text:
\section{Introduction \& preliminaries} \subsection{Motivation}
Results from past few decades show that the
choice of the quantum analogue of classical random walk is highly non-unique.
Some of the most popular models include discrete coined quantum walk
\cite{aharonov2001quantum}, continuous quantum walk \cite{childs2002example},
Szegedy walk \cite{szegedy2004quantum}, open walk \cite{attal2012open2},
staggered quantum walk \cite{portugal2016staggered}, and quantum stochastic walk
\cite{whitfield2010quantum}. They have found applications in designing PageRank
algorithm \cite{loke2017comparing,sanchez2011quantum,paparo2014quantum}, search
algorithms \cite{childs2004spatial,childs2003exponential,shenvi2003quantum, glos2017vertices},
solving triangle problem \cite{magniez2007quantum}, and describing chemical
reactions \cite{chia2016coherent}. All of them outperform their classical
counterpart at least for some large class of graphs. From the algorithmic
perspective, it is crucial to understand the limiting behaviour of the walks.
This includes the propagation speed, which decides about efficiency of the
search algorithm \cite{domino2017spontaneous}, mixing time and relaxing, which
find applications in PageRank algorithm
\cite{sanchez2011quantum,liu2016continuous,nayak2000quantum}, limit theorem
\cite{konno2005new,sadowski2016central}, and trapping \cite{sadowski2016lively}.

A special kind of continuous quantum evolution is quantum stochastic walk, which
generalizes both classical and quantum walk \cite{whitfield2010quantum}. The
model has been investigated in the context of relaxing property
\cite{sanchez2011quantum,liu2016continuous}, propagation speed
\cite{domino2016properties}, and applications to various physics
\cite{chia2016coherent} and computer science \cite{sanchez2011quantum} problems.
The difference comparing to original continuous-time quantum walk is the
Lindblad operators appearance. The choice of Lindblad operators is highly
nontrivial, and in the case were each one-dimensional subspace corresponds to
different vertex, two model are of particular interest: local environment
interaction and global environment
interaction~\cite{whitfield2010quantum,domino2016properties}.

Local environment interaction case has been extensively used and analysed. The model has been analysed in context of relaxation
\cite{liu2016continuous} and application in PageRank algorithm
\cite{sanchez2011quantum}. In particular, it was show that in the case of
undirected graphs, the QSW is always relaxing \cite{liu2016continuous}. However,
the evolution is decohering, and hence it destroys the ballistic propagation
\cite{bringuier2016central}.

The ballistic propagation was one of the basic motivation for the analysis of quantum walk. Fortunately, the global environment interaction QSW is proved to have
ballistic propagation~\cite{domino2016properties}. However, the global environment
interaction suffers from graph topology change~\cite{domino2017spontaneous}, due
to its character called \emph{moralization}. Thanks to the correction scheme it is possible
to bound the QSW with the global interaction to the digraph structure. Such
evolution, called non-moralizing global environment QSW, is verified numerically
to have at least superdiffusive propagation~\cite{domino2017spontaneous}.

The main contribution of this paper is the description of the limiting properties
of various models of quantum stochastic walks. We describe how the
connectivity of the graph influences the convergence or the relaxation. Numerical analysis suggests, at least
in some cases, that the convergence/relaxation appears on all graphs in the sense
of Erd\H{o}s-Renyi random graph model $\mathcal G(n,p)$. Furthermore our results includes the directed graph preservation analysis for directed graphs in the case of non-moralizing evolution.
The main results are collected in table~\ref{tab:results}

\begin{table}\centering
	\begin{tabular}{lp{3.7cm}p{3.7cm}p{3.7cm}}
		\hline
		& local interaction & global interaction & non-moralizing global interaction \\ \hline\hline
		convergence		 & -- unknown in general case&  -- undirected graphs \newline -- counterexample for digraphs  & -- counterexample for undirected graphs\\\hline
		relaxing  & -- strongly connected digraphs \newline
		-- weakly connected digraphs with one sink in condensation graph  & -- never for undirected graphs \newline -- counterexample for digraphs   &  -- counterexample for undirected graphs \\\hline

	\end{tabular}
	\caption{Main analytical results. Case $\omega\in (0,1]$ is only considered here.}
	\label{tab:results}
\end{table}

The paper is organized as follows. In the following part of this section we
provide some basic definitions concerning graph theory and QSW. In Sec.~\ref{sec:local} we analyse the limiting properties of the local
environment interaction case. In Sec.~\ref{sec:global} and
Sec.~\ref{sec:corrected-global} we analyse the limiting properties of the global
environment interaction case, the standard and the non-moralizing respectively.
We conclude our
results in Sec.~\ref{sec:conclusions}.

\begin{comment}

\subsection{Circulant matrices}
\todo{$\omega$ collision}
Circulant matrix $C$ takes the form
\begin{equation}
C=
\begin{bmatrix}
c_0     & c_{n-1} & \dots  & c_{2} & c_{1}  \\
c_{1} & c_0    & c_{n-1} &         & c_{2}  \\
\vdots  & c_{1}& c_0    & \ddots  & \vdots   \\
c_{n-2}  &        & \ddots & \ddots  & c_{n-1}   \\
c_{n-1}  & c_{n-2} & \dots  & c_{1} & c_0 \\
\end{bmatrix}.
\end{equation}
Throughout this paper we consider the real-valued matrices. One can show, that
the eigenvectors of a circulant matrix have the form
\begin{equation}
\ket{C_j} = \frac{1}{\sqrt{n}}\sum_{i=0}^{n-1} \omega_j^i \ket i
\end{equation}
where $\omega_j = \exp \left(\frac{2 \pi \mathrm{i} j}{n}\right)$. Note that
eigenvectors do not depend on the values $c_0,\dots,c_{n-1}$, and hence
arbitrary two circulant matrices commutes. This will play crucial role in our
latter considerations. The eigenvalues of the circulant
matrices has the form \cite{gray2006toeplitz}
\begin{equation}
\lambda_j = \sum_{i=0}^{n-1} c_{i-n+1}\omega_j^i.
\end{equation}
\end{comment}

\subsection{Graph theory terminology} Let $G=(V,E)$ be a digraph with vertex set
$V$ and arc set $E$. The underlying graph $G^u(V,E^u)$ is an undirected simple
graph, for which every arc from $G$ is replaced with an edge. We say that digraph is
weakly connected if its underlying graph is  connected. We say that digraph is
strongly connected, if for arbitrary $v,w\in V$ there is a directed path from
$v$ to $w$. The strongly connected components are the maximal strongly connected
subgraphs. We call $v$ a sink vertex, if its outdegree (number of outgoing arcs)
is zero. We denote $S(G)$ to be a set of sink vertices.

Let $\mathbb G = (\mathbb V, \mathbb E)$ be condensation of $G=(V,E)$, \emph{i.e.\/}{} a
directed graph constructed as follows. We make a partition $\mathbb V$ of vertex
set $V$ in such a way, that each block forms a maximal strongly connected
component in $G$. Then $(\mathbf v, \mathbf w)\in \mathbb E$ iff there exists
$v\in\mathbf v, w\in\mathbf w$ such that $(v,w)\in E$. In other words there is an
arc from one maximal strongly connected component to the other, if there is at
least one arc (in consistent direction) between their elements. Note that
$\mathbb G$ is directed acyclic graph, hence $S(\mathbb G)\neq\emptyset$.
Furthermore, if $G$ is weakly connected, then for each $\mathbf w\in \mathbb V$
there exists $\mathbf s\in  S(\mathbb G)$ such that there exists directed
path from $\mathbf w$ to $\mathbf s$.

The directed moral graph $G_M=(V,E_M)$ of the directed graph $G=(V,E)$ is defined as
follows. For each $v,v'\in V$ we have $(v,v')\in E_M$ iff $(v,v')\in E$ or there
exists $w\in V$ such that $(v,w),(v' ,w)\in E$. In other words, directed moral graph is
constructed by adding edge between the vertices which have common child. Note
that original moral graph is defined as underlying graph of the directed moral graph~\cite{Cowell1999, StudenyPhD}.

In this work we are using Erd\H{o}s-R\'enyi model of random graphs.
The Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$ is a graph with $n$
vertices such that any pair of vertices is connected with probability
$p$~\cite{erdos1960evolution}.

Throughout this paper we will assume, that graph is at least weakly connected.

\subsection{GKSL master equation and quantum stochastic walks}

To define quantum stochastic walks in general, let us start with the
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation
\cite{kossakowski1972quantum,
lindblad1976generators, gorini1976completely}
\begin{equation}
\frac{\textrm{d}}{\textrm{d}t}\varrho =\mathcal{M} [\varrho]= -\mathrm{i} [H, \varrho] +
\sum_{L\in
	\mathcal{L}} \left(L \varrho
L^\dagger - \frac12 \{L^\dagger L, \varrho\} \right),
\label{eq:GKSL-master-equation}
\end{equation}
where $\{A, B\}$ is the anticommutator and $\mathcal{M}$ is the evolution superoperator.
Here $H$ is the Hamiltonian, which describes the evolution of the closed system,
and $\mathcal{L}$ is the collection of Lindblad operators, which describes the evolution
of the open system. This master equation describes general Markov continuous
evolution of mixed quantum states. Note, that in the case of $\mathcal{L}=\emptyset$ and
$H$ being an adjacency matrix of some graph we recover the original continuous
quantum walk, however on mixed states.

The GKSL master equation was used for defining quantum stochastic walks (QSW),
which are a generalization of both classical random walks and quantum walks
\cite{whitfield2010quantum}. Both $H$ and $\mathcal{L}$ correspond to the graph
structure, however one may verify that at least a choice of Lindblad operators
may be non-unique \cite{whitfield2010quantum, domino2016properties}. Suppose we
have a directed graph $G=(V,E)$. Since Hamiltonian $H$ needs to be hermitian, we
always choose adjacency matrix of the underlying graph graph. In the \emph{local
	environment interaction case} each Lindblad operator corresponds to a single
arc, $ \mathcal{L} = \{c_{(v,w)}\ketbra{w}{v}\colon (v,w)\in E\}$ for $c_{(v,w)}\in
\mathbb C_{\neq 0}$. In the \emph{global environment interaction case} we choose a
single Lindblad operator, which is adjacency matrix of directed graph.

To analyse the impact of Lindbladian part, we add the smoothing parameter
$\omega\in[0,1]$
\begin{equation}
\frac{\textrm{d}}{\textrm{d}t}\varrho =-\mathrm{i} (1-\omega) [H, \varrho] +
\omega \sum_{L\in
	\mathcal{L}} \left(L \varrho
L^\dagger - \frac12 \{L^\dagger L, \varrho\} \right),
\label{eq:GKSL-omega}
\end{equation}
In \cite{bringuier2016central} it was shown, by infinite path graph analysis,
that the local interaction case leads to the classical propagation of the walk.
Oppositely, in the case of the global interaction, the ballistic propagation is
obtained for arbitrary middle value $\omega$. Since for $\omega=0$ we recover
continuous quantum walk (CQW) on closed system, we are particularly interested
in $\omega \in(0,1]$ case.

