text:
Children's Charity Merger Revealed
Wednesday, 20th May 2015 at 3:09 pm
Lina Caneva
Two major children's charities, Save the Children and Good Beginnings have merged in a move they say will create one of Australia's largest agencies working to improve the lives of children in disadvantaged communities.
Good Beginnings specialises in providing early intervention and practical parenting programs for children in disadvantaged communities, and Save the Children works around Australia and in more than 120 countries delivering humanitarian aid.
"There are no simple answers or quick fixes to the complex issues that children face in vulnerable communities. That's why we have been working on a 10 year strategy," CEO of Good Beginnings Australia, Jayne Meyer Tucker said.
"Now, by joining with Save the Children an opportunity exists to leverage our joint expertise and local knowledge. Together we can broaden our reach, drive systemic change and become greater advocates for children and their rights."
"Both organisations have recognised the opportunity to work together on a shared bold goal for children in Australia," Save the Children's CEO Paul Ronalds told Pro Bono Australia News.
"I think the (Not for Profit) sector needs to face up to the challenges we have, such as increased stakeholder expectations, the technical challenges of finding and retaining donors and maintaining programs as well as the tough funding environment.
"These issues are hitting us all at once and only agile organisations that adapt to these changing conditions will survive. All organisations need to examine what their value-add is and how this can affect their mission.
"The area we want to be best practice is in early childhood and we asked ourselves what it would take and it was clear that an increase in scale would make enormous sense," he said.
Ronalds said that Save the Children's Australian funding is $20million per year which is part of its global budget of $145 million and Good Beginnings had annual funds totalling $8 million.
"If we combine this we will have close to $30 million that we can do more with, to be effective and efficient and be able to trial innovation," Ronalds said.
"The merger will allow us to invest in the technical skills around early childhood development including an evaluation that can demonstrate our impact. We really need to be clear about the impact of our programs and for policy makers to understand this, otherwise we will not survive.
Ronalds said that the merger decision was not about cutting programs.
"Yes, there will be efficiencies that allow us to be far more effective but we want to expand our programs."
Ronalds said the tipping point for Save the Children was a desire by the organisation to increase its domestic footprint and increase its Australian programs with a strong emphasis on service delivery.
He said the organisation would continue to work with refugee camps around the world and its humanitarian work would continue to be a significant part of the organisation.
Ronalds said the organisation would also continue to work providing services for children at Nauru's detention centre until its contract was up in October. But he said the end of the contract did not have any impact on the organisation's discussions around the merger which began in July 2014.
"There are clear benefits from maximising and sharing resources which is where a lot of NFP merges are derailed."
He said the commitment of Good Beginnings CEO Jayne Meyer Tucker was very focused especially with her bringing it to her board. "Without that focus it wouldn't have happened."
Ronalds said under the merger agreement a number of directors will join the Save the Children Board and the Good Beginnings organisation will become a subsidiary program-brand of Save the Children.
He said the two Sydney offices will become one, with some rationalising of back-office services to allow for general savings.
"Save the Children is 75 per cent Federal Government funded and Good Beginnings is 80 per cent Government funded."
He said both organisations receive State Government funding nationally but both bring significant corporate engagement teams together, which brings an important source of ongoing funding along with the regular donors and those who respond to humanitarian emergencies such as Nepal.
After the last Department of Social Services funding round Ronalds said there was a small net increase for Save the Children.
"The decision to merge however, was made well before the DSS funding decision at Christmas.
"By joining forces we have a unique opportunity to combine our Australian programs for young children, have greater influence on Government policy and the systems that support children, and of course leverage the best of both organisations.
"The two charities have consulted extensively with major stakeholders on the merger and have found strong support for the purposes of the merger."
Community Council of Australia Chair Tim Costello said a lot has been achieved through the reform agenda over recent years, including the introduction of the charity regulator and charities need to keep looking for ways to improve.
"Australia's diverse Not for Profit sector plays a vital role and delivers outstanding value. This merger of Save the Children Australia and Good Beginnings Australia is a commendable reform and I wish to congratulate them on their vision to put children first."
The two charities will officially join forces on July 1, 2015.
Lina Caneva | Editor | @ProBonoNews
Lina Caneva has been a journalist for more than 35 years. She was the editor of Pro Bono Australia News from when it was founded in 2000 until 2018.
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text:
\section{Introduction}

The use of zero-range contact interactions to model real interactions in
atomic systems has a long history \cite{fermi1936}. The interest in these
interaction models arises from the simplifications that can be made to complex
systems \cite{houbiers1997ssa,bruun1998ifg,roth2001esa}. Unfortunately, in
strongly interacting or high density systems, the overly singular nature of
the $\delta$-function often poses a problem. For example, when the
non-regularized zero-range interaction is used variationally in a two
component degenerate Fermi gas, it produces a collapse behavior that is not
seen in experiment
\cite{regal2004orc,kinast2004esr,zwierlein2004cpf,bartenstein2004cmb,bourdel2004esb}%
. One method of avoiding these problems is to use a renormalized interaction.
Ref. \cite{vonstecher2006renorm} does just that, introducing a
density-dependent interaction strength for a zero-range interaction. Our study
applies the hyperspherical K-harmonic method of Ref. \cite{rittenhouse2006hdd}
to this interaction.

The starting point of the K harmonic method describes the degenerate Fermi gas
with a set of $3N-1$ hyperangular coordinates on the surface of a $3N$
dimensional hypersphere of hyperradius $R$, where $N$ is the number of atoms
in the system. For this system, $R$ is simply the rms radius of the gas, but
more generally $R^{2}$ is proportional to the trace of the moment of inertia
tensor \cite{avery,SmirnovShitikova,fano1986aca}. This formulation is a
variational treatment of the $N$-body problem in which the hyperangular
behavior of the system is approximated by that of a non-interacting degenerate
Fermi gas. At first glance this approach might seem non-intuitive, but it is
natural to assume that, in a first approximation, the behavior of the gas will
be determined by its overall spatial extent. This type of approach has been
used in studying Bose-Einstein condensates \cite{bohn_esry_greene_hsbec} and
it has also been applied to finite nuclei \cite{SmirnovShitikova,Timofeyuk04}.
The theoretical approach developed here shares some mathematical kinship with
D-dimensional perturbation theory \cite{dimensionalPT}; for instance, the
$N\rightarrow\infty$ and $D\rightarrow\infty$ limits both result in
wavefunctions perfectly localized in the hyperradius. However, our goals and
motivations differ for the most part from those of Ref. \cite{dimensionalPT}.

The paper is organized as follows: Section II reviews the formulation that
leads to a hyperradial effective potential; Section III shows how to take the
hyperangular matrix element of an operator in the large $N$ limit; Section IV
applies this method to the two component gas with zero-range,
density-dependent interaction of Ref. \cite{vonstecher2006renorm}. Sections
IV(A) and IV(B) examine the resulting effective potential in systems with
positive and negative two-body scattering lengths $a$; Section IV(C) explores
the unitarity regime when $a\rightarrow\pm\infty$; Section IV(D) analyzes the
low energy radial excitation frequency. Section V expands the treatment of
Section IV to an arbitrary number of spin components and briefly examines the
resulting effective potentials. Finally, Section VI summarizes the results and
discusses future avenues of study.

