text:
\section*{APPENDIX}

\maketitle

\begin{abstract}
Several features of an analytic (infinite-dimensional) Grassmannian
of (commensurable) subspaces of a Hilbert space were developed in the context
of integrable PDEs (KP hierarchy). We extended some of those
features  when polarized
separable Hilbert spaces are generalized to a class of polarized
Hilbert modules, in particular the Baker and $\tau$-functions,
which become operator-valued.
Following from Part I we produce a pre-determinant
structure for a class of $\tau$-functions defined in the setting
of the similarity class of projections of a certain Banach *-algebra.
This structure is explicitly derived from the transition map
of a corresponding principal bundle.
The determinant of this map
 gives a generalized, operator-valued $\tau$-function
that takes values in a commutative C*-algebra. We extend to this setting
the operator cross-ratio which had been used to produce the
scalar-valued $\tau$-function, as well
as the associated notion of a Schwarzian
derivative along curves inside the space of similarity classes.
We link directly this cross-ratio with Fay's
trisecant identity for the $\tau$-function (equivalent to the KP
hierarchy).
By restriction to the image of the Krichever map, we use the
Schwarzian to introduce the notion of
operator-valued projective structure on a compact Riemann
surface:
 this allows a deformation inside the Grassmannian
(as it varies its complex structure). Lastly,
we use our identification of the Jacobian of the Riemann surface
in terms of extensions of the Burchnall-Chaundy C*-algebra
(Part I) to describe the KP hierarchy.

\end{abstract}

\med
\textbf{Mathematics Subject Classification (2010)}: 46L08,
53B10, 53C30, 14H70

\med
\textbf{Keywords}: Hilbert module, polarization, tau-function,
projective structure, cross-ratio, Schwarzian derivative, KP hierarchy, Fay trisecant identity.

\section{Introduction}

This paper continues from Part I \cite{DGP2} of this work where we focused on extensions
of the algebra of continuous functions $C(X)$ by compact operators,
classified via
the Calkin algebra using Brown-Douglas-Fillmore extension
theory and KK-theory to obtain results on the K-homology
of $X$ and its Jacobian.
In this Part II, we turn our attention to the connection
on the determinant bundle over the Grassmannian,
 whose curvature is related to Sato's $\tau$-function
in the scalar case, as well as in our
operator-coefficient case \cite{DGP3}, and tie these in with several concepts
developed in Part I in relationship to the KP equation. Accordingly, this Part II, as for Part I, continues
an operator-theoretic approach to the Sato-Segal-Wilson theory \cite{Sato,SW}.

We recall in \S\ref{multflow}
that the classical $\tau$-function can be realized as a
specific cocycle  of the determinant line bundle of
the universal bundle pulled back over the space of restricted
polarizations from the restricted Grassmannian and its dual
\cite{Sato,SW}. The specific cocycle is determined by two
non-vanishing sections. The
underlying cocycle for the universal bundle is used to arrive at
the \emph{pre-determinant} structure of the $\tau$-function which we
refer to as the $\mathfrak T$-\emph{function}; accordingly, its determinant
$\Det(\mathfrak T)$ is the $\tau$-function (see
\cite{DGP3,MSW, Zelikin}). The important structure lies
within the geometry of this determinant bundle, its connection and
its curvature.  But the $\mathfrak T$-function is the cocycle for the
universal bundle over a space of restricted polarizations $\pol$,
relating essentially the same two underlying sections, in this case, of
the associated principal bundle.
Hence the interest is in the
calculation of the geometry, connection, and curvature of the
principal bundle of the universal bundle using the two sections
which are each covariantly constant over two complementary
subbundles of the tangent bundle of the space of restricted
polarizations \cite{DGP3,Zelikin}.

In \cite{DGP1,DGP2} we considered a certain (complex) Banach *-algebra
$A$ modeled on the linear operators of a Hilbert module denoted
$H_{\A}$, where $\A$ is a commutative separable C*-algebra.
Letting $P(A)$ denote the idempotents in $A$, in \cite{DGP3}
we considered the geometry of the space
$\Lambda = \Sim(p,A)$, the similarity class of $p\in P(A)$,
which is closely related
to the Grassmanian $\Gr(p,A)$ of Part I \cite{DGP2}
(see also \cite{DGP1,DGP3}).
 From the transition map
of a principal bundle $V_{\Lambda} \lra \Lambda$, we
 deduced a corresponding pre-determinant
denoted $\T$. We identify in \S\ref{multflow} a relationship between $\T$ and
 $\mathfrak T$
obtained via a diffeomorphism between $\Lambda$ and $\pol$.
In fact, as shown in \cite{DGP3}, certain calculations involving the
connection and curvature
are more straightforward when performed on
$\Lambda$, and that is why we choose to work mainly with predeterminants
 $\T_{\lambda}$ and the
ensuing determinants $\tau_{\lambda}$.

We revisit the construction of the $\tau$-function in terms of a cross-ratio
(\S\ref{opcrossratio}),  and of the Schwarzian derivative that
preserves it (\S\ref{cross-ratio}).
 In doing so, we need to extend to the C*-algebra case
an (abelian) group action on the Grassmannian (\S\ref{group}),
study the pull-back of the universal bundle through it,
compare this to the Poincar\'e bundle \cite{Alvarez}
(\S\ref{poincare}), and relate the operator cross-ratio

 to the projective structure on the Riemann surface associated
to the Schwarzian derivative (\S\ref{proj-struct}).
We then switch (\S\ref{wick})
to Raina's interpretation of the KP hierarchy
(Fay's trisecant identity) in model quantum-field theory \cite{Raina}:
 he reduced the identity to Wick's theorem,
by writing the 4-point function in terms of theta functions and prime forms,
essentially  a (generalized) cross-ratio
on a Riemann surface.
As a result, we  are able to give in \S\ref{k-dv} a KP hierarchy
satisfied on the space of extensions of the Burchnall-Chaundy C* algebra.

We retain the notation of Part I and refer there to the relevant concepts and results obtained.
The Appendix to Part I recalls much of the background material which is used here. The new results obtained
in this Part II are Lemma \ref{gamma-action}, Propositions \ref{T-diagram}, \ref{Det-diagram}, \ref{2-forms}, \ref{lambda-analytic},
\ref{proj-prop}, Theorems \ref{main-2}, \ref{main-1}.

\section{Universal bundles with connection and the $\T$-function}\label{previous}

\subsection{The space of polarizations}\label{polarizations}

Let $A$ be a unital complex Banach(able) algebra with group of units $G(A)$ and space
of idempotents $P(A)$.
Let $H$ be a separable (infinite dimensional) Hilbert space (here as
in Part I we take
$H = L^2(S^{1}, \bC)$). Given a unital separable C*-algebra $\A$, we take the
standard (free countable dimensional) Hilbert module $H_{\A}$ over
$\A$ and consider a \emph{polarization} of $H_{\A}$ given by a pair of
 submodules
$(\hp, \hm)$, such that
\begin{equation}\label{polar1}
H_{\A} = \hp \oplus \hm ~,~\text{and}~ \hp \cap \hm = \{0\}.
\end{equation}
Recall also from \cite{DGP1,DGP2} the (restricted) Banach *-algebra $A =
\cL_J(H_{\A})$
($J$ being a unitary $\A$--module map  satisfying
$J^2 = 1$)
which henceforth we use.
As in \cite{DGP2}, we assume $\A$ to be commutative (and separable). The Gelfand transform
implies there exists a compact
metric space $Y$ such that $Y = \Spec(\A)$ and $\A \cong C(Y)$, and thus
we note that
\begin{equation}\label{aform}
A \cong \{\text{continuous functions}~ Y \lra B
\} = C(Y,B),
\end{equation}
where $B = \cL_J(H)$ corresponds to taking $\A= \bC$.

We recall the Grassmannian $\Gr(p,A)$ and refer to \cite[A.2]{DGP2} for the necessary background.
For a given $p \in P(A)$, there is associated to $\Gr(p,A)$ its
\emph{dual Grassmannian} $\Gr^{*}(p,A)$ \cite[\S3]{DGP3}. Let $\pol$
denote the space of polarizations $(\hp, \hm)$ on $H_{\A}$. Then as shown in
\cite[\S3]{DGP3}, the space $\pol$ can be regarded as a subspace
\begin{equation}
\pol \subset \Gr(p, A) \times \Gr^*(p, A).
\end{equation}
A significant observation is that $\pol$ can be
closely related to the similarity class $\Lambda = \Sim(p,A)$ of $A$
where $\Lambda$ consists of the elements of $\pol$ expressed in terms of
projections.
In fact, $\Lambda$ admits a natural complex analytic structure induced
from that of $A$ via $P(A)$ (see \cite{DG2,DGP3} and Proposition \ref{lambda-analytic} below).
Further, from \cite[Theorem 4.1(3)]{DGP3} there exists
an analytic diffeomorphism
\begin{equation}\label{diffeo-1}
{\phi} : \pol \lra \Lambda \subset P(A).
\end{equation}
\begin{remark}\label{sa-remark-1}
Given that $\Lambda$, as a set of idempotents in the Banach *-algebra $A$, can admit non self-adjoint
elements (see \cite[Remark 2.1]{DGP2}), we note that
it will not be diffeomorphic to $\Gr(p,A)$
under the natural quotient map of equivalence relations $\Pi : P(A) \lra \Gr(A)= P(A)/{\sim}$ . However, restriction of $\Pi$  to the
self-adjoint elements $\Lambda^{\sa}$ of $\Lambda$ can be shown to
establish a bijective
diffeomorphism between $\Lambda^{\sa}$ and $\Gr(p,A)$, and moreover,
there exists a
smooth retraction of $P(A)$ onto $P^{\sa}(A)$ (as can be shown using the technique of \cite[Proof of Proposition 4.6.2]{Bl}).
\end{remark}

\subsection{The universal bundle with connection over $\Lambda$}

Here we briefly describe part of the basic geometry of \cite{DGP3}.
Firstly, let $\pi_{\Lambda} =\Pi \vert \Lambda$ and
$\pi_V=\Pi\vert V(p,A)$.
Let $V_{\Lambda}=\pi_{\Lambda}^*(V(p,A))$; specifically,
\begin{equation}
V_{\Lambda}= \{(r,u) \in {\Lambda} \times V(p,A):
\pi_{\Lambda}(r)=\pi_V(u)\}.
\end{equation}
In \cite{DGP3} we constructed an analytic
principal right $G(pAp)$-bundle with connection
\begin{equation}\label{lambda-principal}
(V_{\Lambda}, \omega_{\Lambda}) \lra \Lambda,
\end{equation}
whose (analytic) $G(pAp)$-valued transition map $t_{\Lambda}$ is given by
the formula
\begin{equation}\label{trans2}
t_{\Lambda}((r,u),(r,v))=t_V(u,v),
\end{equation}
where $t_V$ is the transition map for the $G(pAp)$-bundle $V(p,A) \lra \Gr(p,A)$ (see \cite[A.2]{DGP2}).

By the standard means we have the associated vector bundle with (Koszul)
connection, namely
\emph{the universal bundle with connection over $\Lambda$}
\begin{equation}\label{lambda-universal}
(\gamma_{\Lambda}, \nabla_{\Lambda}) \lra \Lambda,
\end{equation}
for which the curvature operator $R_{\nabla}$ was computed explicitly in \cite[\S 8]{DGP3}.

