text:
\begin{document}

\title[Parabolic vector bundles on Klein surfaces]{Parabolic vector bundles on Klein surfaces}

\author[I. Biswas]{Indranil Biswas}

\address{School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India}

\email{indranil@math.tifr.res.in}

\author[F. Schaffhauser]{Florent Schaffhauser}

\address{Universidad de Los Andes, Departamento de Matemáticas, Carrera 1 \#18A-12, 111 711 Bogot\'a, Colombia \& Universit\'e de Strasbourg, Institut de Recherche Mathématique Avancée, 7 rue Ren\'e Descartes, 67 000 Strasbourg, France.}

\email{schaffhauser@math.unistra.fr}

\subjclass[2010]{14H60, 30F35}

\keywords{Klein surface, real parabolic vector bundle, Fuchsian groups, orbifold fundamental group.}

\date{\today}

\begin{abstract}
Given a discrete subgroup $\Ga$ of finite co-volume of $\PGL(2,\R)$, we define and study
parabolic vector bundles on the quotient $\Sigma$ of the (extended) hyperbolic plane by $\Ga$.
If $\Ga$ contains an orientation-reversing isometry, then the above is equivalent to studying
real and quaternionic parabolic vector bundles on the orientation cover
of degree two of $\Sigma$. We then
prove that isomorphism classes of polystable real and quaternionic parabolic vector bundles are
in a natural bijective correspondence with the equivalence classes of real and quaternionic unitary
representations of $\Gamma$. Similar results are obtained for compact-type real parabolic
vector bundles over Klein surfaces.
\end{abstract}

\maketitle

\section{Introduction}

The study of vector bundles over varieties defined over the field of real numbers was initiated
in the seminal work of Atiyah \cite{AtiyahKR}. The present work investigates parabolic vector bundles
over curves defined over $\mathbb R$.

Holomorphic vector bundles on compact Riemann surfaces is a rich topic that started with the
papers of Weil and Atiyah (\cite{Weil_matrix_div}, \cite{Atiyah_connections}), who gave
necessary and sufficient conditions for such bundles to arise from linear representations of the
fundamental group of the Riemann surface over which
the bundle lives. The notion of \textit{stability}, which had been introduced by Mumford in
order to construct moduli varieties of such bundles (\cite{Mumford_Proc,Mumford_GIT_first_ed}),
was then used by Narasimhan and Seshadri to prove that vector bundles arising from irreducible
\textit{unitary} representations were exactly the stable vector bundles of degree zero
(\cite{NS}).
Seshadri later introduced the notion of parabolic vector bundles (\cite{Se}), and
then proved with Mehta an analog of the Narasimhan-Seshadri theorem for unitary representations
of the fundamental group of a punctured Riemann surface
\cite{MS}. More precisely, they proved that,
given a finite subset $S$ of a compact Riemann surface $X$, irreducible representations of
$\pi_1(X\setminus S)$ in $\U(r)$ correspond bijectively to the stable parabolic vector bundles on
$X$ of rank $r$ and parabolic degree zero with parabolic structure over the points of $S$.

In this paper, we call \textit{Klein surface} a Riemann surface equipped with an anti-holomorphic involution. Just
as compact Riemann surfaces correspond to smooth complex projective curves, compact Klein surfaces correspond to
smooth complex projective curves defined over the field of real numbers. Let then $(X,\si)$ be a compact connected
Klein surface ($\si$ being the given anti-holomorphic involution of the Riemann surface $X$). A real vector bundle
on $(X,\sigma)$ is a holomorphic vector bundle on $X$ equipped with an anti-holomorphic lift $\tau$ of $\sigma$
such that $\tau^2=Id$, and a quaternionic vector bundle on $(X,\sigma)$ is a holomorphic vector bundle on $X$
equipped with an anti-holomorphic lift $\tau$ of $\sigma$ such that $\tau^2=-Id$. Real and quaternionic vector
bundles on Klein surfaces have been intensively investigated (\cite{KW,BHH,Sch_GD}) and our aim here is to study
the parabolic analog of them. We note that real parabolic vector bundles and linear connections on such bundles
have already been studied by S. Amrutiya and L. A. Calvo independently in
\cite{Amrutiya_connections_on_real_parab_vb,Amrutiya_real_parab_vb}, \cite{Calvo} (see
also \cite{GW16} for related results). In the present paper, we will
focus on stable and polystable such bundles, in both the real and quaternionic case, and on unitary connections on
such bundles. Given $(X,\sigma)$ as above ($X$ compact) and a finite subset $S\subset X$ such that $\sigma(S)\,=\,S$,
we set $Y \,:=\, X\setminus S$ and we let $\Ga(Y,\,\si)\,:=\,\pi_1^{\mathrm{orb}}(Y/\si)$ be the orbifold fundamental
group of the quotient of $Y$ by the action of the group $\langle \si\rangle$ generated by the anti-holomorphic
involution $\si\vert_Y$. Then there is a short exact sequence
\begin{equation}\label{fund_ses}
1\,\lra\, \pi_1(Y) \,\lra\, \Ga(Y,\,\si) \,\lra\, \Z/2\Z \,\lra\, 1
\end{equation} and, by definition, a \textit{unitary representation} of this short exact sequence will be a group morphism from $\Ga(Y,\,\si)$ to an extension $\W(r)$ of the form
\begin{equation}\label{extensions_by_unit_gps}
1\longrightarrow \U(r) \longrightarrow \W(r) \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow 1
\end{equation} which induces the identity on $\Z/2\Z$ (see Definition \ref{rep_of_fund_ses}). Recall that, when $\Z/2\Z$ acts on $\U(r)$ by complex conjugation, one has $H^2(\Z/2\Z;\mathcal{Z}(\U(r)))=\{\pm1\}$. Therefore, with respect to that action, there are only two isomorphism classes of extensions of the form $\eqref{extensions_by_unit_gps}$, the trivial class being that of the semi-direct product $\U(r)\rtimes \Z/2\Z$. Representations of $\Ga(Y,\,\si)$ into the trivial
(respectively, non-trivial) type of extension will be called \textit{real} (respectively, \textit{quaternionic}) unitary representations of $\Ga(Y,\,\si)$.
We then prove the following result.

\begin{theorem}\label{MS_corresp_for_real_and_quat}
Let $(X,\si)$ be a compact Klein surface and let $S\subset X$ be a finite subset such that $\si(S)=S$. Let $Y$ be the punctured surface $X\setminus S$, assumed to be of negative Euler characteristic, and let $\Ga(Y,\,\si)$ be the orbifold fundamental group of the quotient surface $Y/\si$. Then there is a bijective correspondence between equivalence classes of real (respectively, quaternionic) unitary representations of $\Ga(Y,\,\si)$ and isomorphism classes of polystable real (respectively, quaternionic) parabolic vector bundles of parabolic degree zero on $(X,\si)$ with parabolic structure at the points of $S$.
\end{theorem}

More precisely, we prove that every polystable real (respectively, quaternionic) parabolic
vector bundle of parabolic degree zero on $(X,\si,S)$ is isomorphic to a parabolic bundle of the
form $\cE(\rho)_\bullet$, where $\rho$ is a real (respectively, quaternionic) unitary
representation of $\Ga(Y,\,\si)$ and $\cE(\rho)_\bullet$ is equipped with its canonical real
(respectively, quaternionic) structure (constructed in Proposition
\ref{construction_of_real_parab_vb}). A similar result is then proved for compact-type real
parabolic vector bundles (see Definition \ref{compact_type_real_and_quat_parab_bundles},
Proposition \ref{construction_of_compact_type_real_parab_vb} and Theorem
\ref{MS_corresp_for_compact_type_real}).

\section{Parabolic vector bundles and stability}\label{parab_vb_and_stability}

\subsection{Real and quaternionic parabolic vector bundles}

Let $X$ be a compact connected Riemann surface.
Let $\si:X\lra X$ be an anti-holomorphic involution of $X$.
Fix a finite subset $S=\{x_1,\cdots,x_d\}\subset\ X$
such that $\sigma(S)= S$. Note that $\sigma$ may not fix $S$ point-wise (compare \cite[p.~212]{MS}). Let $E$ be a holomorphic vector bundle on $X$.

\begin{definition}[{\cite[Definition 1.5]{MS}}]
A \textit{quasi-parabolic structure} on $E$ over $S$ consists of $d$ strictly decreasing filtrations
$$
E_{x_j}\,=E^1_j\supsetneq E^2_j \,\supsetneq\cdots\supsetneq\,
E^{n_j}_j \supsetneq E^{n_j+1}_j \,=0 \quad (1\leq j\leq d),
$$ and a \textit{parabolic structure} on $E$ over $S$ is a
quasi-parabolic structure as above, together with $d$ strictly increasing sequences of real numbers
$$
0\,\leq\,\alpha^1_j\,<\,\alpha^2_j\,<\,
\cdots\,<\, \alpha^{n_j}_j \,<\, 1\ \quad (1\,\leq\, j\,\leq\, d)\, .
$$
The real number $\alpha^i_j$ is called the \textit{parabolic weight} of the subspace $E^i_j$ in
the quasi-parabolic filtration. The \textit{multiplicity} of a parabolic weight
$\alpha^i_j$ at $x_j$ is defined to be the dimension of the complex vector space $E^i_j/E^{i+1}_j$.
A \textit{parabolic vector bundle} on $(X,S)$ is a triple $(E,\{E^i_j\}, \,\{\alpha^i_j\})$ consisting of a vector bundle on $X$ equipped with a parabolic structure over $S$.
\end{definition}

We refer to \cite{MS} for the definition of a morphism of parabolic vector bundles. For
notational convenience, a parabolic vector bundle $(E,\{E^i_j\}, \,\{\alpha^i_j\})$ as above
will also be denoted by $E_{\bullet}$.

