text:
\section{Introduction}

The second $0^+$ state of $^{12}$C which is located near the
 3$\alpha$ breakup threshold and is called the Hoyle
 state~\cite{hoyle} is well known as one of the mysterious
 $0^+$ states in light nuclei.
The understanding of its structure
 has been actually one of the most difficult and challenging
 problems of nuclear structure.
Its small excitation energy of
 7.66 MeV is clearly not easy to explain by the shell model and,
 in fact, even the most advanced modern shell model approach known as the
 no-core shell model, fails by far to reproduce
 its position~\cite{nocore}.

About 40 years ago Morinaga proposed to assign the 3$\alpha$
 linear-chain structure to this Hoyle state~\cite{morinaga}.
However the observed reduced $\alpha$ decay width of this state which is
 larger than the Wigner limit value was pointed out to be contradictory
 to the linear-chain structure which can give at most only one third
 of the Wigner limit value~\cite{suzhorike}.
The large $\alpha$-decay reduced width of the Hoyle state
 in the $^8$Be$(0_1^+)$+$\alpha$ channel was reproduced by
 a full three-body calculation with a semi-microscopic $\alpha$-$\alpha$
 interaction~\cite{hori}, namely an OCM
 ( orthogonality condition model )~\cite{saito} calculation
 for the 3$\alpha$ system.
This 3$\alpha$ calculation contradicted the
 3$\alpha$ chain structure of the Hoyle state and gave as the
 dominant component of the Hoyle state the $^8$Be$(0_1^+)$ + $\alpha$
 structure with relative {\it S} wave between $^8$Be$(0_1^+)$
 and $\alpha$.
Since $^8$Be$(0_1^+)$ consists of two $\alpha$ clusters
 weakly coupled in relative {\it S} wave, the Hoyle state was concluded to
 have a weakly coupled 3$\alpha$ structure in relative {\it S} waves with
 large spatial extent, which is a gas-like structure of $\alpha$ clusters.
A few years later, this understanding of the structure of the Hoyle
 state was reported to be reproduced by full microscopic 3$\alpha$
 calculations by two groups, namely Kamimura and his
 collaborators~\cite{kamimura} and Uegaki and his collaborators~\cite{uegaki}.
These calculations nicely reproduced not only the
 position of the Hoyle state but also other experimental properties
 including inelastic electron form factor and E0 and E2 transition
 properties.
Other states of $^{12}$C below and around the Hoyle
 state were also well described.
The microscopic 3$\alpha$ model
 treatments of $^{12}$C of Refs.~\cite{kamimura,uegaki} were
 later extended to reaction theory in Ref.~\cite{baye}.
Furthermore investigation by antisymmetrized molecular dynamics~\cite{enyo} and
 that by fermionic molecular dynamics~\cite{neff} starting from a
 realistic bare $N$-$N$ force, both of which do not assume alpha
 clustering, have now reproduced all the salient features of the Hoyle
 state~\cite{hoyle}.
Let us also mention that for a product state of only a few bosons,
 the phase fluctuates.
However, we use the word 'condensate' in the same 'loose' sense
 as nuclear physicists are used to say that 'nuclei are superfluid',
 in spite of the fact that there is not a macroscopic number of
 Cooper pairs in nucleus.

Almost 30 years after the first proposal of the $^8$Be$(0_1^+)$
+ $\alpha$ structure for the Hoyle state, this state was reconsidered
in a new light in Ref.~\cite{thsr} involving four members of the present
authors. They presented the following new model wave function
$\Phi_B (3 \alpha)$, called the THSR wave function:
\begin{eqnarray}
   \Phi_B (3 \alpha) &=& {\cal A} \{ \exp
   [-\frac{2}{B^2} (\Vec{X}_1^2 + \Vec{X}_2^2 + \Vec{X}_3^2)] \
   \phi(\alpha_1)\phi(\alpha_2)\phi(\alpha_3) \} \label{eq:THSR_cm}\\
   &=& \exp( -\frac{6}{B^2} \Vec{\xi}_3^2 ) {\cal A} \{
   \exp( -\frac{4}{3B^2} \Vec{\xi}_1^2 - \frac{1}{B^2} \Vec{\xi}_2^2 ) \
   \phi(\alpha_1)\phi(\alpha_2)\phi(\alpha_3) \}, \label{eq:THSR_jacobi}\\
   \Vec{\xi}_1 &=& \Vec{X}_1 - \frac{1}{2} ( \Vec{X}_2 + \Vec{X}_3 ), \
   \Vec{\xi}_2 = \Vec{X}_2 - \Vec{X}_3, \  \Vec{\xi}_3 = \frac{1}{3}
   ( \Vec{X}_1 + \Vec{X}_2 + \Vec{X}_3 ),\label{eq:jacobi_coordinates_12C}
\end{eqnarray}
 where $\phi(\alpha_j)$ and $\Vec{X}_j$ stand for the internal wave
 function and the center-of-mass coordinate of the $j$-th $\alpha$ cluster,
 respectively, and ${\cal A}$ is the antisymmetrizer of the nucleons.
As shown in Eq.~(\ref{eq:THSR_jacobi}), the THSR wave function can
 be regarded as expressing
 the cluster structure where a $^8$Be$(0_1^+)$-like cluster
 ${\cal A} \{ \exp( -(1/B^2) \Vec{\xi}_2^2 ) \phi(\alpha_2)\phi(\alpha_3) \}$
 and the $\alpha_1$ cluster couple via an {\it S}-wave inter-cluster wave
 function $\exp( -(4/3B^2) \Vec{\xi}_1^2 )$.
On the other hand, Equation~(\ref{eq:THSR_cm})
 shows that the THSR wave function represents the state where three
 $\alpha$ clusters occupy the same single 0{\it S}-orbit
 $\exp(-(2/B^2)\Vec{X}^2)$, namely a 3$\alpha$ condensate state which
 is the finite size counterpart of the macroscopic $\alpha$-particle
 condensation in infinite nuclear matter at low density~\cite{roepke}.
What the authors of Ref.~\cite{thsr} proposed was that the
 $^8$Be$(0_1^+)$ + $\alpha$ structure of the Hoyle state can be regarded
 as being a 3$\alpha$ condensate state and that one can expect
 in general the existence of $n \alpha$ condensate states in the
 vicinity of the $n \alpha$ breakup threshold in $4n$ self-conjugate
 nuclei~\cite{yamada04}.
It was soon discovered~\cite{funaki} that the microscopic
 $3 \alpha$ wave functions of both of Refs.~\cite{kamimura,uegaki}
 have overlaps of more than 95\% with a single THSR
 wave function with a large size parameter $B$, implying small
 overlaps between individual $\alpha$ clusters.

This above-mentioned striking fact reported in Ref.~\cite{funaki} means
without doubt that the Hoyle state structure has a strong relation
with the $\alpha$ condensation physics in dilute infinite nuclear
matter. One of the important tasks for the clarification of this
relation is to study the magnitude of the component of the ideal
Bose-Einstein condensation of structureless $\alpha$ particles
which is contained in the Hoyle state wave function.  This was
investigated by the authors of Refs.~\cite{matsumura,yamada}
 who solved the eigenvalue problem of the internal
one-particle density matrix of the Hoyle state wave functions
where the center-of-mass coordinate is eliminated.  As the
internal coordinates for calculating the internal one-particle
density matrix $\rho$ they used the Jacobi coordinates, namely
$\Vec{\xi}_1$ and $\Vec{\xi}_2$ in Eq.~(\ref{eq:jacobi_coordinates_12C}).
They obtained a maximum eigenvalue larger than $0.7$ for the normalized density
 matrix $\rho$ ( Tr $\rho = 1$ ), which means that the corresponding
 eigenfunction which is a $0S$-orbit is occupied to more than
 $70$\% by the three $\alpha$ particles.
This is a large percentage corroborating the almost ideal
 $\alpha$-particle condensation nature of the Hoyle state, that is
 the Hoyle state is describable to very good approximation
 by a product state of these bosons,
 each in the same $0S$ orbit~\cite{matsumura,yamada}.

For the discussion of the Bose-Einstein condensation of a confined
 macroscopic system, one uses in general the one-particle
 density matrix in the laboratory frame where the coordinate
 system consisting of individual particle coordinates is adopted.
However, Pethick and Pitaevskii~\cite{PP} (PP) proposed to use
 the internal one-particle density matrix by eliminating the
 center-of-mass coordinate when one discusses the condensate
 fraction of the system where only the center-of-mass degree of
 freedom is excited but all relative degrees of freedom are kept
 unchanged.
Their proposal was to point out that the conclusion
 of the paper by Wilkin et al.~\cite{wilkin} does not appropriately
 reflect the physics.
Wilkin et al. discussed the lowest excitation of
 a condensate of cold bosonic atoms with attractive interactions
 which rotates with its center of mass in a harmonic
 potential but keeps all
 the internal degrees of freedom as in its ground state
 which shows complete condensation.
It was found that the corresponding one-body density matrix
 in the laboratory frame has many eigenvalues of comparable
 size, and thus the system should be characterized as a
 fragmented condensate~\cite{wilkin}.
On the other hand PP
 claimed that if one uses an internal one-particle density
 matrix by eliminating the center-of-mass coordinate it
 should give a single eigenvalue of the order of
 the number of particles indicating the non-fragmented
 condensate character of the internal part of the system.
Unfortunately PP presented only their idea and did not
 demonstrate explicit results of the diagonalization of their
 definition of the internal one-particle density matrix.

In this paper, following the definition of PP, we construct the
 internal one-particle density matrix of a many-boson system in
 a harmonic trap and then give the explicit analytical form of
 the eigenvalues and eigenfunctions of this density matrix.
We will see that the eigenvalues are fragmented, which is
 contrary to PP's initial objective.
We will discuss that this result does not mean that
 the original idea of PP is incorrect but it means that
 the internal coordinate system which PP adopted is inadequate.
Actually when we construct the internal
 one-particle density matrix by using the internal Jacobi
 coordinates, the resulting eigenvalues show complete
 condensation.
We will give two criteria for the choice of the adequate
 coordinate systems which will give us a unique answer for the
 definition of the internal one-particle density matrix.
One criterion is that in the infinite particle number limit ( $N=\infty$ )
 the internal one-particle density matrix should have the same
 eigenvalues and eigenfunctions as those of the corresponding
 ideal Bose-Einstein  condensate in the laboratory frame.
This is in line with the general wisdom that in the thermodynamic
 limit all macroscopic quantities shall be the same,
 regardless whether considered in the laboratory frame or
 in the internal frame~\cite{blaizot-ripka}.
The other criterion is that the coordinate used in the internal one-particle density matrix should be orthogonal to
 (or maximally independent from) the remaining $( N - 2 )$
 internal coordinates, though the $( N - 2 )$ coordinates, in general, do not need to be mutually orthogonal.
This second criterion is shown to imply the first criterion.
We show that the internal Jacobi coordinate system satisfies
 these two criteria, while the internal coordinate system adopted
 by PP for the construction of the internal one-particle density
 matrix does not.
Furthermore we argue that these two criteria uniquely
 determine the internal one-particle density matrix which is the
 same as that calculated with the Jacobi coordinates.
The results of this paper justify the use of the Jacobi
coordinates in Refs.~\cite{matsumura,yamada} and, hence,
corroborate the almost ideal $\alpha$-particle condensation nature
of the Hoyle state.

The present paper is organized as follows.  In Sec.~II, we
formulate the internal one-particle density matrix with respect
to the internal wave function of a Bose-Einstein condensate with
finite particle number in a harmonic trap.  Then, the analytical
form of the eigenvalues and eigenfunctions is presented for the
internal density matrix and numerical eigenvalues are discussed.
In Sec.~III. we propose two criteria for the choice of proper
coordinate systems for internal one-particle density matrix.
Finally, the summary is given in Sec.~IV.
Appendix~\ref{app:A} serves to present the analytical solution of the eigenvalue
 problem of the density matrix and Appendix~\ref{app:B} is given for
 the explanation of the bosonic symmetry of the Jacobi-type internal
 one-particle density matrix.
The original version of this paper was given in an article on
 the arXiv~\cite{arxiv} and a short version of this paper
 is reported in a letter paper~\cite{letter} with special
 attention to the cold atom community.

\section{Internal One-Particle Density Matrix}

First we consider the one-particle density matrix in the
laboratory frame for an ideal Bose-Einstein condensate with
$N$ spinless bosons in a harmonic potential. The result is
trivial but instructive for studying the nature of the internal
one-particle density matrix, as will be discussed later.

