text:
\section{Introduction}

Phylogenetic trees  are used to represent evolutionary relationships
among some taxa in many fields, such as biology and philology. Unfortunately,
methods to reconstruct phylogenetic trees generally do not work for large
numbers of taxa; so it would be useful  to have a criterion to say if, given a
collection of
phylogenetic trees with overlapping sets of taxa, there exists a (super)tree
``including'' all the trees in the given collection. In this case we say that
the collection is ``compatible''.
The literature on the ``supertree problem'' is wide. We quote only few of the
known results.

One of the first results is due to Colonius and Schultze: in  \cite{C-S} they
gave a criterion to say if, given a finite set $X$, a system of trees ${\cal P}=
\{T_i\}_i$ with leaf sets $L(T_i) \in {X \choose 4}$
can be  amalgamated into a supertree, that is, if there exists a tree $T$ with
$L(T)=X$  such that $T$ restricted to ${L(T_i)}$ is equal to $T_i$.  Obviously,
a tree with leaf set
$\{a,b,c,d\}$ is determined by the partition (called ``quartet'')  of
$\{a,b,c,d\}$ into the cherries; Colonius and Schultze defined three properties,
thinness, transitivity and saturation, that are necessary and sufficient for a
quartet system to be treelike.

In \cite{BBDS} the authors suggest a polynomial time algorithm that,
given a collection of trees,  produces a supertree, if it exists and
under some conditions.

We quote also the papers \cite{BS} and \cite{GSS}, where the authors studied
closure rules among compatible trees, i.e. rules that, given a compatible
collection of trees, determine some other trees not in the original collection.

Finally, in 2012, Gr\"{u}newald
gave a sufficient criterion for a set of binary phylogenetic trees to be
compatible; precisely he proved that, if ${\cal P}$ is a finite collection of
phylogenetic trees and the cardinality of the union of the leaf sets of the
elements of ${\cal P}$ minus $3$ is equal to the sum of the  cardinalities
of the set of the interior edges of the  elements of ${\cal P}$, then ${\cal
P}$
is compatible (see \cite{Gru}).

A possible variant of the supertree problem is to fix the cardinality of the
leaf sets of the trees in the given collection.
In this paper we consider this problem in the case the cardinality of the leaf
set
of every tree in the given collection is $5$.
Obviously a tree with the cardinality of the leaf set equal to $5$ is given by a
 partition (called ``quintet'')  of the leaf set  into the cherries and  the
complementary of the union of the cherries.
We define three properties, analogous to the ones for quartets, that are
necessary and sufficient for a quintet system to be treelike.

\section{Notation and recalls}

\begin{defin}
$\bullet $ Let $Y$ be a set. A partition of $Y$ into $k$ subsets
of cardinality $n_1, \ldots ,n_k$, with $n_1 \geq \ldots \geq n_k$, is said a partition of kind $(n_1, \ldots, n_k)$.

$\bullet$
Let $X$ be set.

A partition of a $4$-subset $Y$ of $X$ is said a {\bf quartet}  (on $Y$) in $X$
if its kind is one of the following: $(2,2)$,  $(4)$.

A partition of a $5$-subset $Y$ of $X$ is said a {\bf quintet}  (on $Y$) in $X$
if its kind is one of the following: $(2,2,1)$, $(3,2)$, $(5)$.

\end{defin}

\begin{nota}
Throughout the paper, $X$ will denote a finite nonempty set.
\end{nota}

\begin{defin}

 Let $T$ be a tree.

 $\bullet$ We denote by $L(T)$ the leaf set of $T$.

 $\bullet$ For any  $S \subset L(T)$, we denote by $T|_S$ the minimal subtree
of $T$ whose vertex set   contains $S$.

$\bullet$
We say that two leaves  $i$ and $j$ of $T$ are  neighbours
if in the path from $i$ to $j$ there is only one vertex of  degree  greater than
$2$;
furthermore, we say that  $C \subset L(T)$ is a {\bf cherry} if any $i,j \in C$
are neighbours.

$\bullet $ We say that a cherry  is {\bf complete}
 if it is not strictly  contained in another cherry.
\end{defin}

\begin{defin}
A {\bf phylogenetic $X$-tree} $(T ,\varphi ) $ is a finite tree $T$ without
vertices of degree $2$ and endowed with a bijective function
$\varphi: X \rightarrow L(T) $.

{\bf The quartet system $S$ in $X$  associated to a phylogenetic $X$-tree} is
the
quartet system defined as follows: for any $a,b,c,d \in X$,

$(a,b \, |\, c,d) \in S$ if and only if $\{a,b\}$ and $\{c,d\}$ are complete
cherries of $T|_{\{a,b,c,d\}}$,

$(a,b , c,d) \in S$ if and only if $T|_{\{a,b,c,d\}}$  is a star tree.

{\bf The quintet system $S$ in $X$  associated to a phylogenetic $X$-tree} is
the
quintet system defined as follows: for any $a,b,c,d,e \in X$,

$(a,b \, |\, c,d\,|\, e) \in S$ if and only if
 $\{a,b\}$ and $\{c,d\}$ are complete cherries of $T|_{\{a,b,c,d,e\}}$,

$(a,b \, |\, c,d, e) \in S$ if and only if
 $\{a,b\}$ and $\{c,d,e\}$ are complete cherries of $T|_{\{a,b,c,d,e\}}$,

$(a,b , c,d,e) \in S$ if and only if $T|_{\{a,b,c,d,e\}}$  is a star tree.

Given a quartet system (respectively  a quintet system) $S$ in $X$ and
 a quartet (respectively a quintet) in  $X$, we  often write either simply ``$P$'' or ``$P$ holds''
instead of writing ``$ P \in S$'' when it   is clear from the context the system
which we are referring to.

\end{defin}

\begin{defin} Let $S$ be a  quartet system  in $X$.

$\bullet$
We say that $S$ is {\bf saturated} if the following implication holds
for any $a_1, a_2, b_1, b_2, x \in X$:

$(a_1, a_2 \, | \, b_1, b_2) \Rightarrow (a_1, x \,|\, b_1, b_2 ) \vee (a_1, a_2
\,|\, b_1, x )$.

$\bullet$
We say that $S$ is {\bf transitive} if the following implication holds
for any $a_1, a_2, b_1, b_2, x \in X$:

$(a_1, x \, | \, b_1, b_2)  \wedge (a_2, x \, | \, b_1, b_2) \Rightarrow (a_1,
a_2 \,|\, b_1, b_2 ) $.

$\bullet $
We say that $S$ is {\bf thin} if,
for any $4$-subset $Y$ of $X$, there exists only one quartet on $Y$ in $S$.
\end{defin}

As we have already said in the introduction,
Colonius and Schultze characterized treelike quartet systems. The statement
we recall here is the one in  \cite{Dresslibro}.

\begin{thm} \label{CS}
 Let $S$ be a quartet system in $X$; we have that  $S$ is the quartet system of
a phylogenetic  $X$-tree
if and only if $S$ is thin, transitive and saturated.
\end{thm}

\begin{nota} Let $a_1, a_2 ,  b_1, b_2 , b_3 \in X$ and
let $Q$ be a  quintet system in $X$.

We write $(a_1, a_2 \,|\, \overline{ b_1, b_2, b_3})$ instead of
  $$ (a_1, a_2 \,|\,  b_1, b_2, b_3) \vee (a_1, a_2 \,|\,  b_1, b_2 \,|\, b_3)
   \vee (a_1, a_2 \,|\,  b_1, b_3 \,|\, b_2)
   \vee (a_1, a_2 \,|\,  b_2, b_3 \,|\, b_1)
  $$

\end{nota}

\begin{defin} Let $Q$ be a  quintet system in $X$.

$\bullet$
We say that $Q$ is {\bf saturated} if the following implications hold
for any $a_i, b_j, c, x \in X$:

\smallskip

(i) $(a_1, a_2 \,|\, b_1, b_2 \,|\,c)  \Rightarrow $
$(a_1, a_2 \,|\, b_1, b_2 \,|\,x)  \vee (a_1, x \,|\, b_1, b_2 \,|\,c)  \vee
(a_1, a_2 \,|\, b_1, x \,|\,c) $,

\smallskip

(ii) $(a_1, a_2 \,|\, b_1, b_2 , b_3 )  \Rightarrow (a_1, x \,|\, b_1, b_2 ,
b_3)  \vee (a_1, a_2 \,|\, \overline{ b_1, b_2, x})$

\smallskip

(iii)  $(a_1, a_2 , a_3, a_4 , a_5 )  \Rightarrow $

$(a_1, a_2 , a_3, a_4 , x)
 \vee (a_1, x \,|\, a_2, a_3 , a_4 )
\vee (a_2, x \,|\, a_1, a_3 , a_4 )
\vee (a_3, x \,|\, a_1, a_2 , a_4 )
\vee (a_4, x \,|\, a_1, a_2 , a_3 )  $

$\bullet$
We say that $Q$ is {\bf transitive} if the following implications hold
for any $a_i, b_j, c_k, x \in X$:

(i)  $(a_1, a_2 \, |\, b_1, x \,| \, c_1) \wedge  (a_1, a_2 \, |\, b_1, x \,| \,
c_2)  \Rightarrow  (a_1, a_2 \, |\, \overline{ c_1, c_2 , b_1})$,

(ii) $(a_1, a_2 \, |\, b_1, x \,| \, c_1) \wedge  (a_1, a_2 \, |\, b_2, x \,| \,
c_1)  \Rightarrow $ $(a_1, a_2 \, |\, b_1, b_2 \,| \, c_1) $

(iii) $(a_1, x \, |\, b_1, b_2 , b_3) \wedge  (a_2, x \, |\, b_1, b_2 , b_3)
\Rightarrow $ $(a_1, a_2 \, |\, b_1, b_2 , b_3) $

(iv) $(a_1, a_2 \, |\, b_1, b_3 , x) \wedge  (a_1, a_2 \, |\, b_2, b_3, x)
\Rightarrow $ $(a_1, a_2 \, |\, b_1, b_2 , b_3) \vee  (a_1, a_2 \, |\, b_1 , b_2
\,|\, b_3)  $

(v) $(a_1, a_2 \, |\, b_1,x,  b_2 ) \wedge  (a_1, a_2 \, |\, b_1, x \,|\, b_3)
\Rightarrow $ $(a_1, a_2 \, |\, b_1, b_2 \,|\, b_3)$

(vi) $(a_1, a_2 \, |\, b_1, b_2 \,|\, x)  \wedge
(a_1, a_2 \, |\, b_1,  b_3,x )    \Rightarrow $ $(a_1, a_2 \, |\, b_1, b_2 \,|\,
b_3)$

$\bullet $
We say that $Q$ is {\bf thin} if,
for any $5$-subset $Y$ of $X$, there exists only one quintet on $Y$ in $Q$ and,
for any $a,b,c,d,x,y \in X$,

(i) $(a,b \,| \,c,x \,|\, d) \wedge (a,c\,| \,b,y \,| \, d)$ is impossible,

(ii)  $(a,b \,| \,c, d,x) \wedge (a,y \,| \,b, c,d)$ is impossible,

(iii) $(a,b \,| \,c,x \,|\, d) \wedge (a,c,d \,| \,b,y)$ is impossible,

(iv)  $(a,x \,| \,b,c, d) \wedge (a,d \,| \,b,c \,|\, y)$ is impossible.

\end{defin}

Both for quartet systems and quintet systems, we will write TTS instead of thin, transitive and saturated.

\section{Characterization of treelike quintet systems}

Our aim is the prove that a quintet system is treelike if and only if
it is TTS.

