text:
\section*{Introduction}
The study of the relation between the combinatorial/analytic torsion  of
a flat vector bundle and the Morse-Smale flow is initiated by Fried \cite{Friedconj} and
S\'anchez-Morgado \cite{MorgadoMorseSmale}.
In this paper, we give a formula relating
\begin{itemize}
	\item  a spectral invariant: the Ray-Singer metric associated
	to  a  flat  vector bundle with a Hermitian metric on a closed Riemannian
	manifold;
	\item  a dynamical  invariant: the Milnor metric which reflects
	the interactions 	between the fixed points and the
	closed orbits of the Morse-Smale flow, and  generalises the
	absolute value at zero point of the Ruelle dynamical zeta function;

	\item a transgressed Euler class:

the Mathai-Quillen current.
\end{itemize}

 \subsection{Background}
 Let $X$ be a connected closed   smooth   manifold. Let $F$ be a complex
 flat vector bundle of rank $r$ on $X$, which amounts to specifying a
representation
$\rho: \pi_1(X)\rightarrow \GL_r(\mathbf{C})$ of the
fundamental group $\pi_1(X)$.
Denote by $H^\cdot(X,F)$ the cohomology of
the sheaf of locally constant sections of $F$, and by $\lambda$ the
determinant line of $H^{\cdot}(X,F)$.

{Assume that  $H^{\cdot}(X,F)=0$ and that $F$ is equipped
with a flat metric, which is equivalent to say that  its holonomy
representation $\rho$ is unitary.
The Reidemeister torsion  \cite{ReidemeisterTorsion,FranzTorsion,dRTorsion}  is a positive real number
defined with the  help of a triangulation on $X$. However, it does not depend on the
triangulation and becomes a topological invariant.  It is the first
invariant that could distinguish closed manifolds such as lens spaces
which are homotopy equivalent but not homeomorphic.}

The analytic torsion was introduced by Ray and Singer
\cite{RSTorsion} as an analytic counterpart of the Reidemeister torsion.
In order to define the analytic torsion one has to choose a
Riemannian metric on $X$. The analytic torsion is a certain
weighted alternating product of regularized determinants of the Hodge
Laplacians  acting on the space of differential forms  with values in
$F$.

The celebrated Cheeger-M\"uller Theorem
\cite{Ch79,Muller78} tells us that the Ray-Singer analytic torsion
coincides with the Reidemeister combinatorial torsion. Bismut-Zhang \cite{BZ92} and M\"uller \cite{Muller2} simultaneously
considered generalisations of this result.
M\"uller \cite{Muller2} extended his result to the case where $F$ is
unimodular, i.e., $|\det
\rho(\gamma)|=1$ for all $\gamma\in \pi_{1}(X)$. Bismut and Zhang \cite[Theorem 0.2]{BZ92}  generalised the original Cheeger-M\"uller
Theorem to arbitrary flat vector bundles with arbitrary Hermitian
metrics.  Subsequently Bismut-Zhang's theorem was generalised to the
equivariant case  by the same authors  \cite{BZ94}, to the family case by
Bismut-Goette  \cite{BG01}, and to
manifolds with arbitrary boundaries by Br\"uning-Ma \cite{BruningMa_2012}.

Let us explain Bismut-Zhang's theorem \cite[Theorem 0.2]{BZ92} in more detail. Indeed, to
formulate their result in the case  where the flat vector bundle is not
necessarily  acyclic or unitarily flat, Bismut and Zhang introduced the so-called Ray-Singer metric, which  is a
metric on $\lambda$  defined as the product of
the analytic torsion with an $L^2$-metric on $\lambda$.
Also they introduced the Milnor metric on $\lambda$ which is  a combinatorial
metric associated to a  Morse-Smale gradient  flow. It generalises the
Reidemeister torsion to  the case where $F$ is neither  acyclic
nor unitarily flat. In this way,
they were able to extend the Cheeger-M\"uller Theorem to a comparison theorem of two metrics on $\lambda$, one is analytic and the other
one is combinatorial.

The study of the relation between the combinatorial/analytic torsion  and the dynamical system
can be traced  back to Milnor \cite{MilnorZcover}.  Fried
\cite{FriedRealtorsion}  showed that on hyperbolic manifolds the analytic torsion of an acyclic unitarily flat vector
bundle is equal to the  value at zero point   of the Ruelle dynamical zeta
function of the geodesic flow.  He conjectured
\cite[p.\,66~Conjecture]{Friedconj} that similar results should hold true for  more general
flows. In \cite{Shfried}, following previous contributions
by
Moscovici-Stanton \cite{MStorsion}, using Bismut's orbital integral formula
\cite{B09}, the  author  affirmed  the Fried
conjecture for  geodesic flows on  closed locally symmetric
manifolds. In \cite{Shen_Yu}, the authors made a further
generalisation to closed locally symmetric orbifolds.

Besides the gradient flow, Morse-Smale flow is the simplest structurally stable dynamical system
which has only two  types of  recurrent behaviors: closed orbits and
fixed points.  Fried \cite[Theorem 3.1]{Friedconj}
proved  his conjecture for the Morse-Smale flows without fixed points.
When compared with Bismut-Zhang's Theorem  \cite[Theorem 0.2]{BZ92}, it seems  nature to ask
whether there is a relation between the torsion invariant (or more generally the
Ray-Singer metric for non acyclic and non unitarily flat vector bundle) and a general Morse-Smale flow which has both
fixed points and closed orbits.

This is one of the motivations  of S\'anchez-Morgado's work
\cite{MorgadoMorseSmale}. He showed that the
heteroclinic orbits have a non trivial contribution in the torsion
invariant, and in this way he constructed  a counterexample to Fried's
conjecture on Seifert manifolds.

In this
paper, we introduce a new Milnor metric,
which indeed contains the heteroclinic  contributions and
generalises  the absolute  value at zero point of the Ruelle dynamical zeta function, and we
give a comparison theorem for the Milnor and Ray-Singer metrics on
$\lambda$. We believe
that in this way we  give  a complete answer in the affirmative to the above  question.

Let us  mention that there is another interpretation of the Ruelle dynamical zeta
function  provided by Dang-Rivi\`ere \cite{DR_MSR}. See also
\cite{DR_MSG, DR_MSI, DR_MSII, DR_MSW} for related works.

 \subsection{A new Milnor metric}
  A vector field $V$ is called Morse-Smale if $V$ generates a flow
  whose  nonwandering set is the union of a finite set $A$ of
  hyperbolic fixed points and a finite set $B$ of hyperbolic prime
  orbits, and if the stable and unstable manifolds of the critical
  elements in $A\coprod B$ intersect transversally.

  Let us take a Hermitian metric $g^{F}$ on $F$. In Section \ref{sSF},  we construct on $\lambda$   a   Milnor metric
$\|\cdot\|^{M,2}_{\lambda, V}$ using  long exact
sequences associated to a Smale
filtration of the Morse-Smale flow.  Note that
the long exact
sequences  encode the information about the interactions
between the critical elements in $A\coprod B$.
 If $V$ is a gradient,
then our Milnor metric is just the classical one as defined in
\cite{MilnorWhiteheadTor} and \cite[Definition 1.9]{BZ92}.

Our first result says that  the  Milnor metric
$\|\cdot\|^{M,2}_{\lambda, V}$ is a generalisation of  the absolute value at
zero point of
the Ruelle dynamical zeta function. For a closed orbit $\gamma\in B$,
let $\ell_{\gamma}\in \mathbf{R}^{*}_{+}$ be its minimal period, and let
$\ind(\gamma)\in \mathbf{N}$  be its index
(see \eqref{indexg}). Take  $\Delta(\gamma)$ to be $1$ if
$\gamma$ is untwist and $-1$ in the contrary  case (see \eqref{Delta}). The  Ruelle dynamical zeta function
 is given by
\begin{align}\label{intoR}
	R_{\rho}(s)=\prod_{\gamma\in
	B}\det\(1-\Delta(\gamma)\rho(\gamma)^{-1}e^{-s
	\ell_\gamma}\)^{(-1)^{\ind(\gamma)}}.
\end{align}
Note that the above definition of the dynamical zeta function
corresponds to  Fried's definition  \cite{Friedconj} associated to
the holonomy of the dual bundle $F^{*}$.

\begin{prop}\label{prop}
	If $V$ does not have any fixed points, and if none of $\Delta(\gamma)$ is an eigenvalue of $\rho(\gamma)$, then
$H^{\cdot}(X,F)=0$, and the norm of the canonical section
$\mathbf{1}\in \mathbf{C}=\lambda$ is given by
\begin{align}
	\|\mathbf{1}\|^{M}_{\lambda,V}=|R_{\rho}(0)|^{-1}.
\end{align}
\end{prop}

\subsection{The main result of the paper}
Let $g^{TX}$ be a metric on $TX$. Let
$\psi(TX,\nabla^{TX})$ be the Mathai-Quillen current  associated to the
Levi-Civita connection $\nabla^{TX}$ (see Section \ref{sMQ}). It is a
current of degree $n-1$ defined
on the total space of the tangent bundle $TX$, which takes values in $o(TX)$, the orientation
bundle of $TX$. Let $\|\cdot\|^{RS,2}_{\lambda}$ be the Ray-Singer metric
on $\lambda$ associated to $(g^{TX},g^{F})$ (see Section \ref{sRS}).
Let $
\theta(F,g^{F})=\Tr\[g^{F,-1}\nabla^{F}g^{F}\]\in \Omega^{1}(X)$.
Our main result  is the following.

\begin{thm}\label{thm1} We have
	\begin{align}
	\log
	\(\frac{\|\cdot\|^{RS,2}_{\lambda}}{\|\cdot\|^{M,2}_{\lambda,V}}\)=-\int_{X}\theta\(F,g^{F}\)V^{*}\psi\(TX,\nabla^{TX}\).\end{align}
\end{thm}

If $V$ does not have any closed orbits, Theorem \ref{thm1} reduces to
\cite[Theorem 0.2]{BZ92}. If $V$ does not have any fixed  points,
and if $F$ is unitarily flat, by Proposition \ref{prop},  then  our
theorem corresponds to \cite[Theorem 3.1]{Friedconj}.

Our proof of Theorem \ref{thm1} is based on a result of Franks
\cite[Proposition 5.1]{Franks1979}, who
constructed  a gradient flow by destroying the
closed orbits of the Morse-Smale flow.
In Section \ref{sCMM}, we first
establish a comparison formula between
our Milnor metric
associated to the original Morse-Smale flow
and the classical one associated to Franks'
gradient flow. In Section \ref{S2}, to obtain Theorem \ref{thm1}, we apply Bismut-Zhang's formula \cite[Theorem
0.2]{BZ92}, which compares the
Ray-Singer metric with the
Milnor metric for Franks' gradient flow.

Recall that $F$ is said to be  unimodular, if its holonomy representation
$\rho$ is unimodular, i.e., $|\det \rho(\gamma)|= 1$ for all
$\gamma\in \pi_{1}(X)$.  This is
equivalent to the fact that there is a Hermitian metric $g^{F}$
such that $\theta\(F,g^{F}\)=0.$
By Theorem \ref{thm1}, we get
\begin{cor}
	If $(F,g^{F})$ is unimodular, then
	\begin{align}
	\|\cdot\|^{RS,2}_{\lambda}=\|\cdot\|^{M,2}_{\lambda,V}.
\end{align}
\end{cor}

\subsection{Organisation  of the paper}
In Section \ref{Sp1}, we
introduce some conventions on the determinant line, the cohomology
of a circle, and also a long exact sequence associated to three manifolds $Y_{1}\subset Y_{2}\subset Y_{3}$.  In Section \ref{S1M}, we recall some background on Morse-Smale
flows. We also introduce the Milnor metric, and we show Proposition
\ref{prop}. In
Section \ref{S2}, we recall the constructions of Mathai-Quillen
current and Ray-Singer metric. We show our main result.
We use the convention $\mathbf{N}=\{0,1,2,\ldots\}$ and $\mathbf{R}_{+}^{*}=(0,\infty)$.

\subsection{Acknowledgement}
We are indebted to  Xiaolong Han and Xiaonan Ma for reading  a
preliminary version of this paper and for useful suggestions. S.S.
would like to thank Nguyen Viet Dang and Gabriel  Rivi\`ere for
fruitful  discussions  on Morse-Smale flows. This paper was written
while J.Y. was visiting the Institut de
Math\'ematiques de Jussieu-Paris Rive Gauche in Spring 2018.
The hospitality of the Institute and the partial financial support
from NSFC under Grant No.~11771411 are gratefully acknowledged.