Let $E$ be a holomorphic vector bundle on $X$ of rank $r$. Let $\overline{E}$ be the
$C^\infty$ complex vector bundle of rank $r$ on $X$ whose underlying real vector bundle of
rank $2r$ is the $C^\infty$ real vector bundle underlying $E$ and whose multiplication by $\sqrt{-1}$
on the fibers coincides with the multiplication by $-\sqrt{-1}$
on the fibers of $E$. The holomorphic structure on $E$ induces a holomorphic
structure on $\sigma^*\overline{E}$, uniquely characterized by the fact that the holomorphic sections of $\si^*\ov{E}$ over U are the anti-holomorphic sections of $E$ on $\si(U)$. Note that a morphism of holomorphic vector bundles $\phi:E\lra F$ gives rise to a morphism of holomorphic vector bundles $\si^*\ov{\phi}:\si^*\ov{E}\lra \si^*\ov{F}$.
Let then $E_{\bullet}\,=(E,\{E^i_j\}, \,\{\alpha^i_j\})$ be a parabolic bundle. For all $x_j\in S$, the filtration $\{E^i_j\}$ of $E_{x_j}$ produces a filtration
of the fiber $(\sigma^*\overline{E})_{\sigma(x_j)}$ using the conjugate linear
identification of $E_{x_j}$ with $(\sigma^*\overline{E})_{\sigma(x_j)}$. For any $j$, the
parabolic weights $\{\alpha^i_j\}_{i=1}^{n_j}$ associated to $\{E^i_j\}_{i=1}^{n_j}$ can be
considered as parabolic weights associated to this filtration of
$(\sigma^*\overline{E})_{\sigma(x_j)}$. The new parabolic vector bundle obtained this way will be denoted by
$\sigma^*\overline{E}_{\bullet}$. We then have the following definition of a real parabolic vector
bundle (equivalent to \cite[Definition 3.1]{Amrutiya_real_parab_vb}; see also \cite[Section~3]{Calvo}), and its immediate generalization to the quaternionic case.

\begin{definition}[Real and quaternionic parabolic vector bundles]
A \textit{real parabolic vector bundle} on $(X,\si,S)$ is a parabolic vector bundle $E_{\bullet}\,=(E,\{E^i_j\},
\,\{\alpha^i_j\})$ on $(X,S)$, together with a holomorphic isomorphism of parabolic vector bundles $\phi:E_{\bullet}\lra \si^*\ov{E}_{\bullet}$ such that $\si^*\ov{\phi}\circ \phi = \Id_E$.
A \textit{quaternionic parabolic vector bundle} on $(X,\si,S)$ is a parabolic vector bundle $E_{\bullet}$ on $(X,S)$, together with a holomorphic isomorphism of parabolic vector bundles $\phi:E_{\bullet}\lra \si^*\ov{E}_{\bullet}$ such that $\si^*\ov{\phi}\circ \phi = -\Id_E$.
\end{definition}

Note that a real or quaternionic parabolic vector bundle in particular satisfies $\si^*\ov{E}_\bullet \simeq E_\bullet$. Conversely, if $E_\bullet$ is a \textit{simple} parabolic vector bundle over $(X,\si)$ that satisfies $\si^*\ov{E}_\bullet \simeq E_\bullet$, then $E_\bullet$ admits either a real or a quaternionic structure. Let now $E^{\vee}_{\bullet}$ be the parabolic dual of $E_{\bullet}$ (see \cite[Section~3]{Yo}, \cite[p.~309]{Bi}). The holomorphic vector bundle underlying the parabolic vector bundle $E^{\vee}_{\bullet}$ will be denoted by $E^{\vee}_0$. Note that $E^{\vee}_0$ is in general only a subsheaf of $E^\vee$ (the dual of $E$ as a vector bundle), and that the inclusion map $E^{\vee}_0\,\hookrightarrow\,E^\vee$ is an isomorphism over $X\setminus S$, but fails
to be an isomorphism over $x_j\in S$ if there is a non-zero parabolic weight of $E_{\bullet}$ at $x_j$. Note that $(E_\bullet^\vee)^\vee$ is canonically isomorphic to $E_\bullet$.
\begin{definition}[Compact-type real parabolic vector bundles]\label{compact_type_real_and_quat_parab_bundles}
A \textit{compact-type real parabolic vector bundle} on $(X,\si,S)$ is a parabolic vector bundle
$E_{\bullet}\,=(E,\{E^i_j\}, \,\{\alpha^i_j\})$ on $(X,S)$, together with a holomorphic isomorphism $\phi:E_\bullet\lra (\si^*\ov{E})_\bullet^\vee$ such that $(\si^*\ov{\phi})^\vee \circ \phi = \Id_E$.
\end{definition}

\begin{remark}\label{on_compact_type_quaternionic}
Note that a compact-type real parabolic vector bundle in particular satisfies $(\si^*\ov{E})_\bullet^\vee \simeq E_\bullet$. Conversely, if $E_\bullet$ is a \textit{simple} parabolic vector bundle over $(X,\si)$ that satisfies $(\si^*\ov{E})_\bullet^\vee \simeq E_\bullet$, then $E_\bullet$ admits a compact-type real structure (because $\mathrm{Aut}(E_\bullet)\simeq \mathcal{Z}(\mathbf{GL}(r;\C)) \,\simeq\,\C^*$ is $2$-divisible). In principle, one could also define compact-type quaternionic parabolic vector bundles (by requiring that $(\si^*\ov{\phi})^\vee \circ \phi = -\Id_E$), but these can in fact only occur for vector bundles with structure group contained in $\mathbf{SL}(2m;\C)$, as $\mathcal{Z}(\mathbf{SL}(2m;\C))$ is no longer $2$-divisible.
\end{remark}

\subsection{Stability}

The \textit{parabolic degree} of a parabolic vector bundle $E_{\bullet}\,=(E,\{E^i_j\},
\,\{\alpha^i_j\})$ is defined (\cite[Definition~1.11]{MS}) to be $$ \text{par-deg}(E_{\bullet})\,=\text{degree}(E)+\sum_{j=1}^d \sum_{i=1}^{n_j}
\alpha^i_j \dim (E^i_j/E^{i+1}_j).$$ Take any holomorphic subbundle $F\subset E$. For each $x_j\in\,
S$, the fiber $F_{x_j}$ has a filtration obtained by intersecting the quasi-parabolic filtration
of $E_{x_j}$ with the subspace $F_{x_j}$. The parabolic weight of a subspace $V\subset\,
F_{x_j}$ in this filtration is the maximum of the numbers $\{\alpha^i_j\mid V\subset E^i_j\cap F_{x_j}\}$. This parabolic structure on $F$ will be denoted by $F_{\bullet}$.

\begin{definition}[Stability for parabolic bundles]
A parabolic vector bundle $E_{\bullet}\,=(E,\{E^i_j\}, \,\{\alpha^i_j\})$ is called
\textit{stable} (respectively, \textit{semistable}) if for all subbundles $F\subsetneq
E$ of positive rank, the inequality
$$
\frac{\text{par-deg}(F_{\bullet})}{\text{rank}(F)}\,<\,
\frac{\text{par-deg}(E_{\bullet})}{\text{rank}(E)}\ \
\left(\text{respectively, } \ \frac{\text{par-deg}(F_{\bullet})}{\text{rank}(F)} \leq\,
\frac{\text{par-deg}(E_{\bullet})}{\text{rank}(E)}\right)
$$
holds \cite[Definition~1.13]{MS}. A parabolic vector bundle $E_{\bullet}$ is called \textit{polystable} if it is semistable and isomorphic to a direct sum of parabolic stable vector bundles (see \cite{Yo}, \cite{Bi} for
direct sums of parabolic vector bundles).
\end{definition}

A stable parabolic bundle is simple, meaning that any holomorphic automorphism of it is a constant
non-zero scalar multiplication (see \cite{MS}). The parabolic bundles appearing in the direct sum defining a polystable parabolic bundle all have the same parabolic slope, equal to that of $E$.

\begin{definition}[Stability for real and quaternionic parabolic bundles]\label{stability_of_real_or_quat_parab_vb}
A real or quaternionic parabolic vector bundle $(E_{\bullet},\phi) \,=((E,\{E^i_j\}, \,\{\alpha^i_j\}),
\phi)$ is called
\textit{stable} (respectively, \textit{semistable}) if for all subbundles $F\subsetneq
E$ of positive rank with $\phi(F)\,=\sigma^*\overline{F}$, the inequality
$$
\frac{\text{par-deg}(F_{\bullet})}{\text{rank}(F)}\,<\,
\frac{\text{par-deg}(E_{\bullet})}{\text{rank}(E)}\ \
\left(\text{respectively, } \ \frac{\text{par-deg}(F_{\bullet})}{\text{rank}(F)} \leq\,
\frac{\text{par-deg}(E_{\bullet})}{\text{rank}(E)}\right)
$$
holds. A real (respectively, quaternionic) parabolic vector bundle $(E_{\bullet},\phi)$ is called
\textit{polystable} if it is semistable and isomorphic to a direct sum of real
(respectively, quaternionic) parabolic stable vector bundles.
\end{definition}

A stable real parabolic vector bundle is not necessarily simple; take for instance the trivial
parabolic structure on a real vector bundle of the form $E\,=\,F\bigoplus\si^*\ov{F}$, with $F$ stable
and $\si^*\ov{F}\,\not\simeq\, F$. However, the automorphism group of such a bundle is either
$\R^*$ or $\C^*$, as follows from adapting the techniques of \cite{Sch_JSG} to the present
parabolic setting.

\begin{definition}[Stability for compact-type real parabolic bundles]
A compact-type real
parabolic vector bundle $(E_{\bullet},\phi) \,=((E,\{E^i_j\}, \,\{\alpha^i_j\}), \phi)$ is called
\textit{stable} (respectively, \textit{semistable}) if for all subbundles $F\subsetneq
E$ of positive rank such that $\phi(F)$ annihilates $\si^*\ov{F}$, the inequality
$$
\frac{\text{par-deg}(F_{\bullet})}{\text{rank}(F)}\,<\,
\frac{\text{par-deg}(E_{\bullet})}{\text{rank}(E)}\ \
\left(\text{respectively, } \ \frac{\text{par-deg}(F_{\bullet})}{\text{rank}(F)} \leq\,
\frac{\text{par-deg}(E_{\bullet})}{\text{rank}(E)}\right)
$$
holds. A compact-type real (respectively, quaternionic) parabolic vector bundle $(E_{\bullet},\tau)$ is called
\textit{polystable} if it is semistable and isomorphic to a direct sum of compact-type real
(respectively, quaternionic) parabolic stable vector bundles.
\end{definition}

We can then state the main result of Section \ref{parab_vb_and_stability}.

\begin{theorem}\label{polystability_over_R_and_over_C}
A real or quaternionic parabolic vector bundle $((E,\{E^i_j\}, \,\{\alpha^i_j\}),
\tau)$ is semistable (respectively, polystable) if and only if the parabolic vector
bundle $(E,\{E^i_j\}, \,\{\alpha^i_j\})$ is semistable (respectively, polystable).