First of all, we need to define the quartet system associated to a quintet system
and the quintet system associated to a quartet system.

\begin{defin}\label{defi:quartet S coming from quintet Q}
 Given a TTS quintet system $Q$ on  $X$,
 let $S$ be the quartet  system  defined as follows: for any $a,b,c,d\in X$,
we have that
 $(a,b\, |\,c,d) \in S$ if and only if there exists $y\in X$ for which at least one of
the following instances
 occurs:

 $ (i) \; (a,b\, |\,c,d\, |\,y) \in Q$,

 $ (ii) \; (a,b\, |\,c,d,y) \in Q$,

 $(iii) \; (a,b\, |\,c,y\, |\,d) \in Q $,

 $(iv) \; (a,b\, |\,d,y\, |\,c) \in Q$,

 $(v) \; (c,d\, |\,b,y\, |\,a) \in Q$,

 $(vi) \; (c,d\, |\,a,y\, |\,b) \in Q$,

 $(vii) \; (a,b,y\, |\,c,d) \in Q$.

We say that $S$ is {\bf the quartet system associated to the quintet system $Q$}.
\end{defin}

\begin{defin}
Let $S$ be a TTS quartet system in $X$.
Let $Q'$ be the quintet system in $X$ defined as follows:

$(a, b\, |\, c,d \,| \, e ) \in Q' $ if and only if $(a, b\, |\, c,d  ) ,
(a, b\, |\, c,e  ),
(a, e\, |\, c,d  ) \in S $

$(a, b\, |\, c,d , e ) \in Q' $ if and only if $(a, b\, |\, c,d  ) ,
(a, b\, |\, c,e  ),
(a, e, c,d  ), (b, c,d  ,e) \ \in S $

$(a_1, a_2, a_3,a_4 , a_5 ) \in Q' $ if and only if $(a_1,...., \hat{a}_i,.... ,
a_5 )  \in S $ for any $ i \in \{1,...., 5\}$.

We say that $Q'$ is {\bf the quintet system associated to the quartet system}
$S$.
\end{defin}

The sketch of the proof of our result is the following: given a TTS quintet
system $Q$, we will show that the associated quartet system $S$ is TTS;
so there exists a phylogenetic $X$-tree $(T,\varphi)$ inducing $S$.
We will show that the quintet system associated to $(T, \varphi)$
is exactly $Q$ and this will end the proof.

First, we have
to state two lemmas which will be useful in the remainder of the paper.

\begin{lem}\label{lem: unione di lemmi}
 Let $Q$ be a TTS  quintet system  in $X$. For any  $a,b,c,d,x,y \in X$, it is
not possible the simultaneous  occurrence of $(A) \; (a,b \, | \, c,x \, |\, d
)$ and any of the following:

$(B) \; (a, c \,|\, b, d \, |\, y)$,

$(C) \; (a , c \, | \, d , y \, |\, b)$,

$(D) \; (b , d \, | \, c , y \, |\, a)$,

$(E) \; (y , c \, |\, a ,b , d)$,

$(F) \; (a , c \, |\, b ,d, y)$,

$(G) \; (a , d \, |\, b ,c , y)$,

$(H) \; (d , y \, |\, a ,b, c)$.
\end{lem}
\begin{proof}
 As $Q$ is saturated, $(A)$ implies at least one of the following cases:

$(A.1)  \; (a,y \, |\, c,x \,| \, d)  \;\wedge \;(y,b \, |\, c,x \,| \, d),  $

$(A.2)  \; (a,b \, |\, c,y \,| \, d) \;\wedge \;(a,b \, |\, x,y \,| \, d),$

$(A.3)  \; (a,b \, |\, c,x \,| \, y)$.

We claim that $(A)\wedge(B)$ cannot hold. As $Q$ is saturated,
$(B)$ implies at least one of the following:

$(B.1)  \; (a,x \, |\, b,d \,| \, y)  \;\wedge \;(x,c \, |\, b,d \,| \, y),  $

$(B.2)  \; (a,c \, |\, x , d \,| \, y)\;\wedge \;(a,c \, |\, b,x \,| \, y),$

$(B.3)  \; (a,c \, |\, b,d \,| \, x)$.

Since $Q$ is thin, each of $(A)\wedge(B.3)$,
$(A.1)\wedge (B.1)$, $(A.1)\wedge (B.2)$,  $(A.2)\wedge (B)$, $(A.3)\wedge (B.2)$ is impossible. As $Q$ is
transitive, $(A)\wedge(A.3)$ implies
$(a, b \, | \, \overline{d, x , y})$ which contradicts $(B.1)$; thus the claim is proved.

We now show that $(A)\wedge (C)$ cannot hold. As $Q$ is saturated, $(C)$ implies at least one of the following:

$(C.1) \; (a, x \, | \, d , y \, | \, b) \; \wedge \;(x, c \, | \, d , y \, |
\,
b) ,$

$(C.2) \; (a, c \, | \, d , x \, | \, b)$,

$(C.3) \; (a, c \, | \, d , y \, | \, x)$.

Since $Q$ is thin, each of  $(A)\wedge (C.2)$,
$(A.1)\wedge (C.1)$, $(A.1)\wedge (C.3)$,  $(A.2)\wedge(C)$ is impossible.
As $Q$ is transitive, $(A)\wedge (A.3)$ implies $ (a,b \, |\, \overline{c, d,y}
)$, which contradicts $(C)$, yielding the claim.

We prove that $(A)\wedge (D)$ cannot hold. As $Q$ is saturated, $(D)$ implies at least one of the following:

$(D.1) \; (x, d \, | \, c , y \, | \, a) \; \wedge \; (b, x \, | \, c , y \, |
\, a)$,

$(D.2) \; (b, d \, | \, c , x \, | \, a)$,

$(D.3) \; (b, d \, | \, c , y \, | \, x) $.

Since $Q$ is thin, it is impossible to have each of $(A)\wedge(D.2)$,
$(A.1)\wedge(D.1)$, $(A.1)\wedge (D.3)$, $(A.2)\wedge (D)$. As $Q$ is
transitive, $(A)\wedge(A.3)$ implies $(a,b \,
|\,
\overline{c, d,y} )$, which contradicts $(D)$, yielding the claim.

We claim that $(A)\wedge(E) $ is impossible.  As $Q$ is saturated, $(E)$ implies

$   (x, c \, | \, a , b , d)\; (E.1) \quad \vee \quad [(y , c\, | \, \overline{
a , b , x}) \: (E.2) \; \wedge \; (y , c \, | \, \overline{ a ,d , x}) \:
(E'.2)] $.

The thinness of $Q$ excludes each of  $(A)\wedge
(E.1)$, $(A.1)\wedge (E'.2)$, $(A.2)\wedge (E)$,  $(A.3)\wedge
(E.2)$, yielding the claim.

We claim that $(A)\wedge (F)$ cannot hold. As $Q$ is saturated, $(F)$ implies at least  one of the following:

$(F.1)\; (a,x \, |\,b,d, y) \; \wedge \; (c,x \, |\, b,d,y)  $,

$(F.2)\; (a,c \, |\, \overline{b,d,x}) $.

As $Q$ is thin, each of $(A)\wedge(F.2)$, $(A.1)\wedge(F.1)$,
$(A.2)\wedge(F.1)$ is impossible. Moreover, since $Q$ is transitive,  $(A.3)\wedge(A)$
implies $(a,b \, |\,
\overline{d,x,y}) $, which contradicts $(F.1)$ by the thinness of $Q$, so the
claim follows.

We claim that $(A)\wedge (G)$ cannot hold. As $Q$ is saturated, $(G)$ implies at least one of the following:

$(G.1)\; (a, x \, | \, b ,c ,y)\; \wedge\;  (d, x \, | \, b ,c ,y)$,

$(G.2)\; (a , d \, | \, \overline{b, c, x})$.

By the thinness of $Q$, each of $(A)\wedge(G.2)$, $(A.1)\wedge
(G.1)$, $(A.2)\wedge (G)$,  $(A.3)\wedge (G.1)$ is impossible, so we get the claim.

We claim that $(A)\wedge(H)$ cannot hold. As $Q$ is saturated, $(H)$ implies at least  one of the following:

$(H.1)\; (d, x \, | \, a, b ,c)$,

$(H.2)\; (d, y \, | \, \overline{a, c, x})$.

By the thinness of $Q$, each of $(A)\wedge(H.1)$,
$(A.1)\wedge(H.2)$,  $(A.2)\wedge (H)$ is impossible. Moreover, since $Q$ is transitive,
$(A.3)\wedge(A)$
implies $(a,b \, |\,
\overline{d,c,y}) $, which contradicts $(H)$ by the thinness of $Q$, so the
claim follows.
\end{proof}

\begin{lem}\label{lem: osservazione nuova}
 Let $Q$ be a TTS  quintet system  in $X$. For any  $a,b,c,d,x,y \in X$, it is
not possible the simultaneous occurence of
 $  (A) \;\;\; (a, c \, | \, b , d , y )   $ and any of the following:

$(B) \;(b , x \, | \, a ,c ,d)$,

$(C) \; (a,b \, |\, c,d \,| \, x)  $.
\end{lem}

\begin{proof}
 As $Q$ is saturated,
 $(A)$ implies at least one of the following:

$(A.1) \; (a , x \, |\, b, d, y) \wedge (c, x \, |\, b, d, y)$,

$(A.2) \; (a ,c \, | \, \overline{b , d, x})$.

and $(B)$ implies  at least one of the following:

 $(B.1)  \;  (b, y  \,|\, a,c ,  d), $

$ (B.2) \; (b, x  \,|\, \overline{ a, d, y})   $.

As $Q$ is thin, each of $(B)\wedge (A.2)$, $(B.1)\wedge (A)$,
$(B.2)\wedge (A.1)$ is impossible, which concludes the proof that
$(A) \wedge (B)$ cannot hold.

Since $Q$ is saturated, from $(C)$ we get at least one of the following:

$(C.1) \; (a , y \, | \, c , d \, |\, x)\; \wedge \;(b , y \, | \, c , d \,
|\,x)$,

$(C.2) \; (a, b \, | \, c , y \, | \, x ) \; \wedge \; (a, b \, | \, d , y \, |
\, x )$,

$(C.3) \; (a , b \, | \, c , d \, | \, y)$.

By the thinness of $Q$, each of  $(C)\wedge(A.2)$,
$(C.1)\wedge(A.1)$, $(C.2)\wedge(A.1)$ cannot hold.
Moreover $(C.3)\wedge (C)$ implies $(a,b \;|\; \overline{d, x,y})$, which contradicts $(A.1)$ and this conludes the proof that
$(A) \wedge (C)$ cannot hold.
\end{proof}

\begin{prop} \label{esistequalsiasi}
Let $Q$ be a TTS quintet system  in $X$.  Let $a_1, a_2, b_1,b_2 \in X$.

There exists $x \in X$ such that at least one of the following holds:

$(1) \; (a_1, a_2 \, |\, b_1, b_2 \, | \, x)  $,

$(2) \;  (a_1, a_2 \, |\, b_1, b_2 , x)  $,

$(3) \; (a_1, a_2 , x \, |\, b_1, b_2 )  $,

$(4) \;(a_1, a_2 \, |\, b_1, x \, |\, b_2)  $,

$(5) \; (a_1, a_2 \, |\, b_2, x \, |\, b_1)  $,

$(6) \;  (a_1, x \, |\, b_1, b_2 \, |\, a_2)  $,

$(7) \;  (a_2, x \, |\, b_1, b_2 \, |\, a_1)  $

if and only if for any $x \in X$ at least one of (1),....., (7) holds.

\end{prop}

\begin{proof}
$\Leftarrow $ Obvious.