In \cite{domino2017spontaneous} it has been notes that the global interaction QSW
suffers for the graph topology change and the resulting
process fails to reproduce the structure of the original graph.
The resulting graph, according to which the system is evolving, is the directed moral graph ~\cite{Cowell1999, StudenyPhD}, of the original one.
This effect is called a spontaneous moralization and to prevent this
a correction scheme based on the system enlargement has been proposed~\cite{domino2017spontaneous}. The scheme consist of following steps:
first we combine with each vertex a subspace of the system of dimension
equal to the indegree of the vertex (in the case of source vertices the subspace
is onedimensional). Next, we choose a family of orthogonal matrices for
new Lindblad operator $\tilde L$, which destroys the spontaneous moralization.
The last step is to add a Hamiltonian $\tilde H_{\mathrm{local}}$ acting
locally on the subspaces corresponding to different vertices. The Hamiltonian
$\tilde H$ corresponding to the graphs structure is a 0-1 matrix, for which zero
values coincides with zero values of $\tilde L$. For details we refer the reader
to the original paper. Since in this model we combine each vertex with some
orthogonal subspaces, a natural measurement is the collection of operations
which projects the state onto the subspaces corresponding to the vertices.
We call this model of evolution as \emph{non-moralizing global environment
interaction QSW}, or simply non-moralizing QSW. Similarly to Eq.~\eqref{eq:GKSL-omega}
we add smoothing parameter $\omega$ and the evolution takes the form
\begin{equation}
\frac{\textrm{d}}{\textrm{d}t}\varrho =-\mathrm{i} (1-\omega) [\tilde H, \varrho] +
\omega  \left(-\mathrm{i}[\tilde H_{\mathrm{local}}, \varrho]+ \tilde L \varrho
\tilde L^\dagger - \frac12 \{\tilde L^\dagger \tilde L, \varrho\} \right).
\end{equation}

Numerical simulation of the QSW with non-moralizing global environment interaction is difficult because of enlarging of the system. With the increase of the input graph density, the size and density of output Lindblad operator $\tilde L$ increases rapidly.

Throughout this paper we analyse the limit behaviour of QSW in all of three
mentioned cases: local environment interaction, global environment interaction,
and corrected non-moralizing environment interaction. We analyse the evolution
in context of convergence and relaxation. We say that evolution is convergent, if
for arbitrary initial state $\varrho_0$ there exists stationary state
$\varrho_\infty$ such that $\varrho_t\xrightarrow{t\to\infty}\varrho_\infty$. We
say that evolution is relaxing, if there exists unique stationary state. In GKSL master equation uniqueness of stationary state is equivalent to
relaxing property, see Theorem 1 from \cite{schirmer_stabilizing_2010}.
Similarly one can define convergence and relaxing in the context of probability
distribution of quantum measurement. Note that relaxation implies convergence, but the opposite does not hold in general.

In the case of QSW $H$ and $\mathcal{L}$ do not depend on time. Henceforth, we can solve
the differential equation analytically: if we choose initial state $\varrho_0$,
then
\begin{equation}
\vecc{\varrho_t} =\exp(tF) \vecc{\varrho_0}, \label{eq:evolution}
\end{equation}
where
\begin{equation}
F = -\mathrm{i} \left(H \otimes \1 - \1 \otimes \con H \right) +
\sum_{L\in\mathcal{L}} \left ( L \otimes \con L - \frac{1}{2}\left( L^\dagger L \otimes
\1
+ \1 \otimes L^\dagger  L\right )\right
),
\end{equation}
and $\vecc{\,\cdot\,}$ denotes the vectorization of the matrix (see eg.
\cite{miszczak2011singular}).
Note that the eigenvalues of $F$ implies the
behaviour of the evolution. If there exists purely imaginary nonzero
eigenvalues, the evolution is non-convergent for some initial state $\varrho_0$,
otherwise it is convergent. If the null-space is one-dimensional, then the
evolution is relaxing. Hence our numerical analysis is mostly based on analysing
the spectrum of $F$.

\section{Convergence of local interaction case} \label{sec:local}

The local environment interaction case is relaxing for undirected
graphs~\cite{liu2016continuous} and arbitrary Hamiltonian. The proof is based on
the Spohn theorem~\cite{Spohn1977}, which requires the self-adjointess of the set of Lindblad
operators, hence its applications is limited to the undirected graphs case.
Nevertheless we show, that the result can be extended to strongly connected
digraphs and weakly connected graphs with single sink vertex in $\mathbb G$ graph. Our proofs utilise the Condition 2. and Condition 3. from
\cite{schirmer_stabilizing_2010}, recalled here as Lemma~\ref{lemma:condition2} and \ref{lemma:condition3}. By interior we mean collection of density matrices with full rank.

\begin{lemma}[\cite{schirmer_stabilizing_2010}] \label{lemma:condition2}
Let $\mathcal H$ be a Hilbert space. If there is no proper subspace $S\subsetneq\mathcal H$, that is invariant under all Lindblad generators $L\in\mathcal{L}$  then
the system has a unique steady state in the interior.
\end{lemma}

\begin{lemma}[\cite{schirmer_stabilizing_2010}] \label{lemma:condition3}
	If there do not exists two orthogonal proper subspaces of $\mathcal H$ that are simultaneously invariant under all Lindblad generators $L\in\mathcal{L}$, then the system has unique fixed point, either at the boundary or in the interior.
\end{lemma}

Using the above lemmas we can prove the following.

\begin{theorem} \label{theorem:local-strongly}
Let $G=(V,E)$ be a strongly connected digraph and let $\mathcal{L}=
\{L_{vw}=c_{(v,w)}\ketbra{w}{v}\colon (v,w)\in E\}$ for some $c_{(v,w)}\in \mathbb C_{\neq 0}$. Then
evolution described by Eq.~(\ref{eq:GKSL-master-equation}) is relaxing for
arbitrary Hamiltonian $H$ with stationary state in the
interior.
\end{theorem}
\begin{proof}

Let $\mathcal H$ be a
Hilbert space  spanned by $\{\ket v: v\in V\}$ and $S\neq \{0\}$ be arbitrary
subspace of $\mathcal H$ invariant under $\mathcal{L}$. Furthermore, suppose that $\ket \psi\in S$ is a
nonzero vector and  $v\in V$ is such that $\braket{v}{\psi}\neq 0$.
Since $G$ is strongly connected, there is a directed path
$(v_1=v,v_2,\dots,v_k,w)$ for arbitrary $w\in V$. Then $c_w \ket w =L_{v_kw}
L_{v_{k-1}v_k}\cdots L_{v_2,v_1}\ket{\psi} \in S$ for some $c_w\in\mathbb C_{\neq
	0}$. Since $\{\ket{w} : w\in V \}$ forms a basis of $\mathcal H$, we have $S=\mathcal H$. By
Lemma~\ref{lemma:condition2} the theorem is true.
\end{proof}

\begin{theorem} \label{theorem:local-one-sink}
Let $G=(V,E)$ be a weakly connected digraph such that
$|S(\mathbb{G})|=|\{\mathbf s\}|=1$ and let $\mathcal{L}= \{c_{(v,w)}\ketbra{w}{v}\colon
(v,w)\in E\}$ for some $c_{(v,w)}\in \mathbb C_{\neq 0}$. Then the evolution described by
Eq.~(\ref{eq:GKSL-master-equation}) is relaxing for arbitrary Hamiltonian $H$.
\end{theorem}
\begin{proof}
Suppose $S_1\neq \{0\},S_2\neq \{0\}$ are two  subspaces of $\mathcal H$ and let
$\ket{\psi_1}\in S_1,\ket{\psi_2}\in S_2$. Similarly to method in
Theorem~\ref{theorem:local-strongly} one can show that there exist
$L_1^1,\dots,L_k^1\in \mathcal{L}$ and $L_1^2,\dots,L_{k'}^2\in \mathcal{L}$ such that
$c_w^1\ket{w} = L_k^1L_{k-1}^1\dots L_1^1 \ket{\psi_1}$ and $c_w^2\ket{w} =
L_{k'}^2L_{k'-1}^2\dots L_1^2 \ket{\psi_2}$ for some $w\in \mathbf s$ and
$c_w^1,c_w^2\in \mathbb C_{\neq 0}$. Hence $S_1$ and $S_2$ are not orthogonal and by
Lemma~\ref{lemma:condition3} the theorem holds.
\end{proof}
Note that no information about the graph structure needs to be encoded in the
Hamiltonian---the graph structure is encoded in the $\mathcal{L}$ only.

The remaining class of weakly connected graphs is those for which $|S(\mathbb
G)|>1$. However in this case one can shown that Hamiltonian has impact on limiting
behaviour of the evolution. In the case of $\omega=1$, one can show that the
evolution is equivalent to the classical one. Because of that, there are different
stationary states in each of the sinks. Hence, the evolution is not relaxing.
However, due to de-cohering character of the evolution, it should be
convergent to the state from the sinks subspace.

Similarly convergence is expected in the case of $\omega\in(0,1)$. In
this case, if the Hamiltonian impact is sufficiently large, one can observe the
relaxation of the evolution.  As an example, we analyse a bidirected path with
vertex set $\{-n,-(n-1),\dots,n-1,n\}$, and edge set $E=\{(i,i+1):i<0\} \cup
\{(i-1,i):i>0\}$ (see Fig.~\ref{fig:bidirected-path}). The Hamiltonian is chosen to be
the adjacency matrix of the underlying graph, \emph{i.e.\/}{} the path graph. In
Fig.~\ref{fig:bidirected-path-threshold} the values of $\omega$ for which the evolution is relaxing are presented. One can see that for each $n$
there is $\omega_t$ such that for $\omega < \omega_t$ evolution is relaxing, and
for $\omega>\omega_t$ is not. As the value of $\omega_t$ decreased with $n$, one can observe that for larger graphs the stronger influence of the coherent part is necessary to ensure the relaxing property.

\begin{figure}\centering
 \begin{tikzpicture}[node distance=2cm]
\tikzset{nodeStyle/.style = {circle,draw,minimum size=3.5em}}

\node[nodeStyle] (A)  {$-n$};
\node[] (C) [right of=A] {$\dots$};
\node[nodeStyle] (D) [right of=C] {$-1$};
\node[nodeStyle] (E) [right of=D] {$0$};
\node[nodeStyle] (F) [right of=E] {$1$};
\node[] (G) [right of=F] {$\dots$};
\node[nodeStyle] (I) [right of=G] {$n$};

\tikzset{EdgeStyle/.style   = {->,>=latex}}
\draw[EdgeStyle] (C) to (A);
\draw[EdgeStyle] (D) to (C);
\draw[EdgeStyle] (E) to (D);
\draw[EdgeStyle] (E) to (F);
\draw[EdgeStyle] (F) to (G);
\draw[EdgeStyle] (G) to (I);

\end{tikzpicture}
\caption{Biderected path graph of size $2n+1$.}\label{fig:bidirected-path}
\end{figure}

\begin{figure}\centering
	\includegraphics[width=0.7\textwidth]{bi-directed_segment_dense.pdf}
	\caption{Values of $\omega$ for which the evolution on bidirected path graph is relaxing on local environment interaction evolution. Note that for each $n$ there is a threshold $\omega_t$, before which the evolution is relaxing. Numerical analysis was performed for $n=10,14,\dots,94$ and for $\omega=0.01,0.03,\dots,0.99$.
	}\label{fig:bidirected-path-threshold}
\end{figure}

We have analysed the graphs from Erd\H{o}s-R\'enyi model $\mathcal G(n,\frac{1}{10})$ with
$|S(\mathbb G)|>1$. For each size $n=10,15,\dots,45$ we have tested 200 graphs with values
$\omega\in\{0.05,0.1,\dots,1\}$. We have ignored the graphs for which $|S(\mathbb G)|>1$ was not satisfied or which were not weakly connected. We have not found any example of graph for which the evolution was non-convergent. All graphs yield
convergent evolution for $\omega=1$.  Moreover, for all graphs it was possible to find such $\omega_t$, that for all $\omega\in(0,\omega_t)$ the evolution is relaxing, while for $(\omega_t,1]$ the evolution is convergent.

\begin{figure}
	\centering
 \begin{tikzpicture}[node distance=2cm]
\tikzset{nodeStyle/.style = {circle,draw,minimum size=3.5em}}

\node[nodeStyle] (A)  {};
\node[nodeStyle] (C) [below of=A] {};
\node[nodeStyle] (B) [left of=C] {};
\node[nodeStyle] (D) [right of=C] {};

\tikzset{EdgeStyle/.style   = {->,>=latex}}
\draw[EdgeStyle] (A) to (B);
\draw[EdgeStyle] (A) to (C);
\draw[EdgeStyle] (A) to (D);

\end{tikzpicture}
\caption{Star graph of size 4, which yields non-relaxing evolution for $\omega\in(0,1]$}\label{fig:multi-sink-nonrelaxing-classical}
\end{figure}

However, we were able to identify graphs which are non-relaxing for all $\omega\in(0,1)$. Let us consider a star graph in Fig.~\ref{fig:multi-sink-nonrelaxing-classical}. One can show, that for arbitrary $\omega\in(0,1]$ the evolution operator $F$ consists of at least 4 zero eigenvalues. Hence, the evolution for such graph is never relaxing. Star graphs of bigger size yield similar properties. On th other hand, we were not able to find any graph, which yields relaxing evolution for all $\omega\in(0,1)$.

