From 6d6aa07ba2a1e17c7d6b470aabd46447efdae1f4 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 11 May 2023 09:54:40 +0200
Subject: More work on the isolated eigenvalues.

---
 2-point.tex | 87 +++++++++++++++++++++++++++++++++++++++++++++++++++----------
 1 file changed, 73 insertions(+), 14 deletions(-)

(limited to '2-point.tex')

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}
-- 
cgit v1.2.3-70-g09d2

