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\title{Depathing maps for circulating state shifts}
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\author{Mau-Hsiang Shih$^1$%
\email{Mau-Hsiang Shih - [email protected]} and
Feng-Sheng Tsai\correspondingauthor$^2$% \email{Feng-Sheng Tsai\correspondingauthor - [email protected]}% } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% %% %%
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\address{%
\iid(1)\iid(,)\iid(2)Department of Mathematics, National Taiwan Normal University, %
88 Sec. 4, Ting Chou Road, Taipei 11677, Taiwan }%
\maketitle
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~~\\
\noindent\textit{Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday.}
\begin{abstract}
We had described a decirculation process which marks
perturbations of network structure that are necessary for nonlinear network dynamics to proceed from one
circulating state (a limit cycle) to another stable state (a limit cycle or a fixed point).
Armed with the decirculation process, a sort of decirculating maps and its structural properties had also been built,
dedicated to showing that circulation breaking
taking place in nonlinear network dynamics can collaborate harmoniously toward the completion of network
structure that generates attractors (equilibrium states).
Here we wish to extend the notion of decirculating maps to the notion of depathing maps.
The extension allows us to reshape network structure not only on the occasion of circulating states but on the occasion of any required path states.
This gives a crucial improvement in generating circulating state shifts more feasibly.
\end{abstract}
\noindent\textbf{Key words.} fixed point, equilibrium, depathing maps, state shifts, nonlinear dynamics\\
\textbf{AMS subject classifications.} 47H10, 37F20, 92B20, 00A71, 68T05, 91E40
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%% use \cite{...} to cite references %% %% \cite{koon} and %% %% \cite{oreg,khar,zvai,xjon,schn,pond} %% %% \nocite{smith,marg,hunn,advi,koha,mouse}%% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% %%%%%%%%%%%%%%%% %% Background %% %% \section{Introduction}
Pattern generation in complex biological systems may be understood by means of the concepts of nonlinear network dynamics \cite{Schoner,StrogatzBook}.
The modeled systems can be formed by large numbers of interacting units whose dynamical properties tend to emerge through the collective interactions of many units. The modeled systems generally reach one of possible multiple stable states (alternative stable states)
\cite{May,Beisner,Scheffer2001},
which have multistability governed by the control parameters assigned to evolutionary network structure.
State shifts between multiple stable states can be induced by the decirculation process \cite{ShihTsai2012},
which marks a quantified determinant of the reshape of network structure that is sufficient for shifts from one circulating state
(a limit cycle) to another stable state (a limit cycle or a fixed point). The decirculation process is generally stated as
``the occurrence of a loop of unit states in the modeled systems leads to a change in network connections,
which feeds back to reinforce interacting units to tend to break the circulation of unit states in this loop."
Armed with the decirculation process,
a sort of decirculating maps and its structural properties are built in \cite{Tsai2013AAA,ShihTsai2013},
dedicated to showing that circulation breaking
taking place in nonlinear network dynamics can collaborate harmoniously toward the completion of network
structure that generates attractors (equilibrium states).
Here we wish to extend the notion of decirculating maps to the notion of depathing maps.
The extension allows us to reshape network structure not only on the occasion of circulating states but on the occasion of any required path states.
Hence it can generate circulating state shifts more feasibly. It reveals the depathing process which is generally stated as ``the occurrence of a path of unit states in the modeled systems leads to a change in network connections,
which feeds back to reinforce interacting units to tend to break the flow of unit states in this path."
Operator construction for path breaking is also put in a section at the end,
displaying the tendency toward path breaking aiming to control nonlinear network dynamics.
\section{Depathing maps}
Let $\{0,1\}^{n}$ denote the binary code consisting of all $01$-strings of fixed-length $n$.
Denote by $\Omega=[x^{0},x^{1},\ldots,x^{p}]$ a path of states in $\{0,1\}^{n}$,
meaning that $p>1$, $x^{0},x^{1},\ldots,x^{p}\in\{0,1\}^{n}$, and $x^{0}\neq x^{i}$ for some $i\in\{1,2,\ldots,p\}$.
Specifically, we call that $\Omega$ is a {loop} if $x^{0}= x^{p}$. For every $i,j=$ $1,2,\ldots,n$, we assign an
integer, denoted $c_{ij}(\Omega)$, according to the rule: \begin{equation}\label{path} c_{ij}(\Omega)=x^{0}_{j}(x^{0}_{i}-x^{1}_{i})+x^{1}_{j} (x^{1}_{i}-x^{2}_{i}) +\cdots+x^{p-1}_{j}(x^{p-1}_{i}-x^{p}_{i}). \end{equation}
We refer to the resulting matrix $C(\Omega)=(c_{ij}(\Omega))$ as the {depathing map} of $\Omega$.
