As a dancewear retailer, we continue to be asked for advice on local classes from new students, parents of young children and also from current students of dance looking to broaden their skills by taking a new class.
In response to these requests and a general need for information, we decided to publish The Malabar Guide to Dance Schools & Studios. We faxed, phoned, emailed and wrote to schools all over Ontario to get as much up-to-date information as we possibly could, organized the data by region and formatted it into a handy reference guide.
This guide is designed to help dance students find classes in their area, while at the same time give some much-deserved promotion to our local schools and studios. Most importantly, the guide is FREE for all dance customers, both here online and in hard copy at our retail stores. It's our way of saying thank you for shopping at Malabar.
We're grateful to have such supportive dancewear suppliers, who have consistently purchased advertising space in our Dance Guides. Their continued support, along will ads booked by local schools over the years, is what allows us to keep producing this free guide.
We hope you find this guide useful and we welcome you to send us any comments or suggestions for our future editions to info@malabar.net.


All steel naya with c 70 cm long reinforced flexible blade.
I would like to find the translation for the marks in the blade and also I would like to find more information about this sword.
The only think I know about nayar is it is a temple sword and also a ceremonial sword from the 18 th century.
Any comment and on it would be welcome.
I Just Wanted to make u aware that these swords were never used for battle,but mainly for ceremonial use in religious ceremonies during Theyyam in Kerala and Bhuta Kola in Karnataka State of India.
I shall get back to you soon about the translation.
Thank you Rajesh,I hope you could translate the marks.Also could you let me know your opinion about age.
Apparently~ These rare swords - known as Temple Swords - are associated with the Nayar or Nair of Malabar of Southern India, believed circa 18th century. They were used for religious and domestic ceremonies.
This example has the classic flexible steel blade and is ~28" in length. They are double-edged but not necessarily combat worthy. Upper edge - near the crescent-moon-shaped tip - has small holes drilled to accept ringlets attached to "charms" or other trinkets such as bells (only two ringlets/trinkets remain on this example). The grip is iron - cylindrical in geometry - with light brass cross guard and langets (the fastening strips for the blade); also present - towards the pommel - are a series of brass discs, the last disc has small holes drilled to accept ringlets attached to "charms" or other trinkets such as bells as on the blade (only three ringlets/trinkets remain on this example). Overall length is ~34-1/2". These are considered very scarce if not rare.
These swords are very similar to - and often mistaken for - the African Ngombe "Execution" Swords common to the Ngombe people of central Zaire, Africa.


This collection of Upper Deck Xponential inserts comes from Reader Tim. I sent him some Yankees relics and I got a ton of shiny goodness.
The group of cards also gave me my first completed page in the Xponential to the 1st Power sub set. Joy!
And if anyone has any Xponential inserts that they are willing to part with here is my needs list.


8aumvjr98736. Ae012299. 746a5ac700286%. Like cuteFlat (≤1cm). Men canvas shoes. 7e0453. Faroot. 35-39. Ydbsc3986.
7 pcs/sets. 8 pieces. Air brush medium. S1060. Kabuki brush. Wholesale eraser suede. Floor repair cream. Wholesale dust brush. Dust shoes cleaning. Foot basin. Approx. 3.4 x 1.4 x 1.2 inch. 23.5x4cm.
Wholesale elderly shoes. Miss rose. Cleaning gloves. Pu bag size: Cross-tied. Shoes tango. Pig. Oxford protector. Brush wood handle shoes. Mini facial brushes. Bristle. Crystals boots. A997d. Plush shoes cleaning gloves. Product: Cloth rags for cleaning. Animals hair brush. Feature 1: Easy breezies.


text:
Destruction Was My Beatrice by Jed Rasula
by Katie Richards | Aug 21, 2015 | Archived Reviews (pre-April 2020), Art, Design & Photography, Biographies & Memoirs, History | 0 comments
[alert variation="alert-info"]Publisher: Basic Books
Formats: Hardcover, eBook, Kindle
Purchase: Powell's | Amazon | IndieBound | Barnes & Noble | iBooks[/alert]
Jed Rasula's Destruction Was My Beatrice takes on the Dada movement of the early 20th century. Rasula, a highly qualified and serious modern scholar has an interesting and slightly unusual take on Dada. Most critics and writers thinking about the movement focus on the art. Likely because Rasula is a professor of English, he does not take on the art and the artifacts in the way that many writers do. Instead, Destruction Was My Beatrice (as the title hints), focuses on the substantial literary output of the movement, with detailed attention to the poetry, journals and correspondence of the many participants and hangers-on of Dada. Rasula also focuses on the personalities and the personal conflicts of the at times touchy artists and egos involved in the movement, separating out distinct personalities and types of work that are associated with the movement.
Readers who are new to Dada, or who don't have some background with early 20th century modern movements may easily find themselves swept away in the tide of names in the book, and even while writings like the noise poems are less controversial today, they can still be confusing to a modern audience. It is sometime hard to identify a specific author, piece of writing, or other artifact, and trace that through the tangled history of Dada, especially as the work is presented in a non-chronological order, based largely on geography. Readers who have some background in the subject, and especially art historians who are looking to broaden their viewpoint, will appreciate the inclusive nature of the text, and the fresh look at a subject and movement, which is often overshadowed by other early 20th century modern movements.
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redpajama_set_name: RedPajamaCommonCrawl


… and we made a mistake!
In the Premium Membership section we had the monthly podcasts as a discounted item – whereas it's actually included in the membership! – Sorry about that.
The proper version is just below in case you missed it.
Also, we would love your feedback about the newsletter and the website which we are going to improve as we go along, adding new resources and items each month.
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text:
Rick Warren, founder and pastor of Saddleback Community Church in Orange County California, addressed the convention of the Islamic Society of North America. Warren stated that Muslims and Christians must work together to combat stereotypes, promote peace and freedom, and solve global problems. Christians and Moslems – faith mates, soul mates and now work mates! Chrislam!
by Geoffrey Grider February 4, 2011
NOW THE END BEGINS SHARE:
Pastors that are really wolves in sheep's clothing
"And unto the angel of the church of the Laodiceans write; These things saith the Amen, the faithful and true witness, the beginning of the creation of God; I know thy works, that thou art neither cold nor hot: I would thou wert cold or hot. So then because thou art lukewarm, and neither cold nor hot, I will spue thee out of my mouth." Revelation 3:14-16 (KJV)
You knew this day was going to come, when the liberal "love gospel" preachers who, desiring to fill their seats and bank accounts, would find a way to merge apostate Christianity with the Devil's religion of Islam. That day is here.
Rick Warren with his new best friend, Christ rejector Cat Stevens, who changed his name to Yusef Isalm
In 2009, Rick Warren, founder and pastor of Saddleback Community Church in Orange County California, addressed the convention of the Islamic Society of North America. Warren stated that Muslims and Christians must work together to combat stereotypes, promote peace and freedom, and solve global problems. Christians and Moslems – faith mates, soul mates and now work mates! Chrislam!
Quote of the Day: "Before we "shake your hand" in responding to your letter, we ask forgiveness of the All-Merciful One and of the Muslim community around the world." – Rick Warren, one of the main signers of the Yale Covenant For Islam and Christian Unity
This weekend, the Memorial Drive Presbyterian Church in Houston along with Christian communities in Atlanta, Seattle, and Detroit will initiate a series of sermons that have been designed to produce an ecumenical reconciliation between Christianity and Islam. In addition to the sermons, the Sunday school lessons will center on the inspired teachings of the Prophet Mohammad. Qurans will be placed in the pews next to the Bibles.
(NOTE: Due to the overwhelming response that this article has generated, the Pastor of Memorial Church has issued a semi-denial of the events we reported. They do not, however, deny supporting and teaching a doctrine called "Jesus in the Koran. Click here to read their semi-denial. So until other information is made available to NTEB, we will run the story as is, along with their side, and you the reader can decide what's going on.)
The concept of Chrislam, now embraced by such preachers as Rick Warren and Robert Schuller, appears to have emerged from a program on the meaning of "love your neighbor" at Grace Fellowship Church in Atlanta, Georgia "In 2001, like most Americans, we were pretty awakened to the true Islamic presence in the world and in the United States," says Jon Stallsmith, the outreach minister at Grace Fellowship. "Jesus says we should love our neighbors. We can't do that without having a relationship with them."
The sign may be Photoshop, but the message it carries is all too real. Chrislam must be stopped now. If your church teaches this heresy – run!
Stallsmith maintains that a rapprochement between Muslims and Christians can be achieved by the fact that Jesus is mentioned twenty-five times in the Quran.
The Chrislam movement has gained impetus by statements from President George W. Bush and that Christians, Jews, and Muslims all worship the same God and by Rick Warren's reference to Isa (the Muslim name for Jesus) in his prayer at the inauguration of President Barack Obama. Only 30 percent of Americans have a favorable view of Muslims, according to a Pew Forum poll. At the same time, more than half the country says they know "not very much" or "nothing at all" about the Islamic faith.
Where Chrislam Was Born: Rick Warren at the ISNA 2009 Conference
"The recent political developments and the fact that we're fighting two wars in Muslim countries should sharpen that need to know how to talk to these guys," Stallsmith insists "We want to find peace, reconciliation around a scriptural understanding of Jesus."
Jesus in the Quran is neither the only-begotten Son of God nor the Messiah who was divinely appointed to restore the House of David. He is rather viewed as a prophet who was appointed by Allah to prepare mankind for the coming of Mohammad."
CHRISLAM UPDATE: On June 26, 2011, churches all across America held Chrislam services in their churches. Click here to read story and see list of participating churches.
Is YOUR CHURCH involved with Chrislam? Find out!
chrislamIslamjesus in the qurankoranprophet mohammadrick warrenrobert schullerSaddleback Community Church
NTEB is run by end times author and editor-in-chief Geoffrey Grider. Geoffrey runs a successful web design company, and is a full-time minister of the gospel of the Lord Jesus Christ. In addition to running NOW THE END BEGINS, he has a dynamic street preaching outreach and tract ministry team in Saint Augustine, FL.
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text:
\section{Introduction}

The aim of this paper is to deduce the algebraic rules for determining the
dynamical charactersitics of a prescribed network consisting of specified
quantum oscillator systems connected by input-output fields \cite{Gardiner},
\cite{Wiseman}. Physical models included cavity systems or local quantum
oscillators with a quantum optical field. The resulting dynamics is linear,
and the analysis is carried out using transfer function techniques \cite{YK1}%
, \cite{YK2}. The rules have been recently deduced in \cite{QFN1} in the
general setting for nonlinear quantum dynamical systems by first
constructing a network Hamiltonian and transfering to the interaction
picture with respect to the free flow of the fields around the network
channels. However it is of interest to restrict to linear systems for two
main reasons. Firstly, the derivation here for linear systems procedes by an
alternative method to the general nonlinear case, and we are able to confirm
the restriction of the nonlinear formula to linear systems yields the same
result. Secondly, linear systems are the most widely studied models in both
classical and quantum dynamical systems theory and so it is natural to
develop these further. There has been recent interest in the development of
coherent, or fully quantum control for linear systems \cite{GJ Series}-\cite
{NJP}\ and this paper contributes by establishing the algebraic rules for
building networks of such devices.

