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authorKarl Berry <karl@freefriends.org>2019-09-14 22:07:09 +0000
committerKarl Berry <karl@freefriends.org>2019-09-14 22:07:09 +0000
commit77c6ab621157215821ffc5d8b0dd655748c262e7 (patch)
tree8c7cc34fd220c9cacf2bfb1f0149419f083805ea /Master/texmf-dist/doc/latex-dev/amsmath
parente1f5126a433abad76994faf305f1df3ad04b6f02 (diff)
latex-dev (15sep19)
git-svn-id: svn://tug.org/texlive/trunk@52096 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex-dev/amsmath')
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/README.md81
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/ams-external.txt547
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/ams-internal.txt482
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amsbsy.pdfbin0 -> 161341 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amscd.pdfbin0 -> 165759 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amsgen.pdfbin0 -> 164359 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amsldoc.pdfbin0 -> 453064 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amsldoc.tex2770
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amsmath.pdfbin0 -> 379697 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amsopn.pdfbin0 -> 129178 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amstext.pdfbin0 -> 193911 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/amsxtra.pdfbin0 -> 104557 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/changes.txt123
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/diffs-m.txt551
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/manifest.txt32
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/subeqn.pdfbin0 -> 101303 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/subeqn.tex202
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/technote.pdfbin0 -> 130493 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/technote.tex233
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/testmath.pdfbin0 -> 428714 bytes
-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex2342
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diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/README.md b/Master/texmf-dist/doc/latex-dev/amsmath/README.md
new file mode 100644
index 00000000000..66a27edf03c
--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/README.md
@@ -0,0 +1,81 @@
+The amsmath bundle for LaTeX
+============================
+
+Release 2019-10-01 pre-release 3
+
+Overview
+--------
+
+The `amsmath` package is an extension package for LaTeX that provides
+additional features to facilitate mathematical typesetting. It has been
+developed by the American Mathematical Society (AMS) and released for general
+use as a service to the mathematical community. A number of smaller
+auxiliary packages are also distributed with the `amsmath` package.
+
+Effective in 2016, maintenance of `amsmath was transferred from AMS to
+the LaTeX Project; as `amsmath` is considered a "required" package, this
+centralizes control over the core LaTeX components.
+
+Installation
+------------
+
+We no longer distribute installation instructions for the various TeX
+implementations. All modern TeX systems include `amsmath` as-standard, and end
+users should in general use the release versions supplied in this way.
+
+Release distribution is carried out only through the CTAN archives.
+
+Documentation
+-------------
+
+The primary documentation for `amsmath` is in provided as `amsldoc.pdf`.
+
+Additional documentation files include:
+
+ - `diffs-m.txt`
+ - `subeqn.pdf`
+ - `technote.pdf`
+ - `testmath.pdf`
+
+which are included in the collection. All of these can be accessed
+easily with most distributions by entering `texdoc` *filename* at the
+command line, or via [TeXdoc Online](http://texdoc.net).
+
+Change logs
+-----------
+
+The file `changes.txt` lists recent changes in reverse chronological order.
+
+The file `diffs-m.txt` contains information on development and changes
+prior to 2000.
+
+Reporting Bugs
+--------------
+
+If you wish to report a problem or bug in the `amsmath` bundle
+please use the [Issue Tracker for LaTeX2e on
+GitHub](https://github.com/latex3/latex2e/issues)
+and follow the guidelines that pop up if you press the "New issue" button.
+
+In particular, to check that you are really seeing a bug, please write
+a short, self-contained document that shows the problem. This should
+include the `latexbug` package, which will warn if your test file is
+not suitable for one or the other reason. See the [CONTRIBUTING
+guide](https://github.com/latex3/latex2e/blob/master/CONTRIBUTING.md)
+for further details, or if you need to obtain the `latexbug` package.
+
+If the bug turns out to be with third-party software then please
+contact the developer, and not us!
+
+License
+-------
+
+The contents of this bundle are distributed under the [LaTeX Project
+Public License](https://www.latex-project.org/lppl/lppl-1-3c/),
+version 1.3c or later.
+
+-----
+
+<p>Copyright (C) 2001-2004, 2007, 2008, 2010, 2011, 2013 American Mathematical Society. <br />
+<p>Copyright (C) 2016-2019 The LaTeX Project and American Mathematical Society. <br />
+
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/ams-external.txt b/Master/texmf-dist/doc/latex-dev/amsmath/ams-external.txt
new file mode 100644
index 00000000000..fa2dbea5561
--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/ams-external.txt
@@ -0,0 +1,547 @@
+%%% list of external commands used by amsmath (not totally correct)
+
+\!
+\"
+\(
+\*
+\,
+\-
+\.
+\/
+\:
+\;
+\=
+\Acute
+\AmS
+\AmSfont
+\And
+\AtBeginDocument
+\Bar
+\Big
+\Bigg
+\Biggr
+\Bigr
+\Breve
+\Check
+\DOTS
+\DOTSB
+\DOTSI
+\DOTSX
+\Ddot
+\DeclareFontEncoding
+\DeclareFontShape
+\DeclareMathAccent
+\DeclareMathDelimiter
+\DeclareMathOperator
+\DeclareMathSymbol
+\DeclareOption
+\DeclareRobustCommand
+\Dot
+\ExecuteOptions
+\GenericError
+\GenericInfo
+\GenericWarning
+\Grave
+\Hat
+\Leftarrow
+\Longleftarrow
+\Longleftrightarrow
+\Longrightarrow
+\MessageBreak
+\MultiIntegral
+\PackageError
+\PackageWarning
+\PackageWarningNoLine
+\PassOptionsToPackage
+\Pr
+\ProcessOptions
+\Relbar
+\RequirePackage
+\Rightarrow
+\Tilde
+\Umathaccent
+\Umathcar
+\Umathchar
+\Umathcharnumdef
+\Umathcode
+\Umathcodenum
+\Umathfractiondelsize
+\Vec
+\Vert
+\XeTeXcharclass
+\[
+\]
+\above
+\abovedisplayskip
+\abovewithdelims
+\accentedsymbol
+\acute
+\addtocounter
+\advance
+\aftergroup
+\allowbreak
+\allowdisplaybreaks
+\alph
+\arabic
+\arccos
+\arcsin
+\arctan
+\arg
+\array
+\arraycolsep
+\arraystretch
+\atop
+\atopwithdelims
+\badness<
+\bar
+\baselineskip
+\baselineskip6
+\begin
+\begingroup
+\belowdisplayskip
+\bgroup
+\big
+\bigcap
+\bigcup
+\bigg
+\biggr
+\bigodot
+\bigoplus
+\bigotimes
+\bigr
+\bigsqcup
+\biguplus
+\bigvee
+\bigwedge
+\binom
+\bmod
+\boldsymbol
+\box
+\boxed
+\breve
+\cases
+\cdotp
+\cdots
+\cfrac
+\char
+\chardef
+\check
+\cleaders
+\colon
+\columnwidth
+\coprod
+\copy
+\cos
+\cosh
+\cot
+\coth
+\cr
+\crcr
+\csc
+\csname
+\csnamen
+\csnamethe
+\dbinom
+\ddddot
+\dddot
+\ddot
+\def
+\deg
+\delimiter
+\delimiterfactor
+\delimitershortfall
+\det
+\dfrac
+\dim
+\displaybreak
+\displayindent
+\displaylimits
+\displaystyle
+\displaywidth
+\divide
+\do
+\dot
+\doteq
+\dots
+\dotsb
+\dotsc
+\dotsi
+\dotsm
+\dotso
+\dp
+\edef
+\egroup
+\else
+\end
+\endalign
+\endaligned
+\endarray
+\endcsname
+\endgather
+\endgroup
+\endmathdisplay
+\endmatrix
+\endmultline
+\endsplit
+\endsubarray
+\eqno
+\eqref
+\everycr
+\everydisplay
+\everymath
+\exp
+\expandafter
+\fbox
+\fi
+\fontchardp
+\fontcharht
+\fontdimen
+\fontencoding
+\fontfamily
+\frac
+\fracwithdelims
+\futurelet
+\gcd
+\gdef
+\genfrac
+\global
+\grave
+\halign
+\halignto
+\hat
+\hbox
+\hdots
+\hdotsfor
+\hfil
+\hfill
+\hfilneg
+\hfuzz
+\hom
+\hookleftarrow
+\hookrightarrow
+\hrule
+\hskip
+\ht
+\ialign
+\idotsint
+\if
+\ifDOTS
+\ifcase
+\ifdim
+\iff
+\iffalse
+\iffalse%initialvalue
+\ifhbox
+\ifinner
+\ifm
+\ifmacrostarts
+\ifmmode
+\ifnum
+\ifnum`
+\ifodd
+\iftrue
+\ifvmode
+\ifx
+\ignorespaces
+\ignorespacesafterend
+\iiiint
+\iiint
+\iint
+\impliedby
+\implies
+\inf
+\injlim
+\int
+\interdisplaylinepenalty
+\intertext
+\intop
+\joinrel
+\jot
+\ker
+\kern
+\l%
+\lVert
+\label
+\lastbox
+\lastkern
+\lastskip
+\lbrace
+\ldots
+\leavevmode
+\left
+\leftarrow
+\leftmargini
+\leftmarginiminus
+\leftroot
+\leqno
+\let
+\lg
+\lhook
+\lim
+\liminf
+\limits
+\lineskip
+\lineskiplimit
+\linewidth
+\llap
+\ln
+\log
+\long
+\longleftarrow
+\longleftrightarrow
+\longmapsto
+\longrightarrow
+\loop
+\lq
+\lvert
+\mapsto
+\mapstochar
+\mathaccent
+\mathaccentV
+\mathalpha
+\mathbin
+\mathchar
+\mathchardef
+\mathchoice
+\mathclose
+\mathcode
+\mathdisplay
+\mathellipsis
+\mathgroup
+\mathhexbox
+\mathinner
+\mathop
+\mathopen
+\mathord
+\mathpalette
+\mathpunct
+\mathrel
+\mathring
+\mathsurround
+\mathxxx
+\mathxxxtest
+\mathxxxx
+\matrix
+\max
+\maxdimen
+\mbox
+\meaning
+\medmuskip
+\medspace
+\min
+\minalignsep
+\mintagsep
+\mkern
+\mod
+\moveleft
+\mskip
+\mspace
+\multicolumn
+\multiply
+\multlinegap
+\multlinegap10pt
+\multlinetaggap
+\multlinetaggap10pt
+\negmedspace
+\negthickspace
+\negthinspace
+\newbox
+\newcommand
+\newcount
+\newcounter
+\newdimen
+\newenvironment
+\newhelp
+\newif
+\newmuskip
+\newskip
+\newtoks
+\noalign
+\nobreak
+\nobreakdash
+\nobreakspace
+\noexpand
+\noindent
+\nointerlineskip
+\nolimits
+\nonscript
+\nonumber
+\normalbaselines
+\normalcolor
+\normalfont
+\normallineskiplimit
+\not
+\notag
+\nottest
+\null
+\nulldelimiterspace
+\number
+\numberwithin
+\of
+\oint
+\ointop
+\omit
+\openup
+\operatorfont
+\operatorname
+\operatornamewithlimits
+\or
+\over
+\overleftarrow
+\overleftrightarrow
+\overline
+\overrightarrow
+\overset
+\overwithdelims
+\par
+\parshape
+\penalty
+\pmatrix
+\pmod
+\pod
+\postdisplaypenalty
+\predisplaypenalty
+\prevdepth
+\primfrac
+\prod
+\projlim
+\protect
+\providecommand
+\qopname
+\quad
+\rVert
+\raise
+\raisetag
+\rangle
+\rbrace
+\rbrack
+\rceil
+\ref
+\refstepcounter
+\relax
+\relaxo
+\relbar
+\renewcommand
+\renewenvironment
+\repeat
+\rfloor
+\rgroup
+\rhook
+\right
+\rightarrow
+\rlap
+\rmoustache
+\romannumeral
+\root
+\rootbox
+\rq
+\rvert
+\scriptfont
+\scriptscriptfont
+\scriptscriptstyle
+\scriptstyle
+\sec
+\selectfont
+\setbox
+\setcounter
+\shoveleft
+\shoveright
+\sideset
+\sim
+\sin
+\sinh
+\skewchar
+\smash
+\space
+\spaceor
+\span
+\spbreve
+\spcheck
+\spdddot
+\spddot
+\spdot
+\sphat
+\split
+\sptilde
+\sqrtsign
+\stepcounter
+\string
+\strut
+\strutbox
+\subarray
+\substack
+\sum
+\sup
+\symoperators
+\tabskip
+\tag
+\taghere
+\tan
+\tanh
+\tbinom
+\text
+\textellipsis
+\textfont
+\textstyle
+\textup
+\tfrac
+\the
+\theequation
+\theparentequation
+\thetag
+\thickmuskip
+\thickmuskip0mu
+\thickspace
+\thinmuskip
+\thinspace
+\tilde
+\tmspace
+\toks
+\typeout
+\u
+\underleftarrow
+\underleftrightarrow
+\underrightarrow
+\underset
+\unhbox
+\unhcopy
+\unpenalty
+\unskip
+\unvbox
+\unvcopy
+\uppercase
+\uproot
+\usefont
+\value
+\varDelta
+\varGamma
+\varLambda
+\varOmega
+\varPhi
+\varPi
+\varPsi
+\varSigma
+\varTheta
+\varUpsilon
+\varXi
+\varinjlim
+\varliminf
+\varlimsup
+\varprojlim
+\vbox
+\vcenter
+\vec
+\vee
+\veqno
+\vert
+\vrule
+\vskip
+\vss
+\vtop
+\wd
+\widehat
+\widetilde
+\xdef
+\xleaders
+\xleftarrow
+\xrightarrow
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/ams-internal.txt b/Master/texmf-dist/doc/latex-dev/amsmath/ams-internal.txt
new file mode 100644
index 00000000000..06a0c97f091
--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/ams-internal.txt
@@ -0,0 +1,482 @@
+%%%% list of internal commands used by amsmath (not totally correct)
+
+\@@above
+\@@abovewithdelims
+\@@atop
+\@@atopwithdelims
+\@@eqno
+\@@italiccorr
+\@@leqno
+\@@over
+\@@overline
+\@@overwithdelims
+\@@sqrt
+\@@undefined
+\@@underline
+\@M
+\@addtoreset
+\@amsmath@err
+\@arrayboxrestore
+\@arrayparboxrestore
+\@backslashchar
+\@badmath
+\@car
+\@cdots
+\@centering
+\@checkend
+\@currentlabel
+\@currenvir
+\@declmathop
+\@depth
+\@display@init
+\@displaytrue
+\@eha
+\@ehc
+\@ehd
+\@elt
+\@empty
+\@emptytoks
+\@envbody
+\@eqnnum
+\@eqnswfalse
+\@eqnswtrue
+\@eqpen
+\@fleqntrue
+\@for
+\@genfrac
+\@getpen
+\@gobble
+\@gobblefour
+\@gobblethree
+\@gobbletwo
+\@gtempa
+\@height
+\@iden
+\@ifdefinable
+\@ifnextchar
+\@ifnotempty
+\@ifpackageloaded
+\@ifpackagewith
+\@ifstar
+\@ifundefined
+\@ignoretrue
+\@ldots
+\@let@token
+\@lign
+\@m
+\@mathmargin
+\@mathmeasure
+\@mathstyle
+\@namedef
+\@ne
+\@nil
+\@nocnterr
+\@nocounterr
+\@nx
+\@onlypreamble
+\@saveprimitive
+\@sptoken
+\@stpelt
+\@tempa
+\@tempb
+\@tempc
+\@tempcnta
+\@tempcntb
+\@tempd
+\@tempdima
+\@tempdimb
+\@tempdimc
+\@temptokena
+\@testopt
+\@totalleftmargin
+\@undefined
+\@width
+\@xp
+\@xp#1
+\DN@
+\DOTS@
+\DOTSCASE@
+\FN@
+\Invalid@@
+\Leftarrowfill@
+\Leftrightarrowfill@
+\Let@
+\Mathstrut@
+\Mathstrutbox@
+\RIfM@
+\Rightarrowfill@
+\SK@@label
+\SK@equationtrue
+\SK@tagform@
+\Umathch@
+\acc@check
+\acc@error
+\accentclass@
+\add@amp
+\add@amps
+\addto@envbody
+\addto@hook
+\align@
+\align@preamble
+\align@recover
+\aligned@a
+\alignedat@a
+\alignedspace@left
+\alignsafe@testopt
+\alignsep@
+\alignsep@-
+\alignsep@<
+\alt@tag
+\ams@DeclareRobustCommand
+\ams@def
+\ams@newcommand
+\ams@renewcommand
+\andhelp@
+\arrowfill@
+\bBigg@
+\begin@stack
+\big@size
+\bigcap@
+\bigcup@
+\bigodot@
+\bigoplus@
+\bigotimes@
+\bigsqcup@
+\biguplus@
+\bigvee@
+\bigwedge@
+\binrel@
+\binrel@@
+\black@
+\bold@false
+\bold@true
+\boldsymboldots@
+\boxz@
+\c@MaxMatrixCols
+\c@MaxMatrixColsc
+\calc@shift@align
+\calc@shift@gather
+\check@mathfonts
+\checkat@false
+\checkat@true
+\cl@@ckpt
+\classnum@
+\cmex@opt
+\collect@@body
+\collect@body
+\column@
+\column@plus
+\coprod@
+\count@
+\csname@@
+\csnamec@
+\csnamecl@
+\csnameemdf@
+\csnamem@gobble
+\csnamemb@
+\ctagsplit@false
+\ctagsplit@true
+\default@tag
+\delayed@
+\df@label
+\df@tag
+\dft@tag
+\dimen@
+\displ@y
+\displ@y@
+\displaywidth@
+\dotsb@
+\dotso@
+\dotsspace@
+\dspbrk@
+\dspbrk@context
+\dspbrk@lvl
+\dt@pfalse
+\dt@ptrue
+\emdf@L
+\emdf@La
+\emdf@R
+\emdf@Ra
+\emdf@U
+\emdf@Ua
+\endmathdisplay@a
+\endmathdisplay@fleqn
+\endmultline@math
+\env@cases
+\env@matrix
+\eqnshift@
+\every@math@size
+\ex@
+\ext@arrow
+\extra@
+\extrap@
+\f@series
+\f@size
+\field@lengths
+\fieldlengths@
+\finsm@sh
+\firstchoice@false
+\firstchoice@true
+\fracwithdelims@
+\frozen@everymath
+\gather@
+\gather@split
+\gdisplaywidth@
+\genfrac@choice
+\genfrac@rule
+\getdsp@pen
+\getmathch@
+\getmathcode@
+\gmeasure@
+\gobble@tag
+\gtest@false
+\gtest@true
+\hdots@for
+\hexnumber@
+\if@display
+\if@eqnsw
+\if@fleqn
+\ifbold@
+\ifcheckat@
+\ifctagsplit@
+\ifdt@p
+\iffirstchoice@
+\ifgtest@
+\ifinalign@
+\ifingather@
+\ifmeasuring@
+\ifshifttag@
+\ifst@rred
+\iftag@
+\iftagsleft@
+\iftagsleft@L
+\ifxxat@
+\ilimits@
+\inalign@false
+\inalign@true
+\incr@eqnum
+\inf@bad
+\ingather@false
+\ingather@true
+\insplit@
+\intdots@
+\intertext@
+\intkern@
+\ints@a
+\ints@b
+\ints@c
+\invalid@tag
+\keybin@
+\keybin@test
+\label@in@display
+\leavevmode@ifvmode
+\leftarrowfill@
+\leftrightarrowfill@
+\leftroot@
+\leftroot@mu
+\lendmultline@
+\lendsplit@
+\lineht@
+\ltx@label
+\m@gobble
+\m@ne
+\m@th
+\m@th$
+\macc@a
+\macc@adjust
+\macc@code
+\macc@depth
+\macc@group
+\macc@kerna
+\macc@kernb
+\macc@nested
+\macc@nested@a
+\macc@nucleus
+\macc@palette
+\macc@set@skewchar
+\macc@skewchar
+\macc@style
+\macc@test
+\macc@tmp
+\macro@
+\macro@@
+\make@df@tag
+\make@df@tag@@
+\make@df@tag@@@
+\make@display@tag
+\makesm@sh
+\maketag@@
+\maketag@@@
+\math@
+\math@bgroup
+\math@cr
+\math@cr@
+\math@cr@@
+\math@cr@@@
+\math@cr@@@align
+\math@cr@@@align@measure
+\math@cr@@@aligned
+\math@cr@@@alignedat
+\math@cr@@@gather
+\math@cr@@[
+\math@egroup
+\mathbin@
+\mathch@
+\mathdisplay@@pop
+\mathdisplay@@push
+\mathdisplay@pop
+\mathdisplay@push
+\mathdisplay@stack
+\mathrel@
+\mathsm@sh
+\matrix@check
+\matrix@error
+\maxcol@width
+\maxcolumn@widths
+\maxfields@
+\mb@b
+\mb@t
+\mb@tb
+\mdots@
+\mdots@@
+\meaning@
+\meaning@@
+\measure@
+\measuring@false
+\measuring@true
+\mmeasure@
+\multline@
+\multline@indent
+\new@ifnextchar
+\newmcodes@
+\newmcodes@m
+\newmcodes@o
+\next@
+\nextii@
+\nextiii@
+\nextiv@
+\nextix@
+\nextv@
+\nextvi@
+\nextvii@
+\nextviii@
+\nfss@text
+\nmlimits@
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+\operator@font
+\operator@fontlim$
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+\p@equation
+\place@tag
+\place@tag@gather
+\plainroot@
+\print@eqnum
+\process@envbody
+\prod@
+\protected@edef
+\push@begins
+\r@@t
+\raise@tag
+\relaxnext@
+\rendmultline@
+\rendsplit@
+\reset@equation
+\reset@strutbox@
+\resetMathstrut@
+\restore@hfuzz
+\restore@math@cr
+\restorealignstate@
+\restorecolumn@
+\restorecounters@
+\rightarrowfill@
+\rightdelim@
+\row@
+\savealignstate@
+\savecolumn@
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+\savefieldlength@
+\saveshift@
+\savetaglength@
+\select@group
+\set@field
+\set@gather@field
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+\setboxz@h
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+\shdots@for
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+\shifttag@false
+\shifttag@true
+\sixt@@n
+\skip@
+\slimits@
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+\split@al@a
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+\split@aligned
+\split@err
+\split@tag
+\split@warning
+\spread@equation
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+\st@rredfalse
+\st@rredtrue
+\start@align
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+\underarrow@
+\uproot@
+\use@mathgroup
+\varlim@
+\varliminf@
+\varlimsup@
+\wdz@
+\x@calc@shift@lc
+\x@calc@shift@lf
+\x@calc@shift@rc
+\x@calc@shift@rf
+\x@lcalc@width
+\x@rcalc@width
+\xatlevel@
+\xxat@true
+\z@
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--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/amsldoc.tex
@@ -0,0 +1,2770 @@
+%
+% Copyright (C) 1995, 2000, 2013 American Mathematical Society.
+% Copyright (C) 2016-2019 LaTeX3 Project and American Mathematical Society.
+%
+% This work may be distributed and/or modified under the
+% conditions of the LaTeX Project Public License, either version 1.3c
+% of this license or (at your option) any later version.
+% The latest version of this license is in
+% https://www.latex-project.org/lppl.txt
+% and version 1.3c or later is part of all distributions of LaTeX
+% version 2005/12/01 or later.
+%
+% This work has the LPPL maintenance status `maintained'.
+%
+% The Current Maintainer of this work is the LaTeX3 Project.
+%
+\documentclass[leqno,titlepage,openany]{amsldoc}[1999/12/13]
+
+\def\MaintainedByLaTeXTeam#1{%
+\begin{center}%
+\fbox{\fbox{\begin{tabular}{@{}l@{}}%
+This file is maintained by the \LaTeX{} Project team.\\%
+Bug reports can be opened (category \texttt{#1}) at\\%
+\url{https://latex-project.org/bugs/}.\end{tabular}}}\end{center}}
+
+\ifx\UndEfiNed\url
+ \ClassError{amsldoc}{%
+ This version of amsldoc.tex must be processed\MessageBreak
+ with a newer version of amsldoc.cls (2.02 or later).}{}
+\fi
+
+\renewcommand\bibname{References}
+
+% Definitions that should be moved to amsldoc.cls
+% when it is next updated
+\makeatletter
+\let\cleardouble@page\cleardoublepage
+\AtBeginDocument{%
+ \ifx\cleardouble@page\cleardoublepage
+ \def\cleardoublepage{\clearpage{\pagestyle{empty}\cleardouble@page}}
+ \fi
+}
+
+% #1 is empty or one of class, environment, option, package
+% could use different font in each case
+\makeatletter
+\def\@category@index#1#2{%
+ \autoindex{%
+ #2\actualchar\string\index#1font{#2}%
+ \ifx\@nil#1\@nil\else\space#1\fi
+ }%
+}
+
+\let\indexpackagefont\textsf
+\let\indexclassfont\textsf
+\let\indexoptionfont\texttt
+\let\indexenvironmentfont\texttt
+\let\indexfont\texttt
+\makeatother
+% For a package that won't be indexed, i.e. amsmath
+\newcommand{\nipkg}{\textsf}
+\def\tikz/{Ti\textit{k}Z}
+% End definitions for amsldoc.cls
+
+\title{User's Guide for the \nipkg{amsmath} Package (Version~2.1)}
+\author{American Mathematical Society, \LaTeX3 Project}
+\date{1999-12-13 (revised 2002-02-25, 2016-11-14, 2018-04-05)}
+\makeatletter
+\def\@thanks{\bigskip\MaintainedByLaTeXTeam{amslatex}}
+\makeatother
+
+% Use the amsmath package and amscd package in order to print
+% examples.
+\usepackage{amsmath}
+\usepackage{amscd}
+
+\usepackage{imakeidx}
+% Need alternate coding for vertical bar to get it into index.
+% Based on \qcbang and \cnbang in amsldoc.cls, refined by David Carlisle.
+% Give it a sort field that will be at the end of the symbols;
+% must avoid " and | which have special meanings.
+\DeclareRobustCommand{\qcvert}{\qc\|\index{{z1}@{\ntt\char'174}}}
+\DeclareRobustCommand{\cnvert}{%
+ \ncn{\|}\index{{z0}@{\ntt\bslchar\char'174}}}
+% Need alternate index sort field for bang to allow & to sort first
+\DeclareRobustCommand{\altcnbang}{%
+ \ncn{\!}\index{"!b@{\ntt\bslchar\qcbang}}}
+\DeclareRobustCommand{\cnamp}{%
+ \index{"!a@\texttt{\&}}}
+
+\makeindex[intoc] % generate index data
+\providecommand{\see}[2]{\textit{see} #1}
+
+% The amsldoc class includes a number of features useful for
+% documentation about TeX, including:
+%
+% ---Commands \tex/, \amstex/, \latex/, ... for uniform treatment
+% of the various logos and easy handling of following spaces.
+%
+% ---Commands for printing various common elements: \cn for command
+% names, \fn for file names (including font-file names), \env for
+% environments, \pkg and \cls for packages and classes, etc.
+
+% Many of the command names used here are rather long and will
+% contribute to poor linebreaking if we follow the \latex/ practice
+% of not hyphenating anything set in tt font; instead we selectively
+% allow some hyphenation.
+\allowtthyphens % defined in amsldoc.cls
+
+\hyphenation{ac-cent-ed-sym-bol add-to-counter add-to-length align-at
+ aligned-at allow-dis-play-breaks ams-art ams-cd ams-la-tex amsl-doc
+ ams-symb ams-tex ams-text ams-xtra bmatrix bold-sym-bol cen-ter-tags
+ eqn-ar-ray idots-int int-lim-its latex med-space neg-med-space
+ neg-thick-space neg-thin-space no-int-lim-its no-name-lim-its
+ over-left-arrow over-left-right-arrow over-right-arrow pmatrix
+ qed-sym-bol set-length side-set small-er tbinom the-equa-tion
+ thick-space thin-space un-der-left-arrow un-der-left-right-arrow
+ un-der-right-arrow use-pack-age var-inj-lim var-proj-lim vmatrix
+ xalign-at xx-align-at}
+
+% Prepare for illustrating the \vec example
+\newcommand{\vect}[1]{\mathbf{#1}}
+
+\newcommand{\booktitle}[1]{\textit{#1}}
+\newcommand{\journalname}[1]{\textit{#1}}
+\newcommand{\seriesname}[1]{\textit{#1}}
+
+% Command to insert and index a particular phrase. Doesn't work for
+% certain kinds of special characters in the argument.
+\newcommand{\ii}[1]{#1\index{#1}}
+
+\newcommand{\vstrut}[1]{\vrule width0pt height#1\relax}
+
+% An environment for presenting comprehensive address information:
+\newenvironment{infoaddress}{%
+ \par\topsep\medskipamount
+ \trivlist\centering
+ \item[]%
+ \begin{minipage}{.7\columnwidth}%
+ \raggedright
+}{%
+ \end{minipage}%
+ \endtrivlist
+}
+
+\newenvironment{eqxample}{%
+ \par\addvspace\medskipamount
+ \noindent\begin{minipage}{.5\columnwidth}%
+ \def\producing{\end{minipage}\begin{minipage}{.5\columnwidth}%
+ \hbox\bgroup\kern-.2pt\vrule width.2pt%
+ \vbox\bgroup\parindent0pt\relax
+% The 3pt is to cancel the -\lineskip from \displ@y
+ \abovedisplayskip3pt \abovedisplayshortskip\abovedisplayskip
+ \belowdisplayskip0pt \belowdisplayshortskip\belowdisplayskip
+ \noindent}
+}{%
+ \par
+% Ensure that a lonely \[\] structure doesn't take up width less than
+% \hsize.
+ \hrule height0pt width\hsize
+ \egroup\vrule width.2pt\kern-.2pt\egroup
+ \end{minipage}%
+ \par\addvspace\medskipamount
+}
+
+% Errors in output shouldn't be \texttt, and they may also be longer
+% than one line, but they should be identified by the "errorbullet".
+% (They were formerly labeled as aubsections.)
+\newenvironment{erroro}[1]{%
+ \par\addvspace\medskipamount
+ \noindent\hangindent\parindent
+ \hbox to\parindent{\errorbullet\hfil}\ignorespaces
+ #1\par\vspace{\smallskipamount}
+}{%
+ \par\addvspace\medskipamount
+}
+
+% The chapters are so short, perhaps we shouldn't call them by the
+% name `Chapter'. We make \chaptername read an argument in order to
+% remove a following \space or "{} " (both possibilities are present
+% in book.cls).
+
+\renewcommand{\chaptername}[1]{}
+\newcommand{\chapnum}[1]{\mdash #1\mdash }
+\makeatletter
+\def\@makechapterhead#1{%
+ \vspace{1.5\baselineskip}%
+ {\parindent \z@ \raggedright \reset@font
+ \ifnum \c@secnumdepth >\m@ne
+ \large\bfseries \chapnum{\thechapter}%
+ \par\nobreak
+ \vskip.5\baselineskip\relax
+ \fi
+ #1\par\nobreak
+ \vskip\baselineskip
+ }}
+\makeatother
+
+% A command for ragged-right parbox in a tabular.
+\newcommand{\rp}{\let\PBS\\\raggedright\let\\\PBS}
+
+% Non-indexed file name
+\newcommand{\nfn}[1]{\texttt{#1}}
+
+% For the examples in the math spacing table.
+%%\newcommand{\lspx}{\mbox{\rule{5pt}{.6pt}\rule{.6pt}{6pt}}}
+%%\newcommand{\rspx}{\mbox{\rule[-1pt]{.6pt}{7pt}%
+%% \rule[-1pt]{5pt}{.6pt}}}
+\newcommand{\lspx}{\mathord{\Rightarrow\mkern-1mu}}
+\newcommand{\rspx}{\mathord{\mkern-1mu\Leftarrow}}
+\newcommand{\spx}[1]{$\lspx #1\rspx$}
+
+% For a list of characters representing document input.
+\newcommand{\clist}[1]{%
+ \mbox{\ntt\spaceskip.2em plus.1em \xspaceskip\spaceskip#1}}
+
+% Fix weird \latex/ definition of rightmark.
+\makeatletter
+\def\rightmark{\expandafter\@rightmark\botmark{}{}}
+% Also turn off section marks.
