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authorJaron Kent-Dobias <jaron@kent-dobias.com>2023-05-11 09:54:40 +0200
committerJaron Kent-Dobias <jaron@kent-dobias.com>2023-05-11 09:54:40 +0200
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treef3df62cf0090d1e21bd2fce08b4a6cd8d6aa38d8 /2-point.tex
parentbb3f5cbd80482791340e9f9bb3d40767fc249a0a (diff)
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More work on the isolated eigenvalues.
Diffstat (limited to '2-point.tex')
-rw-r--r--2-point.tex87
1 files changed, 73 insertions, 14 deletions
diff --git a/2-point.tex b/2-point.tex
index b4cb216..bb1b625 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -392,6 +392,50 @@ What about the average for the Hessian terms?
&=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu(\pmb\sigma_a\mid E_0,\mu_0)\right]\,F(\beta\mid E_1,\mu_1,q,\pmb\sigma_1)
\end{aligned}
\]
+\[
+ \sum_a^m(i\hat{\pmb\sigma}_{\pmb\sigma_a}\cdot\partial_a-\hat\beta_0)H(\pmb\sigma_a)
+ +
+ \sum_b^n(i\hat{\mathbf s}_{\mathbf s_b}\cdot\partial_b-\hat\beta_1)H(\mathbf s_b)
+ +
+ \sum_c^\ell(\mathbf x_c\cdot\partial_{\mathbf s_1})^2H(\mathbf s_1)
+\]
+\begin{align*}
+ &\sum_{ab}^\ell(\mathbf x_a\cdot\partial_{\mathbf s_1})^2(\mathbf x_b\cdot\partial_{\mathbf s_1'})^2\overline{H(\mathbf s_1)H(\mathbf s_1')}\\
+ &=(\mathbf x_a\cdot\mathbf s_1)^2(\mathbf x_b\cdot\mathbf s_1)^2f''''(1)
+ +2(\mathbf x_a\cdot\mathbf s_1)(\mathbf x_b\cdot\mathbf s_1)(\mathbf x_a\cdot\mathbf x_b)f'''(1)
+ +(\mathbf x_a\cdot\mathbf x_b)^2f''(1) \\
+ &=f''(1)\sum_{ab}^\ell A_{ab}
+\end{align*}
+\begin{align*}
+ &\sum_{a}^\ell\sum_b^n(i\hat{\mathbf s}_{\mathbf s_b}\cdot\partial_b-\hat\beta_1)(\mathbf x_a\cdot\partial_{\mathbf s_1})^2\overline{H(\mathbf s_1)H(\mathbf s_b)}\\
+ &=-\hat\beta_1(\mathbf x_a\cdot\mathbf s_b)^2f''(C^{11}_{1b})
+ +i(\hat{\mathbf s}_b\cdot\mathbf s_1)(\mathbf x_a\cdot\mathbf s_b)^2f'''(C^{11}_{1b})
+ +2i(\hat{\mathbf s}_b\cdot\mathbf x_a)(\mathbf x_a\cdot\mathbf s_b)f''(C^{11}_{1b}) \\
+ &=
+ \sum_{a=1}^\ell\sum_{b=2}^n\left[
+ -\hat\beta_1(X^1_{ab})^2f''(C^{11}_{1b})-(X^1_{ab})^2R^{11}_{1b}f'''(C^{11}_{1b})
+ -2X^1_{ab}\hat X_{ab}f''(C^{11}_{1b})
+ \right] \\
+ &=\ell\left[-\hat\beta_1x_1^2\sum_{b=2}^nf''(C^{11}_{1b})
+ -x_1^2\sum_{b=2}^nR^{11}_{1b}f'''(C^{11}_{1b})
+ -x_1\hat x_1\sum_{b=2}^nf''(C^{11}_{1b})
+ \right]
+\end{align*}
+\begin{align*}
+ &\sum_{a}^\ell\sum_b^m(i\hat{\pmb\sigma}_b\cdot\partial_b-\hat\beta_0)(\mathbf x_a\cdot\partial_{\mathbf s_1})^2\overline{H(\mathbf s_1)H(\pmb\sigma_b)}\\
+ &=-\hat\beta_0(\mathbf x_a\cdot\pmb \sigma_b)^2f''(C^{01}_{1b})
+ +i(\hat{\pmb \sigma}_b\cdot\pmb \sigma_1)(\mathbf x_a\cdot\pmb \sigma_b)^2f'''(C^{01}_{1b})
+ +2i(\hat{\pmb \sigma}_b\cdot\mathbf x_a)(\mathbf x_a\cdot\pmb \sigma_b)f''(C^{01}_{1b}) \\
+ &=
+ \sum_{a=1}^\ell\sum_{b=1}^m\left[
+ -\hat\beta_0(X^0_{ab})^2f''(C^{01}_{1b})-(X^0_{ab})^2R^{01}_{1b}f'''(C^{01}_{1b})
+ -2X^0_{ab}\hat X^0_{ab}f''(C^{01}_{1b})
+ \right] \\
+ &=\ell\left[-\hat\beta_0x_0^2f''(q)
+ -x_0^2r_{10}f'''(q)
+ -x_0\hat x_0f''(q)
+ \right]
+\end{align*}
\begin{align}
&\log\det
@@ -523,25 +567,36 @@ What about the average for the Hessian terms?
