
--R 6\|3 cosh(x) + 4\|3 cosh(x) - 12\|3 cosh(x)
--R *
--R sinh(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+ 2
--R (\|3 cosh(x) + \|3 cosh(x) - 6\|3 cosh(x) + 4\|3 )csch(x)
--R +
--R +-+ 6 +-+ 4 +-+ 2 +-+
--R \|3 cosh(x) + \|3 cosh(x) - 6\|3 cosh(x) + 4\|3
--R *
--R +-----------------------------------------+
--R +----------------+ | 3
--R \|- %i csch(x) - 1 |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 6
--R (- 3%i csch(x) - 3%i)sinh(x)
--R +
--R 2 5
--R (- 18%i cosh(x)csch(x) - 18%i cosh(x))sinh(x)
--R +
--R 2 2 2 4
--R ((- 45%i cosh(x) + 3%i)csch(x) - 45%i cosh(x) + 3%i)sinh(x)
--R +
--R 3 2 3
--R (- 60%i cosh(x) + 12%i cosh(x))csch(x) - 60%i cosh(x)
--R +
--R 12%i cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 45%i cosh(x) + 18%i cosh(x) + 12%i)csch(x)
--R +
--R 4 2
--R - 45%i cosh(x) + 18%i cosh(x) + 12%i
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 18%i cosh(x) + 12%i cosh(x) + 24%i cosh(x))csch(x)
--R +
--R 5 3
--R - 18%i cosh(x) + 12%i cosh(x) + 24%i cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 3%i cosh(x) + 3%i cosh(x) + 12%i cosh(x) - 12%i)csch(x)
--R +
--R 6 4 2
--R - 3%i cosh(x) + 3%i cosh(x) + 12%i cosh(x) - 12%i
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R *
--R +--------------+
--R \|%i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 304
--S 305 of 526
t0458:= (a+%i*a*csch(x))^(1/2)
--R
--R
--R +----------------+
--R (218) \|%i a csch(x) + a
--R Type: Expression(Complex(Integer))
--E 305
--S 306 of 526
r0458:= 2*a*atan((-1+%i*csch(x))^(1/2))*_
coth(x)/(-1+%i*csch(x))^(1/2)/(a+%i*a*csch(x))^(1/2)
--R
--R
--R +--------------+
--R 2a coth(x)atan(\|%i csch(x) - 1 )
--R (219) ------------------------------------
--R +--------------+ +----------------+
--R \|%i csch(x) - 1 \|%i a csch(x) + a
--R Type: Expression(Complex(Integer))
--E 306
--S 307 of 526
a0458:= integrate(t0458,x)
--R
--R
--R (220)
--R -
--R +-+
--R \|a
--R *
--R log
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+
--R (- sinh(x) - cosh(x) - %i)\|a
--R /
--R sinh(x) + cosh(x)
--R +
--R +-+
--R \|a
--R *
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3 2
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x) - 2%i cosh(x)
--R +
--R - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R - %i sinh(x) + (- 3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (- 3%i cosh(x) - 2cosh(x) + %i)sinh(x) - %i cosh(x)
--R +
--R 2
--R - cosh(x) + %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R Type: Union(Expression(Complex(Integer)),...)
--E 307
--S 308 of 526
m0458:= a0458-r0458
--R
--R
--R (221)
--R -
--R +-+ +--------------+ +----------------+
--R \|a \|%i csch(x) - 1 \|%i a csch(x) + a
--R *
--R log
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+
--R (- sinh(x) - cosh(x) - %i)\|a
--R /
--R sinh(x) + cosh(x)
--R +
--R +-+ +--------------+ +----------------+
--R \|a \|%i csch(x) - 1 \|%i a csch(x) + a
--R *
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x)
--R +
--R 2
--R - 2%i cosh(x) - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R - %i sinh(x) + (- 3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (- 3%i cosh(x) - 2cosh(x) + %i)sinh(x) - %i cosh(x)
--R +
--R 2
--R - cosh(x) + %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +--------------+
--R - 2a coth(x)atan(\|%i csch(x) - 1 )
--R /
--R +--------------+ +----------------+
--R \|%i csch(x) - 1 \|%i a csch(x) + a
--R Type: Expression(Complex(Integer))
--E 308
--S 309 of 526
d0458:= D(m0458,x)
--R
--R
--R (222)
--R 2 2 6
--R (- 2a csch(x) + 2a coth(x) - 2a)sinh(x)
--R +
--R 2 2
--R (- 12a cosh(x)csch(x) + 12a cosh(x)coth(x) - 12a cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 2 2
--R (- 30a cosh(x) - 2a)csch(x) + (30a cosh(x) + 2a)coth(x)
--R +
--R 2
--R - 30a cosh(x) - 2a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 40a cosh(x) - 8a cosh(x))csch(x)
--R +
--R 3 2 3
--R (40a cosh(x) + 8a cosh(x))coth(x) - 40a cosh(x) - 8a cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 30a cosh(x) - 12a cosh(x) + 12a)csch(x)
--R +
--R 4 2 2 4
--R (30a cosh(x) + 12a cosh(x) - 12a)coth(x) - 30a cosh(x)
--R +
--R 2
--R - 12a cosh(x) + 12a
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 12a cosh(x) - 8a cosh(x) + 24a cosh(x))csch(x)
--R +
--R 5 3 2
--R (12a cosh(x) + 8a cosh(x) - 24a cosh(x))coth(x)
--R +
--R 5 3
--R - 12a cosh(x) - 8a cosh(x) + 24a cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 2a cosh(x) - 2a cosh(x) + 12a cosh(x) - 8a)csch(x)
--R +
--R 6 4 2 2
--R (2a cosh(x) + 2a cosh(x) - 12a cosh(x) + 8a)coth(x)
--R +
--R 6 4 2
--R - 2a cosh(x) - 2a cosh(x) + 12a cosh(x) - 8a
--R *
--R +-----------------------------------------+
--R +-+ | a
--R \|a |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 2 2 2 2 6
--R (2%i a csch(x) - 2%i a coth(x) + 2%i a )sinh(x)
--R +
--R 2 2 2 2 2
--R (12%i a cosh(x)csch(x) - 12%i a cosh(x)coth(x) + 12%i a cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 2 2
--R (30%i a cosh(x) - 2%i a )csch(x)
--R +
--R 2 2 2 2 2 2 2
--R (- 30%i a cosh(x) + 2%i a )coth(x) + 30%i a cosh(x) - 2%i a
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 2 2
--R (40%i a cosh(x) - 8%i a cosh(x))csch(x)
--R +
--R 2 3 2 2 2 3
--R (- 40%i a cosh(x) + 8%i a cosh(x))coth(x) + 40%i a cosh(x)
--R +
--R 2
--R - 8%i a cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 2 2 2 2
--R (30%i a cosh(x) - 12%i a cosh(x) - 8%i a )csch(x)
--R +
--R 2 4 2 2 2 2
--R (- 30%i a cosh(x) + 12%i a cosh(x) + 8%i a )coth(x)
--R +
--R 2 4 2 2 2
--R 30%i a cosh(x) - 12%i a cosh(x) - 8%i a
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3 2 2
--R (12%i a cosh(x) - 8%i a cosh(x) - 16%i a cosh(x))csch(x)
--R +
--R 2 5 2 3 2 2
--R (- 12%i a cosh(x) + 8%i a cosh(x) + 16%i a cosh(x))coth(x)
--R +
--R 2 5 2 3 2
--R 12%i a cosh(x) - 8%i a cosh(x) - 16%i a cosh(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4 2 2 2 2
--R (2%i a cosh(x) - 2%i a cosh(x) - 8%i a cosh(x) + 8%i a )csch(x)
--R +
--R 2 6 2 4 2 2 2 2
--R (- 2%i a cosh(x) + 2%i a cosh(x) + 8%i a cosh(x) - 8%i a )coth(x)
--R +
--R 2 6 2 4 2 2 2
--R 2%i a cosh(x) - 2%i a cosh(x) - 8%i a cosh(x) + 8%i a
--R *
--R +--------------+
--R atan(\|%i csch(x) - 1 )
--R +
--R 2 6
--R (- 2a csch(x) - 2a)sinh(x)
--R +
--R 2 5
--R ((- 12a cosh(x) + 6%i a)csch(x) - 12a cosh(x) + 6%i a)sinh(x)
--R +
--R 2 2 2
--R (- 30a cosh(x) + 30%i a cosh(x) - a)csch(x) - 30a cosh(x)
--R +
--R 30%i a cosh(x) - a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 40a cosh(x) + 60%i a cosh(x) - 4a cosh(x) - 10%i a)
--R *
--R 2
--R csch(x)
--R +
--R 3 2
--R - 40a cosh(x) + 60%i a cosh(x) - 4a cosh(x) - 10%i a
--R *
--R 3
--R sinh(x)
--R +
--R 4 3 2
--R - 30a cosh(x) + 60%i a cosh(x) - 6a cosh(x)
--R +
--R - 30%i a cosh(x) + 8a
--R *
--R 2
--R csch(x)
--R +
--R 4 3 2
--R - 30a cosh(x) + 60%i a cosh(x) - 6a cosh(x)
--R +
--R - 30%i a cosh(x) + 8a
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 3
--R - 12a cosh(x) + 30%i a cosh(x) - 4a cosh(x)
--R +
--R 2
--R - 30%i a cosh(x) + 16a cosh(x) + 4%i a
--R *
--R 2
--R csch(x)
--R +
--R 5 4 3
--R - 12a cosh(x) + 30%i a cosh(x) - 4a cosh(x)
--R +
--R 2
--R - 30%i a cosh(x) + 16a cosh(x) + 4%i a
--R *
--R sinh(x)
--R +
--R 6 5 4 3
--R - 2a cosh(x) + 6%i a cosh(x) - a cosh(x) - 10%i a cosh(x)
--R +
--R 2
--R 8a cosh(x) + 4%i a cosh(x) - 4a
--R *
--R 2
--R csch(x)
--R +
--R 6 5 4 3
--R - 2a cosh(x) + 6%i a cosh(x) - a cosh(x) - 10%i a cosh(x)
--R +
--R 2
--R 8a cosh(x) + 4%i a cosh(x) - 4a
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 6
--R (%i a csch(x) + %i a)sinh(x)
--R +
--R 2 5
--R ((6%i a cosh(x) + 2a)csch(x) + 6%i a cosh(x) + 2a)sinh(x)
--R +
--R 2 2
--R (15%i a cosh(x) + 10a cosh(x) - %i a)csch(x)
--R +
--R 2
--R 15%i a cosh(x) + 10a cosh(x) - %i a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2 2
--R (20%i a cosh(x) + 20a cosh(x) - 4%i a cosh(x) - 8a)csch(x)
--R +
--R 3 2
--R 20%i a cosh(x) + 20a cosh(x) - 4%i a cosh(x) - 8a
--R *
--R 3
--R sinh(x)
--R +
--R 4 3 2
--R 15%i a cosh(x) + 20a cosh(x) - 6%i a cosh(x)
--R +
--R - 24a cosh(x) - 6%i a
--R *
--R 2
--R csch(x)
--R +
--R 4 3 2
--R 15%i a cosh(x) + 20a cosh(x) - 6%i a cosh(x) - 24a cosh(x)
--R +
--R - 6%i a
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 3
--R 6%i a cosh(x) + 10a cosh(x) - 4%i a cosh(x)
--R +
--R 2
--R - 24a cosh(x) - 12%i a cosh(x) + 4a
--R *
--R 2
--R csch(x)
--R +
--R 5 4 3 2
--R 6%i a cosh(x) + 10a cosh(x) - 4%i a cosh(x) - 24a cosh(x)
--R +
--R - 12%i a cosh(x) + 4a
--R *
--R sinh(x)
--R +
--R 6 5 4 3
--R %i a cosh(x) + 2a cosh(x) - %i a cosh(x) - 8a cosh(x)
--R +
--R 2
--R - 6%i a cosh(x) + 4a cosh(x) + 4%i a
--R *
--R 2
--R csch(x)
--R +
--R 6 5 4 3
--R %i a cosh(x) + 2a cosh(x) - %i a cosh(x) - 8a cosh(x)
--R +
--R 2
--R - 6%i a cosh(x) + 4a cosh(x) + 4%i a
--R *
--R +-+
--R \|a
--R *
--R +--------------+ +----------------+
--R \|%i csch(x) - 1 \|%i a csch(x) + a
--R +
--R 2 2 6
--R (- %i a coth(x) csch(x) - a coth(x) )sinh(x)
--R +
--R 2 2 5
--R (- 6%i a cosh(x)coth(x) csch(x) - 6a cosh(x)coth(x) )sinh(x)
--R +
--R 2 2
--R (- 15%i a cosh(x) - %i a)coth(x) csch(x)
--R +
--R 2 2
--R (- 15a cosh(x) - a)coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 20%i a cosh(x) - 4%i a cosh(x))coth(x) csch(x)
--R +
--R 3 2
--R (- 20a cosh(x) - 4a cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 15%i a cosh(x) - 6%i a cosh(x) + 6%i a)coth(x) csch(x)
--R +
--R 4 2 2
--R (- 15a cosh(x) - 6a cosh(x) + 6a)coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R 5 3
--R (- 6%i a cosh(x) - 4%i a cosh(x) + 12%i a cosh(x))
--R *
--R 2
--R coth(x) csch(x)
--R +
--R 5 3 2
--R (- 6a cosh(x) - 4a cosh(x) + 12a cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R 6 4 2
--R (- %i a cosh(x) - %i a cosh(x) + 6%i a cosh(x) - 4%i a)
--R *
--R 2
--R coth(x) csch(x)
--R +
--R 6 4 2 2
--R (- a cosh(x) - a cosh(x) + 6a cosh(x) - 4a)coth(x)
--R *
--R +-----------------------------------------+
--R +-+ | a
--R \|a |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 2 2 2 6
--R (- a coth(x) csch(x) + %i a coth(x) )sinh(x)
--R +
--R 2 2 2 2 5
--R (- 6a cosh(x)coth(x) csch(x) + 6%i a cosh(x)coth(x) )sinh(x)
--R +
--R 2 2 2 2
--R (- 15a cosh(x) + a )coth(x) csch(x)
--R +
--R 2 2 2 2
--R (15%i a cosh(x) - %i a )coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 2 2
--R (- 20a cosh(x) + 4a cosh(x))coth(x) csch(x)
--R +
--R 2 3 2 2
--R (20%i a cosh(x) - 4%i a cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 2 2 2 2
--R (- 15a cosh(x) + 6a cosh(x) + 4a )coth(x) csch(x)
--R +
--R 2 4 2 2 2 2
--R (15%i a cosh(x) - 6%i a cosh(x) - 4%i a )coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3 2 2
--R (- 6a cosh(x) + 4a cosh(x) + 8a cosh(x))coth(x) csch(x)
--R +
--R 2 5 2 3 2 2
--R (6%i a cosh(x) - 4%i a cosh(x) - 8%i a cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4 2 2 2 2
--R (- a cosh(x) + a cosh(x) + 4a cosh(x) - 4a )coth(x) csch(x)
--R +
--R 2 6 2 4 2 2 2 2
--R (%i a cosh(x) - %i a cosh(x) - 4%i a cosh(x) + 4%i a )coth(x)
--R *
--R +--------------+
--R \|%i csch(x) - 1
--R /
--R 2 6 2 5
--R (csch(x) + 1)sinh(x) + (6cosh(x)csch(x) + 6cosh(x))sinh(x)
--R +
--R 2 2 2 4
--R ((15cosh(x) + 1)csch(x) + 15cosh(x) + 1)sinh(x)
--R +
--R 3 2 3 3
--R ((20cosh(x) + 4cosh(x))csch(x) + 20cosh(x) + 4cosh(x))sinh(x)
--R +
--R 4 2 2 4 2
--R (15cosh(x) + 6cosh(x) - 6)csch(x) + 15cosh(x) + 6cosh(x)
--R +
--R - 6
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2 5
--R (6cosh(x) + 4cosh(x) - 12cosh(x))csch(x) + 6cosh(x)
--R +
--R 3
--R 4cosh(x) - 12cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2 6 4
--R (cosh(x) + cosh(x) - 6cosh(x) + 4)csch(x) + cosh(x) + cosh(x)
--R +
--R 2
--R - 6cosh(x) + 4
--R *
--R +-----------------------------------------+
--R +-+ | a
--R \|a |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 6
--R (- %i a csch(x) - %i a)sinh(x)
--R +
--R 2 5
--R (- 6%i a cosh(x)csch(x) - 6%i a cosh(x))sinh(x)
--R +
--R 2 2 2 4
--R ((- 15%i a cosh(x) + %i a)csch(x) - 15%i a cosh(x) + %i a)sinh(x)
--R +
--R 3 2 3
--R (- 20%i a cosh(x) + 4%i a cosh(x))csch(x) - 20%i a cosh(x)
--R +
--R 4%i a cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 15%i a cosh(x) + 6%i a cosh(x) + 4%i a)csch(x)
--R +
--R 4 2
--R - 15%i a cosh(x) + 6%i a cosh(x) + 4%i a
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 6%i a cosh(x) + 4%i a cosh(x) + 8%i a cosh(x))csch(x)
--R +
--R 5 3
--R - 6%i a cosh(x) + 4%i a cosh(x) + 8%i a cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- %i a cosh(x) + %i a cosh(x) + 4%i a cosh(x) - 4%i a)csch(x)
--R +
--R 6 4 2
--R - %i a cosh(x) + %i a cosh(x) + 4%i a cosh(x) - 4%i a
--R *
--R +--------------+ +----------------+
--R \|%i csch(x) - 1 \|%i a csch(x) + a
--R Type: Expression(Complex(Integer))
--E 309
--S 310 of 526
t0459:= (a-%i*a*csch(x))^(1/2)
--R
--R
--R +------------------+
--R (223) \|- %i a csch(x) + a
--R Type: Expression(Complex(Integer))
--E 310
--S 311 of 526
r0459:= 2*a*atan((-1-%i*csch(x))^(1/2))*_
coth(x)/(-1-%i*csch(x))^(1/2)/(a-%i*a*csch(x))^(1/2)
--R
--R
--R +----------------+
--R 2a coth(x)atan(\|- %i csch(x) - 1 )
--R (224) ----------------------------------------
--R +------------------+ +----------------+
--R \|- %i a csch(x) + a \|- %i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 311
--S 312 of 526
a0459:= integrate(t0459,x)
--R
--R
--R (225)
--R +-+
--R \|a
--R *
--R log
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+
--R (sinh(x) + cosh(x) - %i)\|a
--R /
--R sinh(x) + cosh(x)
--R +
--R -
--R +-+
--R \|a
--R *
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R %i sinh(x) + (3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (3%i cosh(x) - 2cosh(x) - %i)sinh(x) + %i cosh(x)
--R +
--R 2
--R - cosh(x) - %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R Type: Union(Expression(Complex(Integer)),...)
--E 312
--S 313 of 526
m0459:= a0459-r0459
--R
--R
--R (226)
--R +-+ +------------------+ +----------------+
--R \|a \|- %i a csch(x) + a \|- %i csch(x) - 1
--R *
--R log
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R +-+
--R (sinh(x) + cosh(x) - %i)\|a
--R /
--R sinh(x) + cosh(x)
--R +
--R -
--R +-+ +------------------+ +----------------+
--R \|a \|- %i a csch(x) + a \|- %i csch(x) - 1
--R *
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R %i sinh(x) + (3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (3%i cosh(x) - 2cosh(x) - %i)sinh(x) + %i cosh(x)
--R +
--R 2
--R - cosh(x) - %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +----------------+
--R - 2a coth(x)atan(\|- %i csch(x) - 1 )
--R /
--R +------------------+ +----------------+
--R \|- %i a csch(x) + a \|- %i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 313
--S 314 of 526
d0459:= D(m0459,x)
--R
--R
--R (227)
--R 2 2 6
--R (- 2a csch(x) + 2a coth(x) - 2a)sinh(x)
--R +
--R 2 2
--R (- 12a cosh(x)csch(x) + 12a cosh(x)coth(x) - 12a cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 2 2
--R (- 30a cosh(x) - 2a)csch(x) + (30a cosh(x) + 2a)coth(x)
--R +
--R 2
--R - 30a cosh(x) - 2a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 40a cosh(x) - 8a cosh(x))csch(x)
--R +
--R 3 2 3
--R (40a cosh(x) + 8a cosh(x))coth(x) - 40a cosh(x) - 8a cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 30a cosh(x) - 12a cosh(x) + 12a)csch(x)
--R +
--R 4 2 2 4
--R (30a cosh(x) + 12a cosh(x) - 12a)coth(x) - 30a cosh(x)
--R +
--R 2
--R - 12a cosh(x) + 12a
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 12a cosh(x) - 8a cosh(x) + 24a cosh(x))csch(x)
--R +
--R 5 3 2
--R (12a cosh(x) + 8a cosh(x) - 24a cosh(x))coth(x)
--R +
--R 5 3
--R - 12a cosh(x) - 8a cosh(x) + 24a cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- 2a cosh(x) - 2a cosh(x) + 12a cosh(x) - 8a)csch(x)
--R +
--R 6 4 2 2
--R (2a cosh(x) + 2a cosh(x) - 12a cosh(x) + 8a)coth(x)
--R +
--R 6 4 2
--R - 2a cosh(x) - 2a cosh(x) + 12a cosh(x) - 8a
--R *
--R +-----------------------------------------+
--R +-+ | a
--R \|a |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 2 2 2 2 6
--R (2%i a csch(x) - 2%i a coth(x) + 2%i a )sinh(x)
--R +
--R 2 2 2 2 2
--R (12%i a cosh(x)csch(x) - 12%i a cosh(x)coth(x) + 12%i a cosh(x))
--R *
--R 5
--R sinh(x)
--R +
--R 2 2 2 2
--R (30%i a cosh(x) - 2%i a )csch(x)
--R +
--R 2 2 2 2 2 2 2
--R (- 30%i a cosh(x) + 2%i a )coth(x) + 30%i a cosh(x) - 2%i a
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 2 2
--R (40%i a cosh(x) - 8%i a cosh(x))csch(x)
--R +
--R 2 3 2 2 2 3
--R (- 40%i a cosh(x) + 8%i a cosh(x))coth(x) + 40%i a cosh(x)
--R +
--R 2
--R - 8%i a cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 2 2 2 2
--R (30%i a cosh(x) - 12%i a cosh(x) - 8%i a )csch(x)
--R +
--R 2 4 2 2 2 2
--R (- 30%i a cosh(x) + 12%i a cosh(x) + 8%i a )coth(x)
--R +
--R 2 4 2 2 2
--R 30%i a cosh(x) - 12%i a cosh(x) - 8%i a
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3 2 2
--R (12%i a cosh(x) - 8%i a cosh(x) - 16%i a cosh(x))csch(x)
--R +
--R 2 5 2 3 2 2
--R (- 12%i a cosh(x) + 8%i a cosh(x) + 16%i a cosh(x))coth(x)
--R +
--R 2 5 2 3 2
--R 12%i a cosh(x) - 8%i a cosh(x) - 16%i a cosh(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4 2 2 2 2
--R (2%i a cosh(x) - 2%i a cosh(x) - 8%i a cosh(x) + 8%i a )csch(x)
--R +
--R 2 6 2 4 2 2 2 2
--R (- 2%i a cosh(x) + 2%i a cosh(x) + 8%i a cosh(x) - 8%i a )coth(x)
--R +
--R 2 6 2 4 2 2 2
--R 2%i a cosh(x) - 2%i a cosh(x) - 8%i a cosh(x) + 8%i a
--R *
--R +----------------+
--R atan(\|- %i csch(x) - 1 )
--R +
--R 2 6
--R (2a csch(x) + 2a)sinh(x)
--R +
--R 2 5
--R ((12a cosh(x) + 6%i a)csch(x) + 12a cosh(x) + 6%i a)sinh(x)
--R +
--R 2 2 2
--R (30a cosh(x) + 30%i a cosh(x) + a)csch(x) + 30a cosh(x)
--R +
--R 30%i a cosh(x) + a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (40a cosh(x) + 60%i a cosh(x) + 4a cosh(x) - 10%i a)
--R *
--R 2
--R csch(x)
--R +
--R 3 2
--R 40a cosh(x) + 60%i a cosh(x) + 4a cosh(x) - 10%i a
--R *
--R 3
--R sinh(x)
--R +
--R 4 3 2
--R 30a cosh(x) + 60%i a cosh(x) + 6a cosh(x)
--R +
--R - 30%i a cosh(x) - 8a
--R *
--R 2
--R csch(x)
--R +
--R 4 3 2
--R 30a cosh(x) + 60%i a cosh(x) + 6a cosh(x) - 30%i a cosh(x)
--R +
--R - 8a
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 3
--R 12a cosh(x) + 30%i a cosh(x) + 4a cosh(x)
--R +
--R 2
--R - 30%i a cosh(x) - 16a cosh(x) + 4%i a
--R *
--R 2
--R csch(x)
--R +
--R 5 4 3
--R 12a cosh(x) + 30%i a cosh(x) + 4a cosh(x)
--R +
--R 2
--R - 30%i a cosh(x) - 16a cosh(x) + 4%i a
--R *
--R sinh(x)
--R +
--R 6 5 4 3
--R 2a cosh(x) + 6%i a cosh(x) + a cosh(x) - 10%i a cosh(x)
--R +
--R 2
--R - 8a cosh(x) + 4%i a cosh(x) + 4a
--R *
--R 2
--R csch(x)
--R +
--R 6 5 4 3
--R 2a cosh(x) + 6%i a cosh(x) + a cosh(x) - 10%i a cosh(x)
--R +
--R 2
--R - 8a cosh(x) + 4%i a cosh(x) + 4a
--R *
--R +------------------+
--R \|- %i a csch(x) + a
--R +
--R 2 2 6
--R (%i a coth(x) csch(x) - a coth(x) )sinh(x)
--R +
--R 2 2 5
--R (6%i a cosh(x)coth(x) csch(x) - 6a cosh(x)coth(x) )sinh(x)
--R +
--R 2 2
--R (15%i a cosh(x) + %i a)coth(x) csch(x)
--R +
--R 2 2
--R (- 15a cosh(x) - a)coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (20%i a cosh(x) + 4%i a cosh(x))coth(x) csch(x)
--R +
--R 3 2
--R (- 20a cosh(x) - 4a cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (15%i a cosh(x) + 6%i a cosh(x) - 6%i a)coth(x) csch(x)
--R +
--R 4 2 2
--R (- 15a cosh(x) - 6a cosh(x) + 6a)coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (6%i a cosh(x) + 4%i a cosh(x) - 12%i a cosh(x))coth(x)
--R *
--R csch(x)
--R +
--R 5 3 2
--R (- 6a cosh(x) - 4a cosh(x) + 12a cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R 6 4 2
--R (%i a cosh(x) + %i a cosh(x) - 6%i a cosh(x) + 4%i a)
--R *
--R 2
--R coth(x) csch(x)
--R +
--R 6 4 2 2
--R (- a cosh(x) - a cosh(x) + 6a cosh(x) - 4a)coth(x)
--R *
--R +-+
--R \|a
--R *
--R +-----------------------------------------+
--R +----------------+ | a
--R \|- %i csch(x) - 1 |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 6
--R (- %i a csch(x) - %i a)sinh(x)
--R +
--R 2 5
--R ((- 6%i a cosh(x) + 2a)csch(x) - 6%i a cosh(x) + 2a)sinh(x)
--R +
--R 2 2
--R (- 15%i a cosh(x) + 10a cosh(x) + %i a)csch(x)
--R +
--R 2
--R - 15%i a cosh(x) + 10a cosh(x) + %i a
--R *
--R 4
--R sinh(x)
--R +
--R 3 2
--R (- 20%i a cosh(x) + 20a cosh(x) + 4%i a cosh(x) - 8a)
--R *
--R 2
--R csch(x)
--R +
--R 3 2
--R - 20%i a cosh(x) + 20a cosh(x) + 4%i a cosh(x) - 8a
--R *
--R 3
--R sinh(x)
--R +
--R 4 3 2
--R - 15%i a cosh(x) + 20a cosh(x) + 6%i a cosh(x)
--R +
--R - 24a cosh(x) + 6%i a
--R *
--R 2
--R csch(x)
--R +
--R 4 3 2
--R - 15%i a cosh(x) + 20a cosh(x) + 6%i a cosh(x)
--R +
--R - 24a cosh(x) + 6%i a
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 3
--R - 6%i a cosh(x) + 10a cosh(x) + 4%i a cosh(x)
--R +
--R 2
--R - 24a cosh(x) + 12%i a cosh(x) + 4a
--R *
--R 2
--R csch(x)
--R +
--R 5 4 3
--R - 6%i a cosh(x) + 10a cosh(x) + 4%i a cosh(x)
--R +
--R 2
--R - 24a cosh(x) + 12%i a cosh(x) + 4a
--R *
--R sinh(x)
--R +
--R 6 5 4 3
--R - %i a cosh(x) + 2a cosh(x) + %i a cosh(x) - 8a cosh(x)
--R +
--R 2
--R 6%i a cosh(x) + 4a cosh(x) - 4%i a
--R *
--R 2
--R csch(x)
--R +
--R 6 5 4 3
--R - %i a cosh(x) + 2a cosh(x) + %i a cosh(x) - 8a cosh(x)
--R +
--R 2
--R 6%i a cosh(x) + 4a cosh(x) - 4%i a
--R *
--R +-+ +------------------+
--R \|a \|- %i a csch(x) + a
--R +
--R 2 2 2 2 6
--R (a coth(x) csch(x) + %i a coth(x) )sinh(x)
--R +
--R 2 2 2 2 5
--R (6a cosh(x)coth(x) csch(x) + 6%i a cosh(x)coth(x) )sinh(x)
--R +
--R 2 2 2 2
--R (15a cosh(x) - a )coth(x) csch(x)
--R +
--R 2 2 2 2
--R (15%i a cosh(x) - %i a )coth(x)
--R *
--R 4
--R sinh(x)
--R +
--R 2 3 2 2
--R (20a cosh(x) - 4a cosh(x))coth(x) csch(x)
--R +
--R 2 3 2 2
--R (20%i a cosh(x) - 4%i a cosh(x))coth(x)
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 2 2 2 2
--R (15a cosh(x) - 6a cosh(x) - 4a )coth(x) csch(x)
--R +
--R 2 4 2 2 2 2
--R (15%i a cosh(x) - 6%i a cosh(x) - 4%i a )coth(x)
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 2 3 2 2
--R (6a cosh(x) - 4a cosh(x) - 8a cosh(x))coth(x) csch(x)
--R +
--R 2 5 2 3 2 2
--R (6%i a cosh(x) - 4%i a cosh(x) - 8%i a cosh(x))coth(x)
--R *
--R sinh(x)
--R +
--R 2 6 2 4 2 2 2 2
--R (a cosh(x) - a cosh(x) - 4a cosh(x) + 4a )coth(x) csch(x)
--R +
--R 2 6 2 4 2 2 2 2
--R (%i a cosh(x) - %i a cosh(x) - 4%i a cosh(x) + 4%i a )coth(x)
--R *
--R +----------------+
--R \|- %i csch(x) - 1
--R /
--R 2 6 2 5
--R (csch(x) + 1)sinh(x) + (6cosh(x)csch(x) + 6cosh(x))sinh(x)
--R +
--R 2 2 2 4
--R ((15cosh(x) + 1)csch(x) + 15cosh(x) + 1)sinh(x)
--R +
--R 3 2 3 3
--R ((20cosh(x) + 4cosh(x))csch(x) + 20cosh(x) + 4cosh(x))sinh(x)
--R +
--R 4 2 2 4 2
--R ((15cosh(x) + 6cosh(x) - 6)csch(x) + 15cosh(x) + 6cosh(x) - 6)
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2 5
--R (6cosh(x) + 4cosh(x) - 12cosh(x))csch(x) + 6cosh(x)
--R +
--R 3
--R 4cosh(x) - 12cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2 6 4
--R (cosh(x) + cosh(x) - 6cosh(x) + 4)csch(x) + cosh(x) + cosh(x)
--R +
--R 2
--R - 6cosh(x) + 4
--R *
--R +-+ +------------------+ +----------------+
--R \|a \|- %i a csch(x) + a \|- %i csch(x) - 1
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 2 6
--R (- %i a csch(x) - %i a)sinh(x)
--R +
--R 2 5
--R (- 6%i a cosh(x)csch(x) - 6%i a cosh(x))sinh(x)
--R +
--R 2 2 2 4
--R ((- 15%i a cosh(x) + %i a)csch(x) - 15%i a cosh(x) + %i a)sinh(x)
--R +
--R 3 2 3
--R (- 20%i a cosh(x) + 4%i a cosh(x))csch(x) - 20%i a cosh(x)
--R +
--R 4%i a cosh(x)
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 2
--R (- 15%i a cosh(x) + 6%i a cosh(x) + 4%i a)csch(x)
--R +
--R 4 2
--R - 15%i a cosh(x) + 6%i a cosh(x) + 4%i a
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 2
--R (- 6%i a cosh(x) + 4%i a cosh(x) + 8%i a cosh(x))csch(x)
--R +
--R 5 3
--R - 6%i a cosh(x) + 4%i a cosh(x) + 8%i a cosh(x)
--R *
--R sinh(x)
--R +
--R 6 4 2 2
--R (- %i a cosh(x) + %i a cosh(x) + 4%i a cosh(x) - 4%i a)csch(x)
--R +
--R 6 4 2
--R - %i a cosh(x) + %i a cosh(x) + 4%i a cosh(x) - 4%i a
--R *
--R +------------------+ +----------------+
--R \|- %i a csch(x) + a \|- %i csch(x) - 1
--R Type: Expression(Complex(Integer))
--E 314
--S 315 of 526
t0460:= 1/(a+%i*a*csch(x))^(1/2)
--R
--R
--R 1
--R (228) -------------------
--R +----------------+
--R \|%i a csch(x) + a
--R Type: Expression(Complex(Integer))
--E 315
--S 316 of 526
r0460:= -(2^(1/2)*atan(2^(1/2)*a^(1/2)/(-a+%i*a*csch(x))^(1/2))+_
2*atan((-a+%i*a*csch(x))^(1/2)/a^(1/2)))*(-a+%i*a*csch(x))^(1/2)*_
(a+%i*a*csch(x))^(1/2)*tanh(x)/a^(3/2)
--R
--R
--R (229)
--R +----------------+
--R +----------------+ +----------------+ \|%i a csch(x) - a
--R - 2tanh(x)\|%i a csch(x) - a \|%i a csch(x) + a atan(-------------------)
--R +-+
--R \|a
--R +
--R -
--R +-+ +----------------+ +----------------+
--R \|2 tanh(x)\|%i a csch(x) - a \|%i a csch(x) + a
--R *
--R +-+ +-+
--R \|2 \|a
--R atan(-------------------)
--R +----------------+
--R \|%i a csch(x) - a
--R /
--R +-+
--R a\|a
--R Type: Expression(Complex(Integer))
--E 316
--S 317 of 526
a0460:= integrate(t0460,x)
--R
--R
--R (230)
--R [
--R -
--R log
--R +-----------------------------------------+
--R +-+ | a
--R \|a |----------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R - a cosh(x) - %i a
--R /
--R sinh(x) + cosh(x)
--R +
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x)
--R +
--R 2
--R - 2%i cosh(x) - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R - %i sinh(x) + (- 3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (- 3%i cosh(x) - 2cosh(x) + %i)sinh(x) - %i cosh(x)
--R +
--R 2
--R - cosh(x) + %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+
--R |2 +-+
--R |- \|a
--R \|a
--R *
--R log
--R 7 6
--R - 2sinh(x) + (- 14cosh(x) - 12%i)sinh(x)
--R +
--R 2 5
--R (- 42cosh(x) - 72%i cosh(x) + 10)sinh(x)
--R +
--R 3 2
--R (- 70cosh(x) - 180%i cosh(x) + 50cosh(x) + 24%i)
--R *
--R 4
--R sinh(x)
--R +
--R 4 3 2
--R - 70cosh(x) - 240%i cosh(x) + 100cosh(x)
--R +
--R 96%i cosh(x) - 16
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R - 42cosh(x) - 180%i cosh(x) + 100cosh(x)
--R +
--R 2
--R 144%i cosh(x) - 48cosh(x) - 12%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R - 14cosh(x) - 72%i cosh(x) + 50cosh(x)
--R +
--R 3 2
--R 96%i cosh(x) - 48cosh(x) - 24%i cosh(x) + 8
--R *
--R sinh(x)
--R +
--R 7 6 5
--R - 2cosh(x) - 12%i cosh(x) + 10cosh(x)
--R +
--R 4 3 2
--R 24%i cosh(x) - 16cosh(x) - 12%i cosh(x) + 8cosh(x)
--R *
--R +-+
--R |2 +-+
--R |- \|a
--R \|a
--R +
--R 7 6
--R - 4sinh(x) + (- 28cosh(x) - 12%i)sinh(x)
--R +
--R 2 5
--R (- 84cosh(x) - 72%i cosh(x) + 20)sinh(x)
--R +
--R 3 2
--R (- 140cosh(x) - 180%i cosh(x) + 100cosh(x) + 28%i)
--R *
--R 4
--R sinh(x)
--R +
--R 4 3 2
--R - 140cosh(x) - 240%i cosh(x) + 200cosh(x)
--R +
--R 112%i cosh(x) - 32
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R - 84cosh(x) - 180%i cosh(x) + 200cosh(x)
--R +
--R 2
--R 168%i cosh(x) - 96cosh(x) - 24%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R - 28cosh(x) - 72%i cosh(x) + 100cosh(x)
--R +
--R 3 2
--R 112%i cosh(x) - 96cosh(x) - 48%i cosh(x) + 16
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R - 4cosh(x) - 12%i cosh(x) + 20cosh(x) + 28%i cosh(x)
--R +
--R 3 2
--R - 32cosh(x) - 24%i cosh(x) + 16cosh(x) + 8%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 8 7
--R - 3sinh(x) + (- 24cosh(x) + 4%i)sinh(x)
--R +
--R 2 6
--R (- 84cosh(x) + 28%i cosh(x))sinh(x)
--R +
--R 3 2 5
--R (- 168cosh(x) + 84%i cosh(x) - 12%i)sinh(x)
--R +
--R 4 3
--R (- 210cosh(x) + 140%i cosh(x) - 60%i cosh(x) + 16)
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 2
--R - 168cosh(x) + 140%i cosh(x) - 120%i cosh(x)
--R +
--R 64cosh(x) + 24%i
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 3
--R - 84cosh(x) + 84%i cosh(x) - 120%i cosh(x)
--R +
--R 2
--R 96cosh(x) + 72%i cosh(x) - 20
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 4
--R - 24cosh(x) + 28%i cosh(x) - 60%i cosh(x)
--R +
--R 3 2
--R 64cosh(x) + 72%i cosh(x) - 40cosh(x) - 16%i
--R *
--R sinh(x)
--R +
--R 8 7 5 4
--R - 3cosh(x) + 4%i cosh(x) - 12%i cosh(x) + 16cosh(x)
--R +
--R 3 2
--R 24%i cosh(x) - 20cosh(x) - 16%i cosh(x) + 8
--R *
--R +-+
--R \|a
--R +
--R 8 7
--R - 2a sinh(x) + (- 16a cosh(x) + 2%i a)sinh(x)
--R +
--R 2 6
--R (- 56a cosh(x) + 14%i a cosh(x) - 2a)sinh(x)
--R +
--R 3 2
--R - 112a cosh(x) + 42%i a cosh(x) - 12a cosh(x)
--R +
--R - 6%i a
--R *
--R 5
--R sinh(x)
--R +
--R 4 3 2
--R - 140a cosh(x) + 70%i a cosh(x) - 30a cosh(x)
--R +
--R - 30%i a cosh(x) + 16a
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 3
--R - 112a cosh(x) + 70%i a cosh(x) - 40a cosh(x)
--R +
--R 2
--R - 60%i a cosh(x) + 64a cosh(x) + 12%i a
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 4
--R - 56a cosh(x) + 42%i a cosh(x) - 30a cosh(x)
--R +
--R 3 2
--R - 60%i a cosh(x) + 96a cosh(x) + 36%i a cosh(x) - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 5
--R - 16a cosh(x) + 14%i a cosh(x) - 12a cosh(x)
--R +
--R 4 3 2
--R - 30%i a cosh(x) + 64a cosh(x) + 36%i a cosh(x)
--R +
--R - 24a cosh(x) - 8%i a
--R *
--R sinh(x)
--R +
--R 8 7 6
--R - 2a cosh(x) + 2%i a cosh(x) - 2a cosh(x)
--R +
--R 5 4 3
--R - 6%i a cosh(x) + 16a cosh(x) + 12%i a cosh(x)
--R +
--R 2
--R - 12a cosh(x) - 8%i a cosh(x)
--R *
--R +-+
--R |2
--R |-
--R \|a
--R /
--R 7 6
--R 4sinh(x) + (28cosh(x) - 4%i)sinh(x)
--R +
--R 2 5
--R (84cosh(x) - 24%i cosh(x) + 4)sinh(x)
--R +
--R 3 2 4
--R (140cosh(x) - 60%i cosh(x) + 20cosh(x) + 12%i)sinh(x)
--R +
--R 4 3 2
--R 140cosh(x) - 80%i cosh(x) + 40cosh(x)
--R +
--R 48%i cosh(x) - 24
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R 84cosh(x) - 60%i cosh(x) + 40cosh(x)
--R +
--R 2
--R 72%i cosh(x) - 72cosh(x) - 16%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R 28cosh(x) - 24%i cosh(x) + 20cosh(x)
--R +
--R 3 2
--R 48%i cosh(x) - 72cosh(x) - 32%i cosh(x) + 16
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R 4cosh(x) - 4%i cosh(x) + 4cosh(x) + 12%i cosh(x)
--R +
--R 3 2
--R - 24cosh(x) - 16%i cosh(x) + 16cosh(x) + 8%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 8 7
--R sinh(x) + (8cosh(x) - 4%i)sinh(x)
--R +
--R 2 6
--R (28cosh(x) - 28%i cosh(x) - 4)sinh(x)
--R +
--R 3 2 5
--R (56cosh(x) - 84%i cosh(x) - 24cosh(x) + 4%i)sinh(x)
--R +
--R 4 3 2
--R 70cosh(x) - 140%i cosh(x) - 60cosh(x)
--R +
--R 20%i cosh(x) + 8
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 3
--R 56cosh(x) - 140%i cosh(x) - 80cosh(x)
--R +
--R 2
--R 40%i cosh(x) + 32cosh(x) + 16%i
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 4
--R 28cosh(x) - 84%i cosh(x) - 60cosh(x)
--R +
--R 3 2
--R 40%i cosh(x) + 48cosh(x) + 48%i cosh(x) - 12
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 5 4
--R 8cosh(x) - 28%i cosh(x) - 24cosh(x) + 20%i cosh(x)
--R +
--R 3 2
--R 32cosh(x) + 48%i cosh(x) - 24cosh(x) - 16%i
--R *
--R sinh(x)
--R +
--R 8 7 6 5
--R cosh(x) - 4%i cosh(x) - 4cosh(x) + 4%i cosh(x)
--R +
--R 4 3 2
--R 8cosh(x) + 16%i cosh(x) - 12cosh(x) - 16%i cosh(x) + 8
--R *
--R +-+
--R \|a
--R /
--R +-+
--R \|a
--R ,
--R
--R -
--R log
--R +-----------------------------------------+
--R +-+ | a
--R \|a |----------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R - a cosh(x) - %i a
--R /
--R sinh(x) + cosh(x)
--R +
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x)
--R +
--R 2
--R - 2%i cosh(x) - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R - %i sinh(x) + (- 3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (- 3%i cosh(x) - 2cosh(x) + %i)sinh(x) - %i cosh(x)
--R +
--R 2
--R - cosh(x) + %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+
--R |2 +-+
--R 2%i |- \|a
--R \|a
--R *
--R atan
--R +-----------------------------------------+
--R +-+ | a
--R %i\|a |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R - %i a sinh(x) - %i a cosh(x) + a
--R /
--R +-+
--R |2 +-+
--R (a sinh(x) + a cosh(x)) |- \|a
--R \|a
--R /
--R +-+
--R \|a
--R ]
--R Type: Union(List(Expression(Complex(Integer))),...)
--E 317
--S 318 of 526
m0460a:= a0460.1-r0460
--R
--R
--R (231)
--R -
--R a
--R *
--R log
--R +-----------------------------------------+
--R +-+ | a
--R \|a |----------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R - a cosh(x) - %i a
--R /
--R sinh(x) + cosh(x)
--R +
--R a
--R *
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x)
--R +
--R 2
--R - 2%i cosh(x) - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R - %i sinh(x) + (- 3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (- 3%i cosh(x) - 2cosh(x) + %i)sinh(x) - %i cosh(x)
--R +
--R 2
--R - cosh(x) + %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+
--R |2 +-+
--R a |- \|a
--R \|a
--R *
--R log
--R 7 6
--R - 2sinh(x) + (- 14cosh(x) - 12%i)sinh(x)
--R +
--R 2 5
--R (- 42cosh(x) - 72%i cosh(x) + 10)sinh(x)
--R +
--R 3 2
--R (- 70cosh(x) - 180%i cosh(x) + 50cosh(x) + 24%i)
--R *
--R 4
--R sinh(x)
--R +
--R 4 3 2
--R - 70cosh(x) - 240%i cosh(x) + 100cosh(x)
--R +
--R 96%i cosh(x) - 16
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R - 42cosh(x) - 180%i cosh(x) + 100cosh(x)
--R +
--R 2
--R 144%i cosh(x) - 48cosh(x) - 12%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R - 14cosh(x) - 72%i cosh(x) + 50cosh(x)
--R +
--R 3 2
--R 96%i cosh(x) - 48cosh(x) - 24%i cosh(x) + 8
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R - 2cosh(x) - 12%i cosh(x) + 10cosh(x) + 24%i cosh(x)
--R +
--R 3 2
--R - 16cosh(x) - 12%i cosh(x) + 8cosh(x)
--R *
--R +-+
--R |2 +-+
--R |- \|a
--R \|a
--R +
--R 7 6
--R - 4sinh(x) + (- 28cosh(x) - 12%i)sinh(x)
--R +
--R 2 5
--R (- 84cosh(x) - 72%i cosh(x) + 20)sinh(x)
--R +
--R 3 2 4
--R (- 140cosh(x) - 180%i cosh(x) + 100cosh(x) + 28%i)sinh(x)
--R +
--R 4 3 2
--R - 140cosh(x) - 240%i cosh(x) + 200cosh(x)
--R +
--R 112%i cosh(x) - 32
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R - 84cosh(x) - 180%i cosh(x) + 200cosh(x)
--R +
--R 2
--R 168%i cosh(x) - 96cosh(x) - 24%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R - 28cosh(x) - 72%i cosh(x) + 100cosh(x)
--R +
--R 3 2
--R 112%i cosh(x) - 96cosh(x) - 48%i cosh(x) + 16
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R - 4cosh(x) - 12%i cosh(x) + 20cosh(x) + 28%i cosh(x)
--R +
--R 3 2
--R - 32cosh(x) - 24%i cosh(x) + 16cosh(x) + 8%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 8 7
--R - 3sinh(x) + (- 24cosh(x) + 4%i)sinh(x)
--R +
--R 2 6
--R (- 84cosh(x) + 28%i cosh(x))sinh(x)
--R +
--R 3 2 5
--R (- 168cosh(x) + 84%i cosh(x) - 12%i)sinh(x)
--R +
--R 4 3 4
--R (- 210cosh(x) + 140%i cosh(x) - 60%i cosh(x) + 16)sinh(x)
--R +
--R 5 4 2
--R - 168cosh(x) + 140%i cosh(x) - 120%i cosh(x)
--R +
--R 64cosh(x) + 24%i
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 3
--R - 84cosh(x) + 84%i cosh(x) - 120%i cosh(x)
--R +
--R 2
--R 96cosh(x) + 72%i cosh(x) - 20
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 4
--R - 24cosh(x) + 28%i cosh(x) - 60%i cosh(x)
--R +
--R 3 2
--R 64cosh(x) + 72%i cosh(x) - 40cosh(x) - 16%i
--R *
--R sinh(x)
--R +
--R 8 7 5 4
--R - 3cosh(x) + 4%i cosh(x) - 12%i cosh(x) + 16cosh(x)
--R +
--R 3 2
--R 24%i cosh(x) - 20cosh(x) - 16%i cosh(x) + 8
--R *
--R +-+
--R \|a
--R +
--R 8 7
--R - 2a sinh(x) + (- 16a cosh(x) + 2%i a)sinh(x)
--R +
--R 2 6
--R (- 56a cosh(x) + 14%i a cosh(x) - 2a)sinh(x)
--R +
--R 3 2
--R (- 112a cosh(x) + 42%i a cosh(x) - 12a cosh(x) - 6%i a)
--R *
--R 5
--R sinh(x)
--R +
--R 4 3 2
--R - 140a cosh(x) + 70%i a cosh(x) - 30a cosh(x)
--R +
--R - 30%i a cosh(x) + 16a
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 3
--R - 112a cosh(x) + 70%i a cosh(x) - 40a cosh(x)
--R +
--R 2
--R - 60%i a cosh(x) + 64a cosh(x) + 12%i a
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 4
--R - 56a cosh(x) + 42%i a cosh(x) - 30a cosh(x)
--R +
--R 3 2
--R - 60%i a cosh(x) + 96a cosh(x) + 36%i a cosh(x) - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 5
--R - 16a cosh(x) + 14%i a cosh(x) - 12a cosh(x)
--R +
--R 4 3 2
--R - 30%i a cosh(x) + 64a cosh(x) + 36%i a cosh(x)
--R +
--R - 24a cosh(x) - 8%i a
--R *
--R sinh(x)
--R +
--R 8 7 6
--R - 2a cosh(x) + 2%i a cosh(x) - 2a cosh(x)
--R +
--R 5 4 3
--R - 6%i a cosh(x) + 16a cosh(x) + 12%i a cosh(x)
--R +
--R 2
--R - 12a cosh(x) - 8%i a cosh(x)
--R *
--R +-+
--R |2
--R |-
--R \|a
--R /
--R 7 6
--R 4sinh(x) + (28cosh(x) - 4%i)sinh(x)
--R +
--R 2 5
--R (84cosh(x) - 24%i cosh(x) + 4)sinh(x)
--R +
--R 3 2 4
--R (140cosh(x) - 60%i cosh(x) + 20cosh(x) + 12%i)sinh(x)
--R +
--R 4 3 2
--R 140cosh(x) - 80%i cosh(x) + 40cosh(x) + 48%i cosh(x)
--R +
--R - 24
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3 2
--R 84cosh(x) - 60%i cosh(x) + 40cosh(x) + 72%i cosh(x)
--R +
--R - 72cosh(x) - 16%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4 3
--R 28cosh(x) - 24%i cosh(x) + 20cosh(x) + 48%i cosh(x)
--R +
--R 2
--R - 72cosh(x) - 32%i cosh(x) + 16
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R 4cosh(x) - 4%i cosh(x) + 4cosh(x) + 12%i cosh(x)
--R +
--R 3 2
--R - 24cosh(x) - 16%i cosh(x) + 16cosh(x) + 8%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 8 7
--R sinh(x) + (8cosh(x) - 4%i)sinh(x)
--R +
--R 2 6
--R (28cosh(x) - 28%i cosh(x) - 4)sinh(x)
--R +
--R 3 2 5
--R (56cosh(x) - 84%i cosh(x) - 24cosh(x) + 4%i)sinh(x)
--R +
--R 4 3 2
--R 70cosh(x) - 140%i cosh(x) - 60cosh(x) + 20%i cosh(x)
--R +
--R 8
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 3 2
--R 56cosh(x) - 140%i cosh(x) - 80cosh(x) + 40%i cosh(x)
--R +
--R 32cosh(x) + 16%i
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 4 3
--R 28cosh(x) - 84%i cosh(x) - 60cosh(x) + 40%i cosh(x)
--R +
--R 2
--R 48cosh(x) + 48%i cosh(x) - 12
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 5 4
--R 8cosh(x) - 28%i cosh(x) - 24cosh(x) + 20%i cosh(x)
--R +
--R 3 2
--R 32cosh(x) + 48%i cosh(x) - 24cosh(x) - 16%i
--R *
--R sinh(x)
--R +
--R 8 7 6 5
--R cosh(x) - 4%i cosh(x) - 4cosh(x) + 4%i cosh(x)
--R +
--R 4 3 2
--R 8cosh(x) + 16%i cosh(x) - 12cosh(x) - 16%i cosh(x) + 8
--R *
--R +-+
--R \|a
--R +
--R +----------------+
--R +----------------+ +----------------+ \|%i a csch(x) - a
--R 2tanh(x)\|%i a csch(x) - a \|%i a csch(x) + a atan(-------------------)
--R +-+
--R \|a
--R +
--R +-+ +-+
--R +-+ +----------------+ +----------------+ \|2 \|a
--R \|2 tanh(x)\|%i a csch(x) - a \|%i a csch(x) + a atan(-------------------)
--R +----------------+
--R \|%i a csch(x) - a
--R /
--R +-+
--R a\|a
--R Type: Expression(Complex(Integer))
--E 318
--S 319 of 526
--d0460a:= D(m0460a,x)
--E 319
--S 320 of 526
m0460b:= a0460.2-r0460
--R
--R
--R (232)
--R -
--R a
--R *
--R log
--R +-----------------------------------------+
--R +-+ | a
--R \|a |----------------------------------------- - a sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R - a cosh(x) - %i a
--R /
--R sinh(x) + cosh(x)
--R +
--R a
--R *
--R log
--R 3 2
--R sinh(x) + (3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (3cosh(x) - 4%i cosh(x) - 1)sinh(x) + cosh(x)
--R +
--R 2
--R - 2%i cosh(x) - cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R - %i sinh(x) + (- 3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (- 3%i cosh(x) - 2cosh(x) + %i)sinh(x) - %i cosh(x)
--R +
--R 2
--R - cosh(x) + %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +----------------+
--R +----------------+ +----------------+ \|%i a csch(x) - a
--R 2tanh(x)\|%i a csch(x) - a \|%i a csch(x) + a atan(-------------------)
--R +-+
--R \|a
--R +
--R +-+
--R |2 +-+
--R 2%i a |- \|a
--R \|a
--R *
--R atan
--R +-----------------------------------------+
--R +-+ | a
--R %i\|a |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R - %i a sinh(x) - %i a cosh(x) + a
--R /
--R +-+
--R |2 +-+
--R (a sinh(x) + a cosh(x)) |- \|a
--R \|a
--R +
--R +-+ +-+
--R +-+ +----------------+ +----------------+ \|2 \|a
--R \|2 tanh(x)\|%i a csch(x) - a \|%i a csch(x) + a atan(-------------------)
--R +----------------+
--R \|%i a csch(x) - a
--R /
--R +-+
--R a\|a
--R Type: Expression(Complex(Integer))
--E 320
--S 321 of 526
--d0460b:= D(m0460b,x)
--E 321
--S 322 of 526
t0461:= 1/(a-%i*a*csch(x))^(1/2)
--R
--R
--R 1
--R (233) ---------------------
--R +------------------+
--R \|- %i a csch(x) + a
--R Type: Expression(Complex(Integer))
--E 322
--S 323 of 526
r0461:= -(2^(1/2)*atan(2^(1/2)*a^(1/2)/(-a-%i*a*csch(x))^(1/2))+_
2*atan((-a-%i*a*csch(x))^(1/2)/a^(1/2)))*(-a-%i*a*csch(x))^(1/2)*_
(a-%i*a*csch(x))^(1/2)*tanh(x)/a^(3/2)
--R
--R
--R (234)
--R -
--R +------------------+ +------------------+
--R 2tanh(x)\|- %i a csch(x) - a \|- %i a csch(x) + a
--R *
--R +------------------+
--R \|- %i a csch(x) - a
--R atan(---------------------)
--R +-+
--R \|a
--R +
--R -
--R +-+ +------------------+ +------------------+
--R \|2 tanh(x)\|- %i a csch(x) - a \|- %i a csch(x) + a
--R *
--R +-+ +-+
--R \|2 \|a
--R atan(---------------------)
--R +------------------+
--R \|- %i a csch(x) - a
--R /
--R +-+
--R a\|a
--R Type: Expression(Complex(Integer))
--E 323
--S 324 of 526
a0461:= integrate(t0461,x)
--R
--R
--R (235)
--R [
--R log
--R +-----------------------------------------+
--R +-+ | a
--R \|a |----------------------------------------- + a sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R a cosh(x) - %i a
--R /
--R sinh(x) + cosh(x)
--R +
--R +-+
--R |2 +-+
--R |- \|a
--R \|a
--R *
--R log
--R 7 6
--R 2sinh(x) + (14cosh(x) - 12%i)sinh(x)
--R +
--R 2 5
--R (42cosh(x) - 72%i cosh(x) - 10)sinh(x)
--R +
--R 3 2
--R (70cosh(x) - 180%i cosh(x) - 50cosh(x) + 24%i)
--R *
--R 4
--R sinh(x)
--R +
--R 4 3 2
--R 70cosh(x) - 240%i cosh(x) - 100cosh(x)
--R +
--R 96%i cosh(x) + 16
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R 42cosh(x) - 180%i cosh(x) - 100cosh(x)
--R +
--R 2
--R 144%i cosh(x) + 48cosh(x) - 12%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R 14cosh(x) - 72%i cosh(x) - 50cosh(x)
--R +
--R 3 2
--R 96%i cosh(x) + 48cosh(x) - 24%i cosh(x) - 8
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R 2cosh(x) - 12%i cosh(x) - 10cosh(x) + 24%i cosh(x)
--R +
--R 3 2
--R 16cosh(x) - 12%i cosh(x) - 8cosh(x)
--R *
--R +-+
--R |2 +-+
--R |- \|a
--R \|a
--R +
--R 7 6
--R - 4sinh(x) + (- 28cosh(x) + 12%i)sinh(x)
--R +
--R 2 5
--R (- 84cosh(x) + 72%i cosh(x) + 20)sinh(x)
--R +
--R 3 2
--R (- 140cosh(x) + 180%i cosh(x) + 100cosh(x) - 28%i)
--R *
--R 4
--R sinh(x)
--R +
--R 4 3 2
--R - 140cosh(x) + 240%i cosh(x) + 200cosh(x)
--R +
--R - 112%i cosh(x) - 32
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R - 84cosh(x) + 180%i cosh(x) + 200cosh(x)
--R +
--R 2
--R - 168%i cosh(x) - 96cosh(x) + 24%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R - 28cosh(x) + 72%i cosh(x) + 100cosh(x)
--R +
--R 3 2
--R - 112%i cosh(x) - 96cosh(x) + 48%i cosh(x) + 16
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R - 4cosh(x) + 12%i cosh(x) + 20cosh(x) - 28%i cosh(x)
--R +
--R 3 2
--R - 32cosh(x) + 24%i cosh(x) + 16cosh(x) - 8%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 8 7
--R 3sinh(x) + (24cosh(x) + 4%i)sinh(x)
--R +
--R 2 6
--R (84cosh(x) + 28%i cosh(x))sinh(x)
--R +
--R 3 2 5
--R (168cosh(x) + 84%i cosh(x) - 12%i)sinh(x)
--R +
--R 4 3 4
--R (210cosh(x) + 140%i cosh(x) - 60%i cosh(x) - 16)sinh(x)
--R +
--R 5 4 2
--R 168cosh(x) + 140%i cosh(x) - 120%i cosh(x)
--R +
--R - 64cosh(x) + 24%i
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 3
--R 84cosh(x) + 84%i cosh(x) - 120%i cosh(x)
--R +
--R 2
--R - 96cosh(x) + 72%i cosh(x) + 20
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 4
--R 24cosh(x) + 28%i cosh(x) - 60%i cosh(x)
--R +
--R 3 2
--R - 64cosh(x) + 72%i cosh(x) + 40cosh(x) - 16%i
--R *
--R sinh(x)
--R +
--R 8 7 5 4
--R 3cosh(x) + 4%i cosh(x) - 12%i cosh(x) - 16cosh(x)
--R +
--R 3 2
--R 24%i cosh(x) + 20cosh(x) - 16%i cosh(x) - 8
--R *
--R +-+
--R \|a
--R +
--R 8 7
--R - 2a sinh(x) + (- 16a cosh(x) - 2%i a)sinh(x)
--R +
--R 2 6
--R (- 56a cosh(x) - 14%i a cosh(x) - 2a)sinh(x)
--R +
--R 3 2
--R - 112a cosh(x) - 42%i a cosh(x) - 12a cosh(x)
--R +
--R 6%i a
--R *
--R 5
--R sinh(x)
--R +
--R 4 3 2
--R - 140a cosh(x) - 70%i a cosh(x) - 30a cosh(x)
--R +
--R 30%i a cosh(x) + 16a
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 3
--R - 112a cosh(x) - 70%i a cosh(x) - 40a cosh(x)
--R +
--R 2
--R 60%i a cosh(x) + 64a cosh(x) - 12%i a
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 4
--R - 56a cosh(x) - 42%i a cosh(x) - 30a cosh(x)
--R +
--R 3 2
--R 60%i a cosh(x) + 96a cosh(x) - 36%i a cosh(x) - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 5
--R - 16a cosh(x) - 14%i a cosh(x) - 12a cosh(x)
--R +
--R 4 3 2
--R 30%i a cosh(x) + 64a cosh(x) - 36%i a cosh(x)
--R +
--R - 24a cosh(x) + 8%i a
--R *
--R sinh(x)
--R +
--R 8 7 6
--R - 2a cosh(x) - 2%i a cosh(x) - 2a cosh(x)
--R +
--R 5 4 3
--R 6%i a cosh(x) + 16a cosh(x) - 12%i a cosh(x)
--R +
--R 2
--R - 12a cosh(x) + 8%i a cosh(x)
--R *
--R +-+
--R |2
--R |-
--R \|a
--R /
--R 7 6
--R 4sinh(x) + (28cosh(x) + 4%i)sinh(x)
--R +
--R 2 5
--R (84cosh(x) + 24%i cosh(x) + 4)sinh(x)
--R +
--R 3 2 4
--R (140cosh(x) + 60%i cosh(x) + 20cosh(x) - 12%i)sinh(x)
--R +
--R 4 3 2
--R 140cosh(x) + 80%i cosh(x) + 40cosh(x)
--R +
--R - 48%i cosh(x) - 24
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R 84cosh(x) + 60%i cosh(x) + 40cosh(x)
--R +
--R 2
--R - 72%i cosh(x) - 72cosh(x) + 16%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R 28cosh(x) + 24%i cosh(x) + 20cosh(x)
--R +
--R 3 2
--R - 48%i cosh(x) - 72cosh(x) + 32%i cosh(x) + 16
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R 4cosh(x) + 4%i cosh(x) + 4cosh(x) - 12%i cosh(x)
--R +
--R 3 2
--R - 24cosh(x) + 16%i cosh(x) + 16cosh(x) - 8%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 8 7
--R - sinh(x) + (- 8cosh(x) - 4%i)sinh(x)
--R +
--R 2 6
--R (- 28cosh(x) - 28%i cosh(x) + 4)sinh(x)
--R +
--R 3 2 5
--R (- 56cosh(x) - 84%i cosh(x) + 24cosh(x) + 4%i)sinh(x)
--R +
--R 4 3 2
--R - 70cosh(x) - 140%i cosh(x) + 60cosh(x)
--R +
--R 20%i cosh(x) - 8
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 3
--R - 56cosh(x) - 140%i cosh(x) + 80cosh(x)
--R +
--R 2
--R 40%i cosh(x) - 32cosh(x) + 16%i
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 4
--R - 28cosh(x) - 84%i cosh(x) + 60cosh(x)
--R +
--R 3 2
--R 40%i cosh(x) - 48cosh(x) + 48%i cosh(x) + 12
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 5
--R - 8cosh(x) - 28%i cosh(x) + 24cosh(x)
--R +
--R 4 3 2
--R 20%i cosh(x) - 32cosh(x) + 48%i cosh(x) + 24cosh(x)
--R +
--R - 16%i
--R *
--R sinh(x)
--R +
--R 8 7 6 5
--R - cosh(x) - 4%i cosh(x) + 4cosh(x) + 4%i cosh(x)
--R +
--R 4 3 2
--R - 8cosh(x) + 16%i cosh(x) + 12cosh(x) - 16%i cosh(x) - 8
--R *
--R +-+
--R \|a
--R +
--R -
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R %i sinh(x) + (3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (3%i cosh(x) - 2cosh(x) - %i)sinh(x) + %i cosh(x)
--R +
--R 2
--R - cosh(x) - %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R /
--R +-+
--R \|a
--R ,
--R
--R log
--R +-----------------------------------------+
--R +-+ | a
--R \|a |----------------------------------------- + a sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R a cosh(x) - %i a
--R /
--R sinh(x) + cosh(x)
--R +
--R -
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R %i sinh(x) + (3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (3%i cosh(x) - 2cosh(x) - %i)sinh(x) + %i cosh(x)
--R +
--R 2
--R - cosh(x) - %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+
--R |2 +-+
--R 2%i |- \|a
--R \|a
--R *
--R atan
--R +-----------------------------------------+
--R +-+ | a
--R %i\|a |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R %i a sinh(x) + %i a cosh(x) + a
--R /
--R +-+
--R |2 +-+
--R (a sinh(x) + a cosh(x)) |- \|a
--R \|a
--R /
--R +-+
--R \|a
--R ]
--R Type: Union(List(Expression(Complex(Integer))),...)
--E 324
--S 325 of 526
m0461a:= a0461.1-r0461
--R
--R
--R (236)
--R a
--R *
--R log
--R +-----------------------------------------+
--R +-+ | a
--R \|a |----------------------------------------- + a sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R a cosh(x) - %i a
--R /
--R sinh(x) + cosh(x)
--R +
--R +-+
--R |2 +-+
--R a |- \|a
--R \|a
--R *
--R log
--R 7 6
--R 2sinh(x) + (14cosh(x) - 12%i)sinh(x)
--R +
--R 2 5
--R (42cosh(x) - 72%i cosh(x) - 10)sinh(x)
--R +
--R 3 2 4
--R (70cosh(x) - 180%i cosh(x) - 50cosh(x) + 24%i)sinh(x)
--R +
--R 4 3 2
--R 70cosh(x) - 240%i cosh(x) - 100cosh(x)
--R +
--R 96%i cosh(x) + 16
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R 42cosh(x) - 180%i cosh(x) - 100cosh(x)
--R +
--R 2
--R 144%i cosh(x) + 48cosh(x) - 12%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R 14cosh(x) - 72%i cosh(x) - 50cosh(x)
--R +
--R 3 2
--R 96%i cosh(x) + 48cosh(x) - 24%i cosh(x) - 8
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R 2cosh(x) - 12%i cosh(x) - 10cosh(x) + 24%i cosh(x)
--R +
--R 3 2
--R 16cosh(x) - 12%i cosh(x) - 8cosh(x)
--R *
--R +-+
--R |2 +-+
--R |- \|a
--R \|a
--R +
--R 7 6
--R - 4sinh(x) + (- 28cosh(x) + 12%i)sinh(x)
--R +
--R 2 5
--R (- 84cosh(x) + 72%i cosh(x) + 20)sinh(x)
--R +
--R 3 2 4
--R (- 140cosh(x) + 180%i cosh(x) + 100cosh(x) - 28%i)sinh(x)
--R +
--R 4 3 2
--R - 140cosh(x) + 240%i cosh(x) + 200cosh(x)
--R +
--R - 112%i cosh(x) - 32
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3
--R - 84cosh(x) + 180%i cosh(x) + 200cosh(x)
--R +
--R 2
--R - 168%i cosh(x) - 96cosh(x) + 24%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4
--R - 28cosh(x) + 72%i cosh(x) + 100cosh(x)
--R +
--R 3 2
--R - 112%i cosh(x) - 96cosh(x) + 48%i cosh(x) + 16
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R - 4cosh(x) + 12%i cosh(x) + 20cosh(x) - 28%i cosh(x)
--R +
--R 3 2
--R - 32cosh(x) + 24%i cosh(x) + 16cosh(x) - 8%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 8 7
--R 3sinh(x) + (24cosh(x) + 4%i)sinh(x)
--R +
--R 2 6
--R (84cosh(x) + 28%i cosh(x))sinh(x)
--R +
--R 3 2 5
--R (168cosh(x) + 84%i cosh(x) - 12%i)sinh(x)
--R +
--R 4 3 4
--R (210cosh(x) + 140%i cosh(x) - 60%i cosh(x) - 16)sinh(x)
--R +
--R 5 4 2
--R 168cosh(x) + 140%i cosh(x) - 120%i cosh(x)
--R +
--R - 64cosh(x) + 24%i
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 3 2
--R 84cosh(x) + 84%i cosh(x) - 120%i cosh(x) - 96cosh(x)
--R +
--R 72%i cosh(x) + 20
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 4 3
--R 24cosh(x) + 28%i cosh(x) - 60%i cosh(x) - 64cosh(x)
--R +
--R 2
--R 72%i cosh(x) + 40cosh(x) - 16%i
--R *
--R sinh(x)
--R +
--R 8 7 5 4
--R 3cosh(x) + 4%i cosh(x) - 12%i cosh(x) - 16cosh(x)
--R +
--R 3 2
--R 24%i cosh(x) + 20cosh(x) - 16%i cosh(x) - 8
--R *
--R +-+
--R \|a
--R +
--R 8 7
--R - 2a sinh(x) + (- 16a cosh(x) - 2%i a)sinh(x)
--R +
--R 2 6
--R (- 56a cosh(x) - 14%i a cosh(x) - 2a)sinh(x)
--R +
--R 3 2
--R (- 112a cosh(x) - 42%i a cosh(x) - 12a cosh(x) + 6%i a)
--R *
--R 5
--R sinh(x)
--R +
--R 4 3 2
--R - 140a cosh(x) - 70%i a cosh(x) - 30a cosh(x)
--R +
--R 30%i a cosh(x) + 16a
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 3
--R - 112a cosh(x) - 70%i a cosh(x) - 40a cosh(x)
--R +
--R 2
--R 60%i a cosh(x) + 64a cosh(x) - 12%i a
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 4
--R - 56a cosh(x) - 42%i a cosh(x) - 30a cosh(x)
--R +
--R 3 2
--R 60%i a cosh(x) + 96a cosh(x) - 36%i a cosh(x) - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 5
--R - 16a cosh(x) - 14%i a cosh(x) - 12a cosh(x)
--R +
--R 4 3 2
--R 30%i a cosh(x) + 64a cosh(x) - 36%i a cosh(x)
--R +
--R - 24a cosh(x) + 8%i a
--R *
--R sinh(x)
--R +
--R 8 7 6
--R - 2a cosh(x) - 2%i a cosh(x) - 2a cosh(x)
--R +
--R 5 4 3
--R 6%i a cosh(x) + 16a cosh(x) - 12%i a cosh(x)
--R +
--R 2
--R - 12a cosh(x) + 8%i a cosh(x)
--R *
--R +-+
--R |2
--R |-
--R \|a
--R /
--R 7 6
--R 4sinh(x) + (28cosh(x) + 4%i)sinh(x)
--R +
--R 2 5
--R (84cosh(x) + 24%i cosh(x) + 4)sinh(x)
--R +
--R 3 2 4
--R (140cosh(x) + 60%i cosh(x) + 20cosh(x) - 12%i)sinh(x)
--R +
--R 4 3 2
--R 140cosh(x) + 80%i cosh(x) + 40cosh(x) - 48%i cosh(x)
--R +
--R - 24
--R *
--R 3
--R sinh(x)
--R +
--R 5 4 3 2
--R 84cosh(x) + 60%i cosh(x) + 40cosh(x) - 72%i cosh(x)
--R +
--R - 72cosh(x) + 16%i
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 4 3
--R 28cosh(x) + 24%i cosh(x) + 20cosh(x) - 48%i cosh(x)
--R +
--R 2
--R - 72cosh(x) + 32%i cosh(x) + 16
--R *
--R sinh(x)
--R +
--R 7 6 5 4
--R 4cosh(x) + 4%i cosh(x) + 4cosh(x) - 12%i cosh(x)
--R +
--R 3 2
--R - 24cosh(x) + 16%i cosh(x) + 16cosh(x) - 8%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 8 7
--R - sinh(x) + (- 8cosh(x) - 4%i)sinh(x)
--R +
--R 2 6
--R (- 28cosh(x) - 28%i cosh(x) + 4)sinh(x)
--R +
--R 3 2 5
--R (- 56cosh(x) - 84%i cosh(x) + 24cosh(x) + 4%i)sinh(x)
--R +
--R 4 3 2
--R - 70cosh(x) - 140%i cosh(x) + 60cosh(x)
--R +
--R 20%i cosh(x) - 8
--R *
--R 4
--R sinh(x)
--R +
--R 5 4 3
--R - 56cosh(x) - 140%i cosh(x) + 80cosh(x)
--R +
--R 2
--R 40%i cosh(x) - 32cosh(x) + 16%i
--R *
--R 3
--R sinh(x)
--R +
--R 6 5 4
--R - 28cosh(x) - 84%i cosh(x) + 60cosh(x)
--R +
--R 3 2
--R 40%i cosh(x) - 48cosh(x) + 48%i cosh(x) + 12
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 5 4
--R - 8cosh(x) - 28%i cosh(x) + 24cosh(x) + 20%i cosh(x)
--R +
--R 3 2
--R - 32cosh(x) + 48%i cosh(x) + 24cosh(x) - 16%i
--R *
--R sinh(x)
--R +
--R 8 7 6 5
--R - cosh(x) - 4%i cosh(x) + 4cosh(x) + 4%i cosh(x)
--R +
--R 4 3 2
--R - 8cosh(x) + 16%i cosh(x) + 12cosh(x) - 16%i cosh(x) - 8
--R *
--R +-+
--R \|a
--R +
--R -
--R a
--R *
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R %i sinh(x) + (3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (3%i cosh(x) - 2cosh(x) - %i)sinh(x) + %i cosh(x)
--R +
--R 2
--R - cosh(x) - %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +------------------+ +------------------+
--R 2tanh(x)\|- %i a csch(x) - a \|- %i a csch(x) + a
--R *
--R +------------------+
--R \|- %i a csch(x) - a
--R atan(---------------------)
--R +-+
--R \|a
--R +
--R +-+ +------------------+ +------------------+
--R \|2 tanh(x)\|- %i a csch(x) - a \|- %i a csch(x) + a
--R *
--R +-+ +-+
--R \|2 \|a
--R atan(---------------------)
--R +------------------+
--R \|- %i a csch(x) - a
--R /
--R +-+
--R a\|a
--R Type: Expression(Complex(Integer))
--E 325
--S 326 of 526
--d0461a:= D(m0461a,x)
--E 326
--S 327 of 526
m0461b:= a0461.2-r0461
--R
--R
--R (237)
--R a
--R *
--R log
--R +-----------------------------------------+
--R +-+ | a
--R \|a |----------------------------------------- + a sinh(x)
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R a cosh(x) - %i a
--R /
--R sinh(x) + cosh(x)
--R +
--R -
--R a
--R *
--R log
--R 3 2
--R - sinh(x) + (- 3cosh(x) - 2%i)sinh(x)
--R +
--R 2 3
--R (- 3cosh(x) - 4%i cosh(x) + 1)sinh(x) - cosh(x)
--R +
--R 2
--R - 2%i cosh(x) + cosh(x) + 2%i
--R *
--R +-----------------------------------------+
--R | a
--R |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R 3 2
--R %i sinh(x) + (3%i cosh(x) - 1)sinh(x)
--R +
--R 2 3
--R (3%i cosh(x) - 2cosh(x) - %i)sinh(x) + %i cosh(x)
--R +
--R 2
--R - cosh(x) - %i cosh(x) + 2
--R *
--R +-+
--R \|a
--R /
--R 4 3 2 2
--R sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) - 1)sinh(x)
--R +
--R 3 4 2
--R (4cosh(x) - 2cosh(x))sinh(x) + cosh(x) - cosh(x)
--R +
--R +-+
--R |2 +-+
--R 2%i a |- \|a
--R \|a
--R *
--R atan
--R +-----------------------------------------+
--R +-+ | a
--R %i\|a |-----------------------------------------
--R | 2 2
--R \|sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1
--R +
--R %i a sinh(x) + %i a cosh(x) + a
--R /
--R +-+
--R |2 +-+
--R (a sinh(x) + a cosh(x)) |- \|a
--R \|a
--R +
--R +------------------+ +------------------+
--R 2tanh(x)\|- %i a csch(x) - a \|- %i a csch(x) + a
--R *
--R +------------------+
--R \|- %i a csch(x) - a
--R atan(---------------------)
--R +-+
--R \|a
--R +
--R +-+ +------------------+ +------------------+
--R \|2 tanh(x)\|- %i a csch(x) - a \|- %i a csch(x) + a
--R *
--R +-+ +-+
--R \|2 \|a
--R atan(---------------------)
--R +------------------+
--R \|- %i a csch(x) - a
--R /
--R +-+
--R a\|a
--R Type: Expression(Complex(Integer))
--E 327
--S 328 of 526
--d0461b:= D(m0461b,x)
--E 328
--S 329 of 526
t0462:= 1/(3+5*%i*csch(x))
--R
--R
--R %i
--R (238) - --------------
--R 5csch(x) - 3%i
--R Type: Expression(Complex(Integer))
--E 329
--S 330 of 526
r0462:= 1/3*x+5/6*%i*atan(-3/4+5/4*%i*tanh(1/2*x))
--R
--R
--R 5 5 1 3 1
--R (239) - %i atan(- %i tanh(- x) - -) + - x
--R 6 4 2 4 3
--R Type: Expression(Complex(Fraction(Integer)))
--E 330
--S 331 of 526
a0462:= integrate(t0462,x)
--R
--R
--R (240)
--R - 5log(3sinh(x) + 3cosh(x) + %i) + 5log(sinh(x) + cosh(x) + 3%i) + 4x
--R ---------------------------------------------------------------------
--R 12
--R Type: Union(Expression(Complex(Integer)),...)
--E 331
--S 332 of 526
m0462:= a0462-r0462
--R
--R
--R (241)
--R 5 5
--R - -- log(3sinh(x) + 3cosh(x) + %i) + -- log(sinh(x) + cosh(x) + 3%i)
--R 12 12
--R +
--R 5 5 1 3
--R - - %i atan(- %i tanh(- x) - -)
--R 6 4 2 4
--R Type: Expression(Complex(Fraction(Integer)))
--E 332
--S 333 of 526
d0462:= D(m0462,x)
--R
--R
--R (242)
--R 1 2 2 1 2 1 1 2
--R (- sinh(x) + - cosh(x)sinh(x) + - cosh(x) - -)tanh(- x)
--R 3 3 3 3 2
--R +
--R 4 4 1 1 2 2
--R (- sinh(x) + - cosh(x))tanh(- x) - - sinh(x) - - cosh(x)sinh(x)
--R 3 3 2 3 3
--R +
--R 1 2 1
--R - - cosh(x) + -
--R 3 3
--R /
--R 2 10 2 10
--R (sinh(x) + (2cosh(x) + -- %i)sinh(x) + cosh(x) + -- %i cosh(x) - 1)
--R 3 3
--R *
--R 1 2
--R tanh(- x)
--R 2
--R +
--R 6 2 12 6 2
--R - %i sinh(x) + (-- %i cosh(x) - 4)sinh(x) + - %i cosh(x) - 4cosh(x)
--R 5 5 5
--R +
--R 6
--R - - %i
--R 5
--R *
--R 1
--R tanh(- x)
--R 2
--R +
--R 2 10 2 10
--R - sinh(x) + (- 2cosh(x) - -- %i)sinh(x) - cosh(x) - -- %i cosh(x) + 1
--R 3 3
--R Type: Expression(Complex(Fraction(Integer)))
--E 333
--S 334 of 526
t0463:= 1/(a+b*csch(x))
--R
--R
--R 1
--R (243) -------------
--R b csch(x) + a
--R Type: Expression(Integer)
--E 334
--S 335 of 526
r0463:= x/a+2*b*atanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/a/(a^2+b^2)^(1/2)
--R
--R
--R x
--R b tanh(-) - a +-------+
--R 2 | 2 2
--R - 2b atanh(-------------) + x\|b + a
--R +-------+
--R | 2 2
--R \|b + a
--R (244) ---------------------------------------
--R +-------+
--R | 2 2
--R a\|b + a
--R Type: Expression(Integer)
--E 335
--S 336 of 526
a0463:= integrate(t0463,x)
--R
--R
--R (245)
--R b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b + a
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 2 3 2 3 3 2
--R (2a b + 2a )sinh(x) + (2a b + 2a )cosh(x) + 2b + 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R - a
--R +
--R +-------+
--R | 2 2
--R x\|b + a
--R /
--R +-------+
--R | 2 2
--R a\|b + a
--R Type: Union(Expression(Integer),...)
--E 336
--S 337 of 526
m0463:= a0463-r0463
--R
--R
--R (246)
--R b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b + a
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 2 3 2 3 3 2
--R (2a b + 2a )sinh(x) + (2a b + 2a )cosh(x) + 2b + 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R - a
--R +
--R x
--R b tanh(-) - a
--R 2
--R 2b atanh(-------------)
--R +-------+
--R | 2 2
--R \|b + a
--R /
--R +-------+
--R | 2 2
--R a\|b + a
--R Type: Expression(Integer)
--E 337
--S 338 of 526
d0463:= D(m0463,x)
--R
--R
--R (247)
--R 2 2 x 2
--R (b sinh(x) + 2b cosh(x)sinh(x) + b cosh(x) - b)tanh(-)
--R 2
--R +
--R x 2
--R (4b sinh(x) + 4b cosh(x))tanh(-) - b sinh(x) - 2b cosh(x)sinh(x)
--R 2
--R +
--R 2
--R - b cosh(x) + b
--R /
--R 2 2 2
--R a b sinh(x) + (2a b cosh(x) + 2b )sinh(x) + a b cosh(x)
--R +
--R 2
--R 2b cosh(x) - a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 2 2 2 2 2
--R - 2a sinh(x) + (- 4a cosh(x) - 4a b)sinh(x) - 2a cosh(x)
--R +
--R 2
--R - 4a b cosh(x) + 2a
--R *
--R x
--R tanh(-)
--R 2
--R +
--R 2 2 2
--R - a b sinh(x) + (- 2a b cosh(x) - 2b )sinh(x) - a b cosh(x)
--R +
--R 2
--R - 2b cosh(x) + a b
--R Type: Expression(Integer)
--E 338
--S 339 of 526
t0464:= 1/(a+b*csch(x))^2
--R
--R
--R 1
--R (248) ------------------------------
--R 2 2 2
--R b csch(x) + 2a b csch(x) + a
--R Type: Expression(Integer)
--E 339
--S 340 of 526
r0464:= x/a^2-2*b^3*atanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/a^2/_
(a^2+b^2)^(3/2)+4*b*atanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/_
a^2/(a^2+b^2)^(1/2)-b^2*cosh(x)/a/(a^2+b^2)/(b+a*sinh(x))
--R
--R
--R (249)
--R x
--R b tanh(-) - a
--R 3 3 4 2 2 2
--R ((- 2a b - 4a b)sinh(x) - 2b - 4a b )atanh(-------------)
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R +-------+
--R 2 3 2 3 2 | 2 2
--R ((a b + a )x sinh(x) - a b cosh(x) + (b + a b)x)\|b + a
--R /
--R +-------+
--R 3 2 5 2 3 4 | 2 2
--R ((a b + a )sinh(x) + a b + a b)\|b + a
--R Type: Expression(Integer)
--E 340
--S 341 of 526
a0464:= integrate(t0464,x)
--R
--R
--R (250)
--R 3 3 2 3 3 4 2 2
--R (a b + 2a b)sinh(x) + ((2a b + 4a b)cosh(x) + 2b + 4a b )sinh(x)
--R +
--R 3 3 2 4 2 2 3 3
--R (a b + 2a b)cosh(x) + (2b + 4a b )cosh(x) - a b - 2a b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b + a
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 2 3 2 3 3 2
--R (2a b + 2a )sinh(x) + (2a b + 2a )cosh(x) + 2b + 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R - a
--R +
--R 2 3 2
--R (a b + a )x sinh(x)
--R +
--R 2 3 3 2 3
--R ((2a b + 2a )x cosh(x) + (2b + 2a b)x + 2b )sinh(x)
--R +
--R 2 3 2 3 2 3 2 3
--R (a b + a )x cosh(x) + ((2b + 2a b)x + 2b )cosh(x) + (- a b - a )x
--R +
--R 2
--R - 2a b
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R /
--R 3 2 5 2 3 2 5 2 3 4
--R (a b + a )sinh(x) + ((2a b + 2a )cosh(x) + 2a b + 2a b)sinh(x)
--R +
--R 3 2 5 2 2 3 4 3 2 5
--R (a b + a )cosh(x) + (2a b + 2a b)cosh(x) - a b - a
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R Type: Union(Expression(Integer),...)
--E 341
--S 342 of 526
m0464:= a0464-r0464
--R
--R
--R (251)
--R 2 3 4 3
--R (a b + 2a b)sinh(x)
--R +
--R 2 3 4 4 3 2 2
--R ((2a b + 4a b)cosh(x) + 3a b + 6a b )sinh(x)
--R +
--R 2 3 4 2 4 3 2 5 2 3
--R (a b + 2a b)cosh(x) + (4a b + 8a b )cosh(x) + 2b + 3a b
--R +
--R 4
--R - 2a b
--R *
--R sinh(x)
--R +
--R 4 3 2 2 5 2 3 4 3 2
--R (a b + 2a b )cosh(x) + (2b + 4a b )cosh(x) - a b - 2a b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b + a
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 2 3 2 3 3 2
--R (2a b + 2a )sinh(x) + (2a b + 2a )cosh(x) + 2b + 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R - a
--R +
--R 2 3 4 3
--R (2a b + 4a b)sinh(x)
--R +
--R 2 3 4 4 3 2 2
--R ((4a b + 8a b)cosh(x) + 6a b + 12a b )sinh(x)
--R +
--R 2 3 4 2 4 3 2 5 2 3
--R (2a b + 4a b)cosh(x) + (8a b + 16a b )cosh(x) + 4b + 6a b
--R +
--R 4
--R - 4a b
--R *
--R sinh(x)
--R +
--R 4 3 2 2 5 2 3 4 3 2
--R (2a b + 4a b )cosh(x) + (4b + 8a b )cosh(x) - 2a b - 4a b
--R *
--R x
--R b tanh(-) - a
--R 2
--R atanh(-------------)
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 2 2 3 2
--R (a b cosh(x) + 2a b )sinh(x)
--R +
--R 2 2 2 3 4 2 2 2 2 3
--R (2a b cosh(x) + 4a b cosh(x) + 2b - 2a b )sinh(x) + a b cosh(x)
--R +
--R 3 2 4 2 2 3
--R 2a b cosh(x) + (2b - a b )cosh(x) - 2a b
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R /
--R 4 2 6 3 4 2 6 3 3 5 2
--R (a b + a )sinh(x) + ((2a b + 2a )cosh(x) + 3a b + 3a b)sinh(x)
--R +
--R 4 2 6 2 3 3 5 2 4 4 2 6
--R ((a b + a )cosh(x) + (4a b + 4a b)cosh(x) + 2a b + a b - a )
--R *
--R sinh(x)
--R +
--R 3 3 5 2 2 4 4 2 3 3 5
--R (a b + a b)cosh(x) + (2a b + 2a b )cosh(x) - a b - a b
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R Type: Expression(Integer)
--E 342
--S 343 of 526
d0464:= D(m0464,x)
--R
--R
--R (252)
--R 3 3 5 6
--R (2a b + 2a b)sinh(x)
--R +
--R 3 3 5 2 4 4 2 5
--R ((8a b + 8a b)cosh(x) + 7a b + 8a b )sinh(x)
--R +
--R 3 3 5 2 2 4 4 2 5
--R (11a b + 12a b)cosh(x) + (24a b + 28a b )cosh(x) + 9a b
--R +
--R 3 3 5
--R 10a b - 4a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 3 5 3 2 4 4 2 2
--R (4a b + 8a b)cosh(x) + (26a b + 36a b )cosh(x)
--R +
--R 5 3 3 5 6 2 4 4 2
--R (24a b + 32a b - 8a b)cosh(x) + 4b + 2a b - 12a b
--R *
--R 3
--R sinh(x)
--R +
--R 3 3 5 4 2 4 4 2 3
--R (- 4a b + 2a b)cosh(x) + (4a b + 20a b )cosh(x)
--R +
--R 5 3 3 5 2 6 2 4 4 2
--R (18a b + 38a b - 4a b)cosh(x) + (8b + 12a b - 20a b )cosh(x)
--R +
--R 5 3 3 5
--R - 2a b - 10a b + 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 3 5 2 4 4 2 4 3 3 3
--R - 4a b cosh(x) + (- 9a b + 4a b )cosh(x) + 20a b cosh(x)
--R +
--R 6 2 4 4 2 2 3 3 2 4 4 2
--R (4b + 18a b - 8a b )cosh(x) - 16a b cosh(x) - a b + 4a b
--R *
--R sinh(x)
--R +
--R 3 3 6 2 4 5 5 3 3 4
--R - a b cosh(x) - 4a b cosh(x) + (- 3a b + 4a b )cosh(x)
--R +
--R 2 4 3 5 3 3 2 2 4 5 3 3
--R 8a b cosh(x) + (2a b - 5a b )cosh(x) - 4a b cosh(x) + a b + 2a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 4 2 6 4 2 3 3 5 5
--R - 2a b sinh(x) + (- 8a b cosh(x) - 2a b + 8a b)sinh(x)
--R +
--R 4 2 2 3 3 5 2 4 4 2
--R (- 10a b cosh(x) + (- 8a b + 24a b)cosh(x) + 8a b + 28a b )
--R *
--R 4
--R sinh(x)
--R +
--R 3 3 5 2 2 4 4 2 5
--R (- 4a b + 24a b)cosh(x) + (24a b + 72a b )cosh(x) + 16a b
--R +
--R 3 3 5
--R 28a b - 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 2 4 3 3 5 3
--R 10a b cosh(x) + (16a b + 8a b)cosh(x)
--R +
--R 2 4 4 2 2 5 3 3 5 6
--R (32a b + 56a b )cosh(x) + (40a b + 64a b - 8a b)cosh(x) + 8b
--R +
--R 4 2
--R - 18a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 2 5 3 3 4 2 4 4 2 3
--R 8a b cosh(x) + 22a b cosh(x) + (24a b + 8a b )cosh(x)
--R +
--R 5 3 3 2 6 2 4 4 2
--R (32a b + 36a b )cosh(x) + (16b + 8a b - 16a b )cosh(x)
--R +
--R 5 3 3
--R - 8a b - 10a b
--R *
--R sinh(x)
--R +
--R 4 2 6 3 3 5 2 4 4 2 4
--R 2a b cosh(x) + 8a b cosh(x) + (8a b - 4a b )cosh(x)
--R +
--R 5 3 6 2 4 4 2 2
--R 8a b cosh(x) + (8b + 8a b + 2a b )cosh(x)
--R +
--R 5 3 3
--R (- 8a b - 8a b )cosh(x)
--R *
--R x
--R tanh(-)
--R 2
--R +
--R 3 3 5 6
--R (- 2a b - 2a b)sinh(x)
--R +
--R 3 3 5 2 4 4 2 5
--R ((- 8a b - 8a b)cosh(x) - 7a b - 8a b )sinh(x)
--R +
--R 3 3 5 2 2 4 4 2 5
--R (- 11a b - 12a b)cosh(x) + (- 24a b - 28a b )cosh(x) - 9a b
--R +
--R 3 3 5
--R - 10a b + 4a b
--R *
--R 4
--R sinh(x)
--R +
--R 3 3 5 3 2 4 4 2 2
--R (- 4a b - 8a b)cosh(x) + (- 26a b - 36a b )cosh(x)
--R +
--R 5 3 3 5 6 2 4 4 2
--R (- 24a b - 32a b + 8a b)cosh(x) - 4b - 2a b + 12a b
--R *
--R 3
--R sinh(x)
--R +
--R 3 3 5 4 2 4 4 2 3
--R (4a b - 2a b)cosh(x) + (- 4a b - 20a b )cosh(x)
--R +
--R 5 3 3 5 2 6 2 4 4 2
--R (- 18a b - 38a b + 4a b)cosh(x) + (- 8b - 12a b + 20a b )cosh(x)
--R +
--R 5 3 3 5
--R 2a b + 10a b - 2a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 3 5 2 4 4 2 4 3 3 3
--R 4a b cosh(x) + (9a b - 4a b )cosh(x) - 20a b cosh(x)
--R +
--R 6 2 4 4 2 2 3 3 2 4 4 2
--R (- 4b - 18a b + 8a b )cosh(x) + 16a b cosh(x) + a b - 4a b
--R *
--R sinh(x)
--R +
--R 3 3 6 2 4 5 5 3 3 4 2 4 3
--R a b cosh(x) + 4a b cosh(x) + (3a b - 4a b )cosh(x) - 8a b cosh(x)
--R +
--R 5 3 3 2 2 4 5 3 3
--R (- 2a b + 5a b )cosh(x) + 4a b cosh(x) - a b - 2a b
--R /
--R 5 3 7 6
--R (a b + a b)sinh(x)
--R +
--R 5 3 7 4 4 6 2 5
--R ((4a b + 4a b)cosh(x) + 6a b + 6a b )sinh(x)
--R +
--R 5 3 7 2 4 4 6 2 3 5
--R (6a b + 6a b)cosh(x) + (20a b + 20a b )cosh(x) + 13a b
--R +
--R 5 3 7
--R 11a b - 2a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 3 7 3 4 4 6 2 2
--R (4a b + 4a b)cosh(x) + (24a b + 24a b )cosh(x)
--R +
--R 3 5 5 3 7 2 6 4 4 6 2
--R (36a b + 32a b - 4a b)cosh(x) + 12a b + 4a b - 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 3 7 4 4 4 6 2 3
--R (a b + a b)cosh(x) + (12a b + 12a b )cosh(x)
--R +
--R 3 5 5 3 7 2
--R (34a b + 32a b - 2a b)cosh(x)
--R +
--R 2 6 4 4 6 2 7 3 5 5 3 7
--R (28a b + 16a b - 12a b )cosh(x) + 4a b - 6a b - 9a b + a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 4 6 2 4 3 5 5 3 3
--R (2a b + 2a b )cosh(x) + (12a b + 12a b )cosh(x)
--R +
--R 2 6 4 4 6 2 2
--R (20a b + 16a b - 4a b )cosh(x)
--R +
--R 7 3 5 5 3 2 6 4 4 6 2
--R (8a b - 4a b - 12a b )cosh(x) - 4a b - 2a b + 2a b
--R *
--R sinh(x)
--R +
--R 3 5 5 3 4 2 6 4 4 3
--R (a b + a b )cosh(x) + (4a b + 4a b )cosh(x)
--R +
--R 7 3 5 5 3 2 2 6 4 4 3 5
--R (4a b + 2a b - 2a b )cosh(x) + (- 4a b - 4a b )cosh(x) + a b
--R +
--R 5 3
--R a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 6 2 8 6
--R (- 2a b - 2a )sinh(x)
--R +
--R 6 2 8 5 3 7 5
--R ((- 8a b - 8a )cosh(x) - 12a b - 12a b)sinh(x)
--R +
--R 6 2 8 2 5 3 7 4 4
--R (- 12a b - 12a )cosh(x) + (- 40a b - 40a b)cosh(x) - 26a b
--R +
--R 6 2 8
--R - 22a b + 4a
--R *
--R 4
--R sinh(x)
--R +
--R 6 2 8 3 5 3 7 2
--R (- 8a b - 8a )cosh(x) + (- 48a b - 48a b)cosh(x)
--R +
--R 4 4 6 2 8 3 5 5 3 7
--R (- 72a b - 64a b + 8a )cosh(x) - 24a b - 8a b + 16a b
--R *
--R 3
--R sinh(x)
--R +
--R 6 2 8 4 5 3 7 3
--R (- 2a b - 2a )cosh(x) + (- 24a b - 24a b)cosh(x)
--R +
--R 4 4 6 2 8 2
--R (- 68a b - 64a b + 4a )cosh(x)
--R +
--R 3 5 5 3 7 2 6 4 4 6 2 8
--R (- 56a b - 32a b + 24a b)cosh(x) - 8a b + 12a b + 18a b - 2a
--R *
--R 2
--R sinh(x)
--R +
--R 5 3 7 4 4 4 6 2 3
--R (- 4a b - 4a b)cosh(x) + (- 24a b - 24a b )cosh(x)
--R +
--R 3 5 5 3 7 2
--R (- 40a b - 32a b + 8a b)cosh(x)
--R +
--R 2 6 4 4 6 2 3 5 5 3 7
--R (- 16a b + 8a b + 24a b )cosh(x) + 8a b + 4a b - 4a b
--R *
--R sinh(x)
--R +
--R 4 4 6 2 4 3 5 5 3 3
--R (- 2a b - 2a b )cosh(x) + (- 8a b - 8a b )cosh(x)
--R +
--R 2 6 4 4 6 2 2 3 5 5 3 4 4
--R (- 8a b - 4a b + 4a b )cosh(x) + (8a b + 8a b )cosh(x) - 2a b
--R +
--R 6 2
--R - 2a b
--R *
--R x
--R tanh(-)
--R 2
--R +
--R 5 3 7 6
--R (- a b - a b)sinh(x)
--R +
--R 5 3 7 4 4 6 2 5
--R ((- 4a b - 4a b)cosh(x) - 6a b - 6a b )sinh(x)
--R +
--R 5 3 7 2 4 4 6 2 3 5
--R (- 6a b - 6a b)cosh(x) + (- 20a b - 20a b )cosh(x) - 13a b
--R +
--R 5 3 7
--R - 11a b + 2a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 3 7 3 4 4 6 2 2
--R (- 4a b - 4a b)cosh(x) + (- 24a b - 24a b )cosh(x)
--R +
--R 3 5 5 3 7 2 6 4 4 6 2
--R (- 36a b - 32a b + 4a b)cosh(x) - 12a b - 4a b + 8a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 3 7 4 4 4 6 2 3
--R (- a b - a b)cosh(x) + (- 12a b - 12a b )cosh(x)
--R +
--R 3 5 5 3 7 2
--R (- 34a b - 32a b + 2a b)cosh(x)
--R +
--R 2 6 4 4 6 2 7 3 5 5 3 7
--R (- 28a b - 16a b + 12a b )cosh(x) - 4a b + 6a b + 9a b - a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 4 6 2 4 3 5 5 3 3
--R (- 2a b - 2a b )cosh(x) + (- 12a b - 12a b )cosh(x)
--R +
--R 2 6 4 4 6 2 2
--R (- 20a b - 16a b + 4a b )cosh(x)
--R +
--R 7 3 5 5 3 2 6 4 4 6 2
--R (- 8a b + 4a b + 12a b )cosh(x) + 4a b + 2a b - 2a b
--R *
--R sinh(x)
--R +
--R 3 5 5 3 4 2 6 4 4 3
--R (- a b - a b )cosh(x) + (- 4a b - 4a b )cosh(x)
--R +
--R 7 3 5 5 3 2 2 6 4 4 3 5 5 3
--R (- 4a b - 2a b + 2a b )cosh(x) + (4a b + 4a b )cosh(x) - a b - a b
--R Type: Expression(Integer)
--E 343
--S 344 of 526
t0465:= 1/(a+b*csch(x))^3
--R
--R
--R 1
--R (253) ----------------------------------------------
--R 3 3 2 2 2 3
--R b csch(x) + 3a b csch(x) + 3a b csch(x) + a
--R Type: Expression(Integer)
--E 344
--S 345 of 526
r0465:= x/a^3-b^3*(a^2-2*b^2)*atanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/_
a^3/(a^2+b^2)^(5/2)-6*b^3*atanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/_
a^3/(a^2+b^2)^(3/2)+6*b*atanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/_
a^3/(a^2+b^2)^(1/2)+1/2*b^3*cosh(x)/a^2/(a^2+b^2)/(b+a*sinh(x))^2+_
3/2*b^4*cosh(x)/a^2/(a^2+b^2)^2/(b+a*sinh(x))-_
3*b^2*cosh(x)/a^2/(a^2+b^2)/(b+a*sinh(x))
--R
--R
--R (254)
--R 2 5 4 3 6 2
--R (- 4a b - 10a b - 12a b)sinh(x)
--R +
--R 6 3 4 5 2 7 2 5 4 3
--R (- 8a b - 20a b - 24a b )sinh(x) - 4b - 10a b - 12a b
--R *
--R x
--R b tanh(-) - a
--R 2
--R atanh(-------------)
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 2 4 4 2 6 2
--R (2a b + 4a b + 2a )x sinh(x)
--R +
--R 2 4 4 2 5 3 3 5
--R ((- 3a b - 6a b )cosh(x) + (4a b + 8a b + 4a b)x)sinh(x)
--R +
--R 5 3 3 6 2 4 4 2
--R (- 2a b - 5a b )cosh(x) + (2b + 4a b + 2a b )x
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R /
--R 5 4 7 2 9 2 4 5 6 3 8 3 6
--R (2a b + 4a b + 2a )sinh(x) + (4a b + 8a b + 4a b)sinh(x) + 2a b
--R +
--R 5 4 7 2
--R 4a b + 2a b
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R Type: Expression(Integer)
--E 345
--S 346 of 526
a0465:= integrate(t0465,x)
--R
--R
--R (255)
--R 2 5 4 3 6 4
--R (2a b + 5a b + 6a b)sinh(x)
--R +
--R 2 5 4 3 6 6 3 4 5 2 3
--R ((8a b + 20a b + 24a b)cosh(x) + 8a b + 20a b + 24a b )sinh(x)
--R +
--R 2 5 4 3 6 2
--R (12a b + 30a b + 36a b)cosh(x)
--R +
--R 6 3 4 5 2 7 2 5 4 3 6
--R (24a b + 60a b + 72a b )cosh(x) + 8b + 16a b + 14a b - 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 5 4 3 6 3
--R (8a b + 20a b + 24a b)cosh(x)
--R +
--R 6 3 4 5 2 2
--R (24a b + 60a b + 72a b )cosh(x)
--R +
--R 7 2 5 4 3 6 6 3 4 5 2
--R (16b + 32a b + 28a b - 24a b)cosh(x) - 8a b - 20a b - 24a b
--R *
--R sinh(x)
--R +
--R 2 5 4 3 6 4 6 3 4 5 2 3
--R (2a b + 5a b + 6a b)cosh(x) + (8a b + 20a b + 24a b )cosh(x)
--R +
--R 7 2 5 4 3 6 2
--R (8b + 16a b + 14a b - 12a b)cosh(x)
--R +
--R 6 3 4 5 2 2 5 4 3 6
--R (- 8a b - 20a b - 24a b )cosh(x) + 2a b + 5a b + 6a b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b + a
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 2 3 2 3 3 2
--R (2a b + 2a )sinh(x) + (2a b + 2a )cosh(x) + 2b + 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R - a
--R +
--R 2 4 4 2 6 4
--R (2a b + 4a b + 2a )x sinh(x)
--R +
--R 2 4 4 2 6 5 3 3 5
--R (8a b + 16a b + 8a )x cosh(x) + (8a b + 16a b + 8a b)x
--R +
--R 5 3 3
--R 8a b + 14a b
--R *
--R 3
--R sinh(x)
--R +
--R 2 4 4 2 6 2
--R (12a b + 24a b + 12a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R ((24a b + 48a b + 24a b)x + 24a b + 42a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2
--R (8b + 12a b - 4a )x + 12b + 18a b - 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 2 4 4 2 6 3
--R (8a b + 16a b + 8a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3 2
--R ((24a b + 48a b + 24a b)x + 24a b + 42a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2
--R ((16b + 24a b - 8a )x + 24b + 36a b - 24a b )cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R (- 8a b - 16a b - 8a b)x - 16a b - 34a b
--R *
--R sinh(x)
--R +
--R 2 4 4 2 6 4
--R (2a b + 4a b + 2a )x cosh(x)
--R +
--R 5 3 3 5 5 3 3 3
--R ((8a b + 16a b + 8a b)x + 8a b + 14a b )cosh(x)
--R +
--R 6 2 4 6 6 2 4 4 2 2
--R ((8b + 12a b - 4a )x + 12b + 18a b - 12a b )cosh(x)
--R +
--R 5 3 3 5 5 3 3
--R ((- 8a b - 16a b - 8a b)x - 16a b - 34a b )cosh(x)
--R +
--R 2 4 4 2 6 2 4 4 2
--R (2a b + 4a b + 2a )x + 6a b + 12a b
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R /
--R 5 4 7 2 9 4
--R (2a b + 4a b + 2a )sinh(x)
--R +
--R 5 4 7 2 9 4 5 6 3 8 3
--R ((8a b + 16a b + 8a )cosh(x) + 8a b + 16a b + 8a b)sinh(x)
--R +
--R 5 4 7 2 9 2 4 5 6 3 8
--R (12a b + 24a b + 12a )cosh(x) + (24a b + 48a b + 24a b)cosh(x)
--R +
--R 3 6 5 4 9
--R 8a b + 12a b - 4a
--R *
--R 2
--R sinh(x)
--R +
--R 5 4 7 2 9 3 4 5 6 3 8 2
--R (8a b + 16a b + 8a )cosh(x) + (24a b + 48a b + 24a b)cosh(x)
--R +
--R 3 6 5 4 9 4 5 6 3 8
--R (16a b + 24a b - 8a )cosh(x) - 8a b - 16a b - 8a b
--R *
--R sinh(x)
--R +
--R 5 4 7 2 9 4 4 5 6 3 8 3
--R (2a b + 4a b + 2a )cosh(x) + (8a b + 16a b + 8a b)cosh(x)
--R +
--R 3 6 5 4 9 2 4 5 6 3 8
--R (8a b + 12a b - 4a )cosh(x) + (- 8a b - 16a b - 8a b)cosh(x)
--R +
--R 5 4 7 2 9
--R 2a b + 4a b + 2a
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R Type: Union(Expression(Integer),...)
--E 346
--S 347 of 526
m0465:= a0465-r0465
--R
--R
--R (256)
--R 4 5 6 3 8 6
--R (2a b + 5a b + 6a b)sinh(x)
--R +
--R 4 5 6 3 8 3 6 5 4 7 2 5
--R ((8a b + 20a b + 24a b)cosh(x) + 12a b + 30a b + 36a b )sinh(x)
--R +
--R 4 5 6 3 8 2
--R (12a b + 30a b + 36a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2 7 4 5 6 3
--R (40a b + 100a b + 120a b )cosh(x) + 26a b + 61a b + 68a b
--R +
--R 8
--R - 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 5 6 3 8 3
--R (8a b + 20a b + 24a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2
--R (48a b + 120a b + 144a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 8 3 6
--R (72a b + 172a b + 196a b - 24a b)cosh(x) + 24a b + 44a b
--R +
--R 5 4 7 2
--R 32a b - 48a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 5 6 3 8 4
--R (2a b + 5a b + 6a b)cosh(x)
--R +
--R 3 6 5 4 7 2 3
--R (24a b + 60a b + 72a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 2
--R (68a b + 166a b + 194a b - 12a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 9 4 5
--R (56a b + 116a b + 108a b - 72a b )cosh(x) + 8b - 24a b
--R +
--R 6 3 8
--R - 55a b + 6a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 6 5 4 7 2 4
--R (4a b + 10a b + 12a b )cosh(x)
--R +
--R 2 7 4 5 6 3 3
--R (24a b + 60a b + 72a b )cosh(x)
--R +
--R 8 3 6 5 4 7 2 2
--R (40a b + 92a b + 100a b - 24a b )cosh(x)
--R +
--R 9 2 7 4 5 6 3 8 3 6
--R (16b + 16a b - 12a b - 72a b )cosh(x) - 8a b - 16a b
--R +
--R 5 4 7 2
--R - 14a b + 12a b
--R *
--R sinh(x)
--R +
--R 2 7 4 5 6 3 4 8 3 6 5 4 3
--R (2a b + 5a b + 6a b )cosh(x) + (8a b + 20a b + 24a b )cosh(x)
--R +
--R 9 2 7 4 5 6 3 2
--R (8b + 16a b + 14a b - 12a b )cosh(x)
--R +
--R 8 3 6 5 4 2 7 4 5 6 3
--R (- 8a b - 20a b - 24a b )cosh(x) + 2a b + 5a b + 6a b
--R *
--R log
--R 2 2 2 2 2
--R a sinh(x) + (2a cosh(x) + 2a b)sinh(x) + a cosh(x)
--R +
--R 2 2
--R 2a b cosh(x) + 2b + a
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 2 3 2 3 3 2
--R (2a b + 2a )sinh(x) + (2a b + 2a )cosh(x) + 2b + 2a b
--R /
--R 2 2
--R a sinh(x) + (2a cosh(x) + 2b)sinh(x) + a cosh(x) + 2b cosh(x)
--R +
--R - a
--R +
--R 4 5 6 3 8 6
--R (4a b + 10a b + 12a b)sinh(x)
--R +
--R 4 5 6 3 8 3 6 5 4 7 2 5
--R ((16a b + 40a b + 48a b)cosh(x) + 24a b + 60a b + 72a b )sinh(x)
--R +
--R 4 5 6 3 8 2
--R (24a b + 60a b + 72a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2 7 4 5 6 3
--R (80a b + 200a b + 240a b )cosh(x) + 52a b + 122a b + 136a b
--R +
--R 8
--R - 24a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 5 6 3 8 3
--R (16a b + 40a b + 48a b)cosh(x)
--R +
--R 3 6 5 4 7 2 2
--R (96a b + 240a b + 288a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 8 3 6
--R (144a b + 344a b + 392a b - 48a b)cosh(x) + 48a b + 88a b
--R +
--R 5 4 7 2
--R 64a b - 96a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 5 6 3 8 4
--R (4a b + 10a b + 12a b)cosh(x)
--R +
--R 3 6 5 4 7 2 3
--R (48a b + 120a b + 144a b )cosh(x)
--R +
--R 2 7 4 5 6 3 8 2
--R (136a b + 332a b + 388a b - 24a b)cosh(x)
--R +
--R 8 3 6 5 4 7 2 9 4 5
--R (112a b + 232a b + 216a b - 144a b )cosh(x) + 16b - 48a b
--R +
--R 6 3 8
--R - 110a b + 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 3 6 5 4 7 2 4
--R (8a b + 20a b + 24a b )cosh(x)
--R +
--R 2 7 4 5 6 3 3
--R (48a b + 120a b + 144a b )cosh(x)
--R +
--R 8 3 6 5 4 7 2 2
--R (80a b + 184a b + 200a b - 48a b )cosh(x)
--R +
--R 9 2 7 4 5 6 3 8 3 6
--R (32b + 32a b - 24a b - 144a b )cosh(x) - 16a b - 32a b
--R +
--R 5 4 7 2
--R - 28a b + 24a b
--R *
--R sinh(x)
--R +
--R 2 7 4 5 6 3 4
--R (4a b + 10a b + 12a b )cosh(x)
--R +
--R 8 3 6 5 4 3
--R (16a b + 40a b + 48a b )cosh(x)
--R +
--R 9 2 7 4 5 6 3 2
--R (16b + 32a b + 28a b - 24a b )cosh(x)
--R +
--R 8 3 6 5 4 2 7 4 5 6 3
--R (- 16a b - 40a b - 48a b )cosh(x) + 4a b + 10a b + 12a b
--R *
--R x
--R b tanh(-) - a
--R 2
--R atanh(-------------)
--R +-------+
--R | 2 2
--R \|b + a
--R +
--R 4 4 6 2 3 5 5 3 5
--R ((3a b + 6a b )cosh(x) + 8a b + 14a b )sinh(x)
--R +
--R 4 4 6 2 2 3 5 5 3 2 6
--R (12a b + 24a b )cosh(x) + (38a b + 71a b )cosh(x) + 28a b
--R +
--R 4 4 6 2
--R 46a b - 12a b
--R *
--R 4
--R sinh(x)
--R +
--R 4 4 6 2 3 3 5 5 3 2
--R (18a b + 36a b )cosh(x) + (68a b + 134a b )cosh(x)
--R +
--R 2 6 4 4 6 2 7 3 5 5 3
--R (92a b + 158a b - 36a b )cosh(x) + 32a b + 34a b - 58a b
--R *
--R 3
--R sinh(x)
--R +
--R 4 4 6 2 4 3 5 5 3 3
--R (12a b + 24a b )cosh(x) + (56a b + 116a b )cosh(x)
--R +
--R 2 6 4 4 6 2 2
--R (108a b + 198a b - 36a b )cosh(x)
--R +
--R 7 3 5 5 3 8 2 6 4 4
--R (80a b + 102a b - 116a b )cosh(x) + 12b - 14a b - 74a b
--R +
--R 6 2
--R 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 4 4 6 2 5 3 5 5 3 4
--R (3a b + 6a b )cosh(x) + (20a b + 44a b )cosh(x)
--R +
--R 2 6 4 4 6 2 3
--R (52a b + 106a b - 12a b )cosh(x)
--R +
--R 7 3 5 5 3 2
--R (64a b + 98a b - 68a b )cosh(x)
--R +
--R 8 2 6 4 4 6 2 7 3 5 5 3
--R (24b - 4a b - 109a b + 6a b )cosh(x) - 16a b - 22a b + 24a b
--R *
--R sinh(x)
--R +
--R 3 5 5 3 5 2 6 4 4 4
--R (2a b + 5a b )cosh(x) + (8a b + 20a b )cosh(x)
--R +
--R 7 3 5 5 3 3 8 2 6 4 4 2
--R (16a b + 30a b - 10a b )cosh(x) + (12b + 10a b - 32a b )cosh(x)
--R +
--R 7 3 5 5 3 2 6 4 4
--R (- 16a b - 32a b + 5a b )cosh(x) + 6a b + 12a b
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R /
--R 7 4 9 2 11 6
--R (2a b + 4a b + 2a )sinh(x)
--R +
--R 7 4 9 2 11 6 5 8 3 10 5
--R ((8a b + 16a b + 8a )cosh(x) + 12a b + 24a b + 12a b)sinh(x)
--R +
--R 7 4 9 2 11 2
--R (12a b + 24a b + 12a )cosh(x)
--R +
--R 6 5 8 3 10 5 6 7 4 9 2 11
--R (40a b + 80a b + 40a b)cosh(x) + 26a b + 48a b + 18a b - 4a
--R *
--R 4
--R sinh(x)
--R +
--R 7 4 9 2 11 3
--R (8a b + 16a b + 8a )cosh(x)
--R +
--R 6 5 8 3 10 2
--R (48a b + 96a b + 48a b)cosh(x)
--R +
--R 5 6 7 4 9 2 11 4 7 6 5 8 3
--R (72a b + 136a b + 56a b - 8a )cosh(x) + 24a b + 32a b - 8a b
--R +
--R 10
--R - 16a b
--R *
--R 3
--R sinh(x)
--R +
--R 7 4 9 2 11 4 6 5 8 3 10 3
--R (2a b + 4a b + 2a )cosh(x) + (24a b + 48a b + 24a b)cosh(x)
--R +
--R 5 6 7 4 9 2 11 2
--R (68a b + 132a b + 60a b - 4a )cosh(x)
--R +
--R 4 7 6 5 8 3 10 3 8 5 6 7 4
--R (56a b + 88a b + 8a b - 24a b)cosh(x) + 8a b - 4a b - 30a b
--R +
--R 9 2 11
--R - 16a b + 2a
--R *
--R 2
--R sinh(x)
--R +
--R 6 5 8 3 10 4
--R (4a b + 8a b + 4a b)cosh(x)
--R +
--R 5 6 7 4 9 2 3
--R (24a b + 48a b + 24a b )cosh(x)
--R +
--R 4 7 6 5 8 3 10 2
--R (40a b + 72a b + 24a b - 8a b)cosh(x)
--R +
--R 3 8 5 6 7 4 9 2 4 7 6 5 10
--R (16a b + 8a b - 32a b - 24a b )cosh(x) - 8a b - 12a b + 4a b
--R *
--R sinh(x)
--R +
--R 5 6 7 4 9 2 4 4 7 6 5 8 3 3
--R (2a b + 4a b + 2a b )cosh(x) + (8a b + 16a b + 8a b )cosh(x)
--R +
--R 3 8 5 6 9 2 2 4 7 6 5 8 3
--R (8a b + 12a b - 4a b )cosh(x) + (- 8a b - 16a b - 8a b )cosh(x)
--R +
--R 5 6 7 4 9 2
--R 2a b + 4a b + 2a b
--R *
--R +-------+
--R | 2 2
--R \|b + a
--R Type: Expression(Integer)
--E 347
--S 348 of 526
d0465:= D(m0465,x)
--R
--R
--R (257)
--R 5 5 7 3 9 9
--R (5a b + 11a b + 6a b)sinh(x)
--R +
--R 5 5 7 3 9 4 6 6 4 8 2 8
--R ((30a b + 66a b + 36a b)cosh(x) + 29a b + 68a b + 42a b )sinh(x)
--R +
--R 5 5 7 3 9 2
--R (72a b + 159a b + 90a b)cosh(x)
--R +
--R 4 6 6 4 8 2 3 7 5 5 7 3
--R (156a b + 366a b + 228a b )cosh(x) + 74a b + 173a b + 105a b
--R +
--R 9
--R - 18a b
--R *
--R 7
--R sinh(x)
--R +
--R 5 5 7 3 9 3
--R (82a b + 184a b + 120a b)cosh(x)
--R +
--R 4 6 6 4 8 2 2
--R (326a b + 770a b + 510a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 2 8 4 6
--R (342a b + 816a b + 528a b - 72a b)cosh(x) + 98a b + 208a b
--R +
--R 6 4 8 2
--R 92a b - 102a b
--R *
--R 6
--R sinh(x)
--R +
--R 5 5 7 3 9 4
--R (30a b + 75a b + 90a b)cosh(x)
--R +
--R 4 6 6 4 8 2 3
--R (304a b + 736a b + 600a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 2
--R (594a b + 1467a b + 1098a b - 108a b)cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 9 3 7
--R (384a b + 870a b + 504a b - 360a b )cosh(x) + 64a b + 94a b
--R +
--R 5 5 7 3 9
--R - 33a b - 213a b + 18a b
--R *
--R 5
--R sinh(x)
--R +
--R 5 5 7 3 9 5
--R (- 30a b - 54a b + 36a b)cosh(x)
--R +
--R 4 6 6 4 8 2 4
--R (60a b + 180a b + 390a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 3
--R (430a b + 1144a b + 1212a b - 72a b)cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 2
--R (546a b + 1368a b + 1134a b - 468a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 9 10
--R (216a b + 402a b + 102a b - 678a b + 36a b)cosh(x) + 16b
--R +
--R 2 8 4 6 6 4 8 2
--R - 14a b - 84a b - 186a b + 78a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 5 7 3 9 6
--R (- 40a b - 79a b + 6a b)cosh(x)
--R +
--R 4 6 6 4 8 2 5
--R (- 116a b - 242a b + 132a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 4
--R (30a b + 171a b + 753a b - 18a b)cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 3
--R (304a b + 924a b + 1316a b - 264a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 9 2
--R (256a b + 628a b + 570a b - 789a b + 18a b)cosh(x)
--R +
--R 10 2 8 4 6 6 4 8 2 9
--R (48b + 24a b - 120a b - 594a b + 132a b )cosh(x) - 16a b
--R +
--R 3 7 5 5 7 3 9
--R - 22a b - 53a b + 115a b - 6a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 5 7 3 7
--R (- 18a b - 36a b )cosh(x)
--R +
--R 4 6 6 4 8 2 6
--R (- 94a b - 214a b + 18a b )cosh(x)
--R +
--R 3 7 5 5 7 3 5
--R (- 126a b - 288a b + 252a b )cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 4
--R (6a b + 144a b + 804a b - 54a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 3
--R (112a b + 412a b + 774a b - 396a b )cosh(x)
--R +
--R 10 2 8 4 6 6 4 8 2 2
--R (48b + 108a b + 90a b - 654a b + 54a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 2 8 4 6
--R (- 24a b - 54a b - 228a b + 180a b )cosh(x) + 6a b + 4a b
--R +
--R 6 4 8 2
--R 64a b - 18a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 5 7 3 8 4 6 6 4 7
--R (- 3a b - 6a b )cosh(x) + (- 24a b - 60a b )cosh(x)
--R +
--R 3 7 5 5 7 3 6
--R (- 58a b - 163a b + 36a b )cosh(x)
--R +
--R 2 8 4 6 6 4 5
--R (- 48a b - 114a b + 228a b )cosh(x)
--R +
--R 3 7 5 5 7 3 4
--R (78a b + 411a b - 72a b )cosh(x)
--R +
--R 10 2 8 4 6 6 4 3
--R (16b + 88a b + 204a b - 276a b )cosh(x)
--R +
--R 3 7 5 5 7 3 2
--R (- 18a b - 249a b + 60a b )cosh(x)
--R +
--R 4 6 6 4 3 7 5 5 7 3
--R (- 18a b + 108a b )cosh(x) - 2a b + 4a b - 18a b
--R *
--R sinh(x)
--R +
--R 4 6 6 4 8 3 7 5 5 7
--R (- a b - 4a b )cosh(x) + (- 6a b - 24a b )cosh(x)
--R +
--R 2 8 4 6 6 4 6
--R (- 10a b - 40a b + 18a b )cosh(x)
--R +
--R 9 3 7 5 5 5
--R (- 8a b - 14a b + 72a b )cosh(x)
--R +
--R 2 8 4 6 6 4 4
--R (18a b + 78a b - 30a b )cosh(x)
--R +
--R 9 3 7 5 5 3
--R (8a b + 14a b - 72a b )cosh(x)
--R +
--R 2 8 4 6 6 4 2 3 7 5 5 2 8
--R (- 6a b - 32a b + 22a b )cosh(x) + (6a b + 24a b )cosh(x) - 2a b
--R +
--R 4 6 6 4
--R - 5a b - 6a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 6 4 8 2 9
--R (- 6a b - 12a b )sinh(x)
--R +
--R 6 4 8 2 5 5 7 3 9 8
--R ((- 36a b - 72a b )cosh(x) - 22a b - 46a b + 24a b)sinh(x)
--R +
--R 6 4 8 2 2
--R (- 84a b - 168a b )cosh(x)
--R +
--R 5 5 7 3 9 4 6 6 4 8 2
--R (- 120a b - 252a b + 120a b)cosh(x) - 16a b - 28a b + 156a b
--R *
--R 7
--R sinh(x)
--R +
--R 6 4 8 2 3
--R (- 84a b - 168a b )cosh(x)
--R +
--R 5 5 7 3 9 2
--R (- 232a b - 490a b + 240a b)cosh(x)
--R +
--R 4 6 6 4 8 2 3 7 5 5
--R (- 60a b - 120a b + 696a b )cosh(x) + 56a b + 130a b
--R +
--R 7 3 9
--R 362a b - 48a b
--R *
--R 6
--R sinh(x)
--R +
--R 5 5 7 3 9 3
--R (- 128a b - 272a b + 240a b)cosh(x)
--R +
--R 6 4 8 2 2
--R (- 36a b + 1188a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 2 8
--R (288a b + 612a b + 1440a b - 144a b)cosh(x) + 136a b
--R +
--R 4 6 6 4 8 2
--R 244a b + 300a b - 228a b
--R *
--R 5
--R sinh(x)
--R +
--R 6 4 8 2 5 5 5 7 3 9 4
--R (84a b + 168a b )cosh(x) + (180a b + 390a b + 120a b)cosh(x)
--R +
--R 4 6 6 4 8 2 3
--R (292a b + 544a b + 912a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 2
--R (648a b + 1254a b + 2052a b - 144a b)cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 9
--R (552a b + 984a b + 1092a b - 600a b )cosh(x) + 112a b
--R +
--R 3 7 5 5 7 3 9
--R 88a b - 90a b - 378a b + 24a b
--R *
--R 4
--R sinh(x)
--R +
--R 6 4 8 2 6 5 5 7 3 9 5
--R (84a b + 168a b )cosh(x) + (328a b + 724a b + 24a b)cosh(x)
--R +
--R 4 6 6 4 8 2 4
--R (528a b + 1116a b + 228a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 3
--R (800a b + 1480a b + 1112a b - 48a b)cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 2
--R (880a b + 1528a b + 1308a b - 480a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 9 10
--R (384a b + 384a b - 144a b - 876a b + 24a b)cosh(x) + 32b
--R +
--R 2 8 4 6 6 4 8 2
--R - 80a b - 232a b - 204a b + 84a b
--R *
--R 3
--R sinh(x)
--R +
--R 6 4 8 2 7 5 5 7 3 6
--R (36a b + 72a b )cosh(x) + (200a b + 458a b )cosh(x)
--R +
--R 4 6 6 4 8 2 5
--R (396a b + 936a b - 72a b )cosh(x)
--R +
--R 3 7 5 5 7 3 4
--R (552a b + 1086a b - 6a b )cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 3
--R (688a b + 1168a b + 468a b - 72a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 2
--R (480a b + 576a b - 36a b - 546a b )cosh(x)
--R +
--R 10 2 8 4 6 6 4 8 2 9
--R (96b - 144a b - 540a b - 480a b + 72a b )cosh(x) - 48a b
--R +
--R 3 7 5 5 7 3
--R - 48a b + 46a b + 94a b
--R *
--R 2
--R sinh(x)
--R +
--R 6 4 8 2 8 5 5 7 3 7
--R (6a b + 12a b )cosh(x) + (48a b + 120a b )cosh(x)
--R +
--R 4 6 6 4 8 2 6
--R (128a b + 356a b - 36a b )cosh(x)
--R +
--R 3 7 5 5 7 3 5
--R (192a b + 468a b - 168a b )cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 4
--R (264a b + 468a b - 120a b + 36a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 3
--R (256a b + 352a b - 24a b - 24a b )cosh(x)
--R +
--R 10 2 8 4 6 6 4 8 2 2
--R (96b - 48a b - 360a b - 276a b - 12a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 2 8 4 6
--R (- 96a b - 144a b - 12a b + 72a b )cosh(x) + 24a b + 52a b
--R +
--R 6 4
--R 34a b
--R *
--R sinh(x)
--R +
--R 5 5 7 3 8 4 6 6 4 7
--R (2a b + 8a b )cosh(x) + (12a b + 48a b )cosh(x)
--R +
--R 3 7 5 5 7 3 6
--R (24a b + 90a b - 24a b )cosh(x)
--R +
--R 2 8 4 6 6 4 5
--R (40a b + 88a b - 72a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 4
--R (48a b + 72a b - 42a b + 24a b )cosh(x)
--R +
--R 10 2 8 4 6 3
--R (32b + 16a b - 52a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 2
--R (- 48a b - 96a b - 50a b - 8a b )cosh(x)
--R +
--R 2 8 4 6 6 4
--R (24a b + 48a b + 24a b )cosh(x)
--R *
--R x
--R tanh(-)
--R 2
--R +
--R 5 5 7 3 9 9
--R (- 5a b - 11a b - 6a b)sinh(x)
--R +
--R 5 5 7 3 9 4 6 6 4 8 2 8
--R ((- 30a b - 66a b - 36a b)cosh(x) - 29a b - 68a b - 42a b )sinh(x)
--R +
--R 5 5 7 3 9 2
--R (- 72a b - 159a b - 90a b)cosh(x)
--R +
--R 4 6 6 4 8 2 3 7 5 5 7 3
--R (- 156a b - 366a b - 228a b )cosh(x) - 74a b - 173a b - 105a b
--R +
--R 9
--R 18a b
--R *
--R 7
--R sinh(x)
--R +
--R 5 5 7 3 9 3
--R (- 82a b - 184a b - 120a b)cosh(x)
--R +
--R 4 6 6 4 8 2 2
--R (- 326a b - 770a b - 510a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 2 8 4 6
--R (- 342a b - 816a b - 528a b + 72a b)cosh(x) - 98a b - 208a b
--R +
--R 6 4 8 2
--R - 92a b + 102a b
--R *
--R 6
--R sinh(x)
--R +
--R 5 5 7 3 9 4
--R (- 30a b - 75a b - 90a b)cosh(x)
--R +
--R 4 6 6 4 8 2 3
--R (- 304a b - 736a b - 600a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 2
--R (- 594a b - 1467a b - 1098a b + 108a b)cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 9 3 7
--R (- 384a b - 870a b - 504a b + 360a b )cosh(x) - 64a b - 94a b
--R +
--R 5 5 7 3 9
--R 33a b + 213a b - 18a b
--R *
--R 5
--R sinh(x)
--R +
--R 5 5 7 3 9 5
--R (30a b + 54a b - 36a b)cosh(x)
--R +
--R 4 6 6 4 8 2 4
--R (- 60a b - 180a b - 390a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 3
--R (- 430a b - 1144a b - 1212a b + 72a b)cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 2
--R (- 546a b - 1368a b - 1134a b + 468a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 9 10
--R (- 216a b - 402a b - 102a b + 678a b - 36a b)cosh(x) - 16b
--R +
--R 2 8 4 6 6 4 8 2
--R 14a b + 84a b + 186a b - 78a b
--R *
--R 4
--R sinh(x)
--R +
--R 5 5 7 3 9 6
--R (40a b + 79a b - 6a b)cosh(x)
--R +
--R 4 6 6 4 8 2 5
--R (116a b + 242a b - 132a b )cosh(x)
--R +
--R 3 7 5 5 7 3 9 4
--R (- 30a b - 171a b - 753a b + 18a b)cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 3
--R (- 304a b - 924a b - 1316a b + 264a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 9 2
--R (- 256a b - 628a b - 570a b + 789a b - 18a b)cosh(x)
--R +
--R 10 2 8 4 6 6 4 8 2 9
--R (- 48b - 24a b + 120a b + 594a b - 132a b )cosh(x) + 16a b
--R +
--R 3 7 5 5 7 3 9
--R 22a b + 53a b - 115a b + 6a b
--R *
--R 3
--R sinh(x)
--R +
--R 5 5 7 3 7 4 6 6 4 8 2 6
--R (18a b + 36a b )cosh(x) + (94a b + 214a b - 18a b )cosh(x)
--R +
--R 3 7 5 5 7 3 5
--R (126a b + 288a b - 252a b )cosh(x)
--R +
--R 2 8 4 6 6 4 8 2 4
--R (- 6a b - 144a b - 804a b + 54a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 3
--R (- 112a b - 412a b - 774a b + 396a b )cosh(x)
--R +
--R 10 2 8 4 6 6 4 8 2 2
--R (- 48b - 108a b - 90a b + 654a b - 54a b )cosh(x)
--R +
--R 9 3 7 5 5 7 3 2 8 4 6 6 4
--R (24a b + 54a b + 228a b - 180a b )cosh(x) - 6a b - 4a b - 64a b
--R +
--R 8 2
--R 18a b
--R *
--R 2
--R sinh(x)
--R +
--R 5 5 7 3 8 4 6 6 4 7
--R (3a b + 6a b )cosh(x) + (24a b + 60a b )cosh(x)
--R +
--R 3 7 5 5 7 3 6
--R (58a b + 163a b - 36a b )cosh(x)
--R +
--R 2 8 4 6 6 4 5
--R (48a b + 114a b - 228a b )cosh(x)
--R +
--R 3 7 5 5 7 3 4
--R (- 78a b - 411a b + 72a b )cosh(x)
--R +
--R 10 2 8 4 6 6 4 3
--R (- 16b - 88a b - 204a b + 276a b )cosh(x)
--R +
--R 3 7 5 5 7 3 2 4 6 6 4
--R (18a b + 249a b - 60a b )cosh(x) + (18a b - 108a b )cosh(x)
--R +
--R 3 7 5 5 7 3
--R 2a b - 4a b + 18a b
--R *
--R sinh(x)
--R +
--R 4 6 6 4 8 3 7 5 5 7
--R (a b + 4a b )cosh(x) + (6a b + 24a b )cosh(x)
--R +
--R 2 8 4 6 6 4 6 9 3 7 5 5 5
--R (10a b + 40a b - 18a b )cosh(x) + (8a b + 14a b - 72a b )cosh(x)
--R +
--R 2 8 4 6 6 4 4
--R (- 18a b - 78a b + 30a b )cosh(x)
--R +
--R 9 3 7 5 5 3 2 8 4 6 6 4 2
--R (- 8a b - 14a b + 72a b )cosh(x) + (6a b + 32a b - 22a b )cosh(x)
--R +
--R 3 7 5 5 2 8 4 6 6 4
--R (- 6a b - 24a b )cosh(x) + 2a b + 5a b + 6a b
--R /
--R 8 5 10 3 12 9
--R (2a b + 4a b + 2a b)sinh(x)
--R +
--R 8 5 10 3 12 7 6 9 4 11 2
--R ((12a b + 24a b + 12a b)cosh(x) + 18a b + 36a b + 18a b )
--R *
--R 8
--R sinh(x)
--R +
--R 8 5 10 3 12 2
--R (30a b + 60a b + 30a b)cosh(x)
--R +
--R 7 6 9 4 11 2 6 7 8 5 10 3
--R (96a b + 192a b + 96a b )cosh(x) + 66a b + 126a b + 54a b
--R +
--R 12
--R - 6a b
--R *
--R 7
--R sinh(x)
--R +
--R 8 5 10 3 12 3
--R (40a b + 80a b + 40a b)cosh(x)
--R +
--R 7 6 9 4 11 2 2
--R (210a b + 420a b + 210a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 5 8
--R (312a b + 600a b + 264a b - 24a b)cosh(x) + 126a b
--R +
--R 7 6 9 4 11 2
--R 210a b + 42a b - 42a b
--R *
--R 6
--R sinh(x)
--R +
--R 8 5 10 3 12 4
--R (30a b + 60a b + 30a b)cosh(x)
--R +
--R 7 6 9 4 11 2 3
--R (240a b + 480a b + 240a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 2
--R (594a b + 1152a b + 522a b - 36a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 4 9
--R (528a b + 912a b + 240a b - 144a b )cosh(x) + 132a b
--R +
--R 6 7 8 5 10 3 12
--R 150a b - 90a b - 102a b + 6a b
--R *
--R 5
--R sinh(x)
--R +
--R 8 5 10 3 12 5
--R (12a b + 24a b + 12a b)cosh(x)
--R +
--R 7 6 9 4 11 2 4
--R (150a b + 300a b + 150a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 3
--R (576a b + 1128a b + 528a b - 24a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 2
--R (870a b + 1560a b + 510a b - 180a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 12
--R (492a b + 648a b - 168a b - 312a b + 12a b)cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 11 2
--R 72a b - 6a b - 198a b - 90a b + 30a b
--R *
--R 4
--R sinh(x)
--R +
--R 8 5 10 3 12 6
--R (2a b + 4a b + 2a b)cosh(x)
--R +
--R 7 6 9 4 11 2 5
--R (48a b + 96a b + 48a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 4
--R (294a b + 582a b + 282a b - 6a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 3
--R (704a b + 1312a b + 512a b - 96a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 12 2
--R (696a b + 1044a b + 6a b - 336a b + 6a b)cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 11 2
--R (240a b + 96a b - 480a b - 288a b + 48a b )cosh(x)
--R +
--R 2 11 4 9 6 7 8 5 10 3 12
--R 16a b - 64a b - 122a b + 10a b + 50a b - 2a b
--R *
--R 3
--R sinh(x)
--R +
--R 7 6 9 4 11 2 6
--R (6a b + 12a b + 6a b )cosh(x)
--R +
--R 6 7 8 5 10 3 5
--R (72a b + 144a b + 72a b )cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 4
--R (282a b + 546a b + 246a b - 18a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 3
--R (456a b + 768a b + 168a b - 144a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 11 2 2
--R (288a b + 252a b - 342a b - 288a b + 18a b )cosh(x)
--R +
--R 2 11 4 9 6 7 8 5 10 3 3 10
--R (48a b - 120a b - 312a b - 72a b + 72a b )cosh(x) - 24a b
--R +
--R 5 8 7 6 9 4 11 2
--R - 6a b + 54a b + 30a b - 6a b
--R *
--R 2
--R sinh(x)
--R +
--R 6 7 8 5 10 3 6
--R (6a b + 12a b + 6a b )cosh(x)
--R +
--R 5 8 7 6 9 4 5
--R (48a b + 96a b + 48a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 4
--R (132a b + 246a b + 96a b - 18a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 3
--R (144a b + 192a b - 48a b - 96a b )cosh(x)
--R +
--R 2 11 4 9 6 7 8 5 10 3 2
--R (48a b - 48a b - 222a b - 108a b + 18a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 4 9 6 7
--R (- 48a b - 48a b + 48a b + 48a b )cosh(x) + 12a b + 18a b
--R +
--R 10 3
--R - 6a b
--R *
--R sinh(x)
--R +
--R 5 8 7 6 9 4 6 4 9 6 7 8 5 5
--R (2a b + 4a b + 2a b )cosh(x) + (12a b + 24a b + 12a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 4
--R (24a b + 42a b + 12a b - 6a b )cosh(x)
--R +
--R 2 11 4 9 6 7 8 5 3
--R (16a b + 8a b - 32a b - 24a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 2
--R (- 24a b - 42a b - 12a b + 6a b )cosh(x)
--R +
--R 4 9 6 7 8 5 5 8 7 6 9 4
--R (12a b + 24a b + 12a b )cosh(x) - 2a b - 4a b - 2a b
--R *
--R x 2
--R tanh(-)
--R 2
--R +
--R 9 4 11 2 13 9
--R (- 4a b - 8a b - 4a )sinh(x)
--R +
--R 9 4 11 2 13 8 5 10 3 12
--R ((- 24a b - 48a b - 24a )cosh(x) - 36a b - 72a b - 36a b)
--R *
--R 8
--R sinh(x)
--R +
--R 9 4 11 2 13 2
--R (- 60a b - 120a b - 60a )cosh(x)
--R +
--R 8 5 10 3 12 7 6 9 4
--R (- 192a b - 384a b - 192a b)cosh(x) - 132a b - 252a b
--R +
--R 11 2 13
--R - 108a b + 12a
--R *
--R 7
--R sinh(x)
--R +
--R 9 4 11 2 13 3
--R (- 80a b - 160a b - 80a )cosh(x)
--R +
--R 8 5 10 3 12 2
--R (- 420a b - 840a b - 420a b)cosh(x)
--R +
--R 7 6 9 4 11 2 13 6 7
--R (- 624a b - 1200a b - 528a b + 48a )cosh(x) - 252a b
--R +
--R 8 5 10 3 12
--R - 420a b - 84a b + 84a b
--R *
--R 6
--R sinh(x)
--R +
--R 9 4 11 2 13 4
--R (- 60a b - 120a b - 60a )cosh(x)
--R +
--R 8 5 10 3 12 3
--R (- 480a b - 960a b - 480a b)cosh(x)
--R +
--R 7 6 9 4 11 2 13 2
--R (- 1188a b - 2304a b - 1044a b + 72a )cosh(x)
--R +
--R 6 7 8 5 10 3 12 5 8
--R (- 1056a b - 1824a b - 480a b + 288a b)cosh(x) - 264a b
--R +
--R 7 6 9 4 11 2 13
--R - 300a b + 180a b + 204a b - 12a
--R *
--R 5
--R sinh(x)
--R +
--R 9 4 11 2 13 5
--R (- 24a b - 48a b - 24a )cosh(x)
--R +
--R 8 5 10 3 12 4
--R (- 300a b - 600a b - 300a b)cosh(x)
--R +
--R 7 6 9 4 11 2 13 3
--R (- 1152a b - 2256a b - 1056a b + 48a )cosh(x)
--R +
--R 6 7 8 5 10 3 12 2
--R (- 1740a b - 3120a b - 1020a b + 360a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 13
--R (- 984a b - 1296a b + 336a b + 624a b - 24a )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 12
--R - 144a b + 12a b + 396a b + 180a b - 60a b
--R *
--R 4
--R sinh(x)
--R +
--R 9 4 11 2 13 6
--R (- 4a b - 8a b - 4a )cosh(x)
--R +
--R 8 5 10 3 12 5
--R (- 96a b - 192a b - 96a b)cosh(x)
--R +
--R 7 6 9 4 11 2 13 4
--R (- 588a b - 1164a b - 564a b + 12a )cosh(x)
--R +
--R 6 7 8 5 10 3 12 3
--R (- 1408a b - 2624a b - 1024a b + 192a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 13 2
--R (- 1392a b - 2088a b - 12a b + 672a b - 12a )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 12
--R (- 480a b - 192a b + 960a b + 576a b - 96a b)cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 11 2 13
--R - 32a b + 128a b + 244a b - 20a b - 100a b + 4a
--R *
--R 3
--R sinh(x)
--R +
--R 8 5 10 3 12 6
--R (- 12a b - 24a b - 12a b)cosh(x)
--R +
--R 7 6 9 4 11 2 5
--R (- 144a b - 288a b - 144a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 4
--R (- 564a b - 1092a b - 492a b + 36a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 3
--R (- 912a b - 1536a b - 336a b + 288a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 12 2
--R (- 576a b - 504a b + 684a b + 576a b - 36a b)cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 11 2
--R (- 96a b + 240a b + 624a b + 144a b - 144a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 12
--R 48a b + 12a b - 108a b - 60a b + 12a b
--R *
--R 2
--R sinh(x)
--R +
--R 7 6 9 4 11 2 6
--R (- 12a b - 24a b - 12a b )cosh(x)
--R +
--R 6 7 8 5 10 3 5
--R (- 96a b - 192a b - 96a b )cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 4
--R (- 264a b - 492a b - 192a b + 36a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 3
--R (- 288a b - 384a b + 96a b + 192a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 11 2 2
--R (- 96a b + 96a b + 444a b + 216a b - 36a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 5 8 7 6
--R (96a b + 96a b - 96a b - 96a b )cosh(x) - 24a b - 36a b
--R +
--R 11 2
--R 12a b
--R *
--R sinh(x)
--R +
--R 6 7 8 5 10 3 6
--R (- 4a b - 8a b - 4a b )cosh(x)
--R +
--R 5 8 7 6 9 4 5
--R (- 24a b - 48a b - 24a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 4
--R (- 48a b - 84a b - 24a b + 12a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 3
--R (- 32a b - 16a b + 64a b + 48a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 2
--R (48a b + 84a b + 24a b - 12a b )cosh(x)
--R +
--R 5 8 7 6 9 4 6 7 8 5 10 3
--R (- 24a b - 48a b - 24a b )cosh(x) + 4a b + 8a b + 4a b
--R *
--R x
--R tanh(-)
--R 2
--R +
--R 8 5 10 3 12 9
--R (- 2a b - 4a b - 2a b)sinh(x)
--R +
--R 8 5 10 3 12 7 6 9 4 11 2
--R ((- 12a b - 24a b - 12a b)cosh(x) - 18a b - 36a b - 18a b )
--R *
--R 8
--R sinh(x)
--R +
--R 8 5 10 3 12 2
--R (- 30a b - 60a b - 30a b)cosh(x)
--R +
--R 7 6 9 4 11 2 6 7 8 5 10 3
--R (- 96a b - 192a b - 96a b )cosh(x) - 66a b - 126a b - 54a b
--R +
--R 12
--R 6a b
--R *
--R 7
--R sinh(x)
--R +
--R 8 5 10 3 12 3
--R (- 40a b - 80a b - 40a b)cosh(x)
--R +
--R 7 6 9 4 11 2 2
--R (- 210a b - 420a b - 210a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 5 8 7 6
--R (- 312a b - 600a b - 264a b + 24a b)cosh(x) - 126a b - 210a b
--R +
--R 9 4 11 2
--R - 42a b + 42a b
--R *
--R 6
--R sinh(x)
--R +
--R 8 5 10 3 12 4
--R (- 30a b - 60a b - 30a b)cosh(x)
--R +
--R 7 6 9 4 11 2 3
--R (- 240a b - 480a b - 240a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 2
--R (- 594a b - 1152a b - 522a b + 36a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 4 9 6 7
--R (- 528a b - 912a b - 240a b + 144a b )cosh(x) - 132a b - 150a b
--R +
--R 8 5 10 3 12
--R 90a b + 102a b - 6a b
--R *
--R 5
--R sinh(x)
--R +
--R 8 5 10 3 12 5
--R (- 12a b - 24a b - 12a b)cosh(x)
--R +
--R 7 6 9 4 11 2 4
--R (- 150a b - 300a b - 150a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 3
--R (- 576a b - 1128a b - 528a b + 24a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 2
--R (- 870a b - 1560a b - 510a b + 180a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 12 3 10
--R (- 492a b - 648a b + 168a b + 312a b - 12a b)cosh(x) - 72a b
--R +
--R 5 8 7 6 9 4 11 2
--R 6a b + 198a b + 90a b - 30a b
--R *
--R 4
--R sinh(x)
--R +
--R 8 5 10 3 12 6
--R (- 2a b - 4a b - 2a b)cosh(x)
--R +
--R 7 6 9 4 11 2 5
--R (- 48a b - 96a b - 48a b )cosh(x)
--R +
--R 6 7 8 5 10 3 12 4
--R (- 294a b - 582a b - 282a b + 6a b)cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 3
--R (- 704a b - 1312a b - 512a b + 96a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 12 2
--R (- 696a b - 1044a b - 6a b + 336a b - 6a b)cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 11 2 2 11
--R (- 240a b - 96a b + 480a b + 288a b - 48a b )cosh(x) - 16a b
--R +
--R 4 9 6 7 8 5 10 3 12
--R 64a b + 122a b - 10a b - 50a b + 2a b
--R *
--R 3
--R sinh(x)
--R +
--R 7 6 9 4 11 2 6
--R (- 6a b - 12a b - 6a b )cosh(x)
--R +
--R 6 7 8 5 10 3 5
--R (- 72a b - 144a b - 72a b )cosh(x)
--R +
--R 5 8 7 6 9 4 11 2 4
--R (- 282a b - 546a b - 246a b + 18a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 3
--R (- 456a b - 768a b - 168a b + 144a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 11 2 2
--R (- 288a b - 252a b + 342a b + 288a b - 18a b )cosh(x)
--R +
--R 2 11 4 9 6 7 8 5 10 3 3 10
--R (- 48a b + 120a b + 312a b + 72a b - 72a b )cosh(x) + 24a b
--R +
--R 5 8 7 6 9 4 11 2
--R 6a b - 54a b - 30a b + 6a b
--R *
--R 2
--R sinh(x)
--R +
--R 6 7 8 5 10 3 6
--R (- 6a b - 12a b - 6a b )cosh(x)
--R +
--R 5 8 7 6 9 4 5
--R (- 48a b - 96a b - 48a b )cosh(x)
--R +
--R 4 9 6 7 8 5 10 3 4
--R (- 132a b - 246a b - 96a b + 18a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 3
--R (- 144a b - 192a b + 48a b + 96a b )cosh(x)
--R +
--R 2 11 4 9 6 7 8 5 10 3 2
--R (- 48a b + 48a b + 222a b + 108a b - 18a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 4 9 6 7 10 3
--R (48a b + 48a b - 48a b - 48a b )cosh(x) - 12a b - 18a b + 6a b
--R *
--R sinh(x)
--R +
--R 5 8 7 6 9 4 6 4 9 6 7 8 5 5
--R (- 2a b - 4a b - 2a b )cosh(x) + (- 12a b - 24a b - 12a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 4
--R (- 24a b - 42a b - 12a b + 6a b )cosh(x)
--R +
--R 2 11 4 9 6 7 8 5 3
--R (- 16a b - 8a b + 32a b + 24a b )cosh(x)
--R +
--R 3 10 5 8 7 6 9 4 2
--R (24a b + 42a b + 12a b - 6a b )cosh(x)
--R +
--R 4 9 6 7 8 5 5 8 7 6 9 4
--R (- 12a b - 24a b - 12a b )cosh(x) + 2a b + 4a b + 2a b
--R Type: Expression(Integer)
--E 348
--S 349 of 526
t0466:= 1/(a+b*csch(x)^2)
--R
--R
--R 1
--R (258) --------------
--R 2
--R b csch(x) + a
--R Type: Expression(Integer)
--E 349
--S 350 of 526
r0466:= x/a-b^(1/2)*atan((a-b)^(1/2)*tanh(x)/b^(1/2))/a/(a-b)^(1/2)
--R
--R
--R +-------+
--R +-+ tanh(x)\|- b + a +-------+
--R - \|b atan(-----------------) + x\|- b + a
--R +-+
--R \|b
--R (259) -------------------------------------------
--R +-------+
--R a\|- b + a
--R Type: Expression(Integer)
--E 350
--S 351 of 526
a0466:= integrate(t0466,x)
--R
--R
--R (260)
--R [
--R +-----+
--R | b
--R |-----
--R \|b - a
--R *
--R log
--R 2 2 2
--R (4a b - 4a )sinh(x) + (8a b - 8a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (4a b - 4a )cosh(x) + 8b - 12a b + 4a
--R *
--R +-----+
--R | b
--R |-----
--R \|b - a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 4a b - 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (8a b - 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2 2
--R (4a b - 2a )cosh(x) + 8b - 8a b + a
--R /
--R 4 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2
--R (6a cosh(x) + 4b - 2a)sinh(x)
--R +
--R 3 4
--R (4a cosh(x) + (8b - 4a)cosh(x))sinh(x) + a cosh(x)
--R +
--R 2
--R (4b - 2a)cosh(x) + a
--R +
--R 2x
--R /
--R 2a
--R ,
--R +-------+
--R | b
--R +-------+ (2b - 2a) |- -----
--R | b \| b - a
--R - |- ----- atan(----------------------------------------------------) + x
--R \| b - a 2 2
--R a sinh(x) + 2a cosh(x)sinh(x) + a cosh(x) + 2b - a
--R --------------------------------------------------------------------------]
--R a
--R Type: Union(List(Expression(Integer)),...)
--E 351
--S 352 of 526
m0466a:= a0466.1-r0466
--R
--R
--R (261)
--R +-----+
--R +-------+ | b
--R \|- b + a |-----
--R \|b - a
--R *
--R log
--R 2 2 2
--R (4a b - 4a )sinh(x) + (8a b - 8a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (4a b - 4a )cosh(x) + 8b - 12a b + 4a
--R *
--R +-----+
--R | b
--R |-----
--R \|b - a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 4a b - 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (8a b - 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2 2
--R (4a b - 2a )cosh(x) + 8b - 8a b + a
--R /
--R 4 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2
--R (6a cosh(x) + 4b - 2a)sinh(x)
--R +
--R 3 4
--R (4a cosh(x) + (8b - 4a)cosh(x))sinh(x) + a cosh(x)
--R +
--R 2
--R (4b - 2a)cosh(x) + a
--R +
--R +-------+
--R +-+ tanh(x)\|- b + a
--R 2\|b atan(-----------------)
--R +-+
--R \|b
--R /
--R +-------+
--R 2a\|- b + a
--R Type: Expression(Integer)
--E 352
--S 353 of 526
d0466a:= D(m0466a,x)
--R
--R
--R (262)
--R 4 3 2 2
--R b sinh(x) + 4b cosh(x)sinh(x) + (6b cosh(x) + 2b)sinh(x)
--R +
--R 3 4 2
--R (4b cosh(x) + 4b cosh(x))sinh(x) + b cosh(x) + 2b cosh(x) + b
--R *
--R 2
--R tanh(x)
--R +
--R 4 3 2 2
--R - b sinh(x) - 4b cosh(x)sinh(x) + (- 6b cosh(x) + 2b)sinh(x)
--R +
--R 3 4 2
--R (- 4b cosh(x) + 4b cosh(x))sinh(x) - b cosh(x) + 2b cosh(x) - b
--R /
--R 2 4 2 3
--R (a b - a )sinh(x) + (4a b - 4a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((6a b - 6a )cosh(x) + 4b - 6a b + 2a )sinh(x)
--R +
--R 2 3 2 2
--R ((4a b - 4a )cosh(x) + (8b - 12a b + 4a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R (a b - a )cosh(x) + (4b - 6a b + 2a )cosh(x) + a b - a
--R *
--R 2
--R tanh(x)
--R +
--R 4 3
--R - a b sinh(x) - 4a b cosh(x)sinh(x)
--R +
--R 2 2 2
--R (- 6a b cosh(x) - 4b + 2a b)sinh(x)
--R +
--R 3 2 4
--R (- 4a b cosh(x) + (- 8b + 4a b)cosh(x))sinh(x) - a b cosh(x)
--R +
--R 2 2
--R (- 4b + 2a b)cosh(x) - a b
--R Type: Expression(Integer)
--E 353
--S 354 of 526
m0466b:= a0466.2-r0466
--R
--R
--R (263)
--R -
--R +-------+
--R +-------+ | b
--R \|- b + a |- -----
--R \| b - a
--R *
--R +-------+
--R | b
--R (2b - 2a) |- -----
--R \| b - a
--R atan(----------------------------------------------------)
--R 2 2
--R a sinh(x) + 2a cosh(x)sinh(x) + a cosh(x) + 2b - a
--R +
--R +-------+
--R +-+ tanh(x)\|- b + a
--R \|b atan(-----------------)
--R +-+
--R \|b
--R /
--R +-------+
--R a\|- b + a
--R Type: Expression(Integer)
--E 354
--S 355 of 526
d0466b:= D(m0466b,x)
--R
--R
--R (264)
--R 4 3 2 2
--R b sinh(x) + 4b cosh(x)sinh(x) + (6b cosh(x) + 2b)sinh(x)
--R +
--R 3 4 2
--R (4b cosh(x) + 4b cosh(x))sinh(x) + b cosh(x) + 2b cosh(x) + b
--R *
--R 2
--R tanh(x)
--R +
--R 4 3 2 2
--R - b sinh(x) - 4b cosh(x)sinh(x) + (- 6b cosh(x) + 2b)sinh(x)
--R +
--R 3 4 2
--R (- 4b cosh(x) + 4b cosh(x))sinh(x) - b cosh(x) + 2b cosh(x) - b
--R /
--R 2 4 2 3
--R (a b - a )sinh(x) + (4a b - 4a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((6a b - 6a )cosh(x) + 4b - 6a b + 2a )sinh(x)
--R +
--R 2 3 2 2
--R ((4a b - 4a )cosh(x) + (8b - 12a b + 4a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R (a b - a )cosh(x) + (4b - 6a b + 2a )cosh(x) + a b - a
--R *
--R 2
--R tanh(x)
--R +
--R 4 3
--R - a b sinh(x) - 4a b cosh(x)sinh(x)
--R +
--R 2 2 2
--R (- 6a b cosh(x) - 4b + 2a b)sinh(x)
--R +
--R 3 2 4
--R (- 4a b cosh(x) + (- 8b + 4a b)cosh(x))sinh(x) - a b cosh(x)
--R +
--R 2 2
--R (- 4b + 2a b)cosh(x) - a b
--R Type: Expression(Integer)
--E 355
--S 356 of 526
t0467:= 1/(a+b*csch(x)^2)^2
--R
--R
--R 1
--R (265) -------------------------------
--R 2 4 2 2
--R b csch(x) + 2a b csch(x) + a
--R Type: Expression(Integer)
--E 356
--S 357 of 526
r0467:= x/a^2+1/2*(a-2*b)*b^(1/2)*atan((a-b)^(1/2)*_
tanh(x)/b^(1/2))/a^2/(a-b)^(3/2)-2*b^(1/2)*atan((a-b)^(1/2)*_
tanh(x)/b^(1/2))/a^2/(a-b)^(1/2)+1/2*b*_
tanh(x)/a/(a-b)/(b+(a-b)*tanh(x)^2)
--R
--R
--R (266)
--R +-------+
--R 2 2 2 2 +-+ tanh(x)\|- b + a
--R ((- 2b + 5a b - 3a )tanh(x) + 2b - 3a b)\|b atan(-----------------)
--R +-+
--R \|b
--R +
--R 2 2 2 2 +-------+
--R ((2b - 4a b + 2a )x tanh(x) + a b tanh(x) + (- 2b + 2a b)x)\|- b + a
--R /
--R 2 2 3 4 2 2 2 3 +-------+
--R ((2a b - 4a b + 2a )tanh(x) - 2a b + 2a b)\|- b + a
--R Type: Expression(Integer)
--E 357
--S 358 of 526
a0467:= integrate(t0467,x)
--R
--R
--R (267)
--R [
--R 2 4 2 3
--R (2a b - 3a )sinh(x) + (8a b - 12a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((12a b - 18a )cosh(x) + 8b - 16a b + 6a )sinh(x)
--R +
--R 2 3 2 2
--R ((8a b - 12a )cosh(x) + (16b - 32a b + 12a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R (2a b - 3a )cosh(x) + (8b - 16a b + 6a )cosh(x) + 2a b - 3a
--R *
--R +-----+
--R | b
--R |-----
--R \|b - a
--R *
--R log
--R 2 2 2
--R (4a b - 4a )sinh(x) + (8a b - 8a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (4a b - 4a )cosh(x) + 8b - 12a b + 4a
--R *
--R +-----+
--R | b
--R |-----
--R \|b - a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 4a b - 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (8a b - 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2 2
--R (4a b - 2a )cosh(x) + 8b - 8a b + a
--R /
--R 4 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2
--R (6a cosh(x) + 4b - 2a)sinh(x)
--R +
--R 3 4
--R (4a cosh(x) + (8b - 4a)cosh(x))sinh(x) + a cosh(x)
--R +
--R 2
--R (4b - 2a)cosh(x) + a
--R +
--R 2 4 2 3
--R (4a b - 4a )x sinh(x) + (16a b - 16a )x cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 2
--R ((24a b - 24a )x cosh(x) + (16b - 24a b + 8a )x + 8b - 4a b)sinh(x)
--R +
--R 2 3
--R (16a b - 16a )x cosh(x)
--R +
--R 2 2 2
--R ((32b - 48a b + 16a )x + 16b - 8a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 2 4 2 2 2 2
--R (4a b - 4a )x cosh(x) + ((16b - 24a b + 8a )x + 8b - 4a b)cosh(x)
--R +
--R 2
--R (4a b - 4a )x + 4a b
--R /
--R 3 4 4 3 4 3
--R (4a b - 4a )sinh(x) + (16a b - 16a )cosh(x)sinh(x)
--R +
--R 3 4 2 2 2 3 4 2
--R ((24a b - 24a )cosh(x) + 16a b - 24a b + 8a )sinh(x)
--R +
--R 3 4 3 2 2 3 4
--R ((16a b - 16a )cosh(x) + (32a b - 48a b + 16a )cosh(x))sinh(x)
--R +
--R 3 4 4 2 2 3 4 2 3 4
--R (4a b - 4a )cosh(x) + (16a b - 24a b + 8a )cosh(x) + 4a b - 4a
--R ,
--R
--R 2 4 2 3
--R (- 2a b + 3a )sinh(x) + (- 8a b + 12a )cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R ((- 12a b + 18a )cosh(x) - 8b + 16a b - 6a )sinh(x)
--R +
--R 2 3 2 2
--R ((- 8a b + 12a )cosh(x) + (- 16b + 32a b - 12a )cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R (- 2a b + 3a )cosh(x) + (- 8b + 16a b - 6a )cosh(x) - 2a b + 3a
--R *
--R +-------+
--R | b
--R +-------+ (2b - 2a) |- -----
--R | b \| b - a
--R |- ----- atan(----------------------------------------------------)
--R \| b - a 2 2
--R a sinh(x) + 2a cosh(x)sinh(x) + a cosh(x) + 2b - a
--R +
--R 2 4 2 3
--R (2a b - 2a )x sinh(x) + (8a b - 8a )x cosh(x)sinh(x)
--R +
--R 2 2 2 2 2 2
--R ((12a b - 12a )x cosh(x) + (8b - 12a b + 4a )x + 4b - 2a b)sinh(x)
--R +
--R 2 3
--R (8a b - 8a )x cosh(x)
--R +
--R 2 2 2
--R ((16b - 24a b + 8a )x + 8b - 4a b)cosh(x)
--R *
--R sinh(x)
--R +
--R 2 4 2 2 2 2
--R (2a b - 2a )x cosh(x) + ((8b - 12a b + 4a )x + 4b - 2a b)cosh(x)
--R +
--R 2
--R (2a b - 2a )x + 2a b
--R /
--R 3 4 4 3 4 3
--R (2a b - 2a )sinh(x) + (8a b - 8a )cosh(x)sinh(x)
--R +
--R 3 4 2 2 2 3 4 2
--R ((12a b - 12a )cosh(x) + 8a b - 12a b + 4a )sinh(x)
--R +
--R 3 4 3 2 2 3 4
--R ((8a b - 8a )cosh(x) + (16a b - 24a b + 8a )cosh(x))sinh(x)
--R +
--R 3 4 4 2 2 3 4 2 3 4
--R (2a b - 2a )cosh(x) + (8a b - 12a b + 4a )cosh(x) + 2a b - 2a
--R ]
--R Type: Union(List(Expression(Integer)),...)
--E 358
--S 359 of 526
m0467a:= a0467.1-r0467
--R
--R
--R (268)
--R 2 2 3 4
--R (2a b - 5a b + 3a )sinh(x)
--R +
--R 2 2 3 3
--R (8a b - 20a b + 12a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 3 2 2 3
--R ((12a b - 30a b + 18a )cosh(x) + 8b - 24a b + 22a b - 6a )
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 3
--R (8a b - 20a b + 12a )cosh(x)
--R +
--R 3 2 2 3
--R (16b - 48a b + 44a b - 12a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 4
--R (2a b - 5a b + 3a )cosh(x)
--R +
--R 3 2 2 3 2 2 2 3
--R (8b - 24a b + 22a b - 6a )cosh(x) + 2a b - 5a b + 3a
--R *
--R 2
--R tanh(x)
--R +
--R 2 2 4 2 2 3
--R (- 2a b + 3a b)sinh(x) + (- 8a b + 12a b)cosh(x)sinh(x)
--R +
--R 2 2 2 3 2 2 2
--R ((- 12a b + 18a b)cosh(x) - 8b + 16a b - 6a b)sinh(x)
--R +
--R 2 2 3 3 2 2
--R ((- 8a b + 12a b)cosh(x) + (- 16b + 32a b - 12a b)cosh(x))sinh(x)
--R +
--R 2 2 4 3 2 2 2 2
--R (- 2a b + 3a b)cosh(x) + (- 8b + 16a b - 6a b)cosh(x) - 2a b
--R +
--R 2
--R 3a b
--R *
--R +-----+
--R +-------+ | b
--R \|- b + a |-----
--R \|b - a
--R *
--R log
--R 2 2 2
--R (4a b - 4a )sinh(x) + (8a b - 8a )cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (4a b - 4a )cosh(x) + 8b - 12a b + 4a
--R *
--R +-----+
--R | b
--R |-----
--R \|b - a
--R +
--R 2 4 2 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2 2 2
--R (6a cosh(x) + 4a b - 2a )sinh(x)
--R +
--R 2 3 2 2 4
--R (4a cosh(x) + (8a b - 4a )cosh(x))sinh(x) + a cosh(x)
--R +
--R 2 2 2 2
--R (4a b - 2a )cosh(x) + 8b - 8a b + a
--R /
--R 4 3
--R a sinh(x) + 4a cosh(x)sinh(x)
--R +
--R 2 2
--R (6a cosh(x) + 4b - 2a)sinh(x)
--R +
--R 3 4
--R (4a cosh(x) + (8b - 4a)cosh(x))sinh(x) + a cosh(x)
--R +
--R 2
--R (4b - 2a)cosh(x) + a
--R +
--R 2 2 3 4
--R (4a b - 10a b + 6a )sinh(x)
--R +
--R 2 2 3 3
--R (16a b - 40a b + 24a )cosh(x)sinh(x)
--R +
--R 2 2 3 2 3 2 2
--R (24a b - 60a b + 36a )cosh(x) + 16b - 48a b + 44a b
--R +
--R 3
--R - 12a
--R *
--R 2
--R sinh(x)
--R +
--R 2 2 3 3
--R (16a b - 40a b + 24a )cosh(x)
--R +
--R 3 2 2 3
--R (32b - 96a b + 88a b - 24a )cosh(x)
--R *
--R sinh(x)
--R +
--R 2 2 3 4
--R (4a b - 10a b + 6a )cosh(x)
--R +
--R 3 2 2 3 2 2 2 3
--R (16b - 48a b + 44a b - 12a )cosh(x) + 4a b - 10a b + 6a
--R *
--R 2
--R tanh(x)
--R +
--R 2 2 4 2 2 3
--R (- 4a b + 6a b)sinh(x) + (- 16a b + 24a b)cosh(x)sinh(x)
--R +
--R 2 2 2 3 2 2 2
--R ((- 24a b + 36a b)cosh(x) - 16b + 32a b - 12a b)sinh(x)
--R +
--R 2 2 3 3 2 2
--R ((- 16a b + 24a b)cosh(x) + (- 32b + 64a b - 24a b)cosh(x))
--R *
--R sinh(x)
--R +
--R 2 2 4 3 2 2 2 2
--R (- 4a b + 6a b)cosh(x) + (- 16b + 32a b - 12a b)cosh(x) - 4a b
--R +
--R 2
--R 6a b
--R *
--R +-------+
--R +-+ tanh(x)\|- b + a
--R \|b atan(-----------------)
--R +-+
--R \|b
--R +
--R 3 2 2 2
--R (8b - 12a b + 4a b)sinh(x)
--R +
--R 3 2 2
--R (16b - 24a b + 8a b)cosh(x)sinh(x)
--R +
--R 3 2 2 2 2 2
--R (8b - 12a b + 4a b)cosh(x) + 4a b - 4a b
--R *
--R 2
--R tanh(x)
--R +
--R 2 4 2 3
--R - 2a b sinh(x) - 8a b cosh(x)sinh(x)
--R +
--R 2 2 2 2 2
--R (- 12a b cosh(x) - 8a b + 4a b)sinh(x)
--R +
--R 2 3 2 2
--R (- 8a b cosh(x) + (- 16a b + 8a b)cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 2
--R - 2a b cosh(x) + (- 8a b + 4a b)cosh(x) - 2a b
--R *
--R tanh(x)
--R +
--R 3 2 2 3 2
--R (- 8b + 4a b )sinh(x) + (- 16b + 8a b )cosh(x)sinh(x)
--R +
--R 3 2 2 2
--R (- 8b + 4a b )cosh(x) - 4a b
--R *
--R +-------+
--R \|- b + a
--R /
--R 3 2 4 5 4
--R (4a b - 8a b + 4a )sinh(x)
--R +
--R 3 2 4 5 3
--R (16a b - 32a b + 16a )cosh(x)sinh(x)
--R +
--R 3 2 4 5 2 2 3 3 2 4 5
--R ((24a b - 48a b + 24a )cosh(x) + 16a b - 40a b + 32a b - 8a )
--R *
--R 2
--R sinh(x)
--R +
--R 3 2 4 5 3
--R (16a b - 32a b + 16a )cosh(x)
--R +
--R 2 3 3 2 4 5
--R (32a b - 80a b + 64a b - 16a )cosh(x)
--R *
--R sinh(x)
--R +
--R 3 2 4 5 4
--R (4a b - 8a b + 4a )cosh(x)
--R +
--R 2 3 3 2 4 5 2 3 2 4 5
--R (16a b - 40a b + 32a b - 8a )cosh(x) + 4a b - 8a b + 4a
--R *
--R 2
--R tanh(x)
--R +
--R 3 2 4 4 3 2 4 3
--R (- 4a b + 4a b)sinh(x) + (- 16a b + 16a b)cosh(x)sinh(x)
--R +
--R 3 2 4 2 2 3 3 2 4 2
--R ((- 24a b + 24a b)cosh(x) - 16a b + 24a b - 8a b)sinh(x)
--R +
--R 3 2 4 3 2 3 3 2 4
--R ((- 16a b + 16a b)cosh(x) + (- 32a b + 48a b - 16a b)cosh(x))
--R *
--R sinh(x)
--R +
--R 3 2 4 4 2 3 3 2 4 2 3 2
--R (- 4a b + 4a b)cosh(x) + (- 16a b + 24a b - 8a b)cosh(x) - 4a b
--R +
--R 4
--R 4a b
--R *
--R +-------+
--R \|- b + a
--R Type: Expression(Integer)
--E 359
--S 360 of 526
d0467a:= D(m0467a,x)
--R
--R
--R (269)
--R 2 8 2 7
--R (b - 2a b)sinh(x) + (8b - 16a b)cosh(x)sinh(x)
--R +
--R 2 2 2 6
--R ((28b - 56a b)cosh(x) - 4b )sinh(x)
--R +
--R 2 3 2 5
--R ((56b - 112a b)cosh(x) - 24b cosh(x))sinh(x)
--R +
--R 2 4 2 2 2 4
--R ((70b - 140a b)cosh(x) - 60b cosh(x) - 10b + 4a b)sinh(x)
--R +
--R 2 5 2 3 2
--R ((56b - 112a b)cosh(x) - 80b cosh(x) + (- 40b + 16a b)cosh(x))
--R *
--R 3
--R sinh(x)
--R +
--R 2 6 2 4 2 2
--R (28b - 56a b)cosh(x) - 60b cosh(x) + (- 60b + 24a b)cosh(x)
--R +
--R 2
--R - 4b
--R *
--R 2
--R sinh(x)
--R +
--R 2 7 2 5 2 3
--R (8b - 16a b)cosh(x) - 24b cosh(x) + (- 40b + 16a b)cosh(x)
--R +
--R 2
--R - 8b cosh(x)
--R *
--R sinh(x)
--R +
--R 2 8 2 6 2 4
--R (b - 2a b)cosh(x) - 4b cosh(x) + (- 10b + 4a b)cosh(x)
--R +
--R 2 2 2
--R - 4b cosh(x) + b - 2a b
--R *
--R 4
--R tanh(x)
--R +
--R 2 8 2 7
--R (- 2b + 2a b)sinh(x) + (- 16b + 16a b)cosh(x)sinh(x)
--R +
--R 2 2 2 6
--R ((- 56b + 56a b)cosh(x) + 8b - 8a b)sinh(x)
--R +
--R 2 3 2 5
--R ((- 112b + 112a b)cosh(x) + (48b - 48a b)cosh(x))sinh(x)
--R +
--R 2 4 |
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Spectral Theory, Discrete Dirac Operator, Hausdorff dimension
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\begin{document}
\title{{Sparse 1D discrete Dirac operators II: Spectral properties}}
\author{S. L. Carvalho\thanks{
Supported by FAPESP under grant \#2010/10236-3. Email: \texttt{[email protected]}},
\ C. R. de Oliveira\thanks{Partially supported by CNPq.
Email: \texttt{[email protected]}} \ \\
\small Departamento de Matem\'atica -- UFSCar, \\
\small S\~{a}o Carlos, SP, 13560-970 Brazil \\
\author{ R. A. Prado\thanks{
Email:\texttt{\ [email protected]}}} \\ R. A. Prado\thanks{Partially
supported by PROPe/UNESP and FUNDUNESP. Email: \texttt{[email protected]}}
\\
\small Departamento de Matem\'{a}tica, Estat\'{\i}stica e Computa\c{c}\~ao -- UNESP,
\\ \small Presidente Prudente, SP,
19060-900 Brazil}
\date{}
\maketitle
\begin{abstract}
We study spectral properties of some discrete Dirac operators
with nonzero potential only at some sparse and suitably randomly distributed
positions. As observed in the corresponding Schr\"o\-ding\-er operators, we
determine the
Hausdorff dimension of its spectral measure and identify a sharp spectral
transition from point to singular continuous.
\end{abstract}
\section{Introduction\label{2}}
\setcounter{equation}{0} \setcounter{theorem}{0}
Sparse potentials have played a special role in the context of discrete (also called
``tight-binding'') Schr\"o\-ding\-er operators, particularly due to the possibility of
rather detailed spectral and dynamical analysis, which makes such investigations
especially interesting (see, for instance,
\cite{CMW,CombesMantica,JitLast,Krutikov,MWGA,Tcherem,Zla}). Pearson \cite{Pearson}
was the first to recognize the
utility of this kind of
potential on the construction of Schr\"o\-ding\-er operators with singular
continuous spectrum (see Chapter~13 in~\cite{ISTQD} for dynamical interpretations
of different spectral types). There are many works that deal with sparse potentials,
related to distinct sparseness conditions and `barriers' with distinct growing rate
(see the above mentioned references and~\cite{L2} for a brief discussion and a
collection of results).
In the works \cite{OP,OP1}, two of the present authors have proposed a discrete
version of
the unidimensional Dirac operator, now defined on $l^2(\mathbb{Z},\mathbb{C}^2)$,
which can be considered a relativistic version of the usual discrete Schr\"o\-ding\-er
operator (with Planck's constant $\hbar\equiv 1$)
\begin{equation}\label{HNRe}
(H_S\psi)_n=-\frac{1}{2m}(\Delta\psi)_n+V_n\psi_n=\frac{1}{2m}\left(
\psi_{n+1}+\psi_{n-1}-2\psi_n\right)+V_n\psi_n\; ,
\end{equation}
\noindent
where $(V_n)$ is the potential given by a real sequence and $m>0$ denotes the
particle mass. Explicitly, the action of the discrete Dirac operator is given by
\begin{equation}\label{MDir}
(H_D)(m,c)=H_0(m,c)+V\,{\mathrm I_2}=c\mathcal{B}+mc^2\sigma_3+V\,{\mathrm I_2}\;,
\end{equation}
\noindent
where $c>0$ represents the speed of light,
\begin{equation*}
\mathcal{B}=\left(\begin{array}{cc}
0 & d^\ast \\
d & 0
\end{array}\right),
\end{equation*}
\noindent
$\sigma_3$ is the Pauli matrix $\left(\begin{array}{cc}1 & 0\\0 & -1
\end{array}\right)$, ${\mathrm I_2}$ is the $2\times2$ identity matrix, and~$d$ is
the finite difference operator (a discrete version of the first derivative)
\begin{equation*}
(d\psi)_n=\psi_{n+1}-\psi_n\;,
\end{equation*}
\noindent
with its adjoint given by $(d^\ast\psi)_n=\psi_n-\psi_{n-1}$; thus, $H_0(m,c)$
is a bounded self-adjoint operator with the mass parameter taking values $m\ge0$,
an important difference with respect to nonrelativistic models.
The resulting Dirac evolution equation can be written in the compact form
\begin{eqnarray}\label{HRel}
i\frac{\partial\Psi_n}{\partial t}=(H_D(m,c)\Psi)_n=\left(\begin{array}{cc} mc^2
+V_n & cd^\ast \\ cd & -mc^2 +V_n\end{array}\right)\Psi_n\;,
\end{eqnarray}
\noindent
with the ``spinorial'' wave function $\Psi=(\Psi_n)$, with
$\Psi_n=\left(\begin{array}{c}\psi_{1,n} \\
\psi_{2,n}
\end{array}\right)$, where $\psi_{1,n}$ and $\psi_{2,n}$ are the solutions associated,
respectively, with
positive and negative energy values (see, for instance, Chapter~3 in~\cite{Sa}
or~\cite{thaller}). It is possible to show \cite{OP,OP1} that the nonrelativistic
limit ($c\rightarrow\infty$)
of such Dirac operator is the corresponding Schr\"o\-ding\-er operator
\eqref{HNRe}, as one would expect.
In a previous work \cite{deOP1}, some lower bounds of the dynamics generated by
$H_D(m,c)$ with sparse potentials have been obtained. In the present work we
continue our investigations on the discrete Dirac operator with sparse potentials by
presenting a detailed analysis of spectral properties of~\eqref{MDir} defined in
$l^2(\mathbb{Z}_+,\mathbb{C}^2)$ (here
$\mathbb{Z}_+$ represents the set of all nonnegative integers~$n\ge0$), satisfying
the boundary condition,
\begin{equation}
\psi_{2,-1}\cos\phi-\psi_{1,0}\sin\phi=0\,,
\label{bouconphi}
\end{equation}
with $\phi\in[0,\pi]$, and subject to randomly sparse perturbations
composed of infinitely many vertical `barriers' whose distances from consecutive
barriers
are rapidly growing. These barriers
may assume distinct sizes, that grow, diminish or remain constant.
The
randomness in the distribution will play a fundamental role in the
determination of the exact Hausdorff dimension of the spectral measure. We will
consider a set $\{a_j\}_{j\ge 1}$ of a rapidly increasing sequence of natural
numbers, so that the potential $V_n=0$ if $n\notin\{a_j\}_{j\ge 1}$ and nonzero
$V_{a_j}$; see ahead for precise statements.
We, nevertheless, emphasize the results obtained by Zlato\v{s} \cite{Zla} in the
context of Schr\"o\-ding\-er operators, who
has proved the
existence of a transition between pure point and singular continuous spectra
for a class of sparse operators such that $\{V_{a_j}\}_{j=1}^\infty$ is a constant
sequence. Depending on the sparseness and
on the intensity of the barriers, it is possible
to have pure point or singular continuous spectrum, and with an explicitly
determination of the Hausdorff dimension of the spectral measure.
A similar deterministic model was proposed in~\cite{MWGA}, but with off-diagonal
perturbations.
Carvalho \textit{et. al.} \cite{CMW} extended this operator to the strip
$\Lambda:=\mathbb{Z}_+\times\{0,1,\ldots,L-1\}$ of width $L$ in the $\mathbb{Z}^
2_+$ plane, obtaining the Hausdorff dimension associated with its spectral
measure with arbitrary precision.
In \cite{CMW1}, the same authors have dealt with a modification of the
(unidimensional) Schr\"o\-ding\-er model proposed in~\cite{MWGA}, in which they have
considered some randomness in the position
of the barriers, and this has allowed the determination of the exact Hausdorff
dimension of the spectral measure.
Hence, by taking into account our previous study~\cite{deOP1}, it is natural to try
to extend these spectral results obtained for Schr\"o\-ding\-er operators to the Dirac
model. In order to achieve our goal, we apply and extend to this
new situation the main techniques discussed in~\cite{CMW,CMW1}. This work is to be
considered a natural continuation of~\cite{deOP1}.
Despite some results obtained here are valid for a rather general class of
potentials, we deal essentially with the sequence $(V_n(\omega))$ of
barriers given by
\begin{eqnarray}
V_n = \left\{ \begin{array}
{c@{\quad \mathrm{if} \quad}l}
v & n=a_j^{\omega} \in \mathcal{A} \, ,\\
0 & n \not\in \mathcal{A} \, ,
\end{array} \right.
\label{Ve}
\end{eqnarray}
\noindent
where $v\neq 0, -\infty<v<\infty$ and $\mathcal{A}=(a_j^\omega)_{j\ge 1}$
is a random sequence of natural numbers written in the form $a_j^\omega = a_j +
\omega_j$ such that
$a_j$ satisfies
\begin{eqnarray}
a_j-a_{j-1} \ge 2 \;, \qquad \qquad j=2,3,\ldots
\label{spacon1}
\end{eqnarray}
\noindent
and
\begin{eqnarray*}
\lim_{j\rightarrow\infty}\frac{a_{j+1}}{a_j} = \beta >1 \;;
\end{eqnarray*}
\noindent
$\omega=(\omega_1,\omega_2,\ldots)$
represents a sequence of independent random variables defined on a probability
space $(\Xi,\mathcal{B},\nu)$, so that there is some $\eta>0$ with~$\omega_j$
uniformly distributed over the finite set
$\left\{0,1,2,\ldots,\left[j^\eta\right]\right\}$, for all~$j
$ (here, $[x]$ denotes the integer part of $x\in\mathbb{R}$).
Condition~\eqref{spacon1} makes each barriers placed at a unique point,
with $\beta$ the ``sparseness parameter.'' In order to simplify our analysis, we
follow~\cite{MWGA} and restrict the
separation between barriers by the identity
\begin{eqnarray}\label{spacon}
a_j - a_{j-1} = \beta^j \;, \qquad \qquad j=2,3,\ldots
\end{eqnarray}
\noindent
with $a_1+1=\beta \ge 2$ an integer (we might have considered any real
number $\beta>1$
and rewrite~\eqref{spacon} as $a_j - a_{j-1} = \left[\beta^j\right]$,
$j=2,3,\ldots$). Summing up, the distance between (average) consecutive nonzero
potential positions grows exponentially with an additional power-law randomness;
thus, there is no superpositions among the barriers, for any $\eta>0$ (this
is an improvement with respect to Zlato\v{s}'s work that has considered $\eta=1$).
\DEFI \label{defiHvphi} Denote by $H_{v,\phi}(m,c)$ the Dirac operator~\eqref{MDir}
acting on $l^2(\mathbb{Z}_+,\mathbb{C}^2)$, with potential $(V_n(\omega))_{n\ge 0}$
satisfying \eqref{Ve}--\eqref{spacon}, and the
$\phi $-boundary condition~\eqref{bouconphi} at~$n=-1$. This will be the main object
of study here.
\DEFF
%\OBSI Note that we whereas the distances between the average values (in
%the distribution $\nu$) of the positions of the barriers grow
% exponentially (thanks to condition~\eqref{spacon}), the uncertainty of these
%positions (due to $\omega_j\in\{0,\dots,[j^\eta]\}$) grows only as a power law with
%the sparseness index; so, there is no superpositions among the barriers.
%\OBSF
Besides the verification of a huge amount of technical details and suitable
adaptations, the main difficulties we have found in such extension to the (Dirac)
relativistic framework were:
\begin{itemize}
\item The nonlinearities in the Dirac transfer matrices entries (as functions of
energy and potential values) when compared with the Schr\"o\-ding\-er case; see
Eq.~\eqref{mt}.
\item In contrast to the block-Jacobi matrices in~\cite{CMW}, the spectral matrix
$\Omega$ in the Dirac case (see
Proposition~\ref{TAC}) reflects the impossibility of decoupling the upper and lower
spinor components.
\item Finding explicit expressions for the Green's function; see Eq.~\eqref{FGreen}
(it has
involved a trial and error process while handling the exact matrix form and
indices!).
\item Finding a formula for variation of parameters (Lemma~\ref{LFVP}) was
nontrivial and it also involved a trial and error process, and this was fundamental
for an appropriate version of the Jitomirskaya-Last inequalities for Dirac operators
(Theorem~\ref{JLineq}).
\item Checking the validity of the spectral criteria of Last-Simon (see
Proposition~\ref{PROPO} and Corollary~\ref{C4.4a} for examples of application of
these
criteria), whose long details are not reported here.
\end{itemize}
We note that in the process of adaptation of known results to the relativistic
setting, we will avoid repeating the arguments and just present references when a
proof turns out to be quite similar to the corresponding one for Schr\"o\-ding\-er
operators (e.g., the spectral criteria of Last-Simon just mentioned).
\OBSI
\label{rank1}The operator $H_{v,\phi}(m,c)$ with $\phi $-phase
boundary condition~\eqref{bouconphi} may be treated as a rank-one
perturbation of the same Dirac operator with Dirichlet boundary condition
\begin{equation}
\psi_{2,-1}=0\;.
\label{boucond}
\end{equation}
As a matter of fact, if $H_{v,0}(m,c)$ represents the operator which satisfies
\eqref{boucond}, we have
\begin{equation*}
H_{v,\phi }(m,c)=H_{v,0}(m,c)+E_{0} \tan \phi\;, \label{Eqdir}
\end{equation*}
where $E_{0}$ is an operator on $l^{2}(\mathbb{Z}_{+},\mathbb{C}^2)$
given by $(E_0\Psi)_n=\left(\begin{array}{c} 0 \\ \psi_{2,n}\delta_{n,0}
\end{array}\right)$, $\delta_{n,0}=1$ if $n=0$ and zero otherwise.
\OBSF
\subsection{Main Results}In this subsection we state the main results and
conclusions of this work. Their proofs will be the discussed in the next sections.
Such results can be shortly described as follows: determination of the essential
spectrum
of~$H_{v,\phi}(m,c)$ along with the exact Hausdorff dimension of the spectral
measure, and proving the existence of a spectral transition.
What happens with the essential spectrum of $H_{0,0}(m,c)$ (obtained in
Proposition \ref{SUPO}) when we consider the
sparse barriers? This question is settled by the following theorem, whose proof is
presented in Subsection \ref{SFN2}.
\begin{theorem}
\label{tee} Let $H_{v,\phi}(m,c)$ be the Dirac operator in
Definition~\ref{defiHvphi} and let
\begin{equation}\label{Hprime}
H^\prime_{0,0}(m,c)=H_{0,0}(m,c)+v\delta_0{\mathrm I_2}\;,
\end{equation}
\noindent
where $(\delta_0\psi)_n=\delta_{n,0}\psi_n$ for any $\psi\in l^2(\mathbb{Z}_+,
\mathbb{C})$. Then,
$\sigma_{\mathrm{ess}}(H_{v,\phi}(m,c))=\sigma(H^\prime_{0,0}(m,c))$.
\end{theorem}
For the Hausdorff dimension, we have
\begin{theorem}
\label{thethe1} Let~$\Omega$ be the spectral
matrix measure of $H_{v,\phi}(m,c)$, given by Definition \ref{defiHvphi}, and the sparseness parameter~$\beta\ge2$. Then $\Omega$ is
absolutely continuous with respect to the spectral measure~$\rho$ introduced in~\eqref{TBor}. Given $\varepsilon >0$ and a
closed interval of
energies
\begin{equation}
L\subset I= \left[-\sqrt{m^2c^4+4c^2},-mc^2\right]\cup\left[mc^2,\sqrt{m^2c^4
+4c^2}\right]\;,
\label{INTI}
\end{equation}
for almost every $\phi\in[0,\pi]$ and almost every~$\omega\in\Xi$, the spectral
measure $\rho$ restrict to~$L$ has
Hausdorff dimension
\begin{equation}
\alpha _{\rho}(E)=\max\left\{0,1-\frac{\ln r}{\ln \beta }\right\}\;,
\label{HDim}
\end{equation}
with $r=r(v,E)$ given by~\eqref{erre}.
\end{theorem}
With respect to the existence of spectral transitions, now we state a result
parallel to Theorem~2.4 in~\cite{CMW1}.
\begin{theorem}
\label{tmarwre} Write
\begin{equation}
I_{1}\equiv \left\{ E \in \sigma(H_{0,0}(m,c))\setminus A:r<\beta\right\}
\label{I1}
\end{equation}
\noindent with $\beta \in \mathbb{N}$, $\beta \geq 2$ and $A$ as in~\eqref{defAeq}.
Then, for $\nu$-almost every $\omega\in\Xi$, there exists a set $A_1$ of Lebesgue
measure zero such that:
(a) the spectrum of~$H_{v,\phi}(m,c)$
restricted to the set $I_1\backslash A_1$ is purely singular continuous;
(b) the spectrum of~$H_{v,\phi}(m,c)$ is pure point when restricted to
$\sigma(H_{0,0}(m,c))
\setminus I_1$ for almost every $\phi \in \lbrack 0,\pi ]$.
\end{theorem}
Theorem~\ref{tmarwre} shows that there exists a sharp transition between
singular continuous and pure point spectrum. Note from~\eqref{HDim} that the
condition for the Hausdorff dimension to be positive is the same for the existence
of singular continuous spectrum, i.e., $\beta>r$. In fact, the set of energies
for which the Hausdorff dimension is zero coincides with the set where the
pure point spectrum is supported. This result, due to Theorem~\ref{tmarwre} is,
nevertheless, far from trivial.
Note also that there is no absolutely continuous spectrum for this class of
sparse potentials.
The proofs of Theorems \ref{thethe1} and \ref{tmarwre} are both presented in
Subsection~\ref{HDST}. Despite some obvious differences due to the
intrinsic nature of the Schr\"o\-ding\-er and Dirac
tight-binding models, if we compare the results just stated with those obtained in
\cite{CMW,CMW1,
Zla} for sparse potentials satisfying relations
\eqref{Ve}--\eqref{spacon}, we have got quite similar statements; this does not seem
to be clear from the actions of the corresponding operators and the different forms
of their transfer matrices. In any event, recall
that only in the relativistic case it is meaningful to consider~$m=0$.
\section{The Spectrum of $H_{v,\phi}(m,c)$
\label{SFN}}
\setcounter{equation}{0} \setcounter{theorem}{0}
\subsection{Green function and the spectral matrix}
\label{SFN1}
It is well known that the spectral properties of any other self-adjoint operator
are directly related to the behavior of the Green function
$\mathcal{G}(z)$, for $z=E+i\varepsilon$, in the limit $\varepsilon\downarrow 0$.
