text:
\section{Introduction}\label{section:intro}

In \cite[Question 7.9]{EV} Etnyre and Van Horn-Morris ask the following question:
\begin{question}\cite[Question 7.9]{EV}
\label{question:main}
If a symmetric mapping class $\phi\in{\rm Mod}(S)$ admits a positive factorization, then is $\phi$ a lift of a quasi-positive braid ?
\end{question}

This is a profound question connecting three important objects in topology; (1) symmetric mapping classes, (2) positive factorizations, and (3) quasipositive brads.
We describe each here.

Let $S$ be a compact, oriented surface with non-empty boundary, which is a special (see Section~\ref{section:BHtheory}) cyclic branched covering of a disk $D$ branched at $n$-points.
A mapping class $\phi \in {\rm Mod}(S)$ is called \emph{symmetric} (fiber-preserving) \cite[p.65]{B} if $\phi$ is a lift of an element of the braid group $B_{n}={\rm Mod}(D^{2} \setminus \{n \mbox{ points}\})$.
Symmetric mapping class groups were introduced and studied by Birman and Hilden in a series of papers culminating in \cite{BH}.
As Margalit and Winarski say in \cite{MW}, the Birman-Hilden theory has had influence on many areas of mathematics, from low-dimensional topology, to geometric group theory, to representation theory, to algebraic geometry and more.

A \emph{positive factorization} of a mapping class $\phi \in {\rm Mod}(S)$ is a factorization of $\phi$ into positive (right-handed) Dehn twists about simple closed curves.
In contact and symplectic geometry, positive factorizations of  mapping classes play an important role due to the following fact:
A contact $3$-manifold is Stein fillable if and only if it is supported by an open book whose monodromy admits a positive factorization \cite{AO, GI, LP}.

A {\em quasipositive} braid in $B_n$ is a braid which factorizes into positive half twists about proper simple arcs in the $n$-punctured disk.
Quasipositive knots and links are introduced and studied by Rudolph in a series of papers.
Rudolph showed \cite{R} that a quasipositive knot can be realized as an intersection (transverse $\mathbb C$-link) of the unit sphere in $\mathbb C^2$ with an algebraic complex curve in $\mathbb C^2$. Boileau and Orevkov \cite{BO} proved the converse that any knot arising as such an intersection must be quasipositive.

In this paper, we give partial answers to the question of Etnyre and Van Horn-Morris.

Let $S$ be a compact, oriented surface with non-empty boundary.
Let $D$ be a disk.
Suppose $\pi: S \to D$ is a special $k$-fold cyclic branched covering of the disk $D$ branched at $n$ points. The meaning of ``special'' will be made clear in Section~\ref{section:BHtheory}.
In \cite{BH} Birman and Hilden show that there is a well-defined injective homomorphism $\Psi: B_n \to {\rm Mod}(S)$ whose image is the symmetric mapping class group ${\rm SMod}(S)$ which is defined in Section~\ref{section:BHtheory}.

Using the homomorphism $\Psi:B_n \to {\rm SMod}(S)$, Etnyre and Van Horn-Morris in \cite{EV} consider various submonoids in $B_n$.
\begin{eqnarray*}
P(n)
&:=&
\left\{ b \in B_n \ | \ b \mbox{ is a positive braid } \right\} \\
QP(n)
&:=&
\left\{ b \in B_n \ | \ b \mbox{ is a quasi-positive braid } \right\} \\
Dehn^+(n, k)
&:=&
\Psi^{-1}(Dehn^+(S)) \\
Tight^+(n, k)
&:=&
\Psi^{-1}(Tight^+(S)) \\
RV(n)
&:=&
\left\{ b \in B_n \ | \ b \mbox{ is a right-veering braid }
\right\} \\
Veer^+(n, k)
&:=&
\Psi^{-1}(Veer^+(S))
\end{eqnarray*}
Here, a braid $b \in B_{n}$ is \emph{positive} if it is a product of standard generators $\sigma_{1},\ldots,\sigma_{n-1}$, and is \emph{quasi-positive} if it is a product of conjugates of $\sigma_1$.
We have $P(2)={\it QP}(2)$ and $P(n) \subsetneq QP(n)$ for $n>2$.
The set $Dehn^{+}(S) \subset {\rm Mod}(S)$ is a monoid generated by positive Dehn twists, $Tight^{+}(S)\subset {\rm Mod}(S)$ is a monoid consisting of  monodromies supporting tight contact structures, and $Veer^{+}(S)\subset {\rm Mod}(S)$ is a monoid consisting of right-veering mapping classes.
One can see that $\Psi(b)$ is right-veering if and only if $b$ is right-veering (see \cite[Section 7]{IK2} for the definition(s) of right-veering braids).

\begin{proposition}
We have $Veer^+(n, k) = RV(n)$ for all $n$ and $k$.
\end{proposition}

Etnyre and Van Horn-Morris observe that \cite[Lemma 3.1]{HKP} implies the following:
\begin{proposition}\label{subset} \cite[p.355]{EV}
For all $n \geq 2$ and $k \geq 2$ we have ${\it QP}(n) \subset {\it Dehn}^+(n, k)$.
\end{proposition}

We prove Proposition~\ref{subset} in Section~\ref{section:BHtheory}.

In summary, we have;
$$
P(n) \subset QP(n) \subset Dehn^+(n, k) \subset Tight(n,k) \subset Veer^+(n, k) = RV(n) \subsetneq B_n.
$$

In \cite[Example 2.9]{HKM}, the strictness of the inclusion $QP(3)\subset Veer^+(3, 2)$ is shown.

\begin{proposition}
In general, both the inclusions $Dehn^+(n, k) \subset Tight(n,k) \subset Veer^+(n, k)$ are strict as we show below:
\end{proposition}

\begin{proof}
Let $\beta = (\sigma_{1}\sigma_{2})^{6} \sigma_{1}^{-13} \in B_{3}$.
By \cite[Theorem 1.2]{HKM}
the braid $\beta$ is in $Tight(3, 2)$ since the fractional Dehn twist coefficient of $\Psi(\beta) \in {\rm Mod}(S)$ is one.
However, $\beta$ is not in $Dehn^{+}(3,2)$ since its exponent sum is negative which means $b \not\in QP(3)=Dehn^+(3, 2)$, where the equality $QP(3)=Dehn^+(3, 2)$ will be proved in Theorem~\ref{theorem:main2}. (See \cite[Corollary 3.6]{HKM} for a better criterion for $b \not \in QP(n)$ than a naive exponent sum argument.)

To see $Tight(n,k) \subsetneq Veer^+(n, k)$, recall \cite[Proposition 3.1]{PL} (for the open book $(D^2, id)$) and \cite[Proposition 3.14]{IK2} (for general open books) that every braid with respect to an open book is transversely isotopic to a right-veering braid after suitable positive stabilizations.
Let $\beta \in B_{n}$ be a right-veering braid that is a stabilization of $\sigma_1^{-1} \in B_2$. Then for each $k\geq 2$, $\beta$ is in $Veer^{+}(n,k)$ but not in $Tight(n,k)$.
\end{proof}

With these terminologies, Question 1.1 of Etnyre and Van Horn-Morris is equivalent to the following:

\begin{question}\cite[Question 7.9]{EV}
Do we have ${\it QP}(n) = {\it Dehn}^+(n, k)$?
\end{question}

In \cite{EV} they say ``the answer is almost certainly no''.
Thus our goal can be set to find sufficient conditions for ${\it QP}(n) \supset {\it Dehn}^+(n, k)$.

\subsection{Motivation}
Our particular branched covering $\pi:S\to D$ is closely related to the cyclic branched covering of the standard contact 3-sphere $(S^{3},\xi_{std}=\ker\alpha_{std})$.
Let $K=\widehat{b}$ be a transverse knot in $(S^{3},\xi_{std})$ represented by the closure $\widehat b$ of an $n$-braid $b\in B_n$ with respect to the open book $(D^{2},id)$. Let $p: M_{K, k} \rightarrow S^{3}$ be the $k$-fold cyclic branched covering, branched along $K$. Then $M_{K,k}$ is equipped with a contact structure $\xi_{K,k}$ that is
a perturbation of the kernel of the pull-back $p^{*}(\alpha_{std})$. Such a contact structure is supported by the open book $(S,\Psi(b))$. Thus, $(M_{K,k}, \xi_{K,k}) \simeq (M_{(S,\Psi(b))}, \xi_{(S,\Psi(b))})$.

Let $B^{4} (\subset {\mathbb C}^4)$ be the unit complex ball giving a Stein filling of $(S^{3},\xi_{std})$.
If the braid $b \in B_n$ is quasi-positive, a factorization of $b$ as a product of positive half twists gives rise to an immersed Seifert surface of $K=\widehat b$ with ribbon intersections as shown in \cite[Figure 9]{EV}.
Pushing this surface into the interior of $B^{4}$ we have a properly  embedded symplectic surface $\Sigma$ in $B^{4}$ such that $\Sigma \cap \partial B^{4} = \partial \Sigma = K$.
Let $W$ be the $k$-fold cyclic branched cover of $B^{4}$ branched along $\Sigma$. Then $W$ gives a Stein filling of $(M_{K,k},\xi_{K,k})$

On the other hand, a factorization of $b\in B_n$ into positive half twists induces a factorization of $\Psi(b) \in {\rm Mod}(S)$ into positive Dehn twists  (Proposition~\ref{subset}), to which one can associate a Legendrian surgery diagram (see Section~\ref{section:BHtheory} and \cite[Fig. 12]{HKP}).
Let $X$ be the 4-dimensional handlebody obtained by attaching 2-handles to $B^4$ according to the Legendrian surgery diagram.
By \cite{Eli}, \cite[Proposition 2.3]{G} the manifold $X$ is
a Stein filling of $(M_{(S,\Psi(b))}, \xi_{(S,\Psi(b))})$.

In fact, these two constructions of Stein fillings give the same manifold. Namely, the two 4-manifolds $X$ and $W$ are diffeomorphic, which follows from the proof of  \cite[Claim 2.1]{IT}.
Thus, the branched covering construction behaves nicely not only for contact structures but also for Stein fillings.

With the above discussion in mind, we may extend Question \ref{question:main} to the following question about Stein filling and $\mathbb{Z} \slash k\mathbb{Z}$-action:

Assume that $b\in {\it Dehn}^+(n, k)$.
Then $(M_{K,k},\xi_{K,k})\simeq (M_{(S,\Psi(b))}, \xi_{(S,\Psi(b))})$ is Stein fillable because
$\Psi(b) \in {\it Dehn}^+(S)$ and ${\it Dehn}^+(S)$ is contained in the monoid  $Stein(S)$ of Stein fillable open books \cite{Eli, GI}.
The contact manifold $(M_{K,k},\xi_{K,k})$ also admits an $\mathbb{Z}\slash k\mathbb{Z}$-action as a contactomorphism with the quotient space $(S^{3},\xi_{std})$. Our question is:

{\em Can we find a Stein filling $X$ coming from the above construction; namely, can the $\mathbb{Z} \slash k\mathbb{Z}$-action on $(M_{K,k},\xi_{K,k})$ extend to a holomorphic $\mathbb{Z} \slash k\mathbb{Z}$-action on some $X$ with the quotient space $B^4$? }

This new question suggests that, even if Question \ref{question:main} may have negative answer in general as Etnyre and Van Horn-Morris expect in \cite{EV}, it is worth trying to find sufficient conditions for ${\it QP}(n) \supset {\it Dehn}^+(n, k)$ to be hold.

\subsection{Main results}
The following are our main results that give sufficient conditions for ${\it QP}(n) \supset {\it Dehn}^+(n, k)$.

