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\begin{document}
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\title[Title (or Abbreviation of the title)]
{Title of the paper} 
\author{Name(s) of the author(s)}    %%%  by our standard, for multiple authors, the names of the authors must be written in alphabetical order %%%  
\address{Complete address of the first author}
\email{email address of the first author} %%(if possible)}
\urladdr{URL address (if need be)}
\address{Complete address of the second author}
\email{email address of the second author}  %%(if possible)}
\urladdr{URL address (if need be)}
\address{etc.}

\dedicatory{Dedication (if need be)}

\begin{abstract}
Text of the abstract (not exceeding 13 lines, references and big formulas in abstract should be avoided).
\end{abstract}

\subjclass{Mathematics Subject Classification code(s)}
\keywords{Modulus function, summability method, \ldots}
\date{xxxx (will be added by the editors)}

\thanks{https://doi.org/10.12697/ACUTM.xxxx.yy.zz (will be added later)}
\thanks{Corresponding author: (in the case of more than one author)}  %% For thanks we suggest to use the section Acknowledgement(s) at the end of the paper  %%

\maketitle

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




\section{Introduction}
Overview of the area. Relation to the current study. Contribution of the paper. The paper should not exceed 25 pages. We refer to the literature by numbers, like \cite{C}--\cite{MKZK} or sometimes including name (see Connor~\cite{C}). All items in the References should be referred to. A digital paper should be given together with the link to its DOI, like in \cite{K1}, \cite{KV}. The link opens when clicking on DOI. Where available, the URLs of other papers are welcome (see \cite{O1}).
 
\section{The section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\N=\{1,2,\dots\}$ and let $\K$ be the field of real numbers $\R$ or complex numbers $\C$. 
We specify the domains of indices by the symbols $\inf$, $\sup$, $\lim$ and $\sum$ only if they are different from $\N$.
We use also the notation $\R^+=[0,\infty)$.

\begin{definition}
A function $\phi:\R^+\to\R^+$ is called a {\it modulus function} (or, simply, a {\it modulus}), if
\begin{enumerate}
\item[(M1)] $\phi(t)=0 \iff t=0$,
\item[(M2)] $\phi(t+u)\le\phi(t)+\phi(u)\ \;(t,u\in\R^+)$,
\item[(M3)] $\phi$ is non-decreasing,
\item[(M4)] $\phi$ is continuous from the right at $0$.
\end{enumerate}
\end{definition}

\section{The next section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The first subsection}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The most common summability method is  matrix method defined by 
an infinite scalar matrix $A=(a_{nk})$. A well-known example of a regular matrix method is Ces\`aro method $C_1$ defined by the matrix $C_1=(c_{nk})$, 
where, for any $n\in\N$, 
$$
c_{nk}=
\begin{cases}
   n^{-1} &\text{if} \ k\le n, \\ 
   0 &\text{otherwise}.
\end{cases}
$$

\begin{theorem}
Let $p\ge 1$ and let $A=(a_{nk})$ be a non-negative infinite matrix. Suppose that  
$\lambda\subset s$ is a solid AK space   with respect to absolutely monotone F-seminorm $g_{_{\lambda}}$. 
If the matrix $A$ is row-finite $($i.e., for any $n\in\N$ there exists an index $k_n$ with $a_{nk}=0\quad(k>k_n))$ and 
\begin{equation}\label{eq-1}
((a_{nk})^{1/p})_{n\in\N}\in\lambda\quad(k\in\N),
\end{equation} 
then $(U\!{\lambda}^p[A],g_{_{\lambda,\tilde{A}}}^p)$ is the AK space, where
\begin{align*}
g_{_{\lambda,\tilde{A}}}^p(\bu)&=g_{_\lambda}\left(\tilde{A}^{1/p}\left(|\bu^2|^p\right)\right),\\
\tilde{A}^{1/p}\left(|\bu^2|^p\right)&=\left(\sup_i\left(\sum_ka_{nk}|u_{ki}|^p\right)^{1/p}\right)_{n\in\N}.
\end{align*} 
\end{theorem}
\begin{proof}
To prove the equality $\lim_m \bu^{2[m]}=\bu^2$ in $U\!{\lambda}^p[A]$, we use the inequality 
\begin{equation}\label{eq-2}
\begin{aligned}
&g_{_{\lambda,\tilde{A}}^p}\left(\bu^2-\bu^{2[m]}\right)\\
&\le\sum_{n=1}^s g_{_{\lambda}}\left(\sup_i \left(\sum_{k=m+1}^\infty a_{nk}|u_{ki}|^p\right)^{1/p}e^n\right)\\
&\quad +g_{_{\lambda}}\left(\left(\overbrace{0,\dots,0}^{s},\sup_i\left(\sum_{k=m+1}^\infty a_{s+1,k}|u_{ki}|^p\right)^{1/p},\dots\right)\right)\\
&=G_{sm}^1+G_{sm}^2.
\end{aligned}
\end{equation}
By \eqref{eq-2}, using also \eqref{eq-1}, we have $\dots$.
\end{proof}

