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

The achievement of Bose-Einstein condensation in diluted atomic vapors of
$^{87}$Rb and $^{23}$Na~\cite{CWK-02} and the impressive progress in the
experimental manipulation of cold atoms in optical lattices (see, e.g.,
Ref.~\cite{BDZ-08} and references therein) have provided a great opportunity
to investigate the interplay between quantum and statistical behaviors in
particle systems.  In these systems, phase transitions are phenomena of great
interest, see, e.g.,
Refs.~\cite{DRBOKS-07,GBMHS-02,PWMMFCSHB-04,KWW-05,HSBBD-06,FWMGB-06,CFFFI-09}.

Phase transitions related to the formation of the Bose-Einstein condensation
in interacting Bose gases at a nonzero temperature, as the one reported in
Ref.~\cite{DRBOKS-07}, are essentially driven by thermal fluctuations, giving
rise to a {\em classical\/} critical behavior, see, e.g., Ref.~\cite{PV-02}.
Quantum fluctuations play a dominant role at $T=0$ transitions, where the
low-energy properties show a {\em quantum\/} critical behavior with a peculiar
interplay between quantum and thermal fluctuations at low $T$, see, e.g.,
Ref.~\cite{Sachdev-book}.  Quantum Mott insulator to superfluid transitions
have been observed in experiments with ultracold atomic gases loaded in
optical
lattices~\cite{GBMHS-02,PWMMFCSHB-04,KWW-05,HSBBD-06,FWMGB-06,CFFFI-09}.

A common feature of the above-mentioned experimental realizations is the
presence of a trapping potential $V(r)$ coupled to the particle density.  For
example, atomic gases loaded in optical lattices are generally described by
the Bose-Hubbard (BH) model~\cite{JBCGZ-98}
\begin{equation}
H_{\rm BH} = -{J\over 2}\sum_{\langle ij\rangle}
(b_i^\dagger b_j + b_j^\dagger b_i)
    + \sum_i[(\mu + V(r_i))n_i + U n_i(n_i-1)],
\label{bhm}
\end{equation}
where $\langle ij\rangle$ is the set of nearest-neighbor sites and $n_i\equiv
b_i^\dagger b_i$ is the particle density operator.
Far from the origin the potential $V(r)$
diverges, therefore $\langle n_i\rangle$ vanishes and the particles are
trapped.  The inhomogeneity due to the trapping potential strongly
affects the phenomenology of quantum transitions in homogeneous systems.
Correlation functions are not expected to develop a diverging length scale in
the presence of a trap.  Therefore, a theoretical description of how critical
correlations develop in systems subjected to confining potentials is of great
importance for experimental investigations.

We consider the trapping power-law potential
\begin{equation}
V(r) = v^p |r|^p \equiv (|\vec{r}|/l)^p,
\label{potential}
\end{equation}
where $v$ and $p$ are positive constants and $l\equiv 1/v$ is the trap size,
coupled to the particle number.  Experiments are usually set up with
a harmonic potential, i.e., $p=2$.

Let us consider the case in which the system parameters are tuned to values
corresponding to the critical regime of the unconfined system.  In the
presence of a {\em confining potential}, the critical behavior of the {\em
  unconfined} homogeneous system can be observed around the middle of the trap
only in a window where the length scale $\xi$ of the critical modes is much
smaller than the trap size, but sufficiently large to show the universal
scaling behavior.  If $\xi$ is large, but not much smaller than the trap size,
the critical behavior gets somehow distorted by the trap, although it may
still show universal effects controlled by the universality class of the phase
transition of the unconfined system.  In this paper we investigate this issue
within quantum transitions.

In Ref.~\cite{CV-09} we have shown that the critical behavior of trapped
systems at {\em classical\/} continuous transitions can be cast in the form of
a trap-size scaling (TSS), resembling the finite-size scaling theory for
homogeneous systems~\cite{FB-72,GKMD-08}, but characterized by a further
nontrivial {\em trap critical exponent\/} $\theta$, which describes how the
length scale $\xi$ depends on the trap size at criticality, i.e., $\xi\sim
l^\theta$, and which can be estimated using renormalization-group (RG)
arguments.  TSS was supported by numerical results for some lattice gas
models.

In the present paper, we extend the study of the effects of trapping
potentials to {\em quantum\/} critical behaviors at $T=0$ transitions.  We put
forward a TSS scenario to describe how critical correlations develop in large
traps, analogous to the one outlined in Ref.~\cite{CV-09} for {\em classical}
continuous transitions. We then check the validity of the TSS scenario within
quantum transitions by determining the trap-size dependence in a few quantum
models.  As a theoretical laboratory, we consider a quantum XY chain in an
external space-dependent transverse field, acting as a trap for the spinless
fermions of its quadratic Hamiltonian representation.  We present exact and
very accurate numerical results which fully support TSS.  Moreover, we discuss
TSS at the Mott insulator to superfluid transition in the BH model.  In
particular, we present analytical results for the low-density Mott
transition in the hard-core limit of the one-dimensional BH
model.  The results again support TSS, although the corresponding TSS
functions show peculiar behaviors, like discontinuities in the scaling
particle density, which are related to the quantum nature of the transition.

The paper is organized as follows.  In Sec.~\ref{trapss} we outline the main
ideas of the TSS theory.  In Sec.~\ref{XY} we study the XY chain in a
space-dependent transverse field, which gives rise to an inhomogeneity
analogous to the one arising from a trapping potential in particle systems.
We show how TSS emerges by analytical and very accurate numerical results.  In
Sec.~\ref{BHmodel} we consider the BH model (\ref{bhm}) in a confining
potential. We discuss the general features of the Mott insulator to superfluid
transitions, and present the RG arguments to derive the corresponding trap
exponent $\theta$.  We then present results for the TSS at the low-density
Mott transition in the one-dimensional hard-core BH
model.  Finally, in Sec.~\ref{conclusions} we draw our conclusions.

\section{Trap-size scaling at quantum transitions}
\label{trapss}

Quantum phase transitions arise from a nonanaliticity of the ground-state
energy, where the gap $\Delta=E_1-E_0$ vanishes.  In addition,
continuous quantum transitions have a diverging length scale $\xi$, which
gives rise to peculiar scaling properties, see, e.g., Ref.~\cite{Sachdev-book}.

In a standard general scenario, a quantum $T=0$ transition of a homogeneous
$d$-dimensional system is generally characterized by one relevant parameter
$\mu$, with critical value $\mu_c$ and RG dimension $y_\mu\equiv 1/\nu$, and a
given dynamic critical exponent $z$.  Interesting physical examples are the
Mott insulator to superfluid transitions in BH models.  The quantum critical
behavior is characterized by a vanishing energy scale, for example the gap
$\Delta$, and a diverging length scale $\xi$, behaving respectively as
$\Delta\sim |\bar{\mu}|^{z\nu}$ and $\xi\sim |\bar{\mu}|^{-\nu}$, where
$\bar{\mu}=\mu-\mu_c$, thus $\Delta\sim \xi^{-z}$.

The confining potential gives rise to a space inhomogeneity, thus changing the
scaling behavior of the homogeneous system, which could be only recovered in
the limit of large trap size (keeping fixed the other parameters of the
system).  We want to describe how the critical correlations develop in a
trap. Our starting point is a scaling Ansatz which extends the scaling laws of
homogeneous systems at quantum transitions, to allow for the presence of a
confining potential like (\ref{potential}). We write the scaling law of the
singular part of the free energy density
at the quantum transition as
\begin{equation}
F(\mu,T,v,x) = b^{-(d+z)} F(\bar{\mu} b^{y_\mu},T b^z, v b^{y_v},x/b),
\label{sfreee}
\end{equation}
where $b$ is any positive number, $y_v$ is the RG dimension of the trap
parameter $v$, cf.\ Eq.~(\ref{potential}), and $x$ is the distance from the
middle of the trap. We are neglecting irrelevant scaling fields, because they
do not affect the asymptotic behaviors.  Then, fixing $ v b^{y_v}=1$ and
introducing the trap size $l=v^{-1}$, we obtain the following trap-size
scaling (TSS)
\begin{equation}
F = l^{-\theta (d+z)} {\cal F}(\bar{\mu} l^{\theta/\nu},
Tl^{\theta z},xl^{-\theta}),
\label{freee}
\end{equation}
where $\nu\equiv 1/y_\mu$ and $\theta \equiv 1/y_v$ is the {\em trap
  exponent}.  $\theta$ depends on the universality class of the quantum
transition. It can be computed by evaluating the RG dimensions of the
corresponding perturbation.

Analogously, one can derive the TSS of other observables.  Any
low-energy scale at $T=0$, and in particular the gap, is expected to
behave as
\begin{equation}
\Delta = l^{-\theta z} {\cal D}(\bar{\mu} l^{\theta/\nu}),
\label{Deltasca}
\end{equation}
with ${\cal D}(y)\sim y^{z\nu}$ for $y\to 0$ to match the scaling $\Delta\sim
\bar{\mu}^{z\nu}$ in the absence of the trap.  The correlation length $\xi$
around the middle of the trap, or any generic length scale associated with the
critical modes, behaves as
\begin{eqnarray}
\xi = l^{\theta} {\cal X}(\bar{\mu} l^{\theta/\nu},Tl^{\theta z}),
\label{xisca}
\end{eqnarray}
where ${\cal X}(y,0)\sim y^{-\nu}$ for $y\to 0$.  This implies that at the
$T=0$ quantum critical point the trap induces a finite length scale: $\xi\sim
l^{\theta}$.  In the case of a
generic local operator $O(x)$, with RG dimension $y_o$, we expect that
its expectation value and equal-time correlator behave as
\begin{eqnarray}
&&\langle O(x) \rangle  = l^{-\theta y_o}
{\cal O}(\bar{\mu} l^{\theta/\nu},Tl^{\theta z},xl^{-\theta}),
\label{psca}\\
&&\langle O(x) O(0) \rangle_c
= l^{- 2 \theta y_o} {\cal G}_O(\bar{\mu} l^{\theta/\nu},
Tl^{\theta z},xl^{-\theta}),
\label{gpsca}
\end{eqnarray}
where $x$ measures the distance from the origin, i.e., the middle of the trap.
In the above scaling formulae we have neglected scaling
corrections due to irrelevant perturbations, and possible analytic
contributions.

When $p\to \infty$ the effect of the trapping potential is equivalent to
confine a homogeneous system in a box of size $L=2l$ with open boundary
conditions, whose behavior can be described by a standard finite-size
scaling~\cite{FB-72,PV-02}. Therefore we must have that $\theta\to 1$ when
$p\to \infty$.

The TSS theory outlined in this section provides an effective theoretical
framework to describe quantum critical behaviors in confined systems.
However, it is important to check the validity of its scaling Ansatz, because
the quantum nature of the phenomenon may lead to subtle effects.  For this
purpose, in the following sections, we study the trap-size dependence at some
specific quantum transitions of one-dimensional systems in the presence of a
space-dependent confining potential.