\section{Convergence of global interaction case} \label{sec:global}

\subsection{Undirected graphs} We start this section with providing the general
result for the commuting operators. The result generalizes the case of quantum
stochastic walk with global interaction for all undirected graphs and for some
directed graphs~\cite{domino2016properties}, including circulant matrices.

\begin{proposition}\label{theorem:commuting}
Let us consider Eq.~\eqref{eq:evolution} in the case of commuting
Lindbladian operators $\mathcal{L}$ and Hamiltonian $H$. Then the evolution
operation is of the form
\begin{equation}
(U\otimes \bar U)\exp (tD_{F_{\omega}})(U\otimes \bar U)^\dagger,
\end{equation}
where
\begin{equation}
D_{F_{\omega}}= -\mathrm{i} (1-\omega) (D_H\otimes \1
-\1\otimes
 D_H) +
\omega \sum_{L\in\mathcal{L}}(D_L\otimes \bar D_L - \frac{1}{2} \bar D_L D_L\otimes
\1-\frac{1}{2} \1
\otimes
 D_L \bar  D_L)  \label{eq:diagonal-commuting}
\end{equation}
is a diagonal matrix. Here we assume that $U$ is a unitary operator and $D_H,D_L$ are
diagonal operators such that $H=UD_HU^\dagger$ and $L=UD_LU^\dagger$.
\end{proposition}
\begin{proof}
The proof comes directly from the eigendecompositions of the operators. Since
all operators commute, it is possible to find common eigendecomposition with the
same unitary matrix. By this we can easily find the result.
\end{proof}
One can note the global interaction quantum stochastic walk on undirected graphs
is a special case of the evolution described in the above theorem. In the walk
model the difference comes from the size of $\mathcal{L}$, where we choose only single
Lindbladian operator. Hence we prove a result concerning undirected graphs.

\begin{theorem} \label{theorem:undirected}
The stationary states in the GKSL master equation evolution for which
$\mathcal{L}=\{H\}$  are precisely the stationary states of the pure continuous quantum
evolution.  The evolution is convergent for $\omega\in(0,1]$, but
not relaxing iff the system size is greater than one.
\end{theorem}
\begin{proof} By the model construction  we have $\mathcal{L}={H}$. Hence the
formula Eq.~\eqref{eq:diagonal-commuting} simplifies to
\begin{equation}
D_{F_{\omega}}=-\mathrm{i} (1-\omega) (D\otimes \1
-\1\otimes
 D) +
\omega (D \otimes D - \frac{1}{2} D^2\otimes
\1-\frac{1}{2} \1
\otimes
 D ^2).
\end{equation}
Here we assume $H=UDU^\dagger$. Since $H$ is hermitian, operator $D$ is a
real-valued diagonal matrix. The diagonal entries of operator $D_{F_{\omega}}$
are eigenvalues which characterize the evolution. We have
\begin{equation}
\begin{split}
\bra {i,j} D_{F_{\omega}}\ket {i,j} &=  -\mathrm{i}(1-\omega) (\bra i D \ket i - \bra
j D\ket j)+\omega ( \bra i D \ket i \bra j D \ket j  -
\frac{1}{2}(\bra{i}D^2\ket i+\bra{j}D^2\ket j)) =\\
&= -\mathrm{i}(1-\omega) (\bra i D
\ket i - \bra j D\ket j)- \frac\omega 2 (\bra i D \ket i - \bra j D\ket j)^2.
\end{split}
\end{equation}

Note $-\mathrm{i} (1-\omega ) (\bra i D \ket i - \bra j D \ket j)$ corresponds to purely Hamiltonian evolution, and hence to continuous quantum walk. Since 0-eigenvalues of $F_\omega$ correspond to 0-eigenvalues of $H$ Hamiltonian part of the system, which furthermore correspond to the stationary states of the continuous-time quantum walk, we obtained the first part of the theorem.

Note that there are no purely imaginary eigenvalues od $F_\omega$. Hence, we have that the evolution is convergent. Since the set of stationary states of continuous-time quantum walk of size $n$ has at least $n$ elements, we obtain that QSW with global interaction is never relaxing.
\end{proof}
Note that the result from the above theorem implies that we can generate the
stationary states from the continuous quantum walk by adding the same
Lindbladian operator.

\begin{remark}
	Global interaction case QSW is convergent, but not relaxing for arbitrary undirected graph with number of vertices greater than one 1 and for arbitrary $\omega \in (0,1]$. Furthermore, the stationary states are precisely those from CQW.
\end{remark}

In the next section we show that Theorem~\ref{theorem:undirected} cannot be
generalized for directed graphs.

\subsection{Directed graphs} In this section we
provide an example of a directed graph for which we do not necessary obtain
a stationary state. It has been proven that the evolution converges for arbitrary
initial state iff all nonzero eigenvalues of $F_{\omega}$ have negative real
part \cite[Theorem 5.4]{rivas2012open}. We found an example of a digraph which do not satisfy
the condition, and provide an exemplary initial state which results in
non-convergent evolution. One should note, that it is possible (and more probable)
to find non-convergent states.

\begin{theorem}
Let us take the evolution for which the only Lindbladian operator is an
adjacency matrix of the directed graph and the Hamiltonian is an adjacency
matrix of the underlying graph. Then there exists a directed graph $G$ and an
initial state $\varrho_0$ for which the evolution is non-convergent for an
arbitrary value of the smoothing parameter $\omega\in(0,1]$.
\end{theorem}
\begin{proof}
\begin{figure}
\centering
\begin{tikzpicture}

\def 5 {8}
\def 2cm {2.5cm}
\def \margin {8}

\foreach 5 in {7,...,0}
{
	\node[draw, circle] (5) at ({360/5 * (5 - 1)}:2cm) {$5$};
	\draw[<->, >=latex] ({360/5 * (5 - 1)+\margin}:2cm)
	arc ({360/5 * (5 - 1)+\margin}:{360/5 * (5)-\margin}:2cm);
}
\draw [<-,>=latex] (0)  edge (2);
\draw [<-,>=latex] (1)  edge (3);
\draw [<-,>=latex] (2)  edge (4);
\draw [<-,>=latex] (3)  edge (5);
\draw [<-,>=latex] (4)  edge (6);
\draw [<-,>=latex] (5)  edge (7);
\draw [<-,>=latex] (6)  edge (0);
\draw [<-,>=latex] (7)  edge (1);

\end{tikzpicture}
\caption{An example of strongly connected directed graph, for which the global interaction case evolution is not convergent.}\label{fig:example-small-directed}
\end{figure}
As an example we choose a circulant graph of size $4k$ for $k>1$ and with extra jump every two vertices. An example for $k=2$ is presented in
Fig.~\ref{fig:example-small-directed}. The graph and its underlying graph are
circulant matrices. Therefore, we can use Eq.~\eqref{eq:diagonal-commuting}
to find out that there exists one eigenvalue of the form $2(1-\omega)\mathrm{i}$ with
corresponding eigenvector $\ket{C_{k}}\overline {\ket{C_{2k}}}$, where
$\overline{(\cdot)}$ denotes the element-wise conjugation and $\ket {C_i}$  is
the $i$-th eigenvector of a circulant matrix of the form
\begin{equation}
\ket {C_i}= \frac{1}{2\sqrt k}\sum_{j=0}^{4k} \exp \left(\frac{2\pi \mathrm{i}
ij}{4k-1}\right) \ket i.
\end{equation}
We need to find a matrix which is not orthogonal to
$\ketbra{C_k}{C_{2k}}$. Our exemplary initial state is
\begin{equation}
\begin{split}
\varrho_0  &= \frac{1}{2}(\ket{C_k}+\ket{C_{2k}})(\bra{C_k}+\bra{C_{2k}}).
\end{split}
\end{equation}
The $\varrho_t$ takes the form
\begin{equation}
\varrho_t = \frac{1}{2}(\ketbra{C_k}{C_k} + \ketbra{C_{2k}}{C_{2k}} +
e^{2\mathrm{i}(1-\omega)t}\ketbra{C_k}{C_{2k}}+ e^{-2\mathrm{i}(1-\omega)t}\ketbra{C_{2k}}{C_k}).
\end{equation}
Since $\varrho_t$ is periodic with period $\frac{\pi}{(1-\omega)}$, we obtain
the result.
\end{proof}
Note, that for different $t$ we can obtain different state in the sense of
possible measurement output. For example we have $\bra 0 \varrho_0\ket 0 =
\frac{1}{2k}$, but at the same time we have $\bra 0 \varrho
(\frac{\pi}{2(1-\omega)}) \ket 0 = 0$.

\begin{remark}
The evolution for which the only Lindbladian operator is an adjacency matrix of
the directed graph and the Hamiltonian is an adjacency matrix of the underlying
graph, does not converge in general, even in the sense of the canonical
measurement probability distribution.
\end{remark}

Circulant graphs provide an infinite collection of directed graphs for which the convergence does not hold. Note that the example used in the proof is strongly connected directed graph. This shows, that the convergence in the local interaction case does not imply the convergence in the global interaction case.

Contrary, it is very difficult to find a directed graph for which the convergence does not hold. We have made numerical analysis for Erd\H{o}s-R\'enyi model $\mathcal{G}(n,0.5)$ for $n=10,15,\dots,45$ and for $\omega=0.05,0.1,\dots,1$. We have guaranteed that graph is weakly connected by ignoring such cases if they appear. For each $n$ we chose 200 graphs and we have not found any graph which is not convergent. At the same time it is possible to find relaxing evolution. We were not able to identify graph properties causing such behaviour. From all 1600 graphs, 1599 were relaxing for all value of $\omega$, and there was one graph of size 10 which was relaxing for $\omega\in\{0.05,\dots,0.95\}$ and convergent for $\omega= 1$. Therefore, relaxing property is statistically common for directed graphs in the global environment interaction case.

