amslatex package amsmath update from latex-tds (27aug06)

git-svn-id: svn://tug.org/texlive/trunk@2050 c570f23f-e606-0410-a88d-b1316a301751

38 files changed, 23639 insertions, 0 deletions

diff --git a/Master/texmf-dist/doc/latex/amsmath/amsbsy.pdf b/Master/texmf-dist/doc/latex/amsmath/amsbsy.pdf
new file mode 100644
index 00000000000..2348790f37b
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/amsmath/amsbsy.pdf
diff --git a/Master/texmf-dist/doc/latex/amsmath/amscd.pdf b/Master/texmf-dist/doc/latex/amsmath/amscd.pdf
new file mode 100644
index 00000000000..29ea7dce315
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/amsmath/amscd.pdf
diff --git a/Master/texmf-dist/doc/latex/amsmath/amsgen.pdf b/Master/texmf-dist/doc/latex/amsmath/amsgen.pdf
new file mode 100644
index 00000000000..c49f976b59f
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/amsmath/amsgen.pdf
diff --git a/Master/texmf-dist/doc/latex/amsmath/amsldoc.pdf b/Master/texmf-dist/doc/latex/amsmath/amsldoc.pdf
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index 00000000000..d715f46c381
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diff --git a/Master/texmf-dist/doc/latex/amsmath/amsmath.pdf b/Master/texmf-dist/doc/latex/amsmath/amsmath.pdf
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diff --git a/Master/texmf-dist/doc/latex/amsmath/amsopn.pdf b/Master/texmf-dist/doc/latex/amsmath/amsopn.pdf
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diff --git a/Master/texmf-dist/doc/latex/amsmath/amstext.pdf b/Master/texmf-dist/doc/latex/amsmath/amstext.pdf
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diff --git a/Master/texmf-dist/doc/latex/amsmath/amsxtra.pdf b/Master/texmf-dist/doc/latex/amsmath/amsxtra.pdf
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diff --git a/Master/texmf-dist/doc/latex/amsmath/diffs-m.txt b/Master/texmf-dist/doc/latex/amsmath/diffs-m.txt
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+++ b/Master/texmf-dist/doc/latex/amsmath/diffs-m.txt
@@ -0,0 +1,552 @@
+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 Hellstr�m.
+
+---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; see
+http://ctan.tug.org/info/compan-rev/ch8.pdf).
+
+---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/amsmath/subeqn.pdf b/Master/texmf-dist/doc/latex/amsmath/subeqn.pdf
new file mode 100644
index 00000000000..4b769ea1ff0
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/amsmath/subeqn.pdf
diff --git a/Master/texmf-dist/doc/latex/amsmath/technote.pdf b/Master/texmf-dist/doc/latex/amsmath/technote.pdf
new file mode 100644
index 00000000000..d3ec2677db0
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/amsmath/technote.pdf
diff --git a/Master/texmf-dist/doc/latex/amsmath/testmath.pdf b/Master/texmf-dist/doc/latex/amsmath/testmath.pdf
new file mode 100644
index 00000000000..53bc768ff7d
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/amsmath/testmath.pdf
diff --git a/Master/texmf-dist/source/latex/amsmath/00readme.txt b/Master/texmf-dist/source/latex/amsmath/00readme.txt
new file mode 100644
index 00000000000..080ed622537
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/00readme.txt
@@ -0,0 +1,123 @@
+00readme.txt for amsmath 2.0 [2004/08/06] American Mathematical Society
+
+See manifest.txt for a list of all the files in the distribution.
+
+See install.txt for installation instructions. [Installing from an
+archive file such as ftp://ftp.ams.org/pub/tex/amsmath.zip is
+recommended.]
+
+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 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.
+
+In order to use amsmath you need to have TeX installed first. TeX is not
+an AMS product. For information on getting TeX see one of the following:
+
+  http://www.tug.org/
+  http://www.ams.org/tex/tex-resources.html
+
+The primary documentation for amsmath is in
+
+  amsldoc.pdf
+
+Additional documentation files include:
+
+  diffs-m.txt
+  subeqn.pdf
+  technote.pdf
+  testmath.pdf
+
+which are included in the collection.  Additional documentation can be
+found in the amsmath FAQ:
+
+  http://www.ams.org/tex/amsmath-faq.html
+
+For technical support:
+
+  American Mathematical Society
+  Technical Support
+  Publications Technical Group
+  P. O. Box 6248
+  Providence, RI 02940-6248
+  Phone: 800-321-4AMS (321-4267) (USA/Canada) or 401-455-4080
+  tech-support@ams.org
+
+For submitting bug reports, however, users are encouraged to use the
+standard LaTeX bug reporting system: Issue the command
+
+  latex latexbug.tex
+
+and follow the resulting instructions (selecting the "amslatex" category
+when the list of possible categories is shown).
+
+========================================================================
+RECENT CHANGES (REVERSE ORDER)
+
+---amsmath.faq - 2004/08/06
+Removed from distribution; replaced by on-line FAQ at
+http://www.ams.org/tex/amsmath-faq.html
+
+---amsmath.dtx 2.13 2000/07/18:
+
+After the numbering patches in 2.11, \notag failed in certain
+circumstances: introduce some more auxiliary functions to sort things
+out, and redefine \nonumber.
+
+---amstext.dtx 2.01 2000/06/29:
+
+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.
+
+---amsmath.dtx 2.12 2000/06/06:
+
+Fix transposed lines in 2.11 patch.
+
+---amsdtx.dtx - 2000/06/02:
+
+Move to the amscls distribution.
+
+---amsmath.dtx 2.11 2000/06/02:
+
+Prevent "Arithmetic overflow" error by guarding against divide-by-zero
+in \x@calc@shift@lc (align environment).
+
+---amsmath.dtx 2.10 2000/05/25:
+
+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.
+
+---amsmath.dtx 2.09 2000/04/21:
+
+Ensure good catcodes for " etc.
+
+---amsmath.dtx 2.08 2000/03/16:
+
+Fixed erroneous tag placement on split with fleqn/tbtags options.
+
+---amsmath.dtx 2.07 2000/03/15:
+
+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.
+
+---amsmath.dtx 2.06 2000/03/10:
+
+Change \MathAccent to \mathaccentV so \DeclareMathAccent won't give an
+error when redefining an accent.
+
+---amsmath.dtx 2.05 2000/01/06:
+
+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/source/latex/amsmath/ams-m1.ins b/Master/texmf-dist/source/latex/amsmath/ams-m1.ins
new file mode 100644
index 00000000000..a03642e51a7
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/ams-m1.ins
@@ -0,0 +1,69 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "ams-m1.ins",
+%%%   version   = "1.05",
+%%%   date      = "2000/05/25",
+%%%   time      = "16:28:35 EDT",
+%%%   checksum  = "13352 69 258 2600",
+%%%   author    = "American Mathematical Society",
+%%%   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@math.ams.org (Internet)",
+%%%   supported = "yes",
+%%%   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.",
+%%%   keywords  = "latex, amsmath, ams-latex, installation",
+%%%   abstract  = "This is an installation file that will generate
+%%%                loadable package and class files when run
+%%%                through \LaTeX{}.",
+%%%   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.",
+%%% }
+%%% ====================================================================
+\def\batchfile{ams-m1.ins}
+\input docstrip.tex
+\preamble
+\endpreamble
+
+\keepsilent
+
+\csname newtoks\endcsname\filesmsg
+\def\dofile#1#2#3{\generateFile{#1}{f}{\from{#2}{#3}}%
+  \filesmsg\expandafter{\the\filesmsg
+    \Msg{* \space\space#1}}}
+
+%    Math packages
+\dofile{amsmath.sty}{amsmath.dtx}{}
+\dofile{amsgen.sty}{amsgen.dtx}{}
+\dofile{amstext.sty}{amstext.dtx}{}
+\dofile{amsopn.sty}{amsopn.dtx}{}
+\dofile{amsbsy.sty}{amsbsy.dtx}{}
+\dofile{amscd.sty}{amscd.dtx}{}
+\dofile{amsxtra.sty}{amsxtra.dtx}{}
+
+\ifToplevel{
+\Msg{******************************************************************}
+\Msg{*}
+\Msg{* To produce the User's Guide for amsmath, run amsldoc.tex through}
+\Msg{* LaTeX.}
+\Msg{*}
+\Msg{******************************************************************}
+}
+
+\endinput
diff --git a/Master/texmf-dist/source/latex/amsmath/amsbsy.dtx b/Master/texmf-dist/source/latex/amsmath/amsbsy.dtx
new file mode 100644
index 00000000000..30c4dee1bbf
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amsbsy.dtx
@@ -0,0 +1,251 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsbsy.dtx",
+%%%   version   = "1.2d",
+%%%   date      = "1999/11/29",
+%%%   time      = "14:26:42 EST",
+%%%   author    = "American Mathematical Society",
+%%%   copyright = "Copyright 1995 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",
+%%%   checksum  = "08988 251 1238 9625",
+%%%   email     = "tech-support@ams.org (Internet)",
+%%%   codetable = "ISO/ASCII",
+%%%   keywords  = "latex, bold symbol, pmb",
+%%%   supported = "yes",
+%%%   abstract  = "This is a \LaTeX{} package that provides a command
+%%%                for producing bold math symbols when appropriate
+%%%                fonts exist, and a `poor man's bold' command that
+%%%                can be applied when no appropriate bold font
+%%%                is available.",
+%%%   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.",
+%%% }
+%%% ====================================================================
+%\iffalse
+%<*driver>
+\documentclass{amsdtx}
+\usepackage{amsbsy}
+\begin{document}
+\title{The \pkg{amsbsy} package}
+\author{Frank Mittelbach\and Rainer Sch\"opf\and Michael Downes}
+\date{Version \fileversion, \filedate}
+\hDocInput{amsbsy.dtx}
+\end{document}
+%</driver>
+%\fi
+%
+% \maketitle
+%
+% \MakeShortVerb\|
+%
+% \section{Introduction}
+%
+%    The package \texttt{amsbsy}, first written in 1989, implements a
+%    few commands for producing \textbf{bold} characters in the `normal'
+%    \emph{math version}.
+% \begin{quote}\em
+%    Note: It is recommended nowadays to use the \pkg{bm} package, which
+%    became available in 1997.
+% \end{quote}
+%
+%    If we have bold fonts which contain the character in question then
+%    we will use these fonts to produce the wanted character. But
+%    sometimes math fonts are only available in a certain weight
+%    (e.g.\ the AMS symbol fonts).
+% \DescribeMacro\pmb
+%    For these cases we provide a command which is called \verb+\pmb+ (an
+%    acronym for \textbf{p}oor \textbf{m}an's \textbf{b}old) with one argument.
+%    The bolder weight is achieved by copying the argument three times in
+%    slightly different positions which is better than nothing but no
+%    match for a real bold font.
+%
+% \DescribeMacro\boldsymbol
+%    There also exists the \verb+\boldsymbol+ command which is better in
+%    all cases where bold fonts exists. This command will internally
+%    switch to the corresponding `bold' \emph{math version} typeset its
+%    argument in this version.
+%
+%    Both commands will preserve the nature of their arguments, i.e.\ if
+%    they get a relational atom their result will again be a relation as
+%    far as \tex/'s mathspacing is concerned.
+%
+%    Since it is good policy to make at least a small test we try to
+%    typeset the infinity sign ($\infty$) first with \verb+\pmb+ and then
+%    with \verb+\boldsymbol+.
+% \[
+%           \pmb{\infty} = \boldsymbol{\infty} \quad ?
+% \]
+% \StopEventually{}
+%
+%    Standard package info.
+%    \begin{macrocode}
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must December 1994 or later
+\ProvidesPackage{amsbsy}[1999/11/29 v1.2d]
+%    \end{macrocode}
+%
+% \section{The implementation}
+%
+%    We need some functions from the \pkg{amsgen} package.
+%    \begin{macrocode}
+\RequirePackage{amsgen}
+%    \end{macrocode}
+%
+%\begin{macro}{\boldsymbol}
+%    In implementing boldsymbol, we must take into account \tex/'s
+%    limitation of only 16 mathgroups (math families, in Knuth's
+%    terminology). If we wanted to maintain mathgroups for both the bold
+%    and non-bold version of each math font, it would not take long to
+%    run out of mathgroups. Therefore what we do instead for a bold
+%    symbol is embed it in an \verb|\hbox|; inside that \verb|\hbox|,
+%    when we start another math formula, we can change all the
+%    mathgroups to their bold equivalents.
+%
+%    However, to get the correct math style inside the hbox (display,
+%    text, script or scriptscript) we have to use \verb|\mathchoice|.
+%    Since \verb|\mathversion{bold}| has a lot of overhead, and
+%    \verb|\mathchoice| typesets the argument text four times, we would
+%    rather not put the \verb|\mathversion| command inside each
+%    \verb|\hbox| in the \verb|\mathchoice|; on the other hand,
+%    \verb|\mathversion| gives an error message if it's used in math
+%    mode. Therefore if we want to execute \verb|\mathversion{bold}|
+%    before starting the \verb|\mathchoice| we have to temporarily
+%    disable the \verb|\@nomath| error. (The error message is intended
+%    to keep people from accidentally emboldening a preceding part of a
+%    math formula, since only the mathgroups defined at the end of a
+%    math formula will determine the fonts used in that formula; but we
+%    are going to typeset our bold symbol not in the current formula but
+%    in an embedded formula, so that this danger doesn't apply here.)
+%    \begin{macrocode}
+\DeclareRobustCommand{\boldsymbol}[1]{%
+%    \end{macrocode}
+%    Start a group to localize the change of \verb|\@nomath|:
+%    \begin{macrocode}
+  \begingroup
+%    \end{macrocode}
+%    Disable \verb|\@nomath| so that we don't have to leave math
+%    mode before executing \verb|\mathversion|:
+%    \begin{macrocode}
+  \let\@nomath\@gobble \mathversion{bold}%
+%    \end{macrocode}
+%    \cs{math@atom} is a test macro which looks at its argument and
+%    produces a math atom of the proper class.
+%    \begin{macrocode}
+  \math@atom{#1}{%
+%    \end{macrocode}
+%    Although it is tempting to use \verb|\text| here, to save some
+%    main memory, that caused a bug in the past due to some internal
+%    interactions with \verb|\mathversion|.
+%    \begin{macrocode}
+  \mathchoice%
+    {\hbox{$\m@th\displaystyle#1$}}%
+    {\hbox{$\m@th\textstyle#1$}}%
+    {\hbox{$\m@th\scriptstyle#1$}}%
+    {\hbox{$\m@th\scriptscriptstyle#1$}}}%
+%    \end{macrocode}
+%    End the group we started earlier.
+%    \begin{macrocode}
+  \endgroup}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\math@atom}
+%    The macro \verb+\math@atom+ looks at its argument and produce a
+%    correct math atom, i.e.\ a primitive like \verb+\mathopen+.
+%    Until the day we have a real implementation for all cases we use
+%    the \verb+\binrel@+ command from \amstex/ which can distinguish
+%    between binary, relation and ord atoms.
+%    \begin{macrocode}
+\def\math@atom#1#2{%
+   \binrel@{#1}\binrel@@{#2}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\pmb}
+%    Poor man's bold command, works by typesetting multiple copies of
+%    the given argument with small offsets.
+%    \begin{macrocode}
+\DeclareRobustCommand{\pmb}{%
+  \ifmmode\else \expandafter\pmb@@\fi\mathpalette\pmb@}
+%    \end{macrocode}
+%
+%    \cs{pmb@@} is called by \cn{pmb} in the non-math-mode case.
+%    Discard the first two arguments which are for the math-mode case.
+%    \begin{macrocode}
+\def\pmb@@#1#2#3{\leavevmode\setboxz@h{#3}%
+   \dimen@-\wdz@
+   \kern-.5\ex@\copy\z@
+   \kern\dimen@\kern.25\ex@\raise.4\ex@\copy\z@
+   \kern\dimen@\kern.25\ex@\box\z@
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\newdimen\pmbraise@
+%    \end{macrocode}
+%    Note: because of the use of \cs{mathpalette}, if \cs{pmb@} is applied to a
+%    single math italic character (or a single character from some other
+%    slanted math font), the italic correction will be added. This will
+%    cause subscripts to fall too far away from the character in some
+%    cases, e.g., $\pmb{T}_1$ or $\pmb{\mathcal{T}}_1$.
+%    \begin{macrocode}
+\def\pmb@#1#2{\setbox8\hbox{$\m@th#1{#2}$}%
+  \setboxz@h{$\m@th#1\mkern.5mu$}\pmbraise@\wdz@
+  \binrel@{#2}%
+  \dimen@-\wd8 %
+  \binrel@@{%
+    \mkern-.8mu\copy8 %
+    \kern\dimen@\mkern.4mu\raise\pmbraise@\copy8 %
+    \kern\dimen@\mkern.4mu\box8 }%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macrocode}
+\def\binrel@#1{\begingroup
+  \setboxz@h{\thinmuskip0mu
+    \medmuskip\m@ne mu\thickmuskip\@ne mu
+    \setbox\tw@\hbox{$#1\m@th$}\kern-\wd\tw@
+    ${}#1{}\m@th$}%
+%    \end{macrocode}
+%    The \cn{noexpand} here should be unnecessary, but just in case
+%    \ldots
+%    \begin{macrocode}
+  \edef\@tempa{\endgroup\let\noexpand\binrel@@
+    \ifdim\wdz@<\z@ \mathbin
+    \else\ifdim\wdz@>\z@ \mathrel
+    \else \relax\fi\fi}%
+  \@tempa
+}
+%    \end{macrocode}
+%    For completeness, assign a default value for \cs{binrel@@}.
+%    \begin{macrocode}
+\let\binrel@@\relax
+%    \end{macrocode}
+%
+%    The usual \cs{endinput} to ensure that random garbage at the end of
+%    the file doesn't get copied by \fn{docstrip}.
+%    \begin{macrocode}
+\endinput
+%    \end{macrocode}
+%
+% \CheckSum{131}
+% \Finale
diff --git a/Master/texmf-dist/source/latex/amsmath/amscd.dtx b/Master/texmf-dist/source/latex/amsmath/amscd.dtx
new file mode 100644
index 00000000000..7dd765109d1
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amscd.dtx
@@ -0,0 +1,333 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amscd.dtx",
+%%%   version   = "2.0",
+%%%   date      = "1999/11/29",
+%%%   time      = "14:30:07 EST",
+%%%   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",
+%%%   checksum  = "07428 333 1152 12117",
+%%%   email     = "tech-support@ams.org (Internet)",
+%%%   codetable = "ISO/ASCII",
+%%%   keywords  = "latex, commutative diagram",
+%%%   supported = "yes",
+%%%   abstract  = "This is a \LaTeX{} package that adapts the
+%%%                commutative diagram macros of AMS-\TeX{} for
+%%%                use in \LaTeX{}",
+%%%   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.",
+%%% }
+%%% ====================================================================
+%\iffalse
+%<*driver>
+\documentclass{amsdtx}
+\usepackage{amscd}
+%    Dummy citation to avoid error in an example.
+\bibcite{fre:cichon}{C}
+\begin{document}
+\title{The \pkg{amscd} package}
+\author{Frank Mittelbach\and Rainer Sch\"opf\and Michael Downes}
+\date{Version \fileversion, \filedate}
+\hDocInput{amscd.dtx}
+\end{document}
+%</driver>
+%\fi
+%
+% \maketitle
+%
+% \MakeShortVerb\|
+% \newcommand{\operatorname}[1]{\mathop{\mathrm{#1}}}
+% \newcommand{\per}{\operatorname{per}}
+% \newcommand{\cov}{\operatorname{cov}}
+% \newcommand{\non}{\operatorname{non}}
+% \newcommand{\cf}{\operatorname{cf}}
+% \newcommand{\add}{\operatorname{add}}
+% \newcommand{\End}{\operatorname{End}}
+%
+% \section{Introduction}
+%
+%    The \pkg{amscd} package provides a \env{CD} environment that
+%    emulates the commutative diagram capabilities of \amstex/ version
+%    2.x. This means that only simple rectangular diagrams are
+%    supported, with no diagonal arrows or more exotic features. Many
+%    users will be better served by one of the more powerful diagram
+%    packages such as \pkg{diagram}, \pkg{xypic}, or \pkg{kuvio}.
+%
+%    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{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}
+% \end{verbatim}
+%    (assuming \cn{End} is defined as an `operator name'.
+%
+%    Another example:
+%
+%    We will make liberal use of Cichon's Diagram \cite{fre:cichon}:
+%    \begin{equation}
+%    \begin{CD}
+%    \cov(\mathcal{L}) @>>> \non(\mathcal{K}) @>>> \cf(\mathcal{K}) @>>>
+%    \cf(\mathcal{L})\\
+%    @VVV @AAA @AAA @VVV\\
+%    \add(\mathcal{L}) @>>> \add(\mathcal{K}) @>>> \cov(\mathcal{K}) @>>>
+%    \non(\mathcal{L})
+%    \end{CD}\end{equation}
+% \begin{verbatim}
+% \begin{equation}\begin{CD}
+% \cov(\mathcal{L}) @>>> \non(\mathcal{K}) @>>> \cf(\mathcal{K}) @>>>
+% \cf(\mathcal{L})\\
+% @VVV @AAA @AAA @VVV\\
+% \add(\mathcal{L}) @>>> \add(\mathcal{K}) @>>> \cov(\mathcal{K}) @>>>
+% \non(\mathcal{L})
+% \end{CD}\end{equation}
+% \end{verbatim}
+%
+% \StopEventually{}
+%
+%    Standard package info.
+%    \begin{macrocode}
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must December 1994 or later
+\ProvidesPackage{amscd}[1999/11/29 v2.0]
+%    \end{macrocode}
+%    \begin{macrocode}
+\RequirePackage{amsgen}
+%    \end{macrocode}
+%      Better not to redefine \verb|\math@cr| if it is already defined,
+%      because for CD use only we want to omit the part of the code
+%      related to \verb|\dspbrk@lvl| (see \fn{amsmath.sty}).
+%
+%    [mjd,1999/11/04] These definitions have gone somewhat obsolete; but
+%    we had probably better leave them as they are for backward
+%    compatibility.
+%    \begin{macrocode}
+\@ifundefined{math@cr}{%
+  \def\math@cr{{\ifnum0=`}\fi
+    \@ifstar{\global\@eqpen\@M\math@cr@}%
+          {\global\@eqpen\interdisplaylinepenalty \math@cr@}}
+%    \end{macrocode}
+%    The following section merely duplicates some code from the
+%    \pkg{amsmath} package, in case the \pkg{amscd} package is used by
+%    itself. For documentation of the code refer to \fn{amsmath.dtx}.
+%    \begin{macrocode}
+  \def\math@cr@{\new@ifnextchar[\math@cr@@{\math@cr@@[\z@]}}
+  \def\math@cr@@[#1]{\ifnum0=`{\fi}\math@cr@@@
+    \noalign{\vskip#1\relax}}
+  \def\restore@math@cr{\def\math@cr@@@{\cr}}
+}{}
+\restore@math@cr
+%    \end{macrocode}
+%    [mjd,1999/11/04] These definitions too are somewhat obsolete;
+%    but we had probably better leave them as they are for backward
+%    compatibility.
+%    \begin{macrocode}
+\@ifundefined{rightarrowfill@}{
+  \def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
+    $#1\copy\z@\mkern-6mu\cleaders
+    \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
+    \mkern-6mu\mathord\rightarrow$}
+  \def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
+    $#1\mathord\leftarrow\mkern-6mu\cleaders
+    \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
+    \mkern-6mu\box\z@$}
+  \def\leftrightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
+    $#1\mathord\leftarrow\mkern-6mu\cleaders
+    \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
+    \mkern-6mu\mathord\rightarrow$}
+}{}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\atdef@#1{\expandafter\def\csname\space @\string#1\endcsname}
+\@ifundefined{Iat}{%
+  \DeclareRobustCommand{\Iat}{\FN@\at@}
+}{}
+\begingroup \catcode`\@=\active
+%    \end{macrocode}
+%    Define math \qc{\@} to replicate its mathcode-dictated behavior.
+%    This is in case \qc{\@} occurs outside of \env{CD}.
+%    \begin{macrocode}
+\csname if\string @compatibility\endcsname
+\else \fam=\mathcode`\@
+  \xdef @{\mathchar\number\fam\space }
+\fi
+\gdef\CDat{\let @=\Iat}
+\endgroup
+\mathcode`\@="8000 % make @ pseudo-active in math
+\def\at@{\let\next@\at@@
+ \ifcat\noexpand\next a\else
+ \ifcat\noexpand\next0\else
+ \ifcat\noexpand\next\relax\else
+ \let\next@\at@@@\fi\fi\fi\next@}
+\def\at@@#1{\expandafter
+  \ifx\csname\space @\string#1\endcsname\relax
+    \DN@{\at@@@#1}%
+  \else
+    \DN@{\csname\space @\string#1\endcsname}%
+  \fi\next@}%
+%    \end{macrocode}
+%    The following items should be defined only if they are not
+%    already defined, either to leave the package name untouched (in
+%    the case of \cs{PackageError}) or to avoid redundant allocation
+%    of token or dimen registers.
+%    \begin{macrocode}
+\@ifundefined{default@tag}{%
+  \def\default@tag{%
+    \def\tag{\PackageError{amscd}{\protect\tag\space not allowed
+      here}\@eha}}%
+}{}%
+\@ifundefined{at@@@}{%
+  \def\at@@@{\PackageError{amscd}{\Invalid@@ @}{\the\athelp@}\char64\relax}
+}{}
+\@ifundefined{athelp@}{\csname newhelp\endcsname\athelp@
+{Only certain combinations beginning with @ make sense to me.^^J%
+I'll assume you wanted @@ for a printed @.}}{}
+\@ifundefined{minaw@}{\newdimen\minaw@}{}
+\@ifundefined{bigaw@}{\newdimen\bigaw@}{}
+%    \end{macrocode}
+%
+%      Assignment of a couple of dimensions, and initialization of
+%      \verb=\ampersand@=. We check to see if we need to define
+%      \verb=\minaw@= and \verb=\bigaw@=.
+%    \begin{macrocode}
+\minaw@11.111pt
+\newdimen\minCDarrowwidth
+\minCDarrowwidth2.5pc
+\newif\ifCD@
+\let\ampersand@\relax
+%    \end{macrocode}
+%
+%      Added \verb|\restore@math@cr\default@tag| to fix line numbering
+%      problems, 7-JUN-1991. Suggested by I.~Zakharevich.
+%    \begin{macrocode}
+\newenvironment{CD}{%
+  \CDat
+  \bgroup\relax\iffalse{\fi\let\ampersand@&\iffalse}\fi
+  \CD@true\vcenter\bgroup\let\\\math@cr\restore@math@cr\default@tag
+  \tabskip\z@skip\baselineskip20\ex@
+  \lineskip3\ex@\lineskiplimit3\ex@\halign\bgroup
+  &\hfill$\m@th##$\hfill\crcr
+}{%
+  \crcr\egroup\egroup\egroup
+}
+%    \end{macrocode}
+%
+% \begin{macro}{\CD@check}
+%    This check is used by all macros that must not appear outside the
+%    \env{CD} environment. The first argument is the symbol to be used
+%    after \verb+@+, the second one the action to be performed.
+%
+%    Once again we use the trick of defining a temporary control
+%    sequence \verb+\next@+ and calling it after the final \verb+\fi+.
+%    This is not absolutely necessary, but it ensures that the
+%    conditional text is processed in one and the same column
+%    of the enclosing alignment.
+%    \begin{macrocode}
+\def\CD@check#1#2{\ifCD@\DN@{#2}\else
+  \DN@{\PackageError{amscd}{@\protect#1 not
+    allowed outside of the CD environment}\@eha}%
+  \fi\next@}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macrocode}
+\atdef@>#1>#2>{\ampersand@
+  \ifCD@ \global\bigaw@\minCDarrowwidth \else \global\bigaw@\minaw@ \fi
+  \setboxz@h{$\m@th\scriptstyle\;{#1}\;\;$}%
+  \ifdim\wdz@>\bigaw@\global\bigaw@\wdz@\fi
+%    \end{macrocode}
+%      If \verb|#2| is empty we can save some work.
+%    \begin{macrocode}
+  \@ifnotempty{#2}{\setbox\@ne\hbox{$\m@th\scriptstyle\;{#2}\;\;$}%
+    \ifdim\wd\@ne>\bigaw@\global\bigaw@\wd\@ne\fi}%
+ \ifCD@\enskip\fi
+   \mathrel{\mathop{\hbox to\bigaw@{\rightarrowfill@\displaystyle}}%
+     \limits^{#1}\@ifnotempty{#2}{_{#2}}}%
+ \ifCD@\enskip\fi \ampersand@}
+%
+\atdef@<#1<#2<{\ampersand@
+  \ifCD@ \global\bigaw@\minCDarrowwidth \else \global\bigaw@\minaw@ \fi
+  \setboxz@h{$\m@th\scriptstyle\;\;{#1}\;$}%
+  \ifdim\wdz@>\bigaw@ \global\bigaw@\wdz@ \fi
+  \@ifnotempty{#2}{\setbox\@ne\hbox{$\m@th\scriptstyle\;\;{#2}\;$}%
+    \ifdim\wd\@ne>\bigaw@ \global\bigaw@\wd\@ne \fi}%
+  \ifCD@\enskip\fi
+    \mathrel{\mathop{\hbox to\bigaw@{\leftarrowfill@\displaystyle}}%
+      \limits^{#1}\@ifnotempty{#2}{_{#2}}}%
+  \ifCD@\enskip\fi \ampersand@}
+%    \end{macrocode}
+%
+%      Variants of the above two arrows, using \verb|(| and \verb|)|
+%      characters instead of \verb|<| and \verb|>| characters, are
+%      provided for those whose keyboards don't have the latter.
+%    \begin{macrocode}
+\begingroup \catcode`\~=\active \lccode`\~=`\@
+\lowercase{%
+  \global\atdef@)#1)#2){~>#1>#2>}
+  \global\atdef@(#1(#2({~<#1<#2<}
+}% end lowercase
+\endgroup
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\atdef@ A#1A#2A{\CD@check{A..A..A}{\llap{$\m@th\vcenter{\hbox
+  {$\scriptstyle#1$}}$}\Big\uparrow
+  \rlap{$\m@th\vcenter{\hbox{$\scriptstyle#2$}}$}&&}}
+%
+\atdef@ V#1V#2V{\CD@check{V..V..V}{\llap{$\m@th\vcenter{\hbox
+  {$\scriptstyle#1$}}$}\Big\downarrow
+  \rlap{$\m@th\vcenter{\hbox{$\scriptstyle#2$}}$}&&}}
+%
+\atdef@={\CD@check={&\enskip\mathrel
+  {\vbox{\hrule\@width\minCDarrowwidth\vskip2\ex@\hrule\@width
+  \minCDarrowwidth}}\enskip&}}
+%
+\atdef@|{\CD@check|{\Big\Vert&&}}
+%
+\atdef@\vert{\CD@check\vert{\Big\Vert&&}}
+%
+\atdef@.{\CD@check.{&&}}
+%    \end{macrocode}
+%
+%    The \cn{pretend} command has weird syntax that doesn't fit well
+%    with standard \latex/ syntax so we leave it undone, at least for
+%    now. [mjd,1994/10/27]
+%    \begin{macrocode}
+%\def\pretend#1\haswidth#2{\setboxz@h{$\m@th\scriptstyle{#2}$}\hbox
+% to\wdz@{\hfill$\m@th\scriptstyle{#1}$\hfill}}
+%    \end{macrocode}
+%
+%    The usual \cs{endinput} to ensure that random garbage at the end of
+%    the file doesn't get copied by \fn{docstrip}.
+%    \begin{macrocode}
+\endinput
+%    \end{macrocode}
+%
+% \CheckSum{459}
+% \Finale
diff --git a/Master/texmf-dist/source/latex/amsmath/amsdtx.cls b/Master/texmf-dist/source/latex/amsmath/amsdtx.cls
new file mode 100644
index 00000000000..96a50814c8f
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amsdtx.cls
@@ -0,0 +1,517 @@
+%%
+%% This is file `amsdtx.cls',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amsdtx.dtx  (with options: `amsdtx')
+%% 
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsdtx.dtx",
+%%%   version   = "2.06",
+%%%   date      = "2004/08/06",
+%%%   time      = "12:18:49 EDT",
+%%%   checksum  = "19929 1192 3506 39423",
+%%%   author    = "Michael J. Downes, updated by David M. Jones",
+%%%   copyright = "Copyright 1996, 1999, 2004
+%%%                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,
+%%%                Publications Technical Group,
+%%%                201 Charles Street,
+%%%                Providence, RI 02904,
+%%%                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, amslatex, ams-latex, user documentation",
+%%%   supported = "yes",
+%%%   abstract  = "This is the source for two document classes, amsdtx
+%%%                and amsldoc, used to produce AMS user documentation
+%%%                or technical documentation.",
+%%%   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)
+[1995/06/01]% LaTeX date must be June 1995 or later
+\ProvidesClass{amsdtx}[2004/08/06 v2.06]
+\DeclareOption*{\PassOptionsToClass{\CurrentOption}{book}}
+\ProcessOptions
+\LoadClass{book}
+\IfFileExists{url.sty}{%
+    \RequirePackage{url}\relax
+    \@gobble
+}{%
+    \@firstofone
+}
+{
+    \DeclareRobustCommand{\url}[1]{%
+        \def\@tempa{#1}%
+        \texttt{\urlsetup $\expandafter\strip@prefix\meaning\@tempa$}%
+    }%
+    \def\urlsetup{%
+        \check@mathfonts \textfont\@ne\the\font \textfont\z@\the\font
+        \urlfix +\urlfix\=\urlfix\:\urlfix\-\urlfix\.\urlfix\,\urlfix\;%
+        \urlbreak\&\urlbreak\/\urlbreak\?%
+    }%
+    \def\urlbreak#1{%
+        \mathcode`#1="8000
+        \begingroup \lccode`\~=`#1 \lowercase{\endgroup \edef~}%
+        {\mathchar\number`#1\penalty\hyphenpenalty}%
+    }%
+    \def\urlfix#1{%
+        \mathcode`#1=`#1\relax
+    }%
+}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\providecommand{\qq}[1]{\textquotedblleft#1\/\textquotedblright}
+\providecommand{\mdash}{\textemdash\penalty\exhyphenpenalty}
+\providecommand{\ndash}{\textendash\penalty\exhyphenpenalty}
+\let\@xp\expandafter
+\newcommand*{\autoindex}[1]{%
+    \index{#1\ifcodeline@index\encapchar usage\fi}%
+}
+\newcommand{\ntt}{\normalfont\ttfamily}
+\newcommand*{\indexcs}[1]{%
+    \@xp\@xp\@xp\@indexcs\@xp\@nobslash\string#1\@nil
+}
+\def\@indexcs#1\@nil{%
+    \autoindex{#1\actualchar
+        \string\verb\quotechar*\verbatimchar
+            \@xp\@bothoftwo\string\ #1\@empty
+        \verbatimchar
+    }%
+}
+\def\@bothoftwo#1#2{#1#2}
+\def\category@index#1#2{%
+    {\ntt#2}%
+    \@category@index{#1}{#2}%
+}
+\def\@category@index#1#2{%
+    \autoindex{%
+        #2\actualchar\string\texttt{#2}%
+        \ifx\@nil#1\@nil\else\space#1\fi
+    }%
+}
+\chardef\bslchar=`\\ % p. 424, TeXbook
+\newcommand{\addbslash}{\expandafter\@addbslash\string}
+\def\@addbslash#1{\bslchar\@nobslash#1}
+\newcommand{\nobslash}{\expandafter\@nobslash\string}
+\def\@nobslash#1{\ifnum`#1=\bslchar\else#1\fi}
+\def\@boxorbreak{%
+    \leavevmode
+    \ifmmode\hbox\else\ifdim\lastskip=\z@\penalty9999 \fi\fi
+}
+\DeclareRobustCommand{\cs}[1]{%
+    \@boxorbreak{%
+        \ntt
+        \addbslash#1\@empty
+        \@xp\@xp\@xp\@indexcs\@xp\@nobslash\string#1\@nil
+    }%
+}
+\let\cn\cs
+\DeclareRobustCommand{\cls}{\category@index{class}}
+\DeclareRobustCommand{\pkg}{\category@index{package}}
+\DeclareRobustCommand{\opt}{\category@index{option}}
+\DeclareRobustCommand{\env}[1]{{\ntt#1}\SpecialEnvIndex{#1}}
+\DeclareRobustCommand{\fn}{\category@index{}}
+\DeclareRobustCommand{\bst}{\category@index{\string\BibTeX{} style}}
+\DeclareRobustCommand{\cnt}{\category@index{counter}}
+\DeclareRobustCommand{\fnt}{\category@index{font}}
+\def\allowtthyphens{\begingroup
+  \fontencoding{OT1}\fontfamily{cmtt}%
+  \expandafter\let\csname OT1+cmtt\endcsname\relax
+  \try@load@fontshape
+  \endgroup
+  \expandafter\let\csname OT1+cmtt\endcsname\@empty
+}
+\hfuzz2pc
+\vbadness9999 \hbadness5000
+\def\AmS{{\protect\usefont{OMS}{cmsy}{m}{n}%
+  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}
+\def\latex/{{\protect\LaTeX}}
+\def\amslatex/{{\protect\AmS-\protect\LaTeX}}
+\def\tex/{{\protect\TeX}}
+\def\amstex/{{\protect\AmS-\protect\TeX}}
+\def\bibtex/{{Bib\protect\TeX}}
+\def\makeindx/{MakeIndex}
+\def\xypic/{XY\mbox{-}pic}
+\newcommand{\Textures}{\textit{Textures}}
+\def\<#1>{\textit{$\langle$#1\/$\rangle$}}
+\def\@listi{%
+  \leftmargin\leftmargini
+  \topsep 3\p@ \@plus2\p@ \@minus\p@
+  \parsep \p@ \@plus\p@ \itemsep\parsep
+}
+\let\@listI\@listi
+\@listi
+\def\@listii{%
+  \leftmargin\leftmarginii
+  \labelwidth\leftmarginii \advance\labelwidth-\labelsep
+  \topsep\p@\@plus\p@ \@minus\p@
+  \parsep\z@skip \itemsep\z@skip
+}
+\def\@listiii{%
+  \leftmargin\leftmarginiii
+  \labelwidth\leftmarginiii \advance\labelwidth-\labelsep
+  \topsep\z@skip \parsep\z@skip \itemsep\z@skip
+}
+\newenvironment{histnote}{%
+  \trivlist\item[\hspace{\labelsep}\bfseries Historical Note:]%
+}{%
+  \endtrivlist
+}
+\def\@starttoc#1{\begingroup
+  \let\autoindex\@gobble
+  \makeatletter
+  \@input{\jobname.#1}\if@filesw
+  \expandafter\newwrite\csname tf@#1\endcsname
+             \immediate\openout
+                 \csname tf@#1\endcsname \jobname.#1\relax
+  \fi \global\@nobreakfalse \endgroup}
+\newcommand*{\secref}[1]{\S\ref{#1}}
+\newcommand{\qc}[1]{}% check for prior definition
+\edef\qc#1{\noexpand\protect\expandafter\noexpand\csname qc \endcsname
+  \noexpand\protect#1}
+\@namedef{qc }#1#2{\begingroup\ntt
+  \ifx\ #2\char`\ \else\escapechar\m@ne\string#2\fi\endgroup}
+\DeclareRobustCommand{\qcat}{\qc\@}%
+\DeclareRobustCommand{\qcamp}{\qc\&}%
+\DeclareRobustCommand{\qcbang}{\qc\!}%
+\DeclareRobustCommand{\arg}[1]{{\ntt\##1}}
+\def\ncn#1{{\let\index\@gobble\cn{#1}}}
+
+\DeclareRobustCommand{\cnbang}{%
+  \ncn{\!}\index{"!@{\ntt\bslchar\qcbang}}}
+\DeclareRobustCommand{\cnat}{%
+  \ncn{\!}\index{"@@{\ntt\bslchar\qcat}}}
+\def\5{\penalty500 }
+\newcommand{\errexa}{\par\noindent\textit{Example}:\ }
+\newcommand{\errexpl}{\par\noindent\textit{Explanation}:\ }
+\renewcommand\chapter{\par \@afterindentfalse
+                    \secdef\@chapter\@schapter}
+\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
+                       \if@mainmatter
+                         \refstepcounter{chapter}%
+                         \typeout{\@chapapp\space\thechapter.}%
+                         \addcontentsline{toc}{chapter}%
+                                   {\protect\numberline{\thechapter}#1}%
+                       \else
+                         \addcontentsline{toc}{chapter}{#1}\fi
+                    \else
+                      \addcontentsline{toc}{chapter}{#1}
+                    \fi
+                    \chaptermark{#1}%
+                    \addtocontents{lof}{\protect\addvspace{10\p@}}%
+                    \addtocontents{lot}{\protect\addvspace{10\p@}}%
+                    \if@twocolumn
+                      \@topnewpage[\@makechapterhead{#2}]%
+                    \else
+                      \@makechapterhead{#2}%
+                      \@afterheading
+                    \fi}
+\def\@makechapterhead#1{%
+  \vspace{1.5\baselineskip}%
+  {\parindent \z@ \raggedright \reset@font
+    \ifnum \c@secnumdepth >\m@ne
+      \large\bfseries \@chapapp\space\thechapter
+      \par\nobreak
+      \vskip.5\baselineskip\relax
+    \fi
+    #1\par\nobreak
+    \vskip\baselineskip
+  }}
+\def\@schapter#1{\if@twocolumn
+                   \@topnewpage[\@makeschapterhead{#1}]%
+                 \else
+                   \@makeschapterhead{#1}%
+                   \@afterheading
+                 \fi}
+\def\@makeschapterhead#1{%
+  \vspace*{1.5\baselineskip}%
+  {\parindent \z@ \raggedright
+    \reset@font
+    \large \bfseries  #1\par\nobreak
+    \vskip\baselineskip
+  }}
+\def\@removefromreset#1#2{\let\@tempb\@elt
+   \expandafter\let\expandafter\@tempa\csname c@#1\endcsname
+   \def\@elt##1{\expandafter\ifx\csname c@##1\endcsname\@tempa\else
+         \noexpand\@elt{##1}\fi}%
+   \expandafter\edef\csname cl@#2\endcsname{\csname cl@#2\endcsname}%
+   \let\@elt\@tempb}
+\@removefromreset{footnote}{chapter}
+\def\ps@headings{%
+  \let\@oddfoot\@empty\let\@evenfoot\@empty
+  \def\@evenhead{\thepage\hfil{\footnotesize\leftmark{}{}}}%
+  \def\@oddhead{{\footnotesize\rightmark{}{}}\hfil\thepage}%
+  \let\@mkboth\markboth
+  \def\chaptermark##1{%
+    \markboth {\uppercase{\ifnum \c@secnumdepth >\m@ne
+      \if@mainmatter
+        \@chapapp\ \thechapter. \ \fi
+      \fi
+        ##1}}{}}%
+  \def\sectionmark##1{%
+    \markright {\uppercase{\ifnum \c@secnumdepth >\z@
+        \thesection. \ \fi
+        ##1}}}}
+\renewcommand\maketitle{\par
+  \begingroup
+    \renewcommand\thefootnote{\fnsymbol{footnote}}%
+    \def\@makefnmark{\hbox to\z@{$\m@th^{\@thefnmark}$\hss}}%
+    \long\def\@makefntext##1{\parindent 1em\noindent
+            \hbox to1.8em{\hss$\m@th^{\@thefnmark}$}##1}%
+    \if@twocolumn
+      \ifnum \col@number=\@ne
+        \@maketitle
+      \else
+        \twocolumn[\@maketitle]%
+      \fi
+    \else
+      \newpage
+      \global\@topnum\z@   % Prevents figures from going at top of page.
+      \@maketitle
+    \fi
+    \thispagestyle{plain}\@thanks
+  \endgroup
+  \setcounter{footnote}{0}%
+  \let\thanks\relax
+  \let\maketitle\relax\let\@maketitle\relax
+  \gdef\@thanks{}\gdef\@author{}\gdef\@title{}}
+\def\@maketitle{%
+  \newpage
+  \null
+  \vskip 2em%
+  \begin{center}%
+    {\LARGE \@title \par}%
+    \vskip 1.5em%
+    {\large
+      \lineskip .5em%
+      \begin{tabular}[t]{c}%
+        \@author
+      \end{tabular}\par}%
+    \vskip 1em%
+    {\large \@date}%
+  \end{center}%
+  \par
+  \uppercase\expandafter{\expandafter\toks@\expandafter{\@title}}%
+  \edef\@tempa{\noexpand\markboth{\the\toks@}{\the\toks@}}%
+  \@tempa
+  \vskip 1.5em}
+\renewcommand\tableofcontents{%
+    \if@twocolumn
+      \@restonecoltrue\onecolumn
+    \else
+      \@restonecolfalse
+    \fi
+    \subsection*{\contentsname}%
+    \@starttoc{toc}%
+    \if@restonecol\twocolumn\fi
+    }
+\renewcommand\section{\@startsection {section}{1}{\z@}%
+                                   {-.6\baselineskip \@plus -3\p@}%
+                                   {.4\baselineskip}
+                          {\reset@font\Large\bfseries}}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+                                     {-.3\baselineskip\@plus -2\p@}%
+                                     {.2\baselineskip}%
+                          {\reset@font\large\bfseries}}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+                                     {-.2\baselineskip\@plus -2\p@}%
+                                     {.2\baselineskip}%
+                                     {\reset@font\normalsize\bfseries}}
+\renewenvironment{theindex}{%
+  \if@twocolumn \@restonecolfalse \else \@restonecoltrue \fi
+  \columnseprule \z@ \columnsep 35\p@
+  \let\autoindex\@gobble
+  \twocolumn[\section*{\indexname}]%
+  \addcontentsline{toc}{chapter}{\indexname}%
+  \@mkboth{\uppercase{\indexname}}{\uppercase{\indexname}}%
+  \thispagestyle{plain}\parindent\z@
+  \parskip\z@ \@plus .3\p@\relax
+  \let\item\@idxitem
+}{%
+  \if@restonecol\onecolumn\else\clearpage\fi
+}
+\renewcommand{\thesection}{\arabic{section}}
+\long\def\@makecaption#1#2{%
+  \addvspace\abovecaptionskip
+  \begingroup
+    \countdef\@parcycles=8 % local count register
+    \@parcycles\z@
+    \@setpar{\advance\@parcycles\@ne \ifnum\@parcycles>999
+      \@@par\@parcycles\z@\fi
+      \ifhmode \unskip\hskip\parfillskip\penalty-\@M\fi}%
+    \@hangfrom{\textbf{#1.} }\vadjust{\penalty\m@ne}#2%
+  \endgroup
+  \ifhmode\unpenalty\fi\par
+  \ifnum\lastpenalty=\m@ne % only one line in the caption
+    \unpenalty \setbox\@tempboxa\lastbox
+    \nointerlineskip
+    \hbox to\hsize{\hfill\unhbox\@tempboxa\unskip\hfill}%
+  \fi
+  \nobreak\vskip\belowcaptionskip
+}
+\renewenvironment{table}{%
+  \belowcaptionskip\abovecaptionskip \abovecaptionskip\z@skip
+  \@float{table}%
+}{%
+  \end@float
+}
+\renewenvironment{table*}{%
+  \belowcaptionskip\abovecaptionskip \abovecaptionskip\z@skip
+  \@dblfloat{table}%
+}{%
+  \end@dblfloat
+}
+\setcounter{topnumber}{4}\setcounter{bottomnumber}{4}
+\setcounter{totalnumber}{4}\setcounter{dbltopnumber}{4}
+\renewcommand{\topfraction}{.97}\renewcommand{\bottomfraction}{.97}
+\renewcommand{\textfraction}{.03}\renewcommand{\floatpagefraction}{.9}
+\renewcommand{\dbltopfraction}{.97}
+\renewcommand{\dblfloatpagefraction}{.9}
+\setlength{\floatsep}{8pt plus6pt}
+\setlength{\textfloatsep}{10pt plus8pt}
+\setlength{\intextsep}{8pt plus6pt}
+\setlength{\dblfloatsep}{8pt plus6pt}
+\setlength{\dbltextfloatsep}{10pt plus8pt}
+\setlength{\@fptop}{0pt}\setlength{\@fpsep}{8pt}%
+\setlength{\@fpbot}{0pt plus 1fil}
+\setlength{\@dblfptop}{0pt}\setlength{\@dblfpsep}{8pt}%
+\setlength{\@dblfpbot}{0pt plus 1fil}
+
+\pagestyle{headings}
+\newcommand{\arrayargpatch}{%
+  \let\@oldarray\@array
+  \edef\@array[##1]##2{\catcode\number`\|=\number\catcode`\|
+    \catcode\number`\@=\number\catcode`\@ \relax
+    \let\noexpand\@array\noexpand\@oldarray
+    \noexpand\@array[##1]{##2}}%
+  \catcode`\|=12 \catcode`\@=12 \relax
+}
+\newenvironment{ctab}{%
+  \par\topsep\medskipamount
+  \trivlist\centering
+  \item[]%
+  \arrayargpatch
+  \begin{tabular}%
+}{%
+  \end{tabular}%
+  \endtrivlist
+}
+\RequirePackage{doc}
+\def\AltMacroFont{\MacroFont}
+\def\SpecialMainEnvIndex#1{%
+    \@bsphack
+    \special@index{%
+        #1\actualchar\string\texttt{#1} environment\encapchar main%
+    }%
+    \special@index{%
+        environments:\levelchar#1%
+        \actualchar\string\texttt{#1}%
+        \encapchar main}%
+    \@esphack
+}
+\def\SpecialEnvIndex#1{%
+    \@bsphack
+    \index{#1\actualchar\string\texttt{#1} environment\encapchar usage}%
+    \index{%
+        environments:\levelchar#1%
+        \actualchar\string\texttt{#1}%
+        \encapchar usage}%
+    \@esphack
+}
+\def\DescribeOption{%
+    \leavevmode
+    \@bsphack
+    \begingroup
+        \MakePrivateLetters
+    \Describe@Option
+}
+\def\Describe@Option#1{%
+    \endgroup
+    \marginpar{\raggedleft\PrintDescribeOption{#1}}%
+    \SpecialOptionIndex{#1}%
+    \@esphack
+    \ignorespaces
+}
+\def\PrintDescribeOption#1{\strut \MacroFont #1\ }
+\def\option{%
+    \let\SpecialMainEnvIndex\SpecialMainOptionIndex
+\begingroup
+   \catcode`\\12
+   \MakePrivateLetters \m@cro@ \iffalse}
+\def\SpecialMainOptionIndex#1{%
+    \@bsphack
+    \special@index{#1\actualchar\string\texttt{#1} option\encapchar main}%
+    \@esphack
+}
+\def\SpecialOptionIndex{\@category@index{option}}
+\def\hDocInput#1{\MakePercentIgnore
+  \begingroup
+  \begingroup \lccode`\~=`\@
+  \lowercase{\endgroup\long\def ~}##1##{%
+    \catcode`\==12 \skipfileheader{##1}}%
+  \catcode`\@=\active \catcode`\==14 % comment
+  \def\filename{#1}%
+  \@@input#1 \MakePercentComment}
+\def\skipfileheader#1#2 {\endgroup
+  \hGetFileInfo#2 version = "??" date = "??"\@nil
+  \begingroup\catcode`\==9 \catcode`\ =9 \futurelet\0\endgroup
+}
+\long\def\hGetFileInfo#1 version = "#2"#3 date = "#4"#5\@nil{%
+  \def\fileversion{#2}\def\filedate{#4}}
+\DoNotIndex{\@xp,\@nx,\@empty,\newcommand,\renewcommand}
+\DoNotIndex{\newenvironment,\renewenvironment,\providecommand}
+\DoNotIndex{\if,\fi,\ifnum,\fi,\@let@token,\futurelet,\fsa@n}
+\DoNotIndex{\ignorespaces,\@tempcnta,\@tempcntb,\count@}
+\DoNotIndex{\toks@,\@ne,\advance}
+\DoNotIndex{\!,\/,\?,\@,\^,\_}
+\DoNotIndex{\@@par,\@M,\@auxout,\@bsphack,\@esphack,\@depth,\@ehc}
+\DoNotIndex{\@for,\@flushglue,\@gobble,\@gobbletwo,\@height,\@idxitem}
+\DoNotIndex{\@ifnextchar,\@ifstar,\@ifundefined,\@input,\@latexerr}
+\DoNotIndex{\@makeschapterhead,\@namedef,\@nameuse,\@nil}
+\DoNotIndex{\@nobreakfalse,\@restonecolfalse,\@restonecoltrue}
+\DoNotIndex{\@tempa,\@tempb,\@tempc,\@tempf,\@temptokena,\@themark,\@width}
+\DoNotIndex{\active,\aindex,\baselineskip,\begin,\begingroup,\box}
+\DoNotIndex{\c@page,\catcode,\chapter,\char,\chardef,\closeout}
+\DoNotIndex{\CodelineIndex,\sp,\sb,\label,\leavevmode,\mark}
+\DoNotIndex{\mark,\newinsert,\newwrite,\newtoks,\xdef}
+\DoNotIndex{\columnsep,\columnseprule,\columnwidth,\csname,\def}
+\DoNotIndex{\dimen,\do,\DocInput,\documentstyle,\edef,\em}
+\DoNotIndex{\EnableCrossrefs,\end,\endcsname,\endgroup,\endinput}
+\DoNotIndex{\everypar,\expandafter,\filedate,\fileversion}
+\DoNotIndex{\footnotesize,\gdef,\global,\glossary,\hangindent}
+\DoNotIndex{\if@filesw,\else,\fi}
+\DoNotIndex{\if@nobreak,\if@twocolumn,\if@twoside,\fi,\fi,\fi}
+\DoNotIndex{\hsize,\hskip}
+\DoNotIndex{\ifhmode,\ifmmode,\ifodd,\ifvmode,\ifx,\fi,\fi,\fi,\fi,\fi}
+\DoNotIndex{\ifcase,\ifdim,\ifeof,\iffalse,\iftrue,\fi,\fi,\fi,\fi,\fi}
+\DoNotIndex{\ifcat,\fi}
+\DoNotIndex{\immediate,\insert,\item,\jobname,\long}
+\DoNotIndex{\let,\lineskip,\marginparsep,\marginparwidth,\maxdimen}
+\DoNotIndex{\makeatletter,\noexpand,\openout,\protect,\rlap}
+\DoNotIndex{\min,\newpage,\nobreak,\normalbaselineskip}
+\DoNotIndex{\normallineskip,\p@,\par,\parfillskip,\parindent,\parskip}
+\DoNotIndex{\penalty,\relax,\section,\sin,\sloppy,\space,\string}
+\DoNotIndex{\tableofcontents,\the,\thepage,\thispagestyle,\toks,\tt}
+\DoNotIndex{\twocolumn,\uppercase,\vbox,\vrule,\vskip,\vss}
+\DoNotIndex{\write,\z@,\z@skip}
+\endinput
+%%
+%% End of file `amsdtx.cls'.
diff --git a/Master/texmf-dist/source/latex/amsmath/amsgen.dtx b/Master/texmf-dist/source/latex/amsmath/amsgen.dtx
new file mode 100644
index 00000000000..dddaadd9af8
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amsgen.dtx
@@ -0,0 +1,396 @@
+%%% ====================================================================
+%%%  @LaTeX-file{
+%%%     filename  = "amsgen.dtx",
+%%%     version   = "2.0",
+%%%     date      = "1999/11/30",
+%%%     time      = "12:33:33 EST",
+%%%     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",
+%%%     checksum  = "07569 396 1613 14368",
+%%%     email     = "tech-support@ams.org (Internet)",
+%%%     codetable = "ISO/ASCII",
+%%%     keywords  = "latex, amslatex, ams-latex",
+%%%     supported = "yes",
+%%%     abstract  = "This is part of the AMS-\LaTeX{} distribution.
+%%%                  It contains some general internal macros shared
+%%%                  by several different files in AMS-\LaTeX{}.",
+%%%     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.",
+%%%  }
+%%% ====================================================================
+% \iffalse
+%<*driver>
+\documentclass{amsdtx}
+\begin{document}
+\title{The \pkg{amsgen} package}
+\author{American Mathematical Society\\Michael Downes}
+\date{Version \fileversion, \filedate}
+\hDocInput{amsgen.dtx}
+\end{document}
+%</driver>
+% \fi
+%
+% \maketitle
+%
+% \MakeShortVerb\|
+%
+% \section{Introduction}
+%    This is an internal package for storing common functions
+%    that are shared by more than one package in the \amslatex/
+%    distribution. Some of these might eventually make it into the \latex/
+%    kernel.
+%
+% \StopEventually{}
+%
+%    Standard package info.
+%    Using \cs{ProvidesFile} rather than \cs{ProvidesPackage} because
+%    the latter, when input by, e.g, \cls{amsbook}, results in
+%    \texttt{LaTeX warning: You have requested document class `amsbook',
+%    but the document class provides `amsgen'.}
+%    \begin{macrocode}
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must December 1994 or later
+\ProvidesFile{amsgen.sty}[1999/11/30 v2.0]
+%    \end{macrocode}
+%
+% \section{Implementation}
+%    Some general macros shared by \fn{amsart.dtx}, \fn{amsmath.dtx},
+%    \fn{amsfonts.dtx}, \dots
+%
+% \begin{macro}{\@saveprimitive}
+%    The \pkg{amsmath} package redefines a number of \tex/ primitives.
+%    In case some preceding package also decided to redefine one of
+%    those same primitives, we had better do some checking to make
+%    sure that we are able to save the primitive meaning for internal
+%    use. This is handled by the \cs{@saveprimitive} function. We
+%    follow the example of \cs{@@input} where the primitive meaning is
+%    stored in an internal control sequence with a |@@| prefix.
+%    Primitive control sequences can be distinguished by the fact that
+%    \cs{string} and \cs{meaning} return the same information.
+%    \begin{macrocode}
+\providecommand{\@saveprimitive}[2]{\begingroup\escapechar`\\\relax
+  \edef\@tempa{\string#1}\edef\@tempb{\meaning#1}%
+  \ifx\@tempa\@tempb \global\let#2#1%
+  \else
+%    \end{macrocode}
+%    Check to see if \arg{2} was already given the desired primitive
+%    meaning somewhere else.
+%    \begin{macrocode}
+    \edef\@tempb{\meaning#2}%
+    \ifx\@tempa\@tempb
+    \else
+      \@latex@error{Unable to properly define \string#2; primitive
+      \noexpand#1no longer primitive}\@eha
+    \fi
+  \fi
+  \endgroup}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\@xp}
+% \begin{macro}{\@nx}
+%    Shorthands for long command names.
+%    \begin{macrocode}
+\let\@xp=\expandafter
+\let\@nx=\noexpand
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%
+%  \begin{macro}{\@emptytoks}
+%    A token register companion for \cs{@empty}. Saves a little main mem and
+%    probably makes initializations such as |\toks@{}| run faster too.
+%    \begin{macrocode}
+\newtoks\@emptytoks
+%    \end{macrocode}
+%  \end{macro}
+%
+% \begin{macro}{\@oparg}
+%    Use of \cs{@oparg} simplifies some constructions where a macro
+%    takes an optional argument in square brackets. We can't use
+%    \cn{newcommand} here because this function might be previously
+%    defined by the \pkg{amsmath} package in a loading sequence such as
+% \begin{verbatim}
+% \usepackage{amsmath,amsthm}
+% \end{verbatim}
+%    \begin{macrocode}
+\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\@ifempty}
+% \begin{macro}{\@ifnotempty}
+%    |\@ifnotempty| and |\@ifempty| use category 11 |@| characters to
+%    test whether the argument is empty or not, since these are highly
+%    unlikely to occur in the argument. As with \cn{@oparg}, there is a
+%    possibility that these commands were defined previously in
+%    \fn{amsmath.sty}.
+%    \begin{macrocode}
+\long\def\@ifempty#1{\@xifempty#1@@..\@nil}
+\long\def\@xifempty#1#2@#3#4#5\@nil{%
+  \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
+%    \end{macrocode}
+%    \cs{@ifnotempty} is a shorthand that makes code read better when
+%    no action is needed in the empty case. At a cost of double
+%    argument-reading---so for often-executed code, avoiding
+%    \cs{@ifnotempty} might be wise.
+%    \begin{macrocode}
+\long\def\@ifnotempty#1{\@ifempty{#1}{}}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%    Some abbreviations to conserve token mem.
+%    \begin{macrocode}
+\def\FN@{\futurelet\@let@token}
+\def\DN@{\def\next@}
+\def\RIfM@{\relax\ifmmode}
+\def\setboxz@h{\setbox\z@\hbox}
+\def\wdz@{\wd\z@}
+\def\boxz@{\box\z@}
+\def\relaxnext@{\let\@let@token\relax}
+%    \end{macrocode}
+%
+% \begin{macro}{\new@ifnextchar}
+%    This macro is a new version of \latex/'s \verb+\@ifnextchar+,
+%    macro that does not skip over spaces.
+%    \begin{macrocode}
+\long\def\new@ifnextchar#1#2#3{%
+%    \end{macrocode}
+%    By including the space after the equals sign, we make it possible
+%    for \cs{new@ifnextchar} to do look-ahead for any token, including a
+%    space!
+%    \begin{macrocode}
+  \let\reserved@d= #1%
+  \def\reserved@a{#2}\def\reserved@b{#3}%
+  \futurelet\@let@token\new@ifnch
+}
+%
+\def\new@ifnch{%
+  \ifx\@let@token\reserved@d \let\reserved@b\reserved@a \fi
+  \reserved@b
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\@ifstar}
+%    There will essentially never be a space before the \qc{\*}, so
+%    using \cs{@ifnextchar} is unnecessarily slow.
+%    \begin{macrocode}
+\def\@ifstar#1#2{\new@ifnextchar *{\def\reserved@a*{#1}\reserved@a}{#2}}
+%    \end{macrocode}
+% \end{macro}
+%
+%    The hook \cs{every@size} was changed to \cs{every@math@size} in the
+%    December 1994 release of \latex/ and its calling procedures changed.
+%    If \cs{every@math@size} is undefined it means the user has an older
+%    version of \latex/ so we had better define it and patch a couple of
+%    functions (\cs{glb@settings} and \cs{set@fontsize}).
+%    \begin{macrocode}
+\@ifundefined{every@math@size}{%
+%    \end{macrocode}
+%    Reuse the same token register; since it was never used except for
+%    the purposes that are affected below, this is OK.
+%    \begin{macrocode}
+\let\every@math@size=\every@size
+\def\glb@settings{%
+     \expandafter\ifx\csname S@\f@size\endcsname\relax
+       \calculate@math@sizes
+     \fi
+     \csname S@\f@size\endcsname
+      \ifmath@fonts
+%       \ifnum \tracingfonts>\tw@
+%          \@font@info{Setting up math fonts for
+%                  \f@size/\f@baselineskip}\fi
+        \begingroup
+          \escapechar\m@ne
+          \csname mv@\math@version \endcsname
+          \globaldefs\@ne
+          \let \glb@currsize \f@size
+          \math@fonts
+        \endgroup
+        \the\every@math@size
+      \else
+%       \ifnum \tracingfonts>\tw@
+%        \@font@info{No math setup for \f@size/\f@baselineskip}%
+%       \fi
+      \fi
+}
+%    \end{macrocode}
+%    Remove |\the\every@size| from \cs{size@update}.
+%    \begin{macrocode}
+\def\set@fontsize#1#2#3{%
+    \@defaultunits\@tempdimb#2pt\relax\@nnil
+    \edef\f@size{\strip@pt\@tempdimb}%
+    \@defaultunits\@tempskipa#3pt\relax\@nnil
+    \edef\f@baselineskip{\the\@tempskipa}%
+    \edef\f@linespread{#1}%
+    \let\baselinestretch\f@linespread
+      \def\size@update{%
+        \baselineskip\f@baselineskip\relax
+        \baselineskip\f@linespread\baselineskip
+        \normalbaselineskip\baselineskip
+        \setbox\strutbox\hbox{%
+          \vrule\@height.7\baselineskip
+                \@depth.3\baselineskip
+                \@width\z@}%
+%%%     \the\every@size
+        \let\size@update\relax}%
+  }
+}{}% end \@ifundefined test
+%    \end{macrocode}
+%
+% \begin{macro}{\ex@}
+%    The \cs{ex@} variable provides a small unit of space for use in
+%    math-mode constructions, that varies according to the current type
+%    size. For example, the \cn{pmb} command uses \cs{ex@} units.
+%    Since a macro or mu unit solution for the \meta{dimen} \cs{ex@} won't
+%    work without changing a lot of current code in the \pkg{amsmath}
+%    package, we set \cs{ex@} through the \cs{every@math@size} hook.
+%    The value of \cs{ex@} is scaled nonlinearly in a range of roughly
+%    0.5pt to 1.5pt, by the function \cs{compute@ex@}.
+%    \begin{macrocode}
+\newdimen\ex@
+\addto@hook\every@math@size{\compute@ex@}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \cs{compute@ex@} computes \cs{ex@} as a nonlinear scaling from 10pt
+%    to current font size (\cs{f@size}). Using .97 as the multiplier makes 1
+%    |ex@| $\approx$ .9pt when the current type size is 8pt and 1 |ex@|
+%    $\approx$ 1.1pt when the current type size is 12pt.
+%
+%    The formula is essentially
+% \begin{displaymath}
+% \newcommand{\points}{\mbox{pt}}
+% \newcommand{\floor}[1]{\lfloor#1\rfloor}
+%      1\points \pm (1\points - (.97)^{\floor{\vert 10 - n\vert}})
+% \end{displaymath}
+%    where $n$ = current type size, but adjusted to differentiate
+%    half-point sizes as well as whole point sizes, and there is a
+%    cutoff for extraordinarily large values of \cs{f@size} ($>$ 20pt)
+%    so that the value of \cs{ex@} never exceeds 1.5pt.
+%
+%    \begin{macrocode}
+\def\compute@ex@{%
+  \begingroup
+  \dimen@-\f@size\p@
+  \ifdim\dimen@<-20\p@
+%    \end{macrocode}
+%    Never make \cs{ex@} larger than 1.5pt.
+%    \begin{macrocode}
+    \global\ex@ 1.5\p@
+  \else
+%    \end{macrocode}
+%    Adjust by the reference size and multiply by 2 to allow for
+%    half-point sizes.
+%    \begin{macrocode}
+    \advance\dimen@10\p@ \multiply\dimen@\tw@
+%    \end{macrocode}
+%    Save information about the current sign of \cs{dimen@}.
+%    \begin{macrocode}
+    \edef\@tempa{\ifdim\dimen@>\z@ -\fi}%
+%    \end{macrocode}
+%    Get the absolute value of \cs{dimen@}.
+%    \begin{macrocode}
+    \dimen@ \ifdim\dimen@<\z@ -\fi \dimen@
+    \advance\dimen@-\@m sp % fudge factor
+%    \end{macrocode}
+%    Here we use \cs{vfuzz} merely as a convenient scratch register
+%    \begin{macrocode}
+    \vfuzz\p@
+%    \end{macrocode}
+%    Multiply in a loop.
+%    \begin{macrocode}
+    \def\do{\ifdim\dimen@>\z@
+      \vfuzz=.97\vfuzz
+      \advance\dimen@ -\p@
+%\message{\vfuzz: \the\vfuzz, \dimen@: \the\dimen@}%
+      \@xp\do \fi}%
+    \do
+    \dimen@\p@ \advance\dimen@-\vfuzz
+    \global\ex@\p@
+    \global\advance\ex@ \@tempa\dimen@
+  \fi
+  \endgroup
+%\typeout{\string\f@size: \f@size}\showthe\ex@
+}
+%    \end{macrocode}
+%    Tests of the \cs{compute@ex@} function yield the following results:
+%
+% \begin{center}\begin{tabular}{rlrl}
+%    \cs{f@size}& \cs{ex@}&   \cs{f@size}& \cs{ex@}\\
+%    10&          1.0pt&      9&           0.94089pt\\
+%    11&          1.05911pt&  8.7&         0.91266pt\\
+%    12&          1.11473pt&  8.5&         0.91266pt\\
+%    14.4&        1.23982pt&  8.4&         0.88527pt\\
+%    17.28&       1.36684pt&  8&           0.88527pt\\
+%    20.74&       1.5pt&      7&           0.83293pt\\
+%    19.5&        1.4395pt&   6&           0.78369pt\\
+%                       &&    5&           0.73737pt\\
+%                       &&    1&           0.57785pt
+% \end{tabular}\end{center}
+%
+%  \begin{macro}{\@addpunct}
+%    Use of the \cs{@addpunct} function allows ending punctuation in
+%    section headings and elsewhere to be intelligently omitted
+%    when punctuation is already present.
+%    \begin{macrocode}
+\def\@addpunct#1{\ifnum\spacefactor>\@m \else#1\fi}
+%    \end{macrocode}
+%  \end{macro}
+%
+%  \begin{macro}{\frenchspacing}
+%    Change \cn{frenchspacing} to ensure that \cs{@addpunct} will
+%    continue to work properly even when `french' spacing is in effect.
+%    \begin{macrocode}
+\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
+  \sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
+%    \end{macrocode}
+%  \end{macro}
+%
+% \subsection{Miscellaneous}
+%    \begin{macrocode}
+\def\nomath@env{\@amsmath@err{%
+  \string\begin{\@currenvir} allowed only in paragraph mode%
+}\@ehb% "You've lost some text"
+}
+%    \end{macrocode}
+%     A trade-off between main memory space and hash size; using
+%     \verb+\Invalid@@+ saves 14 bytes of main memory for each use of
+%     \verb+\Invalid@+, at the cost of one control sequence name.
+%     \verb+\Invalid@+ is currently used about five times and
+%     \verb+\Invalid@@+ is used by itself in some other instances,
+%     which means that it saves us more memory than \verb+\FN@+,
+%     \verb+\RIfM@+, and some of the other abbreviations above.
+%    \begin{macrocode}
+\def\Invalid@@{Invalid use of \string}
+%    \end{macrocode}
+%
+%    The usual \cs{endinput} to ensure that random garbage at the end of
+%    the file doesn't get copied by \fn{docstrip}.