\section{Linear Quantum Markov Models}

The dynamical evolution of a quantum system is determined by a family of
unitaries $\left\{ V\left( t,s\right) :t\geq s\right\} $ satisfying the
propagation law $V\left( t_{3},t_{2}\right) V\left( t_{2},t_{1}\right)
=V\left( t_{3},t_{1}\right) $ where $t_{3}\geq t_{2}\geq t_{1}$. The
evolution of a state from time $s$ to a later time $t$ being then given by $%
\psi \left( t\right) =V\left( t,s\right) \psi \left( s\right) $. In a Markov
model we factor the underlying Hilbert space as $\frak{h}\otimes \mathcal{E}$
representing the system and its environment respectively and the unitary $%
V\left( t,s\right) $ couples the system specifically with the degrees of
freedom of the environment acting between times $s$ and $t$. For a bosonic
environment, we introduce input processes $b_{i}\left( t\right) $ for $%
i=1,\cdots ,n$ with the canonical commutation relations, \cite{Gardiner},
\begin{equation}
\left[ b_{i}\left( t\right) ,b_{j}^{\dag }\left( s\right) \right] =\delta
_{ij}\,\delta \left( t-s\right) .
\end{equation}
It is convenient to assemble these into the following column vectors of
length $n$
\begin{equation}
\mathbf{b}^{\text{in}}\left( t\right) =\left(
\begin{array}{c}
b_{1}\left( 1\right) \\
\vdots \\
b_{n}\left( t\right)
\end{array}
\right) .
\end{equation}
A Markov evolution can be described equivalently by the
chronological-ordered and Wick-ordered expressions
\begin{equation*}
V\left( t,s\right) =\;\vec{T}\exp -i\int_{s}^{t}\Upsilon \left( \tau \right)
d\tau \equiv \;:\exp -i\int_{s}^{t}\Upsilon _{\text{Wick}}\left( \tau
\right) d\tau :
\end{equation*}
where the stochastic Hamiltonian is (with $E_{ij}^{\dag }=E_{ji}$ and $%
K^{\dag }=K$)
\begin{equation*}
\Upsilon \left( t\right) =\sum_{i,j=1}^{n}E_{ij}\otimes b_{i}^{\dag }\left(
t\right) b_{j}\left( t\right) +\sum_{i=1}^{n}F_{i}\otimes b_{i}^{\dag
}\left( t\right) +\sum_{j=1}^{n}F_{j}^{\dag }\otimes b_{j}\left( t\right)
+K\otimes 1,
\end{equation*}
and the Wick-ordered generator is given by \cite{G Wong-Zakai}
\begin{eqnarray*}
-i\Upsilon _{\text{Wick}}\left( t\right) &=&\sum_{i,j=1}^{n}(S_{ij}-\delta
_{ij})\otimes b_{i}^{\dag }\left( t\right) b_{j}\left( t\right)
+\sum_{i=1}^{n}L_{i}\otimes b_{i}^{\dag }\left( t\right) \\
&&-\sum_{j=1}^{n}L_{i}^{\dag }S_{ij}\otimes b_{j}\left( t\right) -\left(
\frac{1}{2}\sum_{i=1}^{n}L_{i}^{\dag }L_{i}-iH\right) \otimes 1.
\end{eqnarray*}
The Wick-ordered coefficients are given by the Stratonovich-Ito conversion
formulae, see appendix,
\begin{equation}
S=\frac{1-\frac{i}{2}E}{1+\frac{i}{2}E},\quad L=-i\frac{1}{1+\frac{i}{2}E}%
F,\quad H=E_{00}+\frac{1}{2}\func{Im}F\frac{1}{1+\frac{i}{2}E}F^{\dag }.
\label{Strat-Ito}
\end{equation}
Note that $H$ is selfadjoint, and that $S$ is a unitary matrix whose entries
are operators on $\frak{h}$:$\sum_{k=1}^{n}S_{ik}S_{jk}^{\dag }=\delta
_{ij}=\sum_{k=1}^{n}S_{ki}^{\dag }S_{kj}$. In fact, we may write $S=e^{-iJ}$
with $J=2\arctan \dfrac{E}{2}$.

In differential form we have
\begin{eqnarray*}
\frac{d}{dt}V\left( t,s\right) &=&\;-i:\Upsilon _{\text{Wick}}\left(
t\right) V\left( t,s\right) : \\
&\equiv &\sum_{i,j=1}^{n}b_{i}^{\dag }\left( t\right) (S_{ij}-\delta
_{ij})V\left( t,s\right) b_{j}\left( t\right) +\sum_{i=1}^{n}b_{i}^{\dag
}\left( t\right) L_{i}V\left( t,s\right) \\
&&-\sum_{j=1}^{n}L_{i}^{\dag }S_{ij}V\left( t,s\right) b_{j}\left( t\right)
-\left( \frac{1}{2}\sum_{i=1}^{n}L_{i}^{\dag }L_{i}-iH\right) V\left(
t,s\right) .
\end{eqnarray*}
Note that all the creators appear on the left and all annihilators on the
right. This equation can be interpreted as a quantum stochastic differential
equation \cite{Gardiner}, \cite{HP}, \cite{partha}.

We sketch the system plus field as a two port device having an input and an
output port.

\begin{center}
\setlength{\unitlength}{.04cm}
\begin{picture}(120,45)
\label{pic1}
\thicklines
\put(45,10){\line(0,1){20}}
\put(45,10){\line(1,0){30}}
\put(75,10){\line(0,1){20}}
\put(45,30){\line(1,0){30}}
\thinlines
\put(48,20){\vector(-1,0){45}}
\put(120,20){\vector(-1,0){20}}
\put(120,20){\line(-1,0){48}}
\put(50,20){\circle{4}}
\put(70,20){\circle{4}}
\put(100,26){input, ${\bf b}^{\rm in}$}
\put(48,35){system}
\put(-10,26){output, ${\bf b}^{\rm out}$}
\end{picture}

Figure 1: input-output component
\end{center}

The output fields are defined by $b_{i}^{\text{out}}\left( t\right) =V\left(
t,0\right) ^{\dag }b_{i}\left( t\right) V\left( t,0\right) $ and we have the
input-output relation
\begin{equation*}
b_{i}^{\text{out}}\left( t\right) =\sum_{j=1}^{n}S_{ij}\left( t\right)
b_{j}\left( t\right) +L_{i}\left( t\right) ,
\end{equation*}
where $S_{ij}\left( t\right) =V\left( t,0\right) ^{\dag }S_{ij}V\left(
t,0\right) $ and $L_{i}\left( t\right) =V\left( t,0\right) ^{\dag
}L_{i}V\left( t,0\right) $. More compactly, $\mathbf{b}^{\text{out}}\left(
t\right) =S\left( t\right) \mathbf{b}^{\text{in}}\left( t\right) +L\left(
t\right) $.

Let $X$ be a fixed operator of the system and set $X\left( t,t_{0}\right)
=V\left( t,t_{0}\right) ^{\dag }XV\left( t,t_{0}\right) $, then we obtain
the Heisenberg-Langevin equation
\begin{eqnarray*}
\frac{d}{dt}X\left( t,t_{0}\right) &=&V\left( t,t_{0}\right) ^{\dag }\frac{1%
}{i}[X,\Upsilon \left( t\right) ]V\left( t,t_{0}\right) \\
&=&b_{i}^{\dag }\left( t\right) V\left( t,t_{0}\right) ^{\dag }\left(
S_{ki}^{\dag }XS_{kj}-\delta _{ij}X\right) V\left( t,t_{0}\right)
b_{j}\left( t\right) \\
&&+b_{i}^{\dag }\left( t\right) V\left( t,t_{0}\right) ^{\dag }S_{ki}^{\dag
} \left[ X,L_{k}\right] V\left( t,t_{0}\right) \\
&&+V\left( t,t_{0}\right) ^{\dag }[L_{i}^{\dag },X]S_{ij}V\left(
t,t_{0}\right) b_{j}\left( t\right) \\
&&+V\left( t,t_{0}\right) ^{\dag }\left\{ \frac{1}{2}L_{k}^{\dag }\left[
X,L_{k}\right] +\frac{1}{2}[L_{k}^{\dag },X]L_{k}-i\left[ X,H\right]
\right\} V\left( t,t_{0}\right) .
\end{eqnarray*}
Note that the final term does not involve the input noises, and that the
expression in braces is a Lindbladian. In the special case where $S=1$, this
equation reduces to the class of Heisenberg-Langevin equations introduced by
Gardiner \cite{Gardiner}.

\subsection{Linear Models}

We consider a quantum mechanical system consisting of a family of harmonic
oscillators $\left\{ a_{j}:j=1,\cdots ,m\right\} $ with canonical
commutation relations $\left[ a_{j},a_{k}\right] =0=\left[ a_{j}^{\dag
},a_{k}^{\dag }\right] $ and $\left[ a_{j},a_{k}^{\dag }\right] =\delta
_{jk} $. We collect into column vectors:
\begin{equation}
\mathbf{a}=\left(
\begin{array}{c}
a_{1} \\
\vdots \\
a_{m}
\end{array}
\right) .
\end{equation}

Our interest is in the general linear open dynamical system and here we make
several simplifying assumptions:

\begin{itemize}
\item[1)]  The $S_{jk}$ are scalars.

\item[2)]  The $L_{j}^{\prime }s$ are linear, i.e., there exist constants $%
c_{jk}$ such that $L_{j}\equiv \sum_{k}c_{jk}a_{k}$.

\item[3)]  $H$ is quadratic, i.e., there exist constants $\omega _{jk}$ such
that $H=\sum_{jk}a_{j}^{\dag }\omega _{jk}a_{k}$.
\end{itemize}

The complex damping is $\frac{1}{2}L^{\dag }L+iH=-\mathbf{a}^{\dag }A\mathbf{%
a}$ where $A=-\frac{1}{2}C^{\dag }C-i\Omega $ with $C=\left( c_{jk}\right) $
and $\Omega =\left( \omega _{jk}\right) $. The Heisenberg-Langevin equations
for $\mathbf{a}\left( t\right) =V\left( t,0\right) \mathbf{a}V\left(
t,0\right) $ and input-output relations then simplify down to
\begin{eqnarray}
\mathbf{\dot{a}}\left( t\right) &=&A\mathbf{a}\left( t\right) -C^{\dag }S%
\mathbf{b}(t), \\
\mathbf{b}^{\text{out}}\left( t\right) &=&S\mathbf{b}\left( t\right) +C%
\mathbf{a}\left( t\right) .
\end{eqnarray}
These linear equations are amenable to Laplace transform techniques \cite
{YK1},\cite{YK2}. We define for $\func{Re}s>0$
\begin{equation}
\hat{C}\left( s\right) =\int_{0}^{\infty }e^{-st}C\left( t\right) dt,
\end{equation}
where $C$ is now any of our stochastic processes. Note that $\widehat{%
\mathbf{\dot{a}}}\left( s\right) =s\mathbf{\hat{a}}\left( s\right) -\mathbf{a%
}$. We find that
\begin{eqnarray*}
\mathbf{\hat{a}}\left( s\right) &=&-\left( sI_{m}-A\right) ^{-1}C^{\dag }S%
\mathbf{\hat{b}}^{\text{in}}\left( s\right) +\left( sI_{m}-A\right) ^{-1}%
\mathbf{a}, \\
\mathbf{\hat{b}}^{\text{out}}\left( s\right) &=&S\mathbf{\hat{b}}^{\text{in}%
}\left( s\right) +C\mathbf{\hat{a}}\left( s\right) .
\end{eqnarray*}

The operator $\mathbf{\hat{a}}\left( s\right) $ can be eliminated entirely
to give
\begin{equation}
\mathbf{\hat{b}}^{\text{out}}\left( s\right) =\Xi \left( s\right) \mathbf{%
\hat{b}}^{\text{in}}\left( s\right) +\xi \left( s\right) \mathbf{a}
\end{equation}
where the \textit{transfer matrix function} is
\begin{equation}
\Xi \left( s\right) =S-C\left( sI_{m}-A\right) ^{-1}C^{\dag }S
\end{equation}
and $\xi \left( s\right) =C\left( sI_{m}-A\right) ^{-1}$.

\bigskip

As an example, consider a single mode cavity coupling to the input field via
$L=\sqrt{\gamma }a,$ and with Hamiltonian $H=\omega a^{\dag }a$. This
implies $K=\frac{\gamma }{2}+i\omega $ and $C=\sqrt{\gamma }$. If the output
picks up an additional phase $S=e^{i\phi }$, the corresponding transfer
function is then computed to be
\begin{equation}
\Xi _{cavity}\left( s\right) =e^{i\phi }\,\frac{s+i\omega -\frac{\gamma }{2}%
}{s+i\omega +\frac{\gamma }{2}}.
\end{equation}

\subsection{The Transfer Matrix Function}

The models we consider are therefore determined completely by the matrices $%
\left( S,C,\Omega \right) $ with $S\in \mathbb{C}^{n\times n},C\in \mathbb{C}%
^{n\times m}$ and $\Omega \in \mathbb{C}^{m\times m}$. We shall use the
convention $\left[
\begin{tabular}{l|l}
$A$ & $B$ \\ \hline
$C$ & $D$%
\end{tabular}
\right] \left( s\right) =D+C\left( s-A\right) ^{-1}B$ for matrices $A\in
\mathbb{C}^{m\times m},B\in \mathbb{C}^{m\times n},C\in \mathbb{C}^{n\times
m}$ and $D\in \mathbb{C}^{n\times n}$, and write the transfer matrix
function as

\begin{equation}
\Xi \left( s\right) =\left[
\begin{tabular}{r|r}
$A$ & $-C^{\dag }S$ \\ \hline
$C$ & $S$%
\end{tabular}
\right] \left( s\right) ,  \label{TF}
\end{equation}
where $A=-\frac{1}{2}C^{\dag }C-i\Omega $. We note the decomposition
\begin{equation*}
\Xi =\left[ I_{n}-C\left( sI_{m}-A\right) ^{-1}C^{\dag }\right] S\equiv
\left[
\begin{tabular}{r|r}
$A$ & $-C^{\dag }$ \\ \hline
$C$ & $I_{n}$%
\end{tabular}
\right] S.
\end{equation*}
In the simplest case of a single cavity mode we have
\begin{equation*}
\Xi _{cavity}\left( s\right) =\left[
\begin{tabular}{r|r}
$-\frac{\gamma }{2}-i\omega $ & $-\sqrt{\gamma }e^{i\phi }$ \\ \hline
$\sqrt{\gamma }$ & $e^{i\phi }$%
\end{tabular}
\right] \left( s\right) .
\end{equation*}

\begin{lemma}
For each $\omega \in \mathbb{R}$, the transfer function $\Xi \left( i\omega
\right) \equiv \Xi \left( 0^{+}+i\omega \right) $ is unitary whenever it
exists.
\end{lemma}

\begin{proof}
The decomposition follows immediately from $\left( \ref{TF}\right) $. We
have then for instance
\begin{multline*}
\Xi \left( 0^{+}+i\omega \right) \Xi \left( 0^{+}+i\omega \right) ^{\dag }=
\left[ I-C\frac{1}{\frac{1}{2}C^{\dag }C+i\Omega ^{\prime }}C^{\dag }\right] %
\left[ I-C\frac{1}{\frac{1}{2}C^{\dag }C-i\Omega ^{\prime }}C^{\dag }\right]
\\
=I-C\frac{1}{\frac{1}{2}C^{\dag }C+i\Omega ^{\prime }}\left\{ \frac{1}{2}%
C^{\dag }C+i\Omega ^{\prime }+\frac{1}{2}C^{\dag }C-i\Omega ^{\prime
}-C^{\dag }C\right\} \frac{1}{\frac{1}{2}C^{\dag }C-i\Omega ^{\prime }}%
C^{\dag },
\end{multline*}
where $\Omega ^{\prime }=\Omega +\omega $. The term in braces however
vanishes identically, leaving $\Xi \left( 0^{+}+i\omega \right) \Xi \left(
0^{+}+i\omega \right) ^{\dag }=I$. The relation $\Xi \left( 0^{+}+i\omega
\right) \Xi \left( 0^{+}+i\omega \right) ^{\dag }=I$ is similarly
established.
\end{proof}