(If $\Omega$ is a loop, then the depathing map $C(\Omega)$ is equivalent to the decirculating map defined in
\cite{Tsai2013AAA,ShihTsai2013},
where we have explained why the terminology is used in connection with circulation breaking.)
For example, let
$\Omega=[1111100000,0011111000,$ $ 0000111110, 0111110000,0001111100]$. Then \begin{eqnarray*} C(\Omega) &=&\left( \begin{array}{rrrrrrrrrr}
1 & 1 & 1 & 1 &1 & 0 & 0 & 0 &0 & ~~0 \\ 1 & 2 & 2 & 2 & 1 & 0 & -1 & -1 &-1 & 0 \\ 0 & 1 & 2 & 2 & 1 & 1 & 0 & -1 &-1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 &-1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 \\ -1 & -1 & -1 & -1 & -1 & 0& 0 & 0 &0 & 0 \\ -1 & -2 & -2 & -2 & -1 & 0 & 1 & 1 &1 & 0 \\ 0 & -1 & -2 & -2 & -1 & -1& 0 & 1 &1 & 0 \\ 0 & 0 & -1 & -1 & 0 & 0 & 0 & 1 &1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 \\ \end{array}
\right).
\end{eqnarray*}
Consider the dynamical system of $n$ coupled units modeled by the equation \cite{ShihTsai2012,ShihTsai2009}:
\begin{equation}\label{dynamics}
\end{equation}
where $x(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))\in\{0,1\}^{n}$ is the vector of {unit states} at time $t$,
$A=(a_{ij})\in M_{n}(\Real)$ is the {coupling matrix} of the $n$ coupled units,
$s(t)\subset\{1,2,\ldots,n\}$ denotes the units that adjust their states at time $t$,
and $H_{A}(\cdot,s(t))$ is a function whose $i$th component is defined by $$ [H_{A}(x,s(t))]_{i}=\bbbone\left(\sum_{j=1}^{n}a_{ij}x_{j}-b_{i}\right)\quad\mbox{if}~i\in s(t), $$ otherwise $[H_{A}(x,s(t))]_{i}=x_{i}$,
where $b_{i}\in\Real$ is the threshold of unit $i$ and
the function $\bbbone$ is the Heaviside function: $\bbbone(u)=1$ for $u\geq 0$, otherwise 0,
which describes an instantaneous unit pulse.
The dynamical system generates the vector of unit states according to $(\ref{dynamics})$,
resulting in the {phase flow} $x(t)$, $t=0,1,\ldots.$ With the depathing map $C(\Omega)$,
we are bound to consider the linear functional $A\To\langle A, C(\Omega)\rangle$
on the Hilbert space $M_{n}(\Real)$ of all real $n\times n$ matrices endowed with
the Hilbert-Schmidt inner product $\langle\cdot,\cdot\rangle$. \begin{theorem}\label{th1}
Let $\Omega=[x^{0},x^{1},\ldots,x^{p}]$ be a path of states in $\{0,1\}^{n}$.
