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| ==Approximate Confidence Bounds for the Eyring-Weibull==
| | #REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]] |
| <br>
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| ===Bounds on the Parameters===
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| <br>
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| From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta }</math> is a positive parameter, <math>\ln (\widehat{\beta })</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters are estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
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| & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}
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| \end{align}</math>
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| <br>
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| :also:
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| <br>
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| ::<math>\begin{align}
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| & {{A}_{U}}= & \widehat{A}+{{K}_{\alpha }}\sqrt{Var(\widehat{A})} \\
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| & {{A}_{L}}= & \widehat{A}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})}
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| \end{align}</math>
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| <br>
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| :and:
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| <br>
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| ::<math>\begin{align}
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| & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
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| & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
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| \end{align}</math>
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| <br>
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| The variances and covariances of <math>\beta ,</math> <math>A,</math> and <math>B</math> are estimated from the Fisher matrix (evaluated at <math>\widehat{\beta },</math> <math>\widehat{A},</math> <math>\widehat{B})</math> as follows:
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| <br>
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| ::<math>\left[ \begin{matrix}
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| Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{A}) & Cov(\widehat{\beta },\widehat{B}) \\
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| Cov(\widehat{A},\widehat{\beta }) & Var(\widehat{A}) & Cov(\widehat{A},\widehat{B}) \\
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| Cov(\widehat{B},\widehat{\beta }) & Cov(\widehat{B},\widehat{A}) & Var(\widehat{B}) \\
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| \end{matrix} \right]={{\left[ \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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| ===Confidence Bounds on Reliability===
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| <br>
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| The reliability function for the Eyring-Weibull model (ML estimate) is given by:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-{{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}}}}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]}}}}</math>
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| <br>
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| :Setting:
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| <br>
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| ::<math>\widehat{u}=\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)+\ln (V)+\widehat{A}-\frac{\widehat{B}}{V} \right]</math>
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| <br>
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| The reliability function now becomes:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-e\widehat{^{u}}}}</math>
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| <br>
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| The next step is to find the upper and lower bounds on <math>\widehat{u}</math> :
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| <br>
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| ::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| ::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| :where:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial A} \right)}^{2}}Var(\widehat{A}) \\
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| & & +{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial A} \right)Cov(\widehat{\beta },\widehat{A}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial A} \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) \\
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| & & +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) \\
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| & & -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| The upper and lower bounds on reliability are:
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| <br>
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| ::<math>\begin{align}
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| & {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
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| & {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}
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| \end{align}</math>
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| <br>
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| where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated using Eqns (EyrExpu) and (EyrExpl).
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| ===Confidence Bounds on Time===
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| <br>
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| The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:
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| <br>
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| ::<math>\begin{align}
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| & \ln (R)= & -{{\left( \widehat{T}\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \\
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| & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}+\ln V+\widehat{A}-\frac{\widehat{B}}{V} \right)
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln V-\widehat{A}+\frac{\widehat{B}}{V}</math>
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| <br>
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| where
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| <br>
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| ::<math></math>
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| <br>
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| The upper and lower bounds on <math>\widehat{u}</math> are then estimated from:
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| <br>
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| ::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| ::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| :where:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial A} \right)}^{2}}Var(\widehat{A}) \\
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| & & +{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial A} \right)Cov(\widehat{\beta },\widehat{A}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial A} \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta }) \\
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| & & +Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B}) \\
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| & & +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}}Cov(\widehat{\beta },\widehat{A})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B}) \\
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| & & -\frac{2}{V}Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| The upper and lower bounds on time are then found by:
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| <br>
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
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| & {{T}_{L}}= & {{e}^{{{u}_{L}}}}
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| \end{align}</math>
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| <br>
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| where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated using Eqns. (EyrTimeu) and (EyrTimel).
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| <br>
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