\bigskip

Whenever appropriate, we may determine $\Xi $ from its (unitary) values on
the imaginary axis by using the Hilbert transform
\begin{equation*}
\Xi \left( s\right) =\frac{1}{2\pi i}PV\int_{-\infty }^{\infty }\frac{\Xi
\left( i\omega \right) }{\omega +is}d\omega .
\end{equation*}

In general, the real and imaginary parts of $A$ need not commute - that is, $%
\left[ C^{\dag }C,\Omega \right] $ need to be identically zero. However,
when this does occur we recover a multi-mode version of the cavity situation.

\begin{lemma}
If $A$ is a function of $C^{\dag }C$ then
\begin{equation*}
\Xi \left( s\right) =\frac{s+\tilde{A}^{\dag }}{s-\tilde{A}^{\dag }}S,
\end{equation*}
where $\tilde{A}$ is a function of $CC^{\dag }$ and $\Xi $ may be
analytically continued into the whole complex plane.
\end{lemma}

\begin{proof}
Here we must have $A=-\frac{1}{2}C^{\dag }C-i\varepsilon \left( C^{\dag
}C\right) $ where $\varepsilon $ is a real valued function. We set $\tilde{A}%
=-\frac{1}{2}CC^{\dag }-i\varepsilon \left( CC^{\dag }\right) $. From the
identity $Cf\left( C^{\dag }C\right) C^{\dag }=CC^{\dag }f\left( CC^{\dag
}\right) $ for suitable analytic functions $f$, we have
\begin{equation*}
\left( s-\tilde{A}\right) \Xi \left( s\right) =\left( s-\tilde{A}\right) %
\left[ I-\frac{1}{s-\tilde{A}}CC^{\dag }\right] S=\left( s-\tilde{A}%
-CC^{\dag }\right) S
\end{equation*}
however, $-\tilde{A}-CC^{\dag }=\tilde{A}^{\dag }$, and this gives the
result.

The hermitean matrices $C^{\dag }C$ and $CC^{\dag }$ will have the same set
of eigenvalues: to see this, suppose that $\phi $ is a non-zero unit
eigenvector of $CC^{\dag }$ with eigenvalue $\gamma $, then $\psi =\gamma
^{-1/2}C^{\dag }\phi $ is a unit eigenvector of $C^{\dag }C$ with the same
eigenvalue, conversely, every eigenvector $\psi $ of $C^{\dag }C$ with
non-zero eigenvalue $\gamma $ gives rise to a nonzero eigenvector $\phi
=\gamma ^{-1/2}C\psi $ of $CC^{\dag }$.

Let $CC^{\dag }$ have the spectral form $\sum_{k}\gamma _{k}E_{k}$ with real
eigenvalues $\gamma _{k}$ and corresponding eigenprojectors $E_{k}$, then we
have
\begin{equation*}
\Xi \left( s\right) =\sum_{k}\frac{s-\frac{1}{2}\gamma _{k}+i\varepsilon _{k}%
}{s+\frac{1}{2}\gamma _{k}+i\varepsilon _{k}}E_{k}S,
\end{equation*}
where $\varepsilon _{k}=\varepsilon \left( \gamma _{k}\right) $. In
particular, the rational fraction is of modulus unity for imaginary $s\left(
=i\omega \right) $ and we may write
\begin{equation*}
\Xi \left( 0^{+}+i\omega \right) =\sum_{k}e^{i\phi _{k}\left( \omega \right)
}E_{k}S
\end{equation*}
where $\phi _{k}\left( \omega \right) =\arg \frac{i\left( \omega
+\varepsilon _{k}\right) -\gamma _{k}/2}{i\left( \omega +\varepsilon
_{k}\right) +\gamma _{k}/2}$. Note that $\Xi \left( 0^{+}+i\omega \right) $
is clearly unitary and the limit $\omega \rightarrow 0$ is well-defined.
This limit will equal $-S$ in the special case that $K$ is selfadjoint
(i.e., $\varepsilon \equiv 0$). $\Xi $ may be analytically continued into
the negative-real part of the complex plane. The poles of $\Xi $ then form
the resolvent set of $\tilde{K}$, and the zeroes being the complex
conjugates.
\end{proof}

\bigskip

\section{Introducing Connections}

The situation depicted in the figure below is one where (some of) the output
channels are fed back into the system as an input. Prior to the connection
between output port(s) $s_{\mathsf{i}}$ and input port(s) $r_{\mathsf{i}}$
being made, we may model the component as having the total input $\mathbf{b}%
^{\text{in}}=\left(
\begin{array}{c}
\mathbf{b}_{\mathsf{i}}^{\text{in}} \\
\mathbf{b}_{\mathsf{e}}^{\text{in}}
\end{array}
\right) $ and total output $\mathbf{b}^{\text{out}}=\left(
\begin{array}{c}
\mathbf{b}_{\mathsf{i}}^{\text{out}} \\
\mathbf{b}_{\mathsf{e}}^{\text{out}}
\end{array}
\right) $ where the $\mathbf{b}_{j}^{\text{in}}$ and $\mathbf{b}_{j}^{\text{%
out}}$ may be multi-dimensional noises (we in fact only require the
multiplicities to agree for $j=\mathsf{i},\mathsf{e}$ respectively).

\begin{center}
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\begin{picture}(80,28)
\label{pic2}

\thicklines

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\put(35,10){$s_{\sf e}$}
\put(35,15){$s_{\sf i}$}
\put(42,10){$r_{\sf e}$}
\put(42,15){$r_{\sf i}$}

\end{picture}%

figure 2: A quantum system with feedback
\end{center}

The transfer matrix function takes the general form
\begin{equation*}
\Xi \equiv \left[
\begin{tabular}{l|ll}
$A$ & $-\sum_{j}C_{j}^{\dag }S_{j\mathsf{i}}$ & $-\sum_{j}C_{j}^{\dag }S_{j%
\mathsf{e}}$ \\ \hline
$C_{\mathsf{i}}$ & $S_{\mathsf{ii}}$ & $S_{\mathsf{ie}}$ \\
$C_{\mathsf{e}}$ & $S_{\mathsf{ei}}$ & $S_{\mathsf{ee}}$%
\end{tabular}
\right] .
\end{equation*}
When we make the connection, we impose the various constraints $b_{r_{%
\mathsf{i}}\left( k\right) }^{\text{in}}\left( t\right) =b_{s_{\mathsf{i}%
}\left( j\right) }^{\text{out}}\left( t-\tau \right) $ where output field
labelled $s_{\mathsf{i}}\left( j\right) $ is to be connected to the input
field $r_{\mathsf{i}}\left( k\right) $ where $\tau >0$ is the time delay. We
assume the idealized situation of instantaneous feedback $\tau \rightarrow
0^{+}$. To avoid having to match up the labels of the internal channels, it
is more convenient to introduce a fixed labelling and write
\begin{equation*}
\mathbf{b}_{\mathsf{i}}^{\text{out}}\left( t^{-}\right) =\eta \mathbf{b}_{%
\mathsf{i}}^{\text{in}}\left( t\right)
\end{equation*}
where $\eta $ is the adjacency matrix:
\begin{equation*}
\eta _{sr}=\left\{
\begin{array}{cc}
1, & \text{if }\left( s,r\right) \text{ is an internal channel,} \\
0, & \text{otherwise}
\end{array}
\right.
\end{equation*}
The model with the connections is then a reduction of the original and the
remaining external fields are the input $\mathbf{b}_{\mathsf{e}}^{\text{in}}$
and output $\mathbf{b}_{\mathsf{e}}^{\text{out}}$.

\begin{theorem}
Let $\left( \eta -S_{\mathsf{ii}}\right) $ be invertible. The feedback
system described above has input-output relation $\mathbf{\hat{b}}_{\mathsf{e%
}}^{\text{out}}=\Xi _{\mathrm{red}}\mathbf{\hat{b}}_{\mathsf{e}}^{\text{in}%
}+\xi _{\mathrm{red}}\mathbf{a}$ and the reduced transfer matrix function
\begin{equation*}
\Xi _{\mathrm{red}}\equiv \left[
\begin{tabular}{r|r}
$A_{\mathrm{red}}$ & $-C_{\mathrm{red}}^{\dag }S_{\mathrm{red}}$ \\ \hline
$C_{\mathrm{red}}$ & $S_{\mathrm{red}}$%
\end{tabular}
\right] ,\quad \xi _{\mathrm{red}}\equiv C_{\mathrm{red}}\frac{1}{s-A_{%
\mathrm{red}}},
\end{equation*}
where
\begin{eqnarray}
S_{\mathrm{red}} &=&S_{\mathsf{ee}}+S_{\mathsf{ei}}\left( \eta -S_{\mathsf{ii%
}}\right) ^{-1}S_{\mathsf{ie}},  \notag \\
C_{\mathrm{red}} &=&S_{\mathsf{ei}}\left( \eta -S_{\mathsf{ii}}\right)
^{-1}C_{\mathsf{i}}+C_{\mathsf{e}},  \notag \\
A_{\mathrm{red}} &=&A-\sum_{j=\mathsf{i},\mathsf{e}}C_{j}^{\dag }S_{j\mathsf{%
i}}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}C_{\mathsf{i}}.
\end{eqnarray}
\end{theorem}

\begin{proof}
The dynamical equations can be written as
\begin{eqnarray*}
\mathbf{\dot{a}}\left( t\right) &=&A\mathbf{a}\left( t\right)
-\sum_{j,k}C_{j}^{\dag }S_{jk}\mathbf{b}_{k}^{\text{in}}\left( t\right) , \\
\mathbf{b}_{j}^{\text{out}}\left( t\right) &=&\sum_{k=\mathsf{i},\mathsf{e}%
}S_{jk}\mathbf{b}_{k}^{\text{in}}\left( t\right) +C_{j}\mathbf{a}\left(
t\right) .
\end{eqnarray*}
Now the constraint $\eta \mathbf{b}_{\mathsf{i}}^{\text{in}}=\mathbf{b}_{%
\mathsf{i}}^{\text{out}}$ implies that
\begin{equation*}
\mathbf{b}_{\mathsf{i}}^{\text{in}}\left( t\right) =\left( \eta -S_{\mathsf{%
ii}}\right) ^{-1}(S_{\mathsf{ie}}\mathbf{b}_{\mathsf{e}}^{\text{in}}\left(
t\right) +C_{\mathsf{i}}\mathbf{a}\left( t\right) ),
\end{equation*}
and so
\begin{eqnarray*}
\mathbf{\dot{a}}\left( t\right) &=&[A-\sum_{j=\mathsf{i},\mathsf{e}%
}C_{j}^{\dag }S_{j\mathsf{i}}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}C_{%
\mathsf{i}}]\mathbf{a}\left( t\right) \\
&&-\sum_{j=\mathsf{i},\mathsf{e}}C_{j}^{\dag }\left( S_{j\mathsf{e}}+S_{j%
\mathsf{i}}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{\mathsf{ie}}\right)
\mathbf{b}_{\mathsf{e}}^{\text{in}}\left( t\right)
\end{eqnarray*}
or
\begin{equation*}
\mathbf{\hat{a}}\left( s\right) =-\frac{1}{s-A_{\mathrm{red}}}\sum_{j=%
\mathsf{i},\mathsf{e}}C_{j}^{\dag }\left( S_{j\mathsf{e}}+S_{j\mathsf{i}%
}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{\mathsf{ie}}\right) \mathbf{%
\hat{b}}_{\mathsf{e}}\left( s\right) +\frac{1}{s-A_{\mathrm{red}}}\mathbf{a},
\end{equation*}
with $A_{\mathrm{red}}$ as above. Consequently,
\begin{eqnarray*}
\mathbf{\hat{b}}_{\mathsf{e}}^{\text{out}} &=&S_{\mathsf{ei}}\mathbf{\hat{b}}%
_{\mathsf{i}}^{\text{in}}+S_{\mathsf{ee}}\mathbf{\hat{b}}_{\mathsf{e}}^{%
\text{in}}+C_{\mathsf{e}}\mathbf{\hat{a}} \\
&=&S_{\mathrm{red}}\mathbf{\hat{b}}_{\mathsf{e}}^{\text{in}}+C_{\mathrm{red}}%
\mathbf{\hat{a}} \\
&=&\Xi _{\mathrm{red}}\mathbf{\hat{b}}_{\mathsf{e}}^{\text{in}}+\xi _{%
\mathrm{red}}\mathbf{a}
\end{eqnarray*}
where
\begin{eqnarray*}
\Xi _{\mathrm{red}} &=&S_{\mathrm{red}}-\sum_{j=\mathsf{i},\mathsf{e}}C_{%
\mathrm{red}}\frac{1}{s-A_{\mathrm{red}}}C_{j}^{\dag }\left( S_{j\mathsf{e}%
}+S_{j\mathsf{i}}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{\mathsf{ie}%
}\right) , \\
\xi _{\mathrm{red}} &=&C_{\mathrm{red}}\frac{1}{s-A_{\mathrm{red}}},
\end{eqnarray*}
and \ $S_{\mathrm{red}}$, $C_{\mathrm{red}}$ are as in the statement of the
theorem.