\section{Hyperspherical coordinates}

The hyperspherical formulation starts with the K harmonics description given
in Ref. \cite{rittenhouse2006hdd}. We briefly restate this formulation in
order to make this article self contained. We begin by considering $N$
identical fermionic atoms of mass $m$ in a spherically symmetric oscillator
trap with oscillator frequency $\omega$ distributed equally in two spin
substates. The governing Hamiltonian is%
\begin{equation}
H=-\dfrac{\hbar^{2}}{2m}\sum_{i=1}^{N}\nabla_{i}^{2}+\dfrac{1}{2}m\omega
^{2}\sum_{i=1}^{N}r_{i}^{2}+\sum_{i>j}U_{int}\left(  \vec{r}_{ij}\right)
\label{Ham1}%
\end{equation}
where $\vec{r}_{i}$ is a trap centered vector describing the position of the
$i$th atom and $\vec{r}_{ij}=\vec{r}_{i}-\vec{r}_{j}$ is the separation vector
between atoms $i$ and $j$. Transforming into hyperspherical coordinates this
Hamiltonian becomes%
\begin{equation}
H=-\dfrac{\hbar^{2}}{2M}\left(  \dfrac{1}{R^{3N-1}}\dfrac{\partial}{\partial
R}R^{3N-1}\dfrac{\partial}{\partial R}-\dfrac{\mathbf{\Lambda}^{2}}{R^{2}%
}\right)  +\dfrac{1}{2}M\omega^{2}R^{2}+V_{int}\left(  R,\Omega\right)
\label{Ham2}%
\end{equation}
where $M=Nm$ and the generalized angular momentum operator $\mathbf{\Lambda}$
is defined by \cite{avery}%
\begin{align}
\mathbf{\Lambda}^{2}  &  =-\sum_{i>j}\Lambda_{ij}^{2},\label{hypermom}\\
\text{where }\Lambda_{ij}  &  =x_{i}\dfrac{\partial}{\partial x_{j}}%
-x_{j}\dfrac{\partial}{\partial x_{i}}.
\end{align}
The hyperradius $R$ is given as%
\begin{equation}
R\equiv\left(  \dfrac{1}{N}\sum_{i=1}^{N}r_{i}^{2}\right)  ^{1/2}.
\label{hyperr}%
\end{equation}
The remaining $3N-1$ degrees of freedom are defined by angular coordinates
$2N$ of which are the normal spherical polar angles for each atom $\left(
\phi_{1},\theta_{1},\phi_{2},\theta_{2},\ldots,\phi_{N},\theta_{N}\right)  $.
The remaining $N-1$ hyperangles are defined in the convention of
\cite{SmirnovShitikova} as%
\begin{align}
\tan\alpha_{i}  &  =\dfrac{\sqrt{\sum_{j=1}^{i}r_{j}^{2}}}{r_{i+1}%
},\label{hypera}\\
i  &  =1,2,3,...,N-1.\nonumber
\end{align}
Alternatively we may write this as%
\begin{align}
r_{n}  &  =\sqrt{N}R\cos\alpha_{n-1}\prod\limits_{j=n}^{N-1}\sin\alpha
_{j}\label{hypera2}\\
0  &  \leq\alpha_{j}\leq\dfrac{\pi}{2},\text{ }j=1,2,...,N-1\nonumber
\end{align}
where we define $\cos\alpha_{0}\equiv1$ and $\prod\limits_{j=N}^{N-1}%
\sin\alpha_{j}\equiv1$. Collectively the full set of hyperangles are referred
to as $\Omega$. For notational simplicity we have rewritten the interaction as
a function of the hyperradius and hyperangles.%
\begin{equation}
V_{int}\left(  R,\Omega\right)  =\sum_{i>j}U_{int}\left(  \vec{r}_{ij}\right)
\label{Vint}%
\end{equation}
We now make the assumption that the wave function for this system is
approximately separable into hyperradial and hyperangular parts,
\begin{equation}
\Psi=F\left(  R\right)  \Phi_{\lambda}\left(  \Omega,\sigma_{1},\sigma
_{2},\ldots,\sigma_{N}\right)  . \label{sep}%
\end{equation}
where $F\left(  R\right)  $ is an undetermined hyperradial function, $\left(
\sigma_{1},\sigma_{2},\ldots,\sigma_{N}\right)  $ are the spin coordinates for
the $N$ atoms and $\Phi_{\lambda}\left(  \Omega,\sigma_{1},\sigma_{2}%
,\ldots,\sigma_{N}\right)  $ is an eigenfunction of the hyperangular momentum
operator $\mathbf{\Lambda}^{2}$ which obeys the eigenvalue equation
$\mathbf{\Lambda}^{2}\Phi_{\lambda}\left(  \Omega\right)  =\lambda\left(
\lambda+3N-2\right)  \Phi_{\lambda}\left(  \Omega\right)  $ with $\lambda$ an
integer. We note that $R$ is completely symmetric under all permutations of
atomic space and spin coordinates, whereby all of the permutational symmetry
must be contained in $\Phi_{\lambda}$. Thus we will assume that $\Phi
_{\lambda}\left(  \Omega\right)  $ has the lowest value of $\lambda$ allowed
by the fermionic symmetry constraints. Eq. \ref{sep} can be viewed as a trial
wavefunction whose hyperradial behavior will be variationally optimized later.
To employ the variational principle we must consider the matrix element%
\[
\left\langle \Phi_{\lambda}\left\vert V_{int}\left(  R,\Omega\right)
\right\vert \Phi_{\lambda}\right\rangle
\]
where the integral is taken over all hyperangular coordinates. We now have
created an effective 1D Schr\"{o}dinger equation in the hyperradius%
\begin{equation}
\left(  \dfrac{-\hbar^{2}}{2M}\dfrac{\partial^{2}}{\partial R^{2}}%
+V_{eff}\left(  R\right)  \right)  R^{\left(  3N-1\right)  /2}F\left(
R\right)  =ER^{\left(  3N-1\right)  /2}F\left(  R\right)  \label{Ham3}%
\end{equation}
where the first derivative terms in Eq. \ref{Ham2} have been removed by
multiplying $F\left(  R\right)  $ by $R^{\left(  3N-1\right)  /2}$ and the
effective potential is given by%
\begin{equation}
V_{eff}\left(  R\right)  =\dfrac{\left(  3N-1\right)  \left(  3N-3\right)
}{8MR^{2}}+\dfrac{\lambda\left(  \lambda+3N-2\right)  }{2MR^{2}}+\dfrac{1}%
{2}M\omega^{2}R^{2}+\left\langle \Phi_{\lambda}\left\vert V_{int}\left(
R,\Omega\right)  \right\vert \Phi_{\lambda}\right\rangle . \label{Veff1}%
\end{equation}
In order to calculate $V_{eff}$ we must first specify the function
$\Phi_{\lambda}\left(  \Omega\right)  $. For $N$-body systems having
completely filled shells (magic numbers), to which we restrict this study,
this is given in Ref. \cite{rittenhouse2006hdd} as the Slater determinant:%
\begin{equation}
\Phi_{\lambda}\left(  \Omega,\sigma_{1},\sigma_{2},\ldots,\sigma_{N}\right)
=\dfrac{1}{\sqrt{N!}G\left(  R\right)  }\sum_{P}\left(  -1\right)
^{p}P\left[  \prod\limits_{j=1}^{N}\psi_{n_{j}\ell_{j}m_{j}}\left(  \vec
{r}_{j}\right)  \left\langle \sigma_{j}|m_{s_{j}}\right\rangle \right]  .
\label{phi}%
\end{equation}
Here the sum is over all possible permutations of the $N$ spatial and spin
coordinates. For brevity of notation the spin coordinates will be omitted,
i.e. we abbreviate $\Phi_{\lambda}\left(  \Omega\right)  =\Phi_{\lambda
}\left(  \Omega,\sigma_{1},\sigma_{2},\ldots,\sigma_{N}\right)  $. Here
$\psi_{n_{j}\ell_{j}m_{j}}\left(  \vec{r}_{j}\right)  $ is the spatial wave
function of the $j$th atom given by
\begin{equation}
r\psi_{n\ell m}\left(  \vec{r}\right)  =B_{\ell nl}\left(  \dfrac{r}%
{l}\right)  ^{\ell+1}L_{n}^{\ell+1/2}\left(  \dfrac{r^{2}}{l^{2}}\right)
\exp\left(  -r^{2}/2l^{2}\right)  Y_{\ell m}\left(  \omega\right)
\label{NIsol}%
\end{equation}
where $Y_{\ell m}\left(  \omega\right)  $ is a normal 3D spherical harmonic of
the solid angle $\omega$ and $l$ is the length scale of the oscillator
functions. $G\left(  R\right)  $ is the nodeless hyperradial solution to the
non-interacting $N$ particle Schr\"{o}dinger equation:
\begin{equation}
R^{\left(  3N-1\right)  /2}G\left(  R\right)  =A_{N}\exp\left(  -R^{2}%
/2\mathcal{L}^{2}\right)  \left(  R/\mathcal{L}\right)  ^{\lambda+3N/2-1/2}
\label{oscRfun}%
\end{equation}
with $\mathcal{L}=l/\sqrt{N}$ and the spin ket $\left\vert m_{s}\right\rangle
$ allows for different spin components. While $\Phi_{\lambda}$ is constructed
from independent-particle oscillator functions, it is completely independent
of the length scale given by $l$. To simplify the overall behavior we will
only consider filled energy shells, the so called magic numbers, of atoms. We
will also be particularly interested in the large $N$ limit of the system.
Finally, to simplify the procedure we rescale $R$ and $E$ in Eq. \ref{Ham3}
their values $R_{NI}\equiv\sqrt{\left\langle R^{2}\right\rangle _{NI}}$ and
$E_{NI}$ for the noninteracting $N$-particle oscillator:%
\begin{align}
E  &  =E_{NI}E^{\prime}\label{ERrescale}\\
R  &  =\sqrt{\left\langle R^{2}\right\rangle _{NI}}R^{\prime},\nonumber
\end{align}
which introduces the dimensionless variables of energy ($E^{\prime}$) and
hyperradius ($R^{\prime}$). Here the non-interacting energy $E_{NI}$ and
average hyperradius squared $\left\langle R^{2}\right\rangle _{NI}$ are given
explicitly by%
\begin{align*}
E_{NI}  &  =\left(  \lambda+\dfrac{3N}{2}\right)  \hbar\omega\\
\left\langle R^{2}\right\rangle _{NI}  &  =\left(  \dfrac{\lambda}{N}%
+\dfrac{3}{2}\right)  l_{0}^{2}.
\end{align*}
Here $l_{0}=\sqrt{\hbar/m\omega}$ is the one particle oscillator length. Under
this rescaling the effective Schrodinger equation becomes%
\begin{equation}
\left(  \dfrac{-1}{2m^{\ast}}\dfrac{\partial^{2}}{\partial R^{\prime2}}%
+\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}}\right)  R^{\prime\left(
3N-1\right)  /2}F\left(  R^{\prime}\right)  =E^{\prime}R^{\prime\left(
3N-1\right)  /2}F\left(  R^{\prime}\right)  \label{Hrescal}%
\end{equation}
with $m^{\ast}=mE_{NI}N\left\langle R^{2}\right\rangle _{NI}/\hbar^{2}$. Exact
values of $\lambda$ and $N$ for the $n$th filled shell are written in Refs.
\cite{roth2001esa,rittenhouse2006hdd} , the large $N$ limit of interest here
is approximated by%
\[
\lambda\rightarrow\dfrac{\left(  3N\right)  ^{4/3}}{4}%
\]
giving the effective potential as%
\begin{equation}
\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}}\rightarrow\dfrac
{1}{2R^{\prime2}}+\dfrac{1}{2}R^{\prime2}+\dfrac{\left\langle \Phi_{\lambda
}\left\vert V_{int}\left(  R^{\prime},\Omega\right)  \right\vert \Phi
_{\lambda}\right\rangle }{E_{NI}}. \label{Veffrescale}%
\end{equation}
Here $V_{int}\left(  R^{\prime},\Omega\right)  $ is the interaction potential
of Eq. \ref{Vint} written in terms of the rescaled hyperradius. We now need a
method for calculating the interaction matrix element in the large $N$ limit.