\section{Trap-size scaling in the quantum XY chain}
\label{XY}

The quantum XY chain in a transverse field is a standard theoretical
laboratory for issues related to quantum transitions, see, e.g.,
Ref.~\cite{Sachdev-book}. In this model, an inhomogeneity analogous to
the one arising from a trapping potential in particle systems can be
achieved by considering a space-dependent transverse field, i.e.,
\begin{eqnarray}
H_{\rm XY} = -  \sum_i {1\over 2} [ (1+\gamma) \sigma^x_i \sigma^x_{i+1}
+ (1-\gamma) \sigma^y_i \sigma^y_{i+1}]
- \mu \sigma^z_i  - V(x_i) \sigma^z_i,
\label{Isc}
\end{eqnarray}
where $\sigma^i$ are the Pauli matrices, $0<\gamma\le1$ and $V(x) = v^p |x|^p
\equiv (|x|/l)^p$,  where $v$ and $p$ are positive constants and $l = v^{-1}$
is its length scale.  This model can be mapped into a quadratic Hamiltonian of
spinless fermions by a Jordan-Wigner transformation,
\begin{eqnarray}
&&H =
\sum [ c_{i}^\dagger A_{ij} c_{j}  + {1\over 2}
(c_{i}^\dagger B_{ij} c_{j}^\dagger + {\rm H.c.})], \nonumber \\
&&A_{ij} = 2 \delta_{ij} - \delta_{i+1,j} - \delta_{i,j+1}
+ 2 Q(x_i) \delta_{ij}, \nonumber \\
&&B_{ij} = - \gamma \left( \delta_{i+1,j} - \delta_{i,j+1} \right),
\label{sfi} \\
&&Q(x)= \bar{\mu}+V(x),\quad \bar{\mu}\equiv \mu-1.
\nonumber
\end{eqnarray}
In this picture $\mu$ plays the role of {\em chemical potential} for the
$c$-{\em particles}, and the space-dependent field $V(x)$ acts as a {\em trap}
for the $c$-particles, making their {\em local density} $\langle
n_i\rangle\equiv \langle c^\dagger_i c_i \rangle$ vanish at large distance.
In the following, $l\equiv v^{-1}$ will be considered the {\em trap size}.

In the absence of the trap, the model undergoes a quantum transition at
$\bar{\mu}=0$ in the two-dimensional Ising universality class, separating a
quantum paramagnetic phase for $\bar{\mu}>0$ from a quantum ferromagnetic
phase for $\bar{\mu}<0$.  Around $\bar{\mu}=0$, the quantum critical behavior
shows a diverging length scale, $\xi\sim |\bar{\mu}|^{-\nu}$, and a vanishing
energy scale, $\Delta\sim \xi^{-z}$, where $z$ and $\nu$ are universal
critical exponents: $z=1$ and $\nu\equiv 1/y_\mu=1$ ($y_\mu$ is the RG
dimension of $\mu$).

In the presence of the confining potential, the critical behavior can be
observed only in the limit of large trap size. The trap exponent $\theta$
controlling such limit can be determined by analyzing the RG properties of the
corresponding perturbation at the critical point, which can be represented by
$P_V = \int d^d x\,V(x) \phi(x)^2$, where $\phi(x)$ is the order-parameter
field of the $\phi^4$ theory which describes the behavior of the critical
modes~\cite{CV-09,PV-02}.  Since the RG dimensions of the potential $V(x)$ and
the energy operator $\phi^2$, respectively $y_V=py_v-p$ and
$y_{\phi^2}=d+z-y_\mu$, are related by $y_V + y_{\phi^2} = d+z$, we obtain $p
y_v - p = y_\mu$, and therefore, since $z=1$ and $y_\mu=1$,
\begin{equation}
\theta \equiv 1/y_v =  p/(p+1).
\label{theta}
\end{equation}
Notice that $\theta\to 1$ when $p\to\infty$; this was to be expected, since
when $p\to \infty$ the system becomes equivalent to a homogeneous chain with
$-l\le x \le l$ and open boundary conditions.

We mention that some issues related to the presence of a space-dependent
transverse field in the quantum XY chain model have been addressed in
Refs.~\cite{PKT-07,ZD-08,EIP-09,CKT-09}.

\subsection{The trap-size scaling limit}
\label{tsslim}

The Hamiltonian (\ref{sfi}) can be solved by exact numerical diagonalization,
even in the presence of the trapping potential, following, for example,
Ref.~\cite{LSM-61}.  We look for new canonical fermionic variables
$\eta_k = g_{ki} c_i^\dagger+ h_{ki} c_i$ which diagonalize the
Hamiltonian, i.e., such that
\begin{eqnarray}
H = \sum_k \omega_k \eta_k^\dagger \eta_k,\qquad \omega_k\ge 0,
\label{diagH}
\end{eqnarray}
and the ground state satisfies $\eta_k |0\rangle =0$ for any $k$.  The
diagonalization is achieved by introducing two sets of orthonormal vectors
$\phi_k$ and $\psi_k$ with components $\phi_{ki}=g_{ki} + h_{ki}$ and
$\psi_{ki}=g_{ki}-h_{ki}$ respectively, satisfying the equations
\begin{equation}
(A+B)\phi_k = \omega_k \psi_k, \qquad
(A-B)\psi_k = \omega_k \phi_k,
\label{abpp}
\end{equation}
where the matrices $A$ and $B$ have been defined in Eq.~(\ref{sfi}).
Thus, we have
\begin{equation}
(A-B)(A+B) \phi_k = \omega_k^2 \phi_k.
\label{phieq}
\end{equation}
Once the vectors $\phi_k$ have been computed, one can easily determine
$\psi_k$ and reconstruct the original $c$-operators as
\begin{equation}
c_i = {1\over 2} \sum_k (\phi_{ki} + \psi_{ki}) \eta_k +
(\phi_{ki} - \psi_{ki}) \eta_k^\dagger,
\label{cieta}
\end{equation}
from which one can compute the correlation functions of the $c$-operators, by
straightforward calculations.

In order to show how TSS emerges, we consider the continuum limit of
Eq.~(\ref{phieq}).  A similar approach was developed in Ref.~\cite{PKT-07} to
study the critical behavior of the quantum XY chain in the presence of a
gradient perturbation.  By rewriting the discrete differences in terms of
derivatives, near the critical point $\bar{\mu}=0$ and for sufficiently small
values of $k$ (this is required by the smoothness hypothesis underlying the
continuum limit), we obtain
\begin{equation}
\left[  4 Q(x)^2 + 4 \gamma \partial_x Q(x)  -
4 \gamma^2 \partial_x^2 - 4 Q(x)\partial_x^2
- 2 \partial_x Q(x) \partial_x - 2 \partial_x^2 Q(x) + ...
\right] \phi_k(x)   =\omega_k^2 \phi_k(x).
\label{schreq}
\end{equation}
This equation has a nontrivial TSS limit: by rescaling
\begin{eqnarray}
&&x = \gamma^{1/(1+p)} l^{p/(1+p)} X, \nonumber\\
&&\bar{\mu} = \gamma^{p/(1+p)} l^{-p/(1+p)} \mu_r, \label{isresc}\\
&&\omega_k = 2 \gamma^{p/(1+p)} l^{-p/(1+p)} \Omega_{k},
\nonumber
\end{eqnarray}
and keeping only the leading terms in the large-$l$ limit (and for small
$\omega_k$), we obtain
\begin{equation}
\left( \mu_r + X^p - \partial_{X} \right)
\left( \mu_r + X^p + \partial_{X} \right) \phi_k(X) = \Omega_{k}^2 \phi_k(X).
\label{trapscaleq}
\end{equation}
Analogously, we obtain the equation for the function $\psi_k(x)$, i.e.,
\begin{equation}
\left( \mu_r + X^p - \partial_X \right)
\psi_k(X) = \Omega_k \phi_k(X).
\label{trapscaleq2}
\end{equation}
It is worth noting that the dependence on $\gamma$ in Eqs.~(\ref{trapscaleq})
and (\ref{trapscaleq2}) disappears, showing the universality of TSS with
respect to changes of the values of $\gamma$ (of course, excluding the value
$\gamma=0$).  One can easily check that the next-to-leading terms in the
large-trap limit give rise to $O(l^{-\theta})$ scaling corrections.
Corrections due to irrelevant perturbations already present in the homogeneous
system are expected to be more suppressed: the exponent associated with the
leading scaling corrections in generic systems belonging to the
two-dimensional Ising universality class is $\omega=2$~\cite{CCCPV-00,PV-02},
thus leading to $O(l^{-2\theta})$ corrections in TSS.

The solution of Eqs.~(\ref{trapscaleq}) and (\ref{trapscaleq2}) allows us to
determine the critical correlations of the original $c$-operators, using
Eq.~(\ref{cieta}).  One can easily determine the large-distance behavior of
the functions $\phi_k(x)$ and $\psi_k(x)$: they decay exponentially as
$\sim\exp(- a |x|^{p+1})$ apart from prefactors.  Their
large-distance decay determines the large-distance behavior of the scaling
correlation functions of the $c$ operators.

A natural question concerns the robustness of the TSS results with respect to
variation of the trapping potential from a simple power law.  For example, let
us consider a potential given by
\begin{equation}
U(x) = (|x|/l)^p + b (|x|/l)^q, \qquad q>p,
\label{vpq}
\end{equation}
where $b>0$ is a constant. A simple analysis based on the rescalings
(\ref{isresc}) shows that the TSS limit is determined by the
smallest power law, while the higher power gives rise to
$O(l^{-(q-p)/(1+p)})$ scaling corrections.

\subsection{Trap-size scaling of observables}
\label{tsso}

In the following we report results obtained by exact diagonalization
at fixed trap size, and compare them with the TSS derived in
Sec.~\ref{trapss}.  The exact numerical diagonalization in the
presence of the trap is done on a chain of length $L$ with open
boundary conditions, with $L$ large enough to have negligible
finite-size effects.

\subsubsection{Trap-size scaling of low-energy scales}
\label{gap}

\begin{figure}[tbhp]
\begin{center}
\leavevmode
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYD,p=2.eps}
\hbox to \hsize {\hss}
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYD,p=4.eps}
\caption{(Color online) TSS of the energy differences $\Delta_1\equiv
  E_1-E_0$ and
  $\Delta_2\equiv E_2-E_0$ for $p=2$ (above) and $p=4$ (below) vs.\
  $\mu_r\equiv \gamma^{-\theta} l^{\theta} \bar{\mu}$, for several values of
  $\gamma$ and for $l\ge10$.  Numerical diagonalization results clearly
  approach universal TSS functions in the large-$l$ limit (represented by full
  lines and obtained by extrapolations), with $O(l^{-\theta})$ scaling
  corrections (larger at small $\gamma$ and for higher levels).  }
\label{XYTSS}
\end{center}
\end{figure}

The universal TSS limit obtained after the rescalings (\ref{isresc}) implies
the following asymptotic behavior for any low-energy scale:
\begin{equation}
\Delta \approx \gamma^{\theta} l^{-\theta} {\cal D}(\mu_r),
\quad
\mu_r \equiv \gamma^{-\theta} l^{\theta} \bar{\mu},
\label{deltaisi}
\end{equation}
which is approached with $O(l^{-\theta})$ scaling corrections.  This proves
the scaling behavior (\ref{Deltasca}) obtained by RG arguments.

In Fig.~\ref{XYTSS} we plot the differences among the first few energy levels,
$\Delta_1=E_1-E_0$ and $\Delta_2=E_2-E_0$, obtained by numerical
diagonalization, for trapping potentials with $p=2$ and $p=4$ and for
several values of $\mu$ and $\gamma$.  The numerical results
clearly approaches a universal TSS behavior in the large-$l$ limit, as
predicted by Eq.~(\ref{deltaisi}).  The large-$|\mu_r|$ behavior of the
scaling functions ${\cal D}_i$ related to $\Delta_i$ are ${\cal
  D}_1\sim2\mu_r$ for $\mu_r\to+\infty$, ${\cal D}_1\to0$ for
$\mu_r\to-\infty$, and ${\cal D}_2\sim2|\mu_r|$ for $\mu_r\to\pm\infty$; they
are consistent with the behaviors of the corresponding quantities in the
absence of the trap.  The vanishing of ${\cal D}_1$ for $\mu_r\to -\infty$ is
related to the degeneracy of the ground state in the quantum ferromagnetic
phase.