+%    \begin{macrocode}
+\endinput
+%    \end{macrocode}
+%
+% \CheckSum{337}
+% \Finale
diff --git a/Master/texmf-dist/source/latex/amsmath/amsldoc.cls b/Master/texmf-dist/source/latex/amsmath/amsldoc.cls
new file mode 100644
index 00000000000..0e8d142cbf9
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amsldoc.cls
@@ -0,0 +1,451 @@
+%%
+%% This is file `amsldoc.cls',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amsdtx.dtx  (with options: `amsldoc')
+%% 
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsdtx.dtx",
+%%%   version   = "2.06",
+%%%   date      = "2004/08/06",
+%%%   time      = "12:18:49 EDT",
+%%%   checksum  = "19929 1192 3506 39423",
+%%%   author    = "Michael J. Downes, updated by David M. Jones",
+%%%   copyright = "Copyright 1996, 1999, 2004
+%%%                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,
+%%%                Publications Technical Group,
+%%%                201 Charles Street,
+%%%                Providence, RI 02904,
+%%%                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, amslatex, ams-latex, user documentation",
+%%%   supported = "yes",
+%%%   abstract  = "This is the source for two document classes, amsdtx
+%%%                and amsldoc, used to produce AMS user documentation
+%%%                or technical documentation.",
+%%%   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)
+[1995/06/01]% LaTeX date must be June 1995 or later
+\ProvidesClass{amsldoc}[2004/08/06 v2.06]
+\DeclareOption*{\PassOptionsToClass{\CurrentOption}{book}}
+\ProcessOptions
+\LoadClass{book}
+\IfFileExists{url.sty}{%
+    \RequirePackage{url}\relax
+    \@gobble
+}{%
+    \@firstofone
+}
+{
+    \DeclareRobustCommand{\url}[1]{%
+        \def\@tempa{#1}%
+        \texttt{\urlsetup $\expandafter\strip@prefix\meaning\@tempa$}%
+    }%
+    \def\urlsetup{%
+        \check@mathfonts \textfont\@ne\the\font \textfont\z@\the\font
+        \urlfix +\urlfix\=\urlfix\:\urlfix\-\urlfix\.\urlfix\,\urlfix\;%
+        \urlbreak\&\urlbreak\/\urlbreak\?%
+    }%
+    \def\urlbreak#1{%
+        \mathcode`#1="8000
+        \begingroup \lccode`\~=`#1 \lowercase{\endgroup \edef~}%
+        {\mathchar\number`#1\penalty\hyphenpenalty}%
+    }%
+    \def\urlfix#1{%
+        \mathcode`#1=`#1\relax
+    }%
+}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\providecommand{\qq}[1]{\textquotedblleft#1\/\textquotedblright}
+\providecommand{\mdash}{\textemdash\penalty\exhyphenpenalty}
+\providecommand{\ndash}{\textendash\penalty\exhyphenpenalty}
+\let\@xp\expandafter
+\def\actualchar{@}
+\def\quotechar{"}
+\def\levelchar{!}
+\def\encapchar{|}
+\def\verbatimchar{+}
+\newif\ifcodeline@index
+\newcommand*{\autoindex}[1]{%
+    \index{#1\ifcodeline@index\encapchar usage\fi}%
+}
+\newcommand{\ntt}{\normalfont\ttfamily}
+\newcommand*{\indexcs}[1]{%
+    \@xp\@xp\@xp\@indexcs\@xp\@nobslash\string#1\@nil
+}
+\def\@indexcs#1\@nil{%
+    \autoindex{#1\actualchar
+        \string\verb\quotechar*\verbatimchar
+            \@xp\@bothoftwo\string\ #1\@empty
+        \verbatimchar
+    }%
+}
+\def\@bothoftwo#1#2{#1#2}
+\def\category@index#1#2{%
+    {\ntt#2}%
+    \@category@index{#1}{#2}%
+}
+\def\@category@index#1#2{%
+    \autoindex{%
+        #2\actualchar\string\texttt{#2}%
+        \ifx\@nil#1\@nil\else\space#1\fi
+    }%
+}
+\chardef\bslchar=`\\ % p. 424, TeXbook
+\newcommand{\addbslash}{\expandafter\@addbslash\string}
+\def\@addbslash#1{\bslchar\@nobslash#1}
+\newcommand{\nobslash}{\expandafter\@nobslash\string}
+\def\@nobslash#1{\ifnum`#1=\bslchar\else#1\fi}
+\def\@boxorbreak{%
+    \leavevmode
+    \ifmmode\hbox\else\ifdim\lastskip=\z@\penalty9999 \fi\fi
+}
+\DeclareRobustCommand{\cs}[1]{%
+    \@boxorbreak{%
+        \ntt
+        \addbslash#1\@empty
+        \@xp\@xp\@xp\@indexcs\@xp\@nobslash\string#1\@nil
+    }%
+}
+\let\cn\cs
+\DeclareRobustCommand{\cls}{\category@index{class}}
+\DeclareRobustCommand{\pkg}{\category@index{package}}
+\DeclareRobustCommand{\opt}{\category@index{option}}
+\DeclareRobustCommand{\env}{\category@index{environment}}
+\DeclareRobustCommand{\fn}{\category@index{}}
+\DeclareRobustCommand{\bst}{\category@index{\string\BibTeX{} style}}
+\DeclareRobustCommand{\cnt}{\category@index{counter}}
+\DeclareRobustCommand{\fnt}{\category@index{font}}
+\def\allowtthyphens{\begingroup
+  \fontencoding{OT1}\fontfamily{cmtt}%
+  \expandafter\let\csname OT1+cmtt\endcsname\relax
+  \try@load@fontshape
+  \endgroup
+  \expandafter\let\csname OT1+cmtt\endcsname\@empty
+}
+\hfuzz2pc
+\vbadness9999 \hbadness5000
+\def\AmS{{\protect\usefont{OMS}{cmsy}{m}{n}%
+  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}
+\def\latex/{{\protect\LaTeX}}
+\def\amslatex/{{\protect\AmS-\protect\LaTeX}}
+\def\tex/{{\protect\TeX}}
+\def\amstex/{{\protect\AmS-\protect\TeX}}
+\def\bibtex/{{Bib\protect\TeX}}
+\def\makeindx/{MakeIndex}
+\def\xypic/{XY\mbox{-}pic}
+\newcommand{\Textures}{\textit{Textures}}
+\def\<#1>{\textit{$\langle$#1\/$\rangle$}}
+\def\@listi{%
+  \leftmargin\leftmargini
+  \topsep 3\p@ \@plus2\p@ \@minus\p@
+  \parsep \p@ \@plus\p@ \itemsep\parsep
+}
+\let\@listI\@listi
+\@listi
+\def\@listii{%
+  \leftmargin\leftmarginii
+  \labelwidth\leftmarginii \advance\labelwidth-\labelsep
+  \topsep\p@\@plus\p@ \@minus\p@
+  \parsep\z@skip \itemsep\z@skip
+}
+\def\@listiii{%
+  \leftmargin\leftmarginiii
+  \labelwidth\leftmarginiii \advance\labelwidth-\labelsep
+  \topsep\z@skip \parsep\z@skip \itemsep\z@skip
+}
+\newenvironment{histnote}{%
+  \trivlist\item[\hspace{\labelsep}\bfseries Historical Note:]%
+}{%
+  \endtrivlist
+}
+\def\@starttoc#1{\begingroup
+  \let\autoindex\@gobble
+  \makeatletter
+  \@input{\jobname.#1}\if@filesw
+  \expandafter\newwrite\csname tf@#1\endcsname
+             \immediate\openout
+                 \csname tf@#1\endcsname \jobname.#1\relax
+  \fi \global\@nobreakfalse \endgroup}
+\newcommand{\gloss}[1]{}
+\newcommand*{\secref}[1]{\S\ref{#1}}
+\newcommand{\qc}[1]{}% check for prior definition
+\edef\qc#1{\noexpand\protect\expandafter\noexpand\csname qc \endcsname
+  \noexpand\protect#1}
+\@namedef{qc }#1#2{\begingroup\ntt
+  \ifx\ #2\char`\ \else\escapechar\m@ne\string#2\fi\endgroup}
+\DeclareRobustCommand{\qcat}{\qc\@}%
+\DeclareRobustCommand{\qcamp}{\qc\&}%
+\DeclareRobustCommand{\qcbang}{\qc\!}%
+\DeclareRobustCommand{\arg}[1]{{\ntt\##1}}
+\newcommand{\openbox}{\leavevmode
+  \hbox to.77778em{%
+  \hfil\vrule
+  \vbox to.675em{\hrule width.6em\vfil\hrule}%
+  \vrule\hfil}}
+\newcommand{\qedsymbol}{\openbox}
+\def\mail{\texttt}
+\def\*#1{\def\@tempa{#1}\def\@tempb{*}%
+  \ifx\@tempa\@tempb \expandafter\index
+  \else #1\index{#1}\fi}
+\def\ncn#1{{\let\index\@gobble\cn{#1}}}
+
+\DeclareRobustCommand{\cnbang}{%
+  \ncn{\!}\index{"!@{\ntt\bslchar\qcbang}}}
+\DeclareRobustCommand{\cnat}{%
+  \ncn{\!}\index{"@@{\ntt\bslchar\qcat}}}
+\def\5{\penalty500 }
+\newenvironment{error}{%
+  \begingroup\catcode`\\=12 \expandafter\endgroup\errora
+}{%
+  \endtrivlist
+}
+\newcommand{\errora}[1]{%
+  \trivlist
+  \item[\hskip\labelsep\errorbullet\enspace
+    \ntt\frenchspacing\def\@tempa{#1}%
+    \expandafter\strip@prefix\meaning\@tempa]\leavevmode\par
+}
+\newcommand{\errorbullet}{\rule[-.5pt]{2.5pt}{7.5pt}%
+  \rule[-.5pt]{5pt}{2.5pt}\kern-2.5pt%
+  \rule[4.5pt]{2.5pt}{2.5pt}}
+\newcommand{\errexa}{\par\noindent\textit{Example}:\ }
+\newcommand{\errexpl}{\par\noindent\textit{Explanation}:\ }
+\renewcommand\frontmatter{\clearpage
+            \@mainmatterfalse\pagenumbering{roman}}
+\renewcommand\mainmatter{\clearpage
+       \@mainmattertrue\pagenumbering{arabic}}
+\renewcommand\backmatter{\clearpage \@mainmatterfalse}
+\renewcommand\part{\clearpage
+                 \thispagestyle{plain}%
+                 \if@twocolumn
+                     \onecolumn
+                     \@tempswatrue
+                   \else
+                     \@tempswafalse
+                 \fi
+                 \hbox{}\vfil
+                 \secdef\@part\@spart}
+\def\@part[#1]#2{%
+    \ifnum \c@secnumdepth >-2\relax
+      \refstepcounter{part}%
+      \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
+    \else
+      \addcontentsline{toc}{part}{#1}%
+    \fi
+    \markboth{}{}%
+    {\centering
+     \interlinepenalty \@M
+     \reset@font
+     \ifnum \c@secnumdepth >-2\relax
+       \Large\bfseries \partname~\thepart
+       \par
+       \vskip 20\p@
+     \fi
+     \Large \bfseries #2\par}%
+    \@endpart}
+\def\@spart#1{%
+    {\centering
+     \interlinepenalty \@M
+     \reset@font
+     \Large \bfseries #1\par}%
+    \@endpart}
+\def\@endpart{\vfil\newpage
+              \if@twoside
+                \hbox{}%
+                \thispagestyle{empty}%
+                \newpage
+              \fi
+              \if@tempswa
+                \twocolumn
+              \fi}
+\renewcommand\chapter{\par \@afterindentfalse
+                    \secdef\@chapter\@schapter}
+\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
+                       \if@mainmatter
+                         \refstepcounter{chapter}%
+                         \typeout{\@chapapp\space\thechapter.}%
+                         \addcontentsline{toc}{chapter}%
+                                   {\protect\numberline{\thechapter}#1}%
+                       \else
+                         \addcontentsline{toc}{chapter}{#1}\fi
+                    \else
+                      \addcontentsline{toc}{chapter}{#1}
+                    \fi
+                    \chaptermark{#1}%
+                    \addtocontents{lof}{\protect\addvspace{10\p@}}%
+                    \addtocontents{lot}{\protect\addvspace{10\p@}}%
+                    \if@twocolumn
+                      \@topnewpage[\@makechapterhead{#2}]%
+                    \else
+                      \@makechapterhead{#2}%
+                      \@afterheading
+                    \fi}
+\def\@makechapterhead#1{%
+  \vspace{1.5\baselineskip}%
+  {\parindent \z@ \raggedright \reset@font
+    \ifnum \c@secnumdepth >\m@ne
+      \large\bfseries \@chapapp\space\thechapter
+      \par\nobreak
+      \vskip.5\baselineskip\relax
+    \fi
+    #1\par\nobreak
+    \vskip\baselineskip
+  }}
+\def\@schapter#1{\if@twocolumn
+                   \@topnewpage[\@makeschapterhead{#1}]%
+                 \else
+                   \@makeschapterhead{#1}%
+                   \@afterheading
+                 \fi}
+\def\@makeschapterhead#1{%
+  \vspace*{1.5\baselineskip}%
+  {\parindent \z@ \raggedright
+    \reset@font
+    \large \bfseries  #1\par\nobreak
+    \vskip\baselineskip
+  }}
+\def\@removefromreset#1#2{\let\@tempb\@elt
+   \expandafter\let\expandafter\@tempa\csname c@#1\endcsname
+   \def\@elt##1{\expandafter\ifx\csname c@##1\endcsname\@tempa\else
+         \noexpand\@elt{##1}\fi}%
+   \expandafter\edef\csname cl@#2\endcsname{\csname cl@#2\endcsname}%
+   \let\@elt\@tempb}
+\@removefromreset{footnote}{chapter}
+\def\ps@headings{%
+  \let\@oddfoot\@empty\let\@evenfoot\@empty
+  \def\@evenhead{\thepage\hfil{\footnotesize\leftmark{}{}}}%
+  \def\@oddhead{{\footnotesize\rightmark{}{}}\hfil\thepage}%
+  \let\@mkboth\markboth
+  \def\chaptermark##1{%
+    \markboth {\uppercase{\ifnum \c@secnumdepth >\m@ne
+      \if@mainmatter
+        \@chapapp\ \thechapter. \ \fi
+      \fi
+        ##1}}{}}%
+  \def\sectionmark##1{%
+    \markright {\uppercase{\ifnum \c@secnumdepth >\z@
+        \thesection. \ \fi
+        ##1}}}}
+\renewcommand\section{\@startsection {section}{1}{\z@}%
+                                   {-.6\baselineskip \@plus -3\p@}%
+                                   {.4\baselineskip}
+                         {\reset@font\normalsize\bfseries}}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+                                     {-.3\baselineskip\@plus -2\p@}%
+                                     {.2\baselineskip}%
+                         {\reset@font\normalsize\bfseries}}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+                                     {-.2\baselineskip\@plus -2\p@}%
+                                     {.2\baselineskip}%
+                                     {\reset@font\normalsize\bfseries}}
+\renewenvironment{theindex}{%
+  \if@twocolumn \@restonecolfalse \else \@restonecoltrue \fi
+  \columnseprule \z@ \columnsep 35\p@
+  \let\autoindex\@gobble
+  \twocolumn[\@makeschapterhead{\indexname}]%
+  \addcontentsline{toc}{chapter}{\indexname}%
+  \@mkboth{\uppercase{\indexname}}{\uppercase{\indexname}}%
+  \thispagestyle{plain}\parindent\z@
+  \parskip\z@ \@plus .3\p@\relax
+  \let\item\@idxitem
+}{%
+  \if@restonecol\onecolumn\else\clearpage\fi
+}
+\long\def\@makecaption#1#2{%
+  \addvspace\abovecaptionskip
+  \begingroup
+    \countdef\@parcycles=8 % local count register
+    \@parcycles\z@
+    \@setpar{\advance\@parcycles\@ne \ifnum\@parcycles>999
+      \@@par\@parcycles\z@\fi
+      \ifhmode \unskip\hskip\parfillskip\penalty-\@M\fi}%
+    \@hangfrom{\textbf{#1.} }\vadjust{\penalty\m@ne}#2%
+  \endgroup
+  \ifhmode\unpenalty\fi\par
+  \ifnum\lastpenalty=\m@ne % only one line in the caption
+    \unpenalty \setbox\@tempboxa\lastbox
+    \nointerlineskip
+    \hbox to\hsize{\hfill\unhbox\@tempboxa\unskip\hfill}%
+  \fi
+  \nobreak\vskip\belowcaptionskip
+}
+\renewenvironment{table}{%
+  \belowcaptionskip\abovecaptionskip \abovecaptionskip\z@skip
+  \@float{table}%
+}{%
+  \end@float
+}
+\renewenvironment{table*}{%
+  \belowcaptionskip\abovecaptionskip \abovecaptionskip\z@skip
+  \@dblfloat{table}%
+}{%
+  \end@dblfloat
+}
+\setcounter{topnumber}{4}\setcounter{bottomnumber}{4}
+\setcounter{totalnumber}{4}\setcounter{dbltopnumber}{4}
+\renewcommand{\topfraction}{.97}\renewcommand{\bottomfraction}{.97}
+\renewcommand{\textfraction}{.03}\renewcommand{\floatpagefraction}{.9}
+\renewcommand{\dbltopfraction}{.97}
+\renewcommand{\dblfloatpagefraction}{.9}
+\setlength{\floatsep}{8pt plus6pt}
+\setlength{\textfloatsep}{10pt plus8pt}
+\setlength{\intextsep}{8pt plus6pt}
+\setlength{\dblfloatsep}{8pt plus6pt}
+\setlength{\dbltextfloatsep}{10pt plus8pt}
+\setlength{\@fptop}{0pt}\setlength{\@fpsep}{8pt}%
+\setlength{\@fpbot}{0pt plus 1fil}
+\setlength{\@dblfptop}{0pt}\setlength{\@dblfpsep}{8pt}%
+\setlength{\@dblfpbot}{0pt plus 1fil}
+
+\pagestyle{headings}
+\AtBeginDocument{\catcode`\|=\active }
+\def\activevert{\verb|}
+\expandafter\gdef\expandafter\dospecials\expandafter
+  {\dospecials \do\|}%
+\expandafter\gdef\expandafter\@sanitize\expandafter
+  {\@sanitize \@makeother\|}
+\begingroup\catcode`\|=\active \gdef|{\protect\activevert{}}\endgroup
+\newcommand{\arrayargpatch}{%
+  \let\@oldarray\@array
+  \edef\@array[##1]##2{\catcode\number`\|=\number\catcode`\|
+    \catcode\number`\@=\number\catcode`\@ \relax
+    \let\noexpand\@array\noexpand\@oldarray
+    \noexpand\@array[##1]{##2}}%
+  \catcode`\|=12 \catcode`\@=12 \relax
+}
+\newenvironment{ctab}{%
+  \par\topsep\medskipamount
+  \trivlist\centering
+  \item[]%
+  \arrayargpatch
+  \begin{tabular}%
+}{%
+  \end{tabular}%
+  \endtrivlist
+}
+\endinput
+%%
+%% End of file `amsldoc.cls'.
diff --git a/Master/texmf-dist/source/latex/amsmath/amsldoc.tex b/Master/texmf-dist/source/latex/amsmath/amsldoc.tex
new file mode 100644
index 00000000000..abb532cccb8
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amsldoc.tex
@@ -0,0 +1,2620 @@
+%%% ====================================================================
+%%%  @LaTeX-file{
+%%%     filename        = "amsldoc.tex",
+%%%     version         = "2.09",
+%%%     date            = "2004/04/06",
+%%%     time            = "15:13:33 EDT",
+%%%     checksum        = "39446 2620 11111 93831",
+%%%     author          = "American Mathematical Society",
+%%%     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)",
+%%%     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.",
+%%%     supported       = "yes",
+%%%     keywords        = "latex, amsmath",
+%%%     abstract        = "This file is the source file for the user's
+%%%                        guide of the \LaTeX{} package amsmath .",
+%%%     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[leqno,titlepage,openany]{amsldoc}[1999/12/13]
+
+\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
+
+\title{User's Guide for the \pkg{amsmath} Package (Version~2.0)}
+\author{American Mathematical Society}
+\date{1999-12-13 (revised 2002-02-25)}
+
+%    Use the amsmath package and amscd package in order to print
+%    examples.
+\usepackage{amsmath}
+\usepackage{amscd}
+
+\makeindex % 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
+}
+
+%    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}
+
+\DeclareMathOperator{\ix}{ix}
+\DeclareMathOperator{\nul}{nul}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\xxx}{xxx}
+
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\frontmatter
+
+\maketitle
+\pagestyle{headings}
+\tableofcontents
+\cleardoublepage % for better page number placement
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mainmatter
+\chapter{Introduction}
+
+The \pkg{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 \pkg{amsmath} package is normally
+provided along with it. Upgrading when a newer version of the
+\pkg{amsmath} package is released can be done via
+\url{http://www.ams.org/tex/amsmath.html} or
+\url{ftp://ftp.ams.org/pub/tex/}.
+
+This documentation describes the features of the \pkg{amsmath} package
+and discusses how they are intended to be used. It also covers some
+ancillary packages:
+\begin{ctab}{ll}
+\pkg{amsbsy}& \pkg{amstext}\\
+\pkg{amscd}& \pkg{amsxtra}\\
+\pkg{amsopn}
+\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{http://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{instr-l} and
+\url{http://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 \pkg{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 \pkg{amsmath} package is distributed together with some small
+auxiliary packages:
+\begin{description}
+\item[\pkg{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 \pkg{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.
+
+\chapter{Options for the \pkg{amsmath} package}\label{options}
+
+The \pkg{amsmath} package has the following options:
+\begin{description}
+
+\item[\opt{centertags}] (default) For a split equation, place equation
+numbers\index{equation numbers!vertical placement} vertically centered
+on the total height of the equation.
+
+\item[\opt{tbtags}] `Top-or-bottom tags': For a split equation, place
+equation numbers\index{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}
+
+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}".
+
+The \pkg{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{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}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{Displayed equations}
+
+\section{Introduction}
+The \pkg{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*       flalign     flalign*
+  multline     multline*     alignat     alignat*
+  split
+\end{verbatim}
+(Although the standard \env{eqnarray} environment remains available,
+it is better to use \env{align} or \env{equation}+\env{split} instead.)
+
+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 \cn{\\}; you can also
+override\index{equation numbers!overriding} it 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. 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 \pkg{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 \pkg{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.
+
+In the structures that do alignment (\env{split}, \env{align} and
+variants), relation symbols have an \verb'&' before them but not
+after\mdash unlike \env{eqnarray}. Putting the \verb'&' after the
+relation symbol will interfere with the normal spacing; it has to go
+before.
+
+\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}&
+  a_{12}& =b_{12}\\
+a_{21}& =b_{21}&
+  a_{22}& =b_{22}+c_{22}
+\end{flalign*}
+\end{verbatim}
+\producing
+\begin{flalign*}
+a_{11}& =b_{11}&
+  a_{12}& =b_{12}\\
+a_{21}& =b_{21}&
+  a_{22}& =b_{22}+c_{22}
+\end{flalign*}
+\end{eqxample}
+\end{table}
+
+\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}.}
+
+\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).
+
+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}.
+
+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 \pkg{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}: 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}
+
+\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
+\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 \verb'[t]' or \verb'[b]' argument to specify vertical
+positioning.
+
+\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 \pkg{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.
+
+\section{Adjusting tag placement}
+
+Placing equation numbers can be a rather complex problem in multiline
+displays. The environments of the \pkg{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. There is a
+\cn{raisetag} command 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}|. 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>|]| command to get extra vertical
+space between lines in all the \pkg{amsmath} displayed equation
+environments, as is usual in \latex/.  When the \pkg{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 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 multi-line equation, then you might put
+\cn{allowdisplaybreaks}\texttt{[1]} in the preamble of your document. An
+optional argument 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: 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}
+
+\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 \pkg{amsmath} package provides a command\index{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{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).
+
+\subsection{Subordinate numbering sequences}
+
+The \pkg{amsmath} package provides also a \env{subequations}
+environment\index{equation numbers!subordinate numbering} to make it
+easy to number equations in a particular 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}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Miscellaneous mathematical features}
+
+\section{Matrices}\label{ss:matrix}
+
+The \pkg{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 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 must resort to \env{array}.)
+
+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 must be provided; there are no |p|,|b|,|B|,|v|,|V|
+versions of \env{smallmatrix}. 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, 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 \pkg{amsmath} package slightly extends the set of math
+spacing\index{horizontal space!in math mode} 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{\,}&
+  \cnbang& \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}
+
+\section{Nonbreaking dashes}
+
+A command \cn{nobreakdash} is provided to suppress the possibility
+of a linebreak after the following hyphen or dash. For example, if you
+write `pages 1\ndash 9' as |pages 1\nobreakdash--9| then a linebreak 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 linebreak 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 \pkg{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.
+
+\section{Roots}
+
+In ordinary \latex/ the placement of root indices is sometimes not so
+good: $\sqrt[\beta]{k}$ (|\sqrt||[\beta]{k}|).  In the
+\pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
+the position of the root:
+\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 \pkg{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 (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 \pkg{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
+\pkg{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 $\binom{n}{k}$
+\pkg{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 (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 \pkg{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}.
+
+\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 \pkg{amsmath} package \cn{smash} has optional
+arguments |t| and |b|, 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}
+
+\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.
+\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 \pkg{amsmath} package they are.
+
+\subsection{Vertical bar notations}
+
+The \pkg{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
+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
+double-bar command \ncn{\|}\index{"|@\verb"*+"\"|+}. This immediately
+rules out the use of \qc{|}
+and \ncn{\|}\index{"|@\verb"*+"\"|+} 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 $\lVert v\rVert$.
+
+\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
+\pkg{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
+space!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.
+
+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}.
+
+The following operator names are predefined:
+\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}
+\begin{ctab}{rlrl}
+\cn{varlimsup}&  $\displaystyle\varlimsup$&
+  \cn{varinjlim}&  $\displaystyle\varinjlim$\\
+\cn{varliminf}&  $\displaystyle\varliminf$&
+  \cn{varprojlim}& $\displaystyle\varprojlim$
+\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 \pkg{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}
+
+\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 |..._{\text{word or phrase}}|, which is slightly
+easier than the \cn{mbox} equivalent: |..._{\mbox{\scriptsize| |word|
+|or| |phrase}}|. The other advantage is the more descriptive name.
+\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}
+
+\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!multi-line}\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 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}
+
+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
+\pkg{kuvio} or \xypic/, 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{tlc}. 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).
+
+\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}
+
+\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.
+
+The \pkg{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 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 \pkg{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). For the reader's convenience, the set of error
+messages discussed here overlaps somewhat with the set in that chapter,
+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|.
+
+There is also a section discussing some output errors, i.e., instances
+where the printed document has something wrong but there was no \latex/
+error during typesetting.
+
+\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}{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}
+
+\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}{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{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
+\fn{cmex10} for the \pkg{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}
+
+\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}
+
+\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 \pkg{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 linebreak 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}
+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}
+a+b\\
+{[f,g]}\\
+m+n
+\end{array}
+\end{verbatim}
+
+\end{error}
+
+\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 linebreak inside a
+\cn{left}-\cn{right} pair of delimiters in a \env{multline} or
+\env{split} environment:
+\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 of fixed size (\cn{bigl} \cn{bigr}
+\cn{biggl} \cn{biggr} \dots; see \secref{bigdel}); or (2)~use null
+delimiters to break up the \cn{left}-\cn{right} pair into parts for each
+line:
+\begin{verbatim}
+AAA\left(BBB\right.\\
+  \left.CCC\right)
+\end{verbatim}
+The latter solution may result in mismatched delimiter sizes;
+ensuring that they match requires using \cn{vphantom} in the line
+that has the smaller delimiter (or possibly \cn{smash} in the line that
+has the larger delimiter). In the argument of \cn{vphantom} put a copy
+of the tallest element that occurs in the other line, e.g.,
+\begin{verbatim}
+xxx \left(\int_t yyy\right.\\
+  \left.\vphantom{\int_t} zzz ... \right)
+\end{verbatim}
+\end{error}
+
+\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 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 \pkg{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 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}
+
+\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 \pkg{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}
+
+\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}
+
+\section{Warning messages}
+
+\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
+\pkg{amsmath} package is used because their syntax is foreign to \latex/
+and \pkg{amsmath} provides native \latex/ equivalents. See
+\fn{technote.tex} for further information.
+\end{error}
+
+\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}
+
+\section{Wrong output}
+
+\subsection{Section numbers 0.1, 5.1, 8.1 instead of 1, 2, 3}
+\label{numinverse}
+
+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}
+
+\subsection{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
+\secref{numinverse}.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Additional information}
+
+\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 \pkg{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 \pkg{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 \pkg{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 \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}
+
+Questions or comments regarding \pkg{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}
+
+\section{Of possible interest}\label{a:possible-interest}
+Information about obtaining AMSFonts or other \tex/-related
+software from the AMS Internet archive \fn{e-math.ams.org}
+can be obtained by sending a request through electronic mail to:
+\mail{webmaster@ams.org}.
+
+Information about obtaining the \pkg{amsmath} distribution on diskette
+from the AMS is available from:
+\begin{infoaddress}
+American Mathematical Society\\
+Customer Services\\
+P. O. Box 6248\\
+Providence, RI 02940\\[3pt]
+Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4000\\
+Internet: \mail{cust-serv@ams.org}
+\end{infoaddress}
+
+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 clearing-house of 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.
+
+There is a Usenet newsgroup called \fn{comp.text.tex} that is a fairly
+good source of information about \latex/ and \tex/ in general. If you
+don't know about reading newsgroups, check with your local system
+administrator to see if newsgroup service is available at your site.
+
+\begin{thebibliography}{9}
+\addcontentsline{toc}{chapter}{Bibliography}
+
+\bibitem{amsfonts}\booktitle{AMSFonts version \textup{2.2}\mdash user's guide},
+Amer. Math. Soc., Providence, RI, 1994; distributed
+with the AMSFonts package.
+
+\bibitem{instr-l}\booktitle{Instructions for preparation of
+papers and monographs\mdash \amslatex/},
+Amer. Math. Soc., Providence, RI, 1996, 1999.
+
+\bibitem{amsthdoc}\booktitle{Using the \pkg{amsthm} Package},
+Amer. Math. Soc., Providence, RI, 1999.
+
+\bibitem{tlc} 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
+    \pkg{amsmath} package unless you refer to the errata for Chapter
+    8\mdash file \fn{compan.err}, distributed with \LaTeX{}.}]
+
+% 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{http://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{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.
+
+\end{thebibliography}
+
+\begin{theindex}
+
+  \item {\normalfont  \ttfamily  \bslchar \qcbang  }, 11
+  \item \verb*+\,+, 11
+  \item \verb*+\:+, 11
+  \item \verb*+\;+, 11
+  \item \verb*+\\+, 3, 5, 6, 8, 9, 24, 26
+  \item \verb*+\\*+, 9
+  \item \verb*+\|+, 16
+
+  \indexspace
+
+  \item \verb*+\above+, 14, 29
+  \item \verb*+\abovewithdelims+, 14
+  \item \verb*+\accentedsymbol+, 2
+  \item \texttt{accents} package, 12
+  \item \verb*+\addtocounter+, 10
+  \item \verb*+\addtolength+, 5
+  \item \texttt{align} environment, 3, 5--7, 28--30
+  \item \texttt{alignat} environment, 7, 24, 26, 29
+  \item \texttt{aligned} environment, 7, 9
+  \item \texttt{alignedat} environment, 7, 9
+  \item \verb*+\allowdisplaybreaks+, 8, 9
+  \item \texttt{amsart} class, 1
+  \item \texttt{amsbook} class, 1
+  \item \texttt{amsbsy} package, 1, 2
+  \item \texttt{amscd} package, 1, 2, 20
+  \item \texttt{amsfonts} package, 21
+  \item AMSFonts collection, 25
+  \item \texttt{amsmath} package, i, 1--3, 5, 8--18, 22, 23, 25, 26,
+		28--32
+  \item \texttt{amsopn} package, 1, 2
+  \item \texttt{amssymb} package, 22
+  \item \texttt{amstext} package, 1, 2
+  \item \texttt{amsthm} package, 1, 32
+  \item \texttt{amsxtra} package, 1, 2, 12
+  \item \verb*+\arccos+, 18
+  \item \verb*+\arcsin+, 18
+  \item \verb*+\arctan+, 18
+  \item \verb*+\arg+, 18
+  \item \texttt{array} environment, 8, 10, 26
+  \item arrows
+    \subitem extensible, 13
+    \subitem in commutative diagrams, 20
+  \item \verb*+\atop+, 14, 29
+  \item \verb*+\atopwithdelims+, 14
+
+  \indexspace
+
+  \item BaKoMa fonts, 25
+  \item \cn{big}, \cn{Big}, \cn{bigg}, \dots\ delimiters, 16
+  \item \verb*+\biggl+, 27
+  \item \verb*+\biggr+, 27
+  \item \verb*+\bigl+, 27
+  \item \verb*+\bigr+, 27
+  \item \verb*+\binom+, 14
+  \item binomials, 14
+  \item \texttt{bm} package, 2, 23
+  \item \texttt{Bmatrix} environment, 10
+  \item \texttt{bmatrix} environment, 10
+  \item \verb*+\bmod+, 18
+  \item \verb*+\boldsymbol+, 22, 23
+  \item \verb*+\boxed+, 13
+
+  \indexspace
+
+  \item \verb*+\C+, 22
+  \item \texttt{cases} environment, 8
+  \item \verb*+\cases+, 28
+  \item \texttt{CD} environment, 2, 21
+  \item \verb*+\cdots+, 12
+  \item \texttt{centertags} option, 2
+  \item \verb*+\cfrac+, 15
+  \item \texttt{cmbsy5}, 25
+  \item \texttt{cmex}, 25
+  \item \texttt{cmex10}, 25
+  \item \texttt{cmex7}, 25
+  \item \texttt{cmmib5}, 25
+  \item \texttt{comp.text.tex}, 31
+  \item \texttt{compan.err}, 32
+  \item continued fractions, 15
+  \item \verb*+\cos+, 18
+  \item \verb*+\cosh+, 18
+  \item \verb*+\cot+, 18
+  \item \verb*+\coth+, 18
+  \item \verb*+\csc+, 18
+
+  \indexspace
+
+  \item \verb*+\dbinom+, 14
+  \item \verb*+\ddddot+, 12
+  \item \verb*+\dddot+, 12
+  \item \verb*+\ddot+, 12
+  \item \verb*+\DeclareMathOperator+, 2, 17, 18
+  \item \verb*+\deg+, 18
+  \item \verb*+\det+, 18
+  \item \verb*+\dfrac+, 14
+  \item \texttt{diffs-m.txt}, 30
+  \item \verb*+\dim+, 18
+  \item \verb*+\displaybreak+, 8, 9
+  \item displayed   equations, 3
+  \item displayed equations
+    \subitem centering, 3
+  \item \verb*+\displaylimits+, 20
+  \item \texttt{displaymath} environment, 3
+  \item \verb*+\displaystyle+, 14
+  \item \verb*+\documentclass+, 2
+  \item \verb*+\dot+, 12
+  \item dots, \see{ellipsis dots}{11}
+  \item \verb*+\dotsb+, 12
+  \item \verb*+\dotsc+, 12
+  \item \verb*+\dotsi+, 12
+  \item \verb*+\dotsm+, 12
+  \item \verb*+\dotso+, 12
+
+  \indexspace
+
+  \item \texttt{e-math.ams.org}, 31
+  \item ellipsis dots
+    \subitem in matrices, 11
+  \item \texttt{eqnarray} environment, 1, 3, 26, 30
+  \item \verb*+\eqref+, 10
+  \item \texttt{equation} environment, 1, 3, 5, 7, 24, 30
+  \item equation numbers
+    \subitem cross-references, 10
+    \subitem hierarchy, 9
+    \subitem left or right placement, 3
+    \subitem overriding, 3
+    \subitem subordinate numbering, 10
+    \subitem vertical placement, 2
+  \item \texttt{equation*} environment, 3
+  \item equations, \see{displayed equations}{3}
+  \item \texttt{euscript} package, 21
+  \item \verb*+\exp+, 18
+
+  \indexspace
+
+  \item \verb*+\fbox+, 13
+  \item \texttt{fleqn} option, 3, 5
+  \item \texttt{fntguide.tex}, 21
+  \item \verb*+\frac+, 14, 15
+  \item fractions, 14
+  \item \verb*+\fracwithdelims+, 2
+  \item function names, \see{operator names}{17}
+
+  \indexspace
+
+  \item \texttt{gather} environment, 5--7, 24, 28--30
+  \item \texttt{gathered} environment, 7, 9
+  \item \verb*+\gcd+, 18
+  \item \verb*+\genfrac+, 14
+
+  \indexspace
+
+  \item \verb*+\hat+, 12
+  \item \verb*+\hdotsfor+, 11
+  \item \verb*+\hom+, 18
+  \item horizontal space
+    \subitem around operator names, 17
+    \subitem in math mode, 11
+
+  \indexspace
+
+  \item \verb*+\idotsint+, 20
+  \item \verb*+\iiiint+, 20
+  \item \verb*+\iiint+, 20
+  \item \verb*+\iint+, 20
+  \item \verb*+\inf+, 18
+  \item \verb*+\injlim+, 18
+  \item integrals
+    \subitem multiple, 20
+    \subitem placement of limits, 2
+  \item \verb*+\intertext+, 9
+  \item \texttt{intlimits} option, 2, 20
+
+  \indexspace
+
+  \item \verb*+\ker+, 18
+  \item \texttt{kuvio} package, 20
+
+  \indexspace
+
+  \item \verb*+\label+, 6, 10
+  \item \verb*+\langle+, 16
+  \item \verb*+\ldots+, 12
+  \item \verb*+\left+, 15, 16, 27
+  \item \verb*+\leftroot+, 13
+  \item \texttt{leqno} option, 3, 5
+  \item \verb*+\lg+, 18
+  \item \verb*+\lim+, 1, 2, 17, 18
+  \item \verb*+\liminf+, 18
+  \item limits, \see{subscripts and superscripts}{2}
+  \item \verb*+\limits+, 20
+  \item \verb*+\limsup+, 18
+  \item \verb*+\ln+, 18
+  \item \verb*+\log+, 17, 18
+  \item \verb*+\lVert+, 16
+  \item \verb*+\lvert+, 16
+
+  \indexspace
+
+  \item math fonts, 21
+  \item math symbols, \see{math fonts}{21}
+  \item \verb*+\mathbb+, 21, 22
+  \item \verb*+\mathbf+, 21, 22
+  \item \verb*+\mathcal+, 21
+  \item \verb*+\mathfrak+, 21
+  \item \verb*+\mathit+, 21
+  \item \verb*+\mathrm+, 21
+  \item \verb*+\mathscr+, 21
+  \item \verb*+\mathsf+, 21
+  \item \verb*+\mathtt+, 21
+  \item matrices, 10
+    \subitem ellipsis dots, 11
+  \item \texttt{matrix} environment, 10
+  \item \verb*+\matrix+, 28
+  \item \verb*+\max+, 18
+  \item \verb*+\mbox+, 18
+  \item \verb*+\medspace+, 11
+  \item Metafont source files, 25
+  \item \verb*+\min+, 18
+  \item \verb*+\mod+, 18
+  \item \verb*+\mspace+, 11
+  \item \texttt{multline} environment, 3, 5, 27, 30
+  \item \verb*+\multlinegap+, 5
+
+  \indexspace
+
+  \item \texttt{namelimits} option, 2
+  \item \verb*+\negmedspace+, 11
+  \item \verb*+\negthickspace+, 11
+  \item \verb*+\negthinspace+, 11
+  \item \verb*+\newcommand+, 21
+  \item \verb*+\nobreakdash+, 12
+  \item \texttt{nointlimits} option, 2
+  \item \verb*+\nolimits+, 19, 20
+  \item \texttt{nonamelimits} option, 2
+  \item \texttt{nosumlimits} option, 2
+  \item \verb*+\notag+, 3, 5
+  \item \verb*+\numberwithin+, 9, 29, 30
+
+  \indexspace
+
+  \item operator names, 17
+  \item \verb*+\operatorname+, 18
+  \item \verb*+\operatorname*+, 18
+  \item \verb*+\over+, 14, 29
+  \item \verb*+\overleftarrow+, 13
+  \item \verb*+\overleftrightarrow+, 13
+  \item \verb*+\overrightarrow+, 13
+  \item \verb*+\overset+, 13
+  \item \verb*+\overwithdelims+, 14
+
+  \indexspace
+
+  \item page breaks, 8
+  \item \verb*+\pagebreak+, 8
+  \item \texttt{pmatrix} environment, 10
+  \item \verb*+\pmatrix+, 28
+  \item \verb*+\pmb+, 22
+  \item \verb*+\pmod+, 18
+  \item \verb*+\pod+, 18
+  \item PostScript fonts, 25
+  \item \verb*+\Pr+, 18
+  \item \verb*+\projlim+, 18
+
+  \indexspace
+
+  \item \verb*+\qquad+, 11
+  \item \verb*+\quad+, 11
+
+  \indexspace
+
+  \item \verb*+\R+, 22
+  \item \verb*+\raisetag+, 8
+  \item \verb*+\rangle+, 16
+  \item \verb*+\ref+, 10
+  \item \texttt{reqno} option, 3, 5
+  \item \verb*+\right+, 15, 16, 27
+  \item \verb*+\rVert+, 16
+  \item \verb*+\rvert+, 16
+
+  \indexspace
+
+  \item \verb*+\scriptscriptstyle+, 14
+  \item \verb*+\scriptstyle+, 14
+  \item \verb*+\sec+, 18
+  \item \verb*+\setcounter+, 9, 10
+  \item \verb*+\setlength+, 5
+  \item \verb*+\shoveleft+, 5
+  \item \verb*+\shoveright+, 5
+  \item \verb*+\sideset+, 13, 19
+  \item \verb*+\sin+, 1, 2, 17, 18
+  \item \verb*+\sinh+, 18
+  \item \texttt{smallmatrix} environment, 11
+  \item \verb*+\smash+, 15, 27
+  \item \verb*+\sphat+, 12
+  \item \texttt{split} environment, 3, 5, 6, 9, 24, 27--30
+  \item \verb*+\sptilde+, 12
+  \item \verb*+\stackrel+, 13
+  \item \texttt{subarray} environment, 19
+  \item \texttt{subeqn.tex}, 3
+  \item \texttt{subequations} environment, 10
+  \item subscripts and superscripts, 13
+    \subitem multi-line, 19
+    \subitem on sums, 19
+    \subitem placement, 2
+  \item \verb*+\substack+, 19
+  \item \verb*+\sum+, 20, 22
+  \item \texttt{sumlimits} option, 2
+  \item \verb*+\sup+, 18
+  \item superscripts, \see{subscripts and superscripts}{19}
+
+  \indexspace
+
+  \item \texttt{tabular} environment, 26
+  \item \verb*+\tag+, 3
+  \item \verb*+\tag*+, 3
+  \item \verb*+\tan+, 18
+  \item \verb*+\tanh+, 18
+  \item \verb*+\tbinom+, 14
+  \item \texttt{tbtags} option, 2
+  \item \texttt{technote.tex}, 14, 28--30
+  \item \texttt{testmath.tex}, 3
+  \item \tex/ Users Group, 31
+  \item \verb*+\text+, 2, 7--9, 18
+  \item text fragments inside math, 9, 18
+  \item \verb*+\textstyle+, 14
+  \item \verb*+\tfrac+, 14
+  \item \verb*+\theequation+, 9, 10
+  \item \verb*+\thickspace+, 11
+  \item \verb*+\thinspace+, 11
+  \item \journalname{TUGboat}, 31
+
+  \indexspace
+
+  \item \verb*+\underleftarrow+, 13
+  \item \verb*+\underleftrightarrow+, 13
+  \item \verb*+\underrightarrow+, 13
+  \item \verb*+\underset+, 13
+  \item \verb*+\uproot+, 13
+  \item \verb*+\usepackage+, 2, 25
+
+  \indexspace
+
+  \item \verb*+\value+, 10
+  \item \verb*+\varDelta+, 23
+  \item \verb*+\varGamma+, 23
+  \item \verb*+\varinjlim+, 18
+  \item \verb*+\varLambda+, 23
+  \item \verb*+\varliminf+, 18
+  \item \verb*+\varlimsup+, 18
+  \item \verb*+\varOmega+, 23
+  \item \verb*+\varPhi+, 23
+  \item \verb*+\varPi+, 23
+  \item \verb*+\varprojlim+, 18
+  \item \verb*+\varPsi+, 23
+  \item \verb*+\varSigma+, 23
+  \item \verb*+\varTheta+, 23
+  \item \verb*+\varUpsilon+, 23
+  \item \verb*+\varXi+, 23
+  \item \texttt{Vmatrix} environment, 10
+  \item \texttt{vmatrix} environment, 10
+  \item \verb*+\vphantom+, 27
+
+  \indexspace
+
+  \item \verb*+\xleftarrow+, 13
+  \item \verb*+\xrightarrow+, 13
+
+\end{theindex}
+
+\end{document}
diff --git a/Master/texmf-dist/source/latex/amsmath/amsmath.dtx b/Master/texmf-dist/source/latex/amsmath/amsmath.dtx
new file mode 100644
index 00000000000..9810bc283f0
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amsmath.dtx
@@ -0,0 +1,5829 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsmath.dtx",
+%%%   version   = "2.13",
+%%%   date      = "2000/07/18",
+%%%   time      = "08:16:41 EDT",
+%%%   author    = "American Mathematical Society",
+%%%   copyright = "Copyright 1995, 2000 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,
+%%%                Publications Technical Group,
+%%%                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",
+%%%   checksum  = "38084 5829 21569 202795",
+%%%   email     = "tech-support@ams.org (Internet)",
+%%%   codetable = "ISO/ASCII",
+%%%   keywords  = "latex, amslatex, math, amsmath",
+%%%   supported = "yes",
+%%%   abstract  = "This is a \LaTeX{} package that provides a variety of
+%%%                extra mathematical features, largely derived from
+%%%                AMS-\TeX{}.",
+%%%   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.",
+%%% }
+%%% ====================================================================
+%
+%\iffalse
+%<*driver>
+\documentclass{amsdtx}
+\raggedbottom
+\begin{document}
+\title{The \pkg{amsmath} package}
+\author{Frank Mittelbach\and Rainer Sch\"opf\and Michael Downes\and
+  David M. Jones}
+\date{Version \fileversion, \filedate}
+\providecommand{\histnote}{}
+\renewenvironment{histnote}{%
+  \trivlist\item[\hspace{\labelsep}\bfseries Historical Note:]%
+}{%
+  \endtrivlist
+}
+\hDocInput{amsmath.dtx}
+\end{document}
+%</driver>
+%\fi
+%
+% \maketitle
+%
+% \MakeShortVerb\|
+%
+% \section{Introduction}
+%
+%    A \latex/ package named \pkg{amstex} was created in 1988--1989 by
+%    adapting \fn{amstex.tex} for use within \latex/. The \pkg{amsmath}
+%    package is the successor of the \pkg{amstex} package. It was
+%    substantially overhauled to integrate it with \latex/2e, which
+%    arrived on the scene in 1994. It provides more or less the same
+%    features, but there are quite a few organizational differences as
+%    well as some new features and bug fixes. For example, the
+%    \pkg{amstex} package automatically loaded the \pkg{amsfonts}
+%    package, but the \pkg{amsmath} package does not. At the present
+%    time (November 1999) user-level documentation of the commands
+%    provided here is found in the AMSmath Users' Guide,
+%    \fn{amsldoc.tex}.
+%
+% \StopEventually{}
+%
+%    Standard file identification.
+%    \begin{macrocode}
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must be December 1994 or later
+\ProvidesPackage{amsmath}[2000/07/18 v2.13 AMS math features]
+%    \end{macrocode}
+%
+% \section{Catcode defenses}
+%
+%    Some packages change the catcode of characters that are essential
+%    in low-level \tex/ syntax. Any package that does so does not
+%    qualify as a PWWO package (\qq{Plays Well With Others}) because it
+%    can cause other packages to fail if they are loaded later. \LaTeX{}
+%    is partly to blame for this because it fails to provide adequate
+%    built-in safeguards in the package loading mechanisms. In the
+%    absence of such safeguards, we will provide them here.
+%    \begin{macrocode}
+\edef\@temp{\catcode 96=\number\catcode 96 }
+\catcode\string `\`=12
+\def\do#1{\catcode\number`#1=\number\catcode`#1}
+\edef\@temp{%
+  \noexpand\AtEndOfPackage{%
+    \@temp
+    \do\"\do\'\do\(\do\)\do\*\do\+\do\,\do\-\do\.%
+    \do\/\do\<\do\=\do\>\do\[\do\]\do\^\do\_\relax
+  }%
+}
+\@temp
+\def\do#1{\catcode\number`#1=12 }
+\do\"\do\'\do\(\do\)\do\*\do\+\do\,\do\-\do\.
+\do\/\do\<\do\=\do\>\do\[\do\]
+\catcode`\^=7 \catcode`\_=8
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Declare some options}
+%
+%    Handling of limits on integrals, sums, operatornames.
+%    \begin{macrocode}
+\DeclareOption{intlimits}{\let\ilimits@\displaylimits}
+\DeclareOption{nointlimits}{\let\ilimits@\nolimits}
+\DeclareOption{sumlimits}{\let\slimits@\displaylimits}
+\DeclareOption{nosumlimits}{\let\slimits@\nolimits}
+\DeclareOption{namelimits}{\PassOptionsToPackage{namelimits}{amsopn}}
+\DeclareOption{nonamelimits}{%
+  \PassOptionsToPackage{nonamelimits}{amsopn}}
+%    \end{macrocode}
+%
+%    The following two switches might have been defined already by the
+%    documentclass, but it doesn't hurt to re-execute the \cs{newif}'s.
+%    \begin{macrocode}
+\newif\ifctagsplit@
+\newif\iftagsleft@
+%    \end{macrocode}
+%    Right or left placement of equation numbers.
+%    \begin{macrocode}
+\DeclareOption{leqno}{\tagsleft@true}
+\DeclareOption{reqno}{\tagsleft@false}
+\DeclareOption{centertags}{\ctagsplit@true}
+\DeclareOption{tbtags}{\ctagsplit@false}
+%    \end{macrocode}
+%
+%    The \opt{cmex10} option is an escape hatch for people who don't
+%    happen to have sizes 7--9 of the \fn{cmex} fonts available to them
+%    yet. (Strictly speaking they are considered part of a minimum
+%    \latex/ distribution now, i.e., all \LaTeXe{} users should have
+%    them, without needing to get the AMSFonts distrib.)
+%    \begin{macrocode}
+\DeclareOption{cmex10}{%
+  \ifnum\cmex@opt=\@ne \def\cmex@opt{0}%
+  \else \def\cmex@opt{10}\fi
+}
+%    \end{macrocode}
+%    To help things work out better with various package loading orders
+%    of \pkg{amsmath} and \pkg{amsfonts}, we establish a variable to
+%    communicate the status of the cmex font definition. If the
+%    \pkg{amsfonts} package was loaded first this variable might be
+%    already defined, in which case we want to preserve its value.
+%    \begin{macrocode}
+\@ifundefined{cmex@opt}{\def\cmex@opt{7}}{}
+%    \end{macrocode}
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Flush-left equations [DMJ]}
+%
+%    The left margin of math enviroments is controlled by
+%    \cs{@mathmargin}.  This can be set to \cs{@centering} to
+%    implement the default behaviour, i.e., centered equations, and to
+%    something else to implement the flushleft style.
+%
+%    In theory, all that's needed to activate the flushleft mode
+%    in the AMS document classes is something like this:
+% \begin{verbatim}
+% \DeclareOption{fleqn}{%
+%   \AtBeginDocument{\@mathmargin30pt\relax}%
+% }
+% \end{verbatim}
+%    (In fact, unless the document class wants to specify the
+%    \cs{@mathmargin}, it doesn't need to do anything with the
+%    \opt{fleqn} option.)
+%    \begin{macrocode}
+\newif\if@fleqn
+%
+\newskip\@mathmargin
+\@mathmargin\@centering
+%
+\DeclareOption{fleqn}{%
+    \@fleqntrue
+    \@mathmargin = -1sp
+    \let\mathindent=\@mathmargin
+    \AtBeginDocument{%
+        \ifdim\@mathmargin= -1sp
+            \@mathmargin\leftmargini minus\leftmargini
+        \fi
+    }%
+}
+%    \end{macrocode}
+%
+%    DMJ: This ensures that \cs{@mathmargin} is given some sort of
+%    sensible default if the class doesn't specify one, while still
+%    allowing a user to override the default value in their document
+%    preamble.  (Incidentally, I'm initializing \cs{@mathmargin} to
+%    \cs{leftmargini} for compatibility with \fn{fleqn.clo}, but I'm
+%    not at all convinced that's the right thing to do.)
+%
+%    The next question is what happens when amsmath is used with
+%    one of the standard classes.  Unfortunately, \latex/ implements
+%    \opt{fleqn} somewhat clumsily; instead of paramaterizing the
+%    definitions of the math structures (as I've attempted to do
+%    here), \fn{fleqn.clo} declares a dimen \cn{mathindent} that is
+%    much like my \cs{@mathmargin} and then redefines \cn\[, \cn\],
+%    \cn{equation}, and \cn{eqnarray}.  This means that things could
+%    get rather messy in 2.09 compatibility mode, since \fn{fleqn.clo}
+%    might be loaded after \fn{amsmath.sty}, which could cause a real
+%    mess.
+%
+%    [mjd,1999/07/07]: Let \cs{mathindent} = \cs{@mathmargin} as
+%    envisioned by DMJ. Compatibility-mode documents will all use the
+%    \pkg{amstex} package, not \pkg{amsmath}. There is a remote chance
+%    of a problem if someone makes an assignment to \cs{mathindent} in a
+%    way that implicitly assumes it is a dimen register (inasmuch as it
+%    has now become a skip register), and the string ``plus'' follows in
+%    the input stream, but if someone's document croaks in that way, I
+%    think they will just have to bite the bullet and fix it. The
+%    alternative is to penalize a lot of other users with a known
+%    handicap.
+%
+%    \begin{macrocode}
+\DeclareOption{?}{}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\ExecuteOptions{nointlimits,sumlimits,namelimits,centertags}
+%    \end{macrocode}
+%    The \cs{par} after \cs{ProcessOptions} is to ensure the correct
+%    line number on screen if an error occurs during option processing;
+%    otherwise the lookahead for a \qc{\*} option would result in \tex/
+%    showing the following line instead.
+%    \begin{macrocode}
+\ProcessOptions\par
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\@ifpackagewith{amsmath}{?}{%
+  \typeout{^^J%
+Documentation for the amsmath package is found in amsldoc.dvi^^J%
+(or .pdf or .tex).^^J%
+^^J%
+See also http://www.ams.org/tex/amslatex.html.^^J%
+^^J%
+Note: Using the first edition of The LaTeX Companion (1994) without^^J%
+errata as a guide for amsmath use is not recommended.^^J%
+  }%
+}{%
+  \typeout{%
+For additional information on amsmath, use the \lq ?\rq\space option.%
+  }%
+}
+%    \end{macrocode}
+%
+%    Processing to handle the \opt{cmex10} option is a little tricky
+%    because of different possible loading orders for \pkg{amsmath} and
+%    \pkg{amsfonts}. The package \pkg{amsmath} sets the \cs{cmex@opt}
+%    flag to 7 or 10, and the package \pkg{amsfonts} sets the flag to 1
+%    or 0.
+%    \begin{macrocode}
+\ifnum\cmex@opt=7 \relax
+  \DeclareFontShape{OMX}{cmex}{m}{n}{%
+    <-8>cmex7<8>cmex8<9>cmex9%
+    <10><10.95><12><14.4><17.28><20.74><24.88>cmex10%
+  }{}%
+  \expandafter\let\csname OMX/cmex/m/n/10\endcsname\relax
+\else
+  \ifnum\cmex@opt=\z@ % need to override cmex7 fontdef from amsfonts
+%    \end{macrocode}
+%    Force reloading of the OMX/cmex font definition file.
+%    \begin{macrocode}
+    \begingroup
+    \fontencoding{OMX}\fontfamily{cmex}%
+    \expandafter\let\csname OMX+cmex\endcsname\relax
+    \try@load@fontshape
+    \endgroup
+%    \end{macrocode}
+%    The \fn{cmex10} font gets special preload handling in the building
+%    of the \latex/ format file, need an extra bit here to work around
+%    that.
+%    \begin{macrocode}
+    \expandafter\let\csname OMX/cmex/m/n/10\endcsname\relax
+    \def\cmex@opt{10}%
+  \fi
+\fi
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Call some other packages}
+%
+%    The \pkg{amstext} package provides the \cn{text} command. The
+%    \pkg{amsbsy} package provides \cn{boldsymbol} and \cn{pmb}. (Since
+%    1997 it is usually better to use the \pkg{bm} package instead; but
+%    I think we have to keep \pkg{amsbsy} here for backward
+%    compatibility [mjd,1999/11/19].) The \pkg{amsopn} package provides
+%    \cn{DeclareMathOperator}.
+%    \begin{macrocode}
+\RequirePackage{amstext}[1995/01/25]
+\RequirePackage{amsbsy}[1995/01/20]
+\RequirePackage{amsopn}[1995/01/20]
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Miscellaneous}
+%
+%  \begin{macro}{\@amsmath@err}
+%    Defining this error function saves main mem.
+%    \begin{macrocode}
+\def\@amsmath@err{\PackageError{amsmath}}
+%    \end{macrocode}
+%  \end{macro}
+%
+% \begin{macro}{\AmS}
+%    The \cs{AmS} prefix can be used to construct the combination
+%    |\AmS-\LaTeX|.
+%    \begin{macrocode}
+\providecommand{\AmS}{{\protect\AmSfont
+  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}
+%    \end{macrocode}
+%    In \cn{AmSfont} we call cmsy directly in lieu of trying to access
+%    it through the math fonts setup (e.g. |\the\textfont2|) because
+%    math fonts can't be relied on to be properly set up if we are not
+%    inside a math formula. This means that if this command is used in a
+%    document where CM fonts are not wanted, then a font substitution
+%    will need to be declared, e.g.:
+% \begin{verbatim}
+% \DeclareFontShape{OMS}{cmsy}{m}{n}{ <-> sub * xxx/m/n }{}
+% \end{verbatim}
+%    where |xxx| is some alternate font family.
+%    Taking the first letter of \cs{f@series} will produce |b| or |m|
+%    for the most common values (|b,bx,m|). It may produce nonsense for
+%    more unusual values of \cs{f@series}, so for safety's sake we have
+%    an additional \cs{if} test. We want to avoid setting the series to
+%    |bx| because in a standard \latex/ installation the combination
+%    |cmsy/bx/n| does not have a font definition, and the user
+%    would get a font substitution warning on screen.
+%    \begin{macrocode}
+\newcommand{\AmSfont}{%
+  \usefont{OMS}{cmsy}{\if\@xp\@car\f@series\@nil bb\else m\fi}{n}}
+%    \end{macrocode}
+% \end{macro}
+%
+%  \begin{macro}{\@mathmeasure}
+%    The function |\@mathmeasure| takes three arguments; the third arg
+%    is typeset as a math formula in an hbox, using arg |#2| as the
+%    mathstyle, and the result is left in the box named by the first
+%    arg. It is assumed that we are already in math mode, so we can turn
+%    off |\everymath| (in particular, |\check@mathfonts|).
+%    \begin{macrocode}
+\def\@mathmeasure#1#2#3{\setbox#1\hbox{\frozen@everymath\@emptytoks
+  \m@th$#2#3$}}
+%    \end{macrocode}
+%  \end{macro}
+%
+%    The \cs{inf@bad} constant is for testing overfull boxes.
+%    \begin{macrocode}
+\@ifundefined{inf@bad}{%
+  \newcount\inf@bad \inf@bad=1000000 \relax
+}{}
+%    \end{macrocode}
+%
+%\subsection{Math spacing commands}
+%
+%    Here we fill in some gaps in the set of spacing commands, and make them
+%    all work equally well in or out of math.
+%    We want all these commands to be robust but declaring them all with
+%    \cs{DeclareRobustCommand} uses up an control sequence name per
+%    command; to avoid this, we define a common command \cs{tmspace}
+%    (text-or-math space) which carries the robustness burden for all of
+%    them. The standard \cs{relax} before the \cs{ifmmode} is not
+%    necessary because of the \cs{protect} added by
+%    \cs{DeclareRobustCommand}.
+%    \begin{macrocode}
+\DeclareRobustCommand{\tmspace}[3]{%
+  \ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
+\renewcommand{\,}{\tmspace+\thinmuskip{.1667em}}
+\let\thinspace\,
+\renewcommand{\!}{\tmspace-\thinmuskip{.1667em}}
+\let\negthinspace\!
+\renewcommand{\:}{\tmspace+\medmuskip{.2222em}}
+\let\medspace\:
+\newcommand{\negmedspace}{\tmspace-\medmuskip{.2222em}}
+\renewcommand{\;}{\tmspace+\thickmuskip{.2777em}}
+\let\thickspace\;
+\newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}}
+%    \end{macrocode}
+%
+%  \begin{macro}{\mspace}
+%    And while we're at it, why don't we provide an equivalent of
+%    \cn{hspace} for math mode use. This allows use of |mu| units in
+%    (for example) constructing compound math symbols.
+%    \begin{macrocode}
+\newcommand{\mspace}[1]{\mskip#1\relax}
+%    \end{macrocode}
+%  \end{macro}
+%
+% \subsection{Vertical bar symbols}
+%    Add left/right specific versions of \cn{vert}, \cn{Vert}. Don't
+%    assume the delimiter codes are the CM defaults.
+%    \begin{macrocode}
+\def\@tempa#1#2\@nil{%
+  \ifx\delimiter#1\@tempcnta#2\relax\else\@tempcnta\z@\fi
+}
+\@xp\@tempa\vert\@empty\@nil
+\ifnum\@tempcnta>\z@
+  \advance\@tempcnta "4000000
+  \xdef\lvert{\delimiter\number\@tempcnta\space }
+  \advance\@tempcnta "1000000
+  \xdef\rvert{\delimiter\number\@tempcnta\space }
+\else
+  \ifx\@@undefined\lvert
+    % Fall back to cmex encoding since we don't know what else to do.
+    \DeclareMathDelimiter{\lvert}
+      {\mathopen}{symbols}{"6A}{largesymbols}{"0C}
+    \DeclareMathDelimiter{\rvert}
+      {\mathclose}{symbols}{"6A}{largesymbols}{"0C}
+  \fi
+\fi
+\@xp\@tempa\Vert\@empty\@nil
+\ifnum\@tempcnta>\z@
+  \advance\@tempcnta "4000000
+  \xdef\lVert{\delimiter\number\@tempcnta\space }
+  \advance\@tempcnta "1000000
+  \xdef\rVert{\delimiter\number\@tempcnta\space }
+\else
+  \ifx\@@undefined\lVert
+    \DeclareMathDelimiter{\lVert}
+      {\mathopen}{symbols}{"6B}{largesymbols}{"0D}
+    \DeclareMathDelimiter{\rVert}
+      {\mathclose}{symbols}{"6B}{largesymbols}{"0D}
+  \fi
+\fi
+%    \end{macrocode}
+%
+% \subsection{Fractions}
+%    Bury the generalized fraction primitives \cs{over}, \cs{atop},
+%    etc., because of their bizarre syntax, which is decidedly out of
+%    place in a \latex/ document.
+%    \begin{macrocode}
+\@saveprimitive\over\@@over
+\@saveprimitive\atop\@@atop
+\@saveprimitive\above\@@above
+\@saveprimitive\overwithdelims\@@overwithdelims
+\@saveprimitive\atopwithdelims\@@atopwithdelims
+\@saveprimitive\abovewithdelims\@@abovewithdelims
+%    \end{macrocode}
+%
+%    If someone insists on using \cs{over}, give a warning the first
+%    time and then resurrect the old definition. Laissez-faire policy.
+%    \begin{macrocode}
+\DeclareRobustCommand{\primfrac}[1]{%
+  \PackageWarning{amsmath}{%
+Foreign command \@backslashchar#1;\MessageBreak
+\protect\frac\space or \protect\genfrac\space should be used instead%
+\MessageBreak
+  }
+  \global\@xp\let\csname#1\@xp\endcsname\csname @@#1\endcsname
+  \csname#1\endcsname
+}
+\renewcommand{\over}{\primfrac{over}}
+\renewcommand{\atop}{\primfrac{atop}}
+\renewcommand{\above}{\primfrac{above}}
+\renewcommand{\overwithdelims}{\primfrac{overwithdelims}}
+\renewcommand{\atopwithdelims}{\primfrac{atopwithdelims}}
+\renewcommand{\abovewithdelims}{\primfrac{abovewithdelims}}
+%    \end{macrocode}
+%
+%    \cn{frac} calls \cn{@@over} directly instead of via \cn{genfrac}, for
+%    better speed because it is so common. \cn{tfrac} and \cn{dfrac} are
+%    abbreviations for some commonly needed mathstyle overrides. To
+%    conserve csnames we avoid making \cn{dfrac} and \cn{tfrac} robust
+%    (\cn{genfrac} is itself robust).
+%    \begin{macrocode}
+\DeclareRobustCommand{\frac}[2]{{\begingroup#1\endgroup\@@over#2}}
+\newcommand{\dfrac}{\genfrac{}{}{}0}
+\newcommand{\tfrac}{\genfrac{}{}{}1}
+%    \end{macrocode}
+%    The \cn{binom} command for binomial notation works like \cn{frac}
+%    and has similar variants. Note that we do not use \cs{z@} in
+%    \cn{dbinom} and \cn{tbinom} because they are not top-level robust
+%    like \cn{binom}, and so the \cs{z@} with the potentially
+%    problematic \qc{\@} character would become visible when writing one
+%    of those commands to a \fn{.toc} file.
+%    \begin{macrocode}
+\DeclareRobustCommand{\binom}{\genfrac()\z@{}}
+\newcommand{\dbinom}{\genfrac(){0pt}0}
+\newcommand{\tbinom}{\genfrac(){0pt}1}
+%    \end{macrocode}
+%
+%  \begin{macro}{\genfrac}
+%    This command provides access to \tex/'s generalized fraction
+%    primitives. Args: \arg{1} left delim, \arg{2} right delim, \arg{3}
+%    line thickness, \arg{4} mathstyle override, \arg{5} numerator,
+%    \arg{6} denominator. But we only read the first four args at first,
+%    in order to give us a moment to select the proper generalized
+%    fraction primitive. Any of those four args could be empty, and when
+%    empty the obvious defaults are selected (no delimiters, default
+%    line thickness (normally .4pt), and no mathstyle override).
+%    \begin{macrocode}
+\DeclareRobustCommand{\genfrac}[4]{%
+  \def\@tempa{#1#2}%
+  \edef\@tempb{\@nx\@genfrac\@mathstyle{#4}%
+    \csname @@\ifx @#3@over\else above\fi
+    \ifx\@tempa\@empty \else withdelims\fi\endcsname}
+  \@tempb{#1#2#3}}
+%    \end{macrocode}
+%    \cs{@genfrac} takes the preceding arguments and reads the
+%    numerator and denominator. Note that there's no convenient way to
+%    make the numerator and denominator \emph{contents}
+%    displaystyle, through this interface.
+%
+%    Args: \arg{1} mathstyle, \arg{2} fraction primitive,
+%    \arg{3} delimiters and rule thickness,
+%    \arg{4} numerator, \arg{5} denominator.
+%    \begin{macrocode}
+\def\@genfrac#1#2#3#4#5{{#1{\begingroup#4\endgroup#2#3\relax#5}}}
+%    \end{macrocode}
+%  \end{macro}
+%
+%    Empty mathstyle arg: no change; 0 = displaystyle, 1 = textstyle, 2
+%    = scriptstyle, 3 = scriptscriptstyle.
+%    \begin{macrocode}
+\def\@mathstyle#1{%
+  \ifx\@empty#1\@empty\relax
+  \else\ifcase#1\displaystyle % case 0
+    \or\textstyle\or\scriptstyle\else\scriptscriptstyle\fi\fi}
+%    \end{macrocode}
+%
+% \subsection{Sums and Integrals}
+%    Default value for sum limits is \cs{displaylimits}, see option
+%    `nosumlimits'.
+%
+%    We redefine all the cumulative operator symbols to use
+%    \cs{slimits@} so that switching between \cs{displaylimits} and
+%    \cs{nolimits} can be controlled by package options. Also add
+%    \cs{DOTSB} for the benefit of the dots lookahead. But we'd better
+%    make sure \cn{coprod} and the others are simple mathchars; if not,
+%    the attempted changes will probably fail miserably.
+%
+%    \begin{macrocode}
+\begingroup
+\edef\@tempa{\string\mathchar"}
+\def\@tempb#1"#2\@nil{#1"}
+\edef\@tempc{\expandafter\@tempb\meaning\coprod "\@nil}
+\ifx\@tempa\@tempc
+  \global\let\coprod@\coprod
+  \gdef\coprod{\DOTSB\coprod@\slimits@}
+  \global\let\bigvee@\bigvee
+  \gdef\bigvee{\DOTSB\bigvee@\slimits@}
+  \global\let\bigwedge@\bigwedge
+  \gdef\bigwedge{\DOTSB\bigwedge@\slimits@}
+  \global\let\biguplus@\biguplus
+  \gdef\biguplus{\DOTSB\biguplus@\slimits@}
+  \global\let\bigcap@\bigcap
+  \gdef\bigcap{\DOTSB\bigcap@\slimits@}
+  \global\let\bigcup@\bigcup
+  \gdef\bigcup{\DOTSB\bigcup@\slimits@}
+  \global\let\prod@\prod
+  \gdef\prod{\DOTSB\prod@\slimits@}
+  \global\let\sum@\sum
+  \gdef\sum{\DOTSB\sum@\slimits@}
+  \global\let\bigotimes@\bigotimes
+  \gdef\bigotimes{\DOTSB\bigotimes@\slimits@}
+  \global\let\bigoplus@\bigoplus
+  \gdef\bigoplus{\DOTSB\bigoplus@\slimits@}
+  \global\let\bigodot@\bigodot
+  \gdef\bigodot{\DOTSB\bigodot@\slimits@}
+  \global\let\bigsqcup@\bigsqcup
+  \gdef\bigsqcup{\DOTSB\bigsqcup@\slimits@}
+\fi
+\endgroup
+%    \end{macrocode}
+%
+% \subsection{Roots and radicals}
+%
+%    This root stuff needs syntax work and implementation work. Surely
+%    something more compact can be done?? [mjd, 1994/09/05]
+%    \begin{macrocode}
+\newcommand{\leftroot}{\@amsmath@err{\Invalid@@\leftroot}\@eha}
+\newcommand{\uproot}{\@amsmath@err{\Invalid@@\uproot}\@eha}
+\newcount\uproot@
+\newcount\leftroot@
+\renewcommand{\root}{\relaxnext@
+  \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else
+   \ifx\@let@token\leftroot\let\next@\nextiii@\else
+   \let\next@\plainroot@\fi\fi\next@}%
+  \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}%
+  \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else
+   \DN@.{\FN@\nextv@}\fi\next@.}%
+  \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else
+   \let\next@\plainroot@\fi\next@}%
+  \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}%
+   \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}%
+  \def\nextvii@{\ifx\@let@token\@sptoken
+   \DN@. {\FN@\nextviii@}\else
+   \DN@.{\FN@\nextviii@}\fi\next@.}%
+  \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else
+   \let\next@\plainroot@\fi\next@}%
+  \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}%
+  \bgroup\uproot@\z@\leftroot@\z@\FN@\next@}
+\def\plainroot@#1\of#2{\setbox\rootbox\hbox{%
+ $\m@th\scriptscriptstyle{#1}$}%
+ \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}}
+ {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup}
+%    \end{macrocode}
+%
+% \changes{v2.0}{1999/06/17}{Normalize @@sqrt to sqrtsign}
+
+%    Name change from \cs{@@sqrt} to \cs{sqrtsign} happened in the
+%    1995/12/01 release of \latex/. If we were to assume that
+%    \cs{sqrtsign} is defined then someone with the 1995/06/01 release
+%    of \latex/ would have trouble using this package.
+%    \begin{macrocode}
+\@ifundefined{sqrtsign}{\let\sqrtsign\@@sqrt}{}
+\def\r@@t#1#2{\setboxz@h{$\m@th#1\sqrtsign{#2}$}%
+ \dimen@\ht\z@\advance\dimen@-\dp\z@
+ \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}%
+ \advance\dimen@ by1.667\wd\@ne
+ \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox
+ \mkern-10mu\mkern\leftroot@ mu\boxz@}
+%    \end{macrocode}
+%
+% \subsection{Et cetera}
+%
+% \changes{v2.0}{1999/06/18}{Leave Greek cap letters unaltered}
+%
+%    Specific names for the variant italic cap Greek letters are not
+%    defined by \latex/. If no preceding package defined these, we will
+%    define them now.
+%    \begin{macrocode}
+\@ifundefined{varGamma}{%
+  \DeclareMathSymbol{\varGamma}{\mathord}{letters}{"00}
+  \DeclareMathSymbol{\varDelta}{\mathord}{letters}{"01}
+  \DeclareMathSymbol{\varTheta}{\mathord}{letters}{"02}
+  \DeclareMathSymbol{\varLambda}{\mathord}{letters}{"03}
+  \DeclareMathSymbol{\varXi}{\mathord}{letters}{"04}
+  \DeclareMathSymbol{\varPi}{\mathord}{letters}{"05}
+  \DeclareMathSymbol{\varSigma}{\mathord}{letters}{"06}
+  \DeclareMathSymbol{\varUpsilon}{\mathord}{letters}{"07}
+  \DeclareMathSymbol{\varPhi}{\mathord}{letters}{"08}
+  \DeclareMathSymbol{\varPsi}{\mathord}{letters}{"09}
+  \DeclareMathSymbol{\varOmega}{\mathord}{letters}{"0A}
+}{}
+%    \end{macrocode}
+%
+%    \amstex/ redefines \cn{overline} as shown here, for reasons that
+%    are probably less important in \latex/: Make it read its argument
+%    as a macro argument rather than a ``math field'' (\emph{The
+%    \tex/book}, Chapter 26), to avoid problems when something that is
+%    apparently a single symbol is actually a non-simple macro (e.g.,
+%    \cn{dag}) \emph{and} is given as a single-token argument without
+%    enclosing braces.
+%    \begin{macrocode}
+\@saveprimitive\overline\@@overline
+\DeclareRobustCommand{\overline}[1]{\@@overline{#1}}
+%    \end{macrocode}
+%
+%    The \cs{boxed} command is specifically intended to put a box around
+%    an equation or piece of an equation. (Not including the equation
+%    number.) This isn't trivial for end-users to do it properly
+%    with \cs{fbox} so we provide a command for them.
+%    \begin{macrocode}
+\newcommand{\boxed}[1]{\fbox{\m@th$\displaystyle#1$}}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\newcommand{\implies}{\DOTSB\;\Longrightarrow\;}
+\newcommand{\impliedby}{\DOTSB\;\Longleftarrow\;}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\And{\DOTSB\;\mathchar"3026 \;}
+%    \end{macrocode}
+%
+%    The command \cn{nobreakdash} is designed only for use before a
+%    hyphen or dash (|-|, |--|, or |---|).
+%    Setting the hyphen in a box and then unboxing it means that the
+%    normal penalty will not be added after it---and if the penalty is
+%    not there a break will not be taken (unless an explicit penalty
+%    or glue follows, thus the final \verb=\nobreak=).
+%    \begin{macrocode}
+\newcommand{\nobreakdash}{\leavevmode
+  \toks@\@emptytoks \def\@tempa##1{\toks@\@xp{\the\toks@-}\FN@\next@}%
+  \DN@{\ifx\@let@token-\@xp\@tempa
+       \else\setboxz@h{\the\toks@\nobreak}\unhbox\z@\fi}%
+  \FN@\next@
+}
+%    \end{macrocode}
+%
+%    \cs{colon} is for a colon in math that resembles a text colon:
+%    small space on the left, larger space on the right. The \qc{\:}
+%    character by itself is treated as a \cs{mathrel} i.e. large, equal
+%    spacing on both sides.
+%    \begin{macrocode}
+\renewcommand{\colon}{\nobreak\mskip2mu\mathpunct{}\nonscript
+  \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Ellipsis dots}
+%
+%    We can't use \cs{newif} for \cs{ifgtest@} because we want
+%    to include \cs{global} in the definitions of
+%    \cs{gtest@true} and \cs{gtest@false}.
+%    \begin{macrocode}
+\let\ifgtest@\iffalse                              % initial value
+\def\gtest@true{\global\let\ifgtest@\iftrue}
+\def\gtest@false{\global\let\ifgtest@\iffalse}
+\let\DOTSI\relax
+\let\DOTSB\relax
+\let\DOTSX\relax
+{\uccode`7=`\\ \uccode`8=`m \uccode`9=`a \uccode`0=`t \uccode`!=`h
+ \uppercase{%
+  \gdef\math@#1#2#3#4#5#6\math@{\gtest@false\ifx 7#1\ifx 8#2%
+  \ifx 9#3\ifx 0#4\ifx !#5\xdef\meaning@{#6}\gtest@true
+  \fi\fi\fi\fi\fi}}}
+{\uccode`7=`c \uccode`8=`h \uccode`9=`\"
+ \uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\gtest@false
+  \ifx 7#1\ifx 8#2\ifx 9#5\gtest@true\xdef\meaning@{9#6}\fi\fi\fi}}}
+\newcount\classnum@
+\def\getmathch@#1.#2\getmathch@{\classnum@#1 \divide\classnum@4096
+ \ifcase\number\classnum@\or\or\gdef\thedots@{\dotsb@}\or
+ \gdef\thedots@{\dotsb@}\fi}
+{\uccode`4=`b \uccode`5=`i \uccode`6=`n
+ \uppercase{\gdef\mathbin@#1#2#3{\relaxnext@
+  \def\nextii@##1\mathbin@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+  \gtest@false\DN@##1\mathbin@{}%
+ \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
+{\uccode`4=`r \uccode`5=`e \uccode`6=`l
+ \uppercase{\gdef\mathrel@#1#2#3{\relaxnext@
+  \def\nextii@##1\mathrel@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+ \gtest@false\DN@##1\mathrel@{}%
+ \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
+{\uccode`5=`m \uccode`6=`a \uccode`7=`c
+ \uppercase{\gdef\macro@#1#2#3#4\macro@{\gtest@false
+  \ifx 5#1\ifx 6#2\ifx 7#3\gtest@true
+  \xdef\meaning@{\macro@@#4\macro@@}\fi\fi\fi}}}
+\def\macro@@#1->#2\macro@@{#2}
+\newcount\DOTSCASE@
+{\uccode`6=`\\ \uccode`7=`D \uccode`8=`O \uccode`9=`T \uccode`0=`S
+ \uppercase{\gdef\DOTS@#1#2#3#4#5{\gtest@false\DN@##1\DOTS@{}%
+  \ifx 6#1\ifx 7#2\ifx 8#3\ifx 9#4\ifx 0#5\let\next@\DOTS@@
+  \fi\fi\fi\fi\fi
+  \next@}}}
+{\uccode`3=`B \uccode`4=`I \uccode`5=`X
+ \uppercase{\gdef\DOTS@@#1{\relaxnext@
+  \def\nextii@##1\DOTS@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+  \DN@{\FN@\nextii@}%
+  \ifx 3#1\global\DOTSCASE@\z@\else
+  \ifx 4#1\global\DOTSCASE@\@ne\else
+  \ifx 5#1\global\DOTSCASE@\tw@\else\DN@##1\DOTS@{}%
+  \fi\fi\fi\next@}}}
+{\uccode`5=`\\ \uccode`6=`n \uccode`7=`o \uccode`8=`t
+ \uppercase{\gdef\not@#1#2#3#4{\relaxnext@
+  \def\nextii@##1\not@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+ \gtest@false\DN@##1\not@{}%
+ \ifx 5#1\ifx 6#2\ifx 7#3\ifx 8#4\DN@{\FN@\nextii@}\fi\fi\fi
+ \fi\next@}}}
+\def\keybin@{\gtest@true
+ \ifx\@let@token+\else\ifx\@let@token=\else
+ \ifx\@let@token<\else\ifx\@let@token>\else
+ \ifx\@let@token-\else\ifx\@let@token*\else\ifx\@let@token:\else
+   \gtest@false\fi\fi\fi\fi\fi\fi\fi}
+%    \end{macrocode}
+%    Patch to ensure \cs{@ldots} is defined. (Name changed to
+%    \cn{mathellipsis} in Dec 94 release of \latex/.)
+%    \begin{macrocode}
+\@ifundefined{@ldots}{\def\@ldots{\mathellipsis}}{}
+%    \end{macrocode}
+%    Reiterate the standard definition of \cs{ldots} to keep it from
+%    being clobbered by the redefinition of \cs{dots}.
