%**	This is a pure LaTeX 2e file [article class modified for transparencies]
%**	Filename:	tex
%**	Last update:	June 10, 2004
%**     Final version:
%**	Title:
%**
%**	Purpose:
%**
%**
%**
%**
%**
%**	Author (and responsible for all errors):

%***************************************************************************%
%**                                                                       **%
%**	 Article style of BOZHIDAR ZAKHARIEV ILIEV: 		 	  **%
%**                                                                       **%
%**	 Laboratory of Mathematical Modeling in Physics			  **%
%**	 Institute for Nuclear Research and Nuclear Energy                **%
%**	 Bulgarian Academy of Sciences                                    **%
%**	 Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria           **%
%**	 E-mail address: bozho@inrne.bas.bg				  **%
%**      URL: http://theo.inrne.bas.bg/~bozho/	        		  **%
%**                                                                       **%
%**                                                                       **%
%**			(May be used freely by everybody)                 **%
%**                                                                       **%
%***************************************************************************%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                                                       %%
%% 		Beginning of \CharacterTable                             %%
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%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z  %%
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%%   Minus         \-     Point         \.     Solidus       \/          %%
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\typeout{}
\typeout{????????????????????????????????????????????????????????????????}
\typeout{}
\typeout{This is the file of the article}
\typeout{"" }
\typeout{ by Bozhidar Zakhariev Iliev. }
\typeout{Its initial draft version was written during the period}
\typeout{ }
\typeout{}
\typeout{vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv}
\typeout{}

%====>>	BEGINNING OF EXTERNAL FILES	====>>	====>>	====>>	====>>

% Beginning of file
%\begin{filecontents}{}
%
%
%\end{filecontents}
% End of file


% Beginning of file action.bbl (bibliography)
% \begin{filecontents}{.bbl}
% \end{filecontents}
% End of file act-rep.bbl (bibliography)


% Beginning of file Bozho-LaTeX-Style.sty
\begin{filecontents}{Bozho-LaTeX-Style.sty}
%***************************************************************************%
%**     BEGINNING of LaTeX2e style package of BOZHIDAR ZAKHARIEV ILIEV    **%
%**	Filename: Bozho-LaTeX-Style.sty                                            **%
%**	Author (and responsible for all errors):                          **%
%***************************************************************************%
%**                                                                       **%
%**	  BOZHIDAR ZAKHARIEV ILIEV:					  **%
%**                                                                       **%
%**	  Laboratory of Mathematical Modeling in Physics		  **%
%**	  Institute for Nuclear Research and Nuclear Energy               **%
%**	  Bulgarian Academy of Sciences                                   **%
%**	  Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria          **%
%**	  E-mail address: bozho@inrne.bas.bg                              **%
%**       URL: http://theo.inrne.bas.bg/~bozho/		       		  **%
%**                                                                       **%
%**                                                                       **%
%**			(May be used freely by everybody)                 **%
%**                                                                       **%
%***************************************************************************%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 		Beginning of \CharacterTable                             %%
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%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z  %%
%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z  %%
%%   Digits        \0\1\2\3\4\5\6\7\8\9                                  %%
%%   Exclamation   \!     Double quote  \"     Hash (number) \#          %%
%%   Dollar        \$     Percent       \%     Ampersand     \&          %%
%%   Acute accent  \'     Left paren    \(     Right paren   \)          %%
%%   Asterisk      \*     Plus          \+     Comma         \,          %%
%%   Minus         \-     Point         \.     Solidus       \/          %%
%%   Colon         \:     Semicolon     \;     Less than     \<          %%
%%   Equals        \=     Greater than  \>     Question mark \?          %%
%%   Commercial at \@     Left bracket  \[     Backslash     \\          %%
%%   Right bracket \]     Circumflex    \^     Underscore    \_          %%
%%   Grave accent  \`     Left brace    \{     Vertical bar  \|          %%
%%   Right brace   \}     Tilde         \~}                              %%
%%                                                                       %%
%% 		End of \CharacterTable                                   %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



% ======================== identification ===================================
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
\ProvidesPackage{bozhomac}[2004/05/26
			   BOZHIDAR Z. ILIEV's LaTeX2e style package]


% ========================  initial code ====================================

% ---------> Beginning of Package Loading part

%	Some definitions need (or are modified by) the following packages
%	AMS-LaTeX, AMSFonts

%	Beginning of full AMS-LaTeX package
\RequirePackage{amsmath}[2000/07/18]% it subsumes: amstext, amsbsy, amsopn
% \RequirePackage{amstext}[2000/06/29]%	main AMS math package
% \RequirePackage{amsbsy}[1999/11/29]%	bold symbols
% \RequirePackage{amsopn}[1999/12/14]%	operator names
% \RequirePackage{amsintx}[1996/10/28]%	dummy (at present)
\RequirePackage{amscd}[1999/11/29]%	Commutative Diagrams
\RequirePackage{amsxtra}[1999/11/15]%	extra math symbols
\RequirePackage{upref}[1999/11/29]%	upright references in \emph mode
\RequirePackage{amsthm}[2000/06/02]%	theorem environment
%	End of full AMS-LaTeX

%	Beginning of full AMSFonts
\RequirePackage{amsfonts}[1995/02/01]%	fonts
\RequirePackage{amssymb}[1995/02/01]%	bold symbols
% \RequirePackage{eucal}[1995/01/06]
% \RequirePackage{cmmib57}[1995/01/26]
%	End of full AMSFonts

% ---------> End of Package Loading part


% =================== declaration of options ===============================

% =================== execution of options =================================



% =================== main code ============================================


%++++++++++++++++++++++++++++++++++++++----------------------------------------
%	BEGINNING of theorem-like environments (BOZHO'S DEFINITIONS):
%++++++++++++++++++++++++++++++++++++++----------------------------------------

%	\swapnumbers
	\theoremstyle{plain} % default
%	\theorembodyfont{\rmfamily\itshape} % requires amstex package