\section{Operator matrix elements in the $N\rightarrow\infty$ limit}

In this section we develop a method for calculating hyperangular matrix
elements of an operator in the large $N$ limit, e.g.,
\begin{equation}
\tilde{O}\left(  R^{\prime}\right)  =\int\Phi_{\lambda}\left(  \Omega\right)
O\left(  R^{\prime},\Omega\right)  \Phi_{\lambda}\left(  \Omega\right)
d\Omega. \label{Otil}%
\end{equation}
Here $O\left(  R^{\prime},\Omega\right)  $ is a general operator that is a
function of the rescaled hyperradius and the hyperangles. To allow us to
integrate over all of the $3N$ dimensions of the space, we multiply both sides
of Eq. \ref{Otil} by a $\delta$-function in the hyperradius and integrate.%
\begin{equation}
\tilde{O}\left(  R_{0}^{\prime}\right)  =\int\delta\left(  R^{\prime}%
-R_{0}^{\prime}\right)  \Phi_{\lambda}\left(  \Omega\right)  O\left(
R^{\prime},\Omega\right)  \Phi_{\lambda}\left(  \Omega\right)  d\Omega
dR^{\prime}. \label{OtilR0}%
\end{equation}
To create the $\delta$-function we consider a function of the form
\begin{equation}
R^{\prime\left(  3N-1\right)  /2}G_{N}\left(  R^{\prime}\right)  =A_{N}%
\exp\left(  -\dfrac{R^{\prime2}N\left\langle R^{2}\right\rangle _{NI}}%
{2l_{0}^{2}R_{0}^{\prime2}}\right)  \left(  \dfrac{\sqrt{N\left\langle
R^{2}\right\rangle _{NI}}R^{\prime}}{l_{0}R_{0}^{\prime}}\right)
^{\lambda+3N/2-1/2} \label{limfun}%
\end{equation}
where $A_{N}$ is a normalization constant and $l_{0}=\sqrt{\hbar/m\omega}$ is
the oscillator length. In the limit where $N\rightarrow\infty$ we see that
using this definition we have that%
\[
\lim\limits_{N\rightarrow\infty}\left[  R^{\prime\left(  3N-1\right)  /2}%
G_{N}\left(  R^{\prime}\right)  \right]  ^{2}=\delta\left(  R^{\prime}%
-R_{0}^{\prime}\right)  .
\]
From this we make the substitution in Eq. \ref{OtilR0}
\begin{equation}
\tilde{O}\left(  R_{0}^{\prime}\right)  =\lim\limits_{N\rightarrow\infty}%
\int\left[  R^{\prime\left(  3N-1\right)  }G_{N}\left(  R^{\prime}\right)
\right]  ^{2}\Phi_{\lambda}\left(  \Omega\right)  O\left(  R^{\prime}%
,\Omega\right)  \Phi_{\lambda}\left(  \Omega\right)  d\Omega dR^{\prime}.
\label{OtilR02}%
\end{equation}
Referring to Eq. \ref{phi} and remembering that the K-harmonic $\Phi_{\lambda
}\left(  \Omega\right)  $ is \emph{independent} of the oscillator length
scale, it follows that the wave function $G_{N}\left(  R^{\prime}\right)
\Phi_{\lambda}\left(  \Omega\right)  $ is merely a Slater determinant of
non-interacting single particle oscillator states with oscillator length
\begin{equation}
l_{eff}=R_{0}^{\prime}l_{0}. \label{leff}%
\end{equation}
Further, Ref. \cite{avery} gives that $R^{\prime\left(  3N-1\right)
}dR^{\prime}d\Omega$ is the full volume element for the $3N$ dimensional
space. All of this implies that in the large $N$ limit, the hyperangular
operator expectation value $\left\langle \Phi_{\lambda}\left\vert O\left(
R,\Omega\right)  \right\vert \Phi_{\lambda}\right\rangle $ is approximated by
the full expectation value of the operator for a trial wavefunction consisting
of a Slater determinant of non-interacting oscillator states, i.e.%
\begin{equation}
\tilde{O}\left(  R_{0}^{\prime}\right)  =\left\langle D_{l_{eff}}\left(
\vec{r}_{1},\vec{r}_{2},...,\vec{r}_{N}\right)  \left\vert O\left(  R^{\prime
},\Omega\right)  \right\vert D_{l_{eff}}\left(  \vec{r}_{1},\vec{r}%
_{2},...,\vec{r}_{N}\right)  \right\rangle _{3N} \label{DetME}%
\end{equation}
where $D_{l_{eff}}\left(  \vec{r}_{1},\vec{r}_{2},...,\vec{r}_{N}\right)  $ is
a Slater determinant of oscillator states with oscillator length $l_{eff}$ and
the subscript $3N$ is to indicate that the matrix element is taken over all
$3N$ spatial and $N$ spin degrees of freedom.

\section{Renormalized zero-range interactions}

To show the utility of the result in the previous section, we will apply it to
the density-dependent renormalized zero-range interactions presented in Ref.
\cite{vonstecher2006renorm}, in which a zero-range interaction is used whose
strength is dependent on the density of the gas.
\begin{equation}
U_{int}\left(  \vec{r}_{ij}\right)  =\dfrac{4\pi\hbar^{2}}{m}\dfrac
{\zeta\left(  k_{f}\left(  \vec{r}_{i}\right)  a\right)  }{k_{f}\left(
\vec{r}_{i}\right)  }\delta\left(  \vec{r}_{ij}\right)  \label{renormint}%
\end{equation}
where $a$ is the two-body s-wave scattering length and the fermi wave number
$k_{f}=k_{f}\left(  \vec{r}\right)  =\left(  6\pi^{2}\rho^{\left(  1\right)
}\left(  \vec{r}\right)  \right)  ^{1/3}$ is defined in terms of the single
spin component density, $\rho^{\left(  1\right)  }\left(  \vec{r}\right)  $.
We approximate the dimensionless renormalized function $\zeta\left(
k_{f}a\right)  $ from Ref. \cite{vonstecher2006renorm} with%
\begin{align}
\zeta\left(  k_{f}a\right)   &  =A+B\arctan\left(  Ck_{f}a-D\right)
\label{gkfa}\\
&  \text{where}\nonumber\\
A  &  =0.3949\nonumber\\
B  &  =1.1375\nonumber\\
C  &  =\dfrac{1+\tan^{2}\left(  \dfrac{A}{B}\right)  }{B}=0.9942\nonumber\\
D  &  =\tan\left(  \dfrac{A}{B}\right)  =0.3618.\nonumber
\end{align}
Two of the fitting parameters $A$ and $B$ are found by fitting the asymptotic
behavior of $\zeta\left(  k_{f}a\right)  $ as $k_{f}a\rightarrow\pm\infty$,
which are given in ref. \cite{vonstecher2006renorm} by%
\begin{align*}
\lim\limits_{k_{f}a\rightarrow\infty}\zeta\left(  k_{f}a\right)   &  =2.1817\\
\lim\limits_{k_{f}a\rightarrow-\infty}\zeta\left(  k_{f}a\right)   &  =-1.3919
\end{align*}
The constants $C$ and $D$ in Eq. \ref{gkfa} are determined by matching the
Fermi pseudo-potential in the $\left\vert k_{f}a\right\vert \ll1$ limit
\cite{fermi1936,roth2001esa}, i.e.
\begin{equation}
\dfrac{4\pi\hbar^{2}}{m}\dfrac{\zeta\left(  k_{f}a\right)  }{k_{f}}%
\rightarrow\dfrac{4\pi\hbar^{2}a}{m}. \label{Fermipseudo}%
\end{equation}
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=3in]{fig1.eps}
\end{center}
\caption{The density-dependent interaction strength function $\zeta(k_{f}a)$
is shown plotted versus $k_{f}a$.}%
\label{zetaeff}%
\end{figure}Fig. \ref{zetaeff} shows the behavior of this interaction as a
function of $k_{f}a$.