$\Rightarrow$ Suppose, contrary to our claim, that
there exists $y \in X$ such that no one of (1),...., (7) holds (with $y$ instead
of $x$).
So we must have at least one of the following:

$(8) \;(a_1, a_2 ,  b_1, b_2 , y)  $

$ (9)  \;(a_i , b_j \, |\, a_l, b_r , y)  $ for  some $i,l,j,r$ with
$\{i,l\}=\{1,2\}= \{j,r\}$,

$(10)  \;(a_i, y \, |\, a_l, b_1, b_2 )  $ for   some $i,l$ with
$\{i,l\}=\{1,2\}$,

$(11) \;(b_i, y \, |\, b_l, a_1, a_2)  $ for   some $i,l$ with
$\{i,l\}=\{1,2\}$,

$(12)  \;(a_i, b_j \, |\, a_l, b_r \, |\, y)  $ for   some $i,l,j,r$ with
$\{i,l\}=\{1,2\}= \{j,r\}$,

$(13)  \;(a_i, y \, |\, a_l, b_j \, |\, b_r)  $ for   some $i,l,j,r$
with$\{i,l\}=\{1,2\}= \{j,r\}$,

$(14) \; (b_j, y \, |\, b_r, a_i \, |\, a_l)  $ for   some $i,l,j,r$
with$\{i,l\}=\{1,2\}= \{j,r\}$.

 \medskip

 \underline{Suppose (1) holds.}
By the saturation of $Q$, it  implies at least one of the following:

$(1.1) \;  (a_1,y \,|\, b_1,b_2\,|\; x) \wedge (a_2,y \,|\, b_1,b_2\,|\; x), $

$(1.2) \; (a_1 ,a_2 \,|\, b_1,y \, | \, x ) \wedge
(a_1 ,a_2 \,|\, b_2,y \, | \, x ), $

$(1.3) \;  (a_1 ,a_2 \,|\, b_1 , b_2 \,|\, y ) $.

$\bullet $ Suppose  (8) holds.
Since $Q$ is saturated, (8) implies $(a_1,a_2, b_1,b_2, x) \vee
(a_i ,x \,|\, a_j , b_1, b_2 ) \vee (b_i ,x \,|\, b_j , a_1, a_2 )  $ for some
$i,j$ with   $\{i,j\}=\{1,2\}$, which contradicts (1).

$\bullet $ Suppose  (9) holds. We can suppose that $i=j=1$ and $l=r=2$ in (9),
so $ (a_1 , b_1 \, |\, a_2, b_2 , y)  $ holds. We get a contradiction by
Lemma \ref{lem: osservazione nuova}, case (C).

$\bullet $ Suppose  (10) holds. We can suppose that $i=1$ in (10), so $ (a_1 , y
\, |\, a_2, b_1 , b_2)  $ holds, but, by the thinness of $Q$, this is impossible.

$\bullet $ Suppose  (11) holds.
This case is analogous to the previous case (by swapping $(a_1, a_2)$ with
$(b_1, b_2)$).

 $\bullet $ Suppose  (12) holds. We can suppose that $i=j=1$ in (12), so $ (a_1
,b_1 \, |\, a_2, b_2 \,|\, y)  $ holds.
By the saturation of $Q$,  this implies at least one of the following:

$(12. 1)  \;  (a_1, x  \,|\, a_2,b_2\, |\,y)  \wedge  (b_1, x  \,|\, a_2,b_2\,
|\,y),$

$ (12.2) \;   (a_1, b_1  \,|\, a_2,x \,|\, y )  \wedge (a_1, b_1  \,|\, b_2,x
\,|\, y ),
   $

   $ (12.3)   \;  (a_1, b_1 \,|\, a_2,b_2 \, |\,x)    $.

 Observe that (1.1) contradicts  both (12.1) and (12.2),
 (1.2) contradicts both (12.1) and (12.2),
 (1.3) contradicts (12), and, finally, (12.3) contradicts (1).

$\bullet $ Suppose  (13) holds.
We can suppose $i=r=1$, so $ (y, a_1  \, |\, a_2, b_2 \,|\, b_1)  $ holds.
   We get a contradiction by Lemma \ref{lem: unione di lemmi}, case $(B)$.

$\bullet $ Suppose  (14) holds.
This case is analogous to  the previous case (by swapping $(a_1, a_2)$ with
$(b_1, b_2)$).

 \underline{Suppose (2) holds.}
By the saturation of $Q$, it  implies at least one of:

$(2.1) \; (a_1,y \,|\, b_1,b_2, x) \wedge (a_2,y \,|\, b_1,b_2, x) , $

$(2.2) \; (a_1 ,a_2 \,|\, \overline{ b_1, b_2 ,  y} )   $.

$\bullet $ Suppose  (8) holds.
By the saturation of $Q$, it  implies:

$  (a_i ,x  \,|\, b_1,b_2, a_j) \;\quad \vee \quad\; (b_r,x \,|\, a_1,a_2, b_l)
\;  \quad \vee \quad
 \; (a_1 ,a_2 , b_1, b_2 ,  x )   $

for some $i,j$ with $\{i,j\} =\{1,2\}$ and
some $r,l$ with $\{r,l\} =\{1,2\}$.
All the possibilities contradict (2).

$\bullet $ Suppose  (9) holds. We can suppose $i=j=1$.
So $ (a_1 ,b_1 \, |\, a_2, b_2 , y)  $ holds.
By the saturation of $Q$, it  implies at least one of the following:

$(9.1) \;  (a_1,x \,|\, a_2,b_2, y) \wedge (b_1,x \,|\, a_2,b_2, y)  $,

$(9.2) \; (a_1 ,b_1 \,|\, \overline{ a_2, b_2 ,  x} )   $.

Observe that (2.2) contradicts (9) and (9.2) contradicts (2). Moreover (2.1)
contradicts (9.1).

$\bullet $ Suppose  (10) holds.  Since $Q$ is thin, we get a contradiction.

$\bullet $ Suppose  (11) holds.
We can suppose $i=1$, and we get a contradiction by Lemma \ref{lem: osservazione
nuova}, case $(B)$.

$\bullet $ Suppose  (12) holds. We get a contradiction by Lemma \ref{lem:
osservazione nuova}, case (C).

$\bullet $ Suppose  (13) holds.
We can suppose $i=r=1$, so $ (y, a_1  \, |\, a_2, b_2 \,|\, b_1)  $ holds.
 We get a contradiction by Lemma \ref{lem: unione di lemmi}, case $(F)$.

$\bullet $ Suppose  (14) holds. We can suppose $j=l=1$, so
$ (y, b_1  \, |\, a_2, b_2  \, |\,  a_1 ) $ holds. We get a contradiction by Lemma \ref{lem: unione di lemmi}, case $(G)$.

 \underline{Suppose (3) holds.}
This case is analogous to the case where (2) holds (swap $(a_1 , a_2)$ with
$(b_1 , b_2)$).

 \underline{Suppose (4) holds.}
 By the saturation of $Q$, it implies at least one of the following:

$(4. 1)  \;  (a_1, y  \,|\, b_1, x \, |\,b_2)
  \; \wedge   (a_2, y  \,|\, b_1, x \, |\,b_2)   $,

$ (4.2) \;   ( a_1 , a_2 \,|\, b_1, y \,|\, b_2 )   $,

$ (4.3)   \;  (a_1 ,a_2 \,|\, b_1, x  \, |\,y)  $.

$\bullet $ Suppose  (8) holds.  By the saturation of $Q$, condition
(8) implies  $ ( a_1, a_2 , b_1, b_2, x)  \vee  ( a_1, x \, |\,  a_2 , b_1, b_2)
\vee  ( a_2, x \, |\,  a_1 , b_1, b_2) \vee  ( b_1, x \, |\,  a_1 , a_2, b_2)
\vee  ( b_2, x \, |\,  a_1 , a_2, b_1)  $, which contradicts (4).

$\bullet $ Suppose  (9) holds.

First case: $ i=j=1$ in (9).  So we have $ ( a_1, b_1  \, |\,
 a_2, b_2, y) $. By Lemma \ref{lem: unione di lemmi}, case $(F)$ we get a
contradiction.

Second case:  $ i=r=1$ in (9).
So we have $ ( a_1, b_2  \, |\,   a_2, b_1, y) $. We get a contradiction
by Lemma \ref{lem: unione di lemmi}, case $(G)$.

 $\bullet $ Suppose  (10) holds.
 We can suppose $i=1$ in (10). Since $Q$ is thin, (4) contradicts (10).

 $\bullet $ Suppose  (11) holds.

 First suppose that $i=1$ in (11). This is impossible by Lemma \ref{lem:
unione di lemmi}, case $(E)$.

 Now suppose that $i=2$ in (11). This is impossible by Lemma \ref{lem: unione di
lemmi}, case $(H)$.

 $\bullet $ Suppose  (12) holds. We can suppose $i=j=1$. By Lemma
\ref{lem: unione di lemmi}, case $(B)$ we get a contradiction.

 $\bullet $ Suppose  (13) holds.
 We can suppose $i=1$.
 If $ r=1$ we get a contradiction by Lemma \ref{lem: unione di lemmi}, case
$(C)$.
 If $r=2$ we get a contradiction by the thinness of $Q$.

 $\bullet $ Suppose  (14) holds.
 We can suppose $l=1$.
 If $j=1$  we get a contradiction by Lemma \ref{lem: unione di lemmi}, case
$(D)$.
 If $j=2$ we get a contradiction by Lemma \ref{lem: unione di lemmi}, case $(C)$.

 \underline{Suppose (5) holds.}  This case is analogous to the case
 where (4) holds (swap $b_1$ with $b_2$).

 \underline{Suppose (6) holds.}  This case is analogous to the case
 where (4) holds (swap $a_1$ with $b_1$ and  $a_2$ with $b_2$).

 \underline{Suppose (7) holds.}  This case is analogous to the case
 where (6) holds (swap $a_1$ with $a_2$).
 \end{proof}

\bigskip

The next goal is to prove that a quartet system  $S$ as in Definition
\ref{defi:quartet S coming from quintet Q} is in fact
TTS.

\begin{prop}\label{prop: quartets induced by quintets are thin}
 A quartet system  $S$ associated to a TTS quintet system $Q$
 as in Definition \ref{defi:quartet S coming from quintet Q} is thin.
\end{prop}
\begin{proof}
 Assume by contradiction that $(a,b\, |\,c,d)\wedge (a,c\,
|\,b,d)$ holds; by  Proposition \ref{esistequalsiasi},
 the hypothesis that $(a,b\, |\,c,d)$ holds
 is equivalent to
 say that, for any $y\in X$, we have:
 \begin{align} \label{conditions one}
  (a,b\, |\,\overline{c,d,y}) \lor (c,d\, |\,\overline{a,b,y}) .
 \end{align}
 Moreover, by Definition \ref{defi:quartet S coming from quintet Q},
 the fact that $(a,c\, |\,b,d)$ means that there exists $x\in X$ such that
 \begin{align} \label{conditions two}
  (a,c\, |\,\overline{b,d,x}) \lor (b,d\, |\,\overline{a,c,x}) .
 \end{align}
If we choose $y=x$ in \eqref{conditions one}, we get a contradiction with \eqref{conditions two} by  the thinness of $Q$.
\end{proof}

\begin{prop}\label{prop: quartets induced by quintets are transitive}
 A quartet system  $S$ associated to a TTS quintet system  $Q$
 as in Definition \ref{defi:quartet S coming from quintet Q} is transitive.
\end{prop}
\begin{proof}
 The goal is to prove $(a_1 , a_2 \, |\, b_1 , b_2 ) \wedge (a_2 , a_3 \, |\,
b_1 , b_2 ) \Rightarrow (a_1 , a_3 \, |\, b_1 , b_2 )$.
 Recall that, by Proposition \ref{esistequalsiasi},
 $(a_1 , a_2 \, |\, b_1 , b_2 )$ means that, for every $x\in X$, at least one of the
following conditions must hold:

 $(1.1) \; (a_1 , a_2 \, |\, b_1 , b_2 \, |\,x )$,

 $(1.2) \;(a_1  , a_2 \, |\, b_1 , b_2 , x) $,

 $(1.3) \; (a_1 , a_2 , x\, |\, b_1 , b_2 )$,

 $(1.4) \; (a_1 ,  a_2 \, |\, b_1 , x \, |\, b_2 )$,

 $(1.5) \; ( a_1 , a_2 \, |\, b_2 , x \, |\, b_1)$,

 $(1.6) \; ( a_1 , x \, |\, b_1 , b_2 \, |\, a_2)$,

 $(1.7) \; ( a_2  , x \, |\, b_1 , b_2 \, |\, a_1 ) $.

