\documentclass[reqno,A4paper]{amsart}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{enumerate}
\usepackage{mathrsfs} %%\mathcal classic,  \mathscr decorative
%\usepackage[mathscr]{eucal} %%instead of ``mathrsfs''  you can use this package ``eucal'':
                             %%\mathcal classic,  \mathscr simple, but it enables to switch
                             %%\mathcal to the bold by using command \boldsymbol
\usepackage{eqlist}
\usepackage{array}

\setlength{\textwidth}{150mm}
\setlength{\textheight}{206mm}
\setlength{\oddsidemargin}{5mm}
\setlength{\evensidemargin}{5mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Options of setting environments for theorems, definitions, etc.
%and their counters -- a hint.

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
%[section] at the end
%subordinates the counter ``theorem'' to the counter ``section''
\newtheorem{prop}[theorem]{Proposition}
%[theorem] after setting the counter ``prop'':
%the counter ``prop'' joins counter ``theorem''
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corol}{Corollary}[theorem]
%the counter ``corol'' is subordinate to the counter ``theorem''
\newtheorem*{illust}{Illustration}
%* the environment ``illust'' is not counted

%Theorem, Proposition, Lemma, Corollary, etc. must be ``under'' \theoremstyle{plain}
%``counting'' is free

\theoremstyle{definition}
\newtheorem{definition}{Definition}
%the counter ``definition'' does not join any counter and
%is not subordinated to any counter
\newtheorem{remark}{\textup{Remark}} %\textup for ``Remark'' is required
\newtheorem{example}{\textit{Example}} %\textit for ``Example'' is required
\newtheorem*{acknowledgement}{\textup{Acknowledgement}}
%\textup and * for Acknowledgement is required

%Definition, Remark, Example, Acknowledgement, etc.
%must be ``under'' \theoremstyle{definition}
%``counting'' is free (except Acknowledgement)


%\newtheorem{name of a new (just setting) counter}{text which will appear
%in the output}[name of an existing counter to which
%the new counter will be subordinated]

%\newtheorem{name of a new (just setting) counter}[name of an existing counter
%which the new counter joins]{text which will appear in the output}


\numberwithin{equation}{section}
%subordinates the counter ``equation'' to the counter ``section''

\renewcommand{\labelenumi}{\textup{(\arabic{enumi})}}
%or \roman \Roman \alph \Alph instead of \arabic; \textup is required
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%These comands are inclued into our style Math. Slovaca:
\newcommand{\dd}{\mathrm{d}}    %%differential
\newcommand{\e}{\mathrm{e}}     %%a base of the natural logarithm
\newcommand{\ii}{\mathrm{i}}    %%complex unit

\newcommand{\NN}{\mathbb{N}}    %%natural numbers
\newcommand{\ZZ}{\mathbb{Z}}    %%integers
\newcommand{\QQ}{\mathbb{Q}}    %%rational numbers
\newcommand{\RR}{\mathbb{R}}    %%real numbers
\newcommand{\CC}{\mathbb{C}}    %%complex numbers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title[A brief pattern for writing a paper in \LaTeX 2e]%
{A brief pattern for writing a paper in \LaTeX 2e correctly, comfortably and elegantly}
\author[Kvetoslava Dvorakova \and Flora Sch\"on]%
{Kvetoslava Dvo\v{r}\'akov\'a* \and Flora Sch\"on**}

\newcommand{\acr}{\newline\indent}

\address{\llap{*\,}Mathematical Institute\acr
                   Slovak Academy of Sciences\acr
                   \v{S}tef\'anikova 49\acr
                   SK--814 73 Bratislava\acr
                   SLOVAKIA}
\email{dvorak@mat.savba.sk}

\address{\llap{**\,}Department of Mathematics\acr
                    Comenius University\acr
                    Mlynsk\'a Dolina\acr
                    SK--842 15 Bratislava\acr
                    SLOVAKIA}
\email{Flora.Schon@fmph.uniba.sk}

%%\acr is not required (if you do not need to see a column);
%%in our style \\ makes a column automatically

\thanks{This work was supported by ...... Grant No. ...., etc. (optional)}

\subjclass[2010]{Primary 06B05, 06D35; Secondary 35R30, 39A10} %Secondary is optional
\keywords{MV-algebra, effect algebra, Navier-Stokes equation, oscillatory solution,
asymptotic behaviour}

\begin{abstract}
A brief abstract without bibliography marks [.]:
Having an MV-algebra, we can restrict
its binary operation addition
only to the pairs of orthogonal elements.
The resulting structure is known as an effect algebra,
precisely distributive lattice effect algebra.
Basic algebras were introduced as a generalization of MV-algebras.
Hence, there is a natural question
what an effect-like algebra can be reached
by the above mentioned construction
if an MV-algebra is replaced by a basic algebra.
This is answered in the paper and
properties of these effect-like algebras are studied.
\end{abstract}