\settocdepth{subsection}
\section{Preliminary} \label{Sp1}
This section is organised as follows. In Section \ref{sDL}, we
introduce our convention on the determinant line. In Section
\ref{sCS1}, we give a metric on the determinant line of  the
cohomology of $\mathbb{S}^{1}$. This is  our model case near the
closed orbits of a flow. In Section \ref{Fp}, we explain a long exact
sequence associated to a triple  of manifolds $Y_{1}\subset
Y_{2}\subset Y_{3}$.

\subsection{The determinant line}\label{sDL}
Let $W$ be a complex finite dimensional vector space. We denote by
$W^*$ the dual space. If $\dim W=1$, we write $W^{-1}=W^{*}$.  Take
$\Lambda^\cdot(W)=\oplus_{j=0}^{\dim W}\Lambda^{j}(W)$ to be the exterior
algebra.

Set
\begin{align}
\det W =\Lambda^{\dim W} (W).
\end{align}
Clearly, $\det W$ is a complex line. If $W=0$, then
\begin{align}
	\det
W=\mathbf{\mathbf{C}}.
\end{align}
Let $E^{\cdot}=\oplus_{i\in \mathbf{Z}} E^{i}$ be a finite dimensional  $\mathbf{Z}$-graded
space. Put
\begin{align}
	\det E^{\cdot}= \bigotimes_{i\in \mathbf{Z}} \(\det E^{i}\)^{(-1)^{i}}.
\end{align}

For $m\in \mathbf{N}$, let
\begin{align}
	(C^{\cdot},d): 0\to C^{0}\to C^{1}\to\cdots\to C^{m}\to 0
\end{align}
be a complex of finite vector
spaces. By \cite{KM76} or
\cite[(1.5)]{BGS1}, we
have the canonical isomorphism of lines
\begin{align}\label{detCdetH}
	\det C^{\cdot}\simeq \det H^{\cdot}(C^{\cdot},d).
\end{align}
If $H^{\cdot}(C^{\cdot},d)={0}$, we can choose $s_{j}^{k}\in C^{k}$
such that $\{s_{j}^{k}\}_{j=1}^{k_{j}}$ projects to a basis of
$C^{k}/dC^{k-1}$, then
\begin{align}\label{Maja1}
	\(\wedge_{j} s^{0}_{j}\)\otimes \(\wedge_{j} ds^{0}_{j}\wedge (
	\wedge_{j}s^{1}_{j})\)^{-1} \otimes\cdots \otimes\(\wedge_{j}
	ds^{m-1}_{j}\)^{(-1)^m}\in \det C^{\cdot}
\end{align}
defines a canonical element of $\det C^\cdot$. Under
\eqref{detCdetH}, this element maps to $\mathbf{1}\in \mathbf{C}=\det
H^{\cdot}(C^{\cdot},d)$.

\subsection{The cohomology of $\mathbb{S}^{1}$}\label{sCS1}
Let $\mathbb{S}^{1}=\mathbf{R}/\mathbf{Z}$ be an oriented circle.
Let $F$ be a flat vector bundle of rank $r$ on
$\mathbb{S}^{1}$. Let $A\in \GL_{r}(\mathbf{C})$ be
the holonomy of $F$ along the oriented circle.
The canonical triangulation on $\mathbb{S}^{1}$ induces a
complex of simplicial cochains with values in $F$,\footnote{The cochain  complex $\(C^{\cdot}(\mathbb{S}^{1},F),d\)$
is the dual of the chain complex $\(C_{\cdot}(\mathbb{S}^{1},F^{*}),d\)$.
This explains why $A^{-1}$ appears in the complex \eqref{CS1}.}
\begin{align}\label{CS1}
\(C^{\cdot}(\mathbb{S}^{1},F),d\):		\xymatrixcolsep{3pc}\xymatrix{
0  \ar[r] &\mathbf{C}^{r}
\ar[r]^{A^{-1}-1} &\mathbf{C}^{r} \ar[r]          &0.}
\end{align}
By \eqref{detCdetH}, the canonical element $
\mathbf{1}\in \mathbf{C}=\det C^{\cdot}(\mathbb{S}^{1},F)$ defines  an element
\begin{align}\label{eqsigmaa}
\sigma_{A}\in 	\det H^{\cdot}\(\mathbb{S}^{1},F\).
\end{align}
We equip $\det H^{\cdot}\(\mathbb{S}^{1},F\)$ with a metric
$\|\cdot\|^{2}_{\det H^{\cdot}\(\mathbb{S}^{1},F\)}$ such that
\begin{align}\label{nsa}
	\|\sigma_{A}\|_{\det H^{\cdot}\(\mathbb{S}^{1},F\)}=1
\end{align}

If $1$ is not an eigenvalue of $A$, then $H^{\cdot}(\mathbb{S}^{1},F)=0$.
By \eqref{Maja1},  the norm of the canonical element
$\mathbf{1}\in \mathbf{C}=\det H^{\cdot}(\mathbb{S}^{1},F)$ is given by
\begin{align}\label{sa}
	\|\mathbf{1}\|_{\det
	H^{\cdot}\(\mathbb{S}^{1},F\)}=\left|\det\(1-A^{-1}\)\right|^{-1}.
\end{align}

\begin{re}
	Equation \eqref{sa} is just Proposition \ref{prop} for  the 	rotation flow on  	$\mathbb{S}^{1}$.
\end{re}

\begin{re}
	Since the flat vector bundle is not necessarily unimodular, i.e.,
	$|\det(A)|$ is
	not necessarily equal to $1$,  the choice of the
	orientation on $\mathbb{S}^{1}$ is very important.
\end{re}

\subsection{A fusion principle}\label{Fp}
Let $Y_{1}\subset Y_{2}\subset Y_{3}$ be three compact smooth
manifolds with boundaries of
the same dimension such that $Y_{1}\subset \mathring{Y_{2}}$ and
$Y_{2}\subset \mathring{Y_{3}}$. Let $F$ be a flat vector bundle on
$Y_{3}$ of rank $r$. Denote by $H^{\cdot}(Y_{3},Y_{2},F)$,
$H^{\cdot}(Y_{3},F)$,\ldots, the corresponding relative or absolute
cohomologies  with coefficients in  $F$.

As in \cite[(0.16)]{BruningMa_2012}, we have a long exact sequence
\begin{align}\label{Les}
\cdots\to 	H^{i}(Y_{3},Y_{2},F)\to H^{i}(Y_{3},Y_{1},F)\to
	H^{i}(Y_{2},Y_{1},F)\to \cdots.
\end{align}
By \eqref{detCdetH} and \eqref{Les}, we get an isomorphism of lines
\begin{align}\label{Maja3}
	f_{21,32	}:\det H^{\cdot}(Y_{2},Y_{1},F)\otimes \det 		H^{\cdot}(Y_{3},Y_{2},F)\simeq \det H^{\cdot}(Y_{3},Y_{1},F).
\end{align}
Using the other triples  $(\varnothing,Y_{1},Y_{2})$ and
$(\varnothing,Y_{1},Y_{3})$, we get similar isomorphisms
\begin{align}
f_{1,21}:	\det H^{\cdot}(Y_{1},F)\otimes \det
	H^{\cdot}(Y_{2},Y_{1},F)\simeq \det H^{\cdot}(Y_{2},F),
\end{align}
and
\begin{align}\label{Maja4}
f_{1,31}:	\det H^{\cdot}(Y_{1},F)\otimes \det
	H^{\cdot}(Y_{3},Y_{1},F)\simeq \det H^{\cdot}(Y_{3},F).
\end{align}

By \eqref{Maja3}-\eqref{Maja4}, we see that
$f_{2,32}\circ\(f_{1,21}\otimes\mathrm{id}\) $ and $f_{1,31}\circ \(\mathrm{id} \otimes f_{21,32}\)$
define  two isomorphisms
\begin{align}\label{FYYY}
	\det H^{\cdot}(Y_{1},F)\otimes \det H^{\cdot}(Y_{2},Y_{1},F)\otimes \det
	H^{\cdot}(Y_{3},Y_{2},F)	\simeq  \det H^{\cdot}(Y_{3},F).
\end{align}

\begin{prop}\label{propFp}
	There is $\mu\in \mathbf{C}$ with $| \mu |=1$ such that
	\begin{align}\label{mabu}
f_{2,32}	\circ\(f_{1,21}\otimes\mathrm{id}\) = \mu \times
	f_{1,31}\circ \(\mathrm{id} \otimes f_{21,32}\).
	\end{align}
\end{prop}
\begin{proof}
	Let us give a short analytic proof. Take  a Riemannian metric on $Y_{3}$ such that  $\partial Y_{1}$ and $\partial Y_{2}$ are totally
	geodesic. Take also a Hermitian metric on $F$. By abuse of
	notation denote by
	$\|\cdot\|^{RS,2}$  the Ray-Singer
	metrics\footnote{We will not use the explicit definition  of this metric.
	We refer the reader to \cite{BruningMa_2012} for the  detail. }
	on $\det H^{\cdot}(Y_{1},F)$, $\det
	H^{\cdot}(Y_{2},Y_{1},F),\ldots$, or the induced  metric on the
	lines likes
	\begin{align}
		\Big(\det H^{\cdot}(Y_{1},F)\otimes \det H^{\cdot}(Y_{2},Y_{1},F)\otimes \det
	H^{\cdot}(Y_{3},Y_{2},F)	\Big)^{-1}\otimes  \det
	H^{\cdot}(Y_{3},F).
	\end{align}

	By \cite[Theorem 0.3]{BruningMa_2012},  we have
	\begin{align}\label{f121}
		\|f_{2,32} \circ\(f_{1,21}\otimes\mathrm{id}\)
		\|^{RS,2}
	=\|f_{2,32}\|^{RS,2}\|f_{1,21}\|^{RS,2}=2^{r\chi(\partial
	Y_{1})+r\chi\(\partial Y_{2}\)},
	\end{align}
	and also
	\begin{align}\label{f131}
		\|f_{1,31}\circ \(\mathrm{id} \otimes f_{21,32}\)\|^{RS,2}
	=\|f_{1,31}\|^{RS,2}\|f_{21,32}\|^{RS,2}=2^{r\chi(\partial
	Y_{1})+r\chi\(\partial Y_{2}\)},
	\end{align}
	where $r$ is the rank of $F$ and $\chi\(\partial Y_{1}\),\chi\(\partial
	Y_{2}\)$ are the Euler characteristic numbers of the boundaries  $\partial Y_{1}, \partial Y_{2}$.
	By \eqref{f121} and \eqref{f131}, we get \eqref{mabu}.
\end{proof}
\begin{re}
	Instead of using the analytic Ray-Singer metric and \cite[Theorem
	0.3]{BruningMa_2012}, 	we can also use the Milnor
	metric \cite[Definition 1.6]{BruningMa_2012} and 	\cite[(3.71)]{BruningMa_2012}.
\end{re}

\section{Milnor metric}\label{S1M}
This section is organised as follows. In Sections \ref{sMS} and
\ref{sDZf}, we recall
the definitions of Morse-Smale flow and the associated Ruelle dynamical zeta
function. In Section \ref{sMSMF}, we recall  some results due to Franks
\cite{Franks1979,Franks1982} on the construction of  a new
gradient  flow  by destroying  the closed orbits of the original Morse-Smale flow.  In Section
\ref{sSF}, using the Smale filtration, we introduce the Milnor
metric. In Section  \ref{sCMM}, we establish a comparison  formula for the two Milnor
metrics,  one is associated to the Morse-Smale flow and the other is
associated to the gradient  flow constructed by Franks.
\subsection{Morse-Smale flow}\label{sMS}
Let $X$ be a connected closed smooth manifold of dimension $m$. Let $V$ be a
 vector field on $X$, and let
$\(\phi_{t}\)_{t\in \mathbf{R}}$ be the
 flow generated
 by $V$.