\section{Convergence of non-moralizing global interaction case}\label{sec:corrected-global}
\subsection{Non-convergent in the space state}

The non-moralizing model has been introduced in \cite{domino2017spontaneous} as a
provide an similar example of a graph and initial state such that it does not
converge. Similarly we found a digraph, for which an evolution operator
$S_{t,\omega}$ has an imaginary part. Again, it is easy to find a state for
which
the evolution is non-convergent.

\begin{theorem} \label{th:global-nonmoralizing}
Let us take the non-moralizing evolution described in
\cite{domino2017spontaneous}. Then there exists a directed graph $G$ and initial
state $\varrho_0$ for which the evolution is periodic in time for an arbitrary
value of the smoothing parameter $\omega\in(0,1]$.
\end{theorem}

\begin{proof}
\begin{figure}
\centering
\begin{tikzpicture}
\node[draw,circle] (0) {$v_0$};
\def 5 {5}
\def 2cm {2cm}
\def \margin {8}

\def 5 {1}
\node[draw, circle] (5) at ({360/5 * (5 - 1)}:2cm) {$v_5$};
\def 5 {2}
\node[draw, circle] (5) at ({360/5 * (5 - 1)}:2cm) {$v_5$};
\def 5 {3}
\node[draw, circle] (5) at ({360/5 * (5 - 1)}:2cm) {$v_3$};
\def 5 {4}
\node[draw, circle] (5) at ({360/5 * (5 - 1)}:2cm) {$v_4$};
\def 5 {5}
\node[draw, circle] (5) at ({360/5 * (5 - 1)}:2cm) {$v_5$};

\draw [-,>=latex] (0)  edge (1);
\draw [-,>=latex] (0)  edge (2);
\draw [-,>=latex] (0)  edge (3);
\draw [-,>=latex] (0)  edge (4);
\draw [-,>=latex] (0)  edge (5);
\draw [-,>=latex] (5)  edge (4);
\end{tikzpicture}
\caption{An example of graph for which non-moralizing global interaction evolution is not convergent.}\label{fig:example-big-directed}
\end{figure}

Let us take a graph presented in Fig.~\ref{fig:example-big-directed}. Using the
scheme presented in \cite{domino2017spontaneous}, new graph will consist of 5
copies of vertex $v_0$, two copies of vertices $v_4$ and $v_5$, and single copy of
other vertices. As the orthogonal matrices we choose the Fourier matrices and
in order to remove the premature localization  we choose the rotating Hamiltonian $\tilde
H_{\textrm{rot}} $ constructed from the Hamiltonians of the form
\begin{equation}
 \begin{bmatrix}
0 & \mathrm{i} &  &  \\
-\mathrm{i} & \ddots & \ddots   & \\
& \ddots & \ddots & \mathrm{i} & \\
&  & -\mathrm{i} & 0 & \\
\end{bmatrix}.
\end{equation}

Let us choose two eigenvectors of the rotating Hamiltonian
\begin{gather}
\ket{\lambda ^1_{\tilde H_{\textrm{rot}}}} = \frac{1}{2\sqrt 3}\ket{v_0^0}
-\frac{\mathrm{i}}{2}\ket{v_0^1}- \frac{1}{\sqrt 3}\ket{v_0^2} + \frac{\mathrm{i}}{2}
\ket{v_0^3}+\frac{1}{2\sqrt 3}\ket{v_0^4},\\
\ket{\lambda^2_{\tilde H_{\textrm{rot}}}} = \frac{1}{2\sqrt 3}\ket{v_0^0}
-\frac{\mathrm{i}}{2}\ket{v_0^1}- \frac{1}{\sqrt 3}\ket{v_0^2} + \frac{\mathrm{i}}{2}
\ket{v_0^3}+\frac{1}{2\sqrt 3}\ket{v_0^4}.\\
\end{gather}
One can show that the vectors $\ket{\lambda ^1_{\tilde
H_{\textrm{rot}}},\bar\lambda ^1_{\tilde H_{\textrm{rot}}}},\ket{\lambda
^1_{\tilde
H_{\textrm{rot}}},\bar\lambda ^2_{\tilde H_{\textrm{rot}}}},\ket{\lambda
^2_{\tilde
H_{\textrm{rot}}},\bar\lambda ^1_{\tilde H_{\textrm{rot}}}},\ket{\lambda
^2_{\tilde
H_{\textrm{rot}}},\bar\lambda ^2_{\tilde H_{\textrm{rot}}}}$ are the
eigenvectors of the increased evolution operator $\tilde S_{t,\omega}$ for
arbitrary
$\omega\in(0,1]$. Corresponding eigenvalues are respectively $0,-2\mathrm{i}\sqrt
3\omega,2\mathrm{i}\sqrt 3\omega,0 $. Similarly to the example presented in the
previous
section, the
state
\begin{equation}
\tilde \varrho_0 = \frac{1}{2}(\ket{\lambda ^1_{\tilde
H_{\textrm{rot}}}}+\ket{\lambda ^2_{\tilde
H_{\textrm{rot}}}})(\bra{\lambda ^1_{\tilde
H_{\textrm{rot}}}}+\bra{\lambda ^2_{\tilde
H_{\textrm{rot}}}})
\end{equation}
is the required initial state. The state after time $t$ takes the form
\begin{equation}
\tilde \varrho_t = \frac{1}{2}(\ket{\lambda ^1_{\tilde
H_{\textrm{rot}}}}\bra{\lambda ^1_{\tilde
H_{\textrm{rot}}}}+e^{-2\mathrm{i} t\sqrt{3}\omega}\ket{\lambda ^1_{\tilde
H_{\textrm{rot}}}}\bra{\lambda ^2_{\tilde
H_{\textrm{rot}}}}+e^{2\mathrm{i} t\sqrt{3}\omega}\ket{\lambda ^2_{\tilde
H_{\textrm{rot}}}}\bra{\lambda ^1_{\tilde
H_{\textrm{rot}}}}+\ket{\lambda ^2_{\tilde
H_{\textrm{rot}}}}\bra{\lambda ^2_{\tilde
H_{\textrm{rot}}}}).
\end{equation}
The function $\tilde\varrho_t$ is periodic with period
$\frac{\pi}{\sqrt3\omega}$, hence we obtained the result.
\end{proof}

\subsection{Convergence in the sense of the measurement}

The example from Theorem~\ref{th:global-nonmoralizing}, the probability
distribution obtained from the measurement in the canonical basis of the system
changes in time. However, when we analyse the canonical measurement from
\cite{domino2017spontaneous}, where as the measurement operators we choose the
projections onto the subspaces corresponding to different vertices, we can
observe that the probability does not change---in this case the probability
of measuring  vertex $v_0$ for each time point is one.

We have performed numerical analysis for graphs of size $n=9$ from Erd\H{o}s-R\'enyi $\mathcal G(n,0.5)$ model for $\omega\in\{0,0.1,\ldots,1\}$. We have checked 200 random graphs and the non-moralizing QSW on each of them was convergent.

\begin{conjecture}
Let us choose the non-moralizing evolution model. Let $\Pi(\varrho_0,t)$
denotes
the probability distribution of canonical
measurement onto the subspaces of
vertices in time $t$ with the initial state $\varrho_0$. Then for arbitrary
$\varrho_0$ there exists probability distribution $\Pi_\infty$ such that
\begin{equation}
\lim_{t\to\infty}\Pi(\varrho_0,t) = \Pi_\infty.
\end{equation}
\end{conjecture}

Note, that the probability distribution may be nonunique. To see that let us
analyse the graph presented in
Fig.~\ref{fig:big-evolution-different-measurements}. We choose
$\omega=\frac{1}{2}$ and two initial states
$\tilde\varrho_0=\ketbra{v_6^0}{v_6^0}$ and
$\tilde\varrho_1=\ketbra{v_7^0}{v_7^0}$. We have found the limiting probability
distribution
\begin{gather}
\Pi_0 \coloneqq\lim\limits_{t\to\infty}\Pi(\varrho_0,t), \\
\Pi_1 \coloneqq\lim\limits_{t\to\infty}\Pi(\varrho_1,t).
\end{gather}
The probability distributions differs, for example $\Pi_0(v_0) = 0.666616$ and
$\Pi_1(v_0)=0.11897$.

\begin{figure}
\centering

 \begin{tikzpicture}[node distance=2cm]
  \tikzset{nodeStyle/.style = {circle,draw,minimum size=2.5em}}

  \node[nodeStyle] (A)  {$v_7$};
  \node[nodeStyle] (D) [below right of=A] {$v_2$};
  \node[nodeStyle] (B) [above right of=D] {$v_4$};
  \node[nodeStyle] (E) [below right of=B] {$v_1$};
  \node[nodeStyle] (C) [above right of=E] {$v_5$};
  \node[nodeStyle] (F) [below of=D] {$v_6$};
  \node[nodeStyle] (G) [below of=E] {$v_3$};

  \tikzset{EdgeStyle/.style   = {}}
  \draw[EdgeStyle] (A) to (D);
  \draw[EdgeStyle] (D) to (B);
  \draw[EdgeStyle] (B) to (E);
  \draw[EdgeStyle] (C) to (E);
  \draw[EdgeStyle] (D) to (F);
  \draw[EdgeStyle] (E) to (G);
  \draw[EdgeStyle] (D) to (E);
  \draw[EdgeStyle] (F) to (E);
  \draw[EdgeStyle] (D) to (G);

  \end{tikzpicture}

\caption{Graph for which there exists two different stationary states in the
sense of the natural measurement. The states can be obtained by starting in
vertices $v_7$ and $v_6$.}\label{fig:big-evolution-different-measurements}
\end{figure}

\subsection{The digraph structure observance}
\label{sec:digraph-structure-observance}

In \cite{domino2017spontaneous} it was
suggested, that the non-moralizing global environment interaction QSW can be
applied for modelling the walk on directed graphs. Moreover, an example
suggesting that the original global interaction evolution, where Lindbladian
operator is an adjacency matrix of the directed graph, does not preserve the
digraph structure have been provided. Let us take
the graph from Fig.~\ref{fig:intuitive-graph} and let us analyse the graph
without the Hamiltonian. One can find that
\begin{equation}
\frac{1}{4}(\ket{v_1}-\ket{v_2})(\bra{v_1}-\bra{v_2}) +
\frac{1}{2}\ketbra{v_3}{v_3}\label{eq:unintuitive}
\end{equation}
is proper stationary state. There is a nonzero probability of measuring the
state in vertex $v_1$ and $v_2$. Thus, one can see that the superoperator constructed
using a directed graphs can lead to the evolution on some other structure, namely the
directed moral graphs. This behaviour, expected in the case when the Hamiltonian part is present,
is surprising in the pure Lindbladian case.

For the purpose of quantifying this behaviour we introduce the following property
of the evolution on directed graphs.

\begin{property}[Digraph structure observance]
	Let us assume that for an arbitrary vertex there is path to
	some sink vertex. We say that the evolution on a directed graphs
	has \emph{digraph structure observance} property if an arbitrary
	initial state converges to the state spanned by vectors corresponding
	to the sink vertices from the condensation of the graph.
\end{property}

We require that the evolution on directed graph, at least for Lindbladian part,
should have the digraph structure observance property.

One should note that digraph structure observance can be quantified for any directed
graph using additional sink vertices attached to the orginal vertices.

In the case of the example in Fig.~\ref{fig:lindbladWalkGraphExample}
arbitrary state should converge to the state $\ketbra{v_3}{v_3}$.

\begin{figure}
\centering
\subfigure[\label{fig:intuitive-graph}]{\begin{minipage}[b]{.35\linewidth}
\centering
 \begin{tikzpicture}[node distance=2cm]
  \tikzset{nodeStyle/.style = {circle,draw,minimum size=2.5em}}

  \node[nodeStyle] (A)  {$v_1$};
  \node[nodeStyle] (C) [below right of=A] {$v_3$};
  \node[nodeStyle] (B) [above right of=C] {$v_2$};

  \tikzset{EdgeStyle/.style   = {->,>=latex}}
  \draw[EdgeStyle] (A) to (C);
  \draw[EdgeStyle] (B) to (C);

  \end{tikzpicture}

  \end{minipage}}
 \hspace{1cm}
\subfigure[\label{fig:true-graph}]{\begin{minipage}[b]{.35\linewidth}
\centering
 \begin{tikzpicture}[node distance=2cm]
  \tikzset{nodeStyle/.style = {circle,draw,minimum size=2.5em}}

  \node[nodeStyle] (A)  {$v_1$};
  \node[nodeStyle] (C) [below right of=A] {$v_3$};
  \node[nodeStyle] (B) [above right of=C] {$v_2$};

  \tikzset{EdgeStyle/.style   = {->,>=latex}}
  \draw[EdgeStyle] (A) to (C);
  \draw[EdgeStyle] (B) to (C);

  \tikzset{additionalEdgeStyle/.style = {dashed}}
  \draw[additionalEdgeStyle] (A) -- (B);
  \end{tikzpicture}

  \end{minipage}}

    \caption{Visualisation of a directed graph \subref{fig:intuitive-graph}
    and
    its spontaneous moralization
    \subref{fig:true-graph}.
    }\label{fig:lindbladWalkGraphExample}
\end{figure}

The unintuitive stationary state in Eq.~\eqref{eq:unintuitive} comes from the
spontaneous moralization of the graph \cite{domino2017spontaneous}. Since the
moralization of the graph was corrected, it is necessary to check whether the
improper stationary state still occurs in the non-moralizing evolution. This
can be achieved by analysing the stationary states.