\subsection{The $\T$-function}\label{opcrossratio}

For given parallel (covariantly constant) sections $\a_p, \be_p$ of \eqref{lambda-principal}, we
defined the \emph{$\T$-function} in terms of the transition map $t_{\Lambda}$ of
\eqref{trans2} by
\begin{equation}\label{T-FUNCTION-1}
\T(r)= t_{\Lambda}(\alpha_p(r),\beta_p(r)),
\end{equation}
which can be expressed
 more conveniently as $\T= t_{\Lambda}(\alpha_p,\beta_p)$ (see \cite[(8.1)]{DGP3}).

Returning to the space of polarizations $\pol$, there are several closely
 associated objects
as described in \cite{DGP3,Zelikin}. Firstly, there is the principal
bundle with connection
\begin{equation}\label{pol-principal}
(V_{\pol}, \omega_{\pol})  \lra \pol,
\end{equation}
and associated universal (vector) bundle with connection
\begin{equation}\label{pol-universal}
(\gamma_{\pol}, \nabla_{\pol})  \lra \pol.
\end{equation}
Observe that there is a splitting
\begin{equation}\label{pmsplit}
d = \del_{+} + \del_{-},
\end{equation}
of the exterior derivative of $\omega_{\Lambda}$ induced in the
following way.
For a given polarization $\mathcal P =
(\hp, \hm) \in \pol$, the exterior derivative of $\omega_{\pol}$
splits as
$d = \del_{+} + \del_{-}$ where $\del_{+}$ (respectively, $\del_{-}$)
denotes the covariant
derivative in the directions from $\hp$ (respectively, $\hm$).
The induced splitting \eqref{pmsplit}
thus follows from the analytic diffeomorphism in \eqref{diffeo-1}
for which $\omega_{\Lambda} = ({\phi}^{-1})^* \omega_{\pol}$.

In \cite{Zelikin}, the function $\mathfrak T$ constructed via
operator cross-ratio, is likewise shown to be derived from the transition
map $t_{\pol}$ for
$V_{\pol} \lra \pol$. This is closely related to our $\T$-function, and in
 \cite{DGP3}, we showed that the
geometry of $(V_{\Lambda} , \omega_{\Lambda}) \lra \Lambda$ is also closely
related to that of  $(V_{\pol} , \omega_{\pol}) \lra \pol$,
where local coordinates for $\Lambda$ and $\pol$ can be expressed
in terms of an \emph{operator cross-ratio} (see \S\ref{cross-ratio} below).
 In particular,
with reference to \cite[p. 47]{Zelikin}
and recalling the analytic diffeomorphism
${\phi} : \pol \lra \Lambda$ in
\eqref{diffeo-1},
we have in terms of the parallel sections
$\a_p$ and $\be_p$, the relationship
\begin{equation}\label{T-relation-1}
\phi^* \T = \phi^*(t_{\Lambda}(\alpha_p,\beta_p)) = \be_p^{-1}
\a_p = \mathfrak{T}.
\end{equation}

Consider a pair of polarizations $(\hp, \hm), (\kp,
\km) \in \pol$. Let $\hpm$ and $\kpm$ be `coordinatized' via maps
$P_{\pm} : \hpm \lra \hmp$, and $Q_{\mp}: \kpm \lra \kmp$. The composite map
\begin{equation}\label{composite}
\hp \ovsetl{\km} \kp \ovsetl{\hm} \hp,
\end{equation}
allows us to take
 the operator cross--ratio \cite{Zelikin} (cf. \cite{DGP3}):
\begin{equation}\label{tauop1}
\mathfrak{T} (\hp, \hm; \kp, \km) = (P_{-}P_{+} -
1)^{-1}(P_{-}Q_{+} - 1) (Q_{-}Q_{+} - 1)^{-1} (Q_{-}P_{+} - 1).
\end{equation}
 For this construction there is no essential algebraic change in
generalizing from polarized Hilbert spaces to polarized Hilbert
modules. The principle here is that the transitions between charts
define endomorphisms of $W \in \Gr(p,A)$ that will become the
transition functions of the universal bundle $\gamma_{\pol} \lra
\pol$.
The main properties of $\T$ and $\mathfrak T$ is that they are
pre-determinants
for various classes of $\tau$-functions, as we shall see in \S\ref{multflow}.

\subsection{Trace-class operators and the determinant}\label{trace-class}

An equivalent, operator, description leading to
the functions $\T$ and $\mathfrak{T}$ above, can be obtained
along the lines of e.g. \cite{MSW,SW,Zelikin}. Here, as in \cite{DGP2},
we make use of the nested sequence
of Schatten ideals definable in $\cL(H_{\A})$ (see
e.g. \cite{Smith}). Suppose $(\hp, \hm), (\kp, \km) \in \pol$ are such
that $\hp$ is the graph of a linear map $S : \kp \lra \km$ and
$\hm$ is the graph of a linear map $T: \km \lra \kp$~. Then on
$H_{\A}$ we consider the identity map $\hp \oplus \hm \lra \kp
\oplus \km$, as represented in the block form
\begin{equation}\label{tauop2}
\bmatrix a &b \\ c &d
\endbmatrix
\end{equation}
where $a: \hp \lra \kp,~ d: \hp \lra \km$ are zero--index Fredholm
operators, and $b: \hp \lra \kp,\\ ~ c: \hp \lra \km$ are in
$\mathcal K(H_{\A})$ (the compact operators), such that $S =
ca^{-1}$ and $T = bd^{-1}$. Initially, one considers the operator
$1 - ST = 1 - ca^{-1}bd^{-1}$. In particular, with a view to
defining a generalized determinant leading to \emph{an
operator--valued $\tau$-function}, we consider cases where
ST is of \emph{trace class}. If we take $b,c$ to be
Hilbert--Schmidt operators (as for the case $\A = \bC$ as in
\cite{MSW,SW,Zelikin}), then $ST$ is of trace--class, the
operator $(1 - ST)$ is $\mathfrak{T}(\hp, \hm; \kp,
\km)$ above, and the $\tau$-function is defined as
\begin{equation}\label{tauop3}
\Det ~\mathfrak{T}(\hp, \hm; \kp, \km) \otimes 1_{\A}
= \Det (1 - ca^{-1}bd^{-1}) \otimes 1_{\A} = \Det (\alpha_p \beta^{-1}_p).
\end{equation}
Starting from the universal bundle $\gamma_{\Gr} \lra \Gr(p,A)$,
then with respect to an `admissible basis' (for defining determinants) in the Stiefel bundle $V(p,A) \lra \Gr(p,A)$
(see \cite{DG2} and \cite[A.2]{DGP2}), the
$\tau$-function in \eqref{tauop3} is equivalently derived from the
canonical section of the determinant line bundle $\Det(\gamma^*_{\Gr}) \lra \Gr(p,A)$
(cf. \cite{MSW,SW,Zelikin}).

\section{Relationship between the predeterminants and
$\tau$-functions}\label{multflow}

\subsection{Relationship between the predeterminants}\label{group}

We start with the following lemma:
\begin{lemma}\label{gamma-action}
Let $\Gamma \subset G(pAp)$ be a group acting on the subspace $\hp$. Then
there exists a natural action of $\Gamma$ on the spaces $\Gr(p,A)$,
$\Lambda= \Sim(p,A)$ and $\pol$.
\end{lemma}

\begin{proof}
Firstly, $\hp$
is a splitting subspace for $H_{\A}$ and thus determines
the polarization $H_{\A} = \hp \oplus \hm$, and likewise
for any other polarizing pair $(\kp, \km)$.

We recall from \cite[\S4]{DG2} that $\Gr$ is a functor, and if
$g$ is a linear automorphism of $\hpm$, it thus defines an element of
$G(pAp)$. For $p \in P(A)$, let $\hat{p} = 1 -p$.
Then $g + \hat{p} \in G(A)$, and therefore this term defines an inner
automorphism of $A$ taking $\Lambda$ to
itself, fixing $p$ and inducing an analytic diffeomorphism of $\Gr(p,A)$
with itself. Thus by functorial properties
of inner automorphisms and the functorial properties of $\Gr$, the group
$\Gamma$ acting on $\hp$, viewed as a subgroup of
$G(pAp)$, induces actions on $\Lambda$ and $\Gr(p,A)$.

Further, given an action of $\Gamma$ on $\Lambda$, there is an induced action
on the space of polarizations $\pol \subset
\Gr(p,A) \times \Gr^*(p,A)$, since $\Lambda$ simply consists of
the elements of $\pol$ viewed as projections.
\end{proof}
The induced action of $\Gamma$ gives
rise to following commutative diagram
\begin{equation}\label{Gamma-pic}
\begin{CD}
\Gamma \times \pol  @> \mu_{\pol} >> \pol
\\ @V \bar{\phi} VV   @VV {\phi}  V \\
\Gamma \times \Lambda     @> \mu_{\Lambda} >> \Lambda
\\ @V \bar{\pi}_{\Lambda} VV   @VV \pi_{\Lambda} V \\
\Gamma \times \Gr(p,A)     @> \mu_{\Gr} >> \Gr(p,A)
\end{CD}
\end{equation}
where we note that the action of $\Gamma$ as a subgroup of $G(pAp)$ on
$\Gr(p,A)$ is essentially trivial since $G(pAp)$ is
the structure group of the principal bundle $V(p,A) \lra \Gr(p,A)$.
An example of such a group $\Gamma$ is given by the group of multiplication
operators $\Gamma_{+}(\A)$ in \cite{DGP2} (in particular $\Gamma_{+}$ in
\cite{SW}
 for $\A = \bC$).

 In the following we will make use of the various maps appearing in
\eqref{Gamma-pic} besides recalling the role of canonical sections
for the universal bundles $\gamma_{\Gr} \lra \Gr(p,A)$ (the associated vector bundle
to $V(p,A) \lra \Gr(p, A)$, see \cite[A.2]{DGP2}) and
$\gamma_{\Lambda} \lra \Lambda$, introduced in \cite[\S7]{DGP3}.
Let $S_{p}$ be the canonical section of $\gamma_{\Gr}$ and set $\wti{S}_{p}
= \mu_{\Gr}^*(S_{p})$. Likewise, let $S_{\lambda}'$ be the canonical
section of $\gamma_{\Lambda}$ and set  $\wti{S}'_{\lambda}
= \mu_{\Lambda}^*(S_{\lambda}')$. The following implements the sections
functor $H^0$.

\begin{proposition}\label{T-diagram}
Let $W \in \Gr(p,A)$ and $\lambda \in \Lambda$. In terms of the maps
in \eqref{Gamma-pic} the following diagram is
commutative where the horizontal maps
$\rho_1, \rho_2$ are homomorphisms
and the vertical maps $\bar{\pi}^*_{\Lambda},
 (\bar{\pi}^*_{\Lambda})_{\res}$ are isomorphisms:
\begin{equation}
\begin{CD}
H^{0}(\Gamma \times \Gr(p,A), ~\mu^*_{\Gr} \gamma_{\Gr}) @> \rho_1
>> H^{0}(\Gamma \times \{W \}, ~\mu^*_{\Gr} \gamma_{\Gr}\vert \Gamma
\times \{W \})
\\ @V \bar{\pi}^*_{\Lambda} V \cong V   @VV
(\bar{\pi}^*_{\Lambda})_{\res} \cong V\\
H^{0}(\Gamma \times \Lambda, ~\mu^*_{\Lambda} \gamma_{\Lambda}) @>
\rho_2
>> H^{0}(\Gamma \times \{\lambda\}, ~\mu^*_{\Lambda} \gamma_{\Lambda}
\vert \Gamma \times \{\lambda\})
\end{CD}
\end{equation}
In particular, we have $ \bar{\pi}^*_{\Lambda}(\wti{S}_{\lambda}) =
\wti{S}'_{p}$.
\end{proposition}

\begin{proof}
The commutativity follows by applying the sections functor $H^{0}$
to the lower square in \eqref{Gamma-pic}. Since
 we have $\gamma_{\Lambda} \cong \pi^*_{\Lambda}
\gamma_{\Gr}$, it follows that
\begin{equation}
\begin{aligned}
(\mu_{\Gr} \circ \bar{\pi}_{\Lambda})^* \gamma_{\Gr} &\cong
(\pi_{\Lambda} \circ \mu)^* \gamma_{\Gr} \\
&\cong \mu_{\Lambda}^*( \pi^*_{\Lambda} \gamma_{\Gr})
\cong \mu_{\Lambda}^*\gamma_{\Lambda}.
\end{aligned}
\end{equation}
Thus $\bar{\pi}_{\Lambda}^*(\mu_{\Gr}^* \gamma_{\Gr}) \cong
\mu_{\Lambda}^* \gamma_{\Lambda}$, and likewise for the restriction
to $\Gamma \times \{W\}$.
\end{proof}