 Similarly, by Proposition \ref{esistequalsiasi},
 $(a_2 , a_3 \, |\, b_1 , b_2 )$ means that, for every $x\in X$, at least one of the
following conditions holds:

  $(2.1)\; (a_2 , a_3 \, |\, b_1 , b_2 \, |\,x ) $,

  $(2.2)\; (a_2 , a_3 \, |\, b_1 , b_2 , x) $,

  $(2.3)\;(a_2 , a_3 , x\, |\, b_1 , b_2 ) $,

  $(2.4)\;(a_2 , a_3 \, |\, b_1 , x \, |\, b_2 ) $,

  $(2.5)\;( a_2 , a_3 \, |\, b_2 , x \, |\, b_1) $,

  $(2.6)\;( a_2 , x \, |\, b_1 , b_2 \, |\, a_3) $,

  $(2.7)\;( a_3 , x \, |\, b_1 , b_2 \, |\, a_2 )$.

First we are going to show that, for any $k\in\{1,\ldots , 7\}$,
if there exists $x $ such that
$(1.k)\wedge(2.k) $ holds, then $ (a_1 , a_3 \, |\, b_1 , b_2 )$ holds; then we prove that if there exists $x $ such that one of
the remaining pairings holds, then we get either $(a_1 , a_3 \, |\, b_1 , b_2 )$ or a
contradiction. Observe that, by symmetry, it is sufficient to consider the cases $(1.k)\wedge(2.j) $ with $j >k$.

\underline{Suppose $(1.h)\wedge(2.h)$} for some  $x\in X$ and $h\in\{1,\ldots ,
5\}$; then $(a_1 , a_3 \, |\, b_1 , b_2 ) $ follows from the
transitivity of $Q$ and  Definition \ref{defi:quartet S coming from quintet
Q}.

\underline{Assume $(1.6)\wedge(2.6) $};
since $Q$ is saturated, $(1.6)$ implies that

  $(x , a_3 \, |\, b_1 , b_2 \, |\, a_2 )\; (1.6.1) \quad\vee\quad (a_1 , x \,
|\, b_1 , a_3 \, |\, a_2 )\;(1.6.2)
  \quad\vee\quad (a_1 , x \, |\, b_1 , b_2 \, |\, a_3 ) \;(1.6.3) $

and $(2.6)$ implies that

  $(a_1 , x   \, |\, b_1 , b_2 \, |\, a_3 )\;(2.6.1)\quad\vee\quad (a_2 , x   \,
|\, b_1 , a_1 \, |\, a_3 )\;(2.6.2)
  \quad\vee\quad (a_2 , x   \, |\, b_1 , b_2 \, |\, a_1 ) \;(2.6.3)$.

Each of $(1.6.1)\wedge(2.6)$, $(2.6.3)\wedge(1.6)$,
$(1.6.2)\wedge(2.6.2)$ contradicts the thinness of $Q$; moreover, by Definition \ref{defi:quartet S coming from quintet Q},  each of $(1.6.3)$ and  $(2.6.1)$  implies
$(a_1 ,a_3 \, |\, b_1 , b_2 ) $ and this allows us to conclude.

\underline{The case $(1.7)\wedge(2.7)$} can be recovered from the previous one
by swapping $a_1$ with $a_3$.

\underline{The case $(1.1)\wedge(2.2) $} is
impossible by Lemma \ref{lem: unione di lemmi}, case $(E)$.

\underline{Assume $(1.1)\wedge(2.3)$};
since $Q$ is saturated, from $(2.3)$ we get at least one of the following:

  $(2.3.1)\;(b_1 , a_1 \, |\, a_2 , a_3  ,  x   ) $,

  $(2.3.2)\;(b_1 , b_2 \, |\, x   , a_3  ,  a_1 ) $,

  $(2.3.3)\;(b_1 , b_2 \, |\, x   , a_3 \, |\, a_1 ) $,

  $(2.3.4)\;(b_1 , b_2 \, |\, x   , a_1 \, |\, a_3 ) $,

  $(2.3.5)\;(b_1 , b_2 \, |\, a_1 , a_3 \, |\, x   ) $.

The occurrence of $(1.1) \wedge (2.3.1)$ is impossible
by Lemma \ref{lem: unione di lemmi}, case $(F)$.

Each of $(2.3.2)$, $(2.3.3)$, $(2.3.4)$, $(2.3.5)$ allows to
conclude, by
 Definition \ref{defi:quartet S coming from quintet Q}, that $(a_1 , a_3 \,|\, b_1 , b_2 )$ holds.

\underline{The case $(1.1)\wedge(2.4) $} is
impossible by Lemma \ref{lem: unione di lemmi}, case $(D)$; swapping
$b_1$ with $b_2$, we also get that \underline{the case $(1.1)\wedge(2.5) $} is
impossible.

\underline{Suppose $(1.1)\wedge(2.6) $};
since $Q$ is saturated, $(1.1)$ implies

  $(a_1 , a_3 \, |\, b_1 , b_2 \, |\, x   )\;(1.1.1)\quad\vee\quad
  (a_1 , a_2 \, |\, b_1 , a_3 \, |\, x   )\;(1.1.2)\quad\vee\quad
  (a_1 , a_2 \, |\, b_1 , b_2 \, |\, a_3 ) \;(1.1.3)$

and  $(2.6)$ implies one of    $(2.6.1)$, $(2.6.2)$, $(2.6.3)$  above.
From $(1.1.1)$ as well as from $(1.1.3)$ and  from $(2.6.1)$
one can deduce,  by means of Definition
\ref{defi:quartet S
coming from quintet Q}, that $(a_1 , a_3 \, |\, b_1 , b_2 ) $ holds;
each  of $(2.6.3)\wedge (1.1)$ and $(1.1.2)\wedge (2.6.2)$
contradicts the thinness of $Q$ and this allows us to conclude.

\underline{Suppose $(1.1)\wedge(2.7) $};
since $Q$ is saturated, $(1.1)$ implies one of    $(1.1.1)$, $(1.1.2)$, $(1.1.3)$ and $(2.7)$ implies

  $(a_1 , a_3 \, |\, b_1 , b_2 \, |\, a_2   )\;(2.7.1)\quad\vee\quad
  (a_3 , x \, |\, b_1 , a_1 \, |\, a_2   )\;(2.7.2)\quad\vee\quad
  (a_3 , x \, |\, b_1 , b_2 \, |\, a_1 ) \;(2.7.3)$.

From $(1.1.1)$ as well as from $(1.1.3)$  and
from  $(2.7.3)$ one can deduce that
$(a_1 , a_3 \, |\, b_1 , b_2 ) $ holds. Finally,
the case $(2.7.1)\wedge(1.1)$, as well as the case $(1.1.2)\wedge(2.7.2)$, contradicts the thinness of $Q$ and so we can conclude.

\underline{The case $(1.2)\wedge(2.3) $} is impossible by the thinness of $ Q$.

\underline{The case $(1.2)\wedge(2.4) $} is
impossible by  Lemma \ref{lem: unione di lemmi}, case $(E)$; swapping
$b_1$ with $b_2$, we also have that \underline{the case $(1.2)\wedge(2.5) $}
cannot hold.

\underline{The case $(1.2)\wedge(2.6)$} is impossible by the thinness of $Q$.

\underline{The case $(1.2)\wedge(2.7) $} cannot hold by Lemma \ref{lem:
unione di lemmi}, case $(H)$.

\underline{The case $(1.3)\wedge (2.4) $} cannot hold by Lemma
\ref{lem: unione di lemmi}, case $(G)$.

\underline{The case $(1.3)\wedge(2.5) $}
is analogous to the previous one swapping $b_1$ with $b_2$.

\underline{Assume $(1.3)\wedge(2.6) $};
since $Q$ is saturated,  $(1.3)$ implies

  $(b_1 , a_3 \, |\, a_1 , a_2 , x   )\;(1.3.1)\quad\vee\quad
  (b_1 , b_2 \, |\, \overline{  a_1 , a_2, a_3 }  )\;(1.3.2)$

and,
similarly, $(2.6)$ implies one of $(2.6.k)$, $k=1,2,3$.
From $(2.6.1)$, as well as from $(1.3.2)$,
one can deduce  that $(a_1 , a_3 \, |\, b_1  , b_2 ) $ holds.
Moreover, $(2.6.3)\wedge(1.3)$, as well as  $(2.6.2)\wedge(1.3.1)$, contradicts the thinness of $Q$ and so we conclude.

\underline{Assume $(1.3)\wedge(2.7) $}. Since $Q$ is saturated,
 $(1.3)$ implies one of $(1.3.1)$, $(1.3.2)$ and $(2.7) $ implies
one of $(2.7.1)$, $(2.7.2)$, $(2.7.3)$. Observe that each of
$(1.3.2)$, $(2.7.1)$, $(2.7.3)$ implies $(a_1 , a_3 \, |\, b_1  , b_2 ) $. Moreover,   $(1.3.1) \wedge (2.7.2)$
contradicts the thinness of $Q$.

\underline{Cases $(1.4)\wedge (2.5) $, $(1.4)\wedge (2.6) $, $(1.4)\wedge (2.7) $}  are
impossible by  Lemma \ref{lem: unione di lemmi}, respectively case $(D)$, case $(B)$, case $(C)$.

\underline{Assume $(1.5)\wedge (2.6)$};
then by swapping $b_1$ with $b_2$ one gets back to the case
$(1.4)\wedge(2.6)$.

\underline{Suppose $(1.5)\wedge (2.7) $};
then by swapping $b_1$ with $b_2$ one gets back to the case
$(1.4)\wedge(2.7)$.

\underline{Assume $(1.6)\wedge (2.7) $};
then, by transitivity of $Q$, one gets  $(a_1 , a_3 \, |\, b_1 , b_2 \, |\,
a_2 )$, and by Definition \ref{defi:quartet S coming from quintet Q}
one can deduce $(a_1 , a_3 \, |\, b_1 , b_2 ) $.
\end{proof}

\begin{prop}\label{prop: quartets induced by quintets are saturated}
 A quartet system  $S$ associated to a TTS quintet system $Q$
 as in  Definition \ref{defi:quartet S coming from quintet Q} is saturated.
\end{prop}
\begin{proof}
 Suppose that $(a_1 , a_2 \, |\, b_1  , b_2) $ and fix $x\in X$.
 By Definition \ref{defi:quartet S coming from quintet Q}
 there exists  $z \in X$ such that at least one of the following holds:

  $(1)\; (z\, |\,a_1 , a_2 \, |\, b_1 , b_2 )$,

  $(2)\;(a_1  , a_2 \, |\, b_1 ,  b_2 , z)$,

  $(3)\;(a_1 , a_2 \, |\, b_1 , z \, |\, b_2 )$,

  $(4)\;(a_1 , a_2 \, |\, b_2 , z \, |\, b_1 )$,

  $(5)\;( b_1 , b_2 \, |\, a_1 , z \, |\, a_2)$,

  $(6)\;( b_1 , b_2 \, |\, a_2 , z \, |\, a_1)$,

  $(7)\;( b_1 , b_2 \, |\,a_1 , a_2  , z )$.

