text:
\section{Introduction}
The quantum Zeno effect is the basic principle that repeated or continual external interventions on a quantum dynamical system may lead to a
constraining of the system to a sub-dynamics. The original formulation by
Misra and Sudarshan \cite{MS77} dealt with a system initially in an unstable system $|\psi _{0}\rangle $ which is repeatedly measured to see if it is in this
state (a von Neumann projective measurement with orthogonal projection $P_{\mathtt{z}}=|\psi _{0}\rangle \langle \psi _{0}|$). At each stage the survival
probability $p(t)=|\langle \psi _{0}|e^{-itH}\psi _{0}\rangle |^{2}$ is $1+O(t^{2})$ for small $t$, so for $N$ measurements at time separations $t/N$
the probability of measuring the same state on all $N$ attempts is $p(t/N)^{N}$
which converges to $1$ as $N\uparrow \infty $. The state is in principle ``frozen'' as $|\psi _{0}\rangle $, up to a phase.

More generally the projection $P_{\mathtt{z}}$ may be of rank greater than one, in which case we find that the state is constrained to remain the Zeno subspace
\begin{equation*}
\mathfrak{h}_{\mathrm{Zeno}}=P_{\mathtt{z}} \, \mathfrak{h}
\end{equation*}
where $\mathfrak{h}$ is the total Hilbert space of states. Typically, we still have a reduced evolution on the Zeno subspace governed by the Zeno
Hamiltonian $H_{\mathrm{Zeno}}=P_{\mathtt{z}}HP_{\mathtt{z}}$. (The unitary $e^{-itP_{\mathtt{z}}HP_{\mathtt{z}}}$ on $\mathfrak{h}_{\mathrm{Zeno}}$ sometimes being referred to as a ``non-abelian phase'').
One can obtain results of the form
\begin{equation*}
\lim_{N\rightarrow \infty }\left[ P_{\mathtt{z}}e^{-iHt/N}\right] ^{N}=P_{\mathtt{z}}e^{-itP_{\mathtt{z}}HP_{\mathtt{z}}}
\end{equation*}

The intervention does not necessarily have to take the form of a
measurement, see the authoritative review by Facchi and Pascazio \cite{FP}. An alternative is to apply stroboscopic unitary kicks and
take the limit as the time between kicks vanishes. Another possibility is to
apply a strong perturbation $kV$ to obtain a result of the form
\begin{equation*}
\lim_{k\rightarrow \infty }e^{+ikVt}e^{-i\left( H+kV\right) t}P_{\mathtt{z}}=P_{\mathtt{z}}e^{-itP_{\mathtt{z}}HP_{\mathtt{z}}}
\end{equation*}
where $P_{\mathtt{z}}$ is projection onto the kernel of $V$. This is a case of the
adiabatic theorem.

Despite the fact that quantum Zeno effect is typically described in terms of
open systems concepts such as quantum measurement, external interaction or
environmental coupling, it is rarely if ever formulated mathematically in
these terms. In practice, on never makes direct measurements on quantum
mechanical systems, but instead couples them to external quantum fields
which are then monitored continually. A complete model would include the
probing field \cite{Bel80}-\cite{Bel92a}. Similarly, the coupling of the environment requires an open
systems model.

\begin{figure}[htbp]
	\centering
		\includegraphics[width=1.00\textwidth]{Zeno0.eps}
	\caption{In a Zeno limit we find that a quantum state $\vert \Psi \rangle$ will have component perpendicular to a Zeno subspace $\mathfrak{h}_{\mathrm{Zeno}}$ which so strongly coupled to the environment that it effectively disappears in real time, leaving the system constrained to the Zeno subspace. The component may be adiabatically eliminated.}
	\label{fig:Zeno}
\end{figure}

In this paper we start in the general framework of an open quantum Markov
model for the system undergoing a quantum stochastic evolution in
conjunction with a Boson field environment \cite{HP,partha,Gardiner}. We shall equate the limit
procedure leading to a Zeno dynamics as adiabatic elimination. This choice
is a matter of convenience as we at once have access to powerful results due
to Luc Bouten and Andrew Silberfarb \cite{Bouten_Silberfarb}, and later with Ramon van Handel
\cite{Bouten_Handel_Silberfarb}, on the
adiabatic elimination procedure in the framework of quantum stochastic
models. This is a significantly richer theory than for closed systems. In
addition we get clear conditions on when a candidate subspace will be a Zeno
subspace for a sequence of models. We can the readily develop the theory of
how we actually probe the Zeno sub-dynamics by for instance quantum
trajectories methods for the constrained dynamics \cite{Belavkin} - \cite{BHJ07}.

\subsection{Open Quantum Systems: The ``$S,L,H$'' Formalism}

In figure \ref{input-plant-output component} below, we sketch an open quantum Markov
 system as a black box with a Bosonic input driving field
and a Bosonic output. The quantum
mechanical system (plant) will have underlying Hilbert space $\mathfrak{h}$
while the input will be a continuous quantum field with Fock space $\mathfrak{F}$. The coupled model will have joint Hilbert space $\mathfrak{h}\otimes \mathfrak{F}$,
which is also the space on which the output observables act.

We shall outline below the theory of quantum stochastic evolutions due to Hudson-Parthasarathy.
This shows that the unitary dynamics of the combined system and its Bosonic environment are specified by a triple of coefficients
\begin{equation}
\mathbf{G} \sim (S,L,H) .
\end{equation}

We shall in turns refer to $\mathbf{G}$ as the Hudson-Parthasarathy (HP) parameters, the generating coefficients, or just the $SLH$ for the model.

\begin{figure}[h]
\centering
\begin{picture}(120,55)
\setlength{\unitlength}{.04cm
\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}
\put(36,35){$\mathbf{G} \sim (S,L,H)$}
\put(48,2){system}
\put(10,26){output}
\end{picture}
\caption{input-system-output component}
\label{input-plant-output component}
\end{figure}

The input-plant-output model can be summarized as
\begin{eqnarray*}
\mathbf{plant~dynamics} &:&j_{t}\left( X\right) =U\left( t\right) ^{\ast
}\left( X\otimes I\right) U\left( t\right) ; \\
\mathbf{output~process} &:&B_{\mathrm{out},i}\left( t\right) =U\left(
t\right) ^{\ast }\left( I\otimes B_{i}\left( t\right) \right) U\left(
t\right) .
\end{eqnarray*}
where $X$ is an arbitrary plant observable and $B_{i}\left( t\right) $ is
the $i$th input field's annihilator process.

In the following we shall specify to the category of model where $U\left(
\cdot \right) $ is a unitary family of operators on $\mathfrak{h}\otimes
\mathfrak{F} $, satisfying a differential equation of the form \cite{HP,partha}
\begin{eqnarray}
dU\left( t\right) = \left\{ \sum_{ij}\left( S_{ij}-\delta _{ij}\right)
\otimes d\Lambda _{ij}\left( t\right) +\sum_{i}L_{i}\otimes dB_{i}^{\ast
}\left( t\right) \right.  \notag \\
\left. -\sum_{ij}L_{i}^{\ast }S_{ij}\otimes dB_{j}\left( t\right) +K\otimes
dt\right\} U\left( t\right) ,\quad U(0)=I,  \label{HP_QSDE}
\end{eqnarray}
Formally, we can introduce input process $b_{i}\left( t\right) $ for $i=1,\cdots ,n$ satisfying singular commutation relations
$[ b_{ i}(t),b_{j}(t^{\prime })^{\ast }]=\delta _{ij}\delta (t-t^{\prime })$, so that the
processes appearing in are
\begin{eqnarray*}
\Lambda _{ij}\left( t\right) &\triangleq &\int_{0}^{t}b_{i}\left( t^{\prime
}\right) ^{\ast }b_{ j}(t^{\prime })dt^{\prime }, \\
B_{i}\left( t\right) ^{\ast } &\triangleq &\int_{0}^{t}b_{i}\left( t^{\prime
}\right) ^{\ast }dt^{\prime },\quad B_{j}\left( t\right) \triangleq
\int_{0}^{t}b_{ j}(t^{\prime })dt^{\prime }.
\end{eqnarray*}
More exactly, the are rigorously defined as creation and annihilation field
operators on the Boson Fock space $\mathfrak{F}$ over $L_{\mathbb{C
^{n}}^{2}\left( \mathbb{R}\right) $. The increments in \ are understood to
be future pointing in the Ito sense. We have the following table of
non-vanishing products
\begin{eqnarray*}
d\Lambda _{ij}d\Lambda _{kl} &=&\delta _{jk}d\Lambda _{il},\qquad d\Lambda
_{ij}dB_{k}^{\ast }=\delta _{jk}dB_{i}^{\ast } \\
dB_{i}d\Lambda _{kl} &=&\delta _{ik}dB_{l},\qquad dB_{i}dB_{k}^{\ast
}=\delta _{ij}dt.
\end{eqnarray*}

Necessary and sufficient conditions for unitarity \cite{HP,partha} are that we can collect the coefficients of (\ref{HP_QSDE}) to form a triple $(S,L,H)$,
which we call the \emph{Hudson-Parthasarathy (HP) or $SLH$ parameters}, consisting of a unitary matrix $S$, a column vector $L$, and a self-adjoint operator $H$,
\begin{equation*}
S=\left[
\begin{array}{ccc}
S_{11} & \cdots & S_{1n} \\
\vdots & \ddots & \vdots \\
S_{n1} & \cdots & S_{nn}
\end{array}
\right] , \quad L=\left[
\begin{array}{c}
L_{1} \\
\vdots \\
L_{n}
\end{array}
\right] ,
\end{equation*}
with $S_{ij},L_{i},H$ are all operators on $\mathfrak{h}$, and where
\begin{equation*}
K\equiv -\frac{1}{2}\sum_{i}L_{i}^{\ast }L_{i}-iH.
\end{equation*}

We shall refer to $U(t)$ as the unitary determined by the parameters $\left(
S,L,H\right) $. In differential form, the input-plant-output model then
becomes \cite{HP,partha}

\bigskip

\noindent \textbf{plant dynamical (Heisenberg) equation:}
\begin{eqnarray*}
dj_{t}\left( X\right) &=&j_{t}\left( \mathscr{L}X\right)
dt+\sum_{i}j_{t}\left( \mathscr{M}_{i}X\right) dB_{i}^{\ast }\left( t\right)
\\
&&+\sum_{i}j_{t}\left( \mathscr{N}_{i}X\right) dB_{i}\left( t\right)
+\sum_{j,k}j_{t}\left( \mathscr{S}_{jk}X\right) d\Lambda _{jk}\left(
t\right) ;
\end{eqnarray*}

\bigskip

\noindent \textbf{input-output relations:}

\begin{equation*}
dB_{\mathrm{out},i}\left( t\right) =j_{t}\left( S_{ik}\right) dB_{k}\left(
t\right) +j_{t}\left( L_{i}\right) dt.
\end{equation*}

Here
\[
\mathscr{L}X=\frac{1}{2}\sum_{i}L_{i}^{\ast }\left[ X,L_{i}\right] +\frac{1}{2}\sum_{i}\left[ L_{i}^{\ast },X\right] L_{i}-i\left[ X,H\right]
\]
is the Lindbladian, $\mathscr{M}_i X = S_{ji}^\ast [X,L_j ], \mathscr{N}_i X= [L_k^\ast , X ] S_{ki}$ and $\mathscr{S}_{ik} X = S_{ji}^\ast XS_{jk} $
$-\delta_{ik} X$.

\section{Zenofiability}

The essential idea is that there exists a subspace $\mathfrak{h}_{\mathrm{Zeno}}$ of $\mathfrak{h}$ to which the system may end up restricted (though
entangled with the environment) due to the continuous open system dynamics.
We decompose $\mathfrak{h}$ as
\begin{equation}
\mathfrak{h}=\mathfrak{h}_{\mathrm{Zeno}}\oplus \mathfrak{h}_{\mathrm{Fast}}
\label{fs_decomp}
\end{equation}
where the subspace $\mathfrak{h}_{\mathrm{Fast}}=\mathfrak{h}\ominus \mathfrak{h}_{\mathrm{Zeno}}$ contains the vector states that are most
strongly coupled to the environment so that any drift out of $\mathfrak{h}_{\mathrm{Zeno}}$ will decay to zero almost instantaneously.

To make these concepts mathematically rigorous, we introduce a strength parameter $k>0$ for the coupling and make the generating operators $k
$-dependent according to $\mathbf{G} (k)$ with $SLH$ operators of the form
\begin{eqnarray}
S(k) &=&S\text{, \ \ (independent of }k) ;  \notag \\
L\left( k\right)  &=&kL^{\left( 1\right) }+L^{\left( 0\right) };  \notag \\
H\left( k\right)  &=&k^{2}H^{\left( 2\right) }+kH^{\left( 1\right)
}+H^{\left( 0\right) },  \label{eq:scaling1}
\end{eqnarray}
leading to the open unitary dynamics $U(t,k)$.

