text:
\section{Introduction}
The notion of \emph{String Attractor} has been recently introduced and studied in \cite{prezzaArXiv2017,KempaP18} to find a common principle underlying the main techniques constituting the fields of dictionary-based compression.
It is a subset of the text's positions such that all distinct factors have an occurrence crossing at least one of the string attractor's elements.
From one hand the problem of finding the smallest string attractor of a word has been proved to be NP-complete, on the other hand  dictionary compressors can be interpreted as algorithms approximating the smallest string attractor for a given word \cite{KempaP18}. Moreover, approssimation rates with respect to the smallest string attractor can be derived for most known compressors. In particular compressors based on the Burrows-Wheeler Transform and the dictionary-based compressors are considered.

The Burrows-Wheeler Transform (BWT) is a reversible transformation that was introduced in 1994 in the field of Data Compression and it also largely used for self-indexing data structures. It has several combinatorial properties that make it a versatile tool in several contexts and applications \cite{PrezzaPSR19,LouzaTGZ18,MaReSc,RestivoRosoneTCS2011,MantaciRRSV17}.

Dictionary-based compressors are mainly based on a technique originated in two theoretical papers of Ziv and Lempel \cite{Ziv77auniversal,Ziv78}. Such compressors, that are able to combine compression power and
compression/decompression speed, are based on a paper in which combinatorial properties of word factorization are explored \cite{LZ1976}.

In this paper we explore the notion of string attractor from a combinatorial point of view. In particular, we compute the size of a smallest string attractor for infinite families of words that are well known in the field of Combinatorics on Words: standard Sturmian words (and some their extension to bigger alphabets), Thue-Morse words and de Brujin words. In particular, we show that the size of the smallest string attractor for standard Sturmian words is $2$ and it contains two consecutive positions. For the de Brujin words the size of the smallest string attractor grows asintotically as $\frac{n}{\log n}$, where $n$ is the length of the word. We show a string attractor of size $\log n$ for Thue-Morse words and we conjecture that this size is minimum.  From the results presented in the paper, we believe that the distribution of the position in the smallest string attractor of a word, in addition to its size, can provide some interesting information about the combinatorial properties of the word itself. For this reason the notion of string attractor can provide hints for defining new methods and measures to investigate the combinatorial complexity of the words.

\section{Preliminaries}
Let $\Sigma =\{a_1, a_2, \ldots, a_\sigma\}$ be a finite ordered alphabet with $a_1< a_2< \ldots < a_\sigma$, where $<$ denotes the standard lexicographic order. We denote by $\Sigma^*$ the set of words over $\Sigma$.
Given a finite word $w = w_1w_2 \cdots w_n \in \Sigma^*$ with each $w_i \in \Sigma$, the length of $w$, denoted $|w|$, is equal to $n$.

Given a finite word $w=w_1w_2\cdots w_{n}$ with each $w_i \in \Sigma$, a \emph{factor} of a word $w$ is written as $w[i,j] = w_i \cdots w_j$ with $1\leq i \leq j \leq n$. A factor of type $w[1,j]$ is called a \emph{prefix}, while a factor of type $w[i,n]$ is called a \emph{suffix}.
We also denote by $w[i]$ the $i$-th letter in $w$ for any $1\leq i \leq n$.

We denote by $\tilde{w}$ the reversal of $w$, given by $\tilde{w} = w_n \cdots w_2w_1$.
If $w$ is a word that has the property of reading the same in either direction,
i.e. if $w =\tilde{w}$, then $w$ is called a \emph{palindrome}.

We say that two words $x,y\in \Sigma^*$ are {\em conjugate}, if $x=uv$ and $y=vu$, where $u,v\in \Sigma^*$.
Conjugacy between words is an equivalence relation over $\Sigma^*$.

Given a finite word $w$, $w^k$ denotes the word obtained by concatenating $k$ copies of $w$.  A nonempty word $w \in \Sigma^+$ is {\em primitive} if $w=u^h$ implies $w=u$ and $h=1$.
A word $x$ is {\em periodic} if there exists a positive integer $p$ such that $x[i]=x[j]$ if $i=j\mod p$. The integer $p$ is called \emph{period} of $x$.

A {\em Lyndon word} is a primitive word which is the minimum in its conjugacy class, with respect to the lexicographic order relation. We call \emph{Lyndon conjugate} of a primitive word $w$ the conjugate of $w$ that is a Lyndon word.

The Burrows-Wheeler Transform is a permutation $bwt(v)$ of the symbols in $v$, obtained as the concatenation of the last symbol of each conjugate in the list of the lexicographically sorted conjugates of $v$.

The LZ factorization of a word $w$ is its factorization $s = p_1 \cdots p_z$ built left to right in a greedy way by the following rule: each new factor (also called an LZ77 phrase) $p_i$ is either the leftmost occurrence of a letter in $w$ or the longest prefix of $p_i \cdots p_z$ which occurs, as a factor, in $p_1 \cdots p_{i-1}$.

\section{String Attractor of a word}

In this section we describe the notion of string attractor that is a combinatorial object introduced in
\cite{prezzaArXiv2017,KempaPrezzaArXiv2017} to obtain a unifying framework for dictionary compressors.

\begin{definition}
A {\em string attractor} of a word $w \in \Sigma^n$ is a set of $\gamma$ positions $\Gamma = \{j_1, \ldots, j_\gamma \}$ such
that every factor $w[i, j]$ has an occurrence $w[i',j'] = w[i,j]$ with $j_k \in [i', j']$, for some $j_k \in \Gamma$.
\end{definition}

Simply put, a string attractor for a word $w$ is a set of positions in $w$ such that all distinct factors of $w$ have an occurrence crossing at least one of the attractor's elements.

Note that, trivially, any set that contains a string attractor for $w$, is a string attractor for $w$ as well. Note also that a word can have different string attractors that are not included into each other. We are interested in finding a {\em smallest string attractor}, i.e. a string attractor with a minimum number of elements. We denote by $\gamma^*(w)$ the size of the smallest string attractor for $w$. Note that all the factors made of a single letters should be covered, and therefore $\gamma^*(w)\geq |\Sigma|$.

\begin{example}
Let $w=adcbaadcbadc$ be a word on the alphabet $\Sigma=\{a,b,c,d\}$. A string attractor for $w$ is for instance $\Gamma=\{1,4,6,8,11\}$. Note that, in order to have a string attractor,  position $1$ can be removed from $\Gamma$, since all the factors that cross position $1$ have a different occurrence that crosses a different position in $\Gamma$. Therefore $\Gamma'=\{4,6,8,11\}$ is also a string attractor for $w$ with a smaller number of elements. The positions of $\Gamma'$ are underlined in $$w=adc\underline{b}a\underline{a}d\underline{c}ba\underline{d}c.$$ $\Gamma'$ is also a smallest string attractor since $|\Gamma|=|\Sigma|$. Then  $\gamma^*(w)=4$.
Remark that the sets $\{3,4,5,11\}$ and $\{3,4,6,7,11\}$ are also string attractors for $w$. It is easy to verify that the set $\Delta=\{1,2,3,4\}$ is not a string attractor since the factor $aa$ does not intersect any position in $\Delta$.
\label{exp:gamma}
\end{example}

The following two propositions, proved in \cite{prezzaArXiv2017}, are useful to derive a lower bound on the value of $\gamma^*$.

\begin{proposition}
Let $\Gamma$ be a string attractor for the word $w$. Then, $w$ contains at most $|\Gamma|k$ distinct factors of length $k$, for every $1\leq k \leq |w|$.\label{prop:kfactors}
\end{proposition}

\begin{proposition}\label{prop:repeatedfactor}
Let $w\in \Sigma^*$ and let $r$ be the length of its longest repeated factor. Then it holds $\gamma^*(w)\geq \frac{|w|-r}{r+1}$.
\end{proposition}

The following proposition gives an upper bound for $\gamma^*$ of a concatenation of words, when $\gamma^*$ of the single words are known.

\begin{proposition}\label{prop:concatenate}
Let $u$ and $v$ two words, then $\gamma^*(uv)\leq \gamma^*(u)+\gamma^*(v)+1$.
\end{proposition}

\begin{example}
The bound defined in the previous proposition is tight. In fact, let $u=\underline{b}aa\underline{a}ba$ and $v=c\underline{d}cc\underline{c}d$ be two words in which the positions of the smallest string attractors are underlined. If we consider $uv=\underline{b}aa\underline{a}ba\underline{c}\underline{d}cc\underline{c}d$, the underlined positions represent one of the smallest string attractors for $uv$, as one can verify.
\end{example}

The following proposition gives an upper and lower bound for $\gamma^*$, when a power of a given word is considered.

\begin{proposition}\label{p-power}
Let $w=u^n$. Then $\gamma^*(u)\leq \gamma^*(u^n)\leq \gamma^*(u)+1$.

\end{proposition}

\begin{example}
The upper bound given by Proposition \ref{p-power} is tight. In fact consider the word $u=abbaab$. It is easy to check that the only smallest string attractors for $u$ are $\Gamma_1^*=\{2,4\}$ and $\Gamma_2^*=\{3,5\}$.
In order to find the smallest string attractor for $u^2=abbaababbaab$, we remark that neither $\Gamma_1$ nor $\Gamma_2$ (neither any string attractor obtained from them by moving some position from the first to the second occurrence of $u$) cover all the new factors that appears after the concatenation.
In particular $aba$ is not covered by $\Gamma_1^{'}$, and $bab$ is not covered by $\Gamma_2'$. A way to get the smallest string attractor for $u^2$ is to add to $\Gamma_1$ or $\Gamma_2$, the position corresponding either to the end of the first occurrence of $u$ or the beginning of the second occurrence. For instance, $\Gamma^*=\{2,4,6\}$ is a smallest string attractor for $u^2$.
\end{example}

\begin{example}
Remark that $\gamma^*(u^n)$ can be equal to $\gamma^*(u)$ although different point for the string attractor could be chosen. For instance, let $u=a\underline{b}\underline{a}b\underline{c}bc$ be a word whose smallest string attractor is $\{2,3,5\}$ (the underlined letters). Then $u^2= \underline{ab}abcb\underline{c}ababcbc$ has a string attractor $\{3,6,7\}$ of cardinality $3$. Remark that $\{2,3,5\}$ is not a string attractor for $u^2$.

\end{example}
A straightforward consequence of Proposition \ref{p-power} is the following:

\begin{corollary}\label{cor:conjAttractor}
If $u$ and $v$ are conjugate words, then $|\gamma^*(u)-\gamma^*(v)| \leq 1$.
\end{corollary}

\begin{example}
Consider the word $w=babbaaa$. Then a smallest string attractor for $w$ is $\{3,5\}$, i.e. $\gamma^*(w)=2$. Consider its conjugate $u=a\underline{b}ab\underline{b}a\underline{a}$. Its smallest string attractor is $\{2,4,6\}$, i.e. $\gamma^*(u)=3$. Note that the Lyndon word does not have necessarily the smallest $\gamma^*$ among the conjugates. In fact, for instance, the Lyndon conjugate of $w$ is $aaababb$, and it is easy to verify that one of the smallest string attractor is
$\{3,4,6\}$.
\end{example}

\section{Approximating a string attractor via compressors}

In \cite{KempaP18} the authors show that many of the most well-known compression schemes reducing the text's size by exploiting its repetitiveness can induce string attractors whose sizes are bounded by the repetitiveness measures associated to such compressors. In particular, straight-line programs, Run-Length Burrows-Wheeler transform, macro schemes, collage systems, and the compact directed acyclic word
graph are considered. Here we report some result related to the Burrows-Wheeler Transform, collage systems, and Lempel-Ziv 77 (that is a particular macro-scheme) that provide upper bounds on the size of the smallest string attractor for a given word.  Such bounds will be used in next sections to compute the string attractors for known families of finite words.

