%% Version 1.06
%% revised July 1, 2020
%% the number for Theorems and other theorem-like environments was changed, so that everything is numbered with respect to its same type of statement, yet only within the section
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Version 1.05
%% revised Dec. 9, 2019
%% added a new citation command to get author and year without parentheses
%%%%
% Version 1.04
% changed the Remark environment to not affect theorem numbering
% added more space between paragraphs
% increased the font size for math environment
% added new theorem environments
% theorems with header changed to Theorem Section.Number. (Header Title) 
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\@twosidetrue\@mparswitchtrue\@twocolumnfalse
\newif\if@openright
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\newif\if@restonecol
%
%
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\newdimen\tempdimen
\newdimen\tablewidth
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\newdimen\tablewd
%

%
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\DeclareOption{final}{\setlength\overfullrule{0pt}}

\ExecuteOptions{letterpaper,12pt,oneside,onecolumn,final}
\ProcessOptions
%

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\hoffset=-0.069in
\voffset=0.18in %0.2

%
\newcommand\appendixname{Appendix}
\newcommand\abstractname{Abstract}
\newcommand\bibname{}%{REFERENCES}%
\newcommand\contentsname{Contents}
\newcommand\indexname{Index}
\newcommand\figurename{Figure}
\newcommand\tablename{Table}
\newcommand\partname{Part}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Font Defs. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\email#1{\it #1\par}
\def\http#1{\it http://#1\par}
\def\url#1{\it #1\par}
\def\received#1{Received:~#1}%\par}
\def\revised#1{Revised:~#1\par}
\def\accepted#1{Accepted:~#1}%\par}
\def\comby#1{\par\addvspace{1\baselineskip}Communicated\ by~#1\par}
%

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\def\bibfont{\fontsize{12}{15}\selectfont}
\def\rhfont{\footnotesize\itshape{}}
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\def\titlefont{\fontsize{16}{20}\bfseries\boldmath\selectfont\centering{}}
\def\authorfont{\fontsize{14 pt}{12}\bfseries\boldmath\selectfont\centering{}}
\def\abstractfont{\fontsize{14}{16}}
\def\affiliationfont{\fontsize{10}{12}\upshape}
\def\datefont{\footnotesize}
\def\sectionfont{\fontsize{14 pt}{12}\bfseries\boldmath\rightskip2pc plus1fill{}}
%\def\subsectionfont{\bfseries\boldmath\itshape\rightskip2pc plus1fill{}}
\def\subsectionfont{\bfseries\boldmath\rightskip2pc plus1fill{}}
\def\subsubsectionfont{\itshape\rightskip2pc plus1fill{}}
\def\paragraphfont{\bfseries\boldmath}
\def\subparagraphfont{\itshape}

\def\figurecaptionfont{\footnotesize}
\def\tablefont{\footnotesize}
\def\tablecaptionfont{\footnotesize}
\def\tabnotefont{\normalfont\footnotesize}
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%
%%%%%%%%%%%%%%%%%%%%%%%% Fonts %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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\DeclareRobustCommand\mit{\@fontswitch\relax\mathnormal}
\DeclareOldFontCommand{\bi}{\bfseries\itshape}{\@nomath\bi}
\renewcommand{\rmdefault}{cmr}%
%

\renewcommand\normalsize{%
\@setfontsize\normalsize\@xiipt{15}%\@xiiipt
 %  \@setfontsize\normalsize\@xpt{13}%\@xiiipt
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   \abovedisplayshortskip\abovedisplayskip
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\normalsize
%

