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Dissipative Properties of ω-Order Preserving Partial Contraction Mapping in Semigroup of Linear Operator ()

*C*

_{0}-semigroup of

*ω*-order preserving partial contraction mapping (

*ω*-

*OCP*) in semigroup of linear operator. The purpose of this paper is to establish some dissipative properties on

_{n}*ω*-

*OCP*which have been obtained in the various theorems (research results) and were proved.

_{n}Share and Cite:

*Advances in Pure Mathematics*,

**9**, 544-550. doi: 10.4236/apm.2019.96026.

1. Introduction

Suppose X is a Banach space,
${X}_{n}\subseteq X$ a finite set,
${\left(T\left(t\right)\right)}_{t\ge 0}$ the C_{0}-semigroup that is strongly continuous one-parameter semigroup of bounded linear operator in X. Let ω-OCP_{n} be ω-order-preserving partial contraction mapping (semigroup of linear operator) which is an example of C_{0}-semigroup. Furthermore, let
$Mm\left(\mathbb{N}\right)$ be a matrix,
$L\left(X\right)$ a bounded linear operator on X,
${P}_{n}$ a partial transformation semigroup,
$\rho \left(A\right)$ a resolvent set,
$F\left(x\right)$ a duality mapping on X and A is a generator of C_{0}-semigroup. Taking the importance of the dissipative operator in a semigroup of linear operators into cognizance, dissipative properties characterized the generator of a semigroup of linear operator which does not require the explicit knowledge of the resolvent.

This paper will focus on results of dissipative operator on ω-OCP_{n} on Banach space as an example of a semigroup of linear operator called C_{0}-semigroup.

Yosida [1] proved some results on differentiability and representation of one-parameter semigroup of linear operators. Miyadera [2] , generated some strongly continuous semigroups of operators. Feller [3] , also obtained an unbounded semigroup of bounded linear operators. Balakrishnan [4] introduced fractional powers of closed operators and semigroups generated by them. Lumer and Phillips [5] , established dissipative operators in a Banach space and Hille & Philips [6] emphasized the theory required in the inclusion of an elaborate introduction to modern functional analysis with special emphasis on functional theory in Banach spaces and algebras. Batty [7] obtained asymptotic behaviour of semigroup of operator in Banach space. More relevant work and results on dissipative properties of ω-Order preserving partial contraction mapping in semigroup of linear operator could be seen in Engel and Nagel [8] , Vrabie [9] , Laradji and Umar [10] , Rauf and Akinyele [11] and Rauf et al. [12] .

2. Preliminaries

Definition 2.1 (C_{0}-Semigroup) [9]

C_{0}-Semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space.

Definition 2.2 (ω-OCP_{n}) [11]

Transformation $\alpha \in {P}_{n}$ is called ω-order-preserving partial contraction mapping if $\forall x\mathrm{,}y\in \text{Dom}\alpha \mathrm{:}x\le y$ $\Rightarrow \alpha x\le \alpha y$ and at least one of its transformation must satisfy $\alpha y=y$ such that $T\left(t+s\right)=T\left(t\right)T\left(s\right)$ whenever $t,s>0$ and otherwise for $T\left(0\right)=I$.

Definition 2.3 (Subspace Semigroup) [8]

A subspace semigroup is the part of A in Y which is the operator ${A}_{\mathrm{*}}$ defined by ${A}_{\mathrm{*}}y=Ay$ with domain $D\left({A}_{\mathrm{*}}\right)=\left\{y\in D\left(A\right)\cap Y\mathrm{:}Ay\in Y\right\}$.

Definition 2.4 (Duality set)

Let X be a Banach space, for every $x\in X$, a nonempty set defined by $F\left(x\right)=\left\{{x}^{\mathrm{*}}\in {X}^{\mathrm{*}}\mathrm{:}\left(x\mathrm{,}{x}^{\mathrm{*}}\right)={\Vert x\Vert}^{2}={\Vert {x}^{\mathrm{*}}\Vert}^{2}\right\}$ is called the duality set.

Definition 2.5 (Dissipative) [9]

A linear operator $\left(A\mathrm{,}D\left(A\right)\right)$ is dissipative if each $x\in X$, there exists ${x}^{\mathrm{*}}\in F\left(x\right)$ such that $Re\left(Ax\mathrm{,}{x}^{\mathrm{*}}\right)\le 0$.