\begin{defin}
A fixed point $x\in X$ of the flow $\phi_{\cdot}$ is called hyperbolic if there is a splitting
\begin{align}
	T_{x}X= E^{u}_{x}\oplus E^{s}_{x},
\end{align}
and there exist $C>0,\theta>0$ and a Riemannian metric on $X$
such that for  $v\in E^{u}_{x}$,  $v'\in E^{s}_{x}$, and $t>0$, we have
\begin{align}\label{eqMS}
&\left|\phi_{- t,*}v\right|\l
Ce^{-\theta|t|}\left|v\right|,&\left|\phi_{ t,*}v'\right|\l
Ce^{-\theta|t|}\left|v'\right|.
\end{align}
The unstable and stable manifolds of the  hyperbolic fixed point $x$ are defined by
\begin{align}
	&W^{u}_{x}=\Big\{y\in X: \lim_{t\to -
	\infty}d_{X}(\phi_{t}(y),x)=0\Big\},&W^{s}_{x}= \Big\{y\in X: \lim_{t\to
	\infty}d_X(\phi_{t}(y),x)=0 \Big\},
\end{align}
where $d_{X}$ denotes the Riemannian distance. The index $\ind(x)\in \mathbf{N}$ of $x$ is defined by
\begin{align}
	\ind(x)=\dim E^{u}_{x}.
\end{align}
\end{defin}

\begin{defin}
A  closed orbit $\gamma$ of the flow $\phi_{\cdot}$ is called
hyperbolic,
if there is  a $\phi_{t}$-invariant continuous  splitting
\begin{align}\label{eq:AF1}
TX|_{\gamma}=\mathbf{R}V\oplus E^{u}_{\gamma}\oplus E^{s}_{\gamma},
\end{align}
such that \eqref{eqMS} holds.  The associated unstable and
stable manifolds are defined by
\begin{align}
\begin{aligned}
	W^{u}_{\gamma}=\bigcup_{x\in \gamma} \Big\{y\in X: \lim_{t\to
	-\infty}d_{X}(\phi_{t}(y),\phi_{t}(x))=0 \Big\},\\ W^{s}_{\gamma}=\bigcup_{x\in \gamma} \Big\{y\in X: \lim_{t\to
	+\infty}d_{X}(\phi_{t}(y),\phi_{t}(x))=0 \Big\}.
	\end{aligned}
\end{align}
The index $\ind(\gamma)\in \mathbf{N}$ of $\gamma$ is defined by
\begin{align}\label{indexg}
	\ind(\gamma)=\dim E^{u}_{\gamma}.
	\end{align}
\end{defin}

Denote by $A$ the set of fixed points and by $B$ the set of
prime closed orbits.

\begin{defin}
	A vector field $V$ or a flow $\phi_{\cdot}$ is called Morse-Smale if
	\begin{itemize}
		\item the sets $A$ and $B$ are finite and contain only hyperbolic elements;
       \item the nonwandering set of
$\phi_{\cdot}$ is equal to
$	A\cup \bigcup_{\gamma\in B}\gamma$;
     \item the stable  and unstable manifold of any critical
	 element in $A\coprod B$  intersect transversally.
	\end{itemize}
\end{defin}

In the sequel , we assume
that $V$ is a
Morse-Smale vector field.

\subsection{Ruelle dynamical zeta function}\label{sDZf}
 For $\gamma\in B$, denote by $\ell_{\gamma}\in \mathbf{R}^{*}_{+}$ its
 minimal period.
 A closed orbit $\gamma\in B$
 is called untwist if $E^{u}_{\gamma}$ is orientable along $\gamma$, and is
 called twist otherwise. Put
 \begin{align}\label{Delta}
	 \Delta(\gamma)=\left\{
	 \begin{array}{cl}
		 1,&\hbox{if } \gamma \hbox{ is untwist},\\
		 -1,&\hbox{if } \gamma \hbox{ is twist}.
	\end{array}
	 \right.
 \end{align}

 Let $\rho:\pi_{1}(X)\to \mathrm{GL}_{r}(\mathbf{C})$ be a representation of the fundamental group of $X$. If
 $\gamma\in B$, denote by $\rho(\gamma)$
 the holonomy along $\gamma$. Clearly, $\rho(\gamma)$ is well
 defined up to a conjugation.

\begin{defin} The twist Ruelle dynamical zeta function is a meromorphic
	function on $\mathbf{C}$ defined by
\begin{align}\label{rrs}
	R_{\rho}(s)=\prod_{\gamma\in
	B}\det\(1-\Delta(\gamma)\rho(\gamma)^{-1}e^{-s
	\ell_\gamma}\)^{(-1)^{\ind(\gamma)}}.
\end{align}
This definition of the Ruelle dynamical zeta function
corresponds to  Fried's definition  \cite{Friedconj} associated to
the  holonomy of the  dual bundle  $F^{*}$.
\end{defin}

\subsection{Frank's Morse function}\label{sMSMF}
 We follow \cite[Section 1]{Franks1979}. Let $\mathbb{D}^{r}$ be the
$r$-dimensional open unit ball of center $0\in
\mathbf{R}^{r}$.
 A fixed point $x\in A$ of index $p$ is said to be
 of standard form if there
 is a system of
 coordinates $(y_{1},\ldots, y_{m})\in \mathbb{D}^{m}$ on a neighborhood
 of $x$ such that $x$ is represented by $0$ and
 \begin{align}
	 V=\sum_{i=1}^{p}y_{i}\frac{\partial}{\partial
	 y_{i}}-\sum_{i=p+1}^{m}y_{i}\frac{\partial}{\partial y_{i}}.
\end{align}

For closed orbits we must distinguish the following four cases  in
establishing the standard forms. Assume $\gamma\in B$ such that
$\ind(\gamma)=p$.
\begin{description}
	\item[Case (1)] Suppose that $TM|_{\gamma}$ is orientable and that  $\gamma$ is
	untwist. In this case, $\gamma$ is said to be of standard form, if there
	is a  system of coordinates $(t,y_{1},\ldots,
 y_{m-1})\in \mathbb{S}^{1}\times \mathbb{D}^{m-1}$
 on a tubular neighborhood
	$U_{\gamma}$	of $\gamma$ such that $\gamma$ is represented by
 $(t,0)\in  \mathbb{S}^{1}\times \mathbb{D}^{m-1}$ and
\begin{align}\label{eqV}
	 V=\frac{\partial}{\partial t}+\sum_{i=1}^{p}y_{i}\frac{\partial}{\partial
	 y_{i}}-\sum_{i=p+1}^{m-1}y_{i}\frac{\partial}{\partial y_{i}}.
\end{align}
	\item[Case (2)] Suppose that $TM|_{{\gamma}}$ is orientable and
	that   $\gamma$ is
	twist. In this case, $\gamma$ is said to be of standard form, if
	 $U_{\gamma}$ and $V$ can be obtained
	from Case (1) by  the identification
	\begin{align}\label{enid}
	(t,
	x_{1},\ldots,x_{m-1})\sim 	(t+1/2,
	-x_{1},x_{2},\ldots, x_{p}, -x_{p+1}, x_{p+2},\ldots,x_{m-1}).
	\end{align}
	\item[Case (3)]  Suppose that   $TM|_{{\gamma}}$ is not orientable  and
 that  $\gamma$ is
	untwist. In this case, $\gamma$ is said to be of standard form, if
	 $U_{\gamma}$ and $V$ can be obtained
	from Case (1) by  the identification
	\begin{align}
	(t,
	x_{1},\ldots,x_{m-1})\sim 	(t+1/2,
	x_{1},\cdots,x_{p},-x_{p+1},x_{p+1},\ldots,x_{m-1}).	\end{align}
	\item[Case (4)]  Suppose that  $TM|_{{\gamma}}$ is not orientable
	and that 	 $\gamma$ is
	twist. In this case,  $\gamma$ is said to be of standard form, if
	 $U_{\gamma}$ and $V$ can be obtained
	from Case (1) by  the identification
	\begin{align}
		(t,
	x_{1},\ldots,x_{m-1})\sim (t+1/2,
	-x_{1},x_{2},\cdots,x_{m-1}).
	\end{align}
\end{description}
Note that in \cite[Section 1]{Franks1979} the author assumed that $M$ is orientable, so only the first two cases appear.

The following three propositions are  \cite[Proposition 1.6,
Theorem 2.2, Proposition 5.1]{Franks1979}.
Their proofs can be generalised to the non orientable case
with some evident modifications. We omit the details.

 \begin{prop}\label{propV01}
	 For any Morse-Smale vector field $V$, there is a family of Morse-Smale
	 vector fields $(V_{\ell})_{0\l \ell\l 1}$ such that $V_{0}=V$ and that the critical
	 elements of
	 $V_{1}$ are all of
standard forms and are precisely the same as the critical
	 elements of $V$. Moreover, $V_{0}$ and $V_{1}$ are topologically
	 conjugated, i.e., there is a homeomorphism carrying the  orbits of
	 $V_{0}$ to those
of $V_{1}$ and preserving their orientations.
\end{prop}

The relation between the Morse-Smale flow and the  Morse function is
summarised in the following two propositions. The first one is due to Smale \cite{S61}.

\begin{prop}
	If $V$ is a Morse-Smale vector field whose flow has fixed points
	in standard form and no closed orbits, then $V$ is a  gradient of
a certain Morse function with respect to some Riemannian metric.
\end{prop}

To state the following  proposition, let us introduce some notation. For $x,y\in A$  such that $\ind(y)=\ind(x)-1$, then $W^{u}_{x}\cap W^{u}_{y}$
consists of a finite set $\Gamma(x,y)=\{a_{\cdot}\}$ of integral curves of
$V$ such that $a_{-\infty}=x$ and $a_{\infty}=y$ along
which $W^{u}_{x}$ and $W^{s}_{y}$ intersect transversally. Define
$n(a)=\pm1$ as in  \cite[(1.28)]{BZ92}, whose definition does not
require the manifold to be orientable.

\begin{prop}\label{propfV}
	For some small neighborhood $U=\cup_{\gamma\in B}U_{\gamma}$ of
	closed orbits $\cup_{\gamma\in B}\gamma$, there is
	a 	Morse function $f$ on $M$ whose gradient vector field $\nabla f$
	with respect to a certain Riemannian metric is
	Morse-Smale, such that
	\begin{itemize}
		\item on $M\backslash U$, we have
	\begin{align}\label{eqdf=V}
		\nabla f=V,
	\end{align}
	\item on each $U_{\gamma}$, the Morse function  $f$ has only two  critical points $x_{\gamma},x'_{\gamma}$ of
 index $\ind(\gamma)+1$ and $\ind(\gamma)$ respectively.
	\end{itemize}
	  Also, $\Gamma(x_{\gamma},x'_{\gamma})$ consists of two
 integral curves  $a_{\gamma},a'_{\gamma}$ (see Figure \ref{fig:1})
 such that their composition
 $a_{\gamma}\circ  (a'_{\gamma})^{-1}$ and the
 closed orbit $\gamma$ lie
 in the same freely  homotopy class of
 loops on $X$ and  that
 \begin{align}\label{nana}
	 n(a_{\gamma})n(a'_{\gamma})=-\Delta(\gamma).
\end{align}
\end{prop}

\begin{figure}[htbp]
\centering
\includegraphics[width=3.5in]{MS}
\caption{A closed orbit  and  integral curves }\label{fig:1}
\end{figure}

\begin{re}\label{reK}We recall the essential step in the
	construction of the gradient $\nabla f$ given by Franks \cite[Proposition
	8.5]{Franks1982}. For simplicity, assume that	there is only one
	closed orbit 	$\gamma$ and it is of index $p$ and in standard
	form of Case (1).
For $\epsilon>0$ small enough, let $\rho\in
C^{\infty}_{c}(\mathbb{D}^{m-1},[0,1])$
be a cutoff function, which is equal to
$1$ on $|y|<\epsilon$ and to $0$ on $|y|>2\epsilon$. Put
\begin{align}
		V_{1}=\left\{
		\begin{array}{lr}
		\big((1-\rho)+\rho\sin\(2\pi t\)\big)\frac{\partial}{\partial t}+\sum_{i=1}^{p}y_{i}\frac{\partial}{\partial
	 y_{i}}-\sum_{i=p+1}^{m-1}y_{i}\frac{\partial}{\partial y_{i}}, &\hbox{ in
	 } 	 U_{\gamma},\\
	 V,& \hbox{ outside } U_{\gamma}.
		\end{array}\right.
	\end{align}
It is easy to see that the
	nonwandering set of $V_{1}$ in $U_{\gamma}$ consists of two
	points $(1,0),(\frac{1}{2},0)\in
	\mathbb{S}^{1}\times \mathbb{D}^{m-1}$.
	Then, by a small perturbation on $V_{1}$,  we  get a Morse-Smale gradient vector
	field  $\nabla f$ which
	has the desired transversality and other properties.