The argument consists in showing that any of the  items above implies either $(a_1 , a_2
\, |\, b_1 , x)$ or  $(a_1 , x\, |\, b_1 , b_2)$; since it is repetitive, and uses essentially only Definition
\ref{defi:quartet S coming from quintet Q}, we only give a sample of the whole
argument.

Suppose that $(4)$ holds; then, as $Q$ is saturated, we have:

 $ (a_1 , a_2 \, |\, b_2 , z \, |\, x)\;(4.1)\quad \vee \quad (a_1 , x \, |\,
b_2  , z \, |\, b_1)\;(4.2) \quad \vee \quad (a_1 , a_2 \, |\, b_2 , x \, |\,
b_1)\;(4.3) $.

By $(iv)$ of Definition \ref{defi:quartet S coming
from quintet Q}
with $a=a_1, \, b=a_2 ,\, c=x,\, d=b_2 ,\, y=z $, condition $(4.1)$  entails  $(a_1 , a_2 \,
|\, x , b_2) $; since
also $(a_1 , a_2 \, |\, b_1 , b_2) \in S$, the hypothesis that $S$ is transitive
allows us to get $(a_1 , a_2 \, |\, b_1 , x)$.

By $(iv)$ of Definition \ref{defi:quartet S coming from quintet Q}
with $a=a_1, \, b=x ,\, c=b_1,\, d=b_2 , \, y=z $, case $(4.2)$  entails $(a_1 , x \,|\, b_1 ,b_2)$.

By $(iv)$ of Definition \ref{defi:quartet S coming
from quintet Q}
with $a=a_1, \, b=a_2 ,\, c=b_1,\, d=x ,\, y=b_2 $, case $(4.3)$  entails $(a_1 , a_2 \, |\, b_1 , x)$ as wanted.
\end{proof}

\begin{rem} \label{Q'S}
Let $S$ be the quartet system of a phylogenetic $X$-tree $(T, \varphi)$
 and let $Q'$ be the quintet system in $X$ associated to $S$. Then $Q'$ is the
quintet system of $(T, \varphi)$.
\end{rem}

\begin{rem}  \label{abcdinS} Let
$Q$ be a TTS quintet system in $X$;
call $S$ the quartet system associated to $Q$. We have that  $(a,b,c,d) \in S$
if and only if
\begin{equation} \label{ma} (a,b,c,d,x) \in Q \vee (a,x \,| \, b, c,d) \in Q
\vee (b,x \,| \, a, c,d) \in Q
 \vee (c,x \,| \, a,b,d) \in Q  \vee (d,x \,| \, a, b, c) \in Q  \end{equation}
  for any $x \in X$.
 By Proposition \ref{esistequalsiasi} this holds if and only if
there exists  $x \in X$ such  that (\ref{ma}) holds.
\end{rem}

\begin{prop} \label{Q'=Q}
Let $Q$ be a TTS quintet system in $X$;
call $S$ the quartet system associated to $Q$ and
$Q'$ the quintet system associated to $S$. Then $Q=Q'$.
\end{prop}

\begin{proof}
$\bullet $ First we prove that every partition of  $Q'$ of kind $(2,2,1)$    or
of kind $(2,3)$  is also an element of  $ Q$.

Let $(a,b \, |\, c,d \,|\, e) \in Q'$. Suppose that  $(a,b \, |\, c,d \,|\, e)
\not\in Q$.
Thus one of the following conditions holds:

1) $(a,b,c,d,e) \in Q$; by the definition of $S$, this would imply $(a,b,c,d)
\in S $, which is absurd since, by the definition of $Q'$,
we have that $(a,b \, |\, c,d ) \in S$ (and $S$ is thin).

2) $(x,y\,| \, z, w,u) \in Q$ for $\{x,y,z,w,u\} =\{a,b,c,d,e\}$; by the
definition of $S$, this would imply $(x,z ,w,u) \in S$, which is absurd since,
by the definition of $Q'$,
we have that a partition of kind $(2,2) $ of   $\{x,z,w,u\}$  is in $S$ (and $S$
is thin).

3) A partition of kind $(2,2,1) $
 of   $\{a,b,c,d,e\}$, different from $(a,b \, |\, c,d \,|\, e) $,  is in $Q$;
up to swapping $a$ with $b$ or $c$ with $d$ or $\{a,b\}$ with $\{c,d\}$, we can
suppose  $(a,c\,| \, b, d \,| \, e) \in Q$ or $(a,e\,| \, c, d \,| \, b) \in Q$
or $(a,e\,| \, b, d \,| \, c) \in Q$.

By the definition of $S$, $(a,c\,| \, b, d \,| \, e) \in Q$ would imply $(a,c\,
|\, b,d) \in S$, which is absurd since, by the definition of $Q'$,
we have that $(a,b \, |\, c,d ) \in S$ (and $S$ is thin).

By the definition of $S$, $(a,e\,| \, c, d \,| \, b) \in Q$ would imply $(a,e\,
|\, b,c) \in S$, which is absurd since, by the definition of $Q'$,
we have that $(a,b \, |\, c,e ) \in S$ (and $S$ is thin).

By the definition of $S$, $(a,e\,| \, b, d \,| \, c) \in Q$ would imply $(a,c\,
|\, b,d) \in S$, which is absurd since, by the definition of $Q'$,
we have that $(a,b \, |\, c,d ) \in S$ (and $S$ is thin).

Let $(a,b \, |\, c,d , e) \in Q'$. Suppose that  $(a,b \, |\, c,d , e) \not\in
Q$.
Thus one of the following conditions holds:

1) $(a,b,c,d,e) \in Q$; by Remark \ref{abcdinS}, this would imply $(a,b,c,d)
\in S $, which is absurd since, by the definition of $Q'$,
we have that $(a,b \, |\, c,d ) \in S$ (and $S$ is thin).

2) A partition of kind $(2,3) $
 of   $\{a,b,c,d,e\}$, different from $(a,b \, |\, c,d , e) $,  is in $Q$; up to
making a permutation of   $\{a,b\}$ or of $\{c,d,e\}$, we can suppose  $(a,c\,|
\, b, d , e)
 \in Q$ or $(c,d\,| \, a , b, e) \in Q$.

  The condition $(a,c\,| \, b, d , e) \in Q$ would imply $(a,c \, |\, b,d)
   \in S$, which is absurd since, by the definition of $Q'$, we have that
    $(a,b \, |\, c,d) \in S$

  The condition $(c,d\,| \, a , b, e)  \in Q$ would imply $(c,d \, |\, b,e)
   \in S$, which is absurd since, by the definition of $Q'$, we have that
    $(b ,c,d,e) \in S$.

3) A partition of kind $(2,2,1) $
 of   $\{a,b,c,d,e\}$  is in $Q$;
 up to making a permutation of   $\{a,b\}$ or of $\{c,d,e\}$, we can suppose
 $(a,b\,| \, c, d \,| \, e) \in Q$ or $(a,c\,| \,  d ,e \,| \, b) \in Q$ or
$(a,c\,| \, b, d \,| \, e) \in Q$ or $(c,d\,| \, b, e \,| \, a) \in Q$.

  The condition   $(a,b\,| \, c, d \,| \, e) \in Q$   would imply $(b,e \, |\,
c,d)
   \in S$, which is absurd since, by the definition of $Q'$, we have that
    $(b , c,d,e) \in S$.

  The condition   $(a,c\,| \,  d ,e \,| \, b) \in Q$   would imply $(a,c \, |\,
b,d)
   \in S$, which is absurd since, by the definition of $Q'$, we have that
    $(a,b \,|\, c,d) \in S$.

  The condition   $(a,c\,| \, b, d  \,| \, e) \in Q$   would imply $(a,c \, |\,
b,d)
   \in S$, which is absurd since, by the definition of $Q'$, we have that
    $(a,b \,|\, c,d) \in S$.

  The condition   $(c,d\,| \, b, e \,| \, a) \in Q$  would imply $(c,d \, |\,
b,e)
   \in S$, which is absurd since, by the definition of $Q'$, we have that
    $(c,d,b,e) \in S$.

$\bullet $ Let us  prove  that every partition of  $Q$ of kind $(2,2,1)$ or of
kind $(2,3)$     is also an element of  $ Q'$.

Let $(a,b \, |\, c,d \,|\, e) \in Q$.
By the definition of $Q'$, we have  that   $(a,b \, |\, c,d \,|\, e) \in Q'$ if
and only if  $(a,b \, |\, c,d ) \in S \wedge (a,b \, |\, c,e) \in S \wedge (a,e
\, |\, c,d ) \in S  $
and this follows from the fact that $(a,b \, |\, c,d \,|\, e) \in Q$ and the
definition
of $S$.

Let $(a,b \, |\, c,d , e) \in Q$.
By the definition of $Q'$, we have  that   $(a,b \, |\, c,d , e) \in Q'$ if and
only if  $(a,b \, |\, c,d ) \in S \wedge (a,b \, |\, c,e) \in S \wedge (a,e ,
c,d ) \in S
\wedge (b , c,d ,e) \in S  $
and this follows from the fact that $(a,b \, |\, c,d , e) \in Q$ and the
definition
of $S$.
\end{proof}

\begin{thm} Let $Q$ be a quintet system in $X$; we have that  $Q$ is the quintet
system of a phylogenetic $X$-tree
if and only if $Q$ is TTS.
\end{thm}

\begin{proof}
$\Rightarrow$ Very easy to prove.

$\Leftarrow$
By Propositions \ref{prop: quartets induced by quintets are thin}, \ref{prop:
quartets induced by quintets are saturated} and \ref{prop: quartets induced by
quintets are transitive}, the quartet system $S$ associated to $Q$ is TTS. So,
by Theorem \ref{CS},
there exists an $X$-tree $(T, \varphi)$  whose quartet system is $S$. Let $Q'$
be the quintet system associated to $S$.  By Remark \ref{Q'S}, we have
that $Q'$ is the quintet system associated to $(T, \varphi)$. By Proposition
\ref{Q'=Q},   we have that $Q=Q'$, hence $Q$
 is the quintet system associated to $(T, \varphi)$.
\end{proof}