\newtheorem{Axm}{Axiom}[section]			% [ AXIOM ]

\newtheorem{Post}{Postulate}[section]			% [ PRINCIPLE ]

\newtheorem{Prin}{Principle}[section]			% [ PRINCIPLE ]

\newtheorem{Thm}{Theorem}[section]			% [ THEOREM ]

\newtheorem{Prop}{Proposition}[section]			% [ PROPOSITION ]

\newtheorem{Lem}{Lemma}[section]     			% [ LEMMA ]

\newtheorem{Cor}{Corollary}[section] 			% [ COROLLARY ]

\newtheorem{Crit}{Criterion}[section] 			% [ Criterion ]

%	Versions
% to produce variants of the theorems-like ... numbering Redefine \theVar... !!!
% \theoremstyle{plain}	% plain, definition, or remark
\newtheorem{VarThm}{Theorem}			% [THEOREM ]
\newtheorem{VarAxm}{Axiom}			% [ AXIOM ]
\newtheorem{VarPost}{Postulate}			% [ POSTULATE ]
\newtheorem{VarPrin}{Principle}			% [ PRINCIPLE ]
\newtheorem{VarProp}{Proposition}		% [ PROPOSITION ]
\newtheorem{VarLem}{Lemma}		     	% [ LEMMA ]
\newtheorem{VarCor}{Corollary}			% [ COROLLARY ]
\newtheorem{VarCrit}{Criterion} 		% [ Criterion ]

	\theoremstyle{definition}	%*********************************
%	\theorembodyfont{\rmfamily\slshape} % requires amstex package

\newtheorem{Defn}{Definition}[section]			% [ DEFINITION ]

\newtheorem{Conj}{Conjecture}[section] 			% [ CONJECTURE ]

\newtheorem{Asse}{Assertion}[section]			% [ ASSERTION ]

\newtheorem{Conc}{Conclusion}[section]			% [ CONCLUSION ]

\newtheorem{Exmp}{Example}[section]			% [ EXAMPLE ]

\newtheorem{Exrc}{Exercise}[section]			% [ EXERCISE ]

\newtheorem{Prob}{Problem}[section] 			% [ PROBLEM ]

%	Versions
\newtheorem{VarDefn}{Definition}		% [ DEFINITION ]
\newtheorem{VarConj}{Conjecture} 		% [ CONJECTURE ]
\newtheorem{VarAsse}{Assertion}			% [ ASSERTION ]
\newtheorem{VarConc}{Conclusion}		% [ CONCLUSION ]
\newtheorem{VarExmp}{Example}			% [ EXAMPLE ]
\newtheorem{VarExrc}{Exercise}			% [ EXERCISE ]
\newtheorem{VarProb}{Problem} 			% [ PROBLEM ]

	\theoremstyle{remark}	%*********************************
%	\theorembodyfont{\rmfamily\upshape} % requires amstex package

\newtheorem{Rem}{Remark}[section] 			% [ REMARK ]

\newtheorem{Comm}{Comments}[section] 			% [ COMMENT ]

\newtheorem{Note}{Note}[section] 			% [ NOTE ]

\newtheorem{Case}{Case}[section] 			% [ CASE ]

%	Versions
\newtheorem{VarRem}{Remark} 			% [ REMARK ]
\newtheorem{VarComm}{Comments} 			% [ COMMENT ]
\newtheorem{VarNote}{Note} 			% [ NOTE ]
\newtheorem{VarCase}{Case} 			% [ CASE ]

%	\theoremstyle{remark}	%*********************************
 %	\theorembodyfont{\sffamily\upshape}  requires amstex package

\newtheorem{Alg}{Algorithm}[section] 			% [ ALGORITHM ]

\newtheorem{Claim}{Claim}[section] 			% [ CLAIM ]

\newtheorem{Ack}{Acknowledgment}[section]		% [ ACKNOWLEDGMENT ]

%	Versions
\newtheorem{VerAlg}{Algorithm} 			% [ ALGORITHM ]
\newtheorem{VerClaim}{Claim} 			% [ CLAIM ]
\newtheorem{VerAck}{Acknowledgment}		% [ ACKNOWLEDGMENT ]

%++++++++++++++++++++++++++++++++++++++----------------------------------------
%	END of theorem-like environments (BOZHO'S DEFINITIONS):
%++++++++++++++++++++++++++++++++++++++----------------------------------------


%  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
%  BEGINNING of New and Renew commands and environments (BOZHO'S DEFINITIONS):

% use to produce exceptional versions of existing environments
%	REDEFINE the corresponding counters and there appearance!!
\newenvironment{Version}[1][Choose a NAME and FONT!]%
		{\vspace{0.0ex}\noindent{#1}}%
		{\vspace{1.2ex}}

% {Proof}: an alternative is the AMS (amsthm package) "proof" environment
%	   but it often does not end correctly.
\newenvironment{Proof}[1][Proof.]%
		{\vspace{0.0ex}\noindent\textit{#1\hspace*{0.5em}}}%
		{\hspace*{0pt}\hfill\QEDsymbol\vspace{1.2ex}}
% Q.E.D.  symbols. The AMS package `amsthm` provides \qed producing \qedsymbol
\providecommand{\qedsymbol}{$\square$}  % Requires amssymb package
    \newcommand{\QEDsymbol}{$\square$}	% Requires amssymb package
\providecommand{\qed}{\hfill\qedsymbol}
%\newcommand{\QED}[1][6]{\hfill\hbox{\vrule width #1pt height #1pt depth 0pt}}
\newcommand{\QED}{\hspace*{0pt}\hfill Q.E.D.}

%	Matrices, vectors, vector spaces:
\newcommand{\Mat}[1]{\boldsymbol{#1}}	% Matrix of a vector or linear map
\newcommand{\Vect}[1]{\boldsymbol{#1}}	% Vector quantity
\providecommand{\openone}{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}
					% Unit Matrix (Taken from RevTeX)