We remark that by the above construction, we can find  a	family of vector fields $(V_{\epsilon})_{0\l\epsilon\l1}$ connecting $V$ and
	$\nabla f$ such that near $\{1/4\}\times \mathbb{D}^{m-1}$, for
	any $ \epsilon\in [0,1]$, $V_{\epsilon}$ does not vanish.
	Similar remark holds for $\gamma$ in standard forms of Cases (2)-(4).
	In Section \ref{s33}, we will use this fact to simplify the proof of our main theorem.
\end{re}

\subsection{Smale filtration and Milnor metric}\label{sSF}
 Following \cite[Definition 9.10]{Franks1982},
let
\begin{align}\label{eqSmale}
	\varnothing=X^{0}\subset X^{1}\subset\cdots \subset X^{N}=X
\end{align}
be a Smale filtration on $X$ associated to $V$. Note that each
$X^{p}\subset X$ is a submanifold with boundary, and can be constructed by the sublevel set  of a smooth Lyapunov function. Also, we
have
\begin{itemize}
	\item on each $\partial X^{p}$, $V$ does not vanish and points toward the inside of $X^{p}$;
	\item  there is only one critical element $c\in A\coprod
	B$ contained in $X^{p+1}\backslash
	X^{p}$ and
	\begin{align}\{c\}=\bigcap_{t\in \mathbf{R}}\phi_{t}\(X^{p+1}\backslash
	X^{p}\).
	\end{align}
\end{itemize}

Let $F$ be a flat vector bundle on $X$ induced by the representation
$\rho$.
Let $H^{\cdot}(X,F)$ be the cohomology of the sheaf of locally constant
sections of $F$. Put
\begin{align}\label{eqlambda}
	\lambda=\det H^{\cdot}(X,F).
\end{align}
We use the notation in Section \ref{Fp}. By \eqref{FYYY}, we get an isomorphism
 \begin{align}\label{eqSS}
 	\sigma_{V}:\bigotimes_{p=0}^{N-1}\det
	H^{\cdot}(X^{p+1},X^{p},F)\simeq \lambda.
 \end{align}
By Proposition \ref{propFp}, up to a multiplication  by a unit in $\mathbf{C}$, the
morphism $\sigma_{V}$ does not depend on the way that the
cohomologies are fused.

By \cite[Theorem 9.11]{Franks1982} (see also \cite[Section 2]{MorgadoMorseSmale}), if the critical element  $c\in
X^{p+1}\backslash X^{p}$ is a fixed
point, then
\begin{align}\label{E1x}
	H^{q}\(X^{p+1},X^{p},F\)=\left\{ \begin{array}{ll}
	F_{c},& q=\ind (c),\\
	0,&\hbox{otherwise},
\end{array}\right.
\end{align}
and if  the critical element  $c\in
X^{p+1}\backslash X^{p}$ is a closed orbit, then
\begin{align}\label{E1g}
	H^{q}\(X^{p+1},X^{p},F\)=\left\{ \begin{array}{ll}
	H^{q-\ind (c)}\big(c,o(E^{u}_{c})\otimes F|_{c}\big),& q-\ind( c)=0
	\hbox{ or }1,\\
	0,&\hbox{otherwise,}
\end{array}\right.
\end{align}
where $o(E_{c}^{u})$ is the orientation line bundle of $E^{u}_{c}$ along the
closed orbit $c$.

We equip $\det
	H^{\cdot}\(\gamma,o(E^{u}_{\gamma})\otimes
	F|_{\gamma}\)$ with the metric $\|\cdot\|^{2}_{\det
	H^{\cdot}\(\gamma,o(E^{u}_{\gamma})\otimes
	F|_{\gamma}\)}$ defined  in \eqref{nsa}.
Let $g^{F}$ be a Hermitian metric on $F$. By \eqref{eqSS}-\eqref{E1g}, the
restriction  $g^{F}|_{A}$
and  the metric $\|\cdot\|^{2}_{\det
	H^{\cdot}\(\gamma,o(E^{u}_{\gamma})\otimes
	F|_{\gamma}\)}$ induce a metric
$\|\cdot\|^{M}_{\lambda,V}$ on
$\lambda$. By Proposition \ref{propFp},
this metric  does not depend on the way that the
cohomologies are fused.

\begin{defin}
	The  metric $\|\cdot\|^{M}_{\lambda,V}$ on
$\lambda$ is called the Milnor metric associated to
$V$.
\end{defin}

\begin{re}
	If $V$ is a Morse-Smale gradient vector field, then
	$\|\cdot\|^{M}_{V}$ coincides with the one constructed by
	Bismut-Zhang \cite[Definition 1.9]{BZ92}. In fact, there is a small difference with Bismut-Zhang's
	construction, where  they used a filtration \cite[(1.37)]{BZ92} induced
	by sublevel sets of a nice 	Morse function.  Using Proposition
	\ref{propFp}, we can deduce  that the two constructions coincide.
\end{re}

\begin{re}
	The Milnor metric for
	general Morse-Smale flow 	does not depend on the Smale
filtration \eqref{eqSmale}. We will not give the direct proof  since it is a consequence of our main Theorem \ref{thm1}.
\end{re}

Let us restate and prove Proposition \ref{prop}.

\begin{prop}\label{Prop}
	If $V$ does not have any fixed points, and
if none of $\Delta(\gamma)$ is an  eigenvalue of $\rho(\gamma)$, then
$H^{\cdot}(X,F)=0$, and the norm of the canonical section
$\mathbf{1}\in \mathbf{C}=\det H^{\cdot}(X,F)$ is given by
\begin{align}\label{eqReMi}
	\|\mathbf{1}\|^{M}_{\lambda,V}=|R_{\rho}(0)|^{-1}.
\end{align}
\end{prop}
\begin{proof}
	By \eqref{Les}, \eqref{E1x}, and \eqref{E1g}, we can deduce  that  $H^{\cdot}(X,F)=0$. By
	\eqref{sa}, \eqref{rrs}, and \eqref{eqSS}, we get
	\eqref{eqReMi}.
\end{proof}

\subsection{A comparison formula for  Milnor metrics}\label{sCMM}
In this section, we assume that all the critical elements of
$V$ are in standard forms, and that $f$ is chosen as in
Proposition 	\ref{propfV}.

Let $\det \tau({a'_{\gamma}})\in \det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}$ be the canonical element induced by the
	parallel transport  with respect to the flat connection	along the
	integral curve $a_{\gamma}'$ (see Figure \ref{fig:1}). Let $\|\cdot\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}$ be the metric on $\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}$ induced by $g^{F}_{x_{\gamma}}$ and
		$g^{F}_{x'_{\gamma}}$.

\begin{prop}\label{propMM}
	The following identity holds,
	\begin{align}\label{eqMM}
		\log\(\frac{\|\cdot\|^{M,2}_{\lambda,\nabla
		f}}{\|\cdot\|^{M,2}_{\lambda,V}}\)=\sum_{\gamma\in
		B}(-1)^{\ind(\gamma)}\log
		\left\|\det\tau(a_{\gamma}')\right\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}.
	\end{align}
\end{prop}
\begin{proof}
	We refine the filtration \eqref{eqSmale} by the new critical
	points of $f$. By Propositions \ref{propFp} and \ref{propfV}, and by
	\eqref{eqSS}, the following diagram commutes
	 \begin{align}\label{eqDia}
		\begin{aligned}
			\xymatrix{
	\bigotimes_{x\in A}  \big(\!\det F_{x}\big)^{ (-1)^{\ind (x)}}
		\bigotimes_{\gamma\in B} \left\{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}\right\}^{(-1)^{\ind (\gamma)}}
		\ar[d]
		\ar[r]^-{\sigma_{\nabla f}} &\lambda\ar[d]
\\
		\bigotimes_{x\in A}\!\big(\!\det
F_{x}\big)^{(-1)^{\ind (x)}} \!
	\bigotimes_{\gamma \in B}\Big\{\!\det
	H^{\cdot}\(\gamma,o(E^{u}_{\gamma})\otimes
	F|_{\gamma}\)\!\Big\}^{(-1)^{\ind
	(\gamma)}}\ar[r]^-{\sigma_V}         &\lambda}
	\end{aligned}
	\end{align}
where the first vertical arrow  is induced by \eqref{eqsigmaa} and
the second vertical arrow is a multiplication  by a unit in $\mathbf{C}$.
	The Milnor metric $\|\cdot\|_{\lambda,\nabla f}^{M,2}$ is
	obtained   from the metric $g^{F}|_{A \cup
	\{x_{\gamma},x'_{\gamma}:\gamma\in B\}}$ via $\sigma_{\nabla f}$.
	By \eqref{eqDia}, we get
	\eqref{eqMM}.
\end{proof}

\section{An extension of  Bismut-Zhang's Theorem to Morse-Smale
flow}\label{S2}
This section is organised as follows. In Sections
\ref{sec:Bere}-\ref{sRS},  following \cite{BZ92}, we recall the
constructions of the Berezin integral, the  Mathai-Quillen current and the Ray-Singer metric.
In Section \ref{s33}, we restate and prove our main theorem.
\subsection{Berezin integral}\label{sec:Bere}
Let $E$ be a real Euclidean space of dimension $n$ with
the scalar product $\<,\>$, and let $W$ be a real vector space of finite dimension.

Suppose temporarily that $E$ is oriented and that $e_1,\ldots, e_n$
is an oriented orthogonal basis of $E$. Let $e^1,\ldots,e^n$ be the corresponding dual basis of $E^*$. We define $\int^B$ to be the linear map from $\Lambda^\cdot(W^*)\widehat{\otimes}\Lambda^\cdot(E^*)$ into $\Lambda^\cdot(W^*)$, such that if $\alpha\in \Lambda^\cdot(W^*), \beta\in \Lambda^\cdot(E^*)$,
\begin{align}\label{eq:defBere}
& \int^B\alpha\beta=0, \hbox{ if } \deg \beta<n,&\int^B\alpha
e^1\wedge\cdots\wedge e^n=\frac{(-1)^{n(n+1)/2}}{\pi^{\frac{n}{2}}}\alpha.
\end{align}
More generally, if $o(E)$ is the orientation line of $E$, then $\int^B$ defines a linear map from $\Lambda^\cdot(W^*)\widehat{\otimes}\Lambda^\cdot(E^*)$ into $\Lambda^\cdot(W^*)\widehat{\otimes} o(E)$, which is called a Berezin integral.

Let $A\in \End^{\mathrm{anti}}(E)$ be an antisymmetric endomorphism of $E$. We identify $A$ with
\begin{align}\label{eq:ALam}
 \dot{A}=\frac{1}{2}\sum_{1\l i,j\l n}\<e_i,Ae_j\>e^i\wedge e^j\in \Lambda^2(E^*).
\end{align}
By definition, the Pfaffian $\mathrm{Pf}[A]\in o(E)$ of $A$ is given by
\begin{align}\label{eq:PfA}
\mathrm{Pf}\[\frac{A}{2\pi}\]=\int^B\exp\(-\frac{\dot{A}}{2}\).
\end{align}
Clearly, $\mathrm{Pf}[A]$ vanishes if $n$ is odd.

\subsection{Mathai-Quillen formalism }\label{sMQ}
Recall that  $X$ is a connected closed smooth    manifold of dimension $m$. Let
$E$ be a Euclidean vector bundle of rank $n$ on  $X$ with a Euclidean
metric $g^{E}$ and a metric connection  $\nabla^{E}$. Let $R^{E}=\(\nabla^{E}\)^{2}\in
\Omega^{2}(X,\End^{\rm anti}(E))$ be its curvature.  Denote by $o(E)$  the orientation line
bundle of $E$.
The Euler form of
$(E,\nabla^{E})$ is given by
\begin{align}
	e\(E,\nabla^{E}\)=\mathrm{Pf}\[\frac{R^{E}}{2\pi}\] \in
	\Omega^{n}(X,o(E)).
\end{align}
Clearly, $e\(E,\nabla^{E}\)=0$,  if $n$ is odd.

Let $\mathcal{E}$ be the total space of $E$, and let $\pi: \mathcal{E}\to X$ be the
natural projection. We will use the formalism of the Berezin integral developed in
Section \ref{sec:Bere} with $W = T\mathcal{E}$. If $\omega$ is a smooth section of
 $\Lambda^{\cdot}(T^{*}\mathcal{E})\otimes\pi^{*}\Lambda^{\cdot}(E^{*})$
 over $\mathcal{E}$, then $\int^{B}\omega$ is a smooth section of $\Lambda^{\cdot}
 (T^{*} \mathcal{E})
 \otimes \pi^{*}o(E)$ over $\mathcal{E}$.