With regards to the maps in Proposition \ref{T-diagram}, we next introduce
the following:
\begin{definition}\label{def-T-1}
For $W \in \Gr(p,A)$, we denote by $\mathfrak{T}_W$ \emph{the
 $\mathfrak{T}$-function of the point $W$ over $\Gamma$}, as defined
to be the image of the section $\wti{S}_{p} = \mu^*_{\Gr}(S_{p})$ under $\rho_1$.
\end{definition}

\begin{definition}\label{def-T-2}
For $\lambda \in \Lambda$, we denote by $\T_{\lambda}$ \emph{the
 $\T$-function of the point $\lambda$ over $\Lambda$}, as defined
to be the image of the
section $\wti{S}'_{\lambda} = \mu^*_{\Lambda}(S'_{\lambda})$ under $\rho_2$.
\end{definition}
Then it is easily seen from Proposition \ref{T-diagram} that we have
\begin{equation}\label{T-relation}
(\bar{\pi}^*_{\Lambda})_{\res}(\mathfrak{T}_W) = \T_{\lambda}.
\end{equation}
Recalling the analytic diffeomorphism ${\phi}: \pol \lra
\Lambda$ in \eqref{diffeo-1}, the relationship \eqref{T-relation} is
another way of interpreting our observation that ${\phi}^* \T
= \mathfrak{T}$.

In terms of the group multiplication $m: \Gamma \times \Gamma \lra \Gamma$
in $\Gamma$, we have the following
commutative diagram
\begin{equation}\label{Gamma-times}
\begin{CD}
\Gamma \times
\Gamma \times \Lambda @> {m \times \ID} >> \Gamma \times \Lambda
\\ @VVV  @VVV \\
\Gamma \times
\Gamma \times \Gr(p,A) @> {m \times \ID} >> \Gamma \times \Gr(p,A)
\end{CD}
\end{equation}

\begin{itemize}
\item[(1)]
For $W \in \Gr(p,A)$, let $\mu_W : \Gamma \times \{W\} \lra
\Gr(p,A)$ denote the map induced by $\mu_{\Gr}$ in
\eqref{Gamma-pic}.

\item[(2)]
For $\lambda \in \Lambda$, let $\mu_{\lambda} : \Gamma \times
\{\lambda\} \lra \Lambda$ denote the map induced by $\mu_{\Lambda}$
in \eqref{Gamma-pic}.
\end{itemize}
Thus for each point $W \in \Gr(p,A)$ (respectively, $\lambda \in
\Lambda$) we obtain vector bundles over $\Gamma \times \Gamma$
associated with $W$ (respectively, $\lambda$), as given by
\begin{equation}\label{gamma-cross1}
\begin{aligned}
\mathfrak{E}_W &= (1 \times \mu_{W})^*
(\mu_{\Gr}^* \gamma_{\Gr}) =
m^*(\mu_{\Gr}^* \gamma_{\Gr}), \\
\mathfrak{E}_{\lambda}&= (1 \times \mu_{\lambda})^*(\mu_{\Lambda}^*
\gamma_{\Lambda})
= m^*(\mu_{\Lambda}^* \gamma_{\Lambda}).
\end{aligned}
\end{equation}

\subsection{The Poincar\'{e} bundles and the
$\tau$-function}\label{poincare}

Returning to the $\tau$-functions considered in Part I
\cite{DGP2}, we next make
some corresponding observations for determinants using admissible
bases in $V(p,A)$
(cf. \cite{Alvarez}).
Here, for ease of notation, we  set $\Gamma_1 = \Gamma_{+}(\A)$,
the action we are concerned with (cf Lemma \ref{gamma-action}).
In this case it will be convenient to use the
following notation:
\begin{equation}\label{Dets}
\Det_{\Gr}= \Det(\gamma^*_{\Gr})~ \text{and} ~ \Det_{\Lambda}=
\Det(\gamma^*_{\Lambda}),
\end{equation}
noting that $\Det_{\Lambda} = \pi_{\Lambda}^* \Det_{\Gr}$. These we
 will pull-back by maps
\begin{equation}
\begin{aligned}
\hat{\mu}_{\Gr}&: J_{\A}(X) \times \Gr(p,A) \lra \Gr(p,A), \\
\hat{\mu}_{\Lambda}&: J_{\A}(X) \times \Lambda \lra \Lambda,
\end{aligned}
\end{equation}
where we recall the space $J_{\A}(X)$ of monomorphisms $\bA \otimes
\A \lra B_W$ with respect to the spectral
curve $X$
(see \cite[Appendix A.4]{DGP2}). In keeping with some standard
algebraic-geometric terminology (cf. \cite{Alvarez}), the line bundles
\begin{equation}\label{poincare-0}
\begin{aligned}
\hat{\mu}^*_{\Gr} \Det_{\Gr} &\lra J_{\A}(X) \times \Gr(p,A), \\
\hat{\mu}^*_{\Lambda} \Det_{\Lambda} & \lra J_{\A}(X) \times \Lambda,
\end{aligned}
\end{equation}
are  referred to as \emph{Poincar\'{e} bundles}.
Next we pull-back the maps in \eqref{poincare-0} along the map $\Gamma_1
\lra J_{\A}(X)$ in \cite[A.19]{DGP2} to
obtain
\begin{equation}\label{poincare-1}
\begin{aligned}
\mu^*_{\Gr} \Det_{\Gr} &\lra \Gamma_1 \times \Gr(p,A), \\
\mu^*_{\Lambda} \Det_{\Lambda} &\lra \Gamma_1 \times \Lambda.
\end{aligned}
\end{equation}
\begin{remark}
By incorporating the group multiplication, we have in a similar
way to \eqref{gamma-cross1} the following
Poincar\'{e} bundles over
${\Gamma}_1 \times {\Gamma}_1$ associated with $W \in \Gr(p,A)$
(respectively, $\lambda \in
{\Lambda}$)
\begin{equation}\label{poincare-2}
\begin{aligned}
\mathfrak{B}_W &= (1 \times \mu_{W})^*(\mu_{\Gr}^* \Det_{\Gr}) =
m^*(\mu_{\Gr}^* \Det_{\Gr}), \\
\mathfrak{B}_{\lambda}&= (1 \times \mu_{\lambda})^*(\mu_{\Lambda}^*
\Det_{\Lambda}) = m^*(\mu_{\Lambda}^* \Det_{\Lambda}).
\end{aligned}
\end{equation}
\end{remark}
The next step is to give the analogous statement to Proposition
\ref{T-diagram}
for determinants and thus relate the $\tau$-functions corresponding
to respective points
of $\Gr(p,A)$ and $\Lambda$. Specifically, for fixed $W \in \Gr(p,A)$,
 let us set
$\wti{\cL}_{\tau}(W) = \mu^*_{\Gr} \Det_{\Gr}\vert \Gamma_1 \times \{W
\}$, and likewise, for fixed $\lambda \in \Lambda$, let us set
$\wti{\cL}_{\tau}(\lambda) = \mu^*_{\Lambda} \Det_{\Lambda}\vert
\Gamma_1 \times \{\lambda \}$.

Let $Q_{p}$ be the canonical section of $\Det_{\Gr}$ and set
$\wti{Q}_{p} = \mu_{\Gr}^*(Q_{p})$. Likewise, we take $Q_{\lambda}'$ to be
the canonical section of $\Det_{\Lambda}$ and set $\wti{Q}'_{\lambda} =
\mu_{\Lambda}^*(Q_{\lambda}')$.
Motivated by \cite[\S5]{Alvarez} and \cite{SW}, we introduce a
$\tau$-function $\tilde{\tau}_W$ associated to $W \in \Gr(p,A)$
defined by $\tilde{\tau}_W = \rho_1(\wti{Q}_p)$, and associated
to $\lambda \in \Lambda$, we likewise
define $\tilde{\tau}_{\lambda}= \rho_2(\wti{Q}_{\lambda}')$.

\begin{proposition}\label{Det-diagram}
Let $W \in \Gr(p,A)$ and $\lambda \in \Lambda$. With regards to the
maps in \eqref{Gamma-pic} we have the following commutative diagram
in which the horizontal maps $\rho_1, \rho_2$ are homomorphisms
and the vertical maps
$\bar{\pi}^*_{\Lambda},(\bar{\pi}^*_{\Lambda})_{\res} $
are isomorphisms:
\begin{equation}
\begin{CD}
H^{0}(\Gamma_1 \times \Gr(p,A), ~\mu^*_{\Gr} \Det_{\Gr}) @> \rho_1
>> H^{0}(\Gamma_1 \times \{W \}, ~\wti{\cL}_{\tau}(W))
\\ @V \bar{\pi}^*_{\Lambda} V \cong V   @VV (\bar{\pi}^*_{\Lambda})_{\res}V\\
H^{0}(\Gamma_1 \times \Lambda, ~\mu^*_{\Lambda} \Det_{\Lambda}) @>
\rho_2
>> H^{0}(\Gamma_1 \times \{\lambda\}, ~ \wti{\cL}_{\tau}(\lambda))
\end{CD}
\end{equation}
In particular, we have $\tilde{\tau}_W = \Det~ \mathfrak{T}_W =
\rho_1(\wti{Q}_p)$ and $\tilde{\tau}_{\lambda} = \Det~ \T_{\lambda}
= \rho_2(\wti{Q}_{\lambda}')$.
\end{proposition}
\begin{proof}
This follows immediately from Proposition \ref{T-diagram} when the
coefficients are in the respective determinant bundle, and from the
definitions of the $\mathfrak{T}_W$ and $\T_{\lambda}$-functions
once their respective determinants are taken.
\end{proof}

We also recall the notion of `transverse subspace' from \cite{DGP2,SW}.
We note that one way of characterizing $W$'s transverse
to $H_-$ is that the orthogonal projection
$W\longrightarrow H_+$ should be an isomorphism; this
property is not preserved in general under multiplication by
an element of the group $\Gamma_1= \Gamma_+(\A)$,
but it is preserved over a dense subset
which we denote by $\Gamma_1^W = \Gamma_+^W(\A)$
(see \cite[Appendix A.5]{DGP2}).
Likewise, we set $\Gamma_1^{\lambda} =\Gamma_+^{\lambda}(\A)$.