%	Mathematical Abbreviations:
\newcommand{\id}{\mspace{2mu}\mathsf{i}\mspace{-1mu}\mathsf{d}}% identity map
\newcommand{\const}{\mathrm{const}}	%  constant quantity
\newcommand{\iu}   {\mathrm{i}}		%  Imaginary Unit
\newcommand{\e}{\ensuremath{\mathrm{e}}}%  base of natural logarithms
\newcommand{\Lied}{\mathcal{L}}         % Lie Derivative

%	Field (in algebraic sense)
\newcommand{\field}[1][K]{\ensuremath{\mathbb{#1}}}

%	(Lie) groups and algebras
\newcommand{\G}{\mathrm{G}}		% Group
\newcommand{\g}{\mathfrak{g}}		% Lie algebra of a group G
%
\DeclareMathOperator{\Int}{Int}	% Internal automorphisms x\mapsto axa^{-1}
\DeclareMathOperator{\Ad}{Ad}	% Adjoint representation of a Lie group
\DeclareMathOperator{\ad}{ad}	%   ... the differential of \Ad ...
% \newcommand{\exp}{\mathrm{exp}}	% exponential mapping; LaTeX predefined
\DeclareMathOperator{\Lie}{Lie}	% Lie algebra of ...
%
\DeclareMathOperator{\GL}{\mathrm{GL}}	% General Linear (Group)
\DeclareMathOperator{\gl}{\mathfrak{gl}}% General Linear (Group) Lie algebra
%
\DeclareMathOperator{\Ug}{\mathrm{U}}	% Unitary (Group)
\DeclareMathOperator{\ug}{\mathfrak{u}}	% Unitary (Group) Lie algebra
\DeclareMathOperator{\SU}{\mathrm{SU}}	% Special Unitary (Group)
\DeclareMathOperator{\su}{\mathfrak{su}}% Special Unitary (Group) Lie algebra
%
\DeclareMathOperator{\Og}{\mathrm{O}}	% Orthogonal (Group)
\DeclareMathOperator{\og}{\mathfrak{o}}	% Orthogonal (Group) Lie algebra
\DeclareMathOperator{\SO}{\mathrm{SO}}	% Special Orthogonal (Group)
\DeclareMathOperator{\so}{\mathfrak{so}}% Special Orthogonal (Group) Lie algebra

%	Ordinary and Partial differentials:
\newcommand{\ordinary}{\mathrm{d}}	% ordinary differential
\newcommand{\od}      {\ordinary}       % Ordinary Differential (abbreviation)
% the command \partial is predefined    % Partial Differential
\newcommand{\pd}      {\partial}        % Partial Differential  (abbreviation)
\DeclareMathOperator{\odo}{\mathrm{d}}  % Ordinary Differential Operator
\DeclareMathOperator{\pdo}{\partial}    % Partial Differential Operator
\newcommand{\Id}[1][\,]{#1\mathrm{d}}	% Integral Differential

%	left index(es) in \mathnormal:
\newcommand{\lindex}[3][{}]{{}\mspace{3mu}{\vphantom{#1}}%
	_{#2}^{#3}\mspace{-0.8mu}{#1}}
%	left index(es) in \mathrm:
\newcommand{\lindexrm}[3][{}]{{}\mspace{3mu}{\vphantom{#1}}%
	_{\mathrm{#2}}^{\mathrm{#3}}\mspace{-0.8mu}{#1}}
%	right indexes with space bellow/over them
\newcommand{\Sprindex}[3][]{{#1}_{\hphantom{#3}#2}^{#3}}% Sup: space bellow it
\newcommand{\Sbrindex}[3][]{{#1}_{#2}^{\hphantom{#2}#3}}% Sub: space over it

%	Differential geometry
%	Fields (in differential geometric sense; particular sections}
\newcommand{\Fields}[1]{\ensuremath{\mathfrak{#1}}}
%       Algebra (as an algebraic structure)
\newcommand{\algebra}[1]{\ensuremath{\boldsymbol{#1}}}
%	Dimensions with respect to some (algebraic) field
\newcommand{\dimR}[1][R]{\dim_{\field[#1]}}% Real dimension
\newcommand{\dimC}[1][C]{\dim_{\field[#1]}}% Complex dimension
\newcommand{\dimK}[1][K]{\dim_{\field[#1]}}% Dimension with resp. to \field K
%	Christoffel symbols
\newcommand{\Christoffel}[3][]{\genfrac{\{}{\}}{0pt}{}{#2}{#3}_{#1}}


%	Physical Abbreviations:
\newcommand{\ih}{\mathrm{i}\hbar}% imaginary unit times Planck constant/{2\pi}
\newcommand{\iih}{\frac{1}{\ih}} % the inverse to \ih (see previous command)
			      % =1/(imaginary unit times Planck constant/2\pi)

%	Additional description of the title page
   \newcommand{\ShortTitle}[1]{\textbf{\textsl{Short title:} #1}}
   \newcommand{\Subject}[1]   {\textit{\textbf{Subject Classes:\\}#1}}
   \newcommand{\MSC}[2][2000]{\begin{tabular}{@{}c@{}}%
	\textit{\textbf{#1 MSC numbers:}}\\  \itshape #2    \end{tabular} }
   \newcommand{\PACS}[2][2003]{\begin{tabular}{@{}c@{}}%
	\textit{\textbf{#1 PACS numbers:}}\\ \itshape #2    \end{tabular} }
   \newcommand{\KeyWords}[1]  {\textit{\textbf{Key-Words:\\}#1}}

%	Nice formatting of (subject) index
\newcommand{\ff}[1]{#1ff}% e.g., "64ff" in the Index ("and following pages")
\newcommand{\nn}[1]{#1n} % e.g., "64n"  in the Index (footNote item)
\newcommand{\defined}[2][{}]{\textbf{#2}#1{}}%
				% Index page(s) where a concept is defined
% use "see{\emph{also} ...}" to obtain the version "see also"
% of the "see" command for MakeIndex