+%    \begin{macrocode}
+\DeclareRobustCommand{\ldots}{%
+  \ifmmode \mathellipsis \else \textellipsis \fi
+}
+\DeclareRobustCommand{\dots}{%
+  \ifmmode \@xp\mdots@\else \@xp\textellipsis \fi
+}
+\def\tdots@{\leavevmode\unskip\relaxnext@
+ \DN@{$\m@th\@ldots\,
+   \ifx\@let@token,\,$\else\ifx\@let@token.\,$\else
+   \ifx\@let@token;\,$\else\ifx\@let@token:\,$\else
+   \ifx\@let@token?\,$\else\ifx\@let@token!\,$\else
+     $ \fi\fi\fi\fi\fi\fi}%
+  \ \FN@\next@}
+\def\mdots@{\FN@\mdots@@}
+\def\mdots@@{\gdef\thedots@{\dotso@}%
+ \ifx\@let@token\boldsymbol \gdef\thedots@\boldsymbol{\boldsymboldots@}%
+ \else\ifx,\@let@token \gdef\thedots@{\dotsc}%
+ \else\ifx\not\@let@token \gdef\thedots@{\dotsb@}%
+ \else\keybin@
+ \ifgtest@\gdef\thedots@{\dotsb@}%
+ \else\xdef\meaning@{\meaning\@let@token..........}%
+   \xdef\meaning@@{\meaning@}%
+  \@xp\math@\meaning@\math@
+  \ifgtest@
+   \@xp\mathch@\meaning@\mathch@
+   \ifgtest@\@xp\getmathch@\meaning@\getmathch@\fi
+  \else\@xp\macro@\meaning@@\macro@
+  \ifgtest@
+   \@xp\not@\meaning@\not@\ifgtest@\gdef\thedots@{\dotsb@}%
+  \else\@xp\DOTS@\meaning@\DOTS@
+  \ifgtest@
+   \ifcase\number\DOTSCASE@\gdef\thedots@{\dotsb@}%
+    \or\gdef\thedots@{\dotsi}\else\fi
+  \else\@xp\math@\meaning@\math@
+  \ifgtest@\@xp\mathbin@\meaning@\mathbin@
+  \ifgtest@\gdef\thedots@{\dotsb@}%
+  \else\@xp\mathrel@\meaning@\mathrel@
+  \ifgtest@\gdef\thedots@{\dotsb@}%
+  \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+ \thedots@}
+%    \end{macrocode}
+%
+%    The \qc{\=} character is necessary in the two \cs{let} assignments
+%    in \cs{boldsymboldots@}, because the symbol we are making
+%    bold might be an \qc{\=} sign.
+%    \begin{macrocode}
+\def\boldsymboldots@#1{%
+  \bold@true\let\@let@token=#1\let\delayed@=#1\mdots@@
+  \boldsymbol#1\bold@false}
+%    \end{macrocode}
+%
+%    The definition of \cs{@cdots} is merely the \fn{plain.tex}
+%    definition of \cs{cdots}.
+%    \begin{macrocode}
+\def\@cdots{\mathinner{\cdotp\cdotp\cdotp}}
+\newcommand{\dotsi}{\!\@cdots}
+\let\dotsb@\@cdots
+%    \end{macrocode}
+%
+%    If any new right delimiters are defined, they would need to be
+%    added to the definition of \cs{rightdelim@} in order for \cn{dots}
+%    to work properly in all cases.
+%    \begin{macrocode}
+\def\rightdelim@{\gtest@true
+ \ifx\@let@token)\else
+ \ifx\@let@token]\else
+ \ifx\@let@token\rbrack\else
+ \ifx\@let@token\}\else
+ \ifx\@let@token\rbrace\else
+ \ifx\@let@token\rangle\else
+ \ifx\@let@token\rceil\else
+ \ifx\@let@token\rfloor\else
+ \ifx\@let@token\rgroup\else
+ \ifx\@let@token\rmoustache\else
+ \ifx\@let@token\right\else
+ \ifx\@let@token\bigr\else
+ \ifx\@let@token\biggr\else
+ \ifx\@let@token\Bigr\else
+ \ifx\@let@token\Biggr\else\gtest@false
+ \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
+\def\extra@{%
+ \rightdelim@\ifgtest@
+ \else\ifx\@let@token$\gtest@true
+ \else\xdef\meaning@{\meaning\@let@token..........}%
+ \@xp\macro@\meaning@\macro@\ifgtest@
+ \@xp\DOTS@\meaning@\DOTS@
+ \ifgtest@
+ \ifnum\DOTSCASE@=\tw@\gtest@true\else\gtest@false
+ \fi\fi\fi\fi\fi}
+\newif\ifbold@
+\def\dotso@{\relaxnext@
+ \ifbold@
+  \let\@let@token\delayed@
+  \def\nextii@{\extra@\@ldots\ifgtest@\,\fi}%
+ \else
+  \def\nextii@{\DN@{\extra@\@ldots\ifgtest@\,\fi}\FN@\next@}%
+ \fi
+ \nextii@}
+%    \end{macrocode}
+%    Why not save some tokens? (space vs. time).
+%    \begin{macrocode}
+\def\extrap@#1{%
+ \DN@{#1\,}%
+ \ifx\@let@token,\else
+ \ifx\@let@token;\else
+ \ifx\@let@token.\else\extra@
+ \ifgtest@\else
+ \let\next@#1\fi\fi\fi\fi\next@}
+%    \end{macrocode}
+%
+%    The \cn{cdots} command.
+%    \begin{macrocode}
+\DeclareRobustCommand{\cdots}{\DN@{\extrap@\@cdots}\FN@\next@}
+\let\dotsb\cdots
+\let\dotsm\cdots
+\DeclareRobustCommand{\dotso}{\relax
+  \ifmmode \DN@{\extrap@\@ldots}%
+  \else \let\next@\tdots@\fi
+  \FN@\next@}
+\DeclareRobustCommand{\dotsc}{%
+  \DN@{\ifx\@let@token;\@ldots\,%
+       \else \ifx\@let@token.\@ldots\,%
+       \else \extra@\@ldots \ifgtest@\,\fi
+       \fi\fi}%
+  \FN@\next@}
+\renewcommand{\longrightarrow}{%
+  \DOTSB\protect\relbar\protect\joinrel\rightarrow}
+\renewcommand{\Longrightarrow}{%
+  \DOTSB\protect\Relbar\protect\joinrel\Rightarrow}
+\renewcommand{\longleftarrow}{%
+  \DOTSB\leftarrow\protect\joinrel\protect\relbar}
+\renewcommand{\Longleftarrow}{%
+  \DOTSB\Leftarrow\protect\joinrel\protect\Relbar}
+\renewcommand{\longleftrightarrow}{\DOTSB\leftarrow\joinrel\rightarrow}
+\renewcommand{\Longleftrightarrow}{\DOTSB\Leftarrow\joinrel\Rightarrow}
+\renewcommand{\mapsto}{\DOTSB\mapstochar\rightarrow}
+\renewcommand{\longmapsto}{\DOTSB\mapstochar\longrightarrow}
+\renewcommand{\hookrightarrow}{\DOTSB\lhook\joinrel\rightarrow}
+\renewcommand{\hookleftarrow}{\DOTSB\leftarrow\joinrel\rhook}
+\renewcommand{\iff}{\DOTSB\;\Longleftrightarrow\;}
+%    \end{macrocode}
+%    The \cn{doteq} command formerly used \cs{buildrel}; we avoid that
+%    because it requires `\cn{over}' as part of its syntax. Use 0pt
+%    instead of \cs{z@} for robustitude.
+%    \begin{macrocode}
+\renewcommand{\doteq}{%
+  \DOTSB\mathrel{\mathop{\kern0pt =}\limits^{\textstyle.}}}
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Integral signs}
+%
+%    The straightforward \cs{ifinner} test to see if the current math
+%    context is non-display, fails if, for instance, we are typesetting
+%    a multiline display within an \cs{halign}, with the pieces going
+%    into constructions like
+%    \begin{verbatim}
+%    $\displaystyle...$
+%    \end{verbatim}
+%    So we need a better test to find out if we are `in a display'. We
+%    therefore create \cs{if@display}.
+%
+%    \begin{macrocode}
+\newif\if@display
+\everydisplay\@xp{\the\everydisplay \@displaytrue}
+%    \end{macrocode}
+%
+%    Default value for integral limits is \cs{nolimits}, see the
+%    definition of the `nointlimits' option.
+%    \begin{macrocode}
+\renewcommand{\int}{\DOTSI\intop\ilimits@}
+\renewcommand{\oint}{\DOTSI\ointop\ilimits@}
+\def\intkern@{\mkern-6mu\mathchoice{\mkern-3mu}{}{}{}}
+\def\intdots@{\mathchoice{\@cdots}%
+ {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
+ {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
+ {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}
+%
+\newcommand{\iint}{\DOTSI\protect\MultiIntegral{2}}
+\newcommand{\iiint}{\DOTSI\protect\MultiIntegral{3}}
+\newcommand{\iiiint}{\DOTSI\protect\MultiIntegral{4}}
+\newcommand{\idotsint}{\DOTSI\protect\MultiIntegral{0}}
+%    \end{macrocode}
+%
+%    If the \cs{limits} option is applied, use \cs{mathop} and fudge
+%    the left-hand space a bit to make the subscript visually centered.
+%
+%    \verb'#1' is the multiplicity.
+%    \begin{macrocode}
+\newcommand{\MultiIntegral}[1]{%
+  \edef\ints@c{\noexpand\intop
+    \ifnum#1=\z@\noexpand\intdots@\else\noexpand\intkern@\fi
+    \ifnum#1>\tw@\noexpand\intop\noexpand\intkern@\fi
+    \ifnum#1>\thr@@\noexpand\intop\noexpand\intkern@\fi
+    \noexpand\intop
+    \noexpand\ilimits@
+  }%
+  \futurelet\@let@token\ints@a
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\ints@a{%
+  \ifx\limits\@let@token \ints@b
+  \else \ifx\displaylimits\@let@token \ints@b
+  \else\ifx\ilimits@\displaylimits \ints@b
+  \fi\fi\fi
+  \ints@c
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\ints@b{%
+  \mkern-7mu\mathchoice{\mkern-2mu}{}{}{}%
+  \mathop\bgroup
+    \mkern7mu\mathchoice{\mkern2mu}{}{}{}%
+    \let\ilimits@\egroup
+}%
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Size dependent definitions}
+%
+%    We now define all stuff which has to change whenever a new math
+%    size is to be activated. \latex/ provides a hook called
+%    |\every@math@size| to support such a need. All assignments in the
+%    |\every@math@size| hook that need to take outside effect should be
+%    global.
+%
+% \subsection{Struts for math}
+%
+%    The various kinds of struts could use some analysis and perhaps
+%    consolidation.
+%
+%    For example perhaps the \cn{bBigg} delimiters could use
+% \begin{verbatim}
+% 1.2\ht\strutbox (1.8, 2.4, 3.0)
+% \end{verbatim}
+%    instead of
+% \begin{verbatim}
+% 1.0\big@size (1.5, 2.0, 2.5)
+% \end{verbatim}
+%    since \cs{strut} is reset with every size change [mjd, 1994/10/07].
+%    But this change would introduce the possibility of changed line
+%    and page breaks in existing documents, so would need to be
+%    handled with care.
+%
+% \begin{macro}{\Mathstrut@}
+% \begin{macro}{\Mathstrutbox@}
+% \begin{macro}{\resetMathstrut@}
+%    Here comes the code for Spivak's |\Mathstrut@|.
+%    \begin{macrocode}
+\newbox\Mathstrutbox@
+\setbox\Mathstrutbox@=\hbox{}
+\def\Mathstrut@{\copy\Mathstrutbox@}
+%    \end{macrocode}
+%    The setting of the height and depth of the |\Mathstrutbox@| is done
+%    in the |\every@math@size| hook since it depends on the height of a
+%    paren. As \cs{every@math@size} is triggered by |$| after a font
+%    size change, we want to avoid using another math formula |$...$| to
+%    measure the math paren height; instead we go through the mathcode
+%    of the \qc{\(} character. We assume that the mathcode has a leading
+%    hex digit 4 indicating `open delimiter'; this allows us to make a
+%    relatively simple function to get the correct font and character
+%    position.
+%    \begin{macrocode}
+\def\resetMathstrut@{%
+  \setbox\z@\hbox{%
+    \mathchardef\@tempa\mathcode`\(\relax
+    \def\@tempb##1"##2##3{\the\textfont"##3\char"}%
+    \expandafter\@tempb\meaning\@tempa \relax
+  }%
+%    \end{macrocode}
+%    These height and depth assignments are implicitly global.
+%    \begin{macrocode}
+  \ht\Mathstrutbox@\ht\z@ \dp\Mathstrutbox@\dp\z@
+}
+\addto@hook\every@math@size{\resetMathstrut@}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\strut@}
+% \begin{macro}{\strutbox@}
+%    Next follows a special internal strut which is supposed to match
+%    the height and the depth of a normal |\strut| minus
+%    |\normallineskiplimit| according to M. Spivak.
+%
+%    This should really go into the definition of \cs{size@update}, and
+%    then the box reset could be local; but \cs{size@update} doesn't
+%    have any hook and is handled in such a way that it cannot even be
+%    changed except by changing \cs{set@fontsize}. So instead we put
+%    \cs{reset@strutbox@} into \cs{every@math@size} and make it global.
+%    Then because of some complications in the way \cs{glb@settings} and
+%    \cs{check@mathfonts} work, we have to re-invoke it at the beginning
+%    of every environment that might use \cs{strut@}. Fortunately this
+%    can be achieved (more or less) through the \cs{spread@equation}
+%    hook. [mjd,2000/03/10]
+%    \begin{macrocode}
+\newbox\strutbox@
+\def\strut@{\copy\strutbox@}
+\def\reset@strutbox@{%
+  \global\setbox\strutbox@\hbox{%
+    \lower.5\normallineskiplimit
+       \vbox{\kern-\normallineskiplimit\copy\strutbox}}}
+\addto@hook\every@math@size{\reset@strutbox@}
+\AtBeginDocument{\reset@strutbox@}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Big delimiters}
+%
+%    We are now going to redefine the plain \tex/ commands \cn{big},
+%    \cn{bigl}, etc., to produce different results in different sizes.
+%    Actually we only have to define \cn{big}, \cn{Big}, etc., since
+%    they are used to construct the directional versions \cn{bigl},
+%    \cn{bigr}, and the rest.
+%
+% \begin{macro}{\big}
+% \begin{macro}{\Big}
+% \begin{macro}{\bigg}
+% \begin{macro}{\Bigg}
+%     To save token space we put everything into the common macro
+%    |\bBigg@|. The macros are now simply a call to |\bBigg@| with a
+%    factor to determine the correct height of the delimiter as an
+%    argument. This code should better go into a future version of
+%    the \latex/ kernel; the macro |\n@space| is then superfluous (since
+%    it is only used once) and should be removed to avoid wasting hash
+%    table space unnecessarily.
+%    \begin{macrocode}
+\renewcommand{\big}{\bBigg@\@ne}
+\renewcommand{\Big}{\bBigg@{1.5}}
+\renewcommand{\bigg}{\bBigg@\tw@}
+\renewcommand{\Bigg}{\bBigg@{2.5}}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\bBigg@}
+%    Now we tackle the macro which has to do the real work. It
+%    actually has two arguments, the factor and the wanted delimiter.
+%    \begin{macrocode}
+\def\bBigg@#1#2{%
+%    \end{macrocode}
+%    We start with an extra set of braces because we want
+%    constructions like |\def\bigl{\mathopen\big}| to work without the
+%    overhead of extra arguments.
+%    \begin{macrocode}
+  {\@mathmeasure\z@{\nulldelimiterspace\z@}%
+     {\left#2\vcenter to#1\big@size{}\right.}%
+   \box\z@}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\big@size}
+%    |\big@size| needs to be set to 1.2 times the height of a math
+%    paren. This height is already recorded in |\Mathstrutbox@|.
+%    \begin{macrocode}
+\addto@hook\every@math@size{%
+  \global\big@size 1.2\ht\Mathstrutbox@
+  \global\advance\big@size 1.2\dp\Mathstrutbox@ }
+\newdimen\big@size
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Math accents}
+%
+%    We want to change the leading digit of math accents to be
+%    \cs{accentclass@} so that it can vary according to certain internal
+%    purposes.
+%    \begin{macrocode}
+\def\accentclass@{7}
+\def\noaccents@{\def\accentclass@{0}}
+%    \end{macrocode}
+%
+%    There are a few \meta{math alphabet}s in the standard fonts where
+%    we have to change the extra macros because the standard definitions
+%    don't account for these accent problems. The first is for the
+%    \cs{mathit} command.
+%    \begin{macrocode}
+\DeclareFontEncoding{OML}{}{\noaccents@}
+%    \end{macrocode}
+%    The next one corrects the \cs{cal} alphabet.
+%    \begin{macrocode}
+\DeclareFontEncoding{OMS}{}{\noaccents@}
+%    \end{macrocode}
+%
+%    Triple and quadruple dot accents.
+%    \begin{macrocode}
+\newcommand{\dddot}[1]{%
+  {\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
+   \hbox{\normalfont ...}\vss}}}}
+\newcommand{\ddddot}[1]{%
+  {\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
+   \hbox{\normalfont....}\vss}}}}
+%    \end{macrocode}
+%
+%    The following code deals with support for compound accents.
+%    By redefining \cs{set@mathaccent} we ensure that
+%    \cn{DeclareMathAccent} will define accent commands to run our
+%    \cs{mathaccentV} function instead of the primitive \cs{mathaccent}.
+%    \begin{macrocode}
+\def\set@mathaccent#1#2#3#4{%
+  \xdef#2{\@nx\protect\@nx\mathaccentV
+    {\@xp\@gobble\string#2}\hexnumber@#1#4}%
+}
+%    \end{macrocode}
+%
+%    We redefine the standard math accent commands to
+%    call \cs{mathaccentV}, using the mathgroup/encoding-number
+%    information embedded in their previous definitions. If the
+%    definition of an accent command does not have the expected form, we
+%    leave the accent command alone, but give a warning. For widehat and
+%    widetilde, we need to avoid clobbering the definitions done by the
+%    \pkg{amsfonts} package. Arbitrating the contention between
+%    \pkg{amsmath} and \pkg{amsfonts} to allow doubling a widetilde
+%    accent looks tricky, so for the time being [mjd,1999/07/19] we just
+%    leave \cn{widehat} and \cn{widetilde} alone. As a result, if the
+%    \pkg{amsmath} package is loaded on top of a vanilla \latex/
+%    documentclass, everything runs through with no warnings. If a
+%    Lucida Math or other math fonts package is loaded in addition to
+%    \pkg{amsmath}, there are greater difficulties, but those are
+%    addressed elsewhere.
+%    \begin{macrocode}
+\def\@tempa#1{\@xp\@tempb\meaning#1\@nil#1}
+\def\@tempb#1>#2#3 #4\@nil#5{%
+  \@xp\ifx\csname#3\endcsname\mathaccent
+    \@tempc#4?"7777\@nil#5%
+  \else
+    \PackageWarningNoLine{amsmath}{%
+      Unable to redefine math accent \string#5}%
+  \fi
+}
+\def\@tempc#1"#2#3#4#5#6\@nil#7{%
+  \chardef\@tempd="#3\relax\set@mathaccent\@tempd{#7}{#2}{#4#5}}
+\@tempa{\hat}
+\@tempa{\check}
+\@tempa{\tilde}
+\@tempa{\acute}
+\@tempa{\grave}
+\@tempa{\dot}
+\@tempa{\ddot}
+\@tempa{\breve}
+\@tempa{\bar}
+\@tempa{\vec}
+\@ifundefined{mathring}{%
+  \DeclareMathAccent{\mathring}{\mathalpha}{operators}{"17}
+}{%
+  \@tempa{\mathring}
+}
+%%\@tempa\widetilde
+%%\@tempa\widehat
+%    \end{macrocode}
+%
+%    Regression testing of amsmath 2.0 showed that in some documents
+%    there occurred fragments of the form
+% \begin{verbatim}
+% \hat\mathcal{G}
+% \end{verbatim}
+%    This is not at all correct syntax for the argument of a \latex/
+%    command but it produced the intended result anyway because of the
+%    internal syntax of the \cs{mathaccent} primitive. With
+%    \cs{mathaccentV}, it will yield an error message. We therefore do a
+%    special check for such syntax problems in order to make the error
+%    message more informative.
+%    \begin{macrocode}
+\newcommand{\acc@check}{}
+\newcommand{\acc@error}{}
+\def\acc@check{\@ifnextchar\@empty\relax\acc@error}
+%    \end{macrocode}
+%    We put most of the tokens in a separate macro so they do not get
+%    scanned unless they are actually needed.
+%    \begin{macrocode}
+\def\acc@error{%
+  \@amsmath@err{%
+    Improper argument for math accent:\MessageBreak
+    Extra braces must be added to prevent wrong output%
+  }\@ehc
+}
+%    \end{macrocode}
+%
+%    For \cs{mathaccentV} part of the processing is dependent on the
+%    depth of nesting of math accent commands. We introduce a dedicated
+%    counter for this instead of using chardef because we want to
+%    increment/decrement it during processing, and incrementing a
+%    chardef integer is more work.
+%    \begin{macrocode}
+\newcount\macc@depth
+%    \end{macrocode}
+%
+%    Provide this function in case it is not already available.
+%    \begin{macrocode}
+\long\def\@gobblethree#1#2#3{}
+%    \end{macrocode}
+%
+%    The \cs{mathaccentV} function first counts the number of nested
+%    math accents by setting the argument in a throw-away box. (This is
+%    not as risky as such an operation would normally be because the
+%    argument is generally either a simple math symbol or a nested math
+%    accent call with a simple math symbol at the bottom of the
+%    nesting.)
+%
+%    There are two benefits from counting the nesting levels first
+%    before doing anything else: (1) we can fall back to a simple
+%    \cs{mathaccent} call if the nesting depth is 1, and (2) if the
+%    nesting depth is greater than 1, we would like to be able to tell
+%    when we have reached the lowest level, because at that point we
+%    want to save the argument for later use and place an accent on top
+%    of a phantom copy.
+%
+%    When we have multiple accents, they will be placed on top of the
+%    invisible box, followed by some suitable kerns, then a visible copy
+%    of the nucleus. To see why, let us look at what goes wrong with a
+%    double application of the \cs{mathaccent} primitive. The standard
+%    definition of \cs{hat} is \verb'\mathaccent"705E', so
+%    \verb'\hat{\hat{F}}' expands to
+%\begin{verbatim}
+%\mathaccent"705E{\mathaccent"705E{F}}
+%\end{verbatim}
+%    The result of this operation is
+%\begin{verbatim}
+%\vbox(12.11111+0.0)x7.81946
+%.\hbox(6.94444+0.0)x0.0, shifted 1.40973
+%..\OT1/cmr/m/n/10 ^
+%.\kern-4.30554
+%.\vbox(9.47221+0.0)x7.81946
+%..\hbox(6.94444+0.0)x0.0, shifted 2.24309
+%...\OT1/cmr/m/n/10 ^
+%..\kern-4.30554
+%..\hbox(6.83331+0.0)x7.81946
+%...\OML/cmm/m/it/10 F
+%\end{verbatim}
+%    \tex/ starts by constructing a vbox with the hat character on top
+%    of the F. Then it puts another hat character on top of the vbox;
+%    but without skew information, because that is only applied by
+%    \cs{mathaccent} when the base object is a simple symbol. So the
+%    first accent is skewed to the correct position but all later
+%    accents are not. By the way, the actual width of the F in the above
+%    example is less than 7.81946; the box in which it is packed was
+%    automatically lengthened by the width of the F's italic correction
+%    (without actually putting in a kern for it).
+%
+%    To get the second accent shifted farther to the right we
+%    artificially increase the width of the innermost box and add
+%    a compensating kern afterward. Furthermore, to get proper placement
+%    of a following subscript or superscript, we take the base symbol
+%    out, leaving a phantom in its place, and print it by itself
+%    following the kern. We then need to increase the kern amount to
+%    move the base character backward under the accents again.
+%    Here is what the results look like:
+%\begin{verbatim}
+%\vbox(12.11111+0.0)x9.48618
+%.\hbox(6.94444+0.0)x0.0, shifted 2.24309
+%..\OT1/cmr/m/n/10 ^
+%.\kern-4.30554
+%.\vbox(9.47221+0.0)x9.48618
+%..\hbox(6.94444+0.0)x0.0, shifted 2.24309
+%...\OT1/cmr/m/n/10 ^
+%..\kern-4.30554
+%..\hbox(6.83331+0.0)x9.48618
+%...\hbox(6.83331+0.0)x7.81946
+%...\kern 1.66672
+%\kern -9.48618
+%\OML/cmm/m/it/10 F
+%\end{verbatim}
+%
+%    Much of this implementation is based on code from the \pkg{accents}
+%    package of Javier Bezos. I added the test to revert to a simple
+%    \cs{mathaccent} when accents are not nested, and some other
+%    refinements to reduce the number of kerns used (to conserve box
+%    memory) and the number of cycles through \cs{mathchoice} (to make
+%    things run a little faster). It was all rather difficult and my
+%    first two attempts had serious bugs but I hope and believe that
+%    this version will do better. [mjd,2000/03/15]
+%
+%    The \qq{V} in \cs{mathaccentV} is just an indication that it takes
+%    five arguments. It is important that the name includes
+%    \texttt{mathaccent}, otherwise \cs{DeclareMathAccent} will balk at
+%    redefining one of our accent commands, for example when an
+%    alternative math font package is loaded.
+%    \begin{macrocode}
+\def\mathaccentV#1#2#3#4#5{%
+  \ifmmode
+    \gdef\macc@tmp{\macc@depth\@ne}%
+    \setbox\z@\hbox{%
+      \let\mathaccentV\macc@test
+      \let\use@mathgroup\@gobbletwo \let\select@group\@gobblethree
+      \frozen@everymath{}$#5$%
+    }%
+    \macc@tmp
+    \ifnum\macc@depth=\@ne
+      \global\let\macc@nucleus\@empty
+      \mathaccent"\accentclass@
+    \else
+      \@xp\macc@nested
+    \fi
+    #2#3#4{#5}%
+    \macc@nucleus
+  \else
+    \@xp\nonmatherr@\csname#1\endcsname
+  \fi
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\macc@test#1#2#3#4{\xdef\macc@tmp{\macc@tmp\advance\macc@depth\@ne}}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\macc@group{-1}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\macc@nested#1#2#3#4{%
+  \begingroup
+  \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
+  \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
+  \macc@set@skewchar\relax
+  \let\mathaccentV\macc@nested@a
+  \macc@nested@a\relax#1#2#3{#4}%
+  \endgroup
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\macc@palette\mathpalette
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\macc@nested@a#1#2#3#4#5{%
+%    \end{macrocode}
+%    This test saves some work that would otherwise be always repeated
+%    fourfold thanks to \cs{mathchoice}.
+%    \begin{macrocode}
+  \ifnum\macc@group=\mathgroup
+  \else \macc@set@skewchar\relax \edef\macc@group{\the\mathgroup}%
+  \fi
+  \mathchardef\macc@code "\accentclass@ #2#3#4\relax
+  \macc@palette\macc@a{#5}%
+}
+%    \end{macrocode}
+%
+%    The reason that \cs{macc@set@skewchar} takes an argument is so that
+%    it can serve as a direct substitute for \cs{math@egroup}, in
+%    addition to being used separately.
+%
+%    Setting a skewchar with this method works for symbols of variable
+%    mathgroup (class 7, letters and numbers) but not necessarily for
+%    special symbols like \cn{partial} or \cs{xi} whose mathgroup
+%    doesn't change; fortunately the most commonly used ones come from
+%    mathgroup one, which is the fall-back mathgroup for skewchar.
+%    \begin{macrocode}
+\def\macc@set@skewchar#1{%
+  \begingroup
+  \ifnum\mathgroup=\m@ne \let\@tempa\@ne
+  \else
+    \ifnum\skewchar\textfont\mathgroup=\m@ne \let\@tempa\@ne
+    \else \let\@tempa\mathgroup
+    \fi
+  \fi
+  \count@=\skewchar\textfont\@tempa
+  \advance\count@"7100
+  \edef\@tempa{\endgroup
+    \mathchardef\noexpand\macc@skewchar=\number\count@\relax}%
+  \@tempa
+  #1%
+}
+%    \end{macrocode}
+%
+%    Arg1 is math-style, arg2 is accent base object. We assume that math
+%    style doesn't change within the nested group of accents; this means
+%    we can set \cs{macc@style} only once and redefine \cs{macc@palette}
+%    to use it, in order to run \cs{mathchoice} only once instead of
+%    multiplying the calls exponentially as the nesting level increases.
+%    \begin{macrocode}
+\def\macc@a#1#2{%
+  \begingroup
+  \let\macc@style#1\relax
+  \def\macc@palette##1{##1\macc@style}%
+  \advance\macc@depth\m@ne
+  \ifnum\macc@depth=\z@
+    \gdef\macc@nucleus{#2}%
+%    \end{macrocode}
+%    Extra \cs{@empty} tokens are to prevent low-level \tex/ errors from
+%    the potential syntactic error that \cs{acc@check} checks for.
+%    \begin{macrocode}
+    \setbox\z@\hbox{$#1#2\@empty{}\macc@skewchar$}%
+    \setbox\tw@\hbox{$#1#2\@empty\macc@skewchar$}%
+    \dimen@\tw@\wd\tw@ \advance\dimen@-\tw@\wd\z@
+    \xdef\macc@kerna{\the\dimen@\relax}%
+    \setbox4\hbox{$#1#2\acc@check\@empty$}%
+    \global\setbox\@ne\hbox to\wd4{}%
+    \ht\@ne\ht4 \dp\@ne\dp4
+    \xdef\macc@kernb{\the\wd4\relax}%
+    \mathaccent\macc@code{\box\@ne\kern\macc@kerna}%
+  \else
+    \mathaccent\macc@code{\let\macc@adjust\@empty #1#2\@empty}%
+    \macc@adjust
+  \fi
+  \endgroup
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\macc@adjust{%
+  \dimen@\macc@kerna\advance\dimen@\macc@kernb
+  \kern-\dimen@
+}
+%    \end{macrocode}
+%
+%    The commands \cs{Hat}, \cs{Tilde}, \ldots, are supported as
+%    synonyms of \cs{hat}, \cs{tilde}, \ldots, for backward
+%    compatibility.
+%    \begin{macrocode}
+\def\Hat{\hat}
+\def\Check{\check}
+\def\Tilde{\tilde}
+\def\Acute{\acute}
+\def\Grave{\grave}
+\def\Dot{\dot}
+\def\Ddot{\ddot}
+\def\Breve{\breve}
+\def\Bar{\bar}
+\def\Vec{\vec}
+%    \end{macrocode}
+%
+%    This error message about math mode is used several times so we make
+%    an abbreviation for it.
+%    \begin{macrocode}
+\def\nonmatherr@#1{\@amsmath@err{\protect
+  #1 allowed only in math mode}\@ehd}
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Mods, continued fractions, etc.}
+%
+%    The commands \cn{bmod}, \cn{pmod}, \cn{pod}, \cn{mod} aren't
+%    currently robust. [mjd, 1994/09/05]
+%    \begin{macrocode}
+\renewcommand{\bmod}{\nonscript\mskip-\medmuskip\mkern5mu\mathbin
+  {\operator@font mod}\penalty900
+  \mkern5mu\nonscript\mskip-\medmuskip}
+\newcommand{\pod}[1]{\allowbreak
+  \if@display\mkern18mu\else\mkern8mu\fi(#1)}
+\renewcommand{\pmod}[1]{\pod{{\operator@font mod}\mkern6mu#1}}
+\newcommand{\mod}[1]{\allowbreak\if@display\mkern18mu
+  \else\mkern12mu\fi{\operator@font mod}\,\,#1}
+%    \end{macrocode}
+%
+%    Continued fractions. The optional arg l or r controls horizontal
+%    placement of the numerators. The |\kern-\nulldelimiterspace|
+%    is needed in the definition if we want the right-hand sides of the
+%    fraction rules to line up. The \cs{strut} keeps the numerator of
+%    a subsidiary cfrac from coming too close to the fraction rule above
+%    it.
+%    \begin{macrocode}
+\newcommand{\cfrac}[3][c]{{\displaystyle\frac{%
+  \strut\ifx r#1\hfill\fi#2\ifx l#1\hfill\fi}{#3}}%
+  \kern-\nulldelimiterspace}
+%    \end{macrocode}
+%
+%    \cn{overset} and \cn{underset} put symbols above, respectively
+%    below, a symbol that is not a \cs{mathop} and therefore does not
+%    naturally accept limits. \cs{binrel@@} uses information collected
+%    by \cs{binrel@} to make the resulting construction be of type
+%    mathrel or mathbin if the base symbol is either of those types.
+%    \begin{macrocode}
+\newcommand{\overset}[2]{\binrel@{#2}%
+  \binrel@@{\mathop{\kern\z@#2}\limits^{#1}}}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\newcommand{\underset}[2]{\binrel@{#2}%
+  \binrel@@{\mathop{\kern\z@#2}\limits_{#1}}}
+%    \end{macrocode}
+%
+%    \cn{sideset} allows placing `adscript' symbols at the four
+%    corners of a \cs{mathop}, \emph{in addition to} limits. Left-side
+%    adscripts go into arg \arg{1}, in the form |_{...}^{...}|, and
+%    right-side adscripts go into arg \arg{2}.
+%
+%    As currently written [mjd, 1995/01/21] this is pretty haphazard.
+%    In order to really make it work properly in full generality we'd
+%    have to read and measure the top and bottom limits and use
+%    mathchoice to always get the right mathstyle for each piece,
+%    etc., etc.
+%    \begin{macrocode}
+\newcommand{\sideset}[3]{%
+  \@mathmeasure\z@\displaystyle{#3}%
+%    \end{macrocode}
+%    Use a global box assignment here since the depth override is
+%    implicitly global. Then move the constructed box to a local box
+%    register (2) to ensure it won't get destroyed during the next two
+%    mathmeasure statements. This precaution may be more extreme than
+%    necessary in practice.
+%    \begin{macrocode}
+  \global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@
+  \setbox\tw@\box\@ne
+  \@mathmeasure4\displaystyle{\copy\tw@#1}%
+  \@mathmeasure6\displaystyle{#3\nolimits#2}%
+  \dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@
+  \hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}%
+}
+%    \end{macrocode}
+%
+% \begin{macro}{\smash}
+%    We add to the \cn{smash} command an optional argument
+%    denoting the part of the formula to be smashed.
+%    \begin{macrocode}
+\renewcommand{\smash}[1][tb]{%
+  \def\mb@t{\ht}\def\mb@b{\dp}\def\mb@tb{\ht\z@\z@\dp}%
+  \edef\finsm@sh{\csname mb@#1\endcsname\z@\z@ \box\z@}%
+  \ifmmode \@xp\mathpalette\@xp\mathsm@sh
+  \else \@xp\makesm@sh
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Extensible arrows}
+%
+%    The minus sign used in constructing these arrow fills is smashed so
+%    that superscripts above the arrows won't be too high. This
+%    primarily affects the \cn{xleftarrow} and \cn{xrightarrow} arrows.
+%
+%    \begin{macrocode}
+\mathchardef\std@minus\mathcode`\-\relax
+\mathchardef\std@equal\mathcode`\=\relax
+%    \end{macrocode}
+%    In case some alternative math fonts are loaded later:
+%    \begin{macrocode}
+\AtBeginDocument{%
+  \mathchardef\std@minus\mathcode`\-\relax
+  \mathchardef\std@equal\mathcode`\=\relax
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\relbar{\mathrel{\mathpalette\mathsm@sh\std@minus}}
+\def\Relbar{\mathrel\std@equal}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\arrowfill@#1#2#3#4{%
+  $\m@th\thickmuskip0mu\medmuskip\thickmuskip\thinmuskip\thickmuskip
+   \relax#4#1\mkern-7mu%
+   \cleaders\hbox{$#4\mkern-2mu#2\mkern-2mu$}\hfill
+   \mkern-7mu#3$%
+}
+\def\leftarrowfill@{\arrowfill@\leftarrow\relbar\relbar}
+\def\rightarrowfill@{\arrowfill@\relbar\relbar\rightarrow}
+\def\leftrightarrowfill@{\arrowfill@\leftarrow\relbar\rightarrow}
+\def\Leftarrowfill@{\arrowfill@\Leftarrow\Relbar\Relbar}
+\def\Rightarrowfill@{\arrowfill@\Relbar\Relbar\Rightarrow}
+\def\Leftrightarrowfill@{\arrowfill@\Leftarrow\Relbar\Rightarrow}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\overarrow@#1#2#3{\vbox{\ialign{##\crcr#1#2\crcr
+ \noalign{\nointerlineskip}$\m@th\hfil#2#3\hfil$\crcr}}}
+\renewcommand{\overrightarrow}{%
+  \mathpalette{\overarrow@\rightarrowfill@}}
+\renewcommand{\overleftarrow}{%
+  \mathpalette{\overarrow@\leftarrowfill@}}
+\newcommand{\overleftrightarrow}{%
+  \mathpalette{\overarrow@\leftrightarrowfill@}}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\underarrow@#1#2#3{%
+ \vtop{\ialign{##\crcr$\m@th\hfil#2#3\hfil$\crcr
+ \noalign{\nointerlineskip\kern1.3\ex@}#1#2\crcr}}}
+\newcommand{\underrightarrow}{%
+  \mathpalette{\underarrow@\rightarrowfill@}}
+\newcommand{\underleftarrow}{%
+  \mathpalette{\underarrow@\leftarrowfill@}}
+\newcommand{\underleftrightarrow}{%
+  \mathpalette{\underarrow@\leftrightarrowfill@}}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+%\newcommand{\xrightarrow}[2][]{\ext@arrow 0359\rightarrowfill@{#1}{#2}}
+\def\ext@arrow#1#2#3#4#5#6#7{%
+  \mathrel{\mathop{%
+%    \end{macrocode}
+%    Measure the superscript and subscript.
+%    \begin{macrocode}
+    \setbox\z@\hbox{#5\displaystyle}%
+    \setbox\tw@\vbox{\m@th
+      \hbox{$\scriptstyle\mkern#3mu{#6}\mkern#4mu$}%
+      \hbox{$\scriptstyle\mkern#3mu{#7}\mkern#4mu$}%
+      \copy\z@
+    }%
+    \hbox to\wd\tw@{\unhbox\z@}}%
+%    \end{macrocode}
+%    We don't want to place an empty subscript since that will produce
+%    too much blank space below the arrow.
+%    \begin{macrocode}
+  \limits
+    \@ifnotempty{#7}{^{\if0#1\else\mkern#1mu\fi
+                       #7\if0#2\else\mkern#2mu\fi}}%
+    \@ifnotempty{#6}{_{\if0#1\else\mkern#1mu\fi
+                       #6\if0#2\else\mkern#2mu\fi}}}%
+}
+%    \end{macrocode}
+%
+%    Some extensible arrows to serve as mathrels and taking
+%    sub/superscripts. These commands are robust because they take an
+%    optional argument.
+%    \begin{macrocode}
+\newcommand{\xrightarrow}[2][]{\ext@arrow 0359\rightarrowfill@{#1}{#2}}
+\newcommand{\xleftarrow}[2][]{\ext@arrow 3095\leftarrowfill@{#1}{#2}}
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Array-related environments}
+% \subsection{Remarks}
+%
+%    Because these environments can be nested within the equation
+%    structures that allow \cn{tag}, there is some cross-influence in
+%    the internal workings of the \cn{\\} command.
+%
+% \subsection{The \cn{substack} command}
+%
+%    The \cn{substack} command can be used to set subscripts
+%    and superscripts that consist of several lines. Usage:
+% \begin{verbatim}
+% X_{\substack{a=1\\b=2}}
+% \end{verbatim}
+%
+% \changes{v2.0}{1999/06/17}{Removed environment definitions: Sb, Sp}
+%
+% \begin{environment}{subarray}
+%    The \env{subarray} environment makes a small-size array suitable
+%    for use in a subscript or superscript. At the moment the supported
+%    arguments are not the full possibilities of \env{array} but only
+%    |c| or |l| for centered or left-aligned. And only one column.
+%    \begin{macrocode}
+\newenvironment{subarray}[1]{%
+%    \end{macrocode}
+%    Note: The predecessors of \env{subarray} (\env{Sb} and \env{Sp},
+%    inherited from \amstex/) used \cs{vbox} instead of \cs{vcenter}.
+%    But when a multiline subscript is placed in \cs{limits} position
+%    \cs{vcenter} is no worse than \cs{vbox}, and when it is placed
+%    in the \cs{nolimits} position (e.g., for an integral), \cs{vcenter}
+%    provides clearly better positioning than \cs{vbox}.
+%    \begin{macrocode}
+  \vcenter\bgroup
+%    \end{macrocode}
+%    Use \cs{Let@} to set the proper meaning of the \cn{\\} and \cn{\\*}
+%    commands. And restore the meaning of \cs{math@cr@@@} to \cs{cr}
+%    (see above) in case \env{subarray} is used inside one of the more
+%    complicated alignment macros where the meaning of \cs{math@cr@@@}
+%    is different. Similarly, call \cs{default@tag} to ensure that a
+%    line break here doesn't get an equation number!
+%    \begin{macrocode}
+  \Let@ \restore@math@cr \default@tag
+%    \end{macrocode}
+%    Set the line spacing to be the same as \cs{atop} (when \cs{atop}
+%    occurs in \cs{textstyle} or smaller), cf \textit{The \tex/book},
+%    Appendix G.
+%    \begin{macrocode}
+  \baselineskip\fontdimen10 \scriptfont\tw@
+  \advance\baselineskip\fontdimen12 \scriptfont\tw@
+  \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
+  \lineskiplimit\lineskip
+%    \end{macrocode}
+%    Start the \cs{vbox} \cs{halign} structure that encloses the
+%    contents. Notice that we never get \cs{scriptscriptstyle}. That
+%    would require a \cs{mathchoice} (ugh).
+%    \begin{macrocode}
+  \ialign\bgroup\ifx c#1\hfil\fi
+    $\m@th\scriptstyle##$\hfil\crcr
+}{%
+  \crcr\egroup\egroup
+}
+%    \end{macrocode}
+% \end{environment}
+%
+% \begin{macro}{\substack}
+%    The \cn{substack} command is just an abbreviation for the
+%    most common use of \env{subarray}.
+%    \begin{macrocode}
+\newcommand{\substack}[1]{\subarray{c}#1\endsubarray}
+%    \end{macrocode}
+% \end{macro}
+%
+% \subsection{Matrices}
+%
+% \begin{environment}{smallmatrix}
+%    \env{smallmatrix} is again an alignment, this time in a centered
+%    box. The opening incantations are basically the same as those in
+%    \cs{multilimits@}, followed by the alignment itself. A remark:
+%    the baselineskip (|9\ex@|) used in \amstex/ is too large for
+%    use in text with the usual baselineskip of $12$ or $13$ points; we
+%    change it here to |6\ex@| and also adjust the \cs{lineskip}
+%    and \cs{lineskiplimit} slightly to compensate. (MJD)
+%    \begin{macrocode}
+\newenvironment{smallmatrix}{\null\,\vcenter\bgroup
+  \Let@\restore@math@cr\default@tag
+  \baselineskip6\ex@ \lineskip1.5\ex@ \lineskiplimit\lineskip
+  \ialign\bgroup\hfil$\m@th\scriptstyle##$\hfil&&\thickspace\hfil
+  $\m@th\scriptstyle##$\hfil\crcr
+}{%
+  \crcr\egroup\egroup\,%
+}
+%    \end{macrocode}
+% \end{environment}
+%
+% \begin{environment}{matrix}
+%    The \env{matrix} environment is just an \env{array} that provides
+%    up to ten centered columns, so that users don't have to give the
+%    col-spec argument explicitly---unless they want some of the columns
+%    noncentered, that is. The maximum number of columns is actually not
+%    fixed at ten but given by the counter |MatrixCols|, and can
+%    therefore be increased by changing that counter.
+%
+%    The extra space of \cn{arraycolsep} that \env{array} adds on each
+%    side is a waste so we remove it here (perhaps we should instead
+%    remove it from \env{array} in general, but that's a harder task).
+%
+%    TODO: Think about re-implementing \cn{matrix} to get rid of the
+%    \cs{c@MatrixCols} limit and have hard-wired preamble that doesn't
+%    have to be rebuilt each time.
+%
+%    We must use \cn{renewenvironment} for \env{matrix} and
+%    \env{pmatrix} because \latex/ doesn't kill the definitions found in
+%    \fn{plain.tex}, even though it probably should because of their
+%    foreign syntax.
+%    \begin{macrocode}
+\renewenvironment{matrix}{%
+  \matrix@check\matrix\env@matrix
+}{%
+  \endarray \hskip -\arraycolsep
+}
+%    \end{macrocode}
+% \end{environment}
+%
+%  \begin{macro}{\env@matrix}
+%
+%    \begin{macrocode}
+\def\env@matrix{\hskip -\arraycolsep
+  \let\@ifnextchar\new@ifnextchar
+  \array{*\c@MaxMatrixCols c}}
+%    \end{macrocode}
+%  \end{macro}
+%
+% \begin{macro}{\c@MaxMatrixCols}
+%    \begin{macrocode}
+\newcount\c@MaxMatrixCols \c@MaxMatrixCols=10
+%    \end{macrocode}
+% \end{macro}
+%
+%  \begin{macro}{\matrix@check}
+%    For various reasons, authors sometimes use the Plain \TeX{} form of
+%    \cn{matrix} or \cn{pmatrix} in \LaTeX{} documents. If they later
+%    add an invocation of the \pkg{amsmath} package to their document,
+%    the Plain \TeX{} syntax would lead to rather unintelligible error
+%    messages. The \cs{matrix@check} function does some checking to
+%    forestall that problem.
+%    \begin{macrocode}
+\def\matrix@check#1{%
+  \@xp\ifx\csname\@currenvir\endcsname#1%
+  \else\matrix@error#1%
+%    \end{macrocode}
+%    This error recovery is not that good but is better than the
+%    infinite loop that can result from calling \cs{array} without a
+%    matching \cs{endarray}. (The array setup leaves \cs{par} empty.)
+%    \begin{macrocode}
+    \@xp\@gobble
+  \fi
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+%  \begin{macro}{\matrix@error}
+%
+%    \begin{macrocode}
+\def\matrix@error#1{%
+  \@amsmath@err{%
+Old form `\string#1' should be \string\begin{\@xp\@gobble\string#1}%
+  }{%
+`\string#1{...}' is old Plain-TeX syntax whose use is
+ill-advised in LaTeX.%
+  }%
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+%    \begin{macrocode}
+\renewenvironment{pmatrix}{%
+  \left(%
+  \matrix@check\pmatrix\env@matrix
+}{
+  \endmatrix\right)%
+}
+\newenvironment{bmatrix}{\left[\env@matrix}{\endmatrix\right]}
+\newenvironment{Bmatrix}{%
+  \left\lbrace\env@matrix
+}{%
+  \endmatrix\right\rbrace
+}
+\newenvironment{vmatrix}{\left\lvert\env@matrix}{\endmatrix\right\rvert}
+\newenvironment{Vmatrix}{\left\lVert\env@matrix}{\endmatrix\right\rVert}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\hdots\@ldots
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\newcommand{\hdotsfor}[1]{%
+  \ifx[#1\@xp\shdots@for\else\hdots@for\@ne{#1}\fi}
+\newmuskip\dotsspace@
+\def\shdots@for#1]{\hdots@for{#1}}
+\def\hdots@for#1#2{\multicolumn{#2}c%
+  {\m@th\dotsspace@1.5mu\mkern-#1\dotsspace@
+   \xleaders\hbox{$\m@th\mkern#1\dotsspace@.\mkern#1\dotsspace@$}%
+           \hfill
+   \mkern-#1\dotsspace@}%
+   }
+%    \end{macrocode}
+%
+% \begin{environment}{cases}
+%    The easiest way to produce the \env{cases} environment is to base
+%    it on the \env{array} environment. We must use
+%    \cn{renewenvironment} to override the definition of \cn{cases} that
+%    \latex/ (unwisely) leaves in place from \fn{plain.tex}.
+%    \begin{macrocode}
+\renewenvironment{cases}{%
+  \matrix@check\cases\env@cases
+}{%
+  \endarray\right.%
+}
+\def\env@cases{%
+  \let\@ifnextchar\new@ifnextchar
+  \left\lbrace
+  \def\arraystretch{1.2}%
+  \array{@{}l@{\quad}l@{}}%
+}
+%    \end{macrocode}
+% \end{environment}
+%
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Equation sub-numbering}
+%
+%    \begin{macrocode}
+\newcounter{parentequation}% Counter for ``parent equation''.
+%    \end{macrocode}
+%
+%    We can't assume \cs{ignorespacesafterend} is defined since it was
+%    not there in the earliest releases of \latex/ 2e. And we need to
+%    include the \cs{global} for the same reason.
+%    \begin{macrocode}
+\@ifundefined{ignorespacesafterend}{%
+  \def\ignorespacesafterend{\global\@ignoretrue}%
+}{}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\newenvironment{subequations}{%
+%    \end{macrocode}
+%    Before sending down the `equation' counter to the subordinate
+%    level, add 1 using standard \cn{refstepcounter}.
+%    \begin{macrocode}
+  \refstepcounter{equation}%
+%    \end{macrocode}
+%    Define \cn{theparentequation} equivalent to current
+%    \cn{theequation}. \cn{edef} is necessary to expand the current
+%    value of the equation counter. This might in rare cases cause
+%    something to blow up, in which case the user needs to add
+%    \cn{protect}.
+%    \begin{macrocode}
+  \protected@edef\theparentequation{\theequation}%
+  \setcounter{parentequation}{\value{equation}}%
+%    \end{macrocode}
+%    And set the equation counter to 0, so that the normal incrementing
+%    processes in the various equation environments will produce the
+%    desired results.
+%    \begin{macrocode}
+  \setcounter{equation}{0}%
+  \def\theequation{\theparentequation\alph{equation}}%
+  \ignorespaces
+}{%
+  \setcounter{equation}{\value{parentequation}}%
+  \ignorespacesafterend
+}
+%    \end{macrocode}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Equation numbering}
+%
+%    In the multiline equation environments provided here, the task
+%    of equation numbering is linked to the task of line breaking
+%    in the sense that it is the \cn{\\} command that marks where an
+%    equation number for the current line will be processed and added to
+%    the page.
+%
+%    Provide a convenient way to specify that equations should be
+%    numbered within sections.
+%    \begin{macrocode}
+\newcommand{\numberwithin}[3][\arabic]{%
+  \@ifundefined{c@#2}{\@nocounterr{#2}}{%
+    \@ifundefined{c@#3}{\@nocnterr{#3}}{%
+      \@addtoreset{#2}{#3}%
+      \@xp\xdef\csname the#2\endcsname{%
+        \@xp\@nx\csname the#3\endcsname .\@nx#1{#2}}}}%
+}
+%    \end{macrocode}
+%
+%    To make references to equation numbers easier, we provide
+%    \cn{eqref}.  We almost don't need \cn{textup}, except that
+%    \cs{tagform@} doesn't supply the italic correction.
+%    \begin{macrocode}
+\newcommand{\eqref}[1]{\textup{\tagform@{\ref{#1}}}}
+%    \end{macrocode}
+%
+% \subsection{Preliminary macros}
+%
+%    The following macros implement the \latex/ syntax for the
+%    \cn{\\} command, i.e. the possibility to add an asterisk to
+%    inhibit a page break, or an optional argument to denote additional
+%    vertical space. They are modelled more or less after the
+%    corresponding macros for \latex/'s \env{eqnarray} and \env{array}
+%    environments.
+%
+%    [We can perhaps use the eqnarray mechanism if we change it so that
+%    it also uses \cs{openup}.]
+%
+% \begin{macro}{\dspbrk@lvl}
+%    We begin by defining the \cs{dspbrk@lvl} counter. This counter
+%    records the desirability of a break after the current row, as a
+%    number between $0$ and $4$. Its default value is $-1$ meaning that
+%    no explicit \cn{displaybreak} command was given, and the default
+%    \cs{interdisplaylinepenalty} is to be used.
+%    \begin{macrocode}
+\newcount\dspbrk@lvl
+\dspbrk@lvl=-1
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\interdisplaylinepenalty}
+%    We set the \cs{interdisplaylinepenalty} to $10000$.
+%    \begin{macrocode}
+\interdisplaylinepenalty\@M
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\allowdisplaybreaks}
+%    The \cn{allowdisplaybreaks} command. Since this is intended for use
+%    outside displayed formulas (typically in the preamble), it does not
+%    need to use \cs{new@ifnextchar}.
+%    \begin{macrocode}
+\newcommand{\allowdisplaybreaks}[1][4]{%
+  \interdisplaylinepenalty\getdsp@pen{#1}\relax
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\getdsp@pen}
+%    Modelled after \latex/'s \cs{@getpen}.  We use higher numbers
+%    than would normally be provided by \cs{@lowpenalty},
+%    \cs{@medpenalty}, and \cs{@highpenalty}, since display
+%    breaks are almost always less desirable.
+%    \begin{macrocode}
+\def\getdsp@pen#1{%
+  \ifcase #1\@M \or 9999 \or 6999 \or 2999 \or \z@\fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\displaybreak}
+% \begin{macro}{\dspbrk@}
+% \begin{macro}{\dspbrk@context}
+% \begin{macro}{\nogood@displaybreak}
+%    For breaks in a certain row of a alignment.
+%    \begin{macrocode}
+\newcommand{\displaybreak}{\new@ifnextchar[\dspbrk@{\dspbrk@[4]}}
+\chardef\dspbrk@context=\sixt@@n
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\dspbrk@[#1]{%
+  \ifmeasuring@
+  \else
+    \ifcase\dspbrk@context % case 0 --- OK
+      \global\dspbrk@lvl #1\relax
+    \or                    % case 1 --- inside a box
+      \nogood@displaybreak
+    \else                  % other cases --- outside of a display
+      \@amsmath@err{\Invalid@@\displaybreak}\@eha
+    \fi
+  \fi
+}
+%    \end{macrocode}
+%
+%    This is the value of \cn{displaybreak} when it occurs inside some
+%    structure where it will not work.
+%    \begin{macrocode}
+\def\nogood@displaybreak{%
+  \@amsmath@err{\protect
+\displaybreak\space cannot be applied here}%
+{One of the enclosing environments creates an
+  unbreakable box\MessageBreak
+(e.g., split, aligned, gathered, ...).}%
+}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\math@cr}
+%    The macro \cs{math@cr} ends a row inside one of the equation
+%    environments, i.e., this is the internal name of the \cn{\\}
+%    commands in these environments. As usual for this kind of macro
+%    inside of alignments we insert a special brace into \tex/'s input
+%    stream. The initial \cs{relax} is needed to trigger entry into the
+%    \textit{u} template of the current column if the author ended the
+%    current row with an empty column (i.e., the mathcr was immediately
+%    preceded by an ampersand).
+%    \begin{macrocode}
+\def\math@cr{\relax\iffalse{\fi\ifnum0=`}\fi
+%    \end{macrocode}
+%    The first step is now to check whether an asterisk follows.
+%    \cs{@eqpen} is used to hold the penalty value to be put on
+%    the vertical list.
+%    Then we call up \cs{math@cr@} which performs the next step.
+%    If an asterisk is read page breaking is inhibited.
+%    \begin{macrocode}
+  \@ifstar{\global\@eqpen\@M\math@cr@}%
+%    \end{macrocode}
+%    Otherwise we have to check the \cs{dspbrk@lvl} value.
+%    \begin{macrocode}
+          {\global\@eqpen
+             \ifnum\dspbrk@lvl <\z@ \interdisplaylinepenalty
+              \else -\@getpen\dspbrk@lvl \fi
+           \math@cr@}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\math@cr@}
+%    The purpose of \cs{math@cr@} is to check whether an optional
+%    argument follows.  If not it provides \cs{z@} as default
+%    value.
+%    \begin{macrocode}
+\def\math@cr@{\new@ifnextchar[\math@cr@@{\math@cr@@[\z@]}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\math@cr@@}
+%    \cs{math@cr@@} closes the special brace opened in
+%    \cs{math@cr}, and calls \cs{math@cr@@@} which is supposed
+%    the `real' row ending command.  The meaning of this macro depends
+%    on the environment in which it is used.
+%    \begin{macrocode}
+\def\math@cr@@[#1]{\ifnum0=`{\fi \iffalse}\fi\math@cr@@@
+%    \end{macrocode}
+%    Finally we put the additional space onto the vertical list.
+%    \begin{macrocode}
+  \noalign{\vskip#1\relax}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\Let@}
+%    \cs{Let@} is called by all environments where \cn{\\}
+%    ends a row of an alignment.
+%    \begin{macrocode}
+\def\Let@{\let\\\math@cr}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\restore@math@cr}
+%    We mentioned already that the exact meaning of \cs{math@cr@@@}
+%    depends on the current environment.  Since it is often a simple
+%    \cs{cr} we provide \cs{restore@math@cr} to reset it.
+%    \begin{macrocode}
+\def\restore@math@cr{\def\math@cr@@@{\cr}}
+%    \end{macrocode}
+%    This is also the default case.
+%    \begin{macrocode}
+\restore@math@cr
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\intertext}
+% \begin{macro}{\intertext@}
+%    The \cn{intertext} command is used for inserting text between the
+%    rows of an alignment. It might better be done as an environment,
+%    but the \cs{begingroup} from \cn{begin} would cause the
+%    \cs{noalign} to fail.
+%    \begin{macrocode}
+\newcommand{\intertext}{\@amsmath@err{\Invalid@@\intertext}\@eha}
+%    \end{macrocode}
+%    \cs{intertext@} is called by all environments that allow the use of
+%    the \cn{intertext} command.
+%    \begin{macrocode}
+\def\intertext@{%
+  \def\intertext##1{%
+%    \end{macrocode}
+%    If current mode is not vmode, the most likely reason is that the
+%    writer forgot the \cn{\\} that is supposed to precede
+%    \cn{intertext}. All right, then, let's try adding it our ownself.
+%    But, to be slightly careful: \cn{\\} does a futurelet, and it's
+%    slightly dangerous to allow a letted token to barge around loose in
+%    our internal code when it has been let to a conditional token
+%    like \cs{fi}. So let's interpose something in front of the \cs{fi}
+%    for the futurelet to take instead. (And careful again: it has to be
+%    something evanescent, not (e.g.) \cs{relax} which would cause the
+%    next halign cell to fire up and keep \cs{noalign} from working.)
+%    \begin{macrocode}
+    \ifvmode\else\\\@empty\fi
+    \noalign{%
+      \penalty\postdisplaypenalty\vskip\belowdisplayskip
+      \vbox{\normalbaselines
+%    \end{macrocode}
+%    We need to do something extra if the outside environment is a list
+%    environment. I don't see offhand an elegant way to test ``are we
+%    inside any list environment'' that is both easy and reliable (for
+%    example, checking for zero \cs{@totalleftmargin} wouldn't catch the
+%    case where \cs{@totalleftmargin} is zero but \cs{linewidth} is less
+%    than \cs{columnwidth}), so it seems to me checking \cs{linewidth}
+%    is the best practical solution.
+%    \begin{macrocode}
+        \ifdim\linewidth=\columnwidth
+        \else \parshape\@ne \@totalleftmargin \linewidth
+        \fi
+        \noindent##1\par}%
+      \penalty\predisplaypenalty\vskip\abovedisplayskip%
+    }%
+}}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% \subsection{Implementing tags and labels}
+%
+%    In this section we describe some of the macros needed to make the
+%    \cn{tag} command work in various places. We start by defining a
+%    help text to be used when a \cn{tag} command is used somewhere
+%    it should not appear.
+%
+% \begin{macro}{\tag@help}
+%    This is the default error help text provided when \cn{tag}
+%    generates an error message.
+%    Note that \cs{newhelp} generates a control sequence name
+%    from the string given as its argument so that a leading
+%    backslash is provided automatically.
+%    \begin{macrocode}
+\newhelp\tag@help
+  {tag cannot be used at this point.\space
+   If you don't understand why^^Jyou should consult
+   the documentation.^^JBut don't worry: just continue, and I'll
+   forget what happened.}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\gobble@tag}
+%    This macro is to be used when \cn{tag} should silently
+%    skip its argument.
+%    It is made to handle the \qc{\*}-form of \cn{tag} as well.
+%    \begin{macrocode}
+\def\gobble@tag{\@ifstar\@gobble\@gobble}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\invalid@tag}
+%    \cs{invalid@tag} is a macro that should be used whenever
+%    \cn{tag} appears in an illegal place.
+%    It sets up \cs{tag@help} (as defined above) as help message,
+%    prints its argument as error message, and skips \cn{tag}'s
+%    argument.
+%    \begin{macrocode}
+\def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\dft@tag}
+% \begin{macro}{\default@tag}
+%    \cs{dft@tag} provides a convenient way to disallow the
+%    use of \cn{tag} at certain points.
+%    One simply has to write
+%    \begin{verbatim}
+%\let\tag\dft@tag
+%    \end{verbatim}
+%    and the \cn{tag} command will produce an error message,
+%    with a suitable error help text, and discard its argument.
+%    \begin{macrocode}
+\def\dft@tag{\invalid@tag{\string\tag\space not allowed here}}
+%    \end{macrocode}
+%    Since this is used several times we provide an abbreviation for
+%    it.
+%    \begin{macrocode}
+\def\default@tag{\let\tag\dft@tag}
+%    \end{macrocode}
+%    Since this is also the default, i.e.\ the \cn{tag} command
+%    should not be used except in special places, we issue a
+%    \cs{default@tag} command.
+%    \begin{macrocode}
+\default@tag
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%    Now that we have taken care of the case that \cn{tag} is not
+%    allowed we will provide some macros to process tags appropriately.
+%    As the user documentation states, a \cn{tag} command (without
+%    the asterisk typesets its argument according to the document
+%    styles' conventions, whereas a \cn{tag*} command typesets its
+%    argument exactly as given. We define therefore the following
+%    interface:
+%
+% \begin{macro}{\maketag@@}
+% \begin{macro}{\maketag@@@}
+% \begin{macro}{\tagform@}
+%    \cn{tag} is supposed to call \cs{maketag@@} which checks
+%    whether an asterisk follows. If this is the case it calls up
+%    \cs{maketag@@@} which sets its argument `as is'. Otherwise
+%    \cs{tagform@} is called to do the job. (This macro is to be
+%    defined appropriately by the document style.)
+%    \begin{macrocode}
+\def\maketag@@{\@ifstar\maketag@@@\tagform@}
+%    \end{macrocode}
+%    We define \cs{maketag@@@} to use the normal font of the document
+%    text (since this is the usual practice for numbering of document
+%    elements) and to put a box around the tag. Furthermore we use
+%    \cs{m@th} for exceptional cases where the tag involves a
+%    superscript or some such math. (Probably from an explicit use of
+%    \cs{tag*} rather than from the automatic numbering.)
+%    \begin{macrocode}
+\def\maketag@@@#1{\hbox{\m@th\normalfont#1}}
+%    \end{macrocode}
+%    We use the following default definition for \cs{tagform@}
+%    that puts only parentheses around the tag.
+%    \begin{macrocode}
+\def\tagform@#1{\maketag@@@{(\ignorespaces#1\unskip\@@italiccorr)}}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%    We need to insinuate \cs{tagform@} into \cs{@eqnnum} in case
+%    \env{eqnarray} is used (probably in a document that was originally
+%    written without use of the \pkg{amsmath} package).
+%    \begin{macrocode}
+\iftagsleft@
+  \def\@eqnnum{\hbox to1sp{}\rlap{\normalfont\normalcolor
+    \hskip -\displaywidth\tagform@\theequation}}
+\else
+  \def\@eqnnum{{\normalfont\normalcolor \tagform@\theequation}}
+\fi
+%    \end{macrocode}
+%
+% \begin{macro}{\thetag}
+%    Sometimes one needs to set a literal tag according to the rules of
+%    the document style. To achieve this we provide the \cn{thetag}
+%    command. It typesets its argument by calling \cs{tagform@} on
+%    it.
+%    \begin{macrocode}
+\newcommand{\thetag}{\leavevmode\tagform@}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\df@tag}
+% \begin{macro}{\make@df@tag}
+% \begin{macro}{\make@df@tag@@}
+% \begin{macro}{\make@df@tag@@@}
+%    Sometimes it is necessary for a \cn{tag} command to store a tag
+%    in a safe place and to process it later, e.g., for a tag in a row
+%    of an alignment where the tag can only be typeset when the
+%    \cn{\\} at the end of the row was seen. Such a tag is stored in
+%    the macro \cs{df@tag} (for `deferred tag'). For this purpose we
+%    provide the \cs{make@df@tag} macro. It is built very similar to
+%    the \cs{maketag@@} macro above.
+%    \begin{macrocode}
+\let\df@tag\@empty
+\def\make@df@tag{\@ifstar\make@df@tag@@\make@df@tag@@@}
+%    \end{macrocode}
+%    \cs{make@df@tag} sets \cs{@currentlabel} and defines
+%    \cs{df@tag} appropriately.
+%
+%    To simplify the task of tracking \cs{tag} and \cs{label}
+%    commands inside math display environments, we defer \cs{label}
+%    commands until the tag is typeset, similar to the way that
+%    \cs{tag}s themselves are deferred.  This allows arbitrary
+%    placement of \cs{label} and \cs{tag} commands and also means we
+%    only increment the \cs{equation} counter when we really need to,
+%    thus avoiding the \cs{setb@ck} nonsense that used to be required.
+%
+%    \begin{macrocode}
+\def\make@df@tag@@#1{%
+  \gdef\df@tag{\maketag@@@{#1}\def\@currentlabel{#1}}}
+%    \end{macrocode}
+%    Autogenerated number:
+%    \begin{macrocode}
+\def\make@df@tag@@@#1{\gdef\df@tag{\tagform@{#1}%
+  \toks@\@xp{\p@equation{#1}}\edef\@currentlabel{\the\toks@}}}
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%    \end{macro}
+%    \end{macro}
+%
+%    \begin{macro}{\ltx@label}
+%    \begin{macro}{\label@in@display}
+%    \begin{macro}{\df@label}
+%    Next, we store the default definition of \cs{label} in
+%    \cs{ltx@label} and then define a new version of \cs{label} for use
+%    in math display environments. \cs{label@in@display} merely issues a
+%    warning message if there is already a pending label (which will be
+%    discarded) and then stores the label in \cs{df@label}.
+%    \begin{macrocode}
+\let\ltx@label\label
+%
+\def\label@in@display{%
+    \ifx\df@label\@empty\else
+        \@amsmath@err{Multiple \string\label's:
+            label '\df@label' will be lost}\@eha
+    \fi
+    \gdef\df@label
+}
+%    \end{macrocode}
+%    In case there is an enumerate inside a minipage inside an equation,
+%    we need to reset \cn{label} to its normal value:
+%    \begin{macrocode}
+\toks@\@xp{\@arrayparboxrestore \let\label\ltx@label}%
+\edef\@arrayboxrestore{\the\toks@}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\df@label\@empty
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%    \end{macro}
+%
+%    \begin{macro}{\make@display@tag}
+%    Now we define a macro to process \cs{tag} and \cs{label} commands
+%    in various display environments.  If the |@eqnsw| switch is set,
+%    then we should supply an equation number; otherwise, if the
+%    |@tag| switch is set, we should use the tag stored in
+%    \cs{df@tag}.  Finally, we process any pending \cs{label}s.
+%
+%    TODO: Arguably, \cs{make@display@tag} should issue a warning
+%    message if there is a \cs{label} but neither a tag nor an
+%    equation number.  Also, it would probably be worthwhile to
+%    explore whether \cs{iftag@} could be done away with and replaced
+%    by checks to see if \cs{df@tag} is empty or not.
+%    \begin{macrocode}
+\def\make@display@tag{%
+  \if@eqnsw \incr@eqnum \print@eqnum
+  \else \iftag@ \df@tag \global\let\df@tag\@empty \fi
+  \fi
+%    \end{macrocode}
+%    Need to check the \cs{ifmeasuring@} flag otherwise the \cs{write}
+%    node from \cn{label} might be discarded in a temp box and clearing
+%    \cs{df@label} will keep it from being reiterated on the real
+%    typesetting pass.
+%    \begin{macrocode}
+  \ifmeasuring@
+  \else
+    \ifx\df@label\@empty
+    \else
+      \@xp\ltx@label\@xp{\df@label}%
+      \global\let\df@label\@empty
+    \fi
+  \fi
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    Now we define the special versions of \cn{tag} used within the
+%    \env{align} environments.
+%
+% \begin{macro}{\tag@in@align}
+%    The \cn{tag} command may only appear once in a row of
+%    an alignment.  Therefore we first check the switch |tag@|
+%    that is set to false at the begin of every row.
+%    If this switch is true a \cn{tag} was already given in this
+%    row and we define \cs{next@} to expand to a call to
+%    \cs{invalid@tag}.
+%    \begin{macrocode}
+\def\tag@in@align{%
+    \relax
+    \iftag@
+        \DN@{\invalid@tag{Multiple \string\tag}}%
+    \else
+%    \end{macrocode}
+%    Otherwise we set the |tag@| switch.  But there is more to
+%    be done: we must also prevent the automatic generation of a
+%    tag.  Therefore we also reset the |@eqnsw|.
+%    \begin{macrocode}
+    \global\tag@true
+%    \end{macrocode}
+%    Changed to \cs{nonumber}, since that seems to be all that's
+%    required.---dmj, 1994/12/21
+%    \begin{macrocode}
+    \nonumber
+%    \end{macrocode}
+%    Within a row of an \env{align} environment the \cn{tag}
+%    command must not typeset the tag immediately since its
+%    position can be determined only later.
+%    Therefore we use the \cs{make@df@tag} macro defined
+%    earlier.
+%    Finally we call \cs{next@} to process the argument
+%    that follows.
+%    \begin{macrocode}
+        \let\next@\make@df@tag
+    \fi
+    \next@
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\raisetag}
+%    Usage: \cn{raisetag} \meta{dimen}
+%
+%    This will modify the vertical placement of the tag of the current
+%    equation by \meta{dimen}. Note that according to the current uses
+%    of \cs{raise@tag} in e.g., \cs{place@tag@gather}, no adjustment
+%    occurs if the tag falls in its normal position; i.e., \cn{raisetag}
+%    has no effect unless the tag has already been shifted off-line.
+%
+%    \begin{macrocode}
+\newcommand{\raisetag}[1]{\skip@#1\relax
+  \xdef\raise@tag{\vskip\iftagsleft@\else-\fi\the\skip@\relax}%
+}
+%    \end{macrocode}
+%    \cn{raise@tag} will be reemptied at the beginning of each equation,
+%    which might occur at a |\begin{xxx}| or \cn{\\}.
+%    \begin{macrocode}
+\let\raise@tag\@empty
+%    \end{macrocode}
+%    \end{macro}
+%
+% \begin{macro}{\notag}
+%    For consistency we provide \cn{notag}, equivalent to
+%    \cn{nonumber}. The alternative would have been to rename
+%    \cn{tag} as \cn{number} to go along with \cn{nonumber},
+%    but of course \cs{number} is a \tex/ primitive that should not
+%    be redefined.
+%    \begin{macrocode}
+\newcommand{\notag}{\nonumber}
+%    \end{macrocode}
+% \end{macro}
+%
+%  \begin{macro}{\nonumber}
+%    Need to add some additional code to \cn{nonumber} to deal with some
+%    complications related to nested environments.
+%    \begin{macrocode}
+\renewcommand{\nonumber}{%
+  \if@eqnsw
+    \ifx\incr@eqnum\@empty \addtocounter{equation}\m@ne \fi
+  \fi
+  \let\print@eqnum\@empty \let\incr@eqnum\@empty
+  \global\@eqnswfalse
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\print@eqnum{\tagform@\theequation}
+\def\incr@eqnum{\refstepcounter{equation}\let\incr@eqnum\@empty}
+%    \end{macrocode}
+%  \end{macro}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Multiline equation environments}
+%
+% \subsection{Remarks}
+%
+%    In late 1994 David M. Jones did a thorough overhaul of these
+%    environments so that the number placement and a few other aspects
+%    are substantially improved over the original versions that were
+%    ported essentially unchanged from \fn{amstex.tex} in 1989. Most of
+%    the commentary in this section is DMJ's, and comments of any
+%    significance that I added are marked by my initials and date
+%    [mjd, 1995/01/11].
+%
+% \subsection{Preliminaries}
+%
+% \begin{macro}{\ifinalign@}
+% \begin{macro}{\ifingather@}
+%    We define two switches that are set to true in certain
+%    alignments: |inalign@| and |ingather@| inside of
+%    the \env{align} and \env{gather} environments.
+%    These switches are needed to control certain actions that
+%    depend on the surrounding conditions, more specifically:
+%    on the setting already done by the surrounding environments.
+%    \begin{macrocode}
+\newif\ifinalign@
+\newif\ifingather@
+%    \end{macrocode}
+% \begin{histnote}
+% Removed the \cs{ifinany@} test [mjd,1999/06/28] since it was mainly
+% used for the purpose now handled by \cs{spread@equation}.
+% \changes{v1.2g}{1999/06/28}{Removed ifinany@}
+% \end{histnote}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\@arrayparboxrestore}
+%    Here we must reset a few additional parameters.
+%    \begin{macrocode}
+\@xp\def\@xp\@arrayparboxrestore\@xp{\@arrayparboxrestore
+  \ingather@false\inalign@false \default@tag
+  \let\spread@equation\@spread@equation
+  \let\reset@equation\@empty
+  \def\print@eqnum{\tagform@\theequation}%
+  \def\incr@eqnum{\refstepcounter{equation}\let\incr@eqnum\@empty}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\iftag@}
+%    The switch |tag@| is set to false at the beginning of every
+%    row and set to true by a \cn{tag} command.
+%    This allows us to check whether there is more than one tag on
+%    a row.
+%    \begin{macrocode}
+\newif\iftag@
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\ifst@rred}
+%    The switch |st@rred| is set to true by all starred
+%    environments
+%    and set to false by the unstarred versions.
+%    \begin{macrocode}
+\newif\ifst@rred
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\ifmeasuring@}
+%    All display environments get typeset twice---once during a
+%    ``measuring'' phase and then again during a ``production'' phase;
+%    \cs{ifmeasuring@} will be used to determine which case we're in,
+%    so we can take appropriate action.
+%
+%    \begin{macrocode}
+\newif\ifmeasuring@
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macro}{\ifshifttag@}
+%    \cs{ifshifttag@} is used by \env{gather} to communicate between
+%    \cs{calc@shift@gather} and \cs{place@tag@gather} whether an
+%    equation tag should be shifted to a separate line.  It's also
+%    used by \env{multline}.
+%    \begin{macrocode}
+\newif\ifshifttag@
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\row@}
+%    \begin{macrocode}
+\newcount\row@
+%    \end{macrocode}
+%    \end{macro}
+%
+% \begin{macro}{\column@}
+%    The counter \cs{column@} is used by the alignment macros to
+%    keep track of the current column.
+%
+%    \begin{macrocode}
+\newcount\column@
+%    \end{macrocode}
+% \end{macro}
+%
+%  \begin{macro}{\column@plus}
+%    \cs{\column@plus} is a useful abbreviation.