\subsubsection{Trap-size scaling of the particle density and its correlators}
\label{density}

Other interesting quantities are the particle density and its correlation
function. At the quantum critical point, the particle density is expected to
behave analogously to the energy density in the standard Ising
model~\cite{CV-09}, which presents leading analytic contributions arising from
the analytic part of the free energy.  At $\bar{\mu}=0$ and in the middle of
the trap, we expect
\begin{equation}
\langle n_0 \rangle \equiv  \langle c_0^\dagger c_0 \rangle =
\rho_c(\gamma) + a_n \gamma^{\theta/p} l^{-\theta} + \dots,
\label{nobeh}
\end{equation}
where we used the fact that the RG dimension of the particle density operator
is $y_n=d+z-y_\mu=1$, and the rescalings (\ref{isresc}) to guess the
dependence on $\gamma$ of the amplitude of the scaling term.  $a_n$ is a
constant that does not depend on $\gamma$.  $\rho_c(\gamma)$ is the critical
value of the homogeneous system without trap, for example $\rho_c(1) =
0.18169011...$~\cite{Pfeuty}.  Our numerical results for several values of
$\gamma$ support the scaling behavior (\ref{nobeh}).  Scaling corrections are
(consistent with) integer powers of $l^{-\theta}$.

We have also studied the dependence on the distance from the middle of the
trap, which must be an even analytic function of $x$ at fixed trap size, but
it may present a nontrivial scaling for large $l$.  Analogously to the scaling
at the origin, we expect it to be given by a sum of an analytic behavior and a
nonanalytic term in the trap size, such as
\begin{equation}
\langle n_x - n_0 \rangle  =
\rho_{\rm ns}(x/l) + a_n l^{-\theta}{\cal R}(xl^{-\theta}),
\label{rhoscal}
\end{equation}
where $\rho_{\rm ns}$ is a nonsingular term which we expect to be a
function of $x/l$.  Let us compare this Ansatz with the results from
exact diagonalization.  At small $x$, the latter are consistent with
the expansion
\begin{equation}
\langle n_x - n_0 \rangle \approx b_2(l) x^2 + b_4(l) x^4 + ...,
\label{nxno}
\end{equation}
with
\begin{equation}
b_2(l) = b_{21} l^{-2} \ln l  +  b_{22} l^{-2} + ...,
\qquad b_4(l) = b_{41} l^{-10/3} + ...
\label{nxres2}
\end{equation}
for $p=2$ (dots indicate terms that are more suppressed in the large-$l$ limit
by power laws), and
\begin{equation}
 b_2(l) = b_{21} l^{-12/5} +  b_{22} l^{-16/5} +...,
\qquad b_4(l) = b_{41} l^{-4}\ln l + b_{42} l^{-4} + ...
\label{nxres4}
\end{equation}
for $p=4$. These results should be compared with the expansion of
Eq.~(\ref{rhoscal}) in powers of $x$. They are in substantial agreement; the
only difference is given by the presence of logarithms.  However, this should
not be surprising, because logarithmic corrections to power laws are peculiar
of the two-dimensional Ising universality class, see, e.g., the behavior of
the specific heat, and arise~\cite{Wegner-76} from integer resonances of the
RG dimensions of different perturbations, see also Ref.~\cite{CHPV-02}. Note
that the logarithms appearing in the expansion (\ref{nxno}) show up in the
$x^2$ term for $p=2$ and in the $x^4$ term for $p=4$, where the trap-size
power-law expansions of the two terms in Eq.~(\ref{rhoscal}) present an
accidental degeneracy, just because $3\theta=2$ for $p=2$ and $5\theta=4$ for
$p=4$.  The large-distance behavior of the particle density turns out to be
dominated by the nonuniversal analytic term, indeed at large distance we find
\begin{equation}
\langle n_x \rangle  \approx {\gamma^2 \over 8}
\left(x\over l\right)^{-2p},
\label{ldist}
\end{equation}
while the scaling nonanalytic part is expected to be exponentially suppressed,
as suggested by the large-distance behavior of the solutions of
Eq.~(\ref{trapscaleq}).

\begin{figure}[tbp]
\begin{center}
\leavevmode
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYrhocc,p=2.eps}
\hbox to \hsize {\hss}
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYrhocc,p=4.eps}
\caption{(Color online) TSS of $G_n(x)$, cf.\ Eq.~(\ref{nxco}),
 at $\bar{\mu}=0$ vs.\ $X\equiv \gamma^{-\theta/p}
  l^{-\theta} x$, for $p=2$ (above) and
  $p=4$ (below).  Numerical diagonalization results clearly
  approach universal TSS functions in the large-$l$ limit (represented by full
  lines and obtained by extrapolations).  }
\label{TSSGn}
\end{center}
\end{figure}

The static particle-density correlator is not affected by analytic
backgrounds; therefore, according to Eq.~(\ref{gpsca}), we expect that at
$\bar{\mu}=0$ and for $x\ne0$
\begin{equation}
G_n(x) \equiv
    \langle n_0 n_x\rangle - \langle n_0\rangle \langle n_x\rangle
    \approx  \gamma^{2\theta/p} l^{-2\theta}{\cal G}_n(X),
\label{nxco}
\end{equation}
where ${\cal G}_n(X)$ is a universal function.  This is confirmed by the
results of numerical diagonalization, as shown in Fig.~\ref{TSSGn} for
$p=2$ and $p=4$.  At small $X$, ${\cal G}_n(X)\sim 1/X^2$, which is the
behavior in the absence of trap, while at large $X$ it decays very rapidly.

\begin{figure}[tbp]
\begin{center}
\leavevmode
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYrhocc,p=inf.eps}
\caption{(Color online)
The scaling function ${\cal G}_n(X)\equiv l^{2\theta} G_n(x)$
vs.\ $X\equiv x/l^{\theta}$ in the limit $p\to\infty$
where $\theta\to 1$, cf.\ Eq.~(\ref{cfgn}).
}
\label{TSSGninf}
\end{center}
\end{figure}

In the limit $p\to\infty$ the system becomes equivalent to a homogeneous
finite-size chain with $-l\le x\le l$ and open boundary conditions.  Thus, in
this limit we can exploit conformal field theory (CFT), by mapping the
half-plane result for the two-dimensional Ising universality class
\cite{Cardy-84} into the strip, obtaining the correlation
function~\cite{PC-pri}
\begin{equation}
G_n(x) = {\pi^2\over 4l^2} {\cos(\pi X/2) \over \sin^2(\pi X/2)},
\qquad X=x/l.
\label{cfgn}
\end{equation}
Since $\theta\to 1$ for $p\to\infty$, this result agrees with TSS.  Of
course, the corresponding scaling function ${\cal G}_n(X)\equiv l^2G_n(x)$ is
defined for $|X|\le 1$, and vanishes for $X=\pm 1$. It is shown in
Fig.~\ref{TSSGninf}.

\subsubsection{Trap-size scaling of two-point correlation function}
\label{corrfunc}

We have also computed the static correlation function of the spin operator
$\sigma^x$ at $T=0$ and $\mu_c$, following essentially the method of
Ref.~\cite{YR-96}.  According to Eq.~(\ref{gpsca}), $G_s(x)$
behaves as
\begin{equation}
G_s(x) \equiv \langle \sigma^x_0 \sigma^x_x \rangle =
a_s l^{-\theta\eta}{\cal G}_s(X),
\label{sxco}
\end{equation}
where we used the fact that the RG dimension of the spin operator $\sigma^x$
is $\eta/2$ with $\eta=1/4$. The constant $a_s$ is not universal, it is
expected to depend on $\gamma$.  Our results from numerical diagonalization
are perfectly consistent with Eq.~(\ref{sxco}): we find that, introducing a
$\gamma$- (and $p$-) dependent normalization $c$, determined
phenomenologically, the combination $c l^{\theta\eta} G_s(x)$ plotted vs.\
$X=\gamma^{-1/(1+p)} l^{-\theta}x$, for different values of $x$ (not too
small, e.g., $x>4$), $l$ (large), and $\gamma$, falls on a single curve; see
Fig.\ \ref{fig:XYcxxscal}.
\begin{figure}[tbp]
\begin{center}
\leavevmode
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYcxx,p=2.eps}
\hbox to \hsize {\hss}
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYcxx,p=4.eps}
\caption{(Color online) TSS of the two-point function $G_s(x)$,
  cf.\ Eq.~(\ref{sxco}), for $p=2$ (above) and $p=4$ (below).
  The universal TSS functions are represented by full lines and
  obtained by extrapolations in the large-$l$ limit.}
\label{fig:XYcxxscal}
\end{center}
\end{figure}
${\cal G}_s(X)$ diverges as $X^{-1/4}$ for
$X\to 0$ (corresponding to the behavior in the absence of the trap) and decay
rapidly at large $X$, as $\exp(-ax^{(p+1)})$, in agreement with the large
distance behavior of the solutions of Eq.~(\ref{trapscaleq}).

One may also consider the susceptibility and second moment correlation length
defined as
\begin{eqnarray}
&&\chi = \int dr \,G_s(r) \approx c_\chi l^{(d-\eta)\theta}+B
=a_\chi l^{3\theta/4}+B,
\label{chis}\\
&&\xi^2 = {1\over 2\chi} \int dr\,r^2\,G_s(r)
\approx a_{\xi^2} l^{2\theta},
\label{xi2}
\end{eqnarray}
where $B$ is a background constant term already present in the homogeneous
system without trap~\cite{PV-02}.
Since the normalization of $G_s(x)$ cancels out, $\xi$ must scale as
$x$, i.e.,
\begin{equation}
\xi = a_\xi \gamma^{\theta/p}  l^{\theta} [ 1 + O(l^{-\theta})],
\label{xissca}
\end{equation}
with $a_\xi$ dependent on $p$ but not on $\gamma$.  The above scaling
behaviors have been verified by numerical computations.

Finally, the $p=\infty$ limit of the correlation function (\ref{sxco}) can be
computed using CFT, analogously to the density-density correlation function
(\ref{cfgn}), obtaining~\cite{PC-pri}
\begin{eqnarray}
&&G_s(x) =  \left[ {\pi^2\over l^2 16 Y^2 \sin^2 (\pi X/4)}\right]^{1/8}
{\sqrt{1+Y}-\sqrt{1-Y}\over \sqrt{2}},
\nonumber\\
&&Y = \sqrt{1 - \tan^2( \pi X/4)},\qquad X = x/l.
\label{gspinfty}
\end{eqnarray}
The corresponding scaling curve ${\cal G}_s(X) \equiv l^{1/4} G_s(x)$ is
plotted in Fig.~\ref{fig:XYcxxscalinf}.
\begin{figure}[tbp]
\begin{center}
\leavevmode
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYcxx,p=inf.eps}
\caption{(Color online) The scaling function
   ${\cal G}_s(X)\equiv l^{\theta\eta} G_s(x)$
   vs.\ $X\equiv x/l^{\theta}$ in the limit $p\to\infty$
   where $\theta\to 1$, cf.\ Eq.~(\ref{gspinfty}).}
\label{fig:XYcxxscalinf}
\end{center}
\end{figure}

\subsubsection{Trap-size scaling of quantum entanglement}
\label{entang}

We now discuss quantum entanglement in the presence of the confining
potential.

We divide the chain in two parts of length $l_A$ and $L-l_A$. At $\bar{\mu}=0$,
in the absence of the trap and for open boundary conditions, the entanglement
entropy is given by~\cite{CC-04}
\begin{equation}
S(l_A;L)\approx {c\over 6} \ln \left[L
\sin( \pi l_A/L)\right] + E
\label{ccfo}
\end{equation}
where $c=1/2$ is the central charge.  The constant $E$ depends on $\gamma$:
\begin{equation}
E = {a\over 2} + {1\over 12} \ln (2\gamma/\pi)
\label{bcoe}
\end{equation}
where $a=0.478558...$~\cite{CCD-07}.

