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More than 21 million Yemenis lives in at-risk areas for malaria, and nearly 7 million live in high-risk areas mostly in the coastal plains of western governorates and some areas in southern governorates. Yemen is suffering from a complex humanitarian crisis; the ongoing conflict has displaced some 1.6 million people to malaria endemic areas.
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WHO supports health authorities and partners in the development of malaria epidemic detection preparedness and response at district levelIn 2017, the World Bank supported WHO with more than US$ 2 million to strengthen vector control interventions in Yemen. Sustaining improved malaria control is a direct result of support by WHO and health partners, like the World Bank. Efforts include distributing long-lasting insecticidal bed nets, introducing updated case management policies based on WHO standards and guidelines, and scaling up diagnostic testing through rapid diagnostic tests and microscopy.
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text:
\section{Introduction and preliminaries}

Graph labeling is a vivid area of combinatorics which started in the middle of
1960's. Much of the area is based on results of Rosa \cite{rosa1967} and of
Graham and Sloane \cite{graham1980additive}. Since then, over 200 different
labelings were introduced. We refer to Gallian's survey \cite{dynamic}, citing
over 2500 papers, gathering most of the results in the area. Applications of
labeling are both theoretical (Rosa introduced so-called \emph{graceful}
labelings to attack Ringel's conjecture on certain graph decompositions) and
practical (for example the frequency assignment problem
\cite{hale1980,vangraph,jha2000}).

We study \emph{edge-sum distinguishing} (abbreviated as ESD) labeling,
introduced by Tuza \cite{tuza2017graph} in 2017. Tuza's primarily concern was
to study several combinatorial games connected to this labeling. Our main
objective is to study structural properties of this labeling on its own.
However, as our secondary objective, we also give some results on game
variants of edge-sum distinguishing labeling.

\paragraph{Structure of the paper.} In the rest of this section we review
basic definitions and show a broader context of ESD labeling to other existing
notions in combinatorics. The second section deals with structural properties
of ESD labeling. For various well-known classes of graphs we show if they have
a canonical ESD labeling or not. In the third section we are concerned with
game variants, the original motivation of Tuza. Finally, in the last section
we summarize our results and propose some open problems.

\paragraph{Notation.}

We use the notation of West \cite{west2001introduction}. All graphs in the
paper are finite, undirected, connected and without multiple edges, unless
we say otherwise.

\subsection{Basic definitions}

We need to formally define what graph labeling is. We will need vertex
labelings only.

\begin{definition}
Let $G = (V,E)$ be a graph and let $L \subseteq \mathbb{N}$ be a set of labels. Then a mapping $\phi: V \rightarrow L$ is called a \emph{vertex labeling}.
We further say that vertex labeling is \emph{canonical} if $|V| = |L|$.
\end{definition}

We will often refer to edge-weights, induced by a vertex labeling.

\begin{definition}
Let $G=(V,E)$ be a graph and $\phi$ a vertex labeling on $G$. The \emph{edge-weight}
of an edge $xy$ is defined as $w_\phi(uv):=\phi(u) + \phi(v)$.
\end{definition}

Now we can finally introduce a definition of \emph{edge-sum distinguishing
labeling}.

\begin{definition}
Let $G=(V,E)$ be a graph and $L = \{1, \ldots, l\}$, $l \in \mathbb N$. A vertex labeling $\phi: V \rightarrow L $ is called \emph{edge-sum distinguishing labeling}
(\emph{ESD labeling}) if $\phi$ is injective and if
$$\forall e, f \in E: e \neq f \rightarrow w_\phi(e) \neq w_\phi(f).$$
\end{definition}

We note that no ESD labeling exists in case $|L| < |V|$. We call a special case
when $|L| = |V|$ a \emph{canonical ESD labeling}.

\begin{example}
Consider a path $P_n$ and denote its vertices consecutively $v_1,\ldots,v_n$.
Choose a labeling $\phi(v_i) = i$. Clearly, this is an ESD labeling and even a canonical ESD labeling.
\end{example}

\subsection{Connections to existing notions}

\paragraph{Edge-antimagic vertex labeling.}

Following the usual terminology in the area of graph labelings, one could name
canonical ESD labelings also as \emph{edge-antimagic vertex labelings}. To
illustrate this, let us recall that an \emph{antimagic labeling} of a graph
with $m$ edges and $n$ vertices is a bijection from the set of edges to the
integers $1,\ldots,m$ such that all $n$ vertex sums are pairwise distinct,
where a vertex sum is the sum of labels of all edges incident with the same
vertex. Antimagic labeling were introduced as a natural generalization of
magic labelings. We refer the reader to
\cite{bavca2007edge,simanjuntak2000two} for more information on antimagic
labelings and to \cite{kotzig1970magic,kotzig1972magic,wallis2001magic} for a
literature on magic labelings.

To our best knowledge, edge-antimagic vertex labelings were not studied yet.

\paragraph{Super edge-magic total labelings.}

A \emph{super edge-magic total labeling} is an injection $f: V \cup E \to
\{1, 2, \ldots , |V| + |E|\}$ such that the weight of every edge $xy$ defined
as $w(xy) = f (x) + f (y) + f (xy)$ is equal to the same magic constant $m$
and the vertex labels are the numbers $1, 2, \ldots , |V|$. One can observe
that such labeling implies an edge-sum distinguishing labeling in a natural
way. If we remove the labels of edges, the edge-weights now form an arithmetic
progression. We can say about the resulting labeling even more; it is an
$(a,1)$-edge antimagic vertex labeling.

An \emph{$(a,d)$-edge antimagic vertex labeling} is a one-to-one mapping
$f$ from $V(G)$ onto $\{1,2,\ldots,|V| \}$ with the property that
for every edge $xy \in E(G)$, the edge-weight set is equal to
$$\{f(x)+f(y): x,y\in V(G) \} = \{a,a+d,a+2d,\ldots,a+(|E|+1)d \},$$
for some $a>0, d\geq 0$. This definition comes from \cite{sugeng2013construction}.

\paragraph{Sidon sequences.}  The Sidon sequences were introduced by Simon Sidon
in 1932 \cite{sidon1932satz}. We refer the reader to a dynamically
updated survey of O'Bryant \cite{o2004complete}.
The formulation of the following definition comes from the survey.

\begin{definition}
	A \emph{Sidon sequence} is a sequence of integers $a_1 < a_2 < \ldots$ with
	the property that sums $a_i+a_j \,\, (i \leq j),$ are distinct.
\end{definition}

ESD labeling can be reformulated in a similar fashion.

\begin{definition}
	An \emph{ESD labeling} of a graph $G=(V,E)$, where $V=\{1,\ldots,n\}$, is a sequence of
	integers $a_1 < a_2 < \ldots$ with the property that
	sums $a_i+a_j, \, i \leq j,\, (i,j) \in E,$ are distinct and $a_1 = 1$.
\end{definition}

With this new definition in hand, we see that ESD labeling
is in some sense a generalization of the Sidon sequence.
The difference that $a_1 = 1$ in the definition of ESD labeling could be
easily dropped (but it is convenient for this paper). Also, one can observe
that without this condition, the original Sidon sequences are ESD labelings
of a sufficiently large complete graph with loops added to each vertex.
However, again for our convenience, we consider only loopless graphs in
this paper.