If $A\in M_{n}(\Real)$ and $b\in\Real^{n}$ satisfy \begin{equation}\label{eq005}
\langle A, C(\Omega)\rangle\geq \left\langle b,x^{0}-x^{p}\right\rangle,
\end{equation}
any updating $s(t)\subset\{1,2,\ldots,n\}$, $t=0,1,\ldots,$
the resulting phase flow $x(t)$ of $(\ref{dynamics})$ cannot behave in
$$
x(T)=x^{0}, x(T+1)=x^{1}, \ldots, x(T+p)=x^{p} $$
for each $T=0,1,\ldots.$ \end{theorem}
\begin{proof}
For any 01-string $x=x_{1}x_{2}\cdots x_{n}$ we define \begin{eqnarray*}
&\1(x)=\{i;~x_{i}=1,~1\leq i\leq n\},&\\ &\0(x)=\{i;~x_{i}=0,~1\leq i\leq n\}.& \end{eqnarray*}
Suppose, by contradiction, that there exist $b\in\Real^{n}$, $x(0)\in\{0,1\}^{n}$, $s(t)\subset\{1,2,\ldots,$ $n\}$,
$t=0,1,\ldots,$ and $T\geq 0$
such that $x(T)=x^{0}, x(T+1)=x^{1}, \ldots, x(T+p)=x^{p}$. Let \begin{eqnarray*} &\Lambda^{+}=\{t;~\0(x(t))\cap\1(x(t+1))\neq\emptyset,~T\leq t<T+p\},&\\ &\Lambda^{-}=\{t;~\1(x(t))\cap\0(x(t+1))\neq\emptyset,~T\leq t<T+p\}.& \end{eqnarray*}
Then $\Lambda^{+}\neq\emptyset$ and $\Lambda^{-}\neq\emptyset$. Indeed, if $\Lambda^{+}=\emptyset$ or $\Lambda^{-}=\emptyset$, then $$ x(T)=x(T+1)=\cdots=x(T+p), $$
contradicting the path assumption $x(T)\neq x(T+p)$. According to $(\ref{path})$, we have
\begin{eqnarray}\label{eq001}
\left(\sum_{0\leq m< p}x^{m}_{j}x^{m}_{i}-\sum_{0\leq m< p}x^{m}_{j}x^{m+1}_{i}\right)\nonumber\\
&=&\sum_{0\leq m<
p}\left(\sum_{i,j}a_{ij}x^{m}_{j}x^{m}_{i}-\sum_{i,j}a_{ij}x^{m }_{j}x^{m+1}_{i}\right)\\
&=&\sum_{0\leq m< p}\left(\langle Ax(T+m),x(T+m)\rangle-\langle Ax(T+m),x(T+m+1)\rangle \right)\nonumber\\
&=&\sum_{0\leq m< p}\langle Ax(T+m),x(T+m)-x(T+m+1)\rangle.\nonumber
\end{eqnarray}
Since $\0(x(t))\cap\1(x(t+1))\subset s(t)$ and
$\1(x(t))\cap\0(x(t+1))\subset s(t)$ for each $t=0,1,\ldots,$ we conclude from $(\ref{dynamics})$ that
\begin{eqnarray}\label{eq002}
&&\sum_{0\leq m< p}\langle Ax(T+m),x(T+m)-x(T+m+1)\rangle\nonumber\\ &<&-\sum_{t\in\Lambda^{+}}\sum_{j\in\0(x(t))\cap\1(x(t+1))}b_{j }+\sum_{t\in\Lambda^{-}}\sum_{j\in\1(x(t))\cap\0(x(t+1))}b_{j}\\ &=&\sum_{0\leq m< p}\left\langle b,x(T+m)-x(T+m+1)\right\rangle\nonumber\\ &=&\left\langle b,x(T)-x(T+p)\right\rangle.\nonumber \end{eqnarray}
Combining $(\ref{eq001})$ and $(\ref{eq002})$ shows that $\langle A,C(\Omega)\rangle<\left\langle
b,x^{0}-x^{p}\right\rangle$,
contradicting $(\ref{eq005})$, and that completes the proof. \end{proof}
\section{Operator control on path breaking}
Denote by $\Omega=[x^{0},x^{1},\ldots,x^{p}]$ a {path} of states in $\{0,1\}^{n}$.
For each $m=0,1,\ldots,p$, we say that the state $x^{m}$ is in the position $m$ of the path $\Omega$.
For each 01-string $x=x_{1}x_{2}\cdots x_{n}$, let \begin{eqnarray*}
&\1(x)=\{i;~x_{i}=1,~1\leq i\leq n\},&\\ &\0(x)=\{i;~x_{i}=0,~1\leq i\leq n\}.& \end{eqnarray*}
Let us recall that the {symmetric difference} of two sets $U$ and $V$ is the set $U\triangle V$,
each of whose elements belongs to $U$ but not to $V$, or belongs to $V$ but not to $U$.