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               \itemsep \parsep}%
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%
\newcommand\footnotesize{%
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               \itemsep \parsep}%
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%
\newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt\@ixpt}
\newcommand\tiny{\@setfontsize\tiny\@vpt\@vipt}
\newcommand\large{\@setfontsize\large\@xiipt{14}}%
\newcommand\Large{\@setfontsize\Large\@xivpt{16}}%%
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\newcommand\Huge{\@setfontsize\Huge\@xxvpt{30}}
%
% We will declare the deafult font for the article to times.
% These three commands make up the entire times.sty package.
\renewcommand{\sfdefault}{phv}
\renewcommand{\rmdefault}{ptm}
\renewcommand{\ttdefault}{pcr}
% This line enables Times now - so that all class options can see the correct font families
\normalfont\selectfont
%
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%
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%
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%
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\@settopoint\textwidth
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%\advance\textheight-3pt
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%
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%
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%
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%
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%\skip\@mpfootins = \skip\footins
%
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%\setlength\fboxrule{.4\p@}
%%
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%
%
\DeclareMathSizes{28}{28}{25}{20}
\DeclareMathSizes{24}{24}{21}{18}
\DeclareMathSizes{22}{22}{19}{17}
\DeclareMathSizes{20}{20}{17}{15}
\DeclareMathSizes{18}{18}{15}{10}
\DeclareMathSizes{12}{12}{8}{5}
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\def\bm#1{\mathchoice
	{\mbox{\boldmath$#1$}}%
          {\mbox{\boldmath$#1$}}%
          {\mbox{\boldmath$\scriptstyle#1$}}%
          {\mbox{\boldmath$\scriptscriptstyle#1$}}}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%% Article FM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\def\title#1{\gdef\@title{{#1}}}
\newcommand\maketitle{\par
  \begingroup
   \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
    \def\@makefnmark{\rlap{\@textsuperscript{\normalfont\@thefnmark}}}%
    \long\def\@makefntext##1{\noindent
	\@textsuperscript{\normalfont\@thefnmark}{{##1}}}%
      %\newpage
      \global\@topnum\z@
      \@maketitle
    \thispagestyle{plain}\@thanks
  \endgroup
  \setcounter{footnote}{0}%
  \global\let\thanks\relax
  \global\let\maketitle\relax
  \global\let\@maketitle\relax
  \global\let\@thanks\@empty
  \global\let\@author\@empty
  \global\let\@title\@empty
  \global\let\title\relax
  \global\let\author\relax
}
\def\@maketitle{%
  %\newpage
  \null
  \vspace*{-10pt}
  \vspace*{-29pt}% to have the bylines on the beginning of the page
% \begin{center}%       %center for catchline command
  \@clinebuf
  \vspace*{22pt}
  \begin{center}%       %flush left for catchline
  \let\footnote \thanks
  {\titlefont\@title \par}%
  \vspace*{29pt}
  \@aabuffer\par
  \end{center}%
}
%
\def\thebepage{\thepage}
\def\theeepage{\pageref{eof}}
%

\newenvironment{history}{%
	\addvspace{-2pt plus-2pt minus-2pt}
   \authorfont\begin{center}\vskip-\lastskip
}{%
   \end{center}
	\addvspace{15pt plus2pt minus2pt}
}

\def\pub#1#2{\par%
	\addvspace{-2pt plus-2pt minus-2pt}
	{\centering{
	\authorfont #1\par
	#2\par}\par}
	\addvspace{15pt plus2pt minus2pt}}%
%
\def\@clinebuf{}
\def\catchline#1#2#3#4#5{\expandafter\def\expandafter\@clinebuf\expandafter
	{\@clinebuf\catchlinefont
	\noindent Applications and Applied Mathematics: An International Journal (AAM)\par
	%\noindent \copyright\ World Scientific Publishing Company\par
	}\relax\par
	}