\begin{proposition}
Text of the proposition.
\end{proposition}

\begin{corollary}
Text of the corollary.
\end{corollary}

\begin{example}
Text of the example.
\end{example}

\begin{remark}
Text of the remark.
\end{remark}

\section*{Acknowledgements} %(if need be)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The research of the first (second) author was (partially) supported by $\cdots$. The authors thank the anonymous referee(s) for $\cdots$.



\begin{thebibliography}{9}
%References should be listed in alphabetical order; 
%abbreviations of journal names should follow "Mathematical Reviews". 

\bibitem{AGL}
V.~Abramov, S.~Groote, and P.~Lätt,  \emph{Algebra with ternary cyclic relations, representations and quark model}, Proc. Est. Acad. Sci. 72 (2023), 61–76.

\bibitem{C}
J.~Connor, \emph{A topological and functional analytic approach to statistical convergence}. In: W.~O. Bray et al. (eds) Analysis of Divergence, 
Appl. Numer. Harmon. Anal., 403--413, Birkh\"auser, Boston, 1999.
\bibitem{KKM} 
M.~Kilp, U.~Knauer, and A.~V.~Mikhalev, \emph{Monoids, Acts and Categories}, Expositions in Mathematics {\bf 29}, Walter de Gruyter, Berlin, 2000.

\bibitem{K1}
C.~Kızılateş, \emph{A new generalization of Fibonacci hybrid and Lucas hybrid numbers},
Chaos Solitons Fractals  {\bf 130} (2020), 109449, 5 pp. \href{https://doi.org/10.1016/j.chaos.2019.109449}{DOI}


\bibitem{KV}
G. Kudryavtseva and V. Laan, \emph{Globalization of partial actions of semigroupsemp}, arXiv:2206.06808v2 [math.RA] (2022), 15~pp. \href{https://doi.org/10.48550/arXiv.2206.06808}{DOI}

\bibitem{MKZK}
V.~L.~Makarov, L.~V.~Kantorovich, V.~I.~Zhiyanov, and A.~G.~Khovanskii, \emph{The principle of differential optimization in application to a one-product dynamic model of the economy}, 
Sibirsk. Mat. Zh. {\bf 19} (1978), 1053--1064. (Russian)
\bibitem{ME}
\emph{Mathematics Encyclopedia}, M.~S.~Shapiro (ed), Doubleday, Garden City, New York, 1977. 
\bibitem{O1} 
E.~Oja, \emph{On bounded approximation properties of Banach spaces}, Banach Center Publ. {\bf 91} (2010), 219--231. \href{https://www.impan.pl/en/publishing-house/banach-center-publications/all/91//86365/on-bounded-approximation-properties-of-banach-spaces}{URL}
%\bibitem{O2} 
%E.~Oja, \emph{Inner and outer inequalities with applications to approximation properties}, Trans. Amer. Math. %
%Soc. {\bf 363} (2011), 5827--5846.

\bibitem{OP}
E.~Oja and M.~P\~oldvere, \emph{Norm-preserving extensions of functionals and denting points of convex sets},
Math. Z. {\bf 258} (2008), 333--345.



\end{thebibliography}

\end{document}