+\let\sectionmark\@gobble
+\renewcommand{\chaptermark}[1]{%
+ \uppercase{\markboth{\rhcn#1}{\rhcn#1}}}
+\newcommand{\rhcn}{\thechapter. }
+\makeatother
+
+% Include down to \section but not \subsection, in toc:
+\setcounter{tocdepth}{1}
+
+% Subheadings for the bibliography
+\newif\iffirstbibsubhead \firstbibsubheadtrue
+\newcommand{\bibsubhead}[1]{%
+ \iffirstbibsubhead \firstbibsubheadfalse
+ \else \addvspace{\medskipamount}
+ \fi
+ \item[]\hspace*{-\leftmargin}\textbf{#1}\par
+ \vspace{\smallskipamount}
+}
+
+\DeclareMathOperator{\ix}{ix}
+\DeclareMathOperator{\nul}{nul}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\xxx}{xxx}
+
+\usepackage{hyperref}
+\hypersetup{pdfborder=0 0 0}
+\makeatletter
+\let\oldcs\cs
+\def\cs#1{\texorpdfstring{\oldcs{#1}}{\@backslashchar\@backslashchar#1}}
+\let\cn\cs
+\makeatother
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\frontmatter
+
+\maketitle
+
+\pagestyle{headings}
+\tableofcontents
+\cleardoublepage % for better page number placement
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mainmatter
+\chapter{Introduction}
+
+\index{amsmath@\texttt{amsmath} package|(}
+
+The \nipkg{amsmath} package is a \LaTeX{} package that provides
+miscellaneous enhancements for improving the information structure and
+printed output of documents that contain mathematical formulas. Readers
+unfamiliar with \LaTeX{} should refer to \cite{lamport}. If you have an
+up-to-date version of \LaTeX{}, the \nipkg{amsmath} package is normally
+provided along with it. Upgrading when a newer version of the
+\nipkg{amsmath} package is released can be done via
+\url{http://mirror.ctan.org/macros/latex/required/amsmath.zip}.
+
+This documentation describes the features of the \nipkg{amsmath} package
+and discusses how they are intended to be used. It also covers some
+ancillary packages:
+\begin{ctab}{lll}
+\pkg{amsbsy}& \pkg{amsopn}& \pkg{amsxtra}\\
+\pkg{amscd}& \pkg{amstext}\\
+\end{ctab}
+These all have something to do with the contents of math formulas. For
+information on extra math symbols and math fonts, see \cite{amsfonts}
+and \url{https://www.ams.org/tex/amsfonts.html}. For documentation of the
+\pkg{amsthm} package or AMS document classes (\cls{amsart},
+\cls{amsbook}, etc.\@) see \cite{amsthdoc} or \cite{amshandbk} and
+\url{https://www.ams.org/tex/author-info.html}.
+
+If you are a long-time \latex/ user and have lots of mathematics in what
+you write, then you may recognize solutions for some familiar problems
+in this list of \nipkg{amsmath} features:
+\begin{itemize}
+
+\item A convenient way to define new `operator name' commands analogous
+to \cn{sin} and \cn{lim}, including proper side spacing and automatic
+selection of the correct font style and size (even when used in
+sub- or superscripts).
+
+\item Multiple substitutes for the \env{eqnarray} environment to make
+various kinds of equation arrangements easier to write.
+
+\item Equation numbers automatically adjust up or down to avoid
+overprinting on the equation contents (unlike \env{eqnarray}).
+
+\item Spacing around equals signs matches the normal spacing in the
+\env{equation} environment (unlike \env{eqnarray}).
+
+\item A way to produce multiline subscripts as are often used with
+summation or product symbols.
+
+\item An easy way to substitute a variant equation number for a given
+equation instead of the automatically supplied number.
+
+\item An easy way to produce subordinate equation numbers of the form
+(1.3a) (1.3b) (1.3c) for selected groups of equations.
+
+\end{itemize}
+
+The \nipkg{amsmath} package is distributed together with some small
+auxiliary packages:
+\begin{description}
+\item[\nipkg{amsmath}] Primary package, provides various features for
+ displayed equations and other mathematical constructs.
+
+\item[\pkg{amstext}] Provides a \cn{text} command for
+ typesetting a fragment of text inside a display.
+
+\item[\pkg{amsopn}] Provides \cn{DeclareMathOperator} for defining new
+ `operator names' like \cn{sin} and \cn{lim}.
+
+\item[\pkg{amsbsy}] For backward compatibility this package continues
+to exist but use of the newer \pkg{bm} package that comes with \LaTeX{}
+is recommended instead.
+
+\item[\pkg{amscd}] Provides a \env{CD} environment for simple
+ commutative diagrams (no support for diagonal arrows).
+
+\item[\pkg{amsxtra}] Provides certain odds and ends such as
+ \cn{fracwithdelims} and \cn{accentedsymbol}, for compatibility with
+ documents created using version 1.1.
+
+\end{description}
+
+The \nipkg{amsmath} package incorporates \pkg{amstext}, \pkg{amsopn}, and
+\pkg{amsbsy}. The features of \pkg{amscd} and \pkg{amsxtra}, however,
+are available only by invoking those packages separately.
+
+The independent \pkg{mathtools} package \cite{mt} provides some
+enhancements to \nipkg{amsmath}; \pkg{mathtools} loads \nipkg{amsmath}
+automatically, so there is no need to separately load \nipkg{amsmath}
+if \pkg{mathtools} is used. Some \pkg{mathtools} facilities
+will be noted below as appropriate.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Options for the \nipkg{amsmath} package}\label{options}
+
+The \nipkg{amsmath} package has the following options:
+\index{options!amsmath@\texttt{amsmath} package options|(}
+\begin{description}
+
+\item[\opt{centertags}] (default) For an equation containing a
+\env{split} environment, place equation
+numbers\index{equationn@equation numbers!vertical placement}
+vertically centered on the total height of the equation.
+
+\item[\opt{tbtags}] `Top-or-bottom tags': For an equation containing
+a \env{split} environment, place equation
+numbers\index{equationn@equation numbers!vertical placement} level with the last
+(resp.\ first) line, if numbers are on the right (resp.\ left).
+
+\item[\opt{sumlimits}] (default) Place the subscripts and
+superscripts\index{subscripts and superscripts!placement}\relax
+\index{limits|see{subscripts and superscripts}} of summation symbols
+above and below, in displayed equations. This option also affects other
+symbols of the same type\mdash $\prod$, $\coprod$, $\bigotimes$,
+$\bigoplus$, and so forth\mdash but excluding integrals (see below).
+
+\item[\opt{nosumlimits}] Always place the subscripts and superscripts of
+summation-type symbols to the side, even in displayed equations.
+
+\item[\opt{intlimits}] Like \opt{sumlimits}, but for
+integral\index{integrals!placement of limits} symbols.
+
+\item[\opt{nointlimits}] (default) Opposite of \opt{intlimits}.
+
+\item[\opt{namelimits}] (default) Like \opt{sumlimits}, but for certain
+`operator names' such as $\det$, $\inf$, $\lim$, $\max$, $\min$, that
+traditionally have subscripts%
+\index{subscripts and superscripts!placement} placed underneath when
+they occur in a displayed equation.
+
+\item[\opt{nonamelimits}] Opposite of \opt{namelimits}.
+
+\end{description}
+
+\goodbreak
+
+\begin{description}
+\item[\opt{alignedleftspaceyes}]
+\item[\opt{alignedleftspaceno}]
+\item[\opt{alignedleftspaceyesifneg}]
+\end{description}
+
+To use one of these package options, put the option name in the optional
+argument of the \cn{usepackage} command\mdash e.g.,
+\verb"\usepackage[intlimits]{amsmath}".
+For AMS document classes and any other classes that preload \nipkg{amsmath}
+desired options must be specified with the \cn{documentclass}\mdash
+e.g.,\\
+\verb+\documentclass[intlimits,tbtags,reqno]{amsart}+.
+
+The \nipkg{amsmath} package also recognizes the following options which
+are normally selected (implicitly or explicitly) through the
+\cn{documentclass} command, and thus need not be repeated in the option
+list of the \cn{usepackage}|{amsmath}| statement.
+\begin{description}
+
+\item[\opt{leqno}] Place equation numbers on the left.%
+\index{equationn@equation numbers!left or right placement}
+
+\item[\opt{reqno}] Place equation numbers on the right.
+
+\item[\opt{fleqn}] Position equations at a fixed indent from the left
+margin rather than centered in the text column.%
+\index{displayed equations!centering}
+
+\end{description}
+
+Three options have been added to control the space to the left of
+\env{aligned} and \env{gathered} environments. Prior to the 2017 release
+a thin space was added to the left but not the right of these constructs.
+This appears to have been an accidental feature of the implementation and
+was typically corrected by prefixing the environments with \altcnbang.
+
+The new default behavior is aimed to ensure that the environments do
+not have a thin space added in most cases, and that existing documents
+using |\!\begin{aligned}| continue to work as before.
+
+\begin{description}
+\item[\opt{alignedleftspaceyes}] Always add \cn{\,} to the left of \env{aligned} and \env{gathered}.
+\item[\opt{alignedleftspaceno}] Never add \cn{\,} to the left of \env{aligned} and \env{gathered}.
+\item[\opt{alignedleftspaceyesifneg}] Only add \cn{\,} if the environment is prefixed by negative space. (New default behavior.)
+\end{description}
+
+\index{options!amsmath@\texttt{amsmath} package options|)}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Displayed equations}
+
+\section{Introduction}
+The \nipkg{amsmath} package provides a number of additional displayed
+equation structures\index{displayed equations}%
+\index{equations|see{displayed equations}} beyond the ones
+provided in basic \latex/. The augmented set includes:
+\begin{verbatim}
+ equation equation* align align*
+ gather gather* alignat alignat*
+ multline multline* flalign flalign*
+ split
+\end{verbatim}
+(Although the standard \env{eqnarray} environment remains available,
+it is better to use \env{align} or \env{equation}+\env{split} instead.
+Within \env{eqnarray}, spacing\index{horizontal spacing} around signs
+of relation is not the preferred mathematical spacing, and is
+inconsistent with that spacing as it appears in other environments.
+Long lines in this environment may result in misplaced or overprinted
+equation numbers. This environment also does not support the use of
+\cn{qed} or \cn{qedhere} as provided by theorem packages.)
+
+Except for \env{split}, each environment has both starred and unstarred
+forms, where the unstarred forms have automatic numbering using
+\latex/'s \env{equation} counter. You can suppress the number on any
+particular line by putting \cn{notag} before the end of that line;
+\cn{notag} should not be used outside a display environment as it will
+mess up the numbering. You can also
+override\index{equationn@equation numbers!overriding} a number with a tag
+of your own using \cn{tag}|{|\<label>|}|, where \<label> means arbitrary
+text such as |$*$| or |ii| used to \qq{number} the equation.
+A tag can reference a different tagged display by use of
+|\tag{\ref{|\<label>|}|\<modifier>|}| where \<modifier> is optional.
+If you are using \pkg{hyperref}, use \cn{ref*};
+use of the starred form of \cn{ref} prevents a reference to a modified
+tag containing a nested link from linking to the original display.
+
+There is also a
+\cn{tag*} command that causes the text you supply to be typeset
+literally, without adding parentheses around it. \cn{tag} and \cn{tag*}
+can also be used within the unnumbered versions of all the \nipkg{amsmath}
+alignment structures. Some examples of the use of \cn{tag} may be found
+in the sample files \fn{testmath.tex} and \fn{subeqn.tex} provided with
+the \nipkg{amsmath} package.
+
+The \env{split} environment is a special subordinate form that is used
+only \emph{inside} one of the others. It cannot be used inside
+\env{multline}, however. \env{split} supports only one alignment (|&|)
+column; if more are needed, \env{aligned} or \env{alignedat} should be
+used. The width of a \env{split} structure is the full line width.
+
+In the structures that do alignment (\env{split}, \env{align} and
+variants), relation symbols have an |&| before them but not
+after\mdash unlike \env{eqnarray}. Putting the |&| after the
+relation symbol will interfere with the normal spacing; it has to go
+before.
+
+In all multiline environments, lines are divided by \cn{\\}. The \cn{\\}
+should \emph{not} be used to end the last line. Using it there will
+result in unwanted extra vertical space following the display.
+
+In \emph{all} math environments (inline or display), blank lines
+(equivalent to \cn{par}) are not permitted, and will result in an error.
+
+\begin{table}[p]
+\caption[]{Comparison of displayed equation environments (vertical lines
+indicating nominal margins)}\label{displays}
+\renewcommand{\theequation}{\arabic{equation}}
+\begin{eqxample}
+\begin{verbatim}
+\begin{equation*}
+a=b
+\end{equation*}
+\end{verbatim}
+\producing
+\begin{equation*}
+a=b
+\end{equation*}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{equation}
+a=b
+\end{equation}
+\end{verbatim}
+\producing
+\begin{equation}
+a=b
+\end{equation}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{equation}\label{xx}
+\begin{split}
+a& =b+c-d\\
+ & \quad +e-f\\
+ & =g+h\\
+ & =i
+\end{split}
+\end{equation}
+\end{verbatim}
+\producing
+\begin{equation}\label{xx}
+\begin{split}
+a& =b+c-d\\
+ & \quad +e-f\\
+ & =g+h\\
+ & =i
+\end{split}
+\end{equation}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{multline}
+a+b+c+d+e+f\\
++i+j+k+l+m+n
+\end{multline}
+\end{verbatim}
+\producing
+\begin{multline}
+a+b+c+d+e+f\\
++i+j+k+l+m+n
+\end{multline}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{gather}
+a_1=b_1+c_1\\
+a_2=b_2+c_2-d_2+e_2
+\end{gather}
+\end{verbatim}
+\producing
+\begin{gather}
+a_1=b_1+c_1\\
+a_2=b_2+c_2-d_2+e_2
+\end{gather}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{align}
+a_1& =b_1+c_1\\
+a_2& =b_2+c_2-d_2+e_2
+\end{align}
+\end{verbatim}
+\producing
+\begin{align}
+a_1& =b_1+c_1\\
+a_2& =b_2+c_2-d_2+e_2
+\end{align}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{align}
+a_{11}& =b_{11}&
+ a_{12}& =b_{12}\\
+a_{21}& =b_{21}&
+ a_{22}& =b_{22}+c_{22}
+\end{align}
+\end{verbatim}
+\producing
+\begin{align}
+a_{11}& =b_{11}&
+ a_{12}& =b_{12}\\
+a_{21}& =b_{21}&
+ a_{22}& =b_{22}+c_{22}
+\end{align}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{flalign*}
+a_{11} + b_{11}& = c_{11}&
+ a_{12}& =b_{12}\\
+b_{21}& = c_{21}&
+ a_{22}& =b_{22}+c_{22}
+\end{flalign*}
+\end{verbatim}
+\producing
+\begin{flalign*}
+a_{11} + b_{11}& = c_{11}&
+ a_{12}& =b_{12}\\
+b_{21}& = c_{21}&
+ a_{22}& =b_{22}+c_{22}
+\end{flalign*}
+\end{eqxample}
+\end{table}
+
+%% ------------------------------------------------------------------ %%
+\enlargethispage{1\baselineskip}
+
+\section{Single equations}
+
+The \env{equation} environment is for a single equation with an
+automatically generated number. The \env{equation*} environment is the
+same except for omitting the number.%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\footnote{Basic \latex/ doesn't provide an \env{equation*} environment,
+but rather a functionally equivalent environment named
+\env{displaymath}.}
+The wrapper \verb+\[ ... \]+ is equivalent to \env{equation*}.%
+\index{[@\verb+\[ ... \]+}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Split equations without alignment}
+
+The \env{multline} environment is a variation of the \env{equation}
+environment used for equations that don't fit on a single line. The
+first line of a \env{multline} will be at the left margin and the last
+line at the right margin, except for an indention on both sides in the
+amount of \cn{multlinegap}. Any additional lines in between will be
+centered independently within the display width (unless the \opt{fleqn}
+option is in effect).%
+\index{options!behavior of particular options}
+
+Like \env{equation}, \env{multline} has only a single equation number
+(thus, none of the individual lines should be marked with \cn{notag}).
+The equation number is placed on the last line (\opt{reqno} option) or
+first line (\opt{leqno} option); vertical centering as for \env{split}
+is not supported by \env{multline}.%
+\index{options!behavior of particular options}
+
+It's possible to force one of the middle lines to the left or right with
+commands \cn{shoveleft}, \cn{shoveright}. These commands take the entire
+line as an argument, up to but not including the final \cn{\\}; for
+example
+\begin{multline}
+\framebox[.65\columnwidth]{A}\\
+\framebox[.5\columnwidth]{B}\\
+\shoveright{\framebox[.55\columnwidth]{C}}\\
+\framebox[.65\columnwidth]{D}
+\end{multline}
+\begin{verbatim}
+\begin{multline}
+\framebox[.65\columnwidth]{A}\\
+\framebox[.5\columnwidth]{B}\\
+\shoveright{\framebox[.55\columnwidth]{C}}\\
+\framebox[.65\columnwidth]{D}
+\end{multline}
+\end{verbatim}
+
+The value of \cn{multlinegap} can be changed with the usual \latex/
+commands \cn{setlength} or \cn{addtolength}.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Split equations with alignment}
+
+Like \env{multline}, the \env{split} environment is for \emph{single}
+equations that are too long to fit on one line and hence must be split
+into multiple lines. Unlike \env{multline}, however, the \env{split}
+environment provides for alignment among the split lines, using |&| to
+mark alignment points. Unlike the other \nipkg{amsmath} equation
+structures, the \env{split} environment provides no numbering, because
+it is intended to be used \emph{only inside some other displayed
+ equation structure}, usually an \env{equation}, \env{align}, or
+\env{gather} environment, which provides the numbering. For example:
+\begin{equation}\label{e:barwq}\begin{split}
+H_c&=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}
+ \sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\
+&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
+ \Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].
+ \kern-2em % adjust equation body to the right [mjd,13-Nov-1994]
+\end{split}\end{equation}
+
+\begin{verbatim}
+\begin{equation}\label{e:barwq}\begin{split}
+H_c&=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}
+ \sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\
+&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
+ \Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].
+\end{split}\end{equation}
+\end{verbatim}
+
+The \env{split} structure should constitute the entire body of the
+enclosing structure, apart from commands like \cn{label} that produce no
+visible material.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Equation groups without alignment}
+
+The \env{gather} environment is used for a group of consecutive
+equations when there is no alignment desired among them; each one is
+centered separately within the text width (see Table~\ref{displays}).
+Equations inside \env{gather} are separated by a \cn{\\} command.
+Any equation in a \env{gather} may consist of a \verb'\begin{split}'
+ \dots\ \verb'\end{split}' structure\mdash for example:
+\begin{verbatim}
+\begin{gather}
+ first equation\\
+ \begin{split}
+ second & equation\\
+ & on two lines
+ \end{split}
+ \\
+ third equation
+\end{gather}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Equation groups with mutual alignment}
+
+The \env{align} environment is used for two or more equations when
+vertical alignment is desired; usually binary relations such as equal
+signs are aligned (see Table~\ref{displays}).
+
+To have several equation columns side-by-side, use extra ampersands
+to separate the columns:
+\begin{align}
+x&=y & X&=Y & a&=b+c\\
+x'&=y' & X'&=Y' & a'&=b\\
+x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b
+\end{align}
+%
+\begin{verbatim}
+\begin{align}
+x&=y & X&=Y & a&=b+c\\
+x'&=y' & X'&=Y' & a'&=b\\
+x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b
+\end{align}
+\end{verbatim}
+Line-by-line annotations on an equation can be done by judicious
+application of \cn{text} inside an \env{align} environment:
+\begin{align}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+ && \text{by \eqref{eq:C}}\\
+ & = y'\circ y^* && \text{by \eqref{eq:D}}\\
+ & = y(0) y' && \text {by Axiom 1.}
+\end{align}
+%
+\begin{verbatim}
+\begin{align}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+ && \text{by \eqref{eq:C}}\\
+ & = y'\circ y^* && \text{by \eqref{eq:D}}\\
+ & = y(0) y' && \text {by Axiom 1.}
+\end{align}
+\end{verbatim}
+A variant environment \env{alignat} allows the horizontal space between
+equations to be explicitly specified. This environment takes one argument,
+the number of \qq{equation columns} (the number of pairs of right-left
+aligned columns; the argument is the number of pairs): count the maximum
+number of \verb'&'s in any row, add~1 and divide by~2.
+\begin{alignat}{2}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+ &\quad& \text{by \eqref{eq:C}}\\
+ & = y'\circ y^* && \text{by \eqref{eq:D}}\\
+ & = y(0) y' && \text {by Axiom 1.}
+\end{alignat}
+%
+\begin{verbatim}
+\begin{alignat}{2}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+ &\quad& \text{by \eqref{eq:C}}\\
+ & = y'\circ y^* && \text{by \eqref{eq:D}}\\
+ & = y(0) y' && \text {by Axiom 1.}
+\end{alignat}
+\end{verbatim}
+The environment \env{flalign} (``full length alignment'') stretches the
+space between the equation columns to the maximum possible width, leaving
+only enough space at the margin for the equation number, if present.
+\begin{flalign}
+x&=y & X&=Y\\
+x'&=y' & X'&=Y'\\
+x+x'&=y+y' & X+X'&=Y+Y'
+\end{flalign}
+%
+\begin{verbatim}
+\begin{flalign}
+x&=y & X&=Y\\
+x'&=y' & X'&=Y'\\
+x+x'&=y+y' & X+X'&=Y+Y'
+\end{flalign}
+\end{verbatim}
+%
+\begin{flalign*}
+x&=y & X&=Y\\
+x'&=y' & X'&=Y'\\
+x+x'&=y+y' & X+X'&=Y+Y'
+\end{flalign*}
+%
+\begin{verbatim}
+\begin{flalign*}
+x&=y & X&=Y\\
+x'&=y' & X'&=Y'\\
+x+x'&=y+y' & X+X'&=Y+Y'
+\end{flalign*}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Alignment building blocks}
+
+Like \env{equation}, the multi-equation environments \env{gather},
+\env{align}, and \env{alignat} are designed to produce a structure
+whose width is the full line width. This means, for example, that one
+cannot readily add parentheses around the entire structure. But
+variants\index{ed env@\texttt{-ed} environments|(}
+\env{gathered}, \env{aligned}, and \env{alignedat} are provided whose
+total width is the actual width of the contents; thus they can be used
+as a component in a containing expression. E.g.,
+\begin{equation*}
+\left.\begin{aligned}
+ B'&=-\partial\times E,\\
+ E'&=\partial\times B - 4\pi j,
+\end{aligned}
+\right\}
+\qquad \text{Maxwell's equations}
+\end{equation*}
+\begin{verbatim}
+\begin{equation*}
+\left.\begin{aligned}
+ B'&=-\partial\times E,\\
+ E'&=\partial\times B - 4\pi j,
+\end{aligned}
+\right\}
+\qquad \text{Maxwell's equations}
+\end{equation*}
+\end{verbatim}
+Like the \env{array} environment, these \texttt{-ed} variants also take
+an optional\index{options!positioning of \texttt{-ed} environments}
+|[t]|\index{t option@\texttt{t} (top) option},
+|[b]|\index{b option@\texttt{b} (bottom) option} or the default
+|[c]|\index{c option@\texttt{c} (center) option}
+argument to specify vertical
+positioning. For maximum interoperability, do not insert a
+space\index{options!space before \texttt{[}}
+or line break\index{line break} before the option. See
+also the note on page~\pageref{breaknote} regarding page breaking
+within the \texttt{-ed} environments.
+
+\qq{Cases} constructions like the following are common in
+mathematics:
+\begin{equation}\label{eq:C}
+P_{r-j}=
+ \begin{cases}
+ 0& \text{if $r-j$ is odd},\\
+ r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
+ \end{cases}
+\end{equation}
+and in the \nipkg{amsmath} package there is a \env{cases} environment to
+make them easy to write:
+\begin{verbatim}
+P_{r-j}=\begin{cases}
+ 0& \text{if $r-j$ is odd},\\
+ r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
+ \end{cases}
+\end{verbatim}
+Notice the use of \cn{text} (cf.~\secref{text}) and the nested math
+formulas. \env{cases} is set in \cn{textstyle}. If \cn{displaystyle}
+is wanted, it must be requested explicitly; \pkg{mathtools} provides
+a \env{dcases} environment for this purpose.
+
+The |-ed|\index{ed env@\texttt{-ed} environments|)} and \env{cases}
+environments must appear within an enclosing math environment, which can
+be either in text, between |$...$|, or in any of the display environments.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Adjusting tag placement}
+
+Placing equation numbers can be a rather complex problem in multiline
+displays. The environments of the \nipkg{amsmath} package try hard to
+avoid overprinting an equation number on the equation contents, if
+necessary moving the number down or up to a separate line. Difficulties
+in accurately calculating the profile of an equation can occasionally
+result in number movement that doesn't look right.
+A \cn{raisetag} command is provided to adjust the vertical position of the
+current equation number, if it has been shifted away from its normal
+position. To move a particular number up by six points, write
+|\raisetag{6pt}|. (At the end of a display, this also shifts up the
+text following the display.)
+This kind of adjustment is fine tuning like line
+breaks and page breaks, and should therefore be left undone until your
+document is nearly finalized, or you may end up redoing the fine tuning
+several times to keep up with changing document contents.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Vertical spacing and page breaks in multiline displays}
+
+You can use the \cn{\\}|[|\<dimension>|]|%
+\index{options!extra vertical space after \cn{\\}}
+command to get extra vertical
+space between lines in all the \nipkg{amsmath} displayed equation
+environments, as is usual in \latex/. Do not type a space between the%
+\index{options!space before \texttt{[}}
+\cn{\\} and the following |[|; \emph{only} for display environments
+defined by \nipkg{amsmath} the space is interpreted to mean that the
+bracketed material is part of the document content.
+
+When the \nipkg{amsmath} package is
+in use \ii{page breaks} between equation lines are normally disallowed;
+the philosophy is that page breaks in such material should receive
+individual attention from the author. To get an individual page break
+inside a particular displayed equation, a \cn{displaybreak} command is
+provided. \cn{displaybreak} is best placed immediately before the
+\cn{\\} where it is to take effect. Like \latex/'s \cn{pagebreak},
+\cn{displaybreak} takes an optional argument%
+\index{options!behavior of particular options} between 0 and 4 denoting
+the desirability of the pagebreak. |\displaybreak[0]| means \qq{it is
+ permissible to break here} without encouraging a break;
+\cn{displaybreak} with no optional argument is the same as
+|\displaybreak[4]| and forces a break.
+
+If you prefer a strategy of letting page breaks fall where they may,
+even in the middle of a multiline equation, then you might put
+\cn{allowdisplaybreaks}\texttt{[1]} in the preamble of your document. An
+optional argument\index{options!behavior of particular options}
+1\ndash 4 can be used for finer control: |[1]| means
+allow page breaks, but avoid them as much as possible; values of 2,3,4
+mean increasing permissiveness. When display breaks are enabled with
+\cn{allowdisplaybreaks}, the \cn{\\*} command can be used to prohibit a
+pagebreak after a given line, as usual.
+
+\begin{bfseries}
+Note:\label{breaknote} Certain equation environments wrap their contents
+in an unbreakable box, with the consequence that neither \cn{displaybreak}
+nor \cn{allowdisplaybreaks} will have any effect on them. These include
+\env{split}, \env{aligned}, \env{gathered}, and \env{alignedat}.
+\end{bfseries}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Interrupting a display}
+
+The command \cn{intertext} is used for a short interjection of one or
+two lines of text\index{text fragments inside math} in the middle of a
+multiple-line display structure (see also the \cn{text} command in
+\secref{text}). Its salient feature is preservation of the alignment,
+which would not happen if you simply ended the display and then started
+it up again afterwards. \cn{intertext} may only appear right after a
+\cn{\\} or \cn{\\*} command. Notice the position of the word \qq{and} in
+this example.
+\begin{align}
+ A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\
+ A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\
+\intertext{and}
+ A_3&=\mathcal{N}(\lambda;\omega).
+\end{align}
+\begin{verbatim}
+\begin{align}
+ A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\
+ A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\
+\intertext{and}
+ A_3&=\mathcal{N}(\lambda;\omega).
+\end{align}
+\end{verbatim}
+
+The \pkg{mathtools} package provides a command \cn{shortintertext} that
+is intended for use when the interjected text is only a few words; it uses
+less vertical space than \cn{intertext}. This
+is most effective when equation numbers are on the right.%
+\index{equationn@equation numbers!left or right placement}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Equation numbering}
+
+\subsection{Numbering hierarchy}
+In \latex/ if you wanted to have equations numbered within
+sections\mdash that is, have
+equation numbers (1.1), (1.2), \dots, (2.1), (2.2),
+\dots, in sections 1, 2, and so forth\mdash you could redefine
+\cn{theequation} as suggested in the \latex/ manual \cite[\S6.3, \S
+C.8.4]{lamport}:
+\begin{verbatim}
+\renewcommand{\theequation}{\thesection.\arabic{equation}}
+\end{verbatim}
+
+This works pretty well, except that the equation counter won't be reset
+to zero at the beginning of a new section or chapter, unless you do it
+yourself using \cn{setcounter}. To make this a little more convenient,
+the \nipkg{amsmath} package provides a
+command\index{equationn@equation numbers!hierarchy} \cn{numberwithin}.
+To have equation numbering tied to section numbering, with automatic
+reset of the equation counter, write
+\begin{verbatim}
+\numberwithin{equation}{section}
+\end{verbatim}
+As its name implies, the \cn{numberwithin} command can be applied to
+any counter, not just the \texttt{equation} counter.
+
+\subsection{Cross references to equation numbers}
+
+To make cross-references to equations easier, an \cn{eqref}
+command\index{equationn@equation numbers!cross-references} is provided. This
+automatically supplies the parentheses around the equation number. I.e.,
+if \verb'\ref{abc}' produces 3.2 then \verb'\eqref{abc}' produces
+(3.2). The parentheses around an \cn{eqref} equation number will be
+set in upright type regardless of type style of the context.
+
+\subsection{Subordinate numbering sequences}
+
+The \nipkg{amsmath} package provides also a wrapper environment,
+\env{subequations}\index{equationn@equation numbers!subordinate numbering}, to make it
+easy to number equations in a particular \cn{align} or similar group with
+a subordinate numbering scheme. For example
+\begin{verbatim}
+\begin{subequations}
+...
+\end{subequations}
+\end{verbatim}
+causes all numbered equations within that part of the document to be
+numbered (4.9a) (4.9b) (4.9c) \dots, if the preceding numbered
+equation was (4.8). A \cn{label} command immediately after
+\verb/\begin{subequations}/ will produce a \cn{ref} of the parent
+number 4.9, not 4.9a. The counters used by the subequations
+environment are \verb/parentequation/ and \verb/equation/ and
+\cn{addtocounter}, \cn{setcounter}, \cn{value}, etc., can be applied
+as usual to those counter names. To get anything other than lowercase
+letters for the subordinate numbers, use standard \latex/ methods for
+changing numbering style \cite[\S6.3, \S C.8.4]{lamport}. For example,
+redefining \cn{theequation} as follows will produce roman numerals.
+\begin{verbatim}
+\begin{subequations}
+\renewcommand{\theequation}{\theparentequation \roman{equation}}
+...
+\end{verbatim}
+
+\subsection{Numbering style}
+
+The default equation number is set in \cn{normalfont}. This means that
+in bold section headings, bold is suppressed; a workaround for this is
+to use \verb+(\ref{+\ldots\verb+})+ rather than \verb+\eqref{+\ldots\verb+}+.
+
+If smaller type is specified for a numbered display, the size of the
+equation number will also be small. The default size can be ensured
+throughout a document by applying this patch in the preamble:
+\begin{verbatim}
+\makeatletter
+\renewcommand{\maketag@@@}[1]{\hbox{\m@th\normalsize\normalfont#1}}%
+\makeatother
+\end{verbatim}
+(This modification may be included in a future version of
+\nipkg{amsmath}.)
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Miscellaneous mathematical features}
+
+\section{Matrices}\label{ss:matrix}
+
+The \nipkg{amsmath} package provides some environments for
+matrices\index{matrices} beyond the basic \env{array} environment of
+\latex/. The \env{pmatrix}, \env{bmatrix}, \env{Bmatrix}, \env{vmatrix}
+and \env{Vmatrix} have (respectively) $(\,)$, $[\,]$, $\lbrace\,\rbrace$,
+$\lvert\,\rvert$, and $\lVert\,\rVert$ delimiters\index{delimiters}
+built in. For naming consistency there is a
+\env{matrix} environment sans delimiters. This is not entirely redundant
+with the \env{array} environment; the matrix environments all use more
+economical horizontal spacing than the rather prodigal spacing of the
+\env{array} environment. Also, unlike the \env{array} environment, you
+don't have to give column specifications for any of the matrix
+environments; by default you can have up to 10 centered columns.%
+\footnote{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+More precisely: The maximum number of columns in a matrix is determined
+by the counter |MaxMatrixCols| (normal value = 10), which you can change
+if necessary using \latex/'s \cn{setcounter} or \cn{addtocounter}
+commands.%
+}\space%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+(If you need left or right alignment in a column or other special
+formats you may use \env{array}, or the \pkg{mathtools}
+package which provides \verb+*+ variants%
+\index{matrix@\indexenvironmentfont{matrix} environment!\verb+*+ variants}
+of these environments with
+an optional argument to specify left or right alignment.)
+
+To produce a small matrix suitable for use in text, there is a
+\env{smallmatrix} environment (e.g.,
+\begin{math}
+\bigl( \begin{smallmatrix}
+ a&b\\ c&d
+\end{smallmatrix} \bigr)
+\end{math})
+that comes closer to fitting within a single text line than a normal
+matrix. Delimiters\index{delimiters} must be provided. (The
+\pkg{mathtools} package provides |p|,|b|,|B|,|v|,|V| versions of
+\env{smallmatrix},%
+\index{smallmatrix@\indexenvironmentfont{smallmatrix} environment!variant delimiters}
+as well as |*| variants%
+\index{smallmatrix@\indexenvironmentfont{smallmatrix} environment!\verb+*+ variants}
+as described above.)