\maketitle

\section{Options of setting environments for theorems, definitions, etc.\\
         and their counters}

\begin{remark}
\quad

1. formula. \emph{Subordinate counting}:
\begin{verbatim}
\newtheorem{new counter}{text}[existing counter]
\end{verbatim}
\emph{new counter}: a name of the new (just setting) counter and environment
\newline
\emph{text}: a text which appears in the output
\newline
\emph{existing counter}:
a name of the existing counter to which the new counter will be subordinate

\medskip
2. formula. \emph{Joint counting:}
\begin{verbatim}
\newtheorem{new counter}[existing counter]{text}
\end{verbatim}
\emph{new counter}: a name of the new (just setting) counter and environment
\newline
\emph{existing counter}: a name of the existing counter which the new counter joins
\newline
\emph{text}: a text which appears in the output

\medskip
3. formula. \emph{Independent counting:}
\begin{verbatim}
\newtheorem{new counter}{text}
\end{verbatim}
\emph{new counter}: a name of the new (just setting) counter and environment
\newline
\emph{text}: a text which appears in the output

\medskip
4. formula. \emph{No counting:}
\begin{verbatim}
\newtheorem*{new counter}{text}
\end{verbatim}
\emph{new counter}: a name of the new (just setting) counter and environment
\newline
\emph{text}: a text which appears in the output
\end{remark}


\section{Examples of some environments:
\\
definitions, theorems, \dots, and itemization}


Effect algebras were introduced by D. J. Foulis and M. K. Bennett \cite{2}.
An effect algebra is a partial algebra which serves as a generalization
of the set of Hilbert-space effects, i.e. self-adjoint operators
on a Hilbert space (see e.g. \cite{1} for the motivation in full details).
For reader's convenience, we recall the definition of effect algebra.


\begin{theorem}
Let $\mathcal{A}=(A;\oplus,\neg,0)$ be a basic algebra. The induced lattice
weak effect algebra $\mathcal{E}(A)=(A;+,0,1)$ is commutative if and only if
$\mathcal{A}$ is quasi-commutative.
\end{theorem}

\begin{proof}
Assume that $a+b$ is defined. Then $a\leq b'$, i.e. $a\leq\neg b$ and
$b\leq\neg a$ in $\mathcal{A}$ and, due to quasi-commutativity of $\mathcal{A}$
we have
$$
a+b=(\neg a)^{b}=(\neg b)^{a}=b+a.
$$
Conversely, if $\mathcal{E}(A)$ is commutative, $x,y\in A$ and $y\leq x$, then
$\neg x\leq\neg y$ thus $y+x'$, $x'+y$ are defined and $(\neg y)^{(\neg x)}=
(\neg y\vee\neg x)^{(\neg x)}=y\oplus\neg x=y+x'=x'+y=\neg x\oplus y=x^{y}$
thus $\mathcal{A}$ is quasi-commutative.
\end{proof}


Due to the mutual one-to-one correspondence mentioned in Remark 3, we can conclude:

\begin{corol}
A lattice weak effect algebra $\mathcal{E}$ is commutative if and only if the induced
basic algebra $\mathcal{A}(E)$ is quasi-commutative.
\end{corol}


\subsection*{Example of using the environment ``itemize''}


\begin{definition}
An \emph{effect algebra} is a partial algebra $\mathcal{E}=(E;+,0,1)$ of type
$(2,0,0)$ satisfying the axioms
\begin{itemize}
\item[(EA1)] if $x+y$ is defined, then $y+x$ is defined and $x+y=y+x$;
\item[(EA2)] if $x+y$ and $(x+y)+z$ are defined, then $y+z$ and $x+(y+z)$ are defined
and $x+(y+z)=(x+y)+z$;
\item[(EA3)] for each $a\in E$ there exists a unique $b\in E$ such that $a+b=1$;
let us denote this $b$ by $a'$;
\item[(EA4)] if $1+a$ is defined, then $a=0$.
\end{itemize}
\end{definition}

\begin{remark}[Upright marks of items in italic environments]
To avoid italic item marks in an italic environment, use the environment ``itemize''
instead of ``enumerate'' provided item marks are ``hand-made''.
Or use (in the preamble)
\begin{verbatim}
\renewcommand{\labelenumi}{\textup{(\arabic{enumi})}}
or \roman \Roman \alph \Alph instead of \arabic
\end{verbatim}
\end{remark}