2.1. Properties of Dissipative Operator

For dissipative operator $A\mathrm{:}D\left(A\right)\subseteq X\to X$, the following properties hold:

a) $\lambda -A$ is injective for all $\lambda >0$ and

$\Vert {\left(\lambda -A\right)}^{-1}\Vert \le 1/\lambda \Vert y\Vert $ (2.1)

for all y in the range $\text{rg}\left(\lambda -A\right)=\left(\lambda -A\right)D\left(A\right)$.

b) $\lambda -A$ is surjective for some $\lambda >0$ if and only if it is surjective for each $\lambda >0$. In that case, we have $\left(\mathrm{0,}\infty \right)\subset \rho \left(A\right)$, where $\rho \left(A\right)$ is the resolvent of the generator A.

c) A is closed if and only if the range $\text{rg}\left(\lambda -A\right)$ is closed for some $\lambda >0$.

d) If $\text{rg}\left(A\right)\subseteq D\left(A\right)$, that is if A is densely defined, then A is closable. its closure A is again dissipative and satisfies $\text{rg}\left(\lambda -A\right)=\text{rg}\left(\lambda -A\right)$ for all $\lambda >0$.

Example 1

$2\times 2$ matrix $\left[{M}_{m}\left(\mathbb{N}\cup \left\{0\right\}\right)\right]$

Suppose

$A=\left(\begin{array}{cc}1& 2\\ 2& 2\end{array}\right)$

and let $T\left(t\right)={e}^{tA}$, then

${e}^{tA}=\left(\begin{array}{cc}{e}^{t}& {e}^{2t}\\ {e}^{2t}& {e}^{2t}\end{array}\right)$

$3\times 3$ matrix $\left[{M}_{m}\left(\mathbb{N}\cup \left\{0\right\}\right)\right]$

Suppose

$A=\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 2\\ -& 2& 3\end{array}\right)$

and let $T\left(t\right)={e}^{tA}$, then

${e}^{tA}=\left(\begin{array}{ccc}{e}^{t}& {e}^{2t}& {e}^{3t}\\ {e}^{t}& {e}^{2t}& {e}^{2t}\\ I& {e}^{2t}& {e}^{3t}\end{array}\right)$

Example 2

In any $2\times 2$ matrix $\left[{M}_{m}\left(\u2102\right)\right]$, and for each $\lambda >0$ such that $\lambda \in \rho \left(A\right)$ where $\rho \left(A\right)$ is a resolvent set on X.

Also, suppose

$A=\left(\begin{array}{cc}1& 2\\ -& 2\end{array}\right)$

and let $T\left(t\right)={e}^{t{A}_{\lambda}}$, then

${e}^{t{A}_{\lambda}}=\left(\begin{array}{cc}{e}^{t\lambda}& {e}^{2t\lambda}\\ I& {e}^{2t\lambda}\end{array}\right)$

Example 3

Let $X={C}_{ub}\left(\mathbb{N}\cup \left\{0\right\}\right)$ be the space of all bounded and uniformly continuous function from $\mathbb{N}\cup \left\{0\right\}$ to $\mathbb{R}$, endowed with the sup-norm ${\Vert \text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\Vert}_{\infty}$ and let $\left\{T\left(t\right)\mathrm{;}t\ge 0\right\}\subseteq L\left(X\right)$ be defined by

$\left[T\left(t\right)f\right]\left(s\right)=f\left(t+s\right)$

For each $f\in X$ and each $t\mathrm{,}s\in {\mathbb{R}}_{+}$, it is easily verified that $\left\{T\left(t\right)\mathrm{;}t\ge 0\right\}$ satisfies Examples 1 and 2 above.