We now show that $\sum_{j=\mathsf{i},\mathsf{e}}C_{j}^{\dag }[S_{j\mathsf{e}%
}+S_{j\mathsf{i}}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{\mathsf{ie}%
}]=C_{\mathrm{red}}^{\dag }S_{\mathrm{red}}$. Now
\begin{eqnarray*}
\sum_{j=\mathsf{i},\mathsf{e}}C_{j}^{\dag }[S_{j\mathsf{e}}+S_{j\mathsf{i}%
}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{\mathsf{ie}}] &=&C_{\mathsf{i}%
}^{\dag }[S_{\mathsf{ie}}+S_{\mathsf{ii}}\left( \eta -S_{\mathsf{ii}}\right)
^{-1}S_{\mathsf{ie}}]+C_{\mathsf{e}}^{\dag }S_{\mathrm{red}} \\
&=&C_{\mathsf{i}}^{\dag }\eta \left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{%
\mathsf{ie}}+C_{\mathsf{e}}^{\dag }S_{\mathrm{red}},
\end{eqnarray*}
while $C_{\mathrm{red}}^{\dag }S_{\mathrm{red}}=C_{\mathsf{i}}^{\dag }(\eta
^{\dag }-S_{\mathsf{ii}}^{\dag })^{-1}S_{\mathsf{ie}}^{\dag }S_{\mathrm{red}%
}+C_{\mathsf{e}}^{\dag }S_{\mathrm{red}}$. However,
\begin{equation*}
(\eta ^{\dag }-S_{\mathsf{ii}}^{\dag })^{-1}S_{\mathsf{ie}}^{\dag }S_{%
\mathrm{red}}=(\eta ^{\dag }-S_{\mathsf{ii}}^{\dag })^{-1}S_{\mathsf{ei}%
}^{\dag }(S_{\mathsf{ee}}+S_{\mathsf{ei}}\left( \eta -S_{\mathsf{ii}}\right)
^{-1}S_{\mathsf{ie}})
\end{equation*}
and using the identities $S_{\mathsf{ii}}^{\dag }S_{\mathsf{ii}}+S_{\mathsf{%
ei}}^{\dag }S_{\mathsf{ei}}=1$, $S_{\mathsf{ii}}^{\dag }S_{\mathsf{ie}}+S_{%
\mathsf{ei}}^{\dag }S_{\mathsf{ee}}=0$, this reduces to
\begin{eqnarray*}
(\eta ^{\dag }-S_{\mathsf{ii}}^{\dag })^{-1}S_{\mathsf{ie}}^{\dag }S_{%
\mathrm{red}} &=&(\eta ^{\dag }-S_{\mathsf{ii}}^{\dag })^{-1}\left[ -S_{%
\mathsf{ii}}^{\dag }S_{\mathsf{ie}}+(1-S_{\mathsf{ii}}^{\dag }S_{\mathsf{ii}%
})\left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{\mathsf{ie}}\right] \\
&=&(\eta ^{\dag }-S_{\mathsf{ii}}^{\dag })^{-1}\left[ -S_{\mathsf{ii}}^{\dag
}\left( \eta -S_{\mathsf{ii}}\right) +(1-S_{\mathsf{ii}}^{\dag }S_{\mathsf{ii%
}})\right] \left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{\mathsf{ie}} \\
&=&\eta \left( \eta -S_{\mathsf{ii}}\right) ^{-1}S_{\mathsf{ie}}.
\end{eqnarray*}
Therefore $\Xi _{\mathrm{red}}=S_{\mathrm{red}}-\sum_{j=\mathsf{i},\mathsf{e}%
}C_{\mathrm{red}}\dfrac{1}{s-A_{\mathrm{red}}}C_{\mathrm{red}}^{\dag }S_{%
\mathrm{red}}$, as required.

For consistency, we should check that we have $A_{\mathrm{red}}=-\frac{1}{%
\mathsf{e}}C_{\mathrm{red}}^{\dag }C_{\mathrm{red}}-i\Omega _{\mathrm{red}}$
with $\Omega _{\mathrm{red}}$ selfadjoint. Indeed, setting $A=-\frac{1}{2}C_{%
\mathsf{i}}^{\dag }C_{\mathsf{i}}-\frac{1}{2}C_{\mathsf{e}}^{\dag }C_{%
\mathsf{e}}-i\Omega $ and substituting in for $C_{\mathrm{red}}$ and $K_{%
\mathrm{red}}$ we find after some algebra that
\begin{equation*}
\Omega _{\mathrm{red}}=\Omega +\func{Im}\left\{ C_{\mathsf{i}}^{\dag }S_{%
\mathsf{ii}}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}C_{\mathsf{i}}\right\} +%
\func{Im}\left\{ C_{\mathsf{e}}^{\dag }S_{\mathsf{ei}}\left( \eta -S_{%
\mathsf{ii}}\right) ^{-1}C_{\mathsf{i}}\right\} .
\end{equation*}
The manipulation for this is trivial except for the calculation of the term
of the form $\frac{1}{2}C_{\mathsf{i}}^{\dag }XC_{\mathsf{i}}$ where
\begin{eqnarray*}
X &=&1+2S_{\mathsf{ii}}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}-(\eta
^{\dag }-S_{\mathsf{ii}}^{\dag })^{-1}S_{\mathsf{ei}}^{\dag }S_{\mathsf{ei}%
}\left( \eta -S_{\mathsf{ii}}\right) ^{-1} \\
&\equiv &(1-\eta S_{\mathsf{ii}}^{\dag })^{-1}\left[ S_{\mathsf{ii}}\eta
^{\dag }-\eta S_{\mathsf{ii}}^{\dag }\right] \left( 1-S_{\mathsf{ii}}\eta
^{\dag }\right) ^{-1} \\
&=&2i\func{Im}\frac{S_{\mathsf{ii}}\eta ^{\dag }}{1-S_{\mathsf{ii}}\eta
^{\dag }}=2i\func{Im}\left\{ S_{\mathsf{ii}}\left( \eta -S_{\mathsf{ii}%
}\right) ^{-1}\right\}
\end{eqnarray*}
where again we use the identity $S_{\mathsf{ii}}^{\dag }S_{\mathsf{ii}}+S_{%
\mathsf{ei}}^{\dag }S_{\mathsf{ei}}=1$.
\end{proof}

\bigskip

In terms of the parameters $\left( S,L,H\right) $ with $S=\left(
\begin{array}{cc}
S_{\mathsf{ii}} & S_{\mathsf{ie}} \\
S_{\mathsf{ei}} & S_{\mathsf{ee}}
\end{array}
\right) $, $L=\left(
\begin{array}{c}
L_{\mathsf{i}} \\
L_{\mathsf{e}}
\end{array}
\right) =\left(
\begin{array}{c}
C_{\mathsf{i}}\mathbf{a} \\
C_{\mathsf{e}}\mathbf{a}
\end{array}
\right) $ and $H=\mathbf{a}^{\dag }\Omega \mathbf{a}$, we have that the
feedback system is described by the reduced parameters $\left( S_{\mathrm{red%
}},L_{\mathrm{red}},H_{\mathrm{red}}\right) $ where
\begin{eqnarray}
S_{\mathrm{red}} &=&S_{\mathsf{ee}}+S_{\mathsf{ei}}\left( \eta -S_{\mathsf{ii%
}}\right) ^{-1}S_{\mathsf{ie}}  \notag \\
L_{\mathrm{red}} &=&S_{\mathsf{ei}}\left( \eta -S_{\mathsf{ii}}\right)
^{-1}L_{\mathsf{i}}+L_{\mathsf{e}},  \notag \\
H_{\mathrm{red}} &=&H+\func{Im}\left\{ L_{\mathsf{i}}^{\dag }S_{\mathsf{ii}%
}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}L_{\mathsf{i}}\right\} +\func{Im}%
\left\{ L_{\mathsf{e}}^{\dag }S_{\mathsf{ei}}\left( \eta -S_{\mathsf{ii}%
}\right) ^{-1}L_{\mathsf{i}}\right\} .  \notag \\
&&  \label{general feedback law}
\end{eqnarray}
The same equations have been deduced in the nonlinear case by different
arguments \cite{QFN1}. Note the identity $\func{Im}\left\{ L_{\mathsf{i}%
}^{\dag }S_{\mathsf{ii}}\left( \eta -S_{\mathsf{ii}}\right) ^{-1}L_{\mathsf{i%
}}\right\} =\func{Im}\left\{ L_{\mathsf{i}}^{\dag }\left( \eta -S_{\mathsf{ii%
}}\right) ^{-1}L_{\mathsf{i}}\right\} $.

\bigskip

\begin{remark}
Let $U$ be a unitary operator on a fixed Hilbert space $\frak{H}=\frak{H}%
_{1}\oplus \frak{H}_{2}$ which decomposes as $U=\left(
\begin{array}{cc}
U_{11} & U_{12} \\
U_{21} & U_{22}
\end{array}
\right) $. The non-commutative M\"{o}bius transform $\varphi
_{U}^{2\rightarrow 1}$ is the superoperator defined by
\begin{equation*}
\varphi _{U}^{2\rightarrow 1}\left( X\right) =U_{11}+U_{12}\left(
1-XU_{22}\right) ^{-1}XU_{21}
\end{equation*}
defined on the domain of operators $X$ on $\frak{H}_{2}$ for which the
inverse $\left( 1-XU_{22}\right) ^{-1}$ exists. The transform $\varphi
_{U}^{2\rightarrow 1}$ maps unitaries on $\frak{H}_{2}$ in its domain to
unitaries in $\frak{H}_{1}$ \cite{Young}.
\end{remark}

\begin{remark}
In particular, $S_{\mathrm{red}}$ is unitary as it equals $\varphi _{S}^{%
\mathsf{i}\rightarrow \mathsf{e}}\left( \xi \right) $ where $\xi =\eta ^{-1}$
with $\eta $ being unitary. We may expand the geometric series to write
\begin{equation*}
S_{\mathrm{red}}=S_{\mathsf{ee}}+S_{\mathsf{ei}}\xi S_{\mathsf{ie}}+S_{%
\mathsf{ei}}\xi S_{\mathsf{ii}}\xi S_{\mathsf{ie}}+S_{\mathsf{ei}}\xi S_{%
\mathsf{ii}}\xi S_{\mathsf{ii}}\xi S_{\mathsf{ie}}+\cdots =S_{\mathsf{ee}%
}+\sum_{n=0}^{\infty }S_{\mathsf{ei}}\xi \left( S_{\mathsf{ii}}\xi \right)
^{n}S_{\mathsf{ie}}
\end{equation*}
which shows that $S_{\mathrm{red}}$ can be built up from\ contributions from
the various paths through the network. Likewise
\begin{eqnarray*}
L_{\mathrm{red}} &=&L_{\mathsf{e}}+\sum_{n=0}^{\infty }S_{\mathsf{ei}}\xi
\left( S_{\mathsf{ii}}\xi \right) ^{n}L_{\mathsf{i}},\quad \\
H_{\mathrm{red}} &=&H+\sum_{n=0}^{\infty }\func{Im}\left\{ L_{\mathsf{i}%
}^{\dag }\left( S_{\mathsf{ii}}\xi \right) ^{n}L_{\mathsf{i}}\right\}
+\sum_{n=0}^{\infty }\func{Im}\left\{ L_{\mathsf{e}}^{\dag }S_{\mathsf{ei}%
}\xi \left( S_{\mathsf{ii}}\xi \right) ^{n}L_{\mathsf{i}}\right\} .
\end{eqnarray*}
\end{remark}

\section{Systems in Series}

As a very special case of feedback connections we consider the situation of
systems in series. This is referred to as \textit{feedforward} in
engineering.