In a similar way to \cite{SW} (cf. \cite{Alvarez}) we define
operator-valued $\tau$-functions
relative to such subspaces
$W \in \Gr(p,A)$ and $\lambda \in \Lambda$ as follows:
\begin{itemize}
\item[(1)]
Fix a transverse subspace $W \in \Gr(p,A)$ and define
\begin{equation}
{\cL}_{\tau}(W) = \wti{\cL}_{\tau}(W) \vert {\Gamma}_1^W \times \{W \}.
\end{equation}
Let $\sigma_W$ be a constant section
 trivializing ${\cL}_{\tau}(W)$
over ${\Gamma_1^W}$ (it is to ensure the existence
of this section that we take $W$ transverse \cite[Prop. 3.3]{SW}).
Recalling from
Proposition \ref{Det-diagram} that
 we have $\tilde{\tau}_W = \Det~ \mathfrak{T}_W =
\rho_1(\wti{Q}_{p})$, and for $g \in {\Gamma_1^W}$, we define
$\tau_W: \Gamma_1^W
\lra \bC \otimes 1_{\A}$ by
\begin{equation}
\tau_W(g) = \tilde{\tau}_{W}(g) (\sigma_W(g))^{-1},
\end{equation}
which simply recovers the tau-function $\tau_W$ of \cite{DGP2}
(cf. \cite{SW}) with $\A$-valued coefficients.

\item[(2)]
Fix a transverse subspace $\lambda \in \Lambda$ and define
\begin{equation}
{\cL}_{\tau}(\lambda) = \wti{\cL}_{\tau}(\lambda)
\vert {\Gamma}_1^{\lambda} \times \{ \lambda \}.
\end{equation}
Let $\sigma_{\lambda}$ be a constant section
trivializing ${\cL}_{\tau}(\lambda)$
over ${\Gamma_1^\lambda}$. Recalling from Proposition \ref{Det-diagram} that
 we have $\tilde{\tau}_{\lambda}= \Det~ \mathcal{T}_{\lambda} =
\rho_2(\wti{Q}_{\lambda}')$, and for $g \in {\Gamma_1^\lambda}$,
we define $\tau_\lambda:
\Gamma_1 \lra \bC \otimes 1_{\A}$ by
\begin{equation}\label{tau-lambda-1}
\tau_{\lambda}(g) = \tilde{\tau}_{\lambda}(g) (\sigma_{\lambda}(g))^{-1}.
\end{equation}
\end{itemize}
This leads to the straightforward relationship
\begin{equation}\label{tau-lambda}
(\bar{\pi}^*_{\Lambda})_{\res}~ (\tau_W(g)) = \tau_{\lambda}(g).
\end{equation}

At this stage it should be clear from the above results
 that the geometry of
$\Lambda$ and $\Gr(p,A)$, as well as the functions $\tau_{\lambda},
\tau_W$, are  closely related via $\pi_{\Lambda}$ and \eqref{diffeo-1}.
In particular,  by following standard procedures, $(\gamma_{\Lambda},
\nabla_{\Lambda})
\lra \Lambda$ induces the determinant line bundle with its connection
$(\Det_{\Lambda},
\nabla(\Det_{\Lambda}) )\lra \Lambda$, and likewise for
$(\Det_{\Gr}, \nabla(\Det_{\Gr})) \lra \Gr(p,A)$. Here the $\tau$-function
serves as a
`logarithmic potential' for the curvature of the connection, and from
\cite{MSW,Zelikin} (cf. \cite{DGP3}), we have recalling \eqref{pmsplit} for
the curvature 2-forms:
\begin{equation}\label{curv-2-form}
\Omega(\Det_{\Lambda}) = \frac{1}{2\pi \iota} \del_{+}
\del_{-} \log \vert \tau_{\lambda} \vert, ~ \text{and}~
\Omega(\Det_{\Gr}) = \frac{1}{2\pi \iota} \del_{+} \del_{-}
\log \vert \tau_{W} \vert.
\end{equation}
Since the corresponding calculations of the connection and curvature
of $(\gamma_{\Gr}, \nabla_{\Gr}) \lra \Gr(p,A)$ are more straightforward
on passing to
$(\gamma_{\Lambda}, \nabla_{\Lambda}) \lra \Lambda$ (see \cite{DGP3}
for details), we choose to emphasize objects relative to $\Lambda$, such
as $\T_{\lambda}, \tau_{\lambda}$, etc. in the following.

Specifically, we can use the canonical section $S_{\lambda}'$
to lift the action of
$\Gamma_1$ to
the universal bundle $\gamma_{\Lambda}$. Moreover, in order to
give a more explicit expression for
$\T_{\lambda}$, we use parallel sections
$\a_{\lambda}, \be_{\lambda}$ as in
\eqref{T-FUNCTION-1}, and then  $\T_{\lambda}: \Gamma_1 \lra
G(\lambda A \lambda)$ is equivalently
defined by
\begin{equation}\label{T-lambda-1}
\T_\lambda(g)(r) = t_{\Lambda}(g^{-1} \a_{\lambda}(r), \be_{\lambda}(rg)),
\end{equation}
for $g \in \Gamma$. In the following we shall simply drop
the argument in $r$ since this is understood.

On recalling the element $q_{\zeta} \in {\Gamma}_1$ as given by a
map $q_{\zeta}(z)= (1- z\zeta^{-1})$ in \cite[\S4.4]{DGP2},
we proceed, in view of the `transversality' to
define for $g\in \Gamma_1^\lambda$,
\begin{equation}\label{T-lambda-2}
\Psi_{\lambda}(g, \zeta) = \T_{\lambda}(g \cdot q_{\zeta})
(\T_{\lambda}(g))^{-1}.
\end{equation}
Observe that the function $\Psi_{\lambda}$ is a `predeterminant' for a
Baker function $\psi_{\lambda}$
in the following sense. On taking determinants of \eqref{T-lambda-2}, we
obtain
\begin{equation}\label{T-lambda-3}
\psi_{\lambda}(g, \zeta): = \Det (\Psi_{\lambda}(g, \zeta)) =
\tau_{\lambda}(g \cdot q_{\zeta}) (\tau_{\lambda}(g))^{-1},
\end{equation}
which is simply a `lifted-to-$\Lambda$' version of the relationship
between the Baker $\psi_W$ and $\tau_{W}$ functions
under pull-back by $\pi_{\Lambda}$ (see \cite[\S4.4]{DGP2} and \cite{SW}):
\begin{equation}
\psi_{W}(g, \zeta) = \tau_{W}(g \cdot q_{\zeta}) (\tau_{W}(g))^{-1}.
\end{equation}

\section{Applications}

\subsection{Operator cross-ratio and the Schwarzian
derivative}\label{cross-ratio}

Smooth and analytic parametrizations of subspaces of a Banach
space were studied in \cite{DG2,DEG} (cf \cite{GL1}). Using the techniques in question
we can regard
spaces such as $\Lambda$ (and likewise, $\Gr(p,A), \pol$, etc.)
as \emph{analytically} parametrized in terms of analytic maps
$D_0 \lra \Lambda$
to the underlying Banach space (of $\Lambda$), where $D_0 \subset \bC$
denotes the open unit disk. Thus taking $w \in D_0$ as a local parameter,
one can assign (operator) $\Lambda$-valued
functions $\zeta_{\lambda}(w)$ parametrizing $\lambda \in \Lambda$
(and likewise
for, e.g., $W \in \Gr(p,A)$). With this understood, we shall simply
write, as a convention,
$\zeta$ for $\zeta(w)$ and $z$ for $z = f(w)$, etc.,
in the following. We also take $\Hol(D_0,\Lambda)$ to denote the space
of holomorphic (analytic)
$\Lambda$-valued functions on $D_0$.

 Following \cite{Zelikin}, we  assume that
(commensurable) subspaces in $\Gr(p,A)$ are
isomorphic to those of $\Gr^*(p,A)$ and consider a smooth family
of subspaces $\cH(s) \in \Gr(p,A)$ parametrized by one real parameter $s$,
where $\cH(u) \cong \hp$ for $u > 0$, and $\cH(v) \cong \hm$ for
$v \leq 0$, so that
the pair $(\cH(u), \cH(v))$ defines a polarization of $H_{\A}$.
More specifically, consider
an ordering $s_2 < 0 <s_1 < s_3$, and a pair of polarizations
identified with points
$(\hp, \hm), (\kp, \km) \in \pol$:
\begin{equation}\label{pair-pol}
\begin{aligned}
(\cH_2, \cH_1) : &= (\cH(s_2), \cH(s_1)) \cong (\hp, \hm), \\
(\cH, \cH_3) : &= (\cH(0), \cH(s_3)) \cong (\kp, \km).
\end{aligned}
\end{equation}
For spaces such as $\Gr(p,A), \Lambda$ and $\pol$
(we recall that $\Lambda$ and $\pol$ are analytically diffeomorphic),
we define \emph{the operator cross-ratio} (`$\cross$') in terms
of projection ($A$-valued)
affine coordinates as
\begin{equation}\label{cross}
\cross(a,b;c,d) = (a - c)(a - b)^{-1}(b - d)(c -d)^{-1}.
\end{equation}
Let $z$ be such a $\Lambda$-valued variable in $\cH(s)$, and
letting $z_i$ denote variables with respect to
$\cH_i$ ($z$ corresponds to $\cH(0)$), we apply \eqref{cross} to the
polarizing pair in \eqref{pair-pol}
to obtain
\begin{equation}\label{cross-1}
\cross(\cH_2, \cH_1; \cH, \cH_3) = \cross(z_1,z_2;z,z_3)
= (z_1 - z)(z_1 - z_3)^{-1}(z_2 - z)(z_2 - z_3)^{-1},
\end{equation}
which is defined on projection ($A$-valued) analytic coordinates of $\pol$.
As shown in \cite{Zelikin}, this yields a
$\End(\gamma_{\Gr})$-valued $1$-cocycle $\{ \cross \}
\in H^1(\Gr(p,A), \End(\gamma_{\Gr}))$,
and hence under the pullback
\begin{equation}\label{pull-back-1}
\pi^*_{\Lambda}: H^1(\Gr(p,A), \End(\gamma_{\Gr}))
\lra H^1(\Lambda, \End(\gamma_{\Lambda})),
\end{equation}
we regard $\{\cross\}$ as also an $\End(\gamma_{\Lambda})$-valued
1-cocycle on $\Lambda$. Likewise the $\T$-function
can be viewed as a $G(pAp)$-valued $1$-cocycle in terms of the
transition function $t_{\Lambda}$ as
given by \eqref{T-FUNCTION-1}(see \cite[\S8]{DGP3} for details).
The operator cross-ratio is used by Zelikin
 \cite{Zelikin} to introduce an
operator analogous to the Schwarzian derivative  \cite[\S4]{Zelikin} .