A compact-type real
parabolic vector bundle $((E,\{E^i_j\}, \,\{\alpha^i_j\}),
\tau)$ is semistable (respectively, polystable) if and only if the parabolic vector
bundle $(E,\{E^i_j\}, \,\{\alpha^i_j\})$ is semistable (respectively, polystable).
\end{theorem}

\begin{proof}
Let $(E_\bullet\, ,\phi)$ be a real or quaternionic parabolic vector bundle. If $E_\bullet$ is semistable, then $(E_\bullet\, , \phi)$ is evidently semistable. For the converse, note that if $(E_\bullet\, , \phi)$ is real (respectively, quaternionic), then it follows from the uniqueness of the Harder--Narasimhan filtration of the parabolic vector bundle $E_\bullet$, it follows that this filtration is preserved by $\phi$. Consequently, if $(E_\bullet\, ,\phi)$ is semistable, then $E_\bullet$ is semistable. A similar argument applies in the compact-type case.

If $(E_\bullet\, , \phi)$ is a real or quaternionic parabolic vector bundle such that $E_\bullet$ is polystable, then $$E_\bullet \simeq E^{(1)}_\bullet \oplus \ldots \oplus E^{(k)}_\bullet$$ with $E^{(m)}_\bullet$ stable. The issue is that a given $E^{(m)}_\bullet$ in that decomposition may not be a real (respectively, quaternionic) sub-bundle of $(E_\bullet\, ,\phi)$. But, if it is not, then $\si^*\ov{E^{(m)}}_\bullet$ also appears in the decomposition of $E_\bullet$ as a direct sum of stable parabolic bundles, and it is easy to check, adapting the techniques of \cite{Sch_JSG}, that in this case $E^{(m)}\oplus \si^*\ov{E^{(m)}}$ is a stable real parabolic bundle. Conversely, if $(E_\bullet\, , \phi)$ is polystable, then
we know that $E_\bullet$ is semistable because
$(E_\bullet\, , \phi)$ is semistable, therefore that
$E_\bullet$ has a unique maximal polystable parabolic
subbundle $F$ with parabolic slope the same as that of $E_\bullet$
\cite[page 23, Lemma 1.5.5]{HL}. Now, from the uniqueness
of such a maximal polystable parabolic
subbundle, it follows that $F$ is invariant under $\phi$ (here, $\phi$-invariant means that $\phi(F)\subset\si^*\ov{F}$). As $(E_\bullet\, ,\phi)$ is polystable, the subbundle $F$ has a
$\phi$-invariant complement $F'$. But if $F'$ is non-zero, then the maximality of
$F$ is contradicted, because $F'$ also has a unique maximal polystable parabolic
subbundle $F$ with parabolic slope same as that of $E_\bullet$.
Therefore, we conclude that $F'\,=0$, which implies that
$E_\bullet$ is polystable. Again, a similar argument applies in the compact-type case.
\end{proof}

\begin{corollary}\label{geom_stable_bdles}
A stable real (respectively, quaternionic) parabolic vector bundle on $(X,\si,S)$ is, in general, only polystable as a parabolic vector bundle. If it is in fact stable as a parabolic vector bundle, it will be called geometrically stable as a real (respectively, quaternionic) parabolic vector bundle.

A stable compact-type real
parabolic vector bundle on $(X,\si,S)$ is, in general, only polystable as a parabolic vector bundle. If it is in fact stable as a parabolic vector bundle, it will be called geometrically stable as a compact-type real
parabolic vector bundle.
\end{corollary}

\section{Parabolic vector bundles arising from unitary representations}

\subsection{From unitary representations to polystable parabolic vector bundles}

Let $(X,\si)$ be a compact Klein surface and let $S=\{x_1,\,\ldots\, x_d\}$ be a finite subset of $X$ satisfying $\si(S)=S$. Set $Y:=X\setminus S$ and let $\Ga(Y,\,\si)$ be the orbifold fundamental group of the quotient surface $Y/\si$. Recall that $\Ga(Y,\,\si)$ fits in the fundamental short exact sequence \eqref{fund_ses}. Consider the action of $\Z/2\Z$ on the unitary group $\U(r)$ by complex conjugation and recall that $H^2(\Z/2\Z;\cZ(\U(r)))=\{\pm1\}$, where $\cZ(\U(r))\simeq S^1$ is the center of $\U(r)$. We denote by $\W(r)_\pm$ the corresponding extensions $$1\lra\U(r) \lra \W(r)\pm\lra \Z/2\Z\lra 1.$$ In particular, $\W(r)_+\simeq \U(r)\rtimes\Z/2\Z$.

\begin{definition}[Real and quaternionic unitary representations]\label{rep_of_fund_ses}
A \textit{real unitary representation} of $\Ga(Y,\,\si)$ is a group morphism $\rho\,:\,\Ga(Y,\,\si)
\, \lra \,\W(r)_+$ such that the following diagram commutes.
$$
\xymatrix{
1 \ar[r] & \pi_1(Y) \ar[r] \ar[d] & \Ga(Y,\,\si) \ar[r] \ar[d]^{\rho} & \Z/2\Z \ar[r] \ar@{=}[d] & 1\\
1 \ar[r] & \U(r) \ar[r] & \W(r)_+\ar[r] & \Z/2\Z \ar[r] & 1
}$$
A \textit{quaternionic unitary representation} of $\Ga(Y,\,\si)$ is a group morphism $\rho:\Ga(Y,\,\si) \lra \W(r)_-$ such that the following diagram commutes.
$$
\xymatrix{
1 \ar[r] & \pi_1(Y) \ar[r] \ar[d] & \Ga(Y,\,\si) \ar[r] \ar[d]^{\rho} & \Z/2\Z \ar[r] \ar@{=}[d] & 1\\
1 \ar[r] & \U(r) \ar[r] & \W(r)_-\ar[r] & \Z/2\Z \ar[r] & 1
}$$
Two real (respectively, quaternionic) unitary representations $\rho$ and $\rho'$ are called \textit{equivalent} if there is an element $u\in\U(r)$ such that, for all $\ga\,\in\,\Ga(Y,\,\si)$, we have $u\rho(\ga)u^{-1} = \rho'(\ga)$.
\end{definition}

In particular, if $\rho\,:\,\Ga(Y,\,\si)\,\lra\, \W(r)_\pm$ is either a real or a quaternionic representation, $\rho|_{\pi_1(Y)}:\pi_1(Y)\lra \U(r)$ defines a parabolic vector bundle $\cE(\rho)_{\bullet}$ over $X$ (\cite[Section~1]{MS}). The parabolic weights at $x_j$ and their multiplicities are
given by the eigenvalues of the image under $\rho$ of an element of $\pi_1(Y)$ produced by an oriented loop around $x_j$ (this element of $\pi_1(Y)$
is not unique, but its conjugacy class is, so the eigenvalues and their multiplicities are also uniquely determined). By \cite[Proposition~1.12]{MS}, the parabolic vector bundle
$\cE(\rho)_{\bullet}$ is polystable of parabolic degree zero, and it is stable if and only if the representation $\rho|_{\pi_1(Y)}$ is irreducible. Conversely, any polystable parabolic vector bundle of rank $r$ and parabolic degree zero is given by a homomorphism from $\pi_1(Y)$ to $\U(r)$ (\cite{MS}).

\begin{proposition}\label{construction_of_real_parab_vb}
If $\rho\,:\,\Ga(Y,\,\si)\,\lra\, \W(r)_+$ is a real unitary representation of $\Ga(Y,\,\si)$, then the parabolic vector bundle $\cE(\rho)_\bullet$, of parabolic degree $0$, has a canonical real structure, with respect to which it is polystable.

If $\rho\,:\,\Ga(Y,\,\si)\,\lra\, \W(r)_-$ is a quaternionic unitary representation of $\Ga(Y,\,\si)$, then the parabolic vector bundle $\cE(\rho)_\bullet$, of parabolic degree $0$, has a canonical quaternionic structure, with respect to which it is polystable.
\end{proposition}

This is proved by going deeper into the details of the construction of the parabolic vector bundle $\cE(\rho)_\bullet$ over $X$. We do it below in the real case, the quaternionic one being similar (see \cite[Section 2]{Sch_JDG}). Note that, in \cite{MS}, the construction of $\cE(\rho)_\bullet$ is presented from a slightly different point of view, closer to that of \cite{Weil_matrix_div}. One starts with a Fuchsian group of finite co-volume $\Ga\subset \PSL(2,\R)$ and consider the compact Riemann surface $X:=(\mathcal{H}\cup\{\mathrm{cusps\ of}\ \Ga\})/\Ga$, where $\mathcal{H}$ is the real hyperbolic plane, acted upon holomorphically by the discrete group $\Ga$. The finite set $S\subset X$ is the projection to $X$ of the fixed points and cusps of $\Ga$ in $\widehat{\mathcal{H}}:=\mathcal{H}\cup\partial\mathcal{H}$. In particular, each point $x\in S$ has an order $1\leq d_x \leq \infty$, which by definition is the order of the cyclic group $\mathrm{Stab}_\Ga(h)$, for all $h$ lying above $x$ in $\widehat{\mathcal{H}}$. So $d_x=\infty$ if and only if $x$ comes from a cusp of $\Ga$ (in $\partial\mathcal{H}$). The relation with the previous point of view is given by the fact that a generator of $\mathrm{Stab}_\Ga(h)$ comes from the homotopy class of a small loop around $x$ in $X\setminus S$. Note that, on the one hand, if $\Ga$ is cocompact, then $X=\mathcal{H}/\Ga$ can be viewed as an orbifold Riemann surface, with $S$ equal to the set of orbifold singularities of $X$. Parabolic vector bundles over $(X,S)$ then correspond to orbifold vector bundles over $X$ (\cite{Boden,Biswas_Duke}). On the other hand, if $\Ga$ is torsion-free of finite co-volume, so has only cusps as fixed points in $\widehat{\mathcal{H}}$, then $\Ga\simeq \pi_1(X\setminus S)$.