%	Sans Serif capital Latin letters & their possible usage
\newcommand{\Dsf}{\mathsf{D}}		% Derivation along paths of sections
					% or of the tensor algebra
\newcommand{\Esf}{\mathsf{E}}
\newcommand{\Isf}{\mathsf{I}}		% Parallel transport assigned to a
					% transport I along paths
\newcommand{\Psf}{\mathsf{P}}		% Parallel transport along path
\newcommand{\Rsf}{\mathsf{R}}		% parallel transport cuRvature

%	Black board capital Latin letters & their possible usage
\newcommand{\Pbb}{\mathbb{P}}		% Parallelism structure

%	Slashed and backslashed symbols (normal and diagonal variant):
\newcommand{\slashed}[1]{\ensuremath{#1\mspace{-9.2mu}/}}%slashed small symbol
\newcommand{\Slashed}[1]{\ensuremath{#1\mspace{-11.3mu}/}}% for BIG symbols
\newcommand{\backslashed}[1]{\ensuremath{#1\mspace{-9.4mu}\backslash}}% small
\newcommand{\Backslashed}[1]{\ensuremath{#1\mspace{-11.0mu}\backslash}}% BIG
\newcommand{\varslashed}[1]{\ensuremath{#1\mspace{-11.8mu}\diagup}} % var
\newcommand{\varSlashed}[1]{\ensuremath{#1\mspace{-14.9mu}\diagup}} % var
\newcommand{\varbackslashed}[1]{\ensuremath{#1\mspace{-12.4mu}\diagdown}}% var
\newcommand{\varBackslashed}[1]{\ensuremath{#1\mspace{-14.2mu}\diagdown}}% var

%	Latin abbreviations:
\newcommand{\ie} {i.e.\ }		%  That is (Latin: It Est)
\newcommand{\cf} {cf.\ }		%  Compare with (Latin: ConFer)
\newcommand{\eg} {e.g.\ }		%  For example (Latin: Exapli gratia)
\newcommand{\etc}{etc.\ }		%  And so on (Latin: et cetera)
\newcommand{\viz}{viz.\ }		%  That is (Latin: vidilizet)
\newcommand{\vv}{v.v.\ }		%  Inversely (Latin: vice versa)
\newcommand{\vive}{\emph{vice versa}}	%  Inversely (Latin: vice versa)

%	No breakable dashes:
\newcommand{\Ndash}{\nobreakdash--}	     %no breakable dash (Number range)
\newcommand{\ndash}{\nobreakdash-\hspace{0pt}}% no breakable normal dash

%	Special brackets
\newcommand{\slb}{[\![}       % Special Left  Bracket (like black board "[")
\newcommand{\srb}{]\!]}	      % Special Right Bracket (like black board "]")
\newcommand{\clangle}{\mspace{0.0mu}\prec\mspace{-5.3mu}}%calligraphic \langle
\newcommand{\crangle}{\mspace{-5.3mu}\succ\mspace{0.0mu}}%calligraphic \rangle

%	Misc
\newcommand{\bs}[1]{\boldsymbol{#1}}	% Boldface Symbols (AMS style)
\newcommand{\xlrarrow}[2][12mu]{% extensible left-right arrow over an object
    \overset{%
	\xleftarrow{\hphantom{#2}\mspace{-2mu}}%
	\mspace{-#1}\mspace{-#1}%
	\xrightarrow{\mspace{2mu}}%
   	    }%
   	    {{#2}}%
   }% optional argument: equals 1/2 times the overlapping of the 2 arrows
    % Ignore the internal overfull warnings!


%		END of new commands and environments
%  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<


%	>=================================>
%	BEGINNING of Mathematical Operators  (requires amsopn.sty package)

% Set theory and topology
 \DeclareMathOperator{\diam}{diam}	% diameter

% Mappings
 \DeclareMathOperator{\Ker}{Ker}	% Kernel of a mapping
% \DeclareMathOperator{\ker}{ker}	% kernel	; LaTeX predefined
\DeclareMathOperator{\Image}{\mathrm{Im}}% Image of a mapping
 \DeclareMathOperator{\rank}{rank}	% rank
 \DeclareMathOperator{\im}{im}		% image
%\DeclareMathOperator{\Im}{Im}		% Image;LaTeX predefined,Complex numbers
 \DeclareMathOperator{\degree}{deg}	% degree        ; LaTeX predefined
 \DeclareMathOperator{\pr}{pr}		% projection
 \DeclareMathOperator{\End}{End}	% set of Endomorphisms
 \DeclareMathOperator{\Aut}{Aut}	% set of Automorphisms

% Complex numbers:
 \DeclareMathOperator{\Imp}{Im}		% Imaginary part; LaTeX predefined
 \DeclareMathOperator{\Rep}{Re}		% Real part     ; LaTeX predefined

% Matrices/Linear operators
 \DeclareMathOperator{\tr}{tr}		% Trace (spur) of matrices/operators
 \DeclareMathOperator{\Tr}{Tr}		% Trace (spur) of matrices/operators
 \DeclareMathOperator{\Spur}{Sp}	% Spur (trace) of matrices/operators
 \DeclareMathOperator{\diag}{diag}	% Diagonal matrix