Eq. \ref{DetME} implies that%
\[
\left\langle \Phi_{\lambda}\left\vert V_{int}\left(  R^{\prime},\Omega\right)
\right\vert \Phi_{\lambda}\right\rangle =\left\langle D_{l_{efff}}\left(
\vec{r}_{1},\vec{r}_{2},...,\vec{r}_{N}\right)  \left\vert \sum_{i>j}%
U_{int}\left(  \vec{r}_{ij}\right)  \right\vert D_{l_{eff}}\left(  \vec{r}%
_{1},\vec{r}_{2},...,\vec{r}_{N}\right)  \right\rangle _{3N}%
\]
Owing to the exchange anti-symmetry of the determinantal wavefunction and the
orthogonality of the single atom wave functions (see Ref. \cite{cowan1981tas}
for details), this becomes%
\begin{align*}
\left\langle \Phi_{\lambda}\left\vert V_{int}\left(  R^{\prime},\Omega\right)
\right\vert \Phi_{\lambda}\right\rangle  &  =\dfrac{1}{2}\sum_{i,j=1}%
^{N}\left[  \int\left\vert \psi_{i}\left(  \vec{r}_{1}\right)  \right\vert
^{2}U_{int}\left(  \vec{r}_{12}\right)  \left\vert \psi_{j}\left(  \vec{r}%
_{2}\right)  \right\vert ^{2}d^{2}r_{1}d^{3}r_{2}\right. \\
&  -\left.  \delta_{m_{s_{i}}m_{s_{j}}}\int\psi_{i}^{\ast}\left(  \vec{r}%
_{1}\right)  \psi_{j}\left(  \vec{r}_{1}\right)  U_{int}\left(  \vec{r}%
_{12}\right)  \psi_{j}^{\ast}\left(  \vec{r}_{2}\right)  \psi_{i}\left(
\vec{r}_{2}\right)  d^{3}r_{1}d^{3}r_{2}\right]
\end{align*}
Here $\psi_{i}\left(  \vec{r}\right)  $ is the spatial state of an atom in the
$i$th orbital in the spin substate defined by $m_{s_{i}}$ and the sum runs
over all the single particle states in the original determinant. Substituting
Eq. \ref{renormint} for the interaction and using the $\delta$-function to
simplify one of the integrals gives%
\begin{equation}
\left\langle \Phi_{\lambda}\left\vert V_{int}\left(  R^{\prime},\Omega\right)
\right\vert \Phi_{\lambda}\right\rangle =\dfrac{1}{2}\dfrac{4\pi\hbar^{2}}%
{m}\sum_{i,j=1}^{N}\left(  1-\delta_{m_{s_{i}}m_{s_{j}}}\right)
\int\left\vert \psi_{i}\left(  \vec{r}_{1}\right)  \right\vert ^{2}\left\vert
\psi_{j}\left(  \vec{r}_{1}\right)  \right\vert ^{2}\dfrac{\zeta\left(
k_{f}\left(  \vec{r}_{1}\right)  a\right)  }{k_{f}\left(  \vec{r}_{1}\right)
}d^{3}r_{1} \label{Vint1}%
\end{equation}
If we sum over all possible spin projections $\delta_{m_{s_{i}}m_{s_{j}}}$ and
remember that we have assumed an equal distribution of atoms in each spin
substate we arrive at%
\begin{equation}
\left\langle \Phi_{\lambda}\left\vert V_{int}\left(  R^{\prime},\Omega\right)
\right\vert \Phi_{\lambda}\right\rangle =\dfrac{4\pi\hbar^{2}}{m}\int
\dfrac{\zeta\left[  k_{f}\left(  \vec{r}\right)  a\right]  }{k_{f}\left(
\vec{r}\right)  }\left[  \rho_{l_{eff}}^{\left(  1\right)  }\left(  \vec
{r}\right)  \right]  ^{2}d^{3}r. \label{Intelem}%
\end{equation}
Here we have used the definition of the density of a single spin component
\[
\rho_{l_{eff}}^{\left(  1\right)  }\left(  \vec{r}\right)  =\sum_{i}%
^{N/2}\left\vert \psi_{n_{i}\ell_{i}m_{i}}\left(  \vec{r}\right)  \right\vert
^{2}.
\]
In the Thomas-Fermi approximation, which should be exact for non-interacting
oscillator states in the large $N$ limit, this density is given in oscillator
units ($l_{0}=\hbar\omega=1$) by%
\begin{equation}
\rho_{l_{eff}}^{\left(  1\right)  }\left(  \vec{r}\right)  =\dfrac{1}{6\pi
^{2}l_{eff}^{3}}\left(  2\mu\right)  ^{3/2}\left(  1-\dfrac{r^{2}}%
{2l_{eff}^{2}\mu}\right)  ^{3/2} \label{dense}%
\end{equation}
where $\mu=\left(  3N\right)  ^{1/3}$ is the chemical potential at zero
temperature of $N$ non-interacting fermions divided equally between two
different spin substates. We may also note from Eq. \ref{leff} that in
oscillator units, $l_{eff}=R^{\prime}.$ Inserting Eq. \ref{gkfa} and
$k_{f}\left(  \vec{r}\right)  =\left[  6\pi^{2}\rho_{leff}^{\left(  1\right)
}\left(  \vec{r}\right)  \right]  ^{1/3}$ and making a change of variables in
the integral, Eq. \ref{Intelem} becomes%
\begin{align}
\left\langle \Phi_{\lambda}\left\vert V_{int}\left(  R_{0}^{\prime}%
,\Omega\right)  \right\vert \Phi_{\lambda}\right\rangle  &  =\dfrac{64N^{4/3}%
}{3^{2/3}\pi^{2}R^{\prime2}}f\left(  \dfrac{k_{f}^{0}a}{R^{\prime}}\right)
,\label{Vintfinal}\\
\text{where }f\left(  \dfrac{k_{f}^{0}a}{R^{\prime}}\right)   &  \equiv
\int_{0}^{1}y^{6}\sqrt{1-y^{2}}\zeta\left(  \dfrac{k_{f}^{0}a}{R^{\prime}%
}y\right)  dy.\nonumber
\end{align}
Here $k_{f}^{0}$ is the peak Fermi wave number for $N$ non-interacting atoms,
i.e. $k_{f}^{0}=\left[  6\pi^{2}\rho_{l_{0}}^{\left(  1\right)  }\left(
0\right)  \right]  ^{1/3}=\sqrt{2\left(  3N\right)  ^{1/3}}$. Observe that the
only parameter in this expression is $k_{f}^{0}a$ which is dimensionless.
Inserting Eq. \ref{Vintfinal} into Eq. \ref{Veffrescale} now gives the final
effective hyperradial potential in the $N\gg1$ limit.%
\begin{equation}
\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}}\rightarrow\dfrac
{1}{2R^{\prime2}}+\dfrac{1}{2}R^{\prime2}+\dfrac{256}{9\pi^{2}R^{\prime2}%
}f\left(  \dfrac{k_{f}^{0}a}{R^{\prime}}\right)  . \label{Vefffinal}%
\end{equation}

For $\left\vert k_{f}^{0}a\right\vert \ll R^{\prime}$ the integral may be
evaluated exactly using Eq. \ref{Fermipseudo} giving
\begin{equation}
\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}}\rightarrow\dfrac
{1}{2R^{\prime2}}+\dfrac{1}{2}R^{\prime2}+\dfrac{4096k_{f}^{0}a}{2835\pi
^{2}R^{\prime3}} \label{Veffbare}%
\end{equation}
which is exactly the behavior predicted in Ref. \cite{rittenhouse2006hdd}
using non-renormalized zero-range interactions.

\subsection{Repulsive effective interactions, $a>0$}

Here we explore the behavior of the DFG under a repulsive effective potential
where the two-body scattering length, $a$, is positive. The renormalized
description of the interactions used here and in Ref.
\cite{vonstecher2006renorm} is only accurate if the real two-body interactions
are purely repulsive or if the gas is somehow prevented from forming into
molecular dimer states. In other words we can only look at a gas of atoms not
of molecules. Fig. \ref{Veffrepul}, which shows $V_{eff}$ for several positive
two-body scattering lengths, also shows an example of the bare
non-renormalized effective potential. As one would expect, the repulsive
interactions cause the gas to push out against itself and against the trap
walls, which increases the overall energy and size of the gas.
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=3in]{fig2.eps}
\end{center}
\caption{The dimensionless ratio of the effective hyperradial potential to the
non-interacting total energy is plotted as a function of the dimensionless
rescaled hyperradius, for several different repulsive interaction strengths.
The non-interacting limit $k_{f}^{0}a=0$ is shown as the sold curve; the
dashed curves show the renormalized effective potential for (bottom to top)
$k_{f}^{0}a=2$, $k_{f}^{0}a=5$ and $k_{f}^{0}a=50$. Also shown is the
non-renormalized effective potential with $k_{f}^{0}a=5$ (dotted curve).}%
\label{Veffrepul}%
\end{figure}\begin{figure}[ptbptb]
\begin{center}
\includegraphics[width=3in]{fig3.eps}
\end{center}
\caption{The ground state energy of the DFG in units of the non-interacting
energy predicted by the K harmonic method (solid line) is plotted versus
$\arctan\left(  k_{f}^{0}a\right)  /\pi$ and compared with that predicted by
the Hartree-Fock method with 2280 atoms (circles). The dashed line is the
ground state energy predicted by the K harmonic method using the bare Fermi
pseudo-potential.}%
\label{Erepuls}%
\end{figure}\begin{figure}[ptbptbptb]
\begin{center}
\includegraphics[width=3in]{fig4.eps}
\end{center}
\caption{The ground state average squared hyperradius of the two-component DFG
in the large-N limit, divided by the non-interacting value for this quantity,
is plotted versus $\arctan\left(  k_{f}^{0}a\right)  /\pi$. Also shown are the
values predicted by the Hartree-Fock method with 2280 atoms (circles). The
dashed line is the ground state energy predicted by the K harmonic method
using the bare Fermi pseudo-potential.}%
\label{Rrepuls}%
\end{figure}

The true ground state energy would be found by solving Eq. \ref{Hrescal} for
the lowest eigenvalue, but if we examine $m^{\ast}$ in the large $N$ limit one
can see that the second derivative term becomes negligible. The ground state
energy and hyperradius can thus be found by minimizing the effective potential
$V_{eff}\left(  R^{\prime}\right)  .$ Figs. \ref{Erepuls} and \ref{Rrepuls}
show the energy and average squared hyperradius of the minimum of $V_{eff}$ as
functions of $k_{f}^{0}a$, compared to those same values calculated using the
bare non-renormalized effective potential given by Eq. \ref{Veffbare}. Also
shown are the ground state energy and average hyperradius squared predictions
from the Hartree-Fock method using the renormalized interaction. As the
interaction gets stronger the renormalized energies and hyperradii flatten out
and approach a constant in the unitarity limit. This behavior will be examined
more carefully in a Section IV(C). For $k_{f}^{0}a\ll1$ the Fermi
pseudo-potential approximation is in good quantitative agreement with the
renormalized interactions, but diverges dramatically as $k_{f}^{0}%
a\rightarrow\infty$. This dramatizes the breakdown of the non-renormalized
zero-range approximation, which overestimates the interaction strength as the
unitarity regime is approached.

\subsection{Attractive effective interactions, $a<0$}

Fig. \ref{Veffattract} shows the behavior of the effective potential for some
attractive values of the two-body scattering length, along with an example of
the non-renormalized effective potential. The decisive qualitative importance
of the renormalization is now apparent; for attractive non-renormalized
interactions the interaction term in the effective potential will always take
over as $R^{\prime}\rightarrow0$ which creates an inner collapse region where
the ground state energy of the gas diverges toward $-\infty$ and the gas lives
in a metastable outer potential well. In contrast, the renormalized effective
potential has no such collapse phenomenon, and the ground state of the gas is
in a global minimum. This behavior will be discussed further in Section IV(C).