Let $T^{V}\mathcal{E}\subset T\mathcal{E}$ be the vertical subbundle of $T\mathcal{E}$, and
let $T^{H}\mathcal{E}\subset T\mathcal{E}$ be the horizontal  subbundle of $T\mathcal{E}$
with respect to $\nabla^{E}$, so that
\begin{align}\label{splitting}
	T\mathcal{E}=T^{H}\mathcal{E}\oplus  T^{V}\mathcal{E}.
\end{align}
By the identification
$T^{V}\mathcal{E}\simeq \pi^{*}E$,
the
vertical projection with
respect to the splitting \eqref{splitting} induces a section
$P^{E}\in C^{\infty}(\mathcal{E},T^{*}\mathcal{E} \otimes \pi^{*}E)$. Using the metric $g^{E}$, we identify $P^{E}$ with $\dot{P}^{E}\in C^{\infty}(\mathcal{E},T^{*}\mathcal{E} \otimes \pi^{*}E^{*})$.
Let $Y\in C^{\infty}(\mathcal{E},\pi^{*}E)$ be the tautological section.
Write $\widehat{Y}$ the corresponding  section  in
$C^{\infty}(\mathcal{E},\pi^{*}E^{*})$ induced by  $g^{E}$.
Recall that $\dot{R}^{E}\in C^{\infty}(X,\Lambda^{2}(T^{*}X)\otimes \Lambda^{2}(E^{*}))$ is
defined in \eqref{eq:ALam}.

\begin{defin}
	For
$T\geqslant 0$, set
\begin{align}
	A_{T}=\frac{1}{2}\pi^{*}\dot{R}^{E}+\sqrt{T}\dot{P}^{E}+T|Y|^{2}\in C^{\infty}\big(\mathcal{E},\Lambda^{\cdot}(T^{*}\mathcal{E})\otimes \pi^{*}\Lambda^{\cdot}(E^{*})\big).
\end{align}
	Let $(\alpha_{T})_{T\g0}$ and $(\beta_{T})_{T>0}$ be  families  of forms on $\mathcal{E}$ defined
	by
	\begin{align}
		\begin{aligned}
		&\alpha_{T}=\int^{B}\exp\(-A_{T}\)\in
		\Omega^{n}(\mathcal{E},\pi^{*}o(E)),\\
		&\beta_{T}=\int^{B}\frac{\widehat{Y}}{2\sqrt{T}}\exp\(-A_{T}\)\in \Omega^{n-1}(\mathcal{E},\pi^{*}o(E)).
		\end{aligned}
	\end{align}
\end{defin}
Clearly,
\begin{align}
	\alpha_{0}=\pi^{*}e\(E,\nabla^{E}\).
\end{align}
Let us recall \cite[Theorems 3.4, 3.5, and 3.7]{BZ92}.
\begin{thm}
For $T\geqslant 0$, the form $\alpha_{T}$ is closed whose  cohomology class
does not depend on $T$.
For $T>0$,  $\alpha_{T}$ represents the Thom
class of $E$ so that $\pi_{*}\alpha_{T}=1$, and  we have
\begin{align}
&\frac{\partial \alpha_{T}}{\partial T}=-d\beta_{T}.
\end{align}
\end{thm}

We identify $X$ as a submanifold of $\mathcal{E}$ by the zero section.  The normal bundle
to  $X$ in $\mathcal{E}$ is exactly $E$ and the conormal bundle is $E^{*}$.
Let $\delta_{X}$ be the current on $\mathcal{E}$ defined by  the integration  on $X$.
For a current $v$ on $\mathcal{E}$, denote by $\mathrm{WF}(v)\subset T^{*}\mathcal{E}$ its wave front set.

\begin{thm}
Let $K\subset
\mathcal{E}$ be a compact subset of $\mathcal{E}$. There is $C_{K}>0$ such that  for any $u\in
\Omega^{\cdot}(\mathcal{E},\pi^{*}o(TX))$ whose support is contained in $K$
and for any $T\geqslant 1$, we have
\begin{align}\label{eqabT}
&\left|	\int_{\mathcal{E}}u(\alpha_{T}-\delta_{X})\right|\l
\frac{C_{K}}{\sqrt{T}}\|u\|_{C^{1}},&\left|\int_{\mathcal{E}}u \beta_{T}\right|\l
\frac{C_{K}}{T^{3/2}}\|u\|_{C^{1}},
\end{align}
where $\|\cdot\|_{C^1}$ denotes the $C^{1}$-norm.
\end{thm}

 In view of \eqref{eqabT},
 \begin{align}
	 \psi\(E,\nabla^{E}\)=\int_{0}^{\infty}\beta_{T}dT,
	 \end{align}
 is a well defined current of degree $n-1$ on $\mathcal{E}$ with values in $\pi^{*}o(E)$.

\begin{thm}
The current $\psi\(E,\nabla^{E}\)$ is locally
integrable such that
\begin{align}\label{WFphi}
	\mathrm{WF} \big(\psi\(E,\nabla^{E}\)\big)\subset E^{*}.
\end{align}
  The following identity of currents
on $\mathcal{E}$ holds,
\begin{align}
	d\psi\(E,\nabla^{E}\)=\pi^{*}e\(E,\nabla^{E}\)-\delta_{X}.
\end{align}
\end{thm}

 \begin{defin}
The  current  $\psi\(E,\nabla^{E}\)$ is called the Mathai-Quillen
current.
\end{defin}

\begin{re}
	The restriction of $\psi(E,\nabla^{E})$ to the
	sphere bundle  of $E$
	was first constructed in Mathai-Quillen \cite[Section
	7]{MathaiQuillen}.
	If $E=TX$, this restriction  coincides with the transgressed Euler
	class defined by Chern \cite{Chern2}.
\end{re}

\subsection{Ray-Singer metric}\label{sRS}
We use the notation in Section \ref{S1M}. Recall that $F$ is a flat  vector bundle on $X$.
Let $(\Omega^{\cdot}(X,F),d^{X})$ be the de Rham complex with values in
$F$.  By de Rham's theory, its  cohomology is $H^{\cdot}(X,F)$.

Take metrics $g^{TX}$ and $g^{F}$ on $TX$ and $F$. Let
$\<,\>_{\Lambda^{\cdot}(T^{*}X)\otimes F}$  be the induced metric on
$\Lambda^{\cdot}(T^{*}X)\otimes F$.  Let $dv_{X}\in \Omega^{m}(X,,o(TX))$  be the volume form on $X$.
For $s_{1},s_{2}\in \Omega^{\cdot}(X,F)$, set
\begin{align}\label{eqX}
	\<s_{1},s_{2}\>_{\Omega^{\cdot}(X,F)}=\int_{x\in X}\<s_{1}(x),s_{2}(x)\>_{\Lambda^{\cdot}(T^{*}X)\otimes
	F}dv_{X}.
\end{align}
Then \eqref{eqX} defines an $L^{2}$-metric on $\Omega^{\cdot}(X,F)$.

Let $d^{X*}$  be the formal adjoint of $d^{X}$ with respect to the
$L^{2}$-metric $\<\cdot,\cdot\>_{\Omega^{\cdot}(X,F)}$. Put
\begin{align}
	\Box^{X}=d^{X}d^{X*}+d^{X*}d^{X}.
\end{align}
Then $\Box^{X}$ is a formally self-adjoint  second order elliptic
operator acting on $\Omega^{\cdot}(X,F)$. By Hodge theory, we have
\begin{align}\label{Hodge}
	\ker \Box^{X}\simeq H^{\cdot}(X,F).
\end{align}
By \eqref{eqlambda} and \eqref{Hodge}, the restriction of the $L^{2}$-metric $\<\cdot,\cdot\>_{\Omega^{\cdot}(X,F)}$ to $\ker
\Box^{X}$ induces a metric $|\cdot|^{RS,2}_{\lambda}$ on $\lambda$.

Let $(\ker \Box^X)^{\bot}$ be the orthogonal  space to $\ker \Box^X$
in $\Omega^{\cdot}(X,F)$. Then $\Box^{X}$ acts as an invertible
operator on $\(\ker\Box^{X}\)^{\bot}$. Let $\(\Box^{X}\)^{-1}$ be the
inverse of $\Box^{X}$ acting on $\(\ker\Box^X\)^{\bot}$. Let
$N^{\Lambda^{\cdot}(T^{*}X)}$ be the  number operator on
$\Lambda^{\cdot}(T^{*}X)$, which is multiplication by $p$ on
$\Lambda^{p}(T^{*}X)$. For $s\in \mathbf{C}$ such that $\Re(s)>\frac{m}{2}$, set
\begin{align}
	\theta(s)=-\Tr\[(-1)^{N^{\Lambda^{\cdot}(T^{*}X)}}N^{\Lambda^{\cdot}(T^{*}X)}\(\Box^{X}\)^{-s}\].
\end{align}
By a result of Seeley \cite{Seeley66} or by \cite[Theorem 7.10]{BZ92}, $\theta(s)$ extends to a meromorphic function
of $s\in \mathbf{C}$, which is holomorphic at $s=0$.

\begin{defin}
The Ray-Singer metric $\|\cdot\|^{RS,2}_{\lambda}$ on
$\lambda$ is defined  by
\begin{align}
	\|\cdot\|^{RS,2}_{\lambda}=|\cdot|^{RS,2}_{\lambda}\exp\big(\theta'(0)\big).
\end{align}
\end{defin}

Let $\nabla^{TX}$ be the Levi-Civita connection on $(TX,g^{TX})$. Let
 $\psi(TX,\nabla^{TX})$ be the Mathai-Quillen current.
By \eqref{eqMS} and \eqref{WFphi}, for any Morse-Smale vector field
$V$, the pull back $V^{*}\psi(TX,\nabla^{TX})$ is a well
 defined current of degree $(m-1)$ on $X$ with values in $o(TX)$. Set
\begin{align}
	\theta\(F,g^{F}\)=\Tr\[g^{F,-1}\nabla^{F}g^{F}\]\in \Omega^{1}(X).
\end{align}
Then $\theta(F,g^{F})$ is a closed $1$-form.

The main result of Bismut-Zhang \cite[Theorem 0.2]{BZ92} is the following.

\begin{thm}\label{BZ}
	Suppose that  $f$ is a Morse function, whose gradient $\nabla f$ with respect to
some Riemannian metric is Morse-Smale. The following identity holds,
	\begin{align}\label{eqBZ}
	\log
	\(\frac{\|\cdot\|^{RS,2}_{\lambda}}{\|\cdot\|^{M,2}_{\lambda,\nabla f}}\)=-\int_{X}\theta\(F,g^{F}\)\(\nabla f\)^{*}\psi(TX, \nabla ^{TX}).
\end{align}
\end{thm}

\subsection{Proof of the main result}\label{s33}

 We restate our main
result Theorem \ref{thm1}, which is an extension of Theorem \ref{BZ}.
\begin{thm}\label{Thm1}
	Suppose that $V$ is a Morse-Smale flow. The following identity holds,
	\begin{align}
	\log
	\(\frac{\|\cdot\|^{RS,2}_{\lambda}}{\|\cdot\|^{M,2}_{\lambda,V}}\)=-\int_{X}\theta\(F,g^{F}\)V^{*}\psi\(TX,\nabla^{TX}\).
\end{align}
\end{thm}
\begin{proof}
Take 	$\(V_{\ell}\)_{\ell\in [0,1]}$ as in Proposition
\ref{propV01}. By our construction, we have
\begin{align}\label{MVMV1}
	\|\cdot\|^{M,2}_{\lambda,V}=\|\cdot\|^{M,2}_{\lambda,V_{1}}.
\end{align}
Proceeding as in \cite[Proposition 6.1]{BZ92},  we have
\begin{align}\label{XVV1}
	\int_{X}\theta\(F,g^{F}\)V^{*}\psi\(TX,
	\nabla^{TX}\)=\int_{X}\theta\(F,g^{F}\)V^{*}_{1}\psi\(TX, \nabla^{TX}\).
\end{align}
By \eqref{MVMV1} and \eqref{XVV1}, it is enough to show our theorem
for the Morse-Smale vector field $V$ whose  critical elements are all
of standard forms.

Take $f$  as in  Proposition
	\ref{propfV}.
By Proposition \ref{propMM} and Theorem  \ref{BZ}, we have
\begin{multline}\label{LLa}
	\log
	\(\frac{\|\cdot\|^{RS,2}_{\lambda}}{\|\cdot\|^{M,2}_{\lambda,V}}\)=-\int_{X}\theta\(F,g^{F}\)\(\nabla f\)^{*}\psi(TX, \nabla ^{TX})\\
	+\sum_{\gamma\in
		B}(-1)^{\ind(\gamma)}\log
		\left\|\det\tau(a_{\gamma}')\right\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}.
\end{multline}
By \eqref{LLa}, it remains to show
\begin{multline}\label{eqaaa}
	\int_{X}\theta\(F,g^{F}\)\(\nabla
	f\)^{*}\psi\(TX,
	\nabla^{TX}\)-\int_{X}\theta\(F,g^{F}\)V^{*}\psi\(TX,\nabla^{TX}\)\\
=\sum_{\gamma\in
		B}(-1)^{\ind(\gamma)}\log
		\left\|\det\tau(a_{\gamma}')\right\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}.
\end{multline}
We assume all the closed orbits are in standard form
of Case (1).  Cases (2)-(4) can be dealt similarly.