+The above example was produced by
+\begin{verbatim}
+\bigl( \begin{smallmatrix}
+ a&b\\ c&d
+\end{smallmatrix} \bigr)
+\end{verbatim}
+
+\cn{hdotsfor}|{|\<number>|}| produces a row of dots in a
+matrix\index{matrices!ellipsis dots}\index{ellipsis dots!in
+matrices}\index{dots|see{ellipsis dots}} spanning the given number of
+columns. For example,
+\begin{center}
+\begin{minipage}{.3\columnwidth}
+\noindent$\begin{matrix} a&b&c&d\\
+e&\hdotsfor{3} \end{matrix}$
+\end{minipage}%
+\qquad
+\begin{minipage}{.45\columnwidth}
+\begin{verbatim}
+\begin{matrix} a&b&c&d\\
+e&\hdotsfor{3} \end{matrix}
+\end{verbatim}
+\end{minipage}%
+\end{center}
+
+The spacing of the dots can be varied through use of a square-bracket
+option,\index{options!behavior of particular options}
+for example, |\hdotsfor[1.5]{3}|. The number in square brackets
+will be used as a multiplier (i.e., the normal value is 1.0).
+\begin{equation}\label{eq:D}
+\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
+-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
+\hdotsfor[2]{4}\\
+-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix},
+\end{equation}
+\begin{verbatim}
+\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
+-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
+\hdotsfor[2]{4}\\
+-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Math spacing commands}
+
+The \nipkg{amsmath} package slightly extends the set of math
+spacing\index{horizontal spacing} commands, as shown below.
+Both the spelled-out and abbreviated forms of these commands are robust,
+and they can also be used outside of math.
+\begin{ctab}{lll|lll}
+Abbrev.& Spelled out& Example & Abbrev.& Spelled out& Example\\
+\hline
+\vstrut{2.5ex}
+& no space& \spx{}& & no space & \spx{}\\
+\cn{\,}& \cn{thinspace}& \spx{\,}&
+ \altcnbang& \cn{negthinspace}& \spx{\!}\\
+\cn{\:}& \cn{medspace}& \spx{\:}&
+ & \cn{negmedspace}& \spx{\negmedspace}\\
+\cn{\;}& \cn{thickspace}& \spx{\;}&
+ & \cn{negthickspace}& \spx{\negthickspace}\\
+& \cn{quad}& \spx{\quad}\\
+& \cn{qquad}& \spx{\qquad}
+\end{ctab}
+For the greatest possible control over math spacing, use \cn{mspace}
+and `math units'. One math unit, or |mu|, is equal to 1/18 em. Thus to
+get a negative \cn{quad} you could write |\mspace{-18.0mu}|.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Dots}
+
+For preferred placement of ellipsis dots (raised or on-line) in various
+contexts there is no general consensus. It may therefore be considered a
+matter of taste. By using the semantically oriented commands
+\begin{itemize}
+\item \cn{dotsc} for \qq{dots with commas}
+\item \cn{dotsb} for \qq{dots with binary operators/relations}
+\item \cn{dotsm} for \qq{multiplication dots}
+\item \cn{dotsi} for \qq{dots with integrals}
+\item \cn{dotso} for \qq{other dots} (none of the above)
+\end{itemize}
+instead of \cn{ldots} and \cn{cdots}, you make it possible for your
+document to be adapted to different conventions on the fly, in case (for
+example) you have to submit it to a publisher who insists on following
+house tradition in this respect. The default treatment for the various
+kinds follows American Mathematical Society conventions:
+\begin{center}
+\begin{tabular}{@{}l@{}l@{}}
+\begin{minipage}[t]{.54\textwidth}
+\begin{verbatim}
+Then we have the series $A_1, A_2,
+\dotsc$, the regional sum $A_1
++A_2 +\dotsb $, the orthogonal
+product $A_1 A_2 \dotsm $, and
+the infinite integral
+\[\int_{A_1}\int_{A_2}\dotsi\].
+\end{verbatim}
+\end{minipage}
+&
+\begin{minipage}[t]{.45\textwidth}
+\noindent
+Then we have the series $A_1,A_2,\dotsc$,
+the regional sum $A_1+A_2+\dotsb$,
+the orthogonal product $A_1A_2\dotsm$,
+and the infinite integral
+\[\int_{A_1}\int_{A_2}\dotsi.\]
+\end{minipage}
+\end{tabular}
+\end{center}
+For most situations, the undifferentiated \cn{dots} can be used, and
+\nipkg{amsmath} will output the most suitable form based on the
+immediate context; if an inappropriate form results, it can be corrected
+after examining the output.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Nonbreaking dashes}
+
+A command \cn{nobreakdash} is provided to suppress the possibility of a
+line break\index{line break|(} after the following hyphen or dash. For
+example, if you write `pages 1\ndash 9' as |pages 1\nobreakdash--9|
+then a line break will never occur between the dash and the 9. You can
+also use \cn{nobreakdash} to prevent undesirable hyphenations in
+combinations like |$p$-adic|. For frequent use, it's advisable to make
+abbreviations, e.g.,
+\begin{verbatim}
+\newcommand{\p}{$p$\nobreakdash}% for "\p-adic"
+\newcommand{\Ndash}{\nobreakdash--}% for "pages 1\Ndash 9"
+% For "\n dimensional" ("n-dimensional"):
+\newcommand{\n}[1]{$n$\nobreakdash-\hspace{0pt}}
+\end{verbatim}
+The last example shows how to prohibit a line break\index{line break|)}
+after the hyphen but allow normal hyphenation in the following
+word. (It suffices to add a zero-width space after the hyphen.)
+
+%% ------------------------------------------------------------------ %%
+
+\section{Accents in math}
+
+In ordinary \LaTeX{} the placement of the second accent in doubled math
+accents is often poor. With the \nipkg{amsmath} package you
+will get improved placement of the second accent:
+$\hat{\hat{A}}$ (\cn{hat}|{\hat{A}}|).
+
+The commands \cn{dddot} and \cn{ddddot} are available to produce triple
+and quadruple dot accents in addition to the \cn{dot} and \cn{ddot}
+accents already available in \latex/.
+
+To get a superscripted hat or tilde character, load the \pkg{amsxtra}
+package and use \cn{sphat} or \cn{sptilde}. Usage is \verb'A\sphat'
+(note the absence of the \verb'^' character).
+
+
+To place an arbitrary symbol in math accent position, or to get under
+accents, see the \pkg{accents} package by Javier Bezos. (\nipkg{amsmath}
+must be loaded before \pkg{accents}.)
+
+%% ------------------------------------------------------------------ %%
+
+\section{Roots}
+
+In ordinary \latex/ the placement of root indices is sometimes not so
+good: $\sqrt[\beta]{k}$ (|\sqrt||[\beta]{k}|). In the
+\nipkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
+the position of the root:%
+\index{options!adjust positioning}
+\begin{verbatim}
+ \sqrt[\leftroot{-2}\uproot{2}\beta]{k}
+\end{verbatim}
+will move the beta up and to the right:
+$\sqrt[\leftroot{-2}\uproot{2}\beta]{k}$. The negative argument used
+with \cn{leftroot} moves the $\beta$ to the right. The units are a small
+amount that is a useful size for such adjustments.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Boxed formulas}
+
+The command \cn{boxed} puts a box around its
+argument, like \cn{fbox} except that the contents are in math mode:
+\begin{equation}
+\boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))}
+\end{equation}
+\begin{verbatim}
+ \boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Over and under arrows}
+
+Basic \latex/ provides \cn{overrightarrow} and \cn{overleftarrow}
+commands. Some additional over and under arrow commands are provided
+by the \nipkg{amsmath} package to extend the set:
+
+\begin{tabbing}
+\qquad\=\ncn{overleftrightarrow}\qquad\=\kill
+\> \cn{overleftarrow} \> \cn{underleftarrow} \+\\
+ \cn{overrightarrow} \> \cn{underrightarrow} \\
+ \cn{overleftrightarrow}\> \cn{underleftrightarrow}
+\end{tabbing}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Extensible arrows}
+
+\cn{xleftarrow} and \cn{xrightarrow} produce
+arrows\index{arrows!extensible} that extend automatically to accommodate
+unusually wide subscripts or superscripts. These commands take one
+optional argument\index{options!behavior of particular options}
+(the subscript) and one mandatory argument (the
+superscript, possibly empty):
+\begin{equation}
+A\xleftarrow{n+\mu-1}B \xrightarrow[T]{n\pm i-1}C
+\end{equation}
+\begin{verbatim}
+ \xleftarrow{n+\mu-1}\quad \xrightarrow[T]{n\pm i-1}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Affixing symbols to other symbols}
+
+\latex/ provides \cn{stackrel} for placing a
+superscript\index{subscripts and superscripts} above a binary relation.
+In the \nipkg{amsmath} package there are somewhat more general commands,
+\cn{overset} and \cn{underset}, that can be used to place one symbol
+above or below another symbol, whether it's a relation or something
+else. The input |\overset{*}{X}| will place a superscript-size $*$ above
+the $X$: $\overset{*}{X}$; \cn{underset} is the analog for adding a
+symbol underneath.
+
+See also the description of \cn{sideset} in \secref{sideset}.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Fractions and related constructions}
+
+\subsection{The \cn{frac}, \cn{dfrac}, and \cn{tfrac} commands}
+
+The \cn{frac} command, which is in the basic command set of
+\latex/,\index{fractions|(} takes two arguments\mdash numerator and
+denominator\mdash and typesets them in normal fraction form. The
+\nipkg{amsmath} package provides also \cn{dfrac} and \cn{tfrac} as
+convenient abbreviations for |{\displaystyle\frac| |...| |}|
+and\indexcs{textstyle}\relax
+\indexcs{displaystyle} |{\textstyle\frac| |...| |}|.
+\begin{equation}
+\frac{1}{k}\log_2 c(f)\quad\tfrac{1}{k}\log_2 c(f)\quad
+\sqrt{\frac{1}{k}\log_2 c(f)}\quad\sqrt{\dfrac{1}{k}\log_2 c(f)}
+\end{equation}
+\begin{verbatim}
+\begin{equation}
+\frac{1}{k}\log_2 c(f)\;\tfrac{1}{k}\log_2 c(f)\;
+\sqrt{\frac{1}{k}\log_2 c(f)}\;\sqrt{\dfrac{1}{k}\log_2 c(f)}
+\end{equation}
+\end{verbatim}
+
+\subsection{The \cn{binom}, \cn{dbinom}, and \cn{tbinom} commands}
+
+For binomial expressions\index{binomials} such as $\binom{n}{k}$
+\nipkg{amsmath} has \cn{binom}, \cn{dbinom} and \cn{tbinom}:
+\begin{equation}
+2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
+\end{equation}
+\begin{verbatim}
+2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
+\end{verbatim}
+
+\subsection{The \cn{genfrac} command}
+
+The capabilities of \cn{frac}, \cn{binom}, and their variants are
+subsumed by a generalized fraction command \cn{genfrac} with six
+arguments. The last two correspond to \cn{frac}'s numerator and
+denominator; the first two are optional delimiters\index{delimiters} (as
+seen in \cn{binom}); the third is a line thickness override (\cn{binom}
+uses this to set the fraction line thickness to 0\mdash i.e., invisible);
+and the fourth argument is a mathstyle override: integer values 0\ndash 3
+select respectively \cn{displaystyle}, \cn{textstyle}, \cn{scriptstyle},
+and \cn{scriptscriptstyle}. If the third argument is left empty, the line
+thickness defaults to `normal'.
+
+\begin{center}\begin{minipage}{.85\columnwidth}
+\raggedright \normalfont\ttfamily \exhyphenpenalty10000
+\newcommand{\ma}[1]{%
+ \string{{\normalfont\itshape#1}\string}\penalty9999 \ignorespaces}
+\string\genfrac \ma{left-delim} \ma{right-delim} \ma{thickness}
+\ma{mathstyle} \ma{numerator} \ma{denominator}
+\end{minipage}\end{center}
+To illustrate, here is how \cn{frac}, \cn{tfrac}, and
+\cn{binom} might be defined.
+\begin{verbatim}
+\newcommand{\frac}[2]{\genfrac{}{}{}{}{#1}{#2}}
+\newcommand{\tfrac}[2]{\genfrac{}{}{}{1}{#1}{#2}}
+\newcommand{\binom}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}}
+\end{verbatim}
+If you find yourself repeatedly using \cn{genfrac} throughout a document
+for a particular notation, you will do yourself a favor (and your
+publisher) if you define a meaningfully-named abbreviation for that
+notation, along the lines of \cn{frac} and \cn{binom}.
+
+The primitive generalized fraction commands \cs{over}, \cs{overwithdelims},
+\cs{atop}, \cs{atopwithdelims}, \cs{above}, \cs{abovewithdelims} produce
+warning messages if used with the \nipkg{amsmath} package, for reasons
+discussed in \fn{technote.tex}.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Continued fractions}
+
+The continued fraction\index{continued fractions}
+\begin{equation}
+\cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+\cdots
+}}}
+\end{equation}
+can be obtained by typing
+{\samepage
+\begin{verbatim}
+\cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+\dotsb
+}}}
+\end{verbatim}
+}% End of \samepage
+This produces better-looking results than straightforward use of
+\cn{frac}. Left or right placement of any of the numerators is
+accomplished by using \cn{cfrac}|[l]| or \cn{cfrac}|[r]| instead of
+\cn{cfrac}.\index{options!adjust positioning}%
+\index{fractions|)}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Smash options}
+
+The command \cn{smash} is used to typeset a subformula with an
+effective height and depth of zero, which is sometimes
+useful in adjusting the subformula's position with respect to adjacent
+symbols. With the \nipkg{amsmath} package \cn{smash} has optional
+arguments\index{options!behavior of particular options}
+|[t]|\index{t option@\texttt{t} (top) option} and
+|[b]|\index{b option@\texttt{b} (bottom) option} because occasionally it is
+advantageous to be able to \qq{smash} only the top or only the bottom
+of something while retaining the natural depth or height. For example,
+when adjacent radical symbols are unevenly sized or positioned because
+of differences in the height and depth of their contents, \cn{smash}
+can be employed to make them more consistent. Compare
+$\sqrt{x}+\sqrt{y}+\sqrt{z}$ and $\sqrt{x}+\sqrt{\smash[b]{y}}+\sqrt{z}$,
+where the latter was produced by
+\verb"$\sqrt{x}" \verb"+"
+\verb"\sqrt{"\5\verb"\smash[b]{y}}" \verb"+" \verb"\sqrt{z}$".
+
+%% ------------------------------------------------------------------ %%
+
+\section{Delimiters}
+
+\index{delimiters|(}
+
+\subsection{Delimiter sizes}\label{bigdel}
+
+The automatic delimiter sizing done by \cn{left} and \cn{right} has two
+limitations: First, it is applied mechanically to produce delimiters
+large enough to encompass the largest contained item, and second, the
+range of sizes is not even approximately continuous but has fairly large
+quantum jumps. This means that a math fragment that is infinitesimally
+too large for a given delimiter size will get the next larger size, a
+jump of 3pt or so in normal-sized text. There are two or three
+situations where the delimiter size is commonly adjusted, using a set of
+commands that have `big' in their names.%
+\index{delimiters!fixed size|(}
+\begin{ctab}{l|llllll}
+Delimiter&
+ text& \ncn{left}& \ncn{bigl}& \ncn{Bigl}& \ncn{biggl}& \ncn{Biggl}\\
+size&
+ size& \ncn{right}& \ncn{bigr}& \ncn{Bigr}& \ncn{biggr}& \ncn{Biggr}\\
+\hline
+Result\vstrut{5ex}&
+ $\displaystyle(b)(\frac{c}{d})$&
+ $\displaystyle\left(b\right)\left(\frac{c}{d}\right)$&
+ $\displaystyle\bigl(b\bigr)\bigl(\frac{c}{d}\bigr)$&
+ $\displaystyle\Bigl(b\Bigr)\Bigl(\frac{c}{d}\Bigr)$&
+ $\displaystyle\biggl(b\biggr)\biggl(\frac{c}{d}\biggr)$&
+ $\displaystyle\Biggl(b\Biggr)\Biggl(\frac{c}{d}\Biggr)$
+\end{ctab}
+The first kind of situation is a cumulative operator with limits above
+and below. With \cn{left} and \cn{right} the delimiters usually turn out
+larger than necessary, and using the |Big| or |bigg|
+sizes\index{big@\cn{big}, \cn{Big}, \cn{bigg}, \dots\ delimiters}
+instead gives better results:
+\begin{equation*}
+\left[\sum_i a_i\left\lvert\sum_j x_{ij}\right\rvert^p\right]^{1/p}
+\quad\text{versus}\quad
+\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p}
+\end{equation*}
+\begin{verbatim}
+\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p}
+\end{verbatim}
+The second kind of situation is clustered pairs of delimiters where
+\cn{left} and \cn{right} make them all the same size (because that is
+adequate to cover the encompassed material) but what you really want
+is to make some of the delimiters slightly larger to make the nesting
+easier to see.
+\begin{equation*}
+\left((a_1 b_1) - (a_2 b_2)\right)
+\left((a_2 b_1) + (a_1 b_2)\right)
+\quad\text{versus}\quad
+\bigl((a_1 b_1) - (a_2 b_2)\bigr)
+\bigl((a_2 b_1) + (a_1 b_2)\bigr)
+\end{equation*}
+\begin{verbatim}
+\left((a_1 b_1) - (a_2 b_2)\right)
+\left((a_2 b_1) + (a_1 b_2)\right)
+\quad\text{versus}\quad
+\bigl((a_1 b_1) - (a_2 b_2)\bigr)
+\bigl((a_2 b_1) + (a_1 b_2)\bigr)
+\end{verbatim}
+The third kind of situation is a slightly oversize object in running
+text, such as $\left\lvert\frac{b'}{d'}\right\rvert$ where the
+delimiters produced by \cn{left} and \cn{right} cause too much line
+spreading. In that case \ncn{bigl} and \ncn{bigr}\index{big@\cn{big},
+\cn{Big}, \cn{bigg}, \dots\ delimiters} can be used to produce
+delimiters that are slightly larger than the base size but still able to
+fit within the normal line spacing:
+$\bigl\lvert\frac{b'}{d'}\bigr\rvert$.
+
+In ordinary \latex/ \ncn{big}, \ncn{bigg}, \ncn{Big}, and \ncn{Bigg}
+delimiters aren't scaled properly over the full range of \latex/ font
+sizes. With the \nipkg{amsmath} package they are.%
+\index{delimiters!fixed size|)}
+
+\subsection{Vertical bar notations}
+
+The \nipkg{amsmath} package provides commands \cn{lvert}, \cn{rvert},
+\cn{lVert}, \cn{rVert} (compare \cn{langle}, \cn{rangle}) to address the
+%problem of overloading for the vert bar character \qc{\|}. This
+problem of overloading for the vert bar character \qcvert. This
+character is currently used in \latex/ documents to represent a wide
+variety of mathematical objects: the `divides' relation in a
+number-theory expression like $p\vert q$, or the absolute-value
+operation $\lvert z\rvert$, or the `such that' condition in set
+notation, or the `evaluated at' notation $f_\zeta(t)\bigr\rvert_{t=0}$.
+The multiplicity of uses in itself is not so bad; what is bad, however,
+is that fact that not all of the uses take the same typographical
+treatment, and that the complex discriminatory powers of a knowledgeable
+reader cannot be replicated in computer processing of mathematical
+documents. It is recommended therefore that there should be a one-to-one
+correspondence in any given document between the vert bar character
+%\qc{\|} and a selected mathematical notation, and similarly for the
+\qcvert\ and a selected mathematical notation, and similarly for the
+%double-bar command \ncn{\|}\index{"|@\verb"*+"\"|+}. This immediately
+double-bar command \cnvert. This immediately
+rules out the use of \qc{|}
+%and \ncn{\|}\index{"|@\verb"*+"\"|+} for delimiters, because left and right
+and \cnvert\ for delimiters, because left and right
+delimiters are distinct usages that do not relate in the same way to
+adjacent symbols; recommended practice is therefore to define suitable
+commands in the document preamble for any paired-delimiter use of vert
+bar symbols:
+\begin{verbatim}
+\providecommand{\abs}[1]{\lvert#1\rvert}
+\providecommand{\norm}[1]{\lVert#1\rVert}
+\end{verbatim}
+whereupon the document would contain |\abs{z}| to produce $\lvert
+z\rvert$ and |\norm{v}| to produce $\lVert v\rVert$.
+The \pkg{mathools} provides the command \cn{DeclarePairedDelimiter}
+for defining |\abs|-like macros with scaling delimiters.
+\index{delimiters|)}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Operator names}
+
+\section{Defining new operator names}\label{s:opname}
+
+Math functions\index{operator names}\relax \index{function
+names|see{operator names}} such as $\log$, $\sin$, and $\lim$ are
+traditionally typeset in roman type to make them visually more distinct
+from one-letter math variables, which are set in math italic. The more
+common ones have predefined names, \cn{log}, \cn{sin}, \cn{lim}, and so
+forth, but new ones come up all the time in mathematical papers, so the
+\nipkg{amsmath} package provides a general mechanism for defining new
+`operator names'. To define a math function \ncn{xxx} to work like
+\cn{sin}, you write
+\begin{verbatim}
+\DeclareMathOperator{\xxx}{xxx}
+\end{verbatim}
+whereupon ensuing uses of \ncn{xxx} will produce {\upshape xxx} in the
+proper font and automatically add proper spacing\index{horizontal
+spacing!around operator names} on either side when necessary, so that you
+get $A\xxx B$ instead of $A\mathrm{xxx}B$. In the second argument of
+\cn{DeclareMathOperator} (the name text), a pseudo-text mode prevails:
+the hyphen character \qc{\-} will print as a text hyphen rather than a
+minus sign and an asterisk \qc{\*} will print as a raised text asterisk
+instead of a centered math star. (Compare
+\textit{a}-\textit{b}*\textit{c} and $a-b*c$.) But otherwise the name
+text is printed in math mode, so that you can use, e.g., subscripts and
+superscripts there.
+
+%%%%xxx%%%\enlargethispage{1\baselineskip}
+If the new operator should have subscripts and superscripts placed in
+`limits' position above and below as with $\lim$, $\sup$, or $\max$, use
+the \qc{\*} form of the \cn{DeclareMathOperator} command:
+\begin{verbatim}
+\DeclareMathOperator*{\Lim}{Lim}
+\end{verbatim}
+See also the discussion of subscript placement in
+Section~\ref{subplace}.
+
+\goodbreak
+
+The following operator names are predefined:
+
+\vspace{-1\baselineskip}
+\begin{ctab}{@{}rlrlrlrl}
+\cn{arccos}& $\arccos$ &\cn{deg}& $\deg$ & \cn{lg}& $\lg$ & \cn{projlim}& $\projlim$\\
+\cn{arcsin}& $\arcsin$ &\cn{det}& $\det$ & \cn{lim}& $\lim$ & \cn{sec}& $\sec$\\
+\cn{arctan}& $\arctan$ &\cn{dim}& $\dim$ & \cn{liminf}& $\liminf$ &\cn{sin}& $\sin$\\
+\cn{arg}& $\arg$ & \cn{exp}& $\exp$ & \cn{limsup}& $\limsup$ &\cn{sinh}& $\sinh$\\
+\cn{cos}& $\cos$ & \cn{gcd}& $\gcd$ & \cn{ln}& $\ln$ & \cn{sup}& $\sup$\\
+\cn{cosh}& $\cosh$ & \cn{hom}& $\hom$ & \cn{log}& $\log$ & \cn{tan}& $\tan$\\
+\cn{cot}& $\cot$ & \cn{inf}& $\inf$ & \cn{max}& $\max$ & \cn{tanh}& $\tanh$\\
+\cn{coth}& $\coth$ & \cn{injlim}& $\injlim$ &\cn{min}& $\min$\\
+\cn{csc}& $\csc$ & \cn{ker}& $\ker$ & \cn{Pr}& $\Pr$
+\end{ctab}
+\par\nobreak
+\vspace{-1.2\baselineskip}
+\begin{ctab}{rlrl}
+\cn{varinjlim}& $\displaystyle\varinjlim$&
+\cn{varliminf}& $\displaystyle\varliminf$\\
+\cn{varprojlim}& $\displaystyle\varprojlim$&
+\cn{varlimsup}& $\displaystyle\varlimsup$
+\end{ctab}
+
+There is also a command \cn{operatorname} such that using
+\begin{verbatim}
+\operatorname{abc}
+\end{verbatim}
+in a math formula is equivalent to a use of \ncn{abc} defined by
+\cn{DeclareMathOperator}. This may be occasionally useful for
+constructing more complex notation or other purposes. (Use the variant
+\cn{operatorname*} to get limits.)
+
+%% ------------------------------------------------------------------ %%
+
+\section{\cn{mod} and its relatives}
+
+Commands \cn{mod}, \cn{bmod}, \cn{pmod}, \cn{pod} are provided to deal
+with the special spacing conventions of \qq{mod} notation. \cn{bmod} and
+\cn{pmod} are available in \latex/, but with the \nipkg{amsmath} package
+the spacing of \cn{pmod} will adjust to a smaller value if it's used in
+a non-display-mode formula. \cn{mod} and \cn{pod} are variants of
+\cn{pmod} preferred by some authors; \cn{mod} omits the parentheses,
+whereas \cn{pod} omits the \qq{mod} and retains the parentheses.
+\begin{equation}
+\gcd(n,m\bmod n);\quad x\equiv y\pmod b;
+\quad x\equiv y\mod c;\quad x\equiv y\pod d
+\end{equation}
+\begin{verbatim}
+\gcd(n,m\bmod n);\quad x\equiv y\pmod b;
+\quad x\equiv y\mod c;\quad x\equiv y\pod d
+\end{verbatim}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\enlargethispage{1\baselineskip}
+
+\chapter{The \cn{text} command}\label{text}
+
+The main use of the command \cn{text} is for words or
+phrases\index{text fragments inside math} in a display. It is very
+similar to the \latex/ command \cn{mbox} in its effects, but has a
+couple of advantages. If you want a word or phrase of text in a
+subscript, you can type |..._{\textrm{word or phrase}}|, which is slightly
+easier than the \cn{mbox} equivalent: |..._{\mbox{\rmfamily\scriptsize| |word|
+|or| |phrase}}|. Note that the standard \cn{textrm} command will use the
+\nipkg{amsmath} \cn{text} definition, but ensure the \verb+\rmfamily+
+font is used.
+\begin{equation}
+f_{[x_{i-1},x_i]} \text{ is monotonic,}
+\quad i = 1,\dots,c+1
+\end{equation}
+\begin{verbatim}
+f_{[x_{i-1},x_i]} \text{ is monotonic,}
+\quad i = 1,\dots,c+1
+\end{verbatim}
+
+The font used for \cn{text} is the same as that of the surrounding
+environment; i.e., within a theorem, the contents of \cn{text} will
+be set in italic.
+
+If a math expression is included in a \cn{text} string, it must be
+explicitly marked as math (|$...$|).
+\[
+\partial_s f(x) = \frac{\partial}{\partial x_0} f(x)\quad
+ \text{for $x= x_0 + I x_1$.}
+\]
+\begin{verbatim}
+\partial_s f(x) = \frac{\partial}{\partial x_0} f(x)\quad
+ \text{for $x= x_0 + I x_1$.}
+\end{verbatim}
+
+Function names should \emph{not} be entered as \cn{text}. Instead,
+use \cn{mathrm} or \cn{DeclareMathOperator} as appropriate. These are
+fixed entities that should not change depending on outside content
+(such as appearing within a theorem that is set in italic), and in the
+case of declared operators, proper spacing\index{horizontal
+spacing!around operator names} is applied automatically.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Integrals and sums}
+
+\section{Multiline subscripts and superscripts}
+
+The \cn{substack} command can be used to produce a multiline subscript
+or superscript:\index{subscripts and superscripts!multiline}\relax
+\index{superscripts|see{subscripts and superscripts}} for example
+\begin{ctab}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sum_{\substack{
+ 0\le i\le m\\
+ 0<j<n}}
+ P(i,j)
+\end{verbatim}
+\end{minipage}
+&
+$\displaystyle
+\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)$
+\end{ctab}
+A slightly more generalized form is the \env{subarray} environment which
+allows you to specify that each line should be left-aligned instead of
+centered, as here:
+\begin{ctab}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sum_{\begin{subarray}{l}
+ i\in\Lambda\\ 0<j<n
+ \end{subarray}}
+ P(i,j)
+\end{verbatim}
+\end{minipage}
+&
+$\displaystyle
+ \sum_{\begin{subarray}{l}
+ i\in \Lambda\\ 0<j<n
+ \end{subarray}}
+ P(i,j)$
+\end{ctab}
+
+%% ------------------------------------------------------------------ %%
+
+\section{The \cn{sideset} command}\label{sideset}
+
+There's also a command called \cn{sideset}, for a rather special
+purpose: putting symbols at the subscript and
+superscript\index{subscripts and superscripts!on sums} corners of a
+large operator symbol such as $\sum$ or $\prod$. \emph{Note: this
+command is not designed to be applied to anything other than sum-class
+symbols.} The prime
+example is the case when you want to put a prime on a sum symbol. If
+there are no limits above or below the sum, you could just use
+\cn{nolimits}: here's
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+|\sum\nolimits' E_n| in display mode:
+\begin{equation}
+\sum\nolimits' E_n
+\end{equation}
+If, however, you want not only the prime but also something below or
+above the sum symbol, it's not so easy\mdash indeed, without
+\cn{sideset}, it would be downright difficult. With \cn{sideset}, you
+can write
+\begin{ctab}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sideset{}{'}
+ \sum_{n<k,\;\text{$n$ odd}} nE_n
+\end{verbatim}
+\end{minipage}
+&$\displaystyle
+\sideset{}{'}\sum_{n<k,\;\text{$n$ odd}} nE_n
+$
+\end{ctab}
+The extra pair of empty braces is explained by the fact that
+\cn{sideset} has the capability of putting an extra symbol or symbols at
+each corner of a large operator; to put an asterisk at each corner of a
+product symbol, you would type
+\begin{ctab}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sideset{_*^*}{_*^*}\prod
+\end{verbatim}
+\end{minipage}
+&$\displaystyle
+\sideset{_*^*}{_*^*}\prod
+$
+\end{ctab}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Placement of subscripts and limits}\label{subplace}
+
+The default positioning for subscripts depends on the
+base symbol involved. The default for sum-class symbols is
+`displaylimits' positioning: When a sum-class symbol appears
+in a displayed formula, subscript and superscript are placed in `limits'
+position above and below, but in an inline formula, they are placed to
+the side, to avoid unsightly and wasteful spreading of the
+surrounding text lines.
+The default for integral-class symbols is to have sub- and
+superscripts always to the side, even in displayed formulas.
+(See the discussion of the \opt{intlimits} and related
+options\index{options!behavior of particular options} in
+Section~\ref{options}.)
+
+Operator names such as $\sin$ or $\lim$ may have either `displaylimits'
+or `limits' positioning depending on how they were defined. The standard
+operator names are defined according to normal mathematical usage.
+
+The commands \cn{limits} and \cn{nolimits} can be used to override the
+normal behavior of a base symbol:
+\begin{equation*}
+\sum\nolimits_X,\qquad \iint\limits_{A},
+\qquad\varliminf\nolimits_{n\to \infty}
+\end{equation*}
+To define a command whose subscripts follow the
+same `displaylimits' behavior as \cn{sum}, put
+\cn{displaylimits} at the tail end of the definition. When multiple
+instances of \cn{limits}, \cn{nolimits}, or \cn{displaylimits} occur
+consecutively, the last one takes precedence.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Multiple integral signs}
+
+\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral
+signs\index{integrals!multiple} with the spacing between them nicely
+adjusted, in both text and display style. \cn{idotsint} is an extension
+of the same idea that gives two integral signs with dots between them.
+\begin{gather}
+\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A
+f(x,y,z)\,dx\,dy\,dz\\
+\iiiint\limits_A
+f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k)
+\end{gather}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Commutative diagrams}\label{s:commdiag}
+\index{commutative diagrams}
+
+Some commutative diagram commands like the ones in \amstex/ are
+available as a separate package, \pkg{amscd}. For complex commutative
+diagrams authors will need to turn to more comprehensive packages like
+\tikz/\index{TikZ@\tikz/ package} (in particular, \pkg{tikz-cd})
+or \xypic/\index{XY-pic@\xypic/ package},
+but for simple diagrams without diagonal
+arrows\index{arrows!in commutative diagrams} the \pkg{amscd} commands
+may be more convenient. Here is one example.
+\begin{equation*}
+\begin{CD}
+S^{\mathcal{W}_\Lambda}\otimes T @>j>> T\\
+@VVV @VV{\End P}V\\
+(S\otimes T)/I @= (Z\otimes T)/J
+\end{CD}
+\end{equation*}
+\begin{verbatim}
+\begin{CD}
+S^{\mathcal{W}_\Lambda}\otimes T @>j>> T\\
+@VVV @VV{\End P}V\\
+(S\otimes T)/I @= (Z\otimes T)/J
+\end{CD}
+\end{verbatim}
+In the \env{CD} environment the commands |@>>>|,
+|@<<<|, |@VVV|, and |@AAA| give respectively right, left, down, and up
+arrows. For the horizontal arrows, material between the first and second
+|>| or |<| symbols will be typeset as a superscript, and material
+between the second and third will be typeset as a subscript. Similarly,
+material between the first and second or second and third |A|s or |V|s
+of vertical arrows will be typeset as left or right \qq{sidescripts}.