The $N$-particle Hamiltonian in laboratory frame is presented
as
\begin{eqnarray}
 H = {\sum_{i=1}^{N} \frac{1}{2m}\Vec{p}_i^2} +
   \sum_{i=1}^N \frac{1}{2}m\omega^2 \Vec{r}_i^2.
   \label{eq:total_hamiltonian}
\end{eqnarray}
The normalized ground-state wave function of this system is expressed as
a product of identical Gaussian single-particle wave functions,
i.e.
\begin{eqnarray}
\Phi(\{\Vec{r}_i\}_{i=1}^N) &=& \prod_{i=1}^{N}
   \left( \frac{2\nu}{\pi} \right)^{3/4}
   \exp\left(-\nu\Vec{r}_i^2\right) \\
  &=& \left( \frac{2\nu}{\pi} \right)^{3N/4}
   \exp\left(-\nu \sum_{i=1}^{N} \Vec{r}_i^2 \right)
   \label{eq:total_wf}
\end{eqnarray}
where $\nu=m\omega/2\hbar$, and $\{\Vec{r}_i\}_{i=1}^N$ denotes
the set of the coordinates $\Vec{r}_i$ ($i=1\cdots,N$).
The one particle density matrix in the laboratory frame is
defined as
\begin{eqnarray}
{\rho^{(1)}_{\rm Lab}(\Vec{r}, \Vec{r}')} &=& \int d\Vec{r}_2
   d\Vec{r}_3 \cdots d\Vec{r}_N
   \Phi^*(\Vec{r},\Vec{r}_2,\Vec{r}_3,\cdots,\Vec{r}_N)
   \Phi(\Vec{r}',\Vec{r}_2,\Vec{r}_3,\cdots,\Vec{r}_N),\\
  &=& \left(\frac{2\nu}{\pi}\right)^{3/2}
   \exp\left[-\nu(\Vec{r}^2+\Vec{r}'^2)\right].
   \label{eq:one-body_density}
\end{eqnarray}
It is noted that the density matrix is independent of the number
of particles $N$ and is separable with respect to $\Vec{r}$ and
$\Vec{r}^{\prime}$.  The separability originates from the fact
that the Hamiltonian is separable with respect to different
particle operators, $\Vec{r}_i$ and $\Vec{p}_i$ ($i=1,\cdots,N$),
in Eq.~(\ref{eq:total_hamiltonian}), or equivalently from the
fact that the wave function of Eq.~(\ref{eq:total_wf}) is
separable with respect to different particle coordinates,
$\Vec{r}_i$ ($i=1,\cdots,N$).

The nature of the single particle orbits and their occupation
probabilities in the relevant system can be obtained by solving
the eigenvalue problem of the density matrix,
\begin{eqnarray}
\int \rho^{(1)}_{\rm Lab}(\Vec{r},\Vec{r}') \phi(\Vec{r}')
   d\Vec{r}' = \lambda \phi(\Vec{r}),
\end{eqnarray}
where $\phi(\Vec{r})$ is the single particle orbit, and
$\lambda$ is its occupation probability.  This equation can easily
be solved, and we find that the density matrix has only one non zero
eigenvalue $\lambda = 1$ with one eigenfunction, namely, the
zero-node $S$-wave Gaussian
$\phi(\Vec{r})=(2\nu/\pi)^{3/4}\exp(-\nu\Vec{r}^2)$
(or $0S$ harmonic oscillator wave function $\phi_{000}(\Vec{r},\nu)$,
which will be defined later) with 100~\% occupancy ($\lambda=1$).
This means that all particles are condensed in that single orbit,
i.e. an ideal Bose-Einstein condensation is realized in the
laboratory frame.  This feature is independent of the number of
particles $N$.

Next we consider the internal one-particle density matrix for
 the $N$-particle Bose-Einstein condensation in a harmonic trap
 described by the wave function Eq.~(\ref{eq:total_wf})  with the
 total Hamiltonian Eq.~(\ref{eq:total_hamiltonian}).
{\it Internal} means that the density is free from the center-of-mass
 coordinate of the system, as it must be the case, when
 dealing with, e.g., selfbound systems.
In the present paper, two kinds of internal
 coordinate sets are introduced;  1)~coordinates with respect to
 the center of mass of the total system and 2)~Jacobi coordinates.
The former set was
 first considered by Pethick and Pitaevskii~\cite{PP} to define
 the internal one-particle density matrix.  We call it
 Pethick-Pitaevskii-type (PP-type) internal one-particle density
 matrix in the present paper. For the latter set, we call it
 Jacobi-type density matrix.

\subsection{Pethick-Pitaevskii-type internal one-particle density matrix}

As already mentioned in the introduction, Pethick and Pitaevskii proposed to
 consider the internal single particle density matrix for a Bose condensed
 system when only the center-of-mass but no internal degree of freedom
 is excited.
In order to define an internal one-particle density matrix, Pethick
 and Pitaevskii adopted internal coordinates defined with respect to
 the center of mass of the total system~\cite{PP}.
The center-of-mass coordinate $\Vec{R}$ and the coordinate $\Vec{q}_i$
 of particle $i$ relative to the center of mass are given, respectively, by
\begin{eqnarray}
\Vec{R}=\frac{1}{N}\sum_{i=1}^N \Vec{r}_i, \hspace*{5mm}
   \Vec{q}_i=\Vec{r}_i-\Vec{R}. \label{eq:coordinate_cm_rel}
\end{eqnarray}
Since $\sum_{i=1}^N \Vec{q}_i = 0$,  only $N-1$ of the $\Vec{q}_i$ are
 independent.
Hereafter, as the independent internal coordinates, we adopt
 $\Vec{q}_i$ ($i=1,\cdots,N-1$).
We introduce the conjugate momenta $\Vec{\pi}_i$ ($i=1,\cdots,N-1$)
 and $\Vec{P}$ for the coordinates $\Vec{q}_i$ ($i=1,\cdots,N-1$) and $\Vec{R}$, respectively.
Then, the total Hamiltonian in Eq.~(\ref{eq:total_hamiltonian}) is rewritten as
\begin{eqnarray}
&&H = H_{\rm int} + H_{\rm cm},\label{eq:PP_Hamiltonian} \\
&&H_{\rm int}=\frac{1}{2m} \left[ {\left(\frac{N-1}{N}\right)
  \sum_{i=1}^{N-1} \Vec{\pi}_{i}^{2}} - {\frac{2}{N} \sum_{i<i'=1}^{N-1}
  \Vec{\pi}_i\cdot\Vec{\pi}_{i'}} \right]
  + m\omega^2 \left[ \sum_{i=1}^{N-1}{\Vec{q}_i}^2 +
  \sum_{i<i'=1}^{N-1}\Vec{q}_i\cdot\Vec{q}_{i'}\right],
  \label{eq:PP_internal_Hamiltonian} \\
&&H_{\rm cm}=\frac{1}{2Nm}\Vec{P}^2 + \frac{1}{2}Nm\omega^2\Vec{R}^2,
\end{eqnarray}
where $H_{\rm int}$ and $H_{\rm cm}$ denote the internal and
center-of-mass Hamiltonians, respectively.
It should be mentioned that the above Hamiltonian includes the cross terms,
 $\Vec{q}_i\cdot\Vec{q}_{i'}$ and $\Vec{\pi}_i\cdot\Vec{\pi}_{i'}$, which,
 in general, appear so far as one takes non-orthonormal coordinate
 systems~\cite{clobt1,clobt2,clobt3,clobt4}.

By using the relation
\begin{eqnarray}
 \sum_{i=1}^N \Vec{r}_i^2 = N \Vec{R}^2 + \sum_{i,i'=1}^{N-1}
  (\delta_{i,i'} + 1) \Vec{q}_i \cdot \Vec{q}_{i'},
  \label{eq:PPquadra}
\end{eqnarray}
the total wave function in Eq.~(\ref{eq:total_wf}) is expressed as
\begin{eqnarray}
&&\Phi(\{\Vec{r}_i\}_{i=1}^{N}) = \frac{1}{N^{3/2}} \times
  \Phi_{\rm int}(\{\Vec{q}_i\}_{i=1}^{N-1}) \Phi_{\rm cm}(\Vec{R}),
  \label{eq:wf_cm_rel}\\
&&\Phi_{\rm int}(\{\Vec{q}_i\}_{i=1}^{N-1}) =
  \left( \frac{N\times (2\nu)^{N-1}}{\pi^{N-1}} \right)^{3/4}
  \exp\left[-\sum_{i,i'=1}^{N-1} (\delta_{ii'}+1) \nu \Vec{q}_i
  \cdot \Vec{q}_{i'}\right],
  \label{eq:PP_wf_internal} \\
&&\Phi_{\rm cm}(\Vec{R})=\left( \frac{2N\nu}{\pi} \right)^{3/4}
  \exp(-N\nu\Vec{R}^2),
  \label{eq:PP_wf_cm}
\end{eqnarray}
where $\Phi_{\rm int}$ and $\Phi_{\rm cm}$ denote the internal and
center-of-mass wave functions, respectively. The separability of
$\Phi_{\rm int}$ and $\Phi_{\rm cm}$ comes from the fact that the
total Hamiltonian is separable with respect to the internal and
center-of-mass parts [see Eq.~(\ref{eq:PP_Hamiltonian})] or directly
from Eq.~(\ref{eq:PPquadra}) inserted
into Eq.~(\ref{eq:total_wf}).  The factor $1/N^{3/2}$ in
Eq.~(\ref{eq:wf_cm_rel}) drops out when taking into account the
Jacobian $\partial (\{ \Vec{r}_i \}_{i=1}^{N})/\partial
(\{ \Vec{q}_i \}_{i=1}^{N-1},\Vec{R})=N^3$, coming from the
coordinate transformation from the space-fixed system to the
center-of-mass system. It should be noted that both wave functions
$\Phi_{\rm int}$ and $\Phi_{\rm cm}$ are normalized and satisfy
$H_{\rm int}\Phi_{\rm int}=(3/2)(N-1)\hbar\omega\Phi_{\rm int}$
and $H_{\rm cm}\Phi_{\rm cm}=(3/2)\hbar\omega\Phi_{\rm cm}$,
respectively.

According to Pethick and Pitaevskii~\cite{PP}, the internal
one-particle density matrix is defined as,
\begin{eqnarray}
&&\rho_{\rm int, PP}^{(1)}(\Vec{q},\Vec{q}^\prime) =
  \int d\Vec{q}_2 \cdots d\Vec{q}_{N-1}~\rho_{\rm int, PP}
  (\Vec{q},\Vec{q}_2,\cdots,\Vec{q}_{N-1};\Vec{q}^\prime,
  \Vec{q}_2,\cdots,\Vec{q}_{N-1}),
  \label{eq:one_body_den_PP} \\
&&\rho_{\rm int, PP}(\{\Vec{q}_i\}_{i=1}^{N-1},
  \{\Vec{q}_i^\prime\}_{i=1}^{N-1})
  =N^3 \int d\Vec{R}~\Phi^*(\{\Vec{q}_i+\Vec{R}\}_{i=1}^{N})
  \Phi(\{\Vec{q}_i^\prime+\Vec{R}\}_{i=1}^{N}),
  \label{eq:internal_density_PP}
\end{eqnarray}
where the Jacobian factor $N^3$ is inserted in the above
equation, which is missing in Ref.~\cite{PP}. Using the wave
function $\Phi(\{\Vec{r}_i\}_{i=1}^{N})$ in
Eq.~(\ref{eq:total_wf}), the internal one-particle density
matrix (\ref{eq:one_body_den_PP}) is expressed as
\begin{eqnarray}
\rho_{\rm int,PP}^{(1)}(\Vec{q},\Vec{q}^\prime) =
  \left( \frac{N}{N-1} \right)^{3/2}
  \left( \frac{2\nu}{\pi} \right)^{3/2}
  \exp\left[-\frac{3N-2}{2(N-1)}\nu(\Vec{q}^2 +
  {\Vec{q}^\prime}^2)
  + \frac{N-2}{N-1}\nu\Vec{q}\cdot\Vec{q}^\prime \right].
  \label{eq:one_body_den_int}
\end{eqnarray}
It is noted that this density matrix depends on the number
of particles $N$ and contains the cross term
$\Vec{q}\cdot\Vec{q}^\prime$.  The origin of the cross term
comes from the nonseparability of the coordinate $\Vec{q}_1$
from the other $(N-2)$ coordinates $\Vec{q}_i$ ($i=2,\cdots,N-1$)
in the internal wave function of Eq.~(\ref{eq:PP_wf_internal}), or
equivalently from the fact that the internal Hamiltonian of
Eq.~(\ref{eq:PP_internal_Hamiltonian}) is non-separable between
the first particle operators $\Vec{\pi}_1$ and $\Vec{q}_1$ and the
other $(N-2)$ particle operators $\Vec{\pi}_i$ and $\Vec{q}_i$
($i=2,\cdots,N-1$).  The result of Eq.~(\ref{eq:one_body_den_int})
can also be verified with the use of the internal density
matrix in Eq.~(\ref{eq:internal_density_PP}) presented as
follows:
\begin{eqnarray}
\rho_{\rm int, PP}(\{\Vec{q}_i\}_{i=1}^{N-1},
  \{\Vec{q}_i^\prime\}_{i=1}^{N-1}) = \Phi_{\rm int}^*
  (\{\Vec{q}_i\}_{i=1}^{N-1}) \Phi_{\rm int}
  (\{\Vec{q}^\prime_i\}_{i=1}^{N-1}),
  \label{eq:PP_internal_density_int}
\end{eqnarray}
where $\Phi_{\rm int}^*(\{\Vec{q}_i\}_{i=1}^{N-1})$ is the
internal wave function defined in Eq.~(\ref{eq:PP_wf_internal}).