% Fibre bundles
 \DeclareMathOperator{\Sec}{Sec}	% set of Sections of a fibre bundle
 \DeclareMathOperator{\PSec}{PSec}	% set of Sections of ... along Paths
 \DeclareMathOperator{\Morf}{Mor}	% set of Morphisms of a fibre bundle
 \DeclareMathOperator{\PMorf}{PMor}	% set of Morphisms of... along Paths
 \DeclareMathOperator{\MorfSec}{MorSec}	% set of Morphism of the set of
					% Sections of a bundle
 \DeclareMathOperator{\Diff}{Diff}	% set of Diffeomorphisms
 \DeclareMathOperator{\Lift}{Lift}	% set of Liftings
 \DeclareMathOperator{\PLift}{PLift}	% set of Liftings along Paths
 \DeclareMathOperator{\Path}{P}		% set of Paths
 \DeclareMathOperator{\PF}{PF}		% set of Functions along Paths
% \DeclareMathOperator{\pr}{\mathrm{pr}}% projection mapping, defined above
 \DeclareMathOperator{\morf}{mor}	% Morphism fibre bundle of a bundle

% Exponents
%\DeclareMathOperator{\exp}{exp}	% exponent: LaTeX predefined
 \DeclareMathOperator{\Exp}{Exp}	% exponent
 \DeclareMathOperator{\Pexp}{Pexp}	% Path/Time ordered exponent
 \DeclareMathOperator{\Texp}{Texp}	% Time/Path ordered exponent

%	Differential vector operators:
\DeclareMathOperator{\diver}{div}	% divergence
\DeclareMathOperator{\grad}{grad}	% gradient
\DeclareMathOperator{\rot}{rot}		% rotor

%	Ordinary and Partial differentials: 	these are defined above
% \DeclareMathOperator{\odo}{\mathrm{d}}  % Ordinary Differential Operator
% \DeclareMathOperator{\pdo}{\partial}    % Partial Differential Operator

% Misilaneous
 \DeclareMathOperator{\sign}{sign}	% sign of a real number
 \DeclareMathOperator{\sgn} {sgn}	% sign of a real number
 \DeclareMathOperator{\Supp}{Supp}	% Support of a function
 \DeclareMathOperator{\supp}{supp}	% Support of a function

%	<==================================<
%	END of Mathematical Operators  (requires amsopn.sty package)



%	>=================================>
%	BEGINNING of Text representation (lay-out) commands

% Putting a point after section number in the text titles (not in .toc file)
\renewcommand{\@seccntformat}[1]{\csname the#1\endcsname.\hspace{1.0em}}

%	Equation numbering and referring
\numberwithin{equation}{section}
% \renewcommand{\theequation}{\thesection.\arabic{equation}}
% \setcounter{equation} {0}	% put this after every \section command
				% to be used only with the above eqn. No. def

% Equation reference number in (...)
% AMS (amsmath package) provides \eqref{...} in upright/roman font in (...)
%	but it often add unnecessary space at the end of the reference.
\providecommand{\eqref}[1]{{\rmfamily\upshape(\ref{#1})}}
     \newcommand{\eref}[1]{{\rmfamily\upshape(\ref{#1})}}

% Table reference number
\numberwithin{table}{section}
\renewcommand{\thetable}{\arabic{section}.\arabic{table}}

%	<==================================<
%	END of Text representation (lay-out) commands


%***************************************************************************%
%**       END of LaTeX2e style package of BOZHIDAR ZAKHARIEV ILIEV        **%
%***************************************************************************%
\end{filecontents}
% End of file Bozho-LaTeX-Style.sty


% Beginning of file Bozho-LaTeX-Logo.sty
\begin{filecontents}{Bozho-LaTeX-Logo.sty}
%***************************************************************************%
%**     BEGINNING of LaTeX2e logo package of BOZHIDAR ZAKHARIEV ILIEV     **%
%**	Filename: Bozho-LaTeX-Logo.sty                                            **%
%**	Author (and responsible for all errors): Bozho                    **%
%***************************************************************************%
%**                                                                       **%
%**       This file defines the commands \bozho and \BOZHO                **%
%**       which are the logos of BOZHIDAR ZAKHARIEV ILIEV:                **%
%**                                                                       **%
%**	  Laboratory of Mathematical Modeling in Physics		  **%
%**	  Institute for Nuclear Research and Nuclear Energy               **%
%**	  Bulgarian Academy of Sciences                                   **%
%**	  Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria          **%
%**	  E-mail address: bozho@inrne.bas.bg                              **%
%**       URL: http://theo.inrne.bas.bg/~bozho/		       		  **%
%**                                                                       **%
%**                                                                       **%
%**		    (DON'T USE WITHOUT PERMISSION)			  **%
%**                                                                       **%
%***************************************************************************%

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%%   Acute accent  \'     Left paren    \(     Right paren   \)          %%
%%   Asterisk      \*     Plus          \+     Comma         \,          %%
%%   Minus         \-     Point         \.     Solidus       \/          %%
%%   Colon         \:     Semicolon     \;     Less than     \<          %%
%%   Equals        \=     Greater than  \>     Question mark \?          %%
%%   Commercial at \@     Left bracket  \[     Backslash     \\          %%
%%   Right bracket \]     Circumflex    \^     Underscore    \_          %%
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%%   Right brace   \}     Tilde         \~}                              %%
%%                                                                       %%
%% 		End of \CharacterTable                                   %%
%%                                                                       %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\NeedsTeXFormat{LaTeX2e}
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% testing for different font sizes:
%\Huge\BOZHO\hspace{2em}\huge\BOZHO\hspace{1.9em}\LARGE\BOZHO\\
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%***************************************************************************%
%**     	END of LaTeX2e logo package of BOZHIDAR ZAKHARIEV ILIEV   **%
%***************************************************************************%
\end{filecontents}
% End of file Bozho-LaTeX-Logo.sty


% Beginning of file cite.sty
\begin{filecontents}{cite.sty}
%     C I T E . S T Y
%
%     version 3.4  (Jan 1995)
%
%     Compressed, sorted lists of numerical citations: [11-16]
%     see also OVERCITE.STY and DRFTCITE.STY
%
%     Copyright (C) 1989-1995 by Donald Arseneau
%     These macros may be freely transmitted, reproduced, or modified for
%     non-commercial purposes provided that this notice is left intact.
%
%     Instructions follow \endinput.
%  ------------------------------------
%   Handle optional variations:  [verbose,nospace,space],
%   \citeform,\citeleft,\citeright,\citemid,\citepunct
%
%   Set defaults:

%   [ on the left.
\@ifundefined{citeleft}{\let\citeleft=[}{}

%   ] on the right:
\@ifundefined{citeright}{\let\citeright=]}{}

%   , (comma space) before note
\@ifundefined{citemid}{\def\citemid{,\penalty\@medpenalty\ }}{}

%   , (comma thin-space) between entries; [nospace] eliminates the space
\@ifundefined{citepunct}{
   \def\citepunct{,\penalty\@m\hskip.13emplus.1emminus.1em}%
  }{}

%   Each number left as-is:
\@ifundefined{citeform}{\def\citeform{}}{}

%   Do not repeat warnings.  [verbose] reverses
\let\oc@verbo\relax

\@ifundefined{DeclareOption}{}%
{ \toks@={\def\oc@verbo#1#2#3#4{}}
  \DeclareOption{verbose}{\the\toks@}
  \DeclareOption{nospace}{\def\citepunct{,\penalty\@m}}
  \DeclareOption{space}{\def\citepunct{,\penalty\@highpenalty\ }}
  \ProvidesPackage{cite}[1995/01/30 \space  v 3.4]
  \ProcessOptions }

%----------------------
% \citen uses \nocite to ignore spaces after commas, and write the aux file
% \citation. \citen then loops over the citation tags, using \@make@cite@list
% to make a sorted list of numbers.  Finally, \citen executes \@citelist to
% compress ranges of numbers and print the list. \citen can be used by itself
% to give citation numbers without the brackets and other formatting; e.g.,
% "See also ref.~\citen{junk}."
%
\edef\citen{\noexpand\protect \expandafter\noexpand\csname citen \endcsname}

\@namedef{citen }#1{%
\nocite{#1}% ignores spaces, writes to .aux file, returns #1 in \@tempa!!
\@tempcntb\m@ne    % \@tempcntb tracks highest number
\let\@h@ld\relax   % nothing held from list yet
\let\@citea\@empty % no punctuation preceding first
\let\@celt\delimiter % an unexpandable, but identifiable, token
\def\@cite@list{}% % empty list to start
\@for \@citeb:=\@tempa \do{\@make@cite@list}% make a sorted list of numbers
% After sorted citelist is made, execute it to compress citation ranges.
\@tempcnta\m@ne    % no previous number
\let\@celt\@compress@cite \@cite@list % output number list with compression
\@h@ld}% output anything held over

% Aliases:
\let\citenum\citen
\let\citeonline\citen

% For each citation, check if it is defined and if it is a number.
% if a number: insert it in the sorted \@cite@list
% otherwise: output it immediately.
%
\begingroup \catcode`\_=8 % Just in case it was changed
\toks@={
\def\@make@cite@list{%
 \expandafter\let \expandafter\@B@citeB
          \csname b@\@citeb\@extra@b@citeb \endcsname
 \ifx\@B@citeB\relax % undefined: output ? and warning
    \@citea {\bf{?}}\let\@citea\citepunct
    \@warning {Citation `\@citeb' on page \thepage\space undefined}%
    \oc@verbo \global\@namedef{b@\@citeb\@extra@b@citeb}{?}%
 \else %  defined               % remove previous line to repeat warnings
    \ifcat _\ifnum\z@<0\@B@citeB _\else A\fi % a positive number, put in list
       \@tempcnta\@B@citeB \relax
       \ifnum \@tempcnta>\@tempcntb % new highest, add to end (efficiently)
          \edef\@cite@list{\@cite@list \@celt{\@B@citeB}}%
          \@tempcntb\@tempcnta
       \else % arbitrary number: insert appropriately
          \edef\@cite@list{\expandafter\@sort@celt \@cite@list \@gobble @}%
       \fi
    \else % citation is not a number, output immediately
       \@citea \citeform{\@B@citeB}%
       \let\@citea\citepunct
 \fi\fi}
}
\expandafter \endgroup \the\toks@  % restore _ catcode

% Check if each number follows previous and can be put in a range
%
\def\@compress@cite#1{%  % This is executed for each number
  \advance\@tempcnta\@ne % Now \@tempcnta is one more than the previous number
  \ifnum #1=\@tempcnta   % Number follows previous--hold on to it
     \ifx\@h@ld\relax    % first pair of successives
        \edef\@h@ld{\@citea \noexpand\citeform{#1}}%
     \else               % compressible list of successives
        \def\@h@ld{\hbox{--}\penalty\@m \citeform{#1}}%
     \fi % (using \hbox avoids easy \exhyphenpenalty breaks)
  \else   %  non-successor -- dump what's held and do this one
     \@h@ld \@citea \citeform{#1}\let\@h@ld\relax
  \fi \@tempcnta#1\let\@citea\citepunct
}

% \@sort@celt inserts number (\@tempcnta) into list of \@celt{num} (#1{#2})
% \@celt must not be expandable; list should end with two vanishing tokens.
%
\def\@sort@celt#1#2{\ifx \@celt #1% parameters are \@celt {num}
     \ifnum #2<\@tempcnta % number goes later in list
        \@celt{#2}%
        \expandafter\expandafter\expandafter\@sort@celt % continue
     \else % number goes here
        \@celt{\number\@tempcnta}\@celt{#2}% stop comparing
  \fi\fi}


% Make \cite robust. "\cite " is the default \cite.
%
\edef\cite{\noexpand\protect\expandafter\noexpand\csname cite \endcsname}