For every $i=1,2,\ldots,n,$ let \begin{eqnarray}\label{eqq1} &\hspace{-0.5cm}M_{i}(\Omega)=\{m;~i\in \1(x^{m-1})\triangle \ 1(x^{m}), m=1,2,\ldots,p\},&\nonumber\\ &\hspace{-0.5cm}M_{i}(\Omega)^{+}=\{m;~i\in \ 1(x^{m})\setminus \1(x^{m-1}), m=1,2,\ldots,p\},&\\ &\hspace{-0.5cm}M_{i}(\Omega)^{-}=\{m;~i\in \1(x^{m-1})\setminus\1(x^{m}), m=1,2,\ldots,p\}.&\nonumber \end{eqnarray}
Here $M_{i}(\Omega)$ denotes the collection of the positions $m$ of the path $\Omega$ in which unit $i$ changes its state from $x^{m-1}_{i}=0$ to $x^{m}_{i}=1$ or from $x^{m-1}_{i}=1$ to $x^{m}_{i}=0$,
whereas $M_{i}(\Omega)^{+}$ (resp., $M_{i}(\Omega)^{-}$) denotes the collection of the positions $m$
of the path $\Omega$ in which unit $i$ changes its state from $x^{m-1}_{i}=0$ to $x^{m}_{i}=1$
(resp., changes its state from $x^{m-1}_{i}=1$ to $x^{m}_{i}=0$).
For every $i,j=1,2,\ldots,n$, denote by \begin{eqnarray}\label{eq33} \Upsilon_{ij}(\Omega)&=&\sharp(M_{i}(\Omega)^{+}\cap M_{j} (\Omega)^{+})+\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{-})\nonumber\\ &&-\sharp(M_{i}(\Omega)^{+}\cap M_{j} (\Omega)^{-})-\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{+}), \end{eqnarray}
which can be regarded as a measure of {synchronous activity} between units $i,j$, that is,
if units $i,j$ tend to change their states synchronously (resp., asynchronously) in $\Omega$,
then $\Upsilon_{ij}(\Omega)>0$ (resp., $\Upsilon_{ij} (\Omega)<0$).
For every $i,j=1,2,\ldots,n$, denote by \begin{equation}\label{eq34}
\Gamma_{ij}(\Omega)=\min\{\sharp M_{i}(\Omega),\sharp M_{j}(\Omega)\},
\end{equation}
which can be regarded as a measure of {self-sustaining activity} of units $i,j$, that is,
if unit $i$ or $j$ tends to maintain more self-sustaining states in $\Omega$ than unit $i'$ or $j'$, then
$\Gamma_{ij}(\Omega)<\Gamma_{i'j'}(\Omega)$.
We refer to the resulting matrices $\Upsilon(\Omega)=(\Upsilon_{ij} (\Omega))$ and $\Gamma(\Omega)=(\Gamma_{ij}(\Omega))$ as the measure of synchronous activity and the measure of self-sustaining activity derived from the path $\Omega$ of states in $\ {0,1\}^{n}$, respectively.
Let $\overline{\Omega}=[x^{0},x^{1},\ldots,x^{p},x^{0}]$. Denote
by $\langle\cdot,\cdot\rangle$ the Hilbert-Schmidt inner product in $M_{n}(\Real)$, i.e. if $A=(a_{ij})$ and $B=(b_{ij})\in
M_{n}(\Real)$, then $\langle A,B\rangle={\rm tr}(AB^{T})=\sum_{i,j}a_{ij}b_{ij}$. Denote by $I(\Omega)=\1(x^{p})\cap \1(x^{0})$. Let \begin{eqnarray}\label{eq02} \mathcal{D}(\Omega)&=&\{\mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}};~\mathcal{D}_{\mathtt{SY}}=\mat hcal{D}_{\mathtt{SY}}^{T}\in M_{n}(\Real), \mathcal{D}_{\mathtt{SK}}=-\mathcal{D}_{\mathtt{SK}}^{T}\in M_{n}(\Real) \mbox{ with }(\mathcal{D}_{\mathtt{SK}})_{ij}\geq 0 \mbox{ for each }\\
&&(i,j)\in\left(\1(x^{0})\times\1(x^{p})\right)\setminus\left(I(\Omeg a)\times I(\Omega)\right),\mbox{ and }
\ left\langle|\mathcal{D}_{\mathtt{SK}}|,\Gamma(\overline{\Omega })\right\rangle<\left\langle \mathcal{D}_{\mathtt{SY}},\Upsilon(\overline{\Omega})-C([x^{p},x^{0}])\right\rangle\},\nonumber \end{eqnarray}
where
$|\mathcal{D}_{\mathtt{SK}}|=(| (\mathcal{D}_{\mathtt{SK}})_{ij}|)$.
The set $\mathcal{D}(\Omega)$ collects all the combining operation of the operators $\mathcal{D}_{\mathtt{SY}}$ and $\mathcal{D}_{\mathtt{SK}}$,
which will determine a clamp of network modification by
$A+\mathcal{D}_{\mathtt{SY}}+\mathcal{D}_{\mathtt{SK}}$. Fix $\mathcal{D}_{\mathtt{SY}}
+\mathcal{D}_{\mathtt{SK}}\in\mathcal{D}(\Omega)$. Denote by
$I_{1}(\Omega)=\1(x^{p})\cap\0(x^{0})$ and $I_{2} (\Omega)=\0(x^{p})\cap\1(x^{0})$.