\begin{illust}
In an italic environment:
\begin{itemize}
\item[(EA1)] if $x+y$ is defined, then $y+x$ is defined and $x+y=y+x$;
\item[(EA2)] if $x+y$ and $(x+y)+z$ are defined, then $y+z$ and $x+(y+z)$ are defined
and $x+(y+z)=(x+y)+z$;
\item[(EA3)] for each $a\in E$ there exists a unique $b\in E$ such that $a+b=1$;
let us denote this $b$ by $a'$;
\item[(EA4)] if $1+a$ is defined, then $a=0$.
\end{itemize}
\end{illust}

\subsection*{Example of using the environment ``eqlist''}


\begin{remark}[Long marks of items]
For long marks of items, you can use the package ``eqlist'' and subsequently
the environment ``eqlist'' as in the following definition.
\end{remark}


\pagebreak
\begin{definition}
A partial algebra $\mathcal{E}=(E;+,0,1)$ of type $(2,0,0)$ is called
a \emph{weak effect algebra} if it satisfies the following conditions:
\begin{eqlist}
\item[(WEA1)] for each $a\in E$ there exists a unique $b\in E$, denoted as $a'$,
such that $a+b=1=b+a$;
\item[(WEA2)] if $a+1$ or $1+a$ is defined, then $a=0$;
\item[(WEA3)] $a=x+a$ implies $x=0$, $a=x+(y+a)$ implies $x=0=y$;
\item[(WEA4)] if $b=x+a$ and $c=y+b$, then there exists $z\in E$ with $c=z+a$;
\item[(WEA5)] if $b=x+a$, then there exists $y\in E$ with $a'=y+b'$;
\item[(WEA6)] if $b=x+a$ and $a+z$, $b+z$ are defined, then there exists $v\in E$
such that $b+z=v+(a+z)$;
\item[(WEA7)] if $a+b=c$, then $a'=c'+b$.
\end{eqlist}
\end{definition}

\begin{remark}[``eqlist'' in an italic environment]
In an italic environment use:
\begin{verbatim}
\begin{eqlist}[\def\makelabel #1{\textup{#1}}]
\end{verbatim}

You can even use the following:
\begin{verbatim}
\begin{eqlist}[\def\makelabel #1{\textup{(WEA#1)}}]
\item[1]
\item[2]
 ...
\end{eqlist}
\end{verbatim}
\end{remark}

\begin{illust}
In an italic environment we can use:
\begin{eqlist}[\def\makelabel #1{\textup{(WEA#1)}}]
\item[1] for each $a\in E$ there exists a unique $b\in E$, denoted as $a'$,
such that $a+b=1=b+a$;
\item[2] if $a+1$ or $1+a$ is defined, then $a=0$;
\item[3] $a=x+a$ implies $x=0$, $a=x+(y+a)$ implies $x=0=y$;
\item[4] if $b=x+a$ and $c=y+b$, then there exists $z\in E$ with $c=z+a$;
\item[5] if $b=x+a$, then there exists $y\in E$ with $a'=y+b'$;
\item[6] if $b=x+a$ and $a+z$, $b+z$ are defined, then there exists $v\in E$
such that $b+z=v+(a+z)$;
\item[7] if $a+b=c$, then $a'=c'+b$.
\end{eqlist}
\end{illust}


\begin{remark}
By using the environment ``itemize'' (or ``enumerate'') for long marks of items
we have the following unsightly result.
\end{remark}

\begin{definition}
A partial algebra $\mathcal{E}=(E;+,0,1)$ of type $(2,0,0)$ is called
a \emph{weak effect algebra} if it satisfies the following conditions:
\begin{itemize}
\item[(WEA1)] for each $a\in E$ there exists a unique $b\in E$, denoted as $a'$,
such that $a+b=1=b+a$;
\item[(WEA2)] if $a+1$ or $1+a$ is defined, then $a=0$;
\item[(WEA3)] $a=x+a$ implies $x=0$, $a=x+(y+a)$ implies $x=0=y$;
\item[(WEA4)] if $b=x+a$ and $c=y+b$, then there exists $z\in E$ with $c=z+a$;
\item[(WEA5)] if $b=x+a$, then there exists $y\in E$ with $a'=y+b'$;
\item[(WEA6)] if $b=x+a$ and $a+z$, $b+z$ are defined, then there exists $v\in E$
such that $b+z=v+(a+z)$;
\item[(WEA7)] if $a+b=c$, then $a'=c'+b$.
\end{itemize}
\end{definition}