\begin{proof}[Proof of Proposition \ref{construction_of_real_parab_vb}]
In view of \cite[Proposition~1.12]{MS} and Theorem \ref{polystability_over_R_and_over_C}, it suffices to proves that the parabolic vector bundle $\cE(\rho)_\bullet\lra X$ has a real (respectively, quaternionic) structure which is compatible with its parabolic structure; it will then automatically be polystable in the real (respectively, quaternionic) parabolic sense. The real (respectively, quaternionic) structure in question comes from the $\Ga(Y,\,\si)$-equivariant structure of the vector bundle $\mathcal{H}\times\C^r$ over $\mathcal{H}$, associated to the representation $\rho\,:\,\Ga(Y,\,\si)\,\lra\,\W(r)_\pm$, which is defined by $\ga\cdot(h,v) := (\ga\cdot h,\rho(\ga)v)$. If $\ga\,\in\,\Ga(Y,\,\si)\setminus\pi_1(Y)$, then $\ga^2$ is, so it descends to an anti-holomorphic transformation $\cE(\rho)=(\mathcal{H}\times\C^r)/\Ga$, covering the real structure of the base. It is then easy to check that this induced transformation squares to $\Id_{\cE(\rho)}$ (respectively, $-\Id_{\cE(\rho)}$) if $\rho$ is a real (respectively, quaternionic) representation, and that it does not depend on the choice of the element $\ga\,\in\,
\Ga(Y,\,\si)\setminus\pi_1(Y)$; see \cite[Section 2]{Sch_JDG} for details. If we now add the fixed points of $\Ga(Y,\,\si)$, then the parabolic structure of $(\widehat{\mathcal{H}}\times\C^r)/\pi_1(Y)$ over $S$ comes, as we have recalled, from the $\mathrm{Stab}_{\pi_1(Y)}(h)$-equivariant structure of $\widehat{\mathcal{H}}\times\C^r$ around $y\in\widehat{\mathcal{H}}$ (\cite[Section 1]{MS}). It therefore suffices to show that this equivariant structure extends to an equivariant structure for the larger stabilizer subgroup $\mathrm{Stab}_{\Ga(Y,\si)}(h)$, which is indeed the case since $\rho$ is, by assumption, a unitary representation of the full $\Ga(Y,\,\si)$.
\end{proof}

\begin{remark}
When $S$ is contained in the fixed point set $X^\si$ (compare \cite[p.~212]{MS}), we can be more explicit in the proof of Proposition \ref{construction_of_real_parab_vb} for real parabolic vector bundles. Indeed, in this case, the parabolic structure on $E_x$ (for all $x\in S$) comes from a representation of the stabilizer group $\mathrm{Stab}_{\pi_1(Y)}(h)\lra\U(r)$ for some chosen $h$ above $x$ in $\widehat{\mathcal{H}}$. Such a representation sends a generator $\ga_y$ of the cyclic group $\mathrm{Stab}_{\pi_1(Y)}(h)$ to the unitary matrix
$$\begin{pmatrix}
\exp^{2\pi\sqrt{-1} \beta_1} & & 0\\
& \ddots & \\
0 & & \exp^{2\pi\sqrt{-1}\beta_n}
\end{pmatrix}$$ where $(\beta_1,\, \ldots\, ,\beta_n)$ are the (non-necessarily distinct) weights of $\cE(\rho)_\bullet$ at $x$. This representation is $\Z/2\Z$-equivariant with respect to the involution $z\lmt z^{-1}$ on $\mathrm{Stab}_{\pi_1(Y)}(h)$ and complex conjugation on $\U(r)$, so it extends to a group morphism from $\mathrm{Stab}_{\pi_1(Y)}(h)\rtimes \Z/2\Z \simeq \mathrm{Stab}_{\Ga(Y,\si)}(h)$ to $\U(r)\rtimes\Z/2\Z\simeq \W(r)_+$, as claimed.
\end{remark}

We now perform a similar analysis in the compact-type case. This time, we consider the trivial action of $\Z/2\Z$ on the unitary group $\U(r)$. Then we have $H^2(\Z/2\Z;\cZ(\U(r)))=1$, because the Abelian group $\cZ(\U(r))\simeq S^1$ is $2$-divisible (note that this would not happen for $\mathbf{SU}(2m)$ and that this is related to what we were saying in Remark \ref{on_compact_type_quaternionic}, regarding the existence of compact-type quaternionic vector bundles with structure group contained in $\mathbf{SL}(2m;\C)$; see also \cite{Ho}). Therefore, any extension of $\Z/2\Z$ by $\U(r)$ in this case is isomorphic to the direct product $\U(r)\times\Z/2\Z$. In particular, if we look at the analog of Definition \ref{rep_of_fund_ses} in this context, we find that unitary representations of $\Ga(Y,\,\si)$ inducing the identity on $\Z/2\Z$ correspond bijectively to unitary representations of $\Ga(Y,\,\si)$ in the usual sense, i.e.\ group morphisms $\rho\,:\,\Ga(Y,\,\si)\,\lra\,\U(r)$. We then have the following analog of Proposition \ref{construction_of_real_parab_vb}.

\begin{proposition}\label{construction_of_compact_type_real_parab_vb}
Let $\rho\,:\Ga(Y,\,\si)\,\lra\, \U(r)$ be a unitary representation of $\Ga(Y,\,\si)$. Then the parabolic vector bundle $\cE(\rho)_\bullet$ has a canonical compact-type real structure, with respect to which it is polystable.
\end{proposition}

\begin{proof}
It suffices to prove that there is an isomorphism $\phi:\cE(\rho)_\bullet\lra (\si^*\cE(\rho))_\bullet^\vee$ such that $(\si^*\ov{\phi})^\vee\circ\phi = \Id_{\cE(\rho)}$. This follows from the fact that $\mathcal{H}\times \C^r$ descends to a Hermitian vector bundle $\mathcal{V}$ on the open dianalytic surface $Y/\si=\mathcal{H}/\Ga(Y,\,\si)$, which in turn pulls back to an anti-invariant vector bundle $\mathcal{W}$ on the double cover $Y$ of $Z$ (where by anti-invariant we mean that $\si^*\ov{\mathcal{W}}^\vee\simeq \mathcal{W}$ on $Y$). As $\mathcal{V}^\vee\simeq \ov{\mathcal{V}}$ on $Y/\si$, the isomorphism $\si^*\ov{\mathcal{W}}^\vee\simeq \mathcal{W}$ over $Y$ is actually a compact-type real structure, by construction. Finally, $\mathcal{W}$ is just $\cE(\rho)$, since both come from the $\pi_1(Y)$-equivariant vector bundle $\mathcal{H}\times\C^r$, induced on by $\rho:\Ga(Y,\,\si)\,\lra\, \U(r)$. The compatibility with the parabolic structure is checked in the same way as in Proposition \ref{construction_of_real_parab_vb}.
\end{proof}

\subsection{From polystable parabolic vector bundles to unitary representations}

In this section, we prove the main result of this paper (Theorem \ref{MS_corresp_for_real_and_quat}) and its analog for compact-type real bundles (Theorem \ref{MS_corresp_for_compact_type_real}). The strategy is to construct an inverse to the maps $\rho\lmt\cE(\rho)_\bullet$ constructed in Propositions \ref{construction_of_real_parab_vb} and \ref{construction_of_compact_type_real_parab_vb}, thus proving that every polystable real (respectively, quaternionic) parabolic vector
bundle of parabolic degree zero comes from a real (respectively, quaternionic) representation $\rho
\,:\,\Ga(Y,\,\si)\,\lra\,\W(r)_\pm$, and that every polystable compact-type real parabolic vector bundle of parabolic degree zero comes from a unitary representation in the usual sense $\rho\,:
\,\Ga(Y,\,\si)\,\lra\,\U(r)$. Given a polystable real (respectively, quaternionic, respectively, compact-type real) vector bundle $(E_\bullet\, , \phi)$ of parabolic degree zero, such a map is obtained via the holonomy representation of a certain singular unitary connection on $E_\bullet$, namely the Chern connection $A_h$ of an adapted Hermitian-Yang-Mills metric $h$ on $E_\bullet$ (see \cite[Section 2.1]{Biq}). Note that, if we do not impose any compatibility condition on $A_h$ and $\phi$, then we are exactly in the situation of the Mehta-Seshadri theorem (since $E_\bullet$ is polystable in view of Theorem \ref{polystability_over_R_and_over_C}), so this already gives us a holonomy representation $\rho':\pi_1(Y)\lra \U(r)$ such that $\cE(\rho')_\bullet\simeq E_\bullet$ and all that is left to do is incorporate the real (respectively, quaternionic, respectively, compact-type real) structure $\phi$ and show that, when $A_h$ is compatible with $\phi$ in a sense to be made precise, then $\rho'$ extends to a real or quaternionic representation $\rho\,:\,\Ga(Y,\,\si) \,\lra\, \W(r)_\pm$ (respectively, a unitary representation
$\rho\,:\,\Ga(Y,\,\si) \,\lra\, \U(r)$ in the compact-type case), which is what we deal with next, as it has independent interest (Lemma \ref{ext_of_hol_rep}). To simplify the exposition, we focus in what follows on the real case. Unless explicitly stated otherwise, the quaternionic and compact-type real cases are dealt with similarly.