Since $I_{1}(\Omega)\cap I(\Omega)=\emptyset$, $I_{1} (\Omega)\cup
I(\Omega)=\1(x^{p})$, $I_{2}(\Omega)\cap I(\Omega)=\emptyset$, and
$I_{2}(\Omega)\cup I(\Omega)=\1(x^{0})$, we have $$
\left(\1(x^{0})\times\1(x^{p})\right)\setminus\left(I(\Omega)\times I(\Omega)\right) =\left(I(\Omega)\times
I_{1}(\Omega)\right)\cup\left(I_{2}(\Omega)\times\1(x(p))\right), $$
and hence, by $(\ref{eq02})$, \begin{eqnarray}\label{eq41} \ left\langle\mathcal{D}_{\mathtt{SK}},C([x^{p},x^{0}])\right\rang le&=& \sum_{(i,j)\in I_{1}(\Omega)\times \1(x(p))} (\mathcal{D}_{\mathtt{SK}})_{ij}-\sum_{(i,j)\in I_{2} (\Omega)\times\1(x(p))} (\mathcal{D}_{\mathtt{SK}})_{ij}\nonumber\\
&=&\sum_{(i,j)\in I_{1}(\Omega)\times I_{1}(\Omega)} (\mathcal{D}_{\mathtt{SK}})_{ij}
+\sum_{(i,j)\in I_{1}(\Omega)\times I(\Omega)}
(\Omega)\times\1(x(p))}
(\mathcal{D}_{\mathtt{SK}})_{ij}\nonumber\\[0.5em] %&&=-\sum_{(i,j)\in I(\Omega)\times I_{1}(\Omega)} (\mathcal{D}_{\mathtt{SK}})_{ij}-\sum_{(i,j)\in I_{2}
(\Omega)\times\1(x(p))}(\mathcal{D}_{\mathtt{SK}})_{ij}\\ &\leq& 0.
\end{eqnarray}
Furthermore, according to the proof in \cite[Theorem 1] {ShihTsai2013},
the following assertion holds: \begin{equation}\label{eq43} \left\langle \mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}},C(\overline{\Omega})\right\rangle \geq\left\langle \ mathcal{D}_{\mathtt{SY}},\Upsilon(\overline{\Omega})\right\rangl e -\left\langle |\mathcal{D}_{\mathtt{SK}}|,\Gamma(\overline{\Omega})\right\ra ngle. \end{equation}
Combining $(\ref{eq02})$, $(\ref{eq41})$, and $(\ref{eq43})$ shows that \begin{eqnarray*} \left\langle\mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}},C(\Omega)\right\rangle&=& \left\langle \mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}},C(\overline{\Omega})-C([x^{p},x^{0}])\right\rangle\\ &\geq&\left\langle \mathcal{D}_{\mathtt{SY}},\Upsilon(\overline{\Omega})\right\rang le -\left\langle |\mathcal{D}_{\mathtt{SK}}|,\Gamma(\overline{\Omega})\right\ra ngle-\left\langle \mathcal{D}_{\mathtt{SY}},C([x^{p},x^{0}])\right\rangle
-\left\langle\mathcal{D}_{\mathtt{SK}},C([x^{p},x^{0}])\right\ran gle\\
&>&0.
\end{eqnarray*}
%Let $\alpha=\left|\left\langle b,x^{0}-x^{p}\right\rangle-\langle A, C(\Omega)\rangle\right|/\left\langle\mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}},C(\Omega)\right\rangle$.
%Then $\langle A+\alpha(\mathcal{D}_{\mathtt{SY}}
+\mathcal{D}_{\mathtt{SK}}), C(\Omega)\rangle\geq \left\langle b,x^{0}-x^{p}\right\rangle$.
With the notation and the arguments above, we describe operator control on path breaking as follows.
\begin{theorem}\label{th2}
Let $A\in M_{n}(\Real)$ and $b\in\Real^{n}$.
Let $\Omega=[x^{0},x^{1},\ldots,x^{p}]$ be a path of states in $\{0,1\}^{n}$.