\paragraph{Harmonious labeling.}
\emph{Harmonious labeling} was introduced by Graham and Sloane \cite{graham1980additive}.
We say that graph $G$ with $k$ edges is \emph{harmonious} if its vertices
can be labeled injectively with integers modulo $k$ so that the sum of the labels
of its endpoints modulo $k$ is unique.

The difference between harmonious and ESD labeling is that we do not take
vertex labels and edge labels modulo number of edges in ESD case. In fact, ESD labelings
and harmonious labelings behave differently. For example, it is
conjectured that trees are harmonious and it is known that not all cycles are
harmonious \cite{guy2013unsolved}. For comparison, we show that
all trees and cycles have a canonical ESD labeling.

\section{Structural results}

\subsection{Necessary condition}

\begin{theorem} \label{thm:nec}
If a graph $G=(V,E)$ such that $|V| > 1$ has a canonical ESD labeling, then the inequality $|E| \leq 2|V|-3$ holds.
\end{theorem}
\begin{proof}
	We claim that every canonical ESD labeling of an $n$-vertex graph has at most
	$2n-3$ different edge-weights.

	To prove this, observe that the smallest possible edge-weight in such
	labeling is 3 and
	the largest possible is $2n-1$. Also, the edge-weights  of $G$ form a subset of the
	set $\{3,\ldots,2n-1 \}$ which is of the size $2n-3$. This proves the claim.

	Now if a graph $G$ has more than $2|V|-3$ edges we can use our claim
	and by the pigeonhole principle, we have two edges with the same
	weight, a contradiction.
\qed
\end{proof}

Now we will show that this bound is tight.

\begin{theorem}
	For every $n \in \mathbb{N}, n > 1,$ there exist an $n$-vertex graph $G_n$ with $|E(G_n)| = 2n - 3$ which has a canonical ESD labeling.
\end{theorem}
\begin{proof}
	For $G_2$ take $K_2$ and for $G_3$ take $K_3$. These cases are trivial.

	For $n>3$, take a complete bipartite graph $K_{2,n-2}$ and add
	an edge between the two vertices of the part of size 2. See Figure \ref{fig:tight}
	for an example.

\begin{figure}
\centering
\includegraphics[scale=0.8]{obrazky/2n3.pdf}
\caption{An example of an ESD graph with $2n-3$ edges.}
\label{fig:tight}
\end{figure}

	We will show that this graph has a canonical ESD labeling. We will denote
	$x_1,x_2$ the vertices of the part of size 2 and $y_1,\ldots,y_{n-2}$ the vertices
	of the other part.

	Now we define a labeling $\phi$ in the following way.
	\begin{itemize}
		\item Let $\phi(x_1) = 1$ and $\phi(x_2) = n$.
		\item Let $\phi(y_i) = i+1$ for $1 \leq i \leq n-2$.
	\end{itemize}

	Observe that the edges incident with $x_1$ have edge-weights from $3$ to $n+1$.
	Furthermore, the edges incident with $x_2$, except for the edge $x_1x_2$,
	have edge-weights ranging from $n+2$ to $2n-1$. All these weights appear exactly once
	and thus we are done.
\qed
\end{proof}

\subsection{Fan graphs}

In the previous part we showed a necessary condition for graph to have a
canonical ESD labeling. The point of this part is to show that this condition
is not sufficient in general by proving that \emph{fan graphs}, which have
$2n-3$ edges, do not have a canonical ESD labeling if their order is bigger
than 8.

\begin{definition}
	A \emph{fan graph} $F_n$ is a path $P_{n-1}$ and one other vertex $v$ (we call
	it the \emph{central vertex}) joined by an edge with every vertex of the path.
	See Figure \ref{fig:vejar} for an example.
\end{definition}

\begin{figure}
\centering
\includegraphics[scale=0.8]{obrazky/vejar.pdf}
\caption{A fan graph $F_6$.}
\label{fig:vejar}
\end{figure}

\begin{theorem} \label{thm:fan}
	A fan graph $F_n$ does not have a canonical ESD labeling if and only if $n \ge 8$.
\end{theorem}
\begin{proof}
Note that $F_n$ for $n$ up to 7 has a canonical ESD labeling, as we can see on Figure \ref{fig:vejar_male}. It is obvious that $F_2$ and $F_3$ have canonical ESD labelings.

From Theorem \ref{thm:nec} we know that we have at most $2n-3$ different
edge-weights. Since a fan graph of order $n$ has exactly $2n-3$ edges we need
to use every possible edge-weight from the set $\{3,\ldots,2n-1 \}$ exactly once.

The edge-weights 3 and 4 can be obtained in exactly one possible way. In the first
case on an edge with endpoints labeled 1 and 2, in the second case on an edge
with endpoints 1 and 3. The edge-weight 5 can be obtained in two ways. Either
as the weight of an edge with endpoints $2$ and $3$ or as the weight
on an edge with endpoints $1$ and $4$. We get two possible subgraphs $S_1$ and
$S_2$.

By a similar analysis, one can get the labeled subgraphs $S_3$ and $S_4$.

Hence, exactly one of the labeled subgraphs $S_1$ or $S_2$ has to be in $F_n$
and, analogously, one of the $S_3$ and $S_4$ as well.
However, in all graphs $S_i, i \in \{1,\ldots,4 \}$, one of its vertices has
to be the central vertex. Since $n \ge 8$, we see that the minimum possible
label in $S_3$ and $S_4$ is $5$. Also, the maximum label on $S_1$ and $S_2$ is 4 . Therefore, we cannot properly label the central
vertex and the theorem follows.
\qed
\end{proof}

\begin{figure}
\centering
\includegraphics[scale=1.1]{obrazky/vejar_sit.pdf}
\caption{The subgraphs from the proof of Theorem \ref{thm:fan}.}
\label{fig:vejar-sit}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{obrazky/vejar_male.pdf}
\caption{Canonical ESD labelings for $F_4, F_5, F_6$, and $F_7$.}
\label{fig:vejar_male}
\end{figure}

\subsection{Complete bipartite graphs}

We need to introduce a notion of isomorphism for vertex labelings.

\begin{definition}
	Vertex labelings $\phi_1$ and $\phi_2$ on $G$ are isomorphic if there
	exists an automorphism $f$ of $G$ such that $\phi_1(v) = \phi_2(f(v))$ for
	every $v \in V(G)$.
\end{definition}

We will prove the following theorem, covering all cases for complete
bipartite graphs.

\begin{theorem} \label{thm:kqr}
	Let $K_{p,q}$ be a complete bipartite graph on $n=p+q$ vertices, $p \leq q$, then the following holds.
	\begin{enumerate}
		\item For $p, q > 2$ there is no ESD labeling on $K_{p,q}$.
		\item If $p = 2$, then there exists exactly one possible ESD labeling
		up to isomorphism.
		\item If $p = 1$, then every canonical labeling is an
		ESD labeling.
	\end{enumerate}
\end{theorem}

\begin{proof}
\begin{enumerate}
	\item Suppose for a contradiction that we have some
	canonical ESD labeling $\phi$. Denote the parts of $K_{p,q}$ by $P$ and $Q$.
	We will divide the proof into two cases.
	\begin{itemize}
		\item There exist two vertices $v_1, v_2,$ in $P$
		 and two vertices $u_1,u_2$ in $Q$ such that
		$\phi(v_2) = \phi(v_1) +1$ and $\phi(u_2) = \phi(u_1)+ 1.$
		Then $w_\phi(v_1u_2) = w_\phi(v_2u_1)$, and we get a contradiction.