The first theorem, proved in \cite{KempaP18}, states a connection between a string attractor of a word $w$ and the runs of equal letters in the $\texttt{bwt}(w\$)$.
\begin{theorem}
\label{th:prezzaBWT}
Let $w$ be a word and let $r$ be the number of equal-letter runs in the $bwt(w\$)$, where $\$$ is a simbol different from the ones in the alphabet $\Sigma$ and assumed smaller than any symbol in $\Sigma$.
Then, $w$ has a string attractor of size $r$.
\end{theorem}

In particular, in the proof of Theorem \ref{th:prezzaBWT} the string attractor is constructed by considering the position of the symbols in $w$ that correspond, in the output of the transformation, to the first occurrence of a symbol in each run (or, equivalently, the last occurrence of a symbol in  each run).

The following result, proved in \cite{KempaP18}, states the relationship between a string attractor of a word $w$ and the number of phrases in the LZ  parsing of $w$.
\begin{theorem}\label{L_KePrLZ}
Given a word $w$, there exist a string attractor of $w$ of size equal to the number of phrases of its LZ parsing.
\end{theorem}

By using the previous result, a string attractor can be constructed by considering the set of  positions at the end of each phrase.

The following theorem in \cite{KempaP18} gives a connection between a particular class of grammars, called \emph{collage systems} \cite{KIDA2003253}, and string attractors.

\begin{definition}
A collage system is a set of $c$ rules of four possible types:
\begin{itemize}
    \item $X\to a$: nonterminal $X$
    expands to a terminal $a$
\item $X\to AB$: nonterminal $X$ expands to $AB$, with $A$ and $B$ nonterminals different from $X$
\item $X \to R^{\ell}$: nonterminal $X$ expands to nonterminal $R \neq X$
repeated $\ell$ times
\item $X \to K[l,r]$: nonterminal $X$ expands to a substring of the
expansion of nonterminal $K\neq X$.
\end{itemize}
\end{definition}

\begin{theorem}\label{th-PrezzaGr}
Let $G=\{X_i\to a_i, i=1, \ldots g'\} \cup \{X_i\to A_iB_i, i=1,\ldots, g''\}\cup\{Y_i\to Z_i^{l_i}, l_i\geq 2, i=1,\ldots, g'''\}\cup\{W_i\to K_i[l_i\ldots r_i], i=1\ldots, g''''\}$ be a collage system of size $g=g'+g''+g'''+g''''$ generating a word $w$. Then $w$ has a string attractors of size at most $g$.
\end{theorem}

By using this theorem one can easily find the minimum string attractor for a particular class of very ``regular'' strings, as shown in the following corollary.

\begin{corollary}
Let $u \in \Sigma^*$ be a word.
If $u$ is the form of $u=\sigma_{i_1}^{n_1}\sigma_{i_2}^{n_2} \cdots \sigma_{i_k}^{n_k}$ (where all $\sigma_{i_j}$ are different symbols in the alphabet), then  $\Gamma=\{n_1,n_1+n_2, \ldots, n_1+\ldots+n_k\}$ is a string attractor of minimum size for $u$, so $\gamma^*(u)=\sigma$.
\end{corollary}

\section{Minimum size string attractors}

In this section, we analyze the words whose smallest string attractor has size equal to the the size of the alphabet, that is the minimum possible size.
In the following two subsections we distinguish the case of binary words and the case of words over alphabets with more than two letters.

\subsection{Binary words}

In this subsection we focus on an infinite family of finite binary words whose minimum string attractor has size $2$.

Standard Sturmian words is a very well known family of binary words that are the basic bricks used for the construction of infinite Sturmian words, in the  sense  that  every  characteristic Sturmian word is the limit of a sequence of standard words (cf. Chapter 2 of \cite{Lothaire:2005}).
These words have a multitude of characterizations and appear as extremal case in a very great range of contexts \cite{KMP77,CS09}. In this paper two of their characterization are particularly useful, that is a special decomposition into palindrome words and an extremal property on the periods of the word that is closely related to Fine and Wilf's theorem (cf. \cite{deLucaMignosi1994,deLuca1997}). More formally, standard Sturmian words can be defined in the following way which is a natural generalization of the definition of the Fibonacci word. Let $q_0,q_1,\ldots q_n,\ldots$ any sequence of natural integers such that $q_0 \geq 0$ and $q_i > 0$ ($i = 1,\ldots,n$), called \emph{directive sequence}. The sequence $\{s_n\}_{n\geq 0}$ can be defined inductively as follows: $s_0 = b$, $s_1=a$, $s_{n+1}= (s_{n})^{q_{n-1}}s_{n-1}$, for $n > 1$. We denote by $Stand$ the set of all words $s_n$, $n\geq 0$, constructed for any directive sequence of integers.

Furthermore, another characterization of standard Sturmian words is related to the Burrows Wheeler Transform (BWT) since, for binary alphabets, the application of the BWT to standard Sturmian words produces a total clustering of all the instances of any character (cf. \cite{MaReSc}), as reported in the following theorem.

\begin{theorem}[\cite{MaReSc}]\label{th_cluster}
Let $w \in \Sigma^*$. Then $w$ is a conjugate of a word in $Stand$ if and only if $bwt(w) = b^pa^q$ with $gcd(p,q)=1$.
\end{theorem}

In the following theorem, for each standard Sturmian word, we individuate a string attractor, whose positions are strictly related with particular decompositions of such words depending on their periodicity. In particular, we recall that $Stand=\{a,b\}\cup PER\{ab,ba\}$ (cf. \cite{deLucaMignosi1994}), where $PER$ is the set of all words $v$ having two periods $p$ and $q$ such that $gcd(p,q)=1$ and $|v| = p + q - 2$. Given a word $w\in Stand$, we denote by $\pi(w)$ its prefix of length $|w|-2$, belonging to the set $PER$, uniquely defined by using previous equality.  By using a property of words in $PER$ (cf. \cite{deLucaMignosi1994}), $\pi(w)=QxyP=PyxQ$, where $x\neq y$ are characters and $Q$ and $P$ are uniquely determined palindromes. So, a standard Sturmian word $w=\pi(w)ba$ can be decomposed as $w=QxyPba=PyxQba$. We call $PER$ decompositions such  factorizations of $w$.

\begin{theorem}\label{th:string_attract}
By using $PER$ decompositions, a Standard sturmian word can be decomposed as $w=QxyPba=PyxQba$.
For each $w\in Stand$ with $|w|\geq 2$,  let $\eta$ be the length of the longest palindromic proper prefix of $\pi(w)$, the set $\Gamma_1=\{\eta+1,\eta+2\}$ or the set $\Gamma_2=\{|w|-\eta-3,|w|-\eta-2\}$ is a minimum string attractor for $w$.
\end{theorem}

\begin{proof
Let us suppose that $w=\pi(w)ba$.  By using a property of words in $PER$ (cf. \cite{deLucaMignosi1994}), $\pi(w)=QxyP=PyxQ$, where $x\neq y$ are characters and $Q$ and $P$ are uniquely determined palindromes. Let us suppose that $|Q|>|P|$. So, $\eta=|Q|$. Firstly we suppose that $x=b$. This means that $w=QbaPba=PabQba$. From a result in \cite{BersteldeLuca97} $aPabQb$ and $bQbaPa$ are the smallest and the greatest conjugates in the lexicographic order, respectively. By Theorem \ref{th_cluster} and by using an argument similar to the proof of Theorem \ref{th:prezzaBWT}, a string attractor can be constructed by considering the positions corresponding to the end of each run. It is possible to see that such positions correspond to the two characters following the prefix $P$ of length $|w|-\eta-4$. If $x=a$, then $w=QabPba=PbaQba$. In this case $aQabPb$ and $bPbaQa$ are the smallest and the greatest conjugates in the lexicographic order, respectively. In this case the ending positions of each run in the output of $BWT$ correspond to the two characters following the prefix $Q$. So, the positions in the string attractor are $\{\eta+1, \eta+2\}$. The case $w=\pi(w)ab$ can be proved analogously by considering the starting characters of each run in the clustered output of $BWT$. \qed
\end{proof}

\begin{example}
Given the standard Sturmian word $w=ababaababa ab aba ba$, the $PER$ decompositions of $w$ are $ababaababa.ab.aba.ba=aba.ba.ababaababa.ba$, then $\{11,12\}$ is a (smallest) string attractor for $w$, since $\eta=10$. Given $v=abaababaababa$, its $PER$ decompositions are $abaaba.ba.aba.ba=aba.ab.abaaba.ba$ and $\eta=6$. So, $\{4,5\}$ is a string attractor.

\end{example}

\begin{comment}
\begin{corollary}
Let $v$ a conjugate of a standard Sturmian word. Then $v$ has a string attractors of size $2$ in which the positions are consecutive.
\end{corollary}
\end{comment}

Previous theorem shows an infinite family of finite binary words such that the size of the smallest string attractor is minimum. In Example \ref{ex:minimum2} we provide some binary words not belonging to $Stand$ with minimum size of string attractor. An open question is to characterize all the binary words with string attractor of size $2$. Furthermore, we can remark that for the standard Sturmian words one can construct string attractors whose positions are consecutive. This fact could be related to the number of distinct factors appearing in the words. It could be interesting to investigate which classes of binary words have minimum size string attractors containing consecutive positions.

\begin{example}\label{ex:minimum2}
A possible set of smallest string attractor of the words $u=a^nb^m$ or $w=b^na^m$ is $\{n,n+m\}$. Moreover, if we consider the word $u=(ab)^{n_1}(ba)^{n_2}$ or $u=(ba)^{n_1}(ab)^{n_2}$ a smallest string attractor is
$\{2{n_1},2{n_1}+2{n_2}\}$.
\end{example}

\begin{comment}
\begin{theorem}\label{th:costantsize}
Let $w \in \Sigma^*$. Then the \emph{minimum size of string attractor} of $w$ is $\sigma$ if $w$ belongs to the following set of words:
 \emph{standard sturmian} words (cf.~\cite{Loth2}) in the case of binary alphabet;
\end{theorem}

In the binary alphabet, the authors in~\cite{MaReSc} proved the following theorem.

\begin{example}

\begin{itemize}
    \item A possible string attractors of $$u=aabaaabaaabaabaaabaaaba\underline{a}\underline{b}aaabaaabaaab$$ is $\{24,25\}$
    \item a possible string attractors of $$v=aaabaaaabaaaabaaabaaaabaaaabaaa\underline{b}\underline{a}aaabaaaabaaaab$$ is $\{32,33\}$
\end{itemize}

\end{example}
\end{comment}

\subsection{The case of bigger alphabets}
We now analyze the string attractors for words of more of two letters by generalizing the standard Sturmian words.
Numerous generalizations of Sturmian sequences have been introduced for an alphabet with more than $2$ letters. Among them, one natural generalization are the episturmian sequences that are defined by using the palindromic closure property of Sturmian sequences (cf. \cite{PAQUIN2009}).
Here we consider some special prefixes of episturmian sequences that are balanced, called \emph{finite epistandard words} \cite{Paquin2007}. We remark that, a word $w$ is balanced if, for any symbol $a$, the numbers of $a$'s in two factors of $w$ of the same length differ at most by $1$ and it is circularly balanced if each conjugate is balanced.
In the case of a binary alphabet they correspond to the standard Sturmian words.