X_0
=
\begin{subarray}{l}
- \hphantom{[}\begin{array}{ccc}\leftarrow&m&\rightarrow\end{array}\hphantom{\Bigg]}\\
+ \hphantom{[}\begin{array}{ccc}\leftarrow&\ell&\rightarrow\end{array}\hphantom{\Bigg]}\\
\left[
\begin{array}{ccc}
x_0&\cdots&x_0\\
+ 0&\cdots&0\\
\vdots&\ddots&\vdots\\
- x_0&\cdots&x_0
+ 0&\cdots&0
\end{array}
\right]\begin{array}{c}
- \uparrow\\\ell\\\downarrow
+ \\\uparrow\\m-1\\\downarrow
\end{array}\\
\vphantom{\begin{array}{c}n\end{array}}
\end{subarray}
&&
\hat X_0
- =\begin{bmatrix}
- \hat x_0&\cdots&\hat x_0\\
- \vdots&\ddots&\vdots\\
- \hat x_0&\cdots&\hat x_0
- \end{bmatrix}
+ =
+ \begin{subarray}{l}
+ \hphantom{[}\begin{array}{ccc}\leftarrow&\ell&\rightarrow\end{array}\hphantom{\Bigg]}\\
+ \left[
+ \begin{array}{ccc}
+ \hat x_0&\cdots&\hat x_0\\
+ 0&\cdots&0\\
+ \vdots&\ddots&\vdots\\
+ 0&\cdots&0
+ \end{array}
+ \right]\begin{array}{c}
+ \\\uparrow\\m-1\\\downarrow
+ \end{array}\\
+ \vphantom{\begin{array}{c}n\end{array}}
+ \end{subarray}
&&
X_1
=
@@ -549,10 +604,10 @@ What about the average for the Hessian terms?
\hphantom{[}\begin{array}{ccc}\leftarrow&n&\rightarrow\end{array}\hphantom{\Bigg]}\\
\left[
\begin{array}{ccc}
- q&\cdots&q\\
0&\cdots&0\\
+ x_1&\cdots&x_1\\
\vdots&\ddots&\vdots\\
- 0&\cdots&0
+ x_1&\cdots&x_1
\end{array}
\right]\begin{array}{c}
\\\uparrow\\m-1\\\downarrow
@@ -560,15 +615,19 @@ What about the average for the Hessian terms?
\vphantom{\begin{array}{c}n\end{array}}
\end{subarray}
&&
- D^{01}
+ \hat X_1
=\begin{bmatrix}
- d_{01}&\cdots&d_{01}\\
- 0&\cdots&0\\
+ \hat x_1^0&\cdots&\hat x_1^0\\
+ \hat x_1^1&\cdots&\hat x_1^1\\
\vdots&\ddots&\vdots\\
- 0&\cdots&0
+ \hat x_1^1&\cdots&\hat x_1^1
\end{bmatrix}
\end{align}
+\[
+ 2(A-X^TC^{-1}X)^{-1}X^TC^{-1}
+\]
+
\paragraph{Acknowledgements}
\paragraph{Funding information}