+The commands |@=| and \verb'@|' give horizontal and vertical double lines.
+A \qq{null arrow} command |@.| can be used instead of a visible arrow
+to fill out an array where needed.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Using math fonts}
+
+\section{Introduction}
+
+For more comprehensive information on font use in \latex/, see the
+\latex/ font guide (\fn{fntguide.tex}) or \booktitle{The \latex/
+Companion} \cite{tlc2}. The basic set of math font commands\index{math
+fonts}\relax \index{math symbols|see{math fonts}} in \latex/ includes
+\cn{mathbf}, \cn{mathrm}, \cn{mathcal}, \cn{mathsf}, \cn{mathtt},
+\cn{mathit}. Additional math alphabet commands such as
+\cn{mathbb} for blackboard bold, \cn{mathfrak} for Fraktur, and
+\cn{mathscr} for Euler script are available through the packages
+\pkg{amsfonts} and \pkg{euscript} (distributed separately).
+
+%% ------------------------------------------------------------------ %%
+\enlargethispage{1\baselineskip}
+
+\section{Recommended use of math font commands}
+
+If you find yourself employing math font commands frequently in your
+document, you might wish that they had shorter names, such as \ncn{mb}
+instead of \cn{mathbf}. Of course, there is nothing to keep you from
+providing such abbreviations for yourself by suitable \cn{newcommand}
+statements. But for \latex/ to provide shorter names would actually be a
+disservice to authors, as that would obscure a much better alternative:
+defining custom command names derived from the names of the underlying
+mathematical objects, rather than from the names of the fonts used to
+distinguish the objects. For example, if you are using bold to indicate
+vectors, then you will be better served in the long run if you define a
+`vector' command instead of a `math-bold' command:
+\begin{verbatim}
+ \newcommand{\vect}[1]{\mathbf{#1}}
+\end{verbatim}
+you can write |\vect{a} + \vect{b}| to produce $\vect{a} +
+\vect{b}$.
+If you decide several months down the road that you want to use the bold
+font for some other purpose, and mark vectors by a small over-arrow
+instead, then you can put the change into effect merely by changing the
+definition of \ncn{vect}; otherwise you would have to replace all
+occurrences of \cn{mathbf} throughout your document, perhaps even
+needing to inspect each one to see whether it is indeed
+an instance of a vector.
+
+It can also be useful to assign distinct
+command names for different letters of a particular font:
+\begin{verbatim}
+\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}% or use amsfonts package
+\DeclareMathSymbol{\C}{\mathalpha}{AMSb}{"43}
+\DeclareMathSymbol{\R}{\mathalpha}{AMSb}{"52}
+\end{verbatim}
+These statements would define the commands \cn{C} and \cn{R} to produce
+blackboard-bold letters from the `AMSb' math symbols font. If you refer
+often to the\break complex numbers or real numbers in your document, you
+might find this method more convenient than (let's say) defining a
+\ncn{field} command and writing\break |\field{C}|, |\field{R}|. But for
+maximum flexibility and control, define such a \ncn{field} command and
+then define \ncn{C} and \ncn{R} in terms of that
+command:\indexcs{mathbb}
+\begin{verbatim}
+\usepackage{amsfonts}% to get the \mathbb alphabet
+\newcommand{\field}[1]{\mathbb{#1}}
+\newcommand{\C}{\field{C}}
+\newcommand{\R}{\field{R}}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+\enlargethispage{3\baselineskip}
+
+\section{Bold math symbols}
+
+The \cn{mathbf} command is commonly used to obtain bold Latin letters in
+math, but for most other kinds of math symbols it has no effect, or its
+effects depend unreliably on the set of math fonts that are in use. For
+example, writing
+\begin{verbatim}
+\Delta \mathbf{\Delta}\mathbf{+}\delta \mathbf{\delta}
+\end{verbatim}
+produces $\Delta \mathbf{\Delta}\mathbf{+}\delta
+\mathbf{\delta}$; the \cn{mathbf} has no effect on the plus
+sign or the small delta.
+% should be paragraph break; avoid overfull box on next line
+The \nipkg{amsmath} package therefore provides two additional commands,
+\cn{boldsymbol} and \cn{pmb}, that can be applied to other kinds of math
+symbols. \cn{boldsymbol} can be used for a math symbol that remains
+unaffected by \cn{mathbf} if (and only if) your current math font set
+includes a bold version of that symbol. \cn{pmb} can be used as a last
+resort for any math symbols that do not have a true bold version
+provided by your set of math fonts; \qq{pmb} stands for \qq{poor man's
+bold} and the command works by typesetting multiple copies of the symbol
+with slight offsets. The quality of the output is inferior, especially
+for symbols that contain any hairline strokes. When the standard default set of
+\latex/ math fonts are in use (Computer Modern), the only symbols that
+are likely to require \cn{pmb} are large operator symbols like \cn{sum},
+extended delimiter\index{delimiters} symbols, or the extra math symbols
+provided by the \pkg{amssymb} package \cite{amsfonts}.
+
+The following formula shows some of the results that are possible:
+\begin{verbatim}
+A_\infty + \pi A_0
+\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
+ \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
+\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}}
+\end{verbatim}
+\begin{equation*}
+A_\infty + \pi A_0
+\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
+ \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
+\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}}
+\end{equation*}
+If you want to use only the \cn{boldsymbol} command without loading the
+whole \nipkg{amsmath} package, the \pkg{bm} package is recommended (this
+is a standard \latex/ package, not an AMS package; you probably have it
+already if you have a 1997 or newer version of \latex/).
+
+%% ------------------------------------------------------------------ %%
+
+\section{Italic Greek letters}
+
+For italic versions of the capital Greek letters, the following commands
+are provided:
+\begin{ctab}{rlrl}
+\cn{varGamma}& $\varGamma$& \cn{varSigma}& $\varSigma$\\
+\cn{varDelta}& $\varDelta$& \cn{varUpsilon}& $\varUpsilon$\\
+\cn{varTheta}& $\varTheta$& \cn{varPhi}& $\varPhi$\\
+\cn{varLambda}& $\varLambda$& \cn{varPsi}& $\varPsi$\\
+\cn{varXi}& $\varXi$& \cn{varOmega}& $\varOmega$\\
+\cn{varPi}& $\varPi$
+\end{ctab}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Error messages and output problems}
+
+\section{General remarks}
+
+This is a supplement to Chapter~8 of the \latex/ manual \cite{lamport}
+(first edition: Chapter~6). Appendix~B of the
+\textit{Companion}~\cite{tlc2} lists all \latex/ errors including those
+for \nipkg{amsmath} with likely causes for each. For the reader's
+convenience, the set of error messages discussed here overlaps
+somewhat with the set in those references,
+but please be aware that we don't provide exhaustive coverage here.
+The error messages are arranged in alphabetical order, disregarding
+unimportant text such as |! LaTeX Error:| at the beginning, and
+nonalphabetical characters such as \qc{\\}. Where examples are given, we
+show also the help messages that appear on screen when you respond to an
+error message prompt by entering |h|.
+
+Remember that the important error message is the first for a given line.
+When running in nonstop mode, errors accumulate, and the first error
+may create a further error condition from which it is impossible to
+recover. In such a situation processing will stop after 100~errors,
+so the reporting of genuine errors may not be complete, and it may be
+impossible to determine which reported errors are genuine and which
+are not.
+
+A final section discusses some output errors, i.e., instances where
+the printed document has something wrong but there was no \latex/
+error during typesetting.
+
+%% ------------------------------------------------------------------ %%
+%% **********
+\enlargethispage{-2\baselineskip}
+
+\section{Error messages}
+
+\begin{error}{\begin{split} won't work here.}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: \begin{split} won't work here.
+ ...
+
+l.8 \begin{split}
+
+? h
+\Did you forget a preceding \begin{equation}?
+If not, perhaps the `aligned' environment is what you want.
+?
+\end{verbatim}
+\errexpl
+The \env{split} environment does not construct a stand-alone displayed
+equation; it needs to be used within some other environment such as
+\env{equation} or \env{gather}.
+
+\end{error}
+
+\begin{error}{Erroneous nesting of equation structures}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: Erroneous nesting of equation structures;
+(amsmath) trying to recover with `aligned'.
+
+See the amsmath package documentation for explanation.
+Type H <return> for immediate help.
+ ...
+
+l.260 \end{alignat*}
+ \end{equation*}
+\end{verbatim}
+\errexpl
+The structures \env{align}, \env{alignat}, etc., are designed
+for top-level use and for the most part cannot be nested inside some
+other displayed equation structure. The chief exception is that
+\env{align} and most of its variants can be used inside the
+\env{gather} environment.
+\end{error}
+
+\begin{error}{Extra & on this line}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: Extra & on this line.
+
+See the amsmath package documentation for explanation.
+Type H <return> for immediate help.
+ ...
+
+l.9 \end{alignat}
+
+? h
+\An extra & here is so disastrous that you should probably exit
+ and fix things up.
+?
+\end{verbatim}
+\errexpl
+In an \env{alignat} structure the number of alignment points per line
+is dictated by the numeric argument given after |\begin{alignat}|.
+If you use more alignment points in a line it is assumed that you
+accidentally left out a newline command \cn{\\} and the above error is
+issued.
+\end{error}
+
+%% **********
+\enlargethispage{1.5\baselineskip}
+
+\begin{error}{Font OMX/cmex/m/n/7=cmex7 not loadable ...}
+\errexa
+\begin{verbatim}
+! Font OMX/cmex/m/n/7=cmex7 not loadable: Metric (TFM) file not found.
+<to be read again>
+ relax
+l.8 $a
+ b+b^2$
+? h
+I wasn't able to read the size data for this font,
+so I will ignore the font specification.
+[Wizards can fix TFM files using TFtoPL/PLtoTF.]
+You might try inserting a different font spec;
+e.g., type `I\font<same font id>=<substitute font name>'.
+?
+\end{verbatim}
+\errexpl
+Certain extra sizes of some Computer Modern fonts that were formerly
+available mainly through the AMSFonts\index{amsfont@AMSFonts collection}
+distribution are considered part of standard \latex/ (as of June 1994):
+\fn{cmex7}\ndash \texttt{9}, \fn{cmmib5}\ndash \texttt{9}, and
+\fn{cmbsy5}\ndash \texttt{9}. If these extra sizes are missing on your
+system, you should try first to get them from the source where you
+obtained \latex/. If that fails, you could try getting the fonts from
+CTAN (e.g., in the form of Metafont\index{Metafont source files} source
+files, directory \nfn{/tex-archive/fonts/latex/mf}, or in PostScript
+Type 1 format, directory
+\nfn{/tex-archive/fonts/cm/ps-type1/bakoma}\index{BaKoMa fonts}\relax
+\index{PostScript fonts}).
+
+If the font name begins with \fn{cmex}, there is a special option%
+\index{options!behavior of particular options}
+\opt{cmex10} for the \nipkg{amsmath} package that provides a temporary
+workaround. I.e., change the \cn{usepackage} to
+\begin{verbatim}
+\usepackage[cmex10]{amsmath}
+\end{verbatim}
+This will force the use of the 10-point size of the \fn{cmex} font in
+all cases. Depending on the contents of your document this may be
+adequate.
+\end{error}
+
+\enlargethispage{1\baselineskip}
+\begin{error}{Improper argument for math accent}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: Improper argument for math accent:
+(amsmath) Extra braces must be added to
+(amsmath) prevent wrong output.
+
+See the amsmath package documentation for explanation.
+Type H <return> for immediate help.
+ ...
+
+l.415 \tilde k_{\lambda_j} = P_{\tilde \mathcal
+ {M}}
+?
+\end{verbatim}
+\errexpl
+Non-simple arguments for any \LaTeX{} command should be enclosed in
+braces. In this example extra braces are needed as follows:
+\begin{verbatim}
+... P_{\tilde{\mathcal{M}}}
+\end{verbatim}
+\end{error}
+
+\begin{error}{Math formula deleted: Insufficient extension fonts}
+\errexa
+\begin{verbatim}
+! Math formula deleted: Insufficient extension fonts.
+l.8 $ab+b^2$
+
+?
+\end{verbatim}
+\errexpl
+This usually follows a previous error |Font ... not loadable|; see the
+discussion of that error (above) for solutions.
+\end{error}
+
+\enlargethispage{1\baselineskip}
+\begin{error}{Missing number, treated as zero}
+\errexa
+\begin{verbatim}
+! Missing number, treated as zero.
+<to be read again>
+ a
+l.100 \end{alignat}
+
+? h
+A number should have been here; I inserted `0'.
+(If you can't figure out why I needed to see a number,
+look up `weird error' in the index to The TeXbook.)
+
+?
+\end{verbatim}
+\errexpl
+There are many possibilities that can lead to this error. However, one
+possibility that is relevant for the \nipkg{amsmath} package is that you
+forgot to give the number argument of an \env{alignat} environment, as
+in:
+\begin{verbatim}
+\begin{alignat}
+ a& =b& c& =d\\
+a'& =b'& c'& =d'
+\end{alignat}
+\end{verbatim}
+where the first line should read instead
+\begin{verbatim}
+\begin{alignat}{2}
+\end{verbatim}
+
+Another possibility is that you have a left bracket character |[|
+following a line break\index{line break} command \cn{\\} in a multiline
+construction such as \env{array}, \env{tabular}, or \env{eqnarray}.
+This will be interpreted by \latex/ as the beginning of an `additional
+vertical space request \cite[\S C.1.6]{lamport}, even if it occurs on the
+following line and is intended to be part of the contents. For example,
+\begin{verbatim}
+\begin{array}{c}
+a+b\\
+[f,g]\\
+m+n
+\end{array}
+\end{verbatim}
+To prevent the error message in such a case, you can
+add braces as discussed in the \latex/ manual \cite[\S C.1.1]{lamport}:
+\begin{verbatim}
+\begin{array}{c}
+a+b\\
+{[f,g]}\\
+m+n
+\end{array}
+\end{verbatim}
+or precede the bracketed expression by \cn{relax}.
+\end{error}
+
+\goodbreak
+
+\begin{error}{Missing \right. inserted}
+\errexa
+\begin{verbatim}
+! Missing \right. inserted.
+<inserted text>
+ \right .
+l.10 \end{multline}
+
+? h
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+\end{verbatim}
+\errexpl
+This error typically arises when you try to insert a
+line break\index{line break} or |&|\cnamp\ between a
+\cn{left}-\cn{right} pair of delimiters\index{delimiters} in any
+multiline environment, including \env{split}.
+\begin{verbatim}
+\begin{multline}
+AAA\left(BBB\\
+ CCC\right)
+\end{multline}
+\end{verbatim}
+There are two possible solutions: (1)~instead of using \cn{left} and
+\cn{right}, use `big' delimiters\index{delimiters!fixed size} of fixed
+size (\cn{bigl} \cn{bigr} \cn{biggl} \cn{biggr} \dots; see
+\secref{bigdel}); or (2)~use null delimiters\index{delimiters!null} to
+break up the \cn{left}-\cn{right} pair into parts for each line (or cell):
+\begin{verbatim}
+AAA\left(BBB\right.\\
+ \left.CCC\right)
+\end{verbatim}
+The latter solution may result in mismatched
+delimiter\index{delimiters!mismatched sizes} sizes;
+ensuring that they match requires using \cn{vphantom} in the segment
+that has the smaller delimiter (or possibly \cn{smash} in the segment
+that has the larger delimiter). In the argument of \cn{vphantom} put a
+copy of the tallest element that occurs in the other segment, e.g.,
+\begin{verbatim}
+xxx \left(\int_t yyy\right.\\
+ \left.\vphantom{\int_t} zzz ... \right)
+\end{verbatim}
+\end{error}
+
+\goodbreak
+
+\begingroup
+\lccode`M=`M
+\lccode`@=`\}
+\lowercase{\endgroup
+\begin{error}{Missing @ inserted}}
+
+\errexa
+\begin{verbatim}
+! Missing } inserted.
+<inserted text>
+ \right .
+l.10 \end{multline}
+
+? h
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+\end{verbatim}
+\errexpl
+This error is often the result of using \verb+$+ within a multiline
+display environment. Remove such \verb+$+ signs (except when they
+appear within \verb+\text{...}+).
+\end{error}
+
+\begin{error}{Old form `\pmatrix' should be \begin{pmatrix}.}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: Old form `\pmatrix' should be
+ \begin{pmatrix}.
+
+See the amsmath package documentation for explanation.
+Type H <return> for immediate help.
+ ...
+
+\pmatrix ->\left (\matrix@check \pmatrix
+ \env@matrix
+l.16 \pmatrix
+ {a&b\cr c&d\cr}
+? h
+`\pmatrix{...}' is old Plain-TeX syntax whose use is
+ill-advised in LaTeX.
+?
+\end{verbatim}
+\errexpl
+When the \nipkg{amsmath} package is used, the old forms of \cn{pmatrix},
+\cn{matrix}, and \cn{cases} cannot be used any longer because of naming
+conflicts. Their syntax did not conform with standard \LaTeX{} syntax
+in any case.
+\end{error}
+
+\enlargethispage{2\baselineskip}
+\begin{error}{Paragraph ended before \xxx was complete}
+\errexa
+\begin{verbatim}
+Runaway argument?
+
+! Paragraph ended before \multline was complete.
+<to be read again>
+ \par
+l.100
+
+? h
+I suspect you've forgotten a `}', causing me to apply this
+control sequence to too much text. How can we recover?
+My plan is to forget the whole thing and hope for the best.
+?
+\end{verbatim}
+\errexpl
+This might be produced by a blank line between the |\begin| and |\end|.
+Another possibility is a misspelling in the |\end{multline}| command,
+e.g.,
+\begin{verbatim}
+\begin{multline}
+...
+\end{multiline}
+\end{verbatim}
+or by using abbreviations for certain environments, such as |\bal| and
+|\eal| for |\begin{align}| and |\end{align}|:
+\begin{verbatim}
+\bal
+...
+\eal
+\end{verbatim}
+For technical reasons that kind of abbreviation does not work with
+the more complex displayed equation environments of the \nipkg{amsmath} package
+(\env{gather}, \env{align}, \env{split}, etc.; cf.\ \fn{technote.tex}).
+\end{error}
+
+\begin{error}{Runaway argument?}
+See the discussion for the error message
+\texttt{Paragraph ended before \ncn{xxx} was complete}.
+\end{error}
+
+\begin{error}{Unknown option `xxx' for package `yyy'}
+\errexa
+\begin{verbatim}
+! LaTeX Error: Unknown option `intlim' for package `amsmath'.
+...
+? h
+The option `intlim' was not declared in package `amsmath', perhaps you
+misspelled its name. Try typing <return> to proceed.
+?
+\end{verbatim}
+\errexpl
+This means that you misspelled the option\index{options!unknown}
+name, or the package simply does not have an option that you expected
+it to have. Consult the documentation for the given package.
+\end{error}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Warning messages}
+
+\begin{error}{Cannot use `split' here}
+\errexa
+\begin{verbatim}
+Package amsmath Warning: Cannot use `split' here;
+(amsmath) trying to recover with `aligned'
+\end{verbatim}
+\errexpl The \env{split} environment is designed to serve as the entire
+body of an equation, or an entire line of an \env{align} or \env{gather}
+environment. There cannot be any printed material before or
+after it within the same enclosing structure:
+\begin{verbatim}
+\begin{equation}
+\left\{ % <-- Not allowed
+\begin{split}
+...
+\end{split}
+\right. % <-- Not allowed
+\end{equation}
+\end{verbatim}
+\end{error}
+
+\begin{error}{Foreign command \over [or \atop or \above]}
+\errexa
+\begin{verbatim}
+Package amsmath Warning: Foreign command \over; \frac or \genfrac
+(amsmath) should be used instead.
+\end{verbatim}
+\errexpl The primitive generalized fraction commands of \tex/\mdash
+\cs{over}, \cs{atop}, \cs{above}\mdash are deprecated when the
+\nipkg{amsmath} package is used because their syntax is foreign to \latex/
+and \nipkg{amsmath} provides native \latex/ equivalents. See
+\fn{technote.tex} for further information.
+\end{error}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Wrong output}
+
+\begin{erroro}{Section numbers 0.1, 5.1, 8.1 instead of 1, 2, 3}
+%
+This most likely means that you have the arguments for \cn{numberwithin}
+in reverse order:
+\begin{verbatim}
+\numberwithin{section}{equation}
+\end{verbatim}
+That means `print the section number as \textit{equation
+number}.\textit{section number} and reset to 1 every time an equation
+occurs' when what you probably wanted was the inverse
+\begin{verbatim}
+\numberwithin{equation}{section}
+\end{verbatim}
+\end{erroro}
+
+\begin{erroro}{The \cn{numberwithin} command had no effect on equation
+ numbers}
+%
+Are you looking at the first section in your document? Check the section
+numbers elsewhere to see if the problem is the one described in the
+previous item.
+\end{erroro}
+
+\begin{erroro}{Bracketed material disappears at the beginning of
+ \env{aligned} or \env{gathered}}
+%
+For most multiline equation environments, \nipkg{amsmath} disables the
+\latex/ convention to recognize a [bracketed] expression as an option%
+\index{ed env@\texttt{-ed} environments}%
+\index{options!positioning of \texttt{-ed} environments}%
+\index{options!space before \texttt{[}}
+when it occurs at the beginning of an environment or immediately
+following \cn{\\}, provided a space intervenes. This fails at the
+beginning of \env{aligned} and \env{gathered}, and is a bug. Insert
+\cn{relax} before the opening brace to restore the desired result.
+\end{erroro}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Additional information}
+
+\section{Compatibility with other packages}
+
+Several packages have already been mentioned as providing facilities
+lacking in \nipkg{amsmath}.
+
+\begin{itemize}
+\item \pkg{bm} is recommended for specifying bold math symbols.
+\item \pkg{mathtools} provides enhancements to some \nipkg{amsmath}
+ environments as well as additional compatible features, such as
+ \env{rcases} (similar to \env{cases} but with the brace on the
+ right-hand side), \env{multlined} (a subsidiary environment
+ comparable to \env{multline}), and \cn{shortintertext} (more
+ compact spacing than \cn{intertext}). If \pkg{mathtools} is used,
+ \nipkg{amsmath} is loaded automatically, so it is not necessary
+ to load it separately.
+\item \pkg{unicode-math} provides commands for (most?)\ math symbols
+ in Unicode.
+%\item \pkg{empheq}
+\end{itemize}
+
+These packages have known problems when used with \nipkg{amsmath}.
+\begin{itemize}
+\item \pkg{lineno} may omit numbers before display environments.
+\item \pkg{breqn} redefines a number of commands from other packages,
+ so is best loaded after \nipkg{amsmath} and other packages with math
+ facilities.\\ \pkg{breqn} is not yet totally stable, and a number of
+ bugs have been reported via \texttt{tex.stackexchange} \cite{tex-sx}.
+\item \pkg{wasysym}, \pkg{mathabx} and perhaps other font packages also
+ define multiple integrals\index{integrals!multiple} using the same
+ names as \nipkg{amsmath}.
+\end{itemize}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Converting existing documents}
+
+\subsection{Converting from plain \LaTeX{}}
+
+A \LaTeX{} document will typically continue to work the same in most
+respects if \verb'\usepackage{amsmath}' is added in the document
+preamble. By default, however, the \nipkg{amsmath} package suppresses page
+breaks inside multiple-line displayed equation structures such as
+\env{eqnarray}, \env{align}, and \env{gather}. To continue allowing page
+breaks inside \env{eqnarray} after switching to \nipkg{amsmath}, you will
+need to add the following line in your document preamble:
+\begin{verbatim}
+\allowdisplaybreaks[1]
+\end{verbatim}
+To ensure normal spacing around relation symbols, you might also want to
+change \env{eqnarray} to \env{align}, \env{multline}, or
+\env{equation}\slash\env{split} as appropriate.
+
+Most of the other differences in \nipkg{amsmath} usage can be considered
+optional refinements, e.g., using
+\begin{verbatim}
+\DeclareMathOperator{\Hom}{Hom}
+\end{verbatim}
+instead of \verb'\newcommand{\Hom}{\mbox{Hom}}'.
+
+\subsection{Converting from \texorpdfstring{\amslatex/}{amslatex} 1.1}
+See \fn{diffs-m.txt}.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Technical notes}
+The file \fn{technote.tex} contains some remarks on miscellaneous
+technical questions that are less likely to be of general interest.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Getting help}
+
+The \nipkg{amsmath} collection of packages is maintained by the \latex/
+team \cite{ltx-team}. Bugs should be reported using the instructions
+at \url{https://www.latex-project.org/bugs/}.
+Please be careful not to confuse \nipkg{amsmath} with the AMS document
+classes (\cls{amsart}, \cls{amsbook}, and \cls{amsproc}), which
+automatically incorporate \nipkg{amsmath}; bugs in the classes should
+be reported directly to AMS, by email to \mail{tech-support@ams.org}.
+
+%Questions or comments regarding \nipkg{amsmath} and related packages
+%should be sent to:
+%\begin{infoaddress}
+%American Mathematical Society\\
+%Technical Support\\
+%Electronic Products and Services\\
+%P. O. Box 6248\\
+%Providence, RI 02940\\[3pt]
+%Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4080\\
+%Internet: \mail{tech-support@ams.org}
+%\end{infoaddress}
+%If you are reporting a problem you should include
+%the following information to make proper investigation possible:
+%\begin{enumerate}
+%\item The source file where the problem occurred, preferably reduced
+% to minimum size by removing any material that can be removed without
+% affecting the observed problem.
+%\item A \latex/ log file showing the error message (if applicable) and
+% the version numbers of the document class and option files being used.
+%\end{enumerate}
+
+The online forum\index{online Q\,\&\,A forum} \texttt{tex.stackexchange}
+\cite{tex-sx} is a good source of answers to common questions on usage.
+Look first in the archive to see if your question has already been asked
+and answered; if it has not, post your own question.
+
+There is an online discussion group called \fn{comp.text.tex} \cite{ctt}
+that is a fairly good source of information about \latex/ and \tex/ in
+general. This group, formerly a Unix newsgroup, predates
+\texttt{tex.stackexchange} by decades, and the archives can be a useful
+resource for historical investigation.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Of possible interest}\label{a:possible-interest}
+Information about obtaining AMSFonts or other \tex/-related
+software from the AMS web server at \fn{www.ams.org}
+can be obtained by sending an email request to
+\mail{tech-support@ams.org}. This software is also posted on
+CTAN\index{CTAN (Comprehensive \TeX\ Archive Network)} and included
+in distributions based on \TeX\,Live\index{TEX Live@\TeX\,Live}.
+
+The \tex/ Users Group\index{TeX Users@\tex/ Users Group} is a nonprofit
+organization that publishes a journal
+(\journalname{TUGboat}\index{TUGboat@\journalname{TUGboat}}), holds
+meetings, and serves as a clearinghouse for general information about
+\tex/ and \tex/-related software.
+\begin{infoaddress}
+\tex/ Users Group\\
+PO Box 2311\\
+Portland, OR 97208-2311\\
+USA\\[3pt]
+Phone: +1-503-223-9994\\
+Email: \mail{office@tug.org}
+\end{infoaddress}
+Membership in the \tex/ Users Group is a good way to support continued
+development of free \tex/-related software. There are also many local
+\tex/ user groups in other countries; information about contacting a
+local user group can be gotten from the \tex/ Users Group.
+
+\index{amsmath@\texttt{amsmath} package|)}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newpage
+
+\begin{thebibliography}{10}
+\addcontentsline{toc}{chapter}{\bibname}
+
+\raggedright
+
+\bibsubhead{References in print}
+
+\bibitem{mil} George Gr\"atzer, \booktitle{More Math into \latex/}, fifth ed.,
+Springer, New York, 2016.
+% Deal with a line breaking problem
+%\begin{raggedright}
+%\bibitem{mil} G. Gr\"{a}tzer,
+%\emph{Math into \LaTeX{}: An Introduction to \LaTeX{} and AMS-\LaTeX{}}
+% \url{https://www.ams.org/cgi-bin/bookstore/bookpromo?fn=91&arg1=bookvideo&itmc=MLTEX},
+%Birkh\"{a}user, Boston, 1995.\par
+%\end{raggedright}
+
+\bibitem{kn} Donald E. Knuth, \booktitle{The \tex/book},
+Addison-Wesley, Reading, MA, 1984.
+
+\bibitem{lamport} Leslie Lamport, \booktitle{\latex/: A document preparation
+system}, 2nd revised ed., Addison-Wesley, Reading, MA, 1994.
+
+\bibitem{tlc2} Frank Mittelbach, Michel Goossens, et al.,
+\booktitle{The \latex/ companion}, second ed., Addison-Wesley, Reading,
+MA, 2004. This is now also available as an ebook, in both English and
+German; see \url{https://www.latex-project.org/help/books/}.
+The front matter, including the full Table of Contents, can be viewed
+online, from a link on the same page.
+%Michel Goossens, Frank Mittelbach, and Alexander Samarin,
+%\booktitle{The \latex/ companion}, Addison-Wesley, Reading, MA, 1994.
+% [\emph{Note: The 1994 edition is not a reliable guide for the
+% \nipkg{amsmath} package unless you refer to the errata for Chapter
+% 8\mdash file \fn{compan.err}, distributed with \LaTeX{}.}]
+
+\bibitem{msf} Frank Mittelbach and Rainer Sch\"opf,
+\textit{The new font family selection\mdash user
+interface to standard \latex/}, \journalname{TUGboat} \textbf{11},
+no.~2 (June 1990), pp.~297\ndash 305.
+
+\bibitem{jt} Michael Spivak, \booktitle{The joy of \tex/}, 2nd revised ed.,
+Amer. Math. Soc., Providence, RI, 1990.
+
+\bibsubhead{Package documentation}
+
+\bibitem{amshandbk}\booktitle{AMS author handbook}, separate versions
+for journal articles, monographs and proceedings articles, Amer. Math.
+Soc., Providence, RI, 2017;
+\url{https://www.ams.org/publications/authors/tex/author-handbook}.
+
+\bibitem{amsfonts}\booktitle{AMSFonts version \textup{2.2d}\mdash user's guide},
+Amer. Math. Soc., Providence, RI, 2002; distributed
+with the AMSFonts package
+\url{http://mirror.ctan.org/tex-archive/fonts/amsfonts/doc/amsfndoc.pdf}.
+
+\bibitem{amsthdoc}\booktitle{Using the \pkg{amsthm} Package}, version 2.20.3,
+Amer. Math. Soc., Providence, RI, 2017;
+\url{http://mirror.ctan.org/tex-archive/macros/latex/required/amscls/doc/amsthdoc.pdf}.
+
+\bibitem{mt} Morten H{\o}gholm, Lars Madsen, \booktitle{The \pkg{mathtools}
+package}, 2018;
+\url{http://mirror.ctan.org/tex-archive/macros/latex/contrib/mathtools/mathtools.pdf}.
+
+\bibsubhead{Online resources}
+
+\bibitem{ltx-team} The \latex/ Project, \url{https://www.latex-project.org}.
+
+\bibitem{ctt} Online discussion group \texttt{comp.text.tex},
+\url{https://groups.google.com/forum/\#!forum/comp.text.tex}.
+
+\bibitem{tex-sx} Online question and answer forum \texttt{tex.stackexchange},
+\url{https://tex.stackexchange.com}.
+
+\end{thebibliography}
+
+\printindex
+
+\end{document}
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/amsmath.pdf b/Master/texmf-dist/doc/latex-dev/amsmath/amsmath.pdf
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diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/amsopn.pdf b/Master/texmf-dist/doc/latex-dev/amsmath/amsopn.pdf
new file mode 100644
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--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/amsopn.pdf
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diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/amstext.pdf b/Master/texmf-dist/doc/latex-dev/amsmath/amstext.pdf
new file mode 100644
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diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/amsxtra.pdf b/Master/texmf-dist/doc/latex-dev/amsmath/amsxtra.pdf
new file mode 100644
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diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/changes.txt b/Master/texmf-dist/doc/latex-dev/amsmath/changes.txt
new file mode 100644
index 00000000000..1b2d097dd52
--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/changes.txt
@@ -0,0 +1,123 @@
+2019-02-06 Frank Mittelbach <Frank.Mittelbach@latex-project.org>
+
+ * amsmath.dtx (section{Mods, continued fractions, etc.}):
+ Command \overunderset added
+
+#########################
+# 2018-12-01 Release
+#########################
+
+2018-10-09 Frank Mittelbach <Frank.Mittelbach@latex-project.org>
+
+ * amsmath.dtx
+ Don't drop \frozen@everymath
+ Start LR-mode for \thinspace and friends if necessary (github/49)
+ Start LR-mode for \bigl and friends if necessary (github/49)
+ Start LR-mode for \smash if necessary (github/49)
+
+#########################
+# 2018-04-01 Release
+#########################
+
+2017-09-02 David Carlisle <latex-bugs@latex-project.org>
+
+ * amsmath.dtx: ensure that overfull alignments give warnings in more cases
+ https://tex.stackexchange.com/questions/141287/addmargin-enumerate-and-align-miss-an-overfull-hbox
+
+ * amsmath.dtx: move the nulldelimiterspace correction for closing
+ delimiters in \genfrac for extended TeXs to the line before
+ \nulldelimiterspace is locally set to 0pt.