 The key idea  is that, though operators do not
commute, one can take limits within the cross-ratio
along the real parameter $s$, one at a time
for $s_2$, $s_3$, and $s_1$ as in \eqref{pair-pol}, checking at every step an
asymptotic polarization consistency in the process.
We can apply Zelikin's result verbatim for  $z$-curves ($z\in\bC$)
in $\Lambda$, and write (with respect to the 1-dimensional parameter $s$):
\begin{equation}\label{schwarzian}
\Schw_{\Lambda}(z) = (z')^{-1} z''' - \frac{3}{2} ((z')^{-1} z'')^2.
\end{equation}

\begin{remark}\label{schw-deriv}
The Schwarzian derivative,
in the classical case of scalar-valued functions,
 arises naturally (cf. \cite{ZelikinBook},
where it is derived from the Wronskian for a basis
of solutions of a third-order differential equation obtained
by writing the invariance of the cross-ratio under
linear-fractional transformation, differentiating both sides, and
eliminating the three parameters of $\mathrm{PSL}(2)$)
and is in fact the only projectively
invariant 1-cocycle on $\mathrm{Diff}
(\mathbb{R}\mathbb{P}^1)$ \cite{OT1}.
The significance of Zelikin's definition
 \cite{Zelikin} rests partly on the fact
that, for matrix-valued deformations $z(s)$,
he was able  to show that the Schwarzian operator preserves
the operator cross-ratio
(for extensions of linear fractional transformations/cross-ratio to the
operator-valued
setting, along with applications, see e.g. \cite[Ch. 3]{Helton}).
\end{remark}

We now apply properties of this Schwarzian to our setting and proceed to define:
\begin{equation}\label{hol3}
\Hol^{(3)}(D_0,\Lambda) = \{ f \in \Hol(D_0,\Lambda):
\text{values of the derivatives $f^{(n)}$ commute for $n \leq 3$},
f'(z) \neq 0\}.
\end{equation}

Taking  $f \in \Hol^{(3)}(D_0,\Lambda)$ with $z=f(w)$ as before, we next
consider
\begin{equation}
(S_{\Lambda}f)(w,t) = \cross(f(w+ ta), f(w + tb); f(w + tc), f(w+ td)),
\end{equation}
Following \cite{Ahlfors} (cf. \cite{OT1}), we deduce formally
from the vector-valued case that, to second order,
\begin{equation}\label{ahlfors-1}
(S_{\Lambda}f)(w,t) = \cross(a,b;c,d)(1 + \frac{1}{6}(a -b)(c -d)
 (\Schw_{\Lambda}f)(w) t^2 + o(t^2)),
\end{equation}
confirming the Schwarzian $\Schw_{\Lambda}f$ to be
the infinitesimal version of the cross-ratio
that does not change in first order (cf. \cite{Zelikin}).

\begin{remark}\label{proj-remark}
In direct analogy with the classical case \cite{Gunning}
(see also \S\ref{proj-struct})
we may regard those $f \in \Hol^{(3)}(D_0,\Lambda)$ for
which $S_{\Lambda}f = 0$ as \emph{projective transformations}
and in particular, a coordinate system for which
$S_{\Lambda}(z) = 0$ could be taken to define a \emph{projective
structure on $\Lambda$}.
\end{remark}

Let us now recall the function $\T_{\lambda}$ from Definition \ref{def-T-2}.
In view of \eqref{tauop1}, and on setting $\Delta \T_{\lambda}(t)
= (1 + \frac{1}{6}(a -b)(c -d) (\Schw_{\Lambda}f)(w) t^2 + o(t^2))$,
we directly deduce from \eqref{ahlfors-1}
 on applying the $\T_{\lambda}$-function to \eqref{cross-1},
the relationship
\begin{equation}\label{ahlfors-2}
\T_{\lambda}(\cH_2, \cH_1;\cH, \cH_3) ~\Delta \T_{\lambda}(t)
 = S_{\Lambda} (\cH_2, \cH_1; \cH, \cH_3)(t),
\end{equation}
where we have implicitly used the pull-back of
the cross-ratio under $\pi_{\Lambda}$ in \eqref{pull-back-1}
above, along with \eqref{T-relation-1}.
In view of the formulas in \eqref{curv-2-form},
 we next consider the infinitesimal deformation of the
curvature $2$-form of $\nabla (\Det_{\Lambda})$ under an element of
$\Hol^{(3)}(D_0,\Lambda)$:

\begin{proposition}\label{2-forms}
Let $f \in \Hol^{(3)}(D_0,\Lambda)$. Then we have the following `
asymptotic' relationship under $f$ between $2$-forms in
\eqref{curv-2-form}:
\begin{equation}\label{omega-1}
\Omega(\Det_{\Lambda})  - \Omega(\Det_{f(\Lambda)})
= \frac{1}{2\pi \iota} \del_{+}
\del_{-} \log \vert \det (\Delta \T_{\lambda}(t)^{-1}) \vert.
\end{equation}
\end{proposition}

\begin{proof}
 From \eqref{curv-2-form}, we have on applying `$\det$' to
\eqref{ahlfors-2}
the following:
\begin{equation}\label{ahlfors-3}
\begin{aligned}
\Omega(\Det_{\Lambda}) &= \frac{1}{2\pi \iota} \del_{+}
\del_{-} \log \vert \tau_{\lambda} \vert \\
&= \frac{1}{2\pi \iota} \del_{+}
\del_{-} \big[ \log \vert \det (\Delta \T_{\lambda}(t)^{-1}) \vert + \
\log \vert \det ((S_{\Lambda})f(w,t)) \vert \big].
\end{aligned}
\end{equation}
But by definition, $(S_{\Lambda})f(w,t) = f \circ \T_{\lambda}$, and so
 we have
\begin{equation}
\det((S_{\Lambda}f)(w,t)) = \det (f \circ \T_{\lambda}) = \tau_{f(\Lambda)}.
\end{equation}
Thus the right-hand side of \eqref{ahlfors-3} becomes
\begin{equation}
\frac{1}{2\pi \iota} \del_{+}
\del_{-} \big[ \log \vert \det (\Delta \T_{\lambda}(t)^{-1})
\vert + \log \vert  \tau_{f(\Lambda)} \vert  \big],
\end{equation}
from which the result follows in view of \eqref{curv-2-form} (observe
that essentially the same
applies to $\Omega(\Det_{\Gr})$ in \eqref{curv-2-form}).
\end{proof}

\begin{remark}
In view of his definition  of the operator Schwarzian derivative
 on one-dimensional submanifolds of the Grassmannian,
 Zelikin posits, in a speculative manner, that
the operator-valued KP deformations (whose Baker functions
satisfy Riccati's equation,
cf. also \cite{Segal}) might  be studied using the
Schwarzian (of
which he proved that
a quotient of solutions, which defines a projective
structure, also satisfies a Riccati equation); but, quote \cite[p. 51]{Zelikin}: ``Unfortunately,
the trajectories of [the Riccati] fields do not lie in
[the restricted Grassmannian], not even on arbitrarily small time intervals''.
Here we take a different route.  Raina \cite{Raina},
motivated by conformal-field theory, re-wrote Fay's trisecant
identity (the Riemann-surface version of the KP equation)
as a generalized cross-ratio; below (\S\ref{proj-struct}-\ref{k-dv})
we use that strategy, together with the operator
cross-ratio for the Riemann surface, to arrive at the operator KP equations.
\end{remark}

\subsection{Relationship with projective structures on $X$}\label{proj-struct}

We turn to the work in \cite{BR, Raina};
we apply Zelikin's Schwarzian operator (\S\ref{cross-ratio}),
extended to our setting, to the
description of the projective structure on a
compact Riemann surface $X$ given in \cite{BR}.
 We are ultimately  able (\S\ref{wick}) to  re-derive the KP hierarchy
with operator coefficients that was the motivation behind \cite{DGP2}.

Let $K_X$ denote the canonical line bundle of $X$. The space of all
projective structures on $X$
is an affine space for the complex
 vector space $H^0(X, K_X^2)$
of global holomorphic quadratic differentials, in which the classical
Schwarzian $\Schw_{X}$
determines a cocycle (see \cite[pp. 170--172]{Gunning}).

More explicitly, take a covering
$\{U_i\}_{i \in \mathcal I}$ for $X$ and let $\phi_i$ be a
holomorphic function on $\zeta_i\in U_i$, so that the transition
function $h_i$ for $H^0(X, K_X^2)$, satisfies (see e.g. \cite{BR,Gunning})
\begin{equation}\label{schw-1}
h_i = \Schw_{X}(\phi_i) \zeta_i.
\end{equation}
Observe that if $\phi'_i$ is another function
satisfying \eqref{schw-1}, then $\phi'_i(z_i) = \Xi \circ \phi_i(z_i)$
where $\Xi$ denotes a M\"obius transformation.

Referring back to \S\ref{cross-ratio},
as noted in \cite{Zelikin} (in the matrix-valued case),
the definition of cross-ratio and Schwarzian can be extended
to complex deformations in one parameter $s$.
In such a situation we denote the Schwarz operator by $\Schw_{\Lambda}(s)$.
We focus on the deformations taking place along the curve (the compact Riemann
surface) $X$ embedded in the $\Gr(p,A)$ by the Krichever map:
recall
\cite[Appendix \S A.4]{DGP2} that to define the Krichever map we need to
fix a local parameter at a point $\infty$, a
(generic) line bundle and a local trivialization; with these data,
we produce a point $W\in \Gr(p,A)$; the action of $\Gamma_1 = \Gamma_+(\A)$
on $W$ sweeps out $J_{\A}(X)$, so by the Abel map
we have (up to several choices) a holomorphic embedding
$X \hookrightarrow\Lambda$ (see Proposition \ref{proj-prop} below).
In analogy with the classical case \cite{Gunning}
(see also Remark \ref{proj-remark})
we  regard those $z$ for which $\Schw_{\Lambda}(z) = 0$ as projective
coordinates, thereby defining
a projective
structure on the embedded copy of $X$.

The following proposition summarizes certain analytic properties of $\Lambda$
(which like $\Gr(p,A)$ is modeled on a complex Banach space):
\begin{proposition}\label{lambda-analytic}
The space $\Lambda$ is an open and closed holomorphic (Banach)
submanifold of $P(A)$
which is a holomorphic (Banach) submanifold of $A$.
\end{proposition}
\begin{proof}
That $\Lambda$ is an open and closed holomorphic submanifold of $P(A)$
has been shown to be the case in \cite{DG2,DGP3}(cf. \cite{Raeburn}).
In fact, $\Lambda$ is  locally a holomorphic retract of $A$, as seen as
follows. For $x,y \in A$, we define $g(x,y) = xy + (1-x)(1-y)$,
noting that $g(p,p) =1$ and therefore invertible for all $x,y$
in some open subset $U$ of $A$ containing $p$.
We note then for $q \in \Lambda$, that $pg(p,q) = g(p,q)q$,
so for $q \in \Lambda \cap U$, we have $g(p,q)^{-1}pg(p,q) = q$.
Thus $r(x)= g(p,x)^{-1}pg(p,x)$ is a holomorphic retraction of $U$ onto
its overlap with $\Lambda$, on shrinking $U$ further if necessary.
This also shows that $\Lambda$ is a holomorphic submanifold of $A$.
\end{proof}
We summarize the above facts relating to holomorphic/projective structures
in the following proposition which
compares, via $\Schw_{\Lambda}$,
the complex structure induced from $\Lambda$
with the one intrinsic to $X$.

\begin{proposition}\label{proj-prop}
Let $\eta: X \hookrightarrow \Lambda$ be a holomorphic
embedding with respect to the
natural (complex) analytic structure of $\Lambda$. Then
the Schwarzian operator intrinsic to $X$ and the one induced from $\Lambda$,
with respect to the holomorphic
deformation along the embedded curve, correspond.
\end{proposition}

Returning to the cross-ratio class $\{\cross\}
\in H^1(\Lambda, \End(\gamma_{\Lambda}))$ in \S\ref{cross-ratio}
leading to the Schwarzian $\Schw_{\Lambda}$, we see that for functions
$f,h \in \Hol^{(3)}(D_0,\Lambda)$, it is formally
deduced from the classical case that
\begin{equation}
\Schw_{\Lambda}(h \circ f) = (\Schw_{\Lambda}(h)
\circ f)(f')^2 + \Schw_{\Lambda}(f),
\end{equation}
(see e.g. \cite{Ahlfors,Osgood,OT1}). Here the first right-hand summand is the
action of $f$ on a quadratic differential
\begin{equation}
(u \circ f)(z) = u(f(z) \cdot f'(z))^2,
\end{equation}
in terms of the $z$-coordinate above, in turn leading to a transition
 function for the vector bundle
$K^2_{\Lambda}$ in an analogous way to the classical situation.
 We thus have arrived at the novel concept of an \emph{operator-valued
projective structure} induced on a
one-dimensional complex submanifold from a space such as $\Lambda$.