\noindent
{\bf Theorem~\ref{theorem:main2}}. {\em
For $n\leq 4$, ${\it QP}(n) = {\it Dehn}^+(n,2)$. \\
}

\noindent
{\bf Theorem~\ref{theorem:twist}}. {\em
Let $\iota:S\to S$ be a deck transformation.
Let $C$ be a simple closed geodesic curve in $S$ such that $C, \iota(C), \dots, \iota^{e-1}(C)$ are pairwise disjoint and $\iota^e(C)=C$ for some $e\in\{1,\dots,k\}$ that divides $k$.
Let $d, j \in \mathbb{N}$.
Suppose that $b^d \in {\it Dehn}^+(n, k)$
with
$$
\Psi(b^d)= \left(T_C \circ T_{\iota(C)} \circ T_{\iota^2(C)} \circ\dots\circ T_{\iota^{e-1}(C)}\right)^j.$$
Then $b \in QP(n)$ $($and so $b^d \in QP(n)).$ \\
}

This theorem states that Question \ref{question:main} has an affirmative answer for a symmetric mapping class which is a root of symmetric product of positive Dehn twists (see also Corollary \ref{cor:froot}). \\

\noindent
{\bf Theorem \ref{thm:general}.} {\em
Suppose that the subsurfaces $S'_1,\ldots, S'_k \subset S$ are pairwise non-isotopic.
Assume that $[f]\in Dehn^{+}(S')$ is either
\begin{itemize}
\item a non-negative power of a single Dehn twist (i.e., $S'$ is an annulus which is neighborhood of a simple closed geodesic curve), or
\item a pseudo-Anosov map (i.e., $S'$ is a hyperbolic surface).
\end{itemize}
Suppose that $b \in Dehn^+(n,k)$ satisfies
\[ \Psi(b)=[f_1 f_2 \cdots f_{k}] \]
then $b \in QP(n)$.
}

We say that a simple closed curve $C$ is \emph{symmetric} if it is invariant under some deck translation. Question \ref{question:main} can be understood as a question
of the existence of factorizations of elements of $Dehn^+(S) \cap {\rm SMod}(S)$ into positive Dehn twists about symmetric simple closed curves.

A well-known example where a positive factorization coincides with a product of Dehn twists about symmetric simple closed curves may be the daisy relation \cite{EMV}.
In Example \ref{example-daisy} we see that the technical looking assumptions in Theorem \ref{thm:general} can be understood as a generalization of the setting of the daisy relation, and view Theorem \ref{thm:general} as a generalization of the daisy relation.

\subsection{Organization of the paper}

In Section \ref{section:BHtheory} we review results of Birman and Hilden that we need in this paper.

In Section \ref{section:panswer}, we show that the answer to the question is affirmative when the number of branch points $n$ is two (Theorem \ref{theorem:main1}) or the degree of the branched covering is two with $n\leq 4$ branch points (Theorem \ref{theorem:main2}).

In Section \ref{section:QP}, we discuss roots of quasi-positive braids and find conditions that a root of a quasi-positive braid is also quasi-positive.
We obtain results that are used in Section~\ref{section:suff}.

In Section \ref{section:suff},
we prove our main results (Theorems~\ref{theorem:twist} and  \ref{thm:general}).

\section{Birman-Hilden theory}
\label{section:BHtheory}

Throughout the paper, unless otherwise stated, we always assume that the boundary of a surface is non-empty, any homeomorphism of a surface with marked points (punctures) fixes the boundary pointwise and permutes the marked points, and any isotopy of homeomorphisms $\{f_{t}\}$ pointwise fixes  the boundary and the marked points.
We denote by $[f] \in{\rm Mod}(S)$ the isotopy class of a homeomorphism $f \in {\rm Homeo}^+(S)$.
For a simple closed curve $C$ in a surface $S$, we denote by $t_{C} \in {\rm Homeo}^+(S)$ a right-handed Dehn twist about $C$ which is  supported in a neighborhood of $C$, and denote its isotopy class by $T_{C}\in{\rm Mod}(S)$.
A simple closed curve in a surface is called \emph{essential} if it is not homotopic to a point, a puncture, or a boundary component of the surface.

Let $P =\{p_1,\ldots,p_n \}\subset \textrm{Int} (D)$ be $n$ points in the interior of a 2-disk $D$. We may identify the braid group $B_{n}$ with the mapping class group ${\rm Mod}(D\setminus P)$ of the $n$-punctured disk $D \setminus P$.
The fundamental group $\pi_{1}(D\setminus P)$  is the free group of rank $n$ and generated by $\{x_1,\ldots,x_n\}$, where $x_{i}$ is a loop winding once around $p_i$ counter-clockwise.
For $k\in \mathbb{N}$ let $e_k: \pi_{1}(D\setminus P) \rightarrow \mathbb{Z}\slash k\mathbb{Z}$
be a homomorphism defined by $e_{k}(x_i) = 1$ for all $i=1,\ldots,n$.
For $n\geq 2$ and $k\geq 2$, let $$\pi: S \approx S_{n,k} \to D$$ be the $k$-fold cyclic branched covering branched at $P$ such that the associated regular covering $S \setminus \pi^{-1}(P) \rightarrow D \setminus P$ is the $k$-fold cyclic cover corresponding to the subgroup $\ker(e_{k})$ of $\pi_1(D \setminus P)$. We denote by $\widetilde P =\{\widetilde{p_1},\ldots,\widetilde{p_{n}}\}= \pi^{-1}(P) \subset S$ the set of branch points in $S$.
Let $\iota= \iota_{k} : S \to S$ be a generator of the deck transformation group ${\rm Aut}(S, \pi) \simeq \mathbb{Z}/k\mathbb{Z}$.

We visualize $S$ and $\iota$ as follows. The covering $S$ can be viewed as the canonical Seifert surface of the $(n,k)$-torus link represented as the closure of the $n$-braid $(\sigma_{1}\cdots \sigma_{n-1})^{k}$
(see Figure~\ref{torus-link}). The deck transformation $\iota_k$ is the $\frac{2\pi}{k}$ rotation of the surface $S$ about the braid axis through the branch points $\widetilde P$,
and  $\partial S$ is not pointwise fixed by $\iota_k$.

\begin{figure}[htbp]
\includegraphics*[bb=71 564 384 745,width=110mm]{torus-link-new.pdf}
\caption{The Seifert surface $S$ of a $(n,k)$ torus link, where $n=5$ and $k=4$. Hollowed circles are the lifted branch points $\widetilde P$. Cutting $S$ along the dashed arcs gives $k$ sheets of disks. The deck transformation $\iota_k$ takes $i$-th disk to $i+1$st disk.
}
\label{torus-link}
\end{figure}

Let $\beta \in B_{n} \approx {\rm Mod}(D\setminus P)$ and $f_{\beta}:(D,P) \rightarrow (D,P)$ be a homeomorphism representing the braid $\beta$.
Since $\beta(\ker (e_k))= \ker(e_k)$ via the action of $B_{n}$ on $\pi_{1}(D \setminus P)$,
there is a lift $\widetilde{f_{\beta}}: S \rightarrow S$ of $f_{\beta}$ (see \cite[Lemma 5.1]{BH} and \cite[Theorem 1.1]{GW}).
Among the $k$ possible lifts which are related to each other by deck transformations, $\widetilde{f_{\beta}}$ is the unique lift that fixes $\partial S$ pointwise.

When two homeomorphisms $f_{\beta}$ and $f'_{\beta} \in {\rm Homeo}^+(D, P)$ represent the same braid $\beta\in B_n$ we note that an isotopy between $f_{\beta}$ and $f'_{\beta}$  lifts to an isotopy between their lifts $\widetilde{f_{\beta}}$ and $\widetilde{f'_{\beta}}$. Thus, we have a well-defined homomorphism
\[ \Psi :B_{n} \rightarrow {\rm Mod}(S) \]
defined by $\Psi(\beta) = [\widetilde{f_{\beta}}]$.
Let $$\Theta=\Theta_{k}: {\rm Mod}(S) \rightarrow {\rm Mod}(S)$$ be an automorphism defined by $ \Theta([f]) =[\iota_k \circ f \circ \iota_k^{-1}]$.

\begin{definition}
Define
\begin{equation*}
{\rm SMod}(S)  :=  \left\{ [f] \in {\rm Mod}(S) \ \middle| \ f \in {\rm Homeo}^+(S) \mbox{ and } \Theta([f])=[f] \right\}
\end{equation*}
and call it the \emph{symmetric mapping class group}.
An element of ${\rm SMod}(S)$ is called {\em symmetric mapping class}.
\end{definition}

The following fact is attributed to Birman and Hilden.

\begin{proposition}\label{prop-Birman-Hilden}
The homomorphism $\Psi$ is injective and onto ${\rm SMod}(S)$.
\end{proposition}

For the sake of completeness, we include a proof of Proposition~\ref{prop-Birman-Hilden}.

\begin{proof}[Proof of Proposition~\ref{prop-Birman-Hilden}]
The injectivity of $\Psi$ follows from \cite{BH}.
A homeomorphism $f: S \rightarrow S$ is called \emph{fiber-preserving} if $\pi\circ f(p) = \pi \circ f(p')$ for any $p,p' \in S$ with $\pi(p)=\pi(p')$.
Suppose that $f_0, f_1 \in {\rm Homeo}^+(D, P)$ satisfy $\Psi([f_0])=\Psi([f_1])$.
According to \cite[Theorem 1]{BH},
since the lifts $\tilde{f_0}, \tilde{f_1}\in{\rm Homeo}^+(S)$ represent the same element of ${\rm SMod}(S)$ and are fiber-preserving,
there exists a fiber-preserving isotopy $g_t\in {\rm Homeo}^+(S)$ between $\tilde{f_0}$ and $\tilde{f_1}$.
Since $g_t$ is fiber-preserving the composition $\pi\circ g_t\circ \pi^{-1}$ is a well-defined homeomorphism of $(D, P)$ and it gives an isotopy between $f_0$ and $f_1$; hence, $[f_0]=[f_1] \in B_n$.

To see ${\tt Im}(\Psi)\supset{\rm SMod}(S)$, assume that $f\in {\rm Homeo}^+(S)$ satisfies $[f] = [\iota \circ f\circ \iota^{-1}]$.
The same argument as in the proof of \cite[Theorem 3]{BH} with Teichm\"uller's theorem for bordered surfaces \cite[page 59]{A} shows that $[f]$ can be represented by a homeomorphism $f' \in {\rm Homeo}^+(S)$ such that $f' = \iota \circ f'\circ \iota^{-1}$.

For $x \in D$ let $\tilde x \in \pi^{-1}(x) \subset S$ be a pre-image of $x$ under the branched covering map $\pi:S\to D$.
Define a homeomorphism $b \in {\rm Homeo}^+(D)$ by
$$b(x)=\pi(f'(\tilde x)).$$
Since $\pi\circ f'\circ \iota(\tilde x)=\pi\circ\iota\circ f'(\tilde x)=\pi\circ f'(\tilde x)$, the image $b(x)$ is independent of the choice of the pre-image $\tilde x$.

We observe that $b$ acts on the branch set $P$ as a permutation.
Suppose that $p \in P$ and $\tilde p \in \tilde P$ satisfy $\pi(\tilde p)=p$.
Since $\iota$ fixes the branch set $\tilde P$ point-wise we have $\iota(f'(\tilde p))=f'(\iota (\tilde p))=f'(\tilde p)$.
That is, $f'(\tilde p)$ is a fixed point of $\iota$ and $f'(\tilde p) \in \tilde P$.
We get $b(p)= \pi(f'(\tilde p)) \in \pi (\tilde P)=P$ which shows $b(P)=P$.

Since $\pi\circ f' = b \circ \pi$ the map $f'\in{\rm Homeo}^+(S)$ is the unique lift of $b\in{\rm Homeo}^+(D;P)$. We obtain $[f]=[f']=\Psi([b])\in {\tt Im}(\Psi)$.

To see ${\tt Im}(\Psi)\subset{\rm SMod}(S)$, let $\alpha^i_l$ be an simple closed curve that goes through $i$th and $(i+1)$th sheets and $l$th and $l+1$st twisted bands as depicted in Figure~\ref{torus-link}.
Let $t_{i,l}:=t_{\alpha^{i}_{l}}\in{\rm Homeo}^+(S)$ be a positive Dehn twist about $\alpha^i_l$.
It is shown in \cite[Lemma~3.1]{HKP} that for the standard braid generators $\sigma_1,\dots,\sigma_{n-1}$ of the Artin braid group $B_n$ we have
$$\Psi(\sigma_i) =[t_{i,1} \circ t_{i,2} \circ \dots \circ t_{i,k-1}] \in {\rm Mod}(S).$$
For $j=1,\dots,n-1$ and $l=1,\dots, k$ let $\lambda^j_l$  (see Figure~\ref{torus-link}) be properly embedded arcs on the $l$th band between $j$th and $j+1$st sheets such that $\iota(\lambda^j_l)=\lambda^j_{l+1}$.
Let $[\lambda^j_l]$ denote the isotopy class of $\lambda^j_l$ relative to $\partial S$.
We get
$$
\left[\iota^{-1}\circ (t_{i,1} \circ t_{i,2} \circ \dots \circ t_{i,k-1}) \circ\iota\right] \left[\lambda^j_l\right] =
\left[t_{i,1} \circ t_{i,2} \circ \dots \circ t_{i,k-1}\right] \left[\lambda^j_l\right]
$$
for all $j=1,\dots,n-1$ and $l=1,\dots, k$.
Knowing that the surface $S$ cut along the union of arcs $\cup_{j, l} \lambda^j_l$ yields $n$ disjoint disks, the Alexander trick \cite[Proposition 2.8]{FM} implies that $\left[\iota^{-1}\circ (t_{i,1} \circ t_{i,2} \circ \dots \circ t_{i,k-1}) \circ\iota\right]  =
\left[t_{i,1} \circ t_{i,2} \circ \dots \circ t_{i,k-1}\right]$.
This shows that $\Psi(\sigma_i)\in{\rm SMod}(S)$.
\end{proof}

\section{Positive answers for simple cases}
\label{section:panswer}

\begin{theorem}
\label{theorem:main1}
We have
$P(2)={\it QP}(2) = {\it Dehn}^+(2,k)= Tight(2,k) = RV(2)$ for all $k \geq 2$.
\end{theorem}

\begin{proof}
The first equality $P(2)={\it QP}(2)$ is obvious.