The numerical analysis was performed as follows. Let $S$ be a set of all sink
vertices. We start in some vertex with nonzero outdegree. We determine the state
$\varrho_\infty$ for large time value and we verified numerically, whether it is
close to stationary state in the sense of probability distribution of
measurement. Then we compute the cumulated probability of measuring the state in
the sink vertices
\begin{equation}
p_{S}(\varrho_\infty) = \sum_{v\in \bigcup S(\mathbb G)} \bra{v}\varrho_\infty\ket v
\end{equation}
and the second moment of the distance from the sink vertices
\begin{equation}
\mu_{S}(\varrho_\infty) = \sum_{v\in V} d^2(v,S) \bra v
\varrho_\infty \ket v,
\end{equation}
where $d(v,S)$ is the length of the shortest path from $v$ to closest sink
vertex, \emph{i.e.\/}{} $d(v,S)=\min_{s\in \bigcup S(\mathbb G)}d(v,s)$. We say, that the
greater the value of $p_{S}$ and the lower the value of $\mu_{S}$ are, the more
the evolution preserves the graph.

We have analysed the evolution for
$\omega\in(0,1]$. For the purpose of our analysis we have selected four types of
graphs, namely: a path graph, the Petersen graph, the Apollonian graph, and the
Sierpinski triangle. We choose the orientation of the graphs such that each
vertex is either a sink vertex, or there is a path from it to some sink vertex.

The obtained results are presented in Fig.~\ref{fig:model-check}. One can see
that the larger the value of $\omega$ is, the more probability cumulates in the
close neighbourhood of the sink vertices. In these examples for
$\omega\in[0.7,1]$,  $\mu_\textrm{S}$ and $p_\textrm{S}$ respectively decreases
and grows with $\omega$. One can also notice that in the $\omega\to1$ limit the
$p_{\textrm{S}}$ converge to one and $\mu_\textrm{S}$ vanishes. The above
analysis suggests that when $\omega=1$, the directed graph structure is fully
preserved.

We have done further analysis for Erd\H{o}s-R\'enyi graphs $\mathcal G(n,0.2)$ with $n=9,11,12$, for each size 500 samples. In all of
these graphs for $\omega=1$ we have $p_S=1$ and $\mu_S=0$, which suggests that
at least for this extreme value of $\omega$ the digraph structure is observed.

Furthermore, we have searched for the $\omega_0$, for which for all
$\omega>\omega_0$ both measures $p_S$ and $\mu_S$ were monotonic in $\omega$. We were selecting random graphs $\mathcal G(n,0.2)$ with $n=9,11,12$ and next we were calculating $p_S$ and $\mu_S$ for $\omega$ from 1 to 0 with step $-0.02$. When $p_S$ started increasing or $\mu_S$ started decreasing we stopped the calculations and save $\omega_0$.
The statistics of the threshold $\omega_0$ is presented in
Fig.~\ref{fig:omega-treshold}. We have found no graph, for which $\omega_0>0.7$. This may suggest that for $\omega>0.7$ structure of the directed graph is preserved at least for given sizes of the digraphs.However, to provide more information about this behaviour the more detailed analysis is necessary.

\begin{figure}[h!]
	\includegraphics[scale=1]{histogram.pdf}
	\caption{Histogram presents amount of graphs $\mathcal G(n,0.2)$ with $n=9,11,12$ having particular $\omega_0$.}
	\label{fig:omega-treshold}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{sinks.pdf}

    \caption{Top plot shows the second moment of the distance from the vertex to
	the closest sink vertex. Bottom plot shows the probability of measuring the
	only single sink vertex for large time graphs. Sierpinski triangle has 15
	vertices, Apollonian graph has 12 vertices and length of path graph equals
	10.
}\label{fig:model-check}
\end{figure}

\section{Concluding remarks}\label{sec:conclusions} In this article we analysed three cases of QSW:
with local interaction, with global interaction and with non-moralizing global
interaction. As for local interaction case our results generalize the results
from \cite{liu2016continuous}. We prove that the evolution is relaxing
in the case of undirected graphs. We also show that the result can be extended to
arbitrary strongly connected digraph and for weakly connected digraphs with one
sink in the condensation graph. Furthermore, we show that for strongly connected digraphs the
stationary state is located in the interior. At the same time we provide a counterexample demonstrating that the result cannot be extended to all directed graphs.

Global interaction case QSW is convergent, but never relaxing for arbitrary
undirected graph with number of vertices greater than one 1 and for arbitrary
$\omega \in (0,1]$. Furthermore the stationary states are precisely does from
CQW. We show by example, that the result cannot be extended into arbitrary
directed graphs. Surprisingly, applying the Hamiltonian may help relax the
evolution.

In the non-moralizing global interaction case we provide examples demonstrating
that the evolution does not need to be relaxing, or even convergent, even for undirected graphs.
We also  give an example of evolution, for which the evolution is not relaxing,
even in the sense of canonical measurement. However, we conjecture by numerical
analysis, that the evolution is convergent in the sense of the canonical measurement.

For the purpose of analysing digraph structure observance we have introduced sink observance property. This property can be quantified by
analysing the probability of measuring the state in the sink vertex and its
neighbourhood. We argue that the bigger the probability
of measuring the state in sink vertex or its closes neighbourhood, the better
the structure observance.
Since the Lindblad operators corresponds to the directed graph structure, we
expect, that for $\omega$ close to $1$ the probability of measuring the sink
vertex is one. For the global interaction case the digraph structure is not
preserved, even for a very simple example. Fortunately, the digraph observance is
recovered in the non-moralizing global interaction case.

\section*{Acknowledgements} This work has been supported by the Polish National Science
Centre under project number 2011/03/D/ST6/00413.
\bibliographystyle{ieeetr}


text:
Adidas and McDonald's help Don Valley inject £1.5m into Sheffield
12 June 2012 by Tom Hall
Don Valley Stadium has generated £1.5m for Sheffield's economy in the last year, with bookings from corporates including Adidas and McDonald's.
Adidas and McDonalds help Don Valley events generate £1.5m for Sheffield
A new study has shown that the Sheffield International Venues (SIV) facility attracted more than 125,000 visitors from outside the city to the stadium in the 12 months to May 2012
Around 200 commercial events were hosted at the stadium, with Adidas and McDonald's among the corporates hiring the space.
A Royal College of Psychiatry event, the National Bonsai exhibition, property auctions, and team-building days are among the highlights of the past year.
Don Valley also staged the biggest Sheffield Half Marathon on record with nearly 6,000 runners starting and finishing the race at the stadium.
Pat Smith, general manager of Don Valley, said: "It has been another great year. Key to this success has been the adaptability and flexibility of the venue. From national exams to concerts and film shoots to team building, every day is busy with different activities."
Don Valley will host 'Sheftival' on 4 and 5 August to celebrate London 2012. It is being billed as the city's biggest-ever festival of sport and culture and is expected to attract more than 50,000 people to the city's Olympics party.

redpajama_set_name: RedPajamaCommonCrawl


text:
APA > Official Guidelines > List of Individual Standards > (BS) EN 12871: 2013 (updated)
Section Contents
Updates: Standards & Regulations
Moisture Content: Use & Service Classes
(BS) EN 300: 2019
(BS) CEN/TR 14823 : 2003
(BS) EN 12114 : 2000
(BS) EN 12664
(BS) EN 310
(BS) EN 354:2003
(BS) EN ISO 12572:2016
(BS) EN 313-1
(BS) EN 326-2 (updated)
(BS) EN 335-2013
(BS) EN 350: 2016 (updated)
(BS) EN 636 (updated)
(BS) EN 1058
(BS) EN 1087-1
(BS) EN 1156 (updated)
(BS) EN 12369-1
(BS) EN 16733: 2016
EuroCode 5 ((BS) EN 1995-1-1:2004+A2: 2014)
(BS) EN 12871: 2013 (updated)
(BS) EN 12369-2 (updated)
(BS) EN 13986 +A1 2015 (updated)
(BS) EN 15804:2012+A2:2019 (updated)
(BS) EN ISO 12460-3:2015
(BS) EN ISO 12460-5: 2015
PD CEN/TR 12872: 2014
DDCEN/TS 1099
Regulations for Construction Products
Basic Requirements, Construction Works
Standards for US Wood-Based Panels
US Product Standard PS 1-19
Wood-based panels – Determination of performance characteristics for load bearing panels for use in floors, walls and roofs
This standard provides methods for determining performance characteristics for concentrated loads and impact on wood-based panels used for
structural floor and roof decking and
wall sheathing
Additional information is now contained in Annex B, giving proposals for national performance requirements. These should be applied where there are no national requirements. EN 12871 is referenced in the harmonised European standard for wood-based panels, EN 13986.
The following modifications have been made to the original EN 12871:2010:
Examples of national performance requirements are given in the new section, Annex B
Annex B.2.3 includes a new reduction factor for ultimate limit state
New info on soft body impact tests: three impact classes have been defined relating to performance. These can be linked to load categories as suggested in Annex B3
Information on testing of racking behaviour or uniformly distributed loads are not covered in this standard – refer to EN 594 (not given here) or EN 1995-1-1 (Eurocode 5) respectively
The option to increase the span based on the roof slope is included in Annex B.2.4
This revision now incorporates the evaluation methods from EN 1058
The load pad used for point load testing has changed and is now 50 mm x 50 mm in all cases, see Annex A
Panel product specification standards and technical classes for structural decking (floors/roofs) and wall sheathing
Technical Class
Relevant European Standards
OSB/2
EN 636-1
LVL-1
EN 1195 applies to floors but not roofs. However, the test procedure in EN 1195 is also relevant to roofing provided that the apparatus is appropriately modified. Annex A (normative) of EN 12871 gives a modification to EN 1195 to cover this issue.


Canowindra Produce Co Pty Ltd is an Australian owned and operated company based in Canowindra and its surrounding districts. Canowindra Produce has been operating since the early 1930's and is a family owned and run business. Over the many years in operation Canowindra Produce has established very strong ties and successful relationships with many grain growers and major participants within the Australian grain industry. This knowledge ensures we have a good supply of quality Hay and Grain to fulfil market demand and provide the best possible service to our farmers and customers on every level.
We source the highest quality products from all over Canowindra and its surrounding districts (Canowindra, Gooloogong, Titchbourne, Forbes, Eugowra, Nanami). We also have a 'Fodder Bank' for all types of Hay and Grain, this enables you to purchase what you require at a fair price for yourself and the farmer all year round.
We provide farmers with seeds for planting and fertilisers along with free agronomic advice. It enables us to evolve with the farmer throughout the duration of their crop from sowing and growing to harvesting and marketing. It enables us to source the right type of products for your needs.
At Canowindra Produce we don't mind if you want to use your own branded bags when we are making up your order.
Authorised distributers for Auswest, MSM Milling & Incitec Fertilisers.