Example 4

Let $X=C\left[0,1\right]$ and consider the operator $Af=-{f}^{\prime}$ with domain $D\left(A\right)=\left\{f\in {C}^{\prime}\left[0,1\right]:f\left(0\right)=0\right\}$. It is a closed operator whose domain is not dense. However, it is dissipative, since its resolvent can be computed explicitly as

$R\left(\lambda \mathrm{,}A\right)f\left(t\right)={\displaystyle {\int}_{0}^{t}{e}^{-\lambda \left(t-s\right)}f\left(s\right)\text{d}s}$

for $t\in \left[\mathrm{0,1}\right]$, $f\in C\left[\mathrm{0,1}\right]$. Moreover, $\Vert R\left(\lambda \mathrm{,}A\right)\Vert \le \frac{1}{\lambda}$ for all $\lambda >0$. Therefore $\left(A\mathrm{,}D\left(A\right)\right)$ is dissipative.

2.2. Theorem (Hille-Yoshida [9] )

A linear operator
$A\mathrm{:}D\left(A\right)\subseteq X\to X$ is the infinitesimal generator for a C_{0}-semigroup of contraction if and only if

1) A is densely defined and closed,

2) $\left(\mathrm{0,}+\infty \right)\subseteq \rho \left(A\right)$ and for each $\lambda >0$

${\Vert R\left(\lambda \mathrm{,}A\right)\Vert}_{L\left(X\right)}\le \frac{1}{\lambda}$ (2.2)

2.3. Theorem (Lumer-Phillips [5] )

Let X be a real, or complex Banach space with norm $\Vert \text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\Vert $, and let us recall that the duality mapping $F\mathrm{:}X\to {2}^{x}$ is defined by

$F\left(x\right)=\left\{{x}^{*}\in {X}^{*};\left(x,{x}^{*}\right)={\Vert x\Vert}^{2}={\Vert {x}^{*}\Vert}^{2}\right\}$ (2.3)

for each $x\in X$. In view of Hahn-Banach theorem, it follows that, for each $x\in X$, $F\left(x\right)$ is nonempty.

2.4. Theorem (Hahn-Banach Theorem [2] )

Let V be a real vector space. Suppose $p\mathrm{:}V\in \left[\mathrm{0,}+\infty \right]$ is mapping satisfying the following conditions:

1) $p\left(0\right)=0$;

2) $p\left(tx\right)=tp\left(x\right)$ for all $x\in V$ and real of $t\ge 0$; and

3) $p\left(x+y\right)\le p\left(x\right)+p\left(y\right)$ for every $x\mathrm{,}y\in v$.

Assume, furthermore that for each $x\in V$, either both $p\left(x\right)$ and $p\left(-x\right)$ are $\infty $ or that both are finite.

3. Main Results

In this section, dissipative results on ω-OCP_{n} as a semigroup of linear operator were established and the research results(Theorems) were given and proved appropriately:

Theorem 3.1

Let $A\in w\text{-}OC{P}_{n}$ where $A\mathrm{:}D\left(A\right)\subseteq X\to X$ is a dissipative operator on a Banach space X such that $\lambda -A$ is surjective for some $\lambda >0$. Then

1) the part A, of A in the subspace ${X}_{0}=\stackrel{\xaf}{D\left(A\right)}$ is densely defined and generates a constrain semigroup in ${X}_{0}$, and

2) considering X to be a reflexive, A is densely defined and generates a contraction semigroup.

Proof

We recall from Definition 2.3 that

${A}_{*}x=Ax$ (3.1)

for

$x\in D\left({A}_{*}\right)=\left\{x\in x\in D\left(A\right):Ax\in {X}_{0}\right\}=R\left(\lambda ,A\right){X}_{0}$ (3.2)

Since $R\left(\lambda ,A\right)$ exists for $\lambda >0$, this implies that $R{\left(\lambda ,A\right)}_{*}=R\left(\lambda ,{A}_{*}\right)$, hence

$\left(\mathrm{0,}\infty \right)\subset \rho (A*)$

we need to show that $D\left({A}_{\mathrm{*}}\right)$ is dense in ${X}_{0}$.