In the presence of a trapping potential, and for sufficiently large lattice
sizes $L$, we obtain
\begin{equation}
S(L/2;L)\approx {c\over 6} \ln \xi_e  + E,
\label{ccfot}
\end{equation}
which defines an entanglement length $\xi_e$ at the critical point. Note that
$E$ is the same constant appearing in Eq.~(\ref{bcoe}).  The ratio between
$\xi_e$ and the correlation length $\xi$ defined from the correlation function
of $\sigma^x$, cf.\ Eqs.~(\ref{xi2}) and (\ref{xissca}), should be
universal in the large-$l$ limit, thus independent of $\gamma$.  This
implies that  at criticality $\xi_e$ has the asymptotic behavior
\begin{equation}
\xi_e = a_e \gamma^{\theta/p} l^{\theta} [1+ O(l^{-\theta})],
\label{xie}
\end{equation}
with $a_e$ dependent on $p$ but not on $\gamma$.

The above scenario is fully supported by numerical computations, using the
technique of Refs.\ \cite{EIP-09,Peschel}.

\subsection{Trap-size dependence for $\bm{\mu<1}$}
\label{mumin}

Finally, it is worth discussing the trap-size dependence at values of $\mu<1$,
i.e., $\bar{\mu}<0$, within the quantum ordered phase.  In this case the TSS
theory outlined in Secs.~\ref{trapss} and \ref{tsslim} does not apply, because
the system is not critical in the middle of the trap.  Around the middle of
the trap we have a quantum ferromagnetic phase, up to the point
$\bar{\mu}+V(x)=0$, i.e., up to $|x_c|=(-\bar{\mu})^{1/p} l$, around which
critical fluctuations develop.  Then, as before, at larger distances the
system is in the quantum paramagnetic phase and the particle density vanishes
for $|x|\to \infty$.

In Fig.~\ref{deltamin1} we plot the differences of the lowest energy
levels, i.e., $\Delta_1=E_1-E_0$ and $\Delta_2=E_2-E_0$, for a few
values of $\mu<1$ for the harmonic potential, i.e., $p=2$.  $\Delta_1$
vanishes rapidly, apparently as $\Delta_1 \sim \exp(- \kappa l)$,
where the constant $\kappa$ depends on $\mu$.  This behavior is
clearly related to the degeneracy of the ground state in the quantum
ferromagnetic phase in the absence of trapping potential.

It is interesting to note that also $\Delta_2$ appears to vanish in the
large-$l$ limit, although at a much slower rate.  Indeed, our data for $\mu$
not too close to the critical value $\mu=1$ are consistent with the power-law
behavior $\Delta_2\sim l^{-1/2}$.  The vanishing of $\Delta_2$ should be
related to the developing of critical fluctuations at the points $x_c=\pm
(-\bar{\mu})^{1/p} l$ where the sum $\bar{\mu}+V(x)$ vanishes.  Since around
$x_c$ the potential $V(x)=|x|^p/l^p$ can be approximate by a linear potential,
i.e.,
\begin{equation}
V(x)\approx u (x-x_c), \qquad u = p (-\bar{\mu})^{1/p}/l,
\label{vlin}
\end{equation}
the exponent $1/2$ should be related to the RG scaling dimensions of the
parameter $u$ of the linear potential.  Using RG arguments analogous to those
leading to Eq.~(\ref{theta}), we obtain $y_u=2$, see also Ref.~\cite{PKT-07}.
This explanation is further supported by the fact that the data for $\Delta_2$
for different values of $\mu<1$ appear to merge to a unique curve as a
function of $u$ for sufficiently large trap size $l$, as shown in
Fig.~\ref{deltamin1} ($\mu=0.99$ clearly displays crossover effects because it
is very close to the critical value $\mu=1$).

\begin{figure}[tbp]
\begin{center}
\leavevmode
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYD1.eps}
\hbox to \hsize {\hss}
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XYD2.eps}
\vskip1mm
\caption{(Color online)
  Trap-size dependence at $\mu<1$, i.e., in the quantum ferromagnetic phase, of
  the difference of the lowest energy levels for $\mu<1$ and $p=2$:
  $\Delta_1\equiv E_1-E_0$ vs.\ $l$ (above) and $\Delta_2\equiv E_2-E_0$
  vs.\ $u^{1/2}\equiv 2^{1/2}(-\bar{\mu})^{1/4} l^{-1/2}$ (below).
}
\label{deltamin1}
\end{center}
\end{figure}

\section{Trap-size scaling in the Bose-Hubbard model}
\label{BHmodel}

\subsection{General features}
\label{gf}

We now discuss TSS within the Bose-Hubbard (BH) model (\ref{bhm}) at the Mott
insulator to superfluid transitions. A review of results for the BH model can
be found in Ref.~\cite{Sachdev-book}.  In the homogeneous BH model without
trap, the low-energy properties of the transitions driven by the chemical
potential $\mu$ are described by a {\em nonrelativistic\/} U(1)-symmetric
bosonic field theory~\cite{FWGF-89}, whose partition function is given by
\begin{eqnarray}
&&Z = \int [D\phi] \exp \left( - \int_0^{1/T} dt \, d^dx
\,{\cal L}_c \right) ,
\\
&&{\cal L}_c = \phi^* \partial_t \phi + {1\over 2 m}
|\nabla \phi|^2 + r |\phi|^2  + u |\phi|^4,
\label{lb}
\end{eqnarray}
where $r \sim \mu-\mu_c$.  The upper critical dimension of this bosonic theory
is $d=2$. Thus its critical behavior is mean field for $d>2$.  For $d=2$ the
field theory is essentially free (apart from logarithmic corrections), thus
the dynamic critical exponent is $z=2$ and the RG dimension of the
coupling $\mu$ is $y_\mu = 2$.  In $d=1$ the theory turns out to be equivalent
to a free field theory of nonrelativistic spinless fermions, from which one
infers the RG exponents $z=2$ and $y_\mu=2$.

The special transitions at fixed integer density (i.e., fixed $\mu$) belong to
a different universality class, described by a {\em relativistic}
U(1)-symmetric bosonic field theory~\cite{FWGF-89},
whose Lagrangian is
\begin{equation}
{\cal L}_c = |\partial_t \phi|^2 +  v^2
|\nabla \phi|^2 + r |\phi|^2  + u |\phi|^4,
\label{lbo2}
\end{equation}
for which $z=1$ and $y_\mu = 1/\nu_{XY}$ where $\nu_{XY}$ is the correlation
length exponent of the $D=d+1$ XY universality class. Thus $\nu_{XY}=1/2$ for
$d=3$ (i.e., mean-field behavior apart from logarithms), $\nu_{XY}=0.6717(1)$
in the $d=2$ case~\cite{CHPV-06}, and formally $\nu=\infty$ for the
Kosterlitz-Thouless transition~\cite{KT-73} at $d=1$.

In the presence of a confining potential, theoretical and experimental results
have shown the coexistence of Mott insulator and superfluid regions when
varying the total occupancy of the lattice, see, e.g.,
Refs.~\cite{JBCGZ-98,BRSRMDT-02,WATB-04,RM-04,GKTWB-06,FWMGB-06}.  However, at
fixed trap size, the system does not develop a critical behavior with
diverging length scale~\cite{BRSRMDT-02,WATB-04}. Criticality is recovered
only in the limit of large trap size, where the critical behavior can be
described in the framework of the TSS theory.  The trap exponent $\theta$ can
be determined by an analysis of the corresponding RG perturbation, i.e.,
\begin{equation}
\int d^dx\,dt\,V(x) |\phi(x)|^2,
\label{rgpert}
\end{equation}
leading to the relation
\begin{equation}
p y_v - p =  d+z-y_{|\phi|^2} = y_\mu,
\label{yU}
\end{equation}
where $y_\mu$ is the RG dimension of the coupling $r$ of the quadratic term of
Lagrangian (\ref{lb}) or (\ref{lbo2}). Therefore,
\begin{equation}
\theta\equiv {1\over y_v} = {p\over p+y_\mu}.
\label{thetaHB}
\end{equation}
By replacing the corresponding value of $y_\mu$, this relation yields the
value of $\theta$ for each specific transition.

\subsection{TSS at the low-density Mott transition in the
1D hard-core Bose-Hubbard model}
\label{tsshc1dbh}

We now show how TSS emerges at the low-density Mott transition
of the one-dimensional BH model, which is also of experimental relevance in
optical lattices, see, e.g., Refs.~\cite{BDZ-08,PWMMFCSHB-04,KWW-05,CFFFI-09}.
In particular, we consider its hard-core limit $U\to\infty$, which implies
that the particle number is restricted to the values $n_i=0,1$.  In this
limit, analytical results can be obtained by exploiting exact mappings into
the so-called XX chain model in the presence of a space-dependent external
field,
\begin{eqnarray}
H_{\rm XX} = - {1\over 2}\sum_i \left( \sigma^x_i \sigma^x_{i+1}
+ \sigma^y_i \sigma^y_{i+1} \right)
- \mu \sum_i \sigma^z_i  - \sum_i V(x_i) \sigma^z_i
\label{XXH}
\end{eqnarray}
(which is the Hamiltonian (\ref{Isc}) with $\gamma=0$).  The Pauli spin
matrices are related to the boson operators of the Bose-Hubbard Hamiltonian
$H_{\rm BH}$, cf.\ Eq.~(\ref{bhm}), by $\sigma^x_i = b_i^\dagger + b_i$,
$\sigma^y_i = i(b_i^\dagger - b_i)$, $\sigma^z_i = 1-2b_i^\dagger b_i$.
Actually we have that $H_{\rm XX}= 2 H_{\rm BH}(J=1,U\to\infty)$.  One can
then map it into a model of free spinless fermions by a Jordan-Wigner
transformation, given by Eq.~(\ref{sfi}) for $\gamma=0$, see, e.g.,
Ref.~\cite{Sachdev-book}.

In the absence of the trap, the 1D hard-core BH model has three phases: two
Mott insulator phases, for $\mu>1$ with $\langle n_i\rangle=0$ and for
$\mu<-1$ with $\langle n_i\rangle=1$, separated by a gapless superfluid phase
for $|\mu|<1$.  Therefore, there are two quantum transitions at $\mu=\pm 1$,
with $z=2$ and $y_\mu=1/\nu=2$. In the following we consider the low-density
Mott transition at $\mu=1$, for which we can
analytically show the existence of the TSS limit, and compute the
corresponding scaling functions. The results provide a further analytical
check of the TSS theory in quantum transitions, in a case with dynamic
exponent $z\ne 1$, whose quantum critical behavior is not described by a
$d+1$-dimensional conformal field theory.

The trap-size dependence, and corresponding TSS, turns out to be more subtle
at the $\langle n_i\rangle=1$ Mott insulator to superfluid transition, i.e.,
at $\mu=-1$.  This is essentially due to the fact that at $\mu=-1$ there is an
infinite number of level crossings as $l\to\infty$.~\footnote{For $\mu<1$, the
  ground state contains all the $\eta$-fermions diagonalizing the Hamiltonian
  with $\omega_k<0$, cf.\ Eq.\ (\ref{hdiagXX}).  In the presence of the
  trapping potential (\ref{potential}), level crossings of the lowest states
  occur in the $\mu$-$l$ plane separating the region with $N \equiv \sum_i
  \langle b_i^\dagger b_i \rangle =n$ from $ N =n+1$.  Since, in the absence
  of the trap potential and for $\mu<1$, the ground state has a finite {\em
    density\/} $N/L>0$, for fixed $\mu<1$ the lowest states
  show an infinite number of level crossings as $l\to\infty$ (after
  $L\to\infty$) where the gap vanishes.}  Results will be presented
elsewhere.