\@namedef{cite }{\@ifnextchar [{\@tempswatrue\@citex}{\@tempswafalse\@citex[]}}

%  Make \@citex refer to \citen:
%
\def\@citex[#1]#2{\@cite{\citen{#2}}{#1}}%

%  Replacement for \@cite which defines the formatting normally done
%  around the citation list.  Put a penalty before the citation.  Also,
%  adjust the spacing: if no space or if there is extra space due to some
%  punctuation, then change to one inter-word space.  The way to change
%  this is by changing \citeleft, \citemid, and \citeright; but in extreme
%  cases it might be necessary to redefine the whole macro.
%
\def\@cite#1#2{\leavevmode
  \@tempskipa\lastskip \edef\@tempa{\the\@tempskipa}\unskip
  \ifnum\lastpenalty=\z@ \penalty\@highpenalty \fi
  \ifx\@tempa\@zero@skip \spacefactor1001 \fi % if no space before, set flag
  \ifnum\spacefactor>\@m \ \else \hskip\@tempskipa \fi
  \citeleft{#1\if@tempswa \citemid #2\fi}\citeright
  \spacefactor\@m % punctuation in note doesn't affect outside
}

\edef\@zero@skip{\the\z@skip}

% \nocite: This is changed to ignore *ALL* spaces and be robust.  The
% parameter list, with spaces removed, is `returned' in \@tempa, which
% is used by \citen.
%
\edef\nocite{\noexpand\protect\expandafter\noexpand\csname nocite \endcsname}

\@namedef{nocite }#1{%
\edef\@tempa{\@ignsp#1 \! }% remove *ALL* spaces from parameter list
\if@filesw \immediate \write \@auxout {\string \citation {\@tempa}}\fi}

% for ignoring *ALL* spaces in the input.  This presumes there are no
% \outer tokens and no \if-\fi constructs in the parameter.  Spaces inside
% braces are retained.
%
\def\@ignsp#1 {\ifx\!#1\@empty\else #1\expandafter\@ignsp\fi}

\let\nocitecount\relax  % in case \nocitecount was used for drftcite

\@ifundefined{@extra@b@citeb}{\def\@extra@b@citeb{}}{}
%  in case no fancy bib package (chapterbib) defines this

\endinput
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


                      CITE.STY

Modify LaTeX's normal citation mechanism to:

o Sort citation numbers into ascending order; print non-numbers before
  numbers.  Compress lists of three or more consecutive numbers to one
  number range which can be split, with difficulty, after the dash.
  All numbers should be greater than zero.
  E.g., if you used to get [7,5,6,?,4,9,8,Einstein,6], then this style
  will give you [?,Einstein,4-6,6-9].

o Allow, but strongly discourage, line breaks within a series of
  citations.  Each number is separated by a comma and a small space.
  A break at the beginning of an optional note is discouraged also.

o Put a highpenalty before the citation (unless you specifically forbid it
  with ~ ).  Also, adjust the spacing: if there is no space or if there is
  extra space due to some punctuation, then change to one inter-word space.
  E.g.,   Something really stupid\cite{Larry,Curly,Moe}.
  A space inserted here ---------^

o Define \citen to get just the numbers (and write to the aux file)
  without the brackets and extra formatting (`\citen{x}' -> `12').  The
  aliases \citenum and \citeonline are also defined the same as \citen

o `Citation...undefined' warnings are only given once per undefined
  citation tag.  In the text, missing numbers are represented with a
  bold `?' at the first occurrence, and with a normal `?' thenceforth.

o Make \nocite, \cite, and \citen all ignore spaces in the input tags.

Linebreaks are allowed with extra-high penalties (1000) after dashes
and commas; these may have to be taken if TeX sees no other viable
breakpoints.  If you think citations are being split unnecessarily,
try using \sloppy or a sloppypar environment.

Although each \cite command sorts its numbers, better compression
into ranges can usually be achieved by carefully selecting the order
of the \bibitem entries, or the order of initial citations when using
bibtex.  Having the entries presorted will also save processing time,
especially for long lists of numbers.