\begin{center}
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\put(77,10){\circle{2}}

\put(35,11){$s_2$}
\put(65,11){$s_1$}
\put(42,11){$r_2$}
\put(72,11){$r_1$}

\end{picture}%

figure 3: Cascaded systems
\end{center}

The individual transfer functions before the connection $e=\left(
s_{1},r_{2}\right) $ is made are given by $\Xi _{i}=\left[
\begin{tabular}{r|r}
$A_{i}$ & $-C_{i}^{\dag }S_{i}$ \\ \hline
$C_{i}$ & $S_{i}$%
\end{tabular}
\right] $ with $A_{i}=-\frac{1}{2}C_{i}^{\dag }C_{i}-i\Omega _{i}$.and these
may be concatenated to give
\begin{equation*}
\Xi =\left[
\begin{tabular}{l|ll}
$A_{1}+A_{2}$ & $-C_{1}^{\dag }S_{1}$ & $-C_{2}^{\dag }S_{2}$ \\ \hline
$C_{1}$ & $S_{1}$ & $0$ \\
$C_{2}$ & $0$ & $S_{2}$%
\end{tabular}
\right] .
\end{equation*}

To use the formula for the reduced transfer function following connection,
we must first of all identify the internal (eliminated) and external fields:
here
\begin{equation*}
\mathbf{b}^{\text{in}}=\left(
\begin{array}{c}
\mathbf{b}_{\mathsf{i}}^{\text{in}} \\
\mathbf{b}_{\mathsf{e}}^{\text{in}}
\end{array}
\right) =\left(
\begin{array}{c}
\mathbf{b}_{2}^{\text{in}} \\
\mathbf{b}_{1}^{\text{in}}
\end{array}
\right) ,\quad \mathbf{b}^{\text{out}}=\left(
\begin{array}{c}
\mathbf{b}_{\mathsf{i}}^{\text{out}} \\
\mathbf{b}_{\mathsf{e}}^{\text{out}}
\end{array}
\right) \equiv \left(
\begin{array}{c}
\mathbf{b}_{1}^{\text{out}} \\
\mathbf{b}_{2}^{\text{out}}
\end{array}
\right) ,
\end{equation*}
and
\begin{equation*}
\left(
\begin{array}{cc}
S_{\mathsf{ii}} & S_{\mathsf{ie}} \\
S_{\mathsf{ei}} & S_{\mathsf{ee}}
\end{array}
\right) \equiv \left(
\begin{array}{cc}
0 & S_{1} \\
S_{2} & 0
\end{array}
\right) ,\quad L_{\mathsf{i}}\equiv L_{1},L_{\mathsf{e}}\equiv L_{2},
\end{equation*}
with trivially $\eta =1$. The reduced transfer function is then readily
computed to be
\begin{equation*}
\Xi _{\text{series}}=\left[
\begin{tabular}{r|r}
$A_{1}+A_{2}-C_{2}^{\dag }S_{2}C_{1}$ & $-\left( C_{2}^{\dag
}S_{2}+C_{1}^{\dag }\right) S_{1}$ \\ \hline
$C_{2}+S_{2}C_{1}$ & $S_{2}S_{1}$%
\end{tabular}
\right] .
\end{equation*}
Likewise we deduce the relations
\begin{equation}
S=S_{2}S_{1},\quad L=L_{2}+S_{2}L_{1},\quad H=H_{1}+H_{2}+\func{Im}\left\{
L_{2}^{\dag }S_{2}L_{1}\right\} .  \label{special feedback law}
\end{equation}
The same equations have been deduced in the nonlinear case by different
arguments \cite{GJ Series}.

\subsection{Feedforward: Cascades}

If the two systems are truly distinct systems, that is, if they are
different sets of oscillators, then we are in the situation of properly
\textit{cascaded} systems. In this case one would expect that the transfer
function to factor as the ordinary matrix product $\Xi _{\text{series}%
}\equiv \Xi _{2}\Xi _{1}$. We now show that this is indeed the case.

\begin{lemma}
Let $\Xi _{j}$ be transfer functions for $m_{j}$ oscillators coupled to $n$
fields $(j=1,2)$. If we consider the ampliated transfer functions for $%
m_{1}+m_{2}$ oscillators coupled to $n$ fields
\begin{eqnarray*}
\tilde{\Xi}_{1} &=&\left[
\begin{tabular}{c|c}
$\left(
\begin{array}{cc}
A_{1} & 0 \\
0 & 0
\end{array}
\right) $ & $\left(
\begin{array}{c}
-C_{1}^{\dag }S_{1} \\
0
\end{array}
\right) $ \\ \hline
$\left( C_{1},0\right) $ & $S_{1}$%
\end{tabular}
\right] , \\
\tilde{\Xi}_{2} &=&\left[
\begin{tabular}{c|c}
$\left(
\begin{array}{cc}
0 & 0 \\
0 & A_{2}
\end{array}
\right) $ & $\left(
\begin{array}{c}
0 \\
-C_{2}^{\dag }S_{2}
\end{array}
\right) $ \\ \hline
$\left( 0,C_{2}\right) $ & $S_{2}$%
\end{tabular}
\right] ,
\end{eqnarray*}
then
\begin{equation}
\tilde{\Xi}_{\text{series}}=\Xi _{2}\Xi _{1}.
\end{equation}
\end{lemma}

\begin{proof}
We compute this directly,
\begin{eqnarray*}
\tilde{\Xi}_{\text{series}} &=&\left[
\begin{tabular}{c|c}
$\left(
\begin{array}{cc}
A_{1} & 0 \\
-C_{2}^{\dag }S_{2}C_{1} & -A_{2}
\end{array}
\right) $ & $\left(
\begin{array}{c}
-C_{1}^{\dag }S_{1} \\
-C_{2}^{\dag }S_{2}S_{1}
\end{array}
\right) $ \\ \hline
$\left( C_{1},C_{2}\right) $ & $S_{2}S_{1}$%
\end{tabular}
\right] \\
&=&S_{2}S_{1}+\left.
\begin{array}{c}
(C_{1},C_{2})
\end{array}
\right. \left(
\begin{array}{cc}
s-A_{1} & 0 \\
C_{2}^{\dag }S_{2}C_{1} & s-A_{2}
\end{array}
\right) ^{-1}\left(
\begin{array}{c}
-C_{1}^{\dag }S_{1} \\
-C_{2}^{\dag }S_{2}S_{1}
\end{array}
\right) \\
&=&S_{2}S_{1}-\left.
\begin{array}{c}
(C_{1},C_{2})
\end{array}
\right. \left(
\begin{array}{ll}
\frac{1}{s-A_{1}} & 0 \\
-\frac{1}{s-A_{2}}C_{2}^{\dag }S_{2}C_{1}\frac{1}{s-A_{1}} & \frac{1}{s-A_{2}%
}
\end{array}
\right) \left(
\begin{array}{c}
C_{1}^{\dag }S_{1} \\
C_{2}^{\dag }S_{2}S_{1}
\end{array}
\right) \\
&=&\left[ S_{2}-C_{2}\left( s-A_{2}\right) ^{-1}C_{2}^{\dag }S_{2}\right]
\times \left[ S_{1}-C_{1}\left( s-A_{1}\right) ^{-1}C_{1}^{\dag }S_{1}\right]
,
\end{eqnarray*}
giving the result.
\end{proof}

\section{Beam Splitters}

A simple beam splitter is a device performing physical superposition of two
input fields. It is described by a fixed unitary operator $T=\left(
\begin{array}{cc}
\alpha & \beta \\
\mu & \nu
\end{array}
\right) \in U\left( 2\right) $:
\begin{equation*}
\left(
\begin{array}{c}
\mathbf{b}_{1}^{\text{out}} \\
\mathbf{b}_{2}^{\text{out}}
\end{array}
\right) =\left(
\begin{array}{cc}
\alpha & \beta \\
\mu & \nu
\end{array}
\right) \left(
\begin{array}{c}
\mathbf{b}_{1}^{\text{in}} \\
\mathbf{b}_{2}^{\text{in}}
\end{array}
\right) .
\end{equation*}
This is a canonical transformation and the output fields satisfy the same
canonical commutation relations as the inputs. The action of the beam
splitter is depicted in the figure below. On the left we have a traditional
view of the two inputs being split into two output fields. On the right we
have our view of the beam splitter as being a component with two input ports
and two output ports: we have sketched some internal detail to emphasize how
the scattering (superimposing) of inputs how ever we shall usually just draw
this as a ``black box'' component in the following.

\begin{center}
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\put(80,25){\circle{2}}

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\put(7,12){\line(1,1){16}}

\put(88,12){${\bf b}^{\rm in}_2$}
\put(88,27){${\bf b}^{\rm in}_1$}
\put(48,12){${\bf b}^{\rm out}_2$}
\put(48,27){${\bf b}^{\rm out}_1$}

\put(18,8){${\bf b}^{\rm in}_2$}
\put(0,22){${\bf b}^{\rm in}_1$}
\put(17,35){${\bf b}^{\rm out}_2$}
\put(30,22){${\bf b}^{\rm out}_1$}

\put(62,7){\dashbox(22,20)}

\end{picture}

Figure 4: Beam-splitter component.
\end{center}

To emphasize that the beam splitter is an input-output device of exactly the
for we have been considering up to now, let us state that its transfer
matrix function is
\begin{equation*}
\Xi _{\text{beam splitter}}=\left[
\begin{tabular}{c|c}
$0$ & $0$ \\ \hline
$0$ & $T$%
\end{tabular}
\right] \equiv T.
\end{equation*}
Our aim is to describe the effective Markov model for the feedback device
sketched below where the feedback is implemented by means of a beam
splitter. Here we have a component system, called the plant, in-loop and we
assume that it is described by the transfer function $\Xi _{0}=\left[
\begin{tabular}{c|c}
$A_{0}$ & $-C_{0}^{\dag }S_{0}$ \\ \hline
$C_{0}$ & $S_{0}$%
\end{tabular}
\right] $.

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\begin{picture}(120,80)
\label{pic5}

\thicklines
\put(30,40){\line(1,1){20}}

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\put(73,20){\circle{2}}
\put(73,33){plant}

\put(10,55){${\bf b}_{1}^{\rm in}$}
\put(40,74){${\bf b}_{1}^{\rm out}$}
\put(83,54){${\bf b}_{2}^{\rm out}$}

\put(33,14){${\bf b}_{2}^{\rm in}$}

\end{picture}

Figure 5: Feedback using a beam-splitter.
\end{center}

It is more convenient to view this as the network sketched below.

\begin{center}
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\begin{picture}(60,50)
\label{pic6}
\thicklines
\put(26,10){$s_1$}
\put(26,20){$s_2$}
\put(26,40){$r_3$}
\put(31,10){$r_1$}
\put(31,20){$r_2$}
\put(31,40){$s_3$}
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\put(20,5){\line(1,0){20}}
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\put(37,10){\line(1,0){13}}
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\put(50,10){\vector(-1,0){5}}
\put(10,20){\line(0,1){20}}
\put(10,40){\line(1,0){13}}
\put(50,40){\line(-1,0){13}}
\put(50,20){\line(0,1){20}}
\put(2,5){$ {\bf b}_{1}^{\rm out}$}
\put(55,5){$ {\bf b}_{1}^{\rm in}$}
\put(43,43){{\it plant}}
\put(42,14){{\it beam splitter} }
\end{picture}

Figure 6: Network representation.
\end{center}

Here we have the pair of internal edges $\left( s_{2},r_{3}\right) $ and $%
\left( s_{3},r_{2}\right) $. The transfer function for the network is
\begin{equation*}
\Xi _{\text{unconn.}}=\left[
\begin{tabular}{l|lll}
$A_{0}$ & 0 & 0 & $-C_{0}^{\dag }S_{0}$ \\ \hline
0 & $T_{11}$ & $T_{12}$ & 0 \\
0 & $T_{21}$ & $T_{22}$ & 0 \\
$C_{0}$ & 0 & 0 & $S_{0}$%
\end{tabular}
\right]
\end{equation*}
with respect to the labels $\left( 0,s_{1},s_{2},s_{3}\right) $ for the rows
and $\left( 0,r_{1},r_{2},r_{3}\right) $ for the columns. This time the
external fields are $\mathbf{b}_{\mathsf{e}}^{\text{in}}=\mathbf{b}_{1}^{%
\text{in}}$, $\mathbf{b}_{\mathsf{e}}^{\text{out}}=\mathbf{b}_{1}^{\text{out}%
}\equiv T_{11}\mathbf{b}_{1}^{\text{in}}+T_{12}\mathbf{b}_{2}^{\text{in}}$
while the (matched) internal fields are
\begin{equation*}
\mathbf{b}_{\mathsf{i}}^{\text{in}}=\left(
\begin{array}{c}
\mathbf{b}_{2}^{\text{in}} \\
\mathbf{b}_{3}^{\text{in}}
\end{array}
\right) ,\quad \mathbf{b}_{\mathsf{i}}^{\text{out}}=\left(
\begin{array}{c}
\mathbf{b}_{2}^{\text{out}} \\
\mathbf{b}_{3}^{\text{out}}
\end{array}
\right) \equiv \left(
\begin{array}{c}
T_{21}\mathbf{b}_{1}^{\text{in}}+T_{22}\mathbf{b}_{2}^{\text{in}} \\
S_{0}\mathbf{b}_{3}^{\text{in}}+L_{0}
\end{array}
\right) .
\end{equation*}
That is
\begin{equation*}
\begin{tabular}{ll}
$S_{\mathsf{ii}}=\left(
\begin{array}{cc}
T_{22} & 0 \\
0 & S_{0}
\end{array}
\right) ,$ & $S_{\mathsf{ie}}=\left(
\begin{array}{c}
T_{21} \\
0
\end{array}
\right) ,$ \\
$S_{\mathsf{ei}}=\left( T_{12},0\right) ,$ & $S_{\mathsf{ee}}=T_{11},$ \\
$L_{\mathsf{i}}=\left(
\begin{array}{c}
L_{0} \\
0
\end{array}
\right) ,$ & $L_{\mathsf{i}}=0,\quad \eta =\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right) .$%
\end{tabular}
\end{equation*}