Take $x\in D\left(A\right)$ and set ${x}_{n}=nR\left(n,A\right)x$. Then ${x}_{n}\in D\left(A\right)$ and

$\underset{n\to \infty}{\mathrm{lim}}{x}_{n}=\underset{n\to \infty}{\mathrm{lim}}R\left(n,A\right)Ax+x=x,$

since $\Vert R\left(n\mathrm{,}A\right)\Vert \le \frac{1}{n}$. Therefore the operators $nR\left(n\mathrm{,}A\right)$ converge pointwise on

$D\left(A\right)$ to the identity. Since $\Vert nR\left(n\mathrm{,}A\right)\Vert \le 1$ for all $n\in \mathbb{N}$, we obtain the convergence of ${y}_{n}=nR\left(n,A\right)y\to y$ for all $y\in {X}_{0}$. If for each ${y}_{n}$ in $D\left({A}_{\mathrm{*}}\right)$, the density of $D\left({A}_{\mathrm{*}}\right)$ in ${X}_{0}$ is shown which proved (i).

To prove (ii), we need to obtain the density of $D\left(A\right)$.

Let $x\in X$ and define ${x}_{n}=nR\left(n,A\right)x\in D\left(A\right)$. The element $y=nR\left(1,A\right)x$, also belongs to $D\left(A\right)$. Moreover, by the proof of (i) the operators $nR\left(n\mathrm{,}A\right)$ converges towards the identity pointwise on ${X}_{0}=\stackrel{\xaf}{D\left(A\right)}$. It follows that

${y}_{n}=R\left(1,A\right){x}_{n}=nR\left(n,A\right)R\left(1,A\right)x\to y\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{for}\text{\hspace{0.17em}}n\to \infty $

Since X is reflexive and $\left\{{x}_{n}\mathrm{:}n\in \mathbb{N}\right\}$ is bounded, there exists a subsequence, still denoted by ${\left({x}_{n}\right)}_{\left(n\in \mathbb{N}\right)}$, that converges weakly to some $z\in X$. Since ${x}_{n}\in D\left(A\right)$, implies that $z\in \stackrel{\xaf}{D\left(A\right)}$.

On the other hand, the elements ${x}_{n}=\left(1-A\right){y}_{n}$ converges weakly to z, so the weak closedness of A implies that $y\in D\left(A\right)$ and $x=\left(1-A\right)y=z\in \stackrel{\xaf}{D\left(A\right)}$ which proved (ii).

Theorem 3.2

The linear operator $A\mathrm{:}D\left(A\right)\subseteq X\to X$ is a dissipative if and only if for each $x\in D\left(A\right)$ and $\lambda >0$, where $A\in \omega \text{-}OC{P}_{n}$, then we have

$\Vert \left({\lambda}_{1}-A\right)x\Vert \ge \lambda \Vert x\Vert $ (3.3)

Proof

Suppose A is dissipative, then, for each $x\in D\left(A\right)$ and $\lambda >0$, there exists ${x}^{\mathrm{*}}\in F\left(x\right)$ such that $Re\left(\lambda x-Ax\mathrm{,}{x}^{\mathrm{*}}\right)\le 0$. Therefore

$\Vert x\Vert \Vert \lambda x-Ax\Vert \ge \left|\left(\lambda x-Ax\mathrm{,}x\right)\right|\ge Re\left(\lambda x-Ax\mathrm{,}x\right)\ge \lambda {\Vert x\Vert}^{2}$

and this completes the proof. Next, let $x\in D\left(A\right)$ and $\lambda >0$.

Let ${y}_{\lambda}^{\mathrm{*}}\in F\left(\lambda x-Ax\right)$ and let us observe that, by virtue of (3.3), $\lambda x-Ax=0$ $\Rightarrow $ $x=0$.

So, in this case, we clearly have $Re\left({x}^{\mathrm{*}}\mathrm{,}\lambda x-Ax\right)=0.$ Therefore, by assuming that $\lambda x-Ax\ne 0$. As a consequence, ${y}_{\lambda}^{\mathrm{*}}\ne 0$, and thus

${z}_{\lambda}^{*}=\frac{{y}_{\lambda}^{*}}{\Vert {y}_{\lambda}^{*}\Vert}$

lies on the unit ball, i.e. $\Vert {z}_{\lambda}^{*}\Vert =1$. We have $\left(\lambda x-Ax\mathrm{,}{z}_{\lambda}^{\mathrm{*}}\right)=\Vert \lambda x-Ax\Vert \ge \lambda \Vert x\Vert $ $\Rightarrow $ $Re\left(x\mathrm{,}{z}_{\lambda}^{\mathrm{*}}\right)-Re\left(Ax\mathrm{,}{z}_{\lambda}^{\mathrm{*}}\right)\le \lambda \Vert x\Vert -Re\left(Ax\mathrm{,}{z}_{\lambda}^{\mathrm{*}}\right)$ hence