By Proposition \ref{subset} we only need prove ${\it QP}(2) \supset{\it Dehn}^+(2,k)$.
Take a braid $b \in B_2 \setminus {\it QP}(2)$.
There exists a positive integer $m$ such that $b = (\sigma_1)^{-m}$.
Let $T_i = [t_{\alpha^1_i}] \in {\rm Mod}(S)$.
By Lemma 3.1 of \cite{HKP} we have $\Psi(b)=(T_1 \circ T_2 \circ \dots \circ T_{k-1})^{-m}$.
Clearly $(T_1 \circ \dots \circ T_{k-1})^{-m} \notin Veer^+(S)$.
Since $Dehn^+(S) \subset Veer^+(S)$ we have $\Psi(b)\notin Dehn^+(S)$; hence, $b \notin  {\it Dehn}^+(2,k).$
\end{proof}

\begin{theorem}
\label{theorem:main2}
For $n\leq 4$, ${\it QP}(n) = {\it Dehn}^+(n,2)$.
\end{theorem}

The equality $QP(3)=Dehn^+(3, 2)$ has been known and used in the literature. However for sake of completeness, we give a proof of this case along with the case of $n=4$.

\begin{proof}
Since the $n=2$ case is shown in Theorem \ref{theorem:main1} we may assume $n=3$ or $4$.

Let $a_1,a_{2},\delta$ (for the $n=3$ case) and $a_1,a_2,a_3,\delta_1, \delta_{2}$ (for the $n=4$ case) be simple closed curves on $S$ as depicted in Figure~\ref{fig:mcggenerator}.
The Dehn twists about these curves generate ${\rm Mod}(S)$, and $\Psi(\sigma_i)=T_{a_i}$ holds.
\begin{figure}[htbp]
\begin{center}
\includegraphics*[bb=71 644 328 734, width=80mm]{zu_mcggenerator.pdf}
\end{center}
\caption{}
\label{fig:mcggenerator}
\end{figure}

If $b \in Dehn^{+}(n,2)$ then there exist non-boundary parallel simple closed curves $\Gamma_1,\dots,\Gamma_m \subset S$ (possibly $\Gamma_i = \Gamma_j$ for $i\not=j$) and $x, y \geq 0$ such that
\begin{equation*}
\Psi(b) =
\begin{cases}
T_{\Gamma_1} \circ\dots\circ T_{\Gamma_m} \circ T_{\delta}^{x} & (n=3) \\
T_{\Gamma_1} \circ\dots\circ T_{\Gamma_m} \circ T_{\delta_1}^x \circ T_{\delta_2}^y & (n=4).
\end{cases}
\end{equation*}
We may assume that all of the curves $\Gamma_i$ are non-separating since the positive Dehn twist about a separating curve can be written as a product of the positive Dehn twists about non-separating curves.
Since $a_1$ is non-separating $\Gamma_i = f_i(a_1)$ for some $f_i \in {\rm Mod}(S)$. We can write $f_i$ as
\[ f_i = T_{a_{j_l}}^{\varepsilon_l}\circ \cdots \circ T_{a_{j_1}}^{\varepsilon_1} \]
for some $j_p \in \{1,\ldots, n-1\}$ and $\varepsilon_p \in \{\pm 1\}$.
Since the Dehn twist along the boundary does not affect simple closed curves in $S$, the words $T_{\delta}$, $T_{\delta_{1}}$ and $T_{\delta_2}$ are not needed in the expression.
Note that $f_{i} = \Psi(b_{i})$ for $b_{i}=\sigma_{j_l}^{\varepsilon_l}\cdots \sigma_{j_1}^{\varepsilon_1} \in B_{n}$.
Therefore, we have
\[
T_{\Gamma_i} = T_{f(a_1)}  =  f_i \circ T_{a_1} \circ f_i^{-1} = \Psi(b_i) \Psi(\sigma_{1}) \Psi(b_{i}^{-1}) = \Psi(b_{i}\sigma_1 b_{i}^{-1})
\]
and $T_{\Gamma_i} \in \Psi(QP(n))$ for all $i$.
In particular, $\Theta(T_{\Gamma_1} \circ\dots\circ T_{\Gamma_m}) = T_{\Gamma_1} \circ\dots\circ T_{\Gamma_m}$ by Proposition~\ref{prop-Birman-Hilden}.

For the case $n=4$, since $\iota(\delta_1)=\delta_2$, $\Theta(T_{\delta_{1}}) = T_{\delta_{2}}$ and $\Theta(T_{\delta_{2}}) = T_{\delta_{1}}$ which give\begin{eqnarray*}
T_{\Gamma_1} \circ\dots\circ T_{\Gamma_m}\circ T_{\delta_1}^x \circ T_{\delta_2}^y
&=&
\Theta( T_{\Gamma_1} \circ\dots\circ T_{\Gamma_m} \circ T_{\delta_1}^x \circ T_{\delta_2}^y) \\
&=&
\Theta(T_{\Gamma_1} \circ\dots\circ T_{\Gamma_m}) \circ\Theta(T_{\delta_1}^x) \circ \Theta(T_{\delta_2}^y)  \\
&=&
T_{\Gamma_1} \circ\dots\circ T_{\Gamma_m}\circ  T_{\delta_2}^x \circ T_{\delta_1}^y.
\end{eqnarray*}
Therefore, $x=y$.

With the chain relations \cite[Proposition 4.12]{FM} $T_{\delta} = \Psi((\sigma_{1}\sigma_{2})^{6})$ and $T_{\delta_1}^x T_{\delta_2}^x = \Psi((\sigma_1\sigma_2\sigma_3)^{4x})$,
we can conclude $b \in QP(n)$.
\end{proof}

\section{Roots in quasi-positive braids}
\label{section:QP}

In this section we address the following question. Among the results, Corollary~\ref{prop:rootqperidic} plays an important role to prove our main theorems in Section~\ref{section:suff}.

\begin{question}
\label{question:QProot}
Are $QP(n)$ and $Dehn^{+}(n,k)$ closed under taking roots?
That is,

(1) Does $b=x^d \in QP(n)$ for an integer $d\geq 2$ imply $x \in QP(n)$?

(2) Does $x^d \in Dehn^{+}(n,k)$ for an integer $d\geq 2$ imply $x\in Dehn^{+}(n,k)$?
\end{question}

\begin{definition}[Property {\bf(QP-root)}]
We say that a quasi-positive braid $b \in QP(n)$ has the property {\bf (QP-root)} if the following condition is satisfied.
\[ \mbox{ If } b=x^{d} \mbox{ for } x \in B_{n} \mbox{ and } d \geq 2
\mbox{ then }  x \in QP(n). \]
\end{definition}

It is shown  in \cite[Theorem 1.1]{GM} that the $d$-th root of a braid (if it exists) is unique up to conjugacy; namely, $x^{d}=y^{d} \in B_n$ implies that $x$ and $y$ are conjugate to each other. This leads to the following observation.

\begin{proposition}
\label{proposition:root}
For $x, x' \in B_{n}$ assume that $x^{d} = x'^{d}$.
Then $x \in QP(n)$ if and only if $x' \in QP(n)$.
\end{proposition}

Let $b\in B_n$ be a non-periodic reducible braid. According to \cite[Definition 5.1]{GMW}, up to conjugacy, $b$ is in  the following {\em regular form}.

Let $C$ be an essential multi-curve in the $n$-punctured disk $D_n$ such that
\begin{itemize}
\item
$b(C)$ is isotopic to $C$, and
\item
any simple closed curve which has non-zero geometric intersection with $C$ must not be preserved by any power of $b$.
\end{itemize}
Such a multi-curve $C$ is called a {\em canonical reduction system} for $b$ \cite{BLM} and it is unique up to conjugacy.
It always exists for a non-periodic reducible braid \cite{I}.

Let $\mathcal A' $ be the set of outermost curves of $C$.
There could exist punctures in $D_n$ not enclosed by any circle in $\mathcal A'$.
Define a set of curves $\mathcal A$ to be the union of $\mathcal A'$ and one circle around each such puncture of $D_n$.
We may enumerate the elements of $\mathcal A$ as
$$\mathcal{A}=\{ a_{1,1},\ldots, a_{1,r_1}, a_{2,1},\ldots, a_{2,r_2}, \dots, a_{c,1},\ldots, a_{c,r_c}\}$$
so that $b(a_{i,j}) = a_{i,j+1}$ $(j = 1,\ldots,r_i-1)$ and $b(a_{i,r_i}) = a_{i,1}$.
This action of $b$ on $\mathcal A$ gives $m=r_1+r_2+\cdots+r_c$ disjoint braided tubes (see Figure~\ref{figure:redbraid} (i)).
Identifying each tube with a string, we get an $m$-braid $\widehat{b} \in B_m$ which we call the {\em tubular braid} associated to $\mathcal A$.
By the nature of the canonical reduction system, this braid $\widehat b$ is unique up to conjugacy.
\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=120mm, bb=70 533 441 735]{zu_redbraid.pdf}
\end{center}
\caption{ (i) Non-periodic and reducible braid in regular form. (ii) Taking a conjugate the condition (\ref{eqn:regularform}) is  satisfied.}
\label{figure:redbraid}
\end{figure}

The braid closure of $\widehat b$ gives a $c$-component link.
The braid closure of the original braid $b$ can be viewed as a satellite of the $c$-component link.
Let $b_{i,j}$ denote the braiding inside the tube which starts at $a_{i,j}$ and ends at $a_{i,j+1}$ (where the indices $j$ are considered up to modulo $r_{i}$), which we call the \emph{interior braids}.

We say that $b$ is in \emph{regular form} if
\begin{equation}
\label{eqn:regularform}
\mbox{the only non-trivial interior braids are } b_{1,r_1},\ldots, b_{c,r_c}.
\end{equation}
We denote these non-trivial interior braids by $b_{[1]},\ldots,b_{[c]}$.
The condition (\ref{eqn:regularform}) can be realized by shifting non-trivial interior braid $b_{i,j}$ $(j \neq r_i)$ along the closure of the tubular braid $\widehat b$, which is equivalent to taking a conjugate (see Figure \ref{figure:redbraid} (ii)).
In this process, the tubular braid $\widehat{b}$ remains the same and
\begin{equation}
\label{eqn:intbraid}
b_{[i]} = b_{i,1}b_{i,2}\cdots b_{i,r_i} \mbox{ (up to conjugacy).}
\end{equation}

\begin{remark}
In \cite{GMW} a regular form requires one more property that if $b_{[i]}$ and $b_{[j]}$ are conjugate then $b_{[i]} = b_{[j]}$. But we do not use this property in this paper.
\end{remark}

Here is a simple observation.
\begin{lemma}\label{lem}
If a braid $b$ is in regular form with quasi-positive tubular braid and interior braids then $b$ is also quasi-positive.
\end{lemma}

The converse of Lemma~\ref{lem} is not true.
The reducible $4$-braid $(\sigma_{2}\sigma_{3}\sigma_{2}^{-1})(\sigma_{1}\sigma_{2}\sigma_{1}^{-1}) = (\sigma_{2} \sigma_{1}\sigma_{3}\sigma_{2})\sigma_{1}^{-2}$ is quasi-positive and in regular form with the tubular braid $\sigma_1 \in B_2$ and the interior braid $\sigma_{1}^{-2}\in B_2$.

The next proposition gives a criterion of the property {\bf (QP-root)}.

\begin{proposition}\label{prop:reducible}
Let $b$ be a non-periodic reducible, quasi-positive braid. If $b$ has a regular form such that all of its tubular and interior braids are quasi-positive and having the property {\bf (QP-root)}, then $b$ also has the property {\bf (QP-root)}.
\end{proposition}

\begin{proof}
Let $b$ be a non-periodic reducible, quasi-positive braid.
Assume that $b=x^{d}$ for some $x \in B_{n}$ and $d\geq 1$.
We will show that $x$ is quasi-positive.
Suppose that $b$ is in regular form with quasi-positive tubular braid $\widehat b \in B_m$ and quasi-positive interior braids $b_{[1]},\cdots, b_{[c]}$ all of which have the property {\bf (QP-root)}.