Let us discuss the nature of the internal one-particle density
 $\rho_{\rm int}^{(1)}(\Vec{q},\Vec{q}^\prime)$.
First we study the single-particle orbits and their eigenvalues
 obtained by solving the eigenvalue problem for the density matrix,
\begin{eqnarray}
\int \rho_{\rm int,PP}^{(1)}(\Vec{q},\Vec{q}^\prime)
  \phi(\Vec{q}) d\Vec{q}^\prime = \lambda \phi(\Vec{q}).
\end{eqnarray}
This equation can be solved analytically.
There are several ways to solve it.
In Appendix A we give one method.
We can also analyze the description given in Ref.~\cite{gajda}.
The single-particle orbits
 $\phi$ are expressed by the harmonic oscillator wave functions
 $\phi_{nLM}(\Vec{q},\beta_N)$
 with the orbital angular momentum $L$,
 magnetic quantum number $M$ and harmonic oscillator quanta
 $Q=2n+L$ $(n=0,1,2,\cdots)$,
\begin{eqnarray}
&&\phi_{nLM}(\Vec{q},\beta_N) = \left[\frac{(2\beta_N)^{L+3/2}2^{n+L+2}n!}
  {\sqrt{\pi}(2n+2L+1)!!} \right]^{1/2} q^L L_n^{(L+1/2)}(2\beta_N q^2)
  {\exp (-\beta_N q^2)} Y_{LM}(\hat{\Vec{q}}),  \label{eq:harmosci} \\
&&\beta_N=\sqrt{\frac{2N}{N-1}}\nu.
\end{eqnarray}
The eigenvalues or occupation probabilities $\lambda$ are given as
\begin{eqnarray}
\lambda^{(LM)}_{n,N} =
  \left[ \frac{4N}{3N-2+2\sqrt{2N(N-1)}} \right]^{3/2}
  \left[ \frac{N-2}{3N-2+2\sqrt{2N(N-1)}} \right]^{2n+L},
\end{eqnarray}
and satisfy the following completeness relation,
\begin{eqnarray}
\sum_{L=0}^{\infty} \sum_{M=-L}^{L} \sum_{n=0}^{\infty}
  \lambda^{(LM)}_{n,N} =1.
\end{eqnarray}
Then, the internal one-particle density matrix for the $N$ particles
 in Eq.~(\ref{eq:one_body_den_int}) can be presented in terms of
 the wave functions (\ref{eq:harmosci})
\begin{eqnarray}
\rho_{\rm int,PP}^{(1)}(\Vec{q},\Vec{q}^\prime)
  = \sum_{L=0}^{\infty} \sum_{M=-L}^{L}\sum_{n=0}^{\infty}\
    {| \phi_{nLM}(\Vec{q},\beta_N) \rangle}\ \lambda^{(LM)}_{n,N}\
    {\langle  \phi_{nLM}(\Vec{q}^{\prime},\beta_N)|}.
\end{eqnarray}
The occupation probability with respect to the partial wave with quantum
number $L$ is defined as
\begin{eqnarray}
\Lambda^{(L)}_N = \sum_{M=-L}^{L} \sum_{n=0}^{\infty}
  \lambda^{(LM)}_{n,N}.
  \label{eq:PP_occupation_probablity_N_L}
\end{eqnarray}
In the macroscopic limit ($N\rightarrow \infty$), the internal one-particle
density matrix $\rho_{{\rm int,PP},N=\infty}^{(1)}$,  its eigenfunctions
$\phi_{{\rm int,PP},N=\infty}$, and eigenvalues
$\lambda^{(LM)}_{n,N=\infty}$ are given by
\begin{eqnarray}
&&\rho_{{\rm int,PP},N=\infty}^{(1)}(\Vec{q},\Vec{q}^\prime) =
  \left( \frac{2\nu}{\pi} \right)^{3/2}
  \exp\left[-\frac{3}{2}\nu(\Vec{q}^2+{\Vec{q}^\prime}^2) +
  \nu\Vec{q}\cdot\Vec{q}^\prime \right],
  \label{eq:one_body_den_int_infinite} \\
&&\phi_{{\rm int,PP},N=\infty}(\Vec{q})=\phi_{nLM}(\Vec{q},\sqrt{2}\nu),
  \label{eq:one_body_den_int_infinite_eigenfunction} \\
&&\lambda^{(LM)}_{n,N=\infty}=2^3 (3-2\sqrt{2})^{2n+L+3/2},
  \label{eq:one_body_den_int_infinite_eigenvalue} \\
&&\Lambda^{(L)}_{N=\infty} = \sum_{M=-L}^{L}
  \sum_{n=0}^{\infty}\lambda^{(LM)}_{n,N=\infty} =
  (2L+1)(2-\sqrt{2})(3-2\sqrt{2})^L.
  \label{eq:eq:one_body_den_int_infinite_probability}
\end{eqnarray}
We remark that the summed eigenvalues $\Lambda^{(L)}_{N=\infty}$ still
depend on the angular momentum $L$.

The eigenvalue or occupation probability of the PP-type internal
 one-particle density matrix, $\Lambda_N^{(L)}$ of
 Eq.~(\ref{eq:PP_occupation_probablity_N_L}) depends on
 the orbital angular momentum $L$, and the particle number $N$.
In the case of $N=3$, we obtain the occupation
 probabilities, $\Lambda_3^{(S)}=0.804$ for $S$ wave,
 $\Lambda_3^{(P)}=0.173$ for $P$, $\Lambda_3^{(D)}=0.021$ for $D$,
 $\Lambda_3^{(F)}=0.002$ for $F$, and so on.
Increasing the particle number, the $S$-wave occupation probability
 is decreasing, while the higher partial-wave ones are increasing
 (see Fig.~1 in Refs.~\cite{arxiv,letter}).
These results show that the PP-type one-particle density matrix
 leads to a fragmented condensate.
The reason why the PP-type internal density matrix shows the fragmented
 condensate is due to the existence of the cross term
 $\Vec{q}\cdot\Vec{q}^{\prime}$ in Eq.~(\ref{eq:one_body_den_int}).

\subsection{Jacobi-type internal one-particle density matrix}

For the $N$-particle system, we define the $(N-1)$ internal Jacobi
coordinates $\{ \Vec{\xi}_i, (i=1, \cdots, N-1) \}$ and the
center-of-mass coordinate $\Vec{R}$ as follows:
\begin{eqnarray}
&&\Vec{\xi}_i = \Vec{r}_i -{\frac{1}{N-i}\sum_{k=i+1}^{N}\Vec{r}_k},
  \hspace*{5mm}{\rm for}~~{i=1, 2, \cdots, {N-1}}
  \label{eq:Jacobi_coordinates}\\
&&\Vec{R} = {\frac{1}{N}\sum_{i=1}^{N} \Vec{r}_i},
  \label{eq:Jacobi_coordinates_cm}
\end{eqnarray}
where $\Vec{\xi}_1$ denotes the relative coordinate between the first
particle and the remaining $(N-1)$ particles, and
other Jacobi coordinates are self-evident.  Then, the $N$-particle
Hamiltonian in Eq.~(\ref{eq:total_hamiltonian}) can be separated into
the internal and center-of-mass Hamiltonian,
\begin{eqnarray}
&&H = H_{\rm int} + H_{\rm cm},
  \label{eq:Jacobi_Hamiltonian} \\
&&H_{\rm int}= \sum_{i=1}^{N-1} \{ \frac{1}{2\mu_i} \Vec{\pi}_i^2
  + \frac{1}{2} \mu_i {\omega}^2 \Vec{\xi}_i^2 \},
  \hspace*{5mm} \mu_i = \frac{N-i}{N-i+1} m,
  \label{eq:Jacobi_internal_Hamiltonian} \\
&&H_{\rm cm}=\frac{1}{2Nm}\Vec{P}^2 + \frac{1}{2}Nm\omega^2\Vec{R}^2,
  \label{eq:Jacobi_cm_Hamiltonian}
\end{eqnarray}
where $\Vec{\pi}_i$ and $\Vec{P}$ denote the conjugate momenta
corresponding to the coordinates $\Vec{\xi}_i$ and $\Vec{R}$,
respectively.

Since the total Hamiltonian is a sum of decoupled $N$ harmonic oscillator
Hamiltonians ( for $\Vec{R}$ and $(N-1)$ internal Jacobi coordinates ), the
total wave function in Eq.~(\ref{eq:total_wf}) is expressed as
\begin{eqnarray}
\Phi(\{\Vec{r}_i\}_{i=1}^{N}) &=& \Phi_{\rm int}(\{\Vec{\xi}_i\}_{i=1}^{N-1})
  \Phi_{\rm cm}(\Vec{R}),
  \label{eq:Jacobi_wf_cm_rel} \\
\Phi_{\rm int}(\{\Vec{\xi}_i\}_{i=1}^{N-1}) &=& \prod_{i=1}^{N-1}
  \left( \frac{2\nu_i}{\pi} \right)^{3/4} \exp\left(-\nu_i\Vec{\xi}_{i}^2
    \right) \\
  &=& \left( \frac{(2\nu)^{N-1}}{N\times \pi^{N-1}} \right)^{3/4}
    \exp \{ -\sum_{i=1}^{N-1} \nu_i \Vec{\xi}_i^2 \}, \hspace*{5mm}
    \nu_i=\frac{N-i}{N-i+1}\nu,
  \label{eq:Jacobi_wf_internal} \\
\Phi_{\rm cm}(\Vec{R}) &=& \left( \frac{2N\nu}{\pi} \right)^{3/4}
  \exp(-N\nu\Vec{R}^2).
\end{eqnarray}
This expression of $\Phi(\{\Vec{r}_i\}_{i=1}^{N})$ can also be derived by
inserting the relation
\begin{eqnarray}
 \sum_{i=1}^N \Vec{r}_i^2 = N \Vec{R}^2 +
   \sum_{i=1}^{N-1} \frac{\nu_i}{\nu} \Vec{\xi}_i^2,
   \label{eq:Jacobi_quadra}
\end{eqnarray}
into the original expression given in  Eq.~(\ref{eq:total_wf}).  The total
wave function is the product of the internal and center-of-mass wave
functions, $\Phi_{\rm int}$ and $\Phi_{\rm cm}$, which are normalized and
satisfy the relations, $H_{\rm int}\Phi_{\rm int}=(3/2)(N-1)\hbar\omega
\Phi_{\rm int}$ and $H_{\rm cm}\Phi_{\rm cm}=(3/2)\hbar\omega\Phi_{\rm cm}$.
It is noted that the internal wave function is given as a product of
harmonic oscillator wave functions $\phi_{000}(\Vec{\xi}_i,\nu_i)$
$(i=1,\cdots,N-1)$.