$Re\left(Ax\mathrm{,}{z}_{\lambda}^{\mathrm{*}}\right)\le 0$

and $Re\left({z}_{\lambda}^{\mathrm{*}},x\right)\ge \Vert x\Vert -\frac{1}{\lambda}\Vert Ax\Vert $. Now, let us recall that the closed unit ball in ${X}^{\mathrm{*}}$

is weakly-star compact. Thus, the net ${\left({z}_{\lambda}^{\mathrm{*}}\right)}_{\lambda >0}$ has at least one weak-star cluster point ${z}^{\mathrm{*}}\in {X}^{\mathrm{*}}$ with

$\Vert {z}^{\mathrm{*}}\Vert \le 1$ (3.4)

From (3.4), it follows that $Re\left(Ax\mathrm{,}{z}^{\mathrm{*}}\right)\le 0$ and $Re\left(x\mathrm{,}{z}^{\mathrm{*}}\right)\ge \Vert x\Vert $. Since $Re\left(x\mathrm{,}{z}^{\mathrm{*}}\right)\le \left|\left(x\mathrm{,}{z}^{\mathrm{*}}\right)\right|\le \Vert x\Vert $, it follows that $\left(x\mathrm{,}{z}^{\mathrm{*}}\right)=\Vert x\Vert $. Hence ${x}^{\mathrm{*}}=\Vert x\Vert {z}^{\mathrm{*}}\in F\left(x\right)$ and $Re\left(Ax\mathrm{,}{x}^{\mathrm{*}}\right)\le 0$ and this completes the proof.

Proposition 3.3

Let
$A\mathrm{:}D\left(A\right)\subseteq X\to X$ be infinitesimal generator of a C_{0}-semigroup of contraction and
$A\in \omega \text{-}OC{P}_{n}$. Suppose
${X}_{*}=D\left(A\right)$ is endowed with the graph-norm
${\left|\text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\right|}_{D\left(A\right)}\mathrm{:}{X}_{\mathrm{*}}\to \mathbb{N}\cup \left\{0\right\}$ defined by
${\left|u\right|}_{D\left(A\right)}=\Vert u-Au\Vert $ for
$u\in {X}_{\mathrm{*}}$. Then operator
${A}_{\mathrm{*}}\mathrm{:}D\left({A}_{\mathrm{*}}\right)\subseteq {X}_{\mathrm{*}}\to {X}_{\mathrm{*}}$ defined by

$\{\begin{array}{l}D\left({A}_{*}\right)=\left\{x\in {X}_{*};\mathrm{}Ax\in {X}_{*}\right\}\\ {A}_{*}x=Ax,\mathrm{}\text{for}\text{\hspace{0.17em}}x\in D(X*)\end{array}$

is the infinitesimal generator of a C_{0}-semigroup of contractions on
${X}_{\mathrm{*}}$.

Proof

Let
$\lambda >0$ and
$f\in {X}_{\mathrm{*}}$ and let us consider the equation
$\lambda u-Au=F$ Since A generates a C_{0}-semigroup of contraction [6] , it follows that this equation has a unique solution
$u\in D\left(A\right)$.

Since $f\in {X}_{\mathrm{*}}$, we conclude that $Au\in D\left(A\right)$ and thus $u\in D\left({A}_{\mathrm{*}}\right)$.

Thus $\lambda u-{A}_{*}u=f$. On the other hand, we have

$\begin{array}{l}{\left|{\left(\lambda I-{A}_{\mathrm{*}}\right)}^{-1}f\right|}_{D\left(A\right)}=\Vert \left(I-A\right){\left(\lambda I-A\right)}^{-1}f\Vert \\ =\Vert {\left(\lambda I-A\right)}^{-1}\left(I-A\right)f\Vert \le \frac{1}{\lambda}\Vert f-Af\Vert =\frac{1}{\lambda}{\left|f\right|}_{D\left(A\right)}\end{array}$ (3.5)

which shows that ${A}_{\mathrm{*}}$ satisfies condition (ii) in Theorem 2.2. Moreover, it follows that ${A}_{\mathrm{*}}$ is closed in ${X}_{\mathrm{*}}$.