Since $b$ is non-periodic and reducible, the root $x$ is also non-periodic and reducible.
We may assume that $x$ is in regular form with tubular braid $\widehat x \in B_m$ and interior braids $x_{[1]},\ldots, x_{[c']}$.
Since $b=x^{d}$ we see that $c'\leq c$ and $\widehat b$ is  conjugate to $(\widehat x)^d$.
That is, there exists $y \in B_m$ such that $\widehat b = y (\widehat x)^d y^{-1} = (y \widehat x y^{-1})^d$.
By the property {\bf (QP-root)} of $\widehat b$, the tubular braid $y\widehat x y^{-1}$ (hence, $\widehat x$) is quasi-positive.

As for the interior braids, each $b_{[i]}$ is conjugate to a power of some single interior braid $x_{[i']}$.
Moreover, for each $i' \in\{1,\ldots, c'\}$ there exists $i\in\{1,\ldots, c\}$ such that $b_{[i]}$ is conjugate to a power of  $x_{[i']}$.
Therefore, by the property {\bf (QP-root)} of $b_{[1]},\ldots,b_{[c]}$, all the interior braids $x_{[1]},\ldots, x_{[c']}$ are quasi-positive.

By Lemma~\ref{lem} we conclude that $x$ is quasi-positive.
\end{proof}

A classical theorem of Ker\'ej\'art\'o and Eilenberg states that any  root of a periodic $n$-braid is conjugate to either
$(\sigma_1 \sigma_2 \sigma_3 \cdots \sigma_{n-1})^{i}$ or $(\sigma_{1}^{2}\sigma_2 \sigma_3 \cdots \sigma_{n-1})^{i}$ for some $i$. Thus, every periodic braid has the property {\bf (QP-root)}.
This fact and Proposition~\ref{prop:reducible} imply the following corollary.

\begin{corollary}
\label{prop:rootqperidic}
Let $b\in QP(n)$ admitting a factorization $b= T_{a_1}^{N_1} \circ \cdots \circ T_{a_p}^{N_p}\in {\rm Mod}(D_n)=B_n$ where $a_1,\ldots,a_p$ are pairwise disjoint simple closed curves in $D_{n}$ and $N_1, \cdots, N_p >0$.
If $b=x^d$ for some $d\geq 2$ then $x \in QP(n)$.
\end{corollary}

The above results show that Question \ref{question:QProot}, whether a quasi-positive braid has the property {\bf (QP-root)}, is reduced to pseudo-Anosov braids.

The next proposition gives a potential negative answer to  Question \ref{question:QProot}.

\begin{proposition}
\label{prop:counterexample}
Let
$b \in QP(n)$ be a pseudo-Anosov quasi-positive braid.
Let $Ab: B_{n} \rightarrow \mathbb{Z}$ denote the abelianization (exponent sum) map.
If $b=x^d$ for some $d>1$ and $x\in B_n$ with $Ab(x)=1$
then $b$ does not have the property {\bf (QP-root)}.
\end{proposition}
\begin{proof}
Since $Ab(x)=1$, if $x$ is quasi-positive then $x$ must be the positive half twist about a simple arc connecting two distinct punctures of $D_{n}$.
Such a mapping class $x$ is reducible. Hence, $x^d$ is reducible and $b=x^d$ cannot be pseudo-Anosov.
\end{proof}

At this time of writing, we do not know whether a pseudo-Anosov quasi-positive braid that satisfies the assumption of Proposition~\ref{prop:counterexample} exists or not.

\section{Sufficient conditions for quasi-positive}
\label{section:suff}

In this section, we study quasi-positive braids admitting certain  symmetric conditions.
Throughout this section, we fix a hyperbolic structure on $S$ so that the deck transformation $\iota=\iota_k:S \rightarrow S$ is an isometry (see \cite[Theorem 11.6]{FLP}).
Let $S' \subsetneq S$ be a connected subsurface of $S$ that satisfies one of the following conditions.
\begin{itemize}
\item $S'$ is an annular neighborhood of a geodesic simple closed curve in $S$.
\item $S'$ is a hyperbolic surface with geodesic boundary and the inclusion $S' \hookrightarrow S$ is an isometry.
\end{itemize}
Note that the surface $S'_{i}:=\iota^{i-1}(S') \subset S$ also satisfies the same property.
For $f \in {\rm Homeo}^+(S')$ let $\widehat{f} \in{\rm Homeo}^+(S)$ be a homeomorphism extending $f$ such that $\widehat{f}(x)=x$ for $x \in S \setminus S'$.
For $i=1,\ldots,k$ let $$f_i:=\iota^{i-1} \circ \widehat{f} \circ \iota^{-i+1} \in {\rm Homeo}^+(S).$$
Our goal is to study elements $\Psi(b)=[\phi] \in {\rm SMod}(S) \cap Dehn^+(S)$ of the form
\[ \phi=f_{1}f_{2}\cdots f_k \in {\rm Homeo}^+(S)\]
and find sufficient conditions
that guarantees $b \in QP(n)$.

We first study the following special case.

\begin{theorem}
\label{theorem:twist}
Let $C$ be a simple closed geodesic curve in $S$ such that $C, \iota(C), \dots, \iota^{e-1}(C)$ are pairwise disjoint with $\iota^e(C)=C$ for some $e\in\{1,\dots,k\}$ that divides $k$.
Let $d, j \in \mathbb{N}$.
Suppose that $b^d \in {\it Dehn}^+(n, k)$
with
$$
\Psi(b^d)= \left(T_C \circ T_{\iota(C)} \circ T_{\iota^2(C)} \circ\dots\circ T_{\iota^{e-1}(C)}\right)^j.$$
Then $b \in QP(n)$ $($and so $b^d \in QP(n)).$
\end{theorem}

When $e=1$ we get the following.
\begin{corollary}
\label{cor:froot}
Let $j,d\in\mathbb{N}$.
If $b^d \in Dehn^+(n, k)$ with $\Psi(b^d)= T_C^j$ then $b \in QP(n)$.
\end{corollary}

By Proposition~\ref{prop-Birman-Hilden}  it is easy to see that $\Psi(b^d)= T_C^j$ implies $\iota(C)=C$.

\begin{proof}[Proof of Theorem~\ref{theorem:twist}]

Since $C$ is simple and $\pi^{-1}(\pi(C))=C \sqcup \iota(C) \sqcup \cdots \sqcup \iota^{e-1}(C)$, the projection $\pi(C)$ is also simple.

First, we treat an exceptional case where the projection $\pi(C)$ is a simple proper arc in the $n$-punctured disk $D_{n}:=D \setminus P$ connecting two distinct punctures. This can be realized only if $k=2$, $e=1$ and $C$ is a non-separating simple closed curve in $S$.
Let $h \in QP(n)$ be the braid represented by a positive half twist about the arc $\pi(C)$. We have $\Psi(h) = T_{C}$.
Thus, $\Psi(b^{d})=T_{C}^{j} = \Psi(h^{j})$.
Since $\Psi$ is injective (Proposition~\ref{prop-Birman-Hilden}) $b^{d}=h^{j} \in QP(n)$. Let $Ab : B_{n} \rightarrow \mathbb{Z}$ be the abelianization map defined by $Ab (\sigma_i^{\pm 1})=\pm 1$.
Since $Ab (h)=1$ we get $Ab(b) \cdot d = j$ and $\Psi(b^{d}) = \Psi(h^j)= \Psi((h^{Ab(b)})^{d})$.
Proposition \ref{proposition:root} implies that $b \in QP(n)$.

Next, suppose that the projection $\pi(C)$ is a simple closed curve in the punctured disk $D_n$.
Let $k' = \frac{k}{e}$. Since the map $\pi$ restricted to each connected component of $\pi^{-1}(\pi(C))$ is a $k':1$ cover we have
\begin{equation}
\label{eqn:liftC}
\Psi((T_{\pi(C)})^{k'}) = T_C \circ T_{\iota(C)} \circ T_{\iota^2(C)} \circ\dots\circ T_{\iota^{e-1}(C)}.
\end{equation}
Hence,
\[ \Psi(b^{d}) = (T_C \circ T_{\iota(C)} \circ T_{\iota^2(C)} \circ\dots\circ T_{\iota^{e-1}(C)})^j = \Psi( (T_{\pi(C)})^{k' j}).\]
Proposition~\ref{prop-Birman-Hilden} gives $b^{d}= (T_{\pi(C)})^{k'j}$. By Corollary~\ref{prop:rootqperidic}, $b \in QP(n)$.
\end{proof}

Next, we study a more general case.
Recall that $f \in {\rm Homeo}^+(S')$, $S'_{i}:=\iota^{i-1}(S') \subset S$,  and   $f_i:=\iota^{i-1} \circ \widehat{f} \circ \iota^{-i+1} \in {\rm Homeo}^+(S)$ for $i=1,\ldots,k$.

\begin{lemma}\label{lemma:centralizer}
Suppose that $[f]\in {\rm Mod}(S')$ is pseudo-Anosov.
Any centralizer $[g]\in Z([f]) \subset {\rm Mod}(S')$ of $[f]$ is either pseudo-Anosov or periodic
\end{lemma}

\begin{proof}
Suppose that $[f] \in{\rm Mod}(S')$ is pseudo-Anosov and $[g]\in Z([f])$.
Let $\mathcal F$ be the stable foliation of $[f]$.
Since $\mathcal F$ is preserved under $[f]$ we have
$[f][g](\mathcal F) = [g][f](\mathcal F)=[g](\mathcal F)$, which means that the foliation $[g](\mathcal F)$ is either $\mathcal F$ itself or the unstable foliation of $[f]$. In either way, the homeomorphism $g\in{\rm Homeo}^+(S')$ is freely isotopic to a pseudo-Anosov map or a periodic map.
\end{proof}

Here is our main result.
Later in Example~\ref{example-daisy} we see that it is generalizing the so called {\em daisy relation } \cite{EMV} in mapping class group theory.

\begin{theorem}\label{thm:general}
Suppose that the surfaces $S'_1,\ldots, S'_k$ are pairwise non-isotopic.
Assume that $[f]\in Dehn^{+}(S')$ is either
\begin{itemize}
\item a non-negative power of a single Dehn twist (i.e., $S'$ is an annulus which is neighborhood of a simple closed geodesic curve), or
\item a pseudo-Anosov map (i.e., $S'$ is a hyperbolic surface).
\end{itemize}
Suppose that $b \in Dehn^+(n,k)$ satisfies
\[ \Psi(b)=[f_1 f_2 \cdots f_{k}] \]
then $b \in QP(n)$.
\end{theorem}

\begin{proof

There are two cases to consider.

\noindent
{\bf Case 1:} $S'_1,S'_2,\ldots,S'_k$ are pairwise disjoint.

\noindent
Since $[f] \in Dehn^{+}(S')$ we may write $[f_1]=T_{C_1}\cdots T_{C_l}$ for some simple closed curves $C_1,\ldots, C_l \subset S'$.
We get $[f_{j}]=[\iota^{j-1} \widehat{f} \iota^{-j+1}]= T_{\iota^{j-1}(C_1)}\cdots T_{\iota^{j-1}(C_l)}$.
Since $S'_1,S'_2,\ldots,S'_k$ are pairwise disjoint, if $j\neq j'$ then $T_{\iota^{j}(C_{i})}$ and $T_{\iota^{j'}(C_{i'})}$ commute for every $i,i'$ and we have
\begin{eqnarray*}
\Psi(b) & = & [f_{1}\cdots f_{k}] \\
&= &
 (T_{C_1}\cdots T_{C_l})(T_{\iota(C_1)}\cdots T_{\iota(C_{l})}) \cdots (T_{\iota^{k-1}(C_1)}\cdots T_{\iota^{k-1}(C_l)})\\
& = &(T_{C_1} T_{\iota(C_1)}\cdots T_{\iota^{k-1}(C_{1})}) \cdots (T_{C_l} T_{\iota(C_l)}\cdots T_{\iota^{k-1}(C_{l})}).
\end{eqnarray*}
Since $S'_1,S'_2,\ldots,S'_k$ are pairwise disjoint, the projection $\pi(C_i)$ is a simple closed curve in $D_n$, and we can use the second-half argument in the proof of Theorem~\ref{theorem:twist}.
By the formula (\ref{eqn:liftC}) we get $\Psi(T_{\pi(C_{i})})=T_{C_i} T_{\iota(C_i)}\cdots T_{\iota^{k-1}(C_{i})}$.
Proposition~\ref{prop-Birman-Hilden} gives $b= T_{\pi(C_1)}\cdots T_{\pi(C_l)} \in QP(n)$.

