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Anderson localization, density of states, potentials with stationary increments
---------------9811100641922
Content-Type: text/plain; name="amssymb.sty"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="amssymb.sty"

%%
%% This is file `amssymb.sty',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% amssymb.dtx 
%% 
%%% ====================================================================
%%%  @LaTeX-file{
%%%     filename        = "amssymb.dtx",
%%%     version         = "2.2c",
%%%     date            = "1997/05/15",
%%%     time            = "11:10:27",
%%%     author          = "American Mathematical Society",
%%%     copyright       = "Copyright (C) 1996 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        = "01698 351 1072 17688",
%%%     email           = "tech-support@ams.org (Internet)",
%%%     codetable       = "ISO/ASCII",
%%%     keywords        = "latex, amslatex, ams-latex, math symbol,
%%%                        amsfonts, msam, msbm",
%%%     supported       = "yes",
%%%     abstract        = "This is part of the AMSFonts distribution.
%%%                        It is a \LaTeX{} option that defines symbol
%%%                        names for all the math symbols in the fonts
%%%                        MSAM and MSBM, of the AMSFonts (2.0+) 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 be December 1994 or later
\ProvidesPackage{amssymb}[1996/11/03 v2.2b]
\DeclareOption{psamsfonts}{\PassOptionsToPackage{psamsfonts}{amsfonts}}
\ProcessOptions\relax
\RequirePackage{amsfonts}[1995/01/01]
\let\square\relax \let\rightsquigarrow\square \let\lozenge\square
\let\vartriangleright\square \let\vartriangleleft\square
\let\trianglerighteq\square \let\trianglelefteq\square
\DeclareMathSymbol{\boxdot}       {\mathbin}{AMSa}{"00}
\DeclareMathSymbol{\boxplus}      {\mathbin}{AMSa}{"01}
\DeclareMathSymbol{\boxtimes}     {\mathbin}{AMSa}{"02}
\DeclareMathSymbol{\square}       {\mathord}{AMSa}{"03}
\DeclareMathSymbol{\blacksquare}  {\mathord}{AMSa}{"04}
\DeclareMathSymbol{\centerdot}    {\mathbin}{AMSa}{"05}
\DeclareMathSymbol{\lozenge}      {\mathord}{AMSa}{"06}
\DeclareMathSymbol{\blacklozenge} {\mathord}{AMSa}{"07}
\DeclareMathSymbol{\circlearrowright}   {\mathrel}{AMSa}{"08}
\DeclareMathSymbol{\circlearrowleft}    {\mathrel}{AMSa}{"09}
%% In amsfonts.sty:
%%\DeclareMathSymbol{\rightleftharpoons}{\mathrel}{AMSa}{"0A}
\DeclareMathSymbol{\leftrightharpoons}  {\mathrel}{AMSa}{"0B}
\DeclareMathSymbol{\boxminus}     {\mathbin}{AMSa}{"0C}
\DeclareMathSymbol{\Vdash}        {\mathrel}{AMSa}{"0D}
\DeclareMathSymbol{\Vvdash}       {\mathrel}{AMSa}{"0E}
\DeclareMathSymbol{\vDash}        {\mathrel}{AMSa}{"0F}
\DeclareMathSymbol{\twoheadrightarrow}  {\mathrel}{AMSa}{"10}
\DeclareMathSymbol{\twoheadleftarrow}   {\mathrel}{AMSa}{"11}
\DeclareMathSymbol{\leftleftarrows}     {\mathrel}{AMSa}{"12}
\DeclareMathSymbol{\rightrightarrows}   {\mathrel}{AMSa}{"13}
\DeclareMathSymbol{\upuparrows}         {\mathrel}{AMSa}{"14}
\DeclareMathSymbol{\downdownarrows} {\mathrel}{AMSa}{"15}
\DeclareMathSymbol{\upharpoonright} {\mathrel}{AMSa}{"16}
 \let\restriction\upharpoonright
\DeclareMathSymbol{\downharpoonright}   {\mathrel}{AMSa}{"17}
\DeclareMathSymbol{\upharpoonleft}  {\mathrel}{AMSa}{"18}
\DeclareMathSymbol{\downharpoonleft}{\mathrel}{AMSa}{"19}
\DeclareMathSymbol{\rightarrowtail} {\mathrel}{AMSa}{"1A}
\DeclareMathSymbol{\leftarrowtail}  {\mathrel}{AMSa}{"1B}
\DeclareMathSymbol{\leftrightarrows}{\mathrel}{AMSa}{"1C}
\DeclareMathSymbol{\rightleftarrows}{\mathrel}{AMSa}{"1D}
\DeclareMathSymbol{\Lsh}            {\mathrel}{AMSa}{"1E}
\DeclareMathSymbol{\Rsh}            {\mathrel}{AMSa}{"1F}
\DeclareMathSymbol{\rightsquigarrow}  {\mathrel}{AMSa}{"20}
\DeclareMathSymbol{\leftrightsquigarrow}{\mathrel}{AMSa}{"21}
\DeclareMathSymbol{\looparrowleft}  {\mathrel}{AMSa}{"22}
\DeclareMathSymbol{\looparrowright} {\mathrel}{AMSa}{"23}
\DeclareMathSymbol{\circeq}       {\mathrel}{AMSa}{"24}
\DeclareMathSymbol{\succsim}      {\mathrel}{AMSa}{"25}
\DeclareMathSymbol{\gtrsim}       {\mathrel}{AMSa}{"26}
\DeclareMathSymbol{\gtrapprox}    {\mathrel}{AMSa}{"27}
\DeclareMathSymbol{\multimap}     {\mathrel}{AMSa}{"28}
\DeclareMathSymbol{\therefore}    {\mathrel}{AMSa}{"29}
\DeclareMathSymbol{\because}      {\mathrel}{AMSa}{"2A}
\DeclareMathSymbol{\doteqdot}     {\mathrel}{AMSa}{"2B}
 \let\Doteq\doteqdot
\DeclareMathSymbol{\triangleq}    {\mathrel}{AMSa}{"2C}
\DeclareMathSymbol{\precsim}      {\mathrel}{AMSa}{"2D}
\DeclareMathSymbol{\lesssim}      {\mathrel}{AMSa}{"2E}
\DeclareMathSymbol{\lessapprox}   {\mathrel}{AMSa}{"2F}
\DeclareMathSymbol{\eqslantless}  {\mathrel}{AMSa}{"30}
\DeclareMathSymbol{\eqslantgtr}   {\mathrel}{AMSa}{"31}
\DeclareMathSymbol{\curlyeqprec}  {\mathrel}{AMSa}{"32}
\DeclareMathSymbol{\curlyeqsucc}  {\mathrel}{AMSa}{"33}
\DeclareMathSymbol{\preccurlyeq}  {\mathrel}{AMSa}{"34}
\DeclareMathSymbol{\leqq}         {\mathrel}{AMSa}{"35}
\DeclareMathSymbol{\leqslant}     {\mathrel}{AMSa}{"36}
\DeclareMathSymbol{\lessgtr}      {\mathrel}{AMSa}{"37}
\DeclareMathSymbol{\backprime}    {\mathord}{AMSa}{"38}
\DeclareMathSymbol{\risingdotseq} {\mathrel}{AMSa}{"3A}
\DeclareMathSymbol{\fallingdotseq}{\mathrel}{AMSa}{"3B}
\DeclareMathSymbol{\succcurlyeq}  {\mathrel}{AMSa}{"3C}
\DeclareMathSymbol{\geqq}         {\mathrel}{AMSa}{"3D}
\DeclareMathSymbol{\geqslant}     {\mathrel}{AMSa}{"3E}
\DeclareMathSymbol{\gtrless}      {\mathrel}{AMSa}{"3F}
%% in amsfonts.sty
%% \DeclareMathSymbol{\sqsubset}    {\mathrel}{AMSa}{"40}
%% \DeclareMathSymbol{\sqsupset}    {\mathrel}{AMSa}{"41}
\DeclareMathSymbol{\vartriangleright}{\mathrel}{AMSa}{"42}
\DeclareMathSymbol{\vartriangleleft} {\mathrel}{AMSa}{"43}
\DeclareMathSymbol{\trianglerighteq} {\mathrel}{AMSa}{"44}
\DeclareMathSymbol{\trianglelefteq}  {\mathrel}{AMSa}{"45}
\DeclareMathSymbol{\bigstar}    {\mathord}{AMSa}{"46}
\DeclareMathSymbol{\between}    {\mathrel}{AMSa}{"47}
\DeclareMathSymbol{\blacktriangledown}  {\mathord}{AMSa}{"48}
\DeclareMathSymbol{\blacktriangleright} {\mathrel}{AMSa}{"49}
\DeclareMathSymbol{\blacktriangleleft}  {\mathrel}{AMSa}{"4A}
\DeclareMathSymbol{\vartriangle}        {\mathrel}{AMSa}{"4D}
\DeclareMathSymbol{\blacktriangle}      {\mathord}{AMSa}{"4E}
\DeclareMathSymbol{\triangledown}       {\mathord}{AMSa}{"4F}
\DeclareMathSymbol{\eqcirc}       {\mathrel}{AMSa}{"50}
\DeclareMathSymbol{\lesseqgtr}    {\mathrel}{AMSa}{"51}
\DeclareMathSymbol{\gtreqless}    {\mathrel}{AMSa}{"52}
\DeclareMathSymbol{\lesseqqgtr}   {\mathrel}{AMSa}{"53}
\DeclareMathSymbol{\gtreqqless}   {\mathrel}{AMSa}{"54}
\DeclareMathSymbol{\Rrightarrow}  {\mathrel}{AMSa}{"56}
\DeclareMathSymbol{\Lleftarrow}   {\mathrel}{AMSa}{"57}
\DeclareMathSymbol{\veebar}       {\mathbin}{AMSa}{"59}
\DeclareMathSymbol{\barwedge}     {\mathbin}{AMSa}{"5A}
\DeclareMathSymbol{\doublebarwedge} {\mathbin}{AMSa}{"5B}
%% In amsfonts.sty
%%\DeclareMathSymbol{\angle}        {\mathord}{AMSa}{"5C}
\DeclareMathSymbol{\measuredangle}  {\mathord}{AMSa}{"5D}
\DeclareMathSymbol{\sphericalangle} {\mathord}{AMSa}{"5E}
\DeclareMathSymbol{\varpropto}    {\mathrel}{AMSa}{"5F}