Following \cite[Section IV c)]{BZ92},
choose a smooth triangulation $K$ of $X$ such that $A\cap
K^{m-1}=\varnothing$, and such that on $\overline{U}_{\gamma}\simeq \mathbb{S}^{1}\times
\overline{\mathbb{D}}^{m-1}$ the
triangulation is given by the $m$-simplex
$\sigma^{m}_{\gamma}=\(\mathbb{S}^{1}-\{1/4\}\)\times
\overline{\mathbb{D}}^{m-1}$ and $(m-1)$-simplex
$\sigma^{m-1}_{\gamma}=\{1/4\}\times \overline{\mathbb{D}}^{m-1}$.

On each simplex $\sigma\in K^{m}\backslash K^{m-1}$ of maximal degree, choose a
primitive $W_{0,\sigma}\in C^{\infty}(\sigma)$ of $\theta(F,g^{F})$,
such that on $\sigma$ we have
\begin{align}
	dW_{0,\sigma}=\theta\(F,g^{F}\).
\end{align}
Let $W_{0}$ be the locally integrable current  on $X$, such that
for each $\sigma\in  K^{m}\backslash K^{m-1}$ we have
\begin{align}
W_{0}|_{\sigma}=W_{0,\sigma}.
\end{align}
 By our construction of $K$, for $\gamma\in B$, the two points
$x_{\gamma},x_{\gamma}'$, and the integral curve $a_{\gamma}'$ are in the same simplex $\sigma^{m}_{\gamma}\in
K^{m}$, so that
\begin{align}\label{W0W0}
	W_{0}(x_{\gamma}')-W_{0}(x_{\gamma})=\log
		\left\|\det\tau(a_{\gamma}')\right\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}.
\end{align}

Set
\begin{align}
W_{1}=\theta\(F,g^{F}\)-dW_{0}.
\end{align}
Then $W_{1}$ is  a closed
current  of degree $1$ on $X$ such that $\mathrm{Supp}(W_{1})\subset K^{m-1}$. By \cite[(6.16)]{BZ92} and by \eqref{W0W0}, we see that \eqref{eqaaa} is equivalent to
\begin{align}\label{eqLast}
	\int_{X}W_{1} \wedge (\nabla
	f)^{*}\psi\(TX,\nabla^{TX}\)=\int_{X}W_{1}\wedge
	V^{*}\psi\(TX,\nabla^{TX}\).
\end{align}

By \eqref{eqdf=V}, on any simplex in
$K^{m-1}$ other than $\sigma_{\gamma}^{m-1}$, we have $\nabla f=V$.
By Remark \ref{reK}, near $\sigma_{\gamma}^{m-1}$, $\nabla
	f$ and $V$ can be connected by a family of vector fields without
	zero. Using the fact that $\mathrm{Supp}(W_{1})\subset K^{m-1}$,
	and proceeding as  \cite[Proposition 6.4]{BZ92}, we get \eqref{eqLast}.
The proof of our  theorem is completed.
\end{proof}

\def$'${$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


text:
Jaffery Academy Head of School (Senior Section) Jobs in Mombasa, Kenya
Jaffery Academy, Mombasa (JAM) is a welcoming friendly school with a vibrant learning space incorporating the latest technology to engage and inspire students.
The school provides high quality education from the ages of 3 to 18 and follows the UK Curriculum (IGCSE, AS & A Levels).
JAM is seeking to appoint an innovative and inspirational Head of School for its Senior Section, who will be required to work in a fast paced environment with proven and effective leadership and management skills.
Evidence of a strategic approach and philosophy to raising the status of the school and its academic, pastoral, sporting and co-curricular standards is required.
Head of School – Senior Section
Reporting To:
Head of Academy
Direct Reports:
Dean of studies, Deans of subjects, Dean of sports, senior school teachers, laboratory technicians and Librarian.
The Head of School will be responsible for internal organization, management and control of the senior school and provide strong teaching and learning aspects by supporting teachers and students in achieving high academic standards;
Leadership role
In consultation with the Board and the Head of Academy, formulating the overall aims, objectives and policies of the school and ensuring their implementation;
Leading by example, providing educational vision and direction to secure the strong and passionate commitment of staff, parents and students;
Ensuring that staff are aware of current educational developments; Ensuring that all aspects of school performance are monitored and evaluated in a robust, cyclical manner and maintaining a record of self-evaluation and areas for improvement, and of progress made in respect of these;
Allocating, controlling and accounting for those financial and material resources of the school which are under his/her control;
Attend meetings of the Board, provide Board Members with regular reports on school developments and activities and consult with the Head of Academy and the Board wherever appropriate, for example in the formulation of development plans; and support the busy and varied life of the school, attending its events and wider programme of activities.
Human resource management and Professional Development
Leading the selection and appointment of the teaching and non-teaching staff of the senior school;
Deploying and managing all teaching and non-teaching staff of the school and allocating particular duties to them in a manner consistent with their conditions of employment, maintaining a reasonable balance for each teacher between work carried out in school and work carried out elsewhere;
Leading, motivating, supporting, challenging and developing staff at all levels in order to secure and sustain continuous improvement and staff wellbeing and to be committed to personal continuing professional development; and managing staff performance by appraising staff and recommending for rewards to high performers and performance improvement plans to under-performers;
Ensuring high standards of professional practice and quality of teaching and learning of the subject/s through effective dialogue, participating in reciprocal peer review and observation of class teaching practice by the Head of Academy;
Reviewing and evaluating teachers' teaching and learning strategies, methodologies and programme(s); and participating in in-service education and training courses as well as in continuing professional development (CPD) opportunities.
Teaching, Learning and Curriculum
Ensuring that all lessons in the school are delivered in a planned, prepared and delivered as per the approved curriculum;
Determining, organizing and implementing a broad and balanced curriculum for the school, having regard to the needs, experiences, interests, aptitudes and stage of development of the students and the resources available to the school;
Evaluating the standards of teaching and learning in the school, and ensuring that proper standards of professional performance are established and maintained;
Ensuring that improvements in the curriculum are a priority for all students including the gifted, slow and talented learners;
Ensure the highest possible standards of education through the promotion of effective teaching and learning;
Teaching in area of specialization; and ensure that teachers use creative, interactive and engaging teaching methods to encourage students to actively participate in classroom activities and develop holistically.
Ensuring that the progress of students
of the school is monitored and recorded accordingly;
Ensuring continuing effective systems of planning, assessment, recording and reporting, using data and benchmarks to monitor attainment and progress, in every child's learning;
Using this data continuously to support and improve standards;
Ensuring that appropriate feedback is given to parents with regard to records of students' progress and development;
Ensuring that internal and external examinations are prepared accordingly and registrations done in time;
Pay attention to individual student's progress and evaluate their learning abilities;
Ensuring that all students are given assignments that are corrected and marked accordingly; and have a clear understanding of the needs of all students, including those with special educational needs; gifted and talented; disabilities; and be able to use and evaluate distinctive teaching approaches to engage and support them.
Extra -curricular activities
Ensuring that the school participates in games and other co-curricular activities; Participating in school/academic meetings, parent's meetings and whole school training events;
Participating in staff, group or other meetings related to the school for the better organization and administration of the school;
Participating in school assemblies/events;
Ensuring safe school trips for students going for extra-curricular activities.
Discipline and moral development
Ensuring that proper standards of behavior among staff and students are implemented in the school during the school day and whenever the students are engaged in authorized activities outside the school;
maintaining productive working habits and discipline in the classroom;
Observing, assessing and recording students' behavior; and managing and applying appropriate and effective measures in case of misbehavior.
Providing a safe and healthy environment for the wellbeing of staff, students, visitors and neighbours;
Promoting equality and inclusion in all aspects of school life to all teaching and Non-teaching staff Making arrangements for the security, maintenance, development and effective supervision of the school;
Ensuring that child protection and safeguarding of students are given high priority at all times;
Providing pastoral care and support to the teachers and students; and taking on the role of mentor/class teacher, and supporting students on an individual basis through academic or personal difficulties.
Making arrangements for parents to be given regular information about the school curriculum, the progress of their children and other matters affecting the school, so as to promote common understanding;
Creating and maintaining an effective partnership with parents/guardians to support and improve students' achievement and personal development;
Working with parents/guardians to ensure children have access to extended services (g. counselling, spiritual guidance etc.), extra-curricular opportunities, homework and other social and educational experiences;
Qualifications and Key Skills Required
A Bachelor's Degree from a recognised University in Education or equivalent
A Master's Degree in Education or related field is preferred.
Must be registered with Teachers Service Commission.
At least Ten years' teaching experience
Experience in leading a whole school or section
Knowledge of quality assurance and accountability systems
Knowledge and understanding of current trends in teaching and learning practices
Excellent interpersonal and communication skills, and the ability to form effective relationships with all stakeholder groups
Ability to lead strategies successfully
An attractive salary based on the skills and experience of the successful candidate, by negotiation with the School Board.
Pension plan.
Details of contract: A 3-year renewal employment contract based on satisfactory performance which commences on 1st August 2019 (or by negotiation with the successful candidate.)
If you are qualified and motivated to work in a school with an international curriculum, we would like to hear from you.
Kindly submit your application accompanied by a detailed CV and relevant documents to chairman@jafferyacademy.org not later than 15th April 2019.
Find jobs in Kenya. Jobs - Kenya jobs. Search our career portal & find the latest Kenyan job positions, career opportunities & jobs in Kenya.
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redpajama_set_name: RedPajamaCommonCrawl


Insurance is a crucial step in planning for your future and the future of your loved ones.
Let Robert Edgar take the confusion out of your Health and Life Insurance plans. As an independent Insurance Agent and Broker Rob has flexible plans to suit every situation and budget. Give Rob a call today for a no cost or obligation assessment of your families particular situation and needs.
Rob Edgar offers a variety of Life Insurance policies to suit your families needs. Term Life, Whole Life or Universal Life products available. Protect your family in the event of an unfortunate tragedy.
Schedule A No Cost, No Obligation Consultation Visit Today!
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Are Your Diabetes Prescription Costs Spiraling Out Of Control?


text:
One million people show solidarity with Dakota Access Pipeline protesters by checking in on Facebook
under News, social media
People are jumping on social media to show their support for the Dakota Access Pipeline protesters. Over one million people checked in to the Standing Rock Indian Reservation and Standing Rock, North Dakota on Facebook in an effort to bewilder police. While law enforcement claims they're not tracking people via Facebook, supporters continue to check in and show solidarity.
On Monday, a Facebook post challenged people to back pipeline protesters on the ground and "overwhelm and confuse" law enforcement by checking in to Standing Rock, North Dakota on Facebook. Thousands responded, including the Morton County Sheriff's Department, which said on Facebook, "The Morton County Sheriff's Department is not and does not follow Facebook check-ins for the protest camp or any location. This claim/rumor is absolutely false."
Related: Dakota Access Pipeline protesters raise over $1 million on GoFundMe
This statement didn't deter the thousands of people who checked in and continue to check in on Facebook. Sacred Stone Camp says they did not start the Facebook movement but in a statement emailed to Mic said checking in is a "great way to express solidarity" and that there is "no doubt that law enforcement monitor communications and comb social media for incriminating material."
They also noted there are many other actions people can take beyond a Facebook check in. They asked people to consider their own consumption of fossil fuels, and get involved with environmental or indigenous struggles near their own homes. They called on CitiBank, Mizho Bank, and the Bank of Tokyo to deny the Dakota Access Pipeline a $1.1 billion loan. Supporters of the protesters can also contribute to legal defense on FundRazr or donate to Sacred Stone Camp on GoFundMe.
Sacred Stone Camp also called for more people to actually come to Standing Rock, saying "We also need 10,000 to 100,000 people to join us here on the ground. Now."
Via The Guardian and Mic
Images via Fibonacci Blue on Flickr and Sacred Stone Camp Facebook
Dakota Access Pipeline Protest Solidarity Full Width Tall
Over one million people checked in to Standing Rock, North Dakota on Facebook to show solidarity with those protesting the Dakota Access Pipeline.
Standing Rock Solidarity
Law enforcement said they were not tracking people via social media.
Dakota Access Pipeline Protest Solidarity
Yet many people checked in anyway to show support for those protesting the oil pipeline.