\def\No{No.}
%
\def\@aabuffer{}
\def\author#1{\expandafter\def\expandafter\@aabuffer\expandafter
	{\@aabuffer{\authorfont{{#1}}}\relax\par
	\vspace*{3pt}}}
%
\def\address#1{\expandafter\def\expandafter\@aabuffer\expandafter
	{\@aabuffer{\affiliationfont{#1}}\relax\par
	\vspace*{16pt}}}
%
\def\keywords#1{\par
	\vspace*{8pt}
	{\authorfont{\leftskip18pt\rightskip\leftskip
	\noindent{\it Keywords}\/:\ #1\par}}\par}
%
\def\category#1#2{\par
	\vspace*{8pt}
	{\authorfont{\leftskip18pt\rightskip\leftskip
	\noindent{\it Categories}\/:\ #1:\ #2\par}}\par}
%
\newenvironment{abstract}{\par%
	\abstractfont
  	\noindent\leftskip0pt\rightskip0pt
}{%
  \par}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Sectioning %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\setcounter{secnumdepth}{3}
\newcounter {section}
\newcounter {subsection}[section]
\newcounter {subsubsection}[subsection]
\newcounter {paragraph}[subsubsection]
\newcounter {subparagraph}[paragraph]
\renewcommand\thesection      {\arabic{section}}
\renewcommand\thesubsection   {\thesection.\arabic{subsection}}
\renewcommand\thesubsubsection{\thesubsection .\arabic{subsubsection}}
\renewcommand\theparagraph    {\thesubsubsection.\arabic{paragraph}}
\renewcommand\thesubparagraph {\theparagraph.\arabic{subparagraph}}
%
\newcommand\section{\@startsection{section}{1}{\z@}{-18pt \@plus-3pt minus-6pt}{6pt}{\sectionfont}}
\newcommand\subsection{\@startsection{subsection}{2}{\z@}{-18pt \@plus-3pt minus-6pt}{6pt}{\subsectionfont}}
\newcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}{-18pt \@plus-3pt minus-6pt}{6pt}{\subsubsectionfont}}
\newcommand\paragraph{\@startsection{paragraph}{4}{\z@}{12pt \@plus -3pt}{-.5em}{\paragraphfont}}
\newcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}{3.25ex \@plus1ex}{-1em}{\subparagraphfont}}
%
\def\@seccntformat#1{{\upshape{\csname the#1\endcsname.}}\hskip 1em}%.5em}
%
%\gdef\@author{}
%
%%%%%%%%%%%%%%%%%%%%%%%% Table of Contents %%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand\@pnumwidth{1.55em}
\newcommand\@tocrmarg{2.55em}
\newcommand\@dotsep{4.5}
\setcounter{tocdepth}{3}
\newcommand\tableofcontents{%
    \section*{\contentsname
        \@mkboth{%
           \contentsname}{\contentsname}}%
    \@starttoc{toc}%
    }
\newcommand\l@section[2]{%
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    \addpenalty\@secpenalty
    \addvspace{1.0em \@plus\p@}%
    \setlength\@tempdima{1.5em}%
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      \parfillskip -\@pnumwidth
      \leavevmode \bfseries
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    \endgroup
  \fi}
\newcommand\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\newcommand\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\newcommand\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\newcommand\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\newcommand\listoffigures{%
    \section*{\listfigurename
      \@mkboth{\listfigurename}%
              {\listfigurename}}%
    \@starttoc{lof}%
    }
\newcommand\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand\listoftables{%
    \section*{\listtablename
      \@mkboth{%
          \listtablename}{\listtablename}}%
    \@starttoc{lot}%
    }
\let\l@table\l@figure
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lists %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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\@beginparpenalty -\@lowpenalty
\@endparpenalty   -\@lowpenalty
\@itempenalty     -\@lowpenalty
\renewcommand\theenumi{\arabic{enumi}}
\renewcommand\theenumii{\alph{enumii}}
\renewcommand\theenumiii{\roman{enumiii}}
\renewcommand\theenumiv{\Alph{enumiv}}
\newcommand\labelenumi{(\theenumi)}
\newcommand\labelenumii{(\theenumii)}
\newcommand\labelenumiii{\theenumiii.}
\newcommand\labelenumiv{\theenumiv.}
\renewcommand\p@enumii{\theenumi}
\renewcommand\p@enumiii{\theenumi(\theenumii)}