+
+2017-04-14 David Carlisle <latex-bugs@latex-project.org>
+
+ * amscd.dtx: typo fix in ProvidesPackage line.
+
+2016-11-05 David Carlisle <latex-bugs@latex-project.org>
+
+ * alignedleftspace[yes|no|yesifneg] package options
+ * New genfrac implementation for XeTeX and LuaTeX
+ * Delete obsolete install.txt file.
+
+2016-06-28 David Carlisle <latex-bugs@latex-project.org>
+
+ * amsmath.dtx: Add space to prevent runaway argument error with \dots \left...
+
+2016-05-26 David Carlisle <latex-bugs@latex-project.org>
+
+ * amsmath.dtx: ignore spaces at start of \intertext
+
+2016-03-10 David Carlisle <latex-bugs@latex-project.org>
+
+ * amsmath.dtx: Preserve box0 in \resetMathstrut@
+
+2016-03-08 David Carlisle <latex-bugs@latex-project.org>
+
+ * amsopn.dtx: In xetex, and luatex, add version of \newmcodes@
+ that works even if - has a \Umathcode definition.
+
+2016-03-03 David Carlisle <latex-bugs@latex-project.org>
+
+ * amsmath.dtx: missing %
+
+2016-02-20 David Carlisle <latex-bugs@latex-project.org>
+
+ * amsmath.dtx: update to support \Umathchardef and \long macros in \dots.
+
+2013-01-14 amsmath.dtx 2.14
+
+ * amsmath.sty 2.14
+ -- Minimal changes to make amsmath compatible with stix.sty.
+
+2000-07-18 amsmath.dtx 2.13
+ -- After the numbering patches in 2.11, \notag failed in certain
+ circumstances: introduce some more auxiliary functions to sort
+ things out, and redefine \nonumber.
+
+2000-06-29 amstext.dtx 2.01
+ -- Use \f@size instead of \tf@size because they are not
+ necessarily the same and the former is better for putting a few
+ words into a display.
+
+2000-06-06 amsmath.dtx 2.12
+ -- Fix transposed lines in 2.11 patch.
+
+2000-06-02 amsmath.dtx 2.11
+
+ * amsmath.sty 2.11
+ -- Prevent "Arithmetic overflow" error by guarding against
+ divide-by-zero in \x@calc@shift@lc (align environment).
+
+ * amsdtx.dtx
+ -- Moved to the amscls distribution.
+
+2000-05-25 amsmath.dtx 2.10
+ -- Clear up error message for \allowdisplaybreaks[0].
+ -- Make mathdisplay re-entrant by introducing mathdisplay@stack,
+ to clear up numbering problems in unusual circumstances such as
+ \[ \] nested inside minipage inside equation.
+
+2000-04-21 amsmath.dtx 2.09
+ -- Ensure good catcodes for " etc.
+
+2000-03-16 amsmath.dtx 2.08
+ -- Fixed erroneous tag placement on split with fleqn/tbtags options.
+
+2000-03-15 amsmath.dtx 2.07
+ -- Add \reset@strutbox@ to deal with the following bug: After
+ $...\mbox{\Huge $...$}...$, line spacing is wrong in a
+ following "gather" or other environment that uses \strut@.
+ -- Patch to fix bug with intlimits option: too much space in the
+ middle of \iint.
+ -- Overhaul math accents again to fix a couple of bugs reported by
+ Thimm.
+
+2000-03-10 amsmath.dtx 2.06
+ -- Change \MathAccent to \mathaccentV so \DeclareMathAccent won't
+ give an error when redefining an accent.
+
+2000-01-06 amsmath.dtx 2.05
+ -- Fixed incorrect placement of fleqn/reqno equation numbers
+ inside indented lists (displaywidth < columnwidth). Changed the
+ multline/fleqn/leqno indent to match mathmargin when possible
+ instead of always just using multlinetaggap.
+
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/diffs-m.txt b/Master/texmf-dist/doc/latex-dev/amsmath/diffs-m.txt
new file mode 100644
index 00000000000..af3a273f526
--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/diffs-m.txt
@@ -0,0 +1,551 @@
+diffs-m.txt [1999/12/15] American Mathematical Society
+
+This is a synopsis of the principal differences between various versions
+of the amsmath package and its companion packages.
+
+A synopsis of changes in the "AMS-LaTeX" distribution is included at the
+end of this file. See also diffs-c.txt in the amslatex/classes area.
+
+Those upgrading from version 1.2 to version 2.0 will probably be
+interested in the question:
+
+ After I upgrade will anything fail in my pre-existing documents?
+
+Certain of the bug fixes have the potential to affect the output of
+existing documents in the following ways:
+
+---Vertical spacing for some displayed equations will change because
+"abovedisplayshortskip" is now applied when possible.
+
+---Incorrect horizontal spacing for certain math constructs will change,
+e.g., for \iint\nolimits or a subscript on a double-accented letter.
+
+---The default subscript placement in displayed formulas for \varinjlim,
+\varprojlim, \varliminf, \varlimsup is now below rather than to the
+side. To get the old behavior, add \nolimits---e.g.,
+\varliminf\nolimits_{...}.
+
+---Ellipsis dots in text (\ldots or \dots) will now behave the same as
+in standard LaTeX; the surrounding space may therefore change.
+
+During regression testing we found some errors cropping up in documents
+from incorrect usage that had not been caught with version 1.2. Although
+these errors are relatively rare, documents containing such usage will
+need to be corrected before they will run properly with version 2.0:
+
+---Use of two mutually exclusive display structures for the same
+display. In this example the outer \[ \] are redundant:
+
+ \[
+ \begin{align*}
+ ...
+ \end{align*}
+ \]
+
+---Inadequate argument markup for math accents:
+
+ \hat \mathcal{C}
+
+This should have been written with braces surrounding the argument of
+the \hat command: \hat{\mathcal{C}}.
+
+For documents where error messages crop up after the upgrade, there are
+various possible approaches, including:
+
+(a) Change instances of incorrect usage (if applicable).
+
+(b) Put a copy of amsmath.sty version 1.2 into the home directory of the
+document. Then it should continue to work the same as before.
+
+(c) Use the snapshot package in combination with a shell script to
+arrange that version 1.2 of amsmath is found instead of 2.0 when
+processing the document. See ftp://ftp.ams.org/pub/tex/snapshot.*. This
+approach requires a certain amount of technical knowledge.
+
+(d) Downgrade to version 1.2 again.
+
+========================================================================
+Changes in amsmath 2.0 [November 1999]
+
+---The AMS-LaTeX User's Guide (amsldoc.tex) has been turned into the
+ AMSmath User's Guide by moving the amsthm documentation into a
+ separate file (amsthdoc.tex) in the amslatex/classes area.
+
+---Restored normal use of "shortskip" spacing in single equations.
+
+[!! NOTE: This means that page breaks may change in existing documents.]
+
+---Restored normal use of available shrink around binary operators in
+ single equations.
+
+---Fixed poor subscript placement for double-accented letters. Math
+accent handling was reworked in the process in such a way that the
+capitalized forms of the math accent commands are no longer necessary.
+\hat{\hat{X}} will do proper alignment of the second accent. \Hat et
+cetera are retained for compabitility reasons. (Thanks to Javier Bezos
+for donating some code from his accents package.)
+
+[!! NOTE: Improper math accent use will now produce wrong output.]
+
+Suppose that we have \boldH defined in the preamble as
+\newcommand{\boldH}{\mathbf{H}}. Then the following examples illustrate
+recommended usage.
+
+ \hat{\mathbf{H}} OK
+ \hat{\boldH} OK
+ \hat\boldH Works OK, but deprecated by the LaTeX book
+ \hat\mathbf{H} Not OK! Output will be incorrect.
+
+Since the output will be wrong, amsmath gives an error message when the
+last variant is detected.
+
+---The following kinds of mixed structures worked by happenstance with
+ amsmath 1.2, although they should have produced an error message.
+ With amsmath 2.0 they will produce an error or warning message.
+
+ [!! NOTE: Potential error messages]
+
+ % Outer \[ \] is redundant
+
+ \[
+ \begin{align*}
+ ...
+ \end{align*}
+ \]
+
+ % Outer \begin{equation*} ... \end{equation*} is redundant
+
+ \begin{equation*}\begin{alignat*}{2}
+ ...
+ \end{alignat*}
+ \end{equation*}
+
+ % No printed material is allowed between \begin{equation} and
+ % \begin{split}:
+
+ \begin{equation*}\left \{
+ \begin{split}
+ ...
+ \end{split}\right .
+ \end{equation*}
+
+---diff12.tex is replaced by diffs-m.txt and diffs-c.txt, with the
+ latter moving to the amslatex/classes area.
+
+---Added an "?" option which gives some information about the available
+ documentation.
+
+---The amsmath definitions for matrix, pmatrix, and cases are
+ incompatible with the plain-TeX syntax for \pmatrix, \matrix, and
+ \cases, which are used occasionally by some authors in LaTeX
+ documents. In version 2.0 the error messages resulting from the
+ plain-TeX syntax have been made more intelligible.
+
+---If you need the boldsymbol facilities of the amsbsy package, use of
+ the "bm" (boldmath) package is recommended instead for new documents.
+ The amsbsy package is retained for compatibility with existing
+ documents. If you have a recent version of LaTeX (1998 or newer) you
+ already have the bm package; otherwise you may need to get it from
+ CTAN. (It is not an AMS package.)
+
+---\iint\nolimits: Fixed poor placement of subscript (also \iiint,
+\idotsint, etc).
+
+---\varinjlim, \varprojlim, \varliminf, \varlimsup: Default to
+\displaylimits instead of nolimits.
+
+---Fixed alignedat so that it doesn't take the first token of its
+ contents and try to interpret it as a number.
+
+---Leave text version of \dots and \ldots unchanged.
+
+---Changed \numberwithin so it no longer assumes previous definition of
+ \thesomething contained no prefixes.
+
+---\mathbf{\Gamma}: Leave cap Greek letters alone instead of converting
+ them from variable (fam 7) to fixed (fam 0). \mathbf{\gamma} still
+ fails---but that's a math font problem.
+
+---\mathindent is aliased to \@mathmargin so that when the fleqn option
+ is used, users can reset it as the LaTeX book advertises.
+
+---With the showkeys package: Corrected loss of marginal notes when
+labels occurred inside an align environment.
+
+---Carried over some tweaks (\bmod, \right/leftarrowfill) from the 1996
+ version of plain.tex.
+
+---Adjusted spacing of overarrow@ and underarrow@ to allow more room for
+ larger cmsy arrowheads.
+
+---Corrected spurious "overfull \hbox" warnings for \xrightarrow.
+
+---Corrected poor spacing for \xleftarrow.
+
+---Removed the \leavevmode in \smash which caused trouble for some
+ applications (e.g., the crop package) that expected the original
+ LaTeX definition.
+
+---Changed \def to \newcommand/\newenvironment where appropriate.
+
+---Use \sqrtsign instead of \@@sqrt (but include a backward
+ compatibility hack for the latter).
+
+---Left brace in \env{cases} environment would fail to work in
+ certain circumstances: Fix by using \lbrace instead of \{.
+
+---Use \ignorespacesafterend where applicable instead of
+ \global\@ignoretrue.
+
+---Protected commands in the definition of \theequation were
+ incorrectly expanded by the subequations environment when copying
+ the definition to \theparentequation: Fix by using \protected@edef.
+
+---Spurious extra space following punctuation at that end of a display:
+ In \math@cr, \math@cr@@, etc, use \iffalse{\fi instead of simply { to
+ keep an unwanted mathord atom from being added at the end of the
+ current math list; added \endmultline@math to work around a similar
+ sort of problem specific to the `multline' environment.
+
+---Various wrong results or no effect for \displaybreak inside "align",
+ "alignat", etc. Resetting \dspbrk@level in the \everycr of
+ \@display@init cleared most of these up.
+
+---Changed \displaybreak to issue an error message if enclosed in boxing
+ environment that will keep it from having any effect.
+
+---Turned off \displaybreak during the measuring phase of multline.
+
+---Fixed some numbering problems with multline and multline* and error
+ for use of \tag.
+
+---Use \@gtempa instead of \@tempa where assigned globally (standard
+ save stack precaution).
+
+---Save and restore \column@ to keep aligned from screwing up the column
+ counting of a parent align environment. Made \column@ assignments
+ uniformly global, except in a couple of places where the risk of side
+ effects seemed too great.
+
+---Error recovery for misspelled '\end{multline}' was not very helpful.
+ Improve \collect@body along lines suggested by Hellström.
+
+---Fixed up certain number placement problems with align
+
+---Add \alignsafe@testopt to work around a LaTeX bug: when a nested
+ alignment environment that takes an optional argument has an empty
+ first cell (i.e., begins with an &) LaTeX's optional argument
+ scanning screws up the alignment state.
+
+---Add `mathdisplay' environment (internal use) to correct several
+ problems of \[ \] and the `equation' environment arising from the
+ previous implementation as a degenerate case of `gather'.
+
+---It is possible now to use \eeq as an abbreviation for \end{equation}
+(one of the consequences of the preceding change). But still not \eal
+for \end{align} or certain other display structures.
+
+---Reorganize `split' internal calling sequence to go along with the use
+ of `mathdisplay' in \[ \] and `equation'.
+
+---Suppression of LaTeX error messages during measuring phase made
+ primitive TeX errors harder to understand; stop suppressing. (As a
+ result, users will sometimes get redundant error messages now.)
+ Remove obsolete Sb, Sp environments again.
+
+---The file amsintx.dtx was removed from the distribution pending
+ further development.
+
+---Remove the \ifinany@ switch, use \spread@equation instead to achieve
+ the same ends.
+
+---Remove unbracefonts@ function, no longer needed after changes in
+ math accent handling.
+
+---Got rid of the special trap for '^x that had been carried over from
+AMS-TeX.
+
+---Made \overline, \doteq, \longrightarrow, etc, robust.
+
+---Changed \relbar to use \std@minus---otherwise \underleftrightarrow
+works poorly in a \DeclareMathOperator definition.
+
+---Use \new@ifnextchar in matrix, pmatrix, cases so that \\<newline>[x]
+doesn't trigger an error.
+
+---Bug fix for nested use of \label inside minipage inside equation.
+
+---Add minalignsep after every second column in aligned, for consistency
+with align.
+
+========================================================================
+Changes in amsmath 1.2 [January 1995]
+
+---Name change; relation to the "amsfonts" package
+
+In version 1.1 of AMS-LaTeX, the name of this package was "amstex". The
+new name is "amsmath". The amstex package automatically invoked the
+amsfonts package as well; the amsmath package does not.
+
+---Adapted for LaTeX2e
+
+Old method of invocation:
+
+ \documentstyle[amstex]{article}% LaTeX 2.09
+
+New method of invocation:
+
+ \documentclass{article}
+ \usepackage{amsmath}
+
+---Handling of options intlim, nosumlim, nonamelm, righttag, ctagsplt
+
+Old usage:
+
+ \documentstyle[amstex,righttag,intlim]
+
+New usage:
+
+ \usepackage[reqno,intlimits]{amsmath}
+
+Several option names were changed (mainly, they are no longer restricted
+to eight characters):
+
+ intlim -> intlimits
+ nosumlim -> nosumlimits
+ nonamelm -> nonamelimits
+ righttag -> reqno [for consistency with "leqno"]
+ ctagsplt -> centertags
+
+---Character @ is no longer used as a special command prefix
+
+Certain commands in the amstex pkg started with "@" instead of "\". This
+special usage has been discontinued, except for the amscd package, and
+then only within the CD environment. Practically speaking, this means
+primarily that it is no longer necessary to use doubled @@ to get a
+single printed @ character (e.g., in e-mail addresses).
+
+---The @, and @! micro-spacing commands are no longer provided
+
+As a consequence of the @ character change, "@," and "@!" are no longer
+recognized spacing commands. An \mspace command is provided that can be
+used instead; see the AMSmath User's Guide.
+
+---Extensible arrow commands @>>> and @<<<
+
+The commands @>>> and @<<< are no longer available [except in the CD
+environment of the amscd package]; alternative commands \xleftarrow and
+\xrightarrow are provided by the amsmath package. The sub and
+superscripts are given through an optional resp. mandatory argument, for
+example
+
+ X\xleftarrow{b}Y % superscript "b" only
+ X\xrightarrow[\alpha]{\beta}Y % sub alpha, sup beta
+
+---The command \bold has been superseded by the command \mathbf, as the
+latter is now the standard name provided by LaTeX for this purpose.
+
+---The commands \newsymbol, \frak, \Bbb are no longer available because
+the amsfonts package is not loaded by amsmath. In any case those
+commands now have new names---\frak -> \mathfrak, \Bbb -> \mathbb, etc.
+See the amsfonts documentation for further details.
+
+---The amsmath package provides \lvert and \rvert to use for vert-bar
+symbols when they act as delimiters (compare \langle, \rangle for
+angle-bracket delimiters). Commands \lVert and \rVert are likewise
+provided instead of \| for paired-delimiter use of the double-vert
+symbol. Use of | and \| for paired delimiters can lead to incorrect
+spacing in rare cases. Further discussion is found in the AMS-LaTeX
+user's guide (amsldoc.tex).
+
+---The ~ command is now left unchanged by amsmath; the change
+that makes it remove a redundant preceding or following space character
+is now done only by AMS documentclasses (in version 1.1 this was done by
+the amstex package).
+
+---The following rarely used commands are removed to the amsxtra
+package: \accentedsymbol, \sphat, \spcheck, \sptilde, \spdot, \spddot,
+\spdddot, \spbreve, \@))), \@(((.
+
+---The six generalized fraction commands \over, \overwithdelims, \atop,
+\atopwithdelims, \above, \abovewithdelims are deprecated and produce
+warning messages. Their functionality can be obtained through \frac,
+\binom, and \genfrac instead. See amsldoc.tex and technote.tex for
+further details.
+
+---The \fracwithdelims command is gone, as \genfrac provides
+equivalent functionality (and a little more).
+
+---The optional argument of \frac that allowed changing the thickness of
+the fraction line is gone. That functionality is now provided only
+through \genfrac: the need to change the line thickness is so rare in
+practice that it seems better not to burden the ubiquitous \frac command
+with the somewhat time-consuming look-ahead process required for an
+optional argument. I.e., this change will contribute in a small way to
+making your documents run faster.
+
+---The commands \lcfrac and \rcfrac for left or right alignment of
+continued fraction numerators are gone; they are replaced by an optional
+argument of the \cfrac command.
+
+---A new command \DeclareMathOperator is provided for defining new
+operator names:
+
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator*{\xlim}{x-lim}
+
+The * form makes the operator name produce limit-style sub and
+superscripts, like \max or \lim. This is now the preferred method,
+instead of using the old command \operatorname (which is, however,
+retained for now). The command \operatornamewithlimits has been renamed
+\operatorname*. These commands now can also be obtained through a
+separate package amsopn, so it's not necessary to load the entire
+amsmath package just to get that feature.
+
+---Environments "Sb", "Sp" retired.
+
+The environments Sb and Sp for multiline subscripts have been replaced
+by a single command \substack, used as follows:
+
+\sum_{\substack{first line\\second line}}
+
+This command can be used equally well in a subscript or superscript, and
+provides better vertical positioning (as compared to Sb, Sp) if a
+multiline subscript is placed on the side, as perhaps for an integral.
+There is also a slightly generalized alternative, an environment
+subarray that allows you to specify either centering or left alignment
+for the contents:
+
+\sum_{\begin{subarray}{l}first line\\second line\end{subarray}}
+
+---New command \nobreakdash
+
+A command \nobreakdash is provided to suppress the possibility of a
+linebreak after the following hyphen or dash. For example, if you write
+"$p$\nobreakdash-adic" then a linebreak will never be taken between the
+hyphen and the "adic".
+
+---The placement of equation numbers has been substantially improved for
+align, gather, and split. Numbers will now never overlap on top of the
+equation body (as far as we can ascertain), and they are much less
+likely to be shifted up or down unnecessarily when there is actually
+adequate space to leave the number in the normal place. In a few
+remaining cases, an equation number will be shifted unnecessarily
+because technical complications make accurate measurement of the
+available room too difficult; for those cases a \raisetag command is
+provided that allows you to manually adjust the vertical position of the
+equation number.
+
+---The [xx]align[at] family of environments has been thoroughly revised:
+now the xalignat environment is gone, as its function has been merged
+into the align environment (it's no longer necessary to specify how many
+side-by-side structures, as was required for xalignat, because align
+automatically handles any number of them). Similarly, the xxalignat
+environment has been replaced by an environment flalign that doesn't
+require you to specify how many side-by-side structures will be needed.
+Certain numbering problems (presence of an equation number when it
+should be absent, or vice versa) have also been cleared up.
+
+(Thanks to David M. Jones for his substantial [volunteer] work behind
+this change and the preceding one.)
+
+---The alignedat environment was changed to take two arguments rather
+than one: a mandatory argument (as formerly) specifying the number of
+align structures, and a new optional one specifying the placement of the
+environment (parallel to the optional argument of aligned). However,
+aligned is simpler to use, allowing any number of aligned structures
+automatically, and therefore the use of alignedat is deprecated.
+
+---The seldom-used command \oldnos is gone; if you need it, you should
+consult the LaTeX documentation on using fonts (e.g., fntguide.tex) to
+find out how to construct an equivalent command.
+
+---A new subequations environment causes all numbered equation
+environments within its scope to be numbered (4.9a) (4.9b) (4.9c) etc.,
+if the preceding numbered equation was 4.8. A \label command immediately
+after \begin{subequations} will produce a \ref of the parent number
+`4.9', not `4.9a'. The counters used by the subequations environment are
+parentequation and equation and standard uses of \addtocounter,
+\setcounter, etc., are possible with those counter names.
+
+---The amsmath package (unlike amstex) respects the setting provided by
+the documentclass for putting equation numbers on the right or the left.
+[Technical note: a documentclass that has leqno as the default should
+explicitly pass that option to the amsmath package with
+\PassOptionsToPackage in order for it to be effective.]
+
+---The fleqn option for left-aligned instead of centered equations is
+now supported (thanks to David M. Jones).
+
+************************************************************************
+The remainder of this file discusses some changes that do not directly
+pertain to the amsmath package but to the parent distribution known as
+"AMS-LaTeX".
+
+========================================================================
+Changes in AMS-LaTeX 2.0
+
+---The AMS-LaTeX User's Guide (amsldoc.tex) is split up into amsldoc.tex
+and amsthdoc.tex, with the latter moving into the "classes" area.
+
+---The file amslatex.faq was split into amsmath.faq and amsclass.faq.
+
+---The file amslatex.bug was done away with; most of the information was
+merged into diffs-m.txt, diffs-c.txt, amsmath.faq, and amsclass.faq.
+
+========================================================================
+Changes in AMS-LaTeX 1.2
+
+---Adapted for LaTeX2e
+
+AMS-LaTeX 1.0 and 1.1 were designed to work with LaTeX 2.09, which
+became obsolete in 1994 with the appearance of "LaTeX2e". AMS-LaTeX 1.2
+has been extensively adapted to take advantage of various features of
+LaTeX2e. It is no longer a separate TeX format---i.e., it does not need
+to be invoked by a separate "amslatex" command.
+
+---LaTeX 2.09 font commands (\bf, \it, etc)
+
+Pre-existing documents will for the most part continue to work fine
+without requiring any changes. However, depending on the installation
+options used for your copy of AMS-LaTeX 1.1 you may need to update the
+options list of the \documentstyle command by adding newlfont, for
+example:
+
+ \documentstyle[newlfont]{amsart}
+
+---The packages amsfonts, amssymb were moved to the AMSFonts
+distribution. They used to be included in the AMS-LaTeX distribution for
+pragmatic reasons but their natural home is with the AMSFonts
+distribution.
+
+---Package amstex renamed to amsmath
+
+The amstex package has been renamed amsmath. The old name `amstex' came
+from the fact that version 1.0 was a more or less straight port into
+LaTeX of selected portions from the macro package AmS-TeX written by
+Michael Spivak, but the historical origins of the package are likely of
+less interest to the general usership---especially looking to the
+future---than a straightforward declaration in the name `amsmath' of the
+sort of features that are provided.
+
+Another important reason for the name change was to make possible some
+progress that would be out of the question (because of the impact on
+existing documents) if the name `amstex' were retained. For example,
+unlike amstex, amsmath doesn't automatically load the amsfonts package,
+as that package was moved out to the AMSFonts distribution as described
+above. If this change were not accompanied by a name change, many
+existing documents would have to be updated by hand (adding
+\usepackage{amsfonts}) in order to run without error. A frozen version
+of amstex.sty is provided for processing pre-existing documents; it
+won't have active maintainence henceforth except for bug fixes. For new
+documents, authors should use the amsmath package.
+
+Therefore, "\usepackage{amstex}" is incorrect usage---do not rely on the
+1994 edition of `The LaTeX Companion' in this respect (unless you obtain
+the errata for Chapter 8).
+
+---New documentclass amsproc
+
+A new documentclass amsproc was added for collections of articles, such as
+the proceedings of a conference.
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/manifest.txt b/Master/texmf-dist/doc/latex-dev/amsmath/manifest.txt
new file mode 100644
index 00000000000..ee6b8e3238d
--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/manifest.txt
@@ -0,0 +1,32 @@
+README.txt
+amsbsy.dtx
+amsbsy.ins
+amsbsy.pdf
+amscd.dtx
+amscd.ins
+amscd.pdf
+amsgen.dtx
+amsgen.ins
+amsgen.pdf
+amsldoc.pdf
+amsldoc.tex
+amsmath.dtx
+amsmath.ins
+amsmath.pdf
+amsopn.dtx
+amsopn.ins
+amsopn.pdf
+amstex.sty
+amstext.dtx
+amstext.ins
+amstext.pdf
+amsxtra.dtx
+amsxtra.ins
+amsxtra.pdf
+diffs-m.txt
+subeqn.pdf
+subeqn.tex
+technote.pdf
+technote.tex
+testmath.pdf
+testmath.tex
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/subeqn.pdf b/Master/texmf-dist/doc/latex-dev/amsmath/subeqn.pdf
new file mode 100644
index 00000000000..2e84ebe9422
--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/subeqn.pdf
Binary files differ
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/subeqn.tex b/Master/texmf-dist/doc/latex-dev/amsmath/subeqn.tex
new file mode 100644
index 00000000000..c34b3f6952a
--- /dev/null
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/subeqn.tex
@@ -0,0 +1,202 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%% filename = "subeqn.tex",
+%%% version = "1.2c",
+%%% date = "1999/11/29",
+%%% time = "14:49:29 EST",
+%%% checksum = "29866 202 668 6426",
+%%% author = "American Mathematical Society",
+%%% copyright = "Copyright 1999 American Mathematical Society,
+%%% all rights reserved. Copying of this file is
+%%% authorized only if either:
+%%% (1) you make absolutely no changes to your copy,
+%%% including name; OR
+%%% (2) if you do make changes, you first rename it
+%%% to some other name.",
+%%% address = "American Mathematical Society,
+%%% Technical Support,
+%%% Electronic Products and Services,
+%%% P. O. Box 6248,
+%%% Providence, RI 02940,
+%%% USA",
+%%% telephone = "401-455-4080 or (in the USA and Canada)
+%%% 800-321-4AMS (321-4267)",
+%%% FAX = "401-331-3842",
+%%% email = "tech-support@ams.org (Internet)",
+%%% supported = "yes",
+%%% keywords = "latex, amslatex, ams-latex, amstex",
+%%% abstract = "This is part of the AMS-\LaTeX{} distribution.
+%%% It is a sample document illustrating the use of
+%%% the subequations environment and the effect of
+%%% the fleqn option.",
+%%% docstring = "The checksum field above contains a CRC-16
+%%% checksum as the first value, followed by the
+%%% equivalent of the standard UNIX wc (word
+%%% count) utility output of lines, words, and
+%%% characters. This is produced by Robert
+%%% Solovay's checksum utility.",
+%%% }
+%%% ====================================================================
+\documentclass[fleqn]{article}
+
+\title{Test of subequations environment}
+\author{mjd}
+
+\usepackage{amsmath}
+\numberwithin{equation}{section}
+
+\newcommand{\env}[1]{{\normalfont\texttt{#1}}}
+
+\AtEndDocument{\multipasswarning}
+\newcommand{\multipasswarning}{%
+ \clearpage
+ \typeout{%
+ **********************************************************************}
+ \typeout{%
+ Note: This document needs to run through LaTeX three times, instead of}
+ \typeout{%
+ the usual two, to resolve indirect cross-references.}
+ \typeout{%
+ **********************************************************************}
+}
+
+\makeatletter
+% Omit the warning message if three passes have been completed (on
+% first pass \ref{check} is undefined and it is set to 0; on second
+% pass \ref{check} is 0 and set to 1; on third pass it is 1).
+\def\checkref{\begingroup
+ \@ifundefined{r@check}{\let\r@check\relax\def\@currentlabel{0}}{\def\@currentlabel{1}}%
+ \ifnum1=0\expandafter\@firstoftwo\r@check\relax
+ \global\let\multipasswarning\relax
+ \fi
+ \label{check}%
+ \endgroup}
+\makeatother
+
+\begin{document}
+\section{Prime equation numbers}
+
+First an equation.
+\begin{equation}\label{e:previous}
+A=B
+\end{equation}
+That was equation \eqref{e:previous}.
+
+Then the same, with a prime on the number.\checkref
+\begin{equation}
+\tag{\ref{e:previous}$'$}\label{e:prevprime}
+C=D
+\end{equation}
+And that was equation \eqref{e:prevprime}.
+
+Notice, by the way, that when a \verb"\ref" occurs inside a \verb"\tag",
+and that \verb"\tag" is then \verb"\label"'d, a \verb"\ref" for the the
+second \verb"\label" requires \emph{three} runs of \LaTeX{} in order to
+get the proper value. (If you run through the logic of \LaTeX{}'s
+cross-referencing mechanisms as they apply in this case, you will see
+that this is necessary.)
+
+\section{Subnumbered equations}
+
+Here is a,b,c sub-numbering.
+\begin{subequations}
+\begin{eqnarray}
+A&=&B\\
+D&=&C \label{e:middle}\\
+E&=&F
+\end{eqnarray}
+\end{subequations}
+That was produced with the \env{eqnarray} environment; the middle line
+was labeled as \eqref{e:middle}.
+
+An equation following the end of the \verb"subequations" environment
+should revert to normal numbering:
+\begin{equation}\label{e:check}
+H<K
+\end{equation}
+A check on the labeling: that was equation~\eqref{e:check}.
+
+The sub-numbered equations can be spread out through the text, like
+this:
+\begin{subequations}
+\begin{equation}
+A=B
+\end{equation}
+The \verb"subequations" environment can span arbitrary text between
+subsidiary equations. The only restriction is that if there are any
+numbered equations inside the \verb"subequations" environment that break
+out of the subequation numbering sequence, they would have to be handled
+specially.
+\begin{equation}
+D=C \label{e:newmiddle}
+\end{equation}
+More arbitrary text.
+\begin{equation}
+E=F
+\end{equation}
+\end{subequations}
+Label check: the middle one was \eqref{e:newmiddle}
+
+A final equation for a numbering check.
+\begin{equation}\label{e:final}
+G=H
+\end{equation}
+That equation was labeled as \eqref{e:final}.
+
+\section{Tests of \env{align}, \env{gather}, and other
+AMS-\protect\LaTeX{} environments}
+
+The \env{align} environment:
+\begin{subequations}
+\begin{align}
+\label{align:a}A+B&=B+A\\
+\label{align:b}C&=D+E\\
+\label{align:c}E&=F
+\end{align}
+\end{subequations}
+Label check: that was \eqref{align:a}, \eqref{align:b}, and
+\eqref{align:c}.
+
+The \env{align} environment again:
+\begin{subequations}
+\begin{align}
+\label{xalign:a}A+B&=B& B&=B+A\\
+\label{xalign:b}C&=D+E& C\oplus D&=E\\
+\label{xalign:c}E&=F& E'&=F'
+\end{align}
+\end{subequations}
+Label check: that was \eqref{xalign:a}, \eqref{xalign:b}, and
+\eqref{xalign:c}.
+
+The \env{gather} environment. For the third line we refer to one of the
+numbers in the first \env{align} structure.
+\begin{subequations}
+\begin{gather}
+\label{gather:a}A+B=B\\
+\label{gather:b}C=D+E\\
+\tag{\ref{align:c}$'$}\label{gather:c}E=F
+\end{gather}
+\end{subequations}
+Label check: that was \eqref{gather:a},
+\eqref{gather:b}, and
+\eqref{gather:c}.