The singular limit $k\uparrow \infty $ should then separate $\mathfrak{h}_{\mathrm{Fast}}$ out as the fast subspace which may be eliminated to yield a
dynamics on the open system over $\mathfrak{h}_{\mathrm{Zeno}}$. At this stage, the Zeno limit can be reformulated explicitly as an adiabatic
elimination problem with the Zeno subspace $\mathfrak{h}_{\mathrm{Zeno}}$ being the slow space. At this stage we can invoke the results of Bouten, van
Handel and Silberfarb \cite{GvH},\cite{Bouten_Silberfarb},\cite{Bouten_Handel_Silberfarb} on adiabatic elimination for quantum stochastic systems.

For a given operator $X$ on $\mathfrak{h}$, we write
\begin{equation*}
X=\left[
\begin{array}{cc}
X_{\mathtt{zz}} & X_{\mathtt{zf}} \\
X_{\mathtt{fz}} & X_{\mathtt{ff}}
\end{array}
\right] .
\end{equation*}
More generally we use this notation when $X$ is an array of operators on $\mathfrak{h}$. The projections onto $\mathfrak{h}_{\mathrm{Zeno}}$ and
$\mathfrak{h}_{\mathrm{Fast}}$ are denoted respectively by
\[P_{\mathtt{z}}\equiv \left[
\begin{array}{cc}
1 & 0 \\
0 & 0
\end{array}
\right] , \quad
P_{\mathtt{f}}\equiv \left[
\begin{array}{cc}
0 & 0 \\
0 & 1
\end{array}
\right] .
\]

\subsection{Conditions}

\bigskip

\begin{condition}[Scaling]
We assume that the $SLH$ operators in (\ref{eq:scaling1})\ satisfy
\begin{equation}
L^{\left( 1\right) }P_{\mathtt{z}}=0,\quad H^{\left( 1\right) }P_{\mathtt{z}}=0,\quad P_{\mathtt{z}}H^{\left( 1\right) }=0,\quad P_{\mathtt{z}}H^{\left(
2\right) }P_{\mathtt{z}}=0.
\end{equation}
\end{condition}

Equivalently stated we require the following forms with respect to the decomposition:
\begin{eqnarray*}
L^{\left( 1\right) } &\equiv &\left[
\begin{array}{ll}
0 & L_{\mathtt{zf}}^{\left( 1\right) } \\
0 & L_{\mathtt{ff}}^{\left( 1\right) }
\end{array}
\right] , \\
H &\equiv &\left[
\begin{array}{ll}
H_{\mathtt{zz}}^{\left( 0\right) } & H_{\mathtt{zf}}^{\left( 0\right) }+kH_{\mathtt{zf}}^{\left( 1\right) } \\
H_{\mathtt{fz}}^{\left( 0\right) }+kH_{\mathtt{fz}}^{\left( 1\right) } & H_{\mathtt{ff}}^{\left( 0\right) }+kH_{\mathtt{ff}}^{\left( 1\right) }+k^{2}H_
\mathtt{ff}}^{\left( 2\right) }
\end{array}
\right] .
\end{eqnarray*}
In what follows, it is instructive to write
\begin{eqnarray*}
S &=&\left[
\begin{array}{cc}
S_{\mathtt{zz}} & S_{\mathtt{zf}} \\
S_{\mathtt{fz}} & S_{\mathtt{ff}}
\end{array}
\right] , \\
L(k) &=&k\left[
\begin{array}{cc}
0 & L_{\mathtt{zf}}^{(1)} \\
0 & L_{\mathtt{ff}}^{(1)}
\end{array}
\right] +\left[
\begin{array}{cc}
L^{\left( 0\right) }_{\mathtt{zz}} & \ast  \\
L^{(0)}_{\mathtt{fz}} & \ast
\end{array}
\right] , \\
H(k) &=&k^{2}\left[
\begin{array}{cc}
0 & 0 \\
0 & H_{\mathtt{ff}}^{(2)}
\end{array}
\right] +k\left[
\begin{array}{cc}
0 & H_{\mathtt{zf}}^{(1)} \\
H^{\left( 1\right) }_{\mathtt{fz}} & \ast
\end{array}
\right] +\left[
\begin{array}{cc}
H^{(0)}_{\mathtt{zz}} & \ast  \\
\ast  & \ast
\end{array}
\right]
\end{eqnarray*}
where the terms denote as $\ast $ will not contribute to the $k\uparrow
\infty $ limit. We also note that
\begin{equation*}
K\left( k\right) =-\frac{1}{2}L(k)^{\ast }L(k)-iH(k)\equiv k^{2}A+kM+R,
\end{equation*}
where we will have
\begin{equation*}
A=\left[
\begin{array}{ll}
0 & 0 \\
0 & A_{\mathtt{ff}}
\end{array}
\right] ,M=\left[
\begin{array}{ll}
0 & M_{\mathtt{zf}} \\
M_{\mathtt{fz}} & \ast
\end{array}
\right] ,R=\left[
\begin{array}{ll}
R_{\mathtt{zz}} & \ast  \\
\ast  & \ast
\end{array}
\right]
\end{equation*}
with
\begin{eqnarray*}
A_{\mathtt{ff}} &=&-\frac{1}{2}L_{\mathtt{af}}^{\left( 1\right) \ast }L_{\mathtt{af}}^{\left( 1\right) }-iH_{\mathtt{ff}}^{\left( 2\right) } \\
M_{\mathtt{zf}} &=&-\frac{1}{2}L_{\mathtt{cz}}^{\left( 0\right) \ast }L_{\mathtt{cf}}^{\left( 1\right) }-iH_{\mathtt{zf}}^{\left( 1\right) } \\
M_{\mathtt{fz}} &=&-\frac{1}{2}L_{\mathtt{cf}}^{\left( 1\right) \ast }L_{\mathtt{cz}}^{\left( 0\right) }-iH_{\mathtt{fz}}^{\left( 1\right) } \\
R_{\mathtt{zz}} &=&-\frac{1}{2}L_{\mathtt{cf}}^{\left( 0\right) \ast }L_{\mathtt{cz}}^{\left( 0\right) }-iH_{\mathtt{zz}}^{\left( 0\right) }
\end{eqnarray*}
and we introduce a summation convention where repeated indices imply a summation over the values $\mathtt{z}$ and $\mathtt{f}$.

\begin{condition}[Kernel]
We require  $\mathfrak{h}_{\mathrm{\mathtt{z}}}$ to be the kernel space of $A$.
\end{condition}

Equivalently, we require that the operator $A_{\mathtt{ff}}$ is invertible
on $\mathfrak{h}_{\mathrm{\mathtt{z}}}$. Making this assumption we then
introduce the operators
\begin{eqnarray}
\hat{S}_{\mathtt{ab}} &\triangleq &\left( \delta _{\mathtt{ac}}+L_{\mathtt{af}}^{\left( 1\right) }\frac{1}{A_{\mathtt{ff}}}L_{\mathtt{cf}}^{\left(
1\right) \ast }\right) S_{\mathtt{cb}}, \\
\hat{L}_{\mathtt{a}} &\triangleq &L_{\mathtt{az}}^{\left( 0\right) }-L_{\mathtt{af}}^{\left( 1\right) }\frac{1}{A_{\mathtt{ff}}}M_{\mathtt{fz}}, \\
\hat{H} &\triangleq &H_{\mathtt{zz}}^{(0)}+\mathrm{Im}\left\{
M_{\mathtt{zf}}\frac{1}{A_{\mathtt{ff}}}M_{\mathtt{fz}}\right\} .
\end{eqnarray}

With these definitions we can state the final condition.

\begin{condition}[Decoupling]
The operators $\hat{S}_{\mathtt{zf}},\hat{S}_{\mathtt{fz}}$ and $\hat{L}_{\mathtt{f}}$\ vanish.
\end{condition}

\subsection{Asymptotic Open Zeno Dynamics}

We now recall the adiabatic elimination result of Bouten and Silberfarb
which establishes convergence to a limit unitary $\hat{U}(t)$ with reduced
system space $\mathfrak{h}_{\mathrm{Zeno}}$ and $SLH$ operators
\begin{eqnarray}
\mathbf{G}_{\mathrm{Zeno}} \sim \left(
\hat{S}_{\mathtt{zz}},\hat{L}_{\mathtt{z}},\hat{H}\right) .
\end{eqnarray}

\begin{theorem}[Bouten and Silberfarb 2008 \protect\cite{Bouten_Silberfarb}]

Suppose we are given a sequence of bounded operator parameters $\left( S,L\left( k\right) ,H\left( k\right) \right) $ satisfying the scaling,
kernel and decoupling conditions. Then the quantum stochastic process $U_{k}\left( t\right) P_{\mathtt{z}}$ converges strongly to $\hat{U}\left(
t\right) P_{\mathtt{z}}$, that is
\begin{equation*}
\lim_{k\rightarrow \infty }\left\| U_{k}\left( t\right) \psi -U\left(
t\right) \psi \right\| =0
\end{equation*}
for all $\psi \in \mathfrak{h}\otimes \mathfrak{F}$ with $P_{\mathtt{f}}\otimes I_{\mathfrak{F}}\,\psi =0$. The convergence is moreover uniform in the
time coordinate $t$ for compact time sets.
\end{theorem}

The restriction to bounded operators was lifted in a subsequent publication
\cite{Bouten_Handel_Silberfarb}.

\begin{definition}
Let $\left( S(k),L(k),H(k)\right) $ be a sequence of $SLH$ operators based on a given space $\mathfrak{h}$ having the dependence on a scaling parameter
$k>0$ as in equation (\ref{eq:scaling1}). A subspace $\mathfrak{h}_{\mathrm{Zeno}}$ is said to be \textbf{zenofiable} if, with respect to the orthogonal
decomposition $\mathfrak{h}=\mathfrak{h}_{\mathrm{Zeno}}\oplus \mathfrak{h}_{\mathrm{Fast}}$, the scaling, kernel and decoupling conditions are
satisfied. The subspace $\mathfrak{h}_{\mathrm{Zeno}}$ is called the \textbf{Zeno subspace} for the $k\uparrow \infty $ asymptotic dynamics.
The operators $\left(
\hat{S}_{\mathtt{zz}},\hat{L}_{\mathtt{z}},\hat{H}\right) $ are called the \textbf{Zeno scattering matrix}, the \textbf{Zeno coupling (or collapse) operators} and the
\textbf{Zeno Hamiltonian} operators respectively.
\end{definition}

\section{Controllability and Observability Issues}
The issue of controllability and the Zeno effect has been recently addressed by Burgarth \textit{et al.} \cite{Plato} who have shown that for a system with controlled Hamiltonian of the form
$ \sum_{k=1}^n H_k \, u_k (t)$, with deterministic control policies $u_k$ and self-adjoint operators $H_k$, there are some unexpected benefits of constraints. In particular,
for the case dim $\mathfrak{h} < \infty$, they show that the degree of controllability of the original system may be smaller than the degree of controllability of the Zeno
limit. That is, the dimension of the Lie algebra $\mathfrak{L}_{\mathtt{z}}$ generated by
$\{ - i P_{\mathtt{z}}H_1 P_{\mathtt{z}} , \cdots , -i P_{\mathtt{z}} H_n P_{\mathtt{z}} \}$ may be strictly greater than that of $\mathfrak{L}$ generated by $\{ - iH_1 , \cdots , -iH_n \}$,
despite the fact that the Zeno dynamics is constrained to the smaller subspace.

We should also remark that an experiment to obtain a limiting Zeno dynamical behaviour for a cavity mode through interaction with and repeated measurements of Rydberg atom
has been proposed by the Ecole Normale Sup\'{e}rieure group \cite{Zeno_Cavity}.

\subsection{Zeno Master Equation}

Let us prepare the system in an initial state $\eta \in \mathfrak{h}_{\text{Zeno}}$, then for $t\geq 0$ we have the expectation of an observable $X$ on the Zeno subspace given by
\begin{equation*}
\mathbb{E} _{t}( X) = \langle \eta \otimes \Omega | j_t (X) \, \eta \otimes \Omega \rangle  \equiv tr_{\mathtt{z}}\left\{ \rho _{\mathtt{z}}X\right\} ,
\end{equation*}
which introduces the Zeno density matrix $\rho _{\mathtt{z}}$. (The trace is over the Zeno subspace $\mathfrak{h}_{\mathrm{Zeno}}$.) We obtain the Zeno-Ehrenfest equation
\begin{equation*}
\frac{d}{dt}\mathbb{E} _{t} ( X) =\mathbb{E} _{t}^{\text{vac}}( \mathscr{L}_{\text{Zeno}}X) ,
\end{equation*}
where the Zeno Lindbladian is
\begin{equation*}
 \mathscr{L}_{\text{Zeno}}X =  \hat{L}^\ast_{\mathtt{z} } X \hat{L}_{\mathtt{z}}  -\frac{1}{2} X \hat{L}^\ast_{\mathtt{z}} \hat{L}_{\mathtt{z}}
-\frac{1}{2}  \hat{L}^\ast_{\mathtt{z} } \hat{L}_{\mathtt{z} } X
-i\left[ X ,\hat{H} \right] .
\end{equation*}
The equivalent master equation is
\begin{equation}
\frac{d}{dt}\rho _{\mathtt{z}} = \mathscr{D}_{\mathrm{Zeno}} \rho _{\mathtt{z}}  ,
\end{equation}
where $\mathscr{D}_{\mathrm{Zeno}} \rho _{\mathtt{z}}
=\hat{L}_\mathtt{z} \rho_\mathtt{z} \hat{L}^\ast_\mathtt{z}-\frac{1}{2} \rho_\mathtt{z} \hat{L}^\ast_\mathtt{z} \hat{L}_\mathtt{z}
-\frac{1}{2}  \hat{L}^\ast_\mathtt{z} \hat{L}_\mathtt{z}  \rho_\mathtt{z}
+i\left[ \rho_\mathtt{z} ,\hat{H} \right] $.