Then for every operator $\mathcal{D}_{\mathtt{SY}}
+\mathcal{D}_{\mathtt{SK}}\in\mathcal{D}(\Omega)$, there exists $\gamma\geq 0$ such that
$$
\langle A+\gamma(\mathcal{D}_{\mathtt{SY}}
+\mathcal{D}_{\mathtt{SK}}), C(\Omega)\rangle\geq \left\langle b,x^{0}-x^{p}\right\rangle.
$$
Hence, by Theorem \ref{th1}, the dynamical system of $n$ coupled units modeled by the equation
$$
x(t+1)=H_{A+\gamma(\mathcal{D}_{\mathtt{SY}}
+\mathcal{D}_{\mathtt{SK}})}(x(t),s(t)), \quad t=0,1,\ldots, $$
cannot behave in $$
x(T)=x^{0}, x(T+1)=x^{1}, \ldots, x(T+p)=x^{p} $$
for each $T=0,1,\ldots.$ \end{theorem}
In the following, we give an example to construct a sort of operators $\mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}}\in\mathcal{D}(\Omega)$. Let $n=10$ and $p=4$. Let $\Omega=[x^{0},x^{1},x^{2},x^{3},x^{4}]=[1111100000,001 1111000, 0000111110, 0111110000,0001111100]$. Then \begin{eqnarray*} %&I(\Omega)=\1(0001111100)\cup\1(1111100000)=\ {1,2,\ldots,8\},&\\ &&\hspace{-0.8cm}I(\Omega)=\1(x^{p})\cap\1(x^{0})=\{4,5\},\\ &&\hspace{-0.8cm}I_{1}(\Omega)=\1(x^{p})\cap\0(x^{0})=\ {6,7,8\},\\ &&\hspace{-0.8cm}I_{2}(\Omega)=\0(x^{p})\cap\1(x^{0})=\ {1,2,3\}. \end{eqnarray*}
Associate to each $i\in\{1,2,\ldots,n\}$ a real number $\varepsilon_{i}$ such that
\begin{itemize}
\item[1)] for each $i,j\in\{1,2,\ldots,n\}$, if $(\sum_{m}x^{m}_{i})/(M_{i}(\Omega)
+1)>(\sum_{m}x^{m}_{j})/(M_{j}(\Omega)+1)$,
then $\varepsilon_{i}\leq \varepsilon_{j}$; \item[2)] $\sum_{1\leq i\leq n}M_{i}(\Omega)\varepsilon_{i}+\sum_{i\in
I_{2}(\Omega)}\varepsilon_{i}> 0$ \end{itemize}
(see Table 1(a) for a choice of $\varepsilon_{i}$).
We may select $y=(y_{1},y_{2},\ldots,y_{n})\in\Real^{n}$ and $\mathcal{D}_{\mathtt{SY}}=(y_{i}y_{j}
+\delta_{ij}\varepsilon_{i})\in
M_{n}(\Real)$, where $\delta_{ij}=1$ if $i=j$, otherwise $0$, such that exactly one of the following holds:
\begin{itemize}
\item[1)] $y_{i}\geq 0$ for $i\in \1(x^{p})$ and $\sum_{i\in I_{1}(\Omega)} y_{i}\leq\sum_{i\in I_{2}(\Omega)} y_{i}$; \item[2)] $y_{i}\leq 0$ for $i\in \1(x^{p})$ and $\sum_{i\in I_{1}(\Omega)} y_{i}\geq\sum_{i\in I_{2}(\Omega)} y_{i}$
\end{itemize}
(see Table 1(a) for a choice of $y_{i}$ and Table 1(b) for $\mathcal{D}_{\mathtt{SY}}$).
Consider the shift function $\sigma$ on $\{0,1,\ldots, p\}$ given by
$$
\sigma(k)\equiv k+1\mod p+1 $$
for each $k=0,1,\ldots,p.$ Since $(y_{i}y_{j})\in M_{n}(\Real)$ is positive semidefinite, we have
\begin{eqnarray*} \left\langle
\mathcal{D}_{\mathtt{SY}},\Upsilon(\overline{\Omega})-C([x^{p},x^{0}])\right\rangle&=&
\sum_{0\leq m\leq p}\left\langle \mathcal{D}_{\mathtt{SY}}
\left(x^{m}-x^{\sigma(m)}\right),\left(x^{m}-x^{\sigma(m)}\right)\right\rangle\\
&&-\sum_{j\in\1(x^{p})}y_{j}\left(\sum_{i\in I_{1}(\Omega)} y_{i}-\sum_{i\in I_{2}(\Omega)} y_{i}\right)-\sum_{i\in I_{1} (\Omega)}\varepsilon_{i}\\
&\geq&\sum_{0\leq m< p}\sum_{i\in\1\left(x^{m}\right)\triangle \ 1\left(x^{\sigma(m)}\right)}\varepsilon_{i}+\sum_{i\in I_{2} (\Omega)}\varepsilon_{i}\\
&=&\sum_{1\leq i\leq n}M_{i}(\Omega)\varepsilon_{i}+\sum_{i\in I_{2}(\Omega)}\varepsilon_{i}> 0.