In the case of the discrete Dirac operator, the expression of the Green function
$\mathcal{G}(z)$ we introduce here is nothing but the $2\times2$ matrix
$\mathcal{G}(0,0;
z)$, whose elements are given by
\begin{eqnarray}
\mathcal{G}_{m,n}(0,0;z)=\langle e_m\otimes\delta_0,(H_{v,0}-z{\mathrm I_2})^{-1}e_n
\otimes\delta_0\rangle\;,
\label{Magreen}
\end{eqnarray}
\noindent
$m,n=1,2$, where the sequence $\{\delta_n\}_{n\ge 0}$ represents the canonical
basis for $l^2(\mathbb{Z}_+,\mathbb{C})$ , while $e_1=\left(\begin{array}{c} 1\\
0\end{array}\right)$, $e_2=\left(\begin{array}{c} 0\\1 \end{array}\right)$ is
the canonical basis for $\mathbb{C}^2$; thus, the sequence $\{e_1\otimes
\delta_n,e_2\otimes\delta_n\}_{n\ge 0}$ constitutes a basis for $l^2(\mathbb{Z}_+,
\mathbb{C}^2)$, a linear space with scalar product denoted by $\langle\cdot,
\cdot\rangle$.
\OBSI The results presented in this subsection refer to the operator $H_{v,0}(m,
c)$, with Dirichlet boundary condition \eqref{boucond}. See Remark \ref{Hvphi}
for a discussion of the results regarding the operator $H_{v,\phi}(m,c)$.
\OBSF
Explicitly, $\mathcal{G}(z)$ is given by
\begin{eqnarray}
\mathcal{G}(z)=\frac{1}{c}\left(\begin{array}{cc} u_{1,0}^D(z)\chi_{1,0}(z) &
u_{1,0}^D(z)\chi_{2,0}(z)\\ u_{1,0}^D(z)\chi_{2,0}(z) & u_{2,0}^D(z)\chi_{2,0}(z)
\end{array}\right)\;,
\end{eqnarray}
\noindent
where $\chi_n(z)=\left(\begin{array}{c}\chi_{1,n}(z)\\\chi_{2,n}(z)\end{array}
\right)= -u_n^N(z)+m(z)u_n^D(z)$ is a $l^2(\mathbb{Z}_+,\mathbb{C}^2)$ solution
to the Dirac equation
\begin{equation}
H_{v,0}\Psi=z\Psi\;;
\label{eqdir}
\end{equation}
\noindent
for some fixed $z$, the set of all solutions to~\eqref{eqdir} is a linear space
of dimension 2 (the Dirac equation~\eqref{eqdir} is in fact a system of 2
differential
equations of first order), whose appropriate basis is given by the functions
$u^D(z)$ and $u^N(z)$ satisfying the initial conditions
\begin{eqnarray}
\begin{array}{l}
u^D_{2,-1} = 0 \,, \qquad u^D_{1,0} = 1 \,, \\
u^N_{2,-1} = 1 \,, \qquad u^N_{1,0} = 0\,,
\end{array}
\label{DNboucon}
\end{eqnarray}
\noindent
which correspond to Dirichlet and Neumann boundary conditions, respectively;
finally, $m(z)$ is
the well-known Weyl-Titchmarsh function~\cite{CodLev}.
Equation~\eqref{Magreen} is in fact a special case derived from the general
formula we have found:
\begin{eqnarray}
\mathcal{G}(i,j;z)=\frac{1}{c}\left\{
\begin{array}{c@{\qquad \mathrm{if} \qquad}l}\left(\begin{array}{cc}
u_{1,j}^D(z)\chi_{1,i}(z) &
u_{2,j}^D(z)\chi_{1,i}(z)\\ u_{1,j}^D(z)\chi_{2,i}(z) & u_{2,j}^D(z)\chi_{2,i}(z)
\end{array}\right) & j+1\le i\;, \\
\left(\begin{array}{cc}
u_{1,i}^D(z)\chi_{1,j}(z) &
u_{1,i}^D(z)\chi_{2,j}(z)\\ u_{2,i}^D(z)\chi_{1,j}(z) & u_{2,i}^D(z)\chi_{2,j}(z)
\end{array}\right) & j\ge i\;,\\
\left(\begin{array}{cc}
u_{1,i}^D(z)\chi_{1,i}(z) &
u_{1,i}^D(z)\chi_{2,i}(z)\\ u_{1,i}^D(z)\chi_{2,i}(z) & u_{2,i}^D(z)\chi_{2,i}(z)
\end{array}\right) & j= i\;
\end{array} \right.
\label{FGreen}
\end{eqnarray}
\noindent
(note that the matricial function defined above is continuous at $j=i$).
\OBSI
The
matrix $\mathcal{G}(i,j;z)$ is nothing but the integral kernel of the
resolvent operator $(H_{v,0}-z{\mathrm I_2})^{-1}$; therefore, it must satisfy
the identity
\begin{equation*}
\left(\sum_{k=0}^\infty \left(H_{v,0}(m,c)\right)_{ik}\mathcal{G}(k,j;z)\right)_{lm}
=\delta_{ij}\delta_{lm}\;,
\end{equation*}
\noindent
$i,j\in\mathbb{Z}_+$, $l,m=1,2$. This assertion can be checked by direct substitution.
\OBSF
If $\{P_\Gamma\}$ is a family of spectral projections with respect
to the Borel subset $\Gamma\subset \mathbb R$, it follows by the spectral theorem
that
\begin{equation*}
\mathcal{G}(z)=\int_{-\infty}^{\infty}\frac{d\Omega(\lambda)}{\lambda-z}\;,
\end{equation*}
\noindent
where $\Omega_{m,n}=\langle e_m\otimes\delta_0,P_E e_n\otimes\delta_0
\rangle$, $m,n=1,2$ are the elements of the spectral matrix $\Omega$ of $H_{v,0}$.
Since $e_1\otimes
\delta_0=\left(\begin{array}{c} \delta_0\\ 0\end{array}\right)$, $e_2
\otimes\delta_0=\left(\begin{array}{c} 0\\\delta_0 \end{array}\right)$ are
cyclic vectors in $l^2(\mathbb{Z}_+,\mathbb{C}^2)$, it is sufficient to deal
with this spectral measure, since any other will be absolutely continuous
with respect to $(\Omega_{m,n})_{m,n=1}^2$.
Explicitly, we have
\begin{eqnarray*}
\mathcal{G}(z)=\frac{1}{c}\left(\begin{array}{cc} m(z) &-1+m(z)\frac{mc^2-z+V_0}
{c}\\ -1+m(z)\frac{mc^2-z+V_0}{c} & \frac{mc^2-z+V_0}{c}\left(-1+
m(z)\frac{mc^2-z+V_0}{c}\right)
\end{array}\right)\;,
\end{eqnarray*}
\noindent
and the imaginary part of $\mathcal{G}(z)$ is
\begin{eqnarray}
\label{Geze}
\Im\mathcal{G}(z)=\frac{1}{c}\left(\begin{array}{cc} \Im m(z) &-\frac{\Im z}{c}
\Re m(z)+\Re a\Im m(z)\\ -\frac{\Im z}{c}\Re m(z)+\Re a\Im m(z)
& -\frac{\Im z}{c} + \Re m(z)\Im a^2+\Im m(z)\Re a^2
\end{array}\right)\;,
\end{eqnarray}
\noindent
with $a\equiv (mc^2-z+V_0)/c$. Hence, we conclude that the most important spectral
properties of
$H_{v,0}(m,c)$ depend on the boundary behavior of the function $\Im
m(E+i\varepsilon)$ as $\varepsilon\downarrow 0$ (since $\lim_{\varepsilon
\downarrow 0}\Im a=\lim_{\varepsilon\downarrow 0}\Im z=0$).
It follows from the definition of $m(z)$ and Theorem~3.1 in Chapter~9
of~\cite{CodLev} that
\begin{equation}\label{TBor}
m(z)=\int_{-\infty}^\infty\frac{d\rho(\lambda)}{\lambda-z}\;,
\end{equation}
with $\rho(\lambda)$ denoting a nondecreasing function of bounded variation,
continuous to
the right and that satisfies the limits $\lim_{\lambda\rightarrow -\infty}\rho
(\lambda)=0$ and $\lim_{\lambda\rightarrow \infty}$ $\rho(\lambda)=1$. The next
result is crucial to our analysis.
\begin{proposition}\label{TAC}
The spectral matrix measure $\Omega$ is absolutely continuous with respect
to the Borel-Stieltjes measure $\rho$. Furthermore, there exists a symmetric
and nonnegative matrix $Y(E)$ such that
\begin{equation}\label{RadNik}
d\Omega(E)=Y(E)d\rho(E)\;,
\end{equation}
\noindent
with
\begin{eqnarray}\label{Y}
Y(E)=\frac{1}{c}\left(\begin{array}{cc} 1 & \frac{mc^2-E+V_0}
{c}\\ \frac{mc^2-E+V_0}{c} & \left(\frac{mc^2-E+V_0}{c}\right)^2
\end{array}\right)\;.
\end{eqnarray}
\end{proposition}
\begin{proof}
By the Radon-Nikodym theorem, $\Omega$
is absolutely continuous with respect to $\rho$ if, and only if, there exists a
symmetric integrable matrix $Y(E)$ such that~\eqref{RadNik} is valid for
every $E$. Moreover, the elements of the matrix~$Y(E)$ are given by
\begin{equation}\label{Yij}
Y_{ij}(E)=\frac{d\Omega_{ij}(E)}{d\rho(E)}=\lim_{
\epsilon \downarrow 0}\frac{\Im\mathcal{G}_{ij}(E+i\epsilon )}{\Im\, m(
E +i\epsilon )}\;,
\end{equation}
\noindent
where the last equality comes from the Borel-Stieltjes inversion formula
applied to $\Omega$ and $\rho$ (see the Appendix of~\cite{LASEIEU}).
We obtain~\eqref{Y} simply plugging~\eqref{Geze} into~\eqref{Yij}. Observe that
$Y(E)$ is a symmetric integrable matrix (i.e., $Y(E)\in {\LL}^1(\mathbb{R}
,d\rho)$). This concludes the proof of the
proposition.
\end{proof}
\newline
Proposition~\ref{TAC} shows that the most important features of the spectral
matrix measure~$\Omega$, such its continuity or singularity with respect
to Lebesgue and Hausdorff measures, are entirely determined by the
Borel-Stieltjes measure $\rho$.
To deal with the spectral multiplicity issues, we need the following
\DEFI
The sesquilinear form
\begin{equation}
\langle \mathbf{f},\mathbf{g}\rangle =\int_{\mathbb{R}
}\sum_{i,j=1}^{2}f_{i}^{\ast }(E )g_{j}(E)d\Omega_{ij}(E
)\;, \label{FSES}
\end{equation}
defined for $f_{i},g_{i}\in {\LL}^{2}(\mathbb{R},\mathbb{C},d\rho)$ is
nonnegative provided $\Vert \mathbf{f}\Vert ^{2}=\langle \mathbf{f},\mathbf{f}
\rangle \geq 0$ holds in this space.
\DEFF
Let us consider the sesquilinear form~\eqref{FSES}. Suppose that $
\mathbf{f}$ is a simple function, that is, $\mathbf{f}(E )=\displaystyle
\sum_{l=1}^{n}\chi _{B_{l}}(E )(f_{l,1},f_{l,2})$, where $
(f_{l,1},f_{l,2})\in \mathbb{C}^{2}$ and $B_{l}$ are disjoint
Borel sets. For this class of functions, \eqref{FSES} is a nonnegative
sesquilinear form. If $\mathbf{f}$ is such that $f_{i}\in {\LL}^{2}(\mathbb{R},
\mathbb{C},d\rho)$, we can approximate $\mathbf{f}$ by simple
functions in order to obtain $\left\Vert \mathbf{f}\right\Vert \geq 0$. We
have, as a consequence, a separable Hilbert space
${\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega )$ with scalar product~\eqref{FSES}.
Next, since the operator $H_{v,0}(m,c)$ satisfies Weyl's limit point case (see
also~\cite{LASEIEU}), there exists a unitary
transformation
$\tilde{U}:l^{2}(\mathbb{Z}_{+},\mathbb{C}^2)\longrightarrow {\LL}^{2}(\mathbb{R},
\mathbb{C}^2,d\Omega)$ such that $H_{v,0}(m,c)=\tilde{U}^{-1}\tilde{H}_{v,0}
(m,c)\tilde{U}$, where $\tilde{H}_{v,0}(m,c):{\LL}^{2}(\mathbb{R},\mathbb{C}^{2},
d\Omega)\longrightarrow {\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega)$ is a
multiplication operator (see, e.g., Theorem 2.12 of \cite{LASEIEU}).
As for Schr\"o\-ding\-er operators, we have the following result, whose proof is a
direct
extension of Lemma~B.13 of~\cite{LASEIEU}:
\begin{lemma}
\label{lema} The set $\sigma (\rho):=\left\{ E \in \mathbb{R}
:\rho((E -\varepsilon ,E +\varepsilon ))>0\text{ for all }
\varepsilon >0\right\} $ is precisely the spectrum $\sigma (\tilde{H}_{v,0 }
(m,c))$ of the multiplication operator $\tilde{H}_{v,0}(m,c)\mathbf{f}(E)=
E \mathbf{f}(E)$, with domain $\mathcal{D}(\tilde{H}_{v,0}(m,c))=\{\mathbf{f}\in
{\LL}^{2}(\mathbb{R},\mathbb{C}
^{2},d\Omega)$ $\mid E\mathbf{f}(E)\in {\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega)
\}$.
\end{lemma}
Next, there exists a measurable unitary matrix $F(E)$ which diagonalizes $Y(E)$;
that is,
\begin{eqnarray*}\label{F}
Y(E)=F^{-1}(E)\left(\begin{array}{cc} y_1(E) & 0 \\ 0 & y_2(E)\end{array}\right)
F(E)\;,
\end{eqnarray*}
\noindent
where $y_i(E)$, $i=1,2$, are the integrable eigenvalues of $Y(E)$. The matrix
$F(E)$ provides an unitary operator
\begin{equation*}
{\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega )\rightarrow
{\LL}^{2}(\mathbb{R},y_{1}d\rho)\oplus {\LL}^{2}(\mathbb{R},y_{2}d\rho)\;,\qquad
\qquad
\mathbf{f}(E )\mapsto F\mathbf{f}(E )\;,
\end{equation*}
\noindent which leaves $\tilde{H}_{v,0}(m,c)$ invariant (with respect to the
scalar product defined in ${\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega)$). This
permits us to investigate the spectral multiplicity of $\tilde{H}_{v,0}(m,c)$.
\begin{lemma}
\label{Lmult} The spectral multiplicity of $\tilde{H}_{v,0}(m,c)$ is 1.
\end{lemma}
\begin{proof} Evaluating $y_{1}(E)$ and $y_{2}(E)$ explicitly, we obtain $y_1(E)
=0$ and $y_2(E)=(1+a^2)/c$, where $a=(mc^2-E+V_0)/c$. Thus,
\begin{equation*}
{\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega )=\{0\}\oplus {\LL}^{2}(\mathbb{R},y_{2}
d\rho)= {\LL}^{2}(\mathbb{R},y_{2}d\rho)\;.
\end{equation*}
Hence, $\tilde{H}_{v,0}(m,c)$ is equivalent to multiplication by $E$ on
${\LL}^{2}(\mathbb{R},d\rho)$, since $y_2(E)d\rho(E)$ and $d\rho(E)$ are mutually
absolutely continuous.
\end{proof}
\OBSI\label{Hvphi}
It is possible to show that we obtain similar results for the operator
$H_{v,\phi}(m,c)$, i.e., with boundary condition \eqref{bouconphi}. For this
problem, we must adopt as a basis of the space of solutions the functions $u^\phi(z)$
and
$u^{\phi^\ast}(z)$ (with ${\phi^\ast}=\phi+\pi/2$), which satisfy the initial
conditions
\begin{eqnarray*}
\begin{array}{l}
u^\phi_{2,-1} = \sin\phi \,, \qquad u^\phi_{1,0} = \cos\phi \,, \\
u^{\phi^\ast}_{2,-1} = \cos\phi \,, \qquad u^{\phi^\ast}_{1,0} = -\sin\phi\,.
\end{array}
\end{eqnarray*}
The result of Lemma \ref{lema} extends naturally, as well as the result of
Proposition \ref{TAC}, except for the Radon--Nikodym
derivative, now given by
\begin{eqnarray}\label{Y1}
Y(E)=\frac{1}{c}\left(\begin{array}{cc} \cos^2\phi & \cos\phi\left(
\frac{mc^2-E+V_0}{c}+\cos\phi\sin\phi\right)\\ \cos\phi\left(
\frac{mc^2-E+V_0}{c}+\cos\phi\sin\phi\right) &
\left(\frac{mc^2-E+V_0}{c}\cos\phi+\sin\phi\right)^2
\end{array}\right)\;;
\end{eqnarray}
\noindent
the same situation holds for Lemma \ref{Lmult}, with the eigenvalues of \eqref{Y1}
given now by $y_1=0$ and
$y_2=[{\cos^2\phi+\left((mc^2-E+V_0)/c+\sin\phi\right)^2}]/{c}$.
\OBSF
\subsection{The essential spectrum}
\label{SFN2}
Before we deal with the essential spectrum of $H_{v,\phi}(m,c)$, it would be
interesting to characterize the spectrum of the free operator $H_{0,0}(m,c)$
\cite{OP1} and its spectral matrix measure.
\begin{proposition}
The spectrum of the free operator is given by the intervals
\begin{equation}
\sigma({H}_{0,0}(m,c))=
\left[ -\sqrt{m^2c^4+4c^2},-mc^2\right] \cup
\left[mc^2 ,\sqrt{m^2c^4+4c^2}\right] .
\label{SUPO}
\end{equation}
Moreover, the spectral matrix measure $\Omega$ is purely absolutely continuous
(with respect to the Lebesgue measure $\ell$) in these intervals.
\end{proposition}
\begin{proof} In order to prove this proposition, we will determine the exact
behavior of the function $m(E+i\varepsilon)$ as $\varepsilon
\downarrow 0$.
After some manipulations, it follows by~\eqref{HRel} that
\begin{equation*}
\left(-\Delta-\frac{m^2c^4-z^2}{c^2}\right)\psi_{j,n}=0\,,\quad n\in\mathbb{Z}_+,
\end{equation*}
\noindent
where $\psi_{j}, j=1,2,$ are the components of the spinor $\Psi$.
Since $m(z)$ is uniquely defined imposing that $\Psi=-u^N+m(z)u^D$ is $l^2(
\mathbb{Z}_+,\mathbb{C}^2)$, it is simple to obtain
\begin{equation*}
m(z)=-\frac{w}{2}+\sqrt{\frac{w^{2}}{4}-1}\;,
\label{eme}
\end{equation*}
\noindent
with $w=[{m^2c^4-z^2+2c^2}]/{c^2}$. Now, put
\begin{equation*}
\Im\, m(E)=\limsup_{\varepsilon \downarrow 0}\Im\, m(z)\;,\quad z=E +i\varepsilon ,
\end{equation*}
and let $L(\rho )$ be the set of all $E \in
\mathbb{R}$ for which this limit exists. It is known (see Appendix~B
of~\cite{LASEIEU}) that the minimal (or essential) supports $\mathcal{M}$, $
\mathcal{M}_{\mathrm{ac}}$ and $\mathcal{M}_{\mathrm{s}}$ of~$\rho$, the
absolutely continuous part $\rho_{\mathrm{ac}}$ and the singular part
$\rho_{\mathrm{s}}$ of $\rho$, with respect to the Lebesgue measure in $\mathbb{R}$,
are, respectively, given by $E \in L(\rho )$ such that $0<\Im
\,m(E)\leq \infty $, $0<\Im\, \,m(E)<\infty $ and $\Im\, \,m(E)=\infty $. These
criteria can be obtained by de la Vall\'{e}e-Poussin's
decomposition theorem~\cite{Saks}, the Radon-Nikodym theorem and
the following application of Lemma~3 in~\cite{GP}:
\begin{lemma}
\label{RadNikl} If the Radon-Nikodym derivative $(d\rho/d\ell)(E)$ exists and takes
values in~$[0,\infty]$, then $\Im\, m(E)$ also exists and $
(d\rho/d\ell)(E)=(1/\pi)\Im\, m(E)$ ($\ell$ is the Lebesgue measure on $
\mathbb{R}$).
\end{lemma}
Since
\begin{equation*}
\lim_{\varepsilon\downarrow 0}\Im\, m(E +i\varepsilon)=
\begin{cases}
\displaystyle\frac{\sqrt{(E^2-m^2c^4)(m^2c^4+4c^2-E^2)}}{2c^2} & \text{if}
\;\; E^2\in[m^2c^4,m^2c^4+4c^2] ,\\
0 & \text{otherwise}\;,
\end{cases}
\end{equation*}
\noindent
it follows by the above criteria and Proposition~\ref{TAC} that the spectral
matrix measure $\Omega$ is purely absolutely continuous on the interval defined
by~\eqref{SUPO}; moreover, the essential spectrum of $H_{0,0}(m,c)$ coincides
with its own spectrum. This conclude the proof of the proposition. \end{proof}
\OBSI\label{REE}
As we have discussed in Remark~\ref{rank1}, the operator $H_{v,\phi}(m,c)$ with
boundary condition~\eqref{bouconphi} at $n=-1$ may be written
as a rank-one perturbation of $H_{v,0}(m,c)$; hence,
$\sigma_{\mathrm{ess}}(H_{v,\phi}(m
,c))=\sigma_{\mathrm{ess}}(H_{v,0}(m,c))$, by Weyl's criterion (see, for instance,
Section~11.3 of~\cite{ISTQD}). Thus, it is sufficient to deal with $H_{v,0}(m,c)$ in
order
to determine the essential spectrum of $H_{v,\phi}(m,c)$.
\OBSF
We are now ready to prove Theorem~\ref{tee}.
\noindent \begin{proof} (Theorem~\ref{tee}) By taking into account the above
results and Remark \ref{REE}, the proof of Theorem~\ref{tee} reduces to a direct
extension of Theorem~3.13 in~\cite{CFKS} to the discrete Dirac operator.
\end{proof}
The eigenvalues of $H^\prime_{0,0}(m,c)$ are isolated points of the essential
spectrum of $H_{v,0}(m,c)$; hence, they cannot belong to the continuous spectrum
of $H_{v,0}(m,c)$, neither be eigenvalues of infinite multiplicity (since we
have a unidimensional problem). Thus, they must be accumulation points of the
discrete spectrum of $H_{v,0}(m,c)$.
Since the operator $H^\prime_{0,0}(m,c)$ is a
rank-one perturbation of $H_{0,0}(m,c)$, it follows that
its spectrum is the union of the interval~\eqref{SUPO} with the possible addition of
a finite number of isolated points. Thus, it follows by Theorem~\ref{tee} that the
essential
spectrum of $H_{v,\phi}(m,c)$ has the same structure.
\section{Transfer matrices and Pr\"ufer variables}
\label{TMPV}
\zerarcounters
In order to determine the exact Hausdorff dimension of the spectral matrix
measure $\Omega$ and, consequently, the spectral nature of $H_{v,\phi}(m,c)$,
we study the exact
asymptotic behavior of the solutions to the Dirac eigenvalue equation
\begin{equation}\label{eqdirre}
H_{v,\phi}(m,c)\Psi=E\Psi\;,
\end{equation}
\noindent
with $E\in\mathbb{R}$; this is an important step in our approach. It is here
that the concepts of transfer matrix and
Pr\"ufer variables play a fundamental role. What follows is an adaptation of
the material presented in Sections~3 and~4 of~\cite{MWGA} to the Dirac operator
setting.
For $E\in \mathbb{C}$, let
\begin{equation}
T(n,n-1;E)=\left(
\begin{array}{cc}
1+\displaystyle\frac{m^2c^4-(E-V_n)^2}{c^2} & \displaystyle\frac{mc^2+E-V_n}{c}
\vspace{2mm} \\
\displaystyle\frac{mc^2-E+V_n}{c} & 1
\end{array}
\right) \label{mt}
\end{equation}
\noindent be the $2\times 2$ transfer matrix associated with the
$l^{2}(\mathbb{Z}_{+},\mathbb{C}^2)$ solution to the Dirac equation~\eqref{eqdirre}
(see \cite{OP,OP1}).
The equation
\begin{equation*}
\left(
\begin{array}{c}
\psi_{1,n+1} \\
\psi_{2,n}
\end{array}
\right) =T(n,n-1;E)\left(
\begin{array}{c}
\psi_{1,n} \\
\psi_{2,n-1}
\end{array}
\right) \label{uuTuu}
\end{equation*}
\noindent holds for $n\geq 0$, with ${\dbinom{\psi_{1,0}}{\psi_{2,-1}}}={
\dbinom{\cos\phi }{\sin \phi }}$ satisfying~\eqref{bouconphi} for some
$\phi \in \lbrack 0,\pi ]$. Another important tool is the product of the $n+1$
first transfer matrices, denoted by
\begin{equation}
T(n;E)=T(n,n-1;E)T(n-1,n-2;E)\ldots T(0,-1;E)\;.
\label{Tn}
\end{equation}
Given the values~\eqref{Ve} of the potential $V_{n}$ and the sparseness condition
\eqref{spacon}, only two different $2\times 2$ matrices appear on the r.h.s.\
of~\eqref{Tn}: that is,
\begin{equation*}
T_{v}(E)=\left(
\begin{array}{cc}
1+\displaystyle\frac{m^2c^4-(E-v)^2}{c^2} & \displaystyle\frac{mc^2+E-v}{c}
\vspace{2mm} \\
\displaystyle\frac{mc^2-E+v}{c} & 1
\end{array}\right)
\end{equation*}
and
\begin{equation*}
T_{0}(E)=\left(
\begin{array}{cc}
1+\displaystyle\frac{m^2c^4-E^2}{c^2} & \displaystyle\frac{mc^2+E}{c}
\vspace{2mm} \\
\displaystyle\frac{mc^2-E}{c} & 1
\end{array}
\right) \label{TvT0ad}
\end{equation*}
occur depending on the entry $n$ in~\eqref{mt} being or not $a_{j}^{\omega }\in
\mathcal{A}$.
Let $E=\pm\sqrt{m^2c^4+2c^2(1-\cos\varphi)}$, with $\varphi\in[0,\pi)$, be
parametrizations of the intervals~\eqref{SUPO}. Now note that, for such energies,
the free matrix $T_{0}(E)$ is similar to a purely clockwise rotation $R(\varphi)$,
that is,
\begin{equation}
UT_{0}(E)U^{-1}=\left(
\begin{array}{cc}
\cos \varphi & \sin \varphi \vspace{2mm} \\
-\sin \varphi & \cos \varphi
\end{array}
\right) =R(\varphi )\;, \label{R}
\end{equation}
where
\begin{equation*}
U\equiv \left(
\begin{array}{cc}
-\displaystyle\frac{mc^2-E}{c\sin\varphi} & -\displaystyle\frac{1-\cos\varphi}
{\sin\varphi} \vspace{2mm} \\
0 & 1
\end{array}
\right) ~.\; \label{U}
\end{equation*}
\noindent\ Note also that $U$ is not uniquely defined since any other matrix $
U^{\prime }=HU$, with $H$ commuting with $R$, satisfies~\eqref{R}.
Since the product of rotation matrices is also a rotation, we obtain
\begin{equation}
UT(n;E)U^{-1}=R((n-a_{k}^{\omega })\varphi )P(E)R((a_{k}^{\omega }-a_{k-1}^{\omega })
\varphi )\cdots P(E)R((a_{1}^{\omega }+1)\varphi ) \label{UTn}
\end{equation}
\noindent as the conjugation of~\eqref{Tn} by $U^{-1}$, for every $n\in
\mathbb{N}$ and $\omega_{j}\in \{0,, 1 , 2,\ldots ,[j^\eta]\}$, $j\geq 1$; $k$ is an
integer
such that $a_{k}^{\omega }\leq n<a_{k+1}^{\omega }$; $P(E)$ is defined by
{\small{\begin{eqnarray}\label{Pe}
\nonumber P(E)R(\varphi ) & = & UT_{v}(E)U^{-1}\\
&=&\left(
\begin{array}{cc}
1+\frac{vE_+}{2c^2}-\frac{v^2}{c^2} & -\frac{v}{2c^2}\sqrt{\frac{E_+E_-}
{{F}}}\left(E_+-\frac{4c^2}{E_+}-2v
\right)\\
-\frac{v}{2c^2}\sqrt{\frac{E_+{F}}{{E_-}}} & 1-
\frac{vE_+}{2c^2}
\end{array}\right)\;
\end{eqnarray}}}
where $E_+=E+mc^2$, $E_-=E-mc^2$ and $F= m^2c^4+4c^2-E^2$.
Next we consider the following change of
variables, known as {EFGP} transform:
\begin{eqnarray}\label{EFGPT}
\mathbf{v_k}:=\left(\begin{array}{c}
R_k\cos\theta_k^\omega\\
R_k\sin\theta_k^\omega\end{array}\right)=U\left(\begin{array}{c}
\psi_{1,k}\\
\psi_{2,k-1}\end{array}\right)=\left(\begin{array}{c}
\frac{(E-mc^2)\psi_{1,k}/c-(1-\cos\varphi)\psi_{2,k-1}}{\sin\varphi} \\
\psi_{2,k-1}\end{array}\right)\;,
\end{eqnarray}
\noindent
The variables $R_k$ and $\theta_k^\omega$ are called, respectively, Pr\"ufer
radii and angles. Observe that the Pr\"{u}fer angles are random variables, since
$\omega_j$ are randomly distributed.
The Pr\"{u}fer radii, on the other hand, are random variables only as a
function of the Pr\"{u}fer angles, and their dependence on $\omega $ will be
omitted.
\OBSI
The convention we have
employed here differs from the convention in~\cite{KLS}, since
the functions $\cos\theta_k$ and $\sin\theta_k$ are exchanged. Nevertheless,
the behavior of the Pr\"ufer variables are identical in both conventions.
\OBSF
We nonetheless use a slightly different expression of~\eqref{EFGPT}, more
adequate to relation~\eqref{UTn}. Given the vectors
\begin{equation*}
\mathbf{v}_{k}=\left(R_{k-1}\cos \theta_{k}^{\omega},R_{k-1}\sin\theta_{k}^{\omega}
\right), \qquad \tilde{\mathbf{v}}_{k}=\left(R_{k}\cos\tilde\theta_{k}^{\omega},R_{k}
\sin\tilde\theta_{k}^{\omega}\right)\;,
\label{vk}
\end{equation*}
the Pr\"{u}fer variables $\left(R_k,\theta_k^\omega\right)_{k
\ge 0}$ satisfy a recurrence relation induced by
\begin{equation}
\mathbf{v}_{k}=R((a_{k}^{\omega}-a_{k-1}^{\omega})\varphi)\tilde{\mathbf{v}}_
{k-1}
\label{vk1}
\end{equation}
\noindent and
\begin{equation}
\tilde{\mathbf{v}}_{k}=P(E)\mathbf{v}_{k}\;,
\label{vkt1}
\end{equation}
with $\mathbf{v}_{1}=R((a_{1}^{\omega }+1)\varphi)
\tilde{\mathbf{v}}_{0}$,
\begin{eqnarray*}
\tilde{\mathbf{v}}_0(\theta_0)=R_0\left(\begin{array}{c}
\cos\theta_0\\
\sin\theta_0\end{array}\right)=U\left(\begin{array}{c}
\cos\phi\\
\sin\phi\end{array}\right)=\left(\begin{array}{c}
\frac{(E-mc^2)/c\cos\phi-(1-\cos\varphi)\sin\phi}{\sin\varphi} \\
\cos\phi \end{array}\right)\;,
\label{e3.7MW1}
\end{eqnarray*}
\[
R_0^{2}=\left(\frac{((E-mc^2)\cos\phi)/c-(1-\cos\varphi)\sin\phi}
{\sin\varphi}\right)^2+{\cos}^2\phi.
\]
Thus, if $\mathbf{\psi}_n=\left(
\psi_{1,n},\psi_{2,n-1}\right)$ represents a solution to~\eqref{eqdirre}
satisfying the initial conditions $\mathbf{u}_0=\left(\cos\phi,\sin\phi
\right)$, then
\begin{eqnarray*}\label{e3.1MW1}
R(\varphi)\tilde{\mathbf{v}}_k=U\mathbf{u}_{a_{k}+1}\;.
\end{eqnarray*}
By equivalence of norms, the growth of $T(n;E)$ may be controlled by
the euclidean norm (see Section 4 of \cite{MWGA} for details)
\begin{equation*}
\Vert UT(n;E)U^{-1}v_{0}\Vert ^{2}=\Vert UT(a_{N}^{\omega }+1;E)U^{-1}
v_{0}\Vert^{2}=R_{N}^{2}\;, \label{R2N}
\end{equation*}
\noindent where the equality holds for any unit vector $v_{0}=(\cos
\theta_{0},\sin \theta_{0})$ and for each $n$ such that $a_{N}^{\omega }\leq
n<a_{N+1}^{\omega }$. Thus, from equations~\eqref{vk1},~\eqref{vkt1} and~\eqref{Pe},
$R_N^2$
can be written as
\begin{eqnarray}\label{Rnj}
\left(R_N\right)^2=\left(R_0\right)^2\displaystyle\prod_{n=1}^N\left(\frac{R_n}
{R_{n-1}}\right)^2=\left(R_0\right)^2\left(\exp\left\{\frac{1}{N}\sum_{n=1}^N
\ln f\left(\theta^\omega_n,\varphi\right)\right\}\right)^N\;.
\end{eqnarray}
\noindent
with
\begin{eqnarray}\label{efe}
f(\theta^\omega,\varphi):=\left(A(\varphi)\cos\theta^\omega+B(\varphi)\sin
\theta^\omega\right)^2+\left(C(\varphi)\cos\theta^\omega+D(\varphi)\sin
\theta^\omega\right)^2\;,
\end{eqnarray}
\noindent
where $A(\varphi)=1+\frac{v(E+mc^2)}{2c^2}-\frac{v^2}{c^2}$, $D(\varphi)=
1-\frac{v(E+mc^2)}{2c^2}$, \[
B(\varphi)=-
\frac{v}{2c^2}\sqrt{\frac{E^2-m^2c^4}
{m^2c^4+4c^2-E^2}}\left(E+mc^2-\frac{4c^2}{E+mc^2}-2v\right)
\] and \[
C(\varphi)=-\frac{v}{2c^2}\sqrt{\frac{(E+mc^2)(m^2c^4+
4c^2-E^2)}{E-mc^2}}
\] are the entries of $P(E)$.
The Pr\"{u}fer angles $(\theta_{k}^{\omega})_{k\geq 1}$ are, on the other hand,
obtained recursively by
\begin{equation}
\theta _{k}^{\omega }=\tan ^{-1}\left(\frac{C+ D\tan\theta _{k-1}^{\omega }}{A+ B
\tan\theta _{k-1}^{\omega }}
\right) -(\beta ^{k}+\omega_{k}-\omega _{k-1})\varphi \label{PA}
\end{equation}
\noindent for $k>1$, with $\theta _{1}^{\omega }$ given by
$
\theta _{1}^{\omega }=\theta _{0}-(a_{1}+\omega _{1})\varphi$.
Hence, the determination of the exact asymptotic behavior of the sequence
$\left(R_n(\theta_0)\right)_{n\ge 1}$ involves an estimate of the Birkhoff-like
sum
\begin{eqnarray}\label{SomBir}
\frac{1}{N}\sum_{n=1}^N \ln f\left(v,\theta^\omega_n\right)
\end{eqnarray}
\noindent
for $N$ large, which, on the other hand, depends on the distribution properties
of the sequence $\left(\theta_n^\omega\right)_{n\ge 1}$ of the Pr\"ufer
angles. The tool that intertwines these elements is the following theorem
(recall that a sequence $w=(x_n)_{n\ge 1}$ is said to be uniformly
distributed modulo $\pi$ ($\ud\pi$) if it is equally distributed, in
fractional portions, over half open subintervals of $[0,\pi)$).
\begin{theorem}[Theorem~1.1 in~\cite{KN}]\label{T1.1KN}
The sequence $w=(x_n)_{n\ge 1}$ of real numbers $x_n\in[0,\pi)$ is $\ud$
$\pi$
if, and only if, for every continuous real function $h$ defined on the
closed interval $I=[0,\pi]$, we have
\begin{eqnarray}\label{sud2}
\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^Nh(\{x_n\})=\frac{1}{\pi}
\int_0^\pi h(x)dx\;.
\end{eqnarray}
\end{theorem}
Theorem~\ref{T1.1KN} provides a criterion that permits the substitution,
on the asymptotic limit $N\rightarrow\infty$, of the average~\eqref{SomBir}, by the
integral
\begin{eqnarray*}\label{IntBir}
\frac{1}{\pi}\int_0^\pi \ln f(v,\theta)d\theta\;,
\end{eqnarray*}
\noindent
in case the sequence $\left(\theta_n^\omega\right)_{n\ge 1}$ of the
Pr\"ufer angles is $\ud\pi$ and $\ln f(v,\theta)$, with
$f(v,\theta)$ given by~\eqref{efe}, is a periodic Riemann integrable
function of period $\pi$.
\begin{lemma}\label{Lefe}
The function $h(\theta):=\ln f(v,\theta)$ is a periodic Riemann integrable
function of period $\pi$, which average is given by
\begin{eqnarray}\label{e4.17MW}
\frac{1}{\pi}\int_0^\pi h(\theta)d\theta = \ln r(v,E)\;,
\end{eqnarray}
\noindent
with
\begin{equation}\label{erre}
r(v,E)=1+\frac{1}{({m^2c^4+4c^2-E^2})}\,\frac{v^2}{c^2}\left[\frac{(E^2-m^2c^4)^2+4m^2c^6}{(E^2-m^2c^4)
}-4vE+ 2v^2\right]\;.
\end{equation}
\end{lemma}
\begin{proof} The proof of the lemma is an adaptation of
some results presented in Section~4 in~\cite{MWGA}. \end{proof}
Lemma~\ref{Lefe} and Theorem~\ref{T1.1KN} provide a precise estimate
for the asymptotic limit of the ``time average''~\eqref{SomBir} under
the hypothesis of the $\ud$ $\pi$ of the sequence
$\left(\theta_n^\omega\right)_{n\ge 1}$ of Pr\"ufer angles.
\begin{lemma}\label{LRP}
Let $\left(R_n(\theta_0)\right)_{n\ge 1}$ be the sequence of the Pr\"ufer
radii which satisfy the initial conditions $\mathbf{v}_0=(\cos\vartheta,
\sin\vartheta)$. Suppose there is a set $A\subset\mathbb R$ of null Lebesgue
measure so that
the sequence $\left(\theta_n^\omega\right)_{n\ge 1}$ of
the Pr\"ufer angles is $\ud\pi$ for $\varphi\in[0,\pi)\backslash A$. Then,
\begin{eqnarray}
C_{N}^{-1}r^N \le \left(R_{N}(\theta_0)\right)^2\le C_{N} r^N \;,
\label{R2N1}
\end{eqnarray}
\noindent
where $C_{N}$ is a real number such that $C_{N}>1$ and $\lim_{N\rightarrow\infty}
C_{N}^{1/N}=\left(R_{0}\right)^2$, with $r$ given by~\eqref{erre}.
\end{lemma}
\begin{proof} The inequalities~\eqref{R2N1} follow from the
hypotheses of the lemma, by equations~\eqref{Rnj}, \eqref{e4.17MW} and
the estimate
\begin{eqnarray*}\label{Disc}
\left\vert\frac{1}{N}\sum_{n=1}^N\ln f\left(v,\theta_n^\omega\right)-\frac{1}{\pi}
\int_0^\pi \ln f(v,\theta)d\theta\right\vert \le C D^\ast_N\;,
\end{eqnarray*}
\noindent
where
\begin{eqnarray*}
D^\ast_N(\theta)=D^\ast_N(\theta_1,\ldots,\theta_N)=\sup_{0<\vartheta\le\pi}
\left\vert\frac{\text{card}(\{k:\theta _{k}~\text{mod~}\pi
\in \lbrack 0,\theta ),1\leq k\leq N\})}{N}-\vartheta\right\vert
\end{eqnarray*}
is the so-called {\em discrepancy} of the sequence $\left(\theta_n\right)_{n\ge 1}$
(see~\cite{KN} for an ample discussion on discrepancy) and~$C$ is some positive
constant. For the second part of the lemma, we need the
\begin{theorem}[Corollary~1.1 of Chapter~2 in~\cite{KN}]\label{C1.1KN}
A sequence $w$ is $\ud$ $\pi$ if, and only if, $\lim_{N\rightarrow
\infty}D^\ast_N(w)=0$.
\end{theorem}
It follows by the hypothesis of uniform distribution and Theorem~\ref{C1.1KN}
that
\[
\lim_{N\rightarrow\infty}D^\ast_N(\theta^\omega)=0\;;
\] hence, $\lim_{N
\rightarrow\infty}C_{N}^{1/N}=\left(R_{0}(\theta_0)\right)^2$, and the proof of
Lemma~\ref{LRP} is complete\end{proof}
Now we finally deal with the uniform distribution of the sequence
$\left(\theta_n^\omega
\right)_{n\ge 1}$.
\begin{theorem}
\label{thethe} The sequence of Pr\"{u}fer angles $(\theta _{n}^\omega)_{n\geq 1}$
is $\ud$ $\pi $ for all $\varphi/\pi \in \lbrack 0,1]\backslash \mathbb{Q}$
and all $\omega\in \Lambda $, apart from a set with null $\nu $ measure.
\end{theorem}
\begin{proof} The proof is exactly the same of~Theorem~3.2 in
\cite{CMW1}.\end{proof}
\section{Hausdorff dimension of the spectral measure}
\label{EHD}
\setcounter{equation}{0} \setcounter{theorem}{0}
This section is devoted to the determination of the Hausdorff dimension of
the spectral measure of $H_{v,\phi}(m,c)$. At the end, the proofs of
Theorems~\ref{thethe1} and~\ref{tmarwre} are presented.
\subsection{Basic definitions and generalized subordinacy theory}
Firstly we recall some useful definitions. An almost complete description is
found in~\cite{Last}. Given a Borel set $S\subset \mathbb{R}$ and $\alpha \in
\lbrack 0,1]$, consider the number
\begin{equation}
Q_{\alpha ,\delta }(S)=\inf \left\{ \sum_{\nu =1}^{\infty }|b_{\nu
}|^{\alpha }:|b_{\nu }|<\delta ;S\subset \bigcup_{\nu =1}^{\infty }b_{\nu
}\right\} \;, \label{Qhaus}
\end{equation}
with the infimum taken over all covers by intervals of size at most $\delta $.
The limit
\begin{equation}
h^{\alpha }(S)=\lim_{\delta \downarrow 0}Q_{\alpha ,\delta }(S)\;,
\label{Mhaus}
\end{equation}
is called $\alpha $-\textit{dimensional Hausdorff (outer) measure}. The
counting measure,
at $\alpha=0$, and the Lebesgue measure, at $\alpha =1$, are important particular
cases.
It is clear by the definitions \eqref{Qhaus} and \eqref{Mhaus} that $h^{\alpha }(
S)$ is an outer measure on~$\mathbb{R}$ (see~\cite{Falconer}). For $\beta <\alpha
<\gamma $,
\begin{equation*}
\delta ^{\alpha -\gamma }Q_{\gamma ,\delta }(S)\leq Q_{\alpha ,\delta
}(S)\leq \delta ^{\alpha -\beta }Q_{\beta ,\delta }(S)\;,
\end{equation*}
\noindent holds for any $\delta >0$ and $S\subset \mathbb{R}$. So, if $
h^{\alpha }(S)<\infty $, then $h^{\gamma }(S)=0$ for $\gamma >\alpha $; if $
h^{\alpha }(S)>0$, then $h^{\beta }(S)=\infty $ for $\beta <\alpha $. Thus,
for every Borel set $S$, there is a unique $\alpha _{S}$ such that $
h^{\alpha }(S)=0$ if $\alpha >\alpha _{S}$ and $h^{\alpha }(S)=\infty $ if $
\alpha _{S}<\alpha $. The number $\alpha _{S}$ is called the \textit{Hausdorff
dimension} of the set~$S$.
Another useful concept is the \textit{exact dimension} of a measure, taken
from~\cite{RodTay}.
\DEFI
\label{Exactdim} A Borel measure~$\mu$ in $\mathbb{R}$ is said to be of
exact dimension $\alpha $, $\alpha \in \lbrack 0,1]$, if two
requirements hold: (1)~for every $\beta \in \lbrack 0,1]$ with $\beta
<\alpha $ and $S$ a set of dimension $\beta $, $\mu (S)=0$ (which means that
$\mu (S)$ gives zero weight to any set $S$ with $h^{\alpha }(S)=0$); (2)~there is a
set $S_{0}$ of dimension $\alpha $ which supports $\mu $ in the
sense that $\mu (\mathbb{R}\backslash S_{0})=0$.
\DEFF
Finally, we recall the notions of continuity and singularity of a measure
with respect to the Hausdorff measure. Given $\alpha \in \lbrack 0,1]$, a
measure $\mu $ is called $\alpha $-continuous if $\mu (S)=0$ for every set $
S$ with $h^{\alpha }(S)=0$; it is called $\alpha $-singular if it is
supported on some set $S$ with $h^{\alpha }(S)=0$.
\OBSI\label{DSC}
It is possible to reformulate
Definition~\ref{Exactdim} in this context: a measure $\mu $ is said to have
exact dimension $\alpha $ if, for every $\epsilon >0$, it is simultaneously $(\alpha
-\epsilon )$-continuous and $(\alpha +\epsilon )$-singular.
\OBSF
Jitomirskaya and Last \cite{JitLast} extended, to Hausdorff measures, the
Gilbert-Pearson theory of subordinacy \cite{GP} for Lebesgue measures, which
relates the spectral
property of $\rho $ to the rate of growth of the solutions to the Schr\"{o}dinger
equation. Now we describe the extension of these results to Dirac
operators~\eqref{MDir}.
A solution $\Psi$ to~\eqref{eqdirre} is said to be subordinate if
\begin{equation*}
\lim_{l\rightarrow \infty }\frac{\left\Vert \Psi\right\Vert _{l}}{\left\Vert
\Phi\right\Vert _{l}}=0 \label{sub}
\end{equation*}
\noindent holds for any linearly independent solution $\Phi$ to~\eqref{eqdirre},
where $\left\Vert \cdot \right\Vert _{l}$ denotes the $l^{2}(\mathbb{Z}_{+},
\mathbb{C}^2)$-norm truncated at the length $l\in \mathbb{R}$, i.e.,
\begin{equation*}
\left\Vert \Psi\right\Vert _{l}^{2}\equiv
\sum_{n=0}^{[l]}\left[|\psi_{1,n}|^{2}+|\psi_{2,n}|^{2}\right]+(l-[l])\left(|\psi_{1,
[l]+1}|^{2}+|\psi_{2,[l]+1}|^{2}\right)\;,
\end{equation*}
\noindent $\lbrack l]$ the integer part of $l$.
Following Jitomirskaya-Last \cite{JitLast}, for any given $
\epsilon >0$ introduce a length $l(\epsilon)\in (0,\infty )$ by the equality
\begin{equation}
\left\Vert u^D\right\Vert _{l(\epsilon )}\left\Vert
u^N\right\Vert _{l(\epsilon )}=\frac{c}{2\epsilon } \label{eps}
\end{equation}
\noindent (see equation (1.12) in~\cite{JitLast}), where $u^{D}$ and $u^{N}$ are the
solutions to~\eqref{eqdirre} which satisfy the initial conditions
\eqref{DNboucon}.
At most one of the solutions $\left\{ u^D,u^N\right\} $ to~\eqref{eqdirre} belongs
to~$l^{2}(\mathbb{Z}_+,\mathbb{C}^2)$, thanks to the constancy of the Wronskian,
which follows by the Green's identity; that is, for $n\ge -1$,
\[
\sum_{n=0}^N\left(\Psi^
\ast_n(H_D\Phi)_n-(H_D\Psi)_n^\ast\Phi_n\right)=W[\Phi,\Psi](N)-W[\Phi,\Psi](-1)=0,
\]
(where
$W[\Phi,\Psi](n)=c(\phi_{1,n+1}\psi^\ast_{2,n}-\phi_{2,n}\psi^\ast_{1,n+1})$; see
also
Chapter 9 in~\cite{CodLev}). Hence, the left-hand side
of~\eqref{eps} is a monotone increasing function of $l$ which vanishes at $
l=0$ and diverges as $l\rightarrow \infty $. On the other hand, the
right-hand side of~\eqref{eps} is a monotone decreasing function of $
\epsilon $ which diverges as $\epsilon \rightarrow 0$. It is then concluded that the
function $l(\epsilon )$ is a well-defined monotone decreasing and continuous
function of $\epsilon $ which diverges as $\epsilon \rightarrow 0$.
What follows is the version of Jitomirskaya-Last inequalities for the discrete
Dirac operators.
\begin{theorem}\label{JLineq} Let $H_D(m,c)$ be the Dirac
operator~\eqref{MDir} that satisfies the boundary condition~\eqref{boucond}.
Given $\epsilon>0$, we obtain
\begin{equation*}\label{DM}
\frac{5-\sqrt{24}}{m(E +i\epsilon )}\leq \frac{\left\Vert
u^D\right\Vert _{l(\epsilon )}}{\left\Vert u^N\right\Vert
_{l(\epsilon )}}\leq \frac{5+\sqrt{24}}{m(E +i\epsilon )}\;.~
\end{equation*}
\end{theorem}
In order to prove this theorem, we only need the following lemma, which is an
adaptation of Lemma~3.1 in~\cite{JitLast}; with such result at hand, the proof of
Theorem~\ref{JLineq} follows the same lines as the proof of Theorem~1.1
in~\cite{JitLast}.
The function $\chi_n(z)$ in Lemma~\ref{LFVP}
is nothing but the unique $l^2(\mathbb{Z}_+,\mathbb{C}^2)$ solution of the Dirac
equation \eqref{eqdir}, the one associated with the definition of~$m(z)$ (see
Section~\ref{SFN}).
\begin{lemma} [Variation of Parameters] \label{LFVP} For every $n\ge 0$, $\chi_n(z)$
satisfies the identity
\begin{eqnarray}\label{FVP}
\nonumber\left(\begin{array}{c}\chi_{1,n+1}(z)\\ \chi_{2,n}(z)\end{array}\right)
&=&-\left(\begin{array}{c}u^N_{1,n+1}(E)\\ u^N_{2,n}(E)\end{array}\right)+
m(z)\left(\begin{array}{c}u^D_{1,n+1}(E)\\ u^D_{2,n}(E)\end{array}\right)\\
\nonumber &-&\frac{i\epsilon}{c}\left(\begin{array}{c}u^N_{1,n+1}(E)\\ u^N_{2,n}(
E)\end{array}\right)\sum_{k=1}^n\left\{u^D_{1,k+1}(E)\chi_{1,k}(z)+u^D_{2,k}(E)
\chi_{2,k}(z)\right\}\\
&+& \frac{i\epsilon}{c}\left(\begin{array}{c}u^D_{1,n+1}(E)\\ u^D_{2,n}(E)
\end{array}\right)\sum_{k=1}^n\left\{u^N_{1,k+1}(E)\chi_{1,k}(z)+u^N_{2,k}(E)
\chi_{2,k}(z)\right\}\;.
\end{eqnarray}
\end{lemma}
\begin{proof} We denote by $\binom{\tilde{v}_{1,n+1}(E)}{\tilde{v}_{2,
n}(E)}$ the right-hand side of~\eqref{FVP}, and let $\binom{\tilde{v}_{1,0}(E)}
{\tilde{v}_{2,-1}(E)}=\binom{m(z)}{-1}$. Regarding the Wronskian constancy,
it is easy to verify that
\begin{eqnarray*}
\binom{\tilde{v}_{1,n+1}(E)}{\tilde{v}_{2,n}(E)}=\binom{\tilde{v}_{1,n}(E)+\frac{
mc^2-V_n+E}{c}\tilde{v}_{2,n}(E)+\frac{i\epsilon}{c}\chi_{2,n}(z)}{\tilde{v}_{2,
n-1}(E)+\frac{mc^2-V_n-E}{c}\tilde{v}_{1,n}(E)-\frac{i\epsilon}{c}\chi_{1,n}(z)}
\end{eqnarray*}
holds for every $n\ge -1$. Since
\begin{eqnarray*}
\binom{\chi_{1,n+1}(z)}{\chi_{2,n}(z)}=\binom{\chi_{1,n}(z)+\frac{(mc^2-V_n+E+i
\epsilon)\chi_{2,n}(z)}{c}}{\chi_{2,n-1}(z)+\frac{(mc^2-V_n-E-i\epsilon)\chi_{1,n}(
z)}{c}}
\end{eqnarray*}
\noindent
and that $\binom{\tilde{v}_{1,0}(E)}{\tilde{v}_{2,-1}(E)}=\binom{m(z)}{-1}=
\binom{\chi_{1,0}(E)}{\chi_{2,-1}(E)}$, it follows by induction that $\tilde{
v}_n(z)=\chi_n(z)$ for every $n\ge 0$.
\end{proof}
It is a direct consequence of Theorem~\ref{JLineq} that Theorem~1.2
in~\cite{JitLast} and its corollaries also hold true: if $
\rho$ is defined by~\eqref{TBor}, then, with $b=\alpha /(2-\alpha )$,
\begin{equation*}
D_\rho^\alpha(E):=\limsup_{\epsilon \downarrow 0}\frac{\rho\left( (E -\epsilon,E
+\epsilon )\right)}
{(2\epsilon )^{\alpha }}=\infty \label{sup}
\end{equation*}
if, and only if,
\begin{equation*}
\liminf_{l\rightarrow \infty }\frac{\left\Vert u^D\right\Vert _{l}}{
\left\Vert u^N\right\Vert _{l}^{b}}=0~, \label{inf}
\end{equation*}
and $D_\rho^\alpha(E)$ is the so called \textit{Hausdorff upper derivative}
of~$\rho$ at~$E$ (see~\cite{Last} for a detailed discussion of this concept).
Before we proceed, we need some important results regarding the behavior of
the generalized eigenfunction $u^D$. Next we present a version of Theorem~3.10
in~\cite{LS}.
\begin{proposition}\label{PROPO} Fix $\delta>0$. For almost every
$E$ with respect to the measure $\rho$,
\begin{eqnarray}\label{eT3.10LS}
\frac{1}{l}\sum_{n=0}^l \left(\left\vert u_{1,n}^D(E)\right\vert^2+\left\vert
u_{2,n}^D(E)\right\vert^2\right)\le C_E\,(\ln l)^{1+\delta}\;, \quad l\ge 2\;.
\end{eqnarray}
In particular, by taking $\delta=1$,
\begin{equation}\label{e1.13JL}
\limsup_{l\rightarrow\infty}\frac{\Vert u^D(E)\Vert}{l^{1/2}\ln l}<\infty\;.
\end{equation}
\end{proposition}
\begin{proof} Define the function $g_k(E)=2^{-k}\sum_{n=0}^{2^k}
\left(\vert u_{1,n}^D(E)\vert^2+\vert u_{2,n}^D(E)\vert^2\right)$. It follows from
Proposition 3.3 in~\cite{LS} (in fact, from a straightforward adaptation of this
result) that $\int g_k(E)d\rho_{\textrm{ac}}(E)\le C<\infty$; thus, $\sum_{k=0
}^\infty k^{-1-\delta}g_k(E)\in {\LL}^1(\mathbb{R},d\rho)$. This implies the
inequality $g_k(E)\le \widetilde{C}_E k^{1+\delta}$ for almost every $E$
with respect to $\rho$. Let $2^{k-1}\le l\le 2^k$. Then
\begin{eqnarray*}
\frac{1}{l}\sum_{n=0}^l\left(\left\vert u_{1,n}^D(E)\right\vert^2+\left\vert
u_{2,n}^D(E)\right\vert^2\right)\le 2\widetilde{C}_E k^{1+\delta}\le C_E(\ln l)^{1+
\delta}\;,
\end{eqnarray*}
with $C_E=2\left(1+\frac{1}{\ln 2}\right)\widetilde{C}_E$. In particular, by picking
the
particular value $\delta=1$, relation~\eqref{e1.13JL} follows from~\eqref{eT3.10LS}.
This completes the proof of the proposition. \end{proof}
\begin{lemma} Let $u^D(E)$ and $u^N(E)$ be the solutions to~\eqref{eqdirre} that
satisfy the initial conditions~\eqref{DNboucon}. Then
\begin{eqnarray*}
\Vert u^D(E)\Vert_{l}\Vert u^N(E)\Vert_{l} \ge c\,l\;.
\end{eqnarray*}
\label{LCW}
\end{lemma}
\noindent \textit{Proof}. Since the Wronskian is constant, it follows that
\begin{eqnarray*}
W[u^D,(u^N)^\ast](n)=c\left(u^D_{1,n+1}u^N_{2,n}-u^N_{1,n+1}u^D_{2,n}\right)=W[u^D,
(u^N)^\ast](-1)=c
\label{conswro}
\end{eqnarray*}
\noindent
for every $n\ge 0$. Thus,
\begin{eqnarray*}
\nonumber \sum_{n=0}^{l-1}c &=&\sum_{n=0}^{l-1}W[u^D,(u^N)^\ast](n)\le\left\vert\left
\langle
\binom{(u_{1,n+1}^D(E))^\ast}{(u_{2,n}^D(E))^\ast},\binom{u_{2,n}^N(E)}{-u_{1,n+1}^N
(E)}\right\rangle_{l-1}\right\vert \\
&\le &\left\Vert\binom{u_{1,n+1}^D(E)}{u_{2,n}^D(
E)}\right\Vert_{l-1} \left\Vert\binom{u_{1,n+1}^N(E)}{u_{2,n}^N(E)}\right\Vert_{l-
1}\;,
\end{eqnarray*}
\noindent
where we have used Cauchy-Schwarz inequality in the last step. Hence,
\begin{eqnarray*}
\Vert u^D(E)\Vert_{l}\Vert u^N(E)\Vert_{l} \ge c\,l \;,
\end{eqnarray*}
\noindent
which concludes the proof of the lemma. \hfill $\Box$
By taking into account the above adaptations to the Dirac operator, the proofs of
Corollary~\ref{C4.4}(a) and~(b) follow the same lines of the proofs of
Corollaries~4.4 and~4.5 in~\cite{JitLast}, respectively.
\begin{corollary}\label{C4.4}
(a)~Suppose that for some $\alpha \in \lbrack 0,1)$ and every $E$ in some Borel set
$F$, every solution~$\Psi$ to the Dirac eigenvalue equation~\eqref{eqdirre} obeys
\begin{equation*}
\limsup_{l\rightarrow \infty }\frac{\left\Vert \Psi\right\Vert _{l}^{2}}{
l^{2-\alpha }}<\infty ~.
\end{equation*}
Then, the restriction $\rho(F\cap \cdot )$ is $\alpha $-continuous.
(b)~Suppose that
\begin{equation}
\liminf_{l\rightarrow \infty }\frac{\left\Vert u^D(E)\right\Vert _{l}^{2}}{
l^{\alpha }}=0 \label{c4.5a}
\end{equation}
\noindent is satisfied for every $E$ in some Borel set $F$. Then the
restriction $\rho(F\cap \cdot )$ is $\alpha $-singular.
\end{corollary}
Corollary~\ref{C4.4} can be rewritten in terms of the one-dimensional $
2\times 2$ transfer matrices $T(n;E)$ defined by~\eqref{Tn}. This approach, based on
Corollary~3.7 in~\cite{CMW}, is
of particular importance in our problem, since we have obtained in Section
\ref{TMPV} the exact behavior of these matrices for a sparse potential like~\eqref{Ve}.
\begin{corollary}\label{C4.4a}
Suppose that for some $\alpha \in \lbrack 0,1)$ and every $E$ in some Borel set
$A\subset \mathbb R$,
\begin{equation}
\limsup_{l\rightarrow \infty }\frac{1}{l^{2-\alpha }}\sum_{n=0}^{l}\left
\Vert T(n;E)\right\Vert ^{2}<\infty \;, \label{dnor}
\end{equation}
\noindent with $\left\Vert \cdot\right\Vert $ some matrix norm. Then the
restriction $\rho(A\cap \cdot)$ is $\alpha $-continuous.
\end{corollary}
\begin{proof} By choosing $\theta_1=0$ and $\theta_2=\pi/2$, it follows
from a straightforward adaptation of Theorem~2.3 in~\cite{KLS} that
there exists a
constant $D$ such that
\begin{equation*}
\left\Vert T(n-1;E)\right\Vert \ge C\max \left\{
R_{n}(0),R_{n}(\pi/2)\right\} \;, \label{lr}
\end{equation*}
\noindent
where $R_n(\theta)$ is the Pr\"{u}fer radius at $n$ starting from the initial
condition $\mathbf{v}_{\theta }=\dbinom{\cos \theta }{\sin \theta }$; explicitly,
$C=\sqrt{\frac{(E+mc^2)(2c+E-mc^2)}{(E-mc^2)(2c-E-mc^2)}}$. Since
\begin{eqnarray*}
R_{n}^{2}(\theta _{1(2)}) &=& \frac{4c^2(E-mc^2)}{(E+mc^2)(4c^2+m^2c^4-E^2)}\left
\vert u_{1,n}^{D(N)}\right\vert^2+\frac{4c^2}{(4c^2+m^2c^4-E^2)}\left\vert u_{2,n
-1}^{D(N)}\right\vert^2 \\
&-& \frac{4c(E-mc^2)}{4c^2+m^2c^4-E^2}\Re\left\{u_{1,n}^{D(N)}(u_{2,n-1}^{D(N)})^
\ast\right\}
\end{eqnarray*}
(see~\eqref{EFGPT}), we obtain the inequality
\begin{equation*}
D\left[\left\vert u_{1,n}^{D(N)}\right\vert^2+\left\vert u_{2,n-1}^{D(N)}\right
\vert^2\right] \leq R_{n}^{2}(\theta _{1(2)})\;,
\end{equation*}
with $D=\frac{2c(E-mc^2)(2c-E-mc^2)}{(E+mc^2)(4c^2+m^2c^4-E^2)}$. Thus,
\begin{equation}
\sum_{n=0}^{l}\left\Vert T(n;E)\right\Vert ^{2}\geq C_1\max
\left\{\left\Vert u^D(E)\right\Vert _{l+1}^{2},\left\Vert u^N(E)\right\Vert
_{l+1}^{2}\right\}\;, \label{c4.4a}
\end{equation}
\noindent
$C_1=C\times D$. Hypothesis~\eqref{dnor}, together
with~\eqref{c4.4a}, imply Corollary~\ref{C4.4a}.\end{proof}
Note that the growth of the norm of the transfer matrix gives exactly the
growth of the increasing solution. This fact will be of great importance later on.
\subsection{Hausdorff dimension and spectral transition}
\label{HDST}
The last step in the determination of the Hausdorff dimension of the
measure $\rho$ is the following extension of Proposition~3.9 in~\cite{CMW}:
\begin{proposition}
\label{l2.1} Let $\mathcal{A=}\left( a_{n}\right) _{n\geq 1}$ be given
by~\eqref{spacon}, $E\in \mathbb{R}$ and assume that the sequence $\left(
\theta _{n}^\omega\right) _{n\geq1}$ of Pr\"{u}fer angles~\eqref{PA} is $\ud$ $\pi $
for every $\theta _{0}\in \lbrack 0,\pi )$,
almost every $\varphi\in\lbrack 0,\pi )$ (w.r.t. Lebesgue measure) and almost
every $\omega\in\Xi$. Then, there
is a generalized eigenfunction $\Psi$ (i.e., $\Psi$ satisfies $H_{v,\phi}(m,c)
\Psi=E\Psi$ and the phase boundary condition~\eqref{bouconphi}) such that
\begin{equation}
C_{n}^{-1}r^{n/2}\leq R_n(\phi )\leq C_{n}r^{n/2}
\label{lru}
\end{equation}
holds with~$r$ given by~\eqref{erre} and $C_{n}^{1/n}\searrow R_0(\phi)$
as $n\rightarrow \infty $. In addition, there exists a subordinate solution
$\Phi$ (with $\alpha ^{\ast }$-phase
boundary condition) for energy $E$ such that, for all sufficiently
large $n$, the Pr\"{u}fer radius associated with $\Phi$ satisfies
\begin{equation}
\left\vert R_n(\alpha ^{\ast })\right\vert \leq \tilde{C}
_{n}r_{j}^{-n/2}\; \label{dj}
\end{equation}
with $\tilde{C}_{n}^{1/n}\searrow R_0(\alpha^\ast)$ as $n\rightarrow \infty $.
\end{proposition}
\begin{proof} The proof of the proposition follows the same steps
of the proof of Proposition~3.9 in~\cite{CMW}. We, nevertheless, trace them out.
The first part of the proposition is simply Lemma~\ref{LRP}. The second part
follows the same steps of Lemma 3.6 in~\cite{CMW1}, which adapts the results
presented in Theorem~8.1 in~\cite{LS} (the latter gives a sufficient condition
for the existence of a subordinate solution to the Dirac equation
\eqref{eqdirre}).
The inequalities~\eqref{lru} and the results of Theorem~2.3 in~\cite{KLS} imply
that
\begin{equation}
C_{n}^{-1}r^{n/2}\leq t_{n}\leq C_{n}r^{n/2}~, \label{rtr}
\end{equation}
\noindent
with $t_n:=\Vert T(a_n+1;E)\Vert$ and $r$ given by~\eqref{erre}. From now on,
the proof of the existence of a subordinate solution, as well as the
determination of its asymptotic behavior, follow from the proof of Lemma 3.6
in~\cite{CMW1}. \end{proof}
\OBSI
The corresponding modifications of the known results for Schr\"o\-ding\-er operators,
employed in the proof of Proposition~\ref{l2.1} to the Dirac
setting studied in this work, are straightforward, being therefore omitted.
\OBSF
Finally, we are able to prove our main result.
\noindent\begin{proof}(Theorem~\ref{thethe1}) We base the arguments on the proof of
Theorem~3.11 in~\cite{CMW}. First, introduce the notation
\begin{equation}\label{defAeq}
A=-\sqrt{m^2c^4+2c^2
(1-\cos(\mathbb{Q}\pi))} \cup \sqrt{m^2c^4+2c^2(1-\cos(\mathbb{Q}\pi))},
\end{equation} which will also be used in other occasions ahead.
Theorem~\ref{thethe} implies that the sequence
$\left(\theta_{n}^{\omega}\right) _{n\geq 0}$ of Pr\"{u}fer angles is $\ud$ $\pi$
for every $\theta _{0}\in \lbrack 0,\pi ]$, every $E\in {\LL}^\prime\equiv L\setminus
A$ and almost every $\omega\in\Xi$. We obtain from~\eqref{rtr} the estimates
\begin{equation*}
\left\Vert T(k;E)\right\Vert \leq C_{n}r^{n/2}\leq
C_{n}^{\prime }a_{n}^{\gamma/2}\leq C_{n}^{\prime \prime }k^{\gamma/2}\;,
\end{equation*}
\noindent which hold for every $E\in {\LL}^\prime$ and every $a_n^\omega\le k
<a_{n+1}^\omega$, with $\gamma\equiv \ln r/\ln \beta $, $C_n^{\prime\prime}>0$ and
$\lim_{n\rightarrow \infty }\left( C_{n}^{\prime \prime }\right) ^{1/n}<\infty$.
It follows by the constancy of $\left\Vert T(k;E)\right\Vert $
on $[a_{n}+1,a_{n+1}]$ (since the free matrix $T_0$ is equivalent to a rotation;
see Section~\ref{TMPV}) that
\begin{equation}
\sum_{k=0}^{l}\left\Vert T(k;E)\right\Vert ^{2}\leq c\,l^{1+\gamma} \label{somatra}
\end{equation}
\noindent
holds for some $c>0$ and every $E\in {\LL}^\prime$.
The application of Proposition~\ref{l2.1} together with~\eqref{INTI}
guarantees, for these values of $E$, the existence of a subordinate solution
$\Phi^{\mathrm{sub}}$ such that its sequence of Pr\"ufer radii satisfies the
estimate
\begin{equation*}
\left\vert R_n\left(\theta_{\alpha^\ast}\right)\right\vert\le C_n^{\prime\prime
\prime}\,a_n^{-\gamma/2}~,
\end{equation*}
for some constant $C_n^{\prime\prime\prime}>0$.
Since every solution to~\eqref{eqdirre} has constant modulus
on the interval $[a_{n}+1,a_{n+1}]$, we have
\begin{equation}
\left\Vert \Phi^{\mathrm{sub}}\right\Vert _{l}^{2}\leq c^{\prime
}\,l^{1-\gamma}\;, \label{modusub}
\end{equation}
\noindent for some $c^{\prime }>0$.
Since the restriction of the measure $\rho$ to~$I$ is supported on the set of
those~$E$ for which $\Phi^{\mathrm{sub}}$ satisfies the
boundary condition~$\phi $ (thanks to the fact that $\rho$ has no absolutely
continuous part; see Theorem~1 in~\cite{GP}), we have $u^D=\Phi^{\mathrm{sub}}$
for a.e.\ $E\in I$ with respect to $\rho$ and for a.e.\ boundary condition
$\phi $ (by the theory of rank-one
perturbation; see Theorem~1.3 in~\cite{JitLast}).
Thus, by~\eqref{somatra} and~\eqref{modusub}, on the intervals~\eqref{SUPO} one has
\begin{equation*}
\limsup_{l\rightarrow \infty }\frac{1}{l^{2-\alpha }}\sum_{k=0}^{l}\left
\Vert T(k;E)\right\Vert ^{2}<\infty \label{c4.4c}
\end{equation*}
\noindent and
\begin{equation*}
\liminf_{l\rightarrow \infty }\frac{\left\Vert u^D(E)\right\Vert
_{l}^{2}}{l^{\alpha ^{\prime }}}=0\;, \label{c4.5b}
\end{equation*}
\noindent provided $2-\alpha=1+\gamma+\varepsilon$ and $\alpha^{\prime }=1-\gamma
+\varepsilon$, respectively, where $\varepsilon$ is an arbitrary positive
number.
It follows, by Corollary~\ref{C4.4a}, that the spectral
measure~$\rho$ is simultaneously $(1-\gamma-\varepsilon)$-continuous and
$(1-\gamma+\varepsilon)$-singular. Since $\varepsilon$ is arbitrary, we have
from Remark~\ref{DSC} that the restriction $\rho(I^\prime\cap\cdot)$ has exact
Hausdorff dimension given by~\eqref{HDim}, where $I^\prime\subseteq {\LL}^\prime$.
Finally, from the theory of rank one perturbations (more specifically, Theorem~8.1
of \cite{Si}), we know that $\rho\left(A\right)=0$ (see \eqref{defAeq}) holds for
almost
every~$\phi $; therefore, for almost every~$\phi $, the restriction $\rho(I\cap
\cdot )$ has~\eqref{HDim} as its Hausdorff dimension. This concludes the proof
of the theorem.
\end{proof}
\OBSI
We have checked that it is possible
to extend the results of~\cite{GP} and~\cite{KP} to the Dirac setting, in the same
way as exposed
here for the results of generalized subordinacy discussed in~\cite{JitLast}.
\OBSF
Despite the obtained exact Hausdorff dimension of the spectral measure
$\rho$, what can we infer from the spectral nature of the operator $H_{v,\phi}
(m,c)$ on the interval~\eqref{SUPO}? This is a tricky question, since we are
induced to think that for $\alpha_\rho=0$, the spectrum is simply dense pure
point and for $\alpha_\rho>0$, the spectrum is singular continuous. There are,
however, some examples in the literature \cite{DRJLS,CMW1,Zla} where the
spectrum is singular continuous with null Hausdorff dimension.
Note that Theorem~\ref{tmarwre} settles the problem. Now we present its proof.
\noindent \begin{proof} (Theorem~\ref{tmarwre}) We must prove, for fixed $v$ and
$\beta$, the inclusion
\begin{equation*}
I_1\setminus A_1\subseteq\left\{E\in\mathbb{R}:\sum_{n=0}^
\infty\Vert T(n;E)\Vert^{-2}=\infty\right\}
\end{equation*}
\noindent
($A_1$ is a set with zero Lebesgue measure), since (a)~follows directly from
an adaptation of Theorem~2.1 in~\cite{SS}. Now, if
\begin{equation*}
[I_1\setminus A_1]^C\subseteq\left\{E\in\mathbb{R}:
\sum_{n=1}^{\infty }\beta ^{n}(t_{n})^{2}\left( \sum_{m=n}^{\infty
}(t_{m}^j)^{-2}\right) ^{2}<\infty \right\}\;,
\end{equation*}
\noindent
then (b)~follows from an adaptation of Lemma~4.1 in~\cite{CMW1} and by
Proposition~4.2 in~\cite{MWGA}.
We pick~$n$ such that $a_N^\omega \le n <a_{N+1}^\omega$ is valid for some $N\in
\mathbb{N}$. We obtain from~\eqref{spacon} and~\eqref{R},
\begin{eqnarray}
\!\!\!\!\!\!\!\!\! \sum_{m\le n}\Vert T(m;E)\Vert_{U}^{-2} =
\sum_{k\le N+1}\Vert T(a_k^\omega;E)\Vert_{U}^{-2}\beta^{k+1} + \Vert
T(a_N^\omega;E)\Vert_{U}^{-2}\sum_{a_{N+1}^\omega\le m < n}1 \,,
\label{MTM}
\end{eqnarray}
\noindent with $\Vert \cdot \Vert_U:=\Vert U\cdot U^{-1}\Vert$ (see Section 4
of \cite{MWGA}).
We have by Theorem~2.3 in~\cite{KLS} the inequalities
\begin{eqnarray}
\Vert T(m;E)\Vert^{-2}\le C^2\Vert T(a_k^\omega;E)
\Vert^{-2}_{U} \le C^2\left(\max_{i\in\{1,2\}}R_{k}(\theta_{i})
\right)^{-2}
\label{MTM1}
\end{eqnarray}
\noindent
and
\begin{eqnarray}
\Vert T(m;E)\Vert^{-2}\ge C^{-2}\Vert T(a_k^\omega;E)\Vert^{-2}_{U} \ge
\tilde C^{-2}\left(\max_{i\in\{1,2\}}R_{k}(\theta_{i})\right)^{-2}\;,
\label{MTM2}
\end{eqnarray}
\noindent
for every $a_k^\omega\le m<a_{k+1}^\omega$, with $C$ as in the proof of Corollary
\ref{C4.4a} and
\[
\tilde C=C/|\sin(\theta_{1}-\theta_{2})/2|
\] ($\theta_{1
}$, $\theta_{2} \in [0,\pi)$ represent a pair of initial Pr\"ufer angles
such that $0<|\theta_{1}-\theta_{2}|<\pi/2$; see Lemma 2.2 of \cite{KLS}).
Introduce
\begin{eqnarray}
S_{N,M}^{\pm}(\beta,\varphi)\equiv \sum_{k=N}^MC_{k}^{\pm}\left(\beta^{k+1}
\pm 2k\right)r^{-k} \;;
\label{SNM}
\end{eqnarray}
\noindent
(the term $2k$ is due to the uncertainty associated with the position of the
$k$-th barrier: $\beta^{k+1}-2k\le a_k^\omega \le \beta^{k+1}+2k-1$ for all
$\omega\in\Xi$), with $r$ given by~\eqref{erre}.
It follows by Lemma~\ref{LRP}, and equations~\eqref{MTM}, \eqref{MTM2}
and~\eqref{SNM},
\begin{eqnarray}
\tilde{C}^{-2}S^-_{N+1,M}\le\sum_{m=n}^{a_M+1} \Vert T(m;E) \Vert^{-2}\le
C^{2}S_{N,M}^{+}
\;,
\label{theinq}
\end{eqnarray}
for every $M\ge N+1$ and $E\in B$, with
\begin{eqnarray}\label{Be}
B\equiv -\sqrt{m^2c^4+2c^2(1-\cos([0,\pi]\setminus\mathbb{Q}\pi))} \cup \sqrt{
m^2c^4+2c^2(1-\cos([0,\pi]\setminus\mathbb{Q}\pi))}\;,
\end{eqnarray}
and the set $2\cos(\mathbb{Q}\pi\cap[0,\pi))$ represents the ``energies'' where the
sequence $(\theta^{\omega}_n)_{n\ge 0}$ is not $\ud$ $\pi$; see the proof of
Theorem~\ref{thethe}.
It is clear by Lemma~\ref{LRP}, and equations~\eqref{erre}, \eqref{theinq},
and~\eqref{Be}
that
\begin{eqnarray*}
B \subseteq \left\{ E: \lim_{n\to\infty}\Vert
T(n;E) \Vert = \infty \right\} \;,
\end{eqnarray*}
\noindent
i.e., $B$ belongs to the complement of the essential support of the absolutely
continuous part of the spectrum of $H_{v,\phi}(m,c)$, $\Sigma_{\mathrm{ac}}$,
according to Theorem 1.1 of \cite{LS}. Hence, except for the set $A_1 =
(\mathbb{Q}\pi\cap[0,\pi)) \cup A^*$ of Lebesgue zero measure ($A^*$ is some
set of Lebesgue zero measure presented in the definition of~$\Sigma_{\mathrm{ac}}$),
the set
\begin{eqnarray}
B_1\equiv -\sqrt{m^2c^4+2c^2(1-\cos([0,\pi]\setminus A_1))} \cup \sqrt{
m^2c^4+2c^2(1-\cos([0,\pi]\setminus A_1))}%\;
\label{Be1}
\end{eqnarray}
\noindent
belongs to the singular spectrum. It follows from~\eqref{SNM} and the left-hand
member of~\eqref{theinq}, for $M\rightarrow\infty$, that
\begin{eqnarray*}
\sum_{n=0}^{\infty} \Vert
T(n;E) \Vert^{-2} = \infty
\end{eqnarray*}
\noindent
if $\beta>r$. Therefore, we conclude from~\eqref{Be1} and Theorem 2.2 of
\cite{KLS} that the essential spectrum of $H_{v,\phi}(m,c)$ is purely singular
continuous when restricted to $I\backslash A_1$, and this proves part~(a) of the
theorem.
We prove part~(b) by assuming that~$\beta<r$. By introducing
\begin{eqnarray}
S = \sum_{m=n}C_{m}r^{-m}\;,
\label{ESSE}
\end{eqnarray}
\noindent
it follows by Lemma~\ref{LRP} and Eqs.~\eqref{MTM},~\eqref{MTM1}, that
\begin{eqnarray}
\sum_{m=n}^\infty \Vert T(a_m+1;E) \Vert^{-2} \le C^2 S \;.
\label{ESSE1}
\end{eqnarray}
Finally, we have from Lemma~\ref{LRP} and
Eqs.~\eqref{MTM1},~\eqref{MTM2},~\eqref{ESSE},~\eqref{ESSE1} that
\begin{eqnarray}
\nonumber \sum_{n=1}^\infty \left(\beta^n+\omega_n-\omega_{n-1}\right) \Vert T(a_n+
1;E) \Vert^2\left(\sum_{m=n}^\infty \Vert T(a_m+1;E) \Vert^{-2}\right)^2 & & \\
\nonumber \le C^\prime\sum_{n=1}^\infty \left(\beta^n+2n\right)C_{n}r^nS^2 & & \\
\le C^{\prime\prime}\sum_{n=1}^\infty \left(\beta^n+2n\right)(C_{n})^3r^{-n} & &
\label{Epp}
\end{eqnarray}
\noindent
converges for every $E\in I^C$. It follows by Lemma~3.6 in~\cite{CMW1}
and~\eqref{Epp} above, that if $E\in I^c\setminus A_1$, then the Dirac equation
\eqref{eqdirre} have a $l^2(\mathbb{Z}_+,\mathbb{C}^2)$ solution. Since, by
Theorem~\ref{tee}, $I^c\in\sigma_{\mathrm{ess}}(H_{v,\phi}(m,c))$ for every $\phi\in
[0,\pi]$, the hypothesis of Proposition 4.2 of \cite{MWGA} is fulfilled and
consequently $H_{v,\phi}(m,c)$ have only dense pure point spectrum in $I^C$ for
almost every $\phi\in[0,\pi]$. This concludes the proof of part~(b) of the
theorem.\end{proof}
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\end{document}
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\iteman{ZMATH 1993f.01840}
\itemau{Danckwerts, R.; Vogel, D.}
\itemti{Testing of hypothesis - misunderstandings and perspects. Das Testen von Hypothesen - Missverstaendnisse und Perspektiven.}
\itemso{Didakt. Math. (1993) v. 21(1) p. 51-65. [ISSN 0343-5334]}
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http://ftp.de.freesbie.org/ctan/info/ptex-manual/jfm.tex | freesbie.org | CC-MAIN-2020-10 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-10/segments/1581875146744.74/warc/CC-MAIN-20200227160355-20200227190355-00277.warc.gz | 54,128,296 | 6,243 | \documentclass[a4paper,11pt,nomag]{jsarticle}
\usepackage[textwidth=42zw,lines=40,truedimen,centering]{geometry}
\usepackage{amsmath}
\usepackage[defaultsups]{newpxtext}
\usepackage[zerostyle=c,straightquotes]{newtxtt}
\usepackage{newpxmath}
% common
\usepackage{ptex-manual}
\usepackage{shortvrb}
\MakeShortVerb*{|}
\def\size#1{\mathit{#1}}
\def\code#1{\texttt{#1}}
% logos
\def\pTeX{p\kern-.10em\TeX}\def\upTeX{u\pTeX}
\title{\emph{JFMファイルフォーマット}}
\author{ASCII Corporation \& Japanese \TeX\ Development Community}
\begin{document}
\maketitle
\emph{JFM (Japanese Font Metric)}は，
\pTeX で和文フォントを扱うためのフォントメトリックであり，
オリジナルの\TeX のTFM (\TeX\ Font Metric)に相当する．
\pTeX と同じく株式会社アスキーによって開発され，
この文書も\pTeX に付属していたものであるが，ここでは2018年に
日本語\TeX 開発コミュニティによって拡張されたJFMフォーマットに
基づいて説明する．
なお，\pTeX の内部コードをUnicode化した\upTeX でも，
JFMフォーマットの仕様は全く同じであり，ただ
\node{char\_type}テーブルに文字コードを格納するときに
JISコードを用いる（\pTeX の場合）か，
UCS-4の下位3バイトを用いる（\upTeX の場合）かだけが異なる．
\section{JFMファイルの構成}
JFMファイルのフォーマットは，
基本的にはTFMファイルのフォーマットに準拠しており，
TFMを拡張した形になっている．
ここでは，主にその拡張部分について説明を行い，
その他の部分に関しては，
\TeX\ the program等のTFMの説明を参照してもらいたい．
JFMファイル全体の構成は，
表\ref{構成}（\pageref{構成}ページ）に示すとおりである．
ここでTFMと異なるのは次の点である．
\begin{enumerate}
\item \node{char\_type}のテーブルが付け加えられたこと．
\item \node{exten}の換わりに\node{glue}のテーブルが設けられたこと．
\item 2に関連して，\node{lig\_kern}から\node{glue\_kern}テーブルへ変更されたこと．
\item これらに伴い，先頭のファイル内の各部分を規定するパラメータ表が変更されている．
また，オリジナルのTFMとの区別のために$\size{id}$を付加しており，
先頭の半ワード（2バイト）が横組用は11，縦組用は9である
\footnote{欧文TFMの半ワードは$\size{lf}$すなわちファイルサイズであり，
11や9になることはない．}．
\end{enumerate}
最初の7ワードは半ワード（＝2バイト）ずつに区切られ，
JFMファイルを構成する14個の要素のサイズが収められている
\footnote{欧文TFMでは12個だが，JFMでは$\size{id}$と$\size{nt}$が増え，
$\size{ne}$が$\size{ng}$に置き換わったため14個である．}．
これらの値は，すべて$2^{15}$よりも小さい非負の値で，
次の条件を満たしていなければならない：
\begin{align*}
\size{bc} &= 0 \\
0 &\leq \size{ec} \leq 255 \\
\size{lf} &= 7+\size{nt}+\size{lh}+(\size{ec}-\size{bc}+1)+\size{nw}+\size{nh}
+\size{nd}+\size{ni}+\size{nl}+\size{nk}+\size{ng}+\size{np}
\end{align*}
ここで，$\size{bc}$と$\size{ec}$は最小・最大の文字タイプ番号，
$\size{nt}$は\node{char\_type}テーブルに登録された文字の数（文字タイプ0も含む），
$\size{nl}$と$\size{ng}$はそれぞれは\node{glue\_kern}テーブルと
\node{glue}テーブルのサイズであり，その他の値はTFMを踏襲する．
JFMファイルでもTFMファイルと同じく，拡張子は\code{.tfm}が用いられる．
\subsection{\node{char\_type}テーブル}
\pTeX では欧文\TeX よりはるかに多くの文字を扱う必要があるが，
そのほとんどは漢字であり，それらは全て同一の寸法（全角幅）を持つ．
また，括弧や句読点などの約物も種類が増えるが，こちらも
幾つかのパターンに分類すれば済む（例えば ``「'' と ``（'' は同様に扱える）．
そこで，JFMフォーマットでは，
同一の文字幅，高さ，前後に挿入されるグルー等，「その文字が持つ属性全てが
同じもの」を1つの\emph{文字タイプ} (\node{char\_type})として，
欧文フォントの1文字と同様にして扱うようにしている．
そして，文字コードと文字タイプとの対応付けを，
この\node{char\_type}テーブルを使って行う．
このテーブルの各エントリーは1ワード（＝4バイト）で構成され，
上位3バイトに文字コード（符合位置），下位1バイトに文字タイプを持つ
\footnote{これが日本語\TeX{}開発コミュニティによって2018年1月に導入された
新仕様である；オリジナルのアスキーによる仕様では，「上位半ワードに
文字コード，下位半ワードに文字タイプを持つ」というものであった．
\pTeX{}では内部処理にJISが用いられ，JFMで扱う文字コードは2バイトが
上限だったため十分であったが，\upTeX{}でBMP超えの3バイトの文字を
JFMで扱うことを目的に仕様拡張した．}．
文字コードは，それが16進数24bit（3バイト）で\code{0xABcdef}と表されるとき，
\node{char\_type}テーブルには\code{cd ef AB}として格納される
\footnote{オリジナルの仕様では，下位半ワードに文字タイプを
格納していたが，文字タイプの上限は255なので，実はその上位バイトは
常に\code{00}であった．このことを利用し，日本語\TeX{}開発コミュニティの
新仕様では「この\code{00}が実は文字コードの上位だった」ということに
して，3バイト（U+10000以上）の文字コードで不足する1バイトを確保した．
これにより，新仕様はオリジナルの仕様の上位互換であることが保証されている．}．
テーブル内にはコードの値の順番に収められていなければならない．
またこのテーブルの先頭には，デフォルトのインデックスとして
文字コード及び文字タイプの項が0のものが，
必ず1つ存在しなければならず，このテーブルに登録されていない文字は，
文字タイプが0として扱う．
つまり，このデフォルト以外の文字幅，カーン等の属性を持つキャラクタ
のコードとタイプが2番目以降のエントリーとして存在しなければならない．
\subsection{\node{char\_info}テーブル}
\node{char\_type}をインデックスとしてこのテーブルを
参照することにより，各\node{char\_type}の属性を検索する．
各テーブルへのインデックス等の情報を次の順番でパッキング
して1ワードに収めてある．
\begin{description}\itemindent=2zw
\item[\node{width\_index} (8bits)]
\node{width\_table}へのインデックス
\item[\node{height\_index} (4bits)]
\node{height\_table}へのインデックス
\item[\node{depth\_index} (4bits)]
\node{depth\_table}へのインデックス
\item[\node{italic\_index} (6bits)]
\node{italic\_table}へのインデックス
\item[\node{tag} (2bits)]
\node{remainder}をどのような目的で使うかを示す．
\begin{description}\itemindent=1zw
\item[$\size{tag}=0$]
\node{remainder}の項は無効であり使用しないことを示す．
\item[$\size{tag}=1$]
\node{remainder}の項が\node{glue\_kern}への有効なインデックスであることを示す．
\item[$\size{tag}=2, 3$]
JFMでは使用していない．
\end{description}
\item[\node{remainder} (8bits)]
\end{description}
JFMでは$\size{bc}$は必ずゼロ\footnote{前節にある通り，文字コード及び
文字タイプの項が0のものが必ず1つ存在するため．}なので，
1つのJFMに含まれる\node{char\_info}は全部で$\size{ec}+1$ワードになる．
% [TODO] char_info のサイズが ec+1 ということは，
% 文字タイプは 0 から ec まで連続しなければならないのか？
% （char_type テーブル内の文字コードの順序については説明があるが，
% 文字タイプの連続性についてはどこにも明記されていない気がする…）
\subsection{\node{glue\_kern}テーブル}
特定の文字タイプの組み合せ時に挿入すべき\node{glue}又は\node{kern}を
簡単なプログラム言語によって指定する．
各命令は，以下の4バイトで構成される．
\begin{description}\itemindent=2zw
\item[第1バイト] (\node{skip\_byte})
\begin{itemize}
\item 128より大きいとき\\
現在のワードが\node{char\_info}から示された最初のワード
である場合は，実際の\node{glue\_kern}プログラムが
\node{glue\_kern}[$256\times\size{op\_byte}+\size{remainder}$]から
収められている（すなわち，再配置されている）ことを示す
\footnote{「再配置」は，サイズが256を超える大きな
\node{glue\_kern}テーブルを格納するための方策であり，
欧文TFMの\node{lig\_kern}テーブルにおけるそれと同様である．
2018年2月に日本語\TeX{}開発コミュニティによって，
\pTeX{}及びpPLtoTFで新たにサポートされた．}．
最初のワードでない場合（すなわち，既に再配置先あるいは
プログラムのステップを開始した後のワードである場合）は，
その場でプログラムを終了する．
\item 128のとき\\
このワードを実行してプログラムを終了する．
\item 128より小さいとき\\
このワードを実行した後，次のステップまでスキップする
ワード数を示す
\footnote{「スキップ」(SKIP)は，元々アスキーの公式ページ
\texttt{http://ascii.asciimw.jp/pb/ptex/tfm/jfm.html}に文書化
されてはいたが，実際には(p)PLtoTFの\node{glue\_kern}テーブル内で
SKIP命令を使用することができず，\pTeX{}もやはりSKIP命令を
サポートしていなかった．
2018年2月の日本語\TeX{}開発コミュニティの改修により，
新たにサポートが開始された．}．
\end{itemize}
\item[第2バイト] (\node{char\_type})
\begin{itemize}
\item 次の文字の文字タイプが，
このバイトで示す文字タイプ
\footnote{ここに文字タイプが格納されるため，
文字タイプの上限は255なのである．}と同じ場合，
第3バイトの処理を実行し，プログラム終了．
\item そうでなければ次のステップへ．
\end{itemize}
\item[第3バイト] (\node{op\_byte})\\
この値によってグルーを扱うかカーンを扱うかを規定する．
\begin{itemize}
\item 127以下の場合\node{glue}[$\size{remainder}\times 3$]のグルーを挿入．
\item 128以上の場合\node{kern}[$\size{remainder}$]のカーンを挿入．
\end{itemize}
\item[第4バイト] (\node{remainder})\\
第3バイトにより規定される
\node{glue}または\node{kern}へのインデックスを示す．
\end{description}
\subsection{\node{glue}テーブル}
自然長，伸び長，縮み長の3ワードで1つのグルーを構成する
（したがって，$ng$は必ず3の倍数となる）．
各値は，$\mathrm{design size}\times2^{-20}$を単位として表す．
\begin{description}\itemindent=2zw
\item[第1ワード] width
\item[第2ワード] stretch
\item[第3ワード] shrink
\end{description}
\subsection{\node{param}テーブル}
一応，以下のように定義されている．
\begin{description}\itemindent=2zw
\item[\mbox{param[1]}]
文字の傾き (italic slant)．
\item[\mbox{param[2][3][4]}]
和文文字間に挿入するグルー(|\kanjiskip|)のデフォルト値．
\item[\mbox{param[5]}]
\pTeX{}でzhで参照される寸法．
\item[\mbox{param[6]}]
\pTeX{}でzwで参照される寸法．
\item[\mbox{param[7][8][9]}]
和文文字と欧文文字間に挿入するグルー(|\xkanjiskip|)のデフォルト値．
\end{description}
\begin{dangerous}
このように書かれているが，実際には
\pTeX のzwは「文字タイプ0の文字の幅」，
\pTeX のzhは「文字タイプ0の文字の高さと深さの和」である．
明示的に|\fontdimen|で取得する場合を除くと，JFMの\node{param}テーブルの
値が用いられる状況は限られている．
% [TODO] 例えばいつ？
% ptex-manual によると，\accent で和文文字をアクセントにした場合の
% 上下位置補正に \fontdimen5 の値が用いられるらしい．他には？
\end{dangerous}
\DeleteShortVerb{|}
\begin{table}[tbp]\small
\caption{JFMファイルの構成\label{構成}}
\begin{minipage}[b]{2in}
\begin{tabular}{|c|c|} \hline
\hbox to.8in{\hfil$\size{id}$\hfil} &
\hbox to.8in{\hfil$\size{nt}$\hfil} \\ \hline
$\size{lf}$ & $\size{lh}$ \\ \hline
$\size{bc}$ & $\size{ec}$ \\ \hline
$\size{nw}$ & $\size{nh}$ \\ \hline
$\size{nd}$ & $\size{ni}$ \\ \hline
$\size{nl}$ & $\size{nk}$ \\ \hline
$\size{ng}$ & $\size{np}$ \\ \hline
\multicolumn{2}{|c|}{} \\
\multicolumn{2}{|c|}{\node{header}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{} \\
\multicolumn{2}{|c|}{\node{char\_type}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{char\_info}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{width}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{height}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{depth}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{italic}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{glue\_kern}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{kern}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{glue}} \\
\multicolumn{2}{|c|}{}\\ \hline
\multicolumn{2}{|c|}{}\\
\multicolumn{2}{|c|}{\node{param}} \\
\multicolumn{2}{|c|}{}\\ \hline
\end{tabular}
\end{minipage}
\begin{minipage}[b]{3.3in}
\noindent
\begin{tabular}{l}
$\size{id}=$ JFM\_ID number. ($=11$ for yoko, $9$ for tate) \\
$\size{nt}=$ number of words in the character type table. \\
$\size{lf}=$ length of the entire file, in words. \\
$\size{lh}=$ length of the header data, in words. \\
$\size{bc}=$ smallest character type in the font. ($=0$ for JFM) \\
$\size{ec}=$ largest character type in the font. \\
$\size{nw}=$ number of words in the width table. \\
$\size{nh}=$ number of words in the height table. \\
$\size{nd}=$ number of words in the depth table. \\
$\size{ni}=$ number of words in the italic correction table. \\
$\size{nl}=$ number of words in the glue/kern table. \\
$\size{nk}=$ number of words in the kern table. \\
$\size{ng}=$ number of words in the glue table. \\
$\size{np}=$ number of font parameter words. \\
\end{tabular}
\end{minipage}
\end{table}
\MakeShortVerb*{|}
\section{JPLファイル}
TFMはバイナリ形式であるが，これをプロパティ（特性）という概念を使って
テキスト形式で視覚化したものがPL (Property List)ファイルである．
同様に，JFMをテキスト形式で視覚化したものが
\emph{JPL (Japanese Property List)}ファイルである．
JPLファイルでもPLファイルと同じく，拡張子は\code{.pl}が用いられる．
\section{JFMを扱うプログラム}
\pTeX と\upTeX ，あるいはそれらが生成したDVIを扱うプログラムが
JFMを扱うのは当然であるが，ここではJFMおよび関連するフォントフォーマットを
扱うことに特化したプログラムの主なものを挙げる．
%\subsection{pPLtoTF, upPLtoTF}
%\subsection{pTFtoPL, upTFtoPL}
%\subsection{chkdvifont}
%\subsection{jfmutil}
%\subsection{makejvf}
\end{document}
|
http://drorbn.net/AcademicPensieve/Classes/18-327-Topology/ThingsYouShouldKnow.tex | drorbn.net | CC-MAIN-2020-24 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-24/segments/1590347390442.29/warc/CC-MAIN-20200526015239-20200526045239-00362.warc.gz | 34,337,463 | 3,618 | \documentclass[11pt]{article}
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%%%
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\newtheorem{thm}{Theorem}
\newtheorem{dfn}[thm]{Definition}
%%%
%%% Some Definitions
%%%
\def\Z{\mathbb{Z}}
\def\R{\mathbb{R}}
\def\Q{\mathbb{Q}}
\def\N{\mathbb{N}}
\def\C{\mathcal{C}}
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\noindent
{\tiny \url{http://drorbn.net/18-327/ThingsYouShouldKnow.html}}
\newline{\footnotesize
\href{http://www.math.toronto.edu/~drorbn/Copyleft/}{\textcopyleft} $\mid$
\href{http://www.math.toronto.edu/~drorbn/}{Dror Bar-Natan}:
\href{http://www.math.toronto.edu/~drorbn/classes/index.html}{Classes}:
\href{http://www.math.toronto.edu/~drorbn/classes/index.html#1819}{2018-19}:
\href{http://www.math.toronto.edu/~drorbn/classes/18-327-Topology/}{MAT327}:
}
\hfill{\LARGE\bf Things You Should Already Know}
\vskip -3mm
\noindent\rule{\textwidth}{1pt}
\vspace{-5mm}
\begin{multicols}{2} \raggedcolumns
I will assume that you are familiar with all of the terms and symbols on this handout. Our first tutorials will go over everything here, just in case something is missing.
\section{Basic Set Theory}
In the following, $A,B,X,Y$ are sets, $I$ is an indexing set and $\set{A_\a}{\a \in I}$ and $\set{B_\a}{\a \in I}$ are families of sets indexed by $I$.
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item
Empty set:
$\emptyset$,
the set with no elements.
\item
Subset:
$A \sse B$ means ``$x \in A \implies x \in B$''
\item
Union:
$A \cup B \defas \set{x}{x \in A \text{ or } x \in B}$
\item
Intersection:
$A \cap B \defas \set{x}{x \in A \text{ and } x \in B}$
\item
Complement:
If $A \sse X$, then $X \setminus A \defas \set{x}{x \in X \text{ and } x \notin A}$
\item
Indexed union:
$\bigcup_{\a \in I} A_\a \defas \set{x}{\exists \a \in I, \; x \in A_\a}$
\item
Indexed intersection:
$\bigcap_{\a \in I} A_\a \defas \set{x}{\forall \a \in I, \, x \in A_\a}$
\item
Cartesian product of two sets:
$X \times Y \defas \set{(x,y)}{x \in X, \; y \in Y}$
\item Powers of sets: $Y^X$ is the set of all function $f\colon X\to Y$.
\item
The power set of $X$:
$\P(X) \defas \set{A}{A \sse X}\leftrightarrow \{0,1\}^X$.
\end{itemize}
\section{Functions}
In the following, let $X$ and $Y$ be sets, and let $f: X \to Y$ be a function.
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item $X$ is the \ul{domain} of $f$.
\item $Y$ is the \ul{target space} or \ul{codomain} of $f$.
\item $f(X) \!=\! \set{f(x)}{x \in X} \!\sse\! Y$ is the \ul{range} or \ul{image} of $f$.
\item $f$ is \ul{injective} (or \ul{one-to-one}, or an \ul{injection})
\[
\forall a, b \in X, \quad f(a) = f(b) \implies a = b.
\]
\item $f$ is \ul{surjective} (or \ul{onto}, or a \ul{surjection}) if its range is its entire codomain.
\item $f$ is \ul{bijective} (or a \ul{bijection}) if it is both injective and a surjective.
\item The composition of two injective functions is again injective.
\item The composition of two surjective functions is again surjective.
\item The composition of two bijective functions is again bijective.
\item Given a subset $B \sse Y$, the \ul{preimage} of $B$ is the set $f^{-1}(B) \defas \set{x \in X}{f(x) \in B}$.
\item If $f$ is an injection with range $Y$, then its inverse function $f^{-1}: Y \to X$ is (1) a function; and (2) bijective.
\end{itemize}
\section{De Morgan's Laws and some Further Relations}
The following two expressions are generalized versions of what are called De Morgan's Laws. They describe how unions and intersections interact with complementation.
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item $\ds{X \setminus \left( \bigcup_{\a \in I} A_\a \right) = \bigcap_{\a \in I} (X \setminus A_\a )}$
\item $\ds{X \setminus \left( \bigcap_{\a \in I} A_\a \right) = \bigcup_{\a \in I} (X \setminus A_\a)}$
\end{itemize}
\
\no The following are elementary facts about how functions interact with operations on subsets of their domains, codomains and ranges. Throughout the following, let $X$ and $Y$ be sets, let $f: X \to Y$ be a function, and let $A,B \sse X$ and $C,D \sse Y$.
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item $A \sse B$ implies $f(A) \sse f(B)$
\item $C \sse D$ implies $f^{-1}(C) \sse f^{-1}(D)$
\item $f(A \cup B) = f(A) \cup f(B)$
\item $f^{-1}(C \cup D) = f^{-1}(C) \cup f^{-1}(D)$
\item $f(A \cap B) \sse f(A) \cap f(B)$
\item $f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$
\item $f(A) \setminus f(B) \sse f(A \setminus B)$
\item $f^{-1}(C \setminus D) = f^{-1}(C) \setminus f^{-1}(D)$
\item $f(X \setminus f^{-1}(Y \setminus C)) \sse C$
\item $A \sse f^{-1}(f(A))$, (with equality if $f$ is injective)
\item $f(f^{-1}(C)) \sse C$, (with equality if $f$ is surjective)
\item $f^{-1}(Y \setminus C) = X \setminus f^{-1}(C)$
\end{itemize}
\section{Countability and Uncountability}
We will spend some time on this in class, but I do expect these words to be familiar to you.
\begin{dfn}
A set $A$ is said to be \ul{countably infinite} if there exists a bijection $f: \N \to A$. A set $A$ is said to be \ul{countable} if it is finite or countably infinite. If $A$ is infinite but not countably infinite, $A$ is said to be \ul{uncountable}.
\end{dfn}
The following theorem gives some equivalent conditions for being countable:
\begin{thm} For an infinite set $A$, the following are equivalent:
\begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item $A$ is countable.
\item There is an injection $f: A \to \N$.
\item There is a surjection $g: \N \to A$.
\end{enumerate}
\end{thm}
\no\textbf{Fact}: The following sets are countable:
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item $\N, \Z, \Q$, the set of algebraic numbers.
\item Any infinite subset of a countable set.
\item The Cartesian product of two countable sets (and, inductively, the Cartesian product of a finite number of countable sets).
\item The union of finitely many countable sets.
\item The union of a countable collection of countable sets.
\item The countable union of some countable sets and some finite sets.
\end{itemize}
\no\textbf{Fact}: The following sets are uncountable:
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item $\R$, $\R \setminus \Q$ (the irrational numbers), the set of non-algebraic numbers (i.e. the set of transcendental numbers), $\R^n$.
\item Any superset of an uncountable set.
\item The power set of any infinite set (countable or otherwise), e.g. $\P(\N)$.
\item The set $\N^\N$ of functions from $\N$ to $\N$.
\end{itemize}
\section{\texorpdfstring{Selected Basic Facts About $\R$}{Selected Basic Facts About R}}
First recall: $\N \subset \Z \subset \Q \subset \R$. (For us, $0 \notin \N$.)
\vspace*{\baselineskip}
\no\textbf{Fact}: Between any two distinct real numbers:
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item There are infinitely many rational numbers.
\item There are infinitely many irrational numbers.
\end{itemize}
\no\textbf{Fact}: Here are some useful facts from calculus:
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item $\ds{\bigcup_{n \in \N} [\tfrac{1}{n}, 1] = (0,1]}$.
\item $\ds{\bigcup_{n \in \N} [0, n] = [0, \infty)}$.
\item $\ds{\sum_{n \in \N} 2^{-n} = 1}$.
\end{itemize}
\section{Acknowledgement}
This document was modified from a document by \href{http://boolesrings.org/mpawliuk/}{Micheal Pawliuk} and by \href{http://www.math.toronto.edu/~ivan/}{Ivan Khatchatourian} and used with their permission.
Thanks, Micheal and Ivan!
\end{multicols}
\end{document} |
https://anarhija.info/library/parigi-francia-un-compagno-in-detenzione-preventiva-nel-contesto-della-mobilitazione-it.tex | anarhija.info | CC-MAIN-2022-27 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-27/segments/1656103850139.45/warc/CC-MAIN-20220630153307-20220630183307-00785.warc.gz | 153,858,556 | 5,092 | \documentclass[DIV=12,%
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\title{Parigi [Francia] : Un compagno in detenzione preventiva nel contesto della mobilitazione contro la legge sul lavoro (07\Slash{}12\Slash{}2016)}
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{\usekomafont{title}{\huge Parigi [Francia] : Un compagno in detenzione preventiva nel contesto della mobilitazione contro la legge sul lavoro (07\Slash{}12\Slash{}2016)\par}}%
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Il compagno anarchico arrestato mercoledì 7 in Bretagna è passato in tribunale [\emph{NdT: in una forma di procedimento comparabile al “giudizio immediato” dell’ordinamento giudiziario italiano}] l’indomani, nella 23esima aula del Palazzo di giustizia di Parigi.
È accusato di danneggiamento a un Ufficio di collocamento, una struttura della Camera di Commercio e dell’Industria, un supermercato Franprix e un concessionario della Jaguar durante una manifestazione spontanea che si è svolta la sera del 14 aprile fra il decimo e il diciannovesimo arrondissement di Parigi.
Gli sbirri hanno identificato il compagno solo durante l’estate e hanno avuto delle difficoltà a trovarlo, anche se lui non si nascondeva affatto. Per un mandato di arresto spiccato nel mese di agosto, ci è voluta una ricerca sul Registro dei conti in banca [\emph{Fichier des comptes bancaires – Ficoba}], che elenca tutti i conti bancari aperti in Francia (per poter per esempio localizzare gli ultimi prelievi) e delle ricerche sulla localizzazione del suo cellulare.
Il compagno ha rifiutato di essere giudicato in “giudizio immediato” e la Procuratrice ha quindi chiesto, col pretesto della sua fedina penale e di una possibile recidiva, che venisse messo in carcerazione preventiva. Il giudice ha accettato questa richiesta ed il compagno è attualmente alla prigione di Fleury-Mérogis [\emph{NdT: nella regione di Parigi}].
Tra l’altro, questo tribunale ha ben mostrato, se mai ce ne fosse ancora bisogno, il volto della giustizia. Tutti gli imputati erano dei poveri, i cui espedienti per uscire dalla miseria e\Slash{}o la dipendenza da diverse sostanze venivano sistematicamente considerati dalla Procuratrice e dal giudice come prove a carico. Anche i pochi tentativi di inchinarsi non hanno portato alla clemenza.
Il compagno era in forma, ha avuto un atteggiamento degno di fronte a questi servi del potere e ha fatto sapere che si esprimerà presto. Una piccola perla del discorso della Procuratrice a proposito del compagno : “Il signore si dice anarchico, ciascuno ha diritto ad avere un’opinione, le idee anarchiche sono quello che sono, ma non giustificano per niente i fatti che gli sono contestati”.
\textbf{Il processo si terrà il 19 gennaio alle 13.30 nell’aula 23 del Palazzo di giustizia di Parigi.}
Eravamo in tanti nelle strade durante le manifestazioni di questa primavera. Quei vetri rotti (quelli della Jaguar in particolare!) ci hanno rallegrato per parecchi giorni, come un piccolo raggio di sole in questo grigiume.
Che ogni giorno di detenzione del nostro compagno e di ogni altro prigioniero porti con sé degli atti di rivolta contro questo mondo!
La solidarietà è l’attacco!
\emph{Alcuni\Slash{}e anarchici\Slash{}e}
\bigskip
\bigskip
\textbf{Pariz [Francuska]: Anarhist pritvoren zbog prosvjeda protiv zakona o radu (11.12.2016.)}
Anarhistički drug, uhapšen u srijedu 7. u Bretanji, odveden je na sud dan kasnije, u 23. sudnicu pariške palače pravde.
Optužen je za nanošenje štete Zavodu za zapošljavanje, jednoj zgradi Gospodarske i Industrijske komore, jednom marketu Franprix i jednom koncesionaru Jaguara, tokom spontanog prosvjeda uvečer 14. aprila, između 10. i 19. pariškog okruga.
Panduri su identificirali druga tek tokom ljeta i imali su poteškoća pri pronalasku, mada se on nije uopće skrivao. Za izvršenje naloga za hapšenje izdatog u augustu, bilo je potrebno pretražiti Registar bankovnih računa [\emph{Fichier des comptes bancaires – Ficoba}], to jest popis svih tekućih računa u Francuskoj (kako bi se na primjer lokaliziralo posljednje podizanje novca), i lokalizirati njegov mobitel.
Drug je odbio da mu bude smjesta suđeno te je zato tužiteljica zatražila, pod izlikom njegovog kriminalnog dosjea i moguće iteracije, zatvorski pritvor. Sudac je prihvatio zahtjev i drug se trenutno nalazi u zatvoru Fleury-Mérogis [\emph{na pariškom području, nap.prev. na tal}.].
Između ostalog, navedeni je sud dobro pokazao, ako je uopće još potrebno, lice pravde. Svi optuženi bili su siromasi čije su dosjetke za izlaz iz bijede i\Slash{}ili ovisnosti o različitim sredstvima, tužiteljica i sudac sustavno smatrali kao dokaze protiv njih. Čak ni par pokušaja klanjanja nisu izazvali milost.
Drug je bio u formi, zadržao je ponosan stav naspram tih lakeja moći i rekao nam da će se ubrzo izjasniti. Jedan mali biser iz govora tužiteljice o drugu: “Gospodin se izjašnjava kao anarhist, svatko ima pravo na svoje mišljenje, anarhističke ideje su to što jesu, ali one uopće ne opravdavaju djela za koja se tereti”.
Suđenje će se odviti 19. januara u 13:30 sati u 23. sudnici Palaće pravde u Parizu.
Bilo nas je mnogo na ulicama tokom proljetnih prosvjeda. Ona razbijena stakla (nadasve koncesionara Jaguara!) su nas veselila dugo dana, kao mala zraka sunca u ovom sivilu.
Neka svaki dan zatvora našeg druga i svakog drugog zatvorenika donese sa sobom činove pobune protiv ovog svijeta!
Solidarnost je napad!
\textbf{\emph{Neki\Slash{}e anarhisti\Slash{}ce}}
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Parigi [Francia] : Un compagno in detenzione preventiva nel contesto della mobilitazione contro la legge sul lavoro (07\Slash{}12\Slash{}2016)
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\href{https://attaque.noblogs.org/post/2016/12/11/parigi-un-compagno-in-detenzione-preventiva-nel-contesto-della-mobilitazione-contro-la-legge-sul-lavoro/}{attaque.noblogs.org}
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\centering
\small
\hrule
\[
\begin{array}{rcll}
\tau^n & \in & \mathrm{Tyc} & n\textrm{ is arity}: \Omega^n\Rightarrow \Omega\\
\\
\mathfrak{C}^s & ::= &
\alpha~|~\mathfrak{C}^\lambda(\vv{\mathfrak{C}^s})~|~\vec{\rho}(\mathfrak{C}^s)
& \textrm{semantic monotype}\\
\mathfrak{C}^\lambda & ::= &
\lambda\vec{\alpha}.\mathfrak{C}^s~|~\tau^n & \textrm{semantic tycon}\\
\mathfrak{T} & ::= &
\mathsf{typ}(\mathfrak{C}^s)~|~\forall\vec{\alpha}.\mathfrak{C}^s
& \textrm{semantic type expression}\\
\mathfrak{C}^{nf} & ::= &
\alpha~|~\tau^n(\vv{\mathfrak{C}^{nf}}) & \textrm{normal form monotypes}\\
% Why both the C^s and \lambda forms? Is the C^s form necessary at
% this point?
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\]
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\caption{Semantic type system}
\label{fig:semtypesystem}
\end{figure}
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https://sea.theanarchistlibrary.org/library/crimethinc-tujuh-mitos-tentang-polisi-id.tex | theanarchistlibrary.org | CC-MAIN-2021-25 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-25/segments/1623487608702.10/warc/CC-MAIN-20210613100830-20210613130830-00357.warc.gz | 474,610,448 | 6,936 | \documentclass[DIV=12,%
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\title{Tujuh Mitos Tentang Polisi}
\date{2011\Slash{}10\Slash{}25}
\author{CrimethInc.}
\subtitle{}
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\textbf{Polisi menjalankan hukum yang sah.} Pada kenyataannya, rata-rata polisi bukanlah ahli hukum; dia mungkin tahu protokol dalam departemennya, tapi tidak begitu banyak mengetahui tentang hukum yang sebenarnya. Hal tersebut menunjukan bahwa penegakan hukum mereka lakukan dengan gertakan, improvisasi, dan ketidakjujuran. Polisi berbohong dalam hal-hal mendasar: “saya baru mendapatkan laporan bahwa seseorang dengan ciri-ciri seperti anda membuat kejahatan di sekitar tempat ini. boleh kaish lihat KTPnya?”
\begin{quote}
Ini bukan berarti kita harus menerima hukum yang sah tanpa berpikir juga. Seluruh sistem yudikatf melindungi mereka yang memiliki hak istimewa orang kaya dan berkuasa. Taat hukum tidak selalu berarti benar secara moral-bisa saja itu merupakan hal yang amoral. Sebelumnya perbudakan adalah hal legal, sedangkan membantu budak yang kabur adalah hal ilegal. Begitu pula dengan Nazi, merea berkuasa di Jerman melalui pemilihan yang demokratis dan lolos secara hukum melalui alur yang sudah ditentukan. Kita harus memiliki kekuatan hati nurani untuk melakukan apa yang menurut kita adalah hal yang terbaik, terlepas dari hukum dan intimidasi polisi.
\end{quote}
\textbf{Polisi adalah pekerja, sama seperti kita; mereka seharusnya teman kita.} Sayangnya, ada jarak yang jauh antara “seharusnya” dan “adalah.” Tugas dari poisi adalah untuk melayani kepentingan penguasa; siapapun yang tidak pernah memiliki pengalaman buruk dengan mereka berarti memiliki hak istimewa, penurut, atau dua-duanya. Polisi zaman sekarang tahu persis mereka akan terlibat dalam apa saat bergabung ke dalam satuan-orang berseragam tidak hanya menolong kucing di atas pohon. Ya, mungkin kebanyakan dari mereka melakukan pekerjaan tersebut karena tekanan ekonomi, tapi membutuhkan gaji bukanlah alasan untuk menggusur rumah warga, melecehkan orang berkulit gelap, atau menyemprotkan merica ke demonstran. Mereka yang hati nuraninya bisa dibeli adalah musuh potensial semua orang, bukan kawan.
Dongeng ini lebih persuasif jika ditulis dalam istilah strategis: contohnya, “Setiap revolusi berhasil pada saat pasukan bersenjata menolak untuk berperang dengan sesamanya; untuk itu kita seharusnya fokus untuk merayu polisi agar memihak kepada kita.” Tapi polisi tidak sperti pekerja lainnya; mereka sendiri yang memilih untuk mencari uang dengan cara mempertahankan tatanan saat ini, jadi sangatlah kecil kemungkinannya untuk bersimpati kepada mereka yang ingin mengubah para polisi. Dalam kontes ini, lebih masuk akal untuk menentang polisi dari pada bersolidaritas dengan mereka. Selagi mereka melayani tuannya, mereka tidak mungkin menjadi teman kita; dengan mencela institusi polisi dan menurunkan moral petugas, kita mendorong mereka untuk mencari mata pencaharian lain sehingga suatu hari kita bisa menemukan tujuan yang sama dengan mereka.
\textbf{Mungkin memang ada apel yang busuk, tapi beberapa polisi adalah orang yang baik.} Mungkin beberapa polisi memang memiliki niat baik, tetapi sekali lagi, selama mereka mematuhi perintah daripada hati nurani mereka, mereka tidak dapat dipercaya.
Ada sesuatu yang bisa diceritakan untuk memahami sifat sistematis dari institusi, daripada menghubungkan setiap ketidakadilan dengan kekurangan individu. Ingat kisah pria yang tersiksa oleh kutu, lalu berhasil menangkap satu dengan jarinya? Dia mengamatinya untuk waktu yang lama sebelum menempatkannya kembali di lehernya di tempat di mana dia menangkapnya. Teman-temannya bingung, bertanya mengapa dia melakukan hal seperti itu. “Bukan itu yang menggigit aku,” katanya.
\textbf{Polisi bisa memenangkan semua konfontasi, jadi kita tidak seharusnya memusuhi mereka.} Dengan segala senjata, peralatan, dan pengawasan mereka, polisi kelihatan tidak terkalahkan, tapi itu hanya ilusi. Mereka dibatasi oleh segala kendala yang tak terlihat-Birokrasi, opini publik, komunikasi yang terganggu, sistem peradilan yang kelebihan beban. Jika mereka tidak memiliki kendaraan atau fasilitas yang tersedia untuk mengangkut dan memproses sejumlah besar tahanan, misalnya, mereka tidak dapat melakukan penangkapan massal.
Itulah sebabnya kenapa massa yang beraneka ragam yang ditembaki gas air mata bisa menahan pasukan polisi yang lebih banyak, lebih terorganisir, dan lebih lengkap peralatannya; perselisihan antara kerusuhan sosial dan militer mungkin tidak berjalan sesuai dengan aturan keterlibatan militer. Mereka yang telah mempelajari polisi, yang bisa memprediksi apa yang mereka siapkan dan apa yang bisa dan tidak dapat mereka lakukan, seringkali bisa mengakali mereka.
Kemenangan kecil seperti itu sangat menginspirasi mereka yang setiap hari gerah diinjak kekerasan polisi. Dalam ketidaksadaran kolektif masyarakat kita, polisi adalah benteng terakhir dari realitas, kekuatan yang memastikan bahwa segala sesuatunya tetap sebagaimana adanya; melawan mereka dan menang, bagaimanapun untuk sementara, menunjukan bahwa kenyataan bisa dinegosiasikan.
\textbf{Polisi hanyalah pengalihan terhadap musuh yang sebenarnya, tidak sepadan untuk mendapat kemarahan atau perhatian kita.} Sayangnya, tirani bukan hanya masalah politisi atau eksekutif; mereka tidak akan berdaya tanpa ada yang melakukan permintaan mereka. Saat kita melawan aturan mereka, kita juga melawan kepatuhan yang membuat mereka tetap berkuasa, dan cepat atau lambat kita pasti akan melawan beberapa dari mereka yang tunduk. Meskipun demikian, memang benar bahwa polisi tidak lebih integral dengan hierarki daripada dinamika penindasan di komunitas kita sendiri; mereka hanyalah manifestasi luar, dalam skala yang lebih besar, dari fenomena yang sama. Jika kita ingin melawan dominasi di mana-mana, daripada berspesialisasi dalam memerangi bentuk-bentuk tertentu dari dominasi itu sementara membiarkan yang lain tidak tertandingi, kita harus siap menghadapinya baik di jalan maupun di kamar tidur kita sendiri; kita tidak bisa berharap untuk menang di satu sisi tanpa bertarung di sisi lain. Kita tidak boleh memuja konfrontasi dengan musuh berseragam, sementara melupakan ketidakseimbangan kekuatan di barisan kita sendiri-kita juga tidak boleh puas jika semata mengelola detail penindasan kita sendiri dengan cara yang tidak hierarkis.
\textbf{Kita perlu polisi untuk menjaga kita.} Menurut garis pemikiran ini, bahkan jika kita mungkin bercita-cita untuk hidup dalam masyarakat tanpa polisi di masa depan, kita membutuhkan mereka hari ini, karena orang tidak siap untuk hidup bersama secara damai tanpa penegak bersenjata. Seolah-olah ketidakseimbangan sosial dan ketakutan yang dipertahankan oleh kekerasan polisi adalah perdamaian! Mereka yang berpendapat bahwa polisi terkadang melakukan hal-hal baik menanggung beban untuk membuktikan bahwa hal-hal baik yang sama tidak dapat dicapai setidaknya dengan cara lain.
Bagaimanapun, ini bukan seolah-olah masyarakat bebas polisi tiba-tiba muncul dalam semalam hanya karena seseorang menyemprotkan cat “Persetan dengan Polisi” di dinding. Perjuangan panjang yang dibutuhkan untuk membebaskan komunitas kita dari represi polisi mungkin akan terus berlanjut selama itu membutuhkan kita untuk belajar hidup berdampingan secara damai; komunitas yang tidak dapat menyelesaikan konfliknya sendiri tidak bisa berharap untuk menang melawan kekutan penjajahan yang lebih kuat. Sementara itu, penentangan terhadap polisi harus dilihat sebagai penolakan terhadap salah satu sumber kekerasan paling mengerikan yang menindas, bukan sebuah pernyataan bahwa tanpa polisi tidak akan ada kekerasan yang menindas. Tapi jika kita bisa mengalahkan dan membubarkan polisi, kita pasti bisa membela diri dari ancaman yang kurang terorganisir.
\textbf{Melawan polisi itu kekerasan — itu membuat kamu tidak lebih baik dari mereka.} Menurut aliran pemikiran ini, kekerasan secara inheren merupakan bentuk dominasi, dan karenanya tidak sejalan dengan melawan dominasi. Mereka yang terlibat dalam kekerasan memainkan permainan yang sama seperti penindas mereka, sehingga kalah sejak awal.
Itu sangat menyederhanakan permasalahan dan berbahaya. Apakah wanita yang membela dirinya dari pemerkosa tidak lebih baik dari pemerkosa? Apakah budak yang memberontak tidak lebih baik dari pemilik budak? Kita tahu sesuatu yang disebut membela diri. Dalam beberapa kasus, kekerasan memaksakan ketidakseimbangan kekuasaan; dalam kasus lain, hal itu menantang pemaksaan tersebut. Bagi orang-orang yang masih memiliki keyakinan pada sistem otoriter atau Tuhan, mengikuti aturan — baik legal maupun moral — adalah prioritas utama, apa pun taruhannya: mereka percaya bahwa mereka akan diberi penghargaan karena melakukannya, terlepas dari apa yang terjadi pada orang lain sebagai hasilnya. Apapu sebutan untuk orang-orang seperti itu, konservatif atau pasifis, pada akhirnya hanya ada sedikit perbedaan. Di sisi lain, bagi kita yang bertanggung jawab atas diri kita sendiri, pertanyaan terpenting adalah apa yang akan membuat dunia menjadi tempat yang lebih baik. Terkadang ini mungkin termasuk kekerasan.
\textbf{Polisi juga manusia, dan berhak mendapatkan penghormatan yang sama karena semua makhluk hidup.} Intinya bukanlah bahwa mereka pantas menderita atau kita harus membawa mereka ke pengadilan. Intinya adalah bahwa, dalam istilah pragmatis murni, mereka tidak boleh diizinkan untuk membinasakan orang atau memaksakan tatanan sosial yang tidak adil. Meskipun dapat membangkitkan semangat mereka yang telah menghabiskan hidup mereka di bawah tumit penindasan untuk merenungkan akhirnya menang dari penindas mereka, pembebasan bukanlah masalah menuntut balas dendam tetapi menjadikannya tidak perlu. Oleh karena itu, meskipun kadang-kadang bahkan perlu untuk membakar polisi, ini tidak boleh dilakukan atas dasar semangat balas dendam diri sendiri, tetapi dari tempat kepedulian dan belas kasih-jika bukan karena polisi itu sendiri, setidaknya untuk semua yang jika tidak melakukan itu mereka akan menderita di tangan polisi.
\bigskip
Mendelegitimasi polisi tidak hanya bermanfaat bagi mereka yang menjadi target, tetapi juga bagi keluarga petugas polisi dan petugas polisi itu sendiri. Tidak hanya petugas polisi memiliki tingkat kekerasan dalam rumah tangga dan pelecehan anak yang tidak proporsional, mereka juga lebih mungkin untuk terbunuh, bunuh diri, dan berjuang melawan kecanduan daripada kebanyakan sektor masyarakat. Apa pun yang mendorong petugas polisi untuk berhenti dari pekerjaannya adalah demi kepentingan mereka, serta demi kepentingan orang yang mereka cintai dan masyarakat pada umumnya. Mari kita ciptakan dunia di mana tidak ada yang menindas atau tertindas, di mana tidak ada yang harus hidup dalam ketakutan.
\bigskip
\begin{quote}
“Cari tahu apa yang akan diterima oleh setiap orang secara diam-diam dan kamu akan menemukan ukuran pasti ketidakadilan dan kesalahan yang akan mereka lakukan, dan ini akan berlanjut sampai mereka dilawan dengan kata-kata atau pukulan, atau keduanya.”
\emph{-Frederick Douglass}
\end{quote}
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CrimethInc.
Tujuh Mitos Tentang Polisi
2011\Slash{}10\Slash{}25
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\title{Anarchy and Organization: The Debate at the 1907 International Anarchist Congress}
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\section{Amédée Dunois: Anarchism and Organization}
It is not long since our comrades were almost unanimous in their clear hostility towards any idea of organization. The question we are dealing with today would, then, have raised endless protests from them, and its supporters would have been vehemently accused of a hidden agenda and authoritarianism.
They were times when anarchists, isolated from each other and even more so from the working class, seemed to have lost all social feeling; in which anarchists, with their unceasing appeals for the spiritual liberation of the individual, were seen as the supreme manifestation of the old individualism of the great bourgeois theoreticians of the past.
Individual actions and individual initiative were thought to suffice for everything; and they applauded [Ibsen’s play] “An Enemy of the People” when it declared that a man alone is the most powerful of all. But they did not think of one thing: that Ibsen’s concept was never that of a revolutionary, in the sense that we give this word, but of a moralist primarily concerned with establishing a new moral elite within the very breast of the old society.
In past years, generally speaking, little attention was paid to studying the concrete matters of economic life, of the various phenomena of production and exchange, and some of our people, whose race has not yet disappeared, went so far as to deny the existence of that basic phenomenon — the class struggle — to the point of no longer distinguishing in the present society, in the manner of the pure democrats, anything except differences of opinion, which anarchist propaganda had to prepare individuals for, as a way of training them for theoretical discussion.
In its origins, anarchism was nothing more than a concrete protest against opportunist tendencies and social democracy’s authoritarian way of acting; and in this regard it can be said to have carried out a useful function in the social movement of the past twenty-five years. If socialism as a whole, as a revolutionary idea, has survived the progressive bourgeoisification of social democracy, it is undoubtedly due to the anarchists.
Why have anarchists not been content to support the principle of socialism and federalism against the bare-faced deviations of the [social democratic] cavaliers of the conquest of political power? Why has time brought them to the ambition of re-building a whole new ideology all over again, faced with parliamentary and reformist socialism?
We cannot but recognize it: this ideological attempt was not always an easy one. More often than not we have limited ourselves to consigning to the flames that which social democracy worshipped, and to worshipping that which burned. That is how unwittingly and without even realizing it, so many anarchists were able to lose sight of the essentially practical and working class nature of socialism in general and anarchism in particular, neither of which have ever been anything other than the theoretical expression of the spontaneous resistance of the workers against the oppression by the bourgeois regime. It happened to the anarchists as it happened to German philosophical socialism before 1848 — as we can read in the [Marx \& Engels’] \emph{Communist Manifesto} — which prided itself on being able to remain “in contempt of all class struggles,” defending “not the interests of the proletariat, but the interests of Human Nature, of Man in general, who belongs to no class, has no reality, who exists only in the misty realm of philosophical fantasy”.
Thus, many of our people came back curiously towards idealism on the one hand and individualism on the other. And there was renewed interest in the old 1848 themes of justice, liberty, brotherhood and the emancipatory omnipotence of the Idea of the world. At the same time the Individual was exalted, in the English manner, against the State and any form of organization came, more or less openly, to be viewed as a form of oppression and mental exploitation.
Certainly, this state of mind was never absolutely unanimous. But that does not take away from the fact that it is responsible, for the most part, for the absence of an organized, coherent anarchist movement. The exaggerated fear of alienating our own free wills at the hands of some new collective body stopped us above all from uniting.
It is true that there existed among us “social study groups”, but we know how ephemeral and precarious they were: born out of individual caprice, these groups were destined to disappear with it; those who made them up did not feel united enough, and the first difficulty they encountered caused them to split up. Furthermore, these groups do not seem to have ever had a clear notion of their goal. Now, the goal of an organization is at one and the same time thought and action. In my experience, however, those groups did not act at all: they disputed. And many reproached them for building all those little chapels, those talking shops.
What lies at the root of the fact that anarchist opinion now seems to be changing with regard to the question of organization?
There are two reasons for this:
The first is the example from abroad. There are small permanent organizations in England, Holland, Germany, Bohemia, Romandie and Italy which have been operating for several years now, without the anarchist idea having visibly suffered for this. It is true that in France we do not have a great deal of information on the constitution and life of these organizations; it would be desirable to investigate this.
The second cause is much more important. It consists of the decisive evolution that the minds and practical habits of anarchists have been undergoing more or less everywhere for the last seven years or so, which has led them to join the workers’ movement actively and participate in the people’s lives.
In a word, we have overcome the gap between the pure idea, which can so easily turn into dogma, and real life.
The basic result of this has been that we have become less and less interested in the sociological abstractions of yore and more and more interested in the practical movement, in action. Proof is the great importance that revolutionary syndicalism and anti-militarism, for example, have acquired for us in recent years.
Another result of our participation in the movement, also very important, has been that theoretical anarchism itself has gradually sharpened itself and become alive through contact with real life, that eternal fountain of thought. Anarchism in our eyes is no longer a general conception of the world, an ideal for existence, a rebellion of the spirit against everything that is foul, impure and beastly in life; it is also and above all a revolutionary theory, a concrete programme of destruction and social re-organization. Revolutionary anarchism — and I emphasize the word “revolutionary” — essentially seeks to participate in the spontaneous movement of the masses, working towards what Kropotkin so neatly called the “Conquest of Bread”
Now, it is only from the point of view of revolutionary anarchism that the question of anarchist organization can be dealt with.
The enemies of organization today are of two sorts.
Firstly, there are those who are obstinately and systematically hostile to any sort of organization. They are the individualists. There can be found among them the idea popularized by Rousseau that society is evil, that it is always a limitation on the independence of the individual. The smallest amount of society possible, or no society at all: that is their dream, an absurd dream, a romantic dream that brings us back to the strangest follies of Rousseau’s literature.
Do we need to say and to demonstrate that anarchism is not individualism, then? Historically speaking, anarchism was born, through the development of socialism, in the congresses of the International, in other words, from the workers’ movement itself. And in fact, logically, anarchy means society organized without political authority. I said \emph{organized}. On this point all the anarchists — Proudhon, Bakunin, those of the Jura Federation, Kropotkin — are in agreement. Far from treating organization and government as equal, Proudhon never ceased to emphasize their incompatibility: “The producer is incompatible with government,” he says in the \emph{General Idea of the Revolution in the 19\textsuperscript{th} Century}, “organization is opposed to government”.
Even Marx himself, whose disciples now seek to hide the anarchist side to his doctrine, defined anarchy thus: “All Socialists understand by Anarchy the following: that once the goal of the proletarian movement — the abolition of classes — is reached, the power of the State — which serves to maintain the large producing majority under the yoke of a small exploiting minority — disappears and the functions of government are transformed into simple administrative functions”. In other words, anarchy is not the negation of organization but only of the governing function of the power of the State.
No, anarchism is not individualist, but basically federalist. Federalism is essential to anarchism: it is in fact the very essence of anarchism. I would happily define anarchism as complete federalism, the universal extension of the idea of the free contract.
After all, I cannot see how an anarchist organization could damage the individual development of its members. No one would be forced to join, just as no one would be forced to leave once they had joined. So what is an anarchist federation? Several comrades from a particular region, Romandie for example, having established the impotence of isolated forces, of piecemeal action, agree one fine day to remain in continuing contact with each other, to unite their forces with the aim of working to spread communist, anarchist and revolutionary ideas and of participating in public events through their collective action. Do they thus create a new entity whose designated prey is the individual? By no means. They very simply, and for a precise goal, band together their ideas, their will and their forces, and from the resulting collective potentiality, each gains some advantage.
But we also have, as I said earlier, another sort of adversary. They are those who, despite being supporters of workers’ organizations founded on an identity of interests, prove to be hostile — or at least indifferent — to any organization based on an identity of aspirations, feelings and principles; they are, in a word, the [pure] syndicalists.
Let us examine their objections. The existence in France of a workers’ movement with a revolutionary and almost anarchist outlook is, in that country, currently the greatest obstacle that any attempt at anarchist organization risks foundering on — I do not wish to say being wrecked on. And this important historical fact imposes certain precautions on us, which do not affect, in my opinion, our comrades in other countries.
The workers’ movement today, the syndicalists observe, offers anarchists an almost unlimited field of action. Whereas idea-based groups, little sanctuaries into which only the initiated may enter, cannot hope to grow indefinitely, the workers’ organization, on the other hand, is a widely accessible association; it is not a temple whose doors are closed, but a public arena, a forum open to all workers without distinction of sex, race or ideology, and therefore perfectly adapted to encompassing the whole proletariat within its flexible and mobile ranks.
Now, the syndicalists continue, it is there in the workers’ unions that anarchists must be. The workers’ union is the living bud of the future society; it is the former which will pave the way for the latter. The error is made in staying within one’s own four walls, among the other initiates, chewing the same questions of doctrine over and over again, always moving within the same circle of ideas. We must not, under any pretext, separate ourselves form the people, for no matter how backward and limited the people may be, it is they, and not the ideologue, who are the indispensable driving force of every social revolution. Do we perhaps, like the social democrats, have any interests we wish to promote other than those of the great working mass? Party, sect or factional interests? Is it up to the people to come to us or is it we who must go to them, living their lives, earning their trust and stimulating them with both our words and our example into resistance, rebellion, revolution?
This is how the syndicalists talk. But I do not see how their objections have any value against our project to organize ourselves. On the contrary. I see clearly that if they had any value, it would also be against anarchism itself, as a doctrine that seeks to be distinct from syndicalism and refuses to allow itself to become absorbed into it.
Organized or not, anarchists (by which I mean those of our tendency, who do not arbitrarily separate anarchism from the proletariat) do not by any means expect that they are entitled to act in the role of ‘supreme saviours”, as the song goes. We willingly assign pride of place in the field of action to the workers’ movement, convinced as we have been for so long that the emancipation of the workers will be at the hands of those concerned or it will not be.
In other words, in our opinion the syndicate must not just have a purely corporative, trade function as the Guesdist socialists intend it, and with them some anarchists who cling to now outdated formulae. The time for pure corporativism is ended: this is a fact that could in principle be contrary to previous concepts, but which must be accepted with all its consequences. Yes, the corporative spirit is tending more and more towards becoming an anomaly, an anachronism, and is making room for the spirit of class. And this, mark my words, is not thanks to Griffuelhes, nor to Pouget — it is a result of action. In fact it is the needs of action that have obliged syndicalism to lift up its head and widen its conceptions. Nowadays the workers’ union is on the road to becoming for proletarians what the State is for the bourgeoisie: the political institution par excellence; an essential instrument in the struggle against capital, a weapon of defence or attack according to the situation.
Our task as anarchists, the most advanced, the boldest and the most uninhibited sector of the militant proletariat, is to stay constantly by its side, to fight the same battle among its ranks, to defend it against itself, not necessarily the least dangerous enemy. In other words, we want to provide this enormous moving mass that is the modern proletariat, I will not say with a philosophy and an ideal, something that could seem presumptuous, but with a goal and the means of action.
Far be it from us therefore the inept idea of wanting to isolate ourselves from the proletariat; that would be, we know only too well, to reduce ourselves to the impotence of proud ideologies, of abstractions empty of any ideal. Organized or not organized, then, the anarchists will remain true to their role of educators, stimulators and guides of the working masses. And if we are today of a mind to associate into groups in neighbourhoods, towns, regions or countries, and to federate these groups, it is above all in order to give our union action greater strength and continuity.
What is most often missing in those of us who fight within the world of labour, is the feeling of being supported. Social democratic syndicalists have behind them the constant organized power of the party from which they sometimes receive their watchwords and at all times their inspiration. Anarchist syndicalists on the other hand are abandoned unto themselves and, outside the union, do not have any real links between them or to their other comrades; they do not feel any support behind them and they receive no help. So, we wish to create this link, to provide this constant support; and I am personally convinced that our union activities cannot but benefit both in energy and in intelligence. And the stronger we are — and we will only become strong by organizing ourselves — the stronger will be the flow of ideas that we can send through the workers’ movement, which will thus become slowly impregnated with the anarchist spirit.
But will these groups of anarchist workers, which we would hope to see created in the near future, have no other role than to influence the great proletarian masses indirectly, by means of a militant elite, to drive them systematically into heroic resolutions, in a word to prepare the popular revolt? Will our groups have to limit themselves to perfecting the education of militants, to keep the revolutionary fever alive in them, to allow them to meet each other, to exchange ideas, to help each other at any time?
In other words, will they have their own action to carry out directly?
I believe so.
The social revolution, whether one imagines it in the guise of a general strike or an armed insurrection, can only be the work of the masses who must benefit from it. But every mass movement is accompanied by acts whose very nature — dare I say, whose technical nature — implies that they be carried out by a small number of people, the most perspicacious and daring sector of the mass movement. During the revolutionary period, in each neighbourhood, in each town, in each province, our anarchist groups will form many small fighting organizations, who will take those special, delicate measures which the large mass is almost always unable to do. It is clear that the groups should even now study and establish these insurrectional measures so as not to be, as has often happened, surprised by events.
Now for the principal, regular, continuous aim of our groups. It is (you will by now have guessed) anarchist propaganda. Yes, we will organize ourselves above all to spread our theoretical ideas, our methods of direct action and universal federalism.
Until today our propaganda has been made only or almost only on an individual basis. Individual propaganda has given notable results, above all in the heroic times when anarchists were compensating for the large number they needed with a fever of proselytism that recalled the primitive Christians. But is this continuing to happen? Experience obliges me to confess that it is not.
It seems that anarchism has been going through a sort of crisis in recent years, at least in France. The causes of this are clearly many and complex. It is not my task here to establish what they are, but I do wonder if the total lack of agreement and organization is not one of the causes of this crisis.
There are many anarchists in France. They are much divided on the question of theory, but even more so on practice. Everyone acts in his own way whenever he wants; in this way the individual efforts are dispersed and often exhausted, simply wasted. Anarchists can be found in more or less every sphere of action: in the workers’ unions, in the anti-militarist movement, among anti-clericalist free thinkers, in the popular universities, and so on, and so forth. What we are missing is a specifically anarchist movement, which can gather to it, on the economic and workers’ ground that is ours, all those forces that have been fighting in isolation up till now.
This specifically anarchist movement will spontaneously arise from our groups and from the federation of these groups. The might of joint action, of concerted action, will undoubtedly create it. I do not need to add that this organization will by no means expect to encompass all the picturesquely dispersed elements who describe themselves as followers of the anarchist ideal; there are, after all, those who would be totally inadmissible. It would be sufficient for the anarchist organization to group together, around a programme of concrete, practical action, all the comrades who accept our principles and who want to work with us, according to our methods.
Let me make it clear that I do not wish to go into specifics here. I am not dealing with the theoretical side of the organization. The name, form and programme of the organization to be created will be established separately and after reflection by the supporters of this organization.
\section{Errico Malatesta: Anarchism, Individualism and Organization}
I have listened attentively to everything that has been said before me on the problem of organization and I have the distinct impression that what separates us is the different meaning we give words. Let us not squabble over words. But as far as the basic problem is concerned, I am convinced that we are in total agreement.
All anarchists, whatever tendency they belong to, are individualists in some way or other. But the opposite is not true; not by any means. The individualists are thus divided into two distinct categories: one which claims the right to full development for all human individuality, their own and that of others; the other which only thinks about its own individuality and has absolutely no hesitation in sacrificing the individuality of others. The Tsar of all the Russias belongs to the latter category of individualists. We belong to the former.
Ibsen writes that the most powerful man in the world is the one who is most alone! Absolutely absurd! Doctor Stockmann himself, whom Ibsen has pronounce this maxim, was not even isolated in the full sense of the word; he lived in a constituted society, not on Robinson Crusoe’s island. Man “alone” cannot carry out even the smallest useful, productive task; and if someone needs a master above him it is exactly the man who lives in isolation. That which frees the individual, that which allows him to develop all his faculties, is not solitude, but association.
In order to be able to carry out work that is really useful, co-operation is indispensable, today more than ever. Without doubt, the association must allow its individual members full autonomy and the federation must respect this same autonomy for its groups. We are careful not to believe that the lack of organization is a guarantee of freedom. Everything goes to show that it is not.
An example: there are certain French newspapers whose pages are closed to all those whose ideas, style or simply person have the misfortune to be unwelcome in the eyes of the editors. The result is: the editors are invested with a personal power which limits the freedom of opinion and expression of comrades. The situation would be different if these newspapers belonged to all, instead of being the personal property of this or that individual: then all opinions could be freely debated.
There is much talk of authority, of authoritarianism. But we should be clear what we are speaking of here. We protest with all our heart against the authority embodied in the State, whose only purpose is to maintain the economic slavery within society, and we will never cease to rebel against it. But there does exist a simply moral authority that arises out of experience, intelligence and talent, and despite being anarchists there is no one among us who does not respect this authority.
It is wrong to present the “organizers”, the federalists, as authoritarians; but it is equally quite wrong to imagine the “anti-organizers”, the individualists, as having deliberately condemned themselves to isolation.
For me, I repeat, the dispute between individualists and organizers is a simple dispute over words, which does not hold up to careful examination of the facts. In the practical reality, what do we see? That the individualists are at times “organizers” for the reason that the latter too often limit themselves to preaching organization without practicing it. On the other hand, one can come across much more effective authoritarianism in those groups who noisily proclaim the “absolute freedom of the individual”, than in those that are commonly considered authoritarian because they have a bureau and take decisions.
In other words, everyone organizes themselves — organizers and anti-organizers. Only those who do little or nothing can live in isolation, contemplating. This is the truth; why not recognize it.
If proof be needed of what I say: in Italy all the comrades who are currently active in the struggle refer to my name, both the “individualists” and the “organizers”, and I believe that they are all right, as whatever their reciprocal differences may be, they all practice collective action nonetheless.
Enough of these verbal disputes; let us stick to action! Words divide and actions unite. It is time for all of us to work together in order to exert an effective influence on social events. It pains me to think that in order to free one of our own people from the clutches of the hangman it was necessary for us to turn to other parties instead of our own. Ferrer would not then owe his freedom to masons and bourgeois free thinkers if the anarchists, gathered together in a powerful and feared International, had been able to conduct themselves the worldwide protest against the criminal infamy of the Spanish government.
Let us ensure that the Anarchist International finally becomes a reality. To enable us to appeal quickly to all our comrades, to struggle against the reaction and to act, when the time is right, with revolutionary initiative, there must be an International!
\section{Emma Goldman}
I, too, am in favour of organization in principle. However, I fear that sooner or later this will fall into exclusivism.
Dunois has spoken against the excesses of individualism. But these excesses have nothing to do with true individualism, as the excesses of communism have nothing to do with real communism\dots{} I, too, will accept anarchist organization on just one condition: that it be based on the absolute respect for all individual initiatives and not obstruct their development or evolution.
The essential principle of anarchy is individual autonomy. The International will not be anarchist unless it wholly respects this principle.
\section{Max Baginski}
An error that is too often made is believing that individualism rejects organization. The two terms are, on the contrary, inseparable. Individualism more specifically means working for inner mental liberation of the individual, while organization means association between conscious individuals with a goal to reach or an economic need to satisfy. We must not however forget that a revolutionary organization requires particularly energetic and conscious individuals.
The accusation that anarchy is destructive rather than constructive and that accordingly anarchy is opposed to organization is one of the many falsehoods spread by our adversaries. They confuse today’s institutions with organization and thus cannot understand how one can fight the former and favour the latter. The truth is, though, that the two are not identical.
The State is generally considered to be the highest form of organization. But is it really a true organization? Is it not rather an arbitrary institution cunningly imposed on the masses?
Industry, too, is considered an organization; yet nothing is further from the truth. Industry is piracy of the poor at the hands of the rich.
We are asked to believe that the army is an organization, but careful analysis will show that it is nothing less than a cruel instrument of blind force.
Public education: are not the universities and other scholastic institutions perhaps models of organization, which offer people fine opportunities to educate themselves? Far from it: schools, more than any other institution, are nothing more than barracks, where the human mind is trained and manipulated in order to be subjected to the various social and mental phantoms, and thus rendered capable of continuing this system of exploitation and oppression of ours.
Instead, organization as we understand it is something different. It is based on freedom. It is a natural, spontaneous grouping of energies to guarantee beneficial results to humanity.
It is the harmony of organic development that produces the variety of colours and forms, the combination that we so admire in a flower. In the same way, the organized activity of free human beings imbued with the spirit of solidarity will result in the perfection of social harmony, which we call anarchy. Indeed, only anarchy makes the non-authoritarian organization of common interests possible, since it abolishes the antagonism that exists between individuals and classes.
In the current situation, the antagonism of economic and social interests produces an unceasing war between social units and represents an insurmountable obstacle on the road to collective well-being.
There exists an erroneous conviction that organization does not encourage individual freedom and that, on the contrary, it causes a decay of individual personality. The reality is, however, that the true function of organization lies in personal development and growth.
Just as the cells of an animal, through reciprocal co-operation, express latent powers in the formation of the complete organism, so the individual reaches the highest level of his development through co-operation with other individuals.
An organization, in the true sense of the word, cannot be the product of a union of pure nothingness. It must be made up of self-conscious and intelligent persons. In fact, the sum of the possibilities and activities of an organization is represented by the expression of the single energies.
It follows logically that the greater the number of strong, self-conscious individuals in an organization, the lesser the danger of stagnation and the more intense its vital element.
Anarchism supports the possibility of organization without discipline, fear or punishment, without the pressure of poverty: a new social organism that will end the terrible struggle for the means of subsistence, the vicious struggle that damages man’s best qualities and continually widens the social abyss. In short, anarchism struggles for a form of social organization that will ensure well-being for all.
The embryo of this organization can be found in the type of syndicalism that has freed itself from centralization, bureaucracy and discipline, that encourages autonomous, direct action by its members.
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Library.Anarhija.Net
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Various Authors
Anarchy and Organization: The Debate at the 1907 International Anarchist Congress
1907
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Retrieved on August 20, 2011 from \href{http://robertgraham.wordpress.com/anarchy-and-organization-the-debate-at-the-1907-international-anarchist-congress/}{robertgraham.wordpress.com}
Translations by Nestor McNab are taken from \emph{Studies for a Libertarian Alternative: The International Anarchist Congress, Amsterdam, 1907,} published by the Anarchist Communist Federation in Italy (Federazione dei Comunisti Anarchici — FdCA); paperback edition available from AK Press.
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\title{FE Bookstore}
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\author{Fifth Estate Collective}
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The FE BOOKSERVICE is located in the same place as the Fifth Estate Newspaper, both of which are located at 4403 Second Avenue, Detroit MI 48201—telephone (313) 831–6800. The hours we are open vary considerably, so it’s always best to give us a call before coming down.
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HOW TO ORDER BY MAIL:
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\begin{quote}
1) List the title of the book, quantity wanted, and the price of each;
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\begin{quote}
2) add 10\% for mailing—not less than \$.63 (which is the minimum charge for 4\textsuperscript{th} Class book rate postage);
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\begin{quote}
3) total;
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\begin{quote}
4) write all checks or money orders to: The Fifth Estate. Mail to Fifth Estate Books, 4403 Second Avenue, Detroit MI 48201.
\end{quote}
\textbf{NEW ARRIVALS}
AGAINST DOMESTICATION by Jacques Camatte
Camatte has emerged from the “Hegelian thickets” many have accused him of dwelling in and written a fairly intelligible essay containing many of his basic themes: the domestication of humans by capital, repressive consciousness, the superfluousness of the proletariat, establishment of a human community and others. Several of us who have read it recommend it to those interested in what constitutes at least a portion of the theoretical underpinnings of the Fifth Estate.
Black Thumb 24 pages \$1.50
TELOS: A QUARTERLY JOURNAL OF RADICAL THOUGHT
No. 51 Spring 1982
No. 52 Summer 1982
These people are as behind schedule as the FE! Both issues have appeared since we last published with the Summer number having arrived in late September. No. 51 is an excellent issue on the European peace movement and the Polish crisis. Also “Black Market Technology in the USSR” and “Facing the War Psychosis,” more comments on the controversial Castoriadis article. Plus many reviews, notes and other comments. No. 52 is a “Special Issue on Social Movements” with a special section on “Ecology and the Welfare State,” plus other reviews, notes and commentaries.
240 pp. each \$5.00 each
BROKEN IMAGES: ESSAYS ON CHINESE CULTURE AND POLITICS by Simon Leys
Essays on China’s leading writers as well as portraits of Mao, Chaing Ch’ing and Chiang Kai-shek with first-hand accounts of everyday life in China. All reflect Leys’ Orwellian ability to see events from the worm’s eye view of common humanity (or so says the book jacket).
St. Martins 156 pp. Hardcover reduced \$4
THE CHAIRMAN’S NEW CLOTHES: MAO \& THE CULTURAL REVOLUTION by Simon Leys
A diary of the Cultural Revolution by a firsthand observer whose essays from the period stand above the slavish adulation which typified most writing about the events. Leys notes the emergence of the bureaucratic state as being solidified in a movement ostensibly against bureaucracy. By the author of CHINESE SHADOWS.
St. Martins 260 pp. Hardcover reduced \$4
WE WANT TO RIOT, NOT TO WORK: THE 1981 BRIXTON UPRISINGS by the WWTRNTW Collective
Personal accounts of the riots that swept England last summer, along with suggestions of their social implications.
48 pp. \$1.50
THE GERMAN GUERRILLA: TERROR, REACTION, AND RESISTANCE by Jean Marcel Bougereau
Beginning with an interview with a German urban guerrilla, life underground is explored. The question of the legitimacy of violence, the authoritarianism of armed groups and what the need for constant secrecy produces among those involved in guerrilla activity.
Cienfuegos\Slash{}Soil of Liberty 106 pp. \$3.50.
ANARCHISM AND THE NATIONAL LIBERATION STRUGGLE Alfredo M. Bonanno; revised edition
A rejection of national liberation movements whose goal is the establishment of new nation states, but recognizes the uniqueness of different people and cultures. Additional notes from Bakunin and Rudolph Rocker.
Bratach Dubh 24 pages \$1.25
THE STRUGGLE AGAINST FASCISM BEGINS WITH THE STRUGGLE AGAINST BOLSHEVISM by Otto Ruhle
Written in 1939, Ruhle observes the Soviet Union and the communist movement after 20 years of counter-revolution. He states “[the] essential characteristics of fascism were and are existing in bolshevism. Fascism is merely a copy of bolshevism.”
Bratach Dubh Editions 20 pages \$1.25
BOOKS ON THE RUSSIAN \& HUNGARIAN REVOLUTIONS
November is the 65\textsuperscript{th} anniversary of the Russian Revolution of 1917 and of the Bolshevik counter-revolution. The same month is the 26\textsuperscript{th} anniversary of the Hungarian rebellion of 1956.
BOLSHEVIKS \& WORKERS CONTROL by Maurice Brinton
An excellent chronology and analysis of the Bolshevik betrayal of the revolution from the seizure of the factories to the crushing of the Kronstadt Commune.
Black \& Red 100 pp. \$1.95
HISTORY OF THE MAKHNOVIST MOVEMENT by Peter Arshinov
Exciting account of the anarchist\Slash{}communist peasant revolution in the Ukraine, with telling revelations about the Nature of Bolshevik military and social policy.
Black \& Red 284 pp. \$2.95
A CRITIQUE OF STATE SOCIALISM by Richard Warren and Michael Bakunin A large format comic which intersperses classic Bakunin quotes with a fast-paced history of socialism from Babeuf’s Conspiracy through the Bolshevik counter-revolution up until the current day crop of leftist and would-be rulers. As a little cartoon figure says on the cover, “What a boring title\dots{},” but it masks a thoughtful, informative, and utterly devastating critique of state socialism, much of it out of the mouths of socialist politicians themselves.
Cienfuegos Press 44 pages \$2.95
THE RUSSIAN TRAGEDY by Alexander Berkman
Two years in his native Russia provided both the background material for this analysis of the revolution and its betrayal by the communists. Contains three articles, originally published separately as pamphlets in 1922, “The Russian Tragedy,” “The Russian Revolution and the Communist Party,” and “The Kronstadt Rebellion.”
Cienfuegos Press 112 pp. \$4.50
THE POVERTY OF STATISM: A DEBATE by Fabbri, Rocker, Bukharin
Contains Nikolai Bukharin’s officially-sponsored attack on anarchism published in the Soviet Union in 1922, and Luigi Fabbri’s reply published in Italy the same year. Also, two articles by Rudolf Rocker, “Anarchism and Sovietism,” and “Marx and Anarchism.”
Cienfuegos Press \$3.50
THE GUILLOTINE AT WORK by Gregory Petrovich Maximoff
Develops the theme that the stalinist terror of the 1930s, the bureaucratisation of Russian society, the imperialist escapades, through to today’s lack of human rights in Russia and other East European countries are not aberrations in the development of socialist society, but rather a logical development in marxist philosophy and action. It serves one main purpose: “to dispel the aura which Lenin’s disciples have bestowed on him by showing that Lenin was primarily concerned with attaining power and holding on to it as a dictator by means of terror.”
Black Thorn Books 337 pp. \$9.20
HUNGARY ’56 by Andy Anderson
“All workers, socialists, even communists, must at last understand that a bureaucratic state has nothing to do with Socialism.” (Nemsetor, 15 January 1957)
Black \& Red 137 pp. \$1.25
MANUAL FOR REVOLUTIONARY LEADERS by Michael Velli
Advice to would-be leaders of the proletariat from the mouths of experts on how to have the workers put your gang in power. Somehow, though, it all falls apart.
Black \& Red 288 pp. \$2.50
THE WANDERING OF HUMANITY by Jacques Camatte
Shatters the old commonplaces: “Communism is not a new mode of production; it is the affirmation of a new community\dots{}[Women and men] will not gain mastery over production, but will create new relations among themselves which will determine an entirely different activity.” “Revolution does not emerge from one or another part of our being\dots{}Our revolution as a project to reestablish community was necessary from the moment when ancient communities were destroyed\dots{}”
Black \& Red 64 pp. \$1.00
ON ORGANIZATION by Camatte and Collu
Argues that the establishment of capital within material existence “and therefore within the social community” is accompanied by the disappearance of the traditional personal capitalist, the proletariat, and the theory of the proletariat. “This is only another way of saying that capital has succeeded in establishing its real domination. To accomplish this, capital had to absorb the movement which negates it, the proletariat, and establish a unity in which the proletariat is merely an object of capital. This unity can be destroyed only by a crisis, such as those described by Marx. It follows that all forms of working class political organization have disappeared. In their place, gangs confront one another in an obscene competition, veritable rackets rivaling each other in what they peddle but identical in their essence.”
Anonymous 40 pp. 50 cents
ANARCHISM AND MARXISM by Daniel Guerin
“\dots{}Anarchism and marxism, at the start, drank at the same proletarian spring\dots{}” “As the libertarian historian A.E. Kaminski wrote in his excellent book on Bakunin, a synthesis of anarchism and marxism is not only necessary but inevitable. ‘History,’ he adds, ‘makes her compromises herself.’”
Cienfuegos Press \$1.25
THE REPRODUCTION OF DAILY LIFE by Fredy Perlman
Discusses the mechanism by which human beings continue to reproduce the conditions of our own immiseration. “Men who were much but had little now have much but are little.”
Black \& Red 24 pp. \$.25
THE STRIKE IN GDANSK, AUGUST 14–31 1980 Edited and translated by Andrzej Tymowski
This short history chronicles the strike which marked the beginning of the Polish explosion. Contains accounts of the event taken from strike bulletins, Solidarity newspapers and interviews. Also, a critical Afterword on Solidarity since the strike.
Don’t Hold Back 50 pp. \$2.75
THE WILHELMSHAVEN REVOLT by Icarus
The story of the revolutionary movement in the German Navy in 1918–1919.
Cienfuegos Press 32 pp. \$1.25
SITUATIONIST INTERNATIONAL ANTHOLOGY Edited and translated by Ken Knabb
Contains over eighty texts—leaflets, articles, internal documents, film scripts, etc. With notes, bibliography and index. “The situationist destruction of present conditioning is already at the same time the construction of situations. It is the liberation of the inexhaustible energies trapped in a petrified daily life. With the advent of unitary urbanism, present city planning (that geology of lies) will be replaced by a technique for defending the permanently threatened conditions of freedom, and individuals—who do not yet exist as such—will begin freely constructing their own history.”—S.I., 1961
Bureau of Public Secrets 406 pp. \$10.00
FOR A CRITIQUE OF THE POLITICAL ECONOMY OF THE SIGN by Jean Baudrillard
Attempts an analysis of the sign form in the same way that Marx’s critique of political economy sought an analysis of the commodity form: as the commodity is at the same time both exchange value and use value, the sign is both signifier and signified. Thus, it necessitates an analysis on two levels, with the author confronting all of the conceptual obstacles of semiology in order to provide the same radical critique that Marx developed of classical political economy.
Telos Press 214 pp. \$4.50
THE MIRROR OF PRODUCTION by Jean Baudrillard
Examines the lessons of Marxism which has created a productivist model and a fetishism of labor. Asserts that Marxism reflects “all of Western metaphysics” and that it remains within the restrictive context of political economy whence it was born.
Telos Press \$3.95
TELOS: A QUARTERLY JOURNAL OF RADICAL THOUGHT—NUMBER 46 (Winter 1980–81)
We missed bringing you this issue when it originally appeared, but thought it contained enough interesting material to offer it at this time. Contains a “Special Symposium on the Crisis of the Left” as well as a controversial article by Cornelius Castoriadis, “Facing the War,” in which he sounds (according to some) more like Kissinger than a radical theorist. Castoriadis’ book by the same name, interestingly enough, has achieved widespread notoriety in France. This issue also contains Carlo’s “The Crisis of the State in the Thirties” as well as other articles and reviews.
Telos Press \$5.00
TELOS NUMBER 50 (Winter 1981–82)
Contains another “Special Symposium,” this one on “The Role of Intellectuals in the 1980s,” as well as “The Roots of Re-Armament” which contains a brief rebuttal of Castoriadis and an interesting discussion of the European disarmament movement, “Empire vs. Civil Society: Poland 1980–81” by Andrew Arato and John Zerzan’s “Anti-Work and the Struggle for Control” and Tim Luke’s rebuttal which also appear in the Fifth Estate.
Telos Press \$5.00
THE ATOMIC STATE AND THE PEOPLE WHO HAVE TO LIVE IN IT: Campaign Against the Model West Germany
Number 7 Discusses the qualitatively new form of totalitarian technological state power which is emerging as modern police techniques and nuclearism converge.
44 pp. 25 cents
THE POLITICAL ECONOMY OF HUMAN RIGHTS: Vol. I—The Washington Connection and Third World Fascism; Vol. II—After the Cataclysm: Postwar Indochina and the Reconstruction of Imperial Ideology by Noam Chomsky and Edward S. Herman
A devastating critique of U.S. foreign policy in the Third World and the “free world” media propaganda system which obscures the U.S. support for Third World dictatorships. “The basic fact is that the United States has organized under its sponsorship and protection a neo-colonial system of client states ruled mainly by terror and serving the interests of a small local and foreign business and military elite. The fundamental belief or ideological pretense, is that the United States is dedicated to furthering the cause of democracy and human rights throughout the world, though it may occasionally err in the pursuit of this objective.”
South End Press \$7.50 each sold separately, \$15.00 as a set
FOUR ARGUMENTS FOR THE ELIMINATION OF TELEVISION by Jerry Mander
“Television suburbanizes and commoditizes human beings, who are then easier to control. Meanwhile, those who control television consolidate their power\dots{} Television aids the creation of societal conditions which produce autocracy; it also creates the appropriate mental patterns for it and simultaneously dulls all awareness that this is happening.” Argues that television is not reformable, that its problems are inherent in technology itself and are so dangerous—to health and sanity, to autonomous and democratic forms of life—that it must be eliminated entirely.
Morrow 371 pages \$5.95
UNIONS AGAINST REVOLUTION by G. Munis and J. Zerzan
Contains articles on unionism, the title article by the left communist theoretician Grandizio Munis, and John Zerzan’s “Organized Labor Versus ‘The Revolt Against Work,’” “To attribute a useful function to unions in the revolutionary process is as unthinkable as seeing revolutionary potential in the stock market\dots{}”—from the text by Munis
Black \& Red 96 pp. \$1.00
ANARCHIST REVIEW NUMBER 5
Large-format (8.5 x 11) 120 page anthology of articles, reviews, photos, cartoons, news. Includes the Cienfuegos Press News, “Some Thoughts on Organization,” “Anarchists in the Mexican Revolution,” “Occult Authoritarians,” “Anarchists in Fiction,” “The Libertarian Movement in the Netherlands,” “Do-It-Yourself Radio Station,” several articles by Errico Malatesta, and much more.
Cienfuegos Press \$5.50
ANARCHY COMICS NUMBER 1
International comics by Kinney, Mavrides, Spain and others. How to destroy civilization and build a new cooperative one with tinkertoys.
Last Gasp \$1.25
ANARCHY COMICS NUMBER 2
International comics and better than ever. Starring America’s sweethearts, the Picto Family, the Political Bizarros, and Anarchie, Ludehead and Moronica. See Moronica’s rich father get knee-capped by the Red Brigades.
Last Gasp \$1.25
ANARCHY COMICS NUMBER 3
Seventeen cartoonists from Europe and North America. Contains the Decaying Meat\Slash{}Capitalism Diagram, a punk anarchist’s journey to the future, Anarchy in the Alsace, Wildcat, Roman Spring, and more.
Last Gasp \$2.00
SEX-POL: ESSAYS 1929–34 by Wilhelm Reich, ed. by Lee Baxandall
This volume contains the first complete translations of Reich’s writings from his marxist period, which ended with his expulsion from the German Communist Party. Although they are flawed by the attempt to enforce a stiff ideology onto Reich’s desire for sexual and ultimately total human liberation, they still remain valuable. Actually, several of Reich’s most important writings such as The Mass Psychology of Fascism and The Sexual Revolution were also published during this same period and used marxist terminology as well. However, with his rejection of both stalinism and marxism, he subsequently revised his later editions to provide a much more readable text, excised of the original sterile marxist language. Even with its drawbacks, texts like “The Imposition of Sexual Morality” and the other essays make the collection well worth reading.
Random House hardcover, originally \$10.00, Now \$4.00
LIMITS OF THE CITY by Murray Bookchin
A new edition of Bookchin’s criticism of the modern megalopolis as compared to ancient and medieval villages. Also posits a vision of a harmonized urban and rural life in a future society.
Harper, 148 pp. Originally \$4.50, now \$2.00
OUR ROOTS ARE STILL ALIVE: THE STORY OF THE PALESTINIAN PEOPLE by The Peoples Press Palestine Book Project
Although flawed politically by its leftist fawning before Arab nationalism and the PLO, the book’s solid data regarding the sordid history of Zionism and its effects on the Palestinians far outweighs the defects. Even left and labor Zionism appears as reactionary when presented here.
The Guardian 180 pp. \$5.95
\textbf{BACK IN STOCK!}
LOVE \& RAGE: ENTRIES IN A PRISON DIARY by Carl Harp
Since his death at the hands of the prison officials, these writings take on an even more Powerful meaning. The title of the book is appropriate—Carl oscillated between feelings of inspiration and solidarity on the one hand, and near despair and outrage on the other. Not surprising: you come away from this slim volume wondering how anyone could maintain any spirit all in the face of such absolute degradation and injustice, let alone reflect upon it and write it down.
Pulp Press 73 pp. \$3.95
GOD \& THE STATE by Michael Bakunin
Bakunin’s classic work with a new introduction and index of persons by Paul Avrich.
Dover 89 pp. \$2.50
THE PEOPLE ARMED: DURRUTI by Abel Paz
An exciting biography of a worker who becomes a pistolero for the anarchist movement, robbing banks and assassinating politicians. When the Spanish revolution commences Durruti serves as a militia leader until his untimely death.
Free Life Editions 323 pp. Hardback \$5.00
\textbf{ALSO NEW}
THE ESSENTIAL WORKS OF ANARCHISM edited by Marshall S. Shatz
A rather standard, but valuable collection of basic anarchist works from the classics of Bakunin and Kropotkin to anarchism in practice in Russia and Spain through to anarchist themes in the modern world articulated by Guerin, Daniel Cohn-Bendit and Paul Goodman.
Bantam 600 pp. \$1.95
BULLDOZER (THE ONLY VEHICLE FOR PRISON REFORM)—Number 4 Spring
Articles on “rehabilitation,” the hole, sexual harassment in prison. Free to prisoners from The Fifth Estate or direct from P.O. Box 5052, Sta. A, Toronto, Ont. M5W 1W4.
46 pages \$1.00
(Issues 1 and 2 are available for 50 cents each and no. 3 for \$1)
PROTEST WITHOUT ILLUSIONS by Vernon Richards
This excellent collection contains many articles written for the anarchist journal Freedom by a participant in the sit-downs and marches against the Bomb in England during the late ‘50s and early ‘60s. This is anything but dry history, since while the author commented on the events of the day he raised important criticisms and discussions on the nature of the activity taking place, discussions which have a direct relationship to the questions facing those of us who oppose the nuclear state today: the problem of the media, the notion of pressure groups which attempt to influence the activities of the state, the problem of organizing around fear of war rather than a vision of a new society, the political illusion, etc. This book should be read by all who are interested in this question. “There are no short cuts to peace. There are no compromise solutions between the rulers and the ruled. The day when we will be in a position to influence government we shall also have the strength to dispense with governments. Until we can put short term prospects in their proper perspective we shall continue to overlook the long term aims which alone can ensure a world at peace\dots{}” (1959)
Freedom Press 168 pp. \$4.25
The following books are being sold at reduced prices:
LAND \& LIBERTY: ANARCHIST INFLUENCES IN THE MEXICAN REVOLUTION by Ricardo Flores Magon (compiled and introduced by David Poole)
“Ricardo Flores Magon cannot be ignored either in the context of the Mexican Revolution or the world anarchist movement.”
Cienfuegos Press 156 pp. Originally \$4.70, reduced to \$2.00
THE CHRISTIE FILE by Stuart Christie
Stuart’s autobiography mirrors the turbulent period of the ‘60s and ‘70s, and includes his account of his imprisonment for his plot to assassinate Franco, his involvement with the Angry Brigade, and a broad sketch of the English anarchist movement.
Cienfuegos Press\Slash{}Partisan Press Originally \$9.95, now \$7.95
REVOLUTION IN SEATTLE by Harvey O’Connor
A memoir by a participant in the Seattle General Strike of 1919.
Left Bank Books, Orig. \$7.50, now \$5.00
\textbf{FREE PUBLICATIONS}
Still available free upon request with book orders or by sending postage to cover: “The origins of the Anarchist Movement in China” by Internationalist, “To Serve the Rich” (a cookbook printed in 1974 by the Eat the Rich Gang; available in any quantity with postage), “Shays’ Rebellion” (story of early American armed rebellion; available in any quantity with postage), “under the Polish Volcano”—critique of Polish events published in London, “To Libertarians” (poster explaining the plight of Spanish political prisoners). If you ask for all of the poster and pamphlets, please enclose an extra 25 cents to cover additional postage; 63 cents will cover it for requests without book orders.
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\begin{center}
Fifth Estate Collective
FE Bookstore
\bigskip
\href{https://www.fifthestate.org/archive/310-fall-1982/fe-bookstore}{\texttt{https://www.fifthestate.org/archive/310-fall-1982/fe-bookstore}}
Fifth Estate \#310, Fall 1982
\bigskip
\textbf{fifthestate.anarchistlibraries.net}
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http://edshare.soton.ac.uk/2291/3/448-1999-3.tex | soton.ac.uk | CC-MAIN-2021-43 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-43/segments/1634323585768.3/warc/CC-MAIN-20211023193319-20211023223319-00300.warc.gz | 20,666,366 | 2,625 | \documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\begin{document}
\parindent=0pt
\textbf{Question}
In this question YOU MAY ASSUME
\begin{description}
\item{(i)}
that small changes $df$ in the function $f(S,t)$ are related to small
changes in $S$ and $t$ by Taylor's theorem so that
$$df = f_SdS +f_tdt + \frac{1}{2} f_{SS}dS^2 + f_{St}dSdt +
\frac{1}{2} f_{tt} dt^2+ \cdots$$
\item{(ii)}
that $S$ follows the lognormal random walk
$$\frac{dS}{S} = rdt + \sigma dX$$
where $r$ and $\sigma$ are constants and $X$ is a random variable,
\item{(iii)}
that $dX^2 \to dt$ as $dt \to 0$.
\end{description}
\begin{description}
%Question 3a
\item{(a)}
Derive It\^{o}'s lemma in the form
$$df=\sigma S f_S dX + \left ( f_t + rSf_S + \frac{1}{2} \sigma^2 S^2
f_{SS} \right ) dt$$
and comment briefly on whether or not your derivation is rigorous.
%Question 3b
\item{(b)}
Denote the fair value of an option by $V(S,t)$. By constructing a
portfolio $\Pi = V - \Delta S$ where $\Delta$ is to be determined,
show that $V$ satisfies the Black-Scholes equation
$$V_t + \frac{1}{2} \sigma^2 S^2 V_{SS} + r SV_S - rV =0.$$
%Question 3c
\item{(c)}
A PERPETUAL option is one whose value does not depends upon time. Find
the most general solution for the value of a perpetual option and show
that the value of a perpetual Put is given by
$$V=AS^{-2r/\sigma^2}$$
where $A$ is a constant that depends on the specific details of the
option.
\end{description}
\newpage
\textbf{Answer}
\begin{description}
%Question 3a
\item{(a)}
We have, by Taylor's theorem:-
$$df=f_SdS+f_tdt+\frac{1}{2}f_{SS}dS^2+ f_{St}dSdt+
\frac{1}{2}f_{tt}dt^2+\cdots$$
and
\begin{eqnarray*} dS &= &S\mu dt+ S\sigma dX\\
\Rightarrow \ dS^2 & = & S^2\mu^2dt^2+2S^2\mu\sigma dtdX+S^2\sigma^2dX^2
\end{eqnarray*}
But now as $dt\to 0$, $dX^2\to dt$
\begin{eqnarray*}
\Rightarrow dS^2 & = & S^2\sigma^2dt +2\sigma \mu S^2(dt)^{3/2}+
S^2\mu^2dt^2\\
& = & S^2\sigma^2dt+O(dt^{3/2})\\
\Rightarrow df & = & f_S[S\mu dt+ S\sigma dX]+f_tdt+
\frac{1}{2}(f_{SS}\sigma^2 S^2 dt\\
& & + O(dt^{3/2}))+ O(dt^{3/2})
\end{eqnarray*}
and so, to leading order,
\begin{eqnarray*}
df & = & f_S[S\mu dt+ S\sigma dX]+f_tdt+\frac{1}{2}\sigma^2 S^2 f_{SS}
dt\\
& = & S\sigma F_S dX +(F_t+\mu Sf_S+ \frac{1}{2}\sigma^2 S^2f_{SS})dt
\ \ - \ \ \rm{ITO's\ lemma}
\end{eqnarray*}
The derivation is not very rigorous at all - it started from Taylor's
theorem which is valid for smooth functions - and S follows a random
walk!
%Question 3b
\item{(b)}
Now consider the portfolio $\Pi=V-S\Delta$.
We have
\begin{eqnarray*}
d\Pi & = & dV-\Delta dS = S\sigma V_S dX+ \left ( V_t+\mu SV_S+
\frac{1}{2} \sigma^2 S^2 V_{SS} \right ) dt\\
& & - \Delta (\mu Dst+\sigma
SdX)\\
d\Pi & = & (S\sigma V_S-\Delta S \sigma)dX\\
& & + \left ( V_t+\mu SV_s +
\frac{1}{2} \sigma^2 S^2 V_{SS}- \Delta S\mu \right ) dt
\end{eqnarray*}
All the randomness in $\Pi$ may thus be eliminated by choosing
$\Delta=V_S$, in which case we find that
\begin{eqnarray*}
d\Pi & = & \left ( V_t + \mu SV_S + \frac{1}{2}\sigma^2 S^2 V_{SS}
-S\mu V_S \right ) dt\\
& = & \left ( V_t +\frac{1}{2} \sigma^2 S^2 V_{SS} \right ) st.
\end{eqnarray*}
We now appeal to arbitrage: presumably the option must be neither more
nor less valuable than a risk free investment, otherwise one or the
other would never be used. So the above must be equal to the return in
time $dt$ of an amount $\Pi$ invested in a risk free portfolio. Thus
\begin{eqnarray*}
r\Pi dt & = & \left ( V_t + \frac{1}{2} \sigma^2 S^2 V_{SS} \right )
dt\\
r(V-S\Delta) & = & V_t +\frac{1}{2} \sigma^2 S^2 V_{SS}\\
r(V-SV_S) & = & V_t + \frac{1}{2}\sigma^2 S^2 V_{SS} \ \ \Rightarrow \
\rm{Black-Scholes}\\
\end{eqnarray*}
$$V_t +\frac{1}{2}\sigma^2 S^2 V_{SS} +rSV_S -rV = 0.$$
%Question 3c
\item{(c)}
For a perpetual option we have $V_t=0$ $\Rightarrow\ V=V(S)$ only.
$\Rightarrow$ Black-Scholes becomes
$$\frac{1}{2} \sigma^2 S^2 V_{SS} +rSV_S -rV =0$$
This is Euler's equation, so for solution try $V=S^k$
\begin{eqnarray*}
\frac{1}{2} \sigma^2 S^2 k(k-1)S^{k-2} +rSkS^{k-1} -rS^k & = & 0\\
S^k \left ( \frac{1}{2} \sigma^2 k(k-1) +rk -r \right ) & = & 0
\end{eqnarray*}
So for a solution we need $\frac{1}{2} \sigma^2 k(k-1) +(k-1)r=0$.
$\begin{array}{llr}
\Rightarrow & \rm{either} \ \ & k=1\\
& \rm{or} & \frac{1}{2} \sigma^2 k +r=0\\
\Rightarrow & & k=-2r/\sigma^2
\end{array}$
Thus the most general solution for a perpetual option is
$$V=AS+BS^{-2r/\sigma^2}$$
where $A$ and $B$ are arbitrary constants.
Now consider an American (or European) Put which is perpetual! Clearly
as $S\to\infty$ the option becomes more and more worthless, since the
chance of exercising it becomes less and less. $\Rightarrow A=0$
$\Rightarrow$ for some $\overline{A}$
$$V+\overline{A}S^{-2r/\sigma^2}$$
\end{description}
\end{document} |
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\title{It’s been four years since Zoé died}
\date{April 2013}
\author{}
\subtitle{}
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{\usekomafont{title}{\huge It’s been four years since Zoé died\par}}%
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{\usekomafont{date}{April 2013\par}}%
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Four years. Difficult to know if it’s four years already or just four years. Four years and a long mourning which has only just begun several years after her death, after those who justice found necessary to punish directly for the accident which cost Zoé her life finished the prison sentences which they had been assigned, after those who remained outside prison were no longer listened in on, tailed, photographed, filmed, intimidated. But anyway, that’s another story. After those who believe that seeing a friend die is not enough have satisfied themselves with our pain, have been sated enough by our sadness to leave again with a full belly and head held high, proud to have restored order and justice. This order and this justice which are after our friendships and our loves, and that seek to destroy them, because our friendships and loves are, among other things, born of our desires and our potential to create a space in which to grow and develop. Without passion, theory is but a dead letter. And cynicism is nothing revolutionary.
Zoé is not a martyr of “the cause.” She did not die for an ideology, for the people, or for “the revolution.” Zoé died of being free, or of wanting to be. She died of being in love, of being a friend, being DIY, a feminist, a traveler, of being dynamic, intelligent, radical, generous\dots{}. Zoé died of being who she was. Someone who did not want to suffer further or to endure the greyness, and who acted accordingly, for herself and for others. Someone who did not want to adapt to a world that horrified her, and for whom to merely be indignant was not enough. Zoé was not a hero, just someone who made choices. The choice to refuse, to resist, to not be indifferent to that which surrounded her and to how it surrounded her, to not let herself become absorbed in the tranquil decomposition of everyday life, to not want to stay at the window, railing against those whose attempts to make the world radically better have unfortunately failed. These are some choices that she and others have paid for dearly, here as elsewhere, today as yesterday.
There was a storm of sadness and anger that swept the days and the weeks that followed Zoé’s death. The sadness of losing a friend, the anger over not being able to be sad, of not having the respite. An anger directed against those who make their business on our deaths and our suffering, against this necrophagia elevated to the status of a social model. Yes, this sadness and this anger also have their toll, but the disaster would be much worse if we were used to it, if the ghosts of the struggle erased the sincerity of the continuing fight, if the feelings and affects were relegated to the limbo of ideology. We have been profoundly affected because we are in touch with the world.
A part of us has been devoured, and the monster is always hungry, always wanting more. But it is necessary to pry the memory from its jaws, to pull it out, in order to not forget. Not to erect monuments, because death is nothing glorious, but in order to prevent these passions and this love for freedom that animated Zoé from being swallowed in turn.
Four years have passed, but little water has passed under the bridge. And this water should not flow, because it is our lives, our deaths, and our struggles which slip away with it.
For a world without prisons or borders.
For a free and difficult life, toward an existence without exploitation or domination.
Solidarity to everyone, imprisoned or not, who struggle every day by any means necessary against what destroys them.
A thought for Mauricio Morales, who died in Santiago, Chile, the same month of May, 2009, for those close to him and all those who had to endure the Bombs Case, and whose stories have resonated in a number of heads here.
To Zoé.
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Library.Anarhija.Net
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\strut
\vfill
\begin{center}
It’s been four years since Zoé died
April 2013
\bigskip
Retrieved on May 2013 from \href{http://www.non-fides.fr/?It-s-been-four-years-since-Zoe}{www.non-fides.fr}
from non-fides.fr, translated by waronsociety.
\bigskip
\textbf{lib.anarhija.net}
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|
https://www.zentralblatt-math.org/matheduc/en/?id=47283&type=tex | zentralblatt-math.org | CC-MAIN-2019-39 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-39/segments/1568514574588.96/warc/CC-MAIN-20190921170434-20190921192434-00213.warc.gz | 1,074,524,007 | 1,527 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 2011b.00584}
\itemau{Van den Heuvel-Panhuizen, M.; Buys, K.}
\itemti{Young children learn measurement and geometry. A learning-teaching trajectory with intermediate attainment targets for the lower grades in primary school.}
\itemso{Freudenthal Institut, Utrecht University, Utrecht (ISBN 90-74684-25-4). 356 p. (2005).}
\itemab
Summary: This book provides a longitudinal view of children's learning trajectory in the domain of measurement and geometry in the lower grades of primary school. The learning-teaching trajectory that is described in the book is meant as a framework for instructional decision making. The description offers a rationale for designing and choosing measurement and geometry activities and helps to understand how these activities are related to one another. As such, the trajectory may contribute to a coherent primary school program for measurement and geometry. In addition, the trajectory helps to identify and stimulate children's measurement and geometric development. The intermediate attainment goals that are included in the learning-teaching trajectory serve as benchmarks for assessment. The many key activities from classroom practice that are described in the book and which are illustrated with video clips on the cd-rom that accompanies the book, make this trajectory description very useful for the improvement of mathematics classroom practice. The description may help teachers to enrich their use of textbooks and can give support to their professional development.
\itemrv{~}
\itemcc{F72 G22 D42 C72}
\itemut{primary education; measurement; elementary geometry; teaching-learning processes}
\itemli{}
\end |
http://www.ifa.hawaii.edu/~reipurth/newsletter/newsletter181.tex | hawaii.edu | CC-MAIN-2020-16 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-16/segments/1585370491857.4/warc/CC-MAIN-20200328104722-20200328134722-00460.warc.gz | 255,523,155 | 77,356 | \documentclass[]{article}
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& \\
\multicolumn{2}{|c|}{\LARGE\bf THE\hspace*{1cm}STAR\hspace*{1cm}FORMATION\hspace*{1cm}NEWSLETTER} \\ [0.3cm]
\multicolumn{2}{|c|}{\large\em An electronic publication dedicated to early stellar evolution and molecular clouds} \\ [0.3cm]
{\hspace*{0.8cm} No. 181 --- 23 Nov 2007 } & \multicolumn{1}{r|}{Editor: Bo Reipurth ([email protected])\hspace*{0.8cm}} \\ [-0.1cm]
& \\ \hline
\end{tabular}
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%\begin{center}
%{\Large\em From the Editor}
%\end{center}
%\vspace*{0.6cm}
\def\v3{\,{\vspace{0.3cm}}}
\def\v4{\,{\vspace{0.4cm}}}
\def\v5{\,{\vspace{0.5cm}}}
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\begin{center}
{\Large\em Abstracts of recently accepted papers}
\end{center}
\vspace*{0.6cm}
{\large\bf{ X-ray flaring from the young stars in Cygnus OB2 }}
{\bf{ J. F. Albacete Colombo$^{1,2}$, M. Caramazza$^1$, E. Flaccomio$^1$, G. Micela$^1$,
and S. Sciortino$^1$ }}
$^1$ { INAF - Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy } \\
$^2$ { Centro Universitario Regional Zona Atlantica (CURZA) - Univ. nacional del COMAHUE, Monsenor Esandi y Ayacucho (8500), Viedma (Rio Negro), Argentina }
{E-mail contact: facundo {\em at} astropa.unipa.it }
{\emph{Aims.} We characterize individual and ensemble properties of
X-ray flares from stars in the Cygnus OB2 and ONC star-forming
regions.
\emph{Methods.} We analyzed X-ray lightcurves of 1003 Cygnus OB2
sources observed with Chandra for 100 ks and of 1616 ONC sources
detected in the "Chandra Orion Ultra-deep Project" 850 ks
observation. We employed a binning-free maximum likelihood method to
segment the light-curves into intervals of constants signal and
identified flares on the basis of both the amplitude and the
time-derivative of the source luminosity. We then derived and
compared the flare frequency and energy distribution of Cygnus OB2 and
ONC sources. The effect of the length of the observation on these
results was investigated by repeating the statistical analysis on five
100 ks-long segments extracted from the ONC data.
\emph{Results.} We detected 147 and 954 flares from the Cygnus OB2 and
ONC sources, respectively. The flares in Cygnus OB2 have decay times
ranging from $\simless$0.5 to about 10 h. The flare energy
distributions of all considered flare samples are described at high
energies well by a power law with index $\alpha=-(2.1\pm0.1)$. At low
energies, the distributions flatten, probably because of detection
incompleteness. We derived average flare frequencies as a function of
flare energy. The flare frequency is seen to depend on the source's
intrinsic X-ray luminosity, but its determination is affected by the
length of the observation. The slope of the high-energy tail of the
energy distribution is, however, affected little. A comparison of
Cygnus OB2 and ONC sources, accounting for observational biases, shows
that the two populations, known to have similar X-ray emission levels,
have very similar flare activity.
\emph{Conclusions.} Studies of flare activity are only comparable if
performed consistently and taking the observation length into
account. Flaring activity does not vary appreciably between the age of
the ONC ($\sim$1 Myr) and that of Cygnus OB2 ($\sim$2 Myr). The slope
of the distribution of flare energies is consistent with the
micro-flare explanation of the coronal heating. }
{ Published by Astronomy \& Astrophysics (Vol. 474, p. 495) }
\v5
%%--------SubmissionID=1267----------------
%% Title
{\large\bf{The Spitzer c2d Survey of Large, Nearby, Interstellar Clouds.~X \\
The Chamaeleon~II Pre-Main Sequence Population as Observed With IRAC and MIPS}}
%% Authors
{\bf{ Juan M. Alcal\'a$^{1}$, Loredana Spezzi$^{1, 4}$, Nicholas Chapman$^{2}$, Neal J. Evans II$^{3}$, Tracy L. Huard$^{5}$, Jes K. J{\o}rgensen$^{6}$, Bruno Mer\'{i}n$^{7, 10}$, Karl R. Stapelfeldt$^{9}$, Elvira Covino$^{1}$, Antonio Frasca$^{4}$, Davide~Gandolfi$^{4}$ and Isa Oliveira$^{8, 10}$}}
%% Institutions
$^1$ {INAF-OA Capodimonte, via Moiariello 16, 80131, Naples, Italy} \\
$^2$ {Astronomy Department, University of Maryland, College Park, MD~20742, USA} \\
$^3$ {Astronomy Department, University of Texas at Austin, 1 University Station C1400, Austin, TX~78712-0259, USA} \\
$^4$ {INAF-OA Catania, via S. Sofia, 78, 95123 Catania, Italy} \\
$^5$ {Smithsonian Astrophysical Observatory, 60 Garden Street, MS42, Cambridge, MA~0213, USA} \\
$^6$ {Argelander-Institut f\"ur Astronomie, University of Bonn, Auf dem H\"ugel 71, 53121 Bonn, Germany} \\
$^7$ {Research and Scientific Support Dept. (ESTEC), European Space Agency, Keplerlaan, 1, PO Box 299, 2200 AG Noordwijk, The Netherlands} \\
$^8$ {Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA} \\
$^9$ {Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA} \\
$^10$ {Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands}
%% Email
{E-mail contact: alcala {\em at} oacn.inaf.it}
%% LATEX COMMANDS
%% Abstract body
{We discuss the results from the combined IRAC and MIPS c2d Spitzer Legacy
survey observations and complementary optical and near infrared data of
the Chamaeleon~II (Cha~II) dark cloud. We perform a census of the young
population of Cha~II, in a mapped area of $\sim$1.75~deg$^2$, and study
the spatial distribution and properties of the cloud members and candidate
pre-main sequence (PMS) objects and their circumstellar matter.
Our census of PMS objects and candidates in ChaII is complete down to the
sub-stellar regime ($M \approx 0.03~M_{\odot}$), at the assumed cloud
distance of 178~pc. The population consists of 51 certified and
11 candidate PMS objects, most of them located in the Eastern part of
the cloud, but approximately following the dust emission lanes of the
c2d extinction map. From the analysis of the volume density of the PMS
objects and candidates we find two tight groups of objects with volume
densities higher than 25~M$_{\odot}$~pc$^{-3}$ and 5-10 members each.
These groups correlate well in space with the regions of high extinction.
A multiplicity fraction of about 13$\pm$3\% is observed for objects
with separations 0.8" $< \theta <$ 6.0" (142 - 1065 AU). No evidence for
variability in the IRAC bands between the two epochs of the c2d data set,
$\Delta t \sim$6 hours, is detected.
Using the results of masses and ages from a companion paper, we estimate
the star formation efficiency to be 1-4\%, consistent with the estimates
for Taurus and Lupus, but significantly lower than for Cha~I. This might
mean that different star-formation activities in the Chamaeleon clouds
reflect a different history of star formation.
We also find that the Cha~II cloud is turning some 6-7 $M_{\odot}$ into
stars every Myr, which is low in comparison with the star formation
rate in other c2d clouds. On the other hand, the disk fraction of
70-80\% that we estimate in Cha~II is much higher than in other star
forming regions and indicates that the population in this cloud is dominated
by objects with active accretion, with only a minority being systems with
passive and debris disks. The circumstellar envelope/disk properties of
the PMS objects and candidates are also investigated. Finally, the Cha~II
outflows are discussed, with particular regard to the discovery of a new
Herbig-Haro outflow, HH~939, driven by the classical T~Tauri star Sz~50.}
% Journal
{ Accepted by ApJ}
%% Preprints URL
http://peggysue.as.utexas.edu/SIRTF/PAPERS/pap94.pub.pdf
\v5
{\large\bf{Self-gravitating fragmentation of eccentric accretion disks}}
%% Here comes the author(s) of the paper, please indicate within $^...$
%% the number which corresponds to the institute of each author.