We'd all be a little wiser and a lot richer if we knew some basic things about the cars we drive. Yet we rely on so called "expert mechanics" to diagnose problems with our vehicles and then fix them. We are forced to trust (hope) that the mechanics we deal with are honest enough to tell us the truth about our cars. Sadly, this is rarely the case. In fact, most mechanics go out of their way to treat their customers like mushrooms – keeping them in the dark and feeding them crap. Many mechanics know some basic truths about cars that they would never want any of us "common folk" to know. Instead, they prefer to deal with ignorant people that they can sell lies to and charge a lot of money. Fortunately, a little knowledge goes a long way. Here are 12 things about your car that your mechanic does not want you to know.
The most common reason for the check-engine light being lit up on the dashboard of a car is because the gas cap is not screwed on tight enough. Sad but true. A sensor in the car responds when the gas cap is not secured tight enough and extra oxygen seeps through the gas line. The check-engine light almost always goes off once the cap has been tightened and/or the entire tank of gas has been used and is empty. Remember this the next time the check-engine light comes on the dashboard of your car. Before you panic, give the gas cap a few extra turns with your hand. You could save yourself a big headache and expensive bill from a mechanic.
Mechanics love to tell people that their car battery has died. However, most problems with car batteries are actually caused by the wires connected to the battery and not the battery itself. Over time, the point where the wires are connected to the car battery become corroded and this impacts the amount of power getting from the battery to the car. The fix for this problem? A can of Coca-Cola. Take some Coke and pour it on the area where the wires connect to the car battery and the Coke will eat away at the corrosion. Some "CLR" also works well. This is a quick and easy home fix that need not involve a mechanic. And it can save you on the cost of a new car battery. Also, many car batteries that are running low on power can be recharged. They needn't be replaced.
So, you've brought your car to the garage and paid a little extra to have the tires rotated. After all, the mechanic advised that the tires should be rotated to ensure the tread on them wears properly. Only problem is that when you get the car back, you can't tell if the tires have been rotated at all. They look the same as they did when you brought the car to the garage. This is because few garages actually rotate the car tires. They lie and tell you they did while taking an extra $50 out of your pocket. Furthermore, there is a raging debate about whether rotating the tires is really needed. Some studies show that tires wear out just as fast whether they are rotated or not. To ensure that your car tires are actually rotated, take a piece of chalk and mark them before bringing the car to a garage. Write "FL" on the "front left" tire; "RR" on the "rear right" tire, etc. And remember, your tires are supposed to be rotated front-to-back, not side-to-side.
Most mechanics will tell you that your car needs an oil change every 3,000 miles. Not true. Most car experts agree that the oil in a car should be changed once every 7,500 miles at the earliest, although many folks in the auto industry will tell you that an oil change every 10,000 miles or 16,000 kilometers is perfectly acceptable. What is more important than an oil change is for drivers to ensure that their car does not run out of oil. Adding oil and keeping the oil topped up in your car is much more important than changing the oil. While mechanics will tell you that dirty oil is bad for your car engine, the reality is that engine oil burns off and evaporates over time, meaning that dirty oil is not that big of an issue for cars today. But the oil does get lower over time, so buying a litre and pouring it in yourself is advisable.
Do you hear the brakes on your car screeching? Is it freaking you out? It shouldn't be. The biggest problem with brake pads, and the main reason they screech, is rust. And this rust can be quickly and easily scraped, brushed, or blasted off. You needn't shell out a lot of money for new brake pads. In fact, most people can scrape the rust off their brake pads themselves with the help of a YouTube video to guide them. Most garages will take the rust off of brake pads for not a lot of money. The trick is getting the mechanic to tell you that the problem is rust on the pads. Mechanics will almost always say that you need a whole new set of brake pads. Don't believe them.
Unless you live in northern Canada, chances are you don't need winter tires. The fact is that a good set of all-season tires will do most people just fine. Yet the automotive industry is making a killing by convincing people that they need two sets of tires – one for spring and summer and another for fall and winter driving. Mechanics are making a good deal of money simply taking off one set of tires and putting on another. Many garages also charge a fee to store people's summer tires in the winter and winter tires in the summer. This is a racket, plain and simple. Tests have shown that all-season tires have sufficient tread to stop vehicles in most winter driving conditions and that they work great during the summer as well. Winter tires are helpful, but are only truly necessary in the most extreme winter driving conditions. Unless you're planning to drive exclusively on icy roads, one set of all-season tires will suffice.
So your local garage is offering free bumper-to-bumper inspections of cars, trucks, and any other vehicle they can get their hands on. You think you'll take advantage and swing by to get your car looked at. After all, the inspection is free. Not so fast: what the garage is doing is providing an incentive for people to bring their cars in so that the mechanics can find problems with them and then charge a fortune to fix them. It's a way for mechanics to drum up business for themselves. We guarantee that you'll never bring your car in for a free inspection and hear a mechanic say "Looks great. I don't see any problems. Take care, now." More likely that the mechanic will come back to you with a laundry list of needed repairs and upgrades. Don't fall for this scam. Bring your car into a garage when it needs a repair only. Avoid anything involving a garage that is allegedly free.
It seems like places are popping up all over that offer quick lube jobs. People can pull in with their car and in 15 to 20 minutes have their engine oil replaced, the engine parts lubricated, and the tires filled with air. Wow! What a waste of time and money. The reason these types of places are popping up all over the place is because they are a cheap and easy way for people to make money off unnecessary services. We've already gone over when cars should really have their oil changed, and most people are capable of putting air in their own tires. Which brings us to the lube jobs. Most cars never need to have their engine parts lubricated. They are either already lubricated or don't require lubrication. When you pay for a lube job, you're paying to have a teenager smear gunk around under the hood of your car. Save your money, people.
The mechanic comes out to meet you in the waiting room and tells you that the brake pads on your car are 50% worn and should be replaced. You agree and give the go-ahead to have the brake pads swapped out. The only thing is that brake pads don't actually need to be replaced until they are 90% worn. A brake pad that is 50% worn is only half way through its usable life. Keep the brake pads until they are at least 90% worn. The problem with brakes is that people see them as a safety issue and don't want to take any chances with them. Mechanics prey on this fear. Trust yourself. if there are any problems with your brakes, you will know it right away and can get them checked. Keep your brake pads as long as you can; it'll save you money in the long run.
Given a choice, most drivers would avoid unsightly rust on their car. After all, a rusting car looks junky. However, over time, some rust on a vehicle is inevitable – especially on cars that are seven years old or older. And here's the rub: there is no evidence that rustproofing a car actually prevents rust. To the contrary, there are some studies that show rustproofing can actually lead to more rust on a car. This is because ice and road salt tend to stick to the rustproof coating that is sprayed underneath a car. Rustproofing is popular in the northern U.S. and in Canada, but it provides no real benefit to your car and often ends up ensuring that a bunch of unwanted ice and salt gets trapped underneath a car, causing more damage in the long run.
As we've established, swapping out brake pads is big business for mechanics. This is because replacing brake pads is relatively easy and a lot of money can be charged for the repair. In reality, changing brake pads is so easy that most people can do it themselves. There are plenty of websites and videos online that show how to quickly, easily, and safely change a vehicle's brake pads. While many people shy away from doing this job themselves for fear they will not do it properly and it could be a safety concern, there really is no reason to worry. People will know instantly if they have not put their brake pads on properly. In only a half hour, people can change their brake pads themselves and save money in the process. But again, only do this when the brake pads are 90% worn.
There are many books published each year that detail the number of labor hours that garages should charge for each repair they undertake on a vehicle. Trust us when we say that your mechanic does not want you to know this. That is because most mechanics want the freedom to charge their customers as many hours and as much money as possible for the work they perform on a car. Unsurprisingly, they rarely charge the standard rate. Arming yourself with information on the correct amount of time and money that should be charged for each repair will empower you and give you leverage over the mechanic you're dealing with. The next time you are at a garage, ask for a breakdown of the labor hours you're being charged for, then ask to see the labor hours in the guide. If the mechanic doesn't want to show you, or starts making excuses, or if the hours don't match, it's time to take your business someplace else. That mechanic you are dealing with is neither trustworthy nor reputable.
Little things make a big difference, and this is especially true when it comes to cars.
Nothing is more important than safety when it comes to motor vehicles.
There was no shortage of new luxury sports cars manufactured in 2015.
Not every car that is designed, manufactured and marketed to the public is a winner.
There have been many famous cars featured in movies and television shows over the years.
Consumer Reports has released its annual list of their favorite 2016 model year vehicles.


text:
The Samara Bl g
Saturday, February 23, 2013 View Count = 4009
Restoring Parliamentary Supremacy
In today's Redesigning Parliament post, professor and author Maxwell A. Cameron says it's time to enforce/create rules that remove power from the Prime Minister's Office and ensure it stays in the hands of individual MPs.
How might we redesign parliament for the 21st century? We can begin by asking what we expect or want from parliament. My own view is that parliamentary government is not well understood and, therefore, it might be wise to elucidate the principles and aims that are already inherent in our institutions, and then see if we can make them work better.
The essence of our system of government is captured by the phrase "parliamentary supremacy." Today, however, parliament is anything but supreme.
Parliamentary supremacy is the idea that parliament is the source of all legislation. Where our system and presidentialism differ is that the executive branch is selected by the parliament. In principle, this means that the executive is embedded within a larger, and more important, collegial body. But over time the tail has come to wag the dog. Power has gradually been concentrated—first in the cabinet, and now, increasingly, in the office of the Prime Minister. The main source of this inversion of power is the modern political party.
Today, parliament is a pathetic semblance of its former self. It is an electoral college that chooses the government, which then governs with scarcely any concern for parliamentary debate or procedure. MPs are well-trained "yes" men and women who do what their party leaders tell them. They have little say over committee assignments, few free votes, and, what is worse, they devote precious little effort to legislation.
The inversion of power has come about because parties—which are increasingly PR firms devoted to permanent campaigns for office—control MPs from the moment they are nominated until the day they retire from public life. MPs are more accountable to their party leaders than to the voters. And voters are fed up.
So, if our aim is to restore parliament to some semblance of its former glory, what kinds of measure might we adopt? To restore the supremacy of parliament would require the empowerment of ordinary MPs. There are many measures that could move us in this direction. They include more free votes, more opportunities for ordinary MPs to submit legislation, more power in the hands of MPs to decide committee assignments and distribute other perks, more power to select the cabinet and remove the leader, and more influence in decisions like prorogation.
The corollary would be less power in the hands of the Prime Minister. Less power to use confidence votes to discipline the house and caucus, restrictions on the power to dissolve the house, elimination of the routine use of time limits to shut down debate, tough rules to stop the abuse of omnibus laws, less influence over perks, assignments, and appointments, and less control over nominations.
The bottom line: parliamentary supremacy, the cornerstone of our system, demands that we clip the wings of the office of the Prime Minister. The alternative is a slide toward elective autocracy.
Maxwell A. Cameron is a Professor of Political Science and Director of the Centre for the Study of Democratic Institutions at UBC. He is the author of Democracy and Authoritarianism in Peru (St. Martin's 1994), and co-author of The Making of NAFTA: How the Deal Was Done (Cornell, 2000). Between March and June 2006, Cameron served as political advisor to Lloyd Axworthy, Chief of the Electoral Observation Mission of the Organization of American States in Peru. In 2011 he convened a public conversation at UBC on "Why don't (more) good people enter politics?"
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text:
Josse Lieferinxe fue un pintor de finales del siglo XV y comienzos del siglo XVI (activo entre 1493 y 1503). Ha sido identificado con el maestro anónimo denominado Maestro de San Sebastián. Puede considerársele como pintor flamenco, aunque también incluírsele entre los maestros de la escuela de Provenza, cuyos más prominentes miembros (Barthélemy d'Eyck y Enguerrand Quarton) también procedían del norte.

Su apellido parece sugerir su procedencia de la localidad de Liefringem, cerca de Enghien, en la diócesis de Cambrai (Hainaut), por entonces parte de los Países Bajos de los Habsburgo. Con el nombre de Josse Lieferinxe y el gentilicio "picardo" (de la Picardía) aparece documentado trabajando en la región de Provenza (su primera mención, en 1493) y a finales del siglo XV y comienzos del siglo XVI en Aviñón y Marsella. En 1503 contrae matrimonio con Michelle, hija de Jean Changenet, el más importante de los pintores de Aviñón, en cuyo taller Lieferinxe es posible que hubiera madurado su estilo. Su última mención con vida es de 1505, y para 1508 ya es mencionado como fallecido.