\DeclareMathSymbol{\smallsmile}   {\mathrel}{AMSa}{"60}
\DeclareMathSymbol{\smallfrown}   {\mathrel}{AMSa}{"61}
\DeclareMathSymbol{\Subset}       {\mathrel}{AMSa}{"62}
\DeclareMathSymbol{\Supset}       {\mathrel}{AMSa}{"63}
\DeclareMathSymbol{\Cup}          {\mathbin}{AMSa}{"64}
 \let\doublecup\Cup
\DeclareMathSymbol{\Cap}          {\mathbin}{AMSa}{"65}
 \let\doublecap\Cap
\DeclareMathSymbol{\curlywedge}   {\mathbin}{AMSa}{"66}
\DeclareMathSymbol{\curlyvee}     {\mathbin}{AMSa}{"67}
\DeclareMathSymbol{\leftthreetimes} {\mathbin}{AMSa}{"68}
\DeclareMathSymbol{\rightthreetimes}{\mathbin}{AMSa}{"69}
\DeclareMathSymbol{\subseteqq}    {\mathrel}{AMSa}{"6A}
\DeclareMathSymbol{\supseteqq}    {\mathrel}{AMSa}{"6B}
\DeclareMathSymbol{\bumpeq}       {\mathrel}{AMSa}{"6C}
\DeclareMathSymbol{\Bumpeq}       {\mathrel}{AMSa}{"6D}
\DeclareMathSymbol{\lll}          {\mathrel}{AMSa}{"6E}
 \let\llless\lll
\DeclareMathSymbol{\ggg}          {\mathrel}{AMSa}{"6F}
 \let\gggtr\ggg
\DeclareMathSymbol{\circledS}     {\mathord}{AMSa}{"73}
\DeclareMathSymbol{\pitchfork}    {\mathrel}{AMSa}{"74}
\DeclareMathSymbol{\dotplus}      {\mathbin}{AMSa}{"75}
\DeclareMathSymbol{\backsim}      {\mathrel}{AMSa}{"76}
\DeclareMathSymbol{\backsimeq}    {\mathrel}{AMSa}{"77}
\DeclareMathSymbol{\complement}   {\mathord}{AMSa}{"7B}
\DeclareMathSymbol{\intercal}     {\mathbin}{AMSa}{"7C}
\DeclareMathSymbol{\circledcirc}  {\mathbin}{AMSa}{"7D}
\DeclareMathSymbol{\circledast}   {\mathbin}{AMSa}{"7E}
\DeclareMathSymbol{\circleddash}  {\mathbin}{AMSa}{"7F}
%%   Begin AMSb declarations
\DeclareMathSymbol{\lvertneqq}    {\mathrel}{AMSb}{"00}
\DeclareMathSymbol{\gvertneqq}    {\mathrel}{AMSb}{"01}
\DeclareMathSymbol{\nleq}         {\mathrel}{AMSb}{"02}
\DeclareMathSymbol{\ngeq}         {\mathrel}{AMSb}{"03}
\DeclareMathSymbol{\nless}        {\mathrel}{AMSb}{"04}
\DeclareMathSymbol{\ngtr}         {\mathrel}{AMSb}{"05}
\DeclareMathSymbol{\nprec}        {\mathrel}{AMSb}{"06}
\DeclareMathSymbol{\nsucc}        {\mathrel}{AMSb}{"07}
\DeclareMathSymbol{\lneqq}        {\mathrel}{AMSb}{"08}
\DeclareMathSymbol{\gneqq}        {\mathrel}{AMSb}{"09}
\DeclareMathSymbol{\nleqslant}    {\mathrel}{AMSb}{"0A}
\DeclareMathSymbol{\ngeqslant}    {\mathrel}{AMSb}{"0B}
\DeclareMathSymbol{\lneq}         {\mathrel}{AMSb}{"0C}
\DeclareMathSymbol{\gneq}         {\mathrel}{AMSb}{"0D}
\DeclareMathSymbol{\npreceq}      {\mathrel}{AMSb}{"0E}
\DeclareMathSymbol{\nsucceq}      {\mathrel}{AMSb}{"0F}
\DeclareMathSymbol{\precnsim}     {\mathrel}{AMSb}{"10}
\DeclareMathSymbol{\succnsim}     {\mathrel}{AMSb}{"11}
\DeclareMathSymbol{\lnsim}        {\mathrel}{AMSb}{"12}
\DeclareMathSymbol{\gnsim}        {\mathrel}{AMSb}{"13}
\DeclareMathSymbol{\nleqq}        {\mathrel}{AMSb}{"14}
\DeclareMathSymbol{\ngeqq}        {\mathrel}{AMSb}{"15}
\DeclareMathSymbol{\precneqq}     {\mathrel}{AMSb}{"16}
\DeclareMathSymbol{\succneqq}     {\mathrel}{AMSb}{"17}
\DeclareMathSymbol{\precnapprox}  {\mathrel}{AMSb}{"18}
\DeclareMathSymbol{\succnapprox}  {\mathrel}{AMSb}{"19}
\DeclareMathSymbol{\lnapprox}     {\mathrel}{AMSb}{"1A}
\DeclareMathSymbol{\gnapprox}     {\mathrel}{AMSb}{"1B}
\DeclareMathSymbol{\nsim}         {\mathrel}{AMSb}{"1C}
\DeclareMathSymbol{\ncong}        {\mathrel}{AMSb}{"1D}
\DeclareMathSymbol{\diagup}       {\mathord}{AMSb}{"1E}
\DeclareMathSymbol{\diagdown}     {\mathord}{AMSb}{"1F}
\DeclareMathSymbol{\varsubsetneq}   {\mathrel}{AMSb}{"20}
\DeclareMathSymbol{\varsupsetneq}   {\mathrel}{AMSb}{"21}
\DeclareMathSymbol{\nsubseteqq}     {\mathrel}{AMSb}{"22}
\DeclareMathSymbol{\nsupseteqq}     {\mathrel}{AMSb}{"23}
\DeclareMathSymbol{\subsetneqq}     {\mathrel}{AMSb}{"24}
\DeclareMathSymbol{\supsetneqq}     {\mathrel}{AMSb}{"25}
\DeclareMathSymbol{\varsubsetneqq}  {\mathrel}{AMSb}{"26}
\DeclareMathSymbol{\varsupsetneqq}  {\mathrel}{AMSb}{"27}
\DeclareMathSymbol{\subsetneq}      {\mathrel}{AMSb}{"28}
\DeclareMathSymbol{\supsetneq}      {\mathrel}{AMSb}{"29}
\DeclareMathSymbol{\nsubseteq}      {\mathrel}{AMSb}{"2A}
\DeclareMathSymbol{\nsupseteq}      {\mathrel}{AMSb}{"2B}
\DeclareMathSymbol{\nparallel}      {\mathrel}{AMSb}{"2C}
\DeclareMathSymbol{\nmid}           {\mathrel}{AMSb}{"2D}
\DeclareMathSymbol{\nshortmid}      {\mathrel}{AMSb}{"2E}
\DeclareMathSymbol{\nshortparallel} {\mathrel}{AMSb}{"2F}
\DeclareMathSymbol{\nvdash}         {\mathrel}{AMSb}{"30}
\DeclareMathSymbol{\nVdash}         {\mathrel}{AMSb}{"31}
\DeclareMathSymbol{\nvDash}         {\mathrel}{AMSb}{"32}
\DeclareMathSymbol{\nVDash}         {\mathrel}{AMSb}{"33}
\DeclareMathSymbol{\ntrianglerighteq}{\mathrel}{AMSb}{"34}
\DeclareMathSymbol{\ntrianglelefteq}{\mathrel}{AMSb}{"35}
\DeclareMathSymbol{\ntriangleleft}  {\mathrel}{AMSb}{"36}
\DeclareMathSymbol{\ntriangleright} {\mathrel}{AMSb}{"37}
\DeclareMathSymbol{\nleftarrow}     {\mathrel}{AMSb}{"38}
\DeclareMathSymbol{\nrightarrow}    {\mathrel}{AMSb}{"39}
\DeclareMathSymbol{\nLeftarrow}     {\mathrel}{AMSb}{"3A}
\DeclareMathSymbol{\nRightarrow}    {\mathrel}{AMSb}{"3B}
\DeclareMathSymbol{\nLeftrightarrow}{\mathrel}{AMSb}{"3C}
\DeclareMathSymbol{\nleftrightarrow}{\mathrel}{AMSb}{"3D}
\DeclareMathSymbol{\divideontimes}  {\mathbin}{AMSb}{"3E}
\DeclareMathSymbol{\varnothing}     {\mathord}{AMSb}{"3F}
\DeclareMathSymbol{\nexists}        {\mathord}{AMSb}{"40}
\DeclareMathSymbol{\Finv}           {\mathord}{AMSb}{"60}
\DeclareMathSymbol{\Game}           {\mathord}{AMSb}{"61}
%% In amsfonts.sty:
%%\DeclareMathSymbol{\mho}          {\mathord}{AMSb}{"66}
\DeclareMathSymbol{\eth}            {\mathord}{AMSb}{"67}
\DeclareMathSymbol{\eqsim}          {\mathrel}{AMSb}{"68}
\DeclareMathSymbol{\beth}           {\mathord}{AMSb}{"69}
\DeclareMathSymbol{\gimel}          {\mathord}{AMSb}{"6A}
\DeclareMathSymbol{\daleth}         {\mathord}{AMSb}{"6B}
\DeclareMathSymbol{\lessdot}        {\mathbin}{AMSb}{"6C}
\DeclareMathSymbol{\gtrdot}         {\mathbin}{AMSb}{"6D}
\DeclareMathSymbol{\ltimes}         {\mathbin}{AMSb}{"6E}
\DeclareMathSymbol{\rtimes}         {\mathbin}{AMSb}{"6F}
\DeclareMathSymbol{\shortmid}       {\mathrel}{AMSb}{"70}
\DeclareMathSymbol{\shortparallel}  {\mathrel}{AMSb}{"71}
\DeclareMathSymbol{\smallsetminus}  {\mathbin}{AMSb}{"72}
\DeclareMathSymbol{\thicksim}       {\mathrel}{AMSb}{"73}
\DeclareMathSymbol{\thickapprox}    {\mathrel}{AMSb}{"74}
\DeclareMathSymbol{\approxeq}       {\mathrel}{AMSb}{"75}
\DeclareMathSymbol{\succapprox}     {\mathrel}{AMSb}{"76}
\DeclareMathSymbol{\precapprox}     {\mathrel}{AMSb}{"77}
\DeclareMathSymbol{\curvearrowleft} {\mathrel}{AMSb}{"78}
\DeclareMathSymbol{\curvearrowright}{\mathrel}{AMSb}{"79}
\DeclareMathSymbol{\digamma}        {\mathord}{AMSb}{"7A}
\DeclareMathSymbol{\varkappa}       {\mathord}{AMSb}{"7B}
\DeclareMathSymbol{\Bbbk}           {\mathord}{AMSb}{"7C}
\DeclareMathSymbol{\hslash}         {\mathord}{AMSb}{"7D}
%% In amsfonts.sty:
%%\DeclareMathSymbol{\hbar}         {\mathord}{AMSb}{"7E}
\DeclareMathSymbol{\backepsilon}    {\mathrel}{AMSb}{"7F}
\endinput
%%
%% End of file `amssymb.sty'.
---------------9811100641922
Content-Type: text/plain; name="thmdefs.sty"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="thmdefs.sty"