\subsection{Zeno Quantum Trajectories for Vacuum Input}

We now consider possible continual measurements of the output field. We set
\begin{enumerate}
\item  Homodyne $Z( t) =B_{\text{in}}( t) +B_{\text{in}}^{\ast }( t) ;$

\item  Number counting $Z( t) = \Lambda_{\text{in}}( t) .$
\end{enumerate}

In both cases, the family $\left\{ Z( t) :t\geq 0\right\} $ is self-commuting, as is the set of observables
\begin{equation}
Y( t) =V( t) ^{\ast }( I_{\text{sys}}\otimes
Z( t) ) V( t)
\end{equation}
which constitute the actual measured process. We note the non-demolition property $\left[ j_{t}(X),Y_{s}\right] =0$ for all $t\leq s$.

\begin{figure}[htbp]
	\centering
		\includegraphics[width=1.00\textwidth]{zeno1.eps}
	\caption{We may still measure the environment and obtain a conditioned state for the system within the Zeno subspace $\mathfrak{h}_{\mathrm{Zeno}}$. }
	\label{fig:Zeno1}
\end{figure}

The aim of filtering theory is to obtain a tractable expression for the least-squares estimate of $j_{t}( X) $ given the output observations $Y( \cdot ) $ up to time $t$. Mathematically, this
is the conditional expectation
\begin{equation*}
\pi _{t}( X) =\mathbb{E}_{\eta \Omega }\left[ j_{t}(
X) |\mathcal{Y}_{t}\right]
\end{equation*}
onto the measurement algebra $\mathcal{Y}_{t}$ generated by $Y( s) $ for $s\leq t$, with the fixed state $\eta \otimes \Omega$.

The Zeno filters are given respectively by
\begin{equation}
d\hat{\pi}_{t}(X)=\hat{\pi}_{t}(\mathcal{L}_{\text{Zeno}}X)dt+\mathcal{\hat{H}}_{t}\left( X\right) d\hat{I}(t)
\end{equation}
where we have
\begin{equation*}
\begin{tabular}{l|l|l|}
& Homodyne & Number Counting \\ \hline
$\mathcal{\hat{H}}_{t}\left( X\right) $ & $\hat{\pi}_{t}(X\hat{L}+\hat{L}^{\ast }X)-\hat{\pi}_{t}(X)\hat{\pi}_{t}(\hat{L}+\hat{L}^{\ast })$ &
$dY(t)-\hat{\pi}_{t}(\hat{L}+\hat{L}^{\ast })dt$ \\ \hline
$d\hat{I}(t)$ & $\hat{\pi}_{t}(\hat{L}^{\ast }X\hat{L})/\hat{\pi}_{t}(\hat{L}^{\ast }\hat{L})-\hat{\pi}_{t}(X)$ & $dY(t)-\hat{\pi}_{t}(\hat{L}^{\ast }\hat{L})dt$ \\ \hline
\end{tabular} .
\end{equation*}

The stochastic processes $\hat{I}$ are the innovations and correspond to the difference between observations $dY$ and the expectations given the present conditioned state.

Alternatively we may introduce the conditioned Zeno state $\hat{\varrho} (t)$ defined by $\hat{\pi}_t (X)  \equiv tr_{\mathtt{z}}\left\{ \hat{ \varrho }_{\mathtt{z}}X\right\}$
for any bounded operator $X$ on the Zeno subspace.
From the filter equation we deduce the Zeno stochastic master equations
\begin{eqnarray}
\hat{\varrho} (t) = \mathscr{D}_{\mathrm{Zeno} }\hat{\varrho} (t) \, dt
+ \hat{\mathcal{G} } ( \hat{\varrho} (t)) \, d\hat{I} (t),
\end{eqnarray}
where
\begin{equation*}
\begin{tabular}{l|l|l|}
& Homodyne & Number Counting \\ \hline
$\mathcal{\hat{G}}_{t}\left( \varrho \right) $ & $ \hat{L} \varrho + \varrho \hat{L}^{\ast }-tr_{\mathtt{z}}\left\{   \varrho (\hat{L} +\hat{L}^\ast ) \right\}
\varrho$ &
$dY(t)-  tr_{\mathtt{z}}\left\{   \varrho  (\hat{L}+\hat{L}^{\ast }) ) \right\} \, dt$ \\ \hline
$d\hat{I}(t)$ & $ \hat{L} \varrho \hat{L}^{\ast } / tr_{\mathtt{z}}\left\{ \hat{ \varrho } \hat{L}^\ast \hat{L} \right\} -\varrho $ &
$dY(t)-tr_{\mathtt{z}}\left\{   \varrho   \hat{L}^\ast \hat{L} \right\}  dt$ \\ \hline
\end{tabular} .
\end{equation*}

\subsection{Zeno Dynamics Within Networks}
Let $\mathbf{G}_\alpha (k)$ be a collection of zenofiable models with state spaces $\mathfrak{h}_\alpha$,
Zeno spaces $\mathfrak{h}_{\mathrm{Zeno}, \alpha}$, and limit Zeno generators $\mathbf{G}_{\mathrm{Zeno}, \alpha}$.

It is possible to form a quantum feedback network \cite{GJ09}, \cite{GoughJamesIEEE09} by connecting the output fields of
some of these components as the inputs to others. In the limit of instantaneous feedback/forward we obtain a single
effective $SLH$ model for the remaining inputs. The resulting model will again be zenofiable with network Zeno subspace
\[
\mathfrak{h}_{\mathrm{Zeno}} = \bigotimes_\alpha \mathfrak{h}_{\mathrm{Zeno}, \alpha}.
\]
The consistency of the construction follows from earlier work which establishes that the feedback reduction procedure for
determining network models commutes with the procedure for adiabatic elimination \cite{GNW}, \cite{GN}. We obtain the same Zeno dynamics by first
reducing the components systems to their Zeno dynamics and then performing the network interconnections.

\section{Examples}
In this section we discuss some well-known examples from the perspective of Zeno dynamics and control theory.
\subsection{No scattering, and trivial damping}
Let us set $S=I$, $L^{(1)}=0$, $L^{(0)}_{\mathtt{fz}}=0$ and $H^{(1)}_{\mathtt{zf}} =H^{(1)\ast}_{\mathtt{fz}}=0$. In this case the only
damping of significance is that of the Zeno component.
Then we have
$A_{\mathtt{ff}} = -i H^{(2)}_{\mathtt{ff}}$ and we require that $H^{(2)}_{\mathtt{ff}}$ is invertible on $\mathfrak{h}_{\mathtt{z}}$.
It is easy to see that the decoupling conditions now apply and we obtain the open Zeno dynamics with $(\hat{S}=I, \hat{L}= L^{(0)}_{\mathtt{zz}}, \hat{H})$
where the Zeno Hamiltonian is
\[
\hat{H}=H_{\mathtt{zz}}^{\left( 0\right) },
\]
and we note that
\[
\left[
\begin{array}{ll}
\hat{H} & 0 \\
0 & 0
\end{array}
\right] \equiv P_{\mathtt{z}} H(k) P_{\mathtt{z}} .
\]

The conditions are however still met if we take the off-diagonal terms $H^{(1)}_{\mathtt{zf}} =H^{(1)\ast}_{\mathtt{fz}}$ to be non-zero, and here we have
the Zeno Hamiltonian
\begin{equation*}
\hat{H}=H_{\mathtt{zz}}^{\left( 0\right) }-H_{\mathtt{zf}
}^{(1)}\frac{1}{H_{\mathtt{ff}}^{(2)}}H_{\mathtt{fz}}^{\left( 1\right) }
\end{equation*}
Now $\hat H$ is the shorted version (Schur complement) of
$H(1)=\left[
\begin{array}{cc}
H_{\mathtt{zz}}^{\left( 0\right) } & H_{\mathtt{zf}}^{(1)} \\
H_{\mathtt{fz}}^{\left( 1\right) } & H_{\mathtt{ff}}^{(2)}
\end{array}
\right] $.
Equivalently, $\hat H$ is the the limit $k \uparrow \infty$ of shorted version of $H(k)$.

\subsubsection{Qubit Limit}

Let us consider a cavity consisting of a single photon mode with annihilator
$a$, so that $\left[ a,a^{\ast }\right] =I$. The number states $|n\rangle $, $\left( n=0,1,\cdots \right) $, span the infinite dimensional Hilbert space.
The following model due to Mabuchi \cite{Mabuchi_2012} shows how a large Kerr non-linearity
leads to a Zeno dynamics where we are restricted to the ground and first
excited state of the mode, and so have an effective qubit dynamics. We
consider the $n=2$ input model with
\begin{eqnarray*}
\left[ S(k)\right] _{jk} &=&\delta _{jk}I, \\
\left[ L\left( k\right) \right] _{j} &=&\sqrt{\kappa _{j}}e^{i\omega
t}a,\quad (j=1,2) \\
H\left( k\right)  &=&k^{2}\chi _{0}a^{\ast 2}a^{2}+\Delta a^{\ast }a -i \sqrt{\kappa _{1}} \left( \alpha \left( t\right) a^{\ast }-\alpha ^{\ast
}\left( t\right) a\right) .
\end{eqnarray*}
In the model we are in a rotating frame with frequency $\omega $ and the cavity is detuned from this frequency by an amount $\Delta $. There is a
Kerr non-linearity of strength $\chi \left( k\right) =\chi _{0}k^{2}$ which will be the large parameter. We have two input fields with damping rate
$\kappa _{j} \, (j=1,2)$, and the first input introduces a coherent driving field $\alpha \left( t\right) $.

We have $A\equiv \chi _{0}a^{\ast 2}a^{2}=\chi _{0}N\left( N-1\right) $ where $N=a^{\ast }a$ is the number operator. The kernel space of $A$ is
therefore
\begin{equation*}
\mathfrak{h}_{\mathrm{\mathtt{z}}}=\text{span }\left\{ |0\rangle ,|1\rangle
\right\} .
\end{equation*}
For this situation we have $P_{\mathtt{z}}=|0\rangle \langle 0|+|1\rangle \langle 1|$, and we find $L_{\mathtt{fz}}^{(0)}=0$ since
$P_{\mathtt{f}}aP_{\mathtt{z}}\equiv 0$. The zenofiability conditions are then satisfied and we have
\begin{equation*}
H_{\mathtt{zz}}^{(0)}=P_{\mathtt{z}}\Delta a^{\ast }aP_{\mathtt{z}}\equiv
\Delta \sigma ^{\ast }\sigma
\end{equation*}
where $\sigma \triangleq P_{\mathtt{z}}aP_{\mathtt{z}}\equiv |0\rangle
\langle 1|$. We then have that
\begin{eqnarray*}
\left[ \hat{S}_{\mathtt{zz}}\right] _{jk} &=&\delta _{jk}\,I_{\mathtt{z}}, \\
\left[ \hat{L}_{\mathtt{z}}\right] _{j} &=&\sqrt{\kappa _{j}}e^{i\omega
t}\sigma , \\
\hat{H} &=&\Delta \sigma ^{\ast }\sigma -i \sqrt{\kappa _{1}} \left(
\alpha \left( t\right) \sigma ^{\ast }-\alpha ^{\ast }\left( t\right) \sigma
\right) .
\end{eqnarray*}
The system is then completely controllable through the policy $\alpha$, and observable through
quadrature measurement (homodyning with $B_{\mathrm{out},1} (t) -B_{\mathrm{out},1} (t)^\ast$, and $-i B_{\mathrm{out} ,1} (t) +i B_{\mathrm{out},1} (t)^\ast$)
and by photon counting.