Lee Stetson, left, as naturalist and poet John Muir and Joe Wiegand as President Teddy Roosevelt appear in a scene from "National Parks Adventure 3D." Now showing at the Imax Theater, the film re-creates the camping trip in Yosemite Valley that led to the creation of the national park system 100 years ago. Wiegand will visit today for a screening that includes a program with him in character.
› What: Joe Wiegand's Teddy Roosevelt Show.
› When: 6-7:30 p.m. today, April 14.
› Where: Imax Theater, 201 Chestnut St.
› Admission: $8 Tennessee Aquarium members, $11.95 nonmembers.
Joe Wiegand talks about his career portrayals of Teddy Roosevelt and his connections to Chattanooga in an interview.
Joe Wiegand, whose cross-country portrayals of Teddy Roosevelt landed him a film role, will be in Chattanooga today, April 14, for an appearance at the Imax Theater, which is showing the movie he stars in.
Wiegand will introduce a special screening of "National Parks Adventure 3D," then deliver a rousing reprisal of Roosevelt afterward. Wiegand plays the 26th president in the film, which offers an overview of the 100-year history of national parks and Roosevelt's role in protecting what would become 230 million acres of public lands.
Audiences will relive the famous three-day camping trip Roosevelt spent in Yosemite with naturalist and poet John Muir. That 1903 expedition helped change the course of history, fueling Roosevelt's determination to protect the "temples of nature" for future generations.
As part of his Teddy Roosevelt Show, guests can pose for photos with Wiegand in character, be entertained by Roosevelt's classic wisdom, learn presidential history and discover the roots of the park system.
Wiegand began his portrayal of Roosevelt in 2004 at the pinnacle of a 25-year career in public policy and political campaigns. He began his travels across America with the show in 2008, a year that marked Roosevelt's 150th birthday and the 100th anniversary of the final full year of his presidency.
Wiegand has delivered performances at the White House, in all 50 states and on the History Channel. He was recommended to the filmmakers by his friend Lee Stetson, who plays Muir in the film.
"Clambering atop Glacier Point and camping among the great sequoias of Mariposa Grove was a transformational experience for me as an actor and an historian," he says of the filming.
Wiegand's Teddy Roosevelt Show has been praised for its authenticity and energy. In it, he relates stories of Roosevelt's childhood in New York City, "an asthmatic boy with a passion for nature and all its creatures and critters," says Wiegand.
"I'll relate a youth spent in outdoor adventure and natural science and a young adulthood overcoming great personal tragedy. I'll tell stories from the Northwoods of Maine to the Badlands of the Dakota Territory. I'll answer any questions and be happy to pose for photos. Finally, the children will surely enjoy the stories about Mrs. Roosevelt and me rearing six children at the White House. Mrs. Roosevelt said she reared seven children there, and that I was her worstest and most terrible child."
Wiegand is a 1987 political science graduate of the University of the South in Sewanee, Tenn. His Chattanooga visit comes just before National Park Week, April 16-24.


Free printable pdf grammar worksheets quizzes and games from a to z for. 1000 ideas about english grammar worksheets on pinterest free printable pdf quizzes and games from a to z for. English grammar worksheets free pdf ebook download from larisa 72. Elementary grammar worksheets pdf english learning pinterest pdf. English grammar worksheets free pdf ebook download from larisa school of language nikolaev ukraine over one hundred pages.


Joris represents lenders and corporate borrowers on a wide variety of cross-border finance transactions with an emphasis on asset-based lending. He also has been involved in the sale and purchase of non-performing loans and the issuance of bonds.
Besides, Joris is recognised in the market as a Fin-tech expert and advises both incumbents and Fin-techs on a wide array of regulatory matters with a focus on payments and online lending.


text:
Squad adds Canadian Armed Forces, new maps, weapons and vehicles
By:Semir Omerovic - Ghost
Published: 16:09, 27 July 2019
Offworld Industries
Offworld Industries' realistic tactical shooter Squad has introduced Canadian Armed Forces as a playable faction in the latest update. Furthermore, the update also adds two new maps, several weapons, three new vehicles and much more.
Squad, the early access tactical shooter from the developer and publisher Offworld Industries' has received a major content addition in the latest update. Squad's expansion will add the Canadian Armed Forces faction, as well as additional weapons, vehicles, and more.
Canadian Armed Forces faction began life as a community mod but was later picked up by the Offworld Industries team to become the first full expansion available as a free update for all Squad owners.
Offworld Industries Director of Customer Experience, Dustin Ross, said that the team immediately recognized the quality of the mod and it was a no-brainer to support the creators so they can continue to create similar content. "We knew the moment we saw Canadian Armed Forces that it was going to be something special. It was only natural for us to reach out and give them the support they needed to continue creating quality work," Ross said.
Furthermore, the update also brings two Canadian maps, Quebec and Nanisivik, featuring snowy environments built for intense combined-arms combat. You can take a closer look at these map in the trailer above.
As for the weapons, we're looking at a Canadian variant of the M203 grenade launcher, C7A2 assault rifle with 3.4x optic and finally, an M16A3 equivalent The C7 Colt Canada version of the Armalite AR-15.
Also, you'll get to use C9A3 carbine with C79A2 scope or EoTech Holosight. And the last but not least, there's a new bolt-action medium-range sniper C14 Timberwolf and M2 Gustaf Recoilless rifle.
The vehicles which are also included in the update are the new main battle tank Leopard 2A6M, the Kodiak LAV 6.0 and the M-Gator Logistics MATV, a light utility vehicle.
You can find more details about the latest Squad update on the official website .
Squad is free to play on Steam until Sunday, 28 July 2019 1 PM PDT.
Tactical shooter Squad is free to play for a limited time
Hardcore shooter Squad from the developer Offworld Industries is free to play on Steam until 18 November 2019. The game is also available at a discounted price during the free weekend.
The Division 2 open beta has been kind of extended by 24 hours
Ubisoft initially stated that The Division 2 open beta would run from 01 to 04 March 2019, then announced a 24-hour extension and finally corrected themselves, saying that it was always the intention to keep the beta running until 05 March.
Fortnite wrecks PUBG in March, becomes biggest f2p console game
Bean counters at Superdata have revealed that Fortnite: Battle Royale has officially become the biggest free-to-play console game of all time, dismantling PUBG in the process. Of course, many would argue that this outcome was inevitable.
Anthem will launch with raid matchmaking according to Mark Darrah
Anthem is similar to Destiny in many ways - it's a looter shooter in a sci-fi setting, has a greedy publisher, will be a microtransaction machine and so on. One major difference is that BioWare will give it raid matchmaking from get go.
Respawn were relieved folks didn't believe Apex Legends leak
Today we have a different kind of leak story as one Reddit user posted a reminder of the Apex Legends map leak from almost a year ago, which nobody believed at the time and even Respawn said they were relieved to see folks blow it off.
Fortnite's comet might actually flatten Dusty Depot (Updated)
Speculation about the impact point of Fortnite's comet has been going on for a month now and while everyone thought it's going to flatten Tilted Towers, there are clues bubbling to the surface that it may be headed for Dusty Depot instead.
Epic tease superhero themed Season 4 for Fortnite: Battle Royale
Epic seem to be aiming for a superhero flavoured Season 4 to their massively popular Fortnite Battle Royale. After teasing the Flash looking fellow, the company rolled out another Season 4 poster, this time with a Wonder Woman lookalike.
Apex Legends Season 2 has officially begun, patch notes are in
Respawn have officially started the long-awaited Season 2 of Apex Legends and in case you haven't been following, we're getting a new legend called Wattson, Kings Canyon changes, Flyers and Leviathans, the L-Star EMG, and a bunch more.
Battlefield V Trial by Fire update introduces first Elites
Battlefield V Trial by Fire update, scheduled to drop on Tuesday - 30 April 2019 is bringing the first two Elites named Seamus and Ernst. Elites are Battlefield V's unique looking soldiers with special voices, clothes and melee weapons.
Anthem remains in first place in UK physical sales chart
Much has been said about EA and BioWare's looter shooter Anthem but if the physical sales are anything to go by - it's not doing half bad as it managed to hold onto first place in spite of three newcomers to UK's top 10 all-formats chart.


A very attractive shrub with ovate purple-red leaves tinged with pink and cream on spiny stems. Its pale yellow flowers in spring are tinged with red and followed by dark red glossy berries. In autumn the leaves turn a stunning scarlet red before dropping for the winter. A bright addition to the garden, happy in most sunny or part shaded position. Easy, Hardy. Height up to 1 metre. Can be used as a low hedge, or a specimen shrub. Works well in cottage, formal or coastal gardens. Grows in any fertile, well drained soils.
Add good quality compost to existing soil when planting, top mulch in the autumn. Prune in late spring or autumn if required.
Just to let you know my shrubs arrived yesterday and while one container was severely damaged, when I opened it up, the shrubs were not at all damaged and I am really pleased with all of them so far. I am really looking forward to completing my woodland and shrubbery garden this year. Once my shrubs begin to grow, I will send you some photos later on. Meanwhile many thanks for your excellent service and healthy plants. I may well be ordering more from you later.


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Snapchat 'selfies': The case of disappearing data Jennifer Charteris, Sue Gregory, Yvonne Masters University of New England, Armidale, Australia Little has been written about the impact of ephemeral messaging technologies such as Snapchat, Wickr and iDelete on learner identities. The authors explore how disappearing social media may enable young people to take up a range of discourses and... This paper argues that collectively these discourses represent a model of "edu-business" in English language teaching that reconciles the dichotomies between nationalism and post-nationalism, individualism and collectivism and public and private interests in a neoliberal world.
THE MAKING OF NEOLIBERAL TURKEY Download The Making Of Neoliberal Turkey ebook PDF or Read Online books in PDF, EPUB, and Mobi Format. Click Download or Read Online button to THE MAKING OF NEOLIBERAL TURKEY book pdf for free now.


Windsor Hall's tour guide coordinator, Kristin, took Bob for the afternoon. I was so happy for someone to want to take him out; he's been so good lately yet I haven't had work for him. He was perfectly suited for today's outing which included both running and meeting students. I was a little anxious about sending him on site without me but I was not surprised that he did well.
Kristin writes, "we ran for an hour and except for the occasional squirrel he was very good. After that we went to Windsor and he was very sweet with the kids especially the manager's two year old. After that we walked around West Lafayette for 2ish hours and ran a bit more. Overall it was a blast, and I would love to take him again."


Beginning with an incessantly ringing cell phone in a quiet café, a stranger at the next table who has had enough, and a dead man with a lot of loose ends, this wildly imaginative comedy by MacArthur "Genius" Grant recipient, Sarah Ruhl, examines how we memorialize the dead - and how remembering changes us.
It is the odyssey of a woman forced to confront her own assumptions about morality, redemption, and the need to connect in a technologically obsessed world.
Presented In association with Purdue North Central. Directed by Bethany Lee.


Coimbra. There's a song that says Coimbra is more beautiful when you say goodbye to the city. As it is an old city in the centre of Portugal, it's absolutely charming and so it's always beautiful. Everytime no just on the goodbye.
I may be a little suspect to talk about it because i grew up in Coimbra, i did my Law Degree and my Master there in one of the oldest universities of Europe and i've lived there for the past 22 years until i moved to Oporto (my dad and my cat still live there), but truth is Coimbra breathes history and beautiful corners.
The restaurant has a cozy concept of tapas, so you can stay there with your friends or your love all night long eating and talking.
So, when some years ago this restaurant Tapa nas Costas opened in the downtown i tried it with curiosity. And loved it. I mean, it has been one of my favorite restaurants since then in Coimbra. Plus, it was one of the first places where me and my boyfriend had dinner, and where he gave me a sunflowers bouquet. Yes, sunflowers are my favorite blooms.
Anyways, the restaurant has a cozy concept of tapas, so you can stay there with your friends or your love all night long eating and talking. At this point you are probably asking yourself about the food. Well, i love all the dishes, but my favorite are the chicken focaccia, grilled cheese and off course nachos. I'm obsessed with mexican food, i love it.
Thus, if you happen to be in Coimbra or if you're planing to visit the city i would highly recommend you to eat at Tapas nas Costas. You'll love it.