\subsection*{Examples of using the environment ``enumerate''}
\quad

Actually we show that:
\begin{enumerate}
\item up to isomorphism, every one-generated free $F_p(1)$ algebra is
a relative MV-subalgebra of the cyclic free MV-algebra $F(1)$, for any $p$;
\item up to isomorphism, the set of one-generated free $F_p(1)$ algebras,
$p$ varying in the set of all positive integers, forms a directed
system in the category of relative MV-algebras;
\item up to isomorphism, each one-generated free $F_p(1)$ algebra is
a retractive subalgebra of $F(1)$, in the category of relative $MV$-algebras;
\item there is a family $\mathcal{D}=\{D_{p}\}_{p\in \NN}$
of finite sequences of elements of $Q\cap [0,1]$ (sub-Farey sequences),
such that each element $D_p\in \mathcal{D}$ allows us to cut out a relative
MV-subalgebra of $F(1)$, which is isomorphic to  $F_p(1)$.
\end{enumerate}
We shall refer to \cite{CDM} for any unexplained notion on
MV-algebras and, for a better readability of the paper, we confine
to Appendix the results, useful for our aims, which essentially
concern with elementary properties of the integer numbers.

\subsubsection*{Example of using locally redefined environment ``enumerate''}


\begin{prop}
For $x,y\in [c,d]$ we have
\emph{(by using ``enumerate'')}
\begin{enumerate}
  \renewcommand{\labelenumi}{\textup{(\alph{enumi})}}
  \item $x\dagger y= (x\oplus c^*y)\wedge d$;
  \item $\neg x =c\oplus x^*d$.
\end{enumerate}
\end{prop}

Or by using ``itemize'':

\begin{prop}
For $x,y\in [c,d]$ we have
\emph{(by using ``itemize'')}
\begin{itemize}
  \item[(a)] $x\dagger y= (x\oplus c^*y)\wedge d$;
  \item[(b)] $\neg x =c\oplus x^*d$.
\end{itemize}
\end{prop}

\begin{remark}
Redefinitions of the environments ``enumerate'' and ``itemize'' placed in preambles
``work'' in the whole of a paper.
\end{remark}

\begin{example}[Options of redefining labels]
\begin{verbatim}
 \renewcommand{\labelenumi}{\textup{(\alph{enumi})}}
 \renewcommand{\labelitemi}{(+)}
\end{verbatim}
Levels:
\begin{verbatim}
\labelenumi, \labelenumii, \labelenumiii, \labelenumiv
\labelitemi, \labelitemii, \labelitemiii, \labelitemiv
\end{verbatim}
A typeface can be set by using one of the following commands:
\begin{verbatim}
\arabic, \roman, Roman, \alph, \Alph
for ``itemize'' only a common mark for a level
\end{verbatim}

\begin{itemize}
  \item $x\dagger y= (x\oplus c^{*}y)\wedge d$;
  \item $\neg x =c\oplus x^{*}d$.
\end{itemize}

\begin{itemize}
 \renewcommand{\labelitemi}{$\star$}
  \item $x\dagger y= (x\oplus c^{*}y)\wedge d$;
  \item $\neg x =c\oplus x^{*}d$.
\end{itemize}

\begin{itemize}
 \renewcommand{\labelitemi}{(+)}
  \item $x\dagger y= (x\oplus c^{*}y)\wedge d$;
  \item $\neg x =c\oplus x^{*}d$.
\end{itemize}
\end{example}



\begin{remark}
In an ``upshape'' text we can use for item marks an ``ordinary'' expression:
\newline
If a weak effect algebra is commutative, then clearly (W7) implies (W5) thus
the axiom system can be simplified.

In an italic text we are forced to provide the upright script in the following way:
\begin{verbatim}
\thetag{W7},  or  \textup{(W7)},  or  \emph{(W7)},
\end{verbatim}
see the following corollary.
\end{remark}

\begin{corol}\label{corW}
A partial algebra $\mathcal{E}=(E;+,0,1)$ is a commutative weak effect algebra if
and only if it satisfies the axioms \emph{(W1)--(W4)}, \thetag{W6}, \textup{(W7)}
and
$$
x+y=y+x.
$$
\end{corol}


\section{Display structures, tagging and aligning multi-line structures}


Let $b(\cdot,\cdot,\cdot)$ be the continuous trilinear form on $V_{1,q}$ defined by
$$
b(u,v,w) =\sum_{i,j=1}^{n}\int_{\Omega }u_{i}
\frac{\partial v_{j}}{\partial x_{i}}w_{j} \, \dd x
\qquad \text{for all} \quad u,v,w\in V_{1,q}.
$$

