Commit 6f691b6b by Conor McCoid

Tetra: minor restatement of sign of denom lemma

parent 7e43e7ed
 ... @@ -1095,7 +1095,7 @@ for $J = \set{i_j}_{j=0}^m$ and $\Gamma = \set{\gamma_j}_{j=1}^m$. ... @@ -1095,7 +1095,7 @@ for $J = \set{i_j}_{j=0}^m$ and $\Gamma = \set{\gamma_j}_{j=1}^m$. Then $$\vec{h}(J|\Gamma) \cdot \vec{e}_\eta = \frac{\begin{vmatrix} X_J^\top \vec{e}_\eta & X_J^\top I_\Gamma \end{vmatrix} }{ \begin{vmatrix} \vec{1} & X_J^\top I_\Gamma \end{vmatrix}}.$$ Then $$\vec{h}(J|\Gamma) \cdot \vec{e}_\eta = \frac{\begin{vmatrix} X_J^\top \vec{e}_\eta & X_J^\top I_\Gamma \end{vmatrix} }{ \begin{vmatrix} \vec{1} & X_J^\top I_\Gamma \end{vmatrix}}.$$ \begin{lemma} \begin{lemma} Suppose $\sign(\vec{h}(J \setminus \set{i} | \Gamma) \cdot \vec{e}_\eta) \neq \sign(\vec{h}(J \setminus \set{j} | \Gamma) \cdot \vec{e}_\eta)$, then Suppose $\sign(\vec{h}(J \setminus \set{i} | \Gamma) \cdot \vec{e}_\eta) \neq \sign(\vec{h}(J \setminus \set{j} | \Gamma) \cdot \vec{e}_\eta)$ and an intersection $\vec{h}(J \setminus \set{i,j} | \Gamma \setminus \set{\gamma})$ was calculated for some $\gamma \in \Gamma$, then \begin{equation*} \begin{equation*} \sign \left ( \begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \right ) = \sign \left ( \begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \right ) = \sign \left ( \begin{vmatrix} X_{J \setminus \set{i}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \sign \left ( \begin{vmatrix} X_{J \setminus \set{i}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} ... @@ -1127,13 +1127,13 @@ We expand the numerator of this expression: ... @@ -1127,13 +1127,13 @@ We expand the numerator of this expression: \begin{align*} \begin{align*} & \left ( \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} + \sum\limits_{k=0}^{m-1} (-1)^k \vec{x}_i^\top \vec{e}_{\gamma_k} \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix} \right ) \begin{vmatrix} X_{J \setminus \set{i}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\ & \left ( \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} + \sum\limits_{k=0}^{m-1} (-1)^k \vec{x}_i^\top \vec{e}_{\gamma_k} \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix} \right ) \begin{vmatrix} X_{J \setminus \set{i}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\ & - \left ( \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} + \sum\limits_{k=0}^{m-1} (-1)^k \vec{x}_j^\top \vec{e}_{\gamma_k} \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix} \right ) \begin{vmatrix} X_{J \setminus \set{j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\ & - \left ( \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} + \sum\limits_{k=0}^{m-1} (-1)^k \vec{x}_j^\top \vec{e}_{\gamma_k} \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix} \right ) \begin{vmatrix} X_{J \setminus \set{j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\ = & \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} & = \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} \begin{vmatrix} 1 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 1 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \begin{vmatrix} 1 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 1 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} + \begin{vmatrix} \vec{x}_i^\top I_\Gamma \vec{w} & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ \vec{x}_j^\top I_\Gamma \vec{w} & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\ + \begin{vmatrix} \vec{x}_i^\top I_\Gamma \vec{w} & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ \vec{x}_j^\top I_\Gamma \vec{w} & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\ = & \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} & = \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} \begin{vmatrix} 1 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 1 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \begin{vmatrix} 1 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 1 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} + \begin{vmatrix} 0 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 0 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ -X_{J \setminus \set{i,j}}^\top I_\Gamma \vec{w} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\ + \begin{vmatrix} 0 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 0 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ -X_{J \setminus \set{i,j}}^\top I_\Gamma \vec{w} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\ = & \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} \begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix}. & = \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} \begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix}. \end{align*} \end{align*} where $w_k = (-1)^k \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix}.$ where $w_k = (-1)^k \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix}.$ To prove the last equality note that To prove the last equality note that ... @@ -1160,8 +1160,6 @@ Therefore, the sign of $\begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\ ... @@ -1160,8 +1160,6 @@ Therefore, the sign of$\begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\ \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} \begin{vmatrix} \vec{1} & X_{J \setminus \set{j}}^\top I_\Gamma \end{vmatrix} \right ). \begin{vmatrix} \vec{1} & X_{J \setminus \set{j}}^\top I_\Gamma \end{vmatrix} \right ). \end{equation*} \end{equation*} Note that since both $J \setminus \set{i}$ and $J \setminus \set{j}$ are indices of intersections their common parent $J \setminus \set{i,j}$ also indexes an intersection. Thus all elements of the right hand side are calculated in the previous step of the algorithm. \end{proof} \end{proof} \subsection{Algorithm for the intersection of n-dimensional simplices} \subsection{Algorithm for the intersection of n-dimensional simplices} ... ...
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