In the fermion representation the Hamiltonian can be easily diagonalized:
introducing new canonical fermionic variables $\eta_k=\sum_i \varphi_{ki}
c_i$, $k=0$, $1$, $\dots$, where $\varphi_{ki}$ satisfies
\begin{equation}
A_{ij}\varphi_{kj} = \omega_k \varphi_{kj}
\label{xxeq}
\end{equation}
and $A_{ij}$ is the matrix defined in Eq.~(\ref{sfi}),
we obtain
\begin{equation}
H = \sum_k \omega_k \eta_k^\dagger \eta_k.
\label{hdiagXX}
\end{equation}
The ground state
contains all $\eta$-fermions with $\omega_k<0$, therefore the gap is
\begin{equation}
\Delta= \min_k |\omega_k|.
\label{deltaBH}
\end{equation}
A nontrivial TSS limit around $\bar{\mu}\equiv\mu-1=0$, i.e., at the
transition between a low-density superfluid and the empty vacuum state (named
$\langle n_i\rangle=0$ Mott phase above), is obtained (for small $|\omega_k|$)
by rescaling
\begin{equation}
x = l^{p/(2+p)} X, \quad \bar{\mu}
= l^{-2p/(2+p)} \mu_r, \quad \omega_k = l^{-2p/(2+p)} \Omega_k.
\label{rescBH}
\end{equation}
Indeed, neglecting terms which are suppressed in the large-$l$ limit,
Eq.~(\ref{xxeq}) becomes
\begin{equation}
\left(2 X^p - \partial_X^2\right)\varphi_k(X) =
(\Omega_k - 2\mu_r) \varphi_k(X).
\label{trapscaleqxx}
\end{equation}
This shows that $\theta=p/(2+p)$, in agreement with Eq.~(\ref{thetaHB}).

Moreover, this implies that any energy scale, and in particular the gap
$\Delta=E_1-E_0$, must behave as
\begin{equation}
\Delta \approx l^{-2\theta} {\cal D}(\mu_r),
\qquad \mu_r=l^{2\theta} \bar{\mu},
\label{deltaxxmu1}
\end{equation}
which agrees with the RG scaling equation (\ref{Deltasca}), since $z=2$ and
$\nu=1/2$. For $p=2$, by solving Eq.~(\ref{trapscaleqxx}), we obtain
\begin{equation}
{\cal D}(\mu_r)= \min_k| 2^{3/2} (k + 1/2) + 2\mu_r|,\quad
k=0,1,\dots;
\label{dmur}
\end{equation}
${\cal D}(\mu_r)$ is a triangle wave for $\mu_r\le0$ and it is linear for
$\mu_r\ge -1/\sqrt{2}$.

The TSS of the particle density in the middle of the trap is obtained by
computing
\begin{equation}
\langle n_0 \rangle = l^{-\theta} \sum_k |\varphi_k(0)|^2 \langle
\eta^\dagger_k \eta_k\rangle,
\label{n0exp}
\end{equation}
where $\varphi_k(X)$ are the normalized
eigenfunctions of Eq.~(\ref{trapscaleqxx}); $\langle \eta^\dagger_k
\eta_k\rangle=1$ if $\Omega_k<0$ and 0 otherwise; since
$\varphi_k(X)=(-1)^k\varphi_k(-X)$, only even $k$s contribute.  For $p=2$ we
obtain the sum
\begin{equation}
l^\theta \langle n_0\rangle \equiv
(2^{1/4}/\sqrt{\pi}) \sum {[(2j-1)!!]^2/(2j)!}
\label{n0xx}
\end{equation}
over integer $j\ge 0$ satisfying $2^{1/2}(2j+1/2) + \mu_r <0$.  Again, this
result agrees with the TSS theory, taking into account that the RG dimension
of the particle density is $y_n=d+z-y_\mu=1$.  Results from numerical
diagonalization at $p=2$ are shown in Fig.~\ref{XXfig}; they fully support the
above TSS behaviors.  Note the peculiar plateaux and the discontinuities in
the particle density at negative values of the scaling variable $\mu_r\equiv
l^{2\theta} \bar{\mu}$.  For $\mu_r\to-\infty$, $\langle n_0\rangle \approx
\sqrt{|2\bar\mu|}/\pi$, which matches the critical behavior for $\bar{\mu}<0$
in the absence of the trap~\cite{Sachdev-book}.
\begin{figure}[tbp]
\begin{center}
\leavevmode
\includegraphics*[scale=\twocolumn@sw{0.3}{0.45}]{XXDrho0,p=2.eps}
\caption{(Color online)
  TSS of the gap (below) and the particle density in the middle of the
  trap (above) for $p=2$ and several values of $l\ge10$ and $\bar\mu$.
  The   lines correspond to Eqs.~(\ref{dmur}) and (\ref{n0xx}).
  Scaling corrections turn out to be very small. }
\label{XXfig}
\end{center}
\end{figure}

Numerical results for higher power laws of the potential turn out to be in
full agreement with the predictions of TSS and are qualitatively similar to
the results for $p=2$, including those for $p\to\infty$ which effectively
corresponds to the finite-size scaling of the homogeneous system without
confining potential.

\section{Conclusions}
\label{conclusions}

We have developed a trap-size scaling (TSS) theory for confined
particle systems at quantum transitions. We propose an extension of the
scaling laws of the homogeneous system to allow for the confining potential in
the large trap-size limit.  The quantum critical behavior of the confined
system is cast in the form of a TSS, resembling finite-size scaling
theory~\cite{FB-72,PV-02,GKMD-08}, with a nontrivial trap critical exponent
$\theta$, which describes how the length scale $\xi$ of the critical modes
diverges with increasing trap size, i.e., $\xi\sim l^{\theta}$,
at the quantum critical point. This provides
a theoretical framework to describe the quantum critical behavior of confined
systems.

We have shown by explicit analytical computation how TSS emerges in
the quantum XY chain, where an inhomogeneity analogous to the one
arising from a trapping potential in particle systems can be achieved
by considering a space-dependent transverse field.

Moreover, we have discussed this issue within the Bose-Hubbard model with a
confining potential, which is relevant for the description of cold atomic
gases in optical lattices.  In particular, we have presented some analytical
results for the low-density Mott transition in the hard-core limit of the
one-dimensional BH model. The results again support TSS, although the
corresponding TSS functions show a peculiar behavior, like discontinuities in
the scaling particle density, which are clearly related to the quantum nature
of the transition.

\medskip
Helpful discussions with P. Calabrese are gratefully acknowledged.

redpajama_set_name: RedPajamaArXiv


text:
Grand Prix mixed bag for Kerry and Craine
Figure skater Brendan Kerry
At the second of six ISU Grand Prix events in Regina, Canada, figure skaters Brendan Kerry and Kailani came away with a mixed bag of results in a topsy turvy event that included Olympic and world medallists.
Ranked 15th in the world, Kerry was aiming for a two-quad short program and three quad free program, testing his technical arsenal in Skate Canada's power-packed men's field, which including Olympic, World and European medallists.
For Kerry, the short did not go well, with the opening quad popping to a double and errors on the following quad salchow and triple axel leaving him trailing the field in 12th and well below his personal best.
In the free program Kerry had it all to do and shook off the previous day's disappointment, landing three quads and being one of only two men to do so along with winner Shoma Uno from Japan.
The back end of his free program that includes the usually reliable triple jumps mostly popped out to doubles and whilst Kerry climbed to 9th in the free, his overall result only improved to 11th.
"The men's event here in Canada was a very, very interesting one. A lot of great skates and a lot of not so great skates," Kerry said.
"Unfortunately, I went out and delivered the worst possible short program I could have. The long was also not ideal in not delivering the triples at the end of the program. However, I think that there is always something good to take away from each skate headed towards Pyeong Chang."
"For instance, it was awesome to be able to go and throw down three quads in the free skate. I made a lot of silly mistakes on the easier elements. But that's sport - everyone has their good and bad days. Onwards and upwards from here on out," he said.
The men's Skate Canada title went to Shoma Uno, silver to USA's Jason Brown and bronze to Alexander Samarin of Russia. An epic melt down in the free by Canada's Sochi silver medallist and three-time world champion Patrick Chan saw him slip from second to 4th after a technically weak final performance.
In the ladies, Kailani Craine was a surprise invitee to the prestigious Grand Prix circuit two weeks ago and came into the field ranked near the tail.
Her trademark enthusiasm and infectious performance quality saw the debut GP skater deliver a blistering short program, which included her first successful attempt at a triple/triple this season placing ninth ahead of 2016 World Junior Champion Marin Honda from Japan and two Canadians.
In an attacking free program, Craine went after the jumps but was penalised for under rotations, finishing the event a creditable 10th overall.
The three-time national champion was initially disappointed in her free program score.
"The free program skate wasn't exactly what I wanted it to be. I had higher standards and expectations for myself but at the same time I'm proud of myself and the way that I've represented my country," she said.
"I believe that I benefit so much from different experiences and emotions, and this competition will make me a much stronger skater for upcoming competitions and this Olympic Season."
"I feel really great about my season so far and I was so excited to be invited to my first Senior Grand Prix. I hit a lot of new goals that I set for this competition including hitting a triple triple combo in the second half of my short program, and also no edge calls."
"My short program was technically up to date to be competitive in this field and I needed to get the triple triple combo out there before Four Continents. I was hoping for a much higher score but regardless I have gained much more confidence with this element."
The ladies Skate Canada gold was won in emphatic style by Kaetlyn Osmond from Canada, silver to Maria Sotskova (Russia) and bronze to Ashley Wagner (USA).
Grand Prix season beckons for figure skaters
Figure skater Kailani Craine. Photo: Getty Images
Australian champions Brendan Kerry and Kailani Craine open their 2017 Grand Prix account this coming weekend at Skate Canada in Regina, Saskatchewan.
For Craine, the invitation to compete in her first Grand Prix follows her stunning victory at the Olympic qualifier in Germany last month and was a welcome surprise.
"Honestly, I am so grateful to be competing with the top ladies in the world. I'm not sure how I expect to feel when I am there since it is a new experience for me, but I really just want to show how hard I've been working, and how much I enjoy being out on the ice," Craine said from her," Los Angeles training base on the weekend.
The pressure at the top end of the Ladies event is intense with Canadian and US champions Kaetlyn Osmond and Karen Chen among a hot field with top Russians and Japanese skaters all vying for the Grand Prix podium.
The four-time national champion has bagged a gold and silver in the Challenger events this season but knows that she has stepped up the international ladder.
"This season has been going really well as I've medalled a fair few times and picked up some personal best scores."
"However, I can always improve on my skating and my goal for Skate Canada is to lay out a more technically difficult short program. For my free program I feel like I can make a personal best score as I've been working on areas that need to be improved since Nebelhorn."
"My short program I kept from last season because I feel like it deserves an Olympic moment and it highlights the happiness that I feel when I skate."
"I am completely in love with my free program (to Moulin Rouge) this season. My choreographer and I spent a lot of time making the cuts and storyline just right. I am kind of challenging something cryptic and ancient, like an awakening. This program really brings out an inner intensity and fire that I needed to feel for this season."
Brendan Kerry: Photo: Getty Images
Brendan Kerry is no stranger to the Grand Prix circuit and top echelon of the men's field. Skate Canada will be his fifth career GP in a field that includes Canadian Sochi silver medallist Patrick Chan and 2017 World silver medallist Shoma Uno from Japan.
The technical advance in quad jumps for the men since the Sochi Olympics has been profound and was highlighted again on the weekend at the first Grand Prix in Moscow when USA's Nathan Chen overcame Olympic champion Yuzuru Hanyu with two quads in the short and four in the free program. Hanyu, who won the free program portion of the event with three quads had to settle for silver to the new US men's star.
Kerry, who is coming off a successful Challenger series with two bronze medals, is upping his quad content.
"In Regina, I want to hit a two-quad short and then an overall PB score."
"As far as the level of skating that the top-ten is currently at, the only mental preparation for me is to focus on myself and my skating - not the other competitors."
"The aim is to reach the goals I've set for myself rather than focus on a finish I can't control."
Kerry finished a career-high fifteenth at the World Championships in March and is knocking on the door of the top 12.
Leaving his relatively safe character performers from last season behind, Kerry has taken a more serious-minded approach to his music choices for the Olympic season with two different choreographers in Mark Pillay for the short and Nikolai Morozov for the free.
"The inspiration behind the short to 'Everyone Wants To Rule The World' was to just go out and do something completely different, be more serious and to play to my skating strengths."
"The free program to Pink Floyd was Nikolai's idea and I love it because I feel as though it's super outside of the realm of skating. It's edgy and different," Kerry said.
Skate Canada opens with the Ladies Short Program on Friday 1pm local time (Saturday, 6am AEDT). The men's event begins at 6pm local time (Saturday, 11am AEDT). The final free programs will be competed the following day. Results can be found here
By Belinda Noonan
owia.org
Aussies book two more figure skating berths for PyeongChang
Kailani Craine (centre) wins gold at Nebelhorn Trophy and Olympic qualifier. Photo: Emjo Eyes Instagram
Australia has secured two more Figure Skating qualification spots at the 2018 PyeongChang Olympic Winter Games with an individual gold medal from Kailani Craine and a Pairs bronze medal at the Nebelhorn Trophy in Oberstdorf, Germany.
19-year-old Craine skated to gold on Saturday night, securing not only the Ladies title but an Olympic berth for Australia.
After dominating the Short Program the night before, Craine continued her strong form impressing judges with her Free Skate to 'Moulin Rouge'.
The routine featured five triple jumps, three level-four spins and level-four footwork, skills that picked up the Newcastle native 109.43 points.
A combined total of 167.45 points from both skates saw Craine take the gold by only 0.44 points.
"I felt really good in my free program today. I felt really prepared for this competition. It wasn't the best free program that I've done, but it was enough to get me first place and to qualify an Olympic spot," Craine said.
"I don't think anything really can compare with what the Olympic Games are going to be like. It's been a dream since I started skating, so I really just want to be able to skate my best, but I want to soak up every single moment."
Rounding out the podium was Matilda Algotsson of Sweden and Alexia Paganini of Switzerland who earned the silver and bronze medals respectively, as well as 2018 qualification spots for their countries.
In Friday night's Pairs event, World silver medallists Evgenia Tarasova and Vladimir Morozov of Russia took out the top spot on the podium ahead of Germany's European Champions Aliona Savchenko and Bruno Massot.
Harley Windsor and Katia Alexandrovskaya of Australia stunned the world with a performance of a lifetime to take the bronze. The Australians and four more couples earned an Olympic spot each for their country.
Skating to 'The Mask', the pair executed a perfect triple twist, side by side triple Salchow and triple toe-double toe-double toe as well a throw triple flip and Salchow.
The Junior World Champions secured a huge personal best, scoring 125.80 points for the Free Skating and 190.31 points overall, securing a ticket to the PyeongChang 2018 Winter Olympics for Australia.
"Obviously it is very exciting for us. We skated really, really well and we also qualified (an Olympic spot) for Australia. So that was a huge step for us," 20-year-old Windsor said.
With pairs from Russia, Germany, Canada and USA having already booked their spot to the Games, the Australians will be joined by new pairs from Austria, Czech Republic, Peoples Democratic Republic of Korea and Israel on the Olympic program.
Australia is now qualified in three figure skating disciplines with Men, Pairs and Ladies, also resulting in eligibility into the Olympic Team Event for the top ten nations in the world.
Australian Men's champion Brendan Kerry earned Australia the men's spot with his 15th palce at the 2017 World Championships in March.
For the Olympic Team Event, Australia would unofficially currently be ranked 12th. The top ten nations will be determined on a points basis after the Grand Prix and Junior Grand Prix series has been completed in early December.