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{axiom}{Axiom}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\newtheorem{exercise}{Exercise}[section]
\newtheorem{definition}{Definition}[section]
---------------9811100641922
Content-Type: application/x-tex; name="Amscd.sty"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="Amscd.sty"

%% 
%% This is file `amscd.sty', generated 
%% on <1996/11/1> with the docstrip utility (2.2i).
%% 
%% The original source files were:
%% 
%% amscd.dtx 
%%% ====================================================================
%%%  @LaTeX-file{
%%%     filename        = "amscd.dtx",
%%%     version         = "1.2b",
%%%     date            = "1996/10/28",
%%%     time            = "14:43:47 EST",
%%%     author          = "American Mathematical Society",
%%%     copyright       = "Copyright (C) 1996 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        = "35773 318 1077 11839",
%%%     email           = "tech-support@ams.org (Internet)",
%%%     codetable       = "ISO/ASCII",
%%%     keywords        = "latex, amslatex, ams-latex, commutative diagram",
%%%     supported       = "yes",
%%%     abstract        = "This is part of the AMS-\LaTeX{} distribution.
%%%                        It 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}[1996/10/28 v1.2b]
\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
\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{$\m@th\scriptstyle\;{#2}\;\;$}%
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\atdef@|{\CD@check|{\Big\Vert&&}}
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%% 
%% End of file `amscd.sty'.
---------------9811100641922
Content-Type: application/x-tex; name="Amsmath.sty"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="Amsmath.sty"

%%
%% This is file `amsmath.sty',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% amsmath.dtx 
%% 
%%% ====================================================================
%%% @LaTeX-file{
%%%   filename  = "amsmath.dtx",
%%%   version   = "1.2d",
%%%   date      = "1997/03/20",
%%%   time      = "16:25:04 EST",
%%%   author    = "American Mathematical Society",
%%%   copyright = "Copyright (C) 1996 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  = "45539 4982 18193 174456",
%%%   email     = "tech-support@ams.org (Internet)",
%%%   codetable = "ISO/ASCII",
%%%   keywords  = "latex, amslatex, ams-latex, math, amsmath",
%%%   supported = "yes",
%%%   abstract  = "This is part of the AMS-\LaTeX{} distribution. It
%%%                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 December 1994 or later
\ProvidesPackage{amsmath}[1997/03/20 v1.2d AMS math features]
\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
    \AtBeginDocument{%
        \ifdim\@mathmargin= -1sp
            \@mathmargin\leftmargini
        \fi
    }%
}
\ExecuteOptions{nointlimits,sumlimits,namelimits,centertags}
\ProcessOptions\relax
\ifnum\cmex@opt=7 \relax
  \DeclareFontShape{OMX}{cmex}{m}{n}{%
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    <10><10.95><12><14.4><17.28><20.74><24.88>cmex10%
  }{}%
  \expandafter\let\csname OMX/cmex/m/n/10\endcsname\relax
\else
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    \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}}
\def\AmS{{\protect\AmSfont
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\def\AmSfont{%
  \usefont{OMS}{cmsy}{\if\@xp\@car\f@series\@nil bb\else m\fi}{n}}
\def\pr@m@s{%
  \ifx\@let@token'\DN@##1{\prim@s}\else\let\next@\egroup\fi\next@}
\def\prim@s{\prime\futurelet\@let@token\pr@m@s}
\let\@prime=\prime
\renewcommand{\prime}{{\kern\z@\@prime}}
\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}
\begingroup\catcode`\"=12
\def\@tempa#1{\expandafter\@tempb\meaning#1\relax\relax\relax\relax"0000\@nil#1}
\def\@tempb#1"#2#3#4#5#6\@nil#7{%
  \ifnum"#2=7 \count@"1#3#4#5\relax
    \ifnum\count@<"1000 \else \global\mathchardef#7="0#3#4#5\relax \fi
  \fi}
\@tempa\Gamma \@tempa\Delta \@tempa\Theta \@tempa\Lambda \@tempa\Xi
\@tempa\Pi \@tempa\Sigma \@tempa\Upsilon \@tempa\Phi \@tempa\Psi
\@tempa\Omega
\@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}
}{}
\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\implies{\DOTSB\;\Longrightarrow\;}
\def\impliedby{\DOTSB\;\Longleftarrow\;}
\begingroup \catcode`\"=12 % in case activated by a preceding package
\gdef\And{\DOTSB\;\mathchar"3026 \;}
\@tempcnta=\@xp\@gobble\vert \advance\@tempcnta "4000000
\xdef\lvert{\delimiter\number\@tempcnta\space }
\advance\@tempcnta "1000000
\xdef\rvert{\delimiter\number\@tempcnta\space }
\@tempcnta=\@xp\@gobble\Vert \advance\@tempcnta "4000000
\xdef\lVert{\delimiter\number\@tempcnta\space }
\advance\@tempcnta "1000000
\xdef\rVert{\delimiter\number\@tempcnta\space }
\endgroup % restore "
\@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; %
\protect\frac\space or \protect\genfrac\space should be used instead%
  }
  \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}}
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\DeclareRobustCommand{\binom}{\genfrac()\z@{}}
\newcommand{\dbinom}{\genfrac(){0pt}0}
\newcommand{\tbinom}{\genfrac(){0pt}1}
\DeclareRobustCommand{\genfrac}[4]{%
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  \edef\@tempb{\@nx\@genfrac\@mathstyle{#4}%
    \csname @@\ifx @#3@over\else above\fi
    \ifx\@tempa\@empty \else withdelims\fi\endcsname}
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\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
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\def\colon{\nobreak\mskip2mu\mathpunct{}\nonscript
  \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
\begingroup \catcode`\"=12
\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{\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@
}
\def\leftroot{\@amsmath@err{\Invalid@@\leftroot}\@eha}
\def\uproot{\@amsmath@err{\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@}
\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%
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{\uccode`7=`c \uccode`8=`h \uccode`9=`\"
 \uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\gtest@false
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\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@{}%
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{\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@{}%
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{\uccode`5=`m \uccode`6=`a \uccode`7=`c
 \uppercase{\gdef\macro@#1#2#3#4\macro@{\gtest@false
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\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@{}%
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  \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
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{\uccode`5=`\\ \uccode`6=`n \uccode`7=`o \uccode`8=`t
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\def\keybin@{\gtest@true
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\@ifundefined{@ldots}{\def\@ldots{\mathellipsis}}{}
\DeclareRobustCommand{\dots}{\relax
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\def\tdots@{\leavevmode\unskip\relaxnext@
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   \ifx\@let@token;\,$\else\ifx\@let@token:\,$\else
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\def\mdots@{\FN@\mdots@@}
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   \xdef\meaning@@{\meaning@}%
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 \thedots@}
\def\boldsymboldots@#1{\bold@true\let\@let@token=#1\let\delayed@=#1\mdots@@
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\def\@cdots{\mathinner{\cdotp\cdotp\cdotp}}
\def\dotsi{\!\@cdots}
\let\dotsb@\@cdots
\def\rightdelim@{\gtest@true
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 \ifx\@let@token\rceil\else
 \ifx\@let@token\rfloor\else
 \ifx\@let@token\rgroup\else
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\def\extra@{%
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 \@xp\macro@\meaning@\macro@\ifgtest@
 \@xp\DOTS@\meaning@\DOTS@
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\newif\ifbold@
\def\dotso@{\relaxnext@
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  \def\nextii@{\DN@{\extra@\@ldots\ifgtest@\,\fi}\FN@\next@}%
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 \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\iff{\DOTSB\;\Longleftrightarrow\;}
\def\doteq{\DOTSB\mathrel{\mathop{\kern\z@ =}\limits^{\textstyle.}}}
\newif\if@display
\everydisplay\@xp{\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@
}
\newbox\Mathstrutbox@
\setbox\Mathstrutbox@=\hbox{}
\def\Mathstrut@{\copy\Mathstrutbox@}
\begingroup \catcode`\"=12
\gdef\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@
}
\endgroup
\addto@hook\every@math@size{\resetMathstrut@}
\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{%
  {\@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@}
\begingroup \catcode`\"=12
\def\@tempa#1#2\@nil{\def\@tempc{#1}}\def\@tempb{\mathaccent}
\expandafter\@tempa\hat \relax\relax\@nil
\ifx\@tempb\@tempc
  \def\@tempa#1\@nil{#1}%
  \def\@tempb#1{\afterassignment\@tempa\mathchardef\@tempc=}%
  \def\@tempe#1"{}
  \def\do#1"#2{}
  \def\@tempd#1#2{\expandafter\@tempb#1\@nil
    \ifnum\@tempc<"1000
      \edef\@tempc{"\@nx\accentclass@
        \ifnum\@tempc<"100 0\fi
        \@xp\@tempe\meaning\@tempc\space}%
    \else
      \edef\@tempc{"\@nx\@nx\@nx\accentclass@
        \@xp\do\meaning\@tempc\space}%
    \fi
    \xdef#1{\mathaccent\@tempc}%
    \toks@{%
      \relax\ifmmode \else\DN@##1##2{\nonmatherr@{#2}}\@xp\next@\fi
      \mathaccent@}%
    \xdef#2{\the\toks@{\@tempc}}%
  }
  \@tempd\hat\Hat \@tempd\check\Check \@tempd\tilde\Tilde
  \@tempd\acute\Acute \@tempd\grave\Grave \@tempd\dot\Dot
  \@tempd\ddot\Ddot \@tempd\breve\Breve \@tempd\bar\Bar
\fi
\endgroup
\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
 \@xp\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}
\def\nonmatherr@#1{\@amsmath@err{\protect
  #1 allowed only in math mode}\@ehd}
\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
\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\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\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{%
  \@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}[2][tb]{%
  \def\smash@{#1}%
  \ifmmode\@xp\mathpalette\@xp\mathsm@sh\else
        \@xp\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@}}
\newcommand{\xrightarrow}[2][]{%
  \mathrel{\mathop{%
    \setbox\z@\vbox{\m@th
      \hbox{$\scriptstyle\;{#1}\;\;$}%
      \hbox{$\m@th\scriptstyle\;{#2}\;\;$}%
    }%
    \hbox to\ifdim\wd\z@>\minaw@\wd\z@\else\minaw@\fi{%
      \rightarrowfill@\displaystyle}}%
  \limits^{#2}\@ifnotempty{#1}{_{#1}}}%
}
\newcommand{\xleftarrow}[2][]{%
  \mathrel{\mathop{%
    \setbox\z@\vbox{\m@th
      \hbox{$\scriptstyle\;\;{#1}\;$}%
      \hbox{$\m@th\scriptstyle\;\;{#2}\;\;$}%
    }%
    \hbox to\ifdim\wd\z@>\minaw@\wd\z@\else\minaw@\fi{%
      \leftarrowfill@\displaystyle}}%
  \limits^{#2}\@ifnotempty{#1}{_{#1}}}%
}
\@ifundefined{minaw@}{\newdimen\minaw@\minaw@11pt}{}
\newcommand{\Sb}{\PackageError{amsmath}%
  {Environment `Sb' is obsolete; use `substack' instead}%
  {The \protect\\protect\ used to separate lines in a `Sb' environment can
   cause problems if `Sb' is embedded in some aligning environments.}}
\newcommand{\Sp}{\PackageError{amsmath}%
  {Environment `Sp' is obsolete; use `substack' instead}%
  {The \protect\\protect\ used to separate lines in a `Sp' environment can
   cause problems if `Sp' is embedded in some aligning environments.}}
\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\,%
}
\newcount\c@MaxMatrixCols \c@MaxMatrixCols=10
\renewenvironment{matrix}{%
  \hskip -\arraycolsep\array{*\c@MaxMatrixCols c}%
}{%
  \endarray \hskip -\arraycolsep
}
\renewenvironment{pmatrix}{\left(\matrix}{\endmatrix\right)}
\newenvironment{bmatrix}{\left[\matrix}{\endmatrix\right]}
\newenvironment{Bmatrix}{\left\lbrace\matrix}{\endmatrix\right\rbrace}
\newenvironment{vmatrix}{\left\lvert\matrix}{\endmatrix\right\rvert}
\newenvironment{Vmatrix}{\left\lVert\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}{%
  \left\{\def\arraystretch{1.2}%
  \array{@{}l@{\quad}l@{}}%
}{%
  \endarray\right.%
}
\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\numberwithin#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}{%
  \@ifundefined{c@#2}{\@nocnterr{#2}}{%
  \@addtoreset{#1}{#2}%
  \toks@\@xp\@xp\@xp{\csname the#1\endcsname}%
  \@xp\xdef\csname the#1\endcsname
    {\@xp\@nx\csname the#2\endcsname
     .\the\toks@}}}}
\def\eqref#1{\textup{\tagform@{\ref{#1}}}}
\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{\@amsmath@err{\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
  \@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{\@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
\def\thetag{\leavevmode\tagform@}
\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
}