\subsection{No scattering, but non-trivial damping}

We consider the case where $S=I$,  $L_{\mathtt{fz}}^{(0)}=0$ and  $L_{\mathtt{ff}}^{(1)}=0$, but  $L_{\mathtt{zf}}^{(1)}\neq 0$. The decoupling
conditions are automatically satisfied, so this set-up is zenofiable with Zeno provided that $A_{\mathtt{ff}}$, which is now given by
\begin{equation*}
A_{\mathtt{ff}}\equiv -\frac{1}{2}L_{\mathtt{zf}}^{\left( 1\right) \ast }L_{\mathtt{zf}}^{\left( 1\right) }-iH_{\mathtt{ff}}^{\left( 2\right) },
\end{equation*}
is invertible. If so the Zeno $SLH$ takes the simplified form
\begin{eqnarray*}
\hat{S}_{\mathtt{zz}} &\equiv &I_{\mathtt{z}}+L_{\mathtt{zf}}^{\left(
1\right) }\frac{1}{A_{\mathtt{ff}}}L_{\mathtt{zf}}^{\left( 1\right) \ast },
\\
\hat{L}_{\mathtt{z}} &\equiv &L_{\mathtt{zz}}^{\left( 0\right) }-L_{\mathtt{zf}}^{\left( 1\right) }\frac{1}{A_{\mathtt{ff}}}M_{\mathtt{fz}}, \\
\hat{H} &\equiv &H_{\mathtt{zz}}^{(0)}+\mathrm{Im}\left\{ M_{\mathtt{zf}} \frac{1}{A_{\mathtt{ff}}}M_{\mathtt{fz}}\right\} ,
\end{eqnarray*}
where now
\begin{eqnarray*}
M_{\mathtt{zf}} &\equiv &-\frac{1}{2}L_{\mathtt{zz}}^{\left( 0\right) \ast
}L_{\mathtt{zf}}^{\left( 1\right) }-iH_{\mathtt{zf}}^{\left( 1\right) }, \\
M_{\mathtt{fz}} &\equiv &-\frac{1}{2}L_{\mathtt{zf}}^{\left( 1\right) \ast
}L_{\mathtt{zz}}^{\left( 0\right) }-iH_{\mathtt{fz}}^{\left( 1\right) }.
\end{eqnarray*}

\subsubsection{Alkali Atom}

Consider a model for an atomic electron which has two energy states, a ground state $|g\rangle $ and an excited state $|e\rangle $ spanning
$\mathfrak{h}_{\text{level}}=\mathbb{C}^{2}$, and an intrinsic spin one-half with states
$|+\rangle $ and $|-\rangle $ spanning $\mathfrak{h}_{\text{spin}}=\mathbb{C}^{2}$.
The total Hilbert space is then the 4-dimensional $\mathfrak{h}=\mathfrak{h}_{\text{level}}\otimes \mathfrak{h}_{\text{spin}}$. Taking $n=3$ inputs, one for
each spatial coordinate, the scaled model is \cite{BHJ07}
\begin{eqnarray*}
\left[ L(k)\right] _{j} &=&k\sqrt{\gamma }|g\rangle \langle e|\otimes \sigma
_{j}\text{, \ \ }j=1,2,3, \\
H\left( k\right)  &=&k^{2}\Delta |e\rangle \langle e|\otimes
I+\sum_{j=1,2,3}I\otimes \mathcal{B}_{j}\sigma _{j}
\end{eqnarray*}
where $\Delta >0$ has the interpretation of a detuning frequency for the
excited state, and $\mathcal{B}_{j}$ as the $j$ component of an external
magnetic field. We find that
\begin{equation*}
A=-\left( \frac{3}{2}\gamma +i\Delta \right) |e\rangle \langle e|\otimes I
\end{equation*}
so if we set $P=|g\rangle \langle g|\otimes I$, we have a zenofiable model
with 2-dimensional Zeno space
\begin{equation*}
\mathfrak{h}_{\text{Zeno}}=|g\rangle \otimes \mathfrak{h}_{\text{spin}}\equiv \text
span}\left\{ |g\rangle \otimes |+\rangle ,|g\rangle \otimes |-\rangle
\right\} .
\end{equation*}
The Zeno $SLH$ is
\begin{eqnarray*}
\left[ \hat{S}\right] _{jk} &=&|g\rangle \langle g|\otimes \left\{ \delta
_{jk}-\frac{\gamma }{\frac{3}{2}\gamma +i\Delta }\sigma _{j}\sigma
_{k}\right\} , \\
\left[ \hat{L}\right] _{j} &=&0, \\
\hat{H} &=&\sum_{j=1,2,3}|g\rangle \langle g|\otimes \mathcal{B}_{j}\sigma
_{j}.
\end{eqnarray*}

In this case the Zeno dynamics is undamped $(\hat{L}=0)$ so filtering is not possible for vacuum input. However the Zeno system (effectively the spin) is
controllable through the magnetic field.

\subsubsection{$\Lambda $-systems}

Consider a three level atom with ground states $|g1\rangle $,$|g2\rangle $
and an excited state $|e\rangle $ with Hilbert space $\mathfrak{h}_{\text{level}}=\mathbb{C}^{3}$. The atom is contained in a cavity with quantum mode $a$
with Hilbert space $\mathfrak{h}_{\text{mode}}$ where $\left[ a,a^{\ast }\right]
=1$ and $a$ annihilates a photon of the cavity mode. The combined system and
cavity has Hilbert space $\mathfrak{h}=\mathfrak{h}_{\text{level}}\otimes \mathfrak{h}_{\text{mode}}$, and consider the following \cite{DuanKimble}, \cite{BHJ07},
\begin{eqnarray*}
L\left( k\right)  &=&k\sqrt{\gamma }I\otimes a, \\
H(k) &=&ik^{2}\mathsf{g}\left\{ |e\rangle \langle g1|\otimes a-|g1\rangle
\langle e|\otimes a^{\ast }\right\} +ik\left\{ |e\rangle \langle g2|\otimes
\alpha -|g2\rangle \langle e|\otimes \alpha ^{\ast }\right\} .
\end{eqnarray*}
Here the cavity is lossy and leaks photons with decay rate $\gamma $, we
also have a transition from $|e\rangle $ to $|g1\rangle $ with the emission
of a photon into the cavity, and a scalar field $\alpha $ driving the
transition from $|e\rangle $ to $|g2\rangle $. We see that
\begin{equation*}
A\equiv -\frac{1}{2}\gamma I\otimes a^{\ast }a+\mathsf{g}\left\{ |e\rangle
\langle g1|\otimes a-|g1\rangle \langle e|\otimes a^{\ast }\right\} .
\end{equation*}
and that $A$ has a 2-dimensional kernel space spanned by the pair of states
\begin{equation*}
|\Psi _{1}\rangle =|g1\rangle \otimes |0\rangle ,\quad |\Psi _{2}\rangle
=|g2\rangle \otimes |0\rangle .
\end{equation*}
The Zeno subspace is then the span of $|\Psi _{1}\rangle $ and $|\Psi
_{2}\rangle $, and the resulting $SLH$ operators are
\begin{eqnarray*}
\hat{S} &=&|\Psi _{1}\rangle \langle \Psi _{1}|-|\Psi _{2}\rangle \langle
\Psi _{2}|  \equiv I-2 \sigma^\ast \sigma , \\
\hat{L} &=&-\frac{\gamma \alpha }{\mathsf{g}}|\Psi _{1}\rangle \langle \Psi _{2}|  \equiv -\frac{\gamma \alpha }{\mathsf{g}}  \sigma , \\
\hat{H} &=&0 ,
\end{eqnarray*}
where $\sigma = |\Psi _{1}\rangle \langle \Psi _{2}| $.
Here the Zeno dynamics has a vanishing Zeno Hamiltonian, but is partially
observable through filtering as $\hat{L}\neq 0$.

\section{Oscillator Models}
The original treatment of adiabatic elimination problem for quantum stochastic problems dealt with
systems coupled to oscillator components which decayed rapidly to their ground states \cite{GvH}.
Decomposing the system Hilbert space as $\mathfrak{h} = \mathfrak{h}_{\mathrm{slow}} \otimes \mathfrak{h}_{\mathrm{osc}}$,
one can establish the conditions for adiabatic elimination of the oscillators lead to the Zeno subspace
\begin{equation}
\mathfrak{h}_{\mathrm{Zeno}} = \mathfrak{h}_{\mathrm{slow}} \otimes \vert 0 \rangle ,
\label{eq:osc_zeno}
\end{equation}
where $\vert 0 \rangle $ is the ground state of the oscillatory system. The effective Zeno
subspace obviously being the slow space.

We consider an open model described by the generators $\mathbf{G} (k)$ with
\begin{eqnarray}
S(k) &=& \hat S \otimes I,  \notag \\
L(k) &=&k\sum_{i} \hat C_{i}\otimes a_{i}+ \hat G\otimes I,  \notag \\
K(k) &=&k^{2}\sum_{ij}\hat A_{ij}\otimes a_{i}^{\ast }a_{j}+k\sum_{i}\hat Z_{i} \otimes
a_{i}^{\ast }+k\sum_{i} \hat X_{i}\otimes a_{i}+ \hat R \otimes I,
\label{scaled parameters}
\end{eqnarray}
where $k$ is a positive scaling parameter and $\hat S , \hat C_{i}, \hat G , \hat A_{ij} , \hat X_{i} , \hat Z_{i} , \hat R
$ are bounded operators on $\mathfrak{h}_{\mathrm{slow}}$ with $\hat A=\left[ \hat A_{ij}\right]
$ invertible with bounded inverse. Here $a_{i}$ is the annihilator corresponding to the
$i$th local oscillator, say with $i=1,\cdots ,m$.

As $k\rightarrow \infty $ the oscillators become increasingly strongly coupled to the driving noise field and in this limit we would like to
consider them as being permanently relaxed to their joint ground state. The oscillators are then the fast degrees of freedom of the system, with the
auxiliary space $\mathfrak{h}_{\mathrm{slow}}$ describing the slow degrees.

We say that $\hat A$ is \textit{strictly Hurwitz stable} if
\begin{equation*}
\mathrm {Re}\langle \psi |\hat A \psi \rangle <0\text{, \ for all }\psi \neq 0.
\label{Hurwitz}
\end{equation*}

It is shown in \cite{GNW} that $A = \sum_{ij}\hat A_{ij}\otimes a_{i}^{\ast }a_{j}$ will have kernel space
given by (\ref{eq:osc_zeno}) whenever
$\hat A$ is strictly Hurwitz. Assuming that $\hat A$ is Hurwitz, we then obtain a Zeno limit with $\mathbf{G}_{\mathrm{Zeno}}$
\begin{eqnarray}
\hat{S} &=&(I+\hat C{\hat A}^{-1} {\hat C}^{\ast }) \hat S,  \notag \\
\hat{L} &=&\hat G- \hat C \hat A^{-1} \hat Z,  \notag \\
\hat{K} &=&\hat R- \hat X \hat A^{-1} \hat Z,  \label{Limit parameters}
\end{eqnarray}
where we drop the ``$\otimes |0 \rangle \langle 0 |_{\mathrm{osc}}$'' for convenience.

\subsection{Quantum Linear Models}
In principle, the slow degrees of freedom may also correspond to
an assembly of oscillators too, and we may take the overall model (fast and slow oscillators) to be
linear. The limit dynamics will then be linear, with the model coefficients determined by the limit theorems.
These will in fact be similar in structure to singular perturbation results for linear control systems \cite{KKOR}.
In this case it is known that the strict Hurwitz property of $\hat A$ (now a matrix with scalar entries)
guarantees that there exist a finite constant $k_0$ such the total linear model is stable for $k >k_0$.
This justifies the neglect of parasitic modes in engineering modelling \cite{Khalil}. For quantum linear passive systems,
this will also be the case, and the non-Zeno part of the dynamics may be safely ignored.

Specifically, we may take the slow system to be comprised of $m$ quantum oscillators with annihilators $b_1 , \cdots , b_r$
and we obtain a linear model with fast oscillators $a_i$ and slow (Zeno!) oscillators $b_i$ by taking
\begin{eqnarray*}
\hat G = \sum_i (G^+_i b_i^\ast +G^-_i \, b_i ),\\
\hat Z_j = \sum_i (Z^+_{ij} b_i^\ast + Z^-_{ij} \, b_i ),\\
\hat X_j = \sum_i (X^+_{ij} b_i^\ast + X^-_{ij} \, b_i ),\\
\hat R_j = \sum_i (R^{++}_{ij} b_i^\ast b_j^\ast  + R^{--}_{ij} \, b_i b_j +R^{+-}_{ij} \, b^\ast_i b_j  ),
\end{eqnarray*}
and the $\hat A_{ij}$ scalars.

From the quantum stochastic calculus \cite{HP} the oscillators satisfy linear Heisenberg equations that take the form
\begin{eqnarray*}
d b (t) = \Gamma_1 b \, dt + \Gamma_2 z + \Phi \, dB_{\mathrm{in}} (t) , \\
\frac{1}{k^2} dz(t) = \Gamma_3 b \, dt + \Gamma_4 z + \Psi \, dB_{\mathrm{in}} (t) ,
\end{eqnarray*}
where we set $z \equiv ka$. The coefficient matrices $\Gamma_j \, (j=1,2,3,4), \Phi , \Psi$ are independent of
the scaling parameter $k$. In particular $\Gamma_4 \equiv \hat A$.

We are interested primarily in the stability of these equations and it is sufficient to look at the averages of the operators
where the environment state is vacuum, for which the input processes are martingales, and here we have
\begin{eqnarray}
\frac{d}{dt} \bar b (t) = \Gamma_1 \bar b (t) + \Gamma_2 \bar z  (t) , \\ \notag
\frac{1}{k^2} \frac{d}{dt} \bar z(t) = \Gamma_3 \bar b (t) + \Gamma_4 \bar z (t) ,
\label{eq:Gammas}
\end{eqnarray}

\begin{proposition}
If $\hat A$ is invertible, then the necessary and
sufficient conditions for the existence of a $k_0 <\infty$ such that the system (\ref{eq:Gammas}) is asymptotically stable for $k>k_0$ are that
both $\hat A$ and $\Gamma_0$ are strictly Hurwitz.
\end{proposition}

This is a consequence of the following result \cite{Chow_Kokotovic}.

\begin{lemma} If $\Gamma_4 \equiv \hat A$ is invertible, then for $k$ sufficiently large the eigenvalues for the system of linear dynamical equations (\ref{eq:Gammas})
are $\sigma( \Gamma_0 ) (1 +O( \frac{1}{k^2}) )$, and $ k \sigma( \Gamma_4 ) (1 +O( \frac{1}{k^2}) )$, where $\Gamma_0 = \Gamma_1 - \Gamma_2 \Gamma_4^{-1} \Gamma_3$.
\end{lemma}

(Here $\sigma(X)$ denotes the spectrum of a matrix $X$.) The strict Hurwitz property for $\hat A$ is essential for the existence of the Zeno dynamical limit
for the system of oscillators $b_i$. The additional requirement that $\Gamma_0$ also be Hurwitz is required for stability of the limit system, compare with (\ref{Limit parameters}).