\renewcommand\p@enumiv{\p@enumiii\theenumiii}
\newcommand\labelitemi{$\m@th\bullet$}
\newcommand\labelitemii{\normalfont\bfseries --}
\newcommand\labelitemiii{$\m@th\ast$}
\newcommand\labelitemiv{$\m@th\cdot$}
%
%
\def\@listI{\leftmargin\leftmargini
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            \itemsep0\p@}
%
\let\@listi\@listI
\@listi
%
\def\@listii {\leftmargin\leftmarginii
              \labelwidth\leftmarginii
              \advance\labelwidth-\labelsep
              \topsep    3.5\p@ \@plus2\p@ \@minus\p@
              \parsep    \z@
              \itemsep\z@}
\def\@listiii{\leftmargin\leftmarginiii
              \labelwidth\leftmarginiii
              \advance\labelwidth-\labelsep
              \topsep 3.5\p@ \@plus\p@\@minus\p@
              \parsep \z@
             % \partopsep \p@ \@plus\z@ \@minus\p@
              \itemsep0\p@}%
\def\@listiv {\leftmargin\leftmarginiv
              \labelwidth\leftmarginiv
              \advance\labelwidth-\labelsep}
\def\@listv  {\leftmargin\leftmarginv
              \labelwidth\leftmarginv
              \advance\labelwidth-\labelsep}
\def\@listvi {\leftmargin\leftmarginvi
              \labelwidth\leftmarginvi
              \advance\labelwidth-\labelsep}
%
\def\enummax#1{\setbox\tempbox=\hbox{#1\hskip\labelsep}%
  \expandafter\global\csname leftmargin\romannumeral\the\@enumdepth\endcsname\wd\tempbox}
%
\def\enumerate{\@ifnextchar[{\@enumerate}{\@enumerate[\csname label\@enumctr\endcsname]}}
%
\def\@enumerate[#1]{\ifnum \@enumdepth >3 \@toodeep\else
      \advance\@enumdepth \@ne\edef\@enumctr{enum\romannumeral\the\@enumdepth}%
      \enummax{#1}\list
      {\csname label\@enumctr\endcsname}{\usecounter
        {\@enumctr}\def\makelabel##1{\hss\llap{##1}}}\fi}
%
\let\Item\item
\newenvironment{enumeroman}{%
  \def\theenumi{\roman{enumi}}\def\theenumii{\alph{enumii}}%
  \def\labelenumi{(\theenumi)}\def\labelenumii{(\theenumii)}%
		\let\item\Item
  \begin{enumerate}%
}{%
  \end{enumerate}}
%
\newenvironment{alphlist}{%
  \def\theenumi{\alph{enumi}}\def\theenumii{\roman{enumii}}%
  \def\labelenumi{(\theenumi)}\def\labelenumii{(\theenumii)}%
		\let\item\Item
  \begin{enumerate}%
}{%
  \end{enumerate}}
%
\newenvironment{arabiclist}{%
	\def\theenumi{\arabic{enumi}}\def\theenumii{\alph{enumii}}
	\def\labelenumi{(\theenumi)}\def\labelenumii{(\theenumii)}%
	\let\item\Item
	\begin{enumerate}
	}{%
	\end{enumerate}}
%
\newenvironment{romanlist}{%
	\def\theenumi{\roman{enumi}}\def\theenumii{\alph{enumii}}
	\def\labelenumi{(\theenumi)}\def\labelenumii{(\theenumii)}%
	\let\item\Item
	\begin{enumerate}
	}{%
	\end{enumerate}}
%
\newenvironment{itemlist}{%
	\def\labelenumi{\labelitemi}
	\let\item\Item
	\begin{enumerate}
	}{%
	\end{enumerate}}
%
\newenvironment{description}
               {\list{}{\labelwidth\z@ \itemindent-\leftmargin
                        \let\makelabel\descriptionlabel}}
               {\endlist}
\newcommand*\descriptionlabel[1]{\hspace\labelsep
                                \normalfont\bfseries #1}
%
\newenvironment{unnumlist}{%
	\let\item\Item
  	\leftmargini2pc
   	\ifnum \@enumdepth >3 \@toodeep\else
      	\advance\@enumdepth \@ne
      	\list{}{\itemindent-2pc\topsep6pt
              \def\makelabel##1{\hss\llap{##1}}}%
   \fi
}{%
  \endlist}
%
%%%%%%%%%%%%%%%%%%%%%%%%%% Poetry %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newenvironment{verse}
               {\let\\\@centercr
                \list{}{\itemsep      \z@
                        \itemindent   -1.5em%
                        \listparindent\itemindent
                        \rightmargin  \leftmargin
                        \advance\leftmargin 1.5em}%
                \item\relax}
               {\endlist}
\let\poetry\verse
\let\endpoetry\endverse
%
%%%%%%%%%%%%%%%%%%%%%%%% Extract %%%%%%%%%%%%%%%%%%%%%%%
%
\newenvironment{quotation}
               {\list{}{\listparindent 1.5em%
                        \itemindent    \listparindent