+%    \begin{macrocode}
+\def\column@plus{%
+    \global\advance\column@\@ne
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+%    \begin{macro}{\maxfields@}
+%    \begin{macrocode}
+\newcount\maxfields@
+%    \end{macrocode}
+%    \end{macro}
+%
+%  \begin{macro}{\add@amp}
+%  \begin{macro}{\add@amps}
+%    \begin{macrocode}
+\def\add@amp#1{\if m#1&\@xp\add@amp\fi}
+\def\add@amps#1{%
+    \begingroup
+    \count@#1\advance\count@-\column@
+    \edef\@tempa{\endgroup
+      \@xp\add@amp\romannumeral\number\count@ 000q}%
+    \@tempa
+}
+%    \end{macrocode}
+%  \end{macro}
+%  \end{macro}
+%
+% \begin{macro}{\andhelp@}
+%    The help text stored in \cs{andhelp@} is used for errors
+%    generated by too many \qc{\&} characters in a row.
+%    \begin{macrocode}
+\newhelp\andhelp@
+{An extra & here is so disastrous that you should probably exit^^J
+and fix things up.}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macro}{\eqnshift@}
+%    \cs{eqnshift@} is used by \env{align} and \env{gather} as the
+%    indentation of the lines of the environment from the left margin.
+%    \begin{macrocode}
+\newdimen\eqnshift@
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\alignsep@}
+%    \begin{macrocode}
+\newdimen\alignsep@
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\tagshift@}
+%    \begin{macrocode}
+\newdimen\tagshift@
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\mintagsep}
+%    \cs{mintagsep} is the minimum allowable separation between an
+%    equation and its tag.  We set it to half a quad in
+%    \cs{textfont}2, which is \tex/'s built-in value.
+%    \begin{macrocode}
+\newcommand{\mintagsep}{.5\fontdimen6\textfont\tw@}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\minalignsep}
+%    This should probably be a skip register [mjd,1999/06/18]
+%    \begin{macrocode}
+\newcommand{\minalignsep}{10pt}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\tagwidth@}
+%    \begin{macrocode}
+\newdimen\tagwidth@
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\totwidth@}
+%    \begin{macrocode}
+\newdimen\totwidth@
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\lineht@}
+%    The dimen register \cs{lineht@} is used to keep track of the
+%    height (or depth, if tags are on the right) of a row in an
+%    alignment.
+%    \begin{macrocode}
+\newdimen\lineht@
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macro}{\tag@width}
+%    \begin{macro}{\savetaglength@}
+%    \begin{macro}{\shift@tag}
+%    \begin{macro}{\tag@shifts}
+%    \begin{macrocode}
+\def\tag@width#1{%
+    \ifcase\@xp#1\tag@lengths\fi
+}
+
+\def\savetaglength@{%
+    \begingroup
+        \let\or\relax
+        \xdef\tag@lengths{\tag@lengths\or \the\wdz@}%
+    \endgroup
+}
+
+\def\shift@tag#1{%
+    \ifcase\@xp#1\tag@shifts\fi\relax
+}
+
+\let\tag@shifts\@empty
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%    \end{macro}
+%    \end{macro}
+%
+%    \begin{macro}{\saveshift@}
+%    \begin{macrocode}
+\def\saveshift@#1{%
+    \begingroup
+        \let\or\relax
+        \xdef\tag@shifts{\or#1\tag@shifts}%
+    \endgroup
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%  \begin{macro}{\spread@equation}
+%    This does the line-spacing adjustment that is normally wanted for
+%    displayed equations. We also call \cs{reset@strutbox@} here because
+%    otherwise a preceding font size change might leave \cs{strutbox@}
+%    with wrong contents. This is a less-than-ideal solution but
+%    probably good enough for now, until the situation can be
+%    overhauled.
+%    \begin{macrocode}
+\def\spread@equation{\reset@strutbox@
+  \openup\jot \let\spread@equation\@empty}
+\let\@spread@equation\spread@equation
+%    \end{macrocode}
+%  \end{macro}
+%
+% \begin{macro}{\displ@y}
+% \begin{macro}{\displ@y@}
+% \begin{macro}{\@display@init}
+%    \cs{displ@y} is from \fn{plain.tex}, with
+%    \cs{interdisplaylinepenalty} changed to \cs{@eqpen}. Also we
+%    transplanted most of its internal organs to \cs{@display@init} to
+%    support \cs{displ@y@} and other possibilities. Don't try to make
+%    sense of these naming conventions! They are a narrowly calculated
+%    mishmash of Knuth/Spivak/Lamport/Mittelbach precedents. The reason
+%    for not cleaning them up and forcing all names to a consistent
+%    scheme is that then in principle we'd have to do it everywhere else
+%    too. And we programmers are paranoid about the side effects of name
+%    changes.
+%    \begin{macrocode}
+\def\displ@y{\@display@init{}}
+\def\@display@init#1{%
+    \global\dt@ptrue \spread@equation
+    \everycr{%
+        \noalign{%
+            #1%
+            \ifdt@p
+                \global\dt@pfalse
+                \vskip-\lineskiplimit
+                \vskip\normallineskiplimit
+            \else
+                \penalty\@eqpen \global\dspbrk@lvl\m@ne
+            \fi
+        }%
+    }%
+}
+%    \end{macrocode}
+%    \cs{displ@y@} is nearly the same; it additionally sets the |tag@|
+%    switch and the \cs{column@} and \cs{dspbrk@lvl} counters to their
+%    default values. The argument is normally a bit of code to empty out
+%    \cs{raise@tag}, but in \env{multline} we don't want that to happen
+%    in \cs{everycr}.
+%    \begin{macrocode}
+\def\displ@y@{\@display@init{%
+  \global\column@\z@ \global\dspbrk@lvl\m@ne
+  \global\tag@false \global\let\raise@tag\@empty
+}}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\black@}
+%    This macro is made to produce an overfull box message and
+%    possibly (depending on the value of \cs{overfullrule})
+%    a rule in the margin if the total width of an alignment
+%    is larger than the value of \cs{displaywidth}.
+%    \begin{macrocode}
+\def\black@#1{%
+    \noalign{%
+        \ifdim#1>\displaywidth
+            \dimen@\prevdepth
+            \nointerlineskip
+            \vskip-\ht\strutbox@
+            \vskip-\dp\strutbox@
+            \vbox{\noindent\hbox to#1{\strut@\hfill}}%
+            \prevdepth\dimen@
+        \fi
+    }%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macro}{\savecounters@}
+%    \begin{macro}{\restorecounters@}
+%    These are used during the measuring phase of the various display
+%    math environments to save and restore the values of all \latex/
+%    counters. We make these local to a group, so nested environments
+%    works.
+%
+%    Changed \cn{stepcounter} to |\csname c@...\endcsname| to avoid
+%    overhead of ifundefined test [mjd, 1995/01/20].
+%    \begin{macrocode}
+\def\savecounters@{%
+    \begingroup
+        \def\@elt##1{%
+          \global\csname c@##1\endcsname\the\csname c@##1\endcsname}%
+        \xdef\@gtempa{%
+            \cl@@ckpt
+            \let\@nx\restorecounters@\@nx\@empty
+        }%
+    \endgroup
+    \let\restorecounters@\@gtempa
+}
+%
+\let\restorecounters@\@empty
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%
+%    \begin{macro}{\savealignstate@}
+%    \begin{macro}{\restorealignstate@}
+%    These are used to save the values of various parameters that are
+%    shared by \env{align} and \env{gather} when the former is used
+%    inside the latter.
+%    \begin{macrocode}
+\def\savealignstate@{%
+    \begingroup
+        \let\or\relax
+        \xdef\@gtempa{%
+            \global\totwidth@\the\totwidth@
+            \global\row@\the\row@
+            \gdef\@nx\tag@lengths{\tag@lengths}%
+            \let\@nx\restorealignstate@\@nx\@empty
+        }%
+    \endgroup
+    \let\restorealignstate@\@gtempa
+}
+
+\let\restorealignstate@\@empty
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%
+%  \begin{macro}{\savecolumn@}
+%  \begin{macro}{\restorecolumn@}
+%
+%    \begin{macrocode}
+\def\savecolumn@{%
+  \edef\restorecolumn@{%
+    \global\column@\number\column@
+    \let\@nx\restorecolumn@\@nx\@empty
+  }%
+}
+\let\restorecolumn@\@empty
+%    \end{macrocode}
+%  \end{macro}
+%  \end{macro}
+%
+% \subsection{Scanning the environment's body}
+%
+%    Several of the math alignment macros must scan their body twice:
+%    once to determine how wide the columns are and then to actually
+%    typeset them. This means that we must collect all text in this body
+%    before calling the environment macros.
+%
+%    \begin{macro}{\@envbody}
+%    We start by defining a token register to contain the body.
+%    \begin{macrocode}
+\newtoks\@envbody
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macro}{\addto@envbody}
+%    Then we define a macro to add something (i.e.\ its argument) to the
+%    token register \cs{@envbody}.
+%    \begin{macrocode}
+\def\addto@envbody#1{\global\@envbody\@xp{\the\@envbody#1}}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\collect@body}
+%    The macro \cs{collect@body} starts the scan for the |\end{...}|
+%    command of the current environment. It takes a macro name as
+%    argument. This macro is supposed to take the whole body of the
+%    environment as its argument. For example, |\begin{align}| would
+%    call |\collect@body\@align| if |@align#1{...}| is the macro that
+%    sets the alignment with body \arg{1}.
+%
+%    \begin{macrocode}
+\def\collect@body#1{%
+  \@envbody{\@xp#1\@xp{\the\@envbody}}%
+  \edef\process@envbody{\the\@envbody\@nx\end{\@currenvir}}%
+  \@envbody\@emptytoks \def\begin@stack{b}%
+%    \end{macrocode}
+%
+%    If we simply called \cs{collect@@body} directly,
+%    the error message for a \cn{par} token (usually from a blank line)
+%    would be
+% \begin{verbatim}
+% ! Paragraph ended before \collect@@body was complete.
+% \end{verbatim}
+%    But we use a little finesse to get a more intelligible error
+%    message:
+% \begin{verbatim}
+% ! Paragraph ended before \multline* was complete.
+% \end{verbatim}
+%    In order to avoid using up csnames unnecessarily we use the actual
+%    environment name as the name of the temporary function that is
+%    \cs{let} to \cs{collect@@body}; but then in order to preserve the
+%    theoretical possibility of nesting for environments that use
+%    \cs{collect@body} (not currently required by any \pkg{amsmath}
+%    environment [mjd,1999/06/23]), we do the \cs{let} inside a group.
+%    \begin{macrocode}
+  \begingroup
+  \@xp\let\csname\@currenvir\endcsname\collect@@body
+%    \end{macrocode}
+%    This small twist eliminates the need for \cs{expandafter}'s in
+%    \cs{collect@@body}.
+%    \begin{macrocode}
+  \edef\process@envbody{\@xp\@nx\csname\@currenvir\endcsname}%
+  \process@envbody
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+%  \begin{macro}{\push@begins}
+%    When adding a piece of the current environment's contents to
+%    \cs{@envbody}, we scan it to check for additional \cn{begin}
+%    tokens, and add a `b' to the stack for any that we find.
+%    \begin{macrocode}
+\def\push@begins#1\begin#2{%
+  \ifx\end#2\else b\@xp\push@begins\fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\collect@@body}
+%    \cs{collect@@body} takes two arguments: the first will consist of
+%    all text up to the next \cn{end} command, the second will be the
+%    \cn{end} command's argument. If there are any extra \cn{begin}
+%    commands in the body text, a marker is pushed onto a stack by the
+%    \cs{push@begins} function. Empty state for this stack means that we
+%    have reached the \cn{end} that matches our original \cn{begin}.
+%    Otherwise we need to include the \cn{end} and its argument in the
+%    material that we are adding to our environment body accumulator.
+%
+% \begin{histnote}
+%    In a former implementation, the error messages resulting from a
+%    typo in the environment name were unsatisfactory, because it was
+%    matching of the environment name that was used to determine the end
+%    of our environment body, instead of counting begin-end pairs.
+%    Thanks to Lars Hellstr\"{o}m for a suggestion that led to this
+%    improvement. [mjd,1999/06/23]
+% \end{histnote}
+%    \begin{macrocode}
+\def\collect@@body#1\end#2{%
+  \edef\begin@stack{\push@begins#1\begin\end \@xp\@gobble\begin@stack}%
+  \ifx\@empty\begin@stack
+    \endgroup
+    \@checkend{#2}%
+    \addto@envbody{#1}%
+  \else
+    \addto@envbody{#1\end{#2}}%
+  \fi
+  \process@envbody % A little tricky! Note the grouping
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \subsection{Simple aligning environments}
+%
+%  \begin{macro}{\math@cr@@@aligned}
+%    From tabskip we get an extra space of minalignsep after every
+%    second column; but when this falls at the right edge of the whole
+%    aligned structure, we need to cancel that space.
+%    \begin{macrocode}
+\def\math@cr@@@aligned{%
+  \ifodd\column@ \let\next@\@empty
+  \else \def\next@{&\kern-\alignsep@}%
+  \fi
+  \next@ \cr
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+%    \begin{macro}{\start@aligned}
+%    The \env{aligned} and \env{alignedat} environments are identical
+%    except that the latter takes a mandatory argument to specify the
+%    number of align structures, while the former allows any number of
+%    align structures automatically (the use of \env{alignedat} is
+%    deprecated).  So, they will be defined in terms of
+%    \cs{start@aligned}, which will take two arguments.  The first
+%    argument specifies the placement of the environments; it is
+%    either |c|, |t|, or |b|.  The second is the number of align
+%    structures; a value of~$-1$ means that an arbitrary number are
+%    allowed.
+%    \begin{macrocode}
+\newcommand{\start@aligned}[2]{%
+    \RIfM@\else
+        \nonmatherr@{\begin{\@currenvir}}%
+    \fi
+    \savecolumn@ % Assumption: called inside a group
+%    \end{macrocode}
+%    The \cs{null} here is to keep the \cs{,} glue from causing the
+%    invocation of the clause in \tex/'s built-in tag placement
+%    algorithm that can cause an equation to be shifted all the way over
+%    to the margin.
+%    \begin{macrocode}
+    \null\,%
+    \if #1t\vtop \else \if#1b \vbox \else \vcenter \fi \fi \bgroup
+        \maxfields@#2\relax
+        \ifnum\maxfields@>\m@ne
+            \multiply\maxfields@\tw@
+%    \end{macrocode}
+%    Introduced new \cs{math@cr@@@} so we can provide standard error
+%    message for too many \qc{\&}'s in \env{alignedat}.
+%    \begin{macrocode}
+            \let\math@cr@@@\math@cr@@@alignedat
+            \alignsep@\z@skip
+        \else
+            \let\math@cr@@@\math@cr@@@aligned
+            \alignsep@\minalignsep
+        \fi
+%    \end{macrocode}
+%    Reset the meaning of \cn{\\}.
+%    \begin{macrocode}
+        \Let@ \chardef\dspbrk@context\@ne
+%    \end{macrocode}
+%    Restore the default definition of \cn{tag} (error message), in
+%    case \env{aligned} is used inside, e.g., a \env{gather}
+%    environment that accepts \cn{tag}.
+%    \begin{macrocode}
+        \default@tag
+        \spread@equation % no-op if already called
+%    \end{macrocode}
+%    Finally we start the alignment itself. For \env{aligned} we add
+%    \cs{minalignsep} after every second column to mimic the
+%    behavior of \env{align}. For \env{alignedat} the user has to
+%    specify interalign space explicitly.
+%    \begin{macrocode}
+        \global\column@\z@
+        \ialign\bgroup
+           &\column@plus
+            \hfil
+            \strut@
+            $\m@th\displaystyle{##}$%
+            \tabskip\z@skip
+           &\column@plus
+            $\m@th\displaystyle{{}##}$%
+            \hfil
+            \tabskip\alignsep@
+            \crcr
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\math@cr@@@aligned}
+%    \cs{math@cr@@@aligned} checks to make sure the user hasn't put in
+%    too many \qc{\&}s in \env{alignedat}.  Since \env{alignedat}
+%    doesn't use \cs{displ@y@}, we also reset \cs{column@} here.  Note
+%    than in \env{aligned}, \cs{column@} will increase without bound,
+%    since it never gets reset, but this is harmless.
+%    \begin{macrocode}
+\def\math@cr@@@alignedat{%
+    \ifnum\column@>\maxfields@
+        \begingroup
+          \measuring@false
+          \@amsmath@err{Extra & on this line}%
+            {\the\andhelp@}% "An extra & here is disastrous"
+        \endgroup
+    \fi
+    \global\column@\z@
+    \cr
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%  \begin{macro}{\alignsafe@testopt}
+%    Testing for an optional argument can be really, really tricky in
+%    certain complicated contexts. This we discovered by getting some
+%    bug reports for uses of \env{aligned}. So here is a safer
+%    form of \latex/'s \cs{@testopt} function.
+%    \begin{macrocode}
+\def\alignsafe@testopt#1#2{%
+  \relax\iffalse{\fi\ifnum`}=0\fi
+  \@ifnextchar[%
+    {\let\@let@token\relax \ifnum`{=\z@\fi\iffalse}\fi#1}%
+    {\let\@let@token\relax \ifnum`{=\z@\fi\iffalse}\fi#1[#2]}%
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+% \begin{environment}{aligned}
+%    The \env{aligned} environment takes an optional argument that
+%    indicates its vertical position in relation to surrounding
+%    material: |t|, |c|, or |b| for top, center, or bottom.
+%    \begin{macrocode}
+\newenvironment{aligned}{%
+  \let\@testopt\alignsafe@testopt
+  \aligned@a
+}{%
+  \crcr\egroup
+  \restorecolumn@
+  \egroup
+}
+\newcommand{\aligned@a}[1][c]{\start@aligned{#1}\m@ne}
+%    \end{macrocode}
+% \end{environment}
+%
+% \begin{environment}{alignedat}
+%    To get a top or bottom positioned \env{alignedat} structure, you
+%    would write something like
+% \begin{verbatim}
+% \begin{alignedat}[t]{3}
+% \end{verbatim}
+%
+%    \begin{macrocode}
+\newenvironment{alignedat}{%
+  \let\@testopt\alignsafe@testopt
+  \alignedat@a
+}{%
+  \endaligned
+}
+\newcommand{\alignedat@a}[1][c]{\start@aligned{#1}}
+%    \end{macrocode}
+% \end{environment}
+%
+% \begin{environment}{gathered}
+%    The \env{gathered} environment is for several lines that are
+%    centered independently.
+%    \begin{macrocode}
+\newenvironment{gathered}[1][c]{%
+    \RIfM@\else
+        \nonmatherr@{\begin{gathered}}%
+    \fi
+    \null\,%
+    \if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi \bgroup
+        \Let@ \chardef\dspbrk@context\@ne \restore@math@cr
+        \spread@equation
+        \ialign\bgroup
+            \hfil\strut@$\m@th\displaystyle##$\hfil
+            \crcr
+}{%
+  \endaligned
+}
+%    \end{macrocode}
+% \end{environment}
+%
+%
+%    \subsection{The \env{gather} environment}
+%
+% \begin{macro}{\start@gather}
+%    \begin{macrocode}
+\def\start@gather#1{%
+    \RIfM@
+        \nomath@env
+        \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+    \else
+        $$%
+        #1%
+        \ifst@rred \else \global\@eqnswtrue \fi
+        \let\next@\gather@
+    \fi
+    \collect@body\next@
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{environment}{gather}
+%    \begin{environment}{gather*}
+%    \begin{macrocode}
+\newenvironment{gather}{%
+  \start@gather\st@rredfalse
+}{%
+  \math@cr \black@\totwidth@ \egroup
+  $$\ignorespacesafterend
+}
+
+\newenvironment{gather*}{%
+  \start@gather\st@rredtrue
+}{%
+  \endgather
+}
+%    \end{macrocode}
+%    \end{environment}
+%    \end{environment}
+%
+%    \begin{macro}{\gather@}
+%    \begin{macrocode}
+\def\gather@#1{%
+    \ingather@true \let\split\insplit@
+    \let\tag\tag@in@align \let\label\label@in@display
+    \chardef\dspbrk@context\z@
+    \intertext@ \displ@y@ \Let@
+    \let\math@cr@@@\math@cr@@@gather
+    \gmeasure@{#1}%
+    \global\shifttag@false
+    \tabskip\z@skip
+    \global\row@\@ne
+    \halign to\displaywidth\bgroup
+        \strut@
+        \setboxz@h{$\m@th\displaystyle{##}$}%
+        \calc@shift@gather
+        \set@gather@field
+        \tabskip\@centering
+       &\setboxz@h{\strut@{##}}%
+        \place@tag@gather
+        \tabskip \iftagsleft@ \gdisplaywidth@ \else \z@skip \span\fi
+        \crcr
+        #1%
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\gmeasure@}
+%    \begin{macrocode}
+\def\gmeasure@#1{%
+    \begingroup
+        \measuring@true
+        \totwidth@\z@
+        \global\let\tag@lengths\@empty
+        \savecounters@
+        \setbox\@ne\vbox{%
+            \everycr{\noalign{\global\tag@false
+              \global\let\raise@tag\@empty \global\column@\z@}}%
+            \let\label\@gobble
+            \halign{%
+                \setboxz@h{$\m@th\displaystyle{##}$}%
+                \ifdim\wdz@>\totwidth@
+                    \global\totwidth@\wdz@
+                \fi
+               &\setboxz@h{\strut@{##}}%
+                \savetaglength@
+                \crcr
+                #1%
+                \math@cr@@@
+            }%
+        }%
+        \restorecounters@
+        \if@fleqn
+            \global\advance\totwidth@\@mathmargin
+        \fi
+        \iftagsleft@
+            \ifdim\totwidth@>\displaywidth
+                \global\let\gdisplaywidth@\totwidth@
+            \else
+                \global\let\gdisplaywidth@\displaywidth
+            \fi
+        \fi
+    \endgroup
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\math@cr@@@gather}
+%    Modified \cs{math@cr@@@gather} so that it always puts in the
+%    final field, which needs to be done under the new method for
+%    determining tag placement.  This is probably more efficient
+%    anyway.
+%
+%    \begin{macrocode}
+\def\math@cr@@@gather{%
+    \ifst@rred\nonumber\fi
+   &\relax
+    \make@display@tag
+    \ifst@rred\else\global\@eqnswtrue\fi
+%    \end{macrocode}
+%    We advance \cs{row@} here, rather than at the beginning of the
+%    preamble, because otherwise the \env{split} environment will
+%    cause \cs{row@} to be advanced twice instead of once.
+%    \begin{macrocode}
+    \global\advance\row@\@ne
+    \cr
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\calc@shift@gather}
+%    \cs{calc@shift@gather} has must make two decisions: (1) whether the
+%    equation tag for the current line should be put on a separate
+%    line and (2) what the distance between the equation and the
+%    equation tag should be.  We implement \tex/'s built-in
+%    tag-placement as well as possible, with one improvement: the
+%    minimum separation between tag and equation is now a
+%    user-settable parameter.
+%
+%    [1995/01/17] Added a check to make sure that the width of the tag
+%    on the current line is $>0$ before testing to see if tagwidth +
+%    linewidth + mintagsep $>$ displaywidth. Since an imbedded align
+%    shows up as line with width \cn{displaywidth}, and even lines
+%    without a tag get processed as if an empty tag were present, the
+%    result was that the empty tag assigned to the line containing the
+%    align was being shifted downwards, creating extra space after the
+%    align.
+%    \begin{macrocode}
+\def\calc@shift@gather{%
+    \dimen@\mintagsep\relax
+    \tagwidth@\tag@width\row@\relax
+%    \end{macrocode}
+%    If we're in \opt{fleqn} mode, there is no flexibility about
+%    placement of the equation, so all we can do is see if there's
+%    room for the tag in the given margin.
+%    \begin{macrocode}
+    \if@fleqn
+        \global\eqnshift@\@mathmargin
+        \ifdim\tagwidth@>\z@
+            \advance\dimen@\tagwidth@
+            \iftagsleft@
+                \ifdim\dimen@>\@mathmargin
+                    \global\shifttag@true
+                \fi
+            \else
+                \advance\dimen@\@mathmargin
+                \advance\dimen@\wdz@
+                \ifdim\dimen@>\displaywidth
+                   \global\shifttag@true
+                \fi
+            \fi
+        \fi
+    \else
+        \global\eqnshift@\displaywidth
+        \global\advance\eqnshift@-\wdz@
+        \ifdim\tagwidth@>\z@
+            \multiply\dimen@\tw@
+            \advance\dimen@\wdz@
+            \advance\dimen@\tagwidth@
+            \ifdim\dimen@>\displaywidth
+                \global\shifttag@true
+            \else
+                \ifdim\eqnshift@<4\tagwidth@
+                    \global\advance\eqnshift@-\tagwidth@
+                \fi
+            \fi
+        \fi
+        \global\divide\eqnshift@\tw@
+        \iftagsleft@
+            \global\eqnshift@-\eqnshift@
+            \global\advance\eqnshift@\displaywidth
+            \global\advance\eqnshift@-\wdz@
+        \fi
+        \ifdim\eqnshift@<\z@
+            \global\eqnshift@\z@
+        \fi
+    \fi
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\place@tag@gather}
+%    \begin{macro}{\set@gather@field}
+%    \begin{macrocode}
+\def\place@tag@gather{%
+    \iftagsleft@
+        \kern-\gdisplaywidth@
+        \ifshifttag@
+            \rlap{\vbox{%
+                \normalbaselines
+                \boxz@
+                \vbox to\lineht@{}%
+                \raise@tag
+            }}%
+            \global\shifttag@false
+        \else
+            \rlap{\boxz@}%
+        \fi
+    \else
+        \ifdim\totwidth@>\displaywidth
+            \dimen@\totwidth@
+            \advance\dimen@-\displaywidth
+            \kern-\dimen@
+        \fi
+        \ifshifttag@
+            \llap{\vtop{%
+                \raise@tag
+                \normalbaselines
+                \setbox\@ne\null
+                \dp\@ne\lineht@
+                \box\@ne
+                \boxz@
+            }}%
+            \global\shifttag@false
+        \else
+            \llap{\boxz@}%
+        \fi
+    \fi
+}
+%
+\def\set@gather@field{%
+    \iftagsleft@
+        \global\lineht@\ht\z@
+    \else
+        \global\lineht@\dp\z@
+    \fi
+    \kern\eqnshift@
+    \boxz@
+    \hfil
+}
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%
+%
+% \subsection{The \env{align} family of environments}
+%
+%    The \env{align}, \env{flalign}, \env{alignat}, \env{xalignat},
+%    and \env{xxalignat} environments are virtually
+%    identical, and thus will share much code.  We'll refer to the
+%    environments generically as ``\env{align}'' and will
+%    distinguish between them explicitly
+%    only when necessary.
+%
+% \begin{macro}{\ifxxat@}
+% \begin{macro}{\ifcheckat@}
+% \begin{macro}{\xatlevel@}
+%    The \cs{xatlevel@} macro will be used, informally speaking, to
+%    distinguish between the \env{alignat} and \env{xalignat}, and
+%    \env{xxalignat} environments.
+%
+%    \begin{macrocode}
+\newif\ifxxat@
+
+\newif\ifcheckat@
+
+\let\xatlevel@\@empty
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%    \end{macro}
+%
+% \begin{macro}{\start@align}
+%    \cs{start@align} will be called by all of the \env{align}-like
+%    environments.  The first argument will be the \cs{xatlevel@},
+%    i.e., 0, 1, or~2; the second argument will be either
+%    \cs{st@rredtrue} or \cs{st@rredfalse}.  The third argument will
+%    be the number of aligned
+%    structures in the environment (either as supplied by the user, or
+%    $-1$ to indicate that checking shouldn't be done).  After
+%    performing the appropriate error detection and initialization,
+%    \cs{start@align} calls \cs{align@}.
+%
+%    Note that the \cs{equation} counter is no longer stepped at the
+%    beginning of these environments.
+%
+%    TODO: Implement \cs{shoveleft} and \cs{shoveright} for
+%    \env{align}.
+%    \begin{macrocode}
+\def\start@align#1#2#3{%
+    \let\xatlevel@#1% always \z@, \@ne, or \tw@
+    \maxfields@#3\relax
+    \ifnum\maxfields@>\m@ne
+        \checkat@true
+        \ifnum\xatlevel@=\tw@
+            \xxat@true
+        \fi
+        \multiply\maxfields@\tw@
+    \else
+        \checkat@false
+    \fi
+    \ifingather@
+        \iffalse{\fi\ifnum0=`}\fi
+        \DN@{\vcenter\bgroup\savealignstate@\align@#2}%
+    \else
+        \ifmmode
+          \if@display
+             \DN@{\align@recover}%
+          \else
+            \nomath@env
+            \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+          \fi
+        \else
+            $$%
+            \let\split\insplit@
+            \DN@{\align@#2}%
+        \fi
+    \fi
+    \collect@body\next@
+}
+%    \end{macrocode}
+%
+%    With version 1.2 of \pkg{amsmath}, it was possible to use
+%    \env{align*} and relatives in certain wrong contexts without
+%    getting an error, e.g.
+% \begin{verbatim}
+% \begin{equation*}
+% \begin{align*}
+% ...
+% \end{align*}
+% \end{equation*}
+% \end{verbatim}
+%
+%    For backward compatibility we therefore give only a warning for
+%    this condition instead of a full error, and try to recover using
+%    the \env{aligned} environment. The alignment of the material may be
+%    adversely affected but it will at least remain readable.
+%    \begin{macrocode}
+\def\align@recover#1#2#3{%
+  \endgroup
+  \@amsmath@err{%
+Erroneous nesting of equation structures;\MessageBreak
+trying to recover with `aligned'%
+  }\@ehc
+  \begin{aligned}\relax#1\end{aligned}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{environment}{align}
+%    \begin{environment}{align*}
+%    \begin{environment}{flalign}
+%    \begin{environment}{flalign*}
+%    \begin{environment}{alignat}
+%    \begin{environment}{alignat*}
+%    \begin{environment}{xalignat}
+%    \begin{environment}{xalignat*}
+%    \begin{environment}{xxalignat}
+%    The definitions of the various \env{align} environments are quite
+%    straight-forward.
+%
+%    \begin{macrocode}
+\newenvironment{alignat}{%
+  \start@align\z@\st@rredfalse
+}{%
+  \endalign
+}
+\newenvironment{alignat*}{%
+  \start@align\z@\st@rredtrue
+}{%
+  \endalign
+}
+\newenvironment{xalignat}{%
+  \start@align\@ne\st@rredfalse
+}{%
+  \endalign
+}
+\newenvironment{xalignat*}{%
+  \start@align\@ne\st@rredtrue
+}{%
+  \endalign
+}
+\newenvironment{xxalignat}{%
+  \start@align\tw@\st@rredtrue
+}{%
+  \endalign
+}
+\newenvironment{align}{%
+  \start@align\@ne\st@rredfalse\m@ne
+}{%
+  \math@cr \black@\totwidth@
+  \egroup
+  \ifingather@
+    \restorealignstate@
+    \egroup
+    \nonumber
+    \ifnum0=`{\fi\iffalse}\fi
+  \else
+    $$%
+  \fi
+  \ignorespacesafterend
+}
+\newenvironment{align*}{%
+  \start@align\@ne\st@rredtrue\m@ne
+}{%
+  \endalign
+}
+\newenvironment{flalign}{%
+  \start@align\tw@\st@rredfalse\m@ne
+}{%
+  \endalign
+}
+\newenvironment{flalign*}{%
+  \start@align\tw@\st@rredtrue\m@ne
+}{%
+  \endalign
+}
+%    \end{macrocode}
+%    \end{environment}
+%    \end{environment}
+%    \end{environment}
+%    \end{environment}
+%    \end{environment}
+%    \end{environment}
+%    \end{environment}
+%    \end{environment}
+%    \end{environment}
+%
+%
+%  \begin{macro}{\align@}
+%    TODO: Some of these sets of initializations show up in multiple
+%    places.  It might be worth making an abbreviation for them.
+%
+%    \begin{macrocode}
+\def\align@#1#2{%
+    \inalign@true \intertext@ \Let@ \chardef\dspbrk@context\z@
+    \ifingather@\else\displ@y@\fi
+    \let\math@cr@@@\math@cr@@@align
+    \ifxxat@\else \let\tag\tag@in@align \fi
+    \let\label\label@in@display
+    #1% set st@r
+    \ifst@rred\else \global\@eqnswtrue \fi
+    \measure@{#2}%
+    \global\row@\z@
+    \tabskip\eqnshift@
+    \halign\bgroup
+        \span\align@preamble\crcr
+        #2%
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\math@cr@@@align}
+%    \begin{macrocode}
+\def\math@cr@@@align{%
+  \ifst@rred\nonumber\fi
+  \if@eqnsw \global\tag@true \fi
+  \global\advance\row@\@ne
+  \add@amps\maxfields@
+  \omit
+  \kern-\alignsep@
+  \iftag@
+    \setboxz@h{\@lign\strut@{\make@display@tag}}%
+    \place@tag
+  \fi
+  \ifst@rred\else\global\@eqnswtrue\fi
+  \global\lineht@\z@
+  \cr
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\math@cr@@@align@measure}
+%    \begin{macrocode}
+\def\math@cr@@@align@measure{%
+   &\omit
+    \global\advance\row@\@ne
+    \ifst@rred\nonumber\fi
+    \if@eqnsw \global\tag@true \fi
+    \ifnum\column@>\maxfields@
+        \ifcheckat@
+            \begingroup
+              \measuring@false
+              \@amsmath@err{Extra & on this line}%
+                {\the\andhelp@}% "An extra & here is disastrous"
+            \endgroup
+        \else
+            \global\maxfields@\column@
+        \fi
+    \fi
+    \setboxz@h{\@lign\strut@{%
+        \if@eqnsw
+            \stepcounter{equation}%
+            \tagform@\theequation
+        \else
+            \iftag@\df@tag\fi
+        \fi
+    }}%
+    \savetaglength@
+    \ifst@rred\else\global\@eqnswtrue\fi
+    \cr
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macro}{\field@lengths}
+%    \begin{macro}{\savefieldlength@}
+%    \begin{macro}{\fieldlengths@}
+%    \begin{macrocode}
+\let\field@lengths\@empty
+
+\def\savefieldlength@{%
+    \begingroup
+        \let\or\relax
+        \xdef\field@lengths{%
+            \field@lengths
+            \ifnum\column@=0
+                \or
+            \else
+                ,%
+            \fi
+            \the\wdz@
+        }%
+    \endgroup
+}
+
+\def\fieldlengths@#1{%
+    \ifcase\@xp#1\field@lengths\fi
+}
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%    \end{macro}
+%
+%  \begin{macro}{\maxcolumn@widths}
+%    \cs{maxcolumn@widths} will be used to hold the widths of the
+%    fields of the \env{alignat} environment.  The widths will be
+%    separated by the token \cn{or}, making it easy to extract a given
+%    width using \cn{ifcase}.
+%    \begin{macrocode}
+\let\maxcolumn@widths\@empty
+%    \end{macrocode}
+%  \end{macro}
+%
+%  \begin{macro}{\maxcol@width}
+% \cs{maxcol@width} $n$ = maximum width of $n$th column of the current
+%    \env{alignat} (i.e., the $n$th field of \cs{maxcolumn@widths}.)
+%    It expands to a \<dimen>, so it can be used as the right-hand
+%    side of a \<variable assignment> or \<arithmetic> statement.
+%    It's argument can be any \<number>, \<integer variable> or macro
+%    that expands to one of these.  [Check to make sure this is true.]
+%
+%    This is subtler than it looks.
+%    \begin{macrocode}
+\def\maxcol@width#1{%
+    \ifcase\@xp#1\maxcolumn@widths\fi\relax
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+%    Now comes the real fun.  A typical \env{align} environments looks
+%    something like this, where the vertical bars mark the edges of
+%    the fields of the underlying \cs{halign}:
+%    \[
+%    \makeatletter\tabskip\@centering\offinterlineskip
+%    \halign to\displaywidth{%^^A
+%       &$\strut$\vrule\hfil$\m@th\displaystyle{\@lign#}$\vrule
+%      \tabskip1pt&\vrule$\m@th\displaystyle{\@lign#}$\hfil\vrule
+%      \tabskip\@centering\cr
+%    \omit\small\hfil 1\hfil &\omit\small\hfil 2 &\omit\small\hfil
+%    3\hfil &\omit\small\hfil 4\hfil & \omit\small\hfil 5\hfil
+%    &\omit\small\hfil 6\hfil\cr
+%    \noalign{\vskip8pt\relax}
+%    V_i + q_i v_j & =v_i , &  X_i & = x_i - q_i x_j,
+%     &  U_i & = u_i,\qquad
+%      \hbox{for $i\ne j$;} %&\omit\hfill \llap{(3)}
+%    \cr
+%    V_j & = v_j, &  X_j & = x_j,
+%      &  U_j & = u_j + \sum_{i\ne j} q_i u_i. %&\omit\hfill \llap{(4)}
+%    \cr}
+%    \]
+%    Note that each align structure consists of two fields, with no
+%    space between them (a small space has been added here to
+%    highlight the boundaries).  Furthermore, the text inside the
+%    odd-numbered fields is flushright, while the text inside the
+%    even-numbered fields is flushleft.  The equation tags (shown on
+%    the right here) can be on either the right or the left.  If there
+%    is not room (in a sense to be defined shortly) for the tag on the
+%    same line as the equation, the tag will be shifted to a separate
+%    line.
+%
+%    Each environment also has a certain number of ``flexible
+%    spaces,'' meaning spaces whose width we are allowed to adjust to
+%    take up the amount of ``free space'' in the line, meaning the
+%    space not taken up by the equation tag and the fields of the
+%    underlying \cs{halign}.
+%
+%    The flexible spaces come in two flavors: interalign spaces and
+%    margin spaces.  If there are $n$ align structures ($n=3$ in the
+%    illustration above), there are $n-1$ interalign spaces, unless we
+%    are in an \env{alignat} environment, in which case there are no
+%    flexible interalign spaces.
+%
+%    The number of margin spaces is a little more complicated:
+%    Normally, there are two, but if we're in \opt{fleqn} mode, there
+%    is only one.  Furthermore, if we're in an \env{xxalignat} or
+%    \env{flalign} environment (corresponding to $\cs{xatlevel@} = 2$,
+%    then there are no flexible margin spaces.
+%
+%    Calculating the interalign and margin spaces is done in two
+%    stages.
+%
+%    First, the total amount of free space is divided uniformly among
+%    all the flexible spaces, without regard for the lengths of the tags
+%    on the various lines. For the non-\opt{fleqn} case, this
+%    corresponds to centering the align structures between the margins.
+%    Note that in \opt{fleqn} mode, the right margin is still allowed to
+%    be larger than \cs{@mathmargin}. This introduces an element of
+%    asymmetry into the appearance of the environment, but it has the
+%    advantage of leaving more space for equation tags in the right
+%    margin. If the right margin were constrained to be equal to the
+%    left margin in this case, tags would need to be shifted to a
+%    separate line more often than would be desirable.
+%
+%    Ordinarily, all flexible spaces will be given the same width.
+%    However, this is not invariably true, since the interalign spaces
+%    are constrained to be at least \cs{minalignsep} wide, while---in
+%    the absence of equation tags, at least---the margin spaces are
+%    allowed to shrink to zero.  As we shall see in a minute, if there
+%    are tags in the environment, then the margins are also bounded
+%    below by \cs{mintagsep}.
+%
+%    Next, we examine each line of the environment that has a tag to
+%    see if there is a gap of at least \cs{mintagsep} between the
+%    equation and its tag.  If there isn't, we attempt to center the
+%    equation between the tag and the opposite margin, leaving a gap
+%    of at least \cs{mintagsep} on either side, in order to preserve
+%    some symmetry, i.e., we want the equation to \emph{look} like
+%    it's centered between the margin and the tag, so we don't want
+%    the margin space to be less than the gap between the tag and the
+%    equation.  (Arguably, it would be better to allow the margin
+%    space to shrink to zero in this case in order to avoid shifting
+%    the tag to a separate line at any cost, but that would require
+%    all of our calculations to be a little more complicated and hence
+%    a little slower.)  Finally, if no values of the interalign spaces
+%    and the margins (with the constraints outlined above) will
+%    produce an acceptable distance between the equation and its tag,
+%    then the tag will be shifted to a separate line.
+%
+%  \begin{macro}{\measure@}
+%    \cs{measure@} collects the various bits of information that we'll
+%    need to perform the calculations outlined above, namely, the
+%    number of align structures in the environment, the natural
+%    lengths of the fields on each row, the maximum widths of each
+%    column, and the widths of the equation tags on each line.  It
+%    also calculates the number of flexible interalign and margin
+%    spaces and computes the initial values of the parameters
+%    \cs{eqnshift@} and \cs{alignsep@}, which correspond to the widths
+%    of the margins and the interalign spaces, respectively.
+%    \begin{macrocode}
+\def\measure@#1{%
+    \begingroup
+        \measuring@true
+        \global\eqnshift@\z@
+        \global\alignsep@\z@
+        \global\let\tag@lengths\@empty
+        \global\let\field@lengths\@empty
+        \savecounters@
+        \global\setbox0\vbox{%
+            \let\math@cr@@@\math@cr@@@align@measure
+            \everycr{\noalign{\global\tag@false
+              \global\let\raise@tag\@empty \global\column@\z@}}%
+            \let\label\@gobble
+            \global\row@\z@
+            \tabskip\z@
+            \halign{\span\align@preamble\crcr
+                #1%
+                \math@cr@@@
+                \global\column@\z@
+                \add@amps\maxfields@\cr
+            }%
+        }%
+        \restorecounters@
+%    \end{macrocode}
+%    It's convenient to have \cs{maxfields@} rounded up to the nearest
+%    even number, so that \cs{maxfields@} is precisely twice the
+%    number of align structures.
+%    \begin{macrocode}
+        \ifodd\maxfields@
+            \global\advance\maxfields@\@ne
+        \fi
+%    \end{macrocode}
+%    It doesn't make sense to have a single align structure in either
+%    \env{flalign} or \env{xxalignat}.  So, we check for that case now
+%    and, if necessary, switch to an \env{align} or \env{alignat}.
+%    Arguably, we should issue a warning message, but why bother?
+%    \begin{macrocode}
+        \ifnum\xatlevel@=\tw@
+            \ifnum\maxfields@<\thr@@
+                \let\xatlevel@\z@
+            \fi
+        \fi
+%    \end{macrocode}
+%    |\box0| now contains the lines of the \cs{halign}.  After the
+%    following maneuver, |\box1| will contain the last line of the
+%    \cs{halign}, which is what we're interested in.  (Incidentally,
+%    the penalty we're removing is the \cs{@eqpen} inserted by
+%    \cs{math@cr}.  Normally, this is \cs{interdisplaylinepenalty},
+%    unless the user has overridden that with a \cs{displaybreak}
+%    command.)
+%    \begin{macrocode}
+        \setbox\z@\vbox{%
+          \unvbox\z@ \unpenalty \global\setbox\@ne\lastbox
+        }%
+%    \end{macrocode}
+%    |\box1| begins with \cs{tabskip} glue and contains alternating
+%    \cs{hbox}es (the fields whose widths we're trying to get) and
+%    \cs{tabskip} glue [need better diagram]:
+%    \begin{verbatim}
+%    \hbox{\tabskip\hbox\tabskip...\hbox\tabskip}\end{verbatim}
+%    In fact, all the \cs{tabskip} glue will be 0pt, because all the
+%    \cs{tabskip}s in an \env{alignat} environment have a natural
+%    width of 0pt, and the \cs{halign} has been set in its natural
+%    width.
+%
+%    One nice result of this is that we can read \cs{totwidth@} off
+%    immediately, since it is just the width of |\box1|, plus
+%    \cs{@mathmargin} if we're in \opt{fleqn} mode.  (Actually, we
+%    also have to take \cs{minalignsep} into account, but we'll do
+%    that later):
+%    \begin{macrocode}
+        \global\totwidth@\wd\@ne
+        \if@fleqn \global\advance\totwidth@\@mathmargin \fi
+%    \end{macrocode}
+%    Now we initialize \cs{align@lengths} and start peeling the boxes
+%    off, one by one, and adding their widths to \cs{align@lengths}.
+%    We stop when we run out of boxes, i.e., when \cs{lastbox} returns
+%    a void box.  We're going to build a list using \cs{or} as a
+%    delimiter, so we want to disable it temporarily.
+%    \begin{macrocode}
+        \global\let\maxcolumn@widths\@empty
+        \begingroup
+          \let\or\relax
+          \loop
+            \global\setbox\@ne\hbox{%
+              \unhbox\@ne \unskip \global\setbox\thr@@\lastbox
+            }%
+          \ifhbox\thr@@
+           \xdef\maxcolumn@widths{ \or \the\wd\thr@@ \maxcolumn@widths}%
+          \repeat
+        \endgroup
+%    \end{macrocode}
+%    Now we calculate the number of flexible spaces and the initial
+%    values of \cs{eqnshift@} and \cs{alignsep@}.
+%    We start by calculating $\cs{displaywidth}-\cs{totwidth@}$,
+%    which gives us the total amount of ``free space'' in a row.
+%    \begin{macrocode}
+        \dimen@\displaywidth
+        \advance\dimen@-\totwidth@
+%    \end{macrocode}
+%    Next we calculate the number of columns of flexible spaces in the
+%    display, which depends on whether we're in \opt{fleqn} mode and
+%    in which particular environment we are in.
+%
+%    We use \cs{@tempcnta} to store the total number of flexible spaces
+%    in the align and \cs{@tempcntb} for the number of interalign
+%    spaces.
+%    \begin{macrocode}
+        \ifcase\xatlevel@
+%    \end{macrocode}
+%    In \env{alignat}, the interalign spaces are under user control,
+%    not ours.  So, we set \cs{alignsep@} and \cs{minalignsep} both
+%    equal to 0pt.  Later, when calculating a new value for
+%    \cs{alignsep@}, we will only save the new value if it is less
+%    than the current value of \cs{alignsep@} (i.e., \cs{alignsep@}
+%    will never increase).  Since the values we calculate will never
+%    be negative, this will ensure that \cs{alignsep@} remains zero in
+%    \env{alignat}.
+%    \begin{macrocode}
+            \global\alignsep@\z@
+            \let\minalignsep\z@
+            \@tempcntb\z@
+%    \end{macrocode}
+%    In \opt{fleqn} mode, the left margin---and hence the right margin
+%    in this case---is fixed.  Otherwise, we divide the free space
+%    equally between the two margins.
+%    \begin{macrocode}
+            \if@fleqn
+                \@tempcnta\@ne
+                \global\eqnshift@\@mathmargin
+            \else
+                \@tempcnta\tw@
+                \global\eqnshift@\dimen@
+                \global\divide\eqnshift@\@tempcnta
+            \fi
+        \or
+%    \end{macrocode}
+%    In an \env{align} or \env{xalignat} environment with $n$ aligned
+%    structures, there are $n-1$ interalign spaces and either 1 or~2
+%    flexible margins, depending on whether we're in \opt{fleqn} mode
+%    or not.
+%    \begin{macrocode}
+            \@tempcntb\maxfields@
+            \divide\@tempcntb\tw@
+            \@tempcnta\@tempcntb
+            \advance\@tempcntb\m@ne
+%    \end{macrocode}
+%    If we are in \opt{fleqn} mode, we fix the left margin and divide
+%    the free space equally among the interalign spaces and the right
+%    margin.
+%    \begin{macrocode}
+            \if@fleqn
+                \global\eqnshift@\@mathmargin
+                \global\alignsep@\dimen@
+                \global\divide\alignsep@\@tempcnta
+            \else
+%    \end{macrocode}
+%    Otherwise, we divide the free space equally among the interalign
+%    spaces and both margins.
+%    \begin{macrocode}
+                \global\advance\@tempcnta\@ne
+                \global\eqnshift@\dimen@
+                \global\divide\eqnshift@\@tempcnta
+                \global\alignsep@\eqnshift@
+            \fi
+        \or
+%    \end{macrocode}
+%    Finally, if we're in an \env{flalign} or \env{xxalignat}
+%    environment, there are no flexible margins and $n-1$ flexible
+%    interalign spaces.
+%    \begin{macrocode}
+            \@tempcntb\maxfields@
+            \divide\@tempcntb\tw@
+            \global\advance\@tempcntb\m@ne
+            \global\@tempcnta\@tempcntb
+            \global\eqnshift@\z@
+            \global\alignsep@\dimen@
+%    \end{macrocode}
+%    If we're in \opt{fleqn} mode, we need to add back the
+%    \cs{@mathmargin} that was removed when \cs{dimen@} was originally
+%    calculated above.
+%    \begin{macrocode}
+            \if@fleqn
+                \global\advance\alignsep@\@mathmargin\relax
+            \fi
+            \global\divide\alignsep@\@tempcntb
+        \fi
+%    \end{macrocode}
+%    Now we make sure \cs{alignsep@} isn't too small.
+%    \begin{macrocode}
+        \ifdim\alignsep@<\minalignsep\relax
+            \global\alignsep@\minalignsep\relax
+            \ifdim\eqnshift@>\z@
+                \if@fleqn\else
+                    \global\eqnshift@\displaywidth
+                    \global\advance\eqnshift@-\totwidth@
+                    \global\advance\eqnshift@-\@tempcntb\alignsep@
+                    \global\divide\eqnshift@\tw@
+                \fi
+            \fi
+        \fi
+        \ifdim\eqnshift@<\z@
+            \global\eqnshift@\z@
+        \fi
+        \calc@shift@align
+%    \end{macrocode}
+%    Next, we calculate the value of \cs{tagshift@}.  This is the glue
+%    that will be inserted in front of the equation tag to make sure
+%    it lines up flush against the appropriate margin.
+%    \begin{macrocode}
+        \global\tagshift@\totwidth@
+        \global\advance\tagshift@\@tempcntb\alignsep@
+        \if@fleqn
+            \ifnum\xatlevel@=\tw@
+                \global\advance\tagshift@-\@mathmargin\relax
+            \fi
+        \else
+            \global\advance\tagshift@\eqnshift@
+        \fi
+        \iftagsleft@ \else
+            \global\advance\tagshift@-\displaywidth
+        \fi
+%    \end{macrocode}
+%    Finally, we increase \cs{totwidth@} by an appropriate multiple of
+%    \cs{minalignsep}.  If the result is greater than
+%    \cs{displaywidth}, it means that at least one line in the
+%    \env{align} is overfull and we will issue an appropriate warning
+%    message (via \cs{bl@ck}) at the end of the environment.
+%    \begin{macrocode}
+        \dimen@\minalignsep\relax
+        \global\advance\totwidth@\@tempcntb\dimen@
+        \ifdim\totwidth@>\displaywidth
+            \global\let\displaywidth@\totwidth@
+        \else
+            \global\let\displaywidth@\displaywidth
+        \fi
+    \endgroup
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+%    The code for calculating the appropriate placement of equation
+%    tags in the \env{align} environments is quite complicated and
+%    varies wildly depending on the settings of the |tagsleft@| and
+%    |@fleqn| switches.  To minimize memory and hash space usage, we
+%    only define the variant appropriate for the current setting of
+%    those switches.
+%
+%    It would be worthwhile to examine this code more closely someday
+%    and see if it could be optimized any.
+%
+%    \paragraph{Tag placement when \cs{tagsleft@true},
+%    \cs{@fleqntrue}.}
+%
+%    We begin with the version of \cs{calc@shift@align} appropriate
+%    for flush-left equations with tags on the left.
+%
+%    \begin{macro}{\calc@shift@align}
+%    This is the simplest case.  Since the left margin is fixed, in
+%    general the only thing to do is check whether there is room for
+%    the tag in the left margin.  The only exception is that if
+%    $\cs{eqnshift@} = 0\,\mathrm{pt}$---meaning that we're in a
+%    \env{flalign} environment and this is the first line with a tag
+%    that we've encountered---then we set $\cs{eqnshift@} =
+%    \cs{@mathmargin}$ and recalculate \cs{alignsep@}.  This is done
+%    by \cs{x@calc@shift@lf}.
+%    \begin{macrocode}
+\iftagsleft@\if@fleqn
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+%    \end{macrocode}
+%    \cs{@tempdima} is initialized to $\cs{@mathmargin} -
+%    \cs{mintagsep}$, which yields the maximum size of a tag that will
+%    not be shifted to another line.
+%    \begin{macrocode}
+            \@tempdima\@mathmargin\relax
+            \advance\@tempdima-\mintagsep\relax
+%    \end{macrocode}
+%    Now we examine each row in turn.  If the width of the tag on the
+%    line is non-positive---meaning either that there is no tag or
+%    else that the user has forced it to have zero width---we mark the
+%    tag to remain unshifted.  Otherwise, we call \cs{x@calc@shift@lf}
+%    to determine whether any adjustments need to be made to
+%    \cs{eqnshift@} and \cs{alignsep@}.  Note the difference in
+%    treatment of zero-width tags between this code and \tex/'s
+%    built-in algorithm: here, a width of zero prohibits the tag from
+%    being shifted, while in \tex/'s built-in algorithm, a width of
+%    zero forces the tag to be shifted.
+%    \begin{macrocode}
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@lf
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\x@calc@shift@lf}
+%    As mentioned above, \cs{x@calc@shift@lf} first checks to see if
+%    the current left margin is set to 0 and, if so, resets it to
+%    \cs{@mathmargin} and recalculates \cs{alignsep@}.  Next, it
+%    checks whether the length of the current tag exceeds the
+%    previously calculated limit and, if so, marks the tag to be
+%    shifted to a separate line.
+%    \begin{macrocode}
+    \def\x@calc@shift@lf{%
+        \ifdim\eqnshift@=\z@
+            \global\eqnshift@\@mathmargin\relax
+            \alignsep@\displaywidth
+            \advance\alignsep@-\totwidth@
+            \global\divide\alignsep@\@tempcntb
+            \ifdim\alignsep@<\minalignsep\relax
+                \global\alignsep@\minalignsep\relax
+            \fi
+        \fi
+        \ifdim\tag@width\row@>\@tempdima
+            \saveshift@1%
+        \else
+            \saveshift@0%
+        \fi
+    }
+\fi\fi
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \paragraph{Tag placement when \cs{tagsleft@false},
+%    \cs{@fleqntrue}.}
+%
+%    Next we consider the case when equations are flush-left, but tags
+%    are on the right.  This case is somewhat more complicated than
+%    the previous one, since we can adjust the right margin by varying
+%    the inter-align separatin. Thus, when a tag is found to be too
+%    close to its equation, we first attempt to decrease
+%    \cs{alignsep@} enough to move the equation off to an acceptable
+%    distance.  Only if that would require a value of \cs{alignsep@}
+%    less than \cs{minalignsep} do we move the tag to a separate line.
+%
+%    \begin{macro}{\calc@shift@align}
+%    This version of \cs{calc@shift@align} differs from the previous
+%    version only in calling \cs{x@calc@shift@rf} rather than
+%    \cs{x@calc@shift@lf}.
+%    \begin{macrocode}
+\iftagsleft@\else\if@fleqn
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@rf
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\x@calc@shift@rf}
+%    To start, we need to know two quantities: the number of align
+%    structures in the current row and the ``effective length'' of the
+%    row, defined as the distance from the left margin to the
+%    right edge of the text assuming that \cs{eqnshift@} and
+%    \cs{alignsep@} are both~0.  To get the number of align
+%    structures, we first count the number of columns by counting the
+%    number of entries in the \cs{fieldlengths@} for the current row.
+%    The effective length is calcuated by \cs{x@rcalc@width} and put
+%    in the temporary register \cs{@tempdimc}, using \cs{@tempdimb} as
+%    an auxiliary variable.
+%    \begin{macrocode}
+    \def\x@calc@shift@rf{%
+        \column@\z@
+        \@tempdimb\z@
+        \@tempdimc\z@
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@rcalc@width
+        }%
+        \begingroup
+%    \end{macrocode}
+%    If there are $n$ columns in the current row, then there are
+%    $\lfloor (n+1)/2 \rfloor$ align structures and $\lfloor (n-1)/2
+%    \rfloor$ interalign spaces.
+%    \begin{macrocode}
+            \advance\column@\m@ne
+            \divide\column@\tw@
+%    \end{macrocode}
+%    If this is smaller than the maximum number of interalign spaces
+%    in the environment, then we need to reduce \cs{@tempcnta} (the
+%    total number of flexible spaces in the current line) by
+%    $\cs{@tempcntb} - \cs{column@}$ and reset \cs{@tempcntb} to
+%    \cs{column@}.
+%    \begin{macrocode}
+            \ifnum\@tempcntb>\column@
+                \advance\@tempcnta-\@tempcntb
+                \advance\@tempcnta\column@
+                \@tempcntb\column@
+            \fi
+%    \end{macrocode}
+%    Next, we add the width of the tag and the (fixed) left margin to
+%    the effective length calculated above.  This can be used to
+%    calculate how much ``free space'' there is in the current line
+%    and thus how much leeway we have to increase the amount of space
+%    between the tag and the equation.
+%    \begin{macrocode}
+            \tagwidth@\tag@width\row@\relax
+            \@tempdima\eqnshift@
+            \advance\@tempdima\@tempdimc\relax
+            \advance\@tempdima\tagwidth@
+%    \end{macrocode}
+%    The first thing to check is whether the tag should be shifted to
+%    a separate line.  To do this, we add the minimum interalign
+%    separation and the \cs{mintagsep} to the value of \cs{@tempdima}
+%    just calculated.  This yields the minimum acceptable length of
+%    the current line.  If that is greater than \cs{displaywidth}, we
+%    mark the tag to be calculated.  Otherwise, we mark the tag to be
+%    kept on the same line and then check to see if the \cs{alignsep@}
+%    needs to be reduced to make room for the tag.
+%    \begin{macrocode}
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \advance\dimen@\@tempdima
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+                \saveshift@0%
+%    \end{macrocode}
+%    Now we perform essentially the same calculation, but using the
+%    current value of \cs{alignsep@} instead of \cs{minalignsep}.
+%    This gives the current length of the line.  If this is greater
+%    than \cs{displaywidth}, we recalculate \cs{alignsep@} to make
+%    room for the tag.
+%    \begin{macrocode}
+                \dimen@\alignsep@\relax
+                \multiply\dimen@\@tempcntb
+                \advance\dimen@\@tempdima
+                \advance\dimen@\tagwidth@
+                \ifdim\dimen@>\displaywidth
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+                    \divide\dimen@\@tempcnta
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                    \else
+                        \global\alignsep@\dimen@
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+\fi\fi
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \paragraph{Tag placement when \cs{tagsleft@false},
+%    \cs{@fleqnfalse}.}
+%
+%    This is similar to the previous case, except for the added
+%    complication that both \cs{alignsep@} and \cs{eqnshift@} can
+%    vary, which makes the computations correspondingly more
+%    complicated.
+%
+%    \begin{macro}{\calc@shift@align}
+%    \begin{macrocode}
+\iftagsleft@\else\if@fleqn\else
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@rc
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\x@calc@shift@rc}
+%    \begin{macrocode}
+    \def\x@calc@shift@rc{%
+        \column@\z@
+        \@tempdimb\z@
+        \@tempdimc\z@
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@rcalc@width
+        }%
+        \begingroup
+            \advance\column@\m@ne
+            \divide\column@\tw@
+            \ifnum\@tempcntb>\column@
+                \advance\@tempcnta-\@tempcntb
+                \advance\@tempcnta\column@
+                \@tempcntb\column@
+            \fi
+            \tagwidth@\tag@width\row@\relax
+            \@tempdima\@tempdimc
+            \advance\@tempdima\tagwidth@
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \ifnum\xatlevel@=\tw@ \else
+                \advance\dimen@\mintagsep\relax
+            \fi
+            \advance\dimen@\@tempdima
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+                \saveshift@0%
+                \dimen@\eqnshift@
+                \advance\dimen@\@tempdima
+                \advance\dimen@\@tempcntb\alignsep@
+                \advance\dimen@\tagwidth@
+                \ifdim\dimen@>\displaywidth
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+                    \divide\dimen@\@tempcnta
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                        \eqnshift@\displaywidth
+                        \advance\eqnshift@-\@tempdima
+                        \advance\eqnshift@-\@tempcntb\alignsep@
+                        \global\divide\eqnshift@\tw@
+                    \else
+                        \ifdim\dimen@<\eqnshift@
+                            \ifdim\dimen@<\z@
+                                \global\eqnshift@\z@
+                            \else
+                                \global\eqnshift@\dimen@
+                            \fi
+                        \fi
+                        \ifdim\dimen@<\alignsep@
+                            \global\alignsep@\dimen@
+                        \fi
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+\fi\fi
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\x@rcalc@width}
+%    \begin{macrocode}
+\iftagsleft@\else
+    \def\x@rcalc@width{%
+        \ifdim\@tempa > \z@
+            \advance\@tempdimc\@tempdimb
+            \ifodd\column@
+                \advance\@tempdimc\maxcol@width\column@
+                \@tempdimb\z@
+            \else
+                \advance\@tempdimc\@tempa\relax
+                \@tempdimb\maxcol@width\column@
+                \advance\@tempdimb-\@tempa\relax
+            \fi
+        \else
+            \advance\@tempdimb\maxcol@width\column@\relax
+        \fi
+    }
+\fi
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \paragraph{Tag placement when \cs{tagsleft@true},
+%    \cs{@fleqnfalse}.}
+%
+%    \begin{macro}{\calc@shift@align}
+%    \begin{macrocode}
+\iftagsleft@\if@fleqn\else
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>\z@
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@lc
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\x@calc@shift@lc}
+%    \begin{macrocode}
+    \def\x@calc@shift@lc{%
+        \column@\z@
+%    \end{macrocode}
+%    \cs{@tempdima} will (eventually) be set to the effective width of
+%    the current row, defined as the distance from the leftmost point
+%    of the current line to the end of the last field of the
+%    \cs{halign}, ignoring any intervening \cs{tabskip}s, plus the
+%    width of the current tag.  That is, it will be the width of the
+%    first non-empty field plus the sum of the maximum widths of all
+%    following fields, plus the tag width.
+%
+%    \cs{@tempdimb} will be the ``indentation'' of leftmost end of
+%    text, ignoring the \cs{tabskip} glue, i.e., it will be the sum of
+%    the maximum widths of any fields to the left of the first
+%    non-empty field, plus whatever empty space there is at the
+%    beginning of the first non-empty field.
+%    \begin{macrocode}
+        \@tempdima\z@ % ``width of equation''
+        \@tempdimb\z@ % ``indent of equation''
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@lcalc@width
+        }%
+        \begingroup
+            \tagwidth@\tag@width\row@\relax
+%    \end{macrocode}
+%    \cs{@tempdima} is now easy to calculate, since it is just
+%    $\cs{totwidth@} - \cs{@tempdimb} + \cs{tagwidth@}$.
+%    \begin{macrocode}
+            \@tempdima\totwidth@
+            \advance\@tempdima-\@tempdimb
+            \advance\@tempdima\tagwidth@
+%    \end{macrocode}
+%    Next, we check to see whether there is room for both the equation
+%    and the tag on the same line, by calculating the minimum
+%    acceptable length of the current row and comparing that to
+%    \cs{displaywidth}.  Note that here we use \cs{@tempcntb}, i.e.,
+%    the number of interalign spaces after the first non-empty align
+%    structure.
+%    \begin{macrocode}
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \ifnum\xatlevel@=\tw@ \else
+                \advance\dimen@\mintagsep\relax
+            \fi
+            \advance\dimen@\@tempdima
+%    \end{macrocode}
+%    If the minimum acceptable width of the current line is greater
+%    than \cs{displaywidth}, we mark the current tag to be shifted to
+%    a separate line.
+%    \begin{macrocode}
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+%    \end{macrocode}
+%    Otherwise, the tag can stay on the same line as the equation, but
+%    we need to check whether it is too close to the equation.  So, we
+%    calculate the distance between the left margin and the left side
+%    of the equation, using the current values of \cs{eqnshift@} and
+%    \cs{alignsep@}.  Note that we use \cs{count@} here, not
+%    \cs{@tempcntb}, as above.
+%    \begin{macrocode}
+                \saveshift@0%
+                \dimen@\alignsep@
+                \multiply\dimen@\count@
+                \advance\dimen@\eqnshift@
+                \advance\dimen@\@tempdimb
+%    \end{macrocode}
+%    If the left margin is less than twice the tag width, we calculate
+%    new values of \cs{eqnshift@} and \cs{alignsep@} to move the
+%    equation further away from the tag.  In particular, we center the
+%    current line between its tag and the right margin.  Note that
+%    although we later will need to transform \cs{dimen@} into a value
+%    suitable for use as \cs{eqnshift@}, for the time being it is more
+%    useful to think of it as the space separating the tag from the
+%    equation.
+%    \begin{macrocode}
+                \ifdim\dimen@<2\tagwidth@
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+%    \end{macrocode}
+%    In certain circumstances we will get a divide-by-zero error here
+%    unless we guard against it. Use of \cs{@tempcnta} is complicated,
+%    sometimes it is assigned globally, sometimes locally. Need to sort
+%    it out one of these days [mjd,2000/06/02].
+%    \begin{macrocode}
+                    \ifnum\@tempcnta>\z@
+                      \divide\dimen@\@tempcnta
+                    \else \dimen@\z@
+                    \fi
+%    \end{macrocode}
+%    As usual, we check to make sure we don't set \cs{alignsep@}
+%    smaller than \cs{minalignsep} and, in any case, that we don't
+%    replace \cs{alignsep@} by a larger value.
+%    \begin{macrocode}
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                        \dimen@\displaywidth
+                        \advance\dimen@-\@tempdima
+                        \advance\dimen@-\@tempcntb\alignsep@
+                        \global\divide\dimen@\tw@
+                    \else
+                        \ifdim\dimen@<\alignsep@
+                            \global\alignsep@\dimen@
+                        \fi
+                    \fi
+%    \end{macrocode}
+%    Next, we calculate an appropriate value of \cs{eqnshift@},
+%    assuming that \cs{dimen@} is the desired separation between the
+%    tag and equation of the current line.  This means that we first
+%    need to adjust \cs{dimen@} if we're in an \env{flalign}
+%    environment.
+%    \begin{macrocode}
+                    \ifnum\xatlevel@=\tw@
+                        \dimen@\mintagsep\relax
+                    \fi
+%    \end{macrocode}
+%    Now we calculate the value of \cs{eqnshift@} needed to produce a
+%    separation of \cs{dimen@} between the equation tag and the
+%    beginning of the equation.  To do this, we need the following
+%    equation to hold:
+%    \[
+%       \cs{eqnshift@} + n\cs{alignsep@} + \cs{@tempdimb}
+%           = \cs{tagwidth@} + \cs{dimen@}
+%    \]
+%    where $n = \cs{count@}$ is the number of interalign spaces before
+%    the first non-empty field of the current line.
+%    \begin{macrocode}
+                    \advance\dimen@\tagwidth@
+                    \advance\dimen@-\@tempdimb
+                    \advance\dimen@-\count@\alignsep@
+%    \end{macrocode}
+%    The value of \cs{eqnshift@} just calculated is the minimum
+%    acceptable value; thus, we save it only if it is larger than the
+%    current value.
+%    \begin{macrocode}
+                    \ifdim\dimen@>\eqnshift@
+                        \global\eqnshift@\dimen@
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{macro}{\x@lcalc@width}
+%    This macro calculates the ``indentation'' of the current row, as
+%    defined above under the description of \cs{x@calc@shift@lc}.
+%    This macro is called for each field of the current line, with
+%    \cs{@tempa} set to the width of the current field.  Ideally, the
+%    loop enclosing \cs{x@lcalc@width} would terminate as soon as
+%    \cs{@tempa} is non-zero, but that would be a bit tricky to
+%    arrange.  Instead, we use \cs{@tempdima} as a flag to signal when
+%    we've encountered the first non-empty field.
+%
+%    \begin{macrocode}
+    \def\x@lcalc@width{%
+        \ifdim\@tempdima = \z@
+%    \end{macrocode}
+%    If the current field is empty (i.e., $\cs{@tempa} =
+%    \mathrm{0\,pt}$, then we increment \cs{@tempdimb} by the width of
+%    the current field).  Otherwise, we set $\cs{@tempdima} =
+%    \mathrm{1\,pt}$ as a signal value and increment \cs{@tempdimb} by
+%    the width of whatever empty space there might be at the left of
+%    the current field.
+%    \begin{macrocode}
+            \ifdim\@tempa > \z@
+                \@tempdima\p@
+                \ifodd\column@
+                    \advance\@tempdimb \maxcol@width\column@
+                    \advance\@tempdimb-\@tempa
+                \fi
+%    \end{macrocode}
+%    In addition, we need to adjust the values of \cs{@tempcnta} and
+%    \cs{@tempcntb} to account for any empty align structures that
+%    might occur at the beginning of the current line.  More
+%    specifically, we first set \cs{count@} equal to the number of
+%    interalign spaces preceding the current field (namely, $\lfloor
+%    (\cs{\column@}-1)/2 \rfloor$), and then subtract \cs{count@} from
+%    both \cs{@tempcnta} and \cs{@tempcntb}.  The rationale is that
+%    for the purposes of adjusting the spacing between the tag and the
+%    equation, the only flexible interalign spaces are those after
+%    the first non-empty align structure, so we need to treat those
+%    different from the ones before the first non-empty align
+%    structure.
+%    \begin{macrocode}
+                \count@\column@
+                \advance\count@\m@ne
+                \divide\count@\tw@
+                \advance\@tempcnta-\count@
+                \advance\@tempcntb-\count@
+            \else
+                \advance\@tempdimb \maxcol@width\column@\relax
+            \fi
+        \fi
+    }
+\fi\fi
+%    \end{macrocode}
+%    \end{macro}
+%
+%  \begin{macro}{\place@tag}
+%    \cs{place@tag} takes care of the placment of tags in the
+%    \env{align} environments.
+%    \begin{macrocode}
+\def\place@tag{%
+    \iftagsleft@
+        \kern-\tagshift@
+        \if1\shift@tag\row@\relax
+            \rlap{\vbox{%
+                \normalbaselines
+                \boxz@
+                \vbox to\lineht@{}%
+                \raise@tag
+            }}%
+        \else
+            \rlap{\boxz@}%
+        \fi
+        \kern\displaywidth@
+    \else
+        \kern-\tagshift@
+        \if1\shift@tag\row@\relax
+%    \end{macrocode}
+%    Added depth to correct vertical spacing of shifted
+%    equation tags.---dmj, 1994/12/29
+%    \begin{macrocode}
+            \llap{\vtop{%
+                \raise@tag
+                \normalbaselines
+                \setbox\@ne\null
+                \dp\@ne\lineht@
+                \box\@ne
+                \boxz@
+            }}%
+        \else
+            \llap{\boxz@}%
+        \fi
+    \fi
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+% \begin{macro}{\align@preamble}
+%    \begin{macrocode}
+\def\align@preamble{%
+   &\hfil
+    \strut@
+    \setboxz@h{\@lign$\m@th\displaystyle{##}$}%
+    \ifmeasuring@\savefieldlength@\fi
+    \set@field
+    \tabskip\z@skip
+   &\setboxz@h{\@lign$\m@th\displaystyle{{}##}$}%
+    \ifmeasuring@\savefieldlength@\fi
+    \set@field
+    \hfil
+    \tabskip\alignsep@
+}
+%    \end{macrocode}
+%  \end{macro}
+%
+%  \begin{macro}{\set@field}
+%    \cs{set@field} increments the column counter, tracks the value of
+%    \cs{lineht@} and finally inserts the box containing the contents
+%    of the current field.
+%    \begin{macrocode}
+\def\set@field{%
+    \column@plus
+    \iftagsleft@
+        \ifdim\ht\z@>\lineht@
+            \global\lineht@\ht\z@
+        \fi
+    \else
+        \ifdim\dp\z@>\lineht@
+            \global\lineht@\dp\z@
+        \fi
+    \fi
+    \boxz@
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%
+% \subsection {The \env{split} environment}
+%
+%    \begin{macro}{\split@err}
+%    A special error function for \env{split} to conserve main mem (at a
+%    cost of string pool/hash size.
+%    \begin{macrocode}
+\edef\split@err#1{%
+    \@nx\@amsmath@err{%
+        \string\begin{split} won't work here%
+    }{%
+        \@xp\@nx\csname
+  Did you forget a preceding \string\begin{equation}?^^J%
+  If not, perhaps the `aligned' environment is what
+  you want.\endcsname}%
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+%    \begin{environment}{split}
+%    If the \env{split} environment occurs inside \env{align} or
+%    \env{gather}, it can make use of the enclosing halign; if it is
+%    called inside a simple equation, we add an implicit `gather'
+%    container.
+%
+%    \begin{macrocode}
+\newenvironment{split}{%
+  \if@display
+    \ifinner
+      \@xp\@xp\@xp\split@aligned
+    \else
+      \ifst@rred \else \global\@eqnswtrue \fi
+    \fi
+  \else \let\endsplit\@empty \@xp\collect@body\@xp\split@err
+  \fi
+  \collect@body\gather@split
+}{%
+      \crcr
+    \egroup
+  \egroup
+  \iftagsleft@ \@xp\lendsplit@ \else \@xp\rendsplit@ \fi
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\split@tag\relax % init
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\gather@split#1#2#3{%
+  \@xp\endgroup \reset@equation % math@cr will handle equation numbering
+  \iftag@
+     \toks@\@xp{\df@tag}%
+     \edef\split@tag{%
+       \gdef\@nx\df@tag{\the\toks@}%
+       \global\@nx\tag@true \@nx\nonumber
+     }%
+  \else \let\split@tag\@empty
+  \fi
+  \spread@equation
+%    \end{macrocode}
+%    The extra vcenter wrapper here is not really a good thing but
+%    without it there are compatibility problems with old documents that
+%    throw in some extra material between \verb'\begin{equation}' and
+%    \verb'\begin{split}' (for example, \verb'\hspace{-1pc}' or
+%    \verb'\left\{'). [mjd,1999/09/20]
+%    \begin{macrocode}
+  \vcenter\bgroup
+    \gather@{\split@tag  \begin{split}#1\end{split}}%
+    \def\endmathdisplay@a{%
+      \math@cr \black@ \totwidth@ \egroup
+      \egroup
+    }%
+}
+%    \end{macrocode}
+%
+%    \end{environment}
+%
+% \begin{macro}{\insplit@}
+%    \begin{macrocode}
+\def\insplit@{%
+  \global\setbox\z@\vbox\bgroup
+    \Let@ \chardef\dspbrk@context\@ne \restore@math@cr
+    \default@tag % disallow use of \tag here
+    \ialign\bgroup
+      \hfil
+      \strut@
+      $\m@th\displaystyle{##}$%
+     &$\m@th\displaystyle{{}##}$%
+      \hfill % Why not \hfil?---dmj, 1994/12/28
+      \crcr
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\rendsplit@}
+%    Moved the box maneuvers inside the \cs{ifinalign@}, since that is
+%    the only place they are needed.---dmj, 1994/12/28
+%
+%    TODO: Explore interaction of tag-placement algorithm with
+%    \env{split}.  Is there any way for \env{split} to pass the
+%    relevant information out to the enclosing \env{gather} or
+%    \env{align}?
+%    \begin{macrocode}
+\def\rendsplit@{%
+    \ifinalign@
+%    \end{macrocode}
+%    Changed |\box9| into a \cs{vtop} here for better spacing.