\let\df@label\@empty
\def\make@display@tag{%
    \if@eqnsw
        \refstepcounter{equation}%
        \tagform@\theequation
    \else
        \iftag@
            \df@tag
            \global\let\df@tag\@empty
        \fi
    \fi
    \ifx\df@label\@empty\else
        \ltx@label{\df@label}%
        \global\let\df@label\@empty
    \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@
}
\def\raisetag#1{\skip@#1\relax
  \xdef\raise@tag{\vskip\iftagsleft@\else-\fi\the\skip@\relax}%
}
\let\raise@tag\@empty
\def\notag{\nonumber}
\newif\ifinany@
\newif\ifinalign@
\newif\ifingather@
\newif\iftag@
\newif\ifst@rred
\newif\ifmeasuring@
\newif\ifshifttag@
\newcount\row@
\newcount\column@
\def\column@plus{%
    \global\advance\column@\@ne
}
\newcount\maxfields@
\def\add@amps#1{%
    \begingroup
        \count@#1
        \DN@{}%
        \loop
            \ifnum\count@>\column@
                \edef\next@{&\next@}%
                \advance\count@\m@ne
        \repeat
    \@xp\endgroup
    \next@
}
\newhelp\andhelp@
{An extra & here is so disastrous that you should probably exit^^J
and fix things up.}
\newdimen\eqnshift@
\newdimen\alignsep@
\newdimen\tagshift@
\def\mintagsep{.5\fontdimen6\textfont2}
\def\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\displ@y{\@display@init{}}
\def\@display@init#1{%
    \global\dt@ptrue
    \openup\jot\m@th
    \everycr{%
        \noalign{%
            #1%
            \ifdt@p
                \global\dt@pfalse
                \vskip-\lineskiplimit
                \vskip\normallineskiplimit
            \else
                \penalty\@eqpen
            \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\@tempa{%
            \cl@@ckpt
            \let\@nx\restorecounters@\@nx\@empty
        }%
    \endgroup
    \let\restorecounters@\@tempa
}
\let\restorecounters@\@empty
\def\savealignstate@{%
    \begingroup
        \let\or\relax
        \xdef\@tempa{%
            \global\totwidth@\the\totwidth@
            \global\row@\the\row@
            \gdef\@nx\tag@lengths{\tag@lengths}%
            \let\@nx\restorealignstate@\@nx\@empty
        }%
    \endgroup
    \let\restorealignstate@\@tempa
}

\let\restorealignstate@\@empty
\newtoks\@envbody
\def\addto@envbody#1{\@envbody\@xp{\the\@envbody#1}}
\def\collect@body#1{%
    \@envbody{}%
    \def\process@envbody{%
        \@xp#1\@xp{\the\@envbody}%
    }%
    \@xp\let\csname\@currenvir\endcsname\collect@@body
    \csname\@currenvir\endcsname
}
\def\collect@@body#1\end#2{%
    \def\@tempa{#2}%
    \ifx\@tempa\@currenvir
        \addto@envbody{#1}%
        \@xp\edef\csname\@currenvir\endcsname{%
            \@nx\process@envbody\@nx\end{\@tempa}%
            }%
    \else
        \addto@envbody{#1\end{#2}}%
    \fi
    \csname\@currenvir\endcsname
}
\newcommand{\start@aligned}[2]{%
    \RIfM@\else
        \nonmatherr@{\begin{\@currenvir}}%
    \fi
    \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
        \else
            \restore@math@cr
        \fi
        \Let@
        \default@tag
        \ifinany@\else\openup\jot\fi
        \column@\z@
        \ialign\bgroup
           &\column@plus
            \hfil
            \strut@
            $\m@th\displaystyle{##}$%
           &\column@plus
            $\m@th\displaystyle{{}##}$%
            \hfil
            \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
    \column@\z@
    \cr
}
\newenvironment{aligned}[1][c]{%
    \start@aligned{#1}\m@ne
}{%
    \crcr\egroup\egroup
}
\newcommand{\alignedat}[2][c]{%
        \start@aligned{#1}%
}
\let\endalignedat\endaligned
\newcommand{\gathered}[1][c]{%
    \RIfM@\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
}
\let\endgathered\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@
}
\def\gather{\start@gather\st@rredfalse}

\@namedef{gather*}{\start@gather\st@rredtrue}
\def\gather@#1{%
    \ingather@true
    \inany@true
    \let\tag\tag@in@align
    \let\label\label@in@display
    \displaybreak@
    \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\endgather{%
        \math@cr
        \black@\totwidth@
    \egroup
    $$%
    \global\@ignoretrue
}

\@xp\let\csname endgather*\endcsname\endgather
\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@
        {\ifnum0=`}\fi
        \DN@{\vcenter\bgroup\savealignstate@\align@#2}%
    \else
        \ifmmode
            \nomath@env
            \DN@{\@namedef{end\@currenvir}{}\@gobble}%
        \else
            $$%
            \DN@{\align@#2}%
        \fi
    \fi
    \collect@body\next@
}
\def\alignat{\start@align\z@\st@rredfalse}
\@namedef{alignat*}{\start@align\z@\st@rredtrue}
\def\xalignat{\start@align\@ne\st@rredfalse}
\@namedef{xalignat*}{\start@align\@ne\st@rredtrue}
\def\xxalignat{\start@align\tw@\st@rredtrue}
\def\align{\start@align\@ne\st@rredfalse\m@ne}
\@namedef{align*}{\start@align\@ne\st@rredtrue\m@ne}
\def\flalign{\start@align\tw@\st@rredfalse\m@ne}
\@namedef{flalign*}{\start@align\tw@\st@rredtrue\m@ne}
\def\align@#1#2{%
    \inany@true
    \inalign@true
    \displaybreak@
    \intertext@
    \ifingather@\else\displ@y@\fi
    \Let@
    \let\math@cr@@@\math@cr@@@align
    \ifxxat@\else
        \let\tag\tag@in@align
    \fi
    \let\label\label@in@display
    #1% set st@r
    \ifst@rred\else
        \global\@eqnswtrue
    \fi
    \measure@{#2}%
    \global\row@\z@
    \tabskip\eqnshift@
    \halign\bgroup
        \span\align@preamble\crcr
        #2%
}
\def\endalign{%
        \math@cr
        \black@\totwidth@
    \egroup
    \ifingather@
        \restorealignstate@
        \egroup
        \nonumber
        \ifnum0=`{\fi}%
    \else
        $$%
    \fi
    \global\@ignoretrue
}
\@xp\let\csname endalign*\endcsname\endalign
\let\endxalignat\endalign
\@xp\let\csname endxalignat*\endcsname\endalign
\let\endxxalignat\endalign
\let\endalignat\endalign
\@xp\let\csname endalignat*\endcsname\endalign
\let\endflalign\endalign
\@xp\let\csname endflalign*\endcsname\endalign
\def\math@cr@@@align{%
    \kern-\alignsep@
    \ifst@rred\nonumber\fi
    \if@eqnsw \global\tag@true \fi
    \global\advance\row@\@ne
    \iftag@
        \add@amps\maxfields@
        \omit
        \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
        \eqnshift@\z@
        \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@@@
                \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
        \setbox0\vbox{%
            \unvbox0
            \unpenalty
            \global\setbox1\lastbox
        }%
        \global\totwidth@\wd1
        \if@fleqn
           \global\advance\totwidth@\@mathmargin
        \fi
        \global\let\maxcolumn@widths\@empty
        \begingroup
            \let\or\relax
            \loop
               \setbox1\hbox{%
                   \unhbox1
                   \unskip
                   \global\setbox0\lastbox
               }%
                \ifhbox0
                    \xdef\maxcolumn@widths{ \or \the\wd0 \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
                \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
                \advance\alignsep@\@mathmargin\relax
            \fi
            \global\divide\alignsep@\@tempcntb
        \fi
        \ifdim\alignsep@<\minalignsep\relax
            \global\alignsep@\minalignsep\relax
            \ifdim\eqnshift@>\z@
                \if@fleqn\else
                    \eqnshift@\displaywidth
                    \advance\eqnshift@-\totwidth@
                    \advance\eqnshift@-\@tempcntb\alignsep@
                    \global\divide\eqnshift@\tw@
                \fi
            \fi
        \fi
        \ifdim\eqnshift@<\z@
            \global\eqnshift@\z@
        \fi
        \calc@shift@align
        \tagshift@\totwidth@
        \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
        \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
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\begin{document}
\title[Potentials with Stationary Increments]{Random Schr\"{o}dinger Operators with Potentials Having Stationary Increments}
\author{Werner Kirsch}
\address{Werner Kirsch\\
Institut f\"{u}r Mathematik and Sonderforschungsbereich 237\\
Ruhr-Universit\"{a}t Bochum\\
D-44780 Bochum, Germany}
\email{werner.kirsch@mathphys.ruhr-uni-bochum.de}
\author{Stanislav Molchanov}
\address{Stanislav Molchanov\\
University of North Carolina\\
Charlotte NC 28223, USA}
\email{smolchan@math.uncc.edu}
\thanks{.}
\subjclass{Primary 05C38, 15A15; Secondary 05A15, 15A18}
\keywords{Anderson localization, density of states, potentials with stationary increments}
\maketitle
\begin{abstract}We study one dimensional Schr\"{o}dinger operators $H_{\omega}=-\frac{d^{2}%
}{dx^{2}}+V_{\omega}$ where $V_{\omega}$ is a stochastic process with
stationary, independent increments.
\end{abstract}


\section{Introduction}

In this paper we study spectral properties of one dimensional Schr\"{o}%
\-dinger operators
\begin{equation}
H_{\omega}=-\frac{d^{2}}{dx^{2}}+V_{\omega} \label{gl1}%
\end{equation}
on $L^{2}([0,\infty])$ (resp. $L^{2}(\mathbb{R})$) and its analog on
$l^{2}(\mathbb{Z})$. The potentials $V_{\omega}$ we consider are stochastic
processes with stationary and independent increments. Thus Brownian motion is
a typical example for the continuous case (i.e. $L^{2}$) Our potentials differ
from the models commonly considered so far in that they are not stationary
(ergodic). Consequently the general theory of ergodic potentials like non
randomness of the spectra, existence of the density of states, positivity of
Lyapunov exponents etc. is not applicable.