\begin{lemma}\label{ext_of_hol_rep}
Let $(E_\bullet\, , \phi)$ be a real parabolic vector bundle of parabolic degree zero on the compact Klein surface $(X,\si,S)$. Let $h$ be an adapted Hermitian-Yang-Mills metric on $E_\bullet$ and let $A_h$ be the associated Chern connection. If $A_h$ is compatible with $\phi$ in the sense that the pullback of the singular connection $\si^*A_h$ on $\si^*\ov{E}_\bullet$ under the isomorphism $\phi:E_\bullet \lra\si^*\ov{E}_\bullet$ coincides with $A_h$, then the holonomy representation $\rho':\pi_1(X\setminus S)\lra \U(r)$ extends to a group morphism $\rho:\pi_1^{\mathrm{orb}}((X\setminus S)/\si)\lra \U(r)\rtimes\Z/2\Z$.
\end{lemma}

\begin{proof}
Since $E_\bullet$ has parabolic degree $0$ and $h$ is an adapted Hermitian-Yang-Mills metric on $E_\bullet$, the restriction of the Chern connection $A_h$ to $E|_{X\setminus S}$ is flat (this follows for instance from the Gauss-Bonnet formula \cite[Proposition 2.9]{Biq} and the fact that the curvature of the Chern connection of an adapted Hermitian-Yang-Mills metric is constant over $X\setminus S$, by definition of Yang-Mills equations). So we get a holonomy representation $\rho':\pi_1(X\setminus S)\lra\U(r)$. As the connection $A_h$ is compatible with the real structure $\phi$, we can apply the general properties of parallel transport for invariant connections (\cite[Section 4.1]{Sch_JDG}) and show that $\rho'$ extends to a group morphism $\rho:\pi_1^{\mathrm{orb}}((X\setminus S)/\si)\lra \U(r)\rtimes\Z/2\Z$.
\end{proof}

The final step is then to ensure the existence of an invariant adapted Hermitian-Yang-Mills metric on $(E_\bullet\, , \phi)$. Indeed, after that, it is a simple matter to check that the Chern connection of an invariant adapted Hermitian is itself invariant (this follows from the uniqueness of the Chern connection), which puts us in the situation of Lemma \ref{ext_of_hol_rep}. By invariant metric on $(E_\bullet\, , \phi)$, we mean here an adapted Hermitian metric $h$ on $E_\bullet$ such that the isomorphism $\phi:E_\bullet \lra \si^*\ov{E}_\bullet$ becomes an isometry when $\si^*\ov{E}_\bullet$ is equipped with the metric $\si^*\ov{h}$ induced by $h$. Equivalently, we may use $\phi$ to pull back the metric $\si^*\ov{h}$ to $E_\bullet$~: this sets up a $\Z/2\Z$-action on the space of metrics, and an invariant metric is then just a fixed point of that action. Note that this action is induced by the (anti-holomorphic) involution $\si:Y\lra Y$ and the real structure $\phi:E_\bullet\lra \si^*\ov{E}_\bullet$, which may also be seen as a lift $\tau:E_\bullet \lra E_\bullet$ of the real structure $\si:Y\lra Y$. The existence, under the assumption that $(E_\bullet\, , \phi)$ is polystable, of an invariant adapted Hermitian-Yang-Mills metric on $E_\bullet$ then follows from the work of C. Simpson (\cite[Theorem 1 and Corollary 3.6]{Simpson_JAMS_1988} and \cite[Theorem 6]{Simpson_JAMS_1990}). Strictly speaking, in \cite{Simpson_JAMS_1988}, Simpson considers equivariant bundles over $Y$ when the latter is equipped with a finite group of holomorphic transformations, but since anti-holomorphic transformations of $Y$ also act on the set of Hermitian metrics on a complex vector bundle $E$ over $Y$, his proof extends directly to our setting. Other approaches to the existence of adapted Hermitian-Yang-Mills connections on polystable parabolic vector bundles can be found in \cite{Biq,Poritz,Nasatyr_Steer_orbi_approach}, that could presumably also be used to construct invariant Hermitian-Yang-Mills connections on polystable real parabolic vector bundles. Note that one advantage of Simpson's approach is that it does not require using the polystability of the underlying parabolic vector bundle (Theorem \ref{polystability_over_R_and_over_C}), only its polystability in the real sense (also, Simpson considers parabolic Higgs bundles, which are more general than the parabolic vector bundles we study here). We sum up our discussion in the following statement.

\begin{theorem}\label{existence_of_HYM_metrics}
Let $(E_\bullet\, , \phi)$ be a real parabolic vector bundle over the open Klein surface $(Y,\si)$. Then $(E_\bullet\, , \phi)$ is polystable if and only if it admits an invariant, adapted Hermitian-Yang-Mills metric. In particular, if $E_\bullet$ has vanishing parabolic degree, then the Chern connection of such a metric $h$ is flat and invariant. So, by Lemma \ref{ext_of_hol_rep}, its holonomy representation $\rho_h:\pi_1(Y)\lra \U(r)$ extends to a real unitary representation $\widetilde{\rho_h}:\pi_1^{\mathrm{orb}}(Y/\si)\lra \U(r)\rtimes\Z/2\Z$.
\end{theorem}

We have thus proved Theorem \ref{MS_corresp_for_real_and_quat}, as well as the following result for compact-type real parabolic vector bundles.

\begin{theorem}\label{MS_corresp_for_compact_type_real}
Let $(X,\si)$ be a compact Klein surface and let $S\subset X$ be a finite subset such that $\si(S)=S$. Let $Y$ be the punctured surface $X\setminus S$, assumed to be of negative Euler characteristic, and let $\Ga(Y,\,\si)$ be the orbifold fundamental group of the quotient surface $Y/\si$. Then there is a bijective correspondence between conjugacy classes of unitary representations of $\Ga(Y,\,\si)$ and isomorphism classes of polystable compact-type parabolic vector bundles of parabolic degree zero on $(X,\si)$ with parabolic structure at the points of $S$.
\end{theorem}

Finally, the following result follows directly from Theorem \ref{existence_of_HYM_metrics} and the notion of geometrically stable real or quaternionic (respectively, compact-type real) parabolic vector bundle (see Corollary \ref{geom_stable_bdles}).

\begin{theorem}
Geometrically stable real and quaternionic parabolic vector bundles on $(X,\si,S)$ correspond bijectively to real and quaternionic unitary representations $\rho:\pi_1^{\mathrm{orb}}((X\setminus S)/\si) \lra \W(r)_\pm$ such that $\rho|_{\pi_1(X\setminus S)}: \pi_1(X\setminus S) \lra \U(r)$ is irreducible.

Geometrically stable compact-type real parabolic vector bundles on $(X,\si,S)$ correspond bijectively to unitary representations $\rho:\pi_1^{\mathrm{orb}}((X\setminus S)/\si) \lra \U(r)$ such that $\rho|_{\pi_1(X\setminus S)}: \pi_1(X\setminus S) \lra \U(r)$ is irreducible.
\end{theorem}

\section*{Acknowledgments}

The authors wish to thank the referee for helpful comments.
I.~Biswas is supported by a J. C. Bose Fellowship. F.~Schaffhauser was supported
by \textit{Convocatoria 2018-2019 de la Facultad de Ciencias
(Uniandes), Programa de investigaci\'on “Geometr\'ia y Topolog\'ia de los Espacios de
M\'odulos”}, the \textit{European Union’s Horizon 2020 research and innovation programme under
grant agreement No 795222} and the \textit{University of Strasbourg Institute of Advanced Study
(USIAS)}, and U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS:
Geometric structures And Representation varieties” (the GEAR Network).


text:
\section{Generalised Strata}
\label{sec:GenStrata}

We begin by briefly recalling the notions from \cite{LMS} and \cite{strataEC} in the generality that we here require.

\subsection{Strata with residue conditions}
\label{subsec:rescond}
To obtain a recursive structure on the boundary of $\LMS$, recall the definition of \emph{generalised stratum}, introduced in \cite[\S 4]{strataEC} to cover the case of a level of an enhanced level graph.
More precisely, we allow differentials on disconnected surfaces:
denote by $\mu_i=(m_{i,1},\dots,m_{i,n_i})\in\bZ^{n_i}$ the type of a differential,
i.e., we require that $\sum_{j=1}^{n_i} m_{i,j}=2g_i-2$ for some $g_i \in \bZ$
and $i=1,\ldots,k$.
Then we define, for a tuple $\bfg = (g_1,\ldots,g_k)$
of genera and a tuple $\bfn = (n_1,\ldots,n_k)$ together with
$\bfmu = (\mu_1,\ldots, \mu_k)$, the disconnected stratum
\[\omoduli[\bfg,\bfn](\bfmu) \= \prod_{i=1}^k \omoduli[g_i,n_i](\mu_i)\,.\]
Note that the projectivized stratum $\bP\omoduli[\bfg,\bfn](\bfmu)$ is the quotient
by the diagonal action of $\bC^*$, not the quotient by the action of $(\bC^*)^k$.

Moreover, we consider subspaces of these \emph{cut out by residue conditions}.
More precisely, denote by $H_p \subseteq \cup_{i=1}^k \{(i, 1),\cdots (i, n_i)\}$ the subset of the marked points such that $m_{i,j} < -1$.
Now consider vector spaces
$\frakR$ of the following special shape, modelled on the global residue condition
from \cite{BCGGM}:
for $\lambda$ a partition of $H_p$, with parts denoted
by $\lambda^{(k)}$, and a subset $\lambda_{\frakR}$ of the parts of $\lambda$, we define the $\bC$-vector space
\[\frakR\coloneqq \Bigl\{ r= (r_{i,j})_{(i,j) \in H_p} \in \bC^{H_p}
\quad \text{and} \sum_{(i,j) \in \lambda^{(k)}} r_{i,j}=0 \quad
\text{for all} \quad \lambda^{(k)} \in \lambda_{\frakR} \Bigr\}\,.\]
We denote the subspace of surfaces with residues in~$\frakR$ by
$\Romod[\bfmu][\bfg,\bfn][\frakR]$.