Let  $T\in\mathbb{R}$, $T>0$, be fixed and $t\in [0,T]$. Let $\Omega$ be a bounded,
open domain in $\mathbb{R}^{n}$, $n=2,3$, whose boundary $\Gamma =\partial \Omega$
is Lipschitz continuity. For the functional setting of the problem (1)--(2)
we take a Banach space $L^{q}(\Omega)$ and a Sobolev space
\begin{equation}
W_{0}^{1,q}(\Omega)=\bigl\{u\in L^{q}(\Omega ):\
\partial_{i}u\in L^{q}(\Omega),\ \,u/_{\Gamma }=0\bigr\}.
\end{equation}

We define the next spaces, where $H$ and $V$ are Hilbert spaces and  $V_{1,q}$
is a Banach space.

\subsection*{The environments ``align'',  ``align*'' and ``aligned''}

\subsubsection*{Automatic tagging}

\begin{align}
H &=\bigl\{u\in L^{2}(\Omega ):\ \nabla \cdot u=0,\ \, u/_{\Gamma }=0\bigr\},
\\
 V &=\bigl\{u\in H_{0}^{1}(\Omega ):\ \nabla \cdot u=0\bigr\},
\nonumber\\
V_{1,q} &=\bigl\{u\in W_{0}^{1,q}(\Omega ):\ \nabla \cdot u=0\bigr\},
\end{align}

\subsubsection*{Author's tagging}

\begin{align}
H &=\bigl\{u\in L^{2}(\Omega ):\ \nabla \cdot u=0,\ \, u/_{\Gamma }=0\bigr\},
\notag\\ % or \nonumber\\
 V &=\bigl\{u\in H_{0}^{1}(\Omega ):\ \nabla \cdot u=0\bigr\},
\tag{$\alpha$}
\\
V_{1,q} &=\bigl\{u\in W_{0}^{1,q}(\Omega ):\ \nabla \cdot u=0\bigr\},
\tag{A}
\end{align}


\subsubsection*{Mixed tagging}

\begin{align}
H &=\bigl\{u\in L^{2}(\Omega ):\ \nabla \cdot u=0,\ \, u/_{\Gamma }=0\bigr\},
\nonumber\\  %or \notag\\
 V &=\bigl\{u\in H_{0}^{1}(\Omega ):\ \nabla \cdot u=0\bigr\},
\tag{$\ast$}
\\
V_{1,q} &=\bigl\{u\in W_{0}^{1,q}(\Omega ):\ \nabla \cdot u=0\bigr\},
\end{align}

\subsubsection*{No tagging}

\begin{align*}
H &=\bigl\{u\in L^{2}(\Omega ):\ \nabla \cdot u=0,\ \, u/_{\Gamma }=0\bigr\},
\\
 V &=\bigl\{u\in H_{0}^{1}(\Omega ):\ \nabla \cdot u=0\bigr\},
\\
V_{1,q} &=\bigl\{u\in W_{0}^{1,q}(\Omega ):\ \nabla \cdot u=0\bigr\},
\end{align*}


\subsubsection*{One tag by using the construction ``aligned'' within ``equation''}


\subsubsection*{Automatic:}
\begin{equation}
\begin{aligned}
H &=\bigl\{u\in L^{2}(\Omega ):\ \nabla \cdot u=0,\ \, u/_{\Gamma }=0\bigr\},
\\
 V &=\bigl\{u\in H_{0}^{1}(\Omega ):\ \nabla \cdot u=0\bigr\},
\\
V_{1,q} &=\bigl\{u\in W_{0}^{1,q}(\Omega ):\ \nabla \cdot u=0\bigr\},
\end{aligned}
\end{equation}


\subsubsection*{Author's:}
\begin{equation}
\begin{aligned}
H &=\bigl\{u\in L^{2}(\Omega ):\ \nabla \cdot u=0,\ \, u/_{\Gamma }=0\bigr\},
\\
 V &=\bigl\{u\in H_{0}^{1}(\Omega ):\ \nabla \cdot u=0\bigr\},
\\
V_{1,q} &=\bigl\{u\in W_{0}^{1,q}(\Omega ):\ \nabla \cdot u=0\bigr\},
\end{aligned}
\tag{A}
\end{equation}


\begin{remark}
The command
\begin{verbatim}
\numberwithin{equation}{name of existing counter}
\end{verbatim}
subordinates the counter ``equation'' to the existing counter which name
is within the 2nd braces.
Usually, it is the counter ``section''.
\end{remark}