We are led to consider such random potentials by various physical motivations.
Our first one is the Burgers equation with a random stationary force term
$f(x,{\omega})$, i.e.
\begin{equation}
\frac{\partial u}{\partial t}+uu_{x}=\eta u_{xx}+f(x,{\omega}) \label{gl2}%
\end{equation}
The standard Cole-Hopf substitution $u=-2\eta(\ln\varphi)^{\prime}$ reduces
Burgers's equation (\ref{gl2}) to the (linear) heat equation
\begin{equation}
\frac{\partial\varphi}{\partial t}=\eta\frac{\partial^{2}\varphi}{\partial
x^{2}}+\frac1\eta V_{\omega}(x)\varphi\label{gl3}%
\end{equation}
where
\begin{equation}
V_{\omega}(x)=\int_{0}^{x}f(y,{\omega})\,dy \label{gl4}%
\end{equation}
If $f(x,{\omega})$ is \textit{stationary} the random potential $V_{\,{\omega}%
}$ (\ref{gl4}) in equation (\ref{gl3}) will have \textit{stationary
increments}. The solution $\varphi(t)=e^{-tH_{\omega}}\varphi_{0}$ of
(\ref{gl3}) is consequently intimately connected with the spectral properties
of the operator (\ref{gl1}). For recent papers on Burgers equation we refer to
\cite{burgers1}, \cite{burger2}, \cite{burgers3}, \cite{burgers4} and
references therein.

Another motivation to study potentials with stationary increments comes from
quantum mechanics itself. If we consider an electric field $E_{\omega}(x)$
which is stationary then - as above - the corresponding random potential
$V_{\omega}(x)$ will have stationary increments.

In the following we will describe a few basic examples of potentials with
stationary increments

\begin{itemize}
\item [-]\textbf{Model I:} We start with the discrete case, i.e. the Hilbert
space $l^{2}(\mathbb{Z}_{+})$. The free Hamiltonian is defined as usual by
\begin{equation}
h_{0}\ u(n)=-u(n+1)-u(n-1) \label{gl5}%
\end{equation}
(for $n>0$ and $h_{0}\ u(0)=-u(1)$).

Let $\xi_{n},\ n\in\mathbb{Z}^{+}$ be a sequence of independent, identically
distributed random variables. We set
\begin{equation}
V_{\omega}(n)=\sum_{j=0}^{n}\xi_{j} \label{gl6}%
\end{equation}
This model can be considered on $l^{2}(\mathbb{Z})$ instead if we take
$\xi_{n},n<0$ independent from each other and from the $\xi_{n},n\geq0$ and
with the same distribution as $\xi_{0}$. For $n<0$ we set $V_{\omega}%
(n)=\sum_{j=-1}^{n}\xi_{\smallskip j}$.

\item[-] \textbf{Model II:} Next we describe an analogous model on
$L^{2}(\mathbb{R}_{+})$ by setting
\begin{equation}
V_{\omega}(x)=\sum_{j=0}^{[x]}\xi_{j} \label{gl7}%
\end{equation}
where again the $\xi_{n}$ are independent, identically distributed random
variables and $[x]$ is the largest integer not exceeding $x$. This process has
independent, stationary increments in the sense that $V_{\omega}%
(x+n+1)-V_{\omega}(x+n)$ are independent identically distributed random
variables for $n\in\mathbb{Z}_{+}$.

In what follows we will always assume $E(\xi_{0})=0$ Moreover we suppose that
$E(\xi_{0}^{2})<\infty$ , without loss we set $E(\xi_{0}^{2})=1\smallskip$.

\item[-] \textbf{Model III:} Perhaps the prototype of a process with
stationary increments is the Brownian motion $\beta_{s},s\geq0$, so we
consider
\begin{equation}
V_{\omega}(x)=\beta_{x}({\omega}) \label{gl8}%
\end{equation}
with e.g. $\beta_{0}=0$. This model (like Model II) can be extended to all of
$\mathbb{R}$ by adding an independent copy of the process $V_{\omega}$ to the
left of the origin, i.e. by setting
\[
V_{\omega}(x)=\left\{
\begin{array}
[c]{ll}%
\beta_{x}({\omega}) & \text{for }x\geq0\\
\widetilde{\beta}_{-x}({\omega}) & \text{for }x\leq0
\end{array}
\right.
\]
where $\widetilde{\beta}$ is another Brownian motion with $\widetilde{\beta
}_{0}=0$ independent of $\beta$. An analogous method works for model II.\smallskip
\end{itemize}

In the following, we consider the above models on the halfline $\mathbb{R}%
_{+}$ (resp. $\mathbb{Z}_{+}$) with Dirichlet boundary conditions at $0$
unless otherwise stated. Nevertheless, analogous results can be proven for the
line in very much the same way.

We remark that for model II
\[
\left|  V_{\omega}(x)\right|  \leq\sum_{i=0}^{[x]}\xi_{i}\leq\sqrt{x\sum
_{i=0}^{[x]}\xi_{i}^{2}}\leq cx\sqrt{\frac{1}{[x]+1}\sum_{i=0}^{[x]}\xi
_{i}^{2}}%
\]
Since $\frac{1}{[x]+1}\sum_{i=0}^{[x]}\xi_{i}^{2}\rightarrow E(\xi_{0}^{2})$
a.s. by the ergodic theorem, we have therefore $\left|  V_{\omega}(x)\right|
\leq C_{0}+C_{1}\left|  x\right|  $ almost surely. It follows that the
operator $H_{\omega}=H_{0}+V_{\omega}$ is essentially self adjoint (see
e.g.\cite{RSII}). The law of iterated logarithm (see e.g. \cite{KarS} or
\cite{S}) allows a similar conclusion for model III (see also the discussion
in section 3).

All the above models have the property that $E\left(  V_{\omega}(x)\right)
=0$ for all $x$. This is a serious restriction, since $E\left(  V_{\omega
}(x)\right)  \neq0$ is equivalent to adding a homogeneous electric field to
the potential. It is known that this additional term can change pure point
spectrum to absolutely continous spectrum in the stationary case
\cite{SixAuthors}. We will consider the more complicated case $E\left(
V_{\omega}(x)\right)  \neq0$ in a subsequent publication \cite{KiMo}.

Part of this work was done when S.M. was visiting Ruhr-Universit\"{a}t Bochum
and the SFB 237 and when W.K. was visiting University of North Carolina in
Charlotte. We gratefully acknowledge the hospitality and financial support of
these institutions.

\section{The integrated density of states}

The integrated density of states is an important characteristic of quantum
mechanical disordered systems (for a review from the mathematical point of
view see \cite{KCourse}). For \textit{stationary} potentials (in the one
dimensional case) it is defined by the following procedure:

Let $H_{L}$ denote the random Hamiltonian $H_{\omega}$ restricted to the
Hilbert Space $L^{2}([0,L])$ (resp. $L^{2}(\{0,\ldots,L-1\})$) with
appropriate (say Dirichlet) boundary conditions. By $E_{n}(H_{L})$ we denote
the $n$-th eigenvalue of $H_{L}$ (arranged in increasing order counting
multiplicity). We set
\begin{equation}
N_{L}(E)=\#\left\{  n\left|  E_{n}(H_{L})\leq E\right.  \right\}  \label{gl31}%
\end{equation}
Then (for a huge class of stationary potentials)
\begin{equation}
\frac1LN_{L}(E)\rightarrow N(E). \label{gl32}%
\end{equation}
The limit $N(E)$ is a nonrandom function which we call the integrated density
of state. This convergence is a.s. convergence at each continuity point $E$ of
$N$, i.e. vague convergence of the corresponding Borel measures.

Let us try to do a similar procedure for potentials with independent
increments. We consider the discrete case (model I) first. We may assume that
$E(\xi_{j})=0$ and $E(\xi_{j}^{2})=1$. Since $h_{0}$ is a bounded operator, in
fact $\left\|  h_{0}\right\|  =2$, we have $E_{n}(H_{L})\leq E_{n}(V_{\omega
})+2$ and $E_{n}(H_{L})\geq E_{n}(V_{\omega})-2$, where $E_{n}(V_{\omega})$
are the eigenvalues of the multiplication operator $V_{\omega}$. Because
$V_{\omega}(k)=\sum_{j=0}^{k}\xi_{j}=:S_{k}$ the eigenvalues $E_{n}(V_{\omega
})$ are just the values of $S_{k}$, properly ordered. Thus
\begin{align}
\frac{1}{L+1}N_{L}(E)  &  \leq\frac{1}{L+1}\#\left\{  E_{n}(V_{\omega})\left|
E_{n}(V_{\omega})\leq E+2\right.  \right\} \nonumber\\
&  \leq\frac{1}{L+1}\#\left\{  S_{k}\left|  S_{k}\leq E+2\right.
,k=0,\ldots,L\right\} \label{NLAbsch1}\\
&  =\frac{1}{L+1}\#\left\{  S_{k}\left|  \frac{1}{\sqrt{L}}S_{k}\leq\frac
{E+2}{\sqrt{L}}\right.  ,k=0,\ldots,L\right\} \nonumber
\end{align}
In the same way we prove:
\begin{equation}
\frac{1}{L+1}N_{L}(E)\geq\frac{1}{L+1}\#\left\{  S_{k}\left|  \frac{1}%
{\sqrt{L}}S_{k}\leq\frac{E-2}{\sqrt{L}}\right.  ,k=0,\ldots,L\right\}
\label{NLAbsch2}%
\end{equation}
Since $\frac{1}{\sqrt{L}}S_{k}$ converges in law to a standard Brownian motion
$\beta_{S\text{ }}$ by Donsker's theorem (see e.g.\cite{KarS}, \cite{S}) we
have that
\begin{equation}
\frac{1}{L+1}\#\left\{  S_{k}\left|  \frac{1}{\sqrt{L}}S_{k}\leq a\right.
,k=0,\ldots,L\right\}  \overset{\mathcal{L}}{\Rightarrow}\int_{0}^{1}%
\chi_{_{(-\infty,a]}}(\beta_{s})\ ds \label{gl34}%
\end{equation}
Consequently
\begin{equation}
\frac{1}{L+1}N_{L}(E)\Rightarrow\int_{0}^{1}\chi_{_{(-\infty,0]}}(\beta
_{s})\ ds \label{gl35}%
\end{equation}
Equation (\ref{gl35}) has a number of ''strange'' aspects. First of all the
limit of $\frac{1}{L+1}N_{L}(E)$ is \textit{not} independent of the random
parameters. This is to be expected because the random field $V_{\omega}$ is
not ergodic. But what is surprising, indeed, is the fact that the limit is
\textit{independent} of the energy, which means that this quantity is
physically meaningless. In mathematical terms it means that the scaling and/or
the normalization chosen is incorrect! In fact, an inspection of formulas
(\ref{NLAbsch1} and \ref{NLAbsch2}) above leads us to define
\begin{equation}
M_{L}(E)=\#\left\{  E_{n}(H_{L})\leq\sqrt{L}E\right\}  =N_{L}(\sqrt{L}E)
\label{gl36}%
\end{equation}
since the expected value of the potential $\left|  V_{\omega}(L)\right|  $ is
of the order $\sqrt{L}.$The reasoning above leads us to the following result:

\begin{theorem}
If $V_{\omega}(n)=\sum_{j=0}^{n}\xi_{j}$,$\ \xi_{i}$ being independent,
identically distributed with $E(\xi_{i})=0,E(\xi_{i}^{2})=1$ and if $H_{L}$ is
the restriction of $H_{\omega}=H_{0}+V_{\omega}$ on $l^{2}(\mathbb{Z)}$ to the
interval$\left\{  0,1,\ldots,L\right\}  $ then
\[
\frac{1}{L+1}M_{L}(E)\overset{\mathcal{L}}{\Rightarrow}\int_{0}^{1}%
\chi_{(-\infty,E)}(\beta_{s})ds
\]
where $\beta_{s}$ is a standard Brownian motion.
\end{theorem}

Let us now look at the continuous case, more precisely at model II.
Introducing Dirichlet boundary conditions at the integers $n$ changes the
operator $H_{\omega}$ on $\left[  0,L\right]  $ to the direct sum of the
operators $H_{\omega}$ on $\left[  n,n+1\right]  $ (with Dirichlet boundary
conditions at each endpoint)$.$ Let us denote the latter operator by
$H_{\omega}^{(n)}.$ Since the potential is constant between $n$ and $n+1$ the
eigenvalues of $H_{\omega}^{(n)}$ are given by $m^{2}+S_{n},$ with $S_{n}%
=\sum_{i=0}^{n}\xi_{i}.$ Since introducing boundary conditions changes the
resolvent by a finite rank operator the quantity $N_{L}(E)$ (or $M_{L}(E)$)
are changed by at most $O(L)$. Let us estimate the expectation $N_{L}(E)$ for
$E=0$. Our somewhat formal manipulations will be justified below:
\begin{align}
\mathbf{E}\left(  N_{L}(0)\right)   &  =\sum_{n=0}^{L-1}\ \mathbf{E}\left(
\#\left\{  m\,|\ m^{2}+S_{n}<0\right\}  \right)  +O(L)\label{Ncont}\\
&  =\sum_{n=0}^{L-1}\ \mathbf{E}\left(  \#\left\{  m\,|\ m<\sqrt{\left(
S_{n}\right)  _{+}}\right\}  \right)  +O(L)\\
&  =\sum_{n=0}^{L-1}\mathbf{E}\left(  \sqrt{\left(  S_{n}\right)  _{+}%
}\right)  +O(L)\\
&  =\sum_{n=0}^{L-1}\ n^{\frac14}\ \mathbf{E}\left(  \sqrt{\left(
\frac1{\sqrt{n}}S_{n}\right)  _{+}}\right)  +O(L)\\
&  \approx O\left(  L^{\frac54}\right)
\end{align}

From this considerations we learn that $\frac1LN_{L}$ or $\frac1LM_{L}$ have
no chance to converge. It is rather $\frac1{L^{\frac54}}M_{L}$ that is
suggested by our computation. In fact, we have

\begin{theorem}
If $V_{\omega}(x)=\sum_{j=0}^{n}\xi_{j}$,$\ \xi_{i}$ being independent,
identically distributed with $E(\xi_{i})=0,E(\xi_{i}^{2})=1$ and if $H_{L}$ is
the restriction of $H_{\omega}=H_{0}+V_{\omega}$ on $L^{2}(\mathbb{R)}$ to the
interval $[0,L]$ then
\begin{equation}
\frac{1}{L^{\frac{5}{4}}}M_{L}(E)\overset{\mathcal{L}}{\Rightarrow}\int
_{0}^{1}\sqrt{\left(  \beta_{s}-E\right)  _{+}}ds \label{KonMc}%
\end{equation}
where $\beta_{s}$ is a standard Brownian motion.
\end{theorem}

\begin{proof}
As in the above consideration we obtain
\begin{eqnarray}
\frac1{L^{\frac54}}M_L(E) &=&\frac1{L^{\frac54}}\,\sum_{n=0}^{L-1}\sqrt{%

\left( S_n-\sqrt{L}E\right) _{+}}+o(1)  \notag\\
&=&\frac1L\,\sum_{n=0}^{L-1}\,\,\,\sqrt{\left( \frac1{\sqrt{L}%

}S_n-E\right) _{+}}+o(1)  \label{RiemSum}
\end{eqnarray}
By Donsker's theorem (see e.g. \cite{KarS} or \cite{S}) the stochastic
process $X_s$ on $[0,1]$ defined by $X_{\frac nL}=\frac1{\sqrt{L}}S_n$ and
by linear interpolation in between converges weakly to standard Brownian
motion. The sum in \ref{RiemSum} is actually a Riemann sum for $\int_0^1\,%

\sqrt{\left( X_s-E\right) _{+}}\,ds$ . Consequently the whole term converges
weakly to $\int_0^1\sqrt{\left( \beta_s-E\right) _{+}}ds$.
\end{proof}

\noindent We have an analog for the above result for the Brownian motion
potential (Model~III):

\begin{theorem}
If $V_{\omega}(x)=\beta_{x}$ is the Brownian motion potential then
\begin{equation}
\frac{1}{L^{\frac{5}{4}}}M_{L}(E)\overset{\mathcal{L}}{\Rightarrow}\int
_{0}^{1}\sqrt{\left(  \beta_{s}-E\right)  _{+}}\,\,ds
\end{equation}
\end{theorem}

\begin{proof}
The proof is similar to the above case. Defining $$Y_n=\sup_{0\leq s\leq
1}\,\left| \beta_s-\beta_n\right| $$ we get as above:
\begin{equation*}
\frac1{L^{\frac54}}M_L(E)=\,\sum_{n=0}^{L-1}\,\,\frac1L\,\,\sqrt{\left(
\frac1{\sqrt{L}}\beta_n-E\pm\frac{Y_n}{\sqrt{L}}\right) _{+}}+o(1)
\end{equation*}
Since $\frac{Y_n}{\sqrt{L}}\rightarrow0$ almost surely, the above sum
converges to $$\int_0^1\sqrt{\left( \beta_s-E\right) _{+}}\,ds.$$
\end{proof}

\section{Localization}

\noindent In this section we prove Anderson localization for the model I-III.
Since the proofs for model I and II are similar to the one for model III but
more standard we give the details of the proof only in the latter case. We
also restrict ourselves to the case of the half line, the case of
$L^{2}\left(  \mathbf{R}\right)  $ being similar.

\begin{theorem}
a) Model I: If the distribution of the random variable $\xi_{0}$ in $\left(
\ref{gl6}\right)  $ is absolutely continuous (w.r.t. Lebesgue measure) with a
bounded density, then the operator
\[
h_{\omega}=h_{0}+\sum_{j=0}^{n}\xi_{j}\;\quad\text{ on }l^{2}\left(
\mathbf{N}\right)
\]
has pure point spectrum. The same holds for $h_{\omega}$ on $l^{2}\left(
\mathbf{Z}\right)  .$\newline b) Model II: If the distribution of the random
variable $\xi_{0}$ in $\left(  \ref{gl7}\right)  $ is absolutely continuous
with a bounded density, then the operator
\[
H_{\omega}=H_{0}+\sum_{j=0}^{\left[  x\right]  }\xi_{j}\quad\quad\text{on
}L^{2}\left(  \mathbf{R}_{+}\right)
\]
has pure point spectrum. The same holds for $H_{\omega}$ on $L^{2}\left(
\mathbf{R}\right)  .$\newline c) Model III: The operator $H_{\omega}%
=H_{0}+\beta_{x}$ with a Brownian motion $\beta$ on $L^{2}\left(
\mathbf{R}_{+}\right)  $ has pure point spectrum. The same is true for
$H_{\omega}$ on $L^{2}\left(  \mathbf{R}\right)  $ if $\beta_{x}$ for $x\leq0$
is a Brownian motion independent of the Brownian motion $\beta_{x}$ on
$\mathbf{R}_{+}.$
\end{theorem}