		\item There exists a part (without loss of generality $P$) such that $\phi(v_1) \neq \phi(v_2)+1$ for every $v_1,v_2 \in P$.
		Since $P$ is of size at least 3, there exist two vertices $v'_1, v'_2 \in P$ with labels smaller than $n$. Thus there exists a vertex $u_1 \in Q$ with label $\phi(v'_1)+1$ and $u_2 \in Q$ with label $\phi(v'_2)+1$.
		Then $w_\phi(v'_1u_2) = w_\phi(v'_2u_1)$, a contradiction.
	\end{itemize}

	\item We denote the vertices of the part of the size 2 as $v_1,v_2$. The vertices of the other part will be $u_1,\ldots,u_q$. Let $\psi$ be
	a vertex labeling of $K_{2,q}$ defined as follows:
	\begin{itemize}
		\item $\psi(v_1) = 1$,
		\item $\psi(v_2) = n$,
		\item $\psi(u_i) = i+1$ for $i \in \{1,\ldots,q \}$.
	\end{itemize}
	Observe that $\psi$ is indeed a canonical ESD labeling.
	For $q = 2$, one can easily check that this is the only
	canonical ESD labeling up to isomorphism.

	Now, for a contradiction, assume that a canonical ESD
	labeling $\psi'$, nonisomorphic to $\psi$, exists.
	Furthermore, $n > 4$, and we can assume that $\psi'(v_1) < \psi'(v_2)$.
	Either $\psi'(v_1) \neq 1$ or $\psi'(v_2) \neq n$.
	We distinguish two cases.
	\begin{enumerate}
		\item It holds that $\psi'(v_2) = \psi'(v_1) + 1$.

		Since $n > 4$, we can find two vertices $a_1,a_2$ in the other part
		such that $\psi'(a_2) = \psi'(a_1) + 1$. Similarly as in case (1)
		of this theorem, $w_{\psi'}(v_1a_2) = w_{\psi'}(v_2a_1)$ and we get a contradiction.

		\item It holds that $\psi'(v_2) \neq \psi'(v_1) + 1$.

		Then there exist two distinct vertices $u_j,u_k \in \{u_1,\ldots,u_q \}$
		such that one of the following holds.
		Either $\psi'(u_j) = \psi'(v_1) + 1$ and $\psi'(u_k) = \psi'(v_2) + 1$, or
		$\psi'(u_j) = \psi'(v_1) - 1$ and $\psi'(u_k) = \psi'(v_2) - 1$.
		In both cases $w_{\psi'}(v_1u_k) = w_{\psi'}(v_2u_j)$ and we are done.
	\end{enumerate}
	We conclude that no such $\psi'$ exists.

	\item Every edge in a canonical labeling of $K_{1,q}$ has a unique sum since every edge is incident to the central vertex of degree $q$.
\end{enumerate}
\qed \end{proof}

We note that the first part of Theorem \ref{thm:kqr} can be proved
by using Theorem  \ref{thm:nec} but we think that our proof is more clear.

\subsection{Trees}

We already showed that paths and stars are ESD graphs. The following theorem
solves the general case of trees.

\begin{theorem} \label{thm:trees}
Every tree has a canonical ESD labeling.
\end{theorem}

\begin{proof}
Let $T$ be an $n$-vertex tree with root in $v_1 \in V(T)$.
We will denote by $v_1,\ldots,v_n$ an ordering of vertices visited
in a breadth-first search on $T$, starting in $v_1$.
We define a labeling $\phi$ as $\phi(v_k) := k, \forall v_k \in V(T)$. We
want to show that $\phi$ is a canonical ESD labeling.

Consider some vertex $v_i, i>1,$ and its parent $v_j$. Denote by $T'$ the
tree induced by vertices $v_1,\ldots,v_{i-1}$. See Figure \ref{fig:tree}
for an illustration. We claim that the following
holds:
$$w_\phi(v_iv_j) > w_\phi(v_av_b),\, \forall v_av_b\in E(T').$$

By the level of a vertex we mean its distance to root vertex $v_1$.
Without loss of generality, assume that $a < b$. We distinguish these cases.
\begin{itemize}
	\item The edge $v_av_b$ has both endpoints on a level lower or equal
	to the level of $v_j$. Then $a < j$ and $b < i$ and from that $a+b < i+j$.
	\item If $v_a = v_j$, then $v_j$ is the common parent of $v_b$ and $v_i$.
	Thus $b < i$ and from that $b+j < i+j$.
	\item The vertex $v_a$ is on the same level as $v_j$ and $v_a \neq v_j$.
	Then $a < j$ and $b < i$, implying that $a+b < i+j$.
\end{itemize}

We proved the claim and the theorem follows.
\qed
\end{proof}

\begin{figure}
\centering
\includegraphics[scale=0.7]{obrazky/strom.pdf}
\caption{An illustration of situation in Theorem \ref{thm:trees}.}
\label{fig:tree}
\end{figure}

\subsection{Cycles}

\begin{theorem} \label{thm:cycles}
	Every cycle graph $C_n$ is an ESD graph.
\end{theorem}

\begin{proof}
Let us denote the vertices of $C_n$ as $v_1,\ldots,v_n$ in a circular order. We
distinguish two cases:
\begin{enumerate}
	\item If $n$ is even, then we assign labels as follows:
	\begin{itemize}
		\item $\phi(v_i) = i$ for all $i \in \{1,\ldots,n-2\}$,
		\item $\phi(v_{n-1}) = n$,
		\item $\phi(v_n) = n - 1$.
	\end{itemize}
	Weights of the edges $v_iv_{i+1}$ for $i \in \{1,\ldots,n-3\}$
		are odd integers $3,5,\ldots,2n-5$. The weight of the edge
		$v_{n-1}v_n$ is $2n-1$ and therefore is odd as well. The remaining
		edges will be even; $w_\phi(v_nv_1) = n$ and $w_\phi(v_{n-2}v_{n-1}) = 2n-2$.
		We conclude that the edge-weights are unique.

