%**********************************************************************
%   Sample paper  for Banach Center Publ.   April 3, 2023
%**********************************************************************

%%% IMPORTANT: In case of any problems contact publ@impan.pl  %%%

% Using pdflatex is preferred

\documentclass{bcp}
\usepackage{amsmath,amsthm}
\usepackage{amssymb}

%% Add only when there are figures:
\usepackage{graphicx}


%% For the letters of the Polish alphabet, we are using
%%\usepackage[T1]{fontenc}
%% E.g. the name "Zoladz" is coded \.Zo{\l}\k{a}d\'z

%% For itemized environments, you can use
\usepackage{enumitem}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%% Your numbering system for theorems etc. may be different from the one suggested below. %%%

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}


%%% A numbered theorem with a fancy name: %%%
\newtheorem{maintheorem}[theorem]{Main Theorem}

%%% Definitions, remarks and examples are NOT italicized:  %%%

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}

%%% An unnumbered object: %%%
\newtheorem*{xrem}{Remark}

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%%%% Put your macros here:

\newcommand{\length}{\operatorname{length}}



\begin{document}

\keywords{put your keywords here, separated by commas.}
\mathclass{Primary ???; Secondary ???.}


\abbrevauthors{A. Kowalska and R. {\L}ab\k{e}cki}
\abbrevtitle{Sample paper for bcp}

\title{Sample paper for BCP}


\author{Anna Kowalska}
\address{Institute of Mathematics, Polish Academy of Sciences, 
00-656 Warszawa, Poland\\
ORCID: 0000-0002-2345-6789\quad\ E-mail: kowalska@impan.gov.pl}

\author{Ryszard {\L}ab\k{e}cki}
\address{Institute of Mathematics, A. Mickiewicz University, 
61-614 Pozna\'n, Poland\\
ORCID: 0000-0003-2346-9876\quad\ E-mail: labecki@amu.edu.pl}

\maketitlebcp


\begin{abstract}
This is a sample paper for Banach Center Publications.
\end{abstract}


\section{Introduction} 
Papers  for Banach Center
Publications  should be
prepared using the bcp.cls style file. In case of any problems contact publ@impan.pl. 
The publisher will create page proofs for final review by the author.

 The affiliations  and e-mail addresses of all authors must be provided. Please provide the ORCID number if available.  

Please check whether your article has an abstract, 2020 Mathematics Subject Classification (with one code declared as ``primary'') and a list of key words and phrases.
 
Put all acknowledgments, including those concerning grants, into an unnumbered ``Acknowledgments'' subsection just before the references.

In the running heads, the authors' first names are replaced by initials, and
the titles are strongly abbreviated. Provide an abbreviation of the title of
no more than 40 characters. Write ``A. Kowalska et al.'' in the running head if there are three authors or more.

\section{Theorems}
The statements of theorems,  propositions etc. are set in
italics. In definitions, only the term being defined is emphasized. Remarks
and examples are set in Roman type. 



\begin{definition}
A system $S$ is said to be \emph{self-extensional} if 
\begin{enumerate}[label=\rm(\roman*),leftmargin=*,widest=iii]
\item first item,
\item second item,
\item third item.
\end{enumerate}
\end{definition}




\begin{theorem}[Identity Principle, see also {\cite[Theorem 5]{HillDow}}]\label{T:1}
  If $A=B$, then
\[
  C=D.
\]
\end{theorem}

\begin{proof}  Observe that
\begin{align}
\big(\tfrac{1}{2}(y+\length_iA_i)\big)^2
&= \binom{a+b}{c-d}\notag\\
&\quad + \Big(\prod_{i=1}^n A_i\Big)^2 + \bigg(\frac{u}{v}\bigg)^n\notag\\
&\overset{\alpha}= 
\begin{cases}
\sqrt[3]{2/\!\sin x} &\text{if $x \in (0, \pi)$,}\\
0 &\text{otherwise.}
\end{cases}\label{E:1}
\end{align}
Now apply induction on $n$ to \eqref{E:1}.
\end{proof}

\begin{xrem}
Theorem~\ref{T:1} was independently proved in \cite{Kow}.
\end{xrem}



\begin{maintheorem} 
Here comes the statement of a numbered theorem with a fancy name.
\end{maintheorem}

For many examples of codes of  multiline formulas, see
\[
  \texttt{https://www.impan.pl/en/publishing-house/for-authors}.
\]

The \texttt{eqnarray} construction leads to well-known mistakes---if you have learnt it, just forget it.

Do not leave ``overflows'' in formulas; if the formula is too wide, break it yourself into lines or, e.g., shorten it by introducing some symbols. 

Do not re-invent {\LaTeX}; before using your own construction or creating a new symbol
look up Gr\"atzer \cite{Gratzer}---most probably, your intended construction or symbol is already there.

Add small spaces \texttt{\textbackslash,} only exceptionally, e.g. before differentials.


\section{Sectioning}
Here is an example of a subsection:


\subsection{A subheading} This paragraph is included only to illustrate
the appearance of a~subheading.

\subsubsection{A subsubheading} This paragraph illustrates a subsubheading.




\section{Figures}
Preferably, figures should be prepared as pdf, jpg or eps files.
All figures will be printed black and white; colours will only appear in the online version.

Avoid very thin lines and very small lettering.

Check whether all fonts used are embedded.



\begin{figure}[h]
\caption{A figure caption}
\end{figure}



\subsection*{Acknowledgments}

Research of A. F. Nowak was partially supported by NSF grant No. XXX.




\begin{thebibliography}{4}

%% Use the widest label as parameter.

%% Arrange the reference items in alphabetical order.

%% Reference items can be numbered or have labels of your choice, as below.

%% Abbreviations of journal names should follow MathSciNet.

%% Only the title is italicized; boldface is not used.

%% Our software will add links to many articles (those having doi numbers).
  
%% Do not give the issue number unless the issues are paginated separately.


\bibitem{Gratzer}
G. Gr\"atzer,
\emph{More Math into \LaTeX},
4th ed., Springer, Berlin, 2007.


\bibitem{HillDow}
R. Hill and A. Dow,
\emph{A ground-breaking achievement},
J.~Differential Equations 15 (1982), 197--211.

\bibitem{Kow}
J. Kowalski,
\emph{A very interesting paper},
in: Algebra, Analysis and Beyond (Nowhere, 1973),
E.~Fox et al. (eds.),
Lecture Notes in Math. 867, Springer, Berlin, 1974, 115--124.

\bibitem{Nov}
A. S. Novikov,
\emph{Another fascinating article},
Uspekhi Mat. Nauk 23 (1980), no.~3, 112--134 (in Russian);
English transl.: Russian Math. Surveys 23 (1980), 572--595.

\bibitem{Russ}
B. Russell,
\emph{A new theorem},
arXiv:2012.9876 (2020).


\end{thebibliography}


\end{document}