From a modelling perspective, the Zeno dynamics is a model reduction of the total assembly of fast and slow oscillators. In control applications, one would want to
have the performance under the total dynamics approximated arbitrarily closely by the Zeno model by taking $k$ sufficiently large. In this way any further
feedback construction \cite{GJ09} would be robust against the modelling errors introduced by reducing to the Zeno description. To this end we remark again that
feedback in SLH models is compatible with model reduction through adiabatic elimination \cite{GNW,GN}.

redpajama_set_name: RedPajamaArXiv


New from HiFiMan! The Susvara is a planar magnetic headphone with an ultra thin diaphram - less than a millionth of a meter thin! The Susvara also utilizes "Stealth Magnet" technology.
In Sanskrit, SUSVARA means all the world's harmonious and melodious sounds. As HiFiMan's finest planar phone, it was initially named named Edition 6 and received high praise at several prestigious audio events. In consideration of Edition 6's acoustic performance that has surpassed any product in the HE or EDITION lines, it deserved a dedicated model name and that is: SUSVARA. It has achieved huge strides in terms of openness, transparency, and detail, so that it now sounds almost like a 'junior' version of the Shangri-La electrostatic. HIFIMAN SUSVARA is available now to order.


text:
TRP scam: Former CEO of BARC sent in police custody
Dasgupta was arrested by the Mumbai crime branch from Pune district on Thursday and produced before the court on Friday.
A magistrate's court here on Friday remandedPartho Dasgupta, a former CEO of the rating agency Broadcast Audience Research Council (BARC), in police custody till December 28 in the alleged fake TRP scam.
The police sought his remand for further probe, saying his custodial interrogation was required to understand his role in the scam. Dasgupta's lawyer Kamlesh Ghumre argued that the offences of cheating and criminal breach of trust for which he has been booked were not made out against him.
The probe was harassment and use of pressuring tactics by police, the lawyer alleged. After hearing both the sides, the court sent the accused in the custody of crime branch till December 28.
Dasgupta is the fifteenth person to be arrested in the case related to alleged rigging of Television Rating Points (TRP) by some TV channels. The court on Thursday had granted bail to another accused in the case, Ramil Ramgarhia, a former chief operating officer of the BARC.
The probe was over and therefore his detention was not needed, the court observed. Most of the accused in the case are now out on bail.
Mumbai police began the probe after the BARC, a rating agency, filed a complaint about the rigging of TRP by some channels. TRP, measured by recording viewership data at sample households, is crucial as it helps TV channels attract advertisers. It was alleged that some of these households were being bribed to tune into certain channels to ramp up their ratings.
Tags BARC BARC India TRP Scam TRP scam case
DRDO lab develops bike-based ambulance
Coronavirus news highlights updates: Over 4 million Britons...
US Capitol shut down temporarily out of caution over fire...

redpajama_set_name: RedPajamaCommonCrawl


text:
Hřib červený (Hortiboletus rubellus (Krombh.) Simonini, Vizzini et Gelardi 2015) je méně běžná teplomilná houba z čeledi hřibovitých. Prosperuje i v lokalitách ovlivněných činností člověka a v důsledku variabilního zbarvení může být snadno zaměněn s několika příbuznými druhy.

Synonyma

 Boletus rubellus Krombh. 1836
 Boletus versicolor Rostk. 1844
 Xerocomellus rubellus (Krombh.) Šutara 2008
 Xerocomus rubellus (Krombh.) Quél. 1896
 Xerocomus versicolor (Rostk.) Gilbert 1931

 hřib červený
 hřib různobarevný
 suchohřib červený
 suchohřib pestrý
 poddoubník červený

Taxonomie
V současnosti je hřib červený klasifikován jako samostatný druh. Dříve byl ale některými mykology považován za poddruh nebo formu hřibu žlutomasého - babky (Xerocomellus chrysenteron).

Vzhled

Makroskopický
Klobouk dosahuje 35-65 (až 90) milimetrů. V mládí je sytě, živě až tmavě červený, postupně - především na světlejších stanovištích - bledne až do pleťových odstínů. Povrch má v mládí sametový, později plstnatý až krupičkatý.

Rourky a póry jsou v mládí světle až sytě žluté, později se barví do žlutoolivova.

Třeň může být ve spodní části zesílený, zbarvený bývá načervenale, někdy až červeně, nahoře světleji - do žluta nebo žlutooranžova.

Dužnina poškození modrá až zelená, zejména v případě starších plodnic. Je měkká, nakyslé chuti a snadno hnije. Voní nenápadně. Houba je jedlá, ale kvalitativně na nízké úrovni - srovnatelná se zástupci skupiny babek.

Mikroskopický
Výtrusný prach je hnědoolivový, medově hnědé spory dosahují 9 - 14 × 5 - 6 μm.

Výskyt
Hřib červený je v některých krajích vzácný, v jiných hojný. Jde o teplomilný druh, který se vyskytuje v listnatých a smíšených lesích, případně na stanovištích ovlivněných činností člověka, jako jsou návsi, parky a hráze rybníků. Preferuje duby a lípy, fruktifikuje od července do října.

Rozšíření
Roste v Evropě, Severní Americe (USA) a v Austrálii (Nový Zéland). Z evropských zemí to jsou: Belgie, Česká republika, Dánsko, Chorvatsko, Estonsko, Francie, Gibraltar, Irsko, Itálie, Lucembursko, Maďarsko, Německo, Nizozemí, Norsko, Polsko, Rakousko, Slovensko, Slovinsko, Spojené království, Španělsko a Švédsko.

V rámci chráněných území České republiky je hřib červený známý mimo jiné na následujících lokalitách:

 Hraniční meandry Odry (okres Karviná)
 Luční (okres Tábor)

Záměna
Plodnice, které nejsou vybarvené sytě červeně, mohou být zaměněny za:
 hřib Engelův (Hortiboletus engelii) - báze třeně se na řezu zbarvuje oranžovočerveně
 hřib mokřadní (Xerocomellus ripariellus) - preferuje výrazně vlhká stanoviště, báze třeně na řezu výrazně modrá
 hřib meruňkový (Rheubarbariboletus armeniacus)
 hřib Markův (Xerocomellus marekii)
 hřib políčkatý (Xerocomellus cisalpinus) - vzácný druh, klobouk bez červeného zbarvení, báze třeně výrazně modrá

Obecně se od nich odlišuje obvykle nepopraskanou pokožkou klobouku. Určení se provádí mikroskopicky - hřib červený má užší hyfy pokožky klobouku.

Ochrana
Hřib červený je sice jedlý (kvalitativně dosahuje nízké úrovně podobně jako hřib žlutomasý, snadno plesniví), ale s ohledem na rozšíření by neměl být sbírán ke konzumním účelům.

Odkazy

Reference

Externí odkazy

Jedlé houby
Hřibovité

redpajama_set_name: RedPajamaWikipedia


text:
22 July 2017 – Newbury
Horse Racing Tips - Betinfo24 blog > BetInfo24 Blog – News | Tips | Updates > Horse Racing tips > 22 July 2017 – Newbury
July 21, 2017 SteveHorse Racing tipsLeave a comment0
Newbury 15:00 - Bet365 Hardwood Stakes
There is a nice card at Newbury today, including the lottery that is the Weatherby's Super Sprint, but I think this race is the best of the lot. It's a Group 3 over six furlongs and it has attracted an impressive looking field.
The clear favourite at 10/3 is the Charlie Hills trained Magical Memory. This Zebedee gelding was a good winner of a conditions stakes last time out with Andrea Atzeni on board, where he stayed on very well inside the final furlong. He was completely outclassed prior to that at Royal Ascot where he was never really in the race after being held up through the early stages. He is far better than that, however, and the last time today's jockey Frankie Dettori rode him, he finished second behind the excellent Tasleet in a Group 2 at York. The going was soft that day, so I think he is going to appreciate the good to soft conditions here today more than most and has an excellent chance if running true to form.
Gifted Master, trained by Hugo Palmer, is almost certain to lead this race through the early stages and is he back racing over six furlongs for the first time since May last year. He has two Group 3 wins to his name over six furlongs and his full form over this distance reads 31112. Lady Macapa is the only runner likely to challenge him for the lead, and if Oisin Murphy gets to dictate things at the front, I have a feeling Gifted Master may be very tough to pass, and at a price of 9/1, he represents excellent each way value.
Koropick is the second runner in the race for Hugo Palmer, and he was a winner last time out in a Group 3 on the all weather track at Newcastle. He ran a reasonable in the Juddmonte (a Group 1) last autumn, finishing midfield of ten, and showing that he is comfortable on the turf. Jack Mitchell has ridden him a few times and takes over the ride again today and this three-year-old who is joint highest on the official ratings looks to have a great chance. The two-pound weight allowance he gets from his elders is a plus, and if Mitchell can keep him in a handy position throughout, he definitely has a chance.
The Richard Hannon trained Tupi is a consistent performer at this level, and he won a Listed race with Ryan Moore doing the steering at Doncaster in April. That was also on good to soft ground and he won very easily that day, so I think a return to similar conditions is good form him and I would not be surprised to see him put in a big performance.
Gifted Master - 1 point each way @ 9/1
Premier League Review: Sunderland
Premier League Preview 2017/2018: Arsenal


After his transfer from Tate, Pensacola High senior James Kitt had six sacks and three fumble recoveries as part of an aggressive Tiger defense in 2018.
Decorated Pensacola High School defensive lineman James Kitt will play Division I college football next season after signing with the University of Central Arkansas on National Signing Day.
The incoming PHS senior class surprised Kitt Wednesday during his signing ceremony by presenting him with a framed No. 40 jersey, the number Kitt donned during his one year with the Tigers after transferring from Tate. The 6-foot-3, 255-pound play-maker said he didn't expect it, but that's the kind of bond he and his teammates built during Kitt's brief tenure with the school.
"You could tell we were all dedicated throughout the season together," Kitt said. "Dedicated and determined."
Kitt, an interior lineman, transferred to PHS following his junior year and became an immediate force with 28 solo tackles, 23 assisted tackles, six sacks, 11 tackles for loss, two forced fumbles and three fumble recoveries.
Kitt was named to the Subway All-Star West roster in December.
"He was a big part of a very aggressive defense that we played this year," said new PHS head coach Todd Leonard, who took the job in January. "On the football field he's a very physical presence. He has the ability to be incredibly physical at the next level of play."
Leonard said Tate's coaching staff deserves credit for developing Kitt prior to his senior year. He thinks added visibility at PHS helped Kitt secure a spot at Central Arkansas.
"Certainly we're not going to take credit for placing him, fully, at the Division I level, a lot of that goes to Tate as well," Leonard said. "We just took him, got him some opportunities and here he is today signing."
Kitt said Central Arkansas head coach Nathan Brown and others on Brown's staff would call and catch up with Kitt throughout the past year, something that meant a lot to the PHS senior.
"They were the only school keeping in touch with me after the transfer and before the transfer," Kitt said. "They were always checking on me, on my grades, how my family was, contacting my mother and all that. So I feel right at home."
The Central Arkansas Bears play in Conway, Arkansas, in the Southland Conference. About nine hours of driving distance will separate Kitt from his family in Pensacola, but he said he thinks he'll be alright in Conway.
"They have a few chicken spots out there," he said. "A Dave & Buster's. I'll be alright."


※ Required travel time is approximate.
UT-Kashiwa Campus - Kashiwanoha Campus Station "SHUTTLE BUS"
(Apr.1,2014-Mar.31,2015) except Sat, Sun, National Holidays 6min.
National Cancer Center via Kashiwanoha Park 25min.
Using the Joban Expressway, get off at the Kashiwa I.C. for Chiba direction (go right after the toll gate) on to the rout 16. Change to the right lane soon. Drive 500 m. and turn right at the first traffic light. 5min.
-> Shinkamagaya Sta. -> Tobu Noda Line -> Kashiwa Sta.
limited express -> Keisei Funabashi Sta. -> Walk -> Funabashi Sta. -> Tobu Noda Line -> Kashiwa Sta.