\end{eqnarray*}
For such a choice of $\mathcal{D}_{\mathtt{SY}}$, let $\varepsilon>0$ such that
\begin{equation}\label{eq30} \gamma_{\Omega}=\left(\left\langle \mathcal{D}_{\mathtt{SY}},\Upsilon(\overline{\Omega})-C([x^{p},x^{0}])\right\rangle-\varepsilon\right)/||\Gamma(\overline {\Omega})||\geq0, \end{equation} where $||\Gamma(\overline{\Omega})||=\left\langle \ Gamma(\overline{\Omega}),\Gamma(\overline{\Omega})\right\ran gle^{\frac{1}{2}}$.
Then for any choice of $S\in M_{n}(\Real)$ with $S=-S^{T}$ and $S_{ij}\geq 0$ for each
$
(i,j)\in\left(\1(x^{0})\times\1(x^{p})\right)\setminus\left(I(\Omega)\ times
I(\Omega)\right)$, we set $\mathcal{D}_{\mathtt{SK}}=\alpha S$, where $\alpha\in\Real$ is such that
\begin{equation}\label{eq05} ||\mathcal{D}_{\mathtt{SK}}||=\left\langle \ mathcal{D}_{\mathtt{SK}},\mathcal{D}_{\mathtt{SK}}\right\rangl e^{\frac{1}{2}} =\alpha\left\langle S,S\right\rangle^{\frac{1}{2}}\leq \gamma_{\Omega} \end{equation}
(see Table 1(c) for a choice of $\mathcal{D}_{\mathtt{SK}}$). Thus, by $(\ref{eq30})$ and $(\ref{eq05})$, we have
$$ \left\langle |\mathcal{D}_{\mathtt{SK}}|,\Gamma(\overline{\Omega})\right\ra ngle<\left\langle \mathcal{D}_{\mathtt{SY}},\Upsilon(\overline{\Omega})-C([x^{p},x^{0}])\right\rangle. $$ Hence $\mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}}\in\mathcal{D}(\Omega)$ (see Table 1(d) for a choice of $\mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}}$).
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Competing interests}
The authors declare that they have no competing interests. \section*{Author's contributions}
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Acknowledgements}
\ifthenelse{\boolean{publ}}{\small}{}
This work was supported by the National Science Council of Taiwan. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Tables}
\subsection*{Table 1 - Operator construction} (a) Choose $\varepsilon_{i}$ and $y_{i}$.
%such that $\sum_{1\leq i\leq n}M_{i}(\Omega)\varepsilon_{i} +\sum_{i\in I_{2}(\Omega)}\varepsilon_{i}=6.2.$
\par \mbox{} \par
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{11}{|c|}{$\Omega=[1111100000,0011111000, 0000111110, 0111110000,0001111100]$}\\ \hline $i$&1&2&3&4&5&6&7&8&9&10\\\hline $M_{i}(\Omega)$&1&3&3&2&0&1&3&3&2&0\\\hline $\sum_{m}x^{m}_{i}$&1&2&3&4&5&4&3&2&1&0\\\hline $(\sum_{m}x^{m}_{i})/(M_{i}(\Omega) +1)$&1/2&1/2&3/4&4/3&5&2&3/4&1/2&1/3&0\\\hline %$(\sum_{m}x^{m}_{i}+1)/(M_{i}(\Omega) +1)$&1&3/4&1&5/3&6&5/2&1&3/4&2/3&1\\\hline %$\sum_{m}x^{m}_{i}-M_{i}(\Omega)$&0&-1&0&2&5&3&0&-1&-1&0\\\hline $\varepsilon_{i}$&0.8&0.5&0.5&$-0.7$&$-1.8$&$-1$&$-0.3$&0.5&1.2&1.5\\\hline $y_{i}$&3&1&$-1$&0&0.5&1&1&1&0&$-1$ \\\hline \end{tabular} } \subsection*{}