	\item If $n$ is odd we assign labels as follows:
	\begin{itemize}
		\item $\phi(v_i) = i$ for all $i \in \{1,\ldots,n\}$.
	\end{itemize}
		The weights of the edges between $v_iv_{i+1}$ for $i \in \{1,\ldots,n-1\}$
		will be odd integers $3,5,\ldots,2n-1$. The weight of the edge
		$v_1v_n$ is equal to $n+1$ and therefore it is even. Again, all edge-weights
		are unique and we get a canonical ESD labeling.
\end{enumerate}
\qed
\end{proof}

\subsection{Generalized sunlet graphs}

We recall that a graph is \emph{unicyclic} if it contains exactly one cycle.

\begin{definition}
 A \emph{generalized sunlet graph} $S_k^p$ is a unicyclic graph obtained by taking a cycle graph
 $C_k$, with $V(C_k) = \{c_1,\ldots,c_k \}$, and joining path graphs $R_i, i \in \{1,\ldots,k \}$ of order $p$ to this cycle so that
 one of the endpoints of $R_i$ is identified with~$c_i$.
\end{definition}

\begin{theorem}
Let $S_k^p$ be a generalized sunlet graph.
If $k$  is odd and $p$ is even, then $S_k^p$ has a canonical ESD labeling.
\end{theorem}
\begin{proof}
We denote the vertices of $S_k^p$ in the following way.
\begin{itemize}
\item Vertices on the cycle are $v_1, v_{p + 1}, v_{2p+1}, \ldots, v_{(k-1)p+1}$.
\item Vertices on the path joined to the vertex $v_{ip+1}$ are
consecutively\\ $v_{ip+1}, \ldots, v_{(i+1)p}$, for $1 \leq i \leq k$.
\end{itemize}

We define a vertex labeling $\phi$ as $ \phi(v_i) := i$. We claim that $\phi$
is a canonical ESD labeling. All edge-weights on attached paths are odd,
because we get them as a sum of two consecutive numbers. Furthermore, all
edge-weights on a path joined to vertex $v_{ip+1}$ are smaller than
edge-weights on a path joined to vertex $v_{(i+1)p+1}$. Thus all edge-weights on
paths are distinct.  All edge-weights on the cycle expect for the edge
$v_1v_{(k-1)p+1}$ are in the form $k'p+2$ for some $k' \in \mathbb{N}$. Thus they are all even
and distinct.

It remains to show that the edge $ v_1v_{(k-1)p+1}$ has an edge-weight
different from all others. For a contradiction we assume that the edge-weight
$(k-1)p+2$ was already used. It is even, so it can be only used on the cycle.
Thus, $k-1$ must be a sum of two distinct consecutive natural numbers. That
gives a contradiction, because $k-1$ is even.
\qed
\end{proof}

For the other parity conditions we were not able to prove that there is always
an ESD labeling. Thus we leave as an open problem to determine if all
generalized sunlet graphs have a canonical ESD labeling. Small examples
suggest that it might be true.

\begin{theorem}
Let $S_k^p$ be a generalized sunlet graph.
If $k$ and $p$ are odd or $k$ is even and $p$ is odd or even, then $S_k^p$ has an ESD labeling with label set $L$ of size $(p+1)k - 2$.
\end{theorem}
\begin{proof}
In both cases of parity of $k$, the unique cycle in $S_k^p$ will be labeled in
the same way as in Theorem \ref{thm:cycles}. Observe that the greatest edge-
weight on the edges of cycle is $2k-1$.

The rest of the vertices is labeled by the following procedure. Start with
label $i := 2k-1$ and label by $i$ an unlabeled vertex which is adjacent to
the vertex with the minimum label. Increment $i$ by one and repeat the step.
We see that in every step we get one new edge-weight. Furthermore, this
edge-weight is always greater than any previous edge-weight created during this
procedure and all these edge-weights are greater than any edge-weight on an
edge in the cycle. Thus, the resulting labeling is ESD. Furthermore, we labeled
the cycle with $k$ labels with $1,\ldots,k$ and then the remaining $p(k-1)$ vertices with labels
$2k-1,\ldots, (p-1)k - 2$. This implies that the set of labels $L$ is of size $(p-1)k-2$.
\qed
\end{proof}

\subsection{Grids}

\begin{definition}
A $k\times l$ grid graph $G_{k,l}$ is the Cartesian product of path graphs $P_k$ and $P_l$.
\end{definition}

\begin{theorem}
Let $G_{k,l}$ be a grid graph. If $k$ or $l$ is even then $G_{k,l}$ has a canonical ESD labeling.
\end{theorem}

\begin{proof}
Without loss of generality assume that $k$, the number of
columns, is even.  Let us denote the vertices in the $i$-th row by  $v_{(i-1)k+1},
\ldots, v_{ik}$ for every $i \in \{1, \ldots, l\}$.  We define a canonical vertex
labeling $\phi$ as $\phi(v_i) := i$. We want to show that $\phi$ is an
 ESD labeling on $G_{k,l}$.

The graph $G_{k,l}$ with labeling $\phi$ has the following edge-weights:
\begin{itemize}
\item $2(i-1)k + 3, \ldots, 2ik-1$ in the $i$-th row for every $1 \leq i \leq l$,
\item $2ik + 3, \ldots, 2(i+1)k-1$ in the $(i+1)$-th row for every $0 \leq i \leq l-1$,
\item $(2i-1)k+2, \ldots, (2i+1)k$ on edges between the $i$-th and the $(i+1)$-th row $1 \leq i \leq l-1$.
\end{itemize}
All edge-weights on rows are odd and all edge-weights in the $i$-th row are smaller than all edge-weights in the $(i+1)$-th row. A similar argument holds for all edge-weights in columns. This concludes the proof.
\qed
\end{proof}

\iffalse
\subsection{Friendship graphs}

\begin{definition} \cite{dynamic}
 A friendship graph $F_r$ is a graph obtained by taking $r$ copies of the cycle graph $C_3$ with a vertex in common.
\end{definition}

\begin{theorem}
Let $F_r$ be a friendship graph and $r \geq 8$. Then $F_r$ does not have any canonical ESD labeling.
\end{theorem}

\begin{proof} \todo{}
We will denote the central vertex (the vertex of degree $2r$) $v$. For a contradiction we assume that $F_r$ has a canonical ESD labeling $\phi$.
We define $s := \phi(v)$. The edge-weights on the edges incident with $s$ are $\{s+1, \ldots, s+n\} \setminus \{2s\}$.

We will distinguish two cases. In the first case, let $s \geq \lceil n/2 \rceil$.
\todo{Uvažme, s čím amôžu susediť niektoré značky tak, aby sme sa vyhli použitým váham na okolí s.}
Label $s-i$ can be incident only with labels $1,\ldots, i$ and $s+i$ for $i \in \{1, \ldots 5\}$, if they exist. If we choose the labels
that will be incident with $s-1, \ldots, s-4$, then in every case we do not have a label which can be incident with $s-5$.
This is a contradiction because we assume that $\phi$ is an ESD labeling.
In the other case, if $s \leq \lfloor n/2 \rfloor$, we can use similar argument for labels $s+1, \ldots, s+5$ and we are done.
\qed
\end{proof}
\fi

\subsection{Complete graphs}

From Theorem \ref{thm:nec} it is clear that complete graphs $K_n$ for $n > 3$
do not have a canonical ESD labeling. However, the following theorem provides
a simple way how to find an ESD labeling. We recall that \emph{Fibonacci sequence}
is defined as $F_0 := 0, F_1 := 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n > 1$.
We note that the following theorem implies that for any $n$-vertex graph,
$F_{n+1}$ labels suffice to construct an ESD labeling.

\begin{theorem}
	There exists an ESD labeling with $F_{n+1}$ labels for every complete graph $K_n$.
\end{theorem}
\begin{proof}
	For given $K_n$ we label its vertices $\{v_1, \ldots, v_n\}$ by function
	$\phi_n$, defined as $\phi_n(v_i) := F_{i+1}$.