The Jacobi-type internal density matrix writes
\begin{eqnarray}
\rho_{\rm int, J}(\{\Vec{\xi}_i\}_{i=1}^{N-1},
  \{\Vec{\xi}_i^\prime\}_{i=1}^{N-1}) = \Phi_{\rm int}^*
  (\{\Vec{\xi}_i\}_{i=1}^{N-1}) \Phi_{\rm int}
  (\{\Vec{\xi}^\prime_i\}_{i=1}^{N-1}).
\end{eqnarray}
Then, the Jacobi-type one-particle density matrix is defined with respect to
$\Vec{\xi}_1$ and $\Vec{\xi}_1^{\prime}$ as
\begin{eqnarray}
\rho_{\rm int,J}^{(1)}(\Vec{\xi},\Vec{\xi}^\prime) &=&
  \int d\Vec{\xi}_2 \cdots d\Vec{\xi}_{N-1}~\rho_{\rm int, J}(\Vec{\xi},
   \Vec{\xi}_2,\cdots,\Vec{\xi}_{N-1};\Vec{\xi}^\prime,\Vec{\xi}_2,\cdots,
   \Vec{\xi}_{N-1}),\label{eq:Jacobi_one_body_den_int_general} \\
  &=& \left( \frac{N-1}{N} \right)^{3/2} \left( \frac{2\nu}{\pi} \right)^{3/2}
   \exp\left[-\frac{N-1}{N}\nu(\Vec{\xi}^2+{\Vec{\xi}^\prime}^2)\right].
  \label{eq:Jacobi_one_body_den_int}
\end{eqnarray}
This choice of the coordinate $\Vec{\xi}_1$ for the internal density matrix
 is natural, because the single particle orbit should be defined with respect
 to the relative coordinate between one particle and the other remaining
 $N-1$ particles in the Jacobi coordinate system.
The compatibility between the bosonic symmetry of the system
 and the above definition of the internal
 one-particle density matrix, where it may seem that one special coordinate
 is singled out, is explained in Appendix~\ref{app:B}.
The eigenvalue equation of the one-particle density matrix can easily be solved
 analytically.
We find that the density matrix has only one non-zero eigenvalue $\lambda=1$
 with corresponding eigenfunction $\phi_{{\rm int,J},N}$ which is the $0S$
 harmonic oscillator wave function with 100~\% occupancy ($\lambda=1$),
\begin{eqnarray}
&&\phi_{{\rm int,J},N}(\Vec{\xi})=\delta_{L0}\delta_{M0} \phi_{000}
  (\Vec{\xi},(N-1)\nu/N),
  \label{eq:Jacobi_single_orbit}\\
&&\lambda^{(LM)}_{N}=\delta_{L0}\delta_{M0},\\
&&\Lambda^{(L)}_{N}=\sum_{M=-L}^{L} \lambda^{(LM)}=\delta_{L0}.
  \label{eq:Jacobi_occupation_prob}
\end{eqnarray}
This means that all particles are condensed in the single $0S$ particle state,
 although the size parameter in the state [Eq.~(\ref{eq:Jacobi_single_orbit})]
 depends on $N$ and is slightly different from that in the eigenfunction
 $\phi_{000}(\Vec{r},\nu)$ in laboratory frame, discussed in the beginning of
 Sec.~II.
In the macroscopic limit ($N\rightarrow \infty$), the internal
 one-particle density matrix $\rho_{{\rm int,PP},N=\infty}^{(1)}$,
 its eigenfunction $\phi_{{\rm int,PP},N=\infty}$, and eigenvalues
 $\lambda^{(LM)}_{n,N=\infty}$ are given by
\begin{eqnarray}
&&\rho_{{\rm int},J,N=\infty}^{(1)}(\Vec{\xi},\Vec{\xi}^\prime) =
  \left( \frac{2\nu}{\pi} \right)^{3/2}
  \exp\left[-\nu(\Vec{\xi}^2+{\Vec{\xi}^\prime}^2)\right],
  \label{eq:Jacobi_one_body_den_int_infinite} \\
&&\phi_{{\rm int,J},N=\infty}(\Vec{\xi}) = \delta_{L0}
  \delta_{M0} \phi_{000}(\Vec{\xi},\nu),
  \label{eq:Jacobi_one_body_den_int_infinite_wf}\\
&&\lambda^{(LM)}_{N=\infty}=\delta_{L0}\delta_{M0},
  \label{eq:Jacobi_one_body_den_int_infinite_eigenvalue}\\
&&\Lambda^{(L)}_{N=\infty}=\sum_{M=-L}^{L}\lambda^{(LM)}_{N=\infty}
  =\delta_{L0} .
\end{eqnarray}

The Jacobi-type one-particle density matrix for finite particle
 number is separable with respect to the coordinates $\Vec{\xi}$
 and $\Vec{\xi}^{\prime}$.
This leads to the only one non-zero
 eigenvalue $\lambda = 1$ which is the same as for an ideal
 Bose-Einstein condensate system in the laboratory system.
Also the internal density matrix (\ref{eq:Jacobi_one_body_den_int_infinite}) becomes
 identical to the one in the laboratory system, Eq.~(8), for $N=\infty$.
The separability of the internal density matrix comes from the
 separability of the internal wave function in
 Eq.~(\ref{eq:Jacobi_wf_internal}) with respect to different
 Jacobi coordinates, which originates from the separability
 of the internal Hamiltonian in
 Eq.~(\ref{eq:Jacobi_internal_Hamiltonian}) with respect to
 different Jacobi coordinates.
This feature has it origin in the
 fact that the Jacobi coordinates form an orthogonal coordinate
 system.

\section{Criterion for the Choice of Adequate Internal Coordinates}

\subsection{Convergence to Bose-Einstein Condensation in the Macroscopic Limit}
\label{subsec:convergence_BEC_macro}

In the previous section we learned that the outcome of the
diagonalization of the internal density matrix depends on the
choice of the internal coordinates. This is a serious problem for
treating condensates in internal self-bound systems such as
$\alpha$-particle condensates in nuclear systems or small droplets of
superfluid $^4$He, because only internal degrees of freedom are relevant
in these systems.

In order to overcome the difficulty, we gave a criterion for the choice
 of the internal coordinates:~In the macroscopic limit
 ($N\rightarrow \infty$) the internal density matrix should have the same
 eigenvalues and eigenfunctions as those of the ideal Bose-Einstein
 condensate in the laboratory frame.
This is a very physical boundary condition.
For understanding this physical condition, consideration of
 the following situation in the macroscopic limit may be helpful:~In
 the laboratory frame, the center-of-mass motion of the present
 system is described by the wave function $\Phi_{\rm cm}(\Vec{R})$ of
 Eq.~(\ref{eq:PP_wf_cm}).
In the macroscopic limit, the center-of-mass
 coordinate should be at the coordinate origin of the laboratory frame,
 because the probability of finding the center-of-mass coordinate at
 position $\Vec{R}$ is given by $|\Phi_{\rm cm}(\Vec{R})|^2 =
 (2N \nu/\pi)^{3/2} \exp( -2N \nu\Vec{R}^2 )$ which becomes a
 delta function $\delta (\Vec{R})$ in the limit of $N\rightarrow \infty$.
Thus the internal coordinate $\Vec{q}_i$ should have the same meaning
 as the position of the $i$-th particle coordinate $\Vec{r}_i$ in the
 laboratory frame.
Also the first Jacobi internal coordinate
 $\Vec{\xi}_1$ should have the same meaning as the position of the
 first particle coordinate $\Vec{r}_1$ because $\Vec{\xi}_1 =
 (N/(N-1)) \Vec{q}_1 \rightarrow \Vec{r}_1$ for $N\rightarrow \infty$.

As shown in Eqs.~(\ref{eq:one_body_den_int_infinite_eigenfunction}) and
 (\ref{eq:one_body_den_int_infinite_eigenvalue}), the PP-type internal
 density matrix in the macroscopic limit does not satisfy the condition,
 while the density matrix of the Jacobi-type fulfills this condition
 as given in Eqs.~(\ref{eq:Jacobi_one_body_den_int_infinite_wf}) and
 (\ref{eq:Jacobi_one_body_den_int_infinite_eigenvalue}).
As already mentioned,
 the reason why the PP-type internal density matrix does not satisfy the
 physical boundary condition is due to the fact that it exhibits a
 nonvanishing cross term or correlation term $\Vec{q}\cdot\Vec{q}^{\prime}$
 in the case of $N=\infty$, originating from the pseudo two-body interaction
 terms in the internal Hamiltonian.
On the other hand, in the Jacobi-type
 internal density matrix, the pseudo two-body interaction terms disappear.

These results mean that one should take internal coordinates which do
not produce any correlation in the internal one-particle density matrix
in the macroscopic limit. Otherwise, unphysical situations occur like for
the PP-type internal density matrix. One choice fulfilling the physical
condition is the one of the internal Jacobi coordinates. Of course, there
are many sets of internal coordinates which satisfy the physical
condition. For example, they may be those satisfying the following two
conditions:~(1)~The relative coordinate $\Vec{\xi}$ between one particle
and the other remaining particles should be used as the coordinate of the
internal one-particle density matrix $\rho_{\rm int}^{(1)}$, and (2)~the
internal one-particle density matrix be separable with respect to
$\Vec{\xi}$ and $\Vec{\xi}^{\prime}$ in the macroscopic limit,
$\rho_{\rm int}^{(1)}(\Vec{\xi},\Vec{\xi}^{\prime})\rightarrow
(2\nu/\pi)^{3/2}\exp[-\nu(\Vec{\xi}^2+{\Vec{\xi}^{\prime}}^2)]$ for
$N\rightarrow \infty$.
Among the coordinates satisfying the two conditions, Jacobi coordinates
 are very convenient and useful to describe the internal
 Hamiltonian and thus the internal one-particle density matrix.
We will see in the next section that this is in fact
 the only choice.

\subsection{Orthogonality of the Coordinate $\Vec{\xi}_1$ with respect to
 the $(N-2)$ Internal Coordinates}

In the previous section, a criterion was discussed for the choice of
 the adequate coordinate systems of the internal density matrix.
Here we present the other criterion that the coordinate of the
 internal one-body density matrix, $\Vec{\xi}_1$, should be orthogonal to
 the remaining $(N-2)$ internal coordinates, where the $( N - 2 )$ coordinates
 do not need to be mutually orthogonal.
This is called ``maximal independence of the coordinate $\Vec{\xi}_1$ from
 the rest of $(N-2)$ internal coordinates''.

The coordinate of the internal one-particle density matrix should be
 either the one particle coordinate measured from the total center-of-mass
 or the one particle coordinate measured from the center-of-mass of the
 other $(N-1)$ particles.
The PP-type internal one-particle density matrix
 adopts the former type coordinate $\Vec{q}_1$ while the Jacobi-type
 internal one-particle density matrix adopts the latter type coordinate
 $\Vec{\xi}_1$, coinciding in the macroscopic number.
These two types of coordinates are essentially the same
 since they are related by $\Vec{q}_1 = ((N-1)/N) \Vec{\xi}_1$.
In constructing the internal one-particle density matrix, we integrate
 the internal density matrix over the other $(N-2)$ internal coordinates.
In order for the internal one-particle density matrix to have maximum
 information on the one-particle degree-of-freedom of the system, these
 $(N-2)$ internal coordinates should be maximally independent from the
 coordinate $\Vec{q}_1$ or $\Vec{\xi}_1$.
Since this requirement seems
 natural, we will study in this subsection its consequence.
We will see below that this requirement implies the criterion proposed
 in the preceding subsection.

The requirement that the coordinate $\Vec{q}_1$ or $\Vec{\xi}_1$ is
 maximally independent from the other $(N-2)$ internal coordinates means
 mathematically that coordinate $\Vec{q}_1$ or $\Vec{\xi}_1$
 should be orthogonal to the other $(N-2)$ internal coordinates.
The Jacobi coordinates are just such coordinates, since they constitute
 an orthogonal coordinate system.
On the other hand the internal
 coordinates $\{ \Vec{q}_i, \ i= 1 \sim (N-1) \}$ do not satisfy this
 requirement since they constitute a non-orthogonal coordinate system.
For the sake of self-containedness we recall here the meaning of
 orthogonality between coordinates.
Two coordinates $\Vec{\beta}$ and
 $\Vec{\gamma}$ are defined to be mutually orthogonal when their
 expansion coefficients of linear combination with respect to the
 $N$ particle coordinates $\{ \Vec{r}_i,\ (i=1 \sim N) \}$,
 $\{ C_i(\Vec{\beta}),\ (i=1 \sim N) \}$ and $\{ C_i(\Vec{\gamma}),\ (i=1 \sim N) \}$,
\begin{eqnarray}
 \Vec{\beta} = \sum_{i=1}^N C_i(\Vec{\beta}) \Vec{r}_i, \ \
 \Vec{\gamma} = \sum_{i=1}^N C_i(\Vec{\gamma}) \Vec{r}_i,
\end{eqnarray}
are mutually orthogonal, $\sum_{i=1}^N C_i(\Vec{\beta}) C_i(\Vec{\gamma}) = 0$.
This definition can be stated as follows.
To any coordinate $\Vec{\delta}$ we associate an $N$-dimensional
 number vector $\Vec{C}(\Vec{\delta})$ like $\Vec{C}(\Vec{\beta}) =
 \{ C_i(\Vec{\beta}),\ (i=1 \sim N) \}$ for $\Vec{\beta}$ and
 $\Vec{C}(\Vec{\gamma}) = \{ C_i(\Vec{\gamma}),\ (i=1 \sim N) \}$
 for $\Vec{\gamma}$.
If $\Vec{\beta} \ne \Vec{\gamma}$,
 $\Vec{C}(\Vec{\beta}) \ne \Vec{C}(\Vec{\gamma})$.  $\Vec{\beta}$ and
 $\Vec{\gamma}$ are said to be orthogonal when
 $\Vec{C}(\Vec{\beta}) \cdot \Vec{C}(\Vec{\gamma}) = 0$.
Needless to say, the total center-of-mass coordinate $\Vec{R}$ is
 orthogonal to any kind of internal coordinates $\Vec{\delta}_{\rm int}$,
 i.e.~$\Vec{C}(\Vec{R}) \cdot \Vec{C}(\Vec{\delta}_{\rm int}) = 0$.