redpajama_set_name: RedPajamaCommonCrawl


I remember that I've seen a post somewhere in the forums that explained what are the hidden values behind visible player personality (e.g. 'fairly ambitious' means he has at least X for ambition). Could someone help me to find it?
Lack of ambition i suppose which means the player isn't likely to develop as fast as he could.


text:
Q: How to loop same functionality for each table row? I want to calculate total price based on cargo price. It works for the first row. But if there have multiple rows it doesn't work . How can I loop same functionality for each row  on table? can someone please help me ?

   function calcCargo(){
            var price = parseFloat(document.querySelector('.price').value);
            var total = document.querySelector('.total');
            var cargo = document.querySelector(".cargo");
            var totalCargoAndPrice = price + parseFloat(cargo.value);
            total.value = totalCargoAndPrice;
            console.log(totalCargoAndPrice);
        }

        Price
        Quantity
        Cargo Price
        Total

          <input type="number" class="form-control price" value="15" name="price[]" />

          <input type="number" class="form-control" value="1" />

          <input type="number" class="form-control cargo" value="0" onchange="calcCargo()" />

          <input type="number" class="form-control total" disabled="disabled" id="t-{{ $item->id }}" placeholder="0" />

          <input type="number" class="form-control price" value="15" name="price[]" />

          <input type="number" class="form-control" value="1" />

          <input type="number" class="form-control cargo" value="0" onchange="calcCargo()" />

          <input type="number" class="form-control total" disabled="disabled" id="t-{{ $item->id }}" placeholder="0" />

A: Try like this:

function calcCargo(e) {
  var row = e.target.parentElement.parentElement;
  var price = parseFloat(row.querySelector('.price').value);
  var total = row.querySelector('.total');
  var cargo = row.querySelector(".cargo");
  var totalCargoAndPrice = price + parseFloat(cargo.value);
  total.value = totalCargoAndPrice;
  console.log(totalCargoAndPrice);
}

        Price
        Quantity
        Cargo Price
        Total

          <input type="number" class="form-control price" value="15" name="price[]" />

          <input type="number" class="form-control" value="1" />

          <input type="number" class="form-control cargo" value="0" onchange="calcCargo(event)" />

          <input type="number" class="form-control total" disabled="disabled" id="t-{{ $item->id }}" placeholder="0" />

          <input type="number" class="form-control price" value="15" name="price[]" />

          <input type="number" class="form-control" value="1" />

          <input type="number" class="form-control cargo" value="0" onchange="calcCargo(event)" />

          <input type="number" class="form-control total" disabled="disabled" id="t-{{ $item->id }}" placeholder="0" />

By passing an event object into the calcCargo function, you can identify which input element fired the event, so you can find that row and update only the fields inside it.

A: This does not work because document.querySelector( gets you the first found element. (Explains why its working for the first row!).
JS
function calcCargo(target){
  const row = target.closest("tr");
  var price = parseFloat(row.querySelector('.price').value);
  var total = row.querySelector('.total');
  var cargo = row.querySelector(".cargo");
  var totalCargoAndPrice = price + parseFloat(cargo.value);
  total.value = totalCargoAndPrice;
}

HTML

        Price
        Quantity
        Cargo Price
        Total

          <input type="number" class="form-control price" value="15" name="price[]" />

          <input type="number" class="form-control" value="1" />

          <input type="number" class="form-control cargo" value="0" onchange="calcCargo(this)" />

          <input type="number" class="form-control total" disabled="disabled" id="t-{{ $item->id }}" placeholder="0" />

          <input type="number" class="form-control price" value="15" name="price[]" />

          <input type="number" class="form-control" value="1" />

          <input type="number" class="form-control cargo" value="0" onchange="calcCargo(this)" />

          <input type="number" class="form-control total" disabled="disabled" id="t-{{ $item->id }}" placeholder="0" />

A: // You can do it multiple wasys, but here is one of them

// first you need to collect every table-cell that contains the vaklue you need to summarize.
const cargoItems = document.querySelectorAll('td .cargo')
// this is the way you get all values and now you need just iterate over this input nodes and get the values. And the most elegant way to do that - use Array.prototypre.reduce.call()

// it may looks quizzically, but html nodes collection has no personal reduce method, that can helps us to itertate over the data. Let's do it

// in this case every htmlNode becomes an rec - receiver, and variable that will accumulate all values is a 'acc'. 0 set as default value, that increments untill your cargo inputs ends
const summ = Array.prototypre.reduce.call(cargoItems, (acc, rec) => {
  return acc + parseFloat(rec.value)
}, 0)

// show result
console.log(summ)

redpajama_set_name: RedPajamaStackExchange


text:
L'île de North Sentinel  est l'une des îles Andaman dans le golfe du Bengale (océan Indien). Elle se situe à l'ouest de la partie sud de l'île Andaman du Sud, et a une superficie de . Officiellement, l'île est administrée par l'Inde depuis 1947 et fait partie du territoire des îles Andaman-et-Nicobar. Sa population présente depuis des millénaires, les « Sentinelles », est considérée comme une des dernières tribus sur la planète totalement coupée du monde moderne. L'île est défendue par les guerriers Sentinelles qui n'hésitent pas à tuer les intrus avec leurs flèches et leurs lances ; elle est considérée comme un territoire souverain sous sa protection par l'Inde, qui en interdit strictement l'approche depuis 1996 par la mise en place d'un cordon de sécurité, entourant de plusieurs miles le pourtour de l'île.

Géographie

Située à l'est de l'Océan Indien, l'île possède une superficie de  (environ  de long pour  de large). L'île se situe à  à la partie sud de l'île d'Amandan. Son point culminant atteint les . Elle est entourée d'une vaste barrière de corail et est en grande partie recouverte d'arbres et de divers végétaux typiques des régions tropicales.

Population

Sentinelles

L'île de  est le territoire d'une ancienne tribu autochtone, les Sentinelles, dont la population actuelle est estimée à environ  (les chiffres peuvent varier selon les sources) vivant sur l'île depuis environ . Les Sentinelles sont vraisemblablement liés aux Jarawa-Onge, avec lesquels ils auraient perdu tout contact. Ils sont parmi les derniers humains à rester pratiquement isolés de la civilisation moderne et rejettent tout contact avec le monde extérieur. Le seul contact avec les autorités indiennes a été filmé : les Sentinelles ont alors accepté de récupérer des noix de coco disposées sur la plage par plusieurs individus venus de l'extérieur, la rencontre s'est faite à distance mais sans violence apparente.

Parce qu'il n'y a jamais eu de traité avec les habitants de l'île, ni de trace écrite d'une occupation physique durant laquelle la population de l'île aurait cédé sa souveraineté, le statut de l'île n'est pas clairement défini par le droit international, elle peut être considérée comme une entité souveraine sous protection de l'Inde. Elle est, , l'une des régions autonomes de l'Inde.

Le gouvernement territorial (îles Andaman-et-Nicobar) a déclaré qu'il n'avait aucune intention d'interférer avec le mode de vie ou d'habitat des Sentinelles.

Tsunami de 2004
Bien que l'île ait sans doute gravement souffert des effets du tsunami de , les Sentinelles ont survécu. La confirmation a été obtenue quelques jours plus tard, quand un hélicoptère du gouvernement indien a observé plusieurs d'entre eux, qui ont tiré des flèches vers l'appareil afin de le repousser.

Conflits avec les braconniers
Plus d'une dizaine de pêcheurs illégaux birmans ont été arrêtés par les garde-côtes indiens dans les îles Andaman en août-. Certains pêchaient devant l'île de , près de la réserve jarawa. La tribu des Sentinelles attaque ceux qui s'approchent de son île, deux pêcheurs avaient ainsi été tués en 2006.

Mais selon l'ONG , les braconniers indiens locaux, à la différence des birmans, sont rarement inquiétés par les autorités. Ils capturent les tortues et le concombre de mer pour alimenter le marché chinois et chassent illégalement dans la forêt des Jarawa sur l'île Andaman voisine, en y pénétrant généralement par la route illégale qui traverse le territoire de la tribu.  a demandé à plusieurs reprises aux autorités locales de fermer cette route, mais sans succès (à la date du ). Des sources locales laissent entendre que la situation est bien plus grave que ne le laissent supposer les arrestations de braconniers birmans, la plupart des braconniers n'étant pas même détectés par les autorités, bien qu'ils mettent en danger les ressources en poissons et gibier vitales pour les Jarawas et les Sentinelles qui sont tous chasseurs-cueilleurs. Les braconniers sont aussi une source de microbes dangereux (grippe ou rougeole par exemple contre lesquelles les tribus autochtones ne sont pas immunisées). Les Sentinelles sont selon  particulièrement en danger car ils vivent dans un isolement total du reste du monde.