We focus on the circularly balanced epistandard words defined in  Theorem~\ref{Theo:BalEpistClusterized} (see \cite{Paquin2007,RestivoRosoneTCS2011}), that can be built via the \emph{iterated palindromic closure} function. The \emph{iterated palindromic closure} function \cite{Justin2005}, denoted by $Pal$, is defined recursively as follows. Set $Pal(\varepsilon) = \varepsilon$ and, for any word $w$ and letter $x$, define $Pal(wx)~=~(Pal(w)x)^{(+)}$, where $w^{(+)}$, the \emph{palindromic right-closure} of $w$, is the (unique) shortest palindrome having $w$ as a prefix (see \cite{deLuca1997}).
Circularly balanced epistandard words (up to letter permutation), like the standard Sturmian words, are perfectly clustered words under application of the BWT, i.e. the BWT produces a new word that has the minimum number of clusters (\cite{RestivoRosoneTCS2011}, see also \cite{puglisiSimpson2008,RestivoRosoneTCS2009} for more details about the perfectly clustered words on more letters).

\begin{theorem}\label{Theo:BalEpistClusterized}
 Any
circularly balanced finite epistandard word $t$
belongs to one of the following three families (up to letter permutation):
\begin{enumerate}[label=(\roman*)]
\item \label{type1}   $t = p a_2$, with  $p = Pal(a_1^m a_k a_{k-1}\cdots a_3)$,  where $k\geq 3$ and $m\geq 1$;
\item \label{type2}  $t =  p a_2$, with  $p = Pal(a_1 a_k a_{k-1} \cdots a_{k-\ell} a_1 a_{k-\ell-1}a_{k-\ell-2} \cdots a_3)$, where $0 \leq \ell \leq k -4$ and $k\geq 4$;
\item \label{type3} $t = Pal(a_1 a_k a_{k-1} \cdots a_2)$,  where $k\geq 3$.
\end{enumerate}
\end{theorem}
We observe that the words of the last family of Theorem~\ref{Theo:BalEpistClusterized}  correspond to the {\em Fraenkel's sequence}, that are words related to a important conjecture \cite{FRAENKEL1973}.

For each epistandard word, we find a possible string attractor and show that its size is $\sigma$.

\begin{theorem}\label{th:costantsizeMoreTwo}
If $w$ is a \emph{circularly balanced  epistandard} words, then the \emph{minimum size of string attractor} of $w$ is $\sigma$.
\end{theorem}

The proof of the theorem uses similar arguments as in Theorem \ref{th:string_attract} and the notion of palindromic closure. It will be detailed in the full version of the paper.

\begin{example}
We consider some circularly balanced epistandard words and their corresponding Lyndon words.
A smallest string attractor of
\begin{itemize}
    \item (type \ref{type1}) $w\,=\,(aaa\underline{a}\underline{d}aaaa\underline{c}aaaadaaaa)\underline{b}$, obtained by $p\,=\,Pal(a^4dc)$, is $\{4, 5, 10, 20 \}$,  whereas for  its Lyndon conjugate $aaaa\underline{b}aaa\underline{a}\underline{d}aaaa\underline{c}aaaad$ it is $\{5, 9, 10, 15 \}$.
    \item (type \ref{type2} for $k=4$ and $\ell=0$) $u=(a\underline{d}a\underline{c}ada\underline{a}dacada)\underline{b}$, obtained by $p=Pal(adca)$, is $\{2, 4, 8, 15 \}$ and of its Lyndon conjugate $a\underline{a}\underline{d}a\underline{c}ada\underline{b}adacad$ is $\{2, 3, 5, 9 \}$.
    \item (type \ref{type2} for $k=5$ and $\ell=0$) $u'=(a\underline{e}a\underline{a}ea\underline{d}aeaaea\underline{c}aeaaeadaeaaea)\underline{b}$, obtained by $p=Pal(aeadc)$, is $\{2, 4, 7, 14, 28 \}$ and of its Lyndon conjugate  $aaea\underline{b}aea\underline{a}\underline{e}a\underline{d}aeaaea\underline{c}aeaaeadae$ is $\{5, 9, 10, 12, 19\}$
    \item (type \ref{type2} for $k=5$ and $\ell=1$) $u''=(a\underline{e}a\underline{d}aea\underline{a}eadaea\underline{c}aeadaeaaeadaea)\underline{b}$, obtained by $p=Pal(aedac)$, is $\{2, 4, 8, 15, 30\}$ and of its Lyndon conjugate $aaeadaea\underline{b}aeadaea\underline{a}\underline{e}a\underline{d}aea\underline{c}aeadae$ is $\{9, 17, 18, 20,  24 \}$.
    \item (type \ref{type3}) $v=c\underline{a}\underline{c}\underline{b}cac$, obtained by $p=Pal(cab)$, is $\{2,3,4\}$ and of its Lyndon conjugate
    $acbcacc$ is $\{3, 5, 6\}$.
    \end{itemize}
Note that, in this example,
one has an attractor for each symbol $a$ of the alphabet and the position of such attractor in the epistandard word coincides with the position where the last occurrence letter $a \in p$ appears during the palindromic right-closure.
Note also that, in the case of Lyndon words, for each letter $a$, one has an attractor at the position of the last occurrence of $a$ in the run of the output of the BWT.
 \end{example}\label{ex:epist}

Unfortunately, the authors in \cite{RestivoRosoneTCS2011} show that there exist words that do not belong to the families in Theorem~\ref{Theo:BalEpistClusterized} that are perfectly clusterized via BWT, for instance the perfectly clustered word $u = a\underline{b}bbbb\underline{a}\underline{c}ac$ is not a finite epistandard word and a possible string attractor is $\{2,7,8\}$, whereas the perfectly clustered word $v = aacaabaac$ is a finite epistandard but it is not a balanced and a possible string attractor is $\{2,3,6\}$.
The characterization of all perfectly clustered word via BWT is out the scope of this paper.

It remains open the problem of characterizing all words on an alphabet on more of two letters that have the smallest string attractor having size equal to the cardinality of the alphabet.

\section{String Attractors in Thue-Morse Words}

In this section we consider the problem of finding a smallest string attractor for the family of finite binary Thue-Morse words. Thue-Morse words are a sequence of words obtained by the iterated application of a morphism as described below.

\begin{definition}
Let us consider the alphabet $\Sigma=\{a,b\}$ and the morphism $\varphi:\Sigma^*\mapsto \Sigma^*$ such that $\varphi(a)=ab$ and $\varphi(b)=ba$. Let us denote by $t_n=\varphi^n(a)$ the $n$-th iterate of the morphism $\varphi$ that is called the $n$-th Thue-Morse word.
\end{definition}

It is easy to verify that, for each $n\geq 1$ the $n$-th Thue Morse word has length $2^n$.  The $n$-th Thue-Morse words for $n=3,4,5$ can be found in Figure \ref{fig:addmove}.

By using a result in \cite{Brlek89} on the enumeration of factors in Thue-Morse words the following lower bound is proved.

\begin{proposition}
Let $t_n=\varphi^n(a)$ be the $n$-th Thue-Morse word with $n>2$. Then $\gamma^*(t_n)\geq 3$.
\end{proposition}

The following proposition provides the recursive structure of the $n$-th Thue-Morse words by using the rules of a context-free grammar.

\begin{proposition}\label{pr-ThMo}
The $n$-th Thue-Morse word is obtained by the following grammar:
$$\{A_0\to a, B_0 \to  b\}\bigcup_{i=1}^{n-1}\{ A_i \to A_{i-1} B_{i-1}, B_i \to B_{i-1} A_{i-1}\}\bigcup\{A_n\to A_{n-1}B_{n-1}\}$$
by taking as axiom the non terminal symbol $A_n$.
\end{proposition}

The grammar described in Proposition \ref{pr-ThMo} contains $2n+1$ rules for the Thue Morse word of length $2^n$. Therefore, by using a result proved in \cite{KempaP18}, reported here as Theorem \ref{th-PrezzaGr}, it is possible to construct a string attractor for $t_n$ having size $2n+1$.

\begin{comment}
Moreover, another upper bound on the size of the smallest string attractor for the word $t_n$ can be derived from the following proposition showing the output produced by $BWT$ when applied on $t_n$ \cite{MRS_Words03}.

\begin{proposition}
Given the $n$-th Thue-Morse word, $BWT(t_n)=v\overline{v^R}$, where $v=1^{2^{n-2}}0^{2^{n-3}}1^{2^{n-4}}\cdots 1^{2^{0}}0$ and $v^R$ denotes the reversal of $v$.
\end{proposition}

From the results shown in \cite{KempaP18}, we can derive that:

\begin{proposition}
Let $t_n$ be the $n$-th Thue-Morse word of length $2^n$ and $\gamma^{*}(t_n)$ be the size of the smallest string attractor for $t_n$, one has that $\gamma^*(t_n)\leq 2n$.\todo{sicuro che si può applicare la BWT visto che questo è il risultato della BWT circolare?}
\end{proposition}
\end{comment}

In the last part of this section we exhibit a string attractor $\Gamma_n$ for $t_n$ of size $n$. Our conjecture is that $\gamma^*(t_n)=n$.

\begin{theorem}\label{th:string_attractorTM}
A string attractor of the $n$-th Thue Morse word, with $n\geq 3$ is $$\Gamma_n=\{2^{n-1}+1\}\bigcup_{i=2}^n\{3\cdot 2^{i-2}\}$$
\end{theorem}

\begin{lemma}\label{lem:trequarti}
Let $x$ be a factor of $t_n=\varphi^n(a)$, with $n\geq3$. Then, $t_{n+1}$ admits an occurrence of $x$ crossing a position in the set $\bigcup_{i=2}^{n+1}\{3\cdot 2^{n+1-i}\}$.
\end{lemma}

\begin{figure}[t!]
    \centering
    \includegraphics[width=\textwidth]{addmove_ab.pdf}
    \caption{String attractors $\Gamma_n$ for the word $t_n=\varphi^n(a)$, with $n=3,4,5$ (the positions in $\Gamma_n$ are in bold), i.e. $\Gamma_3=\{3,5,6\}$, $\Gamma_4=\{3,6,9,12\}$, $\Gamma_5=\{3,6,12,17,24\}$. Note that in $\Gamma_4$ the position $9$ is obtained by {\sc ADD($9$)} operation, the position $12$ is obtained by the operation {\sc MOVE($5,3\cdot 4$)}. In $\Gamma_5$ the position $17$ is obtained by {\sc ADD($17$)} operation, the position $24$ is obtained by the operation {\sc MOVE($9,3\cdot 8$)}.}
    \label{fig:addmove}
\end{figure}

\begin{proof}[Theorem \ref{th:string_attractorTM}]
The thesis is proved by induction on $n$. If $n=3$, it is easy to check (see Fig. \ref{fig:addmove}) that $\Gamma_3=\{3,5,6\}$ is a string attractor for $t_3$.