\subsection*{The environments ``alignat'', ``alignat*'' and ``alignedat''}
\quad

Now, our optimal control problem, in the coefficient of kinematic viscosity
$a=a\left(|\nabla u|\right)$ in the modification of Navier-Stokes equations, is:
\begin{equation}
\inf_{a\in Q}\Bigl\{\textstyle J(a):\
J(a)=\int_{0}^{T}|u(a)(t)-z(t)|_{L^{2}(\Omega) }^{2}\,\dd t\Bigl\}
\footnote{In set notations we use the textstyle size for big operators.}
\tag{$\mathcal{P}$}
\end{equation}
subject to
\begin{alignat*}{3}
\frac{\partial u}{\partial t}-\nu_{0}\triangle u-\nu_{1}\nabla \cdot
\bigl(a\left(|\nabla u|\right) \nabla u\bigr)
+\sum_{i=1}^{n}u_{i}D_{i}u+\nabla p
&=0\qquad &&\text{in}\quad &&\Omega\times (0,T),
\\
\operatorname{div} u&=0\qquad &&\text{in}\quad &&\Omega \times (0,T),
\\
u &=0\qquad &&\text{on}\quad &&\Gamma \times (0,T),
\\
u(x,0) &=u_{0}\qquad &&\text{in}\quad &&\Omega,
\end{alignat*}
where $u(x,t) =\bigl(u_{1}(x,t),\dots ,u_{n}(x,t)\bigr)$  is velocity,  $p=p(x,t)$
is pressure, $u_{0}=u_{0}(x)$ is initial velocity.


\begin{remark}
For more (\emph{number}) aligned blocks we can use the environment ``alignat''
which has the following construction.
\begin{verbatim}
\begin{alignat}{number}
 ...
\end{alignat}
\end{verbatim}
odd \&'s  align parts of lines within a given block in this marked place,
\newline
even \&'s separate blocks.
\end{remark}

\begin{example}[``alignat'']
\begin{alignat}{2}
vs(n) &\geq 2^{\frac{3(n-4)}{4}} \cdot 12,&&
\tag{W}\\
s(n) &\geq 2^{\frac{3n}{4}-2} \cdot 31\qquad &&\text{if} \quad n \equiv 0 \pmod{4},
\\
s(n) &\geq 2^{\frac{3(n-1)}{4}-2} \cdot 3\qquad &&\text{if}\quad n \equiv 1 \pmod{4}.
\notag
\end{alignat}

\begin{alignat}{3}
          \Delta^iz_n&>0  \qquad &&\text{for} \quad n\geq n_0,\ \;&0&\leq i\leq p,\\
\noalign{\leftline{\text{and}}}
(-1)^{p+i}\Delta^iz_n&>0, \qquad &&\text{for} \quad n \geq 1,\ \; &p+1&\leq i\leq m-1
\end{alignat}
\end{example}


\begin{example}[``alignat*'']
\begin{alignat*}{3}
\mathbf{r}_l &= (0,\dots,0), \quad &  \mathbf{i}_l &= (0,0) \qquad &&(1 \leq l\leq L'),
\\
\mathbf{r}_l &= \mathbf{r}^*_{l-L'}, \quad & \mathbf{i}_l &=\mathbf{i}^*_{l-L'}
\qquad &&(L'+1 \leq l \leq L-3),
\\
\mathbf{r}_l &= (0,\dots,0), \quad & \mathbf{i}_l &= (q-1,0) \qquad &&(l = L-2),
\\
\mathbf{r}_l &= (0,\dots,0), \quad & \mathbf{i}_l &= (i_1,0) \qquad &&(l = L-1),
\\
\mathbf{r}_l &= (0,\dots,0), \quad & \mathbf{i}_l &= (0,0) \qquad &&(l= L),
\end{alignat*}
\end{example}



\begin{example}[``alignedat'' within ``equation'']
\begin{equation}
\begin{alignedat}{4}
N&\equiv 0\pmod{9}&&\iff &S(N)&\equiv 0\pmod{9} \qquad &&\text{if}\quad w_{i} = 1,
\\
N&\equiv 0\pmod{11}&&\iff &S(N)&\equiv 0\pmod{11} \qquad &&\text{if}\quad
w_{i} = (-1)^{i},
\\
N&\equiv 0\pmod{7}&&\iff & S(N)&\equiv 0\pmod{7} \qquad &&\text{if}\quad
w_{i} \in \{1, 3, 2, -1, -3, -2\}.
\end{alignedat}
\end{equation}
\end{example}