{\bf{ R.D. Alexander$^{1,2}$, P.J. Armitage$^{1,3}$, J. Cuadra$^1$ \& M.C. Begelman$^{1,3}$}}
%% Here you write your institute name(s) and address(es),
%% the number in $^..$ indicates your author number, for example:
$^1$ {JILA, University of Colorado, Boulder, CO 80309-0440, USA} \\
$^2$ {Leiden Observatory, Universiteit Leiden, Niels Bohrweg 2, 2300 RA, Leiden, the Netherlands} \\
$^3$ {Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309-0391, USA}
%% Here you may write the e-mail address of one or more of the authors
%% who will act as contact person for preprint requests etc, for example:
{E-mail contact: [email protected]}
%% IF YOU USE ANY PERSONAL LATEX COMMANDS IN YOUR ABSTRACT,
%% PLEASE INCLUDE THEIR DEFINITIONS HERE!
%% Within the following brackets you place your text:
{We consider the effects of eccentricity on the fragmentation of
gravitationally unstable accretion disks, using numerical
hydrodynamics. We find that eccentricity does not affect the overall
stability of the disk against fragmentation, but significantly alters
the manner in which such fragments accrete gas. Variable tidal forces
around an eccentric orbit slow the accretion process, and suppress the
formation of weakly-bound clumps. The ``stellar'' mass function
resulting from the fragmentation of an eccentric disk is found to have
a significantly higher characteristic mass than that from a
corresponding circular disk. We discuss our results in terms of the
disk(s) of massive stars at $\simeq0.1$pc from the Galactic Center,
and find that the fragmentation of an eccentric accretion disk, due to
gravitational instability, is a viable mechanism for the formation of
these systems.}
% Here you write which journal accepted your paper, for example:
{ Accepted for publication in ApJ. }
%% If preprints are available on the WWW you can give the web
%% direction here.
Preprint available at {\tt
http://www.strw.leidenuniv.nl/$\sim$rda/publications.html} or {\tt
arXiv:0711.0759}
\v5
%%--------SubmissionID=1242----------------
%% Title
{\large\bf{Accretion and ejection properties of embedded protostars:
the case of HH26, HH34 and HH46 IRS}}
%% Authors
{\bf{ Simone Antoniucci$^{1}$, Brunella Nisini$^{1}$, Teresa Giannini$^{1}$ and Dario Lorenzetti$^{1}$}}
%% Institutions
$^1$ {INAF-Osservatorio Astronomico di Roma, Via di Frascati 33, I-00040 Monteporzio Catone, Italy}
%% Email
{E-mail contact: antoniucci {\em at} oa-roma.inaf.it}
%% LATEX COMMANDS
%% Abstract body
{We present the results of a spectroscopic analysis on three young
embedded sources (HH26 IRS, HH34 IRS and HH46 IRS) belonging to
different star-forming regions and displaying well developed jet
structures. The aim is to investigate the source accretion and
ejection properties and their connection. We used VLT-ISAAC near-IR
medium resolution ($R\sim9000$) spectra ($H$ and $K$ bands) to derive,
in a self-consistent way, parameters like the star luminosity, the
accretion luminosity and the mass accretion rate. Mass ejection rates
have also been estimated from the analysis of different emission
features. The spectra present several emission lines but no
photospheric features in absorption, indicating a large veiling in
both $H$ and $K$ bands. In addition to features commonly observed in
jet driving sources ([Fe II], H$_2$, H I, CO), we detect a number of
emission lines due to permitted atomic transitions, such as Na I and
Ti I that are only 2-5 times weaker than the Br$\gamma$ line. Some of
these features remain unidentified. Emission from Na I 2.2$\mu$m
doublet is observed along with CO(2-0) band-head emission, indicating
a common origin in an inner gaseous disc heated by accretion. We find
that accretion provides about 50\% and 80\% of the bolometric
luminosity in HH26 IRS and HH34 IRS, as expected for accreting young
objects. Mass accretion and loss rates spanning $10^{-6}$--$10^{-8}$
M$_{\odot}$\,yr$^{-1}$ have been measured. The derived
$\dot{M}_\mathrm{loss}/\dot{M}_\mathrm{acc}$ is $\sim$0.01 for HH26
IRS and HH34 IRS, and $>$0.1 for HH46 IRS. These numbers are in the
range of values predicted by MHD jet launching models and found in the
most active classical T Tauri stars. Comparison with other
spectroscopic studies performed on Class Is seems to indicate that
Class Is actually having accretion-dominated luminosities are a
limited number. Although the analysed sample is small, we can
tentatively define some criteria to characterise such sources: they
have $K$-band veiling larger than 2 and in the majority of the cases
present IR features of CO and Na I in emission, although these do not
directly correlate with the accretion luminosity. Class Is with
massive jets have high $L_{\mathrm{acc}}/L_{\mathrm{bol}}$ ratios but
not all the identified accretion-dominated objects present a jet. As
suggested by the SEDs of our three objects, the accretion-dominated
objects could be in an evolutionary transition phase between Class 0
and I. Studies of the kind presented here but on larger samples of
possible candidates should be performed in order to test and refine
these criteria.}
% Journal
{ Accepted by A\&A}
%% Preprints URL
http://arxiv.org/abs/0710.5609
\v5
{\large\bf{ An H$_2$CO 6 cm Maser Pinpointing a Possible Circumstellar Torus in IRAS 18566+0408 }}
{\bf{ E. Araya$^{1,2}$, P. Hofner$^{1,2}$, M. Sewilo$^3$, W. M. Goss$^2$, H. Linz$^4$, S. Kurtz$^5$,
L. Olmi$^{6,7}$, E. Churchwell$^3$, L. F. Rodríguez$^5$ and G. Garay$^8$ }}
$^1$ { Department of Physics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA } \\
$^2$ { National Radio Astronomy Observatory, Socorro, NM 87801, USA } \\
$^3$ { Department of Astronomy, University of Wisconsin-Madison, Madison, WI 53706, USA } \\
$^4$ { Max-Planck-Institut f\"ur Astronomie, D-69117 Heidelberg, Germany } \\
$^5$ { Centro de Radioastronom\'{\i}a y Astrof\'{\i}sica, Universidad Nacional Aut\'onoma de M\'exico,
58089 Morelia, Michoac\'an, Mexico } \\
$^6$ { Istituto di Radioastronomia, INAF, Sezione di Firenze, I-50125 Florence, Italy } \\
$^7$ { Department of Physics, University of Puerto Rico at Rio Piedras, P.O. Box 23343, San Juan, PR 00931, Puerto Rico } \\
$^8$ { Departamento de Astronom\'{\i}a, Universidad de Chile, Casilla 36-D, Santiago, Chile }
{ We report observations of 6 cm, 3.6 cm, 1.3 cm, and 7 mm radio continuum, conducted with the Very Large
Array, toward IRAS 18566+0408, one of the few sources known to harbor H$_2$CO 6 cm maser emission. Our
observations reveal that the emission is dominated by an ionized jet at centimeter wavelengths. Spitzer
IRAC images from GLIMPSE support this interpretation, given the presence of 4.5 $\mu$m excess emission
at approximately the same orientation as the centimeter continuum. The 7 mm emission is dominated by
thermal dust from a flattened structure almost perpendicular to the ionized jet; thus, the 7 mm emission
appears to trace a torus associated with a young massive stellar object. The H$_2$CO 6 cm maser is
coincident with the center of the torus-like structure. Our observations rule out radiative pumping
via radio continuum as the excitation mechanism for the H$_2$CO 6 cm maser in IRAS 18566+0408. }
{ Published by The Astrophysical Journal (Vol. 669, p. 1050) }
\vspace{0.3cm}
{\large\bf{Spatially Resolved Molecular Hydrogen Emission in the Inner 200AU
Environments of Classical T Tauri Stars$^1$}}
%% Here comes the author(s) of the paper, please indicate within $^...$
%% the number which corresponds to the institute of each author.
{\bf{Tracy L. Beck$^{2,3}$, Peter J. McGregor$^4$, Michihiro Takami$^{5,6}$ \&
Tae-Soo Pyo$^5$}}
%% Here you write your institute name(s) and address(es),
%% the number in $^..$ indicates your author number, for example:
$^2$ Gemini North Observatory, 670 N. A'ohoku Pl. Hilo, HI 96720, USA\\
$^3$ Current Address: Space Telescope Science Institute, 3700 San Martin Dr.,
Baltimore, MD 21218, USA\\
$^4$ Research School of Astronomy \& Astrophysics,, Australian National
University, Private Bag PO, Weston Creek, ACT 2611, Australia\\
$^5$ Subaru Telescope, National Astronomical Observatory of Japan, 650 North
A`oh\=ok\=u Place, Hilo, HI 96720\\
$^6$ Institute of Astronomy \& Astrophysics, Academia Sinica, P.O. Box 23-
141, Taipei 10617, Taiwan, R. O. C.
%% Here you may write the e-mail address of one or more of the authors
%% who will act as contact person for preprint requests etc, for example:
{E-mail contact: tbeck {\em at} stsci.edu}
%% IF YOU USE ANY PERSONAL LATEX COMMANDS IN YOUR ABSTRACT,
%% PLEASE INCLUDE THEIR DEFINITIONS HERE!
%% Within the following brackets you place your text:
We present 2.0-2.4$\mu$m integral field spectroscopy at adaptive optics
spatial resolution ($\sim$0.''1) obtained with the Near-infrared Integral Field
Spectrograph (NIFS) at Gemini North Observatory of six Classical T Tauri stars: T
Tau, DG Tau, XZ Tau, HL Tau, RW Aur and HV Tau C. In all cases, the {\it v}=1-0
S(1) (2.12$\mu$m) emission is detected at spatially extended distances from
the central stars. HL Tau, T Tau and HV Tau C have H$_2$ emission that
extends to projected distances of more than $\sim$200 AU from the stars. The
bulk of the H$_2$ emission is typically not coincident with the location of
continuum flux. The observed morphologies vary between emission that is
spatially continuous but decreasing away from the star (HV Tau C, DG Tau), and
H$_2$ that shows discrete knots and arcs with 0.''2-0.''3 ($\sim$28-42 AU)
spatial extents (T Tau, XZ Tau). Multiple transitions detected in the K-band
spectra show that H$_2$ level populations are typical of gas in thermal
equilibrium with excitation temperatures in the 1800K-2300 K range. Two
dimensional maps of the extinction and H$_2$ excitation temperature were
estimated from the high signal-to-noise (S/N) observations of T Tau using
maps of the spatially resolved {\it v}=1-0 Q(3)/{\it v}=1-0 S(1) and {\it v}=2-1
S(1)/{\it v}=1-0 S(1) line ratios. These maps show that A$_v$ and T$_{ex}$
values in the vicinity of T Tau vary on a significant level on spatial scales of less
than 100 AU. Three of the stars have H$_2$ velocity profiles that are centered
at the stellar radial velocity, and three show velocity shifts with respect to the
system. RW Aur exhibits high and low velocity H$_2$ emission, and the
observed morphology and kinematics of the high velocity material are consistent
with atomic emission from its red-shifted micro-jet. The morphologies
detected in H$_2$ from DG Tau and HV Tau C resemble emission seen in
scattered light from stars with inclined or edge-on circumstellar disks. The
locations and relative brightnesses of H$_2$ knots detected within $\sim$1$''$
of T Tau has varied on a timescale of $\sim$3 years. Each of the stars studied
here show observed excitation temperatures, spatial extents, and kinematics of
the H$_2$ that are most consistent with shock excited emission from the inner
regions of the known Herbig-Haro energy flows or from wide-angle winds
encompassing the outflows rather than predominantly from UV or X-ray
stimulated emission from the central stars. The data presented in this study
highlights the sensitivity of adaptive optics-fed integral field spectroscopy for
spatially resolving emission line structures in the environments of bright young
stars.
% Here you write which journal accepted your paper, for example:
{ Accepted by the Astrophysical Journal }
{The full resolution paper is available at:
http://www.astro.sunysb.edu/tracy/pubs/Beck07.pdf}
\vspace{0.3cm}
{\large\bf{$\gamma$-ray production in young open clusters: Berk 87, Cyg OB2 and Westerlund 2 }}
{\bf{W. Bednarek$^1$ }}
$^1$ {Department of Experimental Physics, University of L\'od\'z, ul. Pomorska 149/153, 90-236 L\'od\'z, Poland }
{E-mail contact: bednar {\em at} fizwe4.phys.uni.lodz.pl}
{Young open clusters are sites of cosmic ray acceleration as indicated by recent detections of the
TeV $\gamma$-ray sources in the directions of two open clusters (Cyg OB2 and Westerlund 2). In fact,
up to now a few different scenarios for acceleration of particles inside open clusters have been
considered, i.e. shocks in massive star winds, pulsars and their nebulae, supernova shocks, massive
compact binaries. Here we consider in detail the radiation processes due to both electrons and hadrons
accelerated inside the open cluster. As a specific scenario, we apply the acceleration process at the
shocks arising in the winds of Wolf–Rayet (WR) type stars. Particles diffuse through the medium of the
open cluster during the activity time of the acceleration scenario defined by the age of the WR star.
They interact with the matter and radiation, at first inside the open cluster and, later in the dense
surrounding clouds. We calculate the broad-band spectrum in different processes for three exemplary
open clusters (Berk 87, Cyg OB2, Westerlund 2) for which the best observational constraints on the
spectra are at present available. It is assumed that the high-energy phenomena, observed from the X-ray
up to the GeV–TeV $\gamma$-ray energies, are related to each other. We conclude that the most likely
description of the radiation processes in these objects is achieved in the hybrid (leptonic–hadronic)
model in which leptons are responsible for the observed X-ray and GeV $\gamma$-ray emission and hadrons
are responsible for the TeV $\gamma$-ray emission. }
{ Published by Monthly Notices of the Royal Astronomical Society (Vol. 382, p. 367) }
\v5
{\large\bf{Envelope instability in giant planet formation }}
{\bf{Omar G. Benvenuto$^{1,2}$, Adri\'an Brunini$^{1,2}$ and Andrea Fortier$^{1,2}$}}
$^1$ {Facultad de Ciencias Astron\'omicas y Geof\'{\i}sicas, Universidad Nacional de La Plata, 1900 La Plata, Argentina } \\
$^2$ {Instituto de Astrof\'{\i}sica de La Plata (IALP), 1900 La Plata, Argentina }
{ We compute the growth of isolated gaseous giant planets for several values of the density of the protoplanetary disk, several distances from the central star and two values for the (fixed) radii of accreted planetesimals. Calculations were performed in the frame of the core instability mechanism and the solids accretion rate adopted is that corresponding to the oligarchic growth regime. We find that for massive disks and/or for protoplanets far from the star and/or for large planetesimals, the planetary growth occurs smoothly. However, notably, there are some cases for which we find an envelope instability in which the planet exchanges gas with the surrounding protoplanetary nebula. The timescale of this instability shows that it is associated with the process of planetesimals accretion. The presence of this instability makes it more difficult the formation of gaseous giant planets. }
{ Published by Icarus (Vol. 191, p. 394) }
\v5
{\large\bf{ An Imaging Survey for Extrasolar Planets around 45 Close, Young Stars with the Simultaneous Differential Imager at the Very Large Telescope and MMT }}
{\bf{ Beth A. Biller$^1$, Laird M. Close$^1$, Elena Masciadri$^2$, Eric Nielsen$^1$, Rainer Lenzen$^3$,
Wolfgang Brandner$^3$, Donald McCarthy$^1$, Markus Hartung$^{4,5}$, Stephan Kellner$^6$,
Eric Mamajek$^7$, Thomas Henning$^3$, Douglas Miller$^1$, Matthew Kenworthy$^1$, and Craig Kulesa$^1$ }}
$^1$ { Steward Observatory, University of Arizona, Tucson, AZ 85721, USA } \\
$^2$ { Observatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Florence, Italy } \\
$^3$ { Max-Planck-Institut f\"ur Astronomie, K\"onigstuhl 17, 69117 Heidelberg, Germany } \\
$^4$ { European Southern Observatory, Alonso de Cordova 3107, Santiago 19, Chile } \\
$^5$ { Departamento de Astronom\'{\i}a, Universidad de Chile, Casilla 36-D, Santiago, Chile } \\
$^6$ { W. M. Keck Observatory, 65-1120 Mamalahoa Highway, Kamuela, HI 96743, USA } \\
$^7$ { Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA }
{ We present the results of a survey of 45 young ($\simless$ 250 Myr),
close ($\simless$ 50 pc) stars with the Simultaneous Differential
Imager (SDI) implemented at the VLT and the MMT for the direct
detection of extrasolar planets. As part of the survey, we observed 54
objects, consisting of 45 close, young stars; two more distant ($<$150
pc), extremely young ($\le$10 Myr) stars; three stars with known
radial velocity planets; and four older, very nearby ($\le$20 pc)
solar analogs. Our SDI devices use a double Wollaston prism and a quad
filter to take images simultaneously at three wavelengths surrounding
the 1.62 $\mu$m methane absorption bandhead found in the spectrum of
cool brown dwarfs and gas giant planets. By differencing adaptive
optics–corrected images in these filters, speckle noise
from the primary star is significantly attenuated, resulting in photon
(and flat-field)–noise–limited data. In our VLT
data, we achieved H-band contrasts $\simgreat$ 10 mag (5 $\sigma$) at a
separation of 0.5$^{\prime\prime}$ from the primary star on 45\% of
our targets and H-band contrasts $\simgreat$ 9 mag at a separation of
0.5$^{\prime\prime}$ on 80\% of our targets. With these contrasts, we
can image (5 $\sigma$ detection) a 7 M$_J$ planet 15 AU from a 70 Myr
K1 star at 15 pc or a 7.8 M$_J$ planet at 2 AU from a 12 Myr M star at
10 pc. We detected no candidates with S/N $>$2 $\sigma$ which behaved
consistently like a real object. From our survey null result, we can
rule out (with 93\% confidence) a model planet population where N(a)
$\propto$ constant out to a distance of 45 AU. }
{ Published by The Astrophysical Journal Supplement Series (Vol. 173, p. 143) }
\v5
%%--------SubmissionID=1246----------------
%% Title
{\large\bf{A massive protostellar core with an infalling envelope}}
%% Authors
{\bf{ Stephan M. Birkmann$^{1}$, Oliver Krause$^{1}$, Martin Hennemann$^{1}$, Thomas Henning$^{1}$, Jrgen Steinacker$^{1,2}$ and Dietrich Lemke$^{1}$}}
%% Institutions
$^1$ {Max-Planck-Institut f\"ur Astronomie (MPIA), K\"onigstuhl 17, 69117 Heidelberg, Germany} \\
$^2$ {Zentrum fr Astronomie der Universit\"at Heidelberg (ZAH), Mnchhofstr. 12-14, 69120 Heidelberg, Germany}
%% Email
{E-mail contact: birkmann {\em at} mpia.de}
%% LATEX COMMANDS
%% Abstract body
{Context: Due to the short timescales involved and observational difficulties, our knowledge of the earliest phases of massive star formation remains incomplete.\\
Aims: We aim to explore the physical conditions during the initial phases of high-mass star formation and to detect a genuine massive (mass M $>$ 8 M$_\odot$) protostar at an early evolutionary stage.\\
Methods: We have launched a multi-wavelength study of young and massive star-forming regions that were identified by the ISOPHOT Serendipity Survey (ISOSS) performed with the ISO space telescope. The follow-up observations include ground-based near-infrared imaging and (sub)mm continuum and molecular line measurements (both single-dish and interferometric), as well as mid- to far-infrared measurements with the Spitzer Space Telescope. The combined spectrophotometric data are used to determine source temperatures T and masses M.\\
Results: ISOSS J23053+5953 is a massive (M $\sim$ 900 M$_\odot$, luminosity L $\sim$ 2100 L$_\odot$) and cold (T $\sim$ 17 K) star-forming region with two protostellar/protocluster candidates (T $\leq$ 20 K and T $\sim$ 17.5 K, M $\sim$ 200 M$_\odot$ each). The low temperatures are strongly confined by the spectrophotometric Spitzer data in the FIR. Interferometric observations reveal that the colder core (SMM2) has a mass of M = 26 M$_\odot$ within a region of $8700\times 5600$ AU and drives an outflow. It also shows signs of infall in both single-dish and interferometric measurements, and its luminosity can be explained by accretion. We also detect a large-scale jet that is traced by H2 emission.\\
Conclusions: The cold mm-core ISOSS J23053+5953 SMM2 is a promising candidate for a high-mass protostar in an early evolutionary stage and one of the few objects showing both infall signatures and jets as a sign of accretion.}
% Journal
{ Accepted by Astronomy and Astrophysics (Vol. 474, p. 883) }
%% Preprints URL
http://dx.doi.org/10.1051/0004-6361:20077197
\v5
{\large\bf{ Stability and nonlinear adjustment of vortices in Keplerian flows }}
{\bf{ G. Bodo$^1$, A. Tevzadze$^2$, G. Chagelishvili$^2$, A. Mignone$^{1,3}$, P. Rossi$^1$,
and A. Ferrari$^3$ }}
$^1$ { INAF Osservatorio Astronomico di Torino, Strada dell'Osservatorio 20, 10025 Pino Torinese, Italy } \\
$^2$ { E. Kharadze Georgian National Astrophysical Observatory,, 2a Kazbegi Ave. Tbilisi 0160, Georgia } \\
$^3$ { Dipartimento di Fisica Generale dell'Universit\`a di Torino, via Pietro Giuria 1, 10125 Torino, Italy }
{E-mail contact: bodo {\em at} to.astro.it }
{\emph{Aims.} We investigate the stability, nonlinear development and equilibrium structure of vortices in a background shearing Keplerian flow
\emph{Methods.} We make use of high-resolution global two-dimensional compressible hydrodynamic simulations. We introduce the concept of nonlinear adjustment to describe the transition of unbalanced vortical fields to a long-lived configuration.
\emph{Results.} We discuss the conditions under which vortical perturbations evolve into long-lived
persistent structures and we describe the properties of these equilibrium vortices. The properties of
equilibrium vortices appear to be independent from the initial conditions and depend only on the local
disk parameters. In particular we find that the ratio of the vortex size to the local disk scale height
increases with the decrease of the sound speed, reaching values well above the unity. The process of spiral density wave generation by the vortex, discussed in our previous work, appear to maintain its efficiency also at nonlinear amplitudes and we observe the formation of spiral shocks attached to the vortex. The shocks may have important consequences on the long term vortex evolution and possibly on the global disk dynamics.
\emph{Conclusions.} Our study strengthens the arguments in favor of anticyclonic vortices as the candidates for the promotion of planetary formation. Hydrodynamic shocks that are an intrinsic property of persistent vortices in compressible Keplerian flows are an important contributor to the overall balance. These shocks support vortices against viscous dissipation by generating local potential vorticity and should be responsible for the eventual fate of the persistent anticyclonic vortices. Numerical codes have be able to resolve shock waves to describe the vortex dynamics correctly. }
{ Published by Astronomy \& Astrophysics (Vol. 475, p. 51) }
\v5
%%--------SubmissionID=1257----------------
%% Title
{\large\bf{Intermediate to low-mass stellar content of Westerlund 1}}
%% Authors
{\bf{ Wolfgang Brandner$^{1,2}$, J. Simon Clark$^{3}$, Andrea Stolte$^{2}$, Rens Waters$^{4}$, Ignacio Negueruela$^{5}$ and Simon P. Goodwin$^{6}$}}
%% Institutions
$^1$ {Max-Planck-Institute for Astronomy, Heidelberg, Germany} \\
$^2$ {University of California, Los Angeles, USA} \\
$^3$ {Open University, UK} \\
$^4$ {Sterrenkundig Instituut, Amsterdam, The Netherlands} \\
$^5$ {Universidad de Alicante, Spain} \\
$^6$ {University of Sheffield, UK}
%% Email
{E-mail contact: brandner {\em at} mpia.de}
%% LATEX COMMANDS
%% Abstract body
{We have analysed near-infrared NTT/SofI observations of the starburst
cluster Westerlund 1, which is among the most massive young clusters
in the Milky Way. A comparison of colour-magnitude diagrams with
theoretical main-sequence and pre-main sequence evolutionary tracks
yields improved extinction and distance estimates of A$_{Ks}$ =
1.13$\pm$0.03 mag and d = 3.55$\pm$0.17 kpc (DM = 12.75$\pm$0.10
mag). The pre-main sequence population is best fit by a Palla \&
Stahler isochrone for an age of 3.2 Myr, while the main sequence
population is in agreement with a cluster age of 3 to 5 Myr. An
analysis of the structural parameters of the cluster yields that the
half-mass radius of the cluster population increases towards lower
mass, indicative of the presence of mass segregation. The cluster is
clearly elongated with an eccentricity of 0.20 for stars with masses
between 10 and 32 Msun, and 0.15 for stars with masses in the range 3
to 10 Msun. We derive the slope of the stellar mass function for stars
with masses between 3.4 and 27 Msun. In an annulus with radii between
0.75 and 1.5 pc from the cluster centre, we obtain a slope of $\Gamma
= -1.3$. Closer in, the mass function of Westerlund 1 is shallower
with $\Gamma = -0.6$. The extrapolation of the mass function for stars
with masses from 0.08 to 120 Msun yields an initial total stellar mass
of $\approx$52,000 Msun, and a present-day mass of 20,000 to 45,000
Msun (about 10 times the stellar mass of the Orion Nebula Cluster, and
2 to 4 times the mass of the NGC 3603 young cluster), indicating that
Westerlund 1 is the most massive starburst cluster identified to date
in the Milky Way.}
% Journal
{ Accepted by A\&A}
%% Preprints URL
http://arxiv.org/abs/0711.1624
\v5
{\large\bf{Embedded star clusters and the formation of the Oort cloud II. The effect of the primordial solar nebula }}
{\bf{R. Brasser$^{1,2}$, M.J. Duncan$^1$ and H.F. Levison$^3$ }}
$^1$ {Department Physics and Astronomy, Stirling Hall, Queen's University, Kingston, K7L 3N6, Ontario, Canada } \\
$^2$ {Canadian Institute for Theoretical Astrophysics, 60 St. George Street, Toronto, M5S 3H8, Ontario, Canada } \\
$^3$ {Southwest Research Institute, 1050 Walnut Street, Boulder, 80302 CO, USA }
{This paper deals with Oort cloud formation while the Sun was in an
embedded cluster and surrounded by its primordial nebula. This work is
a continuation of Brasser et al. [Brasser, R., Duncan, M., Levison,
H., 2006. Icarus 184, 59–82], building on the model
presented therein, and adding the aerodynamic drag and gravitational
potential of the primordial solar nebula. Results are presented of
numerical simulations of comets subject to the gravitational influence
of the Sun, Jupiter, Saturn, star cluster and primordial solar nebula;
some of the simulations included the gravitational influence of Uranus
and Neptune as well. The primordial solar nebula was approximated by
the minimum-mass Hayashi model [Hayashi, C., Nakozawa, K., Nakagawa,
Y., 1985. In: Black, D.C., Matthews, M.S. (Eds.). Protostars and
Planets II. Univ. of Arizona Press, Tucson, AZ] whose inner and outer
radii have been truncated at various distances from the Sun. A comet
size of 1.7 km was used for most of our simulations. In all of our
simulations, the density of the primordial solar nebula decayed
exponentially with an e-folding time of 2 Myr. It turns out that when
the primordial solar nebula extends much beyond Saturn or Neptune,
virtually no material will end up in the Oort cloud (OC) during this
phase. Instead, the majority of the material will be on circular
orbits inside of Jupiter if the inner edge of the disk is well inside
Jupiter's orbit. If the disk's inner edge is beyond Jupiter's orbit,
most comets end up on orbits in exterior mean-motion resonances with
Saturn when Uranus and Neptune are not present. In those cases where
the outer edge of the disk is close to Saturn or Neptune, the fraction
of material that ends up in the subsequently formed OC is much less
than that found in Brasser et al. [Brasser, R., Duncan, M., Levison,
H., 2006. Icarus 184, 59–82] for the same cluster
densities. This implies that for comets of roughly 2 km in size, the
presence of the primordial solar nebula hinders OC formation. A
byproduct of some of our simulations are endresults with a substantial
fraction of the comets in the Uranus–Neptune scattered
disk. A subsequent followup of this material is planned for the near
future. In order to determine the effect of the size of the comets on
OC formation efficiency, a set of runs with the same initial
conditions but different cometary radii have been performed as well,
from which it is determined that the threshold comet size to begin
producing significant Oort clouds is roughly 20 km. This implies that
the presence of the primordial solar nebula acts as a size-sorting
mechanism, with large bodies unaffected by the gas drag and ending up
in the OC while small bodies remain trapped in the planetary region,
in the models studied. }
{ Published by Icarus (Vol. 191, p. 413) }
\v5
%%--------SubmissionID=1249----------------
%% Title
{\large\bf{Coagulation, fragmentation and radial motion of solid particles in protoplanetary disks}}
%% Authors
{\bf{ Frithjof Brauer$^{1}$, Cornelis P. Dullemond$^{1}$ and Thomas Henning$^{1}$}}
%% Institutions
$^1$ {Max-Planck-Institut f\"ur Astronomie, K\"onigstuhl 17, 69117 Heidelberg, Germany}
%% Email
{E-mail contact: brauer {\em at} mpia.de}
%% LATEX COMMANDS
%% Abstract body
{The growth of solid particles towards meter sizes in
protoplanetary disks has to circumvent at least two hurdles,
namely the rapid loss of material due to radial drift and particle
fragmentation due to destructive collisions. In this paper, we
present the results of numerical simulations with more and more
realistic physics involved. Step by step, we include various
effects, such as particle growth, radial/vertical particle motion
and dust particle fragmentation in our simulations. We demonstrate
that the initial dust-to-gas ratio is essential for the particles
to overcome the radial drift barrier. If this value is increased
by a factor of 2 compared with the canonical value for the
interstellar medium, km-sized bodies can form in the inner disk
($<2$~AU) within $10^4$~yrs. However, we find that solid particles
get destroyed through collisional fragmentation. Only with the
unrealistically high-threshold velocities needed for fragmentation
to occur ($>30$~m/s), particles are able to grow to larger sizes
in disks with low $\alpha$ values. We also find that less than 5\%
of the small dust grains remain in the disk after 1~Myrs due to
radial drift, no matter whether fragmentation is included in the
simulations or not. In this paper, we also present considerable
improvements to existing algorithms for dust-particle coagulation,
which speed up the coagulation scheme by a factor of $\sim 10^4$.}
% Journal
{ Accepted by A\&A}
%% Preprints URL
\v5
{\large\bf{ Continuum Observations at 3 and 12 mm of the High-Mass Protostellar Jet IRAS 16547-4247 }}
{\bf{ Kate J. Brooks$^1$, Guido Garay$^2$, Maxim Voronkov$^1$ and Luis F. Rodr\'{\i}guez$^3$ }}
$^1$ { Australia Telescope National Facility, P.O. Box 76, Epping NSW 1710, Australia } \\
$^2$ { Departamento de Astronom\'{\i}a, Universidad de Chile, Casilla 36-D, Santiago, Chile } \\
$^3$ { Centro de Radioastronom\'{\i}a y Astrof\'{\i}sica, UNAM, Apdo. Postal 3-72, Morelia,
Michoac\'an, 58089, Mexico }
{ Continuum data at 25 and 88 GHz toward the luminous young stellar object IRAS 16547-4247 (G343.126-0.062) have been obtained with the Australia Telescope Compact Array. The triple emission source identified previously at lower frequencies has been detected at 25 GHz. For frequencies between 1.4 and 25 GHz, the flux density of the central continuum source is well fitted with a power-law dependence that is consistent with thermal emission from a jet. The two outer lobes are radio Herbig-Haro objects exhibiting thermal and nonthermal synchrotron emission. At 88 GHz, one unresolved emission source was detected, centered on the radio jet. At this frequency the emission does not arise from the jet but from the dusty molecular envelope within which the jet is embedded. }
{ Published by The Astrophysical Journal (Vol. 669, p. 459) }
\clearpage
{\large\bf{ Grain Alignment and Polarized Emission from Magnetized T Tauri Disks }}
{\bf{ Jungyeon Cho$^1$ and A. Lazarian$^2$ }}
$^1$ { Department of Astronomy and Space Science, Chungnam National University, Daejeon, South Korea } \\
$^2$ { Astronomy Department, University of Wisconsin, Madison, WI 53706, USA }
{E-mail contact: jcho {\em at} cnu.ac.kr }
{ The structure of magnetic fields within protostellar disks may be studied via polarimetry provided that
grains are aligned with respect to the magnetic field within the disks. We explore the alignment of dust
grains by radiative torque in T Tauri disks and provide predictions for polarized emission for disks viewed at different wavelengths and viewing angles. We show that the alignment is especially efficient in the outer parts of the disks. In the presence of a magnetic field, these aligned grains produce polarized emission in infrared wavelengths. We consider a simple disk model and provide predictions for polarization that are available to the present-day instruments that do not resolve the disks and will be available to future instruments that will resolve the disks. We find that the polarized emission drops for wavelengths shorter than
$\sim$ 10 $\mu$m. Between $\sim$10 and $\sim$100 $\mu$m, the polarized emission is dominated by the emission from the surface layer, and the degree of polarization can be as large as $\sim$10\% for unresolved disks. We find that the degree of polarization at these wavelengths is very sensitive to the size distribution of dust grains in the disk surface layer, which should allow for the testing of the predicted grain-size
distributions. The degree of polarization in the far-infrared/submillimeter wavelengths is sensitive to the size distribution of dust grains in the disk interior. When we take a Mathis-Rumpl-Nordsieck–type
distribution with maximum grain size of 500-1000 $\mu$m, the degree of polarization is around the
2\%-3\% level at wavelengths larger than $\sim$100 $\mu$m. Our study indicates that multifrequency
infrared polarimetric studies of protostellar disks can provide good insights into the details of their magnetic structure. }
{ Published by The Astrophysical Journal (Vol. 669, p. 1085) }
\v5
%%--------SubmissionID=1259----------------
%% Title
{\large\bf{Circular polarimetry reveals helical magnetic fields in the young stellar object HH 135-136}}
%% Authors
{\bf{ Antonio Chrysostomou$^{1,2}$, Phillip W. Lucas$^{1}$ and James H. Hough$^{1}$}}
%% Institutions
$^1$ {Centre for Astrophysics Research, University of Hertfordshire, Hatfield, HERTS AL10 9AB, UK} \\
$^2$ {Present address: Joint Astronomy Centre, 660 N. A'ohoku Place, Hilo, HI 96720, USA}
%% Email
{E-mail contact: a.chrysostomou {\em at} herts.ac.uk}
%% LATEX COMMANDS
%% Abstract body
{Magnetic fields are believed to have a vital role in regulating and
shaping the flow of material onto and away from protostars during
their initial mass accretion phase. It is becoming increasingly
accepted that bipolar outflows are generated and collimated as
material is driven along magnetic field lines and centrifugally
accelerated off a rotating accretion disk. However, the precise role
of the magnetic field is poorly understood and evidence for its shape
and structure has not been forthcoming. Here we report imaging
circular polarimetry in the near-infrared and Monte Carlo modelling
showing that the magnetic field along the bipolar outflow of the HH
135-136 young stellar object is helical. The field retains this shape
for large distances along the outflow, so the field structure can also
provide the necessary magnetic pressure for collimation of the
outflow. This result lends further weight to the hypothesis-central to
any theory of star formation-that the outflow is an important
instrument for the removal of high-angular-momentum material from the
accretion disk, thereby allowing the central protostar to increase its
mass.}
% Journal
{ Accepted by Nature, Vol 450, 71-73}
%% Preprints URL
\v5
{\large\bf{Outward Transport of High-Temperature Materials Around the Midplane of the Solar Nebula }}
{\bf{Fred J. Ciesla$^1$ }}
$^1$ {Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road, NW, Washington, DC 20015, USA }
{E-mail contact: fciesla {\em at} ciw.edu }
{he Stardust samples collected from Comet 81P/Wild 2 indicate that large-scale mixing occurred in the solar nebula, carrying materials from the hot inner regions to cooler environments far from the Sun. Similar transport has been inferred from telescopic observations of protoplanetary disks around young stars. Models for protoplanetary disks, however, have difficulty explaining the observed levels of transport. Here I report the results of a new two-dimensional model that shows that outward transport of high-temperature materials in protoplanetary disks is a natural outcome of disk formation and evolution. This outward transport occurs around the midplane of the disk. }
{ Published by Science (Vol. 318, p. 613) }
\v5
{\large\bf{The response of self-gravitating protostellar discs to slow reduction in cooling time-scale: the fragmentation boundary revisited }}
{\bf{C. J. Clarke$^1$, E. Harper-Clark$^2$ and G. Lodato$^3$ }}
$^1$ {Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK } \\
$^2$ {Department of Astronomy and Astrophysics, 50 St George Street, Toronto, Ontario, Canada M5S 3H4, Canada } \\
$^3$ {Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK }
{E-mail contact: cclarke {\em at} ast.cam.ac.uk}
{A number of previous studies of the fragmentation of self-gravitating protostellar discs have involved
suites of simulations in which radiative cooling is modelled in terms of a cooling time-scale ($t_{cool}$)
which is parametrized as a simple multiple ($\beta_{cool}$) of the local dynamical time-scale. Such
studies have delineated the `fragmentation boundary' in terms of a critical value of $\beta_{cool}(\beta_{crit}$) such that the disc fragments if $\beta_{cool} < \beta_{crit}$. Such an approach however begs the
question of how in reality a disc could ever be assembled in a state with $\beta_{cool} < \beta_{crit}$.
Here we adopt the more realistic approach of effecting a gradual reduction in $\beta_{cool}$, as might
correspond to changes in thermal regime due to secular changes in the disc density profile. We find that
the effect of gradually reducing $\beta_{cool}$ (on a time-scale longer than $t_{cool}$) is to stabilize
the disc against fragmentation, compared with models in which $\beta_{cool}$ is reduced rapidly
(over less than $t_{cool}$). We therefore conclude that the ability of a disc to remain in a
self-regulated, self-gravitating state (without fragmentation) is partly dependent on the disc's thermal
history, as well as its current cooling rate. Nevertheless, the effect of a slow reduction in $t_{cool}$
appears only to lower the fragmentation boundary by about a factor of 2 in $t_{cool}$ and thus only
permits maximum `$\alpha$' values (which parametrize the efficiency of angular momentum transfer in the
disc) that are about a factor of 2 higher than determined hitherto. Our results therefore do not
undermine the notion that there is a fundamental upper limit to the heating rate that can be delivered
by gravitational instabilities before the disc is subject to fragmentation. An important implication of
this work, therefore, is that self-gravitating discs can enter into the regime of fragmentation via
secular evolution and it is not necessary to invoke rapid (impulsive) events to trigger fragmentation. }
{ Published by Monthly Notices of the Royal Astronomical Society (Vol. 381, p. 1543) }
\v5
{\large\bf{ Stringent Criteria for Stable and Unstable Planetary Orbits in Stellar Binary Systems }}
{\bf{ M. Cuntz$^1$, J. Eberle$^1$, and Z. E. Musielak$^1$ }}
$^1$ { Department of Physics, University of Texas at Arlington, Arlington, TX 76019-0059, USA }
{E-mail contact: cuntz {\em at} uta.edu }
{ The existence of planets in stellar binary (and higher order)
systems has now been confirmed by many observations. The stability of
planetary orbits in these systems has been extensively studied, but no
precise stability criteria have so far been introduced. Therefore,
there is an urgent need for developing stringent mathematical criteria
that allow us to precisely determine whether a planetary orbit in a
binary system is stable or unstable. In this Letter, such criteria are
defined using the concept of Jacobi's integral and Jacobi's
constant. These criteria are used to contest previous results on
planetary orbital stability in binary systems. }
{ Published by The Astrophysical Journal Letters (Vol. 669, p. L105) }
\clearpage
{\large\bf{ A near-infrared interferometric survey of debris disk stars I:
Probing the hot dust content around $\epsilon$ Eridani and $\tau$ Ceti with CHARA/FLUOR }}
{\bf{ E. Di Folco$^1$, O. Absil$^{2,3}$, J.-C. Augereau$^2$, A. M\'erand$^4$, V. Coud\'e du Foresto$^5$,
F. Th\'evenin$^6$, D. Defr\`ere$^3$, P. Kervella$^5$, T. A. ten Brummelaar$^4$, H. A. McAlister$^4$,
S. T. Ridgway$^{7,4}$, J. Sturmann$^4$, L. Sturmann$^4$, and N. H. Turner$^4$ }}
$^1$ { Observatoire de Gen\`eve, Universit\'e de Gen\`eve, Chemin des Maillettes 51, 1290 Sauverny,
Switzerland } \\
$^2$ { Laboratoire d'Astrophysique de l'Observatoire de Grenoble, UMR CNRS/UJF 5571, BP 53, 38041 Grenoble Cedex 9, France } \\
$^3$ { Institut d'Astrophysique et de G\'eophysique, Universit\'e de Li\`ege, 17 All\'ee du Six Ao\^ut,
4000 Li\`ege, Belgium } \\
$^4$ { Center for High Angular Resolution Astronomy, Georgia State University, PO Box 3969, Atlanta, Georgia 30302-3965, USA } \\
$^5$ { LESIA, UMR8109, Observatoire de Paris-Meudon, 5 place J. Janssen, 92195 Meudon, France } \\
$^6$ { Laboratoire Cassiop\'ee, CNRS, Observatoire de la C\^ote d'Azur, BP 4229, 06304 Nice Cedex 4,
France } \\
$^7$ { National Optical Astronomical Observatory, 950 North Cherry Avenue, Tucson, AZ 85719, USA }
{E-mail contact: emmanuel.difolco {\em at} obs.unige.ch }
{ \emph{Context.} The quest for hot dust in the central region of debris disks requires high resolution
and high dynamic range imaging. Near-infrared interferometry is a powerful means to directly detect faint
emission from hot grains.
\emph{Aims.} We probed the first 3 AU around $\tau$ Ceti and $\epsilon$ Eridani with the CHARA array
(Mt Wilson, USA) in order to gauge the 2 $\mu$m excess flux emanating from possible hot dust grains in
the debris disks and to also resolve the stellar photospheres.
\emph{Methods.} High precision visibility amplitude measurements were performed with the FLUOR single
mode fiber instrument and telescope pairs on baselines ranging from 22 to 241 m of projected length.
The short baseline observations allow us to disentangle the contribution of an extended structure from
the photospheric emission, while the long baselines constrain the stellar diameter.
\emph{Results.} We have detected a resolved emission around $\tau$ Cet, corresponding to a spatially
integrated, fractional excess flux of $0.98\pm0.21 \times 10^{-2}$ with respect to the photospheric flux
in the $K^{\prime}$-band. Around $\epsilon$ Eri, our measurements can exclude a fractional excess of
greater than $0.6\times 10^{-2}$ ($3\sigma$). We interpret the photometric excess around $\tau$ Cet as
a possible signature of hot grains in the inner debris disk and demonstrate that a faint, physical or
background, companion can be safely excluded. In addition, we measured both stellar angular diameters
with an unprecedented accuracy: $\Theta_{\rm LD}(\tau\,{\rm Cet})= 2.015 \pm 0.011$ mas and
$\Theta_{\rm LD}(\epsilon\,{\rm Eri})=2.126 \pm 0.014$ mas. }
{ Published by Astronomy \& Astrophysics (Vol. 475, p. 243) }
\v5
%%--------SubmissionID=1260----------------
%% Title
{\large\bf{The Initial Cluster Mass Function of Super Star Clusters in Irregular and Spiral Galaxies}}
%% Authors
{\bf{ Jayce D. Dowell$^{1}$, Brent A. Buckalew$^{2}$ and Jonathan C. Tan$^{3}$}}
%% Institutions
$^1$ {Dept. of Astronomy, Indiana University, Bloomington, IN 47405, USA} \\
$^2$ {Dept. of Physics, Embry-Riddle Aeronautical Univ., Prescott, AZ 86301, USA} \\
$^3$ {Dept. of Astronomy, Univ. of Florida, Gainesville, FL 32611, USA}
%% Email
{E-mail contact: jt {\em at} astro.ufl.edu}
%% LATEX COMMANDS
%% Abstract body
{The initial cluster mass function (ICMF) is a fundamental property of
star formation in galaxies. To gauge its universality, we measure and
compare the ICMFs in irregular and spiral galaxies. Our sample of
irregular galaxies is based on thirteen nearby galaxies selected from
a volume-limited sample from the fifth data release of the Sloan
Digital Sky Survey (SDSS). The extinctions, ages, and masses were
determined by comparing their u'g'i'z' magnitudes to those generated
from starburst models. Completeness corrections were performed using
Monte Carlo simulations in which artificial clusters were inserted
into each galaxy. We analyzed three nearby spiral galaxies with SDSS
data in exactly the same way to derive their ICMF based on a similar
number of young, massive clusters as the irregular galaxy ICMF. We
find that the ICMFs of irregular and spiral galaxies for masses
$>3x10^4 M_\odot$ are statistically indistinguishable. For clusters
more massive than $3x10^4 M_\odot$, the ICMF of the irregular galaxies
is reasonably well fit by a power law $dN(M)/dM ~ M^{-a_M}$ with $a_M
= 1.88 \pm 0.09$. Similar results were obtained for the ICMF of the
spiral galaxy sample but with $a_M = 1.75 \pm 0.06$. We discuss the
implications of our result for theories of star cluster formation,
which appears to be independent of metallicity and galactic shear
rate.}
% Journal
{ Accepted by Astronomical Journal}
%% Preprints URL
http://arxiv.org/abs/astro-ph/0611586
\v5
{\large\bf{The minimum gap-opening planet mass in an irradiated circumstellar accretion disc }}
{\bf{Richard G. Edgar$^1$, Alice C. Quillen$^1$ and Jaehong Park$^1$ }}
$^1$ {Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA }
{E-mail contact: rge21 {\em at} pas.rochester.edu }
{ We consider the minimum mass planet, as a function of radius, that is capable of opening a gap in
an $\alpha$-accretion disc. We estimate that a half-Jupiter mass planet can open a gap in a disc
with accretion rate $\dot{M} \simless 10^{-8}$ M$_\odot$ yr$^{-1}$ for viscosity parameter $\alpha$ = 0.01,
and solar mass and luminosity. The minimum mass is approximately proportional to
$\dot{M}^{0.48}\alpha^{0.8}M^{0.42}_\star L^{-0.08}_\star$. This estimate can be used to rule out the
presence of massive planets in gapless accretion discs. We identify two radii at which an inwardly migrating planet may become able to open a gap and so slow its migration; the radius at which the heating from viscous dissipation is similar to that from stellar radiation in a flared disc, and the radius at which the disc becomes optically thin in a self-shadowed disc. In the inner portions of the disc, we find that the minimum planet mass required to open a gap is only weakly dependent on radius. If a migrating planet is unable to open a gap by the time it reaches either of the transition radii, then it is likely to be lost on to the star. If a gap-opening planet cuts off disc accretion allowing the formation of a central hole or clearing in the disc then we would estimate that the clearing radius would approximately be proportional to the stellar mass. }
{ Published by Monthly Notices of the Royal Astronomical Society (Vol. 381, p. 1280) }
\v5
%%--------SubmissionID=1255----------------
%% Title
{\large\bf{Variations in Stellar Clustering with Environment: Dispersed Star Formation and the Origin of Faint Fuzzies}}
%% Authors
{\bf{ Bruce G. Elmegreen$^{1}$}}
%% Institutions
$^1$ {IBM Watson Research Center}
%% Email
{E-mail contact: bge {\em at} us.ibm.com}
%% LATEX COMMANDS
%% Abstract body
{The observed increase in star formation efficiency with average
cloud density, from several percent in whole giant molecular clouds
to $\sim30$\% or more in cluster-forming cores, can be understood as
the result of hierarchical cloud structure if there is a
characteristic density as which individual stars become well
defined. Also in this case, the efficiency of star formation
increases with the dispersion of the density probability
distribution function (pdf). Models with log-normal pdf's illustrate
these effects. The difference between star formation in bound
clusters and star formation in loose groupings is attributed to a
difference in cloud pressure, with higher pressures forming more
tightly bound clusters. This correlation accounts for the observed
increase in clustering fraction with star formation rate and with
the observation of Scaled OB Associations in low pressure
environments. ``Faint fuzzie'' star clusters, which are bound but
have low densities, can form in regions with high Mach numbers and
low background tidal forces. The proposal by Burkert, Brodie \&
Larsen (2005) that faint fuzzies form at large radii in galactic
collisional rings, satisfies these constraints.}
% Journal
{ Accepted by ApJ (vol 672, Jan 10 2008)}
%% Preprints URL
astro-ph/0710.5788
\v5
{\large\bf{ Expanded Very Large Array Observations of the 6035 MHz OH Masers in ON 1 }}
{\bf{ Vincent L. Fish$^1$ }}
$^1$ { Jansky Fellow, National Radio Astronomy Observatory, Socorro, NM, USA }
{E-mail contact: vfish {\em at} nrao.edu }
{ This Letter reports on initial Expanded Very Large Array (EVLA) observations of the 6035 MHz masers in ON 1. The EVLA data are of good quality, lending confidence in the new receiver system. Nineteen maser features, including six Zeeman pairs, are detected. The overall distribution of 6035 MHz OH masers is similar to that of the 1665 MHz OH masers. The spatial resolution is sufficient to unambiguously determine that the magnetic field is strong (about -10 mG) at the location of the blueshifted masers in the north, consistent with Zeeman splitting detected in 13441 MHz OH masers in the same velocity range. Left- and right-circularly polarized ground-state features dominate in different regions in the north of the source, which may be due to a combination of magnetic field and velocity gradients. The combined distribution of all OH masers toward the south is suggestive of a shock structure of the sort previously seen in W3(OH). }
{ Published by The Astrophysical Journal Letters (Vol. 669, p. L81) }
\v5
{\large\bf{ The AU Microscopii Debris Disk: Multiwavelength Imaging and Modeling }}
{\bf{ Michael P. Fitzgerald$^{1,2}$, Paul G. Kalas$^{1,2}$, Gaspard Duchene$^3$, Christophe Pinte$^3$,
and James R. Graham$^{1,2}$ }}
$^1$ { Department of Astronomy, 601 Campbell Hall, University of California, Berkeley, CA 94720, USA } \\
$^2$ { National Science Foundation Center for Adaptive Optics, University of California, Santa Cruz, CA 95064, USA } \\
$^3$ { Laboratoire d'Astrophysique, Observatoire de Grenoble, BP 53, F-38041 Grenoble Cedex 9, France }
{E-mail contact: fitz {\em at} astro.berkeley.edu }
{ Debris disks around main-sequence stars are produced by the destruction of unseen parent bodies.
AU Microscopii (GJ 803) is a compelling object to study in the context of disk evolution across different
spectral types, as it is an M dwarf whose nearly edge-on disk may be directly compared to that of its
A5 V sibling $\beta$ Pic. We resolve the disk from 8-60 AU in the near-IR JHK' bands at high resolution
with the Keck II Telescope and adaptive optics, and develop a data reduction technique for the removal
of the stellar point-spread function. We measure a blue color across the near-IR bands, and confirm the
presence of substructure in the inner disk. Some of the structural features exhibit wavelength-dependent
positions. Recent measurements of the scattered-light polarization indicate the presence of porous grains.
The scattering properties of these porous grains have a strong effect on the inferred structure of the
disk relative to the majority of previously modeled grain types. Complementing prior work, we use a Monte
Carlo radiative transfer code to compare a relatively simple model of the distribution of porous grains
to a broad data set, simultaneously fitting midplane surface brightness profiles and the spectral energy
distribution. Our model confirms that the large-scale architecture of the disk is consistent with detailed
models of steady state grain dynamics. A belt of parent bodies from 35–40 AU produces dust that is then
swept outward by stellar wind and radiation. We infer the presence of very small grains in the region
exterior to the belt, down to sizes of $\sim$0.05 $\mu$m. These sizes are consistent with stellar
mass-loss rates $\dot{M_\star} \ll 10^2 \dot{M_\odot}$. }
{ Published by The Astrophysical Journal (Vol. 670, p. 536) }
\v5
{\large\bf{ A Ring of Warm Dust in the HD 32297 Debris Disk }}
{\bf{ Michael P. Fitzgerald$^{1,2}$, Paul G. Kalas$^{1,2}$ and James R. Graham$^{1,2}$ }}
$^1$ { Department of Astronomy, University of California, Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA } \\
$^2$ { National Science Foundation Center for Adaptive Optics, University of California, Santa Cruz, CA 95064, USA }
{ We report the detection of a ring of warm dust in the near-edge-on disk surrounding HD 32297 with the
Gemini North Michelle mid-infrared imager. Our N'-band image shows elongated structure consistent with
the orientation of the scattered-light disk. The F$_ν$(11.2 $\mu$m) = 49.9 $\pm$ 2.1 mJy flux is
significantly above the 28.2 $\pm$ 0.6 mJy photosphere. Subtraction of the stellar point-spread function
reveals a bilobed structure with peaks 0.5$^{\prime\prime}$–0.6$^{\prime\prime}$ from the star. The disk
is detected out to the sensitivity limit at $\sim$1$^{\prime\prime}$, and the flux in each lobe is
symmetric to within 10\%. An analysis of the stellar component of the spectral energy distribution (SED)
suggests a spectral type later than A0, in contrast to commonly cited literature values. We fit
three-dimensional, single-size grain models of an optically thin dust ring to our image and the SED using
a Markov chain Monte Carlo algorithm in a Bayesian framework. The best-fit effective grain sizes are
submicron, suggesting the same dust population is responsible for the bulk of the scattered light. The
inner boundary of the warm dust is located 0.5$^{\prime\prime}$–0.7$^{\prime\prime}$ ($\sim$65 AU)
from the star, which is approximately cospatial with the outer boundary of the scattered-light asymmetry
inward of 0.5$^{\prime\prime}$. The addition of a separate component of larger, cooler grains that provide
a portion of the 60 $\mu$m flux improves both the fidelity of the model fit and consistency with the
slopes of the scattered-light brightness profiles. The peak vertical optical depths in our models
[$\sim (0.3–1) \times 10^{-2}$] imply that grain-grain collisions likely play a significant role in dust
dynamics and evolution. Submicron grains can survive radiation pressure blowout if they are icy and
porous. Similarly, the inferred warm temperatures (130–200 K) suggest that ice sublimation may play a role in truncating the inner disk. }
{ Published by The Astrophysical Journal (Vol. 670, p. 557) }
\vspace{0.3cm}
{\large\bf{A 1.3 cm wavelength radio flare from a deeply embedded source in the Orion BN/KL region}}
%% Here comes the author(s) of the paper, please indicate within $^...$
%% the number which corresponds to the institute of each author.
{\bf{Jan Forbrich$^{1,2}$, Karl M. Menten$^1$ \ and Mark J. Reid$^2$}}
%% Here you write your institute name(s) and address(es),
%% the number in $^..$ indicates your author number, for example:
$^1$ {Max-Planck-Institut f\"ur Radioastronomie, Auf dem H\"ugel 69, D-53121 Bonn, Germany} \\
$^2$ {Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, U.S.A.}
%% Here you may write the e-mail address of one or more of the authors
%% who will act as contact person for preprint requests etc, for example:
{E-mail contact: jforbrich {\em at} cfa.harvard.edu}
%% IF YOU USE ANY PERSONAL LATEX COMMANDS IN YOUR ABSTRACT,
%% PLEASE INCLUDE THEIR DEFINITIONS HERE!
%% Within the following brackets you place your text:
{Aims: Our aim was to measure and characterize the short-wavelength
radio emission from young stellar objects (YSOs) in the Orion Nebula
Cluster and the BN/KL star-forming region. Methods: We used the NRAO
Very Large Array at a wavelength of 1.3~cm and we studied archival
X-ray, infrared, and radio data. Results: During our observation, a
strong outburst (flux increasing $>$10 fold) occurred in one of the 16
sources detected at a wavelength of 1.3~cm, while the others remained
(nearly) constant. This source does not have an infrared counterpart,
but has subsequently been observed to flare in X-rays. Curiously, a
very weak variable double radio source was found at other epochs near
this position, one of whose components is coincident with it. A very
high extinction derived from modeling the X-ray emission and the
absence of an infrared counterpart both suggest that this source is
very deeply embedded.}
% Here you write which journal accepted your paper, for example:
{ Accepted by A\&A }
%% If preprints are available on the WWW you can give the web
%% direction here.
{ http://arxiv.org/abs/0711.2017 }
\vspace{0.3cm}
{\large\bf{ A Low-Mass H$_2$ Component to the AU Microscopii Circumstellar Disk }}
{\bf{ Kevin France$^1$, Aki Roberge$^2$, Roxana E. Lupu$^3$, Seth Redfield$^{4}$ and
Paul D. Feldman$^3$ }}
$^1$ { Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, USA } \\
$^2$ { Exoplanets and Stellar Astrophysics Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA } \\
$^3$ { Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA } \\
$^4$ { Department of Astronomy, University of Texas, Austin, TX 78712, USA }
{E-mail contact: france {\em at} cita.utoronto.ca }
{ We present a determination of the molecular gas mass in the AU Microscopii circumstellar disk. Direct
detection of a gas component to the AU Mic disk has proven elusive, with upper limits derived from
ultraviolet absorption line and submillimeter CO emission studies. Fluorescent emission lines of H$_2$,
pumped by the O VI $\lambda$1032 resonance line through the C–X (1–1) Q(3) $\lambda$1031.87 \AA\
transition, are detected by the Far Ultraviolet Spectroscopic Explorer. These lines are used to derive
the H$_2$ column density associated with the AU Mic system. The derived column density is in the range
N(H$_2$) = 1.9 $\times$ 10$^{17}$ to 2.8 $\times$ 10$^{15}$ cm$^{-2}$, roughly 2 orders of magnitude
lower than the upper limit inferred from absorption line studies. This range of column densities reflects
the range of H$_2$ excitation temperature consistent with the observations, T(H$_2$) = 800-2000 K, derived
from the presence of emission lines excited by O VI in the absence of those excited by Ly$\alpha$.
Within the observational uncertainties, the data are consistent with the H$_2$ gas residing in the disk.
The inferred N(H$_2$) range corresponds to H$_2$-to-dust ratios of $\simless \frac{1}{30}$ : 1 and a
total M(H$_2$) = 4.0 $\times$ 10$^{-4}$ to 5.8 $\times$ 10$^{-6}$ M$_\oplus$. We use these results to
predict the intensity of the associated rovibrational emission lines of H$_2$ at infrared wavelengths
covered by ground-based instruments, HST NICMOS, and the Spitzer IRS. }
{ Published by The Astrophysical Journal (Vol. 668, p. 1174) }
\vspace{0.3cm}
%%--------SubmissionID=1250----------------
%% Title
{\large\bf{SWAS Observations of Water in Molecular Outflows}}
%% Authors
{\bf{ Jonathan Franklin$^{1}$, Ronald L. Snell$^{1}$, Michael J. Kaufman$^{2}$, Gary J. Melnick$^{3}$, David A. Neufeld$^{4}$, David J. Hollenbach$^{5}$ and Edwin A. Bergin$^{6}$}}
%% Institutions
$^1$ {Department of Astronomy, University of Massachusetts, 710 North Pleasant Street, Amherst, MA 01003, USA} \\
$^2$ {Department of Physics, San Jose State University, One Washington
Square, San Jose, CA 95192, USA} \\
$^3$ {Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge MA 02138, USA} \\
$^4$ {Department of Physics and Astronomy, Johns Hopkins University,
3400 North Charles Street, Baltimore, MD 21218, USA} \\
$^5$ {NASA Ames Research Center, Moffett Field, CA 94035, USA} \\
$^6$ {Department of Astronomy, University of Michigan, 825 Dennison
Building, Ann Arbor, MI 48109, USA}
%% Email
{E-mail contact: snell {\em at} astro.umass.edu}
%% LATEX COMMANDS
%% Abstract body
{We present detections of the ground-state $1_{10}\rightarrow1_{01}$
transition of ortho-H$_2$O at 557 GHz in 18 molecular outflows based
on data from the {\it Submillimeter Wave Astronomy Satellite} (SWAS).
These results are combined with ground-based observations
of the J=1-0 transitions of $^{12}$CO and $^{13}$CO obtained at
the {\it Five College Radio Astronomy Observatory} (FCRAO). Data
from {\it Infrared Space Observatory} (ISO) for a subset of the
outflows are also discussed. Assuming the SWAS water line emission
originates from the same gas traced by CO emission, we find that the
outflowing gas in most outflows has an ortho-H$_2$O abundance relative
to H$_2$ of between about 10$^{-7}$ and 10$^{-6}$. Analysis of the
water abundance as a function of outflow velocity reveals a strong
dependence. The abundance of ortho-H$_2$O increases with velocity
and at the highest outflow velocities some of the outflows have
relative ortho-H$_2$O abundances of order 10$^{-4}$. However the mass
of very high velocity gas with such elevated H$_2$O abundances
represents less that 1\% of the total outflow gas mass. The ISO LWS
observations of high-J rotational lines of CO and the 179.5 $\mu$m
transition of ortho-H$_2$O provide evidence for a warmer outflow
component than required to produce either the SWAS or FCRAO lines.
The ISO line flux ratios can be reproduced with C-shock models with
shock velocities of order 25 km s$^{-1}$ and preshock densities of
order 10$^5$ cm$^{-3}$; these C-shocks have post-shock relative water
abundances greater than 10$^{-4}$. The mass associated with the ISO
emission is also quite small compared with the total outflow mass, and
is similar to that responsible for the highest velocity water emission
detected by SWAS. Although the gas responsible for the ISO emission
has elevated levels of water, the bulk of the outflowing gas has an
abundance of ortho-H$_2$O well below what would be expected if the gas
has passed through a C-shock with shock velocities greater than
10 km s$^{-1}$. Gas-phase water can be depleted in the post-shock gas
due to freeze-out onto grain mantles, however the rate of freeze-out is
too slow to explain our results. Therefore we believe that only a small
fraction of the outflowing molecular gas has passed through shocks
strong enough to fully convert the gas-phase oxygen to water. This
result has implications for the acceleration mechanism of the molecular
gas in these outflows.}
% Journal
{ Accepted by Astrophysical Journal (Feb. 2008)}
%% Preprints URL
http://arxiv.org/abs/0711.2055
\v5
%%--------SubmissionID=1262----------------
%% Title
{\large\bf{Spitzer IRS Spectra and Envelope Models of Class I Protostars in Taurus}}
%% Authors
{\bf{ Elise Furlan$^{1,2}$, Melissa McClure$^{3}$, Nuria Calvet$^{4}$, Lee Hartmann$^{4}$, Paola D' Alessio$^{5}$, William J. Forrest$^{3}$, Dan M. Watson$^{3}$, Keven I. Uchida$^{1}$, Ben Sargent$^{3}$, Joel D. Green$^{3}$ and Terry L. Herter$^{1}$}}
%% Institutions
$^1$ {Center for Radiophysics and Space Research, Space Sciences Building, Cornell University, Ithaca, NY 14853, USA} \\
$^2$ {NASA Astrobiology Institute and Department of Physics and Astronomy, UCLA, 430 Portola Plaza, Los Angeles, CA 90095, USA} \\
$^3$ {Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA} \\
$^4$ {Department of Astronomy, The University of Michigan, 500 Church St., 830 Dennison Bldg., Ann Arbor, MI 48109, USA} \\
$^5$ {Centro de Radioastronom{\'\i}a y Astrof{\'\i}sica, UNAM, Apartado Postal 3-72 (Xangari), 58089 Morelia, Michoac\'an, M\'exico}
%% Email
{E-mail contact: furlan {\em at} astro.ucla.edu}
%% LATEX COMMANDS
%% Abstract body
{We present {\it Spitzer} Infrared Spectrograph spectra of 28 Class I
protostars in the Taurus star-forming region. The 5 to 36 $\mu$m
spectra reveal excess emission from the inner regions of the envelope
and accretion disk surrounding these predecessors of low-mass stars,
as well as absorption features due to silicates and ices. Together
with shorter- and longer-wavelength data from the literature, we
construct spectral energy distributions and fit envelope models to 22
protostars of our sample, most of which are well-constrained due to
the availability of the IRS spectra. We infer that the envelopes of
the Class I objects in our sample cover a wide range in parameter
space, particularly in density and centrifugal radius, implying
different initial conditions for the collapse of protostellar cores.}
% Journal
{ Accepted by Astrophysical Journal Supplement}
%% Preprints URL
\v5
{\large\bf{ Results from Droxo: I. The variability of fluorescent Fe 6.4 keV emission in the young star Elias 29. High-energy electrons in the star's accretion tubes? }}
{\bf{ G. Giardino$^1$, F. Favata$^2$, I. Pillitteri$^3$, E. Flaccomio$^3$, G. Micela$^3$,
and S. Sciortino$^3$ }}
$^1$ { Astrophysics Division - Research and Science Support Department of ESA, ESTEC, Postbus 299, 2200 AG Noordwijk, The Netherlands } \\
$^2$ { ESA - Planning and Community Coordination Office, Science Programme, 8-10 rue Mario Nikis, 75738 Paris Cedex 15, France } \\
$^3$ { INAF - Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy }
{E-mail contact: ggiardin {\em at} rssd.esa.int }
{ \emph{Aims.} We study the variability of the Fe 6.4 keV emission line from the Class I young stellar object Elias 29 in the $\rho$ Oph cloud.
\emph{Methods.} We analysed the data from Elias 29 collected by XMM-Newton during a nine-day, nearly
continuous observation of the $\rho$ Oph star-forming region (the Deep Rho-Oph X-ray Observation,
named DROXO). The data were subdivided into six homogeneous time intervals, and the six resulting spectra were individually analysed.
\emph{Results.} We detect significant variability in the equivalent width of the Fe 6.4 keV emission line
from Elias 29. The 6.4 keV line is absent during the first time interval of observation and appears at
its maximum strength during the second time interval (90 ks after Elias 29 undergoes a strong flare).
The X-ray thermal emission is unchanged between the two observation segments, while line variability is
present at a 99.9\% confidence level. Given the significant line variability in the absence of variations
in the X-ray ionising continuum and the weakness of the photoionising continuum from the star's thermal
X-ray emission, we suggest that the fluorescence may be induced by collisional ionisation from an (unseen)
population of non-thermal electrons. We speculate on the possibility that the electrons are accelerated
in a reconnection event of a magnetically confined accretion loop, connecting the young star to its circumstellar disk. }
{ Published by Astronomy \& Astrophysics (Vol. 475, p. 891) }
\v5
%%--------SubmissionID=1269----------------
%% Title
{\large\bf{The relationship between the prestellar core mass function
and the stellar initial mass function}}
%% Authors
{\bf{ Simon Goodwin$^{1}$, Dave Nutter$^{2}$, Pavel Kroupa$^{3}$, Derek Ward-Thompson$^{2}$ and Anthony Whitworth$^{2}$}}
%% Institutions
$^1$ {Sheffield} \\
$^2$ {Cardiff} \\
$^3$ {Bonn}
%% Email
{E-mail contact: S.Goodwin {\em at} sheffield.ac.uk}
%% LATEX COMMANDS
%% Abstract body
{Stars form from dense molecular cores, and the mass function of these
cores (the CMF) is often found to be similar to the form of the
stellar initial mass function (IMF). This suggests that the form of
the IMF is the result of the form of the CMF. However, most stars are
thought to form in binary and multiple systems, therefore the
relationship between the IMF and the CMF cannot be trivial. We test
two star formation scenarios - one in which all stars form as binary
or triple systems, and one in which low-mass stars form in a
predominantly single mode. We show that from a log-normal CMF, similar
to those observed, and expected on theoretical grounds, the model in
which all stars form as multiples gives a better fit to the IMF.}
% Journal
{ Accepted by Astronomy and Astrophysics}
%% Preprints URL
http://de.arxiv.org/abs/0711.1749
\v5
{\large\bf{ Survival of icy grains in debris discs }}
{\bf{ A. Grigorieva$^1$, Ph. Th\'ebault$^{1,2}$, P. Artymowicz$^3$, and A. Brandeker$^1$ }}
$^1$ { Stockholm Observatory, SCFAB, 10691 Stockholm, Sweden } \\
$^2$ { Observatoire de Paris, Section de Meudon, 92195 Meudon Principal Cedex, France } \\
$^3$ { University of Toronto at Scarborough, 1265 Military Trail, Toronto, Ontario, M1C 1A4, Canada }
{E-mail contact: anja {\em at} astro.su.se }
{\emph{ Aims.} We put theoretical constraints on the presence and survival of icy grains in debris discs. Particular attention is paid to UV sputtering of water ice, which has so far not been studied in detail in this context.
\emph{Methods.} We present a photosputtering model based on available experimental and theoretical studies. We quantitatively estimate the erosion rate of icy and ice-silicate grains, under the influence of both sublimation and photosputtering, as a function of grain size, composition and distance from the star. The effect of erosion on the grain's location is investigated through numerical simulations coupling the grain size to its dynamical evolution.
\emph{Results.} Our model predicts that photodesorption efficiently
destroy ice in optically thin discs, even far beyond the sublimation
snow line. For the reference case of $\beta$ Pictoris, we find that
only $\simgreat$5 mm grains can keep their icy component for the age of
the system in the 50-150 AU region. When taking into account the
collisional reprocessing of grains, we show that the water ice
survival on grains improves (grains down to $\simeq$20 $\mu$m might be
partially icy). However, estimates of the amount of gas
photosputtering would produce on such a hypothetical population of big
icy grains lead to values for the O I column density that strongly
exceed observational constraints for $\beta$ Pic, thus ruling out the
presence of a significant amount of icy grains in this system. Erosion
rates and icy grains survival timescales are also given for a set of
11 other debris disc systems. We show that, with the possible
exception of M stars, photosputtering cannot be neglected in
calculations of icy grain lifetimes. }
{ Published by Astronomy \& Astrophysics (Vol. 475, p. 755) }
\v5
%%--------SubmissionID=1254----------------
%% Title
{\large\bf{First images of 6.7-GHz methanol masers in DR21(OH) and DR21(OH)N}}
%% Authors
{\bf{ Lisa Harvey-Smith$^{1,2}$, Rebeca Soria-Ruiz$^{1}$, Ana Duarte-Cabral$^{1,3,5}$ and R. J. Cohen$^{4}$}}
%% Institutions
$^1$ {Joint Institute for VLBI in Europe, Postbus 2, 7990 AA Dwingeloo, The Netherlands} \\
$^2$ {School of Physics, University of Sydney, 2006 NSW, Australia} \\
$^3$ {Departamento de F\'{i}sica da Faculdade de Ci\^{e}ncias da Universidade do Porto, Rua do Campo Alegre 687, 4169 - 007 Porto, Portugal} \\
$^4$ {University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, United Kingdom} \\
$^5$ {Jodrell Bank Centre for Astrophysics, University of Manchester, Alan Turing Building Oxford Road, Manchester, M13 9PL, United Kingdom}
%% Email
{E-mail contact: lhs {\em at} usyd.edu.au}
%% LATEX COMMANDS
%% Abstract body
{The first images of 6.7-GHz methanol masers in the massive
star-forming regions DR21(OH) and DR21(OH)N are presented. By
measuring the shapes, radial velocities and polarization properties of
these masers it is possible to map out the structure, kinematics and
magnetic fields in the molecular gas that surrounds newly-formed
massive stars. The intrinsic angular resolution of the observations
was 43 mas ($\sim$~100 AU at the distance of DR21), but structures far
smaller than this were revealed by employing a non-standard mapping
technique. By plotting the positions of the Gaussian-fitted maser
emission centroids in each velocity channel, the internal velocity
gradients of the masers were investigated at very high spectral and
spatial resolution. This technique was used in an attempt to identify
the physical structure (e.g. disc, outflow, shock) associated with the
methanol masers. Two distinct star-forming centres were identified. In
DR21(OH) the masers had a linear morphology, and the individual maser
spots each displayed an internal velocity gradient in the same
direction as the large-scale structure. They were detected at the same
position as the OH 1.7-GHz ground-state masers, close to the centre of
an outflow traced by CO and class I methanol masers. The shape and
velocity gradients of the masers suggests that they probably delineate
a shock. In DR21(OH)N the methanol masers trace an arc with a
double-peaked profile and a complex velocity gradient. This velocity
gradient closely resembles that of a Keplerian disc. The masers in the
arc are 4.5$\%$ linearly polarized, with a polarization angle that
indicates that the magnetic field direction is roughly perpendicular
to the large-scale magnetic field in the region (indicated by lower
angular resolution measurements of the CO and dust polarization). The
origin and nature of these maser structures is considered within the
context of what is already known about the region. The suitability of
channel-by-channel centroid mapping is discussed as an improved and
viable means to maximise the information gained from the data.}
% Journal
{ Accepted by MNRAS}
%% Preprints URL
\clearpage
%%--------SubmissionID=1253----------------
%% Title
{\large\bf{Star Spot Induced Radial Velocity Variability in LkCa 19}}
%% Authors
{\bf{ Marcos Huerta$^{1,2}$, Christopher M. Johns-Krull$^{1}$, L. Prato$^{3}$, Patrick Hartigan$^{1}$ and D. T. Jaffe$^{4}$}}
%% Institutions
$^1$ {Physics and Astronomy Department, Rice University, Houston, Texas 77005, USA} \\
$^2$ {Department of Astronomy, University of Florida, Gainesville, Florida 32611, USA} \\
$^3$ {Lowell Observatory, Flagstaff, AZ 86001, USA} \\
$^4$ {Department of Astronomy, University of Texas at Austin, Austin, Texas 78712, USA}
%% Email
{E-mail contact: marcosh {\em at} astro.ufl.edu}
%% LATEX COMMANDS
%% Abstract body
{We describe a new radial velocity survey of T Tauri stars and present
the first results. Our search is motivated by an interest in
detecting massive young planets, as well as investigating the origin
of the brown dwarf desert. As part of this survey, we discovered
large-amplitude, periodic, radial velocity variations in the spectrum
of the weak line T Tauri star LkCa 19. Using line bisector analysis
and a new simulation of the effect of star spots on the photometric
and radial velocity variability of T Tauri stars, we show that our
measured radial velocities for LkCa19 are fully consistent with
variations caused by the presence of large star spots on this rapidly
rotating young star. These results illustrate the level of
activity-induced radial velocity noise associated with at least some
very young stars. This activity-induced noise will set lower limits
on the mass of a companion detectable around LkCa 19, and similarly
active young stars.}
% Journal
{ Accepted by ApJ}
%% Preprints URL
http://arxiv.org/abs/0711.2505
\v5
{\large\bf{ Lithium abundances of very low mass members of Chamaeleon I }}
{\bf{ C. M. S. Johnas$^1$, E. W. Guenther$^2$, V. Joergens$^3$, A. Schweitzer$^1$, and P. H. Hauschildt$^1$ }}
$^1$ { Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany } \\
$^2$ { Th\"uringer Landessternwarte Tautenburg, 07778 Tautenburg, Germany } \\
$^3$ { Max-Planck-Institut f\"ur Astronomie, K\"onigstuhl 17, 69117 Heidelberg, Germany }
{E-mail contact: cjohnas {\em at} hs.uni-hamburg.de }
{ \emph{Aims.} We present the first study of the lithium abundances of very low mass objects in
Chamaeleon I close to the hydrogen burning mass limit based on atmospheric models and high-resolution
spectroscopic observations. The studied objects, Cha H$\alpha$ 2, 3, 4, 5, 6 and 8, are very young brown
dwarf candidates and very low mass stars on the verge of lithium depletion.
\emph{Methods.} For this analysis, we have computed a new ``GAIA-cond'' class model grid over effective
temperatures from 2600 K to 3100 K, surface gravities from log(g) = 3.5 to 5.5, and lithium abundances
from log$\epsilon$ = 0.0 to 3.7, for two different line profile setups introduced in previous work.
Calculated synthetic spectra are compared with high-resolution UVES / VLT echelle spectra of the objects.
\emph{Results.} We find good descriptions of the lithium resonance doublet lines at 6708 \AA and of the
surrounding pseudo-continuum and determine a consistent set of lithium abundances (log($\epsilon$) = 1.55).
However, the derived lithium abundances are lower than the meteoritic one (log($\epsilon$) = 3.31) and
that of higher mass stars in Cha I (log($\epsilon$) = 3.1/3.4 for LTE-/non-LTE-calculations). By modeling
the TiO-line, we demonstrate that veiling does not make the lithium lines appear weaker. We can also rule
out that the results are spoiled by the presence of spots.
\emph{Conclusions.} A possible explanation for these results would be that the objects are either more
massive, or much older, than previously thought, so that the lithium depletion has already started.
Although the uncertainties of the masses and ages are large, they are not large enough as to explain the
observed lithium depletion. Therefore, the most likely explanation is either a lack of understanding of
the details of the formation of the lithium line, or a lack of understanding of the internal structure
of the very young low-mass objects. }
{ Published by Astronomy \& Astrophysics (Vol. 475, p. 667) }
\v5
{\large\bf{ IRAC Observations of CO J = 4 $\rightarrow$ 3 High-Velocity Cloud in the 30 Doradus Complex in the Large Magellanic Cloud }}
{\bf{ Hak-Sub Kim$^{1,2}$, Sungeun Kim$^{1,3}$, Jih-Yong Bak$^1$, Mario Garcia$^1$, Bernard Brandl$^4$,
Kecheng Xiao$^3$, Wilfred Walsh$^3$, R. Chris Smith$^5$, and Soyoung Youn$^1$ }}
$^1$ { ARCSEC, Dept. Astronomy and Space Science, Sejong Univ., KwangJin-gu, KunJa-dong 98, Seoul, 143-747, Korea } \\
$^2$ { Current address: Department of Astronomy, Yonsei University, Shinchon, Seodaemungu, Seoul, Korea } \\
$^3$ { Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS-12, Cambridge, MA 02138, USA } \\
$^4$ { Leiden Observatory, Sterrewacht Leiden, P.O. Box 9513, Neils Bohrweg 2, RA Leiden, 2300, Netherlands } \\
$^5$ { NOAO, 950 North Cherry Avenue, Tucson, AZ 85719, USA }
{E-mail contact: [email protected] }
{ We present the results of $^{12}$CO J = 2 $\rightarrow$ 1 observations of the X-ray–bright giant shell
complex 30 Doradus in the Large Magellanic Cloud (LMC) using the Antarctic Submillimeter Telescope and
Remote Observatory (AST/RO). This is the one of the largest H II complexes in the Local Group. We compare
the $^{12}$CO J = 2 $\rightarrow$ 1 observations against previously made $^{12}$CO J = 4 $\rightarrow$ 3
observations and analyze the spatial distribution of young stellar objects (YSOs) within the cloud using
the Spitzer IRAC observations of the 30 Doradus complex. Both peaks of $^{12}$CO J = 2 $\rightarrow$ 1
and J = 4 $\rightarrow$ 3 emitting clouds coincide with the densest region of the filaments in which
multiple shells are colliding. The YSOs are clustered in the southern ridge of the warm and dense
molecular gas clouds traced by $^{12}$CO J = 4 $\rightarrow$ 3, indicating a filamentary structure of
star formation throughout 30 Doradus. We also find an excess of Class I YSO candidates close to the
clouds, which likely represent the most recent phase of star formation in this region. This is a region
where the triggered star formation has actually occurred, and newly formed stars may have produced such
a high-velocity outflow through interacting with the surrounding molecular cloud material. }
{ Published by The Astrophysical Journal (Vol. 669, p. 1003) }
\vspace{0.3cm}
%%--------SubmissionID=1248----------------
%% Title
{\large\bf{Clumpy photon-dominated regions in Carina.
I. [CI] and mid-$J$ CO lines in two $4'\times4'$ fields.}}
%% Authors
{\bf{ C.\,Kramer$^{1}$, M.\,Cubick$^{1}$, M.\,Roellig$^{2}$, Y.\,Yonekura$^{3}$, M.\,Aravena$^{2}$, F.\,Bensch$^{2}$, F.\,Bertoldi$^{2}$, L.\,Bronfman$^{4}$, M.\,Fujishita$^{5}$, Y.\,Fukui$^{5}$, U.U.\,Graf$^{1}$, M.\,Hitschfeld$^{1}$, N.\,Honingh$^{1}$, S.\,Ito$^{5}$, H.\,Jakob$^{1}$, K.\,Jacobs$^{1}$, U.\,Klein$^{2}$, B.-C.\,Koo$^{6}$, J.\,May$^{4}$, M.\,Miller$^{1}$, Y.\,Miyamoto$^{5}$, N.\,Mizuno$^{5}$, T.\,Onishi$^{5}$, Y.-S.\,Park$^{6}$, J.L.\,Pineda$^{2}$, D.\,Rabanus$^{1}$, H.\,Sasago$^{5}$, R.\,Schieder$^{1}$, R.\,Simon$^{1}$, J.\,Stutzki$^{1}$, N.\,Volgenau$^{1}$ and H.\,Yamamoto$^{5}$}}
%% Institutions
$^1$ {KOSMA, I. Physikalisches Institut, Universit\"at zu K\"oln,
Z\"ulpicher Stra\ss{}e 77, D-50937 K\"oln, Germany} \\
$^2$ {Argelander-Institut f\"ur Astronomie, Auf dem H\"ugel 71,
D-53121 Bonn, Germany} \\
$^3$ {Department of Physical Science, Osaka Prefecture University,
Osaka 599-8531, Japan} \\
$^4$ {Departamento de Astronom\'{i}a, Universidad de Chile, Casilla 36-D, Santiago, Chile} \\
$^5$ {Department of Astrophysics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan} \\
$^6$ {Seoul National University, Seoul 151-742, Korea}
%% Email
{E-mail contact: kramer {\em at} ph1.uni-koeln.de}
%% LATEX COMMANDS
%% Abstract body
{The Carina region is an excellent astrophysical laboratory for
studying the feedback mechanisms of newly born, very massive stars
within their natal giant molecular clouds (GMCs) at only 2.35~kpc
distance. We use a clumpy PDR model to analyse the observed
intensities of atomic carbon and CO and to derive the excitation
conditions of the gas. The NANTEN2-4m submillimeter telescope was used
to map the [CI] $^3P_1-^3P_0$, $^3P_2-^3P_1$ and CO 4--3, 7--6 lines
in two $4'\times4'$ regions of Carina where molecular material
interfaces with radiation from the massive star clusters. One region
is the northern molecular cloud near the compact OB cluster Tr\,14,
and the second region is in the molecular cloud south of $\eta$Car and
Tr\,16. These data were combined with $^{13}$CO SEST spectra,
HIRES/IRAS $60\,\mu$m and $100\,\mu$m maps of the FIR continuum, and
maps of 8$\,\mu$m IRAC/Spitzer and MSX emission. We used the HIRES
far-infrared dust data to create a map of the FUV field heating the
gas. The northern region shows an FUV field of a few $10^3$ in Draine
units while the field of the southern region is about a factor 10
weaker. While the IRAC 8$\,\mu$m emission lights up at the edges of
the molecular clouds, CO and also [CI] appear to trace the H$_2$ gas
column density. The northern region shows a complex velocity and
spatial structure, while the southern region shows an edge-on PDR with
a single Gaussian velocity component. We constructed models
consisting of an ensemble of small spherically symmetric PDR clumps
within the $38''$ beam (0.43~pc), which follow canonical power-law
mass and mass-size distributions. We find that an average local clump
density of $2\,10^5$\,cm$^{-3}$ is needed to reproduce the observed
line emission at two selected interface positions. Stationary, clumpy
PDR models reproduce the observed cooling lines of atomic carbon and
CO at two positions in the Carina Nebula.}
% Journal
{ Accepted by A\&A}
%% Preprints URL
http://arxiv.org/abs/0711.1320
\v5
{\large\bf{ Obscured clusters. I. GLIMPSE 30 - A young Milky Way star cluster hosting Wolf-Rayet stars }}
{\bf{ R. Kurtev$^1$, J. Borissova$^1$, L. Georgiev$^2$, S. Ortolani$^3$, and V. D. Ivanov$^4$ }}
$^1$ { Departamento de Fis\'{\i}ca y Astronom\'{\i}a, Facultad de Ciencias, Universidad de Valpara\'{\i}so,
Av. Gran Breta\~na 644, Playa Ancha, Casilla 5030, Valpara\'{\i}so, Chile } \\
$^2$ { Instituto de Astronomia, Universidad Nacional Aut\'onoma de M\'exico, Apartado Postal 70-254,
CD Universitaria, CP 04510, Mexico DF, Mexico } \\
$^3$ { Universit\'a di Padova, Dipartimento di Astronomia, Vicolo dell'Osservatorio 5, 35122 Padova, Italy } \\
$^4$ { European Southern Observatory, Ave. Alonso de Cordova 3107, Casilla 19, Santiago 19001, Chile }
{E-mail contact: radostin.kurtev {\em at} uv.cl }
{ \emph{Context.} Young massive clusters are usually deeply embedded in dust and gas. They represent
excellent astrophysical laboratories for the study of massive stars. Clusters with Wolf-Rayet (WR) stars
are of special importance, since this enables us to study a coeval WR population at a uniform metallicity
and known age.
\emph{Aims.} We started a long-term project to search the inner Milky Way for hidden star clusters and
to study them in detail. GLIMPSE 30 (G30) is one of these clusters. It is situated near the Galactic
plane ( $l=298.^\circ756$, $b=-0.^\circ408$) and we determine its physical parameters and
investigate its high-mass stellar content especially WR stars.
\emph{Methods.} Our analysis is based on SOFI/NTT $J_{\rm S}HK_{\rm S}$ imaging and low resolution
($R\sim2000$) spectroscopy of the brightest cluster members in the K atmospheric window. For the age
determination we applied isochrone fits for MS and Pre-MS stars. We derived stellar parameters of the WR
stars candidates using a full nonLTE modeling of the observed spectra.
\emph{Results.} Using a variety of techniques we found that G30 is
very young cluster, with age t $\approx$ 4 Myr. The cluster is located
in the Carina spiral arm, it is deeply embedded in dust and suffers
reddening of $A_{V} \sim 10.5\pm 1.1$ mag. The distance to the object
is $d=7.2\pm0.9$ kpc. The mass of the cluster members down to 2.35
M$_\odot$ is $\sim$ 1600 M$_\odot$. The cluster's MF for the mass
range of 5.6 to 31.6 M$_\odot$ shows a slope of $\Gamma=-1.01\pm
0.03$. The total mass of the cluster obtained by this MF down to 1
M$_\odot$ is about $3\,\times\,10^3$ M$_\odot$. The spectral analysis
and the models allow us to conclude that at least one Ofpe/WN and two
WR stars can be found in G30. The WR stars are of the WN6-7 hydrogen
rich type with progenitor masses of more than 60 M$_\odot$.
\emph{Conclusions.} G30 is a new member of the family of young
Galactic clusters hosting WR stars. It is a factor of two to three
less massive than some of the youngest super-massive star clusters
like Arches, Quintuplet and the Central cluster and is their smaller
analog. }
{ Published by Astronomy \& Astrophysics (Vol. 475, p. 209) }
\v5
%%--------SubmissionID=1239----------------
%% Title
{\large\bf{Multi-line (sub)millimetre observations of the high-mass
proto cluster IRAS 05358+3543}}
%% Authors
{\bf{ S. Leurini$^{1,2}$, H. Beuther$^{3}$, P. Schilke$^{1}$, F. Wyrowski$^{1}$, Q. Zhang$^{4}$ and K.M. Menten$^{1}$}}
%% Institutions
$^1$ {MPIfR, Auf dem Hgel 69, 53121 Bonn, Germany} \\
$^2$ {ESO, Karl-Schwarzschild-Strasse 2, 85748 Garching bei Mnchen, Germany} \\
$^3$ {MPIA.Knigstuhl 17, 69117 Heidelberg, Germany} \\
$^4$ {CfA, 60 Garden Street, Cambridge, MA 02138, USA}
%% Email
{E-mail contact: sleurini {\em at} eso.org}
%% LATEX COMMANDS
%% Abstract body
{Since most high- and intermediate-mass protostars are at great
distance and form in clusters, high linear resolution observations are
needed to investigate their physical properties. To study the gas in
the innermost region around the protostars in the proto-cluster IRAS
05358+3543, we observed the source in several transitions of methanol
and other molecular species with the Plateau de Bure Interferometer
and the Submillimeter Array, reaching a linear resolution of 1100
AU. We determine the kinetic temperature of the gas around the
protostars through an LVG and LTE analysis of their molecular
emission; the column densities of CH$_3$OH, CH$_3$CN and SO$_2$ are
also derived. Constrains on the density of the gas are estimated for
two of the protostellar cores. We find that the dust condensations are
in various evolutionary stages. The powerhouse of the cluster, mm1a,
harbours a hot core with T$\sim$220 (75$<$T$<$330) K. A double-peaked
profile is detected in several transitions toward mm1a, and we found a
velocity gradient along a linear structure which could be
perpendicular to one of the outflows from the vicinity of mm1a. Since
the size of the double-peaked emission is less than 1100 AU, we
suggest that mm1a might host a massive circumstellar disk. The other
sources are in earlier stages of star formation. The least active
source, mm3, could be a starless massive core, since it is cold
(T$<$20 K), with a large reservoir of accreting material (M$\sim
1$9M$_\odot$), but no molecular emission peaks on it.}
% Journal
{ Accepted by A\&A (Vol. 475, p. 925)}
%% Preprints URL
http://arxiv.org/abs/0710.4238
\v5
{\large\bf{ Discovery of an M9.5 Candidate Brown Dwarf in the TW Hydrae Association: DENIS J124514.1-442907 }}
{\bf{ Dagny L. Looper$^{1,2}$, Adam J. Burgasser$^3$, J. Davy Kirkpatrick$^4$, and Brandon J. Swift$^{1,5}$ }}
$^1$ { Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA } \\
$^2$ { Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaii under cooperative agreement NCC 5-538 with the National Aeronautics and Space Administration, Office of Space Science, Planetary Astronomy Program } \\
$^3$ { Massachusetts Institute of Technology, Kavli Institute for Astrophysics and Space Research, Building 37, Room 664B, 77 Massachusetts Avenue, Cambridge, MA 02139, USA } \\
$^4$ { Infrared Processing and Analysis Center, M/S 100-22, California Institute of Technology, Pasadena, CA 91125, USA } \\
$^5$ { Currently at Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA }
{ We report the discovery of a fifth candidate substellar system in the $\sim$5–10 Myr TW Hydrae
association: DENIS J124514.1-442907. This object has a NIR spectrum remarkably similar to that of
2MASS J1139511-315921, a known TW Hydrae brown dwarf, with low surface gravity features such as a
triangular-shaped H band, deep H$_2$O absorption, weak alkali lines, and weak hydride bands. We find an
optical spectral type of M9.5 and estimate a mass of $\simless$ 24 M$_{Jup}$, assuming an age
of $\sim$ 5–10 Myr. While the measured proper motion for DENIS J124514.1-442907 is inconclusive as a
test for membership, its position in the sky is coincident with the TW Hydrae association. A more
accurate proper-motion measurement, higher resolution spectroscopy for radial velocity, and a parallax
measurement are needed to derive the true space motion and to confirm its membership. }
{ Published by The Astrophysical Journal Letters (Vol. 669, p. L97) }
\v5
{\large\bf{ The Stellar Population of the Chamaeleon I Star-forming Region }}
{\bf{ K. L. Luhman$^1$ }}
$^1$ { Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park,
PA 16802, USA }
{E-mail contact: kluhman {\em at} astro.psu.edu }
{ I present a new census of the stellar population in the Chamaeleon I star-forming region. Using optical
and near-IR photometry and follow-up spectroscopy, I have discovered 50 new members of Chamaeleon I,
expanding the census of known members to 226 objects. Fourteen of these new members have spectral types
later than M6, which doubles the number of known members that are likely to be substellar. I have
estimated extinctions, luminosities, and effective temperatures for the known members, used these data
to construct an H-R diagram for the cluster, and inferred individual masses and ages with the theoretical
evolutionary models of Baraffe and Chabrier. The distribution of isochronal ages indicates that star
formation began 3-4 and 5-6 Myr ago in the southern and northern subclusters, respectively, and has
continued to the present time at a declining rate. The IMF in Chamaeleon I reaches a maximum at a mass
of 0.1-0.15 M$_\odot$ and thus closely resembles the IMFs in IC 348 and the Orion Nebula Cluster. In
logarithmic units where the Salpeter slope is 1.35, the IMF is roughly flat in the substellar regime
and shows no indication of reaching a minimum down to a completeness limit of 0.01 M$_\odot$. The low-mass
stars are more widely distributed than members at other masses in the northern subcluster, but this is
not the case in the southern subcluster. Meanwhile, the brown dwarfs have the same spatial distribution
as the stars out to a radius of 3$^\circ$ (8.5 pc) from the center of Chamaeleon I. }
{ Published by The Astrophysical Journal Supplement Series (Vol. 173, p. 104) }
\v5
{\large\bf{Transient growth and coupling of vortex and wave modes in self-gravitating gaseous discs }}
{\bf{G. R. Mamatsashvili$^1$ and G. D. Chagelishvili$^1$ }}
$^1$ {E. Kharadze Georgian National Astrophysical Observatory, 2a Kazbegi Ave, Tbilisi 0160, Georgia }
{E-mail contact: g.mamatsashvili {\em at} astro-ge.org }
{ Linear transient phenomena induced by flow non-normality in thin self-gravitating astrophysical discs are studied using the shearing sheet approximation. The considered system includes two modes of perturbations: vortex and (spiral density) wave. It is shown that self-gravity considerably alters the vortex mode dynamics; its transient (swing) growth may be several orders of magnitude stronger than in the non-self-gravitating case and two to three times larger than the transient growth of the wave mode. Based on this finding, we comment on the role of vortex mode perturbations in a gravitoturbulent state. We also describe the linear coupling of the perturbation modes, caused by the differential character of disc rotation. The coupling is asymmetric: vortex mode perturbations are able to excite wave mode perturbations, but not vice versa. This asymmetric coupling lends additional significance to the vortex mode as a participant in spiral density waves and shock manifestations in astrophysical discs. }
{ Published by Monthly Notices of the Royal Astronomical Society (Vol. 381, p. 809) }
\v5
{\large\bf{A HST study of the environment of the Herbig Ae/Be star
LkH$\alpha$~233 and its bipolar jet}}
%% Here comes the author(s) of the paper, please indicate within $^...$
%% the number which corresponds to the institute of each author.
{\bf{ Stanislav Melnikov$^{1,2}$, Jens Woitas$^1$, Jochen
Eisl\"offel$^1$, Francesca Bacciotti$^3$, Ugo~Locatelli$^4$ \ and Tom Ray$^5$}}
%% Here you write your institute name(s) and address(es),
%% the number in $^..$ indicates your author number, for example:
$^1$ {Th\"uringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany} \\
$^2$ {Ulugh Beg Astronomical Institute, Astronomical str. 33, 700052 Tashkent, Uzbekistan} \\
$^3$ {I.N.A.F. - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy}
$^4$ {Dipartimento di Matematica, Universit\`a degli Studi di Roma ``Tor Vergata'', Via della Ricerca Scientifica 1, 00133 Roma, Italy}
$^5$ {Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland}
%% Here you may write the e-mail address of one or more of the authors
%% who will act as contact person for preprint requests etc, for example:
{E-mail contact: melnikov {\em at} tls-tautenburg.de}
%% IF YOU USE ANY PERSONAL LATEX COMMANDS IN YOUR ABSTRACT,
%% PLEASE INCLUDE THEIR DEFINITIONS HERE!
%% Within the following brackets you place your text:
{\textit{Context.} LkH$\alpha$~233 is a Herbig Ae/Be star with a
collimated bipolar jet. As such, it may be a high-mass analogue to the
classical T Tauri stars and their outflows.
\textit{Aims.} We investigate optical forbidden lines along the
LkH$\alpha$~233 jet to determine physical parameters of this jet
(electron density $n_{\mathrm{e}}$, hydrogen ionisation fraction
$x_{\mathrm{e}}$, electron temperature $T_e$). The knowledge of these
parameters allows us a direct comparison of a jet from a Herbig star
with those from T Tauri stars.
\textit{Methods.} We present the results of HST/STIS and WFPC2
observations of LkH$\alpha$~233 and its environment. These are the
first observations of this object with a spatial resolution of $\le
0.''1$ at optical wavelengths. Our STIS data provide spectroscopic
maps that allow us to reconstruct high angular resolution images of
the bipolar jet from LkH$\alpha$~233 covering the first
$\approx$~2000~AU from the star in the blueshifted outflow lobe and
$\approx$~4000~AU in the redshifted lobe. These maps are analysed with
a diagnostic code that yields $n_{\mathrm{e}}$, $x_{\mathrm{e}}$,
$T_e$, and mass density $n_{\mathrm{H}}$ within the jet.
\textit{Results.} The WFPC2 images in broad-band filters clearly show
the presence of a dark lane caused either by a circumstellar disk or a
dust torus. The circumstellar environment of LkH$\alpha$~233 can be
interpreted as a conical cavity that was cleared by a bipolar jet. In
this interpretation, the maximum of the optical and near-infrared
brightness distribution is not coincident with the star itself which
is, in fact, deeply extincted. In the blueshifted lobe $n_e$ is close
to or above the critical density for [SII] lines ($2.5\times10^4
\mathrm{cm}^{-3}$) in the first arcsecond and decreases with distance
from the source. The ionisation $x_e \approx 0.2 - 0.6$ gently rises
for the first 500~AU of the flow and shows two re-ionisation events
further away from the origin. The electron temperature
$T_{\mathrm{e}}$ varies along the flow between $10^4\,\mathrm{K}$ and
$3\times10^4\,\mathrm{K}$. The $n_{\mathrm{H}}$ is between
$3\times10^3$ and $10^5\,\mathrm{cm}^{-3}$, and the mass flux
$\dot{M}\approx 10^{-8} - 10^{-7}
\mathrm{M}_{\odot}\mathrm{yr}^{-1}$. The (radial) outflow velocities
are $\approx 80 - 160 \,\mathrm{km}\,\mathrm{s}^{-1}$ and appear to
increase with distance from the source. In the redshifted lobe the
excitation conditions are quite different: $T_{\mathrm{e}}$,
$n_{\mathrm{e}}$, $x_{\mathrm{e}}$, and $n_{\mathrm{H}}$ are all lower
than in the blueshifted lobe, but of the same order of magnitude.
\textit{Conclusions.} All these derived parameters are just beyond or
at the upper limits of those observed for classical T Tauri star
jets. This may indicate that the flows from the higher mass Herbig
stars are indeed scaled-up examples of the same phenomenon as in T
Tauri stars.}
% Here you write which journal accepted your paper, for example:
{ Accepted by Astron. Astroph. }
http://www.arcetri.astro.it/$\sim$starform/publ2007.htm
\v5
{\large\bf{ The distance to the Orion Nebula }}
{\bf{ K. M. Menten$^1$, M. J. Reid$^2$, J. Forbrich$^{1,2}$, and A. Brunthaler$^1$ }}
$^1$ { Max-Planck-Institut f\"ur Radioastronomie, Auf dem H\"ugel 69, 53121 Bonn, Germany } \\
$^2$ { Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA }
{E-mail contact: kmenten {\em at} mpifr-bonn.mpg.de }
{ We have used the Very Long Baseline Array to measure the trigonometric parallax of several member
stars of the Orion Nebula Cluster showing non-thermal radio emission. We have determined the distance
to the cluster to be $414\pm 7$ pc. Our distance determination allows for an improved calibration of
luminosities and ages of young stars. We have also measured the proper motions of four cluster stars
which, when accurate radial velocities are measured, will put strong constraints on the origin of the
cluster. }
{ Published by Astronomy \& Astrophysics (Vol. 474, p. 515) }
\v5
{\large\bf{ Dynamics of the Giant Planets of the Solar System in the Gaseous Protoplanetary Disk and
Their Relationship to the Current Orbital Architecture }}
{\bf{ Alessandro Morbidelli$^1$, Kleomenis Tsiganis$^2$, Aur\'elien Crida$^3$, Harold F. Levison$^4$,
and R. Gomes$^5$ }}
$^1$ { Observatoire de la C\^ote d'Azur, B.P. 4229, 06304 Nice Cedex 4, France } \\
$^2$ { Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece } \\
$^3$ { Department of Physics, University of T\"ubingen, T\"ubingen, Germany } \\
$^4$ { Southwest Research Institute, Boulder, CO 80302, USA } \\
$^5$ { National Observatory, Rio de Janeiro, Brazil }
{E-mail contact: morby {\em at} obs-nice.fr }
{ We study the orbital evolution of the four giant planets of our solar system in a gas disk. Our investigation extends the previous works by Masset \& Snellgrove and Morbidelli \& Crida, which focused on the dynamics of the Jupiter-Saturn system. The only systems we found to reach a steady state are those in which the planets are locked in a quadruple mean-motion resonance (i.e., each planet is in resonance with its neighbor). In total, we found six such configurations. For the gas-disk parameters found in Morbidelli \& Crida, these configurations are characterized by a negligible migration rate. After the disappearance of the gas, and in the absence of planetesimals, only two of these six configurations (the least compact ones) are stable for a time of hundreds of millions of years or more. The others become unstable on a timescale of a few Myr. Our preliminary simulations show that, when a planetesimal disk is added beyond the orbit of the outermost planet, the planets can evolve from the most stable of these configurations to their current orbits in a fashion qualitatively similar to that described in Tsiganis et al. }
{ Published by The Astronomical Journal (Vol. 134, p. 1790) }
\v5
{\large\bf{The dynamics of Jupiter and Saturn in the gaseous protoplanetary disk }}
{\bf{Alessandro Morbidelli$^1$ and Aur\'elien Crida$^1$ }}
$^1$ {Observatoire de la C\^ote d'Azur, B.P. 4229, 06304 Nice Cedex 4, France }
{We study the possibility that the mutual interactions between Jupiter and Saturn prevented Type II
migration from driving these planets much closer to the Sun. Our work extends previous results by Masset
and Snellgrove [Masset, F., Snellgrove, M., 2001. Mon. Not. R. Astron. Soc. 320, L55–L59], by exploring
a wider set of initial conditions and disk parameters, and by using a new hydrodynamical code that
properly describes for the global viscous evolution of the disk. Initially both planets migrate towards
the Sun, and Saturn's migration tends to be faster. As a consequence, they eventually end up locked in
a mean motion resonance. If this happens in the 2:3 resonance, the resonant motion is particularly stable,
and the gaps opened by the planets in the disk may overlap. This causes a drastic change in the torque
balance for the two planets, which substantially slows down the planets' inward migration. If the gap
overlap is substantial, planet migration may even be stopped or reversed. As the widths of the gaps
depend on disk viscosity and scale height, this mechanism is particularly efficient in low viscosity,
cool disks. The initial locking of the planets in the 2:3 resonance is a likely outcome if Saturn formed
at the edge of Jupiter's gap, but also if Saturn initially migrated rapidly from further away. We also
explore the possibility of trapping in other resonances, and the subsequent evolutions. We discuss the
compatibility of our results with the initial conditions adopted in Tsiganis et al. [Tsiganis, K., Gomes,
R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459–461] and Gomes et al. [Gomes, R., Levison,
H.F., Tsiganis, K., Morbidelli, A., 2005. Nature 435, 466–469] to explain the current orbital architecture
of the giant planets and the origin of the Late Heavy Bombardment of the Moon. }
{ Published by Icarus (Vol. 191, p. 158) }
\v5
%%--------SubmissionID=1241----------------
%% Title
{\large\bf{A SCUBA survey of bright-rimmed clouds}}
%% Authors
{\bf{ L.K. Morgan$^{1,2}$, M.A. Thompson$^{3}$, J.S. Urquhart$^{4}$ and G.J. White$^{1,5,6}$}}
%% Institutions
$^1$ {CAPS, The University of Kent, Canterbury, Kent CT2 7NR, U.K.} \\
$^2$ {NRAO, Green Bank Telescope, P.O.Box 2, Green Bank, WV 24944, U.S.A.} \\
$^3$ {Centre for Astrophysics Research, Science and Technology Research Institute, University of Hertfordshire, College Lane, Hatfield AL10 9AB, U.K.} \\
$^4$ {School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, U.K.} \\
$^5$ {Space Physics Division, Room 1.71, Space Science \& Technology Division, CCLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, U.K.} \\
$^6$ {Dept. of Physics \& Astronomy, The Open University, Walton Hall, Milton Keynes MK7 6AA, U.K.}
%% Email
{E-mail contact: lmorgan {\em at} nrao.edu}
%% LATEX COMMANDS
%% Abstract body
{Bright-rimmed clouds (BRCs) are potential examples of triggered star
formation regions, in which photoionisation driven shocks caused by
the expansion of HII regions induce protostellar collapse within the
clouds.} {The main purpose of the paper is to establish the level of
star formation occuring within a known set of BRCs. A secondary aim is
to determine the extent, if any, to which this star formation has been
promulgated by the process of photoionisation triggering.} {A primary
set of observations is presented obtained with submillimeter SCUBA
observations and archival data from near-IR and mid- to far-IR have
been explored for relevant observations and incorporated where
appropriate.} {SCUBA observations show a total of 47 dense cores
within the heads of 44 observed BRCs drawn from a catalogue of IRAS
sources embedded within HII regions, supportive of the scenario
proposed by RDI models. The physical properties of these cores
indicate star formation across the majority of our sample. This star
formation appears to be predominately in the regime of intermediate to
high mass and may indicate the formation of clusters. IR observations
indicate the association of early star forming sources with our
sample. A fundamental difference appears to exist between different
morphological types of BRC, which may indicate a different
evolutionary pathway toward star formation in the different types of
BRC.} {Bright-rimmed clouds are found to harbour star formation in
its early stages. Different evolutionary scenarios are found to exist
for different morphological types of BRC. The morphology of a BRC is
described as type `A', moderately curved rims, type `B', tightly
curved rims, and `C', cometary rims. `B' and `C' morphological types
show a clear link between their associated star formation and the
strength of the ionisation field within which they are embedded.
An analysis of the mass function of potentially induced star-forming
regions indicate that radiatively-driven implosion of molecular clouds
may contribute significantly toward the intermediate to high-mass
stellar mass function.}
% Journal
{ Accepted by A\&A}
%% Preprints URL
http://www.gb.nrao.edu/$\sim$lmorgan/Preprints/SCUBA${\_}$BRC${\_}$Survey.pdf
\v5
%%--------SubmissionID=1265----------------
%% Title
{\large\bf{Interacting Jets from Binary Protostars}}
%% Authors
{\bf{ G.C. Murphy$^{1}$, T. Lery$^{2}$, S. O'Sullivan$^{3}$, D. Spicer,$^{4}$, F. Bacciotti$^{5}$ and A. Rosen$^{6}$}}
%% Institutions
$^1$ {Laboratoire d'Astrophysique de Grenoble, CNRS, Universite Joseph Fourier, B.P. 53, F-38041 Grenoble, France} \\
$^2$ {European Science Foundation, 1 quai Lezay-Marnesia, BP 90015, 67080 Strasbourg, France} \\
$^3$ {UCD School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland} \\
$^4$ {NASA/GSFC Laboratory for Solar and Space Physics, Mail Stop 612.1, Greenbelt, MD 20771, USA} \\
$^5$ {INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Florence, Italy} \\
$^6$ {Max-Planck-Institut fur Radioastronomie, Auf dem Huegel 69, D-53121 Bonn}
%% Email
{E-mail contact: gmurphy {\em at} obs.ujf-grenoble.fr}
%% LATEX COMMANDS
%% Abstract body
{We investigate potential models that could explain why multiple
proto-stellar systems predominantly show single jets. During their
formation, stars most frequently produce energetic outflows and
jets. However, binary jets have only been observed in a very small
number of systems. We model numerically 3D binary jets for various
outflow parameters. We also model the propagation of jets from a
specific source, namely L1551 IRS 5, known to have two jets, using
recent observations as constraints for simulations with a new MHD
code. We examine their morphology and dynamics, and produce synthetic
emission maps. We find that the two jets interfere up to the stage
where one of them is almost destroyed or engulfed into the second
one. We are able to reproduce some of the observational features of
L1551 such as the bending of the secondary jet. While the effects of
orbital motion are negligible over the jets dynamical timeline, their
interaction has significant impact on their morphology. If the jets
are not strictly parallel, as in most observed cases, we show that the
magnetic field can help the collimation and refocusing of both of the
two jets.}
% Journal
{ Accepted by Astronomy and Astrophysics}
%% Preprints URL
http://arxiv.org/abs/0711.3144
\v5
{\large\bf{ Turbulent Torques on Protoplanets in a Dead Zone }}
{\bf{ Jeffrey S. Oishi$^{1,2}$, Mordecai-Mark Mac Low$^2$ and Kristen Menou$^3$ }}
$^1$ { Department of Astronomy, University of Virginia, P.O. Box 3818, Charlottesville, VA 22903, USA } \\
$^2$ { Department of Astrophysics, American Museum of Natural History, Central Park West at 81st Street, New York, NY 10024-5192, USA } \\
$^3$ { Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA }
{E-mail contact: joishi {\em at} amnh.org }
{ Migration of protoplanets in their gaseous host disks may be largely responsible for the observed orbital distribution of extrasolar planets. Recent simulations have shown that the magnetorotational turbulence thought to drive accretion in protoplanetary disks can affect migration by turning it into an orbital random walk. However, these simulations neglected the disk's ionization structure. Low ionization fraction near the midplane of the disk can decouple the magnetic field from the gas, forming a dead zone with reduced or no turbulence. Here, to understand the effect of dead zones on protoplanetary migration, we perform numerical simulations of a small region of a stratified disk with magnetorotational turbulence confined to thin active layers above and below the midplane. Turbulence in the active layers exerts decreased, but still measurable, gravitational torques on a protoplanet located at the disk midplane. We find a decrease of 2 orders of magnitude in the diffusion coefficient for dead zones with dead-to-active surface density ratios approaching realistic values in protoplanetary disks. This torque arises primarily from density fluctuations within a distance of one scale height of the protoplanet. Turbulent torques have correlation times of only $\sim$ 0.3 orbital periods and apparently time-stationary distributions. These properties are encouraging signs that stochastic methods can be used to determine the orbital evolution of populations of protoplanets under turbulent migration. Our results indicate that dead zones may be dynamically distinct regions for protoplanetary migration. }
{ Published by The Astrophysical Journal (Vol. 670, p. 805) }
\v5
{\large\bf{ Unveiling the nature and interaction of the intermediate/high-mass YSOs in IRAS 20343+4129 }}
{\bf{ Aina Palau$^{1,2}$, R. Estalella$^2$, P. T. P. Ho$^{3,4}$, H. Beuther$^5$, and M. T. Beltr\'an$^2$ }}
$^1$ { Laboratorio de Astrof\'{\i}sica Espacial y F\'{\i}sica Fundamental, INTA, Apartado 78, 28691
Villanueva de la Ca\~nada, Madrid, Spain } \\
$^2$ { Departament d'Astronomia i Meteorologia, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Catalunya, Spain } \\
$^3$ { Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA } \\
$^4$ { Academia Sinica, Institute of Astronomy and Astrophysics, PO Box 23-141, Taipei, 106, Taiwan } \\
$^5$ { Max-Planck-Institut for Astronomy, Koenigstuhl 17, 69117 Heidelberg, Germany }
{E-mail contact: apalau {\em at} laeff.inta.es }
{\emph{Context.} IRAS 20343+4129 was suggested to harbor one of the most massive and embedded stars in
the Cygnus OB2 association, IRS 1, which seemed to be associated with a north-south molecular outflow.
However, the dust emission peaks do not coincide with the position of IRS 1, but lie on either side of
another massive Young Stellar Object (YSO), IRS 3, which is associated with centimeter emission.
\emph{Aims.} The goal of this work is to elucidate the nature of IRS 1 and IRS 3, and study their
interactions with the surrounding medium.
\emph{Methods.} The Submillimeter Array (SMA) was used to observe with high angular resolution the 1.3 mm
continuum and CO (2-1) emission of the region, and we compared this millimeter emission with the infrared
emission from 2MASS.
\emph{Results.} Faint millimeter dust continuum emission was detected
toward IRS 1, and we derived an associated gas mass of $\sim$0.8
$M_{\odot}$. The IRS 1 Spectral Energy Distribution (SED) agrees with
IRS 1 being an intermediate-mass Class I source of about 1000
$L_{\odot}$, whose circumstellar material is producing the observed
large infrared excess. We have discovered a high-velocity CO (2-1)
bipolar outflow in the east-west direction, which is clearly
associated with IRS 1. Its outflow parameters are similar to those of
intermediate-mass YSOs. Associated with the blue large-scale CO (2-1)
outflow lobe, detected with single-dish observations, we only found
two elongated low-velocity structures on either side of IRS 3. The
large-scale outflow lobe is almost completely resolved out by the
SMA. Our detected low-velocity CO structures are coincident with
elongated H$_2$ emission features. The strongest millimeter continuum
condensations in the region are found on either side of IRS 3, where
the infrared emission is extremely weak. The CO and H$_2$ elongated
structures follow the border of the millimeter continuum emission that
is facing IRS 3. All these results suggest that the dust is associated
with the walls of an expanding cavity driven by IRS 3, estimated to be
a B2 star from both the centimeter and the infrared continuum
emission.
\emph{Conclusions.} IRS 1 seems to be an intermediate-mass Class I YSO
driving a molecular outflow in the east-west direction, while IRS 3 is
most likely a more evolved intermediate/high-mass star that is driving
a cavity and accumulating dust in its walls. Within and beyond the
expanding cavity, the millimeter continuum sources can be sites of
future low-mass star formation. }
{ Published by Astronomy \& Astrophysics (Vol. 474, p. 911) }
\v5
%%--------SubmissionID=1264----------------
%% Title
{\large\bf{Stellar contents and star formation in the young star cluster Be 59}}
%% Authors
{\bf{ A. K. Pandey$^{1,2,3}$, Saurabh Sharma$^{3}$, K. Ogura$^{4}$, D. K. Ojha$^{2}$, W. P. Chen$^{1}$, B. C. Bhatt$^{5}$ and S. K. Ghosh$^{2}$}}
%% Institutions
$^1$ {Institute of Astronomy, National Central University, Chung-Li 32054, Taiwan} \\
$^2$ {Tata Institute of Fundamental Research, Mumbai - 400 005, India} \\
$^3$ {Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital, 263 129, India} \\
$^4$ {Kokugakuin University, Higashi, Shibuya-ku, Tokyo 150-8440, Japan} \\
$^5$ {CREST, Indian Institute of Astrophysics, Hosakote 562 114, India}
%% Email
{E-mail contact: pandey {\em at} aries.ernet.in}
%% LATEX COMMANDS
%% Abstract body
{We present $UBVI_C$ CCD photometry of the young open cluster Be 59
with the aim to study the star formation scenario in the cluster. The
radial extent of the cluster is found to be $\sim$ 10 arcmin (2.9
pc). The interstellar extinction in the cluster region varies between
$E(B-V) \simeq$ 1.4 to 1.8 mag. The ratio of total-to-selective
extinction in the cluster region is estimated as $3.7\pm0.3$. The
distance of the cluster is found to be $1.00\pm0.05$ kpc. Using
near-infrared colours and slitless spectroscopy, we have identified
young stellar objects (YSOs) in the open cluster Be 59 region. The
ages of these YSOs range between $<1$ Myr to $\sim$ 2 Myr, whereas the
mean age of the massive stars in the cluster region is found to be
$\sim$ 2 Myr. There is evidence for second generation star formation
outside the boundary of the cluster, which may be triggered by massive
stars in the cluster. The slope of the initial mass function,
$\Gamma$, in the mass range $2.5 < M/M_\odot \le 28$ is found to be
$-1.01\pm0.11$ which is shallower than the Salpeter value (-1.35),
whereas in the mass range $1.5 < M/M_\odot \le 2.5$ the slope is
almost flat. The slope of the K-band luminosity function is estimated
as $0.27\pm0.02$, which is smaller than the average value ($\sim$0.4)
reported for young embedded clusters. Approximately $32\%$ of
H$\alpha$ emission stars of Be 59 exhibit NIR excess indicating that
inner disks of the T-Tauri star (TTS) population have not
dissipated. The MSX and IRAS-HIRES images around the cluster region
are also used to study the emission from unidentified infrared bands
and to estimate the spatial distribution of optical depth of warm and
cold interstellar dust.}
% Journal
{ Accepted by MNRAS}
%% Preprints URL
astro-ph 0710.5429
\vspace{0.3cm}
{\large\bf{ Do We Need to Know the Temperature in Prestellar Cores? }}
{\bf{ Ya. Pavlyuchenkov$^1$, Th. Henning$^1$ and D. Wiebe$^2$ }}
$^1$ { Max Planck Institute for Astronomy, K\"onigstuhl 17, D-69117 Heidelberg, Germany } \\
$^2$ { Institute of Astronomy of the Russian Academy of Sciences, 48 Pyatnitskaya Street, Moscow 119017, Russia }
{E-mail contact: pavyar {\em at} mpia.de }
{ Molecular line observations of starless (prestellar) cores combined with a chemical evolution
modeling and radiative transfer calculations are a powerful tool to study the earliest stages of star
formation. However, conclusions drawn from such a modeling may noticeably depend on the assumed thermal
structure of the cores. The assumption of isothermality, which may work well in chemo-dynamical studies,
becomes a critical factor in molecular line formation simulations. We argue that even small temperature
variations, which are likely to exist in starless cores, can have a nonnegligible effect on the
interpretation of molecular line data and derived core properties. In particular, ``chemically pristine''
isothermal cores (low depletion) can have centrally peaked C$^{18}$O and C$^{34}$S radial intensity
profiles, while having ringlike intensity distributions in models with a colder center and/or warmer
envelope assuming the same underlying chemical structure. Therefore, derived molecular abundances
based on oversimplified thermal models may lead to a misinterpretation of the line data. }
{ Published by The Astrophysical Journal Letters (Vol. 669, p. L101) }
\vspace{0.3cm}
{\large\bf{ Molecular Line Radiative Transfer in Protoplanetary Disks: Monte Carlo Simulations versus Approximate Methods }}
{\bf{ Ya. Pavlyuchenkov$^1$, D. Semenov$^1$, Th. Henning$^1$, St. Guilloteau$^2$, V. Pi\'etu$^3$,
R. Launhardt$^1$ and A. Dutrey$^2$ }}
$^1$ { Max Planck Institute for Astronomy, K\"onigstuhl 17, D-69117 Heidelberg, Germany } \\
$^2$ { Laboratoire d'Astrophysique de Bordeaux, OASU, Universit\'e Bordeaux 1, CNRS, 2 rue de l'Observatoire, BP 89, F-33270 Floirac, France } \\
$^3$ { IRAM, 300 rue de la Piscine, F-38406 Saint Martin d'H\`eres, France }
{ We analyze the line radiative transfer in protoplanetary disks using
several approximate methods and a well-tested accelerated Monte Carlo
code. A low-mass flaring disk model with uniform as well as stratified
molecular abundances is adopted. Radiative transfer in low and high
rotational lines of CO, C$^{18}$O, HCO$^+$, DCO$^+$, HCN, CS, and
H$_2$CO is simulated. The corresponding excitation temperatures,
synthetic spectra, and channel maps are derived and compared to the
results of the Monte Carlo calculations. A simple scheme that
describes the conditions of the line excitation for a chosen molecular
transition is elaborated. We find that the simple LTE approach can
safely be applied for the low molecular transitions only, while it
significantly overestimates the intensities of the upper lines. In
contrast, the full escape probability (FEP) approximation can safely
be used for the upper transitions (J$_{up}$ $\simgreat$ 3), but it is
not appropriate for the lowest transitions because of the maser
effect. In general, the molecular lines in protoplanetary disks are
partly subthermally excited and require more sophisticated approximate
line radiative transfer methods. We analyze a number of approximate
methods, namely, LVG, vertical escape probability (VEP), and vertical
one ray (VOR) and discuss their algorithms in detail. In addition, two
modifications to the canonical Monte Carlo algorithm that allow a
significant speed up of the line radiative transfer modeling in
rotating configurations by a factor of 10–50 are
described. }
{ Published by The Astrophysical Journal (Vol. 669, p. 1262) }
\vspace{0.3cm}
{\large\bf{ Protoplanetary dynamics – I. Dynamical modes of isothermal protoplanets }}
{\bf{ B. Pecnik$^{1,2,3}$ and G. Wuchterl$^{1,4}$ }}
$^1$ { Astrophysikalisches Institut und Universit\"ats-Sternwarte, 07745 Jena, Germany } \\
$^2$ { Max-Planck-Institut f\"ur extraterrestrishe Physik, Postfach 1312, 85748 Garching, Germany } \\
$^3$ { Faculty of Natural Sciences and Kinesiology, University of Split, 21000 Split, Croatia } \\
$^4$ { Th\"uringer Landessternwarte Tautenburg, 07778 Tautenburg, Germany}
{E-mail contact: bonnie {\em at} pmfst.hr }
{ This paper is the first in a series of publications which investigate the stability properties of the complete set of isothermal protoplanetary equilibrium solutions. To perform a non-linear stability analysis, we introduce a fluid dynamics numerical model. We inspect the entire solution set and find five basic dynamical modes: oscillation, pulsation, transition, ejection and collapse. }
{ Published by Monthly Notices of the Royal Astronomical Society (Vol. 381, p. 640) }
\v5
{\large\bf{ Laser Guide Star Adaptive Optics Integral Field Spectroscopy of a Tightly Collimated Bipolar Jet from the Herbig Ae Star LkH$\alpha$ 233 }}
{\bf{ Marshall D. Perrin$^{1,2}$ and James R. Graham$^1$ }}
$^1$ { Astronomy Department, University of California, Berkeley, CA 94720-3411, USA } \\
$^2$ { Current address: Department of Astronomy, University of California at Los Angeles, Los Angeles, CA 90095, USA }
{E-mail contact: mperrin {\em at} astro.berkeley.edu }
{ We have used the integral field spectrograph OSIRIS and laser guide star adaptive optics at Keck
Observatory to obtain high angular resolution (0.06$^{\prime\prime}$), moderate spectral resolution
(R $\simeq$ 3800) images of the bipolar jet from the Herbig Ae star LkH$\alpha$ 233, seen in near-IR
[Fe II] emission at 1.600 and 1.644 $\mu$m. This jet is narrow and tightly collimated, with an opening
angle of only 9$^\circ$, and has an average radial velocity of $\sim$100 km s$^{-1}$. The jet and
counterjet are asymmetric, with the redshifted jet much clumpier than its counterpart at the angular
resolution of our observations. The observed properties are in general similar to jets seen around T Tauri
stars, although it has a relatively large mass flux of $1.2 \pm 0.3 \times 10^{-7}$ M$_\odot$ yr$^{-1}$,
near the high end of the observed mass flux range around T Tauri stars. We also spatially resolve an
inclined circumstellar disk around LkH$\alpha$ 233, which obscures the star from direct view. By
comparison with numerical radiative transfer disk models, we estimate the disk midplane to be inclined
i = $65^\circ \pm 5^\circ$ relative to the plane of the sky. Since the star is seen only in scattered
light at near-infrared wavelengths, we detect only a small fraction of its intrinsic flux. Because
previous estimates of its stellar properties did not account for this, either LkH$\alpha$ 233 must be
located closer than previously believed, or its true luminosity must be greater than previously supposed,
consistent with its being a $\sim$4 M$_\odot$ star near the stellar birth line. }
{ Published by The Astrophysical Journal (Vol. 670, p. 499) }
\v5
%%--------SubmissionID=1251----------------
%% Title
{\large\bf{Tracing the origins of permitted emission lines in RU Lupi down to AU scales}}
%% Authors
{\bf{ L. Podio$^{1}$, P. J. V. Garcia$^{2, 3}$, F. Bacciotti$^{1}$, S. Antoniucci$^{4}$, B. Nisini$^{4}$, C. Dougados$^{5}$ and M. Takami$^{6}$}}
%% Institutions
$^1$ {INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy} \\
$^2$ {Centro de Astrof\'{\i}sica da Universidade do Porto, Rua das Estrelas s/n, P-4150-762 Porto, Portugal} \\
$^3$ {Departamento de Engenharia F\'{\i}sica, Faculdade de Engenharia da Universidade do Porto, Portugal} \\
$^4$ {INAF-Osservatorio Astronomico di Roma, Via di Frascati 33, I-00040 Monte Porzio Catone, Italy} \\
$^5$ {Laboratoire d'Astrophysique de Grenoble, BP 53, 38041 Grenoble Cedex, France} \\
$^6$ {Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan, R.O.C.}
%% Email
{E-mail contact: lindapod {\em at} arcetri.astro.it}
%% LATEX COMMANDS
\newcommand{\lam}{$\lambda$}
%% Abstract body
{{\em Context.}
Most of the observed emission lines and continuum excess from
young accreting low mass stars (Classical T Tauri stars -- CTTSs)
take place in the star-disk or inner disk region. These regions
have a complex emission topology still largely unknown. \\
{\em Aims.}
{In this paper the magnetospheric accretion and inner wind
contributions to the observed permitted He and H near infrared (NIR) lines
of the bright southern CTTS RU Lupi are investigated for the first time. \\
{\em Methods.}
Previous optical observations of RU Lupi showed a large
H$\alpha$ profile, due to the emission from a wind in the line wings,
and a micro-jet detected in forbidden lines.
We extend this analysis to NIR lines through seeing-limited
high spectral resolution spectra taken with VLT/ISAAC, and
adaptive optics (AO) aided narrow-band imaging and low spectral resolution
spectroscopy with VLT/NACO. Using spectro-astrometric analysis
we investigate the presence of extended emission down to very
low spatial scales (a few AU). \\
{\em Results.}
The HeI \lam10830 line presents a P Cygni profile whose
absorption feature indicates the presence of an inner stellar wind.
Moreover the spectro-astrometric analysis evidences the presence of an
extended emission superimposed to the absorption feature
and likely coming from the micro-jet detected in the optical.
On the contrary, the origin of the Hydrogen Paschen and Brackett lines
is difficult to address. We tried tentatively to explain the observed
line profiles and flux ratios with both accretion and wind
models showing the limits of both approaches.
The lack of spectro-astrometric signal indicates that the HI emission is
either compact or symmetric.
Our analysis confirms the sensitivity of the HeI line to the presence of
faint extended emission regions in the close proximity of the star.}
% Journal
{ Accepted by A\&A}
%% Preprints URL
http://arxiv.org/abs/0711.2596
\v5
%%--------SubmissionID=1261----------------
%% Title
{\large\bf{Observable Consequences of Planet Formation Models in Systems with Close-in Terrestrial Planets}}
%% Authors
{\bf{ Sean N. Raymond$^{1}$, Rory Barnes$^{2}$ and Avi M. Mandell$^{3}$}}
%% Institutions
$^1$ {CASA, University of Colorado} \\
$^2$ {LPL, University of Arizona} \\
$^3$ {NASA Goddard}
%% Email
{E-mail contact: rayray.sean {\em at} gmail.com}
%% LATEX COMMANDS
\def\mearth{{\rm\,M_\oplus}}
%% Abstract body
{To date, two planetary systems have been discovered with close-in,
terrestrial-mass planets ($< 5-10 \mearth$). Many more such discoveries are
anticipated in the coming years with radial velocity and transit
searches. Here we investigate the different mechanisms that could form ``hot
Earths'' and their observable predictions. Models include: 1) {\it in situ}
accretion; 2) formation at larger orbital distance followed by inward ``type
1'' migration; 3) formation from material being ``shepherded'' inward by a
migrating gas giant planet; 4) formation from material being shepherded by
moving secular resonances during dispersal of the protoplanetary disk; 5)
tidal circularization of eccentric terrestrial planets with close-in
perihelion distances; and 6) photoevaporative mass loss of a close-in giant
planet. Models 1-4 have been validated in previous work. We show that tidal
circularization can form hot Earths, but only for relatively massive planets
($> 5 \mearth$) with very close-in perihelion distances ($<$ 0.025 AU),
and even then the net inward movement in orbital distance is at most only
0.1-0.15 AU. For planets of less than $\sim 70 \mearth$, photoevaporation can
remove the planet's envelope and leave behind the solid core on a Gyr
timescale, but only for planets inside 0.025-0.05 AU. Using two quantities
that are observable by current and upcoming missions, we show that these
models each produce unique signatures, and can be observationally
distinguished. These observables are the planetary system architecture
(detectable with radial velocities, transits and transit-timing) and the bulk composition of transiting close-in terrestrial planets (measured by transits via the planet's
radius).}
% Journal
{ Accepted by MNRAS}
%% Preprints URL
http://adsabs.harvard.edu/abs/2007arXiv0711.2015R
\v5
{\large\bf{Optical spectroscopic classification and membership of young M dwarfs in star-forming regions }}
{\bf{F. C. Riddick$^{1,2}$, P. F. Roche$^2$, and P. W. Lucas$^3$ }}
$^1$ {Department of Astronomy \& Astrophysics, Penn State University, 525 Davey Lab, University Park, PA 16802, USA } \\
$^2$ {Astrophysics, Department of Physics, University of Oxford, DWB, Keble Road, Oxford OX1 3RH, UK } \\
$^3$ {Centre for Astrophysics, University of Hertfordshire, College Lane, Hatfield, Herts AL10 9AB, UK }
{E-mail contact: fionariddick {\em at} gmail.com }
{The spectral type is a key parameter in calibrating the temperature which is required to estimate the mass of young stars and brown dwarfs. We describe an approach developed to classify low-mass stars and brown dwarfs in the Trapezium Cluster using red optical spectra, which can be applied to other star-forming regions. The classification uses two methods for greater accuracy: the use of narrow-band spectral indices which rely on the variation of the strength of molecular lines with spectral type and a comparison with other previously classified young, low-mass objects in the Chamaeleon I star-forming region. We have investigated and compared many different molecular indices and have identified a small number of indices which work well for classifying M-type objects in nebular regions. The indices are calibrated for young, pre-main-sequence objects whose spectra are affected by their lower surface gravities compared with those on the main sequence. Spectral types obtained are essentially independent of both reddening and nebular emission lines.
Confirmation of candidate young stars and brown dwarfs as bona fide cluster members may be accomplished with moderate resolution spectra in the optical region by an analysis of the strength of the gravity-sensitive Na doublet. It has been established that this feature is much weaker in these very young objects than in field dwarfs. A sodium spectral index is used to estimate the surface gravity and to demonstrate quantitatively the difference between young (1–2 Myr) objects, and dwarf and giant field stars. }
{ Published by Monthly Notices of the Royal Astronomical Society (Vol. 381, p. 1067) }
\v5
{\large\bf{An optical spectroscopic HR diagram for low-mass stars and brown dwarfs in Orion }}
{\bf{F. C. Riddick$^{1,2}$, P. F. Roche$^2$, and P. W. Lucas$^3$ }}
$^1$ {Department of Astronomy \& Astrophysics, Penn State University, 525 Davey Lab, University Park, PA 16802, USA } \\
$^2$ {Astrophysics, Department of Physics, University of Oxford, DWB, Keble Road, Oxford OX1 3RH, UK } \\
$^3$ {Centre for Astrophysics, University of Hertfordshire, College Lane, Hatfield, Herts AL10 9AB, UK }
{E-mail contact: fionariddick {\em at} gmail.com }
{The masses and temperatures of young low-mass stars and brown dwarfs in star-forming regions are not yet well established because of uncertainties in the age of individual objects and the spectral type–temperature scale appropriate for objects with ages of only a few Myr. Using multi-object optical spectroscopy, 45 low-mass stars and brown dwarfs in the Trapezium Cluster in Orion have been classified and 44 of these confirmed as bona fide cluster members. The spectral types obtained have been converted to effective temperatures using a temperature scale intermediate between those of dwarfs and giants, which is suitable for young pre-main-sequence objects. The objects have been placed on a Hertzsprung–Russell (HR) diagram overlaid with theoretical isochrones. The low-mass stars and the higher mass substellar objects are found to be clustered around the 1 Myr isochrone, while many of the lower mass substellar objects are located well above this isochrone. An average age of 1 Myr is found for the majority of the objects. Assuming coevality of the sources and an average age of 1 Myr, the masses of the objects have been estimated and range from 0.018 to 0.44 M$_\odot$. The spectra also allow an investigation of the surface gravity of the objects by measurement of the sodium doublet equivalent width. With one possible exception, all objects have low gravities, in line with young ages, and the Na indices for the Trapezium objects lie systematically below those of young stars and brown dwarfs in Chamaeleon, suggesting that the 820 nm Na index may provide a sensitive means of estimating ages in young clusters. }
{ Published by Monthly Notices of the Royal Astronomical Society (Vol. 381, p. 1077) }
\v5
%%--------SubmissionID=1258----------------
%% Title
{\large\bf{Dusty disks at the bottom of the IMF}}
%% Authors
{\bf{ Alexander Scholz$^{1}$ and Ray Jayawardhana$^{2}$}}
%% Institutions
$^1$ {SUPA, School of Physics \& Astronomy, University of St. Andrews, North Haugh, St. Andrews,
KY16 9SS, United Kingdom} \\
$^2$ {Department of Astronomy \& Astrophysics, University of Toronto, 50 St. George Street, Toronto,
ON M5S 3H4, Canada}
%% Email
{E-mail contact: as110 {\em at} st-andrews.ac.uk}
%% LATEX COMMANDS
%% Abstract body
{'Isolated planetary mass objects' (IPMOs) have masses close to or
below the Deuterium-burning mass limit ($\sim 15\,M_\mathrm{Jup}$) --
at {\it the bottom of the stellar initial mass function}. We present
an exploratory survey for disks in this mass regime, based on a
dedicated observing campaign with the Spitzer Space Telescope. Our
targets include the full sample of spectroscopically confirmed IPMOs
in the $\sigma$\,Orionis cluster, a total of 18 sources. In the mass
range $8\ldots 20\,M_\mathrm{Jup}$, we identify 4 objects with
$>3\sigma$ colour excess at a wavelength of 8.0$\,\mu m$, interpreted
as emission from dusty disks. We thus establish that a substantial
fraction of IPMOs harbour disks with lifetimes of at least 2-4\,Myr
(the likely age of the cluster), indicating an origin from core
collapse and fragmentation processes. The disk frequency in the IPMO
sample is $29\pm ^{16}_{13}$\% at 8.0\,$\mu m$, very similar to what
has been found for stars and brown dwarfs ($\sim 30$\%). The object
SOri\,70, a candidate $3\,M_\mathrm{Jup}$ object in this cluster,
shows IRAC colours in excess of the typical values for field T dwarfs
(on a 2$\sigma$ level), possibly due to disk emission or low
gravity. This is a new indication for youth and thus an extremely low
mass for SOri\,70.}
% Journal
{ Accepted by ApJL}
%% Preprints URL
http://arxiv.org/abs/0711.2510
\v5
{\large\bf{ The Spitzer Survey of the Small Magellanic Cloud: Discovery of Embedded Protostars in the H II Region NGC 346 }}
{\bf{ Joshua D. Simon$^1$, Alberto D. Bolatto$^{2,3}$, Barbara A. Whitney$^4$, Thomas P. Robitaille$^5$,
Ronak Y. Shah$^6$, David Makovoz$^7$, Snezana Stanimirovi\'c$^8$, Rodolfo H. Barb\'a$^9$ and
M\'onica Rubio$^{10}$ }}
$^1$ { Department of Astronomy, California Institute of Technology, 1200 E. California Boulevard, MS 105-24, Pasadena, CA 91125, USA } \\
$^2$ { Department of Astronomy, University of Maryland, College Park, MD 20742, USA } \\
$^3$ { Department of Astronomy, University of California at Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA } \\
$^4$ { Space Science Institute, 4750 Walnut Street, Suite 205, Boulder, CO 80301, USA } \\
$^5$ { SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, KY16 9SS, UK } \\
$^6$ { Institute for Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA } \\
$^7$ { Spitzer Science Center, California Institute of Technology, 1200 E. California Boulevard, MS 220-6, Pasadena, CA 91125, USA } \\
$^8$ { Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, WI 53706, USA } \\
$^9$ { Departamento de F\'{\i}sica, Universidad de La Serena, Benavente 980, La Serena, Chile } \\
$^{10}$ { Departamento de Astronom\'{\i}a, Universidad de Chile, Casilla 36-D, Santiago, Chile }
{E-mail contact: jsimon {\em at} astro.caltech.edu }
{ We use Spitzer Space Telescope observations from the Spitzer Survey of the Small Magellanic Cloud
(S$^3$MC) to study the young stellar content of N66, the largest and brightest H II region in the SMC.
In addition to large numbers of normal stars, we detect a significant population of bright, red infrared
sources that we identify as likely to be young stellar objects (YSOs). We use spectral energy distribution
(SED) fits to classify objects as ordinary (main-sequence or red giant) stars, asymptotic giant branch
stars, background galaxies, and YSOs. This represents the first large-scale attempt at blind source
classification based on Spitzer SEDs in another galaxy. We firmly identify at least 61 YSOs, with another
50 probable YSOs; only one embedded protostar in the SMC was reported in the literature prior to the
S$^3$MC. We present color selection criteria that can be used to identify a relatively clean sample of
YSOs with IRAC photometry. Our fitted SEDs indicate that the infrared-bright YSOs in N66 have stellar
masses ranging from 2 to 17 M$_\odot$, and that approximately half of the objects are stage II protostars,
with the remaining YSOs roughly evenly divided between stage I and stage III sources. We find evidence
for primordial mass segregation in the H II region, with the most massive YSOs being preferentially
closer to the center than lower mass objects. Despite the low metallicity and dust content of the SMC,
the observable properties of the YSOs appear consistent with those in the Milky Way. Although the YSOs
are heavily concentrated within the optically bright central region of N66, there is ongoing star
formation throughout the complex, and we place a lower limit on the star formation rate of
3.2 $\times 10^{-3}$ M$_\odot$ yr$^{-1}$ over the last $\sim$1 Myr. }
{ Published by The Astrophysical Journal (Vol. 669, p. 327) }
\v5
{\large\bf{The proper motion of the Arches cluster with Keck Laser-Guide Star Adaptive Optics}}
%% Here comes the author(s) of the paper, please indicate within $^...$
%% the number which corresponds to the institute of each author.
{\bf{ A. Stolte$^1$, A. M. Ghez$^{1,2}$, M. Morris$^1$, J. R. Lu$^1$, W. Brandner$^3$ \ and K. Matthews$^4$ }}
%% Here you write your institute name(s) and address(es),
%% the number in $^..$ indicates your author number, for example:
$^1$ {Division of Astronomy and Astrophysics, UCLA, Los Angeles, CA 90095-1547, USA} \\
$^2$ {Institute of Geophysics and Planetary Physics, UCLA, Los Angeles, CA 90095, USA} \\
$^3$ {Max-Planck-Institut for Astronomy, K\"onigstuhl 17, 69117 Heidelberg, Germany} \\
$^4$ { Caltech Optical Observatories, California Institute of Technology, MS 320-47, Pasadena, CA 91225, USA}
%% Here you may write the e-mail address of one or more of the authors
%% who will act as contact person for preprint requests etc, for example:
{E-mail contact: astolte {\em at} astro.ucla.edu}
%% IF YOU USE ANY PERSONAL LATEX COMMANDS IN YOUR ABSTRACT,
%% PLEASE INCLUDE THEIR DEFINITIONS HERE!
%% Within the following brackets you place your text:
{We present the first measurement of the proper motion of the
young, compact Arches cluster near the Galactic center from
near-infrared adaptive optics (AO) data
taken with the recently commissioned
laser-guide star (LGS) at the Keck 10-m telescope.
The excellent astrometric accuracy achieved with LGS-AO
provides the basis for a detailed comparison with
VLT/NAOS-CONICA data taken 4.3 years earlier. Over the 4.3 year
baseline, a spatial displacement of the Arches cluster with respect
to the field population is measured to be $24.0 \pm 2.2$ mas,
corresponding to a proper motion of $5.6 \pm 0.5$ mas/yr
or $212 \pm 29$ km/s at a distance of 8 kpc. In combination with
the known line-of-sight velocity of the cluster, we derive a 3D space
motion of $232 \pm 30$ km/s of the Arches relative to the field.
The large proper motion of the Arches
cannot be explained with any of the closed orbital families
observed in gas clouds in the bar potential of the inner Galaxy, but
would be consistent with the Arches being on a transitional trajectory between
x1 and x2 orbits. We investigate a cloud-cloud collision as the possible
origin for the Arches cluster.
The integration of the cluster orbit in the potential of the inner Galaxy
suggests that the cluster passes within 10 pc of the supermassive black
hole only if its true GC distance is very close to its projected distance.
A contribution of young stars from the Arches cluster to the young
stellar population in the inner few parsecs of the GC thus appears
increasingly unlikely.
The measurement of the 3D velocity and orbital analysis provides the first
observational evidence that Arches-like clusters do not spiral into the GC.
This confirms that no progenitor clusters to the nuclear cluster are
observed at the present epoch.}
% Here you write which journal accepted your paper, for example:
{ Accepted by ApJ }
%% If preprints are available on the WWW you can give the web
%% direction here.
{\sl Preprints are available at} http://xxx.lanl.gov/abs/0706.4133
\v5
%%--------SubmissionID=1266----------------
%% Title
{\large\bf{A Jet Associated With the Classical T Tauri Star RY Tauri}}
%% Authors
{\bf{ Gilbert St-Onge$^{1}$ and Pierre Bastien$^{2}$}}
%% Institutions
$^1$ {Dorval Astronomy Club, 1335 Chemin Bord-du-Lac, Dorval, QC Canada} \\
$^2$ {D\'epartement de physique and Observatoire du Mont-M\'egantic, Universit\'e de Montr\'eal, Montr\'eal, QC Canada}
%% Email
{E-mail contact: bastien {\em at} astro.umontreal.ca}
%% LATEX COMMANDS
%% Abstract body
{High spatial resolution images have been taken on the Gemini North
telescope of the classical T Tauri star RY Tauri. The H$\alpha$ image,
with the continuum properly subtracted, shows a jet extending out to
at least 31$"$ at a position angle of about 295$^{\circ}$ from the
star. A counterjet extending to at least 3.5$'$ in the opposite
direction is also visible. The knots in the inner part of the jet are
probably less than 10 years old. This new Herbig-Haro bipolar jet has
been labeled HH 938. Comparison with previous HST images revealed a
probable tangential motion of the brightest knot by about 165
km/s. The orientation of the disk, as given by the average position
angle of published polarization measurements, is $\approx 20^{\circ}$,
or almost perpendicular to the direction of the jet. Because of the
proximity of its inner knots to the star and their very young
dynamical age, this jet is a very good candidate to study jet emission
mechanisms.}
% Journal
{ Accepted by The Astrophysical Journal}
%% Preprints URL
http://www.journals.uchicago.edu/ApJ/journal/preprints/ApJ21551.preprint.pdf
\v5
%%--------SubmissionID=1244----------------
%% Title
{\large\bf{\emph{Spitzer} imaging of the jet driving the NGC\,2264\,G outflow}}
%% Authors
{\bf{ Paula S. Teixeira$^{1,2,3}$, Carolyn M$^{\textrm c}$Coey$^{4}$, Michael Fich$^{4}$ and Charles J. Lada$^{1}$}}
%% Institutions
$^1$ {Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,} \\
$^2$ {Departamento de F\'{\i}sica da Faculdade de Ci\^encias da \hbox{Universidade} de Lisboa, Ed. C8, Campo Grande, 1749-016, \hbox{Lisboa},Portugal} \\
$^3$ {Laborat\'orio Associado Instituto D. Luiz - SIM, Universidade de Lisboa, Campo Grande, 1749-016, \hbox{Lisboa}, Portugal} \\
$^4$ {Department of Physics \& Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada}
%% Email
{E-mail contact: pteixeira {\em at} cfa.harvard.edu}
%% LATEX COMMANDS
%% Abstract body
{We present new infrared imaging of the NGC\,2264\,G protostellar
outflow region, obtained with the InfraRed Array Camera (IRAC)
on-board the \emph{Spitzer} Space Telescope. A jet in the red outflow
lobe (eastern lobe) is clearly detected in all four IRAC bands and,
for the first time, is shown to continuously extend over the entire
length of the red outflow lobe traced by CO observations. The
redshifted jet also extends to a deeply embedded Class\,0 source,
VLA\,2, confirming previous suggestions that it is the driving source
of the outflow (G\'omez et a. 1994). The images show that the
easternmost part of the redshifted jet exhibits what appear to be
multiple changes of direction. To understand the redshifted jet
morphology we explore several mechanisms that could generate such
apparent changes of direction. From this analysis, we conclude that
the redshifted jet structure and morphology visible in the IRAC images
can be largely, although not entirely, explained by a slowly
precessing jet (period $\approx$ 8000\,yr) that lies mostly on the
plane of the sky. It appears that the observed changes in the
redshifted jet direction may be sufficient to account for a
significant fraction of the broadening of the outflow lobe observed in
the CO emission.}
% Journal
{ Accepted by MNRAS}
%% Preprints URL
http://www.cfa.harvard.edu/~pteixeir/publications.html\\
http://xxx.lanl.gov/abs/0711.1340
\v5
{\large\bf{The formation of star clusters – II. 3D simulations of magnetohydrodynamic turbulence in molecular clouds }}
{\bf{David A. Tilley$^1$ and Ralph E. Pudritz$^{2,3}$ }}
$^1$ {Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA } \\
$^2$ {Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1, Canada } \\
$^3$ {Origins Institute, McMaster University, Hamilton, Ontario, Canada L8S 4M1, Canada }
{E-mail contact: dtilley {\em at} nd.edu }
{We present a series of decaying turbulence simulations that represent a cluster-forming clump within a
molecular cloud, investigating the role of magnetic fields on the formation of potential star-forming
cores. We present an exhaustive analysis of numerical data from these simulations that include a
compilation of all of the distributions of physical properties that characterize bound cores – including
their masses, radii, mean densities, angular momenta, spins, magnetizations and mass-to-flux ratios.
We also present line maps of our models that can be compared with observations. Our simulations range
between 5 and 30 Jeans masses of gas, and are representative of molecular cloud clumps with masses
between 100 and 1000 M$_\odot$. The field strengths in the bound cores that form tend to have the same
ratio of gas pressure to magnetic pressure, $\beta$, as the mean $\beta$ of the simulation. The cores
have mass-to-flux ratios that are generally less than that of the original cloud, and so a cloud that is
initially highly supercritical can produce cores that are slightly supercritical, similar to that seen
by Zeeman measurements of molecular cloud cores. Clouds that are initially only slightly supercritical
will instead collapse along the field lines into sheets, and the cores that form as these sheets fragment
have a different distribution of masses than what is observed. The spin rates of these cores
(wherein 20–40 per cent of cores have $\Omega t_{ff} \ge 0.2$) suggests that subsequent fragmentation into
multiple systems is likely. The sizes of the bound cores that are produced are typically 0.02–0.2 pc and
have densities in the range $10^4 - 10^5$ cm$^{−3}$ in agreement with observational surveys. Finally, our
numerical data allow us to test theoretical models of the mass spectrum of cores, such as the turbulent
fragmentation picture of Padoan \& Nordlund. We find that while this model gets the shape of the
core mass spectrum reasonably well, it fails to predict the peak mass in the core mass spectrum. }
{ Published by Monthly Notices of the Royal Astronomical Society (Vol. 382, p. 73) }
\v5
%%--------SubmissionID=1263----------------
%% Title
{\large\bf{Debris disks around Sun-like stars}}
%% Authors
{\bf{ D. E. Trilling$^{1}$, G. Bryden$^{2}$, C. A. Beichman$^{2}$, G. H. Rieke$^{1}$, K. Y. L. Su$^{1}$, J. A. Stansberry$^{1}$, M. Blaylock$^{1}$, K. R. Stapelfeldt$^{2}$, J. W. Beeman$^{3}$ and E. E. Haller$^{4}$}}
%% Institutions
$^1$ {University of Arizona} \\
$^2$ {JPL} \\
$^3$ {Lawrence Berkeley National Lab.} \\
$^4$ {UC-Berkeley}
%% Email
{E-mail contact: trilling {\em at} as.arizona.edu}
%% LATEX COMMANDS
%% Abstract body
{We have observed nearly 200 FGK stars at 24 and 70 microns with the
Spitzer Space Telescope. We identify excess infrared emission,
including a number of cases where the observed flux is more than 10
times brighter than the predicted photospheric flux, and interpret
these signatures as evidence of debris disks in those systems. We
combine this sample of FGK stars with similar published results to
produce a sample of more than 350 main sequence AFGKM stars. The
incidence of debris disks is 4.2$^{+2.0}_{-1.1}$\% at 24 microns for a
sample of 213 Sun-like (FG) stars and 16.4$^{+2.8}_{-2.9}$\% at 70
microns for 225 Sun-like (FG) stars. We find that the excess rates for
A, F, G, and K stars are statistically indistinguishable, but with a
suggestion of decreasing excess rate toward the later spectral types;
this may be an age effect. The lack of strong trend among FGK stars of
comparable ages is surprising, given the factor of 50 change in
stellar luminosity across this spectral range. We also find that the
incidence of debris disks declines very slowly beyond ages of 1
billion years.}
% Journal
{ Accepted by ApJ}
%% Preprints URL
http://xxx.lanl.gov/abs/0710.5498
\v5
{\large\bf{ Multiple Sources Toward the High-Mass Young Star S140 IRS 1 }}
{\bf{ Miguel A. Trinidad$^1$, Jos\'e M. Torrelles$^2$, Luis F. Rodr\'{\i}guez$^3$ and Salvador Curiel$^4$ }}
$^1$ { Departamento de Astronom\'{\i}a, Universidad de Guanajuato, A.P. 144, Guanajuato, Guanajuato 36240, Mexico } \\
$^2$ { Instituto de Ciencias del Espacio, and Institut d'Estudis Espacials de Catalunya, Facultat de F\'{\i}sica, Planta 7a, Universitat de Barcelona, Avenida Diagonal 647, 08028 Barcelona, Spain} \\
$^3$ { Centro de Radioastronom\'{\i}a y Astrof\'{\i}sica, Universidad Nacional Aut\'onoma de M\'exico, A. P. 3-72, Xangari, 58089 Morelia, Michoac\'an, Mexico } \\
$^4$ { Instituto de Astronom\'{\i}a, Universidad Nacional Aut\'onoma de M\'exico, A.P. 70-264, D.F. 04510, Mexico }
{ S140 IRS 1 is a remarkable source where the radio source at the center of the main bipolar molecular
outflow in the region is elongated perpendicular to the axis of the outflow, an orientation opposite to
that expected if the radio source is a thermal jet exciting the outflow. We present results of 1.3 cm
continuum and H$_2$O maser emission observations made with the Very Large Array in its A configuration
toward this region. In addition, we also present results of continuum observations at 7 mm and reanalyze
observations at 2, 3.5, and 6 cm (previously published). IRS 1A is detected at all wavelengths, showing
an elongated structure. Three water maser spots are detected along the major axis of the radio source
IRS 1A. We have also detected a new continuum source at 3.5 cm (IRS 1C) located $\simeq 0.6^{\prime\prime}$
northeast of IRS 1A. The presence of these two young stellar objects (IRS 1A and 1C) could explain the
existence of the two bipolar molecular outflows observed in the region. In addition, we have also detected three continuum clumps (IRS 1B, 1D, and 1E) located along the major axis of IRS 1A. We discuss two possible models to explain the nature of IRS 1A: a thermal jet and an equatorial wind. }
{ Published by The Astronomical Journal (Vol. 134, p. 1870) }
\v5
{\large\bf{ Formation of Protoplanets from Massive Planetesimals in Binary Systems }}
{\bf{ Yusuke Tsukamoto$^1$ and Junichiro Makino$^2$ }}
$^1$ { Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan } \\
$^2$ { Division of Theoretical Astronomy, National Astronomical Observatory, 2-2-1 Osawa, Mitaka, Tokyo 181-8588, Japan }
{E-mail contact: tukamoto {\em at} margaux.astron.s.u-tokyo.ac.jp }
{ More than half of all stars reside in binary or multiple star systems, and many planets have been found in binary systems. From a theoretical point of view, however, whether or not the planetary formation proceeds in a binary system is a very complex problem, because secular perturbation from the companion star can easily stir up the eccentricity of the planetesimals and cause high-velocity, destructive collisions between planetesimals. Early stages of the planetary formation process in binary systems have been studied by a restricted three-body approach with gas drag, and it is commonly accepted that accretion of planetesimals can proceed due to orbital phasing by gas drag. However, the gas drag becomes less effective as the planetesimals become more massive. Therefore, it is uncertain whether the collision velocity remains small and planetary accretion can proceed once the planetesimals become massive. We performed N-body simulations of planetary formation in binary systems, starting from massive planetesimals of size $\sim$100–500 km. We found that the eccentricity vectors of planetesimals quickly converge to the forced eccentricity due to the coupling of the perturbation of the companion and the mutual interaction of planetesimals, if the initial disk model is sufficiently wide in radial distribution. This convergence decreases the collision velocity, and as a result accretion can proceed much in the same way as in isolated systems. The basic processes of the planetary formation, such as runaway and oligarchic growth and final configuration of the protoplanets, are essentially the same in binary systems and single star systems, at least in the late stage, where the effect of gas drag is small. }
{ Published by The Astrophysical Journal (Vol. 669, p. 1316) }
\v5
{\large\bf{ Evaporation and condensation of spherical interstellar clouds. Self-consistent models with saturated heat conduction and cooling }}
{\bf{ W. Vieser$^{1,2}$ and G. Hensler$^1$ }}
$^1$ { Institute of Astronomy, University of Vienna, Türkenschanzstr. 17, 1180 Vienna, Austria } \\
$^2$ { Christoph-Probst-Gymnasium, Talhofstr. 7, 82205 Gilching, Germany }
{E-mail contact: hensler {\em at} astro.univie.ac.at }
{ \emph{Aims.} The fate of interstellar clouds embedded in a hot tenuous medium depends on whether the clouds suffer from evaporation or whether material condensates onto the clouds. The knowledge of the evaporation or condensation rates of interstellar clouds at rest is therefore of prime importance for their further evolution. Analytic solutions for the rate of evaporative mass loss from an isolated spherical cloud embedded in a hot tenuous gas are deduced by Cowie \& McKee (1977, ApJ, 211, 135). Their approach is limited to the integration of the time-independent energy conservation equation for the heat-conductive interface. Therefore it is crucial to test the validity of the analytical results for more realistic interstellar conditions. This requires that the full hydrodynamical equations must be treated taking the whole cloud into account with a sufficiently large hot-gas reservoir.
\emph{Methods.} By two-dimensional numerical simulations in an Eulerian, explicit hydrodynamical grid the evolution of interstellar clouds with different internal density structures and surrounded by a hot plasma is simulated. Self-gravity, interstellar heating and cooling effects and heat conduction by electrons are added. We use the classical thermal conductivity of a fully ionized hydrogen plasma proposed by Spitzer and a saturated heat flux according to Cowie and McKee in regions where the mean free path of the electrons is large compared to the temperature scaleheight.
\emph{Results.} Using pure hydrodynamics and taking only the classical heat flux into account, we can reproduce the Cowie and McKee analytical results. If we allow for heat flux saturation the evaporation rate is reduced, but in the simulations even to about one order of magnitude below the predicted saturated one. This happens because the saturated heat flux is density dependent and due to the mixing of the two phases, the warm cloud material on one side and the hot intercloud medium, also the density distribution changes drastically there during the simulation. And this cannot be considered in the analytical study. This main result still holds if we add self gravity or choose another cloud density structure while keeping the cloud radius and temperature of the cloud edge constant. As a further issue the evolution changes, however, totally for more realistic conditions when interstellar heating and cooling effects stabilize the self-gravity. The clouds' evaporation then turns into condensation, because the additional energy input due to heat conduction can be transported away from the interface and radiated off very efficiently from the cloud's inner parts.
\emph{Conclusions.} The assumption of pure classical heat conduction is invalid for the description of the evolution of interstellar clouds in a hot tenuous gas. The consideration of a limited saturated heat flux is inevitable for this kind of simulations and leads to a dramatic decrease of the evaporation rate. And even more realisticly with radiative cooling heat conduction leads to condensation in contradiction to analytical predictions which require evaporation. This has two consequences: Interstellar clouds are stabilized against evaporation. On the other hand this provides an efficient way to accrete and mix intercloud material into clouds. }
{ Published by Astronomy \& Astrophysics (Vol. 475, p. 251) }
\v5
{\large\bf{$\rm{NH_3}$ Observations of the Infrared Dark Cloud
G28.34+0.06}}
%% Here comes the author(s) of the paper, please indicate within $^...$
%% the number which corresponds to the institute of each author.
{\bf{Y. Wang$^{1,2}$, Q. Zhang$^{2}$, T. Pillai$^{2,3}$, F. Wyrowski$^3$, Y. Wu$^{1}$}}
%% Here you write your institute name(s) and address(es),
%% the number in $^..$ indicates your author number, for example:
$^1${Astronomy Department and CAS-PKU Joint Beijing
Astrophysics Center, Peking University, Beijing
100871, P.R.China}
$^2$Harvard Smithsonian Center for Astrophysics\\ 60 Garden Street\\
Cambridge, Massachusetts 02138, USA
$^3${Max-Planck-Institut f{\"u}r Radioastronomie, Auf
dem H{\"u}gel 69, D-53121 Bonn, Germany}
%% Here you may write the e-mail address of one or more of the authors
%% who will act as contact person for preprint requests etc, for example:
{E-mail contact: [email protected]}
%% IF YOU USE ANY PERSONAL LATEX COMMANDS IN YOUR ABSTRACT,
%% PLEASE INCLUDE THEIR DEFINITIONS HERE!
\newcommand{\lsim}{${\raisebox{-.9ex}{$\stackrel{\textstyle<}{\sim}$}}$ }
\def\nh3{$\rm{NH_3}$}
\def\NH3{$\rm{NH_3}$}
\def\msolar{M$_\odot$}
\def\msun{M$_\odot$}
\def\lsolar{L$_\odot$}
\def\lsun{L$_\odot$}
\def\kms-1{km~s$^{-1}$}
%% Within the following brackets you place your text:
{We present observations of the \nh3 (J,K) = (1,1) and (2,2)
inversion transitions toward the infrared dark cloud G28.34+0.06,
using the Very Large Array. Strong \nh3 emission is found to
coincide well with the infrared absorption feature in this cloud.
The northern region of G28.34+0.06 is dominated by a compact clump
(P2) with a high rotation temperature (29 K), large line width
(4.3 km s$^{-1}$), and is associated with strong water maser (240
Jy) and a 24 $\mu$m point source with far IR luminosity of $10^3$
\lsun. We infer that P2 has embedded massive protostars although
it lies in the 8 $\mu$m absorption region. The southern region has
filamentary structures. The rotation temperature in the southern
region decreases with the increase of the integrated \nh3
intensity, which indicates an absence of strong internal heating
in these clumps. In addition, the compact core P1 in the south has
small line width (1.2 km s$^{-1}$) surrounded by extended
emissions with larger line width (1.8 km s$^{-1}$), which suggests
a dissipation of turbulence in the dense part of the cloud. Thus,
we suggest that P1 is at a much earlier evolutionary stage than
P2, possibly at a stage that begins to form a cluster with massive stars.
}
% Here you write which journal accepted your paper, for example:
{ Accepted by ApJ Letters }
%% If preprints are available on the WWW you can give the web
%% direction here.
Preprint available at {\tt arXiv:0711.1999 or
http://cfa-www.harvard.edu/$\sim$qzhang/pub.html}
\vspace{3cm}
\begin{center}
{\Large\em Abstracts of recently accepted major reviews}
\end{center}
%%--------SubmissionID=1238----------------
%% Title
{\large\bf{Toward Understanding Massive Star Formation}}
%% Authors
{\bf{ Hans Zinnecker$^{1}$ and Harold W. Yorke$^{2}$}}
%% Institutions
$^1$ {Astrophysikalisches Institut Potsdam,
An der Sternwarte 16, D-14482 Potsdam, Germany} \\
$^2$ {Jet Propulsion Laboratory,
California Institute
of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109 USA}
%% Email
{E-mail contact: hzinnecker {\em at} aip.de}
%% LATEX COMMANDS
%% Abstract body
{Although fundamental for astrophysics, the processes that
produce massive stars are not
well understood. Large distances, high extinction, and short
timescales of critical evolutionary phases
make observations of these processes challenging. Lacking good
observational guidance, theoretical
models have remained controversial. This review offers a basic
description of the collapse of a
massive molecular core and a critical discussion of the three
competing concepts of massive star
formation:
\begin{itemize}
\item monolithic collapse in isolated cores
\item competitive accretion in a protocluster environment
\item stellar collisions and mergers in very dense systems
\end{itemize}
We also review the observed outflows, multiplicity, and
clustering properties of massive stars, the
upper initial mass function and the upper mass limit.
We conclude that high-mass star formation is
not merely a scaled-up version of low-mass star formation
with higher accretion rates, but partly a
mechanism of its own, primarily owing to the role of stellar
mass and radiation pressure in
controlling the dynamics.}
% Journal
{ Accepted by Annual Reviews of Astronomy and Astrophysics 2007, Vol. 45, pp. 481-563}
Preprints URL:
http://www.arxiv.org/abs/0707.1279
or complimentary one-time access through the ARAA website at
http://arjournals.annualreviews.org/eprint/FSuGwrDxwMpnBkXwWyhX/full/10.1146/annurev.astro.44.051905.092549.
\clearpage
\begin{center}
{\Large\em Dissertation Abstracts}
%\end{center}
%% Title
{\Large\bf{Polycyclic Aromatic Hydrocarbons in Disks\\ around Young Solar-type Stars}}
\vspace*{0.3cm}
%% Author
{\bf{ Vincent Geers }}
%% Institution
{Leiden Observatory, Leiden University}
%% Current Address
{P.O. Box 9513, NL-2300RA, Leiden, The Netherlands}
%% New Address
{Address as of 1 November 2007: Dept.\ of Astronomy \& Astrophysics, University of Toronto, 50 St.\ George St., Toronto, Ontario, M5S 3H4, Canada}
%% Email
{Electronic mail: vcgeers {\em at} astro.utoronto.ca}
%% Adviser
{Ph.D dissertation directed by: Prof. dr. Ewine van Dishoeck}
%% Month/Year
{Ph.D degree awarded: October 2007}
\vspace*{0.2cm}
\end{center}
%% LATEX COMMANDS
%% Body
{\vspace{-0.4cm} This thesis presents a study of the dust around
solar-type young stars, in particular focussing on the Polycyclic
Aromatic Hydrocarbons (PAHs). VLT-ISAAC, -VISIR, -NACO and Spitzer
mid-infrared spectroscopy and imaging surveys are presented and
combined with 3D radiative transfer models to constrain the presence
and location of PAH emission toward embedded young stellar objects and
circumstellar disks around young solar-type stars. The following main
questions are addressed. What happens to PAHs in the embedded phase of
a forming star? Are PAHs present in low-mass young star systems? Does
the PAH emission originate from the envelope or from the disk? What do
they tell us about disk structure and evolution and grain growth? What
can we say about the evolution of PAHs during star formation and their
typical size? In Chapter 2, we present a survey with Spitzer of PAH
features in a sample of intermediate and low-mass stars with disks,
and compare the results with model predictions of PAH emission from
flaring disks. In Chapter 3, we present VISIR images and a spectrum
of IRS~48, a young M-type star with very strong PAH features, which
appears to have a 60 AU radius gap in the disk as seen in large grains
at 18.9 $\mathrm{\mu m}$ but with PAHs originating from inside the
gap. In Chapter 4, we present an ISAAC, VISIR and NACO survey of the
spatial extent of PAH features in protoplanetary disks, and compare
with model predictions. In Chapter 5, we present an ISAAC and Spitzer
survey of PAH features toward embedded young stars, and compare the
results with model predictions. The main conclusions of this thesis
can be summarized as follows:
\begin{itemize}
\item PAHs are shown to be present in several T Tauri disks, but at an abundance 10-100 times lower than standard interstellar values. The detection rate of only 11-14\% is small compared to that toward intermediate-mass stars ($\sim$54\%). At our average derived PAH abundance, PAH emission features around stars with $T_{\mathrm{eff}} \leq 4200$ K fall below the Spitzer IRS detection limit. The 11.2 $\mathrm{\mu m}$ PAH feature is most easily detected, with the 7.7 and 8.6 $\mathrm{\mu m}$ bands readily masked by silicate emission.
\item High spatial resolution spectroscopy confirms that the PAH features detected toward young stars are directly associated with the circumstellar disk and not due to the presence of a tenuous envelope.
\item A new class of disks with weak mid-IR continuum emission and very strong PAH features is found. This class represents a small percentage ($\sim$5\%) of the total population of disks surveyed. Among disks around low-mass stars with PAH detections, it represents a large fraction. This is partially due to a detection effect, where the lower disk continuum between 5--15 $\mathrm{\mu m}$ due to absence of dust results in higher feature-to-continuum ratios for the PAH features. These disks are believed to harbour gaps and/or holes with strong PAH emission originating at, or inwards from, the outer edge of the gap. This evidence for separation of small and large grains implies that their populations evolve differently.
\item PAHs are not detected toward the majority ($\geq$ 97\%) of a sample of 80 embedded sources. Comparison with model calculations show that this detection rate is consistent with a PAH abundance at least 20--50x lower than in the ISM. Variability in luminosity, UV excess and/or envelope mass can change this conclusion to a typical factor of 10--20. In these cold dense environments, two possibilities for lowering the abundance of a species are recognized: coagulation or dust growth and freeze-out of the PAHs onto larger grains. Thus, PAHs likely enter the protoplanetary disks frozen out on grains.
\end{itemize}}
%% URL
http://hdl.handle.net/1887/12414
\clearpage
\begin{center}
{\Large\em New Jobs}
\end{center}
\vspace*{0.6cm}
%% Title
{\Large\bf{Postdoctoral Research Position -- Star Formation}}
\vspace*{0.5cm}
%% Body
{Applications are invited for a postdoctoral research position at the
University of Michigan, to start on Sept. 1 2008 or earlier. The successful
candidate will work with Prof. Edwin Bergin on observational
studies of the physics and chemistry of star forming regions.
Applicants should have experience in observational studies
of star-forming cores and/or protoplanetary disks at centimeter, millimeter,
submillimeter and/or infrared wavelengths. In particular, a demonstrated expertise in molecular line observations of dense molecular regions is desired.
Dr. Bergin is the principal investigator of HEXOS - {\em Herschel observations of EXtra-Ordinary Sources: The Orion and Sgr B2 Molecular Clouds} - a Herschel Guaranteed Time Key Program. The HEXOS program consists of velocity revolved full spectral scans from 480--1250 GHz and 1490--1910 GHz of several sources in these two clouds.
The successful applicant will become a member of the KP team and work with Dr. Bergin and team members on the reduction, analysis, and archiving of HEXOS data. She/he may also work on theoretical analyses of these data in concert with Dr. Bergin and other members of the KP team. Herschel is an ESA mission with NASA participation with a launch date that is currently late 2008 with nominal science operations a few months thereafter.
The position is for two years, with extension to a
third year possible, and includes research support.
Applicants with a Ph.D. should
send a curriculum vitae, a description of research
interests, and a list of publications, and should arrange for three
letters of recommendation to be sent directly to the address below.
Please indicate the preferred starting date. Applications received
prior to 20 January 2008 will receive first consideration.
Women and Minorities are encouraged to apply. The University of Michigan is an equal opportunity/affirmative action employer.
Department of Astronomy\\
University of Michigan\\
500 Church St.\\
825 Dennison Building\\
Ann Arbor, MI 48109-1042\\
U.S.A.\\
Email Submission Address: ebergin\_AT\_umich.edu\\
Email Inquiries: ebergin\_AT\_umich.edu\\
Department Web page: http://www.astro.lsa.umich.edu}
\vspace{1.5cm}
%% Title
{\Large\bf{Tenure Track Assistant Professor in Observational Astronomy (Star Formation) ~-~Stony Brook University}}
\vspace*{0.5cm}
%% Body
{The Department of Physics and Astronomy at Stony Brook seeks applications for a tenure track assistant professor in observational astronomy, with a focus on galactic and/or extragalactic star formation.
The ideal candidate will be able to make use of such facilities as
ALMA, Herschel, and JWST. The current interests of the astronomy group can be found at
http://www.astro.sunysb.edu/
The department is especially interested in recruiting highly qualified women and minority candidates at early stages in their careers and is sensitive to the issues of dual careers. Stony Brook faculty and students conduct experimental and theoretical research in a broad range of topics in physics and astronomy both in facilities on our campus and at laboratories around the world. The successful candidate will have a Ph.D. degree in astronomy, physics, or a related field and relevant postdoctoral experience and publications. S/he will be expected to carry out an independent research program and to attract federal grant support for it. S/he will contribute to the teaching activities in the department at both the undergraduate and graduate levels.
The application process will be conducted electronically. Interested
candidates should go to the web page
http://www.physics.sunysb.edu/Physics/recruit/AST-2008/apply.php
Instructions are given there for uploading all application materials as pdf files to a secure web site.
By 31 December 2007 each candidate should have submitted all
application materials electronically, including the names,
institutions, and email addresses for three references. The candidate
should also have arranged for each of these three references to submit
letters electronically. Each candidate will be notified by email when
his/her application file is begun and again when it is complete.
For further information or for alternative submission, please
email pam.burris {\em at} stonybrook.edu.
Equal Opportunity/Affirmative Action Employer.}
\vspace{1.5cm}
%% Title
{\Large\bf{Joint Yale--Universidad de Chile Postdoctoral Position in Star Formation}}
\vspace*{0.5cm}
%% Body
{The Department of Astronomy at the Universidad de Chile and the Yale Astronomy Department invite applications for a joint Yale--U. de Chile post-doctoral research position in observational star formation. The successful applicant will collaborate with Professors Guido Garay, Diego Mardones and Leonardo Bronfman (Universidad de Chile) and with Prof.~H\'ector Arce (Yale University) on studies of star forming regions using millimeter, sub-millimeter and infrared data as well as radiative transfer codes.
The post-doctoral fellow is expected to spend 3/4 of the time in Chile and 1/4 of the time at Yale, thus he or she will be able to have PI status for the 10\% observing time reserved for Chilean astronomers on all astronomical facilities in Chile. The successful applicant will also have direct access to Yale resources including the WIYN telescope, the SMARTS telescopes, and the Palomar-QUEST large-area survey.
Postdoctoral positions are awarded for a two-year period, renewable for a third, and offer competitive salary and benefits, and travel and research funds. Candidates must hold a Ph.D. in astronomy or related field by date of appointment. The selected candidates are expected to start their position no later than July 2008.
Applications consisting of a cover letter, curriculum vitae, publication list, and a brief (2-3 page) description of research interests and accomplishments should arrive by December 21, 2007. Applicants should also arrange for three letters of recommendation to arrive by the same date. Email submission of all materials, including letters, to susan.delong {\em at} yale.edu. Yale University particularly encourages applications from women and members of underrepresented minority groups.
AAE/EOE}
\vspace{1.5cm}
%% Title
{\Large\bf{Postdoctoral Position in Numerical Simulations of \\ Star and/or Planet Formation
~ \\ {\large Astrophysics Group, University of Exeter}}}
\vspace*{0.3cm}
%% Body
{We are looking for a postdoctoral researcher in the fields of star or planet formation, to work with Professor Matthew Bate in the Astrophysics Group. The Group's work focuses on both theoretical and observational star and planet formation. Our research was rated excellent in the last UK Research Assessment Exercise. Exeter is also the coordinating node of the EC-funded Marie Curie Research Training Network, CONSTELLATION, on the Origin of the Initial Mass Function, a consortium of 12 institutions across Europe.
You will have an appropriate first degree and a PhD (or equivalent). You should also have a strong record of publication in astrophysical hydrodynamical, magnetohydrodynamical and/or radiative transfer numerical simulations. You will be expected to make use of the University of Exeter's supercomputer which will be available from January 2008.
The position is a fixed-term, three-year appointment (associate research fellow/research fellow) with a starting salary up to \pounds 30,913 pa on scale \pounds 22,332 to \pounds 35,837 pa (depending on level of appointment), 36.5 hours per week.
The closing date for receipt of applications is 18 January, 2007.
Informal enquiries can be made by e-mailing mbate {\em at} astro.ex.ac.uk. An application form is available on request, and applicants should send this, along with a description of their current research and future plans (3 pages), a brief curriculum vitae including the names and contact details of three references, and a list of refereed publications to Professor Matthew Bate, School of Physics, University of Exeter, EX4 4QL, United Kingdom. Electronic submissions may be made to the above email address.
Equal Opportunities Employer}
\vspace{2cm}
{\Large\bf Star Formation Postdoctoral Fellowship}\\
UNIVERSITY OF FLORIDA\\
Attention: Jonathan Tan\\
Dept. of Astronomy\\
211 Bryant Space Science Center\\
PO Box 112 055\\
Gainesville, FL 32611-2055, USA\\
Tel: 352 392 2052 ext 254\\
Fax: 352 392 5089\\
Email Inquiries: (Jonathan Tan) jt {\em at} astro.ufl.edu\\
URL1: http://www.astro.ufl.edu/starformation.html\\
URL2: http://www.astro.ufl.edu/theory
The Department of Astronomy at the University of Florida (UF) invites applications for a Star Formation Postdoctoral Fellowship. The successful applicant will submit a research proposal to work in one or more of the following areas: local Galactic star and star cluster formation, global star formation activity of disk galaxies, star formation near AGN and the Galactic Center, and star formation in the early universe. Both observational and theoretical applications will be considered. Research proposals with potential for collaborations with the UF star formation and/or theory groups are encouraged.
UF faculty involved in star formation research include Steve Eikenberry, Elizabeth Lada, Jonathan Tan, and Charlie Telesco. Theoretical work includes analytic and numerical calculations of massive star and star cluster formation, giant molecular cloud formation, and galactic scale star formation (Tan). Observational programs involve a wide range of star formation topics and facilities, including GEMINI-FLAMINGOS-2 surveys of embedded star clusters (Lada) and the Galactic Center (Eikenberry), and a variety of Gran-Telescopio Canarias (10m optical/IR) projects.
The position is an annual appointment, renewable for up to three years
based on satisfactory performance, starting in or around Aug. 2008,
with a salary of \$45,000. Applicants should have a recent PhD.,
preferably in a star formation related field. Further information is
available from Jonathan Tan (jt {\em at} astro.ufl.edu).
Application materials (CV, bibliography, statement of research
interests and plans [no more than 3 pages plus 2 pages for figures],
and three letters of reference [candidates are responsible for having
their reference letters sent to UF]) should be mailed to the above
address or emailed to Jonathan Tan (jt @ astro.ufl.edu) by 15th
January 2008.
The University of Florida is an Equal Opportunity Institution.
\vspace{4cm}
{\Large\bf Postdoctoral Research in Extrasolar Planet Detection and/or
Planet Formation Theory}\\
Email submissions/inquiries: eford+postdoc08 {\em at} astro.ufl.edu\\
Postal addreess for submissions:\\
Prof. Eric B. Ford\\
University of Florida\\
211 Bryant Space Science Center\\
P.O. Box 112055\\
Gainesville, FL 32611\\
United States
The University of Florida's Astronomy Department invites applications for a Postdoctoral Research Associate. The successful candidate will collaborate with Eric Ford on research relating to extrasolar planets and/or planet formation. Possible research programs include formation and evolution of planetary systems, statistical analyses of planetary systems and planet searches, and observational techniques for studying planetary systems. The successful applicant will be encouraged to develop new initiatives in-line with their own research interests and to participate in the department's intellectual activities. Closely related research at UF currently includes exoplanet searches, planetary dynamics, star formation, planetary atmospheres, and instrumentation. UF has deployed a large computer cluster and is a partner in the 10.4m Gran Telescopio Canarias Observatory.
Position includes a competitive salary, health benefits, and research funds. The anticipated start date is Fall 2008. The appointment is is renewable annually based on satisfactory performance, needs of the Department and College, and available funding. Ph.D. in relevant field by starting date required.
Interested applicants should submit a CV, publications list, summary statement of research accomplishments, interests and plans, and names and email addresses of three references. Candidates are responsible for ensuring that their references send a letter of recommendation directly. All application materials should be mailed or emailed to Eric Ford ([email protected]). For full consideration, applications should be received no later than January 4, 2008. Women and underrepresented minorities are strongly encouraged to apply.
The University of Florida is an Equal Opportunity Institution
\vspace{1.5cm}
{\Large\bf Postdoctoral Position in Extrasolar Planet Searches}\\
University of Florida \\
Department of Astronomy \\
211 Bryant Space Science Center\\
P.O.Box 112055 \\
Gainesville, FL 32611, USA\\
Email Inquiries: jge {\em at} astro.ufl.edu\\
URL: http://www.astro.ufl.edu/~jge
Applications are invited for a postdoctoral position to work with Prof. Jian Ge on Doppler planet
search projects using new generation multi-object Doppler instruments with 120 object
capability at the Sloan Digital Sky Survey (SDSS) 2.5 meter telescope and single object high
precision (~1m/s) Doppler instruments at KPNO 2.1 meter and Li Jiang 2.4meter telescopes. The
planet survey at the SDSS telescope, called Multi-object APO Radial Velocity Exoplanet Large-
area Survey (MARVELS), is part of the SDSS III survey program in 2008-2014.
Ph.D. in Astronomy or Physics is required by start date. The successful applicant will work on
developing optimized codes for data reduction and analysis, participate in observations, data
reduction and analysis, and publications. There are also opportunities to be involved in other
areas of research such as project development, proposal writing, and new red and near IR
Doppler instrument development, and supervising students. Expertise in IDL data reduction
software development and also stellar astrophysics is preferred. This is a one-year appointment
and is renewable for up to two additional years, contingent upon performance and the
continuation of funding. Salary will be commensurate with qualifications and experience.
To apply, please submit a curriculum vita, a statement of research interests, and have three letters
of reference sent to Professor Jian Ge at the address above. For full consideration, complete
applications should be received by Dec. 15th, 2007.
University of Florida is committed to affirmative action, equal opportunity and the diversity of
its work force.
\vspace{1.5cm}
{\Large\bf Two Postdoctoral Positions in the SDSS-III planet survey}\\
University of Florida \\
Department of Astronomy \\
211 Bryant Space Science Center\\
P.O.Box 112055 \\
Gainesville, FL 32611, USA\\
Email Inquiries: jge {\em at} astro.ufl.edu\\
URL: http://www.astro.ufl.edu/$\sim$jge
Two postdoctoral positions will be available at University of Florida to work on the Multi-object
APO Radial Velocity Exoplanet Large-area Survey (MARVELS) using the Sloan Digital Sky
Survey (SDSS) 2.5-meter telescope as part of the SDSS III survey program in 2008-2014.
MARVELS will use new generation multiple object Doppler instruments with 120 object
capability to search a total of tens of thousands of V=8-12 FGK main sequence, subgiant and
giant stars for detecting and characterizing hundreds of giant planets.
Ph.D. in Astronomy, Physics, or a related field is required by the start date. The successful
applicants will take a lead or get involved in developing the MARVELS data pipeline, selecting
targets, designing survey plates, scheduling observations, processing survey data, archiving and
distributing survey data. Expertise in IDL programming and data processing, and also stellar
astrophysics is preferred. The positions are expected to be 2-year commitments but are subject to
annual renewal. Salary will be commensurate with qualifications and experience.
To apply, please submit a curriculum vita, a statement of research interests, and have three letters
of reference sent to Professor Jian Ge at the address above. For full consideration, complete
applications should be received by Dec. 15th, 2007.
University of Florida is committed to affirmative action, equal opportunity and the diversity of
its work force.
\vspace{1.5cm}
{\Large\bf Postdoctoral Position in Theoretical Star Formation Studies}
The University of Toronto invites applications for a postdoctoral position in star formation studies, to work on the theory or simulation of star formation, protostellar disks or outflows, or the interaction of stars with the interstellar medium.
The start date is flexible, but preference will be given to applicants who can start relatively soon. The initial appointment of up to two years may be renewed for an additional year contingent on performance and funding. Salary and benefits are competitive. The successful applicant will benefit from proximity to the Canadian Inst. for Theoretical Astrophysics, as well as access to local computing resources (including an existing 10 teraflop machine with an anticipated hundredfold upgrade).
Please send a curriculum vitae, a brief description of scientific qualifications and interests (not to exceed three pages), and reqest three letters of recommendation. Electronic submission to Chris Matzner )matzner {\em at} astro.utoronto.ca) is preferred. Applications will be considered as soon as they are received.
\clearpage
\begin{center}
{\Large\em New Books}
\end{center}
\vspace*{0.6cm}
\begin{centering}
{\Large\bf From Suns to Life }
\vspace{0.2cm}
{\large\bf Edited by
M. Gargaud, P. Claeys, P. L\'opez-Garc\'\i a,
H. Martin,\\ T. Montmerle, R. Pascal, J. Reisse}
\vspace{0.1cm}
\end{centering}
\vspace{0.5cm}
This review emerged from several interdisciplinary meetings and schools gathering a group of astronomers, geologists, biologists, and chemists, attempting to share their specialized knowledge around a common question: how did life emerge on Earth? Their ultimate goal was to provide some kind of answer as a prerequisite to an even more demanding question: is life universal? The resulting state-of-the-art articles were written by twenty-five scientists telling a not-so linear story, but on the contrary, highlighting problems, gaps, and controversies. Needless to say, this approach yielded no definitive answers to both questions. However, by adopting a chronological approach to the question of the emergence of life on Earth, the only place where we know for sure that life exists; it was possible to break down this question into several sub-topics that can be addressed by the different disciplines.
The main chapters of this review present the formation and evolution of the solar system; the building of a habitable planet; prebiotic chemistry, biochemistry, and the emergence of life; the environmental context of the early Earth; and the ancient fossil record and early evolution. The concluding chapter provides the highlights of the review and presents the different points of view about the universality of life. Two pedagogical chapters are included; one on chronometers, another in the form of a frieze, which summarizes in graphical form the present state of knowledge about the chronology of the emergence of life on Earth, before the Cambrian explosion. The chapters are
\vspace{0.2cm}
From the Arrow of Time to the Arrow of Life {\em M. Gargaud \& J. Reisse}
Dating Methods and Corresponding Chronometers in Astrobiology {\em M. Gargaud et al.}
Solar System Formation and Early Evolution: the First 100 Million Years {\em T. Montmerle et al.}
Building of a Habitable Planet {\em H. Martin et al.}
Prebiotic Chemistry - Biochemistry - Emergence of Life (4.4-2 Ga) {\em R. Pascal et al.}
Environmental Context {\em H. Martin et al.}
Ancient Fossil Record and Early Evolution (ca. 3.8 to 0.5 Ga) {\em P. L\'opez-Garc\'\i a et al.}
A Synthetic Interdisciplinary ``Chronological Frieze'': An Attempt {\em D. Despois \& M. Gargaud}
Life on Earth... And Elsewhere? {\em T. Montmerle et al. }
\v5
Springer verlag
ISBN 0-387-45082-7, hardbound, 370 pages, 2006\\
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{\Large\em Meetings}
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Announcement of the ESO workshop:
{\Large\bf Star Formation across the Milky Way Galaxy}
{\bf Santiago de Chile, March 3-6, 2008}
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Motivation:
Star-formation in the Milky Way is an ubiquitous phenomenon. It occurs on many different scales and in diverse environments ranging from isolated cores, to small groups and modest associations, up to massive clusters and super star clusters. Our knowledge about the onset, dominant modes and typical outcomes of star formation is, however, in general biased by the limited observational accessibility of star formation sites at their various distances and locations within the galaxy.
Ongoing large scale surveys like GLIMPSE, SCUBA, ATLASGAL and UKIDSS trace gas, dust, and young stellar populations across our galaxy and provide new insight in the galactic distribution of star-forming regions and young clusters, and the spatial and environmental variation of the star formation history, efficiency and the initial mass function down to sub-stellar masses. A revised picture of galactic star-formation is slowly emerging. This is required in order to understand the physics of young stellar objects, and star-formation at large, which are key science topics for future projects like ALMA and E-ELT.
We therefore want to gather an up-to-date and comprehensive view of galactic star-formation by tracing ongoing and recent star-formation across the Milky Way. The workshop aims to link communities that usually focus on specific scales and environments, and we will discuss star-formation activity spatially spanning from the solar neighborhood, nearby star forming regions and OB associations, to spiral arms, to the galactic disk, around the central bar and bulge, towards the galactic center.
Our ultimate goal is to identify similarities, differences and the dominant modes of the star-formation process and its typical outcomes across the Milky Way and beyond.
Topics and Invited Speakers:
The workshop will be grouped around highlight talks that cover progressively the spatial scale, i.e. starting from local star formation towards increasing distances. The spatial coverage of the Milky Way will be complemented by topical sessions that will highlight overarching concepts and observations. Invited Speakers include:
Fred Adams, Michigan, USA
Joao Alves, Granada, Spain
John Bally, Colorado, USA
Nate Bastian, London, UK
Leo Blitz, Berkeley, USA
Bruce Elmegreen, Yorktown Heights, USA
Mark Gieles, ESO/Chile
Preben Grosbol, ESO/Garching
Phil Lucas, Hertfordshire, UK
Piero Madau, Santa Cruz, USA, TBC
Tom Megeath, Toledo, USA
Jorge Melnick, ESO/Chile
Thierry Montmerle, Grenoble, France
Sergei Nayakshin, Leicester, UK
Livia Origlia, Bologna, Italy
Francesco Palla, Florence, Italy
Fred Schuller, Bonn, Germany
Andrea Stolte, Los Angeles, USA
Hans Zinnecker, U Potsdam, Germany
Please see the conference web-site for details and registration information
http://www.sc.eso.org/santiago/science/MilkyWayStarFormation/
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FIRST ANNOUNCEMENT
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{\Large\bf{Astronomical Polarimetry 2008}}\\
{\large\bf Science from Small to Large Telescopes\\
6-11 July 2008
Fairmont Le Manoir Richelieu, La Malbaie, Quebec, Canada}
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We are pleased to announce the convening of a Conference on uv - optical - infrared - mm/submm (OIM) Astronomical Polarimetry, in the wonderful Charlevoix region by the St-Lawrence River in July 2008. It is located about 400 km (4 1/2 hours drive) from Montreal, and 150 km (2 hours drive) from the Quebec airport.
The aim of the Conference is to bring together workers in all areas of
OIM astronomical polarimetry to discuss the most recent results in this
exciting and crucial field, and to consider the potential for polarimetry
with telescopes of all sizes. The meeting will concentrate on
ground-based polarization measurements, and will include a session
devoted to new and novel instrumentation. The remaining sessions
will be organized according to the astronomical source rather than to
wavelength regime or specific technique. Neither Radio polarimetry nor
Solar polarimetry are within the conference remit, but each will be the
subject of review talks which will set the scene for two of the conference
sessions.
If you are interested in attending this meeting, please let us know.
Note that registration pages on the conference web site will open for
business around 7-Jan-2008. *** Emails sent to the conference address
(given at the bottom of this note) will give a useful indication of the
likely interest. Space may be limited, so we would strongly encourage you
to do this at this point.
SCIENCE AREAS
Sessions will be divided into two, with approximately 80 percent of the time
guaranteed for current results and 20 percent for presentations on future
directions, facilities etc. Proceedings, including posters, will be
published. Details of the division between oral and poster presentations
will be given in the second announcement. The following science areas will be covered:
Techniques and Instrumentation \\
Theory and Modelling \\
Solar system \\
Interstellar Dust and Gas \\
Star Formation \\
Circumstellar Disks and Extrasolar Planets \\
Stars and Stellar Magnetism \\
Galaxies, Radio Galaxies and AGN \\
High-redshift and Cosmological Polarimetry
DATES AND DEADLINES
Second Announcement and Web site opens for registration: 7-Jan-2008 \\
Commencement of Registration: 7-Jan-2008 \\
Third Announcement: 1-Apr-2008 \\
End of Early Registration: 23-May-2008 \\
Abstract Deadline: 1-May-2008 \\
Late Registration Deadline: 6-June-2008
SCIENCE ORGANIZING COMMITTEE
Andy Adamson (Joint Astronomy Centre);
Colin Aspin (Institute for Astronomy) ;
Stefano Bagnulo (Armagh Observatory, Northern Ireland);
Pierre Bastien (Universite de Montreal; chair);
Chris Davis (Joint Astronomy Centre);
Martin Giard (Centre d'Etude Spatiale des Rayonnements, Toulouse);
Martin Houde (University of Western Ontario);
Jim Hough (University of Herfordshire);
Anny-Chantal Levasseur-Regourd (Univ. P. and M. Curie, Paris VI);
Nadine Manset (Canada-France Hawaii Telescope);
Franois Mnard (Obs Grenoble);
Motohide Tamura (National Astrophysical Observatory of Japan);
Doug Whittet (Rensselaer Polytechnic Institute)
Conference web site: www.astro.umontreal.ca/astropol2008 ~~~--~~~
Contact email address: pol2008 {\em at} astro.umontreal.ca}
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%% Title
{\Large\bf The Universe under the Microscope - Astrophysics at High Angular Resolution}
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{\large\bf 21-25 April 2008 -- Physikzentrum Bad Honnef, Germany}
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High angular resolution techniques at infrared and centimeter to millimeter wavelengths have become of ever increasing importance for astrophysical research in the past decade. These new techniques will enable us to address issues such as directly measuring the properties of exoplanets, imaging the surfaces of stars, examining stellar dynamics in extremely dense cluster cores, disentangling the processes at the bottom of black hole accretion flows in the jet launching region, or testing general relativity in the strong gravity regime near the event horizon of supermassive black holes. This conference aims at an interdisciplinary approach by bringing together astrophysicists from the three great branches of the field, instrumentation, observation, and theory, to discuss the current state of research and the possibilities offered by the next-generation instruments.