Figs. \ref{Eattract} and \ref{Rattract} show the ground state energy and
average hyperradius squared of the system compared to the non-renormalized
values. \begin{figure}[ptb]
\begin{center}
\includegraphics[width=3in]{fig5.eps}
\end{center}
\caption{The effective potential in units of the non-interacting energy is
plotted versus the hyperradius in units of $\sqrt{\left\langle R^{2}%
\right\rangle _{NI}}$, for several interaction strengths. The non-interacting
limit $k_{f}^{0}a=0$ is shown as the sold curve and the dashed curves show the
renormalized effective potential for (top to bottom) $k_{f}^{0}a=-1$ and
$k_{f}^{0}a=-5$. Also shown is the non-renormalized effective potential with
$k_{f}^{0}a=-1$ (dotted curve).}%
\label{Veffattract}%
\end{figure}\begin{figure}[ptbptb]
\begin{center}
\includegraphics[width=3in]{fig6.eps}
\end{center}
\caption{The ground state energy of the DFG, predicted by the K harmonic
method in units of the non-interacting energy (solid line) is shown as a
function of $\arctan\left(  k_{f}^{0}a\right)  $/$\pi$ as a solid line. Also
shown is that predicted by the Hartree-Fock method for 2280 atoms (circles).
The dashed line is the ground state energy predicted by the K harmonic method
using the bare Fermi pseudopotential.}%
\label{Eattract}%
\end{figure}\begin{figure}[ptbptbptb]
\begin{center}
\includegraphics[width=3in]{fig7.eps}
\end{center}
\caption{The ground state average hyperradius squared of the DFG in units of
the non-interacting average hyperradius squared predicted by the K harmonic
method (solid line) is shown plotted against $\arctan\left(  k_{f}%
^{0}a\right)  /\pi$ compared with that predicted by the Hartree-Fock method
with 2280 atoms (circles). The dashed line is the ground state energy
predicted by the K harmonic method using the bare Fermi pseudopotential.}%
\label{Rattract}%
\end{figure}The effects of renormalization for attractive interaction are even
more striking than in the repulsive interaction case. Without renormalization
the metastable region of the effective potential disappears for $k_{f}%
^{0}a<-1.21$ \cite{rittenhouse2006hdd} and the gas has no barrier to prevent
it from falling into the central collapse region. With renormalization, as
$k_{f}^{0}a\rightarrow-\infty$ the energy and average hyperradius squared go
towards a fixed value. Figs. \ref{Eattract} and \ref{Rattract} also show the
ground state energy and average squared hyperradius predictions from the
Hartree-Fock method. For $\left\vert k_{f}^{0}a\right\vert \ll1$ the
non-renormalized and renormalized values are in good agreement, but as
$k_{f}^{0}a\rightarrow-1.21$ the Fermi-pseudopotential diverges away from the
renormalized interaction results. In fact, Ref. \cite{rittenhouse2006hdd}
predicts a collapse of the gas for non-renormalized interactions at $k_{f}%
^{0}a=-1.21$. Just before the point of collapse the ground state energy is
predicted to be $E=\sqrt{5}E_{NI}/3=0.745E_{NI}$. Not only does the
renormalized effective potential cut off the collapse behavior, it also allows
the gas to reach a lower energy than it would be able to without the density
dependence. In other words, if the interaction coefficient in $U_{int}\left(
\vec{r}_{ij}\right)  $ were not density-dependent, but merely involved a cut
off as $k_{f}^{0}a\rightarrow-\infty,$ the gas would not be able to reach the
unitarity energy before collapsing.

\subsection{Unitarity regime}

In this section we explore the behavior of $V_{eff}$ in the strong interaction
regime, i.e. $a\rightarrow\pm\infty$. Examining Eq. \ref{Vintfinal} shows that
the unitarity limit is when $\left\vert k_{f}^{0}a\right\vert \gg R^{\prime}%
$.
\begin{align*}
\dfrac{\left\langle \Phi_{\lambda}\left\vert V_{int}\left(  R_{0}^{\prime
},\Omega\right)  \right\vert \Phi_{\lambda}\right\rangle }{E_{NI}}  &
\rightarrow\dfrac{256\zeta_{\pm}}{9\pi^{2}R^{\prime2}}\int_{0}^{1}y^{6}%
\sqrt{1-y^{2}}dy\\
&  =\dfrac{5\zeta_{\pm}}{9\pi R^{\prime2}}%
\end{align*}
where $\zeta_{\pm}$ is the maximum ($+$) or minimum ($-$) value acquired by
the interaction function $\zeta\left(  k_{f}a\right)  $. This gives a total
effective potential of%
\begin{align}
\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}}  &  \rightarrow\dfrac{1}%
{2}R^{\prime2}+\dfrac{1/2+5\zeta_{\pm}/9\pi}{R^{\prime2}}\label{VeffUnit}\\
&  =\left\{
\begin{array}
[c]{ccc}%
\dfrac{1}{2}R^{\prime2}+\dfrac{0.886}{R^{\prime2}} & \text{for} & k_{f}%
^{0}a\gg R^{\prime}\\
&  & \\
\dfrac{1}{2}R^{\prime2}+\dfrac{0.254}{R^{\prime2}} & \text{for} & -k_{f}%
^{0}a\gg R^{\prime}%
\end{array}
\right.  .\nonumber
\end{align}
The hyperradius is a \emph{collective} coordinate so that, as $R\rightarrow0$,
all of the atoms in the system are forced to the center of the trap, which
increases the density of the system. Thus, for small hyperradii, we expect
$V_{eff}$ to act like Eq. \ref{VeffUnit}. In fact, the dashed curves in Figs.
\ref{Veffrepul} and \ref{Veffattract} show that as $R^{\prime}\rightarrow0$
the renormalized effective potential curves start to behave the same,
independently of $a$. Alternatively if the two-body scattering length
approaches $-\infty$, e.g. near a resonance, then we can expect the
interaction to approach Eq. \ref{VeffUnit} for all hyperradii.

In the case where $k_{f}^{0}a\rightarrow\pm\infty$ the effective potential
takes on the form of Eq. \ref{VeffUnit}. Minimization of $V_{eff}$ as a
function of $R^{\prime}$ gives a ground state energy:%
\begin{align}
\dfrac{E}{E_{NI}}  &  =\sqrt{1+10\zeta_{\pm}/9\pi}\label{Eunit}\\
&  =\left\{
\begin{array}
[c]{ccc}%
1.331 & for & a\rightarrow\infty\\
0.712 & for & a\rightarrow-\infty
\end{array}
\right.  .\nonumber
\end{align}
The average hyperradius of the gas is described by the value of the
hyperradius at this minimum which is given by%
\begin{align}
R_{\min}^{\prime}  &  =\left(  1+10\zeta_{\pm}/9\pi\right)  ^{1/4}%
\label{Runit}\\
&  =\left\{
\begin{array}
[c]{ccc}%
1.154 & for & a\rightarrow\infty\\
0.844 & for & a\rightarrow-\infty
\end{array}
\right. \nonumber
\end{align}
At first glance this may seem strange, one might expect the behavior to be
smooth across a resonance, and the energy to connect smoothly from the
$a\rightarrow-\infty$ limit to the $a\rightarrow\infty$ limit
\cite{astrakharchik2004esf}. But the density-dependent renormalization used
here only applies to a degenerate Fermi gas of atoms and does not allow for
the incorporation of higher order correlations, i.e. the formation of diatomic
molecules. Presumably there is another branch in the renormalization that will
match continuously with the $a\rightarrow-\infty$ limit (for a more complete
discussion see section II of Ref. \cite{vonstecher2006renorm}).

Another quantity of interest is the chemical potential of the interacting gas
at unitarity, given by
\[
\mu_{u}=\dfrac{\hbar^{2}k_{f}^{2}\left(  0\right)  }{2m}\left(  1+\beta
\right)  ,
\]
where $\beta$ is a universal parameter. From the single spin component density
given in Eq. \ref{dense} we find that the interacting peak Fermi wavenumber is%
\begin{equation}
k_{f}\left(  0\right)  =\dfrac{k_{f}^{0}}{R_{\min}^{\prime2}}. \label{unitwn}%
\end{equation}
Further, from Eqs. \ref{Eunit} and \ref{Runit}, the ratio of the chemical
potential of the interacting unitarity-limit gas to that of the
non-interacting gas can be written in terms of the rescaled hyperradius as:%
\begin{equation}
\dfrac{\mu_{u}}{\mu}=\dfrac{E}{E_{NI}}=R_{\min}^{\prime2}. \label{unitchem}%
\end{equation}
Solution of Eqs. \ref{unitwn} and \ref{unitchem} in the $a\rightarrow-\infty$
limit yields%
\[
\beta=10\zeta_{-}/9\pi=-0.49.
\]
This value of $\beta$ coincides, not surprisingly, with the value predicted by
the renormalized Hartree-Fock calculation of Ref. \cite{vonstecher2006renorm}.
Even though neither Ref. \cite{vonstecher2006renorm} nor our present treatment
explicitly incorporates Cooper-type fermion pairing, this unitarity limit
$\beta$ is in fair agreement with quantum Monte Carlo estimates that have
obtained $\beta=-0.58$ (Ref. \cite{astrakharchik2004esf}.) and $-0.56$
(Ref.\cite{chang2004qmc} ).

\subsection{Breathing mode excitations}

Refs. \cite{bohn_esry_greene_hsbec,rittenhouse2006hdd} showed that one of the
strengths of the K harmonic method is its ability to predict the lowest radial
excitation, the breathing mode. With the effective potential derived here,
this frequency is found by simply examining the second order Taylor series
about the minimum in $V_{eff}\left(  R^{\prime}\right)  $ and comparing the
resulting Hamiltonian to that of an oscillator. The approximate Hamiltonian is
given by%
\begin{equation}
H=\dfrac{-1}{2m^{\ast}}\dfrac{d^{2}}{dR^{\prime2}}+\dfrac{E_{GS}}{E_{NI}%
}+\dfrac{1}{2E_{NI}}\left.  \dfrac{\partial^{2}V_{eff}}{\partial R^{\prime2}%
}\right\vert _{R^{\prime}=R_{\min}^{\prime}}\left(  R^{\prime}-R_{\min
}^{\prime}\right)  ^{2} \label{Htayor}%
\end{equation}
where $E_{GS}$ is the ground state energy of the system and $R_{\min}^{\prime
}$ is the hyperradius that minimizes $V_{eff}$ scaled by $\sqrt{\left\langle
R^{2}\right\rangle _{NI}}$. this can be recast as the oscillator Hamiltonian%
\begin{equation}
H_{ho}=\dfrac{-1}{2m^{\ast}}\dfrac{d^{2}}{dR^{\prime2}}+\dfrac{E_{GS}}{E_{NI}%
}+\dfrac{1}{2}m^{\ast}\omega_{0}^{\prime2}\left(  R^{\prime}-R_{\min}^{\prime
}\right)  ^{2}. \label{Hho}%
\end{equation}
Here $\omega_{0}^{\prime}$ is given by the second derivative of the $V_{eff}$
at the minimum:%
\begin{subequations}
\begin{equation}
\omega_{0}^{\prime}=\sqrt{\dfrac{1}{m^{\ast}E_{NI}}\left.  \dfrac{\partial
^{2}V_{eff}}{\partial R^{\prime2}}\right\vert _{R^{\prime}=R_{\min}^{\prime}}%
}. \label{BM1}%
\end{equation}
This parameter is the breathing mode frequency in units of $E_{NI}/\hbar$.
Taking $m^{\ast}\rightarrow E_{NI}^{2}/\hbar^{2}\omega^{2}$ in the large $N$
limit gives the breathing mode frequency, $\omega_{0}$:%
\end{subequations}
\begin{equation}
\omega_{0}=\omega\sqrt{\dfrac{1}{E_{NI}}\left.  \dfrac{\partial^{2}V_{eff}%
}{\partial R^{\prime2}}\right\vert _{R^{\prime}=R_{\min}^{\prime}}}
\label{BM2}%
\end{equation}
where $\omega$ is the oscillator frequency of the trap. Fig. \ref{breath}
shows the breathing mode frequency in units of the oscillator frequency
compared to that calculated using non-renormalized interactions. Of course the
breathing mode frequency in the non-interacting limit is $\omega_{0}=2\omega$,
but surprisingly, the frequency turns over and returns to the non-interacting
value as $k_{f}^{0}a\rightarrow\pm\infty$. Upon inserting the second
derivative of Eq. \ref{VeffUnit}, the effective potential as $k_{f}%
^{0}a\rightarrow\pm\infty$ we obtain
\begin{equation}
\omega_{0}=\omega\sqrt{\dfrac{1}{E_{NI}}\left(  1+\left.  3\dfrac
{1+10\zeta_{\pm}/9\pi}{R^{\prime4}}\right\vert _{R^{\prime}=R_{\min}^{\prime}%
}\right)  }. \label{BMunit}%
\end{equation}
When the minimum hyperradius from Eq. \ref{Runit} is plugged in, this implies
that the unitarity limits for the breathing mode frequency are both
$\omega_{0}=2\omega$. This unitarity behavior has also been predicted in Ref.
\cite{werner2006ugi}.\begin{figure}[ptb]
\begin{center}
\includegraphics[width=3in]{fig8.eps}
\end{center}
\caption{The breathing mode frequency $\omega_{0}$ is shown in units of the
trap frequency $\omega$ versus $\arctan\left(  k_{f}^{0}a\right)  /\pi$ . The
solid curve shows the breathing mode predicted using the renormalized
interaction while the dashed curve shows the prediction based on the bare
Fermi pseudo-potential.}%
\label{breath}%
\end{figure}