(b) Construct the operator
$\mathcal{D}_{\mathtt{SY}}=(y_{i}y_{j} +\delta_{ij}\varepsilon_{i})$. \par \mbox{} \par \mbox{ \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline 9.8 & 3 & -3 & 0 & 1.5& 3 & 3 & 3 & 0 &-3\\\hline 3 &1.5& -1 & 0 & 0.5& 1 & 1 & 1 &0 & -1\\\hline
-3 &-1 &1.5& 0 &-0.5& -1 & -1 &-1 & 0& 1 \\\hline 0 &0 &0 &-0.7& 0 &0 &0 &0 & 0 & 0\\\hline
1.5& 0.5& -0.5& 0 &-1.55& 0.5& 0.5& 0.5& 0 &-0.5\\\hline 3& 1 &-1& 0&0.5& 0 & 1 & 1 & 0 &-1\\\hline
3 & 1 & -1 & 0&0.5& 1 & 0.7& 1 & 0 &-1\\\hline 3 & 1& -1 &0&0.5& 1 & 1 & 1.5& 0 &-1\\\hline 0 & 0 &0 &0 & 0& 0&0 & 0 & 1.2& 0 \\\hline -3 &-1 & 1 &0 &-0.5& -1 &-1&-1 &0 & 2.5\\ \hline
\end{tabular} }
\subsection*{}
(c) Construct the operator $\mathcal{D}_{\mathtt{SK}}=\alpha S$ satisfying $(\ref{eq30})$ and $(\ref{eq05})$.
\par \mbox{} \par
\mbox{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\hline
0 & 0 & -0.015& 0 & 0.005& 0 & 0.02& 0.01& -0.015& -0.01\\\hline 0 & 0 & 0.01& 0 & 0.015& 0.01& 0.015& 0.015& 0.01&
-0.005\\\hline
0.015& -0.01& 0 & 0.015& 0 & 0.01& 0.01& 0.01& 0.005& 0.01\\\hline
0 & 0 & -0.015& 0 & -0.005& 0.005& 0.01& 0.01& 0 & -0.01\\\hline -0.005& -0.015& 0& 0.005& 0 & 0.005& 0.01& 0.005& 0.015& 0\\\hline
0 & -0.01& -0.01& -0.005& -0.005& 0 & -0.01& 0.015& 0.01& -0.02\\\hline
-0.02& -0.015& -0.01& -0.01& -0.01& 0.01& 0 & 0.005& 0.015& 0.005\\\hline
-0.01& -0.015& -0.01& -0.01& -0.005& -0.015& -0.005& 0 &0.005& 0.015\\\hline
0.015& -0.01& -0.005& 0 & -0.015& -0.01& -0.015& -0.005& 0 & 0\\\hline
0.01& 0.005& -0.01& 0.01& 0 & 0.02& -0.005& -0.015& 0 &0\\ \hline
\end{tabular} }
\subsection*{}
(d) Construct the combining operation
$\mathcal{D}_{\mathtt{SY}}+\mathcal{D}_{\mathtt{SK}}$. \par \mbox{}
\par \mbox{
9.8& 3 & -3.015& 0& 1.505& 3 & 3.02& 3.01& -0.015& -3.01\\\hline 3 & 1.5& -0.99& 0 & 0.515& 1.01& 1.015& 1.015& 0.01&
-1.005\\\hline
-2.985& -1.01& 1.5& 0.015& -0.5& -0.99& -0.99& -0.99& 0.005& 1.01\\\hline
0 & 0 &-0.015& -0.7& -0.005& 0.005& 0.01& 0.01& 0 & -0.01\\\hline 1.495& 0.485& -0.5& 0.005&-1.55& 0.505& 0.51& 0.505& 0.015& -0.5\\\hline
3& 0.99& -1.01& -0.005& 0.495& 0 & 0.99& 1.015& 0.01& -1.02\\\hline
2.98& 0.985& -1.01& -0.01& 0.49& 1.01& 0.7& 1.005& 0.015& -0.995\\\hline
2.99& 0.985& -1.01& -0.01& 0.495& 0.985& 0.995& 1.5& 0.005& -0.985\\\hline
0.015& -0.01& -0.005& 0& -0.015& -0.01& -0.015& -0.005& 1.2& 0\\\hline
-2.99& -0.995& 0.99& 0.01& -0.5& -0.98& -1.005& -1.015& 0 & 2.5\\\hline \end{tabular} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Additional Files %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section*{Additional Files}
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