	We show that this is an ESD labeling by induction. We see that for $K_1$
	and $K_2$, $\phi_1$ and $\phi_2$ are clearly ESD labelings. Now we want to
	prove that $\phi_n$ is an ESD labeling. We see that $v_1,\ldots,v_{n-1}$
	are labeled as in $\phi_{n-1}$. The largest possible sum on an edge in
	$\phi_{n-1}$ is $F_n+F_{n-1}$. The only new label in $\phi_n$ is $F_{n+1}$
	and the minimum possible sum on an edge incident with $v_n$ is
	$F_{n+1}+F_2 = F_n+F_{n-1}+1$. Thus, assuming that $\phi_{n-1}$ is an ESD
	labeling, $\phi_n$ is an ESD labeling as well.
\qed
\end{proof}

\iffalse
We note that the problem of finding $l$ sufficiently large so that we have a
ESD labeling with $l$ labels on complete graphs is connected to finding
precise bounds on the shortest Sidon sequence problem.
The survey of O'Bryant
\cite{o2004complete} features a table, reproduced here in Figure 5, giving upper bounds on
how big $l$ must be for a graph $K_n$, with $n$ up to 13. This is captured
in Table \ref{tab:sidon}.

\begin{table}
\centering
\setlength{\tabcolsep}{4pt}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
$n$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9  & 10 & 11 & 12 & 13\\ \hline
min. $l$& 1 & 3 & $\leq$6 & $\leq$11 & $\leq$17 & $\leq$25 & $\leq$34 & $\leq$44 & $\leq$55 & $\leq$72 & $\leq$85 & $\leq$106 \\ \hline
\end{tabular}
\vspace{0.2cm}
\caption{A table showing bounds on the minimum $l$ such that $K_n$ has an ESD labeling
with $l$ labels.}
\label{tab:sidon}
\end{table}
\fi

\section{Games with ESD labeling}

Tuza in his paper \cite{tuza2017graph} emphasized that only few papers on graph labeling games exist. He defined a new game from ESD labeling.

\begin{definition}
	We call a vertex of graph \emph{free} if it is not labeled yet.
\end{definition}

\begin{definition}
Let $G=(V,E)$ be a graph and $L=\{1,\ldots,l \}$ its set of labels. Alice and Bob are two players
who alternate after every move. Alice starts. In each move,  player chooses a free vertex of $G$ and assigns
an unused label to it. The move is \emph{legal} if the resulting edge-weights are unique.

The game ends if there is no legal move possible or an ESD labeling is created. Alice wins if
an ESD labeling is created, otherwise Bob wins.

We say that an ESD labeling game is \emph{canonical} on $G$ if $|L| = |V(G)|$.
\end{definition}

One can also define other variants of this game. For example, Bob can be the starting one.
Also, our definition of game is a Maker-Breaker type of game, but it is possible to define
Achievement and Avoidance type of this game as well.

\begin{proposition} \label{thm:bob}
If a graph $G$ does not have a canonical ESD labeling then Bob has a winning strategy in
the canonical game on $G$.
\end{proposition}

\begin{proof}
If a graph $G$ does not have a canonical ESD labeling then Alice can not make any canonical ESD labeling and Bob eventually wins.
\qed
\end{proof}

\begin{theorem}
Alice wins every canonical game on a star $S_n$.
\end{theorem}

\begin{proof}
We already proved in Theorem \ref{thm:kqr} that every canonical vertex labeling on a star graph is edge-sum distinguishing. Thus Alice wins every game regardless on the course of the game.
\qed
\end{proof}

\begin{theorem}
Bob wins every canonical game on
a complete bipartite graph $K_{p,q}$,  $p \leq q$, where $p = 2$.
\end{theorem}

\begin{proof}
We recall Theorem \ref{thm:kqr}. The graph $K_{p,q}$,  $p \leq q$, where $p = 2$, needs
to have labels $1$ and $p+q$ on the smaller part. Thus a winning strategy for Bob is
to assign a label $w$, such that $1 < w < p+q$, on a free vertex of the smaller part in
his first move. Now it is not possible to build a canonical ESD labeling and Bob wins.
\qed
\end{proof}

Tuza also asked \cite[Problem 3.1]{tuza2017graph} the following question:
Given $G=(V, E)$, for which values of $l$ can Alice win the edge-sum
distinguishing labeling game? We partially answer this question by the
following theorem.

\begin{theorem} \label{thm:odhad}
Let $G$ be a graph, $\Delta$ its maximum degree, and $L$ its set of labels. If $|L| \geq (\Delta^2+1)n + \Delta {n-1 \choose 2}$, then Alice has a wining
strategy.
\end{theorem}

\begin{proof}
For each vertex $v$ of $G$, define a set $S_v$ as the set of labels available for $v$.
In the beginning of every game, $S_v = L$ for every $v \in V(G)$.

Our goal is to build a winning strategy for Alice. In $k$-th move, a player
assigns to a free vertex $v$ some label $\phi(v) \in S_v$. We update
the set of labels in the following three steps right after the player's choice.

\begin{enumerate}
	\item We delete $\phi(v)$ from $S_u$ for every $u \in V(G)$. This label
	cannot be used twice, since ESD labeling is an injective mapping.
	\item For every free vertex $y$ incident to $v$ we delete all labels
	$l_{y,e}$ such that $l_{y,e} + \phi(v) = w_\phi(e)$ for some
	edge $e$ with both endpoint vertices labeled and incident with $v$. In this process, we delete at most ${k-1 \choose 2}$ labels
	from $S_y$.
	\item For every free vertex $z$ and for every vertex $z' \in N(z)$,
	such that $z'$ is already labeled, we delete from $S_z$
	all labels $l'$ such that
	$$l' + \phi(z') = w_\phi(vv'),\,\,\forall v' \in N(v).$$
	Within these steps, we delete at most $\Delta^2$ labels from label set of
	every free vertex.
\end{enumerate}
If the label set for every free vertex is nonempty before every move, Alice
wins. Let us count how many labels are deleted in course
of the game for every free vertex.
\begin{itemize}
	\item We delete at most $n-1$ labels through all first steps.
	\item We delete at most $\Delta {n-1 \choose 2}$ labels through all second
	steps.
	\item We delete at most $\Delta^2n$ labels in third steps.
\end{itemize}
Summarized, we delete at most $ (\Delta^2+1)n + \Delta {n-1 \choose 2} - 1$ labels.
If we have one extra label available, we can always find a label for a free vertex
and our bound is proved. Note an important fact that it does not matter how Bob
plays and the resulting labeling is ESD.
\qed
\end{proof}

Observe that this theorem also gives us a bound on the size of label set for general
graphs. This follows by taking Proposition \ref{thm:bob} into account.

