Whether supremum and partial derivative can be interchanged?
Recently, I study replica method derived from statistical physics. I have a confusion on following equation
begin{align}
mathcal{F}=-lim_{Mrightarrow infty}frac{1}{M}lim_{taurightarrow 0}frac{partial }{partial tau} underset{mathbf{Q},tilde{mathbf{Q}}}{supinf}left{alpha^{-1}G(mathbf{Q})-text{tr}left{mathbf{Q}tilde{mathbf{Q}}right}-frac{1}{M}log mathbb{E}_{mathbf{X}}left{exp left[text{tr}(mathbf{X}tilde{mathbf{Q}}mathbf{X}^T)right]right}right}
end{align}
where $mathbf{Q}=(c-q)mathbf{I}_{tau+1}+qboldsymbol{ee}^T$ and $tilde{mathbf{Q}}=(tilde{c}-tilde{q})mathbf{I}_{tau+1}+tilde{q}boldsymbol{ee}^T$ with $boldsymbol{e}in mathbb{R}^{tau+1}$ denoting a column vector whose elements are all 1.
In the most of paper about replica method, they always change the sup, inf and partial derivative like following
begin{align}
mathcal{F}=-lim_{Mrightarrow infty}frac{1}{M} underset{mathbf{Q},tilde{mathbf{Q}}}{supinf}left{
lim_{taurightarrow 0}frac{partial }{partial tau}left[
alpha^{-1}G(mathbf{Q})-text{tr}left{mathbf{Q}tilde{mathbf{Q}}right}-frac{1}{M}log mathbb{E}_{mathbf{X}}left{exp left[text{tr}(mathbf{X}tilde{mathbf{Q}}mathbf{X}^T)right]right}
right]
right}
end{align}
What I have thought?
As far as I know, I consider that both $sup$ and $inf$ are one limit. Generally, $lim$ and partial derivative can be interchanged.
I hope you can help me. This is extremely important for me. Thank you!
partial-derivative supremum-and-infimum
asked Nov 20 at 2:55
Qiuyun
778
add a comment |
Recently, I study replica method derived from statistical physics. I have a confusion on following equation
begin{align}
mathcal{F}=-lim_{Mrightarrow infty}frac{1}{M}lim_{taurightarrow 0}frac{partial }{partial tau} underset{mathbf{Q},tilde{mathbf{Q}}}{supinf}left{alpha^{-1}G(mathbf{Q})-text{tr}left{mathbf{Q}tilde{mathbf{Q}}right}-frac{1}{M}log mathbb{E}_{mathbf{X}}left{exp left[text{tr}(mathbf{X}tilde{mathbf{Q}}mathbf{X}^T)right]right}right}
end{align}
where $mathbf{Q}=(c-q)mathbf{I}_{tau+1}+qboldsymbol{ee}^T$ and $tilde{mathbf{Q}}=(tilde{c}-tilde{q})mathbf{I}_{tau+1}+tilde{q}boldsymbol{ee}^T$ with $boldsymbol{e}in mathbb{R}^{tau+1}$ denoting a column vector whose elements are all 1.
In the most of paper about replica method, they always change the sup, inf and partial derivative like following
begin{align}
mathcal{F}=-lim_{Mrightarrow infty}frac{1}{M} underset{mathbf{Q},tilde{mathbf{Q}}}{supinf}left{
lim_{taurightarrow 0}frac{partial }{partial tau}left[
alpha^{-1}G(mathbf{Q})-text{tr}left{mathbf{Q}tilde{mathbf{Q}}right}-frac{1}{M}log mathbb{E}_{mathbf{X}}left{exp left[text{tr}(mathbf{X}tilde{mathbf{Q}}mathbf{X}^T)right]right}
right]
right}
end{align}
What I have thought?
As far as I know, I consider that both $sup$ and $inf$ are one limit. Generally, $lim$ and partial derivative can be interchanged.