\noindent
{\bf Case 2:} $S'_1 \cap S'_p \neq \emptyset$ for some $p\in\{2,\dots,k\}$.

\noindent
First we note that
$[f_1 f_2 \cdots f_{k}]=\Psi(b) \in {\rm SMod}(S)$ implies that
\[
[f_1f_2\cdots f_k]  = [\iota( f_1 f_2 \cdots f_k) \iota^{-1}] = [f_{2}f_3\cdots f_k f_1] = \dots = [f_k f_1\cdots f_{k-1}].
\]
In particular, we have
\begin{equation}
\label{eqn:center}
\Psi(b) [f_i] = [f_i] \Psi(b) \mbox{ for every } i\in\{1,\dots,k\}.
\end{equation}

Let $\nu$ be the minimal subsurface of $S$ such that
\begin{itemize}
\item $\nu$ contains $S'_1 \cup S'_2 \cup \cdots \cup S'_k$ and
\item $\partial \nu$ is geodesic.
\end{itemize}
Since $\iota(S'_1 \cup S'_2 \cup \cdots \cup S'_k) = S'_1 \cup S'_2 \cup \cdots \cup S'_k$ we have $\iota(\nu)=\nu$.
In particular, the boundary $\partial \nu$ is a multi-curve invariant under $\iota$.

For simplicity, we put $\psi:= f_1 f_2 \cdots f_k \in {\rm Homeo}^+(S)$.

\begin{claim}
\label{claim:subsurface}
$\psi(S'_i)$ is isotopic to $S'_i$.
\end{claim}

We will prove Claim~\ref{claim:subsurface} after the completion of the proof of Theorem~\ref{thm:general}.

By Claim~\ref{claim:subsurface}
there is a homeomorphism $\phi \in {\rm Homeo}^+(S)$ which is isotopic to $\psi$ and preserves $S'_i$ setwise.
Although $\phi$ may permute components of $\partial S'_{i}$, we may assume that there exists $d_0>0$ such that $\phi^{d_0}=id$ on $\partial S'_1 \cup \partial S'_2 \cup \cdots \cup \partial S'_k$.
Let
\begin{equation}\label{def of phi_i}
\phi_i := \phi^{d_0}|_{S'_{i}} \in {\rm Homeo}^+(S'_i).
\end{equation}

\begin{claim}
\label{claim:btwist}
For each $i=1,\ldots,k$ there is $d_i>0$ such that $\phi_i^{d_i}$ is isotopic to a  product of Dehn twists about the boundary components of $S'_i$. Namely,
\begin{equation}\label{eqn}
[\phi_i^{d_i}] = \prod_{C \subset \partial S'_i} T_{C}^{N(C)} \in {\rm Mod}(S'_i) \quad \mbox{ for some } N(C) \in\mathbb{Z}.
\end{equation}
Moreover, the exponent $N(C)=0$ if $C$ is essential in the minimal subsurface $\nu$.
Therefore, if $S'$ is an annulus then $[\phi_i^{d_i}] = id$.
\end{claim}

We will prove the claim after the completion of the proof of Theorem~\ref{thm:general}.

Let $d'$ be the least common multiple of $d_1,\ldots,d_k$ found in Claim~\ref{claim:btwist}. Put $d=d_{0}d'$.
As a consequence of Claim~\ref{claim:btwist} the map $\phi^{d}|_{S_{i}} = \phi_{i}^{d'} \in {\rm Homeo}^+(S'_i)$ is isotopic to a product of Dehn twists along common components of $\partial S'_i$ and  $ \partial \nu$.
Let $C_{1},\ldots,C_m$ denote the boundary components of $\nu$.
We obtain
\begin{equation}\label{phi_d}
[\phi^{d}] = [T_{C_{1}}^{N(1)}T_{C_{2}}^{N(2)}\cdots T_{C_{m}}^{N(m)}].
\end{equation}
Since $[f] \in Dehn^{+}(S')$ we see that $[\phi]=[f_1\cdots f_k]$ is right-veering and $N(i)\geq 0$ for all $i=1,\dots,m$.

We define an equivalence relation $C_{i} \sim C_{i'}$ if $\pi(C_{i})=\pi(C_{i'}) \subset D_{n}$ and let
$\mathcal{C} = \{C_{1},\ldots,C_m\} \slash \sim$.
For $C \in \{C_1, C_2,\dots, C_m\}$ let $[C]\in\mathcal C$ be its equivalence class and $e(C)\in\mathbb{N}$ the smallest positive integer such that $\iota^{e(C)}(C)=C$.
Since $\iota(\partial \nu)=\partial\nu$ we note that $C, \iota(C), \dots, \iota^{e(C)-1}(C) \subset \partial \nu = C_1 \sqcup C_2\sqcup \cdots \sqcup  C_m$.
Put $$T_{[C]} := T_{C}T_{\iota(C)}\cdots T_{\iota^{e(C)-1}(C)}.$$
The fact $[\phi] \in {\rm SMod}(S)$ implies that $N(i)=N(j)$ if $C_{i} \sim C_j$. We may define non-negative integers  $N([C_i]):=N(i)$.
The description (\ref{phi_d}) can be restated as
\[
[\phi^{d}] = \prod_{[C] \in \mathcal{C}} T_{[C]}^{N([C])} \in {\rm Mod}(S).
\]

Recall that the projection $\pi(C)$ is a simple closed curve or a simple proper arc in $D_n$ joining distinct punctures.
This is because $\pi^{-1}\pi(C) \subset \partial \nu$ is a simple closed multi-curve.

If $\pi(C)$ is a simple closed curve, put $b_C := (T_{\pi(C)})^{\frac{k}{e(C)}} \in QP(n)$.
Then by (\ref{eqn:liftC}) we have $\Psi(b_C)=\Psi((T_{\pi(C)})^{\frac{k}{e(C)}}) = T_{[C]}$.

If $\pi(C)$ is a simple arc, i.e., $k=2$ and $e(C)=1$, put $b_C := h_{\pi(C)} \in QP(n)$ the positive half twist about $\pi(C)$.
Then we have $\Psi(b_C)= \Psi(h_{\pi(C)}) = T_C = T_{[C]}$.

For either case, we have
\[
\Psi(b^{d}) = [\phi^{d}] = \Psi \left( \prod_{[C] \in \mathcal{C}} (b_C)^{N([C])} \right).
\]
For distinct $[C]$ and $[C']$ the projections $\pi(C)$ and $\pi(C')$ are disjoint because $C_1,\dots,C_m$ are pairwise disjoint.
Hence, by Proposition~\ref{prop-Birman-Hilden} the braid $b^{2d}$ is a product of positive Dehn twists about pairwise disjoint simple closed curves.
(We consider $b^{2d}$ rather than $b^d$ so that if $\pi(C)$ is an arc the half twist about $\pi(C)$ becomes  a Dehn twist about a closed curve enclosing $\pi(C)$.)
Corollary \ref{prop:rootqperidic} shows that $b \in QP(n)$.
\end{proof}

\begin{proof}[Proof of Claim \ref{claim:subsurface}]

Suppose that $S'$ is an annulus which is a neighborhood of a simple closed geodesic curve $C$ and $[f] = (T_C)^m$.
By (\ref{eqn:center}) we have $(T_{\psi(C)})^m = (\psi \circ T_C \circ\psi^{-1})^m= \psi \circ (T_C)^m \circ\psi^{-1} = (T_C)^m$, and $\psi(C)$ is isotopic to $C$.
Hence, $\psi(S'_i)$ is isotopic to $S'_i$.

Next, suppose that $[f]\in {\rm Mod}(S')$ is pseudo-Anosov.
The property (\ref{eqn:center}) implies that  $f_{i}\psi(S'_i)$ is isotopic to $\psi f_{i}(S'_i) = \psi(S'_i)$. Then either $\psi(S'_i)$ is isotopic to $S'_i$, or by isotopy one can make $\psi(S'_i)$ and $ S'_i$ disjoint.
The latter possibility cannot be realized by the following reason.
Take an essential simple closed curve $\alpha \subset S'_i$.
If we can make $\psi(S'_i)$ disjoint from $S'_i$  by isotopy, then  $\psi(\alpha)$ can also be disjoint from $S'_{i}$.
However, by (\ref{eqn:center})
$\alpha = f_i f_i^{-1}\psi^{-1} \psi(\alpha)=f_i \psi^{-1} f_i^{-1}(\psi(\alpha)) = f_i \psi^{-1}(\psi(\alpha)) = f_{i}(\alpha)$.
This contradicts the assumption that $[f]$ is pseudo-Anosov.
\end{proof}

\begin{proof}[Proof of Claim~\ref{claim:btwist}]

In the case where $S'$ is an annulus,  (\ref{eqn}) is a direct consequence of Claim \ref{claim:subsurface}.

Assume that $[f]\in{\rm Mod}(S')$ is pseudo-Anosov and $S'$ is not an annulus.
By the symmetry, it is enough to prove the statement (\ref{eqn}) for the case $i=1$.
Recall that $S'_1\neq S'_p$ and $S'_1\cap S'_p \neq \emptyset$.
Let $D$ be a connected component of $S'_1 \cap S'_p$.
Since $\phi^{d_0}=id$ on $\partial S'_1 \cup \partial S'_2 \cup \cdots \cup \partial S'_k$ and $\partial D \subset (\partial S'_1 \cup \partial S'_2 \cup \cdots \cup \partial S'_k)$ we have $\phi^{d_{0}}(D)=D$ and $\phi^{d_0}=id$ on $\partial D$.

Since $\partial S'_1$ is geodesic $D$ cannot be a bigon or an annulus.
Let $\Gamma = \partial (S'_1 \setminus D)$. Then $\Gamma$ is a simple closed multi-curve in $S'_1$.
Since $S'_1$ is not an annulus
$\Gamma$ contains an arc $\gamma$ which is essential in $S'_1$.
Since $\phi^{d_0}=id$ on $\partial D$ the curve $\phi_{1}(\gamma)=\phi^{d_0}(\gamma)$ is isotopic to $\gamma$; hence,  the mapping class $[\phi_{1}] \in{\rm Mod}(S')$ cannot be pseudo-Anosov.

On the other hand, $[f]=[f_1|_{S'_1}]\in {\rm Mod}(S')$ is pseudo-Anosov, and by (\ref{eqn:center}) $[\phi_1] \in Z([f]) \subset {\rm Mod}(S')$.
Since $[\phi_{1}]$ is not pseudo-Anosov, Lemma~ \ref{lemma:centralizer} shows that $[\phi_1]$ is periodic.
Namely, there is $d_1 >0$ such that $[\phi_1^{d_1}]$ is a product of Dehn twists about the boundary components of $S'_1$ and we obtain (\ref{eqn}).

Next we show the second statement of the claim.
The surface $S'$ is either annular or hyperbolic.
Suppose that a boundary component $C$ of $S'_1$ is an essential curve in the surface $\nu$.
Since $C$ is not a boundary component of $\nu$ and $\nu$ is the minimal surface containing $S'_1\cup\cdots\cup S'_k$, there must exist $p \neq 1$ such that $C \cap S'_p \neq \emptyset.$

Note that $C$ cannot be a boundary component of $S'_p$ (that is, one side of $C$ is $S'_1$ and the other side of $C$ is $S'_p$)  because in such a case the quotient space $S \slash \iota$ cannot be a topological disk.

If $C$ is not isotopic to a boundary component of $S'_p$ then the descriptions of $[\phi_1^{d_1}]$ and $[\phi_p^{d_p}]$ in (\ref{eqn}) and the definition of $\phi_i$ in (\ref{def of phi_i}) show that $N(C)=0$.

If $C$ transversely intersects a boundary component, say $C'$, of $S'_p$
we also get $N(C)=0$, because otherwise (\ref{def of phi_i}) and (\ref{eqn}) show that $\phi(C')\not\subset S'_p$ which contradicts Claim~\ref{claim:subsurface}.
\end{proof}

We close the paper with an example which shows that Theorem~\ref{thm:general} can be viewed as a generalization of the {\em daisy relation} found in \cite{EMV}.

\begin{example}\label{example-daisy}
Let $F$ be a sphere with $k+1$ holes ($k\geq 3$) that is obtained as the $k$-fold cyclic branched covering $\pi_{F}:F \rightarrow A$ of an annulus $A$ branched at one point.
Let $a_0,\ldots,a_{k}$ be the boundary components of $F$.
Let $\iota_F:F \rightarrow F$ be a deck transformation defined by  a $2\pi \slash k$ rotation of $F$ about the unique branch point such that $\iota_F(a_{0})=a_{0}$ and $\iota_F(a_i)=a_{i+1}$ for $i=1,\dots,k$. See the left picture of Figure~\ref{fig:daisy}.
Let $x_i$ ($i=1,\dots,k$) be a simple closed curves on $F$ enclosing $x_0$ and $x_i$ such that $\iota_F(x_{i})= x_{i+1}$.
According to the \emph{daisy relation} as stated in \cite{EMV} we have
\begin{equation}\label{daisy-1}
T_{x_1}\cdots T_{x_{k}} = T_{a_0}^{k-2}T_{a_1}\cdots T_{a_k} \in {\rm Mod}(F).
\end{equation}
(The case of $k=3$ yields the famous lantern relation.)