How to uncover, articulate, and spread your company's core values. Rally your team, boost performance, and recruit people who share your guiding principles.
Your values help you face ambiguity, find clarity, make tough decisions, and take actions that serve your purpose. This is true for individuals. It's also true for organizations.
Values-driven companies achieve better brand and business performance. This is validated by a growing body of research. You can also see it by noting which employers are most admired by the public. From Apple to Zappos, these firms are guided by clear values.
This is why many startup firms identify a list of core values at an early stage of organizational development. As the company grows and matures, however, that list of early aspirational values doesn't always bear out into reality – doesn't always reflect the actual values that shape how the organization operates. Values can't be willed into practice.
Identifying a true, authentic list of core values – the ones your team already lives and works by – takes considerable time and effort. But it can be done, and it's worth doing. This guide will introduce what values are and how they support business, share the story of how our company, Recruiting Social, uncovered our core values, and provide instructions and templates for your own values-defining project.
A value is a belief that guides your choices and actions; a principle that conveys what is right and what is wrong.
In a business context, core values are the highest values that guide a firm's actions, unite its employees, and define its brand.
These guiding principles are typically communicated in writing as a core values statement.
Performance Driven: Demonstrating initiative in supporting the goals and overall strategy of the company.
Always Improving: Commitment to LEAN, creative problem solving and continual improvement.
Team Focused: Developing a culture of high-functioning and supportive teams.
Honorable and Respectful: Contributing to a work environment that honors and respects the needs of each stakeholder.
Sustainably and Socially Conscious: Involvement in practices that decrease our negative environmental impact and increase our positive community impact.
Clear core values can help an organization hire the right people, dismiss the right people, secure high-value customers, navigate through crises, remain accountable to commitments, and serve stakeholders better.
This is why values deserve a central place in how your organization functions. "Your company's core values should be incorporated into all your processes, including the hiring process, and recruiters and hiring managers should be able to articulate how those qualities individually apply to candidates," says Y-Vonne Hutchinson, founder of ReadySet, a workforce and diversity consulting firm.
But before you can build values-based processes, you need to determine what, exactly, your core values are.
Recruiting Social's core values shape the employee experience.
Recruiting Social is a recruiting services company with offices in Los Angeles, Vancouver, and Toronto. Talent teams across North America work with us through our on-demand and contingency search services to source candidates, build pipelines, manage job requisitions, and recruit people. We donate ten percent of our profits to education support for marginalized youth.
Hiring. Our employer branding emphasizes our values, and we follow a values-based recruitment process. When we interview candidates, we ask behavioral questions focused on values-alignment and score answers using a values interview scorecard.
Onboarding. New team members get oriented to our values on their first day and learn how those values drive how we operate, from our emphasis on candidate experience ("people first") to our volunteer time off policy ("giving back").
Employee engagement. During team meetings and Slack conversations, we regularly reflect on our actions and how they connect to our values.
Learning and development. The internal training we provide, and the external training employees enroll in, supports our values. For example, several team members have taken design-thinking courses in support of our "innovation" and "people first" values.
Feedback. Our 360-degree feedback process is built around our values, and we use a values-based review tool called Threads.
Recruiting Social uses a values-based interview question scorecard as part of its hiring process.
It was the fall of 2015. The Recruiting Social team was poised to grow. We recognized that, for the sake of our existing team members, sustainability of our service business, and delivery on our brand promise, we needed to understand the traits and beliefs that defined a successful RSer. It would have been easy enough to sit down for thirty minutes and scribble out a list of aspirational, good-sounding "values," but from our work with many different employers, we knew that approach seldom resulted in more than platitudes and slogans disconnected from the actual workplace experience.
1. Project brief and plan. In consultation with senior team members, we outlined our objective, or why we were doing this: "Why? To clarify and update what Recruiting Social stands for and what we want to accomplish, so we can hold ourselves accountable internally (as a company, as teams, and as individual team members) and externally (to our clients, candidates, vendors, partner companies, stakeholders, followers, and the broader public." Then, we listed out the steps we'd take to accomplish the objective, and scoped out how much time it would take and who would need to be involved.
2. Discovery. The first and largest step involved research. We performed 15-minute interviews with every member of our small team, plus two clients. Interviewees were asked to identify attributes of our team and workplace and share examples of how they experienced them. Once interviews were completed, we held a 45-minute team workshop to collaboratively explore the insights uncovered in the one-on-one conversations.
3. Distillation. The completed research amounted to, essentially, a big list of words. These were grouped into themes and validated against the feedback collected from clients. Then the themes were ranked based on how often they came up in the research process.
4. Drafts. A clunky, unpolished draft of the values statement was quickly shared with select team members for feedback. That feedback was incorporated into revisions, and the process repeated three more times as we wordsmithed and refined what would become Recruiting Social's seven core values.
5. Unveiling. The big reveal was anticlimactic. Because they'd been involved in the research and drafting process, and because the final values statement reflected the actual day-to-day experience of working at our company, there was nothing to be surprised about.
What constraints do you need to work within?
Where will the work appear, and in what format?
Hold a kickoff and planning meeting. Invite people you believe should be involved in the project and use the meeting to finalize requirements, participants, roles, and timeline.
Conduct research interviews. You'll want to speak with a representative sample of employees and leaders. At the beginning of the interview, tell the interviewee that there are no right or wrong answers, it's okay to not have an answer to a question, and all responses will remain confidential. Then, ask questions to understand the attributes, behaviors, and experiences that shape the organizational culture.
What are our strengths as an organization? Why are they strengths?
What are our weaknesses as an organization? Why are they weaknesses?
Tell me about an action the organization took, or a decision it made, that you believe was right. Why do you believe it chose to do that?
Tell me about a crisis or challenge the organization faced. How did our team respond to the situation? Why did they respond that way?
Walk me through a meaningful moment you experienced while working here. What happened? Why was that moment meaningful?
If you could choose 3 words to describe the team at our organization, what would they be?
At the end of the day, what one thing will the team at our organization be remembered for?
Is there anything I haven't asked that you'd like to add?
Facilitate a team workshop. Invite employees and leaders to participate in a 45-minute workshop to further explore the ideas that emerged during the interviews.
What are values? How we do what we do – the beliefs that guide how we act.
Why are we doing this? Help us focus, help us make the right decisions, help hire the right people, and help us fulfill our purpose as an organization. So we can know who we are, who we want to be, and help us hold ourselves accountable to the things we commit to.
What we've already done: Individual interviews. You answered questions about the company.
What we're doing today: Group discussion. 5 similar questions, but this time we get to respond to each other, share ideas, and find agreement as a group.
Questions before we dive in?
Next steps: We'll write draft core values, and share with you to validate.
The affinity process helps organize research findings into named themes.
Identify themes. You've collected research in the form of comments from employees and other stakeholders. Now it's time to synthesize that information so it can guide the drafting of your organization's core values. Use the "affinity process" to organize research findings into named themes.
Gather the comments. Transcribe them to individual Post-It notes on a whiteboard, or paste them into individual cells in a spreadsheet column.
Group the comments. Read the comments and look for connections and patterns. Group together comments based on affinities they seem to share. You may also choose to group together groups, creating a hierarchy of categories.
Label the groups. Write a name or phrase that describes each group's affinity.
Write the first draft. Use the labels to start your values statement. Produce a rough draft. Don't worry about wordsmithing and refining just yet, as you will go through several rounds of feedback and revision. Your aim with the first draft is to develop an original and memorable statement.
Focus on strengths: Don't assign importance to something you're not good at as an organization!
Speak to your audience: Who are your values for? Use words that mean something to them.
Evoke emotion: As humans, we give importance and ascribe meaning to how we feel.
Limit the number: A long list of values will be difficult to recall. Keep it in the three-to-nine range.
Collect feedback and perform revisions. Share your rough draft with internal stakeholders and ask for feedback. Encourage them to structure their input using the "I like, I wish, I wonder" technique, which helps prompt constructive feedback.
Perform revisions. Make revisions based on the feedback you receive, and share the new draft with stakeholders for new feedback. Repeat the process until a final draft of the values statement takes shape.
Communicate the core values. Share the final values statement with employees and explain its intended use. The reveal should be anticlimactic; though the statement might be original in its wording, the essence of each value should be obvious.
Circulate the values statement. Reproduce the core values statement in different formats and circulate far and wide among internal stakeholders.
Your process and templates workbook includes interview questions, a workshop agenda, core values examples, and everything you need to start your project.
Plus, you'll also get Recruiting Social's easily-customizable interview scorecard.


LEAWOOD, Kan.--(BUSINESS WIRE)-- Tortoise Energy Capital Corp. (NYS: TYY) today announced that as of April 30, 2013, the company's unaudited total assets were approximately $1.0 billion and its unaudited net asset value was $637.0 million, or $31.95 per share.
As of April 30, 2013, the company was in compliance with its asset coverage ratios under the Investment Company Act of 1940 (the 1940 Act) and basic maintenance covenants. The company's asset coverage ratio under the 1940 Act with respect to senior securities representing indebtedness was 655 percent, and its coverage ratio for preferred shares was 467 percent. For more information on calculation of coverage ratios, please refer to the company's most recent applicable prospectus.
The company issued 28,100 shares of common stock under its at-the-market equity offering program for gross proceeds of approximately $0.9 million during the month of April 2013.
Set forth below is a summary of the company's unaudited balance sheet at April 30, 2013, and a summary of its top 10 holdings.
19.94 million common shares currently outstanding.
About Tortoise Energy Capital Corp.
Tortoise Energy Capital Corp. provides financing for master limited partnerships (MLPs) in the energy infrastructure sector, focusing on crude oil and refined petroleum products MLPs and natural gas and natural gas liquids pipelines MLPs. Tortoise Energy Capital Corp. seeks to provide its stockholders a high level of total return with an emphasis on current distributions.
Tortoise Capital Advisors, L.L.C. is an investment manager specializing in listed energy investments. As of March 31, 2013, the adviser had approximately $11.4 billion of assets under management in NYSE-listed closed-end investment companies, an open-end fund and other accounts. For more information, visit www.tortoiseadvisors.com.
The article Tortoise Energy Capital Corp. Provides Unaudited Balance Sheet Information and Asset Coverage Ratio Update as of April 30, 2013 originally appeared on Fool.com.


Nate Crown practices in the firm's Real Estate Group. Although Nate's experience covers the full spectrum of commercial real estate transactions (from acquisitions/dispositions to financing to development), his practice focuses largely on leasing and construction matters. Nate has significant experience representing both landlords and tenants in connection with office, medical, retail, and industrial leases. And in the construction context, Nate's experience includes representing developers in the negotiation and drafting of contractor and architect agreements.


Vidya Vox Family, Husband, Son, Daughter, Father, Mother, Childrens, Marriage,Wedding Photos, Vidya Vox Age Weight, Body Measurements, Sizes, Vidya Vox Education Qualifications, School, College, Contact Information, Vidya Vox Remuneration, Salary, Income, Debut, Controversies, Vidya Vox Profile, Biography, Biodata, Vidya Vox Girl and Boy friends Affairs, Facebook and Twitter and Instagram Links, and more.
Vidya Vox Favorites, Hobbies, Social Links, Money Factor, Etc.
Vidya Vox was born on September 27, 1991 Age 26 years old in Chennai, Tamil Nadu, India. Vidya Vox is a Youtuber and an American singer. She holds a degree in psychology from George Washington University. She learned carnatic classical music since she was a girl and had many amazing gurus throughout her journey, including D.K. Nagarajan, brother of the late D.K. Pattamal. Vidya started making videos on YouTube, which opened a door she never dreamed of. She is currently learning Hindustani Classical and attending Western singing classes. Vidya regularly performs as singer of the Shankar Tucker Band. She has performed in incredible venues around the world, including the White House, NCPA and Webster Hall. Vidya has also performed in numerous festivals, including Festivals Des Artes in Reunion. in the Netherlands, and INK Women.
Vidya was never inclined to music at first. In fact, his inclination was more towards medicine. It was later during her graduation that she decided to pursue music as a full-time career. At that time, YouTube was at its peak and gave the opportunity to all those who wanted to launch their own channel. Vidya launched her own YouTube channel called 'Vidya Vox' in March 2014. On her channel, she publishes mashups of Western and Indian music videos. Alternatively, she sometimes publishes her own songs originally created. His channel has collected more than 3 million subscribers so far and his videos have received more than 300 million views. Vidya Vox's songs have been very popular lately and her mashups have caught the attention of the world. . She plays live with her own band around the world and regularly performs as a singer with Shankar Tucker's band.


text:
debs, theodore, 1864-1945 (3)
bogart, guy, b. 1883 (2)
wilson, oliver c. (1)
famous hoosiers (3)
law & legal affairs (1)
press, socialist (1)
socialist party of the united states of america (1)
All fields: inefficiency
Wilson, Oliver C. 1918-08-22
Debs, Eugene V. (Eugene Victor), 1855-1926; Meetings; Public speaking; Socialist Party of the United States of America;
Letter to Theodore Debs on Socialist Party Campaign and Organization Fund letterhead, 08/22/1918. Says that he is sending Theodore Debs a tentative itinerary for Eugene V. Debs' tour. Writes that he thinks that he has allowed Eugene V. Debs to get...
Bogart, Guy, 1918-04-15
Debs, Eugene V. (Eugene Victor), 1855-1926;
Letter to Theodore Debs on Los Angeles, California Local Socialist Party letterhead, 4/15/1918. Says that Cameron King and the executive committee have handled the transfer of Eugene V. Debs's property with great inefficiency. Mentions that Philip...
Bogart, Guy 1918-04-15
Debs, Eugene V. (Eugene Victor), 1855-1926; Law & legal affairs; Press, Socialist
Letter to Theodore Debs on Los Angeles, California Local Socialist Party letterhead, 04/15/0918. Says that Cameron King and the executive committee have handled the transfer of Eugene V Debs's property with great inefficiency. Mentions that Philip...