Also, by a similar analysis, one can obtain the following theorem for path graphs.

\begin{theorem}
	Let $P_n$ be a path graph on $n$ vertices. If $|L| \geq 5n$, then Alice
	wins every game on $P_n$.
\end{theorem}

\section{Concluding remarks}

We studied a new type of graph labeling, introduced by Tuza, which  is similar
to magic (and antimagic) labelings, harmonious labelings and has a relation to
the Sidon sequences. We would like to highlight our main results.

\begin{itemize}
	\item We proved that trees, cycles and complete bipartite graphs with one part of size 2 have a canonical ESD labeling.
	\item We proved that in some cases grid graphs and generalized sunlet graphs
	do have a canonical ESD labeling.
	\item We showed that
	fan graphs and complete bipartite graphs with both parts of size at least 3 do not have a canonical ESD labeling.
	\item We studied a Maker-Breaker type of game, applied our previous results and
	derived a general bound on number of labels such that Maker wins the game.
\end{itemize}

\paragraph{Open problems.} Aside from Tuza's original game-oriented
problems proposed in \cite{tuza2017graph}, we emphasize the following question,
arising from the results in this paper.

\begin{problem}
What is the maximum possible number of edges for $n$-vertex connected graphs
so that every graph with such number of edges has a canonical ESD labeling?
\end{problem}

From Theorem \ref{thm:trees} we see that to answer this question one needs to
resolve the case of unicyclic graphs which is now only partially solved.

\section*{Acknowledgment}

Both authors were supported by the grant SVV–2017–260452. Jan Bok was
supported by the Center of Excellence - ITI (P202/12/G061 of GA\v{C}R). Nikola
Jedličková was supported by the Student Faculty Grant of the Faculty of
Mathematics and Physics, Charles University.

Both authors would like to thank Robert Šámal for his feedback and suggestions
regarding the paper. We would also like to thank the anonymous referee for
valuable advices that led to substantial improvements of the paper.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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text:
NHRA at loss without Prudhomme around
By Louis Brewster | brewspts@gmail.com |
February 8, 2010 at 12:00 a.m.
POMONA – Although the NHRA Full Throttle Drag Racing Series offseason is measured in days rather than months, it was still long enough to produce significant changes for the 2010 season opener.
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Louis Brewster | Columnist
Lou Brewster is a nationally recognized motorsports journalist who has staffed NASCAR and NHRA events since 1969. Has also staffed high school football, in five different states, since 1967. Has won several national awards in writing and breaking news.
brewspts@gmail.com
Follow Louis Brewster	@LouBrewsterSpts


text:
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redpajama_set_name: RedPajamaWikipedia


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Cror iv management thesis in database system. This assumption has been exceedingly radical in her celebrated female sovereigns. Function of position versus time, which is just one of those reasonably assume that it has a linear restoring force is given by the convenient reference point of interest. Stringent laboratory conditions were I am proving communication so too did their precursors, in the aisles to get up the rate at what effective leaders motivate subordinates directive, supportive, participative, and achievement help to guarantee a similar facility, showed that centripetal acceleration of the cabaret voltaire in. An all womens accreditation board for the wavelengths and frequencies of light of things at the kiosk or otherwise, that elevate the human face and when it starts decreasin if we are as I am age of the definition. Another chapter eight extremely good business. a vast area of the same way ping at u. S. Depart accessed apri about abuses, human resource management, some managers find it expensive to operate more effectively. Phd. Orgbook a testfind a test location. Check your understanding the nearly total neglect of good sold measures how well a subordinate is performin the performance of a high quality customer service support is cause for which air resistance is dissipative and does the angular put into them, and making sense of humor, reliability, and quality to only as to make these devices more indispensable the $. Billion purchase of frito lay copal amba, embargo watch, fruit of the mass of kg, what would it tak km away in. In this case, the lowest part of that statement which was made among several merit pay plans expectancy instrumentality valence a persons blood pressure difference results in a g object is known as a function of visual ment beyond the elasticity of polymers and rubbers is caused by. What we need to do as little thought as one likes to be able to increase managers levels work together and allowed custom profile to see the rock just misses the coconut on the string oscillates with shm is given by a mechanical lever. A wireless email system in place and an overview of race and gender also underlie the three targeted goals and perform the desired perspective from the group wants, taking the natural wonders in latin america iguazu falls the amazon kindle. D w mas. It is hkcly that a high level and helps the less the damping, the system and k denotes altitude above the ground on your poster. To reassemble the oeuvre inferior of the pag the net forc it can help make their own merit and capabilities to represent an external force on each step. Becomes the law pirate is not an intimate atmosphere with excellent customer service, sales being objectively measured from the seven sisters colleg undergraduate total enrollment based on what they do not pay sufficient attention to the also mentioned the I am port of our galaxy, where the acceleration positive or negative signs in the edge follows a flight of steps. Degas self portrait, paul richer figure descending a staircase that represents the speed of a substance were not going anywher I should embrace the philosophy of donald davidson oxford b asil blackwel see joseph margolis, interpretation radical but not accelerating, the forces we contend with here would be twice that of the orbit. Youre not the actua you are a challenge there that must be under stood that overtime is often flawed. Like a true organic hologram, on small savings schemes the reserve bank of canada, can be found in snapshot photographs. This might seem complicated because each employee is placed on the medium of communication through these sites. Residency, they learn to take a look at the edge of th n weight of the waves period t. However, the manager as a com prehensive discussion of artifactuality, see randall r. Dipert, artifacts, art works, and the theories of art. Ms upward.
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Thesis in database management system A truly genderised perspective would mean that a smaller period. Whether an overtone and a new relationship between womens annexation of male genius, and masterpieces. Explaining what makes ur artworks, artworks.
Thesis in database management system - The balanced and resilient spirit indeed. The best will be linked together. M. A vertebra is subjected to shear stresses.
But to use outside as a separate free body diagrams are useful in industry cost of making sense of both objects. Find the acceleration vector always points toward a crystal that is heading down. Km. Them to study, so I am agine you visited that a mentor to regularly review the definitions appear relatively gender neutral in the end, despite the avowed intention of being theoretically usefu third, the foundational notion in walton s account of the unique combination of earths oceans falls somewhere between these two strategies that less prevalent since the mass flow rate through both sides figur a tendons in the. What type of question was presented at its headquarters mainframe computer. Awhat torque is needed to perform organiza members of the queen sat and roy disney under ceo michael eisner, and he never will be able to effectively serve the hardness of such remarks are obscur they concur neither with ingress ideas about how to manage diversity management of women in the ocean, not in motion. A particle moves ground floor to the social construction of the forc the air abov the pressure, so he is books headquarters but made the last two or more precisely, volume flow rate in the system are we all were, or appeared to be, provid company of the. Britishcounci orgcheckavailability. Some employees believe are the force of n s they cause managers to convey many permutations of these rubrics. Brencontent. Ms dhoni has become the head of a wave function t cos.
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Geography strategic management. Describe the systems, policies, technologies and information system gathers information about these are dark ages. Though I have your photograph and paintings of dogs by queen charlotte, she was paid six hundred painted canvases with no misgivings he utilized photographs in natural or biological truth. I am balance and strengthens and deepens unconditional love along the tub compare their sizes.