\subsection*{Functions with cases}

$$
h^{*}(v) =
\begin{cases}
h(v), & v\in A,\\
h(b)+(i-1)2n, & v=b_{i}\in B_{i},\\
h(c)+(x-i)2n, & v=c_{i}.
\end{cases}
$$

$$
\Gamma_{u,\kappa}=
\begin{cases}
0 &  \text{if}\ \, h(u,\kappa) \ \, \text{is odd},\\
2^{\ell(u,\kappa)+1}& \text{otherwise}.
\end{cases}
$$


\begin{equation}
\bigl|t\bigl([0,x)\times[0,y)\bigr)\bigr|-xy=
\begin{cases}
x-xy &\text{for}\ \,(x,y)\in A_1,\\
y-(1/q)-xy &\text{for}\ \,(x,y)\in A_2,\\
y+x-1-xy &\text{for}\ \,(x,y)\in A_3.
\end{cases}
\end{equation}


\begin{align}
Ax(t)&=
\begin{cases}
Ax(T), & t \in [T_0, T],\\
p(t) x(t-\tau) + \frac{9(1 -p)}{10},& t \geq T.
\end{cases}
\\[2mm]
Bx (t)&=
\begin{cases}
Bx(T)  &\text{for}\ \, t \in [T_0, T],\\
-\int^\infty_t \frac 1{r(s)}
\bigl(\int^\infty_s q(u) G\bigl(x(h(u))\bigr)\,\dd u\bigr)\,\dd s &
\\
-\int^\infty_t \frac{F(S)}{r(s)}\,\dd s, & \text{for}\ \,t \geq T.
\end{cases}
\end{align}

\section{About labels and refs}

\begin{remark}
In this section, there are shown some possibilities to put labels
into theorems, lemmas, \dots and equations and subsequently to refer to them in a text.
These constructions are not obligatory.
\end{remark}

As the nonlinear operator $\mathbf{A}$ is monotone and $z(0)=0$, we find
\begin{equation}\label{For11}
|z(t)|^{2}+2\nu_{0}\int_{0}^{t}\|z(s)\|^{2}\,\dd s
\leq 2 \int_{0}^{t} \bigl|b\bigl(z(s),u_{1}(s),z(s)\bigr)\bigr|\,\dd s
\end{equation}
We have also inequality
\begin{equation}
\|v\|_{L^{3}(\Omega)}
\leq |v|^{\frac{1}{2}}\|v\|_{L^{6}(\Omega)}^{\frac{1}{2}}
\end{equation}
(see \cite[pp.~86]{10}). As a consequence of these two last relations we have
\begin{equation}\label{For13}
\begin{aligned}
|b(z,u_{1},z)| &\leq \|u_{1}\|_{L^{6}(\Omega)}\|z\|\|z\|_{L^{3}(\Omega)}
\\
&\leq \|u_{1}\|_{L^{6}(\Omega)}\|z\||z|^{\frac{1}{2}}\|z\|_{L^{6}(\Omega)}^{\frac{1}{2}}
\\
&\leq \|u_{1}\|_{L^{6}(\Omega)}|z|^{\frac{1}{2}}\|z\|^{\frac{3}{2}}
\\
&\leq \varepsilon\|z\|^{2}+C_{\varepsilon}|z|^{2}\|u_{1}\|_{L^{6}(\Omega)}^{4}.
\end{aligned}
\end{equation}
If we put $\varepsilon=2\nu_{0}$ in the above inequality \eqref{For13}, then
from \eqref{For11}, we shall have
\begin{equation*}
|z(t)|^{2}\leq C_{\varepsilon} \int_{0}^{t} |z(s)|^{2}\|u_{1}\|_{L^{6}(\Omega)}^{4}.
\end{equation*}

In this paper, sufficient conditions are obtained, so that every solution of
\begin{equation}\label{en1}
\Delta^m(y_{n}-p_{n}y_{\tau(n)}) + q_{n}G(y_{\sigma(n)})-u_nH(y_{\alpha(n)})=f_{n},
\end{equation}
oscillates or tends to zero or $\pm \infty$ as $n\to\infty$,
where $\Delta$ is the forward difference operator given
by $\Delta x_{n}=x_{n+1}-x_{n}$, $p_{n}$, $q_{n}$, $u_{n}$ and $f_{n}$
are infinite sequences of real numbers with $q_{n}>0$, $u_{n} \geq 0$,
$G,H \in C(\RR ,\RR)$.