{\small

redpajama_set_name: RedPajamaArXiv


text:
Expect Better : GW2
MMOland, the one place where you always get a second chance to make a first impression. And a third. As many as you want, really...
GW2's "Battle For Lion's Arch" update dropped four days ago and it's been bouncing ever since. Every day is Patch Day as ArenaNet scramble to get the wheels back on Scarlet's tumbril for the Living Story Season One Grand Finale. Jeromai has a very detailed guide to the core events and Ravious takes a considered look at what works and what doesn't.
If there's a single take-away it has to be that these events simply need better testing. Or maybe just testing... A rolling bi-weekly schedule is clearly and evidently incompatible with current resources. There may be multiple teams working separate development cycles for individual content drops but however many teams there are and however long each is being given to prepare it's plainly not enough.
The patience, the stoicism even, of GW2 players is striking. The reaction to game-breaking bugs and show-stopping design failures tends to be a verbal shrug and a flip comment: "What did you expect? This is GW2". There's a devastating gulf between what players now expect from ANet, based on what they've been given this first year-and-a-half, and what they thought they'd be getting down those five long years of rhetoric, "Iteration, Iteration, Iteration" and "It'll be done when it's done".
Anyone fancy a game of 5-a-side?
It says a huge amount for how invested players are in the game that they not only put up with these repeated and predictable failures to provide content that works first time round but come back, over and over again, for more. It also says plenty about how strong the foundation that underpins these rickety, makeshift sideshows must be that it keeps everyone from open revolt.
The opening three days of the current update risked pushing the patience of every player close to the edge and some duly slipped over. While tempers can sometimes, understandably, get a little frayed when a large-scale event isn't going well, this is perhaps the first time I've had to wait through twenty or thirty minutes of increasingly vicious and intolerant argument before an event has even started. For a whole evening, starting every hour, on the hour, that gets wearing.
It isn't that the mechanics of the events, as first received, were too challenging. They were neither difficult to understand, nor to learn, nor to execute. It's not even that the events that appeared on Tuesday were particularly poorly-designed. It's that they just didn't work.
Scarlet's Knights either dropped barrow-loads of loot, which ANet didn't like, or none at all, which players liked even less. Moreover, some players got the barrow-loads for the same event where other players got the none at all and no-one could really figure out why. Even when the Knights were downed, which wasn't as often as it should have been in the opinion of many, and people managed to get in to the airship to fight Scarlet, the signature battle could also stall, bugging into an unwinnable state. And then there was Six Minutes To Knightfall.
Six Minutes To Knightfall is an achievement that requires you to "Defeat all three assault knights within six minutes after the event starts". The semantic inexactitude of that description itself led to some argument as players debated whether it meant "down all three knights in the first six minutes of the event" or "down all three knights in the same six-minute window at any point during the event". Then there was the question of whether an individual player had to down all three knights in the same event or whether downing a different knight inside six minutes in three separate events would count.
The debate raged in map chat, on the forums and especially on Reddit. It seemed every player looking to get the achievement had a theory and map chat rang with the competing demands, pleas and threats of Commanders, would-be community leaders and flat-out crazed conspiracy theorists, all wanting to try out their own particular arcane plan of action. Meanwhile, players who had already done the Three Knights event several times and seen it fail just wanted to get in the damn airship and forget about the blasted achievement...
It all got very fraught indeed and wasn't a lot of fun for anyone and to add meta-insult to injured feelings, many claimed it made no difference what anyone did because the Knightfall achievement was bugged anyway. Some people supposedly got it when they killed their very first Knight while others didn't get it when they had followed the exact same sequence that had worked for someone else. Or so they claimed. Who could possibly know the truth of it?
Bit late to be recruiting now, don't you think?
With all this rumbling away in the background, the bulk of the playerbase took comfort in the familiar and found that during trying times the old ways often work best. Successful Knight runs eschewed fancy tactics in favor of The Zerg, focusing the force of the entire map population on each knight in turn, a tactic that seemed as good as any and better than most. Even if it didn't give any loot at least it opened the doors to somewhere that did.
So, once all the theories had been tested and proved worthless, that's how things might have settled, had ANet not had other ideas. Friday brought a patch that dictated terms. You will not mount a map-wide mega-zerg. You will split into three acceptable zergs of fifty players or fewer. You will co-operate and address this event appropriately. Then, if you're good and do it the way you're supposed to, we will give you loot. At the end. When you've demonstrated you can behave yourselves. And not before.
It felt a little like teacher coming back into the classroom and restoring order on finding the senior pupils he'd left in charge for five minutes weren't up to the task. Not angry. Just disappointed. It felt a lot more like a fuming DM waving the rulebook at a bunch of D&D players who'd just found a way to brute-force a clever encounter he'd spent hours designing. Either way, it didn't feel much like the way an MMO with a metacritic score of 90 ought to feel.
Green on Red.
That aside, the event is now more fun, and more reliable. Well, it was yesterday. Haven't done it yet this morning and there was another patch overnight. Anything could have happened. If it had been like this from the beginning how happy we'd all have been. Instead we feel like we've been schooled for making the best of a bad situation that was none of our doing in the first place. That sentiment was very evident in map chat last night and a sour taste will linger.
It's not really an issue of content or design. Those are factors you either like or you don't. It's a matter of competency and of respect. On balance I like The Living Story. While it has huge problems, particularly in pacing, as a means of imparting plot and narrative to an MMO it does have merit. Unfortunately the two-week roll-out is too ambitious to support it, particularly whenever an update includes a large-scale Event and it's the playerbase that cops the fallout from ANet's over-reaching ambition every time.
There's a thread on the forum in which two players succinctly sum up the extreme positions existing within the current playerbase:
Full Thread Here
I wholeheartedly agree with Darkobra. This isn't a beta. Nor is it a public Test server. This is the live game. We all understand there will be some bugs. We all know there will be unexpected outcomes. We all expect some post-update tweaking. This is none of that. This is riveters still working on the hull when the ship's already far out to sea.
Moreover, it was never the players who demanded, or even requested, this accelerated pace. If there's ever been a consensus it's that we want more solid additions to the game, added at a rate we can manage. Over the year players have adjusted to the breakneck schedule and developers have found ways to make temporary content hang around a while longer but what's been achieved is more of a grudging acceptance than a gleeful assimilation.
Fortunately there's something of a lull on the way. The next update, in ten days' time, is a coda, trailed as modest, where we all get to take a deep breath and reflect. Almost immediately after that WvW Season 2 (now renamed The Tourney and running under what looks to be an improved ruleset) begins. That should keep everyone busy for a while until Living Story Season Two arrives at some yet-unspecified later date.
Curiosity counts for much and GW2's aggressive update schedule, combined with zero-limit barriers to re-entry, has worked well enough in keeping players on-board until now, but player patience won't carry buggy content indefinitely. Soon ESO will be upon us with WildStar not far behind. Supposedly competition sharpens. Let's hope so. Some sharpening is sorely needed.
Labels: Events, GW2, Living Story, MMO, Scarlet
João Carlos March 8, 2014 at 1:38 PM
"Soon ESO will be upon us with WildStar not far behind. Supposedly competition sharpens."
It is a supposition. ESO is a traditional themepark with traditional quests and pvp is zerg versus zerg. It is not competition to GW2 and I fear it will have the same doom that War had.
Wildstar, I don't tested it. But I have no hopes about it.
Maybe we have EQN next year. But take note they are trying to make a better GW2, including living events.
Bhagpuss March 8, 2014 at 2:40 PM
I think you're placing far too much importance on the actual nature of the MMO in question. Whether it's sandbox, themepark or hybrid has a lot less impact than the simple fact that its new because currently GW2 is the go-to MMO for bored casuals, the natural launch-day target market for every new MMO of the last five years or more.
If ESO was launching as a F2P of some kind you'd see a large short-term impact in most current MMOs but it is possible that the box-price-plus-sub model will put some dampeners on that. Added to that, GW2's ace-in-the-hole is that you can't ever actually quit, you can only not be playing right now. There's a very interesting forum thread where a player wants ANet to delete his account (for very good reasons, in my opinion) and an ANet rep explains that it's something they are very reluctant to do and which requires an internal investigation of the particular circumstances of each case. It's an obviously advantageous commercial position but morally very dubious, I would say, and possibly not even legal in certain jurisdictions. I might do a bit of research on that, if I can be bothered.
Anyway, because of that GW2 is more heavily protected from any long-term impact from new MMOs than most established games but if you think there won't be some short-term impact you must not have noticed the visible drop in population, especially in WvW, each time ESO does a big weekend beta event.
Anyway, we'll see soon enough. Some time in April, isn't it?
Attic March 11, 2014 at 3:18 PM
The implied opinion that GW2 PvP isn't zerg vs zerg is absolutely hilarious.
Electrolux March 10, 2014 at 12:48 PM
ESO looks like a pretty coordinated attack on GW2s WvW. The combat looks hilarious though, I'll pass. I don't think they actually took a step back and just looked at people fighting in that game.
The living story is too fast and too terribad for me to do anything with but ignore. I tried to buy into it but clicking on bonfires was a step too far towards perfect retardination and caused me to walk away, still struggling to return.
Who in ANet thought that what I may want to do after I come home from work is just click on a bonfire? Those things you make, you do realise we are supposed to actually do them right? I mean in real life. Actually do them. On purpose and everything, they don't do themselves. Hmm now there's an idea. GW2 Achievements should be like EVEs skill system. You just select the ones you want to 'do' while you log off and do Anything Else In The Entire World.
Bhagpuss March 10, 2014 at 8:22 PM
Yes, it looks very much like WvW, doesn't it? I wish we had ESO's moveable siege engines!
You put your finger right on the problem for MMO devs, I think. When I come home from work I'm quite often in the mood to do something fairly mindless and relaxing like wandering around clicking on bonfires. Content that one player finds relaxing, though, another will find insulting to their intelligence, just like one player's stimulating challenge is another's frustrating road-block.
I'm glad I only have to play these things and write about them for my own amusement. I'd hate to have my mortgage payments depending on trying to keep enough players happy to stay in business.
This is true. I think it's a problem beyond dev control when somebody clicks on bonfires but resents every minute of it and ragequits when not doing that was an easily available alternative.
It's not like it was the first time I'd been presented with a laundry list of mindless tat to do. It was ,however, the first time I had to absolutely do it tonight which I feel was the subtle difference that caused an out-of-proportion emotional response to an Internet Goblins vs Wizards Game.
I honestly don't know what force it is that suppresses my natural bloody-minded disobedience when I am logged in, where is it when I need it :)
Every Day Is Not Like Sunday : GW2
Journeys and Tourneys : Landmark, GW2
Same River, Different Water : Everquest, GW2
As One Door Closes... : EQN Landmark
One Of These Things Is Not Like The Others : GW2
What Shall We Play Next?
Late To The Party : GW2
Aftermath : GW2
15, Going On 85 : Everquest
Grandfather's Silver Doesn't Come Free! : SOE, EQN...
Enquiring Minds Must Know (Caution - Spoilers - Th...
Learning To Listen : GW2
Why I'm Not Playing Landmark
Living Story - First Annual Report : GW2
Build Or Bust? : GW2

redpajama_set_name: RedPajamaCommonCrawl


text:
Tag: Twitter
Journalism, The everyday
Cats and kings on Twitter
May 17, 2013 May 20, 2013 by clareodeaLeave a Comment on Cats and kings on Twitter
I'm not just a late adopter, I'm a reluctant one. If I'd been around in the 1920s, I probably would only have only learned the steps for the Charleston in 1931. It took a social media course at work for me to finally 'join the conversation' through gritted teeth last May. Turns out, like many arranged marriages, it was a good match after all.
Some see the service only as a glorified link-sharing platform or a place to let off steam, but it is much more than that. Twitter is a fantastic shortcut to good quality information. Through Twitter you borrow the eyes and ears of the people you find most interesting, important or fun. I think of it as a never-ending group discussion, like sitting around with a bunch of people making scrapbooks from a pile of global content. They're pasting news reports, research findings, events, reviews, blog posts, videos, photos, personal observations and witty one-liners into their scrapbooks, pointing out each item to you as they go along. It's a way of sharing passions and you can join or leave the feast at any time. For me that's been invigorating, it has fed into my work on the journalism side as well as my creative writing.
Without Twitter I would not have:
1. Entered a flash fiction story for the National Flash Fiction Day Anthology yesterday because I wouldn't have heard of it without following Irish writer @NualaNiC (Nuala Ní Chonchúir)
2. Interviewed @AlaindeBotton in Basel on Wednesday (all arranged last minute on Twitter, more about that next week)
3. Written articles about women's issues in Switzerland in response to Anne Marie Slaughter's @SlaughterAM having-it-all essay.
Twitter is all things to all (wo)men. The Twitterati I follow fall into three broad categories – news, countries and writing.
Twitter is a fabulous resource for people interested in a particular region or country. It has helped me feel much more present in Ireland. From small things, like hearing a pub I used to work in burned down, to big things like the current abortion debate, I get a sense of being around again. Don't worry there's lots of good stuff too, like the pictures of sunrises in Sandycove posted by @blathnaidhealy.
There's a thrill to be had following a dynamic developing story on Twitter. You don't have to wait for the reporter to come out of the court for his or her piece to camera. Follow the right person (like BBC Africa Correspondent Andrew Harding @BBCAndrewH at the Pistorius trial) and you can get the action line by line as it happens.
Of course with an unfolding story, information has to be handled with care. One tweet during the Boston manhunt said it all (I'm paraphrasing here): "Faced with this barrage of confusing and conflicting information, I just wish there was a printed summary of all the verified facts available the next day."
There's so much to say about the writing community on Twitter that I'll have to put it in another post. For emerging writers Twitter is Open University meets support group. Check it out for yourself!