Let us suppose that $\Gamma_{n}$ is a string attractor for $t_{n}$. We show that a string attractor for $t_{n+1}$ can be obtained by applying to $\Gamma_n$ the following two operations:
\begin{itemize}
    \item {\sc ADD($2^{n}+1$)}, that adds the new position $2^{n}+1$
    \item {\sc MOVE($2^{n-1}+1$,$3\cdot 2^{n-1}$)} that replaces the position $2^{n-1}+1$ with $3\cdot 2^{n-1}$.
\end{itemize}
Such operations are described, for $n=3,4,5$, in Fig.  \ref{fig:addmove}.
Let $x$ be a factor of $t_{n+1}$. If $x$ is also a factor of $t_n$, then by Lemma \ref{lem:trequarti} has at least an occurrence crossing a position in the set
$\Delta=\bigcup_{i=2}^{n+1}\{3\cdot 2^{n+1-i}\}$. We can suppose that $x$ is factor of $t_{n+1}$ that does not appear in $t_n$ and that does not cross any position in $\Delta$. It means that $x$ has to be factor of $u_n v_{n+1}$. In particular, such a factor exists. In fact, let $a$ and $b$ be the first and the last character of $u_n$, then $av_nb$ has only one occurrence in $t_{n+1}$. If follows from the fact that $t_{n+1}=u_n v_n v_n u_n v_n u_n u_n v_n$ and that the Thue-Morse words has no overlapping factors. Moreover, the occurrence of $av_n b$ crosses the position $2^{n}+1$.\qed
\end{proof}

\begin{comment}
\begin{figure}
$$
\begin{array}{l}
t_2=0110                                                             \\
t_3=01101001                                                         \\
t_4=0110100110010110                                                 \\
t_5=01101001100101101001011001101001                                 \\
t_6=0110100110010110100101100110100110010110011010010110100110010110 \\
\end{array}
$$

$$
\begin{array}{|l|l|l|}
\hline
T_2                                                             & 0\ 2                 & SI \\
\hline
T_3                                                         & 1\ 4\ 6               & SI \\
\hline
T_4                                                 & 1\ 5\ 8\ 12            & SI \\
\hline
T_5                                 & 3\ 8\ 12\ 17\ 23        & SI \\
\hline
T_6 & 8\ 15\ 26\ 32\ 40\ 47    & SI \\
\hline
T_6 & 8\ 15\ 21\ 26\ 40\ 47    & SI \\
\hline
T_6 & 8\ 15\ 21\ 26\ 32\ 40\ 47 & SI \\
\hline
T_6 & 8\ 21\ 26\ 32\ 40\ 47    & NO \\
\hline
T_6 & 8\ 15\ 21\ 26\ 32\ 47    & NO\\
\hline
\end{array}
$$
\caption{Thue-Morse. Attrattori indicizzati da 0.}
\end{figure}
\end{comment}

\section{Attractors in de Brujin words}

A {\em de Bruijn} sequence (or words) $B$ of order $k$ on an alphabet $\Sigma$ of size $\sigma$, is a circular sequence in which every possible length-$k$ string on $\Sigma$ occurs exactly once as a substring.

De Brujin words are widely studied in combinatorics on words, and all of them can be constructed by considering all the Eulerian walks on de Brujin graphs. All the de Brujin sequences of order $k$ over an alphabet of size $\sigma$ have length $\sigma^k$.
For instance the (circular) word $w=aaaababbbbabaabb$ is a de Brujin word of order 4 over the alphabet $\{a,b\}$. In fact one can verify that all strings of length 4 over $\{a,b\}$ appear as factor of $w$ just once.

Since we are here interested to linear and not to cyclic words, it is easy to verify that in order to have linear words containing all the $k$-length factors exactly once, it is sufficient to consider any linearization of the circular de Brujin word of order $k$ (that is, we cut the circular word in any position to get a linear word) and concatenate it with a word equal to its own prefix of length $k-1$. Therefore its length is $\sigma^k+k-1$.  We call such words {\em linear de Brujin sequences (or words)}.  For instance the linear de Brujin word corresponding to the circular one in the above example is the word  $w'=aaaababbbbabaabbaaa$.
Remark that the length of $w'$ is $2^k+k-1$.

In \cite{LZ1976} the following two theorems are proved.

\begin{theorem}\label{th-dBupper}
The number of phrases $c(n)$ in a LZ parsing of a sequence of length $n$ over an alphabet of size $\sigma$ satisfies:
$$c(n)\leq \frac{n}{(1-\epsilon_n)\log n}$$
where $\epsilon_n=2\frac{1+\log(\log (\sigma n))}{\log n}$.
\end{theorem}

\begin{theorem}\label{th-dBLower}
Let $B$ be a de Brujin sequence of order $k$ and length $n+k-1$ over an alphabet of size $\sigma$ ($n=\sigma^k$). Then $$c(B)\geq \frac{n}{\log n}$$
\end{theorem}
By combining Theorem \ref{th-dBupper}, \ref{th-dBLower} and Lemma \ref{L_KePrLZ}, we get both upper and lower bounds for a smallest string attractor for de Brujin sequences.

\begin{proposition}
Let $B$ be a de Brujin sequence of order $k$ and length $n+k-1$ over an alphabet of size $\sigma$ ($n=\sigma^k$). Then the cardinality $\gamma^*$ of a smallest string attractor for $B$ satisfies:
$$\frac{n}{\log n}\leq \gamma^* \leq \frac{n}{(1-\epsilon_n) \log n}+1$$
where $\epsilon_n=2\frac{1+\log(\log (\sigma n))}{\log n}$.
\end{proposition}

This means that $\gamma^*$ for a de Brujin word of length $n$ grows asintotically as $\frac{n}{\log n}$, corresponding to the worst case for the size of a smallest string attractor of any word in $\Sigma^*$.

Notice that the lower bound is somehow intuitively expected, since all the words of length $k$ appear only once in $B$, therefore two consecutive positions in any string attractor cannot be farthest than $k$.

For instance one can verify that for
$w'=aaaababbbbabaabbaaa$ a smallest string attractor  is $\{4, 8, 12, 16 \}$.

\section{Conclusion and Open Problems}
In this paper we have studied the notion of string attractor from a combinatorial point of view. We have given an explicit construction of the string attractor for the infinite families of standard Sturmian words and Thue-Morse words. For standard Sturmian words, by using their combinatorial properties, the construction gives a smallest string attractor whose size is $2$. String attractors of minimum size can be also constructed for circularly balanced epistandard words. It is open the question to characterize all the words whose smallest string attractor has size equal to the cardinality of the alphabet. For Thue-Morse words, the size of the attractor is logarithmic with respect to the length of the word. We conjecture that such size is minimum and we leave it as an open problem.
Based on the results presented in the paper two research directions could be explored. Some standard Sturmian words and Thue-Morse words are generated by morphisms. It could be interesting to find which properties of the morphism determine a smallest string attractor of constant size. Moreover, we plan to study how the distribution of the positions in the smallest string attractor is related to the combinatorial structure of the words.

Finally, the size of smallest string attractor could be used to define a new function to measure the complexity of infinite words. It could be interesting to investigate how such a measure is related with other known complexity measures, such as the factor complexity.

\begin{comment}
\begin{enumerate}
\item Dobbiamo considerare la nozione di String Attractor SA per una stringa o per una parola circolare (nell'impotesi che vogliamo estenderla al caso infinito)?
\item Nel caso infinito: given an infinite word $x$, $SA(x)\subset \mathbb{N}$. We can define $$s_x(n)=|SA(x)\cap [1,n]|,$$ and we can consider $\limsup \frac{s_x(n)}{n}$.
\item Nel caso finito, possiamo considerare le relazioni con:
\begin{enumerate}
\item Distinct parsing of a text
\item Minimal Unique Factor (shortest factor having a unique occurrence)
\item Periodicity
\end{enumerate}
\item Possiamo considerare inoltre i seguenti problemi:
\begin{enumerate}
\item Have a word and its reverse the same minimum SA? What about palindromic word?
\item Palindromic String Attractor. Could we define a SA such that the set of positions is crossed by a factor or its reverse?
\item Can we always find two different minimal SA that are disjoint?
\item How many minimal SA there exist for a string?
\end{enumerate}
\end{enumerate}
\end{comment}

\section*{Acknowledgements}
S. Mantaci, G. Rosone and M. Sciortino are partially supported by the project MIUR-SIR CMACBioSeq (``Combinatorial methods for analysis and compression of biological sequences'') grant n.~RBSI146R5L.