Although the question of whether to leave the European Union was put to voters in 2016, Brexit never stopped requiring choices. The referendum verdict did not settle the strategic decisions about Britain's future relationship with its continental neighbours, still less provide methods for enacting such decisions.
Even now that negotiations are in their final stages, the dilemma of a post-Brexit trajectory – alignment or divergence – is unresolved. Downing Street is signalling that a deal is close. Ministers have been invited to peruse a provisional text. This is a spin campaign to build domestic momentum behind the prime minister more than a definitive breakthrough. The EU side sounds more cautious because progress is measured differently in the UK and Brussels. Theresa May chalks up a personal victory if she manages to persuade her cabinet to agree on a policy. That doesn't mean that the policy will be welcome across the Channel.
The sticking point has not changed for months: how to ensure a frictionless Irish border when the UK wants to leave the single market and the customs union. The solution, as was long ago apparent, is to stay in a customs union after all. That will be the basis of any withdrawal agreement that Mrs May puts before parliament, with the promise of an upgrade to a wider trade agreement, to be finalised in the transition period. But transition ends in 2020 and free trade agreements usually take longer than two years to negotiate. The missing element is a mechanism for the UK to extricate itself from what will be sold to Conservative MPs as a "temporary" customs union.
Hard Brexiters demand an escape hatch so that they can embark on the journey of deregulation and dismantling of European social protections that was their reason for wanting to leave the EU in the first place. But any such clause would frighten off former remainers who envisage a more pro-European future. Labour MPs could never endorse a Brexit that gave succour to the neo-Thatcherite ambitions of the Tory right. The EU, meanwhile, will not agree to anything that allows the UK to coast along with near-current levels of European market access, then pivot into a rival set of global trading arrangements with the Irish border issue still unresolved.
For all the talk of an imminent deal, and the theatre of inviting cabinet ministers to see the work in progress, a definitive text that is acceptable both to Brussels and a majority in parliament still eludes Mrs May. That is because she has never had the courage to choose between irreconcilable propositions: a future harmonised with the EU or founded in hostility to it. She claims to want the former, but fails to repudiate the faction of her party that is steeped in the latter.
If the prime minister is lucky she will alight on some formula that postpones once again the critical strategic choice or, more likely, smuggles it into the small print of the withdrawal agreement. She might effectively lock the UK into an open-ended trade regime, obviously inferior to EU membership, while telling Tory Brexit fundamentalists that their ideological dreams might yet come true. Under the circumstances, pulling off that manoeuvre would probably feel like success to Mrs May, but, in all probability, it would not be a good deal for the country.


Dresden Edition to be launched at the end of this year.
Plenty to do: André Nowoisky from the Production facility in Dresden shows Mr. Monopoly the ropes.
The exclusive Dresden edition of Monopoly will hit stores just in time for Christmas. The Volkswagen Transparent Factory is to be allocated at least one square on the board. All Dresdeners can vote on which streets and other sights should appear in the classic board game. Only 10,000 copies of this limited edition version will be produced.
"I took the opportunity to get him trained up at the high point station"
Employee André Nowoisky was astonished when a special guest stopped by the Transparent Factory. Mr. Monopoly, the well-known mascot – complete with mustache, top hat, and tailcoat – was allowed to get up close and personal with the e-Golf. "I took the opportunity to get him trained up at the high point station," recalls André.


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Well, Brent drove the van to church today - actually, it went OK. Traffic volume is down on Sundays and he managed quite well. So! no surprises!
The sacrament meeting was in Greek this morning, but we had English translation which worked well. RS took the form of a Visiting Teacher Conference with some inspiring talks from the president and the visiting teaching leader. The latter, spoke in English first and after each "paragraph" translated it into Greek. That was interesting as she spoke very fast and there was no way we could pick up much of it so it was just as well that we got the English first. Following that, there was a pot-luck lunch. There are quite a few Philippino sister there and one of them made the best spring rolls. They also had sausage tools, potato salad, Greek sala and homemade pizza. For dessert, there was a large chocolate mousse cake!!! and a baked caramel custard. It was very yummy.
The rest of the day was free time and we study Greek for a bit and Brent caught up on some sleep (he is doing better though).


text:
Romania Guide
Top ten places to see in Romania
The definitive list
Romania - Travel & Leisure
Traveling in Romania gives you the opportunity to see many interesting and wonderful places. The country offers something for everyone: beautiful landscapes, museums, spas, old cities, castles, restaurants, shopping, movies, malls and more. Those with an adventurous spirit will not run out of things to do.
It is difficult to create just one list, as there are many great places to visit in Romania:
Danube Delta: The waters of the Danube form the largest and best preserved of Europe's deltas and flow into the Black Sea. The Danube Delta hosts over 300 species of birds as well as 45 freshwater fish species in its numerous lakes and marshes. It is a designated UNESCO heritage site.
The Sighişoara Citadel is in the old historic center of the town of Sighişoara. It was built in the 12th century by Saxon colonists and was designated as a UNESCO World Heritage Site in 1999. It is a unique window into the past history and culture of the Transylvanian. It was the birthplace of Vlad III the Impaler, also known as Dracula.
The Black Church from Brasov has a turbulent and interesting history. It was built between 1385 and 1477 on the site of an earlier church (destroyed by Mongol invasions in 1242). The construction of the Marienkirche (the German name of this place) was affected by extensive damage caused by Turkish raids in 1421. The church was renamed in 1689, when Hapsburg invaders torched and leveled most of the town – the church was heavily damaged and its wall was blackened from the fires. Restoration took almost 100 years and most of the church's artifacts date from this time.
Rasnov citadel is located at about 15 km from the city of Braşov and is about the same distance from the Bran Castle. In 14th century, German documents used the name Rasnov, but the modern German name of Rosenau is based on a popular etymology, being influenced by the German word "Rose". In Râşnov, the citadel was built around the year 1215 by the Teutonic Knights. The citadel was conquered only once in its history, around the year 1600 by Gabriel Báthory.
Bran Castle, situated near Bran, is a national monument and landmark in Romania. Commonly known as "Dracula's Castle", it is marketed as the home of the titular character in Bram Stoker'sDracula, which has led to persistent myths that it was once the home of Vlad Ţepeş, ruler of Wallachia, who is said to have been Dracula himself. The castle is now a museum open to tourists, displaying art and furniture collected by Queen Maria. Tourists can see the interior individually or by a guided tour. At the bottom of the hill is a small open air museum and park presenting traditional Romanian peasant structures (cottages, barns, etc.) and objects from across the country.
Histria citadel was first settled by Greek traders in 657 BC and is considered Romania's oldest town. It eventually surpassed Constanta as being the primary port in the region. The city was abandoned in the seventh century AD and its ruins were discovered in 1914. You can visit the remains of the citadel, and in summer you may have a chance to see the archaeological work in progress.
Painted churches in Moldavia and the Olt Valley. There are seven churches in northern Moldavia with their exterior walls uniquely painted with 15th and 16thth century. Here you can find tombs of the first Romanian rulers, the Govora Monastery, the Arnota Monastery or the Horezu Monastery. In the area of Horezu you will find exquisite ceramics and pottery, adorned with beautiful and traditional decoration unique to the region.
Peles Castle in Sinaia: this Castle is considered by many to be one of the most beautiful castles in all of Europe. It was the final resting place for several Romanian monarchs including King Carol I, who died here in 1914. The building of the castle began in 1873 under the direct order of the Viennese architect Wilhem Doderer and was continued in 1876 by his assistant, Johann Schultz de Lemberg. The location for the castle was chosen by the German prince Carol I de Hohenzollern, who was to become a king and it draws its name from the river which passes through the courtyard.
The Merry Cemetery (Cimitirul Vesel) is a cemetery in the village of Săpânţa, in Maramureş county. It is famous for its colorful tombstones with native paintings describing, in an original and poetic manner, the persons that are buried there as well as scenes from their lives. The Merry Cemetery became an open-air museum and a national tourist attraction. This cemetery diverges from the prevalent belief that death is something solemn. These inscriptions imply death is a moment filled with joy and anticipation for a better life.
Sibiu is one of the largest cities in Transylvania and has a population of about 150,000. It is about 282 km north-west of Bucharest. It is one of the most important cultural and religious centers in Romania as well a major transportation hub in central Romania. The city used to be the centre of the Transylvanian Saxons in Romania until World War II. Sibiu was designated the European Capital of Culture in 2007 together with Luxembourg. The architecture was greatly influenced by the German colonists who lived and worked here. In Sibiu, you can visit the Brukenthal Museu, the Lutheran Church built in 14th century, the Roman Catholic Church and the Traditional Art Museum. Around Sibiu, there are also salty lakes in Ocna Sibiului or a famous spa in Balea Lake. Around Sibiu in the area known as Marginimea Sibiului, you can find many hostels and you can see and experience traditional Romanian life.
For more information about different places to visit in Romania, visit: http://www.brasovtravelguide.ro/en/.
Getting around in Bucharest: Public transport
Trains in Romania: Ticket types and advice
By plane to and from Romania: Airports and air travel
Driving in Romania: Rentals and road tips
Leisure in Romania: Holidays, weather and sports
Romania's coast: Beaches and resorts
Great skiing: Romanian ski resorts
Alternative tourism: Exploring the countryside
Romania Guide Services Classifieds Forums Articles


text:
University of Miami Miller School of Medicine phlebotomist Mayra Fernandez prepares to take a blood sample from study participant Julio Li, on Sept. 2, in Miami., Florida. Miami was one of 89 cities around the U.S. that's tested the Moderna COVID-19 vaccine, which is expected to be available in Canada shortly. (AP Photo/Taimy Alvarez)
Moderna vaccine coming soon for Canadians but no hard date set as yet
By Terry Haig |
Posted: Tuesday, December 15, 2020 14:30
Last Updated: Tuesday, December 15, 2020 16:09
Prime Minister Justin Trudeau said today that the Moderna COVID-19 vaccine will be available in Canada by the end of the month–if it passes regulatory muster.
If it is approved, the prime minister said, Canada will receive up to 168,000 doses before the end of December.
"As with the early shipments of the Pfizer vaccine, this moves us even further forward on getting Canadians protected as quickly as possible," Trudeau said.
"We are very, very well positioned."
His announcement followed comments by Health Canada's chief medical adviser on Monday that the country was on track to approve Moderna's vaccine soon–though Dr. Supriya Sharma provided no hard date.
Sharma told reporters that Moderna provided the final clinical data on Friday to Health Canada, which has been reviewing the company's vaccine since October.
Health Canada's chief medical advisor, Dr. Supriya Sharma, who is at the centre of the vaccine approval process, says Canadians will soon have access to Moderna's vaccine. (CBC News)
All that's left now, she said, is data on the manufacturing plants, which the department expects to receive by the end of the week.
"It does look promising and it does look positive," Sharma said.
The pandemic has killed more than 13,500 people in Canada and infected nearly 470,000 others.
Canada, which already has enough vaccines secured to protect a population four times its size–the most of any country in the world–ordered an additional 20 million doses from Moderna earlier this month, bringing its confirmed order commitment to 40 million doses from the Massachusetts-based pharmaceutical giant.
Canada approved the Pfizer-BioNTech vaccine last week and what officials are calling the largest mass vaccination campaign in Canada's history began yesterday when long-term care residents in Quebec City and Montreal and health-care workers in Toronto became the first persons in the country to receive their vaccinations
Vaccinations are scheduled today at a long-term facility in Ottawa and in Manitoba and Alberta tomorrow.
Procurement Minister Anita Anand said Monday she anticipates receiving 30,000 Pfizer-BioNTech doses both this week and next.
The federal government has said about 249,000 doses will arrive by the end of the month.
Both the Pfizer-BioNTech and the Moderna vaccines require two doses–21 days apart for Pfizer and 28 days apart for Moderna.
Special freezers for COVID-19 for Pfizer-BioNTech vaccines are loaded onto a transport aircraft in Ottawa on Saturday. (Matthew Tower/Canadian Forces Combat Camera/Canadian Armed Forces via Reuters)
However, the Pfizer-BioNTech vaccine must be stored at temperatures below -70C while Moderna's stays stable at -20C for up to six months.
That difference prompted Canada's three northern territories–the Northwest Territories, Nunavut and Yukon–to wait for the Moderna vaccine.
Pfizer and Moderna together are expected to deliver 60 million doses of vaccine to Canada by next fall.
Ottawa has signed deals with seven pharmaceutical companies to deliver millions of doses at a cost of $1 billion.
Sharma said Monday Health Canada was still reviewing AstraZeneca's application after the company realized some patients in its clinical trial had not been given full doses of the vaccine.
She said Johnson & Johnson only submitted its application two weeks ago for Canadian approval and the teams are now conducting preliminary reviews.
Both the Pfizer-BioNTech and Moderna vaccines use messenger RNA technology, or mRNA, which directs cells in the body to make proteins to prevent or fight disease.
The U.S. Food and Drug Administration is expected to approve the use of the Moderna vaccine this Friday.
With files from CBC News, The Canadian Press, RCI
Categories: Health, International, Internet, Science & Technology, Society
Tags: Anita Anand, AstraZeneca, Health Canada, Johnson & Johnson, Moderna vaccine, mRNA technology, Pfizer-BioNTech vaccine, Supriya Sharma
« McGill leads trials of drug to treat COVID-19
Pandemic drives maple sugar shack dining rooms out of business »