\begin{remark}
It would be interesting to apply the
operator Schwarzian derivative
to compare the deformations
of $X$ inside $\Lambda$ with the deformations of $X$ parametrized by
$H^0(X,K^2_X)$, especially since the former should be
unobstructed (Krichever's map can be applied to any Riemann surface,
and locally it should be possible to make consistent
choices of a defining quintuple, cf. \cite{SW}).
However, the tangent bundle to $\Lambda$,
whose first cohomology gives the deformations, is
of infinite rank, and in order to define a canonical line bundle
over $\Lambda$ requires different techniques (cf \cite{BG}), as
 $\Lambda$ is an infinite-dimensional
Banach manifold which is a holomorphic submanifold of $A$, thus in
general, since we are in infinite dimensions, the top
exterior power is not formed in the usual way. However, in our
specific situation, where  $A$ is the restricted algebra, we have a group
transforming the restricted frames in $V(p,A)$
(``admissible bases'' in \cite[\S3]{SW})
on which the determinant is defined, and a central extension
of it ($\mathcal{E}$ in \cite[\S3]{SW}) where
the function $g$  giving the retraction
in Proposition \ref{lambda-analytic} takes values, so
that coordinate transformations in effect have derivatives which have
values in that central extension
($\mathcal{E}$) as well.
With this in hand,  we would need to restrict the
line bundle thus obtained to $X$
and see if each projective structure of $X$ can
be extended away from it, not only along the one-dimensional
deformations controlled by the Schwarzian derivative.
\end{remark}

\subsection{The Riemann theta function and Wick's theorem}\label{wick}

Our next main observation concerns (generalized) projective structures on
$X$  using a `correlation function'
approach motivated by {Wick's theorem} and a (generalized) cross-ratio
(see \cite{BR,Raina}). The point of this subsection is to
implement Raina's rendition of Fay's trisecant identity in terms
of the cross-ratio we developed
(\S\ref{cross-ratio}), and obtain the KP hierarchy in
\S\ref{wick} below, in our extension-group model.
Here we utilize
the Burchnall-Chaundy C*-algebra $\uA$ and the extension
group $\Ext(\uA)$ from \cite[\S4]{DGP2}.

\begin{theorem}\label{main-2}

For genus $g_X \geq 2$,
the action of the group $\Gamma_1 =\Gamma_{+}(\A)$ on $\Ext(\uA)$
corresponds to translating the theta function of $X$ on the Jacobian.
\end{theorem}

\begin{proof}
Let $\mathfrak{L}_{\theta} \lra J_{\A}(X)$ be the holomorphic
 line bundle whose sections are theta
functions $\theta[\xi](z)$ of characteristic $\xi$ (see e.g. \cite{Fay,GH})
taken as $\bC \otimes 1_{\A}$-valued.
Similar to before, consider a $\Lambda$-parametrization of the
surjective homomorphism of
\cite[A(19)]{DGP2}, giving a commutative diagram
\begin{equation}
\begin{CD}
\Gamma_1 \times \Lambda @> \tau_{\lambda}
>> \bC \otimes 1_{\A}
\\ @VVV   @VVV\\
J_{\A}(X) \times \Lambda @>
\theta_{\lambda}[\xi]
>> \bC \otimes 1_{\A}
\end{CD}
\end{equation}
The commutativity of this diagram reveals
that $\tau_{\lambda}$ is `proportional' to
$\theta_{\lambda}[\xi]$ following the classical case
 (cf. \cite{Raina,SW}), and hence from
\cite[Theorem 4.5]{DGP2}, $\Ext(\uA)$ parametrizes a family
of (translations of the)
theta function(s) via extensions by the compact operators.

Next, consider an (operator)-valued spinor
field $\Vec{\psi}$ on $X$,
and points $a,b \in X$ suitably chosen to lie in the same coordinate patch.
In \cite{Raina} it is shown that the Fay trisecant identity
(see \eqref{tris-1}) is equivalent to \emph{Wick's theorem},
 which in terms of a left-hand-side `correlation function' below
(see \cite{BR,Raina}),
 can be expressed in the form
\begin{equation}\label{tris2}
\langle \Vec{\psi}^*(b_1) \Vec{\psi}(a_1) \Vec{\psi}^*(b_2)
\Vec{\psi}(a_2) \rangle =
\det \bmatrix S_{\a}(b_1, a_1) & S_{\a}(b_1, a_2) \\ S_{\a}(b_2, a_1) &
S_{\a}(b_2, a_2)
\endbmatrix
\end{equation}
where $S_{\a}$ is the Szeg\"{o} kernel of a theta prime-form \cite{Fay}.
Moreover, in view of \cite[\S5]{BR} the `correlation function' of
\eqref{tris2}
defines a projective connection and hence a projective structure
\cite{Gunning}).
\end{proof}

\begin{remark}
In terms of the Cauchy kernels of flat vector bundles of arbitrary
rank on $X$ (for $g_X \geq 1$), explicit
formulas in \cite{Ball3} interpolate homomorphisms of these bundles
 leading to generalizations of the Fay
trisecant identity. We note that flat rank-2 bundles over $X$
(for $g_X \geq 2$) have been related \cite{BR}
to quadratic differentials (in both cases
the moduli spaces have the same dimension, $3g-3$).
It is possible this may lead to further
applications of operator
 projective structures on $X$
as well.
\end{remark}

\subsection{The KP tau-function and trisecant identity}\label{k-dv}

We briefly recall the concept of the KP-hierarchy starting
 from \cite[Appendix \S A4]{DGP2}.
Consider a formal pseudodifferential operator of the form
\begin{equation}
L= \del + a_0\del^{-1} + a_1\del^{-2} + \cdots,
\end{equation}
where as in \cite[\S A4]{DGP2} we take the
coefficients $a_i = a_i(t_1, t_2, \ldots)$
 to be $\A$-valued functions.
Next, let us set $P^{(k)} = (L^k)_{+}$. Then for each $k \in \mathbb{N}$,
 the \emph{KP hierarchy} (see e.g. \cite{Krich1,Sato,Segal,SW}) consists
of partial differential equations of the type
\begin{equation}
\frac{\del L}{\del t_k} = [P^{(k)}, L],
\end{equation}
according to which
\begin{equation}
\frac{\del}{\del t_k}P^{(\ell)} - \frac{\del}{\del t_{\ell}}P^{(k)} +
 [P^{(\ell)}, P^{(k)}] = 0.
\end{equation}
Since we are using $\A$-valued coefficients, we choose to denote this
hierarchy by $\KP(\A)$. Once more we apply the extension
group $\Ext(\uA)$ and establish the following for the
$\KP(\A)$ flows with respect to the $\Gamma_1$-action:

\begin{theorem}\label{main-1}
The $\KP(\A)$ flows evolve on
the group $\Ext(\uA)$ via extensions by compact operators.
\end{theorem}

\begin{proof}
Recalling the $z$-coordinate of \S\ref{cross-ratio}, we set
 $[z] = (z, z^2/2, z^3/3, \ldots)$.
Then from the operator-valued  $\tau$-function $\tau_{\lambda}:
\Gamma_1 \lra \bC \otimes 1_{\A}$
in \S\ref{poincare-1}
(cf. \cite{DGP2}), it follows that
(cf. \cite{SW,Sato})
\begin{equation}\label{kdv-tau}
\tau_{\lambda}(t + [z]) = \tau_{\lambda}(t_1 + z, t_2 + z^2/2, t_3 + z^3/3,
\ldots),
\end{equation}
that satisfies $\tau_{\lambda}(t + [-z]) = \tau_{\lambda}(t - [z])$.
Further, on setting $C(z_i, z_j, z_k, z_{\ell}) =
(z_i - z_j)(z_k - z_{\ell})$ in $\A$-valued variables, for
$0 \leq i,j,k, \ell \leq 3$, we formally deduce from references
\cite{Fay,Miwa} the
\emph{Fay trisecant identity}:
\begin{equation}\label{tris-1}
\begin{matrix}
C(z_0, z_1, z_2, z_3) \tau_{\lambda}(t + [z_0] +[z_1]) \tau_{\lambda}
 (t + [z_2] +[z_3]) +\\
C(z_0, z_2, z_3, z_1) \tau_{\lambda} (t + [z_0] +[z_2]) \tau_{\lambda}
 (t + [z_3] +[z_1]) + \\
C(z_0, z_3, z_1, z_2) \tau_{\lambda}(t + [z_0] +[z_3]) \tau_{\lambda}
(t + [z_1] +[z_2]) = &0.
\end{matrix}
\end{equation}
from the analogous identity for $\tau_W$ and then using \eqref{tau-lambda}.

 From \cite[Theorem 4.5]{DGP2}, elements of $\Ext(\uA)$, extensions
by the Burchnall-Chaundy C*-algebra $\uA$ of the compact operators,
lead to the map
\begin{equation}
\Upsilon^{-1}_{\A}: \Ext(\uA) \lra \Gamma_1
\end{equation}
(the inverse of the map $\Upsilon_{\A}$ as in the proof of \cite[Theorem 4.5,
 (4.20)]{DGP2}),
which yields a family of $\tau$-functions, one for each such extension.
Hence, in each case a corresponding trisecant identity as in \eqref{tris-1}
follows. But \eqref{tris-1}
has been shown to be equivalent to the KP-hierarchy (see e.g. \cite{Miwa}).
Hence we conclude that
the KP-hierarchy, implemented by the $\Gamma_1$-action,
flows on these extensions by compact operators as derived from $\Ext(\uA)$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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GPS-enhanced tsunami warning system
Briefing by: AccessScience Editors
Last reviewed:2016
DOI:https://doi.org/10.1036/1097-8542.BR0307161
A regional or local tsunami warning system that uses both seismic and GPS stations could broadcast more prompt and accurate tsunami warnings and could potentially reduce casualties in the event of a major earthquake, according to a report in Geophysical Research Letters (February 2016). The warning system would not require any new instrumentation, as it would use instruments already deployed. See also: Earthquake; Satellite navigation systems; Seismic risk; Tsunami
GPS station in the San Francisco Bay area, California. (USGS)
At present, tsunami warning systems rely on seismic data to determine the location, depth, and magnitude of an underwater earthquake and then use this data to calculate the potential tsunami hazard and arrival time, without knowing for certain whether a tsunami will occur. Following the detection of an earthquake, it can take up to 20 minutes to issue a tsunami warning. This lag is particularly a problem when the epicenter is close to the coastline. Seismic methods have been fairly accurate in providing tsunami warnings for M8 and lower-magnitude earthquakes but less so for earthquakes greater than M8, such as the 2011 Tōhoku, Japan M9 earthquake, in which a fault 500 km long and 200 km wide ruptured over 3 minutes and a major tsunami was generated that reached the coast in less than 20 minutes. See also: Earthquake early warning; Fault and fault structures; Seismographic instrumentation; Seismology
In an underwater earthquake, the vertical movement of the seafloor at a fault pushes up a large volume of water, producing a series of waves that travel 970 km/h or faster. GPS measures fault motion by observing from satellites the positions of ground stations. In an earthquake, the seafloor deformation is calculated by determining how the stations have moved. Using GPS, tsunami warning maps could be generated within 2 minutes after the seismic detection of an earthquake and the estimated height of the tsunami waves could be calculated in less than 2 minutes, at which point a warning would be issued. See also: Geodesy
Links to Primary Literature
A. Hoechner et al., Instant tsunami early warning based on real-time GPS – Tohoku 2011 case study, Nat. Hazards Earth Syst. Sci., 13: 1285–1292, 2013 DOI: https://doi.org/10.5194/nhess-13-1285-2013
D. Melgar et al., Local tsunami warnings: Perspectives from recent large events, Geophys. Res. Lett., 43, 2016 DOI: https://doi.org/10.1002/2015GL067100
Cascadia EarthScope Earthquake & Tsunami Education Program: GPS as an Essential Component of Cascadia Earthquake Early Warning [VIDEO]
IRIS/NSF: Tsunamis Generated by Megathrust Earthquakes [VIDEO]
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Clayton, CA – Contra Costa County Public Works will conduct slide removal and shoulder repair activities on Marsh Creek Road, between Deer Valley Road and the Clayton City limit.
The repair work will begin today and last through Monday, February 25th from 9am to 4pm each day, weather permitting. Message boards will be placed at Deer Valley Road and in Clayton, advising motorists of work ahead. Expect delays during the repair work.