\noindent As mentioned above we give the proof for model III on $\mathbf{R}%
_{+}$. The proof consists of two steps. In the first step we prove that the
resolvent is square integrable in the sense
\begin{equation}
\underset{\varepsilon>0}{sup}\int\left|  R_{E+i\varepsilon}\left(  a,x\right)
\right|  \text{ }dx<\infty\label{loc1}%
\end{equation}
for any $a\in\mathbf{R}_{+}$ where $R_{E+i\varepsilon}\left(  x,y\right)  $ is
the kernel of the resolvent of $H_{\omega}.$ To prove this we employ the
paper\cite{KMP}. We will give details below. To deduce localization from
$\left(  \ref{loc1}\right)  $ we need a procedure of Simon-Wolff-type
\cite{SW}. For this we refer to the paper of Kotani and Simon\cite{KS}. They
prove localization from $\left(  \ref{loc1}\right)  $ if the Pr\"{u}fer angle
$\phi$ at a point $a>0$ has a conditional distribution with a bounded density.
To prove this we use the technique Kotani-Simon use in example 2 of section 3
of their work\cite{KS}. We refer to this paper for details. Let us now prove
$\left(  \ref{loc1}\right)  $. In\cite{KMP} the following theorem is proved:

\begin{theorem}
\label{KMP-Thm}Suppose the potential $V$ satisfies,
\begin{equation}
V\left(  x\right)  \geq-c\left(  1+x\right)  ^{-\alpha} \label{loc1.2}%
\end{equation}
for some $\alpha<2$ and denote by $R_{z}$ the resolvent kernel of the operator
$H=H_{0}+V$ on $L^{2}\left(  \mathbf{R}_{+}\right)  $ (with some boundary
condition at 0) for the energy $z\in\mathbf{C}$. Assume there are sequences
$V_{n},x_{n}\rightarrow\infty$ and $h_{n}>0$ with $\sqrt{V_{n}}h_{n}%
\rightarrow\infty$ such that:
\[
V\left(  x\right)  \geq V_{n}\quad\quad\text{for }x\in\left[  x_{n}%
-h_{n},x_{n}+h_{n}\right]
\]
If
\[
\sum x_{n+1}^{\frac{3+\alpha}{2}}e^{-\gamma\sqrt{V_{n}}h_{n}}<\infty
\,\quad\quad\text{for a }\gamma<1
\]
then
\[
\underset{\varepsilon>0}{sup}\int_{a}^{\infty}\left|  R_{e+i\varepsilon
}\left(  a,x\right)  \right|  ^{2}dx<\infty
\]
\end{theorem}

\begin{proof}
The theorem is essentially proved in\cite{KMP}. The only difference being
that\cite{KMP} assume $V\left( x\right) \geq0,$ i.e. $\alpha=0.$ In our
more general case we have to change the estimate of
\begin{equation*}
N_L:=\#\left\{ \text{eigenvalues of }H_L\leq E\right\}
\end{equation*}
in the lemma of section 5 in \cite{KMP}. We may estimate
\begin{equation*}
N_L\leq C\int_{x_n}^{x_{n+1}}\left( 1+V_{_{-}}\left( x\right) \right) \text{
}dx
\end{equation*}
$V_{-}$ is the negative part of $V.$ By \ref{loc1.2} we obtain
\begin{equation*}
N_L\leq C_2x_{n+1}^\alpha\left( x_{n+1}-x_n\right) \leq
C_2x_{n+1}^{1+\alpha}
\end{equation*}
Inserting this bound in \cite{KMP} we obtain the desired result.
\end{proof}

\noindent Now we prove that the Brownian motion potential $V\left(  x\right)
=\beta_{x}$ satisfies the assumptions of \ref{KMP-Thm}. \newline First, it is
an immediate consequence of the law of iterated logarithm (see e.g
\cite{KarS}or\cite{S}) that with probability one:
\begin{equation}
\left|  V\left(  x\right)  \right|  =\left|  \beta_{x}\right|  \leq\left|
x\right|  ^{\frac12+\varepsilon} \label{Wurzelverhalten}%
\end{equation}
for any $\varepsilon>0$ and all $x$ large enough. Hence $\left(
\ref{loc1.2}\right)  $ is satisfied for $\alpha=\frac12+\varepsilon$. Moreover
we prove:

\begin{proposition}
With probability one there is a sequence $x_{n}\uparrow\infty$ such that
\[
\beta_{y}\geq C\sqrt{x_{k}}\quad\quad\text{for }x_{k}\leq y\leq x_{k}+1
\]
\end{proposition}

\begin{remark}
Assertion and proof of the about proposition are very much in the spirit of
the law of iterated logarithm.
\end{remark}

\begin{proof}
The set ${\Omega}_0=\{\beta|$ $\beta_x\leq\left| x\right| ^{\frac23}$
for $x$ large$\}$ has probability one (see \ref{Wurzelverhalten}). Look at
the points $t_n=2^n$ and define
\begin{equation*}
A_m:=\left\{ \beta|\text{ }\beta_{2^{m+1}}-\beta_{2^m+1}\geq2^{\frac{m+1}%

2}\text{ and }\underset{0\leq s\leq1}{sup}\left| \beta_{2^{m+1}+s}-\beta
_{2^{m+1}}\right| \leq2^{\frac{m-1}2}\right\}
\end{equation*}
These events are independent and satisfy
\begin{eqnarray*}
P\left( A_m\right)  &=&P\left( \beta_{2^{m+1}}-\beta_{2^m+1}\geq2^{\frac{%

m+1}2}\right) P\left( \underset{0\leq s\leq1}{sup}\left| \beta_s-\beta
_0\right| \leq2^{\frac{m-1}2}\right)  \\
&\geq&\frac12P\left( \frac{\beta_{2^{m+1}}-\beta_{2^m+1}}{\sqrt{%

2^{m+1}-2^m-1}}\geq\frac{2^{\frac12}}{\sqrt{1-2^{-m}}}\right)  \\
&\geq&\frac12P\left( \beta_1-\beta_0\geq2\right) \geq C
\end{eqnarray*}
if m is large enough. Hence $\sum P\left( A_m\right) =\infty$, which
implies that $P$-almost surely there exists a sequence $n_k\uparrow\infty$
such that $\beta\in A_{n_k}$. Without loss we may assume that $%

n_{k+1}>2\,n_k$. Consequently with probability one:
\begin{eqnarray*}
\beta_{2^{n_{k+1}}} &\geq&\left( \beta_{2^{n_{k+1}}}-\beta
_{2^{n_k}+1}\right) +\beta_{2^{n_k}+1} \\
&\geq&2^{\frac{n_{k+1}}2}-\left( 2^{n_k}+1\right) ^{\frac23} \\
&\geq&2^{\frac{n_{k+1}}2}-2\cdot2^{\frac23n_k} \\
&=&\left( 1-2\cdot2^{\frac23n_k-\frac{_{n_{k+1}}}2}\right) 2^{\frac{n_{k+1}%

}2} \\
&\geq&\frac122^{\frac{_{n_{k+1}}}2}
\end{eqnarray*}
for large $k$. Moreover, for $0\leq s\leq1$%

\begin{eqnarray*}
\beta_{2^{n_{k+1}}+s} &\geq&\left( \beta_{2^{n_{k+1}}+s}-\beta
_{2^{n_{k+1}}}\right) +\left( \beta_{2^{n_{k+1}}}-\beta_{2^{n_k}+1}\right)
+\beta_{2^{n_k}+1} \\
&\geq&2^{\frac{n_{k+1}}2}-\left( 2^{n_k}+1\right) ^{\frac23}-2^{\frac{%

n_{k+1}-2}2} \\
&=&2^{\frac{n_{k+1}}2}\left( 1-2\cdot2^{\frac23n_k-\frac{n_{k+1}}2}-\frac
12\right)  \\
&\geq&\frac13\cdot2^{\frac{n_{k+1}}2}
\end{eqnarray*}
($k$ large enough). This proves the theorem.
\end{proof}

\begin{thebibliography}{99}
\bibitem{burger2}Albeverio S., Molchanov S., Surgailis D.: Stratified
structure of the Universe and Burgers' equation - a probabilistic approach,
Probab. Theory Relat. Fields 100, 457-484 (1994)

\bibitem {SixAuthors}Bentosela F., Carmona R., Duclos P., Simon B., Souillard
B., Weder R.: Schr\"{o}dinger operators with electric field and random or
deterministic potential, Commun. Math. Phys. 88, 387-397 (1983)

\bibitem {burgers4}Bulinskij A., Molchanov S.: Asymptotic normality of a
solution of Burgers Equation with Random Initial Data, Theory Probab. Appl 36,
217-236 (1991)

\bibitem {KarS}Karatzas I., Shreve S.E.: Brownian Motion and Stochastic
Calculus, Springer 1994

\bibitem {KCourse}Kirsch W.: Random Schr\"{o}dinger Operators --- A Course,
in: Holden H., Jensen A., Lecture Notes inPhysics 345, Springer 1989, p. 264-370

\bibitem {burgers3}Kirsch W., Kutzelnigg A.: Time Asymptotic for Solutions of
the Burgers Equation with a Periodic Force, Preprint

\bibitem {KiMo}Kirsch W., Molchanov S., in preparation

\bibitem {KMP}Kirsch W., Molchanov S., Pastur L.: One dimensional
Schr\"{o}dinger operators with high potential barriers, Oper. Theory Adv.
Appl. 57, 163--170 (1992)

\bibitem {KS}Kotani S., Simon B.: Localization in general one-dimensional
random systems II. Continuum Schr\"{o}dinger operators, Commun. Math. Phys.
112, 103-119 (1987)

\bibitem {RSII}Reed M., Simon B.: Methods of Modern Mathematical Physics II,
Academic Press 1975

\bibitem {S}Simon B.: Functional Integration and Quantum Physics, Academic
Press 1978

\bibitem {burgers1}Sinai Ya.: Two Results Concerning Asymptotic Behavior of
Solutions of the Burgers Equation with Force, J. Stat. Phys. 64, 1-12 (1991)
\end{thebibliography}
\end{document}
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       {\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
\long\def\@gobblethree#1#2#3{}
\if@compatibility
  \let\@obsolete\@gobblethree
\else
  \def\@obsolete#1#2#3{\PackageWarning{#1}{%
    Obsolete command \protect#2; \protect#3 should be used instead}}%
\fi
\def\@subst@obsolete#1#2#3{\@obsolete{#1}#2#3\gdef#2{#3}#2}
\begingroup \catcode`\"=12
\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}
\@ifpackageloaded{latexsym}{\@tempswafalse}{\@tempswatrue}
\if@tempswa
  \global\let\Box\square
  \global\let\Diamond\lozenge
  \global\let\leadsto\rightsquigarrow
  \global\let\lhd\@@undefined
  \global\let\unlhd\@@undefined
  \global\let\rhd\@@undefined
  \global\let\unrhd\@@undefined
  \DeclareMathSymbol\lhd     {\mathbin}{AMSa}{"43}
  \DeclareMathSymbol\unlhd   {\mathbin}{AMSa}{"45}
  \DeclareMathSymbol\rhd     {\mathbin}{AMSa}{"42}
  \DeclareMathSymbol\unrhd   {\mathbin}{AMSa}{"44}
\xdef\Join{\mathrel{\mathchar"0\hexnumber@\symAMSb 6F\mkern-13.8mu%
  \mathchar"0\hexnumber@\symAMSb 6E}}
\fi
\endgroup
\endinput
%%
%% End of file `amsfonts.sty'.
---------------9811100641922
Content-Type: application/x-tex; name="graphicx.sty"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="graphicx.sty"