Topics:
High mass star formation
Optical and infrared interferometry
Physics of jets and accretion flows
Discs around young stars
Sub-millimeter interferometry/VLBI
Supermassive black holes
The center of the Milky Way
The central 100 pcs of galaxies
Detailed information can be found on: {\large\bf https://www.ph1.uni-koeln.de/AHAR08}
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{The Star Formation Newsletter is a vehicle for fast distribution of
information of interest for astronomers working on star formation
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{\bf Latex macros for submitting abstracts and dissertation abstracts
(by e-mail to [email protected]) are appended to each issue of
the newsletter. You can also submit via the Newsletter web interface
at http://www2.ifa.hawaii.edu/star-formation/index.cfm }. \\
The Star Formation Newsletter is available on the World Wide Web at
http://www.ifa.hawaii.edu/users/reipurth or at
http://www.eso.org/gen-fac/pubs/starform/ .
}}}
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https://theanarchistlibrary.org/library/petr-kropotkin-advice-to-those-about-to-emigrate.tex | theanarchistlibrary.org | CC-MAIN-2021-21 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-21/segments/1620243990929.24/warc/CC-MAIN-20210512131604-20210512161604-00008.warc.gz | 576,025,658 | 8,433 | \documentclass[DIV=12,%
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\title{Advice to Those About to Emigrate}
\date{1893}
\author{Pëtr Kropotkin}
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\section{Editor’s Preface}
Peter Kropotkin (1842–1921) was one of the greatest anarchist theoreticians of his time. Although he admired the directly democratic and non-authoritarian practices of the traditional peasant village commune, he was never an advocate of small and isolated communal experimentalism. Many people, upon reading his works, have been inspired to found such communities, both in his own time as well as the hippies of the 1960s (a period when Kropotkin’s major works were epublished and influential). Kropotkin did not consider such ventures were likely to be successful or useful in achieving wider revolutionary goals. His friend, Elisee Reclus, who had been involved in such a venture in South America in his youth, was even more hostile to small communal experiments. It is a pity that some of the founders of the many hippy communes in the 1960s (nearly all of which faded rather quickly) did not read Kropotkin more carefully. Unfortunately, they made the same mistakes as many anarchists, communists and socialists had made a century before them. In the anarchist press today one still finds adverts for prospective small and isolated anarchist colonies. Also, many commentaries about Kropotkin still misrepresent him as having had a vision of society consisting of unfederated and independent village-like settlements and of advocating small communal experiments as a means of achieving an anarchist society. The following speech and two ‘open’ letters, which have not been in print for a century, clearly show, that although not emotionally opposed to such ventures, he was highly sceptical about their chances of success and generally believed them to be a drain upon the energies of the anarchist movement. Despite his warnings, these articles also contain much good and practical advice to those who are still tempted to found small experimental communes in the wilderness, or perhaps, those tempted in some future era to colonise space.
\begin{quote}
\emph{Graham Purchase}
\end{quote}
\clearpage
\section{Advice to Those About to Emigrate}
\emph{In these days when Home Colonisation is seriously discussed, and is even tried, in England as an outlet for the populations of our congested towns, the following letters will be of much interest to our readers. A comrade in New South Wales, writing to Kropotkin for suggestions and advice, says:}
\begin{quote}
“As you are probably aware, the Labour movement in Australia has advanced tremendously during the last four or five years. The reason, I believe, lies in the increased agitation in the minds of the people through the late strikes here and also in England and America. The Labor Party here got the worst of it in the last three big strikes, yet the importance of those strikes as factors in educating people’s minds cannot be overlooked — eg. direct results of defeat of the Maritime Strike were the formation of Labor Electoral Leagues all over New South Wales, and the sending in of thirty-four — Labor members into Parliament: result of last year’s Shearer’s Strike in Queensland has been the beginning of the New Australia movement about which I write.
The New Australia Movement is a proposal to all healthy and intelligent men and women to leave Australia and to go to a certain part of South America, there to establish Co-operative Settlements on Socialistic principles. The idea of this movement originated with Mr Lane, editor of the best Socialistic labor paper in Queensland. Three agents of the Association are at present in Argentina (S.A.), prospecting there for the best land for the settlement, and they have already found a site for it on the banks of the River Niger. In Australia we have five or six agents, Mr. Lane included, organising groups in different parts of the country, and the result has been better than we expected. We have already from five to six hundred members, and the first batch of settlers sails for Argentina some time in January.
It may seem strange that while thousands of men are emigrating annually from different parts of the world to Australia — the so-called working man’s Paradise — men should be found in Australia willing to leave behind the country which they have helped to raise up into a nation and to go to a foreign country which, perhaps, is no better than Australia. But this is not a case of “It’s better where we are not.” There is more than one reason why it would be better to establish the settlement in Argentina but I will cite only one: Capitalistic opposition would be too strong here in Australia. Capital is organised here stronger than ever it was before; it rules the Governments here. Again the motto of Socialists is “the world is my country, and we are going to act up to it. We’ll have no distinctions either in nationalities or in religions. All men are welcome — provided they are physically and morally healthy, and not afraid to work or to think.”
\end{quote}
\emph{To which Kropotkin replies:}
The fact that men and women, who have made Australia what it is, are compelled to migrate from it, speaks volumes in itself. “Make the land, be the dung which renders it productive, build the centres of civilisation which render it valuable — and go away!” That is the true picture of modern capitalist management. The same here, the same at the antipodes — always the same!
Every time I see men and women of energy, enterprise and initiative, starting similar colonies, I feel sorry You know how much Russia has lost of her best elements, those that had the capacity of being dissatisfied and of revolting against bad conditions, because she had at her very doors Siberia, whereto the lovers of freedom could go and escape for a few years all the curses of the State — military service, bureaucracy, functionaries and their despotism.
What would become of the European revolutionary movement if most women and men of strong individuality — most of those ready to rebel — went to settle in distant lands, trying to make colonies there? Is there not work enough in each land for every one who wishes to work for the modification of the atrocious conditions of the present time? Are there not at hand enough opportunities for exercising the spirit of Solidarity which inspires the Communist? Do we not want here, in every great and small city, that communist spirit put into practice and radiating from small groups, however limited in extent, so as to make it permeate the whole society?
The longer we all live, the more we see that the very limited communist solidarity which is practised among all revolutionary, and especially all Anarchist groups exercises a much more powerful effect than if it were practised, even to its full extent, somewhere on the boundaries of the civilised world! Remember the change produced in all Russian society by Nihilism. Compare the manners, the habits of life at the time of Turgenev’s “On the Eve” with present manners and habits. Not to mention that, besides the propaganda by example, which is carried on more or less here by all who have broken with old forms, there is going on hand in hand a propaganda of general socialist principles, Socialist agitation, and Socialistic enlightening of the masses; and this is what prepares the way for Communism on a grand scale in the cities of the civilised world itself.
Besides, when I recollect the numerous colonies, which have been started over the last 50 years, and the number of men and women, some of whom I knew personally, whose unflinching energies and perseverance I cannot but admire, and yet see the failures on record, I cannot but think that there is some great cause at work against such colonies.
These causes I imagine to be two, and I recommend them to your most careful consideration: First, the colonies are usually not numerous enough. If you are a small family, united by bonds of common education and thousands of family bonds, you may succeed. If you are more than that, you must be numerous: 2000 souls will succeed better than 200, on account of the variety there would be of characters, aptitudes, inclinations. The individual and the individual’s personality more easily disappear in a group of 2000 than in a group of 200 or 20. It is extremely difficult to keep 50 or 100 persons in continuous full agreement. For 2000 or 10,000 this is \textbf{not} required. They only need agree as to some advantageous methods of common work, and are free otherwise to live in their own way.
The second difficulty is this: Peasants no doubt succeed in founding such colonies because, in their mother country, the conditions are so bad that, after 2 or 3 years of very hard work, they feel better off than before. Their colonies only disintegrate when they (through some special conditions) fall from bad to worse.
But most Communistic colonies are composed chiefly of men who are put, in the colony; into worse material conditions that their previous ones. However bad the present conditions, the worker in a civilised country, \textbf{if he is permitted by the exploiters to work}, and if he is an average worker, has certain conditions of life, which in most cases he does not find in the colony, where 5, 10, often more, years he has to fight against the most crushing difficulties.
In the colony he works hard, and has none of the trifles which civilisation gives, and which we all like so much, and he has no prospect of having them. He also feels less personal liberty in his actions — it is always the case in small communities — and he is deprived of the higher stimuli which he has in his mother country — even of the struggle in a large arena which every active nature likes.
That is why, I have long since thought, that if I were one of those who start colonies, I should never go into the wilderness. A Communist Colony? Well, the best spot for it is near London or near Paris! And even if it started without, or with very little, capital or land, I am persuaded that the privations one would have to impose upon himself to make such a colony thrive in a London suburb would be much less than the privations one must endure to make a colony thrive in Argentina
I have read a good deal about the first steps of colonists in America, both in records and private letters; I saw many colonists on the fertile plains of the middle Amur in Siberia, so I have some idea what these privations are, and I am firmly persuaded that if 20 of 200 persons had endured like privations in starting a Communist farm near London — they would be prosperous now.
Of course the chief thing in such a case would be not to undertake agriculture in the way it was practiced 2000 years ago, but the agriculture which is required now ie., gardening and most intensive culture, combined with \textbf{handicraft}.
When I saw at Harrow (NE London suburb) what is obtained from a horrid, heavy clay by intensive labor a Labor which is still a plaything in comparison with the labor a colony has to face in unbroken countries — I always thought that if I were a born “colonist” I should try to colonise here, not in South America.
Reasoned, intensive gardening to grow all sorts of vegetables (and perhaps to attempt intensive culture of wheat) — guided by the experience of real gardeners and in accordance with the advice readily received from neighbours; that alone might give nearly the whole of the colony’s food, and pay the rent, as well as permit the concern to increase gradually — even if one half of the colony’s adults were compelled to work all the year round in a factory (or, still better, one half of the year only), to earn the necessary money; while the other half obtained from the land, by intensive culture, all that is required for living. And such a colony close to a big city would have the advantage of not cutting itself off from the civilised world; it would be part of it, and would enjoy some of its joys, which are so attractive for one who has a taste for learning or art. A lecture, good music, a good library would be within reach of the colonist, not to say that be would remain in contact with the Communists who carry on the active work of propaganda and agitation amidst the old world; he might even join in whenever he liked.
I am persuaded that if a Communist colony can live together in our present society it can only live near a big city. But, even in its best, it will only be a refuge for those who have abandoned the battle, which has to be fought — face to face with the enemy\dots{}I need not tell you that, if the colony is to have any chance of success, it ought to have no directors, no superintendents, no balloting, no voting whatsoever These, and the intrigues they give rise to, have always been the stumbling blocks of the colonies. Are the new settlers less intelligent, less capable than a Russian village \emph{mir} that goes to settle in Siberia? The Russian peasants live without authority, agree at their meetings for common work, and are intelligent enough not to have authorities or ballots, and to arrive at unanimity in their decisions. Are the Australians inferior to them in any way that they need rulers?
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Pëtr Kropotkin
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1893
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Retrieved on February 26\textsuperscript{th}, 2009 from \href{http://flag.blackened.net/daver/anarchism/kropotkin/emmadvice.html}{flag.blackened.net}
Freedom: March 1893 p.14, Reprinted in Small Communal Experiments and Why They Fail, published by Jura Books
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Project Gutenberg's Tom Swift and his Submarine Boat, by Victor Appleton
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.net
Title: Tom Swift and his Submarine Boat
or, Under the Ocean for Sunken Treasure
Author: Victor Appleton
Posting Date: July 13, 2008 [EBook \#949]
Release Date: June, 1997
[Last updated on June 6, 2013]
Language: English
Character set encoding: ASCII
*** START OF THIS PROJECT GUTENBERG EBOOK TOM SWIFT AND HIS SUBMARINE BOAT ***
Produced by Anthony Matonac
TOM SWIFT AND HIS SUBMARINE BOAT
or
Under the Ocean for Sunken Treasure
by
VICTOR APPLETON
CONTENTS
I News of a Treasure Wreck
II Finishing the Submarine
III Mr. Berg Is Astonished
IV Tom Is Imprisoned
V Mr. Berg Is Suspicious
VI Turning the Tables
VII Mr. Damon Will Go
VIII Another Treasure Expedition
IX Captain Weston's Advent
X Trial of the Submarine
XI On the Ocean Bed
XII For a Breath of Air
XIII Off for the Treasure
XIV In the Diving Suits
XV At the Tropical Island
XVI ``We'll Race You For It!''
XVII The Race
XVIII The Electric Gun
XIX Captured
XX Doomed to Death
XXI The Escape
XXII At the Wreck
XXIII Attacked by Sharks
XXIV Ramming the Wreck
XXV Home with the Gold
TOM SWIFT AND HIS SUBMARINE BOAT
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.
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\gutchapter{Chapter One}
News of a Treasure Wreck
There was a rushing, whizzing, throbbing noise in the air. A great body, like that of some immense bird, sailed along, casting a grotesque shadow on the ground below. An elderly man, who was seated on the porch of a large house, started to his feet in alarm.
``Gracious goodness! What was that, Mrs. Baggert?'' he called to a motherly-looking woman who stood in the doorway. ``What happened?''
``Nothing much, Mr. Swift,'' was the calm reply ``I think that was Tom and Mr. Sharp in their airship, that's all. I didn't see it, but the noise sounded like that of the Red Cloud.''
``Of course! To be sure!'' exclaimed Mr. Barton Swift, the well-known inventor, as he started down the path in order to get a good view of the air, unobstructed by the trees. ``Yes, there they are,'' he added. ``That's the airship, but I didn't expect them back so soon. They must have made good time from Shopton. I wonder if anything can be the matter that they hurried so?''
He gazed aloft toward where a queerly-shaped machine was circling about nearly five hundred feet in the air, for the craft, after swooping down close to the house, had ascended and was now hovering just above the line of breakers that marked the New Jersey seacoast, where Mr. Swift had taken up a temporary residence.
``Don't begin worrying, Mr. Swift,'' advised Mrs. Baggert, the housekeeper. ``You've got too much to do, if you get that new boat done, to worry.''
``That's so. I must not worry. But I wish Tom and Mr. Sharp would land, for I want to talk to them.''
As if the occupants of the airship had heard the words of the aged inventor, they headed their craft toward earth. The combined aeroplane and dirigible balloon, a most wonderful traveler of the air, swung around, and then, with the deflection rudders slanted downward, came on with a rush. When near the landing place, just at the side of the house, the motor was stopped, and the gas, with a hissing noise, rushed into the red aluminum container. This immediately made the ship more buoyant and it landed almost as gently as a feather.
No sooner had the wheels which formed the lower part of the craft touched the ground than there leaped from the cabin of the Red Cloud a young man.
``Well, dad!'' he exclaimed. ``Here we are again, safe and sound. Made a record, too. Touched ninety miles an hour at times---didn't we, Mr. Sharp?''
``That's what,'' agreed a tall, thin, dark-complexioned man, who followed Tom Swift more leisurely in his exit from the cabin. Mr. Sharp, a veteran aeronaut, stopped to fasten guy ropes from the airship to strong stakes driven into the ground.
``And we'd have done better, only we struck a hard wind against us about two miles up in the air, which delayed us,'' went on Tom. ``Did you hear us coming, dad?''
``Yes, and it startled him,'' put in Mrs. Baggert. ``I guess he wasn't expecting you.''
``Oh, well, I shouldn't have been so alarmed, only I was thinking deeply about a certain change I am going to make in the submarine, Tom. I was day-dreaming, I think, when your ship whizzed through the air. But tell me, did you find everything all right at Shopton? No signs of any of those scoundrels of the Happy Harry gang having been around?'' and Mr. Swift looked anxiously at his son.
``Not a sign, dad,'' replied Tom quickly. ``Everything was all right. We brought the things you wanted. They're in the airship. Oh, but it was a fine trip. I'd like to take another right out to sea.''
``Not now, Tom,'' said his father. ``I want you to help me. And I need Mr. Sharp's help, too. Get the things out of the car, and we'll go to the shop.''
``First I think we'd better put the airship away,'' advised Mr. Sharp. ``I don't just like the looks of the weather, and, besides, if we leave the ship exposed we'll be sure to have a crowd around sooner or later, and we don't want that.''
``No, indeed,'' remarked the aged inventor hastily. ``I don't want people prying around the submarine shed. By all means put the airship away, and then come into the shop.''
In spite of its great size the aeroplane was easily wheeled along by Tom and Mr. Sharp, for the gas in the container made it so buoyant that it barely touched the earth. A little more of the powerful vapor and the Red Cloud would have risen by itself. In a few minutes the wonderful craft, of which my readers have been told in detail in a previous volume, was safely housed in a large tent, which was securely fastened.
Mr. Sharp and Tom, carrying some bundles which they had taken from the car, or cabin, of the craft, went toward a large shed, which adjoined the house that Mr. Swift had hired for the season at the seashore. They found the lad's father standing before a great shape, which loomed up dimly in the semi-darkness of the building. It was like an immense cylinder, pointed at either end, and here and there were openings, covered with thick glass, like immense, bulging eyes. From the number of tools and machinery all about the place, and from the appearance of the great cylinder itself, it was easy to see that it was only partly completed.
``Well, how goes it, dad?'' asked the youth, as he deposited his bundle on a bench. ``Do you think you can make it work?''
``I think so, Tom. The positive and negative plates are giving me considerable trouble, though. But I guess we can solve the problem. Did you bring me the galvanometer?''
``Yes, and all the other things,'' and the young inventor proceeded to take the articles from the bundles he carried.
Mr. Swift looked them over carefully, while Tom walked about examining the submarine, for such was the queer craft that was contained in the shed. He noted that some progress had been made on it since he had left the seacoast several days before to make a trip to Shopton, in New York State, where the Swift home was located, after some tools and apparatus that his father wanted to obtain from his workshop there.
``You and Mr. Jackson have put on several new plates,'' observed the lad after a pause.
``Yes,'' admitted his father. ``Garret and I weren't idle, were we, Garret?'' and he nodded to the aged engineer, who had been in his employ for many years.
``No; and I guess we'll soon have her in the water, Tom, now that you and Mr. Sharp are here to help us,'' replied Garret Jackson.
``We ought to have Mr. Damon here to bless the submarine and his liver and collar buttons a few times,'' put in Mr. Sharp, who brought in another bundle. He referred to an eccentric individual who had recently made an airship voyage with himself and Tom, Mr. Damon's peculiarity being to use continually such expressions as: ``Bless my soul! Bless my liver!''
``Well, I'll be glad when we can make a trial trip,'' went on Tom. ``I've traveled pretty fast on land with my motorcycle, and we certainly have hummed through the air. Now I want to see how it feels to scoot along under water.''
``Well, if everything goes well we'll be in position to make a trial trip inside of a month,'' remarked the aged inventor. ``Look here, Mr. Sharp, I made a change in the steering gear, which I'd like you and Tom to consider.''
The three walked around to the rear of the odd-looking structure, if an object shaped like a cigar can be said to have a front and rear, and the inventor, his son, and the aeronaut were soon deep in a discussion of the technicalities connected with under-water navigation.
A little later they went into the house, in response to a summons from the supper bell, vigorously rung by Mrs. Baggert. She was not fond of waiting with meals, and even the most serious problem of mechanics was, in her estimation, as nothing compared with having the soup get cold, or the possibility of not having the meat done to a turn.
The meal was interspersed with remarks about the recent airship flight of Tom and Mr. Sharp, and discussions about the new submarine. This talk went on even after the table was cleared off and the three had adjourned to the sitting-room. There Mr. Swift brought out pencil and paper, and soon he and Mr. Sharp were engrossed in calculating the pressure per square inch of sea water at a depth of three miles.
``Do you intend to go as deep as that?'' asked Tom, looking up from a paper he was reading.
``Possibly,'' replied his father; and his son resumed his perusal of the sheet.
``Now,'' went on the inventor to the aeronaut, ``I have another plan. In addition to the positive and negative plates which will form our motive power, I am going to install forward and aft propellers, to use in case of accident.''
``I say, dad! Did you see this?'' suddenly exclaimed Tom, getting up from his chair, and holding his finger on a certain place in the page of the paper.
``Did I see what?'' asked Mr. Swift.
``Why, this account of the sinking of the treasure ship.''
``Treasure ship? No. Where?''
``Listen,'' went on Tom. ``I'll read it: 'Further advices from Montevideo, Uruguay, South America, state that all hope has been given up of recovering the steamship Boldero, which foundered and went down off that coast in the recent gale. Not only has all hope been abandoned of raising the vessel, but it is feared that no part of the three hundred thousand dollars in gold bullion which she carried will ever be recovered. Expert divers who were taken to the scene of the wreck state that the depth of water, and the many currents existing there, due to a submerged shoal, preclude any possibility of getting at the hull. The bullion, it is believed, was to have been used to further the interests of a certain revolutionary faction, but it seems likely that they will have to look elsewhere for the sinews of war. Besides the bullion the ship also carried several cases of rifles, it is stated, and other valuable cargo. The crew and what few passengers the Boldero carried were, contrary to the first reports, all saved by taking to the boats. It appears that some of the ship's plates were sprung by the stress in which she labored in a storm, and she filled and sank gradually.' There! what do you think of that, dad?'' cried Tom as he finished.
``What do I think of it? Why, I think it's too bad for the revolutionists, Tom, of course.''
``No; I mean about the treasure being still on board the ship. What about that?''
``Well, it's likely to stay there, if the divers can't get at it. Now, Mr. Sharp, about the propellers---''
``Wait, dad!'' cried Tom earnestly.
``Why, Tom, what's the matter?'' asked Mr. Swift in some surprise.
``How soon before we can finish our submarine?'' went on Tom, not answering the question.
``About a month. Why?''
``Why? Dad, why can't we have a try for that treasure? It ought to be comparatively easy to find that sunken ship off the coast of Uruguay. In our submarine we can get close up to it, and in the new diving suits you invented we can get at that gold bullion. Three hundred thousand dollars! Think of it, dad! Three hundred thousand dollars! We could easily claim all of it, since the owners have abandoned it, but we would be satisfied with half. Let's hurry up, finish the submarine, and have a try for it.''
``But, Tom, you forget that I am to enter my new ship in the trials for the prize offered by the United States Government.''
``How much is the prize if you win it?'' asked Tom.
``Fifty thousand dollars.''
``Well, here's a chance to make three times that much at least, and maybe more. Dad, let the Government prize go, and try for the treasure. Will you?''
Tom looked eagerly at his father, his eyes shining with anticipation. Mr. Swift was not a quick thinker, but the idea his son had proposed made an impression on him. He reached out his hand for the paper in which the young inventor had seen the account of the sunken treasure. Slowly he read it through. Then he passed it to Mr. Sharp.
``What do you think of it?'' he asked of the aeronaut.
``There's a possibility,'' remarked the balloonist ``We might try for it. We can easily go three miles down, and it doesn't lie as deeply as that, if this account is true. Yes, we might try for it. But we'd have to omit the Government contests.''
``Will you, dad?'' asked Tom again.
Mr. Swift considered a moment longer.
``Yes, Tom, I will,'' he finally decided. ``Going after the treasure will be likely to afford us a better test of the submarine than would any Government tests. We'll try to locate the sunken Boldero.''
``Hurrah!'' cried the lad, taking the paper from Mr. Sharp and waving it in the air. ``That's the stuff! Now for a search for the submarine treasure!''
\gutchapter{Chapter Two}
Finishing the Submarine
``What's the matter?'' cried Mrs. Baggert, the housekeeper, hurrying in from the kitchen, where she was washing the dishes. ``Have you seen some of those scoundrels who robbed you, Mr. Swift? If you have, the police down here ought to---''
``No, it's nothing like that,'' explained Mr. Swift. ``Tom has merely discovered in the paper an account of a sunken treasure ship, and he wants us to go after it, down under the ocean.''
``Oh, dear! Some more of Captain Kidd's hidden hoard, I suppose?'' ventured the housekeeper. ``Don't you bother with it, Mr. Swift. I had a cousin once, and he got set in the notion that he knew where that pirate's treasure was. He spent all the money he had and all he could borrow digging for it, and he never found a penny. Don't waste your time on such foolishness. It's bad enough to be building airships and submarines without going after treasure.'' Mrs. Baggert spoke with the freedom of an old friend rather than a hired housekeeper, but she had been in the family ever since Tom's mother died, when he was a baby, and she had many privileges.
``Oh, this isn't any of Kidd's treasure,'' Tom assured her. ``If we get it, Mrs. Baggert, I'll buy you a diamond ring.''
``Humph!'' she exclaimed, as Tom began to hug her in boyish fashion. ``I guess I'll have to buy all the diamond rings I want, if I have to depend on your treasure for them,'' and she went back to the kitchen.
``Well,'' went on Mr. Swift after a pause, ``if we are going into the treasure-hunting business, Tom, we'll have to get right to work. In the first place, we must find out more about this ship, and just where it was sunk.''
``I can do that part,'' said Mr. Sharp. ``I know some sea captains, and they can put me on the track of locating the exact spot. In fact, it might not be a bad idea to take an expert navigator with us. I can manage in the air all right, but I confess that working out a location under water is beyond me.''
``Yes, an old sea captain wouldn't be a bad idea, by any means,'' conceded Mr. Swift. ``Well, if you'll attend to that detail, Mr. Sharp, Tom, Mr. Jackson and I will finish the submarine. Most of the work is done, however, and it only remains to install the engine and motors. Now, in regard to the negative and positive electric plates, I'd like your opinion, Tom.''
For Tom Swift was an inventor, second in ability only to his father, and his advice was often sought by his parent on matters of electrical construction, for the lad had made a specialty of that branch of science.
While father and son were deep in a discussion of the apparatus of the submarine, there will be an opportunity to make the reader a little better acquainted with them. Those of you who have read the previous volumes of this series do not need to be told who Tom Swift is. Others, however, may be glad to have a proper introduction to him.
Tom Swift lived with his father, Barton Swift, in the village of Shopton, New York. The Swift home was on the outskirts of the town, and the large house was surrounded by a number of machine shops, in which father and son, aided by Garret Jackson, the engineer, did their experimental and constructive work. Their house was not far from Lake Carlopa, a fairly large body of water, on which Tom often speeded his motor-boat.
In the first volume of this series, entitled ``Tom Swift and His Motor-Cycle,'' it was told how he became acquainted with Mr. Wakefield Damon, who suffered an accident while riding one of the speedy machines. The accident disgusted Mr. Damon with motor-cycles, and Tom secured it for a low price. He had many adventures on it, chief among which was being knocked senseless and robbed of a valuable patent model belonging to his father, which he was taking to Albany. The attack was committed by a gang known as the Happy Harry gang, who were acting at the instigation of a syndicate of rich men, who wanted to secure control of a certain patent turbine engine which Mr. Swift had invented.
Tom set out in pursuit of the thieves, after recovering from their attack, and had a strenuous time before he located them.
In the second volume, entitled ``Tom Swift and His Motor-Boat,'' there was related our hero's adventures in a fine craft which was recovered from the thieves and sold at auction. There was a mystery connected with the boat, and for a long time Tom could not solve it. He was aided, however, by his chum, Ned Newton, who worked in the Shopton Bank, and also by Mr. Damon and Eradicate Sampson, an aged colored whitewasher, who formed quite an attachment for Tom.
In his motor-boat Tom had more than one race with Andy Foger, a rich lad of Shopton, who was a sort of bully. He had red hair and squinty eyes, and was as mean in character as he was in looks. He and his cronies, Sam Snedecker and Pete Bailey, made trouble for Tom, chiefly because Tom managed to beat Andy twice in boat races.
It was while in his motor-boat, Arrow, that Tom formed the acquaintance of John Sharp, a veteran balloonist. While coming down Lake Carlopa on the way to the Swift home, which had been entered by thieves, Tom, his father and Ned Newton, saw a balloon on fire over the lake. Hanging from a trapeze on it was Mr. Sharp, who had made an ascension from a fair ground. By hard work on the part of Tom and his friends the aeronaut was saved, and took up his residence with the Swifts.
His advent was most auspicious, for Tom and his father were then engaged in perfecting an airship, and Mr. Sharp was able to lend them his skill, so that the craft was soon constructed.
In the third volume, called ``Tom Swift and His Airship,'' there was set down the doings of the young inventor, Mr. Sharp and Mr. Damon on a trip above the clouds. They undertook it merely for pleasure, but they encountered considerable danger, before they completed it, for they nearly fell into a blazing forest once, and were later fired at by a crowd of excited people. This last act was to effect their capture, for they were taken for a gang of bank robbers, and this was due directly to Andy Foger.
The morning after Tom and his friends started on their trip in the air, the Shopton Bank was found to have been looted of seventy-five thousand dollars. Andy Foger at once told the police that Tom Swift had taken the money, and when asked how he knew this, he said he had seen Tom hanging around the bank the night before the vault was burst open, and that the young inventor had some burglar tools in his possession. Warrants were at once sworn out for Tom and Mr. Damon, who was also accused of being one of the robbers, and a reward of five thousand dollars was offered.
Tom, Mr. Damon and Mr. Sharp sailed on, all unaware of this, and unable to account for being fired upon, until they accidentally read in the paper an account of their supposed misdeeds. They lost no time in starting back home, and on the way got on the track of the real bank robbers, who were members of the Happy Harry gang.
How the robbers were captured in an exciting raid, how Tom recovered most of the stolen money, and how he gave Andy Foger a deserved thrashing for giving a false clue was told of, and there was an account of a race in which the Red Cloud (as the airship was called) took part, as well as details of how Tom and his friends secured the reward, which Andy Foger hoped to collect.
Those of you who care to know how the Red Cloud was constructed, and how she behaved in the air, even during accidents and when struck by lightning, may learn by reading the third volume, for the airship was one of the most successful ever constructed.
When the craft was finished, and the navigators were ready to start on their first long trip, Mr. Swift was asked to go with them. He declined, but would not tell why, until Tom, pressing him for an answer, learned that his father was planning a submarine boat, which he hoped to enter in some trials for Government prizes. Mr. Swift remained at home to work on this submarine, while his son and Mr. Sharp were sailing above the clouds.
On their return, however, and after the bank mystery had been cleared up, Tom and Mr. Sharp, aided Mr. Swift in completing the submarine, until, when the present story opens, it needed but little additional work to make the craft ready for the water.
Of course it had to be built near the sea, as it would have been impossible to transport it overland from Shopton. So, before the keel was laid, Mr. Swift rented a large cottage at a seaside place on the New Jersey coast and there, after, erecting a large shed, the work on the Advance, as the under-water ship was called, was begun.
It was soon to be launched in a large creek that extended in from the ocean and had plenty of water at high tide. Tom and Mr. Sharp made several trips back and forth from Shopton in their airship, to see that all was safe at home and occasionally to get needed tools and supplies from the shops, for not all the apparatus could be moved from Shopton to the coast.
It was when returning from one of these trips that Tom brought with him the paper containing an account of the wreck of the Boldero and the sinking of the treasure she carried.
Until late that night the three fortune-hunters discussed various matters.
``We'll hurry work on the ship,'' said Mr. Swift at length. ``Tom, I wonder if your friend, Mr. Damon, would care to try how it seems under water? He stood the air trip fairly well.''
``I'll write and ask him,'' answered the lad. ``I'm sure he'll go.''
Securing, a few days later, the assistance of two mechanics, whom he knew he could trust, for as yet the construction of the Advance was a secret, Mr. Swift prepared to rush work on the submarine, and for the next three weeks there were busy times in the shed next to the seaside cottage. So busy, in fact, were Tom and Mr. Sharp, that they only found opportunity for one trip in the airship, and that was to get some supplies from the shops at home.
``Well,'' remarked Mr. Swift one night, at the close of a hard day's work, ``another week will see our craft completed. Then we will put it in the water and see how it floats, and whether it submerges as I hope it does. But come on, Tom. I want to lock up. I'm very tired to-night.''
``All right, dad,'' answered the young inventor coming from the darkened rear of the shop. ``I just want to---''
He paused suddenly, and appeared to be listening. Then he moved softly back to where he had come from.
``What's the matter?'' asked his father in a whisper. ``What's up, Tom?''
The lad did not answer Mr. Swift, with a worried look on his face, followed his son. Mr. Sharp stood in the door of the shop.
``I thought I heard some one moving around back here,'' went on Tom quietly.
``Some one in this shop!'' exclaimed the aged inventor excitedly. ``Some one trying to steal my ideas again! Mr. Sharp, come here! Bring that rifle! We'll teach these scoundrels a lesson!''
Tom quickly darted back to the extreme rear of the building. There was a scuffle, and the next minute Tom cried out:
``What are you doing here?''
``Ha! I beg your pardon,'' replied a voice. ``I am looking for Mr. Barton Swift.''
``My father,'' remarked Tom. ``But that's a queer place to look for him. He's up front. Father, here's a man who wishes to see you,'' he called.
``Yes, I strolled in, and seeing no one about I went to the rear of the place,'' the voice went on. ``I hope I haven't transgressed.''
``We were busy on the other side of the shop, I guess,'' replied Tom, and he looked suspiciously at the man who emerged from the darkness into the light from a window. ``I beg your pardon for grabbing you the way I did,'' went on the lad, ``but I thought you were one of a gang of men we've been having trouble with.''
``Oh, that's all right,'' continued the man easily. ``I know Mr. Swift, and I think he will remember me. Ah, Mr. Swift, how do you do?'' he added quickly, catching sight of Tom's father, who, with Mr. Sharp, was coming to meet the lad.
``Addison Berg!'' exclaimed the aged inventor as he saw the man's face more plainly. ``What are you doing here?''
``I came to see you,'' replied the man. ``May I have a talk with you privately?''
``I---I suppose so,'' assented Mr. Swift nervously. ``Come into the house.''
Mr. Berg left Tom's side and advanced to where Mr. Swift was standing. Together the two emerged from the now fast darkening shop and went toward the house.
``Who is he?'' asked Mr. Sharp of the young inventor in a whisper.
``I don't know,'' replied the lad; ``but, whoever he is, dad seems afraid of him. I'm going to keep my eyes open.''
\gutchapter{Chapter Three}
Mr. Berg is Astonished
Following his father and the stranger whom the aged inventor had addressed as Mr. Berg, Tom and Mr. Sharp entered the house, the lad having first made sure that Garret Jackson was on guard in the shop that contained the submarine.
``Now,'' said Mr. Swift to the newcomer, ``I am at your service. What is it you wish?''
``In the first place, let me apologize for having startled you and your friends,'' began the man. ``I had no idea of sneaking into your workshop, but I had just arrived here, and seeing the doors open I went in. I heard no one about, and I wandered to the back of the place. There I happened to stumble over a board---''
``And I heard you,'' interrupted Tom.
``Is this one of your employees?'' asked Mr. Berg in rather frigid tones.
``That is my son,'' replied Mr. Swift.
``Oh, I beg your pardon.'' The man's manner changed quickly. ``Well, I guess you did hear me, young man. I didn't intend to bark my shins the way I did, either. You must have taken me for a burglar or a sneak thief.''
``I have been very much bothered by a gang of unscrupulous men,'' said Mr. Swift, ``and I suppose Tom thought it was some of them sneaking around again.''
``That's what I did,'' added the lad. ``I wasn't going to have any one steal the secret of the submarine if I could help it.''
``Quite right! Quite right!'' exclaimed Mr. Berg. ``But my purpose was an open one. As you know, Mr. Swift, I represent the firm of Bentley \& Eagert, builders of submarine boats and torpedoes. They heard that you were constructing a craft to take part in the competitive prize tests of the United States Government, and they asked me to come and see you to learn when your ship would be ready. Ours is completed, but we recognize that it will be for the best interests of all concerned if there are a number of contestants, and my firm did not want to send in their entry until they knew that you were about finished with your ship. How about it? Are you ready to compete?''
``Yes,'' said Mr. Swift slowly. ``We are about ready. My craft needs a few finishing touches, and then it will be ready to launch.''
``Then we may expect a good contest on your part,'' suggested Mr. Berg.
``Well,'' began the aged inventor, ``I don't know about that.''
``What's that?'' exclaimed Mr. Berg.
``I said I wasn't quite sure that we would compete,'' went on Mr. Swift. ``You see, when I first got this idea for a submarine boat I had it in mind to try for the Government prize of fifty thousand dollars.''
``That's what we want, too,'' interrupted Mr. Berg with a smile.
``But,'' went on Tom's father, ``since then certain matters have come up, and I think, on the whole, that we'll not compete for the prize after all.''
``Not compete for the prize?'' almost shouted the agent for Bentley \& Eagert. ``Why, the idea! You ought to compete. It is good for the trade. We think we have a very fine craft, and probably we would beat you in the tests, but---''
``I wouldn't be too sure of that,'' put in Tom. ``You have only seen the outside of our boat. The inside is better yet.''
``Ah, I have no doubt of that,'' spoke Mr. Berg, ``but we have been at the business longer than you have, and have had more experience. Still we welcome competition. But I am very much surprised that you are not going to compete for the prize, Mr. Swift. Very much surprised, indeed! You see, I came down from Philadelphia to arrange so that we could both enter our ships at the same time. I understand there is another firm of submarine boat builders who are going to try for the prize, and I want to arrange a date that will be satisfactory to all. I am greatly astonished that you are not going to compete.''
``Well, we were going to,'' said Mr. Swift, ``only we have changed our minds, that's all. My son and I have other plans.''
``May I ask what they are?'' questioned Mr. Berg.
``You may,'' exclaimed Tom quickly; ``but I don't believe we can tell you. They're a secret,'' he added more cordially.
``Oh, I see,'' retorted Mr. Berg. ``Well, of course I don't wish to penetrate any of your secrets, but I hoped we could contest together for the Government prize. It is worth trying for I assure you---fifty thousand dollars. Besides, there is the possibility of selling a number of submarines to the United States. It's a fine prize.''
``But the one we are after is a bigger one,'' cried Tom impetuously, and the moment he had spoken he wished he could recall the words.
``Eh? What's that?'' exclaimed Mr. Berg. ``You don't mean to say another government has offered a larger prize? If I had known that I would not have let my firm enter into the competition for the bonus offered by the United States. Please tell me.''
``I'm sorry,'' went on Tom more soberly. ``I shouldn't have spoken. Mr. Berg, the plans of my father and myself are such that we can't reveal them now. We are going to try for a prize, but not in competition with you. It's an entirely different matter.''
``Well, I guess you'll find that the firm of Bentley \& Eagert are capable of trying for any prizes that are offered,'' boasted the agent. ``We may be competitors yet.''
``I don't believe so,'' replied Mr. Swift.
``We may,'' repeated Mr. Berg. ``And if we do, please remember that we will show no mercy. Our boats are the best.''
``And may the best boat win,'' interjected Mr. Sharp. ``That's all we ask. A fair field and no favors.''
``Of course,'' spoke the agent coldly. ``Is this another son of yours?'' he asked.
``No but a good friend,'' replied the aged inventor. ``No, Mr. Berg, we won't compete this time. You may tell your firm so.''
``Very good,'' was the other's stiff reply. ``Then I will bid you good night. We shall carry off the Government prize, but permit me to add that I am very much astonished, very much indeed, that you do not try for the prize. From what I have seen of your submarine you have a very good one, almost as good, in some respects, as ours. I bid you good night,'' and with a bow the man left the room and hurried away from the house.
\gutchapter{Chapter Four}
Tom is Imprisoned
``Well, I must say he's a cool one,'' remarked Tom, as the echoes of Mr. Berg's steps died away. ``The idea of thinking his boat better than ours! I don't like that man, dad. I'm suspicious of him. Do you think he came here to steal some of our ideas?''
``No, I hardly believe so, my son. But how did you discover him?''
``Just as you saw, dad. I heard a noise and went back there to investigate. I found him sneaking around, looking at the electric propeller plates. I went to grab him just as he stumbled over a board. At first I thought it was one of the old gang. I'm almost sure he was trying to discover something.''
``No, Tom. The firm he works for are good business men, and they would not countenance anything like that. They are heartless competitors, however, and if they saw a legitimate chance to get ahead of me and take advantage, they would do it. But they would not sneak in to steal my ideas. I feel sure of that. Besides, they have a certain type of submarine which they think is the best ever invented, and they would hardly change at this late day. They feel sure of winning the Government prize, and I'm just as glad we're not going to have a contest.''
``Do you think our boat is better than theirs?''
``Much better, in many respects.''
``I don't like that man Berg, though,'' went on Tom.
``Nor do I,'' added his father. ``There is something strange about him. He was very anxious that I should compete. Probably he thought his firm's boat would go so far ahead of ours that they would get an extra bonus. But I'm glad he didn't see our new method of propulsion. That is the principal improvement in the Advance over other types of submarines. Well, another week and we will be ready for the test.''
``Have you known Mr. Berg long, dad?''
``Not very. I met him in Washington when I was in the patent office. He was taking out papers on a submarine for his firm at the same time I got mine for the Advance. It is rather curious that he should come all the way here from Philadelphia, merely to see if I was going to compete. There is something strange about it, something that I can't understand.''
The time was to come when Mr. Swift and his son were to get at the bottom of Mr. Berg's reasons, and they learned to their sorrow that he had penetrated some of their secrets.
Before going to bed that night Tom and Mr. Sharp paid a visit to the shed where the submarine was resting on the ways, ready for launching. They found Mr. Jackson on guard and the engineer said that no one had been around. Nor was anything found disturbed.
``It certainly is a great machine,'' remarked the lad as he looked up at the cigar-shaped bulk towering over his head. ``Dad has outdone himself this trip.''
``It looks all right,'' commented Mr. Sharp. ``Whether it will work is another question.''
``Yes, we can't tell until it's in the water,'' conceded Tom. ``But I hope it does. Dad has spent much time and money on it.''
The Advance was, as her name indicated, much in advance of previous submarines. There was not so much difference in outward construction as there was in the means of propulsion and in the manner in which the interior and the machinery were arranged.
The submarine planned by Mr. Swift and Tom jointly, and constructed by them, with the aid of Mr. Sharp and Mr. Jackson, was shaped like a Cigar, over one hundred feet long and twenty feet in diameter at the thickest part. It was divided into many compartments, all water-tight, so that if one or even three were flooded the ship would still be useable.
Buoyancy was provided for by having several tanks for the introduction of compressed air, and there was an emergency arrangement so that a collapsible aluminum container could be distended and filled with a powerful gas. This was to be used if, by any means, the ship was disabled on the bottom of the ocean. The container could be expanded and filled, and would send the Advance to the surface.
Another peculiar feature was that the engine-room, dynamos and other apparatus were all contained amidships. This gave stability to the craft, and also enabled the same engine to operate both shafts and propellers, as well as both the negative forward electrical plates, and the positive rear ones.
These plates were a new idea in submarine construction, and were the outcome of an idea of Mr. Swift, with some suggestions from his son.
The aged inventor did not want to depend on the usual screw propellers for his craft, nor did he want to use a jet of compressed air, shooting out from a rear tube, nor yet a jet of water, by means of which the creature called the squid shoots himself along. Mr. Swift planned to send the Advance along under water by means of electricity.
Certain peculiar plates were built at the forward and aft blunt noses of the submarine. Into the forward plate a negative charge of electricity was sent, and into the one at the rear a positive charge, just as one end of a horseshoe magnet is positive and will repel the north end of a compass needle, while the other pole of a magnet is negative and will attract it. In electricity like repels like, while negative and positive have a mutual attraction for each other.
Mr. Swift figured out that if he could send a powerful current of negative electricity into the forward plate it would pull the boat along, for water is a good conductor of electricity, while if a positive charge was sent into the rear plate it would serve to push the submarine along, and he would thus get a pulling and pushing motion, just as a forward and aft propeller works on some ferry boats.
But the inventor did not depend on these plates alone. There were auxiliary forward and aft propellers of the regular type, so that if the electrical plates did not work, or got out of order, the screws would serve to send the Advance along.
There was much machinery in the submarine. There were gasolene motors, since space was too cramped to allow the carrying of coal for boilers. There were dynamos, motors and powerful pumps. Some of these were for air, and some for water. To sink the submarine below the surface large tanks were filled with water. To insure a more sudden descent, deflecting rudders were also used, similar to those on an airship. There were also special air pumps, and one for the powerful gas, which was manufactured on board.
Forward from the engine-room was a cabin, where meals could be served, and where the travelers could remain in the daytime. There was also a small cooking galley, or kitchen, there. Back of the engine-room were the sleeping quarters and the storerooms. The submarine was steered from the forward compartment, and here were also levers, wheels and valves that controlled all the machinery, while a number of dials showed in which direction they were going, how deep they were, and at what speed they were moving, as well as what the ocean pressure was.
On top, forward, was a small conning, or observation tower, with auxiliary and steering and controlling apparatus there. This was to be used when the ship was moving along on the surface of the ocean, or merely with the deck awash. There was a small flat deck surrounding the conning tower and this was available when the craft was on the surface.
There was provision made for leaving the ship when it was on the bed of the ocean. When it was desired to do this the occupants put on diving suits, which were provided with portable oxygen tanks. Then they entered a chamber into which water was admitted until it was equal in pressure to that outside. Then a steel door was opened, and they could step out. To re-enter the ship the operation was reversed. This was not a new feature. In fact, many submarines to-day use it.
At certain places there were thick bull's-eye windows, by means of which the under-water travelers could look out into the ocean through which they were moving. As a defense against the attacks of submarine monsters there was a steel, pointed ram, like a big harpoon. There were also a bow and a stern electrical gun, of which more will be told later.
In addition to ample sleeping accommodations, there were many conveniences aboard the Advance. Plenty of fresh water could be carried, and there was an apparatus for distilling more from the sea water that surrounded the travelers. Compressed air was carried in large tanks, and oxygen could be made as needed. In short, nothing that could add to the comfort or safety of the travelers had been omitted. There was a powerful crane and windlass, which had been installed when Mr. Swift thought his boat might be bought by the Government. This was to be used for raising wrecks or recovering objects from the bottom of the ocean. Ample stores and provisions were to be carried and, once the travelers were shut up in the Advance, they could exist for a month below the surface, providing no accident occurred.
All these things Tom and Mr. Sharp thought of as they looked over the ship before turning in for the night. The craft was made immensely strong to withstand powerful pressure at the bottom of the ocean. The submarine could penetrate to a depth of about three miles. Below that it was dangerous to go, as the awful force would crush the plates, powerful as they were.
``Well, we'll rush things to-morrow and the next day,'' observed Tom as he prepared to leave the building. ``Then we'll soon see if it works.''
For the next week there were busy times in the shop near the ocean. Great secrecy was maintained, and though curiosity seekers did stroll along now and then, they received little satisfaction. At first Mr. Swift thought that the visit of Mr. Berg would have unpleasant results, for he feared that the agent would talk about the craft, of which he had so unexpectedly gotten a sight. But nothing seemed to follow from his chance inspection, and it was forgotten.
It was one evening, about a week later, that Tom was alone in the shop. The two mechanics that had been hired to help out in the rush had been let go, and the ship needed but a few adjustments to make it ready for the sea.
``I think I'll just take another look at the water tank valves,'' said Tom to himself as he prepared to enter the big compartments which received the water ballast. ``I want to be sure they work properly and quickly. We've got to depend on them to make us sink when we want to, and, what's more important, to rise to the surface in a hurry. I've got time enough to look them over before dad and Mr. Sharp get back.''
Tom entered the starboard tank by means of an emergency sliding door between the big compartments and the main part of the ship. This was closed by a worm and screw gear, and once the ship was in the water would seldom be used.
The young inventor proceeded with his task, carefully inspecting the valves by the light of a lantern he carried. The apparatus seemed to be all right, and Tom was about to leave when a peculiar noise attracted his attention. It was the sound of metal scraping on metal, and the lad's quick and well-trained ear told him it was somewhere about the ship.
He turned to leave the tank, but as he wheeled around his light flashed on a solid wall of steel back of him. The emergency outlet had been closed! He was a prisoner in the water compartment, and he knew, from past experience, that shout as he would, his voice could not be heard ten feet away. His father and Mr. Sharp, as he was aware, had gone to a nearby city for some tools, and Mr. Jackson, the engineer, was temporarily away. Mrs. Baggert, in the house, could not hear his cries.
``I'm locked in!'' cried Tom aloud. ``The worm gear must have shut of itself. But I don't see how that could be. I've got to get out mighty soon, though, or I'll smother. This tank is airtight, and it won't take me long to breath up all the oxygen there is here. I must get that slide open.''
He sought to grasp the steel plate that closed the emergency opening. His fingers slipped over the smooth, polished surface. He was hermetically sealed up---a captive! Blankly he set his lantern down and leaned hopelessly against the wall of the tank.
``I've got to get out,'' he murmured.
As if in answer to him he heard a voice on the outside, crying:
``There, Tom Swift! I guess I've gotten even with you now! Maybe next time you won't take a reward away from me, and lick me into the bargain. I've got you shut up good and tight, and you'll stay there until I get ready to let you out.''
``Andy Foger!'' gasped Tom. ``Andy Foger sneaked in here and turned the gear. But how did he get to this part of the coast? Andy Foger, you let me out!'' shouted the young inventor; and as Andy's mocking laugh came to him faintly through the steel sides of the submarine, the imprisoned lad beat desperately with his hands on the smooth sides of the tank, vainly wondering how his enemy had discovered him.
\gutchapter{Chapter Five}
Mr. Berg is Suspicious
Not for long did the young inventor endeavor to break his way out of the water-ballast tank by striking the heavy sides of it. Tom realized that this was worse than useless. He listened intently, but could hear nothing. Even the retreating footsteps of Andy Foger were inaudible.
``This certainly is a pickle!'' exclaimed Tom aloud. ``I can't understand how he ever got here. He must have traced us after we went to Shopton in the airship the last time. Then he sneaked in here. Probably he saw me enter, but how could he know enough to work the worm gear and close the door? Andy has had some experience with machinery, though, and one of the vaults in the bank where his father is a director closed just like this tank. That's very likely how he learned about it. But I've got to do something else besides thinking of that sneak, Andy. I've got to get out of here. Let's see if I can work the gear from inside.''
Before he started, almost, Tom knew that it would be impossible. The tank was made to close from the interior of the submarine, and the heavy door, built to withstand the pressure of tons of water, could not be forced except by the proper means.
``No use trying that,'' concluded the lad, after a tiring attempt to force back the sliding door with his hands. ``I've got to call for help.''
He shouted until the vibrations in the confined space made his ears ring, and the mere exertion of raising his voice to the highest pitch made his heart beat quickly. Yet there came no response. He hardly expected that there would be any, for with his father and Mr. Sharp away, the engineer absent on an errand, and Mrs. Baggert in the house some distance off, there was no one to hear his calls for help, even if they had been capable of penetrating farther than the extent of the shed, where the under-water craft had been constructed.
``I've got to wait until some of them come out here,'' thought Tom. ``They'll be sure to release me and make a search. Then it will be easy enough to call to them and tell them where I am, once they are inside the shed. But---'' He paused, for a horrible fear came over him. ``Suppose they should come---too late?'' The tank was airtight. There was enough air in it to last for some time, but, sooner or later, it would no longer support life. Already, Tom thought, it seemed oppressive, though probably that was his imagination.
``I must get out!'' he repeated frantically. ``I'll die in here soon.''
Again he tried to shove back the steel door. Then he repeated his cries until he was weary. No one answered him. He fancied once he could hear footsteps in the shed, and thought, perhaps, it was Andy, come back to gloat over him. Then Tom knew the red-haired coward would not dare venture back. We must do Andy the justice to say that he never realized that he was endangering Tom's life. The bully had no idea the tank was airtight when he closed it. He had seen Tom enter and a sudden whim came to him to revenge himself.
But that did not help the young inventor any. There was no doubt about it now---the air was becoming close. Tom had been imprisoned nearly two hours, and as he was a healthy, strong lad, he required plenty of oxygen. There was certainly less than there had been in the tank. His head began to buzz, and there was a ringing in his ears.
Once more he fell upon his knees, and his fingers sought the small projections of the gear on the inside of the door. He could no more budge the mechanism than a child could open a burglar-proof vault.
``It's no use,'' he moaned, and he sprawled at full length on the floor of the tank, for there the air was purer. As he did so his fingers touched something. He started as they closed around the handle of a big monkey wrench. It was one he had brought into the place with him. Imbued with new hope be struck a match and lighted his lantern, which he had allowed to go out as it burned up too much of the oxygen. By the gleam of it he looked to see if there were any bolts or nuts he could loosen with the wrench, in order to slide the door back. It needed but a glance to show him the futility of this.
``It's no go,'' he murmured, and he let the wrench fall to the floor. There was a ringing, clanging sound, and as it smote his ears Tom sprang up with an exclamation.
``That's the thing!'' he cried. ``I wonder I didn't think of it before. I can signal for help by pounding on the sides of the tank with the wrench. The blows will carry a good deal farther than my voice would.'' Every one knows how far the noise of a boiler shop, with hammers falling on steel plates, can be heard; much farther than can a human voice.
Tom began a lusty tattoo on the metal sides of the tank. At first he merely rattled out blow after blow, and then, as another thought came to him, he adopted a certain plan. Some time previous, when he and Mr. Sharp had planned their trip in the air, the two had adopted a code of signals. As it was difficult in a high wind to shout from one end of the airship to the other, the young inventor would sometimes pound on the pipe which ran from the pilot house of the Red Cloud to the engine-room. By a combination of numbers, simple messages could be conveyed. The code included a call for help. Forty-seven was the number, but there had never been any occasion to use it.
Tom remembered this now. At once he ceased his indiscriminate hammering, and began to beat out regularly---one, two, three, four---then a pause, and seven blows would be given. Over and over again he rang out this number---forty seven---the call for help.
``If Mr. Sharp only comes back he will hear that, even in the house,'' thought poor Tom ``Maybe Garret or Mrs. Baggert will hear it, too, but they won't know what it means. They'll think I'm just working on the submarine.''
It seemed several hours to Tom that he pounded out that cry for aid, but, as he afterward learned, it was only a little over an hour. Signal after signal he sent vibrating from the steel sides of the tank. When one arm tired he would use the other. He grew weary, his head was aching, and there was a ringing in his ears; a ringing that seemed as if ten thousand bells were jangling out their peals, and he could barely distinguish his own pounding.
Signal after signal he sounded. It was becoming like a dream to him, when suddenly, as he paused for a rest, he heard his name called faintly, as if far away.
``Tom! Tom! Where are you?''
It was the voice of Mr. Sharp. Then followed the tones of the aged inventor.
``My poor boy! Tom, are you still alive?''
``Yes, dad! In the starboard tank!'' the lad gasped out, and then he lost his senses. When he revived he was lying on a pile of bagging in the submarine shop, and his father and the aeronaut were bending over him.
``Are you all right, Tom?'' asked Mr. Swift.
``Yes---I---I guess so,'' was the hesitating answer. ``Yes,'' the lad added, as the fresh air cleared his head. ``I'll be all right pretty soon. Have you seen Andy Foger?''
``Did he shut you in there?'' demanded Mr. Swift.
Tom nodded.
``I'll have him arrested!'' declared Mr. Swift. ``I'll go to town as soon as you're in good shape again and notify the police.''
``No, don't,'' pleaded Tom. ``I'll take care of Andy myself. I don't really believe he knew how serious it was. I'll settle with him later, though.''
``Well, it came mighty near being serious,'' remarked Mr. Sharp grimly. ``Your father and I came back a little sooner than we expected, and as soon as I got near the house I heard your signal. I knew what it was in a moment. There were Mrs. Baggert and Garret talking away, and when I asked them why they didn't answer your call they said they thought you were merely tinkering with the machinery. But I knew better. It's the first time we ever had a use for `forty-seven,' Tom.''
``And I hope it will be the last,'' replied the young inventor with a faint smile. ``But I'd like to know what Andy Foger is doing in this neighborhood.''
Tom was soon himself again and able to go to the house, where he found Mrs. Baggert brewing a big basin of catnip tea, under the impression that it would in some way be good for him. She could not forgive herself for not having answered his signal, and as for Mr. Jackson, he had started for a doctor as soon as he learned that Tom was shut up in the tank. The services of the medical man were canceled by telephone, as there was no need for him, and the engineer came back to the house.
Tom was fully himself the next day, and aided his father and Mr. Sharp in putting the finishing touches to the Advance. It was found that some alteration was required in the auxiliary propellers, and this, much to the regret of the young inventor, would necessitate postponing the trial a few days.
``But we'll have her in the water next Friday,'' promised Mr. Swift.
``Aren't you superstitious about Friday?'' asked the balloonist.
``Not a bit of it,'' replied the aged inventor. ``Tom,'' he added, ``I wish you would go in the house and get me the roll of blueprints you'll find on my desk.''
As the lad neared the cottage he saw, standing in front of the place, a small automobile. A man had just descended from it, and it needed but a glance to show that he was Mr. Addison Berg.
``Ah, good morning, Mr. Swift,'' greeted Mr. Berg. ``I wish to see your father, but as I don't wish to lay myself open to suspicions by entering the shop, perhaps you will ask him to step here.''
``Certainly,'' answered the lad, wondering why the agent had returned. Getting the blueprints, and asking Mr. Berg to sit down on the porch, Tom delivered the message.
``You come back with me, Tom,'' said his father. ``I want you to be a witness to what he says. I'm not going to get into trouble with these people.''
Mr. Berg came to the point at once.
``Mr. Swift,'' he said, ``I wish you would reconsider your determination not to enter the Government trials. I'd like to see you compete. So would my firm.''
``There is no use going over that again,'' replied the aged inventor. ``I have another object in view now than trying for the Government prize. What it is I can't say, but it may develop in time---if we are successful,'' and he looked at his son, smiling the while.
Mr. Berg tried to argue, but it was of no avail. Then he changed his manner, and said:
``Well, since you won't, you won't, I suppose. I'll go back and report to my firm. Have you anything special to do this morning?'' he went on to Tom.
``Well, I can always find something to keep me busy,'' replied the lad, ``but as for anything special---''
``I thought perhaps you'd like to go for a trip in my auto,'' interrupted Mr. Berg. ``I had asked a young man who is stopping at the same hotel where I am to accompany me, but he has unexpectedly left, and I don't like to go alone. His name was---let me see. I have a wretched memory for names, but it was something like Roger or Moger.''
``Foger!'' cried Tom. ``Was it Andy Foger?''
``Yes, that was it. Why, do you know him?'' asked Mr. Berg in some surprise.
``I should say so,'' replied Tom. ``He was the cause of what might have resulted in something serious for me,'' and the lad explained about being imprisoned in the tank.
``You don't tell me!'' cried Mr. Berg. ``I had no idea he was that kind of a lad. You see, his father is one of the directors of the firm by whom I am employed. Andy came from home to spend a few weeks at the seaside, and stopped at the same hotel that I did. He went off yesterday afternoon, and I haven't seen him since, though he promised to go for a ride with me. He must have come over here and entered your shop unobserved. I remember now he asked me where the submarine was being built that was going to compete with our firm's, and I told him. I didn't think he was that kind of a lad. Well, since he's probably gone back home, perhaps you will come for a ride with me, Tom.''
``I'm afraid I can't go, thank you,'' answered the lad. ``We are very busy getting our submarine in shape for a trial. But I can imagine why Andy left so hurriedly. He probably learned that a doctor had been summoned for me, though, as it happened, I didn't need one. But Andy probably got frightened at what he had done, and left. I'll make him more sorry, when I meet him.''
``Don't blame you a bit,'' commented Mr. Berg. ``Well, I must be getting back.''
He hastened out to his auto, while Tom and his father watched the agent.
``Tom, never trust that man,'' advised the aged inventor solemnly.
``Just what I was about to remark,'' said his son. ``Well, let's get back to work. Queer that he should come here again, and it's queer about Andy Foger.''
Father and son returned to the machine shop, while Mr. Berg puffed away in his auto. A little later, Tom having occasion to go to a building near the boundary line of the cottage property which his father had hired for the season, saw, through the hedge that bordered it, an automobile standing in the road. A second glance showed him that it was Mr. Berg's machine. Something had gone wrong with it, and the agent had alighted to make an adjustment.
The young inventor was close to the man, though the latter was unaware of his presence.
``Hang it all!'' Tom heard Mr. Berg exclaim to himself. ``I wonder what they can be up to? They won't enter the Government contests, and they won't say why. I believe they're up to some game, and I've got to find out what it is. I wonder if I couldn't use this Foger chap?''
``He seems to have it in for this Tom Swift,'' Mr. Berg went on, still talking to himself, though not so low but that Tom could hear him. ``I think I'll try it. I'll get Andy Foger to sneak around and find out what the game is. He'll do it, I know.''
By this time the auto was in working order again, and the agent took his seat and started off.
``So that's how matters lie, eh?'' thought Tom. ``Well, Mr. Berg, we'll be doubly on the lookout for you after this. As for Andy Foger, I think I'll make him wish he'd never locked me in that tank. So you expect to find out our `game,' eh, Mr. Berg? Well, when you do know it, I think it will astonish you. I only hope you don't learn what it is until we get at that sunken treasure, though.''
But alas for Tom's hopes. Mr. Berg did learn of the object of the treasure-seekers, and sought to defeat them, as we shall learn as our story proceeds.
\gutchapter{Chapter Six}
Turning the Tables
When the young inventor informed his father what he had overheard Mr. Berg saying, the aged inventor was not as much worried as his son anticipated.
``All we'll have to do, Tom,'' he said, ``is to keep quiet about where we are going. Once we have the Advance afloat, and try her out, we can start on our voyage for the South American Coast and search for the sunken treasure. When we begin our voyage under water I defy any one to tell where we are going, or what our plans are. No, I don't believe we need worry about Mr. Berg, though he probably means mischief.''
``Well, I'm going to keep my eyes open for him and Andy Foger,'' declared Tom.
The days that followed were filled with work. Not only were there many unexpected things to do about the submarine, but Mr. Sharp was kept busy making inquiries about the sunken treasure ship. These inquiries had to be made carefully, as the adventurers did not want their plans talked of, and nothing circulates more quickly than rumors of an expedition after treasure of any kind.
``What about the old sea captain you were going to get to go with us?'' asked Mr. Swift of the balloonist one afternoon. ``Have you succeeded in finding one yet?''
``Yes; I am in communication with a man I think will be just the person for us. His name is Captain Alden Weston, and he has sailed all over the world. He has also taken part in more than one revolution, and, in fact, is a soldier of fortune. I do not know him personally, but a friend of mine knows him, and says he will serve us faithfully. I have written to him, and he will be here in a few days.''
``That's good. Now about the location of the wreck itself. Have you been able to learn any more details?''
``Well, not many. You see, the Boldero was abandoned in a storm, and the captain did not take very careful observations. As nearly as it can be figured out the treasure ship went to the bottom in latitude forty-five degrees south, and longitude twenty-seven east from Washington. That's a pretty indefinite location, but I hope, once we get off the Uruguay coast, we can better it. We can anchor or lay outside the harbor, and in the small boat we carry go ashore and possibly gain more details. For it was at Montevideo that the shipwrecked passengers and sailors landed.''
``Does Captain Weston know our object?'' inquired Tom.
``No, and I don't propose to tell him until we are ready to start,'' replied Mr. Sharp. ``I don't know just how he'll consider a submarine trip after treasure, but if I spring it on him suddenly he's less likely to back out. Oh, I think he'll go.''
Somewhat unexpectedly the next day it was discovered that certain tools and appliances were needed for the submarine, and they had been left in the house at Shopton, where Eradicate Sampson was in charge as caretaker during the absence of Mr. Swift and his son and the housekeeper.
``Well, I suppose we'll have to go back after them,'' remarked Tom. ``We'll take the airship, dad, and make a two-days' trip of it. Is there anything else you want?''
``Well, you might bring a bundle of papers you'll find in the lower right hand drawer of my desk. They contain some memoranda I need.''
Tom and Mr. Sharp had become so used to traveling in the airship that it seemed no novelty to them, though they attracted much attention wherever they went. They soon had the Red Cloud in readiness for a flight, and rising in the air above the shop that contained the powerful submarine, a craft utterly different in type from the aeroplane, the nose of the airship was pointed toward Shopton.
They made a good flight and landed near the big shed where the bird of the air was kept. It was early evening when they got to the Swift homestead, and Eradicate Sampson was glad to see them.
Eradicate was a good cook, and soon had a meal ready for the travelers. Then, while Mr. Sharp selected the tools and other things needed, and put them in the airship ready for the start back the next morning, Tom concluded he would take a stroll into Shopton, to see if he could see his friend, Ned Newton. It was early evening, and the close of a beautiful day, a sharp shower in the morning having cooled the air.
Tom was greeted by a number of acquaintances as he strolled along, for, since the episode of the bank robbery, when he had so unexpectedly returned with the thieves and the cash, the lad was better known than ever.
``I guess Ned must be home,'' thought our hero as he looked in vain for his chum among the throng on the streets. ``I've got time to take a stroll down to his house.''
Tom was about to cross the street when he was startled by the sound of an automobile horn loudly blown just at his side. Then a voice called:
``Hey, there! Git out of the way if you don't want to be run over!''
He looked up, and saw a car careening along. At the wheel was the red-haired bully, Andy Foger, and in the tonneau were Sam Snedecker and Pete Bailey.
``Git out of the way,'' added Sam, and he grinned maliciously at Tom.
The latter stepped back, well out of the path of the car, which was not moving very fast. Just in front of Tom was a puddle of muddy water. There was no necessity for Andy steering into it, but he saw his opportunity, and a moment later one of the big pneumatic tires had plunged into the dirty fluid, spattering it all over Tom, some even going as high as his face.
``Ha! ha!'' laughed Andy. ``Maybe you'll get out of my way next time, Tom Swift.''
The young inventor was almost speechless from righteous anger. He wiped the mud from his face, glanced down at his clothes, which were all but ruined, and called out:
``Hold on there, Andy Foger! I want to see you!'' for he thought of the time when Andy had shut him in the tank.
``Ta! ta!'' shouted Pete Bailey.
``See you later,'' added Sam.
``Better go home and take a bath, and then sail away in your submarine,'' went on Andy. ``I'll bet it will sink.''
Before Tom could reply the auto had turned a corner. Disgusted and angry, he tried to sop up some of the muddy water with his handkerchief. While thus engaged he heard his name called, and looked up to see Ned Newton.
``What's the matter? Fall down?'' asked his chum.
``Andy Foger,'' replied Tom.
``That's enough,'' retorted Ned. ``I can guess the rest. We'll have to tar and feather him some day, and ride him out of town on a rail. I'd kick him myself, only his father is a director in the bank where I work, and I'd be fired if I did. Can't afford any such pleasure. But some day I'll give Andy a good trouncing, and then resign before they can discharge me. But I'll be looking for another job before I do that. Come on to my house, Tom, and I'll help you clean up.''
Tom was a little more presentable when he left his chum's residence, after spending the evening there, but he was still burning for revenge against Andy and his cronies. He had half a notion to go to Andy's house and tell Mr. Foger how nearly serious the bully's prank at the submarine had been, but he concluded that Mr. Foger could only uphold his son. ``No, I'll settle with him myself,'' decided Tom.
Bidding Eradicate keep a watchful eye about the house, and leaving word for Mr. Damon to be sure to come to the coast if he again called at the Shopton house, Tom and Mr. Sharp prepared to make their return trip early the next morning.
The gas tank was filled and the Red Cloud arose in the air. Then, with the propellers moving at moderate speed, the nose of the craft was pointed toward the New Jersey coast.
A few miles out from Shopton, finding there was a contrary wind in the upper regions where they were traveling, Mr. Sharp descended several hundred feet. They were moving over a sparsely settled part of the country, and looking down, Tom saw, speeding along a highway, an automobile.
``I wonder who's in it?'' he remarked, taking down a telescope and peering over the window ledge of the cabin. The next moment he uttered a startled exclamation.
``Andy Foger, Sam Snedecker and Pete Bailey!'' he cried. ``Oh, I wish I had a bucket of water to empty on them.''
``I know a better way to get even with them than that,'' said Mr. Sharp.
``How?'' asked Tom eagerly.
``I'll show you,'' replied the balloonist. ``It's a trick I once played on a fellow who did me an injury. Here, you steer for a minute until I get the thing fixed, then I'll take charge.''
Mr. Sharp went to the storeroom and came back with a long, stout rope and a small anchor of four prongs. It was carried to be used in emergencies, but so far had never been called into requisition. Fastening the grapple to the cable, the balloonist said:
``Now, Tom, they haven't seen you. You stand in the stern and pay out the rope. I'll steer the airship, and what I want you to do is to catch the anchor in the rear of their car. Then I'll show you some fun.''
Tom followed instructions. Slowly he lowered the rope with the dangling grapple. The airship was also sent down, as the cable was not quite long enough to reach the earth from the height at which they were. The engine was run at slow speed, so that the noise would not attract the attention of the three cronies who were speeding along, all unconscious of the craft in the air over their heads. The Red Cloud was moving in the same direction as was the automobile.
The anchor was now close to the rear of Andy's car. Suddenly it caught on the tonneau and Tom called that fact to Mr. Sharp.
``Fasten the rope at the cleat,'' directed the balloonist.
Tom did so, and a moment later the aeronaut sent the airship up by turning more gas into the container. At the same time he reversed the engine and the Red Cloud began pulling the touring car backward, also lifting the rear wheels clear from the earth.
A startled cry from the occupants of the machine told Tom and his friend that Andy and his cronies were aware something was wrong. A moment later Andy, looking up, saw the airship hovering in the air above him. Then he saw the rope fast to his auto. The airship was not rising now, or the auto would have been turned over, but it was slowly pulling it backward, in spite of the fact that the motor of the car was still going.
``Here! You let go of me!'' cried Andy. ``I'll have you arrested if you damage my car.''
``Come up here and cut the rope,'' called Tom leaning over and looking down. He could enjoy the bully's discomfiture. As for Sam and Pete, they were much frightened, and cowered down on the floor of the tonneau.
``Maybe you'll shut me in the tank again and splash mud on me!'' shouted Tom.
The rear wheels of the auto were lifted still higher from the ground, as Mr. Sharp turned on a little more gas. Andy was not proof against this.
``Oh! oh!'' he cried. ``Please let me down, Tom. I'm awful sorry for what I did! I'll never do it again! Please, please let me down! Don't! You'll tip me over!''
He had shut off his motor now, and was frantically clinging to the steering wheel.
``Do you admit that you're a sneak and a coward?'' asked Tom, ``rubbing it in.''
``Yes, yes! Oh, please let me down!''
``Shall we?'' asked Tom of Mr. Sharp.
``Yes,'' replied the balloonist. ``We can afford to lose the rope and anchor for the sake of turning the tables. Cut the cable.''
Tom saw what was intended. Using a little hatchet, he severed the rope with a single blow. With a crash that could be heard up in the air where the Red Cloud hovered, the rear wheels of the auto dropped to the ground. Then came two loud reports.
``Both tires busted!'' commented Mr. Sharp dryly, and Tom, looking down, saw the trio of lads ruefully contemplating the collapsed rubber of the rear wheels. The tables had been effectually turned on Andy Foger. His auto was disabled, and the airship, with a graceful sweep, mounted higher and higher, continuing on its way to the coast.
\gutchapter{Chapter Seven}
Mr. Damon Will Go
``Well, I guess they've had their lesson,'' remarked Tom, as he took an observation through the telescope and saw Andy and his cronies hard at work trying to repair the ruptured tires. ``That certainly was a corking good trick.''
``Yes,'' admitted Mr. Sharp modestly. ``I once did something similar, only it was a horse and wagon instead of an auto. But let's try for another speed record. The conditions are just right.''
They arrived at the coast much sooner than they had dared to hope, the Red Cloud proving herself a veritable wonder.
The remainder of that day, and part of the next, was spent in working on the submarine.
``We'll launch her day after to-morrow,'' declared Mr. Swift enthusiastically. ``Then to see whether my calculations are right or wrong.''
``It won't be your fault if it doesn't work,'' said his son. ``You certainly have done your best.''
``And so have you and Mr. Sharp and the others, for that matter. Well, I have no doubt but that everything will be all right, Tom.''
``There!'' exclaimed Mr. Sharp the next morning, as he was adjusting a certain gage. ``I knew I'd forget something. That special brand of lubricating oil. I meant to bring it from Shopton, and I didn't.''
``Maybe I can get it in Atlantis,'' suggested Tom, naming the coast city nearest to them. ``I'll take a walk over. It isn't far.''
``Will you? I'll be glad to have you,'' resumed the balloonist. ``A gallon will be all we'll need.''
Tom was soon on his way. He had to walk, as the roads were too poor to permit him to use the motor-cycle, and the airship attracted too much attention to use on a short trip. He was strolling along, when from the other side of a row of sand dunes, that lined the uncertain road to Atlantis, he heard some one speaking. At first the tones were not distinct, but as the lad drew nearer to the voice he heard an exclamation.
``Bless my gold-headed cane! I believe I'm lost. He said it was out this way somewhere, but I don't see anything of it. If I had that Eradicate Sampson here now I'd---bless my shoelaces I don't know what I would do to him.''
``Mr. Damon! Mr. Damon!'' cried Tom. ``Is that you?''
``Me? Of course it's me! Who else would it be?'' answered the voice. ``But who are you. Why, bless my liver! If it isn't Tom Swift!'' he cried. ``Oh, but I'm glad to see you! I was afraid I was shipwrecked! Bless my gaiters, how are you, anyhow? How is your father? How is Mr. Sharp, and all the rest of them?''
``Pretty well. And you?''
``Me? Oh, I'm all right; only a trifle nervous. I called at your house in Shopton yesterday, and Eradicate told me, as well as he could, where you were located. I had nothing to do, so I thought I'd take a run down here. But what's this I hear about you? Are you going on a voyage?''
``Yes.''
``In the air? May I go along again? I certainly enjoyed my other trip in the Red Cloud. That is, all but the fire and being shot at. May I go?''
``We're going on a different sort of trip this time,'' said the youth.
``Where?''
``Under water.''
``Under water? Bless my sponge bath! You don't mean it!''
``Yes. Dad has completed the submarine he was working on when we were off in the airship, and it will be launched the day after to-morrow.''
``Oh, that's so. I'd forgotten about it. He's going to try for the Government prize, isn't he? But tell me more about it. Bless my scarf-pin, but I'm glad I met you! Going into town, I take it. Well, I just came from there, but I'll walk back with you. Do you think---is there any possibility---that I could go with you? Of course, I don't want to crowd you, but---''
``Oh, there'll be plenty of room,'' replied the young inventor. ``In fact, more room than we had in the airship. We were talking only the other day about the possibility of you going with us, but we didn't think you'd risk it.''
``Risk it? Bless my liver! Of course I'll risk' it! It can't be as bad as sailing in the air. You can't fall, that's certain.''
``No; but maybe you can't rise,'' remarked Tom grimly.
``Oh, we won't think of that. Of course, I'd like to go. I fully expected to be killed in the Red Cloud, but as I wasn't. I'm ready to take a chance in the water. On the whole, I think I prefer to be buried at sea, anyhow. Now, then, will you take me?''
``I think I can safely promise,'' answered Tom with a smile at his friend's enthusiasm.
The two were approaching the city, having walked along as they talked. There were still some sand dunes near the road, and they kept on the side of these, nearest the beach, where they could watch the breakers.
``But you haven't told me where you are going,'' went on Mr. Damon, after blessing a few dozen objects. ``Where do the Government trials take place?''
``Well,'' replied the lad, ``to be frank with you, we have abandoned our intention of trying for the Government prize.''
``Not going to try for it? Bless my slippers! Why not? Isn't fifty thousand dollars worth striving for? And, with the kind of a submarine you say you have, you ought to be able to win.''
``Yes, probably we could win,'' admitted the young inventor, ``but we are going to try for a better prize.''
``A better one? I don't understand.''
``Sunken treasure,'' explained Tom. ``There's a ship sunk off the coast of Uruguay, with three hundred thousand dollars in gold bullion aboard. Dad and I are going to try to recover that in our submarine. We're going to start day after to-morrow, and, if you like, you may go along.''
``Go along! Of course I'll go along!'' cried the eccentric man. ``But I never heard of such a thing. Sunken treasure! Three hundred thousand dollars in gold! My, what a lot of money! And to go after it in a submarine! It's as good as a story!''
``Yes, we hope to recover all the treasure,'' said the lad. ``We ought to be able to claim at least half of it.''
``Bless my pocketbook!'' cried Mr. Damon, but Tom did not hear him. At that instant his attention was attracted by seeing two men emerge from behind the sand dune near which he and Mr. Damon had halted momentarily, when the youth explained about the treasure. The man looked sharply at Tom. A moment later the first man was joined by another, and at the sight of him our hero could not repress an exclamation of alarm. For the second man was none other than Addison Berg.
The latter glanced quickly at Tom, and then, with a hasty word to his companion, the two swung around and made off in the opposite direction to that in which they had been walking.
``What's the matter?'' asked Mr. Damon, seeing the young inventor was strangely affected.
``That---that man,'' stammered the lad.
``You don't mean to tell me that was one the Happy Harry gang, do you?''
``No. But one, or both of those men, may prove to be worse. That second man was Addison Berg, and he's agent for a firm of submarine boat builders who are rivals of dad's. Berg has been trying to find out why we abandoned our intention of competing for the Government prize.''
``I hope you didn't tell him.''
``I didn't intend to,'' replied Tom, smiling grimly, ``but I'm afraid I have, however. He certainly overheard what I said. I spoke too loud. Yes, he must have heard me. That's why he hurried off so.''
``Possibly no harm is done. You didn't give the location of the sunken ship.''
``No; but I guess from what I said it will be easy enough to find. Well, if we're going to have a fight for the possession of that sunken gold, I'm ready for it. The Advance is well equipped for a battle. I must tell dad of this. It's my fault.''
``And partly mine, for asking you such leading questions in a public place,'' declared Mr. Damon. ``Bless my coat-tails, but I'm sorry! Maybe, after all, those men were so interested in what they themselves were saying that they didn't understand what you said.''
But if there had been any doubts on this score they would have been dissolved had Tom and his friend been able to see the actions of Mr. Berg and his companion a little later. The plans of the treasure-hunters had been revealed to their ears.
\gutchapter{Chapter Eight}
Another Treasure Expedition
While Tom and Mr. Damon continued on to Atlantis after the oil, the young inventor lamenting from time to time that his remarks about the real destination of the Advance had been overheard by Mr. Berg, the latter and his companion were hastening back along the path that ran on one side of the sand dunes.
``What's your hurry?'' asked Mr. Maxwell, who was with the submarine agent. ``You turned around as if you were shot when you saw that man and the lad. There didn't appear to be any cause for such a hurry. From what I could hear they were talking about a submarine. You're in the same business. You might be friends.''
``Yes, we might,'' admitted Mr. Berg with a peculiar smile; ``but, unless I'm very much mistaken, we're going to be rivals.''
``Rivals? What do you mean?''
``I can't tell you now. Perhaps I may later. But if you don't mind, walk a little faster, please. I want to get to a long-distance telephone.''
``What for?''
``I have just overheard something that I wish to communicate to my employers, Bentley \& Eagert.''
``Overheard something? I don't see what it could be, unless that lad---''
``You'll learn in good time,'' went on the submarine agent. ``But I must telephone at once.''
A little later the two men had reached a trolley line that ran into Atlantis, and they arrived at the city before Mr. Damon and Tom got there, as the latter had to go by a circuitous route. Mr. Berg lost no time in calling up his firm by telephone.
``I have had another talk with Mr. Swift,'' he reported to Mr. Bentley, who came to the instrument in Philadelphia.
``Well, what does he say?'' was the impatient question. ``I can't understand his not wanting to try for the Government prize. It is astonishing. You said you were going to discover the reason, Mr Berg, but you haven't done so.''
``I have.''
``What is it?''
``Well, the reason Mr. Swift and his son don't care to try for the fifty thousand dollar prize is that they are after one of three hundred thousand dollars.''
``Three hundred thousand dollars!'' cried Mr. Bentley. ``What government is going to offer such a prize as that for submarines, when they are getting almost as common as airships? We ought to have a try for that ourselves. What government is it?''
``No government at all. But I think we ought to have a try for it, Mr. Bentley.''
``Explain.''
``Well, I have just learned, most accidentally, that the Swifts are going after sunken treasure---three hundred thousand dollars in gold bullion.''
``Sunken treasure? Where?
``I don't know exactly, but off the coast of Uruguay,'' and Mr. Berg rapidly related what he had overheard Tom tell Mr. Damon. Mr. Bentley was much excited and impatient for more details, but his agent could not give them to him.
``Well,'' concluded the senior member of the firm of submarine boat builders, ``if the Swifts are going after treasure, so can we. Come to Philadelphia at once, Mr. Berg, and we'll talk this matter over. There is no time to lose. We can afford to forego the Government prize for the chance of getting a much larger one. We have as much right to search for the sunken gold as the Swifts have. Come here at once, and we will make our plans.''
``All right,'' agreed the agent with a smile as he hung up the receiver. ``I guess,'' he murmured to himself, ``that you won't be so high and mighty with me after this, Tom Swift. We'll see who has the best boat, after all. We'll have a contest and a competition, but not for a government prize. It will be for the sunken gold.''
It was easy to see that Mr. Berg was much pleased with himself.
Meanwhile, Tom and Mr. Damon had reached Atlantis, and had purchased the oil. They started back, but Tom took a street leading toward the center of the place, instead of striking for the beach path, along which they had come.
``Where are you going?'' asked Mr. Damon.
``I want to see if that Andy Foger has come back here,'' replied the lad, and he told of having been shut in the tank by the bully.
``I've never properly punished him for that trick,'' he went on, ``though we did manage to burst his auto tires. I'm curious to know how he knew enough to turn that gear and shut the tank door. He must have been loitering near the shop, seen me go in the submarine alone, watched his chance and sneaked in after me. But I'd like to get a complete explanation, and if I once got hold of Andy I could make him talk,'' and Tom clenched his fist in a manner that augured no good for the squint-eyed lad. ``He was stopping at the same hotel with Mr. Berg, and he hurried away after the trick he played on me. I next saw him in Shopton, but I thought perhaps he might have come back here. I'm going to inquire at the hotel,'' he added.
Andy's name was not on the register since his hasty flight, however, and Tom, after inquiring from the clerk and learning that Mr. Berg was still a guest at the hostelry, rejoined Mr. Damon.
``Bless my hat!'' exclaimed that eccentric individual as they started back to the lonely beach where the submarine was awaiting her advent into the water. ``The more I think of the trip I'm going to take, the more I like it.''
``I hope you will,'' remarked Tom. ``It will be a new experience for all of us. There's only one thing worrying me, and that is about Mr. Berg having overheard what I said.''
``Oh, don't worry about that. Can't we slip away and leave no trace in the water?''
``I hope so, but I must tell dad and Mr. Sharp about what happened.''
The aged inventor was not a little alarmed at what his son related, but he agreed with Mr. Damon, whom he heartily welcomed, that little was to be apprehended from Berg and his employers.
``They know we're after a sunken wreck, but that's all they do know,'' said Tom's father. ``We are only waiting for the arrival of Captain Alden Weston, and then we will go. Even if Bentley \& Eagert make a try for the treasure we'll have the start of them, and this will be a case of first come, first served. Don't worry, Tom. I'm glad you're going, Mr Damon. Come, I will show you our submarine.''
As father and son, with their guest, were going to the machine shop, Mr. Sharp met them. He had a letter in his hand.
``Good news!'' the balloonist cried. ``Captain Weston will be with us to-morrow. He will arrive at the Beach Hotel in Atlantis, and wants one of us to meet him there. He has considerable information about the wreck.''
``The Beach Hotel,'' murmured Tom. ``That is where Mr. Berg is stopping. I hope he doesn't worm any of our secret from Captain Weston,'' and it was with a feeling of uneasiness that the young inventor continued after his father and Mr. Damon to where the submarine was.
\gutchapter{Chapter Nine}
Captain Weston's Advent
``Bless my water ballast, but that certainly is a fine boat!'' cried Mr. Damon, when he had been shown over the new craft. ``I think I shall feel even safer in that than in the Red Cloud.''
``Oh, don't go back on the airship!'' exclaimed Mr Sharp. ``I was counting on taking you on another trip.''
``Well, maybe after we get back from under the ocean,'' agreed Mr. Damon. ``I particularly like the cabin arrangements of the Advance. I think I shall enjoy myself.''
He would be hard to please who could not take pleasure from a trip in the submarine. The cabin was particularly fine, and the sleeping arrangements were good.
More supplies could be carried than was possible on the airship, and there was more room in which to cook and serve food. Mr. Damon was fond of good living, and the kitchen pleased him as much as anything else.
Early the next morning Tom set out for Atlantis, to meet Captain Weston at the hotel. The young inventor inquired of the clerk whether the seafaring man had arrived, and was told that he had come the previous evening.
``Is he in his room?'' asked Tom.
``No,'' answered the clerk with a peculiar grin. ``He's an odd character. Wouldn't go to bed last night until we had every window in his room open, though it was blowing quite hard, and likely to storm. The captain said he was used to plenty of fresh air. Well, I guess he got it, all right.''
``Where is he now?'' asked the youth, wondering what sort of an individual he was to meet.
``Oh, he was up before sunrise, so some of the scrubwomen told me. They met him coming from his room, and he went right down to the beach with a big telescope he always carries with him. He hasn't come back yet. Probably he's down on the sand.''
``Hasn't he had breakfast?''
``No. He left word he didn't want to eat until about four bells, whatever time that is.''
``It's ten o'clock,'' replied Tom, who had been studying up on sea terms lately. ``Eight bells is eight o'clock in the morning, or four in the afternoon or eight at night, according to the time of day. Then there's one bell for every half hour, so four bells this morning would be ten o'clock in this watch, I suppose.''
``Oh, that's the way it goes, eh?'' asked the clerk. ``I never could get it through my head. What is twelve o'clock noon?''
``That's eight bells, too; so is twelve o'clock midnight. Eight bells is as high as they go on a ship. But I guess I'll go down and see if I can meet the captain. It will soon be ten o'clock, or four bells, and he must be hungry for breakfast. By the way, is that Mr. Berg still here?''
``No; he went away early this morning. He and Captain Weston seemed to strike up quite an acquaintance, the night clerk told me. They sat and smoked together until long after midnight, or eight bells,'' and the clerk smiled as he glanced down at the big diamond ring on his little finger.
``They did?'' fairly exploded Tom, for he had visions of what the wily Mr. Berg might worm out of the simple captain.
``Yes. Why, isn't the captain a proper man to make friends with?'' and the clerk looked at Tom curiously.
``Oh, yes, of course,'' was the hasty answer. ``I guess I'll go and see if I can find him---the captain, I mean.''
Tom hardly knew what to think. He wished his father, or Mr. Sharp, had thought to warn Captain Weston against talking of the wreck. It might be too late now.
The young inventor hurried to the beach, which was not far from the hotel. He saw a solitary figure pacing up and down, and from the fact that the man stopped, every now and then, and gazed seaward through a large telescope, the lad concluded it was the captain for whom he was in search. He approached, his footsteps making no sound on the sand. The man was still gazing through the glass.
``Captain Weston?'' spoke Tom.
Without a show of haste, though the voice must have startled him, the captain turned. Slowly he lowered the telescope, and then he replied softly:
``That's my name. Who are you, if I may ask?''
Tom was struck, more than by anything else, by the gentle voice of the seaman. He had prepared himself, from the description of Mr. Sharp, to meet a gruff, bewhiskered individual, with a voice like a crosscut saw, and a rolling gait. Instead he saw a man of medium size, with a smooth face, merry blue eyes, and the softest voice and gentlest manner imaginable. Tom was very much disappointed. He had looked for a regular sea-dog, and he met a landsman, as he said afterward. But it was not long before our hero changed his mind regarding Captain Weston.
``I'm Tom Swift,'' the owner of that name said, ``and I have been sent to show you the way to where our ship is ready to launch.'' The young inventor refrained from mentioning submarine, as it was the wish of Mr Sharp to disclose this feature of the voyage to the sailor himself.
``Ha, I thought as much,'' resumed the captain quietly. ``It's a fine day, if I may be permitted to say so,'' and he seemed to hesitate, as if there was some doubt whether or not he might make that observation.
``It certainly is,'' agreed the lad. Then, with a smile he added: ``It is nearly eight bells.''
``Ha!'' exclaimed the captain, also smiling, but even his manner of saying ``Ha!'' was less demonstrative than that of most persons. ``I believe I am getting hungry, if I may be allowed the remark,'' and again he seemed asking Tom's pardon for mentioning the fact.
``Perhaps you will come back to the cabin and have a little breakfast with me,'' he went on. ``I don't know what sort of a galley or cook they have aboard the Beach Hotel, but it can't be much worse than some I've tackled.''
``No, thank you,'' answered the youth. ``I've had my breakfast. But I'll wait for you, and then I'd like to get back. Dad and Mr. Sharp are anxious to meet you.''
``And I am anxious to meet them, if you don't mind me mentioning it,'' was the reply, as the captain once more put the spyglass to his eye and took an observation. ``Not many sails in sight this morning,'' he added. ``But the weather is fine, and we ought to get off in good shape to hunt for the treasure about which Mr. Sharp wrote me. I believe we are going after treasure,'' he said; ``that is, if you don't mind talking about it.''
``Not in the least,'' replied Tom quickly, thinking this a good opportunity for broaching a subject that was worrying him. ``Did you meet a Mr. Berg here last night, Captain Weston?'' he went on.
``Yes. Mr. Berg and I had quite a talk. He is a well-informed man.''
``Did he mention the sunken treasure?'' asked the lad, eager to find out if his suspicions were true.
``Yes, he did, if you'll excuse me putting it so plainly,'' answered the seaman, as if Tom might be offended at so direct a reply. But the young inventor was soon to learn that this was only an odd habit with the seaman.
``Did he want to know where the wreck of the Boldero was located?'' continued the lad. ``That is, did he try to discover if you knew anything about it?''
``Yes,'' said Mr. Weston, ``he did. He pumped me, if you are acquainted with that term, and are not offended by it. You see, when I arrived here I made inquiries as to where your father's place was located. Mr. Berg overheard me, and introduced himself as agent for a shipbuilding concern. He was very friendly, and when he said he knew you and your parent, I thought he was all right.''
Tom's heart sank. His worst fears were to be realized, he thought.
``Yes, he and I talked considerable, if I may be permitted to say so,'' went on the captain. ``He seemed to know about the wreck of the Boldero, and that she had three hundred thousand dollars in gold aboard. The only thing he didn't know was where the wreck was located. He knew it was off Uruguay somewhere, but just where he couldn't say. So he asked me if I knew, since he must have concluded that I was going with you on the gold-hunting expedition.''
``And you do know, don't you?'' asked Tom eagerly.
``Well, I have it pretty accurately charted out, if you will allow me that expression,'' was the calm answer. ``I took pains to look it up at the request of Mr. Sharp.''
``And he wanted to worm that information out of you?'' inquired the youth excitedly.
``Yes, I'm afraid he did.''
``Did you give him the location?''
``Well,'' remarked the captain, as he took another observation before closing up the telescope, ``you see, while we were talking, I happened to drop a copy of a map I'd made, showing the location of the wreck. Mr. Berg picked it up to hand to me, and he looked at it.''
``Oh!'' cried Tom. ``Then he knows just where the treasure is, and he may get to it ahead of us. It's too bad.''
``Yes,'' continued the seaman calmly, ``Mr. Berg picked up that map, and he looked very closely at the latitude and longitude I had marked as the location of the wreck.''
``Then he won't have any trouble finding it,'' murmured our hero.
``Eh? What's that?'' asked the captain, ``if I may be permitted to request you to repeat what you said.''
``I say he won't have any trouble locating the sunken Boldero,'' repeated Tom.
``Oh, but I think he will, if he depends on that map,'' was the unexpected reply. ``You see,'' explained Mr. Weston, ``I'm not so simple as I look. I sensed what Mr. Berg was after, the minute he began to talk to me. So I fixed up a little game on him. The map which I dropped on purpose, not accidentally, where he would see it, did have the location of the wreck marked. Only it didn't happen to be the right location. It was about five hundred miles out of the way, and I rather guess if Mr. Berg and his friends go there for treasure they'll find considerable depth of water and quite a lonesome spot. Oh, no, I'm not as easy as I look, if you don't mind me mentioning that fact; and when a scoundrel sets out to get the best of me, I generally try to turn the tables on him. I've seen such men as Mr. Berg before. I'm afraid, I'm very much afraid, the sight he had of the fake map I made won't do him much good. Well, I declare, it's past four bells. Let's go to breakfast, if you don't mind me asking you,'' and with that the captain started off up the beach, Tom following, his ideas all a whirl at the unlooked-for outcome of the interview.
\gutchapter{Chapter Ten}
Trial of the Submarine
Tom felt such a relief at hearing of Captain Weston's ruse that his appetite, sharpened by an early breakfast and the sea air, came to him with a rush, and he had a second morning meal with the odd sea captain, who chuckled heartily when he thought of how Mr Berg had been deceived.
``Yes,'' resumed Captain Weston, over his bacon and eggs, ``I sized him up for a slick article as soon as I laid eyes on him. But he evidently misjudged me, if I may be permitted that term. Oh, well, we may meet again, after we secure the treasure, and then I can show him the real map of the location of the wreck.''
``Then you have it?'' inquired the lad eagerly.
Captain Weston nodded, before hiding his face behind a large cup of coffee; his third, by the way.
``Let me see it?'' asked Tom quickly. The captain set down his cup. He looked carefully about the hotel dining-room. There were several guests, who, like himself, were having a late breakfast.
``It's a good plan,'' the sailor said slowly, ``when you're going into unknown waters, and don't want to leave a wake for the other fellow to follow, to keep your charts locked up. If it's all the same to you,'' he added diffidently, ``I'd rather wait until we get to where your father and Mr. Sharp are before displaying the real map. I've no objection to showing you the one Mr. Berg saw,'' and again he chuckled.
The young inventor blushed at his indiscretion. He felt that the news of the search for the treasure had leaked out through him, though he was the one to get on the trail of it by seeing the article in the paper. Now he had nearly been guilty of another break. He realized that he must be more cautious. The captain saw his confusion, and said:
``I know how it is. You're eager to get under way. I don't blame you. I was the same myself when I was your age. But we'll soon be at your place, and then I'll tell you all I know. Sufficient now, to say that I believe I have located the wreck within a few miles. I got on the track of a sailor who had met one of the shipwrecked crew of the Boldero, and he gave me valuable information. Now tell me about the craft we are going in. A good deal depends on that.''
Tom hardly knew what to answer. He recalled what Mr. Sharp had said about not wanting to tell Captain Weston, until the last moment, that they were going in a submarine, for fear the old seaman (for he was old in point of service though not in years) might not care to risk an under-water trip. Therefore Tom hesitated. Seeing it, Captain Weston remarked quietly:
``I mean, what type is your submarine? Does it go by compressed air, or water power?''
``How do you know it's a submarine?'' asked the young inventor quickly, and in some confusion.
``Easy enough. When Mr. Berg thought he was pumping me, I was getting a lot of information from him. He told me about the submarine his firm was building, and, naturally, he mentioned yours. One thing led to another until I got a pretty good idea of your craft. What do you call it?''
``The Advance.''
``Good name. I like it, if you don't mind speaking of it.''
``We were afraid you wouldn't like it,'' commented Tom.
``What, the name?''
``No, the idea of going in a submarine.''
``Oh,'' and Captain Weston laughed. ``Well, it takes more than that to frighten me, if you'll excuse the expression. I've always had a hankering to go under the surface, after so many years spent on top. Once or twice I came near going under, whether I wanted to or not, in wrecks, but I think I prefer your way. Now, if you're all done, and don't mind me speaking of it, I think we'll start for your place. We must hustle, for Berg may yet get on our trail, even if he has got the wrong route,'' and he laughed again.
It was no small relief to Mr Swift and Mr. Sharp to learn that Captain Weston had no objections to a submarine, as they feared he might have. The captain, in his diffident manner, made friends at once with the treasure-hunters, and he and Mr. Damon struck up quite an acquaintance. Tom told of his meeting with the seaman, and the latter related, with much gusto, the story of how he had fooled Mr. Berg.
``Well, perhaps you'd like to come and take a look at the craft that is to be our home while we're beneath the water,'' suggested Mr. Swift and the sailor assenting, the aged inventor, with much pride, assisted by Tom, pointed out on the Advance the features of interest. Captain Weston gave hearty approval, making one or two minor suggestions, which were carried out.
``And so you launch her to-morrow,'' he concluded, when he had completed the inspection ``Well, I hope it's a success, if I may be permitted to say so.''
There were busy times around the machine shop next day. So much secrecy had been maintained that none of the residents, or visitors to the coast resort, were aware that in their midst was such a wonderful craft as the submarine. The last touches were put on the under-water ship; the ways, leading from the shop to the creek, were well greased, and all was in readiness for the launching. The tide would soon be at flood, and then the boat would slide down the timbers (at least, that was the hope of all), and would float in the element meant to receive her. It was decided that no one should be aboard when the launching took place, as there was an element of risk attached, since it was not known just how buoyant the craft was. It was expected she would float, until the filled tanks took her to the bottom, but there was no telling.
``It will be flood tide now in ten minutes,'' remarked Captain Weston quietly, looking at his watch. Then he took an observation through the telescope. ``No hostile ships hanging in the offing,'' he reported. ``All is favorable, if you don't mind me saying so,'' and he seemed afraid lest his remark might give offense.
``Get ready,'' ordered Mr. Swift. ``Tom, see that the ropes are all clear,'' for it had been decided to ease the Advance down into the water by means of strong cables and windlasses, as the creek was so narrow that the submarine, if launched in the usual way, would poke her nose into the opposite mud bank and stick there.
``All clear,'' reported the young inventor.
``High tide!'' exclaimed the captain a moment later, snapping shut his watch.
``Let go!'' ordered Mr. Swift, and the various windlasses manned by the inventor, Tom and the others began to unwind their ropes. Slowly the ship slid along the greased ways. Slowly she approached the water. How anxiously they all watched her! Nearer and nearer her blunt nose, with the electric propulsion plate and the auxiliary propeller, came to the creek, the waters of which were quiet now, awaiting the turn of the tide.
Now little waves lapped the steel sides. It was the first contact of the Advance with her native element.
``Pay out the rope faster!'' cried Mr. Swift.
The windlasses were turned more quickly. Foot by foot the craft slid along until, with a final rush, the stern left the ways and the submarine was afloat. Now would come the test. Would she ride on an even keel, or sink out of sight, or turn turtle? They all ran to the water's edge, Tom in the lead.
``Hurrah!'' suddenly yelled the lad, trying to stand on his head. ``She floats! She's a success! Come on! Let's get aboard!''
For, true enough, the Advance was riding like a duck on the water. She had been proportioned just right, and her lines were perfect. She rode as majestically as did any ship destined to sail on the surface, and not intended to do double duty.
``Come on, we must moor her to the pier,'' directed Mr. Sharp. ``The tide will turn in a few minutes and take her out to sea.''
He and Tom entered a small boat, and soon the submarine was tied to a small dock that had been built for the purpose.
``Now to try the engine,'' suggested Mr. Swift, who was almost trembling with eagerness; for the completion of the ship meant much to him.
``One moment,'' begged Captain Weston. ``If you don't mind, I'll take an observation,'' he went on, and he swept the horizon with his telescope. ``All clear,'' he reported. ``I think we may go aboard and make a trial trip.''
Little time was lost in entering the cabin and engine-room, Garret Jackson accompanying the party to aid with the machinery. It did not take long to start the motors, dynamos and the big gasolene engine that was the vital part of the craft. A little water was admitted to the tanks for ballast, since the food and other supplies were not yet on board. The Advance now floated with the deck aft of the conning tower showing about two feet above the surface of the creek. Mr. Swift and Tom entered the pilot house.
``Start the engines,'' ordered the aged inventor, ``and we'll try my new system of positive and negative electrical propulsion.''
There was a hum and whir in the body of the ship beneath the feet of Tom and his father. Captain Weston stood on the little deck near the conning tower.
``All ready?'' asked the youth through the speaking tube to Mr. Sharp and Mr. Jackson in the engine-room.
``All ready,'' came the answer.
Tom threw over the connecting lever, while his father grasped the steering wheel. The Advance shot forward, moving swiftly along, about half submerged.
``She goes! She goes!'' cried Tom.
``She certainly does, if I may be permitted to say so,'' was the calm contribution of Captain Weston. ``I congratulate you.''
Faster and faster went the new craft. Mr. Swift headed her toward the open sea, but stopped just before passing out of the creek, as he was not yet ready to venture into deep water.
``I want to test the auxiliary propellers,'' he said. After a little longer trial of the electric propulsion plates, which were found to work satisfactorily, sending the submarine up and down the creek at a fast rate, the screws, such as are used on most submarines, were put into gear. They did well, but were not equal to the plates, nor was so much expected of them.
``I am perfectly satisfied,'' announced Mr. Swift as he once more headed the boat to sea. ``I think, Captain Weston, you had better go below now.''
``Why so?''
``Because I am going to completely submerge the craft. Tom, close the conning tower door. Perhaps you will come in here with us, Captain Weston, though it will be rather a tight fit.''
``Thank you, I will. I want to see how it feels to be in a pilot house under water.''
Tom closed the water-tight door of the conning tower. Word was sent through the tube to the engine-room that a more severe test of the ship was about to be made. The craft was now outside the line of breakers and in the open sea.
``Is everything ready, Tom?'' asked his father in a quiet voice.
``Everything,'' replied the lad nervously, for the anticipation of being about to sink below the surface was telling on them all, even on the calm, old sea captain.
``Then open the tanks and admit the water,'' ordered Mr. Swift.
His son turned a valve and adjusted some levers. There was a hissing sound, and the Advance began sinking. She was about to dive beneath the surface of the ocean, and those aboard her were destined to go through a terrible experience before she rose again.
\gutchapter{Chapter Eleven}
On the Ocean Bed
Lower and lower sank the submarine. There was a swirling and foaming of the water as she went down, caused by the air bubbles which the craft carried with her in her descent. Only the top of the conning tower was out of water now, the ocean having closed over the deck and the rounded back of the boat. Had any one been watching they would have imagined that an accident was taking place.
In the pilot house, with its thick glass windows, Tom, his father and Captain Weston looked over the surface of the ocean, which every minute was coming nearer and nearer to them.
``We'll be all under in a few seconds,'' spoke Tom in a solemn voice, as he listened to the water hissing into the tanks.
``Yes, and then we can see what sort of progress we will make,'' added Mr. Swift. ``Everything is going fine, though,'' he went on cheerfully. ``I believe I have a good boat.''
``There is no doubt of it in my mind,'' remarked Captain Weston, and Tom felt a little disappointed that the sailor did not shout out some such expression as ``Shiver my timbers!'' or ``Keel-haul the main braces, there, you lubber!'' But Captain Weston was not that kind of a sailor, though his usually quiet demeanor could be quickly dropped on necessity, as Tom learned later.
A few minutes more and the waters closed over the top of the conning tower. The Advance was completely submerged. Through the thick glass windows of the pilot house the occupants looked out into the greenish water that swirled about them; but it could not enter. Then, as the boat went lower, the light from above gradually died out, and the semi-darkness gave place to gloom.
``Turn on the electrics and the searchlight, Tom,'' directed his father.
There was the click of a switch, and the conning tower was flooded with light. But as this had the effect of preventing the three from peering out into the water, just as one in a lighted room cannot look out into the night, Tom shut them off and switched on the great searchlight. This projected its powerful beams straight ahead and there, under the ocean, was a pathway of illumination for the treasure-seekers.
``Fine!'' cried Captain Weston, with more enthusiasm than he had yet manifested. ``That's great, if you don't mind me mentioning it. How deep are we?''
Tom glanced at a gage on the side of the pilot tower.
``Only about sixty feet,'' he answered.
``Then don't go any deeper!'' cried the captain hastily. ``I know these waters around here, and that's about all the depth you've got. You'll be on the bottom in a minute.''
``I intend to get on the bottom after a while,'' said Mr. Swift, ``but not here. I want to try for a greater distance under water before I come to rest on the ocean's bed. But I think we are deep enough for a test. Tom, close the tank intake pipes and we'll see how the Advance will progress when fully submerged.''
The hissing stopped, and then, wishing to see how the motors and other machinery would work, the aged inventor and his son, accompanied by Captain Weston, descended from the conning tower, by means of an inner stairway, to the interior of the ship. The submarine could be steered and managed from below or above. She was now floating about sixty-five feet below the surface of the bay.
``Well, how do you like it?'' asked Tom of Mr. Damon, as he saw his friend in an easy chair in the living-room or main cabin of the craft, looking out of one of the plate-glass windows on the side.
``Bless my spectacles, it's the most wonderful thing I ever dreamed of!'' cried the queer character, as he peered at the mass of water before him. ``To think that I'm away down under the surface, and yet as dry as a bone. Bless my necktie, but it's great! What are we going to do now?''
``Go forward,'' replied the young inventor.
``Perhaps I had better make an observation,'' suggested Captain Weston, taking his telescope from under his arm, where he had carried it since entering the craft, and opening it. ``We may run afoul of something, if you don't mind me mentioning such a disagreeable subject.'' Then, as he thought of the impossibility of using his glass under water, he closed it.
``I shall have little use for this here, I'm afraid,'' he remarked with a smile. ``Well, there's some consolation. We're not likely to meet many ships in this part of the ocean. Other vessels are fond enough of remaining on the surface. I fancy we shall have the depths to ourselves, unless we meet a Government submarine, and they are hardly able to go as deep as we can. No, I guess we won't run into anything and I can put this glass away.''
``Unless we run into Berg and his crowd,'' suggested Tom in a low voice.
``Ha! ha!'' laughed Captain Weston, for he did not want Mr. Swift to worry over the unscrupulous agent. ``No, I don't believe we'll meet them, Tom. I guess Berg is trying to work out the longitude and latitude I gave him. I wish I could see his face when he realizes that he's been deceived by that fake map.''
``Well, I hope he doesn't discover it too soon and trail us,'' went on the lad. ``But they're going to start the machinery now. I suppose you and I had better take charge of the steering of the craft. Dad will want to be in the engine-room.''
``All right,'' replied the captain, and he moved forward with the lad to a small compartment, shut off from the living-room, that served as a pilot house when the conning tower was not used. The same levers, wheels and valves were there as up above, and the submarine could be managed as well from there as from the other place.
``Is everything all right?'' asked Mr Swift as he went into the engine-room, where Garret Jackson and Mr. Sharp were busy with oil cans.
``Everything,'' replied the balloonist. ``Are you going to start now?''
``Yes, we're deep enough for a speed trial. We'll go out to sea, however, and try for a lower depth record, as soon as there's enough water. Start the engine.''
A moment later the powerful electric currents were flowing into the forward and aft plates, and the Advance began to gather way, forging through the water.
``Straight ahead, out to sea, Tom,'' called his father to him.
``Aye, aye, sir,'' responded the youth.
``Ha! Quite seaman-like, if you don't mind a reference to it,'' commented Captain Weston with a smile. ``Mind your helm, boy, for you don't want to poke her nose into a mud bank, or run up on a shoal.''
``Suppose you steer?'' suggested the lad. ``I'd rather take lessons for a while.''
``All right. Perhaps it will be safer. I know these waters from the top, though I can't say as much for the bottom. However, I know where the shoals are.''
The powerful searchlight was turned, so as to send its beams along the path which the submarine was to follow, and then, as she gathered speed, she shot ahead, gliding through the waters like a fish.
Mr. Damon divided his time between the forward pilot-room, the living-apartment, and the place where Mr. Swift, Garret Jackson and Mr. Sharp were working over the engines. Every few minutes he would bless some part of himself, his clothing, or the ship. Finally the old man settled down to look through the plate-glass windows in the main apartment.
On and on went the submarine. She behaved perfectly, and was under excellent control. Some times Tom, at the request of his father, would send her toward the surface by means of the deflecting rudder. Then she would dive to the bottom again. Once, as a test, she was sent obliquely to the surface, her tower just emerging, and then she darted downward again, like a porpoise that had come up to roll over, and suddenly concluded to seek the depths. In fact, had any one seen the maneuver they would have imagined the craft was a big fish disporting itself.
Captain Weston remained at Tom's side, giving him instructions, and watching the compass in order to direct the steering so as to avoid collisions. For an hour or more the craft was sent almost straight ahead at medium speed. Then Mr. Swift, joining his son and the captain, remarked:
``How about depth of water here, Captain Weston?''
``You've got more than a mile.''
``Good! Then I'm going down to the bottom of the sea! Tom, fill the tanks still more.
``Aye, aye, sir,'' answered the lad gaily. ``Now for a new experience!''
``And use the deflecting rudder, also,'' advised his father. ``That will hasten matters.''
Five minutes later there was a slight jar noticeable.
``Bless my soul! What's that?'' cried Mr. Damon. ``Have we hit something?''
``Yes,'' answered Tom with a smile.
``What, for gracious sake?''
``The bottom of the sea. We're on the bed of the ocean.''
\gutchapter{Chapter Twelve}
For a Breath of Air
They could hardly realize it, yet the depth-gage told the story. It registered a distance below the surface of the ocean of five thousand seven hundred feet---a little over a mile. The Advance had actually come to rest on the bottom of the Atlantic.
``Hurrah!'' cried Tom. ``Let's get on the diving suits, dad, and walk about on land under water for a change.''
``No,'' said Mr. Swift soberly. ``We will hardly have time for that now. Besides, the suits are not yet fitted with the automatic air-tanks, and we can't use them. There are still some things to do before we start on our treasure cruise. But I want to see how the plates are standing this pressure.''
The Advance was made with a triple hull, the spaces between the layers of plates being filled with a secret material, capable of withstanding enormous pressure, as were also the plates themselves. Mr. Swift, aided by Mr. Jackson and Captain Weston, made a thorough examination, and found that not a drop of water had leaked in, nor was there the least sign that any of the plates had given way under the terrific strain.
``She's as tight as a drum, if you will allow me to make that comparison,'' remarked Captain Weston modestly. ``I couldn't ask for a dryer ship.''
``Well, let's take a look around by means the searchlight and the observation windows, and then we'll go back,'' suggested Mr. Swift. ``It will take about two days to get the stores and provisions aboard and rig up the diving suits; then we will start for the sunken treasure.''
There were several powerful searchlights on the Advance, so arranged that the bow, stern or either side could be illuminated independently. There were also observation windows near each light.
In turn the powerful rays were cast first at the bow and then aft. In the gleams could be seen the sandy bed of the ocean, covered with shells of various kinds. Great crabs walked around on their long, jointed legs, and Tom saw some lobsters that would have brought joy to the heart of a fisherman.
``Look at the big fish!'' cried Mr. Damon suddenly, and he pointed to some dark, shadowy forms that swam up to the glass windows, evidently puzzled by the light.
``Porpoises,'' declared Captain Weston briefly, ``a whole school of them.''
The fish seemed suddenly to multiply, and soon those in the submarine felt curious tremors running through the whole craft.
``The fish are rubbing up against it,'' cried Tom. ``They must think we came down here to allow them to scratch their backs on the steel plates.''
For some time they remained on the bottom, watching the wonderful sight of the fishes that swam all about them.
``Well, I think we may as well rise,'' announced Mr. Swift, after they had been on the bottom about an hour, moving here and there. ``We didn't bring any provisions, and I'm getting hungry, though I don't know how the others of you feel about it.''
``Bless my dinner-plate, I could eat, too!'' cried Mr. Damon. ``Go up, by all means. We'll get enough of under-water travel once we start for the treasure.''
``Send her up, Tom,'' called his father. ``I want to make a few notes on some needed changes and improvements.''
Tom entered the lower pilot house, and turned the valve that opened the tanks. He also pulled the lever that started the pumps, so that the water ballast would be more quickly emptied, as that would render the submarine buoyant, and she would quickly shoot to the surface. To the surprise of the lad, however, there followed no outrushing of the water. The Advance remained stationary on the ocean bed. Mr. Swift looked up from his notes.
``Didn't you hear me ask you to send her up, Tom?'' he inquired mildly.
``I did, dad, but something seems to be the matter,'' was the reply.
``Matter? What do you mean?'' and the aged inventor hastened to where his son and Captain Weston were at the wheels, valves and levers.
``Why, the tanks won't empty, and the pumps don't seem to work.''
``Let me try,'' suggested Mr. Swift, and he pulled the various handles. There was no corresponding action of the machinery.
``That's odd,'' he remarked in a curious voice ``Perhaps something has gone wrong with the connections. Go look in the engine-room, and ask Mr. Sharp if everything is all right there.''
Tom made a quick trip, returning to report that the dynamos, motors and gas engine were running perfectly.
``Try to work the tank levers and pumps from the conning tower,'' suggested Captain Weston. ``Sometimes I've known the steam steering gear to play tricks like that.''
Tom hurried up the circular stairway into the tower. He pulled the levers and shifted the valves and wheels there. But there was no emptying of the water tanks. The weight and pressure of water in them still held the submarine on the bottom of the sea, more than a mile from the surface. The pumps in the engine-room were working at top speed, but there was evidently something wrong in the connections. Mr. Swift quickly came to this conclusion.
``We must repair it at once,'' he said. ``Tom, come to the engine-room. You and I, with Mr. Jackson and Mr. Sharp, will soon have it in shape again.''
``Is there any danger?'' asked Mr. Damon in a perturbed voice. ``Bless my soul, it's unlucky to have an accident on our trial trip.''
``Oh, we must expect accidents,'' declared Mr. Swift with a smile. ``This is nothing.''
But it proved to be more difficult than he had imagined to re-establish the connection between the pumps and the tanks. The valves, too, had clogged or jammed, and as the pressure outside the ship was so great, the water would not run out of itself. It must be forced.
For an hour or more the inventor, his son and the others, worked away. They could accomplish nothing. Tom looked anxiously at his parent when the latter paused in his efforts.
``Don't worry,'' advised the aged inventor. ``It's got to come right sooner or later.''
Just then Mr. Damon, who had been wandering about the ship, entered the engine-room.
``Do you know,'' he said, ``you ought to open a window, or something.''
``Why, what's the matter?'' asked Tom quickly, looking to see if the odd man was joking.
``Well, of course I don't exactly mean a window,'' explained Mr. Damon, ``but we need fresh air.''
``Fresh air!'' There was a startled note in Mr. Swift's voice as he repeated the words.
``Yes, I can hardly breathe in the living-room, and it's not much better here.''
``Why, there ought to be plenty of fresh air,'' went on the inventor. ``It is renewed automatically.''
Tom jumped up and looked at an indicator. He uttered a startled cry.
``The air hasn't been changed in the last hour!'' he exclaimed. ``It is bad. There's not enough oxygen in it. I notice it, now that I've stopped working. The gage indicates it, too. The automatic air-changer must have stopped working. I'll fix it.''
He hurried to the machine which was depended on to supply fresh air to the submarine.
``Why, the air tanks are empty!'' the young inventor cried. ``We haven't any more air except what is in the ship now!''
``And we're rapidly breathing that up,'' added Captain Weston solemnly.
``Can't you make more?'' cried Mr. Damon. ``I thought you said you could make oxygen aboard the ship.''
``We can,'' answered Mr. Swift, ``but I did not bring along a supply of the necessary chemicals. I did not think we would be submerged long enough for that. But there should have been enough in the reserve tank to last several days. How about it, Tom?''
``It's all leaked out, or else it wasn't filled,'' was the despairing answer. ``All the air we have is what's in the ship, and we can't make more.''
The treasure-seekers looked at each other. It was an awful situation.
``Then the only thing to do is to fix the machinery and rise to the surface,'' said Mr. Sharp simply. ``We can have all the air we want, then.''
``Yes, but the machinery doesn't seem possible of being fixed,'' spoke Tom in a low voice.
``We must do it!'' cried his father.
They set to work again with fierce energy, laboring for their very lives. They all knew that they could not long remain in the ship without oxygen. Nor could they desert it to go to the surface, for the moment they left the protection of the thick steel sides the terrible pressure of the water would kill them. Nor were the diving suits available. They must stay in the craft and die a miserable death---unless the machinery could be repaired and the Advance sent to the surface. The emergency expanding lifting tank was not yet in working order.
More frantically they toiled, trying every device that was suggested to the mechanical minds of Tom, his father, Mr. Sharp or Mr. Jackson, to make the pumps work. But something was wrong. More and more foul grew the air. They were fairly gasping now. It was difficult to breathe, to say nothing of working, in that atmosphere. The thought of their terrible position was in the minds of all.
``Oh, for one breath of fresh air!'' cried Mr. Damon, who seemed to suffer more than any of the others. Grim death was hovering around them, imprisoned as they were on the ocean's bed, over a mile from the surface.
\gutchapter{Chapter Thirteen}
Off for the Treasure
Suddenly Tom, after a moment's pause, seized a wrench and began loosening some nuts.
``What are you doing?'' asked his father faintly, for he was being weakened by the vitiated atmosphere.
``I'm going to take this valve apart,'' replied his son. ``We haven't looked there for the trouble. Maybe it's out of order.''
He attacked the valve with energy, but his hands soon lagged. The lack of oxygen was telling on him. He could no longer work quickly.
``I'll help,'' murmured Mr. Sharp thickly. He took a wrench, but no sooner had he loosened one nut than he toppled over. ``I'm all in,'' he murmured feebly.
``Is he dead?'' cried Mr. Damon, himself gasping.
``No, only fainted. But he soon will be dead, and so will all of us, if we don't get fresh air,'' remarked Captain Weston. ``Lie down on the floor, every one. There is a little fairly good air there. It's heavier than the air we've breathed, and we can exist on it for a little longer. Poor Sharp was so used to breathing the rarified air of high altitudes that he can't stand this heavy atmosphere.''
Mr. Damon was gasping worse than ever, and so was Mr. Swift. The balloonist lay an inert heap on the floor, with Captain Weston trying to force a few drops of stimulant down his throat.
With a fierce determination in his heart, but with fingers that almost refused to do his bidding, Tom once more sought to open the big valve. He felt sure the trouble was located there, as they had tried to locate it in every other place without avail.
``I'll help,'' said Mr. Jackson in a whisper. He, too, was hardly able to move.
More and more devoid of oxygen grew the air. It gave Tom a sense as if his head was filled, and ready to burst with every breath he drew. Still he struggled to loosen the nuts. There were but four more now, and he took off three while Mr. Jackson removed one. The young inventor lifted off the valve cover, though it felt like a ton weight to him. He gave a glance inside.
``Here's the trouble!'' he murmured. ``The valve's clogged. No wonder it wouldn't work. The pumps couldn't force the water out.''
It was the work of only a minute to adjust the valve. Then Tom and the engineer managed to get the cover back on.
How they inserted the bolts and screwed the nuts in place they never could remember clearly afterward, but they managed it somehow, with shaking, trembling hands and eyes that grew more and more dim.
``Now start the pumps!'' cried Tom faintly. ``The tanks will be emptied, and we can get to the surface.''
Mr. Sharp was still unconscious, nor was Mr. Swift able to help. He lay with his eyes closed. Garret Jackson, however, managed to crawl to the engine-room, and soon the clank of machinery told Tom that the pumps were in motion. The lad staggered to the pilot house and threw the levers over. An instant later there was the hissing of water as it rushed from the ballast tanks. The submarine shivered, as though disliking to leave the bottom of the sea, and then slowly rose. As the pumps worked more rapidly, and the sea was sent from the tank in great volumes, the boat fairly shot to the surface. Tom was ready to open the conning tower and let in fresh air as soon as the top was above the surface.
With a bound the Advance reached the top. Tom frantically worked the worm gear that opened the tower. In rushed the fresh, life-giving air, and the treasure-hunters filled their lungs with it.
And it was only just in time, for Mr. Sharp was almost gone. He quickly revived, as did the others, when they could breathe as much as they wished of the glorious oxygen.
``That was a close call,'' commented Mr. Swift. ``We'll not go below again until I have provided for all emergencies. I should have seen to the air tanks and the expanding one before going below. We'll sail home on the surface now.''
The submarine was put about and headed for her dock. On the way she passed a small steamer, and the passengers looked down in wonder at the strange craft.
When the Advance reached the secluded creek where she had been launched, her passengers had fully recovered from their terrible experience, though the nerves of Mr. Swift and Mr. Damon were not at ease for some days thereafter.
``I should never have made a submerged test without making sure that we had a reserve supply of air,'' remarked the aged inventor. ``I will not be caught that way again. But I can't understand how the pump valve got out of order.''
``Maybe some one tampered with it,'' suggested Mr. Damon. ``Could Andy Foger, any of the Happy Harry gang, or the rival gold-seekers have done it?''
``I hardly think so,'' answered Tom. ``The place has been too carefully guarded since Berg and Andy once sneaked in. I think it was just an accident, but I have thought of a plan whereby such accidents can be avoided in the future. It needs a simple device.''
``Better patent it,'' suggested Mr. Sharp with a smile.
``Maybe I will,'' replied the young inventor. ``But not now. We haven't time, if we intend to get fitted out for our trip.''
``No; I should say the sooner we started the better,'' remarked Captain Weston. ``That is, if you don't mind me speaking about it,'' he added gently, and the others smiled, for his diffident comments were only a matter of habit.
The first act of the adventurers, after tying the submarine at the dock, was to proceed with the loading of the food and supplies. Tom and Mr. Damon looked to this, while Mr. Swift and Mr. Sharp made some necessary changes to the machinery. The next day the young inventor attached his device to the pump valve, and the loading of the craft was continued.
All was in readiness for the gold-seeking expedition a week later. Captain Weston had carefully charted the route they were to follow, and it was decided to move along on the surface for the first day, so as to get well out to sea before submerging the craft. Then it would sink below the surface, and run along under the water until the wreck was reached, rising at times, as needed, to renew the air supply.
With sufficient stores and provisions aboard to last several months, if necessary, though they did not expect to be gone more than sixty days at most, the adventurers arose early one morning and went down to the dock. Mr. Jackson was not to accompany them. He did not care about a submarine trip, he said, and Mr. Swift desired him to remain at the seaside cottage and guard the shops, which contained much valuable machinery. The airship was also left there.
``Well, are we all ready?'' asked Mr. Swift of the little party of gold-seekers, as they were about to enter the conning tower hatchway of the submarine.
``All ready, dad,'' responded his son.
``Then let's get aboard,'' proposed Captain Weston. ``But first let me take an observation.''
He swept the horizon with his telescope, and Tom noticed that the sailor kept it fixed on one particular spot for some time.
``Did you see anything?'' asked the lad.
``Well, there is a boat lying off there,'' was the answer. ``And some one is observing us through a glass. But I don't believe it matters. Probably they're only trying to see what sort of an odd fish we are.''
``All aboard, then,'' ordered Mr. Swift, and they went into the submarine. Tom and his father, with Captain Weston, remained in the conning tower. The signal was given, the electricity flowed into the forward and aft plates, and the Advance shot ahead on the surface.
The sailor raised his telescope once more and peered through a window in the tower. He uttered an exclamation.
``What's the matter?'' asked Tom.
``That other ship---a small steamer---is weighing anchor and seems to be heading this way,'' was the reply.
``Maybe it's some one hired by Berg to follow us and trace our movements,'' suggested Tom.
``If it is we'll fool them,'' added his father. ``Just keep an eye on them, captain, and I think we can show them a trick or two in a few minutes.''
Faster shot the Advance through the water. She had started on her way to get the gold from the sunken wreck, but already enemies were on the trail of the adventurers, for the ship the sailor had noticed was steaming after them.
\gutchapter{Chapter Fourteen}
In the Diving Suits
There was no doubt that the steamer was coming after the submarine. Several observations Captain Weston made confirmed this, and he reported the fact to Mr. Swift.
``Well, we'll change our plans, then,'' said the inventor. ``Instead of sailing on the surface we'll go below. But first let them get near so they may have the benefit of seeing what we do. Tom, go below, please, and tell Mr. Sharp to get every thing in readiness for a quick descent. We'll slow up a bit now, and let them get nearer to us.''
The speed of the submarine was reduced, and in a short time the strange steamer had overhauled her, coming to within hailing distance.
Mr. Swift signaled for the machinery to stop and the submarine came to a halt on the surface, bobbing about like a half-submerged bottle. The inventor opened a bull's-eye in the tower, and called to a man on the bridge of the steamer:
``What are you following us for?''
``Following you?'' repeated the man, for the strange vessel had also come to a stop. ``We're not following you.''
``It looks like it,'' replied Mr. Swift. ``You'd better give it up.''
``I guess the waters are free,'' was the quick retort. ``We'll follow you if we like.''
``Will you? Then come on!'' cried the inventor as he quickly closed the heavy glass window and pulled a lever. An instant later the submarine began to sink, and Mr. Swift could not help laughing as, just before the tower went under water, he had a glimpse of the astonished face of the man on the bridge. The latter had evidently not expected such a move as that.
Lower and lower in the water went the craft, until it was about two hundred feet below the surface. Then Mr. Swift left the conning tower, descended to the main part of the ship, and asked Tom and Captain Weston to take charge of the pilot house.
``Send her ahead, Tom,'' his father said. ``That fellow up above is rubbing his eyes yet, wondering where we are, I suppose.''
Forward shot the Advance under water, the powerful electrical plates pulling and pushing her on the way to secure the sunken gold.
All that morning a fairly moderate rate of speed was maintained, as it was thought best not to run the new machinery too fast.
Dinner was eaten about a quarter of a mile below the surface, but no one inside the submarine would ever have known it. Electric lights made the place as brilliant as could be desired, and the food, which Tom and Mr. Damon prepared, was equal to any that could have been served on land. After the meal they opened the shutters over the windows in the sides of the craft, and looked at the myriads of fishes swimming past, as the creatures were disclosed in the glare of the searchlight.
That night they were several hundred miles on their journey, for the craft was speedy, and leaving Tom and Captain Weston to take the first watch, the others went to bed.
``Bless my soul, but it does seem odd, though, to go to bed under water, like a fish,'' remarked Mr. Damon. ``If my wife knew this she would worry to death. She thinks I'm off automobiling. But this isn't half as dangerous as riding in a car that's always getting out of order. A submarine for mine, every time.''
``Wait until we get to the end of this trip,'' advised Tom. ``I guess you'll find almost as many things can happen in a submarine as can in an auto,'' and future events were to prove the young inventor to be right.
Everything worked well that night, and the ship made good progress. They rose to the surface the next morning to make sure of their position, and to get fresh air, though they did not really need the latter, as the reserve supply had not been drawn on, and was sufficient for several days, now that the oxygen machine had been put in running order.
On the second day the ship was sent to the bottom and halted there, as Mr. Swift wished to try the new diving suits. These were made of a new, light, but very strong metal to withstand the pressure of a great depth.
Tom, Mr. Sharp and Captain Weston donned the suits, the others agreeing to wait until they saw how the first trial resulted. Then, too, it was necessary for some one acquainted with the machinery to remain in the ship to operate the door and water chamber through which the divers had to pass to get out.
The usual plan, with some changes, was followed in letting the three out of the boat, and on to the bottom of the sea. They entered a chamber in the side of the submarine, water was gradually admitted until it equaled in pressure that outside, then an outer door was opened by means of levers, and they could step out.
It was a curious sensation to Tom and the others to feel that they were actually walking along the bed of the ocean. All around them was the water, and as they turned on the small electric lights in their helmets, which lights were fed by storage batteries fastened to the diving suits, they saw the fish, big and little, swarm up to them, doubtless astonished at the odd creatures which had entered their domain. On the sand of the bottom, and in and out among the shells and rocks, crawled great spider crabs, big eels and other odd creatures seldom seen on the surface of the water. The three divers found no difficulty in breathing, as there were air tanks fastened to their shoulders, and a constant supply of oxygen was fed through pipes into the helmets. The pressure of water did not bother them, and after the first sensation Tom began to enjoy the novelty of it. At first the inability to speak to his companions seemed odd, but he soon got so he could make signs and motions, and be understood.
They walked about for some time, and once the lad came upon a part of a wrecked vessel buried deep in the sand. There was no telling what ship it was, nor how long it had been there, and after silently viewing it, they continued on.
``It was great!'' were the first words Tom uttered when he and the others were once more inside the submarine and had removed the suits. ``If we can only walk around the wreck of the Boldero that way, we'll have all the gold out of her in no time. There are no life-lines nor air-hose to bother with in these diving suits.''
``They certainly are a success,'' conceded Mr. Sharp.
``Bless my topknot!'' cried Mr. Damon. ``I'll try it next time. I've always wanted to be a diver, and now I have the chance.''
The trip was resumed after the diving chamber had been closed, and on the third day Captain Weston announced, after a look at his chart, that they were nearing the Bahama Islands.
``We'll have to be careful not to run into any of the small keys,'' he said, that being the name for the many little points of land, hardly large enough to be dignified by the name of island. ``We must keep a constant lookout.''
Fortune favored them, though once, when Tom was steering, he narrowly avoided ramming a coral reef with the submarine. The searchlight showed it to him just in time, and he sheered off with a thumping in his heart.
The course was changed from south to east, so as to get ready to swing out of the way of the big shoulder of South America where Brazil takes up so much room, and as they went farther and farther toward the equator, they noticed that the waters teemed more and more with fish, some beautiful, some ugly and fear-inspiring, and some such monsters that it made one shudder to look at them, even through the thick glass of the bulls-eye windows.
\gutchapter{Chapter Fifteen}
At the Tropical Island
It was on the evening of the fourth day later that Captain Weston, who was steering the craft, suddenly called out:
``Land ho!''
``Where away?'' inquired Tom quickly, for he had read that this was the proper response to make.
``Dead ahead,'' answered the sailor with a smile. ``Shall we make for it, if I may be allowed the question?''
``What land is it likely to be?'' Mr. Swift wanted to know.
``Oh, some small tropical island,'' replied the seafaring man. ``It isn't down on the charts. Probably it's too small to note. I should say it was a coral island, but we may be able to find a spring of fresh water there, and some fruit.''
``Then we'll land there,'' decided the inventor. ``We can use some fresh water, though our distilling and ice apparatus does very well.''
They made the island just at dusk, and anchored in a little lagoon, where there was a good depth of water.
``Now for shore!'' cried Tom, as the submarine swung around on the chain. ``It looks like a fine place. I hope there are cocoanuts and oranges here. Shall I get out the electric launch, dad?''
``Yes, you may, and we'll all go ashore. It will do us good to stretch our legs a bit.''
Carried in a sort of pocket on the deck of the submarine was a small electric boat, capable of holding six. It could be slid from the pocket, or depression, into the water without the use of davits, and, with Mr. Sharp to aid him, Tom soon had the little craft afloat. The batteries were already charged, and just as the sun was going down the gold-seekers entered the launch and were soon on shore.
They found a good spring of water close at hand, and Tom's wish regarding the cocoanuts was realized, though there were no oranges. The lad took several of the delicious nuts, and breaking them open poured the milk into a collapsible cup he carried, drinking it eagerly. The others followed his example, and pronounced it the best beverage they had tasted in a long time.
The island was a typical tropical one, not very large, and it did not appear to have been often visited by man. There were no animals to be seen, but myriads of birds flew here and there amid the trees, the trailing vines and streamers of moss.
``Let's spend a day here to-morrow and explore it,'' proposed Tom, and his father nodded an assent. They went back to the submarine as night was beginning to gather, and in the cabin, after supper, talked over the happenings of their trip so far.
``Do you think we'll have any trouble getting the gold out of the wrecked vessel?'' asked Tom of Captain Weston, after a pause.
``Well, it's hard to say. I couldn't learn just how the wreck lays, whether it's on a sandy or a rocky bottom. If the latter, it won't be so hard, but if the sand has worked in and partly covered it, we'll have some difficulties, if I may be permitted to say so. However, don't borrow trouble. We're not there yet, though at the rate we're traveling it won't be long before we arrive.''
No watch was set that night, as it was not considered necessary. Tom was the first to arise in the morning, and he went out on the deck for a breath of fresh air before breakfast.
He looked off at the beautiful little island, and as his eye took in all of the little lagoon where the submarine was anchored he uttered a startled cry.
And well he might, for, not a hundred yards away, and nearer to the island than was the Advance, floated another craft---another craft, almost similar in shape and size to the one built by the Swifts. Tom rubbed his eyes to make sure he was not seeing double. No, there could be no mistake about it. There was another submarine at the tropical island.
As he looked, some one emerged from the conning tower of the second craft. The figure seemed strangely familiar. Tom knew in a moment who it was---Addison Berg. The agent saw the lad, too, and taking off his cap and making a mocking bow, he called out:
``Good morning! Have you got the gold yet?''
Tom did not know what to answer. Seeing the other submarine, at an island where he had supposed they would not be disturbed, was disconcerting enough, but to be greeted by Berg was altogether too much, Tom thought. His fears that the rival boat builders would follow had not been without foundation.
``Rather surprised to see us, aren't you?'' went on Mr. Berg, smiling.
``Rather,'' admitted Tom, choking over the word.
``Thought you'd be,'' continued Berg. ``We didn't expect to meet you so soon, but we're glad we did. I don't altogether like hunting for sunken treasure, with such indefinite directions as I have.''
``You---are going to---'' stammered Tom, and then he concluded it would be best not to say anything. But his talk had been heard inside the submarine. His father came to the foot of the conning tower stairway.
``To whom are you speaking, Tom?'' he asked.
``They're here, dad,'' was the youth's answer.
``Here? Who are here?''
``Berg and his employers. They've followed us, dad.''
\gutchapter{Chapter Sixteen}
``We'll Race You For It''
Mr. Swift hurried up on deck. He was accompanied by Captain Weston. At the sight of Tom's father, Mr. Berg, who had been joined by two other men, called out:
``You see we also concluded to give up the trial for the Government prize, Mr. Swift. We decided there was more money in something else. But we still will have a good chance to try the merits of our respective boats. We hurried and got ours fitted up almost as soon as you did yours, and I think we have the better craft.''
``I don't care to enter into any competition with you,'' said Mr. Swift coldly.
``Ah, but I'm afraid you'll have to, whether you want to or not,'' was the insolent reply.
``What's that? Do you mean to force this matter upon me?''
``I'm afraid I'll have to---my employers and I, that is. You see, we managed to pick up your trail after you left the Jersey coast, having an idea where you were bound, and we don't intend to lose you now.''
``Do you mean to follow us?'' asked Captain Weston softly.
``Well, you can put it that way if you like,'' answered one of the two men with Mr. Berg.
``I forbid it!'' cried Mr. Swift hotly. ``You have no right to sneak after us.''
``I guess the ocean is free,'' continued the rascally agent.
``Why do you persist in keeping after us?'' inquired the aged inventor, thinking it well to ascertain, if possible, just how much the men knew.
``Because we're after that treasure as well as you,'' was the bold reply. ``You have no exclusive right to it. The sunken ship is awaiting the first comer, and whoever gets there first can take the gold from the wreck. We intend to be there first, but we'll be fair with you.''
``Fair? What do you mean?'' demanded Tom.
``This: We'll race you for it. The first one to arrive will have the right to search the wreck for the gold bullion. Is that fair? Do you agree to it?''
``We agree to nothing with you,'' interrupted Captain Weston, his usual diffident manner all gone. ``I happen to be in partial command of this craft, and I warn you that if I find you interfering with us it won't be healthy for you. I'm not fond of fighting, but when I begin I don't like to stop,'' and he smiled grimly. ``You'd better not follow us.''
``We'll do as we please,'' shouted the third member of the trio on the deck of the other boat, which, as Tom could see, was named the Wonder. ``We intend to get that gold if we can.''
``All right. I've warned you,'' went on the sailor, and then, motioning to Tom and his father to follow, he went below.
``Well, what's to be done?'' asked Mr. Swift when they were seated in the living-room, and had informed the others of the presence of the rival submarine.
``The only thing I see to do is to sneak away unobserved, go as deep as possible, and make all haste for the wreck,'' advised the captain. ``They will depend on us, for they have evidently no chart of the wreck, though of course the general location of it may be known to them from reading the papers. I hoped I had thrown them off the track by the false chart I dropped, but it seems they were too smart for us.''
``Have they a right to follow us?'' asked Tom.
``Legally, but not morally. We can't prevent them, I'm afraid. The only thing to do is to get there ahead of them. It will be a race for the sunken treasure, and we must get there first.''
``What do you propose doing, captain?'' asked Mr. Damon. ``Bless my shirt-studs, but can't we pull their ship up on the island and leave it there?''
``I'm afraid such high-handed proceedings would hardly answer,'' replied Mr. Swift. ``No, as Captain Weston says, we must get there ahead of them. What do you think will be the best scheme, captain?''
``Well, there's no need for us to forego our plan to get fresh water. Suppose we go to the island, that is, some of us, leaving a guard on board here. We'll fill our tanks with fresh water, and at night we'll quietly sink below the surface and speed away.''
They all voted that an excellent idea, and little time was lost putting it into operation.
All the remainder of that day not a sign of life was visible about the Wonder. She lay inert on the surface of the lagoon, not far away from the Advance; but, though no one showed himself on the deck, Tom and his friends had no doubt but that their enemies were closely watching them.
As dusk settled down over the tropical sea, and as the shadows of the trees on the little island lengthened, those on board the Advance closed the Conning tower. No lights were turned on, as they did not want their movements to be seen, but Tom, his father and Mr. Sharp took their positions near the various machines and apparatus, ready to open the tanks and let the submarine sink to the bottom, as soon as it was possible to do this unobserved.
``Luckily there's no moon,'' remarked Captain Weston, as he took his place beside Tom. ``Once below the surface and we can defy them to find us. It is odd how they traced us, but I suppose that steamer gave them the clue.''
It rapidly grew dark, as it always does in the tropics, and when a cautious observation from the conning tower did not disclose the outlines of the other boat, those aboard the Advance rightly concluded that their rivals were unable to see them.
``Send her down, Tom,'' called his father, and with a hiss the water entered the tanks. The submarine quickly sank below the surface, aided by the deflecting rudder.
But alas for the hopes of the gold-seekers. No sooner was she completely submerged, with the engine started so as to send her out of the lagoon and to the open sea, than the waters all about were made brilliant by the phosphorescent phenomenon. In southern waters this frequently occurs. Millions of tiny creatures, which, it is said, swarm in the warm currents, give an appearance of fire to the ocean, and any object moving through it can plainly be seen. It was so with the Advance. The motion she made in shooting forward, and the undulations caused by her submersion, seemed to start into activity the dormant phosphorus, and the submarine was afloat in a sea of fire.
``Quick!'' cried Tom. ``Speed her up! Maybe we can get out of this patch of water before they see us.''
But it was too late. Above them they could hear the electric siren of the Wonder as it was blown to let them know that their escape had been noticed. A moment later the water, which acted as a sort of sounding-board, or telephone, brought to the ears of Tom Swift and his friends the noise of the engines of the other craft in operation. She was coming after them. The race for the possession of three hundred thousand dollars in gold was already under way. Fate seemed against those on board the Advance.
\gutchapter{Chapter Seventeen}
The Race
Directed by Captain Weston, who glanced at the compass and told him which way to steer to clear the outer coral reef, Tom sent the submarine ahead, signaling for full speed to the engine-room, where his father and Mr. Sharp were. The big dynamos purred like great cats, as they sent the electrical energy into the forward and aft plates, pulling and pushing the Advance forward. On and on she rushed under water, but ever as she shot ahead the disturbance in the phosphorescent water showed her position plainly. She would be easy to follow.
``Can't you get any more speed out of her?'' asked the captain of the lad.
``Yes,'' was the quick reply; ``by using the auxiliary screws I think we can. I'll try it.''
He signaled for the propellers, forward and aft, to be put in operation, and the motor moving the twin screws was turned on. At once there was a perceptible increase to the speed of the Advance.
``Are we leaving them behind?'' asked Tom anxiously, as he glanced at the speed gage, and noted that the submarine was now about five hundred feet below the surface.
``Hard to tell,'' replied the Captain. ``You'd have to take an observation to make sure.''
``I'll do it,'' cried the youth. ``You steer, please, and I'll go in the conning tower. I can look forward and aft there, as well as straight up. Maybe I can see the Wonder.''
Springing up the circular ladder leading into the tower, Tom glanced through the windows all about the small pilot house. He saw a curious sight. It was as if the submarine was in a sea of yellowish liquid fire. She was immersed in water which glowed with the flames that contained no heat. So light was it, in fact, that there was no need of the incandescents in the tower. The young inventor could have seen to read a paper by the illumination of the phosphorus. But he had something else to do than observe this phenomenon. He wanted to see if he could catch sight of the rival submarine.
At first he could make out nothing save the swirl and boiling of the sea, caused by the progress of the Advance through it. But suddenly, as he looked up, he was aware of some great, black body a little to the rear and about ten feet above his craft.
``A shark!'' he exclaimed aloud. ``An immense one, too.''
But the closer he looked the less it seemed like a shark. The position of the black object changed. It appeared to settle down, to be approaching the top of the conning tower. Then, with a suddenness that unnerved him for the time being, Tom recognized what it was; it was the underside of a ship. He could see the plates riveted together, and then, as he noted the rounded, cylindrical shape, he knew that it was a submarine. It was the Wonder. She was close at hand and was creeping up on the Advance. But, what was more dangerous, she seemed to be slowly settling in the water. Another moment and her great screws might crash into the Conning tower of the Swifts' boat and shave it off. Then the water would rush in, drowning the treasure-seekers like rats in a trap.
With a quick motion Tom yanked over the lever that allowed more water to flow into the ballast tanks. The effect was at once apparent. The Advance shot down toward the bottom of the sea. At the same time the young inventor signaled to Captain Weston to notify those in the engine-room to put on a little more speed. The Advance fairly leaped ahead, and the lad, looking up through the bull's-eye in the roof of the conning tower, had the satisfaction of seeing the rival submarine left behind.
The youth hurried down into the interior of the ship to tell what he had seen, and explain the reason for opening the ballast tanks. He found his father and Mr. Sharp somewhat excited over the unexpected maneuver of the craft.
``So they're still following us,'' murmured Mr. Swift. ``I don't see why we can't shake them off.''
``It's on account of this luminous water,'' explained Captain Weston. ``Once we are clear of that it will be easy, I think, to give them the slip. That is, if we can get out of their sight long enough. Of course, if they keep close after us, they can pick us up with their searchlight, for I suppose they carry one.''
``Yes,'' admitted the aged inventor, ``they have as strong a one as we have. In fact, their ship is second only to this one in speed and power. I know, for Bentley \& Eagert showed me some of the plans before they started it, and asked my opinion. This was before I had the notion of building a submarine. Yes, I am afraid we'll have trouble getting away from them.''
``I can't understand this phosphorescent glow keeping up so long,'' remarked Captain Weston. ``I've seen it in this locality several times, but it never covered such an extent of the ocean in my time. There must be changed conditions here now.''
For an hour or more the race was kept up, and the two submarines forged ahead through the glowing sea. The Wonder remained slightly above and to the rear of the other, the better to keep sight of her, and though the Advance was run to her limit of speed, her rival could not be shaken off. Clearly the Wonder was a speedy craft.
``It's too bad that we've got to fight them, as well as run the risk of lots of other troubles which are always present when sailing under water,'' observed Mr Damon, who wandered about the submarine like the nervous person he was. ``Bless my shirt-studs! Can't we blow them up, or cripple them in some way? They have no right to go after our treasure.''
``Well, I guess they've got as much right as we have,'' declared Tom. ``It goes to whoever reaches the wreck first. But what I don't like is their mean, sneaking way of doing it. If they went off on their own hook and looked for it I wouldn't say a word. But they expect us to lead them to the wreck, and then they'll rob us if they can. That's not fair.''
``Indeed, it isn't,'' agreed Captain Weston, ``if I may be allowed the expression. We ought to find some way of stopping them. But, if I'm not mistaken,'' he added quickly, looking from one of the port bull's-eyes, ``the phosphorescent glow is lessening. I believe we are running beyond that part of the ocean.''
There was no doubt of it, the glow was growing less and less, and ten minutes later the Advance was speeding along through a sea as black as night. Then, to avoid running into some wreck, it was necessary to turn on the searchlight.
``Are they still after us?'' asked Mr. Swift of his son, as he emerged from the engine-room, where he had gone to make some adjustments to the machinery, with the hope of increasing the speed.
``I'll go look,'' volunteered the lad. He climbed up into the conning tower again, and for a moment, as he gazed back into the black waters swirling all about, he hoped that they had lost the Wonder. But a moment later his heart sank as he caught sight, through the liquid element, of the flickering gleams of another searchlight, the rays undulating through the sea.
``Still following,'' murmured the young inventor. ``They're not going to give up. But we must make 'em---that's all.''
He went down to report what he had seen, and a consultation was held. Captain Weston carefully studied the charts of that part of the ocean, and finding that there was a great depth of water at hand, proposed a series of evolutions.
``We can go up and down, shoot first to one side and then to the other,'' he explained. ``We can even drop down to the bottom and rest there for a while. Perhaps, in that way, we can shake them off.''
They tried it. The Advance was sent up until her conning tower was out of the water, and then she was suddenly forced down until she was but a few feet from the bottom. She darted to the left, to the right, and even doubled and went back over the course she had taken. But all to no purpose. The Wonder proved fully as speedy, and those in her seemed to know just how to handle the submarine, so that every evolution of the Advance was duplicated. Her rival could not be shaken off.
All night this was kept up, and when morning came, though only the clocks told it, for eternal night was below the surface, the rival gold-seekers were still on the trail.
``They won't give up,'' declared Mr. Swift hopelessly.
``No, we've got to race them for it, just as Berg proposed,'' admitted Tom. ``But if they want a straightaway race we'll give it to 'em. Let's run her to the limit, dad.''
``That's what we've been doing, Tom.''
``No, not exactly, for we've been submerged a little too much to get the best speed out of our craft. Let's go a little nearer the surface, and give them the best race they'll ever have.''
Then the race began; and such a contest of speed as it was! With her propellers working to the limit, and every volt of electricity that was available forced into the forward and aft plates, the Advance surged through the water, about ten feet below the surface. But the Wonder kept after her, giving her knot for knot. The course of the leading submarine was easy to trace now, in the morning light which penetrated ten feet down.
``No use,'' remarked Tom again, when, after two hours, the Wonder was still close behind them. ``Our only chance is that they may have a breakdown.''
``Or run out of air, or something like that,'' added Captain Weston. ``They are crowding us pretty close. I had no idea they could keep up this speed. If they don't look out,'' he went on as he looked from one of the aft observation windows, ``they'll foul us, and---''
His remarks were interrupted by a jar to the Advance. She seemed to shiver and careened to one side. Then came another bump.
``Slow down!'' cried the captain, rushing toward the pilot house.
``What's the matter?'' asked Tom, as he threw the engines and electrical machines out of gear. ``Have we hit anything?''
``No. Something has hit us,'' cried the captain. ``Their submarine has rammed us.''
``Rammed us!'' repeated Mr. Swift. ``Tom, run out the electric cannon! They're trying to sink us! We'll have to fight them. Run out the stern electric gun and we'll make them wish they'd not followed us.''
\gutchapter{Chapter Eighteen}
The Electric Gun
There was much excitement aboard the Advance. The submarine came to a stop in the water, while the treasure-seekers waited anxiously for what was to follow. Would they be rammed again? This time, stationary as they were, and with the other boat coming swiftly on, a hole might be stove through the Advance, in spite of her powerful sides.
They had not long to wait. Again there came a jar, and once more the Swifts' boat careened. But the blow was a glancing one and, fortunately, did little damage.
``They certainly must be trying to sink us,'' agreed Captain Weston. ``Come, Tom, we'll take a look from the stern and see what they're up to.''
``And get the stern electric gun ready to fire,'' repeated Mr. Swift. ``We must protect ourselves. Mr. Sharp and I will go to the bow. There is no telling what they may do. They're desperate, and may ram us from in front.''
Tom and the captain hurried aft. Through the thick plate-glass windows they could see the blunt nose of the Wonder not far away, the rival submarine having come to a halt. There she lay, black and silent, like some monster fish waiting to devour its victim.
``There doesn't appear to be much damage done back here,'' observed Tom. ``No leaks. Guess they didn't puncture us.''
``Perhaps it was due to an accident that they rammed us,'' suggested the captain.
``Well, they wouldn't have done it if they hadn't followed us so close,'' was the opinion of the young inventor. ``They're taking too many chances. We've got to stop 'em.''
``What is this electric gun your father speaks of?''
``Why, it's a regular electric cannon. It fires a solid ball, weighing about twenty-five pounds, but instead of powder, which would hardly do under water, and instead of compressed air, which is used in the torpedo tubes of the Government submarines, we use a current of electricity. It forces the cannon ball out with great energy.''
``I wonder what they will do next?'' observed the captain, peering through a bull'seye.
``We can soon tell,'' replied the youth. ``We'll go ahead, and if they try to follow I'm going to fire on them.''
``Suppose you sink them?''
``I won't fire to do that; only to disable them. They brought it on themselves. We can't risk having them damage us. Help me with the cannon, will you please, captain?''
The electric cannon was a long, steel tube in the after part of the submarine. It projected a slight distance from the sides of the ship, and by an ingenious arrangement could be swung around in a ball and socket joint, thus enabling it to shoot in almost any direction.
It was the work of but a few minutes to get it ready and, with the muzzle pointing toward the Wonder, Tom adjusted the electric wires and inserted the solid shot.
``Now we're prepared for them!'' he cried. ``I think a good plan will be to start ahead, and if they try to follow to fire on them. They've brought it on themselves.''
``Correct,'' spoke Captain Weston.
Tom hurried forward to tell his father of this plan.
``We'll do it!'' cried Mr. Swift. ``Go ahead, Mr. Sharp, and we'll see if those scoundrels will follow.''
The young inventor returned on the run to the electric cannon. There was a whir of machinery, and the Advance moved forward. She increased her speed, and the two watchers in the stern looked anxiously out of the windows to see what their rivals would do.
For a moment no movement was noticeable on the part of the Wonder. Then, as those aboard her appeared to realize that the craft on which they depended to pilot them to the sunken treasure was slipping away, word was given to follow. The ship of Berg and his employers shot after the Advance.
``Here they come!'' cried Captain Weston. ``They're going to ram us again!''
``Then I'm going to fire on them!'' declared Tom savagely.
On came the Wonder, nearer and nearer. Her speed was rapidly increasing. Suddenly she bumped the Advance, and then, as if it was an unavoidable accident, the rear submarine sheered off to one side.
``They're certainly at it again!'' cried Tom, and peering from the bull's-eye he saw the Wonder shoot past the mouth of the electric cannon. ``Here it goes!'' he added.
He shoved over the lever, making the proper connection. There was no corresponding report, for the cannon was noiseless, but there was a slight jar as the projectile left the muzzle. The Wonder could be seen to heel over.
``You hit her! You hit her!'' cried Captain Weston. ``A good shot!''
``I was afraid she was past me when I pulled the lever,'' explained Tom. ``She went like a flash.''
``No, you caught her on the rudder,'' declared the captain. ``I think you've put her out of business. Yes, they're rising to the surface.''
The lad rapidly inserted another ball, and recharged the cannon. Then he peered out into the water, illuminated by the light of day overhead, as they were not far down. He could see the Wonder rising to the surface. Clearly something had happened.
``Maybe they're going to drop down on us from above, and try to sink us,'' suggested the youth, while he stood ready to fire again. ``If they do---''
His words were interrupted by a slight jar throughout the submarine.
``What was that?'' cried the captain.
``Dad fired the bow gun at them, but I don't believe he hit them,'' answered the young inventor.
``I wonder what damage I did? Guess we'll go to the surface to find out.''
Clearly the Wonder had given up the fight for the time being. In fact, she had no weapon with which to respond to a fusillade from her rival. Tom hastened forward and informed his father of what had happened.
``If her steering gear is out of order, we may have a chance to slip away,'' said Mr. Swift ``We'll go up and see what we can learn.''
A few minutes later Tom, his father and Captain Weston stepped from the conning tower, which was out of water, on to the little flat deck a short distance away lay the Wonder, and on her deck was Berg and a number of men, evidently members of the crew.
``Why did you fire on us?'' shouted the agent angrily.
``Why did you follow us?'' retorted Tom.
``Well, you've broken our rudder and disabled us,'' went on Berg, not answering the question. ``You'll suffer for this! I'll have you arrested.''
``You only got what you deserved,'' added Mr. Swift. ``You were acting illegally, following us, and you tried to sink us by ramming my craft before we retaliated by firing on you.''
``It was an accident, ramming you,'' said Berg. ``We couldn't help it. I now demand that you help us make repairs.''
``Well, you've got nerve!'' cried Captain Weston, his eyes flashing. ``I'd like to have a personal interview with you for about ten minutes. Maybe something besides your ship would need repairs then.''
Berg turned away, scowling, but did not reply. He began directing the crew what to do about the broken rudder.
``Come on,'' proposed Tom in a low voice, for sounds carry very easily over water. ``Let's go below and skip out while we have a chance. They can't follow now, and we can get to the sunken treasure ahead of them.''
``Good advice,'' commented his father. ``Come, Captain Weston, we'll go below and close the conning tower.''
Five minutes later the Advance sank from sight, the last glimpse Tom had of Berg and his men being a sight of them standing on the deck of their floating boat, gazing in the direction of their successful rival. The Wonder was left behind, while Tom and his friends were soon once more speeding toward the treasure wreck.
\gutchapter{Chapter Nineteen}
Captured
``Down deep,'' advised Captain Weston, as he stood beside Tom and Mr. Swift in the pilot house. ``As far as you can manage her, and then forward. We'll take no more chances with these fellows.''
``The only trouble is,'' replied the young inventor, ``that the deeper we go the slower we have to travel. The water is so dense that it holds us back.''
``Well, there is no special need of hurrying now,'' went on the sailor. ``No one is following you, and two or three days difference in reaching the wreck will not amount to anything.''
``Unless they repair their rudder, and take after us again,'' suggested Mr. Swift.
``They're not very likely to do that,'' was the captain's opinion. ``It was more by luck than good management that they picked us up before. Now, having to delay, as they will, to repair their steering gear, while we can go as deep as we please and speed ahead, it is practically impossible for them to catch up to us. No, I think we have nothing to fear from them.''
But though danger from Berg and his crowd was somewhat remote, perils of another sort were hovering around the treasure-seekers, and they were soon to experience them.
It was much different from sailing along in the airship, Tom thought, for there was no blue sky and fleecy clouds to see, and they could not look down and observe, far below them, cities and villages. Nor could they breathe the bracing atmosphere of the upper regions.
But if there was lack of the rarefied air of the clouds, there was no lack of fresh atmosphere. The big tanks carried a large supply, and whenever more was needed the oxygen machine would supply it.
As there was no need, however, of remaining under water for any great stretch of time, it was their practice to rise every day and renew the air supply, also to float along on the surface for a while, or speed along, with only the conning tower out, in order to afford a view, and to enable Captain Weston to take observations. But care was always exercised to make sure no ships were in sight when emerging on the surface, for the gold-seekers did not want to be hailed and questioned by inquisitive persons.
It was about four days after the disabling of the rival submarine, and the Advance was speeding along about a mile and a half under water. Tom was in the pilot house with Captain Weston, Mr. Damon was at his favorite pastime of looking out of the glass side windows into the ocean and its wonders, and Mr. Swift and the balloonist were, as usual, in the engine-room.
``How near do you calculate we are to the sunken wreck?'' asked Tom of his companion.
``Well, at the calculation we made yesterday, we are within about a thousand miles of it now. We ought to reach it in about four more days, if we don't have any accidents.''
``And how deep do you think it is?'' went on the lad.
``Well, I'm afraid it's pretty close to two miles, if not more. It's quite a depth, and of course impossible for ordinary divers to reach. But it will be possible in this submarine and in the strong diving suits your father has invented for us to get to it. Yes, I don't anticipate much trouble in getting out the gold, once we reach the wreck of course---''
The captain's remark was not finished. From the engine-room there came a startled shout:
``Tom! Tom! Your father is hurt! Come here, quick!''
``Take the wheel!'' cried the lad to the captain. ``I must go to my father.'' It was Mr. Sharp's voice he had heard.
Racing to the engine-room, Tom saw his parent doubled up over a dynamo, while to one side, his hand on a copper switch, stood Mr. Sharp.
``What's the matter?'' shouted the lad.
``He's held there by a current of electricity,'' replied the balloonist. ``The wires are crossed.''
``Why don't you shut off the current?'' demanded the youth, as he prepared to pull his parent from the whirring machine. Then he hesitated, for he feared he, too, would be glued fast by the terrible current, and so be unable to help Mr. Swift.
``I'm held fast here, too,'' replied the balloonist. ``I started to cut out the current at this switch, but there's a short circuit somewhere, and I can't let go, either. Quick, shut off all power at the main switchboard forward.''
Tom realized that this was the only thing to do. He ran forward and with a yank cut out all the electric wires. With a sigh of relief Mr. Sharp pulled his hands from the copper where he had been held fast as if by some powerful magnet, his muscles cramped by the current. Fortunately the electricity was of low voltage, and he was not burned. The body of Mr. Swift toppled backward from the dynamo, as Tom sprang to reach his father.
``He's dead!'' he cried, as he saw the pale face and the closed eyes.
``No, only badly shocked, I hope,'' spoke Mr. Sharp. ``But we must get him to the fresh air at once. Start the tank pumps. We'll rise to the surface.''
The youth needed no second bidding. Once more turning on the electric current, he set the powerful pumps in motion and the submarine began to rise. Then, aided by Captain Weston and Mr. Damon, the young inventor carried his father to a couch in the main cabin. Mr. Sharp took charge of the machinery.
Restoratives were applied, and there was a flutter of the eyelids of the aged inventor.
``I think he'll come around all right,'' said the sailor kindly, as he saw Tom's grief. ``Fresh air will be the thing for him. We'll be on the surface in a minute.''
Up shot the Advance, while Mr. Sharp stood ready to open the conning tower as soon as it should be out of water. Mr. Swift seemed to be rapidly reviving. With a bound the submarine, forced upward from the great depth, fairly shot out of the water. There was a clanking sound as the aeronaut opened the airtight door of the tower, and a breath of fresh air came in.
``Can you walk, dad, or shall we carry you?'' asked Tom solicitously.
``Oh, I---I'm feeling better now,'' was the inventor's reply. ``I'll soon be all right when I get out on deck. My foot slipped as I was adjusting a wire that had gotten out of order, and I fell so that I received a large part of the current. I'm glad I was not burned. Was Mr. Sharp hurt? I saw him run to the switch, just before I lost consciousness.''
``No, I'm all right,'' answered the balloonist. ``But allow us to get you out to the fresh air. You'll feel much better then.''
Mr. Swift managed to walk slowly to the ladder leading to the conning tower, and thence to the deck. The others followed him. As all emerged from the submarine they uttered a cry of astonishment.
There, not one hundred yards away, was a great warship, flying a flag which, in a moment, Tom recognized as that of Brazil. The cruiser was lying off a small island, and all about were small boats, filled with natives, who seemed to be bringing supplies from land to the ship. At the unexpected sight of the submarine, bobbing up from the bottom of the ocean, the natives uttered cries of fright. The attention of those on the warship was attracted, and the bridge and rails were lined with curious officers and men.
``It's a good thing we didn't come up under that ship,'' observed Tom. ``They would have thought we were trying to torpedo her. Do you feel better, dad?'' he asked, his wonder over the sight of the big vessel temporarily eclipsed in his anxiety for his parent.
``Oh, yes, much better. I'm all right now. But I wish we hadn't disclosed ourselves to these people. They may demand to know where we are going, and Brazil is too near Uruguay to make it safe to tell our errand. They may guess it, however, from having read of the wreck, and our departure.''
``Oh, I guess it will be all right,'' replied Captain Weston. ``We can tell them we are on a pleasure trip. That's true enough. It would give us great pleasure to find that gold.''
``There's a boat, with some officers in it, to judge by the amount of gold lace on them, putting off from the ship,'' remarked Mr. Sharp.
``Ha! Yes! Evidently they intend to pay us a formal visit,'' observed Mr. Damon. ``Bless my gaiters, though. I'm not dressed to receive company. I think I'll put on my dress suit.''
``It's too late,'' advised Tom. ``They'll be here in a minute.''
Urged on by the lusty arms of the Brazilian sailors, the boat, containing several officers, neared the floating submarine rapidly.
``Ahoy there!'' called an officer in the bow, his accent betraying his unfamiliarity with the English language. ``What craft are you?''
``Submarine, Advance, from New Jersey,'' replied Tom. ``Who are you?''
``Brazilian cruiser San Paulo,'' was the reply. ``Where are you bound?'' went on the officer.
``On pleasure,'' answered Captain Weston quickly. ``But why do you ask? We are an American ship, sailing under American colors. Is this Brazilian territory?''
``This island is---yes,'' came back the answer, and by this time the small boat was at the side of the submarine. Before the adventurers could have protested, had they a desire to do so, there were a number of officers and the crew of the San Paulo on the small deck.
With a flourish, the officer who had done the questioning drew his sword. Waving it in the air with a dramatic gesture, he exclaimed:
``You're our prisoners! Resist and my men shall cut you down like dogs! Seize them, men!''
The sailors sprang forward, each one stationing himself at the side of one of our friends, and grasping an arm.
``What does this mean?'' cried Captain Weston indignantly. ``If this is a joke, you're carrying it too far. If you're in earnest, let me warn you against interfering with Americans!''
``We know what we are doing,'' was the answer from the officer.
The sailor who had hold of Captain Weston endeavored to secure a tighter grip. The captain turned suddenly, and seizing the man about the waist, with an exercise of tremendous strength hurled him over his head and into the sea, the man making a great splash.
``That's the way I'll treat any one else who dares lay a hand on me!'' shouted the captain, who was transformed from a mild-mannered individual into an angry, modern giant. There was a gasp of astonishment at his feat, as the ducked sailor crawled back into the small boat. And he did not again venture on the deck of the submarine.
``Seize them, men!'' cried the gold-laced officer again, and this time he and his fellows, including the crew, crowded so closely around Tom and his friends that they could do nothing. Even Captain Weston found it impossible to offer any resistance, for three men grabbed hold of him but his spirit was still a fighting one, and he struggled desperately but uselessly.
``How dare you do this?'' he cried.
``Yes,'' added Tom, ``what right have you to interfere with us?''
``Every right,'' declared the gold-laced officer.
``You are in Brazilian territory, and I arrest you.''
``What for?'' demanded Mr. Sharp.
``Because your ship is an American submarine, and we have received word that you intend to damage our shipping, and may try to torpedo our warships. I believe you tried to disable us a little while ago, but failed. We consider that an act of war and you will be treated accordingly. Take them on board the San Paulo,'' the officer went on, turning to his aides. ``We'll try them by court-marital here. Some of you remain and guard this submarine. We will teach these filibustering Americans a lesson.''
\gutchapter{Chapter Twenty}
Doomed to Death
There was no room on the small deck of the submarine to make a stand against the officers and crew of the Brazilian warship. In fact, the capture of the gold-seekers had been effected so suddenly that their astonishment almost deprived them of the power to think clearly.
At another command from the officer, who was addressed as Admiral Fanchetti, several of the sailors began to lead Tom and his friends toward the small boat.
``Do you feel all right, father?'' inquired the lad anxiously, as he looked at his parent. ``These scoundrels have no right to treat us so.''
``Yes, Tom, I'm all right as far as the electric shock is concerned, but I don't like to be handled in this fashion.''
``We ought not to submit!'' burst out Mr. Damon. ``Bless the stars and stripes! We ought to fight.''
``There's no chance,'' said Mr. Sharp. ``We are right under the guns of the ship. They could sink us with one shot. I guess we'll have to give in for the time being.''
``It is most unpleasant, if I may be allowed the expression,'' commented Captain Weston mildly. He seemed to have lost his sudden anger, but there was a steely glint in his eyes, and a grim, set look around his month that showed his temper was kept under control only by an effort. It boded no good to the sailors who had hold of the doughty captain if he should once get loose, and it was noticed that they were on their guard.
As for Tom, he submitted quietly to the two Brazilians who had hold of either arm, and Mr. Swift was held by only one, for it was seen that he was feeble.
``Into the boat with them!'' cried Admiral Fanchetti. ``And guard them well, Lieutenant Drascalo, for I heard them plotting to escape,'' and the admiral signaled to a younger officer, who was in charge of the men guarding the prisoners.
``Lieutenant Drascalo, eh?'' murmured Mr. Damon. ``I think they made a mistake naming him. It ought to be Rascalo. He looks like a rascal.''
``Silenceo!'' exclaimed the lieutenant, scowling at the odd character.
``Bless my spark plug! He's a regular fire-eater!'' went on Mr. Damon, who appeared to have fully recovered his spirits.
``Silenceo!'' cried the lieutenant, scowling again, but Mr. Damon did not appear to mind.
Admiral Fanchetti and several others of the gold-laced officers remained aboard the submarine, while Tom and his friends were hustled into the small boat and rowed toward the warship.
``I hope they don't damage our craft,'' murmured the young inventor, as he saw the admiral enter the conning tower.
``If they do, we'll complain to the United States consul and demand damages,'' said Mr. Swift.
``I'm afraid we won't have a chance to communicate with the consul,'' remarked Captain Weston.
``What do you mean?'' asked Mr. Damon. ``Bless my shoelaces, but will these scoundrels---''
``Silenceo!'' cried Lieutenant Drascalo quickly. ``Dogs of Americans, do you wish to insult us?''
``Impossible; you wouldn't appreciate a good, genuine United States insult,'' murmured Tom under his breath.
``What I mean,'' went on the captain, ``is that these people may carry the proceedings off with a high hand. You heard the admiral speak of a court-martial.''
``Would they dare do that?'' inquired Mr. Sharp.
``They would dare anything in this part of the world, I'm afraid,'' resumed Captain Weston. ``I think I see their plan, though. This admiral is newly in command; his uniform shows that. He wants to make a name for himself, and he seizes on our submarine as an excuse. He can send word to his government that he destroyed a torpedo craft that sought to wreck his ship. Thus he will acquire a reputation.''
``But would his government support him in such a hostile act against the United States, a friendly nation?'' asked Tom.
``Oh, he would not claim to have acted against the United States as a power. He would say that it was a private submarine, and, as a matter of fact, it is. While we are under the protection of the stars and stripes, our vessel is not a Government one,'' and Captain Weston spoke the last in a low voice, so the scowling lieutenant could not hear.
``What will they do with us?'' inquired Mr. Swift.
``Have some sort of a court-martial, perhaps,'' went on the captain, ``and confiscate our craft. Then they will send us back home, I expect for they would not dare harm us.''
``But take our submarine!'' cried Tom. ``The villains---''
``Silenceo!'' shouted Lieutenant Drascalo and he drew his sword.
By this time the small boat was under the big guns of the San Paulo, and the prisoners were ordered, in broken English, to mount a companion ladder that hung over the side. In a short time they were on deck, amid a crowd of sailors, and they could see the boat going back to bring off the admiral, who signaled from the submarine. Tom and his friends were taken below to a room that looked like a prison, and there, a little later, they were visited by Admiral Fanchetti and several officers.
``You will be tried at once,'' said the admiral. ``I have examined your submarine and I find she carries two torpedo tubes. It is a wonder you did not sink me at once.''
``Those are not torpedo tubes!'' cried Tom, unable to keep silent, though Captain Weston motioned him to do so.
``I know torpedo tubes when I see them,'' declared the admiral. ``I consider I had a very narrow escape. Your country is fortunate that mine does not declare war against it for this act. But I take it you are acting privately, for you fly no flag, though you claim to be from the United States.''
``There's no place for a flag on the submarine,'' went on Tom. ``What good would it be under water?''
``Silenceo!'' cried Lieutenant Drascalo, the admonition to silence seeming to be the only command of which he was capable.
``I shall confiscate your craft for my government,'' went on the admiral, ``and shall punish you as the court-martial may direct. You will be tried at once.''
It was in vain for the prisoners to protest. Matters were carried with a high hand. They were allowed a spokesman, and Captain Weston, who understood Spanish, was selected, that language being used. But the defense was a farce, for he was scarcely listened to. Several officers testified before the admiral, who was judge, that they had seen the submarine rise out of the water, almost under the prow of the San Paulo. It was assumed that the Advance had tried to wreck the warship, but had failed. It was in vain that Captain Weston and the others told of the reason for their rapid ascent from the ocean depths---that Mr. Swift had been shocked, and needed fresh air. Their story was not believed.
``We have heard enough!'' suddenly exclaimed the admiral. ``The evidence against you is over-whelming---er---what you Americans call conclusive,'' and he was speaking then in broken English. ``I find you guilty, and the sentence of this court-martial is that you be shot at sunrise, three days hence!''
``Shot!'' cried Captain Weston, staggering back at this unexpected sentence. His companions turned white, and Mr. Swift leaned against his son for support.
``Bless my stars! Of all the scoundrelly!'' began Mr. Damon.
``Silenceo!'' shouted the lieutenant, waving his sword.
``You will be shot,'' proceeded the admiral. ``Is not that the verdict of the honorable court?'' he asked, looking at his fellow officers. They all nodded gravely.
``But look here!'' objected Captain Weston. ``You don't dare do that! We are citizens of the United States, and---''
``I consider you no better than pirates,'' interrupted the admiral. ``You have an armed submarine---a submarine with torpedo tubes. You invade our harbor with it, and come up almost under my ship. You have forfeited your right to the protection of your country, and I have no fear on that score. You will be shot within three days. That is all. Remove the prisoners.''
Protests were in vain, and it was equally useless to struggle. The prisoners were taken out on deck, for which they were thankful, for the interior of the ship was close and hot, the weather being intensely disagreeable. They were told to keep within a certain space on deck, and a guard of sailors, all armed, was placed near them. From where they were they could see their submarine floating on the surface of the little bay, with several Brazilians on the small deck. The Advance had been anchored, and was surrounded by a flotilla of the native boats, the brown-skinned paddlers gazing curiously at the odd craft.
``Well, this is tough luck!'' murmured Tom. ``How do you feel, dad?''
``As well as can be expected under the circumstances,'' was the reply. ``What do you think about this, Captain Weston?''
``Not very much, if I may be allowed the expression,'' was the answer.
``Do you think they will dare carry out that threat?'' asked Mr. Sharp.
The captain shrugged his shoulders. ``I hope it is only a bluff,'' he replied, ``made to scare us so we will consent to giving up the submarine, which they have no right to confiscate. But these fellows look ugly enough for anything,'' he went on.
``Then if there's any chance of them attempting to carry it out,'' spoke Tom, ``we've got to do something.''
``Bless my gizzard, of course!'' exclaimed Mr. Damon. ``But what? That's the question. To be shot! Why, that's a terrible threat! The villains---''
``Silenceo!'' shouted Lieutenant Drascalo, coming up at that moment.
\gutchapter{Chapter Twenty-One}
The Escape
Events had happened so quickly that day that the gold-hunters could scarcely comprehend them. It seemed only a short time since Mr. Swift had been discovered lying disabled on the dynamo, and what had transpired since seemed to have taken place in a few minutes, though it was, in reality, several hours. This was made manifest by the feeling of hunger on the part of Tom and his friends.
``I wonder if they're going to starve us, the scoundrels?'' asked Mr. Sharp, when the irate lieutenant was beyond hearing. ``It's not fair to make us go hungry and shoot us in the bargain.''
``That's so, they ought to feed us,'' put in Tom. As yet neither he nor the others fully realized the meaning of the sentence passed on them.
From where they were on deck they could look off to the little island. From it boats manned by natives were constantly putting off, bringing supplies to the ship. The place appeared to be a sort of calling station for Brazilian warships, where they could get fresh water and fruit and other food.
From the island the gaze of the adventurers wandered to the submarine, which lay not far away. They were chagrined to see several of the bolder natives clambering over the deck.
``I hope they keep out of the interior,'' commented Tom. ``If they get to pulling or hauling on the levers and wheels they may open the tanks and sink her, with the Conning tower open.''
``Better that, perhaps, than to have her fall into the hands of a foreign power,'' commented Captain Weston. ``Besides, I don't see that it's going to matter much to us what becomes of her after we're---''
He did not finish, but every one knew what he meant, and a grim silence fell upon the little group.
There came a welcome diversion, however, in the shape of three sailors, bearing trays of food, which were placed on the deck in front of the prisoners, who were sitting or lying in the shade of an awning, for the sun was very hot.
``Ha! Bless my napkin-ring!'' cried Mr. Damon with something of his former gaiety. ``Here's a meal, at all events. They don't intend to starve us. Eat hearty, every one.''
``Yes, we need to keep up our strength,'' observed Captain Weston.
``Why?'' inquired Mr. Sharp.
``Because we're going to try to escape!'' exclaimed Tom in a low voice, when the sailors who had brought the food had gone. ``Isn't that what you mean, captain?''
``Exactly. We'll try to give these villains the slip, and we'll need all our strength and wits to do it. We'll wait until night, and see what we can do.''
``But where will we escape to?'' asked Mr. Swift. ``The island will afford no shelter, and---''
``No, but our submarine will,'' went on the sailor.
``It's in the possession of the Brazilians,'' objected Tom.
``Once I get aboard the Advance twenty of those brown-skinned villains won't keep me prisoner,'' declared Captain Weston fiercely. ``If we can only slip away from here, get into the small boat, or even swim to the submarine, I'll make those chaps on board her think a hurricane has broken loose.''
``Yes, and I'll help,'' said Mr. Damon.
``And I,'' added Tom and the balloonist.
``That's the way to talk,'' commented the captain. ``Now let's eat, for I see that rascally lieutenant coming this way, and we mustn't appear to be plotting, or he'll be suspicious.''
The day passed slowly, and though the prisoners seemed to be allowed considerable liberty, they soon found that it was only apparent. Once Tom walked some distance from that portion of the deck where he and the others had been told to remain. A sailor with a gun at once ordered him back. Nor could they approach the rails without being directed, harshly enough at times, to move back amidships.
As night approached the gold-seekers were on the alert for any chance that might offer to slip away, or even attack their guard, but the number of Brazilians around them was doubled in the evening, and after supper, which was served to them on deck by the light of swinging lanterns, they were taken below and locked in a stuffy cabin. They looked helplessly at each other.
``Don't give up,'' advised Captain Weston. ``It's a long night. We may be able to get out of here.''
But this hope was in vain. Several times he and Tom, thinking the guards outside the cabin were asleep, tried to force the lock of the door with their pocket-knives, which had not been taken from them. But one of the sailors was aroused each time by the noise, and looked in through a barred window, so they had to give it up. Slowly the night passed, and morning found the prisoners pale, tired and discouraged. They were brought up on deck again, for which they were thankful, as in that tropical climate it was stifling below.
During the day they saw Admiral Fanchetti and several of his officers pay a visit to the submarine. They went below through the opened conning tower, and were gone some time.
``I hope they don't disturb any of the machinery,'' remarked Mr. Swift. ``That could easily do great damage.''
Admiral Fanchetti seemed much pleased with himself when he returned from his visit to the submarine.
``You have a fine craft,'' he said to the prisoners. ``Or, rather, you had one. My government now owns it. It seems a pity to shoot such good boat builders, but you are too dangerous to be allowed to go.''
If there had been any doubt in the minds of Tom and his friends that the sentence of the court-martial was only for effect, it was dispelled that day. A firing squad was told off in plain view of them, and the men were put through their evolutions by Lieutenant Drascalo, who had them load, aim and fire blank cartridges at an imaginary line of prisoners. Tom could not repress a shudder as he noted the leveled rifles, and saw the fire and smoke spurt from the muzzles.
``Thus we shall do to you at sunrise to-morrow,'' said the lieutenant, grinning, as he once more had his men practice their grim work.
It seemed hotter than ever that day. The sun was fairly broiling, and there was a curious haziness and stillness to the air. It was noticed that the sailors on the San Paulo were busy making fast all loose articles on deck with extra lashings, and hatch coverings were doubly secured.
``What do you suppose they are up to?'' asked Tom of Captain Weston.
``I think it is coming on to blow,'' he replied, ``and they don't want to be caught napping. They have fearful storms down in this region at this season of the year, and I think one is about due.''
``I hope it doesn't wreck the submarine,'' spoke Mr. Swift. ``They ought to close the hatch of the conning tower, for it won't take much of a sea to make her ship considerable water.''
Admiral Fanchetti had thought of this, however, and as the afternoon wore away and the storm signs multiplied, he sent word to close the submarine. He left a few sailors aboard inside on guard.
``It's too hot to eat,'' observed Tom, when their supper had been brought to them, and the others felt the same way about it. They managed to drink some cocoanut milk, prepared in a palatable fashion by the natives of the island, and then, much to their disgust, they were taken below again and locked in the cabin.
``Whew! But it certainly is hot!'' exclaimed Mr. Damon as he sat down on a couch and fanned himself. ``This is awful!''
``Yes, something is going to happen pretty soon,'' observed Captain Weston. ``The storm will break shortly, I think.''
They sat languidly about the cabin. It was so oppressive that even the thought of the doom that awaited them in the morning could hardly seem worse than the terrible heat. They could hear movements going on about the ship, movements which indicated that preparations were being made for something unusual. There was a rattling of a chain through a hawse hole, and Captain Weston remarked:
``They're putting down another anchor. Admiral Fanchetti had better get away from the island, though, unless he wants to be wrecked. He'll be blown ashore in less than no time. No cable or chain will hold in such storms as they have here.''
There came a period of silence, which was suddenly broken by a howl as of some wild beast.
``What's that?'' cried Tom, springing up from where he was stretched out on the cabin floor.
``Only the wind,'' replied the captain. ``The storm has arrived.''
The howling kept up, and soon the ship began to rock. The wind increased, and a little later there could be heard, through an opened port in the prisoners' cabin, the dash of rain.
``It's a regular hurricane!'' exclaimed the captain. ``I wonder if the cables will hold?''
``What about the submarine?'' asked Mr. Swift anxiously.
``I haven't much fear for her. She lies so low in the water that the wind can't get much hold on her. I don't believe she'll drag her anchor.''
Once more came a fierce burst of wind, and a dash of rain, and then, suddenly above the outburst of the elements, there sounded a crash on deck. It was followed by excited cries.
``Something's happened!'' yelled Tom. The prisoners gathered in a frightened group in the middle of the cabin. The cries were repeated, and then came a rush of feet just outside the cabin door.
``Our guards! They're leaving!'' shouted Tom.
``Right!'' exclaimed Captain Weston. ``Now's our chance! Come on! If we're going to escape we must do it while the storm is at its height, and all is in confusion. Come on!''
Tom tried the door. It was locked.
``One side!'' shouted the captain, and this time he did not pause to say ``by your leave.'' He came at the portal on the run, and his shoulder struck it squarely. There was a splintering and crashing of wood, and the door was burst open.
``Follow me!'' cried the valiant sailor, and Tom and the others rushed after him. They could hear the wind howling more loudly than ever, and as they reached the deck the rain dashed into their faces with such violence that they could hardly see. But they were aware that something had occurred. By the light of several lanterns swaying in the terrific blast they saw that one of the auxiliary masts had broken off near the deck.
It had fallen against the chart house, smashing it, and a number of sailors were laboring to clear away the wreckage.
``Fortune favors us!'' cried Captain Weston. ``Come on! Make for the small boat. It's near the side ladder. We'll lower the boat and pull to the submarine.''
There came a flash of lightning, and in its glare Tom saw something that caused him to cry out.
``Look!'' he shouted. ``The submarine. She's dragged her anchors!''
The Advance was much closer to the warship than she had been that afternoon. Captain Weston looked over the side.
``It's the San Paulo that's dragging her anchors, not the submarine!'' he shouted. ``We're bearing down on her! We must act quickly. Come on, we'll lower the boat!''
In the rush of wind and the dash of rain the prisoners crowded to the accommodation companion ladder, which was still over the side of the big ship. No one seemed to be noticing them, for Admiral Fanchetti was on the bridge, yelling orders for the clearing away of the wreckage. But Lieutenant Drascalo, coming up from below at that moment, caught sight of the fleeing ones. Drawing his sword, he rushed at them, shouting:
``The prisoners! The prisoners! They are escaping!''
Captain Weston leaped toward the lieutenant.
``Look out for his sword!'' cried Tom. But the doughty sailor did not fear the weapon. Catching up a coil of rope, he cast it at the lieutenant. It struck him in the chest, and he staggered back, lowering his sword.
Captain Weston leaped forward, and with a terrific blow sent Lieutenant Drascalo to the deck.
``There!'' cried the sailor. ``I guess you won't yell `Silenceo!' for a while now.''
There was a rush of Brazilians toward the group of prisoners. Tom caught one with a blow on the chin, and felled him, while Captain Weston disposed of two more, and Mr. Sharp and Mr. Damon one each. The savage fighting of the Americans was too much for the foreigners, and they drew back.
``Come on!'' cried Captain Weston again. ``The storm is getting worse. The warship will crash into the submarine in a few minutes. Her anchors aren't holding. I didn't think they would.''
He made a dash for the ladder, and a glance showed him that the small boat was in the water at the foot of it. The craft had not been hoisted on the davits.
``Luck's with us at last!'' cried Tom, seeing it also. ``Shall I help you, dad?''
``No; I think I'm all right. Go ahead.''
There came such a gust of wind that the San Paulo was heeled over, and the wreck of the mast, rolling about, crashed into the side of a deck house, splintering it. A crowd of sailors, led by Admiral Fanchetti, who were again rushing on the escaping prisoners, had to leap back out of the way of the rolling mast.
``Catch them! Don't let them get away!'' begged the commander, but the sailors evidently had no desire to close in with the Americans.
Through the rush of wind and rain Tom and his friends staggered down the ladder. It was hard work to maintain one's footing, but they managed it. On account of the high side of the ship the water was comparatively calm under her lee, and, though the small boat was bobbing about, they got aboard. The oars were in place, and in another moment they had shoved off from the landing stage which formed the foot of the accommodation ladder.
``Now for the Advance!'' murmured Captain Weston.
``Come back! Come back, dogs of Americans!'' cried a voice at the rail over their heads, and looking up, Tom saw Lieutenant Drascalo. He had snatched a carbine from a marine, and was pointing it at the recent prisoners. He fired, the flash of the gun and a dazzling chain of lightning coming together. The thunder swallowed up the report of the carbine, but the bullet whistled uncomfortable close to Tom's head. The blackness that followed the lightning shut out the view of everything for a few seconds, and when the next flash came the adventurers saw that they were close to their submarine.
A fusillade of shots sounded from the deck of the warship, but as the marines were poor marksmen at best, and as the swaying of the ship disconcerted them, our friends were in little danger.
There was quite a sea once they were beyond the protection of the side of the warship, but Captain Weston, who was rowing, knew how to manage a boat skillfully, and he soon had the craft alongside the bobbing submarine.
``Get aboard, now, quick!'' he cried.
They leaped to the small deck, casting the rowboat adrift. It was the work of but a moment to open the conning tower. As they started to descend they were met by several Brazilians coming up.
``Overboard with 'em!'' yelled the captain. ``Let them swim ashore or to their ship!''
With almost superhuman strength he tossed one big sailor from the small deck. Another showed fight, but he went to join his companion in the swirling water. A man rushed at Tom, seeking the while to draw his sword, but the young inventor, with a neat left-hander, sent him to join the other two, and the remainder did not wait to try conclusions. They leaped for their lives, and soon all could be seen, in the frequent lightning flashes, swimming toward the warship which was now closer than ever to the submarine.
``Get inside and we'll sink below the surface!'' called Tom. ``Then we don't care what happens.''
They closed the steel door of the conning tower. As they did so they heard the patter of bullets from carbines fired from the San Paulo. Then came a violent tossing of the Advance; the waves were becoming higher as they caught the full force of the hurricane. It took but an instant to sever, from within, the cable attached to the anchor, which was one belonging to the warship. The Advance began drifting.
``Open the tanks, Mr. Sharp!'' cried Tom. ``Captain Weston and I will steer. Once below we'll start the engines.''
Amid a crash of thunder and dazzling flashes of lightning, the submarine began to sink. Tom, in the conning tower had a sight of the San Paulo as it drifted nearer and nearer under the influence of the mighty wind. As one bright flash came he saw Admiral Fanchetti and Lieutenant Drascalo leaning over the rail and gazing at the Advance.
A moment later the view faded from sight as the submarine sank below the surface of the troubled sea. She was tossed about for some time until deep enough to escape the surface motion. Waiting until she was far enough down so that her lights would not offer a mark for the guns of the warship, the electrics were switched on.
``We're safe now!'' cried Tom, helping his father to his cabin. ``They've got too much to attend to themselves to follow us now, even if they could. Shall we go ahead, Captain Weston?''
``I think so, yes, if I may be allowed to express my opinion,'' was the mild reply, in strange contrast to the strenuous work in which the captain had just been engaged.
Tom signaled to Mr. Sharp in the engine-room, and in a few seconds the Advance was speeding away from the island and the hostile vessel. Nor, deep as she was now, was there any sign of the hurricane. In the peaceful depths she was once more speeding toward the sunken treasure.
\gutchapter{Chapter Twenty-Two}
At the Wreck
``Well,'' remarked Mr. Damon, as the submarine hurled herself forward through the ocean, ``I guess that firing party will have something else to do to-morrow morning besides aiming those rifles at us.''
``Yes, indeed,'' agreed Tom. ``They'll be lucky if they save their ship. My, how that wind did blow!''
``You're right,'' put in Captain Weston. ``When they get a hurricane down in this region it's no cat's paw. But they were a mighty careless lot of sailors. The idea of leaving the ladder over the side, and the boat in the water.''
``It was a good thing for us, though,'' was Tom's opinion.
``Indeed it was,'' came from the captain. ``But as long as we are safe now I think we'd better take a look about the craft to see if those chaps did any damage. They can't have done much, though, or she wouldn't be running so smoothly. Suppose you go take a look, Tom, and ask your father and Mr. Sharp what they think. I'll steer for a while, until we get well away from the island.''
The young inventor found his father and the balloonist busy in the engine-room. Mr. Swift had already begun an inspection of the machinery, and so far found that it had not been injured. A further inspection showed that no damage had been done by the foreign guard that had been in temporary possession of the Advance, though the sailors had made free in the cabins, and had broken into the food lockers, helping themselves plentifully. But there was still enough for the gold-seekers.
``You'd never know there was a storm raging up above,'' observed Tom as he rejoined Captain Weston in the lower pilot house, where he was managing the craft. ``It's as still and peaceful here as one could wish.''
``Yes, the extreme depths are seldom disturbed by a surface storm. But we are over a mile deep now. I sent her down a little while you were gone, as I think she rides a little more steadily.''
All that night they speeded forward, and the next day, rising to the surface to take an observation, they found no traces of the storm, which had blown itself out. They were several hundred miles away from the hostile warship, and there was not a vessel in sight on the broad expanse of blue ocean.
The air tanks were refilled, and after sailing along on the surface for an hour or two, the submarine was again sent below, as Captain Weston sighted through his telescope the smoke of a distant steamer.
``As long as it isn't the Wonder, we're all right,'' said Tom. ``Still, we don't want to answer a lot of questions about ourselves and our object.''
``No. I fancy the Wonder will give up the search,'' remarked the captain, as the Advance was sinking to the depths.
``We must be getting pretty near to the end of our search ourselves,'' ventured the young inventor.
``We are within five hundred miles of the intersection of the forty-fifth parallel and the twenty-seventh meridian, east from Washington,'' said the captain. ``That's as near as I could locate the wreck. Once we reach that point we will have to search about under water, for I don't fancy the other divers left any buoys to mark the spot.''
It was two days later, after uneventful sailing, partly on the surface, and partly submerged, that Captain Weston, taking a noon observation, announced:
``Well, we're here!''
``Do you mean at the wreck?'' asked Mr. Swift eagerly.
``We're at the place where she is supposed to lie, in about two miles of water,'' replied the captain. ``We are quite a distance off the coast of Uruguay, about opposite the harbor of Rio de La Plata. From now on we shall have to nose about under water, and trust to luck.''