+%    \begin{macrocode}
+        \global\setbox9 \vtop{%
+            \unvcopy\z@
+            \global\setbox8 \lastbox
+            \unskip
+        }%
+        \setbox\@ne\hbox{%
+            \unhcopy8
+            \unskip
+            \global\setbox\tw@\lastbox
+            \unskip
+            \global\setbox\thr@@\lastbox
+        }%
+        \ifctagsplit@
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vcenter{\vbox{\moveleft\wd\thr@@\boxz@}}%
+            }%
+        \else
+            \global\setbox7 \hbox{\unhbox\tw@\unskip}%
+%    \end{macrocode}
+%    Added \cs{add@amps} to make sure we put the last line of the
+%    \env{split} into the proper column of an \env{align} environment
+%    with multiple align structures.---dmj, 1994/12/28
+%
+%    Special care has to be taken in this case because the \env{split}
+%    turns into two lines of the \env{align} instead of just one.  So,
+%    we have to make sure that the first line produced by the
+%    \env{split} doesn't upset our bookkeeping, hence we call
+%    \cs{savetaglength@} to insert 0\,pt as the tag for this
+%    pseudo-line, and we advance the \cs{row@} counter and reset
+%    \cs{lineht@} afterwards.  It would be nice if we could just
+%    replace the \cs{crcr} by \cs{math@cr@@@}, but that would cause
+%    problems with the tag processing.
+%    \begin{macrocode}
+            \gdef\split@{%
+                \global\@tempcnta\column@
+               &\setboxz@h{}%
+                \savetaglength@
+                \global\advance\row@\@ne
+                \vbox{\moveleft\wd\thr@@\box9}%
+                \crcr
+                \noalign{\global\lineht@\z@}%
+                \add@amps\@tempcnta
+                \box\thr@@
+               &\box7
+            }%
+        \fi
+    \else
+        \ifctagsplit@
+            \gdef\split@{\vcenter{\boxz@}}%
+        \else
+%    \end{macrocode}
+%    Changed to just \cs{boxz@}, otherwise last line gets centered
+%    rather than aligned properly with respect to the rest of the
+%    lines.  But this means that we can't see inside of the last line
+%    to decide whether the tag needs to be moved.  Will have to think
+%    about this.---dmj, 1994/12/28
+%    \begin{macrocode}
+            \gdef\split@{%
+                \boxz@
+%                \box9
+%                \crcr
+%                \hbox{\box\thr@@\box7}%
+            }%
+        \fi
+    \fi
+    \aftergroup\split@
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\lendsplit@}
+%    \begin{macrocode}
+\def\lendsplit@{%
+    \global\setbox9\vtop{\unvcopy\z@}%
+    \ifinalign@
+%    \end{macrocode}
+%    Moved following two boxes inside the \cs{ifinalign@}, since they
+%    are only used in that case.  In fact, if we just kept track of
+%    the width of the first column, we could dispense with this
+%    entirely.  Surely that would be more efficient than all these box
+%    copies.---dmj, 1994/12/28
+%    \begin{macrocode}
+        \setbox\@ne\vbox{%
+            \unvcopy\z@
+            \global\setbox8\lastbox
+        }%
+        \setbox\@ne\hbox{%
+            \unhcopy8%
+            \unskip
+            \setbox\tw@\lastbox
+            \unskip
+            \global\setbox\thr@@\lastbox
+        }%
+        \ifctagsplit@
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vcenter{\vbox{\moveleft\wd\thr@@\box9}}%
+            }%
+        \else
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vbox{\moveleft\wd\thr@@\box9}%
+            }%
+        \fi
+    \else
+        \ifctagsplit@
+            \gdef\split@{\vcenter{\box9}}%
+        \else
+            \gdef\split@{\box9}%
+        \fi
+    \fi
+    \aftergroup\split@
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%    With \pkg{amsmath} 1.2 it was possible to put things like
+%    \verb'\left\{' between \verb'\begin{equation}' and
+%    \verb'\begin{split}' without getting any error message. For
+%    backward compatibility we try to avoid a fatal error in this case
+%    and instead attempt recovery with \env{aligned}.
+%    \begin{macrocode}
+\def\split@aligned#1#2{%
+   \iffalse{\fi\ifnum0=`}\fi
+   \collect@body\split@al@a}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\split@al@a#1#2#3{%
+  \split@warning
+  \endgroup
+%    \end{macrocode}
+%    If the \opt{fleqn} and \opt{tbtags} options are both in effect then
+%    we will need to add an optional argument on the \env{aligned}
+%    environment.
+%    \begin{macrocode}
+  \toks@{\begin{aligned}}%
+  \if@fleqn \split@al@tagcheck \fi
+%    \end{macrocode}
+%    The \cs{relax} here is to prevent \cs{@let@token} from being left
+%    equal to an ampersand if that happens to be the first thing in the body.
+%    \begin{macrocode}
+  \the\toks@\relax#1\end{aligned}%
+  \ifnum0=`{\fi\iffalse}\fi
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\split@al@tagcheck{%
+   \ifctagsplit@
+   \else
+     \iftagsleft@ \toks@\@xp{\the\toks@ [t]}%
+     \else \toks@\@xp{\the\toks@ [b]}%
+     \fi
+   \fi
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\split@warning{%
+  \PackageWarning{amsmath}{%
+Cannot use `split' here;\MessageBreak trying to recover with `aligned'}%
+}
+%    \end{macrocode}
+%
+% \subsection{The \env{multline} environment}
+%
+%    In the original \amstex/, \cn{multlinegap} is a macro with an
+%    argument that resets an internal dimension (one with an \qc{\@}
+%    character in its name). Here, to save control sequence names, we
+%    define \cn{multlinegap} to be the dimension itself and the
+%    documentation instructs users to use \cn{setlength} if they
+%    need to change it.
+% \begin{macro}{\multlinegap}
+% \begin{macro}{\multlinetaggap}
+%    Changed \cs{multlinegap} and \cs{multlinetaggap} to skip
+%    registers.  Also changed name to \cs{multlinetaggap} from
+%    \cs{multlinetaggap@}.
+%    \begin{macrocode}
+\newskip\multlinegap
+\multlinegap10pt
+\newskip\multlinetaggap
+\multlinetaggap10pt
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%    \begin{macro}{\start@multline}
+%    \begin{macrocode}
+\def\start@multline#1{%
+    \RIfM@
+        \nomath@env
+        \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+    \else
+        $$%
+        #1%
+        \ifst@rred
+            \nonumber
+        \else
+            \global\@eqnswtrue
+        \fi
+        \let\next@\multline@
+    \fi
+    \collect@body\next@
+}
+%    \end{macrocode}
+%    \end{macro}
+%
+% \begin{environment}{multline}
+% \begin{environment}{multline*}
+%    \begin{macrocode}
+\newenvironment{multline}{%
+  \start@multline\st@rredfalse
+}{%
+  \iftagsleft@ \@xp\lendmultline@ \else \@xp\rendmultline@ \fi
+  \ignorespacesafterend
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\newenvironment{multline*}{\start@multline\st@rredtrue}{\endmultline}
+%    \end{macrocode}
+% \end{environment}
+% \end{environment}
+%
+% \begin{macro}{\multline@}
+%    \begin{macrocode}
+\def\multline@#1{%
+    \Let@
+%    \end{macrocode}
+%
+%    For multline neither \cs{displ@y} no \cs{displ@y@} is quite right;
+%    we want to advance the row number and (I suppose?) the
+%    display-pagebreak level, but we only want to do tag-related stuff
+%    once before the first line, not repeat it for every line. (Recall
+%    that the arg of \cs{@display@init} goes into \cs{everycr}.)
+%    \begin{macrocode}
+    \@display@init{\global\advance\row@\@ne \global\dspbrk@lvl\m@ne}%
+    \chardef\dspbrk@context\z@
+    \restore@math@cr
+%    \end{macrocode}
+%    The \env{multline} environment is somewhat unusual, in that
+%    \cs{tag} and \cs{label} are enabled only during the measuring
+%    phase and disabled during the production phase.
+%    Here we disable \cs{tag} and \cs{label}; \cs{mmeasure@} will
+%    re-enable them temporarily.
+%    \begin{macrocode}
+    \let\tag\tag@in@align
+    \global\tag@false \global\let\raise@tag\@empty
+    \mmeasure@{#1}%
+    \let\tag\gobble@tag \let\label\@gobble
+    \tabskip \if@fleqn \@mathmargin \else \z@skip \fi
+    \totwidth@\displaywidth
+    \if@fleqn
+        \advance\totwidth@-\@mathmargin
+    \fi
+    \halign\bgroup
+        \hbox to\totwidth@{%
+%    \end{macrocode}
+%    In order to get the spacing of the last line right in fleqn
+%    mode, we need to play a little game here.  Normally the
+%    stretchability of the \cs{hskip} here will be suppressed by the
+%    \cs{hfil} at the end of the template, except inside the last line,
+%    when that \cs{hfil} will be removed by the \cs{hfilneg} in
+%    \cs{lendmultline@}.
+%    \begin{macrocode}
+            \if@fleqn
+                \hskip \@centering \relax
+            \else
+                \hfil
+            \fi
+            \strut@
+            $\m@th\displaystyle{}##\endmultline@math
+            \hfil
+        }%
+        \crcr
+%    \end{macrocode}
+%    In \opt{fleqn} mode, it's the \cs{tabskip} of \cs{@mathmargin}
+%    that needs to be removed in the first line, not the \cs{hfil} at
+%    the beginning of the template.
+%    \begin{macrocode}
+        \if@fleqn
+            \hskip-\@mathmargin
+            \def\multline@indent{\hskip\@mathmargin}% put it back
+        \else
+            \hfilneg
+            \def\multline@indent{\hskip\multlinegap}%
+        \fi
+        \iftagsleft@
+            \iftag@
+                \begingroup
+                    \ifshifttag@
+                        \rlap{\vbox{%
+                                \normalbaselines
+                                \hbox{%
+                                    \strut@
+                                    \make@display@tag
+                                }%
+                                \vbox to\lineht@{}%
+                                \raise@tag
+                        }}%
+%    \end{macrocode}
+%    If the equation tag doesn't fit on the same line with the first
+%    line of the display, we'll indent the first line by
+%    \cn{multlinegap}.  This is a change from \pkg{amstex}, where the
+%    first line would have been flush against the left margin in this
+%    case.  A corresponding change will be made in \cs{rendmultline@}.
+%    \begin{macrocode}
+                        \multline@indent
+                    \else
+                        \setbox\z@\hbox{\make@display@tag}%
+                        \dimen@\@mathmargin \advance\dimen@-\wd\z@
+                        \ifdim\dimen@<\multlinetaggap
+                          \dimen@\multlinetaggap
+                        \fi
+                        \box\z@ \hskip\dimen@\relax
+                    \fi
+                \endgroup
+            \else
+                \multline@indent
+            \fi
+        \else
+            \multline@indent
+        \fi
+    #1%
+}
+%    \end{macrocode}
+%
+%    An extra level of indirection for the closing \verb'$' in multline
+%    allows us to avoid getting an extra thinmuskip from a final
+%    mathpunct in the equation contents, when equation numbers are on
+%    the right. If we did not use this workaround, the sequence of
+%    elements for a final comma would be, e.g.,
+% \begin{verbatim}
+% ... ,<hskip><box containing equation number>
+% \end{verbatim}
+%    which is equivalent to a sequence \verb'<mathpunct><mathord>' as
+%    far as the automatic math spacing is concerned.
+%    \begin{macrocode}
+\def\endmultline@math{$}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\lendmultline@}
+%    Bug fix: changed \cs{crcr} to \cs{math@cr} so that \cs{@eqpen}
+%    gets reset properly if \cs{displaybreak} is used on the
+%    penultimate line of an \env{align}.
+%    \begin{macrocode}
+\def\lendmultline@{%
+        \hfilneg
+        \hskip\multlinegap
+        \math@cr
+    \egroup
+    $$%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\rendmultline@}
+%    \begin{macrocode}
+\def\rendmultline@{%
+    \iftag@
+        $\let\endmultline@math\relax
+            \ifshifttag@
+                \hskip\multlinegap
+%    \end{macrocode}
+%    Added depth to correct vertical spacing of shifted equation
+%    tags.---dmj, 1994/12/29
+%    \begin{macrocode}
+                \llap{\vtop{%
+                    \raise@tag
+                    \normalbaselines
+                    \setbox\@ne\null
+                    \dp\@ne\lineht@
+                    \box\@ne
+                    \hbox{\strut@\make@display@tag}%
+                }}%
+            \else
+                \hskip\multlinetaggap
+                \make@display@tag
+            \fi
+    \else
+        \hskip\multlinegap
+    \fi
+    \hfilneg
+%    \end{macrocode}
+%    Use \cs{math@cr} rather than just \cs{crcr} so that \cs{@eqpen}
+%    gets reset properly if \cs{displaybreak} is used.
+%    \begin{macrocode}
+        \math@cr
+    \egroup$$%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\mmeasure@}
+%    \begin{macrocode}
+\def\mmeasure@#1{%
+    \begingroup
+        \measuring@true
+%    \end{macrocode}
+%    We use \cs{begin/endgroup} rather than |{}| in this definition of
+%    \cn{label} because the latter would create an extra (wasteful of
+%    main mem) null box in the current math list. [mjd, 1995/01/17]
+%    \begin{macrocode}
+        \def\label##1{%
+          \begingroup\measuring@false\label@in@display{##1}\endgroup}%
+        \def\math@cr@@@{\cr}%
+        \let\shoveleft\@iden \let\shoveright\@iden
+        \savecounters@
+        \global\row@\z@
+        \setbox\@ne\vbox{%
+            \global\let\df@tag\@empty
+            \halign{%
+                \setboxz@h{\@lign$\m@th\displaystyle{}##$}%
+                \iftagsleft@
+                    \ifnum\row@=\@ne
+                        \global\totwidth@\wdz@
+                        \global\lineht@\ht\z@
+                    \fi
+                \else
+                    \global\totwidth@\wdz@
+                    \global\lineht@\dp\z@
+                \fi
+                \crcr
+                #1%
+                \crcr
+            }%
+        }%
+        \ifx\df@tag\@empty\else\global\tag@true\fi
+        \if@eqnsw\global\tag@true\fi
+        \iftag@
+            \setboxz@h{%
+                \if@eqnsw
+                    \stepcounter{equation}%
+                    \tagform@\theequation
+                \else
+                    \df@tag
+                \fi
+            }%
+            \global\tagwidth@\wdz@
+            \dimen@\totwidth@
+            \advance\dimen@\tagwidth@
+            \advance\dimen@\multlinetaggap
+            \iftagsleft@\else
+                \if@fleqn
+                    \advance\dimen@\@mathmargin
+                \fi
+            \fi
+            \ifdim\dimen@>\displaywidth
+                \global\shifttag@true
+            \else
+                \global\shifttag@false
+            \fi
+        \fi
+        \restorecounters@
+    \endgroup
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%    \begin{macro}{\shoveleft}
+%    \begin{macro}{\shoveright}
+%    \cs{shoveleft} and \cs{shoveright} need to do slightly different
+%    things depending on whether tags are on the left or the right and
+%    whether we're in \opt{fleqn} mode.  For compactness of code, we
+%    make the appropriate decisions at ``compile'' time rather than at
+%    load time.
+%
+%    TODO: Investigate making \cs{shoveright} behave ``properly''(?) if
+%    used on the first line of a \env{multline} and make \cs{shoveleft}
+%    behave properly if used on the last line of a \env{multline}. But
+%    in his \fn{amstex.doc} Spivak indicates those commands should never
+%    be used on a first or last line. Perhaps better to leave the
+%    question open unless/until real-life examples turn up.
+%    \begin{macrocode}
+\iftagsleft@
+    \def\shoveright#1{%
+        #1%
+        \hfilneg
+        \hskip\multlinegap
+    }
+\else
+    \def\shoveright#1{%
+        #1%
+        \hfilneg
+        \iftag@
+            \ifshifttag@
+                \hskip\multlinegap
+            \else
+                \hskip\tagwidth@
+                \hskip\multlinetaggap
+            \fi
+        \else
+            \hskip\multlinegap
+        \fi
+    }
+\fi
+
+\if@fleqn
+    \def\shoveleft#1{#1}%
+\else
+    \iftagsleft@
+        \def\shoveleft#1{%
+            \setboxz@h{$\m@th\displaystyle{}#1$}%
+            \setbox\@ne\hbox{$\m@th\displaystyle#1$}%
+            \hfilneg
+            \iftag@
+                \ifshifttag@
+                    \hskip\multlinegap
+                \else
+                    \hskip\tagwidth@
+                    \hskip\multlinetaggap
+                \fi
+            \else
+                \hskip\multlinegap
+            \fi
+            \hskip.5\wd\@ne
+            \hskip-.5\wdz@
+            #1%
+        }
+    \else
+        \def\shoveleft#1{%
+            \setboxz@h{$\m@th\displaystyle{}#1$}%
+            \setbox\@ne\hbox{$\m@th\displaystyle#1$}%
+            \hfilneg
+            \hskip\multlinegap
+            \hskip.5\wd\@ne
+            \hskip-.5\wdz@
+            #1%
+        }
+    \fi
+\fi
+%    \end{macrocode}
+%    \end{macro}
+%    \end{macro}
+%
+%    \subsection{The \env{equation} environment}
+%
+%    Rewritten from the ground up for version 2.0 to fix no-shrink and
+%    no-shortskips bugs [mjd,2000/01/06].
+%
+%    Standard \latex/ provides three environments for one-line equations:
+%    \cn{[}\cn{]}, \env{equation}, and \env{displaymath}. We add
+%    \env{equation*} as a synonym for \env{displaymath}.
+%    \begin{macrocode}
+\@saveprimitive\leqno\@@leqno
+\@saveprimitive\eqno\@@eqno
+\def\eqno{\@@eqno\let\eqno\relax\let\leqno\relax}
+\def\leqno{\@@leqno\let\leqno\relax\let\eqno\relax}
+%
+\let\veqno=\@@eqno
+\iftagsleft@ \let\veqno=\@@leqno \fi
+%    \end{macrocode}
+%
+%    Support for the \pkg{showkeys} package: provide no-op definitions
+%    for a couple of SK functions, if they are not already defined. Then
+%    we can just call them directly in our code without any extra fuss.
+%    If the \pkg{showkeys} package is loaded later, our trivial
+%    definitions will get overridden and everything works fine.
+%    \begin{macrocode}
+\@ifundefined{SK@@label}{%
+  \let\SK@@label\relax \let\SK@equationtrue\relax
+}{}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\reset@equation\@empty
+%    \end{macrocode}
+%
+%    Cf \cs{tag@in@align}. This is a bit of a mess though. Could use
+%    some work. [mjd,1999/12/21]
+%    \begin{macrocode}
+\let\alt@tag\@empty
+\def\tag@in@display#1#{\relax\tag@in@display@a{#1}}
+\def\tag@in@display@a#1#2{%
+  \iftag@
+    \invalid@tag{Multiple \string\tag}\relax
+  \else
+    \global\tag@true \nonumber \reset@equation \st@rredtrue
+    \if *\string#1%
+      \gdef\alt@tag{\def\SK@tagform@{#2\@gobble}%
+        \ifx\SK@@label\relax \let\tagform@\SK@tagform@ \fi
+      }%
+      \make@df@tag@@{#2}%
+    \else
+      \make@df@tag@@@{#2}%
+    \fi
+  \fi
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\restore@hfuzz\@empty
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\mathdisplay#1{%
+  \ifmmode \@badmath
+  \else
+    $$\def\@currenvir{#1}%
+%    \end{macrocode}
+%    Allow use of \cn{displaybreak}.
+%    \begin{macrocode}
+    \let\dspbrk@context\z@
+%    \end{macrocode}
+%    Although in some cases simpler label handling would seem to be
+%    sufficient, always using \cs{label@in@display} makes it easier to
+%    support the \pkg{showkeys} package.
+%    \begin{macrocode}
+    \let\tag\tag@in@display \let\label\label@in@display \SK@equationtrue
+    \global\let\df@label\@empty \global\let\df@tag\@empty
+    \global\tag@false
+    \let\mathdisplay@push\mathdisplay@@push
+    \let\mathdisplay@pop\mathdisplay@@pop
+    \if@fleqn
+%    \end{macrocode}
+%    Turn off overfull box messages temporarily\mdash otherwise there
+%    would be unwanted extra ones emitted during our measuring
+%    operations.
+%    \begin{macrocode}
+      \edef\restore@hfuzz{\hfuzz\the\hfuzz\relax}%
+      \hfuzz\maxdimen
+%    \end{macrocode}
+%    Initially set the equation body in a box of displaywidth. Then if
+%    the box is not overfull, as we find by checking \cs{badness}, we
+%    have acquired useful information for the subsequent processing.
+%    \begin{macrocode}
+      \setbox\z@\hbox to\displaywidth\bgroup
+        \let\split@warning\relax \restore@hfuzz
+        \everymath\@emptytoks \m@th $\displaystyle
+    \fi
+  \fi
+}
+%    \end{macrocode}
+%
+%    Arg 1 is not currently used. I thought it might come in handy for
+%    error messages.
+%    \begin{macrocode}
+\def\endmathdisplay#1{%
+  \ifmmode \else \@badmath \fi
+  \endmathdisplay@a
+  $$%
+%    \end{macrocode}
+%    I guess the following code means this structure is non-reentrant.
+%    But there is plenty of scope for tricky bugs here; suppressing them
+%    by brute force at least makes it possible to get things working
+%    correctly for normal use. [mjd,2000/01/06]
+%    \begin{macrocode}
+  \global\let\df@label\@empty \global\let\df@tag\@empty
+  \global\tag@false \global\let\alt@tag\@empty
+  \global\@eqnswfalse
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\endmathdisplay@a{%
+  \if@eqnsw \gdef\df@tag{\tagform@\theequation}\fi
+  \if@fleqn \@xp\endmathdisplay@fleqn
+  \else \ifx\df@tag\@empty \else \veqno \alt@tag \df@tag \fi
+    \ifx\df@label\@empty \else \@xp\ltx@label\@xp{\df@label}\fi
+  \fi
+  \ifnum\dspbrk@lvl>\m@ne
+    \postdisplaypenalty -\@getpen\dspbrk@lvl
+    \global\dspbrk@lvl\m@ne
+  \fi
+}
+%    \end{macrocode}
+%
+%    A boolean variable: Was that last box overfull or not? A value of 0
+%    means yes, it was overfull.
+%    \begin{macrocode}
+\let\too@wide\@ne
+%    \end{macrocode}
+%
+%    Special handling is needed for flush-left equations. We need to
+%    measure the equation body (found in box 0 after we close it with
+%    the \cs{egroup}). Then after a fairly normal test to see if it fits
+%    within the available space, we need to consider overlapping into
+%    the displayindent area if displayindent is nonzero (as in an
+%    indented list). If there is an equation number we may have to shift
+%    it by hand to a separate line when there is not enough room;
+%    we can no longer take advantage of the automatic shifting provided
+%    by the \cn{leqno}, \cn{eqno} primitives.
+%
+%    We initially add \cs{@mathmargin} glue at the end of box 0 to get
+%    an accurate overfull test. If \cs{@mathmargin} contains any shrink
+%    then we cannot reliably tell whether the box will be overfull or
+%    not simply by doing hand calculations from the actual width of the
+%    equation body. We have to actually set the box and find out what
+%    happens.
+%
+%    On the other hand if we put the \cs{@mathmargin} glue at the
+%    beginning of the box it's awkward to remove it afterwards. So we
+%    first put it in at the end and later we will move it to the
+%    beginning as needed.
+%
+%    \begin{macrocode}
+\def\endmathdisplay@fleqn{%
+  $\hfil\hskip\@mathmargin\egroup
+%    \end{macrocode}
+%    We need to save the information about whether box 0 was overfull in
+%    a variable, otherwise it will disappear in the next setbox
+%    operation. And we couldn't set the equation number box earlier than
+%    now, because the body of the equation might have contained a
+%    \cs{tag} command (well, it could have been done, but this way
+%    we can reuse the tag-handling code from elsewhere).
+%    \begin{macrocode}
+  \ifnum\badness<\inf@bad \let\too@wide\@ne \else \let\too@wide\z@ \fi
+  \ifx\@empty\df@tag
+  \else
+    \setbox4\hbox{\df@tag
+      \ifx\df@label\@empty \else \@xp\ltx@label\@xp{\df@label}\fi
+    }%
+  \fi
+  \csname emdf@%
+    \ifx\df@tag\@empty U\else \iftagsleft@ L\else R\fi\fi
+  \endcsname
+}
+%    \end{macrocode}
+%
+%    For an unnumbered flush-left equation we hope first that the
+%    the contents fit within displaywidth. If not we need to fall back
+%    on a more complicated reboxing operation.
+%    \begin{macrocode}
+\def\emdf@U{%
+  \restore@hfuzz
+  \ifodd\too@wide % not too wide: just need to swap the glue around
+    \hbox to\displaywidth{\hskip\@mathmargin\unhbox\z@\unskip}%
+  \else % M+B > displaywidth
+    \emdf@Ua
+  \fi
+}
+%    \end{macrocode}
+%
+%    Some notation: $M$ \cs{@mathmargin}, $B$ the width of the equation
+%    body, $I$ \cs{displayindent}, $D$ \cs{displaywidth}, $N$ the width
+%    of the equation number (aka the tag), $S$ \cs{mintagsep}, $C$
+%    \cs{columnwidth}. If $M+B > \mbox{displaywidth}$, and if we assume
+%    $M$ contains shrink, then the only solution left is to encroach
+%    into the displayindent space.
+%    \begin{macrocode}
+\def\emdf@Ua{%
+  \hbox to\columnwidth{%
+    \ifdim\displayindent>\z@
+      \hskip\displayindent minus\displayindent
+    \fi
+    \hskip\@mathmargin \unhbox\z@ \unskip
+  }%
+  \displayindent\z@ \displaywidth\columnwidth
+}
+%    \end{macrocode}
+%
+%    Find out first if the tag fits in ideal position. If so we can just
+%    plunk down box 2. Otherwise we need to do something more complicated.
+%    \begin{macrocode}
+\def\emdf@R{%
+  \setbox\tw@\hbox to\displaywidth{%
+    \hskip\@mathmargin \unhcopy\z@\unskip\hfil\hskip\mintagsep\copy4
+  }%
+  \restore@hfuzz
+  \ifnum\badness<\inf@bad \box\tw@ \else \emdf@Ra \fi
+}
+%    \end{macrocode}
+%
+%    We shift the equation number to line 2 if it does not fit within
+%    \cs{displaywidth}. Note that we do not first attempt to let the
+%    equation body shift leftward into the \cs{displayindent} space. If
+%    that is desired it will have to be done by hand by adding negative
+%    space at the beginning of the equation body. I don't expect this to
+%    arise very often in practice since most of the time
+%    \cs{displayindent} is zero anyway.
+%    \begin{macrocode}
+\def\emdf@Ra{%
+  \skip@\displayindent minus\displayindent
+  \displayindent\z@ \displaywidth\columnwidth
+  \spread@equation \everycr{}\tabskip\z@skip
+  \halign{\hbox to\displaywidth{##}\cr
+    \relax
+    \ifdim\skip@>\z@ \hskip\skip@ \fi
+    \hskip\@mathmargin\unhbox\z@\unskip\hfil\cr
+    \noalign{\raise@tag}%
+    \hfil\box4 \cr}%
+}
+%    \end{macrocode}
+%
+%    Find out first if the tag fits in ideal position. If so we can just
+%    plunk down box 2. Otherwise we need to do something more
+%    complicated.
+%    \begin{macrocode}
+\def\emdf@L{%
+%    \end{macrocode}
+%    Calculate the difference between $M$ and $N+S$. If the latter is
+%    greater, we don't want to add any extra glue between the number and
+%    the equation body. Otherwise the amount that we want to add is
+%    \verb'x minus x' where $x=M-(N+S)$. I.e., the distribution of
+%    spaces across the line is $N,S,x minus x,B,hfil$.
+%    \begin{macrocode}
+  \@tempdima\@mathmargin
+  \advance\@tempdima-\wd4 \advance\@tempdima-\mintagsep
+  \skip@\@tempdima minus\@tempdima
+  \setbox\tw@\hbox to\displaywidth{%
+    \copy4\hskip\mintagsep
+    \ifdim\skip@>\z@ \hskip\skip@\fi
+    \unhcopy\z@\unskip
+  }%
+  \restore@hfuzz
+  \ifnum\badness<\inf@bad \box\tw@ \else \emdf@La \fi
+}
+%    \end{macrocode}
+%
+%    If the equation body and equation number will not fit on the same
+%    line, we put the number on line 1 and the body on line 2, with the
+%    body positioned as for an unnumbered equation.
+%    \begin{macrocode}
+\def\emdf@La{%
+  \spread@equation \everycr{}\tabskip\z@skip
+  \halign{\hbox to\displaywidth{##}\cr
+    \box4 \hfil \cr
+    \noalign{\raise@tag}%
+    \hskip\@mathmargin\unhbox\z@\unskip\hfil\cr}%
+}
+%    \end{macrocode}
+%
+%    If someone has \verb'\[ \]' nested inside a minipage environment
+%    nested inside a numbered equation, the mathdisplay variables that
+%    are global will get out of whack unless we take extra care. So we
+%    make a stack and push all the variables before entering mathdisplay
+%    and pop them afterwards. But we can save a little work by not doing
+%    this at the top level, only at inner levels.
+%    \begin{macrocode}
+\newtoks\mathdisplay@stack
+\let\mathdisplay@push\@empty
+\def\mathdisplay@@push{%
+  \begingroup
+  \toks@\@xp{\df@label}\@temptokena\@xp{\df@tag}%
+  \toks8\@xp{\alt@tag}%
+  \edef\@tempa{%
+    \global\if@eqnsw\@nx\@eqnswtrue\else\@nx\@eqnswfalse\fi
+    \global\iftag@\@nx\tag@false\else\@nx\tag@true\fi
+    \gdef\@nx\df@label{\the\toks@}\gdef\@nx\df@tag{\the\@temptokena}%
+    \gdef\@nx\alt@tag{\the\toks8}%
+    \global\mathdisplay@stack{\the\mathdisplay@stack}%
+  }%
+  \global\mathdisplay@stack\@xp{\@tempa}
+  \endgroup
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\mathdisplay@pop\@empty
+\def\mathdisplay@@pop{\the\mathdisplay@stack}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\renewenvironment{equation}{%
+  \incr@eqnum
+  \mathdisplay@push
+  \st@rredfalse \global\@eqnswtrue
+  \mathdisplay{equation}%
+}{%
+  \endmathdisplay{equation}%
+  \mathdisplay@pop
+  \ignorespacesafterend
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\newenvironment{equation*}{%
+  \mathdisplay@push
+  \st@rredtrue \global\@eqnswfalse
+  \mathdisplay{equation*}%
+}{%
+  \endmathdisplay{equation*}%
+  \mathdisplay@pop
+  \ignorespacesafterend
+}
+%    \end{macrocode}
+%
+%    Note: \latex/ defines the \env{displaymath} environment in
+%    terms of \cn{[} and \cn{]}.
+%    \begin{macrocode}
+\DeclareRobustCommand{\[}{\begin{equation*}}
+\DeclareRobustCommand{\]}{\end{equation*}}
+%    \end{macrocode}
+%
+%    The usual \cs{endinput} to ensure that random garbage at the end of
+%    the file doesn't get copied by \fn{docstrip}.
+%    \begin{macrocode}
+\endinput
+%    \end{macrocode}
+%
+% \section{Credits}
+%
+%    Much of the code for the \pkg{amsmath} package had its orgin in
+%    \fn{amstex.tex}, written by Michael Spivak. The initial work of
+%    porting \fn{amstex.tex} to \fn{amstex.sty} was done in 1988--1989
+%    by Frank Mittelbach and Rainer Sch\"opf. In 1994 David M. Jones
+%    added the support for the \opt{fleqn} option and did extensive
+%    improvements to the \env{align[at]} family of environments and to
+%    the equation number handling in general. Michael Downes at the AMS
+%    served as coordinator for the efforts of Mittelbach, Sch\"opf, and
+%    Jones, and has contributed various bug fixes and additional
+%    refinements over time.
+%
+%    Versions 1.0 and 1.1 of the package carried the name \pkg{amstex}
+%    instead of \pkg{amsmath}, to indicate its origins; the name was
+%    changed in 1994 to make it user-oriented rather than
+%    history-oriented.
+%
+% \changes{v1.2b}{1995/02/15}{Added kern -alignsep@ to math@cr@@@align}
+% \changes{v1.2b}{1995/02/15}{Modified one piece of measure@}
+% \changes{v1.2b}{1995/02/15}{Cleaned out some obsolete commentary}
+% \changes{v1.2b}{1995/02/15}{Changed kern tagshift@ to kern -tagshift@
+%   in the reqno case of place@tag}
+%
+% \changes{v1.2c}{1996/11/01}{Changed `over etc from error to warning}
+%
+% \changes{v1.2d}{1997/03/20}{%
+%   Removed dependency on mixed-case fd file names}
+%
+% \CheckSum{6699}
+% \Finale
diff --git a/Master/texmf-dist/source/latex/amsmath/amsopn.dtx b/Master/texmf-dist/source/latex/amsmath/amsopn.dtx
new file mode 100644
index 00000000000..fd60965d788
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amsopn.dtx
@@ -0,0 +1,254 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsopn.dtx",
+%%%   version   = "2.01",
+%%%   date      = "1999/12/14",
+%%%   time      = "17:30:59 EST",
+%%%   checksum  = "46904 254 1037 9431",
+%%%   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  = "operator name, amsopn, DeclareMathOperator",
+%%%   supported = "yes",
+%%%   abstract  = "This is a \LaTeX{} package that provides a
+%%%                DeclareMathOperator command for defining named
+%%%                operators like sin and lim.",
+%%%   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.",
+%%% }
+%%% ====================================================================
+%
+%\iffalse
+%<*driver>
+\documentclass{amsdtx}
+\begin{document}
+\title{The \pkg{amsopn} package}
+\author{Michael Downes}
+\date{Version \fileversion, \filedate}
+\hDocInput{amsopn.dtx}
+\end{document}
+%</driver>
+%\fi
+%
+% \maketitle
+%
+% \MakeShortVerb\|
+%
+% \section{Introduction}
+%
+%    The \pkg{amsopn} package provides a command
+%    \cn{DeclareMathOperator} for defining new `math operator names'
+%    similar to the standard function names \cn{sin}, \cn{lim},
+%    \cn{max}, etc.
+%
+% \StopEventually{}
+%
+%    Standard file identification.
+%    \begin{macrocode}
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must December 1994 or later
+\ProvidesPackage{amsopn}[1999/12/14 v2.01 operator names]
+%    \end{macrocode}
+%
+%    What \cs{nolimits@} does is keep a \cn{limits} typed by the user
+%    from having any effect. This is used for operatornames whose
+%    standard usage is never to have limits.
+%    \begin{macrocode}
+\def\nolimits@{\@ifnextchar\limits{\nolimits\@gobble}{\nolimits}}
+%    \end{macrocode}
+%
+%    In operator names, it is sometimes desired to have text-mode
+%    punctuation characters such as |*-/:'|.
+%    Because the body of an
+%    operator name is set in math mode, these few punctuation
+%    characters will not come out right (wrong symbol/and or wrong
+%    spacing). The purpose of \cs{newmcodes@} is to make them act like
+%    their normal text versions.
+%
+%    Where practical, we use decimal numbers to cut down main mem
+%    usage (\qc{\"} not needed).
+%    \begin{macrocode}
+\begingroup \catcode`\"=12
+\gdef\newmcodes@{\mathcode`\'39\mathcode`\*42\mathcode`\."613A%
+%    \end{macrocode}
+%    Define \cs{std@minus} for \cs{relbar} use; otherwise there are
+%    problems with arrows constructed with relbar.
+%    \begin{macrocode}
+  \ifnum\mathcode`\-=45 \else
+    \mathchardef\std@minus\mathcode`\-\relax
+  \fi
+  \mathcode`\-45\mathcode`\/47\mathcode`\:"603A\relax}
+\endgroup
+%    \end{macrocode}
+%
+%    The command \cs{operatorname} prints its argument as a `math
+%    operator' like \cs{sin} or \cs{det}, with proper font and spacing.
+%    \begin{macrocode}
+\DeclareRobustCommand{\operatorname}{%
+  \@ifstar{\qopname\newmcodes@ m}%
+          {\qopname\newmcodes@ o}}%
+%    \end{macrocode}
+%    In the interior of the \cs{mathop} we need a null object (we choose
+%    a zero kern for minimum waste of main mem) in order to guard
+%    against the case where \arg{3} is a single letter; \tex/ will seize
+%    it and center it on the math axis if there is nothing else inside
+%    the \cs{mathop} atom.
+%    \begin{macrocode}
+\DeclareRobustCommand{\qopname}[3]{%
+  \mathop{#1\kern\z@\operator@font#3}%
+  \csname n#2limits@\endcsname}
+%    \end{macrocode}
+%
+%  \begin{macro}{\DeclareMathOperator}
+%    The command \cn{DeclareMathOperator} defines the first argument to
+%    be an operator name whose text is the second argument. The star
+%    form means that the operator name should take limits (like \cn{max}
+%    or \cn{lim}).
+%    \begin{macrocode}
+\newcommand{\DeclareMathOperator}{%
+  \@ifstar{\@declmathop m}{\@declmathop o}}
+%    \end{macrocode}
+%  \end{macro}
+%
+%    In the basic set of operator names (below) we did not use
+%    \cn{DeclareRobustCommand} because of the hash table cost. But we
+%    use it here to minimize the chances of trouble, since we are
+%    producing a user-defined command.
+%    \begin{macrocode}
+\long\def\@declmathop#1#2#3{%
+  \@ifdefinable{#2}{%
+    \DeclareRobustCommand{#2}{\qopname\newmcodes@#1{#3}}}}
+%
+\@onlypreamble\DeclareMathOperator
+\@onlypreamble\@declmathop
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\arccos{\qopname\relax o{arccos}}
+\def\arcsin{\qopname\relax o{arcsin}}
+\def\arctan{\qopname\relax o{arctan}}
+\def\arg{\qopname\relax o{arg}}
+\def\cos{\qopname\relax o{cos}}
+\def\cosh{\qopname\relax o{cosh}}
+\def\cot{\qopname\relax o{cot}}
+\def\coth{\qopname\relax o{coth}}
+\def\csc{\qopname\relax o{csc}}
+\def\deg{\qopname\relax o{deg}}
+\def\det{\qopname\relax m{det}}
+\def\dim{\qopname\relax o{dim}}
+\def\exp{\qopname\relax o{exp}}
+\def\gcd{\qopname\relax m{gcd}}
+\def\hom{\qopname\relax o{hom}}
+\def\inf{\qopname\relax m{inf}}
+\def\injlim{\qopname\relax m{inj\,lim}}
+\def\ker{\qopname\relax o{ker}}
+\def\lg{\qopname\relax o{lg}}
+\def\lim{\qopname\relax m{lim}}
+\def\liminf{\qopname\relax m{lim\,inf}}
+\def\limsup{\qopname\relax m{lim\,sup}}
+\def\ln{\qopname\relax o{ln}}
+\def\log{\qopname\relax o{log}}
+\def\max{\qopname\relax m{max}}
+\def\min{\qopname\relax m{min}}
+\def\Pr{\qopname\relax m{Pr}}
+\def\projlim{\qopname\relax m{proj\,lim}}
+\def\sec{\qopname\relax o{sec}}
+\def\sin{\qopname\relax o{sin}}
+\def\sinh{\qopname\relax o{sinh}}
+\def\sup{\qopname\relax m{sup}}
+\def\tan{\qopname\relax o{tan}}
+\def\tanh{\qopname\relax o{tanh}}
+%    \end{macrocode}
+%
+% \begin{macro}{\operator@font}
+% \begin{macro}{\operatorfont}
+%    This command is provided to allow the document styles to decide in
+%    which way math operators like `max' or `log' are typeset. The
+%    default is to set them in \meta{math group} zero of the current
+%    math version. The original name was \cs{operator@font}, retained
+%    for compatibility; the second name was added to make it
+%    more accessible so that users can call this font for use in special
+%    constructs that are not ordinary operator names but conceptually
+%    related.
+%    \begin{macrocode}
+\def\operator@font{\mathgroup\symoperators}
+\def\operatorfont{\operator@font}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%    For backwards compatibility we keep this old command name for the
+%    time being:
+%    \begin{macrocode}
+\def\operatornamewithlimits{\operatorname*}
+%    \end{macrocode}
+%
+%    These macros use \cs{mathpalette}s so that they will change size
+%    in script and scriptscript styles, though it's hard to imagine they
+%    will ever be used there (the arrows, particularly, look bad in
+%    subscript sizes). Notice that the use of \cs{ex@} means that the
+%    vertical spacing may not be optimal in script and scriptscript
+%    sizes. Unfortunately \tex/ provides no easy way to do math mode
+%    vertical spacing that varies with current math style like mu units.
+%    \begin{macrocode}
+\def\varlim@#1#2{%
+  \vtop{\m@th\ialign{##\cr
+    \hfil$#1\operator@font lim$\hfil\cr
+    \noalign{\nointerlineskip\kern1.5\ex@}#2\cr
+    \noalign{\nointerlineskip\kern-\ex@}\cr}}%
+}
+\def\varinjlim{%
+  \mathop{\mathpalette\varlim@{\rightarrowfill@\textstyle}}\nmlimits@
+}
+\def\varprojlim{%
+  \mathop{\mathpalette\varlim@{\leftarrowfill@\textstyle}}\nmlimits@
+}
+\def\varliminf{\mathop{\mathpalette\varliminf@{}}\nmlimits@}
+\def\varliminf@#1{\@@underline{\vrule\@depth.2\ex@\@width\z@
+   \hbox{$#1\m@th\operator@font lim$}}}
+\def\varlimsup{\mathop{\mathpalette\varlimsup@{}}\nmlimits@}
+\def\varlimsup@#1{\@@overline{\hbox{$#1\m@th\operator@font lim$}}}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\nmlimits@\displaylimits
+\DeclareOption{namelimits}{\let\nmlimits@\displaylimits}
+\DeclareOption{nonamelimits}{\let\nmlimits@\nolimits}
+\ProcessOptions\relax
+%    \end{macrocode}
+%
+%    If we don't load the \pkg{amsgen} package then the use of \cs{ex@}
+%    in \cs{varlim@} will lead to trouble.
+%    \begin{macrocode}
+\RequirePackage{amsgen}\relax
+%    \end{macrocode}
+%
+%    The usual \cs{endinput} to ensure that random garbage at the end of
+%    the file doesn't get copied by \fn{docstrip}.
+%    \begin{macrocode}
+\endinput
+%    \end{macrocode}
+%
+% \changes{v1.2a}{1995/02/20}{Fixed operatorname
+%   (shouldn't have same limits behavior as operatorname*)}
+% \CheckSum{300}
+% \Finale
diff --git a/Master/texmf-dist/source/latex/amsmath/amstext.dtx b/Master/texmf-dist/source/latex/amsmath/amstext.dtx
new file mode 100644
index 00000000000..0ef49f67a64
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amstext.dtx
@@ -0,0 +1,314 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amstext.dtx",
+%%%   version   = "2.01",
+%%%   date      = "2000/06/29",
+%%%   time      = "08:34:33 EDT",
+%%%   checksum  = "09860 314 1647 12368",
+%%%   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  = "latex, math, text",
+%%%   abstract  = "This is a \LaTeX{} package that defines a
+%%%                \cn{\text} macro, which makes it easy to
+%%%                incorporate fragments of text inside a displayed
+%%%                equation or a sub or superscript. Font sizes
+%%%                are automatically scaled in sub/superscripts.",
+%%%   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.",
+%%% }
+%%% ====================================================================
+% \iffalse
+%<*driver>
+\documentclass{amsdtx}
+\usepackage{amstext}
+\begin{document}
+\title{The \pkg{amstext} package}
+\author{Frank Mittelbach \and Rainer Sch\"opf}
+\date{Version \fileversion, \filedate}
+\hDocInput{amstext.dtx}
+\end{document}
+%</driver>
+% \fi
+%
+% \maketitle
+%
+% \MakeShortVerb\|
+%
+% \section{Introduction}
+%
+%    This style file implements the \AmSTeX{} macro \cn{text} for use
+%    with the new font selection scheme.  The \cn{text} macro is a
+%    sophisticated command which allows the user to insert ``normal text''
+%    into math formulas without worrying about correct sizes in sub- or
+%    superscripts. It can also be used in ordinary text; there it
+%    produces an unbreakable unit similar to \cn{mbox}.
+%
+%    Here is an example demonstrating some of its features:
+% \[
+%^^A mathbin spacing is nonscript glue, disappears in superscript
+%    x^{2\,\times\,\text{size of $y$}}
+%    \leq
+%    z_{i_{\text{upper bound of the array}}}
+% \]
+%    This was produced by
+% \begin{verbatim}
+% \[
+%    x^{2\,\times\,\text{size of $y$}}
+%    \leq
+%    z_{i_{\text{upper bound of the array}}}
+% \]
+% \end{verbatim}
+%
+%    Additionally this style file redefines an internal \fn{plain.tex}
+%    macro called \cs{mathhexbox} so that commands like \cn{dag}
+%    or \cn{P} will change sizes if used in math subscripts.
+%
+% \StopEventually{}
+%    Package information.
+%    \begin{macrocode}
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must be December 1994 or later
+\ProvidesPackage{amstext}[2000/06/29 v2.01]
+%    \end{macrocode}
+%
+% \section{The implementation}
+%    We need a few tools from \fn{amsgen.sty}.
+%    \begin{macrocode}
+\RequirePackage{amsgen}
+%    \end{macrocode}
+%
+% \begin{macro}{\text}
+%    Now we come to the \cn{text} macro which is used to place
+%    ordinary text inside of math formulas. If it is used outside
+%    math it will produce an unbreakable unit of text.
+%    \begin{macrocode}
+\DeclareRobustCommand{\text}{%
+  \ifmmode\expandafter\text@\else\expandafter\mbox\fi}
+%    \end{macrocode}
+%    At the present time (late 1994) the \latex/ internal function
+%    |\nfss@text| is used in |\ref|, in font commands like |\textbf|,
+%    and in a few text symbol definitions like |\$| and |\pounds|. By
+%    equating |\nfss@text| to |\text| we give it the ability of
+%    |\text| to change sizes properly if used in a subscript.
+%    \begin{macrocode}
+\let\nfss@text\text
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\text@}
+%    If \cn{text} is encountered inside math mode the macro
+%    \cs{text@} is called.  It has one mandatory argument, the text
+%    which should be produced.  Since we do not know in which math
+%    style we are currently in we call \cs{mathchoice} to tyeset
+%    our text in all four possible styles.
+%    \begin{macrocode}
+\def\text@#1{{\mathchoice
+%    \end{macrocode}
+%    To save token space we call a macro \cs{textdef@} which takes
+%    three arguments: the current math style, the corresponding size
+%    macro and the text to typeset possibly with some additional
+%    information for typesetting.
+%    \begin{macrocode}
+  {\textdef@\displaystyle\f@size{#1}}%
+%    \end{macrocode}
+%    The other three cases are similar except for the
+%    \cs{iffirstchoice} switch which we set to false. This is
+%    done to prevent \LaTeX{} macros like \cn{ref} or \cn{index}
+%    from writing their arguments more than once.
+%    \begin{macrocode}
+  {\textdef@\textstyle\f@size{\firstchoice@false #1}}%
+  {\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
+  {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
+%    \end{macrocode}
+%    Here we need to check whether a math size-change occurred inside
+%    the argument of \cn{text}. If so, restore
+%    \begin{macrocode}
+  \check@mathfonts
+  }%
+}
+%    \end{macrocode}
+%    The macros \cs{f@size}, \cs{sf@size} and \cs{ssf@size} hold the
+%    sizes which should be used when we are loading a new font for use
+%    in \cs{textfont}, \cs{scriptfont} and \cs{scriptscriptfont}. There
+%    is some question whether we should use use \cs{tf@size} or
+%    \cs{f@size} for the main size, but since the primary purpose of the
+%    \cn{text} macro is to switch back to text within a display, it
+%    seems that \cs{f@size} is the better choice. (Indeed it could be
+%    said that the \cn{text} actually provides two different functions:
+%    one for escaping out of math mode in a display to print some words,
+%    and the other for handling math objects that are named by a
+%    fragment of text, when \cn{operatorname} isn't the right choice.
+%    For the latter \cs{tf@size} might be more correct but for the
+%    former \cs{f@size} is clearly better.)
+% \end{macro}
+%
+% \begin{macro}{\textdef@}
+%    To typeset the argument of \cn{text} correctly we have to make
+%    several actions. We start by placing everything inside an
+%    \cs{hbox}.  But this is not enough: we need one extra level of
+%    grouping.  These extra braces are necessary because of the new
+%    font selection scheme which might produce an \cs{aftergroup}
+%    to globally restore some font values after the current group. To
+%    prevent any damage by this mechanism we add the braces thereby
+%    bringing the token inserted by \cs{aftergroup} inside the
+%    \cs{hbox}.\footnote{The mechanism will not produce a second
+%    \cs{aftergroup}. For more details see the technical documentation
+%    for NFSS2.}
+%    \begin{macrocode}
+\def\textdef@#1#2#3{\hbox{{%
+%    \end{macrocode}
+%    Since text typeset inside an \cs{hbox} always stays in the size
+%    of the text surrounding the formula we have to adjust this for
+%    script and scriptscript sizes.  For any math formula inside this
+%    argument this will be achieved by setting \cs{everymath} to
+%    the first argument of \cs{textdef@} since this argument
+%    contains the math size in the current typeset case of
+%    \cs{mathchoice}.
+%    Since \latex/ also knows about \cn{parbox}es and the
+%    \env{minipage} environment it might be neccessary to adjust
+%    \cs{everydisplay} too but this has to be tested further.
+%    \begin{macrocode}
+                    \everymath{#1}%
+%    \end{macrocode}
+%    The next line of code changes locally (i.e.\ inside the current
+%    \cs{hbox}) the value of \cs{f@size}. This macro holds the
+%    size for typesetting ordinary text (e.g.\ loading or selecting a
+%    new font via \cs{selectfont}). By changing it to a smaller
+%    value a following \cs{selectfont} will switch to the wanted
+%    size.
+%    \begin{macrocode}
+                    \let\f@size#2\selectfont
+%    \end{macrocode}
+%    Now we simply call the third argument and close all open groups.
+%    \begin{macrocode}
+                    #3}}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\iffirstchoice@}
+%    Here is the switch that we use to decide if \cn{ref} etc.\
+%    should print its warnings.  The default is true since
+%    normally these warnings shouldn't be suppressed.
+%    \begin{macrocode}
+\newif\iffirstchoice@
+\firstchoice@true
+%    \end{macrocode}
+% \end{macro}
+%
+% \subsection{Re-definition of \latex/ macros to work with \cn{text}}
+%
+%    If a counter-changing command occurs inside the argument of
+%    \cn{text}, we don't want the counter to be changed four times
+%    because \cn{stepcounter} and \cn{addtocounter} have global effect.
+%    So we add the \cs{iffirstchoice@} test to make the counter
+%    operations execute only once.
+%
+% \begin{macro}{\stepcounter}
+%    Use \cs{def} rather than \cn{renewcommand*} because the star-form
+%    (for non-\cs{long} definitions) doesn't work with the June 1994
+%    release of \latex/.
+%    \begin{macrocode}
+\def\stepcounter#1{%
+  \iffirstchoice@
+     \addtocounter{#1}\@ne
+     \begingroup \let\@elt\@stpelt \csname cl@#1\endcsname \endgroup
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\addtocounter}
+%    \begin{macrocode}
+\def\addtocounter#1#2{%
+  \iffirstchoice@
+  \@ifundefined {c@#1}{\@nocounterr {#1}}%
+    {\global \advance \csname c@#1\endcsname #2\relax}%
+  \fi}
+%    \end{macrocode}
+% \end{macro}
+%
+%    For \cn{ref}, \cn{pageref}, and indeed anything else that issues a
+%    warning or error, \cn{text} will produce four copies of the
+%    warning/error message. To suppress the last three copies, we change
+%    \cs{GenericInfo}, \cs{GenericWarning}, \cs{GenericError}.
+%    \begin{macrocode}
+\let\m@gobble\@empty
+\@xp\let\csname m@gobble4\endcsname\@gobblefour
+\long\@xp\def\csname m@gobble6\endcsname#1#2#3#4#5#6{}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\toks@{%
+  \csname m@gobble\iffirstchoice@\else 4\fi\endcsname
+  \protect}
+\edef\GenericInfo{\the\toks@
+  \@xp\@nx\csname GenericInfo \endcsname}
+%
+\edef\GenericWarning{\the\toks@
+  \@xp\@nx\csname GenericWarning \endcsname}
+%
+\toks@{%
+  \csname m@gobble\iffirstchoice@\else 6\fi\endcsname
+  \protect}
+\edef\GenericError{\the\toks@
+  \@xp\@nx\csname GenericError \endcsname}
+%    \end{macrocode}
+%
+%    At one time \cn{label}, \cs{@wrindex} and \cs{@wrglossary} were
+%    changed here too to use the \cs{iffirstchoice@} test but it seems
+%    that was a mistake because those are non-immediate writes.
+%    Something like
+% \begin{verbatim}
+% \text{something \index{foo}}
+% \end{verbatim}
+%    within a math formula would therefore \emph{lose the index term} if
+%    the surrounding context was not displaystyle. (Unlikely in
+%    practice, but not impossible.) [mjd,1994/12/09]
+%
+%\subsection{Applications of \cn{text}}
+%
+% \begin{macro}{\mathhexbox}
+%    We start with an re-definition of the \fn{plain.tex} macro
+%    \cs{mathhexbox}. (Although M. Spivak in \amstex/ uses the name
+%    \cs{mathhexbox@} for this purpose, I [FMi] don't see any reason
+%    to use a new name since the new definition is superior, has the
+%    same syntax and is used for the same purpose.)
+%    \begin{macrocode}
+\begingroup \catcode`\"=12
+\gdef\mathhexbox#1#2#3{\text{$\m@th\mathchar"#1#2#3$}}
+\endgroup
+%    \end{macrocode}
+%    This redefinition means that now symbols like \S, \P, \dag, \ldots,
+%    which are defined via \cs{mathhexbox} in \fn{plain.tex} or
+%    elsewhere now correctly change sizes if they are used in math mode.
+% \end{macro}
+%
+%    The usual \cs{endinput} to ensure that random garbage at the end of
+%    the file doesn't get copied by \fn{docstrip}.
+%    \begin{macrocode}
+\endinput
+%    \end{macrocode}
+%
+% \CheckSum{129}
+% \Finale
diff --git a/Master/texmf-dist/source/latex/amsmath/amsxtra.dtx b/Master/texmf-dist/source/latex/amsmath/amsxtra.dtx
new file mode 100644
index 00000000000..b0f19da7d07
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/amsxtra.dtx
@@ -0,0 +1,136 @@
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsxtra.dtx",
+%%%   version   = "1.2c",
+%%%   date      = "1999/11/15",
+%%%   time      = "14:15:41 EST",
+%%%   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",
+%%%   checksum  = "41382 136 533 5124",
+%%%   email     = "tech-support@ams.org (Internet)",
+%%%   codetable = "ISO/ASCII",
+%%%   keywords  = "latex, accentedsymbol, fracwithdelims, sphat",
+%%%   supported = "yes",
+%%%   abstract  = "This package contains some seldom-used commands
+%%%                ported from AMS-TeX in 1989.",
+%%%   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.",
+%%% }
+%%% ====================================================================
+%\iffalse
+%<*driver>
+\documentclass{amsdtx}
+\begin{document}
+\title{The \pkg{amsxtra} package}
+\author{American Mathematical Society\\Michael Downes}
+\date{Version \fileversion, \filedate}
+\hDocInput{amsxtra.dtx}
+\end{document}
+%</driver>
+%\fi
+%
+% \maketitle
+%
+% \MakeShortVerb\|
+%
+% \section{Introduction}
+%    The \pkg{amsxtra} package contains a few rarely used commands that
+%    were present in version 1.1 of the \pkg{amstex} package:
+%    \cn{fracwithdelims}, {accentedsymbol}, \cn{sphat}, \cn{spcheck},
+%    \cn{sptilde}, \cn{spdot}, \cn{spddot}, \cn{spdddot}, \cn{spbreve}.
+%    If any of these commands were used in existing documents, the
+%    \pkg{amsxtra} package provides a way to get them. But for 99\% of
+%    documents using the \pkg{amstex} package these commands are just
+%    wasted overhead.
+%
+% \StopEventually{}
+%
+%    \begin{macrocode}
+\ProvidesPackage{amsxtra}[1999/11/15 v1.2c]
+%    \end{macrocode}
+%    We need to ensure that either the \pkg{amstex} package or the
+%    \pkg{amsmath} package is loaded. This means we need something a
+%    little more complicated than \cn{RequirePackage}.
+%    \begin{macrocode}
+\@ifpackageloaded{amstex}{}{%
+  \RequirePackage{amsmath}[1996/10/01 v1.2]}
+%    \end{macrocode}
+%
+%    Superscripted accent characters.
+%    \begin{macrocode}
+\def\sphat{^{\mathchoice{}{}%
+ {\,\,\smash[b]{\hbox{\lower4\ex@\hbox{$\m@th\widehat{\null}$}}}}%
+ {\,\smash[b]{\hbox{\lower3\ex@\hbox{$\m@th\hat{\null}$}}}}}}
+\def\spcheck{^\vee}
+\def\sptilde{^\sim}
+\def\spdot{^{\hbox{\raise\ex@\hbox{\normalfont .}}}}
+\def\spddot{^{\hbox{\raise\ex@\hbox{\normalfont ..}}}}
+\def\spdddot{^{\hbox{\raise\ex@\hbox{\normalfont...}}}}
+\def\spbreve{^{\!\smash[b]{\hbox{\lower4\ex@\hbox{\u{}}}}}}
+%    \end{macrocode}
+%
+%    The \cn{fracwithdelims} command is skipped if the base package is
+%    \pkg{amsmath}; it is only needed if the base package is
+%    \pkg{amstex}.
+%    \begin{macrocode}
+\@ifpackageloaded{amstex}{%
+\def\fracwithdelims#1#2{\new@ifnextchar[{\fracwithdelims@#1#2}%
+  {\fracwithdelims@#1#2[\@empty]}}
+\def\fracwithdelims@#1#2[#3]#4#5{\relax
+  \ifx\@empty#3{#4\overwithdelims#1#2#5}\else
+  {#4\abovewithdelims#1#2#3\relax#5}\fi}
+}{}
+%    \end{macrocode}
+%
+%    Perhaps |\accentedsymbol| should use |\newsavebox|, but I don't
+%    know of any compelling reason at the moment.[mjd,5-Sep-1994]
+%    |\accentedsymbol| can only produce a single size of
+%    the given symbol, so you'd need two different versions for main
+%    size and subscript size.
+%    \begin{macrocode}
+\def\accentedsymbol#1#2{%
+  \expandafter\newbox
+    \csname\expandafter\@gobble\string#1@box\endcsname
+  \expandafter\setbox
+    \csname\expandafter\@gobble\string#1@box\endcsname
+      \hbox{$\m@th#2$}%
+  \newcommand{#1}{%
+    \expandafter\copy
+      \csname\expandafter\@gobble\string#1@box\endcsname{}}%
+}
+%    \end{macrocode}
+%
+%    Change the definition of \qc{\~} to automatically remove a
+%    preceding space as well as a following space, if present.
+%    \begin{macrocode}
+\def~{\nobreakspace}
+\DeclareRobustCommand{\nobreakspace}{\unskip\nobreak\ \ignorespaces}
+%    \end{macrocode}
+%
+%    The usual \cs{endinput} to ensure that random garbage at the end of
+%    the file doesn't get copied by \fn{docstrip}.
+%    \begin{macrocode}
+\endinput
+%    \end{macrocode}
+%
+% \CheckSum{113}
+% \Finale
diff --git a/Master/texmf-dist/source/latex/amsmath/install.txt b/Master/texmf-dist/source/latex/amsmath/install.txt
new file mode 100644
index 00000000000..fb4b835f282
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/install.txt
@@ -0,0 +1,108 @@
+install.txt for amsmath 2.16 [2004/08/05]
+
+If you have a relatively up-do-date TeX system that complies with the
+TeX Directory Structure (TDS) standard, installing according to the
+instructions in section I below is the easiest method.  See
+
+    http://www.tug.org/tds/tds.pdf
+
+for details on the TDS standard.  If your TeX system does not comply
+with the  TDS standard, please contact the manufactuer of your TeX
+system for details on how to install third-party TeX packages.
+
+The first step is to identify the TEXMF folder that contains files used
+by LaTeX. To do this, look for the file article.cls. If you find, e.g.,
+
+  /usr/local/share/texmf/tex/latex/base/article.cls
+
+then we will define TEXMF to mean /usr/local/share/texmf (everything
+up to the /tex/latex/base part). On a Windows system the file location
+will probably be something like
+
+  c:\sw\texmf\tex\latex\base\article.cls
+
+Then we define TEXMF to mean c:\sw\texmf. In the following instructions,
+wherever you see "TEXMF" substitute the folder name determined in this
+way.
+
+If this terminology does not make sense to you, seek assistance from
+a friend or colleague, or from the company that you bought your TeX
+system from, or contact us at
+
+  tech-support@ams.org
+  800-321-4AMS (321-4267) (USA/Canada)
+  401-455-4080
+
+========================================================================
+I. Installing the easy way.
+
+1. Download the following archive file:
+
+   ftp://ftp.ams.org/pub/tex/amsmath.zip
+
+2. View the following files to assess the effects of the upgrade on
+your system:
+
+  00readme.txt
+  install.txt (this file)
+  diffs-m.txt
+
+3. Backup the old version of AMS-LaTeX, preferably by moving it out of the
+TEXMF tree.
+
+4. Extract files from the .zip archive using your favorite unzip tool,
+setting the extraction target directory to the texmf root directory.
+
+5. To check the end result, you should find that you now have amsmath.sty
+in the following location
+
+  TEXMF/tex/latex/amsmath/amsmath.sty
+
+Some TeX systems also use a database to keep track of TeX input files.
+If this database is not updated, your TeX system might not find the
+newly installed files.  Consult your TeX system's documentation to
+determine how to update this database.
+
+6. Some files will appear in both the source area and another area.
+This is because the source form and the in-use form are identical.
+As distributed, the files in the source/latex/ams* areas comprise a
+complete copy of the distribution from which all the other files can
+be regenerated if need be. If you need to conserve disk space, you
+could delete the files in the source area, since they are not needed
+for normal use.
+
+========================================================================
+II. Installing the not-so-easy way.
+
+Download ftp://ftp.ams.org/pub/tex/amsmath.zip.
+
+Make the following directories, if not already present:
+
+  TEXMF/source/latex/amsmath
+  TEXMF/tex/latex/amsmath
+  TEXMF/doc/latex/amsmath
+
+Backup your old AMS-LaTeX files if they are installed, preferably moving them
+out of the TEXMF tree.
+
+******** revise -- pdf file amsldoc will be included in distrib
+
+Create a temporary directory and generate documentation files (these assume
+a UNIX TeX installation):
+
+  - make directory /tmp/amsmath
+  - unzip -d /tmp/amscls amsmath.zip
+  - cd /tmp/amsmath
+  - Run latex on the file ams-m1.ins to produce .cls and .sty files
+  - Run latex on *.tex
+  - [optional] Run latex on *.dtx (technical documentation)
+  - move *.dvi, *.pdf, *.tex to TEXMF/doc/latex/amsmath
+  - [optional] move *.dtx to TEXMF/doc/latex/amsmath
+  - [optional] convert .dvi to .ps or .pdf
+  - move *.sty and *.cls to TEXMF/tex/latex/amsmath
+  - delete /tmp/amsmath
+
+Some TeX systems also use a database to keep track of TeX input files.
+If this database is not updated, your TeX system might not find the
+newly installed files.  Consult your system's documentation to determine
+how to update this database.
diff --git a/Master/texmf-dist/source/latex/amsmath/manifest.txt b/Master/texmf-dist/source/latex/amsmath/manifest.txt
new file mode 100644
index 00000000000..2ea274a32d7
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/manifest.txt
@@ -0,0 +1,38 @@
+
+texmf/doc/latex/amsmath/
+  amsldoc.pdf
+  diffs-m.txt
+  subeqn.pdf
+  technote.pdf
+  testmath.pdf
+
+texmf/source/latex/amsmath/
+  00readme.txt
+  ams-m1.ins
+  amsbsy.dtx
+  amscd.dtx
+  amsdtx.cls
+  amsgen.dtx
+  amsldoc.cls
+  amsldoc.tex
+  amsmath.dtx
+  amsopn.dtx
+  amstex.sty
+  amstext.dtx
+  amsxtra.dtx
+  diffs-m.txt
+  install.txt
+  manifest.txt
+  subeqn.tex
+  technote.tex
+  testmath.tex
+
+texmf/tex/latex/amsmath/
+  amsbsy.sty
+  amscd.sty
+  amsgen.sty
+  amsmath.sty
+  amsopn.sty
+  amstex.sty
+  amstext.sty
+  amsxtra.sty
diff --git a/Master/texmf-dist/source/latex/amsmath/subeqn.tex b/Master/texmf-dist/source/latex/amsmath/subeqn.tex
new file mode 100644
index 00000000000..d0fa7d9b023
--- /dev/null
+++ b/Master/texmf-dist/source/latex/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}{\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/source/latex/amsmath/technote.tex b/Master/texmf-dist/source/latex/amsmath/technote.tex
new file mode 100644
index 00000000000..aede450621e
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/technote.tex
@@ -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/source/latex/amsmath/testmath.tex b/Master/texmf-dist/source/latex/amsmath/testmath.tex
new file mode 100644
index 00000000000..069944ee9d0
--- /dev/null
+++ b/Master/texmf-dist/source/latex/amsmath/testmath.tex
@@ -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}
+
+\bibitem{dihe:newdir}
+W.~Diffie and E.~Hellman, \emph{New directions in cryptography}, IEEE
+Transactions on Information Theory \textbf{22} (1976), no.~5, 644--654.
+
+\bibitem{fre:cichon}
+D.~H. Fremlin, \emph{Cichon's diagram}, 1983/1984, presented at the
+S{\'e}minaire Initiation {\`a} l'Analyse, G. Choquet, M. Rogalski, J.
+Saint Raymond, at the Universit{\'e} Pierre et Marie Curie, Paris, 23e
+ann{\'e}e.
+
+\bibitem{gouja:lagrmeth}
+I.~P. Goulden and D.~M. Jackson, \emph{The enumeration of directed
+closed {E}uler trails and directed {H}amiltonian circuits by
+{L}angrangian methods}, European J. Combin. \textbf{2} (1981), 131--212.
+
+\bibitem{hapa:graphenum}
+F.~Harary and E.~M. Palmer, \emph{Graphical enumeration}, Academic
+Press, 1973.
+
+\bibitem{imlelu:oneway}
+R.~Impagliazzo, L.~Levin, and M.~Luby, \emph{Pseudo-random generation
+from one-way functions}, Proc. 21st STOC (1989), ACM, New York,
+pp.~12--24.
+
+\bibitem{komiyo:unipfunc}
+M.~Kojima, S.~Mizuno, and A.~Yoshise, \emph{A new continuation method
+for complementarity problems with uniform p-functions}, Tech. Report
+B-194, Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information
+Sciences.
+
+\bibitem{komiyo:lincomp}
+\bysame, \emph{A polynomial-time algorithm for a class of linear
+complementarity problems}, Tech. Report B-193, Tokyo Inst. of
+Technology, Tokyo, 1987, Dept. of Information Sciences.
+
+\bibitem{liuchow:formalsum}
+C.~J. Liu and Yutze Chow, \emph{On operator and formal sum methods for
+graph enumeration problems}, SIAM J. Algorithms Discrete Methods
+\textbf{5} (1984), 384--438.
+
+\bibitem{mami:matrixth}
+M.~Marcus and H.~Minc, \emph{A survey of matrix theory and matrix
+inequalities}, Complementary Series in Math. \textbf{14} (1964), 21--48.
+
+\bibitem{miyoki:lincomp}
+S.~Mizuno, A.~Yoshise, and T.~Kikuchi, \emph{Practical polynomial time
+algorithms for linear complementarity problems}, Tech. Report~13, Tokyo
+Inst. of Technology, Tokyo, April 1988, Dept. of Industrial Engineering
+and Management.
+
+\bibitem{moad:quadpro}
+R.~D. Monteiro and I.~Adler, \emph{Interior path following primal-dual
+algorithms, part {II}: Quadratic programming}, August 1987, Working
+paper, Dept. of Industrial Engineering and Operations Research.
+
+\bibitem{ste:sint}
+E.~M. Stein, \emph{Singular integrals and differentiability properties
+of functions}, Princeton Univ. Press, Princeton, N.J., 1970.
+
+\bibitem{ye:intalg}
+Y.~Ye, \emph{Interior algorithms for linear, quadratic and linearly
+constrained convex programming}, Ph.D. thesis, Stanford Univ., Palo
+Alto, Calif., July 1987, Dept. of Engineering--Economic Systems,
+unpublished.
+
+\end{thebibliography}
+
+\end{document}
+\endinput
diff --git a/Master/texmf-dist/tex/latex/amsmath/amsbsy.sty b/Master/texmf-dist/tex/latex/amsmath/amsbsy.sty
new file mode 100644
index 00000000000..0e64d1b2e28
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/amsmath/amsbsy.sty
@@ -0,0 +1,98 @@
+%%
+%% This is file `amsbsy.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amsbsy.dtx 
+%% 
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsbsy.dtx",
+%%%   version   = "1.2d",
+%%%   date      = "1999/11/29",
+%%%   time      = "14:26:42 EST",
+%%%   author    = "American Mathematical Society",
+%%%   copyright = "Copyright 1995 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",
+%%%   checksum  = "08988 251 1238 9625",
+%%%   email     = "tech-support@ams.org (Internet)",
+%%%   codetable = "ISO/ASCII",
+%%%   keywords  = "latex, bold symbol, pmb",
+%%%   supported = "yes",
+%%%   abstract  = "This is a \LaTeX{} package that provides a command
+%%%                for producing bold math symbols when appropriate
+%%%                fonts exist, and a `poor man's bold' command that
+%%%                can be applied when no appropriate bold font
+%%%                is available.",
+%%%   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
+\ProvidesPackage{amsbsy}[1999/11/29 v1.2d]
+\RequirePackage{amsgen}
+\DeclareRobustCommand{\boldsymbol}[1]{%
+  \begingroup
+  \let\@nomath\@gobble \mathversion{bold}%
+  \math@atom{#1}{%
+  \mathchoice%
+    {\hbox{$\m@th\displaystyle#1$}}%
+    {\hbox{$\m@th\textstyle#1$}}%
+    {\hbox{$\m@th\scriptstyle#1$}}%
+    {\hbox{$\m@th\scriptscriptstyle#1$}}}%
+  \endgroup}
+\def\math@atom#1#2{%
+   \binrel@{#1}\binrel@@{#2}}
+\DeclareRobustCommand{\pmb}{%
+  \ifmmode\else \expandafter\pmb@@\fi\mathpalette\pmb@}
+\def\pmb@@#1#2#3{\leavevmode\setboxz@h{#3}%
+   \dimen@-\wdz@
+   \kern-.5\ex@\copy\z@
+   \kern\dimen@\kern.25\ex@\raise.4\ex@\copy\z@
+   \kern\dimen@\kern.25\ex@\box\z@
+}
+\newdimen\pmbraise@
+\def\pmb@#1#2{\setbox8\hbox{$\m@th#1{#2}$}%
+  \setboxz@h{$\m@th#1\mkern.5mu$}\pmbraise@\wdz@
+  \binrel@{#2}%
+  \dimen@-\wd8 %
+  \binrel@@{%
+    \mkern-.8mu\copy8 %
+    \kern\dimen@\mkern.4mu\raise\pmbraise@\copy8 %
+    \kern\dimen@\mkern.4mu\box8 }%
+}
+\def\binrel@#1{\begingroup
+  \setboxz@h{\thinmuskip0mu
+    \medmuskip\m@ne mu\thickmuskip\@ne mu
+    \setbox\tw@\hbox{$#1\m@th$}\kern-\wd\tw@
+    ${}#1{}\m@th$}%
+  \edef\@tempa{\endgroup\let\noexpand\binrel@@
+    \ifdim\wdz@<\z@ \mathbin
+    \else\ifdim\wdz@>\z@ \mathrel
+    \else \relax\fi\fi}%
+  \@tempa
+}
+\let\binrel@@\relax
+\endinput
+%%
+%% End of file `amsbsy.sty'.
diff --git a/Master/texmf-dist/tex/latex/amsmath/amscd.sty b/Master/texmf-dist/tex/latex/amsmath/amscd.sty
new file mode 100644
index 00000000000..56a4e958001
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/amsmath/amscd.sty
@@ -0,0 +1,171 @@
+%%
+%% This is file `amscd.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amscd.dtx 
+%% 
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amscd.dtx",
+%%%   version   = "2.0",
+%%%   date      = "1999/11/29",
+%%%   time      = "14:30:07 EST",
+%%%   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",
+%%%   checksum  = "07428 333 1152 12117",
+%%%   email     = "tech-support@ams.org (Internet)",
+%%%   codetable = "ISO/ASCII",
+%%%   keywords  = "latex, commutative diagram",
+%%%   supported = "yes",
+%%%   abstract  = "This is a \LaTeX{} package that adapts the
+%%%                commutative diagram macros of AMS-\TeX{} for
+%%%                use in \LaTeX{}",
+%%%   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
+\ProvidesPackage{amscd}[1999/11/29 v1.2d]
+\RequirePackage{amsgen}
+\@ifundefined{math@cr}{%
+  \def\math@cr{{\ifnum0=`}\fi
+    \@ifstar{\global\@eqpen\@M\math@cr@}%
+          {\global\@eqpen\interdisplaylinepenalty \math@cr@}}
+  \def\math@cr@{\new@ifnextchar[\math@cr@@{\math@cr@@[\z@]}}
+  \def\math@cr@@[#1]{\ifnum0=`{\fi}\math@cr@@@
+    \noalign{\vskip#1\relax}}
+  \def\restore@math@cr{\def\math@cr@@@{\cr}}
+}{}
+\restore@math@cr
+\@ifundefined{rightarrowfill@}{
+  \def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
+    $#1\copy\z@\mkern-6mu\cleaders
+    \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
+    \mkern-6mu\mathord\rightarrow$}
+  \def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
+    $#1\mathord\leftarrow\mkern-6mu\cleaders
+    \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
+    \mkern-6mu\box\z@$}
+  \def\leftrightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
+    $#1\mathord\leftarrow\mkern-6mu\cleaders
+    \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
+    \mkern-6mu\mathord\rightarrow$}
+}{}
+\def\atdef@#1{\expandafter\def\csname\space @\string#1\endcsname}
+\@ifundefined{Iat}{%
+  \DeclareRobustCommand{\Iat}{\FN@\at@}
+}{}
+\begingroup \catcode`\@=\active
+\csname if\string @compatibility\endcsname
+\else \fam=\mathcode`\@
+  \xdef @{\mathchar\number\fam\space }
+\fi
+\gdef\CDat{\let @=\Iat}
+\endgroup
+\mathcode`\@="8000 % make @ pseudo-active in math
+\def\at@{\let\next@\at@@
+ \ifcat\noexpand\next a\else
+ \ifcat\noexpand\next0\else
+ \ifcat\noexpand\next\relax\else
+ \let\next@\at@@@\fi\fi\fi\next@}
+\def\at@@#1{\expandafter
+  \ifx\csname\space @\string#1\endcsname\relax
+    \DN@{\at@@@#1}%
+  \else
+    \DN@{\csname\space @\string#1\endcsname}%
+  \fi\next@}%
+\@ifundefined{default@tag}{%
+  \def\default@tag{%
+    \def\tag{\PackageError{amscd}{\protect\tag\space not allowed
+      here}\@eha}}%
+}{}%
+\@ifundefined{at@@@}{%
+  \def\at@@@{\PackageError{amscd}{\Invalid@@ @}{\the\athelp@}\char64\relax}
+}{}
+\@ifundefined{athelp@}{\csname newhelp\endcsname\athelp@
+{Only certain combinations beginning with @ make sense to me.^^J%
+I'll assume you wanted @@ for a printed @.}}{}
+\@ifundefined{minaw@}{\newdimen\minaw@}{}
+\@ifundefined{bigaw@}{\newdimen\bigaw@}{}
+\minaw@11.111pt
+\newdimen\minCDarrowwidth
+\minCDarrowwidth2.5pc
+\newif\ifCD@
+\let\ampersand@\relax
+\newenvironment{CD}{%
+  \CDat
+  \bgroup\relax\iffalse{\fi\let\ampersand@&\iffalse}\fi
+  \CD@true\vcenter\bgroup\let\\\math@cr\restore@math@cr\default@tag
+  \tabskip\z@skip\baselineskip20\ex@
+  \lineskip3\ex@\lineskiplimit3\ex@\halign\bgroup
+  &\hfill$\m@th##$\hfill\crcr
+}{%
+  \crcr\egroup\egroup\egroup
+}
+\def\CD@check#1#2{\ifCD@\DN@{#2}\else
+  \DN@{\PackageError{amscd}{@\protect#1 not
+    allowed outside of the CD environment}\@eha}%
+  \fi\next@}
+\atdef@>#1>#2>{\ampersand@
+  \ifCD@ \global\bigaw@\minCDarrowwidth \else \global\bigaw@\minaw@ \fi
+  \setboxz@h{$\m@th\scriptstyle\;{#1}\;\;$}%
+  \ifdim\wdz@>\bigaw@\global\bigaw@\wdz@\fi
+  \@ifnotempty{#2}{\setbox\@ne\hbox{$\m@th\scriptstyle\;{#2}\;\;$}%
+    \ifdim\wd\@ne>\bigaw@\global\bigaw@\wd\@ne\fi}%
+ \ifCD@\enskip\fi
+   \mathrel{\mathop{\hbox to\bigaw@{\rightarrowfill@\displaystyle}}%
+     \limits^{#1}\@ifnotempty{#2}{_{#2}}}%
+ \ifCD@\enskip\fi \ampersand@}
+\atdef@<#1<#2<{\ampersand@
+  \ifCD@ \global\bigaw@\minCDarrowwidth \else \global\bigaw@\minaw@ \fi
+  \setboxz@h{$\m@th\scriptstyle\;\;{#1}\;$}%
+  \ifdim\wdz@>\bigaw@ \global\bigaw@\wdz@ \fi
+  \@ifnotempty{#2}{\setbox\@ne\hbox{$\m@th\scriptstyle\;\;{#2}\;$}%
+    \ifdim\wd\@ne>\bigaw@ \global\bigaw@\wd\@ne \fi}%
+  \ifCD@\enskip\fi
+    \mathrel{\mathop{\hbox to\bigaw@{\leftarrowfill@\displaystyle}}%
+      \limits^{#1}\@ifnotempty{#2}{_{#2}}}%
+  \ifCD@\enskip\fi \ampersand@}
+\begingroup \catcode`\~=\active \lccode`\~=`\@
+\lowercase{%
+  \global\atdef@)#1)#2){~>#1>#2>}
+  \global\atdef@(#1(#2({~<#1<#2<}
+}% end lowercase
+\endgroup
+\atdef@ A#1A#2A{\CD@check{A..A..A}{\llap{$\m@th\vcenter{\hbox
+  {$\scriptstyle#1$}}$}\Big\uparrow
+  \rlap{$\m@th\vcenter{\hbox{$\scriptstyle#2$}}$}&&}}
+\atdef@ V#1V#2V{\CD@check{V..V..V}{\llap{$\m@th\vcenter{\hbox
+  {$\scriptstyle#1$}}$}\Big\downarrow
+  \rlap{$\m@th\vcenter{\hbox{$\scriptstyle#2$}}$}&&}}
+\atdef@={\CD@check={&\enskip\mathrel
+  {\vbox{\hrule\@width\minCDarrowwidth\vskip2\ex@\hrule\@width
+  \minCDarrowwidth}}\enskip&}}
+\atdef@|{\CD@check|{\Big\Vert&&}}
+\atdef@\vert{\CD@check\vert{\Big\Vert&&}}
+\atdef@.{\CD@check.{&&}}
+\endinput
+%%
+%% End of file `amscd.sty'.
diff --git a/Master/texmf-dist/tex/latex/amsmath/amsgen.sty b/Master/texmf-dist/tex/latex/amsmath/amsgen.sty
new file mode 100644
index 00000000000..9d39357a496
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/amsmath/amsgen.sty
@@ -0,0 +1,160 @@
+%%
+%% This is file `amsgen.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amsgen.dtx 
+%% 
+%%% ====================================================================
+%%%  @LaTeX-file{
+%%%     filename  = "amsgen.dtx",
+%%%     version   = "2.0",
+%%%     date      = "1999/11/30",
+%%%     time      = "12:33:33 EST",
+%%%     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",
+%%%     checksum  = "07569 396 1613 14368",
+%%%     email     = "tech-support@ams.org (Internet)",
+%%%     codetable = "ISO/ASCII",
+%%%     keywords  = "latex, amslatex, ams-latex",
+%%%     supported = "yes",
+%%%     abstract  = "This is part of the AMS-\LaTeX{} distribution.