Cas de John Chau
John Allen Chau, missionnaire évangélique américain âgé de , a été encerclé et tué par les autochtones le  alors qu'il tentait d'interférer directement avec les habitants de l'île.

Après une première tentative avortée le , John Chau a soudoyé des pêcheurs qui l'ont transporté à proximité de l'île de , puis a continué son voyage seul à l'aide d'un canoë. Selon les témoins, il a reçu une volée de flèches à peine débarqué sur l'île mais a continué à marcher. Les pêcheurs ont ensuite vu les tribaux nouer une corde autour de son cou et traîner son corps jusque dans la jungle. Les pêcheurs ayant pris peur, se sont enfuis. Ces derniers sont revenus le matin suivant et ont aperçu son corps sans vie sur la plage.

Selon , John Chau avait dans l'idée d'évangéliser les Sentinelles.

Notes et références

Sources

 Présentation du peuple Jarawa par l'ONG

Liens externes
 Benjamin Hoffmann, L'île de la Sentinelle (Gallimard, roman, 2022). Le roman retrace l'histoire des relations entre les Sentinelles et le monde extérieur, depuis Marco Polo jusqu'à John Chau.
  George Weber,
  , article d'
  Fiche de renseignements sur les populations autochtones des îles Andaman, par l'administration des îles Andaman-et-Nicobar

  Administration de l'Inde les îles Andaman-et-Nicobara finalement décidé d'une politique de non-interférence

Île dans les Andaman et Nicobar
Île dans l'océan Indien
Micronation

redpajama_set_name: RedPajamaWikipedia


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text:
Disrupting the Relationship between the Anti-Violence Movement and the PIC
Carceral Feminism,Chez Rumpf,non-reformist reforms,perfect victims,prison culture,rape crisis centers
By Chez Rumpf | Chicago PIC Teaching Collective
Two weeks ago, more than 1200 people attended the National Sexual Assault Conference (NSAC) in Chicago. I attended all three days of the conference and had the privilege to co-facilitate a workshop with my friend and fellow Chicago PIC Teaching Collective member Jane Hereth. Jane and I both have worked in the sexual assault and domestic violence fields, respectively, and we saw the NSAC as an opportunity to bridge our anti-violence work with our work to disrupt the PIC. Drawing on the groundbreaking work of groups like INCITE! Women of Color Against Violence, Communities Against Rape and Abuse (CARA), and Creative Interventions, we developed a workshop to explore alternatives to the PIC for sexual assault survivors.
Overall, our workshop was very well received. It was incredibly thrilling to spend one and a half hours with a group of about 50 advocates and activists passionately critiquing the PIC. We examined the various ways the criminal legal system fails sexual assault survivors and does not ensure safety, justice, and empowerment for those who turn to it. We explored how the PIC is rooted in a deep history of oppression and systematically targets marginalized communities and individuals, while privileged groups profit. In short, we articulated how the PIC is part of the problem and how severely broken it is as a solution to violence.
Several participants shared encouraging feedback and ideas after the workshop. It was clear to me that we had touched on a topic that advocates and activists in the anti-violence movement eagerly want to engage. People seemed to be yearning for a space to have a critical conversation about the relationship between the anti-violence movement and the criminal legal system. I left the workshop feeling energized and hopeful that we've reached a historical moment where the mainstream anti-violence movement is ready to seriously face the contradiction of relying on a violent, oppressive institution as a solution to sexual violence and to engage the challenging and at times scary work of envisioning and developing alternative responses to sexual assault, such as restorative justice and community accountability practices.
Shortly after our workshop concluded, I attended a thought-provoking workshop on the Prison Rape Elimination Act (PREA). Representatives from Just Detention International and the Pennsylvania Coalition Against Rape provided a very helpful and informative workshop about the prevalence of sexual assault in jails, prisons, and youth detention centers.
The co-facilitators discussed the tremendous impact sexual assault has on incarcerated people. Incarcerated survivors typically are unable to access any resources or support to deal with the trauma they experience, and they often have regular, ongoing contact with the person or people who raped them, since perpetrators usually are other incarcerated people or prison guards. The edited volume Inside This Place, Not of It: Narratives from Women's Prisons provides a number of chilling accounts from women who were sexually assaulted by prison guards while incarcerated, as well as the retribution they faced when they sought help. In short, incarcerated survivors are incredibly vulnerable to continued sexual assault and, given their controlled living circumstances, have very limited power to protect themselves.
The workshop co-facilitators also did an excellent job of explaining how the PREA is creating an opportunity for rape crisis centers to provide counseling services to people who are sexually assaulted while incarcerated, as well as education and training to corrections staff. The staff education and training piece likely will be particularly challenging. According to a Bureau of Justice Statistics report, "Sexual Victimization Reported by Former State Prisoners, 2008," about half of the respondents who reported sexual abuse while in prison identified correctional facility staff as their perpetrators.
A main take-away from the workshop is that incarcerated people who are sexually assaulted have a similar profile as the population of survivors that rape crisis centers already are serving. The co-facilitators stated very directly that survivors of rape in prison need the same quality of care that is available to survivors in the community. In driving home this point, the co-facilitators humanized incarcerated people. Throughout the workshop, co-facilitators and attendees were not thinking of incarcerated people as a bunch of numbers or statistics. We also were not thinking of incarcerated people as "bad" and "dangerous" criminals who need to be locked up. Rather, we were thinking of incarcerated people as human beings who are at risk of violence and deserve support and care.
My hope grew. Yes! Another workshop at the NSAC that explicitly critiqued the PIC and drew attention to the violence and oppression it perpetrates!
But in addition to hopeful, I also felt wary. Something wasn't sitting right with me. Why did I feel so conflicted about rape crisis centers collaborating with corrections facilities to provide much needed education and crisis counseling? Isn't it a good thing to expand options and resources for incarcerated people? It took me a while to sort through these questions and my unexpected reaction to the workshop.
I realized that I was uncomfortable with the overall framework for the conversation. I want to be very clear that I don't think this was an intentional framing of the conversation but rather something that emerged as participants asked questions and thought through the practical challenges of providing services inside jails and prisons, such as funding restrictions that prohibit agencies from providing services to anyone convicted of a felony. Underlying almost all comments and questions was the distinction between "the good prisoner" and "the bad prisoner." The key question seemed to be: How can rape crisis centers help the "the good prisoners" (meaning those who are not "career criminals") who are raped by "the bad prisoners" (meaning those who are dangerous "predators" and target anyone who seems weak)?
This type of binary thinking – between the good/worthy/deserving individual and the bad/unworthy/underserving individual – is a hallmark of U.S. social policy. For instance, we distinguish between groups of welfare recipients. On the one hand, there are people who "deserve" non-stigmatized forms of assistance, such as Social Security and unemployment benefits, because they have "earned" it. On the other hand, we label people who receive stigmatized forms of assistance, such as TANF and food stamps, as "getting a handout from the government." Not surprisingly, these distinctions typically are made along racial and class lines, with whites and middle- and upper-class folks falling into the "deserving" category and people of color and low-income and poor folks falling into the "undeserving" category. We draw similar distinctions in our criminal legal system, with more privileged groups having access to resources that help to evade this system and less privileged groups experiencing targeting by police and disproportionate involvement at all stages of the criminal legal system.
Simply put, this type of binary thinking is dangerous. It cannot grasp the complex reality of social problems such as racism and poverty. This binary thinking pushes us to develop simplistic solutions to social problems that focus on individuals while leaving larger systems of inequality relatively intact.
In her recent book Arrested Justice: Black Women, Violence, and America's Prison Nation, Beth E. Richie explains how this binary thinking has constrained the mainstream anti-violence movement. (Quick plug: Hear Dr. Richie read portions of Arrested Justice at Women and Children First next Thursday, September 13th). She shows how the mainstream movement traded a complex analysis of gender-based violence that considers how racial and class inequality complicate this violence for the more simplistic "everywoman analysis." According to the "everywoman analysis," all women are at equal risk of becoming victims of interpersonal gender-based violence. As Richie argues, "everywoman" quickly became white, heterosexual, middle- and upper-class women in the eyes of legislators and the public. We thus have developed a very narrow mainstream understanding of what counts as gender-based violence. Subsequently, mainstream responses (i.e. calling the police and mandatory arrest policies) often do not work for victims and survivors who do not fit this narrow view. According to Richie (2012):
…the assumed race and class neutrality of gender violence – to some extent ignoring the issue of sexuality – led to the erasure from the dominant view of the victimization of lesbians, women of color in low-income communities, and other marginalized groups. State violence and harmful public policies could not fit into the everywoman analytical paradigm of the male violence that focused on individual men. The loss of focus on these groups seriously compromised the transgressive and transformative potential of the anti-violence movement's radical social critique of power, various patriarchies, economic exploitation, and heterosexism. (P.91)
As rape crisis centers take on the challenging work of providing services inside jails and prisons, we must learn from the lessons that Dr. Richie highlights. We must very carefully identify our overarching framework for anti-violence work and reclaim the movement's transgressive and transformative potential. We have to push past binary distinctions between "good prisoners" who are worthy of help and "bad prisoners" who are nothing more than "predators." We have to question what are the anti-violence movement's ultimate goals with regard to the criminal legal system. Presently, arrest, conviction, and incarceration generally are seen as desired, albeit rare, victories in the mainstream anti-violence movement. What follows from that perspective is that tougher laws are part of the solution. But we know the horrors of the PIC. We know the trauma and oppression it perpetuates. Given that reality, how can we, as anti-violence advocates and activists, in good conscience rely on the criminal legal system as a solution to sexual violence?
I think the overarching framework for the anti-violence movement must be a commitment ultimately to disrupt the relationship between the movement and the PIC. As Critical Resistance co-founder Dr. Ruth Wilson Gilmore advised during her talk last September at UIC, we must commit to "nonreformist reforms" – that is reforms that "do not expand the scope or life of the PIC in any way."
I hope the anti-violence movement can hold on to this big picture focus while developing immediate responses to incarcerated survivors of sexual assault. While partnerships between rape crisis centers and correctional facilities may provide positive interventions in the short-term, these partnerships have the potential to deepen the relationship between the anti-violence movement and the PIC in the long-term. The key is to figure out how to provide these positive interventions in the service of disrupting this relationship. Our task is to figure out not only how to help "good" people who are harmed in prisons but how to end prisons. Given the energy I felt and the critical dialogues I participated in at the NSAC, I'm hopeful that the mainstream anti-violence movement is up for this challenge.
Previous Against Innocence: Race, Gender, and the Politics of Safety Next Heteropatriarchy Kills: Challenging Gender Violence in a Prison Nation


text:
Airport Ulsan
Ulsan Airport is an important nationally operating commercial airport in the East Asian Republic of Korea. Regular scheduled flights to and from Seoul-Gimpo Airport to the capital, from where connecting flights are available then another.
URL: Ulsan (USN)
Service telephone: +82 1661-2626
Address: Ulsan Airport, 1103, Saneop-ro, Buk-gu, ULSAN 44238, Republic of Korea
Operating company: Korea Airports Corporation(KAC)
Parking: Yes, paid short and long-term parking.
Nearby cities: Ulsan (7 km), Kyŏngju (31 km), Kimhae (59 km), Kyŏngsan (61 km), Pusan (62 km), Taegu (75 km), Chinhae (80 km), Masan (83 km)
Directions by car: The airport is located about 10 km north east of the city centre.
Alternative spelling: Улсан, Ульсан, Ουλσάν, أولسان, אולסן-קוריאה הדרומית, اولسان, ウルサン, 울산광역시, उल्सान, 蔚山広域市, 蔚山, 蔚山广域市
The airport went into operation in 1970 and is located about 10 km or 7 miles north east of downtown, near the village of Songjeong-dong. It is easy to reach via highway 7.
35°35'36"N / 129°21'7"E
Pohang (KPO) 44 km
Busan - Gimhae (PUS) 59 km
Daegu (TAE) 70 km
Chinju Sacheon (HIN) 129 km
Tsushima (TSJ) 145 km
Yosu (RSU) 178 km
Samchok (SUK) 207 km
Cheongju (CJJ) 208 km
Iki (IKI) 208 km
Gwangju (KWJ) 236 km


text:
Q: Why is \W meta character not working properly in this Javascript regex Hello I'm having trouble understanding this regex.
'^.*(?=.{8,})(?=.*[a-z])(?=.*[A-Z])(?=.*[\d\W]).*$'

It should match ZXC!"#zxc123, which it does in an online JS regex tester, but when i try it in the application i am working on, it wont match that. It seems to require the letter W, which i assume comes from the non-word selector \W.
So the case is, that it works in an online tester, but not with actual JS.
I've set up an example on Fiddle, so you can see for yourself.