+
+The next \env{subequations} environment encompasses two separate
+equations. A \env{split} environment:
+\begin{subequations}
+\begin{equation}
+\label{split:x}
+\begin{split}
+A&=B+C+F\\
+&=G
+\end{split}
+\end{equation}
+and a \env{multline} environment:
+\begin{multline}\label{multline:x}
+A[B]C[D]E[F]G[[H[I]J[K]L[M]N]]=\\
+H[I]J[K]L[M]N[O]P[Q]R[S]T[U]V[W]X[Y]Z
+\end{multline}
+\end{subequations}
+Label check: That was \eqref{split:x} and \eqref{multline:x}.
+
+\end{document}
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/technote.pdf b/Master/texmf-dist/doc/latex-dev/amsmath/technote.pdf
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@@ -0,0 +1,233 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%% filename = "technote.tex",
+%%% version = "2.0",
+%%% date = "1999/11/15",
+%%% time = "15:07:20 EST",
+%%% checksum = "58506 233 1647 12005",
+%%% author = "American Mathematical Society",
+%%% copyright = "Copyright 1995, 1999 American Mathematical Society,
+%%% all rights reserved. Copying of this file is
+%%% authorized only if either:
+%%% (1) you make absolutely no changes to your copy,
+%%% including name; OR
+%%% (2) if you do make changes, you first rename it
+%%% to some other name.",
+%%% address = "American Mathematical Society,
+%%% Technical Support,
+%%% Electronic Products and Services,
+%%% P. O. Box 6248,
+%%% Providence, RI 02940,
+%%% USA",
+%%% telephone = "401-455-4080 or (in the USA and Canada)
+%%% 800-321-4AMS (321-4267)",
+%%% FAX = "401-331-3842",
+%%% email = "tech-support@ams.org (Internet)",
+%%% supported = "yes",
+%%% keywords = "amsmath, latex",
+%%% abstract = "This is part of the AMS-\LaTeX{} distribution. It
+%%% is a document discussing some technical issues in the
+%%% contents and interface of AMS-\LaTeX{} packages."
+%%% docstring = "The checksum field above contains a CRC-16 checksum
+%%% as the first value, followed by the equivalent of
+%%% the standard UNIX wc (word count) utility output of
+%%% lines, words, and characters. This is produced by
+%%% Robert Solovay's checksum utility.",
+%%% }
+%%% ====================================================================
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must December 1994 or later
+\documentclass{amsdtx}
+
+\title{Technical notes on the \pkg{amsmath} package}
+\author{American Mathematical Society\\Michael Downes}
+\date{1999/10/29}
+
+\markboth
+ {Technical notes on the \pkg{amsmath} package}
+ {Technical notes on the \pkg{amsmath} package}
+\renewcommand{\sectionmark}[1]{}
+
+\providecommand{\etc}[1]{etc.}
+
+\providecommand{\begend}[1]{%
+ {\ntt \symbol{92}begin\symbol{123}#1\symbol{125}}
+ \ldots\ {\ntt\symbol{92}end\symbol{123}#1\symbol{125}}%
+}
+
+\MakeShortVerb{\|}
+
+\begin{document}
+\maketitle
+
+\section{Introduction}
+
+These notes are miscellaneous remarks on some technical questions
+applicable to version 2.0 of \amslatex/.
+
+\section{Deprecated and disallowed commands}
+
+Certain commands that were in the \pkg{amstex} package (the predecessor,
+in \amslatex/ 1.0 and 1.1, of the \pkg{amsmath} package) were moved into
+\pkg{amsxtra} because they seemed to be little-used relics:
+\cn{accentedsymbol}, \qq{sup accents} (\cn{sptilde}, \cn{sphat}, etc.).
+
+Certain other commands\mdash e.g., \cn{over}, \cn{pmatrix}, and
+\cn{cases}\mdash are changed by the \pkg{amsmath} package to produce
+warning messages or error messages, depending on their history; these
+are discussed in more detail in the following sections.
+
+\section{Why do \cn{matrix}, \cn{pmatrix}, and \cn{cases} stop working
+when I add the \pkg{amsmath} package?}
+
+If you used the \fn{plain.tex} versions of \cn{matrix}, \cn{pmatrix}, or
+\cn{cases} in a document and then later converted the document to use
+the \pkg{amsmath} package (or one of the AMS documentclasses, which
+automatically call the \pkg{amsmath} package internally), the instances
+of those commands will produce error messages. The problem is that when
+\latex/ was originally created, it adopted most of its mathematics
+features straight from \fn{plain.tex}. But in the case of \cn{matrix},
+\cn{pmatrix}, \cn{cases} this was a mistake\mdash the \fn{plain.tex}
+syntax for them is decidedly non-\latex/ in style, for example the fact
+that they use \cs{cr} instead of \cn{\\} to mark line breaks, and they
+don't use \cn{begin} and \cn{end}. In basic \latex/ this mistake will be
+perpetuated at least until \latex/3 appears, in order to avoid breaking
+existing documents. But no existing documents that were written with the
+\pkg{amsmath} package have that syntactic problem, as \pkg{amsmath}
+provides proper \latex/-syntax versions of \cn{matrix} and the others.
+The possibility of optionally allowing the \fn{plain.tex} variants to
+make document conversion easier seems ill-advised since those variants
+are so blatantly wrong in a \latex/ context. The \env{array} environment
+ought to have been used instead.
+
+\section{Why do \cn{over}, \cn{atop}, \cn{above} [\dots{\ntt withdelims}]
+ give a warning when I add the \pkg{amsmath} package?}
+
+Using the six generalized fraction commands \cn{over},
+\cn{overwithdelims}, \cn{atop}, \cn{atopwithdelims}, \cn{above},
+\cn{abovewithdelims} is not really a good idea in \latex/, for reasons
+explained below. I construe \latex/'s provision of \cs{frac}, and the
+lack of any mention in the \latex/ book of the primitive fraction
+commands, as an implicit injunction against their use, although I don't
+think Lamport actually spent a lot of time pondering the issue, and the
+basic \latex/ version of \cn{frac} provides access only to \cn{over},
+not to \cn{atop}, \cn{above}, or the \verb'withdelims' variants. The
+\pkg{amsmath} package, however, provides a \cn{genfrac} command that
+gives user-level access to all six of the generalized fraction
+primitives in a way that conforms to the syntactic conventions followed
+by all other \latex/ commands.
+
+Not only is the unusual syntax of the \tex/ primitives rather out of
+place in \latex/, but furthermore that syntax seems to be responsible
+for one of the most significant flaws in \tex/'s mathematical
+typesetting capabilities: the fact that the current mathstyle at any
+given point in a math formula cannot be determined until the end of the
+formula, because of the possibility that a following generalized
+fraction command will change the mathstyle of the \emph{preceding}
+material. To cite two of the worst side effects: \cn{mathchoice} must
+actually typeset all four of its arguments, instead of being able to
+immediately select only one; and, were it possible to always know the
+current math style at a given point, math font selection would be
+greatly simplified and the upper limit of 16 different math font
+\cn{fam}s would never be a problem as \cn{text,script[script]font}
+assignments for any \cn{fam} could take immediate effect and therefore
+could be changed arbitrarily often within a single formula. More
+concretely, math font selection difficulties are responsible for many of
+the more convoluted passages in the source code of \latex/'s NFSS (that
+does font loading on demand) and of the \pkg{amsmath} package, and by
+extension it has historically been responsible for significant delays in
+making new features available to end users and for making those features
+more prone to bugs.
+
+There are additional bad consequences following from the syntax of those
+generalized fraction commands that only become evident when you do some
+writing of nontrivial macros for math use. For example, as things
+currently stand you cannot measure the size of any object in math
+without going through \cn{mathchoice} and \emph{leaving and reentering
+math mode} via \verb'\hbox{$' (which then introduces complications
+regarding \cn{everymath} and \cn{mathsurround}). And it seems that
+uncertainty about the current mathstyle is the only barrier to allowing
+the use of mu units with \cn{vrule}, to make vertical struts in
+constructing compound symbols or notation. And so on and so forth.
+
+\section{The \opt{fleqn} option and \cn{mathindent}}
+
+Strictly speaking, the \pkg{amsmath} package doesn't use \cn{mathindent}
+to control the left indent of displayed equations when the \opt{fleqn}
+option is in effect: it uses an internal parameter \cs{@mathmargin}
+instead. However, for compatibility with existing \latex/ documentation,
+\pkg{amsmath} turns \cn{mathindent} into an alias for \cs{@mathmargin}.
+There is a small risk here: In the plain \latex/ implementation,
+\cn{mathindent} is a dimen register, but with \pkg{amsmath}
+\cs{@mathmargin} is a skip register and, by association, so is
+\cn{mathindent}. If any package or documentclass uses \cn{mathindent} in
+a way that depends on it being a dimen register, when used in
+conjunction with the \pkg{amsmath} package it may be vulnerable to a
+well-known pitfall having to do with the primitive \tex/ lookahead for a
+\texttt{plus} or \texttt{minus} key word. However if the standard
+\latex/ commands \cn{setlength} and \cn{addtolength} are used to modify
+\cn{mathindent} then this problem will not arise.
+
+\section{Why can't I use abbreviations for \protect\begend{align}?}
+
+Authors often like to use abbreviations such as |\beq| |\eeq| for
+|\begin{equation}| |\end{equation}|. For some environments defined by
+the \pkg{amsmath} package, such as \env{align}, \env{gather},
+\env{multline}, and others of the same general type, this does not work:
+An attempt to define |\bal| |\eal| as shorthand for |\begin{align}|
+|\end{align}| will fail with a puzzling error message. This has to do
+with unfortunately nontrivial technical complications: the given
+environments must read their contents as a delimited macro argument
+because they do multipass processing of the contents using algorithms
+inherited from Spivak's \fn{amstex.tex}. The obvious
+solution\mdash substitution of different algorithms that do box shuffling
+instead of token shuffling for the multipass calculations\mdash would
+require rewriting these display environments from the ground up; while
+that is a worthy goal, it was beyond the original scope of the
+\amslatex/ project. Work is under way on an auxiliary package called
+\pkg{breqn} that addresses not only this problem but a number of others;
+at the time of this writing, however [September 1999] it has only
+progressed as far as a beta release.
+
+Some workarounds:
+\begin{itemize}
+\item |\def\bal#1\eal{\begin{align}#1\end{align}}|
+\item Define |\newcommand{\env}[2]{\begin{#1}#2\end{#1}}| and then use
+ |\env{align}{...}|
+\end{itemize}
+
+\section{The \pkg{upref} package}
+
+The reason for splitting out the \pkg{upref} package instead of
+automatically incorporating it in the \cls{amsart} and \cls{amsbook}
+classes is this: It involves low-level surgery on an important \latex/
+command. This means that if ever this command changes in the future (as
+it did between versions 2.09 and 2e of \latex/) we have a maintenance
+problem. And the benefit that \pkg{upref} provides is something that
+most end users don't care much about.
+
+\section{The \pkg{amsintx} package}
+
+After a very preliminary trial release the \pkg{amsintx} package was
+withdrawn to await further development. It is still considered a worthy
+project but has simply not yet made it to the head of the priority queue
+in the AMS \latex/ development program, as it has had to contend with a
+number of other equally worthy projects for development time.
+
+\section{Hyphenation in the documentation}
+
+Hyphenation was allowed for certain long command names in
+\fn{amsldoc.tex}; this presented technical difficulties because \latex/
+normally deactivates hyphenation for tt fonts. The method chosen to
+reinstate hyphenation was to turn off the encoding-specific function
+\cs{OT1+cmtt} that disables the \cs{hyphenchar} for tt fonts; see the
+definition of \cn{allowtthyphens} in \fn{amsdtx.dtx}. Then a list of all
+tt words in the document was gathered (from the \fn{.idx} file, produced
+by the \cn{cn}, \cn{fn}, \cn{pkg}, etc. commands) and \cn{showhyphens}
+was applied to this list. The result was another list in the resulting
+\tex/ log, containing those words in a form suitable for the argument of
+\cn{hyphenation}. That list was then edited by hand to overrule
+undesirable hyphenations; words with acceptable hyphenations were
+dropped from the list, as they don't need to be repeated there.
+
+\end{document}
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/testmath.pdf b/Master/texmf-dist/doc/latex-dev/amsmath/testmath.pdf
new file mode 100644
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diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex b/Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex
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@@ -0,0 +1,2342 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%% filename = "testmath.tex",
+%%% version = "2.0",
+%%% date = "1999/11/15",
+%%% time = "15:09:17 EST",
+%%% checksum = "07762 2342 7811 82371",
+%%% author = "American Mathematical Society",
+%%% copyright = "Copyright 1995, 1999 American Mathematical Society,
+%%% all rights reserved. Copying of this file is
+%%% authorized only if either:
+%%% (1) you make absolutely no changes to your copy,
+%%% including name; OR
+%%% (2) if you do make changes, you first rename it
+%%% to some other name.",
+%%% address = "American Mathematical Society,
+%%% Technical Support,
+%%% Electronic Products and Services,
+%%% P. O. Box 6248,
+%%% Providence, RI 02940,
+%%% USA",
+%%% telephone = "401-455-4080 or (in the USA and Canada)
+%%% 800-321-4AMS (321-4267)",
+%%% FAX = "401-331-3842",
+%%% email = "tech-support@ams.org (Internet)",
+%%% codetable = "ISO/ASCII",
+%%% keywords = "latex, amsmath, examples, documentation",
+%%% supported = "yes",
+%%% abstract = "This is a test file containing extensive examples of
+%%% mathematical constructs supported by the amsmath
+%%% package.",
+%%% docstring = "The checksum field above contains a CRC-16
+%%% checksum as the first value, followed by the
+%%% equivalent of the standard UNIX wc (word
+%%% count) utility output of lines, words, and
+%%% characters. This is produced by Robert
+%%% Solovay's checksum utility.",
+%%% }
+%%% ====================================================================
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must December 1994 or later
+\documentclass[draft]{article}
+\pagestyle{headings}
+
+\title{Sample Paper for the \pkg{amsmath} Package\\
+File name: \fn{testmath.tex}}
+\author{American Mathematical Society}
+\date{Version 2.0, 1999/11/15}
+
+\usepackage{amsmath,amsthm}
+
+% Some definitions useful in producing this sort of documentation:
+\chardef\bslash=`\\ % p. 424, TeXbook
+% Normalized (nonbold, nonitalic) tt font, to avoid font
+% substitution warning messages if tt is used inside section
+% headings and other places where odd font combinations might
+% result.
+\newcommand{\ntt}{\normalfont\ttfamily}
+% command name
+\newcommand{\cn}[1]{{\protect\ntt\bslash#1}}
+% LaTeX package name
+\newcommand{\pkg}[1]{{\protect\ntt#1}}
+% File name
+\newcommand{\fn}[1]{{\protect\ntt#1}}
+% environment name
+\newcommand{\env}[1]{{\protect\ntt#1}}
+\hfuzz1pc % Don't bother to report overfull boxes if overage is < 1pc
+
+% Theorem environments
+
+%% \theoremstyle{plain} %% This is the default
+\newtheorem{thm}{Theorem}[section]
+\newtheorem{cor}[thm]{Corollary}
+\newtheorem{lem}[thm]{Lemma}
+\newtheorem{prop}[thm]{Proposition}
+\newtheorem{ax}{Axiom}
+
+\theoremstyle{definition}
+\newtheorem{defn}{Definition}[section]
+
+\theoremstyle{remark}
+\newtheorem{rem}{Remark}[section]
+\newtheorem*{notation}{Notation}
+
+%\numberwithin{equation}{section}
+
+\newcommand{\thmref}[1]{Theorem~\ref{#1}}
+\newcommand{\secref}[1]{\S\ref{#1}}
+\newcommand{\lemref}[1]{Lemma~\ref{#1}}
+
+\newcommand{\bysame}{\mbox{\rule{3em}{.4pt}}\,}
+
+% Math definitions
+
+\newcommand{\A}{\mathcal{A}}
+\newcommand{\B}{\mathcal{B}}
+\newcommand{\st}{\sigma}
+\newcommand{\XcY}{{(X,Y)}}
+\newcommand{\SX}{{S_X}}
+\newcommand{\SY}{{S_Y}}
+\newcommand{\SXY}{{S_{X,Y}}}
+\newcommand{\SXgYy}{{S_{X|Y}(y)}}
+\newcommand{\Cw}[1]{{\hat C_#1(X|Y)}}
+\newcommand{\G}{{G(X|Y)}}
+\newcommand{\PY}{{P_{\mathcal{Y}}}}
+\newcommand{\X}{\mathcal{X}}
+\newcommand{\wt}{\widetilde}
+\newcommand{\wh}{\widehat}
+
+\DeclareMathOperator{\per}{per}
+\DeclareMathOperator{\cov}{cov}
+\DeclareMathOperator{\non}{non}
+\DeclareMathOperator{\cf}{cf}
+\DeclareMathOperator{\add}{add}
+\DeclareMathOperator{\Cham}{Cham}
+\DeclareMathOperator{\IM}{Im}
+\DeclareMathOperator{\esssup}{ess\,sup}
+\DeclareMathOperator{\meas}{meas}
+\DeclareMathOperator{\seg}{seg}
+
+% \interval is used to provide better spacing after a [ that
+% is used as a closing delimiter.
+\newcommand{\interval}[1]{\mathinner{#1}}
+
+% Notation for an expression evaluated at a particular condition. The
+% optional argument can be used to override automatic sizing of the
+% right vert bar, e.g. \eval[\biggr]{...}_{...}
+\newcommand{\eval}[2][\right]{\relax
+ \ifx#1\right\relax \left.\fi#2#1\rvert}
+
+% Enclose the argument in vert-bar delimiters:
+\newcommand{\envert}[1]{\left\lvert#1\right\rvert}
+\let\abs=\envert
+
+% Enclose the argument in double-vert-bar delimiters:
+\newcommand{\enVert}[1]{\left\lVert#1\right\rVert}
+\let\norm=\enVert
+
+\begin{document}
+\maketitle
+\markboth{Sample paper for the {\protect\ntt\lowercase{amsmath}} package}
+{Sample paper for the {\protect\ntt\lowercase{amsmath}} package}
+\renewcommand{\sectionmark}[1]{}
+
+\section{Introduction}
+
+This paper contains examples of various features from \AmS-\LaTeX{}.
+
+\section{Enumeration of Hamiltonian paths in a graph}
+
+Let $\mathbf{A}=(a_{ij})$ be the adjacency matrix of graph $G$. The
+corresponding Kirchhoff matrix $\mathbf{K}=(k_{ij})$ is obtained from
+$\mathbf{A}$ by replacing in $-\mathbf{A}$ each diagonal entry by the
+degree of its corresponding vertex; i.e., the $i$th diagonal entry is
+identified with the degree of the $i$th vertex. It is well known that
+\begin{equation}
+\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
+\quad i=1,\dots,n
+\end{equation}
+where $\mathbf{K}(i|i)$ is the $i$th principal submatrix of
+$\mathbf{K}$.
+\begin{verbatim}
+\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
+\end{verbatim}
+
+Let $C_{i(j)}$ be the set of graphs obtained from $G$ by attaching edge
+$(v_iv_j)$ to each spanning tree of $G$. Denote by $C_i=\bigcup_j
+C_{i(j)}$. It is obvious that the collection of Hamiltonian cycles is a
+subset of $C_i$. Note that the cardinality of $C_i$ is $k_{ii}\det
+\mathbf{K}(i|i)$. Let $\wh X=\{\hat x_1,\dots,\hat x_n\}$.
+\begin{verbatim}
+$\wh X=\{\hat x_1,\dots,\hat x_n\}$
+\end{verbatim}
+Define multiplication for the elements of $\wh X$ by
+\begin{equation}\label{multdef}
+\hat x_i\hat x_j=\hat x_j\hat x_i,\quad \hat x^2_i=0,\quad
+i,j=1,\dots,n.
+\end{equation}
+Let $\hat k_{ij}=k_{ij}\hat x_j$ and $\hat k_{ij}=-\sum_{j\not=i} \hat
+k_{ij}$. Then the number of Hamiltonian cycles $H_c$ is given by the
+relation \cite{liuchow:formalsum}
+\begin{equation}\label{H-cycles}
+\biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det
+\wh{\mathbf{K}}(i|i),\qquad i=1,\dots,n.
+\end{equation}
+The task here is to express \eqref{H-cycles}
+in a form free of any $\hat x_i$,
+$i=1,\dots,n$. The result also leads to the resolution of enumeration of
+Hamiltonian paths in a graph.
+
+It is well known that the enumeration of Hamiltonian cycles and paths in
+a complete graph $K_n$ and in a complete bipartite graph $K_{n_1n_2}$
+can only be found from \textit{first combinatorial principles}
+\cite{hapa:graphenum}. One wonders if there exists a formula which can
+be used very efficiently to produce $K_n$ and $K_{n_1n_2}$. Recently,
+using Lagrangian methods, Goulden and Jackson have shown that $H_c$ can
+be expressed in terms of the determinant and permanent of the adjacency
+matrix \cite{gouja:lagrmeth}. However, the formula of Goulden and
+Jackson determines neither $K_n$ nor $K_{n_1n_2}$ effectively. In this
+paper, using an algebraic method, we parametrize the adjacency matrix.
+The resulting formula also involves the determinant and permanent, but
+it can easily be applied to $K_n$ and $K_{n_1n_2}$. In addition, we
+eliminate the permanent from $H_c$ and show that $H_c$ can be
+represented by a determinantal function of multivariables, each variable
+with domain $\{0,1\}$. Furthermore, we show that $H_c$ can be written by
+number of spanning trees of subgraphs. Finally, we apply the formulas to
+a complete multigraph $K_{n_1\dots n_p}$.
+
+The conditions $a_{ij}=a_{ji}$, $i,j=1,\dots,n$, are not required in
+this paper. All formulas can be extended to a digraph simply by
+multiplying $H_c$ by 2.
+
+\section{Main Theorem}
+\label{s:mt}
+
+\begin{notation} For $p,q\in P$ and $n\in\omega$ we write
+$(q,n)\le(p,n)$ if $q\le p$ and $A_{q,n}=A_{p,n}$.
+\begin{verbatim}
+\begin{notation} For $p,q\in P$ and $n\in\omega$
+...
+\end{notation}
+\end{verbatim}
+\end{notation}
+
+Let $\mathbf{B}=(b_{ij})$ be an $n\times n$ matrix. Let $\mathbf{n}=\{1,
+\dots,n\}$. Using the properties of \eqref{multdef}, it is readily seen
+that
+
+\begin{lem}\label{lem-per}
+\begin{equation}
+\prod_{i\in\mathbf{n}}
+\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat x_i\biggr)
+=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)\per \mathbf{B}
+\end{equation}
+where $\per \mathbf{B}$ is the permanent of $\mathbf{B}$.
+\end{lem}
+
+Let $\wh Y=\{\hat y_1,\dots,\hat y_n\}$. Define multiplication
+for the elements of $\wh Y$ by
+\begin{equation}
+\hat y_i\hat y_j+\hat y_j\hat y_i=0,\quad i,j=1,\dots,n.
+\end{equation}
+Then, it follows that
+\begin{lem}\label{lem-det}
+\begin{equation}\label{detprod}
+\prod_{i\in\mathbf{n}}
+\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat y_j\biggr)
+=\biggl(\prod_{\,i\in\mathbf{n}}\hat y_i\biggr)\det\mathbf{B}.
+\end{equation}
+\end{lem}
+
+Note that all basic properties of determinants are direct consequences
+of Lemma ~\ref{lem-det}. Write
+\begin{equation}\label{sum-bij}
+\sum_{j\in\mathbf{n}}b_{ij}\hat y_j=\sum_{j\in\mathbf{n}}b^{(\lambda)}
+_{ij}\hat y_j+(b_{ii}-\lambda_i)\hat y_i\hat y
+\end{equation}
+where
+\begin{equation}
+b^{(\lambda)}_{ii}=\lambda_i,\quad b^{(\lambda)}_{ij}=b_{ij},
+\quad i\not=j.
+\end{equation}
+Let $\mathbf{B}^{(\lambda)}=(b^{(\lambda)}_{ij})$. By \eqref{detprod}
+and \eqref{sum-bij}, it is
+straightforward to show the following
+result:
+\begin{thm}\label{thm-main}
+\begin{equation}\label{detB}
+\det\mathbf{B}=
+\sum^n_{l =0}\sum_{I_l \subseteq n}
+\prod_{i\in I_l}(b_{ii}-\lambda_i)
+\det\mathbf{B}^{(\lambda)}(I_l |I_l ),
+\end{equation}
+where $I_l =\{i_1,\dots,i_l \}$ and $\mathbf{B}^{(\lambda)}(I_l |I_l )$
+is the principal submatrix obtained from $\mathbf{B}^{(\lambda)}$
+by deleting its $i_1,\dots,i_l $ rows and columns.
+\end{thm}
+
+\begin{rem}
+Let $\mathbf{M}$ be an $n\times n$ matrix. The convention
+$\mathbf{M}(\mathbf{n}|\mathbf{n})=1$ has been used in \eqref{detB} and
+hereafter.
+\end{rem}
+
+Before proceeding with our discussion, we pause to note that
+\thmref{thm-main} yields immediately a fundamental formula which can be
+used to compute the coefficients of a characteristic polynomial
+\cite{mami:matrixth}:
+\begin{cor}\label{BI}
+Write $\det(\mathbf{B}-x\mathbf{I})=\sum^n_{l =0}(-1)
+^l b_l x^l $. Then
+\begin{equation}\label{bl-sum}
+b_l =\sum_{I_l \subseteq\mathbf{n}}\det\mathbf{B}(I_l |I_l ).
+\end{equation}
+\end{cor}
+Let
+\begin{equation}
+\mathbf{K}(t,t_1,\dots,t_n)
+=\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
+-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
+\hdotsfor[2]{4}\\
+-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix},
+\end{equation}
+\begin{verbatim}
+\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
+-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
+\hdotsfor[2]{4}\\
+-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
+\end{verbatim}
+where
+\begin{equation}
+D_i=\sum_{j\in\mathbf{n}}a_{ij}t_j,\quad i=1,\dots,n.
+\end{equation}
+
+Set
+\begin{equation*}
+D(t_1,\dots,t_n)=\frac{\delta}{\delta t}\eval{\det\mathbf{K}(t,t_1,\dots,t_n)
+}_{t=1}.
+\end{equation*}
+Then
+\begin{equation}\label{sum-Di}
+D(t_1,\dots,t_n)
+=\sum_{i\in\mathbf{n}}D_i\det\mathbf{K}(t=1,t_1,\dots,t_n; i|i),
+\end{equation}
+where $\mathbf{K}(t=1,t_1,\dots,t_n; i|i)$ is the $i$th principal
+submatrix of $\mathbf{K}(t=1,t_1,\dots,t_n)$.
+
+Theorem ~\ref{thm-main} leads to
+\begin{equation}\label{detK1}
+\det\mathbf{K}(t_1,t_1,\dots,t_n)
+=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}t^{n-\envert{I}}
+\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}
+^{(\lambda t)}(\overline{I}|\overline I).
+\end{equation}
+Note that
+\begin{equation}\label{detK2}
+\det\mathbf{K}(t=1,t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}
+\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}
+^{(\lambda)}(\overline{I}|\overline{I})=0.
+\end{equation}
+
+Let $t_i=\hat x_i,i=1,\dots,n$. Lemma ~\ref{lem-per} yields
+\begin{multline}
+\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
+\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
+=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
+\sum_{I\subseteq\mathbf{n}-\{l \}}
+(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)
+\det\mathbf{A}^{(\lambda)}
+(\overline I\cup\{l \}|\overline I\cup\{l \}).
+\label{sum-ali}
+\end{multline}
+\begin{verbatim}
+\begin{multline}
+\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
+\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
+=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
+\sum_{I\subseteq\mathbf{n}-\{l \}}
+(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)
+\det\mathbf{A}^{(\lambda)}
+(\overline I\cup\{l \}|\overline I\cup\{l \}).
+\label{sum-ali}
+\end{multline}
+\end{verbatim}
+
+By \eqref{H-cycles}, \eqref{detprod}, and \eqref{sum-bij}, we have
+\begin{prop}\label{prop:eg}
+\begin{equation}
+H_c=\frac1{2n}\sum^n_{l =0}(-1)^{l}
+D_{l},
+\end{equation}
+where
+\begin{equation}\label{delta-l}
+D_{l}=\eval[2]{\sum_{I_{l}\subseteq \mathbf{n}}
+D(t_1,\dots,t_n)}_{t_i=\left\{\begin{smallmatrix}
+0,& \text{if }i\in I_{l}\quad\\% \quad added for centering
+1,& \text{otherwise}\end{smallmatrix}\right.\;,\;\; i=1,\dots,n}.
+\end{equation}
+\end{prop}
+
+\section{Application}
+\label{lincomp}
+
+We consider here the applications of Theorems~\ref{th-info-ow-ow} and
+~\ref{th-weak-ske-owf} to a complete
+multipartite graph $K_{n_1\dots n_p}$. It can be shown that the
+number of spanning trees of $K_{n_1\dots n_p}$
+may be written
+\begin{equation}\label{e:st}
+T=n^{p-2}\prod^p_{i=1}
+(n-n_i)^{n_i-1}
+\end{equation}
+where
+\begin{equation}
+n=n_1+\dots+n_p.
+\end{equation}
+
+It follows from Theorems~\ref{th-info-ow-ow} and
+~\ref{th-weak-ske-owf} that
+\begin{equation}\label{e:barwq}
+\begin{split}
+H_c&=\frac1{2n}
+\sum^n_{{l}=0}(-1)^{l}(n-{l})^{p-2}
+\sum_{l _1+\dots+l _p=l}\prod^p_{i=1}
+\binom{n_i}{l _i}\\
+&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
+\biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\end{split}
+\end{equation}
+\begin{verbatim}
+... \binom{n_i}{l _i}\\
+\end{verbatim}
+and
+\begin{equation}\label{joe}
+\begin{split}
+H_c&=\frac12\sum^{n-1}_{l =0}
+(-1)^{l}(n-l )^{p-2}
+\sum_{l _1+\dots+l _p=l}
+\prod^p_{i=1}\binom{n_i}{l _i}\\
+&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}
+\left(1-\frac{l _p}{n_p}\right)
+[(n-l )-(n_p-l _p)].
+\end{split}
+\end{equation}
+
+The enumeration of $H_c$ in a $K_{n_1\dotsm n_p}$ graph can also be
+carried out by Theorem ~\ref{thm-H-param} or ~\ref{thm-asym}
+together with the algebraic method of \eqref{multdef}.
+Some elegant representations may be obtained. For example, $H_c$ in
+a $K_{n_1n_2n_3}$ graph may be written
+\begin{equation}\label{j:mark}
+\begin{split}
+H_c=&
+\frac{n_1!\,n_2!\,n_3!}
+{n_1+n_2+n_3}\sum_i\left[\binom{n_1}{i}
+\binom{n_2}{n_3-n_1+i}\binom{n_3}{n_3-n_2+i}\right.\\
+&+\left.\binom{n_1-1}{i}
+\binom{n_2-1}{n_3-n_1+i}
+\binom{n_3-1}{n_3-n_2+i}\right].\end{split}
+\end{equation}
+
+\section{Secret Key Exchanges}
+\label{SKE}
+
+Modern cryptography is fundamentally concerned with the problem of
+secure private communication. A Secret Key Exchange is a protocol
+where Alice and Bob, having no secret information in common to start,
+are able to agree on a common secret key, conversing over a public
+channel. The notion of a Secret Key Exchange protocol was first
+introduced in the seminal paper of Diffie and Hellman
+\cite{dihe:newdir}. \cite{dihe:newdir} presented a concrete
+implementation of a Secret Key Exchange protocol, dependent on a
+specific assumption (a variant on the discrete log), specially
+tailored to yield Secret Key Exchange. Secret Key Exchange is of
+course trivial if trapdoor permutations exist. However, there is no
+known implementation based on a weaker general assumption.
+
+The concept of an informationally one-way function was introduced
+in \cite{imlelu:oneway}. We give only an informal definition here:
+
+\begin{defn} A polynomial time
+computable function $f = \{f_k\}$ is informationally
+one-way if there is no probabilistic polynomial time algorithm which
+(with probability of the form $1 - k^{-e}$ for some $e > 0$)
+returns on input $y \in \{0,1\}^{k}$ a random element of $f^{-1}(y)$.
+\end{defn}
+In the non-uniform setting \cite{imlelu:oneway} show that these are not
+weaker than one-way functions:
+\begin{thm}[\cite{imlelu:oneway} (non-uniform)]
+\label{th-info-ow-ow}
+The existence of informationally one-way functions
+implies the existence of one-way functions.
+\end{thm}
+We will stick to the convention introduced above of saying
+``non-uniform'' before the theorem statement when the theorem
+makes use of non-uniformity. It should be understood that
+if nothing is said then the result holds for both the uniform and
+the non-uniform models.
+
+It now follows from \thmref{th-info-ow-ow} that
+
+\begin{thm}[non-uniform]\label{th-weak-ske-owf} Weak SKE
+implies the existence of a one-way function.
+\end{thm}
+
+More recently, the polynomial-time, interior point algorithms for linear
+programming have been extended to the case of convex quadratic programs
+\cite{moad:quadpro,ye:intalg}, certain linear complementarity problems
+\cite{komiyo:lincomp,miyoki:lincomp}, and the nonlinear complementarity
+problem \cite{komiyo:unipfunc}. The connection between these algorithms
+and the classical Newton method for nonlinear equations is well
+explained in \cite{komiyo:lincomp}.