+%%%                  It contains some general internal macros shared
+%%%                  by several different files in AMS-\LaTeX{}.",
+%%%     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
+\ProvidesFile{amsgen.sty}[1999/11/30 v2.0]
+\providecommand{\@saveprimitive}[2]{\begingroup\escapechar`\\\relax
+  \edef\@tempa{\string#1}\edef\@tempb{\meaning#1}%
+  \ifx\@tempa\@tempb \global\let#2#1%
+  \else
+    \edef\@tempb{\meaning#2}%
+    \ifx\@tempa\@tempb
+    \else
+      \@latex@error{Unable to properly define \string#2; primitive
+      \noexpand#1no longer primitive}\@eha
+    \fi
+  \fi
+  \endgroup}
+\let\@xp=\expandafter
+\let\@nx=\noexpand
+\newtoks\@emptytoks
+\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
+\long\def\@ifempty#1{\@xifempty#1@@..\@nil}
+\long\def\@xifempty#1#2@#3#4#5\@nil{%
+  \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
+\long\def\@ifnotempty#1{\@ifempty{#1}{}}
+\def\FN@{\futurelet\@let@token}
+\def\DN@{\def\next@}
+\def\RIfM@{\relax\ifmmode}
+\def\setboxz@h{\setbox\z@\hbox}
+\def\wdz@{\wd\z@}
+\def\boxz@{\box\z@}
+\def\relaxnext@{\let\@let@token\relax}
+\long\def\new@ifnextchar#1#2#3{%
+  \let\reserved@d= #1%
+  \def\reserved@a{#2}\def\reserved@b{#3}%
+  \futurelet\@let@token\new@ifnch
+}
+\def\new@ifnch{%
+  \ifx\@let@token\reserved@d \let\reserved@b\reserved@a \fi
+  \reserved@b
+}
+\def\@ifstar#1#2{\new@ifnextchar *{\def\reserved@a*{#1}\reserved@a}{#2}}
+\@ifundefined{every@math@size}{%
+\let\every@math@size=\every@size
+\def\glb@settings{%
+     \expandafter\ifx\csname S@\f@size\endcsname\relax
+       \calculate@math@sizes
+     \fi
+     \csname S@\f@size\endcsname
+      \ifmath@fonts
+        \begingroup
+          \escapechar\m@ne
+          \csname mv@\math@version \endcsname
+          \globaldefs\@ne
+          \let \glb@currsize \f@size
+          \math@fonts
+        \endgroup
+        \the\every@math@size
+      \else
+      \fi
+}
+\def\set@fontsize#1#2#3{%
+    \@defaultunits\@tempdimb#2pt\relax\@nnil
+    \edef\f@size{\strip@pt\@tempdimb}%
+    \@defaultunits\@tempskipa#3pt\relax\@nnil
+    \edef\f@baselineskip{\the\@tempskipa}%
+    \edef\f@linespread{#1}%
+    \let\baselinestretch\f@linespread
+      \def\size@update{%
+        \baselineskip\f@baselineskip\relax
+        \baselineskip\f@linespread\baselineskip
+        \normalbaselineskip\baselineskip
+        \setbox\strutbox\hbox{%
+          \vrule\@height.7\baselineskip
+                \@depth.3\baselineskip
+                \@width\z@}%
+%%%     \the\every@size
+        \let\size@update\relax}%
+  }
+}{}% end \@ifundefined test
+\newdimen\ex@
+\addto@hook\every@math@size{\compute@ex@}
+\def\compute@ex@{%
+  \begingroup
+  \dimen@-\f@size\p@
+  \ifdim\dimen@<-20\p@
+    \global\ex@ 1.5\p@
+  \else
+    \advance\dimen@10\p@ \multiply\dimen@\tw@
+    \edef\@tempa{\ifdim\dimen@>\z@ -\fi}%
+    \dimen@ \ifdim\dimen@<\z@ -\fi \dimen@
+    \advance\dimen@-\@m sp % fudge factor
+    \vfuzz\p@
+    \def\do{\ifdim\dimen@>\z@
+      \vfuzz=.97\vfuzz
+      \advance\dimen@ -\p@
+      \@xp\do \fi}%
+    \do
+    \dimen@\p@ \advance\dimen@-\vfuzz
+    \global\ex@\p@
+    \global\advance\ex@ \@tempa\dimen@
+  \fi
+  \endgroup
+}
+\def\@addpunct#1{\ifnum\spacefactor>\@m \else#1\fi}
+\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
+  \sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
+\def\nomath@env{\@amsmath@err{%
+  \string\begin{\@currenvir} allowed only in paragraph mode%
+}\@ehb% "You've lost some text"
+}
+\def\Invalid@@{Invalid use of \string}
+\endinput
+%%
+%% End of file `amsgen.sty'.
diff --git a/Master/texmf-dist/tex/latex/amsmath/amsmath.sty b/Master/texmf-dist/tex/latex/amsmath/amsmath.sty
new file mode 100644
index 00000000000..8d22f7aeb0f
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/amsmath/amsmath.sty
@@ -0,0 +1,2670 @@
+%%
+%% This is file `amsmath.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amsmath.dtx 
+%% 
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsmath.dtx",
+%%%   version   = "2.13",
+%%%   date      = "2000/07/18",
+%%%   time      = "08:16:41 EDT",
+%%%   author    = "American Mathematical Society",
+%%%   copyright = "Copyright 1995, 2000 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,
+%%%                Publications Technical Group,
+%%%                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",
+%%%   checksum  = "38084 5829 21569 202795",
+%%%   email     = "tech-support@ams.org (Internet)",
+%%%   codetable = "ISO/ASCII",
+%%%   keywords  = "latex, amslatex, math, amsmath",
+%%%   supported = "yes",
+%%%   abstract  = "This is a \LaTeX{} package that provides a variety of
+%%%                extra mathematical features, largely derived from
+%%%                AMS-\TeX{}.",
+%%%   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 be December 1994 or later
+\ProvidesPackage{amsmath}[2000/07/18 v2.13 AMS math features]
+\edef\@temp{\catcode 96=\number\catcode 96 }
+\catcode\string `\`=12
+\def\do#1{\catcode\number`#1=\number\catcode`#1}
+\edef\@temp{%
+  \noexpand\AtEndOfPackage{%
+    \@temp
+    \do\"\do\'\do\(\do\)\do\*\do\+\do\,\do\-\do\.%
+    \do\/\do\<\do\=\do\>\do\[\do\]\do\^\do\_\relax
+  }%
+}
+\@temp
+\def\do#1{\catcode\number`#1=12 }
+\do\"\do\'\do\(\do\)\do\*\do\+\do\,\do\-\do\.
+\do\/\do\<\do\=\do\>\do\[\do\]
+\catcode`\^=7 \catcode`\_=8
+\DeclareOption{intlimits}{\let\ilimits@\displaylimits}
+\DeclareOption{nointlimits}{\let\ilimits@\nolimits}
+\DeclareOption{sumlimits}{\let\slimits@\displaylimits}
+\DeclareOption{nosumlimits}{\let\slimits@\nolimits}
+\DeclareOption{namelimits}{\PassOptionsToPackage{namelimits}{amsopn}}
+\DeclareOption{nonamelimits}{%
+  \PassOptionsToPackage{nonamelimits}{amsopn}}
+\newif\ifctagsplit@
+\newif\iftagsleft@
+\DeclareOption{leqno}{\tagsleft@true}
+\DeclareOption{reqno}{\tagsleft@false}
+\DeclareOption{centertags}{\ctagsplit@true}
+\DeclareOption{tbtags}{\ctagsplit@false}
+\DeclareOption{cmex10}{%
+  \ifnum\cmex@opt=\@ne \def\cmex@opt{0}%
+  \else \def\cmex@opt{10}\fi
+}
+\@ifundefined{cmex@opt}{\def\cmex@opt{7}}{}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newif\if@fleqn
+\newskip\@mathmargin
+\@mathmargin\@centering
+\DeclareOption{fleqn}{%
+    \@fleqntrue
+    \@mathmargin = -1sp
+    \let\mathindent=\@mathmargin
+    \AtBeginDocument{%
+        \ifdim\@mathmargin= -1sp
+            \@mathmargin\leftmargini minus\leftmargini
+        \fi
+    }%
+}
+\DeclareOption{?}{}
+\ExecuteOptions{nointlimits,sumlimits,namelimits,centertags}
+\ProcessOptions\par
+\@ifpackagewith{amsmath}{?}{%
+  \typeout{^^J%
+Documentation for the amsmath package is found in amsldoc.dvi^^J%
+(or .pdf or .tex).^^J%
+^^J%
+See also http://www.ams.org/tex/amslatex.html.^^J%
+^^J%
+Note: Using the first edition of The LaTeX Companion (1994) without^^J%
+errata as a guide for amsmath use is not recommended.^^J%
+  }%
+}{%
+  \typeout{%
+For additional information on amsmath, use the \lq ?\rq\space option.%
+  }%
+}
+\ifnum\cmex@opt=7 \relax
+  \DeclareFontShape{OMX}{cmex}{m}{n}{%
+    <-8>cmex7<8>cmex8<9>cmex9%
+    <10><10.95><12><14.4><17.28><20.74><24.88>cmex10%
+  }{}%
+  \expandafter\let\csname OMX/cmex/m/n/10\endcsname\relax
+\else
+  \ifnum\cmex@opt=\z@ % need to override cmex7 fontdef from amsfonts
+    \begingroup
+    \fontencoding{OMX}\fontfamily{cmex}%
+    \expandafter\let\csname OMX+cmex\endcsname\relax
+    \try@load@fontshape
+    \endgroup
+    \expandafter\let\csname OMX/cmex/m/n/10\endcsname\relax
+    \def\cmex@opt{10}%
+  \fi
+\fi
+\RequirePackage{amstext}[1995/01/25]
+\RequirePackage{amsbsy}[1995/01/20]
+\RequirePackage{amsopn}[1995/01/20]
+\def\@amsmath@err{\PackageError{amsmath}}
+\providecommand{\AmS}{{\protect\AmSfont
+  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}
+\newcommand{\AmSfont}{%
+  \usefont{OMS}{cmsy}{\if\@xp\@car\f@series\@nil bb\else m\fi}{n}}
+\def\@mathmeasure#1#2#3{\setbox#1\hbox{\frozen@everymath\@emptytoks
+  \m@th$#2#3$}}
+\@ifundefined{inf@bad}{%
+  \newcount\inf@bad \inf@bad=1000000 \relax
+}{}
+\DeclareRobustCommand{\tmspace}[3]{%
+  \ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
+\renewcommand{\,}{\tmspace+\thinmuskip{.1667em}}
+\let\thinspace\,
+\renewcommand{\!}{\tmspace-\thinmuskip{.1667em}}
+\let\negthinspace\!
+\renewcommand{\:}{\tmspace+\medmuskip{.2222em}}
+\let\medspace\:
+\newcommand{\negmedspace}{\tmspace-\medmuskip{.2222em}}
+\renewcommand{\;}{\tmspace+\thickmuskip{.2777em}}
+\let\thickspace\;
+\newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}}
+\newcommand{\mspace}[1]{\mskip#1\relax}
+\def\@tempa#1#2\@nil{%
+  \ifx\delimiter#1\@tempcnta#2\relax\else\@tempcnta\z@\fi
+}
+\@xp\@tempa\vert\@empty\@nil
+\ifnum\@tempcnta>\z@
+  \advance\@tempcnta "4000000
+  \xdef\lvert{\delimiter\number\@tempcnta\space }
+  \advance\@tempcnta "1000000
+  \xdef\rvert{\delimiter\number\@tempcnta\space }
+\else
+  \ifx\@@undefined\lvert
+    % Fall back to cmex encoding since we don't know what else to do.
+    \DeclareMathDelimiter{\lvert}
+      {\mathopen}{symbols}{"6A}{largesymbols}{"0C}
+    \DeclareMathDelimiter{\rvert}
+      {\mathclose}{symbols}{"6A}{largesymbols}{"0C}
+  \fi
+\fi
+\@xp\@tempa\Vert\@empty\@nil
+\ifnum\@tempcnta>\z@
+  \advance\@tempcnta "4000000
+  \xdef\lVert{\delimiter\number\@tempcnta\space }
+  \advance\@tempcnta "1000000
+  \xdef\rVert{\delimiter\number\@tempcnta\space }
+\else
+  \ifx\@@undefined\lVert
+    \DeclareMathDelimiter{\lVert}
+      {\mathopen}{symbols}{"6B}{largesymbols}{"0D}
+    \DeclareMathDelimiter{\rVert}
+      {\mathclose}{symbols}{"6B}{largesymbols}{"0D}
+  \fi
+\fi
+\@saveprimitive\over\@@over
+\@saveprimitive\atop\@@atop
+\@saveprimitive\above\@@above
+\@saveprimitive\overwithdelims\@@overwithdelims
+\@saveprimitive\atopwithdelims\@@atopwithdelims
+\@saveprimitive\abovewithdelims\@@abovewithdelims
+\DeclareRobustCommand{\primfrac}[1]{%
+  \PackageWarning{amsmath}{%
+Foreign command \@backslashchar#1;\MessageBreak
+\protect\frac\space or \protect\genfrac\space should be used instead%
+\MessageBreak
+  }
+  \global\@xp\let\csname#1\@xp\endcsname\csname @@#1\endcsname
+  \csname#1\endcsname
+}
+\renewcommand{\over}{\primfrac{over}}
+\renewcommand{\atop}{\primfrac{atop}}
+\renewcommand{\above}{\primfrac{above}}
+\renewcommand{\overwithdelims}{\primfrac{overwithdelims}}
+\renewcommand{\atopwithdelims}{\primfrac{atopwithdelims}}
+\renewcommand{\abovewithdelims}{\primfrac{abovewithdelims}}
+\DeclareRobustCommand{\frac}[2]{{\begingroup#1\endgroup\@@over#2}}
+\newcommand{\dfrac}{\genfrac{}{}{}0}
+\newcommand{\tfrac}{\genfrac{}{}{}1}
+\DeclareRobustCommand{\binom}{\genfrac()\z@{}}
+\newcommand{\dbinom}{\genfrac(){0pt}0}
+\newcommand{\tbinom}{\genfrac(){0pt}1}
+\DeclareRobustCommand{\genfrac}[4]{%
+  \def\@tempa{#1#2}%
+  \edef\@tempb{\@nx\@genfrac\@mathstyle{#4}%
+    \csname @@\ifx @#3@over\else above\fi
+    \ifx\@tempa\@empty \else withdelims\fi\endcsname}
+  \@tempb{#1#2#3}}
+\def\@genfrac#1#2#3#4#5{{#1{\begingroup#4\endgroup#2#3\relax#5}}}
+\def\@mathstyle#1{%
+  \ifx\@empty#1\@empty\relax
+  \else\ifcase#1\displaystyle % case 0
+    \or\textstyle\or\scriptstyle\else\scriptscriptstyle\fi\fi}
+\begingroup
+\edef\@tempa{\string\mathchar"}
+\def\@tempb#1"#2\@nil{#1"}
+\edef\@tempc{\expandafter\@tempb\meaning\coprod "\@nil}
+\ifx\@tempa\@tempc
+  \global\let\coprod@\coprod
+  \gdef\coprod{\DOTSB\coprod@\slimits@}
+  \global\let\bigvee@\bigvee
+  \gdef\bigvee{\DOTSB\bigvee@\slimits@}
+  \global\let\bigwedge@\bigwedge
+  \gdef\bigwedge{\DOTSB\bigwedge@\slimits@}
+  \global\let\biguplus@\biguplus
+  \gdef\biguplus{\DOTSB\biguplus@\slimits@}
+  \global\let\bigcap@\bigcap
+  \gdef\bigcap{\DOTSB\bigcap@\slimits@}
+  \global\let\bigcup@\bigcup
+  \gdef\bigcup{\DOTSB\bigcup@\slimits@}
+  \global\let\prod@\prod
+  \gdef\prod{\DOTSB\prod@\slimits@}
+  \global\let\sum@\sum
+  \gdef\sum{\DOTSB\sum@\slimits@}
+  \global\let\bigotimes@\bigotimes
+  \gdef\bigotimes{\DOTSB\bigotimes@\slimits@}
+  \global\let\bigoplus@\bigoplus
+  \gdef\bigoplus{\DOTSB\bigoplus@\slimits@}
+  \global\let\bigodot@\bigodot
+  \gdef\bigodot{\DOTSB\bigodot@\slimits@}
+  \global\let\bigsqcup@\bigsqcup
+  \gdef\bigsqcup{\DOTSB\bigsqcup@\slimits@}
+\fi
+\endgroup
+\newcommand{\leftroot}{\@amsmath@err{\Invalid@@\leftroot}\@eha}
+\newcommand{\uproot}{\@amsmath@err{\Invalid@@\uproot}\@eha}
+\newcount\uproot@
+\newcount\leftroot@
+\renewcommand{\root}{\relaxnext@
+  \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else
+   \ifx\@let@token\leftroot\let\next@\nextiii@\else
+   \let\next@\plainroot@\fi\fi\next@}%
+  \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}%
+  \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else
+   \DN@.{\FN@\nextv@}\fi\next@.}%
+  \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else
+   \let\next@\plainroot@\fi\next@}%
+  \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}%
+   \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}%
+  \def\nextvii@{\ifx\@let@token\@sptoken
+   \DN@. {\FN@\nextviii@}\else
+   \DN@.{\FN@\nextviii@}\fi\next@.}%
+  \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else
+   \let\next@\plainroot@\fi\next@}%
+  \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}%
+  \bgroup\uproot@\z@\leftroot@\z@\FN@\next@}
+\def\plainroot@#1\of#2{\setbox\rootbox\hbox{%
+ $\m@th\scriptscriptstyle{#1}$}%
+ \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}}
+ {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup}
+
+\@ifundefined{sqrtsign}{\let\sqrtsign\@@sqrt}{}
+\def\r@@t#1#2{\setboxz@h{$\m@th#1\sqrtsign{#2}$}%
+ \dimen@\ht\z@\advance\dimen@-\dp\z@
+ \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}%
+ \advance\dimen@ by1.667\wd\@ne
+ \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox
+ \mkern-10mu\mkern\leftroot@ mu\boxz@}
+\@ifundefined{varGamma}{%
+  \DeclareMathSymbol{\varGamma}{\mathord}{letters}{"00}
+  \DeclareMathSymbol{\varDelta}{\mathord}{letters}{"01}
+  \DeclareMathSymbol{\varTheta}{\mathord}{letters}{"02}
+  \DeclareMathSymbol{\varLambda}{\mathord}{letters}{"03}
+  \DeclareMathSymbol{\varXi}{\mathord}{letters}{"04}
+  \DeclareMathSymbol{\varPi}{\mathord}{letters}{"05}
+  \DeclareMathSymbol{\varSigma}{\mathord}{letters}{"06}
+  \DeclareMathSymbol{\varUpsilon}{\mathord}{letters}{"07}
+  \DeclareMathSymbol{\varPhi}{\mathord}{letters}{"08}
+  \DeclareMathSymbol{\varPsi}{\mathord}{letters}{"09}
+  \DeclareMathSymbol{\varOmega}{\mathord}{letters}{"0A}
+}{}
+\@saveprimitive\overline\@@overline
+\DeclareRobustCommand{\overline}[1]{\@@overline{#1}}
+\newcommand{\boxed}[1]{\fbox{\m@th$\displaystyle#1$}}
+\newcommand{\implies}{\DOTSB\;\Longrightarrow\;}
+\newcommand{\impliedby}{\DOTSB\;\Longleftarrow\;}
+\def\And{\DOTSB\;\mathchar"3026 \;}
+\newcommand{\nobreakdash}{\leavevmode
+  \toks@\@emptytoks \def\@tempa##1{\toks@\@xp{\the\toks@-}\FN@\next@}%
+  \DN@{\ifx\@let@token-\@xp\@tempa
+       \else\setboxz@h{\the\toks@\nobreak}\unhbox\z@\fi}%
+  \FN@\next@
+}
+\renewcommand{\colon}{\nobreak\mskip2mu\mathpunct{}\nonscript
+  \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
+\let\ifgtest@\iffalse                              % initial value
+\def\gtest@true{\global\let\ifgtest@\iftrue}
+\def\gtest@false{\global\let\ifgtest@\iffalse}
+\let\DOTSI\relax
+\let\DOTSB\relax
+\let\DOTSX\relax
+{\uccode`7=`\\ \uccode`8=`m \uccode`9=`a \uccode`0=`t \uccode`!=`h
+ \uppercase{%
+  \gdef\math@#1#2#3#4#5#6\math@{\gtest@false\ifx 7#1\ifx 8#2%
+  \ifx 9#3\ifx 0#4\ifx !#5\xdef\meaning@{#6}\gtest@true
+  \fi\fi\fi\fi\fi}}}
+{\uccode`7=`c \uccode`8=`h \uccode`9=`\"
+ \uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\gtest@false
+  \ifx 7#1\ifx 8#2\ifx 9#5\gtest@true\xdef\meaning@{9#6}\fi\fi\fi}}}
+\newcount\classnum@
+\def\getmathch@#1.#2\getmathch@{\classnum@#1 \divide\classnum@4096
+ \ifcase\number\classnum@\or\or\gdef\thedots@{\dotsb@}\or
+ \gdef\thedots@{\dotsb@}\fi}
+{\uccode`4=`b \uccode`5=`i \uccode`6=`n
+ \uppercase{\gdef\mathbin@#1#2#3{\relaxnext@
+  \def\nextii@##1\mathbin@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+  \gtest@false\DN@##1\mathbin@{}%
+ \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
+{\uccode`4=`r \uccode`5=`e \uccode`6=`l
+ \uppercase{\gdef\mathrel@#1#2#3{\relaxnext@
+  \def\nextii@##1\mathrel@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+ \gtest@false\DN@##1\mathrel@{}%
+ \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
+{\uccode`5=`m \uccode`6=`a \uccode`7=`c
+ \uppercase{\gdef\macro@#1#2#3#4\macro@{\gtest@false
+  \ifx 5#1\ifx 6#2\ifx 7#3\gtest@true
+  \xdef\meaning@{\macro@@#4\macro@@}\fi\fi\fi}}}
+\def\macro@@#1->#2\macro@@{#2}
+\newcount\DOTSCASE@
+{\uccode`6=`\\ \uccode`7=`D \uccode`8=`O \uccode`9=`T \uccode`0=`S
+ \uppercase{\gdef\DOTS@#1#2#3#4#5{\gtest@false\DN@##1\DOTS@{}%
+  \ifx 6#1\ifx 7#2\ifx 8#3\ifx 9#4\ifx 0#5\let\next@\DOTS@@
+  \fi\fi\fi\fi\fi
+  \next@}}}
+{\uccode`3=`B \uccode`4=`I \uccode`5=`X
+ \uppercase{\gdef\DOTS@@#1{\relaxnext@
+  \def\nextii@##1\DOTS@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+  \DN@{\FN@\nextii@}%
+  \ifx 3#1\global\DOTSCASE@\z@\else
+  \ifx 4#1\global\DOTSCASE@\@ne\else
+  \ifx 5#1\global\DOTSCASE@\tw@\else\DN@##1\DOTS@{}%
+  \fi\fi\fi\next@}}}
+{\uccode`5=`\\ \uccode`6=`n \uccode`7=`o \uccode`8=`t
+ \uppercase{\gdef\not@#1#2#3#4{\relaxnext@
+  \def\nextii@##1\not@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+ \gtest@false\DN@##1\not@{}%
+ \ifx 5#1\ifx 6#2\ifx 7#3\ifx 8#4\DN@{\FN@\nextii@}\fi\fi\fi
+ \fi\next@}}}
+\def\keybin@{\gtest@true
+ \ifx\@let@token+\else\ifx\@let@token=\else
+ \ifx\@let@token<\else\ifx\@let@token>\else
+ \ifx\@let@token-\else\ifx\@let@token*\else\ifx\@let@token:\else
+   \gtest@false\fi\fi\fi\fi\fi\fi\fi}
+\@ifundefined{@ldots}{\def\@ldots{\mathellipsis}}{}
+\DeclareRobustCommand{\ldots}{%
+  \ifmmode \mathellipsis \else \textellipsis \fi
+}
+\DeclareRobustCommand{\dots}{%
+  \ifmmode \@xp\mdots@\else \@xp\textellipsis \fi
+}
+\def\tdots@{\leavevmode\unskip\relaxnext@
+ \DN@{$\m@th\@ldots\,
+   \ifx\@let@token,\,$\else\ifx\@let@token.\,$\else
+   \ifx\@let@token;\,$\else\ifx\@let@token:\,$\else
+   \ifx\@let@token?\,$\else\ifx\@let@token!\,$\else
+     $ \fi\fi\fi\fi\fi\fi}%
+  \ \FN@\next@}
+\def\mdots@{\FN@\mdots@@}
+\def\mdots@@{\gdef\thedots@{\dotso@}%
+ \ifx\@let@token\boldsymbol \gdef\thedots@\boldsymbol{\boldsymboldots@}%
+ \else\ifx,\@let@token \gdef\thedots@{\dotsc}%
+ \else\ifx\not\@let@token \gdef\thedots@{\dotsb@}%
+ \else\keybin@
+ \ifgtest@\gdef\thedots@{\dotsb@}%
+ \else\xdef\meaning@{\meaning\@let@token..........}%
+   \xdef\meaning@@{\meaning@}%
+  \@xp\math@\meaning@\math@
+  \ifgtest@
+   \@xp\mathch@\meaning@\mathch@
+   \ifgtest@\@xp\getmathch@\meaning@\getmathch@\fi
+  \else\@xp\macro@\meaning@@\macro@
+  \ifgtest@
+   \@xp\not@\meaning@\not@\ifgtest@\gdef\thedots@{\dotsb@}%
+  \else\@xp\DOTS@\meaning@\DOTS@
+  \ifgtest@
+   \ifcase\number\DOTSCASE@\gdef\thedots@{\dotsb@}%
+    \or\gdef\thedots@{\dotsi}\else\fi
+  \else\@xp\math@\meaning@\math@
+  \ifgtest@\@xp\mathbin@\meaning@\mathbin@
+  \ifgtest@\gdef\thedots@{\dotsb@}%
+  \else\@xp\mathrel@\meaning@\mathrel@
+  \ifgtest@\gdef\thedots@{\dotsb@}%
+  \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+ \thedots@}
+\def\boldsymboldots@#1{%
+  \bold@true\let\@let@token=#1\let\delayed@=#1\mdots@@
+  \boldsymbol#1\bold@false}
+\def\@cdots{\mathinner{\cdotp\cdotp\cdotp}}
+\newcommand{\dotsi}{\!\@cdots}
+\let\dotsb@\@cdots
+\def\rightdelim@{\gtest@true
+ \ifx\@let@token)\else
+ \ifx\@let@token]\else
+ \ifx\@let@token\rbrack\else
+ \ifx\@let@token\}\else
+ \ifx\@let@token\rbrace\else
+ \ifx\@let@token\rangle\else
+ \ifx\@let@token\rceil\else
+ \ifx\@let@token\rfloor\else
+ \ifx\@let@token\rgroup\else
+ \ifx\@let@token\rmoustache\else
+ \ifx\@let@token\right\else
+ \ifx\@let@token\bigr\else
+ \ifx\@let@token\biggr\else
+ \ifx\@let@token\Bigr\else
+ \ifx\@let@token\Biggr\else\gtest@false
+ \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
+\def\extra@{%
+ \rightdelim@\ifgtest@
+ \else\ifx\@let@token$\gtest@true
+ \else\xdef\meaning@{\meaning\@let@token..........}%
+ \@xp\macro@\meaning@\macro@\ifgtest@
+ \@xp\DOTS@\meaning@\DOTS@
+ \ifgtest@
+ \ifnum\DOTSCASE@=\tw@\gtest@true\else\gtest@false
+ \fi\fi\fi\fi\fi}
+\newif\ifbold@
+\def\dotso@{\relaxnext@
+ \ifbold@
+  \let\@let@token\delayed@
+  \def\nextii@{\extra@\@ldots\ifgtest@\,\fi}%
+ \else
+  \def\nextii@{\DN@{\extra@\@ldots\ifgtest@\,\fi}\FN@\next@}%
+ \fi
+ \nextii@}
+\def\extrap@#1{%
+ \DN@{#1\,}%
+ \ifx\@let@token,\else
+ \ifx\@let@token;\else
+ \ifx\@let@token.\else\extra@
+ \ifgtest@\else
+ \let\next@#1\fi\fi\fi\fi\next@}
+\DeclareRobustCommand{\cdots}{\DN@{\extrap@\@cdots}\FN@\next@}
+\let\dotsb\cdots
+\let\dotsm\cdots
+\DeclareRobustCommand{\dotso}{\relax
+  \ifmmode \DN@{\extrap@\@ldots}%
+  \else \let\next@\tdots@\fi
+  \FN@\next@}
+\DeclareRobustCommand{\dotsc}{%
+  \DN@{\ifx\@let@token;\@ldots\,%
+       \else \ifx\@let@token.\@ldots\,%
+       \else \extra@\@ldots \ifgtest@\,\fi
+       \fi\fi}%
+  \FN@\next@}
+\renewcommand{\longrightarrow}{%
+  \DOTSB\protect\relbar\protect\joinrel\rightarrow}
+\renewcommand{\Longrightarrow}{%
+  \DOTSB\protect\Relbar\protect\joinrel\Rightarrow}
+\renewcommand{\longleftarrow}{%
+  \DOTSB\leftarrow\protect\joinrel\protect\relbar}
+\renewcommand{\Longleftarrow}{%
+  \DOTSB\Leftarrow\protect\joinrel\protect\Relbar}
+\renewcommand{\longleftrightarrow}{\DOTSB\leftarrow\joinrel\rightarrow}
+\renewcommand{\Longleftrightarrow}{\DOTSB\Leftarrow\joinrel\Rightarrow}
+\renewcommand{\mapsto}{\DOTSB\mapstochar\rightarrow}
+\renewcommand{\longmapsto}{\DOTSB\mapstochar\longrightarrow}
+\renewcommand{\hookrightarrow}{\DOTSB\lhook\joinrel\rightarrow}
+\renewcommand{\hookleftarrow}{\DOTSB\leftarrow\joinrel\rhook}
+\renewcommand{\iff}{\DOTSB\;\Longleftrightarrow\;}
+\renewcommand{\doteq}{%
+  \DOTSB\mathrel{\mathop{\kern0pt =}\limits^{\textstyle.}}}
+\newif\if@display
+\everydisplay\@xp{\the\everydisplay \@displaytrue}
+\renewcommand{\int}{\DOTSI\intop\ilimits@}
+\renewcommand{\oint}{\DOTSI\ointop\ilimits@}
+\def\intkern@{\mkern-6mu\mathchoice{\mkern-3mu}{}{}{}}
+\def\intdots@{\mathchoice{\@cdots}%
+ {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
+ {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
+ {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}
+\newcommand{\iint}{\DOTSI\protect\MultiIntegral{2}}
+\newcommand{\iiint}{\DOTSI\protect\MultiIntegral{3}}
+\newcommand{\iiiint}{\DOTSI\protect\MultiIntegral{4}}
+\newcommand{\idotsint}{\DOTSI\protect\MultiIntegral{0}}
+\newcommand{\MultiIntegral}[1]{%
+  \edef\ints@c{\noexpand\intop
+    \ifnum#1=\z@\noexpand\intdots@\else\noexpand\intkern@\fi
+    \ifnum#1>\tw@\noexpand\intop\noexpand\intkern@\fi
+    \ifnum#1>\thr@@\noexpand\intop\noexpand\intkern@\fi
+    \noexpand\intop
+    \noexpand\ilimits@
+  }%
+  \futurelet\@let@token\ints@a
+}
+\def\ints@a{%
+  \ifx\limits\@let@token \ints@b
+  \else \ifx\displaylimits\@let@token \ints@b
+  \else\ifx\ilimits@\displaylimits \ints@b
+  \fi\fi\fi
+  \ints@c
+}
+\def\ints@b{%
+  \mkern-7mu\mathchoice{\mkern-2mu}{}{}{}%
+  \mathop\bgroup
+    \mkern7mu\mathchoice{\mkern2mu}{}{}{}%
+    \let\ilimits@\egroup
+}%
+\newbox\Mathstrutbox@
+\setbox\Mathstrutbox@=\hbox{}
+\def\Mathstrut@{\copy\Mathstrutbox@}
+\def\resetMathstrut@{%
+  \setbox\z@\hbox{%
+    \mathchardef\@tempa\mathcode`\(\relax
+    \def\@tempb##1"##2##3{\the\textfont"##3\char"}%
+    \expandafter\@tempb\meaning\@tempa \relax
+  }%
+  \ht\Mathstrutbox@\ht\z@ \dp\Mathstrutbox@\dp\z@
+}
+\addto@hook\every@math@size{\resetMathstrut@}
+\newbox\strutbox@
+\def\strut@{\copy\strutbox@}
+\def\reset@strutbox@{%
+  \global\setbox\strutbox@\hbox{%
+    \lower.5\normallineskiplimit
+       \vbox{\kern-\normallineskiplimit\copy\strutbox}}}
+\addto@hook\every@math@size{\reset@strutbox@}
+\AtBeginDocument{\reset@strutbox@}
+\renewcommand{\big}{\bBigg@\@ne}
+\renewcommand{\Big}{\bBigg@{1.5}}
+\renewcommand{\bigg}{\bBigg@\tw@}
+\renewcommand{\Bigg}{\bBigg@{2.5}}
+\def\bBigg@#1#2{%
+  {\@mathmeasure\z@{\nulldelimiterspace\z@}%
+     {\left#2\vcenter to#1\big@size{}\right.}%
+   \box\z@}}
+\addto@hook\every@math@size{%
+  \global\big@size 1.2\ht\Mathstrutbox@
+  \global\advance\big@size 1.2\dp\Mathstrutbox@ }
+\newdimen\big@size
+\def\accentclass@{7}
+\def\noaccents@{\def\accentclass@{0}}
+\DeclareFontEncoding{OML}{}{\noaccents@}
+\DeclareFontEncoding{OMS}{}{\noaccents@}
+\newcommand{\dddot}[1]{%
+  {\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
+   \hbox{\normalfont ...}\vss}}}}
+\newcommand{\ddddot}[1]{%
+  {\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
+   \hbox{\normalfont....}\vss}}}}
+\def\set@mathaccent#1#2#3#4{%
+  \xdef#2{\@nx\protect\@nx\mathaccentV
+    {\@xp\@gobble\string#2}\hexnumber@#1#4}%
+}
+\def\@tempa#1{\@xp\@tempb\meaning#1\@nil#1}
+\def\@tempb#1>#2#3 #4\@nil#5{%
+  \@xp\ifx\csname#3\endcsname\mathaccent
+    \@tempc#4?"7777\@nil#5%
+  \else
+    \PackageWarningNoLine{amsmath}{%
+      Unable to redefine math accent \string#5}%
+  \fi
+}
+\def\@tempc#1"#2#3#4#5#6\@nil#7{%
+  \chardef\@tempd="#3\relax\set@mathaccent\@tempd{#7}{#2}{#4#5}}
+\@tempa{\hat}
+\@tempa{\check}
+\@tempa{\tilde}
+\@tempa{\acute}
+\@tempa{\grave}
+\@tempa{\dot}
+\@tempa{\ddot}
+\@tempa{\breve}
+\@tempa{\bar}
+\@tempa{\vec}
+\@ifundefined{mathring}{%
+  \DeclareMathAccent{\mathring}{\mathalpha}{operators}{"17}
+}{%
+  \@tempa{\mathring}
+}
+%%\@tempa\widetilde
+%%\@tempa\widehat
+\newcommand{\acc@check}{}
+\newcommand{\acc@error}{}
+\def\acc@check{\@ifnextchar\@empty\relax\acc@error}
+\def\acc@error{%
+  \@amsmath@err{%
+    Improper argument for math accent:\MessageBreak
+    Extra braces must be added to prevent wrong output%
+  }\@ehc
+}
+\newcount\macc@depth
+\long\def\@gobblethree#1#2#3{}
+\def\mathaccentV#1#2#3#4#5{%
+  \ifmmode
+    \gdef\macc@tmp{\macc@depth\@ne}%
+    \setbox\z@\hbox{%
+      \let\mathaccentV\macc@test
+      \let\use@mathgroup\@gobbletwo \let\select@group\@gobblethree
+      \frozen@everymath{}$#5$%
+    }%
+    \macc@tmp
+    \ifnum\macc@depth=\@ne
+      \global\let\macc@nucleus\@empty
+      \mathaccent"\accentclass@
+    \else
+      \@xp\macc@nested
+    \fi
+    #2#3#4{#5}%
+    \macc@nucleus
+  \else
+    \@xp\nonmatherr@\csname#1\endcsname
+  \fi
+}
+\def\macc@test#1#2#3#4{\xdef\macc@tmp{\macc@tmp\advance\macc@depth\@ne}}
+\def\macc@group{-1}
+\def\macc@nested#1#2#3#4{%
+  \begingroup
+  \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
+  \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
+  \macc@set@skewchar\relax
+  \let\mathaccentV\macc@nested@a
+  \macc@nested@a\relax#1#2#3{#4}%
+  \endgroup
+}
+\let\macc@palette\mathpalette
+\def\macc@nested@a#1#2#3#4#5{%
+  \ifnum\macc@group=\mathgroup
+  \else \macc@set@skewchar\relax \edef\macc@group{\the\mathgroup}%
+  \fi
+  \mathchardef\macc@code "\accentclass@ #2#3#4\relax
+  \macc@palette\macc@a{#5}%
+}
+\def\macc@set@skewchar#1{%
+  \begingroup
+  \ifnum\mathgroup=\m@ne \let\@tempa\@ne
+  \else
+    \ifnum\skewchar\textfont\mathgroup=\m@ne \let\@tempa\@ne
+    \else \let\@tempa\mathgroup
+    \fi
+  \fi
+  \count@=\skewchar\textfont\@tempa
+  \advance\count@"7100
+  \edef\@tempa{\endgroup
+    \mathchardef\noexpand\macc@skewchar=\number\count@\relax}%
+  \@tempa
+  #1%
+}
+\def\macc@a#1#2{%
+  \begingroup
+  \let\macc@style#1\relax
+  \def\macc@palette##1{##1\macc@style}%
+  \advance\macc@depth\m@ne
+  \ifnum\macc@depth=\z@
+    \gdef\macc@nucleus{#2}%
+    \setbox\z@\hbox{$#1#2\@empty{}\macc@skewchar$}%
+    \setbox\tw@\hbox{$#1#2\@empty\macc@skewchar$}%
+    \dimen@\tw@\wd\tw@ \advance\dimen@-\tw@\wd\z@
+    \xdef\macc@kerna{\the\dimen@\relax}%
+    \setbox4\hbox{$#1#2\acc@check\@empty$}%
+    \global\setbox\@ne\hbox to\wd4{}%
+    \ht\@ne\ht4 \dp\@ne\dp4
+    \xdef\macc@kernb{\the\wd4\relax}%
+    \mathaccent\macc@code{\box\@ne\kern\macc@kerna}%
+  \else
+    \mathaccent\macc@code{\let\macc@adjust\@empty #1#2\@empty}%
+    \macc@adjust
+  \fi
+  \endgroup
+}
+\def\macc@adjust{%
+  \dimen@\macc@kerna\advance\dimen@\macc@kernb
+  \kern-\dimen@
+}
+\def\Hat{\hat}
+\def\Check{\check}
+\def\Tilde{\tilde}
+\def\Acute{\acute}
+\def\Grave{\grave}
+\def\Dot{\dot}
+\def\Ddot{\ddot}
+\def\Breve{\breve}
+\def\Bar{\bar}
+\def\Vec{\vec}
+\def\nonmatherr@#1{\@amsmath@err{\protect
+  #1 allowed only in math mode}\@ehd}
+\renewcommand{\bmod}{\nonscript\mskip-\medmuskip\mkern5mu\mathbin
+  {\operator@font mod}\penalty900
+  \mkern5mu\nonscript\mskip-\medmuskip}
+\newcommand{\pod}[1]{\allowbreak
+  \if@display\mkern18mu\else\mkern8mu\fi(#1)}
+\renewcommand{\pmod}[1]{\pod{{\operator@font mod}\mkern6mu#1}}
+\newcommand{\mod}[1]{\allowbreak\if@display\mkern18mu
+  \else\mkern12mu\fi{\operator@font mod}\,\,#1}
+\newcommand{\cfrac}[3][c]{{\displaystyle\frac{%
+  \strut\ifx r#1\hfill\fi#2\ifx l#1\hfill\fi}{#3}}%
+  \kern-\nulldelimiterspace}
+\newcommand{\overset}[2]{\binrel@{#2}%
+  \binrel@@{\mathop{\kern\z@#2}\limits^{#1}}}
+\newcommand{\underset}[2]{\binrel@{#2}%
+  \binrel@@{\mathop{\kern\z@#2}\limits_{#1}}}
+\newcommand{\sideset}[3]{%
+  \@mathmeasure\z@\displaystyle{#3}%
+  \global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@
+  \setbox\tw@\box\@ne
+  \@mathmeasure4\displaystyle{\copy\tw@#1}%
+  \@mathmeasure6\displaystyle{#3\nolimits#2}%
+  \dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@
+  \hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}%
+}
+\renewcommand{\smash}[1][tb]{%
+  \def\mb@t{\ht}\def\mb@b{\dp}\def\mb@tb{\ht\z@\z@\dp}%
+  \edef\finsm@sh{\csname mb@#1\endcsname\z@\z@ \box\z@}%
+  \ifmmode \@xp\mathpalette\@xp\mathsm@sh
+  \else \@xp\makesm@sh
+  \fi
+}
+\mathchardef\std@minus\mathcode`\-\relax
+\mathchardef\std@equal\mathcode`\=\relax
+\AtBeginDocument{%
+  \mathchardef\std@minus\mathcode`\-\relax
+  \mathchardef\std@equal\mathcode`\=\relax
+}
+\def\relbar{\mathrel{\mathpalette\mathsm@sh\std@minus}}
+\def\Relbar{\mathrel\std@equal}
+\def\arrowfill@#1#2#3#4{%
+  $\m@th\thickmuskip0mu\medmuskip\thickmuskip\thinmuskip\thickmuskip
+   \relax#4#1\mkern-7mu%
+   \cleaders\hbox{$#4\mkern-2mu#2\mkern-2mu$}\hfill
+   \mkern-7mu#3$%
+}
+\def\leftarrowfill@{\arrowfill@\leftarrow\relbar\relbar}
+\def\rightarrowfill@{\arrowfill@\relbar\relbar\rightarrow}
+\def\leftrightarrowfill@{\arrowfill@\leftarrow\relbar\rightarrow}
+\def\Leftarrowfill@{\arrowfill@\Leftarrow\Relbar\Relbar}
+\def\Rightarrowfill@{\arrowfill@\Relbar\Relbar\Rightarrow}
+\def\Leftrightarrowfill@{\arrowfill@\Leftarrow\Relbar\Rightarrow}
+\def\overarrow@#1#2#3{\vbox{\ialign{##\crcr#1#2\crcr
+ \noalign{\nointerlineskip}$\m@th\hfil#2#3\hfil$\crcr}}}
+\renewcommand{\overrightarrow}{%
+  \mathpalette{\overarrow@\rightarrowfill@}}
+\renewcommand{\overleftarrow}{%
+  \mathpalette{\overarrow@\leftarrowfill@}}
+\newcommand{\overleftrightarrow}{%
+  \mathpalette{\overarrow@\leftrightarrowfill@}}
+\def\underarrow@#1#2#3{%
+ \vtop{\ialign{##\crcr$\m@th\hfil#2#3\hfil$\crcr
+ \noalign{\nointerlineskip\kern1.3\ex@}#1#2\crcr}}}
+\newcommand{\underrightarrow}{%
+  \mathpalette{\underarrow@\rightarrowfill@}}
+\newcommand{\underleftarrow}{%
+  \mathpalette{\underarrow@\leftarrowfill@}}
+\newcommand{\underleftrightarrow}{%
+  \mathpalette{\underarrow@\leftrightarrowfill@}}
+\def\ext@arrow#1#2#3#4#5#6#7{%
+  \mathrel{\mathop{%
+    \setbox\z@\hbox{#5\displaystyle}%
+    \setbox\tw@\vbox{\m@th
+      \hbox{$\scriptstyle\mkern#3mu{#6}\mkern#4mu$}%
+      \hbox{$\scriptstyle\mkern#3mu{#7}\mkern#4mu$}%
+      \copy\z@
+    }%
+    \hbox to\wd\tw@{\unhbox\z@}}%
+  \limits
+    \@ifnotempty{#7}{^{\if0#1\else\mkern#1mu\fi
+                       #7\if0#2\else\mkern#2mu\fi}}%
+    \@ifnotempty{#6}{_{\if0#1\else\mkern#1mu\fi
+                       #6\if0#2\else\mkern#2mu\fi}}}%
+}
+\newcommand{\xrightarrow}[2][]{\ext@arrow 0359\rightarrowfill@{#1}{#2}}
+\newcommand{\xleftarrow}[2][]{\ext@arrow 3095\leftarrowfill@{#1}{#2}}
+\newenvironment{subarray}[1]{%
+  \vcenter\bgroup
+  \Let@ \restore@math@cr \default@tag
+  \baselineskip\fontdimen10 \scriptfont\tw@
+  \advance\baselineskip\fontdimen12 \scriptfont\tw@
+  \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
+  \lineskiplimit\lineskip
+  \ialign\bgroup\ifx c#1\hfil\fi
+    $\m@th\scriptstyle##$\hfil\crcr
+}{%
+  \crcr\egroup\egroup
+}
+\newcommand{\substack}[1]{\subarray{c}#1\endsubarray}
+\newenvironment{smallmatrix}{\null\,\vcenter\bgroup
+  \Let@\restore@math@cr\default@tag
+  \baselineskip6\ex@ \lineskip1.5\ex@ \lineskiplimit\lineskip
+  \ialign\bgroup\hfil$\m@th\scriptstyle##$\hfil&&\thickspace\hfil
+  $\m@th\scriptstyle##$\hfil\crcr
+}{%
+  \crcr\egroup\egroup\,%
+}
+\renewenvironment{matrix}{%
+  \matrix@check\matrix\env@matrix
+}{%
+  \endarray \hskip -\arraycolsep
+}
+\def\env@matrix{\hskip -\arraycolsep
+  \let\@ifnextchar\new@ifnextchar
+  \array{*\c@MaxMatrixCols c}}
+\newcount\c@MaxMatrixCols \c@MaxMatrixCols=10
+\def\matrix@check#1{%
+  \@xp\ifx\csname\@currenvir\endcsname#1%
+  \else\matrix@error#1%
+    \@xp\@gobble
+  \fi
+}
+\def\matrix@error#1{%
+  \@amsmath@err{%
+Old form `\string#1' should be \string\begin{\@xp\@gobble\string#1}%
+  }{%
+`\string#1{...}' is old Plain-TeX syntax whose use is
+ill-advised in LaTeX.%
+  }%
+}
+\renewenvironment{pmatrix}{%
+  \left(%
+  \matrix@check\pmatrix\env@matrix
+}{
+  \endmatrix\right)%
+}
+\newenvironment{bmatrix}{\left[\env@matrix}{\endmatrix\right]}
+\newenvironment{Bmatrix}{%
+  \left\lbrace\env@matrix
+}{%
+  \endmatrix\right\rbrace
+}
+\newenvironment{vmatrix}{\left\lvert\env@matrix}{\endmatrix\right\rvert}
+\newenvironment{Vmatrix}{\left\lVert\env@matrix}{\endmatrix\right\rVert}
+\let\hdots\@ldots
+\newcommand{\hdotsfor}[1]{%
+  \ifx[#1\@xp\shdots@for\else\hdots@for\@ne{#1}\fi}
+\newmuskip\dotsspace@
+\def\shdots@for#1]{\hdots@for{#1}}
+\def\hdots@for#1#2{\multicolumn{#2}c%
+  {\m@th\dotsspace@1.5mu\mkern-#1\dotsspace@
+   \xleaders\hbox{$\m@th\mkern#1\dotsspace@.\mkern#1\dotsspace@$}%
+           \hfill
+   \mkern-#1\dotsspace@}%
+   }
+\renewenvironment{cases}{%
+  \matrix@check\cases\env@cases
+}{%
+  \endarray\right.%
+}
+\def\env@cases{%
+  \let\@ifnextchar\new@ifnextchar
+  \left\lbrace
+  \def\arraystretch{1.2}%
+  \array{@{}l@{\quad}l@{}}%
+}
+\newcounter{parentequation}% Counter for ``parent equation''.