$(document).ready(function(){
    var inputField = $('#regex');

    inputField.on('input', function(e){
        var inputValue = $(e.currentTarget).val();

        MatchString(inputValue);
    });

    function MatchString(str){
        var matches = str.match('^.*(?=.{8,})(?=.*[a-z])(?=.*[A-Z])(?=.*[\d\W]).*$');

        if(matches){
            $('#output').text('Matches')
        }
        else{
            $('#output').text('Not matching')
        }
    }

    MatchString(inputField.val());
});

<input type="text" id="regex" value='ZXC!"#zxc!"#'/>

I've tested the regex on regex101
I really hope for a explanation, because this seems very odd to me.
Thanks in advance

A: When I removed the ' from your regex, it worked fine.
^.*(?=.{8,})(?=.*[a-z])(?=.*[A-Z])(?=.*[\d\W]).*$

A: It is because your syntax is not right for regex. In JS regex should be wrapped as /.../ where / is called regex delimiter.
Also your regex can be refactored for efficiency as this:
/^(?=.*?[a-z])(?=.*?[A-Z])(?=.*?[\d\W]).{8,}$/

Updated JSFiddle

A: Note that inside strings "\d" will be escaped to "d" and "\W" will be escaped to "W" due to backslash substitution rules for strings (see the string literal documentation on MDN).
So if you want the backslash character you need to escape it:
str.match('^.*(?=.{8,})(?=.*[a-z])(?=.*[A-Z])(?=.*[\\d\\W]).*$');

Alternatively you can use a regex literal:
str.match(/^.*(?=.{8,})(?=.*[a-z])(?=.*[A-Z])(?=.*[\d\W]).*$/);


text:
Search Statutes: 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997
Citator
Statutes, Constitution,
& Laws of Florida
Florida Statutes
Search Statutes
Florida Constitution
Legislative & Executive Branch Lobbyists
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Joint Administrative Procedures Committee (JAPC)
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Joint Legislative Budget Commission (JLBC)
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Office of Program Policy Analysis & Government Accountability (OPPAGA)
Commission on Ethics
Economic and Demographic Research (EDR)
Florida Historic Capitol
Historical Committees
Florida Legislative Committee on Intergovernmental Relations (LCIR)
Joint Legislative Committee on Everglades Oversight (JCEO)
Joint Legislative Sunset Committee (JCSC)
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Legislative Employment
Legistore
Select Year: 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997
The Florida Statutes
The 2018 Florida Statutes
Title XXVIII
NATURAL RESOURCES; CONSERVATION, RECLAMATION, AND USE Chapter 373
WATER RESOURCES View Entire Chapter
F.S. 373.1391
373.1391 Management of real property.—
(1)(a) Lands titled to the governing boards of the districts shall be managed and maintained, to the extent practicable, in such a way as to ensure a balance between public access, general public recreational purposes, and restoration and protection of their natural state and condition. Except when prohibited by a covenant or condition described in s. 373.056(2), lands owned, managed, and controlled by the district may be used for multiple purposes, including, but not limited to, agriculture, silviculture, and water supply, as well as boating and other recreational uses.
(b) Whenever practicable, such lands shall be open to the general public for recreational uses. General public recreational purposes shall include, but not be limited to, fishing, hunting, horseback riding, swimming, camping, hiking, canoeing, boating, diving, birding, sailing, jogging, and other related outdoor activities to the maximum extent possible considering the environmental sensitivity and suitability of those lands. These public lands shall be evaluated for their resource value for the purpose of establishing which parcels, in whole or in part, annually or seasonally, would be conducive to general public recreational purposes. Such findings shall be included in management plans which are developed for such public lands. These lands shall be made available to the public for these purposes, unless the district governing board can demonstrate that such activities would be incompatible with the purposes for which these lands were acquired. The department in its supervisory capacity shall ensure that the districts provide consistent levels of public access to district lands, consistent with the purposes for which the lands were acquired.
(c) In developing or reviewing land management plans when a dispute arises that has not been resolved by a water management district's final agency action, that dispute must be resolved under chapter 120.
(d) For any fee simple acquisition of a parcel which is or will be leased back for agricultural purposes, or for any acquisition of a less-than-fee interest in lands that is or will be used for agricultural purposes, the district governing board shall first consider having a soil and water conservation district created pursuant to chapter 582 manage and monitor such interest.
(2) Interests in real property acquired by the districts under this section with funds other than those appropriated under the Florida Forever Act may be used for permittable water resource development and water supply development purposes under the following conditions: the minimum flows and levels of priority water bodies on such lands have been established; the project complies with all conditions for issuance of a permit under part II of this chapter; and the project is compatible with the purposes for which the land was acquired.
(3) Each district is encouraged to use volunteers to provide land management and other services. Volunteers shall be covered by liability protection and workers' compensation in the same manner as district employees, unless waived in writing by such volunteers or unless such volunteers otherwise provide equivalent insurance.
(4) Each water management district is authorized and encouraged to enter into cooperative land management agreements with state agencies or local governments to provide for the coordinated and cost-effective management of lands to which the water management districts, the Board of Trustees of the Internal Improvement Trust Fund, or local governments hold title. Any such cooperative land management agreement must be consistent with any applicable laws governing land use, management duties, and responsibilities and procedures of each cooperating entity. Each cooperating entity is authorized to expend such funds as are made available to it for land management on any such lands included in a cooperative land management agreement.
(5) The following additional uses of lands acquired pursuant to the Florida Forever program and other state-funded land purchase programs shall be authorized, upon a finding by the governing board, if they meet the criteria specified in paragraphs (a)-(e): water resource development projects, water supply development projects, stormwater management projects, linear facilities, and sustainable agriculture and forestry. Such additional uses are authorized where:
(a) Not inconsistent with the management plan for such lands;
(b) Compatible with the natural ecosystem and resource values of such lands;
(c) The proposed use is appropriately located on such lands and where due consideration is given to the use of other available lands;
(d) The using entity reasonably compensates the titleholder for such use based upon an appropriate measure of value; and
(e) The use is consistent with the public interest.
A decision by the governing board pursuant to this subsection shall be given a presumption of correctness. Moneys received from the use of state lands pursuant to this subsection shall be returned to the lead managing agency.
(6) The districts have the authority to adopt rules that specify: allowable activities on district-owned lands; the amount of fees, licenses, or other charges for users of district-owned lands; the application and reimbursement process for payments in lieu of taxes; the use of volunteers for management activities; and the processes related to entering into or severing cooperative land management agreements. Rules promulgated pursuant to the subsection shall become effective only after submitted to the President of the Senate and Speaker of the House of Representatives for review by the Legislature not later than 30 days prior to the next regular session. In its review, the Legislature may reject, modify, or take no action relative to such rules. The districts shall conform such rules to changes made by the Legislature, or, if no action is taken, such rules shall become effective.
History.—s. 34, ch. 99-247; s. 14, ch. 2000-170; s. 16, ch. 2008-229; s. 37, ch. 2015-229.
Copyright © 1995-2019 The Florida Legislature • Privacy Statement • Contact Us


1) Heat the oil in a large saute pan over medium heat. Add the onions, garlic, and peppers saute until cooked.
2) Add white wine and reduce by half.
3) Add Tomatoes, Parsley, and Basil. Simmer for 8-10 minutes.
4) Add mussels and cover with aluminum foil, cook for 10-12 minutes.
5) Remove mussels from pan then season sauce until it reaches desired flavor. Pour sauce over mussels to serve. Sprinkle Parmesan cheese and basil to garnish.


text:
Liam Neeson: Aslan not just Jesus
Is Aslan really meant to be Christ?
Actor Liam Neeson, voice of "Aslan in the Chronicles of Narnia" film series, believes the magical lion represents more than Christ. Neeson, who was raised Catholic in Ireland, believes Aslan is all great spiritual leaders rolled into one figure. He said this at a press conference to promote latest Narnia movie, Voyage of the Dawn Treader
"Yes, Aslan symbolizes a Christ-like figure, but he also symbolizes for me Mohammed, Buddha and all the great spiritual leaders and prophets over the centuries," he said.
A key figure in C.S. Lewis' camp was quick to debate the point in the Catholic News Service story linked above.
Walter Hooper, C.S. Lewis's former secretary and a trustee of his estate, disagreed with Neeson's assessment.
"Lewis would have simply denied that," said Hooper, an American Catholic who lives in Oxford, England.
"He (Lewis) wrote that the 'whole Narnian story is about Christ,'" he told Catholic News Service in a Dec. 2 telephone interview. "It is nothing whatever to do with Islam. Lewis could not have been clearer.
What say you literary and movie fans? Comment on Neeson's reservations when it comes to Jesus. Is he just trying to be politically correct? Does he have a point?


text:
Tahoe City's Darla Sue Farnell Loses Life in Small Plane Crash
By John_Hamilton on March 4, 2014, 8:12 am Comments Off on Tahoe City's Darla Sue Farnell Loses Life in Small Plane Crash
Truckee, CA…The Placer County Sheriff's Office has notified the next-of-kin and can now release the name of the woman who died in a plane crash near Truckee today. She was Darla Sue Farnell, 55, of the Tahoe City area. The man who was critically injured is Ray Richard Farnell, 70, also of the Tahoe City area.
On March 3, 2014 at 11:00 am the Placer County Sheriff's Office received information from the Oakland Air Traffic Control Center of a possible plane crash approximately 3-4 miles south by south east of the Truckee Airport. Placer County Sheriff's Deputies, Truckee Police Department, Truckee Fire Department, North Tahoe Fire, California Highway Patrol and the Nevada County Sheriff's Office responded to the area in an attempt to locate the plane. CHP helicopter H24 and a Care Flight helicopter were also dispatched to help search.
At about 11:55 am CHP helicopter H24 located the crashed plane approximately six miles south by south east from the Truckee Airport in the Juniper Creek area of Placer County California. The single engine plane was located in a mountainous area in approximately 4-5 feet of snow, making it difficult to reach by conventional means. Snowmobile teams and two snowcats were already staged in the area and started heading to the crash site from the east and west sides of the mountain.
At 12:50 am snowmobile search teams reached the crash site discovering a male survivor and a deceased female believed to be the only two on board the plane. Team members provided medical aid to the critically injured male until a snowcat with paramedics arrived. The male was extricated from the crash and transported by snowcat to an awaiting ambulance. He was then taken to a local area hospital. The current condition of the injured person is not known.
Placer County Sheriff's investigators and Tahoe Nordic Search and Rescue members are still at the scene of the crash to recover the decedent. Investigators from the FAA will begin investigating the crash tomorrow.
The names of the victims are not being released pending next of kin notification.
Tahoe City's Darla Sue Farnell Loses Life in Small Plane Crash added by John_Hamilton on March 4, 2014, 8:12 am