The United States Federal government has more than 2,7 million employees, ranging in roles from secret service agents to financial analysts to IT specialists to scientists. These people work in a variety of federal agencies, each of which have a strong and clear mission to provide security, services, or assistance to US citizens.
This adds up to a major challenge in talent management. Unlike corporate America, the Federal Government has another challenge: the true measure of "success" is often less obvious. These agencies measure their success through service levels and attainment of critical goals. So as a result, the Federal Government has a very difficult time aligning "performance" with compensation and promotion. In fact, only 40% of employees believe that compensation and promotion is tied to performance and only 7% believe that compensation and promotion are directly and strongly linked to performance.
As this data shows, people are interested in recognition, job satisfaction, career development, work environment, quality of live, and inspiration. These are very similar motivators to the things which drive high engagement and retention in the private sector.
Unfortunately, as this data (and other data) shows, employees do not feel that they have enough career mobility, fair opportunities, or leadership. When compared against corporate organizations, the government falls short in the areas of career development, succession management, performance management, and employer brand.
The Office of Personnel and Management (OPM) understands these issues. While the solutions are not necessarily easy to implement, the Federal Government is well aware of the issues. Many agencies (The NRC and NASA, for example) have implemented highly competitive talent management, career development, and recruiting programs to deal with these issues.
But many of the large agencies have highly diverse workforces: combinations of technical, security, administrative, support, and service roles. We are currently working on a project with the Department of Energy, for example, to build a new set of career models for the "21st Century Energy Demands" of the United States. Such programs can go a long way to building an integrated solution.
The bottom line, of course, is that two things really matter in talent management: leadership and integration. Do the leaders of the organizations understand the "Business of Talent?" And are the organizations integrated enough to take a wholistic view of talent from end to end?
We are working hard on research and services with the Federal government in 2008 and 2009, call us or watch for more information on this important part of all of our lives.


text:
Der Haslsteiner Sitz liegt in Dünzling, heute einem Gemeindeteil des niederbayerischen Marktes Bad Abbach im Landkreis Kelheim. Die ursprüngliche Burganlage liegt unmittelbar westlich der Ortskirche St. Martin. Sie wird als "untertägige mittelalterliche und frühneuzeitliche Befunde im Bereich der ehem. Burg ("Haslsteiner Sitz") mit Friedhofsbefestigung sowie der Kath. Kirche und ehem. Burgkapelle St. Martin in Dünzling, darunter die Spuren von Vorgängerbauten bzw. älteren Bauphasen sowie der abgegangenen Kapelle St. Sebastian" unter der Aktennummer D-2-7138-0201 als Bodendenkmal im Bayernatlas aufgeführt.

Beschreibung
Das sich unmittelbar westlich der Ortskirche befindliche Haus (Saalhaupter Straße 2) mit einer Länge von 15 m und einer Breite von 11 m enthält im Kern noch immer den Wohnturm aus dem 12. Jahrhundert. Bei einer Renovierung in den 1960er Jahren wurden vermauerte Fensterchen und hofseitige Fensternischen mit eingelassenen Steinbänken mit Holzsitzbrettern gefunden. Der ursprüngliche Eingang mit einem romanischen Türbogen, der mittels Holzbalkenriegel verschlossen werden konnte,  lag im ersten Stock; er konnte nur über eine Leiter oder eine außen liegende Stiege erreicht werden. Das Burgareal macht in Nord-Süd-Richtung 35 und in Ost-West-Richtung 60 m aus. Es umsfasste den bis 1950 bestehenden Friedhof und die St.-Martins-Kirche und war mit einer Mauer umschlossen. Der Friedhof war befestigt, wie eine Rechnung von 1717 über Bauausgaben für den "Freyhof Thurn" belegt. Bis 1733, als die Martinskirche verlängert wurde, führte ein Übergang von dem Sitz zu der Westempore der Kirche. Die abgegangene, aber 1712 erwähnte "Sebastian-kürchen" gehörte vermutlich auch zu der Friedhofsbefestigung. Durch Dünzling verlief eine bedeutsame Altstraße von Regensburg nach Langquaid und weiter nach Landshut.

Geschichte
Dünzling gehörte zum großen Teil dem Kloster St. Emmeram, die Vogtei übten bis zum Tod des Niclas von Abensberg die Abensberger und danach die Wittelsbacher aus. Als erster Dienstmann des Klosters tritt 1028 ein "Rihhere de Tunzelingin" auf. Dieser "Richeri" wird 1031 auch als Besitzer von zwei Huben im St. Emmeraner Güterverzeichnis genannt. 1190 hat ein "Sigbert de Tunzelingen" neben weiteren Besitzungen auch den "Thurn" vom Kloster als Lehen inne. Bis Anfang des 13. Jahrhunderts werden hier 18 Dienstmannen des Klosters genannt, die aber nicht alle in der Kleinburg wohnen konnten, sondern auf den früheren Fronhof und den sogenannten Stöderlsitz verteilt waren. Zwischen 1210 und 1217 bezeugt Irnfried von Dünzling zusammen mit seinem Ritter Heinrich eine Tradition. 1256 werden Dietrich von Dünzling und sein Sohn Herrand genannt. Am 24. Mai 1271 erscheinen noch ein Konrad von Dünzling und seine Frau Elisabeth. Mit diesem Konrad starben die Dünzlinger vermutlich aus, denn nun treten als Lehensträger die Poikamer auf. 1324 wird ein Wernt der Poikamer genannt, der seinen väterlichen Besitz zwischen seiner Stiefmutter Heilwig und deren Kindern und sich und seinen Kindern teilt. Diesem folgt sein Sohn Werent der Poikamer von Dünzling, der 1350 erstmals genannt wird. Auf diesen folgen sein Sohn Gebhard und sein Enkel Konrad (er erhielt 1466 von Abt Michael Teuer den Sitz zu Dünzling erneut zum Lehen), mit welchem der Dünzlinger Zweig der Poikamer ausstarb. In der Folge wechselten die Lehensträger in rascher Folge. Am 14. November 1577 verkauft Erhard Vogel zu Tünzling den nach St. Emmeram "lehnbaren Sütz und Purgstall alda an Frauen Elisabethen Marschalkhin zu Pappenheim". Lehensträger wurde dadurch Philipp von Pappenheim zu Bibrach, der den Besitz mit Burg, Sedelhof, mehreren Sölden und großem Waldbesitz zwischen 1577 und 1591 im Namen seiner Frau Elisabeth, einer geborenen Schallei, innehatte. Ab 1662 wurde der Sitz nicht mehr an Adelige, sondern an Bauern verliehen; der dazugehörige Sedelsitz und die Sölden wurden aus dem Gutskomplex herausgelöst und vom Kloster auf Erbrecht vergeben. Das Gebäude ist heute in Privatbesitz.

Literatur
 Johann Auer: Befestigungen und Burgen im Landkreis Kelheim vom Neolithikum bis zum Spätmittelalter. Verlag der Weltenburger Akademie Aventinum e.V.,  Abensberg 2008, S. 183–186.

Weblinks

Ehemalige Burganlage im Landkreis Kelheim
Niederungsburg in Niederbayern
Abgegangenes Bauwerk im Landkreis Kelheim
Bauwerk in Bad Abbach
Bodendenkmal in Bad Abbach
Erbaut im 12. Jahrhundert

redpajama_set_name: RedPajamaWikipedia


text:
Q: MySQL replication recovery I can't sync the slave server after master server crashed. Could you please advise if the recovery is possible (without data/transaction lost) or do I have to create a new slave server?
Please note that I stooped slave after this error:
mysql> SHOW SLAVE STATUS \G
*************************** 1. row ***************************
               Slave_IO_State:
                  Master_Host: 192.168.1.1
                  Master_User: slave
                  Master_Port: 3306
                Connect_Retry: 60
              Master_Log_File: mysql-bin.000010
          Read_Master_Log_Pos: 940132528
               Relay_Log_File: cb1s-relay-bin.000001
                Relay_Log_Pos: 4
        Relay_Master_Log_File: mysql-bin.000010
             Slave_IO_Running: No
            Slave_SQL_Running: No
              Replicate_Do_DB:
          Replicate_Ignore_DB:
           Replicate_Do_Table:
       Replicate_Ignore_Table:
      Replicate_Wild_Do_Table:
  Replicate_Wild_Ignore_Table:
                   Last_Errno: 0
                   Last_Error:
                 Skip_Counter: 0
          Exec_Master_Log_Pos: 940132528
              Relay_Log_Space: 107
              Until_Condition: None
               Until_Log_File:
                Until_Log_Pos: 0
           Master_SSL_Allowed: No
           Master_SSL_CA_File:
           Master_SSL_CA_Path:
              Master_SSL_Cert:
            Master_SSL_Cipher:
               Master_SSL_Key:
        Seconds_Behind_Master: NULL
Master_SSL_Verify_Server_Cert: No
                Last_IO_Errno: 1236
                Last_IO_Error: Got fatal error 1236 from master when reading data from binary log: 'Client requested master to start replication from impossible position; the first event 'mysql-bin.000010' at 940132528, the last event read from './mysql-bin.000010' at 4, the last byte read from './mysql-bin.000010' at 4.'
               Last_SQL_Errno: 0
               Last_SQL_Error:
  Replicate_Ignore_Server_Ids:
             Master_Server_Id: 1
1 row in set (0.00 sec)

Logs:
120711 13:04:24 [Note] Slave I/O thread: connected to master 'user@192.168.1.1:3306',replication started in log 'mysql-bin.000010' at position 940132528
120711 13:04:24 [ERROR] Error reading packet from server: Client requested master to start replication from impossible position; the first event 'mysql-bin.000010' at 940132528, the last event read from './mysql-bin.000010' at 4, the last byte read from './mysql-bin.000010' at 4. ( server_errno=1236)
120711 13:04:24 [ERROR] Slave I/O: Got fatal error 1236 from master when reading data from binary log: 'Client requested master to start replication from impossible position; the first event 'mysql-bin.000010' at 940132528, the last event read from './mysql-bin.000010' at 4, the last byte read from './mysql-bin.000010' at 4.', Error_code: 1236
120711 13:04:24 [Note] Slave I/O thread exiting, read up to log 'mysql-bin.000010', position 940132528
120711 13:04:24 [Note] Slave SQL thread initialized, starting replication in log 'mysql-bin.000010' at position 940132528, relay log './cb1s-relay-bin.000026' position: 940132674
120711 13:05:08 [Note] Error reading relay log event: slave SQL thread was killed
120711 13:06:20 [Note] Slave SQL thread initialized, starting replication in log 'mysql-bin.000010' at position 940132528, relay log './cb1s-relay-bin.000027' position: 4

A: You don't have to create a new slave. You just would need to replay the master's data and re-set the replication coordinates on the slave (CHANGE MASTER TO ...) as described in the MySQL documentation.
You might get the data off the master without stopping the database by specifying the --master-data=2 option to your mysqldump call - this would write the replication coordinates of your dump into the mysqldump output.
Note that running mysqldump could still mean a downtime for your database-backed application as all tables would be read-locked for the duration of the dump.
Another option would be to just reset the replication coordinates as suggested by David Schwartz in his comment to your question and replay all table's data to the slave(s) using pt-table-sync from the Percona Toolkit to ensure integrity.

redpajama_set_name: RedPajamaStackExchange


At this moment Dukla Prague U21 from Czech Republic Championship U21 on last 34 matches played, wins 14 times (41%), draws 6 times (17%) and loses 14 times (41%).
They scored 2.59 goals/match and and take approximative 2.47 goals/match.
In same manner, Trinec U21 from Czech Republic Championship U21 on last 35 matches played, wins 11 times (31%), draws 5 times (14%) and loses 19 times (54%).
They scored 1.4 goals/match and and take approximative 2.63 goals/match.