With her air tanks filled to their capacity, and Tom having seen that the oxygen machine and other apparatus was in perfect working order, the submarine was sent below on her search. Though they were in the neighborhood of the wreck, the adventurers might still have to do considerable searching before locating it. Lower and lower they sank into the depths of the sea, down and down, until they were deeper than they had ever gone before. The pressure was tremendous, but the steel sides of the Advance withstood it.
Then began a search that lasted nearly a week. Back and forth they cruised, around in great circles, with the powerful searchlight focused to disclose the sunken treasure ship. Once Tom, who was observing the path of light in the depths from the conning tower, thought he had seen the remains of the Boldero, for a misty shape loomed up in front of the submarine, and he signaled for a quick stop. It was a wreck, but it had been on the ocean bed for a score of years, and only a few timbers remained of what had been a great ship. Much disappointed, Tom rang for full speed ahead again, and the current was sent into the great electric plates that pulled and pushed the submarine forward.
For two days more nothing happened. They searched around under the green waters, on the alert for the first sign, but they saw nothing. Great fish swam about them, sometimes racing with the Advance. The adventurers beheld great ocean caverns, and skirted immense rocks, where dwelt monsters of the deep. Once a great octopus tried to do battle with the submarine and crush it in its snaky arms, but Tom saw the great white body, with saucer-shaped eyes, in the path of light and rammed him with the steel point. The creature died after a struggle.
They were beginning to despair when a full week had passed and they were seemingly as far from the wreck as ever. They went to the surface to enable Captain Weston to take another observation. It only confirmed the other, and showed that they were in the right vicinity. But it was like looking for a needle in a haystack, almost, to find the sunken ship in that depth of water.
``Well, we'll try again,'' said Mr. Swift, as they sank once more beneath the surface.
It was toward evening, on the second day after this, that Tom, who was on duty in the conning tower, saw a black shape looming up in front of the submarine, the searchlight revealing it to him far enough away so that he could steer to avoid it. He thought at first that it was a great rock, for they were moving along near the bottom, but the peculiar shape of it soon convinced him that this could not be. It came more plainly into view as the submarine approached it more slowly, then suddenly, out of the depths in the illumination from the searchlight, the young inventor saw the steel sides of a steamer. His heart gave a great thump, but he would not call out yet, fearing that it might be some other vessel than the one containing the treasure.
He steered the Advance so as to circle it. As he swept past the bows he saw in big letters near the sharp prow the word, Boldero.
``The wreck! The wreck!'' he cried, his voice ringing through the craft from end to end. ``We've found the wreck at last!''
``Are you sure?'' cried his father, hurrying to his son, Captain Weston following.
``Positive,'' answered the lad. The submarine was slowing up now, and Tom sent her around on the other side. They had a good view of the sunken ship. It seemed to be intact, no gaping holes in her sides, for only her plates had started, allowing her to sink gradually.
``At last,'' murmured Mr. Swift. ``Can it be possible we are about to get the treasure?''
``That's the Boldero, all right,'' affirmed Captain Weston. ``I recognize her, even if the name wasn't on her bow. Go right down on the bottom, Tom, and we'll get out the diving suits and make an examination.''
The submarine settled to the ocean bed. Tom glanced at the depth gage. It showed over two miles and a half. Would they be able to venture out into water of such enormous pressure in the comparatively frail diving suits, and wrest the gold from the wreck? It was a serious question.
The Advance came to a stop. In front of her loomed the great bulk of the Boldero, vague and shadowy in the flickering gleam of the searchlight. As the gold-seekers looked at her through the bull's-eyes of the conning tower, several great forms emerged from beneath the wreck's bows.
``Deep-water sharks!'' exclaimed Captain Weston, ``and monsters, too. But they can't bother us. Now to get out the gold!''
\gutchapter{Chapter Twenty-Three}
Attacked by Sharks
For a few minutes after reaching the wreck, which had so occupied their thoughts for the past weeks, the adventurers did nothing but gaze at it from the ports of the submarine. The appearance of the deep-water sharks gave them no concern, for they did not imagine the ugly creatures would attack them. The treasure-seekers were more engrossed with the problem of getting out the gold.
``How are we going to get at it?'' asked Tom, as he looked at the high sides of the sunken ship, which towered well above the comparatively small Advance.
``Why, just go in and get it,'' suggested Mr. Damon. ``Where is gold in a cargo usually kept, Captain Weston? You ought to know, I should think. Bless my pocketbook!''
``Well, I should say that in this case the bullion would be kept in a safe in the captain's cabin,'' replied the sailor. ``Or, if not there, in some after part of the vessel, away from where the crew is quartered. But it is going to be quite a problem to get at it. We can't climb the sides of the wreck, and it will be impossible to lower her ladder over the side. However, I think we had better get into the diving suits and take a closer look. We can walk around her.''
``That's my idea,'' put in Mr. Sharp. ``But who will go, and who will stay with the ship?''
``I think Tom and Captain Weston had better go,'' suggested Mr. Swift. ``Then, in case anything happens, Mr. Sharp, you and I will be on board to manage matters.''
``You don't think anything will happen, do you, dad?'' asked his son with a laugh, but it was not an easy one, for the lad was thinking of the shadowy forms of the ugly sharks.
``Oh, no, but it's best to be prepared,'' answered his father.
The captain and the young inventor lost no time in donning the diving suits. They each took a heavy metal bar, pointed at one end, to use in assisting them to walk on the bed of the ocean, and as a protection in case the sharks might attack them. Entering the diving chamber, they were shut in, and then water was admitted until the pressure was seen, by gauges, to be the same as that outside the submarine. Then the sliding steel door was opened. At first Tom and the captain could barely move, so great was the pressure of water on their bodies. They would have been crushed but for the protection afforded by the strong diving suits.
In a few minutes they became used to it, and stepped out on the floor of the ocean. They could not, of course, speak to each other, but Tom looked through the glass eyes of his helmet at the captain, and the latter motioned for the lad to follow. The two divers could breathe perfectly, and by means of small, but powerful lights on the helmets, the way was lighted for them as they advanced.
Slowly they approached the wreck, and began a circuit of her. They could see several places where the pressure of the water, and the strain of the storm in which she had foundered, had 'opened the plates of the ship, but in no case were the openings large enough to admit a person. Captain Weston put his steel bar in one crack, and tried to pry it farther open, but his strength was not equal to the task. He made some peculiar motions, but Tom could not understand them.
They looked for some means by which they could mount to the decks of the Boldero, but none was visible. It was like trying to scale a fifty-foot smooth steel wall. There was no place for a foothold. Again the sailor made some peculiar motions, and the lad puzzled over them. They had gone nearly around the wreck now, and as yet had seen no way in which to get at the gold. As they passed around the bow, which was in a deep shadow from a great rock, they caught sight of the submarine lying a short distance away. Light streamed from many hull's-eyes, and Tom felt a sense of security as he looked at her, for it was lonesome enough in that great depth of water, unable to speak to his companion, who was a few feet in advance.
Suddenly there was a swirling of the water, and Tom was nearly thrown off his feet by the rush of some great body. A long, black shadow passed over his head, and an instant later he saw the form of a great shark launched at Captain Weston. The lad involuntarily cried in alarm, but the result was surprising. He was nearly deafened by his own voice, confined as the sound was in the helmet he wore. But the sailor, too, had felt the movement of the water, and turned just in time. He thrust upward with his pointed bar. But he missed the stroke, and Tom, a moment later, saw the great fish turn over so that its mouth, which is far underneath its snout, could take in the queer shape which the shark evidently thought was a choice morsel. The big fish did actually get the helmet of Captain Weston inside its jaws, but probably it would have found it impossible to crush the strong steel. Still it might have sprung the joints, and water would have entered, which would have been as fatal as though the sailor had been swallowed by the shark. Tom realized this and, moving as fast as he could through the water, he came up behind the monster and drove his steel bar deep into it.
The sea was crimsoned with blood, and the savage creature, opening its mouth, let go of the captain. It turned on Tom, who again harpooned it. Then the fish darted off and began a wild flurry, for it was dying. The rush of water nearly threw Tom off his feet, but he managed to make his way over to his friend, and assist him to rise. A confident look from the sailor showed the lad that Captain Weston was uninjured, though he must have been frightened. As the two turned to make their way back to the submarine, the waters about them seemed alive with the horrible monsters.
It needed but a glance to show what they were, Sharks! Scores of them, long, black ones, with their ugly, undershot mouths. They had been attracted by the blood of the one Tom had killed, but there was not a meal for all of them off the dying creature, and the great fish might turn on the young inventor and his companion.
The two shrank closer toward the wreck. They might get under the prow of that and be safe. But even as they started to move, several of the sea wolves darted quickly at them. Tom glanced at the captain. What could they do? Strong as were the diving suits, a combined attack by the sharks, with their powerful jaws, would do untold damage.
At that moment there seemed some movement on board the submarine. Tom could see his father looking from the conning tower, and the aged inventor seemed to be making some motions. Then Tom understood. Mr. Swift was directing his son and Captain Weston to crouch down. The lad did so, pulling the sailor after him. Then Tom saw the bow electric gun run out, and aimed at the mass of sharks, most of whom were congregated about the dead one. Into the midst of the monsters was fired a number of small projectiles, which could be used in the electric cannon in place of the solid shot. Once more the waters were red with blood, and those sharks which were not killed swirled off. Tom and Captain Weston were saved. They were soon inside the submarine again, telling their thrilling story.
``It's lucky you saw us, dad,'' remarked the lad, blushing at the praise Mr. Damon bestowed on him for killing the monster which had attacked the captain.
``Oh, I was on the lookout,'' said the inventor. ``But what about getting into the wreck?''
``I think the only way we can do it will be to ram a hole in her side,'' said Captain Weston. ``That was what I tried to tell Tom by motions, but he didn't seem to understand me.''
``No,'' replied the lad, who was still a little nervous from his recent experience. ``I thought you meant for us to turn it over, bottom side up,'' and he laughed.
``Bless my gizzard! Just like a shark,'' commented Mr. Damon.
``Please don't mention them,'' begged Tom. ``I hope we don't see any more of them.''
``Oh, I fancy they have been driven far enough away from this neighborhood now,'' commented the captain. ``But now about the wreck. We may be able to approach it from above. Suppose we try to lower the submarine on it? That will save ripping it open.''
This was tried a little later, but would not work. There were strong currents sweeping over the top of the Boldero, caused by a submerged reef near which she had settled. It was a delicate task to sink the submarine on her decks, and with the deep waters swirling about was found to be impossible, even with the use of the electric plates and the auxiliary screws. Once more the Advance settled to the ocean bed, near the wreck.
``Well, what's to be done?'' asked Tom, as he looked at the high steel sides.
``Ram her, tear a hole, and then use dynamite,'' decided Captain Weston promptly. ``You have some explosive, haven't you, Mr. Swift?''
``Oh, yes. I came prepared for emergencies.''
``Then we'll blow up the wreck and get at the gold.''
\gutchapter{Chapter Twenty-Four}
Ramming the Wreck
Fitted with a long, sharp steel ram in front, the Advance was peculiarly adapted for this sort of work. In designing the ship this ram was calculated to be used against hostile vessels in war time, for the submarine was at first, as we know, destined for a Government boat. Now the ram was to serve a good turn.
To make sure that the attempt would be a success, the machinery of the craft was carefully gone over. It was found to be in perfect order, save for a few adjustments which were needed. Then, as it was night, though there was no difference in the appearance of things below the surface, it was decided to turn in, and begin work in the morning. Nor did the gold-seekers go to the surface, for they feared they might encounter a storm.
``We had trouble enough locating the wreck,'' said Captain Weston, ``and if we go up we may be blown off our course. We have air enough to stay below, haven't we, Tom?''
``Plenty,'' answered the lad, looking at the gages.
After a hearty breakfast the next morning, the submarine crew got ready for their hard task. The craft was backed away as far as was practical, and then, running at full speed, she rammed the wreck. The shock was terrific, and at first it was feared some damage had been done to the Advance, but she stood the strain.
``Did we open up much of a hole?'' anxiously asked Mr. Swift.
``Pretty good,'' replied Tom, observing it through the conning tower bull's-eyes, when the submarine had backed off again. ``Let's give her another.''
Once more the great steel ram hit into the side of the Boldero, and again the submarine shivered from the shock. But there was a bigger hole in the wreck now, and after Captain Weston had viewed it he decided it was large enough to allow a person to enter and place a charge of dynamite so that the treasure ship would be broken up.
Tom and the captain placed the explosive. Then the Advance was withdrawn to a safe distance. There was a dull rumble, a great swirling of the water, which was made murky; but when it cleared, and the submarine went back, it was seen that the wreck was effectively broken up. It was in two parts, each one easy of access.
``That's the stuff!'' cried Tom. ``Now to get at the gold!''
``Yes, get out the diving suits,'' added Mr. Damon. ``Bless my watch-charm, I think I'll chance it in one myself! Do you think the sharks are all gone, Captain Weston?''
``I think so.''
In a short time Tom, the captain, Mr. Sharp and Mr. Damon were attired in the diving suits, Mr. Swift not caring to venture into such a great depth of water. Besides, it was necessary for at least one person to remain in the submarine to operate the diving chamber.
Walking slowly along the bottom of the sea the four gold-seekers approached the wreck. They looked on all sides for a sight of the sharks, but the monster fish seemed to have deserted that part of the ocean. Tom was the first to reach the now disrupted steamer. He found he could easily climb up, for boxes and barrels from the cargo holds were scattered all about by the explosion. Captain Weston soon joined the lad. The sailor motioned Tom to follow him, and being more familiar with ocean craft the captain was permitted to take the lead. He headed aft, seeking to locate the captain's cabin. Nor was he long in finding it. He motioned for the others to enter, that the combined illumination of the lamps in their helmets would make the place bright enough so a search could be made for the gold. Tom suddenly seized the arm of the captain, and pointed to one corner of the cabin. There stood a small safe, and at the sight of it Captain Weston moved toward it. The door was not locked, probably having been left open when the ship was deserted. Swinging it back the interior was revealed.
It was empty. There was no gold bullion in it.
There was no mistaking the dejected air of Captain Weston. The others shared his feelings, but though they all felt like voicing their disappointment, not a word could be spoken. Mr. Sharp, by vigorous motions, indicated to his companions to seek further.
They did so, spending all the rest of the day in the wreck, save for a short interval for dinner. But no gold rewarded their search.
Tom, late that afternoon, wandered away from the others, and found himself in the captain's cabin again, with the empty safe showing dimly in the water that was all about.
``Hang it all!'' thought the lad, ``we've had all our trouble for nothing! They must have taken the gold with them.''
Idly he raised his steel bar, and struck it against the partition back of the safe. To his astonishment the partition seemed to fall inward, revealing a secret compartment. The lad leaned forward to bring the light for his helmet to play on the recess. He saw a number of boxes, piled one upon the other. He had accidentally touched a hidden spring and opened a secret receptacle. But what did it contain?
Tom reached in and tried to lift one of the boxes. He found it beyond his strength. Trembling from excitement, he went in search of the others. He found them delving in the after part of the wreck, but by motions our hero caused them to follow him. Captain Weston showed the excitement he felt as soon as he caught sight of the boxes. He and Mr. Sharp lifted one out, and placed it on the cabin floor. They pried off the top with their bars.
There, packed in layers, were small yellow bars; dull, gleaming, yellow bars! It needed but a glance to show that they were gold bullion. Tom had found the treasure. The lad tried to dance around there in the cabin of the wreck, nearly three miles below the surface of the ocean, but the pressure of water was too much for him. Their trip had been successful.
\gutchapter{Chapter Twenty-Five}
Home With the Gold
There was no time to be lost. They were in a treacherous part of the ocean, and strong currents might at any time further break up the wreck, so that they could not come at the gold. It was decided, by means of motions, to at once transfer the treasure to the submarine. As the boxes were too heavy to carry easily, especially as two men, who were required to lift one, could not walk together in the uncertain footing afforded by the wreck, another plan was adopted. The boxes were opened and the bars, a few at a time, were dropped on a firm, sandy place at the side of the wreck. Tom and Captain Weston did this work, while Mr. Sharp and Mr. Damon carried the bullion to the diving chamber of the Advance. They put the yellow bars inside, and when quite a number had been thus shifted, Mr. Swift, closing the chamber, pumped the water out and removed the gold. Then he opened the chamber to the divers again, and the process was repeated, until all the bullion had been secured.
Tom would have been glad to make a further examination of the wreck, for he thought he could get some of the rifles the ship carried, but Captain Weston signed to him not to attempt this.
The lad went to the pilot house, while his father and Mr. Sharp took their places in the engine-room. The gold had been safely stowed in Mr. Swift's cabin.
Tom took a last look at the wreck before he gave the starting signal. As he gazed at the bent and twisted mass of steel that had once been a great ship, he saw something long, black and shadowy moving around from the other side, coming across the bows.
``There's another big shark,'' he observed to Captain Weston. ``They're coming back after us.''
The captain did not speak. He was staring at the dark form. Suddenly, from what seemed the pointed nose of it, there gleamed a light, as from some great eye.
``Look at that!'' cried Tom. ``That's no shark!''
``If you want my opinion,'' remarked the sailor, ``I should say it was the other submarine---that of Berg and his friends---the Wonder. They've managed to fix up their craft and are after the gold.''
``But they're too late!'' cried Tom excitedly. ``Let's tell them so.''
``No,'' advised the captain. ``We don't want any trouble with them.''
Mr. Swift came forward to see why his son had not given the signal to start. He was shown the other submarine, for now that the Wonder had turned on several searchlights, there was no doubt as to the identity of the craft.
``Let's get away unobserved if we can,'' he suggested. ``We have had trouble enough.''
It was easy to do this, as the Advance was hidden behind the wreck, and her lights were glowing but dimly. Then, too, those in the other submarine were so excited over the finding of what they supposed was the wreck containing the treasure, that they paid little attention to anything else.
``I wonder how they'll feel when they find the gold gone?'' asked Tom as he pulled the lever starting the pumps.
``Well, we may have a chance to learn, when we get back to civilization,'' remarked the captain.
The surface was soon reached, and then, under fair skies, and on a calm sea, the voyage home was begun. Part of the time the Advance sailed on the top, and part of the time submerged.
They met with but a single accident, and that was when the forward electrical plate broke. But with the aft one still in commission, and the auxiliary screws, they made good time. Just before reaching home they settled down to the bottom and donned the diving suits again, even Mr. Swift taking his turn. Mr. Damon caught some large lobsters, of which he was very fond, or, rather, to be more correct, the lobsters caught him. When he entered the diving chamber there were four fine ones clinging to different parts of his diving suit. Some of them were served for dinner.
The adventurers safely reached the New Jersey coast, and the submarine was docked. Mr. Swift at once communicated with the proper authorities concerning the recovery of the gold. He offered to divide with the actual owners, after he and his friends had been paid for their services, but as the revolutionary party to whom the bullion was intended had gone out of existence, there was no one to officially claim the treasure, so it all went to Tom and his friends, who made an equitable distribution of it. The young inventor did not forget to buy Mrs. Baggert a fine diamond ring, as he had promised.
As for Berg and his employers, they were, it was learned later, greatly chagrined at finding the wreck valueless. They tried to make trouble for Tom and his father, but were not successful.
A few days after arriving at the seacoast cottage, Tom, his father and Mr. Damon went to Shopton in the airship. Captain Weston, Garret Jackson and Mr Sharp remained behind in charge of the submarine. It was decided that the Swifts would keep the craft and not sell it to the Government, as Tom said they might want to go after more treasure some day.
``I must first deposit this gold,'' said Mr. Swift as the airship landed in front of the shed at his home. ``It won't do to keep it in the house over night, even if the Happy Harry gang is in jail.''
Tom helped him take it to the bank. As they were making perhaps the largest single deposit ever put in the institution, Ned Newton came out.
``Well, Tom,'' he cried to his chum, ``it seems that you are never going to stop doing things. You've conquered the air, the earth and the water.''
``What have you been doing while I've been under water, Ned?'' asked the young inventor.
``Oh, the same old thing. Running errands and doing all sorts of work in the bank.''
Tom had a sudden idea. He whispered to his father and Mr. Swift nodded. A little later he was closeted with Mr. Prendergast, the bank president. It was not long before Ned and Tom were called in.
``I have some good news for you, Ned,'' said Mr. Prendergast, while Tom smiled. ``Mr. Swift er---ahem---one of our largest depositors, has spoken to me about you, Ned. I find that you have been very faithful. You are hereby appointed assistant cashier, and of course you will get a much larger salary.''
Ned could hardly believe it, but he knew then what Tom had whispered to Mr. Swift. The wishes of a depositor who brings much gold bullion to a bank can hardly be ignored.
``Come on out and have some soda,'' invited Tom, and when Ned looked inquiringly at the president, the latter nodded an assent.
As the two lads were crossing the street to a drug store, something whizzed past them, nearly running them down.
``What sort of an auto was that?'' cried Tom.
``That? Oh, that was Andy Foger's new car,'' answered Ned. ``He's been breaking the speed laws every day lately, but no one seems to bother him. It's because his father is rich, I suppose. Andy says he has the fastest car ever built.''
``He has, eh?'' remarked Tom, while a curious look came into his eyes. ``Well, maybe I can build one that will beat his.''
And whether the young inventor did or not you can learn by reading the fifth volume of this series, to be called ``Tom Swift and His Electric Runabout; Or, The Speediest Car on the Road.''
``Well, Tom, I certainly appreciate what you did for me in getting me a better position,'' remarked Ned as they left the drug store. ``I was beginning to think I'd never get promoted. Say, have you anything to do this evening? If you haven't, I wish you'd come over to my house. I've got a lot of pictures I took while you were away.''
``Sorry, but I can't,'' replied Tom.
``Why, are you going to build another airship or submarine?''
``No, but I'm going to see--- Oh, what do you want to know for, anyhow?'' demanded the young inventor with a blush. ``Can't a fellow go see a girl without being cross-questioned?''
``Oh, of course,'' replied Ned with a laugh. ``Give Miss Nestor my regards,'' and at this Tom blushed still more. But, as he said, that was his own affair.
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Algebraic Structures & Prof. Dmitry Kerner & \setRTL יום א 18:00 - 16:00 בקרייטמן-זלוטובסקי(חדש) {[}34{]} חדר 116\newline
יום ד 14:00 - 12:00 בבנין 90 (מקיף ז') {[}90{]} חדר 237\\
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Fundamentals of Measure Theory & Prof. Ilan Hirshberg & \setRTL יום ב 16:00 - 14:00 בגולדברגר {[}28{]} חדר 304\newline
יום ד 16:00 - 14:00 בבנין 90 (מקיף ז') {[}90{]} חדר 144\\
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Infinitesimal Calculus 3 & Dr. Inna Entova-Aizenbud & \setRTL יום א 16:00 - 14:00 בצוקר, גולדשטיין-גורן {[}72{]} חדר 213\newline
יום ג 13:00 - 12:00 בקרייטמן-זלוטובסקי(חדש) {[}34{]} חדר 303\newline
יום ה 16:00 - 14:00 בגוטמן {[}32{]} חדר 111\\
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Logic & Dr. Moshe Kamensky & \setRTL יום ב 12:00 -- 10:00 בגוטמן {[}32{]} \emph{חדר 309}
יום ד 12:00 -- 10:00 בגולדברגר {[}28{]} \emph{חדר 103}\\
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Ordinary Differential Equations & Prof. Victor Vinnikov & \setRTL יום ג 12:00 - 10:00 בבנין 90 (מקיף ז') {[}90{]} חדר 234\newline
יום ה 10:00 - 08:00 בגוטמן {[}32{]} חדר 309\\
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Probability & Prof. Ariel Yadin & \setRTL יום א 11:00 - 09:00 בצוקר, גולדשטיין-גורן {[}72{]} חדר 488\newline
יום ג 18:00 - 16:00 בצוקר, גולדשטיין-גורן {[}72{]} חדר 489\\
\hline
Theory of Numbers & Dr. Ishai Dan-Cohen & \setRTL יום ב 16:00 - 12:00 בבנין 90 (מקיף ז') {[}90{]} חדר 223\newline
יום ה 12:00 - 10:00 בגוטמן {[}32{]} חדר 207\\
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Game Theory & Prof. Daniel Berend & \setRTL יום א 12:00 - 11:00 בקרייטמן-זלוטובסקי(חדש) {[}34{]} חדר 205\newline
יום ד 18:00 - 16:00 בבנין 90 (מקיף ז') {[}90{]} חדר 144\\
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Basic Concepts in Topology and Geometry & Dr. Michael Brandenbursky & \setRTL יום א 12:00 - 10:00\newline
יום ג 12:00 - 10:00\\
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Basic Concepts in Modern Analysis & Dr. Izhar Oppenheim & \setRTL יום א 14:00 - 12:00 בבנין 90 (מקיף ז') {[}90{]} חדר 225\newline
יום ד 18:00 - 16:00 בגולדברגר {[}28{]} חדר 204\\
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Commutative Algebra & Prof. Amnon Yekutieli & \setRTL יום ד 12:00-14:00\\
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https://ru.theanarchistlibrary.org/library/narodnaya-samooborona-gosudarstvennaya-armiya-protiv-opolcheniya-istoricheskij-opyt.tex | theanarchistlibrary.org | CC-MAIN-2021-25 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-25/segments/1623488273983.63/warc/CC-MAIN-20210621120456-20210621150456-00443.warc.gz | 439,066,306 | 7,692 | \documentclass[DIV=12,%
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\title{Государственная армия против ополчения. Исторический опыт}
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\author{Народная самооборона}
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Сторонники государственности как один убеждают, что без государственной централизованной армии любое общество обречено на покорение соседями. Мол, армия всегда и везде эффективнее ополчения. Особенно данный тезис любим марксистами, которые доказывают тем самым необходимость государства после революции. Мол, как без государства и армии защищаться от контрреволюции и капиталистической интервенции?
Однако данный тезис не выдерживает никакой проверки историческими фактами. Любой человек, знакомый с историей, поймёт, насколько данные утверждения несостоятельны. Вот лишь несколько примеров, от античности до наших дней.
– Греко-персидские войны. Греческие ополчения строились на принципах выборности командующих территориальными органами самоуправления (филами), и выборности главнокомандующего (стратега) народным собранием (экклесией). Как и со всеми должностными лицами, они были подконтрольны выбравшим их людям, и подчинялись воле народного собрания. Им противостояла централизованная армия державы Ахеменидов. Армия Ахеменидов была довольно развитой для своего времени. Благодаря военным новвоведениям, персам удалось одолеть и покорить множество сильнейших противников, включив их в свою державу. По некоторым оценкам, держава Ахеменидов на момент начала войн с греками включала в себя до половины всего населения Земли. И вот эта огромная армия, с бесконечными ресурсами для ведения войны – материальными и людскими, выстроенная на основах централизации и подчинения низших высшим, обладающая весьма сильными воинами и талантливыми полководцами, направилась в маленькую Грецию. И хотя греческие войска сильно уступали персам в количестве, итог нам известен. Все вторжения персов провалились, и были разгромлены ополчениями греческих городов. Великая глупость утверждать, что солдат, которого под страхом наказания гонят на бойню, смысла которого он не понимает, эффективнее, нежели свободные граждане, сами принимающие решения о войне и самостоятельно вручающие руководство военными действиями выбранному ими человеку, которому еще отвечать перед ними и всем народом.
– Германские варвары, у которых институт государственности еще толком не сложился, громившие образцовое государство Римской Империи – пожалуй, еще один замечательный пример “эффективности” государственной армии над ополчениями. Военная демократия, когда воины-мужчины племени самостоятельно принимали решения на собраниях, стала настоящим могильщиком деградирующего разваливающегося института римской государственности. С другой стороны, возросшая роль войны, покорения новых земель и захвата всё новых богатств привели к сильному возвышению первоначально выборных военных вождей и их дружины. Со временем происходит постепенный рост власти вождя и дружины, что приводит к созданию феодальной системы.
– Швейцария. Это вообще потрясающе. В средние века данная страна сумела добиться независимости в борьбе с окружающими её империями и королевствами. Швейцария представляла из себя образцовую прямую демократию (хотя экономическое неравенство, как водится, обрекло её на перерождение во власть богатых, но это вопрос из экономической сферы, а не военной). Страна была разделена на независимые кантоны, без централизованной армии и управления. Все решения принимались на общих сходах населения кантонов, которые отправляли гонца со своей волей на общее собрание таких гонцов. Гонцы делились мнениями кантонов, после чего возвращались и докладывали на местах мнения других земель. Таким образом принимались решения. Но в отсутствии центрального правительства и армии приходилось договариваться – ни один кантон, будучи не согласен с решениями иных кантонов, не мог быть принужден к подчинению и исполнению этого решения. Такая вот идеальная конфедерация. Интересна здесь история борьбы за независимость этой демократии. В ряде войн крестьянские ополчения разбивали вдребезги армии прекрасно вооруженных рыцарей. В итоге, пришлось тем согласиться с независимостью Швейцарии, вольный строй которой обеспечил дальнейшее экономическое процветание конфедерации.
– Чеченские крестьяне, долгое время подчинявшиеся дагестанским князьям, с возникновением огнестрельного оружия сумели организовать эффективное ополчение и добиться независимости от дагестанских феодалов. После чего в Чечне была организована примитивная родовая демократия, сумевшая наладить даже массовое производство огнестрельного оружия и долгое время сопротивляться пришедшим в регион российским имперским войскам.
– Буры. Буры никакой демократией не являлись. Но вот военная система их строилась по вполне демократической системе ополчения. Никакой армии у буров вообще не было. Вместо этого каждый бурский мужчина был обязан хранить дома оружие. В случае войны вооруженные буры формировали снизу вверх ополчение, самостоятельно выбирая командующих. Эффективность данной системы стала очевидна во время второй англо-бурской войны. Небольшие бурские республики противостояли великой империи, контролировавшей половину мира, и обладающей огромной централизованной армией и колоссальными ресурсами для ведения войны. Буры стали настоящим кошмаром для англичан. В итоге им удалось победить буров, но для этого пришлось нагнать на бурские земли такое количество солдат, которое превышало количество всего бурского населения. И лишь когда половина населения оказалась в концлагерях, англичанам удалось одержать победу.
– Анархистские ополчения в гражданских войнах XX века. Здесь речь идёт о махновцах и испанских анархистах. Махновцы, которые долгое время эффективно воевали то с белыми, то с красными, спасли красную Москву от Деникина, и потерпели поражение лишь после перехода превосходящих сил красных к тактике “выжженной земли”, в ходе многочисленных боёв показали себя вполне боеспособной силой. Тоже самое касается и испанских анархистов. Подавившие фашистский мятеж вооруженных военных в Барселоне в 1936 году, анархисты сформировали многочисленные ополчения, испытывающие, правда, проблемы со снабжением, ввиду непростых отношений с республиканским правительством и коммунистами, в чьих руках был контроль над распределением оружия в республиканском лагере. Более того, анархисты, воюя с фашистами, подвергались давлению и атакам коммунистов с тыла. Тем не менее, когда встала опасность взятия Мадрида в 1936 году, и республиканские власти спешно эвакуировались из города, передать оборону города планировалось именно анархисту Буэнавентурре Дуррути, прибывшему в покидаемую республиканцами столицу с отрядом анархистов. Правда, Дуррути был убит то ли сталинским агентом, то ли фашистским снайпером, и планам не суждено было сбыться. Тем не менее, единственная значительная победа республики в этой войне, в битве под Гвадалахарой, где были разбиты итальянские фашистские части, была одержана в значительной степени именно анархистскими частями под руководством анархиста Меры Санса.
– В наше время также можно встретить эффективные примеры ополчений, успешно воюющих с государственными армиями. Сапатисты в Мексике, выбившие себе автономию у центрального правительства, или сирийские курды, представляющие одну из наиболее боеспособных сил в гражданской войне, убедительно опровергают тезис об обязательной низкой боевой эффективности ополчений. Также можно было бы упомянуть некоторые маоистские повстанческие движения, чья структура также может основываться на крестьянском самоуправлении (подчиненном, правда, иерархической партийной структуре).
В действительности, ополчения имеют ряд преимуществ над централизированной армией. Но нельзя говорить, что армия обязательно более эффективна, нежели ополчения, или же что ополчения обязательно более эффективны. Эффективность и боеспособность формирований зависит не только и не столько от структуры и устройства, но и от множества иных факторов.
Анархисты выступают вовсе не за немедленное уничтожение любых вооруженных структур. Анархисты выступают за уничтожение таких вооруженных структур, которые были бы оторваны от народа и подчинялись бы какому-либо определенному центру. Организованная вооруженная сила служит надежнейшим инструментом для подавления и подчинения населения, организации властной иерархии и институтов господства. Также, как политическая и экономическая власть, военная власть должна находиться в руках всего населения. Всеобщее вооружение, регулярные военные сборы, выборность и подконтрольность командиров – вот предлагаемая анархистами военная структура общества. Военная организация не должна не то что стоять над народом, она вообще не должна отличаться от народа. Сам народ и должен составлять военную организацию общества.
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% the page would be odd on an even one.
\ifthispageodd{\strut\thispagestyle{empty}\clearpage}{}
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\thispagestyle{empty}
\begin{center}
Библиотека Анархизма
\smallskip
Антикопирайт
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Народная самооборона
Государственная армия против ополчения. Исторический опыт
2018-03-20
\bigskip
Скопировано 2018-05-01 с \url{https://naroborona.info/2018/03/20/gosudarstvennaya-armiya-protiv-opolcheniya-istoricheskij-opyt/}
\bigskip
\textbf{ru.theanarchistlibrary.org}
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https://artifacts.opnfv.org/jenkins-cntt-tox-rc1-1343/rc1/anuketreferenceconformanceforopenstackrc1.tex | opnfv.org | CC-MAIN-2023-06 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2023-06/segments/1674764500215.91/warc/CC-MAIN-20230205032040-20230205062040-00058.warc.gz | 119,799,278 | 46,753 | %% Generated by Sphinx.
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\title{Anuket Reference Conformance for OpenStack (RC1)}
\date{May 11, 2022}
\release{}
\author{Anuket}
\newcommand{\sphinxlogo}{\vbox{}}
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\begin{document}
\pagestyle{empty}
\sphinxmaketitle
\pagestyle{plain}
\sphinxtableofcontents
\pagestyle{normal}
\phantomsection\label{\detokenize{index::doc}}
\sphinxAtStartPar
This is the Reference Conformance for OpenStack based infrastructure
(RC\sphinxhyphen{}1)
\chapter{Release Information}
\label{\detokenize{index:release-information}}
\sphinxAtStartPar
\sphinxstylestrong{Bundle: 6}
\sphinxAtStartPar
\sphinxstylestrong{Version: 0}
\sphinxAtStartPar
\sphinxstylestrong{Release Date: 4th January 2022}
\chapter{Bundle/Version History}
\label{\detokenize{index:bundle-version-history}}
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Bundle.Version
&\sphinxstyletheadfamily
\sphinxAtStartPar
Date
&\sphinxstyletheadfamily
\sphinxAtStartPar
Note
\\
\hline
\sphinxAtStartPar
0.0
&
\sphinxAtStartPar
10th January 2020
&
\sphinxAtStartPar
First Initial Draft
\\
\hline
\sphinxAtStartPar
2.0
&
\sphinxAtStartPar
15th May 2020
&
\sphinxAtStartPar
Baldy Release
\\
\hline
\sphinxAtStartPar
3.0
&
\sphinxAtStartPar
25th Sep 2020
&
\sphinxAtStartPar
Baraque Release
\\
\hline
\sphinxAtStartPar
4.0
&
\sphinxAtStartPar
29th Jan 2021
&
\sphinxAtStartPar
Elbrus Release
\\
\hline
\sphinxAtStartPar
5.0
&
\sphinxAtStartPar
1st July 2021
&
\sphinxAtStartPar
Kali Release
\\
\hline
\sphinxAtStartPar
6.0
&
\sphinxAtStartPar
4th Jan 2022
&
\sphinxAtStartPar
Lakelse Release
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\chapter{Overall Status}
\label{\detokenize{index:overall-status}}
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Chapter
&\sphinxstyletheadfamily
\sphinxAtStartPar
Status
\\
\hline
\sphinxAtStartPar
Chapter 01 \sphinxhyphen{} Introduction
&
\sphinxAtStartPar
SME Feedback
\\
\hline
\sphinxAtStartPar
Chapter 02 \sphinxhyphen{} NFVI Conformance
Requirements
&
\sphinxAtStartPar
SME Feedback
\\
\hline
\sphinxAtStartPar
Chapter 03 \sphinxhyphen{} NFVI Test Cases and
Traceability to Requirements
&
\sphinxAtStartPar
SME Feedback
\\
\hline
\sphinxAtStartPar
Chapter 04 \sphinxhyphen{} NFVI Testing
Cookbook
&
\sphinxAtStartPar
SME Feedback
\\
\hline
\sphinxAtStartPar
Chapter 05 \sphinxhyphen{} VNF Conformance
Requirements
&
\sphinxAtStartPar
SME Feedback
\\
\hline
\sphinxAtStartPar
Chapter 06 \sphinxhyphen{} VNF Test Cases and
Traceability to Requirements
&
\sphinxAtStartPar
SME Feedback
\\
\hline
\sphinxAtStartPar
Chapter 07 \sphinxhyphen{} VNF Testing Cookbook
&
\sphinxAtStartPar
SME Feedback
\\
\hline
\sphinxAtStartPar
Chapter 08 \sphinxhyphen{} Gap analysis and
Development
&
\sphinxAtStartPar
SME Feedback
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\section{Introduction}
\label{\detokenize{chapters/chapter01:introduction}}\label{\detokenize{chapters/chapter01::doc}}
\subsection{Synopsis}
\label{\detokenize{chapters/chapter01:synopsis}}
\sphinxAtStartPar
Ensure an implementation of the Anuket Reference Architecture (RA), such
as the Reference Implementation (RI), meets industry driven quality
assurance standards for conformance, verification and validation. The
OPNFV Verified Program (OVP), by Linux Foundation Networking (LFN),
overseen by the Compliance Verification Committee (CVC), will provide
tracking and governance for RC.
\sphinxAtStartPar
For the purpose of this chapter, NFVI+VNF testing will be performed to
evaluate \sphinxstylestrong{Conformance} (i.e. adherence) to, and demonstrated
proficiency with, all aspects of software delivery. More specifically,
Conformance includes:
\begin{itemize}
\item {}
\sphinxAtStartPar
Verified implementations of NFVI+VNF match expected design
requirements
\item {}
\sphinxAtStartPar
Clearly stated guidelines for test, badging, and lifecycle management
\item {}
\sphinxAtStartPar
Inclusion of Operational run\sphinxhyphen{}books for 3rd party supplier
instantiation and validations of NFVI+VNF
\item {}
\sphinxAtStartPar
Evidence, through test results, confirming delivered code matches
industrial requirements
\item {}
\sphinxAtStartPar
Interoperability testing with Reference VNFs, ensuring integration
stability and life\sphinxhyphen{}cycle management of the Reference VNF on the
target implementation.
\end{itemize}
\sphinxAtStartPar
In summary, NFVI+VNF \sphinxstylestrong{Conformance} testing will be performed for
\sphinxstylestrong{Verification} and \sphinxstylestrong{Validation} purposes, defined further as:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Verification} will be used to indicate conformance to design
requirement specifications. Accomplished with Requirement
Traceability and Manifest Reviews to ensure the NFVI is delivered per
implementation specifications.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Validations} is used to indicate that testing performed confirms
the NFVI+VNF meets the expected, or desired outcome, or behaviour.
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{All Terms utilized throughout this chapter are intended to align with
CVC definitions, and their use through CVC documentation, guidelines,
and standards.}
\subsection{Overview}
\label{\detokenize{chapters/chapter01:overview}}
\sphinxAtStartPar
\sphinxstylestrong{Chapter Purpose}
\sphinxAtStartPar
This chapter includes process flow, logistics, and requirements which
must be satisfied to ensure Network Function Virtualisation
Infrastructure (NFVI) meets the design, feature, and capability
expectations of RM and RA. Upstream projects will define
features/capabilities, test scenarios, and test cases which will be used
to augment OVP test harnesses for infrastructure verification purposes.
Existing processes, communication mediums, and related technologies will
be utilized where feasible. Ultimately, test results of certified
NFVI+VNF will reduce the amount of time and cost it takes each operator
to on\sphinxhyphen{}board and maintain vendor provided VNFs.
\sphinxAtStartPar
\sphinxstylestrong{Objective}
\sphinxAtStartPar
Perform NFVI+VNF Verification and Validations using Anuket reference
architecture, leveraging the existing Anuket and CVC Intake and
Validation Process to onboard and validate new test projects for NFVI
compliance. Upstream projects will define features/capabilities, test
scenarios, and test cases to augment existing OVP test harnesses to be
executed via the OVP Ecosystem.
\sphinxAtStartPar
\sphinxstylestrong{Test Methodology}
\begin{itemize}
\item {}
\sphinxAtStartPar
Verification test to make sure if the OpenStack services have been
deployed and configured correctly
\item {}
\sphinxAtStartPar
Manifest Verifications (Termed Compliance by CVC) will ensure the
NFVI is compliant, and delivered for testing, with hardware and
software profile specifications defined by the RM and RA.
\item {}
\sphinxAtStartPar
Empirical Validation with Reference Golden VNFs (Termed Validation by
CVC) will ensure the NFVI runs with a set of VNF Families, or
Classes, to mimic production\sphinxhyphen{}like VNF connectivity, for the purposes
of interoperability checks.
\item {}
\sphinxAtStartPar
Candidate VNF Validation (Termed Validation \& Performance by CVC)
will ensure complete interoperablity of VNF behaviour on the NFVI
leverage VVP/VNFSDK test suites. Testing ensures VNF can be spun up,
modified, or removed, on the target NFVI (aka Interoperability).
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Different Distributions}
\sphinxAtStartPar
The three step methodology described above of verifying Manifest
compliance, executing Empirical Golden VNF transactions, and performing
Interoperability Testing is the same validation process regardless of
the Distribution used to establish a cloud topology, and the components
and services used in the client software stack.
\subsubsection{Terminology}
\label{\detokenize{chapters/chapter01:terminology}}
\sphinxAtStartPar
Terminology in this document will follow \sphinxhref{https://cntt.readthedocs.io/en/latest/common/glossary.html}{Glossary}.
\subsection{Scope}
\label{\detokenize{chapters/chapter01:scope}}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{rc1_scope}.png}
\caption{RC1 Scope}\label{\detokenize{chapters/chapter01:id1}}\end{figure}
\sphinxAtStartPar
This document covers the realisation aspects of conformance of both NFVI
and VNFs. The document will cover the following topics:
\begin{itemize}
\item {}
\sphinxAtStartPar
Identify in details the Requirements for conformance Framework.
\item {}
\sphinxAtStartPar
Identify in details the Requirement of Test Cases (and mapping them
to requirements from The Reference Model and The OpenStack Based
Reference Architecture ).
\item {}
\sphinxAtStartPar
Analysis of existing community projects.
\item {}
\sphinxAtStartPar
Propose an E2E Framework for conformance of NFVI and VNFs.
\item {}
\sphinxAtStartPar
Playbook of instructions, user manuals, steps of how to perform
verification and conformance for both NFVI and VNFs using the
proposed E2E Framework.
\item {}
\sphinxAtStartPar
Gap analysis to identify where the Gaps are in the industry (tooling,
test cases, process, etc).
\item {}
\sphinxAtStartPar
Identify development efforts needed to address any gaps identified.
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Not in Scope}
\begin{itemize}
\item {}
\sphinxAtStartPar
Functional testing / validation of the application provided by the
VNF is outside the scope of this work.
\item {}
\sphinxAtStartPar
ONAP is not used in the process flow for NFVI verifications, or
validations.
\item {}
\sphinxAtStartPar
Upgrades to VNFs, and the respective processes of verifying upgrade
procedures and validating (testing) the success and compatibility of
upgrades is not in scope.
\end{itemize}
\subsection{Relation to other communities}
\label{\detokenize{chapters/chapter01:relation-to-other-communities}}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{rc1_relation}.png}
\caption{RC\sphinxhyphen{}1 Relations other communities}\label{\detokenize{chapters/chapter01:id2}}\end{figure}
\subsection{Principles and Guidelines}
\label{\detokenize{chapters/chapter01:principles-and-guidelines}}
\sphinxAtStartPar
The objectives of the verification program are to deliver a validated
implementation of reference architecture which satisfies infrastructure
needs for VNF\sphinxhyphen{}developer teams, leveraging the OVP ecosystem as the
vehicle for delivering validated NFVI.
\sphinxAtStartPar
These core principles will guide NFV verification deliverables:
\subsubsection{Overarching Objectives and Goals}
\label{\detokenize{chapters/chapter01:overarching-objectives-and-goals}}\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
Deliver verified implementation of reference architecture which
satisfies infrastructure needs for VNF\sphinxhyphen{}developer teams.
\item {}
\sphinxAtStartPar
All accomplished with augmentation to the current OVP ecosystem.
\item {}
\sphinxAtStartPar
Increase probability VNFs will on\sphinxhyphen{}board and function with minimal
problems, or issues, during initial instantiation of VNF.
\item {}
\sphinxAtStartPar
Test Harnesses will be portable, or compatible, across all
RAs/Distributions which already conform to standard interfaces and
services.
\end{enumerate}
\subsection{Best Practices}
\label{\detokenize{chapters/chapter01:best-practices}}
\sphinxAtStartPar
The following best practices have been adopted to ensure verification
and validation procedures are repeatable with consistent quality in test
results, and RI conformances:
\begin{itemize}
\item {}
\sphinxAtStartPar
Standardized test methodology / flow, Test Plan, and Test Case Suites
\item {}
\sphinxAtStartPar
Integration with Anuket Upstream Projects and OVP flow (code, docs,
cert criteria, etc.)
\item {}
\sphinxAtStartPar
Leverage Network and Service Models, with identified VNF\sphinxhyphen{}specific
parameters
\item {}
\sphinxAtStartPar
Standardized conformance criteria
\item {}
\sphinxAtStartPar
Define Anuket RA as scenarios, and have all test cases for the RA be
involved in OVP
\item {}
\sphinxAtStartPar
Add test cases from operators, which operators already tested in
their environment
\end{itemize}
\subsection{Verification methodologies}
\label{\detokenize{chapters/chapter01:verification-methodologies}}
\sphinxAtStartPar
Perform VNF interoperability verifications against an implementation of
Anuket reference architecture, leveraging existing Anuket Intake
Process. Upstream projects will define features/capabilities, test
scenarios, and test cases to augment existing OVP test harnesses to be
executed via the OVP Ecosystem.
\sphinxAtStartPar
3rd Party test platforms may also be leveraged, if desired.
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{RC_certifying_methodlogy_25Nov2019}.jpg}
\caption{Conformance Methodology}\label{\detokenize{chapters/chapter01:id3}}\end{figure}
\subsection{Assumptions \& Dependencies}
\label{\detokenize{chapters/chapter01:assumptions-dependencies}}
\sphinxAtStartPar
\sphinxstylestrong{Assumptions} NFVI+VNF testing will be considered \sphinxstylestrong{Testable} if the
follow qualifiers are present in a test execution, and subsequent
result:
\begin{itemize}
\item {}
\sphinxAtStartPar
Ability to perform Conformance, or Verification of Artifacts to
ensure designs (RM/RA/RI) are delivered per specification
\item {}
\sphinxAtStartPar
Ability to Control (or manipulate), manifestations of RM/RA/RI for
the purposes to adjust the test environment, and respective cases,
scenarios, and apparatus, to support actual test validations
\item {}
\sphinxAtStartPar
Ability to monitor, measure, and report, Validations performed
against a target, controlled system under test
\end{itemize}
\sphinxAtStartPar
In addition, respective Entrance criteria is a prerequisite which needs
to be satisfied for NFVI+VNF to be considered \sphinxstylestrong{Testable}.
\sphinxAtStartPar
\sphinxstylestrong{Dependencies} NFVI+VNF verification will rely upon test harnesses,
test tools, and test suites provided by Anuket projects, including
dovetaill, yardstick, and Bottleneck. These upstream projects will be
reviewed semi\sphinxhyphen{}annually to verify they are still healthy and active
projects. Over time, the projects representing the conformance process
may change, but test parity is required if new test suites are added in
place of older, stale projects.
\begin{itemize}
\item {}
\sphinxAtStartPar
NFVI+VNF verifications will be performed against well defined
instance types consisting of a HW and SW Profile, Configured Options,
and Applied Extensions (See image.)
\end{itemize}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{RC_NFVI_VNF_Instance_Type_25Nov2019}.jpg}
\caption{Instance Type}\label{\detokenize{chapters/chapter01:id4}}\end{figure}
\sphinxAtStartPar
\sphinxstylestrong{NFVI+VNF Instance Type:}
\begin{itemize}
\item {}
\sphinxAtStartPar
Standard compute flavours to be tested are defined in
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter04.html\#virtual-network-interface-specifications}{Virtual Network Interface Specifications}
\item {}
\sphinxAtStartPar
Performance profiles come in the form of Basic, Network Intensive,
and Compute intensive. Refer to
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter02.html\#analysis}{Analysis}
for details on these profiles.
\end{itemize}
\subsection{Results Collation \& Presentation}
\label{\detokenize{chapters/chapter01:results-collation-presentation}}
\sphinxAtStartPar
Test suites will be categorized as functional or performance based.
Results reporting will be communicated as a boolean (pass/fail). The
pass/fail determination for performance\sphinxhyphen{}based test cases will be made by
comparing results against a baseline. Example performance\sphinxhyphen{}based metrics
include, but are not limited to: resource utilization, response times,
latency, and sustained throughput per second (TPS).
\sphinxAtStartPar
\sphinxstylestrong{Placeholder to document where results will be posted (e.g. Dovetail
dashboards.)}
\subsection{Governance}
\label{\detokenize{chapters/chapter01:governance}}\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
Conformance badges will be presented by the CVC
\item {}
\sphinxAtStartPar
CVC will maintain requirements for conformance
\end{enumerate}
\section{NFVI Conformance Requirements}
\label{\detokenize{chapters/chapter02:nfvi-conformance-requirements}}\label{\detokenize{chapters/chapter02::doc}}
\subsection{Synopsis}
\label{\detokenize{chapters/chapter02:synopsis}}
\sphinxAtStartPar
This chapter mainly covers the overall requirement for the Reference
Conformance test. The Conformance tests are conducted on the Cloud
Infrastructure and VNF level. Conformance on the Cloud Infrastructure
makes sure the SUT follows RM \& RA requirements, and conformance on VNF
makes sure that the VNF can be deployed and sufficiently work on the
Cloud Infrastructure that has passed the conformance test.
\sphinxAtStartPar
All Terms utilized throughout this chapter are intended to align with
CVC definitions, and their use through CVC documentation, guidelines,
and standards. This chapter will outline the Requirements, Process, and
Automation, needed to deliver the Cloud Infrastructure conformance.
\subsection{Introduction}
\label{\detokenize{chapters/chapter02:introduction}}
\sphinxAtStartPar
The Cloud Infrastructure refers to the physical and virtual resources
(compute, storage and network) on which virtual network functions (VNFs)
are deployed. Due to the differences in the API and under\sphinxhyphen{}layer
configuration and capability matrix, multiple vendor VNF deployments on
the shared Cloud Infrastructure becomes hard to predict, and requires s
amount of cross vendor interoperability tests. With the combined effort
from operators and vendors, define the RA and RM, that standardises the
under layer configuration, capability and API, so as to provide the
upper layer VNF with a ‘common cloud infrastructure’. Based on this,
Anuket also provides RC for conformance tests of SUT against RA \& RM
requirements. SUT passes the conformance test will be identified as NFVI
that can fit into the common requirements.
\sphinxAtStartPar
In the meantime, RC also provides conformance test for VNF. The
intention is to make sure VNF that passes RC test cases can be deployed
on any NFVI which also passes RC without any conformance and
interoperability issue.
\subsection{Methodology}
\label{\detokenize{chapters/chapter02:methodology}}
\sphinxAtStartPar
The Cloud Infrastructure is consumed or used by VNFs via APIs exposed by
Virtualised Infrastructure Manager (VIM). The resources created by VIM
on the NFVI use the underlying physical hardware (compute, storage and
network) either directly or indirectly. Anuket recommends RA1 to be used
as a reference architecture for the Cloud Infrastructure conformance.
This would provide a set of standard interfaces to create resources on
the Cloud Infrastructure. Below step by step process illustrates the
NFVI conformance methodology:
\begin{itemize}
\item {}
\sphinxAtStartPar
SUT (Anuket RI1 or commercial NFVI) is deployed on hardware for
conformance test.
\item {}
\sphinxAtStartPar
A set of tests run on SUT to determine the SUT readiness for
conformance process.
\item {}
\sphinxAtStartPar
Golden KPIs are taken as a reference.
\item {}
\sphinxAtStartPar
A set of tests are run on the SUT (target for conformance).
\item {}
\sphinxAtStartPar
KPIs obtained from the SUT are collected and submitted to conformance
portal.
\item {}
\sphinxAtStartPar
The SUT KPIs are reviewed and compared with Golden KPIs to determine
if the conformance badge is to be provided to SUT or not.
\end{itemize}
\sphinxAtStartPar
Based on a NFVI passing RC test and getting the conformance badge, VNF
conformance test can be further conducted. Such test will leverage
existing Anuket Intake Process. Upstream projects will define
features/capabilities, test scenarios, and test cases to augment
existing OVP test harnesses to be executed via the OVP Ecosystem.
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{RC_CertificationMethodology}.jpg}
\caption{Conformance Methodology}\label{\detokenize{chapters/chapter02:id1}}\end{figure}
\sphinxAtStartPar
Conformance methodologies to be implemented, from a process perspective
include:
\begin{itemize}
\item {}
\sphinxAtStartPar
Engineering package validations will be performed against targeted
infrastructure/architecture.
\item {}
\sphinxAtStartPar
Configuration settings/features/capabilities will be baseline.
\item {}
\sphinxAtStartPar
Entrance Criteria Guidelines will be satisfied prior to RC test
(i.e. Supplier needs to submit/agree/conform)
\begin{itemize}
\item {}
\sphinxAtStartPar
Conform to Anuket RM
\item {}
\sphinxAtStartPar
Conform to Anuket RA
\item {}
\sphinxAtStartPar
Submit standard documentation
\item {}
\sphinxAtStartPar
Adhere to security compliance
\end{itemize}
\item {}
\sphinxAtStartPar
Exit Criteria Guidelines will be satisfied prior to issuance of
Anuket compliance badges.
\item {}
\sphinxAtStartPar
Verification decisions will be based on data. Test harness is
compatible, or conforms to testing against standard interfaces and
services.
\item {}
\sphinxAtStartPar
Leverage test harnesses from existing open source projects where
practical, and applicable.
\end{itemize}
\subsection{Conformance Strategy \& Vehicle}
\label{\detokenize{chapters/chapter02:conformance-strategy-vehicle}}
\sphinxAtStartPar
In order to begin the Conformance process, the Cloud Infrastructure
needs to be validated and expected to be in a required state. This state
would be determined by running tests as described in RI. Once the target
Cloud Infrastructure passes these tests, it would become a candidate for
the Cloud Infrastructure Conformance. If the Cloud Infrastructure fails
the tests, it will not be moved to the next workflow for Conformance.
The Cloud Infrastructure+VNF conformance consist of a three part process
for Compliance, Validation, and Performance. Adherence to Security
standards are equally important and addressed in
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter07.html}{Security}.
\sphinxAtStartPar
The three part conformance process includes NFVI Manifest conformance,
Empirical Baseline measurements against targeted VNF families, and
Candidate VNF validation. More specifically,
\begin{itemize}
\item {}
\sphinxAtStartPar
NFVI conformance: NFVI is the SUT, ensuring NFVI is compliant with
specs of RM and RA accomplished with Manifest test
\item {}
\sphinxAtStartPar
Empirical Validation with Reference VNF (Validation): NFVI is the
SUT, ensuring NFVI runs with Golden VNFs and is instrumented to
objectively validate resources through consumption and measurement
\item {}
\sphinxAtStartPar
Candidate VNF Conformance (Validation \& Performance): VNF is the SUT,
ensuring VNFs operate with RM and RA leveraging VVP/CVP/VFN SDK Test
Suites
\item {}
\sphinxAtStartPar
Security: Ensures NFVI+VNF is free from known security
vulnerabilities, utilizing industry standard cyber security
frameworks (Refer to Chapter 7 Security for additional
test/verification details) Validations are performed against an
Infrastructure Profile Catalog, VNF performance profile, and targeted
VNF class or family for baseline measurements.
\end{itemize}
\sphinxAtStartPar
The Infrastructure Profile Catalog contains the following attributes:
\begin{itemize}
\item {}
\sphinxAtStartPar
Profile is a collection of (limited) options offered by the
infrastructure to the VNF
\begin{itemize}
\item {}
\sphinxAtStartPar
Capabilities
\item {}
\sphinxAtStartPar
Metrics
\item {}
\sphinxAtStartPar
Compute flavors
\item {}
\sphinxAtStartPar
Interface options
\item {}
\sphinxAtStartPar
Storage extensions
\item {}
\sphinxAtStartPar
Acceleration capabilities
\end{itemize}
\item {}
\sphinxAtStartPar
Profiles are offered to VNFs as an instance types with predefined
compute flavors.
\begin{itemize}
\item {}
\sphinxAtStartPar
A particular set of options is an instance type
\item {}
\sphinxAtStartPar
Compute flavors: .tiny, .small etc as defined in
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter04.html\#profiles}{Profiles}
\end{itemize}
\item {}
\sphinxAtStartPar
NFVI performance profiles, for which NFVI validations will support
and be verified against, are defined as basic and network intensive.
Details for each of these profiles can be found in
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter02.html\#analysis}{Analysis}.
\end{itemize}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{RM-ch04-node-profiles}.png}
\caption{NFVI Profiles}\label{\detokenize{chapters/chapter02:id2}}\end{figure}
\subsection{Profiles Reference}
\label{\detokenize{chapters/chapter02:profiles-reference}}
\sphinxAtStartPar
Different vendors have different types of VNFs to serve different
use\sphinxhyphen{}cases. A VNF like Broadband Network Gateway (BNG) would require high
networking throughput whereas a VNF like Mobility Management Entity
(MME) would require high computing performance. As such, BNG would
require high KPI values for network throughput and MME would require
high CPU performance KPIs like Index Score, Instructions Per Second
(IPS) etc. The target NFVI to cater these needs would have different
characteristics. Depending on VNF’s requirements, the NFVI can be
categorized into below profiles:
\begin{itemize}
\item {}
\sphinxAtStartPar
Basic (B) profile for standard computing and
\item {}
\sphinxAtStartPar
Network intensive (N) profile offering predictable computing
performance along with low latency and high networking throughput
Similarly, different NFVI vendors may specialize in different
hardware profiles and some may specialize in both VNFs and NFVI.
\end{itemize}
\sphinxAtStartPar
To cater to different needs from multiple NFVI vendors, Anuket allows
different types of NFVI Conformance based on their types of
profile \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter02.html\#analysis}{Analysis}
\begin{itemize}
\item {}
\sphinxAtStartPar
Certify Vendor NFVI Hardware solution: This allows for Conformance of
only NFVI.
\item {}
\sphinxAtStartPar
Certify Vendor NFVI Hardware and Software Solution: This
allows for Conformance for NFVI running a particular VNF.
\end{itemize}
\subsection{Compliance, Verification, and Conformance}
\label{\detokenize{chapters/chapter02:compliance-verification-and-conformance}}
\sphinxAtStartPar
The below set of steps define the compliance, verification and
Conformance process for NFVI
\begin{itemize}
\item {}
\sphinxAtStartPar
Based on VNF’s requirements, the Cloud Infrastructure profile is
selected \sphinxhyphen{} B, N
\item {}
\sphinxAtStartPar
The Cloud Infrastructure readiness is checked for Conformance.
\item {}
\sphinxAtStartPar
The test VNFs are on\sphinxhyphen{}boarded using automation scripts on the NFVI.
\item {}
\sphinxAtStartPar
VNF on\sphinxhyphen{}boarding is validated by running functional tests to ensure
that the on\sphinxhyphen{}boarding is successful.
\item {}
\sphinxAtStartPar
VNF performance tests are executed and NFVI KPIs are recorded during
the tests.
\item {}
\sphinxAtStartPar
KPI comparison is run to compare NFVI KPIs with Golden KPIs, which
serve as a reference for NFVI Conformance.
\item {}
\sphinxAtStartPar
If NFVI KPIs meet Golden KPIs, NFVI is certified and granted a
Conformance badge.
\item {}
\sphinxAtStartPar
If NFVI KPIs do not meet Golden KPIs, no Conformance is provided.
\end{itemize}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{RC_Ref_NFVI_Profiles}.jpg}
\caption{Reference NFVI Profiles Implementation}\label{\detokenize{chapters/chapter02:id3}}\end{figure}
\subsection{Entry \& Exit Criteria}
\label{\detokenize{chapters/chapter02:entry-exit-criteria}}
\sphinxAtStartPar
\sphinxstylestrong{Entry criteria}: Before entering into NFVI Conformance, NFVI needs to
satisfy the following requirements as entry pass:
\begin{itemize}
\item {}
\sphinxAtStartPar
Design \& Requirements
\begin{itemize}
\item {}
\sphinxAtStartPar
Design, Configuration, Features, SLAs, and Capability
documentation complete
\item {}
\sphinxAtStartPar
Users stories / Adherence to Anuket Model principles and
guidelines
\item {}
\sphinxAtStartPar
Chosen Reference Architecture matches the Architecture from the
product catalog
\end{itemize}
\item {}
\sphinxAtStartPar
Environment
\begin{itemize}
\item {}
\sphinxAtStartPar
Lab assets/resources and respective software revision levels are
specified, with confirmation of compatibility across external
systems
\item {}
\sphinxAtStartPar
Tenant needs identified
\item {}
\sphinxAtStartPar
All connectivity, network, image, VMs, delivered with successful
pairwise tests
\item {}
\sphinxAtStartPar
Lab instrumented for proper monitoring
\item {}
\sphinxAtStartPar
Lab needs to be setup according to RA1/RA2 as defined by Anuket
specifications and should be in the NFVI required state.
\end{itemize}
\item {}
\sphinxAtStartPar
Planning \& Delivery
\begin{itemize}
\item {}
\sphinxAtStartPar
Kickoff / Acceptance Criteria reviews performed
\item {}
\sphinxAtStartPar
Delivery commitments, timelines, and cadence accepted
\item {}
\sphinxAtStartPar
Confirm backward compatibility across software/flavor revision
levels
\end{itemize}
\item {}
\sphinxAtStartPar
Data/VNFs/Security
\begin{itemize}
\item {}
\sphinxAtStartPar
Images, Heat Templates, Preload Sheets available
\item {}
\sphinxAtStartPar
Images uploaded to tenant space
\item {}
\sphinxAtStartPar
External system test data needs identified
\item {}
\sphinxAtStartPar
Owners (NFVI, VNF, PTL, etc) documented
\item {}
\sphinxAtStartPar
Security Compliance Satisfied (Refer to Anuket specification
Chapter XXXX Security for additional tests, scans, and
vulnerabilities validations)
\end{itemize}
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Exit criteria}: NFVI Conformance testing should complete with
following exit criteria:
\begin{itemize}
\item {}
\sphinxAtStartPar
All mandatory test cases should pass.
\item {}
\sphinxAtStartPar
Test results collated, centralized, and normalized, with a final
report generated showing status of the test scenario/case (e.g. Pass,
Fail, Skip, Measurement Success/Fail, etc), along with trace\sphinxhyphen{}ability
to a functional, or non\sphinxhyphen{}functional, requirement.
\end{itemize}
\subsection{Framework Requirements}
\label{\detokenize{chapters/chapter02:framework-requirements}}
\sphinxAtStartPar
The NFVI Conformance framework deals with the process of testing NFVI in
below three areas:
\begin{itemize}
\item {}
\sphinxAtStartPar
Compliance: The Cloud Infrastructure needs to comply to Anuket
RA1/RA2.
\item {}
\sphinxAtStartPar
Validation: Validation deals with the ability of NFVI to respond to
Cloud APIs and interfaces.
\item {}
\sphinxAtStartPar
Performance: Performance deals with running tests on NFVI depending
on the NFVI profile and collecting KPIs.
\end{itemize}
\sphinxAtStartPar
The Cloud Infrastructure KPIs are compared with Golden KPIs, which serve
as a reference for the Cloud Infrastructure Conformance. If the Cloud
Infrastructure KPIs meet Golden KPIs, The Cloud Infrastructure is
certified and granted a Conformance badge. If the Cloud Infrastructure
KPIs do not meet Golden KPIs, no Conformance badge is provided.
\subsubsection{Best Practices (General)}
\label{\detokenize{chapters/chapter02:best-practices-general}}
\sphinxAtStartPar
The NFVI Conformance framework will be guided by the following core
principles:
\begin{itemize}
\item {}
\sphinxAtStartPar
Implementing, and adhering to, Standardized Test Methodology / flow,
Test Plan, and Test Case Suites, which promotes scalability using
repeatable processes.
\item {}
\sphinxAtStartPar
Integration with Automated Tool\sphinxhyphen{}Chains, such as XTesting or Dovetail,
for continuous deployment, validation, and centralization of test
harnesses and results visualization.
\item {}
\sphinxAtStartPar
Alliance and execution of OVP flows and methodologies, which supports
common structures for code, artifact generation and repository,
Conformance criteria, etc.)
\item {}
\sphinxAtStartPar
Where possible, leveraging ONAP Network and Service Models, with
identified VNF\sphinxhyphen{}specific parameters
\item {}
\sphinxAtStartPar
Utilizing Standard Conformance criteria.
\item {}
\sphinxAtStartPar
Defining reference architecture (RA) as scenarios, and having all
test cases for the RA be involved in OVP
\item {}
\sphinxAtStartPar
Add test cases from operators, which operators already tested in
their environment
\end{itemize}
\subsubsection{Testing}
\label{\detokenize{chapters/chapter02:testing}}
\sphinxAtStartPar
Testing for NFVI Conformance falls under three broad categories \sphinxhyphen{}
Compliance, Validation and Performance. Target NFVI for Conformance
needs to pass all these tests in order to obtain the Conformance badge.
\paragraph{Test Categories}
\label{\detokenize{chapters/chapter02:test-categories}}
\sphinxAtStartPar
The following five test categories have been identified as \sphinxstylestrong{minimal
testing required} to verify NFVI interoperability to satisfy the needs
of VNF developer teams.
\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
Baremetal validation: To validate control and compute nodes hardware
\item {}
\sphinxAtStartPar
VNF Interoperability: After VNFs are on\sphinxhyphen{}boarded, Openstack resources
like Tenant, Network (L2/L3), CPU Pining, security policies, Affinity
anti\sphinxhyphen{}affinity roles and flavors etc. would be validated.
\item {}
\sphinxAtStartPar
Compute components: Validate VMs status and connectivity result after
performing each of listed steps. Best candidate for this testing
would be identify compute node that holds VMs which has L2 and L3
connectivity.
\item {}
\sphinxAtStartPar
Control plane components: Validations for RabbitMQ, Ceph, MariaDB
etc. and OpenStack components like Nova/Glance/Heat etc. APIs.
\item {}
\sphinxAtStartPar
Security: Validation for use RBAC roles and user group policies. See
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_cert/RC1/chapters/chapter07.html}{VNF Testing Cookbook} for complete list.
\end{enumerate}
\sphinxAtStartPar
The following \sphinxstylestrong{Optional Test Categories} which can be considered by
the Operator, or Supplier, for targeted validations to complement
required testing for Conformance:
\begin{itemize}
\item {}
\sphinxAtStartPar
On\sphinxhyphen{}Boarding (MANO agnostic)
\item {}
\sphinxAtStartPar
VNF Functional Testing
\item {}
\sphinxAtStartPar
Charging / Revenue Assurance Verification
\item {}
\sphinxAtStartPar
MicroServices Support
\item {}
\sphinxAtStartPar
Closed Loop Testing
\item {}
\sphinxAtStartPar
VNF Coexistence (ETSI NFV\sphinxhyphen{}TST001 “Noisy Neighbor”)
\item {}
\sphinxAtStartPar
VNF Interactions with Extended NFVi Topology
\item {}
\sphinxAtStartPar
VNF Interactions with Complex NFVi (Akraino)
\item {}
\sphinxAtStartPar
Scalability Testing
\item {}
\sphinxAtStartPar
HA Testing
\item {}
\sphinxAtStartPar
Fault Recovery Testing
\item {}
\sphinxAtStartPar
PM/KPI/Service Assurance Testing
\end{itemize}
\paragraph{Test Harnesses}
\label{\detokenize{chapters/chapter02:test-harnesses}}
\sphinxAtStartPar
In addition to General Best Practices for NFVI Conformance, the
following Quality Engineering (QE) standards will be applied when
defining and delivering test scenarios for Conformance:
\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
Standardized test methodologies / flows capturing requirements from
RA’s, goals and scenarios for test execution, and normalizing test
results.
\item {}
\sphinxAtStartPar
Establishing, and leveraging, working test\sphinxhyphen{}beds which can be
referenced in subsequent test scenario designs.
\item {}
\sphinxAtStartPar
Leveraging standardized cloud\sphinxhyphen{}based facilities such as storage, IAM,
etc.
\item {}
\sphinxAtStartPar
Test Script libraries need to enable Data\sphinxhyphen{}Driven testing of
On\sphinxhyphen{}Boarding, Instantiation, etc.
\item {}
\sphinxAtStartPar
Standards base Test Plan and Test Case suite needs to include sample
VNFs, CSAR, and Automated Test Cases.
\item {}
\sphinxAtStartPar
Documentation needs to be dynamic, and consumable.
\item {}
\sphinxAtStartPar
Harnesses need to apply a “Just add Water” deployment strategy,
enabling test teams to readily implement test harnesses which
promotes Conformance scalability.
\end{enumerate}
\paragraph{Test Results}
\label{\detokenize{chapters/chapter02:test-results}}
\sphinxAtStartPar
\sphinxstylestrong{Categorization}. Test suites will be categorized as Functional or
Performance based.
\sphinxAtStartPar
\sphinxstylestrong{Results.} Test results reporting will be communicated as a boolean
(pass/fail), or Measurements Only.
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Functional Pass/Fail} signals the assertions set in a test script
verify the Functional Requirements (FR) has met its stated objective
as delivered by the developer. This will consist of both positive
validation of expected behavior, as well as negative based testing
when to confirm error handling is working as expected.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Performance\sphinxhyphen{}based Pass/Fail} determination will be made by
comparing Non\sphinxhyphen{}Functional (NFR) NFVI KPIs (obtained after testing)
with the Golden KPIs. Some of the examples of performance KPIs
include, but not limited to: TCP bandwidth, UDP throughput, Memory
latency, Jitter, IOPS etc. See \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter04.html}{Infrastructure Capabilities, Measurements and Catalogue}
for a complete list of metrics and requirements.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Measurement Results}. Baseline Measurements will be performed when
there are no benchmark standards to compare results, or established
FRs/NFRs for which to gauge application / platform behavior in an
integrated environment, or under load conditions. In these cases,
test results will be executed to measure the application, platform,
then prepare FRs/NFRs for subsequent enhancements and test runs.
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Collation | Portal}. The following criteria will be applied to the
collation and presentation of test\sphinxhyphen{}runs seeking NFVI Conformance:
\begin{itemize}
\item {}
\sphinxAtStartPar
RA number and name (e.g. RA\sphinxhyphen{}1 OpenStack)
\item {}
\sphinxAtStartPar
Version of software tested (e.g. OpenStack Ocata)
\item {}
\sphinxAtStartPar
Normalized results will be collated across all test runs
(i.e. centralized database)
\item {}
\sphinxAtStartPar
Clear time stamps of test runs will be provided.
\item {}
\sphinxAtStartPar
Identification of test engineer / executor.
\item {}
\sphinxAtStartPar
Traceability to requirements.
\item {}
\sphinxAtStartPar
Summarized conclusion if conditions warrant test Conformance (see
Badging Section).
\item {}
\sphinxAtStartPar
Portal contains links to Conformance badge(s) received.
\end{itemize}
\subsubsection{Badging}
\label{\detokenize{chapters/chapter02:badging}}
\sphinxAtStartPar
\sphinxstylestrong{Defined}. \sphinxstyleemphasis{Badging} refers to the granting of a Conformance badge by
the OVP to Suppliers/Testers of Anuket NFVI upon demonstration the
testing performed confirms:
\begin{itemize}
\item {}
\sphinxAtStartPar
NFVI adheres to Anuket RA/RM requirements.
\item {}
\sphinxAtStartPar
Anuket certified VNFs functionally perform as expected (i.e. test
cases pass) on NFVI with acceptable levels of stability and
performance.
\end{itemize}
\sphinxAtStartPar
The below figure shows the targeted badge for NFVI.
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{NFVI-badge}.jpg}
\caption{NFVI badge}\label{\detokenize{chapters/chapter02:id4}}\end{figure}
\sphinxAtStartPar
\sphinxstylestrong{Specifics}. More specifically, suppliers of NFVI testing seeking
infrastructure Conformance are required to furnish the following:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Category
&\sphinxstyletheadfamily
\sphinxAtStartPar
OVP/CVC Expectation
&\sphinxstyletheadfamily
\sphinxAtStartPar
Supporting
Artifact(s)
\\
\hline
\sphinxAtStartPar
Lab
&
\sphinxAtStartPar
Verification that the
delivered test lab
conforms to RI\sphinxhyphen{}x lab
requirements for
topology, \# of nodes,
network fabric, etc
&
\sphinxAtStartPar
Bare\sphinxhyphen{}metal H/W
Validations
\\
\hline
\sphinxAtStartPar
Compliance
&
\sphinxAtStartPar
Verification that the
installed software
conforms to RM/RA
requirements for
required components
and configured
options and
extensions, etc
&
\sphinxAtStartPar
Manifest S/W
Validations
\\
\hline
\sphinxAtStartPar
Validation
&
\sphinxAtStartPar
FR Validation of
Component and API
functional behavior
meets requirements
specified in RM/RA\sphinxhyphen{}x
requirements
documents
&
\sphinxAtStartPar
API \& Platform Test
Results
\\
\hline
\sphinxAtStartPar
Performance
&
\sphinxAtStartPar
NFR Validation of
Component, Interface,
and API, results are
within tolerance, or
achieve baseline
measurements
&
\sphinxAtStartPar
Performance Test
Results
\\
\hline
\sphinxAtStartPar
Results Reporting
&
\sphinxAtStartPar
Published of Test
Results into
centralized and
common repository and
reporting portal
&
\sphinxAtStartPar
Normalized Results
per Standards
\\
\hline
\sphinxAtStartPar
Release Notes
&
\sphinxAtStartPar
Supplier provides
concluding remarks,
links to artifacts,
and demonstration of
having met exit
criteria for testing
&
\sphinxAtStartPar
Release Notes
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
\sphinxstylestrong{Conformance Process}
\sphinxAtStartPar
Conformance and issuance of NFVI badges will be as follows:
\begin{itemize}
\item {}
\sphinxAtStartPar
NFVI supplier utilizes, or installs a target RM/RA\sphinxhyphen{}x in a RI lab.
\item {}
\sphinxAtStartPar
Required artifacts are submitted/supplied to the OVP, demonstrating
proper Lab Installation, Compliance, Validation, Performance, and
Release of Results \& Known Issues.
\item {}
\sphinxAtStartPar
Artifact validations will be corroborated and confirmed by the OVP.
with direct comparison between measured results and documented
FRs/NFRs for applications, hardware and software configuration
settings, and host systems.
\item {}
\sphinxAtStartPar
All OVP inquiries, requests for re\sphinxhyphen{}tests, or reformatting /
re\sphinxhyphen{}uploading of results data are closed.
\end{itemize}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{NFVI_certifying_vendor_swhw_solutions}.jpg}
\caption{NFVI Badges}\label{\detokenize{chapters/chapter02:id5}}\end{figure}
\subsection{NFVI Test Cases Requirements}
\label{\detokenize{chapters/chapter02:nfvi-test-cases-requirements}}
\sphinxAtStartPar
The objective of this chapter is to describe the requirements for NFVI
test cases as derived from the reference model and architecture for the
LFN\sphinxhyphen{}based compliance program. This set of requirements eventually
determines the scope of the compliance program and the corresponding
list of test cases included in the compliance program. In particular,
this chapter extends the generic list of NFVI test case requirements
which is provided in Section Test Case Selection Requirements
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter08.html\#multi-cloud-interactions-model}{Multi\sphinxhyphen{}Cloud Interactions Model}
of the reference model.
\subsubsection{Generic Requirements on Test Cases}
\label{\detokenize{chapters/chapter02:generic-requirements-on-test-cases}}
\sphinxAtStartPar
All test cases must fulfill the generic requirements listed in Section
\sphinxtitleref{Test Case Selection
Requirements :ref:\textasciigrave{}ref\_model/chapters/chapter08:multi\sphinxhyphen{}cloud interactions model}
of the reference model.
\sphinxAtStartPar
In addition, for test cases targeting the NFVI compliance program, the
following requirements must be met:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Ref
erence
&\sphinxstyletheadfamily
\sphinxAtStartPar
Description
\\
\hline
\sphinxAtStartPar
x
&
\sphinxAtStartPar
All NFVI test cases \sphinxstyleemphasis{must} be automated. Once the
pre\sphinxhyphen{}conditions of a test case are met, i.e., the system
under test is configured and in a state according to the
pre\sphinxhyphen{}conditions of the particular test case, no manual steps
must be required to run a test case to completion.
\\
\hline
\sphinxAtStartPar
x
&
\sphinxAtStartPar
All NFVI test cases \sphinxstyleemphasis{must} be implemented using publicly
available open source tools. This enables access to test
tools and test case implementations to all interested
parties and organizations.
\\
\hline
\sphinxAtStartPar
x
&
\sphinxAtStartPar
All NFVI test cases \sphinxstyleemphasis{must} be integrated and run in the
Anuket CI/CD pipeline. This requirement ensures that test
cases are functionally correct, reliable, mature and pass
on the NFVI reference implementation.
\\
\hline
\sphinxAtStartPar
x
&
\sphinxAtStartPar
All NFVI test cases \sphinxstyleemphasis{must} treat the NFVI platform as a
black box. In particular, test cases must not perform
actions on or change the state of the system under test
outside the scope of well\sphinxhyphen{}defined APIs as listed by RA1.
This requirement ensures applicability of test cases across
different implementations: reference implementations as
well as commercial implementations.
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\subsubsection{Requirement Types}
\label{\detokenize{chapters/chapter02:requirement-types}}
\sphinxAtStartPar
The compliance and Conformance program intends to validate four
different types of requirements and system properties:
\begin{itemize}
\item {}
\sphinxAtStartPar
API compliance: This is the most relevant type of test case,
validating the functional correctness of the system under test. API
compliance test cases exercise only the specific well\sphinxhyphen{}defined APIs
described in the reference architecture (see \sphinxtitleref{Interfaces and
APIs :doc:\textasciigrave{}ref\_arch/openstack/chapters/chapter05}).
\item {}
\sphinxAtStartPar
Performance: Test cases covering this type of requirement measure
specific performance characteristics of the system under test as
defined in the reference model, the corresponding reference
architectures and in sections further below in this chapter.
\item {}
\sphinxAtStartPar
Resilience: Test cases covering this type of requirement measure
specific resilience characteristics of the system under test as
defined in the reference model, the corresponding reference
architectures and in sections further below in this chapter.
\item {}
\sphinxAtStartPar
Hardware configuration: Validation of the bare\sphinxhyphen{}metal hardware itself
in terms of specs and configuration should be included in the scope
of the compliance test suite eventually. This validation step ensures
that the underlying hardware is correctly configured according to
Anuket hardware specification (TODO: add reference to updated “Pharos
specs”). The purpose of this validation is to act as a pre\sphinxhyphen{}flight
check before performing the extensive compliance test suite.
Moreover, by validating key hardware configuration aspects, it
ensures comparability of performance\sphinxhyphen{}related test results.
\end{itemize}
\sphinxAtStartPar
The extend to which these different types of requirements are included
in the compliance and Conformance test suite is subject to the
availability of test cases. See Section NFVI Test Cases
Requirements below.
\subsubsection{Profile Catalog}
\label{\detokenize{chapters/chapter02:profile-catalog}}
\sphinxAtStartPar
Section Infrastructure Profiles Catalogue
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter04.html\#profiles-and-workload-flavours}{Profiles and Workload Flavours}
of the reference model defines two software profiles, targeting two
different use cases:
\begin{itemize}
\item {}
\sphinxAtStartPar
Basic
\item {}
\sphinxAtStartPar
Network intensive
\end{itemize}
\sphinxAtStartPar
The test cases selected for validating compliance of the two profiles
must cover the functional and non\sphinxhyphen{}functional requirements as listed in
Section Instance Capabilities Mapping
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter04.html\#virtual-network-interface-specifications}{Virtual Network Interface Specifications}
and Section \sphinxtitleref{Instance Performance Measurement Mapping
:ref:\textasciigrave{}ref\_model/chapters/chapter04:storage extensions}
of the reference model.
\sphinxAtStartPar
TODO: what actually needs to be done here is to reference the table from
chapter 4.2.5 and mark for which of those requirements test cases are
actually available in the set of test tools available to us.
\subsubsection{Software \& Hardware Reference}
\label{\detokenize{chapters/chapter02:software-hardware-reference}}
\sphinxAtStartPar
The LFN\sphinxhyphen{}based compliance and Conformance program comprises three
distinct types of NFVI deployment and runtime environments:
\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
A reference implementation deployed in the CI/CD environment,
\item {}
\sphinxAtStartPar
A commercial NFVI product deployed in a vendor’s internal development
and testing environment, and
\item {}
\sphinxAtStartPar
A reference implementation of a commercial NFVI product deployed in a
3rd party lab providing testing and Conformance services.
\end{enumerate}
\sphinxAtStartPar
The test tooling, harnesses and corresponding test cases which are part
of the compliance and Conformance test suite must be capable of running
across all of those environments. This results in the following list of
requirements:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Ref
erence
&\sphinxstyletheadfamily
\sphinxAtStartPar
Description
\\
\hline
\sphinxAtStartPar
x
&
\sphinxAtStartPar
NFVI test cases \sphinxstyleemphasis{must not} interact with remote (Internet)
services apart from downloading container or VM images. In
particular, test tools and test cases must not
automatically upload test data to any system or service run
by LFN or GSMA. The purpose of this requirement is to
protect the confidentially of (intermediate) test data.
\\
\hline
\sphinxAtStartPar
x
&
\sphinxAtStartPar
NFVI test cases \sphinxstyleemphasis{must} support a means of running in an
internal enterprise lab environment. This could be achieved
by either i) natively supporting proxied Internet
connectivity and non\sphinxhyphen{}public DNS servers or ii) by providing
a high\sphinxhyphen{}level description of remote dependencies (e.g.,
container and VM images, network services (DNS), etc.) such
that local mirrors can be set up.
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\subsubsection{Options \& Extensions}
\label{\detokenize{chapters/chapter02:options-extensions}}
\subsubsection{Measurement Criteria}
\label{\detokenize{chapters/chapter02:measurement-criteria}}
\sphinxAtStartPar
Test validations will be corroborated, and confirmed, with direct
comparison between measured results and documented non\sphinxhyphen{}functional
requirements (NFRs) for applications, hardware and software
configuration settings, and host systems. Throughput, latency,
concurrent connections/threads, are all examples of non\sphinxhyphen{}functional
requirements which specify criteria which can be used to judge the
operation of a system, rather than specific behavior of the application
which are defined by functional requirements.
\sphinxAtStartPar
This section attempts to summarize a categorical list of metrics used
for test validations. \sphinxstylestrong{For a complete list of metrics, and
requirements, please refer to Reference Model}
\paragraph{Storage and IOPS}
\label{\detokenize{chapters/chapter02:storage-and-iops}}
\sphinxAtStartPar
\sphinxstylestrong{IOPS} validations for Storage, and/or Storage Extensions, will be
included as part of the final NFVI verification, and validation,
process.
\sphinxAtStartPar
From a definition perspective, IOPS is the standard unit of measurement
for I/O (Input/Output) operations per second. This measurement is a
performance\sphinxhyphen{}based measurement and is usually seen written as\sphinxstylestrong{(1)}:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Total IOPS}: Average number of I/O operations per second.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Read IOPS}: Average number of read I/O operations per second.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Write IOPS}: Average number of write I/O operations per second.
\end{itemize}
\sphinxAtStartPar
For example, if you have a disk that is capable of doing a 100 IOPS, it
means that it is theoretically capable of issuing a 100 read and or
write operations per second. This is in theory. In reality, additional
time is needed to actually process the 100 reads/writes. This additional
time is referred to as “latency”, which reduces the total IOPS that is
calculated, and measured. Latency needs needs to be measured, and
included in the IOPS calculation. Latency will tell us how long it takes
to process a single I/O request, and is generally in the 2 millisecond
(ms) range per IO operation for a physical disk, through 20+ ms, at
which time users will notice an impact in their experience\sphinxstylestrong{(2)}.
\sphinxAtStartPar
Additional factors to consider when measuring IOPS:
\begin{itemize}
\item {}
\sphinxAtStartPar
Take into consideration the percentage of Input (write) vs. Output
(reads) operations, as Writes can be more resource intensive.
\item {}
\sphinxAtStartPar
Determine if Reads were performed from Cache, as this may (will)
result in faster performance, and faster IOPS.
\item {}
\sphinxAtStartPar
Confirm the storage types (Physical, RAID), as storage arrays with
linear, or sequential reading/writing may (will) be slower.
\item {}
\sphinxAtStartPar
Identify the block size used, as using large block sizes vs. small
block sizes can (will) impact IOPS performance.
\item {}
\sphinxAtStartPar
Determine Hard Disk Speeds (HDD in RPMs) used, as the higher the
RPMS, the potential for faster IOPS performance.
\item {}
\sphinxAtStartPar
Quantify the number of disk controllers used to process the number of
requested IO requests.
\item {}
\sphinxAtStartPar
Determine the specific work\sphinxhyphen{}load requirements, as this will dictate
speed, controllers, disk RPM, and latency tolerances.
\end{itemize}
\sphinxAtStartPar
For additional insight, or deeper understanding and reading of IOPS,
refer to the references below.
\subsubsection{Measurement Types}
\label{\detokenize{chapters/chapter02:measurement-types}}
\paragraph{Performance Measurements}
\label{\detokenize{chapters/chapter02:performance-measurements}}
\sphinxAtStartPar
\sphinxstylestrong{Objectives}
\sphinxAtStartPar
The NFVI performance measurements aim at assessing the performance of a
given NFVI implementation on the execution plan (i.e., excluding VIM) by
providing it with a set of significant metrics to be measured.
\sphinxAtStartPar
They should allow validating the performance of any software and/or
hardware NFVI implementation as described in Reference Model.
\sphinxAtStartPar
Of course, they can also be used for other purposes, such as:
\begin{itemize}
\item {}
\sphinxAtStartPar
fine tuning of software and/or hardware NFVI configuration (e.g., the
number of cores dedicated to the DPDK vSwitch)
\item {}
\sphinxAtStartPar
comparing the performances of different software or hardware
technologies (e.g., DPDK vSwitch vs hardware\sphinxhyphen{}offloaded vSwitch)
\item {}
\sphinxAtStartPar
assessing the performance impact of specific features (e.g., with or
without encapsulation)
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Metrics Baseline}
\sphinxAtStartPar
For the purpose of validation, a baseline of the performance metrics is
required for comparison with the results of their measurements on the
NFVI implementation to be validated.
\sphinxAtStartPar
That baseline is a set of threshold values which could be determined by
\sphinxstylestrong{measuring the performance metrics on Reference Implementations}.
\sphinxAtStartPar
The validation can then be based on simple pass/fail test results or on
a grade (e.g., “class” A, B or C) provided by the combination of
pass/fail results for 2 different threshold values of some (or all)
metrics.
\sphinxAtStartPar
\sphinxstylestrong{Metrics Description}
\sphinxAtStartPar
Two categories of metrics are considered depending on whether they are
related to either the VNF domain or the NFVI domain itself:
\begin{itemize}
\item {}
\sphinxAtStartPar
Metrics related to the VNF domain are defined from VNF perspective
(i.e., per VNFC, per vNIC, per vCPU…) and should concern VNF as well
as NFVI actors.
\item {}
\sphinxAtStartPar
Metrics related to the NFVI domain are defined per NFVI node ; their
measurement is based on virtual workloads (i.e., VM or container) in
order to reflect the performance of a NFVI node with a given profile
; they should only concern NFVI actors.
\end{itemize}
\sphinxAtStartPar
The following table contains the list of performance metrics related to
the VNF domain.
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Reference
&\sphinxstyletheadfamily
\sphinxAtStartPar
Name
&\sphinxstyletheadfamily
\sphinxAtStartPar
Unit
&\sphinxstyletheadfamily
\sphinxAtStartPar
Definition/Notes
\\
\hline
\sphinxAtStartPar
vnf.nfvi.perf.001
&
\sphinxAtStartPar
vNIC throughput
&
\sphinxAtStartPar
bits/s
&
\sphinxAtStartPar
Throughput per vNIC
\\
\hline
\sphinxAtStartPar
vnf.nfvi.perf.002
&
\sphinxAtStartPar
vNIC latency
&
\sphinxAtStartPar
second
&
\sphinxAtStartPar
Frame transfer time to
vNIC at the throughput
(vnf.nfvi.perf.001)
\\
\hline
\sphinxAtStartPar
vnf.nfvi.perf.003
&
\sphinxAtStartPar
vNIC delay
variation
&
\sphinxAtStartPar
second
&
\sphinxAtStartPar
Frame Delay Variation
(FDV) to vNIC at the
throughput
(vnf.nfvi.perf.001)
\\
\hline
\sphinxAtStartPar
vnf.nfvi.perf.004
&
\sphinxAtStartPar
vNIC
simultaneous
active flows
&
\sphinxAtStartPar
number
&
\sphinxAtStartPar
Simultaneous active L3/L4
flows per vNIC before a
new flow is dropped
\\
\hline
\sphinxAtStartPar
vnf.nfvi.perf.005
&
\sphinxAtStartPar
vNIC new flows
rate
&
\sphinxAtStartPar
flows/s
&
\sphinxAtStartPar
New L3/L4 flows rate per
vNIC
\\
\hline
\sphinxAtStartPar
vnf.nfvi.perf.006
&
\sphinxAtStartPar
Storage
throughput
&
\sphinxAtStartPar
bytes/s
&
\sphinxAtStartPar
Throughput per virtual
storage unit
\\
\hline
\sphinxAtStartPar
vnf.nfvi.perf.007
&
\sphinxAtStartPar
vCPU capacity
&
\sphinxAtStartPar
test\sphinxhyphen{}specifics
core
&
\sphinxAtStartPar
Compute capacity per vCPU
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
The following table contains the list of performance metrics related to
the NFVI domain.
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Reference
&\sphinxstyletheadfamily
\sphinxAtStartPar
Name
&\sphinxstyletheadfamily
\sphinxAtStartPar
Unit
&\sphinxstyletheadfamily
\sphinxAtStartPar
Definition/Notes
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.001
&
\sphinxAtStartPar
Node network
throughput
&
\sphinxAtStartPar
bits/s
&
\sphinxAtStartPar
Network throughput per
node
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.002
&
\sphinxAtStartPar
Node
simultaneous
active flows
&
\sphinxAtStartPar
number
&
\sphinxAtStartPar
Simultaneous active
L3/L4 flows per node
before a new flow is
dropped
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.003
&
\sphinxAtStartPar
Node new flows
rate
&
\sphinxAtStartPar
flows/s
&
\sphinxAtStartPar
New L3/L4 flows rate
per node
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.004
&
\sphinxAtStartPar
Node storage
throughput
&
\sphinxAtStartPar
bytes/s
&
\sphinxAtStartPar
Storage throughput per
node
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.005
&
\sphinxAtStartPar
Physical core
capacity
&
\sphinxAtStartPar
test\sphinxhyphen{}specifics
core
&
\sphinxAtStartPar
Compute capacity per
physical core usable by
VNFs
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.006
&
\sphinxAtStartPar
Energy
consumption
&
\sphinxAtStartPar
W
&
\sphinxAtStartPar
Energy consumption of
the node without
hosting any VNFC
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.007
&
\sphinxAtStartPar
Network energy
efficiency
&
\sphinxAtStartPar
W/bits/s
&
\sphinxAtStartPar
Energy consumption of
the node at the network
throughput,
(infra.nfvi.perf.001),
normalized to the
measured bit rate
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.008
&
\sphinxAtStartPar
Storage energy
efficiency
&
\sphinxAtStartPar
W/bits/s
&
\sphinxAtStartPar
Energy consumption of
the node at the storage
throughput
(infra.nfvi.perf.004),
normalized to the
measured byte rate
\\
\hline
\sphinxAtStartPar
infra.nfvi.perf.009
&
\sphinxAtStartPar
Compute energy
efficiency
&
\sphinxAtStartPar
W/core
&
\sphinxAtStartPar
Energy consumption of
the node during compute
capacity test
(vnf.nfvi.perf.007 or
infra.nfvi.perf.005),
normalized to the
number of physical
cores usable by VNFs
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
\sphinxstylestrong{MVP Metrics}
\sphinxAtStartPar
The following metrics should be considered as MVP:
\begin{itemize}
\item {}
\sphinxAtStartPar
vnf.nfvi.perf.001,002,006,007
\item {}
\sphinxAtStartPar
infra.nfvi.perf.001,005,006,007,009
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Network Metrics Measurement Test Cases}
\sphinxAtStartPar
The network performance metrics are vnf.nfvi.perf.001\sphinxhyphen{}005 and
infra.nfvi.perf.001\sphinxhyphen{}003,006.
\sphinxAtStartPar
The different possible test cases are defined by each of the 3 following
test traffic conditions.
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Test traffic path across NFVI}
\sphinxAtStartPar
3 traffic path topologies should be considered:
\begin{itemize}
\item {}
\begin{DUlineblock}{0em}
\item[] \sphinxstylestrong{North/South traffic}, between VNFCs within a node and outside
NFVI
\item[] This can be provided by PVP test setup of ETSI GS NFV\sphinxhyphen{}TST009.
\end{DUlineblock}
\item {}
\begin{DUlineblock}{0em}
\item[] \sphinxstylestrong{East/West intra\sphinxhyphen{}node traffic}, between VNFCs within a node
\item[] This can be provided by a V2V (Virtual\sphinxhyphen{}to\sphinxhyphen{}Virtual) test setup
and, in some cases, by PVVP test setup of ETSI GS NFV\sphinxhyphen{}TST009.
\end{DUlineblock}
\item {}
\begin{DUlineblock}{0em}
\item[] \sphinxstylestrong{East/West inter\sphinxhyphen{}node traffic}, between VNFCs in different
nodes
\item[] This can be provided by VPV (Virtual\sphinxhyphen{}Physical\sphinxhyphen{}Virtual) test
setup and, in some cases, by PVVP test setup between 2 nodes.
\end{DUlineblock}
\end{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Test traffic processing by NFVI}
\sphinxAtStartPar
Different processing complexity applicable to the traffic crossing
the NFVI should be considered, including especially (but not
exhaustively):
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{L2 processing} (Ethernet switching), possibly including VLAN
tagging/mapping and encapsulation (e.g., VXLAN)
\item {}
\sphinxAtStartPar
\sphinxstylestrong{L3 processing} (IP routing), possibly including L2 processing
\item {}
\sphinxAtStartPar
\sphinxstylestrong{L4 stateful processing} (e.g., FW, NAT, SFC), also including L3
processing
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Encryption} (e.g., IPSec ESP tunneling)
\end{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Test traffic profile}
\sphinxAtStartPar
Two different test traffic profiles should be considered according to
the two VNF types that must be provided with network connectivity by
the NFVI.
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Forwarded traffic} for L3/L4 forwarding VNF (e.g., PGW, FW)
\sphinxAtStartPar
It is based on ETSI GS NFV\sphinxhyphen{}TST009 and it should be:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{bidirectional UDP traffic} with \sphinxstylestrong{0.001\%} frame loss ratio,
\sphinxstylestrong{300B} average frame size, \sphinxstylestrong{10k} L3/L4 flows,
\item {}
\sphinxAtStartPar
between a \sphinxstylestrong{traffic generator} and a \sphinxstylestrong{traffic receiver}
through a \sphinxstylestrong{L3 forwarding} pseudo\sphinxhyphen{}VNF with sufficient capacity
not to be the test bottleneck.
\end{itemize}
\sphinxAtStartPar
Latency and delay variation measurement should be the 99th
percentile of measured values for one\sphinxhyphen{}way frame transfer
(i.e. from generator to receiver).
\sphinxAtStartPar
The main Anuket test tools candidates for that purpose are
NFVbench and VSPerf.
\begin{quote}
\sphinxAtStartPar
\sphinxstylestrong{Note:}\sphinxstyleemphasis{to be studied whether additional frame sizes and
flows number should be considered}
\end{quote}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Client\sphinxhyphen{}server traffic} for L4/L7 endpoint VNF (e.g., MME, CDN)
\sphinxAtStartPar
It should be:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{bidirectional TCP traffic} with \sphinxstylestrong{1400B} maximum frame
size, \sphinxstylestrong{5k} TCP sessions,
\item {}
\sphinxAtStartPar
between \sphinxstylestrong{2 TCP client\&server endpoints}, one or both as
pseudo\sphinxhyphen{}VNF, with sufficient capacity not to be the test
bottleneck.
\end{itemize}
\sphinxAtStartPar
\sphinxstyleemphasis{Note}: the maximum TCP frame size can be forced by configuring
TCP endpoint link MTU.
\sphinxAtStartPar
The main Anuket test tool candidate for that purpose is Functest
(VMTP and Shaker).
\begin{quote}
\sphinxAtStartPar
\sphinxstylestrong{Note:}\sphinxstyleemphasis{to be studied whether metrics related to latency
and flows for that traffic profile should be considered (how?
with UDP and/or ICMP test traffic in addition?)}
\end{quote}
\end{itemize}
\end{itemize}
\sphinxAtStartPar
The combination of each of those 3 test conditions types and the
different NFVI profiles results in a wide matrix of test cases
(potentially more than 50 cases). Furthermore, these test cases should
be combined with the different metrics resulting in a huge number of
measurements (potentially more than 400 measurements). For the
efficiency of the validation, only the most relevant combinations should
be kept.
\sphinxAtStartPar
This optimization should be based on the following principles:
\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
NFVI domain metrics measurement: on PVP topology only
\item {}
\sphinxAtStartPar
Metrics measurement with forwarded traffic: with no L4 stateful
processing
\item {}
\sphinxAtStartPar
Basic profile metrics measurement: client\sphinxhyphen{}server traffic profile only
\item {}
\sphinxAtStartPar
Flows \& latency related metrics measurement: for PVP only
\end{enumerate}
\sphinxAtStartPar
The following table proposed a possible optimized matrix model of the
test cases against the metrics to be measured.
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|T|T|T|T|T|}
\hline
\sphinxstyletheadfamily &\sphinxstyletheadfamily
\sphinxAtStartPar
\sphinxstylestrong{NFVI
Profiles}
&\sphinxstyletheadfamily
\sphinxAtStartPar
\sphinxstylestrong{B}
&\sphinxstyletheadfamily &\sphinxstyletheadfamily &\sphinxstyletheadfamily &\sphinxstyletheadfamily
\sphinxAtStartPar
\sphinxstylestrong{N}
&\sphinxstyletheadfamily \\
\hline&
\sphinxAtStartPar
\sphinxstylestrong{Test
Cases}
&
\sphinxAtStartPar
V2V \sphinxhyphen{}
L2 \sphinxhyphen{}
SRV
&
\sphinxAtStartPar
VPV \sphinxhyphen{}
L3 \sphinxhyphen{}
SRV
&
\sphinxAtStartPar
PVP \sphinxhyphen{}
L2 \sphinxhyphen{}
SRV
&
\sphinxAtStartPar
PVP \sphinxhyphen{}
L4 \sphinxhyphen{}
SRV
&
\sphinxAtStartPar
PVP \sphinxhyphen{}
L2\sphinxhyphen{}
SRV
&
\sphinxAtStartPar
PVP \sphinxhyphen{}
L2 \sphinxhyphen{}
FWD
\\
\hline&&&&&&&\\
\hline
\sphinxAtStartPar
\sphinxstylestrong{MVP
Metrics}
&
\sphinxAtStartPar
vnf.nfvi.perf.001
&
\sphinxAtStartPar
50Gbps
&
\sphinxAtStartPar
20Gbps
&
\sphinxAtStartPar
20Gbps
&
\sphinxAtStartPar
10Gbps
&
\sphinxAtStartPar
40Gbps
&
\sphinxAtStartPar
40Gbps
\\
\hline&
\sphinxAtStartPar
vnf.nfvi.perf.002
&
\sphinxAtStartPar
n/a (4)
&
\sphinxAtStartPar
n/a (4)
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
0.5ms
\\
\hline&
\sphinxAtStartPar
infra.nfvi.perf.001
&
\sphinxAtStartPar
n/a (1)
&
\sphinxAtStartPar
n/a (1)
&
\sphinxAtStartPar
40Gbps
&
\sphinxAtStartPar
20Gbps
&
\sphinxAtStartPar
60Gbps
&
\sphinxAtStartPar
80Gbps
\\
\hline&
\sphinxAtStartPar
infra.nfvi.perf.007
&
\sphinxAtStartPar
n/a (1)
&
\sphinxAtStartPar
n/a (1)
&
\sphinxAtStartPar
?
W/Gbps
&
\sphinxAtStartPar
?
W/Gbps
&
\sphinxAtStartPar
?
W/Gbps
&
\sphinxAtStartPar
?
W/Gbps
\\
\hline&&&&&&&\\
\hline
\sphinxAtStartPar
\sphinxstylestrong{Non\sphinxhyphen{}MVP
Metrics}
&
\sphinxAtStartPar
vnf.nfvi.perf.003
&
\sphinxAtStartPar
n/a (4)
&
\sphinxAtStartPar
n/a (4)
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
1ms
\\
\hline&
\sphinxAtStartPar
vnf.nfvi.perf.004
&
\sphinxAtStartPar
n/a (4)
&
\sphinxAtStartPar
n/a (4)
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
500k
\\
\hline&
\sphinxAtStartPar
vnf.nfvi.perf.005
&
\sphinxAtStartPar
n/a (4)
&
\sphinxAtStartPar
n/a (4)
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
110kfps
\\
\hline&
\sphinxAtStartPar
infra.nfvi.perf.002
&
\sphinxAtStartPar
n/a (1)
&
\sphinxAtStartPar
n/a (1)
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
1G
\\
\hline&
\sphinxAtStartPar
infra.nfvi.perf.003
&
\sphinxAtStartPar
n/a
(1)
&
\sphinxAtStartPar
n/a
(1)
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
?
&
\sphinxAtStartPar
200kfps
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
\sphinxstyleemphasis{Table notes}:
\begin{itemize}
\item {}
\sphinxAtStartPar
Values are only indicative (see “Metrics Baseline” below)
\item {}
\sphinxAtStartPar
L2/L3/L4 refers to network processing layer
\begin{itemize}
\item {}
\sphinxAtStartPar
L2 for Ethernet switching
\item {}
\sphinxAtStartPar
L3 for IP routing
\item {}
\sphinxAtStartPar
L4 for IP routing with L4 stateful processing (e.g. NAT)
\end{itemize}
\item {}
\sphinxAtStartPar
SRV/FWD refers to the traffic profile (and pseudo\sphinxhyphen{}VNF type implied)
\begin{itemize}
\item {}
\sphinxAtStartPar
SRV for client\sphinxhyphen{}server traffic (and L4/L7 endpoint pseudo\sphinxhyphen{}VNF)
\item {}
\sphinxAtStartPar
FWD for forwarded traffic (and L3/L4 forwarding pseudo\sphinxhyphen{}VNF)
\end{itemize}
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Energy Metrics Measurement Test Cases}
\sphinxAtStartPar
Energy metrics (infra.nfvi.perf.006\sphinxhyphen{}009) should be considered carefully
for NFVI validation since energy consumption may vary a lot across
processor architectures, models and power management features.
\sphinxAtStartPar
They mainly enable to have metrics available regarding NFVI environment
footprint. They also allow energy\sphinxhyphen{}based comparison of different NFVI
software implementations running on a same physical NFVI hardware
implementation.
\sphinxAtStartPar
\sphinxstylestrong{Storage Metrics Measurement Test Cases}
\sphinxAtStartPar
Metric (MVP): vnf.nfvi.perf.006 and infra.nfvi.perf.004,008
\begin{quote}
\sphinxAtStartPar
\sphinxstylestrong{Note:}\sphinxstyleemphasis{to be completed}
\end{quote}
\sphinxAtStartPar
\sphinxstylestrong{Compute Metrics Measurement Test Cases}
\sphinxAtStartPar
The compute performance metrics are vnf.nfvi.perf.007 and
infra.nfvi.perf.004,009.
\sphinxAtStartPar
For normalized results, the compute performance test requires all of the
possible vCPUs available for running workloads to execute workloads. You
need to start as many VMs as needed to force all of the possible CPUs on
the node to run a workload. In this case, the result is normalized:
\begin{itemize}
\item {}
\sphinxAtStartPar
to the number of vCPU, for the vCPU capacity measurements
(vnf.nfvi.perf.007)
\item {}
\sphinxAtStartPar
to the number of physical core usable by VNFs, for the physical core
capacity and compute energy efficiency measurements
infra.nfvi.perf.004,009)
\end{itemize}
\begin{quote}
\sphinxAtStartPar
\sphinxstylestrong{Note:}\sphinxstyleemphasis{to be studied: how to define the different possible test
cases, especially the different workload profiles (i.e., pseudo\sphinxhyphen{}VNF)
to consider}
\end{quote}
\paragraph{Resiliency Measurements}
\label{\detokenize{chapters/chapter02:resiliency-measurements}}
\section{Cloud Infrastructure Test Cases and Traceability to Requirements}
\label{\detokenize{chapters/chapter03:cloud-infrastructure-test-cases-and-traceability-to-requirements}}\label{\detokenize{chapters/chapter03::doc}}
\subsection{Introduction}
\label{\detokenize{chapters/chapter03:introduction}}
\sphinxAtStartPar
The scope of this chapter is to identify and list down test cases based
on requirements defined in \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/README.html}{OpenStack based Reference Architecture}.
This will serve as traceability between test cases and requirements.
\sphinxAtStartPar
Note that each requirement may have one or more test cases associated
with it.
\sphinxAtStartPar
\sphinxstylestrong{must}: Test Cases that are marked as must are considered mandatory
and must pass successfully.
\sphinxAtStartPar
\sphinxstylestrong{should}: Test Cases that are marked as should are expected to be
fulfilled by NFVI but it is up to each service provider to accept an
NFVI targeting reference architecture that is not reflecting on any of
those requirements. The same applies to should not.
\sphinxAtStartPar
\sphinxstylestrong{may}: Test cases that are marked as may are considered optional. The
same applies to may not.
\subsection{Selection Criteria}
\label{\detokenize{chapters/chapter03:selection-criteria}}\begin{quote}
\sphinxAtStartPar
Test cases below are selected based on available test cases in
open\sphinxhyphen{}source tools like FuncTest etc.
\end{quote}
\subsection{Traceability Matrix}
\label{\detokenize{chapters/chapter03:traceability-matrix}}
\sphinxAtStartPar
The following is a Requirements Traceability Matrix (RTM) mapping Test
Case, and/or Test Case Coverage, to RM and RA\sphinxhyphen{}1 requirements (config and
deployment).
\sphinxAtStartPar
The RTM contains RM config (i.e. .conf) requirements listed “per
profile”, followed by RA\sphinxhyphen{}1 requirements. Requirements fall into 8
domains: general(gen), infrastructure(inf), VIM(vim), Interface \&
API(int), Tenants(tnt), LCM(lcm), Assurance(asr), Security(sec).
\sphinxAtStartPar
For detailed information on RM \& RA\sphinxhyphen{}1 NFVI and VNF requirements, please
refer to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_impl/cntt-ri/chapters/chapter03.html}{Cloud Infrastructure + VNF Target State \& Specification}.
\subsubsection{Architecture and OpenStack Requirements}
\label{\detokenize{chapters/chapter03:architecture-and-openstack-requirements}}
\subsubsection{Infrastructure}
\label{\detokenize{chapters/chapter03:infrastructure}}
\subsubsection{VIM}
\label{\detokenize{chapters/chapter03:vim}}
\subsubsection{Interfaces \& APIs}
\label{\detokenize{chapters/chapter03:interfaces-apis}}
\sphinxAtStartPar
The \sphinxhref{https://opendev.org/openstack/devstack-gate}{OpenStack Gates}
verify all changes proposed mostly by running thousands of Tempest tests
completed by Rally scenarios in a few cases. Skipping tests is allowed
in all OpenStack Gates and only failures rate the review \sphinxhyphen{}1 because of
the multiple capabilities and backends selected in the different Gate
jobs. The classical \sphinxhref{https://wiki.opnfv.org/pages/viewpage.action?pageId=29098314}{Functest
containers}
conform to this model which also fits the heterogeneous user
deployments.
\sphinxAtStartPar
From a Anuket Compliance state point, the capabilities are well
described in \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
which allows tuning the test configurations and the test lists to avoid
skipping any test. It results that all tests covering optional
capabilities and all upstream skipped tests due to known bugs are not
executed. All remaining tests must be executed and must pass
successfully.
\sphinxAtStartPar
New \sphinxhref{https://lists.opnfv.org/g/opnfv-tsc/message/5717}{Functest
containers} have
been proposed for Anuket Compliance which simply override the default
test configurations and the default test lists. Any optional capability
or services (e.g. Barbican) can be still verified by the classical
Functest containers.
\sphinxAtStartPar
The next subsections only detail the Tempest tests which must not be
executed from a Compliance state point. The remaining tests have to pass
successfully. They cover all together the API testing requirements as
asked by \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
\sphinxAtStartPar
The following software versions are considered here to verify OpenStack
Wallaby selected by Anuket:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
software
&\sphinxstyletheadfamily
\sphinxAtStartPar
version
\\
\hline
\sphinxAtStartPar
Functest
&
\sphinxAtStartPar
wallaby
\\
\hline
\sphinxAtStartPar
Cinder Tempest plugin
&
\sphinxAtStartPar
1.4.0
\\
\hline
\sphinxAtStartPar
Keystone Tempest plugin
&
\sphinxAtStartPar
0.7.0
\\
\hline
\sphinxAtStartPar
Heat Tempest plugin
&
\sphinxAtStartPar
1.2.0
\\
\hline
\sphinxAtStartPar
Neutron Tempest plugin
&
\sphinxAtStartPar
1.4.0
\\
\hline
\sphinxAtStartPar
Rally OpenStack
&
\sphinxAtStartPar
2.2.1.dev11
\\
\hline
\sphinxAtStartPar
Tempest
&
\sphinxAtStartPar
27.0.0
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{Identity \sphinxhyphen{} Keystone API testing}
\label{\detokenize{chapters/chapter03:identity-keystone-api-testing}}
\sphinxAtStartPar
Keystone API is covered in the OpenStack Gates via
\sphinxhref{https://opendev.org/openstack/tempest}{Tempest} and
\sphinxhref{https://opendev.org/openstack/keystone-tempest-plugin}{keystone\sphinxhyphen{}tempest\sphinxhyphen{}plugin}
as integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}.
\sphinxAtStartPar
According to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
the following test names must not be executed:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\sphinxAtStartPar
.*api.identity.v3.test\_oauth1\_tokens
&
\sphinxAtStartPar
oauth1
\\
\hline
\sphinxAtStartPar
.*scenario.test\_federated\_authentication
&
\sphinxAtStartPar
federation
\\
\hline
\sphinxAtStartPar
.*identity.admin.v2
&
\sphinxAtStartPar
API v2
\\
\hline
\sphinxAtStartPar
.*identity.v2
&
\sphinxAtStartPar
API v2
\\
\hline
\sphinxAtStartPar
.*identity.v3.test\_access\_rules
&
\sphinxAtStartPar
access\_rules
\\
\hline
\sphinxAtStartPar
.*identity.v3.test\_application\_credentials.ApplicationCredentialsV3Test.test\_create\_application\_credential\_access\_rules
&
\sphinxAtStartPar
access\_rules
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
Keystone API is also covered by
\sphinxhref{https://opendev.org/openstack/rally}{Rally}.
\sphinxAtStartPar
Here are the mainline tasks integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}:
\begin{itemize}
\item {}
\sphinxAtStartPar
Authenticate.keystone
\item {}
\sphinxAtStartPar
KeystoneBasic.add\_and\_remove\_user\_role
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_add\_and\_list\_user\_roles
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_and\_list\_tenants
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_and\_delete\_role
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_and\_delete\_service
\item {}
\sphinxAtStartPar
KeystoneBasic.get\_entities
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_update\_and\_delete\_tenant
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_user
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_tenant
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_and\_list\_users
\item {}
\sphinxAtStartPar
KeystoneBasic.create\_tenant\_with\_users
\end{itemize}
\paragraph{Image \sphinxhyphen{} Glance API testing}
\label{\detokenize{chapters/chapter03:image-glance-api-testing}}
\sphinxAtStartPar
Glance API is covered in the OpenStack Gates via
\sphinxhref{https://opendev.org/openstack/tempest}{Tempest} as integrated in
\sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml}{Functest Smoke
CNTT}.
\sphinxAtStartPar
According to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
the following test names must not be executed:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\sphinxAtStartPar
.*image.v1
&
\sphinxAtStartPar
API v1
\\
\hline
\sphinxAtStartPar
.*image.v2.admin.test\_images.ImportCopyImagesTest
&
\sphinxAtStartPar
import\_image
\\
\hline
\sphinxAtStartPar
.*image.v2.test\_images\_negative.ImagesNegativeTest.test\_create\_image\_reserved\_property
&
\sphinxAtStartPar
os\_glance\_reserved
\\
\hline
\sphinxAtStartPar
.*image.v2.test\_images\_negative.ImagesNegativeTest.test\_update\_image\_reserved\_property
&
\sphinxAtStartPar
os\_glance\_reserved
\\
\hline
\sphinxAtStartPar
.*image.v2.test\_images\_negative.ImportImagesNegativeTest.test\_image\_web\_download\_import\_with\_bad\_url
&
\sphinxAtStartPar
web\sphinxhyphen{}downloadimport
\\
\hline
\sphinxAtStartPar
.*image.v2.test\_images.ImportImagesTest
&
\sphinxAtStartPar
import\_image
\\
\hline
\sphinxAtStartPar
.*image.v2.test\_images.MultiStoresImportImages
&
\sphinxAtStartPar
import\_image
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
Glance API is also covered by
\sphinxhref{https://opendev.org/openstack/rally}{Rally}.
\sphinxAtStartPar
Here are the mainline tasks integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}:
\begin{itemize}
\item {}
\sphinxAtStartPar
Authenticate.validate\_glance
\item {}
\sphinxAtStartPar
GlanceImages.create\_and\_delete\_image
\item {}
\sphinxAtStartPar
GlanceImages.create\_and\_list\_image
\item {}
\sphinxAtStartPar
GlanceImages.list\_images
\item {}
\sphinxAtStartPar
GlanceImages.create\_image\_and\_boot\_instances
\end{itemize}
\paragraph{Block Storage \sphinxhyphen{} Cinder API testing}
\label{\detokenize{chapters/chapter03:block-storage-cinder-api-testing}}
\sphinxAtStartPar
Cinder API is covered in the OpenStack Gates via
\sphinxhref{https://opendev.org/openstack/tempest}{Tempest} and
\sphinxhref{https://opendev.org/openstack/cinder-tempest-plugin}{cinder\sphinxhyphen{}tempest\sphinxhyphen{}plugin}
as integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}.
\sphinxAtStartPar
According to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
the following test names must not be executed:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\sphinxAtStartPar
.*test\_incremental\_backup
&
\sphinxAtStartPar
\sphinxurl{https://gerrit.opnfv.org}/gerrit/68881
\\
\hline
\sphinxAtStartPar
.*test\_consistencygroups
&
\sphinxAtStartPar
consistency\_group
\\
\hline
\sphinxAtStartPar
.*test\_backup\_crossproject\_admin\_negative
&
\sphinxAtStartPar
\sphinxurl{https://gerrit.opnfv.org}/gerrit/71011
\\
\hline
\sphinxAtStartPar
.*test\_backup\_crossproject\_user\_negative
&
\sphinxAtStartPar
\sphinxurl{https://gerrit.opnfv.org}/gerrit/71011
\\
\hline
\sphinxAtStartPar
.*test\_volume\_encrypted.TestEncryptedCinderVolumes
&
\sphinxAtStartPar
attach\_encrypted\_volume
\\
\hline
\sphinxAtStartPar
.*test\_encrypted\_volumes\_extend
&
\sphinxAtStartPar
extend\_attached\_encrypted\_volume
\\
\hline
\sphinxAtStartPar
.*test\_group\_snapshots.GroupSnapshotsV319Test.test\_reset\_group\_snapshot\_status
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1770179
\\
\hline
\sphinxAtStartPar
.*test\_multi\_backend
&
\sphinxAtStartPar
multi\sphinxhyphen{}backend
\\
\hline
\sphinxAtStartPar
.*test\_volume\_retype.VolumeRetypeWithMigrationTest
&
\sphinxAtStartPar
multi\sphinxhyphen{}backend
\\
\hline
\sphinxAtStartPar
.*test\_volume\_delete\_cascade.VolumesDeleteCascade.test\_volume\_from\_snapshot\_cascade\_delete
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1677525
\\
\hline
\sphinxAtStartPar
.*test\_volumes\_backup.VolumesBackupsTest.test\_volume\_backup\_create\_get\_detailed\_list\_restore\_delete
&
\sphinxAtStartPar
ceph
\\
\hline
\sphinxAtStartPar
.*test\_volumes\_extend.VolumesExtendAttachedTest.test\_extend\_attached\_volume
&
\sphinxAtStartPar
extend\_attached\_volume
\\
\hline
\sphinxAtStartPar
.*tempest.scenario.test\_volume\_migrate\_attached
&
\sphinxAtStartPar
multi\sphinxhyphen{}backend
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
Cinder API is also covered by
\sphinxhref{https://opendev.org/openstack/rally}{Rally}.
\sphinxAtStartPar
Here are the mainline tasks integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}:
\begin{itemize}
\item {}
\sphinxAtStartPar
Authenticate.validate\_cinder
\item {}
\sphinxAtStartPar
CinderVolumes.create\_and\_delete\_snapshot
\item {}
\sphinxAtStartPar
CinderVolumes.create\_and\_delete\_volume
\item {}
\sphinxAtStartPar
CinderVolumes.create\_and\_extend\_volume
\item {}
\sphinxAtStartPar
CinderVolumes.create\_from\_volume\_and\_delete\_volume
\item {}
\sphinxAtStartPar
CinderQos.create\_and\_list\_qos
\item {}
\sphinxAtStartPar
CinderQos.create\_and\_set\_qos
\item {}
\sphinxAtStartPar
CinderVolumeTypes.create\_and\_list\_volume\_types
\item {}
\sphinxAtStartPar
CinderVolumeTypes.create\_volume\_type\_and\_encryption\_type
\item {}
\sphinxAtStartPar
Quotas.cinder\_update\_and\_delete
\item {}
\sphinxAtStartPar
Quotas.cinder\_update
\end{itemize}
\paragraph{Object Storage \sphinxhyphen{} Swift API testing}
\label{\detokenize{chapters/chapter03:object-storage-swift-api-testing}}
\sphinxAtStartPar
Swift API is covered in the OpenStack Gates via
\sphinxhref{https://opendev.org/openstack/tempest}{Tempest} as integrated in
\sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}.
\sphinxAtStartPar
According to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
the following test names must not be executed:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\sphinxAtStartPar
.*test\_container\_sync.ContainerSyncTest.test\_container\_synchronization
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1317133
\\
\hline
\sphinxAtStartPar
.*test\_container\_sync\_middleware.ContainerSyncMiddlewareTest.test\_container\_synchronization
&
\sphinxAtStartPar
container\_sync
\\
\hline
\sphinxAtStartPar
.*test\_object\_services.ObjectTest.test\_create\_object\_with\_transfer\_encoding
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1905432
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
Swift API is also covered by
\sphinxhref{https://opendev.org/openstack/rally}{Rally}.
\sphinxAtStartPar
Here are the mainline tasks integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}:
\begin{itemize}
\item {}
\sphinxAtStartPar
SwiftObjects.create\_container\_and\_object\_then\_list\_objects
\item {}
\sphinxAtStartPar
SwiftObjects.list\_objects\_in\_containers
\item {}
\sphinxAtStartPar
SwiftObjects.create\_container\_and\_object\_then\_download\_object
\item {}
\sphinxAtStartPar
SwiftObjects.create\_container\_and\_object\_then\_delete\_all
\item {}
\sphinxAtStartPar
SwiftObjects.list\_and\_download\_objects\_in\_containers
\end{itemize}
\paragraph{Networking \sphinxhyphen{} Neutron API testing}
\label{\detokenize{chapters/chapter03:networking-neutron-api-testing}}
\sphinxAtStartPar
Neutron API is covered in the OpenStack Gates via
\sphinxhref{https://opendev.org/openstack/tempest}{Tempest} and
\sphinxhref{https://opendev.org/openstack/neutron-tempest-plugin}{neutron\sphinxhyphen{}tempest\sphinxhyphen{}plugin}
as integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}.
\sphinxAtStartPar
According to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
the following test names must not be executed:
\begin{savenotes}\sphinxatlongtablestart\begin{longtable}[c]{|l|l|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\endfirsthead
\multicolumn{2}{c}%
{\makebox[0pt]{\sphinxtablecontinued{\tablename\ \thetable{} \textendash{} continued from previous page}}}\\
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\endhead
\hline
\multicolumn{2}{r}{\makebox[0pt][r]{\sphinxtablecontinued{continues on next page}}}\\
\endfoot
\endlastfoot
\sphinxAtStartPar
.*admin.test\_agent\_availability\_zone
&
\sphinxAtStartPar
DHCP agent and L3 agent
\\
\hline
\sphinxAtStartPar
.*admin.test\_dhcp\_agent\_scheduler
&
\sphinxAtStartPar
dhcp\_agent\_scheduler
\\
\hline
\sphinxAtStartPar
.*admin.test\_l3\_agent\_scheduler
&
\sphinxAtStartPar
l3\_agent\_scheduler
\\
\hline
\sphinxAtStartPar
.*admin.test\_logging
&
\sphinxAtStartPar
logging
\\
\hline
\sphinxAtStartPar
.*admin.test\_logging\_negative
&
\sphinxAtStartPar
logging
\\
\hline
\sphinxAtStartPar
.*admin.test\_network\_segment\_range
&
\sphinxAtStartPar
network\sphinxhyphen{}segment\sphinxhyphen{}range
\\
\hline
\sphinxAtStartPar
.*admin.test\_ports.PortTestCasesAdmin.test\_regenerate\_mac\_address
&
\sphinxAtStartPar
port\sphinxhyphen{}mac\sphinxhyphen{}address\sphinxhyphen{}regenerate
\\
\hline
\sphinxAtStartPar
.*admin.test\_ports.PortTestCasesResourceRequest
&
\sphinxAtStartPar
port\sphinxhyphen{}resource\sphinxhyphen{}request
\\
\hline
\sphinxAtStartPar
.*admin.test\_routers\_dvr
&
\sphinxAtStartPar
dvr
\\
\hline
\sphinxAtStartPar
.*admin.test\_routers\_flavors
&
\sphinxAtStartPar
l3\sphinxhyphen{}flavors
\\
\hline
\sphinxAtStartPar
.*admin.test\_routers\_ha
&
\sphinxAtStartPar
l3\sphinxhyphen{}ha
\\
\hline
\sphinxAtStartPar
.*test\_floating\_ips.FloatingIPPoolTestJSON
&
\sphinxAtStartPar
floatingip\sphinxhyphen{}pools
\\
\hline
\sphinxAtStartPar
.*test\_floating\_ips.FloatingIPTestJSON.test\_create\_update\_floatingip\_port\_details
&
\sphinxAtStartPar
fip\sphinxhyphen{}port\sphinxhyphen{}details
\\
\hline
\sphinxAtStartPar
.*test\_metering\_extensions
&
\sphinxAtStartPar
metering
\\
\hline
\sphinxAtStartPar
.*test\_metering\_negative
&
\sphinxAtStartPar
metering
\\
\hline
\sphinxAtStartPar
.*test\_networks.NetworksSearchCriteriaTest.test\_list\_validation\_filters
&
\sphinxAtStartPar
filter\sphinxhyphen{}validation
\\
\hline
\sphinxAtStartPar
.*test\_networks.NetworksTestAdmin.test\_create\_tenant\_network\_vxlan
&
\sphinxAtStartPar
vxlan
\\
\hline
\sphinxAtStartPar
.*test\_networks.NetworksTestJSON.test\_create\_update\_network\_dns\_domain
&
\sphinxAtStartPar
dns\sphinxhyphen{}integration
\\
\hline
\sphinxAtStartPar
.*test\_port\_forwardings
&
\sphinxAtStartPar
floating\sphinxhyphen{}ip\sphinxhyphen{}port\sphinxhyphen{}forwarding
\\
\hline
\sphinxAtStartPar
.*test\_port\_forwarding\_negative
&
\sphinxAtStartPar
floating\sphinxhyphen{}ip\sphinxhyphen{}port\sphinxhyphen{}forwarding
\\
\hline
\sphinxAtStartPar
.*test\_ports.PortsTaggingOnCreation
&
\sphinxAtStartPar
tag\sphinxhyphen{}ports\sphinxhyphen{}during\sphinxhyphen{}bulk\sphinxhyphen{}creation
\\
\hline
\sphinxAtStartPar
.*test\_ports.PortsTestJSON.
test\_create\_port\_with\_propagate\_uplink\_status
&
\sphinxAtStartPar
uplink\sphinxhyphen{}status\sphinxhyphen{}propagation
\\
\hline
\sphinxAtStartPar
.*test\_ports.PortsTestJSON.test\_create\_port\_without\_propagate\_uplink\_status
&
\sphinxAtStartPar
uplink\sphinxhyphen{}status\sphinxhyphen{}propagation
\\
\hline
\sphinxAtStartPar
.*test\_ports.PortsTestJSON.test\_create\_update\_port\_with\_dns\_domain
&
\sphinxAtStartPar
dns\sphinxhyphen{}domain\sphinxhyphen{}ports
\\
\hline
\sphinxAtStartPar
.*test\_ports.PortsTestJSON.
test\_create\_update\_port\_with\_dns\_name
&
\sphinxAtStartPar
dns\sphinxhyphen{}integration
\\
\hline
\sphinxAtStartPar
.*test\_ports.PortsTestJSON.test\_create\_update\_port\_with\_no\_dns\_name
&
\sphinxAtStartPar
dns\sphinxhyphen{}integration
\\
\hline
\sphinxAtStartPar
.*test\_revisions.TestRevisions.test\_update\_dns\_domain\_bumps\_revision
&
\sphinxAtStartPar
dns\sphinxhyphen{}integration
\\
\hline
\sphinxAtStartPar
.*test\_revisions.TestRevisions.test\_update\_router\_extra\_attributes\_bumps\_revision
&
\sphinxAtStartPar
l3\sphinxhyphen{}ha
\\
\hline
\sphinxAtStartPar
.*test\_router\_interface\_fip
&
\sphinxAtStartPar
router\sphinxhyphen{}interface\sphinxhyphen{}fip
\\
\hline
\sphinxAtStartPar
.*test\_routers.DvrRoutersTest
&
\sphinxAtStartPar
dvr
\\
\hline
\sphinxAtStartPar
.*test\_routers.HaRoutersTest
&
\sphinxAtStartPar
l3\sphinxhyphen{}ha
\\
\hline
\sphinxAtStartPar
.*test\_routers.RoutersIpV6Test.
test\_extra\_routes\_atomic
&
\sphinxAtStartPar
extraroute\sphinxhyphen{}atomic
\\
\hline
\sphinxAtStartPar
.*test\_routers.RoutersTest.test\_extra\_routes\_atomic
&
\sphinxAtStartPar
extraroute\sphinxhyphen{}atomic
\\
\hline
\sphinxAtStartPar
.*test\_routers\_negative.DvrRoutersNegativeTest
&
\sphinxAtStartPar
dvr
\\
\hline
\sphinxAtStartPar
.*test\_routers\_negative.DvrRoutersNegativeTestExtended
&
\sphinxAtStartPar
dvr
\\
\hline
\sphinxAtStartPar
.*test\_routers\_negative.HaRoutersNegativeTest
&
\sphinxAtStartPar
l3\sphinxhyphen{}ha
\\
\hline
\sphinxAtStartPar
.*test\_security\_groups.RbacSharedSecurityGroupTest
&
\sphinxAtStartPar
rbac\sphinxhyphen{}security\sphinxhyphen{}groups
\\
\hline
\sphinxAtStartPar
.*test\_subnetpool\_prefix\_ops
&
\sphinxAtStartPar
subnetpool\sphinxhyphen{}prefix\sphinxhyphen{}ops
\\
\hline
\sphinxAtStartPar
.*test\_subnetpools.RbacSubnetPoolTest
&
\sphinxAtStartPar
rbac\sphinxhyphen{}subnetpool
\\
\hline
\sphinxAtStartPar
.*test\_subnetp ools\_negative.SubnetPoolsNegativeTestJSON.test\_tenant\_create\_subnetpool\_associate\_shared\_address\_scope
&
\sphinxAtStartPar
rbac\sphinxhyphen{}address\sphinxhyphen{}scope
\\
\hline
\sphinxAtStartPar
.*test\_subnetpools.SubnetPoolsSearchCriteriaTest.test\_list\_validation\_filters
&
\sphinxAtStartPar
filter\sphinxhyphen{}validation
\\
\hline
\sphinxAtStartPar
.*test\_subnets.SubnetsSearchCriteriaTest.test\_list\_validation\_filters
&
\sphinxAtStartPar
filter\sphinxhyphen{}validation
\\
\hline
\sphinxAtStartPar
.*test\_timestamp.TestTimeStamp.
test\_segment\_with\_timestamp
&
\sphinxAtStartPar
standard\sphinxhyphen{}attr\sphinxhyphen{}segment
\\
\hline
\sphinxAtStartPar
.*test\_trunk.TrunkTestInheritJSONBase.test\_add\_subport
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}
/bugs/1863707
\\
\hline
\sphinxAtStartPar
.*test\_trunk.TrunkTestMtusJSON
&
\sphinxAtStartPar
vxlan
\\
\hline
\sphinxAtStartPar
.*test\_trunk\_negative.TrunkTestJSON.test\_create\_subport\_invalid\_inherit\_network\_segmentation\_type
&
\sphinxAtStartPar
vxlan
\\
\hline
\sphinxAtStartPar
.*test\_trunk\_negative.TrunkTestMtusJSON
&
\sphinxAtStartPar
vxlan
\\
\hline
\sphinxAtStartPar
.*test\_qos.QosMinimumBandwidthRuleTestJSON
&
\sphinxAtStartPar
\sphinxurl{https://gerrit.opnfv.org}/gerrit/69105
\\
\hline
\sphinxAtStartPar
.*network.test\_tags
&
\sphinxAtStartPar
tag\sphinxhyphen{}ext
\\
\hline
\sphinxAtStartPar
.*test\_routers.RoutersIpV6Test.test\_create\_router\_set\_gateway\_with\_fixed\_ip
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1676207
\\
\hline
\sphinxAtStartPar
.*test\_routers.RoutersTest.test\_create\_router\_set\_gateway\_with\_fixed\_ip
&
\sphinxAtStartPar
https: //launchpad.net/bugs/1676207
\\
\hline
\sphinxAtStartPar
.*test\_network\_basic\_ops.TestNetworkBasicOps.test\_router\_rescheduling
&
\sphinxAtStartPar
l3\_agent\_scheduler
\\
\hline
\sphinxAtStartPar
.*test\_network\_advanced\_server\_ops.TestNetworkAdvancedServerOps.test\_server\_connectivity\_cold\_migration\_revert
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1836595
\\
\hline
\end{longtable}\sphinxatlongtableend\end{savenotes}
\sphinxAtStartPar
Neutron API is also covered by
\sphinxhref{https://opendev.org/openstack/rally}{Rally}.
\sphinxAtStartPar
Here are the mainline tasks integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}:
\begin{itemize}
\item {}
\sphinxAtStartPar
Authenticate.validate\_neutron
\item {}
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_networks
\item {}
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_ports
\item {}
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_routers
\item {}
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_subnets
\item {}
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_networks
\item {}
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_ports
\item {}
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_routers
\item {}
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_subnets
\item {}
\sphinxAtStartPar
NeutronSecurityGroup.create\_and\_delete\_security\_groups
\item {}
\sphinxAtStartPar
NeutronSecurityGroup.create\_and\_delete\_security\_group\_rule
\item {}
\sphinxAtStartPar
NeutronNetworks.set\_and\_clear\_router\_gateway
\item {}
\sphinxAtStartPar
Quotas.neutron\_update
\end{itemize}
\paragraph{Compute \sphinxhyphen{} Nova API testing}
\label{\detokenize{chapters/chapter03:compute-nova-api-testing}}
\sphinxAtStartPar
Nova API is covered in the OpenStack Gates via
\sphinxhref{https://opendev.org/openstack/tempest}{Tempest} as integrated in
\sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}.
\sphinxAtStartPar
According to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
the following test names must not be executed:
\begin{savenotes}\sphinxatlongtablestart\begin{longtable}[c]{|l|l|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\endfirsthead
\multicolumn{2}{c}%
{\makebox[0pt]{\sphinxtablecontinued{\tablename\ \thetable{} \textendash{} continued from previous page}}}\\
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\endhead
\hline
\multicolumn{2}{r}{\makebox[0pt][r]{\sphinxtablecontinued{continues on next page}}}\\
\endfoot
\endlastfoot
\sphinxAtStartPar
.*admin.test\_agents
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_fixed\_ips
&
\sphinxAtStartPar
neutron
\\
\hline
\sphinxAtStartPar
.*test\_fixed\_ips\_negative
&
\sphinxAtStartPar
neutron
\\
\hline
\sphinxAtStartPar
.*test\_auto\_allocate\_network
&
\sphinxAtStartPar
shared networks
\\
\hline
\sphinxAtStartPar
.*test\_flavors\_microversions.FlavorsV255TestJSON
&
\sphinxAtStartPar
max\_microversion: 2.53
\\
\hline
\sphinxAtStartPar
.*test\_flavors\_microversions.FlavorsV261TestJSON
&
\sphinxAtStartPar
max\_microversion: 2.53
\\
\hline
\sphinxAtStartPar
.*test\_floating\_ips\_bulk
&
\sphinxAtStartPar
nova\sphinxhyphen{}network
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveAutoBlockMigrationV225Test.test\_iscsi\_volume
&
\sphinxAtStartPar
block live migration
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveAutoBlockMigrationV225Test.test\_live\_block\_migration
&
\sphinxAtStartPar
block live migration
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveAutoBlockMigrationV225Test.test\_live\_block\_migration\_paused
&
\sphinxAtStartPar
block live migration
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveAutoBlockMigrationV225Test.test\_volume\_backed\_live\_migration
&
\sphinxAtStartPar
volume\sphinxhyphen{}backed
live migration
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveMigrationTest.test\_iscsi\_volume
&
\sphinxAtStartPar
block live migration
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveMigrationTest.test\_live\_block\_migration
&
\sphinxAtStartPar
block live migration
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveMigrationTest.test\_live\_block\_migration\_paused
&
\sphinxAtStartPar
block live migration
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveMigrationTest.test\_volume\_backed\_live\_migration
&
\sphinxAtStartPar
volume\sphinxhyphen{}backed
live migration
\\
\hline
\sphinxAtStartPar
.*test\_live\_migration.LiveMigrationRemoteConsolesV26Test
&
\sphinxAtStartPar
serial\_console
\\
\hline
\sphinxAtStartPar
.*test\_quotas.QuotasAdminTestV257
&
\sphinxAtStartPar
max\_microversion: 2.53
\\
\hline
\sphinxAtStartPar
.*test\_servers.ServersAdminTestJSON.test\_reset\_network\_inject\_network\_info
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*certificates.test\_certificates
&
\sphinxAtStartPar
cert
\\
\hline
\sphinxAtStartPar
.*test\_quotas\_negative.QuotasSecurityGroupAdminNegativeTest
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1186354
\\
\hline
\sphinxAtStartPar
.*test\_novnc
&
\sphinxAtStartPar
vnc\_console
\\
\hline
\sphinxAtStartPar
.*test\_server\_personality
&
\sphinxAtStartPar
personality
\\
\hline
\sphinxAtStartPar
.*test\_servers.ServerShowV263Test.test\_show\_update\_rebuild\_list\_server
&
\sphinxAtStartPar
certified\_image\_ref
\\
\hline
\sphinxAtStartPar
.*test\_servers\_microversions.ServerShowV254Test
&
\sphinxAtStartPar
max\_microversion: 2.53
\\
\hline
\sphinxAtStartPar
.*test\_servers\_microversions.ServerShowV257Test
&
\sphinxAtStartPar
max\_microversion: 2.53
\\
\hline
\sphinxAtStartPar
.*test\_servers\_negative.ServersNegativeTestJSON.test\_personality\_file\_contents\_not\_encoded
&
\sphinxAtStartPar
personality
\\
\hline
\sphinxAtStartPar
.*test\_server\_actions.ServerActionsTestJSON.test\_change\_server\_password
&
\sphinxAtStartPar
change\_password
\\
\hline
\sphinxAtStartPar
.*test\_server\_actions.ServerActionsTestJSON.test\_get\_vnc\_console
&
\sphinxAtStartPar
vnc\_console
\\
\hline
\sphinxAtStartPar
.*test\_server\_actions.ServerActionsTestJSON.test\_reboot\_server\_soft
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1014647
\\
\hline
\sphinxAtStartPar
.*test\_server\_rescue.ServerBootFromVolumeStableRescueTest
&
\sphinxAtStartPar
stable\_rescue
\\
\hline
\sphinxAtStartPar
.*test\_server\_rescue.ServerStableDeviceRescueTest
&
\sphinxAtStartPar
stable\_rescue
\\
\hline
\sphinxAtStartPar
.*test\_security\_group\_default\_rules
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1311500
\\
\hline
\sphinxAtStartPar
.*test\_security\_groups\_negative.SecurityGroupsNegativeTestJSON.test\_security\_group\_create\_with\_duplicate\_name
&
\sphinxAtStartPar
neutron
\\
\hline
\sphinxAtStartPar
.*test\_security\_groups\_negative.SecurityGroupsNegativeTestJSON.test\_security\_group\_create\_with\_invalid\_group\_description
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1161411
\\
\hline
\sphinxAtStartPar
.*test\_security\_groups\_negative.SecurityGroupsNegativeTestJSON.test\_security\_group\_create\_with\_invalid\_group\_name
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1161411
\\
\hline
\sphinxAtStartPar
.*test\_security\_groups\_negative.SecurityGroupsNegativeTestJSON.test\_update\_security\_group\_with\_invalid\_sg\_des
&
\sphinxAtStartPar
neutron
\\
\hline
\sphinxAtStartPar
.*test\_security\_groups\_negative.SecurityGroupsNegativeTestJSON.test\_update\_security\_group\_with\_invalid\_sg\_id
&
\sphinxAtStartPar
neutron
\\
\hline
\sphinxAtStartPar
.*test\_security\_groups\_negative.SecurityGroupsNegativeTestJSON.test\_update\_security\_group\_with\_invalid\_sg\_name
&
\sphinxAtStartPar
neutron
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata.ServerMetadataTestJSON
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata\_negative.ServerMetadataNegativeTestJSON.test\_delete\_metadata\_non\_existent\_server
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata\_negative.ServerMetadataNegativeTestJSON.test\_metadata\_items\_limit
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata\_negative.ServerMetadataNegativeTestJSON.test\_set\_metadata\_invalid\_key
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata\_negative.ServerMetadataNegativeTestJSON.test\_set\_metadata\_non\_existent\_server
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata\_negative.ServerMetadataNegativeTestJSON.test\_set\_server\_metadata\_blank\_key
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata\_negative.ServerMetadataNegativeTestJSON.test\_set\_server\_metadata\_missing\_metadata
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata\_negative.ServerMetadataNegativeTestJSON.test\_update\_metadata\_non\_existent\_server
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_server\_metadata\_negative.ServerMetadataNegativeTestJSON.test\_update\_metadata\_with\_blank\_key
&
\sphinxAtStartPar
xenapi\_apis
\\
\hline
\sphinxAtStartPar
.*test\_list\_server\_filters.ListServerFiltersTestJSON.test\_list\_servers\_filtered\_by\_ip\_regex
&
\sphinxAtStartPar
\sphinxurl{https://launchpad.net}/bugs/1540645
\\
\hline
\sphinxAtStartPar
.*servers.test\_virtual\_interfaces
&
\sphinxAtStartPar
nova\sphinxhyphen{}network
\\
\hline
\sphinxAtStartPar
.*compute.test\_virtual\_interfaces\_negative
&
\sphinxAtStartPar
nova\sphinxhyphen{}network
\\
\hline
\sphinxAtStartPar
.*compute.test\_networks
&
\sphinxAtStartPar
nova\sphinxhyphen{}network
\\
\hline
\sphinxAtStartPar
.*test\_attach\_volume.AttachVolumeMultiAttach
&
\sphinxAtStartPar
volume\_multiattach
\\
\hline
\sphinxAtStartPar
.*test\_volume\_boot\_pattern
.TestVolumeBootPattern.test\_boot\_server\_from\_encrypted\_volume\_luks
&
\sphinxAtStartPar
attach\_encrypted\_volume
\\
\hline
\sphinxAtStartPar
.*test\_volume\_swap
&
\sphinxAtStartPar
swap\_volume
\\
\hline
\sphinxAtStartPar
.*test\_encrypted\_cinder\_volumes
&
\sphinxAtStartPar
attach\_encrypted\_volume
\\
\hline
\sphinxAtStartPar
.*test\_minbw\_allocation\_placement
&
\sphinxAtStartPar
microversion
\\
\hline
\sphinxAtStartPar
.*test\_volumes\_negative.UpdateMultiattachVolumeNegativeTest.test\_multiattach\_rw\_volume\_update\_failure
&
\sphinxAtStartPar
volume\_multiattach
\\
\hline
\sphinxAtStartPar
.*test\_shelve\_instance.TestShelveInstance.test\_cold\_migrate\_unshelved\_instance
&
\sphinxAtStartPar
shelve\_migrate
\\
\hline
\end{longtable}\sphinxatlongtableend\end{savenotes}
\sphinxAtStartPar
Nova API is also covered by
\sphinxhref{https://opendev.org/openstack/rally}{Rally}.
\sphinxAtStartPar
Here are the mainline tasks integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}:
\begin{itemize}
\item {}
\sphinxAtStartPar
Authenticate.validate\_nova
\item {}
\sphinxAtStartPar
NovaServers.boot\_and\_live\_migrate\_server
\item {}
\sphinxAtStartPar
NovaServers.boot\_server\_attach\_created\_volume\_and\_live\_migrate
\item {}
\sphinxAtStartPar
NovaServers.boot\_server\_from\_volume\_and\_live\_migrate
\item {}
\sphinxAtStartPar
NovaKeypair.boot\_and\_delete\_server\_with\_keypair
\item {}
\sphinxAtStartPar
NovaServers.boot\_server\_from\_volume\_and\_delete
\item {}
\sphinxAtStartPar
NovaServers.pause\_and\_unpause\_server
\item {}
\sphinxAtStartPar
NovaServers.boot\_and\_migrate\_server
\item {}
\sphinxAtStartPar
NovaServers.boot\_server\_and\_list\_interfaces
\item {}
\sphinxAtStartPar
NovaServers.boot\_server\_associate\_and\_dissociate\_floating\_ip
\item {}
\sphinxAtStartPar
NovaServerGroups.create\_and\_delete\_server\_group
\item {}
\sphinxAtStartPar
Quotas.nova\_update
\end{itemize}
\paragraph{Orchestration \sphinxhyphen{} Heat API testing}
\label{\detokenize{chapters/chapter03:orchestration-heat-api-testing}}
\sphinxAtStartPar
Heat API is covered in the OpenStack Gates via
\sphinxhref{https://opendev.org/openstack/heat-tempest-plugin}{heat\sphinxhyphen{}tempest\sphinxhyphen{}plugin}
as integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}
\sphinxAtStartPar
According to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
the following test names must not be executed:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test rejection regular expressions
&\sphinxstyletheadfamily
\sphinxAtStartPar
reasons
\\
\hline
\sphinxAtStartPar
.*functional.test\_lbaasv2
&
\sphinxAtStartPar
lbaasv2
\\
\hline
\sphinxAtStartPar
.*functional.test\_encryption\_vol\_type
&
\sphinxAtStartPar
\sphinxurl{https://storyboard.openstack.org}/\#!/story/2007804
\\
\hline
\sphinxAtStartPar
.*RemoteStackTest.test\_stack\_create\_with\_cloud\_credential
&
\sphinxAtStartPar
\sphinxurl{https://gerrit.opnfv.org}/gerrit/c/functest/+/69926
\\
\hline
\sphinxAtStartPar
.*scenario.test\_aodh\_alarm
&
\sphinxAtStartPar
aodh
\\
\hline
\sphinxAtStartPar
.*tests.scenario.test\_autoscaling\_lb
&
\sphinxAtStartPar
lbaas
\\
\hline
\sphinxAtStartPar
.*scenario.test\_autoscaling\_lbv2
&
\sphinxAtStartPar
lbaasv2
\\
\hline
\sphinxAtStartPar
.*scenario.test\_server\_software\_config
&
\sphinxAtStartPar
\sphinxurl{https://gerrit.opnfv.org/}gerrit/c/functest/+/69926
\\
\hline
\sphinxAtStartPar
.*test\_volumes.VolumeBackupRestoreIntegrationTest
&
\sphinxAtStartPar
\sphinxurl{https://gerrit.opnfv.org/}gerrit/c/functest/+/69931
\\
\hline
\sphinxAtStartPar
.*scenario.test\_octavia\_lbaas
&
\sphinxAtStartPar
octavia
\\
\hline
\sphinxAtStartPar
.*scenario.test\_server\_cfn\_init
&
\sphinxAtStartPar
\sphinxurl{https://gerrit.opnfv.org/}gerrit/c/functest/+/70004
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
Heat API is also covered by
\sphinxhref{https://opendev.org/openstack/rally}{Rally}.
\sphinxAtStartPar
Here are the mainline tasks integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/smoke-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Smoke
CNTT}:
\begin{itemize}
\item {}
\sphinxAtStartPar
Authenticate.validate\_heat
\item {}
\sphinxAtStartPar
HeatStacks.create\_update\_delete\_stack
\item {}
\sphinxAtStartPar
HeatStacks.create\_check\_delete\_stack
\item {}
\sphinxAtStartPar
HeatStacks.create\_suspend\_resume\_delete\_stack
\item {}
\sphinxAtStartPar
HeatStacks.list\_stacks\_and\_resources
\end{itemize}
\subsubsection{Dashboard}
\label{\detokenize{chapters/chapter03:dashboard}}
\sphinxAtStartPar
Horizon is covered in the OpenStack Gates via
\sphinxhref{https://github.com/openstack/tempest-horizon}{tempest\sphinxhyphen{}horizon} as
integrated in \sphinxhref{https://git.opnfv.org/functest/tree/docker/healthcheck/testcases.yaml?h=stable\%2Fwallaby}{Functest
Healthcheck}.
\subsubsection{OpenStack API benchmarking}
\label{\detokenize{chapters/chapter03:openstack-api-benchmarking}}
\sphinxAtStartPar
\sphinxhref{https://opendev.org/openstack/rally}{Rally} is tool and framework
that allows to perform OpenStack API benchmarking.
\sphinxAtStartPar
Here are the Rally\sphinxhyphen{}based test cases proposed by \sphinxhref{https://git.opnfv.org/functest/tree/docker/benchmarking-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Benchmarking
CNTT}:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{rally\_full}:
Functest scenarios iterating 10 times the mainline Rally scenarios
\item {}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_jobs\_cntt-run-5/rally\_jobs\_cntt/rally\_jobs\_cntt.html}{rally\_jobs}:
Neutron scenarios executed in the OpenStack gates
\end{itemize}
\sphinxAtStartPar
At the time of writing, no KPI is defined in
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_arch/openstack/chapters/chapter05.html}{Interfaces and APIs}
which would have asked for an update of the default SLA (maximum failure
rate of 0\%) proposed in \sphinxhref{https://git.opnfv.org/functest/tree/docker/benchmarking-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Benchmarking
CNTT}
\paragraph{Identity \sphinxhyphen{} Keystone API benchmarking}
\label{\detokenize{chapters/chapter03:identity-keystone-api-benchmarking}}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{Functest
rally\_full\_cntt}:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Scenarios
&\sphinxstyletheadfamily
\sphinxAtStartPar
Iterations
\\
\hline
\sphinxAtStartPar
Authenticate.keystone
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.add\_and\_remove\_user\_role
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_add\_and\_list\_user\_roles
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_and\_list\_tenants
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_and\_delete\_role
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_and\_delete\_service
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.get\_entities
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_update\_and\_delete\_tenant
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_user
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_tenant
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_and\_list\_users
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
KeystoneBasic.create\_tenant\_with\_users
&
\sphinxAtStartPar
10
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{Image \sphinxhyphen{} Glance API benchmarking}
\label{\detokenize{chapters/chapter03:image-glance-api-benchmarking}}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{Functest
rally\_full\_cntt}:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Scenarios
&\sphinxstyletheadfamily
\sphinxAtStartPar
Iterations
\\
\hline
\sphinxAtStartPar
Authenticate.validate\_glance
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
GlanceImages.create\_and\_delete\_image
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
GlanceImages.create\_and\_list\_image
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
GlanceImages.list\_images
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
GlanceImages.create\_image\_and\_boot\_instances
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
GlanceImages.create\_and\_deactivate\_image
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
GlanceImages.create\_and\_download\_image
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
GlanceImages.create\_and\_get\_image
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
GlanceImages.create\_and\_update\_image
&
\sphinxAtStartPar
10
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{Block Storage \sphinxhyphen{} Cinder API benchmarking}
\label{\detokenize{chapters/chapter03:block-storage-cinder-api-benchmarking}}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{Functest
rally\_full\_cntt}:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Scenarios
&\sphinxstyletheadfamily
\sphinxAtStartPar
Iterations
\\
\hline
\sphinxAtStartPar
Authenticate.validate\_glance
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_and\_attach\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_and\_list\_snapshots
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_and\_list\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_and\_upload\_volume\_to\_image
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_nested\_snapshots\_and\_attach\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_snapshot\_and\_attach\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.list\_volumes
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_and\_delete\_snapshot
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_and\_delete\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_and\_extend\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumes.create\_from\_volume\_and\_delete\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderQos.create\_and\_get\_qos
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderQos.create\_and\_list\_qos
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderQos.create\_and\_set\_qos
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumeTypes.create\_and\_get\_volume\_type
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumeTypes.create\_and\_list\_volume\_types
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumeTypes.create\_and\_update\_volume\_type
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumeTypes.create\_volume\_type\_and\_encryption\_type
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
CinderVolumeTypes.create\_volume\_type\_add\_and\_list\_type\_access
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
Quotas.cinder\_update\_and\_delete
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
Quotas.cinder\_update
&
\sphinxAtStartPar
10
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{Object Storage \sphinxhyphen{} Swift API benchmarking}
\label{\detokenize{chapters/chapter03:object-storage-swift-api-benchmarking}}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{Functest
rally\_full\_cntt}:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Scenarios
&\sphinxstyletheadfamily
\sphinxAtStartPar
Iterations
\\
\hline
\sphinxAtStartPar
SwiftObjects.create\_container\_and\_object\_then\_list\_objects
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
SwiftObjects.list\_objects\_in\_containers
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
SwiftObjects.create\_container\_and\_object\_then\_download\_object
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
SwiftObjects.create\_container\_and\_object\_then\_delete\_all
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
SwiftObjects.list\_and\_download\_objects\_in\_containers
&
\sphinxAtStartPar
10
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{Networking \sphinxhyphen{} Neutron API benchmarking}
\label{\detokenize{chapters/chapter03:networking-neutron-api-benchmarking}}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{Functest
rally\_full\_cntt}:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Scenarios
&\sphinxstyletheadfamily
\sphinxAtStartPar
Iterations
\\
\hline
\sphinxAtStartPar
Authenticate.validate\_neutron
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_update\_networks
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_update\_ports
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_update\_routers
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_update\_subnets
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_networks
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_ports
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_routers
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_subnets
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_networks
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_ports
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_routers
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_subnets
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronSecurityGroup.create\_and\_delete\_security\_groups
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronSecurityGroup.create\_and\_delete\_security\_group\_rule
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronSecurityGroup.create\_and\_list\_security\_group\_rules
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronSecurityGroup.create\_and\_show\_security\_group
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.set\_and\_clear\_router\_gateway
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_show\_ports
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_show\_routers
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_show\_subnets
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
Quotas.neutron\_update
&
\sphinxAtStartPar
10
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{Functest
rally\_jobs\_cntt}:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Scenarios
&\sphinxstyletheadfamily
\sphinxAtStartPar
Iterations
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_networks
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_ports
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_routers
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_delete\_subnets
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_networks
&
\sphinxAtStartPar
100
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_ports
&
\sphinxAtStartPar
8
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_routers
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_list\_subnets
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_update\_networks
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_update\_ports
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_update\_routers
&
\sphinxAtStartPar
40
\\
\hline
\sphinxAtStartPar
NeutronNetworks.create\_and\_update\_subnets
&
\sphinxAtStartPar
100
\\
\hline
\sphinxAtStartPar
NeutronTrunks.create\_and\_list\_trunks
&
\sphinxAtStartPar
4
\\
\hline
\sphinxAtStartPar
Quotas.neutron\_update
&
\sphinxAtStartPar
40
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{Compute \sphinxhyphen{} Nova API benchmarking}
\label{\detokenize{chapters/chapter03:compute-nova-api-benchmarking}}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{Functest
rally\_full\_cntt}:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Scenarios
&\sphinxstyletheadfamily
\sphinxAtStartPar
Iterations
\\
\hline
\sphinxAtStartPar
Authenticate.validate\_nova
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaKeypair.create\_and\_delete\_keypair
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaKeypair.create\_and\_list\_keypairs
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_and\_bounce\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_and\_delete\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_and\_list\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_and\_rebuild\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.snapshot\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server\_from\_volume
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.list\_servers
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.resize\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_and\_live\_migrate\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server\_attach\_created\_volume\_and\_live\_migrate
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server\_from\_volume\_and\_live\_migrate
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaKeypair.boot\_and\_delete\_server\_with\_keypair
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server\_from\_volume\_and\_delete
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.pause\_and\_unpause\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_and\_migrate\_server
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server\_and\_list\_interfaces
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_and\_get\_console\_url
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server\_and\_attach\_interface
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server\_attach\_volume\_and\_list\_attachments
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_server\_associate\_and\_dissociate\_floating\_ip
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServers.boot\_and\_associate\_floating\_ip
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServerGroups.create\_and\_delete\_server\_group
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServerGroups.create\_and\_get\_server\_group
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
NovaServerGroups.create\_and\_list\_server\_groups
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
Quotas.nova\_update
&
\sphinxAtStartPar
10
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{Orchestration \sphinxhyphen{} Heat API benchmarking}
\label{\detokenize{chapters/chapter03:orchestration-heat-api-benchmarking}}
\sphinxAtStartPar
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-cntt-wallaby-rally\_full\_cntt-run-5/rally\_full\_cntt/rally\_full\_cntt.html}{Functest
rally\_full\_cntt}:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Scenarios
&\sphinxstyletheadfamily
\sphinxAtStartPar
Iterations
\\
\hline
\sphinxAtStartPar
Authenticate.validate\_heat
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
HeatStacks.create\_and\_delete\_stack
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
HeatStacks.create\_and\_list\_stack
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
HeatStacks.create\_update\_delete\_stack
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
HeatStacks.create\_check\_delete\_stack
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
HeatStacks.create\_suspend\_resume\_delete\_stack
&
\sphinxAtStartPar
10
\\
\hline
\sphinxAtStartPar
HeatStacks.list\_stacks\_and\_resources
&
\sphinxAtStartPar
10
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\subsubsection{Dataplane benchmarking}
\label{\detokenize{chapters/chapter03:dataplane-benchmarking}}
\sphinxAtStartPar
\sphinxhref{https://git.opnfv.org/functest/tree/docker/benchmarking-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Benchmarking
CNTT}
offers two benchmarking dataplane test cases leveraging on:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxhref{http://vmtp.readthedocs.io/en/latest}{VMTP}
\item {}
\sphinxAtStartPar
\sphinxhref{http://pyshaker.readthedocs.io/en/latest/}{Shaker}
\end{itemize}
\sphinxAtStartPar
\sphinxhref{http://vmtp.readthedocs.io/en/latest}{VMTP} is a small python
application that will automatically perform ping connectivity, round
trip time measurement (latency) and TCP/UDP throughput measurement on
any OpenStack deployment.
\sphinxAtStartPar
\sphinxhref{http://pyshaker.readthedocs.io/en/latest/}{Shaker} wraps around
popular system network testing tools like iperf, iperf3 and netperf
(with help of flent).
\sphinxhref{http://pyshaker.readthedocs.io/en/latest/}{Shaker} is able to deploy
OpenStack instances and networks in different topologies.
\sphinxhref{http://pyshaker.readthedocs.io/en/latest/}{Shaker} scenario
specifies the deployment and list of tests to execute.
\sphinxAtStartPar
At the time of writing, no KPIs are defined in Anuket specifications
which would have asked for an update of the default SLA proposed in
\sphinxhref{https://git.opnfv.org/functest/tree/docker/benchmarking-cntt/testcases.yaml?h=stable\%2Fwallaby}{Functest Benchmarking
CNTT}
\sphinxAtStartPar
On top of this dataplane benchmarking described in VMTP \& Shaker, we
need to integrate testing as described in \sphinxhref{https://www.etsi.org/deliver/etsi\_gs/NFV-TST/001\_099/009/03.01.01\_60/gs\_NFV-TST009v030101p.pdf}{ETSI GS NFV\sphinxhyphen{}TST 009:
Specification of Networking Benchmarks and Measurement Methods for
NFVI}.
This type of testing is better suited to measure the networking
capabilities of a compute node. The \sphinxhref{https://wiki.opnfv.org/display/SAM/Rapid+scripting}{rapid
scripts} in
conjunction with the \sphinxhref{https://wiki.opnfv.org/pages/viewpage.action?pageId=12387840}{PROX
tool}
offers an open source implementation for this type of testing.
\paragraph{VMTP}
\label{\detokenize{chapters/chapter03:vmtp}}
\sphinxAtStartPar
Here are the
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-wallaby-vmtp-run-8/vmtp/vmtp.json}{scenarios}
executed by \sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-wallaby-vmtp-run-8/vmtp/vmtp.html}{Functest
vmtp}:
\sphinxhyphen{} VM to VM same network fixed IP (intra\sphinxhyphen{}node) \sphinxhyphen{} VM to VM different
network fixed IP (intra\sphinxhyphen{}node) \sphinxhyphen{} VM to VM different network floating IP
(intra\sphinxhyphen{}node) \sphinxhyphen{} VM to VM same network fixed IP (inter\sphinxhyphen{}node) \sphinxhyphen{} VM to VM
different network fixed IP (inter\sphinxhyphen{}node) \sphinxhyphen{} VM to VM different network
floating IP (inter\sphinxhyphen{}node)
\sphinxAtStartPar
Here are all results per scenario:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
protocol
&\sphinxstyletheadfamily
\sphinxAtStartPar
pkt\_size
&\sphinxstyletheadfamily
\sphinxAtStartPar
results
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
64
&
\sphinxAtStartPar
rtt\_avg\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
64
&
\sphinxAtStartPar
rtt\_max\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
64
&
\sphinxAtStartPar
rtt\_min\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
64
&
\sphinxAtStartPar
rtt\_stddev
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
391
&
\sphinxAtStartPar
rtt\_avg\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
391
&
\sphinxAtStartPar
rtt\_max\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
391
&
\sphinxAtStartPar
rtt\_min\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
391
&
\sphinxAtStartPar
rtt\_stddev
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
1500
&
\sphinxAtStartPar
rtt\_avg\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
1500
&
\sphinxAtStartPar
rtt\_max\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
1500
&
\sphinxAtStartPar
rtt\_min\_ms
\\
\hline
\sphinxAtStartPar
ICMP
&
\sphinxAtStartPar
1500
&
\sphinxAtStartPar
rtt\_stddev
\\
\hline
\sphinxAtStartPar
UDP
&
\sphinxAtStartPar
128
&
\sphinxAtStartPar
loss\_rate
\\
\hline
\sphinxAtStartPar
UDP
&
\sphinxAtStartPar
128
&
\sphinxAtStartPar
throughput\_kbps
\\
\hline
\sphinxAtStartPar
UDP
&
\sphinxAtStartPar
1024
&
\sphinxAtStartPar
loss\_rate
\\
\hline
\sphinxAtStartPar
UDP
&
\sphinxAtStartPar
1024
&
\sphinxAtStartPar
throughput\_kbps
\\
\hline
\sphinxAtStartPar
UDP
&
\sphinxAtStartPar
8192
&
\sphinxAtStartPar
loss\_rate
\\
\hline
\sphinxAtStartPar
UDP
&
\sphinxAtStartPar
8192
&
\sphinxAtStartPar
throughput\_kbps
\\
\hline
\sphinxAtStartPar
TCP
&
\sphinxAtStartPar
65536
&
\sphinxAtStartPar
rtt\_ms
\\
\hline
\sphinxAtStartPar
TCP
&
\sphinxAtStartPar
65536
&
\sphinxAtStartPar
throughput\_kbps
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{Shaker}
\label{\detokenize{chapters/chapter03:shaker}}
\sphinxAtStartPar
Here are the
\sphinxhref{http://artifacts.opnfv.org/functest/KDBNITEN317M/functest-opnfv-functest-benchmarking-wallaby-shaker-run-8/shaker/report.json}{scenarios}
executed by Shaker:
\begin{itemize}
\item {}
\sphinxAtStartPar
OpenStack L2
\item {}
\sphinxAtStartPar
OpenStack L3 East\sphinxhyphen{}West
\item {}
\sphinxAtStartPar
OpenStack L3 North\sphinxhyphen{}South
\item {}
\sphinxAtStartPar
OpenStack L3 North\sphinxhyphen{}South Performance
\end{itemize}
\sphinxAtStartPar
Here are all samples:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test
&\sphinxstyletheadfamily
\sphinxAtStartPar
samples
\\
\hline
\sphinxAtStartPar
Bi\sphinxhyphen{}directional
&
\sphinxAtStartPar
ping\_icmp (ms)
\\
\hline
\sphinxAtStartPar
Bi\sphinxhyphen{}directional
&
\sphinxAtStartPar
tcp\_download (Mbits/s)
\\
\hline
\sphinxAtStartPar
Bi\sphinxhyphen{}directional
&
\sphinxAtStartPar
tcp\_upload (Mbits/s)
\\
\hline
\sphinxAtStartPar
Download
&
\sphinxAtStartPar
ping\_icmp (ms)
\\
\hline
\sphinxAtStartPar
Download
&
\sphinxAtStartPar
tcp\_download (Mbits/s)
\\
\hline
\sphinxAtStartPar
Upload
&
\sphinxAtStartPar
ping\_icmp (ms)
\\
\hline
\sphinxAtStartPar
Upload
&
\sphinxAtStartPar
tcp\_upload (Mbits/s)
\\
\hline
\sphinxAtStartPar
Ping
&
\sphinxAtStartPar
ping\_icmp (ms)
\\
\hline
\sphinxAtStartPar
Ping
&
\sphinxAtStartPar
ping\_udp (ms)
\\
\hline
\sphinxAtStartPar
TCP
&
\sphinxAtStartPar
bandwidth (bit/s)
\\
\hline
\sphinxAtStartPar
TCP
&
\sphinxAtStartPar
retransmits
\\
\hline
\sphinxAtStartPar
UDP
&
\sphinxAtStartPar
packets (pps)
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\paragraph{PROX}
\label{\detokenize{chapters/chapter03:prox}}
\sphinxAtStartPar
The generator used with the rapid scripts is PROX with a specific
generator configuration file. When multiple flows are requested, the
generator starts randomizing bits in the source and destination UDP
ports. The number of flows to be generated during each run of the test
is specified in the test files (e.g. TST009\_Throughput.test). Packet
size used during the test is also defined in the test file. IMIX is not
supported yet, but you could take the average packet size of the IMIX
for now. When defining n packet sizes with m different flow sizes, the
test will run n x m times and will produce the results for these n x m
combinations. All throughput benchmarking is done by a generator sending
packets to a reflector. This results in bidirectional traffic which
should be identical (src and dest IP and ports swapped) if all traffic
goes through. The VMs or containers use only 1 vNIC for incoming and
outgoing traffic. Multiple queues can be used. Multiple VMs or
containers can be deployed prior to running any tests. This allows to
use generator\sphinxhyphen{}reflector pairs on the same or different compute nodes, on
the same or different NUMA nodes.
\subsubsection{Opensource VNF onboarding and testing}
\label{\detokenize{chapters/chapter03:opensource-vnf-onboarding-and-testing}}
\sphinxAtStartPar
Running opensource VNFs is a key technical solution to ensure that the
platforms meet Network Functions Virtualization requirements. \sphinxhref{https://git.opnfv.org/functest/tree/docker/vnf/testcases.yaml?h=stable\%2Fwallaby}{Functest
VNF}
offers 5 test cases which automatically onboard and test the following 3
opensource VNFs:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxhref{https://clearwater.readthedocs.io/en/stable/}{Clearwater IMS}
\item {}
\sphinxAtStartPar
\sphinxhref{https://www.vyos.io/}{VyOS vRouter}
\item {}
\sphinxAtStartPar
\sphinxhref{https://www.openairinterface.org/}{OpenAirInterface vEPC}
\end{itemize}
\sphinxAtStartPar
Here are the full list of orchestrators used for all these deployments:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxhref{https://cloudify.co/}{Cloudify}
\item {}
\sphinxAtStartPar
\sphinxhref{https://wiki.openstack.org/wiki/Heat}{Heat}
\item {}
\sphinxAtStartPar
\sphinxhref{https://jaas.ai/}{Juju}
\end{itemize}
\sphinxAtStartPar
The VNF are covered by upstream tests when possible (see
\sphinxhref{https://github.com/Metaswitch/clearwater-live-test}{clearwater\sphinxhyphen{}live\sphinxhyphen{}test})
and by Functest VNF tests in the other cases.
\subsubsection{Tenants}
\label{\detokenize{chapters/chapter03:tenants}}
\subsubsection{LCM}
\label{\detokenize{chapters/chapter03:lcm}}
\subsubsection{Assurance}
\label{\detokenize{chapters/chapter03:assurance}}
\subsubsection{Security}
\label{\detokenize{chapters/chapter03:security}}
\subsubsection{Resilience}
\label{\detokenize{chapters/chapter03:resilience}}
\subsubsection{Bare\sphinxhyphen{}metal validations}
\label{\detokenize{chapters/chapter03:bare-metal-validations}}
\subsection{Test Cases Traceability to Requirements}
\label{\detokenize{chapters/chapter03:test-cases-traceability-to-requirements}}
\subsubsection{RM/RA\sphinxhyphen{}1 Requirements}
\label{\detokenize{chapters/chapter03:rm-ra-1-requirements}}
\sphinxAtStartPar
According to \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_cert/RC1/chapters/chapter04.html}{OpenStack\sphinxhyphen{}based cloud infrastructure Testing Cookbook}
the following test cases must pass as they are for Anuket NFVI Conformance:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
container
&\sphinxstyletheadfamily
\sphinxAtStartPar
test case
&\sphinxstyletheadfamily
\sphinxAtStartPar
criteria
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}healthcheck:wallaby
&
\sphinxAtStartPar
tempest\_horizon
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}smoke\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
tempest\_neutron\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}smoke\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
tempest\_cinder\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}smoke\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
tempest\_keystone\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}smoke\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
rally\_sanity\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}smoke\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
tempest\_full\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}smoke\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
tempest\_scenario\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}smoke\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
tempest\_slow\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}benchmarking\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
rally\_full\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}benchmarking\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
rally\_jobs\_cntt
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}benchmarking\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
vmtp
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}benchmarking\sphinxhyphen{}cntt:wallaby
&
\sphinxAtStartPar
shaker
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}vnf:wallaby
&
\sphinxAtStartPar
cloudify
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}vnf:wallaby
&
\sphinxAtStartPar
cloudify\_ims
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}vnf:wallaby
&
\sphinxAtStartPar
heat\_ims
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}vnf:wallaby
&
\sphinxAtStartPar
vyos\_vrouter
&
\sphinxAtStartPar
PASS
\\
\hline
\sphinxAtStartPar
opnfv/functest\sphinxhyphen{}vnf:wallaby
&
\sphinxAtStartPar
juju\_epc
&
\sphinxAtStartPar
PASS
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\subsubsection{TC Mapping to Requirements}
\label{\detokenize{chapters/chapter03:tc-mapping-to-requirements}}
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
test case
&\sphinxstyletheadfamily
\sphinxAtStartPar
requirements
\\
\hline
\sphinxAtStartPar
tempest\_horizon
&
\sphinxAtStartPar
Horizon testing
\\
\hline
\sphinxAtStartPar
tempest\_neutron\_cntt
&
\sphinxAtStartPar
Neutron API testing
\\
\hline
\sphinxAtStartPar
tempest\_cinder\_cntt
&
\sphinxAtStartPar
Cinder API testing
\\
\hline
\sphinxAtStartPar
tempest\_keystone\_cntt
&
\sphinxAtStartPar
Keystone API testing
\\
\hline
\sphinxAtStartPar
rally\_sanity\_cntt
&
\sphinxAtStartPar
Keystone, Glance, Cinder, Swift, Neutron, Nova and
Heat API testing
\\
\hline
\sphinxAtStartPar
tempest\_full\_cntt
&
\sphinxAtStartPar
Keystone, Glance, Cinder, Swift, Neutron and Nova
API testing
\\
\hline
\sphinxAtStartPar
tempest\_scenario\_cntt
&
\sphinxAtStartPar
Keystone, Glance, Cinder, Swift, Neutron and Nova
API testing
\\
\hline
\sphinxAtStartPar
tempest\_slow\_cntt
&
\sphinxAtStartPar
Keystone, Glance, Cinder, Swift, Neutron and Nova
API testing
\\
\hline
\sphinxAtStartPar
rally\_full\_cntt
&
\sphinxAtStartPar
Keystone, Glance, Cinder, Swift, Neutron, Nova and
Heat API benchmarking
\\
\hline
\sphinxAtStartPar
rally\_jobs\_cntt
&
\sphinxAtStartPar
Neutron API benchmarking
\\
\hline
\sphinxAtStartPar
vmtp
&
\sphinxAtStartPar
Dataplane benchmarking
\\
\hline
\sphinxAtStartPar
shaker
&
\sphinxAtStartPar
Dataplane benchmarking
\\
\hline
\sphinxAtStartPar
cloudify
&
\sphinxAtStartPar
opensource VNF onboarding and testing
\\
\hline
\sphinxAtStartPar
cloudify\_ims
&
\sphinxAtStartPar
opensource VNF onboarding and testing
\\
\hline
\sphinxAtStartPar
heat\_ims
&
\sphinxAtStartPar
opensource VNF onboarding and testing
\\
\hline
\sphinxAtStartPar
vyos\_vrouter
&
\sphinxAtStartPar
opensource VNF onboarding and testing
\\
\hline
\sphinxAtStartPar
juju\_epc
&
\sphinxAtStartPar
opensource VNF onboarding and testing
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\section{OpenStack\sphinxhyphen{}based cloud infrastructure Testing Cookbook}
\label{\detokenize{chapters/chapter04:openstack-based-cloud-infrastructure-testing-cookbook}}\label{\detokenize{chapters/chapter04::doc}}
\subsection{Introduction}
\label{\detokenize{chapters/chapter04:introduction}}
\sphinxAtStartPar
Define the purpose of the chapter which is to:
\begin{itemize}
\item {}
\sphinxAtStartPar
Identify Framework Needs, Goals, and Dependencies
\item {}
\sphinxAtStartPar
Define Opensource Integration (OVP, Functest, CVC, others)
\item {}
\sphinxAtStartPar
Provide Automation Toolchain (list, topology, flow)
\end{itemize}
\subsection{Relevant Community Projects and Initiatives}
\label{\detokenize{chapters/chapter04:relevant-community-projects-and-initiatives}}
\subsubsection{Functest}
\label{\detokenize{chapters/chapter04:functest}}
\sphinxAtStartPar
\sphinxhref{https://functest.readthedocs.io/en/stable-xena/}{Functest} was
initially created to verify OPNFV Installers and Scenarios and then to
publish fair, trustable and public results regarding the status of the
different opensource technologies, especially for Neutron backends
(e.g. Neutron agents, OpenDaylight, OVN, etc.). It has been continuously
updated to offer the best testing coverage for any kind of OpenStack and
Kubernetes deployments including production environments. It also
ensures that the platforms meet Network Functions Virtualization
requirements by running and testing VNFs amongst all tests available.
\sphinxAtStartPar
Functest is driven by a true verification of the platform under test as
opposed to the interoperability programs such as
\sphinxhref{https://refstack.openstack.org/}{RefStack} or \sphinxhref{https://www.opnfv.org/verification}{OPNFV Verification
Program} which select a small
subset of Functional tests passing in many different opensource software
combinations:
\begin{itemize}
\item {}
\sphinxAtStartPar
tests are skipped if an optional support is missing (e.g.
\sphinxhref{https://docs.openstack.org/barbican/latest/}{Barbican} or
networking features such as \sphinxhref{https://docs.openstack.org/networking-bgpvpn/latest/}{BGPVPN
interconnection}
or \sphinxhref{https://docs.openstack.org/networking-sfc/latest/}{Service Function
Chaining})
\item {}
\sphinxAtStartPar
tests are parameterized (e.g. shared vs non\sphinxhyphen{}shared live migration)
\item {}
\sphinxAtStartPar
blacklist mechanisms are available if needed
\end{itemize}
\sphinxAtStartPar
It should be noted that \sphinxhref{https://refstack.openstack.org/}{the RefStack
lists} are included as
they are in Functest in the next 3 dedicated testcases:
\begin{itemize}
\item {}
\sphinxAtStartPar
refstack\_compute (OpenStack Powered Compute)
\item {}
\sphinxAtStartPar
refstack\_object (OpenStack Powered Object Storage)
\item {}
\sphinxAtStartPar
refstack\_platform (OpenStack Powered Platform)
\end{itemize}
\sphinxAtStartPar
Functest also integrates \sphinxhref{https://kubernetes.io/blog/2019/03/22/kubernetes-end-to-end-testing-for-everyone/}{Kubernetes End\sphinxhyphen{}to\sphinxhyphen{}end
tests}
and allows verifying Kubernetes Conformance (see
\sphinxhref{https://build.opnfv.org/ci/job/functest-kubernetes-opnfv-functest-kubernetes-smoke-v1.22-k8s\_conformance-run/25/console}{k8s\sphinxhyphen{}conformance}).
\sphinxAtStartPar
Dovetail (OVP) mostly leverages on Functest but only runs a small part
of Functest (\textasciitilde{}15\% of all functional tests, no benchmarking tests, no VNF
deployment and testing). It’s worth mentioning that Functest is patched
to \sphinxhref{https://github.com/opnfv/dovetail/tree/master/etc/patches/functest/disable-api-validation}{disable API
verification}
which has differed from OpenStack rules for years.
\sphinxAtStartPar
Then Functest conforms with the upstream rules (versions, code quality,
etc.) and especially their
\sphinxhref{https://docs.openstack.org/infra/system-config/devstack-gate.html}{gates}
(a.k.a. the automatic verification prior to any code review) to preserve
the quality between code and deployment. In that case, Functest can be
considered as a smooth and lightweight integration of tests developed
upstream (and the Functest team directly contributes in these projects:
\sphinxhref{https://github.com/openstack/rally-openstack}{Rally},
\sphinxhref{https://github.com/openstack/tempest}{Tempest}, etc.). It’s worth
mentioning that, as opposed to the OpenStack Gates leveraging on
\sphinxhref{https://docs.openstack.org/devstack/latest/}{DevStack}, it can check
the same already deployed SUT over and over even from a \sphinxhref{https://www.raspberrypi.org/}{Raspberry
PI}. Here the testcases can be executed
in parallel vs the same deployment instead of being executed vs
different pools of virtual machines.
\sphinxAtStartPar
Here are the functional tests (\textgreater{}2000) running in OpenStack gates
integrated in Functest Smoke (see \sphinxhref{https://build.opnfv.org/ci/view/functest/job/functest-wallaby-daily/17/}{Functest daily
jobs}
for more details):
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Testcases
&\sphinxstyletheadfamily
\sphinxAtStartPar
Gates
\\
\hline
\sphinxAtStartPar
tempest\_neutron
&
\sphinxAtStartPar
Neutron
\\
\hline
\sphinxAtStartPar
tempest\_cinder
&
\sphinxAtStartPar
Cinder
\\
\hline
\sphinxAtStartPar
tempest\_keystone
&
\sphinxAtStartPar
Keystone
\\
\hline
\sphinxAtStartPar
rally\_sanity
&
\sphinxAtStartPar
General
\\
\hline
\sphinxAtStartPar
refstack\_defcore
&
\sphinxAtStartPar
General
\\
\hline
\sphinxAtStartPar
tempest\_full
&
\sphinxAtStartPar
General
\\
\hline
\sphinxAtStartPar
tempest\_slow
&
\sphinxAtStartPar
General
\\
\hline
\sphinxAtStartPar
tempest\_scenario
&
\sphinxAtStartPar
General
\\
\hline
\sphinxAtStartPar
patrole
&
\sphinxAtStartPar
Patrole
\\
\hline
\sphinxAtStartPar
tempest\_barbican
&
\sphinxAtStartPar
Barbican
\\
\hline
\sphinxAtStartPar
networking\sphinxhyphen{}bgpvpn
&
\sphinxAtStartPar
Networking BGP VPN
\\
\hline
\sphinxAtStartPar
networking\sphinxhyphen{}sfc
&
\sphinxAtStartPar
Networking SFC
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
To complete functional testing, Functest also integrates a few
\sphinxhref{https://docs.openstack.org/developer/performance-docs/methodologies/tools.html}{performance
tools}
(2\sphinxhyphen{}3 hours) as proposed by OpenStack:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Testcases
&\sphinxstyletheadfamily
\sphinxAtStartPar
Benchmarking
\\
\hline
\sphinxAtStartPar
rally\_full
&
\sphinxAtStartPar
Control Plane (API) testing
\\
\hline
\sphinxAtStartPar
rally\_jobs
&
\sphinxAtStartPar
Control Plane (API) testing
\\
\hline
\sphinxAtStartPar
vmtp
&
\sphinxAtStartPar
Data Plane testing
\\
\hline
\sphinxAtStartPar
shaker
&
\sphinxAtStartPar
Data Plane testing
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
And VNFs automatically deployed and tested :
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Testcases
&\sphinxstyletheadfamily
\sphinxAtStartPar
Benchmarking
\\
\hline
\sphinxAtStartPar
cloudify
&
\sphinxAtStartPar
Cloudify deployment
\\
\hline
\sphinxAtStartPar
cloudify\_ims
&
\sphinxAtStartPar
Clearwater IMS deployed via Coudify
\\
\hline
\sphinxAtStartPar
heat\_ims
&
\sphinxAtStartPar
Clearwater IMS deployed via Heat
\\
\hline
\sphinxAtStartPar
vyos\_vrouter
&
\sphinxAtStartPar
VyOS deployed via Cloudify
\\
\hline
\sphinxAtStartPar
juju\_epc
&
\sphinxAtStartPar
OAI deployed via Juju
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
Functest should be considered as a whole as it meets multiple objectives
about the reference implementation:
\begin{itemize}
\item {}
\sphinxAtStartPar
verify all APIs (services, advances, features, etc.) exposed by the
reference implementation
\item {}
\sphinxAtStartPar
compare the reference implementation and local deployments from a
functional standpoint and from OpenStack control plane and dataplane
capabilities
\end{itemize}
\sphinxAtStartPar
It’s worth mentioning that Functest already takes into account the first
Anuket
\sphinxhref{https://git.opnfv.org/functest/tree/functest/ci/config\_patch.yaml\#n2}{profiles}.
Anuket should simply add the next Functest inputs according the
reference implementation:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxhref{https://github.com/opnfv/functest/blob/stable/iruya/functest/utils/env.py\#L17}{Functest
inputs}
\item {}
\sphinxAtStartPar
\sphinxhref{https://github.com/opnfv/functest/blob/stable/iruya/functest/opnfv\_tests/openstack/tempest/custom\_tests/tempest\_conf.yaml}{tempest specific
configuration}
\end{itemize}
\sphinxAtStartPar
Additional links:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxhref{https://functest.readthedocs.io/en/stable-iruya/}{Homepage}
\item {}
\sphinxAtStartPar
\sphinxhref{https://wiki.opnfv.org/pages/viewpage.action?pageId=35291769}{Run Alpine Functest containers
(Iruya)}
\item {}
\sphinxAtStartPar
\sphinxhref{https://wiki.opnfv.org/pages/viewpage.action?pageId=32015004}{Deploy your own Functest CI/CD
toolchains}
\item {}
\sphinxAtStartPar
\sphinxhref{https://build.opnfv.org/ci/view/functest/}{Functest gates}
\end{itemize}
\subsubsection{Yardstick}
\label{\detokenize{chapters/chapter04:yardstick}}
\subsubsection{Bottlenecks}
\label{\detokenize{chapters/chapter04:bottlenecks}}
\subsubsection{Test Tools}
\label{\detokenize{chapters/chapter04:test-tools}}\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
Shaker: \sphinxurl{https://pyshaker.readthedocs.io/en/latest/} (The distributed
data\sphinxhyphen{}plane testing tool built for OpenStack)
\item {}
\sphinxAtStartPar
Sonubuoy: \sphinxurl{https://sonobuoy.io/} It is a diagnostic tool that makes it
easier to understand the state of a Kubernetes cluster by running a
set of plugins (including Kubernetes conformance tests) in an
accessible and non\sphinxhyphen{}destructive manner.
\end{enumerate}
\subsubsection{Scenario Descriptor File (SDF)}
\label{\detokenize{chapters/chapter04:scenario-descriptor-file-sdf}}
\sphinxAtStartPar
As defined by Anuket, Scenario Descriptor File’s (SDF) will be utilized
to relay information from the Scenario Designer (or Test Manager), to
Release Managers, CI Pipeline Owners, and Installer Agents, to define
test scenario content, and specifications.
\sphinxAtStartPar
SDF’s will contain, but not limited to, the following Metadata,
Components, Deployment Options, Deployment Tools, and Hardware
prerequistes:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Metadata}
\begin{itemize}
\item {}
\sphinxAtStartPar
Name
\item {}
\sphinxAtStartPar
History
\item {}
\sphinxAtStartPar
Purpose
\item {}
\sphinxAtStartPar
Owner
\end{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Components}
\begin{itemize}
\item {}
\sphinxAtStartPar
e.g. SDN controllers
\item {}
\sphinxAtStartPar
Versions
\item {}
\sphinxAtStartPar
Optional features, e.g. NFV features
\end{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Deployment Options}
\begin{itemize}
\item {}
\sphinxAtStartPar
Hardware types
\item {}
\sphinxAtStartPar
Virtual deploy
\item {}
\sphinxAtStartPar
HA, NUMA
\end{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Deployment Tools}
\begin{itemize}
\item {}
\sphinxAtStartPar
Supporting installers.
\item {}
\sphinxAtStartPar
Valid options per installer.
\end{itemize}
\end{itemize}
\subsection{OpenStack Testing Cookbook}
\label{\detokenize{chapters/chapter04:openstack-testing-cookbook}}
\sphinxAtStartPar
At the time of writing, the CI description file is hosted in Functest
and only runs the containers listed in RM/RA\sphinxhyphen{}1 Requirements. It will be
completed by the next Anuket mandatory test cases and then a new CI
description file will be proposed in CIRV tree.
\sphinxAtStartPar
Please note the next two points depending on the GNU/Linux distributions
and the network settings:
\begin{itemize}
\item {}
\sphinxAtStartPar
SELinux: you may have to add \textendash{}system\sphinxhyphen{}site\sphinxhyphen{}packages when creating the
virtualenv (“Aborting, target uses selinux but python bindings
(libselinux\sphinxhyphen{}python) aren’t installed!”)
\item {}
\sphinxAtStartPar
Proxy: you may set your proxy in env for Ansible and in systemd for
Docker \sphinxurl{https://docs.docker.com/config/daemon/systemd/\#httphttps-proxy}
\end{itemize}
\sphinxAtStartPar
To deploy your own CI toolchain running Anuket Compliance:
\begin{sphinxVerbatim}[commandchars=\\\{\}]
virtualenv functest \PYGZhy{}\PYGZhy{}system\PYGZhy{}site\PYGZhy{}packages
. functest/bin/activate
pip install ansible
ansible\PYGZhy{}galaxy install collivier.xtesting
ansible\PYGZhy{}galaxy collection install ansible.posix community.general community.grafana kubernetes.core community.docker community.postgresql
git clone https://gerrit.opnfv.org/gerrit/functest functest\PYGZhy{}src
\PYG{o}{(}\PYG{n+nb}{cd} functest\PYGZhy{}src \PYG{o}{\PYGZam{}\PYGZam{}} git checkout \PYGZhy{}b stable/wallaby origin/stable/wallaby\PYG{o}{)}
ansible\PYGZhy{}playbook functest\PYGZhy{}src/ansible/site.cntt.yml
\end{sphinxVerbatim}
\subsubsection{OpenStack API testing configuration}
\label{\detokenize{chapters/chapter04:openstack-api-testing-configuration}}
\sphinxAtStartPar
Here is the default Functest tree as proposed in \sphinxhref{https://wiki.anuket.io/display/HOME/Functest+Wallaby}{Functest
Wallaby}:
\begin{itemize}
\item {}
\sphinxAtStartPar
/home/opnfv/functest/openstack.creds
\item {}
\sphinxAtStartPar
/home/opnfv/functest/images
\end{itemize}
\sphinxAtStartPar
Download the images and fill /home/opnfv/functest/openstack.creds as
proposed in \sphinxhref{https://wiki.anuket.io/display/HOME/Functest+Wallaby}{Functest
Wallaby}
\sphinxAtStartPar
You may have to modify a few Functest env vars according to the SUT (see
env in \sphinxhref{https://wiki.anuket.io/display/HOME/Functest+Wallaby}{Functest
Wallaby}). Be
free to modify functest\sphinxhyphen{}src/ansible/host\_vars/127.0.0.1 at your
convenience and then to reconfigure the toolchain:
\begin{sphinxVerbatim}[commandchars=\\\{\}]
ansible\PYGZhy{}playbook functest\PYGZhy{}src/ansible/site.cntt.yml
\end{sphinxVerbatim}
\subsubsection{Run Anuket OpenStack Testing}
\label{\detokenize{chapters/chapter04:run-anuket-openstack-testing}}
\sphinxAtStartPar
Open \sphinxurl{http://127.0.0.1:8080/job/functest-wallaby-daily/} in a web browser,
login as admin/admin and click on “Build with Parameters” (keep the
default build\_tag value).
\sphinxAtStartPar
If the System under test (SUT) is Anuket compliant, a link to the full
archive containing all test results and artifacts will be printed in
functest\sphinxhyphen{}wallaby\sphinxhyphen{}zip’s console. Be free to download it and then to send
it to any reviewer committee.
\sphinxAtStartPar
To clean your working dir:
\begin{sphinxVerbatim}[commandchars=\\\{\}]
deactivate
rm \PYGZhy{}rf functest\PYGZhy{}src functest
\end{sphinxVerbatim}
\section{VNF Testing Framework Requirements}
\label{\detokenize{chapters/chapter05:vnf-testing-framework-requirements}}\label{\detokenize{chapters/chapter05::doc}}
\subsection{Introduction}
\label{\detokenize{chapters/chapter05:introduction}}
\sphinxAtStartPar
This chapter covers comprehensive VNF Conformance requirements for
enabling required process and steps to provide VNF badging based on
define scope of compliance and validation. This includes end to end test
framework requirements, badging entry and exit criteria, profiles to
reference, different stake holders and Conformance Methodologies by
using certified NFVi under NFVi badging program.