                        \rightmargin   \leftmargin
                        \parsep        \z@ \@plus\p@}%
                \item\relax}
               {\endlist}
\newenvironment{quote}
               {\list{}{\rightmargin\leftmargin}%
                \item\relax}
               {\endlist}
%
\let\extract\quotation
\let\endextract\endquotation
%
\def\source#1{\par\hspace*{\fill}{\it#1}\par}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Maths %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\renewcommand\theequation{\arabic{equation}}
%
\def\bstrut{\vrule width0pt depth5pt}
\def\tstrut{\vrule width0pt height10pt}
\jot=3pt
%
%
\def\refstepeqcnt{\refstepcounter{equation}}
\def\eqnnum{\displaywidth\linewidth\hbox{\@eqnnum}}
%
\def\@equation{\let\topsep\abovedisplayskip\@beginparpenalty\predisplaypenalty\@endparpenalty\postdisplaypenalty
     \refstepeqcnt
     \trivlist \item[]\leavevmode
       \hbox to\linewidth\bgroup\hfill $\m@th% $
       \displaystyle}%
\def\end@equation{$\hfill\eqnnum\egroup\endtrivlist}
%
\renewenvironment{eqnarray}{%
   \stepcounter{equation}%
   \def\@currentlabel{\p@equation\theequation}%
   \global\@eqnswtrue
   \m@th
   \global\@eqcnt\z@
   \tabskip\@centering
   \let\\\@eqncr
   $$\everycr{}\halign to\displaywidth\bgroup
       \hskip\@centering$\displaystyle\tabskip\z@skip{##}$\@eqnsel
      &\global\@eqcnt\@ne\hskip \tw@\arraycolsep \hfil${##}$\hfil
      &\global\@eqcnt\tw@ \hskip \tw@\arraycolsep
         $\displaystyle{##}$\hfil\tabskip\@centering
      &\global\@eqcnt\thr@@ \hb@xt@\z@\bgroup\hss##\egroup
         \tabskip\z@skip
      \cr}
        {%
      \@@eqncr
      \egroup
      \global\advance\c@equation\m@ne
   $$\global\@ignoretrue
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Floats %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\FigName{figure}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\figurename{Figure}
\newcounter{figure}
\renewcommand\thefigure{\@arabic\c@figure}
\def\fps@figure{tbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{\figurename~\thefigure}
\newenvironment{figure}
               {\@float{figure}}
               {\end@float}
\newenvironment{figure*}
               {\@dblfloat{figure}}
               {\end@dblfloat}
%
\InputIfFileExists{psfig.sty}{\typeout{^^Jpsfig.sty inputed...ok}}{\typeout{^^JWarning: psfig.sty could be be found.^^J}}
\InputIfFileExists{epsfsafe.tex}{\typeout{^^Jepsfsafe.tex inputed...ok}}
			{\typeout{^^JWarning: epsfsafe.tex could not be found.^^J}}
%\InputIfFileExists{epsfig.sty}{\typeout{^^Jepsfig.sty inputed...ok}}{\typeout{^^JWarning: epsfig.sty could not be found.^^J}}
\InputIfFileExists{epsf.sty}{\typeout{^^Jepsf.sty inputed...ok}}{\typeout{^^JWarning: epsf.sty could not be found.^^J}}%
%
\def\ArtDir{\jobname/}
%
\def\figurebox#1#2#3{%
        \def\arg{#3}%
        \ifx\arg\empty
        {\hfill\vbox{\hsize#2\hrule\hbox to #2{%
                \vrule\hfill\vbox to #1{\hsize#2\vfill}\vrule}\hrule}\hfill}%
        \else
        \hbox to \hsize{\hfill\epsfbox{\ArtDir#3.eps}\hfill}
        \fi}
%
\def\PSFIG#1#2#3{\par
  {\hbox to \hsize{%
	\hfill{\psfig{file=\ArtDir#1.eps,width=#2,angle=#3}}\hfill}\par}}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%% Tables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcounter{table}
\def\tablename{Table}
\renewcommand\thetable{\@arabic\c@table}
\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{\tablename~\thetable}
\newenvironment{table}
               {\@float{table}}
               {\end@float}
\newenvironment{table*}
               {\@dblfloat{table}}
               {\end@dblfloat}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\def\tableformat#1{\centerline{#1}}
\newdimen\tabledim
\long\def\tbl#1#2{%
\global\setbox\tempbox\hbox{\tablefont #2}%
 \global\tabledim\wd\tempbox
 \tableformat{\vbox{\hsize\tabledim
                   \caption{#1}
                   \box\tempbox}}%
}