Antes de ser identificado por Charles Sterling, que conectó su obra con un documento, su personalidad artística era reconocida como "Maestro de San Sebastián", por un retablo de ocho escenas que representa la vida y milagros de San Sebastián y San Roque, protectores de la peste, que fue encargado en 1497 para la iglesia de Nôtre Dame des Accoulés de Marseille. Bernardino Sismondi, el pintor que recibió inicialmente la comisión, murió antes de terminar la obra. A comienzos del siglo XX el retablo había sido desmontado y sus tablas dispersadas por distintos museos. Otras obras identificadas por su estilo son unos paneles de un retablo de la vida de la Virgen: dos pintados por ambos lados en el Musée Calvet de Aviñón (Circuncisión y Anunciación, y Santa Catalina y San Miguel Matando el Dragón) y unas Bodas de la Virgen en los Musées Royaux de Bruselas; un tercer panel, fragmentario, también pintado por ambos lados, se conserva en el Musée du Louvre.

Obras

Vida y milagros de San Sebastián, 1497.
San Sebastián ante Diocleciano, 1497 (Museo del Hermitage).
San Sebastián destruyendo ídolos paganos, 1497 (Philadelphia Museum of Art).
 San Sebastián intercedeiendo por los enfermos de peste.
Martirio de San Sebastián, 1497, y dos otros paneles (Philadelphia Museum of Art).
 San Sebastián intercede por los enfermos de peste, 1497 (Walters Art Museum, Baltimore).
Peregrinos ante la tumba de San Sebastián (Galleria Nazionale d'Arte Antica, Roma.
El Arcángel San Miguel matando al dragón (Musée du Petit Palais, Aviñón).
Abraham visitado por tres ángeles, ca. 1500 (Kress Collection, Denver Art Museum).
Calvario (Musée du Louvre).
Retablo. Scattered wing panels y otros elementos. (Royal Museum of Fine Arts, Brussels; Musée du Louvre; Musée Calvet, Aviñón).
Ecce Homo (Biblioteca Ambrosiana).
Pietà (Amberes).
Adoración del Niño Jesús, al reverso Santo obispo (Musée du Louvre).
Visitación, al reverso, Santa Lucía (Musée du Louvre).

Notas

Bibliografía
Ring, Grete. "A Century of French Painting, 1400–1500". (London): Phaidon Press, 1949.

Enlaces externos

Web Gallery of Art: Josse Lieferinxe
Lista de obras en Centre for the Study of Fifteenth-Century Painting

Pintores de Flandes del siglo XV
Pintores de Francia del siglo XV
Fallecidos en Marsella

redpajama_set_name: RedPajamaWikipedia


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text:
RICHARD CARRICK
Richard Carrick, a Guggenheim fellow, is a composer, conductor and pianist who writes music of spatial depth and robust stasis, characterized by continual development and the evocation of profound human experiences. Described both as "charming, with exoticism and sheer infectiousness" and "organic and restless" by The New York Times, Carrick's music is influenced by his multicultural background and experiences as well as his commitment to inspire professionals, audiences and youth through composition and live performance.
His CD release, Cycles of Evolution, incorporates pieces commissioned and performed by Musicians of the New York Philharmonic, Either/Or, Sweden's Ensemble Son, Hotel Elefant and String Orchestra of Brooklyn. Carrick conducts or performs on all works on this CD, which includes his 'apocalyptic' multimedia work for performers and video, Prisoner's Cinema. Carrick's first recording also on New World Records, the "rich, beguiling" (The New York Times) extended chamber composition Flow Cycle for Strings, translates psychologist Mihaly Csikszentmihalyi's 'flow' principle into sonic terms. Carrick's improvisation-based disc Stone Guitars garnered critical attention in both the new music and guitar worlds, causing American Record Guide to note 'it may change your perception of electric guitar'.
While on Guggenheim Fellowship in 2015-16 Carrick moved to Rwanda and and penned a new official arrangement of the Rwandan National Anthem, premiered Waterfall for piano and orchestra, and initiated a young composer program in Kigali.
Recent activities include A three premiere tour of Israel including String Quartet #2 - Space:Time (Fromm Foundation Commission) performed by the Mivos Quartet and Nadav Lev, "The Music of Richard Carrick" at Miller Theatre in January 2019, Pacific Rim Festival premiere sandstone(s) for Korean and Western instruments, Enescu Festival premiere as conductor and composer with Dumitrescu's Hyperion Ensemble and Akko for large ensemble in 2017, la touche sonore sous l'eau and Les ciels de Kigali performed internationally by Marilyn Nonken, and Gugak International Workshop Fellow in Seoul.
His music has been programmed and presented internationally at festivals including NYPHIL BIENNIAL, ISCM World Music Days-Switzerland, Library of Congress, Enescu Festival, Pacific Rim Festival, Miller Theatre, Mid-American New Music Festival, and Darmstadt Summer Festival, and performed by musicians including the JACK Quartet, Mivos Quartet, Nieuw Ensemble, Wet Ink Ensemble, New York New Music Ensemble, Sequitur Ensemble, Musica Nova (Tel Aviv), Hotel Elefant, Marilyn Nonken, Taka Kigawa, Margaret Lancaster, Vasko Dukovski, Jennifer Choi, Tony Arnold, Magnus Andersson, Steven Schick, Rohan de Saram, and others.
Carrick is co-founder and co-artistic director of the contemporary music ensemble Either/Or, declared 'first rate' and 'a trustworthy purveyor of fresh sounds' by The New York Times, and winner of the 2015 CMA/ASCAP Award for Adventurous Programming. As conductor and pianist, Carrick has worked closely with many celebrated composers including Helmut Lachenmann, Jonny Greenwood, Chaya Czernowin, Elliott Sharp, George Lewis, Anna Thorvaldsdottir, Iancu Dumitrescu, Robert Ashley, Rebecca Saunders, Karin Rehnqvist and Raphael Cendo. Carrick conducting E/O's ambitious performance of "John Cage Party Pieces" premiered 125 scores by renowned composers from around the world.
Carrick is current Chair of Composition at Berklee College of Music, where he directs the Composer/Performer Ensemble in concert and weekly workshops and teaches composing for dance and alternate approaches for structuring real-time music creation. He has presented masterclasses and lectures throughout the US, Canada, Holland, France, UK, Spain, Sweden, Germany, Israel, Rwanda, Japan and South Korea. Former posts include composition faculty at Columbia and New York Universities, and teaching artist for the New York Philharmonic through which he has mentored hundreds of young composers internationally.
A US/French citizen born in Paris of French-Algerian and British descent, Carrick received his BA from Columbia University, MA and PhD from the University of California-San Diego working with Brian Ferneyhough, and pursued further studies at IRCAM and the Koninklijk Conservatorium in The Hague. Scores distributed by Project Schott New York. www.richardcarrick.com
_Bio by Brad Balliett
Stone Guitars CD
"it may change your
perception of electric guitar."
-American Record Guide
Guggenheim Fellow
in Kigali, Rwanda
The New York Times article:
The Music of Flow
by Richard Carrick
The Flow Cycle for Strings CD
"quietly virtuosic and addictive"
-The Wire
Harmonixity available on CD
New Thread Saxophone Quartet
New Focus Recordings 2019
The Music of Richard Carrick
at Miller Theatre, NYC
2019 Either/Or and Mivos
Cycles of Evolution CD
"uncompromising ... authentic"
-Fanfare
Chair of Composition
National Gugak Center Fellow in Korean Traditional Music
Home News Concerts Works Recordings Multimedia Performer Bio Press Contact
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"spot-on focused and unfaltering"


text:
Sanctuary, Seven Bowls of Wrath
TENT OF WITNESS
November 18, 2020 David R Maas Leave a comment
The destructive forces released by the "last plagues" echo the destruction of the Egyptian army in the Red Sea – Revelation 15:5-8.
Having seen the saints standing victorious on the "sea of glass," John then saw the "sanctuary" and the "Tent of Witness" opened in "heaven," from which the seven angels with the "bowls of wrath" were dispatched to empty the deadly contents of their bowls upon the earth.
The "sanctuary" and the "bowls of wrath" are connected to the "fifth seal," the "souls" of martyrs underneath the "altar" who pleaded for vindication against the "inhabitants of the earth." They were told to wait until the full number of their "fellow servants who should be killed even as they" had been gathered. Here, the completion of the "wrath of God" is about to unfold in response to that plea – (Revelation 6:9-11).
(Revelation 15:5-8) – "And after these things I saw, and the sanctuary of the tent of witness in heaven was opened; and the seven angels who had the seven plagues came out of the sanctuary clothed with a precious stone, pure, bright, and girt about the breasts with girdles of gold. And one of the four living creatures gave to the seven angels seven golden bowls full of the wrath of God who lives to the ages of ages. And the sanctuary was filled with smoke by the glory of God, and by his power; and no one was able to enter the sanctuary, until the seven plagues of the seven angels should be ended."
The ancient tabernacle included the altars of incense and of the whole burnt offerings. Here, the functions of both are combined. The "souls under the altar" correspond to the blood of sacrificial animals that was poured out at the base of the "altar of burnt offering." Similarly, in the prelude to the "seven trumpets," the "prayers of the saints" ascended like "incense" from the "golden altar," then an angel threw coals from the altar onto the earth to release the angels with the "seven trumpets" – (Leviticus 4:7, Revelation 8:3-5, 17:11).
Previously, when the "seventh trumpet" sounded, the "sanctuary" was seen in heaven "opened" with the "Ark of the Covenant." Now, the "sanctuary" is opened once again, however, what is seen is the "tent of the testimony." Considering the Exodus typology, this alludes to the original stone tablet inscribed with the Ten Commandments that was kept in the Ark of the Covenant. The "testimony" of God is His law, which is about to be enforced as the "bowls of wrath" are emptied – (Exodus 16:34, 25:21, 31:18, Revelation 11:15-19).
The image of seven angels arrayed in "linen" alludes to the priestly garb worn by Aaron and his sons as they performed their priestly duties, as well as to their "golden girdles." Likewise, the "seven angels" perform priestly functions in the heavenly "sanctuary" – (Exodus 28:5,28:40-43).
Under the Levitical system, "linen" was a ritually "clean" material, unlike wool and other fabrics produced from animal products. Priests were required to wear "linen garments" before removing the ashes from the altar of the burnt offering. They would then change into another set of "linen" garments to carry the ashes outside the camp, where they would "pour them out." Likewise, any remaining blood from an animal sacrifice was poured out at the base of the altar of burnt offerings – (Exodus 29:12, Leviticus 4:7-12, 6:9-11).
The "seven bowls of wrath" correspond to the basins used by the priests to remove the ashes from the altar. Ashes symbolized the complete offering of an animal to God; nothing remained of the body except ash, which demonstrated its complete destruction – (Exodus 27:3).
So, also, the lives of the martyrs have been fully consumed ("slain") by the persecution of the "beast." They have become "whole burnt offerings" on the heavenly altar. The seven angels carry out what remains – "ashes" – and empty the contents on the persecutors of the saints – the "inhabitants of the earth," the "beast," and "Babylon." The enemies of the "Lamb" will now drink the "wine of the wrath of God prepared unmixed in the cup of his fury" – (Revelation 14:10).
Photo by Jakub Kriz on Unsplash
The image of "the sanctuary filled with smoke" echoes the incident when the Tabernacle was consecrated and filled with the presence of Yahweh. At that time, not even Moses could stand before the sanctuary. Likewise, when Solomon's Temple was consecrated, the "priests could not stand to minister because of the cloud" – (Exodus 40:34-35, 1 Kings 8:10-11).
The picture also echoes the vision from the book of Ezekiel when a man arrayed in "linen" was commanded to take live coals from between the cherubim and cast them into the city in preparation for judgment. Previously, the same passage was alluded to when the "prayers of the saints" ascended to God "like incense" and caused the angel to cast live coals onto the earth – (Ezekiel 10:2-4, Revelation 8:3-5, 15:8-16:1).
Likewise, no one can enter the heavenly "sanctuary" until the "seven last plagues" have been unleashed. The time of the final judgment has arrived, and no one can approach the throne until it is completed.
The "seven bowls of wrath" represent the final "wrath of God" that will be poured out on the "beast" and its allies. The seven "last plagues" correspond to the destruction of the Egyptian army in the Red Sea following Israel's passage through its waters. That is why here the saints are "standing on the sea of glass" and singing the "song of Moses and of the Lamb," all in anticipation of the impending destruction of end-time "Babylon."
[Download PDF copy from OneDrive]
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