I hope you can help me. This is extremely important for me. Thank you!
partial-derivative supremum-and-infimum
asked Nov 20 at 2:55
Qiuyun
778
add a comment |
Recently, I study replica method derived from statistical physics. I have a confusion on following equation
begin{align}
mathcal{F}=-lim_{Mrightarrow infty}frac{1}{M}lim_{taurightarrow 0}frac{partial }{partial tau} underset{mathbf{Q},tilde{mathbf{Q}}}{supinf}left{alpha^{-1}G(mathbf{Q})-text{tr}left{mathbf{Q}tilde{mathbf{Q}}right}-frac{1}{M}log mathbb{E}_{mathbf{X}}left{exp left[text{tr}(mathbf{X}tilde{mathbf{Q}}mathbf{X}^T)right]right}right}
end{align}
where $mathbf{Q}=(c-q)mathbf{I}_{tau+1}+qboldsymbol{ee}^T$ and $tilde{mathbf{Q}}=(tilde{c}-tilde{q})mathbf{I}_{tau+1}+tilde{q}boldsymbol{ee}^T$ with $boldsymbol{e}in mathbb{R}^{tau+1}$ denoting a column vector whose elements are all 1.
In the most of paper about replica method, they always change the sup, inf and partial derivative like following
begin{align}
mathcal{F}=-lim_{Mrightarrow infty}frac{1}{M} underset{mathbf{Q},tilde{mathbf{Q}}}{supinf}left{
lim_{taurightarrow 0}frac{partial }{partial tau}left[
alpha^{-1}G(mathbf{Q})-text{tr}left{mathbf{Q}tilde{mathbf{Q}}right}-frac{1}{M}log mathbb{E}_{mathbf{X}}left{exp left[text{tr}(mathbf{X}tilde{mathbf{Q}}mathbf{X}^T)right]right}
right]
right}
end{align}
What I have thought?
As far as I know, I consider that both $sup$ and $inf$ are one limit. Generally, $lim$ and partial derivative can be interchanged.
I hope you can help me. This is extremely important for me. Thank you!
partial-derivative supremum-and-infimum
asked Nov 20 at 2:55
Qiuyun
778
Recently, I study replica method derived from statistical physics. I have a confusion on following equation
begin{align}
mathcal{F}=-lim_{Mrightarrow infty}frac{1}{M}lim_{taurightarrow 0}frac{partial }{partial tau} underset{mathbf{Q},tilde{mathbf{Q}}}{supinf}left{alpha^{-1}G(mathbf{Q})-text{tr}left{mathbf{Q}tilde{mathbf{Q}}right}-frac{1}{M}log mathbb{E}_{mathbf{X}}left{exp left[text{tr}(mathbf{X}tilde{mathbf{Q}}mathbf{X}^T)right]right}right}
end{align}
where $mathbf{Q}=(c-q)mathbf{I}_{tau+1}+qboldsymbol{ee}^T$ and $tilde{mathbf{Q}}=(tilde{c}-tilde{q})mathbf{I}_{tau+1}+tilde{q}boldsymbol{ee}^T$ with $boldsymbol{e}in mathbb{R}^{tau+1}$ denoting a column vector whose elements are all 1.
In the most of paper about replica method, they always change the sup, inf and partial derivative like following
begin{align}
mathcal{F}=-lim_{Mrightarrow infty}frac{1}{M} underset{mathbf{Q},tilde{mathbf{Q}}}{supinf}left{
lim_{taurightarrow 0}frac{partial }{partial tau}left[
alpha^{-1}G(mathbf{Q})-text{tr}left{mathbf{Q}tilde{mathbf{Q}}right}-frac{1}{M}log mathbb{E}_{mathbf{X}}left{exp left[text{tr}(mathbf{X}tilde{mathbf{Q}}mathbf{X}^T)right]right}
right]
right}
end{align}
What I have thought?
As far as I know, I consider that both $sup$ and $inf$ are one limit. Generally, $lim$ and partial derivative can be interchanged.
I hope you can help me. This is extremely important for me. Thank you!
partial-derivative supremum-and-infimum
partial-derivative supremum-and-infimum
asked Nov 20 at 2:55
Qiuyun
778
asked Nov 20 at 2:55
Qiuyun
778
edited Nov 20 at 7:37
asked Nov 20 at 2:55
Qiuyun
778
asked Nov 20 at 2:55
Qiuyun
778
asked Nov 20 at 2:55
Qiuyun
778
778
add a comment |
add a comment |
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