Let an internal coordinate system
 $\{ \Vec{\eta}_i, \ (i= 1 \sim (N-1)) \}$ be such a system satisfying
 the above-mentioned requirement.
The coordinate $\Vec{\eta}_1$ is either
 $\Vec{q}_1$ or $\Vec{\xi}_1$ and hence we here fix it as $\Vec{\eta}_1
 = \Vec{\xi}_1$ . The orthogonality of $\Vec{\xi}_1$ to the other $(N-2)$
 coordinates $\{ \Vec{\eta}_i, \ (i= 2 \sim (N-1)) \}$ means
 $\Vec{C}(\Vec{\xi}_1) \cdot \Vec{C}(\Vec{\eta}_i) = 0, \
 (i= 2 \sim (N-1))$.
Since the internal Jacobi coordinate system $\{ \Vec{\xi}_i, \ (i= 1 \sim (N-1)) \}$
 is an orthogonal coordinate system, there the relations
 $\Vec{C}(\Vec{\xi}_1) \cdot \Vec{C}(\Vec{\xi}_i) = 0, \  (i= 2 \sim (N-1))$ holds.
Therefore the subspace spanned by the $N$-dimensional vectors
 $\{ \Vec{C}(\Vec{\eta}_i), \ (i= 2 \sim (N-1)) \}$ is identical
 to the subspace spanned by $N$-dimensional vectors
 $\{ \Vec{C}(\Vec{\xi}_i), \ (i= 2 \sim (N-1)) \}$.
We, therefore, have
\begin{eqnarray}
 \Vec{C}(\Vec{\xi}_i) = \sum_{j=2}^{N-1} d_{ij} \Vec{C}(\Vec{\eta}_j),
 \  (i= 2 \sim (N-1)).
\end{eqnarray}
This relation is equivalent to the relation
\begin{eqnarray}
 \Vec{\xi}_i = \sum_{j=2}^{N-1} d_{ij} \Vec{\eta}_j, \  (i= 2 \sim (N-1)).
 \label{eq:xi_d_eta}
\end{eqnarray}
By inserting Eq.~(\ref{eq:xi_d_eta}) into Eq.~(\ref{eq:Jacobi_quadra}) we obtain
\begin{eqnarray}
 \sum_{i=1}^N \Vec{r}_i^2 &=& N \Vec{R}^2 + \sum_{i=1}^{N-1}
    \frac{\nu_i}{\nu} \Vec{\xi}_i^2   \\
  &=& N \Vec{R}^2 + \frac{\nu_1}{\nu} \Vec{\eta}_1^2 +
   \sum_{i,i'= 2}^{N-1} a_{i,i'} \Vec{\eta}_i \cdot \Vec{\eta}_{i'},
   \label{eq:ortho_quadra} \\
 a_{i,i'} &=& \sum_{j=2}^{N-1} \frac{\nu_j}{\nu} d_{j,i} d_{j,i'}.
\end{eqnarray}
It is to be noticed that unlike the Jacobi coordinates the mutual
orthogonality within the $(N-2)$ coordinates
$\{ \Vec{\eta}_i, \ i= 2 \sim (N-1) \}$ does not hold in general.

By using Eq.~(\ref{eq:ortho_quadra}) we can calculate the internal
one-particle density matrix $\rho_{\rm int}^{(1)}$ as follows
\begin{eqnarray}
 \rho_{\rm int}^{(1)}(\Vec{\eta},\Vec{\eta}^\prime) &\propto&
   \int d\Vec{\eta}_2 \cdots d\Vec{\eta}_{N-1}
   \exp\left[ -\nu \left( \frac{\nu_1}{\nu}  \Vec{\eta}^2 + \sum_{i,i'= 2}^{N-1} a_{i,i'} \Vec{\eta}_i \cdot \Vec{\eta}_{i'} \right)\right]
   \nonumber \\
 && \times \exp\left[ -\nu \left( \frac{\nu_1}{\nu} {\Vec{\eta}^\prime}^2 + \sum_{i,i'= 2}^{N-1} a_{i,i'} \Vec{\eta}_i \cdot \Vec{\eta}_{i'} \right) \right] \\
 &\propto& \exp\left[ - \frac{N-1}{N} \nu \left( \Vec{\eta}^2 + {\Vec{\eta}^\prime}^2 \right) \right].
\end{eqnarray}
This result shows that $\rho_{\rm int}^{(1)}(\Vec{\eta},\Vec{\eta}^\prime)$
 is just the same as the Jacobi-type internal one-particle density matrix.
Thus we see that our above requirement concerning the internal coordinates,
 gives us a {\it unique result} for the internal one-particle density matrix.
It is just identical to the Jacobi-type internal one-particle density matrix.

In the previous section \ref{subsec:convergence_BEC_macro}
 we have proposed also another requirement, namely
 that the internal one-particle density matrix should converge to the one-particle density
 matrix in the laboratory frame in the macroscopic limit.
As we discussed in Sec.~\ref{subsec:convergence_BEC_macro}, this requirement
 implies that there should not
 appear any cross terms of $\Vec{\xi}_1$ with the other internal
 coordinates in the internal wave function at least in the macroscopic limit.
Clearly the absence of the cross terms of
 $\Vec{\xi}_1$ with the other internal coordinates is realized only
 when the coordinate $\Vec{\xi}_1$ is orthogonal to all the other
 $(N-2)$ internal coordinates.
Thus the criterion in the
 preceding subsection results from the present requirement of the
 orthogonality of $\Vec{\xi}_1$ to the other $(N-2)$ internal coordinates, where the $( N - 2 )$ coordinates
 generally do not need to be mutually orthogonal.

One may argue that a physical quantity should not depend on the choice of
 the coordinate system.
In the case of the one-particle density matrix this
 argument can be true under the condition that we have extracted maximum
 information on the one-particle degree of freedom of the system.
As already noticed this condition is the same as the requirement that the
 coordinate of the one-particle density matrix is orthogonal to all the other
 coordinates of the system.
This is also true in the macroscopic system where
 we usually adopt the coordinate system composed of individual particle
 coordinates $\{ \Vec{r}_i, \ (i= 1 \sim N) \}$.
The coordinate $\Vec{r}_1$
 of the one-particle density matrix is of course orthogonal to all the other
 coordinates.
If we adopt the coordinate system $\{ \Vec{R}, \Vec{q}_i, \ (i= 1 \sim (N-1)) \}$
 and calculate the one-particle density matrix with
 respect to the coordinate $\Vec{q}_1$ which is practically the same as
 $\Vec{r}_1$ in the macroscopic system by integrating out the remaining
 coordinates $\{ \Vec{R}, \Vec{q}_i, \ (i= 2 \sim (N-1)) \}$, we get the
 result given in Eq.~(\ref{eq:one_body_den_int_infinite}), namely
 $(2\nu/\pi)^{3/2} \exp [-(3/2) \nu(\Vec{q}_1^2+{\Vec{q}_1^\prime}^2) +
 \nu \Vec{q}_1 \cdot \Vec{q}_1^\prime ]$.
This inadequate result is, of course, due to the non-orthogonality of $\Vec{q}_1$ to the other coordinates
 $\{ \Vec{q}_i, \ (i= 2 \sim (N-1)) \}$, as we have already seen earlier.

The PP-type internal one-particle density matrix $\rho_{\rm int,PP}^{(1)}$
 is different from the Jacobi-type one $\rho_{\rm int,J}^{(1)}$.
This is true
 for the non-diagonal elements but for the diagonal elements they are the
 same except for the Jacobian factor
 $\partial(\Vec{q}_1)/\partial(\Vec{\xi}_1) = ((N-1)/N)^3$,
 (which goes to unity in the macroscopic limit)
\begin{eqnarray}
 \rho_{\rm int,PP}^{(1)}(\Vec{q}_1,\Vec{q}_1) = \left(\frac{N}{N-1}\right)^3
  \rho_{\rm int,J}^{(1)}(\Vec{\xi}_1,\Vec{\xi}_1).
  \label{eq:PPJ}
\end{eqnarray}
It is to be noticed that both $\rho_{\rm int,PP}^{(1)}$ and
$\rho_{\rm int,J}^{(1)}$ are normalized,
\begin{eqnarray}
 \int d\Vec{q}_1 \rho_{\rm int,PP}^{(1)}(\Vec{q}_1,\Vec{q}_1) =
 \int d\Vec{\xi}_1 \rho_{\rm int,J}^{(1)} (\Vec{\xi}_1,\Vec{\xi}_1) = 1.
\end{eqnarray}
The equality of Eq.~(\ref{eq:PPJ}) is easily proved by using
 Eqs.~(\ref{eq:one_body_den_int}) and (\ref{eq:Jacobi_one_body_den_int}).
Let us study these relations between $\rho_{\rm int,PP}^{(1)}$ and
 $\rho_{\rm int,J}^{(1)}$ a little more in detail.  First we note the
 relation between the two coordinate systems,
 $\{ \Vec{q}_i, \ (i= 1 \sim (N-1)) \}$ and
 $\{ \Vec{\xi}_i, \ (i= 1 \sim (N-1)) \}$,
\begin{eqnarray}
 \Vec{q}_i = - \sum_{j=1}^{i-1} \frac{1}{N+1-j} \Vec{\xi}_j +
   \frac{N-1}{N+1-i} \Vec{\xi}_i.
\end{eqnarray}
When we calculate $\rho_{\rm int,J}^{(1)}$, we make the following
integration
\begin{eqnarray}
 \rho_{\rm int,J}^{(1)}(\Vec{\xi}_1,\Vec{\xi}_1^\prime) &=&
   \int d\Vec{\xi}_2 \cdots d\Vec{\xi}_{N-1}~\Phi_{\rm int}(\Vec{\xi}_1,
    \Vec{\xi}_2,\cdots,\Vec{\xi}_{N-1}) \Phi_{\rm int}(\Vec{\xi}_1^\prime,
    \Vec{\xi}_2,\cdots,\Vec{\xi}_{N-1}) \\
  &=& \frac{\partial(\Vec{\xi}_2,\cdots,\Vec{\xi}_{N-1})}
      {\partial(\Vec{q}_2,\cdots,\Vec{q}_{N-1})}
   \int d\Vec{q}_2 \cdots d\Vec{q}_{N-1}~{\widehat \Phi}_{\rm int}(\Vec{q}_1,
    \Vec{q}_2,\cdots,\Vec{q}_{N-1}) \nonumber \\
  && \hspace*{2cm} \times {\widehat \Phi}_{\rm int}(\Vec{q}_1^\prime,
    \Vec{q}_2^\prime,\cdots,\Vec{q}_{N-1}^\prime) \\
 \Vec{q}_i^\prime &=& \Vec{q}_i(\Vec{\xi}_1 \rightarrow \Vec{\xi}_1^\prime)
    =  \Vec{q}_i + \frac{1}{N} (\Vec{\xi}_1 - \Vec{\xi}_1^\prime),
\end{eqnarray}
where
\begin{eqnarray}
  &&{\widehat \Phi}_{\rm int}(\Vec{q}_1,\Vec{q}_2,\cdots,\Vec{q}_{N-1}) =
    \Phi_{\rm int}(\Vec{\xi}_1,\Vec{\xi}_2,\cdots,\Vec{\xi}_{N-1}), \\
  &&{\widehat \Phi}_{\rm int}(\Vec{q}_1^\prime,\Vec{q}_2^\prime,\cdots,
    \Vec{q}_{N-1}^\prime) = \Phi_{\rm int}(\Vec{\xi}_1^\prime,
    \Vec{\xi}_2,\cdots,\Vec{\xi}_{N-1}).
\end{eqnarray}
For non-diagonal elements of $\rho_{\rm int,J}^{(1)}(\Vec{\xi}_1,
\Vec{\xi}_1^\prime)$, since $\Vec{\xi}_1 \ne \Vec{\xi}_1^\prime$, it
follows that $\Vec{q}_i^\prime \ne \Vec{q}_i$ for all $i = 1 \sim (N-1)$.
This tells us that $\rho_{\rm int,J}^{(1)}(\Vec{\xi}_1,\Vec{\xi}_1^\prime)$
can not be proportional to $\rho_{\rm int,PP}^{(1)}(\Vec{q}_1,
\Vec{q}_1^\prime)$ because the latter is obtained from
\begin{eqnarray}
  \int d\Vec{q}_2 \cdots d\Vec{q}_{N-1}~{\widehat \Phi}_{\rm int}
    (\Vec{q}_1,\Vec{q}_2,\cdots,\Vec{q}_{N-1}) \
    {\widehat \Phi}_{\rm int}(\Vec{q}_1^\prime, \Vec{q}_2,\cdots,
    \Vec{q}_{N-1}).
\end{eqnarray}
On the other hand, for diagonal elements, $\rho_{\rm int,J}^{(1)}
(\Vec{\xi}_1,\Vec{\xi}_1)$, since $\Vec{\xi}_1 = \Vec{\xi}_1^\prime$, there
follows $\Vec{q}_i^\prime = \Vec{q}_i$ for all $i = 1 \sim (N-1)$.
This tells us that $\rho_{\rm int,J}^{(1)}(\Vec{\xi}_1,\Vec{\xi}_1)$
is now proportional to $\rho_{\rm int,PP}^{(1)}(\Vec{q}_1,\Vec{q}_1)$.
The appearance of the term $(1/N) (\Vec{\xi}_1 - \Vec{\xi}_1^\prime)$
in the relation $\Vec{q}_i^\prime = \Vec{q}_i + (1/N) (\Vec{\xi}_1 -
\Vec{\xi}_1^\prime)$ just stems from the non-orthogonality of
$\Vec{q}_1$ to $\Vec{q}_i \ (i=2 \sim (N-1))$.