In \cite[Prop. 4.2]{strataEC} a modular compactification $\proj\LMS[\bfmu][\bfg,\bfn][\frakR]$ of $\proj \Romod[\bfmu][\bfg,\bfn][\frakR]$ is constructed in analogy to \cite{LMS}.
Consequently, the boundary components are parametrised by \emph{enhanced level graphs}.
More precisely, a \emph{level graph} is defined to be a stable graph together with a level function.
Recall that a \emph{stable graph} is a tuple $\Gamma = (V_i, H_i, E_i, g_i, v_i, \iota_i)_{i=1,\dotsc,k}$ consisting of \emph{vertices} $V_i$, a \emph{genus map} $g_i\colon V_i\to\bZ_{\geq 0}$,
\emph{legs} $H_i$ that are associated to the vertices by a \emph{vertex map} $v_i\colon H_i\to V_i$ and come with an involution $\iota_i\colon H_i\to H_i$, the two-cycles of which form the \emph{edges} $E_i\subseteq H_i\times H_i$ while the fixed points (denoted $H^m_i$) are in bijection with the $n_i$ marked points.
The $k$ graphs $\Gamma_i=(V_i, H_i, E_i, g_i, v_i,\iota_i)$ are required to be connected and satisfy the usual stability conditions.
Moreover, we set $g = \sqcup g_i$, $v = \sqcup v_i$, $E = \sqcup E_i$, $ H=\sqcup H_i$, and $V=\sqcup V_i$.
Note that this data induces a unique bijection $o\colon\bigsqcup H^m_i\to\bfmu$ associating to each marked point $(i,j)$ the order $m_{i,j}$ of the differential.

A \emph{level function} on the vertices is a map
$\ell\colon V\to\bZ$, which we normalise to take values in $\{0,-1,\dotsc,-L\}$ and require that, for all edges $e\in E$,
$\ell(v(e^+)) \geq \ell(v(e^-))$,
where we write $e\eqqcolon(e^+,e^-)\in H\times H$.

Moreover, an \emph{enhancement} is a map $\kappa\colon E\to\bZ_{\geq 0}$ such that $\kappa(e)=0$ if and only if $e$ is horizontal (i.e. $\ell(v(e^+))=\ell(v(e^-))$), subject to the following stability condition:
define the \emph{degree} of a vertex~$v$ in $V$ to be
\[\deg(v) = \sum_{h \in H^m, v(h)=v} o(h) + \sum_{e \in E, v(e^+)=v}  (\kappa(e)-1)
- \sum_{e \in E, v(e^-)=v} (\kappa(e)+1) \,.\]
Then the enhancement is admissible, if
$\deg(v) = 2g(v)-2$ holds for every vertex~$v$ of~$\Gamma$.

The \emph{enhanced level graph} then consists of the triple $(\Gamma, \ell, \kappa)$.
We denote the corresponding boundary component of $\proj\LMS[\bfmu][\bfg,\bfn][\frakR]$
by $D_\Gamma$.

\begin{rem}
\label{poleorders}
Note that $\kappa(e)$ corresponds to the number of \emph{prongs} at $e$ and, for a non-horizontal edge, the associated differential has a zero of order $\kappa(e)-1$ on the top component and a pole of order $-\kappa(e)-1$ on the bottom component (for horizontal edges there is a simple pole on each component). This gives an extension of $o$ to $H$.

Using this identification, the stability condition of \cite[\S 2]{LMS} is simply the requirement that the orders of zeros and poles sum to $2g_{i,j}-2$ on each vertex.
See \cite[\S 2]{LMS} and \cite[\S 3.2]{strataEC} for details.
\end{rem}

To determine, for a generalised stratum $\proj\LMS[\bfmu][\bfg,\bfn][\frakR]$, which enhanced level graphs give non-empty boundary components, we recall the $\frakR$-GRC from \cite[\S 4]{strataEC}:
starting with an enhanced level graph $\Gamma$,
we construct a new \emph{auxiliary level graph} $\widetilde{\Gamma}$
by adding,
for each $\lambda^{(k)}\in \lambda_\frakR$,
a new vertex $v_{\lambda^{(k)}}$ to $\Gamma$ at level $\infty$  and converting a tuple $(i,j)\in \lambda^{(k)}$
into an edge from the marked point $(i,j)$ to  the vertex $v_{\lambda^{(k)}}$.
We then say that $\Gamma$ satisfies the
\emph{$\frakR$-global residue condition ($\frakR$-GRC)} if
the tuple of residues at the legs in $H_p$ belongs to~$\frakR$ and for every
level $L<\infty$ of $\widetilde{\Gamma}$ and every connected component~$Y$
of the subgraph~$\widetilde{\Gamma}_{>L}$ one of
the following conditions holds.
\begin{enumerate}
\item The component~$Y$ contains a marked point with a prescribed
pole that is \emph{not} in $\lambda_\frakR$.
\item The component~$Y$ contains a marked point with a prescribed
pole $(i,j) \in H_p$ and
there is an $r \in \frakR$ with $r_{(i,j)} \neq 0$.
\item Let $e_1,\ldots,e_b$ denote the set of edges where~$Y$
intersects $\widetilde{\Gamma}_{=L}$. Then
\[ \sum_{j=1}^b\Res_{e_j^-}\eta_{v(e^-_j)}\=0\,,\]
where  $v(e^-_j)\in\widetilde{\Gamma}_{=L}$.
\end{enumerate}
By \cite[Prop. 4.2]{strataEC}, the boundary components $D_\Gamma$ of $\proj\LMS[\bfmu][\bfg,\bfn][\frakR]$ are parametri\-sed by enhanced level graphs $(\Gamma, \ell, \kappa)$ satisfying the $\frakR$-GRC.

For applications such as listing all graphs in the boundary of a stratum, it
is convenient to have a purely graph-theoretic criterion in analogy to the one shown for the classical GRC in \cite{MUW}.
The following
proposition strips the tropical language away in the criterion
\cite[Theorem~1]{MUW} and generalises it to meromorphic strata
with residue conditions.

Let~$(\Gamma,\ell,\kappa)$ be an enhanced level graph.

We call a vertex~$v$ of~$\Gamma$ \emph{inconvenient} if $g(v) = 0$,
if it is not adjacent to any edge~$e$ with enhancement $\kappa(e) = 0$
and if it is adjacent to a leg with a \emph{very high enhancement} in
the following precise sense: denote by $\frakp(v)$ the set of half-edges on the vertex $v$ that are poles (in the sense of \autoref{poleorders}).
Then the condition is that there is a $p\in\frakp(v)$ such that
\[o(p) > \sum_{p'\in\frakp(v)} (o(p') - 1) - 1.\]
\begin{prop} \label{prop:realizewithR}
The boundary stratum $D_\Gamma$ of $\Romod[\bfmu][\bfg,\bfn][\frakR]$
associated with the enhanced level graph~$(\Gamma,\ell,\kappa)$ is non-empty if and only if
both of the following conditions are satisfied
\begin{enumerate}

\item For every inconvenient vertex~$v$ of~$\Gamma$ there is
\begin{enumerate}
\item a simple cycle based at~$v$ that does not pass through
any vertex of level smaller than~$v$, or
\item the graph of levels~$\geq \ell(v)$ deprived of the vertex~$v$ has two
components, each of which has a marked pole in $H_p$ whose residue is not constrained to zero for all elements of~$\frakR$.
\end{enumerate}
\item For every horizontal edge~$e$ of~$\Gamma$ there is
\begin{enumerate}
\item a simple cycle based through~$e$ that does not pass through
any vertex of level smaller than~$\ell(e)$, or
\item the graph of levels~$\geq \ell(e)$ deprived of the edge~$e$ has two
components, each of which has a marked pole in $H_p$ whose residue is not constrained
to zero for all elements of~$\frakR$.
\end{enumerate}
\end{enumerate}
\end{prop}
\par
\begin{proof}

Consider first the case that $\widetilde{\Gamma}$ is connected.
We view $\widetilde{\Gamma}$ as an enhanced level graph of an auxiliary stratum $\widetilde{X}$ as follows: we add prongs to the new edges of $\widetilde{\Gamma}$ in accordance with the pole orders of the half-edges on $\Gamma$.
For each vertex $v$ at level $\infty$, we then extend the genus function by $g_v$ setting $2g_v-2$ as the sum of orders of the half-edges on $v$ induced by the newly added prongs, possibly adding an extra simple zero to fix parity issues. Note that all new vertices are of positive genus, so stability is not an issue.
Therefore, for the stratum $\widetilde{X}$, we are now reduced to the situation of \cite[Theorem~1]{MUW}.

In a product of strata, clearly a graph is admissible if and only if each component is admissible.
\end{proof}
See \autoref{subsec:R-GRC} for examples illustrating this criterion.

This criterion allows us to explicitly construct all graphs in a given stratum.
In fact, we can construct all graphs with no horizontal edges recursively from the two-level graphs.

\subsection{Constructing Level Graphs}
Recall the undegeneration maps $\delta_i$ \cite[\S 3.3]{strataEC}, contracting
all level crossing of an enhanced level graph $\Gamma$ without horizontal edges
except for the $i$-th level crossing, yielding a two-level graph.
The component of $\Gamma$ is contained in the product of the components of $\delta_i(\Gamma)$, which may be irreducible.

\begin{definition}\label{mathprofile}
Let $(\Gamma,\ell,\kappa)$ be an enhanced level graph with $L$ levels and without horizontal edges. We define the \emph{profile} of $\Gamma$ to be the tuple $(\delta_1(\Gamma),\dotsc,\delta_L(\Gamma))$.
An \emph{enhanced profile} is a profile together with a choice of irreducible component.
\end{definition}
Note that the association of a profile to a graph is not injective, see \autoref{reducibleprofile}.
It is primarily useful to encode efficiently how graphs degenerate.
By definition, the information of an enhanced profile of $\Gamma$ is equivalent to the data of $\Gamma$.