\subsection{Conformance Methodology}
\label{\detokenize{chapters/chapter05:conformance-methodology}}
\sphinxAtStartPar
It defines the end\sphinxhyphen{}end framework and process required for certifying the
given VNF.
\sphinxAtStartPar
\sphinxstylestrong{End\sphinxhyphen{}End framework}:
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{f3b0c214bc58c44406fd5b801d3dfc91}.png}
\caption{End\sphinxhyphen{}End framework}\label{\detokenize{chapters/chapter05:id1}}\end{figure}
\sphinxAtStartPar
Here, the steps 1\sphinxhyphen{}4 are NFVI related steps are covered in detail in
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_cert/RC1/chapters/chapter02.html}{NFVI Conformance Requirements}.
\sphinxAtStartPar
Step\sphinxhyphen{}5. Interoperability validations for VNF functional testing defined.
\sphinxAtStartPar
Step\sphinxhyphen{}6. Interoperability validations for VNF performance testing defined
(IOPS, connection, threading, resource consumption).
\sphinxAtStartPar
Step\sphinxhyphen{}7. Sending requirements to the VNF requirements projects in terms
of t\sphinxhyphen{}shirt sizes, config settings, required for VNF/orchestration
validation.
\sphinxAtStartPar
\sphinxstylestrong{Conformance flow}:
\sphinxAtStartPar
The entry and exit criteria defined in below section are pre\sphinxhyphen{}requisities
for this flow.
\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
\sphinxstyleemphasis{VNF Vendors} submit the VNF into OVP Lab for Conformance (Fulfilling
the entry criteria is pre\sphinxhyphen{}requisities for this step.)
\item {}
\sphinxAtStartPar
As part of OVP lab, already required test cases, test tools,
eco\sphinxhyphen{}system like MANO and appropriate certified NFVi to be setup as
defined part of entry criteria. This lab could either \sphinxstyleemphasis{OVP 3rd party
lab} or \sphinxstyleemphasis{VNF vendors}.
\end{enumerate}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{f3b0c214bc58c44406fd5b801d3dfc89}.png}
\caption{OVP 3rd party lab}\label{\detokenize{chapters/chapter05:id2}}\end{figure}
\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\setcounter{enumi}{2}
\item {}
\sphinxAtStartPar
Once testing is completed done, test results will be submitted to the
OVP portal for community review along with additional information
such as product name, documentation links, primary company contact
information, etc.
\item {}
\sphinxAtStartPar
\sphinxstyleemphasis{LFN CVC} community team reviewers will review the results submitted
and will approve or reject it based the details provided.
\item {}
\sphinxAtStartPar
If reviewer rejected it, then step 2 and 3 will be ran again to
address the review comments. Otherwise once reviewer approved it,
corresponding VNF will be published into OVP VNF Portal with OVP
badge.
\item {}
\begin{DUlineblock}{0em}
\item[] LFN staff will provide the certificate badge graphics and graphical
usage guidelines.
\item[] The OVP portal will reflect LFN’s disposition and assignment of the
certified VNF badge.
\end{DUlineblock}
\end{enumerate}
\sphinxAtStartPar
Now VNF is ready and \sphinxstyleemphasis{Telco Operators} can start consume it.
\subsubsection{Profiles Reference}
\label{\detokenize{chapters/chapter05:profiles-reference}}
\sphinxAtStartPar
The NFV Infrastructure (NFVI) is the totality of all hardware and
software components which build up the environment in which VNFs are
deployed, managed and executed. It is, therefore, inevitable that
different VNFs would require different capabilities and have different
expectations from it. So one of the main targets of Anuket is to define
an agnostic NFVI and removes any dependencies between VNFs and deployed
Infrastructure (NFVI) and offer NFVI to VNFs in an abstracted way with
defined capabilities and metrics. This would help operators to host
their Telco Workload (VNF) with different traffic types, behaviour and
from any vendor on a unified consistent Infrastructure. so as part of
VNF Conformance, its important to certify the VNF based on profiled
defined in :doc:ref\_model/chapters/chapter02\textasciigrave{}.
\sphinxAtStartPar
In \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_model/chapters/chapter02.html}{Workload Requirements \& Analysis},
following NFVi profiles are proposed as reference:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Basic}: for VNF that can tolerate resource over\sphinxhyphen{}subscription and
variable latency.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Network Intensive}: for VNF that require predictable computing
performance, high network throughput and low network latency.
\end{itemize}
\subsubsection{Protoype VNFs}
\label{\detokenize{chapters/chapter05:protoype-vnfs}}
\sphinxAtStartPar
A portion of the NFVI badging methodology includes Empirical Validation
with Reference Golden VNFs (aka CVC Validation) which will ensure the
NFVI runs with a set of VNF Families, or Classes, to mimic
production\sphinxhyphen{}like VNF connectivity. These tests are to 1) ensure
interoperability checks pass, and 2) there is an established baseline of
VNF behaviors and characters before vendor supplied VNFs are tested and
certified. In other words, empirical validations will confirm
performance and stability between Platform and VNF, such as validating
packet loss is within acceptable tolerances.
\subsection{Badging Requirements}
\label{\detokenize{chapters/chapter05:badging-requirements}}
\sphinxAtStartPar
\sphinxstylestrong{Defined}. \sphinxstyleemphasis{Badging} refers to the granting of a Conformance badge by
the OVP to Suppliers/Testers of Anuket NFVI+VNF upon demonstration the
testing performed confirms:
\begin{itemize}
\item {}
\sphinxAtStartPar
NFVI adheres to Anuket RA/RM requirements.
\item {}
\sphinxAtStartPar
Anuket certified VNFs functionally perform as expected (i.e. test
cases pass) on NFVI with acceptable levels of stability and
performance.
\end{itemize}
\sphinxAtStartPar
Following table shows the bading requirements with scope of mandatory
(must) or optional.
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Requirement id
&\sphinxstyletheadfamily
\sphinxAtStartPar
scope
&\sphinxstyletheadfamily
\sphinxAtStartPar
details
\\
\hline
\sphinxAtStartPar
CVreq.VNF.001
&
\sphinxAtStartPar
must
&
\sphinxAtStartPar
Receive NFVi
badge in lab
setup per RI\sphinxhyphen{}1
standards,
performing h/w
validations,
performing s/w
manifest
validations,
running nfvi
compliance,
validation, and
performance
checks
\\
\hline
\sphinxAtStartPar
CVreq.VNF.002
&
\sphinxAtStartPar
must
&
\sphinxAtStartPar
met all entry
and exit
criteria
\\
\hline
\sphinxAtStartPar
CVreq.VNF.003
&
\sphinxAtStartPar
must
&
\sphinxAtStartPar
run
i
nteroperability
validations,
including
instantiation,
communication /
health, and
removal
\\
\hline
\sphinxAtStartPar
CVreq.VNF.004
&
\sphinxAtStartPar
shall
&
\sphinxAtStartPar
utilize
automation
frameworks to
run all
required tests.
Conformance
process would
improve, if
test framework
satisfy the
required
defined in this
chapter under
\sphinxstyleemphasis{VNF Test
Conformance
platform
requirements}
section
\\
\hline
\sphinxAtStartPar
CVreq.VNF.005
&
\sphinxAtStartPar
must
&
\sphinxAtStartPar
pass all
required tests
\\
\hline
\sphinxAtStartPar
CVreq.VNF.006
&
\sphinxAtStartPar
must
&
\sphinxAtStartPar
prepare release
notes, with
issues known,
their severity
and magnitude,
mitigation plan
\\
\hline
\sphinxAtStartPar
CVreq.VNF.007
&
\sphinxAtStartPar
must
&
\sphinxAtStartPar
publish results
in defined
normalized
output
\\
\hline
\sphinxAtStartPar
CVreq.VNF.008
&
\sphinxAtStartPar
must
&
\sphinxAtStartPar
respond /closed
badging
inquiries
\\
\hline
\sphinxAtStartPar
CVReq.VNF.010
&
\sphinxAtStartPar
optional
&
\sphinxAtStartPar
for bading VNF
supplier can
choose to run
their own test
h
arnesses/suites
to validate VNF
functional and
performance
behaviors and
performance
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\subsubsection{Badging Scope}
\label{\detokenize{chapters/chapter05:badging-scope}}
\sphinxAtStartPar
The VNF badging includes:
\begin{enumerate}
\sphinxsetlistlabels{\arabic}{enumi}{enumii}{}{.}%
\item {}
\sphinxAtStartPar
NFVi Verifications (Compliance): Manifest Verifications will ensure
the NFVI is compliant, and delivered for testing, with hardware and
software profile specifications defined by the Ref Model and Ref
Architecture.
\item {}
\sphinxAtStartPar
Empirical Validation with Reference VNF (Validation): Empirical
Validation with Reference Golden VNFs will ensure the NFVI runs with
a set of VNF Families, or Classes, to mimic production\sphinxhyphen{}like VNFs to
baseline infrastructure conformance.
\item {}
\sphinxAtStartPar
Candidate VNF Validation (Validation \& Performance): Candidate VNF
Validation will ensure complete interoperability of VNF behavior on
the NFVI leveraging VVP/VNFSDK test suites to ensure VNF can be spun
up, modified, or removed, on the target NFVI (aka Interoperability).
\end{enumerate}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{f3b0c214bc58c44406fd5b801d3dfc90}.png}
\caption{Candidate VNF Validation}\label{\detokenize{chapters/chapter05:id3}}\end{figure}
\subsubsection{Entry criteria}
\label{\detokenize{chapters/chapter05:entry-criteria}}
\sphinxAtStartPar
Before entering into the VNF badging process, VNF needs to satisfy the
following requirements as entry criteria:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstyleemphasis{Environment Requirements} : Published details providing evidence
that a RAx compliant lab has been implemented, meeting requirements
set forth in respective RM and RAx documentation for features,
options, and capabilities needed for VNF test validations. Expected
information includes:
\begin{itemize}
\item {}
\sphinxAtStartPar
Lab Flavor
\item {}
\sphinxAtStartPar
Component software rev levels
\item {}
\sphinxAtStartPar
Confirmation of compatibility with external systems
\item {}
\sphinxAtStartPar
Tenant needs identified
\item {}
\sphinxAtStartPar
All connectivity, network, image, VMs, delivered with successful
pairwise tests
\item {}
\sphinxAtStartPar
Lab instrumented for proper monitoring
\end{itemize}
\item {}
\sphinxAtStartPar
\sphinxstyleemphasis{VNF artifact} : VNF cloud (native) image, VNF configurations and
guidelines, automation scripts, etc
\item {}
\sphinxAtStartPar
\sphinxstyleemphasis{NFVi profiles}: List of supporting OVP Certified Anuket compliant
NFVi
\item {}
\sphinxAtStartPar
Completed Security review report
\item {}
\sphinxAtStartPar
Vendor specific test cases and its deployment and usage guidelines
\end{itemize}
\subsubsection{Exit criteria}
\label{\detokenize{chapters/chapter05:exit-criteria}}
\sphinxAtStartPar
VNF Conformance testing should be completed with following exit
criteria:
\begin{itemize}
\item {}
\sphinxAtStartPar
All required test cases should be passed
\item {}
\sphinxAtStartPar
No outstanding high severity issues and other known issues to be
documented
\item {}
\sphinxAtStartPar
Release notes
\item {}
\sphinxAtStartPar
Provided with required installation guide, configuration guide, etc.
\item {}
\sphinxAtStartPar
Test results collated, centralized, and normalized, with a final
report generated showing status of the test scenario/case (e.g. Pass,
Fail, Skip, Measurement Success/Fail, etc), along with traceability
to a functional, or non\sphinxhyphen{}functional, requirement
\end{itemize}
\subsection{VNF Test Conformance platform Requirements}
\label{\detokenize{chapters/chapter05:vnf-test-conformance-platform-requirements}}
\sphinxAtStartPar
Test platform requirements are provided to address test case design,
distribution, execution and result reporting along with required
artifacts and environments in place and are defined based on below
scope.
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{c665a3d13461f67ea8729042cf8d975d}.png}
\caption{Test platform requirements}\label{\detokenize{chapters/chapter05:id4}}\end{figure}
\subsubsection{Standards/Profiles}
\label{\detokenize{chapters/chapter05:standards-profiles}}\begin{itemize}
\item {}
\sphinxAtStartPar
ETSI (TOSCA)
\item {}
\sphinxAtStartPar
GSMA
\item {}
\sphinxAtStartPar
ONAP VNFREQS (HOT)
\end{itemize}
\subsubsection{Test cases}
\label{\detokenize{chapters/chapter05:test-cases}}
\sphinxAtStartPar
Refer \sphinxstyleemphasis{chapter RC\sphinxhyphen{}06} for more details on test case requirements defined
for VNF under Anuket. Platform should support to managed and execute
these test cases.
\sphinxAtStartPar
NOTE: For Conformance, only compliance and verification test cases will
be considered, but in future, it could be extent to validation and
Performance related testing.
\paragraph{Compliance}
\label{\detokenize{chapters/chapter05:compliance}}
\sphinxAtStartPar
Perform compliance check based on
\begin{itemize}
\item {}
\sphinxAtStartPar
TOSCA using ETSI SOL004 \& SOL001
\item {}
\sphinxAtStartPar
OpenStack HOT using ONAP VNFREQS
\item {}
\sphinxAtStartPar
GSMA profile as defined in \sphinxstyleemphasis{chapter RM\sphinxhyphen{}04}.
\end{itemize}
\paragraph{Verification}
\label{\detokenize{chapters/chapter05:verification}}
\sphinxAtStartPar
Perform on\sphinxhyphen{}boarding/ verification life cycle operation (from
instantiation, configuration, update, termination) using MANO supporting
Anuket compliant NFVI.
\paragraph{Validation}
\label{\detokenize{chapters/chapter05:validation}}
\sphinxAtStartPar
Perform various VNF type specific functionality operations on Anuket RA
\& RM compliant NFVI
\paragraph{Performance}
\label{\detokenize{chapters/chapter05:performance}}
\sphinxAtStartPar
Perform various performance related testing and facilitate for
benchmarking the VNF performance on different profile and scenarios.
\subsubsection{Eco\sphinxhyphen{}system MANO/NFVI}
\label{\detokenize{chapters/chapter05:eco-system-mano-nfvi}}
\sphinxAtStartPar
Platform would support to execute various test cases on Anuket RA \& RM
compliant NFVi along with required MANO system supporting these NFVi.
\subsubsection{VNF}
\label{\detokenize{chapters/chapter05:vnf}}
\sphinxAtStartPar
Suppliers of VNFs/CNFs seeking to receive VNF Conformance badges must
first ensure their testing is performed against a compliant RM/RA
architecture supporting all capabilities, features, and services defined
by the respective \sphinxstyleemphasis{RM/RA requirements}. More specifically, the VNF
Supplier must ensure their implementation of the RM/RA receives the NFVI
Conformance badge prior to starting VNF testing. Finally, to receive VNF
Conformance, the test platform will need to support TOSCA and HOT based
VNF distros.
\sphinxAtStartPar
In addition, Platform should be able to perform the required test case
management and executions and produce the result the CVC OVP portal for
Conformance process along with required testing foot print details. So
overall scoped example architecture could be as below:
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{2269537e91994b5b49858734fe73bbb1}.png}
\caption{VNF Test Certification Platform}\label{\detokenize{chapters/chapter05:id5}}\end{figure}
\subsubsection{Test Case Model}
\label{\detokenize{chapters/chapter05:test-case-model}}
\sphinxAtStartPar
As there are more number of VNF at different levels of networking such
as access, transport and core level as well as OSI level L0\sphinxhyphen{}L7. Every
network function provides set of pre\sphinxhyphen{}defined features and
functionalities. So its important to model test cases for every
functionality to identify it uniquely and use it as part of test flow
design.
\sphinxAtStartPar
As part of modeling its very important to capture the following details
\begin{itemize}
\item {}
\sphinxAtStartPar
Test case Name
\item {}
\sphinxAtStartPar
Test case description
\item {}
\sphinxAtStartPar
Virtual Network function Name
\item {}
\sphinxAtStartPar
Network function Feature/functionality name
\item {}
\sphinxAtStartPar
Test case input parameters
\item {}
\sphinxAtStartPar
Test case result attributes
\item {}
\sphinxAtStartPar
Test case version
\end{itemize}
\sphinxAtStartPar
while implementing the test cases, this model would act as specification
and as it captures the input and output, it would help while designing
the test flow which will help to execute set of test cases in
pre\sphinxhyphen{}defined flow.
\subsubsection{Test case management}
\label{\detokenize{chapters/chapter05:test-case-management}}\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Test case} : On\sphinxhyphen{}board/discover, update, disable/enable, delete
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Test suite} : On\sphinxhyphen{}board/discover, update, disable/enable, delete
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Test flow} : design/discover, update, disable/enable, delete
\end{itemize}
\subsubsection{Test Execution management}
\label{\detokenize{chapters/chapter05:test-execution-management}}\begin{itemize}
\item {}
\begin{DUlineblock}{0em}
\item[] \sphinxstylestrong{Run\sphinxhyphen{}time}: One of the common nature of the test environment is
heterogeneous and multiple vendors and open communities would
provide various test tool and environment to support execution of
test cases developed under different run\sphinxhyphen{}times
\item[] (JVM, Python, Shell, Container, Cloud VM, etc)
\end{DUlineblock}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{RPC}: In order to enable the scaling/remote execution, it should
be enabled with required RPC support.
\end{itemize}
\sphinxAtStartPar
When VNF test platform execute the test cases, it captures the
footprints of test case execution along with results, which are made
available to user and integrated system for consuming.
\subsubsection{Test Result management}
\label{\detokenize{chapters/chapter05:test-result-management}}
\sphinxAtStartPar
\sphinxstylestrong{Categorization}. Test suites will be categorized as
Functional/Platform or Performance based.
\sphinxAtStartPar
\sphinxstylestrong{Results.} Test results reporting will be communicated as a boolean
(pass/fail), or Measurements Only.
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Functional Pass/Fail} signals the assertions set in a test script
verify the Functional Requirements (FR) has met its stated objective
as delivered by the developer. This will consist of both positive
validation of expected behavior, as well as negative based testing
when to confirm error handling is working as expected.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Performance\sphinxhyphen{}based Pass/Fail} determination will be made by
comparing Non\sphinxhyphen{}Functional (NFR) KPIs (obtained after testing) with the
Golden KPIs. Some of the examples of performance KPIs include, but
not limited to: TCP bandwidth, UDP throughput, Memory latency,
Jitter, IOPS etc.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Measurement Results}. Baseline Measurements will be performed when
there are no benchmark standards to compare results, or established
FRs/NFRs for which to gauge application / platform behavior in an
integrated environment, or under load conditions. In these cases,
test results will be executed to measure the application, platform,
then prepare FRs/NFRs for subsequent enhancements and test runs.
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Formats}. As part of execution management, system produces the result
in JSON format which can be represented in various form like YAML, CSV,
Table, etc.
\sphinxAtStartPar
\sphinxstylestrong{Search \& Reporting}. Search would help to query the test results
based on various fact such as test case, VNF, date of execution,
environment, etc. and produce the report in various format like
pie\sphinxhyphen{}chart, success rates, etc
\sphinxAtStartPar
\sphinxstylestrong{Collation | Portal}. The following criteria will be applied to the
collation and presentation of test\sphinxhyphen{}runs seeking Conformance:
\begin{itemize}
\item {}
\sphinxAtStartPar
RA number and name (e.g. RA\sphinxhyphen{}1 OpenStack)
\item {}
\sphinxAtStartPar
Version of software tested (e.g. OpenStack Ocata)
\item {}
\sphinxAtStartPar
Normalized results will be collated across all test runs
(i.e. centralized database)
\item {}
\sphinxAtStartPar
Clear time stamps of test runs will be provided.
\item {}
\sphinxAtStartPar
Identification of test engineer / executor.
\item {}
\sphinxAtStartPar
Traceability to requirements.
\item {}
\sphinxAtStartPar
Summarized conclusion if conditions warrant test Conformance (see
Badging Section).
\item {}
\sphinxAtStartPar
Portal contains links to Conformance badge(s) received.
\end{itemize}
\subsubsection{Test Artifact management}
\label{\detokenize{chapters/chapter05:test-artifact-management}}
\sphinxAtStartPar
As part of testing various binaries, configurations, images, scripts
,etc would be used during test cases building or execution and Version
artifact supports such as VNF CSAR.
\subsubsection{Test Scenario management}
\label{\detokenize{chapters/chapter05:test-scenario-management}}
\sphinxAtStartPar
Allow to create repeatable scenario includes test cases, artifacts and
profiles.
\sphinxAtStartPar
It helps to create dynamic testing scenario development and testing from
the existing test cases and flows along with required artifacts and
profiles. It allows to run repeated testing with one or different
profiles.
\subsubsection{Test Profile management}
\label{\detokenize{chapters/chapter05:test-profile-management}}
\sphinxAtStartPar
For every test case execution needs to be configured with required
environments and predefined test input parameter values. This is
provided by means of profile
\sphinxAtStartPar
Profile should be having option to include other profiles to manage the
hierarchy of them.
\sphinxAtStartPar
As part of profile, testing environment URL, credentials and related
security keys are captured and while running the test cases, user would
be able to inputs the required profile in place of actual inputs and
artifacts.
\sphinxAtStartPar
Also helps in Managing System under test configuration and multiple MANO
/ NFVI and related eco system management elements.
\subsubsection{Tenant \& User management}
\label{\detokenize{chapters/chapter05:tenant-user-management}}
\sphinxAtStartPar
Testing involves design, distribution by different user roles and
executed across multiple tenant’s environments.
\subsubsection{Conformance management \& integration}
\label{\detokenize{chapters/chapter05:conformance-management-integration}}
\sphinxAtStartPar
Platform should have integration with OVP Conformance portal for
submitting results with OVP defined format.
\sphinxAtStartPar
It should enable repository of certified VNFs which can be used for
testing validation and performance.
\subsubsection{User \& System interfaces}
\label{\detokenize{chapters/chapter05:user-system-interfaces}}
\sphinxAtStartPar
\sphinxstylestrong{User interface}:
\begin{itemize}
\item {}
\sphinxAtStartPar
CLI
\item {}
\sphinxAtStartPar
Web portal
\end{itemize}
\sphinxAtStartPar
\sphinxstylestrong{Programming interface}:
\begin{itemize}
\item {}
\sphinxAtStartPar
REST API
\item {}
\sphinxAtStartPar
gRPC
\end{itemize}
\subsubsection{Deliverables}
\label{\detokenize{chapters/chapter05:deliverables}}
\sphinxAtStartPar
Platform should be able to get deployed in both container and cloud
environments. so following model deliverables would enable it:
\begin{itemize}
\item {}
\sphinxAtStartPar
Docker image based installation
\item {}
\sphinxAtStartPar
Standalone installation scripts and zip artifact
\end{itemize}
\subsection{VNF Test Cases Requirements}
\label{\detokenize{chapters/chapter05:vnf-test-cases-requirements}}
\subsubsection{Rationale}
\label{\detokenize{chapters/chapter05:rationale}}
\sphinxAtStartPar
Network functions virtualization (NFV) and softwaredefined networking
(SDN) offer service providers increased service agility, OpEx
improvements, and back\sphinxhyphen{}office automation. Disaggregation, the approach
of decoupling the various layers of the stack, from hardware, to
NFVI/VIM software, to dataplane acceleration, SDN controllers, MANO
components, and VNFs, enables multi\sphinxhyphen{}vendor deployments with
best\sphinxhyphen{}of\sphinxhyphen{}breed options at each layer.
\sphinxAtStartPar
The Anuket specifications define the required architecture and model for
NFVI which will help to decouple the various commercial product layers
and it is important to define and certify the VNF and NFVI. Therefore,
in addition to verify general NFVI capabilities based on Anuket
RM/RA/RI, it is also necessary to verify that VNFs can provide
virtualization functions normally based on the Anuket\sphinxhyphen{}compliant NFVI. So
the VNF testing should at least include:
Compliance, verification, validation, Performance. With the improvement
of specifications, the types of tests may continue to add in the future.
\sphinxAtStartPar
In this chapter, the scope and requirements of VNF test cases are
defined as reference for VNF Conformance, which helps to perform the
various compliance and verification (C\&V) testing and submit results to
LFN OVP Conformance portal.
\subsubsection{Assumptions}
\label{\detokenize{chapters/chapter05:assumptions}}
\sphinxAtStartPar
Here lists the assumptions for VNF Conformance: \sphinxhyphen{} NFVI is ready and it
should be an Anuket\sphinxhyphen{}compliant NFVI \sphinxhyphen{} VNF template is ready to deploy and
certificate \sphinxhyphen{} VNF Test environment is ready, the test environment
contains test functions and entities(NFVI, MANO, VNF Test Platform, VNF
Test Tools) to enable controlling the test execution and collecting the
test measurements. \sphinxhyphen{} VNF Test Platform has been integrated with CICD
chain \sphinxhyphen{} VNF test result can be generated with OVP defined format
\subsubsection{Developer Deliverables}
\label{\detokenize{chapters/chapter05:developer-deliverables}}
\sphinxAtStartPar
This section define the developer Deliverables (artifacts),the following
list the expectations and deliverables we expect from developers in
order to achieve the VNF Conformance: \sphinxhyphen{} VNF test cases
model/scripts/programs \sphinxhyphen{} VNF test cases configuration/profile \sphinxhyphen{} VNF test
tools
\subsubsection{Requirement Type}
\label{\detokenize{chapters/chapter05:requirement-type}}
\sphinxAtStartPar
VNF test cases are used to verify whether the virtualization network
functions can be deployed on the Anuket\sphinxhyphen{}compliant NFVI and provide
normal functions and meet performance, security and other requirements.
\sphinxAtStartPar
By running these VNF test cases and analysis the test results, can be
used for VNF compliance, verfication,validation and performance
Conformance and help on Anuket\sphinxhyphen{}compliant NFVI validation and performance
Conformance.
\sphinxAtStartPar
All the VNF test cases should be supported and run by VNF E2E
Conformance and verification Framework and generate outputs, logs to
identify whether the test passed or failed.
\sphinxAtStartPar
Anuket defines the following four category testing which should be
consistent with the VNF test category defined by OVP.
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
VNF Test Case
Category
&\sphinxstyletheadfamily
\sphinxAtStartPar
Requirement
Number
&\sphinxstyletheadfamily
\sphinxAtStartPar
Type
(Measu
rement/Boolean)
&\sphinxstyletheadfamily
\sphinxAtStartPar
Definit
ion/Description
\\
\hline
\sphinxAtStartPar
Compliance
&
\sphinxAtStartPar
VNF.COMPreq.001
&
\sphinxAtStartPar
Boolean
(
i.e. Pass/Fail)
&
\sphinxAtStartPar
Test case
“must”perform a
platform check
against the
Open Stack
requirements
and VNF package
structure and
syntax
requirements
\\
\hline
\sphinxAtStartPar
Verification
&
\sphinxAtStartPar
VN
F.VERIFYreq.001
&
\sphinxAtStartPar
Boolean
(
i.e. Pass/Fail)
&
\sphinxAtStartPar
Test case
“must” perform
on\sphinxhyphen{}boarding/
verification
life cycle
operation
validation
\\
\hline
\sphinxAtStartPar
Validation
&
\sphinxAtStartPar
V
NF.VALIDreq.001
&
\sphinxAtStartPar
Boolean
(
i.e. Pass/Fail)
&
\sphinxAtStartPar
Test case
“must” perform
API validation
tests to verify
operability
\\
\hline
\sphinxAtStartPar
Performance
&
\sphinxAtStartPar
VNF.PERFreq.001
&
\sphinxAtStartPar
Measurement
&
\sphinxAtStartPar
Test case
“must” execute
various
performance
related testing
and facilitate
for
benchmarking
the VNF
performance on
different
profile and
scenarios
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
Note: The four category testing can be gradually supported and in the
future, will also cover secutiry and other test category.
\subsubsection{Interaction Type}
\label{\detokenize{chapters/chapter05:interaction-type}}\begin{itemize}
\item {}
\sphinxAtStartPar
Describe the types of Interactions: Extended Topology, Complex
(Akraino), Functional, HA, Fault, Interoperability
\end{itemize}
\subsubsection{Performance Profiles}
\label{\detokenize{chapters/chapter05:performance-profiles}}
\sphinxAtStartPar
Performance profiles are not in the scope of current release, and in
future it would need to align with \sphinxstyleemphasis{chapter RM\sphinxhyphen{}4} defined measurements.
\subsubsection{VNF Class/Family and Characteristics}
\label{\detokenize{chapters/chapter05:vnf-class-family-and-characteristics}}\begin{itemize}
\item {}
\sphinxAtStartPar
Describe and provide a Table of VNF Class/Family \& Characteristics of
Each
\end{itemize}
\sphinxAtStartPar
The communication network usually consists of three parts: access
network, transmission network/bearer network and core network. Following
are some examples of network elements for each type of network
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Network Type
&\sphinxstyletheadfamily
\sphinxAtStartPar
Network Elements
\\
\hline
\sphinxAtStartPar
Access Network
&
\sphinxAtStartPar
Including mobile access
network, wireless access
network, wired access
network
\\
\hline
\sphinxAtStartPar
Transport network \& Bearer network
&
\sphinxAtStartPar
Including Trunk Optical
Transport Network, Metro
transport network, IP
backbone network, etc.
\\
\hline
\sphinxAtStartPar
Core Network
&
\sphinxAtStartPar
Circuit domain, including
MSC / VLR, GMSC, MGW, NPMSC,
HLR / AUC, NPHLR,
HSS, etc, Packet domain
devices, including MME, SAE
GW, EPC CG, EPC DNS,
PCC, etc; Core network
equipment for IoT private
network, including
PGW/
GGSN, PCRF, HSS/HLR, etc, 5G
core network
element,including
AMF, SMF, UPF, UDM/UDR/AUS
F, PCF, NSSF, NRF, SMSF, etc
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
In addition to the above network elements, there are some other data
communication network element, including FW, DNS, Router, GW, etc|
\sphinxAtStartPar
According to the current level of the entire network virtualization, the
core network already has many VNFs, and also includes some
datacom\sphinxhyphen{}type(data communication) VNFs.
\sphinxAtStartPar
We can also classify VNFs based on the level of VNF operation:
\begin{enumerate}
\sphinxsetlistlabels{\alph}{enumi}{enumii}{}{)}%
\item {}
\sphinxAtStartPar
VNFs that operate at Layer 2 or Layer 3 are primarily involved in
switching or routing packets at these layers. Examples include
vRouter, vBNG, vCE device, or vSwitch.
\item {}
\sphinxAtStartPar
VNFs that operate at Layer 4 through Layer 7 and are involved in
forwarding, dropping, filtering or redirecting packets at Layer 4
through 7. Examples include vFirewall, vADC, vIDS/vIPS, or vWAN
Accelerator.
\item {}
\sphinxAtStartPar
VNFs that are involved in the dataplane forwarding through the
evolved packet core.
\end{enumerate}
\subsubsection{Measurement}
\label{\detokenize{chapters/chapter05:measurement}}
\sphinxAtStartPar
As part of Conformance testing, following measurement would help for
evaluating the badging:
\begin{itemize}
\item {}
\sphinxAtStartPar
VNF type defined as part of \sphinxstyleemphasis{Chapter RM\sphinxhyphen{}02} and its profile used for
testing.
\item {}
\sphinxAtStartPar
Test cases and their test results including the test case outputs,
logs
\item {}
\sphinxAtStartPar
VNF model type (TOSCA/HOT)
\item {}
\sphinxAtStartPar
Test case pass/failed
\item {}
\sphinxAtStartPar
Different NFVi profiles used and LAB reference identifier
\item {}
\sphinxAtStartPar
Test owner (point of contact)
\end{itemize}
\subsubsection{VNF Test Cases}
\label{\detokenize{chapters/chapter05:vnf-test-cases}}
\paragraph{Compliance test cases}
\label{\detokenize{chapters/chapter05:compliance-test-cases}}
\sphinxAtStartPar
Currently, there VNFs can be packaged as HEAT templates or in a CSAR
file using TOSCA and OVP has supported the VNF compliance test
cases(compliance check based on TOSCA using ETSI SOL004 \&
SOL001; OpenStack HOT using ONAP VNFREQS; GSMA profile), all the OVP
supported test case can be found in the following two link:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
Test Cases
&\sphinxstyletheadfamily
\sphinxAtStartPar
Link
\\
\hline
\sphinxAtStartPar
Heat Test Cases
&
\sphinxAtStartPar
\sphinxurl{https://onap.readthedocs.io}/en/latest/submodules/vnfrqts/testcases.git/docs/Appendix.html\#list\sphinxhyphen{}of\sphinxhyphen{}requirements\sphinxhyphen{}with\sphinxhyphen{}associated\sphinxhyphen{}tests
\\
\hline
\sphinxAtStartPar
Tosca Test Cases
&
\sphinxAtStartPar
\sphinxurl{https://onap.readthedocs.io}/en/latest/submodules/vnfsdk/model.git/docs/files/csar\sphinxhyphen{}validation.html
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\sphinxAtStartPar
Above compliance test cases defination can be found
\sphinxurl{https://github.com/onap/vnfsdk-validation/tree/master/csarvalidation/src/main/resources/open-cli-schema}
\sphinxAtStartPar
In order to adapt Anuket specification, more compliance test case will
be added here.
\paragraph{Verification test cases}
\label{\detokenize{chapters/chapter05:verification-test-cases}}
\sphinxAtStartPar
In general, the VNF Manager, in collaboration with the NFV
Orchestrator, the VIM and the EM, is responsible for managing a VNF’s
lifecycle. The lifecycle phases are listed below:
\begin{itemize}
\item {}
\sphinxAtStartPar
VNF on\sphinxhyphen{}boarding, it refers to VNF package onboarding to
service/resouce Orchestrator
\item {}
\sphinxAtStartPar
VNF instantiation, once the VNF is instantiated, its associated VNFCs
have been successfully instantiated and have been allocated necessary
NFVI resources\sphinxhyphen{}
\item {}
\sphinxAtStartPar
VNF scaling/updating, it means the VNF can scale or update by
allocating more or less NFVI resources
\item {}
\sphinxAtStartPar
VNF termination, any NFVI resources consumed by the VNF can be
cleaned up and released.
\end{itemize}
\sphinxAtStartPar
OVP has also supported the lifecycle test
case:https://wiki.lfnetworking.org/display/LN/VNF+Validation+Minimum+Viable+Product?src=contextnavpagetreemode
\paragraph{Validation Test cases}
\label{\detokenize{chapters/chapter05:validation-test-cases}}
\sphinxAtStartPar
From the current situation of operators, there are usually corresponding
functional test specifications for each types of VNFs. Therefore,
different types of VNFs have different functional test cases. Normally,
functional tests for VNFs require the cooperation of surrounding VNFs.
Or use the instruments to simulate the functions of surrounding VNFs for
testing. Therefore, different test cases need to be defined according to
different types of VNFs
\paragraph{Performance Test cases}
\label{\detokenize{chapters/chapter05:performance-test-cases}}
\sphinxAtStartPar
This is the same as what described in validation test cases, the
performance test cases need to be defined according to different types
of VNFs. Combined with the classification of VNF, according to the
protocol level that VNF operates, it can include:
\begin{itemize}
\item {}
\sphinxAtStartPar
VNF data plane benchmarking, like forwarding Performance
Benchmarking,Long duration traffic testing, low misrouting and so on.
\item {}
\sphinxAtStartPar
VNF control plane benchmarking, like throughput
\item {}
\sphinxAtStartPar
VNF user plane benchmarking, like Packet Loss,Latency, Packet Delay
\end{itemize}
\sphinxAtStartPar
ETSI spec has also defined the testing method
\sphinxurl{http://www.etsi.org/deliver/etsi\_gs/NFV-TST/001\_099/001/01.01.01\_60/gs\_nfv-tst001v010101p.pdf}
\section{VNF Test Cases and Traceability to Requirements}
\label{\detokenize{chapters/chapter06:vnf-test-cases-and-traceability-to-requirements}}\label{\detokenize{chapters/chapter06::doc}}
\subsection{Introduction}
\label{\detokenize{chapters/chapter06:introduction}}\begin{itemize}
\item {}
\sphinxAtStartPar
Provide an overview of the purpose for the VNF TC Traceability to RM
Requirements chapter
\item {}
\sphinxAtStartPar
Start with defining requirements
\item {}
\sphinxAtStartPar
Finish with tracing test cases to requirements
\end{itemize}
\subsection{RM/RA1 Requirements}
\label{\detokenize{chapters/chapter06:rm-ra1-requirements}}\begin{itemize}
\item {}
\sphinxAtStartPar
Define RM/RA\sphinxhyphen{}1 Openstack requirements
\end{itemize}
\subsection{Test Case Traceability}
\label{\detokenize{chapters/chapter06:test-case-traceability}}\begin{itemize}
\item {}
\sphinxAtStartPar
Define and provide a table mapping Test Cases to Requirements
\end{itemize}
\section{VNF Testing Cookbook}
\label{\detokenize{chapters/chapter07:vnf-testing-cookbook}}\label{\detokenize{chapters/chapter07::doc}}
\subsection{Introduction}
\label{\detokenize{chapters/chapter07:introduction}}
\sphinxAtStartPar
Define the purpose of the chapter which is to:
\begin{itemize}
\item {}
\sphinxAtStartPar
Identify Framework Needs, Goals, and Dependencies
\item {}
\sphinxAtStartPar
Define Opensource Integration (OVP, Functest, CVC, others)
\item {}
\sphinxAtStartPar
Provide Automation Toolchain (list, topology, flow)
\end{itemize}
\subsection{Relevant Community Projects.}
\label{\detokenize{chapters/chapter07:relevant-community-projects}}
\subsection{VNF Testing Cookbook.}
\label{\detokenize{chapters/chapter07:id1}}
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{rc1_cookbook_vnf}.png}
\caption{VNF\_cookbook}\label{\detokenize{chapters/chapter07:id2}}\end{figure}
\sphinxAtStartPar
As detailed in the RC chapter 05 on E2E VNF test platform requirements,
ONAP VNF Test Platform (VTP) helps to perform the VNF Conformance
process by addressing those requirements. And following sections
provides required guidelines and details for platform and test cases.
\subsubsection{Platform Architecture}
\label{\detokenize{chapters/chapter07:platform-architecture}}
\sphinxAtStartPar
Provides details on the architecture, components and it’s
responsibilities.
\begin{figure}[htbp]
\centering
\capstart
\noindent\sphinxincludegraphics{{f3b0c214bc58c44406fd5b801d3dfc88}.png}
\caption{Platform Architecture}\label{\detokenize{chapters/chapter07:id3}}\end{figure}
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Test Controller}: For every feature supported in VTP, Test
controller provides required REST API along with user authentication
and authorization based on given tenant?.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Agile Test Orchestrator}: Dynamically allows to on\sphinxhyphen{}board and
execute the test cases and test flows across different run\sphinxhyphen{}time
environment on given System under test (SUT) along with required
supported system in place.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Portal \& CLI}: To operate and manage the VTP features, Portal?
provides web 2.0 based graphical user interface along with Command
line interface.
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Test case plug\sphinxhyphen{}ins}: Test cases are on\sphinxhyphen{}boarded into the system as
independent plug\sphinxhyphen{}ins (developed using different programming/scripting
language) and/or plain text yaml file for those supported as profile
in VTP (such as HTTP, SNMP, etc)
\item {}
\sphinxAtStartPar
\sphinxstylestrong{Repository}: Provides version controlled repository for persisting
various aspects of the VTP such as artifacts, results, reports, etc.
\end{itemize}
\sphinxAtStartPar
\sphinxstyleemphasis{? \sphinxhyphen{} Feature in\sphinxhyphen{}progress}
\subsubsection{Platform administrator guide}
\label{\detokenize{chapters/chapter07:platform-administrator-guide}}
\sphinxAtStartPar
Provides detail on installation, configuration, un\sphinxhyphen{}installation
operations.
\sphinxAtStartPar
VTP is provided a script for performing installation and mange the
installed VTP services as below:
\sphinxAtStartPar
\sphinxhref{https://github.com/onap/vnfsdk-refrepo/blob/master/vnfmarket-be/deployment/install/vtp\_install.sh}{vpt\_install.sh}
\sphinxAtStartPar
\sphinxstylestrong{\textendash{}download} : It will download all required artifacts into
/opt/vtp\_stage
\sphinxAtStartPar
\sphinxstylestrong{\textendash{}install} : It will install VTP (/opt/controller) and CLI
(/opt/oclip)
\sphinxAtStartPar
\sphinxstylestrong{\textendash{}start} : It will start VTP controller as tomcat service and CLI as
oclip service
\sphinxAtStartPar
\sphinxstylestrong{\textendash{}verify} : It will verify the setup is done properly by running some
test cases
\sphinxAtStartPar
\sphinxstylestrong{\textendash{}uninstall} : It will stop and uninstall the VTP
\sphinxAtStartPar
\sphinxstylestrong{\textendash{}clean} : It will remove the downloaded artifacts
\sphinxAtStartPar
Customize the download URL as below from latest snapshot or release onap
repository
\sphinxAtStartPar
export
OCLIP\_DOWNLOAD\_URL=“\sphinxurl{https://nexus.onap.org/content/repositories/snapshots/org/onap/cli/cli-zip/5.0.0-SNAPSHOT/cli-zip-5.0.0-20200302.120040-1.zip}”
\sphinxAtStartPar
export
VTP\_DOWNLOAD\_URL=“\sphinxurl{https://nexus.onap.org/content/repositories/snapshots/org/onap/vnfsdk/refrepo/vnf-sdk-marketplace/1.6.4-SNAPSHOT/vnf-sdk-marketplace-1.6.4-20220504.134611-11.war}”
\sphinxAtStartPar
export
CSAR\_VALIDATE\_DOWNLOAD\_URL=“\sphinxurl{https://nexus.onap.org/content/repositories/snapshots/org/onap/vnfsdk/validation/csarvalidation-deployment/1.2.2-SNAPSHOT/csarvalidation-deployment-1.2.2-20200314.160204-18.zip}”
\sphinxAtStartPar
export
CSAR\_VALIDATE\_JAR\_DOWNLOAD\_URL=“\sphinxurl{https://nexus.onap.org/content/repositories/snapshots/org/onap/vnfsdk/validation/validation-csar/1.2.2-SNAPSHOT/validation-csar-1.2.2-20200314.160158-18.jar}”
\subsubsection{Test case development guide}
\label{\detokenize{chapters/chapter07:test-case-development-guide}}
\sphinxAtStartPar
Provides details on how to develop new test cases and packages them for
deploying.
\sphinxAtStartPar
\sphinxhref{https://wiki.onap.org/pages/viewpage.action?pageId=43386304}{More
details}
\subsubsection{Test case model guide}
\label{\detokenize{chapters/chapter07:test-case-model-guide}}
\sphinxAtStartPar
Model the required test cases for various scenario, which could help in
standardizing the test cases for various NF, different compliance and
multiple MANO/NFVi based LCM operations.
\sphinxAtStartPar
\sphinxhref{https://wiki.onap.org/pages/viewpage.action?pageId=43386304}{More
details}
\subsubsection{Test case administrator guide}
\label{\detokenize{chapters/chapter07:test-case-administrator-guide}}
\sphinxAtStartPar
Provides detail on installation, configuration, un\sphinxhyphen{}installation
operations
\sphinxAtStartPar
\sphinxhref{https://wiki.onap.org/pages/viewpage.action?pageId=43386304}{More
details}
\section{Gap analysis and Development}
\label{\detokenize{chapters/chapter08:gap-analysis-and-development}}\label{\detokenize{chapters/chapter08::doc}}
\subsection{Introduction}
\label{\detokenize{chapters/chapter08:introduction}}\begin{itemize}
\item {}
\sphinxAtStartPar
Describe the purpose of this chapter, which includes, but not limited
to:
\item {}
\sphinxAtStartPar
Test Case Gaps (analysis)
\item {}
\sphinxAtStartPar
Automation Gaps
\item {}
\sphinxAtStartPar
OpenStack release Comparisons
\end{itemize}
\subsection{Openstack Release Comparisons}
\label{\detokenize{chapters/chapter08:openstack-release-comparisons}}\begin{itemize}
\item {}
\sphinxAtStartPar
Provide details, preferably in table format, comparing OpenStack
releases based on Wallaby baseline for RI\sphinxhyphen{}1
\end{itemize}
\subsection{Test Case Gaps}
\label{\detokenize{chapters/chapter08:test-case-gaps}}
\sphinxAtStartPar
Anuket has developed many test cases in the different \sphinxhref{https://wiki.opnfv.org/display/testing/TestPerf}{test
projects} which can
quickly improve RC. As listed in \sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_cert/RC1/chapters/chapter02.html}{NFVI Conformance Requirements},
porting all the existing testcases to Xtesting will unify the test case
execution and simplify the test integration as required by RC. Here are all the
related issues:
\begin{itemize}
\item {}
\sphinxAtStartPar
\sphinxhref{https://github.com/cntt-n/CNTT/issues/511}{port VinePerf to
Xtesting}
\item {}
\sphinxAtStartPar
\sphinxhref{https://github.com/cntt-n/CNTT/issues/865}{port NFVbench testcases to
Xtesting}
\end{itemize}
\sphinxAtStartPar
Here are the possible new test cases which could be integrated in the
existing Anuket projects to improve RC:
\begin{savenotes}\sphinxattablestart
\centering
\begin{tabulary}{\linewidth}[t]{|T|T|}
\hline
\sphinxstyletheadfamily
\sphinxAtStartPar
issues
&\sphinxstyletheadfamily
\sphinxAtStartPar
requirements
\\
\hline
\sphinxAtStartPar
\sphinxhref{https://github.com/cntt-n/CNTT/issues/508}{integrate KloudBuster in
Functest}
&
\sphinxAtStartPar
disk benchmarking
\\
\hline
\sphinxAtStartPar
\sphinxhref{https://github.com/cntt-n/CNTT/issues/916}{add tempest\sphinxhyphen{}stress in
Functest}
&
\sphinxAtStartPar
stress testing
\\
\hline
\end{tabulary}
\par
\sphinxattableend\end{savenotes}
\subsection{Framework Gaps}
\label{\detokenize{chapters/chapter08:framework-gaps}}
\sphinxAtStartPar
As proposed in \sphinxhref{https://github.com/cntt-n/CNTT/issues/917}{port VTP test cases to
Xtesting}, VTP selected in
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_cert/RC1/chapters/chapter05.html}{VNF Testing Framework Requirements}
requires small adaptations to fully fulfill the current
\sphinxhref{https://cntt.readthedocs.io/en/latest/ref\_cert/RC1/chapters/chapter02.html}{NFVI Conformance Requirements}.
It seems trivial changes as VTP proposed a REST API but will ensure that both
NFVI and VNF testing can be executed in the same CI toolchain very easily.
\renewcommand{\indexname}{Index}
\printindex
\end{document} |
https://www.cs.ubc.ca/~murphyk/MLbook/figReport-16-Aug-2012/figuresSource/scaleGMM.tex | ubc.ca | CC-MAIN-2023-06 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-06/segments/1674764500094.26/warc/CC-MAIN-20230204044030-20230204074030-00318.warc.gz | 723,177,013 | 1,068 |
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https://www.ph.tum.de/academics/bsc/break/2014s/fk_MA9203_05_testexam.tex | tum.de | CC-MAIN-2021-25 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-25/segments/1623487607143.30/warc/CC-MAIN-20210613071347-20210613101347-00485.warc.gz | 858,176,952 | 6,186 | \newcommand{\authors}{Fabian Kohler, Karolina Stoiber} % Name
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\begin{document}
% Überschrift
\thispagestyle{fancy}
\begin{center}{\textbf{ \Large{Probeklausur}}}\end{center}
\addtocounter{section}{0} %Fügt dem Section-Zähler +x hinzu. Nützlich, falls die Blätter durchnummeriert sind und das erste Übungsblatt z.B. gegliedert ist in Aufgabe 1.1, 1.2 etc. ..
\textbf{Allgemein Hinweise:}\\
Die Arbeitszeit beträgt \textbf{90 Minuten}. Falls nicht anders angegeben, sind alle Lösungen \textbf{ausführlich und nachvollziehbar} zu begründen. Schreiben Sie bitte \textbf{nicht mit Bleistift} und auch \textbf{nicht in roter oder grüner} Farbe. Zum Erreichen der Note 4,0 sind mindestens 50\% der Punkte nötig.
% \section{Stetigkeit [7 Punkte]}
% Sei $X$, metrischer Raum, zusammenh\"angend und $f:X\rightarrow\mathbb{R}$ lokal konstant d.h. zu jedem $x\in X$ exisitiert eine Umgebung $x\in U\subset X$ so dass $f|_U$ konstant. Zeige: $f$ ist konstant. Geben Sie zudem ein Gegenbeispiel an, f\"ur den Fall, dass $X$ nicht zusammenh\"angend ist (eine lokal konstanten Funktion an, die nicht konstant ist).
%\begin{comment}
% \paragraph*{L\"osung}
%Sei $z\in X$ und $A=\{x\in X|f(x)=f(z)\}$. Nach Voraussetzung ist $f$ lokal konstant, also ist $A$ offen. Die Menge $B:=X\setminus A = \{x\in X|f(x)\neq f(z)\} = \{x\in X|f(x)< f(z)\}\cup \{x\in X|f(x)>f(z)\}$ ist auch offen und es gilt $X=A\cup B$. Da $X$ nach Voraussetzung zusammenh\"angend ist, muss gelten $A=\emptyset$ oder $B=\emptyset$. Da aber $z\in A \Rightarrow B=\emptyset \Rightarrow f$ konstant.
%
%Als Gegenbeispiel w\"ahlen wir $X=\mathbb{R}\setminus \{0\}$. Diese Menge ist nicht zusammenh\"angend. Die Funktion $f:\mathbb{R}\setminus \{0\}\rightarrow \mathbb{R},x\mapsto \text{sgn}(x)$ ist dann lokal konstant, aber nicht global konstant.
%\end{comment}
\section{Koordinatentransformation [7 Punkte]}
Sei $U=\mathbb{R^+\times\mathbb{R}}$ und $V=\mathbb{R}^2 \setminus (\mathbb{R}^-_0\times{0})$ und $\Phi:U\rightarrow V$ die Koordinatentransformation
\begin{align*}
\binom{x_1}{x_2}=\Phi(\xi_1,\xi_2)=\binom{\xi_1^2-\xi_2^2}{2\xi_1\xi_2}.
\end{align*}
\begin{itemize}
\item[(a)] Bestimme $D\Phi(\xi)$, das normierte Zweibein $e_{\xi_1}(\xi),e_{\xi_2}(\xi)$ und $D\Phi^{-1}(\Phi(\xi))$.
\item[(b)] Sei $f\in\mathcal{C}\infty(U,\mathbb{R})$ und $\tilde{f}=f\circ\Phi^{-1}:V\rightarrow\mathbb{R}$. Drücke den Gradienten von $\tilde{f}$ durch Ableitungen von $f$ in der Basis $e_{\xi_1},e_{\xi_2}$ aus.
\end{itemize}
\begin{comment}
\paragraph{Lösung}
\begin{itemize}
\item[(a)]
\begin{align*}
\text{D}\Phi(\xi)=2\left(\begin{array}{cc}
\xi_1 & -\xi_2\\
\xi_2 & \xi_1
\end{array}\right)
\end{align*}
\begin{align*}
\text{D}\Phi^{-1}(\Phi(\xi))=\text{D}\Phi(\xi)^{-1}=\frac{1}{2(\xi_1^2+\xi_2^2)}\left(\begin{array}{cc}
\xi_1 & \xi_2\\
-\xi_2 & \xi_1
\end{array}\right)
\end{align*}
\begin{align*}
e_1(\xi)=\frac{1}{\sqrt{\xi_1^2+\xi_2^2}}\left(\begin{array}{c}
\xi_1 \\ \xi_2
\end{array}\right)\\
e_2(\xi)=\frac{1}{\sqrt{\xi_1^2+\xi_2^2}}\left(\begin{array}{c}
-\xi_2 \\ \xi_1
\end{array}\right)
\end{align*}
\item[(b)]Laut Vorlesung ist
\begin{align*}
\binom{\partial_{x_1}}{\partial_{x_2}}\tilde f &= D\Phi(\xi)^{T^{-1}}\binom{\partial_{\xi_1}}{\partial_{\xi_2}}f =\frac{1}{2(\xi_1^2+\xi_2^2)}\left(\begin{array}{cc} \xi_1 & -\xi_2 \\ \xi_2 & \xi_1 \end{array}\right)\binom{\partial_{\xi_1}}{\partial_{\xi_2}}f\\
&=\frac{1}{2\sqrt{\xi_1^2+\xi_2^2}}(e_{\xi_1}\partial_{\xi_1}+e_{\xi_2}\partial_{\xi_2})f
\end{align*}
\end{itemize}
\end{comment}
\section{Differenzierbarkeit [10 Punkte]}
Die Funktion $f:\mathbb{R}^2\rightarrow\mathbb{R}$ sei definiert durch
\begin{align*}
f(x,y) = \left\{
\begin{array}{l l}
\frac{xy}{x^2+y^2} &\quad (x,y) \neq 0\\
0 &\quad (x,y) = (0,0)
\end{array}
\right\}
\end{align*}
Man zeige
\begin{itemize}
\item[(a)] $f$ ist partiell differenzierbar
\item[(b)] $f$ ist nicht stetig
\item[(c)] $f$ ist nicht total differenzierbar
\item[(d)] Wie lautet die Richtungsableitung in Richtung $v=(v_1,v_2)\in\mathbb{R}^2$ im Punkt $(1,0)$?
\end{itemize}
\begin{comment}
\paragraph*{L\"osung}
\begin{itemize}
\item[(a)] F\"ur $(x,y) \neq (0,0)$ ist $f$ als Komposition differenzierbarer Funktionen insbesondere auch partiell differenzierbar. Betrachte also den Fall $(x,y) = (0,0)$
\begin{align*}
\partial_x f(0,0) & = \lim\limits_{(h,0)\rightarrow(0,0)} \frac{f(h,0)-f(0,0)}{h} = \lim\limits_{(h,0)\rightarrow(0,0)} \frac{f(h,0)}{h} \\
&= \lim\limits_{(h,0)\rightarrow(0,0)} \frac{0}{h} = 0
\end{align*}
und analog
\[\partial_y f(0,0) = 0.\]
$f$ ist also auch in $(0,0)$ partiell differenzierbar und damit ist $f$ partiell differenzierbar. [3 Punkte]
\item[(b)] F\"ur $(x,y) \neq (0,0)$ ist $f$ als Komposition stetiger Funktionen stetig. Wir k\"onnen also nur die Unstetigkeit am Nullpunkt zeigen. Wir benutzen daf\"ur Polarkoordinaten $(x,y) = (r\sin\phi,r\cos\phi)$. Dann ist
\begin{align*}
\lim_{(x,y)\rightarrow(0,0)}f(x,y) &= \lim_{r\rightarrow 0}f(r,\phi)\\
& = \lim_{r\rightarrow 0} \frac{r^2 \sin\phi\cos\phi}{r^2}\\
& = \lim_{r\rightarrow 0}\sin\phi\cos\phi
\end{align*}
Es gibt $\phi$ f\"ur die der Grenzwert $\neq 0 = f(0,0)$ ist $\Rightarrow f$ ist nicht stetig in $(0,0)$. \mbox{[3 Punkte]}
\item[(c)] Da $f$ in $(0,0)$ nicht stetig ist, ist $f$ dort auch nicht differenzierbar also $f$ nicht differenzierbar. [1 Punkt]
\item[(d)] Au\ss erhalb von $(0,0)$ ist $f$ differenzierbar. Wir berechnen zun\"achst den Gradienten f\"ur $(x,y)\neq 0$
\begin{align*}
\nabla f(x,y) = \begin{pmatrix}
\frac{y (y^2-x^2)}{(x^2+y^2)^2}\\
\frac{x(x^2-y^2)}{(x^2+y^2)^2}
\end{pmatrix}.
\end{align*}
Dann l\"asst sich die Richtungsableitung einfach berechnen.
\begin{align*}
\partial_v f(1,0) &= \nabla f(1,0) \cdot v\\
&= \begin{pmatrix}
0 \\ 1
\end{pmatrix}
\cdot \begin{pmatrix}
v_1 \\ v_2
\end{pmatrix}\\
&= v_2.
\end{align*}
[3 Punkte]
\end{itemize}
\end{comment}
\section{Taylor und Extrema [10 Punkte]}
Sei $f:\mathbb{R}^2\rightarrow\mathbb{R}$ zweimal stetig differenzierbar mit $f(0,0)=0$, $f$ hat bei $(0,0)$ einen station\"aren Punkt und
\[H_f = \begin{pmatrix}
4 & -1\\
-1 & 4
\end{pmatrix}
\]
Beweisen Sie, es existiert eine Umgebung $U$ von $(0,0)$, sodass f\"ur alle $(x,y)\in U$ gilt $f(x,y) \geq x^2 +y^2$ (Tipp: Taylor-Entwicklung).
\begin{comment}
\paragraph*{L\"osung}
Mit den gegebenen Informationen schreiben wir die Taylorreihe bis zur zweiten Ordnung hin
\begin{align*}
f(x,y) &= 0 + (0,0)(x,y)^T + \tfrac1 2 \left( 4x^2 + 4y^2 -xy -yx\right)+\theta(3)\\
&= \tfrac1 2\left(4x^2+4y^2-2xy\right) + \theta(3)
\end{align*}
Damit gilt $\forall (x,y)\in\mathbb{R}^2\setminus\{(0,0)\}$ wegen $x^2+y^2\geq 2xy \Leftrightarrow -2xy \geq -x^2-y^2$
\begin{align*}
f(x,y) \geq \tfrac1 2 (4x^2+4y^2-x^2-y^2)+\theta(3) = (x^2+y^2)\left(\tfrac3 2 + \tfrac{\theta(3)}{x^2+y^2}\right)
\end{align*}
\end{comment}
\section{Implizite Funktionen [12 Punkte]}
Zeigen Sie, dass es eine Umgebung $U \subset\mathbb{R}^3$ von $(0,0,0)$ gibt , in der das Gleichungssystem
\begin{align*}
\sin (x-y^2+z^3) - \cos (x+y+z) + 1 &= 0\\
\sin (y+x^2-z^3) + \cos (x-y) -1 &= 0
\end{align*}
eindeutig nach $(x,y)$ aufgelöst werden kann (d.h. $(x,y) = h(z)$ mit einer geeigneten Funktion $h$). Berechnen Sie weiterhin die Ableitung von $h$ im Nullpunkt.
\begin{comment}
\paragraph*{L\"osung}
Wir betrachten die stetig differenzierbare Funktion
\begin{align*}
f:\mathbb{R}^3&\rightarrow\mathbb{R}^2\\
(x,y,z)&\mapsto \begin{pmatrix}
\sin (x-y^2+z^3) - \cos (x+y+z) + 1\\
\sin (y+x^2-z^3) + \cos (x-y) -1
\end{pmatrix}
\end{align*}
Es gilt $f(0,0,0) = (0,0)$, der Punkt ist also eine Nullstelle. Nun berechnen wir das partielle Differential
\begin{align*}
D_{xy} f(0,0,0) &= \begin{pmatrix} \cos(x-y^2+z^3)+\sin(x+y+z) & -2y\cos(x-y^2+z^3)+\sin(x+y+z)\\ 2x\cos(y+x^2-z^3)-\sin(x-y) & \cos(y+x^2-z^3)\sin(x-y)\end{pmatrix}\\
&= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\end{align*}
Offensichtlich ist die Einheitsmatrix invertierbar, also k\"onnen wir nach dem Satz \"uber implizite Funktionen die Gleichung an der Stelle $(0,0,0)$ entsprechend aufl\"osen mit einer eindeutig bestimmten Funktion $h(z)$, die sogar differenzierbar ist. Die Ableitung berechnen wir ebenfalls nach dem Satz \"uber implizite Funktionen
\begin{align*}
h'(0) &= - [\text{D}_{xy} f(0,h(0))]^{-1} \text{D}_{z} f(0,h(0))\\
\end{align*}
Das partielle Differential $\text{D}_z f(x,y,z)$ lautet
\begin{align*}
\text{D}_z f(0,0,0) &= \begin{pmatrix} 3z^2\cos(x-y^2+z^3)+\sin(x+y+z)\\ -3z^2\cos(y+x^2-z^3) \end{pmatrix}\\
&= \begin{pmatrix} 0 \\ 0\end{pmatrix}
\end{align*}
und damit erhalten wir f\"ur die Ableitung
\begin{align*}
h'(0) &= - \begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}^{-1} \begin{pmatrix} 0\\ 0 \end{pmatrix}\\
&= \begin{pmatrix} 0\\ 0\end{pmatrix}.
\end{align*}
\end{comment}
\section{Extrema mit Nebenbedingungen [14 Punkte]}
Berechnen Sie diejenigen Punkte auf der Kugeloberfläche
\begin{equation*}
M = \{(x,y,z)\in\mathbb{R}^3| x^2 + y^2 + z^2 = 1\}
\end{equation*}
die von $(1,1,1)$ den kleinsten bzw. größten Abstand haben.
\begin{comment}
\paragraph*{Lösung:}
Gesucht sind die Extrema der Funktion $f(x,y,z)=(x-1)^2+(y-1)^2+(z-1)^2$ unter der Nebenbedingung $g(x,y,z)=x^2+y^2+z^2-1=0$\\
$Dg(x,y,z)=\nabla g(x,y,z)=2(x,y,z)\neq (0,0,0)$,d.h. es genügt die kritischen Punkte der Lagrangeschen Hilfsfunktion zu bestimmen:
\begin{align*}
\nabla f(x,y,z)-\lambda \nabla g(x,y,z)= \left(
\begin{matrix}
2(x-1)\\
2(y-1)\\
2(z-1)
\end{matrix}\right)-\lambda
\left(
\begin{matrix}
2x\\
2y\\
2z
\end{matrix}\right)=\left(
\begin{matrix}
0\\
0\\
0
\end{matrix}\right)
\end{align*}
Zusammen mit der Nebenbedingung liefert dies das Gleichungssystem:
\begin{equation}
\label{a}
1=(1-\lambda)x
\end{equation}
\begin{equation}
\label{b}
1=(1-\lambda)y
\end{equation}
\begin{equation}
\label{c}
1=(1-\lambda)z
\end{equation}
\begin{equation}
\label{d}
1=x^2+y^2+z^2
\end{equation}
Aus \ref{a} bis \ref{c} folgt $x=y=z=(1-\lambda)^{-1}$, in \ref{d} eingesetzt ergibt dies $\frac{3}{(1-\lambda)^2}=1 \Leftrightarrow \lambda = 1\pm \sqrt{3} \Rightarrow x=y=z=\pm \frac{1}{\sqrt{3}}$.\\
Da die Nebenbedingungsmenge kompakt und $f$ stetig ist, existiert ein Minimum in
$$p_1=\frac{1}{\sqrt{3}}(1,1,1)$$ und ein Maximum in
$$p_2=-\frac{1}{\sqrt{3}}(1,1,1),$$ denn
$$f(p_1)=(1-\sqrt{3})^2<(1+\sqrt{3})^2=f(p_2).$$
\end{comment}
\section{Parametrisierung auf Bogenlänge [4 Punkte]}
Geben Sie explizit eine Parametrisierung auf Bogenlänge, $\tilde{\gamma}:\mathbb{R}\rightarrow\mathbb{R}^2,$ der Kettenlinie $\gamma: \mathbb{R}\rightarrow\mathbb{R}^2, \gamma (t)(-t,-\cosh t).$
\begin{comment}
\paragraph*{Lösung}
\begin{equation*}
s(t)=\int_{0}^{t}\sqrt{1+\sinh^2 t'}dt'=\int_0^t\cosh t' dt'=\sinh t.
\end{equation*}
[2Punkte]\\
Mit der Umkehrfunktion $\tilde{t}(s)=\operatorname{arsinh}(s)=\sinh^{-1}(s)$ [1Punkt] ist
\begin{align*}
\tilde{\gamma}(s)=\gamma(\tilde{t}(s))=(-\operatorname{arsinh}(s),-\cosh(\operatorname{arsinh}(s)))=(-\operatorname{arsinh}(s),-\sqrt{1+\sinh(\operatorname{arsinh}(s))^2})\\
=(-\operatorname{arsinh}(s),-\sqrt{1+s^2}).
\end{align*}
[1 Punkt]
\end{comment}
\section{Wegintegrale [7 Punkte]}
Berechnen Sie das Wegintegral $\int_{\gamma}f(x)\text{d}x$:
$$f(x,y,z) = (2z-\sqrt{x^2+y^2},z,z^2)$$, $$\gamma (t) = (t \cos (t), t \sin (t), t), 0 \leq t, \leq 2\pi$$
\begin{comment}
\paragraph*{Lösung}
\begin{equation*}\int_{\gamma}f(x,y,z)d(x,y,z)=\int_{0}^{2\pi}(2t-t,t,t^2)\cdot\left(\begin{matrix}
\cos(t)-t\sin(t)\\ \sin(t)+t\cos(t)\\ 1
\end{matrix}\right)dt=
\end{equation*}
\begin{equation*}
\int_{0}^{2\pi}t\cos(t)dt+\int_{0}^{2\pi}t\sin(t)dt+\int_{0}^{2\pi}t^2\cos(t)dt-\int_{0}^{2\pi}t^2\sin(t)dt+\int_{0}^{2\pi}t^2dt
\end{equation*}
mit partieller Integration $\Rightarrow$\\
\begin{equation*}\frac{8}{3}\pi^3+4\pi^2+2\pi\end{equation*}
\end{comment}
\section{Trennbare Differentialgleichung [8 Punkte]}
\begin{itemize}
\item[(a)] Finden Sie auf ganz $\mathbb{R}$ definierte Lösungen von $yy'=x(1-y^2)$ mit $y(0)=y_0, y_0 \in \mathbb{R}\setminus\{0\}$
\item[(b)] Wie viele konstante Lösungen gibt es?
\item[(c)] Wie viele auf ganz $\mathbb{R}$ definierte Lösungen mit $y(0)=0$ gibt es?
\end{itemize}
\begin{comment}
\paragraph*{Lösung}
\begin{itemize}
\item[(a)] Wir lösen durch Separation der Variablen. Für eine Lösung $y(x)$ mit $y(0)=y_0$ muss gelten
\begin{equation*}
\int_{y_0}^{y(x)}\frac{ydy}{1-y^2}=\int_{0}^{x}x'dx',\text{ also } -\frac{1}{2}\ln|1-y(x)^2|+\frac{1}{2}\ln|1-y_0^2|=\frac{1}{2}x^2.
\end{equation*}
Da die Integration über die Singularitäten $y=\pm 1$ nicht möglich ist, müssen $1-y_0^2$ und $1-y(x)^2$ für alle $x$ das gleiche Vorzeichen haben. Somit ergibt beidseitiges exponenzieren
\begin{equation*}
1-y(x)^2=e^{-x^2}(1-y_0^2).
\end{equation*}
Die rechte Seite ist immer kleiner als 1. Damit $y(x)$ stetig ist, muss also gelten:
\begin{equation*}
y(x)=\begin{cases}
\sqrt{1-e^{-x^2}(1-y_0^2)}, & \text{falls } y_0>0,\\-\sqrt{1-e^{-x^2}(1-y_0^2)}, & \text{falls } y_0<0
\end{cases}
\end{equation*}
denn der Radikant ist in beiden Fällen immer positiv.
\item[(b)] Die rechte Seite der Differentialgleichung ist Null für alle $x$, genau dann, wenn $y=\pm 1$ ist. Somit sind $y(x)=\pm 1$ genau zwei konstante Lösungen.
\item[(c)] Für $y_0=0$ sind $y_\pm (x) = \pm \sqrt{1-e^{-x^2}}$ die einzigen zwei Lösungen auf $\mathbb{R}\setminus\{0\}$ mit $\lim\limits_{0}y_\pm (x)=0.$\\
Für kleine $x$ gilt $y_\pm (x)\approx \pm \sqrt{x^2}=\pm |x|.$ Somit gibt es genau zwei Lösungen
\begin{equation*}
y_1(x)=\begin{cases}
y_+(x) & \text{für } x>0\\0 & \text{für } x=0\\y_-(x) & \text{für } x<0
\end{cases}
\text{ und } y_2(x)=-y_1(x),
\end{equation*}
die auf ganz $\mathbb{R}$ differenzierbar und Lösungen der Differentialgleichung mit $y(0)=0$ sind.
\end{itemize}
\end{comment}
\end{document} |
https://ctan.math.washington.edu/tex-archive/macros/latex/contrib/metre/demo.tex | washington.edu | CC-MAIN-2021-49 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-49/segments/1637964363515.28/warc/CC-MAIN-20211208144647-20211208174647-00531.warc.gz | 267,858,994 | 2,635 | \documentclass[12pt]{report}
\usepackage{metre}
%\usepackage[T1]{fontenc}
\Magnitudo{0}
\parindent 0pt
\parskip 0pt
%
\newenvironment{MetricalScheme}{%
\let\par=\cr \obeylines %
\halign to \hsize \bgroup
\quad\quad\oldstylenums{##}\quad\quad\hfill&%
##\hfill&%
\tabskip 0pt plus 5em minus 5em\quad\quad##\hfill
}{\egroup}
%
\newcommand{\strophe}[1]{%
\noalign{%
\vspace*{4ex}
\quad\quad
\textit{#1}
\vspace*{2ex}
}%
}
%
\begin{document}
{\large\textbf{From:} \textsc{Angelo Paredi} \textit{Thesaurus}}\par
\vspace{5ex}
\textbf{Il giamb\`elego.} \`E un verso composto da un dimetro giambico
e da un trimetro dattilico catalettico \textit{in syllabam}.\par
\vspace{2ex}
Schema:\ \ \metra{\mb\M\s\b\M\s\bm\M\s\b\Bm\c\M\b\b\s\M\b\b\s\Bm}\par
\vspace{2ex}
Preceduto dall'esametro forma la strofa chiamata metro
\textbf{archilocheo secondo}. Orazio lo usa una sola volta
nell'epodo \oldstylenums{13}. Per esempio:
\begin{verse}
\begin{metrica}
H\='orr\-id\-a t\='emp\=est\='a c\={ae}l\='um c\=ontr\='ax\-it \-et \='imbr\=es\\
\quad n\-iv\='esqu\-e d\='ed\=uc\='unt I\-ov\='em;\c n\='unc m\-ar\-e n\='unc s\-il\-u\='{ae}\\
Thr\='e\-ic\-i\='o \-Aqu\-il\='on\-e s\-on\='ant. R\-ap\-i\='am\-us, \-am\='ic\=i,\\
\quad \=occ\='as\-i\='on\=em d\='e d\-i\='e,\c d\='umqu\-e v\-ir\='ent g\-en\-u\='a\\
\='et d\-ec\-et, \='obd\=uct\='a s\=olv\='at\=ur fr\='ont\-e s\-en\='ect\=us.\\
\quad T\=u v\='in\-a T\='orqu\=at\='o m\-ov\='e\c c\='ons\-ul\-e pr\='ess\-a m\-e\='o.\\
\end{metrica}
\end{verse}
%
\vspace{3ex}
%
\begin{verse}
\begin{metrica} \Translatio{.125} \it
H\='orr\-id\-a t\='emp\=est\='a c\={ae}l\='um c\=ontr\='ax\-it \-et \='imbr\=es\\
\quad n\-iv\='esqu\-e d\='ed\=uc\='unt I\-ov\='em;\c n\='unc m\-ar\-e n\='unc s\-il\-u\='{ae}\\
Thr\='e\-ic\-i\='o \-Aqu\-il\='on\-e s\-on\='ant. R\-ap\-i\='am\-us, \-am\='ic\=i,\\
\quad \=occ\='as\-i\='on\=em d\='e d\-i\='e,\c d\='umqu\-e v\-ir\='ent g\-en\-u\='a\\
\='et d\-ec\-et, \='obd\=uct\='a s\=olv\='at\=ur fr\='ont\-e s\-en\='ect\=us.\\
\quad T\=u v\='in\-a T\='orqu\=at\='o m\-ov\='e\c c\='ons\-ul\-e pr\='ess\-a m\-e\='o.\\
\end{metrica}
\end{verse}
%
\newpage
%
%
\FaciesSiglorum{} % suppress the use of italics
\MetraStyle{en}
{\large\textbf{From:} \textsc{Pindaro} \textit{Le Pitiche}}, Fondazione Lorenzo Valla\par
\vspace{5ex}
\begin{center}
\textsc{Pitica VII}\par
\vspace*{1ex}
{\large Schema metrico}\par
\end{center}
\vspace*{15ex}
Metro: strutture miste coriambico-giambiche\par
\begin{MetricalScheme}
\strophe{Strofe antistrofe}%
1& \metra{\m\m\b\m\b\b\b\b\b\b\b\m\m\Cc\S H}& \sigla{3ia_}
2& \metra{\b\m\b\b\m\s\s\s\m\b\m\s\s\s\bm\m\b\m\b\b\m\C}&%
\sigla{_<emiascl cr pros (>>_<<dim\S p)}
3& \metra{\m\m\b\m\m\C}& \sigla{reiz}
4& \metra{\m\m\b\b\m\b\c\b\m\s\s\b\b\b\m\b\b\m\m\C}&%
\sigla{pros pherecr}
5& \metra{\m\m\b\b\m\b\m\Cc\S H}& \sigla{teles}
6& \metra{\b\b\b\m\b\bm\Cc}& \sigla{do}
7& \metra{\bm\b\b\b\m\m\Ccc}& \sigla{reiz}
\strophe{Epodo}%
1& \metra{\m\b\m\b\b\m\s\s\s\m\b\m\b\b\m\C}& \sigla{2hemiascl}
2& \metra{\b\m\m\m\b\m\C}& \sigla{ba cr (pros\S{do})}
3& \metra{\m\m\b\b\m\b\m\C}& \sigla{teles}
4& \metra{\b\b\b\m\b\b\m\C}& \sigla{hemiascl}
5& \metra{\b\b\b\m\b\m\b\b\m\C}& \sigla{dim\S p (tr cho)}
6& \metra{\m\m\m\b\m\b\b\m\C}& \sigla{dim\S p}
7& \metra{\m\m\b\m\m\b\b\m\C}& \sigla{dim\S p (ia cho)}
8& \metra{\b\m\b\b\m\m\Ccc}& \sigla{reiz}
\end{MetricalScheme}
%
%
\newpage
%
%
\begin{center}
\textsc{Pitica VIII}\par
\vspace*{1ex}
{\large Schema metrico}\par
\end{center}
\vspace*{15ex}
Metro: strutture miste coriambico-giambiche\par
\begin{MetricalScheme}
\strophe{Strofe antistrofe}%
1& \metra{\b\b\b\m\b\b\m\b\bm\Cc\S H}& \sigla{glyc}
2& \metra{\m\b\m\b\b\b\b\b\m\Cc\S H}& \sigla{glyc}
3& \metra{\m\m\b\m\b\b\bm\Cc\S H}& \sigla{pros (>>_<<dim\S p)}
4& \metra{\b\m\b\m\s\s\s\m\b\b\m\b\bm\Cc\S H}& \sigla{ia hemiascl}
5& \metra{\m\b\b\m\m\bm\s\s\s\m\b\m\b\b\m\b\bm\Cc}& \sigla{hemiascl glyc}
6& \metra{\bm\m\b\m\b\b\m\m\s\s\s\bm\m\b\m\b\bm\Cc\S H}& \sigla{en pros\S{do}}
7& \metra{\bm\m\b\m\bm\m\n{47}\mb\m\b\m\Ccc}& \sigla{is pros\S{do}}
\strophe{Epodo}%
1& \metra{\b\m\b\m\b\b\m\n{35}\mb\s\s\s\m\m\n{55}\mb\m\b\bm\Cc}&%
\sigla{en pros\S{do}}
2& \metra{\n{36}\bm\m\b\m\b\b\m\s\s\s\m\n{56;76}\mb\m\b\b\m\bm\Cc\S H}&%
\sigla{pros (>>_<<dim\S p) pherecr}
3& \metra{\m\n{57;97}\mb\m\b\b\m\b\m\m\C}& \sigla{hipp}
4& \metra{\m\m\b\b\m\bm\C}& \sigla{reiz}
5& \metra{\m\m\b\b\m\s\s\s\m\b\b\m\s\s\s\b\m\b\m\C}&%
\sigla{_hemiascl cho ia}
6& \metra{\b\m\n{59}\bbm\b\b\m\b\m\s\s\s\m\b\m\b\b\m\b\m\Cc}& \sigla{2glyc}
7& \metra{\m\m\n{\it a}\bm\m\n{\it a}\bm\m\b\n{20}\mb\s\s\s\b\m\b\m\m\Ccc}%
\ \ \numeri{a:60;100}& \sigla{reiz cho (\r epitr\S{tr}) reiz}
\end{MetricalScheme}
%
%
\newpage
%
%
\begin{center}
{\Large Metrische Analyse von Pi.~\textit{O.} 1}\\
\vspace*{2ex}
Rainer Thiel\\
\end{center}
\vspace*{4ex}
\FaciesSiglorum{\it}
\begin{center}
str.\,/\,ant.\vspace{\bigskipamount}
\begin{tabular}{rlr}
1& \metra{\b\m\m\bb\m\b\m\s\m\b\m\b\b\m\m\c}& \sigla{gl pher\c}\\
2& \metra{\bb\b\m\s\b\m\m\s\bb\m\bb\m\bb\m\m\cc}& \sigla{cr ba enopl\cc}\\
3& \metra{\m\b\m\b\m\b\m\cc}& \sigla{lec (= cr ia)\cc}\\
4& \metra{\m\b\m\b\b\m\m\cc}& \sigla{pher\cc}\\
5& \metra{\m\b\m\b\m\b\m\cc}& \sigla{lec (= cr ia)\cc}\\
6& \metra{\m\b\m\s\b\bb\b\m\s\b\m\b\m\s\b\m\bb\m\m\s\b\m\b\m\cc}&%
\sigla{cr 2\,ia reiz ia\cc}\\
7& \metra{\m\b\m\b\s\m\b\m\b\s\m\b\b\m\s\m\b\m\cc}&%
\sigla{2\,tr cho cr\cc}\\
8& \metra{\b\bb\b\bb\s\b\m\b\m\s\b\m\b\m\cc}& \sigla{3\,ia\cc}\\
9& \metra{\b\n{20.\,38}{\bbm}\m\b\m\s\bb\b\m\cc}&%
\sigla{k>\D{}k cr\cc}\\
10& \metra{\b\m\m\s\b\m\b\m\s\bb\b\m\cc}& \sigla{ba ia cr\cc}\\
11& \metra{\n{80}{\mb}\m\b\bb\s\m\b\m\b\m}&\sigla{ia (hypo<<)>\d\ccc}\\
\end{tabular}
\vspace*{2\bigskipamount}
ep.\vspace*{\bigskipamount}
\begin{tabular}{rlr}
1& \metra{\b\m\b\bb\s\m\b\m\s\m\bb\m\s\b\m\m\s\b\m\b\m\cc}&\sigla{ia cr
cho ba ia\cc}\\
2&\metra{\b\m\m\b\b\m\m\c\bb\d{\bb}\m\b\m\cc}&\sigla{pher
\d\cc}\\
3&\metra{\m\b\m\b\m\b\m\s\bb\m\m\b\m\cc}&\sigla{lec (= cr ia)
\d\cc}\\
4&\metra{\b\m\m\b\m\s\m\bb\m\b\m\s\b\m\bb\m\m\cc}&\sigla{3\,\d\cc}\\
5&\metra{\b\b\m\b\s\m\b\b\m\s\s\m\b\m\b\m\cc}&\sigla{wilamow (hypo<<)>\d\cc}\\
6&\metra{\m\m\bb\m\b\m\s\m\b\m\b\m\cc}&\sigla{telesil (= _gl) (hypo<<)>\d\cc}\\
7&\metra{\b\b\b\s\m\b\b\m\s\s\m\b\m\cc}&\sigla{wilamow cr\cc}\\
8&\metra{\b\m\m\s\b\m\m\b\b\m\b\m\s\m\b\b\m\s\b\m\m\ccc}&\sigla{ba gl
cho ba\ccc}\\
\end{tabular}
\end{center}
\end{document}
|
https://www.zentralblatt-math.org/matheduc/en/?id=105031&type=tex | zentralblatt-math.org | CC-MAIN-2019-51 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-51/segments/1575540529955.67/warc/CC-MAIN-20191211045724-20191211073724-00429.warc.gz | 918,830,862 | 1,301 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 1993g.00500}
\itemau{Eisenhart, M.; Borko, H. (Colorado Univ., Boulder (United States)); Underhill, R.; Brown, C.; Agard, P. (Virginia Polytechnic Inst.and State Univ., Blacksburg, VA (United States)); Jones, D. (Kentucky Univ., Lexington (United States))}
\itemti{Conceptual knowledge falls through the cracks: complexities of learning to teach mathematics for understanding. Begriffliches Wissen faellt durch die Maschen: Verflechtungen beim Lernen des Unterrichtens von Mathematik, so dass Verstaendnis entsteht.}
\itemso{J. Res. Math. Educ. (Jan 1993) v. 24(1) p. 8-40. CODEN: JRMEDN [ISSN 0021-8251]}
\itemab
\itemrv{~}
\itemcc{C39 B50}
\itemut{Knowledge; Case Studies; Concept Formation; Preservice Teacher Education; Classroom Observation; Teacher Attitudes; ; Wissen; Fallstudie; Begriffsbildung; Lehrerausbildung; Unterrichtsbeobachtung; Lehrereinstellung}
\itemli{}
\end |
https://bibliotecaanarquista.org/library/charles-fourier-critica-dos-ideais-revolucionarios.tex | bibliotecaanarquista.org | CC-MAIN-2022-40 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-40/segments/1664030337723.23/warc/CC-MAIN-20221006025949-20221006055949-00760.warc.gz | 156,542,202 | 5,384 | \documentclass[DIV=12,%
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\title{Crítica dos Ideais Revolucionários}
\date{1848}
\author{Charles Fourier}
\subtitle{}
% https://groups.google.com/d/topic/comp.text.tex/6fYmcVMbSbQ/discussion
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{\usekomafont{title}{\huge Crítica dos Ideais Revolucionários\par}}%
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\vskip 2em
{\usekomafont{author}{Charles Fourier\par}}%
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\tableofcontents
% start a new right-handed page
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\emph{Uma breve introdução}: Este texto apresenta uma crítica aos ideais da Revolução Francesa - \emph{Liberdade, Igualdade e Fraternidade} - escrita por \textbf{François Marie Charles Fourier} (1772 – 1837), um socialista experimental francês (chamado de socialista utópico pelo jargão marxista). Fourier foi um crítico ferino do economicismo e do capitalismo, e adversário da industrialização, da civilização urbana, do liberalismo e da família baseada no matrimônio e na monogamia. Ele é citado como referência por anarquistas como Pierre-Joseph Proudhon, Joseph Déjacque, Peter Kropotkin, Émile Armand, Paul Goodman, Bob Black, Hakim Bey, pelo antifascista André Breton e por Karl Marx, e serviu de inspiração para o Falanstério do Saí em Santa Catarina e para a Colônia Cecília no Paraná, entre outros.
O caráter jovial com que Fourier realizou algumas de suas críticas fez dele um dos grandes satíricos de todos os tempos. No entanto, não ficou apenas na crítica, assumiu um caráter propositivo. Sugeriu a criação de unidades de produção e consumo - \emph{as falanges} ou \emph{falanstérios} - baseadas em uma forma de cooperativismo integral e auto-suficiente, assim como na livre perseguição do que chamava \emph{paixões} individuais e seu desenvolvimento, o que constituiria um estado que chamava \emph{harmonia}. Neste sentido antecipa a linhagem do socialismo libertário dentro do movimento socialista, mas também em linhas críticas da moral burguesa e cristã, restritiva do desejo e do prazer. Em 1808 Fourier já argumentava abertamente em favor da igualdade de gênero entre homens e mulheres, apesar da palavra feminismo só ter surgido em 1837.
\part{Crítica dos Ideais Revolucionários}
A filosofia estava certa em alardear a \emph{liberdade;} é o principal desejo de todas as criaturas. Mas a filosofia esqueceu que, nas sociedades civilizadas, a liberdade é ilusória se as pessoas comuns não possuem riqueza. Quando as classes assalariadas são pobres, sua independência é tão frágil quanto uma casa sem fundações. O homem livre que carece de riqueza imediatamente mergulha sob o jugo dos ricos. O escravo recém-libertado se assusta com a necessidade de prover sua própria subsistência e se apressa em se vender de volta à escravidão, a fim de escapar dessa nova ansiedade que paira sobre ele como a espada de Dâmocles. Ao dar-lhe liberdade sem riqueza, você simplesmente substitui seu tormento físico por um tormento mental. Ele acha a vida pesada em seu novo estado\dots{} Assim, quando você dá liberdade ao povo, ele deve ser reforçado por dois suportes que são \emph{a garantia de conforto} e \emph{atração industrial\dots{}}
\emph{Igualdade de direitos é} outra quimera, louvável quando considerada abstrata e ridícula do ponto de vista dos meios empregados para introduzi-la na civilização. O primeiro direito dos homens é o direito ao trabalho e o direito ao \emph{mínimo.} Isto é precisamente o que não foi reconhecido em todas as constituições. Sua principal preocupação é com indivíduos privilegiados que não estão necessitando de trabalho. Eles começam com listas pomposas dos eleitos de famílias privilegiadas para as quais a lei garante uma renda de cinquenta ou cem mil francos para a simples tarefa de governar o povo ou sentar em um assento estofado e votar com a maioria em um senado. Se a primeira página das constituições serve para dar aos administradores garantias de afluência e ociosidade, seria bom que a segunda página prestasse atenção ao lote das classes mais baixas, ao \emph{mínimo proporcional} e ao direito ao trabalho, que são omitido em todas as constituições, e ao direito ao prazer, que é garantido apenas pelo mecanismo da série industrial\dots{}
Vamos nos voltar para a \emph{fraternidade.} Nossa discussão aqui será divertida, ao mesmo tempo repugnante e bem estudada. É divertido em vista da imbecilidade das teorias que pretendem estabelecer a fraternidade. É repugnante quando recordamos os horrores que o ideal de fraternidade mascarou. Mas é um problema que merece atenção especial da ciência; pois as sociedades atingirão seu objetivo, e o homem sua dignidade, somente quando a fraternidade universal se tornar um fato estabelecido. Por fraternidade universal queremos dizer um grau de intimidade geral que só pode ser realizado se quatro condições forem satisfeitas:
\begin{quote}
Conforto para o povo e a garantia de um mínimo esplêndido;
A educação e instrução das classes inferiores;
A verdade geral nas relações de trabalho.
A prestação de serviços recíprocos por classes desiguais.
\end{quote}
Uma vez satisfeitas estas quatro condições, os ricos Mondor terão relações verdadeiramente fraternas com Irus que, apesar de sua pobreza, não terão necessidade de um protetor nem motivação para enganar a ninguém, e cuja boa educação lhe permitirá associar-se aos príncipes\dots{} Quanto ao presente, como poderia haver por alguma fraternidade entre sibaritas impregnados de refinamentos e nossos camponeses rudes e famintos, cobertos de farrapos e muitas vezes de insetos e portadores de doenças contagiosas como o tifo, a sarna, a plica e outros frutos da pobreza civilizada? Que tipo de fraternidade poderia ser estabelecida entre essas classes heterogêneas de homens?
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Biblioteca Anarquista
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en.pdf}
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\end{center}
\strut
\vfill
\begin{center}
Charles Fourier
Crítica dos Ideais Revolucionários
1848
\bigskip
De la methode mixte, La Phalange, VII, 1848.
Fonte: Extraído e traduzido de \emph{The Utopian Vision of Charles Fourier}. Selected Texts on Work, Love, and Passionate Attraction. Publicado por Jonathan Cape, 1972. A introdução é inspirada num autor anônimo da Protopia.\forcelinebreak
\bigskip
\textbf{bibliotecaanarquista.org}
\end{center}
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https://21docs.com/users/87121/articles/145474/download_latex | 21docs.com | CC-MAIN-2020-50 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-50/segments/1606141733120.84/warc/CC-MAIN-20201204010410-20201204040410-00016.warc.gz | 169,285,066 | 3,572 | \documentclass{article}
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\begin{document}
\title{Authorea Researcher Spotlight:~Juan de Monasterio}
\author[ ]{Adyam Ghebre}
\author[ ]{Josh Nicholson}
\affil[ ]{}
\vspace{-1em}
\date{}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
Juan de Monasterio is an Applied Mathematics Masters student at the
University of Buenos Aires. Juan's work involves designing and
implementing processes that extract, transform, and load data from a
wide variety of sources (relational, SQL, web services, flat files).
\par\null\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figures/photo-profile-mateo/photo-profile-mateo}
\caption{{Juan de Monasterio, Applied Mathematics Masters student at the
University of Buenos Aires%
}}
\end{center}
\end{figure}
\par\null
\subsection*{\texorpdfstring{You're wrote a chapter on Authorea about
\href{https://www.authorea.com/users/55473/articles/113284-machine-learning-chapter/_show_article}{machine
learning}. Can you describe what it is
about?~}{You're wrote a chapter on Authorea about machine learning. Can you describe what it is about?~}}
{\label{719969}}\par\null
This article is part of my thesis I wrote privately on Authorea;
however, this particular piece sets up the theoretical framework and
tools my advisors and I used for my thesis which asks where do people
from the Gran Chaco region in Argentina, which is known to be endemic of
Chagas disease,~migrate in the long and medium term? We used
telecommunication and banking data to predict, infer or improve
insights.~
\par\null
The key finding was that the datasets we used are rich in geospatial and
social information to predict whether or not a human has lived in a
specific area. Our confirmed hypothesis is that, at an aggregate level,
if one migrates from area A to B, then data from the mobile usage will
show this movement from one area to the other. This, in addition to the
data showing that the outgoing calls are still pointing to the former
areas of where the user used to be (area A). ~These two findings are
very much expected from a sociological point of view. What is surprising
is that this kind of information has a very good predictive rate in
detecting past human mobility, using only data from the present.
\par\null
Finally, we used all of this data and information to build maps of
Argentina that showed which cities and towns that are not in the Gran
Chaco region that had a high volume of outgoing calls to the endemic
area.
\par\null\par\null
\subsection*{Was there a specific reason why you decided to write your
article on Authorea?~ What were some of the features you
liked?}
{\label{261274}}\par\null
Authorea is great. At its heart it is built on latex and git, which are
the best two tools that any scientific researcher needs to use for
publications. These two tools are very good on their own and were the
ones I've been using on my own. However, Authorea adds much more to it
by really enhancing the collaboration aspect to the project.
\par\null
Every author can collaborate on the web-native article in real time on
the same project and Authorea will keep track of the changes itself. It
also allows for non-editing users to add comments on the document. This
really helped me in correcting typos, grammar mistakes and all kind of
things, directly from Authorea's platform. The last nice feature I like
is the direct integration of the git repository with github.com. This
allows me to work offline as well at times where I have no Internet
connection like when traveling.
\par\null
\subsection*{You collaborated with 2 co-authors on this article. Prior to
using Authorea, what issues would you encounter when writing research
with
collaborators?}
{\label{518954}}\par\null
What's great about Authorea is that a user has almost no need to be
technically proficient in latex or git to use the platform. This is very
important for multidisciplinary projects where authors will have
different computer skills. This flexibility allows all of the authors to
work much faster than in other platforms. To add, there is absolutely no
setup needed from the user to start working, which is a huge bonus.
\par\null
\subsection*{When you're not Science-ing, what are you doing for
fun?~}
{\label{878937}}
I like a lot to play football (soccer in the US) and I also love
traveling when I can.~
\selectlanguage{english}
\FloatBarrier
\end{document}
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https://www.apmep.fr/IMG/tex/Corrige_STI_civil_Polynesie_juin_2010.tex | apmep.fr | CC-MAIN-2021-49 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-49/segments/1637964361064.69/warc/CC-MAIN-20211202024322-20211202054322-00000.warc.gz | 711,806,021 | 6,283 | \documentclass[10pt]{article}
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\renewcommand{\labelenumii}{\textbf{\theenumii.}}
\def\Oij{$\left(\text{O}~;~\vect{\imath},~\vect{\jmath}\right)$}
\def\Oijk{$\left(\text{O}~;~\vect{\imath},~\vect{\jmath},~\vect{k}\right)$}
\def\Ouv{$\left(\text{O}~;~\vect{u},~\vect{v}\right)$}
\usepackage{fancyhdr}
\usepackage[frenchb]{babel}
\usepackage[np]{numprint}
% Tapuscrit : Denis Vergès
\begin{document}
\setlength\parindent{0mm}
\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small STI génie mécanique, énergétique, civil}
\lfoot{\small{Polynésie}}
\rfoot{\small{juin 2010}}
\pagestyle{fancy}
\thispagestyle{empty}
\begin{center}\textbf{Durée : 4 heures}
\vspace{0,5cm}
{\Large \textbf{\decofourleft~Corrigé du baccalauréat STI Polynésie juin 2010~\decofourright\\[5pt]Génie mécanique, énergétique, civil}}
\end{center}
\vspace{0,5cm}
\textbf{\textsc{Exercice 1} \hfill 5 points}
\medskip
%Le plan complexe est rapporté à un repère orthonormal direct \Ouv. L'unité graphique est 2~cm.
%On note i le nombre complexe de module 1 et d'argument $\dfrac{\pi}{2}$.
\begin{enumerate}
\item %Soit (E) l'équation de la variable complexe $z$:
%\[ z^2 - 4z + 8 = 0.\]
%Résoudre l'équation (E) dans l'ensemble $\C$ des nombres complexes.
$\Delta = 16 - 4 \times 8 = 16 - 32 = - 16 = (4\text{i})^2$.
Il y a donc deux racines complexes conjuguées :
\[z_{1} = \dfrac{4 + 4\text{i}}{2}
= 2 + 2\text{i}~\text{et}~z_{2} = 2 - 2\text{i}.\]
%On considère les points A, B, C, D et K d'affixes respectives :
%\[a = 2 + 2\text{i}, \quad b = 1 + \text{i}\sqrt{3}, \quad c = 2 - 2\text{i}, \quad d = 3 -\text{i}\sqrt{3}~\text{et} ~ k = 2.\]
\item %Construction du quadrilatère ABCD.
\begin{enumerate}
\item %Déterminer la forme trigonométrique des nombres complexes $a$ et $b$.
On a $a = z_{1} = 2 + 2\text{i}$, donc $|a|^2 = 4 + 4 = 8 = \left(2\sqrt{2} \right)^2$, ce qui implique que $|a| = 2\sqrt{2}$.
En factorisant ce module on peut écrire :
$a = 2\sqrt{2}\left(\dfrac{\sqrt{2}}{2} + \text{i}\dfrac{\sqrt{2}}{2} \right)$.
De même $|b|^2 = 1 + 3 = 4 = 2^2$, d'où $|b| = 2$ et en factorisant ce module :
$b = 2 \left(\dfrac{1}{2} + \text{i}\dfrac{\sqrt{3}}{2} \right)$.
\item %Démontrer que le point K est le milieu du segment [AC] et le milieu du segment [BD].
On a $\dfrac{a + c}{2} = \dfrac{2+ 2\text{i} + 2 - 2\text{i}}{2} = 2 = k$.
De même : $\dfrac{b + d}{2} = \dfrac{1 + \text{i}\sqrt{3} + 3 -\text{i}\sqrt{3}}{2} = 2 = k$, ce qui montre que K est le milieu des segments [AC] et [BD].
\item %Placer les points A, C et K, puis construire les points B et D.
Les points A et C sont symétriques autour de (O$x$). K est donc le point d'intersection de (O$x$) et de la droite (AC).
Le point B appartient au cercle de centre O et de rayon 2 et il a pour abscisse $1$.
Il suffit donc de tracer la perpendiculaire à (O$x$) contenant le point (1~;~0) : elle coupe le cercle précédent au point d'ordonnée positive B. Il suffit ensuite de construire le symétrique de B autour de K : D.
\end{enumerate}
\item %Nature du quadrilatère ABCD.
\begin{enumerate}
\item %Démontrer que les points A, B, C et D appartiennent à un cercle dont on précisera le centre et le rayon.
On calcule AC $ = |c - a| = |2 - 2\text{i)} - 2 - 2\text{i} | = |-4\text{i}| = 4$ et BD $ = |d - b| = |3 -\text{i}\sqrt{3} - 1 - \text{i}| = |2 - 2\text{i}\sqrt{3}| = \sqrt{4 + 12} = \sqrt{16} = 4$.
On a donc KA = KC = KB = KD = 2 : les points A, B, C et D appartiennent au cercle de centre K et de rayon 2.
\item %Démontrer que le quadrilatère ABCD est un rectangle.
On a démontré que le point K est le milieu de [AC] et de [BD] : le quadrilatère ABCD est donc un parallélogramme dont les diagonales ont la même longueur : c'est un rectangle.
\end{enumerate}
\end{enumerate}
\newpage
Figure :
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\begin{center}
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\psgrid[gridlabels=0pt,subgriddiv=1,gridwidth=0.25pt,gridcolor=orange]
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\uput[ur](2,2){A}\uput[d](2,-2){C}\uput[u](1,1.732){B}\uput[dr](3,-1.732){D}\uput[ur](2,0){K}\uput[d](0.5,0){$\vect{u}$}\uput[l](0,0.5){$\vect{v}$}\uput[dl](0,0){O}
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\vspace{1cm}
\textbf{\textsc{Exercice 2} \hfill 4 points}
%\medskip
%Une urne contient 100 boules. Chacune de ces boules porte l'un des numéros 1, 2, 3, 4 ou 5.
%La répartition des boules suivant leur numéro est donnée par le tableau ci-dessous :
%\medskip
%\renewcommand{\arraystretch}{1.5}
%\begin{tabularx}{\linewidth}{|l|*{5}{>{\centering \arraybackslash}X|}}\hline
%Numéro inscrit sur la boule&1&2&3&4&5\\ \hline
%Nombre de boules&15&25&15&35&10\\ \hline
%\end{tabularx}
%\medskip
%
%Un joueur tire au hasard une boule dans cette urne. On admet que tous les tirages sont équiprobables.
\medskip
\begin{enumerate}
\item %Pour tout entier $n$ tel que $1 \leqslant n \leqslant 5$, on note $p_{n}$ la probabilité de tirer une boule numérotée $n$.
%Déterminer $p_{1},~p_{2},~p_{3},~p_{4}$ et $p_{5}$.
$p_{1} = 0,15,~p_{2} = 0,25,~p_{3} = 0,15,~p_{4} = 0,35$ et $p_{5} = 0,10$.
\item %On considère les évènements suivants :
%\setlength\parindent{5mm}
%\begin{itemize}
%\item[$\bullet~$] $A$ : \og La boule tirée porte un numéro inférieur ou égal à 3 \fg{};
%\item[$\bullet~$] $B$ : \og La boule tirée porte un numéro pair \fg.
%\end{itemize}
%\setlength\parindent{0mm}
%
%Déterminer les probabilités des évènements $A,~B,~ A \cap B$ et $A \cup B$.
$p(A) = p_{1} + p_{2} + p_{3} = 0,55$.
$p(B) = p_{2} + p_{4} = 0,6$.
$p(A \cap B) = 0,25$.
$p(A \cup B) = p(A) + p(B) - p(A \cap B) = 0,55 + 0,60 - 0,25 = 0,9$.
\item %Un jeu est défini de la façon suivante : un joueur mise 6~\euro{} puis il tire une boule de l'urne.
%\setlength\parindent{5mm}
%\begin{itemize}
%\item[$\bullet~$] Si le numéro de la boule est impair il reçoit une somme de 11~\euro{};
%\item[$\bullet~$] si le numéro de la boule tirée est pair il ne reçoit rien.
%\end{itemize}
%\setlength\parindent{0mm}
%On désigne par $X$ la variable aléatoire qui à chaque tirage associe le gain (éventuellement négatif) du joueur.
\begin{enumerate}
\item %Déterminer la loi de probabilité de la variable aléatoire $X$
On a le tableau suivant :
\medskip
\begin{tabularx}{\linewidth}{|*{7}{>{\centering \arraybackslash}X|}}\hline
numéro&1&2&3&4&5\\ \hline
$X$&5&$-6$&5&$-6$&5\\ \hline
$p_{i}$&0,15&0,25&0,15&0,35&0,10\\ \hline
\end{tabularx}
\medskip
\item %Calculer l'espérance mathématique E($X$) de la variable aléatoire $X$.
E$(X) = 5(0,15 + 0,15 + 0,2) - 6(0,25 + 0,45) = -1,60$~(\euro).
\item %On modifie la règle du jeu, la mise reste identique.
%\setlength\parindent{5mm}
%\begin{itemize}
%\item[$\bullet~$] Si le numéro de la boule tirée est impair il reçoit la somme de $a$ euros ;
%\item[$\bullet~$] si le numéro de la boule tirée est pair il ne reçoit rien.
%\end{itemize}
%\setlength\parindent{0mm}
%
%Déterminer la valeur du nombre $a$ pour que le jeu soit équitable.
Il suffit de remplaçer dans le calcul précédent 5 par la valeur $a - 6$. D'où :
E$(X) = (6 - a) \times (0,15 + 0,15 + 0,2) - 6(0,25 + 0,45) = 0,4a - 6$.
Le jeu est équitable si l'espérance de gain (et de perte) est nulle.
Donc E$(X) = 0 \iff 0,4a - 6 = 0 \iff a = 15$.
\end{enumerate}
\end{enumerate}
\vspace{1cm}
\textbf{\textsc{Problème} \hfill 11 points}
%\medskip
%On note $f$ la fonction définie sur l'intervalle $]0~;~+ \infty[$ par :
%\[f(x) = \dfrac{2\ln x}{x} - 2x + 4.\]
%On note $\mathcal{C}$ la courbe représentative de la fonction $f$ dans un repère orthonormal \Oij.
%L'unité graphique est 2~cm sur chacun des axes.
\medskip
\textbf{Partie A : Étude d'une fonction auxiliaire}
\medskip
%On note $g$ la fonction définie sur l'intervalle $]0~;~+ \infty[$ par :
%\[g(x) = 1 - \ln x - x^2\]
\begin{enumerate}
\item %Étudier les variations de la fonction $g$ sur l'intervalle $]0~;~+ \infty[$ (les limites ne sont pas demandées).
$g$ somme de fonctions dérivables sur $]0~;~+ \infty[$ est dérivable sur cet intervalle et :
$g'(x) = - \dfrac{1}{x} - 2x < 0$ comme somme de deux termes négatifs. La fonction $g$ est donc décroissante sur $]0~;~+ \infty[$.
\item %Étude du signe de $g$
\begin{enumerate}
\item %Calculer $g(1)$.
$g(1) = 1 - \ln 1 - 1^2 = 1 - 1 = 0.$
\item %En déduire le signe de $g(x)$ sur l'intervalle $]0~;~+ \infty[$.
$g$ étant strictement décroissante sur $]0~;~+ \infty[$ et s'annulant en 1, on en déduit que :
\setlength\parindent{5mm}
\begin{itemize}
\item[$\bullet~$] $g(x) > 0$ si $0 < x < 1$ ;
\item[$\bullet~$] $g(1) = 0$ ;
\item[$\bullet~$] $g(x) < 0$ si $1 < x $.
\end{itemize}
\setlength\parindent{0mm}
\end{enumerate}
\end{enumerate}
\bigskip
\textbf{Partie B : Étude de la fonction} \boldmath $f$ \unboldmath
\begin{enumerate}
\item %Étude des limites
\begin{enumerate}
\item %Déterminer la limite de $f$ en $0$. Que peut-on déduire graphiquement pour la courbe $\mathcal{C}$ ?
On sait que $\displaystyle\lim_{x \to 0} \ln x = - \infty$, donc $\displaystyle\lim_{x \to 0} \dfrac{2\ln x}{x} = - \infty$, donc finalement : $\displaystyle\lim_{x \to 0} f(x) = - \infty$.
Graphiquement : l'axe des ordonnées est asymptote au graphe de la fonction au voisinage de zéro.
\item %Déterminer la limite de $f$ en $+ \infty$.
On sait que $\displaystyle\lim_{x \to + \infty} \dfrac{\ln x}{x} = 0$ et $\displaystyle\lim_{x \to + \infty} - 2x = - \infty$, donc finalement $\displaystyle\lim_{x \to + \infty} f(x) = - \infty$.
\end{enumerate}
\item %Étude d'une asymptote
\begin{enumerate}
\item %Démontrer que la droite $\mathcal{D}$ d'équation $y = -2x + 4$ est asymptote à la courbe $\mathcal{C}$ au voisinage de $+\infty$.
Soit $d$ la fonction définie sur $]0~;~+ \infty[$ par $d(x) = f(x) - (- 2x + 4) = \dfrac{2\ln x}{x}.$
On a vu que $\displaystyle\lim_{x \to + \infty} \dfrac{\ln x}{x} = 0$, donc $\displaystyle\lim_{x \to + \infty} d(x) = 0$.
Ceci montre que la droite $\mathcal{D}$ d'équation $y = -2x + 4$ est asymptote à la courbe $\mathcal{C}$ au voisinage de $+\infty$.
\item %Déterminer la position relative de la courbe $\mathcal{C}$ et de la droite $\mathcal{D}$.
Plus précisèment comme $\ln x > 0$ et $x > 0$ au voisinage de plus l'infini on a donc $\dfrac{2\ln x}{x}> 0$, ce qui signifie que la fonction $d$ est positive, ou encore que la courbe $\mathcal{C}$ est au dessus de la droite $\mathcal{D}$ au voisinage de $+\infty$.
\end{enumerate}
\item %On désigne par $f'$ la dérivée de la fonction $f$ sur l'intervalle $]0~;~+ \infty[$.
\begin{enumerate}
\item %Calculer $f'(x)$ pour tout nombre réel $x$ appartenant à l'intervalle $]0~;~+ \infty[$, puis démontrer que : $f'(x) = \dfrac{2g(x)}{x^2}$.
Le premier terme est un quotient de fonctions dérivables sur $]0~;~+ \infty[$ le dénominateur ne s'annulant pas sur cet intervalle ; $f$ est donc dérivable et sur cet intervalle :
$f'(x) = \dfrac{\frac{2}{x}\times x - 2\ln x}{x^2} - 2 = \dfrac{2 - 2\ln x}{x^2} - 2 = \dfrac{2 - 2\ln x - 2 x^2}{x^2} = \dfrac{2\left(1 - \ln x - x^2 \right)}{x^2} = \dfrac{2g(x)}{x^2}$.
\item %En déduire le sens de variation de la fonction $f$ sur l'intervalle $]0~;~+ \infty[$.
Comme $2 > 0$ et $x^2 > 0$ sur $]0~;~+ \infty[$, le signe de $f'(x)$ est celui de $g(x)$ vu à la partie A.
On a donc
\setlength\parindent{5mm}
\begin{itemize}
\item[$\bullet~$] $f'(x) > 0$ si $0 < x < 1$ ; $f$ est croissante
\item[$\bullet~$] $f'(1) = 0$ ; $f$ a un maximum ;
\item[$\bullet~$] $f'(x) < 0$ si $1 < x $ ; $f$ est décroissante.
\end{itemize}
\setlength\parindent{0mm}
\item %Dresser le tableau de variations de la fonction $f$ sur l'intervalle $]0~;~+ \infty[$.
\medskip
\psset{unit=1cm}
\begin{center}
\begin{pspicture}(7,2.5)
\psframe(7,2.5)
\psline(0,2)(7,2)\psline(1,0)(1,2.5)
\uput[u](0.5,2){$x$}\uput[u](1.15,2){$0$}\uput[u](4,2){$1$}\uput[u](6.6,2){$+\infty$}
\rput(0.5,1){$f(x)$}
\psline{->}(1.4,0.4)(3.5,1.7)\psline{->}(4.5,1.7)(6.5,0.4)
\uput[u](1.4,0){$-\infty$}\uput[d](4,2){2}\uput[u](6.5,0){$-\infty$}
\end{pspicture}
\end{center}
\end{enumerate}
\item %Démontrer qu'il existe une tangente $\mathcal{T}$ à la courbe $\mathcal{C}$ qui est parallèle à la droite $\mathcal{D}$.
La droite $\mathcal{D}$ a un coefficient directeur égal à $-2$. Une tangente à $\mathcal{C}$ au point d'abscisse $x$ est parallèle à $\mathcal{D}$ si son coefficient directeur $f'(x)$ est égal à $-2$, soit :
$f'(x) = - 2 \iff \dfrac{2\left(1 - \ln x - x^2 \right)}{x^2} = - 2 \iff 2 - 2\ln x - 2x^2 = - 2x^2 \iff 2 - 2\ln x = 0 \iff 1 - \ln x = 0 \iff 1 = \Ln x \iff \ln \text{e} = \ln x \iff \text{e} = x$ (d'après la croissance de la fonction $\ln$.
Vérification : $f'(\text{e}) = \dfrac{2\left(1 - \ln \text{e} - \text{e}^2 \right)}{\text{e}^2} = - 2$.
\item %Construire dans le repère \Oij{} les droites $\mathcal{T}$ et $\mathcal{D}$, puis la courbe $\mathcal{C}$.
Voir à la fin.
\end{enumerate}
\bigskip
\textbf{Partie C : Calcul d'une aire}
\begin{enumerate}
\item %Calculer $f(2)$ et en déduire le signe de $f(x)$ sur l'intervalle [1~;~ 2].
$f(2) = \dfrac{2\ln 2}{2} - 2\times 2 + 4 = \ln 2$.
Sur [1~;~2] la fonction $f$ est décroissante de $2$ à $\ln 2$, donc sur [1~;~2], $f(x) > 0$.
\item %On note $F$ la fonction définie sur l'intervalle $]0~;~+ \infty[$ par :
%\[F(x) = (\ln x)^2 - x^2 + 4x.\]
\begin{enumerate}
\item %Démontrer que $F$ est une primitive de la fonction $f$ sur l'intervalle $]0~;~+ \infty[$.
La fonction $F$ est la somme de fonctions dérivables sur $]0~;~+ \infty[$ ; elle est donc dérivable sur cet intervalle et :
$F'(x) = 2\ln x \times \dfrac{1}{x} - 2x + 4 = \dfrac{2\ln x}{x} - 2x + 4 = f(x)$.
$F$ est donc une primitive de $f$ sur $]0~;~+ \infty[$.
\item %On note $\mathcal{A}$ l'aire, exprimée en cm$^2$, du domaine plan compris entre l'axe des abscisses, la courbe $\mathcal{C}$ et les droites d'équation $x = 1$ et $x = 2$.
On a vu que sur [1~ ;~2], $f(x) > 0$, donc l'aire $\mathcal{A}$ est égale en unités d'aire à l'intégrale :
$\mathcal{A} = \displaystyle\int_{1}^2 f(x)\:\text{d}x = \left[F(x) \right]_{1}^2 = F(2) - F(1) = (\ln 2)^2 - 2^2 + 4\times 2 - \left[(\ln 1)^2 - 1^2 + 4\times 1 \right] = (\ln 2)^2 - 4 + 8 + 1 - 4 = 1 + (\ln 2)^2$.
%Déterminer la valeur exacte de $\mathcal{A}$ puis en donner la valeur arrondie au mm$^2$.
D'où $\mathcal{A} \approx \np{1,48045}$~(u. a.)
Comme l'unité d'aire vaut $2 \times 2 = 4$~cm$^2$, on a $\mathcal{A} \approx \np{5,921} \approx 5,92$~(cm$^2$) au mm$^2$ près. Ce que l'on vérifie approximativement sur la figure.
\end{enumerate}
\end{enumerate}
\medskip
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\begin{titlepage}
\vspace*{3cm}
\begin{center}
{\Large \bf Entropy Density for Relativistic Quantum Field Theory}\\[50pt]
Heide Narnhofer \\
Institut f\"ur Theoretische Physik \\
Universit\"at Wien
\vfill
Dedicated to Elliott H. Lieb \\
to his 60th birthday \\
\vfill
{\bf Abstract} \\
\end{center}
We show how the nuclearity condition of Buchholz and Wichmann allows to
define in the ground state a local entropy with the desired properties
despite the fact that local algebras are type III. Generalization to
temperature states is also possible so that thermodynamic functions
also exist in the context of relativistic quantum field theory.
\vfill
\end{titlepage}
\section{Introduction}
Since we know that we have to assign a temperature to the background of the
universe [PW] and we let this temperature change in time in cosmological theories
it is evident that we should not restrict ourselves to consider rigorously
only the ground state of relativistic quantum fields.
Temperature states are either defined as Gibbs states. This only works for
finite systems, if the Hamiltonian has a discrete and sufficiently increasing
spectrum. In the thermodynamic limit we pass to the KMS condition as
requirement for equilibrium states [Ku,MS,HHW]. That this requirement
is the relevant one is justified by stability considerations [HKTP,PuW].
But the first path that led to this description was a variational
problem: KMS states are tangent functionals to the free energy density
[LR].
In [BJu] the KMS condition was taken to be the requirement to be satisfied
by equilibrium states for relativistic theories.
Buchholz and Junglas found an approximation
procedure so that the weak limit state satisfies the KMS condition. But
additional requirements -- the nuclearity -- are necessary. They
essentially replace the condition for Gibbs states, that local
Hamiltonians must have discrete spectrum. What is missing in this approach
is a feeling, how phase transitions can occur. Here the description of
an equilibrium state as tangent plane of a convex functional seems more
promising. But to do so, we must find a reasonable definition of this
functional as energy density minus entropy density.
In this paper we will concentrate on the problems and the possibility to
define an entropy density and we will notice that as in the construction
of KMS states the nuclearity condition is the key property that allows
to estimate local entropy and entropy density.
\section{Local Algebras and Their Entropy}
If we want to get an entropy density corresponding to a state, we have to
assign an entropy to a local algebra$\A_\Lambda$ with appropriate convexity
properties on the state. The entropy density for a space translationally
invariant state is given by $\lim S(\A_\Lambda,\omega)/|\Lambda|$.
The minimal requirement on $S(\A_\Lambda,\omega)$ is that this procedure
works for the ground state and there gives an entropy density 0.
The candidate for a local algebra $\A_\Lambda$ is the algebra over the
double cone $\cO$ with $\cO \cap {\bf R}^3 = \Lambda$. In the example of
a free scalar field theory we have
$$
\A_\Lambda = \{ W(f)|f \in \cL(\cO)\}'' \eqno(2.1)
$$
with
$$
W(f) = \exp\{ i[a^\dg(f) + a(f)]\}
$$
$$
\cL(\cO) = \frac{1}{\sqrt{\omega(p)}} \D(\Lambda) + i\,
\sqrt{\omega(p)} \, \D(\Lambda)
$$
with $\D(\Lambda)$ the space of real test functions with support in
$\Lambda$, the basis of $\cO$. A general quasifree state can be written
as
$$
\omega(W(f)) = \exp[ - \langle f|A|f\rangle] \eqno(2.2)
$$
$A$ an operator on $\cL({\bf R}^3)$. If we calculate the entropy of the
vacuum state restricted to $\A_\Lambda$, which is determined by
$P_0 A P_0$, $P_0$ the projection onto $\cL(\cO)$, then it turns out to be
infinite [A]. We can become more concrete: [BiW] showed that the algebra
of the wedge has as modular automorphism group for the vacuum state
-- which must exist
according to the Reeh-Schlieder theorem -- the boost. Boosts are
evidently weakly asymptotically abelian, therefore the spectrum of the
modular operator $\Delta$ is ${\bf R}_+$ and the algebra of the wedge
is of type III$_1$ [D,C1]. In [H,L] the authors managed to construct a map from
the algebra over the wedge onto the algebra over a double cone, provided
the system is conformally invariant. Thus also the algebra of double cones
is for conformally invariant theories of type III$_1$. Finally [F] took
into account that a reasonable quantum field theory should have a
scaling limit and since in this scaling limit the edge of a double cone
looks like the edge of a wedge he showed that the assumption on the scaling
limit guarantees that the spectrum of the modular operator for the double
cone coincides with that of the wedge so that the algebra of the double
cone is again type III$_1$.
As a consequence we run into problems if we want to calculate the entropy
of a state restricted to a local algebra, since no corresponding density
matrix is available.
But we can use the alternative definition [NT,CNT] that was shown to
coincide for type I algebras with the usual definition. It uses the
relative entropy $S(\omega|\nu)$ which can be defined for arbitrary
states $\omega$, $\nu$ over $C^*$-algebras [Ko].
\paragraph{Definition:} Let $\omega$ be a state over $\A$. Then
$$
S_\A(\omega) = \sup_{\omega = \sum \omega_i,\wh \omega_i =
\omega_i/(\omega_i(1))}
\sum \omega_i(1) S(\omega| \wh \omega_i)_\A \eqno(2.3)
$$
where $\omega$ is decomposed into states $\wh \omega_i$.
\paragraph{Lemma (2.4):} For every hyperfinite type III$_1$ algebra $\A$
$S_\A(\omega) = \infty$ [OP, Lemma 6.9].
\paragraph{Proof:} We consider the realization of the hyperfinite III$_1$
algebra [C1]
$$
\A = \bigotimes_{i=1}^\infty M_i^3 \qquad M_i \; 3 \times 3
\mbox{ matrices}
$$
with the state $\omega$ given as the limit $N \ra \infty$ of local states corresponding
to density matrices
$$
\rho_N = \bigotimes_{i=1}^N \left( \ba{ccc}
e^{-\lambda_1} & & \\
& e^{-\lambda_2} & \\
& & e^{-\lambda_3} \ea \right)
$$
where $\lambda_1$, $\lambda_2$, $\lambda_3$ are incommensurable in the
sense that
$$
\mbox{spec } \ln \Delta = \ol{\{ z_1(\lambda_1 - \lambda_2) + z_2(\lambda_2
- \lambda_3) + z_3(\lambda_3 - \lambda_1); z_i \in Z\}} = {\bf R}.
$$
For this algebra and this state
$$
\M_0 = \bigotimes_{i=0}^\infty \left( \ba{ccc}
a_{i1} && \\
& a_{i2} & \\
&& a_{i3} \ea \right)
$$
form the centralizer. We can use elements of $\M_0$ to give a
decomposition
$$
S_\A(\omega) \geq S_{\M_0}(\omega) = \lim_{N \ra \infty} N \cdot
\sum_{i=1}^3 \lambda_i e^{-\lambda_i} = \infty.
$$
For general states $\vp$ we can use the homogeneity of type III$_1$, i.e.
the following [CS,N]: Given two states $\omega$, $\vp$. Then there
exist sequences of unitaries in $\A$, $\{ U_n\}$ and $\{ V_n\}$ such
that
\beqan
\omega(\cdot) &=& n \lim \vp (U_n \cdot U_n^*) \\
\vp(\cdot) &=& n \lim \omega (V_n \cdot V_n^*).
\eeqan
We pick a decomposition of unity in $\A'$, $1 = \sum x_i$,
$0 \leq x_i \leq 1$, that approaches the optimal decomposition
for $\omega$ with $\omega_i(\cdot) = \langle \Omega|x_i \cdot |\Omega
\rangle$. We denote $\vp_i(\cdot) = \langle \Phi|x_i \cdot |\Phi\rangle$.
Then
\beqan
S(\omega|\omega_i)_\A &=& S(\omega(V_n\cdot V_n^*)|\omega_i(V_n \cdot
V_n^*))_\A \geq S(\vp|\vp_i)_\A \\
&=& S(\vp(U_n \cdot U_n^*)|\vp_i(U_n \cdot U_n^*))_\A
\geq S(\omega|\omega_i)_\A
\eeqan
where we used the semicontinuity of the relative entropy following from
[Ko], [OP].
These inequalities become equalities and imply
$$
S(\omega) \geq S(\vp) \geq S(\omega) = \infty
$$
for all states $\vp$ over $\A$.
We have to look for an alternative definition for a local entropy and this
is offered by [C2], [CNT], [NT]:
\paragraph{Definition (2.5):}
$$
H_\M(\omega,\A) = \sup_{\omega = \sum \omega_i} \sum \omega_i(1)\;
S(\omega|\wh \omega_i)_\A
$$
where now $\omega_i$ has to be a state over $\M \supset \A$.
\paragraph{Corollary (2.6):} If $\M = \B(\Ha)$, i.e. $\omega$ is pure over
$\M$, then
$$
H_\M (\omega,\A) = 0 \qquad \forall \, \A \subset \M.
$$
This holds if $\omega$ is the ground state and therefore the definition
satisfies our requirement.
It might be desirable to consider also $H_{\A_{\bar \Lambda}}(\omega,
\A_\Lambda)$. This should give us information if we construct KMS states
as limit of states, locally arranged on $\A_{\bar\Lambda}$. Especially
we will study in the sequel the following quantity:
Let $\Lambda \subset \bar\Lambda$ correspond to two double cones. Consider
$$
H_\omega(\Lambda,\bar\Lambda) = \sup_{\omega=\sum \omega_i, \omega,
\omega_i \,{\rm states\, over}\, \A(\bar\Lambda)}
\sum \omega_i(1)\; S(\omega|\wh \omega_i)_{\A_\Lambda}.
$$
We will prove that under the nuclearity condition [BW] for the vacuum
$$
\lim_{|\bar\Lambda| \ra \infty} H_\omega(\Lambda,\bar\Lambda) =
H_\M(\omega,\A_\Lambda) = 0
$$
and we will consider whether
$$
\lim_{\Lambda \ra \infty, |\Lambda|/|\bar\Lambda| \ra 1}
\frac{H_\omega(\Lambda,\bar\Lambda)}{|\Lambda|} = 0
$$
so that the difference between $H(\Lambda,\bar\Lambda)$ and
$H_\M(\A_\Lambda)$ only is a surface effect and we will see that this
is true for massive theories. For temperature states and massive theories
we will get
$$
\limsup_{\Lambda \ra \infty,|\Lambda|/|\bar \Lambda| \ra 1}
\frac{H_\omega(\Lambda,\bar \Lambda)}{|\Lambda|} < \infty.
$$
\section{Bounds on the Entropy in the Ground State}
In order to work with $H_\B(\omega,\A)$ we must find some control over
this quantity. Lower bounds are available by trying decompositions.
Simple upper bounds are consequences of the monotonicity
$$
H_\B(\omega,\A) \leq H_{\Un{\B}}(\omega,\bar\A) \quad \mbox{for }
\Un{\B} \subset \B, \; \bar\A \supset \A.
\eqno(3.1)
$$
Therefore
$$
H_\B(\omega,\A) \leq \inf \{ S_\omega(\A),S_\omega(\B)\}.
$$
In our case these estimates are useless, since both
$\B = \A_{\bar\Lambda}$ and $\A = \A_\Lambda$ are type III$_1$.
But from (3.1) we already get a hint how to find an upper bound:
\paragraph{Theorem (3.2):} Let $\A_\Lambda \subset \A_{\bar\Lambda}$
be split in the sense [DL] that there exists a type I algebra $\N$
such that $\A_\Lambda \subset \N \subset \A_{\bar\Lambda}$. If
$S_\omega(\N) < c$, then
$$
H_{\A_{\bar\Lambda}}(\omega,\A_\Lambda) < c.
$$
It is known [BD'AF] that the split property follows from the nuclearity
condition, and this nuclearity condition will also supply us with
explicit upper bounds on the entropy. It is an open problem whether the
split property is also necessary for the entropy to become finite.
We will give some observations that support this suspicion.
\paragraph{Lemma (3.3):} Let $H_\B(\omega,\A) < \infty$. Then to every
$\ve > 0$ there exists an $\bar\omega$, normal with respect to $\omega$,
such that
$$
H_\B(\bar\omega,\A) < \ve
$$
and
$$
\sum |\bar \omega(x_i a_i) - \bar\omega(x_i) \bar\omega(a_i)| < \ve
$$
for all $0 \leq x_i \leq 1$, $a_i \in \A$, $\| a_i \| = 1$, $\sum x_i = 1$,
$x_i \in \B'$.
\paragraph{Proof:} There exists a decomposition $\{y_i,0 \leq i \leq N\}$,
$y_i \in \B'$, such that
$$
H_\B(\omega,\A) \leq \sum_{i=1}^N \omega(y_i) S(\omega|\wh \omega(y_i
\cdot))_\A + \ve.
$$
A refinement can improve only up to $\ve$, such that
$$
\sum_{i,j} \omega(y_{ij}) S(\omega|\wh \omega(y_{ij} \cdot)) \leq
\sum_i \omega(y_i) S(\omega|\wh \omega(y_i \cdot)) + \ve.
$$
Now
$$
\sum_{i,j} \omega(y_{ij}) S(\omega|\wh \omega(y_{ij} \cdot)) =
\sum_i \omega(y_i) S(\omega|\wh \omega(y_i \cdot)) +
\sum_i \omega(y_i) \sum_j \frac{\omega(y_{ij})}{\omega(y_i)}
S(\wh \omega_i | \wh \omega(y_{ij}).
$$
Therefore
$$
\min_i \sum_i \frac{\omega(y_{ij})}{\omega(y_i)}
S(\wh \omega_i | \wh \omega_{ij})_\A < \ve
$$
and we can take the corresponding $\wh \omega_i$ as $\bar \omega$ with
$$
H_\B(\bar\omega,\A) < \ve.
$$
The estimate on $\bar\omega$ follows because for all possible decompositions
[OP]
$$
\ve > \sum \bar\omega (x_i \cdot) S(\bar \omega|\wh{\bar \omega}(x_i
\cdot))_\A \geq \frac{1}{2}
\sum \bar\omega(x_i) \|\bar \omega(\cdot) - \wh{\bar \omega}(x_i \cdot)\|^2.
$$
The condition on $\bar \omega$ should be compared with the following
fact [DL]:
\begin{enumerate}
\item[] $\A_\Lambda$, $\A_{\bar\Lambda}$ is split, if $\A_\Lambda \vee
\A'_{\bar\Lambda}$ is isomorphic to
$\A_\Lambda \otimes \A'_{\bar\Lambda}$,
and this is the case,
if there exists a state with
$\| \bar\omega - \bar \omega \otimes \bar\omega\| < \ve$ (see [DL], remarks
after Def. 1.4 and Prop. 1.5 for the exact statement) (compare with
Theorem 3.7).
\end{enumerate}
Another observation that supports the importance of the split property is
\paragraph{Theorem (3.4):}
$$
H_\B(\omega,\A) \leq S(\omega \otimes \omega|\omega)_{\A \otimes \B'}.
$$
Here $\B'$ is the commutant of $\B$ in the GNS representation corresponding
to $\omega$. $\A \otimes \B'$ is understood to be the algebra built by the
elements $\sum_{n=1}^N a_n b_n$, $a_n \in \A$, $b_n \in \B'$, so far
without topology. On this algebra the state $\omega$ is given by
$$
\omega (\sum a_n b_n) = \sum \langle \Omega|a_n b_n|\Omega\rangle
$$
whereas
$$
\omega \otimes \omega (\sum a_n b_n) = \sum \omega(a_n) \omega(b_n).
$$
The relative entropy is given by [Ko].
\paragraph{Proof:} According to [ST] we can write
$$
H_\B(\omega,\A) = \sup_{\lambda,\C} S(\omega \otimes \mu |\lambda)_{\A
\otimes \C}
$$
with $\C$ an abelian algebra and $\lambda$ a state over $\B \otimes \C$
such that $\left. \lambda\right|_\B = \omega$ and
$\left. \lambda\right|_\C = \mu$.
Due to the monotonicity property of the relative entropy it suffices to
take the supremum over finite--dimensional abelian algebras $\C$. For
such an algebra $\C$ the elements of $\B \otimes \C$ can be written as
$\bigoplus b_i$, $b_i \in \B$, $1 \leq i \leq N$. Also $\lambda =
\bigoplus \omega_i$ where $\omega_i = \omega(\wh b_i \cdot)$ with
$\wh b_i \geq 0 \in \B'$ $\forall \, i$.
Now we construct a completely positive map $\gamma: \C \ra \B' \otimes
\C$ by
$$
\gamma m_i = \sum m_i \wh b_i \otimes {\bf 1}.
$$
It is unity preserving
$$
\gamma(1) = \sum \wh b_i \otimes {\bf 1} = {\bf 1} \otimes {\bf 1}.
$$
Further
$$
\lambda(a \otimes m) = \sum \omega(a \wh b_i) m_i =
(\omega\otimes \mu) (a \otimes \gamma(m))
$$
where $\omega$ is considered as state over $\A \otimes \B'$.
Now we use the monotonicity of the relative entropy under unital
completely positive maps and obtain
\beqan
S(\omega \otimes \mu|\lambda)_{\A \otimes \C} &=&
S((\omega \otimes \omega \otimes \mu) \circ (1 \otimes \gamma)|
(\omega \otimes \mu)
\circ (1 \otimes \gamma))_{\A \otimes \C} \\
& \leq & S(\omega \otimes \omega \otimes \mu | \omega \otimes \mu)_{\A
\otimes \B' \otimes \C} \\
&=& S(\omega \otimes \omega | \omega)_{\A \otimes \B'}.
\eeqan
Taking the supremum over all $\C$ gives (3.4).
\paragraph{Remark (3.5):} Provided $\B$ is type I we get in addition an
inequality chain
$$
S_\omega(\B) \leq S(\omega \otimes \omega | \omega)_{\B \otimes \B'}
\leq 2 S_\omega(\B)
$$
where the left side becomes an equality if $\B$ is abelian and the right
side becomes an equality if $\B$ is a full matrix algebra.
\paragraph{Proof:} Assume $\B$ is abelian and discrete (otherwise
$S_\omega(\B) = \infty$). Then $\omega$ corresponds to a
density $\sum \lambda_i P_i \otimes Q_i$ for $\B \otimes \B'$,
whereas $\omega \otimes \omega$ corresponds to $\sum \lambda_i \lambda_j
P_i \otimes Q_j$. Therefore
\beqan
S(\omega \otimes \omega|\omega) &=& \sum \lambda_i \ln \lambda_i -
\mbox{Tr } \lambda_i P_i \otimes Q_i \ln \sum \lambda_i \lambda_j
P_i \otimes Q_j \\
&=& \sum \lambda_i \ln \lambda_i - \sum \lambda_i \ln \lambda_i^2 =
- \sum \lambda_i \ln \lambda_i = S_\omega (\B).
\eeqan
If instead $\B$ is a full matrix algebra, then $\omega \otimes \omega$
again corresponds to $\sum \lambda_i \lambda_j P_i \otimes Q_j$,
whereas $\omega$ corresponds to the projection
$$
\left|\sum \sqrt{\lambda_i} \, \vp_i \otimes \psi_i\right\rangle
\left\langle \sum \sqrt{\lambda_j}\, \vp_j \otimes \psi_j\right|
$$
with the eigenvectors $\vp_i$ of $P_i$ and $\psi_k$ of $Q_k$.
This leads to
\beqan
S(\omega \otimes \omega|\omega) &=& - \sum \ln (\lambda_i \lambda_j)
\langle \sqrt{\lambda_k}\, \vp_k \otimes \psi_k|
P_i \otimes Q_j| \sqrt{\lambda_\ell} \, \vp_\ell \otimes \psi_\ell\rangle \\
&=& - \sum \lambda_i \ln \lambda_i^2 = 2S.
\eeqan
The general inequality can be obtained by interpolation of the two extremal
cases.
It remains an open problem if $\Ha_\B(\omega,\A)$ can also be bounded
from below by results on the relative entropy. If this were to hold, this
would demonstrate that the split property is necessary to make
$\Ha_\B(\omega,\A)$ finite due to the following
\paragraph{Lemma (3.6):} Let $S(\omega|\vp)_\B < \infty$. Then
$$
\Pi_\omega(\B)'' \simeq \Pi_\vp(\B)''.
$$
\paragraph{Proof:} We use Kosaki's [K] formula for the relative entropy
$$
S(\omega|\vp)_\B = \sup \int \left[ \frac{\omega(1)}{1+t} -
\omega(y^*(t)y(t)) - \frac{1}{t} \vp(x(t)x^*(t))\right] \frac{dt}{t}
$$
where $x(t) = 1 - y(t)$.
To attain the supremum we must choose an $x(t)$ with st-$\lim_{t\ra 0}
\Pi_\vp(x(t)) = 0$ sufficiently fast so that the singularity $1/t^2$ does
not contribute. Now $[\omega(1) - \omega(y^*(t)y(t))]/t =
(\omega(x(t)) + \omega(x^*(t)) - \omega(x^*(t)x(t)))/t$ remains.
We can assume that $0 < x = x^* < 1$ so that we can estimate
$$
\omega(1) - \omega(y^*(t) y(t)) \geq \omega(x(t))
$$
If
$\Pi_\omega \neq \Pi_\vp$ we can find $x(t)$ with st-$\lim_{t \ra 0}
\Pi_\vp(x(t)) = 0$ but with st-$\lim_{t\ra 0} \Pi_\omega(x(t)) \neq 0$.
So either $\lim \omega(x(t)) > 0$ or there exists some operator $a$
such that $\lim \omega(a^* x(t)a) > 0$. But again st-$\lim \Pi_\vp
a^* x(t) a = 0$ and this operator can be used in the calculation of the
relative entropy to make the integral infinite.
We close the section by showing how the split property and estimates on the
relative entropy are related.
\paragraph{Theorem (3.7)} [D,L], [H]: Let $\R$ and $\bar \R$ be factors
acting on a Hilbert space $\Ha$ with a representation $\Pi$ and
$\R \subset \bar \R$. Let $\Omega$ be a vector
cyclic and separating for $\R$, $\bar \R'$, $\bar \R \wedge \R'$. Then
the following conditions are equivalent:
\begin{enumerate}
\item If $\vp_1$ is a state over $\R$ and $\vp_2$ is a state over $\bar \R'$,
then there exists a state $\vp$ such that
$$
\vp(R_1 R'_2) = \vp_1(R_1) \vp_2(R'_2) \qquad \forall \, R_1 \in \R, \qquad
R'_2 \in \bar \R'.
$$
\item There exists a vector $\eta$ cyclic and separating for $\R \vee \bar
\R'$ such that
$$
\langle \eta|R_1 R'_2|\eta\rangle = \langle \Omega|R_1|\Omega\rangle
\langle \Omega|R'_2|\Omega\rangle.
$$
\item There exists a unitary operator $W$ from $\Ha$ to $\Ha \otimes \Ha$
such that
$$
W \Pi(AB') W^* = \Pi(A) \otimes \Pi(B').
$$
\item There exists an intermediate type I factor $\N$
$$
\R \subset \N \subset \bar \R
$$
namely,
$$
\N = W^* \B(\Ha) \otimes 1 \; W.
$$
This property is called split property.
\end{enumerate}
\paragraph{Lemma (3.8):} If $S(\omega \otimes \omega|\omega)_{\R \vee \bar \R'}
< \infty$ for some state $\omega$ over $\bar \R$, then $\R \subset \bar \R$ are split.
\paragraph{Proof:} This is an immediate consequence of (3.6) and (3.7.2).
If therefore $\A_\Lambda \subset \A_{\bar\Lambda}$ are split, we can
consider the type I sequence of algebras $\N_\Lambda$ and can calculate
$\lim [S_{\N_\Lambda}(\omega)]/|\Lambda|$. But neither is $\N_\Lambda$
given very explicitly -- though there exists a method to construct
$\N_\Lambda = \R \vee J_{\bar\R \wedge \R'} \R J_{\bar \R \wedge \R'}$
-- nor does a control over $S_{\N_\Lambda}(\omega)$ seem available.
Instead we will stay with $H_{\A_{\bar\Lambda}}(\omega,A_\Lambda)$ and
relate it to the nuclearity condition of [BW].
\section{The Nuclearity Condition and Entropy Estimates in the Vacuum}
In [BW] the nuclearity condition was considered to find a measure on
the energy level density in finite regions. Take the map
$\A(\Lambda) \ra \B(\Ha)$,
$$
\Theta_{\beta,\Lambda}(A) = e^{-\beta h} \; A|\Omega\rangle,
\qquad h \mbox{ the Hamiltonian} . \eqno(4.1)
$$
This map is nuclear, if there exists a sequence of linear functionals
$\vp_i \in \A(\Lambda)^*$ and vectors $\Phi_i \in \Ha$ such that if
$\Theta_{\beta,\Lambda}$ can be written as
$$
\Theta_{\beta,\Lambda}(A) = \sum_i \vp_i(A) \Phi_i
$$
then
$$
\| \Theta_{\beta,\Lambda}\|_1 =: \inf \sum \|\vp_i\| \|\Phi_i\| < \infty.
\eqno(4.2)
$$
For free massive theories in three + one dimensions this nuclearity
index was estimated in [BW] -- without ambition to become optimal --
and the bound was found
$$
\|\Theta_{\beta,\Lambda}\|_1 \leq \exp\left[ c \left(\frac{r}{\beta}
\right)^3\right] \sum_{i=1}^\infty \ln (1 - e^{-\beta m_i/2}), \qquad
r > \beta
\eqno(4.3)
$$
where $m_i$ are the masses of the theory and $r$ the radius of $\Lambda$,
with the restriction $r > m_i^{-1}$. Similar estimates are found in
[BJ] for free massless field theories and for the free electromagnetic
field
$$
\| \Theta_{\beta,\Lambda}\|_1 \leq \ba{ll}
\exp \left[ c \left( \frac{r}{\beta}\right)^3 \right] & r \geq \beta
\\[10pt]
\exp \left[ c' \left( \frac{r}{\beta}\right) \right] & r \leq \beta
\ea \eqno(4.4)
$$
always for suitable constants $c$.
It seems reasonable that similar nuclearity bounds also hold for
interacting theories.
This nuclearity bound was used to show [BD'AF] that local algebras
$\A_\Lambda \subset \A_{\bar\Lambda}$ with some distance between
$\Lambda$ and $\bar \Lambda$ satisfy the split property. A slight
variation of the arguments allow to find an upper bound on
$\Ha_{\A_{\bar\Lambda}}(\omega,\A_\Lambda)$.
\paragraph{Lemma (4.5)} [BD'AF]: Take $A \in \A_\Lambda$ and
$B' \in \A'_{\bar \Lambda}$ with $\Lambda$, $\bar \Lambda$ balls of
radius $r$ resp. $R$ and $R = r + \beta$. Then one can construct a
continuous function $f$ of the Hamiltonian $h$ such that
$$
\langle \Omega|AB'|\Omega\rangle = \langle \Omega|A f_\beta(h) B'|
\Omega\rangle + \langle \Omega | B'f_\beta(h) A| \Omega\rangle
$$
where $f_\beta$ has almost exponential decrease in the sense that
$$
\lim_{|E| \ra \infty} \exp (|E|^\kappa) |f_\beta(E)| = 0 \qquad
0 \leq \kappa < 1
$$
and $f_\beta(E) = f(E\beta)$, $f(0) = 1/2$.
Under the assumption of the nuclearity condition
$$
\| \Theta_{\beta,\Lambda}\|_s \leq \exp \left[ c r^3 \beta^{-3}
\ln (1 - e^{-\beta m}) \right] = \exp g(r,\beta) \eqno(4.6)
$$
with
$$
\| \Theta_{\beta,\Lambda}\|_s = \inf - \sum \|\vp_i\| \ln \| \vp_i\|,
\qquad \| \Phi_i\| = 1,
$$
we can use this function $f$ to prove:
\paragraph{Theorem (4.6):} Assume that the theory has a mass gap, i.e. 0
is an isolated point in the spectrum of $h$. Then
$$
H_{\A_{\Lambda_R}}(\A_{\Lambda_r}) < \delta
$$
provided $R = r + \beta$ and $\dfrac{r^3}{\beta^{3 + \ve}} < \bar\delta$
so that $\beta(r)$ can be chosen such that with $r \ra \infty$
$\delta \ra 0$ together with $\bar \delta \ra 0$.
\paragraph{Proof:} Following the construction of [BD'AF] we decompose the
map $\Theta_f : \A \ra \Ha$
$$
\Theta_f A = f_\beta(h) A|\Omega\rangle
$$
into
$$
\Theta_k A = P_k f(h) e^{\gamma h} \Theta_\gamma
$$
where $P_k$ is the spectral projection of $h$ for the interval $[k-1,k)$.
We calculate the nuclearity bound for $\Theta_k$ to be
$$
\Theta_1 A = \frac{1}{2} |\Omega\rangle \langle \Omega|A|\Omega\rangle
$$
if the map gap is $ \geq 1$ (otherwise we have to scale the interval
$[k-1,k)$).
Therefore $\| \Theta_1\|_1 = 1$, $\|\Theta_1\|_s = 0$.
Next with $f_k = \sup f(E\beta)$, $E \in [k-1,k)$ we have
$$
\| \Theta_k\|_1 \leq f_k e^{\gamma k + g(r,\gamma)}
$$
resp.
$$
\| \Theta_k\|_s \leq (- f_k \ln f_k + f_k) e^{\gamma k + g(r, \gamma)}.
$$
For every $k$ we optimize $\gamma$ by
\beqan
k &=& c r^3 \gamma^{-3} \left[ - \frac{3}{\gamma} \ln (1-e^{-\gamma m})
+ \frac{e^{-\gamma m}}{1 - e^{-\gamma m}} \right] \\
&\simeq & c r^3 \gamma^{-3} \left[- \frac{3 \ln \gamma m}{\gamma} +
\frac{1}{\gamma m} \right] \qquad \mbox{for large } k
\eeqan
therefore $\gamma$ tends to 0 for large $k$.
$f_k$ and $f_k \ln f_k$ decrease almost exponentially. Therefore
$$
\| \Theta_k \|_s < \exp \left[-k^{\nu/(\nu+1)} \wh c_\beta \right]
\qquad \mbox{for some } \nu < \infty.
$$
This is summable in $k$. In addition the scaling property of $f_\beta$
allows to estimate (for large $\beta$, that correspond to small $k$ and
determine the permitted radius $R$)
$$
\wh c_\beta \approx r^{3/(\nu+1)} \beta^{- (\nu-\ve)/(\nu+1)}
$$
so that if we adjust $\beta$ appropriately to $r$
$$
\lim_{r^3/(\beta^{\nu-\ve/2}) = {\rm const},r \ra \infty} \wh c_\beta(r)
= 0. \eqno(4.7)
$$
We use this result to write
$$
\langle \Omega|AB'|\Omega\rangle =
\langle \Omega|A|\Omega\rangle \langle \Omega|B'|\Omega\rangle
+ \sum \vp_i(A) \psi_i(B')
$$
with $\vp_i$, $\psi_i$ linear functionals over $\A_\Lambda$ resp.
$\A_{\bar \Lambda}$ where $\| \vp_i\| = 1$ and
$- \sum \|\psi_i\| \ln \|\psi_i\| < \ve$ provided according to the previous
estimates $\beta$ is large enough. The left hand side is positive and
therefore also real. Every $\vp_i$ resp. $\psi_i$ can be written as
linear combination of at most four positive linear functionals
$\vp_{i\alpha}$, $\psi_{i\beta}$. For hermitean $A$, $B'$ only the
real combinations remain so that
$$
\langle \Omega|AB'|\Omega\rangle =
\langle \Omega|A|\Omega\rangle \langle \Omega|B|\Omega\rangle
+ \sum \vp_i(A) \psi_i(B') - \sum \bar \vp_i(A) \bar \psi_i(B')
$$
where separately
$$
- \sum \|\psi_i\| \ln \|\psi_i\| < \ve , \qquad
- \sum \|\bar \psi_i\| \ln \|\bar \psi_i\| < \ve, \qquad
\bar \vp_k, \; \vp_i \mbox{ normalized.}
$$
Therefore
$$
\omega + \sum_{k=1} \bar \vp_k \otimes \bar \psi_k = \omega \otimes \omega
+ \sum_{i=1} \vp_i \otimes \psi_i
$$
as positive functionals over $\A_\Lambda \otimes \A'_{\bar \Lambda}$.
Since $\B(\Ha) \supset \A_\Lambda \otimes \A'_{\bar\Lambda }$ we can
extend $\omega + \sum \bar\vp_k \otimes \bar \psi_k$ to a positive
linear functional over $\B(\Ha)$ and therefore also over $\A_{\bar\Lambda}$.
After normalization we can write this functional
$\bar \omega = (1-\ve)\omega + \ve\vp
= \sum_{i=0} \vp_i \otimes \psi_i$ where we set $\psi_0 = (1 - \ve)\omega$,
$\vp_0 = \omega$. Next we use [CNT] that for any $\A \subset \B$
$$
H_\B(\bar \omega,\A) \geq (1-\ve) H_\B(\omega,\A) + \ve H_\B(\vp,\A)
$$
and therefore
$$
H_\B(\omega,\A) \leq \frac{1}{1-\ve} H_\B(\bar\omega,\A).
$$
This will be applied for $\A = \A_\Lambda$, $\B = \A_{\bar \Lambda}$.
We use
$S(\sum \mu_i \vp_i | \sum \mu_i \omega_i) \leq
\sum \mu_i S(\vp_i | \omega_i)$ and
$S(\vp | \lambda \omega) = \lambda \ln \lambda + \lambda S(\vp | \omega)$
to estimate
\beqan
H_{\A_{\bar \Lambda}}(\bar\omega,\A_\Lambda) &=&
\sup_{\sum B_\ell =1,B_\ell \in \A'_{\bar \Lambda}} \sum
[ - \bar\omega(B_\ell) \ln \bar\omega(B_\ell) +
S(\bar \omega| \bar \omega(B_\ell \cdot))_{\A_\Lambda}] \\
&=& \sup_L \sum_\ell \left[ - \sum_k \psi_k(B_\ell) \ln \sum_k \psi_k(B_\ell)
+ S(\sum_k \vp_k \psi_k(1)| \sum_k \vp_k \psi_k(B_\ell))_{\A_\Lambda}\right] \\
&\leq& \sup_L \left( S_L + \sum_{\ell,k} \psi_k(1)
S\left( \vp_k|\vp_k
\frac{\psi_k(B_\ell)}{\psi_k(1)}\right)_{\A_\Lambda}\right)
\eeqan
where we refined the decomposition of $\bar \omega$ given by $B_\ell$ further
into $ \sum_{k,\ell} \vp_k \cdot \psi_k
(B_\ell)$. Then we can continue
$$
= \sup_L (S_L - S_{KL} + S_K) \leq 2S_K = - 2 \sum \|\psi_k\| \ln
\|\psi_k\|
$$
where $S_L$, $S_{KL}$, $S_K$ are the entropies of the abelian models
corresponding to the decompositions $\bar \omega = \sum_\ell \bar \omega
(B_\ell \cdot)$, $\bar \omega = \sum_{k,\ell} \vp_k \psi_k(B_\ell \cdot)$,
$\bar \omega = \sum_k \vp_k \psi_k$.
\paragraph{Remark:} To obtain the desired result it was necessary that in
(4.6) we need $\nu > n$ which asks for $m > 0$.
This does not hold for massless theories. Also to
control $\| \Theta\|_1$ the mass gap was necessary. Otherwise the
estimate (4.7) guarantees
\paragraph{Theorem (4.8):} For massive theories with corresponding
nuclearity bound (4.3)
$$
\lim_{|\Lambda|/|\bar\Lambda| \ra 1,\Lambda \ra {\bf R}^3}
\frac{H_{\A_{\bar\Lambda}}(\omega,\A_\Lambda)}{|\Lambda|} = 0
$$
with $\bar \Lambda$ larger than $\Lambda$ by slightly more than a surface
effect.
\paragraph{Theorem (4.9):} For massless theories with nuclearity bound
(4.6) with $m = n$ we have
$$
\lim_{|\bar \Lambda| \ra \infty} H_{\A_{\bar\Lambda}}(\omega,\A_\Lambda)
= 0.
$$
This follows from the same estimates with considering $\beta \ra \infty$
for fixed $\Lambda$.
\section{Entropy for Temperature States}
Let us first assume that a KMS state exists. We want to evaluate estimates
on the local entropy. Theorem (4.6) can be replaced by
\paragraph{Theorem (5.1):} Let $\omega$ be a KMS state to the temperature
$\beta$. Let $\Omega_\beta$ be the corresponding GNS vector. Assume that
the following nuclearity condition holds: Let
$$
e^{-\gamma h} A|\Omega\rangle = \tau_{i\gamma} A|\Omega\rangle =
\sum \vp_i (A) \Phi_i, \qquad \| \Phi_i\| =1,
$$
for all $A \in \A(\Lambda)$
with $\sum \|\vp_i\| \ln \|\vp_i\| \leq c_\gamma(\Lambda)$.
If $\gamma < \beta/2$, then
$$
H_\M(\omega,\A_\Lambda) \leq c_\gamma(\Lambda).
$$
\paragraph{Remark:} The above estimate seems to be optimal for
$\gamma = \beta/4$, if we check it for lattice systems or the free
massive case (compare the Appendix). Also for lattice systems we lose
control on the nuclearity estimate for $\gamma > \beta/2$, because
$\Omega_\beta$ does not necessarily remain in the domain of
$\tau_{i\gamma}(A)$ (which can be unbounded). So we must be aware that
$c_\gamma \ra \infty$ for $\gamma \ra \beta/2$ [BD'AL].
\paragraph{Proof:} $\omega_\beta$ has to be decomposed by $x_k \in \M'$.
Thus
$$
\langle \Omega_\beta|x_k A|\Omega_\beta \rangle =
\langle \Omega_\beta|y_k \tau_{i\beta/2} A|\Omega_\beta\rangle
= \sum \vp_i(A) \langle \Omega_\beta| \tau_{-i(\beta/2 - \gamma)} y_k
| \Phi_i\rangle
$$
with $x_k \in \M'$, $y_k \in \M$, $\| y_k\| \leq 1$.
The rest is copying the previous proof taking into account that
$$
|\langle \Omega_\beta|\tau_{i\gamma}y|\Phi_i\rangle | \leq
\langle \Omega_\beta| y e^{-2\gamma h}y|\Omega_\beta\rangle^{1/2}
\leq \langle \Omega_\beta | y^2| \Omega_\beta\rangle^{1/2},
\qquad
0 \leq \gamma \leq \beta/2,
$$
because $\langle \Omega_\beta | y e^{-2\gamma h}y|\Omega_\beta\rangle$
is a convex function on $\gamma$ with $f(0) = f(\beta/2)$ and
$f(\gamma) \leq (f(0) + f(\beta/2))/2$ for $0 \leq \gamma \leq \beta/2$.
Again we can be interested in $H_{\A_{\bar\Lambda}}(\A_\Lambda)$ to
control surface effects. We have to modify [BD'AF] slightly:
\paragraph{Lemma (5.2):} Take $A \in \A_\Lambda$ and $B \in
\A'_{\bar\Lambda}$ (in the GNS representation corresponding to
$\omega_\beta$). Then there exists $f_{\beta c}(|h|)$, $h$ implementing
the time evolution in the GNS representation, with almost
exponential decrease such that
$$
\langle \Omega_\beta |AB|\Omega_\beta\rangle =
\langle \Omega_\beta |A f_{\beta c}(|h|)B|\Omega_\beta\rangle
+ \langle \Omega_\beta |B f_{\beta c}(|h|)A|\Omega_\beta\rangle
+ \sum \vp_i(A) \psi_i(B)
$$
with $\sum \| \vp_i\|_s < \infty$, $\| \psi_i\| = 1$.
\paragraph{Proof:} We can mimic the proof of [BD'AF] expressing $f$ as a
contour integral. Under the above assumption the function
$$
g(t) = \frac{1}{2}\langle \Omega_\beta |A e^{iht} B|\Omega_\beta\rangle
+ \frac{1}{2} \langle \Omega_\beta | B e^{iht} A| \Omega_\beta\rangle
$$
is analytic in the strip
$I = \{ z = t + i\gamma, t \in {\bf R}, |\gamma| \leq \beta/2\}$
and with a cut for $|t| > c_{\bar \Lambda}$.
The boundary stems from $B \in \M'$, the cut stems from the fact that
$[\tau_t A,B]$ no longer vanishes for $B \in \A'_{\bar\Lambda} \cap \M$,
if $|t| > c_{\bar\Lambda}$, and $c_{\bar\Lambda}$ depends on the
distance between $\Lambda$ and $\bar\Lambda$ (compare [BD'AF] and
[H]).
Again we make the variable transformation $z = 2w/\beta(w^2+1)$ such that
$z(0) = 0$. Therefore
$$
g(0) = \frac{1}{2\pi i} \oint g(z(w)) \frac{dw}{w}
$$
can be evaluated as integral over the boundary of the analytic region in
$w$. The contribution along the cut in $z$ remains the unit circle as for
the vacuum. The contribution of the boundary in $z$ can be shifted into the
interior $|\gamma| < \beta/2$. In the $w$--plane it corresponds to a
curve of finite length and we can evaluate it by using the nuclearity
condition to
$$
\oint \frac{dw}{w} \langle \Omega_\beta | A e^{iht - \gamma h} B|
\Omega_\beta \rangle =
\sum_i \oint \frac{dw}{w} \wt \vp_i(A) \langle \bar \psi_i
e^{iht} B|\Omega_\beta \rangle
= \sum_i \vp_i(A) \psi_i(B).
$$
$\sum \|\vp_i\|_s < \infty$ follows because $\sum \|\wt \vp_i\|_s
< \infty$ due to
the nuclearity and the rest in the integral is uniformly bounded on the
curve. This allows us to deduce
\paragraph{Theorem (5.3):} Let the nuclearity condition for
$e^{-\gamma h}A\Omega$ be satisfied with
$$
\sum \|\vp_i\| \ln \|\vp_i\| \leq e^{-c_{\bar\Lambda,\Lambda}\gamma^{-n}}
\qquad \forall \, \gamma \leq \beta/4
$$
then
$$
H_{\A_{\bar\Lambda}}(\A_\Lambda) < \infty.
$$
\paragraph{Proof:} This follows by the same arguments as in the vacuum.
Notice that the necessary range of $\gamma$ in (4.6) covers the permitted
region in $\gamma$. Evidently now we have no mass gap, and also
$\gamma$ cannot vary arbitrarily to improve the bound. More control
how $H_{\A_{\bar\Lambda}}(\A_\Lambda)$ tends to $H_\M(\A_\Lambda)$ asks
for controlling the nuclearity condition, but even more how $f_{\beta c}$
tends to $f_\beta = \exp (- \frac{\beta}{2} |h|)$ if the cut moves to
infinity with increasing distance between $\Lambda$ and $|\Lambda|$.
Though we have no necessary conditions on the temperature state, which
are satisfying in the sense that they can be proven for massive free
fields, such that they guarantee the existence of the entropy density,
it seems desirable to find conditions that have to be satisfied in the
vacuum state and then guarantee the existence of temperature states
with finite entropy density. We follow the proposal of [BJu] to construct
KMS states.
\paragraph{Theorem (5.4)} [BJu]: Let $\A_\Lambda \subset \N_{\bar\Lambda}
\subset \A_{\bar\Lambda}$ be split with $\N_{\bar\Lambda}$ an
appropriate type I algebra. Consider the state
$\left. \omega\right|_{\N_{\bar\Lambda}} \otimes \left.\omega
\right|_{\N'_{\bar\Lambda}}$ with GNS vector $\eta$. Define the projector
$E_{\bar \Lambda}$ by
$$
E_{\bar \Lambda} \Ha = \ol{\N_{\bar\Lambda} \eta}
$$
and
$$
\omega_{\beta,\bar\Lambda}(A) = \frac{\mbox{Tr }E_{\bar\Lambda}
e^{-\beta h} E_{\bar\Lambda}A}{\mbox{Tr }E_{\bar\Lambda} e^{-\beta h}
E_{\bar\Lambda}} \qquad \forall \, A \in \ol{\bigcup_\Lambda \A_\Lambda}.
$$
Then w-$\lim_{\bar\Lambda \ra {\bf R}^3} \omega_{\beta,\bar\Lambda}$
satisfies the KMS condition.
We slightly change the definition and consider an imbedding
$$
\A_\Lambda \subset \N_{\bar\Lambda} \subset \A_{\bar\Lambda} \subset
\N_{\wh \Lambda} \subset \A_{\wh \Lambda}
$$
with $\A_{\wh \Lambda} \ra \M$ and such that
$\N_{\wh \Lambda} = \N_{\bar\Lambda} \otimes \N_{\wt \Lambda}$.
We define $\wt \eta$ to be the GNS vector corresponding to
$\left. \omega\right|_{\N_{\bar\Lambda}} \otimes \left.\omega
\right|_{\N_{\wt \Lambda}} \otimes \left.\omega\right|_{\N'_{\wh \Lambda}}$
and $E_{\wh \Lambda} \Ha = \ol{\N_{\wh \Lambda} \wt \eta}$. Again
$$
\lim \omega_{\wt \Lambda}(A) = \lim \frac{\mbox{Tr } E_{\wh \Lambda}
e^{-\beta h} E_{\wh \Lambda} A}{\mbox{Tr } E_{\wh \Lambda} e^{-\beta h}
E_{\wh \Lambda}} \equiv \omega_\beta(A)
$$
satisfies the KMS condition.
We want to use the semicontinuity [CNT]
$$
\lim H_{\omega_n}(A) \geq H_{\lim \omega_n}(A)
$$
and have therefore to estimate $H_{\omega_{\wt \Lambda}}(A)$ by a bound
independent of $\wt \Lambda$.
The trace (for $A \in \A_\Lambda \subset \N_{\wt\Lambda})$ can be written
as
$$
\frac{\sum \langle \vp_i \otimes \psi_j| e^{-\beta h}A| \vp_i \otimes
\psi_j\rangle}{\sum \langle \vp_i \otimes \psi_j|e^{-\beta h}| \vp_i
\otimes \psi_j\rangle}
$$
where
$$
|\vp_i \otimes \psi_j\rangle = |\vp_i\rangle \langle \Omega| \otimes
|\psi_j\rangle \langle \Omega|\;|\wt \eta\rangle =
U_{i1} \otimes V_{j1} |\wt \eta \rangle
$$
where $U_{i1} \in \N_{\bar\Lambda}$ and $V_{j1} \in \N_{\wt \Lambda}$.
If we write (in the interpretation of (4.3)
$\wt \eta = |\Omega\rangle \otimes |\Omega\rangle$, then we can use a
nuclearity condition on $e^{-\gamma h}A|\Omega\rangle$ for
$A \in \N_{\bar\Lambda}$, that is implemented by a nuclearity condition
for $\A_{\bar\Lambda}$, to get
\beqan
\lefteqn{\langle \Omega \otimes \Omega| V_j^* \otimes U_i^* e^{-\beta h}
V_j e^{\gamma h/2} e^{-\gamma h/2} A U_i|\Omega \otimes \Omega\rangle =}
\\
&=& \langle \Omega \otimes \Omega|V_j^* \otimes U_i^* e^{-\beta h} V_j
e^{\gamma h/2}|\Phi_k \otimes \Omega\rangle \vp_k(A U_i).
\eeqan
We assume $|\vp_k(AU_i)| \leq \|\vp_k\|$ with
$\sum \|\vp_k\| \ln \|\vp_k\| < \infty$. For the remaining term we
estimate
\beqan
\lefteqn{|\langle \Omega \otimes \Omega| V_j^* \otimes U_i^* e^{-\beta h}
V_j e^{\gamma h/2} e^{-\gamma h/2} |\Phi_k \otimes \Omega\rangle| \leq}
\\
&\leq& |\langle \Omega \otimes \Omega|V_j^* \otimes U_i^* e^{-\beta h}
U_i \otimes V_j |\Omega \otimes \Omega\rangle|^{1/2}
\cdot \langle \Phi_k \otimes \Omega| e^{\gamma h/2} V_J^* e^{-\beta h}
V_j e^{\gamma h/2} |\Phi_k \otimes \Omega\rangle|^{1/2}.
\eeqan
If we now assume that the $|\wt \Lambda|$ dependence in all terms is the
same, i.e.
$$
\sum_j \langle \Phi_k \otimes \Omega|e^{\gamma h/2} V_J^* e^{-\beta h}
V_j e^{\gamma h/2}|\Phi_k \otimes \Omega\rangle \approx
e^{c|\wt\Lambda| +c_k},
$$
$$
\sum_{i,k} \langle\vp_i \otimes \psi_k | e^{-\beta h}|\vp_i \otimes
\psi_k\rangle
\approx e^{c|\wt \Lambda|+\bar c}
$$
then the estimate on the nuclearity bound for
$\tau_{i \gamma} A|\wt \eta\rangle$ becomes independent of $|\wt \Lambda|$.
This assumption does not seem implausible. $\tau_{i\gamma}A$ should
essentially stay in a finite region and $\wt \eta$ should not be able to
connect it with distant regions. Also $e^{-\gamma} V_j e^\gamma$ should stay
inside of $\wt \Lambda$ up to boundary effects. But evidently the
possibility of phase transitions and long range correlations can be of
importance. So far it is improbable that one can deduce this $|\wt \Lambda|$
behaviour from the given nuclearity condition. Therefore we are unable
to offer a definite answer whether the entropy density of the KMS states
constructed by [BJu] remains finite.
\newpage
\appendix{\section*{Appendix}}
\newcounter{zahler}
\renewcommand{\thesection}{\Alph{zahler}}
\renewcommand{\theequation}{\Alph{zahler}.\arabic{equation}}
\setcounter{zahler}{1}
It remains to verify the necessary nuclearity condition for free field
theories. They can be found in [BD'AL] for several norms. We want to
sketch the proof and
first concentrate on the ground state. Here we succeed for massive
and massless theories. Since the considerations follow closely [BW]
and [BJ].
We write $e^{-\beta h} A|\Omega\rangle = \sum \langle \Phi_n|A|\Omega
\rangle \cdot e^{-\beta h}|\Phi_n\rangle$ with $|\Phi_n\rangle$ an
orthonormal basis that can be chosen to be
\beq
\Phi_n = \prod_{n = \{n_1,\ldots,n_p\}} \frac{(a_i^\dg)^{n_i}}
{\sqrt{n_i}}\, |\Omega\rangle .
\eeq
In [BW] it is shown: if for $G(f;\psi) = \langle e^{a^\dg(f)}\Omega
|\psi\rangle$ $\forall \, f \in L^2({\bf R}^3)$ and
$\psi \in \N = \{ e^{-h} A|\Omega\rangle,\|A\| = 1,A \in \A_\Lambda\}$
we can find a bound such that $|G(f;\psi)| \leq \exp \frac{1}{2} \|
e^{-\gamma h}f - g\|^2$ $\forall \, g \in K(O_r)'$, i.e.
functions with support outside $O_r$.
>From this it can be deduced that
\beq
|G(f;\psi)| \leq \exp \left[ \frac{1}{2} \|Tf\|^2\right]
\eeq
where $T$ is a positive trace class operator with $0 \leq T < 1$ and
eigenvalues $t_k$ corresponding to eigenvectors $e_k$, when we will use
in the construction of $\Phi_n$ $a_i^\dg = a^\dg(e_i)$ and can estimate
$$
|\langle \psi|\Phi_n\rangle | \leq \prod_{k=1}^p (n_k +1) t_k^{n_k},
\qquad n = \{ n_1,\ldots,n_p\}.
$$
So far these results are found in [BW].
Denoting $\lambda_n = \sup_{\psi \in \N} |\langle \psi|\Phi_n\rangle|$
we estimate rather roughly
\beq
- \sum \lambda_n \ln \lambda_n \leq - c \det \frac{1}{(1 - T)^3} \cdot
\mbox{Tr } T \ln T .
\eeq
Again following [BW] and [BJ] we have to estimate
$\det 1/(1-T)^3$ and Tr~$T \ln T$. They split
$$
|G(f;e^{-\beta h}A \Omega)| \leq \left[\frac{1}{2} \|T_+ \frac{1}{2}
(1+J)f\|^2 + \frac{1}{2} \|T_- \frac{1}{2}(1 - J)f\|^2 \right]
$$
with $(Jf)(p) = \ol{f(-p)}$ and $T_+ = E_r^+ e^{-\beta \omega}$ and
$T_- = E_r^- e^{-\beta \omega}$ where $E_r^\pm$ are the projections
onto $\sqrt{\omega}^{\pm 1} \wt f_r$ (compare (2.1)). Then
$$
T = \lim_{n \ra \infty} [(T_+^* T_+)^n + (T_-^* T_-)^n]^{1/2n}.
$$
For estimating Tr~$T \ln T$ a new difficulty arises: Though
$$
\| (A+B)^\alpha\|_1 \leq \|A^\alpha\|_1 + \|B^\alpha\|_1, \qquad
\alpha = \frac{1}{2n},
$$
this does not hold for $T^\alpha \ln T$, because it fails to be an
operator monotonic function for $0 < T \leq 1$. But using its integral
representation one can see
$$
\|(A+B)^\alpha \ln(A+B)\|_1 \leq \|A^\alpha \ln A\|_1 +
\|B^\alpha \ln B\|_1, \qquad \alpha = \frac{1}{2n},
$$
for $A,B < a$ and $\ln a < -1$. If therefore we can prove that
$T_\pm^* T_\pm < 1$, then $(T_\pm^* T_\pm)^n < a$ for $n$ sufficiently
large so that we come into the range of the inequality.
$T_+^* T_+ < 1$ is a consequence of the estimate [BJ]
$$
\det (1-|E e^{-\gamma}|)^3 \leq \exp 3 \sum_{n=1}^\infty \frac{1}{n}
\| E e^{-n\gamma}\|_1 < \infty.
$$
This also shows that $\det 1/(1-T)^3 < \infty$ or more explicitly [BW],
[BJ]
\beq
\det \frac{1}{(1-T)^3} \leq \ba{ll}
\exp \left[ c_1 \frac{r}{\beta} + c_2 \left(\frac{r}{\beta}\right)^3
\right] & m = 0 \\[12pt]
\exp \left[ c\left(\frac{r}{\beta}\right)^3 \sum \ln (1- e^{-\beta m_i/2})
\right] & m_i \neq 0. \ea
\eeq
Following [BW] we estimate $T \ln T$ where $T$ stands for $(T^*_\pm T_\pm)^{1/2}$
by an appropriate decomposition $T = ABC$ and using
$$
\mbox{Tr } T \ln T \leq \| C \ln T\| (\mbox{Tr }AA^\dg)^{1/2}
(\mbox{Tr } BB^\dg)^{1/2}
$$
and $e^{-\beta\omega}$ allows some splitting so that the estimates remain
essentially unchanged.
It remains to evaluate what has to be changed in a temperature state. We
restrict ourselves to the massive case. For the massless case problems
must be anticipated, because, as is argued in [BJu], it can happen that
the temperature state is not locally normal any more. We consider
$\M \vee \M'$ and interpret $\omega_\beta$ as a vacuum state of a free
massive field theory with Weyl algebra $\{\bar W(f_1,f_2)\}''$ of two
classes of particles with an
appropriate Bogoliubov transformation.
Mimicking the above estimates we consider
$$
|G(f_1,f_2; e^{-\gamma H}A)| = |\langle \Omega| e^{\bar a(f_1)+\bar b(f_2)}
e^{-\gamma H} A \Omega\rangle|
$$
where $e^{iHt}$ implements the time evolution in the GNS representation
and use
$$
|G(f_1,f_2;e^{-\gamma H} A )| \leq \exp \left[ \frac{1}{2} \| e^{-\gamma h}
\vec f - \vec g \|^2 \right]
$$
for all $\vec g \in K(O_r)'$. Notice that now $K(O_r)'$ also consists of
elements corresponding to $\M'$. $h$ implements the time evolution in
the the one--particle space, notice that it is not positive definite
anymore but we consider so far only $\vec f$ in the domain of
$e^{-\gamma h}$. We decompose $e^{- \gamma h}\vec f$ into its components
corresponding to $\M'$ which can be completely annihilated by $\vec g$
and those corresponding to $\M$. They are given in $p$--space by
$$
e^{-\gamma\omega(p)} \sqrt{ \frac{1}{1 - e^{-\beta \omega(p)}}} f_1(p)
+ e^{\gamma\omega(p)} \sqrt{ \frac{1}{e^{\beta\omega(p)} - 1}} f_2(p).
$$
If we choose $\gamma = \beta/4$ we get
$$
e^{-\beta\omega(p)/4} \left[
\sqrt{ \frac{1}{1 - e^{-\beta \omega(p)}}} f_1(p) +
\sqrt{ \frac{1}{e^{\beta\omega(p)/2} - e^{-\beta\omega(p)/2}}} f_2
\right] = e^{- \beta\omega(p)/4} A \vec f.
$$
Provided $m > 0$ the operator $A$ is a bounded operator. Therefore we
are back to the previous estimates with the replacement $|G (f;\psi)|
\leq \exp[ \frac{1}{2} \| T A f\|^2]$.
\vfill
\section*{Acknowledgement}
I am grateful to D. Buchholz who turned my interest to this problem and
to W. Thirring and D. Buchholz
who read the manuscript extremely carefully and gave a lot
of hints how to improve.
I am happy to dedicate this paper to Elliott Lieb, especially because I
remember very well when we first met at a workshop at IHES and he taught
me the importance of strong subadditivity of the entropy and how in the
control of the entropy the control of spatial correlations is encoded.
\vfill
\newpage
\bibliographystyle{abbrv}
\begin{thebibliography}{BD'AF}
\bibitem[A]{} H. Araki, Prog. Theor. Phys. {\em 32}, 956 (1964)
\bibitem[BD'AF]{} D. Buchholz, C. D'Antoni, K. Fredenhagen, Commun. Math. Phys.
{\em 111}, 123 (1987)
\bibitem[BD'AL]{} D. Buchholz, C. D'Antoni, R. Longo, Commun. Math. Phys.
{\em 129}, 115 (1990)
\bibitem[BJ]{} D. Buchholz, P. Jacobi, Lett. Math. Phys. {\em 13}, 313 (1987)
\bibitem[BJu]{} D. Buchholz, P. Junglas, Commun. Math. Phys. {\em 121}, 255
(1989)
\bibitem[BW]{} D. Buchholz, E.H. Wichmann, Commun. Math. Phys. {\em 106},
321 (1986)
\bibitem[BiW]{} J.J. Bisognano, E.H. Wichmann, J. Math. Phys. {\em 16}, 985
(1975), J. Math. Phys. {\em 17}, 303 (1976)
\bibitem[C1]{} A. Connes, Ann. Sci. Ecole Norm Sup. {\em 6}, 133 (1973)
\bibitem[C2]{} A. Connes, C.R. Acad. Sci. Paris t {\em 301} I, 1 (1985)
\bibitem[CNT]{} A. Connes, H. Narnhofer, W. Thirring, Commun. Math. Phys.
{\em 112}, 691 (1987)
\bibitem[CS]{} A. Connes, E. St\o rmer, J. Funct. Anal. {\em 28}, 187 (1978)
\bibitem[D]{} W. Driessler, Commun. Math. Phys. {\em 53}, 295 (1974)
\bibitem[DL]{} S. Doplicher, R. Longo, Invent. Math. {\em 73}, 493 (1984)
\bibitem[F]{} K. Fredenhagen, Commun. Math. Phys. {\em 97}, 79 (1985)
\bibitem[H]{} R. Haag, Local Quantum Physics, Springer Berlin (1992)
\bibitem[HHW]{} R. Haag, N.M. Hugenholtz, M. Winnink, Commun. Math. Phys.
{\em 5}, 215 (1967)
\bibitem[HKTP]{} R. Haag, D. Kastler, E.B. Trych-Pohlmeyer, Commun. Math. Phys.
{\em 38}, 173 (1974)
\bibitem[HL]{} P.D. Hislop, R. Longo, Commun. Math. Phys. {\em 84}, 71 (1982)
\bibitem[Ko]{} H. Kosaki, J. Operator Theory {\em 16}, 335 (1986)
\bibitem[Ku]{} R. Kubo, J. Phys. Soc. Jap. {\em 12}, 570 (1957)
\bibitem[LR]{} O. Lanford, D.W. Robinson, Commun. Math. Phys. {\em 9},
327 (1968)
\bibitem[MS]{} D.C. Martin, J. Schwinger, Phys. Rev. {\em 115}, 1342 (1959)
\bibitem[N]{} H. Narnhofer, Vienna preprint (1985)
\bibitem[NT]{} H. Narnhofer, W. Thirring, Fizika {\em 17}, 257 (1985)
\bibitem[OP]{} M. Ohya, D. Petz, Quantum Entropy and its Use, Springer,
Berlin (1993)
\bibitem[PW]{} A.A. Penzias, R.W. Wilson, Astrophys. J. {\em 142}, 419 (1965)
\bibitem[PuW]{} W. Pusz, S.L. Woronowicz, Commun. Math. Phys. {\em 58}, 273 (1978)
\bibitem[ST]{} J.L. Sauvageot, J.P. Thouvenot, Commun. Math. Phys. {\em 145},
411 (1992)
\end{thebibliography}
\end{document}
|
http://siba-sinmemis.unile.it/journals/EJC/Surveys/ds2.tex | unile.it | CC-MAIN-2013-20 | text/x-tex | null | crawl-data/CC-MAIN-2013-20/segments/1368699881956/warc/CC-MAIN-20130516102441-00033-ip-10-60-113-184.ec2.internal.warc.gz | 243,242,414 | 11,247 | \documentclass{article}
\def\here/{in: {\it Games of No Chance}}
%
\parskip 1pt plus 1pt
%
\pagestyle{myheadings}
\markright{\sc the electronic journal of combinatorics \#DS2\hfill}
\def\NP{N\hskip -2pt P}
\font\amsy=msbm10
\newcommand\IZ{\hbox{\amsy\char'132}}
\begin{document}
%\thispagestyle{empty}
%\title[Selected Bibliography]
\title{Combinatorial Games: Selected Bibliography with a Succinct Gourmet
Introduction}
%%Added for DS:
%\section*{\sc \ \ \ the electronic journal of combinatorics 1 (1994),
%\#DS2\hfill}
%%Changed for DS. Note, in particular: \maketitle{} below.
%\address{Aviezri S.\ Fraenkel\\
%Department of Applied Mathematics and Computer Science\\
%Weizmann Institute of Science\\
%Rehovot 76100, Israel}
%\email{[email protected]\\\indent\
%http:/$\!$/www.wisdom.weizmann.ac.il/\tilde fraenkel/fraenkel.html}
\author{Aviezri S.\ Fraenkel\\ \\
Department of Applied Mathematics and Computer Science\\
Weizmann Institute of Science\\
Rehovot 76100, Israel\\
{\tt [email protected]}\\
{\tt http:/$\!$/www.wisdom.weizmann.ac.il/$\sim$fraenkel}\\ \\}
\date{}
\maketitle
\section{What are Combinatorial Games?}
Roughly speaking, the family of {\it combinatorial games\/} consists of
two-player games with perfect information (no hidden information as in
some card games), no chance moves (no dice) and outcome restricted to
(lose,$\,$win), (tie,$\,$tie) and (draw,$\,$draw) for the two players who
move alternately. Tie is an end position such as in tic-tac-toe, where no
player wins, whereas draw is a dynamic tie: any position from which
a player has a nonlosing move, but cannot force a win. Both the easy game
of Nim and the seemingly difficult chess are examples of combinatorial
games. And so is go. The shorter terminology {\it game\/},
{\it games\/} is used below to designate combinatorial games.
\section{Why are Games Intriguing and Tempting?}
Amusing oneself with games may sound like a frivolous occupation. But the
fact is that the bulk of interesting and natural mathematical problems
that are hardest in complexity classes beyond $\NP$, such as
{\sl P}space, {\sl E}xptime and {\sl E}xpspace, are two-player games;
occasionally even one-player games (puzzles) or even zero-player games
(Conway's ``Life"). Some of the reasons for the high complexity of
two-player games are outlined in the next section. Before that we note that in
addition to a natural appeal of the subject, there are applications or
connections to various areas, including complexity, logic, graph and
matroid theory, networks, error-correcting codes, surreal numbers,
on-line algorithms, biology --- and analysis and design of
mathematical and commercial games!\eject
But when the chips are down, it is this ``natural appeal" that lures
both amateurs and professionals to become addicted to the subject. What
is the essence of this appeal? Perhaps the urge to play games is rooted
in our primal beastly instincts; the desire to corner, torture, or at
least dominate our peers. A common expression of these vile cravings
is found in the passions roused by local, national and
international tournaments. An intellectually refined version of these dark
desires, well hidden beneath the fa\c cade of scientific research, is
the consuming drive ``to beat them all", to be more clever than the most
clever, in short --- to create the tools to {\it Math}ter them
all in hot {\it comb\/}inatorial {\it comb\/}at! Reaching this goal is
particularly satisfying and sweet in the context of combinatorial games,
in view of their inherent high complexity.\medskip
With a slant towards artificial intelligence, Pearl wrote that
games ``offer a perfect laboratory for studying complex
problem-solving methodologies. With a few parsimonious rules, one
can create complex situations that require no less insight,
creativity, and expertise than problems actually encountered in
areas such as business, government, scientific, legal, and others.
Moreover, unlike these applied areas, games offer an arena in
which computerized decisions can be evaluated by absolute
standards of performance and in which proven human experts are
both available and willing to work towards the goal of seeing
their expertise emulated by a machine. Last, but not least, games
possess addictive entertaining qualities of a very general appeal.
That helps maintain a steady influx of research talents into the
field and renders games a convenient media for communicating
powerful ideas about general methods of strategic
planning.''\medskip
To further explore the nature of games, we consider, informally, two
subclasses.
\begin{itemize}
\item[(i)] Games People Play ({\it playgames}): games that are challenging
to the point that people will purchase them and play them.
\item[(ii)] Games Mathematicians Play ({\it mathgames}): games that are
challenging to mathematicians or other scientists to play with and
ponder about, but not necessarily to ``the man in the street''. %They
%are usually tractable.
\end{itemize}
Examples of playgames are chess, go, hex, reversi; of mathgames: Nim-type
games, Wythoff games, annihilation games, octal games.
Some ``rule of thumb'' properties, which seem to hold for the majority
of playgames and mathgames are listed below.
\begin{itemize}
\item[I.] {\sf Complexity}. Both playgames and mathgames tend to be
computationally intractable. There are a
few tractable mathgames, such as Nim, but most games still live in
{\it Wonderland\/}: we are wondering about their as yet unknown
complexity. Roughly speaking, however, NP-hardness is a necessary but
not a sufficient condition for being a playgame! (Some games on Boolean
formulas are Exptime-complete, yet none of them seems to have the
potential of commercial marketability.)
\item[II.] {\sf Boardfeel}. None of us may know an exact strategy
from a midgame position of chess, but even a novice, merely by
looking at the board, gets some feel who of the two players is in
a stronger position -- even what a strong or weak next move is.
This is what we loosely call {\it boardfeel}. Our informal
definition of playgames and mathgames suggests that the former do
have a boardfeel, whereas the latter don't. For many mathgames,
such as Nim, a player without prior knowledge of the strategy has
no inkling whether any given position is ``strong'' or ``weak''
for a player. Even when defeat is imminent, only one or two moves
away, the player sustaining it may be in the dark about the
outcome, which will stump him. The player has no boardfeel. (Even
many mathgames, including Nim-type games, can be played,
equivalently, on a board.)
Thus, in the boardfeel sense, simple games are complex and complex
games are simple! This paradoxical property also doesn't seem to have
an analog in the realm of decision problems. The boardfeel is the main
ingredient which makes PlayGames interesting to play.
%Its lack causes mathgames not to be
%played by ``the man in the street'', but the appeal to mathematicians.
%(Of course there
%are exceptions. Some games on Boolean formulas are Exptime-complete,
%and there is a growing number of other games that are Pspace-hard,
%yet none of them seems to have the potential of commercial marketability.
%In the other direction, the exceptions are perhaps rarer: Gale's game
%Bridg-it has been shown to be polynomial by Oliver Gross. Martin Gardner
%has informed me that it is the {\it only\/} polynomial game he knows
%which has enjoyed commercial success.)
\item[III.] {\sf Math Appeal}. Playgames, in addition to being
interesting to play, also have considerable mathematical appeal.
This has been exposed recently by the theory of partizan games
established by Conway and applied to endgames of go by Berlekamp,
students and associates.
On the other hand, mathgames have their own special
combinatorial appeal, of a somewhat different flavor. They appeal to
and are created by mathematicians of various disciplines, who find
special intellectual challenges in analyzing them. As Peter Winkler
called a subset of them: ``games people don't play''.
We might also call them, in a more positive vein, ``games mathematicians
play''. Both classes of games have applications to areas outside game
theory. Examples: surreal numbers (playgames), error correcting codes
(mathgames). Both provide enlightenment through bewilderment, as David
Wolfe and Tom Rodgers put it.
\item[IV.] {\sf Existence}. There are relatively few successful
playgames around. It seems to be hard to invent a playgame
that catches the masses. In contrast, mathgames abound. They appeal to
a large subclass of mathematicians and other scientists, who cherish
producing them and pondering about them. The large proportion of
mathgames-papers in the games bibliography below reflects this phenomenon.
\end{itemize}\medskip
We conclude, inter alia, that for playgames, high complexity is desirable.
Whereas in all respectable walks of life we strive towards solutions or
at least approximate solutions which are polynomial, there are two less
respectable human activities in which high complexity is appreciated.
These are cryptography (covert warfare) and games (overt warfare). The
desirability of high complexity in cryptography --- at least for the
encryptor! --- is clear. We claim that it is also desirable for
playgames.
It's no accident that games and cryptography team up: in both there
are adversaries, who pit their wits against each other! But games
are, in general, considerably harder than cryptography. For the latter,
the problem whether the designer of a cryptosystem has a safe system
can be expressed with two quantifiers only: $\exists$ a cryptosystem such
that $\forall$ attacks on it, the cryptosystem remains unbroken? In
contrast, the decision problem whether White can win if White moves
first in a chess game, has the form: ``$\exists\forall\exists\forall\cdots$
move: White wins?'', expressing the question whether White has an opening
winning move --- with an unbounded number of alternating quantifiers. This
makes games the more challenging and fascinating of the two, besides
being fun! See also the next section.\medskip
Thus, it's no surprise that the skill of playing games,
such as checkers, chess, or go has long been regarded as a distinctive
mark of human intelligence.
\section{Why are Combinatorial Games Hard?}
Existential decision problems, such as graph hamiltonicity and
Traveling Salesperson (Is there a round tour through specified cities
of cost $\leq C$?), involve a single existential quantifier (``Is
there\dots?''). In mathematical terms an existential problem boils down
to finding a path---sometimes even just verifying its existence---in a large
``decision-tree" of all possibilities, that satisfies specified properties.
The above two problems, as well as thousands of other interesting and
important combinatorial-type problems are NP-{\it complete\/}. This means
that they are {\it conditionally intractable}, i.e., the best way to solve
them seems to require traversal of most if not all of the decision tree,
whose size is exponential in the input size of the problem. No essentially
better method is known to date at any rate, and, roughly speaking, if an
efficient solution will ever be found for any NP-complete problem, then
all NP-complete problems will be solvable efficiently.
The decision problem whether White can win if White moves first in a
chess game, on the other hand, has the form: Is there a move of White
such that for {\it every\/} move of Black there is a move of White
such that for {\it every\/} move of Black there is a move of White
$\dots$ such that White can win? Here we have a large number of
alternating existential and universal quantifiers rather than a single
existential one. We are looking for an entire subtree rather than just
a path in the decision tree. Because of this, most nonpolynomial games
are at least {\sl P}space-hard. The problem for generalized chess on an
$n\times n$ board, and even for a number of seemingly simpler mathgames,
is, in fact, Exptime-complete, which is a {\it provable intractability}.
Put in simple language, in analyzing an instance of Traveling
Salesperson, the problem itself is passive: it does not resist your
attempt to attack it, yet it is difficult. In a game, in contrast, there
is your opponent, who, at every step, attempts to foil your effort to
win. It's similar to the difference between an autopsy and surgery.
Einstein, contemplating the nature of physics said, ``Der Allm\"achtige
ist nicht boshaft; Er ist raffiniert" (The Almighty is not mean; He is
sophisticated). NP-complete existential problems are perhaps
sophisticated. But your opponent in a game can be very mean!\medskip
Another manifestation of the high complexity of games is associated with
a most basic tool of a game : its {\it game-graph\/}. It is a directed
graph $G$ whose vertices are the positions of the game, and $(u,v)$
is an edge if and only if there is a move from position $u$ to position
$v$. Since every combination of tokens in the given game is a
{\it single\/} vertex in $G$, the latter has normally exponential size.
This holds, in particular, for both Nim and chess. Analyzing a game
means reasoning about its game-graph. We are thus faced with a problem
that is {\it a priori\/} exponential, quite unlike many present-day
interesting existential problems.
A fundamental notion is the {\it sum\/} (disjunctive compound) of games. A sum
is a finite collection of disjoint games; often very basic, simple games.
Each of the two players, at every turn, selects one of the games and
makes a move in it. If the outcome is not a draw, the sum-game ends when
there is no move left in any of the component games. If the outcome is
not a tie either, then in {\it normal\/} play, the player first unable to
move loses and the opponent wins. The outcome is reversed in {\it
mis\`ere\/} play.
If a game decomposes into a {\it disjoint\/} sum of its components, either
from the beginning (Nim) or after a while (domineering), the potential
for its tractability increases despite the exponential size of the game
graph. As Elwyn Berlekamp remarked, the situation is similar to that in
other scientific endeavors, where we often attempt to decompose a given
system into its functional components. This approach may yield improved
insights into hardware, software or biological systems, human organizations,
and abstract mathematical objects such as groups. %In most cases,
%there are interesting issues concerning the interactions between
%subsystems and their neighbors.
If a game doesn't decompose into a sum of disjoint components, it is
more likely to be intractable (Geography or Poset Games). Intermediate
cases happen when the components are not quite fixed (which explains why
mis\`ere play of sums of games is much harder than normal play) or
not quite disjoint (Welter). Thane Plambeck has recently made
progress with mis\`ere play, and we will be hearing more about this
shortly.
The hardness of games is eased somewhat by the efficient freeware package
``Combinatorial Game Suite'', courtesy of Aaron Siegel.
\section{Breaking the Rules}
As the experts know, some of the most exciting games are obtained by
breaking some of the rules for combinatorial games, such as permitting a
player to pass a bounded or unbounded number of times, i.e., relaxing the
requirement that players play alternately; or permitting a number of
players other than two.
But permitting a payoff function other than
(0,1) for the outcome (lose, win) and a payoff of (${1\over 2},{1\over 2}$)
for either (tie, tie) or (draw, draw) usually, but not always, leads to
games that are not considered to be combinatorial games; or to borderline
cases.
\section{Why Is the Bibliography Vast?}
In the realm of existential problems, such as sorting or Traveling
Salesperson, most present-day interesting decision problems can be
classified into tractable, conditionally intractable, and provably
intractable ones. There are exceptions, to be sure, such as graph
isomorphism, whose complexity is still unknown. But
the exceptions are few. In contrast, most games are still in Wonderland,
as pointed out in \S2(I) above. Only a few games have been classified into
the complexity classes they belong to. Despite recent impressive progress,
the tools for reducing Wonderland are still few and inadequate.
To give an example, many interesting games have a very succinct
input size, so a polynomial strategy is often more difficult to
come by (Richard Guy and Cedric Smith's octal games; Grundy's
game). Succinctness and non-disjointness of games in a sum may be
present simultaneously (Poset games). In general, the alternating
quantifiers, and, to a smaller measure, ``breaking the rules",
add to the volume of Wonderland. We suspect that the large size of
Wonderland, a fact of independent interest, is the main
contributing factor to the bulk of the bibliography on games.
\section{Why Isn't it Larger?}
The bibliography below is a {\sl partial\/} list of books and articles on
combinatorial games and related material. It is partial not only because
I constantly learn of additional relevant material I did not know about
previously, but also because of certain self-imposed restrictions. The
most important of these is that only papers with some original and
nontrivial mathematical content are considered. This excludes most
historical reviews of games and most, but not all, of the work on
heuristic or artificial intelligence approaches to games, especially
the large literature concerning computer chess. I have, however, included
the compendium Levy [1988], which, with its 50 articles and extensive
bibliography, can serve as a first guide to this world. Also some papers
on chess-endgames and clever exhaustive computer searches of some games
have been included.