\bibliographystyle{splncs04}

redpajama_set_name: RedPajamaArXiv


text:
Jan 20, 10:50 PM EST
Women's Colleges Welcome Transgender Students Under Expanded Admission Policies
By Julia Glum @superjulia
Smith College recently announced it will enroll self-identified transgender women. Photo: Reuters
Greta Stacy chose to attend Smith College because she wanted to study chemistry, a field dominated by men, in a place where she felt empowered. At Smith, a private women's college in Northampton, Massachusetts, Stacy found herself on a campus full of diverse women who saw no limits -- they were scientists and athletes and more.
"Every person you see who's in a leadership position, whether that's the student body president or your organic chemistry tutor, these people are women," said 22-year-old Stacy, who was elected student body president last year.
But Smith's population is changing. The liberal arts college announced Saturday that its trustees had voted to expand its admission policy to include self-identified transgender women. Citing the "evolution of female identity," Smith is the seventh women's school to tweak its rules and allow transgender students in recent months, after institutions including Bryn Mawr College in Bryn Mawr, Pennsylvania, and Wellesley College in Boston.
The shift comes amid a national debate on transgender rights and awareness, following transgender teen Leelah Alcorn's suicide last year, former Olympian Bruce Jenner's recent coming out as a woman and the Obama administration's ongoing push to ban so-called conversion therapy for LGBT (lesbian, gay, bisexual and transgender) youth.
On campus, administrators face the challenge of implementing LGBT-friendly policies that may require modifying residence halls, restrooms and traditions without causing controversy.
'One More Opportunity'
Women's college students and alumnae said the schools they've historically fought to keep exclusive are taking the right approach by embracing a more diverse population. "I'm incredibly excited," said 1997 Smith alumna Stephanie Gayle, a 39-year-old writer in Arlington, Massachusetts. "Smith's always been pretty cutting-edge and liberal, and this is just one more opportunity for them to uphold the rights of all women."
Transgender advocates had a landmark year in 2014. Facebook gave users more options for setting the gender that appears on their profiles, "Orange Is The New Black" actress Laverne Cox, who is black and transgender, continued skyrocketing in popularity, and the Amazon series "Transparent" was nominated for two Golden Globe awards.
GLAAD, an advocacy group for gays and lesbians, defines sex as the anatomical classification upon which babies' genders are assigned at birth. Most people grow up to be "cisgender," where their gender matches their assigned sex. Others choose to identify as another gender or no gender at all. Some transition and live as transgender men and transgender women, a process that sometimes includes surgery and therapy.
Under Smith's updated policy, any undergraduate student who selects "female" as their gender on their application is eligible for admission. Scripps College in Claremont, California, and Wellesley in Wellesley, Massachusetts, have similar policies.
Other women's schools have slightly different rules. Simmons College in Boston and Mills College in Oakland, California, accept people who were assigned female at birth no matter how they identify, as well as people who were assigned male at birth and identify as female. Bryn Mawr follows a similar policy and also admits intersex students who live as women. Mount Holyoke College in South Hadley, Massachusetts, has the broadest policy: It accepts everyone except people assigned male who identify as male, Campus Pride says.
All seven schools have gender-neutral restrooms. Wellesley is developing a plan to be "as inclusive and welcoming as possible," said Kris Niendorf, assistant dean of students. Smith is doing the same. Mount Holyoke considers transgender student housing requests on a case-by-case basis, and people are not asked to specify their preferred gender identity in roommate matching, spokeswoman Julia Ferrante said. Mills offers a gender-fluid living community as well as single-gender apartments for men and women. Scripps' website states that it is creating a residential space for people who want gender-neutral housing. At Simmons, housing applications include a question about whether "it is important that my room is an LGBTQA safe space." [In the LGBT community, Q stands for queer and/or questioning, while A can stand for asexual, aromantic, allies and/or advocates.]
These decisions didn't come without controversy. Before Scripps trustees approved the policy tweak, more than 100 alumnae signed a letter concerning "the changes that increasing numbers of transitioning students may necessitate." It referenced Mills College, where the traditional chant "Strong Women! Proud Women! All Women! Mills Women!" was recently changed to "Strong! Proud! All! Mills!" The Scripps alumnae called for the trustees to protect Scripps as a women's space and worried about the erasure of the campus' sisterhood.
"Before we enact this or any policy that potentially evolves the fundamental nature of our women's college, we must have a compelling answer to the question, 'What is a women's college in the 21st century?'" it read.
On Facebook, more than 80 people liked a page called "Keep Smith College Female Forever," which posted a letter by alumna Elizabeth Hungerford, who graduated in 2000. In it, she wrote that she was worried about transgender students acting like men and oppressing women in a place intended to be a sanctuary. "Trans women may retain their penises and secondary sex characteristics," Hungerford wrote. "They may talk loudly over women or on behalf of women while looking and sounding exactly like men."
'They Need The Safe Space'
But other alumnae argued that the policy changes were overdue. Nora Hagerman, a 24-year-old auditor in Milwaukee who graduated from Wellesley in 2013, said transgender women have many of the same experiences with being marginalized that "ciswomen" do. "If anything, they need the safe space even more," she added.
In 2010, 76 percent of Scripps voters approved a student government measure swapping out all the language in its bylaws for gender-neutral terms. Last year, about 60 Bryn Mawr students and alumnae gathered 2,000 signatures on a Change.org petition demanding a more inclusive admission policy.
Ryan McDaniel, a 22-year-old Smith senior studying government, said women's college students tend to be more open-minded about gender. McDaniel, who entered Smith as a woman but now identifies as a man, said he saw this during his first week of school. During icebreaker activities in the dorms, students took turns introducing themselves by name, major and preferred gender pronoun. "People want to go out of their way to support their classmates," he said.
McDaniel said he graduated from a conservative high school and felt liberated by Smith's "culture of understanding." When he came out last year, his friends and professors were accepting and adopted the correct pronouns. The only hangup he had was with one of Smith's therapists, who McDaniel said declined to provide a letter he needed to start hormone replacement therapy.
Justin Kilian, a 19-year-old transgender woman and a member of the transgender student advocacy organization Smith Q&A, said she's happy with the trustees' decision, but hopes administrators are flexible in deciding where transgender students can live. She said she doesn't want them to be unfairly grouped and kept from the general student population, because that essentially negates the progress made by admitting them. "It happens a lot where, when people hear 'trans women,' they don't hear 'women.' They hear 'transgender,'" said Kilian, a student in women, gender and sexuality studies.
Kilian said Smith is still a women's college. She added that it's more important than ever to preserve that setting -- "just women being together," she said.
'Concepts Of Female Identity' Have Evolved
Japan Expands LGBT Accommodations
Russia's Sputnik V Vaccine Offers Higher Omicron Antibodies Than Pfizer: Study
CDC Data Reveals States, Territories With Least Physical Activity
CIA Gives Assessment On Havana Syndrome

redpajama_set_name: RedPajamaCommonCrawl


Juncture Media is an Intellectual Property company that is currently focusing on the creation of video game content. They needed a simple but unique stamp that reflected their brand values. Virtual Monk designed the logo with simplicity and movement to it, with fonts and colors that express speed and forward-thinking ideals.


text:
Burglars ransack hair salon - then return to raid it again
By Mike Laycock Chief reporter
Salon owner Claire Hopwood pictured left, with colleague Hayley Potts outside the burgled business in Badger Hill
BURGLARS have broken into and ransacked a York hair salon - and then raided it a second time and made off with both hairdressing equipment and Christmas decorations.
The intruders used a crowbar to get into Salon 33, in Yarburgh Way, Badger Hill, through the back door on Sunday night and in the early hours of Monday.
Owner Claire Hopwood said they stole a pair of GHD hair straighteners, colourings - both opened and unopened - professional clippers, hairdryers, Christmas decorations and also presents given to her by customers.
She said she had first been alerted to someone having broken in late on Sunday and she went down to the salon where she met police and discovered it had been trashed but without anything being taken.
They left after making the premises secure but she then discovered on Monday morning that the burglars had returned, and this time they had stolen hundreds of pounds worth of equipment.
She said she had been straight out and bought some replacement equipment and tidied up the salon, and was able to re-open it as normal yesterday.
Her daughter Eva Claire Lane urged anyone with any information to contact police, and also appealed for people to watch out for any professional salon items being sold on Facebook.
"Please report them if you have a feeling it may be dodgy," she said, adding that the local community had pulled together to help her mother, and police had been very supportive.
She said officers were also feeling 'pretty confident' they would find those responsible after CCTV footage had been collected from neighbouring shops.
"Whoever has done this went in with a crow-bar through the back door, which is clearly visible from Hull Road, just between the Black Bull pub and the Hull Road roundabout between the hours of 1.30am-5/6am on Monday," she said.
She added that the door had been securely boarded up and the business opened as usual yesterday.
North Yorkshire Police confirmed it had received two reports of break-ins at the salon, the first at 11pm on Sunday, in which damage had been caused, including to the door and door frame, and the second at 9.43am on Monday, in which cash and equipment had been stolen, including hairdryers.
A spokesman said officers had been conducting door to door inquiries and checking local CCTV footage.
He said anyone with information should phone 101, and quote crime reference number NYP- 08122019-0523, or call Crimestoppers anonymously on 0800 555 111.


Another completed project, this time in Belgium. Thanks to AVK Belgium we supplied this complete package.
The package contains Wouter Witzel Double Flanged Butterfly valves in the color Blue RAL 5017 with Auma electric and manual actuation.
Find the perfect product and discover the best solution for your application: www.wouterwitzel.nl.


Looking at these lists, you're likely now realising you have evidence to demonstrate plenty of those skills. If you have a psychology degree, for example, you have a thorough understanding of character traits that can help you to understand an audience. If you have studied English Literature, you have excellent communication skills and the ability to write engaging content.
But it doesn't stop there… maybe your degree isn't directly relevant at all, but you've volunteered at the weekends during university working for a charity, helping them to run successful events and publicise them. Perhaps you've worked on a research project in your degree which involved developing a research proposal, setting the parameters of your study, setting measures to evaluate it, and then testing your hypotheses. If you've completed a dissertation as part of your degree, chances are you have completed this process.
Some vacancies require specialist skills, i.e. engineers, software developers, strategists, etc. and the employer needs someone who can hit the ground running with skills that can only be gained from practical experience or a degree in that area, but for many, you'll see that ScotGrad employers are becoming more flexible in their approach and are open to applicants who can meet the requirements through other means. You'll still need to have a degree to apply to our vacancies, but take advantage of the flexibility.
Remember to think outside the box when it comes to your examples. Be as clear and concise as possible, documenting your approach to the task including tangible outcomes, and be clear about why your examples are relevant.
It's your job to convince the employer you're the right candidate for the job, and the first step is to believe that you are. Take a look at our advice pages and other blogs for more top tips.
Find current vacancies on our graduate placements page.


A U.S. judge has ordered the SEC and Tesla CEO Elon Musk to confer over the next two weeks to try to settle the matter over contempt.
Tesla laid off dozens of in-store sales employees, the electric carmaker said on Monday, as the company looks to reduce its physical footprint across the U.S. and shift operations online.
The staff reductions hit stores in Chicago, Brooklyn, New York and Tampa, Florida, but a spokeswoman declined to share the exact number.
"This is a hard decision, but it necessary to make our cars more affordable. Our sales team has fought on the front lines of advancing our mission and has been our connection to hundreds of thousands of customers along the way," Musk wrote at the time.
Tesla previously reported that vehicle deliveries in the three months through March fell 31 percent year-over-year. The Palo Alto-based company attributed the decline to increased deliveries to Europe and China, which pushed some first quarter shipments to the subsequent three months period.
Morgan Stanley analyst Adam Jones on Monday cut his price target on Tesla to $240 per share after the lower-than-expected delivery number.
"We are increasingly concerned about the impact that investor concerns over Tesla's financial strength and forward looking liquidity position could potentially have on employee morale, customer perceptions and standing with key stakeholders and suppliers," he wrote.
Jonas predicted Tesla would deliver 344,000 vehicles this year, within the company's own expectations but lower than his prior estimates.
Since the start of 2019, Tesla's stock has fallen roughly 11.6 percent to $274.96 per share.


text:
Q: An MDX expression was expected. An empty expression was specified I am getting this message when i am trying to build my mdx query in my ssrs query builder. Any idea????
error:
An MDX expression was expected. An empty expression was specified. (Microsoft SQL Server 2008 R2 Analysis Services)                                                                                Here is my MDX
    with
member [Measures].[Motor: The Car]
as
( [Web Page].[Web Site Structure].[Motor Insurance].[TheCar], [Measures].[Visit Count])
member [Measures].[Motor: YourPolicy]
as
( [Web Page].[Web Site Structure].[Motor Insurance].[YourPolicy], [Measures].[Visit Count])

member [Measures].[Motor:Proposer]
as
( [Web Page].[Web Site Structure].[Motor Insurance].[Proposer], [Measures].[Visit Count])

member [Measures].[Home:About You]
as
( [Web Page].[Web Site Structure].[House Insurance].[AboutYou], [Measures].[Visit Count])
member [Measures].[Motor:About You]
as
( [Web Page].[Web Site Structure].[Motor Insurance].[AboutYou], [Measures].[Visit Count])
member [Measures].[Motor:PricesPage]
as
( [Web Page].[Web Site Structure].[Motor Insurance].[ViewPrices], [Measures].[Visit Count])

member [Measures].[Home:PricesPage]
as
( [Web Page].[Web Site Structure].[House Insurance].[ViewPrices], [Measures].[Visit Count])
member [measures].[Motor:ViewAllPrices]
as
( [Web Page].[Web Site Structure].[Product].&[2].[Motor:AllViewPrices], [Measures].[Visit Count])

member [measures].[Home:ViewAllPrices]
as
( [Web Page].[Web Site Structure].[House Insurance].[Home:AllViewPrices], [Measures].[Visit Count])
member [measures].[Home:YourCover]
as
( [Web Page].[Web Site Structure].[House Insurance].[YourCover], [Measures].[Visit Count])
member [measures].[Home:PolicyHolder]
as
( [Web Page].[Web Site Structure].[House Insurance].[PolicyHolder], [Measures].[Visit Count])
member [measures].[Home:Property]
as
( [Web Page].[Web Site Structure].[House Insurance].[Property], [Measures].[Visit Count])