\section{Multiple spin components}

Next consider what happens when the atoms in the gas are equally distributed
among an arbitrary number $\chi$ of spin substates. First, we assume, in order
to limit parameter space, that the s-wave scattering length between two atoms
in any two different spin states has the same value, $a$. Also we neglect the
possibility of inelastic collisions, e.g. of the type:%
\[
\left\vert m_{1}=\dfrac{3}{2}\right\rangle +\left\vert m_{2}=-\dfrac{3}%
{2}\right\rangle \rightarrow\left\vert m_{1}^{\prime}=\dfrac{1}{2}%
\right\rangle +\left\vert m_{2}^{\prime}=-\dfrac{1}{2}\right\rangle .
\]
To proceed, we must also address the question of what density should go into
the renormalized interactions. A particle in spin state $i$ cannot interact
with any other particle in the same spin state by the zero range
approximation, but the density that determines $k_{f}\left(  r\right)  $ in
the renormalization function $\zeta$ might be chosen in various alternative
ways, and we have not yet developed a unique criterion to specify the
appropriate renormalization in this context. As an initial exploration, we
make the assumption that the density of the component that particle $i$ is
interacting with is the density that modifies the interaction, i.e.%
\[
V\left(  r_{ij}\right)  =\dfrac{4\pi\hbar^{2}\zeta\left[  k_{f}^{\left(
j\right)  }\left(  \vec{r}_{i}\right)  a\right]  }{k_{f}^{\left(  j\right)
}\left(  \vec{r}_{i}\right)  }\delta\left(  \vec{r}_{i}-\vec{r}_{j}\right)
\]
where $k_{f}^{\left(  j\right)  }\left(  \vec{r}\right)  =\left[  6\pi^{2}%
\rho^{\left(  j\right)  }\left(  \vec{r}\right)  \right]  ^{1/3}$ is the Fermi
wave number of the spin component that particle $j$ belongs to.

The derivation following this assumptions the same as that for the
two-component gas, up to Eq. \ref{Vint1}. The only added pieces of information
needed are the common density of each component in the effective trap with
oscillator length $l_{eff}$, the chemical potential, the non-interacting
ground state energy and the average hyperradius squared for the system with
$\chi$ spin substates in the large $N$ limit:%
\begin{align}
\rho_{l_{eff}}^{\left(  1\right)  }\left(  \vec{r}\right)   &  =\dfrac{1}%
{6\pi^{2}l_{eff}^{3}}\left(  2\mu\right)  ^{3/2}\left(  1-\dfrac{r^{2}%
}{2l_{eff}^{2}\mu}\right)  ^{3/2}\label{RNImult}\\
\mu &  =\left(  \dfrac{6N}{\chi}\right)  ^{1/3}\\
E_{NI}  &  =\hbar\omega\dfrac{\left(  6N\right)  ^{4/3}}{\chi^{1/3}%
8}\label{ENI}\\
\left\langle R^{2}\right\rangle _{NI}  &  =\dfrac{\hbar}{m\omega}%
\dfrac{\left(  6N\right)  ^{4/3}}{\chi^{1/3}8N}.
\end{align}
The sum over spin substates in Eq. \ref{Vint1} results in a factor of
$\chi\left(  \chi-1\right)  $, leaving%
\begin{equation}
\left\langle \Phi_{\lambda}\left\vert V_{int}\left(  R^{\prime},\Omega\right)
\right\vert \Phi_{\lambda}\right\rangle =\dfrac{4\pi\hbar^{2}}{m}\dfrac
{\chi\left(  \chi-1\right)  }{2}\int\dfrac{\zeta\left[  k_{f}\left(  \vec
{r}\right)  a\right]  }{k_{f}\left(  \vec{r}\right)  }\left[  \rho_{l_{eff}%
}^{\left(  1\right)  }\left(  \vec{r}\right)  \right]  ^{2}d^{3}r.
\label{Vint2}%
\end{equation}
Another change of integration variables which gives the effective hyperradial
potential as%
\begin{align}
\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}}  &  =\dfrac{1}{2R^{\prime
2}}+\dfrac{1}{2}R^{\prime2}+\left(  \chi-1\right)  \dfrac{64N^{4/3}}%
{3^{2/3}\pi^{2}R^{\prime2}}f\left(  \dfrac{k_{f}^{0}a}{R^{\prime}}\right)
.\label{Veffmulti}\\
f\left(  \dfrac{k_{f}^{0}a}{R^{\prime}}\right)   &  \equiv\int_{0}^{1}%
y^{6}\sqrt{1-y^{2}}\zeta\left(  \dfrac{k_{f}^{0}a}{R^{\prime}}y\right)
dy.\nonumber
\end{align}
Comparison with Eq. \ref{Vefffinal}, the effective potential for the two
component gas, demonstrates that the extra spin components increase the
strength of the interaction by a factor of $\chi-1$.

In the limit where $k_{f}^{0}a\rightarrow-\infty$ the effective potential
limits to%
\begin{equation}
\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}}\rightarrow\dfrac{1}%
{2}R^{\prime2}+\dfrac{1+10\left(  \chi-1\right)  \zeta_{-}/9\pi}{2R^{\prime2}}
\label{Veffmultiunit}%
\end{equation}
Taking $\chi=3$ yields%
\begin{align}
\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}}  &  \rightarrow\dfrac{1}%
{2}R^{\prime2}+\dfrac{1-20\zeta_{-}/9\pi}{2R^{\prime2}}\label{Veff3unit}\\
&  =\dfrac{1}{2}R^{\prime2}+\dfrac{0.00772}{R^{\prime2}}.\nonumber
\end{align}
In this limit the barrier preventing the gas from falling in to the center of
the trap, i.e. $R^{\prime}\rightarrow0$, is \emph{very }weak. The unitarity
energy and average hyperradius are given by%
\begin{align*}
\allowbreak\dfrac{E}{E_{NI}}  &  =\sqrt{1+20\zeta_{-}/9\pi}=0.124\\
\dfrac{R_{\min}}{\sqrt{\left\langle R^{2}\right\rangle _{NI}}}  &  =\left(
1+20\zeta_{-}/9\pi\right)  ^{1/4}=0.352.
\end{align*}
Since the K harmonic method is intrinsically a variational calculation, it is
very possible that a better calculation, for example using the Hartree-Fock
method, might show that the 3 component gas becomes mechanically unstable in
the unitarity limit \cite{heiselberg2001fsl}. In other words the three
component gas might collapse in a manner similar to that of the Bosenova
\cite{bohn_esry_greene_hsbec,shuryak1996mbc,stoof1997mqt,bradley1995ebe,donley2001dca}%
.

With $\chi=4$ the effective potential at $k_{f}^{0}a\rightarrow-\infty$
becomes entirely attractive%
\begin{align}
\dfrac{V_{eff}\left(  R^{\prime}\right)  }{E_{NI}} &  \rightarrow\dfrac{1}%
{2}R^{\prime2}+\dfrac{1-30\zeta_{-}/9\pi}{2R^{\prime2}}\label{Veff4unit}\\
&  =\dfrac{1}{2}R^{\prime2}-\dfrac{\allowbreak0.238}{R^{\prime2}}\nonumber
\end{align}
meaning that the gas is predicted to collapse down toward $R^{\prime
}\rightarrow0$. Presumably some very rich and complex dynamics (cluster
formation, inelastic collisions, etc.) occur during this process, but our K
harmonic trial wave function is too simple to describe these phenomena. When
the local minimum in $V_{eff}\left(  R^{\prime}\right)  $ becomes a saddle
point the gas is no longer mechanically stable and is free to collapse. This
occurs at a critical interaction strength of $k_{f}a_{c}=-0.657$.