We live in a very materialistic society. We are constantly bombarded with advertisements for us to buy things - like or not, we want it! We would go to the ends of the earth to obtain it. Children are constantly asking for us to buy them things. Be is shoes, clothes, bags, toys, you name it, they want it. We as adults too want things, a better job, a better house and more money than our friend. No matter what we persuaded by the pressures of society to compete and get all the materialistic stuff that we can get our hands on. One way to express gratitude in our lives is to write a gratitude journal.
In all spheres of life it seems as if are expected to get, strive and obtain – this is the mantra that we have been and now we feed it to our children. But we forget that we are human beings, we can only do our best and at the same time, we must be content of what we have otherwise things really get out of hand. If one is in a job, we are asked to compete and do our best which is fine but sometimes we must are pushed to the limit. At school, children are asked to do their utmost best in every walks of their education which is perfectly fine, but again they seem to be pushed to the limit. We are constantly seen as successful if we have bigger houses, bigger cars, better mobile phones, go on holidays, spend money on birthday parties etc and even then we are not happy. Why do we have to do it? It's because we live in a very materialistic world! The only way to become free from all this is to express gratitude every day. It only takes a few minutes to say a few words and sentences of gratitude and you will realise that you will feel better.
Alternatively you can write only a few lines of gratitude in a gratitude journal. It should not take more than 5 to 10 minutes of your time and you will feel a lot better. You will realise the importance of life and you will be more content about yourself and all the things you have. You don't have to write assays or paragraphs of gratitude, just a few lines of thankfulness. This will help you become happier and more content about who you are and become from free from the ridiculous rat race of life we are in.
So you see that it only takes a few minutes of your time to express some of these very basic gratitudes and you will find that you are more content with yourself.
There are many ways to express gratitude. You can buy a readymade gratitude journal, you can use a gratitude app on your mobile or you can use a piece of paper- you just have come up with something. But my opinion is that you get a gratitude journal and write your thoughts down. Here's a free gratitude app I found.
Here's my own personal gratitude journal I created and is available on Amazon. I have been keeping a gratitude journal for some time and it has helped me a lot.


Living In An Apartment, Wanting A Pet?
Finding the perfect pet takes a lot more than just looking for the cutest one. It takes a lot of time and consideration as you want to find the one that will suit your lifestyle. For those of you who live in an apartment and would love the company of a pet, here are a few tips in picking "the one". These tips can help you find a pet that makes you happy, your landlord calm, and gives your neighbors peace.
Living in an apartment with a pet requires special considerations for both the pet and owner. For example, some landlords have a specific guideline to pet ownership. Some landlords do not allow their tenants to have pets in their suites, whereas others require a security deposit also known as a pet rent. So make sure you know what your landlord expects if he or she allow pets.
It is also crucial that your new pet matches your lifestyle and living environment. Sometimes, having a dog that is still potty training alone in the apartment is not the best idea. And it probably won't make the landlord very happy either. Some pets require more attention than others.You need a pet that suits your lifestyle which means if you work long hours, you need to find a pet that does not require much attention. Regarding the living environment, some landlords allow their tenants to have a pet, but a certain size. So selecting a certain pet is extremely crucial as you can tell. You must find the pet that suits your life style, and also meets the requirements your landlord put in place for pets.
If you have an animal, you most likely know that it requires a lot of space. Small dogs like terriers or beagles have high energy, which means they like to move around a lot. So a good amount of space is necessary for a happy pet. Make sure that if you do live in an apartment, you get a dog that does not hate to be isolated. Also note that once you find the perfect pet, keep notice of any changes in your pets behavior. Sometimes if you notice your pet is isolating their self from you, something completely new, they could be suffering from a condition, or pet depression. Also note that once you find the perfect pet, keep notice of any changes in your pets behavior. This change you notice in your pet could be linked to a condition your pet is suffering with. There are many pet medication options for your companion to treat pet depression or any other pet medical conditions.
Caring for a pet is a lot of responsibility as you have to meet their needs, and care for them to survive. Pets need space and room to grow, which is why it is necessary to find the proper accommodations. So for those of you who live in an apartment and want a pet, follow the guidelines listed above, and I'm sure you will enjoy having the company of a pet.


As I mentioned in my last post (January 8), over the next few weeks I will dive deeper into some of the sample department "goals" I set out in that article. Today I will focus on this goal: "Build and retain extraordinary team with exceptional people." I always put my "people goal" first because I truly believe that nothing gets done in legal unless you have top talent that is motivated and happy in their jobs. How do you keep and reward people so they stick around? The obvious answer is pay them well, have a good performance bonus program in place, and let them share in equity plans. The problem is, for many reasons, it usually is not fully in your control to make any of these three things happen. For purposes of this article, I am going to assume that you are doing what you can for your team around salary, bonuses and equity and, instead, focus on some low cost ways you can reward/recognize employees.
A common complaint you will hear as in-house counsel is "Why does it take so long for you guys to review my contract?" (Second only to "Why are our contracts so long?") The answer, as you know, is complicated. Legal is a limited resource, typically a small team that reviews hundreds and possibly thousands of contracts in any given year. While a lot of contracts are fairly routine, many involve complicated provisions or transactions with millions of dollars on the line. Sometimes you have to create a contract from scratch, meaning you do not have a form or something to easily model from. Frequently, things like litigation or large M&A deals take up substantial amounts of lawyer time — time that cannot be spent on contracts. Finally, legal will generally prioritize contracts based on the strategic objectives of the business. Deals that better support the strategy/objectives get more attention more quickly.
Ten Things: Your Company Has Just Been Sued. Now What?
It's hard to recall a more disconcerting feeling than getting a copy of a lawsuit filed against your company. If you have no experience with litigation, this can be a panic inducing moment. And no matter how experienced you are handling litigation, your stomach will start to flutter as you read through the allegations.
I was a litigator in private practice and I definitely saw my share of litigation, big and small as in-house counsel. Over the course of that time I developed a standard list of "things to do" when a lawsuit came across my desk. I did this because it's easy to forget some basic things you need to do up-front to put yourself in the best position to defend the claim. Below are ten things to do when your company gets sued (I have added links to additional resources in key spots).
As we head into the holiday season, this is the perfect time to give your anti-bribery program a health check. For those in the U.S., we tend to focus on the Foreign Corrupt Practices Act when thinking about anti-bribery laws. However, if you work for a company that operates globally, you know that many countries have anti-bribery laws and you need to be aware of those requirements as well. Enforcement of the FCPA/anti-bribery laws is not going away. In fact, in my opinion, it will get even more intense over the next few years. Given the level of fines and the reputational risk at stake, it's important to ensure you are taking the right steps to give your employees the tools they need to stay on the right side of the line. At my prior company, we typically used the advent of the holiday season as the time to take a number of steps relating to FCPA/anti-bribery compliance. Below are ten things you can do now to help ensure compliance with anti-bribery laws. In key spots, I have included links to articles or websites with additional information you might find helpful.
Whether you are new to the in-house department or a long-term veteran, the General Counsel or just a basic contract lawyer, there are a number of things that can help make you more successful in your career. I have distilled a lot of hard learned lessons into ten key tips. These are not exhaustive and there are always more, but these are the ten things I consistently taught to my teams over the years.


Are you approaching retirement, but are unsure whether or not you are fully prepared? Trusts, Wills, Powers of Attorney, Health Care Directives - what do you need to be thinking about, and how can you put that estate plan into effect? Whether you have already done some estate planning or are looking to get started, this seminar will assist you with the basics that should be in your estate plan as you head towards retirement. We will also touch on what you need to know in order to maximize your retirement income, and answer these common questions: • Will Social Security be there for me? • How much can I expect to receive? • When should I apply for Social Security? • How can I maximize my benefits? • Will Social Security be enough to live on in retirement? Attorney Adam Kaufman of Henningson & Snoxell is partnering with Bret Larson and Andrew Karlson of Foresters Financial Services, Inc. to provide a FREE informative Seminar on estate planning for retirement. *Foresters Financial Services, Inc. is not affiliated with Adam Kaufman or Henningson & Snoxell LTD. Foresters Financial Services, Inc. does not provide legal, tax or estate planning advice. Should you require these services, please contact a legal, tax or estate planning professional.
Please register at https://www.hennsnoxlaw.com/about-us/events/savvy-social-security-planning-estate-planning-retirement-seminar or by calling our office 763.560.5700.


Leawo Software is committed to protecting your privacy. We will only collect your personal information for defined purposes. We do not distribute or share your personal information beyond what is strictly necessary to fulfill our obligations to you. We may share your information only with partners who adhere to Leawo's commitment to protecting your privacy. Leawo Software will not sell your personal information in any manner whatsoever. You shall have foreseen and agreed to the occurrence of events when you provide your personal information. Undoubtedly, your privacy is a priority at Leawo Software, and we go to great lengths to protect it.
This policy covers the Intel Corporate website and is applicable worldwide. To make sure your personal information is secure, we communicate these guidelines to Leawo employees and strictly enforce privacy safeguards within the company.


Warning! GuruFocus has detected 3 Severe warning signs with ENLK. Click here to check it out.
Crosstex Energy LP is a midstream natural gas master limited partnership that gathers, transports, processes, and markets natural gas and natural gas liquids, primarily in Texas and Louisiana. The company operates gas gathering systems in the Barnett Shale and Louisiana, the largest intrastate pipeline in Louisiana, and large processing operations in South Louisiana.