You get free fitness stuff if you're riding in certain areas of Dallas-Fort Worth. Click here to see).
The company is partnering with Outdoor Voices, Fossil, Snap Kitchen and others, filling Fit Kits with products and offers to help passengers start moving and stick with their fitness-y resolutions.
The loot list includes a $25 gift card to Snap Kitchen and a cap and bottoms from Outdoor Voices. That's only the beginning!
Nifty fitness goodies can be yours for the asking if you take an Uber ride on Thursday, Jan. 19.
1. Download the latest version of the Uber app.
2. Open the app and add promo code FITKITDFW to your account.
3. Starting at 11 a.m. Jan. 19, request the Fit Kit option.
4. When you're connected with a vehicle, you'll receive a free tote, complete with all the goodies inside.
Uber did something similar last week, when the Dallas Cowboys were still in strong contention for the Super Bowl.
More info? Click here, and you'll also find out how to maybe get one early.


text:
Q: ImageReader - discard some frames (decrease FPS) I have application which is using hardware camera to make some analysis in OpenCV. I am using ImageReader to get frames, and then I am passing them to algorithm. But I would like to get only let's say 1 or 2 frames per second from ImageReader.
I was trying to do it like this:
val imageReaderListener = ImageReader.OnImageAvailableListener { reader ->
    reader?.let {
        fps++
        if (fps > 30) {
            val image = it.acquireLatestImage()
            //mage calculations
            fps = 0
            image.close()
        }
    }
}

But unfortunately when ImageReader max images is set to 1 (I need only latest result to be shown to te user, so I do not need to increase this number), camera picture is getting freezed.
So is there any way to decrease FPS going to ImageReader?

redpajama_set_name: RedPajamaStackExchange


We are pleased to announce that Georgia Gubernatorial Candidate Stacey Abrams will be the special guest speaker at the May 4th meeting of the Skidaway Island Democrats.
Like all our recent meetings, the meeting will be held at Messiah Lutheran Church.
Stacey Abrams and her five siblings grew up in Gulfport, Mississippi, and DeKalb County, Georgia, the children of United Methodist ministers. She went on to graduate from Spelman College, the LBJ School of Public Affairs at the University of Texas, and Yale Law School.
In 2010, Stacey became the first woman to lead either party in the Georgia General Assembly and the first African American to lead in the House of Representatives.
Stacey is currently seeking the Democratic Nomination for the 2018 Gubernatorial race.
In order to maintain an accurate count of attendees please reserve your seat in advance.


text:
describe ApplicationHelper::Button::VmRetire do
  describe '#visible?' do
    context "when record is retireable" do
      before do
        @record = FactoryBot.create(:vm_vmware)
        stub_supports(Vm, :retire)
      end

      it_behaves_like "will not be skipped for this record"
    end

    context "when record is not retiretable" do
      before do
        @record = FactoryBot.create(:vm_vmware)
        stub_supports_not(Vm, :retire)
      end

      it_behaves_like "will be skipped for this record"
    end
  end

  describe '#disabled?' do
    context "button should not be disabled" do
      before do
        @record = FactoryBot.create(:vm_vmware)
        allow(@record).to receive(:retired).and_return(true)
      end

      it "does not disable the button" do
        view_context = setup_view_context_with_sandbox({})
        button = described_class.new(view_context, {}, {'record' => @record}, {})
        expect(button.disabled?).to be_falsey
      end
    end
  end
end

redpajama_set_name: RedPajamaGithub


In this short review, we present our current understanding of the role of viruses on honey bee health and address some overarching questions in honey bee virology.
Why Should I Be Concerned about Honey Bee Colony Losses and What Is Colony Collapse Disorder (CCD)?
High annual losses of honey bees, as well as range reductions and local extinctions of wild and native pollinator species, are concerning because bees are important plant pollinators . Approximately one-third of the typical Western diet requires bee pollination, and honey bees (Apis mellifera) are the primary pollinators of numerous food crops, including fruits, nuts, vegetables, and oilseeds . Annually, insect-pollinated crops are valued at approximately US$175 billion worldwide and US$17–US$18 billion in both North America and the European Union . The largest agricultural pollination event in the world occurs each February in the Central Valley of California, where nearly 80% of the world's almonds are produced . This single pollination event requires that over 60% of the commercially managed honey bee colonies in the United States (~1.6 million) be transported to the California almond groves each year (Fig 1) . Since 2006, US honey bee colony losses have averaged 33% annually (increased from ~12% historic level) [6,7]. A small percentage of these losses (3%–8%) are attributed to Colony Collapse Disorder (CCD), which is defined by a specific set of criteria: rapid loss of adult bees, presence of queen and developing bees, delayed pest invasion, and absence of field-diagnosable bee pathogens, including bacterial foulbrood diseases and overwhelming mite infestation [6,8]. Compared to non-CCD affected or healthy colonies, CCD-affected and collapsing colonies have a greater prevalence of pathogens , including viruses such as Israeli acute paralysis virus (IAPV) , Acute bee paralysis virus (ABPV), Kashmir bee virus (KBV) , and Deformed wing virus (DWV) [11,12], though the specific pathogens vary in different sample cohorts (reviewed in ). High annual losses have also been reported in some parts of Europe and Asia, though losses vary by year . Despite high annual losses, US beekeepers have maintained a population of ~2.5 million honey bee colonies by dividing their colonies more frequently to make up for colony deaths . Globally, the stock of managed honey bee colonies has increased by 45% from 1961–2007 but has not met the 300% increase in global demand for pollination services .
Honey bees (Apis mellifera) are eusocial insects that live in large colonies that are composed of sterile female worker bees (~35,000), hundreds of male bees (drones), and a single reproductive female, the queen bee. Beekeepers manage millions of bee colonies for honey production and pollination of numerous fruit, nut, vegetable, and oilseed crops. High losses of bee colonies in North America and some parts of Europe have led researchers to examine the role of multiple stressors, including RNA viruses, on bee health at both the individual and colony levels. Figure photo credits: honey bee and almond flower (middle left) by Christi Heintz, queen bee and attendants (center) and worker bee and honey (middle right) by Kathy K. Garvey, and Lake Sinai virus 2 (LSV2) transmission electron microscope image (lower left) by Sue Brumfield. Noun Project icon credits: bee hive (middle left) by Les vieux garcons, beekeeper (top right) by Luis Prado, and cherry, almonds, and apple by May Irine, Thomas Knopp, and Andrea Bobadilla, respectively.
High annual honey bee colony losses in regions that are vital to global food production are unsustainable; therefore, in order to reduce bee colony losses, it is important to identify the factors most responsible. Multiple factors, including pathogens, agrochemical exposure, lack of quality forage, and reduced habitat, negatively impact bee health (reviewed in [1,13,16]). Honey bee colony health is typically evaluated by estimating the population of the colony, which is a superorganism comprising approximately 35,000 sterile female workers (diploid sisters), hundreds of male bees (haploid drones), and a single reproductive queen bee that lays over 1,000 eggs per day during the summer months. Scientists throughout the globe are investigating the relationship between honey bee colony health and multiple abiotic and biotic factors, including viruses.
What Are the Most Common Viruses Infecting Honey Bees and How Do They Impact Bee Health?
Single-stranded positive sense RNA viruses make up the largest group of honey bee-infecting pathogens. These viruses include the following: the Dicistroviruses, Israeli acute paralysis virus, Kashmir bee virus, Acute bee paralysis virus, and Black queen cell virus; the Iflaviruses, Deformed wing virus, Kakugo virus, Varroa destructor virus-1, Sacbrood virus, and Slow bee paralysis virus; and taxonomically unclassified viruses, Chronic bee paralysis virus, and the recently discovered Lake Sinai viruses, which are a phylogenetically unique, globally distributed group of viruses, including LSV1-7 and other variants (reviewed in [17,18]). RNA viruses are readily detected in bee samples obtained from colonies of varying health and viral presence, and abundance varies by season and geographic location [7,19–21] (reviewed in ). Colony level studies have associated virus (i.e., IAPV, KBV, ABPV, DWV, and LSV2) abundance with CCD-affected or weak colonies, though these associations are not universally observed (reviewed in ). At the individual bee level, virus infections may result in physical deformities, paralysis, or death (reviewed in ). To date, only one double-stranded DNA virus with generally weak pathogenicity (i.e., Apis mellifera filamentous virus) has been described .
How Are Bee Viruses Transmitted?
Honey bee viruses are transmitted both horizontally and vertically (i.e., from queen bees to offspring) . Virus transmission within a colony is enhanced by crowded conditions and the routine transfer of nectar, pollen, and bee bread between colony members via mouth-to-mouth feeding (trophallaxis) . Several viruses have been detected in pollen and honey, and virus infections have been transferred from these substrates to uninfected queen bees and subsequently detected in eggs and progeny .
The ectoparasitic mite Varroa destructor is a major vector of honey bee viruses, including DWV, VDV-1, and IAPV. Varroa destructor mites were first identified in Asia as a parasite of A. cerana (Eastern honey bee) and later introduced into Europe (1960–70s) and North America (1980s) (reviewed in ). Varroa destructor mite infestation is a primary cause of colony loss, as mites feed on developing larvae and adults and may kill enough individual bees to cause colony death . The synergistic negative effects of mite infestation and virus infection are best studied for DWV strains and closely related VDV-1 (84% nucleotide identity). The impact of mite infestation on DWV strain diversity was examined in the Hawaiian islands, where comparative analyses of honey bee samples from colonies located on islands that had experienced two or more years of mite exposure had reduced DWV strain diversity, as compared to samples obtained from colonies located on islands that lacked Varroa destructor mites . Likewise, experiments performed in the United Kingdom demonstrated that, as compared to oral inoculation, mite-mediated transmission reduced DWV virus strain heterogeneity . In addition, increased replication of a DWV/VDV-1 recombinant strain associated with increased rates of wing deformity and shortened abdomens . The interactions between mites, viruses, and the honey bee immune system remain largely uncharacterized. Some studies suggest that mite infestation modulates host immune function [28,29], whereas other studies indicate that immune gene expression is not perturbed at the transcriptional level [27,30]. Although the mechanisms are not fully understood, mite infestation coupled with viral infection is common and often associated with colony death [10,31]; therefore, beekeepers employ various management strategies to reduce mite levels and mite-associated viral infections.
While bee-infecting viruses are primarily studied in honey bees, and thus upon discovery are called "honey bee viruses," many of these viruses infect other bee species [24,32,33]. Intra-species and inter-species transmission of bee viruses occurs in natural settings where viruses are spread via floral resources [24,34]. The first study that examined this phenomena detected bee viruses in 11 non-Apis hymenopteran species obtained near honey bee colonies and implicated inter-species transmission by phylogenetic analyses of viral sequences, which did not cluster by host species . Furthermore, greenhouse studies demonstrated IAPV transmission between honey bees and bumble bees (Bombus impatiens) via shared food sources . Similarly, larger-scale studies have shown that sympatric honey bee and bumble bee populations in Great Britain and the Isle of Man harbored similar virus strains [34,35]. Bumble bees (both managed and native species) and other native or wild pollinators are also very important for pollination of plants in both agricultural and non-agricultural landscapes . Therefore, further investigations of virus host range and the dynamics of intra- and inter-species transmission are required to better understand the role of viruses on bee health, colony losses, range reductions, and local extinctions.
What Are the Mechanisms of Honey Bee Antiviral Defense?
Bees exhibit pathogen defense mechanisms at both the colony and individual levels. Colony level or social immunity includes behaviors such as grooming and removal of infected and/or mite-parasitized larvae . Individual bee immune mechanisms include phagocytosis, melanization, nodulation, autophagy, as well as signal transduction cascades, including the Jak/STAT, Imd, and Toll pathways, which are activated downstream of pathogen recognition receptor binding of pathogen-associated molecular patterns (PAMPs) (reviewed in ). These immune pathways have been implicated in honey bee antiviral defense, but understanding the mechanism(s) of these responses is an active area of research.
RNA interference (RNAi) plays an important role in insect antiviral defense . RNAi is a sequence-specific, post-transcriptional gene and virus silencing mechanism that is induced by double-stranded RNAs (dsRNAs), which are cleaved by a Dicer endonuclease into small inhibitory RNAs (siRNAs) . These siRNAs serve as sequence-specific guides for the RNA-induced silencing complex (RISC) to locate and destroy cognate RNA molecules, including viral RNAs . The role of RNAi in insect antiviral defense has been best characterized in solitary insects including Drosophila melanogaster, because of the availability of mutant fly lines, and Aedes aegypti and Anopheles gambi mosquitos . Numerous studies indicate that RNAi functions in both honey bee gene regulation [40,41] and antiviral defense (reviewed in ). The most definitive studies demonstrated that virus abundance was reduced in infected bees or larvae that were treated with virus-specific dsRNAs or siRNAs  (reviewed in ). Furthermore, small RNAs isolated from naturally infected honey bees were characterized via Northern blot and sequencing . Colony level studies were also performed, though they focused on the effect of virus-specific dsRNAs on colony metrics (i.e., population and honey production) . The use of virus-specific dsRNAs or siRNAs to treat virus infections is currently being investigated; thus, it is important to better understand their impact on bee physiology . Unlike the Drosophila model , treatment of virus infected bees with non-sequence—specific dsRNAs reduced virus abundance [47,48], suggesting that bees may also have a generalized antiviral immune response that is triggered by dsRNA, akin to the interferon response in mammals (reviewed in [17,45]).
In order to better characterize honey bee antiviral defense mechanisms, several studies examined transcriptional responses to virus infection [47,49–51]. These studies suggest immune pathways including RNAi, Toll, Jak/Stat, Imd, and autophagy play a role in honey bee antiviral defense, though antiviral transcriptional responses varied, likely due to differences in experimental factors. Therefore, additional studies that control for age, inoculum route, and virus are required to discern universal, age-dependent antiviral responses and virus-specific antiviral responses. Similar to other co-evolving host—pathogen relationships, honey bee viruses have likely evolved mechanisms to overcome and/or evade bee immune responses. For example, several honey bee infecting Dicistroviruses (i.e., IAPV, KBV, and ABPV) encode a DvExNPGP motif at the 5′ terminus of their genomes, suggesting these honey bee-infecting viruses encode putative viral suppressors of RNAi similar to the Dicistroviruses Cricket paralysis virus and Drosophila C virus . In addition, viruses may manipulate host gene expression to enhance virus entry, replication, and/or exit. A more complete understanding of honey bee antiviral defense and viral counter-measures may lead to the development of strategies that reduce virus infection in bee colonies.
What Is the Future of Honey Bee Virology?
Honey bee virology is a rapidly growing field currently in its infancy. Researchers are employing advanced colony level tools (e.g., imaging and computational image analysis) and molecular level techniques, including high-throughput sequencing, bisulfite sequencing, and qPCR, to investigate the impact of viruses on bee health at the colony, individual, and cellular levels. Cultured cells, both primary cells and immortalized cell lines, provide the opportunity to study host—pathogen and pathogen—pathogen dynamics while minimizing confounding factors (e.g., additional pre-existing conditions, outbred bee populations) inherent to performing in vivo studies that utilize honey bees obtained from managed colonies in natural settings. Recently, an immortalized honey bee cell line (AmE-711) was utilized for the first time to investigate host—virus and virus—virus interactions in mixed infections . Both in vitro and in vivo experiments determined that IAPV outcompeted other viruses (i.e., SBV, DWV, and BQCV) in mixed infections; although, when KBV was in the initial inoculum, it outcompeted IAPV . To date, there are no infectious honey bee virus clones. Though studies that utilize virus preparations from infected pupae and model viruses are informative, both are associated with the advantages and disadvantages of these systems. Therefore, current efforts to produce infectious clones of honey bee viruses that can be quantified and manipulated in vitro will advance the field.
In conclusion, this is an exciting time in honey bee virology. Important research topics include (1) understanding honey bee antiviral responses, including those triggered by dsRNA and siRNA, at the cellular and molecular levels, and identifying viral counter measures; (2) identifying the most pathogenic virus strains and determining what factors govern their virulence and transmission; (3) investigating the role of the bee microbiome on virus infection and bee health; and (4) determining how synergistic variables, including agrochemical exposure and nutritional stress, impact viral pathogenesis. Further investigation of these and other topics will advance our understanding of bee biology, host—pathogen interactions, colony health, and may lead to the development of strategies that limit colony losses.
Klein AM, Vaissière BE, Cane JH, Steffan-Dewenter I, Cunningham SA, Kremen C, et al. Importance of pollinators in changing landscapes for world crops. P R Soc B. 2007;274: 303–313. doi: 10.1098/rspb.2006.3721.
Gallai N, Salles J-M, Settele J, Vaissière BE. Economic valuation of the vulnerability of world agriculture confronted with pollinator decline. Ecological Economics. 2009;68: 810–821. doi: 10.1016/j.ecolecon.2008.06.014.
California ABO. Almond Almanac 2014. http://www.almonds.com/growers/pollination.
National Agricultural Statistics Service NASS. http://www.nass.usda.gov National Agricultural Statistics Service NASS.
Lee KV, Steinhauer N, Rennich K, Wilson ME, Tarpy DR, Caron DM, et al. A national survey of managed honey bee 2013–2014 annual colony losses in the USA. Apidologie. 2015;46: 292–305. doi: 10.1007/s13592-015-0356-z.
Traynor KS, Rennich K, Forsgren E, Rose R, Pettis J. Multiyear survey targeting disease incidence in US honey bees—Online First—Springer. Apidologie. 2016. doi: 10.1007/s13592-016-0431-0.
Johnson RM, Evans JD, Robinson GE, Berenbaum MR. Changes in transcript abundance relating to colony collapse disorder in honey bees (Apis mellifera). P Natl Acad Sci Usa. 2009;106: 14790–14795. doi: 10.1073/pnas.0906970106.
McMenamin AJ, Genersch E. Honey bee colony losses and associated viruses. Current Opinion in Insect Science. Elsevier Inc; 2015;: 1–9. doi: 10.1016/j.cois.2015.01.015.
van der Zee R, Pisa L, Andonov S, Brodschneider R, Charriere J-D, Chlebo R, et al. Managed honey bee colony losses in Canada, China, Europe, Israel and Turkey, for the winters of 2008–9 and 2009–10. J Apicult Res. 2012;51: 110–114. doi: 10.3896/IBRA.1.51.1.12.
McMenamin AJ, Brutscher LM, Glenny W, Flenniken ML. Abiotic and biotic factors affecting the replication and pathogenicity of bee viruses. Current Opinion in Insect Science. 2016. doi: 10.1016/j.cois.2016.04.009.
Brutscher LM, Daughenbaugh KF, Flenniken ML. Antiviral defense mechanisms in honey bees. Current Opinion in Insect Science. 2015. doi: 10.1016/j.cois.2015.04.016.
Chen YP, Siede R. Honey Bee Viruses. Advances in Virus Research. Elsevier; 2007. pp. 33–80. doi: 10.1016/S0065-3527(07)70002-7.
Genersch E, Ohe von der W, Kaatz H, Schroeder A, Otten C, Büchler R, et al. The German bee monitoring project: a long term study to understand periodically high winter losses of honey bee colonies. Apidologie. EDP Sciences; 2010;41: 332–352. doi: 10.1051/apido/2010014.
Nazzi F, Le Conte Y. Ecology of Varroa destructor, the Major Ectoparasite of the Western Honey Bee, Apis mellifera. Annu Rev Entomol. 2015. doi: 10.1146/annurev-ento-010715-023731.
Yang X, Cox-Foster DL. Impact of an ectoparasite on the immunity and pathology of an invertebrate: evidence for host immunosuppression and viral amplification. P Natl Acad Sci Usa. 2005;102: 7470–7475. doi: 10.1073/pnas.0501860102.
Di Prisco G, Annoscia D, Margiotta M, Ferrara R, Varricchio P, Zanni V, et al. A mutualistic symbiosis between a parasitic mite and a pathogenic virus undermines honey bee immunity and health. P Natl Acad Sci Usa. 2016;113: 3203–3208. doi: 10.1073/pnas.1523515113.
Parmentier L, Smagghe G, de Graaf DC, Meeus I. Varroa destructor Macula-like virus, Lake Sinai virus and other new RNA viruses in wild bumblebee hosts (Bombus pascuorum, Bombus lapidarius and Bombus pratorum). J Invertebr Pathol. 2016;: 6–11. doi: 10.1016/j.jip.2015.12.003.
McMahon DP, Fürst MA, Caspar J, Theodorou P, Brown MJF, Paxton RJ. A sting in the spit: widespread cross-infection of multiple RNA viruses across wild and managed bees. J Anim Ecol. 2015. doi: 10.1111/1365-2656.12345.
Nunes FMF, Ihle KE, Mutti NS, Simões ZLP, Amdam GV. The gene vitellogenin affects microRNA regulation in honey bee (Apis mellifera) fat body and brain. J Exp Biol. 2013. doi: 10.1242/jeb.089243.
Desai SD, Eu Y-J, Whyard S, Currie RW. Reduction in deformed wing virus infection in larval and adult honey bees (Apis mellifera L.) by double-stranded RNA ingestion. Insect Mol Biol. Blackwell Publishing Ltd; 2012;21: 446–455. doi: 10.1111/j.1365-2583.2012.01150.x.