Substituting into our reduction formula we obtain
\begin{eqnarray*}
S &=&T_{11}+
\begin{array}{c}
(T_{12},0)
\end{array}
\left(
\begin{array}{cc}
-T_{22} & 1 \\
1 & -S_{0}
\end{array}
\right) ^{-1}\left(
\begin{array}{c}
T_{21} \\
0
\end{array}
\right) \\
&\equiv &T_{11}+T_{12}\left( 1-S_{0}T_{22}\right) ^{-1}S_{0}T_{21}, \\
C &=&
\begin{array}{c}
(T_{12},0)
\end{array}
\left(
\begin{array}{cc}
-T_{22} & 1 \\
1 & -S_{0}
\end{array}
\right) ^{-1}\left(
\begin{array}{c}
0 \\
C_{0}
\end{array}
\right) \\
&\equiv &T_{12}\left( 1-S_{0}T_{22}\right) ^{-1}C_{0}, \\
\Omega &=&\Omega _{0}+\func{Im}
\begin{array}{c}
(0,L_{0}^{\dag })
\end{array}
\left(
\begin{array}{cc}
-T_{22} & 1 \\
1 & -S_{0}
\end{array}
\right) ^{-1}\left(
\begin{array}{c}
0 \\
L_{0}
\end{array}
\right) \\
&\equiv &\Omega _{0}+\func{Im}C_{0}^{\dag }\left( 1-S_{0}T_{22}\right)
^{-1}C_{0}.
\end{eqnarray*}
and so, when the connections are made, the transfer fmatrix function is
\begin{equation*}
\Xi _{\text{conn.}}=\left[
\begin{tabular}{l|l}
$A_{0}-C_{0}^{\dag }S_{0}T_{22}C_{0}$ & $-C_{0}^{\dag
}S_{0}T_{21}-C_{0}^{\dag }S_{0}\left( 1-S_{0}T_{22}\right) ^{-1}T_{22}$ \\
\hline
$T_{12}\left( 1-S_{0}T_{22}\right) ^{-1}C_{0}$ & $T_{11}+T_{12}\left(
1-S_{0}T_{22}\right) ^{-1}S_{0}T_{21}$%
\end{tabular}
\right] ,
\end{equation*}

Note that $S=\varphi _{T}^{2\rightarrow 1}\left( S_{0}\right) $ where $%
\varphi _{T}^{2\rightarrow 1}\left( z\right) =T_{11}+T_{12}\beta \left(
z^{-1}-T_{22}\right) T_{21}$ is the M\"{o}bius transformation in the complex
plane associated with $T$.

If we further set $T=\left(
\begin{array}{cc}
\alpha & \beta \\
\mu & \nu
\end{array}
\right) $, and $x+iy=S_{0}\nu $, then
\begin{gather*}
C^{\dag }C=\left| \frac{\beta }{1-S_{0}\nu }\right| ^{2}C_{0}^{\dag }C_{0}=%
\frac{1-|\nu |^{2}}{|1-S_{0}\nu |^{2}}C_{0}^{\dag }C_{0}\equiv \frac{%
1-x^{2}-y^{2}}{\left( 1-x\right) ^{2}+y^{2}}C_{0}^{\dag }C_{0}, \\
\func{Im}C_{0}^{\dag }\left( 1-S_{0}\nu \right) ^{-1}C_{0}=\func{Im}\left\{
\frac{1}{1-x-iy}\right\} C_{0}^{\dag }C_{0}=\frac{y}{\left( 1-x\right)
^{2}+y^{2}}C_{0}^{\dag }C_{0}.
\end{gather*}
In particular, if we take a single oscillator in-loop with $S_{0}=e^{i\phi
_{0}}$, then we obtain $S\equiv e^{i\phi }$ and the phase is determined by
the M\"{o}bius transformation. If we further have $L_{0}=\sqrt{\gamma _{0}}a$%
, $H_{0}=\omega _{0}a^{\dag }a$, we find that $L\equiv e^{i\delta }\sqrt{%
\gamma }a$ and $H=\omega a^{\dag }a$ where
\begin{equation*}
\gamma =\frac{1-x^{2}-y^{2}}{\left( 1-x\right) ^{2}+y^{2}}\gamma _{0},\quad
\omega =\frac{y}{\left( 1-x\right) ^{2}+y^{2}}\omega _{0},
\end{equation*}
and $\delta $ is a real phase. In the specific case $T=\left(
\begin{array}{cc}
\alpha & \beta \\
\beta & -\alpha
\end{array}
\right) $ with $S_{0}=1,\omega _{0}=0$\ considered by Yanagisawa and Kimura
\cite{YK1}, we have $x=-\alpha $ and $y=0$, therefore we find
\begin{equation*}
\gamma =\frac{1-\alpha }{1+\alpha }\gamma _{0},\quad \omega =0
\end{equation*}
which agrees with their findings.

\bigskip

An alternative computation of $\Xi $ \ is given by the following argument.
We consider the input-output relations
\begin{equation*}
\mathbf{\hat{b}}_{i}^{\text{out}}=\sum_{j=1,2}T_{ij}\mathbf{\hat{b}}_{j}^{%
\text{in}},\quad \mathbf{\hat{b}}_{2}^{\text{in}}=\Xi _{0}\mathbf{\hat{b}}%
_{1}^{\text{out}}+\xi _{0}\mathbf{a}_{0},
\end{equation*}
and eliminating $\mathbf{\hat{b}}_{2}^{\text{out}}\equiv \left( 1-T_{22}\Xi
\right) ^{-1}\left[ T_{21}\mathbf{\hat{b}}_{1}^{\text{in}}+T_{22}\xi _{0}%
\mathbf{a}_{0}\right] $ yields
\begin{equation*}
\mathbf{\hat{b}}_{1}^{\text{out}}=\left[ T_{11}+T_{12}\Xi _{0}\left(
1-T_{22}\Xi _{0}\right) ^{-1}T_{21}\right] \mathbf{\hat{b}}_{1}^{\text{in}%
}+T_{21}\left( 1-\Xi _{0}T_{22}\right) ^{-1}\xi _{0}\mathbf{a}_{0}.
\end{equation*}
That is
\begin{equation*}
\Xi =T_{11}+T_{12}\left( \Xi _{0}^{-1}-T_{22}\right) ^{-1}T_{21}=\varphi
_{T}^{2\rightarrow 1}\left( \Xi _{0}\right) .
\end{equation*}
We remark that if $T_{12}$ and $T_{21}$ are invertible, then we may invert
the M\"{o}bius transformation to get
\begin{equation*}
\Xi _{0}^{-1}=T_{22}+T_{21}\frac{1}{\Xi -T_{11}}T_{12}.
\end{equation*}

To illustrate with a cavity mode in-loop, we take the beam splitter matrix
to be $T=\left(
\begin{array}{cc}
\alpha & \beta \\
\beta & -\alpha
\end{array}
\right) $ with $\alpha ^{2}+\beta ^{2}=1$, and the transfer function $\Xi
_{0}\left( s\right) =\frac{s+i\omega -\gamma /2}{s+i\omega +\gamma /2}$,
then we find
\begin{equation*}
\Xi =\frac{\alpha +\Xi _{0}}{1+\alpha \Xi _{0}}=\frac{s+i\omega -\frac{%
1-\alpha }{1+\alpha }\frac{\gamma }{2}}{s+i\omega +\frac{1-\alpha }{1+\alpha
}\frac{\gamma }{2}}.
\end{equation*}

\section{The Redheffer Star Product}

An important feedback arrangement is shown in the figure below.

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\put(26,26){$s_3$}
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\put(26,56){$r_1$}
\put(31,14){$r_4$}
\put(31,26){$r_3$}
\put(31,44){$s_2$}
\put(31,56){$s_1$}
\end{picture}%

Figure 7: Composite System
\end{center}

We shall now derive the matrices for this system taking component $A$ to be
described $\left(
\begin{array}{cc}
S_{11}^{A} & S_{12}^{A} \\
S_{21}^{A} & S_{22}^{A}
\end{array}
\right) ,$ $\left(
\begin{array}{c}
C_{1}^{A} \\
C_{2}^{A}
\end{array}
\right) ,$ $\Omega _{A}$ and $B$ by $\left(
\begin{array}{cc}
S_{33}^{B} & S_{34}^{B} \\
S_{43}^{B} & S_{44}^{B}
\end{array}
\right) ,$ $\left(
\begin{array}{c}
C_{3}^{B} \\
C_{4}^{B}
\end{array}
\right) ,$ $\Omega _{B}$. The operators of systems $A$ are asumed to commute
with those of $B$. We have two internal channels to eliminate which we can
do in sequence, or simulataneously. We shall do the latter. here we have
\begin{eqnarray*}
\mathsf{S}_{\mathtt{ee}} &=&\left(
\begin{array}{cc}
S_{11}^{A} & 0 \\
0 & S_{44}
\end{array}
\right) ,\mathsf{S}_{\mathtt{ei}}=\left(
\begin{array}{cc}
S_{12}^{A} & 0 \\
0 & S_{43}^{B}
\end{array}
\right) \\
\mathsf{S}_{\mathtt{ie}} &=&\left(
\begin{array}{cc}
S_{21}^{A} & 0 \\
0 & S_{34}^{B}
\end{array}
\right) ,\mathsf{S}_{\mathtt{ii}}=\left(
\begin{array}{cc}
S_{22}^{A} & 0 \\
0 & S_{33}^{B}
\end{array}
\right)
\end{eqnarray*}
and
\begin{equation*}
\mathsf{L}_{\mathtt{e}}=\left(
\begin{array}{c}
L_{1}^{A} \\
L_{4}^{B}
\end{array}
\right) ,\quad \mathsf{L}_{\mathtt{i}}=\left(
\begin{array}{c}
L_{2}^{A} \\
L_{3}^{B}
\end{array}
\right) ,\quad \eta =\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right) .
\end{equation*}
The parameters are therefore
\begin{eqnarray*}
S_{\star } &=&\left(
\begin{array}{cc}
S_{11}^{A} & 0 \\
0 & S_{44}^{B}
\end{array}
\right) +\left(
\begin{array}{cc}
S_{12}^{A} & 0 \\
0 & S_{43}^{B}
\end{array}
\right) \left(
\begin{array}{cc}
-S_{22}^{A} & 1 \\
1 & -S_{33}^{B}
\end{array}
\right) ^{-1}\left(
\begin{array}{cc}
S_{21}^{A} & 0 \\
0 & S_{34}^{B}
\end{array}
\right) \\
&=&\left(
\begin{array}{cc}
S_{11}^{A}+S_{12}^{A}S_{33}^{B}\left( 1-S_{22}^{A}S_{33}^{B}\right)
^{-1}S_{21}^{A} & S_{12}^{A}\left( 1-S_{22}^{A}S_{33}^{B}\right) ^{-1}S_{34}
\\
S_{43}^{B}\left( 1-S_{22}^{A}S_{33}^{B}\right) ^{-1}S_{21}^{A} &
S_{44}+S_{43}^{B}\left( 1-S_{22}^{A}S_{33}^{B}\right) ^{-1}S_{22}^{A}S_{34}
\end{array}
\right) , \\
C_{\star } &=&\left(
\begin{array}{c}
C_{1}^{A} \\
C_{4}^{B}
\end{array}
\right) +\left(
\begin{array}{cc}
S_{12}^{A} & 0 \\
0 & S_{43}^{B}
\end{array}
\right) \left(
\begin{array}{cc}
-S_{22}^{A} & 1 \\
1 & -S_{33}^{B}
\end{array}
\right) ^{-1}\left(
\begin{array}{c}
C_{2}^{A} \\
C_{3}^{B}
\end{array}
\right) \\
&=&\left(
\begin{array}{c}
C_{1}^{A}+S_{12}^{A}S_{33}^{B}\left( 1-S_{22}^{A}S_{33}^{B}\right)
^{-1}C_{2}^{A}+S_{12}^{A}\left( 1-S_{22}^{A}S_{33}^{B}\right) ^{-1}C_{3}^{B}
\\
C_{4}^{B}+S_{43}^{B}\left( 1-S_{22}^{A}S_{33}^{B}\right)
^{-1}C_{2}^{A}+S_{43}^{B}S_{22}^{A}\left( 1-S_{22}^{A}S_{33}^{B}\right)
^{-1}C_{3}^{B}
\end{array}
\right) ,
\end{eqnarray*}
\begin{equation*}
\begin{array}{l}
\Omega _{\star }=\Omega _{A}+\Omega _{B}+\func{Im}\left\{ C_{3}^{B\dag
}\left( 1-S_{33}^{B}S_{22}^{A}\right) ^{-1}C_{3}^{B}+C_{3}^{B\dag }\left(
1-S_{33}^{B}S_{22}^{A}\right) ^{-1}S_{33}^{A}C_{2}^{A}\right. \\
+C_{2}^{A\dag }\left( 1-S_{22}^{A}S_{33}^{B}\right)
^{-1}S_{22}^{A}C_{3}^{B}+C_{2}^{A\dag }\left( 1-S_{22}^{A}S_{33}^{B}\right)
^{-1}C_{2}^{A} \\
+C_{1}^{A\dag }S_{12}^{A}\left( 1-S_{33}^{B}S_{22}^{A}\right)
^{-1}C_{3}^{B}+C_{1}^{A\dag }S_{12}^{A}\left( 1-S_{33}^{B}S_{22}^{A}\right)
^{-1}S_{33}^{B}C_{2}^{B} \\
\left. +C_{4}^{B\dag }S_{43}^{B}\left( 1-S_{22}^{A}S_{33}^{B}\right)
^{-1}S_{22}^{A}C_{3}^{B}+C_{4}^{B\dag }S_{43}^{B}\left(
1-S_{22}^{A}S_{33}^{B}\right) ^{-1}C_{2}^{A}\right\} .
\end{array}
\end{equation*}