%%
%% This is file `graphicx.sty',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% graphicx.dtx  (with options: `package')
%% 
%% graphicx.dtx Copyright (C) 1994      David Carlisle Sebastian Rahtz
%%              Copyright (C) 1995 1996 1997 David Carlisle
%%
%% This file is part of the Standard LaTeX `Graphics Bundle'.
%%
%% It should be distributed *unchanged* and together with all other
%% files in the graphics bundle. The file 00readme.txt contains a list
%% of all of these files.
%%
%% A modified version of this file may be distributed, but it should
%% be distributed with a *different* name. Changed files must be
%% distributed *together with a complete and unchanged* distribution
%% of these files.
%%
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
\ProvidesPackage{graphicx}
          [1997/06/09 v1.0d  Enhanced LaTeX Graphics (DPC,SPQR)]
\DeclareOption*{\PassOptionsToPackage\CurrentOption{graphics}}
\ProcessOptions
\RequirePackage{keyval,graphics}
\define@key{Gin}{bb}
           {\Gin@bboxtrue\Gread@parse@bb#1 \\}
\define@key{Gin}{bbllx}
           {\Gin@bboxtrue\Gin@defaultbp\Gin@llx{#1}}
\define@key{Gin}{bblly}
           {\Gin@bboxtrue\Gin@defaultbp\Gin@lly{#1}}
\define@key{Gin}{bburx}
           {\Gin@bboxtrue\Gin@defaultbp\Gin@urx{#1}}
\define@key{Gin}{bbury}
           {\Gin@bboxtrue\Gin@defaultbp\Gin@ury{#1}}
\define@key{Gin}{hiresbb}[true]{%
  \edef\Gread@BBox{%
    \@percentchar\@percentchar
    \csname if#1\endcsname HiRes\fi
    BoundingBox}}
\let\KV@Gin@natwidth\KV@Gin@bburx
\let\KV@Gin@natheight\KV@Gin@bbury
\define@key{Gin}{viewport}
           {\let\Gin@viewport@code\Gin@viewport\Gread@parse@vp#1 \\}
\define@key{Gin}{trim}
           {\let\Gin@viewport@code\Gin@trim\Gread@parse@vp#1 \\}
\def\Gread@parse@vp#1 #2 #3 #4 #5\\{%
  \Gin@defaultbp\Gin@vllx{#1}%
  \Gin@defaultbp\Gin@vlly{#2}%
  \Gin@defaultbp\Gin@vurx{#3}%
  \Gin@defaultbp\Gin@vury{#4}}%
\def\Gin@viewport{%
  \let\Gin@ollx\Gin@llx
  \let\Gin@olly\Gin@lly
  \let\Gin@ourx\Gin@urx
  \let\Gin@oury\Gin@ury
  \dimen@\Gin@llx\p@\advance\dimen@ \Gin@vurx\p@
                      \edef\Gin@urx{\strip@pt\dimen@}%
  \dimen@\Gin@lly\p@\advance\dimen@ \Gin@vury\p@
                      \edef\Gin@ury{\strip@pt\dimen@}%
  \dimen@\Gin@llx\p@\advance\dimen@ \Gin@vllx\p@
                      \edef\Gin@llx{\strip@pt\dimen@}%
  \dimen@\Gin@lly\p@\advance\dimen@ \Gin@vlly\p@
                      \edef\Gin@lly{\strip@pt\dimen@}}
\def\Gin@trim{%
  \let\Gin@ollx\Gin@llx
  \let\Gin@olly\Gin@lly
  \let\Gin@ourx\Gin@urx
  \let\Gin@oury\Gin@ury
  \dimen@\Gin@llx\p@\advance\dimen@ \Gin@vllx\p@
                      \edef\Gin@llx{\strip@pt\dimen@}%
  \dimen@\Gin@lly\p@\advance\dimen@ \Gin@vlly\p@
                      \edef\Gin@lly{\strip@pt\dimen@}%
  \dimen@\Gin@urx\p@\advance\dimen@ -\Gin@vurx\p@
                      \edef\Gin@urx{\strip@pt\dimen@}%
  \dimen@\Gin@ury\p@\advance\dimen@ -\Gin@vury\p@
                      \edef\Gin@ury{\strip@pt\dimen@}}
\let\Gin@vllx\Gin@llx\let\Gin@vlly\Gin@llx
\let\Gin@vurx\Gin@llx\let\Gin@vury\Gin@llx
\define@key{Gin}{angle}
           {\Gin@esetsize
            \@tempswatrue
            \edef\@tempa{\toks@{\noexpand\Gin@erotate{#1}{\the\toks@}}}%
            \@tempa}
\define@key{Gin}{origin}[c]{%
  \def\Gin@erotate{\Grot@box@kv[origin=#1]}}
\define@key{Gin}{width}{\def\Gin@ewidth{#1}}
\define@key{Gin}{height}{\def\Gin@eheight{#1}}
\define@key{Gin}{totalheight}{%
  \def\Gin@eresize{\totalheight}\def\Gin@eheight{#1}}
\define@key{Gin}{keepaspectratio}[true]{%
  \lowercase{\Gin@boolkey{#1}}{iso}}
\define@key{Gin}{scale}{%
  \if@tempswa
    \edef\@tempa{\toks@{\noexpand\Gscale@box{#1}[#1]{\the\toks@}}}%
    \@tempa
  \else
    \def\Gin@req@sizes{%
      \def\Gin@scalex{#1}\let\Gin@scaley\Gin@exclamation
      \Gin@req@height\Gin@scalex\Gin@nat@height
      \Gin@req@width\Gin@scalex\Gin@nat@width}%
  \fi
  \@tempswatrue}
\define@key{Gin}{draft}[true]{%
  \lowercase{\Gin@boolkey{#1}}{draft}}
\define@key{Gin}{clip}[true]{%
  \lowercase{\Gin@boolkey{#1}}{clip}}
\define@key{Gin}{type}{%
  \def\Ginclude@graphics##1{%
    \begingroup
    \def\Gin@base{##1}%
    \edef\@tempa{{#1}{\Gin@eread}{\Gin@ecom{##1\Gin@eext}}}%
    \expandafter\Gin@setfile\@tempa
    \endgroup}}
\define@key{Gin}{ext}{\def\Gin@eext{#1}}
\let\Gin@eext\@empty
\define@key{Gin}{read}{%
 \def\Gin@eread{#1}%
 \def\@tempa{*}\ifx\@tempa\Gin@eread\def\Gin@eread{\Gin@eext}\fi}
\let\Gin@eread\@empty
\define@key{Gin}{command}{\def\Gin@ecom##1{#1}}
\let\Gin@ecom\@firstofone
\def\Gin@boolkey#1#2{%
  \csname Gin@#2\ifx\relax#1\relax true\else#1\fi\endcsname}
\def\Gin@eresize{\height}
\def\Gin@esetsize{%
  \let\@tempa\Gin@exclamation
  \if@tempswa
    \edef\@tempa{\toks@{\noexpand
             \Gscale@@box\noexpand\Gin@eresize
              {\Gin@ewidth}{\Gin@eheight}{\the\toks@}}}%
    \@tempa
  \else
    \ifx\Gin@ewidth\@tempa
      \ifx\Gin@eheight\@tempa
      \else
         \let\Gin@@eheight\Gin@eheight
         \def\Gin@req@sizes{%
           \Gscale@div\Gin@scaley\Gin@@eheight\Gin@nat@height
           \let\Gin@scalex\Gin@exclamation
           \setlength\Gin@req@height\Gin@@eheight
           \Gin@req@width\Gin@scaley\Gin@nat@width}%
      \fi
    \else
      \ifx\Gin@eheight\@tempa
         \let\Gin@@ewidth\Gin@ewidth
         \def\Gin@req@sizes{%
           \Gscale@div\Gin@scalex\Gin@@ewidth\Gin@nat@width
           \let\Gin@scaley\Gin@exclamation
           \setlength\Gin@req@width\Gin@@ewidth
           \Gin@req@height\Gin@scalex\Gin@nat@height}%
      \else
         \let\Gin@@ewidth\Gin@ewidth
         \let\Gin@@eheight\Gin@eheight
         \def\Gin@nosize##1{%
           \KV@Gin@natwidth\Gin@@ewidth
           \KV@Gin@natheight\Gin@@eheight}%
         \def\Gin@req@sizes{%
           \Gscale@div\Gin@scalex\Gin@@ewidth\Gin@nat@width
           \Gscale@div\Gin@scaley\Gin@@eheight\Gin@nat@height
           \ifGin@iso
             \ifdim\Gin@scaley\p@>\Gin@scalex\p@
               \let\Gin@scaley\Gin@scalex
             \else
               \let\Gin@scalex\Gin@scaley
             \fi
           \fi
           \Gin@req@width\Gin@scalex\Gin@nat@width
           \Gin@req@height\Gin@scaley\Gin@nat@height}%
       \fi
     \fi
  \fi
  \let\Gin@ewidth\Gin@exclamation
  \let\Gin@eheight\Gin@ewidth}
\newdimen\Gin@req@height
\newdimen\Gin@req@width
\let\Gin@outer@scalex\relax
\let\Gin@outer@scaley\relax
\let\Gin@angle\relax
\let\Gin@ewidth\Gin@exclamation
\let\Gin@eheight\Gin@ewidth
\def\Gin@scalex{1}
\let\Gin@scaley\Gin@exclamation
\def\Gin@i{%
 \def\Gin@req@sizes{%
   \Gin@req@height\Gin@nat@height
   \Gin@req@width\Gin@nat@width}%
 \@ifnextchar[\Gin@ii{\Gin@ii[]}}
\def\Gin@ii[#1]#2{%
    \def\@tempa{[}\def\@tempb{#2}%
    \ifx\@tempa\@tempb
      \def\@tempa{\Gin@iii[#1][}%
      \expandafter\@tempa
    \else
     \begingroup
       \@tempswafalse
       \toks@{\Ginclude@graphics{#2}}%
       \setkeys{Gin}{#1}%
       \Gin@esetsize
       \the\toks@
     \endgroup
     \fi}
\def\rotatebox{%
  \@ifnextchar[\Grot@box@kv\Grot@box@std}
\def\Grot@box@std#1#2{%
  \Grot@setangle{#1}%
  \setbox\z@\hbox{{#2}}%
  \Grot@x\z@
  \Grot@y\z@
  \Grot@box}
\def\Grot@box@kv[#1]#2#3{%
  \@begin@tempboxa\hbox{#3}%
    \Grot@x\width \divide\Grot@x\tw@
    \Grot@y\height \advance\Grot@y-\depth \divide\Grot@y\tw@
    \setkeys{Grot}{#1}%
    \setbox\z@\box\@tempboxa
    \Grot@setangle{#2}%
    \Grot@box
  \@end@tempboxa}
\define@key{Grot}{origin}[c]{%
 \@tfor\@tempa:=#1\do{%
    \if l\@tempa \Grot@x\z@\else
    \if r\@tempa \Grot@x\width\else
    \if t\@tempa \Grot@y\height\else
    \if b\@tempa \Grot@y-\depth\else
    \if B\@tempa \Grot@y\z@\fi\fi\fi\fi\fi}}
\define@key{Grot}{x}{\setlength\Grot@x{#1}}
\define@key{Grot}{y}{\setlength\Grot@y{#1}}
\define@key{Grot}{units}{%
  \def\Grot@setangle##1{%
  \dimen@##1\p@
  \dimen@ii#1\p@
  \divide\dimen@ii360\relax
  \divide\dimen@\dimen@ii
  \edef\Grot@angle{\number\dimen@}}}
\let\Gin@erotate\Grot@box@std
\endinput
%%
%% End of file `graphicx.sty'.
---------------9811100641922--