text:
Harry 'Demon' Fairmack
SAMCRO Soldier
0 · 178 views · located in Charming
a character in "MRPP Sons Of Anarchy", as played by anxiousspawncow
SAMCRO
The Sons Of Anarchy Motorcycle Club Redwood Original (SAMCRO)
Harry's Theme:
http://www.youtube.com/watch?v=_zgSUJ9wwWo&feature=share&list=RDGB2oNYc9hCU
Personal Description:
Harry is a large and strong man, standing at a height six ft ten in, with a weight of three hundred pounds of solid muscle. He is a walking mountain of strength. It is common for him to be seen at the gym, maintaining his muscles. His eyes are a vivid green. Count yourself lucky if you ever seen them, because he usually wears shades, but when they come off it's often not good.
Along with his natural qualities, Harry has a lot of tattoos on his large arms and the back of his neck. The theme on his left arm sleeve is of a mystical nature which consists of wizards, castles, and symbols while his right arm sleeve is more of a death and demonic theme. On the back of his neck is a dancing skeleton.
Harry had to grow up before his time when his family were murdered. Beneath his hard protective shell beats the heart of a man who is fully capable of love and caring. He usually only reveals how much he truly cares when he sees other people are in danger. It's not uncommon to see Harry rush into a conflict to defend his club brothers or a club lady if they are threatened, even if that means putting his life in danger. He is extremely loyal to the Sons to the point he will kill or die for them. He enjoys violence, but he tries to keep control of that more dark side of him, especially around the club. If someone messes with his brothers or angers him enough he will happily familiarize them with the 'Demon'. He has quite a temper which unleashed could be quite dangerous, mostly because he allows a more primal side to his nature to take full control of his body. So angering Harry is a lot like poking a stick into a hornet's nest. He will hurt you and usually doesn't feel any remorse for it either, in fact, he might even receive pleasure from it.
People will be shocked to find out that there is another side to the tough biker, that not many people get to see very often. He has a strong dislike for people who think it's alright to hurt innocent children or women. Paedophiles, child-beaters and woman beaters are definitely on his shit list and if he catches them, he will not have a problem killing them. Prospects are usually pretty close to his heart as well because of their age or inexperience in the club. If he sees anyone hurting a child, woman, or prospect it is a trigger that will set him off on a violent path.
Harry was born in Houston, Texas. At the age of seven he witnessed property developers murder his mother, father, and little sister. They were murdered as they refused to sell their house to the property developer. Harry hid when the men arrived, but watched the entire event take place. This image made him the had and brutal killer he is today.
He was then raised by his aunt who paid for him to attend college. He hated college, too many goody goody types around the place. By this time he was already a big guy, but didn't want to play football. He was finally expelled from college when he witnessed the captain of the football team hitting his girl (Captains girl). Harry went mental and put the guy in the hospital, he never played football again and walked with a limp for the rest of his life.
Out of college Harry hated seeing the sorrow in his aunts eyes so he decided to leave Houston. He moved to Arizona looking for work. When he couldn't find any he decided to spend his last night in Arizona at his favourite bar. He was sitting drinking, minding his own business when two Mayans entered the bar and started to beat the Sons of Anarchy SAMREN president. Harry watched, but when the mans ol' lady got punched in the face that was enough, no more watching. He approached the Mayans knocking the first one out with one punch. The second pulled a gun on Harry, which was quickly grabbed away by the big man. Harry beat the Mayan to death in the middle of the bar with his own gun, before making sure the presidents ol' lady was ok. The president told him to attend the SOA clubhouse the following morning, where he was made a prospect.
He did his year and patched in. A few years went by and a gash he used to fuck arrived on the scene with a kid. DNA proved it was Harry's baby. The gash overdosed leaving Harry to raise his daughter. He sheltered her from the MC life, as he believes women have no place in it.
Over the years Harry had killed many Mayans, but this caught up with him tragically. His daughter was with her friends one days when the Mayans took the girls. The three of them were brutally raped, but Harry's daughter was also killed. The girls were dumped at the SOA clubhouse. In his grief and anger Harry went after the Mayans. He entered the clubhouse and killed seven members, including the president and Vice-president. This started a bloody nationwide war between the Mayans and the Sons of Anarchy. This lead to Harry's hatred of Mexicans.
The years went by, truces were made, which Harry saw as bullshit, but he was loyal to SOA so agreed. He Eventually became the VP of the Reno charter (SAMREN). After a few more years he decided it was time for a change. SAMCRO had been taking a beating so he requested a transfer, which was approved by both charters........
So begins...
Harry 'Demon' Fairmack's Story
Seen 0 Times • Page 1 of 1


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You are here: Home / Featured / 6 new cases of COVID-19 are announced in Nova Scotia on Thursday, Jan. 14
6 new cases of COVID-19 are announced in Nova Scotia on Thursday, Jan. 14
January 14, 2021 By Tim Bousquet Leave a Comment
The Halifax Examiner is providing all COVID-19 coverage for free. Please help us continue this coverage by subscribing.
Six new cases of COVID-19 are announced in Nova Scotia today (Thursday, Jan. 14).
Three cases are in Nova Scotia Health's Northern Zone and are close contacts of previously reported cases.
Three cases are in the Central Zone and both are students who travelled in from outside of Atlantic Canada — one is a close contact of a previously reported case, and the other two are related to travel outside of Atlantic Canada, one of whom is a student at Dalhousie who lives on campus
There are 32 known active cases in the province. No one is in hospital with the disease.
Nova Scotia Health labs conducted 2,096 tests yesterday.
Here are the new daily cases and seven-day rolling average since the start of the second wave (Oct. 1):
And here is the active caseload for the second wave:
Yesterday, Public Health issued potential COVID exposure advisories for the Truro Horsemen's Club and for Air Canada flight 614, as follows:
Out of an abundance of caution and given the current testing capacity available, anyone who worked or visited the following locations on the specified dates and times should immediately self-isolate and visit covid-self-assessment.novascotia.ca/ to book a COVID-19 test, regardless of whether or not they have COVID-19 symptoms. You can also call 811 if you don't have online access or if you have other symptoms that concern you.
Anyone present at the below location must self-isolate while waiting for their test result, regardless of whether or not they have COVID-19 symptoms.
Truro Horsemen's Club (288 Main St, Truro) on Jan. 10 between 7:30 a.m. and 1 p.m. It is anticipated that anyone exposed to the virus at this location on the named date may develop symptoms up to, and including, Jan. 24.
Anyone who was on the following flight in the specified rows and seats is asked to continue to self-isolate and immediately visit https://covid-self-assessment.novascotia.ca/en to book a COVID-19 test, regardless of whether or not they have COVID-19 symptoms. You can also call 811 if you don't have online access or if you have other symptoms that concern you.
Air Canada flight 614 travelling on Jan. 7 from Toronto (2:18 p.m.) to Halifax (5:22 p.m.). Passengers in rows 1- 4 seats A, C, and D are asked to continue to self-isolate and immediately visit https://covid-self-assessment.novascotia.ca/en to book a COVID-19 test, regardless of whether or not they have COVID-19 symptoms. All other passengers on this flight should continue to self-isolate as required and monitor for signs and symptoms of COVID-19. It is anticipated that anyone exposed to the virus on this flight on the named date may develop symptoms up to, and including, Jan. 21.
Here is the possible exposure map:
The Halifax Examiner is an advertising-free, subscriber-supported news site. Your subscription makes this work possible; please subscribe.
Some people have asked that we additionally allow for one-time donations from readers, so we've created that opportunity, via the PayPal button below. We also accept e-transfers, cheques, and donations with your credit card; please contact iris "at" halifaxexaminer "dot" ca for details.
Filed Under: Featured, News
About Tim Bousquet
Tim Bousquet is the editor and publisher of the Halifax Examiner. email: [email protected]; Twitter


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PGI INSIGHT: Somalia/Yemen - Volatile onshore security conditions, troop withdrawal to fuel piracy
PGI INSIGHT: Somalia/Yemen – Volatile onshore security conditions, troop withdrawal to fuel piracy
The increased threat of piracy off Somalia and Yemen since the Aris 13 hijacking in March is set to continue due to heightened insecurity in the High Risk Area (HRA).
Pirates have expanded their range of operation, frequency of attacks and acquired more powerful weapons throughout 2017, indicative of the evolving threat in the region.
Pirate groups will likely continue to exploit insecurity in Somalia and the reduced international naval presence in the HRA for the foreseeable future, increasing the risk to international shipping in the region.
Reports of pirate activity in the HRA region have increased by more than five times over the past 12 months, totalling 65 incidents, compared to the same period in 2015-2016. Reports of attempted hijackings sharply increased in the weeks after the hijacking of the Aris 13 bunkering tanker in March and the rate of attempted attacks and suspicious activity has since remained elevated. Most piracy incidents have occurred in the Gulf of Aden and the Bab el Mandeb strait where vessels are most concentrated. However, incidents have been reported hundreds of nautical miles off the Somali coast. Although a commercial vessel has not been successfully hijacked since April, the situation remains volatile.
Evolving threat
Both the territorial range and frequency of attacks have increased since the Aris 13 hijacking, marking a departure from the typical pattern of pirate activity since 2014 when the piracy epidemic ended. The change in modus operandi indicates that pirates are increasing their capabilities as part of renewed efforts to successfully hijack large commercial vessels. Attacks have been reported further out at sea where vessels are more vulnerable, as seen in November, when an attack was recorded 380 nm off the coast of central Somalia. Since May, there have been four reports of piracy-related activity in the Gulf of Oman, an area which has largely been unaffected by piracy since mid-2015. Three of these incidents saw shots fired at commercial vessels, although PGI sources indicate that these may be linked to weapons' smuggling in the region rather than Somali-based pirate groups.
There are also indications that Somali pirates are obtaining more powerful weapons, raising security concerns for shippers. In another incident in November in the southern Somali Basin, pirates used rocket-propelled grenades (RPGs) to attack a container ship in an apparent attempt to hijack the vessel. The incident was the first reported piracy-related RPG attack against a commercial vessel off the Somali coast since 2013. Although there have been a series of RPG attacks in the region against commercial vessels since October 2016, these have been confined to the Bab el Mandeb strait and have been linked to the Yemen-based Houthi militant group rather than pirates.
Drivers of piracy
The growing attacks against commercial vessels in recent months reflect several factors. The initial spike in activity in the weeks following the hijacking of the Aris 13 was likely because the incident alerted pirates to the fact that many vessels were flouting the recommended Best Management Practices (BMP4) guidelines by travelling closer to the Somali coast and not deploying adequate protective measures. Despite the failure to successfully hijack another commercial vessel, reports of suspicious activity and attempted attacks have continued at higher levels compared to the same period in 2016. This trend is indicative of the more permissive security environment for pirate groups in 2017, following the end of NATO's counter-piracy Ocean Shield operation in December 2016, and demonstrates that piracy groups did not disperse after the epidemic.
While BMP4 measures and an international naval presence have served to supress piracy in the region in recent years, the rise in attacks is also symptomatic of the continuation of the underlying causes of Somali piracy. Persistent insecurity along the coastline, the prevalence of illegal fishing, and ongoing conflict in Somalia have deprived many people of adequate economic opportunities. Encouraged by these underlying drivers, pirate groups have remained intact since 2012 and engaged in other illicit activities. The group that conducted the Aris 13 hijacking, for example, has been linked to pirate kingpins Jacfar Saciid Cabdulaahi and Abdinasir Aw-Kombe, who were highly active during the peak of the piracy epidemic.
Onshore security conditions in Somalia remain volatile and are likely to deteriorate in the coming months, as international military forces begin to pull out of the country, leaving a security vacuum, which pirates and other armed groups will seek to exploit. The African Union Mission in Somalia (AMISOM), which is fighting al-Shabaab, began pulling troops out of Somalia in November as part of plans to fully withdraw its 21,000-force by the end of 2020. Although the decade-long counter-insurgency campaign has contained al-Shabaab, the group has maintained the capabilities to contest territory and carry out mass-casualty attacks.
Somali security forces, meanwhile, lack the resources and ability to tackle the country's security challenges independently. Al-Shabaab's resilience indicates that the AMISOM withdrawal could see the security situation deteriorate in Somalia, which will in turn deprive communities of alternative economic opportunities beyond criminality. The drawdown will also further enable the flow of weapons to criminal groups in Somalia, and thus lower the entry costs into piracy.