%
\long\def\ttbl#1#2#3{%
	\parindent\z@
	\noindent
  	\tablecaptionfont
	\caption{#2}%
	\tablewd\hsize
	\advance\tablewd-#1
  	\global\divide\tablewd\tw@
	\ifdim \wd\captionbox<#1
	\centerline{\unhbox\captionbox}
	\else
  	{\leftskip\tablewd% plus1fill{}
	\rightskip\tablewd% plus1fill{}
  	\unhbox\captionbox\par}
	\fi
 	\vskip-8pt
	\centerline{#3}
}%

%
\newenvironment{InTextTable}{%
 	\par\addvspace{16pt plus6pt minus6pt}%
 	\begin{center}
	\tablecaptionfont\noindent\ignorespaces
}{%
   	\end{center}
	\par\addvspace{17pt plus6pt minus6pt}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\tabnotehead{Note}
\def\tabsourcehead{Source}
\newenvironment{tabnote}{%
   \par\addvspace{2pt}
   \setbox\tempbox\vbox\bgroup\hsize\tabledim\parindent12pt
   \tabnotefont
   \@ifnextchar[{\@tabnote}{\@tabnote[\tabnotehead]}%
}{%
		\par\egroup
   \tableformat{\box\tempbox}}
%

\def\@tabnote[#1]{\vskip6pt\noindent\def\@Tempa{#1}\ifx\@Tempa\@empty\hskip0pt\else{\it#1}:\hskip5pt\ignorespaces\fi}
%

\newenvironment{tabfootnote}{%
   \par\addvspace{2pt}
   \setbox\tempbox\vbox\bgroup\hsize\tabledim\parindent12pt
   \tabnotefont
}{%
		\par\egroup
   \tableformat{\box\tempbox}}
%

\def\tabmark#1{\noindent\hbox{\normalsize$\m@th^{#1}$}\ignorespaces}
%
\newenvironment{tabsource}{%
   \let\tabnotehead\tabsourcehead
   \begin{tabnote}
}{%
   \end{tabnote}
}
%
\def\Hline{%
  \noalign{\ifnum0=`}\fi\hrule \@height 2\arrayrulewidth \futurelet
   \@tempa\@xhline}
%
\def\x{@{\extracolsep{\fill}}}
\def\toprule{\Hline\\[-6.5pt]}
\def\colrule{\\[-7pt]\hline\\[-6pt]}
\def\TMcolrule{\\[-9.5pt]\hline\\[-9.5pt]}
\def\botrule{\\[-7pt]\Hline}
\def\crule#1{\\[-7pt]#1\\[-6pt]}
%
%%%%%%%%%%%%%%%%%%%%%%%% Caption %%%%%%%%%%%%%%%%%%%%%%%
%
\newlength\abovecaptionskip
\newlength\belowcaptionskip
\setlength\abovecaptionskip{10\p@}
%\setlength\belowcaptionskip{0\p@}
\setlength\belowcaptionskip{4\p@}

\long\def\@makecaption#1#2{%
  \figurecaptionfont
  \vskip\abovecaptionskip
  \sbox\@tempboxa{#1. #2}%
  \ifdim \wd\@tempboxa >\hsize
    #1. #2\par
  \else
    \global \@minipagefalse
    \hbox to \hsize{\hfil\box\@tempboxa\hfil}%
  \fi
  \vskip\belowcaptionskip}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Footnote %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\renewcommand\footnoterule{%
  \kern-3\p@
  \hrule depth0pt height0pt \@width.2\columnwidth
  \kern2.6\p@}
\newcommand\@makefntext[1]{%
    \parindent 1em%
    \noindent
     \@makefnmark#1}
%
\def\@fnsymbol#1{\ensuremath{\ifcase#1\or *\or \dagger\or \ddagger\or
   	\mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
   	\or \ddagger\ddagger \else\@ctrerr\fi}}
%
\def\thefootnote{\alph{footnote}}
\def\@makefnmark{{$^{\rm \@thefnmark}$}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% THEOREM ENVIRONMENTS %%%%%%%%%%%%%%%%%%%%%%%