 SELECT NON EMPTY { [Measures].[Motor:PricePanel],
 [Measures].[Motor:PricePanel->Compare],
 [Measures].[Motor:PricePanel->MorePanel],
 [Measures].[Home:PricePanel->MorePanel],
 [Measures].[MorePanel->BO+Phone],
 [Measures].[Motor:AY->PricesPage],
 [Measures].[PercentOfVisitsByBrowser],
  [Measures].[Proposer->PricesPage],
  [Measures].[Proposer->TheCar],
   [Measures].[TheCar->YourPolicy],
    [Measures].[PolicyHolder->Property],
     [Measures].[YourCover->PricesPage],
     [Measures].[Property->YourCover],
      [Measures].[YourPolicy->PricesPage],
       [Measures].[BO+Phone],
        [Measures].[Compare],
        [Measures].[AY->Proposer],
        [Measures].[Motor:PricePanel->BO+Phone],
        [Measures].[AY->PolicyHolder],
        [Measures].[Home:PricePanel->BO+Phone],
        [Measures].[Home:PricePanel->Compare],
        [Measures].[Home:PricePanel],
        [Measures].[Home:AY->PricesPage],
        [Measures].[MorePanel],
        [Measures].[Motor:About You]
,[Measures].[Motor:Proposer]
,[Measures].[Motor: The Car]
,[Measures].[Motor: YourPolicy]
,[Measures].[Motor:PricesPage]
,[measures].[Motor:ViewAllPrices]
,[Measures].[Home:About You]
,[measures].[Home:PolicyHolder]
,[measures].[Home:Property]
,[measures].[Home:YourCover]
,[Measures].[Home:PricesPage]
,[measures].[Home:ViewAllPrices]
,[Measures].[BO+Phone] } ON COLUMNS,
        NON EMPTY { ([Timeline].[Financial Date].[Visit Date].ALLMEMBERS *
        [Product].[Products By Category].[Product Category].ALLMEMBERS ) }
         DIMENSION PROPERTIES MEMBER_CAPTION, MEMBER_UNIQUE_NAME ON ROWS
         FROM ( SELECT ( STRTOSET(@ProductProductsByCategory, CONSTRAINED) ) ON COLUMNS
         FROM [Visits]) CELL PROPERTIES VALUE, BACK_COLOR, FORE_COLOR, FORMATTED_VALUE, FORMAT_STRING, FONT_NAME, FONT_SIZE, FONT_FLAGS

A: The syntax looks fine, so I'm guessing @ProductProductsByCategory is the problem.
The string that you pass in to this parameter needs to take the form of a set because you are using the StrToSet function.
I usually remove the 'Constrained' argument because it forces you to list a specific set of tuple(s) enclosed by the squigly braces {}
If you remove that argument you can use an MDX expression that returns a set rather than specifying the set explicitly http://msdn.microsoft.com/en-us/library/ms144782.aspx
In the default value for your parameter you will also need to specify a set. If it's just a member then rather use the StrToMember function.

redpajama_set_name: RedPajamaStackExchange


NCU course codes include a course prefix and number. The course prefix identifies the content area of a course and the number identifies the course-level (e.g., Undergraduate, Master's. etc.). Courses in this catalog section are list by School and content area in numerical order.
Course length varies by course and program. Please refer to the course listing in this catalog to determine the length of a specific course.


text:
/*
 * Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except
 * in compliance with the License. You may obtain a copy of the License at
 *
 * http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software distributed under the License
 * is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express
 * or implied. See the License for the specific language governing permissions and limitations under
 * the License.
 */
/*
 * This code was generated by https://github.com/googleapis/google-api-java-client-services/
 * Modify at your own risk.
 */

package com.google.api.services.monitoring.v3.model;

/**
 * The ListTimeSeries response.
 *
 *
 This is the Java data model class that specifies how to parse/serialize into the JSON that is
 * transmitted over HTTP when working with the Cloud Monitoring API. For a detailed explanation see:
 * https://developers.google.com/api-client-library/java/google-http-java-client/json
 *

 *
 * @author Google, Inc.
 */
@SuppressWarnings("javadoc")
public final class ListTimeSeriesResponse extends com.google.api.client.json.GenericJson {

  /**
   * Query execution errors that may have caused the time series data returned to be incomplete.
   * The value may be {@code null}.
   */
  @com.google.api.client.util.Key
  private java.util.List<Status> executionErrors;

  /**
   * If there are more results than have been returned, then this field is set to a non-empty value.
   * To see the additional results, use that value as page_token in the next call to this method.
   * The value may be {@code null}.
   */
  @com.google.api.client.util.Key
  private java.lang.String nextPageToken;

  /**
   * One or more time series that match the filter included in the request.
   * The value may be {@code null}.
   */
  @com.google.api.client.util.Key
  private java.util.List<TimeSeries> timeSeries;

  /**
   * The unit in which all time_series point values are reported. unit follows the UCUM format for
   * units as seen in https://unitsofmeasure.org/ucum.html. If different time_series have different
   * units (for example, because they come from different metric types, or a unit is absent), then
   * unit will be "{not_a_unit}".
   * The value may be {@code null}.
   */
  @com.google.api.client.util.Key
  private java.lang.String unit;

  /**
   * Query execution errors that may have caused the time series data returned to be incomplete.
   * @return value or {@code null} for none
   */
  public java.util.List<Status> getExecutionErrors() {
    return executionErrors;
  }

  /**
   * Query execution errors that may have caused the time series data returned to be incomplete.
   * @param executionErrors executionErrors or {@code null} for none
   */
  public ListTimeSeriesResponse setExecutionErrors(java.util.List<Status> executionErrors) {
    this.executionErrors = executionErrors;
    return this;
  }

  /**
   * If there are more results than have been returned, then this field is set to a non-empty value.
   * To see the additional results, use that value as page_token in the next call to this method.
   * @return value or {@code null} for none
   */
  public java.lang.String getNextPageToken() {
    return nextPageToken;
  }

  /**
   * If there are more results than have been returned, then this field is set to a non-empty value.
   * To see the additional results, use that value as page_token in the next call to this method.
   * @param nextPageToken nextPageToken or {@code null} for none
   */
  public ListTimeSeriesResponse setNextPageToken(java.lang.String nextPageToken) {
    this.nextPageToken = nextPageToken;
    return this;
  }

  /**
   * One or more time series that match the filter included in the request.
   * @return value or {@code null} for none
   */
  public java.util.List<TimeSeries> getTimeSeries() {
    return timeSeries;
  }

  /**
   * One or more time series that match the filter included in the request.
   * @param timeSeries timeSeries or {@code null} for none
   */
  public ListTimeSeriesResponse setTimeSeries(java.util.List<TimeSeries> timeSeries) {
    this.timeSeries = timeSeries;
    return this;
  }

  /**
   * The unit in which all time_series point values are reported. unit follows the UCUM format for
   * units as seen in https://unitsofmeasure.org/ucum.html. If different time_series have different
   * units (for example, because they come from different metric types, or a unit is absent), then
   * unit will be "{not_a_unit}".
   * @return value or {@code null} for none
   */
  public java.lang.String getUnit() {
    return unit;
  }

  /**
   * The unit in which all time_series point values are reported. unit follows the UCUM format for
   * units as seen in https://unitsofmeasure.org/ucum.html. If different time_series have different
   * units (for example, because they come from different metric types, or a unit is absent), then
   * unit will be "{not_a_unit}".
   * @param unit unit or {@code null} for none
   */
  public ListTimeSeriesResponse setUnit(java.lang.String unit) {
    this.unit = unit;
    return this;
  }

  @Override
  public ListTimeSeriesResponse set(String fieldName, Object value) {
    return (ListTimeSeriesResponse) super.set(fieldName, value);
  }

  @Override
  public ListTimeSeriesResponse clone() {
    return (ListTimeSeriesResponse) super.clone();
  }

}

redpajama_set_name: RedPajamaGithub


HARARE - Harare mayor councillor Herbert Gomba on Sunday presented his spiritual side kneeling before a church leader in Highfield.
Gomba was a congregant at Prophet Ephias Jengeta-led Family Worship Tabernacle Ministries International Church, with congregants describing him as humble.
Speaking to the Daily News, the preacher said the mayor needs support in his work, leading the capital city.
"He is a man bestowed with a huge task, dealing with the affairs of the country's largest city and he needs all the support he can get," Jengeta said.
"He is a humble man and for him to know that there is something called church and God despite his status and busy schedule shows that. We will support him spiritually," added the preacher.
The preacher has hosted several high profile figures at his ministry. Gospel musician Mercy Mutsvene has been leading the preacher's choir.
Jengeta said it is not proper for a church to make noise about the people who would have attended their services.
"Several people come to our church and we are not there to announce that so and so was here.


The Breitling Watch Source Forums • View topic - Just pulled the trigger!
Post subject: Just pulled the trigger!
I was visiting my local Ad yesterday and fell for a Breitling Chronomat in steel and 18K rose gold on the pilot bracelet. Really like the unique nature of the rose gold. Just gives the timepiece a different look. Silver dial with gold sub dials.I got the full gold bezel as opposed to the gold riders. I really hadn't intended on pulling the trigger but couldn't resist. I'm mostly a Rolex guy but this is the second Breitling I've purchased in the last two months. Could be the beginning of a bad habit!
Post subject: Re: Just pulled the trigger!
Be sure to post some photos of that beauty here. So, what was the first one you got 2 weeks ago? True Breitlings are highly addictive, I have been feeding my addiction lately as well.
"Born to Cruise, Forced to Work"
Very nice watch. Post some pictures. Congratulations!!!!!
Sounds like a nice score. I am sitting here talking to my wife and expressing how I feel that Breitlings give you more... bang... Pizzaz? for the buck. I had wanted the gold crown for years and admired Breitlings for just as long and its obvious what I ended up with and havent thought twice about it!
So... Show us some pictures!


It is typical human behavior to try to characterize and typify someone else's behavior and understand them on an intimate level. But human emotions are not robotic at all and you can't just base it around a bunch of mathematical equations, star signs or in this case, a bunch of zodiac sign. Science has consistently proven that heavenly bodies have no influence on us whatsoever.
Every other person with come with their own 'prediction' and 'understanding' of what each zodiac sign might mean. There is no particular definition or standard with which you could hold these predictions accountable. This isn't really the case with science, where everything is agreed upon.
Most people heavily depend on zodiac signs and take them incredibly seriously. They will create their entire life pattern around those signs because it is so much better to entrust your future into the unknown than to actually trust your own instincts.
Astrology has very little to do with reality but because so many people heavily trust them, it has become a profitable industry. Everyone is making 'predictions' in exchange of donations. Here is what the 12 zodiac signs say.
Aries are constantly engaged with their partners in a sort of tug of war which is a desperate struggle for control over the relationship. Their slogan is somehow aligned with conflict because Aries directly translates to "God of War".
These people are paranoid that their partner will end the relationship very early. They're always on edge and desperately try to make sure they are not on the wrong side of the fence from their better half.
They are engaged in a war of wits with their better half. Always teasing them with words in attempts to get a rise out of them. Of course it is all done in the spirit of fun.
They know the relationship is probably not going to last forever so they don't get their hopes up too high. But they do value the time being until which the relationship has been stretching on and will try their best to protect it.
They are social animals and have a large number of friends outside of their immediate relationships. They value each of these relationships with the utmost of sincerity, which sometimes directly gets in the way of their relationship.
They take it to heart the phrase, "Honesty is the best policy" and will maintain the relationship up until the point when they start detecting that their better half has been using lies to get their point across.
They love to maintain their peace. Although they love the occasional companionship every now and then, if it takes too much of their personal space, they would rather call it off. They prefer living single lives for long periods of time.
They really want to learn what's going on inside the minds of their partners. This means they really value all the secrets which they manage to extract and get deeply hurt when its found the secret was intended to be kept away from them.
Hope is what makes us strong. Hope is what powers the world. Hope is what has fuelled humanity and taken them away from the problems of life into the relatively comfortable era of the modern times. A Sagittarius typically seek such contentment from relationships.
Capricorns will chase after you and pursue your love until you make it resoundingly clear you want absolutely nothing to do with them. So while they won't give up they knew you're an option, but it is not an option, the two of them can't fight.
What they're after is exotic beauty. Common isn't going to cut it. So if don't belong to a particular demographic, try to make up for it by adopting their cultural dress codes instead. That could make you look more 'exotic'.
They are extremely ambitious and look toward their partner for help. If the partner is inspiring enough to them. They will make friends with them but if you want something serious, then don't send pictures of you in which you look sick.