We have proved in this section that the internal one-body density matrix is uniquely determined
 for the $0S$ harmonic oscillator wave function.
However, it is noted that this uniqueness holds in the case of a general wave function including the $0S$ harmonic oscillator one.
In fact, Suzuki et al.~\cite{Suzuki1,Suzuki2} have already given the proof of the uniqueness for the more general wave
 function $\Psi$ which is expanded in terms of the correlated Gaussian basis $g$.
The explicit forms of $\Psi$ and $g$ are expressed as follows:
\begin{eqnarray}
&&\Psi=\sum_k C_k {\cal S} g(\Vec{s}^{(k)};A^{(k)},\Vec{x}) \label{eq:SVM}\\
&&g=g(\Vec{s};A,\Vec{x})=\exp\Big[ -\frac{1}{2}\sum_{i,j}A_{ij}\Vec{x}_i \cdot \Vec{x}_j + \sum_i \Vec{s}_i \cdot \Vec{x}_i \Big],
\end{eqnarray}
where $\Vec{x}=\{\Vec{x}_1,\Vec{x}_2,\cdots,\Vec{x}_{N-1}\}$
 and ${\cal S}$ are a set of internal coordinate of $N$-boson system and the symmetrization
 operator acting on the $N$ bosons, respectively.
$C_k$, $A^{(k)}$ and $\Vec{s}^{(k)}$ are the expansion parameters,
 in which $A^{(k)}$ denotes a symmetric positive-defined $(N-1)\times(N-1)$ matrix,
 and $\Vec{s}^{(k)}$ represents $\Vec{s}^{(k)}=\{\Vec{s}_1^{(k)},\Vec{s}_2^{(k)},\cdots,\Vec{s}_{N-1}^{(k)}\}$.
The correlated Gaussian basis $g$ is often used in ab-initio calculations and has succeeded
 in describing structures of many few-body systems~\cite{Suzuki3}.
The uniqueness of the internal one-body density matrix $\rho$ for the wave function $\Psi$ in Eq.~(\ref{eq:SVM}) is
 proved~\cite{Suzuki1,Suzuki2} under the condition that one takes the following
 set of coordinates $\Vec{y}=\{\Vec{y}_1,\Vec{y}_2,\cdots,\Vec{y}_{N-1}\}$ (obtained by a linear transformation from the coordinates $\Vec{x}$)
 as the coordinates of $\Psi$ adopted in calculating $\rho$:~$\Vec{q}_1$
 is chosen as the coordinate ($\Vec{y_1}$) of the internal one-body density matrix and the remaining
 $(N-2)$ internal coordinates ($\Vec{y}_2,\Vec{y}_3,\cdots,\Vec{y}_{N-1}$)
 are orthogonal to $\Vec{q}_1$, although the $(N-2)$ coordinates do not need to be mutually orthogonal.
This requirement for the coordinates $\Vec{y}$ just corresponds to the "maximally independence
 of the coordinate $\Vec{\xi}_1$ ($\Vec{q}_1$) from the rest of $(N-2)$ internal coordinates" as mentioned above.
The proof we gave in this section is specified to the simple system which is composed of $N$ bosons
 in the harmonic oscillator potential and has been discussed by many authors
 (for example, see Refs.~\cite{PP,wilkin,arxiv,letter}).

\section{Summary}

We investigated  the internal one-particle density matrix
in the case of ideal Bose-Einstein condensates with a finite number
( $N$ ) of particles in a harmonic trap.  We calculated
the explicit form of the internal one-particle density matrix
following the definition of Pethick and Pitaevskii  (PP) and
solved  its eigenvalue problem. The result was found to show
a fragmented condensate, contrary to what PP
expected. On the other hand the Jacobi-type  internal one-particle
density matrix gives us complete condensation.
It means that the internal one-particle density matrix is
 different in general for different choices of the internal coordinate system.
In this paper we outlined two physically motivated criteria for
 the choice of the adequate
 coordinate system leading to a unique answer for the internal
 one-particle density matrix.
One criterion is that in the infinite
 particle number ( $N=\infty$ ) limit the internal one-particle density
 matrix should have the same eigenvalues and eigenfunctions as
 those of the corresponding ideal Bose-Einstein condensate in the
 laboratory frame.
The other criterion is that the coordinate of the
 internal one-particle density matrix which is either
 $\Vec{q}_1 = \Vec{r}_1 - \Vec{R}$ or $\Vec{\xi}_1 = (N/(N-1)) \Vec{q}_1$,
 should be maximally independent from  the remaining $( N - 2 )$
 internal coordinates.
Mathematically this criterion means that $\Vec{q}_1$ (or $\Vec{\xi}_1$)
 is orthogonal to the remaining  $( N - 2 )$ internal coordinates,
 though the $( N - 2 )$ coordinates, in general, do not need to be mutually orthogonal.
This second criterion was shown to imply the first criterion.
We saw that the internal Jacobi
 coordinate system satisfies these two criteria while the internal
 coordinate system adopted by Pethick and Pitaevskii for the
 construction of the internal one-particle density matrix does not.
Furthermore we argued that these two criteria {\it uniquely} determine
 the internal one-particle density matrix which is the same as that
 calculated with the Jacobi coordinates.
The results of this paper justify the use of the Jacobi
 coordinates in Refs.~\cite{matsumura,yamada} where the Bose
 condensation of a few $\alpha$-particles in extended states of self-conjugate
 light nuclei was considered.
However, our results are of more general
 interest. For example the number of bosons captured in each site of an optical
 lattice is often very small~\cite{zwerg} and, therefore, our
 analysis surely applies to that situation as well. We also believe that our
 present study has considerably clarified the somewhat controversial issue of
 how to define a Bose condensate of a finite number of particles in their
 internal coordinate system.
This question is particularly, but not only,
 relevant for selfbound Bose systems, like a loosely bound gas of $\alpha$-particles
 in nuclei or nano-droplets of liquid $^{4}$He.

The authors thank Kanto Gakuin University (KGU) and Yukawa Institute for Theoretical
 Physics at Kyoto University, Japan.
Discussions during the KGU Yokohama Autumn School of Nuclear Physics and the YIPQS
 international molecule workshop held in October 2008
 were useful to complete this work.

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The Walk the Light Arts Festival is only a day away and Founder, Laura Allen, was very gracious to take some time and answer a few questions about the festival and organization for ILiveinDallas.com.
Laura Allen is the Founder, Executive Director & Board Chairman of Walk the Light. I am a singer/songwriter, go-getter and mayhem-starter! I fancy myself a creative expressionist; hence my love for promoting art in unconventional ways. I really enjoy antiques, long drives, and coffee. In fact, since I am a Barista on the side, I hold steadfast to the belief that coffee needs to have its own food group in the food pyramid!
Laura Allen: Well, Walk The Light started very organically. A few of us just got together one night and decided we wanted to have a show where all of our friends could come out and just do what we love to do... share their music, share their photography, paintings... and so we just planned it out. We told our friends about it and 100 people came out to just enjoy the night with us! We were so excited. That was in September of 2009. From there, we decided that there really is a need for artists from all genres to have a venue where they can share their art and just do what they love to do. So, we just kept having Walk The Light Arts Festivals and they have kept growing.
I was invited to an information gathering about FUEL and I started talking with the executive directer and his team about Walk The Light. We agreed that it would be good to partner together for this event because their mission and purpose is nearly the same as ours, only they are specifically targeting those in the Philippines. So, noticing that we were similar, we decided to have them as the charity focus to catapult their team moving over to the Philippines to invest in the community of IloIlo.... in the future we hope to partner with them in their endeavor in the Philippines in reaching out to the community.
We've added several new elements to this event including some of the interactive art stations and more artists who are joining to showcase their work. In the past the charity focus has not been as involved in the planning/ preparation process, but this time FUEL has been involved every step of the way and they are helping us run the event, which is so wonderful and truly a great experience.
The staff is entirely made up of young women who volunteer their time outside of their full-time jobs and other committments. I am so thankful for their service and passion that helps to make Walk the Light possible. Also, all of the artists volunteer their time during the festivals so that the proceeds from the event can be given to the charity focus.
We really believe that we are affecting change in our art community by putting on the kind of festivals that we do that provide a venue for artists who otherwise would not have had a community or venue to join. We are also affecting the lives of others by highlighting a charity focus that serves the needs of those less fortunate. We are committed to making these points our main focus as we continue to plan for the years ahead.
The moment we realized we had reached our Kickstarter goal to become a non-profit organization. We raised $2,000 in 28 days. We are so thankful for everyone who so generously gave to help us get started.
Positively. Like I said earlier, we provide a venue for artists that don't have one in a way that is very positive. We also encourage artists and those in the community to use the gifts and talent they have been given to serve those less fortunate through highlighting a charity focus. Oftentimes these charity focuses are local, which really helps to connect the artists and community to the needs of the city.
Oh man! Really big. We hope to be completely funded and expanded to other cities. We are planning and aiming to have city-based festivals. We have some friends in other cities (San Diego, Oklahoma City, etc...) that love Walk the Light and have experienced it in the past, and they have expressed a desire to have Walk the Light come to their city to see local artists and charity focuses supported through this great event.


text:
Hot Air Balloon Flight over Loire Valley in France
France, Centre-Val de Loire
Transfer to take-off site
Traditional Aeronaut's toast and diploma on landing
1-hour flight
Flight duration approx. 1 hour
Starts between 5am and 6am depending on season
Total duration of experience from balloon inflation to end of diploma ceremony is 4 hours
Please wear comfortable clothing, jacket and closed toe shoes
Available every day from April to October
Your ticket will be valid for twelve months from date of issue in order to cover a full flying season(April until October)
Not suitable for people with heart condition
Château de Chenonceau, 37150 Chenonceaux, France
One of the most popular places in France for a balloon trip
Admire the iconic Châteaux of the Loire from above
If there's one essential you must remember for this gift experience in France, it's your camera. You can expect to be glued to the viewfinder as you travel gently through the sky above one of France's most iconic and spectacular landscapes - the Loire Valley. Known as the 'Garden of France' for its fertile fields, vineyards and orchards, the valley is a UNESCO World Heritage Site. It is also dotted with over 300 châteaux, many of which are magnificent examples of architecture.
You'll drift gently over the scenery, with a skilled pilot at the controls, as you join the ranks of the 'Aeronauts'. The Loire Valley is, of course, among the best-known wine regions in the world, let alone France, and you may see people already at work among the vines in the early morning, or maintaining the grounds of the fairy castle-like châteaux. After an hour of sky-sailing, you'll come to a gentle landing where you'll be met with a complimentary glass of champagne, before taking part in a traditional diploma ceremony. A wonderfully
romantic and peaceful way to see this spectacular part of France.
Stunning birds-eye views of many Loire châteaux
The rolling scenery of the Loire is UNESCO-protected
Traditional Aeronauts reception on landing
Full-Day Georgian Wine Tasting Experience for Two Georgia, Tbilisi
Tango Evening Experience in Buenos Aires for Two Several locations available
Delicious Nassau Food Experience in Bahamas for Two The Bahamas, New Providence
Food and Art in Funky Kreuzberg, Berlin for Two Germany, Berlin
Helicopter Flight over Grand Canyon United States, Arizona
VIP Helicopter Tour in Philippines Philippines, Western Visayas

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Omg I wanna shoot myself. Once again, I have packed my weekend with almost no room for rest.
BOTH OF WHICH AT 6.30pm..Omg, need to come late for 1 of them.
3. Possible after-party? HIGH CHANCE..when is there not? NAUGHTY GIRL IS NEXT DOOR.
Damn it…I know Ive got more. Gonna update this again later.
We have to go to naughty girl to finish the bottle toooo!