\section{Summary and prospects}

We have shown that applying the variational hyperspherical treatment of Ref.
\cite{rittenhouse2006hdd} to a density-dependent, zero-range, s-wave
interaction produces a unique, physically intuitive picture of the behavior of
gas. By fixing the hyperangular behavior of the gas, in the large atom number
limit, a simple 1D effective potential in a collective coordinate, the
hyperradius $R$ was produced. The ground state energy and rms radius of the
two component gas predicted by this method are in excellent agreement with
those predicted using the Hartree-Fock method. The ground state energy of the
two component gas goes to a finite, unitarity limit as the two-body scattering
length $a$ diverges to $+\infty$. Due to the inability of our hyperspherical
trial function and the renormalized, density-dependent interaction to describe
two-body bound states, the method only applies (at this level of our
development) to a degenerate Fermi gas of atoms. In the $a\rightarrow-\infty$
the two component gas has a stable ground state and avoids the collapsing
behavior predicted for the bare Fermi pseudo-potential, and has a ground state
energy of $0.73$ times the ground state energy of $N$ non-interacting fermions
in an isotropic oscillator trap. This energy is lower than the minimum energy
that could be reached before collapse, which would be predicted without
density-dependent interactions. In the unitarity limit, the lowest radial
excitation frequency, the breathing mode frequency, for the two component gas
is the \emph{same} as the non-interacting value in both the positive and
negative asymptotic scattering length limits.

The effective potential for a three component gas has a weak hyperradial
barrier in the $a\rightarrow-\infty$ limit, which just barely prevents
collapse of the gas. Because of the variational nature of the K-harmonic
method, a better approximation of the wavefunction, as in the Hartree-Fock
method, might conceivably allow the three component system to collapse. For
four or more spin components the repulsive barrier becomes attractive in the
$a\rightarrow-\infty$ and the gas is predicted to collapse in a manner similar
to the Bosenova
\cite{bohn_esry_greene_hsbec,shuryak1996mbc,stoof1997mqt,bradley1995ebe,donley2001dca}%
. Some uncertainty about this prediction still exists, however, because we
have not yet validated our renormalization procedure for DFGs containing more
than two spin components. Here, the interactions between different spin
components was assumed to be the same for all possible combinations. A more
accurate treatment would include different two-body scattering lengths for
different combinations of spin components, though if all scattering lengths
are large and negative, the prediction of instability would still apply.
Higher-order correlations, such as BEC or BCS pairing are beyond the scope of
this work, and relegated to future publication.

\section*{ACKNOWLEDGMENTS}

We are indebted to Javier von Stecher for extensive discussions, and for
providing some of the Hartree-Fock results that are shown in this paper for
comparison. One of us (CHG) received partial support from the Miller Institute
for Basic Research in Science, University of California Berkeley. This work
was also supported in part by funding from the NSF.

redpajama_set_name: RedPajamaArXiv


Beautifully engraved stock certificate from the International Mercantile Marine Company issued in 1905. This historic document was printed by American Bank Note Company and has an ornate border around it with a vignette of ships in a harbor. This item has the signatures of the Company's Agent and is over 113 years old. The certificate has the names of the Voting Trustees printed on the face. The names are J. P. Morgan, W. J. Pirrie, J. Bruce Ismay, Peter A. B. Widener and Charles Steele. Pre Titanic sinking certificates with the Trustees' names printed on the front are very hard to find. The certificate was issued to Arthur Hill and is endorsed by him on the verso. Stamp cancelled.
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Suddenly, a man holding a baby came up to her. Although he didn't want to be saved, there was no one to save the baby. With no one else willing, Winnie said that she would save the child. She now had a good reason to be rescued. She then made her way to collapsible D and a crew member helped her and the baby into the boat. Once the boat was loaded , Chief Officer Wilde ordered it to be lowered. As it slowly descended to the water, passengers Hugh Woolner and Mauritz Bj�rnstr�m-Steffansson, who had helped load and launch collapsible C, saw that although the boat was mostly full, there was a small space in the bow of the life boat. The two men leaped over the side of the ship. Steffanson landed in it the life boat, but Woolner found himself hanging over the side and Steffanson pulled him in.


text:
Medium Will Now Focus On Rewarding Creators
You probably know Medium, the publishing platform which was founded by Twitter's co-founder Evan Williams. It was geared to make it easy for anyone to write a lengthy post complete with simple formatting tools for easy sharing on Medium or on other social media channels.
Maintaining such a platform is a challenge and this very fact was explained by Ev Williams in a Medium post. The company has decided to make some major changes which he outrightly outlined.
First, Medium will axe a third of its team (50 jobs) in the sales and support departments. This is due to the company's renewed focus of changing its business model to its "original plan."
What was this original plan? Well according to Ev, "our vision when we started in 2012 was ambitious: To build a platform that defined a new model for media on the internet." As noble as the plan was, there were problems in the way. This included the likes of misinformation and the pressure by publishers to put out more content cheaply.
According to Ev, the current system at Medium was "unsustainable and unsatisfying for producers and consumers alike" and that is why they are doing away with it. They had started on various strategies to lure commercial publishers on the platform which according to him was a success. They were able to get the "best publishers on the web" on Medium as well as seeing interests from big brands. The platform is growing too as according to Williams, readers and published posts were up 300% year on year which is impressive.
However, they realized they realized they didn't have a proper platform to drive people to pay for quality content on Medium. Ev realized that the current ad driven digital media system is broken and this is why they are making a change.
This is why Medium is moving to a new model where writers and creators will be rewarded based on their value of their creations. This will not happen soon according to Williams but it will be interesting to see what Medium will come up with as they focus on changing the digital publishing world as we know it.
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redpajama_set_name: RedPajamaCommonCrawl


text:
P30 News
Triangle Shirtwaist Factory Fire Inspires Elizabeth Warren Speech
Posted on July 4, 2020 by P30 News
Category:News
GREENWICH VILLAGE, NY — Democratic presidential candidate Elizabeth Warren drew thousands to Washington Square Park Monday evening — citing in her campaign speech the neighborhood's history for better workers' safety regulations as a touchpoint for her own plan for structural change as president.
Just hours after the Working Families Party endorsed her 2020 campaign, Sen. Warren, of Massachusetts, spoke at length about the 146 women, mostly immigrants, who died in a fire at the Triangle Shirtwaist Factory in 1911 near the square from a lack of safety features and locked doors, leading to workers' safety rights.
"We're not here because of famous arches or famous men. In fact, we're not here because of men at all," Warren said. "We're here because of some hard-working women."
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The 1911 fire "was one of the worst industrial disasters in American history," Warren said.
In her speech, she further detailed her plan to tackle corruption in Washington D.C. — from rewriting the ethics codes for judges and "ending lobbying as we know it," she said.
"I know this change is possible and I know it because America has made big structural change before," she said, referencing Frances Perkins, who became the first woman to serve as a cabinet secretary as the secretary of labor after she watched in horror as women jumped to their deaths from the factory engulfed in flames.
She tied together the country's most pressing issues — climate change, gun safety, and health care — and cited corruption as the stalemate that has "paralyzed" U.S. democracy.
"On the face of it, these three are totally different issues. But despite our being the strongest and wealthiest country in the history of the world, our democracy is paralyzed. And why? Because giant corporations have bought off our government," she said, calling President Donald Trump "corruption in the flesh."
WWE Announces Major Sponsorship Deal With Kmart
WWE SmackDown Preview For Tonight: Cena, Triple H & Cole Appear
Sunny Comments On WWE Return, Ivelisse Velez On Twitter, Dolph Ziggler Shirt
WWE Stars Appearing On MTV Tonight, AJ Lee Works SmackDown Shows, More
Sunny Bumped From WWE HOF Broadcast, Michaels' Speech Heavily Edited


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text:
Marie Vladimirovna Dolgorukovová (rusky Мария Владимировна Долгорукова;(1608 – 17. ledna 1625) byla ruská carevna v letech 1624–1625 coby manželka ruského cara Michaila I. Fjodoroviče.

Život
Marie byla dcerou Vladimira Dolgorukova (1569–1633), jehož rod měl vazby na starou vládnoucí dynastii Rurikovců. Nevěstou cara Michaila Fjodoroviče byla prohlášena 12. července roku 1623, svatební obřad se uskutečnil 19. září roku 1624; ještě v průběhu svatebních oslav onemocněla a již se neuzdravila. Zemřela o několik měsíců později, 17. ledna následujícího roku. Jedna z teorií tvrdí, že byla otrávena, jiná, že carevna zemřela při porodu (což by mohlo znamenat, že si Michail vzal těhotnou ženu; spíše však mohlo jít o předčasný porod či potrat), v každém případě z jejího krátkého manželství s carem Michailem Fjodorovičem nevzešel žádný přeživší potomek.

Reference

Narození v roce 1601
Úmrtí v roce 1625
Ruské carevny
Ženy
Dolgorukovové
Narození v roce 1608