+\@ifundefined{ignorespacesafterend}{%
+  \def\ignorespacesafterend{\global\@ignoretrue}%
+}{}
+\newenvironment{subequations}{%
+  \refstepcounter{equation}%
+  \protected@edef\theparentequation{\theequation}%
+  \setcounter{parentequation}{\value{equation}}%
+  \setcounter{equation}{0}%
+  \def\theequation{\theparentequation\alph{equation}}%
+  \ignorespaces
+}{%
+  \setcounter{equation}{\value{parentequation}}%
+  \ignorespacesafterend
+}
+\newcommand{\numberwithin}[3][\arabic]{%
+  \@ifundefined{c@#2}{\@nocounterr{#2}}{%
+    \@ifundefined{c@#3}{\@nocnterr{#3}}{%
+      \@addtoreset{#2}{#3}%
+      \@xp\xdef\csname the#2\endcsname{%
+        \@xp\@nx\csname the#3\endcsname .\@nx#1{#2}}}}%
+}
+\newcommand{\eqref}[1]{\textup{\tagform@{\ref{#1}}}}
+\newcount\dspbrk@lvl
+\dspbrk@lvl=-1
+\interdisplaylinepenalty\@M
+\newcommand{\allowdisplaybreaks}[1][4]{%
+  \interdisplaylinepenalty\getdsp@pen{#1}\relax
+}
+\def\getdsp@pen#1{%
+  \ifcase #1\@M \or 9999 \or 6999 \or 2999 \or \z@\fi
+}
+\newcommand{\displaybreak}{\new@ifnextchar[\dspbrk@{\dspbrk@[4]}}
+\chardef\dspbrk@context=\sixt@@n
+\def\dspbrk@[#1]{%
+  \ifmeasuring@
+  \else
+    \ifcase\dspbrk@context % case 0 --- OK
+      \global\dspbrk@lvl #1\relax
+    \or                    % case 1 --- inside a box
+      \nogood@displaybreak
+    \else                  % other cases --- outside of a display
+      \@amsmath@err{\Invalid@@\displaybreak}\@eha
+    \fi
+  \fi
+}
+\def\nogood@displaybreak{%
+  \@amsmath@err{\protect
+\displaybreak\space cannot be applied here}%
+{One of the enclosing environments creates an
+  unbreakable box\MessageBreak
+(e.g., split, aligned, gathered, ...).}%
+}
+\def\math@cr{\relax\iffalse{\fi\ifnum0=`}\fi
+  \@ifstar{\global\@eqpen\@M\math@cr@}%
+          {\global\@eqpen
+             \ifnum\dspbrk@lvl <\z@ \interdisplaylinepenalty
+              \else -\@getpen\dspbrk@lvl \fi
+           \math@cr@}}
+\def\math@cr@{\new@ifnextchar[\math@cr@@{\math@cr@@[\z@]}}
+\def\math@cr@@[#1]{\ifnum0=`{\fi \iffalse}\fi\math@cr@@@
+  \noalign{\vskip#1\relax}}
+\def\Let@{\let\\\math@cr}
+\def\restore@math@cr{\def\math@cr@@@{\cr}}
+\restore@math@cr
+\newcommand{\intertext}{\@amsmath@err{\Invalid@@\intertext}\@eha}
+\def\intertext@{%
+  \def\intertext##1{%
+    \ifvmode\else\\\@empty\fi
+    \noalign{%
+      \penalty\postdisplaypenalty\vskip\belowdisplayskip
+      \vbox{\normalbaselines
+        \ifdim\linewidth=\columnwidth
+        \else \parshape\@ne \@totalleftmargin \linewidth
+        \fi
+        \noindent##1\par}%
+      \penalty\predisplaypenalty\vskip\abovedisplayskip%
+    }%
+}}
+\newhelp\tag@help
+  {tag cannot be used at this point.\space
+   If you don't understand why^^Jyou should consult
+   the documentation.^^JBut don't worry: just continue, and I'll
+   forget what happened.}
+\def\gobble@tag{\@ifstar\@gobble\@gobble}
+\def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag}
+\def\dft@tag{\invalid@tag{\string\tag\space not allowed here}}
+\def\default@tag{\let\tag\dft@tag}
+\default@tag
+\def\maketag@@{\@ifstar\maketag@@@\tagform@}
+\def\maketag@@@#1{\hbox{\m@th\normalfont#1}}
+\def\tagform@#1{\maketag@@@{(\ignorespaces#1\unskip\@@italiccorr)}}
+\iftagsleft@
+  \def\@eqnnum{\hbox to1sp{}\rlap{\normalfont\normalcolor
+    \hskip -\displaywidth\tagform@\theequation}}
+\else
+  \def\@eqnnum{{\normalfont\normalcolor \tagform@\theequation}}
+\fi
+\newcommand{\thetag}{\leavevmode\tagform@}
+\let\df@tag\@empty
+\def\make@df@tag{\@ifstar\make@df@tag@@\make@df@tag@@@}
+\def\make@df@tag@@#1{%
+  \gdef\df@tag{\maketag@@@{#1}\def\@currentlabel{#1}}}
+\def\make@df@tag@@@#1{\gdef\df@tag{\tagform@{#1}%
+  \toks@\@xp{\p@equation{#1}}\edef\@currentlabel{\the\toks@}}}
+\let\ltx@label\label
+\def\label@in@display{%
+    \ifx\df@label\@empty\else
+        \@amsmath@err{Multiple \string\label's:
+            label '\df@label' will be lost}\@eha
+    \fi
+    \gdef\df@label
+}
+\toks@\@xp{\@arrayparboxrestore \let\label\ltx@label}%
+\edef\@arrayboxrestore{\the\toks@}
+\let\df@label\@empty
+\def\make@display@tag{%
+  \if@eqnsw \incr@eqnum \print@eqnum
+  \else \iftag@ \df@tag \global\let\df@tag\@empty \fi
+  \fi
+  \ifmeasuring@
+  \else
+    \ifx\df@label\@empty
+    \else
+      \@xp\ltx@label\@xp{\df@label}%
+      \global\let\df@label\@empty
+    \fi
+  \fi
+}
+\def\tag@in@align{%
+    \relax
+    \iftag@
+        \DN@{\invalid@tag{Multiple \string\tag}}%
+    \else
+    \global\tag@true
+    \nonumber
+        \let\next@\make@df@tag
+    \fi
+    \next@
+}
+\newcommand{\raisetag}[1]{\skip@#1\relax
+  \xdef\raise@tag{\vskip\iftagsleft@\else-\fi\the\skip@\relax}%
+}
+\let\raise@tag\@empty
+\newcommand{\notag}{\nonumber}
+\renewcommand{\nonumber}{%
+  \if@eqnsw
+    \ifx\incr@eqnum\@empty \addtocounter{equation}\m@ne \fi
+  \fi
+  \let\print@eqnum\@empty \let\incr@eqnum\@empty
+  \global\@eqnswfalse
+}
+\def\print@eqnum{\tagform@\theequation}
+\def\incr@eqnum{\refstepcounter{equation}\let\incr@eqnum\@empty}
+\newif\ifinalign@
+\newif\ifingather@
+\@xp\def\@xp\@arrayparboxrestore\@xp{\@arrayparboxrestore
+  \ingather@false\inalign@false \default@tag
+  \let\spread@equation\@spread@equation
+  \let\reset@equation\@empty
+  \def\print@eqnum{\tagform@\theequation}%
+  \def\incr@eqnum{\refstepcounter{equation}\let\incr@eqnum\@empty}%
+}
+\newif\iftag@
+\newif\ifst@rred
+\newif\ifmeasuring@
+\newif\ifshifttag@
+\newcount\row@
+\newcount\column@
+\def\column@plus{%
+    \global\advance\column@\@ne
+}
+\newcount\maxfields@
+\def\add@amp#1{\if m#1&\@xp\add@amp\fi}
+\def\add@amps#1{%
+    \begingroup
+    \count@#1\advance\count@-\column@
+    \edef\@tempa{\endgroup
+      \@xp\add@amp\romannumeral\number\count@ 000q}%
+    \@tempa
+}
+\newhelp\andhelp@
+{An extra & here is so disastrous that you should probably exit^^J
+and fix things up.}
+\newdimen\eqnshift@
+\newdimen\alignsep@
+\newdimen\tagshift@
+\newcommand{\mintagsep}{.5\fontdimen6\textfont\tw@}
+\newcommand{\minalignsep}{10pt}
+\newdimen\tagwidth@
+\newdimen\totwidth@
+\newdimen\lineht@
+\def\tag@width#1{%
+    \ifcase\@xp#1\tag@lengths\fi
+}
+
+\def\savetaglength@{%
+    \begingroup
+        \let\or\relax
+        \xdef\tag@lengths{\tag@lengths\or \the\wdz@}%
+    \endgroup
+}
+
+\def\shift@tag#1{%
+    \ifcase\@xp#1\tag@shifts\fi\relax
+}
+
+\let\tag@shifts\@empty
+\def\saveshift@#1{%
+    \begingroup
+        \let\or\relax
+        \xdef\tag@shifts{\or#1\tag@shifts}%
+    \endgroup
+}
+\def\spread@equation{\reset@strutbox@
+  \openup\jot \let\spread@equation\@empty}
+\let\@spread@equation\spread@equation
+\def\displ@y{\@display@init{}}
+\def\@display@init#1{%
+    \global\dt@ptrue \spread@equation
+    \everycr{%
+        \noalign{%
+            #1%
+            \ifdt@p
+                \global\dt@pfalse
+                \vskip-\lineskiplimit
+                \vskip\normallineskiplimit
+            \else
+                \penalty\@eqpen \global\dspbrk@lvl\m@ne
+            \fi
+        }%
+    }%
+}
+\def\displ@y@{\@display@init{%
+  \global\column@\z@ \global\dspbrk@lvl\m@ne
+  \global\tag@false \global\let\raise@tag\@empty
+}}
+\def\black@#1{%
+    \noalign{%
+        \ifdim#1>\displaywidth
+            \dimen@\prevdepth
+            \nointerlineskip
+            \vskip-\ht\strutbox@
+            \vskip-\dp\strutbox@
+            \vbox{\noindent\hbox to#1{\strut@\hfill}}%
+            \prevdepth\dimen@
+        \fi
+    }%
+}
+\def\savecounters@{%
+    \begingroup
+        \def\@elt##1{%
+          \global\csname c@##1\endcsname\the\csname c@##1\endcsname}%
+        \xdef\@gtempa{%
+            \cl@@ckpt
+            \let\@nx\restorecounters@\@nx\@empty
+        }%
+    \endgroup
+    \let\restorecounters@\@gtempa
+}
+\let\restorecounters@\@empty
+\def\savealignstate@{%
+    \begingroup
+        \let\or\relax
+        \xdef\@gtempa{%
+            \global\totwidth@\the\totwidth@
+            \global\row@\the\row@
+            \gdef\@nx\tag@lengths{\tag@lengths}%
+            \let\@nx\restorealignstate@\@nx\@empty
+        }%
+    \endgroup
+    \let\restorealignstate@\@gtempa
+}
+
+\let\restorealignstate@\@empty
+\def\savecolumn@{%
+  \edef\restorecolumn@{%
+    \global\column@\number\column@
+    \let\@nx\restorecolumn@\@nx\@empty
+  }%
+}
+\let\restorecolumn@\@empty
+\newtoks\@envbody
+\def\addto@envbody#1{\global\@envbody\@xp{\the\@envbody#1}}
+\def\collect@body#1{%
+  \@envbody{\@xp#1\@xp{\the\@envbody}}%
+  \edef\process@envbody{\the\@envbody\@nx\end{\@currenvir}}%
+  \@envbody\@emptytoks \def\begin@stack{b}%
+  \begingroup
+  \@xp\let\csname\@currenvir\endcsname\collect@@body
+  \edef\process@envbody{\@xp\@nx\csname\@currenvir\endcsname}%
+  \process@envbody
+}
+\def\push@begins#1\begin#2{%
+  \ifx\end#2\else b\@xp\push@begins\fi
+}
+\def\collect@@body#1\end#2{%
+  \edef\begin@stack{\push@begins#1\begin\end \@xp\@gobble\begin@stack}%
+  \ifx\@empty\begin@stack
+    \endgroup
+    \@checkend{#2}%
+    \addto@envbody{#1}%
+  \else
+    \addto@envbody{#1\end{#2}}%
+  \fi
+  \process@envbody % A little tricky! Note the grouping
+}
+\def\math@cr@@@aligned{%
+  \ifodd\column@ \let\next@\@empty
+  \else \def\next@{&\kern-\alignsep@}%
+  \fi
+  \next@ \cr
+}
+\newcommand{\start@aligned}[2]{%
+    \RIfM@\else
+        \nonmatherr@{\begin{\@currenvir}}%
+    \fi
+    \savecolumn@ % Assumption: called inside a group
+    \null\,%
+    \if #1t\vtop \else \if#1b \vbox \else \vcenter \fi \fi \bgroup
+        \maxfields@#2\relax
+        \ifnum\maxfields@>\m@ne
+            \multiply\maxfields@\tw@
+            \let\math@cr@@@\math@cr@@@alignedat
+            \alignsep@\z@skip
+        \else
+            \let\math@cr@@@\math@cr@@@aligned
+            \alignsep@\minalignsep
+        \fi
+        \Let@ \chardef\dspbrk@context\@ne
+        \default@tag
+        \spread@equation % no-op if already called
+        \global\column@\z@
+        \ialign\bgroup
+           &\column@plus
+            \hfil
+            \strut@
+            $\m@th\displaystyle{##}$%
+            \tabskip\z@skip
+           &\column@plus
+            $\m@th\displaystyle{{}##}$%
+            \hfil
+            \tabskip\alignsep@
+            \crcr
+}
+\def\math@cr@@@alignedat{%
+    \ifnum\column@>\maxfields@
+        \begingroup
+          \measuring@false
+          \@amsmath@err{Extra & on this line}%
+            {\the\andhelp@}% "An extra & here is disastrous"
+        \endgroup
+    \fi
+    \global\column@\z@
+    \cr
+}
+\def\alignsafe@testopt#1#2{%
+  \relax\iffalse{\fi\ifnum`}=0\fi
+  \@ifnextchar[%
+    {\let\@let@token\relax \ifnum`{=\z@\fi\iffalse}\fi#1}%
+    {\let\@let@token\relax \ifnum`{=\z@\fi\iffalse}\fi#1[#2]}%
+}
+\newenvironment{aligned}{%
+  \let\@testopt\alignsafe@testopt
+  \aligned@a
+}{%
+  \crcr\egroup
+  \restorecolumn@
+  \egroup
+}
+\newcommand{\aligned@a}[1][c]{\start@aligned{#1}\m@ne}
+\newenvironment{alignedat}{%
+  \let\@testopt\alignsafe@testopt
+  \alignedat@a
+}{%
+  \endaligned
+}
+\newcommand{\alignedat@a}[1][c]{\start@aligned{#1}}
+\newenvironment{gathered}[1][c]{%
+    \RIfM@\else
+        \nonmatherr@{\begin{gathered}}%
+    \fi
+    \null\,%
+    \if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi \bgroup
+        \Let@ \chardef\dspbrk@context\@ne \restore@math@cr
+        \spread@equation
+        \ialign\bgroup
+            \hfil\strut@$\m@th\displaystyle##$\hfil
+            \crcr
+}{%
+  \endaligned
+}
+\def\start@gather#1{%
+    \RIfM@
+        \nomath@env
+        \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+    \else
+        $$%
+        #1%
+        \ifst@rred \else \global\@eqnswtrue \fi
+        \let\next@\gather@
+    \fi
+    \collect@body\next@
+}
+\newenvironment{gather}{%
+  \start@gather\st@rredfalse
+}{%
+  \math@cr \black@\totwidth@ \egroup
+  $$\ignorespacesafterend
+}
+
+\newenvironment{gather*}{%
+  \start@gather\st@rredtrue
+}{%
+  \endgather
+}
+\def\gather@#1{%
+    \ingather@true \let\split\insplit@
+    \let\tag\tag@in@align \let\label\label@in@display
+    \chardef\dspbrk@context\z@
+    \intertext@ \displ@y@ \Let@
+    \let\math@cr@@@\math@cr@@@gather
+    \gmeasure@{#1}%
+    \global\shifttag@false
+    \tabskip\z@skip
+    \global\row@\@ne
+    \halign to\displaywidth\bgroup
+        \strut@
+        \setboxz@h{$\m@th\displaystyle{##}$}%
+        \calc@shift@gather
+        \set@gather@field
+        \tabskip\@centering
+       &\setboxz@h{\strut@{##}}%
+        \place@tag@gather
+        \tabskip \iftagsleft@ \gdisplaywidth@ \else \z@skip \span\fi
+        \crcr
+        #1%
+}
+\def\gmeasure@#1{%
+    \begingroup
+        \measuring@true
+        \totwidth@\z@
+        \global\let\tag@lengths\@empty
+        \savecounters@
+        \setbox\@ne\vbox{%
+            \everycr{\noalign{\global\tag@false
+              \global\let\raise@tag\@empty \global\column@\z@}}%
+            \let\label\@gobble
+            \halign{%
+                \setboxz@h{$\m@th\displaystyle{##}$}%
+                \ifdim\wdz@>\totwidth@
+                    \global\totwidth@\wdz@
+                \fi
+               &\setboxz@h{\strut@{##}}%
+                \savetaglength@
+                \crcr
+                #1%
+                \math@cr@@@
+            }%
+        }%
+        \restorecounters@
+        \if@fleqn
+            \global\advance\totwidth@\@mathmargin
+        \fi
+        \iftagsleft@
+            \ifdim\totwidth@>\displaywidth
+                \global\let\gdisplaywidth@\totwidth@
+            \else
+                \global\let\gdisplaywidth@\displaywidth
+            \fi
+        \fi
+    \endgroup
+}
+\def\math@cr@@@gather{%
+    \ifst@rred\nonumber\fi
+   &\relax
+    \make@display@tag
+    \ifst@rred\else\global\@eqnswtrue\fi
+    \global\advance\row@\@ne
+    \cr
+}
+\def\calc@shift@gather{%
+    \dimen@\mintagsep\relax
+    \tagwidth@\tag@width\row@\relax
+    \if@fleqn
+        \global\eqnshift@\@mathmargin
+        \ifdim\tagwidth@>\z@
+            \advance\dimen@\tagwidth@
+            \iftagsleft@
+                \ifdim\dimen@>\@mathmargin
+                    \global\shifttag@true
+                \fi
+            \else
+                \advance\dimen@\@mathmargin
+                \advance\dimen@\wdz@
+                \ifdim\dimen@>\displaywidth
+                   \global\shifttag@true
+                \fi
+            \fi
+        \fi
+    \else
+        \global\eqnshift@\displaywidth
+        \global\advance\eqnshift@-\wdz@
+        \ifdim\tagwidth@>\z@
+            \multiply\dimen@\tw@
+            \advance\dimen@\wdz@
+            \advance\dimen@\tagwidth@
+            \ifdim\dimen@>\displaywidth
+                \global\shifttag@true
+            \else
+                \ifdim\eqnshift@<4\tagwidth@
+                    \global\advance\eqnshift@-\tagwidth@
+                \fi
+            \fi
+        \fi
+        \global\divide\eqnshift@\tw@
+        \iftagsleft@
+            \global\eqnshift@-\eqnshift@
+            \global\advance\eqnshift@\displaywidth
+            \global\advance\eqnshift@-\wdz@
+        \fi
+        \ifdim\eqnshift@<\z@
+            \global\eqnshift@\z@
+        \fi
+    \fi
+}
+\def\place@tag@gather{%
+    \iftagsleft@
+        \kern-\gdisplaywidth@
+        \ifshifttag@
+            \rlap{\vbox{%
+                \normalbaselines
+                \boxz@
+                \vbox to\lineht@{}%
+                \raise@tag
+            }}%
+            \global\shifttag@false
+        \else
+            \rlap{\boxz@}%
+        \fi
+    \else
+        \ifdim\totwidth@>\displaywidth
+            \dimen@\totwidth@
+            \advance\dimen@-\displaywidth
+            \kern-\dimen@
+        \fi
+        \ifshifttag@
+            \llap{\vtop{%
+                \raise@tag
+                \normalbaselines
+                \setbox\@ne\null
+                \dp\@ne\lineht@
+                \box\@ne
+                \boxz@
+            }}%
+            \global\shifttag@false
+        \else
+            \llap{\boxz@}%
+        \fi
+    \fi
+}
+\def\set@gather@field{%
+    \iftagsleft@
+        \global\lineht@\ht\z@
+    \else
+        \global\lineht@\dp\z@
+    \fi
+    \kern\eqnshift@
+    \boxz@
+    \hfil
+}
+\newif\ifxxat@
+
+\newif\ifcheckat@
+
+\let\xatlevel@\@empty
+\def\start@align#1#2#3{%
+    \let\xatlevel@#1% always \z@, \@ne, or \tw@
+    \maxfields@#3\relax
+    \ifnum\maxfields@>\m@ne
+        \checkat@true
+        \ifnum\xatlevel@=\tw@
+            \xxat@true
+        \fi
+        \multiply\maxfields@\tw@
+    \else
+        \checkat@false
+    \fi
+    \ifingather@
+        \iffalse{\fi\ifnum0=`}\fi
+        \DN@{\vcenter\bgroup\savealignstate@\align@#2}%
+    \else
+        \ifmmode
+          \if@display
+             \DN@{\align@recover}%
+          \else
+            \nomath@env
+            \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+          \fi
+        \else
+            $$%
+            \let\split\insplit@
+            \DN@{\align@#2}%
+        \fi
+    \fi
+    \collect@body\next@
+}
+\def\align@recover#1#2#3{%
+  \endgroup
+  \@amsmath@err{%
+Erroneous nesting of equation structures;\MessageBreak
+trying to recover with `aligned'%
+  }\@ehc
+  \begin{aligned}\relax#1\end{aligned}%
+}
+\newenvironment{alignat}{%
+  \start@align\z@\st@rredfalse
+}{%
+  \endalign
+}
+\newenvironment{alignat*}{%
+  \start@align\z@\st@rredtrue
+}{%
+  \endalign
+}
+\newenvironment{xalignat}{%
+  \start@align\@ne\st@rredfalse
+}{%
+  \endalign
+}
+\newenvironment{xalignat*}{%
+  \start@align\@ne\st@rredtrue
+}{%
+  \endalign
+}
+\newenvironment{xxalignat}{%
+  \start@align\tw@\st@rredtrue
+}{%
+  \endalign
+}
+\newenvironment{align}{%
+  \start@align\@ne\st@rredfalse\m@ne
+}{%
+  \math@cr \black@\totwidth@
+  \egroup
+  \ifingather@
+    \restorealignstate@
+    \egroup
+    \nonumber
+    \ifnum0=`{\fi\iffalse}\fi
+  \else
+    $$%
+  \fi
+  \ignorespacesafterend
+}
+\newenvironment{align*}{%
+  \start@align\@ne\st@rredtrue\m@ne
+}{%
+  \endalign
+}
+\newenvironment{flalign}{%
+  \start@align\tw@\st@rredfalse\m@ne
+}{%
+  \endalign
+}
+\newenvironment{flalign*}{%
+  \start@align\tw@\st@rredtrue\m@ne
+}{%
+  \endalign
+}
+\def\align@#1#2{%
+    \inalign@true \intertext@ \Let@ \chardef\dspbrk@context\z@
+    \ifingather@\else\displ@y@\fi
+    \let\math@cr@@@\math@cr@@@align
+    \ifxxat@\else \let\tag\tag@in@align \fi
+    \let\label\label@in@display
+    #1% set st@r
+    \ifst@rred\else \global\@eqnswtrue \fi
+    \measure@{#2}%
+    \global\row@\z@
+    \tabskip\eqnshift@
+    \halign\bgroup
+        \span\align@preamble\crcr
+        #2%
+}
+\def\math@cr@@@align{%
+  \ifst@rred\nonumber\fi
+  \if@eqnsw \global\tag@true \fi
+  \global\advance\row@\@ne
+  \add@amps\maxfields@
+  \omit
+  \kern-\alignsep@
+  \iftag@
+    \setboxz@h{\@lign\strut@{\make@display@tag}}%
+    \place@tag
+  \fi
+  \ifst@rred\else\global\@eqnswtrue\fi
+  \global\lineht@\z@
+  \cr
+}
+\def\math@cr@@@align@measure{%
+   &\omit
+    \global\advance\row@\@ne
+    \ifst@rred\nonumber\fi
+    \if@eqnsw \global\tag@true \fi
+    \ifnum\column@>\maxfields@
+        \ifcheckat@
+            \begingroup
+              \measuring@false
+              \@amsmath@err{Extra & on this line}%
+                {\the\andhelp@}% "An extra & here is disastrous"
+            \endgroup
+        \else
+            \global\maxfields@\column@
+        \fi
+    \fi
+    \setboxz@h{\@lign\strut@{%
+        \if@eqnsw
+            \stepcounter{equation}%
+            \tagform@\theequation
+        \else
+            \iftag@\df@tag\fi
+        \fi
+    }}%
+    \savetaglength@
+    \ifst@rred\else\global\@eqnswtrue\fi
+    \cr
+}
+\let\field@lengths\@empty
+
+\def\savefieldlength@{%
+    \begingroup
+        \let\or\relax
+        \xdef\field@lengths{%
+            \field@lengths
+            \ifnum\column@=0
+                \or
+            \else
+                ,%
+            \fi
+            \the\wdz@
+        }%
+    \endgroup
+}
+
+\def\fieldlengths@#1{%
+    \ifcase\@xp#1\field@lengths\fi
+}
+\let\maxcolumn@widths\@empty
+\def\maxcol@width#1{%
+    \ifcase\@xp#1\maxcolumn@widths\fi\relax
+}
+\def\measure@#1{%
+    \begingroup
+        \measuring@true
+        \global\eqnshift@\z@
+        \global\alignsep@\z@
+        \global\let\tag@lengths\@empty
+        \global\let\field@lengths\@empty
+        \savecounters@
+        \global\setbox0\vbox{%
+            \let\math@cr@@@\math@cr@@@align@measure
+            \everycr{\noalign{\global\tag@false
+              \global\let\raise@tag\@empty \global\column@\z@}}%
+            \let\label\@gobble
+            \global\row@\z@
+            \tabskip\z@
+            \halign{\span\align@preamble\crcr
+                #1%
+                \math@cr@@@
+                \global\column@\z@
+                \add@amps\maxfields@\cr
+            }%
+        }%
+        \restorecounters@
+        \ifodd\maxfields@
+            \global\advance\maxfields@\@ne
+        \fi
+        \ifnum\xatlevel@=\tw@
+            \ifnum\maxfields@<\thr@@
+                \let\xatlevel@\z@
+            \fi
+        \fi
+        \setbox\z@\vbox{%
+          \unvbox\z@ \unpenalty \global\setbox\@ne\lastbox
+        }%
+        \global\totwidth@\wd\@ne
+        \if@fleqn \global\advance\totwidth@\@mathmargin \fi
+        \global\let\maxcolumn@widths\@empty
+        \begingroup
+          \let\or\relax
+          \loop
+            \global\setbox\@ne\hbox{%
+              \unhbox\@ne \unskip \global\setbox\thr@@\lastbox
+            }%
+          \ifhbox\thr@@
+           \xdef\maxcolumn@widths{ \or \the\wd\thr@@ \maxcolumn@widths}%
+          \repeat
+        \endgroup
+        \dimen@\displaywidth
+        \advance\dimen@-\totwidth@
+        \ifcase\xatlevel@
+            \global\alignsep@\z@
+            \let\minalignsep\z@
+            \@tempcntb\z@
+            \if@fleqn
+                \@tempcnta\@ne
+                \global\eqnshift@\@mathmargin
+            \else
+                \@tempcnta\tw@
+                \global\eqnshift@\dimen@
+                \global\divide\eqnshift@\@tempcnta
+            \fi
+        \or
+            \@tempcntb\maxfields@
+            \divide\@tempcntb\tw@
+            \@tempcnta\@tempcntb
+            \advance\@tempcntb\m@ne
+            \if@fleqn
+                \global\eqnshift@\@mathmargin
+                \global\alignsep@\dimen@
+                \global\divide\alignsep@\@tempcnta
+            \else
+                \global\advance\@tempcnta\@ne
+                \global\eqnshift@\dimen@
+                \global\divide\eqnshift@\@tempcnta
+                \global\alignsep@\eqnshift@
+            \fi
+        \or
+            \@tempcntb\maxfields@
+            \divide\@tempcntb\tw@
+            \global\advance\@tempcntb\m@ne
+            \global\@tempcnta\@tempcntb
+            \global\eqnshift@\z@
+            \global\alignsep@\dimen@
+            \if@fleqn
+                \global\advance\alignsep@\@mathmargin\relax
+            \fi
+            \global\divide\alignsep@\@tempcntb
+        \fi
+        \ifdim\alignsep@<\minalignsep\relax
+            \global\alignsep@\minalignsep\relax
+            \ifdim\eqnshift@>\z@
+                \if@fleqn\else
+                    \global\eqnshift@\displaywidth
+                    \global\advance\eqnshift@-\totwidth@
+                    \global\advance\eqnshift@-\@tempcntb\alignsep@
+                    \global\divide\eqnshift@\tw@
+                \fi
+            \fi
+        \fi
+        \ifdim\eqnshift@<\z@
+            \global\eqnshift@\z@
+        \fi
+        \calc@shift@align
+        \global\tagshift@\totwidth@
+        \global\advance\tagshift@\@tempcntb\alignsep@
+        \if@fleqn
+            \ifnum\xatlevel@=\tw@
+                \global\advance\tagshift@-\@mathmargin\relax
+            \fi
+        \else
+            \global\advance\tagshift@\eqnshift@
+        \fi
+        \iftagsleft@ \else
+            \global\advance\tagshift@-\displaywidth
+        \fi
+        \dimen@\minalignsep\relax
+        \global\advance\totwidth@\@tempcntb\dimen@
+        \ifdim\totwidth@>\displaywidth
+            \global\let\displaywidth@\totwidth@
+        \else
+            \global\let\displaywidth@\displaywidth
+        \fi
+    \endgroup
+}
+\iftagsleft@\if@fleqn
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \@tempdima\@mathmargin\relax
+            \advance\@tempdima-\mintagsep\relax
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@lf
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+    \def\x@calc@shift@lf{%
+        \ifdim\eqnshift@=\z@
+            \global\eqnshift@\@mathmargin\relax
+            \alignsep@\displaywidth
+            \advance\alignsep@-\totwidth@
+            \global\divide\alignsep@\@tempcntb
+            \ifdim\alignsep@<\minalignsep\relax
+                \global\alignsep@\minalignsep\relax
+            \fi
+        \fi
+        \ifdim\tag@width\row@>\@tempdima
+            \saveshift@1%
+        \else
+            \saveshift@0%
+        \fi
+    }
+\fi\fi
+\iftagsleft@\else\if@fleqn
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@rf
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+    \def\x@calc@shift@rf{%
+        \column@\z@
+        \@tempdimb\z@
+        \@tempdimc\z@
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@rcalc@width
+        }%
+        \begingroup
+            \advance\column@\m@ne
+            \divide\column@\tw@
+            \ifnum\@tempcntb>\column@
+                \advance\@tempcnta-\@tempcntb
+                \advance\@tempcnta\column@
+                \@tempcntb\column@
+            \fi
+            \tagwidth@\tag@width\row@\relax
+            \@tempdima\eqnshift@
+            \advance\@tempdima\@tempdimc\relax
+            \advance\@tempdima\tagwidth@
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \advance\dimen@\@tempdima
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+                \saveshift@0%
+                \dimen@\alignsep@\relax
+                \multiply\dimen@\@tempcntb
+                \advance\dimen@\@tempdima
+                \advance\dimen@\tagwidth@
+                \ifdim\dimen@>\displaywidth
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+                    \divide\dimen@\@tempcnta
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                    \else
+                        \global\alignsep@\dimen@
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+\fi\fi
+\iftagsleft@\else\if@fleqn\else
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@rc
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+    \def\x@calc@shift@rc{%
+        \column@\z@
+        \@tempdimb\z@
+        \@tempdimc\z@
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@rcalc@width
+        }%
+        \begingroup
+            \advance\column@\m@ne
+            \divide\column@\tw@
+            \ifnum\@tempcntb>\column@
+                \advance\@tempcnta-\@tempcntb
+                \advance\@tempcnta\column@
+                \@tempcntb\column@
+            \fi
+            \tagwidth@\tag@width\row@\relax
+            \@tempdima\@tempdimc
+            \advance\@tempdima\tagwidth@
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \ifnum\xatlevel@=\tw@ \else
+                \advance\dimen@\mintagsep\relax
+            \fi
+            \advance\dimen@\@tempdima
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+                \saveshift@0%
+                \dimen@\eqnshift@
+                \advance\dimen@\@tempdima
+                \advance\dimen@\@tempcntb\alignsep@
+                \advance\dimen@\tagwidth@
+                \ifdim\dimen@>\displaywidth
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+                    \divide\dimen@\@tempcnta
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                        \eqnshift@\displaywidth
+                        \advance\eqnshift@-\@tempdima
+                        \advance\eqnshift@-\@tempcntb\alignsep@
+                        \global\divide\eqnshift@\tw@
+                    \else
+                        \ifdim\dimen@<\eqnshift@
+                            \ifdim\dimen@<\z@
+                                \global\eqnshift@\z@
+                            \else
+                                \global\eqnshift@\dimen@
+                            \fi
+                        \fi
+                        \ifdim\dimen@<\alignsep@
+                            \global\alignsep@\dimen@
+                        \fi
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+\fi\fi
+\iftagsleft@\else
+    \def\x@rcalc@width{%
+        \ifdim\@tempa > \z@
+            \advance\@tempdimc\@tempdimb
+            \ifodd\column@
+                \advance\@tempdimc\maxcol@width\column@
+                \@tempdimb\z@
+            \else
+                \advance\@tempdimc\@tempa\relax
+                \@tempdimb\maxcol@width\column@
+                \advance\@tempdimb-\@tempa\relax
+            \fi
+        \else
+            \advance\@tempdimb\maxcol@width\column@\relax
+        \fi
+    }
+\fi
+\iftagsleft@\if@fleqn\else
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>\z@
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@lc
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+    \def\x@calc@shift@lc{%
+        \column@\z@
+        \@tempdima\z@ % ``width of equation''
+        \@tempdimb\z@ % ``indent of equation''
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@lcalc@width
+        }%
+        \begingroup
+            \tagwidth@\tag@width\row@\relax
+            \@tempdima\totwidth@
+            \advance\@tempdima-\@tempdimb
+            \advance\@tempdima\tagwidth@
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \ifnum\xatlevel@=\tw@ \else
+                \advance\dimen@\mintagsep\relax
+            \fi
+            \advance\dimen@\@tempdima
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+                \saveshift@0%
+                \dimen@\alignsep@
+                \multiply\dimen@\count@
+                \advance\dimen@\eqnshift@
+                \advance\dimen@\@tempdimb
+                \ifdim\dimen@<2\tagwidth@
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+                    \ifnum\@tempcnta>\z@
+                      \divide\dimen@\@tempcnta
+                    \else \dimen@\z@
+                    \fi
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                        \dimen@\displaywidth
+                        \advance\dimen@-\@tempdima
+                        \advance\dimen@-\@tempcntb\alignsep@
+                        \global\divide\dimen@\tw@
+                    \else
+                        \ifdim\dimen@<\alignsep@
+                            \global\alignsep@\dimen@
+                        \fi
+                    \fi
+                    \ifnum\xatlevel@=\tw@
+                        \dimen@\mintagsep\relax
+                    \fi
+                    \advance\dimen@\tagwidth@
+                    \advance\dimen@-\@tempdimb
+                    \advance\dimen@-\count@\alignsep@
+                    \ifdim\dimen@>\eqnshift@
+                        \global\eqnshift@\dimen@
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+    \def\x@lcalc@width{%
+        \ifdim\@tempdima = \z@
+            \ifdim\@tempa > \z@
+                \@tempdima\p@
+                \ifodd\column@
+                    \advance\@tempdimb \maxcol@width\column@
+                    \advance\@tempdimb-\@tempa
+                \fi
+                \count@\column@
+                \advance\count@\m@ne
+                \divide\count@\tw@
+                \advance\@tempcnta-\count@
+                \advance\@tempcntb-\count@
+            \else
+                \advance\@tempdimb \maxcol@width\column@\relax
+            \fi
+        \fi
+    }
+\fi\fi
+\def\place@tag{%
+    \iftagsleft@
+        \kern-\tagshift@
+        \if1\shift@tag\row@\relax
+            \rlap{\vbox{%
+                \normalbaselines
+                \boxz@
+                \vbox to\lineht@{}%
+                \raise@tag
+            }}%
+        \else
+            \rlap{\boxz@}%
+        \fi
+        \kern\displaywidth@
+    \else
+        \kern-\tagshift@
+        \if1\shift@tag\row@\relax
+            \llap{\vtop{%
+                \raise@tag
+                \normalbaselines
+                \setbox\@ne\null
+                \dp\@ne\lineht@
+                \box\@ne
+                \boxz@
+            }}%
+        \else
+            \llap{\boxz@}%
+        \fi
+    \fi
+}
+\def\align@preamble{%
+   &\hfil
+    \strut@
+    \setboxz@h{\@lign$\m@th\displaystyle{##}$}%
+    \ifmeasuring@\savefieldlength@\fi
+    \set@field
+    \tabskip\z@skip
+   &\setboxz@h{\@lign$\m@th\displaystyle{{}##}$}%
+    \ifmeasuring@\savefieldlength@\fi
+    \set@field
+    \hfil
+    \tabskip\alignsep@
+}
+\def\set@field{%
+    \column@plus
+    \iftagsleft@
+        \ifdim\ht\z@>\lineht@
+            \global\lineht@\ht\z@
+        \fi
+    \else
+        \ifdim\dp\z@>\lineht@
+            \global\lineht@\dp\z@
+        \fi
+    \fi
+    \boxz@
+}
+\edef\split@err#1{%
+    \@nx\@amsmath@err{%
+        \string\begin{split} won't work here%
+    }{%
+        \@xp\@nx\csname
+  Did you forget a preceding \string\begin{equation}?^^J%
+  If not, perhaps the `aligned' environment is what
+  you want.\endcsname}%
+}
+\newenvironment{split}{%
+  \if@display
+    \ifinner
+      \@xp\@xp\@xp\split@aligned
+    \else
+      \ifst@rred \else \global\@eqnswtrue \fi
+    \fi
+  \else \let\endsplit\@empty \@xp\collect@body\@xp\split@err
+  \fi
+  \collect@body\gather@split
+}{%
+      \crcr
+    \egroup
+  \egroup
+  \iftagsleft@ \@xp\lendsplit@ \else \@xp\rendsplit@ \fi
+}
+\let\split@tag\relax % init
+\def\gather@split#1#2#3{%
+  \@xp\endgroup \reset@equation % math@cr will handle equation numbering
+  \iftag@
+     \toks@\@xp{\df@tag}%
+     \edef\split@tag{%
+       \gdef\@nx\df@tag{\the\toks@}%
+       \global\@nx\tag@true \@nx\nonumber
+     }%
+  \else \let\split@tag\@empty
+  \fi
+  \spread@equation
+  \vcenter\bgroup
+    \gather@{\split@tag  \begin{split}#1\end{split}}%
+    \def\endmathdisplay@a{%
+      \math@cr \black@ \totwidth@ \egroup
+      \egroup
+    }%
+}
+\def\insplit@{%
+  \global\setbox\z@\vbox\bgroup
+    \Let@ \chardef\dspbrk@context\@ne \restore@math@cr
+    \default@tag % disallow use of \tag here
+    \ialign\bgroup
+      \hfil
+      \strut@
+      $\m@th\displaystyle{##}$%
+     &$\m@th\displaystyle{{}##}$%
+      \hfill % Why not \hfil?---dmj, 1994/12/28
+      \crcr
+}
+\def\rendsplit@{%
+    \ifinalign@
+        \global\setbox9 \vtop{%
+            \unvcopy\z@
+            \global\setbox8 \lastbox
+            \unskip
+        }%
+        \setbox\@ne\hbox{%
+            \unhcopy8
+            \unskip
+            \global\setbox\tw@\lastbox
+            \unskip
+            \global\setbox\thr@@\lastbox
+        }%
+        \ifctagsplit@
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vcenter{\vbox{\moveleft\wd\thr@@\boxz@}}%
+            }%
+        \else
+            \global\setbox7 \hbox{\unhbox\tw@\unskip}%
+            \gdef\split@{%
+                \global\@tempcnta\column@
+               &\setboxz@h{}%
+                \savetaglength@
+                \global\advance\row@\@ne
+                \vbox{\moveleft\wd\thr@@\box9}%
+                \crcr
+                \noalign{\global\lineht@\z@}%
+                \add@amps\@tempcnta
+                \box\thr@@
+               &\box7
+            }%
+        \fi
+    \else
+        \ifctagsplit@
+            \gdef\split@{\vcenter{\boxz@}}%
+        \else
+            \gdef\split@{%
+                \boxz@
+            }%
+        \fi
+    \fi
+    \aftergroup\split@
+}
+\def\lendsplit@{%
+    \global\setbox9\vtop{\unvcopy\z@}%
+    \ifinalign@
+        \setbox\@ne\vbox{%
+            \unvcopy\z@
+            \global\setbox8\lastbox
+        }%
+        \setbox\@ne\hbox{%
+            \unhcopy8%
+            \unskip
+            \setbox\tw@\lastbox
+            \unskip
+            \global\setbox\thr@@\lastbox
+        }%
+        \ifctagsplit@
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vcenter{\vbox{\moveleft\wd\thr@@\box9}}%
+            }%
+        \else
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vbox{\moveleft\wd\thr@@\box9}%
+            }%
+        \fi
+    \else
+        \ifctagsplit@
+            \gdef\split@{\vcenter{\box9}}%
+        \else
+            \gdef\split@{\box9}%
+        \fi
+    \fi
+    \aftergroup\split@
+}
+\def\split@aligned#1#2{%
+   \iffalse{\fi\ifnum0=`}\fi
+   \collect@body\split@al@a}
+\def\split@al@a#1#2#3{%
+  \split@warning
+  \endgroup
+  \toks@{\begin{aligned}}%
+  \if@fleqn \split@al@tagcheck \fi
+  \the\toks@\relax#1\end{aligned}%
+  \ifnum0=`{\fi\iffalse}\fi
+}
+\def\split@al@tagcheck{%
+   \ifctagsplit@
+   \else
+     \iftagsleft@ \toks@\@xp{\the\toks@ [t]}%
+     \else \toks@\@xp{\the\toks@ [b]}%
+     \fi
+   \fi
+}
+\def\split@warning{%
+  \PackageWarning{amsmath}{%
+Cannot use `split' here;\MessageBreak trying to recover with `aligned'}%
+}
+\newskip\multlinegap
+\multlinegap10pt
+\newskip\multlinetaggap
+\multlinetaggap10pt
+\def\start@multline#1{%
+    \RIfM@
+        \nomath@env
+        \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+    \else
+        $$%
+        #1%
+        \ifst@rred
+            \nonumber
+        \else
+            \global\@eqnswtrue
+        \fi
+        \let\next@\multline@
+    \fi
+    \collect@body\next@
+}
+\newenvironment{multline}{%
+  \start@multline\st@rredfalse
+}{%
+  \iftagsleft@ \@xp\lendmultline@ \else \@xp\rendmultline@ \fi
+  \ignorespacesafterend
+}
+\newenvironment{multline*}{\start@multline\st@rredtrue}{\endmultline}
+\def\multline@#1{%
+    \Let@
+    \@display@init{\global\advance\row@\@ne \global\dspbrk@lvl\m@ne}%
+    \chardef\dspbrk@context\z@
+    \restore@math@cr
+    \let\tag\tag@in@align
+    \global\tag@false \global\let\raise@tag\@empty
+    \mmeasure@{#1}%
+    \let\tag\gobble@tag \let\label\@gobble
+    \tabskip \if@fleqn \@mathmargin \else \z@skip \fi
+    \totwidth@\displaywidth
+    \if@fleqn
+        \advance\totwidth@-\@mathmargin
+    \fi
+    \halign\bgroup
+        \hbox to\totwidth@{%
+            \if@fleqn
+                \hskip \@centering \relax
+            \else
+                \hfil
+            \fi
+            \strut@
+            $\m@th\displaystyle{}##\endmultline@math
+            \hfil
+        }%
+        \crcr
+        \if@fleqn
+            \hskip-\@mathmargin
+            \def\multline@indent{\hskip\@mathmargin}% put it back
+        \else
+            \hfilneg
+            \def\multline@indent{\hskip\multlinegap}%
+        \fi
+        \iftagsleft@
+            \iftag@
+                \begingroup
+                    \ifshifttag@
+                        \rlap{\vbox{%
+                                \normalbaselines
+                                \hbox{%
+                                    \strut@
+                                    \make@display@tag
+                                }%
+                                \vbox to\lineht@{}%
+                                \raise@tag
+                        }}%
+                        \multline@indent
+                    \else
+                        \setbox\z@\hbox{\make@display@tag}%
+                        \dimen@\@mathmargin \advance\dimen@-\wd\z@
+                        \ifdim\dimen@<\multlinetaggap
+                          \dimen@\multlinetaggap
+                        \fi
+                        \box\z@ \hskip\dimen@\relax
+                    \fi
+                \endgroup
+            \else
+                \multline@indent
+            \fi
+        \else
+            \multline@indent
+        \fi
+    #1%
+}
+\def\endmultline@math{$}
+\def\lendmultline@{%
+        \hfilneg
+        \hskip\multlinegap
+        \math@cr
+    \egroup
+    $$%
+}
+\def\rendmultline@{%
+    \iftag@
+        $\let\endmultline@math\relax
+            \ifshifttag@
+                \hskip\multlinegap
+                \llap{\vtop{%
+                    \raise@tag
+                    \normalbaselines
+                    \setbox\@ne\null
+                    \dp\@ne\lineht@
+                    \box\@ne
+                    \hbox{\strut@\make@display@tag}%
+                }}%
+            \else
+                \hskip\multlinetaggap
+                \make@display@tag
+            \fi
+    \else
+        \hskip\multlinegap
+    \fi
+    \hfilneg
+        \math@cr
+    \egroup$$%
+}
+\def\mmeasure@#1{%
+    \begingroup
+        \measuring@true
+        \def\label##1{%
+          \begingroup\measuring@false\label@in@display{##1}\endgroup}%
+        \def\math@cr@@@{\cr}%
+        \let\shoveleft\@iden \let\shoveright\@iden
+        \savecounters@
+        \global\row@\z@
+        \setbox\@ne\vbox{%
+            \global\let\df@tag\@empty
+            \halign{%
+                \setboxz@h{\@lign$\m@th\displaystyle{}##$}%
+                \iftagsleft@
+                    \ifnum\row@=\@ne
+                        \global\totwidth@\wdz@
+                        \global\lineht@\ht\z@
+                    \fi
+                \else
+                    \global\totwidth@\wdz@
+                    \global\lineht@\dp\z@
+                \fi
+                \crcr
+                #1%
+                \crcr
+            }%
+        }%
+        \ifx\df@tag\@empty\else\global\tag@true\fi
+        \if@eqnsw\global\tag@true\fi
+        \iftag@
+            \setboxz@h{%
+                \if@eqnsw
+                    \stepcounter{equation}%
+                    \tagform@\theequation
+                \else
+                    \df@tag
+                \fi
+            }%
+            \global\tagwidth@\wdz@
+            \dimen@\totwidth@
+            \advance\dimen@\tagwidth@
+            \advance\dimen@\multlinetaggap
+            \iftagsleft@\else
+                \if@fleqn
+                    \advance\dimen@\@mathmargin
+                \fi
+            \fi
+            \ifdim\dimen@>\displaywidth
+                \global\shifttag@true
+            \else
+                \global\shifttag@false
+            \fi
+        \fi
+        \restorecounters@
+    \endgroup
+}
+\iftagsleft@
+    \def\shoveright#1{%
+        #1%
+        \hfilneg
+        \hskip\multlinegap
+    }
+\else
+    \def\shoveright#1{%
+        #1%
+        \hfilneg
+        \iftag@
+            \ifshifttag@
+                \hskip\multlinegap
+            \else
+                \hskip\tagwidth@
+                \hskip\multlinetaggap
+            \fi
+        \else
+            \hskip\multlinegap
+        \fi
+    }
+\fi
+
+\if@fleqn
+    \def\shoveleft#1{#1}%
+\else
+    \iftagsleft@
+        \def\shoveleft#1{%
+            \setboxz@h{$\m@th\displaystyle{}#1$}%
+            \setbox\@ne\hbox{$\m@th\displaystyle#1$}%
+            \hfilneg
+            \iftag@
+                \ifshifttag@
+                    \hskip\multlinegap
+                \else
+                    \hskip\tagwidth@
+                    \hskip\multlinetaggap
+                \fi
+            \else
+                \hskip\multlinegap
+            \fi
+            \hskip.5\wd\@ne
+            \hskip-.5\wdz@
+            #1%
+        }
+    \else
+        \def\shoveleft#1{%
+            \setboxz@h{$\m@th\displaystyle{}#1$}%
+            \setbox\@ne\hbox{$\m@th\displaystyle#1$}%
+            \hfilneg
+            \hskip\multlinegap
+            \hskip.5\wd\@ne
+            \hskip-.5\wdz@
+            #1%
+        }
+    \fi
+\fi
+\@saveprimitive\leqno\@@leqno
+\@saveprimitive\eqno\@@eqno
+\def\eqno{\@@eqno\let\eqno\relax\let\leqno\relax}
+\def\leqno{\@@leqno\let\leqno\relax\let\eqno\relax}
+\let\veqno=\@@eqno
+\iftagsleft@ \let\veqno=\@@leqno \fi
+\@ifundefined{SK@@label}{%
+  \let\SK@@label\relax \let\SK@equationtrue\relax
+}{}
+\let\reset@equation\@empty
+\let\alt@tag\@empty
+\def\tag@in@display#1#{\relax\tag@in@display@a{#1}}
+\def\tag@in@display@a#1#2{%
+  \iftag@
+    \invalid@tag{Multiple \string\tag}\relax
+  \else
+    \global\tag@true \nonumber \reset@equation \st@rredtrue
+    \if *\string#1%
+      \gdef\alt@tag{\def\SK@tagform@{#2\@gobble}%
+        \ifx\SK@@label\relax \let\tagform@\SK@tagform@ \fi
+      }%
+      \make@df@tag@@{#2}%
+    \else
+      \make@df@tag@@@{#2}%
+    \fi
+  \fi
+}
+\let\restore@hfuzz\@empty
+\def\mathdisplay#1{%
+  \ifmmode \@badmath
+  \else
+    $$\def\@currenvir{#1}%
+    \let\dspbrk@context\z@
+    \let\tag\tag@in@display \let\label\label@in@display \SK@equationtrue
+    \global\let\df@label\@empty \global\let\df@tag\@empty
+    \global\tag@false
+    \let\mathdisplay@push\mathdisplay@@push
+    \let\mathdisplay@pop\mathdisplay@@pop
+    \if@fleqn
+      \edef\restore@hfuzz{\hfuzz\the\hfuzz\relax}%
+      \hfuzz\maxdimen
+      \setbox\z@\hbox to\displaywidth\bgroup
+        \let\split@warning\relax \restore@hfuzz
+        \everymath\@emptytoks \m@th $\displaystyle
+    \fi
+  \fi
+}
+\def\endmathdisplay#1{%
+  \ifmmode \else \@badmath \fi
+  \endmathdisplay@a
+  $$%
+  \global\let\df@label\@empty \global\let\df@tag\@empty
+  \global\tag@false \global\let\alt@tag\@empty
+  \global\@eqnswfalse
+}
+\def\endmathdisplay@a{%
+  \if@eqnsw \gdef\df@tag{\tagform@\theequation}\fi
+  \if@fleqn \@xp\endmathdisplay@fleqn
+  \else \ifx\df@tag\@empty \else \veqno \alt@tag \df@tag \fi
+    \ifx\df@label\@empty \else \@xp\ltx@label\@xp{\df@label}\fi
+  \fi
+  \ifnum\dspbrk@lvl>\m@ne
+    \postdisplaypenalty -\@getpen\dspbrk@lvl
+    \global\dspbrk@lvl\m@ne
+  \fi
+}
+\let\too@wide\@ne
+\def\endmathdisplay@fleqn{%
+  $\hfil\hskip\@mathmargin\egroup
+  \ifnum\badness<\inf@bad \let\too@wide\@ne \else \let\too@wide\z@ \fi
+  \ifx\@empty\df@tag
+  \else
+    \setbox4\hbox{\df@tag
+      \ifx\df@label\@empty \else \@xp\ltx@label\@xp{\df@label}\fi
+    }%
+  \fi
+  \csname emdf@%
+    \ifx\df@tag\@empty U\else \iftagsleft@ L\else R\fi\fi
+  \endcsname
+}
+\def\emdf@U{%
+  \restore@hfuzz
+  \ifodd\too@wide % not too wide: just need to swap the glue around
+    \hbox to\displaywidth{\hskip\@mathmargin\unhbox\z@\unskip}%
+  \else % M+B > displaywidth
+    \emdf@Ua
+  \fi
+}
+\def\emdf@Ua{%
+  \hbox to\columnwidth{%
+    \ifdim\displayindent>\z@
+      \hskip\displayindent minus\displayindent
+    \fi
+    \hskip\@mathmargin \unhbox\z@ \unskip
+  }%
+  \displayindent\z@ \displaywidth\columnwidth
+}
+\def\emdf@R{%
+  \setbox\tw@\hbox to\displaywidth{%
+    \hskip\@mathmargin \unhcopy\z@\unskip\hfil\hskip\mintagsep\copy4
+  }%
+  \restore@hfuzz
+  \ifnum\badness<\inf@bad \box\tw@ \else \emdf@Ra \fi
+}
+\def\emdf@Ra{%
+  \skip@\displayindent minus\displayindent
+  \displayindent\z@ \displaywidth\columnwidth
+  \spread@equation \everycr{}\tabskip\z@skip
+  \halign{\hbox to\displaywidth{##}\cr
+    \relax
+    \ifdim\skip@>\z@ \hskip\skip@ \fi
+    \hskip\@mathmargin\unhbox\z@\unskip\hfil\cr
+    \noalign{\raise@tag}%
+    \hfil\box4 \cr}%
+}
+\def\emdf@L{%
+  \@tempdima\@mathmargin
+  \advance\@tempdima-\wd4 \advance\@tempdima-\mintagsep
+  \skip@\@tempdima minus\@tempdima
+  \setbox\tw@\hbox to\displaywidth{%
+    \copy4\hskip\mintagsep
+    \ifdim\skip@>\z@ \hskip\skip@\fi
+    \unhcopy\z@\unskip
+  }%
+  \restore@hfuzz
+  \ifnum\badness<\inf@bad \box\tw@ \else \emdf@La \fi
+}
+\def\emdf@La{%
+  \spread@equation \everycr{}\tabskip\z@skip
+  \halign{\hbox to\displaywidth{##}\cr
+    \box4 \hfil \cr
+    \noalign{\raise@tag}%
+    \hskip\@mathmargin\unhbox\z@\unskip\hfil\cr}%
+}
+\newtoks\mathdisplay@stack
+\let\mathdisplay@push\@empty
+\def\mathdisplay@@push{%
+  \begingroup
+  \toks@\@xp{\df@label}\@temptokena\@xp{\df@tag}%
+  \toks8\@xp{\alt@tag}%
+  \edef\@tempa{%
+    \global\if@eqnsw\@nx\@eqnswtrue\else\@nx\@eqnswfalse\fi
+    \global\iftag@\@nx\tag@false\else\@nx\tag@true\fi
+    \gdef\@nx\df@label{\the\toks@}\gdef\@nx\df@tag{\the\@temptokena}%
+    \gdef\@nx\alt@tag{\the\toks8}%
+    \global\mathdisplay@stack{\the\mathdisplay@stack}%
+  }%
+  \global\mathdisplay@stack\@xp{\@tempa}
+  \endgroup
+}
+\let\mathdisplay@pop\@empty
+\def\mathdisplay@@pop{\the\mathdisplay@stack}
+\renewenvironment{equation}{%
+  \incr@eqnum
+  \mathdisplay@push
+  \st@rredfalse \global\@eqnswtrue
+  \mathdisplay{equation}%
+}{%
+  \endmathdisplay{equation}%
+  \mathdisplay@pop
+  \ignorespacesafterend
+}
+\newenvironment{equation*}{%
+  \mathdisplay@push
+  \st@rredtrue \global\@eqnswfalse
+  \mathdisplay{equation*}%
+}{%
+  \endmathdisplay{equation*}%
+  \mathdisplay@pop
+  \ignorespacesafterend
+}
+\DeclareRobustCommand{\[}{\begin{equation*}}
+\DeclareRobustCommand{\]}{\end{equation*}}
+\endinput
+%%
+%% End of file `amsmath.sty'.
diff --git a/Master/texmf-dist/tex/latex/amsmath/amsopn.sty b/Master/texmf-dist/tex/latex/amsmath/amsopn.sty
new file mode 100644
index 00000000000..76580233765
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/amsmath/amsopn.sty
@@ -0,0 +1,132 @@
+%%
+%% This is file `amsopn.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amsopn.dtx 
+%% 
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsopn.dtx",
+%%%   version   = "2.01",
+%%%   date      = "1999/12/14",
+%%%   time      = "17:30:59 EST",
+%%%   checksum  = "46904 254 1037 9431",
+%%%   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  = "operator name, amsopn, DeclareMathOperator",
+%%%   supported = "yes",
+%%%   abstract  = "This is a \LaTeX{} package that provides a
+%%%                DeclareMathOperator command for defining named
+%%%                operators like sin and lim.",
+%%%   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
+\ProvidesPackage{amsopn}[1999/12/14 v2.01 operator names]
+\def\nolimits@{\@ifnextchar\limits{\nolimits\@gobble}{\nolimits}}
+\begingroup \catcode`\"=12
+\gdef\newmcodes@{\mathcode`\'39\mathcode`\*42\mathcode`\."613A%
+  \ifnum\mathcode`\-=45 \else
+    \mathchardef\std@minus\mathcode`\-\relax
+  \fi
+  \mathcode`\-45\mathcode`\/47\mathcode`\:"603A\relax}
+\endgroup
+\DeclareRobustCommand{\operatorname}{%
+  \@ifstar{\qopname\newmcodes@ m}%
+          {\qopname\newmcodes@ o}}%
+\DeclareRobustCommand{\qopname}[3]{%
+  \mathop{#1\kern\z@\operator@font#3}%
+  \csname n#2limits@\endcsname}
+\newcommand{\DeclareMathOperator}{%
+  \@ifstar{\@declmathop m}{\@declmathop o}}
+\long\def\@declmathop#1#2#3{%
+  \@ifdefinable{#2}{%
+    \DeclareRobustCommand{#2}{\qopname\newmcodes@#1{#3}}}}
+\@onlypreamble\DeclareMathOperator
+\@onlypreamble\@declmathop
+\def\arccos{\qopname\relax o{arccos}}
+\def\arcsin{\qopname\relax o{arcsin}}
+\def\arctan{\qopname\relax o{arctan}}
+\def\arg{\qopname\relax o{arg}}
+\def\cos{\qopname\relax o{cos}}
+\def\cosh{\qopname\relax o{cosh}}
+\def\cot{\qopname\relax o{cot}}
+\def\coth{\qopname\relax o{coth}}
+\def\csc{\qopname\relax o{csc}}
+\def\deg{\qopname\relax o{deg}}
+\def\det{\qopname\relax m{det}}
+\def\dim{\qopname\relax o{dim}}
+\def\exp{\qopname\relax o{exp}}
+\def\gcd{\qopname\relax m{gcd}}
+\def\hom{\qopname\relax o{hom}}
+\def\inf{\qopname\relax m{inf}}
+\def\injlim{\qopname\relax m{inj\,lim}}
+\def\ker{\qopname\relax o{ker}}
+\def\lg{\qopname\relax o{lg}}
+\def\lim{\qopname\relax m{lim}}
+\def\liminf{\qopname\relax m{lim\,inf}}
+\def\limsup{\qopname\relax m{lim\,sup}}
+\def\ln{\qopname\relax o{ln}}
+\def\log{\qopname\relax o{log}}
+\def\max{\qopname\relax m{max}}
+\def\min{\qopname\relax m{min}}
+\def\Pr{\qopname\relax m{Pr}}
+\def\projlim{\qopname\relax m{proj\,lim}}
+\def\sec{\qopname\relax o{sec}}
+\def\sin{\qopname\relax o{sin}}
+\def\sinh{\qopname\relax o{sinh}}
+\def\sup{\qopname\relax m{sup}}
+\def\tan{\qopname\relax o{tan}}
+\def\tanh{\qopname\relax o{tanh}}
+\def\operator@font{\mathgroup\symoperators}
+\def\operatorfont{\operator@font}
+\def\operatornamewithlimits{\operatorname*}
+\def\varlim@#1#2{%
+  \vtop{\m@th\ialign{##\cr
+    \hfil$#1\operator@font lim$\hfil\cr
+    \noalign{\nointerlineskip\kern1.5\ex@}#2\cr
+    \noalign{\nointerlineskip\kern-\ex@}\cr}}%
+}
+\def\varinjlim{%
+  \mathop{\mathpalette\varlim@{\rightarrowfill@\textstyle}}\nmlimits@
+}
+\def\varprojlim{%
+  \mathop{\mathpalette\varlim@{\leftarrowfill@\textstyle}}\nmlimits@
+}
+\def\varliminf{\mathop{\mathpalette\varliminf@{}}\nmlimits@}
+\def\varliminf@#1{\@@underline{\vrule\@depth.2\ex@\@width\z@
+   \hbox{$#1\m@th\operator@font lim$}}}
+\def\varlimsup{\mathop{\mathpalette\varlimsup@{}}\nmlimits@}
+\def\varlimsup@#1{\@@overline{\hbox{$#1\m@th\operator@font lim$}}}
+\let\nmlimits@\displaylimits
+\DeclareOption{namelimits}{\let\nmlimits@\displaylimits}
+\DeclareOption{nonamelimits}{\let\nmlimits@\nolimits}
+\ProcessOptions\relax
+\RequirePackage{amsgen}\relax
+\endinput
+%%
+%% End of file `amsopn.sty'.
diff --git a/Master/texmf-dist/tex/latex/amsmath/amstex.sty b/Master/texmf-dist/tex/latex/amsmath/amstex.sty
new file mode 100644
index 00000000000..f67435c5829
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/amsmath/amstex.sty
@@ -0,0 +1,2043 @@
+%%% ====================================================================
+%%%  @LaTeX-file{
+%%%     filename  = "amstex.sty",
+%%%     version   = "1.2f",
+%%%     date      = "1999/11/15",
+%%%     time      = "14:07:07 EST",
+%%%     checksum  = "05218 2043 3020 71018",
+%%%     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@math.ams.org (Internet)",
+%%%     keywords  = "latex, amslatex, ams-latex, math, amstex, ams-tex",
+%%%     supported = "yes",
+%%%     abstract  = "This package provides a variety of extra
+%%%                  mathematical features, largely derived from
+%%%                  AMS-\TeX{}. It was superseded in 1995 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.",
+%%%  }
+%%% ====================================================================
+%    This file was generated from version 1.2beta of amstex.dtx and then
+%    underwent additional hand-editing.
+\if@compatibility \else \catcode`\%=9 \fi
+%\PackageError{amstex}{%
+%Package `amstex' is obsolete, substituting `amsmath'}{You
+%should use \string\usepackage{amsmath} instead. If
+%that does not give satisfactory results, rename your previous copy
+%of amstex.sty to amstex12beta.sty and call it with
+%\string\usepackage{amstex12beta}.}
+%\catcode`\%=14 \relax \input{amsmath.sty}\endinput
+%
+\ProvidesPackage{amstex}[1999/11/15 v1.2f AMS extra math features]
+\let\@amsfonts=T
+\DeclareOption{noamsfonts}{\let\@amsfonts=F}
+\DeclareOption{psamsfonts}{\let\@amsfonts=P}
+\DeclareOption{intlimits}{\let\ilimits@\displaylimits}
+\DeclareOption{nointlimits}{\let\ilimits@\nolimits}
+\DeclareOption{sumlimits}{\let\slimits@\displaylimits}
+\DeclareOption{nosumlimits}{\let\slimits@\nolimits}
+\DeclareOption{namelimits}{\let\nlimits@\displaylimits}
+\DeclareOption{nonamelimits}{\let\nlimits@\nolimits}
+\newif\ifctagsplit@
+\newif\iftagsleft@
+\DeclareOption{leqno}{\tagsleft@true}
+\DeclareOption{reqno}{\tagsleft@false}
+\DeclareOption{centertags}{\ctagsplit@true}
+\DeclareOption{tbtags}{\ctagsplit@false}
+\newif\if@fleqn
+\newskip\@mathmargin
+\@mathmargin\@centering
+\DeclareOption{fleqn}{%
+    \@fleqntrue
+    \@mathmargin\m@ne sp
+    \AtBeginDocument{%
+        \ifdim\@mathmargin=\m@ne sp
+            \@mathmargin\leftmargini
+        \fi
+    }%
+}
+\DeclareOption{righttag}{\tagsleft@false}
+\DeclareOption{ctagsplt}{\ctagsplit@true}
+\DeclareOption{intlim}{\let\ilimits@\displaylimits}
+\DeclareOption{nosumlim}{\let\slimits@\nolimits}
+\DeclareOption{nonamelm}{\let\nlimits@\nolimits}
+\ExecuteOptions{nointlimits,sumlimits,namelimits,centertags}
+\ProcessOptions
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% In this section \RequirePackage statements were replaced with the
+%% contents of the referenced files.
+%%%     filename        = "amsgen.dtx",
+%%%     version         = "1.2beta",
+%%%     date            = "1994/12/13",
+\providecommand{\@saveprimitive}[2]{\begingroup\escapechar`\\\relax
+  \edef\@tempa{\string#1}\edef\@tempb{\meaning#1}%
+  \ifx\@tempa\@tempb \global\let#2#1%
+  \else
+    \edef\@tempb{\meaning#2}%
+    \ifx\@tempa\@tempb
+    \else
+      \@latex@error{Unable to properly define \string#2; primitive
+      \noexpand#1no longer primitive}\@eha
+    \fi
+  \fi
+  \endgroup}
+\let\@xp=\expandafter
+\let\@nx=\noexpand
+\newtoks\@emptytoks
+\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
+\def\@ifempty#1{\@xifempty#1@@..\@nil}
+\long\def\@xifempty#1#2@#3#4#5\@nil{%
+  \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
+\long\def\@ifnotempty#1{\@ifempty{#1}{}}
+\def\FN@{\futurelet\@let@token}
+\def\DN@{\def\next@}
+\def\RIfM@{\relax\ifmmode}
+\def\setboxz@h{\setbox\z@\hbox}
+\def\wdz@{\wd\z@}
+\def\boxz@{\box\z@}
+\def\relaxnext@{\let\@let@token\relax}
+\def\new@ifnextchar#1#2#3{%
+  \let\@tempe #1\def\@tempa{#2}\def\@tempb{#3}\futurelet
+    \@tempc\new@ifnch}
+\def\new@ifnch{\ifx\@tempc \@tempe \let\@tempd\@tempa
+             \else\let\@tempd\@tempb\fi\@tempd}
+\def\new@ifstar#1#2{\new@ifnextchar *{\def\@tempa*{#1}\@tempa}{#2}}
+\@ifundefined{every@math@size}{%
+\let\every@math@size=\every@size
+\def\glb@settings{%
+     \expandafter\ifx\csname S@\f@size\endcsname\relax
+       \calculate@math@sizes
+     \fi
+     \csname S@\f@size\endcsname
+      \ifmath@fonts
+        \begingroup
+          \escapechar\m@ne
+          \csname mv@\math@version \endcsname
+          \globaldefs\@ne
+          \let \glb@currsize \f@size
+          \math@fonts
+        \endgroup
+        \the\every@math@size
+      \else
+      \fi
+}
+\def\set@fontsize#1#2#3{%
+    \@defaultunits\@tempdimb#2pt\relax\@nnil
+    \edef\f@size{\strip@pt\@tempdimb}%
+    \@defaultunits\@tempskipa#3pt\relax\@nnil
+    \edef\f@baselineskip{\the\@tempskipa}%
+    \edef\f@linespread{#1}%
+    \let\baselinestretch\f@linespread
+      \def\size@update{%
+        \baselineskip\f@baselineskip\relax
+        \baselineskip\f@linespread\baselineskip
+        \normalbaselineskip\baselineskip
+        \setbox\strutbox\hbox{%
+          \vrule\@height.7\baselineskip
+                \@depth.3\baselineskip
+                \@width\z@}%
+%%%     \the\every@size
+        \let\size@update\relax}%
+  }
+}{}% end \@ifundefined test
+\newdimen\ex@
+\addto@hook\every@math@size{\compute@ex@}
+\def\compute@ex@{%
+  \begingroup
+  \dimen@-\f@size\p@
+  \ifdim\dimen@<-20\p@
+    \global\ex@ 1.5\p@
+  \else
+    \advance\dimen@10\p@ \multiply\dimen@\tw@
+    \edef\@tempa{\ifdim\dimen@>\z@ -\fi}%
+    \dimen@ \ifdim\dimen@<\z@ -\fi \dimen@
+    \advance\dimen@-\@m sp % fudge factor
+    \vfuzz\p@
+    \def\do{\ifdim\dimen@>\z@
+      \vfuzz=.97\vfuzz
+      \advance\dimen@ -\p@
+      \@xp\do \fi}%
+    \do
+    \dimen@\p@ \advance\dimen@-\vfuzz
+    \global\ex@\p@
+    \global\advance\ex@ \@tempa\dimen@
+  \fi
+  \endgroup
+}
+\def\@addpunct#1{\ifnum\spacefactor>\@m \else#1\fi}
+\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
+  \sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
+\def\@mathmeasure#1#2#3{\setbox#1\hbox{\frozen@everymath\@emptytoks
+  \m@th$#2#3$}}
+\def\nomath@env{\PackageError{amstex}{%
+  \string\begin{\@currenvir} allowed only in paragraph mode%
+}\@ehb% "You've lost some text"
+}
+\def\Invalid@@{Invalid use of \string}
+\def\nonmatherr@#1{\PackageError{amsfonts}%
+    {\noexpand#1allowed only in math mode}\@ehd}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%     filename        = "amstext.dtx",
+%%%     version         = "1.2beta",
+%%%     date            = "1994/12/13",
+\DeclareRobustCommand{\text}{%
+  \ifmmode\expandafter\text@\else\expandafter\mbox\fi}
+\let\nfss@text\text
+\def\text@#1{\mathchoice
+  {\textdef@\displaystyle\f@size{#1}}%
+  {\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
+  {\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
+  {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
+  \check@mathfonts
+}
+\def\textdef@#1#2#3{\hbox{{%
+                    \everymath{#1}%
+                    \let\f@size#2\selectfont
+                    #3}}}
+\newif\iffirstchoice@
+\firstchoice@true
+\def\stepcounter#1{%
+  \iffirstchoice@
+     \global\expandafter\advance\csname c@#1\endcsname \@ne
+     \begingroup
+         \let\@elt\@stpelt
+         \csname cl@#1\endcsname
+     \endgroup
+  \fi}
+\toks@{\iffirstchoice@\else\@xp\@gobblefour\fi\protect}
+\edef\GenericInfo{\the\toks@
+  \@xp\@nx\csname GenericInfo \endcsname}
+\edef\GenericWarning{\the\toks@
+  \@xp\@nx\csname GenericWarning \endcsname}
+\toks@{\iffirstchoice@\else
+  \@xp\@xp\@xp\@gobblefour
+  \@xp\@gobbletwo\fi\protect}
+\edef\GenericError{\the\toks@
+  \@xp\@nx\csname GenericError \endcsname}
+\def\label#1{\@bsphack\iffirstchoice@\if@filesw {\let\thepage\relax
+   \def\protect{\noexpand\noexpand\noexpand}%
+\xdef\@gtempa{\write\@auxout{\string
+      \newlabel{#1}{{\@currentlabel}{\thepage}}}}}\@gtempa
+   \if@nobreak \ifvmode\nobreak\fi\fi\fi\fi\@esphack}
+\begingroup \catcode`\"=12
+\gdef\mathhexbox#1#2#3{\text{$\m@th\mathchar"#1#2#3$}}
+\endgroup
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%     filename        = "amsfonts.dtx",
+%%%     version         = "1.2beta",
+%%%     date            = "1994/12/13",
+\newbox\Mathstrutbox@
+\setbox\Mathstrutbox@=\hbox{}
+\def\Mathstrut@{\copy\Mathstrutbox@}
+\addto@hook\every@math@size{\setbox\z@\hbox{\normalfont(}%
+  \ht\Mathstrutbox@\ht\z@ \dp\Mathstrutbox@\dp\z@}
+\newbox\strutbox@
+\def\strut@{\copy\strutbox@}
+\addto@hook\every@math@size{%
+  \global\setbox\strutbox@\hbox{\lower.5\normallineskiplimit
+         \vbox{\kern-\normallineskiplimit\copy\strutbox}}}
+\def\big{\bBigg@\@ne}
+\def\Big{\bBigg@{1.5}}
+\def\bigg{\bBigg@\tw@}
+\def\Bigg{\bBigg@{2.5}}
+\def\bBigg@#1#2{%
+   {%
+    \hbox{$\left#2\vcenter to#1\big@size{}\right.%
+           \n@space
+     $}}}
+\addto@hook\every@math@size{%
+  \global\big@size 1.2\ht\Mathstrutbox@
+  \global\advance\big@size 1.2\dp\Mathstrutbox@ }
+\newdimen\big@size
+%% msam, msbm, eufm fonts dealt with at end of file
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%     filename        = "amsbsy.dtx",
+%%%     version         = "1.2beta",
+%%%     date            = "1994/10/27",
+\DeclareRobustCommand{\boldsymbol}[1]{%
+  \begingroup
+  \let\@nomath\@gobble \mathversion{bold}%
+  \math@atom{#1}{%
+  \mathchoice%
+    {\hbox{$\m@th\displaystyle#1$}}%
+    {\hbox{$\m@th\textstyle#1$}}%
+    {\hbox{$\m@th\scriptstyle#1$}}%
+    {\hbox{$\m@th\scriptscriptstyle#1$}}}%
+  \endgroup}
+\def\math@atom#1#2{%
+   \binrel@{#1}\binrel@@{#2}}
+\DeclareRobustCommand{\pmb}{%
+  \ifmmode\else \expandafter\pmb@@\fi\mathpalette\pmb@}
+\def\pmb@@#1#2#3{\leavevmode\setboxz@h{#3}%
+   \dimen@-\wdz@
+   \kern-.5\ex@\copy\z@
+   \kern\dimen@\kern.25\ex@\raise.4\ex@\copy\z@
+   \kern\dimen@\kern.25\ex@\box\z@
+}
+\newdimen\pmbraise@
+\def\pmb@#1#2{\setbox8\hbox{$\m@th#1{#2}$}%
+  \setboxz@h{$\m@th#1\mkern.5mu$}\pmbraise@\wdz@
+  \binrel@{#2}%
+  \dimen@-\wd8 %
+  \binrel@@{%
+    \mkern-.8mu\copy8 %
+    \kern\dimen@\mkern.4mu\raise\pmbraise@\copy8 %
+    \kern\dimen@\mkern.4mu\box8 }%
+}
+\def\binrel@#1{\begingroup
+  \setboxz@h{\thinmuskip0mu
+    \medmuskip\m@ne mu\thickmuskip\@ne mu
+    \setbox\tw@\hbox{$#1\m@th$}\kern-\wd\tw@
+    ${}#1{}\m@th$}%
+  \edef\@tempa{\endgroup\let\noexpand\binrel@@
+    \ifdim\wdz@<\z@ \mathbin
+    \else\ifdim\wdz@>\z@ \mathrel
+    \else \relax\fi\fi}%
+  \@tempa
+}
+\let\binrel@@\relax
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\def\AmSTeX{\protect\AmS-\protect\TeX{}}
+\def\AmS{{\protect\AmSfont
+  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}
+\def\AmSfont{%
+  \usefont{OMS}{cmsy}{\if\@xp\@car\f@series\@nil bb\else m\fi}{n}}
+\def\prim@s{\prime\futurelet\@let@token\pr@m@s}
+\def\pr@m@s{%
+  \ifx\@let@token'\DN@##1{\prim@s}\else\let\next@\egroup\fi\next@}
+\let\@prime=\prime
+\renewcommand{\prime}{{\kern\z@\@prime}}
+\DeclareRobustCommand{\mspace}[3]{%
+  \ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
+\renewcommand{\,}{\mspace+\thinmuskip{.1667em}}
+\let\thinspace\,
+\renewcommand{\!}{\mspace-\thinmuskip{.1667em}}
+\let\negthinspace\!
+\renewcommand{\:}{\mspace+\medmuskip{.2222em}}
+\let\medspace\:
+\newcommand{\negmedspace}{\mspace-\medmuskip{.2222em}}
+\renewcommand{\;}{\mspace+\thickmuskip{.2777em}}
+\let\thickspace\;
+\newcommand{\negthickspace}{\mspace-\thickmuskip{.2777em}}
+\def\implies{\DOTSB\;\Longrightarrow\;}
+\def\impliedby{\DOTSB\;\Longleftarrow\;}
+\begingroup \catcode`\"=12 % in case activated by a preceding package
+\gdef\And{\DOTSB\;\mathchar"3026 \;}
+\gdef\lvert{\delimiter"426A30C }
+\gdef\rvert{\delimiter"526A30C }
+\gdef\lVert{\delimiter"426B30D }
+\gdef\rVert{\delimiter"526B30D }
+\endgroup % restore "
+\DeclareRobustCommand{\frac}{\new@ifnextchar[{\frfrac{}}{\frfrac{}[]}}
+\def\frfrac#1[#2]#3#4{\def\next@{#2}%
+  \ifx\next@\@empty \def\next@{#1{#3\over#4}}%
+  \else \def\next@{{#1{#3\above#2\relax#4}}}%
+  \fi
+  \next@}
+\def\dfrac{\protect\frfrac\displaystyle[]}
+\def\tfrac{\protect\frfrac\textstyle[]}
+\DeclareRobustCommand{\binom}{\frbinom{}}
+\def\frbinom#1#2#3{{#1{#2\atopwithdelims()#3}}}
+\def\dbinom{\protect\frbinom\displaystyle}
+\def\tbinom{\protect\frbinom\textstyle}
+\def\colon{\nobreak\mskip2mu\mathpunct{}\nonscript
+  \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
+\begingroup \catcode`\"=12
+\DeclareMathSymbol{\coprod@}{\mathop}{largesymbols}{"60}
+\gdef\coprod{\DOTSB\coprod@\slimits@}
+\DeclareMathSymbol{\bigvee@}{\mathop}{largesymbols}{"57}
+\gdef\bigvee{\DOTSB\bigvee@\slimits@}
+\DeclareMathSymbol{\bigwedge@}{\mathop}{largesymbols}{"56}
+\gdef\bigwedge{\DOTSB\bigwedge@\slimits@}
+\DeclareMathSymbol{\biguplus@}{\mathop}{largesymbols}{"55}
+\gdef\biguplus{\DOTSB\biguplus@\slimits@}
+\DeclareMathSymbol{\bigcap@}{\mathop}{largesymbols}{"54}
+\gdef\bigcap{\DOTSB\bigcap@\slimits@}
+\DeclareMathSymbol{\bigcup@}{\mathop}{largesymbols}{"53}
+\gdef\bigcup{\DOTSB\bigcup@\slimits@}
+\DeclareMathSymbol{\prod@}{\mathop}{largesymbols}{"51}
+\gdef\prod{\DOTSB\prod@\slimits@}
+\DeclareMathSymbol{\sum@}{\mathop}{largesymbols}{"50}
+\gdef\sum{\DOTSB\sum@\slimits@}
+\DeclareMathSymbol{\bigotimes@}{\mathop}{largesymbols}{"4E}
+\gdef\bigotimes{\DOTSB\bigotimes@\slimits@}
+\DeclareMathSymbol{\bigoplus@}{\mathop}{largesymbols}{"4C}
+\gdef\bigoplus{\DOTSB\bigoplus@\slimits@}
+\DeclareMathSymbol{\bigodot@}{\mathop}{largesymbols}{"4A}
+\gdef\bigodot{\DOTSB\bigodot@\slimits@}
+\DeclareMathSymbol{\bigsqcup@}{\mathop}{largesymbols}{"46}
+\gdef\bigsqcup{\DOTSB\bigsqcup@\slimits@}
+\endgroup
+\def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
+ \hbox{\normalfont ...}\vss}}}}
+\def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
+ \hbox{\normalfont....}\vss}}}}
+\def~{\nobreakspace}
+\DeclareRobustCommand{\nobreakspace}{\unskip\nobreak\ \ignorespaces}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newif\if@display
+\everydisplay\expandafter{\the\everydisplay \@displaytrue}
+\def\int{\DOTSI\intop\ilimits@}
+\def\oint{\DOTSI\ointop\ilimits@}
+\def\intkern@{\mkern-6mu\mathchoice{\mkern-3mu}{}{}{}}
+\def\intdots@{\mathchoice{\@cdots}%
+ {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
+ {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
+ {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}
+
+\def\iint{\DOTSI\protect\ints@\tw@}
+\def\iiint{\DOTSI\protect\ints@\thr@@}
+\def\iiiint{\DOTSI\protect\ints@{4}}
+\def\idotsint{\DOTSI\protect\ints@\z@}
+
+\def\ints@#1{%
+  \mkern-7mu\mathchoice{\mkern-2mu}{}{}{}%
+  \mathop{\mkern7mu\mathchoice{\mkern2mu}{}{}{}%
+    \intop\ifnum#1=\z@\intdots@
+    \else\intkern@\fi
+    \ifnum#1>\tw@\intop\intkern@\fi
+    \ifnum#1>\thr@@\intop\intkern@\fi
+    \intop
+  }\ilimits@
+}
+\def\nolimits@{%
+  \DN@{\nolimits\ifx\@let@token\limits\expandafter\@gobble\fi}%
+  \FN@\next@}
+\begingroup \catcode`\"=12
+\gdef\newmcodes@{\mathcode`\'39\mathcode`\*42\mathcode`\."613A%
+ \mathcode`\-45\mathcode`\/47\mathcode`\:"603A\relax}
+\endgroup
+\DeclareRobustCommand{\operatorname}{%
+  \@ifstar{\qopname\newmcodes@ n}%
+          {\qopname\newmcodes@{no}}}%
+\DeclareRobustCommand{\qopname}[3]{\mathop{#1\kern\z@\operator@font#3}%
+  \csname #2limits@\endcsname}
+\def\arccos{\qopname\relax{no}{arccos}}
+\def\arcsin{\qopname\relax{no}{arcsin}}
+\def\arctan{\qopname\relax{no}{arctan}}
+\def\arg{\qopname\relax{no}{arg}}
+\def\cos{\qopname\relax{no}{cos}}
+\def\cosh{\qopname\relax{no}{cosh}}
+\def\cot{\qopname\relax{no}{cot}}
+\def\coth{\qopname\relax{no}{coth}}
+\def\csc{\qopname\relax{no}{csc}}
+\def\deg{\qopname\relax{no}{deg}}
+\def\det{\qopname\relax n{det}}
+\def\dim{\qopname\relax{no}{dim}}
+\def\exp{\qopname\relax{no}{exp}}
+\def\gcd{\qopname\relax n{gcd}}
+\def\hom{\qopname\relax{no}{hom}}
+\def\inf{\qopname\relax n{inf}}
+\def\injlim{\qopname\relax n{inj\,lim}}
+\def\ker{\qopname\relax{no}{ker}}
+\def\lg{\qopname\relax{no}{lg}}
+\def\lim{\qopname\relax n{lim}}
+\def\liminf{\qopname\relax n{lim\,inf}}
+\def\limsup{\qopname\relax n{lim\,sup}}
+\def\ln{\qopname\relax{no}{ln}}
+\def\log{\qopname\relax{no}{log}}
+\def\max{\qopname\relax n{max}}
+\def\min{\qopname\relax n{min}}
+\def\Pr{\qopname\relax n{Pr}}
+\def\projlim{\qopname\relax n{proj\,lim}}
+\def\sec{\qopname\relax{no}{sec}}
+\def\sin{\qopname\relax{no}{sin}}
+\def\sinh{\qopname\relax{no}{sinh}}
+\def\sup{\qopname\relax n{sup}}
+\def\tan{\qopname\relax{no}{tan}}
+\def\tanh{\qopname\relax{no}{tanh}}
+\def\operator@font{\mathgroup\symoperators}
+\def\operatornamewithlimits{\operatorname*}
+\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
+ \hfil$#1\m@th\operator@font lim$\hfil\crcr
+ \noalign{\nointerlineskip\kern\ex@}#2#1\crcr
+ \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}
+\def\varinjlim{\mathpalette\varlim@\rightarrowfill@}
+\def\varprojlim{\mathpalette\varlim@\leftarrowfill@}
+\def\varliminf{\mathpalette\varliminf@{}}
+\def\varliminf@#1{\mathop{\@@underline{\vrule\@depth.2\ex@\@width\z@
+   \hbox{$#1\m@th\operator@font lim$}}}}
+\def\varlimsup{\mathpalette\varlimsup@{}}
+\def\varlimsup@#1{\mathop{\@@overline
+  {\hbox{$#1\m@th\operator@font lim$}}}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\def\numberwithin#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}{%
+  \@ifundefined{c@#2}{\@nocnterr{#2}}{%
+  \@addtoreset{#1}{#2}%
+  \toks@\expandafter\expandafter\expandafter{\csname the#1\endcsname}%
+  \expandafter\xdef\csname the#1\endcsname
+    {\expandafter\noexpand\csname the#2\endcsname
+     .\the\toks@}}}}
+\def\eqref#1{\textup{\tagform@{\ref{#1}}}}
+\def\bmod{\mskip-\medmuskip\mkern5mu\mathbin
+  {\operator@font mod}\penalty900
+  \mkern5mu\mskip-\medmuskip}
+\def\pod#1{\allowbreak\if@display\mkern18mu\else\mkern8mu\fi(#1)}
+\def\pmod#1{\pod{{\operator@font mod}\mkern6mu#1}}
+\def\mod#1{\allowbreak\if@display\mkern18mu
+  \else\mkern12mu\fi{\operator@font mod}\,\,#1}
+\newcommand{\cfrac}[3][c]{{\displaystyle\frac{%
+  \strut\ifx r#1\hfill\fi#2\ifx l#1\hfill\fi}{#3}}%
+  \kern-\nulldelimiterspace}
+\def\lcfrac{\cfrac[l]}
+\def\rcfrac{\cfrac[r]}
+\def\overset#1#2{\binrel@{#2}%
+  \binrel@@{\mathop{\kern\z@#2}\limits^{#1}}}
+\def\underset#1#2{\binrel@{#2}%
+  \binrel@@{\mathop{\kern\z@#2}\limits_{#1}}}
+\def\sideset#1#2#3{%
+  \setbox\z@\hbox{$\displaystyle{\vphantom{#3}}#1{#3}\m@th$}%
+  \setbox\tw@\hbox{$\displaystyle{#3}#2\m@th$}%
+  \hskip\wd\z@\hskip-\wd\tw@\mathop{\hskip\wd\tw@\hskip-\wd\z@
+  {\vphantom{#3}}#1{#3}#2}}
+\renewcommand{\smash}[2][tb]{%
+  \def\smash@{#1}%
+  \ifmmode\expandafter\mathpalette\expandafter\mathsm@sh\else
+        \expandafter\makesm@sh\fi{#2}}
+\def\finsm@sh{\def\mb@t{\ht\z@\z@}\def\mb@b{\dp\z@\z@}%
+  \def\mb@tb{\mb@t\mb@b}%
+  {\csname mb@\smash@\endcsname}%
+  \leavevmode\boxz@}
+\def\rightarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
+  $#1\copy\z@\mkern-6mu\cleaders
+  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
+  \mkern-6mu\mathord\rightarrow$}
+\def\leftarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
+  $#1\mathord\leftarrow\mkern-6mu\cleaders
+  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
+  \mkern-6mu\box\z@$}
+\def\leftrightarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
+  $#1\mathord\leftarrow\mkern-6mu\cleaders
+  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
+  \mkern-6mu\mathord\rightarrow$}
+\def\overarrow@#1#2#3{\vbox{\ialign{##\crcr#1#2\crcr
+ \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#2#3\hfil$\crcr}}}
+\def\overrightarrow{\mathpalette{\overarrow@\rightarrowfill@}}
+\def\overleftarrow{\mathpalette{\overarrow@\leftarrowfill@}}
+\def\overleftrightarrow{\mathpalette{\overarrow@\leftrightarrowfill@}}
+\def\underarrow@#1#2#3{%
+ \vtop{\ialign{##\crcr$\m@th\hfil#2#3\hfil$\crcr
+ \noalign{\nointerlineskip\kern-.5\ex@}#1#2\crcr}}}
+\def\underrightarrow{\mathpalette{\underarrow@\rightarrowfill@}}
+\def\underleftarrow{\mathpalette{\underarrow@\leftarrowfill@}}
+\def\underleftrightarrow{\mathpalette{\underarrow@\leftrightarrowfill@}}
+\let\ifgtest@\iffalse                              % initial value
+\def\gtest@true{\global\let\ifgtest@\iftrue}
+\def\gtest@false{\global\let\ifgtest@\iffalse}
+\let\DOTSI\relax
+\let\DOTSB\relax
+\let\DOTSX\relax
+{\uccode`7=`\\ \uccode`8=`m \uccode`9=`a \uccode`0=`t \uccode`!=`h
+ \uppercase{%
+  \gdef\math@#1#2#3#4#5#6\math@{\gtest@false\ifx 7#1\ifx 8#2%
+  \ifx 9#3\ifx 0#4\ifx !#5\xdef\meaning@{#6}\gtest@true
+  \fi\fi\fi\fi\fi}}}
+{\uccode`7=`c \uccode`8=`h \uccode`9=`\"
+ \uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\gtest@false
+  \ifx 7#1\ifx 8#2\ifx 9#5\gtest@true\xdef\meaning@{9#6}\fi\fi\fi}}}
+\newcount\classnum@
+\def\getmathch@#1.#2\getmathch@{\classnum@#1 \divide\classnum@4096
+ \ifcase\number\classnum@\or\or\gdef\thedots@{\dotsb@}\or
+ \gdef\thedots@{\dotsb@}\fi}
+{\uccode`4=`b \uccode`5=`i \uccode`6=`n
+ \uppercase{\gdef\mathbin@#1#2#3{\relaxnext@
+  \def\nextii@##1\mathbin@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+  \gtest@false\DN@##1\mathbin@{}%
+ \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
+{\uccode`4=`r \uccode`5=`e \uccode`6=`l
+ \uppercase{\gdef\mathrel@#1#2#3{\relaxnext@
+  \def\nextii@##1\mathrel@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+ \gtest@false\DN@##1\mathrel@{}%
+ \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
+{\uccode`5=`m \uccode`6=`a \uccode`7=`c
+ \uppercase{\gdef\macro@#1#2#3#4\macro@{\gtest@false
+  \ifx 5#1\ifx 6#2\ifx 7#3\gtest@true
+  \xdef\meaning@{\macro@@#4\macro@@}\fi\fi\fi}}}
+\def\macro@@#1->#2\macro@@{#2}
+\newcount\DOTSCASE@
+{\uccode`6=`\\ \uccode`7=`D \uccode`8=`O \uccode`9=`T \uccode`0=`S
+ \uppercase{\gdef\DOTS@#1#2#3#4#5{\gtest@false\DN@##1\DOTS@{}%
+  \ifx 6#1\ifx 7#2\ifx 8#3\ifx 9#4\ifx 0#5\let\next@\DOTS@@
+  \fi\fi\fi\fi\fi
+  \next@}}}
+{\uccode`3=`B \uccode`4=`I \uccode`5=`X
+ \uppercase{\gdef\DOTS@@#1{\relaxnext@
+  \def\nextii@##1\DOTS@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+  \DN@{\FN@\nextii@}%
+  \ifx 3#1\global\DOTSCASE@\z@\else
+  \ifx 4#1\global\DOTSCASE@\@ne\else
+  \ifx 5#1\global\DOTSCASE@\tw@\else\DN@##1\DOTS@{}%
+  \fi\fi\fi\next@}}}
+{\uccode`5=`\\ \uccode`6=`n \uccode`7=`o \uccode`8=`t
+ \uppercase{\gdef\not@#1#2#3#4{\relaxnext@
+  \def\nextii@##1\not@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+ \gtest@false\DN@##1\not@{}%
+ \ifx 5#1\ifx 6#2\ifx 7#3\ifx 8#4\DN@{\FN@\nextii@}\fi\fi\fi
+ \fi\next@}}}
+\def\keybin@{\gtest@true
+  \ifx\@let@token+\else\ifx\@let@token=\else\ifx\@let@token<%
+  \else\ifx\@let@token>\else\ifx\@let@token-\else\ifx\@let@token*%
+  \else\ifx\@let@token:\else\gtest@false\fi\fi\fi\fi\fi\fi\fi}
+\@ifundefined{@ldots}{\def\@ldots{\mathellipsis}}{}
+\DeclareRobustCommand{\dots}{\relax
+  \csname\ifmmode m\else t\fi dots@\endcsname}
+\def\tdots@{\leavevmode\unskip\relaxnext@
+ \DN@{$\m@th\@ldots\,
+   \ifx\@let@token,\,$\else\ifx\@let@token.\,$\else\ifx\@let@token;\,$%
+   \else\ifx\@let@token:\,$\else\ifx\@let@token?\,$%
+   \else\ifx\@let@token!\,$\else$ \fi\fi\fi\fi\fi\fi}%
+ \ \FN@\next@}
+\def\mdots@{\FN@\mdots@@}
+\def\mdots@@{\gdef\thedots@{\dotso@}%
+ \ifx\@let@token\boldsymbol\gdef\thedots@\boldsymbol{\boldsymboldots@}%
+ \else\ifx,\@let@token\gdef\thedots@{\dotsc}%
+ \else\ifx\not\@let@token\gdef\thedots@{\dotsb@}%
+ \else\keybin@
+ \ifgtest@\gdef\thedots@{\dotsb@}%
+ \else\xdef\meaning@{\meaning\@let@token..........}%
+  \xdef\meaning@@{\meaning@}%
+  \expandafter\math@\meaning@\math@
+  \ifgtest@
+   \expandafter\mathch@\meaning@\mathch@
+   \ifgtest@\expandafter\getmathch@\meaning@\getmathch@\fi
+  \else\expandafter\macro@\meaning@@\macro@
+  \ifgtest@
+   \expandafter\not@\meaning@\not@\ifgtest@\gdef\thedots@{\dotsb@}%
+  \else\expandafter\DOTS@\meaning@\DOTS@
+  \ifgtest@
+   \ifcase\number\DOTSCASE@\gdef\thedots@{\dotsb@}%
+    \or\gdef\thedots@{\dotsi}\else\fi
+  \else\expandafter\math@\meaning@\math@
+  \ifgtest@\expandafter\mathbin@\meaning@\mathbin@
+  \ifgtest@\gdef\thedots@{\dotsb@}%
+  \else\expandafter\mathrel@\meaning@\mathrel@
+  \ifgtest@\gdef\thedots@{\dotsb@}%
+  \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+ \thedots@}
+\def\boldsymboldots@#1{%
+ \bold@true\let\@let@token=#1\let\delayed@=#1\mdots@@
+ \boldsymbol#1\bold@false}
+\def\@cdots{\mathinner{\cdotp\cdotp\cdotp}}
+\def\dotsi{\!\@cdots}
+\let\dotsb@\@cdots
+\def\rightdelim@{\gtest@true
+ \ifx\@let@token)\else
+ \ifx\@let@token]\else
+ \ifx\@let@token\rbrack\else
+ \ifx\@let@token\}\else
+ \ifx\@let@token\rbrace\else
+ \ifx\@let@token\rangle\else
+ \ifx\@let@token\rceil\else
+ \ifx\@let@token\rfloor\else
+ \ifx\@let@token\rgroup\else
+ \ifx\@let@token\rmoustache\else
+ \ifx\@let@token\right\else
+ \ifx\@let@token\bigr\else
+ \ifx\@let@token\biggr\else
+ \ifx\@let@token\Bigr\else
+ \ifx\@let@token\Biggr\else\gtest@false
+ \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
+\def\extra@{%
+ \rightdelim@\ifgtest@
+ \else\ifx\@let@token$\gtest@true
+ \else\xdef\meaning@{\meaning\@let@token..........}%
+ \expandafter\macro@\meaning@\macro@\ifgtest@
+ \expandafter\DOTS@\meaning@\DOTS@
+ \ifgtest@
+ \ifnum\DOTSCASE@=\tw@\gtest@true\else\gtest@false
+ \fi\fi\fi\fi\fi}
+\newif\ifbold@
+\def\dotso@{\relaxnext@
+ \ifbold@
+  \let\@let@token\delayed@
+  \def\nextii@{\extra@\@ldots\ifgtest@\,\fi}%
+ \else
+  \def\nextii@{\DN@{\extra@\@ldots\ifgtest@\,\fi}\FN@\next@}%
+ \fi
+ \nextii@}
+\def\extrap@#1{%
+ \DN@{#1\,}%
+ \ifx\@let@token,\else
+ \ifx\@let@token;\else
+ \ifx\@let@token.\else\extra@
+ \ifgtest@\else
+ \let\next@#1\fi\fi\fi\fi\next@}
+\DeclareRobustCommand{\ldots}{\relax
+  \ifmmode \DN@{\extrap@\@ldots}%
+  \else \let\next@\tdots@\fi
+  \FN@\next@}
+\DeclareRobustCommand{\cdots}{\DN@{\extrap@\@cdots}\FN@\next@}
+\let\dotso\ldots
+\let\dotsb\cdots
+\let\dotsm\dotsb
+\DeclareRobustCommand{\dotsc}{%
+  \DN@{\ifx\@let@token;\@ldots\,%
+       \else \ifx\@let@token.\@ldots\,%
+       \else \extra@\@ldots \ifgtest@\,\fi
+       \fi\fi}%
+  \FN@\next@}
+\def\longrightarrow{\DOTSB\relbar\joinrel\rightarrow}
+\def\Longrightarrow{\DOTSB\Relbar\joinrel\Rightarrow}
+\def\longleftarrow{\DOTSB\leftarrow\joinrel\relbar}
+\def\Longleftarrow{\DOTSB\Leftarrow\joinrel\Relbar}
+\def\longleftrightarrow{\DOTSB\leftarrow\joinrel\rightarrow}
+\def\Longleftrightarrow{\DOTSB\Leftarrow\joinrel\Rightarrow}
+\def\mapsto{\DOTSB\mapstochar\rightarrow}
+\def\longmapsto{\DOTSB\mapstochar\longrightarrow}
+\def\hookrightarrow{\DOTSB\lhook\joinrel\rightarrow}
+\def\hookleftarrow{\DOTSB\leftarrow\joinrel\rhook}
+\def\doteq{\DOTSB\buildrel\textstyle.\over=}
+\def\iff{\DOTSB\;\Longleftrightarrow\;}
+\begingroup \catcode`\"=12
+\mathchardef\@tempa="7000
+\ifx\@tempa\Gamma
+  \DeclareMathSymbol{\Gamma}{\mathord}{operators}{"00}
+  \DeclareMathSymbol{\Delta}{\mathord}{operators}{"01}
+  \DeclareMathSymbol{\Theta}{\mathord}{operators}{"02}
+  \DeclareMathSymbol{\Lambda}{\mathord}{operators}{"03}
+  \DeclareMathSymbol{\Xi}{\mathord}{operators}{"04}
+  \DeclareMathSymbol{\Pi}{\mathord}{operators}{"05}
+  \DeclareMathSymbol{\Sigma}{\mathord}{operators}{"06}
+  \DeclareMathSymbol{\Upsilon}{\mathord}{operators}{"07}
+  \DeclareMathSymbol{\Phi}{\mathord}{operators}{"08}
+  \DeclareMathSymbol{\Psi}{\mathord}{operators}{"09}
+  \DeclareMathSymbol{\Omega}{\mathord}{operators}{"0A}
+  \DeclareMathSymbol{\varGamma}{\mathord}{letters}{"00}
+  \DeclareMathSymbol{\varDelta}{\mathord}{letters}{"01}
+  \DeclareMathSymbol{\varTheta}{\mathord}{letters}{"02}
+  \DeclareMathSymbol{\varLambda}{\mathord}{letters}{"03}
+  \DeclareMathSymbol{\varXi}{\mathord}{letters}{"04}
+  \DeclareMathSymbol{\varPi}{\mathord}{letters}{"05}
+  \DeclareMathSymbol{\varSigma}{\mathord}{letters}{"06}
+  \DeclareMathSymbol{\varUpsilon}{\mathord}{letters}{"07}
+  \DeclareMathSymbol{\varPhi}{\mathord}{letters}{"08}
+  \DeclareMathSymbol{\varPsi}{\mathord}{letters}{"09}
+  \DeclareMathSymbol{\varOmega}{\mathord}{letters}{"0A}
+\fi
+\endgroup
+\def\accentclass@{7}
+\def\noaccents@{\def\accentclass@{0}}
+\DeclareFontEncoding{OML}{}{\noaccents@}
+\DeclareFontEncoding{OMS}{}{\noaccents@}
+\edef\@tempa#1#2{\def#1{\mathaccent\string"\noexpand\accentclass@#2 }}
+\@tempa\hat{05E}\@tempa\check{014}\@tempa\tilde{07E}\@tempa\acute{013}
+\@tempa\grave{012}\@tempa\dot{05F}\@tempa\ddot{07F}\@tempa\breve{015}
+\@tempa\bar{016}
+\newcount\skewcharcount@
+\newcount\familycount@
+\def\theskewchar@{\familycount@\@ne
+ \global\skewcharcount@\the\skewchar\textfont\@ne
+ \ifnum\mathgroup>\m@ne\ifnum\mathgroup<16
+  \global\familycount@\the\mathgroup\relax
+  \global\skewcharcount@\the\skewchar\textfont\the\mathgroup\relax\fi\fi
+ \ifnum\skewcharcount@>\m@ne
+  \ifnum\skewcharcount@<128
+  \multiply\familycount@256
+  \global\advance\skewcharcount@\familycount@
+  \global\advance\skewcharcount@28672
+  \mathchar\skewcharcount@\else
+  \global\skewcharcount@\m@ne\fi\else
+ \global\skewcharcount@\m@ne\fi}
+\newcount\pointcount@
+\def\getpoints@#1.#2\getpoints@{\pointcount@#1 }
+\newdimen\accentdimen@
+\newcount\accentmu@
+\def\dimentomu@{\multiply\accentdimen@ 100
+ \expandafter\getpoints@\the\accentdimen@\getpoints@
+ \multiply\pointcount@18
+ \divide\pointcount@\@m
+ \global\accentmu@\pointcount@}
+\def\mathaccent@#1#2{\ifnum\mathgroup=\m@ne\xdef\thefam@{1}\else
+ \xdef\thefam@{\the\mathgroup}\fi
+ \accentdimen@\z@
+ \setboxz@h{\unbracefonts@$\m@th\mathgroup\thefam@\relax#2$}%
+ \ifdim\accentdimen@=\z@\DN@{\mathaccent#1{#2}}%
+  \setbox\@ne\hbox{\unbracefonts@
+    $\m@th\mathgroup\thefam@\relax#2\theskewchar@$}
+  \setbox\tw@\hbox{$\m@th\ifnum\skewcharcount@=\m@ne\else
+   \mathchar\skewcharcount@\fi$}%
+  \global\accentdimen@\wd\@ne\global\advance\accentdimen@-\wdz@
+  \global\advance\accentdimen@-\wd\tw@
+  \global\multiply\accentdimen@\tw@
+  \dimentomu@\global\advance\accentmu@\@ne
+ \else\DN@{{\mathaccent#1{#2\mkern\accentmu@ mu}%
+    \mkern-\accentmu@ mu}{}}\fi
+ \next@}
+\def\unbracefonts@{\let\math@bgroup\@empty\let\math@egroup\@empty}
+\begingroup \catcode`\"=12
+\def\@tempa#1#2{\gdef#1{\RIfM@\DN@{\mathaccent@{"\accentclass@#2 }}%
+  \else\DN@{\nonmatherr@{#1}}\fi\next@}}
+\@tempa\Hat{05E}\@tempa\Check{014}\@tempa\Tilde{07E}\@tempa\Acute{013}
+\@tempa\Grave{012}\@tempa\Dot{05F}\@tempa\Ddot{07F}\@tempa\Breve{015}
+\@tempa\Bar{016}
+\gdef\Vec{\RIfM@\DN@{\mathaccent@{"017E }}\else
+ \DN@{\nonmatherr@\Vec}\fi\next@}
+\endgroup
+\begingroup \catcode`\"=12
+\gdef\@@sqrt#1{\radical"270370 {#1}}
+\endgroup
+\@saveprimitive\overline\@@overline
+\def\overline#1{\@@overline{#1}}
+\def\boxed#1{\fbox{\m@th$\displaystyle#1$}}
+\def\setb@ck{\iffirstchoice@\global\advance\c@equation\m@ne\fi}
+\newcount\dspbrk@lvl
+\dspbrk@lvl=-1
+\interdisplaylinepenalty\@M
+\def\allowdisplaybreaks{%
+  \new@ifnextchar[\allowdspbrks@{\allowdspbrks@[4]}}
+\def\allowdspbrks@[#1]{%
+  \interdisplaylinepenalty\getdsp@pen{#1}}
+\def\getdsp@pen#1{%
+  \ifcase #1\relax \@M
+    \or 9999
+    \or 6999
+    \or 2999
+    \or \z@\fi}
+\def\displaybreak{\PackageError{amstex}{\Invalid@@\displaybreak}\@eha}
+\def\displaybreak@{%
+  \def\displaybreak{\new@ifnextchar[\dspbrk@{\dspbrk@[4]}}}
+\def\dspbrk@[#1]{\global\dspbrk@lvl #1\relax}
+\def\math@cr{{\ifnum0=`}\fi
+  \new@ifstar{\global\@eqpen\@M\math@cr@}%
+          {\global\@eqpen
+             \ifnum\dspbrk@lvl <\z@ \interdisplaylinepenalty
+              \else -\@getpen\dspbrk@lvl \fi
+           \math@cr@}}
+\def\math@cr@{\new@ifnextchar[\math@cr@@{\math@cr@@[\z@]}}
+\def\math@cr@@[#1]{\ifnum0=`{\fi}\math@cr@@@
+  \noalign{\vskip#1\relax}}
+\def\Let@{\let\\\math@cr}
+\def\restore@math@cr{\def\math@cr@@@{\cr}}
+\restore@math@cr
+\def\intertext{\PackageError{amstex}{\Invalid@@\intertext}\@eha}
+\def\intertext@{\def\intertext##1{\noalign{%
+  \penalty\postdisplaypenalty\vskip\belowdisplayskip
+  \vbox{\normalbaselines\noindent##1}%
+  \penalty\predisplaypenalty\vskip\abovedisplayskip}}}
+\newhelp\tag@help
+  {tag cannot be used at this point.\space
+   If you don't understand why^^Jyou should consult
+   the documentation.^^JBut don't worry: just continue, and I'll
+   forget what happened.}
+\def\gobble@tag{\new@ifstar\@gobble\@gobble}
+\def\invalid@tag#1{\PackageError{amstex}{#1}{\the\tag@help}\gobble@tag}
+\def\dft@tag{\invalid@tag{\string\tag\space not allowed here}}
+\def\default@tag{\let\tag\dft@tag}
+\default@tag
+\def\maketag@@{\new@ifstar\maketag@@@\tagform@}
+\def\maketag@@@#1{\hbox{\m@th\normalfont#1}}
+\def\tagform@#1{\maketag@@@{(\ignorespaces#1\unskip)}}
+\def\thetag{\leavevmode\tagform@}
+\def\make@df@tag{\new@ifstar\make@df@tag@@\make@df@tag@@@}
+\def\make@df@tag@@#1{%
+  \def\@currentlabel{#1}%
+  \gdef\df@tag{\maketag@@@{#1}}}
+\def\make@df@tag@@@#1{\@seteqlabel{#1}\gdef\df@tag{\tagform@{#1}}}
+\def\@seteqlabel#1{\toks@\@xp{\p@equation{#1}}%
+  \edef\@currentlabel{\the\toks@}}
+\def\notag{\nonumber}
+\iftagsleft@
+\def\@eqnnum{\hbox to1sp{}\rlap{\normalfont
+    \hskip -\displaywidth\tagform@\theequation}}
+\fi
+
+\let\@adjusttag\@empty
+
+\def\adjusttag#1{\skip@#1\relax\gdef\@adjusttag{#1}}
+
+\def\@movetag{%
+    \ifx\@adjusttag\@empty \else
+        \vskip\@adjusttag\relax
+        \global\let\@adjusttag\@empty
+    \fi
+}
+\def\multilimits@{\bgroup
+  \Let@
+  \restore@math@cr
+  \default@tag
+ \baselineskip\fontdimen10 \scriptfont\tw@
+ \advance\baselineskip\fontdimen12 \scriptfont\tw@
+ \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
+ \lineskiplimit\lineskip
+ \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}
+\def\Sb{_\multilimits@}
+\def\Sp{^\multilimits@}
+\def\endSb{\crcr\egroup\egroup\egroup}
+\let\endSp=\endSb
+\def\smallmatrix{\null\,\vcenter\bgroup
+ \Let@\restore@math@cr\default@tag
+ \baselineskip6\ex@ \lineskip1.5\ex@ \lineskiplimit\lineskip
+ \ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil&&\thickspace\hfil
+ $\m@th\scriptstyle{##}$\hfil\crcr}
+\def\endsmallmatrix{\crcr\egroup\egroup\,}
+\newcount\c@MaxMatrixCols
+\c@MaxMatrixCols=10
+
+\def\matrix{\hskip -\arraycolsep\array{*\c@MaxMatrixCols c}}
+\def\endmatrix{\endarray \hskip -\arraycolsep}
+\@ifundefined{prepnext@tok}{%
+  \def\@array[#1]#2{%
+    \setbox\@arstrutbox\hbox{\vrule\@height\arraystretch\ht\strutbox
+       \@depth\arraystretch\dp\strutbox \@width\z@}%
+    \@mkpream{#2}\edef\@preamble{\ialign \noexpand\@halignto
+    \bgroup \tabskip\z@ \@arstrut \@preamble \tabskip\z@ \cr}%
+    \let\@startpbox\@@startpbox \let\@endpbox\@@endpbox
+    \if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
+    \bgroup \let\par\@empty
+    \let\@sharp##\let\protect\relax
+    \lineskip\z@\baselineskip\z@\@preamble}%
+\begingroup \lccode`\~=`\@ \catcode`\~=\active % just to be sure!