text:
Q: Solving this simple fraction $$-\frac{a}{5}\left(2\times\frac{a^2}{5}\right)^2+5\left(-\frac{a}{5}\right)^3\left(2\times\frac{a^2}{5}\right)
-a^2\left(-\frac{a}{5}\right)\left(2\times\frac{a^2}{5}\right)=0$$
I get
$$\frac{-4a^5}{25}-\frac{2a^5}{125}+\frac{2a^5}{25}$$
Where I multiply the first and third fraction by 5 in the numerator and denominator to get 125 in the denominator
For some reason, I get the wrong answer of
$$-\frac{12a^5}{125}$$
When the answer should be
$$\frac{4a^5}{125}$$
What have I done wrong?

A: we have $$(2\frac{a^2}{5})\left(-\frac{2a^3}{25}-\frac{a^3}{25}+\frac{a^3}{5}\right)$$=$$\frac{2a^2}{5}\cdot \frac{2a^3}{25}=$$ $$\frac{4a^5}{125}=0$$ thus $$a=0$$

A: $$-\frac a5\left(2\times \frac {a^2}5\right)^2=-\frac a5\times \frac {4a^4}{25}=-\frac {4a^5}{125}$$
$$5\left(-\frac a5\right)^3\left(2\times \frac {a^2}5\right)=-\frac {a^3}{25}\left(\frac {2a^2}5\right)=-\frac {2a^5}{125}$$
And $$-a^2\left(-\frac a5\right)\left(2\times \frac {a^2}{5}\right)=\frac {a^3}5\left(2\times\frac {a^2}5\right)=\frac {2a^5}{25}$$

Combining, we get this:$$\color{blue}{-\frac a5\left(\frac {2a^2}5\right)^2}+\color{red}{5\left(-\frac a5\right)^3\left(\frac {2a^2}5\right)}\color{brown}{-a^2\left(-\frac a5\right)\left(\frac {2a^2}5\right)}\\ =\color{blue}{-\frac {4a^5}{125}}\color{red}{-\frac {2a^5}{125}}\color{brown}{+\frac {2a^5}{25}}=-\frac {6a^5}{125}+\frac {2a^5}{25}=\boxed{\frac {4a^5}{125}}$$

For where you went wrong, you put $\frac {-4a^5}{25}$ instead of $-\frac {4a^5}{125}$. Most likely, you forgot to square the denominator of the fraction in $\left(\frac {2a^2}5\right)^2$.

A: You start with an equation but I suppose you are only concerned about the left hand side of the equation.

$$-\frac{a}{5}\left(2\times\frac{a^2}{5}\right)^2+5\left(-\frac{a}{5}\right)^3\left(2\times\frac{a^2}{5}\right)
-a^2\left(-\frac{a}{5}\right)\left(2\times\frac{a^2}{5}\right)\tag{1}$$
  I get $$\frac{-4a^5}{25}-\frac{2a^5}{125}+\frac{2a^5}{25}\tag{2}$$
  (...)
  $$-\frac{12a^5}{125}\tag{3}$$

How to find the error?
Plug in some numbers for the variablee $a$ and compare, e.g $a=5$ in $(1)$ will give
$$-\frac{5}{5}\left(2\times\frac{5^2}{5}\right)^2+5\left(-\frac{5}{5}\right)^3\left(2\times\frac{5^2}{5}\right)
-5^2\left(-\frac{5}{5}\right)\left(2\times\frac{5^2}{5}\right) \\
=-1(10)^2+5(-1)^3(10)-25(-1)(10)\\
=-100-50+250\\=100$$
but $(2)$ will give
$$\frac{-4\cdot 5^5}{25}-\frac{2\cdot 5^5}{125}+\frac{2\cdot  5^5}{25}\\
=-4\cdot 5^3-2\cdot5^2+2\cdot5^3\\
=-500-50+250\\
=300$$
So you introduced an error when you calculated $(2)$ from $(1)$
You can plug in arbitrary numbers for $a$ and use a calculator to verify/disprove your calculation.
If a number you plug in verifies your calculation that does not mean that there is no error in the calculation, maybe other values will disprove the calculation.
But if a number disproves a calculation you can be sure that there is an error in your calculation and try to find it by using the same method for the substeps of your calculation.
So next you check each  term in this step. Ifyou plug in $a=6$ in the first term
$$-\frac{a}{5}\left(2\times\frac{a^2}{5}\right)^2$$
and in the first term of your your result
$$\frac{-4a^5}{25}$$
and use a calculator you will get something like $-248.832$ and $-1244.16$ (You almost always have a calculator. I simply put -6/5*(2*6^2/5)^2 and -4*6^5/25  into Google to get these results).
So there is an error  in transforming this term.
You can often  use such a technique to find errors in your calculations.
If you have access to a CAS like Mathematica you can use this tool to check your intermediate results.