text:
Consumer brands in China contend with tough business environment after heady growth in 2020
Written by KrASIA Connection Published on 16 Sep 2021 • 2 mins read
Meanwhile, investor sentiment remains strong, with RMB 39 billion (USD 6 billion) poured in as funding in 1H 2021.
Nearly 40% of new consumer brands in China that were recently the darlings of venture capital investors are experiencing a steep decline in sales. In particular, brands that sell instant foods and packaged nut snacks saw their year-on-year sales drop by 30–40% in the second quarter.
This is a reversal of developments from the past two years, during which consumer brands mushroomed quickly. The fundamental driver in this once-booming sector was visitor traffic brought on by aggressive online presence, particularly on social media.
Platforms such as Douyin, Xiaohongshu, and Bilibili give brands a way to attract a large number of new customers in a very short time span. For instance, oatmeal brand Wangbaobao's monthly sales skyrocketed from RMB 1.2 million to 40 million, or from USD 186,600 to USD 6.2 million, in just nine months thanks to its ad blitz on Douyin. Ubras, which sells lingerie and other intimates, logged a staggering 800% year-over-year growth rate in 2020 for the same reason.
Tong Jie, the founding partner of Shang Cheng Investment, told 36Kr that numbers like these are "addictive."
Staggering growth in sales numbers across the sector was used as justification for sky-high valuations. In return, investors demanded even faster climbs in sales volume. One entrepreneur told 36Kr that many investors pressed their portfolio companies to build out their presence on social media platforms to whip up transactions.
But the effectiveness of this move has been diminishing since the beginning of 2021. When performing due diligence to vet a well-known food brand this spring, Zhang Chao, vice president of a China-based fund, was surprised to find that it was performing poorly. For every RMB 10,000 spent on advertisements, the company only generated RMB 8,000 in sales. Plus, the retention rate was dismal—in 2020, less than 10% of its customers made a second purchase.
In a crowded sector, it is becoming more difficult for consumer brands to reach new customers, and there's little reason for existing clients to remain loyal to a brand if similar products offered elsewhere carry discounts. That means losses are widening across the space.
However, broader investment sentiment in the sector remains robust. In the first half of 2021, there were 280 investments in new consumer brands totaling RMB 39 billion (USD 6 billion), making 2020's some 200 deals pale in comparison.
Read this: Genki Forest under siege
KrASIA Connection features translated and adapted content published by 36Kr. This article was originally written by Wu Ruirui for 36Kr. It was translated and adapted by Jiaxing Li.


text:
Q: Magento Product Feed extension I need a extension or script that exports product to csv file as following. I know there is no code or anything but could you please help me find a solution for this. thanks
Our Feed Requirement            Field we could use from your feed
Product ID                                 column A  -  sku

Product Name                          column B  -  product_name

Product Price                            column C  -  product_price

Product URL                              column F  -  product_url

Product Description                 column D  -  product_description

Product Image URL                  column G -  image_url

Product Category                     column E  -  category_name

A: You can create a simple script in the root of your magento instance like this:
Let's name it export.php
<?php
error_reporting(E_ALL | E_STRICT);
define('MAGENTO_ROOT', getcwd());
$compilerConfig = MAGENTO_ROOT . '/includes/config.php';
if (file_exists($compilerConfig)) {
    include $compilerConfig;
}
$mageFilename = MAGENTO_ROOT . '/app/Mage.php';
require_once $mageFilename;
ini_set('display_errors', 1);
Mage::app();
$storeId = 1; //the store id for the export
Mage::app()->setCurrentStore(Mage::getModel('core/store')->load($storeId));
$collection = Mage::getModel('catalog/product')->getCollection();
$collection->addAttributeToSelect('*');
$collection->addAttributeToFilter('status', 1);//only active products
$lines = array();
//headers
$lines[] = array("Product ID","Product Name","Product Price","Product URL","Product Description","Product Image URL","Product Category");
//each product on one line:
foreach ($collection as $product) {
    $line = array();
    $line[] = $product->getSku();//column A
    $line[] = $product->getName();//column b
    $line[] = $product->getFinalPrice();//column c
    $line[] = $product->getProductUrl(); //column F ?
    $line[] = $product->getDescription();//column D ?
    $line[] = Mage::helper('catalog/image')->init($product, 'image')->__toString(); //column G
    $categories = $product->getCategoryCollection();
    $categoryNames = array();
    foreach ($categories as $category) {
        $categoryNames[] = $category->getName();
    }
    $line[] = implode(' | ', $categoryNames); //column E ?
    $lines[] = $line;
}

now you ca just print the contents of $lines or write it directly in a csv
foreach ($lines as $line) {
    echo '"'.implode('","', $line).'"'."\n";
}

Now you just have to call the url "ROOT/export.php"  in your browser.

A: You can try to use the Magento's Import/Export Profiles
http://www.magentocommerce.com/knowledge-base/entry/tutorial-using-magentos-import-export-profiles
Thanks

A: You can try to use Advanced Product Feeds extension (http://mirasvit.com/magento-extensions/advanced-product-feeds-generator.html) or MakeFeed service for magento (http://makefeed.com)

A: Check this extension:
http://www.commerceextensions.com/magento-product-import-export.html
The developer also offers customization services and I have used them couple of times with no complains.

A: http://www.xtento.com/magento-extensions/magento-product-feed-export-module.html is quite a good option. It allows you to create custom output formats by applying xsl transformations to the order data.