Recall the $k$-fold cyclic branched covering $\pi:S\to D$ with $n$ branch points and the deck transformation $\iota:S\to S$.
Take an embedding $i:F \hookrightarrow S$ such that $i\circ \iota_F(x) = \iota\circ i(x)$ for all $x \in F$.
Let $C = i(x_{1}), A_{0}=i(a_0)$, and $A = i(a_1)$.
See the right picture in Figure \ref{fig:daisy} for the simplest case $(n,k)=(3,3)$.
\begin{figure}[htbp]
\begin{center}
\includegraphics*[bb=71 553 413 734, width=130mm]{zu_daisy1.pdf}
\end{center}
\caption{Left:  The $(k+1)$ holed sphere $F$ where $k=4$.
Right: The surface $S$ where $(n,k)=(3,3)$.
The curves $C, A, A_{0}$ satisfy the daisy (lantern) relation $T_{C}T_{\iota(C)}T_{\iota^{2}(C)} = T_{A_0}T_{A} T_{\iota(A)}T_{\iota^{2}(A)}$.
The surface $S \setminus i(F)$ is the Bennequin surface of a $(2, 3)$ torus knot, which is $A_0$.}
\label{fig:daisy}
\end{figure}

The daisy relation (\ref{daisy-1}) gives
\begin{equation}\label{eq:daisy}
T_{C}T_{\iota(C)}\cdots T_{\iota^{k-1}(C)} = \left( T_{A_0}^{k-2} \right) \left( T_{A} T_{\iota(A)}\cdots T_{\iota^{k-1}(A)} \right) \in {\rm Mod}(S).
\end{equation}
The left hand side of (\ref{eq:daisy}) is of the form of $[f_1 f_2\cdots f_k]$ which is studied in Theorem~\ref{thm:general}.
One can show, using \cite[Lemma 3.1]{HKP} and the chain relation \cite[Proposition 4.12]{FM} in mapping class group theory, the term $T_{A_0}^{k-2}$ in the right hand side of (\ref{eq:daisy}) is the image of a quasi-positive braid under the homomorphism $\Psi:B_n\to{\rm Mod}(S)$.
We can also see that the link $A \cup \iota(A)\cup\cdots \cup \iota^{k-1}(A)$ is a $(k, k)$ torus link and the term $T_{A} T_{\iota(A)}\cdots T_{\iota^{k-1}(A)}$ is the image under $\Psi$ of a positive full twist of $k$-stranded braid, which is clearly a quasi-positive element in $B_n$.
In this sense, Theorem~\ref{thm:general} can be seen as a generalization of the daisy relation.
\end{example}

\section*{Acknowledgements}
The authors thank Joan Birman, John Etnyre, Amey Kaloti and Rebecca Winarski for useful discussions.
TI was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 15K17540.
KK was partially supported by NSF grant DMS-1206770 and Simons Foundation Collaboration Grants for Mathematicians 426710.

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text:
Dark Ages reenactment is generally considered a sub-branch of Medieval reenactment focussing on the 1st millennium AD beginning with the fall of the Western Roman Empire and ending with the High Middle Ages.

The term 'Dark Ages' is much used. Some groups define it as the period between the collapse of the Western Roman Empire, and the establishment of the main European Christian kingdoms - approx. 400 - 750 AD, whereas others include the Viking period as well (extending as far as 1066 AD). Some people call the later half of this period "Early Medieval".

Most reenactment groups focus on a smaller time period, sometimes restricting their interest to a particular century, or even a specific decade, depending on how authentic the reenactment and encampment is intended to be. Typically Dark Ages reenactment groups focus on a specific time period and culture within this range, such as Vendel Age Vikings or the Picts.

The largest Dark Age event in the UK is the Battle of Hastings reenactment.

See also
Medieval reenactment
The Vikings (reenactment)
Regia Anglorum
Vikings Of Middle England

Medieval reenactment
Medieval-themed fairs by type

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Shanghai Restaurant is a family owned and operated business that has been serving the city of Nampa for over 30 years. We offer a variety of services that include buffet, dine-in, take-out, delivery and banquet facilities.
If you have any questions regarding the services we provide or would like to place a order or make a reservation please give us a call at (208)466-2921.


text:
Neon Trees' Tyler Glenn Denounces Mormon Church in New Music Video, 'Trash' – WATCH
May 1, 2016 by Sean Mandell Leave a Comment
Tyler Glenn, the Neon Trees frontman who in 2014 came out as gay, has issued a shark rebuke of the Mormon Church in his new music video, "Trash."
Rolling Stone reports:
In the clip for the biting electro-pop track, Glenn is first seen drinking straight from a liquor bottle, though alcohol is forbidden by Mormonism. As he sings the song, the musician is surrounded by Mormon imagery, including altered portraits of the church's founder, Joseph Smith. "One man's trash is another man's treasure," he sings during the song's chorus, which he tells Rolling Stone is meant to be a reflection of a "violent relationship."
PREVIOUSLY – Neon Trees Lead Singer Tyler Glenn Opens Up About Coming Out: VIDEO
Glenn reveals that after struggling to reconcile his Mormon faith with his sexuality came to a head 6 months ago when the LDS Church declared that Mormon faithful in same-sex relationships are "apostates."
Glenn said, "I always tried to make being gay and being Mormon work." But in the end, it didn't:
"My entire life and perspective on God, the afterlife, morals and values, my self-worth and my born sexual orientation has been wired within the framework of this religion that doesn't have a place for me." […]
"I served [this church]. I was the square peg trying to fit into the round hole. I believed it till six months ago."
Of the Mormon Church's new anti-gay policy, Glenn adds,
"The big problem here is that they claim it's the only truth. There have been over 40 suicides within the church as a result of this policy. These aren't just grown men and women. Many are children. It's backwards. It's not of God. I needed to make this statement to artfully show the pain of a faith crisis and the darkness of doubt, but also that there's ways to reclaim what is yours."
RELATED – 32 LGBT Mormon Youths Have Killed Themselves Since LDS Church Unveiled Horrific Anti-Gay Policy: VIDEO
This being Glenn's first solo effort, fans can look to expect more from him:
Glenn is promising a full solo album in the future that will address this same faith crisis through a lens of "celebration." He describes his music sans Neon Trees as "more adventurous" and "aggressive." Even though he's exploring a career on his own, Neon Trees are not over, and his band supports his creative endeavor. "I'm not sure if they totally support the content," he adds. "I haven't played or showed them much. I do love them and hope that this record doesn't change things negatively, but other than that, I feel fearless about it."
Watch, below.
Filed Under: Uncategorized Tagged With: Mormon Church, Mormons, music, music video, Religion, Tyler Glenn
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text:
Tag: diversity:workforce diversity
The Workforce Diversity page includes strategies, techniques and resources for attracting, hiring and retaining a diverse workforce. Information will focus on methods of attracting diverse candidates, developing a viable search process and creating and maintaining a positive workplace climate.
Workplace Diversity_ A Social-Ecological Framework and Policy Implications.pdf Abstract: The diversification of the global workforce brings both challenges and opportunities. We focus on diversity defined by membership in traditionally underrepresented groups. To harness the power of diversity, organizations must: increase representation of diverse individuals throughout the organizational hierarchy, attend to the social processes that emerge once diversity is present, and foster an organizational climate that supports the full inclusion of diverse individuals. We review dynamics at multiple levels of analysis that affect organizations in these three realms. Policy recommendations are grounded in the following ecological principles: (1) organizational issues are nested within multiple levels of context, (2) any organizational event can have reverberating effects throughout the system, (3) people's experiences of events shape their reactions and the impact of practices on varied groups shapes organizational consequences, and (4) people are continually adapting to one another and to organizational resources and requirements.
An interesting look at gender, racial, and class diversity within environmental organizations: The Disturbing Lack of D…pdf The full study can be found here: diversity in environmental organizations.pdf
Why do we see a an attrition of women in biological sciences but not in the medical field? This article re-examines some of the reasons hypothesized as to why less women are seen in scientific research fields: Attrition of Women in the Biological Sciences.pdf
The following study is two-fold. The first part gives insights into organizational approaches to diversity and managing a culturally diverse workplace. The second part introduces a "Diversity Perspective Questionnaire" that measures an organization's approach to diversity. Managing a culturally diverse workforce.pdf
Learn how even in workplaces that seek to be inclusive, LGBT employees are affected by a "culture of silence". sound of silence in inclusive organizations.pdf
There have been studies done on what has been termed the "bamboo ceiling" which describes the hinderance of Asian Americans' advancement in the workplace. Many are also familiar with the "glass ceiling" effect describing the same phenomenon in women. However, there has not been many studies that have looked into the intersection between the two. Read the article hereHitting the Ceiling.pdf to learn more about barriers specific to Asian American Women
"Do Millennials Really Want Their Bosses to Call Their Parents?" An interesting article on an often overlooked aspect of workforce diversity, generation. http://blogs.hbr.org/2014/04/do-millennials-really-want-their-bosses-to-call-their-parents/
Workforce Bullying http://stopworkplacebullies.wordpress.com/
In order to ensure that work workplaces are inclusive, faculty may want to consider reflections in lieu of prayers when they are asked to pray at a meeting. This way each person, regardless of religion will feel comfortable in the workplace. http://extension.missouri.edu/staff/diversity-prayer.aspx
Non Tenure Track Women Faculty: Opening the Door Non Tenure Track Women Faculty.pdf
Federal Agencies to Develop a Strategic Plan to Insure a Diverse and Inclusive Workforce: http://1.usa.gov/qzJL5o mmmmm
Share your comments on our CoP Blog http://extensiondeicop.blogspot.com/
Resources for Recruiting Women and Minority Faculty Candidates:http://bit.ly/o5oygJ
Avoid the Craziness When It's Time to Hire
Where to Advertise to Attract Underrepresented Professionals
Sample Diversity-Related Qualification Statements:http://bit.ly/psiWmm
Equity and Social Justice Diversity Links
Multimedia_demo
Challenges and strategies
http://www.diversitycouncil.org/recruiting_diverse_staff.shtml
http://aarjobs.com/
http://aarjobs.com/index.php?option=com_content&view=article&id=28:recruiting-the-finest-gaining-access-to-a-diverse-workforce&catid=43:august-2008-issue
http://www.diversitycouncil.org/toolkit.shtml
http://dlist.sir.arizona.edu/1650/
http://dlist.sir.arizona.edu/1650/01/DiversityKIM.pdf
Diversity Articles
http://www.interplaytheatre.com/Diversity/diversity-articles.html
http://www.diversityworld.com/Diversity/workforce_diversity.htm
Achieving Workforce Diversity
http://www.slb.com/about/guiding_principles/diversity.aspx
http://govinfo.library.unt.edu/npr/library/workforce-diversity.pdf
Workforce Diversity Section Editor: open
As Extension strives to meet the needs of an ever-increasingly diverse society, it must do so by recruiting, hiring and retaining staff that possess the attitude, competencies and qualities that enable them to effectively work across differences. They must have the needed skills for success. As a result, well-thought out recruitment, hiring and retention strategies that attract, support and sustain individuals with a mature diversity lens are a critical component of creating an environment that supports and promotes pluralism and diversity1.
Today, most job postings and descriptions clearly identify the ability to show respect for differences in backgrounds, lifestyles, viewpoints, and needs in reference to areas such as ethnicity, race, gender, creed, and sexual orientation as being essential to the position and the organization. While articulating these as desirable qualities and behaviors is an important first step in recruitment, the challenge often arises in assessing the level of skill and competency during the screening and interview process.
The accompanying materials, Skills for Success©, developed by the Office of Human Resources at Cornell University, can serve a template for developing recruitment materials and interviewing questions that result in attracting and hiring individuals that bring a diverse perspective and are supportive of diversity. Skills for Success©, consist of the following materials:
Staff Skills for Success (pdf) – A broad overview of the skills essential for individual and organizational success and examples of demonstrated behaviors in these areas
Interviewing Candidates for Skills for Success (Short Version – pdf) – A short version of interview questions that address the skill areas
Interviewing Candidates for Skills for Success (Long Version – pdf) – A longer version of targeted interview questions that probe the breadth and depth of candidate's skill level.
Characteristics of Performance Levels – pdf – A matrix that outlines a continuum of the characteristics of individual performance levels in the skill areas identified.
Staff Performance Dialogue Form (pdf) – A standard form to be used as a tool for conducting employee review, staff development and performance improvement discussions.
Each skill area identified in Skills for Success©, can lend itself to diversity and pluralism. The skills, behaviors and interview questions under the heading of inclusiveness specifically and intentionally address and provide a language for articulating desired diversity competencies in recruitment and a process for assessing them in hiring.
In addition to addressing the areas of recruitment and hiring, Skills for Success©, also provides resources for the ongoing support, development and retention of staff. As with other competencies, there is a need for ongoing assessment and development on inclusiveness. The Staff Performance Dialogue Form and the Characteristics of Performance Levels can serve as tools for creating organizational expectations around diversity and pluralism, as well as a means for supporting staff in assessing their skill level and identifying ongoing professional development needs in this area.
For additional information and case studies on Cornell Cooperative Extension's efforts on staffing for diversity check out: http://www.joe.org/joe/1998february/a1.html
By Eduardo González, Jr., Diversity Specialist, Cornell Migrant Program
1The concepts of "diversity" and "pluralism" as defined in Pathway to Diversity: Strategic Plan for the Cooperative Extension System's Emphasis on Diversity (1991) are: Diversity is differences among people with respect to age, class, ethnicity, physical and mental ability, race, sexual orientation, spiritual practice, and other human attributes. Pluralism is an organizational culture that incorporates mutual respect, acceptance, teamwork, and productivity among people who are diverse in the dimensions of human differences listed above as diversity.