If you're looking for high quality and personal service, you've come to the right place. At SprinklerHouz we'll give you the attention and personal service you'll come to expect and enjoy.
We provide Automatic Garden Sprinkler System installation and maintenance for residential and commercial landscape.
Free consultation will be provided upon request in designing your irrigation system. We are currently operating in KL, Seremban, Melaka and sourcing our parts to all over Malaysia .
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text:
Get your ticket now
Versailles Palace & Gardens: Ticket, Audio Guide & Transfer
Paris: Versailles Palace & Gardens Ticket and Audioguide
Popular Tickets in Palace of Versailles
Versailles: Skip-the-Line Entrance and Lunch at Ore
The Paris Pass®: Access to Over 75 Attractions and Tours
Valid: 2 - 6 days
Paris Museum Pass: 2, 4, or 6 Days
Paris: Versailles Palace Tour with Optional Fountain Show
Half Day Versailles Palace & Gardens Tour From Versailles
Versailles Palace & Gardens Tour with Gourmet Lunch
Versailles: Palace Guided Tour with Skip-the-Line Tickets
Versailles Palace Skip-the-Line Entrance and Breakfast
Our most recommended Palace of Versailles Tickets
1. Versailles Palace & Gardens: Ticket, Audio Guide & Transfer
Travel in the comfort of a fully air-conditioned bus from Paris city center to Versailles, home of Louis XVI and Marie Antoinette until the French Revolution in 1789. Your host will be on board the bus and ready to answer any questions you may have. Be transported back to the Golden Age of France with a trip to the Palace of Versailles, a UNESCO World Heritage Site and emblem of French royalty and extravagance. Marvel at the breathtaking royal chateau, transformed by the 'Sun King' Louis XIV in the 1660s from a hunting lodge. Your audio guide, available in 11 languages, will enable you to discover the State apartments of the King and Queen, the Royal Chapel and Hall of Mirrors, the epitome of French opulence; three hundred and fifty-seven mirrors line the gold-adorned hall and crystal chandeliers hang from the ceiling. After explore the famous gardens of Versailles with access to the Grand and Petit Trianon, and the Estate of Marie Antoinette. Depending on the day and option booked, you will also be able to see either the Musical Garden or Fountain Show. After your visit, you will be taken back to Paris by air-conditioned bus.
2. Paris: Versailles Palace & Gardens Ticket and Audioguide
Venture into the grand ball and state rooms of the Palace of Versailles on a full-day tour from Paris. Choose from one of 3 departure points in the French capital for your transfer by comfortable bus. Upon arrival at the UNESCO World Heritage-listed monument, a trilingual host will first escort you to the château. You then will discover more about the site from the informative audio guide available in 11 languages. Delve deep into the heart of court life and learn how the Palace of Versailles grew from a hunting lodge for Louis XIII before his son Louis XIV transformed and expanded it, moving the court and government of France to Versailles in 1682. Explore the dazzling Hall of Mirrors and the opulent State Apartments. Follow in the footsteps of Louis XIV ("The Sun King"), and stroll among gardens designed by André Le Notre at your own pace. From May to October, your ticket includes the legendary Musical Fountains & Gardens Shows. Watch as the water features light up in a dazzling spectacle to complete your day in Versailles.
3. Versailles: Skip-the-Line Entrance and Lunch at Ore
In the recently renovated Pavillon Dufour, ore offers the great French classics, light bites and delicate pastries. Located at the first floor of Pavillon Dufour, the restaurant plan is divided into two main dining rooms, one bar and three salons. One can admire the Palace through the windows overlooking the cour Royale and the cour d'Honneur. In the salons, the subtle position of the mirrors allows the guests to look upon the Palace, even if their back is turned to the windows. Ore - Ducasse au château de Versailles offers contemporary French cuisine and artfully combining great classics and new dishes. Service is relaxed, professional and efficient. After lunch, be guided to the skip-the-line access for the palace and enjoy the rest of your day in Versailles.
4. The Paris Pass®: Access to Over 75 Attractions and Tours
The Paris Pass is an all-inclusive sightseeing package and your key to the city of Paris. The pass offers great value and convenience with admission to more than 75 top attractions, a 1-day hop-on, hop-off bus tour ticket, a Paris Museum Pass giving you access to many of the cities museums and landmarks, and a digital guide to help you plan your stay. You will also get skip-the-ticket-line entrance to many of the sights, saving you time at attractions such as the Centre Pompidou, Arc de Triomphe, Louvre Museum, and more. Head out to the Palace of Versailles, or take a Bateaux River Cruise to see the city from the River Seine. The Paris Pass offers access to over 75 attractions and tours including: Top Attractions and Activities: • Eiffel Tower Elevator Guided Tour (2nd floor) Guided Tours and Experiences: • Ballon de Paris Generali • Parc Asterix Theme Park Ticket Top Attractions and Activities: • Musée du Louvre• Musee d'Orsay• Centre Pompidou, National Museum of Modern Art• Espace Dali • Big Bus – 1-Day Hop-on Hop-off bus tour • Bateaux Parisiens River Cruise• Wine Tasting - Les Caves du Louvre• Tour Montparnasse • Arc du Triomphe• Picasso Museum• Musee Grevin, Wax museum• Châteaux de Versailles • Flyview Package Paris Liberty Walking Tours: • Paris' Hidden Gems in Le Marais – Walking tour• Walking tour along the Seine• Walking Tour in the Parisian Covered Passages• Scandals & Love Affairs at Pere Lachaise - Walking Tour• Stade de France tour Galleries:• Musée Rodin • Musée National de l'Orangerie • Musée national des Arts asiatiques Places of Interest: • Petit Train de Montmartre • Rex Studio• Le Musée Gourmand du Chocolat - Choco- Story • Musée des Égouts de Paris • Musée du Parfum Fragonard Tour• Cité des Sciences et de l'Industrie - La Villette • BAPBAP Brewery Taproom Experience Beer tasting and gifts Historic Buildings: • Panthéon• Conciergerie • Chapelle expiatoire • Château de Rambouillet • Château de Vincennes• Chateaux de Champs-sur-Marne • Château de Pierrefonds • Maison d'Auguste Rodin à Meudon • Musée Condé - Château de Chantilly • Musée et domaine nationaux du Château de Compiègne • Musée national des Châteaux de Malmaison et Bois Préau• Basilique cathedrale de Saint-Denis• Abbaye Royale de Chaalis- Musée Jacquemart André • Sainte Chapelle • Château de Maisons-Laffitte• Villa Savoye• Château de Fontainebleau Museums: • Musée de l'Armée, Tombeau de Napoléon 1er • La Cinémathèque française - Musée du Cinéma • Cite de l'Architecture et du Patrimoine - Musée des Monuments françois • Musée d'art et d'histoire du Judaïsme • Musée de la Musique - La Villette • Musee de l'Ordre de la Liberation • Musée de l'Air et de l'Espace • Musée d'Archéologie nationale de Saint-Germain-en-Laye • Musée de l'Institut du Monde arabe • Musée des Arts décoratifs • Musée des Arts et Métiers • Musée des Plans-reliefs • Musee du quai Branly • Musée Gustave Moreau • Musée national de Céramique de Sèvres • Musée national de la Renaissance - Château d'Ecouen • Musée national du Moyen Age - Themes de Cluny • Musée national Eugène Delacroix • Musée Nissim de Camondo The Paris Pass is fully digital and is accessible to download to your mobile device straight away, together with the digital guide. The Paris Museum Pass is a physical pass and as such must be collected in a redemption location in Paris. Full instructions are included on the digital guide.
5. Paris Museum Pass: 2, 4, or 6 Days
Enjoy priority skip-the-line access to over 60 museums and monuments in and around the city of Paris including the Louvre Museum, Orsay Museum, and Centre Pompidou. The Pass is available for 2, 4, or 6 days with the pick-up point conveniently located a mere 10-minute walk from the Louvre. Benefits of the pass include free admission to many attractions, sights, and museums. In Paris: Musée de l'Armée Tombeau de Napoléon 1er Centre Pompidou Musée national d'Art modern Musée national des Arts asiatiques Guimet Les Arts décoratifs Musée des Arts et métiers Musée de l'Assistance Publique, Hôpitaux de Paris Musée du quai Branly Musée Picasso Paris Chapelle expiatoire La Cinémathèque française Musée du Cinéma Cité des Sciences et de l'Industrie Conciergerie Musée national Eugène Delacroix Musée des Égouts de Paris Musée du Louvre - Musée national de la Marine Musée de la Monnaie Cité de l'Architecture et du Patrimoine Musée national des Monuments français Musée Gustave Moreau Musée national du Moyen Âge, Thermes et hôtel de Cluny Cité de la Musique, Musée de la Musique Crypte archéologique de Notre-Dame Tours de Notre-Dame Musée national de l'Orangerie Musée de l'Ordre de la Libération Musée d'Orsay Panthéon Musée des Plans-reliefs Musée de la Poste Musée Rodin Sainte-Chapelle Outside Paris: Musée de l'Air et de l'Espace Musée d'Archéologie nationale de Saint-Germain-en-Laye Musée national de la Céramique de Sèvres Abbaye royale de Chaalis Musée et domaine nationaux du Château de Compiègne Musée Condé, Château de Chantilly Musée départemental Maurice Denis Château de Fontainebleau Château de Maisons-Laffitte Musée national des Châteaux de Malmaison et Bois-Préau Château de Pierrefonds Musée national de Port-Royal des Champs Château de Rambouillet Musée national de la Renaissance Château d'Ecouen Maison d'Auguste Rodin à Meudon Basilique cathédrale de Saint-Denis Villa Savoye Musée national des Châteaux de Versailles et de Trianon Château de Vincennes
6. Paris: Versailles Palace Tour with Optional Fountain Show
Hidden away from the modern world, this majestic European castle on the outskirts of Paris is a perfectly preserved example of the reckless extravagance of the French monarchy. Experience life at its most luxurious, in a setting of lush rolling lawns that stretch as far as the eye can see. Meet your guide at a central location in Paris and board a private, air-conditioned tour bus to Versailles. You'll skip the lines at the entrance with your pre-booked admission to Versailles, where your guide will lead you through the gardens and palace. With a maximum of 25 people per group, your Versailles tour will be intimate and unhurried. Marvel at the sheer size of the Palace of Versailles, one of the largest and most spectacular castles in the world, boasting 700 rooms! See the incredible array of fine art and the intricate designs of the 1,250 fireplaces as you work your way through the palace's highlights, including the king's and queen's apartments, the chapel, the Coronation Room, the Hall of Mirrors, and the Salon of Abundance. Head out into the sunlight and visit the manicured gardens, covering an area of 1,800 acres. The Musical Gardens show runs every Tuesday, Wednesday, Thursday, and Friday (except Wednesday, July 14) from July 1 to August 31, while the Musical Fountains can be experienced every Saturday and Sunday from May 22 to October 31.
7. Half Day Versailles Palace & Gardens Tour From Versailles
Learn how the glory of the Sun King shone throughout the world on this tour. Your expert English-speaking guide will bring the extravagant lives of the royal family and the French court at Versailles to life. Small groups allow for a relaxed and intimate visit. Your guide will take you on a journey in the footsteps of the Kings and Queens of France. During the reign of Louis XIV - the Sun King - Versailles was the place to be and to be seen. Nowadays visitors from all over the world still come to Versailles to marvel at the splendours of this incredible palace. The skip-the-line access allows you to bypass the long lines that form each day, and your friendly local guide will steer you expertly through the crowds, bringing the famous figures who once inhabited this palace back to life, and making sure you don't miss a thing. Learn how the Kings and Queens lived out their lavish lives in public, how they kept the gossiping courtiers under their thumb, and stole Italy's thunder by making Versailles the artistic center of Europe. Discover a typical day in the life of Marie Antoinette; you'll quickly understand why she hated the palace so much. Her opulent lifestyle here would ultimately lead to her tragic demise under the French Revolution; she left Versailles in 1789 never to return. After visiting the royal apartments inside the palace, your guide will lead you through the spectacular gardens, once the setting of balls, parties and elaborate firework displays. On fountain show days you will discover the amazing waterworks, still running today as they did in the times of Kings and Queens. At the end of your tour your guide will be happy to give you advice on further places to visit or where to enjoy a great meal in this beautiful royal city.
8. Versailles Audio Guided Tour from Paris
Enjoy a day trip from Paris to the Palace of Versailles, an architectural beauty. Cross the western districts of Paris before taking the motorway to the Versailles town, a royal château located in a former village. The Palace of Versailles was the de facto capital of the kingdom of France over a century. Now it is known as the wealthy suburb of Paris. You can discover Versailles on your own as you tour the inside and wander the grounds. You will be given audio guides on arrival to enrich your visit with commentary, and this architectural masterpiece will enchant you through its beauty and historical richness. Visit the Royal State Apartments including the Hall of Mirrors, King's apartments, and Queen's apartment, followed by a leisurely stroll in the Palace gardens.
9. Versailles: Palace Guided Tour with Skip-the-Line Tickets
Learn more about Versailles palace with a 90-minute group tour with no more than 16 people. Avoid the hassle of waiting in long lines with skip-the-line tickets. Meet your local guide at Versailles at the foot of the Equestrian Statue of the Sun King, Louis the 14th. Visit the Royal Apartments. Walk through the palace and learn facinating information about the history of the palace from your expert guide. Following your interior visit of the palace, spend some free time in the gardens, a 2,000-acre (809-hectare) park filled with lovely fountains, statues, and classic formal garden landscaping. See the Musical Gardens show if visiting Versailles on a Tuesday or a Friday. View the Musical Fountains Show if taking the tour on a Saturday or a Sunday (from March to October).
10. Versailles Palace Skip-the-Line Entrance and Breakfast
In the recently renovated Pavillon Dufour, ore offers the great French classics, light bites and delicate pastries. Located at the first floor of Pavillon Dufour, the restaurant plan is divided into two main dining rooms, one bar and three salons. One can admire the Palace through the windows overlooking the cour Royale and the cour d'Honneur. In the salons, the subtle position of the mirrors allows the guests to look upon the Palace, even if their back is turned to the windows. Ore - Ducasse au château de Versailles offers contemporary French cuisine and artfully combining great classics and new dishes. Service is relaxed, professional and efficient. After breakfast, be guided to the skip-the-line access for the palace and enjoy the rest of your day in Versailles.
Top Attractions in Paris
1 Louvre Museum
2 Eiffel Tower
3 Seine River
4 Moulin Rouge
5 Palace of Versailles
6 Paris Catacombs
7 Musée d'Orsay
8 Palais Garnier
9 Sainte-Chapelle
10 Versailles
11 Arc de Triomphe
12 Disneyland Paris
13 Notre Dame Cathedral
14 Eiffel Tower Summit
15 Paris Lido
16 Musee de l'Orangerie
17 Centre Pompidou
18 Disneyland Paris Adventureland
19 Montmartre
20 Army Museum, Paris
1 Paris City Cards
2 Paris Cruises & Boat Tours
3 Paris Hop-on Hop-off Tours
4 Paris Tours
5 Paris River & Harbor Cruises
6 Paris Day Trips
7 Paris Shows & Musicals
8 Paris Tickets
9 Paris Day Trips From
10 Paris Bus & Minivan Tours
11 Paris Family-friendly Activities
12 Paris Zoos & Aquariums
13 Paris Walking Tours
14 Paris Night Tours
15 Paris Bike Tours
16 Paris Cabaret
17 Paris Dinner & Lunch Cruises
18 Paris Wine Tasting & Winery Tours
19 Paris Museum Tickets
20 Paris Top 10
1 Paris
2 Nice
3 Bordeaux
4 Tours, France
5 Marseille
6 Cannes
8 Strasbourg
9 Carcassonne
10 Avignon
11 Saint-Émilion
12 Chamonix-Mont-Blanc
13 Saint-Tropez
14 Antibes
15 Reims
16 Montpellier
17 Lourdes
18 Annecy
19 Amboise
20 Giverny
Other Sightseeing Options in Palace of Versailles
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What people are saying about Palace of Versailles
based on 18,099 reviews
A must have if you plan on visiting any of the historic places who accept the pass. Definitely saves money and time (skip the line with a separate entrance). During our stay and due to construction/covid we needed to a pick a time to enter the Louvre, but all the rest were skip the line.
Paris Museum Pass: 2, 4, or 6 Days Reviewed by Tressa, 12/25/2021
Loved loved it! Our guide was the best and knew so much! I would recommend this to any friend!
Paris: Versailles Palace Tour with Optional Fountain Show Reviewed by Alexandra, 12/28/2021
Our tour was well organized and our group guide was very pleasant and knowledgeable.
From Paris: Full-Day Guided Tour of Versailles with Lunch Reviewed by Miki, 12/31/2021
The ticket office was not far from the Louvre and the staff were very friendly.
Paris Museum Pass: 2, 4, or 6 Days Reviewed by Won Seok, 12/24/2021
We loved the experience. Both guides are great—- Phil and Maribeth.
Paris: Versailles Palace & Gardens Ticket and Audioguide Reviewed by peilin, 12/30/2021