In the October 2016 issue of Plastics Engineering magazine, Mark Clark reveals specific challenges that came with designing BDAS+ , an award-winning multiresin biological emergency response device.
This high-durability disposable is meant for one-time use for potentially extreme conditions. Since it's a portable product, weight was a concern. In the article, "Sophisticated Software Is Helping to Reshape Plastic Part Design," by Geoff Giordano, Clark explains how his team used software to solve for these constraints.
"We wanted to make it tough and durable, and we didn't want to use any more material than was absolutely necessary," Clark told Giordano. He and his project team made use of many tools and techniques, including draft analysis and a considerable amount of FEA (finite element analysis), relying heavily on SolidWorks Simulation. Clark goes on to share some shortcuts and other software used by the project team to develop BDAS+.


text:
'One person responsible for chain's demise'...
'King of retail' Green blamed for collapse of BHS
Terry Murden, Editor | July 25, 2016
Former BHS owner Sir Philip Green's reputation as the "king of retail" was in ruins following his part in the failure of the chain, according to a report from a group of MPs.
Sir Philip contributed to the collapse of the 88-year-old group by extracting huge sums from the company while leaving its pension fund in deficit, the report concludes.
A highly-critical indictment by two Commons select committees says the retail tycoon sought to blame anyone but himself for the company's failure.
The Work and Pensions committee, together with the Business, Innovation and Skills committee also criticised Dominic Chappell, who bought BHS for £1, as well as the "directors, advisers and hangers-on" associated with the deal.
However, MPs felt Sir Philip was ultimately to blame and now had a "moral duty" to compensate 20,000 employees and former employees in the pension scheme who are facing substantial cuts to their benefits.
The 66-page report states that leadership failures and personal greed caused the downfall of the 88-year old chain, calling it "the unacceptable face of capitalism".
The report says Sir Philip's family had accrued "incredible wealth" from their ownership of BHS while paying little in tax.
He failed to invest in the company or tackle the "substantial and unsustainable deficit" in the pension fund.
Bhs store, Edinburgh (pic by Terry Murden)
Frank Field, chairman of the Work and Pensions committee said: "One person alone is ultimately responsible for the BHS disaster.
"While Sir Philip Green signposted blame for 11,000 job losses and a gigantic pension fund hole to every known player, the buck stops with him. His reputation as the king of retail lies in the ruins of BHS."
Field continued: "His family took out of BHS and Arcadia a fortune beyond the dreams of avarice, and he's still to make good his boast of 'fixing' the pension fund.
"What kind of man is it who can count his fortune in billions but does not know what decent behaviour is?"
The committees' report comes days after the Cabinet Office said it would respond to calls for Sir Philip to be stripped of his knighthood.
Bhs, Sir Philip Green Uncategorized No Comments » Print this News
Verizon has agreed to buy online search engine Yahoo for $5 billion, according to US reports Previous News
Next News William Hill facing joint £3 billion takeover bid
FinTech's new champion is focusing on how the sector can make a practical difference to people's lives, writes TERRY MURDEN
Johnson expected to announce Brexit deal
Boris Johnson is expected to make an announcement Boris Johnson is expected to deliver someRead More
Take fishing out of the Brexit talks and get a deal done
SNIB launches with M Squared Lasers deal
DB Live: Covid test stocks 'good value'; economy rebounds
ScottishPower to double green generation


We follow certain syllabus in Music Department which consists both vocal and Instrumental. The children are trained in such a way so that they can choose Music as a career in future.
We follow the syllabus which integrates both Modern and Cultural dance forms. Our children are well trained in both the faculty.
We specialize our students in both Indoor and Outdoor games which include Football, Badminton, Basketball, Table Tennis, Kho- Kho etc. Chess & Carrom board. Nirvanian participate in Inter- School Tournament time to time.
We have Martial Arts from Grade U.K.G. onwards so that the children can protect themselves from the changing and upcoming problems.
It's a personality building process which inculcates Om utterance, Voicing exercises, Speech exercise, facial exercises, yoga, conceptual thinking, writing and presentation. We lay equal emphasis to English Nepali in Dramatics presentation.
Evaluation in Dramatics is done on regular basis.
The sole motive of the incorporation of Dramatics as a subject is to make the students have a creative mind set.
Art & Craft: Play Group, Kindergarten and Grade I to V have this subject in their course of study. Colouring, pasting, drawing, paper craft, clay modeling and activities of the same a kind is carried on. The higher grades go for advanced level.
News Reading: We have formed a NIRVANA NEWS Room which has News Producers and News Readers. These personals are selected from our students. Daily News Reading is done in our School Assembly at two separate levels – Grade II to IV & Grade V to X. The News Reading section operates under the Supervision of our Academic Co-ordinator designated for the work.
News is read simultaneously in English and Nepali. The News Producers & Readers are deployed on rotation basis for a fortnightly from Literary Club.
Current happening in the country & around the world is focused in News Reading.
Activities related to enhancement of students' personality are done on regular basis and their talent is showcased through different events like; Spelling Contest, Game Show, Singing and Dancing Competition, Story Telling, Debate, Quiz, Talk Show, Drama Competition, Talent Show, etc. These events are held as Inter – House Competitions which encourage the students to work in groups.
These events are staged according to the scheduling done in our year planner. Students are encouraged to participate in Inter-School competitions also. Students of all levels are motivated to encash this opportunity.
All classes from P.G onwards are taken for Field Trip to locations matching with their curriculum (English, Nepali, Science & Social Studies). Places of historical & archeological, horticulture, industrial significance are chosen for this purpose. Zoo, Museum, Palace, Markets etc is some favourite ones.
Students belonging to Grade VI onwards are taken for Long Trip at reasonable expense. Such trip termed as Educational Tours/ Excursions are undertaken exclusively under the supervision of a Good Tour Operator.
The clubs are run solely by students from Grade VI to X. under the supervision of Vice Principal and Academic Coordinator.
The managing committee has President, Vice President, Co-ordinator. Treasurer who manage the club affairs. It also has a General Membership quota. All the posts/ designations are filled by students from Grade VI to X. The students who cater interest in those fields are selected for the club. The activities done are competitions, stage performance, Drama etc.


text:
Fowlerville man sentenced to two years' probation in child porn case
Kayla Daugherty
A Fowlerville man who was charged with possessing child pornography was sentenced to two years' probation in a plea deal with prosecutors.
Daniel Delavergne, 46, was arrested by State Police in May 2018 and charged with two counts of using a computer to commit a crime, one count each of unlawful posting of a message and child sexually abusive material.
The MSP Internet Crimes Against Children Task Force began investigating Delavergne in 2018 after receiving a tip. Investigators searched Delavergne's home and seized multiple electronic devices depicting suspected child porn.
In November he pleaded guilty to one count of unlawful posting of a message and one count of using a computer to commit a crime.
Prosecutors dismissed one count of unlawful posting of a message and the child porn charge as part of a plea agreement.
The counts he pleaded guilty to relate to the gathering and distributing of explicit photos of a woman without her consent, according to prosecutors.
Livingston County Assistant Prosecutor Shawn Ryan asked for Delavergne to be sentenced to at least 90 days in jail.
"What this man did preying on this woman and all the images that were found on his computer are sick and despicable," Ryan said.
Delavergne's attorney, Marcus Wilcox, asked Circuit Court Judge Michael Hatty to keep the dismissed count of child porn separate from the sentencing.
Wilcox also argued Delavergne's actions were not predatory and his client was originally trying to help the victim.
"He was not the person who put those pictures on the internet, he was not the person who hacked her phone," Wilcox said.
Delavergne apologized to the woman and asked Hatty "to show mercy" in his sentencing.
"My intentions weren't bad, I know it came out all wrong," he said.
Prior to sentencing Delavergne Hatty told him to expect "serious" jail time if he contacts people on the internet who don't want to be contacted.
"When people get on those computers the intent can't be easily ascertained as when people are talking back and forth... you're responsible for the words you put on there and you're charged for them," Hatty said.
Delavergne will spend the first 30 days of probation in the Livingston County Jail. He had faced up to four years in prison on the original charges.
Contact Kayla Daugherty at 517-552-2848 or kdaugherty@gannett.com. Follow her on Twitter @KayDaugherty92.


text:
Benefits Application Assistance
Prepare for Release
DRCPP
Fortune Faces
Fortune50
Closing Rikers: A New Vision For The New York City Criminal Justice System
08/16/2018 The Fortune Society
Individuals with justice involvement deserve an opportunity to have their cases heard and judged fairly.
They deserve to learn, grow, and change in a supportive environment. Rikers Island is the antithesis of these aims, which is why The Fortune Society is a long-time advocate for its closure. Since our founding, Fortune has believed that incarceration and detention should be options of last resort.
Our latest issue of The Fortune News takes a close look at the many nuanced aspects of closing Rikers. It contextualizes its place in the larger conversation regarding justice reform, which includes bail reform, speedy trial, and alternatives to incarceration. And it sheds light on the work that The Fortune Society has done in all of these efforts, thanks to your support and commitment to our mission.
Support like yours has:
– Driven efforts to close Rikers Island jail complex and replace it with a network of human-centered, modern, and safe facilities;
– Diverted individuals away from the criminal justice system through holistic alternatives to incarceration programs
– Prepared individuals on Rikers Island and in the community for success through career development; and
– Connected individuals with justice involvement and chronic health needs with healthcare.
Since 1967, we have been a voice to the voiceless, encouraging the community to truly embody core values: That people are innocent until proven guilty; that we all deserve and need opportunities to succeed; that we should treat others the way we want to be treated; and that we should not be defined for the rest of our lives by our worst mistake, decision, or action.
More insight from powerful voices awaits you in our latest issue. We hope it affirms your commitment to helping us ensure that everyone, regardless of their past, finds a supportive community to call home.
Employment Services,
Alternatives to Incarceration,
Prepare for Release,
DRCPP,
Legislative Steps Forward for HIV/AIDS Prevention and Testing After Prison
At Fortune, Ron Found Acceptance and Career Success
Passing Generosity Forward: Francisco's Story After Prison


text:
Essex Council integrate new therapeutic training programme as way of helping children in care
As part of the training programme, foster carers will have access to a team of professionals
Image: South Wales Echo
Submitted by Ann Scott
Essex County Council are integrating a new trauma informed therapeutic training programme as a way of renewing their commitment to securing the best outcomes for children in care.
The training will be available to anyone who may come into contact with a child in care - giving them a 'trauma lens' to better understand their experiences and how they can support them. Essex County Council have adapted quickly to the challenges of Covid-19 with the training sessions being switched to online delivery.
The programme has been developed due to the overwhelming evidence that the outcomes for children who have suffered trauma are much better if they live with families and are parented in a trauma informed way.
Fostering can be transformational for children who have experienced a traumatic upbringing but it can be just as life changing for the foster carers themselves.
Janice and Colin, from Rayleigh, Essex, have been fostering for 16 years.
Since receiving trauma informed therapeutic support the couple say that they have started to see how they can not only care for their foster son, but also how they can help repair that trauma he has been through.
"Fostering isn't all hugs and smiles - it can be incredibly challenging. But that certainly makes it more rewarding. You don't need to be special to foster. But it does makes you feel special.
"It is so rewarding to see a child grow in confidence. To go from being so shy, compliant, withdrawn and fearful of asking for anything, even a glass of water, as they were scared of what the reaction might be, to chatting about things and not afraid to ask questions".
As part of the training programme, foster carers will have access to a team of professionals, including mental health coordinators and principal clinical and counselling psychologists.
Foster carers also get a dedicated social worker and access to support networks including peer to peer, 24/hr support line, respite, and lots of training packages, including bespoke training to the individual.
Colin and Janice who have been fostering for 16 years
Essex County Council's consultant clinical psychologist, Barbara Canepa, said: "Moving forward, our aim is to recruit people who want to work therapeutically with children. Our goal is to help give children in care the best possible environment in which to feel safe and secure, and to go on to thrive.
"We see foster carers as the agents of that change, and our role is to give them the knowledge, support and tools with which to heal the trauma and transform the lives of the children and young people they care for."
Call to action - Become an Area Ambassador
Foster carers can be single, married, from a same-sex family or retired. It can be undertaken on a part-time basis alongside a full-time job or offered as a full-time role, both always with the full support of Essex County Council.
Cllr Louise McKinlay, cabinet member for children and families at Essex County Council, said: "All children deserve to feel safe, secure and be nurtured. Our foster carers help build better, brighter futures for hundreds of children across Essex every year. We need more foster carers, like Janice and Colin, to foster on a full and part-time basis".
Janice and Colin believe that a willingness to learn and take on board different ways of working is the key to successfully helping foster children overcome their trauma and be able to move onto a positive future.
"Incorporating trauma informed therapeutic training into everyday life is key to this. Where I would normally react instantly with 'what did you do that for?' I instead pause and ask, 'I wonder why you did that?' I'm much more mindful of my words and think about how my response may trigger a negative reaction. My life is very different to the one I led before fostering, working in the city; it is so much better and has changed me in every way."
Children in care have often experienced trauma in their past and need foster carers who can help them to heal. Essex County Council are encouraging people to watch this video, inspired by foster children from around Essex, to understand how they bring back the light into a child's life and set them on a path for a better future.
Find out more online or by calling 0800 801 530.
Tunbridge Wells student receives prestigious Geography award
Clacton on Sea care centre residents receive first doses of vaccine
Free school meal service in Caerphilly goes from strength to strength