Customization:
~~~~~~~~~~~~~~
There are several commands that you may redefine (using \renewcommand)
to change the formatting of citation lists:

 command       function                   default
----------    -----------------------    ----------------------------
\citeform     reformats every entry      nothing
\citepunct    printed between numbers    comma + penalty + thin space
\citeleft     left delimiter of list     [
\citeright    right delimeter of list    ]
\citemid      printed before note        comma + space

Under LaTeX2e, there are three options for \usepackage{cite}:
[verbose] causes warnings for undefined citations to be repeated each
          time they are used.
[nospace] eliminates the spaces after commas in the number list.
[space] uses a full inter-word space with no penalty after the commas

Some examples:
\renewcommand\citeform[1]{\romannumeral 0#1}} % roman numerals [i,vi]
\renewcommand\citeform[1]{(#1)} % parenthesized numbers [(1)-(5)]
\renewcommand\citeform {\thechapter.}  % by chapter: [2.18-2.21]
\renewcommand\citepunct{,} % no space and no breaks at commas
\renewcommand\citemid{; }  % semicolon before optional note
\renewcommand\citeleft{(}  % parentheses around list
\renewcommand\citeright{)} % parentheses around list

The appearance of the whole citation list is governed by \@cite, so
for more extensive changes to the formatting, redefine \@cite.

Related Note:  cite.sty does not affect the numbering format of the
bibliography; the "[12]" style is still the default.  To change that
format (with or without cite.sty) you can redefine \@biblabel, including
   \renewcommand\@biblabel[1]{#1.}
in your personal style file, or with, for example,
   \makeatletter \renewcommand\@biblabel[1]{(#1)} \makeatother directly
in your document.  If these do not work, your LaTeX and/or document
style are very outdated.

\@extra@b@citeb is a hook for other style files to further specify
citations; for example, to number by chapter (see chapterbib.sty).

See also overcite.sty and drftcite.sty for superscript and draft
(draught) mode citations.

ROBUST!

% Version 1989: Original.
% Version 1991: Ignore spaces after commas in the parameter list. Move most of
% \citen into \@cmpresscites for speed.
% Version 1992: Use \citepunct for commas so it is easier to change.
%
% Version 3.0 (1992):  Rewrite, including sorting.  Make entries like "4th"
% be treated properly as text.
% 3.1: Bug fixes (Joerg-Martin Schwarz also convinced me to use \ifcat)
% 3.2: Supress repetitions of warning messages. Include \@extra@b@citeb hook.
% 3.3: Handle LaTeX2e options. Introduce various customization hooks.
% 3.4: Heuristics to avoid removing \hspace glue before the \cite. Make \nocite
%      ignore spaces in list, simplify. Aliases for \citen. Compatability with
%      amsmath (which defines \over).
%
% Send problem reports to asnd@Reg.triumf.ca
%
% test integrity:
% brackets:  round, square, curly, angle:   () [] {} <>
% backslash, slash, vertical, at, dollar, and: \ / | @ $ &
% hat, grave, acute (apostrophe), quote, tilde, under:   ^ ` ' " ~ _
\end{filecontents}
% End of file cite.sty


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\documentclass[12pt,titlepage,a4paper]{article}
%
\usepackage{bozhomac}	% loads: full AMSLaTeX and AMSFonts
\usepackage{bozhlogo}	% the logo of BOZHO commands \BOZHO and \bozho
% !!!!!!!!!!!!	 when sending files - input the files themselves here!!!
%	other \usepackage commands
% \usepackage{showkeys}	% for draft viewing and printing
\usepackage{cite}	% for condensing sequential cited works




%		formatting the TITLE page	>========>
%
\title{\bfseries	\vspace*{-1.678902345in}
{\huge  }% main title
% \\ \vspace{0.22ex} \\	\vspace{1.1ex}	{\LARGE     }% subtitle
}

\vspace{1.7ex}

\author{
Bozhidar Z.\ Iliev
\thanks{Laboratory of Mathematical Modeling in Physics,
Institute for Nuclear Research and \mbox{Nuclear} Energy,
Bulgarian Academy of Sciences,
Boul.\ Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria}
\thanks{E-mail address: bozho@inrne.bas.bg}
\thanks{URL: http://theo.inrne.bas.bg/$\sim$bozho/}
}

\date{}

% 	Put bellow any additional title page info
%	and (partially) remove it when sending the file
%
%
%		End of the title page           <========<




\listfiles			% lists all of the files being used


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%      Specific settings for this work only      %%%%%%%%%%%%%%
%%%%%%%%
% New commands


  						%%%%%%
%+++++++>>>	Beginning of page layout	+++++++>>>
\pagestyle{myheadings}
\markright{\underline{\itshape\bfseries Bozhidar Z. Iliev:
	\upshape\sffamily\bfseries  SHORT TITLE
}}

%	For maximal and reasonable usage of the a4 paper  for transparencies:
 \topmargin -20mm %-16mm
 \addtolength{\textheight}{63mm}%{33.1mm}
 \addtolength{\textwidth}{64.4mm}
 \oddsidemargin -20.4mm
 \evensidemargin -20.4mm
% \addtolength{\textwidth}{46mm}
% \oddsidemargin -9mm
% \evensidemargin -9mm
%-------<<<	End of page layout		-------<<<

%%%%%%%%								%%%%%%
%%%%%%%%%%%%%%%%    End of the specific settings for this work   %%%%%%%%%%%%%
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%	End of the PREAMBLE  of the BOZHO'S article style   <<--<--<--<--<--<
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\begin{document}		% BEGINNING OF THE DOCUMENT






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%%%%%									%%%%%
%%%%%		actual beginning of the document			%%%%%
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\LARGE \sffamily
% \renewcommand{\theDefn}{\arabic{Defn}}
% \renewcommand{\theequation}{\arabic{equation}}
%  \renewcommand{\hbar}{\bigl(\frac{h}{2\pi}\bigr)}
% \newcommand{\squareNEW}{\pd_\varkappa \pd^\varkappa}
\renewcommand{\thepage}{\Large\bfseries\arabic{page}}

	\begin{center}

{\huge   % main title
TITLE
} \\[2ex]
{\Large\bfseries Bozhidar Z.\ Iliev\\[1ex]
\large
Laboratory of Mathematical Modeling in Physics\\
Institute for Nuclear Research and \mbox{Nuclear} Energy,
Bulgarian Academy of Sciences\\
Boul.\ Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria \\[0.4ex]
E-mail address: bozho@inrne.bas.bg\hfill
URL: http://theo.inrne.bas.bg/$\sim$bozho/
} \\[2ex]
\textbf{Abstract}\\[1ex]
    \begin{minipage}{0.85\textwidth}
%	\\[-2.0ex]
    \end{minipage}
	\end{center}


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%			BEGINNING OF SECTION~\ref{Sect1}
\textbf{1. }



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%			BEGINNING OF SECTION~\ref{Sect2}
\textbf{2. }




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%			BEGINNING OF SECTION~\ref{Sect3}
\textbf{3. }





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\textbf{4. }





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%			BEGINNING OF SECTION~\ref{Sect5}
\textbf{5. }





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\textbf{6. }





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% \section*{Acknowledgments}
%			BEGINNING OF SECTION~\ref{Acknowledgments}

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% \addcontentsline{toc}{section}{References}
% \bibliography{bozhopub,bozhoref}
% \bibliographystyle{unsrt}
% \addcontentsline{toc}{subsubsection}{This article ends at page}
%\addtocontents{toc}{}
%			END OF BIBLIOGRAPHY
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\end{document}

>=>=>=>=>=>
 	   PUT HERE ANY ADDITIONAL APPLICABLE INFORMATION

							 <=<=<=<=<=<=<=<=<=<

=============================================================================
=============================================================================
>>>		I do not want to profit from the Science,                 <<<
>>		but the Science may profit from me!                        <<
>						    Bozho                   <
=============================================================================
=============================================================================