"Automotive Service Technician" means a person engaged in the repair, adjustment and replacement of mechanical and electrical parts of automobiles, light trucks and mini-busses. Technicians would be able to diagnose automotive systems, perform routine component replacment and rebuild or adjust a variety of automotive components. A technician would perform these functions on passenger vehicles, light trucks and mini-buses in the following areas: engines, fuel systems, emission control and exhaust systems, brake systems, steering and suspension systems, manual and automatic transmission/transaxles, differentials, drive axles and drivelines, electrical and electronic systems, air conditioning, heating and cooling systems and trim and body hardware.
The Automotive Service Technician program is a 4 year apprenticeship leading to a Red Seal Endorsement.
You must be a registered apprentice within BC.
Persons wishing to become apprentices must be employed with a business that employs a qualified journeyperson automotive service technician. The Industry Training Authority (ITA) sets up a registration for apprenticeship between the apprentice, the employer and the ITA.
**This outline is for the Harmonized program. If you are in a non-harmonized level there may be some curriculum differences.
For tuition information visit Apprenticeship Tuition Fees.
Students must supply approved safety boots and coveralls.
Income support for apprentices attending technician-training courses is provided by Human Resources Development Canada (HRDC) through regular employment insurance eligibility.
The Automotive Service Technician will write the Red Seal exam on the last date of the level 4 technical training. Successful completion provides the apprentice with the Certificate of Qualification as a Red Seal Automotive Service Technician.
Red Seal certified workers can work anywhere in Canada where their trade is designated. Labour mobility is essential for the Canadian workforce. For skilled tradespeople, the Red Seal is the recognized passport to mobility in trades across Canada.
Applications can be completed online here or over the phone. There is no application fee.


The design of passive or nearly passive houses can pose difficult problems to the architect. Especially in northern Europe this type of building design requires sometimes extreme scales of windows or heat storage masses that can not be estimated easily.
Simulation. Our sophisticated simulation environment allows to accurately capture all aspects of heat generation, heat storage and air exchange. This enables very detailed planning strategies which utilize effects such as boyancy or the control of air exchange through staircases.


I was reading through this article at TheStar.com about sewing being back in fashion. There are some good points made.
Sewing is back. It skipped a generation or two as prices for clothing dropped and it became cheaper and easier to buy rather than make clothes. Now consumers, some exhausted by shopping, others by debt, some rudely awakened by the images of dead bodies pulled from the collapsed factory building in Bangladesh are jumping off the consumer treadmill to take up slow clothing.
This is a good way to look at it. Sewing has a purpose for people again, it helps them achieve their needs and wants in ways that society is not providing to them. This is great stuff.
I like to see people crafting their own means of life, be it making their own food, building their own homes, creating our own clothing, or anything along those lines. It makes us appreciate what it takes to truly be self-sufficient.
No matter what the reason is for your sewing habits, sewing is a productive hobby and it helps us produce things we find useful as opposed to just buying from others, and that is what makes it satisfying for many.


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Home Music and Arts David Gilmour.
David Jon Gilmour, AKA known by his moniker of David Gilmour, he is still a reigning best guitarist, soloist of all time and as of now, David is a singer, songwriter and a former member of the Rock band Pink Floyd that took seriously by stormy performances and universal melodies that were oriented to heaps of attribution by several other amateur and legendary bands such as: Metamorphosis, Airbag and Porcupine Tree, these three bands were mainly influenced by the Floyd's cornerstone, despite the multi generational descent of the bands including Pink Floyd and the others as well, David's touch was to include and still till now a moving rhythm that touched each band's inclination to Rock and Metal sceneries as well; was actually in the beginning of the 1960's when he started gliding from a place to another in a clumsy seeking career finder, where chances appeared for him to play tributes to The Beatles at the time, and where he simultaneously met future Pink Floyd's future band still on the horizon, Syd Barret and Roger Waters.
In the aftermath of his encounter with future guitarists of Pink, David still played music as an amateur in France under a band named Flowers; this little ancillary merriment did not seem to help his career permeate the way he wanted it, so he got back to England still in a connection of his former friend and guitarist from the Flowers; who grumpily goofed around when joining Pink Floyd for some tours, David was called to play instead; it was there he rose to stardom after multiple attempts and tours that recognized his commodity both solo and a member of great disposition to both old and post modern Progressive Rock bands of all time.
David was accredited several awards and creditable accolades for his solo and generously put together performances with Pink, the Rolling Stones magazine ranked him 14 in the list of the greatest guitarists of all time, in 1996; he was proudly introduced into the US Rock and Roll Hall of Fame, and in 2005; the same accolade took place in the UK Music Hall of Fame.
David Jon Gilmour
Legendary Guitarist
A tribute to Marilyn Monroe.