+
+\section{Review}
+\label{computation}
+
+We begin our discussion with the following definition:
+
+\begin{defn}
+
+A function $H\colon \Re^n \to \Re^n$ is said to be
+\emph{B-differentiable} at the point $z$ if (i)~$H$ is Lipschitz
+continuous in a neighborhood of $z$, and (ii)~ there exists a positive
+homogeneous function $BH(z)\colon \Re^n \to \Re^n$, called the
+\emph{B-derivative} of $H$ at $z$, such that
+\[ \lim_{v \to 0} \frac{H(z+v) - H(z) - BH(z)v}{\enVert{v}} = 0. \]
+The function $H$ is \textit{B-differentiable in set $S$} if it is
+B-differentiable at every point in $S$. The B-derivative $BH(z)$ is said
+to be \textit{strong} if
+\[ \lim_{(v,v') \to (0,0)} \frac{H(z+v) - H(z+v') - BH(z)(v
+ -v')}{\enVert{v - v'}} = 0. \]
+\end{defn}
+
+
+\begin{lem}\label{limbog} There exists a smooth function $\psi_0(z)$
+defined for $\abs{z}>1-2a$ satisfying the following properties\textup{:}
+\begin{enumerate}
+\renewcommand{\labelenumi}{(\roman{enumi})}
+\item $\psi_0(z)$ is bounded above and below by positive constants
+$c_1\leq \psi_0(z)\leq c_2$.
+\item If $\abs{z}>1$, then $\psi_0(z)=1$.
+\item For all $z$ in the domain of $\psi_0$, $\Delta_0\ln \psi_0\geq 0$.
+\item If $1-2a<\abs{z}<1-a$, then $\Delta_0\ln \psi_0\geq
+c_3>0$.
+\end{enumerate}
+\end{lem}
+
+\begin{proof}
+We choose $\psi_0(z)$ to be a radial function depending only on $r=\abs{z}$.
+Let $h(r)\geq 0$ be a suitable smooth function satisfying $h(r)\geq c_3$
+for $1-2a<\abs{z}<1-a$, and $h(r)=0$ for $\abs{z}>1-\tfrac a2$. The radial
+Laplacian
+\[\Delta_0\ln\psi_0(r)=\left(\frac {d^2}{dr^2}+\frac
+1r\frac d{dr}\right)\ln\psi_0(r)\]
+has smooth coefficients for $r>1-2a$. Therefore, we may
+apply the existence and uniqueness theory for ordinary differential
+equations. Simply let $\ln \psi_0(r)$ be the solution of the differential
+equation
+\[\left(\frac{d^2}{dr^2}+\frac 1r\frac d{dr}\right)\ln \psi_0(r)=h(r)\]
+with initial conditions given by $\ln \psi_0(1)=0$ and
+$\ln\psi_0'(1)=0$.
+
+Next, let $D_\nu$ be a finite collection of pairwise disjoint disks,
+all of which are contained in the unit disk centered at the origin in
+$C$. We assume that $D_\nu=\{z\mid \abs{z-z_\nu}<\delta\}$. Suppose that
+$D_\nu(a)$ denotes the smaller concentric disk $D_\nu(a)=\{z\mid
+\abs{z-z_\nu}\leq (1-2a)\delta\}$. We define a smooth weight function
+$\Phi_0(z)$ for $z\in C-\bigcup_\nu D_\nu(a)$ by setting $\Phi_
+0(z)=1$ when $z\notin \bigcup_\nu D_\nu$ and $\Phi_
+0(z)=\psi_0((z-z_\nu)/\delta)$ when $z$ is an element of $D_\nu$. It
+follows from \lemref{limbog} that $\Phi_ 0$ satisfies the properties:
+\begin{enumerate}
+\renewcommand{\labelenumi}{(\roman{enumi})}
+\item \label{boundab}$\Phi_ 0(z)$ is bounded above and below by
+positive constants $c_1\leq \Phi_ 0(z)\leq c_2$.
+\item \label{d:over}$\Delta_0\ln\Phi_ 0\geq 0$ for all
+$z\in C-\bigcup_\nu D_\nu(a)$,
+the domain where the function $\Phi_ 0$ is defined.
+\item \label{d:ad}$\Delta_0\ln\Phi_ 0\geq c_3\delta^{-2}$
+when $(1-2a)\delta<\abs{z-z_\nu}<(1-a)\delta$.
+\end{enumerate}
+Let $A_\nu$ denote the annulus $A_\nu=\{(1-2a)\delta<\abs{z-z_\nu}<(1-a)
+\delta \}$, and set $A=\bigcup_\nu A_\nu$. The
+properties (\ref{d:over}) and (\ref{d:ad}) of $\Phi_ 0$
+may be summarized as $\Delta_0\ln \Phi_ 0\geq c_3\delta^{-2}\chi_A$,
+where $\chi _A$ is the characteristic function of $A$.
+\end{proof}
+
+Suppose that $\alpha$ is a nonnegative real constant. We apply
+Proposition~\ref{prop:eg} with $\Phi(z)=\Phi_ 0(z) e^{\alpha\abs{z}^2}$. If
+$u\in C^\infty_0(R^2-\bigcup_\nu D_\nu(a))$, assume that $\mathcal{D}$
+is a bounded domain containing the support of $u$ and $A\subset
+\mathcal{D}\subset R^2-\bigcup_\nu D_\nu(a)$. A calculation gives
+\[\int_{\mathcal{D}}\abs{\overline\partial u}^2\Phi_ 0(z) e^{\alpha\abs{z}^2}
+\geq c_4\alpha\int_{\mathcal{D}}\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2}
++c_5\delta^{-2}\int_ A\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2}.\]
+
+The boundedness, property (\ref{boundab}) of $\Phi_ 0$, then yields
+\[\int_{\mathcal{D}}\abs{\overline\partial u}^2e^{\alpha\abs{z}^2}\geq c_6\alpha
+\int_{\mathcal{D}}\abs{u}^2e^{\alpha\abs{z}^2}
++c_7\delta^{-2}\int_ A\abs{u}^2e^{\alpha\abs{z}^2}.\]
+
+Let $B(X)$ be the set of blocks of $\Lambda_{X}$
+and let $b(X) = \abs{B(X)}$. If $\phi \in Q_{X}$ then
+$\phi$ is constant on the blocks of $\Lambda_{X}$.
+\begin{equation}\label{far-d}
+ P_{X} = \{ \phi \in M \mid \Lambda_{\phi} = \Lambda_{X} \},
+\qquad
+Q_{X} = \{\phi \in M \mid \Lambda_{\phi} \geq \Lambda_{X} \}.
+\end{equation}
+If $\Lambda_{\phi} \geq \Lambda_{X}$ then
+$\Lambda_{\phi} = \Lambda_{Y}$ for some $Y \geq X$ so that
+\[ Q_{X} = \bigcup_{Y \geq X} P_{Y}. \]
+Thus by M\"obius inversion
+\[ \abs{P_{Y}}= \sum_{X\geq Y} \mu (Y,X)\abs{Q_{X}}.\]
+Thus there is a bijection from $Q_{X}$ to $W^{B(X)}$.
+In particular $\abs{Q_{X}} = w^{b(X)}$.
+
+Next note that $b(X)=\dim X$. We see this by choosing a
+basis for $X$ consisting of vectors $v^{k}$ defined by
+\[v^{k}_{i}=
+\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\
+0 &\text{otherwise.} \end{cases}
+\]
+\begin{verbatim}
+\[v^{k}_{i}=
+\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\
+0 &\text{otherwise.} \end{cases}
+\]
+\end{verbatim}
+
+\begin{lem}\label{p0201}
+Let $\A$ be an arrangement. Then
+\[ \chi (\A,t) = \sum_{\B \subseteq \A}
+(-1)^{\abs{\B}} t^{\dim T(\B)}. \]
+\end{lem}
+
+In order to compute $R''$ recall the definition
+of $S(X,Y)$ from \lemref{lem-per}. Since $H \in \B$,
+$\A_{H} \subseteq \B$. Thus if $T(\B) = Y$ then
+$\B \in S(H,Y)$. Let $L'' = L(\A'')$. Then
+\begin{equation}\label{E_SXgYy}
+\begin{split}
+R''&= \sum_{H\in \B \subseteq \A} (-1)^{\abs{\B}}
+t^{\dim T(\B)}\\
+&= \sum_{Y \in L''} \sum_{\B \in S(H,Y)}
+(-1)^{\abs{\B}}t^{\dim Y} \\
+&= -\sum_{Y \in L''} \sum_{\B \in S(H,Y)} (-1)^
+{\abs{\B - \A_{H}}} t^{\dim Y} \\
+&= -\sum_{Y \in L''} \mu (H,Y)t^{\dim Y} \\
+&= -\chi (\A '',t).
+\end{split}
+\end{equation}
+
+\begin{cor}\label{tripleA}
+Let $(\A,\A',\A'')$ be a triple of arrangements. Then
+\[ \pi (\A,t) = \pi (\A',t) + t \pi (\A'',t). \]
+\end{cor}
+
+\begin{defn}
+Let $(\A,\A',\A'')$ be a triple with respect to
+the hyperplane $H \in \A$. Call $H$ a \textit{separator}
+if $T(\A) \not\in L(\A')$.
+\end{defn}
+
+\begin{cor}\label{nsep}
+Let $(\A,\A',\A'')$ be a triple with respect to $H \in \A$.
+\begin{enumerate}
+\renewcommand{\labelenumi}{(\roman{enumi})}
+\item
+If $H$ is a separator then
+\[ \mu (\A) = - \mu (\A'') \]
+and hence
+\[ \abs{\mu (\A)} = \abs{ \mu (\A'')}. \]
+
+\item If $H$ is not a separator then
+\[\mu (\A) = \mu (\A') - \mu (\A'') \]
+and
+\[ \abs{\mu (\A)} = \abs{\mu (\A')} + \abs{\mu (\A'')}. \]
+\end{enumerate}
+\end{cor}
+
+\begin{proof}
+It follows from \thmref{th-info-ow-ow} that $\pi(\A,t)$
+has leading term
+\[(-1)^{r(\A)}\mu (\A)t^{r(\A)}.\]
+The conclusion
+follows by comparing coefficients of the leading
+terms on both sides of the equation in
+Corollary~\ref{tripleA}. If $H$ is a separator then
+$r(\A') < r(\A)$ and there is no contribution
+from $\pi (\A',t)$.
+\end{proof}
+
+The Poincar\'e polynomial of an arrangement
+will appear repeatedly
+in these notes. It will be shown to equal the
+Poincar\'e polynomial
+of the graded algebras which we are going to
+associate with $\A$. It is also the Poincar\'e
+polynomial of the complement $M(\A)$ for a
+complex arrangement. Here we prove
+that the Poincar\'e polynomial is the chamber
+counting function for a real arrangement. The
+complement $M(\A)$ is a disjoint union of chambers
+\[M(\A) = \bigcup_{C \in \Cham(\A)} C.\]
+The number
+of chambers is determined by the Poincar\'e
+polynomial as follows.
+
+\begin{thm}\label{th-realarr}
+Let $\A_{\mathbf{R}}$ be a real arrangement. Then
+\[ \abs{\Cham(\A_{\mathbf{R}})} = \pi (\A_{\mathbf{R}},1). \]
+\end{thm}
+
+\begin{proof}
+We check the properties required in Corollary~\ref{nsep}:
+(i) follows from $\pi (\Phi_{ l},t) = 1$, and (ii) is a
+consequence of Corollary~\ref{BI}.
+\end{proof}
+
+\begin{figure}
+\vspace{5cm}
+\caption[]{$Q(\A_{1}) = xyz(x-z)(x+z)(y-z)(y+z)$}
+\end{figure}
+
+\begin{figure}
+\vspace{5cm}
+\caption[]{$Q(\A_{2})= xyz(x+y+z)(x+y-z)(x-y+z)(x-y-z)$}
+\end{figure}
+
+
+\begin{thm}
+\label{T_first_the_int}
+Let $\phi$ be a protocol for a random pair $\XcY$.
+If one of $\st_\phi(x',y)$ and $\st_\phi(x,y')$ is a prefix of the other
+and $(x,y)\in\SXY$, then
+\[
+\langle \st_j(x',y)\rangle_{j=1}^\infty
+=\langle \st_j(x,y)\rangle_{j=1}^\infty
+=\langle \st_j(x,y')\rangle_{j=1}^\infty .
+\]
+\end{thm}
+\begin{proof}
+We show by induction on $i$ that
+\[
+\langle \st_j(x',y)\rangle_{j=1}^i
+=\langle \st_j(x,y)\rangle_{j=1}^i
+=\langle \st_j(x,y')\rangle_{j=1}^i.
+\]
+The induction hypothesis holds vacuously for $i=0$. Assume it holds for
+$i-1$, in particular
+$[\st_j(x',y)]_{j=1}^{i-1}=[\st_j(x,y')]_{j=1}^{i-1}$. Then one of
+$[\st_j(x',y)]_{j=i}^{\infty}$ and $[\st_j(x,y')]_{j=i}^{\infty}$ is a
+prefix of the other which implies that one of $\st_i(x',y)$ and
+$\st_i(x,y')$ is a prefix of the other. If the $i$th message is
+transmitted by $P_\X$ then, by the separate-transmissions property and
+the induction hypothesis, $\st_i(x,y)=\st_i(x,y')$, hence one of
+$\st_i(x,y)$ and $\st_i(x',y)$ is a prefix of the other. By the
+implicit-termination property, neither $\st_i(x,y)$ nor $\st_i(x',y)$
+can be a proper prefix of the other, hence they must be the same and
+$\st_i(x',y)=\st_i(x,y)=\st_i(x,y')$. If the $i$th message is
+transmitted by $\PY$ then, symmetrically, $\st_i(x,y)=\st_i(x',y)$ by
+the induction hypothesis and the separate-transmissions property, and,
+then, $\st_i(x,y)=\st_i(x,y')$ by the implicit-termination property,
+proving the induction step.
+\end{proof}
+
+If $\phi$ is a protocol for $(X,Y)$, and $(x,y)$, $(x',y)$ are distinct
+inputs in $\SXY$, then, by the correct-decision property,
+$\langle\st_j(x,y)\rangle_{j=1}^\infty\ne\langle
+\st_j(x',y)\rangle_{j=1}^\infty$.
+
+Equation~(\ref{E_SXgYy}) defined $\PY$'s ambiguity set $\SXgYy$
+to be the set of possible $X$ values when $Y=y$.
+The last corollary implies that for all $y\in\SY$,
+the multiset%
+\footnote{A multiset allows multiplicity of elements.
+Hence, $\{0,01,01\}$ is prefix free as a set, but not as a multiset.}
+of codewords $\{\st_\phi(x,y):x\in\SXgYy\}$ is prefix free.
+
+\section{One-Way Complexity}
+\label{S_Cp1}
+
+$\Cw1$, the one-way complexity of a random pair $\XcY$,
+is the number of bits $P_\X$ must transmit in the worst case
+when $\PY$ is not permitted to transmit any feedback messages.
+Starting with $\SXY$, the support set of $\XcY$, we define $\G$,
+the \textit{characteristic hypergraph} of $\XcY$, and show that
+\[
+\Cw1=\lceil\,\log\chi(\G)\rceil\ .
+\]
+
+Let $\XcY$ be a random pair. For each $y$ in $\SY$, the support set of
+$Y$, Equation~(\ref{E_SXgYy}) defined $\SXgYy$ to be the set of possible
+$x$ values when $Y=y$. The \textit{characteristic hypergraph} $\G$ of
+$\XcY$ has $\SX$ as its vertex set and the hyperedge $\SXgYy$ for each
+$y\in\SY$.
+
+
+We can now prove a continuity theorem.
+\begin{thm}\label{t:conl}
+Let $\Omega \subset\mathbf{R}^n$ be an open set, let
+$u\in BV(\Omega ;\mathbf{R}^m)$, and let
+\begin{equation}\label{quts}
+T^u_x=\left\{y\in\mathbf{R}^m:
+ y=\tilde u(x)+\left\langle \frac{Du}{\abs{Du}}(x),z
+\right\rangle \text{ for some }z\in\mathbf{R}^n\right\}
+\end{equation}
+for every $x\in\Omega \backslash S_u$. Let $f\colon \mathbf{R}^m\to
+\mathbf{R}^k$ be a Lipschitz continuous function such that $f(0)=0$, and
+let $v=f(u)\colon \Omega \to \mathbf{R}^k$. Then $v\in BV(\Omega
+;\mathbf{R}^k)$ and
+\begin{equation}
+Jv=\eval{(f(u^+)-f(u^-))\otimes \nu_u\cdot\,
+\mathcal{H}_{n-1}}_{S_u}.
+\end{equation}
+In addition, for $\abs{\wt{D}u}$-almost every $x\in\Omega $ the
+restriction of the function $f$ to $T^u_x$ is differentiable at $\tilde
+u(x)$ and
+\begin{equation}
+\wt{D}v=\nabla (\eval{f}_{T^u_x})(\tilde u)
+\frac{\wt{D}u}{\abs{\wt{D}u}}\cdot\abs{\wt{D}u}.\end{equation}
+\end{thm}
+
+Before proving the theorem, we state without proof three elementary
+remarks which will be useful in the sequel.
+\begin{rem}\label{r:omb}
+Let $\omega\colon \left]0,+\infty\right[\to \left]0,+\infty\right[$
+be a continuous function such that $\omega (t)\to 0$ as $t\to
+0$. Then
+\[\lim_{h\to 0^+}g(\omega(h))=L\Leftrightarrow\lim_{h\to
+0^+}g(h)=L\]
+for any function $g\colon \left]0,+\infty\right[\to \mathbf{R}$.
+\end{rem}
+\begin{rem}\label{r:dif}
+Let $g \colon \mathbf{R}^n\to \mathbf{R}$ be a Lipschitz
+continuous function and assume that
+\[L(z)=\lim_{h\to 0^+}\frac{g(hz)-g(0)}h\]
+exists for every $z\in\mathbf{Q}^n$ and that $L$ is a linear function of
+$z$. Then $g$ is differentiable at 0.
+\end{rem}
+\begin{rem}\label{r:dif0}
+Let $A \colon \mathbf{R}^n\to \mathbf{R}^m$ be a linear function, and
+let $f \colon \mathbf{R}^m\to \mathbf{R}$ be a function. Then the
+restriction of $f$ to the range of $A$ is differentiable at 0 if and
+only if $f(A)\colon \mathbf{R}^n\to \mathbf{R}$ is differentiable at 0
+and
+\[\nabla(\eval{f}_{\IM(A)})(0)A=\nabla (f(A))(0).\]
+\end{rem}
+
+\begin{proof}
+ We begin by showing that $v\in BV(\Omega;\mathbf{R}^k)$ and
+\begin{equation}\label{e:bomb}
+\abs{Dv}(B)\le K\abs{Du}(B)\qquad\forall B\in\mathbf{B}(\Omega ),
+\end{equation}
+where $K>0$ is the Lipschitz constant of $f$. By \eqref{sum-Di} and by
+the approximation result quoted in \secref{s:mt}, it is possible to find
+a sequence $(u_h)\subset C^1(\Omega ;\mathbf{R}^m)$ converging to $u$ in
+$L^1(\Omega ;\mathbf{R}^m)$ and such that
+\[\lim_{h\to +\infty}\int_\Omega \abs{\nabla u_h}\,dx=\abs{Du}(\Omega ).\]
+The functions $v_h=f(u_h)$ are locally Lipschitz continuous in $\Omega
+$, and the definition of differential implies that $\abs{\nabla v_h}\le
+K\abs{\nabla u_h}$ almost everywhere in $\Omega $. The lower semicontinuity
+of the total variation and \eqref{sum-Di} yield
+\begin{equation}
+\begin{split}
+\abs{Dv}(\Omega )\le\liminf_{h\to +\infty}\abs{Dv_h}(\Omega) &
+=\liminf_{h\to +\infty}\int_\Omega \abs{\nabla v_h}\,dx\\
+&\le K\liminf_{h\to +\infty}\int_\Omega
+\abs{\nabla u_h}\,dx=K\abs{Du}(\Omega).
+\end{split}\end{equation}
+Since $f(0)=0$, we have also
+\[\int_\Omega \abs{v}\,dx\le K\int_\Omega \abs{u}\,dx;\]
+therefore $u\in BV(\Omega ;\mathbf{R}^k)$. Repeating the same argument
+for every open set $A\subset\Omega $, we get \eqref{e:bomb} for every
+$B\in\mathbf{B}(\Omega)$, because $\abs{Dv}$, $\abs{Du}$ are Radon measures. To
+prove \lemref{limbog}, first we observe that
+\begin{equation}\label{e:SS}
+S_v\subset S_u,\qquad\tilde v(x)=f(\tilde u(x))\qquad \forall x\in\Omega
+\backslash S_u.\end{equation}
+In fact, for every $\varepsilon >0$ we have
+\[\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}>\varepsilon \}\subset \{y\in
+B_\rho(x): \abs{u(y)-\tilde u(x)}>\varepsilon /K\},\]
+hence
+\[\lim_{\rho\to 0^+}\frac{\abs{\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}>
+\varepsilon \}}}{\rho^n}=0\]
+whenever $x\in\Omega \backslash S_u$. By a similar argument, if $x\in
+S_u$ is a point such that there exists a triplet $(u^+,u^-,\nu_u)$
+satisfying \eqref{detK1}, \eqref{detK2}, then
+\[
+(v^+(x)-v^-(x))\otimes \nu_v=(f(u^+(x))-f(u^-(x)))\otimes\nu_u\quad
+\text{if }x\in S_v
+\]
+and $f(u^-(x))=f(u^+(x))$ if $x\in S_u\backslash S_v$. Hence, by (1.8)
+we get
+\begin{equation*}\begin{split}
+Jv(B)=\int_{B\cap S_v}(v^+-v^-)\otimes \nu_v\,d\mathcal{H}_{n-1}&=
+\int_{B\cap S_v}(f(u^+)-f(u^-))\otimes \nu_u\,d\mathcal{H}_{n-1}\\
+&=\int_{B\cap S_u}(f(u^+)-f(u^-))\otimes \nu_u\,d\mathcal{H}_{n-1}
+\end{split}\end{equation*}
+and \lemref{limbog} is proved.
+\end{proof}
+
+To prove \eqref{e:SS}, it is not restrictive to assume that $k=1$.
+Moreover, to simplify our notation, from now on we shall assume that
+$\Omega = \mathbf{R}^n$. The proof of \eqref{e:SS} is divided into two
+steps. In the first step we prove the statement in the one-dimensional
+case $(n=1)$, using \thmref{th-weak-ske-owf}. In the second step we
+achieve the general result using \thmref{t:conl}.
+
+\subsection*{Step 1}
+Assume that $n=1$. Since $S_u$ is at most countable, \eqref{sum-bij}
+yields that $\abs{\wt{D}v}(S_u\backslash S_v)=0$, so that
+\eqref{e:st} and \eqref{e:barwq} imply that $Dv=\wt{D}v+Jv$ is
+the Radon-Nikod\'ym decomposition of $Dv$ in absolutely continuous and
+singular part with respect to $\abs{\wt{D} u}$. By
+\thmref{th-weak-ske-owf}, we have
+\begin{equation*}
+\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+}
+\frac{Dv(\interval{\left[t,s\right[})}
+{\abs{\wt{D}u}(\interval{\left[t,s\right[})},\qquad
+\frac{\wt{D}u}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+}
+\frac{Du(\interval{\left[t,s\right[})}
+{\abs{\wt{D}u}(\interval{\left[t,s\right[})}
+\end{equation*}
+$\abs{\wt{D}u}$-almost everywhere in $\mathbf{R}$. It is well known
+(see, for instance, \cite[2.5.16]{ste:sint}) that every one-dimensional
+function of bounded variation $w$ has a unique left continuous
+representative, i.e., a function $\hat w$ such that $\hat w=w$ almost
+everywhere and $\lim_{s\to t^-}\hat w(s)=\hat w(t)$ for every $t\in
+\mathbf{R}$. These conditions imply
+\begin{equation}
+\hat u(t)=Du(\interval{\left]-\infty,t\right[}),
+\qquad \hat v(t)=Dv(\interval{\left]-\infty,t\right[})\qquad
+\forall t\in\mathbf{R}
+\end{equation}
+and
+\begin{equation}\label{alimo}
+\hat v(t)=f(\hat u(t))\qquad\forall t\in\mathbf{R}.\end{equation}
+Let $t\in\mathbf{R}$ be such that
+$\abs{\wt{D}u}(\interval{\left[t,s\right[})>0$ for every $s>t$ and
+assume that the limits in \eqref{joe} exist. By \eqref{j:mark} and
+\eqref{far-d} we get
+\begin{equation*}\begin{split}
+\frac{\hat v(s)-\hat
+v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}&=\frac {f(\hat
+u(s))-f(\hat u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\
+&=\frac{f(\hat u(s))-f(\hat
+u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D}u
+}(\interval{\left[t,s\right[}))}%
+{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\
+&+\frac
+{f(\hat u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D}
+u}(\interval{\left[t,s\right[}))-f(\hat
+u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}
+\end{split}\end{equation*}
+for every $s>t$. Using the Lipschitz condition on $f$ we find
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}
+\left\lvert\frac{\hat v(s)-\hat
+v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})} -\frac{f(\hat
+u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)
+\abs{\wt{D}u}(\interval{\left[t,s\right[}))-f(\hat
+u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\right\rvert\\
+\le K\left\lvert
+\frac{\hat u(s)-\hat u(t)}
+ {\abs{\wt{D}u}(\interval{\left[t,s\right[})}
+-\frac{\wt{D}u}{\abs{
+\wt{D}u}}(t)\right\rvert.\end{multline*}
+}% end of group with \multlinegap=0pt
+By \eqref{e:bomb}, the function $s\to
+\abs{\wt{D}u}(\interval{\left[t,s\right[})$ is continuous and
+converges to 0 as $s\downarrow t$. Therefore Remark~\ref{r:omb} and the
+previous inequality imply
+\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0^+}
+\frac{f(\hat u(t)+h\dfrac{\wt{D}u}{\abs{\wt{D}u}}
+(t))-f(\hat u(t))}h\quad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}.\]
+By \eqref{joe}, $\hat u(x)=\tilde u(x)$ for every
+$x\in\mathbf{R}\backslash S_u$; moreover, applying the same argument to
+the functions $u'(t)=u(-t)$, $v'(t)=f(u'(t))=v(-t)$, we get
+\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0}
+\frac{f(\tilde u(t)
++h\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t))-f(\tilde u(t))}{h}
+\qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}\]
+and our statement is proved.
+
+\subsection*{Step 2}
+
+Let us consider now the general case $n>1$. Let $\nu\in \mathbf{R}^n$ be
+such that $\abs{\nu}=1$, and let $\pi_\nu=\{y\in\mathbf{R}^n: \langle
+y,\nu\rangle =0\}$. In the following, we shall identify $\mathbf{R}^n$
+with $\pi_\nu\times\mathbf{R}$, and we shall denote by $y$ the variable
+ranging in $\pi_\nu$ and by $t$ the variable ranging in $\mathbf{R}$. By
+the just proven one-dimensional result, and by \thmref{thm-main}, we get
+\[\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\wt{D}u_y}{\abs{
+\wt{D}u_y}}(t))-f(\tilde u(y+t\nu))}h=\frac{\wt{D}v_y}{\abs{
+\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R}\]
+for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. We claim that
+\begin{equation}
+\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle
+}}(y+t\nu)=\frac{\wt{D}u_y}
+{\abs{\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R}
+\end{equation}
+for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$. In fact, by
+\eqref{sum-ali} and \eqref{delta-l} we get
+\begin{multline*}
+\int_{\pi_\nu}\frac{\wt{D}u_y}{\abs{\wt{D}u_y}}\cdot\abs{\wt{D}u_y
+}\,d\mathcal{H}_{n-1}(y)=\int_{\pi_\nu}\wt{D}u_y\,d\mathcal{H}_{n-1}(y)\\
+=\langle \wt{D}u,\nu\rangle =\frac
+{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}\cdot
+\abs{\langle \wt{D}u,\nu\rangle }=\int_{\pi_\nu}\frac{
+\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}
+(y+\cdot \nu)\cdot\abs{\wt{D}u_y}\,d\mathcal{H}_{n-1}(y)
+\end{multline*}
+and \eqref{far-d} follows from \eqref{sum-Di}. By the same argument it
+is possible to prove that
+\begin{equation}
+\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle
+}}(y+t\nu)=\frac{\wt{D}v_y}{\abs{\wt{D}u_y}}(t)\qquad\abs{
+\wt{D}u_y}\text{-a.e. in }\mathbf{R}\end{equation}
+for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. By \eqref{far-d}
+and \eqref{E_SXgYy} we get
+\[
+\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\langle \wt{D}
+u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(y+t\nu))-f(\tilde
+u(y+t\nu))}{h}
+=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle
+\wt{D}u,\nu\rangle }}(y+t\nu)\]
+for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$, and using again
+\eqref{detK1}, \eqref{detK2} we get
+\[
+\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\langle
+\wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde
+u(x))}{h}=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu
+\rangle }}(x)
+\]
+$\abs{\langle \wt{D}u,\nu\rangle}$-a.e. in $\mathbf{R}^n$.
+
+Since the function $\abs{\langle \wt{D}u,\nu\rangle }/\abs{\wt{D}u}$
+is strictly positive $\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere,
+we obtain also
+\begin{multline*}
+\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\abs{\langle
+\wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\dfrac{\langle \wt{D}
+u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde u(x))}{h}\\
+=\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\frac
+{\langle \wt{D}v,\nu\rangle }{\abs{\langle
+\wt{D}u,\nu\rangle }}(x)
+\end{multline*}
+$\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere in $\mathbf{R}^n$.
+
+Finally, since
+\begin{align*}
+&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}
+\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}
+=\frac{\langle \wt{D}u,\nu\rangle }{\abs{\wt{D}u}}
+=\left\langle \frac{\wt{D}u}{\abs{\wt{D}u}},\nu\right\rangle
+ \qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}^n\\
+&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}
+\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}
+=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\wt{D}u}}
+=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}},\nu\right\rangle
+ \qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}^n
+\end{align*}
+and since both sides of \eqref{alimo}
+are zero $\abs{\wt{D}u}$-almost everywhere
+on $\abs{\langle \wt{D}u,\nu\rangle }$-negligible sets, we conclude that
+\[
+\lim_{h\to 0}\frac{f\left(
+\tilde u(x)+h\left\langle \dfrac{\wt{D}
+u}{\abs{\wt{D}u}}(x),\nu\right\rangle \right)-f(\tilde u(x))}h
+=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}}(x),\nu\right\rangle,
+\]
+$\abs{\wt{D}u}$-a.e. in $\mathbf{R}^n$.
+Since $\nu$ is arbitrary, by Remarks \ref{r:dif} and~\ref{r:dif0}
+the restriction of $f$ to
+the affine space $T^u_x$ is differentiable at $\tilde u(x)$ for $\abs{\wt{D}
+u}$-almost every $x\in \mathbf{R}^n$ and \eqref{quts} holds.\qed
+
+It follows from \eqref{sum-Di}, \eqref{detK1}, and \eqref{detK2} that
+\begin{equation}\label{Dt}
+D(t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\abs{I}-1}\abs{I}
+\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}^{(\lambda)}
+(\overline I|\overline I).
+\end{equation}
+Let $t_i=\hat x_i$, $i=1,\dots,n$. Lemma 1 leads to
+\begin{equation}\label{Dx}
+D(\hat x_1,\dots,\hat x_n)=\prod_{i\in\mathbf{n}}\hat x_i
+\sum_{I\in\mathbf{n}}(-1)^{\abs{I}-1}\abs{I}\per \mathbf{A}
+^{(\lambda)}(I|I)\det\mathbf{A}^{(\lambda)}(\overline I|\overline I).
+\end{equation}
+By \eqref{H-cycles}, \eqref{sum-Di}, and \eqref{Dx},
+we have the following result:
+\begin{thm}\label{thm-H-param}
+\begin{equation}\label{H-param}
+H_c=\frac{1}{2n}\sum^n_{l =1}l (-1)^{l -1}A_{l}
+^{(\lambda)},
+\end{equation}
+where
+\begin{equation}\label{A-l-lambda}
+A^{(\lambda)}_l =\sum_{I_l \subseteq\mathbf{n}}\per \mathbf{A}
+^{(\lambda)}(I_l |I_l )\det\mathbf{A}^{(\lambda)}
+(\overline I_{l}|\overline I_l ),\abs{I_{l}}=l .
+\end{equation}
+\end{thm}
+
+It is worth noting that $A_l ^{(\lambda)}$ of \eqref{A-l-lambda} is
+similar to the coefficients $b_l $ of the characteristic polynomial of
+\eqref{bl-sum}. It is well known in graph theory that the coefficients
+$b_l $ can be expressed as a sum over certain subgraphs. It is
+interesting to see whether $A_l $, $\lambda=0$, structural properties
+of a graph.
+
+We may call \eqref{H-param} a parametric representation of $H_c$. In
+computation, the parameter $\lambda_i$ plays very important roles. The
+choice of the parameter usually depends on the properties of the given
+graph. For a complete graph $K_n$, let $\lambda_i=1$, $i=1,\dots,n$.