John Elway made the leap from specialist to manager look easy but for many of us the transition isn't that simple. Join Elijah Kempton with Assured Flow Solutions at COGA's next Brown Bag Luncheon on November 18th as he shares his experience about his journey from technical specialist to manager. Topics will include, insight into skills required to be a successful manger and lessons learned in the transition from technical specialist.


Rob Chapman shares the importance of wholly invented bands and songs in his creative life, and considers other examples of imaginary culture and the fictitious within our fictions.
Catherine Czerkawska speaks with Cherise Saywell about her fascination with Jean Armour, the greatly underestimated wife of Scots bard Robert Burns, and discusses writing history as fiction, and her own professional journey.
Kevin Clarke speaks with Frances Byrnes about playwriting as an essential dramatic apprenticeship, completing his education as an adult, and how his dramatic subjects choose him, rather than the other way around.


1. Put the flour in a bowl. Add water and knead well to a make a pliable dough.
2. Cover and keep aside for 15 minutes.
3. Divide the dough into equal portions/balls.
4. Dust the surface with flour, roll out each ball into a circle.
5. Rub the top with butter or oil.
6. Sprinkle with mint and salt.
7. Pleat up the circle and twist the dough into a rope.
8. Coil the rope to make a ball. Repeat to make more layers in the paratha.
9. Heat a tawa, place a paratha on it.
10. Dribble butter/oil around the sides and cook on medium heat until golden.
11. Turn and cook the other side the same way.
Feel free to comment or share your thoughts on this Mint Paratha Recipe from Awesome Cuisine.


The EasyRead Time Teacher Alarm Clock is a fully functioning alarm clock that also helps children learn to tell the time using their simple 3 step teaching system.
This fantastic clock features a crescendo alarm, which starts with a gentle beep and gets progressively more urgent until it is turned off. The snooze button disables the alarm for 5 minutes before it starts again, while the built in light sensor controls the brightness of the clock face, which glows gently in the dark and doubles as a night light. Kids can also press the snooze button to increase the brightness of the clock face. The silent sweep movement ensures there's no annoying ticking making it the perfect bedside clock.
Available in rainbow face, and red and blue face.
The EasyRead range also includes a wall clock and wrist watches.
*Requires 3 x AA batteries (not included).


text:
Q: AJAX при коде ответа 200 (ок) заходит в error, а не в success        // base url
       const url = "http://localhost:5002/api/contacts/";

       // send put request to the api
        $.ajax({
            headers: {
                'Accept': 'application/json',
                'Content-Type': 'application/json'
            },

            type: 'DELETE',
            url: url + contactId,
            dataType: 'json',

            success: function () {
                // come back to main page
                window.location.href = '/contacts/list'
            },

            error: function (error) {
                console.log(error);
            }
        });

При этом, вот что выводится в консоли
status: 200
statusCode: ƒ (e)
statusText: "OK"
Код метода в апи
    // DELETE api/contacts/2
    [HttpDelete("{id}")]
    public ActionResult Delete(int id)
    {
        using (var db = new PhonebookDBContext())
        {
            var con = db.Contacts.FirstOrDefault(c => c.Id == id);

            if (con != null)
            {
                db.Contacts.Remove(con);
                db.SaveChanges();

                return Ok();
            }

            return NotFound("Not found contact with a same Id");
        }
    }

A: Нужно убрать строку dataType: 'json', т.к jquery пытался разобрать объект, который не приходит в ответе

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text:
Brama Kujawska w Bydgoszczy – historyczna brama miejska wchodząca w skład murów obronnych Bydgoszczy. Została rozebrana w początkach XIX wieku przez władze pruskie.

Historia

Brama Kujawska zwana także Toruńską, Bernardyńską lub Bosacką wzniesiona została w południowo-wschodniej części miasta u zbiegu ulic: ulicy Długiej i Podwale.
Jej zadaniem była ochrona dróg wylotowych z Bydgoszczy wiodących do Inowrocławia i Torunia.

Ze względu na położenie nieopodal wzgórz szwederowskich jest najlepiej ze wszystkich bydgoskich bram miejskich przedstawiona na ikonografii np. na rysunkach Erika Dahlberga z 1657 r. i Johanna Rudolfa Storna z 1661 r.

Pierwsza wzmianka pisana o Bramie Kujawskiej pochodzi z 1523 r., aczkolwiek można przypuszczać, że wzniesiono ją o wiele wcześniej – w formie drewnianej prawdopodobnie w II połowie XIV wieku, a murowanej – wiek później. Zniszczona wskutek pożaru w 1547 r. została w II połowie XVI wieku odbudowana. Następnie remontowano ją również w 1634 r. i po potopie szwedzkim.

Przed bramą znajdowało się murowane przedbramie, które istniało do połowy XVII w., kiedy zostało zniszczone w czasie budowy nowożytnych fortyfikacji.

Forma i rzut Bramy Kujawskiej jest bardzo podobny do Bramy Poznańskiej, co pozwala postawić hipotezę, iż zostały one zbudowane w tym samym czasie i według tej samej koncepcji architektonicznej. Jednakże w przeciwieństwie do swojej odpowiedniczki Brama Kujawska bardzo długo, bo aż do połowy XVIII w., zachowywała swój pierwotny kształt.

Podczas potopu szwedzkiego budowla broniona była przez bastion ziemny, sprzęgnięty w jeden system obronny z fortyfikacjami bastionowymi zamku. Konstrukcja ta o wymiarach 45×35 m zbudowana była naprzeciw bramy, w miejscu zasypanej na tym odcinku fosy.

Podczas powstania antyszwedzkiego, 21 kwietnia 1656 r. wojska polskie Stefana Czarnieckiego wkroczyły do miasta wyłamując wrota bramy. W trakcie dalszych walk zniszczeniu uległ również bastion bramny podobnie jak bydgoski zamek i jego fortyfikacje.

Gruntowny remont i przebudowa budowli nastąpiła w 1751 r.  Nadbudowano wówczas jedną kondygnację i przykryto dachem na podobieństwo Bramy Poznańskiej.

Podczas Powstania Kościuszkowskiego 2 października  1794 r. przez Bramę Kujawską wtargnęły do miasta wojska polskie pod wodzą generała Henryka Dąbrowskiego. Bramę wyłamał śmiałym atakiem oddział płk Burzyńskiego.

Po zajęciu miasta przez Prusy budowla niepotrzebna już ze względów militarnych pełniła jeszcze jakiś czas rolę rogatki miejskiej, przy której pobierano podatek wjazdowy.
Ostateczna rozbiórka bramy nastąpiła w 1817 r. podobnie jak wielu innych fragmentów murów miejskich w Bydgoszczy.

Charakterystyka
Brama Kujawska była przejazdową wieżą bramną o wymiarach w rzucie: 7,5 × 8,5 m i grubości ścian 1,6 m. Można ją porównać do Bramy Wronieckiej w Poznaniu.
Szerokość przejazdu wynosiła 3,7 m. Przejazd był brukowany. Od strony południowej brama łączyła się z murem obronnym.

Była to budowla piętrowa. Na pierwszej kondygnacji znajdowały się urządzenia do podnoszenia brony oraz komnaty, w których przez pewien czas zamieszkiwał kat miejski. W ścianie południowej i północnej znajdowały się strzelnice. Dach był płaski, zwieńczony flankowanym krenelażem, co tworzyło dodatkowy górny poziom użytkowy. Budowla podzielona była na kondygnacje drewnianymi stropami. Komunikację zapewniała drabina, lub wewnętrzne drewniane schody.

Brama posiadała dwuskrzydłowe wrota, bronę i most zwodzony nad fosą miejską, zastąpiony w II połowie XVIII wieku mostem stałym z poręczami.

Od 1751 r. forma budowli zmieniła się. Zamurowano wówczas wieńczący bramę blankowany krenelaż oraz podwyższono najwyższą kondygnację na co przeznaczono 1000 sztuk cegieł. Budowla otrzymała dwu- lub czterospadowy dach (jak Brama Poznańska), kryty dachówką. W górnej kondygnacji zostały wstawione okna, a cała brama została pomalowana na czerwono, czego dowodzą zachowane rachunki.
W 1756 r. zbudowano ponadto ganek na bramie oraz wstawiono nowe wrota, drzwi, okna i schody.

Badania archeologiczne
Północno-wschodni narożnik bramy został odkryty podczas nadzorów archeologicznych wykopów prowadzonych w 1997 r. Fundamenty w postaci ceglanych murów znaleziono zaledwie 40 cm pod nawierzchnią ulicy. Badania archeologiczne potwierdziły datowanie pozostałości na XVI wiek.

Przypisy

Zobacz też
 Mury miejskie w Bydgoszczy
 Brama Gdańska w Bydgoszczy
 Brama Poznańska w Bydgoszczy
 Zamek w Bydgoszczy
 Ulica Długa w Bydgoszczy
 Fosa miejska (Bydgoszcz)
 Fosa zamkowa (Bydgoszcz)
 Stare Miasto w Bydgoszczy
 Historia Bydgoszczy

Linki zewnętrzne

Bibliografia
 Krzysztof Bartowski. Mury obronne miasta Bydgoszczy. Kronika Bydgoska XIV (1992). Bydgoszcz 1993
 Marian Biskup (red.): Historia Bydgoszczy. Tom I do roku 1920. Warszawa-Poznań: Bydgoskie Towarzystwo Naukowe, 1991
 Jerzy Derenda. Piękna stara Bydgoszcz – tom I z serii Bydgoszcz miasto na Kujawach. Praca zbiorowa. Towarzystwo Miłośników Miasta Bydgoszczy. Bydgoszcz 2006
 Robert Grochowski, Wojciech Siwiak. Baszta pierwsza muru obronnego miasta Bydgoszczy w świetle źródeł archeologicznych, kartograficznych i ikonograficznych. Kronika Bydgoska XXIII (2001). Bydgoszcz 2002
 Robert Grochowski. Bramy, wały i fortyfikacje bastionowe dawnej Bydgoszczy. Kronika Bydgoska XXIII (2001). Bydgoszcz 2002
 Robert Grochowski. Mur obronny miasta Bydgoszczy – przyczynek archeologa. Kronika Bydgoska XXIV (2002). Bydgoszcz 2003
 Robert Grochowski. Baszty muru obronnego miasta Bydgoszczy. Kronika Bydgoska XXVIII (2006). Bydgoszcz 2007

Nieistniejące obiekty budowlane w Bydgoszczy
Mury miejskie w Bydgoszczy
Kujawska
Wyburzone bramy w Polsce

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Find latest Urgent Employment and Recruitment Agency for sale In Victoria, British Columbia and 0 Urgent Employment and Recruitment Agency for sale in Canada. Top Location for Urgent Employment and Recruitment Agency for sale in and around the Victoria area are 1 Victoria and many more. Most of these businesses are listed by 1 Franchise Online Businesses.


text:
Black Beauty By Anna Sewell
Black Beauty By Anna Sewell quantity
Black Beauty By Anna Sewell Story level 4+ (B2+) Audio book with subtitle Black Beauty is an 1877 novel by English author Anna Sewell. The story is narrated in the first person as an autobiographical memoir told by the titular horse named Black Beauty—beginning with his carefree days as a foal on an English farm with his mother, to his difficult life pulling cabs in London, to his happy retirement in the country. Along the way, he meets with many hardships and recounts many tales of cruelty and kindness. ——————————————————– Join this channel to get access to exclusive member-only videos (without Ads): https://www.youtube.comhttps://www.youtube.com/channel/UCrtPxFge-clHUejWD82y_lg/join YT Channel: https://www.youtube.comhttps://youtube.com/c/EnjoyEnglish ———————————– You can download the text scripts of the stories here: https://englishonline.vn ———————————— https://www.youtube.com/hashtag/enjoyenglish https://www.youtube.com/hashtag/blackbeauty https://www.youtube.com/hashtag/annasewell
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Shifting into Overdrive By Kevin Williamson
Simon Decker and the Secret Formula


Dem bones, dem bones, dem Bone beads!
Some of these are not bone at all - but simulated - resin carved and antiqued to look like bone or ivory. We don't have any actual ivory - as we are not big fans of killing elephants, narwhals, walrusii, or unicorns.
Bone Beads: Hand selected, as most of the bone beads we see are rather crudely made, or are indistinguishable from plastic! We prefer to pass on those. Mostly made in India or China, and are probably cow bone.