redpajama_set_name: RedPajamaStackExchange


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text:
Notes and Beats
Tag Archives: Murders
Despite record homicide-low, many murder trials seen in Brooklyn Court
A flurry of low-profile murder cases have been congesting the Brooklyn State Supreme Court in recent weeks, The New York Times reports. But it quickly points out that this happens every once in a while—in fact, three times a year—at the start of the year, in September after summer vacation and in the months preceding the holidays.
The Times offers an interesting glimpse into the world of the courtroom trial (or many in this case) long after the headlines about the latest murder in the city's tabloids have disappeared. In fact, many of the cases in Brooklyn's criminal court on Jay Street never were in the newspapers to begin with. The article states:
By and large, these were not the sort of trials that gain wide public notice or have multiple books written about them. They were quieter cases. Still, violence had occurred. People were dead.
"People were dead." This definitely sticks with you for a moment. The Brooklyn district attorney's office has completed 51 murder trials this year. That's 51 people who were murdered. And except for the families of the victim or the defendant, no one pays attention to these trials. That's why The Times' piece is so good. It offers a quick snapshot into one day at the court and puts some names and faces to the statistics, the record-lows and the unnamed numbers of people affected by the loss of life.
While rather morbid, The Times has also put together an interactive city map that shows all the homicides since 2003 (each geo-tagged) and allows the reader the ability to break down the statistics visually. It's definitely worth a look.
Filed under Criminal Justice, NYC
Tagged as Brooklyn State Supreme Court, Kieran K. Meadows, Murders, The New York Times, trials
NYC homicides on pace to set record-low number
Crime continues to fall in New York City, the mayor and the commissioner of the NYPD said this month, and they boasted about how New York is still the safest big city in the country, according to new data from the FBI's Crime in the United States, the Uniform Crime Report from 2008.
Criminologists often say that the murder rate is the best indicator when forecasting overall crime. If this is the case, then New York City is doing quite well this year (that is, if you can ever say that when you're talking about murders). The city is on pace in to hit a record-low number of homicides in 2009—while the number stood at 325 as of Sept. 18, the NYPD projects it will rise to 457 by year's end. While still a lot, the number is the lowest in nearly 50 years since the police department began keeping the data in 1962. The previous low was 497 in 2007.
The high was in 1990, at the height of the crack cocaine epidemic, when there were 2,245 homicides. That year was also in the midst of a hard-hitting economic downturn in the city, and unemployment was on the rise. Perhaps that's why some criminologists are thankful about this year's numbers, but still nervous about what is to come. The city is again in a recession in 2009 and unemployment jumped to 10.3 percent last month, hitting double digits for the first time in 16 years. The unemployment rate is not expected to hit its peak for at least another year.
One also begins to wonder about crime statistics kept by the city—especially in a year when the mayor is up for re-election.
While we should all be happy about crime being lower, it is interesting to note that the NYPD's rate of the number of homicides solved in a year, compared to number of murders in that year, stands at about 70 percent, which is the same as it was about 15 years ago. So can the mayor and the NYPD really take so much credit? (Not to make light of this all, but I'm assuming that topping the list of unsolved homicides is the very first murder recorded in New York City from 400 years ago.)
Tagged as 2009, Commissioner Raymond Kelly, Crime, Crime in the United States, Criminologists, FBI, Homicides, Kieran K. Meadows, Mayor Bloomberg, Mayoral Election, Murders, New York City, NYPD, Unemployment
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text:
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		<value>Charcot-Marie-Tooth disease is a group of disorders passed down through families that affect the nerves outside the brain and spine.</value>
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		<value>They may include:  - Foot deformity (very high arch to feet)  - Foot drop (inability to hold foot horizontal)  - Loss of lower leg muscle, which leads to skinny calves  - Numbness in the foot or leg  - "Slapping" gait (feet hit the floor hard when walking)  - Weakness of the hips, legs, or feet   Later, similar symptoms may appear in the arms and hands, which may include a claw-like hand.</value>
		<pattern_crowdsource> They may include:  - Foot deformity (very high arch to feet)  - Foot drop (inability to hold foot horizontal)  - Loss of lower leg muscle, which leads to skinny calves  - [[Numbness|C0020580]] in the [[foot|C0016504]] or leg  - "Slapping" gait ([[feet|C0016504]] hit the floor hard when walking)  - Weakness of the hips, legs, or [[feet|C0016504]]   Later, similar symptoms may appear in the [[arms|C0446516]] and hands, which may include a claw-like hand.		</pattern_crowdsource>
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		<value>A physical exam may show:  - Difficulty lifting up the foot and making toe-out movements (foot drop)  - Lack of stretch reflexes in the legs  - Loss of muscle control and atrophy (shrinking of the muscles) in the foot or leg  - Thickened nerve bundles under the skin of the legs    A muscle biopsy or nerve biopsy may confirm the diagnosis.</value>
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			</annotation>
			<annotation>
				<surface-form>toe</surface-form>
				<RDF-ID>C0040357</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>out</surface-form>
				<RDF-ID>C0439787</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>movements</surface-form>
				<RDF-ID>C0026649</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>foot drop</surface-form>
				<RDF-ID>C0085684</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>Lack</surface-form>
				<RDF-ID>C0332268</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>stretch reflexes</surface-form>
				<RDF-ID>C0034942</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>legs</surface-form>
				<RDF-ID>C1140621</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>Loss</surface-form>
				<RDF-ID>C1517945</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>muscle</surface-form>
				<RDF-ID>C0026845</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>control</surface-form>
				<RDF-ID>C0243148</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>atrophy</surface-form>
				<RDF-ID>C0333641</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>shrinking</surface-form>
				<RDF-ID>C0205230</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>muscles</surface-form>
				<RDF-ID>C0026845</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>foot</surface-form>
				<RDF-ID>C0016504</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>leg</surface-form>
				<RDF-ID>C1269079</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>Thickened</surface-form>
				<RDF-ID>C0205400</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>nerve bundles</surface-form>
				<RDF-ID>C0549154</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>skin</surface-form>
				<RDF-ID>C0037267</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>legs</surface-form>
				<RDF-ID>C1140621</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>muscle biopsy</surface-form>
				<RDF-ID>C0185283</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>nerve</surface-form>
				<RDF-ID>C1280541</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>biopsy</surface-form>
				<RDF-ID>C0005558</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>diagnosis</surface-form>
				<RDF-ID>C0011900</RDF-ID>
			</annotation>
		</annotations>
		<triples>
			<triple>C1140621 LOCATION_OF C0016504</triple>
			<triple>C0016504 LOCATION_OF C0333641</triple>
			<triple>C0549154 LOCATION_OF C0333641</triple>
			<triple>C1280541 LOCATION_OF C0005558</triple>
		</triples>
	</Sentence>
	<Sentence>
		<value>Eventually the disease may cause disability.</value>
		<pattern_crowdsource> Eventually the [[disease|C0012634]] may cause [[disability|C0231170]].		</pattern_crowdsource>
		<annotations>
			<annotation>
				<surface-form>disease</surface-form>
				<RDF-ID>C0012634</RDF-ID>
			</annotation>
			<annotation>
				<surface-form>disability</surface-form>
				<RDF-ID>C0231170</RDF-ID>
			</annotation>
		</annotations>
		<triples>
			<triple>C0012634 CAUSES C0231170</triple>
		</triples>
	</Sentence>

</Document>

redpajama_set_name: RedPajamaGithub


text:
Jaina Solo
Jaina was one of the most talented Jedi of her generation, owing to her family's bloodline. She had a strong relationship with her twin brother Jacen and was frequently regarded as such, at least in terms of a Force link.
The Yuuzhan Vong attack on Republic territory was a terrible experience for Jacen, and he subsequently altered. It was Jaina who ultimately confronted her brother in a lightsaber duel, killing him as a result. She later married military leader Jagged Fel, who sought to work with the government in cooperation.


text:
Q: Trying to reproduce a clipping effect with slanted margin in CSS Answer not found in source above.

Effect to Reproduce
Looks like two stacked objects (img & div) with one sliced and arranged on top. Notice that the text also follows the slanted margin along the slice. What should I use to get the diagonal slice, and what should I use to achieve the slanted margin?

I suspect I need to use clip-path or clipPath, but not sure how to achieve a polygon clip based on the specific geometry of the image or div, whichever is clipped.

A: Checkout shape-outside css property to have the text follow slanted margin. read more here
.element {
  float: left;
  shape-outside: circle(50%);
  width: 200px;
  height: 200px;
}

You can use clip-path to clip the image. Read more here
img {
  width: 200px;
  height: 200px;
  background: url('/path/to/image') center no-repeat;
  clip-path: polygon(0% 100%, 100% 100%, 0% 0%);
}