+It follows from \eqref{A-l-lambda} that
+\begin{equation}\label{compl-gr}
+A^{(1)}_l =\begin{cases} n!,&\text{if }l =1\\
+0,&\text{otherwise}.\end{cases}
+\end{equation}
+By \eqref{H-param}
+\begin{equation}
+H_c=\frac 12(n-1)!.
+\end{equation}
+For a complete bipartite graph $K_{n_1n_2}$, let $\lambda_i=0$, $i=1,\dots,n$.
+By \eqref{A-l-lambda},
+\begin{equation}
+A_l =
+\begin{cases} -n_1!n_2!\delta_{n_1n_2},&\text{if }l =2\\
+0,&\text{otherwise }.\end{cases}
+\label{compl-bip-gr}
+\end{equation}
+Theorem ~\ref{thm-H-param}
+leads to
+\begin{equation}
+H_c=\frac1{n_1+n_2}n_1!n_2!\delta_{n_1n_2}.
+\end{equation}
+
+Now, we consider an asymmetrical approach. Theorem \ref{thm-main} leads to
+\begin{multline}
+\det\mathbf{K}(t=1,t_1,\dots,t_n;l |l )\\
+=\sum_{I\subseteq\mathbf{n}-\{l \}}
+(-1)^{\abs{I}}\prod_{i\in I}t_i\prod_{j\in I}
+(D_j+\lambda_jt_j)\det\mathbf{A}^{(\lambda)}
+(\overline I\cup\{l \}|\overline I\cup\{l \}).
+\end{multline}
+
+By \eqref{H-cycles} and \eqref{sum-ali} we have the following asymmetrical
+result:
+\begin{thm}\label{thm-asym}
+\begin{equation}
+H_c=\frac12\sum_{I\subseteq\mathbf{n}-\{l \}}
+(-1)^{\abs{I}}\per\mathbf{A}^{(\lambda)}(I|I)\det
+\mathbf{A}^{(\lambda)}
+(\overline I\cup\{l \}|\overline I\cup\{l \})
+\end{equation}
+which reduces to Goulden--Jackson's formula when $\lambda_i=0,i=1,\dots,n$
+\cite{mami:matrixth}.
+\end{thm}
+
+\section{Various font features of the \pkg{amsmath} package}
+\label{s:font}
+\subsection{Bold versions of special symbols}
+
+In the \pkg{amsmath} package \cn{boldsymbol} is used for getting
+individual bold math symbols and bold Greek letters---everything in
+math except for letters of the Latin alphabet,
+where you'd use \cn{mathbf}. For example,
+\begin{verbatim}
+A_\infty + \pi A_0 \sim
+\mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
+\boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
+\end{verbatim}
+looks like this:
+\[A_\infty + \pi A_0 \sim \mathbf{A}_{\boldsymbol{\infty}}
+\boldsymbol{+} \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}\]
+
+\subsection{``Poor man's bold''}
+If a bold version of a particular symbol doesn't exist in the
+available fonts,
+then \cn{boldsymbol} can't be used to make that symbol bold.
+At the present time, this means that
+\cn{boldsymbol} can't be used with symbols from
+the \fn{msam} and \fn{msbm} fonts, among others.
+In some cases, poor man's bold (\cn{pmb}) can be used instead
+of \cn{boldsymbol}:
+% Can't show example from msam or msbm because this document is
+% supposed to be TeXable even if the user doesn't have
+% AMSFonts. MJD 5-JUL-1990
+\[\frac{\partial x}{\partial y}
+\pmb{\bigg\vert}
+\frac{\partial y}{\partial z}\]
+\begin{verbatim}
+\[\frac{\partial x}{\partial y}
+\pmb{\bigg\vert}
+\frac{\partial y}{\partial z}\]
+\end{verbatim}
+So-called ``large operator'' symbols such as $\sum$ and $\prod$
+require an additional command, \cn{mathop},
+to produce proper spacing and limits when \cn{pmb} is used.
+For further details see \textit{The \TeX book}.
+\[\sum_{\substack{i<B\\\text{$i$ odd}}}
+\prod_\kappa \kappa F(r_i)\qquad
+\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
+\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
+\]
+\begin{verbatim}
+\[\sum_{\substack{i<B\\\text{$i$ odd}}}
+\prod_\kappa \kappa F(r_i)\qquad
+\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
+\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
+\]
+\end{verbatim}
+
+\section{Compound symbols and other features}
+\label{s:comp}
+\subsection{Multiple integral signs}
+
+\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral signs
+with the spacing between them nicely adjusted, in both text and
+display style. \cn{idotsint} gives two integral signs with dots
+between them.
+\begin{gather}
+\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A
+f(x,y,z)\,dx\,dy\,dz\\
+\iiiint\limits_A
+f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k)
+\end{gather}
+
+\subsection{Over and under arrows}
+
+Some extra over and under arrow operations are provided in
+the \pkg{amsmath} package. (Basic \LaTeX\ provides
+\cn{overrightarrow} and \cn{overleftarrow}).
+\begin{align*}
+\overrightarrow{\psi_\delta(t) E_t h}&
+=\underrightarrow{\psi_\delta(t) E_t h}\\
+\overleftarrow{\psi_\delta(t) E_t h}&
+=\underleftarrow{\psi_\delta(t) E_t h}\\
+\overleftrightarrow{\psi_\delta(t) E_t h}&
+=\underleftrightarrow{\psi_\delta(t) E_t h}
+\end{align*}
+\begin{verbatim}
+\begin{align*}
+\overrightarrow{\psi_\delta(t) E_t h}&
+=\underrightarrow{\psi_\delta(t) E_t h}\\
+\overleftarrow{\psi_\delta(t) E_t h}&
+=\underleftarrow{\psi_\delta(t) E_t h}\\
+\overleftrightarrow{\psi_\delta(t) E_t h}&
+=\underleftrightarrow{\psi_\delta(t) E_t h}
+\end{align*}
+\end{verbatim}
+These all scale properly in subscript sizes:
+\[\int_{\overrightarrow{AB}} ax\,dx\]
+\begin{verbatim}
+\[\int_{\overrightarrow{AB}} ax\,dx\]
+\end{verbatim}
+
+\subsection{Dots}
+
+Normally you need only type \cn{dots} for ellipsis dots in a
+math formula. The main exception is when the dots
+fall at the end of the formula; then you need to
+specify one of \cn{dotsc} (series dots, after a comma),
+\cn{dotsb} (binary dots, for binary relations or operators),
+\cn{dotsm} (multiplication dots), or \cn{dotsi} (dots after
+an integral). For example, the input
+\begin{verbatim}
+Then we have the series $A_1,A_2,\dotsc$,
+the regional sum $A_1+A_2+\dotsb$,
+the orthogonal product $A_1A_2\dotsm$,
+and the infinite integral
+\[\int_{A_1}\int_{A_2}\dotsi\].
+\end{verbatim}
+produces
+\begin{quotation}
+Then we have the series $A_1,A_2,\dotsc$,
+the regional sum $A_1+A_2+\dotsb$,
+the orthogonal product $A_1A_2\dotsm$,
+and the infinite integral
+\[\int_{A_1}\int_{A_2}\dotsi\]
+\end{quotation}
+
+\subsection{Accents in math}
+
+Double accents:
+\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad
+\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad
+\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad
+\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad
+\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]
+\begin{verbatim}
+\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad
+\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad
+\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad
+\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad
+\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]
+\end{verbatim}
+This double accent operation is complicated
+and tends to slow down the processing of a \LaTeX\ file.
+
+
+\subsection{Dot accents}
+\cn{dddot} and \cn{ddddot} are available to
+produce triple and quadruple dot accents
+in addition to the \cn{dot} and \cn{ddot} accents already available
+in \LaTeX:
+\[\dddot{Q}\qquad\ddddot{R}\]
+\begin{verbatim}
+\[\dddot{Q}\qquad\ddddot{R}\]
+\end{verbatim}
+
+\subsection{Roots}
+
+In the \pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
+the position of the root index of a radical:
+\begin{verbatim}
+\sqrt[\leftroot{-2}\uproot{2}\beta]{k}
+\end{verbatim}
+gives good positioning of the $\beta$:
+\[\sqrt[\leftroot{-2}\uproot{2}\beta]{k}\]
+
+\subsection{Boxed formulas} The command \cn{boxed} puts a box around its
+argument, like \cn{fbox} except that the contents are in math mode:
+\begin{verbatim}
+\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}
+\end{verbatim}
+\[\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}.\]
+
+\subsection{Extensible arrows}
+\cn{xleftarrow} and \cn{xrightarrow} produce
+arrows that extend automatically to accommodate unusually wide
+subscripts or superscripts. The text of the subscript or superscript
+are given as an optional resp.\@ mandatory argument:
+Example:
+\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
+ \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]
+\begin{verbatim}
+\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
+ \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]
+\end{verbatim}
+
+\subsection{\cn{overset}, \cn{underset}, and \cn{sideset}}
+Examples:
+\[\overset{*}{X}\qquad\underset{*}{X}\qquad
+\overset{a}{\underset{b}{X}}\]
+\begin{verbatim}
+\[\overset{*}{X}\qquad\underset{*}{X}\qquad
+\overset{a}{\underset{b}{X}}\]
+\end{verbatim}
+
+The command \cn{sideset} is for a rather special
+purpose: putting symbols at the subscript and superscript
+corners of a large operator symbol such as $\sum$ or $\prod$,
+without affecting the placement of limits.
+Examples:
+\[\sideset{_*^*}{_*^*}\prod_k\qquad
+\sideset{}{'}\sum_{0\le i\le m} E_i\beta x
+\]
+\begin{verbatim}
+\[\sideset{_*^*}{_*^*}\prod_k\qquad
+\sideset{}{'}\sum_{0\le i\le m} E_i\beta x
+\]
+\end{verbatim}
+
+\subsection{The \cn{text} command}
+The main use of the command \cn{text} is for words or phrases in a
+display:
+\[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad
+y'_k=\delta_k y_{\tau(k)}\]
+\begin{verbatim}
+\[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad
+y'_k=\delta_k y_{\tau(k)}\]
+\end{verbatim}
+
+\subsection{Operator names}
+The more common math functions such as $\log$, $\sin$, and $\lim$
+have predefined control sequences: \verb=\log=, \verb=\sin=,
+\verb=\lim=.
+The \pkg{amsmath} package provides \cn{DeclareMathOperator} and
+\cn{DeclareMathOperator*}
+for producing new function names that will have the
+same typographical treatment.
+Examples:
+\[\norm{f}_\infty=
+\esssup_{x\in R^n}\abs{f(x)}\]
+\begin{verbatim}
+\[\norm{f}_\infty=
+\esssup_{x\in R^n}\abs{f(x)}\]
+\end{verbatim}
+\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\}
+=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\}
+\quad \forall\alpha>0.\]
+\begin{verbatim}
+\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\}
+=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\}
+\quad \forall\alpha>0.\]
+\end{verbatim}
+\cn{esssup} and \cn{meas} would be defined in the document preamble as
+\begin{verbatim}
+\DeclareMathOperator*{\esssup}{ess\,sup}
+\DeclareMathOperator{\meas}{meas}
+\end{verbatim}
+
+The following special operator names are predefined in the \pkg{amsmath}
+package: \cn{varlimsup}, \cn{varliminf}, \cn{varinjlim}, and
+\cn{varprojlim}. Here's what they look like in use:
+\begin{align}
+&\varlimsup_{n\rightarrow\infty}
+ \mathcal{Q}(u_n,u_n-u^{\#})\le0\\
+&\varliminf_{n\rightarrow\infty}
+ \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\
+&\varinjlim (m_i^\lambda\cdot)^*\le0\\
+&\varprojlim_{p\in S(A)}A_p\le0
+\end{align}
+\begin{verbatim}
+\begin{align}
+&\varlimsup_{n\rightarrow\infty}
+ \mathcal{Q}(u_n,u_n-u^{\#})\le0\\
+&\varliminf_{n\rightarrow\infty}
+ \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\
+&\varinjlim (m_i^\lambda\cdot)^*\le0\\
+&\varprojlim_{p\in S(A)}A_p\le0
+\end{align}
+\end{verbatim}
+
+\subsection{\cn{mod} and its relatives}
+The commands \cn{mod} and \cn{pod} are variants of
+\cn{pmod} preferred by some authors; \cn{mod} omits the parentheses,
+whereas \cn{pod} omits the `mod' and retains the parentheses.
+Examples:
+\begin{align}
+x&\equiv y+1\pmod{m^2}\\
+x&\equiv y+1\mod{m^2}\\
+x&\equiv y+1\pod{m^2}
+\end{align}
+\begin{verbatim}
+\begin{align}
+x&\equiv y+1\pmod{m^2}\\
+x&\equiv y+1\mod{m^2}\\
+x&\equiv y+1\pod{m^2}
+\end{align}
+\end{verbatim}
+
+\subsection{Fractions and related constructions}
+\label{fracs}
+
+The usual notation for binomials is similar to the fraction concept,
+so it has a similar command \cn{binom} with two arguments. Example:
+\begin{equation}
+\begin{split}
+\sum_{\gamma\in\Gamma_C} I_\gamma&
+=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\
+&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l}
++\dots+(-1)^k\\
+&=(2-1)^k=1
+\end{split}
+\end{equation}
+\begin{verbatim}
+\begin{equation}
+\begin{split}
+[\sum_{\gamma\in\Gamma_C} I_\gamma&
+=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\
+&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l}
++\dots+(-1)^k\\
+&=(2-1)^k=1
+\end{split}
+\end{equation}
+\end{verbatim}
+There are also abbreviations
+\begin{verbatim}
+\dfrac \dbinom
+\tfrac \tbinom
+\end{verbatim}
+for the commonly needed constructions
+\begin{verbatim}
+{\displaystyle\frac ... } {\displaystyle\binom ... }
+{\textstyle\frac ... } {\textstyle\binom ... }
+\end{verbatim}
+
+The generalized fraction command \cn{genfrac} provides full access to
+the six \TeX{} fraction primitives:
+\begin{align}
+\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}&
+\text{\cn{overwithdelims}: }&
+ \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\
+\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}&
+\text{\cn{atopwithdelims}: }&
+ \genfrac{(}{)}{0pt}{}{n+1}{2}\\
+\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}&
+\text{\cn{abovewithdelims}: }&
+ \genfrac{[}{]}{1pt}{}{n+1}{2}
+\end{align}
+\begin{verbatim}
+\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}&
+\text{\cn{overwithdelims}: }&
+ \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\
+\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}&
+\text{\cn{atopwithdelims}: }&
+ \genfrac{(}{)}{0pt}{}{n+1}{2}\\
+\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}&
+\text{\cn{abovewithdelims}: }&
+ \genfrac{[}{]}{1pt}{}{n+1}{2}
+\end{verbatim}
+
+\subsection{Continued fractions}
+The continued fraction
+\begin{equation}
+\cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+\dotsb
+}}}}}
+\end{equation}
+can be obtained by typing
+\begin{verbatim}
+\cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+\dotsb
+}}}}}
+\end{verbatim}
+Left or right placement of any of the numerators is accomplished by using
+\cn{cfrac[l]} or \cn{cfrac[r]} instead of \cn{cfrac}.
+
+\subsection{Smash}
+
+In \pkg{amsmath} there are optional arguments \verb"t" and \verb"b" for
+the plain \TeX\ command \cn{smash}, because sometimes it is advantageous
+to be able to `smash' only the top or only the bottom of something while
+retaining the natural depth or height. In the formula
+$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ \cn{smash}\verb=[b]= has been
+used to limit the size of the radical symbol.
+\begin{verbatim}
+$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$
+\end{verbatim}
+Without the use of \cn{smash}\verb=[b]= the formula would have appeared
+thus: $X_j=(1/\sqrt{\lambda_j})X_j'$, with the radical extending to
+encompass the depth of the subscript $j$.
+
+\subsection{The `cases' environment}
+`Cases' constructions like the following can be produced using
+the \env{cases} environment.
+\begin{equation}
+P_{r-j}=
+ \begin{cases}
+ 0& \text{if $r-j$ is odd},\\
+ r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
+ \end{cases}
+\end{equation}
+\begin{verbatim}
+\begin{equation} P_{r-j}=
+ \begin{cases}
+ 0& \text{if $r-j$ is odd},\\
+ r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
+ \end{cases}
+\end{equation}
+\end{verbatim}
+Notice the use of \cn{text} and the embedded math.
+
+\subsection{Matrix}
+
+Here are samples of the matrix environments,
+\cn{matrix}, \cn{pmatrix}, \cn{bmatrix}, \cn{Bmatrix}, \cn{vmatrix}
+and \cn{Vmatrix}:
+\begin{equation}
+\begin{matrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{matrix}\quad
+\begin{pmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{pmatrix}\quad
+\begin{bmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{bmatrix}\quad
+\begin{Bmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{Bmatrix}\quad
+\begin{vmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{vmatrix}\quad
+\begin{Vmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{Vmatrix}
+\end{equation}
+%
+\begin{verbatim}
+\begin{matrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{matrix}\quad
+\begin{pmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{pmatrix}\quad
+\begin{bmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{bmatrix}\quad
+\begin{Bmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{Bmatrix}\quad
+\begin{vmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{vmatrix}\quad
+\begin{Vmatrix}
+\vartheta& \varrho\\\varphi& \varpi
+\end{Vmatrix}
+\end{verbatim}
+
+To produce a small matrix suitable for use in text, use the
+\env{smallmatrix} environment.
+\begin{verbatim}
+\begin{math}
+ \bigl( \begin{smallmatrix}
+ a&b\\ c&d
+ \end{smallmatrix} \bigr)
+\end{math}
+\end{verbatim}
+To show
+the effect of the matrix on the surrounding lines of
+a paragraph, we put it here: \begin{math}
+ \bigl( \begin{smallmatrix}
+ a&b\\ c&d
+ \end{smallmatrix} \bigr)
+\end{math}
+and follow it with enough text to ensure that there will
+be at least one full line below the matrix.
+
+\cn{hdotsfor}\verb"{"\textit{number}\verb"}" produces a row of dots in a matrix
+spanning the given number of columns:
+\[W(\Phi)= \begin{Vmatrix}
+\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
+\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
+\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
+\hdotsfor{5}\\
+\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
+\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
+\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
+\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
+\end{Vmatrix}\]
+\begin{verbatim}
+\[W(\Phi)= \begin{Vmatrix}
+\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
+\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
+\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
+\hdotsfor{5}\\
+\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
+\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
+\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
+\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
+\end{Vmatrix}\]
+\end{verbatim}
+The spacing of the dots can be varied through use of a square-bracket
+option, for example, \verb"\hdotsfor[1.5]{3}". The number in square brackets
+will be used as a multiplier; the normal value is 1.
+
+\subsection{The \cn{substack} command}
+
+The \cn{substack} command can be used to produce a multiline
+subscript or superscript:
+for example
+\begin{verbatim}
+\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
+\end{verbatim}
+produces a two-line subscript underneath the sum:
+\begin{equation}
+\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
+\end{equation}
+A slightly more generalized form is the \env{subarray} environment which
+allows you to specify that each line should be left-aligned instead of
+centered, as here:
+\begin{equation}
+\sum_{\begin{subarray}{l}
+ 0\le i\le m\\ 0<j<n
+ \end{subarray}}
+ P(i,j)
+\end{equation}
+\begin{verbatim}
+\sum_{\begin{subarray}{l}
+ 0\le i\le m\\ 0<j<n
+ \end{subarray}}
+ P(i,j)
+\end{verbatim}
+
+
+\subsection{Big-g-g delimiters}
+Here are some big delimiters, first in \cn{normalsize}:
+\[\biggl(\mathbf{E}_{y}
+ \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
+ \biggr)
+\]
+\begin{verbatim}
+\[\biggl(\mathbf{E}_{y}
+ \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
+ \biggr)
+\]
+\end{verbatim}
+and now in \cn{Large} size:
+{\Large
+\[\biggl(\mathbf{E}_{y}
+ \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
+ \biggr)
+\]}
+\begin{verbatim}
+{\Large
+\[\biggl(\mathbf{E}_{y}
+ \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
+ \biggr)
+\]}
+\end{verbatim}
+
+\newpage
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\makeatletter
+
+%% This turns on vertical rules at the right and left margins, to
+%% better illustrate the spacing for certain multiple-line equation
+%% structures.
+\def\@makecol{\ifvoid\footins \setbox\@outputbox\box\@cclv
+ \else\setbox\@outputbox
+ \vbox{\boxmaxdepth \maxdepth
+ \unvbox\@cclv\vskip\skip\footins\footnoterule\unvbox\footins}\fi
+ \xdef\@freelist{\@freelist\@midlist}\gdef\@midlist{}\@combinefloats
+ \setbox\@outputbox\hbox{\vrule width\marginrulewidth
+ \vbox to\@colht{\boxmaxdepth\maxdepth
+ \@texttop\dimen128=\dp\@outputbox\unvbox\@outputbox
+ \vskip-\dimen128\@textbottom}%
+ \vrule width\marginrulewidth}%
+ \global\maxdepth\@maxdepth}
+\newdimen\marginrulewidth
+\setlength{\marginrulewidth}{.1pt}
+\makeatother
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\appendix
+\section{Examples of multiple-line equation structures}
+\label{s:eq}
+
+\textbf{\large Note: Starting on this page, vertical rules are
+added at the margins so that the positioning of various display elements
+with respect to the margins can be seen more clearly.}
+
+\subsection{Split}
+The \env{split} environment is not an independent environment
+but should be used inside something else such as \env{equation}
+or \env{align}.
+
+If there is not enough room for it, the equation number for a
+\env{split} will be shifted to the previous line, when equation numbers are
+on the left; the number shifts down to the next line when numbers are on
+the right.
+\begin{equation}
+\begin{split}
+f_{h,\varepsilon}(x,y)
+&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
+L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
+&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
+&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
+ \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
+ -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
+&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
+ \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
+ \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
+ L_{x,y_\varepsilon(\varepsilon s)}
+ \varphi(x)\,ds\biggr)\biggr]\\
+&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
+\end{split}
+\end{equation}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{equation}
+\begin{split}
+f_{h,\varepsilon}(x,y)
+&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
+L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
+&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
+&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
+ \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
+ -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
+&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
+ \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
+ \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
+ L_{x,y_\varepsilon(\varepsilon s)}
+ \varphi(x)\,ds\biggr)\biggr]\\
+&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
+\end{split}
+\end{equation}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+Unnumbered version:
+\begin{equation*}
+\begin{split}
+f_{h,\varepsilon}(x,y)
+&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
+L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
+&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
+&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
+ \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
+ -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
+&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
+ \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
+ \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
+ L_{x,y_\varepsilon(\varepsilon s)}
+ \varphi(x)\,ds\biggr)\biggr]\\
+&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
+\end{split}
+\end{equation*}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{equation*}
+\begin{split}
+f_{h,\varepsilon}(x,y)
+&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
+L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
+&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
+&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
+ \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
+ -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
+&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
+ \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
+ \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
+ L_{x,y_\varepsilon(\varepsilon s)}
+ \varphi(x)\,ds\biggr)\biggr]\\
+&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
+\end{split}
+\end{equation*}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+If the option \env{centertags} is included in the options
+list of the \pkg{amsmath} package,
+the equation numbers for \env{split} environments will be
+centered vertically on the height
+of the \env{split}:
+{\makeatletter\ctagsplit@true
+\begin{equation}
+\begin{split}
+ \abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)-\int_{\gamma(t)}^a
+ \frac{d\theta}{k(\theta,t)}
+ \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
+&\le C_6\left\lvert \left\lvert f\int_\Omega\left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}
+\end{equation}}%
+Some text after to test the below-display spacing.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+Use of \env{split} within \env{align}:
+{\delimiterfactor750
+\begin{align}
+\begin{split}\abs{I_1}
+ &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
+&\le C_3\left[\int_\Omega\left(\int_{a}^x
+ g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
+&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
+ \left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
+ c\Omega\right]^{1/2}\\
+&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}\label{eq:A}\\
+\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
+ -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
+ \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
+&\le C_6\left\lvert \left\lvert f\int_\Omega
+ \left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}
+\end{align}}%
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{align}
+\begin{split}\abs{I_1}
+ &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
+&\le C_3\left[\int_\Omega\left(\int_{a}^x
+ g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
+&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
+ \left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
+ c\Omega\right]^{1/2}\\
+&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}\label{eq:A}\\
+\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
+ -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
+ \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
+&\le C_6\left\lvert \left\lvert f\int_\Omega
+ \left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}
+\end{align}
+\end{verbatim}
+
+%%%%%%%%%%%%%%%%%%
+
+\newpage
+Unnumbered \env{align}, with a number on the second \env{split}:
+\begin{align*}
+\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
+ &\le C_3\left[\int_\Omega\left(\int_{a}^x
+ g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
+&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
+ \left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
+ c\Omega\right]^{1/2}\\
+&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}\\
+\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
+ -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
+ \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
+&\le C_6\left\lvert \left\lvert f\int_\Omega
+ \left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}\tag{\theequation$'$}
+\end{align*}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{align*}
+\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
+ &\le C_3\left[\int_\Omega\left(\int_{a}^x
+ g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
+&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
+ \left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
+ c\Omega\right]^{1/2}\\
+&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}\\
+\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
+ -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
+ \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
+&\le C_6\left\lvert \left\lvert f\int_\Omega
+ \left\lvert \wt{S}^{-1,0}_{a,-}
+ W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
+ \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
+ (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
+\end{split}\tag{\theequation$'$}
+\end{align*}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+\subsection{Multline}
+Numbered version:
+\begin{multline}\label{eq:E}
+\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
+ -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
+ =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
+ \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
+\end{multline}
+To test the use of \verb=\label= and
+\verb=\ref=, we refer to the number of this
+equation here: (\ref{eq:E}).
+
+\begin{verbatim}
+\begin{multline}\label{eq:E}
+\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
+ -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
+ =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
+ \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
+\end{multline}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Unnumbered version:
+\begin{multline*}
+\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
+ -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
+ =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
+ \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
+\end{multline*}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{multline*}
+\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
+ -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
+ =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
+ \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
+\end{multline*}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\iffalse % bugfix needed, error message "Multiple \tag"
+ % [mjd,24-Jan-1995]
+\newpage
+And now an ``unnumbered'' version numbered with a literal tag:
+\begin{multline*}\tag*{[a]}
+\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
+ -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
+ =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
+ \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
+\end{multline*}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{multline*}\tag*{[a]}
+\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
+ -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
+ =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
+ \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
+\end{multline*}
+\end{verbatim}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The same display with \verb=\multlinegap= set to zero.
+Notice that the space on the left in
+the first line does not change, because of the equation number, while
+the second line is pushed over to the right margin.
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}\tag*{[a]}
+\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
+ -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
+ =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
+ \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
+\end{multline*}}%
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+{\setlength{\multlinegap}{0pt}
+\begin{multline*}\tag*{[a]}
+\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
+ -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
+ =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
+ \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
+\end{multline*}}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\fi % matches \iffalse above [mjd,24-Jan-1995]
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+\subsection{Gather}
+Numbered version with \verb;\notag; on the second line:
+\begin{gather}
+D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
+\seg(a,r)\equiv\{z\in\mathbf{C}\colon
+\Im z= \Im a,\ \abs{z-a}<r\},\notag\\
+c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
+\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
+C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
+\end{gather}
+\begin{verbatim}
+\begin{gather}
+D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
+\seg(a,r)\equiv\{z\in\mathbf{C}\colon
+\Im z= \Im a,\ \abs{z-a}<r\},\notag\\
+c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
+\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
+C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
+\end{gather}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Unnumbered version.
+\begin{gather*}
+D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
+\seg (a,r)\equiv\{z\in\mathbf{C}\colon
+\Im z= \Im a,\ \abs{z-a}<r\},\\
+c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
+ \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
+C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
+\end{gather*}
+Some text after to test the below-display spacing.
+\begin{verbatim}
+\begin{gather*}
+D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
+\seg (a,r)\equiv\{z\in\mathbf{C}\colon
+\Im z= \Im a,\ \abs{z-a}<r\},\\
+c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
+ \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
+C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
+\end{gather*}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+\subsection{Align}
+Numbered version:
+\begin{align}
+\gamma_x(t)&=(\cos tu+\sin tx,v),\\
+\gamma_y(t)&=(u,\cos tv+\sin ty),\\
+\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
+ -\frac\beta\alpha\sin tu+\cos tv\right).
+\end{align}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{align}
+\gamma_x(t)&=(\cos tu+\sin tx,v),\\
+\gamma_y(t)&=(u,\cos tv+\sin ty),\\
+\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
+ -\frac\beta\alpha\sin tu+\cos tv\right).
+\end{align}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Unnumbered version:
+\begin{align*}
+\gamma_x(t)&=(\cos tu+\sin tx,v),\\
+\gamma_y(t)&=(u,\cos tv+\sin ty),\\
+\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
+ -\frac\beta\alpha\sin tu+\cos tv\right).
+\end{align*}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{align*}
+\gamma_x(t)&=(\cos tu+\sin tx,v),\\
+\gamma_y(t)&=(u,\cos tv+\sin ty),\\
+\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
+ -\frac\beta\alpha\sin tu+\cos tv\right).
+\end{align*}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+A variation:
+\begin{align}
+x& =y && \text {by (\ref{eq:C})}\\
+x'& = y' && \text {by (\ref{eq:D})}\\
+x+x' & = y+y' && \text {by Axiom 1.}
+\end{align}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{align}
+x& =y && \text {by (\ref{eq:C})}\\
+x'& = y' && \text {by (\ref{eq:D})}\\
+x+x' & = y+y' && \text {by Axiom 1.}
+\end{align}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+\subsection{Align and split within gather}
+When using the \env{align} environment within the \env{gather}
+environment, one or the other, or both, should be unnumbered (using the
+\verb"*" form); numbering both the outer and inner environment would
+cause a conflict.
+
+Automatically numbered \env{gather} with \env{split} and \env{align*}:
+\begin{gather}
+\begin{split} \varphi(x,z)
+&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
+&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
+\end{split}\\[6pt]
+\begin{align*}
+\zeta^0 &=(\xi^0)^2,\\
+\zeta^1 &=\xi^0\xi^1,\\
+\zeta^2 &=(\xi^1)^2,
+\end{align*}
+\end{gather}
+Here the \env{split} environment gets a number from the outer
+\env{gather} environment; numbers for individual lines of the
+\env{align*} are suppressed because of the star.
+
+\begin{verbatim}
+\begin{gather}
+\begin{split} \varphi(x,z)
+&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
+&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
+\end{split}\\[6pt]
+\begin{align*}
+\zeta^0 &=(\xi^0)^2,\\
+\zeta^1 &=\xi^0\xi^1,\\
+\zeta^2 &=(\xi^1)^2,
+\end{align*}
+\end{gather}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The \verb"*"-ed form of \env{gather} with the non-\verb"*"-ed form of
+\env{align}.
+\begin{gather*}
+\begin{split} \varphi(x,z)
+&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
+&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
+\end{split}\\[6pt]
+\begin{align} \zeta^0&=(\xi^0)^2,\\
+\zeta^1 &=\xi^0\xi^1,\\
+\zeta^2 &=(\xi^1)^2,
+\end{align}
+\end{gather*}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{gather*}
+\begin{split} \varphi(x,z)
+&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
+&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
+\end{split}\\[6pt]
+\begin{align} \zeta^0&=(\xi^0)^2,\\
+\zeta^1 &=\xi^0\xi^1,\\
+\zeta^2 &=(\xi^1)^2,
+\end{align}
+\end{gather*}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+\subsection{Alignat}
+Numbered version:
+\begin{alignat}{3}
+V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
+ & \qquad U_i & = u_i,
+ \qquad \text{for $i\ne j$;}\label{eq:B}\\
+V_j & = v_j, & \qquad X_j & = x_j,
+ & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
+\end{alignat}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{alignat}{3}
+V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
+ & \qquad U_i & = u_i,
+ \qquad \text{for $i\ne j$;}\label{eq:B}\\
+V_j & = v_j, & \qquad X_j & = x_j,
+ & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
+\end{alignat}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Unnumbered version:
+\begin{alignat*}3
+V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
+ & \qquad U_i & = u_i,
+ \qquad \text{for $i\ne j$;} \\
+V_j & = v_j, & \qquad X_j & = x_j,
+ & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
+\end{alignat*}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{alignat*}3
+V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
+ & \qquad U_i & = u_i,
+ \qquad \text{for $i\ne j$;} \\
+V_j & = v_j, & \qquad X_j & = x_j,
+ & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
+\end{alignat*}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+The most common use for \env{alignat} is for things like
+\begin{alignat}{2}
+x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\
+x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\
+x+x' & = y+y' && \qquad \text {by Axiom 1.}
+\end{alignat}
+Some text after to test the below-display spacing.
+
+\begin{verbatim}
+\begin{alignat}{2}
+x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\
+x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\
+x+x' & = y+y' && \qquad \text {by Axiom 1.}
+\end{alignat}
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newpage
+\setlength{\marginrulewidth}{0pt}
+
+\begin{thebibliography}{10}
+
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+\bibitem{ye:intalg}
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+
+\end{thebibliography}
+
+\end{document}
+\endinput