%
\newif\iflogo
\def\prbox{\hbox to \hsize{\hfill\fboxsep0pt\fbox{\phantom{\vrule width5pt height5pt depth0pt}}}\global\logofalse}
%
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Bibliography %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\let\@internalcite\cite
\def\cite{\def\@citeseppen{-1000}%
\def\@cite##1##2{({##1\if@tempswa , ##2\fi})}%   this was \def\@cite##1##2{[{##1\if@tempswa , ##2\fi}]}
	\def\citeauthoryear##1##2{##1 (##2)}\@internalcite}
%
\def\citenop{\def\@citeseppen{-1000}%  added Dec. 9, 2019
\def\@cite##1##2{{##1\if@tempswa , ##2\fi}}%   this was \def\@cite##1##2{[{##1\if@tempswa , ##2\fi}]}
	\def\citeauthoryear##1##2{##1 (##2)}\@internalcite}
%
\def\shortcite{\def\@citeseppen{1000}%
	\def\@cite##1##2{[{##1\if@tempswa , ##2\fi}]}%
\def\citeauthoryear##1##2{##2}\@internalcite}
%
\def\citeauthor#1{\def\@citeseppen{1000}%
	\def\@cite##1##2{{##1\if@tempswa , ##2\fi}}%
\def\citeauthoryear##1##2{##1}\@citedata{#1}}
%
\def\citeyear#1{\def\@citeseppen{1000}%
\def\@cite##1##2{{##1\if@tempswa , ##2\fi}}%
\def\citeauthoryear##1##2{##2}\@citedata{#1}}
%
\def\@citedata#1{\@tempswafalse%
	\if@filesw\immediate\write\@auxout{\string\citation{#1}}\fi
  \def\@citea{}\@cite{\@for\@citeb:=#1\do
    {\@citea\def\@citea{,\penalty\@citeseppen\ }\@ifundefined
       {b@\@citeb}{{\bf ?}\@warning
       {Citation `\@citeb' on page \thepage \space undefined}}%
{\csname b@\@citeb\endcsname}}}{}}
%
\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
 \let\@citea\@empty
  \@cite{\@for\@citeb:=#2\do
    {\@citea\def\@citea{;\penalty\@citeseppen\ }%
     \def\@tempa##1##2\@nil{\edef\@citeb{\if##1\space##2\else##1##2\fi}}%
     \expandafter\@tempa\@citeb\@nil
     \@ifundefined{b@\@citeb}{{\bf ?}\@warning
       {Citation `\@citeb' on page \thepage \space undefined}}%
     {\csname b@\@citeb\endcsname}}}{#1}}
%\def\@biblabel#1{\def\citeauthoryear##1##2{##1, ##2}}
\def\@biblabel#1{}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
\def\@refcitex[#1]#2{\if@filesw\immediate\write\@auxout
	{\string\citation{#2}}\fi
\def\@citea{}\@refcite{\@for\@citeb:=#2\do
	{\@citea\def\@citea{, }\@ifundefined
	{b@\@citeb}{{\bf ?}\@warning
	{Citation `\@citeb' on page \thepage \space undefined}}
	\hbox{\csname b@\@citeb\endcsname}}}{#1}}
 \def\@refcite#1#2{{#1\if@tempswa\typeout
        {WSPC warning: optional citation argument
	ignored: `#2'} \fi}}
 \def\refcite{\@ifnextchar[{\@tempswatrue
	\@refcitex}{\@tempswafalse\@refcitex[]}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newdimen\bibindent
\bibindent=1.5em
\newenvironment{thebibliography}%[1]
%{apalike}
%\bibliographystyle{ieeetr}
% or: plain,unsrt,alpha,abbrv,acm,apalike
{\section*{\bibname}\bibfont%
      \list{}%
           {\leftmargin\parindent
            \itemindent-\parindent
	    \itemsep0pt\parsep0pt
            \usecounter{enumiv}%
            \renewcommand\theenumiv{\arabic{enumiv}}}%
      \sloppy\clubpenalty4000\widowpenalty4000%
      \sfcode`\.=\@m}
     {\def\@noitemerr
       {\@latex@warning{Empty `thebibliography' environment}}%
      \endlist}
\newcommand\newblock{}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Process Options %%%%%%%%%%%%%%%%%%%%%%%%
%
%\pagestyle{plain}
\pagenumbering{arabic}
\onecolumn
\frenchspacing
\flushbottom
\usepackage{amsfonts,amssymb,amsmath}
\usepackage[amsmath,standard,thmmarks]{ntheorem}
\usepackage{color}
\usepackage{graphics}
\usepackage{epstopdf}
\usepackage{graphicx}
\usepackage{tabularx}
%\usepackage{ctable}
\usepackage{figsize}
\usepackage[T1]{fontenc}
\usepackage[font=small,format=hang,tableposition=top,labelsep=period]{caption}
\setlength{\captionmargin}{20pt}
\setlength{\belowcaptionskip}{15pt plus 3pt minus 2pt}
\captionsetup[figure]{labelfont=bf}
\captionsetup[table]{labelfont=bf}
%\usepackage{subcaption}