The results in \cite{tha52,mky} do not hold for a class of equations, where $G$ is either
linear or super linear, i.e., for example, when $G(u)=u$ or $G(u)=u^3$.
Here in this paper an attempt is made to fill this existing gap in literature
and obtain sufficient conditions for oscillation of solutions
of a more general equation \eqref{en1} under the weaker conditions (H4) or (H6).
Moreover, we observe that the existing papers in the literature do not have much
to offer when $p_n$ satisfies (A4), (A6) or (A7). In this direction we find that
the authors in \cite{pari17} have obtained sufficient conditions
for the oscillation of solutions of the equation
\begin{equation}\label{en7}
 \Delta^{m}(y_{n}-p_{n}y_{n-k}) + q_{n}G(y_{n-r})=0
\end{equation}
with (A4) or (A7) and have the following results.

\begin{theorem}\label{thm1.5p}
\emph{(\cite[Theorem 2.6]{pari17})}
Let $p_{n}$ satisfy \thetag{A7}.
If the condition
\begin{equation}\label{h2}
\sum_{n=n_0}^\infty q_n=\infty
\end{equation}
holds, then the following are valid statements.
\begin{itemize}
\item[(i)]  Every solution of \eqref{en7} oscillates, if $m$ is even.
\item[(ii)]  Every solution of \eqref{en7} oscillates or
$\liminf_{n\to\infty}y_n=0$ if $m$ is odd.
\end{itemize}
\end{theorem}

\begin{theorem}\label{thm1.6p}
\emph{(\cite[Theorem 2.7]{pari17})}
Let $p_{n}$ satisfy \thetag{A4}.
If \eqref{h2} holds, then the following statements are true.
\begin{itemize}
\item[(i)] Every solution of \eqref{en7} oscillates for $m$ even.
\item[(ii)] Every solution of \eqref{en7} oscillates or tends to zero
as $n\rightarrow\infty$ if $m$ is odd.
\end{itemize}
\end{theorem}

Theorems \ref{thm1.5p} and \ref{thm1.6p} are very useful for our next considerations.


\begin{remark}
We can refer not only to the environment ``equation'', but also to every particular
line in align, which is tagged and labeled.
\begin{align}
H &=\bigl\{u\in L^{2}(\Omega ):\ \nabla \cdot u=0,\ \, u/_{\Gamma }=0\bigr\},
\nonumber\\
 V &=\bigl\{u\in H_{0}^{1}(\Omega ):\ \nabla \cdot u=0\bigr\},
\tag{$\ast$}\label{al1}
\\
V_{1,q} &=\bigl\{u\in W_{0}^{1,q}(\Omega ):\ \nabla \cdot u=0\bigr\},
\label{al2}
\end{align}
The first line is not tagged, but the lines \eqref{al1} and \eqref{al2}
are tagged and labeled, that is why they can be referred to.
\end{remark}


\begin{thebibliography}{9}


\bibitem{1}
\uppercase{Dvure\v{c}enskij, A.---Pulmannov\'{a}, S.}:
\textit{New Trends in Quantum Structures},
Kluwer Acad. Publ./Ister Sci., Dordrecht/Bratislava, 2000.

\bibitem{2}
\uppercase{Foulis, D. J.---Bennett, M. K.}:
\textit{Effect algebras and unsharp quantum logics},
Found. Phys. \textbf{24} (1994), 1325--1346.


\bibitem{CDM}
CIGNOLI,~R. L. O.---D'OTTAVIANO,~I. M. L.---MUNDICI,~D.:
\textit{Algebraic Foundations of Many-valued Reasoning}.
Trends Log. Stud. Log. Libr.,
Kluwer, Dordrecht, 2000.


\bibitem{10}
\uppercase{Lions, J. L.}:
\textit{Quelques Methodes de Resolution non Linearies},
Dunod, Paris, 1969.

\bibitem{pari17}
PARHI,~N.---TRIPATHY,~A. K.:
\textit{Oscillation of a class of  non-linear neutral difference equations of higher order},
J. Math. Anal. Appl. \textbf{284} (2003), 756--774.

\bibitem{tha52}
THANDAPANI,~E. et al:
\textit{Oscillation of higher order neutral difference equation with a forcing term},
Int. J. Math. Math Sci. \textbf{22} (1999), 147--154.

\bibitem{mky}
YILDIZ,~M. K.---OCALAN,~O.:
\textit{Oscillation results for higher order non-linear neutral delay difference equations},
Appl. Math. Lett. \textbf{20} (2007), 243--247.


\end{thebibliography}


\end{document}