Generating all non-horizontal graphs inside a stratum is thus equivalent to listing all non-empty enhanced profiles.
We do this recursively. For $X$ a generalised stratum, denote by $\BIC(X)$ the (non-horizontal) two-level graphs in the boundary of $X$.
\begin{definition}
Let $X$ be a generalised stratum. For each $\Gamma\in\BIC(X)$ we define:
\begin{enumerate}
\item the generalised strata $\Gamma^\top$ and $\Gamma^\bot$, the top and bottom levels of $\Gamma$;
\item a map $\beta_\Gamma^\top\colon\BIC(\Gamma^\top)\to\BIC(X)$ that associates to a graph $\Gamma'\in\BIC(\Gamma^\top)$ the graph $\delta_0(\Lambda)\in\BIC(X)$ where $\Lambda$ is the graph obtained by clutching $\Gamma'$ to $\Gamma^\bot$;
\item a map $\beta_\Gamma^\bot\colon\BIC(\Gamma^\bot)\to\BIC(X)$ that associates to a graph $\Gamma'\in\BIC(\Gamma^\bot)$ the graph $\delta_1(\Lambda)\in\BIC(X)$ where $\Lambda$ is the graph obtained by clutching $\Gamma^\top$ to $\Gamma'$.
\end{enumerate}
\end{definition}

The following proposition is an immediate consequence of \cite[Prop. 5.1]{strataEC}.

\begin{prop}\label{prop:beta}
Let $X$ be a generalised stratum and $\Gamma\in\BIC(X)$.
\begin{enumerate}
\item The images of $\beta^\top_\Gamma$ and $\beta^\bot_\Gamma$ are disjoint.
\item The profile $(\Gamma',\Gamma)$ is non-empty if and only if $\Gamma'$ is in the image of $\beta^\top_\Gamma$.
\item The profile $(\Gamma,\Gamma')$ is non-empty if and only if $\Gamma'$ is in the image of $\beta^\bot_\Gamma$.
\end{enumerate}
\end{prop}

As a consequence, we may define a partial order $\prec$ on $\BIC(X)$ by defining $\Gamma\prec\Gamma'$ if and only if $(\Gamma',\Gamma)$ is non-empty.

\begin{rem}
The maps $\beta$ are not necessarily injective.
Indeed, whenever a profile $(\Gamma_1,\Gamma_2)$ is reducible, i.e. contains at least two distinct enhanced level graphs $\Lambda_1\neq\Lambda_2$, cutting the bottom level off these three-level graphs gives two distinct BICs $\tilde{\Lambda}_1, \tilde{\Lambda}_2\in\BIC(\Gamma_2^\top)$ (the ``cut'' edges correspond to edges of $\Gamma_2$ and become marked points in $\Gamma_2^\top$ and $\Gamma_2^\bot$) with $\beta_{\Gamma_2}^\top(\tilde{\Lambda}_1)=\beta_{\Gamma_2}^\top(\tilde{\Lambda}_2)=\Gamma_1$. See also \autoref{reducible_degen}.

However, non-injectivity does not imply reducibility. Degenerating a level in different ways may give \emph{isomorphic} graphs, see \autoref{reducible_aut} and \autoref{fig:vgraph}.
\end{rem}

This allows us to recursively compute all profiles.

\begin{prop}\label{prop:gengraphs}
Let $X$ be a generalised stratum.
\begin{enumerate}
\item The enhanced level graphs in $\BIC(X)$ can be listed explicitly.
\item All non-empty profiles in $X$ can be constructed recursively from $\BIC(X)$.
\item All graphs inside a profile can be constructed explicitly from the profile's components.
\end{enumerate}
\end{prop}
\begin{proof}
This is essentially the content of \autoref{sec:AllGraphs}:
the effective construction of BICs is explained in detail in \autoref{subsec:bicgeneration}, in particular \autoref{BICgenalgo}.

The non-empty profiles are constructed recursively: a non-empty profile $(\Gamma_1,\dotsc,\Gamma_l)$ of length $l$ may be extended to a non-empty profile $(\Gamma_0,\Gamma_1,\dotsc,\Gamma_l)$ if and only if $\Gamma_1\prec\Gamma_0$.
Indeed, $\Gamma_1^\top$ is also the top level of any graph $\Lambda$ in $(\Gamma_1,\dotsc,\Gamma_l)$ and thus a preimage $(\beta_{\Gamma_1}^\top)^{-1}(\Gamma_0)$ exists in $\BIC(\Gamma_1^\top)$ and can be clutched to $\Lambda$ to yield a graph in $(\Gamma_0,\Gamma_1,\dotsc,\Gamma_l)$.
Similarly, the profile may be extended to $(\Gamma_1,\dotsc,\Gamma_l,\Gamma_{l+1})$ if and only if $\Gamma_{l+1}\prec\Gamma_l$.

The converse direction is simply squishing of a level.

To get all the graphs we follow the above procedure, noting that we might obtain several graphs in the same profile if $\beta$ is non-injective (we clutch each preimage with each other graph).
\end{proof}

The key observation is that BICs and three-level graphs are sufficient for constructing the entire stratum. In particular, all levels are seen by these.

\begin{rem}\label{leveltypes}
Let $X$ be a generalised stratum. Note that any level $L$ appearing in any graph in $X$ is one of the following three types:
\begin{enumerate}
\item a top level of a BIC,
\item a bottom level of a BIC, or
\item a middle level of a three-level graph.
\end{enumerate}
Indeed, given a graph $\Gamma$, level $l$ of $\Gamma$ remains unchanged by contracting any level crossing not adjacent to $l$. Contracting all non-adjacent levels results either in a BIC (if $l$ is top or bottom level) or in a three-level graph around level $l$.
\end{rem}

While the reducibility is recorded by the three-level graphs, determining the componentes of a profile is not straight-forward.
Indeed, given two non-empty profiles $(\Gamma_1,\Gamma_2)$ and $(\Gamma_2,\Gamma_3)$ the profile $(\Gamma_1,\Gamma_2,\Gamma_3)$ is non-empty, but in the reducible case it is not clear how the reducibility and the individual graphs are related.

\begin{rem}
Determining the reducibility of a profile is a delicate issue and is discussed in detail in \autoref{subsec:degenerations}.
In particular, each of the following may occur:
\begin{enumerate}
\item a degeneration of an irreducible profile can be reducible (\autoref{reducibleprofile});
\item a degeneration of a reducible profile can be irreducible (\autoref{reducible_degen});
\item A reducible profile implies a non-injectivity of one of the maps $\beta$.
\end{enumerate}
The converse of the last statement is false in general, as the non-injectivity may stem from the presence of automorphisms.
\end{rem}

config: arxiv


TITLE: Physics: Columbs Law Over a Distance
QUESTION [0 upvotes]: I have a superposition of charges problem, and the following details are important to problem.  I have a charge $q_1=6\ \mathrm{\mu C}$ located at the origin $(0,0)$, in a Cartesian coordinate system in which the metrics of the set are in meters.  I have as well another charge at the point $(.03,0)$, and the charge has a value of $q_2 = 1.50 \ \mathrm{\mu C}$. Finally I have a charge stationed at the point $(.05,0)$, and the charge has a charge of $q_3 = -2.00 \ \mathrm{\mu C}$
My Attempt
My Question is How to find the net force acting on charge 1.
To attempt the problem I have done the following:
\begin{align*}
F_{net_{q_1}}&= F_{21}+F_{31} \\
F_{net_{q_1}} &= k \ \cdot \ 6.00 \mathrm{\mu C} \ \cdot[\frac{1.50 \mathrm{\mu C}}{(.03 \mathrm{m})^2}-\frac{2.00 \mathrm{\mu C}}{(.02 \mathrm{m})^2}] \\
F_{net_{q_1}} &= 46.7 \mathrm{N}
\end{align*}

REPLY [1 votes]: The net force should be
\begin{align*}
F_{net_{q_1}} &= k \ \cdot \ 6.00 \mathrm{\mu C} \ \cdot[\frac{1.50 \mathrm{\mu C}}{(.03 \mathrm{m})^2}-\frac{2.00 \mathrm{\mu C}}{(.05 \mathrm{m})^2}]
\end{align*}
Use 0.05 instead of 0.02 for the second distance.


TITLE: Do variables without any number beside them have a coefficient?
QUESTION [0 upvotes]: I've seen variables like $x, r, z, q, t$ that don't have any numbers beside them.  Do they even have coefficients, likewise for negative variables without any numbers in front of the negative signs?  I've seen this happen, and it looks like, for these positive variables along, they seem like they have coefficients of one, likewise for negative variables without any numbers beside the negative signs?

REPLY [0 votes]: They have a coefficient of 1, this fact is used when doing integration by parts of some functions such as actan(x) or (log(x))^2. Same applies to a negative value, it has a coefficient of -1. Which you can take out as a factor to create a positive variable.


TITLE: Laplace transform to determine system output
QUESTION [0 upvotes]: I have the following question:
Below, differential equations that describe dynamic systems are given together with system inputs and initial conditions. Use the Laplace transform to determine the system outputs.
$$
\begin{aligned}
\frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}}+3 \frac{\mathrm{d} y}{\mathrm{~d} t}+2 y &=\sigma(t) \\
\sigma(t) &=\left\{\begin{array}{ll}
0, & t<0 \\
1, & t \geq 0
\end{array}\right.\\
\frac{\mathrm{d} y}{\mathrm{~d} t}(0)=y(0) &=0
\end{aligned}
$$
After using the  $\mathcal{L}$-transform and partial fraction decomposition, I come to :
$$
Y(s) = \frac{1}{s+1} - \frac{1}{s+2}$$
which I tranform back to $\mathcal{L}^{-1} = e^{-t} - e^{-2t}$.
The correct answer is : $\frac{1}{2} -e^{-t} + \frac{1}{2} e^{-2t}$.
I am self-taught so I probably missed something very basic here, but after looking up many examples, I don't understand why this is the right answer.
Thanks in advance for help!

REPLY [1 votes]: The Laplace transform of $1$ is $\frac1s$. Probably you'd have taken it as $1$ and so obtained the wrong result.
It is actually, $$Y(s) = \frac{1}{\color{blue}{s}(s^2+3s+2)} = \frac{1}{s(s+1)(s+2)} = \frac1{2s} - \frac1{s+1} + \frac1{2(s+2)}$$
So, $$\boxed{y(t) = \frac12 - e^{-t} + \frac12e^{-2t}}$$


TITLE: the difference between the operators between $\delta$ and $d$
QUESTION [3 upvotes]: In classical mechanics, when talking about the principle of virtual work, what is difference between  $\delta r$ and $dr$? e.g. $W=\int \overrightarrow{F} \cdot \delta \overrightarrow{r}  $  and $W=\int \overrightarrow{F} \cdot d \overrightarrow{r}  $ .
Why can one exchange the place of $d\delta$ and $d$ in derivative calculation? e.g. $d\delta r=\delta d r$?