%%%%%%%%%%%%%%%%%%%%
%%%      defines the theorem formatting
%%%
%%%% code to change the theorem seperator
\makeatletter %
\newtheoremstyle{AAMthm}%
 {\item[\rlap{\vbox{\hbox{\hskip\labelsep \theorem@headerfont
          ##1\ ##2\theorem@separator}\hbox{\strut}}}]}%
  {\item[\rlap{\vbox{\hbox{\hskip\labelsep \theorem@headerfont
          ##1\ ##2\theorem@separator\ (##3)}\hbox{\strut}}}]}
\makeatother
\theoremstyle{AAMthm}  %% this line invokes the new theorem style defined above
\theorembodyfont{\normalfont}%\itshape}
\theoremheaderfont{\normalfont\bfseries}
\theoremseparator{.\vspace{1.3ex}} 
\theorempreskip{\baselineskip}
%\theorempostskip{\baselineskip}
%%%% end of code to change the theorem seperator
%%%%% %%%%%%%%%
%%%%%%%%%%%%%%
%%%%%   this is the old number system, from Version 1.05
%\newtheorem{thm}{Theorem}[section]
%\newtheorem{lem}[thm]{Lemma}%[section]
%\newcommand\lemmabodyfont{}%\itshape}
%\newtheorem{prop}[thm]{Proposition}%[section]
%\newcommand\propositionbodyfont{\upshape}%\itshape}
%\newtheorem{coro}[thm]{Corollary}%[section]
%\newcommand\corollarybodyfont{\upshape}%\itshape}
%\newtheorem{defn}[thm]{Definition}
%\newtheorem{note}[thm]{Note}
%\newtheorem{case}[thm]{Case}
%
%\renewtheorem{example}[thm]{Example}
%
%\renewtheorem{remark}[thm]{Remark}
%%%%%%  this was the old numbering system, from Version 1.05
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%
%% --------------------------------------------------------------
%%   this is the new numbering system, for Version 1.06
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{coro}{Corollary}[section]
\newtheorem{defn}{Definition}[section]
\newtheorem{note}{Note}[section]
\newtheorem{case}{Case}[section]
\renewtheorem{example}{Example}[section]
\renewtheorem{remark}{Remark}[section]
%% this ends the new numbering system, for Version 1.06
%% ----------------------------------------------------------------


% %%%%%%
%%%%%%%
%% this is for theorem environments that should not be numbered
%\theoremstyle{nonumberbreak}
%\theorembodyfont{\normalfont}%\itshape}
%\theoremheaderfont{\normalfont\bfseries}
%\theoremseparator{.\vspace{1.3ex}}
%\theorempreskip{\baselineskip}
%\theorempostskip{\baselineskip}
%\renewtheorem{remark}[thm]{Remark}
%
% This section defines the Proof environment
\theoremstyle{nonumberbreak}
\theoremheaderfont{\bfseries\itshape}
\theorembodyfont{\normalfont}
\theoremseparator{:\vspace{1.3ex}}
\theorempreskip{\baselineskip}
\theorempostskip{\baselineskip}
\theoremsymbol{\ensuremath{_\blacksquare}}
%\theoremsymbol{\rule{1ex}{1ex}}
\renewtheorem{proof}{Proof}
\renewtheorem{Proof}{Proof}
\qedsymbol{\ensuremath{_\blacksquare}}
%%%
%%%
%%%%%%%%%%%%%%%%%%%%%


\def\sloppy{\tolerance=100000\hfuzz=\maxdimen \vfuzz=\maxdimen}
\sloppy
\vbadness=12000
\hbadness=12000
%=============================================================================
% PAGE LAYOUT
%=============================================================================
% All margin dimensions measured from a point one inch from top and side
% of page.
%=============================================================================
\setlength\voffset{0in}
\setlength\hoffset{0in}
\setlength\topmargin{0pt}
\setlength\footskip{24bp}      %    Distance from baseline of box containing foot
                               %    to baseline of last line of text.
\setlength{\textheight}{9in}     % Height of text (including footnotes and figures,
                                      % excluding running head and foot).
\setlength{\textwidth}{6.5in}     % Width of text line
   \setlength\oddsidemargin{0bp}  %   Note that \oddsidemargin = \evensidemargin
    \setlength\evensidemargin{0bp} % 1.25" left margin
    \setlength\marginparwidth{68bp}
%=============================================================================
% FOOTNOTES:
%=============================================================================
\setlength{\footnotesep}{10pt}    % Height of strut placed at the beginning of every
                                   % footnote = height of normal \footnotesize strut,
                                   % so no extra space between footnotes.
\skip\footins 18pt plus 4pt minus 2pt  % Space between last line of text and
                                         % top of first footnote.




\endinput
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%End of file%%%%%%%%%%%%%%%%%%%%%%%%%%%%