\section{Appendix (Stratonovich to It\={o} Conversion)}

It is convenient to introduce integrated fields
\begin{equation*}
B_{i}\left( t\right) \equiv \int_{0}^{t}b_{i}\left( s\right) ds,B_{i}^{\dag
}\left( t\right) \equiv \int_{0}^{t}b_{i}^{\dag }\left( s\right) ds,\Lambda
_{ij}\left( t\right) \equiv \int_{0}^{t}b_{i}^{\dag }\left( s\right)
b_{j}\left( s\right) ds.
\end{equation*}
$B_{i}\left( t\right) $ and $B_{i}^{\dag }\left( t\right) $ are called the
annihilation and creation process, respectively, for the $i$th field and
collectivey are referred to as a quantum Wiener process. $\Lambda
_{ij}\left( t\right) $ is called the gauge process or scattering process
from the $j$th field to the $i$th field. A noncommutative version of the Ito
theory of stochastic integration with respect to these processes can be
built up. The quantum It\={o} table giving the product of infinitesimal
increments of these process is
\begin{equation*}
\begin{tabular}{l|llll}
$\times $ & $dB_{k}$ & $d\Lambda _{kl}$ & $dB_{l}^{\dag }$ & $dt$ \\ \hline
$dB_{i}$ & 0 & $\delta _{ik}dB_{l}$ & $\delta _{il}dt$ & 0 \\
$d\Lambda _{ij}$ & 0 & $\delta _{jk}d\Lambda _{il}$ & $\delta
_{jl}dB_{i}^{\dag }$ & 0 \\
$dB_{j}^{\dag }$ & 0 & 0 & 0 & 0 \\
$dt$ & 0 & 0 & 0 & 0
\end{tabular}
.
\end{equation*}
The Ito equation for the unitary process is then $dV=\left( dG\right) V$
where
\begin{equation*}
dG=\sum_{i,j=1}^{n}(S_{ij}-\delta _{ij})d\Lambda
_{ij}+\sum_{i=1}^{n}L_{i}dB_{i}^{\dag }-\sum_{j=1}^{n}L_{i}^{\dag
}S_{ij}B_{j}-\left( \frac{1}{2}\sum_{i=1}^{n}L_{i}^{\dag }L_{i}-iH\right) dt.
\end{equation*}
The Stratonovich form is $dV=-i\left( dE\right) \circ V$ where
\begin{equation*}
dE=\sum_{i,j=1}^{n}E_{ij}d\Lambda _{ij}+\sum_{i=1}^{n}F_{i}dB_{i}^{\dag
}+\sum_{j=1}^{n}F_{j}^{\dag }dB_{j}\left( t\right) +Kdt.
\end{equation*}
and we define the Stratonovich differential to be $\left( dX\right) \circ
Y=\left( dX\right) Y+\frac{1}{2}\left( dX\right) \left( dY\right) $ with the
last term computed using the It\={o} table. We have the consistency
condition $dV=\left( dG\right) V\equiv -i\left( dE\right) V-\frac{i}{2}%
\left( dE\right) \left( dG\right) V$ or
\begin{equation*}
dG=-idE-\frac{i}{2}\left( dE\right) \left( dG\right) ,
\end{equation*}
and using the table we see that
\begin{eqnarray*}
S-1 &\equiv &-iE-\frac{i}{2}E\left( S-1\right) \\
L &=&-iF-\frac{i}{2}EL \\
-\frac{1}{2}L^{\dag }L-iH &=&-iK-\frac{i}{2}F^{\dag }L
\end{eqnarray*}
which can be solved to give the relations $\left( \ref{Strat-Ito}\right) $.

redpajama_set_name: RedPajamaArXiv


Tea & infusion herbs for the bags, I used mint, black tea & lavender, and camomile.
Machine-sew along 3 sides of the muslin square, as close as possible to the edge.
Trim away any excess material, making sure not to cut through the machine sewing.
Turn the bag inside out, making sure the seams are flat.
Machine sew along the same three sides again enclosing the original seam inside the new seam.
Using a funnel, fill the bags ¾ full.
Write on the label the name of the tea / infusion, punch a hole in the top of the card.
Thread through the twine and wrap around muslin bag, tying a small bow to finish.