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Monsenyor Jaume Camprodon i Rovira (Torelló, 18 de desembre del 1926 - Girona, 26 de desembre de 2016) fou bisbe de Girona entre 1973 i 2001. Va estudiar al seminari de Vic a partir de 1940 i el 1949 fou ordenat prevere. Fins al 1953 va estar en diverses parròquies del bisbat de Vic, quan en aquest any va entrar com a director de l'internat de Sant Miquel dels Sants de Vic i el 1955 va començar a fer de professor al seminari. El 1963 es trasllada a París per seguir estudis de teologia, però sense acabar aquest curs el bisbe de Vic el crida per fer-se càrrec de diferents parròquies i treballar, com havia fet abans, en la relació entre església, obrers i laics. A partir de 1966 fou el rector de la parròquia de Crist Rei de Manresa i des d'allà va treballar per situar aquesta parròquia en la línia del Concili Vaticà II.

El 2 de setembre de 1973 fou nomenat bisbe de Girona i fou consagrat bisbe de Girona el 21 d'octubre del mateix 1973, prenent possessió de la Diòcesi el mateix dia, en substitució de Narcís Jubany, que va passar a l'arquebisbat de Barcelona. En aquest càrrec s'hi va mantenir fins a l'acceptació de la seva dimissió el 30 d'octubre del 2001 i passant a la jubilació el 16 de desembre de 2001, quan fou substituït per Carles Soler i Perdigó. Participà en la redacció del document Arrels cristianes de Catalunya.

Referències

Vegeu també
 Pàgina del Bisbat de Girona

Bisbes de Girona
Torellonencs
Religiosos catalans del sud contemporanis
Morts a Girona
Religiosos osonencs

redpajama_set_name: RedPajamaWikipedia


New London-based start-up Twisted Lingerie has announced the launch of its first collection. Named Xenia, the debut range is available exclusively online. Inspired by the fetish lifestyle, the brand aims to realise its vision of putting the possibility of high quality and well-priced garments out there – the price range starts at £17.


text:
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redpajama_set_name: RedPajamaGithub


TAIPEI ( The China Post) – Fashion icon Karl Lagerfeld has died aged 85, according to media sources. Hitcounter Contact E- mail does not support the hacking of PAY TV. Advice tips inside secrets revealed on how you can get a job in F1.
The FIA has confirmed the details of its new Formula One superlicence points system in an update to the International Sporting Code. 時尚「 老佛爺」 卡爾拉格斐今( 19日) 晚間傳出辭世, 享年85歲。.
Which is the biggest and most Popular sport in the world? Check the rules changing every year by clicking following links.
From in order to be eligible to compete in F1 drivers must accumulate at least 40 points over a three- year period which are allocated based on which championships they have participated in how well they have performed. Update China Visa Rules And Policy in Shanghai. To answer this question we used criteria of 13 factors global base tv viewership numbers money. Highest paid f1 driver 2014.
VisaInChina starts this China visa news center since year. 4 in our list of top 10 highest paying sports in the world.
Formula One ( also Formula 1 or F1) is the highest class of single- seater auto racing sanctioned by the Fédération Internationale de l' Automobile ( FIA) and owned by the Formula One Group. The FIA Formula One World Championship has been one of the premier forms of racing around the world since its inaugural season in 1950. This is a list of the largest sports contracts.
These figures include signing bonuses but exclude options, buyouts, and endorsement deals. This list does not reflect the highest annual salaries or career earnings, only the top 100 largest contracts and thus is limited to athletes in team sports and auto racing. Athletes in individual sports, such as golf, tennis, and boxing are not employed by a.
Subscribe now and save, give a gift subscription or get help with an existing subscription. Many years before Tiger Woods, it was Jack Nicklaus, the greatest golfer of all time secured the 3rd place in 1994 Forbes list with $ 14.
8 million and Arnold Palmer achieved the 4th place the same year with a total income of $ 13. Golf is a gentlemen' s game and it' s a richest game too.
We put Golf at No.


text:
About Kevin Donoghue
About Daniel Fitzsimmons
About Kemmi Alfa
About Jack Hudson
Actions Against The Police
The Law in Civil Actions Against the Police
False Imprisonment Claims Explained
Remedies in Claims Against the Police
How to Calculate Police Abuse Compensation
Time Limits in Actions Against the Police
Funding Options and No Win No Fee Police Claims
Information Police Misconduct Solicitors Need to Help You
Complaints Against the Police
Actions Against the Police FAQs
Client Referrals from Other Solicitors
Police Abuse of Authority for Sexual Gain Compensation Claims
Police Warrant Claims
Maps & Contact Details of Police Forces in England and Wales
Data Protection Claims
Accident Claim Types
Tripping Accidents
Slipping Accidents
Criminal Injuries Compensation Claims
Other Accident Claims
Things To Claim
Professional Negligence Claim Funding Options
What is Professional Negligence?
Professional Negligence FAQs
Three New Year's resolutions for a claim against the police
Kevin Donoghue, Solicitor Director of Donoghue Solicitors
By Kevin Donoghue, Solicitor Director of Donoghue Solicitors
As a solicitor who specialises in helping people claim against the police, I spend a lot of my time dealing with the various police forces in England and Wales.
I am sorry to say that I could use some of my time more productively if those same forces and their solicitors behaved differently.
It is common practice for the police and their solicitors to:
delay progress on cases;
misplace, edit, or redact evidence which may help prove the claim against the police; and
fight losing cases or make 'low-ball' offers.
All these:
cause unnecessary hardship to innocent victims of police misconduct;
increase legal costs (which have to be paid by the taxpayer); and
only serve to damage the reputation of the police in the eyes of the judge and jury who eventually get to hear the claim against the police.
So, with these things in mind, here are three New Year's resolutions for the police forces and their solicitors:
1.Think of the police force's money as your own
I am constantly frustrated by how police forces and their solicitors behave as if money is no object.
In my experience, filing unnecessary applications at court and fighting claims to trial is their preferred approach.
Even in a straightforward claim against the police, the solicitors representing the police forces behave as if they have winnable cases when they know that experienced solicitors like me who deal with no win no fee police claims would not take poor cases.
For example, in the case of Mr. M (read the case report here), the Chief Constable and his solicitors fought this claim against the police until the eve of the trial even though a complaint against the police force had been upheld by the Independent Police Complaints Commission.
By doing so, they wasted valuable funds which could have more properly been spent fighting crime.
2. Treat people making a claim against the police as you would want to be treated
I often come across cases of police misconduct which could have been avoided.
Some clients tell me that they are only making a claim against the police because they feel mistreated, and that they would not be suing the police if the police officers had been more civil with them.
Again, referring to Mr. M, he was (unlawfully) arrested and processed at a police station. The police took £325 from his wallet and returned the empty wallet to him when he was released.
He went to the station to demand his money. A police officer threw the money at him over the desk and asked 'can't you take a joke?'.
Mr. M was rightly outraged at the police officers' conduct, complained, claimed against the police, and won compensation.
But the conduct of the officers involved was made worse by the actions of the police force and its solicitors. They denied liability throughout and refused to make any offers to settle his claim, adding to the stress he already felt about the police's misconduct and having to claim against the police.
Although Mr. M did not have to suffer through a trial, the fact that the police force and its solicitors settled the claim the day before, when no new evidence had been produced, shows that they could have dealt with it earlier and saved him a lot of unnecessary stress.
I wonder how the police force's solicitors would have felt if they, or one of their families, were treated like this?
3. Respect your opponent in the claim against the police
It is not easy to be a solicitor who specialises in helping people claim against the police. You can only become one after years of training, both at law school and 'on the job'. It is hard work that has to be a passion. This type of work does not pay as well as others.
There are no 'fat cats' here.
Recent Government policy has made it harder for solicitors who help people claim against the police. (I blogged here about last year's changes to the funding of personal injury claims, which often include police claims.)
So only specialist solicitors who see a good claim against the police at an early stage can continue to practice in this area of law.
As a result, only the best claims against the police are pursued by expert solicitors.
Police forces and their solicitors know this but I still see examples of their poor conduct more worthy of a 17-year-old insurance clerk than an experienced solicitor.
For example, my clients H and A (read the case report here) were assaulted in their home as a result of a botched search warrant raid.
Not unsurprisingly, the police force's solicitors denied liability, claiming that the officers acted lawfully.
They also claimed that neither client was injured (despite seeing photographs as proof) but made a 'without prejudice' offer to settle of £4,000 which was to be shared between H and A.
As an experienced solicitor who helps people claim against the police on a daily basis, I was confident that the offer was insulting to my clients and that they were telling the truth about their injuries.
I obtained medical evidence, issued proceedings, and eventually settled their claims for £18,000, four times more than the original offer to H, and five times more than the original offer to A.
The police also had to pay full legal costs, which were considerably higher as I had no choice but to issue proceedings due to their denial of liability and ridiculous low offer.
The police force involved and their solicitors clearly thought they could get away with this, and they may have succeeded if they were dealing with someone who does not deal with claims against the police every day. Respecting me, their opponent, would have saved the police a lot of money and time.
New Year's Resolution for Police Force Solicitors
I urge any police forces and their solicitors who read my blog to make a New Year's resolution to learn from these examples.
My clients are innocent victims of police misconduct. By the time they come to me for help with their claim against the police they have suffered enough.
Don't make it worse for them by dragging cases out unnecessarily and making no offers, or insulting ones.
How you would feel if you were the person making the claim against the police?
Wouldn't you like to restore some goodwill by treating them with respect?
If thinking about the innocent victim doesn't get you to change, what about the wasted money that could be spent fighting crime?
Whatever the reason, make 2014 the year you change the way you deal with claims against the police.
Kevin Donoghue is the Solicitor Director of Donoghue Solicitors, a specialist law firm dedicated to helping innocent people claim against the police.
Contact him on 0151 933 1474 or via his firm's website https://www.donoghue-solicitors.co.uk/actions-against-the-police/.
filed under: actions against the police
Tags: actions against the police solicitor, claim against the police, compensation claims against the police, donoghue solicitors, kevin donoghue, new years resolutions, solicitors, suing the police
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