text:
Q: template class method instantiation when a virtual unrelated method in the base class causes compilation failure on MSVC Is the following code legal C++?
MS Visual C++ fails, but gcc and clang are fine: https://godbolt.org/z/vsQOaW
It could be an msvc bug, but wanted to check first:
struct Base {
    virtual void junk() = 0;
};

template <class T>
struct Derived : Base {

    void junk() override {
        T::junkImpl();
    }

    void otherMethod() {
    }
};

template <class T>
struct NotDerived {

    void junk() {
        T::junkImpl();
    }

    void otherMethod() {
    }
};

struct TypeWithJunk {
    void junkImpl() {
    }
};

struct TypeWithoutJunk {};

void reproduce(NotDerived<TypeWithoutJunk>* ndt, Derived<TypeWithoutJunk>* dt) {

    // works - junk is not used, not instantiated
    ndt->otherMethod();

    // fails on MSVC - junk is instantiated even if not used
    dt->otherMethod();
}

A: junk may get instantiated just like the rest of virtual functions because it is required to populate vtable. So all the compilers seem to demonstrate conforming behavior:

17.8.1 Implicit instantiation [temp.inst]
9 … It is unspecified
  whether or not an implementation implicitly instantiates a virtual member function of a class template if
  the virtual member function would not otherwise be instantiated.

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Nanotechnology is exploding. The world of small sizes has never been as vital for so many directions in science and industry. With this new direction comes a greater need for measurement on the atomic scale and new demands for reliable, precise, and multi-purpose instrumentation. NT-MDT enjoys a 20-year history in instrumentation created specifically for nanotechnology research, leading the field in originality, quality, and high tech development.
We strive for next-generation SPM technology, whether it be in pure modularity that allows a university or industrial lab to start with a cost-effective core product and build to a grand, multi-user research center or the ultimate amalgamation of SPM with related technologies that has resulted in ultramicrotomy for nanotomography and spectroscopy-based instruments that meld the world of imaging with the world of chemical analysis. We believe passionately in pushing the envelope for rapid innovation while still delivering superb customer service.
NT-MDT offers expert service and applications development through more than 20 representative offices and distributor centers around the globe. There are two key branch offices opened in Holland and Ireland. In the past five years, our installed base has grown to over 2000 instruments, promoting growth of both lab and research programs world-wide.
Whether you are engaged every day in nano research or just contemplating it, coupling your specific scientific knowledge and expertise with our competence in instrument creation will produce the highest quality research results currently available.
NT-MDT invites everyone to the webinar "Tip-Enhanced Raman Scattering. Approaching 10 nm spatial resolution in Raman imaging" that will take place on March 20-21.
NT-MDT is happy to invite to the webinar "Tip-Enhanced Raman Scattering. Approaching 10 nm spatial resolution in Raman imaging" by Dr. Pavel Dorozhkin.


Just a reminder in case you didn't pick up the holiday schedule in the gym.
Today (Wednesday) will be regular schedule but we will be closed on Thursday Thanksgiving and Friday. We will resume regular 10:00am class on Saturday.
* There will be no yoga.
You are welcome to join us at 8:00am at Point Furman Park on Thanksgiving to do the Turkey Trot fun run.


text:
Anderson Township is one of fourteen townships in Madison County, Indiana, United States. As of the 2010 census, its population was 56,436 and it contained 28,001 housing units.  Aside from the exclaves of Country Club Heights, Woodlawn Heights and River Forest and the town of Edgewood, the entire township is within the city limits of Anderson.

Geography
According to the 2010 census, the township has a total area of , of which  (or 99.76%) is land and  (or 0.27%) is water.

Cities, towns, villages
 Anderson (vast majority)
 Country Club Heights
 Edgewood
 River Forest
 Woodlawn Heights

Communities
 Brentwood at
 Crestlawn at
 Eastern Heights at
 Elmhurst at
 Extension Heights at
 Fairfax at
 Forest Hills at
 Glyn Ellen at
 Grandview at
 Gridley at
 Harmeson Heights at
 Hillcrest at
 Irondale at
 Lowmandale at
 Meadowbrook at
 South Edgewood at
 Western Village at
(This list is based on USGS data and may include former settlements.)

Cemeteries
The township contains these seven cemeteries: Booco, Maplewood, Moss, Pleasant Walk, Saint Marys, Vandeventer and West Maplewood.

Major highways
  Interstate 69
  State Road 9
  State Road 32
  State Road 109

Airports and landing strips
 Community Hospital Airport
 Rolling Hills Convalescent Center Airport

Landmarks
 Hoosier Park Horse Track

Education
 Anderson Community School Corporation

Anderson Township residents may obtain a free library card from the Anderson Public Library in Anderson.

Political districts
 Indiana's 6th congressional district
 State House District 35
 State House District 36
 State House District 37
 State Senate District 25

References

 United States Census Bureau 2008 TIGER/Line Shapefiles
 IndianaMap

External links
 Indiana Township Association
 United Township Association of Indiana
 City-Data.com page for Anderson Township
 Anderson Township History

Townships in Madison County, Indiana
Townships in Indiana


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If you have a garage door that is damaged or showing signs of wear and tear, let us carry out maintenance to restore it to full working order. In addition to our regular repair service, we also offer an emergency 24 hour garage door repairs service to secure your property at short notice – so, if your door is stuck, the cables have come off or the spring has broken, we can help!
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Bollywood's power couple, Deepika Padukone and Ranveer Singh, have been in the news recently for various things. Whether it is their upcoming projects, their holiday pictures or the news of them getting married in November. The duo makes headlines on a day to day basis and we don't blame them as the audience is gaga over this couple. The two have been in a relationship for almost half a decade now and rumours are rife that the couple will officially announce their wedding date sometime next week.
In all this frenzy, a picture of the duo enjoying a cosy dinner with each other has come to the forefront. The couple is seen posing for a selfie with their fans. An eyewitness claims that the couple was with Ranveer's mother when the fans requested them for a picture post their dinner. Ranveer's parents and Deepika's parents have met multiple times now and the couple is often clicked spending time with each other's parents also. Well, we are glad that all's well in their paradise, reports agencies.


text:
April 30, 2013 9:34AM EDT
Canada: Safety of Indigenous Women Under UN Scrutiny
Opportunity for Action to End Violence Against Indigenous Women, Girls
Highway 16, sometimes referred to as "the Highway of Tears" in recognition of the women and girls who have gone missing or been murdered in its vicinity, in northern British Columbia. July 2012.
© Samer Muscati/Human Rights Watch
(Toronto) – The United Nations Human Rights Council adopted today the report on the Universal Periodic Review (UPR) of Canada's human rights record, which included a large set of questions, recommendations, and comments from countries across the world about violence against indigenous women and girls. The attention should spur Canada to take decisive action to address the hundreds of murders and disappearances of indigenous women and girls over the last four decades, Human Rights Watch said today.
"It is not surprising that violence against indigenous women and girls figured so prominently in the discussion of Canada's human rights record," said Liesl Gerntholtz, women's rights director at Human Rights Watch. "It reflects the persistent insecurity faced by women and girls, the urgent need for a public accounting of what has gone wrong for so long, and a robust national plan for addressing it going forward."
In February 2013, Human Rights Watch released "Those Who Take Us Away: Abusive Policing and Failures in Protection of Indigenous Women and Girls in Northern British Columbia, Canada." The report documents the failure of the Royal Canadian Mounted Police (RCMP) in British Columbia to protect indigenous women and girls from violence. It also documents abusive police behavior against indigenous women and girls, including excessive use of force, and physical and sexual assault. The report also found that Canada has inadequate police complaint mechanisms and oversight procedures, including a lack of a mandate for independent civilian investigations into all reported incidents of serious police misconduct.
The report released in Geneva today summarizes Canada's second Universal Periodic Review which took place on April 26, 2013. All UN member countries undergo such a review every four years. During the UPR process, other UN member countries may ask questions and make recommendations about measures to improve the country's human rights situation.
The Netherlands, the United Kingdom, Slovenia, Sweden, and Mexico submitted questions in advance about Canada's response to violence against indigenous women and girls. While recognizing efforts made by Canada on this issue, China, the United States, Estonia, Finland, India, Indonesia, Ireland, Japan, New Zealand, Norway, Slovakia, Togo, and others expressed concerns about the situation during the review and made recommendations to the government.
The development of a national strategy to combat the violence was a recurring recommendation.
The government elaborated on measures in place to address the issue, including the National Center for Missing Persons and Unidentified Remains, police task forces to investigate cases, and community safe plans.
"To be successful, any effort to address violence against indigenous women and girls will need to address the issue of police accountability," Gerntholtz said. "That includes accountability for the police response to reports of violence in the community and accountability for any acts of violence committed by police officers themselves."
In conjunction with the Human Rights Council's review, Canadian government representatives said that Canada had recently approved the requests of three human rights authorities to visit the country, which had been pending for varying lengths of time. James Anaya, UN special rapporteur on the rights of indigenous peoples, said in February 2013 that he was awaiting a response to a February 2012 request to visit Canada to examine and report on the human rights situation.
At a March 2013 hearing of the Inter-American Commission on Human Rights regarding violence against indigenous women and girls, a commissioner noted that the government had not yet responded to a 2012 request to visit. The Committee on the Elimination of Discrimination against Women announced its decision to launch an inquiry into missing and murdered indigenous women and girls in Canada in late fall 2011. Canada's statement at the Human Rights Council on Friday was the first public notice that the committee had received permission to visit.
"Granting human rights authorities permission to visit is an important step forward," Gerntholtz said. "We hope it signals that the government is more receptive to meaningful examination of these issues and will heed the call for a national public inquiry into the murders and disappearances."
The Native Women's Association of Canada documented 582 cases of missing and murdered indigenous women and girls in Canada as of March 2010. Many happened between the 1960s and the 1990s, but 39 percent occurred after 2000. The number of cases is undoubtedly higher today, but comprehensive data is not available since the government cut funding for the organization's database and police forces in Canada do not consistently collect race and ethnicity data.
The federal opposition, human rights organizations, and indigenous groups, including the Native Women's Association of Canada and the Assembly of First Nations, have recommended that the government launch a national commission of inquiry into the overwhelmingly high levels of missing and murdered indigenous women and girls in Canada. Public national inquiries allow for impartial investigation into an issue of national importance. Prime Minister Stephen Harper's conservative government has repeatedly declined to act on this recommendation.
In February the federal government established an all-party committee in Canada's House of Commons to hold hearings on the issue of missing and murdered indigenous women and propose solutions to address root causes of violence. While this is an advance, it is not a substitute for a national commission of inquiry with independent powers beyond those of a parliamentary committee, Human Rights Watch said.
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Started in 2015, the Moving In New Directions (MIND) program began as a grassroots initiative by a group of residents and medical service providers at 682 Warden Avenue, a Toronto Community Housing Corp. (TCHC) building in east Toronto.
This building is home to 224 residents, a vulnerable majority of whom have been housed here following a period of homelessness and addiction issues.
The MIND program started out by running drop-ins twice a week and has since expanded to provide two meals every day, seven days a week.
The MIND program understands the harm of substance use on individuals and the community – these include the transmission of infectious diseases, malnutrition, eviction, conflict, violence, poor health, unremitting poverty, social isolation, and premature death.


