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\usepackage[T1]{fontenc}

\def\bbR{{\mathbb R}}
\def\calD{{\mathcal D}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}

\def\ann{{\operatorname{ann}}}
\def\rad{\operatorname{rad}}
\def\sign{\operatorname{sign}}

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\begin{document} \latintext
\begin{multicols*}{2}

{\bf Def.} Given a v.s.\ $V$, a Partial Quadratic (PQ) $Q$ on $V$ is a symmetric bilinear form $Q$ on a subspace $\calD(Q)\subset V$. For $U\subset\calD(Q)$, denote $\ann_Q(U)\coloneqq\{v\in\calD(Q): Q(U,v)=0\}$.

{\bf Def.} $Q_1+Q_2$ is with $\calD(Q_1+Q_2)=\calD(Q_1)\cap\calD(Q_2)$.

{\bf Def.} Given a linear $\psi\colon V\to W$ and a PQ $Q$ on $W$, the pullback is $(\psi^*Q)(v_1,v_2)=Q(\psi v_1,\psi v_2)$ with $\calD(\psi^*Q)=\phi^{-1}(\calD(Q))$.

{\bf Def.} Given $\phi\colon V\to W$ and a PQ $Q$ on $V$ the pushforward $\phi_*Q$ is with $\calD(\phi_*Q)=\phi(\ann_Q(\calD(Q)\cap\ker\phi))$ and $(\phi_*Q)(w_1,w_2)=Q(v_1,v_2)$, where $v_i$ are s.t.\ $\phi(v_i)=w_i$ and $Q(v_i,\rad Q|_{\ker\phi})=0$.

\parpic[r]{$\xymatrix@C=3mm@R=3mm{
  \bullet \ar[r]^\alpha \ar[d]_\gamma & \bullet \ar[d]^\beta \\
  \bullet \ar[r]_\delta \ar@{.>}[ru] & \bullet
}$}
{\bf Thm(?).} $\psi^*$ and $\phi_*$ are well-defined and functorial, and if $\alpha\act\beta = \gamma\act\delta$, then $\gamma^*\act\alpha_* = \delta_*\act\beta^*$. $\psi^*$ is additive but $\phi_*$ isn't.

{\bf Thm(?).} Over $\bbR$, given $\phi\colon V\to W$ and PQs $Q$ on $V$ and $C$ on $W$,
\[ \sign_V(Q+\phi^*C) = \sign_{\ker\phi}(\iota^*Q) + \sign_W(C+\phi_*Q). \]

\input{PQ.tex}

\end{multicols*}
\end{document}

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