On the other hand, papers on games that break some of the rules of
combinatorial games are included liberally, as long as they are
interesting and retain a combinatorial flavor. These are vague and hard
to define criteria, yet combinatorialists usually recognize a
combinatorial game when they see it. Besides, it is interesting to break
also this rule sometimes!
We have included some references to one-player games, e.g., towers of
Hanoi, $n$-queen problems, 15-puzzle and peg-solitaire, but only few
zero-player games (such as Life and games on ``sand piles''). We have
also included papers on various applications of games, especially when
the connection to games is substantial or the application is interesting
or important.
High-class meetings on combinatorial games, such as in Columbus,
OH (1990), at MSRI (1994, 2000), at BIRS (2005) resulted in books,
or a special issue of a journal -- for the Dagstuhl seminar
(2002). During 1990--2001, {\it Theoretical Computer Science\/}
ran a special Mathematical Games Section whose main purpose was to
publish papers on combinatorial games. TCS still solicits papers
on games. In 2002, {\it INTEGERS---Electronic J.\ of Combinatorial
Number Theory\/} began publishing a Combinatorial Games Section.
The combinatorial games community is growing in quantity and
quality!
\section{The Dynamics of the Literature}
The game bibliography below is very dynamic in nature. Previous versions
have been circulated to colleagues, intermittently, since
the early 1980's. Prior to every mailing updates were prepared, and usually
also afterwards, as a result of the comments received from several
correspondents. The listing can never be ``complete". Thus also the
present form of the bibliography is by no means complete.
Because of its dynamic nature, it is natural that the bibliography
became a ``Dynamic Survey" in the Dynamic Surveys (DS) section of the
{\it Electronic Journal of Combinatorics\/} (ElJC) and {\it The World
Combinatorics Exchange\/} (WCE). The ElJC and WCE are on the World Wide
Web (WWW), and the DS can be accessed at\hfill\break
http://www.combinatorics.org/\hfill\break
(click on ``Surveys'').
The ElJC has mirrors at various locations. Furthermore, the European
Mathematical Information Service (EMIS) mirrors this Journal, as do
all of its mirror sites (currently over forty of them). See\hfill\break
http://www.emis.de/tech/mirrors.html\smallskip
\section{An Appeal}
I ask readers to continue sending to me corrections and
comments; and inform me of significant omissions, remembering, however,
that it is a {\it selected\/} bibliography. I prefer to get reprints,
preprints or URLs, rather than only titles --- whenever possible.
Material on games is mushrooming on the Web. The URLs can be located
using a standard search engine, such as Google.
\section{Idiosyncrasies}
Most of the bibliographic entries refer to items written in English,
though there is a sprinkling of Danish, Dutch, French, German, Japanese,
Slovakian and Russian, as well as some English translations from Russian.
The predominance of English may be due to certain prejudices, but it also
reflects the fact that nowadays the {\it lingua franca\/} of science is
English. In any case, I'm soliciting also papers in languages other than
English, especially if accompanied by an abstract in English.
On the administrative side, Technical Reports, submitted papers and
unpublished theses have normally been excluded; but some exceptions have
been made. Abbreviations of book series and journal names usually follow
the {\it Math Reviews\/} conventions. Another convention is that
de Bruijn appears under D, not B; von Neumann under V, not N, McIntyre
under M not I, etc.
\medskip
Earlier versions of this bibliography have appeared, under the title
``Selected bibliography on combinatorial games and some related
material'', as the master bibliography for the book {\it Combinatorial
Games}, AMS Short Course Lecture Notes, Summer 1990, Ohio State
University, Columbus, OH, {\it Proc.\ Symp.\ Appl.\ Math.}\ {\bf 43}
(R.\ K.\ Guy, ed.), AMS 1991, pp.\ 191--226 with 400 items, and in the
{\it Dynamic Surveys\/} section of the {\it Electronic J.\ of
Combinatorics\/} in November 1994 with 542 items (updated there at odd
times). It also appeared as the master bibliography in {\it Games of
No Chance\/}, Proc.\ MSRI Workshop on Combinatorial Games, July, 1994,
Berkeley, CA (R.\ J.\ Nowakowski, ed.), MSRI Publ.\ Vol.~29, Cambridge
University Press, Cambridge, 1996, pp.\ 493--537, under the present title,
containing 666 items. The version published in the palindromic year
2002 contained the palindromic number 919 of references. It constituted
a growth of $38\%$. It appeared in ElJC and as the master bibliography
in {\it More Games of No Chance\/}, Proc.\ MSRI Workshop on Combinatorial
Games, July, 2000, Berkeley, CA (R.\ J.\ Nowakowski, ed.), MSRI
Publ.\ Vol.~42, Cambridge University Press, Cambridge, pp.\ 475-535.
The current update (mid-2003), in ElJC, contains 1001 items, another
palindrome.
\section{Acknowledgments}
Many people have suggested additions to the bibliography, or
contributed to it in other ways. Ilan Vardi distilled my
{\it Math-mas}ter (\S2) into {\it Math}ter. Among those that contributed more
than two or three items are: Akeo Adachi, Ingo Alth\"ofer, Thomas
Andreae, Eli Bachmupsky, Adriano Barlotti, J\'ozsef Beck, the late
Claude Berge, Gerald E.\ Bergum, H.\ S.\ MacDonald Coxeter, Thomas
S.\ Ferguson, James A.\ Flanigan, Fred Gal\-vin, Martin Gardner,
Alan J.\ Goldman, Solomon W.\ Golomb, Richard K.\ Guy, Shigeki
Iwata, David S.\ Johnson, Victor Klee, Donald E.\ Knuth, Anton
Kotzig, Jeff C.\ Lagarias, Michel Las Vergnas, Hendrik W.\
Lenstra, Hermann Loimer, F.\ Lockwood Morris, Richard J.\
Nowakowski, Judea Pearl, J.\ Michael Robson, David Singmaster,
Wolfgang Slany, Cedric A.\ B.\ Smith, Rastislaw Telg\'arsky, Mark
D. Ward, Y\=ohei Yamasaki and others. Thanks to all and keep up
the game! Special thanks to Mark Ward who went through the entire
file with a fine comb in late 2005, when it contained 1,151 items,
correcting errors and typos. Many thanks also to various anonymous
helpers who assisted with the initial \TeX\ file, to Silvio Levy,
who has edited and transformed it into \LaTeX2e in 1996, and to
Wolfgang Slany, who has transformed it into a BIBTeX file at the
end of the previous millenium, and solved a ``new millenium''
problem encountered when the bibliography grew beyond 999 items.
Keen users of the bibliography will notice that there is a
beginning of MR references, due to Richard Guy's suggestion,
facilitated by his student Alex Fink.
%\newcount\pointnum
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% \noindent\hangindent \thehangindent \hbox to
% \thehangindent{\hfill\the\pointnum .\enspace}\ignorespaces}
\nocite{*}
\sloppy
\bibliographystyle{gb}
\bibliography{ds2}
\label{lastbibl}
\end{document}
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\iteman{ZMATH 2013f.00307}
\itemau{Roegner, Katherine}
\itemti{Cognitive levels and approaches taken by students failing written examinations in mathematics.}
\itemso{Teach. Math. Appl. 32, No. 2, 81-87 (2013).}
\itemab
Summary: A study was conducted at the Technical University Berlin involving students who twice failed the written examination in the first semester course Linear Algebra for Engineers in order to better understand the reasons behind their failure. The study considered student understanding in terms of Bloom's taxonomy and the ways in which students approached problem solving. The results indicate that students rely on lower-order thinking processes and these processes are linked to solution approaches. Thus, by investigating solution strategies in homework sets and in classwork, an instructor can easily identify students at risk of not understanding at the appropriate level. In this contribution, the study is related to the framework set forth by the European Society for Engineering Education (SEFI) mathematics working group.
\itemrv{~}
\itemcc{D55 C35 D75}
\itemut{cognitive processes; thinking skills; at risk students; engineering education; linear algebra; problem solving; understanding; performance}
\itemli{doi:10.1093/teamat/hrt005}
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Schr\"odinger operators, ergodic potentials, quasi-periodic potentials, singular spectrum, Kotani theory
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\begin{document}
\title[Non-Deterministic Ergodic Potentials]{Ergodic Potentials With a Discontinuous Sampling
Function Are Non-Deterministic}
\author[D.\ Damanik, R.\ Killip]{David Damanik and Rowan Killip}
\address{Department of Mathematics 253--37, California Institute of
Technology, Pasadena, CA 91125, USA, E-mail:
\mbox{[email protected]}}
\address{Department of Mathematics, University of California, Los Angeles, CA 90055, USA, E-mail:
\mbox{[email protected]}}
\thanks{D.\ D.\ was supported in part by NSF grant DMS--0227289}
\maketitle
\begin{abstract}
We prove absence of absolutely continuous spectrum for discrete
one-dimensional Schr\"odinger operators on the whole line with
certain ergodic potentials, $V_\omega(n) = f(T^n(\omega))$, where
$T$ is an ergodic transformation acting on a space $\Omega$ and $f: \Omega \to \R$.
The key hypothesis, however, is that $f$ is discontinuous.
In particular, we are able to settle a conjecture of Aubry and Jitomirskaya--Mandel'shtam
regarding potentials generated by irrational rotations on the torus.
The proof relies on a theorem of Kotani, which shows that non-deterministic potentials
give rise to operators that have no absolutely continuous spectrum.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% Section 1
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Consider the discrete quasi-periodic Schr\"odinger operator
$$
[H_\omega \phi](n) = \phi(n+1) + \phi(n-1) + \lambda f(n\alpha + \omega \mod 1) \phi(n)
$$
on $\ell^2(\Z)$ where $\lambda>0$ denotes the coupling constant,
$f: \R/\Z \to \R$ is bounded, $\alpha$ is irrational, and $\omega\in[0,1)$.
When $f$ is continuous, the existence of absolutely continuous spectrum is a difficult
question that has attracted a great deal of attention recently. In this note, we will
show that for discontinuous $f$, there is none.
The absence of absolutely continuous spectrum for such $f$ was
conjectured by Aubry \cite{a} and by Jitomirskaya and Mandel'shtam
\cite{MZ}.
\begin{theorem}\label{T1}
If $f$ has a single {\rm (}non-removable{\rm )} discontinuity, then for all $\omega\in[0,1)$,
the operator $H_\omega$ has no absolutely continuous spectrum.
\end{theorem}
We will actually prove a far more general result, see Theorem~\ref{T2}. Irrational
rotations on $\R/\Z$ can be replaced by a very general dynamical system and the restrictions
on $f$ can be significantly relaxed.
In addition to addressing the conjecture described above, this particular example makes a link
to some recent research, which we will now describe.
If $f$ is nice enough, one expects purely absolutely continuous
spectrum for small $|\lambda|$ and purely point spectrum (with
exponentially decaying eigenfunctions) when~$|\lambda|$ is large.
(The latter scenario is usually referred to as Anderson localization.)
Matters are particularly well understood when $f(x) = \cos (2 \pi x)$;
see Jitomirskaya \cite{j} and the references therein. In this case, the operator $H$
has purely absolutely continuous spectrum (for almost all $\alpha$
and $\omega$) when $|\lambda| < 2$ and localization occurs for $|\lambda| > 2$
(again for almost all $\alpha$ and $\omega$).
Under analyticity assumptions on $f$, localization was shown by
Bourgain and Goldstein, \cite{b2,bg}, for $|\lambda|$ large
enough; while for sufficiently small $\lambda$, Bourgain and
Jitomirskaya, \cite{b2,bj}, proved purely absolutely continuous
spectrum.
It is desirable to prove these results under weaker regularity
assumptions on $f$ (see \cite{b} and \cite{k4} for recent
developments in this direction) and, at the same time, explore the
breakdown of these results once $f$ becomes too singular.
It is known that these results definitely do break down when $f$ takes
only finitely many values. In this case, absence of localization
in almost all cases was shown by Delyon and Petritis in \cite{dp}
and absence of absolutely continuous spectrum in all cases is a
result of Kotani \cite{k1}. Notice that these results hold for all
values of the coupling constant~$\lambda$.
We will now describe the setting for our more general result.
Let $T\!:\!\Omega\to\Omega$ be a homeomorphism of a compact metric space $\Omega$
and let $d\mu$ be a probability measure on $\Omega$ with respect to which $T$ is ergodic.
Given a bounded measurable function $f\!:\!\Omega\to\R$, we associate a potential to
each $\omega\in\Omega$ by
$$
V_\omega(n) = f\bigl( T^n(\omega) \bigr) \quad\text{for all $n\in\Z$}.
$$
The corresponding Schr\"odinger operator is denoted by $H_\omega$:
$$
[H_\omega \phi](n) = \phi(n+1) + \phi(n-1) + V_\omega(n) \phi(n), \qquad \phi\in\ell^2(\Z).
$$
To describe the requirements we make on $f$ we need to introduce the notion of an
essential discontinuity. First, we say that $l\in\R$ is an \textit{essential limit}
of $f$ at $\omega_0$ if there exists a sequence $\{\Omega_k\}$ of sets each of positive measure
such that for any sequence $\{\omega_k\}$ with $\omega_k \in \Omega_k$, both
$\omega_k\to\omega_0$ and $f(\omega_k) \to l$.
If $f$ has more than one essential limit at $\omega_0$, we say that $f$ is \textit{essentially discontinuous}
at this point.
\begin{theorem}\label{T2}
Suppose there is an $\omega_0 \in \Omega$ such that $f$ is essentially discontinuous at $\omega_0$ but is continuous at
all points $T^n \omega_0$, $n < 0$. Then $H_\omega$ has no absolutely continuous spectrum for almost every~$\omega$.
\end{theorem}
\noindent\textit{Remark.} Last and Simon \cite[\S\S 5--6]{ls}
proved the following: if a potential $V$ is the limit (in the
Tychonoff topology) of a sequence of translates of a second
potential, $W$, then the absolutely continuous spectrum of the
operator with potential $V$ contains that of the operator with
potential $W$. This theorem permits one to extend the absence of
absolutely continuous spectrum from almost every $\omega$ to every
$\omega$ in some circumstances. We will use this result in the
proof of Theorem~\ref{T1}.
\medskip
Our strategy will be to revisit a result of Kotani \cite{k1}:
Given an ergodic family of aperiodic potentials that take only
finitely many values, the resulting Schr\"odinger operators have
no absolutely continuous spectrum. The proof of this result rests
on an important consequence of Kotani theory \cite{k3,k2,k1,s}. In
the presence of absolutely continuous spectrum, the potentials are
deterministic in the sense that they are uniquely determined by
their values on a half-line. (The exact meaning of this will be
clarified in the next section.) Kotani proved that when $f$ takes
only finitely many values, the induced potentials are
non-deterministic. We will show that functions $f$ obeying the
assumptions of Theorem~\ref{T2} also give rise to
non-deterministic potentials and so deduce the absence of
absolutely continuous spectrum for the corresponding operators.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% Section 2
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A Short Review of Kotani Theory}
Kotani theory (cf.~\cite{k3,k2,k1,s}) establishes a relation
between the absolutely continuous spectrum of an ergodic family of
Schr\"odinger operators and the set of energies at which the
Lyapunov exponent vanishes. In this section, we recall some
specific results from this theory that are necessary for our proof
of Theorem~\ref{T2}. Essentially, we summarize parts of
\cite{k1}.
The setting is as follows: Let $d\nu$ be a measure on the space of
potentials $[-\lambda,\lambda]^{\Z}$ for which the shift map
$[SV](n)=V(n+1)$ is ergodic. Note that $S$ is a homeomorphism with
respect to the Tychonoff topology.
The system $([-\lambda,\lambda]^\Z,S,d\nu)$ gives rise to an ergodic family
of Schr\"odinger operators in $\ell^2(\Z)$,
$$
[H(V) \phi](n) = \phi(n+1) + \phi(n-1) + V(n) \phi(n), \quad V \in
\supp \; d\nu.
$$
For each energy $E$, there is an associated fundamental solution (or transfer matrix):
$$
U(n,E,V) = \begin{pmatrix} E - V(n) & -1 \\ 1 & 0 \end{pmatrix}
\times \cdots \times \begin{pmatrix} E - V(1) & -1 \\ 1 & 0 \end{pmatrix}.
$$
The multiplicative ergodic theorem shows that there is a non-random function $\gamma\!:\!\R\to [0,\infty)$,
called the \textit{Lyapunov exponent}, such that
$$
\lim_{n \to \infty} \frac1n \log \| U(n,E,V) \| = \gamma(E).
$$
for $\nu$-a.e.\ $V \in [-\lambda,\lambda]^\Z$. We write $N_{d\nu} = \{ E \in \R : \gamma(E) = 0\}$ for the
set of energies at which the Lyapunov exponent vanishes.
\begin{lemma}[Kotani; see \cite{k1}]\label{lemma1}
If $N_{d\nu}$ has zero Lebesgue measure, then $H(V)$ has no
absolutely continuous spectrum for $\nu$-a.e.\ $V$.
\end{lemma}
Given a potential $V \in [-\lambda,\lambda]^\Z$, we will write $V_\pm$ for its restrictions
to $\Z_+ = \{0,1,2,\ldots\}$ and $\Z_-= \{\ldots,-2,-1\}$, respectively.
\begin{lemma}[Kotani; see \cite{k1}]\label{lemma2}
If $N_{d\nu}$ has positive Lebesgue measure, then each of $V_\pm$
determines $V$ uniquely among potentials in $\supp(d\nu)$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% Section 3
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Absence of Absolutely Continuous Spectrum}
In this section we will prove Theorem~\ref{T2} and then use this
to prove Theorem~\ref{T1}.
Let us fix $\lambda>0$ once and for all such that
$[-\lambda,\lambda]$ contains the range of $f$. As in the
introduction, we associate a potential
$V_\omega\in[-\lambda,\lambda]^\Z$ to each $\omega\in\Omega$ by
$V_\omega(n) = f(T^n(\omega))$. We write $d\nu$ for the measure
on $[-\lambda,\lambda]^\Z$ induced by the measure $d\mu$ on
$\Omega$.
When $f$ is continuous, the map $\omega\mapsto V_\omega$ is also continuous and so takes
$\supp(\mu)$ onto $\supp(\nu)$. (Under a continuous map, the image of a compact set is compact,
while the inverse image of a closed set is closed.) When $f$ is not continuous, it is possible for
the support of $d\nu$ to contain potentials that do not correspond to any point $\omega\in\Omega$.
This plays an important role in the proof of Theorem~\ref{T2} which we will now present.
\begin{proof}[Proof of Theorem~\ref{T2}.]
We will show that there are potentials in $\supp(d \nu)$ which
agree on $\Z_-$ but take different values at $n=0$. By
Lemma~\ref{lemma2} this implies that $N_{d\nu}$ has zero Lebesgue
measure. Absence of absolutely continuous spectrum for $\nu$-a.e.
potential then follows by Lemma~\ref{lemma1}.
Let $l$ be an essential limit of $f$ at $\omega_0$. As $f$ is essentially discontinuous at
$\omega_0$, it suffices to construct a potential in $\supp(d\nu)$
that agrees with $V_{\omega_0}$ on $\Z_-$ and takes the value $l$ at $n=0$.
Let $\Omega_k$ be a sequence of sets which exhibit the fact that
$l$ is an essential limit of $f$. Since each has positive
$\mu$-measure, we can find points $\omega_k\in\Omega_k$ so that
$V_{\omega_k}$ is in the support of $\nu$; indeed this is the case
for almost every point in $\Omega_k$.
As $\omega_k \to \omega_0$ and $f$ is continuous at each of the
points $T^n\omega_0$, $n<0$, it follows that $V_{\omega_k}(n)\to
V_{\omega_0}(n)$ for each $n<0$. Moreover, since $f(\omega_k)$
converges to $l$ we also have $V_{\omega_k}(0)\to l$. We can
guarantee convergence of $V_{\omega_k}(n)$ for $n>0$ by passing to
a subsequence because $[-\lambda,\lambda]^\Z$ is compact. Let us
denote this limit potential by~$V$.
As each $V_{\omega_k}$ lies in $\supp(d\nu)$, so does $V$; moreover, $V(0)=l$ and
$V(n)=V_{\omega_0}(n)$ for each $n<0$. The construction of this potential completes
the proof of the theorem for reasons set forth in the opening paragraphs.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T1}.]
This result fits into the framework of Theorem~\ref{T2}, were $\Omega = \R / \Z$,
$d\mu$ is Lebesgue measure on the torus $\R / \Z$, and $T$ is the
rotation by $\alpha$, that is, $T (\omega) = \omega + \alpha \mod 1$. Lastly, we write
$\omega_0$ for the point at which $f$ is discontinuous.
Let us say that $l$ is a limiting value of $f$ at $\omega_0$ if there is
a sequence $\{\omega_k\}$ in $\Omega\setminus\{\omega_0\}$ such that $\omega_k\to\omega_0$
and $f(\omega_k)\to l$. As $f$ has a non-removable discontinuity at $\omega_0$,
it has more than one limiting value at this point. Moreover, since $f$ is continuous away
from $\omega_0$, any limiting value is also an essential limit---simply choose each
$\Omega_k$ to be a suitably small interval around $\omega_k$.
This shows that $f$ has an essential discontinuity at $\omega_0$. As the orbit of $\omega_0$
never returns to this point, $f$ is continuous at each point $T^n\omega_0$, $n\neq0$.
Therefore, Theorem~\ref{T2} is applicable and shows that $H_\omega$ has no absolutely
continuous spectrum for Lebesgue-a.e. $\omega\in[0,1)$.
It remains to show that the absolutely continuous spectrum of
$H_\omega$ is empty for \textit{all} $\omega \in \R / \Z$.
We begin by fixing $\omega_1$ such
that $H_{\omega_1}$ has no absolutely continuous spectrum and such that the orbit
of $\omega_1$ does not meet $\omega_0$; almost all $\omega_1$ have these properties.
Given an arbitrary $\omega\in\R/\Z$, we may choose a sequence of
integers $\{n_i\}$ so that $T^{n_i}(\omega) \to \omega_1$. As $f$
is continuous on the orbit of $\omega_1$, the potentials
associated to $T^{n_i}(\omega)$ converge pointwise to
$V_{\omega_1}$. By the result of Last and Simon described in the
introduction, the absolutely continuous spectrum cannot shrink
under pointwise approximation using translates of a single
potential. Thus, the operator with potential $V_\omega$ cannot
have absolutely continuous spectrum. This concludes the proof.
\end{proof}
\noindent\textit{Remark.} The proof of Theorem~\ref{T1} shows that for functions $f$
on the torus, every isolated (non-removable) discontinuity is essential.
As a result, the arguments presented extend to show absence of absolutely continuous
spectrum for all $\theta$ when $f$ has finitely many discontinuities.
If $f$ has infinitely many discontinuities, these methods still work in most cases;
Theorem~\ref{T2} describes in detail what is needed from $f$.
\begin{thebibliography}{10}
\bibitem{a} S.\ Aubry, Metal-insulator transition in
one-dimensional deformable lattices, in \textit{Bifurcation
Phenomena in Mathematical Physics and Related Topics},
Eds.~C.~Bardos and D.~Bessis, D.~Reidel Publisher Co., Boston,
Dordrecht, London (1980), 163--184
\bibitem{b} K.\ Bjerkl\"ov, Positive Lyapunov exponent for a class
of 1-d quasi-periodic Schr\"odinger equations --- the discrete
case, Preprint (2003)
\bibitem{b2} J.\ Bourgain, Green's function estimates for lattice
Schr\"odinger operators and applications, \textit{Ann.\ Math.\
Stud.}, to appear
\bibitem{bg} J.\ Bourgain and M.\ Goldstein, On nonperturbative localization with quasi-periodic
potential, \textit{Ann.\ of Math.} {\bf 152} (2000), 835--879
\bibitem{bj} J.\ Bourgain and S.\ Jitomirskaya, Absolutely continuous spectrum for 1D quasiperiodic
operators, \textit{Invent.\ Math.} {\bf 148} (2002), 453--463
\bibitem{dp} F.\ Delyon and D.\ Petritis, Absence of localization in a class of Schr\"odinger
operators with quasiperiodic potential, \textit{Commun.\ Math.\
Phys.} {\bf 103} (1986), 441--444
\bibitem{j} S.\ Jitomirskaya, Metal-insulator transition for the almost Mathieu operator,
\textit{Ann.\ of Math.} {\bf 150} (1999), 1159--1175
\bibitem{k4} S.\ Klein, Anderson localization for the discrete one-dimensional quasi-periodic
Schr\"odinger operator with potential defined by a Gevrey-class
function, Preprint (math-ph/0312073, mp-arc/04-17),
\bibitem{k3} S.\ Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random
one-dimensional Schr\"odinger operators, \textit{Stochastic
analysis} (Katata/Kyoto, 1982), 225--247, North-Holland Math.\
Library, 32, North-Holland, Amsterdam, 1984
\bibitem{k2} S.\ Kotani, One-dimensional random Schr\"odinger operators and Herglotz functions,
\textit{Probabilistic methods in mathematical physics}
(Katata/Kyoto, 1985), 219--250, Academic Press, Boston, MA, 1987
\bibitem{k1} S.\ Kotani, Jacobi matrices with random potentials taking finitely many values,
\textit{Rev.\ Math.\ Phys.} {\bf 1} (1989), 129--133
\bibitem{ls} Y.\ Last and B.\ Simon, Eigenfunctions, transfer matrices, and absolutely continuous
spectrum of one-dimensional Schr\"odinger operators,
\textit{Invent.\ Math.} {\bf 135} (1999), 329--367
\bibitem{MZ} V.\ A.~Mandel'shtam and S.~Ya.~Zhitomirskaya,
$1$D-quasiperiodic operators. Latent symmetries, \textit{Commun.\
Math.\ Phys.} {\bf 139} (1991), 589--604
\bibitem{s} B.\ Simon, Kotani theory for one-dimensional stochastic Jacobi matrices, \textit{Commun.\
Math.\ Phys.} {\bf 89} (1983), 227--234
\end{thebibliography}
\end{document}
---------------0402250019345--
|
https://www.zentralblatt-math.org/matheduc/en/?id=17434&type=tex | zentralblatt-math.org | CC-MAIN-2019-22 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-22/segments/1558232257481.39/warc/CC-MAIN-20190524004222-20190524030222-00455.warc.gz | 996,167,890 | 1,460 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 2013d.00772}
\itemau{Robles, P.; Claro, F.}
\itemti{Can there be massive photons? A pedagogical glance at the origin of mass.}
\itemso{Eur. J. Phys. 33, No. 5, 1217-1226 (2012).}
\itemab
Summary: Among the most startling experiences a student encounters is learning that, unlike electrons and other elementary particles, photons have no mass. Under certain circumstances, however, the light quantum behaves as if it did have a finite mass. Starting from Maxwell's equations, we discuss how this arises when light interacts with a charged plasma, or travels along a waveguide. The motion of such photons is analyzed using kinematic concepts of special relativity, and we show how a cutoff frequency for effective propagation appears. Seeing how an environment may yield an apparent dynamic mass to the photon paves the way for later understanding: might the Higgs boson field provide other particles, such as the electron, with a mass? This paper is addressed to mid-level physics students, teachers and lecturers, requiring only a knowledge of classical electromagnetic and special relativity theories.
\itemrv{~}
\itemcc{M50}
\itemut{}
\itemli{doi:10.1088/0143-0807/33/5/1217}
\end |
http://wasp.cs.washington.edu/atomeclipse/slides.tex | washington.edu | CC-MAIN-2017-39 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2017-39/segments/1505818688997.70/warc/CC-MAIN-20170922145724-20170922165724-00675.warc.gz | 369,111,272 | 5,385 | \documentclass[handout]{beamer}
\usepackage{beamerthemesplit}
\title{Introducing Shared-Memory Concurrency}
\subtitle{Race Conditions and Atomic Blocks}
\author{Laura Effinger-Dean}
\date{November 19, 2007}
\begin{document}
\frame{\titlepage}
\section{Concurrency}
\subsection{Why use concurrency?}
\frame
{
\frametitle{Concurrency}
Computation where ``multiple things happen at the same time'' is
inherently more complicated than \emph{sequential} computation.
\begin{itemize}
\item Entirely new kinds of bugs and obligations
\end{itemize}
Two forms of concurrency:
\begin{itemize}
\item \emph{Time-slicing}: only one computation at a time but
\emph{pre-empt} to provide \emph{responsiveness} or \emph{mask I/O
latency}.
\item \emph{True parallelism}: more than one CPU (e.g., the lab
machines have two, the attu machines have 4, ...)
\end{itemize}
Within a program, each computaton becomes a separate \emph{thread}.
}
%% \frame
%% {
%% \frametitle{Example: Processes}
%% The O/S runs multiple processes ``at once''.
%% Why? (Convenience, efficient use of resources, performance)
%% No problem: keep their address spaces separate.
%% But they do communicate/share via files (and pipes).
%% Things can go wrong, e.g., a \emph{race condition}:\\
%% \texttt{~~echo "hi" > someFile}\\
%% \texttt{~~foo=`cat someFile`}\\
%% \texttt{~~\# Can we assume foo holds the string hi??}
%% The O/S provides \emph{synchronization mechanisms} to avoid this
%% \begin{itemize}
%% \item See CSE451; we will focus on \emph{intraprocess} concurrency.
%% \end{itemize}
%% }
%% \frame
%% {
%% \frametitle{Runtime memory layout}
%% We said a running Java or C program had code, a heap, global
%% variables, a stack, and ``what is executing right now'' (in
%% assembly, a \emph{program counter}).
%% C, Java support parallelism similarly (other languages can be different):
%% \begin{itemize}
%% \item One pile of code, global variables, and heap.
%% \item Multiple ``stack + program counter''s --- called \emph{threads}
%% \item Threads can be \emph{pre-empted} whenever by a \emph{scheduler}
%% \item Threads can communicate (or mess each other up) via
%% \emph{shared memory}.
%% \item Various \emph{synchronization mechanisms} control what
%% \emph{thread interleavings} are possible.
%% \begin{itemize}
%% \item ``Do not do your thing until I am done with my thing''
%% \end{itemize}
%% \end{itemize}
%% }
\frame
{
\frametitle{Why do this?}
\begin{itemize}
\item Convenient structure of code
\begin{itemize}
\item Example: web browser. Each ``tab'' becomes a separate
thread.
\item Example: Fairness -- one slow computation only takes some of
the CPU time without your own complicated timer code. Avoids
\emph{starvation}.
\end{itemize}
\item Performance
\begin{itemize}
\item Run other threads while one is reading/writing to disk (or
other slow thing that can happen in parallel)
\item Use more than one CPU at the same time
\begin{itemize}
\item The way computers will get faster over the next 10 years
\item So no parallelism means no faster.
\end{itemize}
\end{itemize}
\end{itemize}
}
\subsection{Communicating between threads}
%% \frame
%% {
%% \frametitle{Simple synchronization}
%% If one thread does nothing of interest to any other thread,
%% why is it running?
%% So threads have to \emph{communicate} and \emph{coordinate}.
%% \begin{itemize}
%% \item Use each others' results; avoid messing up each other's computation.
%% \end{itemize}
%% Simplest two ways not to mess each other up (don't underestimate!):
%% \begin{enumerate}
%% \item Do not access the same memory.
%% \item Do not mutate shared memory.
%% \end{enumerate}
%% Next simplest: One thread does not run until/unless another thread is
%% done
%% \begin{itemize}
%% \item Called a \emph{join}
%% \end{itemize}
%% }
\frame
{
\frametitle{Working in Parallel}
Often you have a bunch of threads running at once and they
\emph{might} need the same mutable memory at the same time but
\emph{probably not}.
Want to be \emph{correct} without sacrificing parallelism.
Example: A bunch of threads processing bank transactions:
\begin{itemize}
\item withdraw, deposit, transfer, currentBalance, ...
\item chance of two threads accessing the same account at the same
time very low, but not zero.
\item want \emph{mutual exclusion} (a way to keep each other out of
the way when there is \emph{contention})
\end{itemize}
%% Another example: Parallel search through an arbitrary graph
}
\subsection{Concurrency in Java/C}
\frame
{
\frametitle{Basics}
C: The POSIX Threads (pthreads) \emph{library}
\begin{itemize}
\item \texttt{\#include <pthread.h>}
%% \item Link with \texttt{-lpthread}
\item \texttt{pthread\_create} takes a function pointer and an
argument for it; runs it as a separate thread.
\item Many types, functions, and macros for threads, locks, etc.
\end{itemize}
Java: Built into the language
\begin{itemize}
\item Subclass \texttt{java.lang.Thread} overriding \texttt{run}
\item Create a Thread object and call its \texttt{start} method
\item Any object can ``be synchronized on'' (later)
\end{itemize}
}
\section{Race conditions}
\subsection{What are race conditions?}
\frame
{
\frametitle{Common bug: race conditions}
There are several new types of bugs that occur in concurrent
programs; \emph{race conditions} are the most fundamental and the
most common.
\begin{itemize}
\item A race condition is when the order of thread execution in a
program affects the program's output.
\item Difficult to identify and fix, because problematic thread
interleavings may be unlikely.
\item \emph{Data races} (common type of race condition) - when
multiple threads access the same location in memory
``simultaneously,'' with at least one access being a write
\end{itemize}
}
\subsection{Example of race conditions}
\frame
{
\frametitle{Example: TwoThreads.java}
\begin{itemize}
\item What is the \emph{intended} output of this program?
\item What \emph{actual} outputs are possible?
\end{itemize}
}
\frame
{
\frametitle{What could go wrong?}
Simultaneous updates lead to race conditions. Suppose two threads
both execute \texttt{i++}. In machine code, this single statement
becomes several operations:
\begin{center}
\begin{tabular}{l@{\hspace{1in}}l}
Thread 1 & Thread 2 \\
\texttt{r1 = i} & \texttt{r2 = i}\\
\texttt{r1 += 1} & \texttt{r2 += 1}\\
\texttt{i = r1} & \texttt{i = r2}
\end{tabular}
\end{center}
If \texttt{i} starts at 0, what is the value of \texttt{i} after execution?
}
\frame
{
\frametitle{What could go wrong?}
Simultaneous updates lead to race conditions.
\begin{center}
\begin{tabular}{l@{\hspace{1in}}l}
Thread 1 & Thread 2 \\
\uncover<2->{\texttt{r1 = i}} & \\
\uncover<2->{\texttt{r1 += 1}} & \\
\uncover<2->{\texttt{i = r1}} & \\
& \uncover<3->{\texttt{r2 = i}} \\
& \uncover<3->{\texttt{r2 += 1}} \\
& \uncover<3->{\texttt{i = r2}}
\end{tabular}
\end{center}
Final value of \texttt{i} is \uncover<4>{2.}
}
\frame
{
\frametitle{What could go wrong?}
Simultaneous updates lead to race conditions.
\begin{center}
\begin{tabular}{l@{\hspace{1in}}l}
Thread 1 & Thread 2 \\
\uncover<2->{\texttt{r1 = i}} & \\
\uncover<2->{\texttt{r1 += 1}} & \\
& \uncover<3->{\texttt{r2 = i}} \\
& \uncover<3->{\texttt{r2 += 1}} \\
\uncover<4->{\texttt{i = r1}} & \\
& \uncover<5->{\texttt{i = r2}}
\end{tabular}
\end{center}
Final value of \texttt{i} is \uncover<6>{1. The first
update to \texttt{i} is lost.}
}
\frame
{
\frametitle{Detecting race conditions}
\begin{itemize}
\item Without calls to \texttt{Thread.sleep()}, our code ran ``so
fast'' that the race condition did not manifest.
\item Forcing a thread to yield control is a good way to encourage
``interesting'' interleavings. \pause
\item BUT:
\begin{itemize}
\item Calling \texttt{sleep} doesn't guarantee that the race
condition will affect the output.
\item In general, programs are large and we don't know where to
look for bugs or if bugs even exist.
\end{itemize}
\end{itemize}
}
\section{Atomic blocks}
\subsection{Using atomic blocks to avoid race conditions}
\frame
{
\frametitle{Avoiding race conditions}
\begin{itemize}
\item We will try to restrict the number of possible thread
interleavings.
\begin{itemize}
\item E.g., in TwoThreads, we got into trouble because the updates
were interleaved.
\end{itemize}
\item Simple limitation is to define \emph{atomic blocks} in which
a thread may assume that no other threads will execute.
\begin{itemize}
\item Lots of variations on terminology: critical sections,
synchronization, etc.
\end{itemize}
\end{itemize}
}
\frame
{
\frametitle{Fixing TwoThreads.java}
\begin{center}
(Demo.)
\end{center}
}
\frame
{
\frametitle{Fixing TwoThreads.java}
Now the following troublesome interleaving is illegal!
\begin{center}
\begin{tabular}{l@{\hspace{1in}}l}
Thread 1 & Thread 2 \\
\texttt{r1 = i} & \\
\texttt{r1 += 1} & \\
& \texttt{r2 = i} \\
& \texttt{r2 += 1} \\
\texttt{i = r1} & \\
& \texttt{i = r2}
\end{tabular}
\end{center}
}
\frame
{
\frametitle{Fixing TwoThreads.java}
Instead, we get:
\begin{center}
\begin{tabular}{l@{\hspace{1in}}l}
Thread 1 & Thread 2 \\
\texttt{atomic \{} \\
\texttt{~~r1 = i} & \\
\texttt{~~r1 += 1} & \\
\texttt{~~i = r1} & \\
\texttt{\}}\\
& \texttt{atomic \{} \\
& \texttt{~~r2 = i} \\
& \texttt{~~r2 += 1} \\
& \texttt{~~i = r2} \\
& \texttt{\}}
\end{tabular}
\end{center}
}
\frame
{
\frametitle{Atomic blocks}
\begin{itemize}
\item Atomic blocks are a common way of fixing race conditions
\item Allows us to think ``single-threaded'' even in a
multi-threaded program
\item How much code should be inside the block?
\begin{itemize}
\item Atomic blocks that are ``too long'' could cause the
program to slow down, as threads sleep waiting for other threads
to finish executing atomically.
\item Atomic blocks that are ``too short'' might miss race
conditions.
\end{itemize}
\end{itemize}
}
\subsection{Example: BankAccount.java}
\frame
{
\frametitle{Familiar example: bank accounts}
\begin{center}
(Demo)
\end{center}
}
\subsection{Example: ProducerConsumer.java}
\frame
{
\frametitle{Producer and consumer threads}
\begin{itemize}
\item One (or more) thread(s) produces values and leaves them in a
buffer to be processed
\item One (or more) thread(s) takes values from the buffer and
consumes them
\item Common application in operating systems, etc.
\end{itemize}
\begin{center}
(Demo.)
\end{center}
}
\frame
{
\frametitle{Limitations of atomic blocks}
In the producer-consumer example, we see:
\begin{itemize}
\item \emph{Busy wait}: threads loop forever waiting for a state
change.
\item \emph{Scheduling fairness}: no guarantee that the threads
will get equal shares of processor time.
\end{itemize}
There are ways to fix both of these problems with atomic blocks,
but we don't have good implementations for Java or C. Instead, we'll
look briefly at what mechanisms Java and pthreads do provide (more
on Wednesday).
}
\section{Real-life mechanisms}
\subsection{Locks}
\frame
{
\frametitle{Atomic blocks don't exist yet}
\begin{itemize}
\item You can't (yet) use atomic blocks in true Java code
(although we ``faked it'').
\begin{itemize}
\item Active area of research---maybe it will be integrated into
Java at some point.
\end{itemize}
\item Instead programmers use \emph{locks} (mutexes) or other
mechanisms, usually to get the behavior of atomic blocks.
\begin{itemize}
\item But misuse of locks will violate the ``all-at-once''
policy.
\item Or lead to other bugs we haven't seen yet.
\end{itemize}
\end{itemize}
}
\frame
{
\frametitle{Lock basics}
A lock is \emph{acquired} and \emph{released} by a thread.
\begin{itemize}
\item At most one thread holds it at any moment.
\item Acquiring it ``blocks'' until the holder releases it and the
blocked thread acquires it.
\begin{itemize}
\item Many threads might be waiting; one will ``win.''
\item The lock-implementor avoids race conditions on lock-acquire.
\end{itemize}
\item So create an atomic block by surrounding with lock acquire
and release.
\item Problems:
\begin{itemize}
\item Easy to mess up (e.g., use two different locks to protect
the same location in memory).
\item \emph{Deadlock}: threads might get stuck forever waiting
for locks that will never be released.
\end{itemize}
\end{itemize}
}
\section{}
\frame
{
\frametitle{Summary}
\begin{itemize}
\item Concurrency introduces bugs that simply don't exist in
sequential programming.
\item Atomic blocks are a useful way of thinking about safety in
concurrent programs.
\item In real code, you'll use locks or other mechanisms, which
eliminate race conditions if used properly but can lead to more bugs
if misused.
\end{itemize}
}
\frame
{
\frametitle{Blatant plug(in)}
\begin{itemize}
\item For the examples we used AtomEclipse, a plugin that I
developed for Eclipse (an IDE for Java (and other languages)).
\item AtomEclipse is designed for students like you to learn about
atomic blocks more easily.
\item Download and try out if you want to play around with the
examples some more:
\texttt{http://wasp.cs.washington.edu/atomeclipse} (linked from the
303 web page)
\item Let me know what you think: \texttt{effinger@cs} or talk to me
after class!
\end{itemize}
}
\end{document} |
http://hgm.nubati.net/cgi-bin/gitweb.cgi?p=xboard.git;a=blob_plain;f=xboard.texi;hb=47253a51e36963f4acc78bac31732e183f5426fa | nubati.net | CC-MAIN-2020-45 | application/x-texinfo | application/x-tex | crawl-data/CC-MAIN-2020-45/segments/1603107889173.38/warc/CC-MAIN-20201025125131-20201025155131-00185.warc.gz | 48,002,293 | 61,312 | \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.
* Basic operation:: Mouse and keyboard functions.
* 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 Basic operation
@chapter Basic operation
@cindex Basic operation
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. In crazyhouse, bughouse or shogi you can
drag and drop pieces to the board from the holdings squares
displayed next to the board.
Old behavior, where right-clicking a square brings up a menu
where you can select what piece to drop on it can still be
selected through the @samp{Drop Menu} option.
Only in Edit Position mode right and middle clicking a square is still used to
put a piece on it, and the piece to drop is selected by sweeping
the mouse vertically with the button held down.
The default function of the right mouse button in other modes is
to display the position the chess program thinks it will end up in.
While moving the mouse vertically with this button pressed
XBoard will step through the principal variation to show how
this position will be reached.
Lines of play displayed in the engine-output window,
or PGN variations in the comment window can similarly
be played out on the board, by right-clicking on them.
Only in Analysis mode, when you walk along a PV,
releasing the mouse button will forward the game upto that point,
like you entered all previous PV moves.
As the display of the PV in that case starts after the first move
a simple right-click will play the move the engine indicates.
In Analysis mode you can also make a move by grabbing the piece
with a double-click of the left mouse button
(or while keeping the Ctrl key pressed).
In this case the move you enter will not be played,
but will be excluded from the analysis of the current position.
(Or included if it was already excluded; it is a toggle.)
This only works for engines that support this feature.
When connected to an ICS, it is possible to call up a graphical
representation of players seeking a game in stead of the chess board,
when the latter is not in use
(i.e. when you are not playing or observing).
Left-clicking the display area will switch between this 'seek graph'
and the chess board.
Hovering the mouse pointer over a dot will show the details of the
seek ad in the message field above the board.
Left-clicking the dot will challenge that player.
Right-clicking a dot will 'push it to the back',
to reveal any dots that were hidden behind it.
Right-clicking off dots will refresh the graph.
Most other XBoard commands are available from the menu bar. The most
frequently used commands also have shortcut keys or on-screen buttons.
These shortcut keystrokes are mostly non-printable characters.
Typing a letter or digit while the board window has focus
will bring up a type-in box with the typed letter already in it.
You can use that to type a move in siuations where it is your
turn to enter a move,
type a move number to call up the position after that move
in the display,
or, in Edit Position mode, type a FEN.
Some rarely used parameters can only be set through options on the
command line used to invoke XBoard.
XBoard uses a settings file, in which it can remember any changes to
the settings that are made through menus or command-line options,
so they will still apply when you restart XBoard for another session.
The settings can be saved into this file automatically when XBoard exits,
or on explicit request of the user.
The default name for the settings file is /etc/xboard/xboard.conf,
but in a standard install this file is only used as a master settings
file that determines the system-wide default settings,
and defers reading and writing of user settings to a user-specific
file like ~/.xboardrc in the user's home directory.
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.
@node Menus
@chapter Menus, buttons, and keys
@cindex Menus
@menu
* File Menu:: Accessing external games and positions.
* Edit Menu:: Altering games, positions, PGN tags or comments.
* View Menu:: Controlling XBoard's shape and looks.
* Mode Menu:: Selecting XBoard's mode.
* Action Menu:: Talking to the chess engine or ICS opponents.
* Engine Menu:: Controlling settings and actions of the engine(s).
* 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{Ctrl-N} 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).
You can also press the @samp{Pick Fixed} button to let XBoard generate
a random number for you.
The thus selected opening position will then persistently be chosen on any following
New Game command until you use this menu to select another.
Selecting position number -1 (or pushing the @samp{Randomize} button)
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.) The shifted @kbd{Alt+V} key is a
keyboard equivalent. 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!
Only sizes bulky (72) and middling (49) have all pieces,
while size petite (33) has most.
These sizes would have to be set at startup through the @code{size}
command-line option when you start up XBoard for such variants to be playable.
You can overrule the default board format of the selected variant,
(e.g. to play suicide chess on a 6 x 6 board),
in this dialog, but normally you would not do that,
and leave them at '-1', which means 'default'.
@item Load Game
@cindex Load Game, Menu Item
Plays a game from a record file. The @kbd{Ctrl-O} 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) also are treated as comments;
however, if you rights-click them in the comment window,
XBoard will shelve the current line, and load the the selected variation,
so you can step through it.
You can later revert to the previous line with the @samp{Revert} command.
This way you can walk quite complex varation trees with XBoard.
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.
Note that it must appear before any FEN tag for XBoard to recognize
variant FENs appropriately.
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 Position
@cindex Load Position, Menu Item
Sets up a position from a position file. A pop-up dialog prompts
you for the file name. The shifted @kbd{Ctrl-O} key is a keyboard
equivalent. 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.
The shifted @kbd{PgDn} key is a keyboard equivalent.
@item Load Previous Position
@cindex Load Previous Position, Menu Item
Loads the previous position from the last position file you
loaded. The shifted @kbd{PgUp} key is a keyboard equivalent.
Not available if the last position was loaded from a pipe.
@item Save Game
@cindex Save Game, Menu Item
Appends a record of the current game to a file.
The @kbd{Ctrl-S} key is a keyboard equivalent.
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 Save Position
@cindex Save Position, Menu Item
Appends a diagram of the current position to a file.
The shifted @kbd{Ctrl+S} key is a keyboard equivalent.
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 Save Selected Games
@cindex Save Selected Games
Will cause all games selected for display in the current Game List
to be appended to a file of the user's choice.
@item Save Games as Book
@cindex Save Games as Book, Menu Item
Creates an opening book from the currently loaded game file,
incorporating only the games currently selected in the Game List.
The book will be saved on the file specified in the @samp{Common Engine}
options dialog.
The value of @samp{Book Depth} specified in that same dialog will
be used to determine how many moves of each game will be added to
the internal book buffer.
This command can take a long time to process,
and the size of the buffer is currently limited.
At the end the buffer will be saved as a Polyglot book,
but the buffer will not be cleared,
so that you can continue adding games from other game files.
@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 @kbd{Ctrl-Q} key is a keyboard equivalent.
@end table
@node Edit Menu
@section Edit Menu
@cindex Menu, Edit
@cindex Edit Menu
@table @asis
@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 @kbd{Ctrl-C}
key is a keyboard equivalent. 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 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 shifted @kbd{Ctrl-C} key
is a keyboard equivalent. 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 Copy Game List
@cindex Copy Game List, Menu Item
Copies the current game list to the clipboard,
and sets the X selection to this text.
A format of comma-separated double-quoted strings is used,
including all tags,
so it can be easily imported into spread-sheet programs.
@item Paste Game
@cindex Paste Game, Menu Item
Interprets the current X selection as a game record and loads it, as
with Load Game. The @kbd{Ctrl-V} key is a keyboard equivalent.
@item Paste Position
@cindex Paste Position, Menu Item
Interprets the current X selection as a FEN position and loads it, as
with Load Position. The shifted @kbd{Ctrl-V} key is a keyboard equivalent.
@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. The @kbd{Ctrl-E} key is a keyboard equivalent.
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.
The shifted @kbd{Ctrl-E} key is a keyboard equivalent.
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 puts a white or black pawn in the square, respectively,
but you can change that to any other piece type by dragging the
mouse down before you release the button.
You will then see the piece on the originally clicked square
cycle through the available pieces
(including those of opposite color),
and can release the button when you see the piece you want.
To alter the side to move, you can click the clock
(the words White and Black above the board)
of the side you want to give the move to.
To clear the board you can click the clock of the side that
alread has the move (which is highlighted in black).
The old behavior with a piece menu can still be configured
with the aid of the @code{pieceMenu} option.
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 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},
PGN variations will also be printed in this window,
and can be promoted to main line by right-clicking them.
@samp{Forward}, and @samp{Backward}.
@item Edit Book
@cindex Edit Book, Menu Item
Pops up a window listing the moves available in the GUI book
(specified in the @samp{Common Engine Settings} dialog)
from the currently displayed position,
together with their weights and (optionally in braces) learn info.
You can then edit this list, and the new list will be stored
back into the book when you press OK.
Note that the listed percentages are neither used, nor updated when
you change the weights; they are just there as an optical aid.
@item Revert
@itemx Annotate
@cindex Revert, Menu Item
@cindex Annotate, Menu Item
If you are examining an ICS game and Pause mode is off,
Revert issues the ICS command @samp{revert}.
In local mode, when you were editing or analyzing a game,
and the @code{-variations} command-line option is switched on,
you can start a new variation by holding the Shift key down while
entering a move not at the end of the game.
Variations can also become the currently displayed line by
clicking a PGN variation displayed in the Comment window.
This can be applied recursively,
so that you can analyze variations on variations;
each time you create a new variation by entering an alternative move
with Shift pressed, or select a new one from the Comment window,
the current variation will be shelved.
@samp{Revert} allows you to return to the most recently shelved variation.
The difference between @samp{Revert} and @samp{Annotate}
is that with the latter,
the variation you are now abandoning will be added as a comment
(in PGN variation syntax, i.e. between parentheses)
to the original move where you deviated, for later recalling.
The @kbd{Home} key is a keyboard equivalent to @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.
The @kbd{End} key is a keyboard equivalent.
@item Backward
@cindex Backward, Menu Item
@cindex <, Button
Steps backward through a series of remembered moves.
The @samp{[<]} button and the @kbd{Alt+LeftArrow} 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{Alt+RightArrow} 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 @kbd{Alt+Home} 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 @kbd{Alt+End} 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.
@end table
@node View Menu
@section View Menu
@cindex Menu, View
@cindex View 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{F2} key is a keyboard equivalent.
@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. The shifted @kbd{Alt+O} key is a keyboard equivalent.
XBoard will display lines of thinking output of the same depth ordered by score,
(highest score on top), rather than in the order the engine produced them.
Usually this amounts to the same, as a normal engine search will only find new PV
(and emit it as thinking output)
when it searches a move with a higher score than the previous variation.
But when the engine is in multi-variation mode this needs not always be true,
and it is more convenient for someone analyzing games to see the moves sorted by score.
The order in which the engine found them is only of interest to the engine author,
and can still be deduced from the time or node count printed with the line.
Right-clicking a line in this window, and then moving the mouse vertically with the
right button kept down, will make XBoard play through the PV listed there.
The use of the board window as 'variation board' will normally end when
you release the right button,
or when the opponent plays a move.
But beware: in Analysis mode, moves thus played out will be added to the game.
The Engine-Output pane for each engine will contain a header displaying the
multi-PV status and a list of excluded moves in Analysis mode,
which are also responsive to right-clicking.
@item Show Move History
@cindex Show Move History, Menu Item
Shows or hides a list of moves of the current game.
The shifted @kbd{Alt+H} key is a keyboard equivalent.
This list allows you to move the display to any earlier position in the game
by clicking on the corresponding move.
@item Show Evaluation Graph
@cindex Show Evaluation Graph, Menu Item
Shows or hides a window which displays a graph of how the engine score(s)
evolved as a function of the move number.
The shifted @kbd{Alt+E} key is a keyboard equivalent.
Clicking on the graph will bring
the corresponding position in the board display.
@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. The shifted @kbd{Alt+G} key is a keyboard equivalent.
@item Tags
@cindex Tags, Menu Item
Pops up a window which shows the PGN (portable game notation)
tags for the current game.
For now this is a duplicate of the @samp{Edit Tags} item in the @samp{Edit} menu.
@item Comments
@cindex Comments, Menu Item
Pops up a window which shows any comments to or variations on the current move.
For now this is a duplicate of the @samp{Edit Comment} item in the @samp{Edit} menu.
@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 Open Chat Window
@cindex Open Chat Window, Menu Item
This menu item opens a window in which you can conduct upto 5 chats
with other ICS users (or channels).
To use the window, write the name of your chat partner, the channel number,
or the words 'shouts', 'whispers', 'cshouts' in the upper field
(closing with <Enter>).
Everything you type in the lowest field will then automatically be sent to
the mentioned party, while everything that party sends to you will
appear in the central text box, rather than appear in the ICS console.
The row of buttons allow you to choose between chat;
to start a new chat, just select an empty button,
and complete the @samp{Chat partner} field.
@item Board
@cindex Board, Menu Item
Summons a dialog where you can customize the look of the chess board.
Here you can specify the directory from which piece images should be taken,
when you don't want to use the built-in piece images
(see @code{pieceImageDirectory} option),
external images to be used for the board squares
(@code{liteBackTextureFile} and @code{darkBackTextureFile} options),
and square and piece colors for the default pieces.
@item Game List Tags
@cindex Game List Tags, Menu Item
a duplicate of the Game List dialog in the Options menu.
@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.
The @kbd{Ctrl-W} key is a keyboard equivalent.
@item Machine Black
@cindex Machine Black, Menu Item
Tells the chess engine to play Black.
The @kbd{Ctrl-B} key is a keyboard equivalent.
@item Two Machines
@cindex Two Machines, Menu Item
Plays a game between two chess engines.
The @kbd{Ctrl-T} key is a keyboard equivalent.
@item Analysis Mode
@cindex Analysis Mode, Menu Item
@cindex null move
XBoard tells the chess engine to start analyzing the current game/position
and shows you the analysis as you move pieces around.
The @kbd{Ctrl-A} key is a keyboard equivalent.
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.
You can now play legal moves to create follow-up positions for the
engine to analyze, while the moves will be remembered as a stored game,
and then step backward through this game to take the moves back.
Note that you can also click on the clocks to set the opposite
side to move (adding a so-called @samp{null move} to the game).
You can also tell the engine to exclude some moves from analysis.
(Engines that do not support the exclude-moves feature will
ignore this, however.)
The general way to do this is to play the move you want to exclude
starting with a double click on the piece.
When you use drag-drop moving, the piece you grab with a double click
will also remain on its square, to show you that you are not really
making the move, but just forbid it from the current position.
Playing a thus excluded move a second time will include it again.
Excluded moves will be listed as text in a header line in the
Engine Output window, and you can also re-include them by
right-clicking them there.
This header line will also contain the words 'best' and 'tail';
right-clicking those will exclude the currently best move,
or all moves not explicitly listed in the header line.
Once you leave the current position all memory of excluded
moves will be lost when you return there.
Selecting this menu item while already in @samp{Analysis Mode} will
toggle the participation of the second engine in the analysis.
The output of this engine will then be shown in the lower pane
of the Engine Output window.
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 Game
@cindex Analyze Game, Menu Item
This option subjects the currently loaded game to automatic
analysis by the loaded engine.
The @kbd{Ctrl-G} key is a keyboard equivalent.
XBoard will start auto-playing the game from the currently displayed position,
while the engine is analyzing the current position.
The game will be annotated with the results of these analyses.
In particlar, the score and depth will be added as a comment,
and the PV will be added as a variation.
Normally the analysis would stop after reaching the end of the game.
But when a game is loaded from a multi-game file
while @samp{Analyze Game} was already switched on,
the analysis will continue with the next game in the file
until the end of the file is reached (or you switch to another mode).
The time the engine spends on analyzing each move can be controlled
through the command-line option @samp{-timeDelay},
which can also be set from the @samp{Load Game Options} menu dialog.
Note: Some chess engines do not support Analysis mode.
@item Edit Game
Duplicate of the item in the Edit menu.
Note that @samp{Edit Game} is the idle mode of XBoard, and can be used
to get you out of other modes. E.g. to stop analyzing, stop a game
between two engines or stop editing a position.
@item Edit Position
Duplicate of the item in the Edit menu.
@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 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 Machine Match
@cindex Machine match, Menu Item
Starts a match between two chess programs,
with a number of games and other parameters set through
the @samp{Match Options} menu dialog.
When a match is already running, selecting this item will make
XBoard drop out of match mode after the current game finishes.
@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{Pause} 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.
The @kbd{F3} key is a keyboard equivalent.
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.).
The @kbd{F4} key is a keyboard equivalent. 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.
The @kbd{F5} key is a keyboard equivalent.
You can also call your
opponent's flag by clicking on his clock.
@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{F6} 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.
The @kbd{F7} key is a keyboard equivalent.
@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.
The @kbd{F8} key is a keyboard equivalent. An aborted
game ends immediately without affecting either player's rating.
@item Resign
@cindex Resign, Menu Item
Resigns the game to your opponent. The @kbd{F9} 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.
The @kbd{F10} key is a keyboard equivalent.
@item Stop Examining
@cindex Stop Examining, Menu Item
Ends your participation in examining a game, by issuing the ICS
unexamine command. ICS mode only.
The @kbd{F11} key is a keyboard equivalent.
@item Upload to Examine
@cindex Upload to Examine, Menu Item
Create an examined game of the proper variant on the ICS,
and send the game there that is currenty loaded in XBoard
(e.g. through pasting or loading from file).
You must be connected to an ICS for this to work.
@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 Engine Menu
@section Engine Menu
@cindex Engine Menu
@cindex Menu, Engine
@table @asis
@item Load Engine
@cindex Load Engine, Menu Item
Pops up a dialog where you can select or specify an engine to be loaded.
You will always have to indicate whether you want to load the engine
as first or second engine, through the ‘Load menitioned engine as’
drop-down list at the bottom of the dialog.
You can even replace engines during a game, without disturbing that game.
(Beware that after loading an engine, XBoard will always be in Edit Game mode,
so you will have to tell the new engine what to do before it does anything!)
When you select an already installed engine from the ‘Select Engine from List’
drop-down list,
all other fields of the dialog will be ignored.
In other cases, you have to specify the engine executable,
possible arguments on the engine command line
(if the engine docs say the engine needs any),
and the directory where the engine should look for its files
(if this cannot be deduced automatically from the specification of the engine executable).
You will also have to specify (with the aid of checkboxes) if the engine is UCI.
If ‘Add this engine to the list’ is ticked (which it is by default),
the engine will be added to the list of installed engines in your settings file,
(provided you save the settings!),
so that next time you can select it from the drop-down list.
You can also specify a ‘nickname’,
under which the engine will then appear in that drop-down list,
and even choose to use that nickname for it in PGN files for engine-engine games.
The info you supply with the checkboxes whether the engine should use GUI book,
or (for variant engines) automatically switch to the current variant when loaded,
will also be included in the list.
For obsolete XBoard engines, which would normally take a long delay to load
because XBoard is waiting for a response they will not give,
you can tick ‘WB protocol v1’ to speed up the loading process.
@item Engine #N Settings
@cindex Engine Settings, Menu Item
@cindex Engine #1 Settings, Menu Item
@cindex Engine #2 Settings, Menu Item
Pop up a menu dialog to alter the settings specific to the applicable engine.
(The second engine is only accessible once it has been used in Two-Machines mode.)
For each parameter the engine allows to be set,
a control element will appear in this dialog that can be used to alter the value.
Depending on the type of parameter (text string, number, multiple choice,
on/off switch, instantaneous signal) the appropriate control will appear,
with a description next to it.
XBoard has no idea what these values mean; it just passes them on to the engine.
How this dialog looks is completely determined by the engine,
and XBoard just passes it on to the user.
Many engines do not have any parameters that can be set by the user,
and in that case the dialog will be empty (except for the OK and cancel buttons).
UCI engines usually have many parameters. (But these are only visible with
a sufficiently modern version of the Polyglot adapter needed to run UCI engines,
e.g. Polyglot 1.4.55b.) For native XBoard engines this is less common.
@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 Move Now
@cindex Move Now, Menu Item
Forces the chess engine to move immediately. Chess engine mode only.
The @kbd{Ctrl-M} key is a keyboard equivalent.
@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.
The @kbd{Ctrl-X} key is a keyboard equivalent.
@item Recently Used Engines
@cindex Recently Used Engines, In Menu
At the bottom of the engine menu there can be a list of names
of engines that you recently loaded through the Load Engine menu dialog
in previous sessions.
Clicking on such a name will load that engine as first engine,
so you won't have to search for it in your list of installed engines,
if that is very long.
The maximum number of displayed engine names is set by the
@code{recentEngines}command-line option.
@end table
@node Options Menu
@section Options Menu
@cindex Menu, Options
@cindex Options Menu
@subsection General Options
@cindex General Options, Menu Item
The following items to set option values appear in the dialog
summoned by the general Options menu item.
@table @asis
@item Absolute Analysis Scores
@cindex Absolute Analysis Scores, Menu Item
Controls if scores on the Engine Output window during analysis
will be printed from the white or the side-to-move point-of-view.
@item Almost Always Queen
@cindex Almost Always Queen, Menu Item
If this option is on, 7th-rank pawns automatically change into
Queens when you pick them up,
and when you drag them to the promotion square and release them there,
they will promote to that.
But when you drag such a pawn backwards first,
its identity will start to cycle through the other available pieces.
This will continue until you start to move it forward;
at which point the identity of the piece will be fixed,
so that you can safely put it down on the promotion square.
If this option is off, what happens depends on the
option @code{alwaysPromoteToQueen},
which would force promotion to Queen when true.
Otherwise XBoard would bring up a dialog
box whenever you move a pawn to the last rank, asking what piece
you want to promote to.
@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.
The shifted @kbd{Ctrl-A} key is a keyboard equivalent.
@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.
The shifted @kbd{Ctrl-F} key is a keyboard equivalent.
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.
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 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 Drop Menu
@cindex Drop Menu, Menu Item
Controls if right-clicking the board in crazyhouse / bughouse
will pop up a menu to drop a piece on the clicked square
(old, deprecated behavior)
or allow you to step through an engine PV
(new, recommended behavior).
@item Enable Variation Trees
@cindex Enable Variation Trees, Menu Item
If this option is on, playing a move in Edit Game or Analyze mode
while keeping the Shift key pressed will start a new variation.
You can then recall the previous line through the @samp{Revert} menu item.
When off, playing a move will truncate the game and append the move
irreversibly.
@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.
The shifted @kbd{Ctrl-H} key is a keyboard equivalent.
@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 Highlight with Arrow
@cindex Highlight with Arrow, Menu Item
Causes the highlighting described in Highlight Last Move to be done
by drawing an arrow between the highlighted squares,
so that it is visible even when the width of the grid lines is set to zero.
@item Move Sound
@cindex Move Sound, Menu Item
Enables the sounding of an audible signal when the computer performs a move.
For the selection of the sound, see @samp{Sound Options}.
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 One-Click Moving
@cindex One-Click Moving, Menu Item
If this option is on, XBoard does not wait for you to click both the
from- and the to-square, or drag the piece, but performs a move as soon
as it is uniqely specified.
This applies to clicking an own piece that only has a single legal move,
clicking an empty square or opponent piece where only one of your pieces
can move (or capture) to.
Furthermore, a double-click on a piece that can only make a single capture
will cause that capture to be made.
Promoting a Pawn by clicking its to-square will suppress the promotion
popup or other methods for selecting an under-promotion,
and make it promote to Queen.
@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 Play Move(s) of Clicked PV
@cindex Play Move(s) of Clicked PV, Menu Item
If this option is on, right-clicking a PV in the Engine Output window
during Analyze mode will cause the first move of that PV to be played.
You could also play more than one (or no) PV move by moving the mouse
to engage in the PV walk such a right-click will start,
to seek out another position along the PV where you want to continue
the analysis, before releasing the mouse button.
@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.
The shifted @kbd{Ctrl-P} key is a keyboard equivalent.
@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 Scores in Move List
@cindex Scores in Move List, Menu Item
If this option is on, XBoard will display the depth and score
of engine moves in the Move List, in the format of a PGN comment.
@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 Show Target Squares
@cindex Show Target Squares, Menu Item
If this option is on, all squares a piece that is 'picked up' with the mouse
can legally move to are highighted with a fat colored dot in the
highlightColor (non-captures) or premoveHighlightColor (captures).
Legality testing must be on for XBoard to know how the piece moves.
@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.
The shifted @kbd{Ctrl-L} key is a keyboard equivalent.
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.)
@item Flash Moves
@itemx Flash Rate
@cindex Flash Moves, Menu Item
@cindex Flash Rate, Menu Item
If this option is non-zero, whenever a move is completed,
the moved piece flashes the specified number of times.
The flash-rate setting determines how rapidly this flashing occurs.
@item Animation Speed
@cindex Animation Speed, Menu Item
Determines the duration (in msec) of an animation step,
when @samp{Animate Moving} is swiched on.
@item Zoom factor in Evaluation Graph
@cindex Zoom factor in Evaluation Graph, Menu Item
Sets the valueof the @code{evalZoom} option,
indicating the factor by which the score interval (-1,1) should be
blown up on the vertical axis of the Evaluation Graph.
@end table
@subsection 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.
The shifted @kbd{Alt+T} key is a keyboard equivalent.
@subsection Common Engine
@cindex Common Engine, Menu Item
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 XBoard/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.
The way book moves are chosen can be influenced through the settings of
book depth and variety.
After both sides have played more moves than the specified depth,
the book will no longer be consulted.
When the variety is set to 50, moves will be played with the probability
specified in the book.
When set to 0, only the move(s) with the highest probability will be played.
When set to 100, all listed moves will be played with equal pobability.
Other settings interpolate between that.
The shifted @kbd{Alt+U} key is a keyboard equivalent.
@subsection 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.
The shifted @kbd{Alt+J} key is a keyboard equivalent.
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.
@subsection ICS Options
@cindex ICS Options, Menu Item
The following options occur in a dialog summoned by the
ICS Options menu item.
@table @asis
@item Auto Kibitz
@cindex Auto Kibitz, Menu Item
Setting this option when playing with or aginst a chess program on an ICS
will cause the last line of thinking output of the engine before its move
to be sent to the ICS in a kibitz command.
In addition, any kibitz message received through the ICS from
an opponent chess program will be diverted to the engine-output window,
(and suppressed in the console),
where you can play through its PV by right-clicking it.
@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 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 Background Observe
@cindex Background Observe, Menu Item
Setting this option will make XBoard suppress display of any boards
from observed games while you are playing.
In stead the last such board will be remembered,
and shown to you when you right-click the board.
This allows you to peek at your bughouse partner's game when you want,
without disturbing your own game too much.
@item Dual Board
@cindex Dual Board, Menu Item
Setting this option in combination with @samp{Background Observe}
will display boards of observed games while you are playing
on a second board next to that of your own game.
@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 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 Seek Graph
@cindex Seek Graph, Menu Item
Setting this option will cause XBoard to display an graph of
currently active seek ads when you left-click the board
while idle and logged on to an ICS.
@item Auto-Refresh Seek Graph
@cindex Auto-Refresh Seek Graph, Menu Item
In combination with the @samp{Seek Graph} option this
will cause automatic update of the seek graph while it is up.
This only works on FICS and ICC,
and requires a lot of bandwidth on a busy server.
@item Premove
@itemx Premove White
@itemx Premove Black
@itemx First White Move
@itemx First Black Move
@cindex Premove, Menu Item
@cindex Premove White, Menu Item
@cindex Premove Black, Menu Item
@cindex First White Move, Menu Item
@cindex First Black Move, 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.
You can also enter premoves for the first white and black moves
of the game.
@item ICS Alarm
@itemx ICS Alarm Time
@cindex ICS Alarm, Menu Item
@cindex ICS Alarm Time, Menu Item
When this option is on, an alarm sound is played when your clock
counts down to the icsAlarmTime in an ICS game.
(By default, the time is 5 seconds, but you can pecify other values
with the Alarm Time spin control.)
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 Colorize Messages
@cindex Colorize Messages, Menu Item
Ticking this options causes various types of ICS messages do be
displayed with different foreground or background colors in the console.
The colors can be individually selected for each type,
through the accompanying text edits.
@end table
@subsection Match Options
@cindex Match Options, Menu Item
Summons a dialog where you can set options important for playing automatic
matches between two chess programs
(e.g. by using the @samp{Machine Match} menu item in the @samp{Mode} menu).
@table @asis
@item Tournament file
@cindex Tournament file, Menu item
To run a tournament, XBoard needs a file to record its progress,
so it can resume the tourney when it is interrupted.
When you want to conduct anything more complex than a simple
two-player match with the currently loaded engines,
(i.e. when you select a list of participants),
you must not leave this field blank.
When you enter the name of an existing tournament file,
XBoard will ignore all other input specified in the dialog,
and will take them from that tournament file.
This resumes an interrupted tournament, or adds another XBoard
agent playing games for it to those that are already doing so.
Specifying a not-yet-existing file will cause XBoard to create it,
according to the tournament parameters specified in the rest of the dialog,
before it starts the tournament on ‘OK’.
Provided that you specify participants;
without participants no tournament file will be made, but other entered values
(e.g. for the file with opening positions) will take effect.
Default: configured by the @code{defaultTourneyName} option.
@item Sync after round
@itemx Sync after cycle
@cindex Sync after round, Menu Item
@cindex Sync after cycle, Menu Item
The sync options, when on, will cause WinBoard to refrain from starting games
of the next round or cycle before all games of the previous round or cycle are finished.
This guarantees correct ordering in the games file,
even when multiple XBoard instances are concurrently playing games for the same tourney.
Default: sync after cycle, but not after round.
@item Select Engine
@itemx Tourney participants
@cindex Select Engine, Menu Item
@cindex Tourney participants, Menu Item
With the Select Engine drop-down list you can pick an engine from your list
of installed engines in the settings file, to be added to the tournament.
The engines selected so far will be listed in the ‘Tourney participants’ memo.
The latter is a normal text edit, so you can use normal text-editing functions
to delete engines you selected by accident, or change their order.
Do not type names yourself there, because names that do not exactly match
one of the names from the drop-down list will lead to undefined behavior.
@item Tourney type
@cindex Tourney type, Menu Item
Here you can specify the type of tournament you want.
XBoard’s intrinsic tournament manager support round-robins (type = 0),
where each participant plays every other participant, and (multi-)gauntlets,
where one (or a few) so-called ‘gauntlet engines’ play an independent set of opponents.
In the latter case, you specify the number of gauntlet engines.
E.g. if you specified 10 engines, and tourney type = 2,
the first 2 engines each play the remaining 8.
A value of -1 instructs XBoard to play Swiss; for this to work an external
pairing engine must be specified through the @code{pairingEngine} option.
Each Swiss round will be considered a tourney cycle in that case.
Default:0
@item Number of tourney cycles
@itemx Default number of Games
@cindex Number of tourney cycles, Menu Item
@cindex Default number of Games, Menu Item
You can specify tourneys where every two opponents play each other multiple times.
Such multiple games can be played in a row,
as specified by the ‘number of games per pairing’,
or by repeating the entire tournament schedule a number of times
(specified by the ‘number of tourney cycles’).
The total number of times two engine meet will be the product of these two.
Default is 1 cycle;
the number of games per pairing is the same as the default number of match games,
stored in your settings file through the @code{defaultMatchGames} option.
@item Save Tourney Games
@cindex Save Tourney Games, Menu Item
File where the tournament games are saved
(duplicate of the item in the @samp{Save Game Options}).
@item Game File with Opening Lines
@itemx File with Start Positions
@itemx Game Number
@itemx Position Number
@itemx Rewind Index after
@cindex Game File with Opening Lines, Menu Item
@cindex File with Start Positions, Menu Item
@cindex Game Number, Menu Item
@cindex Position Number, Menu Item
@cindex Rewind Index after, Menu Item
These items optionally specify the file with move sequences or board positions the tourney
games should start from.
The corresponding numbers specify the number of the game or position in the file.
Here a value -1 means automatic stepping through all games on the file,
-2 automatic stepping every two games.
The Rewind-Index parameter causes a stepping index to reset to one after reaching
a specified value.
A setting of -2 for the game number will also be effective in a tournament without
specifying a game file, but playing from the GUI book instead.
In this case the first (odd) games will randomly select from the book,
but the second (even) games will select the same moves from the book as the previous game.
(Note this leads to the same opening only if both engines use the GUI book!)
Default: No game or position file will be used. The default index if such a file is used is 1.
@item Disable own engine bools be default
@cindex Disable own engine bools be default, Menu Item
Setting this option reverses the default situation for use of the GUI opening book
in tournaments from what it normally is, namely not using it.
So unless the engine is installed with an option to explicitly specify it should
not use the GUI book (i.e. @code{-firstHasOwnBookUCI true}),
it will be made to use the GUI book.
@item Replace Engine
@itemx Upgrade Engine
@cindex Replace Engine, Menu Item
@cindex Upgrade Engine, Menu Item
With these two buttons you can alter the participants of an already running tournament.
After opening the Match Options dialog on an XBoard that is playing for the tourney,
you will see all the tourney parameters in the dialog fields.
You can then replace the name of one engine by that of another
by editing the @samp{participants} field.
(But preserve the order of the others!)
Pressing the button after that will cause the substitution.
With the @samp{Upgrade Engine} button the substitution will only affect future games.
With @samp{Replace Engine} all games the substituted engine has already played will
be invalidated, and they will be replayed with the substitute engine.
In this latter case the engine must not be playing when you do this,
but otherwise there is no need to pause the tournament play
for making a substitution.
@item Clone Tourney
@cindex CloneTourney, Menu Item
Pressing this button after you have specified an existing tournament file
will copy the contents of the latter to the dialog,
and then puts the originally proposed name for the tourney file back.
You can then run a tourney with the same parameters
(possibly after changing the proposed name of the tourney file for the new tourney)
by pressing 'OK'.
@end table
@subsection Load Game Options
@cindex Load Game Options, Menu Item
Summons a dialog where you can set the @code{autoDisplayComment} and
@code{autoDisplayTags} options, (which control popups when viewing loaded games),
and specify the rate at which loaded games are auto-played,
in seconds per move (which can be a fractional number, like 1.6).
You can also set search criteria for determining which games
will be displayed in the Game List for a multi-game file,
and thus be eligible for loading:
@table @asis
@item Elo of strongest player
@itemx Elo of weakest player
@itemx year
@cindex Elo of strongest player, Menu Item
@cindex Elo of weakest player, Menu Item
@cindex year, Menu Item
These numeric fields set thresholds (lower limits) on the Elo rating of the mentioned player,
or the date the game was played.
Defaults: 0
@item Search mode
@cindex Search mode, Menu Item
@cindex find position, Menu Item
@cindex narrow, Menu Item
This setting determines which positions in a game will be considered a match
to the position currently displayed in the board window
when you press the @samp{find position} button in the Game List.
You can search for an exact match,
a position that has all shown material in the same place,
but might contain additional material,
a position that has all Pawns in the same place,
but can have the shown material anywhere,
a position that can have all shown material anywhere,
or a position that has material between certain limits anywhere.
For the latter you have to place the material that must be present
in the four lowest ranks of the board,
and optional additional material in the four highest ranks of the board.
You can request the optional material to be balanced.
The @samp{narrow} button is similar in fuction to the @samp{find position} button,
but only searches in the already selected games,
rather than the complete game file,
and can thus be used to refine a search based on multiple criteria.
@item number of consecutive positions
@cindex number of consecutive positions, Menu Item
When you are searching by material, rather than for an exact match,
this parameter indicates forhowmany consecutive game positions
the same amount of material must be on the board before it is
considered a match.
@item Also match reversed colors
@itemx Also match left-right flipped position
@cindex Also match reversed colors, Menu Item
@cindex Also match left-right flipped position, Menu Item
When looking for matching positions rather than by material,
these settings determine whether mirror images
(in case of a vertical flip in combination with color reversal)
will be also considered a match.
The left-right flipping is only useful after all castling rights
have expired (or in Xiangqi).
@end table
@subsection Save Game Options
@cindex Save Game Options, Menu Item
Summons a dialog where you can specify the files on which XBoard should
automatically save any played or entered games,
(the @code{saveGameFile} option),
or the final position of such games (the @code{savePositionfile} option).
You can also select 'auto-save' without a file name,
in which case XBoard will prompt the user for a file name after each game.
In ICS mode you can limit the auto-saving to your own games
(i.e. suppress saving of observed games).
You can also set the default value for the PGN Event tag that will
be used for each new game you start.
Various options for the format of the game can be specified as well,
such as whether scores and depths of engine games should be saved as comments,
and if a tag with info about the score with which the engine came out of book
should be included.
For Chess, always set the format to PGN, rather than "old save stye"!
@subsection Game List
@cindex Game List Tags, Menu Item
Pops up a dialog where you can select the PGN tags that should appear
on the lines in the game list, and their order.
@subsection Sound Options
@cindex Sound Options, Menu Item
Summons a dialog where you can specify the sounds that should accompany
various events that can occur XBoard.
Most events are only relevant to ICS play,
but the move sound is an important exception.
For each event listed in the dialog,
you can select a standard sound from a menu.
You can also select a user-supplied sound file,
by typing its name into the designated text-edit field first,
and then selecting "Above WAV File" from the menu for the event.
A dummy event has been provided for trying out the sounds with the
"play" button next to it.
The directory with standard sounds, and the external program for playing
the sounds can be specified too, but normally you would not touch these
once XBoard is properly installed.
When a move sound other than 'None' is selected,
XBoard alerts you by playing that 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.
@subsection Save Settings Now
@cindex Save Settings Now, Menu Item
Selecting this menu item causes the current XBoard settings to be
written to the settings file, so they will also apply in future sessions.
Note that some settings are 'volatile', and are not saved,
because XBoard considers it too unlikely that you want those to apply
next time.
In particular this applies to the Chess program names, and all options
giving information on those Chess programs (such as their directory,
if they have their own opening book, if they are UCI or native XBoard),
or the variant you are playing.
Such options would still be understood when they appear in the settings
file in case they were put there with the aid of a text editor, but they
would disappear from the file as soon as you save the settings.
Note that XBoard no longer pays attention to options values specified
in the .Xresources file.
(Specifying key bindings there will still work, though.)
To alter the default of volatile options, you can use the following method:
Rename your ~/.xboardrc settings file (to ~/.yboardrc, say), and create
a new file ~/.xboardrc, which only contains the options
@example
-settingsFile ~/.yboardrc
-saveSettingsFile ~/.yboardrc
@end example
@noindent
This will cause your settings to be saved on ~/.yboardrc in the future,
so that ~/.xboardrc is no longer overwritten.
You can then safely specify volatile options in ~/.xboardrc, either
before or after the settingsFile options.
Note that when you specify persistent options after the settingsFile options
in ~/.xboardrc, you will essentially turn them into volatile options
with the specified value as default, because that value will overrule
the value loaded from the settings file (being read later).
@subsection Save Settings on Exit
@cindex Save Settings on Exit, Menu Item
Setting this option has no immediate effect, but causes the settings
to be saved when you quit XBoard. What happens then is otherwise
identical to what happens when you use select "Save Settings Now",
see there.
@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.
The @kbd{F1} key is a keyboard equivalent. 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 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 Show Last Move
@cindex Show Last Move, Shortcut Key
By hitting @kbd{Enter} the last move will be re-animated.
@item Load Next Game
@cindex Load Next Game, Menu Item
Loads the next game from the last game record file you loaded.
The @kbd{Alt+PgDn} key triggers this action.
@item Load Previous Game
@cindex Load Previous Game, Menu Item
Loads the previous game from the last game record file you
loaded. The @kbd{Alt+PgUp} key triggers this action.
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.
Currently no keystroke is assigned to this ReloadGameProc.
@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.
Currently no keystroke is assigned to this ReloadPositionProc.
@end table
In the Xaw build of XBoard 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>?: MenuItem(Help.About) \n\
Ctrl<Key>y: MenuItem(Action.Accept) \n\
Ctrl<Key>n: MenuItem(Action.Decline) \n\
Ctrl<Key>i: MenuItem(Nothing)
@end example
@noindent
So the key should always be bound to the action 'MenuItem',
with the (hierarchical) name of the menu item as argument.
There are a few actions available for which no menu item exists:
Binding a key to @code{Nothing} makes it do nothing, thus removing
it as a shortcut key. Other such functions that can be bound to keys
are:
@example
AboutGame, DebugProc (switches the -debug option on or off),
LoadNextGame, LoadPrevGame, ReloadGame, ReloadPosition.
@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 editing the settings file
(usually ~/.xboardrc) to alter the value of the setting that was
saved there. Some 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.
@menu
* Chess engine options:: Controlling the chess engine.
* UCI + WB Engine Settings:: Setting some very common engine parameters
* Tournament options:: Running tournaments and matches between engines.
* 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 XBoard/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 -fe or -firstEngine nickname
@cindex fe, option
@cindex firstEngine, option
This is an alternative to the @code{fcp} option for specifying the first engine,
for engines that were already configured (using the @samp{Load Engine} dialog)
in XBoard's settings file.
It will not only retrieve the real name of the engine,
but also all options configured with it.
(E.g. if it is UCI, whether it should use book.)
@item -se or -secondEngine nickname
@cindex se, option
@cindex secondEngine, option
As @code{fe}, but for the second engine.
@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 or -firstInitString
@itemx -secondInitString string
@cindex initString, option
@cindex firstInitString, 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.
Using the character sequence @samp{\n} in the string should work too, though.
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 XBoard (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 an option announced by the engine at startup through the feature commands of the XBoard/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 XBoard would normally omit them
(string = "- -"), or to add variant-specific fields that are not yet supported by XBoard
(e.g. to indicate the number of checks in 3check).
@item -shuffleOpenings
@cindex shuffleOpenings, option
Forces shuffling of the opening setup in variants that normally have a fixed initial position.
Shufflings are symmetric for black and white, and exempt King and Rooks in variants
with normal castling.
Remains in force until a new variant is selected.
@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
on its command line, according to the option @code{adapterCommand}.
@item -fUCCI
@itemx -sUCCI
@itemx -fUSI
@itemx -sUSI
@cindex fUCCI, option
@cindex sUCCI, option
@cindex fUSI, option
@cindex sUSI, option
Options similar to @code{fUCI} and @code{sUCI}, except that they
use the indicated engine with the protocol adapter specified in
the @samp{uxiAdapter} option.
This can then be configured for running an UCCI or USI adapter,
as the need arises.
@item -adapterCommand string
@cindex adapterCommand, option
The string conatins the command that should be issued by XBoard
to start an engine that is accompanied by the @code{fUCI} option.
Any identifier following a percent sign in the command (e.g. %fcp)
will be considered the name of an XBoard option, and be replaced
by the value of that option at the time the engine is started.
For starting the second engine, any leading "f" or "first" in
the option name will first be replaced by "s" or "second",
before finding its value.
Default: 'polyglot -noini -ec "%fcp" -ed "%fd"'
@item -uxiAdapter string
@cindex uxiAdapter, option
Similar to @code{adapterCommand}, but used for engines accompanied
by the @code{fUCCI} or @code{fUSI} option, so you can configure
XBoard to be ready to handle more than one flavor of non-native protocols.
Default: ""
@item -polyglotDir filename
@cindex polyglotDir, option
Gives the name of the directory in which the Polyglot adapter for UCI engines resides.
Default: "".
@item -usePolyglotBook true/false
@cindex usePolyglotBook, option
Specifies if the Polyglot book should be used as GUI book.
@item -polyglotBook filename
@cindex polyglotBook, option
Gives the filename of the opening book.
The book is only used when 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: depends on setting of the option @code{discourageOwnBooks}.
@item -discourageOwnBooks true/false
@cindex discourageOwnBooks, option
When set, newly loaded engines will be assumed to use the GUI book,
unless they explicitly specify differently.
Otherwise they will be assumed to not use the GUI book,
unless the specify differently (e.g. with @code{firstXBook}).
Default: false.
@item -bookDepth n
@cindex bookDepth, option
Limits the use of the GUI book to the first n moves of each side.
Default: 12.
@item -bookVariation n
@cindex bookVariation, option
A value n from 0 to 100 tunes the choice of moves from the GUI books
from totally random to best-only. Default: 50
@item -mcBookMode
@cindex mcBookMode, option
When this volatile option is specified, the probing algorithm of the
GUI book is altered to always select the move that is most under-represented
based on its performance.
When all moves are played in approximately the right proportion,
a book miss will be reported, to give the engine opportunity to
explore a new move.
In addition score of the moves will be kept track of during the session
in a book buffer.
By playing an match in this mode, a book will be built from scratch.
The only output are the saved games, which can be converted to an
actual book later, with the @samp{Save Games as Book} command.
This command can also be used to pre-fill the book buffer
before adding new games based on the probing algorithm.
@item -fn string or -firstPgnName string
@itemx -sn string or -secondPgnName string
@cindex firstPgnName, option
@cindex secondPgnName, option
@cindex fn, option
@cindex sn, option
Indicates the name that should be used for the engine in PGN tags of
engine-engine games.
Intended to allow you to install verions of the same engine with different settings,
and still distinguish them.
Default: "".
@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 XBoard/WinBoard engines,
for those that support the memory feature of the XBoard/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 XBoard/WinBoard engines,
for those that support the memory feature of the XBoard/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: "".
@item -firstChessProgramNames=@{names@}
This option lets you customize the drop-down list of chess engine names
that appears in the @samp{Load Engine} and @samp{Match Options} dialog.
It consists of a list of strings, one per line.
When an engine is loaded, the corresponding line is prefixed with "-fcp ",
and processed like it appeared on the command line.
That means that apart from the engine command,
it can contain any list of XBoard options you want to use with this engine.
(Commonly used options here are -fd, -firstXBook, -fUCI, -variant.)
The value of this option is gradually built as you load new engines
through the @samp{Load Engine} menu dialog, with @samp{Add to list} ticked.
To change it, edit your settings file with a plain text editor.
@end table
@node Tournament options
@section Tournament options
@cindex Tournament Options
@cindex Options, Tournament
@table @asis
@item -defaultMatchGames n
@cindex defaultMatchGames, option
Sets the number of games that will be used for a match between two engines
started from the menu to n. Also used as games per pairing in other tournament
formats. Default: 10.
@item -matchPause n
@cindex matchPause, option
Specifies the duration of the pause between two games of a match or tournament
between engines as n milliseconds.
Especially engines that do not support ping need this option,
to prevent that the move they are thinking on when an opponent unexpectedly
resigns will be counted for the next game, (leading to illegal moves there).
Default: 10000.
@item -tf filename or -tourneyFile filename
@cindex tf, option
@cindex tourneyFile, option
Specifies the name of the tournament file used in match mode
to conduct a multi-player tournament.
This file is a special settings file,
which stores the description of the tournament (including progress info),
through normal options (e.g. for time control, load and save files),
and through some special-purpose options listed below.
@item -tt number or -tourneyType number
@cindex tt, option
@cindex tourneyType, option
Specifies the type of tourney: 0 = round-robin,
N>0 = (multi-)gauntlet with N gauntlet engines,
-1 = Swiss through external pairing engine.
Volatile option, but stored in tourney file.
@item -cy number or -tourneyCycles number
@cindex cy, option
@cindex tourneyCycles, option
Specifies the number of cycles in a tourney.
Volatile option, but stored in tourney file.
@item -participants list
@cindex participants, option
The list is a multi-line text string that specifies engines
occurring in the @code{firstChesProgramNames} list
in the settings file by their (implied or explicitly given) nicknames,
one engine per line.
The mentioned engines will play in the tourney.
Volatile option, but stored in tourney file.
@item -results string
@cindex results, option
The string of +=- characters lists the result of all played games in a toruney.
Games currently playing are listed as *,
while a space indicates a game that is not yet played or playing .
Volatile option, but stored in tourney file.
@item -defaultTourneyName string
@cindex defaultTourneyName, option
Specifies the name of the tournament file XBoard should propose
when the @samp{Match Options} dialog is opened.
Any %y, %M, %d, %h, %m, %s in the string are replaced by the current
year, month, day of the month, hours, minutes, seconds of the current time,
respectively, as two-digit number.
A %Y would be replaced by the year as 4-digit number. Default: empty string.
@item -pairingEngine filename
@cindex pairingEngine, option
Specifies the external program to be used to pair the participants in Swiss tourneys.
XBoard communicates with this engine in the same way as it communicates with Chess engines.
The only commands sent to the pairing engine are “results N string”,
(where N is the number of participants,
and string the results so far in the format of the results option),
and “pairing N”, (where N is the number of the tourney game).
To the latter the pairing engine should answer with “A-B”,
where A and B are participant numbers (in the range 1-N).
(There should be no reply to the results command.) Default: empty string.
@item -afterGame string
@itemx -afterTourney string
@cindex afterGame, option
@cindex afterTourney, option
When non-empty, the given string will be executed as a system command
after each tournament game, orafterthe tourney completes, respectively.
This can be used, for example, to autmatically run a cross-table generator
on the PGN file where games are saved, to update the tourney standings.
Default: ""
@item -syncAfterRound true/false
@itemx -syncAfterCycle true/false
@cindex syncAfterRound, option
@cindex syncAfterCycle, option
Controls whether different instances of XBoard concurrently running the
same tournament will wait for each other.
Defaults: sync after cycle, but not after round.
@item -seedBase number
@cindex seedBase, option
Used to store the seed of the pseudo-random-number generator in the
tourneyFile, so that separate instances of XBoard working on the same
tourney can take coherent 'random' decisions, such as picking an
opening for a given game number.
@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 -seekGraph true/false or -sg
@cindex seekGraph, option
@cindex sg, option
Enables displaying of the seek graph by left-clicking the board when
you are logged on to an ICS and currently idle.
The seek graph show all players currently seeking games on the ICS,
plotted according to their rating and the time control of the game they seek,
in three different colors (for rated, unrated and wild games).
Computer ads are displayed as squares, human ads are dots.
Default: false.
@item -autoRefresh true/false
@cindex autoRefresh, option
Enables automatic updating of the seek graph,
by having the ICS send a running update of all newly placed
and removed seek ads.
This consumes a substantial amount of communication bandwidth,
and is only supported for FICS and ICC.
Default: false.
@item -backgroundObserve true/false
@cindex backgroundObserve, option
When true, boards sent to you by the ICS from other games while you are playing
(e.g. because you are observing them)
will not be automatically displayed.
Only a summary of time left and material of both players will appear
in the message field above the board.
XBoard will remember the last board it has received this way,
and will display it in stead of the position in your own game
when you press the right mouse button.
No other information is stored on such games observed in the background;
you cannot save such a game later, or step through its moves.
This feature is provided solely for the benefit of bughouse players,
to enable them to peek at their partner's game without the need
to logon twice.
Default: false.
@item -dualBoard true/false
@cindex dualBoard, option
In combination with -backgroundObserve true, this option will display
the board of the background game side by side with that of your own game,
so you can have it in view permanently.
Any board or holdings info coming in will be displayed on the secondary
board immediately.
This feature is still experimental and largely unfinished.
There is no animation or highlighting of moves on the secondary board.
Default: false.
@item -disguisePromotedPieces true/false
@cindex disguisePromotedPieces, option
When set promoted Pawns in crazyhouse/bughouse are displayed identical
to primordial pieces of the same type, rather than distinguishable.
Default: true.
@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 -prewhite/-xprewhite or -premoveWhite
@itemx -preblack/-xpreblack or -premoveBlack
@itemx -premoveWhiteText string
@itemx -premoveBlackText string
@cindex prewhite, option
@cindex premoveWhite, option
@cindex preblack, option
@cindex premoveBlack, option
@cindex premoveWhiteText, option
@cindex premoveBlackText, option
Set the menu options for specifying the first move for either color.
@xref{Options Menu}. Defaults: false and empty strings, so no pre-moves.
@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/-xcolorize
@cindex Colors
@cindex colorize, option
@cindex colorizeMessages, 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.
Default: true.
@item -colorShout foreground,background,bold
@itemx -colorSShout foreground,background,bold
@itemx -colorCShout 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 colorCShout, 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.
@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 -soundDirectory directoryname
@cindex soundDirectory, option
@cindex Sounds
This option specifies where XBoard will look for sound files,
when these are not given as an absolute path name.
@item -soundShout filename
@itemx -soundSShout filename
@itemx -soundCShout filename
@itemx -soundChannel filename
@itemx -soundChannel1 filename
@itemx -soundKibitz filename
@itemx -soundTell filename
@itemx -soundChallenge filename
@itemx -soundRequest filename
@itemx -soundSeek filename
@cindex soundShout, option
@cindex soundSShout, option
@cindex soundCShout, option
@cindex soundChannel, option
@cindex soundChannel1, 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.
CShout is synonymous with SShout.
@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).
@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} or @samp{Analyze File}.
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 -onlyOwnGames true/false
@cindex onlyOwnGames, option
Suppresses auto-saving of ICS observed games. Default: false.
@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, XBoard 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 -pgnNumberTag true/false
@cindex pgnNumberTag, option
Include the (unique) sequence number of a tournament game into the saved
PGN file as a 'number' tag.
Default: false.
@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.
Default: true.
@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, c=result comment.
Default: "eprd"
@item -ini or -settingsFile filename
@itemx -saveSettingsFile filename
@itemx @@filename
@cindex saveSettingsFile, option
@cindex SettingsFile, option
@cindex init, option
@cindex at sign, option
When XBoard encounters an option -settingsFile (or -ini for short),
or @@filename, it tries to read the mentioned file,
and substitutes the contents of it (presumaby more command-line options)
in place of the option.
In the case of -ini or -settingsFile, the name of a successfully read
settings file is also remembered as the file to use for saving settings
(automatically on exit, or on user command).
An option of the form @@filename does not affect saving.
The option -saveSettingsFile does specify a name of the file to use
for saving, without reading any options from it, and is thus also effective
when the file did not exist yet.
So the settings will be saved to the file specified in the last
-saveSettingsFile or succesfull -settingsFile / -ini command,
if any, and in /etc/xboard/xboard.conf otherwise.
Usualy the latter is only accessible for the system administrator, though,
and will be used to contain system-wide default setings, amongst which
a -saveSettingsFile and -settingsFile options to specify a settings file
accessible to the individual user, such as ~/.xboardrc in the user's
home directory.
@item -saveSettingsOnExit true/false
@cindex saveSettingsOnExit, option
Controls saving of options on the settings file. @xref{Options Menu}.
Default: true.
@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
@itemx -name
@cindex display, option
@cindex geometry, option
@cindex iconic, option
@cindex resource name, 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 don't 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 -logoSize N
@cindex logoSize, option
This option controls the drawing of player logos next to the clocks.
The integer N specifies the width of the logo in pixels;
the logo height will always be half the width.
When N = 0, no logos will be diplayed.
Default: 0.
@item -firstLogo imagefile
@itemx -secondLogo imagefile
@cindex firstLogo, option
@cindex secondLogo, option
Specify the images to be used as player logos when @code{logoSize}
is non-zero, next to the white and black clocks, respectively.
@item -autoLogo true/false
@item -logoDir filename
@cindex autoLogo, option
@cindex logoDir, option
When @code{autoLogo} is set, XBoard will search for a PNG image file
with the name of the engine or ICS in the directory specified
by @code{logoDir}.
@item -recentEngines number
@itemx -recentEngineList list
@cindex recentEngines, option
@cindex recentEngineList, option
When the number is larger than zero, it determines how many recently
used engines will be appended at the bottom of the @samp{Engines} menu.
The engines will be saved in your settings file as the option
@code{recentEngineList}, by their nicknames,
and the most recently used one will always be sorted to the top.
If the list after that is longer than the specified number,
the last one is discarded.
Changes in the list will only become visible the next session,
provided you saved the settings.
Default: 6.
@item -oneClickMove true/false
@cindex oneClickMove, option
When set, this option allows you to enter moves by only clicking the to-
or from-square, when only a single legal move to or from that square
is possible.
Double-clicking a piece (or clicking an already selected piece)
will instruct that piece to make the only capture it can legally do.
Default: false.
@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 -sweepPromotions true/false
@cindex sweepPromotion, option
Sets the @samp{Almost Always Promote to 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.
Xboard installs with a set of scalable (svg) piece images,
which it scales to any of the requested sizes.
The square size can further be continuously scaled by sizing the board window,
but this only adapts the size of the pieces,
and has no effect on the width of the grid lines or the font choice
(both of which would depend on he selected boardSize).
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 messageFont,
@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 -overrideLineGap n
@cindex overrideLineGap, option
When n >= 0, this forces the width of the black border between squares
to n pixels for any board size. Mostly used to suppress the grid
entirely by setting n = 0, e.g. in xiangqi or just getting a prettier
picture. When n < 0 this the size-dependent width of the grid lines
is used. Default: -1.
@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 -evalZoom factor
@cindex evalZoom, option
The score interval (-1,1) is blown up on the vertical axis of
the Evaluation Graph by the given factor.
Default: 1
@item -evalThreshold n
@cindex evalThreshold, option
Score below n (centiPawn) are plotted as 0 in the Evaluation Graph.
Default: 25
@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 -showTargetSquares true/false
@cindex showTargetSquares, option
Determines whether XBoard can highlight the squares a piece has
legal moves to, when you grab that piece with the mouse.
Default: false.
@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 -highlightMoveWithArrow true/false
@cindex highlight Arrow, option
@cindex highlightMoveWithArrow, option
Sets the Highlight with Arrow 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 -periodic/-xperiodic or -periodicUpdates true/false
@cindex periodic, option
@cindex periodicUpdates, option
Controls updating of current move andnode counts in analysis mode. Default: true.
@item -fSAN
@itemx -sSAN
@cindex fSAN, option
@cindex sSAN, option
Causes the PV in thinking output of the mentioned engine to be converted
to SAN before it is further processed.
Warning: this might lose engine output not understood by the parser,
and uses a lot of CPU power.
Default: the PV is displayed exactly as the engine produced it.
@item -showEvalInMoveHistory true/false
@cindex showEvalInMoveHistory, option
Controls whether the evaluation scores and search depth of engine moves
are displayed with the move in the move-history window.
Default: true.
@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 -messageFont font
@cindex messageFont, option
@cindex Font, message
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 -pid or -pieceImageDirectory dir
@cindex pid, option
@cindex pieceImageDirectory, option
This options control what piece images xboard uses.
XBoard will look in the specified directory for an image in png
or svg format for every piece type, with names like BlackQueen.svg,
WhiteKnight.svg etc.
When neither of these is found (or no valid directory is specified)
XBoard will use the svg piece that was installed with it
(from the source-tree directory @samp{svg}).
Both svg and png images will be scaled by XBoard to the required size,
but the png pieces lose much in quality when scaled too much.
@item -whitePieceColor color
@itemx -blackPieceColor color
@itemx -lightSquareColor color
@itemx -darkSquareColor color
@itemx -highlightSquareColor color
@itemx -preoveHighlightColor color
@itemx -lowTimeWarningColor color
@cindex Colors
@cindex whitePieceColor, option
@cindex blackPieceColor, option
@cindex lightSquareColor, option
@cindex darkSquareColor, option
@cindex highlightSquareColor, option
@cindex premoveHighlightColor, 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
The PieceColor options only work properly if the image files
defining the pieces were pure black & white
(possibly anti-aliased to produce gray scales
and semi-transparancy),
like the pieces images that come with the install.
Their effect on colored pieces is undefined.
The SquareColor option only have an effect
when no board textures are used.
@item -trueColors true/false
@cindex trueColors, option
When set, this option suppresses the effect of the
PieceColor options mentioned above.
This is recommended for images that are already colored.
@item -useBoardTexture true/false
@itemx -liteBackTextureFile filename
@itemx -darkBackTextureFile filename
@cindex useBoardTexture, option
@cindex liteBackTextureFile, option
@cindex darkBackTextureFile, option
Indicate the png image files to be used for drawing the board squares,
and if they should be used rather than using simple colors.
The algorithm for cutting squares out of a given bitmap is such that
the picture is perfectly reproduced when a bitmap the size of
the complete board is given.
Default: false and ""
@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.
@item -autoDisplayComment true/false
@itemx -autoDisplayTags true/false
@cindex -autoDisplayComment, option
@cindex -autoDisplayTags, option
If set to true, these options cause the window with the move comments,
and the window with PGN tags, respectively, to pop up automatically when
such tags or comments are encountered during the replaying a stored or
loaded game. Default: true.
@item -pasteSelection true/false
@cindex -pasteSelection, option
If this option is set to true, the Paste Position and Paste Game
options paste from the currently selected text. If false, they paste
from the clipboard. Default: false.
@item -autoCopyPV true|false
@cindex autoCopyPV, option
When this option is set, the position displayed on the board when
you terminate a PV walk
(initiated by a right-click on board or engine-output window)
will be automatically put on the clipboard as FEN.
Default: false.
@item -dropMenu true|false
@cindex dropMenu, option
This option allows you to emulate old behavior,
where the right mouse button brings up the (now deprecated) drop menu
rather than displaying the position at the end of the principal variation.
Default: False.
@item -pieceMenu true|false
@cindex pieceMenu, option
This option allows you to emulate old behavior,
where the right mouse button brings up the (now deprecated) piece menu
in Edit Position mode.
From this menu you can select the piece to put on the square you
clicked to bring up the menu,
or select items such as @kbd{clear board}.
You can also @kbd{promote} or @kbd{demote} a clicked piece to convert
it into an unorthodox piece that is not directly in the menu,
or give the move to @kbd{black} or @kbd{white}.
@item -variations true|false
@cindex variations, option
When this option is on, you can start new variations in Edit Game or
Analyze mode by holding the Shift key down while entering a move.
When it is off, the Shift key will be ignored.
Default: False.
@item -appendPV true|false
@cindex appendPV, option
When this option is on, right-clicking a PV in the Engine Output window
will play the first move of that PV in Analyze mode,
or as many moves as you walk through by moving the mouse.
Default: False.
@item -absoluteAnalysisScores true|false
@cindex absoluteAnalysisScores, option
When true, scores on the Engine Output window during analysis
will be printed from the white point-of-view, rather than the
side-to-move point-of-view.
Default: False.
@item -scoreWhite true|false
@cindex scoreWhite, option
When true, scores will always be printed from the white point-of-view,
rather than the side-to-move point-of-view.
Default: False.
@item -memoHeaders true|false
@cindex memoHeaders, option
When true, column headers will be displayed in the Engine Output window
for the depth, score, time and nodes data.
Right-clicking on these headers will hide or show the corresponding data.
(Not intended for dynamic use, as already printed data of the current search
will not be affected!)
Defaul: False.
@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 -viewer
@itemx -viewerOptions string
@cindex viewer, option
@cindex viewerOptions, option
Presence of the volatile option @code{viewer} on the command line
will cause the value of the persistent option @code{viewerOptions}
as stored in the settings file to be appended to the command line.
The @code{view} option will be used by desktop associations with
game or position file types, so that @code{viewerOptions} can be
used to configure the exact mode XBoard will start in when it
should act on such a file (e.g. in -ncp mode, or analyzing
with your favorite engine). The options are also automatically
appended when Board is invoked with a single argument not being
an option name, which is then assumed to be the name of a
@code{loadGameFile} or (when the name ends in .fen) a
@code{loadPositionFile}.
Default: "-ncp -engineOutputUp false -saveSettingsOnExit false".
@item -tourneyOptions string
@cindex tourneyOptions, option
When XBoard is invoked with a single argument that is a file
with .trn extension, it will assume this argument to be the value
of a @code{tourneyFile} option,
and apped the value of the persistent option @code{tourneyOptions}
as stored in the settings file to the command line.
Thus the value of @code{tourneyOptions} can be
used to configure XBoard to automatically start running a
tournament when it should act on such a file.
Default: "-ncp -mm -saveSettingsOnExit 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)
makruk Thai Chess (shatranj-like, P promotes on 6th rank)
asean ASEAN Chess (a modernized version of Makruk)
spartan Spartan Chess (black has unorthodox 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 -pieceToCharTable string
@cindex pieceToCharTable, 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 -pieceNickNames string
@cindex pieceNickNames, option
The characters in the string are interpreted the same way as in the
@code{pieceToCharTable} option. But on input, piece-ID letters are
first looked up in the nicknames, and only if not defined there,
in the normal pieceToCharTable. This allows you to have two letters
designate the same piece, (e.g. N as an alternative to H for Horse
in Xiangqi), to make reading of non-compliant notations easier.
Default: ""
@item -colorNickNames string
@cindex colorNickNames, option
The side-to-move field in a FEN will be first matched against the letters
in the string (first character for white, second for black),
before it is matched to the regular 'w' and 'b'.
This makes it easier to read non-compliant FENs,
which, say, use 'r' for white.
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 XBoard saves debug information
(including all communication to and from the engines).
A @kbd{%d} in the given file name (e.g. game%d.debug) will be replaced
by the unique sequence number of a tournament game,
so that the debug output of each game will be written on a separate file.
@item -engineDebugOutput number
@cindex engineDebugOutput, option
Specifies how XBoard 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, XBoard 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 order pauses before and after sending the "quit" command to an engine that must be terminated.
The pause between quit and the previous command is specified in milliseconds.
The pause after quit is used to schedule a kill signal to be sent to the engine process after the
number of specified seconds plus one.
This signal is a different one as the terminiation signal described in the protocol specs
which engines can suppress or ignore, and which is sent directly after the "quit" command.
Setting @code{delayAfterQuit} to -1 will suppress sending of the kill signal.
Default: 0
@item -searchMode n
@cindex searchMode, option
The integer n encodes the mode for the @samp{find position} function.
Default: 1 (= Exact position match)
@item -eloThresholdBoth elo
@itemx -eloThresholdAny elo
@cindex eloThresholdBoth, option
@cindex eloThresholdAny, option
Defines a lower limit for the Elo rating, which has to be surpassed
before a game will be considered when searching for a board position.
Default: 0
@item -dateThreshold year
@cindex dateThreshold, option
Only games not played before the given year will be considered when
searching for a board position
@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{Edit Menu}, @ref{Mode Menu}, and
@ref{Action 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
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
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
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.
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
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,
XBoard 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.
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
You can report bugs and problems with XBoard using
the bug tracker at @code{https://savannah.gnu.org/projects/xboard/}
or by sending mail to @code{<bug-xboard@@gnu.org>}. It can also
be useful to report or discuss bugs in the WinBoard Forum at
@code{http://www.open-aurec.com/wbforum/},
WinBoard development section.
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.
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.
@node Contributors
@chapter Authors and contributors
@cindex Authors
@cindex Contributors
Chris Sears and Dan Sears wrote the original XBoard. They were
responsible for versions 1.0 through 1.2. The color scheme was taken
from Wayne Christopher's @code{XChess} program.
Tim Mann was primarily responsible for XBoard versions 1.3 through
4.2.7, and for WinBoard (a port of XBoard to Microsoft Win32) from its
inception through version 4.2.7.
John Chanak contributed the initial implementation of ICS mode. Evan
Welsh wrote @code{CMail}, and Patrick Surry helped in designing,
testing, and documenting it. Elmar Bartel contributed the new piece
bitmaps introduced in version 3.2. Jochen Wiedmann converted the
documentation to texinfo. Frank McIngvale added click/click moving,
the Analysis modes, piece flashing, ZIICS import, and ICS text
colorization to XBoard. Hugh Fisher added animated piece movement to
XBoard, and Henrik Gram added it to WinBoard. 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.
In a fork from version 4.2.7, 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 continued this fork of the project, producing version
4.3. He 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.
Meanwhile, some work continued on the GNU XBoard project maintained at
savannah.gnu.org, but version 4.2.8 was never released. Daniel
Mehrmann was responsible for much of this work.
Most recently, Arun Persaud worked with H. G. Muller to merge all
the features of the never-released XBoard/WinBoard 4.2.8 of the GNU
XBoard project and the never-released 4.3.16 from H. G.'s fork into a
unified XBoard/WinBoard 4.4, which is now available both from the
savannah.gnu.org 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 -myname <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.
@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.
@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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\begin{center}
\vskip 1cm{\LARGE\bf
Convolution Properties of the Generalized \\
\vskip .02in
Stirling Numbers and the Jacobi-Stirling \\
\vskip .11in
Numbers of the First Kind
}
\vskip 1cm
\large
Jiaqiang Pan\\
School of Biomedical Engineering and Instrumental Science\\
Zhejiang University\\
Hangzhou 210027\\
China\\
\href{mailto:[email protected]} {\tt [email protected]}\\
\end{center}
\vskip .2 in
\begin{abstract}
In this paper, we establish several properties of the unified
generalized Stirling numbers of the first kind, and the
Jacobi-Stirling numbers of the first kind, by means of the
convolution principle of sequences. Obtained results include
generalized Vandermonde convolution for the unified generalized
Stirling numbers of the first kind, triangular recurrence relation
for general Stirling-type numbers of the first kind, and linear
recurrence formula for the Jacobi-Stirling numbers of the first
kind, and so forth, thereby extending and supplementing known
knowledge to the existent literature about these Stirling-type
numbers.
\end{abstract}
\section{Introduction}
We know that \cite{ref1}, if generating functions of two sequences
$\langle a(k)\rangle\triangleq (a(0),a(1),a(2),a(3),\ldots)$ and
$\langle b(k)\rangle\triangleq (b(0),b(1),b(2),a(3),\ldots)$ are
$a(x)$ and $b(x)$ respectively, namely,
$$ a(x)=\sum_{k=0}^\infty
a(k)x^k, \quad b(x)=\sum_{k=0}^\infty b(k)x^k,$$
then product
function $a(x)b(x)$ is generating function of convolution
(sequence) $\langle c(k)\rangle$ of the two sequences $\langle
a(k)\rangle$ and $\langle b(k)\rangle$, where each term of the
sequence $\langle c(k)\rangle$ is calculated by the following
formula:
\begin{equation}\label{e:ConvPrinciple}
c(k)=\sum_{i=0}^k a(i)b(k-i)=\sum_{i=0}^k a(k-i)b(i), \quad
k=0,1,2,3,\ldots.
\end{equation}
For convenience, we occasionally denote the convolution operation
by symbol "$\ast$". For example, equality (\ref{e:ConvPrinciple})
is also expressed as
$$\langle c(k)\rangle=\langle a(k)\rangle\ast\langle
b(k)\rangle=\langle b(k)\rangle\ast\langle a(k)\rangle.$$ In this
paper, we will call this property of sequences \emph{the
convolution principle of sequences}.
The generating functions of several well-known sequences in
combinatorics have product form. \emph{Unified generalized
Stirling numbers of the first kind} and \emph{Jacobi-Stirling
numbers of the first kind} are two examples of such sequences.
The unified generalized Stirling numbers, defined first by Hsu and
Shuie \cite{ref2}, are the connection coefficients of linear
relations between generalized factorial functions. The generalized
factorial functions of a real or complex number $x$ with real
increment $\alpha$, denoted by $(x|\alpha)_n$, are special
polynomials in $x$ of degree $n$, as
\begin{equation}\label{e:GFactorial}
(x|\alpha)_0=1, \quad \mbox{and} \quad
(x|\alpha)_n=x(x-\alpha)\cdots(x-n\alpha+\alpha)=\prod_{i=0}^{n-1}(x-i\alpha),
\quad n=1,2,\ldots.
\end{equation}
Thus, the unified generalized Stirling numbers with real
parameters $\alpha,\beta,\gamma$, denoted by
$S(n,k;\alpha,\beta,\gamma)$, $n,k=0,1,2,\ldots$, are defined as
(see \cite{ref2})
\begin{equation}\label{e:unidefinition0}
S(0,k;\alpha,\beta,\gamma)=\delta_{0,k}, \, \mbox{and} \,
(x|\alpha)_n=\sum_{k=0}^{\infty}S(n,k;\alpha,\beta,\gamma)(x-\gamma|\beta)_k,
\quad n=1,2,\ldots,
\end{equation}
or
\begin{equation}\label{e:unidefinition1}
(x+\gamma|\alpha)_n=\sum_{k=0}^{\infty}S(n,k;\alpha,\beta,\gamma)(x|\beta)_k,
\quad n=1,2,3,\ldots.
\end{equation}
We see from (\ref{e:unidefinition1}) that for any
$\alpha,\beta,\gamma$, when $k>n$, $S(n,k;\alpha,\beta,\gamma)=0$;
and when $k=n$ $S(n,n;\alpha,\beta,\gamma)=1$. Therefore, the
upper limit $\infty$ of the summation in the right side of
equalities (\ref{e:unidefinition0}) and (\ref{e:unidefinition1})
may be replaced by $n$.
The most popular special cases of $S(n,k;\alpha,\beta,\gamma)$ are
the Kronecker delta $\delta_{n,k}$ ($S(n,k;0,0,0)$), the binomial
coefficients $\binom{n}{k}$ ($S(n,k;0,0,1)$), and two kinds of the
classical Stirling numbers $s(n,k)$ and $S(n,k)$ ($S(n,k;1,0,0)$
and $S(n,k;0,1,0)$).
Taking $S(n,k;\alpha,\beta,\gamma)$ $(n,k=0,1,2,\ldots)$ as
entries, we may obtain a $\infty$-dimensional, lower triangular
matrix
$\mathbf{S}_{\alpha,\beta,\gamma}=\big(S(n,k;\alpha,\beta,\gamma)\big)_{n,k=0.1.2.\ldots}$,
named \emph{the Generalized Stirling matrix} with parameters
$\alpha,\beta,\gamma$ \cite{ref7}. We also name the sequence
$$\langle S(n,k;\alpha,\beta,\gamma)\rangle\triangleq
(S(n,0;\alpha,\beta,\gamma), S(n,1;\alpha,\beta,\gamma),
S(n,2;\alpha,\beta,\gamma),S(n,3;\alpha,\beta,\gamma), \ldots)$$
\emph{the $n$-th row sequence of the unified generalized Stirling
numbers $S(n,k;\alpha,\beta,\gamma)$}.
We call $S(n,k;\alpha,0,\gamma)$ \emph{the unified generalized
Stirling numbers of the first kind}. For $S(n,k;\alpha,0,\gamma)$,
\begin{equation}\label{e:unidefinition2}
S(0,k;\alpha,0,\gamma)=\delta_{0,k}, \, \mbox{and}
\,\prod_{i=0}^{n-1}(x+\gamma-i\alpha)
=\sum_{k=0}^{\infty}S(n,k;\alpha,0,\gamma)x^k, \quad
n=1,2,3,\ldots,
\end{equation}
which shows that the (horizontal) generating function,
$\prod_{i=0}^{n-1}(x+\gamma-i\alpha)$, of the $n$-th row sequence
$\langle S(n,k;\alpha,0,\gamma)\rangle$ has product form.
For an excellent account of the unified generalized Stirling
numbers, see \cite{ref2}.
The Jacobi-Stirling numbers of the first kind, $J(n,k;\zeta)$
($n,k=0,1,2,\ldots$, and $\zeta>-1$ is a fixed constant
parameter), are another special case. In this case, the $n$-th row
sequence $\langle J(n,k;\zeta)\rangle$ also has a (horizontal)
generating function of product form, such as
$\prod_{i=0}^{n-1}(x-i(i+\zeta))$. Thus,
\begin{equation}\label{e:JacobiStirling1}
\prod_{i=0}^{n-1}(x-i(i+\zeta))=\sum_{k=0}^\infty J(n,k;\zeta)x^k,
\quad n=1,2,3,\ldots.
\end{equation}
(Note: $J(0,k;\zeta)=\delta_{0,k}, k=0,1,2,\ldots$). We see from
(\ref{e:JacobiStirling1}) that for any $\zeta$, when $k>n$,
$J(n,k;\zeta)=0$; and when $k=n$, $J(n,n;\zeta)=1$. Therefore, the
upper limit $\infty$ of the summation in the right side of
equality (\ref{e:JacobiStirling1}) may be replaced by $n$.
Particularly, we name $J(n,k;1)$ \emph{the Legendre-Stirling
numbers of the first kind}.
For the initial definition, elementary properties (explicit
expressions, triangular recurrence relations, similarity between
the Jacobi-Stirling and classical Stirling numbers, etc.) and
different combinatorial interpretations of special cases of the
Jacobi-Stirling numbers, see
\cite{ref3,ref4,ref5,ref6,ref8}, respectively.
Because the unified generalized Stirling numbers of the first
kind, and the Jacobi-Stirling numbers of the first kind, both have
generating functions of product form, thus it is reasonable to
investigate their several properties by means of the convolution
principle of sequences. In the following sections, we will present
the obtained results, including generalized Vandermonde
convolution for the unified generalized Stirling numbers of the
first kind, triangular recurrence relation for general
Stirling-type numbers of the first kind, and linear recurrence
formulae for the Jacobi-Stirling numbers of the first kind, and so
forth.
\section{Generalized Vandermonde convolution}
For the unified generalized Stirling numbers of the first kind, we
may obtain the following theorem by means of the convolution
principle of sequences.
\begin{theorem}\label{t:CovUGSN1}
Let $r$, $t$ and $n$ be three positive integers, and $n=r+t$. Then
\begin{equation}\label{e:CovUGSN1}
\langle S(n,k;\alpha,0,\gamma)\rangle = \langle
S(r,k;\alpha,0,\gamma)\rangle \ast \langle
S(t,k;\alpha,0,\gamma-r\alpha)\rangle.
\end{equation}
namely for $k=0,1,2,3,\ldots$,
\begin{equation}\label{e:CovUGSN2}
S(n,k;\alpha,0,\gamma)= \sum_{i=0}^k S(r,i;\alpha,0,\gamma)
S(t,k-i;\alpha,0,\gamma-r\alpha).
\end{equation}
We name this convolution formula \emph{the Generalized Vandermonde
convolution}.
\end{theorem}
\begin{proof}
We see from (\ref{e:unidefinition2}) that the generalized
factorial $\prod_{i=0}^{n-1}(x+\gamma-i\alpha)$ is the generating
function of sequence $\langle S(n,k;\alpha,0,\gamma)\rangle$. On
the other hand, $\prod_{i=0}^{n-1}(x+\gamma-i\alpha)$ is a product
of two factorial functions, $\prod_{i=0}^{r-1}(x+\gamma-i\alpha)$
and $\prod_{i=0}^{t-1}(x+\gamma-r\alpha-i\alpha)$, which are
generating functions of sequences $\langle
S(r,k;\alpha,0,\gamma)\rangle$ and $\langle
S(t,k;\alpha,0,\gamma-r\alpha)\rangle$ respectively. Hence,
$\prod_{i=0}^{n-1}(x+\gamma-i\alpha)$ is also the generating
function of convolution $\langle S(r,k;\alpha,0,\gamma)\rangle\ast
\langle S(t,k;\alpha,0,\gamma-r\alpha)\rangle$. Thus, $\langle
S(n,k;\alpha,0,\gamma)\rangle = \langle
S(r,k;\alpha,0,\gamma)\rangle \ast \langle
S(t,k;\alpha,0,\gamma-r\alpha)\rangle$.
\end{proof}
\begin{remark}\label{r:CovUGSN1}
We know that $S(n,k;0,0,1)=\binom{n}{k}$,
$S(r,k;0,0,1)=\binom{r}{k}$ and $S(t,k;0,0,1)=\binom{t}{k}$, (or
$S(n,k;0,0,-1)=(-1)^{n-k}\binom{n}{k}$,
$S(r,k;0,0,-1)=(-1)^{r-k}\binom{r}{k}$ and
$S(t,k;0,0,-1)=(-1)^{t-k}\binom{t}{k}$). In this special case, we
may find that (whether $\gamma=1$ or $\gamma=-1$) formula
(\ref{e:CovUGSN2}) lead to the classical Vandermonde
convolution \cite{ref1} (also named Vandermonde's identity or
Vandermonde formula) as
\begin{equation}\label{e:Binom}
\binom{n}{k}
=\sum_{i=0}^k\binom{r}{i}\binom{t}{k-i}=\sum_{i=0}^k\binom{r}{k-i}\binom{t}{i},
\end{equation}
where $n=r+t$.
\end{remark}
\begin{remark}\label{r:CovUGSN2}
The most simple case of formula (\ref{e:CovUGSN1}) or
(\ref{e:CovUGSN2}) is $\alpha=\gamma=0$. In this case,
$S(n,k;0,0,0)=\delta_{n,k}$, $S(r,k;0,0,0)=\delta_{r,k}$, and
$S(t,k;0,0,0)=\delta_{t,k}$. Thus, we obtain self-convolution
property of the kronecker delta, as
\begin{equation}\label{e:Kronecker1}
\langle\delta_{n,k}\rangle =
\langle\delta_{r,k}\rangle\ast\langle\delta_{t,k}\rangle
\end{equation}
or
\begin{equation}\label{e:Kronecker2}
\delta_{n,k} = \sum_{i=0}^k\delta_{r,i}\delta_{t,k-i}=
\sum_{i=0}^k\delta_{r,k-i}\delta_{t,i}
\end{equation}
where $n=r+s$.
\end{remark}
\begin{remark}\label{r:CovUGSN3}
The unified generalized Stirling number $S(r,k;\alpha,0,\gamma)$
of the first kind is the $(r,k)$-th entry of the generalized
Stirling matrix $\mathbf{S}_{\alpha,0,\gamma}$, and
$S(t,k;\alpha,0,\gamma-r\alpha)$ is the $(t,k)$-th entry of the
generalized Stirling matrix
$\mathbf{S}_{\alpha,0,\gamma-r\alpha}$. We know from
\cite[Theorem 7]{ref7} that, $S(r,k;\alpha,0,\gamma)$ is the scalar product
of the $r$-th row of the matrix $\mathbf{S}_{\alpha,0,0}$ and the
$k$-th column of the matrix $\mathbf{S}_{0,0,\gamma}$; and
$S(t,k;\alpha,0,\gamma-r\alpha)$ is the scalar product of the
$t$-th row of the matrix $\mathbf{S}_{\alpha,0,0}$ and the $k$-th
column of the matrix $\mathbf{S}_{0,0,\gamma-r\alpha}$. Hence,
$S(r,k;\alpha,0,\gamma)$ and $S(t,k;\alpha,0,\gamma-r\alpha)$ in
(\ref{e:CovUGSN2}) may be calculated by using the classical
Stirling numbers $s(n,k)$ of the first kind, and the binomial
coefficients $\binom{n}{k}$ ($n,k=0,1,2,\ldots$), as that
$$
S(r,k;\alpha,0,\gamma)=
\sum_{i=k}^r \gamma^{i-k}\alpha^{r-i}s(r,i)\binom{i}{k},
$$ and
$$
S(t,k;\alpha,0,\gamma-r\alpha)=
\sum_{i=k}^s(\gamma-r\alpha)^{i-k}\alpha^{t-i}s(t,i)\binom{i}{k}.
$$
\end{remark}
\begin{remark}\label{r:CSNF}
For the classical Stirling numbers $s(n,k)$ of the first kind,
namely $S(n,k;1,0,0)$, we have that $$\langle
S(n,k;1,0,0)\rangle=\langle S(r,k;1,0,0)\rangle \ast \langle
S(t,k;1,0,-r)\rangle,$$ or $$ S(n,k;1,0,0)= \sum_{i=0}^k
S(r,k-i;1,0,0)S(t,i;1,0,-r).$$ According to Remark
\ref{r:CovUGSN3}, $$
S(t,i;1,0,-r)=\sum_{j=i}^t(-r)^{j-i}s(t,j)\binom{j}{i}.$$ Finally,
we may write the convolution as
$$s(n,k)=\sum_{i=0}^k\sum_{j=i}^t(-r)^{j-i}\binom{j}{i}s(r,k-i)s(t,j),
\quad n=r+t. $$ This is just the Vandermonde convolution for the
classical Stirling numbers of the first kind.
\end{remark}
\section{Triangular recurrence relations of the Stirling-type numbers of the first kind}
\begin{remark}\label{r:Reccurence1}
We see from Equation (\ref{e:CovUGSN1}) that when $r=n-1$ and
$t=1$,
\begin{equation*}
\langle S(n,k;\alpha,0,\gamma)\rangle = \langle
S(n-1,k;\alpha,0,\gamma)\rangle \ast \langle
S(1,k;\alpha,0,\gamma-(n-1)\alpha)\rangle,
\end{equation*}
namely,
\begin{eqnarray}\label{e:Reccurence1}
S(n,k;\alpha,0,\gamma)=\sum_{i=0}^k
S(n-1,k-i;\alpha,0,\gamma)S(1,i;\alpha,0,\gamma-(n-1)\alpha)
\nonumber\\
=S(n-1,k;\alpha,0,\gamma)S(1,0;\alpha,0,\gamma-(n-1)\alpha)
\nonumber\\
+S(n-1,k-1;\alpha,0,\gamma)S(1,1;\alpha,0,\gamma-(n-1)\alpha)
\nonumber\\
=(\gamma-(n-1)\alpha)S(n-1,k;\alpha,0,\gamma)+S(n-1,k-1;\alpha,0,\gamma)
\end{eqnarray}
This is the triangular recurrence relation of
$S(n,k;\alpha,0,\gamma)$ shown in \cite[Theorem 1]{ref2}.
Therefore, the triangular recurrence relation of the unified
generalized Stirling numbers of the first kind just is a
convolution in essence.
\end{remark}
We may generalize this conclusion to a more general case of
\emph{the Stirling-type numbers of the first kind}, denoted by
$S(n,k;\alpha_i,i=0,1,2,\ldots)$
($\alpha_0,\alpha_1,\alpha_2,\alpha_3,\ldots$ is a given monotonic
non-decreasing or non-increasing sequence). The (horizontal)
generating function of row-sequence $\langle
S(n,k;\alpha_i,i=0,1,2,\ldots)\rangle$ of the Stirling-type
numbers of the first kind is $\prod_{i=0}^{n-1}(x-\alpha_i)$,
namely, $S(0,k;\alpha_i,i=0,1,2,3,\ldots)=\delta_{0,k}$, and
\begin{equation}\label{e:Stirling-type}
\prod_{i=0}^{n-1}(x-\alpha_i)=\sum_{k=0}^n
S(n,k;\alpha_i,i=0,1,2,\ldots)x^k, \quad n=1,2,3,\ldots.
\end{equation}
For the Stirling-type numbers of the first kind, we may obtain
corresponding triangular recurrence relations by means of the
convolution principle of sequences.
\begin{theorem}\label{t:Recurrence}
Let $S(n,k;\alpha_i,i=0,1,2,\ldots)$ be the Stirling-type numbers
of the first kind defined in (\ref{e:Stirling-type}). Then
$S(n,k;\alpha_i,i=0,1,2,\ldots)$ satisfies the following
triangular recurrence relation, namely, for $n,k=1,2,3,\ldots,$
\begin{equation}\label{e:Recurrence2}
S(n,k;\alpha_i,i=0,1,2,\ldots)=-\alpha_{n-1}S(n-1,k;\alpha_i,i=0,1,2,\ldots)\\
+S(n-1,k-1;\alpha_i,i=0,1,2,\ldots)
\end{equation}
\end{theorem}
\begin{proof}
We see from (\ref{e:Stirling-type}) that the generating function
of sequence $\langle S(n,k;\alpha_i,i=0,1,2,\ldots)\rangle$ is
$\prod_{i=0}^{n-1}(x-\alpha_i)$. On the other hand,
$\prod_{i=0}^{n-2}(x-\alpha_i)$ is the generating function of
sequence $\langle S(n-1,k;\alpha_i,i=0,1,2,\ldots)\rangle$, and
$(x-\alpha_{n-1})$ is the generating function of sequence
$(-\alpha_{n-1},1,0,0,0,\ldots)$. Hence, according to the
convolution principle of sequences we have that
\begin{eqnarray*}\label{e:ReccurenceM}
& &S(n,k;\alpha_i,i=0,1,2,\ldots) \\
&=&S(n-1,k;\alpha_i,i=0,1,2,\ldots)\cdot(-\alpha_{n-1})+
S(n-1,k-1;\alpha_i,i=0,1,2,\ldots)\cdot1 \\
&=&-\alpha_{n-1}S(n-1,k;\alpha_i,i=0,1,2,\ldots)
+S(n-1,k-1;\alpha_i,i=0,1,2,\ldots).
\end{eqnarray*}
\end{proof}
This theorem proves that for the most general Stirling-type
numbers of the first kind, exists a triangular recurrence
relation, and the triangular recurrence relation is a convolution
in essence.
\section{Convolution of the Jacobi-Stirling numbers of the first kind}
The Jacobi-Stirling numbers of the first kind, $J(n,k;\zeta)$ are
a special case of the Stirling-type numbers of the first kind, in
which $\alpha_i$ corresponds to $i(i+\zeta)$ ($i=0,1,2,\ldots$).
Because for the Jacobi-Stirling numbers of the first kind,
$\alpha_{n-1}=(n-1)(n+\zeta-1)$, according to
(\ref{e:Recurrence2}), $J(n,k;\zeta)$ satisfy the following
triangular recurrence relation (also see \cite{ref3, ref4, ref6}):
\begin{equation}\label{e:Recurrance3}
J(n,k;\zeta)=-(n-1)(n+\zeta-1)J(n-1,k;\zeta)+J(n-1,k-1;\zeta).
\end{equation}
Furthermore, we may establish several other properties of the
Jacobi-Stirling numbers of the first kind by means of the
convolution principle of sequences, as shown in the subsections
following.
\subsection{Convolution of the degenerate Jacobi-Stirling numbers of the
first kind}
We first investigate $J(n,k;0)$. In this paper, we name $J(n,k;0)$
\emph{the degenerate Jacobi-Stirling numbers of the first kind}.
In fact, they are just so-called \emph{central factorial numbers
of the first kind with even indices} \cite{ref8}.
In this case, the (horizontal) generating function of the $n$-th
row sequence $\langle J(n,k;0)\rangle$ is $\prod_{i=0}^{n-1}
(x-i^2)$, that is,
\begin{equation}\label{e:JS00}
J(0,k;0)=\delta_{0,k}, \quad\mbox{and}\quad \prod_{i=0}^{n-1}
(x-i^2) = \sum_{k=0}^n J(n,k;0)x^k, \quad n=1,2,3,\ldots.
\end{equation}
For the degenerate Jacobi-Stirling numbers of the first kind, we
may obtain the following property by means of the convolution
principle of sequences.
\begin{lemma}\label{l:JS0}
Let $n$ be a given positive integer, and $\langle J(n,k;0)\rangle$
be the $n$-th row sequence of the degenerate Jacobi-Stirling
numbers of the first kind, whose generating function is shown in
(\ref{e:JS00}). Then defining a sequence
$\langle\bar{J}(n,k)\rangle$ derived from $\langle
J(n,k;0)\rangle$ as
$$\langle\bar{J}(n,k)\rangle\triangleq(J(n,0;0),0,J(n,1;0),0,J(n,2;0),0,\ldots),$$
we have that
\begin{equation}\label{e:CJSG0}
\langle\bar{J}(n,k)\rangle = \langle s(n,k)\rangle\ast\langle
(-1)^{n-k}s(n,k)\rangle
\end{equation}
where $s(n,k)$ are the classical Stirling numbers of the first
kind.
\end{lemma}
\begin{proof}
Replacing $x$ in (\ref{e:JS00}) by $y^2$, we have that $$
\prod_{i=0}^{n-1} (y^2-i^2) = \prod_{i=0}^{n-1}
(y-i)\prod_{i=0}^{n-1} (y+i)\\ =\sum_{k=0}^n
J(n,k;0)y^{2k}= \sum_{k=0}^{2n}
\bar{J}(n,k)y^{k}
$$ We know that $\prod_{i=0}^{n-1} (y-i)$ and $\prod_{i=0}^{n-1}
(y+i)$ both are the (horizontal) generating functions of two
sequences $\langle s(n,k)\rangle$ and $\langle
(-1)^{n-k}s(n,k)\rangle$, respectively, Hence according to the
convolution principle of sequences, formula (\ref{e:CJSG0}) holds.
\end{proof}
\begin{theorem}\label{t:JS0}
The degenerate Jacobi-Stirling numbers of the first kind,
$J(n,k;0)$ may be calculated by the classical Stirling numbers
$s(n,k)$ of the first kind, as follows,
\begin{equation}\label{e:CJSG1}
J(n,k;0) = \sum_{i=0}^{2k}(-1)^{n-i}s(n,i)s(n,2k-i), \quad
n,k=0,1,2,\ldots.
\end{equation}
\end{theorem}
\begin{proof}
According to (\ref{e:CJSG0}), we have that $$
\bar{J}(n,k)=\sum_{i=0}^k (-1)^{n-i}s(n,i)s(n,k-i).
$$ Because $J(n,k;0)=\bar{J}(n,2k)$, thus formula (\ref{e:CJSG1})
holds.
\end{proof}
\begin{example}
For example, $$
J(4,2;0)=\sum_{i=0}^4(-1)^{4-i}s(4,i)s(4,4-i)=49,$$
and $$ J(5,2;0)=\sum_{i=0}^4(-1)^{5-i}s(5,i)s(5,4-i)=-820, $$ and
so forth.
\end{example}
\begin{remark}\label{r:Stirling1}
Because $\bar{J}(n,2k+1)\equiv 0$ ($k=0,1,2,\ldots$), from
(\ref{e:CJSG0}) we have the following identity:
\begin{equation}\label{e:Stirling1}
\sum_{i=0}^{2k+1}(-1)^{n-i}s(n,i)s(n,2k+1-i) = 0, \quad
(k=0,1,2,\ldots).
\end{equation}
In fact, this is a trivial identity, for its first $k+1$ terms
corresponding to $i=0$, $i=1$, $\ldots$, $i=k$ are the contrary
numbers of the rest $k+1$ terms corresponding to $i=2k+1$, $i=2k$,
$\ldots$, $i=k+1$, respectively.
\end{remark}
\subsection{Linear recurrence formula of the Legendre-Stirling numbers of the
first kind}
The Jacobi-Stirling numbers of the first kind with $\zeta=1$,
$J(n,k;1)$ also are named \emph{the Legendre-Stirling numbers of
the first kind} \cite{ref3}. For $J(n,k;1)$, we may obtain a
non-homogeneous linear recurrence relation by means of the
convolution principle of sequences.
\begin{theorem}\label{t:LStirling}
Let $n$ be a given non-negative integer. Then the $n$-th row
sequence, $\langle J(n,k;1)\rangle$, of the Legendre-Stirling
numbers of the first kind satisfies the following non-homogeneous
linear recurrence formulae:
\begin{equation}\label{e:LStirling0}
J(n,0;1)=\delta_{n,0}, \quad J(n,1;1)=
\sum_{i=0}^n(-1)^{n-i}s(n,i)s(n,1),
\end{equation}
and for $k=2,3,\ldots,n$,
\begin{equation}\label{e:LStirling1}
J(n,k;1)=-\sum_{i=[\frac{k+1}{2}]}^{k-1}\binom{i}{k-i}J(n,i;1)+
\sum_{i=0}^{k}\sum_{j=i}^n(-1)^{n-j}\binom{j}{i}s(n,j)s(n,k-i),
\end{equation}
where $[\cdot]$ is the floor function, and $s(n,k)$s are the
classical Stirling numbers of the first king.
\end{theorem}
\begin{proof}
We see from (\ref{e:JacobiStirling1}) that for the
Legendre-Stirling numbers $J(n,k;1)$ of the first kind,
\begin{equation}\label{e:LeStirling}
\prod_{i=0}^{n-1}(x-i(i+1))=\sum_{k=0}^nJ(n,k;1)x^k.
\end{equation}
Thus, $J(n,0;1)=0$ for $n=1,2,\ldots$. Besides, we know
$J(0,0;1)=1$. Hence, $J(n,0;1)=\delta_{n,0}$. We note that
$y(y+1)-i(i+1)=(y-i)(y+(i+1))$. Hence, replacing $x$ in
(\ref{e:LeStirling}) with $y(y+1)$ we may express the left side on
(\ref{e:LeStirling}) as product of two factorial functions in $y$,
$\prod_{i=0}^{n-1}(y-i)$ and $\prod_{i=0}^{n-1}(y+1+i)$, which are
the (horizontal) generating functions in $y$ of sequences $\langle
s(n,k)\rangle$ and $\langle S(n,k;-1,0,1)\rangle$ respectively.
Because according to Remark \ref{r:CovUGSN3},
$S(n,k;-1,0,1)=\sum_{i=k}^n(-1)^{n-i}\binom{i}{k}s(n,i)$, by means
of the convolution principle of sequences, we may express the left
side on (\ref{e:LeStirling}) as
$$\sum_{k=0}^n\{\sum_{j=0}^k\sum_{i=j}^n(-1)^{n-i}\binom{i}{j}s(n,k-j)s(n,i)\}y^k$$
On the other hand, now we may express the right side on
(\ref{e:LeStirling}) as $\sum_{j=0}^nJ(n,j;1)y^j(y+1)^j$. In the
latter, coefficients of the terms with monomial $y^k$ are
respectively $J(n,k;1)\binom{k}{0}$, $J(n,k-1;1)\binom{k-1}{1}$,
$J(n,k-2;1)\binom{k-2}{2}$, $\ldots$,
$J(n,k-[\frac{k}{2}];1)\binom{k-[\frac{k}{2}]}{[\frac{k}{2}]}$.
Hence, noting $k=[\frac{k}{2}]+[\frac{k+1}{2}]$ we also may
express the right side as
$$\sum_{k=0}^n\{\sum_{i=[\frac{k+1}{2}]}^k\binom{i}{k-i}J(n,i;1)\}y^k.$$
Because $y$ is arbitrary, by comparison of coefficients on both
sides we obtain that $$
\sum_{j=0}^k\sum_{i=j}^n(-1)^{n-i}\binom{i}{j}s(n,k-j)s(n,i)
=\sum_{i=[\frac{k+1}{2}]}^k\binom{i}{k-i}J(n,i;1).$$ Hence,
(\ref{e:LStirling0}) and (\ref{e:LStirling1}) both hold.
\end{proof}
\begin{example}\label{ex:LStirling}
Substituting $0,720,-1764,1624,-735,175,-21,1$ for the Stirling
numbers of the first kind, $s(7,0)$, $s(7,1)$, $s(7,2)$, $\ldots$,
$s(7,7)$ respectively , we find the Legendre-Stirling numbers of
the first kind, $J(7,0;1)=0$, $J(7,k;1)$ ($k=1,2,\ldots,6$), and
$J(7,7;1)=1$ by using formulae (\ref{e:LStirling0}) and
(\ref{e:LStirling1}), where $J(7,k;1)$ ($k=1,2,\ldots,6$) are
listed as follows,
$$J(7,1;1)=\sum_{j=1}^7(-1)^{7-j}s(7,j)s(7,1)=3628800$$
$$J(7,2;1)=-J(7,1;1)+\sum_{i=0}^{2}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,2-i)=-3110400,$$
$$J(7,3;1)=-2J(7,2;1)+\sum_{i=0}^{3}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,3-i)=808848,$$
$$J(7,4;1)=-3J(7,3;1)-J(7,2;1)+\sum_{i=0}^{4}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,4-i)=-89280,$$
$$J(7,5;1)=-4J(7,4;1)-3J(7,3;1)+\sum_{i=0}^{5}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,5-i)=4648,$$
$$J(7,6;1)=-5J(7,5;1)-6J(7,4;1)-J(7,3;1)+\sum_{i=0}^{6}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,6-i)=-112.$$
(see sequence \seqnum{A191936} in \cite{ref9},
and also \cite[Table 2]{ref3}).
\end{example}
\subsection{Linear recurrence formula of the Jacobi-Stirling numbers of the
first kind}
For general cases of the Jacobi-Stirling numbers $J(n,k;\zeta)$ of
the first kind, we may obtain a similar linear recurrence relation
for its $n$-th row sequence $\langle J(n,k;\zeta)\rangle$, by
means of the convolution principle of sequences.
\begin{theorem}\label{t:GJStirling}
Let $n$ be a given non-negative integer, and $\zeta$ $(>-1)$ be a
real number. Then the $n$-th row sequence $\langle
J(n,k;\zeta)\rangle$ of the Jacobi-Stirling numbers of the first
kind satisfies the following non-homogeneous linear recurrence
formulae:
\begin{equation}\label{e:JStirling0}
J(n,0;\zeta)=\delta_{n,0}, \quad J(n,1;\zeta)=
\sum_{i=0}^n(-1)^{n-i}\zeta^{i-1}s(n,i)s(n,1),
\end{equation}
and for $k=2,3,\ldots,n$,
\begin{equation}\label{e:JStirling1}
\zeta^kJ(n,k;\zeta)=-\sum_{i=[\frac{k+1}{2}]}^{k-1}\zeta^{2i-k}\binom{i}{k-i}J(n,i;\zeta)\\
+\sum_{i=0}^{k}\sum_{j=i}^n(-1)^{n-j}\zeta^{j-i}\binom{j}{i}s(n,j)s(n,k-i),
\end{equation}
where $[\cdot]$ is the floor function, and $s(n,k)$s are the
classical Stirling numbers of the first kind.
\end{theorem}
\begin{proof}
We see from (\ref{e:JacobiStirling1}) that $J(n,0;\zeta)=0$ for
$n=1,2,$. Besides, we know $J(0,0;\zeta)=1$. Hence,
$J(n,0;\zeta)=\delta_{n,0}$. We note that
$y(y+\zeta)-i(i+\zeta)=(y-i)(y+(i+\zeta))$. Hence, replacing $x$
in (\ref{e:JacobiStirling1}) by $y(y+\zeta)$ we may express the
left side on (\ref{e:JacobiStirling1}) as a product of two
factorial functions in $y$, $\prod_{i=0}^{n-1}(y-i)$ and
$\prod_{i=0}^{n-1}(y+\zeta+i)$, which are the (horizontal)
generating functions in $y$ of sequences, $\langle s(n,k)\rangle$
and $\langle S(n,k;-1,0,\zeta)\rangle$, respectively. Noting that
$S(n,k;-1,0,\zeta)=\sum_{i=k}^n(-1)^{n-i}\zeta^{i-k}\binom{i}{k}s(n,i)$,
by means of the convolution principle of sequences, we may express
the left side on (\ref{e:JacobiStirling1}) as
$$\sum_{k=0}^n\{\sum_{j=0}^k\sum_{i=j}^n(-1)^{n-i}\zeta^{i-j}\binom{i}{j}s(n,k-j)s(n,i)\}y^k.$$
On the other hand, we may express the right side of
(\ref{e:JacobiStirling1}) as
$$\sum_{j=0}^nJ(n,j;\zeta)x^j=\sum_{j=0}^nJ(n,j;\zeta)y^j(y+\zeta)^j.$$
In the latter, coefficients of the terms with monomial $y^k$ are
respectively $\zeta^k\binom{k}{0}J(n,k;\zeta)$,
$\zeta^{k-2}\binom{k-1}{1}J(n,k-1;\zeta)$,
$\zeta^{k-4}\binom{k-2}{2}J(n,k-2;\zeta)$, $\ldots$,
$\zeta^{k-2[\frac{k}{2}]}\binom{k-[\frac{k}{2}]}{[\frac{k}{2}]}J(n,k-[\frac{k}{2}];\zeta)$.
Therefore, noting $k=[\frac{k}{2}]+[\frac{k+1}{2}]$, we may
rewrite the sum on the right side as
$$\sum_{k=0}^n\{\sum_{i=[\frac{k+1}{2}]}^k\zeta^{2i-k}\binom{i}{k-i}J(n,i;\zeta)\}y^k.$$
Because $y$ is arbitrary, by comparison of coefficients on both
sides we obtain that $$
\sum_{j=0}^k\sum_{i=j}^n(-1)^{n-i}\zeta^{i-j}\binom{i}{j}s(n,k-j)s(n,i)\\
=\sum_{i=[\frac{k+1}{2}]}^k\zeta^{2i-k}\binom{i}{k-i}J(n,i;\zeta).
$$ Hence, (\ref{e:JStirling0}) and (\ref{e:JStirling1}) both hold.
\end{proof}
\begin{example}\label{ex:JStirling}
Substituting $0,24,-50,35,-10,1$ for the row sequence of the
classical Stirling numbers of the first kind,
($s(5,0),s(5,1),\ldots,s(5,5)$), we may obtain the row sequence of
the Jacobi-Stirling numbers of the first kind, ($J(5,0;\zeta)=0,
J(5,1;\zeta),\ldots,J(5,4;\zeta),J(5,5;\zeta)=1$) according to
(\ref{e:JStirling0}) and (\ref{e:JStirling1}). $J(5,1;\zeta)$,
$\ldots$, $J(5,4;\zeta)$ are listed as follows. $$
J(5,1;\zeta)=\sum_{i=0}^n(-1)^{5-i}\zeta^{i-1}s(5,i)s(5,1),$$
$$\zeta^2J(5,2;\zeta)=-J(5,1;\zeta)+\sum_{i=0}^2\sum_{j=i}^5(-1)^{5-j}\zeta^{j-i}\binom{j}{i}s(5,j)s(5,2-i).$$
$$\zeta^3J(5,3;\zeta)=-2\zeta
J(5,2;\zeta)+\sum_{i=0}^3\sum_{j=i}^5(-1)^{5-j}\zeta^{j-i}\binom{j}{i}s(5,j)s(5,3-i).$$
$$\zeta^4J(5,4;\zeta)=-3\zeta^2J(5,3;\zeta)-J(5,2;\zeta)+\sum_{i=0}^4\sum_{j=i}^5(-1)^{5-j}\zeta^{j-i}\binom{j}{i}s(5,j)s(5,4-i),$$
which then lead to that
$$J(5,1;\zeta)=576+1200\zeta+840\zeta^2+240\zeta^3+24\zeta^4,$$
$$J(5,2;\zeta)=-(820+1030\zeta+404\zeta^2+50\zeta^3),$$
$$J(5,3;\zeta)=273+200\zeta+35\zeta^2,$$ $$
J(5,4;\zeta)=-(30+10\zeta).$$ (see \cite[Table 2]{ref8}).
\end{example}
\begin{remark}\label{r:JSNC}
We may find that the linear recurrence formulae
(\ref{e:JStirling0}) and (\ref{e:JStirling1}) also verify
\cite[Theorem 1]{ref8}, that is, $J(n,k;\zeta)$ is a polynomial in
$\zeta$ of degree $n-k$, the coefficient of the first term with
$\zeta^{n-k}$ is $s(n,k)$, and the last terms (constant term) is
the central factorial numbers $u(n,k)$ of the first kind with even
indices, which is identical to $J(n,k;0)$. By the way, we may see
that the sum of coefficients of the polynomial is $J(n,k;1)$.
\end{remark}
\section{Acknowledgement}
The author would like to thank the referee for his/her very useful
suggestions.
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\newblock published electronically at
\url{http://oeis.org/}.
\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}: Primary
11B73; Secondary 05A15.
\noindent {\it Keywords}: convolution, unified generalized
Stirling numbers of the first kind, Jacobi-Stirling numbers of the
first kind, generalized Vandermonde convolution, triangular
recurrence relation, non-homogeneous linear recurrence relation.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences \seqnum{A191936}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in} \noindent Received April ** 2013; revised version
received September ** 2013.
\bigskip
\hrule
\bigskip
\noindent
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\begin{document}
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\begin{center}
\vskip 1cm{\LARGE\bf Convolution Properties of the Generalized
Stirling Numbers and the Jacobi-Stirling Numbers of the first
kind} \vskip 1cm \large Jiaqiang Pan\\ School of Biomedical
Engineering and Instrumental Science\\ Zhejiang University\\
Hangzhou 210027\\ China\\ \href{mailto:[email protected]} {\tt
[email protected]}\\
\end{center}
\vskip .2 in
\begin{abstract}
In this paper, we establish several properties of the unified
generalized Stirling numbers of the first kind, and the
Jacobi-Stirling numbers of the first kind, by means of the
convolution principle of sequences. Obtained results include
generalized Vandermonde convolution for the unified generalized
Stirling numbers of the first kind, triangular recurrence relation
for general Stirling-type numbers of the first kind, and linear
recurrence formula for the Jacobi-Stirling numbers of the first
kind, and so forth, thereby extending and supplementing known
knowledge to the existent literature about these Stirling-type
numbers.
\end{abstract}
\section{Introduction}
We know that \cite{ref1}, if generating functions of two sequences
$\langle a(k)\rangle\triangleq (a(0),a(1),a(2),a(3),\ldots)$ and
$\langle b(k)\rangle\triangleq (b(0),b(1),b(2),a(3),\ldots)$ are
$a(x)$ and $b(x)$ respectively, namely,
$$ a(x)=\sum_{k=0}^\infty
a(k)x^k, \quad b(x)=\sum_{k=0}^\infty b(k)x^k,$$
then product
function $a(x)b(x)$ is generating function of convolution
(sequence) $\langle c(k)\rangle$ of the two sequences $\langle
a(k)\rangle$ and $\langle b(k)\rangle$, where each term of the
sequence $\langle c(k)\rangle$ is calculated by the following
formula:
\begin{equation}\label{e:ConvPrinciple}
c(k)=\sum_{i=0}^k a(i)b(k-i)=\sum_{i=0}^k a(k-i)b(i), \quad
k=0,1,2,3,\ldots.
\end{equation}
For convenience, we occasionally denote the convolution operation
by symbol "$\ast$". For example, equality (\ref{e:ConvPrinciple})
is also expressed as
$$\langle c(k)\rangle=\langle a(k)\rangle\ast\langle
b(k)\rangle=\langle b(k)\rangle\ast\langle a(k)\rangle.$$ In this
paper, we will call this property of sequences \emph{the
convolution principle of sequences}.
The generating functions of several well-known sequences in
combinatorics have product form. \emph{Unified generalized
Stirling numbers of the first kind} and \emph{Jacobi-Stirling
numbers of the first kind} are two examples of such sequences.
The unified generalized Stirling numbers, defined first by Hsu and
Shuie \cite{ref2}, are the connection coefficients of linear
relations between generalized factorial functions. The generalized
factorial functions of a real or complex number $x$ with real
increment $\alpha$, denoted by $(x|\alpha)_n$, are special
polynomials in $x$ of degree $n$, as
\begin{equation}\label{e:GFactorial}
(x|\alpha)_0=1, \quad \mbox{and} \quad
(x|\alpha)_n=x(x-\alpha)\cdots(x-n\alpha+\alpha)=\prod_{i=0}^{n-1}(x-i\alpha),
\quad n=1,2,\ldots.
\end{equation}
Thus, the unified generalized Stirling numbers with real
parameters $\alpha,\beta,\gamma$, denoted by
$S(n,k;\alpha,\beta,\gamma)$, $n,k=0,1,2,\ldots$, are defined as
(see \cite{ref2})
\begin{equation}\label{e:unidefinition0}
S(0,k;\alpha,\beta,\gamma)=\delta_{0,k}, \, \mbox{and} \,
(x|\alpha)_n=\sum_{k=0}^{\infty}S(n,k;\alpha,\beta,\gamma)(x-\gamma|\beta)_k,
\quad n=1,2,\ldots,
\end{equation}
or
\begin{equation}\label{e:unidefinition1}
(x+\gamma|\alpha)_n=\sum_{k=0}^{\infty}S(n,k;\alpha,\beta,\gamma)(x|\beta)_k,
\quad n=1,2,3,\ldots.
\end{equation}
We see from (\ref{e:unidefinition1}) that for any
$\alpha,\beta,\gamma$, when $k>n$, $S(n,k;\alpha,\beta,\gamma)=0$;
and when $k=n$ $S(n,n;\alpha,\beta,\gamma)=1$. Therefore, the
upper limit $\infty$ of the summation in the right side of
equalities (\ref{e:unidefinition0}) and (\ref{e:unidefinition1})
may be replaced by $n$.
The most popular special cases of $S(n,k;\alpha,\beta,\gamma)$ are
the Kronecker delta $\delta_{n,k}$ ($S(n,k;0,0,0)$), the binomial
coefficients $\binom{n}{k}$ ($S(n,k;0,0,1)$), and two kinds of the
classical Stirling numbers $s(n,k)$ and $S(n,k)$ ($S(n,k;1,0,0)$
and $S(n,k;0,1,0)$).
Taking $S(n,k;\alpha,\beta,\gamma)$ $(n,k=0,1,2,\ldots)$ as
entries, we may obtain a $\infty$-dimensional, lower triangular
matrix
$\mathbf{S}_{\alpha,\beta,\gamma}=\big(S(n,k;\alpha,\beta,\gamma)\big)_{n,k=0.1.2.\ldots}$,
named \emph{the Generalized Stirling matrix} with parameters
$\alpha,\beta,\gamma$ \cite{ref7}. We also name the sequence
$$\langle S(n,k;\alpha,\beta,\gamma)\rangle\triangleq
(S(n,0;\alpha,\beta,\gamma), S(n,1;\alpha,\beta,\gamma),
S(n,2;\alpha,\beta,\gamma),S(n,3;\alpha,\beta,\gamma), \ldots)$$
\emph{the $n$-th row sequence of the unified generalized Stirling
numbers $S(n,k;\alpha,\beta,\gamma)$}.
We call $S(n,k;\alpha,0,\gamma)$ \emph{the unified generalized
Stirling numbers of the first kind}. For $S(n,k;\alpha,0,\gamma)$,
\begin{equation}\label{e:unidefinition2}
S(0,k;\alpha,0,\gamma)=\delta_{0,k}, \, \mbox{and}
\,\prod_{i=0}^{n-1}(x+\gamma-i\alpha)
=\sum_{k=0}^{\infty}S(n,k;\alpha,0,\gamma)x^k, \quad
n=1,2,3,\ldots,
\end{equation}
which shows that the (horizontal) generating function,
$\prod_{i=0}^{n-1}(x+\gamma-i\alpha)$, of the $n$-th row sequence
$\langle S(n,k;\alpha,0,\gamma)\rangle$ has product form.
For an excellent account of the unified generalized Stirling
numbers, see \cite{ref2}.
The Jacobi-Stirling numbers of the first kind, $J(n,k;\zeta)$
($n,k=0,1,2,\ldots$, and $\zeta>-1$ is a fixed constant
parameter), are another special case. In this case, the $n$-th row
sequence $\langle J(n,k;\zeta)\rangle$ also has a (horizontal)
generating function of product form, such as
$\prod_{i=0}^{n-1}(x-i(i+\zeta))$. Thus,
\begin{equation}\label{e:JacobiStirling1}
\prod_{i=0}^{n-1}(x-i(i+\zeta))=\sum_{k=0}^\infty J(n,k;\zeta)x^k,
\quad n=1,2,3,\ldots.
\end{equation}
(Note: $J(0,k;\zeta)=\delta_{0,k}, k=0,1,2,\ldots$). We see from
(\ref{e:JacobiStirling1}) that for any $\zeta$, when $k>n$,
$J(n,k;\zeta)=0$; and when $k=n$, $J(n,n;\zeta)=1$. Therefore, the
upper limit $\infty$ of the summation in the right side of
equality (\ref{e:JacobiStirling1}) may be replaced by $n$.
Particularly, we name $J(n,k;1)$ \emph{the Legendre-Stirling
numbers of the first kind}.
For the initial definition, elementary properties (explicit
expressions, triangular recurrence relations, similarity between
the Jacobi-Stirling and classical Stirling numbers, etc.) and
different combinatorial interpretations of special cases of the
Jacobi-Stirling numbers, see
\cite{ref3,ref4,ref5,ref6,ref8}, respectively.
Because the unified generalized Stirling numbers of the first
kind, and the Jacobi-Stirling numbers of the first kind, both have
generating functions of product form, thus it is reasonable to
investigate their several properties by means of the convolution
principle of sequences. In the following sections, we will present
the obtained results, including generalized Vandermonde
convolution for the unified generalized Stirling numbers of the
first kind, triangular recurrence relation for general
Stirling-type numbers of the first kind, and linear recurrence
formulae for the Jacobi-Stirling numbers of the first kind, and so
forth.
\section{Generalized Vandermonde convolution}
For the unified generalized Stirling numbers of the first kind, we
may obtain the following theorem by means of the convolution
principle of sequences.
\begin{theorem}\label{t:CovUGSN1}
Let $r$, $t$ and $n$ be three positive integers, and $n=r+t$. Then
\begin{equation}\label{e:CovUGSN1}
\langle S(n,k;\alpha,0,\gamma)\rangle = \langle
S(r,k;\alpha,0,\gamma)\rangle \ast \langle
S(t,k;\alpha,0,\gamma-r\alpha)\rangle.
\end{equation}
namely for $k=0,1,2,3,\ldots$,
\begin{equation}\label{e:CovUGSN2}
S(n,k;\alpha,0,\gamma)= \sum_{i=0}^k S(r,i;\alpha,0,\gamma)
S(t,k-i;\alpha,0,\gamma-r\alpha).
\end{equation}
We name this convolution formula \emph{the Generalized Vandermonde
convolution}.
\end{theorem}
\begin{proof}
We see from (\ref{e:unidefinition2}) that the generalized
factorial $\prod_{i=0}^{n-1}(x+\gamma-i\alpha)$ is the generating
function of sequence $\langle S(n,k;\alpha,0,\gamma)\rangle$. On
the other hand, $\prod_{i=0}^{n-1}(x+\gamma-i\alpha)$ is a product
of two factorial functions, $\prod_{i=0}^{r-1}(x+\gamma-i\alpha)$
and $\prod_{i=0}^{t-1}(x+\gamma-r\alpha-i\alpha)$, which are
generating functions of sequences $\langle
S(r,k;\alpha,0,\gamma)\rangle$ and $\langle
S(t,k;\alpha,0,\gamma-r\alpha)\rangle$ respectively. Hence,
$\prod_{i=0}^{n-1}(x+\gamma-i\alpha)$ is also the generating
function of convolution $\langle S(r,k;\alpha,0,\gamma)\rangle\ast
\langle S(t,k;\alpha,0,\gamma-r\alpha)\rangle$. Thus, $\langle
S(n,k;\alpha,0,\gamma)\rangle = \langle
S(r,k;\alpha,0,\gamma)\rangle \ast \langle
S(t,k;\alpha,0,\gamma-r\alpha)\rangle$.
\end{proof}
\begin{rem}\label{r:CovUGSN1}
We know that $S(n,k;0,0,1)=\binom{n}{k}$,
$S(r,k;0,0,1)=\binom{r}{k}$ and $S(t,k;0,0,1)=\binom{t}{k}$, (or
$S(n,k;0,0,-1)=(-1)^{n-k}\binom{n}{k}$,
$S(r,k;0,0,-1)=(-1)^{r-k}\binom{r}{k}$ and
$S(t,k;0,0,-1)=(-1)^{t-k}\binom{t}{k}$). In this special case, we
may find that (whether $\gamma=1$ or $\gamma=-1$) formula
(\ref{e:CovUGSN2}) lead to the classical Vandermonde
convolution \cite{ref1} (also named Vandermonde's identity or
Vandermonde formula) as
\begin{equation}\label{e:Binom}
\binom{n}{k}
=\sum_{i=0}^k\binom{r}{i}\binom{t}{k-i}=\sum_{i=0}^k\binom{r}{k-i}\binom{t}{i},
\end{equation}
where $n=r+t$.
\end{rem}
\begin{rem}\label{r:CovUGSN2}
The most simple case of formula (\ref{e:CovUGSN1}) or
(\ref{e:CovUGSN2}) is $\alpha=\gamma=0$. In this case,
$S(n,k;0,0,0)=\delta_{n,k}$, $S(r,k;0,0,0)=\delta_{r,k}$, and
$S(t,k;0,0,0)=\delta_{t,k}$. Thus, we obtain self-convolution
property of the kronecker delta, as
\begin{equation}\label{e:Kronecker1}
\langle\delta_{n,k}\rangle =
\langle\delta_{r,k}\rangle\ast\langle\delta_{t,k}\rangle
\end{equation}
or
\begin{equation}\label{e:Kronecker2}
\delta_{n,k} = \sum_{i=0}^k\delta_{r,i}\delta_{t,k-i}=
\sum_{i=0}^k\delta_{r,k-i}\delta_{t,i}
\end{equation}
where $n=r+s$.
\end{rem}
\begin{rem}\label{r:CovUGSN3}
The unified generalized Stirling number $S(r,k;\alpha,0,\gamma)$
of the first kind is the $(r,k)$-th entry of the generalized
Stirling matrix $\mathbf{S}_{\alpha,0,\gamma}$, and
$S(t,k;\alpha,0,\gamma-r\alpha)$ is the $(t,k)$-th entry of the
generalized Stirling matrix
$\mathbf{S}_{\alpha,0,\gamma-r\alpha}$. We know from
\cite[Theorem 7]{ref7} that, $S(r,k;\alpha,0,\gamma)$ is the scalar product
of the $r$-th row of the matrix $\mathbf{S}_{\alpha,0,0}$ and the
$k$-th column of the matrix $\mathbf{S}_{0,0,\gamma}$; and
$S(t,k;\alpha,0,\gamma-r\alpha)$ is the scalar product of the
$t$-th row of the matrix $\mathbf{S}_{\alpha,0,0}$ and the $k$-th
column of the matrix $\mathbf{S}_{0,0,\gamma-r\alpha}$. Hence,
$S(r,k;\alpha,0,\gamma)$ and $S(t,k;\alpha,0,\gamma-r\alpha)$ in
(\ref{e:CovUGSN2}) may be calculated by using the classical
Stirling numbers $s(n,k)$ of the first kind, and the binomial
coefficients $\binom{n}{k}$ ($n,k=0,1,2,\ldots$), as that
$$
S(r,k;\alpha,0,\gamma)=
\sum_{i=k}^r \gamma^{i-k}\alpha^{r-i}s(r,i)\binom{i}{k},
$$ and
$$
S(t,k;\alpha,0,\gamma-r\alpha)=
\sum_{i=k}^s(\gamma-r\alpha)^{i-k}\alpha^{t-i}s(t,i)\binom{i}{k}.
$$
\end{rem}
\begin{rem}\label{r:CSNF}
For the classical Stirling numbers $s(n,k)$ of the first kind,
namely $S(n,k;1,0,0)$, we have that $$\langle
S(n,k;1,0,0)\rangle=\langle S(r,k;1,0,0)\rangle \ast \langle
S(t,k;1,0,-r)\rangle,$$ or $$ S(n,k;1,0,0)= \sum_{i=0}^k
S(r,k-i;1,0,0)S(t,i;1,0,-r).$$ According to Remark
\ref{r:CovUGSN3}, $$
S(t,i;1,0,-r)=\sum_{j=i}^t(-r)^{j-i}s(t,j)\binom{j}{i}.$$ Finally,
we may write the convolution as
$$s(n,k)=\sum_{i=0}^k\sum_{j=i}^t(-r)^{j-i}\binom{j}{i}s(r,k-i)s(t,j),
\quad n=r+t. $$ This is just the Vandermonde convolution for the
classical Stirling numbers of the first kind.
\end{rem}
\section{Triangular recurrence relations of the Stirling-type numbers of the first kind}
\begin{rem}\label{r:Reccurence1}
We see from Equation (\ref{e:CovUGSN1}) that when $r=n-1$ and
$t=1$,
\begin{equation*}
\langle S(n,k;\alpha,0,\gamma)\rangle = \langle
S(n-1,k;\alpha,0,\gamma)\rangle \ast \langle
S(1,k;\alpha,0,\gamma-(n-1)\alpha)\rangle,
\end{equation*}
namely,
\begin{eqnarray}\label{e:Reccurence1}
S(n,k;\alpha,0,\gamma)=\sum_{i=0}^k
S(n-1,k-i;\alpha,0,\gamma)S(1,i;\alpha,0,\gamma-(n-1)\alpha)
\nonumber\\
=S(n-1,k;\alpha,0,\gamma)S(1,0;\alpha,0,\gamma-(n-1)\alpha)
\nonumber\\
+S(n-1,k-1;\alpha,0,\gamma)S(1,1;\alpha,0,\gamma-(n-1)\alpha)
\nonumber\\
=(\gamma-(n-1)\alpha)S(n-1,k;\alpha,0,\gamma)+S(n-1,k-1;\alpha,0,\gamma)
\end{eqnarray}
This is the triangular recurrence relation of
$S(n,k;\alpha,0,\gamma)$ shown in \cite[Theorem 1]{ref2}.
Therefore, the triangular recurrence relation of the unified
generalized Stirling numbers of the first kind just is a
convolution in essence.
\end{rem}
We may generalize this conclusion to a more general case of
\emph{the Stirling-type numbers of the first kind}, denoted by
$S(n,k;\alpha_i,i=0,1,2,\ldots)$
($\alpha_0,\alpha_1,\alpha_2,\alpha_3,\ldots$ is a given monotonic
non-decreasing or non-increasing sequence). The (horizontal)
generating function of row-sequence $\langle
S(n,k;\alpha_i,i=0,1,2,\ldots)\rangle$ of the Stirling-type
numbers of the first kind is $\prod_{i=0}^{n-1}(x-\alpha_i)$,
namely, $S(0,k;\alpha_i,i=0,1,2,3,\ldots)=\delta_{0,k}$, and
\begin{equation}\label{e:Stirling-type}
\prod_{i=0}^{n-1}(x-\alpha_i)=\sum_{k=0}^n
S(n,k;\alpha_i,i=0,1,2,\ldots)x^k, \quad n=1,2,3,\ldots.
\end{equation}
For the Stirling-type numbers of the first kind, we may obtain
corresponding triangular recurrence relations by means of the
convolution principle of sequences.
\begin{theorem}\label{t:Recurrence}
Let $S(n,k;\alpha_i,i=0,1,2,\ldots)$ be the Stirling-type numbers
of the first kind defined in (\ref{e:Stirling-type}). Then
$S(n,k;\alpha_i,i=0,1,2,\ldots)$ satisfies the following
triangular recurrence relation, namely, for $n,k=1,2,3,\ldots,$
\begin{equation}\label{e:Recurrence2}
S(n,k;\alpha_i,i=0,1,2,\ldots)=-\alpha_{n-1}S(n-1,k;\alpha_i,i=0,1,2,\ldots)\\
+S(n-1,k-1;\alpha_i,i=0,1,2,\ldots)
\end{equation}
\end{theorem}
\begin{proof}
We see from (\ref{e:Stirling-type}) that the generating function
of sequence $\langle S(n,k;\alpha_i,i=0,1,2,\ldots)\rangle$ is
$\prod_{i=0}^{n-1}(x-\alpha_i)$. On the other hand,
$\prod_{i=0}^{n-2}(x-\alpha_i)$ is the generating function of
sequence $\langle S(n-1,k;\alpha_i,i=0,1,2,\ldots)\rangle$, and
$(x-\alpha_{n-1})$ is the generating function of sequence
$(-\alpha_{n-1},1,0,0,0,\ldots)$. Hence, according to the
convolution principle of sequences we have that
\begin{eqnarray*}\label{e:ReccurenceM}
& &S(n,k;\alpha_i,i=0,1,2,\ldots) \\
&=&S(n-1,k;\alpha_i,i=0,1,2,\ldots)\cdot(-\alpha_{n-1})+
S(n-1,k-1;\alpha_i,i=0,1,2,\ldots)\cdot1 \\
&=&-\alpha_{n-1}S(n-1,k;\alpha_i,i=0,1,2,\ldots)
+S(n-1,k-1;\alpha_i,i=0,1,2,\ldots).
\end{eqnarray*}
\end{proof}
This theorem proves that for the most general Stirling-type
numbers of the first kind, exists a triangular recurrence
relation, and the triangular recurrence relation is a convolution
in essence.
\section{Convolution of the Jacobi-Stirling numbers of the first kind}
The Jacobi-Stirling numbers of the first kind, $J(n,k;\zeta)$ are
a special case of the Stirling-type numbers of the first kind, in
which $\alpha_i$ corresponds to $i(i+\zeta)$ ($i=0,1,2,\ldots$).
Because for the Jacobi-Stirling numbers of the first kind,
$\alpha_{n-1}=(n-1)(n+\zeta-1)$, according to
(\ref{e:Recurrence2}), $J(n,k;\zeta)$ satisfy the following
triangular recurrence relation (also see \cite{ref3, ref4, ref6}):
\begin{equation}\label{e:Recurrance3}
J(n,k;\zeta)=-(n-1)(n+\zeta-1)J(n-1,k;\zeta)+J(n-1,k-1;\zeta).
\end{equation}
Furthermore, we may establish several other properties of the
Jacobi-Stirling numbers of the first kind by means of the
convolution principle of sequences, as shown in the subsections
following.
\subsection{Convolution of the degenerate Jacobi-Stirling numbers of the
first kind}
We first investigate $J(n,k;0)$. In this paper, we name $J(n,k;0)$
\emph{the Degenerate Jacobi-Stirling numbers of the first kind}.
In fact, they are just so-called \emph{central factorial numbers
of the first kind with even indices} \cite{ref8}.
In this case, the (horizontal) generating function of the $n$-th
row sequence $\langle J(n,k;0)\rangle$ is $\prod_{i=0}^{n-1}
(x-i^2)$, that is,
\begin{equation}\label{e:JS00}
J(0,k;0)=\delta_{0,k}, \quad\mbox{and}\quad \prod_{i=0}^{n-1}
(x-i^2) = \sum_{k=0}^n J(n,k;0)x^k, \quad n=1,2,3,\ldots.
\end{equation}
For the degenerate Jacobi-Stirling numbers of the first kind, we
may obtain the following property by means of the convolution
principle of sequences.
\begin{lemma}\label{l:JS0}
Let $n$ be a given positive integer, and $\langle J(n,k;0)\rangle$
be the $n$-th row sequence of the degenerate Jacobi-Stirling
numbers of the first kind, whose generating function is shown in
(\ref{e:JS00}). Then defining a sequence
$\langle\bar{J}(n,k)\rangle$ derived from $\langle
J(n,k;0)\rangle$ as
$$\langle\bar{J}(n,k)\rangle\triangleq(J(n,0;0),0,J(n,1;0),0,J(n,2;0),0,\ldots),$$
we have that
\begin{equation}\label{e:CJSG0}
\langle\bar{J}(n,k)\rangle = \langle s(n,k)\rangle\ast\langle
(-1)^{n-k}s(n,k)\rangle
\end{equation}
where $s(n,k)$ are the classical Stirling numbers of the first
kind.
\end{lemma}
\begin{proof}
Replacing $x$ in (\ref{e:JS00}) by $y^2$, we have that $$
\prod_{i=0}^{n-1} (y^2-i^2) = \prod_{i=0}^{n-1}
(y-i)\prod_{i=0}^{n-1} (y+i)\\ =\sum_{k=0}^n
J(n,k;0)y^{2k}= \sum_{k=0}^{2n}
\bar{J}(n,k)y^{k}
$$ We know that $\prod_{i=0}^{n-1} (y-i)$ and $\prod_{i=0}^{n-1}
(y+i)$ both are the (horizontal) generating functions of two
sequences $\langle s(n,k)\rangle$ and $\langle
(-1)^{n-k}s(n,k)\rangle$, respectively, Hence according to the
convolution principle of sequences, formula (\ref{e:CJSG0}) holds.
\end{proof}
\begin{theorem}\label{t:JS0}
The degenerate Jacobi-Stirling numbers of the first kind,
$J(n,k;0)$ may be calculated by the classical Stirling numbers
$s(n,k)$ of the first kind, as follows,
\begin{equation}\label{e:CJSG1}
J(n,k;0) = \sum_{i=0}^{2k}(-1)^{n-i}s(n,i)s(n,2k-i), \quad
n,k=0,1,2,\ldots.
\end{equation}
\end{theorem}
\begin{proof}
According to (\ref{e:CJSG0}), we have that $$
\bar{J}(n,k)=\sum_{i=0}^k (-1)^{n-i}s(n,i)s(n,k-i).
$$ Because $J(n,k;0)=\bar{J}(n,2k)$, thus formula (\ref{e:CJSG1})
holds.
\end{proof}
\begin{example}
For example, $$
J(4,2;0)=\sum_{i=0}^4(-1)^{4-i}s(4,i)s(4,4-i)=49,$$
and $$ J(5,2;0)=\sum_{i=0}^4(-1)^{5-i}s(5,i)s(5,4-i)=-820, $$ and
so forth.
\end{example}
\begin{rem}\label{r:Stirling1}
Because $\bar{J}(n,2k+1)\equiv 0$ ($k=0,1,2,\ldots$), from
(\ref{e:CJSG0}) we have the following identity:
\begin{equation}\label{e:Stirling1}
\sum_{i=0}^{2k+1}(-1)^{n-i}s(n,i)s(n,2k+1-i) = 0, \quad
(k=0,1,2,\ldots).
\end{equation}
In fact, this is a trivial identity, for its first $k+1$ terms
corresponding to $i=0$, $i=1$, $\ldots$, $i=k$ are the contrary
numbers of the rest $k+1$ terms corresponding to $i=2k+1$, $i=2k$,
$\ldots$, $i=k+1$, respectively.
\end{rem}
\subsection{Linear recurrence formula of the Legendre-Stirling numbers of the
first kind}
The Jacobi-Stirling numbers of the first kind with $\zeta=1$,
$J(n,k;1)$ also are named \emph{the Legendre-Stirling numbers of
the first kind} \cite{ref3}. For $J(n,k;1)$, we may obtain a
non-homogeneous linear recurrence relation by means of the
convolution principle of sequences.
\begin{theorem}\label{t:LStirling}
Let $n$ be a given non-negative integer. Then the $n$-th row
sequence, $\langle J(n,k;1)\rangle$, of the Legendre-Stirling
numbers of the first kind satisfies the following non-homogeneous
linear recurrence formulae:
\begin{equation}\label{e:LStirling0}
J(n,0;1)=\delta_{n,0}, \quad J(n,1;1)=
\sum_{i=0}^n(-1)^{n-i}s(n,i)s(n,1),
\end{equation}
and for $k=2,3,\ldots,n$,
\begin{equation}\label{e:LStirling1}
J(n,k;1)=-\sum_{i=[\frac{k+1}{2}]}^{k-1}\binom{i}{k-i}J(n,i;1)+
\sum_{i=0}^{k}\sum_{j=i}^n(-1)^{n-j}\binom{j}{i}s(n,j)s(n,k-i),
\end{equation}
where $[\cdot]$ is the floor function, and $s(n,k)$s are the
classical Stirling numbers of the first king.
\end{theorem}
\begin{proof}
We see from (\ref{e:JacobiStirling1}) that for the
Legendre-Stirling numbers $J(n,k;1)$ of the first kind,
\begin{equation}\label{e:LeStirling}
\prod_{i=0}^{n-1}(x-i(i+1))=\sum_{k=0}^nJ(n,k;1)x^k.
\end{equation}
Thus, $J(n,0;1)=0$ for $n=1,2,\ldots$. Besides, we know
$J(0,0;1)=1$. Hence, $J(n,0;1)=\delta_{n,0}$. We note that
$y(y+1)-i(i+1)=(y-i)(y+(i+1))$. Hence, replacing $x$ in
(\ref{e:LeStirling}) with $y(y+1)$ we may express the left side on
(\ref{e:LeStirling}) as product of two factorial functions in $y$,
$\prod_{i=0}^{n-1}(y-i)$ and $\prod_{i=0}^{n-1}(y+1+i)$, which are
the (horizontal) generating functions in $y$ of sequences $\langle
s(n,k)\rangle$ and $\langle S(n,k;-1,0,1)\rangle$ respectively.
Because according to Remark \ref{r:CovUGSN3},
$S(n,k;-1,0,1)=\sum_{i=k}^n(-1)^{n-i}\binom{i}{k}s(n,i)$, by means
of the convolution principle of sequences, we may express the left
side on (\ref{e:LeStirling}) as
$$\sum_{k=0}^n\{\sum_{j=0}^k\sum_{i=j}^n(-1)^{n-i}\binom{i}{j}s(n,k-j)s(n,i)\}y^k$$
On the other hand, now we may express the right side on
(\ref{e:LeStirling}) as $\sum_{j=0}^nJ(n,j;1)y^j(y+1)^j$. In the
latter, coefficients of the terms with monomial $y^k$ are
respectively $J(n,k;1)\binom{k}{0}$, $J(n,k-1;1)\binom{k-1}{1}$,
$J(n,k-2;1)\binom{k-2}{2}$, $\ldots$,
$J(n,k-[\frac{k}{2}];1)\binom{k-[\frac{k}{2}]}{[\frac{k}{2}]}$.
Hence, noting $k=[\frac{k}{2}]+[\frac{k+1}{2}]$ we also may
express the right side as
$$\sum_{k=0}^n\{\sum_{i=[\frac{k+1}{2}]}^k\binom{i}{k-i}J(n,i;1)\}y^k.$$
Because $y$ is arbitrary, by comparison of coefficients on both
sides we obtain that $$
\sum_{j=0}^k\sum_{i=j}^n(-1)^{n-i}\binom{i}{j}s(n,k-j)s(n,i)
=\sum_{i=[\frac{k+1}{2}]}^k\binom{i}{k-i}J(n,i;1).$$ Hence,
(\ref{e:LStirling0}) and (\ref{e:LStirling1}) both hold.
\end{proof}
\begin{example}\label{ex:LStirling}
Substituting $0,720,-1764,1624,-735,175,-21,1$ for the Stirling
numbers of the first kind, $s(7,0)$, $s(7,1)$, $s(7,2)$, $\ldots$,
$s(7,7)$ respectively , we find the Legendre-Stirling numbers of
the first kind, $J(7,0;1)=0$, $J(7,k;1)$ ($k=1,2,\ldots,6$), and
$J(7,7;1)=1$ by using formulae (\ref{e:LStirling0}) and
(\ref{e:LStirling1}), where $J(7,k;1)$ ($k=1,2,\ldots,6$) are
listed as follows,
$$J(7,1;1)=\sum_{j=1}^7(-1)^{7-j}s(7,j)s(7,1)=3628800$$
$$J(7,2;1)=-J(7,1;1)+\sum_{i=0}^{2}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,2-i)=-3110400,$$
$$J(7,3;1)=-2J(7,2;1)+\sum_{i=0}^{3}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,3-i)=808848,$$
$$J(7,4;1)=-3J(7,3;1)-J(7,2;1)+\sum_{i=0}^{4}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,4-i)=-89280,$$
$$J(7,5;1)=-4J(7,4;1)-3J(7,3;1)+\sum_{i=0}^{5}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,5-i)=4648,$$
$$J(7,6;1)=-5J(7,5;1)-6J(7,4;1)-J(7,3;1)+\sum_{i=0}^{6}\sum_{j=i}^7(-1)^{7-j}\binom{j}{i}s(7,j)s(7,6-i)=-112.$$
(see sequence \seqnum{A191936} in \cite{ref9},
and also \cite[Table 2]{ref3}).
\end{example}
\subsection{Linear recurrence formula of the Jacobi-Stirling numbers of the
first kind}
For general cases of the Jacobi-Stirling numbers $J(n,k;\zeta)$ of
the first kind, we may obtain a similar linear recurrence relation
for its $n$-th row sequence $\langle J(n,k;\zeta)\rangle$, by
means of the convolution principle of sequences.
\begin{theorem}\label{t:GJStirling}
Let $n$ be a given non-negative integer, and $\zeta$ $(>-1)$ be a
real number. Then the $n$-th row sequence $\langle
J(n,k;\zeta)\rangle$ of the Jacobi-Stirling numbers of the first
kind satisfies the following non-homogeneous linear recurrence
formulae:
\begin{equation}\label{e:JStirling0}
J(n,0;\zeta)=\delta_{n,0}, \quad J(n,1;\zeta)=
\sum_{i=0}^n(-1)^{n-i}\zeta^{i-1}s(n,i)s(n,1),
\end{equation}
and for $k=2,3,\ldots,n$,
\begin{equation}\label{e:JStirling1}
\zeta^kJ(n,k;\zeta)=-\sum_{i=[\frac{k+1}{2}]}^{k-1}\zeta^{2i-k}\binom{i}{k-i}J(n,i;\zeta)\\
+\sum_{i=0}^{k}\sum_{j=i}^n(-1)^{n-j}\zeta^{j-i}\binom{j}{i}s(n,j)s(n,k-i),
\end{equation}
where $[\cdot]$ is the floor function, and $s(n,k)$s are the
classical Stirling numbers of the first kind.
\end{theorem}
\begin{proof}
We see from (\ref{e:JacobiStirling1}) that $J(n,0;\zeta)=0$ for
$n=1,2,$. Besides, we know $J(0,0;\zeta)=1$. Hence,
$J(n,0;\zeta)=\delta_{n,0}$. We note that
$y(y+\zeta)-i(i+\zeta)=(y-i)(y+(i+\zeta))$. Hence, replacing $x$
in (\ref{e:JacobiStirling1}) by $y(y+\zeta)$ we may express the
left side on (\ref{e:JacobiStirling1}) as a product of two
factorial functions in $y$, $\prod_{i=0}^{n-1}(y-i)$ and
$\prod_{i=0}^{n-1}(y+\zeta+i)$, which are the (horizontal)
generating functions in $y$ of sequences, $\langle s(n,k)\rangle$
and $\langle S(n,k;-1,0,\zeta)\rangle$, respectively. Noting that
$S(n,k;-1,0,\zeta)=\sum_{i=k}^n(-1)^{n-i}\zeta^{i-k}\binom{i}{k}s(n,i)$,
by means of the convolution principle of sequences, we may express
the left side on (\ref{e:JacobiStirling1}) as
$$\sum_{k=0}^n\{\sum_{j=0}^k\sum_{i=j}^n(-1)^{n-i}\zeta^{i-j}\binom{i}{j}s(n,k-j)s(n,i)\}y^k.$$
On the other hand, we may express the right side of
(\ref{e:JacobiStirling1}) as
$$\sum_{j=0}^nJ(n,j;\zeta)x^j=\sum_{j=0}^nJ(n,j;\zeta)y^j(y+\zeta)^j.$$
In the latter, coefficients of the terms with monomial $y^k$ are
respectively $\zeta^k\binom{k}{0}J(n,k;\zeta)$,
$\zeta^{k-2}\binom{k-1}{1}J(n,k-1;\zeta)$,
$\zeta^{k-4}\binom{k-2}{2}J(n,k-2;\zeta)$, $\ldots$,
$\zeta^{k-2[\frac{k}{2}]}\binom{k-[\frac{k}{2}]}{[\frac{k}{2}]}J(n,k-[\frac{k}{2}];\zeta)$.
Therefore, noting $k=[\frac{k}{2}]+[\frac{k+1}{2}]$, we may
rewrite the sum on the right side as
$$\sum_{k=0}^n\{\sum_{i=[\frac{k+1}{2}]}^k\zeta^{2i-k}\binom{i}{k-i}J(n,i;\zeta)\}y^k.$$
Because $y$ is arbitrary, by comparison of coefficients on both
sides we obtain that $$
\sum_{j=0}^k\sum_{i=j}^n(-1)^{n-i}\zeta^{i-j}\binom{i}{j}s(n,k-j)s(n,i)\\
=\sum_{i=[\frac{k+1}{2}]}^k\zeta^{2i-k}\binom{i}{k-i}J(n,i;\zeta).
$$ Hence, (\ref{e:JStirling0}) and (\ref{e:JStirling1}) both hold.
\end{proof}
\begin{example}\label{ex:JStirling}
Substituting $0,24,-50,35,-10,1$ for the row sequence of the
classical Stirling numbers of the first kind,
($s(5,0),s(5,1),\ldots,s(5,5)$), we may obtain the row sequence of
the Jacobi-Stirling numbers of the first kind, ($J(5,0;\zeta)=0,
J(5,1;\zeta),\ldots,J(5,4;\zeta),J(5,5;\zeta)=1$) according to
(\ref{e:JStirling0}) and (\ref{e:JStirling1}). $J(5,1;\zeta)$,
$\ldots$, $J(5,4;\zeta)$ are listed as follows. $$
J(5,1;\zeta)=\sum_{i=0}^n(-1)^{5-i}\zeta^{i-1}s(5,i)s(5,1),$$
$$\zeta^2J(5,2;\zeta)=-J(5,1;\zeta)+\sum_{i=0}^2\sum_{j=i}^5(-1)^{5-j}\zeta^{j-i}\binom{j}{i}s(5,j)s(5,2-i).$$
$$\zeta^3J(5,3;\zeta)=-2\zeta
J(5,2;\zeta)+\sum_{i=0}^3\sum_{j=i}^5(-1)^{5-j}\zeta^{j-i}\binom{j}{i}s(5,j)s(5,3-i).$$
$$\zeta^4J(5,4;\zeta)=-3\zeta^2J(5,3;\zeta)-J(5,2;\zeta)+\sum_{i=0}^4\sum_{j=i}^5(-1)^{5-j}\zeta^{j-i}\binom{j}{i}s(5,j)s(5,4-i),$$
which then lead to that
$$J(5,1;\zeta)=576+1200\zeta+840\zeta^2+240\zeta^3+24\zeta^4,$$
$$J(5,2;\zeta)=-(820+1030\zeta+404\zeta^2+50\zeta^3),$$
$$J(5,3;\zeta)=273+200\zeta+35\zeta^2,$$ $$
J(5,4;\zeta)=-(30+10\zeta).$$ (see \cite[Table 2]{ref8}).
\end{example}
\begin{rem}\label{r:JSNC}
We may find that the linear recurrence formulae
(\ref{e:JStirling0}) and (\ref{e:JStirling1}) also verify
\cite[Theorem 1]{ref8}, that is, $J(n,k;\zeta)$ is a polynomial in
$\zeta$ of degree $n-k$, the coefficient of the first term with
$\zeta^{n-k}$ is $s(n,k)$, and the last terms (constant term) is
the central factorial numbers $u(n,k)$ of the first kind with even
indices, which is identical to $J(n,k;0)$. By the way, we may see
that the sum of coefficients of the polynomial is $J(n,k;1)$.
\end{rem}
\section{Acknowledgement}
The author would like to thank the referee for his/her very useful
suggestions.
\begin{thebibliography}{9}
\bibitem{ref1}
Richard~A. Brualdi,
\newblock {\em Introductory Combinatorics}, 5th ed.,
\newblock Pearson Prentice Hall, 2009.
\bibitem{ref2}
Leetsch~C.~Hsu, and Peter~Jau-Shyong~Shiue,
\newblock A unified approach to generalized Stirling numbers,
\newblock {\em Advances in Appl. Math.} {\bf 20} (1998), 366--384.
\bibitem{ref3}
George~E.~Andrews, Wolfgang~Gawronski, and Lance~L.~Littlejohn,
\newblock The Legendre-Stirling numbers,
\newblock {\em Discrete Math.} {\bf 311} (2011),
1255--1272.
\bibitem{ref4}
Eric~S.~Egge,
\newblock Legendre-Stirling permutations,
\newblock {\em European J. Combin.} {\bf 31} (2010),
1735--1750.
\bibitem{ref5}
Pietro~Mongelli,
\newblock Combinatorial interpretations of particular evaluations of
complete and elementary symmetric functions,
\newblock {\em Electron. J. Combin.} {\bf 19} (2012), \#R60.
\bibitem{ref6}
George~E.~Andrews, Eric~S.~Egge, Wolfgang~Gawronski, and
Lance~L.~Littlejohn,
\newblock The Jacobi-Stirling numbers,
\newblock {\em J. Combin. Theory Ser. A} {\bf 120} (2013),
288--303.
\bibitem{ref7}
Jiaqiang~Pan,
\newblock Matrix decomposition of the unified
generalized Stirling numbers and inversion of the generalized
factorial matrices,
\newblock {\em J. Integer Seq.} {\bf 15} (2012),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL15/Pan/pan19.html}{Article 12.6.6}.
\bibitem{ref8}
Yoann~Gelineau, and Jiang~Zeng,
\newblock Combinatorial interpretations of the Jacobi-Stirling numbers,
\newblock {\em Electron. J. Combin.} {\bf 17} (2) (2010),
\href{http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r70}{\#R70}.
\bibitem{ref9}
Neil J.~A. Sloane,
\newblock {\em The On-Line Encyclopedia of Integer Sequences},
\newblock published electronically at
\url{http://oeis.org/}.
\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}: Primary
11B73; Secondary 05A15.
\noindent {\it Keywords}: convolution, unified generalized
Stirling number of the first kind,
Jacobi-Stirling number of the
first kind, generalized Vandermonde convolution, triangular
recurrence relation, non-homogeneous linear recurrence relation.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequence \seqnum{A191936}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in} \noindent
Received April 1 2013;
revised version received September 24 2013.