REPLY [1 votes]: They are not different, they are the same, but "dr" is ossified notation, meaning the integration differential, and physicists often think about infinitesimal increments, so they use different letters to indicate smallness. You sort it out from context.

REPLY [0 votes]: In classical mechanics, $\delta$ is equitemporal variation. $\delta$ and $\mathrm{d}$ are practically the same for constant constraint, but when the constraint is time-varying, they are different.
For example, if a bead is constrained to a moving string, $\delta r$ will be along the string, while $\mathrm{d}r$ won't be.
Conceptually, $\delta$ is variation of a functional, while $\mathrm{d}$ is differential of a function. But in calculations, just change $\mathrm{d}$ to $\delta$, and set $\delta t=0$ and you will get the correct result.


TITLE: Intuitive proof for $(a, b) = (a', b')$.
QUESTION [1 upvotes]: If $p,q,r,s$ are integers such that $ps - qr = \pm 1$, and
$a,b,a',b' $ are integers such that $a' = pa +qb, b' = ra + sb$, prove that $(a, b) = (a', b')$.
Here, the particular value of $ps - qr = \pm 1$
enables us to get integer values.
Can there be an algebraic or intuitive explanation for that?
Have two explanations. My explanation is given for both sequentially. The first explanation is offered as a geometrical / intuitive one.
While the second one uses number theoretic approach, and linear system of equalities.
Request vetting & improvement for the first one
First, needs linear algebra, and orthogonal matrices.
$$\begin{bmatrix}
        a'\\
        b' \\
        \end{bmatrix} =  \begin{bmatrix}
        p & q\\
        r & s \\
        \end{bmatrix}
\begin{bmatrix}
        a\\
        b\\
        \end{bmatrix}
$$
The condition $ps - qr = \pm 1$, states two things about matrix
$$\begin{bmatrix}
        p & q\\
        r & s \\
        \end{bmatrix}
$$
#1. determinant is non-zero; hence invertible matrix.
#2. Also, is an orthogonal matrix.
A function is invertible iff it is bijective. A matrix is only a convenient representation of linear mapping function between two sets, wrt a basis of the domain and co-domain of the function.
#1: So, for linear maps, the determinant of corresponding matrix as non-zero means the mapping is possible in both directions without collapsing of any dimension (row/column).
#2: the area (min. possible contained entity, as order $2$ is the minimum dimension; similarly, for order $3$ matrix, it is volume) is unit as for orthogonal matrix of order $n$ 'that' is given by: $I^n = 1$.
Hence, $(a,b) = (a', b')$.
Think that need some geometrical insight in orthogonal matrices.
Edit ::  in case of orthogonal matrices, they practically arise in rotation or reflection, where the determinant is $\pm 1$.
Can this explanation be modified to take into account that?

Second explanation is equivalent, though needs to ignore the matrix representation. But, needs the linear system of equalities to use the Bezout's lemma. This is a number theoretic approach.
$$a' = pa + qb$$
$$b' = ra + sb$$
any common divisor of $a, b$ will divide both $a', b'$, so $(a,b) ≤ (a',b')$.
Solving for $a, b$ in terms of $a', b'$ gives
$$a = (s/(ps-qr))a' -(q/(ps-qr))b'$$
$$b = (-r/(ps-qr))a' +(p/(ps-qr))b'$$
Since $ps-qr =\pm1$, all the coefficients in this system are integers, so $(a',b')$ is a common divisor of $a$ and of $b$, therefore, $(a',b') \le (a,b)$.

REPLY [1 votes]: Here's a proof that's short but unintuitive.
By multiplying $p, q \text{ and }a' \text{ by } -1$, if necessary, we can assume that $$ps-qr=1.$$
Let's divide $a$ and $b$ on each side by $\gcd(a,b).$
This means that $\gcd(a,b)$ is now $1$ and it only remains to
show that $\gcd(a',b')=1.$  There are integers $u$ and $v$ so that $au+bv=1.$ Hence,
$$(su-rv)a'+(-qu+pv)b'=au+bv=1,$$ completing the proof.
Or, if you want details:
$$(su-rv)(pa+qb)+(-qu+pv)(ra+sb)=$$
$$(psa+qsb-qra-qsb)u+(-pra-qrb+pra+psb)v$$
$$=au+bv=1$$


TITLE: Evaluate $\int_0^{\infty} x^2 e^{-x^2}dx$
QUESTION [3 upvotes]: Evaluate $\int_0^{\infty} x^2 e^{-x^2}dx$
The original problem is :
$$\text{Evaluate} \iint _R ye^{-x^2-y^2}dxdy$$
Where $R=\left\{ (x,y) \vert x\geq0,y\geq0\right\}$
I sub-ed $x=r\cos\theta, y=r\sin\theta$ and this changed to
$\left(\int_0^{\frac{\pi}{2}}\sin\theta d\theta \right)\left(\int_0^{\infty} r^2 e^{-r^2}dx\right)$
So I must calculate  $\int_0^{\infty} r^2 e^{-r^2}dr$, is there an easy way?

REPLY [1 votes]: Let's define the integral $\int_0^{\infty} x^2 e^{-a x^2}dx$, which is equivalent to what be are about to evaluate when $a=1$. We'll use a classical approach of differentiation under the integral sign:
$$\int_0^{\infty} x^2 e^{-a x^2}dx=(-1)\frac{\partial}{\partial a}\int_0^{\infty} e^{-a x^2}dx=-\frac{1}{2}\frac{\partial}{\partial a} \sqrt{\frac{\pi}{a}}=\frac{\sqrt{\pi}}{4a^{3/2}}.$$
Where we used the standard Gaussian integral: $\int_0^{\infty} e^{-a x^2}dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}$
Setting $a=1$ we finally get the result:
$$\int_0^{\infty} x^2 e^{-x^2}dx=\frac{\sqrt{\pi}}{4}.$$


TITLE: Pre composition with f
QUESTION [1 upvotes]: Let $X$, $Y$ be sets and let $f:X \to Y$ be a map.
For any set $T$ and any map $\psi:Y \to T$ from $Y$ to $T$, we may pre-compose $\psi$ with $f$ to get the composite map $\psi \circ f : X \to T$ from $X$ to $T$.
This procedure defines a map $$\Psi_T : Maps(Y,T) \to Maps(X,T),               \psi \mapsto \psi \circ f$$
How do i show that $f:X\to Y$ is surjective if and only if for any set T, the map $\Psi_T$ is injective?

REPLY [0 votes]: If $T$ has only one element, then $Maps(Y,T)$ has also only one element and $\Psi_T$ is injective, no matter how $f$ is. In fact, if $T$ has only one element and $Y$ has more than one element, the statement is false. So let's assume that $T$ has more than one element.
If we assume that $f$ is not surjective then there is some $y_0\in Y$ which is not the image of any element of $X$. Take any function $\psi:Y\to T$ and any element $t_1\in T-\{\psi(y_0)\}$ (at this point we need that $T$ has more than one element), and define
$$\phi(y)=\begin{cases}\psi(y)\text{ if }y\neq y_0\\t_1\text{ if }y=y_0\end{cases}$$
Then $\Psi_T(\psi)=\Psi_T(\phi)$ but $\psi\neq \phi$. Thus, if $\Psi_T$ is injective then $f$ is surjective.
Assume now that $f$ is surjective, and let's show that $\Psi_T$ is injective. For that, suppose that $\Psi_T(\psi)=\Psi_T(\phi)$, or, equivalently, $\psi\circ f=\phi\circ f$. Take any $y\in Y$. Since $f$ is surjective, $y=f(x)$ for some $x\in X$. Then $\psi(y)=\psi(f(x))=\phi(f(x))=\phi(y)$. We conclude that $\psi(y)=\phi(y)$ for every $y\in Y$, that is, $\psi=\phi$.


text:
\section{Conclusions}
\label{sec:conclude}

We consider a family of latent variable Gaussian graphical model whose precision matrix has a sparse plus low-rank structure. We derive parameter error bounds for regularized maximum likelihood estimation. Future work includes extending the framework to other distributions and the application to tasks such as prediction and detection.


text:
(* Copyright 2021 (C) Mihails Milehins *)

theory CZH_ECAT_Conclusions
  imports
    CZH_ECAT_Introduction
    CZH_ECAT_Category
    CZH_ECAT_Small_Category
    CZH_ECAT_Functor
    CZH_ECAT_Small_Functor
    CZH_ECAT_NTCF
    CZH_ECAT_Small_NTCF
    CZH_ECAT_PCategory
    CZH_ECAT_Subcategory
    CZH_ECAT_Simple
    CZH_ECAT_Discrete
    CZH_ECAT_SS
    CZH_ECAT_Parallel
    CZH_ECAT_Comma
    CZH_ECAT_Rel
    CZH_ECAT_Par
    CZH_ECAT_Set
    CZH_ECAT_GRPH
    CZH_ECAT_SemiCAT
    CZH_DG_CAT
    CZH_SMC_CAT
    CZH_ECAT_CAT
    CZH_DG_FUNCT
    CZH_SMC_FUNCT
    CZH_ECAT_FUNCT
    CZH_ECAT_Hom
    CZH_ECAT_Yoneda
    CZH_ECAT_Order
    CZH_ECAT_Small_Order
    CZH_ECAT_Ordinal
    CZH_ECAT_CSimplicial
    CZH_ECAT_Structure_Example
begin
end

subset_name: curated


TITLE: What is the measure of the angle CGF in the given figure?
QUESTION [1 upvotes]: What I could gather:
$$CX=CY=CE=CF=r$$
$$\measuredangle CEB=\measuredangle CFD=90^{\circ}.$$
$$\measuredangle XCE=65^{\circ}$$
$$\measuredangle CEF= \measuredangle CFE$$

Please help me solve this. Thank you.

REPLY [1 votes]: You are almost there. Note that $\measuredangle GCF$ is also $65^\circ$, and $\angle ECF$ is right. But $\triangle ECF$ is isosceles, so what is $\measuredangle CFG$?