redpajama_set_name: RedPajamaWikipedia


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text:
Himanshu Thapa
Tails on Trails
Without a doubt, dogs are the best companions we could ever have, aren't they? They stay with you, they ensure you are okay and will do anything to keep you happy. Ever forgiving, ever-loving, gem of an animal they are. And I experienced their love first hand when I went on a trek in the Himalayas. I was doing my first solo trek, Kedarkantha, in the year 2012, when this black Himalayan furry boy started walking with me. He was extremely friendly, but that's how almost every other dog in the Himalayas are. They are very sweet to you if you show them love and kindness.
The Himalayan dog and I bonding on the trek
THE LOCAL GUIDE
I don't know how and when but he seemed to have become my guide. He would walk ahead, showing me the way usually and then would wait and look back if I stopped. I could see that he was guiding me through the trail. He would rest when I did, drink water where ever he found the water sources and encourage me to hydrate as well. That was really surprising because before that I hadn't really experienced this kind of companionship with any animal.
But being with him made me feel happy and safe. I would try to match his pace, race a little with him and became his friend. The funny part was, I tried drinking water like him and now nine years later, I don't really carry a water bottle on many of my treks. I simply put my mouth into the stream, it is refreshing, you should try it out sometime.
Unfortunately, he didn't accompany me to the summit but was waiting for me when I made my way back to the trail to head back. Of course, seeing such a loyal companion, I bonded with him. I started talking with him asking various questions like "Have you ever been to the top?" and even replied to him with an "if not, you should, it's magical out there". He did look me in the eye when he spoke to me, and it felt like he said that he goes there every other day. It probably was true, I am sure it wasn't his first time up the trail with a human. He was loyal to everyone who would trek up I suppose.
Leading the way on the high Himalaya
He stopped at a small tea shop before the entrance of the village about a kilometre before the end of the trail and looked at me with his puppy dog eyes. I was hungry too, and this good boy definitely deserved a good meal. He ate around four packets of Parle-G and two slices of bread and then wagging his tail as if to say thank you, he took off on the other trail. My fondest memory would be of seeing his black hairy tail at the end. My loyal companion.
A few years later when I started working in the outdoors as a trek leader I met many tails on the trail. I realised how amazing they are, how kind they could be and how some of them could genuinely understand everything you said and just sitting with him would make you feel so much better. There were all kinds of tails-
Some were frisky. Some were sheepdogs. Some use to lead the trail. Some were there just to show their presence. But the one common thing in every other dog is they love to walk on the trails.
This brings me to talk about Casper, my own pet. Unfortunately, until last year he didn't really ever go out into the wild. Of course, I am at fault but every cloud has a silver lining and thanks to the lockdown during Covid I got the time to spend with him.
Casper, my pet at his best behaviour.
We have now been on small hikes in the outskirts of Dehradun a couple of times. He is a big boy of six and I know he should have been out way earlier on the trails but better late than never, right? At least that's what I felt through his actions.
His first hike was an amazing experience for both of us, where we shared a lot of moments of laughter and fun. His first hike gave him the chance to get his first long drive too, where he sat for almost an hour in the taxi. He was very anxious throughout the drive, it was a new experience for him after all. He was used to running on his own, and now suddenly he was moving when he was stationary. Must have been pretty springing for him.
The moment he got off the car I could see him heave a sigh of relief. He was free again, he gave me a glance and simply ran to a certain distance inside the jungle and stopped. There it looked like he was surveying his surrounding for a moment, had a good glance all around as waited for me to come.
EXPLORING TOGETHER
We then started walking ahead, sometimes together, but most of the time he leads the trail. For him everything was new, he was observing everything as much as he could, but I could see that he was excited. I realised I should have taken him out way sooner, but I was happy that I, at last, got the opportunity then.
When he came across a small pond before I could even say no, he was inside the pond. Swimming or struggling I am not sure but he looked like he was enjoying every bit of it. He didn't stop at all, and for the next two hours, we continued walking. We called it a day after that, I didn't want to exhaust him too much on his first outing. But he looks at me eagerly every time I get into a taxi.
Today we had a 3km morning run. He loves running.
Soon we will start our trail running too!
I really hope you guys have already been on a trail with your pet and didn't wait for so long like me. If not, as I said, it's better late than never, so get out as soon as it's safe and get that cute little doggy of yours with you!
It's an amazing experience to be around tails on trails.
Here is one beautiful video I came across on "Tail on Trail"
Benefits of Outdoor Education for Kids in India
Two years of Beyond The Wall!
Exploring Miyar Valley
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Paramount+ Partners with Vivid for Biggest Drone Light Show
It's a bird! It's a plane! It's 600 drones set to light up Sydney Harbour this May!
Paramount+ is taking to the skies on Sunday, 29 May, to deliver audiences the Southern Hemisphere's biggest ever drone show as part of the Vivid Sydney 2022 program.
Now in its 12th year, Vivid Sydney will transform the Harbour City into an illuminating fusion of creativity, innovation, and technology from Friday, 27 May to Saturday, 18 June, with more than 200+ events celebrating the essence of Sydney's soul including its diversity, beauty, resilience, First Nations culture, and vibrant creative community.
This one-of-a-kind show will encapsulate the Paramount+ tagline of 'Unique Stories, Iconic Stars, and a Mountain of Entertainment', through a fleet of 600 drones hovering above the Sydney Harbour, in a playful display, illustrating some of the streaming service's most iconic stars and scenes.
The history-making event will also be available via live stream across Paramount+ and Vivid Sydney social platforms.
On partnering with Vivid Sydney 2022, Regional Vice President Marketing & Growth at Paramount+, Louise Crompton said: "Partnering with the annual Vivid Sydney 2022 and SKYMAGIC for a spectacular light and sound show like no other, secures Paramount+ a special moment in Aussie history: the biggest drone show in the southern hemisphere. We can't wait to share with you which iconic characters from our fleet of big shows, will come to life above Sydney Harbour!"
Minister for Enterprise, Investment and Trade, Tourism and Sport and Western Sydney Stuart Ayres said the Vivid Sydney 2022 program bursts with exciting events, incredible performances and fantastic experiences.
"Paramount+ and SKYMAGIC have wowed audiences across the globe with similar drone shows and I'm looking forward to a blockbuster display at Vivid Sydney 2022," Mr Ayres said. "Vivid Sydney is an open invitation for Sydneysiders and visitors from Australia and the world to come and experience Sydney at its creative best after dark."
Assisting Paramount+ in bringing this record-breaking moment to Australian audiences is SKYMAGIC, which produced Latin America's largest drone show for the local Paramount+ launch in 2021.
SKYMAGIC's Co-Founder and COO Hitomi Uematsu said: "We are delighted to have the opportunity to reunite with Paramount to deliver what will be the largest drone show ever deployed in Australia as part of the Vivid Sydney annual light festival. It also sees SKYMAGIC perform for the first time in this amazing country, and we look forward to bringing some joy and making friends along the way."
Chiew Soon Hooi, Pilot-in-Command commented: "We are really excited to be bringing SKYMAGIC to the skies above Sydney Harbour. We have deployed all over the world, and to have the chance to fly our drone light show offering within such an iconic setting is a real honour. I would like to extend our thanks to The Civil Aviation Safety Authority (CASA) for all of their support to date."
Visit https://www.paramountanz.com.au/ and https://www.vividsydney.com/
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A leading figure in the Australian #MeToo movement, the founder of a blockchain energy trading start-up and an expert in artificial intelligence and customer experience were among the women who were recognised in Sydney last night as part of this year's The Australian Financial Review 100 Women of Influence awards, presented by Qantas.
At a gala dinner at Sydney's Town Hall, 11 women received awards for their achievements across 10 categories, such as young leaders, board and management, public policy and diversity and inclusion. Megan Davis, UNSW Pro-Vice Chancellor, indigenous, was announced the overall winner. Naba Alfayadh won the young leader category.
So far, there are two projects in train – the Catch-Up project is a pilot program aimed at securing futures for vulnerable older women, and Women for Election Australia works with school-age girls to encourage and inspire them to think of politics as a career.


Justice Neuberger provides a robust and spirited defence of the independence and neutrality of the Supreme Court. Obviously this would be of value in adding it to any question on judicial independence and neutrality. It's also worth noting that the two concepts sometimes get confused. I'll post something soon that tries to help you differentiate between the two. Many of you will already be able to do this but it may be worth posting in any case for those of you who do get the 2 mixed up.


Government from 21 Jul, 2009 to 27 Sep, 2009 in Schleswig-Holstein. This government ended with a regular election.
This government arose from a coalition change. See the previous most recently elected government (elected on 20 Feb, 2005) for details.


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Michal Škvarka, Footballer, 27 years old.
Michal Škvarka is a Footballer. Michal Škvarka was born on August 19, 1992 (27 years old) . Michal Škvarka is a Slovak football midfielder who currently plays for the Corgoň Liga club MŠK Žilina. His former club was MFK Zemplín Michalovce. He is a member of the Slovakia national under-19 football team.
When is Michal Škvarka's next birthday?
Michal Škvarka was born on the 19th of August 1992 , which was a Wednesday. Michal Škvarka will be turning 28 in only 217 days from today (15 January, 2020).
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Michal Škvarka is 27 years old.
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Michal Škvarka zodiac sign is leo.
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Birth place: Czechoslovakia, Martin Slovakia
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Salt of the Earth unveils automatic packaging production line
Israel-based low-sodium salt producer Salt of the Earth has introduced a new €1m automatic packaging production line for tablets and granular salt products.
The new line uses rolls instead of traditional premade polyethylene (PE) bags and has been designed to improve quality and expand packaging options. The transition from premade bags to rolled packaging materials is set to yield annual investment savings to the company.
Salt of the Earth's new system improves the packaging quality of its products and enables the expansion of its product range to 10kg and 25kg packaging.
Salt of the Earth CEO Dovik Tal said that the new packaging system is another step towards expanding on the company's sustainable strategy and complies with the highest standards of quality and safety.
"We continue to actively seek ways to reduce energy consumption and resources and, above all, improve product quality, all of which have a direct effect on operating costs," Tal added.
The fully automated packaging line is available with a large storage tank, dual packing machine and conveyors that were custom-adjusted to the former packaging lines, allowing re-use of suitable existing equipment and incorporation of an automatic palletiser robot.
Salt of the Earth managing director Ofer Rokni said that the packaging process no longer requires manual soldering, and raw material is applied through a special weighting system, leading to a decrease in waste.
"Post-production, stacking on palettes is also fully automated to provide employees a safer, healthier working environment," Rokni added.
Image: Salt of the Earth's new packaging production line is designed for tablets and granular salt products. Photo: courtesy of PR Newswire/ Salt of the Earth Ltd.


DES MOINES - The Iowa Barnstormers have partnered with LifeServe to host a "Back to the 90's" themed Blood Drive on Tuesday, May 30 at Wells Fargo Arena as they gear up for their Annual 90's Night game. The Blood Drive will take place between 12:30 p.m. and 6:30 p.m. CDT on the Iowa Barnstormers field.
The first 100 donors at the "Back to the 90's" Blood Drive will receive a free t-shirt, a ticket to the Iowa Barnstormers 90's Night game, and an oil change voucher from United Fleet. No appointment necessary. Donors can enter through the VIP entrance located at the northwest corner of Wells Fargo Arena. Free parking will be available in the lot north of Hy-Vee Hall.
This is a child friendly blood drive. Throughout the event, Barnstormers players will be hanging out on the Iowa Barnstormers field for autographs, pictures, high-fives, and games.
For more information on this event, visit ww.lifeservebloodcenter.org or call the Iowa Barnstormers front office at (515) 633-2255.
LifeServe Blood Center was established in April of 2010 when the Blood Center of Iowa and Siouxland Community Blood Bank joined operations. As one of the 15th largest blood centers in the country, LifeServe Blood Center provides blood and blood products to more than 90 hospitals located across Iowa, Nebraska, and South Dakota. LifeServe is committed to saving lives by providing premier service to volunteer blood donors and access to a safe, quality blood spply for hospitals and patients. For more information, visit www.lifeservebloodcenter.org.