Published in {\it Journal of Integer Sequences},
October 13 2013.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in
\end{document}
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\section*{Blog - Petri net programming (part 6)}
Now we take up the subject of how to modify our simulator to use the rate formulas. Although the explanation of the principle got somewhat involved, the modification to the code will be straightforward.
First, working from the ground up, we need to add a bit of structure to our Transition objectsL a rate constant, and a method to return the firing rate itself. in a labelling. As we said, this will return rateConstant * PROD Falling powers.
The other change is to the logic of a Petri net instance, which is to choose the next rule to fire, using relative probabilities determined by the firing rates returned by he Transitions.
Let's now discuss the algorithm for firing the transitions. We want this to be realistic for low concentrations as well, which means that the stochastic nature should lead to different execution sequences every time. So we won't just assume that the transitions fire at a smooth rate, as would be the case in a direct numerical approximation to the rate equation.
So, given a time interval delta-t, and firing rates Tr for each transition, the \emph{expected} number of firings of T will be Tr * delta-t.
But in our actual simulator, we want the transitions to happen \emph{one at a time}. At least because every time a transition fires, it changes the firing rates of all the transitions. So if the expected number of firings of T is three, we don't want to just force three firings of T -- in the extreme case, the artificiality of this would manifest in the fact that T might consume all of its its tokens after firing once, so that it \emph{couldn't} firing three times in that labelling.
In addition, the following issue is raised here: suppose U is expected to fire three times, and V is expected to fire five times. By what realistic mechanism would we interleave the three firings of U with the 5 firings of V?
Our solution to this is to choose delta-t small enough, so that the probability of any transition firing is ``low.'' Then we just stage a contest, in which we randomly choose a transition (with equal probability among all the transitions), and, if its expected number of firings is 0 {\tt \symbol{60}}= p \guillemotleft{}~1, we flip a biased coin that only gives heads p percent of the time -- if the transition gets heads, it fires, then the labelling gets updated, all the firing rates get re-evaluated, and a new contest begins.
This way we could do empirical experiments, to do with the statistical variations in the runs of the simulator.
How small should delta-t be for this to give a true approximation to ta real stochastic process with the given Petri net?
The crux of our approach here is that we are using the expected number of times e that a transition will fire, as the probability that is will fire in delta-t. Clearly, if e exceeds one this doesn't work at all -- we cannot have a probability of 300\% that a transition will fire.
So to a first approximation, try the following. Let Mr(l) be the maximum of the transition firing rates of of the transitions in labelling l. The let delta-t(l) = 1/Mr(l).
In this time interval, the fastest firing transition T' will have an expected number of firings
T'r * delta-t = Mr * 1/Mr = 1,
and all other transitions will have expected firing rates counts between 0 and 1.
So we \emph{could} use these as probabilities, but it's still not veridical, especially for the transitions with the highest firing rates. This is clear for the fastest transition T -- using this method, T' would fire with probability 1 during delta-t. In reality\_\_\_\_ naturally? sometimes it won't fire, sometimes it would fire twice, etc.
So let eps be a small number between 0 and 1, that will control how good our approximation is -- and let delta-t = eps / Mr.
Then T' will have Mr * delta-t = eps expected firing during delta-t -- and all other expected firings will be less than eps.
It is for small eps that the substitution of probability for expected value becomes valid.
The tradeoff here is that making eps smaller will yield a more veridical simulation, but will increase the running time of the simulation. On the other hand, if it is \emph{too} small, then numerical round issues on the machine will introduce distortions.
Mathematical aside. Why is it --- okay? legitimate? to use expected values as probabilities, for small intervals of time delta-t?
Suppose that the expected number of firings is e. The full statement is that there is a probability distribution P(delta-t) that assigns a probability to each number k, P(delta-t)(k) = probability that there will be k firings during delta-t. e is the mean of this probability distribution.
The exact shape of P(delta-t) depends on the physical characteristics of the underlying process -- which are not determined by the formal structure of the Petri net. Consider the difference in the shape of P(delta-t) that would occur, on the one hand if the firing of a transition \emph{increased} the chance of \_\_\_\_ \textbf{, versus one in which it decreased the chance of our transition firing during the decay window.}
Under the assumption that the firings \_\_\_ independent of each other, P(delta-t) assumes the shape of a Poisson distribution. In this distribution, as delta-t --{\tt \symbol{62}} INF, then P(delta-t) approaches a normal distribution.
But what can we say, in general, about the shape of P(delta-t), as delta-t approaches zero -- which is what we are using for our simulation? By in general, I mean, regardless of the specific nature of the P(delta-t).
What if delta-t were zero? Then there would be a 100\% change that \# of firings = 0, and 0\% change of any other value. So this is the curve towards which P(delta-t) must tend, as delta-t --{\tt \symbol{62}} 0.
Now what if e is a very small positive number?
Then the distribution will look like this:
(0,p), (1,q), with p just under 1, and q just over 0.
The value p at 0 is slightly less than 1, the value q at 1 is a small positive number, and all the other values are \emph{much} smaller than q. As EPS tends to zero, in relation to q, the other terms become negligible. So towards the limit, this becomes a valid approximation:
(0,p), (1,q), p just under 1, q = 1 - p.
where p + q = 1.
Now E(P(delta-t)) = p * 0 + q * 1 = q = probability that the transition fires once during delta-t = probability that it fires at all during delta-t.
E(P(delta-t)) = P(delta-t)(1) = SUMk {\tt \symbol{62}}= 1, INF P(delta-t)(k).
Now, for our transitions, we have that E(\# of firings) = Tr * delta-t = q.
So (under these assumptions) the number we compute, based on rate coefficients, etc., Tr * delta-t is the \emph{probability} that T will fire during delta-t. DONE
\vspace{.5em} \hrule \vspace{.5em}
The modification to our code, then follows as a matter of course from these considerations.
def FireOneRule(labelling):
\begin{verbatim}let M = this.transitions.Max(tr => tr.GetFiringRate(labelling)).\end{verbatim}
Let delta-t = EPS / M.
while (true): let i = random(1, len(this.transitions)) let T = this.transitionsi let q = T.rate(lab) * delta-t let rand = random(0,1) if rand {\tt \symbol{60}}= q: T.fire(lab) return
Code changes.
All the changes can be made as an incremental development of our \_\_\_ previous? Petri net simulator.
Here is the new transition class:
\vspace{.5em} \hrule \vspace{.5em}
Here is the Petri net class:
\vspace{.5em} \hrule \vspace{.5em}
Output will show running sum of delta-t.
Sample runs.
What does our H20 simulation converge to?
Periodic behavior.
Bistable behavior.
Applications / exercises.
\end{document}
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https://mhb.hs-emden-leer.de/latex/Lange/Fertigungstechnik.tex | hs-emden-leer.de | CC-MAIN-2021-25 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-25/segments/1623487647232.60/warc/CC-MAIN-20210619081502-20210619111502-00321.warc.gz | 369,451,077 | 1,800 | \documentclass[11pt]{article}
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%\begin{longtable}{|l|p{0.7\textwidth}|}
\begin{tabularx}{\textwidth}{|X|p{0.58\textwidth}|}
\hline
%\textbf{Studiengang} & 1BaMD, 1BaMDP\\ \hline
\textbf{Modulbezeichnung} & \textbf{Fertigungstechnik}\\ \hline
\textbf{Semester} & 1\\ \hline
\textbf{Dauer} & 1 Semester\\ \hline
\textbf{Art} & Pflichtfach\\ \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} & \\ \hline
\textbf{Verwendbarkeit} & BaMD, BaMDP\\ \hline
\textbf{Prüfungsform und -dauer} & Klausur 2h oder mündliche Prüfung\\ \hline
\textbf{Lehr- und Lernmethoden} & Vorlesung, Labor\\ \hline
\textbf{Modulverantwortlicher} & S. Lange\\ \hline
%\textbf{ModulverantwortlicherLang} & S. Lange\\ \hline
\textbf{Qualifikationsziele} & Die Studierenden kennen die sechs DIN-Hauptgruppen der Fertigungsverfahren und die den Fertigungsverfahren zugrunde liegenden prozess- sowie werkstofftechnologischen Grundlagen.
Die Studierenden sind in der Lage, für Fertigungsaufgaben geeignete Fertigungsverfahren auszuwählen, die Eignung zu bewerten und ihre Auswahl zu begründen.\\ \hline
\textbf{Lehrinhalte} & Vorlesung Fertigungstechnik
Fertigungsverfahren nach DIN 8580; Grundlagen der Ur- und Umformtechnik, trennende Verfahren, Fügetechnik, Beschichtungstechnik, Stoffeigenschaftändern und Wärmebehandlung, Fertigungstechnik im System Fabrikbetrieb
Labor Fertigungstechnik
Versuche zu den Verfahren Urformen, Umformen, Trennen, NC-Programmierung.\\ \hline
\textbf{Literatur} & Klocke, F., König, W.: "'Fertigungsverfahren"' Band 1 bis 5, Springer Verlag
Fritz, A. H., Schulze, G.: "'Fertigungstechnik"', Springer Verlag
Dubbel, H.: "'Taschenbuch für den Maschinenbau"', Springer Verlag\\ \hline
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S. Lange & Vorlesung Fertigungstechnik & 2\\ \hline
S. Lange, L. Krause & Labor Fertigungstechnik & 2\\ \hline
%!!!Dozent2!!! & !!!Titel2!!! & !!!SWS2!!!\\ \hline
%!!!Dozent3!!! & !!!Titel3!!! & !!!SWS3!!!\\ \hline
%!!!Dozent4!!! & !!!Titel4!!! & !!!SWS4!!!\\ \hline
%!!!Dozent5!!! & !!!Titel5!!! & !!!SWS5!!!\\ \hline
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https://mhb.hs-emden-leer.de/latex/Koch/HW_SW_Codesign_PO2017.tex | hs-emden-leer.de | CC-MAIN-2021-25 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-25/segments/1623487636559.57/warc/CC-MAIN-20210618104405-20210618134405-00214.warc.gz | 359,135,915 | 2,263 | \documentclass[10pt]{article}
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%\begin{longtable}{|l|p{0.7\textwidth}|}
\begin{tabularx}{\textwidth}{|X|p{0.64\textwidth}|}
\hline
%\textbf{Studiengang} & 6E, 8EP, 6I, 8BaIP\\ \hline
\textbf{Modulbezeichnung}
(eng.) % HW/SW Codesign
& \textbf{HW/SW Codesign}
(HW/SW Codesign)
\\ \hline
\textbf{Semester} & WPM\\ \hline
\textbf{ECTS-Punkte (Dauer)} & 5 (1 Semester)\\ \hline
\textbf{Art} & Wahlpflichtmodul Zertifikat Technische Informatik\\ \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)} & Hardwarenahe Programmierung\\ \hline
\textbf{Empf.\ Voraussetzungen} & C/C++, Digitaltechnik, Mikrocomputertechnik, VHDL\\ \hline
\textbf{Verwendbarkeit} & BaE, BaEP, BaI, BaIP\\ \hline
\textbf{Prüfungsform und -dauer} & Klausur 1,5h oder mündliche Prüfung oder Studienarbeit\\ \hline
\textbf{Lehr- und Lernmethoden} & Vorlesung, Praktikum\\ \hline
\textbf{Modulverantwortlicher} & C. Koch\\ \hline
%\textbf{ModulverantwortlicherLang} & C. Koch\\ \hline
\multicolumn{2}{|p{0.97\textwidth}|}{
\textbf{Qualifikationsziele}\newline
Ziel der Veranstaltung ist die Zusammenführung der zunächst im Studium getrennten Betrachtung von Hardware- und Software-Systemen zum Aufbau, Entwurf und Analyse moderner eingebetteter Systeme.
Die Studierenden haben hierbei weiterführende Kenntnisse bezüglich eingebetteter Systeme als auch deren Partitionierung erworben und beherrschen grundlegende Methoden zum Design und zur Programmierung eines System-on-Programmable-Chips (SoPC).
}
\\ \hline
\multicolumn{2}{|p{0.97\textwidth}|}{
\textbf{Lehrinhalte}\newline
Die Vorlesung HW/SW Codesign behandelt typische Zielarchitekturen und HW/SW-Komponenten von eingebetteten Standard-Systemen und System-on-Programmable-Chips (SoPC) sowie deren Entwurfswerkzeuge für ein Hardware/Software Codesign.
Hierbei behandelte Zielarchitekturen und Rechenbausteine umfassen Mikrocontroller, DSP (VLIW, MAC), FPGA, ASIC, System-on-Chip als auch hybride Architekturen.
Weitere Stichworte sind: Hardware/Software Performanz, Sequentielle oder parallele Verarbeitung, Multiprozessorsysteme (UMA, NUMA, Cache-Kohärenz), Custom Instruction, Custom Peripherals, IP-Core (Soft-IP-Core, Hard-IP-Core) und Bus-Konzepte eingebetteter Systeme (Gateway, Bridge, Marktübersicht).
}
\\ \hline
\multicolumn{2}{|p{0.97\textwidth}|}{
\textbf{Literatur}\newline
Schaumont, P.: A Practical Introduction to Hardware/Software Codesign, Springer, 2013
Mahr, T: Hardware-Software-Codesign, Vieweg Verlag Wiesbaden, 2007.
Patterson, D.A.: Rechnerorganisation und -entwurf, Elsevier München, 2005
}
\\ \hline
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C. Koch & HW/SW-Codesign & 2\\ \hline
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\title{Beyond Right and Wrong\dots{}}
\date{August 2015}
\author{Conspiracy of Cells of Fire}
\subtitle{For Anarchy}
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{\usekomafont{title}{\huge Beyond Right and Wrong\dots{}\par}}%
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\begin{quote}
\emph{“I don’t believe in rights. Life, which is all a manifestation of incoherent forces, unknown and unknowable, rejects the human artificiality of the right. Rights were born when life was taken away from us. Indeed, originally, humanity had no rights. It lived and that was everything. Today, instead, there are thousands of rights; one could accurately say that everything which we have lost we call a right. I know that I live and that I desire to live. It is most difficult to put this desire into action. I am surrounded by a humanity that wants what everyone else wants. My isolated affirmation is a most serious crime. Laws and morals, in competition, intimidate and persuade me. The “blonde rabbi” [I.e., Christ or Christian values.-translator] has triumphed. One prays, one implores, one curses, but one does not dare. Cowardice, caressed by Christianity, creates morality, and this justifies baseness and begets renunciation. [\dots{}] “Society, on the other hand, modest and clean in appearance, but horribly infected with gangrene throughout its body, makes me vomit, fills me with horror and loathing, kills me.” How I envy the great Bonnot! “Il me faut vivre ma vie!”\footnote{From the known “defense” of the French illegalist Jules Bonnot.} \Slash{} “It is necessary that I live my life””}
\textbf{Bruno Filippi}
\begin{flushright}
\emph{(An Italian anarcho-nihilist who was charged with several armed attacks and was killed by a bomb he carried, on September 7\textsuperscript{th}, 1919, when trying to place it at the “Nobles Club”, headquarters of the wealthiest Italian businessmen)}
\end{flushright}
\end{quote}
The most important and nicest things are spoken in the simplest way. Today, though, the reality of our lives is far from simple. So, we often note that the more complicated the words (and boring, at the same time) that “specialists” of political delivery and the “revolutionary” rhetoric alchemists use, the more uninvited their oversimplifications are. \textbf{The theorist “rebels” construe the world through the tyranny of their “obvious truths”. Their whole rhetorical calcification and their wooden words, that lulls to sleep through their undeviating dogmatism, comes to transfer “social revolution” to an oversimplified version of the eternal fight between the “good” people and the “bad” state, between the “right” and the “wrong” fairytale.}
But if things were that simple, why hasn’t this fairytale, for centuries, come to an end, with the triumph of the “good” and for all of us to “live happily ever after”?
Especially today, when power is not centralist and abstracted to the king’s throne, but, it is spread inside the transparent social factory, both our anarchist words and our actions ought to deepen more and tear up the “religious” missals and the aphorisms of the “good” and “bad” and “right” and “wrong” which lack depth.
Power is not just unfair, bad and malicious, and it is not something which by being ‘denounced’, even violently, will withdraw for the rebels’ rights to be recognised.
\textbf{Power is a social relationship, a social hierarchical organization model, a way of life management.}
In addition to its directorates and its officials, it owns its own preachers, its mentors, its advisers, its jesters, its armed defendants of course, its loyal followers, even its inside objectors – usurpers\dots{}
It’s not just a “bad oligarchic elite” it is an intricate system of relations that defines our everyday lives.
\textbf{We know, of course, that if you cut the snake’s head, the rest of the body, after a few convulsions, stops\dots{} Power, however, has proven to be more like a Hydra.}
\textbf{This is why, while our armed targeting gathers its firepower on the heads of the managers of power and their uniformed mercenaries, our words seek to blow up the social relationships that give rise to power.} Let’s keep in mind that the phrase “no one is indispensable” goes for power as well. If we don’t hit both the heart of the beast (armed attacks against the officials of the power) and the veins of the social machine (criticism and rejection of the submissive mindset), then, maybe, soon, after every attack of ours, we will hear “the king is dead, long live the new king”. \textbf{Because unless slaves, even when they revolt, deeply renounce the mindset of submission, they will soon wish to crown their new king, next to the corpse of the former.}
This is why the \textbf{Conspiracy of Cells of Fire}, \textbf{FAI\Slash{}IRF} and the \textbf{“political” groups in affinity}, that form the stripe of \textbf{black anarchy}, both in the proclamations following our attacks, and in our texts, use heretic, provocative words, that do not comply with the traditions of the revolutionary automatism of the dipole “good – bad”, “fair – unfair”\dots{} We always have the sledgehammer of rudeness available, in order to shatter the window of the good and innocent society and highlight its guilty silence and frustrating passivity.
Of course, through our criticism we do not aim to build the crystal tower of the “revolutionary” self-admiration. This is why we detest the conceit and the arrogance that we sometimes find in our circles, by people who are alternative artists of nothing, not anarchists of \textbf{praxis}.
\textbf{Our aim is to disassemble the stereotypes and the prevailing morality that poison our lives, through the small and large representations of informal power (family, school, work, relationships).}
Our way is a challenge and not a political politeness that caresses the ears of the repressed ones audience, most of whom don’t even bother to read a proclamation. \textbf{An inconvenient truth is more inelegant but it is also more liberating than a pleasant lie\dots{}}
So, it’s not enough to talk about the “rights” of the repressed ones, the proletarians, the “people”\dots{}
First of all, the “fair” and the “unfair” is a moral subjectivity of the impression of reality. There’s no such thing as an objective scale that defines what’s right and what’s wrong.
Power and the capitalistic management of it, along with the armed enforcement of their truth, they have their own think tanks, their own arguments, their own culture, their own civilization, their own suggestions of a way of life. Power does not dominate only supported by the power of its arms’ barrels, but also by its persuasion and its propaganda.
This is why anyone who makes the mistake to talk about the “rights” of the many, will have to be careful because the interpretation of the “right” as a quantitative measurement unit, will not\dots{} prove them right. The right of the many is often the right of the viewers, the consumers, the voters’\dots{} The anarchist struggle is not about counting participation, nor does it have to do with the majority’s right. It is something much bigger than the conflict between the “right” and “wrong”\dots{} It is a constant war between different values, a war that bisects society in two parts. One part is the world of anarchy and the other part is the world of power and organized tedium.
\textbf{In this conflict, several people, who are excluded from the privileges of power, poor and oppressed, stand shoulder to shoulder with their elite rulers’ interests. The mass is usually fond of the mediocre, the immobility of habit, the rigidity of prudence and afraid of the new, the radical, the unknown of insurrection.}
The bureaucratic sector inside the official anarchism and its communist components use the rhetoric of the “common good” and the “justness of the oppressed ones”, thinking there is a \textbf{conscious} proletarian class, which will turn into the basic ingredient of a “social revolution”, as long as it has its ears caressed.
We, for our part, want to set the conditions for the creation of a confrontational anarchist affinity between groups, cells and individuals, which are willing to transfer the experience of rupture with the existent immediately, here and now.
This way, a dangerous enemy in the heart of the beast can be formed, aiming at the \textbf{diffusion of anarchy}.
For this to happen, we have to make the conflict with authority permanent, to create a short circuit in the neurons of the system, to exploit and expand the contradictions of society, to provoke social peace, to qualitatively deepen anarchist thought and aggressively upgrade anarchist action, to challenge law and order. To overcome the moral denunciation of the injustices of authority and to prepare the war against it by promoting the new anarchist urban guerrilla.
\textbf{Here follows the strategic matter between the moral impeachment of the system and the continuous attack.} The biggest part of the anarchist milieu in Greece is usually navigating through the maelstrom of events resulting from short circuits of authority. Occasional demonstrations and sometimes conflicts in an anti-war demonstration, student marches, strikes.
The recent three-year “drought” of social mobilizations caused the “drought” of violent clashes in the streets of the metropolis.
The people didn’t take the streets and anarchists were insufficient in creating their own autonomous collective violent actions. \textbf{This is the result of a conscious and subconscious (because of a habit) strategy, which presents anarchists as the violent reflex of “the sense of justice” of the masses.} There is, namely, a certain timidity for the anarchist attack to be organized and expressed autonomously without moral \textbf{coverage} from the masses. In fact, of course, there is no moral cover in large social protests either, as the mass of protesters is a diverse crowd, from which, some believe in peaceful protest, others are professional walkers and members of unions and parties, others are angry and want to clash, others operate as internal repression, others are not members of some group\dots{} \textbf{The issue is that the strategy of social counter-violence as a moral justification – response of the oppressed, is not defined by us, at a time when authority can set alternative questions and the answers of the masses can come, not as a rupture, but as consent to them.}
So, by waiting for the next social explosion, the next rally, the next big march, we abandon our ideas and actions to luck. \textbf{But even when the social tension takes place, in order for us to get lost in the riots, we look like stowaways who jump in the last car of the train, a train that others drive on different tracks from ours. Even if we derail the train it will soon return on its rails.}
Obviously, in no case do we advocate our absence from the field of metropolitan riots whose context we do not define (student rallies, anti-war mobilizations, large marches), in the name of a supposed anarchist purity. \textbf{Within these mobilizations we can organize attacks against cops, burn banks, destroy cameras, expropriate shops, break the peace in the metropolis.}
\textbf{All these are intense and pleasant moments that, however, when not accompanied by a wider anarchist plan, end up staying isolated moments and beautiful memories, that just wait for the next march to be repeated.} They lose the overall perspective and the potential to sharpen the attack and to make the tension in our lives permanent. This is the result of not only the lack of operational planning, but mainly of \textbf{overall perception}.
The notion of moral vindication of social counter-violence solely in response to systemic anomalies (violence of cops, racist attitudes, employers’ “arbitrariness”, harsh laws etc) incorporates the denouncement of the system (even with violent forms) and prevents the passage from the defensive counter-violence, to the aggressive continuous challenge of anarchist urban guerrilla.
We, on our part, want to articulate and organize a proposal of continuous attack, a complete anarchist plan, an insurrection that does not stop when the masses withdraw from their protests, but continues to feed from its fires, to grow big and to be diffused\dots{}
\textbf{We feel like the hands of our clock have stopped in the moment of attack. We do not now need neither a cause, nor the moral justification. We know that the ugliness of this world is only repealed when one acts.}
\textbf{Our proposal is to create an informal network of anarchist cells that will promote the continuous attack against authority and society.}
\textbf{Many anarchists fear the word “organization” in the way Christians fear the devil.}
Others misunderstand and confuse the meaning of organization with bureaucratic fossils of Marxist centralized organizations, central committees, hierarchies, simple members, \textbf{constipated} rules, obligatory moral guidelines, statutes and enlightened vanguardism\dots{} Others prefer the alternative ways, sureness, adventurism and safety of an anarchist lifestyle, rather than an organized anarchy and a dangerous internal enemy that attacks without looking for pretexts as the causes are more than enough\dots{}
\forcelinebreak
Some will hastily become indignant, saying that organization kills spontaneity, individuality and desires\dots{}
Let’s say, however, what we mean by “anarchist organization”\dots{} Anarchist organization is the living mental and physical coordination of a group of comrades, in order to carry out a certain plan. The more complete that plan is, the more comprehensive is the relationship of the group’s comrades, while the commitment and consistency have as a measure, the power of desires to achieve the plan and not the discipline of a military duty. Each comrade is unique and independent within the group and through the collective life and action of the cell, discovers and releases more of themselves. There is no membership card, but only the individual desire to take part in something genuinely collective.
Of course, organization is not an end in itself, it is the means to get where we want. \textbf{This means that an anarchist organization, an anarchist cell, must keep its procedure under constant review, to develop its relations, to upgrade its actions, to sharpen its theory, so that it comes closer to the purpose of its formation.}
It is only logical that within an anarchist group come up tensions, contradictions, anger and even potential departures. This is because every human relationship is confrontational, sometimes at the level of evolution and sometimes at the level of rupture.
The sure thing is that the existence of informal anarchist organizations and direct action cells fuels anarchist violence against authority.
Every anarchist group is a living outbreak of war against the system. Through discussion, friction and composition within a group of comrades, anarchist action evolves, the threat of an organized internal enemy becomes permanent, the means of attack are upgraded, thought gets sharpened and the plan of the destruction of authority and the social machine is promoted.
\textbf{We know no team can develop those associations of strength in order to decapitate the beast of authority and its products by itself.} Nevertheless, even so, the comrades of the group, through their action, free themselves from the conventions of a world that wants us to be spectators of our lives. \textbf{But if we want to maximize our action, satisfying more and more our desires, we can try to create informal coordinations of individuals, groups and affinity cells which promote the anarchist urban guerrilla.} The creation of such a coordination is not subject, in any case, to the crucible of the quantitative centralism, which crushes the autonomy of each one of us. We are not interested in creating a central super-structure which will cause the creation of central committees and informal hierarchies. We are simply talking about coordinating groups and people looking towards the same direction. We are talking about the coordination of desires that become more dangerous when they are communicated and shared by accomplices.
The basic agreement within such an organization is the desire not to be in a moment of truce with the enemy. Without, therefore, waiting for a favorable wind of social change to act, we decide to arm ourselves and turn our lives into a confrontational reality now. So we do not limit ourselves to the anniversal symbolism (this does not mean that we are absent from the days of wrath and vengeance in memory of our dead), we do not expect fixed appointments, waiting for the state to get out of line causing the people to demonstrate, nor are we satisfied by opportunistic street fights with cops, in order to pretend that we’ve executed our “duty” to the struggle.
This does not make us arrogant to devalue everything from the balcony of an ideological purity. On the contrary, it makes us more prepared to throw ourselves in those battles that we will choose, even in intermediate social struggles, which we think are of interest (i.e. student occupations) without being disoriented by the circumstances.
The compass of organized expression points steadily towards the the intensification of the attack and the diffusion of our theory. \textbf{The words “anarcho-nihilism”, “black anarchy”, “anarchist terrorism” are truly dangerous, when tested in the heat of battle.}
The constant challenging of the enemy through autonomous guerrilla strikes (using the fan of the diversity of means, but with the constant desire of upgrading to armed guerrilla) and organized aggressive intervention to intermediate social struggles are part of the anarchist war. We say again, that the effectiveness of the strategy will not be measured by the figures of participation.
We want to create the possibilities of acting with people who feel stifled in the social cages imposed on them by authority and want to rebel\dots{} \textbf{Our joy is great in any such new meeting with new comrades who bear the sign of complicity. No matter their numbers\dots{} What is important is that the effort is worth it\dots{}}
\emph{“I am not led by the will of the masses. Nor do I mourn for the sorrows of the people. I never accepted the fate of the slave that was prepared for me, I didn’t speak their language, nor imitated their look. I refused to be with the many. My demons never sleep\dots{} I always long for the unsatisfied. And when they set fire to the foundations of society, they don’t daydream on the ashes. They are seeking wildly for the next scarecrow of authority to surrender it to the stake. They do not get comfortable, nor do they rest, they want war with everything that haunts our lives.
\forcelinebreak
They say that whoever loves debris, also loves statues. My demons live in the debris because nobody can hide there. The material of which each of us is made, is revealed there. You will find me among them, where the battle is raging”\dots{}}
\textbf{Conspiracy of Cells of Fire – FAI\Slash{}IRF}
August 2015
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The Anarchist Library (Mirror)
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Conspiracy of Cells of Fire
Beyond Right and Wrong\dots{}
For Anarchy
August 2015
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\href{https://interarma.info/2015/08/21/ellada-pera-apo-to-dikaio-kai-to-adiko-spf/?lang=en}{interarma.info}
Translated by Inter Arma (translation corrected by 325)
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\textbf{usa.anarchistlibraries.net}
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\noindent Classification of irreducible representations of Heisenberg groups and algebras\par}
\bigskip
\centerline{Dmitri Pavlov, University of California, Berkeley, pavlov@math.}
\centerline{Please report any mistakes or misprints to me.}
\section Definitions of Heisenberg groups and algebras
\proclaim Definition. A {\it symplectic vector space\/} is a pair $(V,\omega)$,
where $V$ is a finite-dimensional real vector space and
$\omega$ is a nondegenerate real skew-symmetric bilinear form.
The dimension of a symplectic vector space is always even.
We denote it by~$2n$.
\proclaim Definition. A {\it Heisenberg group\/} for a symplectic vector space~$(V,\omega)$
is the Lie group with the underlying manifold $V\times\R$ and the multiplication
$(u,s)(v,t)=(u+v,s+t+\omega(u,v)/2)$ where $u,v\in V$ and $s,t\in\R$.
The map $t\mapsto(0,t)$ is a Lie group homomorphism from~$\R$ to the Heisenberg group.
Its image coincides with the center of the Heisenberg group.
The dimension of the Heisenberg group equals~$2n+1$.
\proclaim Definition. A {\it Heisenberg algebra\/} for a symplectic vector space~$(V,\omega)$
is the Lie algebra with the underlying vector space $V\oplus\R$ and the commutator
$[(u,s),(v,t)]=(0,\omega(u,v))$ where $u,v\in V$ and $s,t\in\R$.
The map $t\mapsto(0,t)$ is a Lie algebra homomorphism from~$\R$ to the Heisenberg algebra.
Its image coincides with the center of the Heisenberg algebra.
The dimension of the Heisenberg algebra equals~$2n+1$.
\proclaim Theorem. The map $V\oplus\R\to V\times\R$ that sends $(v,t)$
to~$(v,t)$ for every $v\in V$ and $t\in\R$
is the exponential map from a Heisenberg algebra to the corresponding Heisenberg group.
\section Definitions of representations of Heisenberg groups and algebras
Heisenberg groups do not have irreducible finite-dimensional representations
of dimension greater than~1.
This follows from the same result for Lie algebras, which we explain later.
Hence we need to go to the infinite-dimensional case.
Finite-dimensional representations are unitarizable.
In the infinite-dimensional case this is not always true and has to be assumed.
We restrict ourselves to the group of unitary operators on a Hilbert space.
We define a representation of a Heisenberg group as a continuous homomorphism
from the Heisenberg group to the group of the unitary operators.
To talk about continuity we need to put a topology on the group of unitary operators.
One must be very careful about this topology.
For example, if we choose the norm topology, then from Stone's theorem and the corresponding
result for Lie algebras, which we explain later, it follows
that there are no irreducible unitary representations of a Heisenberg group
on an infinite-dimensional Hilbert space.
\proclaim Definition. A {\it representation of a Heisenberg group\/} is a continuous
homomorphism from the Heisenberg group to the topological group of the unitary operators
on a Hilbert space equipped with the strong operator topology.
Here the strong operator topology is the weakest topology on the set of
all bounded operators such that all evaluation maps at points are continuous.
The strong operator topology is weaker than the norm topology.
Hence we can hope to find some irreducible infinite-dimensional unitary representations
of a Heisenberg group.
In fact, there is a family of such representations called Shr\"odinger representations,
as we explain later.
Now we want to define representations of Heisenberg algebras.
Existence of a faithful trace in the finite-dimensional case immediately implies
that there are no irreducible finite-dimensional
Heisenberg algebra representations of dimension greater than~1.
Hence we have to go to the infinite-dimensional case.
It is an easy exercise in functional analysis [Reed and Simon, \S VIII.5, Example~2]
that there are no irreducible representations of a Heisenberg algebra in the Lie
algebra of bounded operators on an infinite-dimensional Hilbert space.
Hence we must allow unbounded operators.
Stone's theorem, which we explain below, tells us that there is a bijective correspondence
between representations of~$\R$ in the group of the unitary operators with the strong topology
and skew-adjoint operators.
Hence it makes sense to look at the Lie algebra of the skew-adjoint operators
defined on a dense subspace.
However, it is unclear how one can define a commutator of two such operators,
in particular, what should be the domain of the commutator.
Here is one possible solution to this problem.
\proclaim Definition. A {\it representation of a Heisenberg algebra\/} is a Lie
algebra homomorphism from the Heisenberg algebra to the Lie algebra of skew-symmetric
endomorphisms of a dense subspace~$D$ of a Hilbert space~$H$.
It does not hurt to point out that continuity of such homomorphism follows from linearity.
The commutator of two skew-symmetric endomorphisms of a dense subspace~$D$
is again an skew-symmetric endomorphism of~$D$ in an obvious way,
hence the target is indeed a Lie algebra.
We remark that the condition of skew-adjointness was relaxed to the condition of skew-symmetry.
Any Lie algebra has the zero operator and if we require it to be skew-adjoint or even closed,
then $D=H$ and we do not want this to happen.
Since we have already mentioned skew-symmetry and skew-adjointness,
let us briefly recall their properties.
\proclaim Definition. An operator~$T$ is {\it skew-symmetric\/} if its adjoint operator~$T^*$
exists and $T\subset-T^*$. Here we write $A\subset B$ whenever $B$ extends~$A$.
If $T$ is skew-symmetric, then its domain is a dense subspace, because
the adjoint operator is defined only for such operators.
Also $T^{**}$ exists and is equal to the closure of~$T$.
We have the following four classes of operators:
\tabskip\parindent
\halign{#\hfil&$#$\hfil\cr
skew-symmetric&T\subset T^{**}\subset-T^*\cr
closed skew-symmetric&T=T^{**}\subset-T^*\cr
essentially skew-adjoint&T\subset T^{**}=-T^*\cr
skew-adjoint&T=T^{**}=-T^*\cr
}
\section Connection between representations of Heisenberg groups and algebras
In the finite-dimensional case the standard functor from the category of finite-dimensional
Lie groups to the category of finite-dimensional Lie algebras maps every finite-dimensional
representation of a Lie group to a finite-dimensional representation
of the corresponding Lie algebra.
In this section we establish an analog of this construction for the infinite-dimensional case.
\proclaim Theorem. (Stone, 1932.)
The map $A\mapsto(t\mapsto\exp(tA))$ establishes a bijective correspondence
between skew-adjoint operators~$A$ on a Hilbert space
and representations of~$\R$ in the group of unitary operators with the strong topology.
\proclaim Theorem. Suppose we have a representation of a Heisenberg group.
Fix an element of the corresponding Heisenberg algebra and take the Lie algebra homomorphism
from~$\R$ to the Heisenberg algebra that sends~1 to this element.
Compose it with the exponential map and then with the representation.
We obtain a continuous homomorphism from~$\R$ to the topological group of unitary operators
with the strong topology.
By Stone's theorem we get an skew-adjoint operator.
Hence we have a mapping~$F$ from the Heisenberg algebra to skew-adjoint operators.
To obtain a Lie algebra homomorphism, denote by~$D$ the intersection of the domains
of all operators in the image of~$F$ and restrict everything to~$D$.
We claim that $D$ is a dense subspace and the restriction of~$F$
is a representation of the Heisenberg algebra.
In the finite-dimensional case there is a functor that goes the other way round:
Every morphism of Lie algebras is mapped to a morphism
of the corresponding simply connected Lie groups.
Moreover, this functor is an equivalence of the categories
of Lie algebras and simply connected Lie groups.
In the infinite-dimensional case this is not true.
Later we give an example of a representation of a Heisenberg algebra
that does not correspond to any representation of the corresponding Heisenberg group.
\section Elementary properties of representations
\proclaim Definition.
A representation of a Heisenberg group on a Hilbert space~$H$
is called {\it irreducible\/} if $H$ is nontrivial and any closed
subspace of~$H$ that is invariant under the action of the group
coincides with~$H$ or with the zero subspace.
\proclaim Proposition. If $F$ is an irreducible representation of a Heisenberg group,
then for all real~$t$ we have $F(0,t)=\exp(ht)I$ for some unique imaginary number~$h$,
where $I$ is the identity operator.
\proclaim Proof. The image of $(0,t)$ commutes with the image of~$F$ for all~$t$.
By the infinite-dimensional Schur's lemma we have $F(0,t)=\lambda(t)I$ for some
continuous homomorphism~$\lambda\colon\R\to\U$,
where $\U$ is the group of complex numbers of norm~1.
Obviously, $\lambda(t)=\exp(ht)$ for some unique imaginary number~$h$.
\proclaim Definition. The number~$h$ defined in the previous proposition for an arbitrary
irreducible representation of a Heisenberg group
is called the {\it parameter\/} of the representation.
If $h=0$, then the representation is called {\it trivial}.
\proclaim Proposition. The map $u\mapsto((v,t)\mapsto\exp(u(v)))$ establishes
a bijective correspondence between $\Hom(V,\I)$ and the set of all isomorphism classes of
trivial irreducible representations of the Heisenberg group
for a symplectic vector space~$(V,\omega)$.
Here $\I:=\{z\in\C\mid\Re z=0\}$ denotes the set of all imaginary numbers.
\proclaim Proof. A trivial representation factors through
the Lie group homomorphism $(v,t)\mapsto v$
from the Heisenberg group to the vector space~$V$ with the additive Lie group structure.
\proclaim Definition.
A representation of a Heisenberg algebra on a Hilbert space~$H$ with a dense subspace~$D$
is called {\it irreducible\/} if $H$ is nontrivial and any closed
subspace~$G$ of~$H$ such that $G\cap D$ is invariant under the action of the algebra
coincides with~$H$ or with the zero subspace.
\proclaim Proposition. If $F$ is an irreducible representation of a Heisenberg algebra,
then for all real~$t$ we have $F(0,t)=htI$ for some unique imaginary number~$h$,
where $I$ is the identity operator.
\proclaim Proof. The image of $(0,t)$ commutes with the image of~$F$ for all~$t$.
By the infinite-dimensional Schur's lemma we have $F(0,t)=\lambda(t)I$ for some
continuous homomorphism~$\lambda\colon\R\to\C$.
Since $\lambda(t)I$ is skew-symmetric, we have
$\lambda(t)=ht$ for some unique imaginary number~$h$.
\proclaim Definition. The number~$h$ defined in the previous proposition for an arbitrary
irreducible representation of a Heisenberg algebra
is called the {\it parameter\/} of the representation.
If $h=0$, then the representation is called {\it trivial}.
\proclaim Proposition. The map $u\mapsto((v,t)\mapsto u(v))$ establishes
a bijective correspondence between $\Hom(V,\I)$ and the set of all isomorphisms classes of
trivial irreducible representations
of the Heisenberg algebra for a symplectic vector space~$(V,\omega)$.
\proclaim Proof. A trivial representation factors through
the Lie algebra homomorphism $(v,t)\mapsto v$ from the Heisenberg algebra
to the vector space~$V$ regarded as an abelian Lie algebra.
\section Schr\"odinger representations of Heisenberg groups and algebras
In this section we define a series of nontrivial irreducible representations
of Heisenberg groups and algebras.
First we need to look deeper into the structure of a symplectic vector space.
\proclaim Definition. A {\it polarization\/} of a symplectic vector space $(V,\omega)$
is a pair~$(W,X)$ of subspaces of~$V$ such that $V=W\oplus X$,
the form $\omega$ vanishes on $W$~and~$X$ and defines a nondegenerate pairing between $W$~and~$X$,
which we denote by~$w\cdot x$.
A {\it Lagrangian subspace\/} is a vector subspace of~$V$ that
appears as a part of a (unique) polarization of~$(V,\omega)$.
It follows that $\omega(w+x,w'+x')=w\cdot x'-w'\cdot x$ for arbitrary $w,w'\in W$ and $x,x'\in X$.
\proclaim Definition. A {\it Schr\"odinger representation\/}
of a Heisenberg group with a nonzero parameter~$h\in\I$
corresponding to a polarization~$(W,X)$ of a symplectic vector space~$(V,\omega)$
is the representation~$R_h$ of the Heisenberg group for the symplectic
vector space~$(V,\omega)$ on the Hilbert space~$L^2(W,\C)$ such that
for arbitrary $w,z\in W$, $x\in X$, $t\in\R$, and $f\in L^2(W,\C)$ we have
$R_h(w+x,t)(f)(z)=\exp(h(t+z\cdot x+w\cdot x/2))f(z+w)$.
\proclaim Theorem. Any Schr\"odinger representation of a Heisenberg group is irreducible.
\proclaim Definition. A {\it Schr\"odinger representation\/}
of a Heisenberg algebra with a nonzero parameter~$h\in\I$
corresponding to a polarization~$(W,X)$ of a symplectic vector space~$(V,\omega)$
is the representation~$S_h$ of the Heisenberg algebra for the symplectic
vector space~$(V,\omega)$ on the Hilbert space~$L^2(W,\C)$ with the
dense subspace~$S(W,\C)$ of Schwartz functions such that
for arbitrary $w,z\in W$, $x\in X$, $t\in\R$, and $f\in S(W,\C)$ we have
$S_h(w,x,t)(f)(z)=h(t+z\cdot x)f(z)+\partial_wf(z)$.
Here $\partial_w$ is the derivation corresponding to the constant
vector field on~$W$ with value~$w$.
\proclaim Theorem. Any Schr\"odinger representation of a Heisenberg algebra is irreducible.
\section Uniqueness of irreducible representations with a given parameter
\proclaim Theorem. (von Neumann, 1931.) Every irreducible representation
of a Heisenberg group with a nonzero parameter~$h$ is unitarily equivalent
to the Schr\"odinger representation with the parameter~$h$.
\proclaim Hypothesis. (Stone, 1930.) Every irreducible representation
of a Heisenberg algebra with a nonzero parameter~$h$ is unitarily equivalent
to the Schr\"odinger representation with the parameter~$h$.
Stone's hypothesis turned out to be wrong.
In fact, Stone gave a sketch of a supposed proof of this hypothesis in his paper.
When von Neumann tried to understand this sketch, he realized
that one needs to put some integrability condition on representations of the Heisenberg algebra.
In the end he obtained his theorem, which he published with a complete proof in~1931.
In the next section we look at the simplest nontrivial case $n=1$ and $|h|=1$.
We give a counterexample to Stone's hypothesis in this case.
Later we discuss a correction to Stone's hypothesis.
\section Examples
The easiest case is $n=0$.
It is obvious that the one-dimensional Schr\"odinger representations
$t\to\exp(ht)I$ and $t\to htI$ are the only nontrivial irreducible representations.
The next case is $n=1$. Assume $|h|=1$.
Fix an irreducible representation~$F$ of the Heisenberg group with these parameters.
Choose $w\in W$ and $x\in X$ such that $w\cdot x=1$.
We have group homomorphisms from~$\R$ to the Heisenberg group:
$s\to(sw,0)$, $t\to(tx,0)$, and $u\to(0,u)$.
Composing with our representation and applying Stone's theorem we see that
$F(sw,0)=\exp(sP)$, $F(tx,0)=\exp(tQ)$, and $F(0,u)=\exp(hu)I$ for some
skew-adjoint operators $P$~and~$Q$.
The images of these three homomorphisms generate the Heisenberg group,
therefore $F$ is determined uniquely by $P$~and~$Q$.
Suppose we are given two arbitrary skew-adjoint operators $P$~and~$Q$.
When do they correspond to some representation of the Heisenberg group?
All relations between the elements of the Heisenberg group are generated
by the three commutator relations between three one-parameter subgroups.
The identity operator commutes with everything, hence two of these relations are always satisfied.
The only nontrivial one is $(sw,0)(tx,0)(sw,0)^{-1}(tx,0)^{-1}=(0,st)$.
Therefore the pair $(P,Q)$ corresponds to a representation of the Heisenberg group
if and only if $\exp(sP)\exp(tQ)\exp(sP)^{-1}\exp(tQ)^{-1}=\exp(hst)I$.
The corresponding example for Heisenberg algebras is almost the same.
Let us point the differences. We have $F(sw,0)=sP$, $F(tx,0)=tQ$, and $F(0,u)=huI$
for some skew-symmetric operators $P$~and~$Q$.
The only nontrivial relation is $[(sw,0),(tx,0)]=(0,st)$.
The pair $(P,Q)$ corresponds to a representation of
the Heisenberg algebra if and only if $[sP,tQ]=hstI$ or, equivalently, $[P,Q]=hI$.
We already know that representations of Heisenberg groups automatically produce
representations of Heisenberg algebras:
If $P$~and~$Q$ are skew-adjoint operators and
$\exp(sP)\exp(tQ)\exp(sP)^{-1}\exp(tQ)^{-1}=\exp(hst)I$, then
$[\hat P,\hat Q]=hI$, where $\hat P$~and~$\hat Q$ are $P$~and~$Q$ restricted
to the dense subspace~$D$ defined earlier.
The converse of this statement is false.
Here is an example of two operators $P$~and~$Q$, which are not only skew-symmetric
but also essentially skew-adjoint (hence they have unique skew-adjoint
extensions $\bar P$~and~$\bar Q$, which are their closures) such that the equality
$\exp(s\bar P)\exp(t\bar Q)\exp(s\bar P)^{-1}\exp(t\bar Q)^{-1}=\exp(hst)I$
does not hold in general.
Suppose $M$ is the Riemann surface of the square root without the origin,
$H=L^2(M,\C)$ and $D$ is a dense subspace of~$H$ consisting of all compactly supported
smooth functions.
Define two essentially skew-adjoint endomorphisms of~$D$ as follows:
$P=\partial_x$ and $Q=hx+\partial_y$.
Here $\partial$ means partial derivative and $hx$ means multiplication by a function.
Obviously $[P,Q]=hI$.
However, the corresponding relation for the exponents does not hold.
This is due to the fact that $\exp(t\partial_x)$ moves a function in the horizontal
direction by $t$ units and $\exp(t\partial_y)$ does the same for the vertical direction.
Hence the commutator $\exp(sP)\exp(tQ)\exp(sP)^{-1}\exp(tQ)^{-1}$ moves the function
to another sheet of the Riemann surface and multiplies it by some other function.
However, the other side $\exp(hst)I$ is just multiplication by constant, it cannot
move the function to another sheet. Hence the equality does not hold.
The idea of this examples is due to Nelson. Reed and Simon give the details in \S VIII.5.
Fuglede in his paper gives a similar example of this kind with weaker conditions on $P$~and~$Q$.
\section Correction to Stone's hypothesis
\proclaim Theorem. (Rellich, 1946; Dixmier, 1958; Kilpi, 1962.)
Every irreducible representation~$F$ of a Heisenberg algebra with a nonzero parameter~$h$
such that for every $w\in W$ and for every $x\in X$ with $w\cdot x=1$
the operator $F(w,0)^2+F(x,0)^2$, which is a symmetric endomorphism of~$D$,
is essentially self-adjoint,
is unitarily equivalent to the Schr\"odinger representation with the parameter~$h$.
\section References
In this survey we touched a large area of mathematics.
We had to omit many interesting results.
For example, Schm\"udgen in his two papers discusses a large class of counterexamples
to Stone's hypothesis and modifications that make it correct.
Putnam's book is a good survey of the area.
Two surveys by Summers and Rosenberg are also useful.
The book by J\o rgensen and Moore is on the same topic but with an emphasis
on Banach spaces and smooth functions instead of Hilbert spaces and analytic functions.
Chapter~11 of the book by Barut and Raczka contains an exposition of G\aa rding's general
representation theory of Lie algebras by unbounded operators.
\medskip
\newcount\icount
\everypar{\advance\icount1\hang\llap{[\the\icount]\enspace}\ignorespaces}\obeylines%
M.~H.~Stone, Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory, Proceedings of the National Academy of Sciences of the United States of America 16 (1930), 172--175.
J.~von Neumann, Die Eindeutigkeit der Schr\"odingerschen Operatoren, Mathematische Annalen, 104 (1931), 570--578.
F.~Rellich, Die Eindeutigkeitssatz f\"ur die L\"osungen der quantenmechanischen Vertauschungsrelationen, Nachrichten der Akademie der Wissenschaften in Gottingen, Mathematisch-Physikalische Klasse (1946), 107--115.
J.~Dixmier, Sur la relation $i(PQ-QP)=1$, Compositio Mathematica 13 (1958), 263--269.
Y.~Kilpi, Zur Theorie der quantenmechanischen Vertauschungsrelationen, Annales Academi\ae{} Scientiarum Fennic\ae{}. Series A I 315 (1962),
E.~Nelson, Analytic vectors, Annals of Mathematics 70 (1959), 572--615.
B.~Fuglede, On the relation $PQ-QP=-iI$, Mathematica Scandinavica 20 (1967), 79--88.
K.~Schm\"udgen, On the Heisenberg Commutation Relation I, Journal of Functional Analysis 50 (1983), 8--49.
K.~Schm\"udgen, On the Heisenberg Commutation Relation II, Publications of the Research Institute for Mathematical Sciences, Kyoto University 19 (1983) 601--671.
S.~J.~Summers, On the Stone-von Neumann uniqueness theorem and its ramifications, John von Neumann and the foundations of quantum physics (Budapest, 1999), 135--152, Vienna Circle Institute Yearbook 8, Kluwer Academic Publishers, Dordrecht, 2001.
J.~Rosenberg, A selective history of the Stone-von Neumann theorem, Operator algebras, quantization, and noncommutative geometry, 331--353, Contemporary Mathematics 365, American Mathematical Society, Providence, RI, 2004.
C.~R.~Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete 36, Springer-Verlag, New York, 1967, $\rm xi+167$~pages.
Palle~E.~T.~J\o rgensen, Robert~T.~Moore, Operator commutation relations. Commutation relations for operators, semigroups, and resolvents with applications to mathematical physics and representations of Lie groups, Mathematics and its Applications, D.~Reidel Publishing Company, Dordrecht, 1984, $\rm xviii+493$~pages.
A.~O.~Barut, R.~Raczka, Theory of group representations and applications, PWN---Polish Scientific Publishers, Warsaw, 1977. $\rm xix+717$~pages.
M.~Reed, B.~Simon, Methods of modern mathematical physics, Volume I, Academic Press, 1980.
\bye
|
https://git.bettercrypto.org/ach-master.git/blobdiff_plain/4b1a97324f070b264ea652fc66c087a183393f4e..a642edd6d667e5fac7420a3a255a9d8aeba019ac:/src/practical_settings.tex | bettercrypto.org | CC-MAIN-2020-34 | text/plain | application/x-tex | crawl-data/CC-MAIN-2020-34/segments/1596439738653.47/warc/CC-MAIN-20200810072511-20200810102511-00049.warc.gz | 326,822,691 | 1,283 | X-Git-Url: https://git.bettercrypto.org/ach-master.git/blobdiff_plain/4b1a97324f070b264ea652fc66c087a183393f4e..a642edd6d667e5fac7420a3a255a9d8aeba019ac:/src/practical_settings.tex
diff --git a/src/practical_settings.tex b/src/practical_settings.tex
index 19f57c9..5cd2485 100644
--- a/src/practical_settings.tex
+++ b/src/practical_settings.tex
@@ -1,57 +1,57 @@
%\newpage
-\section{Recommendations on practical settings}
-\label{section:PracticalSettings}
+%\section{Recommendations on practical settings}
+\label{section:PracticalSettings}
-\subsection{Webservers}
-\input{"./practical_settings/webserver.tex"}
+\section{Webservers}
+\input{"./practical_settings/webserver_generated.tex"}
%\newpage
-\subsection{SSH}
+\section{SSH}
\input{"./practical_settings/ssh.tex"}
%\newpage
-\subsection{Mail Servers}
-\input{"./practical_settings/mailserver.tex"}
+\section{Mail Servers}
+\input{"./practical_settings/mailserver_generated.tex"}
%\newpage
-\subsection{VPNs}
+\section{VPNs}
\input{"./practical_settings/vpn.tex"}
%\newpage
-\subsection{PGP/GPG - Pretty Good Privacy}
+\section{PGP/GPG - Pretty Good Privacy}
\input{"./practical_settings/GPG.tex"}
%\newpage
-\subsection{seclayer-tcp}
-\input{"./practical_settings/seclayer_tcp.tex"}
+%\section{seclayer-tcp}
+%\input{"./practical_settings/seclayer_tcp.tex"}
%\newpage
-\subsection{IPMI, ILO and other lights out management solutions}
+\section{IPMI, ILO and other lights out management solutions}
\input{"./practical_settings/ipmi.tex"}
-%%\subsection{SIP}
+%%\section{SIP}
%%\todo{AK: ask Klaus. Write this section, Klaus??? }
%\newpage
-\subsection{Instant Messaging Systems}
+\section{Instant Messaging Systems}
\input{"./practical_settings/im.tex"}
%\newpage
-\subsection{Database Systems}
+\section{Database Systems}
\input{"./practical_settings/DBs.tex"}
%\newpage
-\subsection{Intercepting proxy solutions and reverse proxies}
-\input{"./practical_settings/proxy_solutions.tex"}
+\section{Intercepting proxy solutions and reverse proxies}
+\input{"./practical_settings/proxy_solutions_generated.tex"}
|
https://svn.osgeo.org/qgis/docs/tags/1.0.0_user_guide/forward.tex | osgeo.org | CC-MAIN-2021-31 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2021-31/segments/1627046154805.72/warc/CC-MAIN-20210804111738-20210804141738-00547.warc.gz | 529,762,483 | 3,365 | % vim: set textwidth=78 autoindent:
\section{Forward}\label{label_forward}
\pagenumbering{arabic}
\setcounter{page}{1}
% when the revision of a section has been finalized,
% comment out the following line:
% \updatedisclaimer
Welcome to the wonderful world of Geographical Information Systems (GIS)!
Quantum GIS (QGIS) is an Open Source Geographic Information System. The project
was born in May of 2002 and was established as a project on SourceForge in June
of the same year. We've worked hard to make GIS software (which is traditionally
expensive proprietary software) a viable prospect for anyone with basic access
to a Personal Computer. QGIS currently runs on most Unix platforms, Windows, and
OS X. QGIS is developed using the Qt toolkit (\url{http://www.trolltech.com})
and C++. This means that QGIS feels snappy to use and has a pleasing, easy-to-
use graphical user interface (GUI).
QGIS aims to be an easy-to-use GIS, providing common functions and features.
The initial goal was to provide a GIS data viewer. QGIS has reached the point
in its evolution where it is being used by many for their daily GIS data viewing
needs. QGIS supports a number of raster and vector data formats, with new
format support easily added using the plugin architecture (see Appendix
\ref{appdx_data_formats} for a full list of currently supported data formats).
QGIS is released under the GNU General Public License (GPL). Developing QGIS
under this license means that you can inspect and modify the source code,
and guarantees that you, our happy user, will always have access to a GIS
program that is free of cost and can be freely modified. You should have
received a full copy of the license with your copy of QGIS, and you also can
find it in Appendix \ref{gpl_appendix}.
\begin{Tip}\caption{\textsc{Up-to-date Documentation}}\index{documentation}
\qgistip{The latest version of this document can always be found at
\url{http://download.osgeo.org/qgis/doc/manual/}, or in the documentation
area of the QGIS website at \url{http://qgis.osgeo.org/documentation/}
}
\end{Tip}
\subsection{Features}\label{label_majfeat}
QGIS offers many common GIS functionalities provided by core features and
plugins. As a short summary they are presented in six categories to gain a
first insight.
\minisec{View data}
You can view and overlay vector and raster data in different formats and
projections without conversion to an internal or common format. Supported
formats include:
\begin{itemize}
\item spatially-enabled PostgreSQL tables using PostGIS, vector formats
\footnote{OGR-supported database formats such as Oracle or mySQL are not yet
supported in QGIS.} supported by the installed OGR library, including ESRI
shapefiles, MapInfo, SDTS and GML.
\item Raster and imagery formats supported by the installed GDAL (Geospatial
Data Abstraction Library) library, such
as GeoTiff, Erdas Img., ArcInfo Ascii Grid, JPEG, PNG,
\item GRASS raster and vector data from GRASS databases (location/mapset),
\item Online spatial data served as OGC-compliant Web Map Service (WMS) or
Web Feature Service (WFS).
\end{itemize}
\minisec{Explore data and compose maps}
You can compose maps and interactively explore spatial data with a friendly
GUI. The many helpful tools available in the GUI include:
\begin{itemize}
\item on the fly projection
\item map composer
\item overview panel
\item spatial bookmarks
\item identify/select features
\item edit/view/search attributes
\item feature labeling
\item change vector and raster symbology
\item add a graticule layer
\item decorate your map with a north arrow scale bar and copyright label
\item save and restore projects
\end{itemize}
\minisec{Create, edit, manage and export data}
You can create, edit, manage and export vector maps in several formats. Raster data
have to be imported into GRASS to be able to edit and export them into other
formats. QGIS offers the following:
\begin{itemize}
\item digitizing tools for OGR supported formats and GRASS vector layer
\item create and edit shapefiles and GRASS vector layer
\item geocode images with the georeferencer plugin
\item GPS tools to import and export GPX format, and convert other GPS
formats to GPX or down/upload directly to a GPS unit
\item create PostGIS layers from shapefiles with the SPIT plugin
\item manage vector attribute tables with the table manager plugin
\end{itemize}
\minisec{Analyse data}
You can perform spatial data analysis on PostgreSQL/PostGIS and other OGR
supported formats using the ftools python plugin. QGIS currently offers
vector analysis, sampling, geoprocessing, geometry and database management
tools. You can also use the integrated GRASS tools, which
include the complete GRASS functionality of more than 300 modules (See
Section \ref{sec:grass}).
\minisec{Publish maps on the internet}
QGIS can be used to export data to a mapfile and to publish them on the
internet using a webserver with UMN MapServer installed. QGIS can also
be used as a WMS or WFS client, and as WMS server.
\minisec{Extend QGIS functionality through plugins}
QGIS can be adapted to your special needs with the extensible
plugin architecture. QGIS provides libraries that can be used to create
plugins. You can even create new applications with C++ or Python!
\begin{itemize}
\item \textbf{Core Plugins}
\\ \\ Add WFS Layer
\\ Add Delimited Text Layer
\\ Coordinate Capture
\\ Decorations (Copyright Label, North Arrow and Scale bar)
\\ Georeferencer
\\ Dxf2Shp Converter
\\ GPS Tools
\\ GRASS integration
\\ Graticules Creator
\\ Interpolation Plugin
\\ OGR Layer Converter
\\ Quick Print
\\ SPIT Shapefile to PostgreSQL/PostGIS Import Tool
\\ Mapserver Export
\\ Python Console
\\ Python Plugin Installer
\\ \item \textbf{Python Plugins}
\\ \\ QGIS offers a growing number of external python plugins that are
provided by the
community. These plugins reside in the the official
PyQGIS repository, and can be easily installed using the python plugin
installer (See Section \ref{sec:plugins}).
\end{itemize}
|
http://dlmf.nist.gov/10.41.E4.tex | nist.gov | CC-MAIN-2014-10 | application/x-tex | null | crawl-data/CC-MAIN-2014-10/segments/1393999654293/warc/CC-MAIN-20140305060734-00085-ip-10-183-142-35.ec2.internal.warc.gz | 54,995,000 | 664 | \[\mathop{K_{{\nu}}\/}\nolimits\!\left(\nu z\right)\sim\left(\frac{\pi}{2\nu}%
\right)^{{\frac{1}{2}}}\frac{e^{{-\nu\eta}}}{(1+z^{2})^{{\frac{1}{4}}}}\sum_{{%
k=0}}^{\infty}(-1)^{k}\frac{U_{k}(p)}{\nu^{k}},\] |
https://www.coaps.fsu.edu/bibliography/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%20346%20ORDER%20BY%20first_author%2C%20author_count%2C%20author%2C%20year%2C%20title&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=10&rowOffset=&wrapResults=1&citeOrder=&citeStyle=APA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg= | fsu.edu | CC-MAIN-2022-33 | application/x-latex | application/x-latex | crawl-data/CC-MAIN-2022-33/segments/1659882570767.11/warc/CC-MAIN-20220808061828-20220808091828-00123.warc.gz | 616,841,964 | 1,426 | %&LaTeX
\documentclass{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{Gouillon_etal2010} Gouillon, F., Morey, S. L., Dukhovskoy, D. S., \& O{\textquoteright}Brien, J. J. (2010). Forced tidal response in the Gulf of Mexico. \textit{J. Geophys. Res.}, \textit{115}(C10).
\end{thebibliography}
\end{document}
|
http://www.robesafe.com/personal/pablo.alcantarilla/bibtex/Alcantarilla10icra1.tex | robesafe.com | CC-MAIN-2017-39 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2017-39/segments/1505818691977.66/warc/CC-MAIN-20170925145232-20170925165232-00220.warc.gz | 547,181,212 | 768 |
@inproceedings{Alcantarilla10icra1,
author = {P.F. Alcantarilla and L.M. Bergasa and F. Dellaert},
title = {Visual Odometry priors for robust {EKF-SLAM}},
booktitle = {IEEE Intl. Conf. on Robotics and Automation (ICRA)},
location = {Anchorage, Alaska, US},
year = {2010}
} |
https://space.mit.edu/CXC/docs/memo_check_vf_pha_1.2.tex | mit.edu | CC-MAIN-2019-04 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2019-04/segments/1547583763149.45/warc/CC-MAIN-20190121050026-20190121072026-00121.warc.gz | 663,592,274 | 2,515 | %+
% Name:
% memo_check_vf_pha_1.2.tex
%
% History:
% 29 Apr 06, created (v1.0), Glenn E. Allen
% 29 Apr 06, unset the bits on input or when check_vf_pha=no (v1.1), GEA
%-
\documentclass{article}
\usepackage{changebar}
\usepackage{cxo-memo-logo}
\usepackage{epsfig}
\usepackage{gea}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 1. Header
\memobasic{
Jonathan McDowell, SDS Group Leader }{
Glenn E.\ Allen, SDS }{
Identifying Potential Cosmic-Ray Background Events for VFAINT Mode
Observations }{
1.2 }{
http://space.mit.edu/CXC/docs/docs.html\#check\_vf }{
/nfs/cxc/h2/gea/sds/docs/memos/memo\_check\_vf\_pha\_1.2.tex }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 2. Introduction
\vspace*{0.3in}
%
In ACIS TIMED VFAINT mode observations the pulse-height information in the
outer sixteen pixels of a 5~pixel $\times$ 5~pixel event island is not used
to compute the summed pulse height.
\begin{changebar}
However, the pulse-height information in these pixels can be helpful in
trying to identify events associated with cosmic rays.
\end{changebar}
This specification describes an algorithm that can be used to search for
such background events. Since the algorithm has not been optimized to
prevent real source events from being flagged as potentially bad, some care
is required when using the algorithm.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3. Changes to acis_process_events
\section{Changes to acis\_process\_events}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Additional Parameters}
\begin{enumerate}
\item
check\_vf\_pha,b,h,``no'',,,``Search VFAINT mode data for potential
background events?''
\item
trail,r,h,0.027,0,1,``Fraction of charge trailed''
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Additional Input}
If the parameter badpixfile is used to specify an input bad-pixel file, then
known bad pixels are excluded from the search.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Additional Output}
If an event is identified as a potential cosmic-ray event, then STATUS bit
23 (of 0--31) is set to one in the output event-data file.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Processing}
If the parameter ${\rm check\_vf\_pha}=``{\rm yes}$,'' then the algorithm
described hereafter is used to identify potential background events. If the
parameter ${\rm check\_vf\_pha}=``{\rm no}$,'' then set the value of STATUS
bit 23 to zero for all events in the output event-data file and do not
perform the following search.
\begin{figure}
\hfil
\hbox{\psfig{file=memo_check_vf_pha_fig2_1.0.eps}}
\hfil
%
\begin{changebar}
\caption{
%
The relative CHIPX and CHIPY coordinates of the twenty five elements
PHAS[$i,j$]$^{1}$ of a 5~pixel $\times$ 5~pixel event island with ${\rm
(CHIPX,CHIPY)} = (x,y)$.
%
\label{fig01}}
\end{changebar}
\end{figure}
\begin{enumerate}
\item
Exit with an error message if
\begin{itemize}
\item
The output file exists and ${\rm clobber}={\rm ``no}$.''
\item
The input file does not exist.
% \item
% The value of the parameter ${\rm trail} < 0$ or $> 1$.
\end{itemize}
If the input data file does not have the keyword ${\rm DATAMODE} = {\rm
`VFAINT'}$, then change check\_vf\_pha to ``no'' and produce a warning
message.
\item
Set the value of STATUS bit 23 to zero for all of the events in the input
event-data file.
\item
For each event in the input event-data file, examine the pulse heights
PHAS[$i,j$]\footnote{If the parameter ${\rm apply\_cti} = {\rm ``yes}$,''
then use PHAS\_ADJ instead of PHAS in the following.} of the outer sixteen
pixels of the 5~pixel $\times$ 5~pixel event island. If
\begin{itemize}
\item
\begin{changebar}
the coordinates of the pixel associated with PHAS[$i,j$]$^{1}$ (see
Fig.~1) are not the coordinates of a pixel listed as bad\footnote{For
the purposes of the VFAINT background algorithm, a pixel is considered
bad if one or more of the STATUS bits 0--4, 11--14 or 16 is set to one
in the input bad-pixel file.} in the input bad-pixel file (if any),
\end{changebar}
\item
the coordinates of the pixel associated with PHAS[$i,j$]$^{1}$ are
greater than or equal to 1 and less than or equal to 1024 and
\item
the pulse height PHAS[$i,j$]$^{1}$ satisfies the condition shown in
Figure~2,
\end{itemize}
then set STATUS bit 23 (of 0--31) to one in the output event-data file for
the event.
\item
Write the output to the output event-data file.
\end{enumerate}
\begin{figure}
\hfil
\hbox{\psfig{file=memo_check_vf_pha_fig1_2.0.eps,width=6.50in}}
\hfil
%
\begin{changebar}
\caption{
%
The pulse-height conditions for each one of the outer sixteen pixels of a
5~pixel $\times$ 5~pixel event island. If one or more of the pulse heights
PHAS[$i,j$]$^{1}$ satisfies these conditions, then the event may have been
caused by a cosmic ray instead of an X ray. Here, $s$ is the split
threshold and $t$ is the value specified by the parameter trail.
%
\label{fig02}}
\end{changebar}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 4. Finish
\end{document}
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\begin{document}
\title{The link between HIV and reduced fertility among women in Sub-Saharan
Africa: Implications for treatment and contraceptive choice.~~ ~}
\author[1]{Virginia Reid}%
\affil[1]{Undergraduate Student at Duke University, Departments of Biology and Global Health}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
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\endgroup
\selectlanguage{english}
\begin{abstract}
The effect of viral infections on female fertility is an essential
relationship\textbf{} to research in order to properly conduct health
interventions and predict population dynamics.~ Particularly in
Sub-Saharan Africa, a relationship between HIV positive status and
fertility will have ramifications for how HIV patients are treated and
what contraceptive methods they should choose.~ This paper investigates
the hypothesis that viral infections reduce fertility in women due to
life history trade-offs and increased susceptibility for co-infections.~
To investigate this hypothesis, I made the prediction that there is a
significant negative correlation between HIV seropositive status and
fertility rate among women of reproductive age. In addition, due to key
recent studies on the association of injectable contraceptive methods
with increased HIV susceptibility, I investigated if injectable
contraceptive use modifies this relationship between HIV positive status
and fertility.~ Using linear regression methods, a significant negative
correlation between HIV positive status and fertility was found.~
~Additionally, when the modifier of injectable contraceptive use was
included in the multiple linear regression analysis, the relationship
between HIV positive status and fertility was no longer found to be
significant.~ These results support the hypothesis that viral infections
may decrease fertility; however, the inclusion of the modifier of use of
injectable contraceptives signals a confounding relationship between
injectable contraceptives and HIV risk.~ The results of this study show
that additional research on the effect of HIV and women's fertility is
needed, and that contraceptive choice cannot be ignored when examining
this relationship.%
\end{abstract}%
\sloppy
~
\subsection*{Introduction ~}
{\label{966618}}
Human Immunodeficiency Virus (HIV) most commonly affects women in their
reproductive years, defined generally as the ages between 15 and 49
(Leeuwen 2006).~ As a result, many HIV-seropositive women consider
having offspring, as do women of reproductive age infected with other
viral infections (Leeuwen 2006). Examining the effect of viral
infections on fertility is essential in order to consider a woman's
fertility needs as a key treatment goal in interventions to remedy viral
disease. More specifically, the link between HIV and fertility is
important to examine in Sub-Saharan Africa because the HIV/AIDS epidemic
and the ``fertility transition'', a shift from high to low birth rates,
co-exist in most regions of that area (Terceria 2003).~ The HIV/AIDS
epidemic in Sub-Saharan Africa has contributed to cyclical poverty,
increased mortality of women and children, high numbers of orphaned
children, deterioration of the health and economic sectors, and reduced
agricultural output (Terceria 2003). Individually, HIV and fertility
have substantial effects on socio-economic development and health
(Shapiro 2015). By uncovering if HIV has a negative effect on fertility
for women in Sub-Saharan Africa, researchers will be able to best craft
interventions and predict mortality and population dynamics .~
HIV is thought to have a negative influence on fertility for several
interacting reasons.~ First, viral infections in general have been shown
to have biological effects which may contribute to declining fertility
rates, likely due to life history trade-offs, meaning that the body must
put significant energy towards fighting off a viral infection, leaving
less energy resources available for successful reproduction.~
Additionally, HIV increases a woman's risk for amenorrhea, miscarriage,
and other co-infections which can make birth more risky or impossible
(Terceria 2003).~ To a lesser extent, it is predicted that knowledge of
one's HIV positive status may contribute to a decreased desire for
pregnancy (Young 2007). ~
Previous studies have shown negative correlations between HIV and
fecundity both at the individual and population level, but given the
wide range of confounding social and environmental factors, causation is
difficult to prove (Population Bulletin of the United Nations 2002).
Most studies which have shown a significant decrease in fertility rates
among women who are infected with HIV focus on community-level analyses,
as confounding variables abound when examining population level and
country-level data, including regional differences in fertility (Fortson
2009). In a population-level study of the effect of HIV on fertility,
Juhn et al. found that HIV contributed to a 20-25\% decrease in
fertility rate among women.~ This decrease was attributed to direct
physiological effects, changes in sexual behavior, and unwillingness to
engage in risky sex (Juhn 2013).~
As shown by this background research, multiple intersecting factors may
contribute to the relationship between HIV and fertility.~ One possible
confounding factor of particular interest to population health
researchers is that of injectable contraceptive use. A recent
meta-analysis conducted in collaboration with the Population Research
Institute reviewed a total of 24 studies published in peer-reviewed
journals and found a significant increased risk of acquiring HIV when
using Depo-Provera and other injectables (Population Reearch Institute
2018). Injectable contraceptives are made to mimic naturally occurring
progesterone, and work by lessening the inflammatory response of the
immune system in order for the body to more easily accept the embryo
during pregnancy (Population Reearch Institute 2018).~ This reduction in
inflammatory response could contribute to the increased susceptibility
for HIV virus to cause infection.~ The possible effect of injectable
contraception methods on HIV risk would have major implications for
international health interventions, as injectable contraceptives are
currently the most-used form of long-acting reversible contraception
used in Sub-Saharan African countries with high rates of HIV/AIDS.~ If
injectable contraceptives do pose an increased risk of HIV, the ethics
of promoting such contraceptives as a solution to unwanted pregnancies
must be immediately considered. Injectable contraceptive use, based on
this information, is an essential modifier to consider when examining
the link between HIV infection and fertility, due to its possible
relationship with both variables.
Based on this prior data, I hypothesize that viral infections reduce
female fertility due to life history trade-offs and increased
susceptibility for co-infections.~ To investigate this hypothesis, I
will make the prediction that there is a there is a significant negative
correlation between HIV seropositive status and fertility rate among
women of reproductive age in Sub-Saharan Africa. In addition, I will
investigate if injectable contraceptive use modifies this relationship
between HIV and fertility.
\par\null
\subsection*{Materials and Methods ~}
{\label{602320}}
All data was collected from the Demographic and Health Surveys (DHS)
public website.~ DHS Surveys are conducted in conjunction with local
governments to collect and share key population health information. Data
from 73 separate DHS Surveys, conducted over the time period of
1986-2016, was initially used, and was eventually parsed down based on
my specifications. Each DHS survey was conducted in a separate year and
country.~ DHS surveys from Sub-Saharan Africa were chosen for this
analysis for multiple reasons; they provided the most complete data on
both fertility and HIV status, and Sub-Saharan Africa is the
geographical area in which most previous studies on the effect of HIV
seropositive status on fertility have been conducted (Population
Bulletin of the United Nations 2002).
The two indicators used to compare the effect of HIV status on fertility
were:
~~~~1. ``HIV Status Among Women Age 15-49'', which the DHS defined as
the percentage of women aged 15-49 in the survey who tested positive to
HIV.
~~~~2. ``Fertility Rate'', which the DHS defined as the total fertility
rate for the three years preceding the survey for age group 15-49
expressed per woman.
Additionally, the third variable, ``Ever Use of Injections'', which was
included as a possible confounding variable in the multiple linear
regression analysis, was defined by the DHS as the percentage of women
surveyed who reported ever having used injectable contraceptives
(Demographic and Health Surveys 2018).
Restrictions were placed on the data set in order to select for relevant
data points.~ Only surveys which contained values in both the ``HIV
Status Among Women Age 15-49'' and ``Fertility Rate'' were selected for
the linear regression, which resulted in a data set containing 55
surveys among 29 countries in Sub-Saharan Africa. A linear regression
analysis was then conducted using R-studio. All statistical tests were
conducted with a significance level of \selectlanguage{greek}α\selectlanguage{english}=0.05.~
Studies of the effect of HIV on fertility are difficult to quantify due
to HIV seropositive status likely having multiple interacting effects on
fertility, including possible biological factors which may lead to
decreased fertility, confounding effects of contraceptive choice, and
behavioral factors which may cause a women to choose not to seek
pregnancy.~ Based on past data on the association of injectable
contraceptives both with fertility and HIV risk, I chose an additional
confounding variable of ``Ever Use of Injections'' for use in a multiple
linear-regression. For the multiple linear regression of HIV and
Fertility Rate + Injectable Contraceptive Use, the data were again
restricted to include only surveys which had values for all three
indicators. This resulted in a data set of 23 DHS Surveys containing all
3 variables. A multiple linear regression was conducted in R-Studio to
find the association of injectable contraceptive use on the linear
relationship between HIV and fertility.
\subsection*{~Results}
{\label{936564}}
\subsection*{}
{\label{936564}}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Screen-Shot-2018-04-24-at-5-44-30-PM/Screen-Shot-2018-04-24-at-5-44-30-PM}
\caption{{Linear regression model of HIV Prevalence Among Women 15-49 and
Fertility Rate. Generated using Demographic and Health Surveys data.
{\label{208822}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Screen-Shot-2018-04-24-at-5-47-25-PM/Screen-Shot-2018-04-24-at-5-47-25-PM}
\caption{{Linear regression model of Ever Use of Injections and HIV Prevalence
Among Women 15-49.~ Generated using Demographic and Health Surveys
Data.~
{\label{775482}}%
}}
\end{center}
\end{figure}
\emph{Linear Correlation between Fertility and HIV in Sub-Saharan
Africa}
Graphical results of the linear regression between Fertility and HIV
Prevalence among women 15-49 are shown in Figure 1.~ The linear
regression resulted in a negative correlation between the variables
Fertility and HIV Prevalence (p-value:4.957e-06, Adjusted R-squared:~
0.315).~
\emph{Multiple Linear Regression Analysis with Modifier ``Ever Use of
Injections''}
The results of the multiple linear regression analysis of the
relationship between HIV and Fertility including the modifier ``Ever Use
of Injections'' resulted in non-significant p-values (p-value for
Fertility vs. HIV Prevalence: 0.128, p-value for Fertility vs. Ever Use
of Injections: 0.371, Adjusted R-squared: 0.2762).
In order to further assess the relationship between these variables, a
simple linear regression of HIV Prevalence vs. Ever Use of Injections
was conducted and resulted in a significant positive correlation
(p-value: 0.0002358, Adjusted R-squared:~ 0.4577). ~This regression is
shown in Figure 2.
\par\null
\subsection*{Discussion}
{\label{544589}}
The link between HIV and fertility represents key missing knowledge in
the global health and development sector (Gregson 1993).~ My research
found a significant negative correlation between HIV-seropositive status
and fertility in women of reproductive age in Sub-Saharan Africa, as
shown in Figure 1.~ This negative correlation supports the hypothesis
that viral infections such as HIV have a negative impact on a woman's
fertility.~ The reasons for this negative correlation occurring in the
Demographic and Health Surveys data may be widely varied and likely
include physiological and behavioral reasons.
One weakness of this analysis is the fact that many confounding
variables likely contribute to the negative correlation between HIV and
fertility, most of which are impossible to control for within the bounds
of available data.~ Thus, it is difficult to pinpoint exactly why this
significant negative correlation between HIV positive status and
fertility was found, but it is unlikely that it was due to a single
variable, and likely involved both biological predictors, such as
life-history trade offs and co-infections, as well as behavioral changes
a HIV-infected individual may make, including making the decision not to
attempt pregnancy.~ For this reason, more research must be done on the
true relationship between viral infections as a physiological predictor
of fertility. ~
\par\null
Additionally, my research found that when the modifier of injectable
contraceptive use was included in my analysis of the relationship
between HIV positive status and fertility, the negative relationship
between these two variables was rendered insignificant.~ This effect
signals that any correlation between fertility and HIV prevalence cannot
be disentangled from both variables' association with injectable
contraceptive use. These results suggest first that there is a negative
association between fertility and injectable contraceptive use, which we
should expect, since injectable contraceptives are a highly effective
form of pregnancy prevention.~ Second, they suggest that there is also
an association between HIV positive status and the modifier of
injectable contraceptive use, supporting past data that injectable
contraceptives may increase susceptibility to HIV through a
physiological response.
This research supports previous findings that viral infections,
particularly HIV, likely lead to reduced fertility among women. However,
the inclusion of the modifying variable of injectable contraceptive use
which led to a non-significant correlation between HIV and fertility
demands that more research on the link between contraceptive choice,
HIV, and fertility be conducted.~ The Population Institute and similar
organizations are conducting meaningful research on the link between
injectable contraceptives and HIV risk, but this research should be put
into the context of HIV's negative correlation with fertility.~ In order
to best treat HIV-seropositive women, we must know if their ability to
reproduce is affected by their viral status.~ With this knowledge,
fertility assistance can be integrated into HIV treatment, and
population health researchers will be able to best predict the effects
of the ``fertility transition'' in Sub-Saharan Africa.~ In a broader
sense, the effect of any viral infection on a woman's fertility will
have significant implications for treatment and intervention. Though
straightforward relationships between variables have not been found in
this study, the results provide an important jumping-off point for
conversations regarding fertility and viral infection.
\par\null
\(\)
\subsection*{Acknowledgements}
{\label{857357}}
I thank Charles Nunn for his guidance on this project, Jordan Anderson
for statistical advice, and my classmates for their feedback as I
constructed my research goals.~ This research was supported by Duke
University's Global Health and Evolutionary Anthropology Departments and
by the course Biology 385- Primate Disease Ecology and Global Health.
\par\null
\subsection*{References}
{\label{743264}}
Demographic and Health Surveys on Fertility, Use of Injections, and HIV
status. (2018).
Depo-Provera and HIV - PRI. (2018).~PRI\emph{.} Retrieved 24 April 2018,
from \url{https://www.pop.org/depo-provera-and-hiv/}
Fortson, J. G. (2009). HIV/AIDS and fertility.\emph{~American Economic
Journal.Applied Economics,~1}(3), 170-194.
~~~~~~~~doi:\url{http://dx.doi.org.proxy.lib.duke.edu/10.1257/app.1.3.170}
Juhn, C., Kalemli-ozcan, S., \& Turan, B. (2013). HIV and fertility in
africa: First evidence from population-based surveys. Journal of
~~~~~~~~Population Economics, 26(3), 835-853. doi:http://
\href{http://dx.doi.org.proxy.lib.duke.edu/10.1007/s00148-012-0456-2}{dx.doi.org.proxy.lib.duke.edu/10.1007/s00148-012-0456-2}
Leeuwen, J.M. Prins, S. Jurriaans, K. Boer, P. Reiss, S. Repping, F. van
der Veen; Reproduction and fertility in human immunodeficiency
~~~~~~~~virus type-1 infection,~\emph{Human Reproduction Update}, Volume
13, Issue 2, 1 March 2007, Pages
197--~~~~~~~~206,~\url{https://doi.org/10.1093/humupd/dml052}
Population Bulletin of the United Nations. (2002). \emph{COMPLETING THE
FERTILITY TRANSITION}. New York: United Nations.
Shapiro, D. (2015). Accelerating Fertility Decline in Sub-Saharan
Africa. \emph{Population Horizons}, \emph{12}(1).
~~~~~~~~\href{http://dx.doi.org/10.1515/pophzn-2015-0002}{http://dx.doi.org/10.1515/pophzn-;2015-0002}
Simon Gregson (2007) Will HIV become a major determinant of fertility in
Sub-Saharan Africa?, The Journal of Development Studies, ~~~~~~~~30:3,
650-679, DOI:
\href{https://doi.org/10.1080/00220389408422331}{10.1080/00220389408422331}
Terceira, N., Gregson, S., Zaba, B., \& Mason, P. (2003). The
contribution of HIV to fertility decline in rural Zimbabwe, 1985-2000.
~~~~~~~~\emph{Population Studies}, \emph{57}(2), 149-164. http://
\href{http://dx.doi.org/10.1080/0032472032000097074}{dx.doi.org/10.1080/0032472032000097074}
Young, A. (2007). In sorrow to bring forth children: fertility amidst
the plague of HIV.~\emph{Journal Of Economic Growth},~\emph{12}(4),
283-327. ~~~~~~~~\url{http://dx.doi.org/10.1007/s10887-007-9021-3}
\par\null
\selectlanguage{english}
\FloatBarrier
\end{document}
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\title{Decembar je rezultat dugogodišnjih društvenih i političkih procesa}
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Pre svega želim da kažem da nisam istoričar. Ja sam aktivista, borac na frontovima anarhističke borbe od kasnih 70-ih. Ne znam koliko je moje poznavanje anarhističke istorije precizno jer je ono proizvod mojih sjećanja i onoga što sam čuo i naučio od drugih drugova tokom svih ovih godina mog učešća u toj borbi.
Koliko mi je poznato prvi anarhisti su se u posleratnom periodu pojavili u ranim 70-im i poslednjim godinama diktature, kao rezultat uticaja pobune iz maja ’68., koja je uglavnom uticala na Grke koji su živjeli u inostranstvu, ali i one koji su živjeli ovdje. Kada kažem uticaj maja ’68. to se takođe odnosi i na ono što mu je prethodilo, situacioniste i druge radikalne pozicije. U tom smislu se rođenje anarhije u Grčkoj, kao pokreta, ne odnosi toliko na tradicionalni anarhizam – čiji je najznačajniji trenutak bila Španska revolucija, a glavni izrazi anarhističke federacije i anarho-sindikalističke organizacije – već uglavnom na antiautoritarne, radikalne političke pokrete 60-ih.
Kao što sam ranije rekao, anarhisti su se u Grčkoj pojavili početkom 70-ih i tada su napravili svoje prve publikacije i analize grčke stvarnosti iz antiautoritarne perspektive.
Prisustvo i učešće anarhističkih drugova u događajima tokom pobune u novembru 1973. bilo je veoma značajno, ne po brojnosti već u pogledu njihovog naročitog političkog doprinosa jer se nisu ograničili na parole protiv diktature. Umjesto toga, oni su usvojili šire političke karakteristike – antikapitalističke i antidržavne. Takođe, oni su se nalazili među nekolicinom koja je započela tu pobunu zajedno sa militantima sa ekstremne ljevice. I bili su toliko vidljivi da su predstavnici institucionalne ljevice osuđivali njihovo prisustvo u događajima tvrdeći da su anarhisti bili provokatori plaćeni od strane hunte, dok su takođe osuđivali i njihove slogane, nazivajući ih stranim (\emph{engl. foreign}) i nepovezanim sa narodnim zahtjevima. U stvarnosti, zvanična ljevica je bila neprijateljski raspoložena prema samoj pobuni i podržavala je takozvanu demokratizaciju, odnosno mirnu tranziciju iz diktature u demokratiju. A pošto tu spontanu pobunu omladine i radnika ’73. nisu mogli zaustaviti, pokušavali su njome manipulisati, a zatim posle pada diktature i politički eksploatisati.
Tokom pobune (iz) ’73. postojale su dvije tendencije. Jedna koja je željela da kontroliše i manipuliše pobunom u kontekstu borbe protiv diktature, a u korist demokratije i protiv američkog uticaja. I tu je bila druga tendencija čiji su važan dio činili anarhisti, koja je pobunu vidjela na širi način, protiv vlasti i kapitalizma. Ove dvije tendencije su nastavile da se sukobe i poslije pada diktature, u periodu koji zovemo metapolitefsi (gr. \emph{μεταπολιτευση}, politička promjena), odnosno nakon što su pukovnici predali vlast političarima. To je bio sukob između onih koji su podržavali građansku demokratiju i onih koji su joj se protivili. Prva tendencija je događaje na Politehničkom fakultetu posmatrala kao pobunu za demokratiju, dok su oni koji su bili protiv režima građanske demokratije događaje na Politehničkom posmatrali kao pobunu za socijalno oslobođenje. Eho ovog sukoba, na neki način, odjekuje i danas.
Dakle, tako su se anarhisti pojavili u Grčkoj i to je bio njihov doprinos\dots{}
Nakon što su pukovnici predali vlast političarima, u Grčkoj su se stvarnosti pojavile dvije glavne snage. Sa jedne strane, radikalne političke i društvene snage koje su osporavale postojeći politički, društveni i ekonomski poredak, i to su izražavali dijelovi omladine kao i radnika. A na drugoj strani su bile političke snage dominacije, od konzervativne desnice koja se nalazila u vladi do njihovih saveznika sa institucionalne ljevice koja je posle pada diktature inkorporirana u politički sistem. Desničarska vlada je pokušavala da suzbije i teroriše, ranije pomenute, radikalne političke i društvene snage. Isto je činila i institucionalna ljevica, sopstvenim sredstvima, kada ih nije mogla kontrolisati ili njima manipulisati. Među tim radikalnim političkim i društvenim snagama nalazili su se anarhisti, koji su bili u sukobu sa tradicionalnim konceptima – čak i onim najradnikalnijim – ljevice, poput onih o centralnoj ulozi radničke klase, hijerarhijskom organizovanju u političke partije, ideji avangarde, viziji o preuzimanju vlasti, i socijalističkoj transformaciji društva “odozgo”.
Bitan momenat socijalnih borbi tokom prvih godina metapolitefsi, krajem 70ih, bila je borba na univerzitetima, izazvana naporima desničarske vlade da nametne reformu obrazovanja. Anarhisti su i u ovoj borbi imali značajno prisustvo, kao i druge grupe i pojedinci sa antiautoritarnim i slobodarskim perspektivama. U velikoj mjeri su ove borbe premašile granice univerziteta, kao i univerzitetske studente kao subjekte, zadobijajući šire radikalne karakteristike i privlačeći prisustvo i učešće još mnogo ljudi, ne samo studenata, već generalno mladih, poput srednjoškolaca, kao i radnika. To je bio važan trenutak u kome su anarhisti proširili svoj uticaj među širim društvenim segmentima koji su se borili.
Otprilike u isto vrijeme, malo poslije tih borbi protiv reforme obrazovanja, anarhisti su, skoro sami, vodili još jednu borbu – solidarnost sa borbama zatvorenika. Tu su pokazali još jednu karakteristiku svog radikalizma: nisu oklijevali da se uključe u pitanja koja su bila tabu tema za društvo – poput pitanja o zatvorima i zatvorenicima – i izražavali su svoju solidarnost sa zatvorenicima boreći se zajedno sa njima za njihove zahtjeve – za ukidanje disciplinskih kazni, denuncijaciju mučenja, i za pravo osuđenika na doživotne kazne da njihove slučajeve razmatraju apelacioni sudovi – ali uvek zadržavajući svoju viziju društva bez ikakvih zatvora.
Veoma važan događaj iz tog perioda koji pokazuje političku i društvenu dinamiku subjekata otpora i, u isto vrijeme, žestinu političke moći, događaj koji je zapravo definisao politička dešavanja tog vremena, bile su demonstracije koje su se desile 17. novembra 1980., na sedmu godišnjicu pobune na Politehnici (demonstracije su se dešavale i dešavaju svake godine na godišnjicu tog događaja). Te godine je vlada zabranila demonstrantima odlazak do ambasade SAD. Omladinske organizacije, kao i studenstske organizacije koje su kontrolisale KKE i PASOK (komunistička i socijalistička partija), povinovale su se toj zabrani; međutim, političke organizacije ekstremne ljevice, koje su u tom periodu bile jake, odlučile su da pokušaju nastaviti demonstracije ispred američke ambasade prkoseći zabrani od strane vlade i policije.
Dakle, u noći 17. novembra 1980., pored zgrade parlamenta, u ulici koja je vodila do ambasade, na hiljade demonstranata se suočilo sa veoma jakim policijskim snagama. Nakon pokušaja prvih redova demonstranata, u kojima su se nalazili pripadnici ekstremne ljevice, da krenu naprijed ka američkoj ambasadi, usledio je masovni napad policijskih snaga u cilju rastjerivanja na hiljade okupljenih. Ali uprkos policijskim napadima pružen je snažan i trajan otpor (koji je trajao satima) od strane nekoliko hiljada ljudi, omladine i radnika, pripadnika ekstremne ljevice, anarhista i autonomista, koji su postavili barikade u centru Atine – barikade za čije je uklanjanje policija koristila oklopna vozila. Tokom ovih sukoba policija je ubila dvojicu demonstranta Jakovosa Kumisa i Stamatina Kanelopulua, obojica pripadnici ekstremno-ljevičarskih organizacija, a na stotine je povređeno, neki ozbiljno. Među povređenima, dvije osobe su ranjene bojevom municijom, jedna od njih u grudi koju je policija upucala ispred Politehnike.
Tokom ovih sukoba mnoge kapitalističke mete su napadnute i opljačkane, poput robnih kuća, zlatara i sličnih. Ovu vrstu napada – koji su bili jedan od prvih izraza gradskog nasilja koje nije bilo striktno ograničeno samo na policiju već su napadani izrazi i simboli bogatstva – osuđivala je čak i ekstremna ljevica čija je politička kultura kao legitimnu metu prepoznavala samo policiju. Ali, nastajao je novi fenomen gradskog nasilja, gdje su demonstranti – pored angažovanja u sukobima sa policijom – takođe uništavali i pljačkali kapitalističke mete, a to je upravo ono što je ljevica osuđivala.
Ovi događaji iz novembra 1980. bili su, kao što smo već pomenuli, izraz političke i društvene dinamike prvih godina metapoliftesi, ali i vrhunac i kraj hegemonije ekstremne ljevice nad tim dinamikama jer ljevica nije uspela da objasni, pod sopstvenim uslovima, obim i oblik tih događaja širem društvu, pa čak ni svojim sledbenicima. Međutim, ovi događaji bili su katalizator pada desničarske vlade, godinu dana kasnije.
Početkom 80ih, kao rezultat velikog napora dijela političkog sistema da kontroliše i manipuliše socijalnim, političkim i klasnim otporima i zahtjevima, došlo je do nove političke promjene, a na vlast je došla socijalistička partija PASOK (oktobar ’81.). To je bilo nešto što je u ono vrijeme izgledalo kao ogromna, istorijska promjena. To je stvorilo mnogo iluzija, neutralizovalo i inkorporiralo u institucije stare militante i obilježilo kraj tih prvih godina metapolitefsi, kraj raznih spontanih društvenih i klasnih borbi koje su se pojavile u prvim godinama nakon pada diktature. Dakle, nakon ove političke promjene, anarhisti koji su bili neprijateljski raspoloženi prema bilo kakvoj vrsti posredovanja i inkorporiranja u institucije, ostali su u izvjesnom smislu sami protiv ove nove vlasti koja je imala mnogo kontrolisanih i izmanipulisanih pristalica, mnogo sledbenika punih iluzija.
PASOK je došao na vlast sa ciljem modernizacije grčkog društva, ukinuti su zakoni koji su bili proizvod perioda građanskog rata – kada je desnica u oružanom sukobu slomila ljevicu (1946-1949) – kao i zakoni iz perioda nakon građanskog rata, a udovoljeno je i nizu zahtjeva sa ljevice. Zahtjevima koji uopšte nisu ugrožavali autoritarnu i klasnu organizaciju društva već su je, naprotiv, modernizovali i ojačali, dovodeći je bliže modelu zapadnoevropskih društava.
Ova politička promjena je značila da je veliki dio ljevice oslabljen i apsorbovan u sistem. Dakle, u stvarnosti, promjena koja se dogodila u tom trenutku takođe je značila da su na kraju anarhisti zajedno sa autonomistima i uopšte antiautoritarci, sada jedini pokušavali da društveno intervenišu, obraćajući se uglavnom mladima, a zatim izvodeći i prve okupacije zgrada (skvotiranja) u Grčkoj, pod uticajem sličnih poduhvata u Zapadnoj Evropi.
Prvi poduhvat skvotiranja koji se desio u Eksarhiji postao je na neko vrijeme epicentar anarhističkih i antiautoritarnih mobilizacija i doveo je do drugih okupacija u Atini i Solunu, ali se nakon nekog vremena našao na udaru represije i iseljen je početkom 1982. godine. Isto se desilo i sa drugim skvotovima. (U ovom trenutku je korisno pomenuti da su se od kasnih 70-ih i posebno početkom 80-ih represivne operacije države sprovodile u cilju erozije i uništavanja pokreta otpora širenjem heroina u društvenim prostorima mladih. Ova operacija je tada bila veoma nova, bez presedana u Grčkoj, a anarhisti su se licem-u-lice sukobili sa svim tim, boreći se protiv toga u društvenim prostorima, mjestima za mlade , ali takođe i unutar skvotova.)
Prve godine vladavine PASOK-a bile su pune vještački uzgajanih težnji za promjenama, promjenama koje naravno nisu bile ni suštinske niti subverzivne. To su bile godine širokog društvenog pristanka na političku vlast protiv koje su anarhisti u velikoj meri stajali sami. Međutim, vrlo brzo je ta politička vlast pokazala svoje pravo okrutno lice i njen duboki klasni karakter protiv nižih društvenih slojeva, kao i svoje represivne ambicije usmjerene protiv onih koji su pružali otpor – anarhista, ljevičara i nepokorne omladine.
Prekretnica, kraj iluzija, desila se 1985. godine, godina koju je obilježilo policijsko ubistvo petnaestogodišnjeg Mihalisa Kaltezasa kome je pucano u potiljak ispred Politehničke škole tokom nereda između anarhista i nepokorne omladine sa jedne strane i policije sa druge, nakon završetka demonstracija obeležavanja 17. novembra te godine. Ovo ubistvo je pokrenulo niz ustaničkih događaja otpora čiji su glavni momenti bili okupacija Hemijskog univerziteta i Polithenike. Štaviše, ono je izazvalo dublju pobunu svijesti i neprijateljske pozicije prema policiji i vlastima što je dovelo do (rađanja) brojnih manifestacija otpora u narednim godinama, jer to nije bilo nešto što je izraženo i iscrpljeno u jednom trenutku već je postalo nasleđe mnogih nasilnih i borbenih trenutaka otpora u godinama koje su slijedile. Ono je formiralo „tradiciju“ sličnih dešavanja; događaja koji bi buknuli ili kao reakcija na državna ubistva, ili kao izrazi solidarnosti sa borbama potlačenih ljudi, poput zatvorenika. U ovim uslovima se, takođe, pojavio i društveno ukorenio novi talas skvotiranja, uglavnom anarhista i antiautoritarnih grupa, proširujući na taj način koliko frontove toliko i uticaj borbe.
Primeri takvih događaja koje možemo pomenuti su na primer sukobi sa policijom i okupacija Politehnike 1990. koja je trajala sedamnaest dana, nakon oslobađajuće presude panduru koji je ubio Kaltezasa\dots{}
\dots{}Obimni društveni sukobi na ulicama Atine 1991. koji su trajali puna dva dana, nakon ubistva nastavnika i militantnog levičara Nikosa Temponerasa od strane paradržavnih siledžija u – od strane učenika – okupiranoj školi u gradu Patrasu.
Ustanak anarhista i omladine u novembru 1995. tokom godišnjice pobune iz 1973., tokom kog je okupirana Politehnička škola u znak solidarnosti sa pobunama zatvorenika koja je izbila tih dana. Ta se pobuna u zatvorima našla pod paljbom propagandne mašinerije države, medija, i odmah je suočena sa neposrednom pretnjom policijske invazije u zatvorske objekte. Država je u pokušaju da uguši pobunu iz ’95. na Politehnici i napadne anarhiste i omladinu – ne samo zbog učešća u otporu u tim trenucima već takođe i zbog svih događaja koje su pokretali tokom prethodnih godina, i događaja koji su pretili da se nastave – pokrenula veliki propagandni napad medijima sa ciljem dobijanja društvenog konsenzusa za policijsku intervenciju i represivne planove. Usledila je invazija policije na okupiranu Politehničku školu u jutro 17. novembra 1995. i uhapšeno je više od 500 ljudi. Ali je cela ta represivna operacija doživela neuspeh: slika koju su želeli da predstave bila je da su anarhisti malobrojni i izolovani, male bande izgrednika – stereotip predstavljen od strane države bio je „50 poznatih nepoznanica“ – ali se ispostavilo suprotno i otkrilo da anarhisti imaju veliki uticaj na mlade. Takođe, država nije uspela u pokušaju da anarhiste teroriše hapšenjima i krivičnim gonjenjima jer je većina optuženih ostala nepokorna pretvarajući suđenja koja bi usledila u još jedno mesto žestokog sukoba sa državom.
U narednim godinama ovaj fenomen odbijanja i otpora od strane anarhista, anti-autoritaraca i nepokorne omladine proširio se društvom i doveo do mnoštva političkih inicijativa, društvenih inicijativa, kontra-informacionih projekata, manifestacija otpora i stvaranja novih samoroganizovanih prostora. Nijedna strategija dominacije nije ostala bez odgovora i osporavanja, ni politika protiv imigranata, ni Olimpijske igre 2004., međunarodni politički i ekonomski samiti, učešća Grčke u vojnim planovima i operacijama Zapada protiv zemalja na Istoku.
Bazirani na političkim i istovremeno organizacionim vrednostima društvene solidarnosti, direktne akcije, ravnopravnosti, anti-hijerarhije i samoorganizovanja, anarhisti nisu oklevali i uspevali su da odgovore, makar u meri u kojoj su mogli, na sve napade države na društvo, čak i njegove najmarginalizovanije delove. Uvek su stajali rame uz rame sa ugnjetenim ljudima i sa onima koji su se borili, odbijajući dileme i prkoseći ucenama kojima se država koristila kako bi zadobila konsenzus. A to su činili jasno i bez obzira na cenu koju bi morali da plate. Dosledno su ostali izvan i protiv svih institucija, izvan i protiv političkog sistema. U vreme kada su drugi, bez obzira na to koliko radikalnim su se činili, usvajali logiku države, anarhisti su stajali usamljeni protiv takvih predloga. Rezultat je bio taj da je levica izgubila svoj uticaj među najradikalnijim delovima društva dok je za anarhiste ista ona stvar za koju se govorilo da je slabost koja će ih dovesti socijalne izolacije, bila i još uvek jeste upravo njihova snaga: činjenica da su ostali izvan političkog sistema i svih institucija. Jer, kada se ljudi pobune oni prevazilaze institucije i ograničenja koja nameću i veoma lako komuniciraju sa anarhistima.
Jedva da smo imali ikakvog novca, radili smo nesebično u malim, fluidnim grupama afiniteta, ali to je naša snaga. I kao što se pokazalo u događajima iz decembra 2008., oni koji su izgubili kontakt sa najradikalnijim i najmilitantnijim izrazima društva nisu bili anarhisti, već, naprotiv, oni koji su flertovali sa idejama i strukturama vlasti, tvrdeći za sebe ulogu predstavnika društvenih subjekata i posrednika društvenih protivrečnosti.
Kroz dugotrajni proces borbe, koji sam ukratko opisao ranije, anarhisti i antiautoritarci uopšte su stekli jako uporište u svesti ljudi, nešto što nije bilo svima očigledno pre Decembra. Jer, iza ideje da je država izgubila dosta društvenog uporišta tokom decembarskih dana, mnogo dublja istina jeste da je ona već izgubila dosta tog uporišta pre decembarskih dešavanja, tokom dužeg vremenskog perioda. A to je nešto što je bilo izraženo na veoma oktrivajući način od prvih trenutaka eksplozije pobune, uz učešće mnogih ljudi u akcijama koje su se do tog trenutka smatrale isključivo akcijama malih grupa anarhista.
U stvarnosti, decembar 2008. ima duboku istorijsku, političku i socijalnu pozadinu koja je povezana sa celokupnom istorijom borbi u poslednjih 30 godina, kao i prisustvom i učešćem anarhista unutar tih borbi. Učešće koje se odlikuje praksom socijalne pobune bez posrednika i bez iluzija o mogućnosti promene unutar postojećeg sistema, predlažući samoorganizovanje protiv svih oblika hijerarhijske organizacije, predlažući društveno kontra-nasilje protiv državnog nasilja, i solidarnost nasuprot individualizaciji i veštačkim podelama stvorenim od strane vlasti.
Ovde možemo govoriti o dinamičnim borbenim praksama, poput sukoba sa policijom, koje je tokom Decembra usvojilo mnogo ljudi, kao i okupacije zgrada (univerziteta, škola, gradskih skupština i mnogih drugih). Ili možemo govoriti o samoorganizovanju kroz otvorene anti-hijerarhijske skupštine koje su stvarane tokom i nakon decembarskih dana. Levica je te prakse izbegavala i potcenjivala, i rezultat je da su ih događaji prevazišli.
Međutim, iako je Decembar rezultat društvenih i političkih procesa koji sežu godinama unazad, i iako ima sličnosti i analogije sa pređašnjim događajima, on ih istovremeno i prevazilazi i odražava nove situacije, potrebe i želje, stvarajući nove perspektive. Posmatrajući razlike aktuelnih dešavanja u odnosu na ranije događaje, vidimo da događaji ovog puta nisu bili ograničeni ili usmereni na određeno mesto, vreme i način. Proširili su se na brojne gradove širom zemlje i uzeli mnogo različitih oblika, manje ili više nasilnih ali uvek antagonističkih prema državi, svaki put zasnovani na inspiraciji i mašti i dovitljivosti ljudi koji su učestvovali.
Osim toga, to je proces koji, zbog svog difuznog i raznovrsnog karaktera, ne izgleda da ima krajnju tačku. Pre izgleda da se obnavlja i nastavlja uzimajući nove oblike noseći obećanje o novim erupcijama socijalnih eksplozija uprkos trenutnom padu nasilnih događaja. Ranije su, takođe, događaji bili fokusirani uglavnom na grčku omladinu ali ono što se tokom Decembra širilo celom zemljom uključivalo je ljude mnogih drugih nacionalnosti, uključujući imigrante i izbeglice.
Dinamične metode borbe i procese samoorganizovanja usvojili su mnogi ljudi, bez predstavnika i bez postavljanja ikakvih zahteva. Decembar ne samo da nastavlja kulturu političkog nasilja, već takođe i polaže novu tradiciju samoorganizovanja – organizovanje ljudi “odozdo” – zasnovanog na direktnim i opipljivim socijalnim potrebama. Sada ovi procesi samoorganizovanja koji predstavljaju oblik nastavka pobune nemaju kao svoj jedini cilj odgovor na ubilačko policijsko nasilje, već odgovor na sve aspekte vlasti, od načina na koji živimo, načina na koji radimo, proizvodimo, trošimo, do pitanja zdravlja, životne sredine, svega. Svaki aspekt vlasti predstavlja borbeni front za ljude koji se samoorganizuju i bore „odozdo“, ne uvek nasilno ali gotovo uvek antagonistički prema državi.
Druga tačka vredna pomena jeste da je pobuna potvrdila neke ideje u okviru antiautoritarnog pokreta, a opovrgla druge. Na primer, mišljenja koja su tvrdila da se sve nalazi pod kontrolom, da su manipulacija i kontrola nad ljudima danas toliko snažni da pobune nisu moguće, ili da je društvo mrtvo, da ne može proizvesti ništa zdravo i da smo mi anarhisti sami protiv države. Decembar je pokazao da je pobuna moguća i, mnogo više, da je moguća socijalna pobuna.
Još jedan aspekt ima veze sa subjektom pobune. Bilo je dosta priče o tome ko su bili oni koji su se pobunili. Uložen je veliki napor od strane medija i predstavnika političkog sistema da odrede subjekte pobune kako bi napisali svoju istoriju; da kontrolišu, čak i posle toga, sve što mogu. Oni tvrde da je to bila pobuna mladih, konkretnije grčke omladine, a posebno srednjoškolaca, na osnovu činjenice da su deo pobune zaista činile mobilizacije srednjoškolaca, koji su, u mnogim slučajevima, išli toliko daleko da su demonstrirali ispred policijskih stanica i napadali ih. Ali to je veoma ograničena i iskrivljena slika pobune. Politički sistem i mediji žele da sakriju širi socijalni, multietnički i klasni karakter pobune. Nisu samo učenici bili na ulicama! A, u svakom slučaju, većina mladih koji su bili na ulicama nisu izašli kao učenici, već kao pobunjenici protiv sveta dominacije, državnog nasilja, vlasti i eksploatacije. Dakle, oni žele da sakriju ono što je očigledno svima koji su bili na ulicama: da su na tim ulicama bili siromašni, plaćeni radnici, nezaposleni, oni koje zovemo isključenima. I veliki broj njih bili su imigranti, oni koji su najjeftinija radna snaga i glavne žrtve ne samo radne eksploatacije već takođe i policijskog nasilja i državne represije.
Shodno tome, subjekt koga je svaki od analitičara predstavljao kao da je imao centralnu ulogu u pobuni, u suštini pokazuje njegov ili njen politički cilj i odražava njihovu subjektivnu percepciju pobune, i njihove buduće ciljeve. Na primer, kada govore o grčkoj omladini i naročito o srednjoškolcima, to je u cilju da ih odvoje kao “dobre” pobunjenike – smatrajući da je njima lakše manipulisati – od “loših”, nekontrolisanih pobunjenika. Međutim, većina ljudi koji su bili na ulicama, u osnovi su pripadali drugoj kategoriji, bili su nekontrolisani, potlačeni ljudi.
Danas se suočavamo sa dve stvari. Prva su represivni potezi države kroz pravosudni sistem i policiju, poput hapšenja, zatvaranja, ljudi koji se drže kao taoci kroz krivična gonjenja, odluke o postavljanju kamera za nadzor posvuda, kažnjavanja zbog nošenja maski i verbalnih vređanja policije, ciljanje na skvotove, samoupravne prostore i generalno samoorganizovane strukture pokreta. Sa druge strane imamo ideološku ofanzivu države sa ciljem da se decembarski pobunjenici podele na “dobre” učenike, sa ciljem da ih inkorporiraju u sistem, i “loše” koji ne mogu ili ne žele biti inkorporirani u sistem i stoga moraju biti izolovani, napadnuti i uništeni represijom.
Ovde treba reći da ako su represiju u osnovi izražavali direktno državni mehanizmi, sa druge strane ideološki rat ne vode samo oni već i druga pomagala poput stranaka institucionalne levice. Dok su sudska i policijska represija odmah vidljive i shvaćene kao nešto što dolazi “spolja”, ideološki rat je podmukliji i generisan je unutar samog pokreta pošto ga ne izražavaju samo oni koji su neprijateljski nastrojeni prema pokretu, već takođe i drugi koji se prikazuju kao prijatelji pokreta i koji selektivno ističu one karakteristike pobune koje im se sviđaju, što znači one karakteristike za koje misle da ih mogu apsorbovati i iskoristiti. A u isto vreme oni klevetaju one karakteristike i subjekte pobune koje ne smatraju prijatnim, nazivajući ih “nepolitičkim”, antisocijalnim, ili čak kriminalnim.
Ovaj ideološki rat ima za cilj da inkorporira, da zastraši one koji nisu inkorporirani i izoluje one koji nastavljaju da se kreću perspektivama pobune.
Ali kriza sistema, koja je u svojoj osnovi kriza njegovog društvenog legitimiteta, značajno ograničava mogućnosti inkorporacije velikog dela ljudi koji reaguju i pružaju otpor. Jednostavnije rečeno, to znači da sve više i više ljudi gubi poverenje u institucije ili zagovornike sistema. Dakle, čak i ako uspeju da inkorporiraju neke, to je razlog zašto ne mogu stvarno ograničiti i zaustaviti uticaj radikalnih ideja.
Oni prema kojima bismo trebali biti sumnjičavi, zbog njihovog korozivnog i podrivačkog stava, upravo su oni koji jednom nogom stoje u starom svetu a drugom sa nama, pričajući o novom svetu. Ovi dvolični neprijatelji pobune su gori čak i od policije i sudija.
Ovde bi trebalo da razjasnimo o kome konkretno govorimo – to se odnosi na one koji igraju određenu ulogu, čak i ne toliko važnu, unutar institucija, a ne generalno na ljude – radnike, komšije, omladinu – sa kojima se susrećemo. Što se tiče ovih drugih, ljudi koje sistem otuđuje i trenira da veruju u institucije, bilo je mnogo lakše komunicirati sa njima naročito u prvim danima pobune zbog toga što su materijalni uslovi i intenzitet ovih događaja bili takvi da se svako kretao od svojih starih pozicija ka novim.
Danas se, kako vreme prolazi, testiraju naše političke i lične sposobnosti da očuvamo te kontakte. Kao i naše strpljenje kada delujemo zajedno sa ljudima različitim od nas, prepoznavajući da imamo mnogo toga da naučimo o tome kako ostati u kontaktu sa svima onima koje smo sreli na ulici u Decembru. A najvažniji način da se danas sretnemo licem u lice, odnosno izvan kontrainformacija, propagandnog materijala, tekstova i flajera, jesu samoorganizovane skupštine. Sa naše strane, podstičemo stvaranje takvih skupština, učestvujemo i intervenišemo u njima. A i tamo smo, takođe, suočeni sa ideološkim ratom koji sam ranije pomenuo. Ali osim toga, postoje predrasude; i predrasude drugih ljudi u vezi nas, i naše predrasude prema ljudima koji nisu u potpunosti odbacili postojeći sistem, iz naivnosti, straha ili samo zato što su navikli na njega.
Ali na pravom smo putu. Odnosi koji su razvijeni između anarhista, antiautoritaraca i drugih delova društva predstavljaju vrtlog, a ishod je nepredvidljiv. Sigurno je to dobra stvar jer ne dopuštamo ponovno uspostavljanje normalnosti i otuđenja. Jer za razliku od vrtloga pobune u kojoj je sve moguće, i nadamo se najboljem, normalnost je stanje u kojem je gotovo sve predvidljivo, a rezultat uglavnom negativan.
Stvari su nepredvidljive, ne samo u pogledu odnosa između anarhista i antiautoritaraca sa drugim ljudima, već i u okviru pokreta. A, uglavnom su nepredvidive i stvari po pitanju odnosa između anarhista, društva i države. Anarhistički\Slash{}antiautoritarni društveni pokret stvara mnogo inicijativa i dela otpora protiv države, neke više dinamične neke manje, neke više socijalne neke manje. Odnosno, ne postoji nikakav centralni organ ili jedno jezgro, već niz većih ili manjih borbenih inicijativa odozdo od kojih se neke međusobno koordinišu, a druge ne. U svakom slučaju, ono što bi po mom mišljenju trebalo izbegavati jeste biti društveno izolovan, biti izolovan među nama, u pokretu, i ostati sam u sukobu sa državom.
Jasno nam je da ako bi se neke stvari koje se rade ovde, radile u SAD ili u Italiji na primer, da bi neki od nas bili mrtvi, a još mnogo više njih bi bilo u zatvorima na mnogo godina. Ova politička ravnoteža moći koja danas postoji – činjenica da postoje i da možemo nastaviti sa ovakvim aktivnostima i da o njima možemo razgovarati – stvarala se tokom 30 godina. Ali naši životi i naša sloboda su uvek ugroženi i nalaze se na meti represivnih mehanizama države. Nakon Decembra, država želi da promeni taj odnos moći, i u nekom trenutku bi ga mogla preokrenuti. Baš kao što je u jednom trenutku, kada je ubijen Aleksis Grigoropulos, u društvu oslobođeno pobunjeničko raspoloženje, u nekom drugom trenutku, na osnovu drugačijeg događaja, može doći do eksplozije državne represije; a anarhisti, kao i ostali borci, mogu biti izloženi ogromnim rizicima.
Istorija pokreta u SAD, u Evropi, i u svetu uči nas i šta možemo uraditi, ali i sa čim se sve možemo suočiti. Imajući dublje razumevanje o tome ko smo i šta želimo, ali takođe i o tome šta je država i šta ona želi da uradi sa nama – da nas zbriše – ono što treba obezbediti jeste da ne budemo izolovani od društva, ali takođe da ne budemo podeljeni unutar pokreta, tako da kao celina ne ostanemo sami protiv države, niti da ijedan pojedinačni drug bude ostavljen sam protiv države. Ali takođe je važno da ne obuzdavamo naš impuls ili kompromitujemo naše unutrašnje želje.
Važno je da delujemo i da se stvari dešavaju, da koristimo našu hrabrost, pa čak i našu ludost\dots{}
Ali nismo do sada ništa rekli o ulozi spontanosti u decembarskim dešavanjima. Spontanost je uvek igrala posebnu ulogu u anarhističkim inicijativama, a tako je ponovo bilo i u Decembru. Bilo je, međutim, i spontanosti društvenih grupa koje su učestvovale u pobuni, spontanosti masa. Prema Kastorijadisu, spontanost je višak “rezultata” u odnosu na “uzroke”. U Decembru su bile izražene spontane snage, snage koje su bile skrivene u masama ljudi i koje ranije nisu bile predvidljive. A te snage su i dalje prisutne u društvu, mnogo više u društvu koje je na svojim kolenima, mnogo više u društvu podeljenom na klase, gušenom sistemskim nasiljem, siromaštvom, očajem, strahom. Ljudima koji žive u takvom društvu preostaju dve mogućnosti: ili pasivno prihvatanje postojeće stvarnosti, koju država želi da predstavi kao jedinu opciju; ili ustanak, koji čak i kada nije vidljiv kao mogućnost ili opcija ne znači da ne postoji ili da neće izbiti.
Postoji još jedna stvar: u današnjim uslovima dominacije države i kapitala na Zapadu, eksplozije pobuna nisu toliko retke, uključujući i gradske nerede, uglavnom omladinskih grupa i obično izazvane slučajevima policijskog nasilja, poput događaja u francuskim predgrađima, ili pobune crnaca u Los Anđelesu 1992. godine. A kao poseban slučaj možemo takođe pomenuti pobunu u Albaniji 1997. godine, iako je ona imala veoma specifične karakteristike. Ali ono što se ovde desilo u Decembru, u odnosu na druge velike ustaničke događaje, jeste da su se politički i društveni subjekti sreli i uzajamno delovali. Anarhisti su se sastali sa društvenim subjektima spremnim na pobunu.
U tom kontekstu, pobuna postaje mnogo opasnija za vlast; odnosno, onda kada nije samo izliv socijalnog besa neke posebne potlačene društvene grupe, već plodno sastajalište dinamike različitih društvenih grupa koje zajedno usmeravaju svoje nasilje protiv izvora svake eksploatacije i ugnjetavanja.
Pobune izbijaju i ne mogu se izbeći. Vlast to zna pa zato više voli da suzbija svaku društvenu grupu pojedinačno i ne dopušta pobunama da uzmu jasne političke karakteristike, ne dopušta im da imaju sveobuhvatnu kritiku postojećeg poretka. Prisustvo i učešće anarhista u Decembru dalo je takve šire političke karakteristike; i u velikoj meri razvijena je subverzivna kritika sistema kao celine.
I to je bilo dobro, i to je dobro za svakog druga ili grupu drugova, gde god se u svetu oni nalazili, da pokušaju da se sastanu sa društvenim grupama koje pate pod tiranijom države i kapitalizma i koje su spremne i imaju želju da uzvrate udarac, tako da se neizbežne pobune šire, a ne ograničavaju.
Ako bismo samo zamislili šta bi se moglo desiti susretanjem političkih subjekata koji svesno žele da svrgnu postojeći poredak sa svim onim društvenim subjektima koje država i kapitalizam guše i koji imaju razloge za pobunu. Samo zamišljanje je dovoljno da se shvati. A to je ono što se u velikoj meri desilo u Grčkoj u decembru 2008. godine.
Alkis,
april 2009.
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Anarhistička biblioteka
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Anti-Copyright
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Alkis
Decembar je rezultat dugogodišnjih društvenih i političkih procesa
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\href{http://www.squathost.com/anar\_gr/gr/s\_alkis.htm\#english}{www.squathost.com}, 2011.
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\textbf{anarhisticka-biblioteka.net}
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\title{Retained faecalith following laparoscopic appendectomy}
\author[1]{Yegi Kim}%
\author[2]{Joseph Kong}%
\author[2]{Evan Williams}%
\author[2]{Satish Warrier}%
\affil[1]{Peninsula Health}%
\affil[2]{Alfred Health}%
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\begin{abstract}
There have been a few case studies showing intra-abdominal abscesses due
to retained faecalith post laparoscopic appendectomy. A 29 years-old
woman presented with right lateral abdominal wall and pelvic collection
due to retained faecalith post interval laparoscopic appendectomy for
perforated appendicitis. She underwent exploration for faecalith
retrieval successful outcome.%
\end{abstract}%
\sloppy
\textbf{Introduction}
Laparoscopic appendectomy is currently the first choice of surgical
approach for an acute appendicitis due to lower rate of surgical site
infections, shorter hospital stay and better cosmetic outcome (1, 2, 3).
Furthermore, an interval appendicectomy is favoured by many surgeons for
treatment of perforated appendicitis. However, one of the complications
from laparoscopic appendectomy includes intra-abdominal abscess
formation secondary to retained faecalith (1,4). Although rare, a
dropped faecalith can occur during appendectomy or due to expulsion from
perforated appendix. We present a case of recurrent intra-abdominal
abscesses from a retained faecalith in the intra-muscular layers of the
iliacus in a patient whose initial diagnosis of an acute appendicitis
was delayed.
\textbf{Case Report}
A 29 years-old woman initially presented with a 2 week-history of
non-migratory right iliac fossa pain with subjective fever. The
patient's history of complaint was not typical of acute appendicitis.
After unremarkable inflammatory markers and pelvic ultrasound that did
not visualise the appendix, she was discharged home with a provisional
diagnosis of pelvic inflammatory disease and was given oral antibiotics
for seven days.
However, she represented to our emergency department (ED) with worsening
abdominal pain, ongoing fever and mild nausea. A computed tomography
(CT) was organised and it showed a perforated appendicitis with
retrocaecal abscess and a calcified appendicolith. She then had an
ultrasound-guided drainage of the abscess and was discharged home with a
view of performing an interval laparoscopic appendectomy. After
discharge, the patient returned to ED for persistent recurrence of a
right pelvic collection involving the right iliacus and lateral
abdominal wall muscle. This was again radiologically drained. A
colonoscopy was performed during this presentation to exclude any
primary tumours before surgery. The entire colon was examined and showed
normal mucosa and appendiceal orifice.
The patient then underwent a semi-emergent laparoscopic appendectomy.
During the procedure, thorough abdominal wash was performed and a drain
tube placed in the right iliac fossa. The patient had an uneventful
post-operative recovery, with normalisation of inflammatory markers and
the drain tube removed. She was then discharged home with oral
antibiotics. During her follow-up in the general surgery clinic, the
histopathology reported chronic appendicitis with granulomas but the
faecalith was not within the specimen. On review, it was noted that she
had intermittent abdominal discomfort and a repeat of C-reactive protein
(CRP) showed that it has elevated to 41. A repeat CT scan was performed
which showed re-accumulation of the right pelvic and lateral abdominal
collection with the retained faecalith (figures 1-2).
Although the abscess was intramuscular, the faecalith and the pelvic
abscess were going through the abdominal wall into the subcutaneous fat
(figure 3). There was a brief discussion with an interventional
radiologist for a hook-wire insertion preoperatively for direct
localisation of the faecalith but given the location of the abscesses,
intraoperative ultrasound was used instead. The patient underwent
exploration and surgical removal of retained faecalith through a right
lateral hip approach (figure 4). A drain was inserted into the cavity
and the wounds were closed with interrupted sutures.
\textbf{Outcome/follow up}
The wound was complicated by formation of seroma, which was managed
conservatively. The patient was reviewed again six weeks post-operation
and remained well with no further complications of her wound or pelvic
abscess drainage.
\textbf{Discussion}
An interval appendectomy for perforated appendicitis with established
abscesses is widely practiced by many surgeons (5). It has been
suggested that an interval appendectomy has many benefits including
reducing the overall rate of adverse events (6). The concept of delaying
surgery would allow the intra-peritoneal contamination and inflammation
to settle down with antibiotics, allowing surgeons to mitigate the risk
of an unfriendly plane due to ongoing sepsis (6). However, the strategy
of an interval appendectomy was not successful in our case due to an
infective nidus, a retained faecalith. This particular complication,
although not common, is a well-known complication after a laparoscopic
appendectomy. The risk of retained faecalith is even higher in a case of
perforated appendicitis (7). The faecalith may drop from the base of the
appendix when it is being resected or when the appendix is being
extracted through the port (7, 8). A study suggests different strategies
to prevent faecalith spillage including gentle manipulation of an
appendix and use of an endoscopic bag to retrieve the appendix (9). We
suspect that in our case the retained faecalith will likely have
migrated from the perforated appendix into the intramuscular layers of
iliacus post percutaneous drainage of the abscess.
There have been different recommendations for the management of a
retained faecalith (1, 10). Black et al. presented a case of an abscess
with faecalith adequately managed with intravenous antibiotics only (4).
Some studies suggest percutaneous drainage and extraction of faecalith
(4, 10). Despite these, surgical removal of faecalith is recommended by
many surgeons (1). We had two rationales for a right lateral hip
approach instead of intra-abdominal approach. Firstly the location of
the abscess was continuous from subcutaneous plane to intraabdominal
plane. Secondly the planes of dissection and retrieving the faecalith
would be difficult if through intra-abdominal. Hence, taking a new
approach through virgin territory has aid with the recovery of the
faecalith. Although she developed seroma post operatively, she recovered
relatively quickly after the operation.
This case highlights the importance of identifying and extracting a
retained faecalith to decrease the morbidity associated with a
persistent infective nidus. Given that the patient had a laparoscopic
procedure with a failed attempt, we have decided on a different surgical
access, right lateral hip approach, with the intention of a midline
laparotomy if not successful. This has led to a successful resolution of
her right pelvic abscess formation with no added surgical morbidity.
\textbf{Acknowledgements}
Nil financial support
\textbf{Ethics}
Patient's informed and signed consent obtained
\textbf{References}
\begin{enumerate}
\tightlist
\item
Gamble L, Saze A. 2016. Chronically retained fecalith following
laparoscopic appendectomy. Surgical infections case reports 1(1):
35-37.
\item
Hori T, Machimoto T, Kadokawa Y, Hata T, Ito T, Kato S, Yasukawa D,
Aisu Y, Kimura Y, Sasaki M, Takamatsu, Kitano T, Hisamori S, Yoshimura
T. 2017. Laparoscopic appendectomy for acute appendicitis: How to
discourage surgeons using inadequate therapy. World Journal of
Gastroenterology 23(32): 5849-5859.
\item
Kim N, Reed W, Abbas M, Katz D. 2004. CT Identification of Abscesses
After Dropped Appendicoliths During Laparoscopic Appendectomy.
American Journal of Roentgenology 182:1203-1205.
\item
Black M, Ha BY, Kang YS, Garland A. 2013. Perihepatic abscess caused
by dropped appendicoliths following laparoscopic appendectomy:
Sonographic finding. Journal of Clincal Ultrasound 1:366--369.
\item
Weiner D, Katz A, Hirschl R, Drongowski R, Coran A. 1995. Interval
appendectomy in perforated appendicitis. Paediatric Surgery
International 10:82-85.
\item
Kim IY. 2016. Minimally Invasive Interval Appendectomy for Perforated
Appendicitis With a Periappendiceal Abscess. Annals of Coloproctology
32(3): 88--89.
\item
Katagiri H, Ishitani M, Sakamoto T, Yoshinaga Y, Kubota T, Miyabe A.
2013. Retained fecaliths after laparoscopic appendectomy disappearing
spontaneous with non-operative management. Edorium Journals
4(11):650-653.
\item
Rasuli P, Friendlich MS, Mahoney JE. 2007. Percutaneous Retrieval of a
Retained Appendicolith. Cardiovascular Interventional Radiology
30(2):342--4.
\item
Strathern DW, Jones BT. 1999. Retained fecalith after laparoscopic
appendectomy. Surg Endosc 13(3):287--9.
\item
Singh AK, Hahn PF, Gervais D, Vijayraghavan G, Mueller PR. 2008.
Dropped appendicolith: CT findings and implications for management.
American Journal of Roentgenology 190(3):707--11.
\end{enumerate}
\textbf{Figures}
Figure 1.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}
\end{center}
\end{figure}
CT abdomen showing the retained faecalith.
Figure2.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
\end{center}
\end{figure}
CT abdomen showing right lateral abdominal wall collection.
Figure 3.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\end{center}
\end{figure}
CT abdomen showing horseshoe shaped collection in right lateral
abdominal wall into the subcutaneous fat.
Figure 4.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}
\end{center}
\end{figure}
Intra-operative view of exploration.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Image-1/Image-1}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Image-2/Image-2}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Image-3/Image-3}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Image-4/Image-4}
\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
\end{document}
|
https://doc.libelektra.org/api/current/latex/doc_decisions_array_md.tex | libelektra.org | CC-MAIN-2020-10 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-10/segments/1581875144708.87/warc/CC-MAIN-20200220070221-20200220100221-00287.warc.gz | 356,878,590 | 1,338 | Currently it is inefficient to detect the length of an array and it is impossible to know if an key (without subkeys) should be an array or not.
For the latter problem different workarounds exist, such as {\ttfamily \#\#\#empty\+\_\+array} in {\ttfamily yajl}.
\begin{DoxyItemize}
\item None
\end{DoxyItemize}
\begin{DoxyItemize}
\item None
\end{DoxyItemize}
\begin{DoxyItemize}
\item {\ttfamily \#\#\#empty\+\_\+array} as in {\ttfamily yajl}, problem\+: does not allow efficient access of first element
\item store length (and not last element), problem\+: needs prepending of {\ttfamily \#\+\_\+...}
\item store element after last element (C++-\/\+Style), problem\+: very unusual style
\item use value and not metadata array, problem\+: is ambiguous
\end{DoxyItemize}
Store length in metadata {\ttfamily array} of key, or keep metadata {\ttfamily array} empty if empty array. For example ({\ttfamily ni syntax}, sections used for metadata)\+:
\begin{DoxyCode}{0}
\DoxyCodeLine{myarray/\#0 = value0}
\DoxyCodeLine{myarray/\#1 = value1}
\DoxyCodeLine{myarray/\#2 = value2}
\DoxyCodeLine{myarray/\#3 = value3}
\DoxyCodeLine{myarray/\#4 = value4}
\DoxyCodeLine{myarray/\#5 = value5}
\DoxyCodeLine{[myarray]}
\DoxyCodeLine{ array = \#5}
\end{DoxyCode}
\begin{DoxyItemize}
\item Is very similar to {\ttfamily binary} metadata.
\item The key alone suffices to know if its an array
\item one can distinguish an array with keys that are called by chance \#0
\end{DoxyItemize}
\begin{DoxyItemize}
\item yajl needs to be fixed
\item metadata library needs to be adapted
\item spec plugin needs to be fixed
\end{DoxyItemize}
\begin{DoxyItemize}
\item \mbox{\hyperlink{doc_decisions_global_validation_md}{Global Validation}}
\end{DoxyItemize}
\href{https://github.com/ElektraInitiative/libelektra/issues/182}{\texttt{ https\+://github.\+com/\+Elektra\+Initiative/libelektra/issues/182}} |
https://authorea.com/users/295218/articles/423870/download_latex | authorea.com | CC-MAIN-2021-17 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-17/segments/1618038887646.69/warc/CC-MAIN-20210419142428-20210419172428-00372.warc.gz | 224,782,921 | 2,529 | \documentclass[10pt]{article}
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\begin{document}
\title{Systematic Mapping Study: Augmenting Personal Software Process Analysis
For Extreme Programming Teams}
\author[1]{Abdul Razzaq}%
\author[2]{Shahbaz Ahmad}%
\author[2]{Asim Khalil}%
\author[3]{Soban Ahmad}%
\affil[1]{Zhejiang University}%
\affil[2]{International Islamic University}%
\affil[3]{BUITEMS}%
\vspace{-1em}
\date{\today}
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\let\center\flushleft
\let\endcenter\endflushleft
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\selectlanguage{english}
\begin{abstract}
The Personal Software Process offers individuals with a self-controlled
structure for doing a job. To improve individual and team ability is a
crucial source of productivity and quality. Measuring an individual's
performance is a challenging task in an agile environment as individuals
work on several projects at the same time. No specific criteria exist,
which gives personal growth in agile XP. This research study is based on
an idea to align the personal software process with agile extreme
programming and propose a new model for an individual's professional
growth measurement. An evidence-based case study is conducted to
accumulate knowledge about the measurement of an individual's
performance in the agile extreme programming team. In this study,
systematic mapping is used to collect issues in existing literature. The
reason for systematic mapping is needed to recap the enhancement and
need to classify the holes also requirements for upcoming studies
related to agile with process improvement. This study supports to
realize the variance between SPS and XP. This scientist mapping makes
mindfulness for the procedure improvement with a mix of SPS and XP. We
also proposed a solution model which we have implemented in our
research.%
\end{abstract}%
\sloppy
\textbf{Hosted file}
\verb`manuscript.pdf` available at \url{https://authorea.com/users/295218/articles/423870-systematic-mapping-study-augmenting-personal-software-process-analysis-for-extreme-programming-teams}
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\title{Race, Class and Sandino’s Politics}
\date{August 1995}
\author{Katherine Hoyt}
\subtitle{}
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\begin{quote}
“The simple folk with whom we talked were all agog over Sandino. He had become ubiquitous. He had been seen here; he had been seen there. At night he had gone stalking along a ridge, god of the universe. Later I found the same mythology was believed everywhere in Nicaragua. At many a low doorstop I sat and talked over a jicara of chicha corn beer, or a glass of yellowish palm wine, and there was no place Sandino had not been seen. He had fired the imagination of the humble people of Nicaragua. In every town, Sandino had his Homer. He was of the constellation of Abd-el-Krim, Robin Hood, Villa, the untamed out-laws who knew only daring and great deeds, imbued ever with the tireless persistence to overcome insurmountable odds and confront successfully overwhelming power. His epos will grow-in Nicaragua, in Latin America, the wide world over. For heroes grow ever more heroic with time.” –Carlton Beals, Banana Gold, (1932)
\end{quote}
FROM 1927 TO 1933, a small man with a rag-tag army kept the U.S. Marines at bay in the northern jungle-covered mountains of Nicaragua. Chilean poet Gabriela Mistral referred to the forces of Augusto C. Sandino (1895–1934) as the “small crazy army,” but she made the remark with respect and admiration; Sandino had become a symbol of resistance to the “colossus of the north” for an entire generation of Latin Americans.
Sandino’s primary objective was to rid his country of the U.S. occupation, which had lasted since 1912. His other aims, however, have been the subject of much controversy. Was Sandino a petit-bourgeois nationalist who merely wanted the Yankees out? Or was he a communist, a Bolshevik who would take property away from the land owners? What were his true aims?
Studies of Sandino and his writings which appeared outside Nicaragua between 1936 and 1979 show an almost total absence of political analysis. (Inside Nicaragua, in these same years under the dictatorship of the Somoza family, no materials on Sandino appeared.) In virtually every case, the authors assumed that Sandino was a nationalist fighting to rid his nation of U.S. Marine occupation and that he had no additional goals. The embarrassingly large number of ideas permeating his writings and the confused way in which they are often presented strengthened the notion that his main role was that of a national liberator.
In the 1970s and 1980s, leaders of the Sandinista National Liberation Front (FSLN), following the example of their founder Carlos Fonseca, began emphasizing one previously overlooked aspect of Sandino’s thought, above all based on class. In the view of the modern-day Sandinista, Sandino used a “class-analysis,” siding invariably with workers, peasants and Indians, in a word, the oppressed — los oprimidos. FSLN writers emphasized that Sandino had two objectives: first, to rid his nation of the Yankee invaders, and second, to make social changes for the benefit of the poor majorities.
A thorough reading of Sandino’s writings, journalist interviews of him, and other studies shows, however, that Sandino used both a race and a class analysis of Nicaraguan society. Yet even the most competent of the interpreters of Sandino give little importance to Sandino’s race analysis.
Sandino lived in a period of intellectual effervescence in Latin America, in which the idea of anti-interventionism was combined with a glorification of the Indian and of Indo-hispanic culture as well as with various forms of populism, anarchism, socialism, communism and spiritualism. Sandino believed that Indo-hispanic unity was necessary to throw off the yoke of oppression from the north but that Indo-America would merely light the fuse for a revolution of all the oppressed peoples of the world.
When this revolution triumphed, injustice would be destroyed and “love, with its favorite daughter, Divine Justice” (Sandino, 1976:214) would rule the earth.
Today the struggle to preserve ethnic identity often seems to lack that second stage — the commitment to struggle for the liberation of all who are oppressed, above and beyond but still including one’s own ethnic group. A new look at the thought of Augusto Sandino in this year of the centennial of his birth can help us understand how that further commitment might be attained.
The U.S. had invaded six Latin American countries (Cuba, Panama, Haiti, Nicaragua, Mexico and the Dominican Republic) in the first third of the 20\textsuperscript{th} century. The violent Mexican revolution was in process of consolidation. Sandino was one of the most important symbols of this period of ferment, one of the sources of that intellectual effervescence while at the same time deriving sustenance from it. As leaders of various movements vied for Sandino’s allegiance, he absorbed what he felt was useful from their ideas and adopted them to the Nicaraguan reality as he perceived it.
\section{Sandino’s Intellectual Development}
Augusto Sandino was born on May 18, 1895, in the province of Masaya. He was the illegitimate son (later recognized) of a medium landholder and a servant woman. His class consciousness emerged in the course of his early poverty-stricken life with his mother at the same time that a legitimate brother lived in comfort.
While Sandino was in Mexico the first time in the early 1920s, he became a Mason; when he went there again in 1929–30 to obtain the Mexican government’s support in his struggle against the United States Marines he rose to the rank of Master Mason.
Also in this second trip, according to historian Donald Hodges, he became a member of the Magnetic-Spiritual School of the Universal Commune headquartered in Argentina. Founded by Basque Joaquin Trincado, the School combined spiritism and Spanish anarchism. It complemented the study Sandino had made in his previous trip to Mexico of the writings of Mexican anarchist Ricardo Flores Magon. (Hodges, 1986: 13)
Living and writing at the same time as Sandino were three actors on the Latin American political scene who shared with him the ideas of what was called indigenismo, a kind of racial mystique glorifying both the Indian and the mestizo person of mixed race. Exponents of indigenismo came from varying political perspectives: from the Mexican Jose Vasconcelos, who with middle age became conservative (much like the revolution he supported); through the Peruvian populist and spiritualist founder of the APRA Party, Victor Raul Haya de la Torre; to Jose Carlos Mariategui, the Peruvian intellectual who combined Marxist social and economic analysis with the agonic Christianity of Spanish philosopher Miguel de Unamuno.
All of these supported social measures in their countries that would improve the living conditions of the poor. They, along with Sandino of Nicaragua, were beginning to combine race with class in analyzing why and how some groups of people dominated and oppressed others. Black socialist anti-colonial leaders in Africa used this analysis, as would Martin Luther King when he expanded his aims from purely civil rights for African-Americans to rights for the poor and to an end to the Vietnam war.
Class analysis was not enough in itself for Sandino or for others who sought to explain the contempt and scorn felt by the “Yanqui” for Indo-Americans. Racial prejudice was also involved. However, racial prejudice alone did not suffice to explain inequities within Nicaragua, where landowners were not always of lighter skin than the peasants whom they hired for the harvesting of their crops. An analysis also based on social class was necessary to explain these inequalities.
Many people find themselves uncomfortable with this combination of race and class. For one, Marxists have called the “race question” the “national question” and have found it difficult to resolve, given their emphasis both on internationalism and on the primary importance of economic class relationships. For another, Liberals who support equal opportunity may have no problem with supporting racial equality but object when questions of economic and social class are introduced, as many did when Martin Luther King expanded his horizons and began to do battle for the poor of all races.
To understand revolutions that have occurred in the Third World and elsewhere, one needs to grasp the ideological currents behind them. If racism is an important part of colonialism, slavery and neo-colonialism, then racial pride must inevitably become a part of the struggle for independence, freedom and national sovereignty. Vasconcelos, Haya de la Torre, Mariategui, and Sandino were determined to forge this kind of pride.
The combination of an understanding of class struggle, drawn from anarchism and Marxism, and of racial conflict between Indo-Hispanics and Anglo-Saxon Yankees was an essential part of Sandino’s ideology. This synthesis must be considered in order to make clear what this jungle general, who meditated long hours, read voraciously and corresponded with the world from his mountain hide-out, intended for Nicaragua and for Latin America.
Sandino spent several years in revolutionary Mexico in the early 1920s working in the oil fields of Tampico which, according to Neil Macaulay, supported some fifty thousand workers and several different radical social doctrines. The long occupation of his country by the U.S. Marines was painful to Sandino. He said:
\begin{quote}
“In about 1925, I began to believe that in Nicaragua everything had become ignominious and that honor had completely vanished among men in that land. At that same time\dots{}I had a circle of spiritualist friends and we daily commented on the submissiveness of our Latin American peoples in the face of the advances through hypocrisy or by force of the murderous Yankee empire. On one of those occasions, I said to my friends that if in Nicaragua there were one hundred men who loved her as I did, our nation would be able to restore her absolute sovereignty, which the Yankee empire has imperiled. My friends answered me saying that there could possibly be that number of men in Nicaragua, or even more, but the difficulty would be in finding them.
It is because of this [Yankee] intervention that the peoples of Central America and Mexico hate us Nicaraguans. I had the opportunity to confirm that in my travels through those countries.
I felt wounded in the depths of my being when they called me ‘sell-out,’ ‘shameless,’ ‘traitor.’” (Sandino, 1976: 53)
\end{quote}
\section{Sandino’s Nationalist Struggle}
Upon hearing the news of the revolt of Liberal Juan B. Sacasa against U.S.-imposed President Adolfo Diaz, Sandino returned to Nicaragua from Mexico in May of 1926. Although he joined the Liberal cause, it is quite certain that by this time he had moved beyond Liberal beliefs.
He began working at the San Albino gold mine, a company owned by Americans, and soon started talking to workers about how they were exploited by the capitalists and by the foreign companies. He told them that they had a right to unions, schools and medical care. “I also explained to them that I wasn’t a communist but rather a socialist.” (Roman, 1979: 49)
Soon Sandino left the gold mine with a small band to join General Jose Maria Moncada’s Constitutionalist forces in Prinzapolca. He asked Moncada to supply his force with arms. Said Moncada:
\begin{quote}
“I saw Sandino for the first time in Prinzapolca. He addressed me, saying that he wanted to go fight in the interior; at the same time he gave me a written statement concerning his ideas, the concluding sentence of which proclaimed that ‘PROPERTY IS THEFT.’” (Somoza, 1936: 85)
\end{quote}
Needless to say, Moncada denied his request for arms and Sandino learned to hide his political convictions. With the help of a group of prostitutes, he was able to retrieve a quantity of rifles that had been dumped into the bay at Puerto Cabezas and arm his men with them. With each victory, Sandino’s small army grew. His headquarters was in San Rafael del Norte where the Arauz family ran the telegraph office, and nineteen-year-old daughter Blanca became one of his most important collaborators.
By April of 1927 Sandino had taken Jinotega and the U.S. Marines had declared Matagalpa neutral territory. But by May, Moncada was negotiating with U.S. representative (later Secretary of State) Henry Stimson to end the fighting with the assurance that he would have the presidency in the 1928 elections.
Sandino was the only one of the chiefs of the Liberal Army to oppose the pact. He returned with his men to San Rafael del Norte and sent a cache of arms into the mountains to be retrieved later. On May 18\textsuperscript{th}, his thirty-second birthday, he married Blanca. A few days later he dared Moncada to come and disarm him: “I am at my post and waiting for you \dots{} I will not sell out nor will I surrender. You will have to defeat me.” (Sandino, 1976: 85)
He also wrote:
\begin{quote}
“We are alone. The cause of Nicaragua has been abandoned. Our enemies from this day forward will not be the forces of the tyrant Diaz, but rather the Marines of the most powerful empire in history. It is against them that we are going to fight\dots{}Those who are married or who have other family obligations should return to their homes.” (Sandino, 1976: 90)
\end{quote}
In July of 1927, Sandino led 800 men to take the city of Ocotal. A Sandinista victory would have been complete if the Marines had not sent airplanes to bomb the city, forcing the guerrillas to flee. The city was wrecked by the bombardment. This bombing from the air of towns and villages by U.S. planes was repeated all over the north. It forced Sandino to change his tactics to “a special system of war that we have taken to calling ‘little war’ (guerrilla).” (Roman, 1979: 146)
Neil Macaulay calls Sandino one of the precursors of modern revolutionary guerrilla warfare and says that 1927 marks the year that, because of the use of airplanes and machine guns, revolutionary fighters were forced to abandon the plains and deserts for the forests and mountains. (Macaulay, 1967: 9–10)
International solidarity with Sandino was enormous. He received messages of support from Nehru and from Madame Sun Yat-Sen in Asia. In Latin America, Diego Rivera, Jose Vasconcelos, Victor Haya de la Torre, Jose Carlos Mariategui and many others were among his supporters. From Europe, French Communist Henri Barbusse sent Sandino a long letter that was a source of great pride to the guerrilla leader:
\begin{quote}
“General, I send you this greeting in personal homage and in that of the proletariat and revolutionary intellectuals of France and Europe\dots{}You, Sandino, general of free men, are performing a historic indelible role.” (Quoted in Macaulay, 1967: 109)
\end{quote}
Sandino placed representatives in several countries to help in obtaining support for the anti-interventionist cause. Probably the most important of these representatives was Honduran poet Froylan Turcios, editor of the journal Ariel. Turcios was geographically close to Sandino in Tegucigalpa and could transmit Sandino’s messages to the outside world.
But when Sandino announced that he would not accept the results of the 1928 elections and would set up a rival government and continue the war rather than negotiate with the Managua government to speed Marine withdrawal, Turcios broke with him. He said Sandino had now gone beyond his original nationalist goal and was considering civil war. Turcios also objected to Sandino’s call for Central American union and Latin American solidarity to revive the dream of Simon Bolivar. (Sandino, 1976: 141–159) Turcios could not comprehend Sandino’s broader vision.
Liberal Party candidate Moncada won the U.S.-supervised elections of 1928. The U.S. Marines stayed on to continue the fight against Sandino and to train a National Guard i.e., to Nicaraguanize the war. Sandino’s struggle was now no longer fought under a Liberal banner; rather he began to emphasize broader objectives than mere Yankee withdrawal. This may have been the result of the influence of Salvadoran Communist Augustin Farabundo Marti, who joined Sandino in the Mountains. Or possibly Sandino had merely peeled off another layer of his true beliefs. Whatever the cause, it resulted in the loss of some supporters.
Communist organizations like the Anti-Imperialist League of the Americas and the Mexican-based Hands off Nicaragua Committee worked very hard to build support for Sandino, but in return expected him to adopt the program of the Communist International. Sandino was himself committed to a broad anti-imperialist front, at the very time that the Comintern was abandoning that strategy for a hard (“third period”) line against social democratic and socialist parties, calling them “social-fascist.” (Cerda, 1983: 91–103; and Baines, 1972: 135–136)
Central to Sandino’s strategy were his continued efforts to gain the support of the moderate revolutionary government of Mexico. He went to Mexico in 1929 but, although his expenses were paid by the government, he was not able to arrange a meeting with President Emilio Portes Gil until early 1930. The results were minimal. With a few guns and a little ammunition, he returned to Nicaragua in May of 1930.
Meanwhile, the Mexican government had been suppressing leftist organizations. Several communist leaders were killed. In January of 1930, the Mexican government broke diplomatic relations with the Soviet Union. The fact that Sandino and his men were receiving two thousand pesos a month from this very same government angered the Communists. (Cerda, 1983: 102)
Nevertheless, in February Sandino did enter into a series of secret talks with Communist Party representatives, apparently agreeing to denounce the Mexican government. But the guerrilla leader soon broke with the Communists — explaining, however, that the rupture was strategic, not philosophical.
The military struggle in Nicaragua continued in 1929 and 1930 with increasing ferocity. The U.S. Marines were heavily involved both in the jungle and in the air. At about this same time Sandino issued his famous Light and Truth Manifesto to his troops:
\begin{quote}
“Many times you will have heard people speak of the final judgement of the world. By final judgement you should understand the destruction of injustice upon the earth and the reign of the Spirit of Light and Truth, that is Love \dots{}. What will happen is the following: The oppressed peoples will break the chains of their humiliation, with which the imperialists of the earth have kept them imprisoned. The trumpets that will be heard will be the horns of war, playing the hymns of the freeing of the oppressed peoples from injustice of the oppressors. The only thing that will be forever destroyed is injustice; what will remain is perfection, that is, love; with its favorite daughter, Divine Justice. We in Nicaragua my brothers, have the honor of being chosen by Divine Justice to begin the judgement of injustice on the earth. Do not be afraid my dear brothers; and be sure, very sure that in a very short time we will have our definitive triumph in Nicaragua which will light the fuse of the ‘proletarian explosion’ against the imperialists of the earth.” (Sandino, 1976: 214)
\end{quote}
This was the kind of inspiration which Sandino used to keep his ragged army fighting year after year. His forces carried out attacks throughout the country in 1931 and, at the end of that year, U.S. charge d’affaires Willard Beaulac considered the situation “as grave or graver than at any time since I have been in Nicaragua.” (Quoted in Crawley, 1979: 76)
The United States, however, was determined to withdraw the Marines on schedule after a new president was elected in 1932 and inaugurated in 1933. “Politicians in Washington were already promising the American nation that the United States would never again become entangled in such a predicament.” (Crawley, 1979: 76)
In 1932, Sandino held the hope that he would be able to take power and install a revolutionary government in Nicaragua. The reality, however, was different. Liberal leader Juan B. Sacasa won the election over Conservative “Yankee puppet” Adolfo Diaz. The “traitor” Moncada finished his term and twenty-four hours after Sacasa was inaugurated [on January 1,1933], the U.S. Marines sailed out of the port of Corinto. Sandino was forced to re-evaluate the situation. He explained in a letter to Aleman Bolanos:
\begin{quote}
“When foreign intervention in Nicaragua has ceased, albeit in appearance alone, the people’s spirit has cooled down. Political and economic intervention is suffered by the people, but they cannot see it — even worse, they do not believe in its existence. This situation placed us in a very difficult position, and in the meantime the government was negotiating a multi-million dollar loan and preparing to blast us to hell and consolidate the political, economic and military intervention in our country. As this government was elected mainly by the Liberals of Leon, our strength would have dwindled in any confrontation; our financial and military resources were exhausted, and our troops could not have sought refuge in Honduras because the war in that country is raging intensely, and Nicaraguan refugees are being murdered there. Nor could we count on El Salvador, where the government is machine-gunning the peasants.” (Quoted in Aleman Bolanos, 1980: 160–1; trans. in Crawley, 1979: 82)
\end{quote}
Sandino realized also that the U.S. would not stop “its intrigue and manipulation substituting for armed intervention another type of intervention that is too subtle to be fought with weapons.” (Roman, 1979: 165) He thereupon decided that the only good course of action was to negotiate the best concessions he could get from Sacasa for whom he still retained a modicum of respect and he did so.
According to the treaty signed February 2, 1933, Sandino was given a large extension of land along the Coco River in the north. The treaty called it “empty land” but in reality thousands of Indians lived there who had been supporters of Sandino’s cause and whom he wanted to organize into agricultural cooperatives. Sandino was allowed to keep one hundred armed men as an emergency force, which the guerrilla leader saw as a sort of insurance policy against violations of the treaty.
However, General Anastasio Somoza, whom the U.S. Marines had left in charge of the National Guard, was furious at the concessions the newly-elected president of Nicaragua had made to Sandino and he continued to harass the Sandinistas. He realized there were three forces in Nicaragua: President Sacasa, the National Guard and Sandino’s small army. Somoza already had his eye on the job of his uncle the President; Sandino, who denounced the newly trained National Guard and repeatedly pledged the loyalty of his forces to Sacasa in the event of a Somoza coup, stood in the way.
A year after the peace treaty had been signed Sandino went to Managua to talk with the President. On the night of February 21, Sandino and several of his supporters ate with Sacasa at his home. When they left the dinner, Sandino along with Generals Francisco Estrada and Juan Pablo Umanzor were taken prisoner by members of the National Guard, driven to the local airfield and shot. Their bodies were never recovered.
A few days later the National Guard massacred the occupants of the Sandinista camps in the north. The era of Sandino’s small crazy army was over. The Somoza era had begun.
\section{Sandino’s Use of Race\Slash{}Class Analysis}
\begin{quote}
“My obsession is to repel with the dignity and pride that is characteristic of our race, any and all domination that, with the cynicism typical of the powerful, the assassins of weak peoples are preparing in my country.” –Augusto Sandino, Letter to Froylan Turcios, September 20,1927
\end{quote}
It was probably in Mexico that Sandino discovered his racial identity. Macaulay states:
\begin{quote}
“Sandino found in Tampico a Mexican nationalism that gloried in Mexico’s Indian heritage \dots{} He began to identify himself with a broad nationality embracing all Americans of Iberian and Indian descent. (1967:53)
\end{quote}
When Sandino returned to his homeland “to search for one hundred men who loved her as he did” he found, according to Jaime Wheelock’s analysis, a nascent working class. Its diverse elements included peasants who had been made landless by the expansion of the cultivation of export crops, as well as workers on the banana plantations and at the great lumber mills of the Atlantic Coast. These workers formed an independent force with homogeneous ideas, interests and demands and they were ready to join an anti-oligarchic struggle. (Wheelock, 1979: 119–120)
Shortly after Moncada laid down his arms and Sandino began his solitary struggle against the Marines, the latter issued his famous Political Manifesto, which reflects his conviction that he and his men were fighting in defense both of their race and of their class:
\begin{quote}
“I am a Nicaraguan and proud that Indian-American blood, more than any other, flows in my veins — blood that contains the mystery of loyal and sincere patriotism. The bond of nationality gives me the right to assume responsibility for my actions with respect to Nicaragua–and indeed to Central America and the whole continent that speaks our language-without caring what names the prophets of doom, cowards and eunuchs may choose to call me. I am a city worker, an artisan as they say in my country, but my ideal is a wide horizon of internationalism, the right to be free and to demand justice even if to win the state of perfection it be necessary to shed one’s own and other’s blood. The oligarchs, that is, the geese that paddle in the muck, will say I’m plebeian. Good enough: my highest honor is having come from the oppressed masses who are the soul and nerve-system of the race \dots{}.
“I swear before country and history that my sword will defend our nation’s dignity, that it will be a sword for the oppressed. I accept the invitation to fight, and will personally provoke it \dots{}. The last of my soldiers, the soldiers of freedom for Nicaragua, may die; but before that, more than a battalion of your blond invaders will have bitten the dust of my wild mountains.
“I will be no supplicant for the mercy of my enemies, who are the enemies of Nicaragua, for I don’t believe anyone has the right to be a demigod. I want to convince indifferent Nicaraguans and Central Americans and the Indo-Hispanic race that up here in the Andean heights, there is a group of patriots who will know how to fight and die like men.” (Quoted in Selser, 1981: 91–92)
\end{quote}
In this manifesto, Sandino appealed for an internationalism based upon race and class. He said his sword would be the weapon of the oppressed against both the traitorous oligarchs and the blonde invaders. To Sandino it seemed that while economic and political motives were the bases for the Yankee invasion, a racial disdain expressed toward those of Indian blood was also present.
But there was a racial factor inside Nicaragua as well. Because of inequities stemming from the conquest, the poorer classes in many Latin American countries were made up mostly of Indians and mestizos and thus were united on class and racial bases against the more purely Spanish oligarchies. These oligarchies frequently joined with the imperialists to oppress the lower classes and sell out the interests of their own countries.
In order to arouse as many people as possible to his cause in Nicaragua and in the rest of Latin America, Sandino glorified both his race and his class origin as something to be proud of. Elsewhere in the Manifesto, he vented his fury on the oligarchs of both the Liberal and Conservative Parties whose interests, he asserted, were not patriotic but rather treasonous.
The interests of the workers and of the people of Indian blood, Sandino felt, were those of the nation as a whole and led them in consequence to the patriotic defense of national sovereignty against the invader.
Carleton Beals, the Nation correspondent who visited Sandino in his jungle hideaway and then visited high U.S. Marine Corps officials as well as U.S. Embassy officials in Managua, agreed with Sandino:
\begin{quote}
“[The imperialist] does not analyze his own deeper motives nor does he see the contradiction between his faith in democracy and his fervent belief in his own race superiority. He never stops to try to reconcile his inner conviction that backward dark peoples are incapable of progress, efficiency, honesty or democracy with his belief that the only possible way for a foreign people to be happy is to be standardized into the mold already created in the United States. Because of his faith in the value of lightness of skin, he hobnobs with the aristocratic Creoles who have exploited and betrayed their countries since the first days of independence.” (Beals, 1932: 295)
\end{quote}
For Sandino the interests of race as well as of class demanded the expulsion of the Yankee invaders from Nicaragua. Further, the unity of all of Latin America was necessary to prevent continued U. S. domination of the area. But in order to achieve this goal, those governments which sold out their countries to the Yankee invaders had to be overthrown, by popular rebellion if necessary. “Tyrants do not represent nations,” Sandino wrote, “and liberty is not conquered with flowers.” (Sandino, 1976: 142–143)
After the Yankee invaders were expelled, Sandino felt the next stage for Nicaragua would be a profound popular social revolution: “the oppressed people will break the chains with which the imperialists of the earth have held them down.” (Quoted in Ortega, 1980: 94) At the time of the 1932 elections Sandino issued a circular, whose realistic possibilities he himself may have doubted but which provide an idea of what his hopes for a popular revolution included:
\begin{quote}
“Our army is prepared to take the reins of national power in order to then proceed to the organization of large cooperatives of Nicaraguan workers and peasants who will exploit our own natural resources in benefit of the Nicaraguan family as a whole.” (Sandino, 1976: 254)
\end{quote}
Sandino also had hopes that, in the future, international solidarity of the oppressed and exploited would triumph over racial prejudice. In an earlier letter to Spanish socialist Luis Araquistain, Sandino had said that “if in the present historical moment our struggle is national and racial, it will become international as colonial and semicolonial peoples learn to unite with peoples of the imperialist metropolies.” (Quoted in Selser, 1981: 110)
Whether Sandino expected this to happen with relation to the United States is not clear. He appeared to remain convinced that North Americans supported their government’s imperialist adventures. A few months after he wrote Araquistain, Sandino sent a letter to a man named Henry Amphlett in which he said:
\begin{quote}
“For a long time I believed that the North American people were not in agreement with the abuse committed in Nicaragua by the government of Calvin Coolidge; but I have become convinced that North Americans in general applaud Coolidge’s intervention in my country and for this reason all North Americans who fall into our hands will have met their end.” (Sandino, 1976: 148)
\end{quote}
This did not of course mean literally “all” North Americans because Beals of the Nation, which opposed Coolidge’s policy, was well received by Sandino at about this same time.
Sandino was a great admirer of Simon Bolivar, telling Spanish journalist Ramon Belausteguigoitia that reading Bolivar’s life always moved him and “had made him cry” (Sandino, 1976: 287) The story of Bolivar tragic, and one with which Sandino could identify readily: Bolivar was able to drive out the Spanish colonialists but was unable to achieve Latin American unity.
For Sandino the Monroe doctrine was the antithesis of Bolivarism and he offered a new modified version of the former:
\begin{quote}
“Speaking of the Monroe Doctrine, they say “America for the Americans.” Fine; that is well put. All of us who are born in the Americas are American. The mistake has been that of the imperialists who have interpreted the Monroe Doctrine as saying “America for the Yankees.” All right, so that the blond beasts do not continue in their mistakes, I reform the phrase as follows: “The United States of North America for the yankees. Latin America for the Indolatins.” (Sandino, 1976: 140)
\end{quote}
Sandino anticipated that the Indolatins — and most especially those of Nicaragua — would “ignite the fuse of the proletarian explosion against the imperialists of the earth” and bring on the reign “of the Spirit of Light and Truth, that is Love,” as he said in his Light and Truth Manifesto quoted above. Here again is presented in these documents the political-spiritual rhetoric that associates the oppressed classes and the brown races with love, truth and light and the blond capitalists and imperialists with oppression and injustice.
Although Sandino came very close to race-hatred it appears that he used it rather as a tool for consciousness raising. It was softened by the compassion he felt for the families of the Marines killed in battle and the respect he expressed for Marines who deserted and asked to join his struggle. (They were not allowed to join, by the way, because only Latin Americans could be members. Sandino, 1976: 133)
Sandino’s statements also were softened by the hopes he expressed for class solidarity with people of other races including those in the old colonial nations. The Indolatin race would light the fuse and then proletarians of all the world would join.
By 1933, when he signed the peace agreement with Sacasa, Sandino knew that the explosion might not come immediately. He anticipated, however, that the depressed world economic situation of that day would hasten its coming. Sandino decided to begin building in his corner of Nicaragua the kind of society he envisaged for the nation and the rest of Latin America.
Sandino had a revolutionary social agenda which he hoped to put in practice in Nicaragua in which the land would belong to the state and would be farmed in cooperatives by mestizo and Indian peasants. Meanwhile, as he waited for the correlation of forces to be right (Sandino had an appreciation for stages in the revolutionary struggle), he organized cooperatives in the area allotted to him in the peace treaty.
Sandino told Jose Roman that he would rather stay and work with the Miskitos, Sumos, and other Indians than accept invitations to travel to Paris and South America, where he felt he would be exhibited like a singer of tango music. (Roman, 1979: 98)
\begin{quote}
“I will stay here along the Coco, the most savage but most beautiful part of our nation, to bring it out of the abandonment it has suffered first by feudo-colonial exploitation and now by that of the capitalists. I want to do what is possible to civilize these Indians who are the marrow of our race.” (Quoted in Roman, 1979: 98)
\end{quote}
By “civilize” Sandino meant health care, education and better living conditions, and not the destruction of Indian culture. But these efforts to improve the lives of the Indians, to farm the land and mine gold cooperatively were too revolutionary for the “vendepatrias” in Managua and, in February, 1934, the experiment came to a bloody end, destroyed by the National Guard of Anastasio Somoza.
Carleton Beals said:
\begin{quote}
“The few people we met were all loyal Sandinistas, fleeing ever deeper into the wilderness\dots{}They were seeking safety, a new patch of ground to clear. But one and all, they vowed never to give up the struggle, and if necessary, pass it on to their children.” (Beals, 1932: 242)
\end{quote}
When the early fighters of the FSLN went into the mountains, they found those old Sandinistas who had survived and they learned from them how to flee from the National Guard and how to live in the jungle.
\section{Hope and Tragedy}
Much of the left of Sandino’s day did not understand the importance he placed on the racial question. Today many involved in struggles for national, ethnic, and racial identity, pride and respect often seem unable to come together around questions of economic oppression. A close reading of Sandino can provide us with an heroic example of someone who never compromised in his endeavors in both those essential struggles.
Many who have examined Sandino’s letters and political manifestoes have concluded that the totality of Sandino’s cause was the Bolivaran struggle for freedom from foreign domination and the quest for Latin American unity against the colossus of the north. This thesis was given credence by Sandino’s refusal to become a part of any sectarian political grouping and instead to accept the aid of all. Sandino went his own way and the Mexican revolutionaries, the APRA of Peru and the Comintern had to decide if they would support him.
Because he was such an important anti-imperialist symbol, the Comintern waited until 1930 before definitively breaking with Sandino. They had repudiated the APRA and Mariategui months previously. Many analysts, both Marxists and non-Marxists, through the years have assumed because of this break with the Comintern that Sandino did not have a revolutionary social agenda. But the break only indicates that his politics were not in accord with those of the Comintern in 1930.
While Sandino emphasized that the first part of his struggle was “national and racial,” the class nature of the cause, for all who cared to look for it, was visible in the social class of Sandino’s supporters, and in the social class of those he condemned repeatedly for having sold out their country to the invaders.
In the last year of his life Sandino alternated between hope for the future of the revolution and his sense of Bolivarian tragedy, of impending doom. If it is true that he cried before the firing squad (out of anger, he said) it must have been because now, for him, the dream was gone, tragedy was real.
\section{References}
\begin{amusebiblio}
Aleman Bolanos, Gustavo, Sandino: El Libertador. San Jose, Costa Rica: Nueva Decada, 1980.
Baines, John M., Revolution in Peru: Mariategui and the Myth. University, AL: University of Alabama Press, 1972.
Beals, Carleton, Banana Gold. Philadelphia: Lippincott, 1932.
Booth, John A., The End and the Beginning: The Nicaraguan Revolution. Boulder: Westview Press, 1985.
Cerda, Rodolfo, Sandino, El APRA y la Internacional Comunista. Lima: EDIMSSA, 1983.
Conrad, Robert Edgar, ed., Sandino: The Testimony of a Nicaraguan Patriot, 1921–1934. Princeton: Princeton University Press, 1990.
Crawley, Eduardo, Dictators Never Die: A Portrait of Nicaragua and the Somoza Dynasty. New York: St. Martin’s Press, 1979.
Hodges, Donald, Intellectual Foundations of the Nicaraguan Revolution. Austin: University of Texas Press, 1992; Sandino’s Communism: Spiritual Politics for the Twenty-First Century. Austin: University of Texas Press, 1986.
Macaulay, Neil, The Sandino Affair. Chicago: Quadrangle Books, 1967.
O’Brien, Conor Cruise, “God and Man in Nicaragua,” The Atlantic Monthly, (August 1986).
Ortega, Humberto, 50 Anos de Lucha Sandinista. Havana: Editorial de Ciencias Sociales, 1980.
Roman, Jose, Maldito Pais. Managua: El Pez y la Serpiente, 1979.
Sandino, Augusto C., El pensamiento vivo de Sandino. Edited by Sergio Ramirez. San Jose, Costa Rica: EDUCA, 1976.
Selser, Gregorio, Sandino. Translated by Cedric Belfrage. New York: Monthly Review Press, 1981.
Somoza Garcia, Anastasio, El verdadero Sandino, o el Calvario de las Segovias. Managua: Robelo, 1936.
Wheelock, Jaime, Imperialismo y dictadura: crisis de una formacion social. Mexico City: Siglo Veintiuno, 1979.
\end{amusebiblio}
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The Anarchist Library
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Katherine Hoyt
Race, Class and Sandino’s Politics
August 1995
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Retrieved on 25\textsuperscript{th} February 2021 from \href{https://againstthecurrent.org/atc057/p2643/}{againstthecurrent.org}
Published in Against The Current \#57
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\begin{document}
\title{The Role of Urine Biochemical Parameters for Predicting Disease Severity
In COVID-19 Patients}
\author[1]{omer erdogan}%
\author[1]{fesih ok}%
\author[1]{Serkan Carkci}%
\author[2]{emrullah durmus}%
\affil[1]{Siirt Training and Research Hospital}%
\affil[2]{Affiliation not available}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
Background: We aimed to determine the importance of urinary biochemical
parameters in predicting the severity of COVID-19 disease. Methods:
Totally 133 individuals diagnosed with COVID-19 in our clinic were
included in the study. The groups were formed according to the severity
of COVID-19 disease (moderate 85, severe 29, and critical 19), and an
additional control group was created from 50 healthy individuals. The
correlation between urine biochemical parameters and the severity of
disease was investigated. Results: Erythrocyturia, proteinuria, and
glucosuria rates were significantly higher in patients than controls. In
patients, the median urine specific gravity (SG) value was found to be
lower (p\textless{}0.001), and median potential of hydrogen (pH) value
was found to be higher compared to the controls (p\textless{}0.001). In
the severe group age, erythrocyturia, proteinuria, and glucosuria were
significantly higher than the non-severe group. On multivariate
analysis, proteinuria (OR: 4.66, 95\%CI 1.02-21.4, p=0.047) and age (OR:
1.06, 95\% CI 1.03-1.10, p\textless{}0.001) were independent predictive
factors for disease severity. Conclusion: Some urine biochemical
parameters especially proteinuria and advanced age may be useful for
predicting the COVID-19 disease severity.%
\end{abstract}%
\sloppy
\textbf{The Role of Urine Biochemical Parameters for Predicting Disease
Severity In COVID-19 Patients}
\textbf{Running Title: Urine biochemical parameters and COVID-19
disease}
\textbf{Keywords:} Urine biochemical parameters, COVID-19, disease
severity, coronavirus
\textbf{Corresponding Author: Omer Erdogan,} M.D. Specialist in Urology
Department, Siirt State Hospital, 56000, Turkey
\emph{[email protected]}
\textbf{Fesih Ok,} M.D. Specialist in Urology Department, Siirt State
Hospital, 56000, Turkey \emph{[email protected]}
\textbf{Serkan Carkci,} M.D. Specialist in Urology Department, Siirt
State Hospital, 56000, Turkey
\emph{[email protected]}
\textbf{Emrullah Durmus,} M.D. Specialist in Urology Department, Siirt
State Hospital, 56000, Turkey\emph{emrullah\[email protected]}
\textbf{Title: The Role of Urine Biochemical Parameters for Predicting
Disease Severity In COVID-19 Patients}
\textbf{Aim:} We aimed to determine the importance of urinary
biochemical parameters in predicting the severity of COVID-19 disease.
\textbf{Methods:} Totally 133 individuals diagnosed with COVID-19 in our
clinic were included in the study. The groups were formed according to
the severity of COVID-19 disease (moderate 85, severe 29, and critical
19), and an additional control group was created from 50 healthy
individuals. The correlation between urine biochemical parameters and
the severity of disease was investigated.
\textbf{Results:} Erythrocyturia, proteinuria, and glucosuria rates were
significantly higher in patients than controls. In patients, the median
urine specific gravity (SG) value was found to be lower
(p\textless{}0.001), and median potential of hydrogen (pH) value was
found to be higher compared to the controls (p\textless{}0.001). In the
severe group age, erythrocyturia, proteinuria, and glucosuria were
significantly higher than the non-severe group. On multivariate
analysis, proteinuria (OR: 4.66, 95\%CI 1.02-21.4, p=0.047) and age (OR:
1.06, 95\% CI 1.03-1.10, p\textless{}0.001) were independent predictive
factors for disease severity.
\textbf{Conclusion:} Some urine biochemical parameters especially
proteinuria and advanced age may be useful for predicting the COVID-19
disease severity.
\textbf{Keywords:~} Urine biochemistry, COVID-19, severe disease,
coronavirus
\textbf{What's known}
\begin{itemize}
\tightlist
\item
COVID-19 disease is a serious problem in society.
\item
The severity of the COVID-19 disease varies in patients.
\item
Parameters to predict the severity of the COVID-19 disease are being
investigated.
\item
Urine analysis can be useful in predicting the severity of Covid-19
disease.
\end{itemize}
\textbf{What's new}
\begin{itemize}
\tightlist
\item
There is one study that supports our work.
\item
COV\selectlanguage{polish}İ\selectlanguage{english}D-19 disease is a serious social problem and it is important to
predict its severity.
\end{itemize}
\textbf{Introduction}
In late 2019, pneumonia epidemic began in Wuhan, China's Hubei Province,
with a primarily unknown cause, which is known to have spread
significantly across the World \textsuperscript{1}. The virus that
caused the disease was initially named as Severe Acute Respiratory
Syndrome Coronavirus 2 (SARS-CoV-2), and later the World Health
Organization described this disease as COVID-19 \textsuperscript{2}. The
disease usually affects individuals between the ages of 30 and 79. About
half of those with COVID-19 disease have mild or indeterminate symptoms.
Significant symptoms in symptomatic patients are fatigue, fever, cough,
muscle pain, and shortness of breath \textsuperscript{2, 3}. Sometimes,
more critical conditions such as acute respiratory distress syndrome
(ARDS) and multi-organ failure can be observed. Patients with these
severe conditions have comorbid diseases, especially hypertension (HT),
diabetes mellitus (DM), and heart diseases\textsuperscript{3}.
Neutrophilia and lymphopenia are the most common laboratory parameters.
Abnormal liver function test findings at different rates have been
reported. Serum procalcitonin levels are generally at normal levels,
while mild increases in C-reactive protein levels can be seen. D-Dimer
levels are high in 30 percent of patients \textsuperscript{4,5}.
Coronaviruses are enveloped RNA viruses that consist of a single chain
and have positive polarity. Therefore, they do not have RNA-dependent
RNA polymerase enzymes, but this enzyme code has been identified in
their genetic makeup. Their surfaces have rod-like
extensions\textsuperscript{6}.
Urine examination is fast, convenient, and economical. It can be used as
an assay to diagnose many diseases such as urinary tract infections
(UTI), kidney diseases, and stone diseases by showing biochemical
parameters of urine \textsuperscript{7,8,9}. So far, one study has been
conducted showing the relationship between the biochemical parameters of
urine and COVID-19 disease \textsuperscript{10}. We aimed to determine
the role of biochemical parameters of urine in predicting the COVID-19
disease severity.
\textbf{Materials and methods}
\textbf{Study design and participants}
Pre-work permits were obtained by the Turkish Ministry of Health and the
local ethics committee of Siirt University (decision no: 2020/05.02).
Patients hospitalized in Siirt State Hospital between April and May 2020
and whose COVID-19 PCR (Polymerase Chain Reaction) test was positive
were included in the study. Also, a control group was formed from 50
healthy individuals. Urine potential of hydrogen (pH), specific gravity
(SG), leukocyte, erythrocyte, protein, nitrite, glucose, and bacteria
were recorded by asking the patients for a full urine examination. The
patient group was divided into four (mild, moderate, severe, and
critical) groups according to the Diagnostic Treatment Program of New
Coronavirus Pneumonia (seventh trial version). Patients with the mild
group were excluded from the study because they were treated,
outpatient.
Patients who were thought to affect the study results, such as chronic
kidney failure, asthma, HT, DM, and chronic obstructive pulmonary
disease (COPD) were excluded. Also, all patient groups had nitrite
negative in urine and no bacteria. Therefore patients with urinary tract
infections were excluded from the study. We compared the patient group
with healthy controls. Besides, the severe and critical groups in the
patient group were combined to determine the severe group, and the
moderate group was defined as the non-severe group.
\textbf{Method~}
After the patients were hospitalized in Siirt State Hospital, about 30
ml of clean mid-flow urine samples were taken from the patients on the
same day. A urine sample was taken from critical COVID-19 patients by
inserting a catheter. Biochemical parameters of urine such as urine
occult blood, urine glucose, nitrite, SG, pH proteinuria, and leukocytes
were tested using a fully automatic urine biochemical analyzer (DIRUI
FUS 200 / H -800, DIRUI IndustrialCo., China). All collected samples
were studied within 2 hours.
\textbf{Statistical Analysis}
All statistical analysis was performed out using SPSS Statistics
software version 26.0 (IBM, Armonk, NY, USA). Continuous variables were
expressed as the appropriate means and standard deviations or medians
and interquartile ranges. Categorical variables were summarized as the
counts and percentages in each category. One-way analysis of variance
(ANOVA), Kruskal--Wallis test, Student's T-test, and Mann Whitney U
tests were applied to continuous variables, and chi-square and Fisher's
exact tests were used for categorical variables. Binary logistic
regression analysis was used to determine the predictive effect of age,
proteinuria, and all other significant factors on disease severity. The
optimal cut-off value of age was calculated by applying the receiver
operating curve (ROC) analysis. A value of P \textless{}0.05 was
considered statistically significant.
\textbf{Results}
In our study, there were 85 (63.9\%) patients in the moderate group, 29
(21.8\%) patients in the severe group, and 19 (14.3\%) patients in the
critical group. For the control group, 50 healthy people without
COVID-19 disease were selected.
\textbf{Urine biochemical parameters analysis of patients' and control
groups}
There was no significant difference found between the patients and
control groups in terms of age (p=0.070) and gender (p=0.125) (Table 1).
The rates of erythrocyturia (p\textless{}0.001), proteinuria (p=0.015),
and glucosuria (p=0.020) were significantly higher in patients than
controls. In the patient group, the median SG value was found to be
significantly lower than the control group (p\textless{}0.001). The
median pH value was found to be significantly higher in the patient
group compared to the control group (p\textless{}0.001) (Table 1).
\textbf{Urine biochemical parameters analysis of three patients'
groups~}
In terms of SG (p=0.334) and pH (p=0.229), there was no statistically
significant difference found between the three patients groups. Patients
in the moderate group had a significantly lower average age than
patients in the severe and critical groups (p\textless{}0.001). The rate
of proteinuria was significantly higher in patients in the severe and
critical groups compared to the moderate group (p\textless{}0.001).
Erythrocyturia ratio was significantly higher in the critical group than
in the moderate group (p\textless{}0.001), but there was no significant
difference found between the severe group and the other two groups
(p\textgreater{}0.05). Proteinuria and glucosuria rates were
significantly higher in severe and critical groups than in the moderate
group (p\textless{}0.001). Proteinuria and glucosuria rates were
significantly higher in the critical group than in the severe group
(p\textless{}0.05).
\textbf{Independent predictive risk factors for disease severity}
The severe and critical groups were determined as severe and the
moderate group as non-severe. Age, erythrocyturia, proteinuria, and
glucosuria were significantly higher parameters in the severe group than
the non-severe group (p\textless{}0.001). On multivariate analysis,
proteinuria (OR: 4.66, 95\%CI 1.02-21.4, p=0.047) and age (OR: 1.06,
95\% CI 1.03-1.10, p\textless{}0.001) were independent predictive
factors for disease severity (Table 2). The optimum cut-off value of age
for predicting severe disease was 59.5 years. AUC of age was 0.828
{[}95\%CI (0.756-0.899); P\textless{}0.001{]}. The highest sensitivity
and specificity were 0.771 and 0.765 for age (Figure 1).
\textbf{Discussion}
COVID-19 disease spread throughout the World, starting in China in 2019
and It was accepted by the World Health Organization as a pandemic. It
is a RNA virus known as coronavirus 2 (SARS-CoV-2) which causes life
losses all over the World. Clinical findings are non-specific and
usually include cough, fever, myalgia, weakness, and
nausea\textsuperscript{2,3}. In patients with high comorbidity, it may
be more severe and cause multi-organ failure \textsuperscript{3}.
According to the latest Coronavirus Pneumonia Diagnosis and Treatment
Program (7th Edition), patients are divided into four groups as mild,
moderate, severe and critical. Clinical findings of patients, blood
values, respiratory count, and blood pressure are useful in determining
the severity of the disease \textsuperscript{11}.
In a study that compared the COVID-19 patients and controls, the
incidence of protein and erythrocyte in the urine of patients was higher
than the controls (p\textless{}0.05). Besides, urine pH and SG were
considerably different from the controls. However, the incidence of
leukocytes in urine did not differ between the patient and the healthy
group. This is because SARS-CoV-2 infection has been linked to
non-bacterial \textsuperscript{10}. In our study, the positive rates of
erythrocyturia (p\textless{}0.001), proteinuria (p=0.015), and
glucosuria (p=0.020) were higher in patients than controls. SG was
considerably lower in the patients than controls. Besides, urine pH
value of patients was significantly higher than controls. However, urine
pH and SG values were similar in the patient groups. The average age,
glucosuria, erythrocyturia, and proteinuria of severe and critical
patients were significantly higher compared to the moderate group.
COVID-19 disease is mostly asymptomatic, and symptomatic patients are
treated by hospitalization. In our study, two critical predictive
factors, advanced age and proteinuria, were found to be correlated with
the COVID-19 disease severity. For this reason, we think that these
patients should be hospitalized and closely follow-up even if they are
asymptomatic. The frequency of ARDS is observed more frequently in
severe patients than non-severe patients \textsuperscript{12}. The
cytokine storm thought to cause the ARDS table can cause multiple organ
failure, affecting the kidney. In the severe and critical group, we
believe that kidney damage caused by this mechanism causes proteinuria.
As a result, urine biochemical parameters have no place in the diagnosis
of COVID-19 disease but are valuable in terms of the disease's
progression. Therefore, we think it would be useful to routinely examine
the biochemical parameters of urine that are easily applicable and
cost-effective in all COVID-19 patients.
\textbf{Acknowledgement}
The authors would like to thank the healthcare staff and the hospital
management who worked intensively during the pandemic process.
\textbf{Statement of Ethics}
All procedures in studies with human participants complied with the
Ethical standards of the Corporate Research Committee and the 1964
Helsinki Declaration and subsequent updates.
\textbf{Funding Sources}
There is no funding source to be declared for this study.
\textbf{Conflicts of Interest}
No conflicts of interest.
\textbf{References}
\begin{enumerate}
\tightlist
\item
He, F., Deng, Y., \& Li, W. (2020). Coronavirus disease 2019: What we
know?.~\emph{Journal of medical virology} . doi: 10.1002/jmv.25766
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Lu, R., Zhao, X., Li, J., Niu, P., Yang, B., Wu, H., \ldots{} \& Bi,
Y. (2020). Genomic characterisation and epidemiology of 2019 novel
coronavirus: implications for virus origins and receptor
binding.~\emph{The Lancet} ,~\emph{395} (10224), 565-574. doi:
10.1016/S0140-6736(20)30251-8
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Wu, Z., \& McGoogan, J. M. (2020). Characteristics of and important
lessons from the coronavirus disease 2019 (COVID-19) outbreak in
China: summary of a report of 72 314 cases from the Chinese Center for
Disease Control and Prevention.~\emph{Jama} ,~\emph{323} (13),
1239-1242. doi: 10.1001/jama.2020.2648
\item
Li, Q., Guan, X., Wu, P., Wang, X., Zhou, L., Tong, Y., \ldots{} \&
Xing, X. (2020). Early transmission dynamics in Wuhan, China, of novel
coronavirus--infected pneumonia.~\emph{New England Journal of
Medicine} . doi: 10.1056/NEJMoa2001316
\item
Wang, D., Hu, B., Hu, C., Zhu, F., Liu, X., Zhang, J., \ldots{} \&
Zhao, Y. (2020). Clinical characteristics of 138 hospitalized patients
with 2019 novel coronavirus--infected pneumonia in Wuhan,
China.~\emph{Jama} ,~\emph{323} (11), 1061-1069. doi:
10.1001/jama.2020.1585
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Zhou, Y., Yang, Y., Huang, J., Jiang, S., \& Du, L. (2019). Advances
in MERS-CoV vaccines and therapeutics based on the receptor-binding
domain.~\emph{Viruses} ,~\emph{11} (1), 60. doi:10.3390/v11010060
\item
Berger, R. E. (2005). The urine dipstick test useful to rule out
infections. A meta-analysis of the accuracy.~\emph{The Journal of
urology} . doi: 10.1016/S0022-5347(01)68458-1
\item
Falbo, R., Sala, M. R., Signorelli, S., Venturi, N., Signorini, S., \&
Brambilla, P. (2012). Bacteriuria screening by automated
whole-field-image-based microscopy reduces the number of necessary
urine cultures.~\emph{Journal of clinical microbiology} ,~\emph{50}
(4), 1427-1429. ~doi: 10.1128/JCM.06003-11
\item
Erdman, P., Anderson, B., Zacko, J. C., Taylor, K., \& Donaldson, K.
(2017). The Accuracy of the Sysmex UF-1000 i in Urine Bacterial
Detection Compared With the Standard Urine Analysis and
Culture.~\emph{Archives of pathology \& laboratory medicine}
,~\emph{141} (11), 1540-1543. doi: 10.5858/arpa.2016-0520-OA
\item
Liu, R., Ma, Q., Han, H., Su, H., Liu, F., Wu, K., \ldots{} \& Zhu, C.
(2020). The value of urine biochemical parameters in the prediction of
the severity of coronavirus disease 2019.~\emph{Clinical Chemistry and
Laboratory Medicine (CCLM)} ,~\emph{1} (ahead-of-print). doi:
10.1515/cclm-2020-0220
\item
Chen, N., Zhou, M., Dong, X., Qu, J., Gong, F., Han, Y., \ldots{} \&
Yu, T. (2020). Epidemiological and clinical characteristics of 99
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Tetro, J. A. (2020). Is COVID-19 receiving ADE from other
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\end{enumerate}
\textbf{Figure 1}. The receiver operating characteristic (ROC) Curve
analysis of age for COVID-19 disease severity.\textbf{Table 1.} The
comparison of demographic and urine biochemical parameters between
patient groups and healthy controls.\textbf{Table 2.} The comparison of
demographic and urine biochemical parameters between patient groups and
independent predictors of disease severity by univariate and
multivariate logistic regression.
\textbf{Hosted file}
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\section{Sencha}
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%D copyright={PRAGMA ADE \& \CONTEXT\ Development Team}]
%C
%C This module is part of the \CONTEXT\ macro||package and is
%C therefore copyrighted by \PRAGMA. See mreadme.pdf for
%C details.
%D The less there is here, the better. After a decade it is time to remove the \type
%D {\pdf*} ones completely. For the moment I keep them commented but even that will
%D go away.
\writestatus{loading}{ConTeXt Backend Macros / PDF}
\registerctxluafile{lpdf-ini}{optimize}
\registerctxluafile{lpdf-nod}{}
\registerctxluafile{lpdf-eng}{optimize}
\registerctxluafile{lpdf-col}{}
\registerctxluafile{lpdf-vfc}{}
\registerctxluafile{lpdf-xmp}{}
\registerctxluafile{lpdf-ano}{}
\registerctxluafile{lpdf-res}{}
\registerctxluafile{lpdf-mis}{}
\registerctxluafile{lpdf-ren}{}
\registerctxluafile{lpdf-grp}{}
\registerctxluafile{lpdf-wid}{}
\registerctxluafile{lpdf-fld}{}
\registerctxluafile{lpdf-mov}{}
\registerctxluafile{lpdf-u3d}{} % this will become a module
\registerctxluafile{lpdf-swf}{} % this will become a module
\registerctxluafile{lpdf-tag}{}
\registerctxluafile{lpdf-fmt}{}
\registerctxluafile{lpdf-pde}{}
\registerctxluafile{lpdf-epa}{}
\registerctxluafile{lpdf-fnt}{}
\registerctxluafile{back-pdp}{}
\registerctxluafile{back-pdf}{} % some code will move to lpdf-*
\loadmarkfile{back-u3d} % this will become a module
\loadmarkfile{back-swf} % this will become a module
\unprotect
%D We will minimize the number of calls to \PDF\ specific primitives and delegate
%D all management and injection of code to the backend.
%D
%D Because we do a lot in \LUA\ and don't want interferences, we nil most of the
%D \PDFTEX\ primitives. Of course one can always use the \type {\pdfvariable},
%D \type {\pdfextension} and \type {\pdffeedback} primitives but it will probably
%D have bad side effects.
\unexpanded\def\pdfextension{\clf_pdfextension}
\def\pdffeedback {\clf_pdffeedback}
%D For the moment we keep this for tikz but hopefully it will at some point use
%D the proper ones. Consider them obsolete:
\unexpanded\def\pdfliteral {\clf_pdfliteral}
\unexpanded\def\pdfobj {\clf_pdfobj}%
\unexpanded\def\pdflastobj {\numexpr\clf_pdflastobj\relax}
\unexpanded\def\pdfrefobj {\clf_pdfrefobj }
\unexpanded\def\pdfrestore {\pdfextension restore}
\unexpanded\def\pdfsave {\pdfextension save}
\unexpanded\def\pdfsetmatrix{\pdfextension setmatrix}
\let\pdfxform \saveboxresource
\let\pdflastxform \lastsavedboxresourceindex
\let\pdfrefxform \useboxresource
%D Here are quick and dirty compression flippers, mostly used when testing something
%D as one can best stick to the defaults that also adapt to specific standards.
\unexpanded\def\nopdfcompression {\clf_setpdfcompression\zerocount\zerocount}
\unexpanded\def\onlypdfobjectcompression{\clf_setpdfcompression\zerocount\plusthree}
\unexpanded\def\maximumpdfcompression {\clf_setpdfcompression\plusnine \plusnine }
\unexpanded\def\normalpdfcompression {\clf_setpdfcompression\plusthree\plusthree}
%D PDF/X (maybe combine the two lua calls)
\setupbackend
[xmpfile=]
\appendtoks
\edef\p_file{\backendparameter{xmpfile}}%
\ifx\empty\p_file\else
\clf_setxmpfile{\p_file}%
\fi
\to \everysetupbackend
%D This will change:
\appendtoks
\clf_setformat
format {\backendparameter\c!format}%
level {\backendparameter\c!level}%
option {\backendparameter\c!option}%
profile {\backendparameter\c!profile}%
intent {\backendparameter\c!intent}%
file {\backendparameter\c!file}%
\relax
\to \everysetupbackend
%D These are the only official methods to add stuff to the resources. If more is
%D needed for third party low level code, it can be added.
\unexpanded\def\pdfbackendsetcatalog #1#2{\clf_lpdf_addtocatalog{#1}{#2}}
\unexpanded\def\pdfbackendsetinfo #1#2{\clf_lpdf_addtoinfo{#1}{#2}}
\unexpanded\def\pdfbackendsetname #1#2{\clf_lpdf_addtonames{#1}{#2}}
\unexpanded\def\pdfbackendsetpageattribute #1#2{\clf_lpdf_addtopageattributes{#1}{#2}}
\unexpanded\def\pdfbackendsetpagesattribute#1#2{\clf_lpdf_addtopagesattributes{#1}{#2}}
\unexpanded\def\pdfbackendsetpageresource #1#2{\clf_lpdf_addtopageresources{#1}{#2}}
\unexpanded\def\pdfbackendsetextgstate #1#2{\clf_lpdf_adddocumentextgstate{#1}{#2}}
\unexpanded\def\pdfbackendsetcolorspace #1#2{\clf_lpdf_adddocumentcolorspace{#1}{#2}}
\unexpanded\def\pdfbackendsetpattern #1#2{\clf_lpdf_adddocumentpattern{#1}{#2}}
\unexpanded\def\pdfbackendsetshade #1#2{\clf_lpdf_adddocumentshade{#1}{#2}}
\def\pdfbackendcurrentresources {\clf_lpdf_collectedresources}
\def\pdfcolor #1{\clf_lpdf_color\numexpr\thecolorattribute{#1}\relax}
%D This is a temporary hack mthat will be removed, improved or somehow can become
%D default.
\def\TransparencyHack{\setupcolors[\c!pagecolormodel=\v!auto]}
%D Just in case one needs this \unknown:
%D
%D \starttyping
%D text \pdfbackendactualtext{Meier}{Müller} text
%D \stoptyping
\unexpanded\def\pdfbackendactualtext#1#2% not interfaced
{\clf_startactualtext{#2}%
#1%
\clf_stopactualtext}
\let\pdfactualtext\pdfbackendactualtext
%D Bah, this is also needed for tikz:
\ifdefined\pdfsavepos\else
\let\pdfsavepos \savepos
\let\pdflastxpos\lastxpos
\let\pdflastypos\lastypos
\fi
\protect \endinput
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