+\lowercase{%
+\gdef\@xexpast#1*#2#3#4\@@{\begingroup
+  \def~{@}\edef\reserved@a{#1}\@tempcnta#2\relax
+  \ifnum\@tempcnta >\z@
+    \@whilenum\@tempcnta >\z@\do
+      {\edef\reserved@a{\reserved@a#3}\advance\@tempcnta \m@ne}%
+    \let\reserved@b\@xexpast
+  \else \let\reserved@b\@xexnoop\fi
+  \expandafter\reserved@b \reserved@a #4\@@
+  \expandafter\endgroup
+  \expandafter\def\expandafter\reserved@a\expandafter{\reserved@a}%
+  \let\@tempa\reserved@a
+}% End of \@xexpast
+}% End of \lowercase
+\endgroup
+}{}% End of \@ifundefined test.
+\def\pmatrix{\left(\matrix}
+\def\endpmatrix{\endmatrix\right)}
+\def\bmatrix{\left[\matrix}
+\def\endbmatrix{\endmatrix\right]}
+\def\vmatrix{\left|\matrix}
+\def\endvmatrix{\endmatrix\right|}
+\def\Vmatrix{\left\|\matrix}
+\def\endVmatrix{\endmatrix\right\|}
+\let\hdots\@ldots
+\def\hdotsfor#1{\if\noexpand#1[\expandafter\shdots@for\else
+                \hdots@for\@ne{#1}\fi}
+\newmuskip\dotsspace@
+\def\shdots@for#1]{\hdots@for{#1}}
+\def\hdots@for#1#2{\multicolumn{#2}c%
+  {\m@th\dotsspace@1.5mu\mkern-#1\dotsspace@
+   \xleaders\hbox{$\m@th\mkern#1\dotsspace@.\mkern#1\dotsspace@$}%
+           \hfill
+   \mkern-#1\dotsspace@}%
+   }
+\def\cases{\left\{\def\arraystretch{1.2}\hskip-\arraycolsep
+  \array{l@{\quad}l}}
+\def\endcases{\endarray\hskip-\arraycolsep\right.}
+\newif\ifinany@
+\newif\ifinalign@
+\newif\ifingather@
+\newcount\and@
+\newif\iftag@
+\newhelp\andhelp@
+{An extra & here is so disastrous that you should probably exit^^J
+and fix things up.}
+\newif\ifst@rred
+\newtoks\@envbody
+\def\addto@envbody#1{\@envbody\expandafter{\the\@envbody#1}}
+\def\collect@body#1{%
+  \@envbody{}%
+  \def\process@envbody{%
+    \expandafter#1\expandafter{\the\@envbody}}%
+  \expandafter\let\csname\@currenvir\endcsname\collect@@body
+  \csname\@currenvir\endcsname}
+\def\collect@@body#1\end#2{%
+  \def\@tempa{#2}%
+  \ifx\@tempa\@currenvir
+    \addto@envbody{#1}%
+    \expandafter\edef\csname\@currenvir\endcsname
+      {\noexpand\process@envbody\noexpand\end{\@tempa}}%
+  \else
+    \addto@envbody{#1\end{#2}}%
+  \fi
+  \csname\@currenvir\endcsname}
+\newenvironment{aligned}{%
+  \relax\ifmmode\else\nonmatherr@{\begin{aligned}}\fi
+  \null\,\new@ifnextchar[{\aligned@}{\aligned@[c]}%
+}{%
+  \crcr\egroup\egroup
+}
+\def\aligned@[#1]{%
+  \if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
+  \bgroup
+   \Let@\restore@math@cr
+   \default@tag
+   \ifinany@\else\openup\jot\fi
+   \ialign\bgroup\hfil\strut@$\m@th\displaystyle{##}$&%
+    $\m@th\displaystyle{{}##}$\hfil\crcr}
+\newtoks\hashtoks@
+\hashtoks@{#}
+\newcount\atcount@
+\def\doat@#1{\toks@{\hfil\strut@$\m@th
+  \displaystyle{\the\hashtoks@}$&$\m@th\displaystyle
+  {{}\the\hashtoks@}$\hfil}%
+ \atcount@#1\relax\advance\atcount@\m@ne
+ \loop\ifnum\atcount@>\z@\toks@\expandafter{\the\toks@&\hfil$\m@th
+  \displaystyle{\the\hashtoks@}$&$\m@th
+  \displaystyle{{}\the\hashtoks@}$\hfil}\advance
+ \atcount@\m@ne\repeat
+ \xdef\preamble@{\the\toks@}\xdef\preamble@@{\preamble@}}
+\newenvironment{alignedat}[2][c]{%
+  \null\,%
+  \if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
+  \bgroup\doat@{#2}\Let@ \restore@math@cr\default@tag
+  \ifinany@\else\openup\jot\fi\ialign\bgroup\span\preamble@@\crcr
+}{%
+  \endaligned
+}
+\newenvironment{gathered}[1][c]{%
+  \relax\ifmmode\else\nonmatherr@{\begin{gathered}}\fi
+  \null\,%
+  \if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
+  \bgroup\Let@\restore@math@cr
+  \ifinany@\else\openup\jot\fi\ialign
+  \bgroup\hfil\strut@$\m@th\displaystyle##$\hfil\crcr
+}{%
+  \endaligned
+}
+\def\tag@in@align{\relax
+  \iftag@\DN@{\invalid@tag{Multiple \string\tag}}\else
+   \global\tag@true
+   \if@eqnsw\global\@eqnswfalse\fi
+   \let\next@\make@df@tag\fi
+  \next@}
+\def\math@cr@@@align{%
+  \if@eqnsw\global\tag@true\fi
+  \iftag@
+   \ifcase\and@
+    \DN@{&&}\else
+    \DN@{&}\fi\else
+    \let\next@\relax\fi
+  \next@
+  \relax
+  \if@eqnsw\tagform@\theequation\stepcounter{equation}\else
+  \iftag@\df@tag\fi\fi
+  \global\@eqnswtrue\cr}
+\def\Tag@{\relax\iftag@\else\firstchoice@true
+  \PackageError{amstex}{%
+  Extra & on this line%
+  }{\the\andhelp@% "An extra & here is disastrous"
+}\fi}
+\newdimen\lwidth@
+\newdimen\rwidth@
+\newdimen\maxlwidth@
+\newdimen\maxrwidth@
+\newdimen\totwidth@
+\def\measure@#1{\lwidth@\z@\rwidth@\z@\maxlwidth@\z@\maxrwidth@\z@
+ \global\and@\z@
+ \setbox\@ne\vbox{%
+   \everycr{\noalign{\global\tag@false\global\and@\z@}}\Let@
+   \let\tag\gobble@tag
+   \let\notag\@empty \let\nonumber\@empty
+   \firstchoice@false
+    \halign{\setboxz@h{$\m@th\displaystyle{\@lign##}$}%
+     \global\lwidth@\wdz@
+     \ifdim\lwidth@>\maxlwidth@\global\maxlwidth@\lwidth@\fi
+     \global\advance\and@\@ne
+     &\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}%
+     \global\rwidth@\wdz@
+     \ifdim\rwidth@>\maxrwidth@\global\maxrwidth@\rwidth@\fi
+     \global\advance\and@\@ne
+     &\Tag@\@gobble{##}\crcr#1\crcr}}%
+ \totwidth@\maxlwidth@\advance\totwidth@\maxrwidth@}
+\def\displ@y{\global\dt@ptrue\openup\jot\m@th
+  \everycr{\noalign{\ifdt@p \global\dt@pfalse
+      \vskip-\lineskiplimit \vskip\normallineskiplimit
+      \else \penalty\@eqpen \fi}}}
+\def\displ@y@{\global\dt@ptrue\openup\jot
+  \everycr{\noalign{\global\tag@false
+                   \global\and@\z@
+                   \global\dspbrk@lvl\m@ne
+                   \ifdt@p\global\dt@pfalse
+      \vskip-\lineskiplimit\vskip\normallineskiplimit\else
+      \penalty\@eqpen\fi}}}
+\def\black@#1{\noalign{\ifdim#1>\displaywidth
+ \dimen@\prevdepth\nointerlineskip
+ \vskip-\ht\strutbox@\vskip-\dp\strutbox@
+ \vbox{\noindent\hbox to#1{\strut@\hfill}}%
+ \prevdepth\dimen@
+ \fi}}
+\def\align{\ifingather@
+  {\ifnum0=`}\fi
+  \DN@{\@nameuse{align (in \string\gather)}}%
+  \else \ifmmode\nomath@env
+    \DN@{\let\endalign\relax\@gobble}\else
+  $$\stepcounter{equation}%
+  \st@rredfalse\let\next@\align@\fi\fi
+ \collect@body\next@}
+\@namedef{align*}{\ifingather@
+  {\ifnum0=`}\fi
+  \DN@{\@nameuse{align* (in \string\gather)}}\else
+   \ifmmode\nomath@env
+    \DN@{\@namedef{endalign*}{}\@gobble}\else
+   $$\st@rredtrue
+  \let\next@\align@\fi\fi
+ \collect@body\next@}
+\newdimen\lineht@
+\def\align@{\inalign@true\inany@true
+ \displaybreak@\intertext@
+ \let\math@cr@@@\math@cr@@@align
+ \let\tag\tag@in@align
+ \global\and@\z@\global\dspbrk@lvl\m@ne
+ \iftagsleft@\DN@{\@nameuse{align }}\else
+  \DN@{\@nameuse{align \space}}\fi\next@}
+\@namedef{align }#1{%
+    \measure@{#1}%
+    \global\and@\z@
+    \global\dspbrk@lvl\m@ne
+    \if@fleqn %%*F
+        \advance\totwidth@\@mathmargin %%*F
+    \fi %%*F
+    \ifdim\totwidth@>\displaywidth
+        \let\displaywidth@\totwidth@
+    \else
+        \let\displaywidth@\displaywidth
+    \fi
+    \ifingather@
+        \everycr{%
+            \noalign{%
+                \global\and@\z@
+                \global\dspbrk@lvl\m@ne
+            }%
+        }%
+    \else
+        \displ@y@
+    \fi
+    \Let@
+    \ifst@rred\else\@seteqlabel\theequation\fi
+    \tabskip\@mathmargin %%*F
+    \dimen@\displaywidth %%*F
+    \if@fleqn %%*F
+        \ifingather@ %%*F
+            \tabskip\z@skip %%*F
+            \advance\dimen@-\@mathmargin %%*F
+        \fi %%*F
+    \fi %%*F
+    \halign to\dimen@\bgroup %%*F
+        \ifst@rred\nonumber\fi
+        \hfil
+        \strut@
+        \setboxz@h{\global\tag@false$\m@th\displaystyle{\@lign##}$}%
+        \global\lwidth@\wdz@
+        \global\lineht@\ht\z@
+        \boxz@
+        \global\advance\and@\@ne
+        \tabskip\z@skip
+       &\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}%
+        \global\rwidth@\wdz@
+        \ifdim\ht\z@>\lineht@
+            \global\lineht@\ht\z@
+        \fi
+        \boxz@
+        \hfil
+        \global\advance\and@\@ne
+        \tabskip\@centering
+       &\kern-\displaywidth@
+        \setboxz@h{\@lign\strut@{##}}%
+        \if@fleqn %%*F
+            \dimen@\@mathmargin %%*F
+            \advance\dimen@\maxlwidth@ %%*F
+            \advance\dimen@-\lwidth@ %%*F
+        \else %%*F
+            \dimen@\displaywidth
+            \advance\dimen@-\totwidth@
+            \divide\dimen@\tw@
+            \advance\dimen@\maxlwidth@
+            \advance\dimen@-\lwidth@
+        \fi %%*F
+        \ifdim\dimen@<\tw@\wdz@
+            \rlap{\vbox{\normalbaselines\boxz@\vbox to\lineht@{}}}%
+        \else
+            \rlap{\boxz@}%
+        \fi
+        \tabskip\displaywidth@
+        \crcr
+        #1%
+}
+\@namedef{align \space}#1{%
+    \measure@{#1}%
+    \if@fleqn %%*F
+        \advance\totwidth@\@mathmargin %%*F
+    \fi %%*F
+    \global\and@\z@
+    \global\dspbrk@lvl\m@ne
+    \ifingather@
+        \everycr{\noalign{\global\and@\z@\global\dspbrk@lvl\m@ne}}%
+    \else
+        \displ@y@
+    \fi
+    \Let@
+    \ifst@rred\else\@seteqlabel\theequation\fi
+    \tabskip\@mathmargin %%*F
+    \dimen@\displaywidth %%*F
+    \if@fleqn %%*
+        \ifingather@ %%*F
+            \tabskip\z@skip %%*F
+            \advance\dimen@-\@mathmargin %%*F
+        \fi %%*F
+    \fi %%*F
+     \halign to\dimen@\bgroup %%*F
+        \ifst@rred\nonumber\fi
+        \hfil
+        \strut@
+        \setboxz@h{\global\tag@false$\m@th\displaystyle{\@lign##}$}%
+        \global\lwidth@\wdz@\boxz@
+        \global\advance\and@\@ne
+        \tabskip\z@skip
+       &\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}%
+        \global\rwidth@\wdz@
+        \boxz@
+        \hfill
+        \global\advance\and@\@ne
+        \tabskip\@centering
+       &\setboxz@h{\@lign\strut@{##}}%
+        \dimen@\displaywidth
+        \advance\dimen@-\totwidth@
+        \divide\dimen@\tw@
+        \advance\dimen@\maxrwidth@
+        \advance\dimen@-\rwidth@
+        \ifdim\dimen@<\tw@\wdz@
+            \llap{\vtop{\@movetag\normalbaselines\null\boxz@}}%%*T
+        \else
+            \llap{\boxz@}%
+        \fi
+        \tabskip\z@skip
+        \crcr#1%
+}
+\def\endalign{\math@cr@@@align
+  \black@\totwidth@
+  \egroup$$\global\@ignoretrue\setb@ck}
+
+\@namedef{endalign*}{\math@cr@@@align\black@\totwidth@
+  \egroup$$\global\@ignoretrue}
+\@namedef{align (in \string\gather)}{%
+  \ifst@rred\stepcounter{equation}\fi
+  \def\endalign{\@nameuse{endalign (in \string\gather)}\ifnum0=`{\fi}}%
+  \vcenter\bgroup\st@rredfalse
+    \collect@body\align@}
+\@namedef{endalign (in \string\gather)}{%
+  \math@cr@@@align\black@\totwidth@
+  \egroup\egroup
+  \ifst@rred\setb@ck\fi% this ifst@rred is set by the enclosing gather
+  \global\tag@false\global\@eqnswfalse}
+\@namedef{align* (in \string\gather)}{%
+  \@namedef{endalign*}{\@nameuse{endalign* (in \string\gather)}%
+                       \ifnum0=`{\fi}}%
+  \vcenter\bgroup\st@rredtrue
+    \collect@body\align@}
+\@namedef{endalign* (in \string\gather)}{%
+  \math@cr@@@align\black@\totwidth@\egroup\egroup
+  \global\tag@false\global\@eqnswfalse}
+\newif\ifxat@
+\def\alignat#1{%
+ \ifmmode\nomath@env\DN@{\let\endalignat\relax\@gobble}\else$$%
+ \st@rredfalse\stepcounter{equation}\@seteqlabel\theequation
+ \DN@{\@nameuse{alignat }#1}\fi
+ \collect@body\next@}
+\@namedef{alignat*}#1{%
+ \ifmmode\nomath@env\DN@{\@namedef{endalignat*}{}\@gobble}\else$$%
+ \st@rredtrue
+ \DN@{\@nameuse{alignat }#1}\fi
+ \collect@body\next@}
+\newif\ifmeasuring@
+\newbox\savealignat@
+\def\math@cr@@@alignat{%
+  \if@eqnsw\global\tag@true\fi
+  \iftag@
+   \multiply\count@\tw@
+   \xdef\next@{}\loop\ifnum\count@>\and@\xdef\next@{&\next@}%
+    \advance\count@\m@ne\repeat\else
+   \let\next@\relax\fi
+  \next@
+  \relax
+  \if@eqnsw\tagform@\theequation\stepcounter{equation}\else
+  \iftag@\df@tag\fi\fi
+  \global\@eqnswtrue\cr}%
+\@namedef{alignat }#1#2{%
+    \inany@true
+    \xat@false
+    \let\tag\tag@in@align
+    \def\math@cr@@@{\count@#1\relax \math@cr@@@alignat}%
+    \displaybreak@
+    \intertext@
+    \displ@y@
+    \ifst@rred
+        \global\@eqnswfalse
+    \else
+        \global\@eqnswtrue
+    \fi
+    \measuring@true
+    \setbox\savealignat@\hbox{%
+        $\m@th\displaystyle\Let@
+        \attag@{#1}%
+        \vbox{\firstchoice@false\halign{\span\preamble@@\crcr#2\crcr}}$%
+    }%
+    \measuring@false
+    \Let@
+    \attag@{#1}%
+    \tabskip\@mathmargin %%*F
+    \halign to\displaywidth\bgroup
+        \span\preamble@@\crcr
+        #2%
+}
+\def\endalignat{\math@cr@@@\black@{\wd\savealignat@}%
+     \egroup\setb@ck$$\global\@ignoretrue}
+\@namedef{endalignat*}{\math@cr@@@\black@{\wd\savealignat@}\egroup
+  $$\global\@ignoretrue}
+\def\xalignat#1{%
+ \ifmmode\nomath@env\DN@{\let\endxalignat\relax\@gobble}\else$$%
+ \st@rredfalse\stepcounter{equation}\@seteqlabel\theequation
+ \DN@{\@nameuse{xalignat }#1}\fi
+ \collect@body\next@}
+\@namedef{xalignat*}#1{%
+ \ifmmode\nomath@env\DN@{\@namedef{endxalignat*}{}\@gobble}\else$$%
+ \st@rredtrue
+ \DN@{\@nameuse{xalignat }#1}\fi
+ \collect@body\next@}
+\let\endxalignat=\endalignat
+
+\expandafter\let\csname endxalignat*\expandafter
+  \endcsname\csname endalignat*\endcsname
+\@namedef{xalignat }#1#2{%
+    \inany@true
+    \xat@true
+    \let\tag\tag@in@align
+    \def\math@cr@@@{\count@#1\relax\math@cr@@@alignat}%
+    \displaybreak@
+    \intertext@
+    \displ@y@
+    \measuring@true
+    \setbox\savealignat@\hbox{%
+        $\m@th\displaystyle\Let@
+        \attag@{#1}%
+        \vbox{%
+            \firstchoice@false
+            \halign{\span\preamble@@\crcr#2\crcr}%
+        }%
+        $%
+    }%
+    \measuring@false
+    \Let@
+    \attag@{#1}%
+    \tabskip\@mathmargin %%*F
+    \halign to\displaywidth\bgroup
+        \span\preamble@@\crcr#2%
+}
+\def\attag@#1{%
+ \let\TAG@\Tag@\let\Tag@0%
+ \ifmeasuring@\def\llap@##1{\setboxz@h{##1}\hbox to\tw@\wdz@{}}%
+  \def\rlap@##1{\setboxz@h{##1}\hbox to\tw@\wdz@{}}\else
+  \let\llap@\llap\let\rlap@\rlap\fi
+ \toks@{\hfil\strut@$\m@th\displaystyle{\@lign\the\hashtoks@}$%
+  \tabskip\z@skip
+  \global\advance\and@\@ne
+  &$\m@th\displaystyle{{}\@lign\the\hashtoks@}$\hfil
+  \ifxat@\tabskip\@centering\fi\global\advance\and@\@ne}%
+ \iftagsleft@
+  \@temptokena{\tabskip\@centering&\Tag@\kern-\displaywidth
+   \rlap@{\@lign
+     \the\hashtoks@
+    }%
+   \global\advance\and@\@ne\tabskip\displaywidth}\else
+  \@temptokena{\tabskip\@centering&\Tag@\llap@{\@lign
+   \the\hashtoks@
+  }\global\advance\and@\@ne\tabskip\z@skip}\fi
+ \atcount@#1\relax\advance\atcount@\m@ne
+ \loop\ifnum\atcount@>\z@
+ \toks@\expandafter{\the\toks@&\hfil$\m@th\displaystyle{\@lign
+  \the\hashtoks@}$\global\advance\and@\@ne
+  \tabskip\z@skip
+  &$\m@th\displaystyle{{}\@lign\the\hashtoks@}$\hfil\ifxat@
+  \tabskip\@centering\fi\global\advance\and@\@ne}\advance\atcount@\m@ne
+ \repeat
+ \xdef\preamble@{\the\toks@\the\@temptokena}%
+ \xdef\preamble@@{\ifst@rred\noexpand\nonumber\fi\preamble@}%
+ \let\Tag@\TAG@}
+\def\xxalignat#1{%
+ \ifmmode\nomath@env\DN@{\let\endxxalignat\relax\@gobble}\else$$%
+ \DN@{\@nameuse{xxalignat }#1}\fi
+ \collect@body\next@}
+\@namedef{xxalignat }#1#2{\inany@true
+ \displaybreak@\intertext@
+ \displ@y\setbox\savealignat@\hbox{$\m@th\displaystyle\Let@
+ \xxattag@{#1}%
+ \vbox{\firstchoice@false\halign{\span\preamble@@\crcr#2\crcr}}$}%
+ \Let@\xxattag@{#1}\tabskip\z@skip\halign to\displaywidth\bgroup
+  \span\preamble@@\crcr#2}
+\expandafter\let\expandafter\endxxalignat\csname endxalignat*\endcsname
+\def\xxattag@#1{\toks@{\tabskip\z@skip\hfil\strut@
+ $\m@th\displaystyle{\the\hashtoks@}$&%
+ $\m@th\displaystyle{{}\the\hashtoks@}$\hfil\tabskip\@centering&}%
+ \atcount@#1\relax\advance\atcount@\m@ne\loop\ifnum\atcount@>\z@
+ \toks@
+  \expandafter
+   {\the\toks@&\hfil$\m@th\displaystyle{\the\hashtoks@}$%
+  \tabskip\z@skip&$\m@th\displaystyle{{}\the\hashtoks@}$\hfil
+  \tabskip\@centering}\advance\atcount@\m@ne\repeat
+ \xdef\preamble@{\the\toks@\tabskip\z@skip}\xdef\preamble@@{\preamble@}}
+\newdimen\gwidth@
+\newdimen\gmaxwidth@
+\def\gmeasure@#1{\gwidth@\z@\gmaxwidth@\z@\setbox\@ne\vbox{\Let@
+ \firstchoice@false\let\tag\gobble@tag
+ \halign{\setboxz@h{$\m@th\displaystyle{##}$}\global\gwidth@\wdz@
+ \ifdim\gwidth@>\gmaxwidth@\global\gmaxwidth@\gwidth@\fi
+ &\@gobble{##}\crcr#1\crcr}}}
+\def\math@cr@@@gather{%
+  \if@eqnsw\global\tag@true\fi
+  \iftag@
+    \DN@{&\relax \if@eqnsw
+                    \tagform@\theequation \stepcounter{equation}%
+                 \else\iftag@\df@tag\fi\fi}%
+  \else
+    \expandafter\@gobble\fi
+  \next@
+  \global\@eqnswtrue\cr}
+\def\gather{%
+ \ifmmode\nomath@env\DN@{\let\endgather\relax\@gobble}\else $$%
+  \ingather@true\inany@true
+  \stepcounter{equation}%
+  \let\tag\tag@in@align\st@rredfalse
+  \displaybreak@\intertext@
+  \displ@y
+  \Let@\@seteqlabel\theequation
+  \let\math@cr@@@\math@cr@@@gather
+  \iftagsleft@\DN@{\@nameuse{gather }}\else
+   \DN@{\@nameuse{gather \space}}\fi\fi
+ \collect@body\next@}
+\@namedef{gather*}{%
+ \ifmmode\nomath@env\DN@{\@namedef{endgather*}{}\@gobble}\else $$%
+  \ingather@true\inany@true
+  \let\tag\tag@in@align\st@rredtrue
+  \displaybreak@\intertext@
+  \displ@y\Let@
+  \let\math@cr@@@\math@cr@@@gather
+  \iftagsleft@\DN@{\@nameuse{gather }}\else
+   \DN@{\@nameuse{gather \space}}\fi\fi
+ \collect@body\next@}
+\newdimen\glineht@
+\@namedef{gather \space}#1{%
+    \gmeasure@{#1}%
+    \if@fleqn %%*F
+        \advance\gmaxwidth@\@mathmargin %%*F
+    \fi %%*F
+    \global\dt@ptrue
+    \tabskip\@mathmargin %%*F
+    \halign to\displaywidth\bgroup
+        \ifst@rred\nonumber\else\global\@eqnswtrue\fi
+        \if@fleqn\else\hfil\fi %%*F
+        \strut@
+        \setboxz@h{\global\tag@false$\m@th\displaystyle{##}$}%
+        \global\gwidth@\wdz@\boxz@
+        \hfil
+        \tabskip\@centering %%*F
+       &\setboxz@h{\strut@{##}}%
+        \dimen@\displaywidth
+        \advance\dimen@-\gwidth@
+        \if@fleqn %%*F
+            \advance\dimen@-\@mathmargin %%*F
+        \else %%*F
+            \divide\dimen@\tw@ %%*B
+        \fi %%*F
+        \ifdim\dimen@<\tw@\wdz@ %%*
+            \llap{\vtop{\@movetag\normalbaselines\null\boxz@}}%%*T %%*
+        \else
+            \llap{\boxz@}% %%*
+        \fi
+        \tabskip\z@skip
+        \crcr
+        #1%
+}
+\def\endgather{\math@cr
+  \black@\gmaxwidth@\egroup $$\setb@ck\global\@ignoretrue}
+
+\@namedef{endgather*}{\math@cr
+  \black@\gmaxwidth@\egroup $$\global\@ignoretrue}
+\@namedef{gather }#1{%
+    \gmeasure@{#1}%
+    \global\dt@ptrue
+    \if@fleqn %%*F
+        \advance\gmaxwidth@\@mathmargin %%*F
+    \fi %%*F
+    \ifdim\gmaxwidth@>\displaywidth
+        \let\gdisplaywidth@\gmaxwidth@
+    \else
+        \let\gdisplaywidth@\displaywidth
+    \fi
+    \tabskip\@mathmargin %%*F
+    \halign to\displaywidth\bgroup
+        \ifst@rred\nonumber\else\global\@eqnswtrue\fi
+        \if@fleqn\else\hfil\fi %%*F
+        \strut@
+        \setboxz@h{\global\tag@false$\m@th\displaystyle{##}$}%
+        \global\gwidth@\wdz@
+        \global\glineht@\ht\z@
+        \boxz@
+        \hfil
+        \tabskip\@centering %%*F
+       &\kern-\gdisplaywidth@
+        \setboxz@h{\strut@{##}}%
+        \if@fleqn %%*F
+            \dimen@\@mathmargin %%*F
+        \else %%*F
+            \dimen@\displaywidth
+            \advance\dimen@-\gwidth@
+            \divide\dimen@\tw@ %%*B
+        \fi %%*F
+        \ifdim\dimen@<\tw@\wdz@ %%*
+            \rlap{\vbox{\normalbaselines\boxz@\vbox to\glineht@{}}}% %%*
+        \else
+            \rlap{\boxz@}% %%*
+        \fi
+        \tabskip\gdisplaywidth@
+        \crcr
+        #1%
+}
+\def\split{%
+ \ifinany@\let\next@\insplit@\else
+  \DN@{\PackageError{amstex}{\string\begin{split} is not allowed
+    here.  Try the `aligned' environment.}\@ehb}\fi
+ \next@}
+\def\insplit@{\global\setbox\z@\vbox\bgroup
+  \Let@ \restore@math@cr \default@tag % disallow use of \tag here
+  \ialign\bgroup \hfil\strut@
+   $\m@th\displaystyle{##}$&$\m@th\displaystyle{{}##}$\hfill\crcr}
+\def\endsplit{\crcr\egroup\egroup
+ \iftagsleft@\expandafter\lendsplit@\else
+  \expandafter\rendsplit@\fi}
+\def\rendsplit@{\global\setbox9 \vbox
+ {\unvcopy\z@\global\setbox8 \lastbox\unskip}%
+ \setbox\@ne\hbox{\unhcopy8 \unskip\global\setbox\tw@\lastbox
+ \unskip\global\setbox\thr@@\lastbox}%
+ \global\setbox7 \hbox{\unhbox\tw@\unskip}%
+ \ifinalign@\ifctagsplit@
+  \gdef\split@{\hbox to\wd\thr@@{}&
+   \vcenter{\vbox{\moveleft\wd\thr@@\boxz@}}}%
+ \else\gdef\split@{&\vbox{\moveleft\wd\thr@@\box9}\crcr
+  \box\thr@@&\box7}\fi
+ \else
+  \ifctagsplit@\gdef\split@{\vcenter{\boxz@}}\else
+  \gdef\split@{\box9\crcr\hbox{\box\thr@@\box7}}\fi
+ \fi
+ \aftergroup\split@}
+\def\lendsplit@{\global\setbox9\vtop{\unvcopy\z@}%
+ \setbox\@ne\vbox{\unvcopy\z@\global\setbox8\lastbox}%
+ \setbox\@ne\hbox{\unhcopy8\unskip\setbox\tw@\lastbox
+  \unskip\global\setbox\thr@@\lastbox}%
+ \ifinalign@\ifctagsplit@
+  \gdef\split@{\hbox to\wd\thr@@{}&
+  \vcenter{\vbox{\moveleft\wd\thr@@\box9}}}%
+  \else
+  \gdef\split@{\hbox to\wd\thr@@{}&\vbox{\moveleft\wd\thr@@\box9}}\fi
+ \else
+  \ifctagsplit@\gdef\split@{\vcenter{\box9}}\else
+  \gdef\split@{\box9}\fi
+ \fi\aftergroup\split@}
+\newdimen\multlinegap
+\multlinegap10pt
+\newdimen\multlinetaggap@
+\multlinetaggap@10pt
+\def\multline{%
+ \ifmmode\nomath@env\DN@{\let\endmultline\relax\@gobble}%
+ \else $$\DN@{\collect@body\multline@}\fi
+ \st@rredfalse\@eqnswtrue \@seteqlabel\theequation
+ \stepcounter{equation}\next@}
+\@namedef{multline*}{%
+ \ifmmode\nomath@env\DN@{\@namedef{endmultline*}{}\@gobble}%
+ \else $$\DN@{\collect@body\multline@}\fi
+ \st@rredtrue\@eqnswfalse \let\@currentlabel\@empty
+ \next@}
+\def\multline@{\inany@true\displaybreak@
+ \global\tag@false
+ \def\tag{\ifst@rred\else
+   \global\advance\c@equation\m@ne\fi\tag@in@align}%
+ \iftagsleft@
+  \expandafter\lmultline@@\else
+  \expandafter\rmultline@@\fi}
+\newdimen\mwidth@
+\def\rmmeasure@#1{%
+ \def\shoveleft##1{##1}\def\shoveright##1{##1}%
+ \setbox\@ne\vbox{\firstchoice@false\Let@\halign{\setboxz@h
+  {$\m@th\@lign\displaystyle{}##$}\global\mwidth@\wdz@
+  \crcr#1\crcr}}}
+\newdimen\mlineht@
+\newif\ifzerocr@
+\newif\ifonecr@
+\def\lmmeasure@#1{\global\zerocr@true\global\onecr@false
+ \everycr{\noalign{\ifonecr@\global\onecr@false\fi
+  \ifzerocr@\global\zerocr@false\global\onecr@true\fi}}%
+  \def\shoveleft##1{##1}\def\shoveright##1{##1}%
+ \setbox\@ne\vbox{\Let@\firstchoice@false\halign{\setboxz@h
+  {$\m@th\@lign\displaystyle{}##$}\ifonecr@\global\mwidth@\wdz@
+  \global\mlineht@\ht\z@\fi\crcr#1\crcr}}}
+\newbox\mtagbox@
+\newdimen\ltwidth@
+\newdimen\rtwidth@
+\def\lmultline@@#1{%
+    \lmmeasure@{#1}%
+    \setbox\mtagbox@\hbox{%
+        \iftag@
+            \df@tag
+        \else
+            \if@eqnsw
+                \global\tag@true
+                \tagform@\theequation
+            \fi
+        \fi
+    }%
+    \dimen@\mwidth@
+    \advance\dimen@\wd\mtagbox@
+    \advance\dimen@\multlinetaggap@
+    \if@fleqn %%*F
+        \advance\dimen@\@mathmargin %%*F
+    \fi %%*F
+    \ifdim\dimen@>\displaywidth
+        \ltwidth@\z@
+    \else
+        \ltwidth@\wd\mtagbox@
+    \fi
+    \lmultline@@@#1
+}
+\def\lmultline@@@{%
+    \displ@y
+    \def\shoveright##1{##1\hfilneg\hskip\multlinegap}%
+    \if@fleqn %%*F
+        \def\shoveleft##1{##1}% %%*F
+    \else %%*F
+        \def\shoveleft##1{%
+                \setboxz@h{$\m@th\displaystyle{}##1$}%
+                \setbox\@ne\hbox{$\m@th\displaystyle##1$}%
+                \hfilneg
+                \iftag@
+                    \ifdim\ltwidth@>\z@
+                        \hskip\ltwidth@
+                        \hskip\multlinetaggap@
+                    \fi
+                \else
+                    \hskip\multlinegap
+                \fi
+                \hskip.5\wd\@ne
+                \hskip-.5\wdz@##1%
+        }%
+    \fi %%*F
+    \if@fleqn %%*F
+        \tabskip\@mathmargin %%*F
+    \else %%*F
+        \tabskip\z@skip %%*
+    \fi %%*F
+    \totwidth@\displaywidth %%*F
+    \if@fleqn %%*F
+        \advance\totwidth@-\@mathmargin %%*F
+    \fi %%*F
+    \halign\bgroup
+        \Let@
+        \let\tag\gobble@tag
+        \hbox to\totwidth@{% %%*F
+            \strut@$\m@th\displaystyle
+            \if@fleqn %%*F
+                \hskip 0pt plus 1000pt %%*F
+            \else %%*F
+                \hfil
+            \fi %%*F
+            {}##\hfil$%
+        }%
+        \crcr
+        \if@fleqn %%*F
+            \hskip-\@mathmargin %%*F
+        \else %%*F
+            \hfilneg
+        \fi %%*F
+        \iftag@
+            \ifdim\ltwidth@>\z@
+                \box\mtagbox@
+                \hskip\multlinetaggap@
+            \else
+                \rlap{%
+                    \vbox{%
+                        \normalbaselines
+                        \hbox{\strut@\box\mtagbox@}%
+                        \vbox to\mlineht@{}%
+                    }%
+                }%
+                \hskip\multlinegap %%*
+            \fi
+        \else
+            \hskip\multlinegap
+        \fi
+}
+\def\rmultline@@#1{%
+    \rmmeasure@{#1}%
+    \setbox\mtagbox@\hbox{%
+        \iftag@
+            \df@tag
+        \else
+            \if@eqnsw
+                \global\tag@true
+                \tagform@\theequation
+            \fi
+        \fi
+    }%
+    \dimen@\mwidth@
+    \advance\dimen@\wd\mtagbox@
+    \advance\dimen@\multlinetaggap@
+    \if@fleqn %%*F
+        \advance\dimen@\@mathmargin %%*F
+    \fi %%*F
+    \ifdim\dimen@>\displaywidth
+        \rtwidth@\z@
+    \else
+        \rtwidth@\wd\mtagbox@
+    \fi
+    \rmultline@@@#1
+}
+\def\rmultline@@@{%
+    \displ@y
+    \def\shoveright##1{%
+        ##1%
+        \hfilneg
+        \iftag@
+            \ifdim\rtwidth@>\z@
+                \hskip\rtwidth@
+                \hskip\multlinetaggap@
+            \fi
+        \else
+            \hskip\multlinegap
+        \fi
+    }%
+    \if@fleqn %%*F
+        \def\shoveleft##1{##1}% %%*F
+    \else %%*F
+        \def\shoveleft##1{%
+            \setboxz@h{$\m@th\displaystyle{}##1$}%
+            \setbox\@ne\hbox{$\m@th\displaystyle##1$}%
+            \hfilneg
+            \hskip\multlinegap
+            \hskip.5\wd\@ne
+            \hskip-.5\wdz@
+            ##1%
+        }%
+    \fi %%*F
+    \if@fleqn %%*F
+        \tabskip\@mathmargin %%*F
+    \else %%*F
+        \tabskip\z@skip %%*
+    \fi %%*F
+    \totwidth@\displaywidth %%*F
+    \if@fleqn %%*F
+        \advance\totwidth@-\@mathmargin %%*F
+    \fi %%*F
+    \halign\bgroup
+        \Let@
+        \let\tag\gobble@tag
+        \hbox to\totwidth@{%
+            \strut@$\m@th\displaystyle
+            \if@fleqn %%*F
+                \hskip 0pt plus 1000pt %%*F
+            \else %%*F
+                \hfil
+            \fi %%*F
+            {}##\hfil$%
+        }%
+        \crcr
+        \if@fleqn %%*F
+            \hskip-\@mathmargin %%*F
+        \else %%*F
+            \hfilneg
+        \fi %%*F
+        \hskip\multlinegap
+}
+\def\endmultline{\iftagsleft@\expandafter\lendmultline@\else
+  \expandafter\rendmultline@\fi\global\@ignoretrue}
+\expandafter\let\csname endmultline*\endcsname=\endmultline
+\def\lendmultline@{\hfilneg\hskip\multlinegap\crcr\egroup$$}
+\def\rendmultline@{%
+    \iftag@
+        \ifdim\rtwidth@>\z@
+            \hskip\multlinetaggap@
+            \box\mtagbox@
+        \else
+            \hskip\multlinegap %%*
+            \llap{%
+                \vtop{%
+                    \@movetag %%*T
+                    \normalbaselines\null\hbox{\strut@\box\mtagbox@}%
+                }%
+            }%
+        \fi
+    \else
+        \hskip\multlinegap
+    \fi
+    \hfilneg
+    \crcr
+    \egroup$$%
+}
+\expandafter\def
+ \expandafter\@arrayparboxrestore
+  \expandafter{\@arrayparboxrestore
+               \inany@false\ingather@false\inalign@false
+   \default@tag}
+\def\equation{\gather\def\\{\PackageError{amstex}{\Invalid@@\\}\@eha}}
+\def\endequation{\endgather}
+\newenvironment{equation*}{%
+  \equation
+}{%
+  \nonumber\endequation
+}
+\newcounter{parentequation}% Counter for ``parent equation''.
+\newenvironment{subequations}{%
+  \refstepcounter{equation}%
+  \begingroup % conservative approach
+  \let\protect\@nx
+  \edef\@tempa{\def\@nx\theparentequation{\theequation}}%
+  \@xp\endgroup\@tempa
+  \setcounter{parentequation}{\value{equation}}%
+  \setcounter{equation}{0}%
+  \def\theequation{\theparentequation\alph{equation}}%
+  \ignorespaces
+}{%
+  \setcounter{equation}{\value{parentequation}}%
+  \global\@ignoretrue
+}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\def\atdef@#1{\expandafter\def\csname\space @\string#1\endcsname}
+\begingroup \catcode`\@=\active
+\gdef @{\Iat}
+\endgroup
+\DeclareRobustCommand{\Iat}{\FN@\at@}
+\def\at@{\let\next@\at@@
+ \ifcat\noexpand\@let@token a\else
+ \ifcat\noexpand\@let@token0\else
+ \ifcat\noexpand\@let@token\relax\else
+ \let\@let@token@\at@@@\fi\fi\fi\next@}
+\def\at@@#1{\expandafter
+  \@ifundefined{ @\string#1}%
+    {\DN@{\at@@@#1}}%
+    {\DN@{\csname\space @\string#1\endcsname}}%
+  \next@}%
+\def\at@@@{\PackageError{amstex}{\Invalid@@ @}%
+  {\the\athelp@}% "I'll assume you wanted @@"
+  \char64\relax}
+\newhelp\athelp@
+{Only certain combinations beginning with @ make sense to me.^^J%
+I'll assume you wanted @@ for a printed @.}
+\atdef@ @{{\atchar}}% to define @@
+\chardef\atchar=64
+\def\makeatother{\catcode64=\active}
+\def\@tempa#1#2#3#4#5#6{%
+  {#1}{#2}\ifnum0#6=0{13}\@xp\@gobblefour
+  \else{#3}\fi\@tempa{#4}{#5}{#6}}%
+\xdef\@currnamestack{\@xp\@tempa\@currnamestack{}{}{0}}
+{\def\@makeother{\noexpand\@makeother\noexpand}
+\xdef\@sanitize{\@sanitize\@makeother\@}}
+{\def\do{\noexpand\do\noexpand}
+\xdef\dospecials{\dospecials\do\@}}
+\atdef@,{\ifmmode\mskip.1\thinmuskip\else \nonmatherr@{@,}\fi}
+\atdef@!{\RIfM@\mskip-.1\thinmuskip\else \nonmatherr@{@!}\fi}
+
+\@ifpackageloaded{amscd}{%
+}{%
+\atdef@>#1>#2>{%
+  \global\bigaw@\minaw@
+  \setboxz@h{$\m@th\scriptstyle\;{#1}\;\;$}%
+  \ifdim\wdz@>\bigaw@ \global\bigaw@\wdz@ \fi
+  \@ifnotempty{#2}{\setbox\@ne\hbox{$\m@th\scriptstyle\;{#2}\;\;$}%
+    \ifdim\wd\@ne>\bigaw@ \global\bigaw@\wd\@ne \fi}%
+   \mathrel{\mathop{\hbox to\bigaw@{\rightarrowfill@\displaystyle}}%
+     \limits^{#1}\@ifnotempty{#2}{_{#2}}}%
+}% end of @>>>
+\atdef@<#1<#2<{%
+  \global\bigaw@\minaw@
+  \setboxz@h{$\m@th\scriptstyle\;\;{#1}\;$}%
+  \ifdim\wdz@>\bigaw@ \global\bigaw@\wdz@ \fi
+  \@ifnotempty{#2}{\setbox\@ne\hbox{$\m@th\scriptstyle\;\;{#2}\;$}%
+    \ifdim\wd\@ne>\bigaw@ \global\bigaw@\wd\@ne \fi}%
+  \mathrel{\mathop{\hbox to\bigaw@{\leftarrowfill@\displaystyle}}%
+      \limits^{#1}\@ifnotempty{#2}{_{#2}}}%
+}% end of @<<<
+}% end of @ifpackageloaded
+\@ifundefined{minaw@}{\newdimen\minaw@}{}
+\minaw@11.111pt
+\@ifundefined{bigaw@}{\newdimen\bigaw@}{}
+\atdef@-{\let\next-%%
+  \DN@{\ifx\next-%%
+    \def\next-{\setbox\z@\hbox{-}\unhbox\z@\FN@\next@}%%
+    \expandafter\next \else\nobreak\fi}%%
+  \next@-}%%
+\def\leftroot{\PackageError{amstex}{\Invalid@@\leftroot}\@eha}
+\def\uproot{\PackageError{amstex}{\Invalid@@\uproot}\@eha}
+\newcount\uproot@
+\newcount\leftroot@
+\def\root{\relaxnext@
+  \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else
+   \ifx\@let@token\leftroot\let\next@\nextiii@\else
+   \let\next@\plainroot@\fi\fi\next@}%
+  \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}%
+  \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else
+   \DN@.{\FN@\nextv@}\fi\next@.}%
+  \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else
+   \let\next@\plainroot@\fi\next@}%
+  \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}%
+   \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}%
+  \def\nextvii@{\ifx\@let@token\@sptoken
+   \DN@. {\FN@\nextviii@}\else
+   \DN@.{\FN@\nextviii@}\fi\next@.}%
+  \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else
+   \let\next@\plainroot@\fi\next@}%
+  \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}%
+  \bgroup\uproot@\z@\leftroot@\z@\FN@\next@}
+\def\plainroot@#1\of#2{\setbox\rootbox\hbox{%
+ $\m@th\scriptscriptstyle{#1}$}%
+ \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}}
+ {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup}
+\def\r@@t#1#2{\setboxz@h{$\m@th#1\@@sqrt{#2}$}%
+ \dimen@\ht\z@\advance\dimen@-\dp\z@
+ \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}%
+ \advance\dimen@ by1.667\wd\@ne
+ \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox
+ \mkern-10mu\mkern\leftroot@ mu\boxz@}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% Fragment from old amsfonts.sty to deal with msam, msbm, eufm fonts;
+%% omit if `noamsfonts' option was specified.
+\ifx F\@amsfonts \endinput \fi
+\DeclareSymbolFont{AMSa}{U}{msa}{m}{n}
+\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}
+\ifx P\@amsfonts
+  \DeclareFontFamily{U}{msa}{}%
+  \DeclareFontShape{U}{msa}{m}{n}{<-6>msam5<6-8>msam7<8->msam10}{}%
+  \DeclareFontFamily{U}{msb}{}%
+  \DeclareFontShape{U}{msb}{m}{n}{<-6>msbm5<6-8>msbm7<8->msbm10}{}%
+  \DeclareFontFamily{U}{euf}{}%
+  \DeclareFontShape{U}{euf}{m}{n}{<-6>eufm5<6-8>eufm7<8->eufm10}{}%
+  \DeclareFontShape{U}{euf}{b}{n}{<-6>eufb5<6-8>eufb7<8->eufb10}{}%
+\fi
+\@ifundefined{yen}{%
+  \edef\yen{\noexpand\mathhexbox{\hexnumber@\symAMSa}55}
+}{}
+\@ifundefined{checkmark}{%
+  \edef\checkmark{\noexpand\mathhexbox{\hexnumber@\symAMSa}58}
+}{}
+\@ifundefined{circledR}{%
+  \edef\circledR{\noexpand\mathhexbox{\hexnumber@\symAMSa}72}
+}{}
+\@ifundefined{maltese}{%
+  \edef\maltese{\noexpand\mathhexbox{\hexnumber@\symAMSa}7A}
+}{}
+\begingroup \catcode`\"=12
+\DeclareMathDelimiter\ulcorner{\mathopen} {AMSa}{"70}{AMSa}{"70}
+\DeclareMathDelimiter\urcorner{\mathclose}{AMSa}{"71}{AMSa}{"71}
+\DeclareMathDelimiter\llcorner{\mathopen} {AMSa}{"78}{AMSa}{"78}
+\DeclareMathDelimiter\lrcorner{\mathclose}{AMSa}{"79}{AMSa}{"79}
+\xdef\widehat#1{\noexpand\@mathmeasure\z@\textstyle{#1}%
+  \noexpand\ifdim\noexpand\wdz@>\tw@ em%
+  \mathaccent"0\hexnumber@\symAMSb 5B{#1}%
+  \noexpand\else\mathaccent"0362{#1}\noexpand\fi}
+\xdef\widetilde#1{\noexpand\@mathmeasure\z@\textstyle{#1}%
+  \noexpand\ifdim\noexpand\wdz@>\tw@ em%
+  \mathaccent"0\hexnumber@\symAMSb 5D{#1}%
+  \noexpand\else\mathaccent"0365{#1}\noexpand\fi}
+\DeclareMathSymbol\dabar@{\mathord}{AMSa}{"39}
+\xdef\dashrightarrow{\mathrel{\dabar@\dabar@
+                              \mathchar"0\hexnumber@\symAMSa 4B}}%
+\xdef\dashleftarrow{\mathrel{\mathchar"0\hexnumber@\symAMSa 4C\dabar@
+                              \dabar@}}%
+\global\let\dasharrow\dashrightarrow
+\global\let\rightleftharpoons\undefined
+\DeclareMathSymbol\rightleftharpoons{\mathrel}{AMSa}{"0A}
+\global\let\angle\undefined
+\DeclareMathSymbol\angle            {\mathord}{AMSa}{"5C}
+\global\let\hbar\undefined
+\DeclareMathSymbol\hbar             {\mathord}{AMSb}{"7E}
+\global\let\sqsubset\undefined
+\DeclareMathSymbol\sqsubset         {\mathrel}{AMSa}{"40}
+\global\let\sqsupset\undefined
+\DeclareMathSymbol\sqsupset         {\mathrel}{AMSa}{"41}
+\global\let\mho\undefined
+\DeclareMathSymbol\mho              {\mathord}{AMSb}{"66}
+\DeclareMathSymbol\square           {\mathord}{AMSa}{"03}
+\DeclareMathSymbol\lozenge          {\mathord}{AMSa}{"06}
+\DeclareMathSymbol\vartriangleright {\mathrel}{AMSa}{"42}
+\DeclareMathSymbol\vartriangleleft  {\mathrel}{AMSa}{"43}
+\DeclareMathSymbol\trianglerighteq  {\mathrel}{AMSa}{"44}
+\DeclareMathSymbol\trianglelefteq   {\mathrel}{AMSa}{"45}
+\DeclareMathSymbol\rightsquigarrow  {\mathrel}{AMSa}{"20}
+\def\@tempa{\not@base\lhd}
+\ifx\lhd\@tempa
+ \global\let\lhd\vartriangleleft
+ \global\let\unlhd\trianglelefteq
+ \global\let\rhd\vartriangleright
+ \global\let\unrhd\trianglerighteq
+ \global\let\Box\square
+ \global\let\Diamond\lozenge
+ \global\let\leadsto\rightsquigarrow
+\xdef\Join{\mathrel{\mathchar"0\hexnumber@\symAMSb 6F\mkern-13.8mu%
+  \mathchar"0\hexnumber@\symAMSb 6E}}
+\fi
+\endgroup
+\DeclareMathAlphabet\mathfrak{U}{euf}{m}{n}
+\SetMathAlphabet\mathfrak{bold}{U}{euf}{b}{n}
+\DeclareSymbolFontAlphabet{\mathbb}{AMSb}
+\DeclareFontEncodingDefaults{\relax}{\def\accentclass@{7}}
+\def\frak{\mathfrak}
+\def\Bbb{\mathbb}
+\def\bold{\mathbf}
+\begingroup \catcode`\"=12 \relax
+\gdef\newsymbol#1#2#3#4#5{%
+  \@ifdefinable#1{%
+     \edef\next@
+       {\ifcase #2 \or
+          \hexnumber@\symAMSa\or
+          \hexnumber@\symAMSb\fi}%
+     \ifx\next@\@empty
+       \PackageError{amsfonts}{\Invalid@@\newsymbol}\@ehd%
+     \else
+      \global\mathchardef#1"#3\next@#4#5
+     \fi}}
+\endgroup
+%%   Some odds and ends
+\def\sphat{^{\mathchoice{}{}%
+ {\,\,\smash[b]{\hbox{\lower4\ex@\hbox{$\m@th\widehat{\null}$}}}}%
+ {\,\smash[b]{\hbox{\lower3\ex@\hbox{$\m@th\hat{\null}$}}}}}}
+\def\spcheck{^\vee}
+\def\sptilde{^\sim}
+\def\spdot{^{\hbox{\raise\ex@\hbox{\normalfont .}}}}
+\def\spddot{^{\hbox{\raise\ex@\hbox{\normalfont ..}}}}
+\def\spdddot{^{\hbox{\raise\ex@\hbox{\normalfont...}}}}
+\def\spbreve{^{\!\smash[b]{\hbox{\lower4\ex@\hbox{\u{}}}}}}
+\def\fracwithdelims#1#2{\new@ifnextchar[{\fracwithdelims@#1#2}%
+  {\fracwithdelims@#1#2[\@empty]}}
+\def\fracwithdelims@#1#2[#3]#4#5{\relax
+  \ifx\@empty#3{#4\overwithdelims#1#2#5}\else
+  {#4\abovewithdelims#1#2#3\relax#5}\fi}
+\def\accentedsymbol#1#2{%
+  \expandafter\newbox
+    \csname\expandafter\@gobble\string#1@box\endcsname
+  \expandafter\setbox
+    \csname\expandafter\@gobble\string#1@box\endcsname
+      \hbox{$\m@th#2$}%
+  \newcommand{#1}{%
+    \expandafter\copy
+      \csname\expandafter\@gobble\string#1@box\endcsname{}}%
+}
+\def~{\nobreakspace}
+\DeclareRobustCommand{\nobreakspace}{\unskip\nobreak\ \ignorespaces}
+%%
+%% End of file `amstex.sty'.
diff --git a/Master/texmf-dist/tex/latex/amsmath/amstext.sty b/Master/texmf-dist/tex/latex/amsmath/amstext.sty
new file mode 100644
index 00000000000..6c3c1692d89
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/amsmath/amstext.sty
@@ -0,0 +1,101 @@
+%%
+%% This is file `amstext.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amstext.dtx 
+%% 
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amstext.dtx",
+%%%   version   = "2.01",
+%%%   date      = "2000/06/29",
+%%%   time      = "08:34:33 EDT",
+%%%   checksum  = "09860 314 1647 12368",
+%%%   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  = "latex, math, text",
+%%%   abstract  = "This is a \LaTeX{} package that defines a
+%%%                \cn{\text} macro, which makes it easy to
+%%%                incorporate fragments of text inside a displayed
+%%%                equation or a sub or superscript. Font sizes
+%%%                are automatically scaled in sub/superscripts.",
+%%%   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 be December 1994 or later
+\ProvidesPackage{amstext}[2000/06/29 v2.01]
+\RequirePackage{amsgen}
+\DeclareRobustCommand{\text}{%
+  \ifmmode\expandafter\text@\else\expandafter\mbox\fi}
+\let\nfss@text\text
+\def\text@#1{{\mathchoice
+  {\textdef@\displaystyle\f@size{#1}}%
+  {\textdef@\textstyle\f@size{\firstchoice@false #1}}%
+  {\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
+  {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
+  \check@mathfonts
+  }%
+}
+\def\textdef@#1#2#3{\hbox{{%
+                    \everymath{#1}%
+                    \let\f@size#2\selectfont
+                    #3}}}
+\newif\iffirstchoice@
+\firstchoice@true
+\def\stepcounter#1{%
+  \iffirstchoice@
+     \addtocounter{#1}\@ne
+     \begingroup \let\@elt\@stpelt \csname cl@#1\endcsname \endgroup
+  \fi
+}
+\def\addtocounter#1#2{%
+  \iffirstchoice@
+  \@ifundefined {c@#1}{\@nocounterr {#1}}%
+    {\global \advance \csname c@#1\endcsname #2\relax}%
+  \fi}
+\let\m@gobble\@empty
+\@xp\let\csname m@gobble4\endcsname\@gobblefour
+\long\@xp\def\csname m@gobble6\endcsname#1#2#3#4#5#6{}
+\toks@{%
+  \csname m@gobble\iffirstchoice@\else 4\fi\endcsname
+  \protect}
+\edef\GenericInfo{\the\toks@
+  \@xp\@nx\csname GenericInfo \endcsname}
+\edef\GenericWarning{\the\toks@
+  \@xp\@nx\csname GenericWarning \endcsname}
+\toks@{%
+  \csname m@gobble\iffirstchoice@\else 6\fi\endcsname
+  \protect}
+\edef\GenericError{\the\toks@
+  \@xp\@nx\csname GenericError \endcsname}
+\begingroup \catcode`\"=12
+\gdef\mathhexbox#1#2#3{\text{$\m@th\mathchar"#1#2#3$}}
+\endgroup
+\endinput
+%%
+%% End of file `amstext.sty'.
diff --git a/Master/texmf-dist/tex/latex/amsmath/amsxtra.sty b/Master/texmf-dist/tex/latex/amsmath/amsxtra.sty
new file mode 100644
index 00000000000..10a65df4a20
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/amsmath/amsxtra.sty
@@ -0,0 +1,80 @@
+%%
+%% This is file `amsxtra.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amsxtra.dtx 
+%% 
+%%% ====================================================================
+%%% @LaTeX-file{
+%%%   filename  = "amsxtra.dtx",
+%%%   version   = "1.2c",
+%%%   date      = "1999/11/15",
+%%%   time      = "14:15:41 EST",
+%%%   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",
+%%%   checksum  = "41382 136 533 5124",
+%%%   email     = "tech-support@ams.org (Internet)",
+%%%   codetable = "ISO/ASCII",
+%%%   keywords  = "latex, accentedsymbol, fracwithdelims, sphat",
+%%%   supported = "yes",
+%%%   abstract  = "This package contains some seldom-used commands
+%%%                ported from AMS-TeX in 1989.",
+%%%   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.",
+%%% }
+%%% ====================================================================
+\ProvidesPackage{amsxtra}[1999/11/15 v1.2c]
+\@ifpackageloaded{amstex}{}{%
+  \RequirePackage{amsmath}[1996/10/01 v1.2]}
+\def\sphat{^{\mathchoice{}{}%
+ {\,\,\smash[b]{\hbox{\lower4\ex@\hbox{$\m@th\widehat{\null}$}}}}%
+ {\,\smash[b]{\hbox{\lower3\ex@\hbox{$\m@th\hat{\null}$}}}}}}
+\def\spcheck{^\vee}
+\def\sptilde{^\sim}
+\def\spdot{^{\hbox{\raise\ex@\hbox{\normalfont .}}}}
+\def\spddot{^{\hbox{\raise\ex@\hbox{\normalfont ..}}}}
+\def\spdddot{^{\hbox{\raise\ex@\hbox{\normalfont...}}}}
+\def\spbreve{^{\!\smash[b]{\hbox{\lower4\ex@\hbox{\u{}}}}}}
+\@ifpackageloaded{amstex}{%
+\def\fracwithdelims#1#2{\new@ifnextchar[{\fracwithdelims@#1#2}%
+  {\fracwithdelims@#1#2[\@empty]}}
+\def\fracwithdelims@#1#2[#3]#4#5{\relax
+  \ifx\@empty#3{#4\overwithdelims#1#2#5}\else
+  {#4\abovewithdelims#1#2#3\relax#5}\fi}
+}{}
+\def\accentedsymbol#1#2{%
+  \expandafter\newbox
+    \csname\expandafter\@gobble\string#1@box\endcsname
+  \expandafter\setbox
+    \csname\expandafter\@gobble\string#1@box\endcsname
+      \hbox{$\m@th#2$}%
+  \newcommand{#1}{%
+    \expandafter\copy
+      \csname\expandafter\@gobble\string#1@box\endcsname{}}%
+}
+\def~{\nobreakspace}
+\DeclareRobustCommand{\nobreakspace}{\unskip\nobreak\ \ignorespaces}
+\endinput
+%%
+%% End of file `amsxtra.sty'.
diff --git a/Master/texmf-dist/tpm/amsmath.tpm b/Master/texmf-dist/tpm/amsmath.tpm
new file mode 100644
index 00000000000..e057d187ba4
--- /dev/null
+++ b/Master/texmf-dist/tpm/amsmath.tpm
@@ -0,0 +1,61 @@
+<!DOCTYPE rdf:RDF SYSTEM "../../support/tpm.dtd">
+<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:TPM="http://texlive.dante.de/">
+  <rdf:Description about="http://texlive.dante.de/texlive/Package/amsmath.zip">
+    <TPM:Name>amsmath</TPM:Name>
+    <TPM:Type>Package</TPM:Type>
+    <TPM:Date>2006/08/26 16:19:36</TPM:Date>
+    <TPM:Version></TPM:Version>
+    <TPM:Creator>karl</TPM:Creator>
+    <TPM:Title>The amsmath package.</TPM:Title>
+    <TPM:Description></TPM:Description>
+    <TPM:Author></TPM:Author>
+    <TPM:Size>3434956</TPM:Size>
+    <TPM:Build/>
+    <TPM:RunFiles size="180640">
+texmf-dist/tex/latex/amsmath/amsbsy.sty
+texmf-dist/tex/latex/amsmath/amscd.sty
+texmf-dist/tex/latex/amsmath/amsgen.sty
+texmf-dist/tex/latex/amsmath/amsmath.sty
+texmf-dist/tex/latex/amsmath/amsopn.sty
+texmf-dist/tex/latex/amsmath/amstex.sty
+texmf-dist/tex/latex/amsmath/amstext.sty
+texmf-dist/tex/latex/amsmath/amsxtra.sty
+texmf-dist/tpm/amsmath.tpm
+    </TPM:RunFiles>
+    <TPM:DocFiles size="2751948">
+texmf-dist/doc/latex/amsmath/amsbsy.pdf
+texmf-dist/doc/latex/amsmath/amscd.pdf
+texmf-dist/doc/latex/amsmath/amsgen.pdf
+texmf-dist/doc/latex/amsmath/amsldoc.pdf
+texmf-dist/doc/latex/amsmath/amsmath.pdf
+texmf-dist/doc/latex/amsmath/amsopn.pdf
+texmf-dist/doc/latex/amsmath/amstext.pdf
+texmf-dist/doc/latex/amsmath/amsxtra.pdf
+texmf-dist/doc/latex/amsmath/diffs-m.txt
+texmf-dist/doc/latex/amsmath/subeqn.pdf
+texmf-dist/doc/latex/amsmath/technote.pdf
+texmf-dist/doc/latex/amsmath/testmath.pdf
+    </TPM:DocFiles>
+    <TPM:SourceFiles size="504746">
+texmf-dist/source/latex/amsmath/00readme.txt
+texmf-dist/source/latex/amsmath/ams-m1.ins
+texmf-dist/source/latex/amsmath/amsbsy.dtx
+texmf-dist/source/latex/amsmath/amscd.dtx
+texmf-dist/source/latex/amsmath/amsdtx.cls
+texmf-dist/source/latex/amsmath/amsgen.dtx
+texmf-dist/source/latex/amsmath/amsldoc.cls
+texmf-dist/source/latex/amsmath/amsldoc.tex
+texmf-dist/source/latex/amsmath/amsmath.dtx
+texmf-dist/source/latex/amsmath/amsopn.dtx
+texmf-dist/source/latex/amsmath/amstext.dtx
+texmf-dist/source/latex/amsmath/amsxtra.dtx
+texmf-dist/source/latex/amsmath/install.txt
+texmf-dist/source/latex/amsmath/manifest.txt
+texmf-dist/source/latex/amsmath/subeqn.tex
+texmf-dist/source/latex/amsmath/technote.tex
+texmf-dist/source/latex/amsmath/testmath.tex
+    </TPM:SourceFiles>
+    <TPM:Provides>Package/amsmath</TPM:Provides>
+  </rdf:Description>
+</rdf:RDF>
+