text:
The term Hinduism
General nature of Hinduism
The five tensile strands
Central conceptions
Veda, Brahmans, and issues of religious authority
Doctrine of atman-brahman
Karma, samsara, and moksha
Dharma and the three paths
Ashramas: the four stages of life
The history of Hinduism
Sources of Hinduism
Indo-European sources
Other sources: the process of "Sanskritization"
The prehistoric period (3rd and 2nd millennia bce)
Indigenous prehistoric religion
Religion in the Indus valley civilization
Survival of archaic religious practices
The Vedic period (2nd millennium–7th century bce)
Challenges to Brahmanism (6th–2nd century bce)
Early Hinduism (2nd century bce–4th century ce)
The rise of the major sects: Vaishnavism, Shaivism, and Shaktism
The development of temples
The spread of Hinduism in Southeast Asia and the Pacific
Questions of influence on the Mediterranean world
The rise of devotional Hinduism (4th–11th century)
Hinduism under Islam (11th–19th century)
The challenge of Islam and popular religion
Temple complexes
Sectarian movements
Bhakti movements
The modern period (from the 19th century)
Hindu reform movements
Brahmo Samaj
Ramakrishna Mission
Aurobindo Ashram
Other reform movements
The struggle for independence
The religious situation after independence
Hinduism outside India
Importance of the Vedas
The components of the Vedas
The Rigveda
Elaborations of text and ritual: the later Vedas
The Yajurveda and Samaveda
The Atharvaveda
The Brahmanas and Aranyakas
Vedic religion
Cosmogony and cosmology
Ethical and social doctrines
The sacred: nature, humanity, and God
Vedic and Brahmanic rites
The Upanishads
Sutras, shastras, and smritis
The Vedangas
Dharma-sutras and Dharma-shastras
Smriti texts
Epics and Puranas
The Ramayana
The Mahabharata
The Bhagavadgita
The Puranas
Cosmogony
Myths of time and eternity
Stories of the gods
Vaishnavism and Shaivism
Vaishnavism
Shaivism
Narratives of culture heroes
Myths of holy rivers and holy places
Philosophical texts
Philosophical sutras and the rise of the Six Schools of philosophy
Tantrism
Tantric traditions and Shaktism
Shaiva Agamas
Vaishnava Samhitas
Shakta Tantras
Nature of Tantric tradition
Tantric and Shakta views of nature, humanity, and the sacred
Tantric ritual and magical practices
Tantric and Shakta ethical and social doctrines
Vernacular literatures
Practical Hinduism
Divination, spirit possession, and healing
Women's religious practices
Rituals, social practices, and institutions
Sacrifice and worship
Domestic rites
Samskaras: rites of passage
Daily offerings
Other private rites
Temple worship
Shaiva rites
Vaishnava rites
Sacred times and festivals
Ritual and social status
Social protest
Renunciants and the rejection of social order
Religious orders and holy men
Sectarian symbols
Cultural expressions: visual arts, theatre, and dance
Types of symbols
Yantra and mandala
Lingam and yoni
Visual theology in icons
Religious principles in sculpture and painting
Religious organization of sacred architecture
Hinduism and the world beyond
Hinduism and religions of Indian origin
Hinduism and Islam
Hinduism and Christianity
Diasporic Hinduism
Most of the texts cited in this survey are Sanskrit texts, which constitute the oldest layer of extant Hindu literature. But the sacred literature of India is not as monolithic as these texts might suggest. Several other essential elements exist: independent sacred literatures in languages other than Sanskrit and material in other languages related to the Sanskrit texts either as sources of material now preserved only in Sanskrit or as new texts originating as translations of Sanskrit texts. Because Sanskrit has been in intimate contact with the mother tongues of India for such a long time, it is often impossible to determine in which of these categories a particular vernacular text belongs.
Indologists usually emphasize the influence of Sanskritic culture on vernacular culture, and indeed this influence was considerable. Sanskritic influence was already in evidence in the earliest Tamil (a principal Dravidian language) literature, perhaps dating from before the Common Era. At this time in South India the orthodox cults were aristocratic in character and were supported by kings and chiefs who gained in prestige by patronizing Brahmans. The Tamils were still primarily devoted to their local traditions, some of which, however, were becoming Sanskritized. The pastoral god Murugan was identified with Skanda, and his mother, the fierce war goddess Korravai, with Durga. Varunan, a sea god who had adopted the name of an old Vedic god but otherwise had few Vedic features, and Mayon, a black god who was a rural divinity with many of the characteristics of Krishna in his pastoral aspect, also are depicted in Tamil literature. The final Sanskritization of the Tamils was brought about through the patronage of the Pallava kings of Kanchipuram, who began to rule in the 4th century ce and who financed the making of many temples and fine religious sculptures. Similar processes took place in the Deccan, Bengal, and other regions.
Sanskritization is a term that refers to a style of text that imitates the customs and manners of the Brahmans. But, although most sacred texts in Sanskrit were composed by Brahmans, many were composed by lower-class authors. Likewise, although some sacred texts in vernacular languages were written by authors of lower castes, many others were written by Brahmans. In addition, because Sanskrit ceased to be spoken as a primary language soon after the Vedas were composed, it is likely that most of the thoughts underlying all subsequent Sanskrit literature emerged first in some other language. The issue is further clouded by the fact that, though Sanskrit texts tend to be written and vernacular traditions are primarily oral, there are important oral traditions in Sanskrit too (including the traditions of the two great Sanskrit epics), and there are important manuscript traditions in some of the non-Sanskritic languages (such as Bengali and Tamil). Indeed, written and oral versions of the epics and Puranas have been, from the very start, in constant symbiosis.
Little relevance, therefore, attaches to the distinction between written and oral traditions. A story is narrated, a process that is designated in Sanskrit by such words as purana (ancient story) and akhyana (illustrative narrative). In the oldest source, the Rigveda, myths are not so much told as alluded to; it is in the later Vedic literature of the Brahmanas that narratives are found, and these are often prejudiced by the liturgical concerns of the authors.
The recitation of certain myths was prescribed for various rituals. The epic Mahabharata states that Vedic stories were narrated "in the pauses of the ritual," probably by Brahmans. The sutas (charioteers and panegyrists), who celebrated the feats of great rulers, were the mythographers of the Kshatriyas (the warrior class). The sutas were popular narrators of myth and legend and developed their own bardic repertoire, which was extended to higher mythology. They—and other wanderers who found ready audiences at sacrifices or places of pilgrimage—disseminated the lore.
Narrators continue to repeat and embroider the ancient stories of gods, sages, and kings. At an early stage their narratives were dramatized and gave rise to the Sanskrit theatre, in which epic mythic themes preponderate, and to the closely related dance, which survives in the now largely South Indian schools of bharata natyam (traditional dance) and the kathakali (narrative dance) of Kerala. Thus, even in Sanskrit literature, oral performance was an essential component, which further facilitated the assimilation of oral vernacular elements.
Of the four primary Dravidian literatures—Tamil, Telugu, Kannada, and Malayalam—the oldest and best-known is Tamil. The earliest preserved Tamil literature, the so-called Sangam poetry anthologies, dates from the 1st century bce. These poems are classified by theme into akam ("interior," primarily love poetry) and puram ("exterior," primarily about war, the poverty of poets, and the deaths of kings).
Tamil devotional poetry was remarkable for a number of reasons. It was composed by both men and women, by people of different castes (including "outcastes"), and in a language that was classical (Tamil already had a long and rich literary heritage) but also vernacular, or spoken. Despite all these factors, the poems were frequently hailed as divine revelation and sometimes as the equal of the Sanskrit Vedas.
The bhakti movement has been traced to Tamil poetry, beginning with the poems of the devotees of Shiva called Nayanars and the devotees of Vishnu called Alvars. The Nayanars, who date from about 800 ce, composed intensely personal and devout hymns addressed to the local manifestations of Shiva. The most famous Nayanar lyricists are Appar, Sambandar, and Chuntarar, whose hymns are collected in the Tevaram (c. 11th century). More or less contemporary were their Vaishnava counterparts, the Alvars Poykai, Putan, Peyar, and Tirumankaiyalvar; and in the 8th century the poetess Andal, as well as Periyalvar, Kulachekarar, Tiruppanalvar, and notably Nammalvar, who is held to be the greatest, composed their works. Shrivaishnavas consider Nammalvar's poems, especially his Tiruvaymoli ("Sacred Utterance"), to be the Tamil Veda.
The devotion of which they sang exemplified the new bhakti movement, which sought a more direct contact between humans and God, carried by a passionate love for the Deity, who would reciprocate by extending his grace to humankind. These saints became the inspiration of theistic systematic religion: the Shaivas for the Shaiva-siddhanta, the Vaishnavas for Vishistadvaita. In Kannada the same movement was exemplified by Basava, whose vachanas ("sayings" or "talks") achieved great popularity.
New literary genres in Dravidian languages continued to evolve into the 17th and 18th centuries, when the Tamil Chittars (name derived from Sanskrit siddha, "perfected one"), who were eclectic mystics, composed poems noted for the power of their naturalistic diction. The Tamil sense and style of these poems belied the Sanskrit-derived title of their authors, a phenomenon that could stand as a symbol of the complex relationship between the vernacular and Sanskrit religious texts.
The main languages derived from Sanskrit are Bengali, Hindi (with its many dialects, of which Maithili is the oldest and Urdu, heavily influenced by Persian and Arabic and written in a Perso-Arabic script, is the most important), Punjabi, Gujarati, Marathi, Oriya, Kashmiri, Sindhi, Assamese, Nepali, Rajasthani, and Sinhalese. Most of these languages began to develop literary traditions about 1000 ce.
Although the earliest texts in Hindi are sometimes attributed to the 13th–14th-century Muslim poet Amir Khosrow, it was not until the 15th century that Hindi literature produced its own great religious lyricists. The earliest of these lyricists were the disciples of Ramananda (c. 140), who was a follower of the philosopher Ramanuja. The most famous of these lyricists is Kabir, a poet and mystic who was the forerunner of Sikhism. Tulsidas, apart from his Ramcharitmanas, composed Ramaite lyrics. Surdas (1483–1563), a follower of the Vallabha school of Vedanta, is known for his Sursagar ("Ocean of the Poems of Sursagar"), a collection of poems based on the stories of the childhood of Krishna found in the Bhagavata-purana. Perhaps the best-known bhakti poems are those of Mira Bai (1503–73), a Rajput princess who composed mostly in a local dialect of Hindi. She wrote passionate love poems to Krishna, whom she regarded as her husband and lover. Her bhajans (devotional songs) are sung by Hindus both privately and in public performances in India and throughout the diaspora. In the Marathi tradition, Namdev (1270?–1350?) celebrated Vishnu, particularly in his manifestation as Vitthoba at the Pandharpur temple; and in the 17th century Tukaram, the greatest poet of this literature, sang of the god of love in numerous hymns.
The importance of these writers is not limited to literature. A small sect, the Kabirpanthis, acknowledges Kabir as its founder, but its importance is less than that of the vigorous new religion (Sikhism) founded by one of Kabir's disciples, Nanak.
Although the earliest Hindu text in Bengali is a mid-15th-century poem about Radha and Krishna, texts in praise of gods and goddesses, known as mangal-kavyas, surely existed in oral versions long before then. In later Bengal Vaishnavism, the emphasis shifts from service and surrender to mutual attachment and attraction between God (i.e., Krishna) and humankind: God is said to yearn for the worshiper's identification with himself, which is his gift to the wholly purified devotee. The mystical and devotional possibilities of the Krishna legend are subordinated to religious practice; the divine sport and wonderful feats of this youthful hero are interpreted symbolically and allegorically. Thus, the highest fruition of bhakti is admission to the eternal sport of Krishna and his beloved Radha, whose sacred love story is explained as the mutual love between God and the human soul. Various gradations of bhakti are distinguished, such as awe, subservience, and parental affection. These are correlated with the persons of the Krishna legend; the highest and most intimate emotion is said to be the love of Radha and her girlfriends for Krishna.
Krishna and Radha, detail of a Kishangarh painting, mid-18th century; in a private collection.
P. Chandra
A particularly rich Bengali tradition concentrated on the love of Radha, who symbolizes the human soul, for Krishna, the supreme god. In this tradition are Chandidas, a 15th-century poet known for his love songs, and the Maithili poet Vidyapati (c. 1400). The single most influential figure, however, was Chaitanya, who in the 16th century renewed Krishnaism. He left no writings but inspired many hagiographies, among the most important of which is the Chaitanya-charitamrita ("Nectar of Chaitanya's Life") by Krishna Das (born 1517).
Chaitanya had a profound and lasting effect on the religious sentiments of the people of Bengal. He propagated the community celebration (sankirtana) of Krishna as the most powerful means of bringing about the proper bhakti attitude. Chaitanya also introduced the worship of God, the director of the senses, through the very activity of the senses, which must be free from all egoism and completely filled with the intense desire (preman) for the satisfaction of the beloved (i.e., Krishna).
Another form of religious lyric are the so-called padas (verses). Govinda Das (1537–1612) is one of the greatest poets in this bhakti genre of poetry in which divine love is symbolized by human love. The songs of Ramprasad Sen (1718–75) similarly honour Shakti as mother of the universe and are still in wide devotional use.
The complex interaction between Sanskrit and non-Sanskrit religious classics may be seen in the development of the epics. The two great Sanskrit epics, the Mahabharata and the Ramayana, and many Puranas (especially the Bhagavata-purana) were rendered in various vernaculars. These works were not literal translations but free versions in which the authors inserted their own emphases, which differed both from the original and from those of other authors. The oldest vernacular version of the Ramayana is the Tamil translation, the Iramavataram by Kampan (c. 12th century), a work of high literary distinction that is suffused with devotion (bhakti). A Telugu rendering was made by Ranganatha about 1300. Several translations in Bengali include some interesting and probably authentic variations from the "official" Rama story by Valmiki, the best-known translation being that of Krittibas Ojha (1450). Equally, if not more, famous is Ramcharitmanas ("Holy Lake of the Acts of Rama").
The Mahabharata was rendered in Bengali about 1600 and into Telugu by Nannaya and Tikkana in the 13th century. The Bhagavata-purana, which was translated frequently (e.g., into Bengali by Maladhar Vasu, 1480), was popular because it gave the canonical account of Krishna's life and especially his boyhood, which is the perennial inspiration of the bhakti poets.
The teacher Jnanadeva (also known as Jnaneshvara; 1275–96) composed a commentary on the Bhagavadgita in Marathi that remains a classic in that literature. His work was continued by Eknath (c. 1600), who also composed bhakti poetry. In the 16th century the Kannada poet Gadugu produced a highly individual version of the Mahabharata. In addition to the literal or not-so-literal translations of the Sanskrit epics, the Tamils composed their own epics, notably Ilanko Atikal's Chilappatikaram ("The Lay of the Anklet") and its sequel, the Manimekhalai ("Jeweled Girdle"). In Telugu there is the great Palnadu epic; Rajasthani has an entire epic cycle about the hero Pabuji. The remaining vernaculars have also produced many epics of their own.
Much of the classical mythology persists today, and its stories have been conveyed to Hindus through traditional means as well as via the mass media. Mythic illustrations remain favourites in Indian calendar art. Television series and motion pictures called "mythological" are extremely popular, perpetuating the ancient stories, and so are "devotionals," in which an example of bhakti is illustrated. The television series Ramayana, for example, was one of the biggest successes in the Indian media. Radio regularly carries bhajans (devotional songs) and classical South Indian songs, the themes of which are often mythic. Narratives from Hindu texts have also been portrayed in popular literature. Many stories became the bases for popular comic books in the 1960s and '70s. An English-language series called Amar Chitra Katha ("Immortal Stories in Picture Form"), for example, was read by millions of people.
Every orthodox Hindu's home has at least one corner set aside as a domestic sanctuary where representations of a chosen deity are placed, and puja (worship) is done with prayers, hymns, flowers, and incense. Richer establishments set aside entire rooms as shrines. New temples have been constructed with modern techniques; one temple in Varanasi (Banaras) contains mirrors onto which are etched the entire Ramcharitmanas. This same poem is the basis of the annual celebration of Ram Lila (the play of Rama) in northern India, in which the entire community participates. The story of Rama was evoked by Mahatma Gandhi when he set the Ram Raj ("Kingdom of Rama") as India's governmental ideal.
On occasion, social protesters have armed themselves with myth to make a point. For example, Karna, an antagonist in the Mahabharata who is berated for his low birth, has been extolled in intellectual circles as a truer champion than the aristocratic heroes. Anti-northern groups in Tamil Nadu revised the story of Rama, whose expedition against the demon Ravana was believed by some to be the "Aryan" invasion of South India, by reversing it to abuse Rama and to glorify Ravana.
On a popular level, people at temples and fairs are continually reacquainted with their mythological heritage by pauranikas, tellers of the ancient stories and heirs of the sutas of 3,000 years ago, and no festival ground is complete without tents where the religious are reminded of their myths by pious speakers, modestly compensated by fees but richly rewarded by the honour in which they are held.
J.A.B. van Buitenen Edward C. Dimock Arthur Llewellyn Basham Wendy Doniger Brian K. Smith Vasudha Narayanan


text:
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BEIN CHANNEL FREQUENCIES All Satellte Update 2023
Satellite TP admin January 5, 2023 0 Comment
BEIN CHANNEL FREQUENCIES: Here is the Bein Channels frequencies and details for the satellites like Es'hail-1 @ 25.5°E, Eutelsat @ 8°W, Eutelsat @ 7.3° W, Es'hail-2 @ 26.0°E etc.beIN Media Group (/ˈbiːɪn/; Arabic: مجموعة بي إن الإعلامية, Majmū'at Bī'in al-I'lāmiyyah) is a state-owned global sport and entertainment network headquartered in Doha, Qatar. beIN distributes entertainment, live sports action, and major international events across 5 continents, in 43 countries, and 7 different languages spanning Europe, North America, Asia, Australia and the Middle East and North Africa (MENA).
beIN Sports was first launched, as a brand, in June 2012 by Al Jazeera Media Network in France. On 1 January 2014, beIN Media Group was established and became the new independent holding company of beIN Sports. In October 2014, it was announced that beIN Media Group had agreed to acquire pay-TV sports channel Setanta Sports Australia, with Setanta being rebranded as beIN Sports Australia.
In 2015, beIN Sports launched an HD channel specially dedicated to football in Spain. The group declared in November 2015 that it would be expanding from sports-only programming to include entertainment and movies as well. It was announced in January 2016 that beIN Media Group secured an agreement with Turner Broadcasting System, allowing it exclusive rights to broadcast a number of Turner-licensed entertainment and news channels across the Middle East and North Africa. beIN Media Group has since signed strategic partnerships with BBC Studios Warner Bros. CBS DreamWorks Animation and Discovery.
It was rumored in November 2015 that beIN Media Group were interested in purchasing Miramax. On 1 March 2016, beIN Media Group announced its full acquisition of Miramax from Filmyard Holdings. beIN announced they would sell at least 49% of the company in June 2019, with Viacom and Lionsgate as the front-liners.
It was announced in August 2016 that beIN Media Group acquired Turkish pay TV service Digiturk. In November 2019, beIN Media acquired exclusive broadcast rights for the 2019 and 2020 FIFA Club World Cup.
Following the Covid-19 crisis, BeIN is implementing a layoff plan for its activities in the Middle East and North Africa.
BEIN CHANNEL FREQUENCIES
Es'hail-1 @ 25.5°E on BEIN New Frequency 2023
Frequency MHz
Symbol Rate
8PSK
Eutelsat @ 8°W – BEIN
Eutelsat @ 7.3° W – BEIN
Es'hail-2 @ 26.0°E – BEIN
bein Biss key today frequency 2023 bein New Frequency 2023 Bein Today Frequency update 2023 Frequency of Bein Sports 1hd 2023
admin January 26, 2023 0 Comment
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