Indeed, as ${\left(\lambda I-A\right)}^{-1}\in L\left({X}_{\mathrm{*}}\right)$, it is closed, and consequently $\lambda I-{A}_{\mathrm{*}}$ enjoys the same property which proves that ${A}_{\mathrm{*}}$ is closed.

Now, let
$x\in {X}_{\mathrm{*}}$,
$\lambda >0$,
$A\in \omega \text{-}OC{P}_{n}$ and let
${x}_{\lambda}=\lambda x-{A}_{*}x$. Clearly
${x}_{\lambda}\in D\left({A}_{\mathrm{*}}\right)$, and in addition
${\mathrm{lim}}_{\lambda \to \infty}{\left|{x}_{\lambda}-x\right|}_{D\left(A\right)}=0$ Thus,
$D\left({A}_{\mathrm{*}}\right)$ is dense in
${X}_{\mathrm{*}}$ by virtue of Theorem 2.2,
${A}_{\mathrm{*}}$ generates a C_{0}-semigroup of contraction on
${X}_{\mathrm{*}}$. Hence the proof.

4. Conclusion

In this paper, it has been established that ω-OCP_{n} possesses the properties of dissipative operators as a semigroup of linear operator, and obtaining some dissipative results on ω-OCP_{n}.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

[1] |
Yosida, K. (1948) On the Differentiability and Representation of One-Parameter Semigroups of Linear Operators. Journal of the Mathematical Society of Japan, 1, 15-21. https://doi.org/10.2969/jmsj/00110015 |

[2] |
Miyadera, I. (1952) Generation of Strongly Continuous Semigroups Operators. Tohoku Mathematical Journal, 4, 109-114. https://doi.org/10.2748/tmj/1178245412 |

[3] |
Feller, W. (1953) On the Generation of Unbounded Semigroup of Bounded Linear Operators. Annals of Mathematics, 58, 166-174. https://doi.org/10.2307/1969826 |

[4] |
Balakrishnan, A.V. (1960) Fractional Powers of Closed Operators and Semigroups Generated by Them. Pacific Journal of Mathematics, 10, 419-437. https://doi.org/10.2140/pjm.1960.10.419 |

[5] |
Lumer, G. and Phillips, R.S. (1961) Dissipative Operators in a Banach Space. Pacific Journal of Mathematics, 11, 679-698. https://doi.org/10.2140/pjm.1961.11.679 |

[6] | Hille, E. and Phillips, R.S. (1981) Functional Analysis and Semigroups. American Mathematical Society, Providence, Colloquium Publications Vol. 31. |

[7] |
Batty, C.J.K. (1994) Asymptotic Behaviour of Semigroup of Operators. Banach Center Publications, 30, 35-52. https://doi.org/10.4064/-30-1-35-52 |

[8] | Engel, K. and Nagel, R. (1999) One-Parameter Semigroup for Linear Evolution Equations. Graduate Texts in Mathematics Vol. 194, Springer, New York. |

[9] | Vrabie, I.I. (2003) C0-Semigroup and Application. Mathematics Studies Vol. 191, Elsevier, North-Holland. |

[10] |
Laradji, A. and Umar, A. (2004) Combinatorial Results for Semigroups of Order Preserving Partial Transformations. Journal of Algebra, 278, 342-359. https://doi.org/10.1016/j.jalgebra.2003.10.023 |

[11] | Rauf, K. and Akinyele, A.Y. (2019) Properties of ω-Order-Preserving Partial Contraction Mapping and Its Relation to C0-Semigroup. International Journal of Mathematics and Computer Science, 14, 61-68. |

[12] |
Rauf, K., Akinyele, A.Y., Etuk, M.O., Zubair, R.O. and Aasa, M.A. (2019) Some Results of Stability and Spectra Properties on Semigroupn of Linear Operator. Advances of Pure Mathematics, 9, 43-51. https://doi.org/10.4236/apm.2019.91003 |

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