Thank you for visiting my site. I have been creating art since I was a child and began painting with gusto in the late 70's. I paint what I see, urban and rural landscapes, portraits of friends and people on the street. I see in blocks of color with details that catch my eye. Body language is an important element in defining my portraits; the way my characters dress and interact with each other, I hope gives the viewer a glimpse into who these people may be.
I don't put much detail in my characters' faces so the observer is drawn into the portrait and mentally fills it in with someone they know.
I lean toward clean and simple lines with detail pulled to create texture. In my early career I was influenced by Alex Katz and Milton Avery's use of spare space and bold shapes. The Japanese aesthetic has been a deep influence on my work, especially the spare, clean lines of their landscapes. I hope that my work expresses, for the viewer, a moment of peace and calm in our over stressed environment.


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Cyber-psi: A Potential for Exploration
Geplaatst op 28 februari, 2015 door Deb Snavely
Psychic Connections
It is said that religion is humanity's attempt to determine a pattern in the randomness of the universe as they experience it. Certainly my own path to Paganism is solidly based on my experiences, which included events for which I had no better label than psychic: telepathic communication with another human, empathic communication with another human, and empathic communication with non-human living beings. One of my great validators, when exploring Pagan belief systems, was discovering that others experienced such events, shared them without prejudice, and held working world-views that included patterns where such events were possible. Given the frequency of the feeling of "coming home" expressed by my Pagan correspondents both online and off, I'm not unique in this regard.
Such psychic events are among the strongest type of emotional connection we feel with others and with our world. That sense of connectedness underlies much of neopagan and earth-based religions, and is often experienced in more than logical ways; our detractors don't call us "tree-huggers" unjustly. In my pre-Pagan days I felt vivid psychic connections: the music-hunger of a grove of redwoods, the emotional turmoil of a sibling's puberty, the voice of a lover declaring his love in so many words—while his lips remained unmoving. Unsurprising to the Pagan community, such experiences are, at best, classed with ghost stories and other presumed fictions in the mundane world we also inhabit.
Viewpoint from center stage (stone wall in foreground bounds the orchestra pit) at the Forest Theater,
Armstrong Redwoods State Park, Guerneville, California, USA.
It was here that my favorite redwood grove pressured me for a musical encore after I tried the acoustics of the Forest Theater during a wintertime visit.
Accepting such psychic—or parapsychic or paranormal—events is nothing new in the neoPagan subculture. I still recall my sense of discovery, of understanding when I first read P. E. I. Bonewits' carefully expounded "switchboard" hypothesis describing the as-yet-undiscovered frequency of electromagnetic or related energies that carry the information we call psychic or paranormal.[1]
My dual studies in Pagan practice and scientific paranormal research have reinforced my view that Bonewits' hypothesis provides a sound if still intuitive basis for the workings of psi phenomena that has yet to be bettered. But I've read little yet, in either Pagan or scientific venues, that discuss the potential of "cyberspace" for exploring the subtler types of psychic connection. In the flurry of connectivity and international networking that characterizes the Net, I find this gap astonishing.
Such online psychic experiments as can be found on the Internet today are largely confined to prediction and intention experiments.[2] However, in an example run by the University of Amsterdam in the early 1990s, one first expended some effort to intuit an image, then described one's image by writing a short essay and answering a series of true–false statements about the image, and finally got to view a photograph (the experimental "target") for comparison, along with a numerical score. To me, this cyber-linked example of remote viewing barely scratches the surface of the technological potential for psychic events.
Body English and Other Red Herrings
Common explanations for empathic experiences lean heavily on the non-verbal communication at which the human animal, like all animals, excels. Those subtle cues collectively called "body English," both facial and structural, can communicate considerable information about the emotional state of someone who speaks not a word. Not only does such unspoken information communicate, it can be communicated unknowingly along a chain of people. In an example of my personal knowledge that dates from 1959, an engineer in a major research firm entered an open-plan office full of other engineers, and found the atmosphere so tangibly bleak that she asked the department secretary what was up. The other woman agreed that things felt dismal but could not account for it. The following day several hundred people were laid off abruptly, in a surprise management decision.
Typical example of a late 1950s open-plan office workspace
In a post-mortem discussion, the two women hypothesized that body English had somehow communicated the impending bad news on an emotional level, producing a negative atmosphere without any concrete information being passed. Although one or both women may have had a psychic (empathic) experience, the body-English explanation is at least equally likely, and is less upsetting to people afraid of psi. As Dr. Charles Tart (academic parapsychological researcher) described more than a decade ago,
Unacknowledged fears of psi can create unconsciously motivated behaviors that inhibit and/or distort the operation of psi in the laboratory. Observations suggest that unacknowledged fear of psi is widespread among parapsychologists, as well as others.[3]
Preferring common-sense explanations for communication events—when circumstances support them—I also tend to assume that I've read body English rather than experienced a psychic phenomena. Even on the telephone, one has the advantage of vocal tones, hesitations, and non- words—the body English of the voice—to offset psychic explanations of otherwise inexplicable communication between individuals. When your mother calls you and you know she's got bad news from the first "Hello," is that psychic or not? Pragmatically speaking, I'd say not. You just know that voice better than you do your own.
Cyber Connections
While I'm not a formal psychic experimenter, my personal grimoire, like many of ours, is as much a magical lab notebook as anything else. I note prophetic dreams, tarot readings of especial significance, and other psychic events that appear to me to include magico–psychic information. By documenting a deja vu event the first time I dream it, I get great personal satisfaction and validation when I see it in the flesh the second time—often years later—and have written evidence when and what I saw. Such documentation is difficult when it comes to psychic events that involve real-time psychic communication. However, cyberspace has given me anecdotal evidence of such subtle events.
Although a long-time techie, I was slow arriving in cyberspace by Silicon Valley standards. I signed onto America Online in early 1993, and discovered an existing, thriving Pagan community there, even at a time when AOL subscribership was limited to 300,000 members in the U.S. and was not yet linked to the larger Internet. Tucked into a corner of the Religion-and-Ethics forum, a small multi-faceted message board overflowed its boundaries daily trying to cover the lumped "metaphysical/new age/occult/pagan" topics. The tolerant fundamentalist-Christian forum moderator had a Pagan sub-moderator who quietly encouraged our successful sub-forum along with his other duties. A weekly chat existed even then, in AOL's People Connection forum (quite unrelated to the R&E forum), entitled White Magick, that provided a real-time forum for Pagans of all stripes discuss topics to their taste… and wherein they routinely bumped noses with the ever-curious and occasionally brimstone-breathing hoi polloi. And in another quadrant of AOL cyberspace, the Tips and Advice forum featured a "Crystal Ball" board for psychic readings—which repeatedly petitioned the Pagan AOL-ers for readers, since AOL and the Crystal Ball board both grew like weeds to produce an ever-impatient backlog.
AOL sign-on screen, early 1990s.
At the time, I was practicing Wicca solely as a solitary, and had no opportunities to exercise psychic experiments in the physical world. The online Pagan community contributed to my growth enormously. I was petitioned (and succumbed) to two pleas: one to read tarot in the Crystal Ball forum, and another to facilitate a Religion-and-Ethics–based Pagan chat—which successfully spun off chat after chat due primarily to AOL's booming population and continued Pagan networking online. (By 1997, there existed an entire Pagan & Wiccan sub-region of the re-titled Religion & Beliefs Forum, with its own conference room, several libraries, and hundreds of message boards.)
For the traditional year and a day, I worked as AOL remote staff in these two capacities, interacting in real time with thousands of people at distances ranging from two to 3,500 miles. The people I "spoke" to covered the spectrum of humanity: Pagans and Christians and Hindus, educated and ignorant and studying, technologically adept and clueless, all ages and sexes and backgrounds. I made connections in all sorts of ways, ways that I'd never dreamed of.
I played at cybersex (another topic!) until interactive pornography began to pall, experiencing an online infatuation that required a physical meeting to put to rest. Before that episode, I'd been at best dubious about the concept of a psychic connection. What I recall most vividly from that time is the dearth of usable information in Pagan literature on the subject of functional psychic management. Bonewits provided me understanding but no guidance; I found some usable hints in Murry Hope[4] and borrowed snippets from a dozen other marginally useless volumes to help me break the psychic link I had inadvertently formed (that online infatuation…).
I quit working the real-time Tarot chat after the night a potential suicide was our last reading of a session. Although AOL legalities demanded I follow procedure in advising the party to get offline and call a crisis hotline, a comment I made during that exchange began an email conversation of some months. Several months later, the other party began to turn his email messages back towards the "might as well be dead." By that time, I knew him fairly well, but my heart was literally in my throat the day I tossed a virtual bucket of cold water on renewed outpourings of self-pity. Thank the gods, he took it well and positively, accepting that his own choices had as much to do with the unfortunate outcome of many past events as did any "uncaring providence."
As an AOL official host, I learned to know within seconds if a new arrival (in a chat room) represented potential trouble or was simply clueless. Evangelical witnessers really interrupt any conversation, but worse if twenty Pagans and Pagan-friendlies then feel obliged to rebut the intrusive advice all at once. The curious but clueless could be gently directed to places where their questions got answered without interrupting expensive online time (in 1994, AOL cost $4/hour after one's subscribed five monthly hours plus telephone bills—if your dial-up connection was a toll call, you could easily run up a thousand dollar total in a single month).
One night after a chat concluded, I exchanged greetings with a regular attendee who usually lurked during the discussions. For no specific reason, I found myself driven to ask whether she was all right—to discover that, while she knew she had exophthalmia ("pop-eye," the result of Graves' disease), she'd awakened the night before with one eye slid out of its socket. Reflexively, she had replaced it even as panicked emotions set in. A trip to her doctor in the morning had reassured her intellectually. Nonetheless, attending the Pagan Discussion Group that evening, she continued to be thoroughly anxious and troubled. In that moment, I stopped wondering whether cyber-empathy was just my pipe dream or wild hypothesis, and began acting on the presumption that empathy and other psychic communication can and does occur with the remote and largely impersonal link provided by the Internet.
Marty Feldman, 1934-1982, had Graves' disease, which caused his exophthalmia.
Since that time, I have successfully "sent energy" down its perhaps peculiar medium to other Pagans, without specific notice, and had at least half of them comment within ten seconds on an unusual sensation of warmth or tingling. Not only has my online experience of empathy led me to attempt this kind of magical working/psychic event, it has helped me learn techniques that work in the everyday world. I now use directed energy for healing efforts, sense the energy flows when my writing is successful, and sense unspoken information that moves in my vicinity.
As yet, I have not seen any evidence of cyber-psychic communications in the research literature. That may be due to limited research budgets, problematic experiment design, the usual infinite delays in publishing research results, or a simple failure of insight by investigators. Nonetheless, cyberspace in all its variety provides another venue for Pagans and psychics wishing to explore this largely unexplored set of magical tools.
This article was originally written in 1995 and revised in 1997. It was significantly edited and updated for the Wiccan Rede Online late in 2014.
About the author: Deb Snavely began her occult studies shortly after her college graduation: divination, healing, herbs, folklore. Initiated into the Wicca in 1995, she currently leads a coven in the south Puget Sound area of Washington state in the Pacific Northwest of the USA. Retired medically from an award-winning technical writing career, she applies her available energies to her work as a priestess and healer.
Bonewits, P.E.I., Real Magic, 3rd edition, Weiser, 1989
Hope, Murry, Practical Techniques of Psychic Self-Defense, St. Martin's Press, 1986
Charles Tart Parapsychology Blog, Library of Articles;
Institute of Noetic Sciences, Psi Arcade (ongoing online experiements);
Princeton Engineering Anomalies Research (PEAR) Laboratory;
Global Consciousness Project;
Journal of Scientific Exploration;
International Consciousness Research Laboratory;
[1] Real Magic, P. E. I. Bonewits, 3rd ed.,1989 Samuel Weiser, Inc.
[2] http://www.noetic.org/research/project/psi-arcade/
[3] "Acknowledging and Dealing with the Fear of Psi," Journal of the American Society for Psychical Research, 1984, volume 78, pp. 133–143, Charles T. Tart, (http://blog.paradigm-sys.com/articles-library/)
[4] Practical Techniques of Psychic Self-Defense, Murry Hope, St. Martin's Press, 1986
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