text:
Hillary Clinton avoids prosecution but called 'extremely careless' by FBI
By Daniel DaleWashington Bureau
Tues., July 5, 2016timer4 min. read
WASHINGTON—A year into the email scandal that has deeply damaged her presidential campaign, Hillary Clinton finally heard the words she insisted would come: "No charges."
But this was no vindication, not close. This was another blow.
In declining to recommend a criminal indictment of Clinton, the director of the Federal Bureau of Investigation simultaneously delivered a scathing indictment of her judgment. Clinton was "extremely careless" in handling "very sensitive, highly classified" information on a home email server, knew better or should have, and may have been hacked by foreign enemies, James Comey said Tuesday.
Comey, critically, also contradicted Clinton's chief political defence: that none of the emails she sent from the account she set up in her New York basement as secretary of state contained classified information at the time she sent them. In fact, 110 emails were classified at the time.
Comey's self-described "unusual statement," delivered at FBI headquarters in Washington, lifted the biggest cloud over Clinton's quest to become the first female president. But it raised further questions about the honesty and wisdom of a Democratic nominee who is already facing a severe trust problem among independent voters.
The ad-ready censure from a top law enforcement official will likely feature in Republican campaigns through the summer. It blows a deep hole in Clinton's suggestion that she had made a minor mistake.
"Even if the public doesn't follow or understand the details of it, it's a simple enough issue that they're able to form a judgment over whether they think it's appropriate. And it's hard for me to imagine that there are many people out there who think there was absolutely nothing wrong with it," said Eric Heberlig, a political science professor at the University of North Carolina at Charlotte.
Comey's decision was fraught with political intrigue. It came less than three hours before Clinton boarded Air Force One for her first campaign appearance with President Barack Obama, three days after Clinton was interviewed by FBI investigators, and a week after former president Bill Clinton had a widely criticized impromptu conversation with Attorney General Loretta Lynch at an airport.
The final decision on prosecution lies with the Justice Department, but Lynch is certain to accept Comey's recommendation. Comey found that Clinton's actions amounted to a series of egregious decision-making errors that "no reasonable prosecutor" would take to trial – because there was no evidence Clinton "intended" to violate laws on handling classified information or was grossly negligent, as the law requires for a conviction.
Even her deletion of some work emails, he said, was not done to conceal them.
"All the cases prosecuted involved some combination of clearly intentional and willful mishandling of classified information, or vast quantities of materials exposed in such a way as to support an inference of intentional misconduct, or indications of disloyalty to the United States, or efforts to obstruct justice. We do not see those things here," Comey said.
He suggested, though, that Clinton's actions would merit consequences if she were still a government employee. And he found that Clinton was not truthful when she repeatedly said she "never sent or received any information that was classified at the time it was sent and received."
Comey said 52 of her email conversations included material classified at the time, including eight "highly sensitive" chains classified "top secret" and 36 classified "secret." The FBI found classified material in three emails Clinton didn't turn over to the State Department for its investigation.
"None of these emails should have been on any kind of unclassified system," Comey said, especially one he said was less secure that Google's public Gmail. Clinton used her unprotected email even while travelling in hostile countries, he said, and hackers gained access to the accounts of people she emailed.
"Given that combination of factors, we assess it is possible that hostile actors gained access to Secretary Clinton's personal email account," Comey said.
Comey, a Republican, was appointed to a 10-year term by Obama in 2013. Mindful of the political optics, he said he did not reveal the contents of his statement with any other branch of the government, including Lynch's.
Trump, who has declared Clinton "guilty as hell," was unsatisfied. "The system is rigged," he wrote on Twitter. "FBI director said Crooked Hillary compromised our national security. No charges. Wow!"
"Our adversaries almost certainly have a blackmail file on Hillary Clinton," he said, without evidence, in a subsequent statement, "and this fact alone disqualifies her from service."
Neither Clinton nor Obama said anything about the matter at their upbeat rally in Charlotte. But a fiery Obama testified at length to Clinton's character, and he called her a "great secretary of state" – a claim he said was broadly accepted fact until the "political machinery" of the election churned into action.
"There has never been any man, or woman, more qualified for this office than Hillary Clinton. And that's the truth," Obama said.
Clinton's only response to Comey was a terse statement from a spokesman. "As the Secretary has long said, it was a mistake to use her personal email and she would not do it again," Brian Fallon said. "We are glad that this matter is now resolved."
It's not over, of course. Within two hours, a Republican Super PAC had released a YouTube ad compiling clips of Clinton's dishonesty on classified material. Soon after, Republican congressmen began calling for the case to be referred to an independent special prosecutor.


text:
Tag: TD Canada Trust
Padovani: TD to keep ATM in Weston for now.
The TD Canada Trust branch at 1979 Weston Road. Note the Rockport rental apartment building in the background that will be home to hundreds of new residents when it opens in the coming months.
Toronto Council candidate Chiara Padovani has managed to wrestle a concession from TD Canada Trust, set to close its 1979 Weston Road branch on September 21. The building's ATM will remain open for 'the time being' after the branch closes. The bank's WiFi hotspot (who knew?) will not continue past the closing date.
Ms Padovani also wrote to the Financial Consumer Agency of Canada; the 120-employee independent agency that according to its own site,
"ensures federally regulated financial entities comply with consumer protection measures, promotes financial education and raises consumers' awareness of their rights and responsibilities."
These are the people who sit on their hands while predatory payday loan companies fill the void left by departing bank branches. Apparently they've educated the public about those things so it's ok. In their reply to Ms Padovani's letter, FCAC alleges that TD provided adequate consultation with the community before announcing the closure. As a result, FCAC won't compel TD to hold a community meeting.
If a letter announcing the closure counts as adequate consultation, then yes, the community was consulted adequately.
Undaunted, Ms Padovani also tackled TD who have relented somewhat by agreeing to keep the ATM open past the closing date. In the meantime, she is working with TD Canada Trust to set up a permanent ATM in the vicinity.
So far I have received confirmation that the ATM will remain at the current location while they search for a permanent home in the vicinity. While TD is hosting sessions on digital banking and financial literacy in the community, they have not committed to installing a WiFi hub to facilitate the use of such services for people who don't have access to the Internet.
I'm committed to continue to advocate for accommodations for the members of the community who will be negatively impacted by the bank's imminent closure.
Access to fair banking and financial services is especially important in Weston, given the increase in predatory lending that has sever consequences on socioeconomic health of our neighbourhood. As a social worker in the community, I've seen far too many hardworking people get trapped in debt through pay day lenders.– Chiara Padovani
Read more from the candidate here.
Read Adam's excellent take on the issues here.
Banking in Canada is regulated federally. Incidentally; crickets from MP Ahmed Hussen's office on the branch's closing.
Author Roy MurrayPosted on August 27, 2018 August 26, 2018 Categories 22 John, Ahmed Hussen, Business, Opinion, Payday Loan Companies, PoliticsTags 1979 Weston Road, Ahmed Hussen, Chiara Padovani, TD Canada Trust3 Comments on Padovani: TD to keep ATM in Weston for now.
Padovani tackles TD Canada Trust closing.
TD Canada Trust in Weston.
Chiara Padovani wants to be York South-Weston's next councillor for Ward 11, replacing long-time incumbent Frances Nunziata. She has become quite active in the ward and among other things is closely monitoring the state of retail in YSW.
She recently promised that she would contact the Financial Consumer Agency regarding the recently announced closure of Weston's TD Canada Trust branch. The closure was announced without any community consultation and Ms Padovani is seeking such a consultation by writing to the FCAC's Commissioner, Lucie Tedesco.
Here is the letter that Ms Padovani sent:
Re: File with reference number 363484
Dear Commissioner Tedesco,
I am writing to request a meeting regarding the closure of the TD Canada Trust Bank located at 1979 Weston Road in Toronto Ontario (Branch Number 335).
This marks the fourth TD Branch closure in York South-Weston, a community that has become the target of predatory lending and payday loans. The bank has failed to fulfill its obligation to consult with the community, and many customers – myself included – did not even receive notice of the bank's closure.
As a resident and local social worker in the community, I cannot stress the importance of providing fair access to financial institutions. Had TD Canada Trust consulted with the community, they would have become aware of the social and economic hardship that this closure will cause to the residents of Weston including but not limited to:
Proximity – The branch is within walking distance of over 10,000 residents. Many of these residents do not own cars and spending nearly $7 on the TTC fare required to get to the new location will be a significant barrier for low-income customers. In Forest Hill, a far less densely populated neighbourhood, there are seven TD Banks within close proximity of each other.
Accessibility – The branch is located in a neighbourhood where many residents are seniors and people with disabilities. These residents rely on this branch because they can access banking services without having to travel far.
Lack of financial alternatives – CIBC, Scotiabank and TD Canada Trust branches have already closed down and left the neighbourhood. Predatory lenders are quickly filling the gap for financial services in York South-Weston. As you know, these institutions trap hard-working families in a cycle of debt and economic violence that will threaten the quality of life for too many in this community.
These are just some of the reasons I ask that the Financial Consumer Agency of Canada convene a public meeting between the community and TD Canada Trust to discuss and resolve the challenges posed by the bank's imminent closure.
Chiara Padovani
I contacted the MP for York South-Weston Ahmed Hussen a couple of weeks ago regarding the closure and a staff member tried to fob me off to the Ottawa office bizarrely claiming that it was not a local matter. After persisting, the staffer said he would inform the minister and call back but never did. Apparently that's not an unusual experience.
If you would like to add your voice to the initiative, contact the FCAC at 1-866-461-3222 and mention file number 363484.
Author Roy MurrayPosted on May 29, 2018 Categories Ahmed Hussen, Business, Frances Nunziata, Muncipal Election, News, Payday Loan CompaniesTags Chiara Padovani, TD Canada Trust, ward 11, York South–Weston5 Comments on Padovani tackles TD Canada Trust closing.
TD Canada Trust closing Weston branch.
TD Canada Trust bids farewell to Weston and hundreds of potential new customers.
TD Canada Trust has officially announced the upcoming closure of its Weston branch at 1979 Weston Road. Adam broke the story last year. The bean counters at TD Canada Trust have decided that closing the Weston branch will save them more money than the potential loss of clients. According to the bank handout regarding the closure,
Prior to making a decision to close a particular branch and move customer accounts to another branch, TD Canada Trust always undertakes a thorough review of the customer impact of the move. If our review indicates that the move may result in some particular concerns for all or a certain group of the affected customers, TD Canada Trust will hold an information session to discuss our plans and how we propose to deal with those concerns.
From September 21, Weston customers who stay with TD will be expected to schlep along to 2547 Weston Road at the 401. It's another example of the banking industry reaping huge profits while abandoning their customers. This also makes room for payday loan companies to fill the void; unconscionable in a community with a fair number of vulnerable residents.
There is a ray of hope via a federal government agency called 'The Financial Consumer Agency of Canada' (FCAC). According to the blurb that came with news of the closure,
"You should know, however, that under the regulations, the Commissioners of the FCAC may require TD Canada Trust representatives to hold and attend a meeting with FCAC representatives and interested parties, in order to exchange views about the closure of a branch IF:
An individual from the area affected by the closure of the branch submits a written request to the FCAC for the meeting; and
2. TD Canada Trust has not adequately consulted the community about the branch closure; and
3. The request is not frivolous or vexatious."
I'd like to know who the bank consulted with before deciding it was ok to close the branch. My wife has an account at the branch and this is the first she'd heard about the closing.
If you feel you haven't been consulted enough about the move, submit your non-frivolous written request to:
Financial Consumer Agency of Canada
427 Laurier Avenue West, 6th Floor
Ottawa ON K1R 1B9
Phone contact for TD Canada Trust: 1-866-222-3456
Phone contact for FCAC: 1-866-461-3222
Perhaps this is where York South-Weston's non-resident MP, Honourable Ahmed Hussen could get involved. Scotiabank's closing was seemingly ignored by Mr. Hussen – a self-described social activist. I contacted his office and a staff member said they would pass my concerns about the latest bank closing to the minister.
Ahmed Hussen MP: 416-656-2526
Author Roy MurrayPosted on May 17, 2018 May 18, 2018 Categories 22 John, Ahmed Hussen, News, OpinionTags Bank Closing, Payday Loans, TD Canada Trust, weston14 Comments on TD Canada Trust closing Weston branch.


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The vast majority of people with a hearing loss do not realize how much this loss affects them. However, family members may observe gradual personality changes and, typically, will suggest that they get help.
"So, what are the consequences of prolonging help," Wohlers asks? "First, is the issue of understanding speech. In simple terms, if you do not use it, you loss it. The brain's auditory cortex is not being stimulated with sound and the brain stops processing the sound into meaningful words which is referred to as 'lack of discrimination.' Many people say 'I can hear you, but I cannot understand what you are saying,'" Wohlers explained.
Most people have lost a significant amount of word understanding before they get a hearing aid and this leads to many problems. Most people believe that a hearing aid will solve the problem, when in reality, the sound can be restored, but the speech discrimination lost is often irreversible.
"Early detection and correction allows people to stay in the game in a world where communication is everything," Wohlers said.
Anyone who thinks they may have hearing loss should see their doctor, get a hearing test, and visit: www.myhearpod.com.
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